{ "dataset": { "theorems": [ { "id": 0, "type": "theorem", "label": "stacks-perfect-lemma-shriek-derived", "categories": [ "stacks-perfect" ], "title": "stacks-perfect-lemma-shriek-derived", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Notation as in", "Cohomology of Stacks,", "Lemmas \\ref{stacks-cohomology-lemma-lisse-etale} and", "\\ref{stacks-cohomology-lemma-lisse-etale-modules}.", "\\begin{enumerate}", "\\item The functor", "$g_! : \\textit{Ab}(\\mathcal{X}_{lisse,\\etale}) \\to", "\\textit{Ab}(\\mathcal{X}_\\etale)$", "has a left derived functor", "$$", "Lg_! :", "D(\\mathcal{X}_{lisse,\\etale})", "\\longrightarrow", "D(\\mathcal{X}_\\etale)", "$$", "which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \\text{id}$.", "\\item The functor $g_! : ", "\\textit{Mod}(\\mathcal{X}_{lisse,\\etale},", "\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}) \\to", "\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_{\\mathcal{X}})$", "has a left derived functor", "$$", "Lg_! :", "D(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})", "\\longrightarrow", "D(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})", "$$", "which is left adjoint to $g^*$ and such that $g^*Lg_! = \\text{id}$.", "\\item The functor $g_! : \\textit{Ab}(\\mathcal{X}_{flat,fppf}) \\to", "\\textit{Ab}(\\mathcal{X}_{fppf})$", "has a left derived functor", "$$", "Lg_! :", "D(\\mathcal{X}_{flat, fppf})", "\\longrightarrow", "D(\\mathcal{X}_{fppf})", "$$", "which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \\text{id}$.", "\\item The functor $g_! :", "\\textit{Mod}(\\mathcal{X}_{flat,fppf},", "\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}) \\to", "\\textit{Mod}(\\mathcal{X}_{fppf}, \\mathcal{O}_{\\mathcal{X}})$", "has a left derived functor", "$$", "Lg_! :", "D(\\mathcal{O}_{\\mathcal{X}_{flat, fppf}})", "\\longrightarrow", "D(\\mathcal{O}_\\mathcal{X})", "$$", "which is left adjoint to $g^*$ and such that $g^*Lg_! = \\text{id}$.", "\\end{enumerate}", "Warning: It is not clear (a priori) that $Lg_!$ on modules agrees", "with $Lg_!$ on abelian sheaves, see", "Cohomology on Sites, Remark", "\\ref{sites-cohomology-remark-when-derived-shriek-equal}." ], "refs": [ "stacks-cohomology-lemma-lisse-etale", "stacks-cohomology-lemma-lisse-etale-modules", "sites-cohomology-remark-when-derived-shriek-equal" ], "proofs": [ { "contents": [ "The existence of the functor $Lg_!$ and adjointness to $g^*$ is", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-existence-derived-lower-shriek}.", "(For the case of abelian sheaves use the constant sheaf $\\mathbf{Z}$", "as the structure sheaves.)", "Moreover, it is computed on a complex $\\mathcal{H}^\\bullet$", "by taking a suitable left resolution", "$\\mathcal{K}^\\bullet \\to \\mathcal{H}^\\bullet$", "and applying the functor $g_!$ to $\\mathcal{K}^\\bullet$.", "Since $g^{-1}g_!\\mathcal{K}^\\bullet = \\mathcal{K}^\\bullet$ by", "Cohomology of Stacks,", "Lemmas \\ref{stacks-cohomology-lemma-lisse-etale-modules} and", "\\ref{stacks-cohomology-lemma-lisse-etale}", "we see that the final assertion holds in each case." ], "refs": [ "sites-cohomology-lemma-existence-derived-lower-shriek", "stacks-cohomology-lemma-lisse-etale-modules", "stacks-cohomology-lemma-lisse-etale" ], "ref_ids": [ 4337, 4165, 4164 ] } ], "ref_ids": [ 4164, 4165, 4433 ] }, { "id": 1, "type": "theorem", "label": "stacks-perfect-lemma-lisse-etale-functorial-derived", "categories": [ "stacks-perfect" ], "title": "stacks-perfect-lemma-lisse-etale-functorial-derived", "contents": [ "With assumptions and notation as in", "Cohomology of Stacks,", "Lemma \\ref{stacks-cohomology-lemma-lisse-etale-functorial}.", "We have", "$$", "g^{-1} \\circ Rf_* = Rf'_* \\circ (g')^{-1}", "\\quad\\text{and}\\quad", "L(g')_! \\circ (f')^{-1} = f^{-1} \\circ Lg_!", "$$", "on unbounded derived categories", "(both for the case of modules and for the case of abelian sheaves)." ], "refs": [ "stacks-cohomology-lemma-lisse-etale-functorial" ], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{X}_\\etale$", "(resp.\\ $\\mathcal{X}_{fppf}$). We first show that the canonical", "(base change) map", "$$", "g^{-1} Rf_*\\mathcal{F} \\longrightarrow Rf'_* (g')^{-1}\\mathcal{F}", "$$", "is an isomorphism. To do this let $y$ be an object of", "$\\mathcal{Y}_{lisse,\\etale}$ (resp.\\ $\\mathcal{Y}_{flat,fppf}$).", "Say $y$ lies over the scheme $V$ such that $y : V \\to \\mathcal{Y}$ is", "smooth (resp.\\ flat). Since $g^{-1}$ is the restriction we find that", "$$", "\\left(g^{-1}R^pf_*\\mathcal{F}\\right)(y) =", "H^p_\\tau(V \\times_{y, \\mathcal{Y}} \\mathcal{X},\\ \\text{pr}^{-1}\\mathcal{F})", "$$", "where $\\tau = \\etale$ (resp.\\ $\\tau = fppf$), see ", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-pushforward-restriction}.", "By", "Cohomology of Stacks, Equation", "(\\ref{stacks-cohomology-equation-pushforward-lisse-etale}) ", "for any sheaf $\\mathcal{H}$ on", "$\\mathcal{X}_{lisse,\\etale}$ (resp.\\ $\\mathcal{X}_{flat,fppf}$)", "$$", "f'_*\\mathcal{H}(y) =", "\\Gamma((V \\times_{y, \\mathcal{Y}} \\mathcal{X})',", "\\ (\\text{pr}')^{-1}\\mathcal{H})", "$$", "An object of $(V \\times_{y, \\mathcal{Y}} \\mathcal{X})'$ can be seen", "as a pair $(x, \\varphi)$ where $x$ is an object of", "$\\mathcal{X}_{lisse,\\etale}$ (resp.\\ $\\mathcal{X}_{flat,fppf}$)", "and $\\varphi : f(x) \\to y$ is a morphism in $\\mathcal{Y}$.", "We can also think of $\\varphi$ as a section of $(f')^{-1}h_y$ over $x$.", "Thus $(V \\times_\\mathcal{Y} \\mathcal{X})'$ is the localization", "of the site $\\mathcal{X}_{lisse,\\etale}$", "(resp. $\\mathcal{X}_{flat,fppf}$) at the sheaf of sets $(f')^{-1}h_y$, see", "Sites, Lemma \\ref{sites-lemma-localize-topos-site}. The morphism", "$$", "\\text{pr}' : (V \\times_{y, \\mathcal{Y}} \\mathcal{X})'", "\\to \\mathcal{X}_{lisse,\\etale}", "\\ (\\text{resp. }", "\\text{pr}' : (V \\times_{y, \\mathcal{Y}} \\mathcal{X})'", "\\to \\mathcal{X}_{flat,fppf})", "$$", "is the localization morphism.", "In particular, the pullback $(\\text{pr}')^{-1}$ preserves", "injective abelian sheaves, see", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-cohomology-on-sheaf-sets}.", "At this point exactly the same argument as in", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-pushforward-restriction}", "shows that", "\\begin{equation}", "\\label{equation-higher-direct-image-lisse-etale}", "R^pf'_*\\mathcal{H}(y) =", "H^p_\\tau((V \\times_{y, \\mathcal{Y}} \\mathcal{X})',", "\\ (\\text{pr}')^{-1}\\mathcal{H})", "\\end{equation}", "where $\\tau = \\etale$ (resp.\\ $\\tau = fppf$). Since $(g')^{-1}$", "is given by restriction we conclude that", "$$", "\\left(R^pf'_*(g')^*\\mathcal{F}\\right)(y) =", "H^p_\\tau((V \\times_{y, \\mathcal{Y}} \\mathcal{X})',", "\\ \\text{pr}^{-1}\\mathcal{F}|_{(V \\times_{y, \\mathcal{Y}} \\mathcal{X})'})", "$$", "Finally, we can apply", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-cohomology-on-subcategory}", "to see that", "$$", "H^p_\\tau((V \\times_{y, \\mathcal{Y}} \\mathcal{X})',", "\\ \\text{pr}^{-1}\\mathcal{F}|_{(V \\times_{y, \\mathcal{Y}} \\mathcal{X})'})", "=", "H^p_\\tau(V \\times_{y, \\mathcal{Y}} \\mathcal{X},\\ \\text{pr}^{-1}\\mathcal{F})", "$$", "are equal as desired; although we omit the verification of the assumptions", "of the lemma we note that the fact that $V \\to \\mathcal{Y}$ is smooth", "(resp.\\ flat) is used to verify the second condition.", "\\medskip\\noindent", "The rest of the proof is formal. Since cohomology of abelian groups and", "sheaves of modules agree we also conclude that ", "$g^{-1} Rf_*\\mathcal{F} = Rf'_* (g')^{-1}\\mathcal{F}$ when $\\mathcal{F}$", "is a sheaf of modules on $\\mathcal{X}_\\etale$", "(resp.\\ $\\mathcal{X}_{fppf}$).", "\\medskip\\noindent", "Next we show that for $\\mathcal{G}$ (either sheaf of modules", "or abelian groups) on", "$\\mathcal{Y}_{lisse,\\etale}$ (resp.\\ $\\mathcal{Y}_{flat,fppf}$)", "the canonical map", "$$", "L(g')_!(f')^{-1}\\mathcal{G} \\to f^{-1}Lg_!\\mathcal{G}", "$$", "is an isomorphism. To see this it is enough to prove for any", "injective sheaf $\\mathcal{I}$ on $\\mathcal{X}_\\etale$", "(resp.\\ $\\mathcal{X}_{fppf}$) that the induced map", "$$", "\\Hom(L(g')_!(f')^{-1}\\mathcal{G}, \\mathcal{I}[n])", "\\leftarrow", "\\Hom(f^{-1}Lg_!\\mathcal{G}, \\mathcal{I}[n])", "$$", "is an isomorphism for all $n \\in \\mathbf{Z}$. (Hom's taken", "in suitable derived categories.) By the adjointness of", "$f^{-1}$ and $Rf_*$, the adjointness of $Lg_!$ and $g^{-1}$, and", "their ``primed'' versions this follows from the isomorphism", "$g^{-1} Rf_*\\mathcal{I} \\to Rf'_* (g')^{-1}\\mathcal{I}$ proved above.", "\\medskip\\noindent", "In the case of a bounded complex $\\mathcal{G}^\\bullet$", "(of modules or abelian groups) on", "$\\mathcal{Y}_{lisse,\\etale}$ (resp.\\ $\\mathcal{Y}_{fppf}$)", "the canonical map", "\\begin{equation}", "\\label{equation-to-show}", "L(g')_!(f')^{-1}\\mathcal{G}^\\bullet \\to f^{-1}Lg_!\\mathcal{G}^\\bullet", "\\end{equation}", "is an isomorphism as follows from the case of a sheaf by the usual arguments", "involving truncations and the fact that the functors", "$L(g')_!(f')^{-1}$ and $f^{-1}Lg_!$ are exact functors of", "triangulated categories.", "\\medskip\\noindent", "Suppose that $\\mathcal{G}^\\bullet$ is a bounded above complex", "(of modules or abelian groups) on", "$\\mathcal{Y}_{lisse,\\etale}$ (resp.\\ $\\mathcal{Y}_{fppf}$).", "The canonical map (\\ref{equation-to-show})", "is an isomorphism because we can use the stupid truncations", "$\\sigma_{\\geq -n}$ (see", "Homology, Section \\ref{homology-section-truncations}) to write", "$\\mathcal{G}^\\bullet$ as a colimit", "$\\mathcal{G}^\\bullet = \\colim \\mathcal{G}_n^\\bullet$", "of bounded complexes. This gives a distinguished triangle", "$$", "\\bigoplus\\nolimits_{n \\geq 1} \\mathcal{G}_n^\\bullet \\to", "\\bigoplus\\nolimits_{n \\geq 1} \\mathcal{G}_n^\\bullet \\to", "\\mathcal{G}^\\bullet \\to \\ldots", "$$", "and each of the functors $L(g')_!$, $(f')^{-1}$, $f^{-1}$, $Lg_!$", "commutes with direct sums (of complexes).", "\\medskip\\noindent", "If $\\mathcal{G}^\\bullet$ is an arbitrary complex", "(of modules or abelian groups) on", "$\\mathcal{Y}_{lisse,\\etale}$ (resp.\\ $\\mathcal{Y}_{fppf}$)", "then we use the canonical truncations $\\tau_{\\leq n}$ (see", "Homology, Section \\ref{homology-section-truncations})", "to write $\\mathcal{G}^\\bullet$ as a colimit of bounded above complexes", "and we repeat the argument of the paragraph above.", "\\medskip\\noindent", "Finally, by the adjointness of", "$f^{-1}$ and $Rf_*$, the adjointness of $Lg_!$ and $g^{-1}$, and", "their ``primed'' versions we conclude that the first", "identity of the lemma follows from the second in full generality." ], "refs": [ "stacks-sheaves-lemma-pushforward-restriction", "sites-lemma-localize-topos-site", "sites-cohomology-lemma-cohomology-on-sheaf-sets", "stacks-sheaves-lemma-pushforward-restriction", "stacks-sheaves-lemma-cohomology-on-subcategory" ], "ref_ids": [ 11609, 8585, 4215, 11609, 11616 ] } ], "ref_ids": [ 4168 ] }, { "id": 2, "type": "theorem", "label": "stacks-perfect-lemma-higher-shriek-quasi-coherent", "categories": [ "stacks-perfect" ], "title": "stacks-perfect-lemma-higher-shriek-quasi-coherent", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Notation as in", "Cohomology of Stacks,", "Lemma \\ref{stacks-cohomology-lemma-lisse-etale}.", "\\begin{enumerate}", "\\item Let $\\mathcal{H}$ be a quasi-coherent", "$\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}$-module ", "on the lisse-\\'etale site of $\\mathcal{X}$. For all $p \\in \\mathbf{Z}$", "the sheaf $H^p(Lg_!\\mathcal{H})$ is a locally quasi-coherent module with", "the flat base change property on $\\mathcal{X}$.", "\\item Let $\\mathcal{H}$ be a quasi-coherent", "$\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}$-module ", "on the flat-fppf site of $\\mathcal{X}$. For all $p \\in \\mathbf{Z}$", "the sheaf $H^p(Lg_!\\mathcal{H})$ is a locally quasi-coherent module with the", "flat base change property on $\\mathcal{X}$.", "\\end{enumerate}" ], "refs": [ "stacks-cohomology-lemma-lisse-etale" ], "proofs": [ { "contents": [ "Pick a scheme $U$ and a surjective smooth morphism $x : U \\to \\mathcal{X}$. By", "Modules on Sites, Definition \\ref{sites-modules-definition-site-local}", "there exists an \\'etale (resp.\\ fppf) covering", "$\\{U_i \\to U\\}_{i \\in I}$ such that each pullback $f_i^{-1}\\mathcal{H}$", "has a global presentation (see", "Modules on Sites, Definition \\ref{sites-modules-definition-global}).", "Here $f_i : U_i \\to \\mathcal{X}$ is the composition", "$U_i \\to U \\to \\mathcal{X}$ which is a morphism of algebraic stacks.", "(Recall that the pullback ``is'' the restriction to $\\mathcal{X}/f_i$, see", "Sheaves on Stacks, Definition \\ref{stacks-sheaves-definition-pullback}", "and the discussion following.)", "After refining the covering we may assume each $U_i$ is an affine scheme.", "Since each $f_i$ is smooth (resp.\\ flat) by", "Lemma \\ref{lemma-lisse-etale-functorial-derived}", "we see that $f_i^{-1}Lg_!\\mathcal{H} = Lg_{i, !}(f'_i)^{-1}\\mathcal{H}$.", "Using", "Cohomology of Stacks,", "Lemma \\ref{stacks-cohomology-lemma-check-lqc-fbc-on-covering}", "we reduce the statement of the lemma to the case where $\\mathcal{H}$", "has a global presentation and where $\\mathcal{X} = (\\Sch/X)_{fppf}$", "for some affine scheme $X = \\Spec(A)$.", "\\medskip\\noindent", "Say our presentation looks like", "$$", "\\bigoplus\\nolimits_{j \\in J} \\mathcal{O} \\longrightarrow", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{O} \\longrightarrow", "\\mathcal{H} \\longrightarrow 0", "$$", "where $\\mathcal{O} = \\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}$", "(resp.\\ $\\mathcal{O} = \\mathcal{O}_{\\mathcal{X}_{flat,fppf}}$).", "Note that the site $\\mathcal{X}_{lisse,\\etale}$", "(resp.\\ $\\mathcal{X}_{flat,fppf}$) has a final object, namely", "$X/X$ which is quasi-compact (see", "Cohomology on Sites, Section \\ref{sites-cohomology-section-limits}).", "Hence we have", "$$", "\\Gamma(\\bigoplus\\nolimits_{i \\in I} \\mathcal{O}) =", "\\bigoplus\\nolimits_{i \\in I} A", "$$", "by Sites, Lemma \\ref{sites-lemma-directed-colimits-sections}. Hence the map", "in the presentation corresponds to a similar presentation", "$$", "\\bigoplus\\nolimits_{j \\in J} A \\longrightarrow", "\\bigoplus\\nolimits_{i \\in I} A \\longrightarrow", "M \\longrightarrow 0", "$$", "of an $A$-module $M$. Moreover, $\\mathcal{H}$ is equal to the restriction", "to the lisse-\\'etale (resp.\\ flat-fppf) site of the quasi-coherent sheaf", "$M^a$ associated to $M$. Choose a resolution", "$$", "\\ldots \\to F_2 \\to F_1 \\to F_0 \\to M \\to 0", "$$", "by free $A$-modules. The complex", "$$", "\\ldots \\mathcal{O} \\otimes_A F_2 \\to \\mathcal{O} \\otimes_A F_1 \\to", "\\mathcal{O} \\otimes_A F_0 \\to \\mathcal{H} \\to 0", "$$", "is a resolution of $\\mathcal{H}$ by free $\\mathcal{O}$-modules because", "for each object $U/X$ of $\\mathcal{X}_{lisse,\\etale}$", "(resp.\\ $\\mathcal{X}_{flat,fppf}$) the structure morphism $U \\to X$", "is flat. Hence by construction the value of $Lg_!\\mathcal{H}$ is", "$$", "\\ldots \\to", "\\mathcal{O}_\\mathcal{X} \\otimes_A F_2 \\to", "\\mathcal{O}_\\mathcal{X} \\otimes_A F_1 \\to", "\\mathcal{O}_\\mathcal{X} \\otimes_A F_0 \\to 0 \\to \\ldots", "$$", "Since this is a complex of quasi-coherent modules on", "$\\mathcal{X}_\\etale$ (resp.\\ $\\mathcal{X}_{fppf}$)", "it follows from", "Cohomology of Stacks,", "Proposition \\ref{stacks-cohomology-proposition-lcq-flat-base-change}", "that $H^p(Lg_!\\mathcal{H})$ is quasi-coherent." ], "refs": [ "sites-modules-definition-site-local", "sites-modules-definition-global", "stacks-sheaves-definition-pullback", "stacks-perfect-lemma-lisse-etale-functorial-derived", "stacks-cohomology-lemma-check-lqc-fbc-on-covering", "sites-lemma-directed-colimits-sections", "stacks-cohomology-proposition-lcq-flat-base-change" ], "ref_ids": [ 14289, 14286, 11628, 1, 4156, 8531, 4173 ] } ], "ref_ids": [ 4164 ] }, { "id": 3, "type": "theorem", "label": "stacks-perfect-lemma-compare-etale-fppf-QCoh", "categories": [ "stacks-perfect" ], "title": "stacks-perfect-lemma-compare-etale-fppf-QCoh", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. The comparison morphism", "$\\epsilon : \\mathcal{X}_{fppf} \\to \\mathcal{X}_\\etale$", "induces a commutative diagram", "$$", "\\xymatrix{", "D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X}) \\ar[r] &", "D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X}) \\ar[r] &", "D(\\mathcal{O}_\\mathcal{X}) \\\\", "D_{\\mathcal{P}_\\mathcal{X}}(", "\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})", "\\ar[r] \\ar[u]^{\\epsilon^*} &", "D_{\\mathcal{M}_\\mathcal{X}}(", "\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})", "\\ar[r] \\ar[u]^{\\epsilon^*} &", "D(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})", "\\ar[u]^{\\epsilon^*}", "}", "$$", "Moreover, the left two vertical arrows are equivalences of triangulated", "categories, hence we also obtain an equivalence", "$$", "D_{\\mathcal{M}_\\mathcal{X}}(", "\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})", "/", "D_{\\mathcal{P}_\\mathcal{X}}(", "\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})", "\\longrightarrow", "D_\\QCoh(\\mathcal{O}_\\mathcal{X})", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since $\\epsilon^*$ is exact it is clear that we obtain a diagram as", "in the statement of the lemma. We will show the middle vertical", "arrow is an equivalence by applying", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-compare-topologies-derived-adequate-modules}", "to the following situation:", "$\\mathcal{C} = \\mathcal{X}$,", "$\\tau = fppf$,", "$\\tau' = \\etale$,", "$\\mathcal{O} = \\mathcal{O}_\\mathcal{X}$,", "$\\mathcal{A} = \\mathcal{M}_\\mathcal{X}$, and", "$\\mathcal{B}$ is the set of objects of $\\mathcal{X}$ lying over", "affine schemes. To see the lemma applies we have to check conditions", "(1), (2), (3), (4). Conditions (1) and (2) are clear from the discussion", "above (explicitly this follows from", "Cohomology of Stacks,", "Proposition \\ref{stacks-cohomology-proposition-lcq-flat-base-change}).", "Condition (3) holds because every scheme has a Zariski", "open covering by affines. Condition (4) follows from", "Descent, Lemma \\ref{descent-lemma-quasi-coherent-and-flat-base-change}.", "\\medskip\\noindent", "We omit the verification that the equivalence of", "categories $\\epsilon^* : ", "D_{\\mathcal{M}_\\mathcal{X}}(", "\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})", "\\to", "D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$", "induces an equivalence of the subcategories of complexes", "with parasitic cohomology sheaves." ], "refs": [ "sites-cohomology-lemma-compare-topologies-derived-adequate-modules", "stacks-cohomology-proposition-lcq-flat-base-change", "descent-lemma-quasi-coherent-and-flat-base-change" ], "ref_ids": [ 4293, 4173, 14631 ] } ], "ref_ids": [] }, { "id": 4, "type": "theorem", "label": "stacks-perfect-lemma-derived-quasi-coherent", "categories": [ "stacks-perfect" ], "title": "stacks-perfect-lemma-derived-quasi-coherent", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "\\begin{enumerate}", "\\item", "Let $\\mathcal{F}^\\bullet$ be an object of", "$D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.", "With $g$ as in", "Cohomology of Stacks,", "Lemma \\ref{stacks-cohomology-lemma-lisse-etale}", "for the lisse-\\'etale site we have", "\\begin{enumerate}", "\\item $g^{-1}\\mathcal{F}^\\bullet$ is in", "$D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$,", "\\item $g^{-1}\\mathcal{F}^\\bullet = 0$ if and only if", "$\\mathcal{F}^\\bullet$ is in", "$D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$,", "\\item $Lg_!\\mathcal{H}^\\bullet$ is in", "$D_{\\mathcal{M}_\\mathcal{X}}(", "\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "for $\\mathcal{H}^\\bullet$ in", "$D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$, and", "\\item the functors $g^{-1}$ and $Lg_!$ define mutually inverse functors", "$$", "\\xymatrix{", "D_\\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar@<1ex>[r]^-{g^{-1}} &", "D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})", "\\ar@<1ex>[l]^-{Lg_!}", "}", "$$", "\\end{enumerate}", "\\item", "Let $\\mathcal{F}^\\bullet$ be an object of", "$D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$. With $g$ as in", "Cohomology of Stacks,", "Lemma \\ref{stacks-cohomology-lemma-lisse-etale}", "for the flat-fppf site we have", "\\begin{enumerate}", "\\item $g^{-1}\\mathcal{F}^\\bullet$ is in", "$D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat, fppf}})$,", "\\item $g^{-1}\\mathcal{F}^\\bullet = 0$ if and only if", "$\\mathcal{F}^\\bullet$ is in", "$D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$,", "\\item $Lg_!\\mathcal{H}^\\bullet$ is in", "$D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$", "for $\\mathcal{H}^\\bullet$ in", "$D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat,fppf}})$, and", "\\item the functors $g^{-1}$ and $Lg_!$ define mutually inverse functors", "$$", "\\xymatrix{", "D_\\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar@<1ex>[r]^-{g^{-1}} &", "D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}) \\ar@<1ex>[l]^-{Lg_!}", "}", "$$", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [ "stacks-cohomology-lemma-lisse-etale", "stacks-cohomology-lemma-lisse-etale" ], "proofs": [ { "contents": [ "The functor $g^{-1}$ is exact, hence (1)(a), (2)(a), (1)(b), and (2)(b)", "follow from Cohomology of Stacks,", "Lemmas \\ref{stacks-cohomology-lemma-quasi-coherent} and", "\\ref{stacks-cohomology-lemma-parasitic-in-terms-flat-fppf}.", "\\medskip\\noindent", "Proof of (1)(c) and (2)(c).", "The construction of $Lg_!$ in Lemma \\ref{lemma-shriek-derived}", "(via Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-existence-derived-lower-shriek}", "which in turn uses", "Derived Categories, Proposition \\ref{derived-proposition-left-derived-exists})", "shows that $Lg_!$ on any object $\\mathcal{H}^\\bullet$ of", "$D(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$ is computed", "as", "$$", "Lg_!\\mathcal{H}^\\bullet = \\colim g_!\\mathcal{K}_n^\\bullet =", "g_! \\colim \\mathcal{K}_n^\\bullet", "$$", "(termwise colimits) where the quasi-isomorphism", "$\\colim \\mathcal{K}_n^\\bullet \\to \\mathcal{H}^\\bullet$", "induces quasi-isomorphisms", "$\\mathcal{K}_n^\\bullet \\to \\tau_{\\leq n} \\mathcal{H}^\\bullet$.", "Since", "$\\mathcal{M}_\\mathcal{X} \\subset", "\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "(resp.\\ $\\mathcal{M}_\\mathcal{X} \\subset \\textit{Mod}(\\mathcal{O}_\\mathcal{X})$)", "is preserved under colimits we see that it suffices to prove (c)", "on bounded above complexes $\\mathcal{H}^\\bullet$ in", "$D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$", "(resp.\\ $D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat,fppf}})$).", "In this case to show that $H^n(Lg_!\\mathcal{H}^\\bullet)$ is", "in $\\mathcal{M}_\\mathcal{X}$ we can argue by induction on the integer", "$m$ such that $\\mathcal{H}^i = 0$ for $i > m$. If $m < n$, then", "$H^n(Lg_!\\mathcal{H}^\\bullet) = 0$ and the result holds. In general", "consider the distinguished triangle", "$$", "\\tau_{\\leq m - 1}\\mathcal{H}^\\bullet \\to \\mathcal{H}^\\bullet \\to", "H^m(\\mathcal{H}^\\bullet)[-m] \\to \\ldots", "$$", "(Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle})", "and apply the functor $Lg_!$. Since $\\mathcal{M}_\\mathcal{X}$", "is a weak Serre subcategory of the module category it suffices to", "prove (c) for two out of three. We have the result for", "$Lg_!\\tau_{\\leq m - 1}\\mathcal{H}^\\bullet$ by induction and we", "have the result for $Lg_!H^m(\\mathcal{H}^\\bullet)[-m]$ by", "Lemma \\ref{lemma-higher-shriek-quasi-coherent}. Whence (c) holds.", "\\medskip\\noindent", "Let us prove (2)(d). By (2)(a) and (2)(b) the functor $g^{-1} = g^*$ induces", "a functor", "$$", "c :", "D_\\QCoh(\\mathcal{O}_\\mathcal{X})", "\\longrightarrow", "D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat, fppf}})", "$$", "see", "Derived Categories, Lemma \\ref{derived-lemma-universal-property-quotient}.", "Thus we have the following diagram of triangulated categories", "$$", "\\xymatrix{", "D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})", "\\ar[rd]^{g^{-1}} \\ar[rr]_q & &", "D_\\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar[ld]^c \\\\", "& D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat, fppf}})", "\\ar@<1ex>[lu]^{Lg_!}", "}", "$$", "where $q$ is the quotient functor, the inner triangle is commutative, and", "$g^{-1}Lg_! = \\text{id}$.", "For any object of $E$ of $D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$", "the map $a : Lg_!g^{-1}E \\to E$ maps to a quasi-isomorphism in", "$D(\\mathcal{O}_{\\mathcal{X}_{flat, fppf}})$. Hence the cone on", "$a$ maps to zero under $g^{-1}$ and by (2)(b) we see that $q(a)$ is", "an isomorphism. Thus $q \\circ Lg_!$ is a quasi-inverse to $c$.", "\\medskip\\noindent", "In the case of the lisse-\\'etale site exactly the same argument as above", "proves that", "$$", "D_{\\mathcal{M}_\\mathcal{X}}(", "\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})", "/", "D_{\\mathcal{P}_\\mathcal{X}}(", "\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})", "$$", "is equivalent to", "$D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$.", "Applying the last equivalence of", "Lemma \\ref{lemma-compare-etale-fppf-QCoh}", "finishes the proof." ], "refs": [ "stacks-cohomology-lemma-quasi-coherent", "stacks-cohomology-lemma-parasitic-in-terms-flat-fppf", "stacks-perfect-lemma-shriek-derived", "sites-cohomology-lemma-existence-derived-lower-shriek", "derived-proposition-left-derived-exists", "derived-remark-truncation-distinguished-triangle", "stacks-perfect-lemma-higher-shriek-quasi-coherent", "derived-lemma-universal-property-quotient", "stacks-perfect-lemma-compare-etale-fppf-QCoh" ], "ref_ids": [ 4171, 4167, 0, 4337, 1964, 2016, 2, 1788, 3 ] } ], "ref_ids": [ 4164, 4164 ] }, { "id": 5, "type": "theorem", "label": "stacks-perfect-lemma-bousfield-colocalization", "categories": [ "stacks-perfect" ], "title": "stacks-perfect-lemma-bousfield-colocalization", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Let $E$ be an object of $D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$.", "There exists a canonical distinguished triangle", "$$", "E' \\to E \\to P \\to E'[1]", "$$", "in $D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$ such that", "$P$ is in $D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$", "and", "$$", "\\Hom_{D(\\mathcal{O}_\\mathcal{X})}(E', P') = 0", "$$", "for all $P'$ in $D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$." ], "refs": [], "proofs": [ { "contents": [ "Consider the morphism of ringed topoi", "$g : \\Sh(\\mathcal{X}_{flat, fppf}) \\longrightarrow \\Sh(\\mathcal{X}_{fppf})$.", "Set $E' = Lg_!g^{-1}E$ and let $P$ be the cone on the adjunction", "map $E' \\to E$. Since $g^{-1}E' \\to g^{-1}E$ is an isomorphism we see that", "$P$ is an object of $D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$ by", "Lemma \\ref{lemma-derived-quasi-coherent} (2)(b).", "Finally, $\\Hom(E', P') = \\Hom(Lg_!g^{-1}E, P') = \\Hom(g^{-1}E, g^{-1}P') = 0$", "as $g^{-1}P' = 0$.", "\\medskip\\noindent", "Uniqueness. Suppose that $E'' \\to E \\to P'$ is a second distinguished", "triangle as in the statement of the lemma. Since $\\Hom(E', P') = 0$", "the morphism $E' \\to E$ factors as $E' \\to E'' \\to E$, see", "Derived Categories, Lemma \\ref{derived-lemma-representable-homological}.", "Similarly, the morphism $E'' \\to E$ factors as $E'' \\to E' \\to E$.", "Consider the composition $\\varphi : E' \\to E'$ of the maps $E' \\to E''$ and", "$E'' \\to E'$. Note that $\\varphi - 1 : E' \\to E'$ fits into the commutative", "diagram", "$$", "\\xymatrix{", "E' \\ar[d]^{\\varphi - 1} \\ar[r] & E \\ar[d]^0 \\\\", "E' \\ar[r] & E", "}", "$$", "hence factors through $P[-1] \\to E$. Since $\\Hom(E', P[-1]) = 0$", "we see that $\\varphi = 1$. Whence the maps $E' \\to E''$ and $E'' \\to E'$", "are inverse to each other." ], "refs": [ "stacks-perfect-lemma-derived-quasi-coherent", "derived-lemma-representable-homological" ], "ref_ids": [ 4, 1758 ] } ], "ref_ids": [] }, { "id": 6, "type": "theorem", "label": "stacks-perfect-proposition-derived-direct-image-quasi-coherent", "categories": [ "stacks-perfect" ], "title": "stacks-perfect-proposition-derived-direct-image-quasi-coherent", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and", "quasi-separated morphism of algebraic stacks.", "The functor $Rf_*$ induces a commutative diagram", "$$", "\\xymatrix{", "D^{+}_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})", "\\ar[r] \\ar[d]^{Rf_*} &", "D^{+}_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})", "\\ar[r] \\ar[d]^{Rf_*} &", "D(\\mathcal{O}_\\mathcal{X})", "\\ar[d]^{Rf_*} \\\\", "D^{+}_{\\mathcal{P}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y}) \\ar[r] &", "D^{+}_{\\mathcal{M}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y}) \\ar[r] &", "D(\\mathcal{O}_\\mathcal{Y})", "}", "$$", "and hence induces a functor", "$$", "Rf_{\\QCoh, *} :", "D^{+}_\\QCoh(\\mathcal{O}_\\mathcal{X})", "\\longrightarrow", "D^{+}_\\QCoh(\\mathcal{O}_\\mathcal{Y})", "$$", "on quotient categories. Moreover, the functor $R^if_\\QCoh$", "of", "Cohomology of Stacks,", "Proposition \\ref{stacks-cohomology-proposition-direct-image-quasi-coherent}", "are equal to $H^i \\circ Rf_{\\QCoh, *}$ with $H^i$ as in", "(\\ref{equation-Hi-quasi-coherent})." ], "refs": [ "stacks-cohomology-proposition-direct-image-quasi-coherent" ], "proofs": [ { "contents": [ "We have to show that $Rf_*E$ is an object of", "$D^{+}_{\\mathcal{M}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y})$ for", "$E$ in $D^{+}_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$.", "This follows from", "Cohomology of Stacks,", "Proposition \\ref{stacks-cohomology-proposition-lcq-flat-base-change}", "and the spectral sequence $R^if_*H^j(E) \\Rightarrow R^{i + j}f_*E$.", "The case of parasitic modules works the same way using", "Cohomology of Stacks, Lemma", "\\ref{stacks-cohomology-lemma-pushforward-parasitic}.", "The final statement is clear from the definition of", "$H^i$ in (\\ref{equation-Hi-quasi-coherent})." ], "refs": [ "stacks-cohomology-proposition-lcq-flat-base-change", "stacks-cohomology-lemma-pushforward-parasitic" ], "ref_ids": [ 4173, 4158 ] } ], "ref_ids": [ 4174 ] }, { "id": 7, "type": "theorem", "label": "stacks-perfect-proposition-derived-pullback-quasi-coherent", "categories": [ "stacks-perfect" ], "title": "stacks-perfect-proposition-derived-pullback-quasi-coherent", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The exact functor $f^*$ induces a commutative diagram", "$$", "\\xymatrix{", "D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X}) \\ar[r] &", "D(\\mathcal{O}_\\mathcal{X}) \\\\", "D_{\\mathcal{M}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y})", "\\ar[r] \\ar[u]^{f^*} &", "D(\\mathcal{O}_\\mathcal{Y}) \\ar[u]^{f^*}", "}", "$$", "The composition", "$$", "D_{\\mathcal{M}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y})", "\\xrightarrow{f^*}", "D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})", "\\xrightarrow{q_\\mathcal{X}}", "D_\\QCoh(\\mathcal{O}_\\mathcal{X})", "$$", "is left derivable with respect to the localization", "$D_{\\mathcal{M}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y}) \\to", "D_\\QCoh(\\mathcal{O}_\\mathcal{Y})$", "and we may define $Lf^*_\\QCoh$ as its left derived functor", "$$", "Lf_\\QCoh^* :", "D_\\QCoh(\\mathcal{O}_\\mathcal{Y})", "\\longrightarrow", "D_\\QCoh(\\mathcal{O}_\\mathcal{X})", "$$", "(see", "Derived Categories,", "Definitions \\ref{derived-definition-right-derived-functor-defined} and", "\\ref{derived-definition-everywhere-defined}). If $f$ is quasi-compact", "and quasi-separated, then $Lf^*_\\QCoh$ and $Rf_{\\QCoh, *}$", "satisfy the following adjointness:", "$$", "\\Hom_{D_\\QCoh(\\mathcal{O}_\\mathcal{X})}(Lf^*_\\QCoh A, B)", "=", "\\Hom_{D_\\QCoh(\\mathcal{O}_\\mathcal{Y})}(A, Rf_{\\QCoh, *}B)", "$$", "for $A \\in D_\\QCoh(\\mathcal{O}_\\mathcal{Y})$ and", "$B \\in D^{+}_\\QCoh(\\mathcal{O}_\\mathcal{X})$." ], "refs": [ "derived-definition-right-derived-functor-defined", "derived-definition-everywhere-defined" ], "proofs": [ { "contents": [ "To prove the first statement, we have to show that $f^*E$ is an object of", "$D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$ for", "$E$ in $D_{\\mathcal{M}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y})$.", "Since $f^* = f^{-1}$ is exact this follows immediately from the fact that", "$f^*$ maps $\\mathcal{M}_\\mathcal{Y}$ into $\\mathcal{M}_\\mathcal{X}$.", "\\medskip\\noindent", "Set $\\mathcal{D} = D_{\\mathcal{M}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y})$.", "Let $S$ be the collection of morphisms in $\\mathcal{D}$", "whose cone is an object of", "$D_{\\mathcal{P}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y})$.", "Set $\\mathcal{D}' = D_\\QCoh(\\mathcal{O}_\\mathcal{X})$.", "Set $F = q_\\mathcal{X} \\circ f^* : \\mathcal{D} \\to \\mathcal{D}'$.", "Then $\\mathcal{D}, S, \\mathcal{D}', F$ are as in", "Derived Categories, Situation \\ref{derived-situation-derived-functor} and", "Definition \\ref{derived-definition-right-derived-functor-defined}.", "Let us prove that $LF(E)$ is defined for any object $E$ of $\\mathcal{D}$.", "Namely, consider the triangle", "$$", "E' \\to E \\to P \\to E'[1]", "$$", "constructed in Lemma \\ref{lemma-bousfield-colocalization}.", "Note that $s : E' \\to E$ is an element of $S$. We claim that $E'$ computes", "$LF$. Namely, suppose that $s' : E'' \\to E$ is another element of $S$, i.e.,", "fits into a triangle $E'' \\to E \\to P' \\to E''[1]$ with $P'$ in", "$D_{\\mathcal{P}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y})$. By", "Lemma \\ref{lemma-bousfield-colocalization} (and its proof)", "we see that $E' \\to E$ factors through $E'' \\to E$. Thus we see that", "$E' \\to E$ is cofinal in the system $S/E$. Hence it is clear that", "$E'$ computes $LF$.", "\\medskip\\noindent", "To see the final statement, write $B = q_\\mathcal{X}(H)$ and", "$A = q_\\mathcal{Y}(E)$.", "Choose $E' \\to E$ as above.", "We will use on the one hand that", "$Rf_{\\QCoh, *}(B) = q_\\mathcal{Y}(Rf_*H)$", "and on the other that", "$Lf^*_\\QCoh(A) = q_\\mathcal{X}(f^*E')$.", "\\begin{align*}", "\\Hom_{D_\\QCoh(\\mathcal{O}_\\mathcal{X})}(Lf^*_\\QCoh A, B)", "& = ", "\\Hom_{D_\\QCoh(\\mathcal{O}_\\mathcal{X})}(q_\\mathcal{X}(f^*E'),", "q_\\mathcal{X}(H)) \\\\", "& = ", "\\colim_{H \\to H'} \\Hom_{D(\\mathcal{O}_\\mathcal{X})}(f^*E', H') \\\\", "& = \\colim_{H \\to H'} \\Hom_{D(\\mathcal{O}_\\mathcal{Y})}(E', Rf_*H') \\\\", "& = \\Hom_{D(\\mathcal{O}_\\mathcal{Y})}(E', Rf_*H) \\\\", "& =", "\\Hom_{D_\\QCoh(\\mathcal{O}_\\mathcal{Y})}(A, Rf_{\\QCoh, *}B)", "\\end{align*}", "Here the colimit is over morphisms $s : H \\to H'$ in", "$D^+_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$", "whose cone $P(s)$ is an object of", "$D^+_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$.", "The first equality we've seen above.", "The second equality holds by construction of the Verdier quotient.", "The third equality holds by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-adjoint}.", "Since $Rf_*P(s)$ is an object of", "$D^+_{\\mathcal{P}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y})$ by", "Proposition \\ref{proposition-derived-direct-image-quasi-coherent}", "we see that $\\Hom_{D(\\mathcal{O}_\\mathcal{Y})}(E', Rf_*P(s)) = 0$.", "Thus the fourth equality holds. The final equality", "holds by construction of $E'$." ], "refs": [ "derived-definition-right-derived-functor-defined", "stacks-perfect-lemma-bousfield-colocalization", "stacks-perfect-lemma-bousfield-colocalization", "sites-cohomology-lemma-adjoint", "stacks-perfect-proposition-derived-direct-image-quasi-coherent" ], "ref_ids": [ 1987, 5, 5, 4249, 6 ] } ], "ref_ids": [ 1987, 1988 ] }, { "id": 9, "type": "theorem", "label": "spaces-more-morphisms-theorem-topological-invariance", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-theorem-topological-invariance", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume $f$ is integral, universally injective and surjective.", "The functor", "$$", "V \\longmapsto V_X = X \\times_Y V", "$$", "defines an equivalence of categories", "$Y_{spaces, \\etale} \\to X_{spaces, \\etale}$." ], "refs": [], "proofs": [ { "contents": [ "The morphism $f$ is representable and a universal homeomorphism, see", "Morphisms of Spaces,", "Section \\ref{spaces-morphisms-section-universal-homeomorphisms}.", "\\medskip\\noindent", "We first prove that the functor is faithful.", "Suppose that $V', V$ are objects of $Y_{spaces, \\etale}$ and", "that $a, b : V' \\to V$ are distinct morphisms over $Y$.", "Since $V', V$ are \\'etale over $Y$ the equalizer", "$$", "E = V' \\times_{(a, b), V \\times_Y V, \\Delta_{V/Y}} V", "$$", "of $a, b$ is \\'etale over $Y$ also. Hence $E \\to V'$ is an \\'etale monomorphism", "(i.e., an open immersion) which is an isomorphism if and only if it is", "surjective. Since $X \\to Y$ is a universal homeomorphism we see that this", "is the case if and only if $E_X = V'_X$, i.e., if and only if $a_X = b_X$.", "\\medskip\\noindent", "Next, we prove that the functor is fully faithful.", "Suppose that $V', V$ are objects of $Y_{spaces, \\etale}$ and", "that $c : V'_X \\to V_X$ is a morphism over $X$. We want to construct", "a morphism $a : V' \\to V$ over $Y$ such that $a_X = c$.", "Let $a' : V'' \\to V'$ be a surjective \\'etale morphism such that $V''$ is", "a separated algebraic space. If we can construct a morphism", "$a'' : V'' \\to V$ such that $a''_X = c \\circ a'_X$, then the two compositions", "$$", "V'' \\times_{V'} V'' \\xrightarrow{\\text{pr}_i} V'' \\xrightarrow{a''} V", "$$", "will be equal by the faithfulness of the functor proved in the first", "paragraph. Hence $a''$ will factor through a unique morphism", "$a : V' \\to V$ as $V'$ is (as a sheaf) the quotient of $V''$ by", "the equivalence relation $V'' \\times_{V'} V''$. Hence we may assume that", "$V'$ is separated. In this case the graph", "$$", "\\Gamma_c \\subset (V' \\times_Y V)_X", "$$", "is open and closed (details omitted). Since $X \\to Y$ is a universal", "homeomorphism, there exists an open and closed subspace", "$\\Gamma \\subset V' \\times_Y V$ such that $\\Gamma_X = \\Gamma_c$.", "The projection $\\Gamma \\to V'$ is an \\'etale morphism whose base", "change to $X$ is an isomorphism. Hence $\\Gamma \\to V'$ is \\'etale,", "universally injective, and surjective, so an isomorphism by", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-etale-universally-injective-open}.", "Thus $\\Gamma$ is the graph of a morphism $a : V' \\to V$ as desired.", "\\medskip\\noindent", "Finally, we prove that the functor is essentially surjective.", "Suppose that $U$ is an object of $X_{spaces, \\etale}$.", "We have to find an object $V$ of $Y_{spaces, \\etale}$", "such that $V_X \\cong U$. Let $U' \\to U$ be a surjective \\'etale morphism", "such that $U' \\cong V'_X$ and $U' \\times_U U' \\cong V''_X$", "for some objects $V'', V'$ of $Y_{spaces, \\etale}$.", "Then by fully faithfulness of the functor we obtain morphisms", "$s, t : V'' \\to V'$ with $t_X = \\text{pr}_0$ and $s_X = \\text{pr}_1$", "as morphisms $U' \\times_U U' \\to U'$. Using that", "$(\\text{pr}_0, \\text{pr}_1) : U' \\times_U U' \\to U' \\times_S U'$", "is an \\'etale equivalence relation, and that $U' \\to V'$ and", "$U' \\times_U U' \\to V''$ are universally injective and surjective", "we deduce that", "$(t, s) : V'' \\to V' \\times_S V'$ is an \\'etale equivalence relation.", "Then the quotient $V = V'/V''$ (see", "Spaces, Theorem \\ref{spaces-theorem-presentation})", "is an algebraic space $V$ over $Y$. There is a morphism", "$V' \\to V$ such that $V'' = V' \\times_V V'$. Thus we obtain a morphism", "$V \\to Y$ (see", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-fpqc-universal-effective-epimorphisms}).", "On base change to $X$ we see that we have a morphism $U' \\to V_X$", "and a compatible isomorphism $U' \\times_{V_X} U' = U' \\times_U U'$, which", "implies that $V_X \\cong U$ (by the lemma just cited once more).", "\\medskip\\noindent", "Pick a scheme $W$ and a surjective \\'etale morphism $W \\to Y$.", "Pick a scheme $U'$ and a surjective \\'etale morphism $U' \\to U \\times_X W_X$.", "Note that $U'$ and $U' \\times_U U'$ are schemes \\'etale over $X$ whose", "structure morphism to $X$ factors through the scheme $W_X$.", "Hence by", "\\'Etale Cohomology,", "Theorem \\ref{etale-cohomology-theorem-topological-invariance}", "there exist schemes $V', V''$ \\'etale over $W$ whose base change to", "$W_X$ is isomorphic to respectively $U'$ and $U' \\times_U U'$.", "This finishes the proof." ], "refs": [ "spaces-morphisms-lemma-etale-universally-injective-open", "spaces-theorem-presentation", "spaces-descent-lemma-fpqc-universal-effective-epimorphisms", "etale-cohomology-theorem-topological-invariance" ], "ref_ids": [ 4973, 8124, 9367, 6383 ] } ], "ref_ids": [] }, { "id": 10, "type": "theorem", "label": "spaces-more-morphisms-theorem-openness-flatness", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-theorem-openness-flatness", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Assume $f$ is locally of finite presentation and that", "$\\mathcal{F}$ is an $\\mathcal{O}_X$-module which is", "locally of finite presentation. Then", "$$", "\\{x \\in |X| : \\mathcal{F}\\text{ is flat over }Y\\text{ at }x\\}", "$$", "is open in $|X|$." ], "refs": [], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_p \\ar[r]_\\alpha &", "V \\ar[d]^q \\\\", "X \\ar[r]^a & Y", "}", "$$", "with $U$, $V$ schemes and $p$, $q$ surjective and \\'etale as in", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}.", "By", "More on Morphisms, Theorem \\ref{more-morphisms-theorem-openness-flatness}", "the set", "$U' = \\{u \\in |U| : p^*\\mathcal{F}\\text{ is flat over }V\\text{ at }u\\}$", "is open in $U$. By", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-flat-module}", "the image of $U'$ in $|X|$ is the set", "of the theorem. Hence we are done because the map $|U| \\to |X|$ is", "open, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-topology-points}." ], "refs": [ "spaces-lemma-lift-morphism-presentations", "more-morphisms-theorem-openness-flatness", "spaces-morphisms-definition-flat-module", "spaces-properties-lemma-topology-points" ], "ref_ids": [ 8159, 13670, 5008, 11822 ] } ], "ref_ids": [] }, { "id": 11, "type": "theorem", "label": "spaces-more-morphisms-theorem-criterion-flatness-fibre", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-theorem-criterion-flatness-fibre", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ and $Y \\to Z$ be morphisms of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume", "\\begin{enumerate}", "\\item $X$ is locally of finite presentation over $Z$,", "\\item $\\mathcal{F}$ an $\\mathcal{O}_X$-module of finite presentation, and", "\\item $Y$ is locally of finite type over $Z$.", "\\end{enumerate}", "Let $x \\in |X|$ and let $y \\in |Y|$ and $z \\in |Z|$ be the images of", "$x$. If $\\mathcal{F}_{\\overline{x}} \\not = 0$, then the following are", "equivalent:", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is flat over $Z$ at $x$ and", "the restriction of $\\mathcal{F}$ to its fibre over $z$", "is flat at $x$ over the fibre of $Y$ over $z$, and", "\\item $Y$ is flat over $Z$ at $y$ and $\\mathcal{F}$ is", "flat over $Y$ at $x$.", "\\end{enumerate}", "Moreover, the set of points $x$ where (1) and (2) hold is open in", "$\\text{Supp}(\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram as in", "Lemma \\ref{lemma-flat-on-fibres-at-point} part (3).", "It follows from the definitions that this reduces to the", "corresponding theorem for the morphisms of schemes", "$U \\to V \\to W$, the quasi-coherent sheaf $a^*\\mathcal{F}$,", "and the point $u \\in U$. Thus the theorem follows from the", "corresponding result for schemes which is", "More on Morphisms,", "Theorem \\ref{more-morphisms-theorem-criterion-flatness-fibre}." ], "refs": [ "spaces-more-morphisms-lemma-flat-on-fibres-at-point", "more-morphisms-theorem-criterion-flatness-fibre" ], "ref_ids": [ 130, 13672 ] } ], "ref_ids": [] }, { "id": 12, "type": "theorem", "label": "spaces-more-morphisms-theorem-criterion-flatness-fibre-Noetherian", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-theorem-criterion-flatness-fibre-Noetherian", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ and $Y \\to Z$ be morphisms of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume", "\\begin{enumerate}", "\\item $X$, $Y$, $Z$ locally Noetherian, and", "\\item $\\mathcal{F}$ a coherent $\\mathcal{O}_X$-module.", "\\end{enumerate}", "Let $x \\in |X|$ and let $y \\in |Y|$ and $z \\in |Z|$ be the images of", "$x$. If $\\mathcal{F}_{\\overline{x}} \\not = 0$, then the following are", "equivalent:", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is flat over $Z$ at $x$ and", "the restriction of $\\mathcal{F}$ to its fibre over $z$", "is flat at $x$ over the fibre of $Y$ over $z$, and", "\\item $Y$ is flat over $Z$ at $y$ and $\\mathcal{F}$ is", "flat over $Y$ at $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram as in", "Lemma \\ref{lemma-flat-on-fibres-at-point} part (3).", "It follows from the definitions that this reduces to the", "corresponding theorem for the morphisms of schemes", "$U \\to V \\to W$, the quasi-coherent sheaf $a^*\\mathcal{F}$,", "and the point $u \\in U$. Thus the theorem follows from the", "corresponding result for schemes which is", "More on Morphisms,", "Theorem \\ref{more-morphisms-theorem-criterion-flatness-fibre-Noetherian}." ], "refs": [ "spaces-more-morphisms-lemma-flat-on-fibres-at-point", "more-morphisms-theorem-criterion-flatness-fibre-Noetherian" ], "ref_ids": [ 130, 13671 ] } ], "ref_ids": [] }, { "id": 13, "type": "theorem", "label": "spaces-more-morphisms-theorem-stein-factorization-Noetherian", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-theorem-stein-factorization-Noetherian", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of algebraic", "spaces over $S$ with $Y$ locally Noetherian.", "There exists a factorization", "$$", "\\xymatrix{", "X \\ar[rr]_{f'} \\ar[rd]_f & & Y' \\ar[dl]^\\pi \\\\", "& Y &", "}", "$$", "with the following properties:", "\\begin{enumerate}", "\\item the morphism $f'$ is proper with connected geometric fibres,", "\\item the morphism $\\pi : Y' \\to Y$ is finite,", "\\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{Y'}$,", "\\item we have $Y' = \\underline{\\Spec}_Y(f_*\\mathcal{O}_X)$, and", "\\item $Y'$ is the normalization of $Y$ in $X$, see", "Morphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-normalization-X-in-Y" ], "proofs": [ { "contents": [ "Let $f = \\pi \\circ f'$ be the factorization of", "Lemma \\ref{lemma-stein-universally-closed}. Note that besides the", "conclusions of Lemma \\ref{lemma-stein-universally-closed} we", "also have that $f'$ is separated", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-compose-after-separated})", "and finite type", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-permanence-finite-type}).", "Hence $f'$ is proper. By", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-pushforward-coherent}", "we see that $f_*\\mathcal{O}_X$ is a coherent $\\mathcal{O}_Y$-module.", "Hence we see that $\\pi$ is finite, i.e., (2) holds.", "\\medskip\\noindent", "This proves all but the most interesting assertion, namely that", "the geometric fibres of $f'$ are connected. It is clear from the", "discussion above that we may replace $Y$ by $Y'$. ", "Then $Y$ is locally Noetherian,", "$f : X \\to Y$ is proper, and $f_*\\mathcal{O}_X = \\mathcal{O}_Y$.", "Let $\\overline{y}$ be a geometric point of $Y$.", "At this point we apply the theorem on formal functions,", "more precisely Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-formal-functions-stalk}.", "It tells us that", "$$", "\\mathcal{O}^\\wedge_{Y, \\overline{y}} =", "\\lim_n H^0(X_n, \\mathcal{O}_{X_n})", "$$", "where $X_n =", "\\Spec(\\mathcal{O}_{Y, \\overline{y}}/\\mathfrak m_{\\overline{y}}^n) \\times_Y X$.", "Note that $X_1 = X_{\\overline{y}} \\to X_n$ is a (finite order) thickening", "and hence the underlying topological space of $X_n$ is equal to that", "of $X_{\\overline{y}}$. Thus, if $X_{\\overline{y}} = T_1 \\amalg T_2$", "is a disjoint union of nonempty open and closed subspaces, then similarly", "$X_n = T_{1, n} \\amalg T_{2, n}$ for all $n$. And this in turn means", "$H^0(X_n, \\mathcal{O}_{X_n})$ contains a nontrivial idempotent $e_{1, n}$,", "namely the function which is identically $1$ on $T_{1, n}$ and", "identically $0$ on $T_{2, n}$. It is clear that $e_{1, n + 1}$", "restricts to $e_{1, n}$ on $X_n$. Hence $e_1 = \\lim e_{1, n}$", "is a nontrivial idempotent of the limit. This contradicts the fact", "that $\\mathcal{O}^\\wedge_{Y, \\overline{y}}$ is a local ring. Thus the", "assumption was wrong, i.e., $X_{\\overline{y}}$ is connected", "as desired." ], "refs": [ "spaces-more-morphisms-lemma-stein-universally-closed", "spaces-more-morphisms-lemma-stein-universally-closed", "spaces-morphisms-lemma-compose-after-separated", "spaces-morphisms-lemma-permanence-finite-type", "spaces-cohomology-lemma-proper-pushforward-coherent", "spaces-cohomology-lemma-formal-functions-stalk" ], "ref_ids": [ 176, 176, 4720, 4818, 11331, 11340 ] } ], "ref_ids": [ 5591 ] }, { "id": 14, "type": "theorem", "label": "spaces-more-morphisms-theorem-stein-factorization-general", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-theorem-stein-factorization-general", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of algebraic", "spaces over $S$. There exists a factorization", "$$", "\\xymatrix{", "X \\ar[rr]_{f'} \\ar[rd]_f & & Y' \\ar[dl]^\\pi \\\\", "& Y &", "}", "$$", "with the following properties:", "\\begin{enumerate}", "\\item the morphism $f'$ is proper with connected geometric fibres,", "\\item the morphism $\\pi : Y' \\to Y$ is integral,", "\\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{Y'}$,", "\\item we have $Y' = \\underline{\\Spec}_Y(f_*\\mathcal{O}_X)$, and", "\\item $Y'$ is the normalization of $Y$ in $X$ (Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-normalization-X-in-Y}).", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-normalization-X-in-Y" ], "proofs": [ { "contents": [ "We may apply Lemma \\ref{lemma-stein-universally-closed} to get the", "morphism $f' : X \\to Y'$.", "Note that besides the", "conclusions of Lemma \\ref{lemma-stein-universally-closed} we", "also have that $f'$ is separated", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-compose-after-separated})", "and finite type", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-permanence-finite-type}).", "Hence $f'$ is proper. At this point we have proved all of the", "statements except for the statement", "that $f'$ has connected geometric fibres.", "\\medskip\\noindent", "It is clear from the discussion that we may replace $Y$ by $Y'$. ", "Then $f : X \\to Y$ is proper and $f_*\\mathcal{O}_X = \\mathcal{O}_Y$.", "Note that these conditions are preserved under flat base change", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-proper}", "and", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-flat-base-change-cohomology}).", "Let $\\overline{y}$ be a geometric point of $Y$. By", "Lemma \\ref{lemma-characterize-geometrically-connected-fibres}", "and the remark just made we reduce to the case where $Y$ is a", "scheme, $y \\in Y$ is a point, $f : X \\to Y$ is a proper algebraic", "space over $Y$ with $f_*\\mathcal{O}_X = \\mathcal{O}_Y$,", "and we have to show the fibre $X_y$ is connected.", "Replacing $Y$ by an affine neighbourhood of $y$ we may", "assume that $Y = \\Spec(R)$ is affine. Then $f_*\\mathcal{O}_X = \\mathcal{O}_Y$", "signifies that the ring map", "$R \\to \\Gamma(X, \\mathcal{O}_X)$ is bijective.", "\\medskip\\noindent", "By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-proper-limit-of-proper-finite-presentation-noetherian}", "we can write $(X \\to Y) = \\lim (X_i \\to Y_i)$ with $X_i \\to Y_i$", "proper and of finite presentation and $Y_i$ Noetherian. For $i$ large", "enough $Y_i$ is affine (Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-limit-is-affine}).", "Say $Y_i = \\Spec(R_i)$. Let $R'_i = \\Gamma(X_i, \\mathcal{O}_{X_i})$.", "Observe that we have ring maps $R_i \\to R_i' \\to R$. Namely, we have", "the first because $X_i$ is an algebraic space over $R_i$ and the second because", "we have $X \\to X_i$ and $R = \\Gamma(X, \\mathcal{O}_X)$. Note that", "$R = \\colim R'_i$ by Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-section}.", "Then ", "$$", "\\xymatrix{", "X \\ar[d] \\ar[r] & X_i \\ar[d] \\\\", "Y \\ar[r] & Y'_i \\ar[r] & Y_i", "}", "$$", "is commutative with $Y'_i = \\Spec(R'_i)$.", "Let $y'_i \\in Y'_i$ be the image of $y$.", "We have $X_y = \\lim X_{i, y'_i}$ because $X = \\lim X_i$,", "$Y = \\lim Y'_i$, and $\\kappa(y) = \\colim \\kappa(y'_i)$.", "Now let $X_y = U \\amalg V$ with $U$ and $V$ open and closed.", "Then $U, V$ are the inverse images of opens $U_i, V_i$ in $X_{i, y'_i}$", "(Limits of Spaces, Lemma \\ref{spaces-limits-lemma-descend-opens}).", "By Theorem \\ref{theorem-stein-factorization-Noetherian} the fibres", "of $X_i \\to Y'_i$ are connected, hence either $U$ or $V$ is empty.", "This finishes the proof." ], "refs": [ "spaces-more-morphisms-lemma-stein-universally-closed", "spaces-more-morphisms-lemma-stein-universally-closed", "spaces-morphisms-lemma-compose-after-separated", "spaces-morphisms-lemma-permanence-finite-type", "spaces-morphisms-lemma-base-change-proper", "spaces-cohomology-lemma-flat-base-change-cohomology", "spaces-more-morphisms-lemma-characterize-geometrically-connected-fibres", "spaces-limits-lemma-proper-limit-of-proper-finite-presentation-noetherian", "spaces-limits-lemma-limit-is-affine", "spaces-limits-lemma-descend-opens", "spaces-more-morphisms-theorem-stein-factorization-Noetherian" ], "ref_ids": [ 176, 176, 4720, 4818, 4917, 11296, 178, 4617, 4578, 4575, 13 ] } ], "ref_ids": [ 5026 ] }, { "id": 15, "type": "theorem", "label": "spaces-more-morphisms-theorem-flatten-module", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-theorem-flatten-module", "contents": [ "Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $X$ be an algebraic space over $B$.", "Let $\\mathcal{F}$ be a quasi-coherent module on $X$.", "Let $U \\subset B$ be a quasi-compact open subspace. Assume", "\\begin{enumerate}", "\\item $X$ is quasi-compact,", "\\item $X$ is locally of finite presentation over $B$,", "\\item $\\mathcal{F}$ is a module of finite type,", "\\item $\\mathcal{F}_U$ is of finite presentation, and", "\\item $\\mathcal{F}_U$ is flat over $U$.", "\\end{enumerate}", "Then there exists a $U$-admissible blowup $B' \\to B$ such that the", "strict transform $\\mathcal{F}'$ of $\\mathcal{F}$ is an", "$\\mathcal{O}_{X \\times_B B'}$-module of finite presentation and", "flat over $B'$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $V$ and a surjective \\'etale morphism $V \\to X$.", "Because strict transform commutes with \\'etale localization", "(Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-strict-transform-local})", "it suffices to prove the result with $X$ replaced by $V$. Hence we", "may assume that $X \\to B$ is representable (in addition to the", "hypotheses of the lemma).", "\\medskip\\noindent", "Assume that $X \\to B$ is representable. Choose an affine scheme $W$ and a", "surjective \\'etale morphism $\\varphi : W \\to B$. Note that $X \\times_B W$", "is a scheme. By the case of schemes", "(More on Flatness, Theorem \\ref{flat-theorem-flatten-module})", "we can find a finite type quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_W$ such that", "(a) $|V(\\mathcal{I})| \\cap |\\varphi^{-1}(U)| = \\emptyset$ and (b)", "the strict transform of $\\mathcal{F}|_{X \\times_B W}$ with respect", "to the blowing up $W' \\to W$ in $\\mathcal{I}$ becomes flat over $W'$", "and is a module of finite presentation. Choose a finite type sheaf of ideals", "$\\mathcal{J} \\subset \\mathcal{O}_B$ as in Lemma \\ref{lemma-push-ideal}.", "Let $B' \\to B$ be the blowing up of $\\mathcal{J}$. We claim that this", "blowup works. Namely, it is clear that $B' \\to B$ is $U$-admissible", "by our choice of ideal $\\mathcal{J}$. Moreover, the base change", "$B' \\times_B W \\to W$ is the blowup of $W$ in", "$\\varphi^{-1}\\mathcal{J} = \\mathcal{I}\\mathcal{I}'$", "(compatibility of blowup with flat base change, see", "Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-flat-base-change-blowing-up}).", "Hence there is a factorization", "$$", "W \\times_B B' \\to W' \\to W", "$$", "where the first morphism is a blowup as well, see", "Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-blowing-up-two-ideals}).", "The restriction of $\\mathcal{F}'$ (which lives on $B' \\times_B X$)", "to $W \\times_B B' \\times_B X$ is the strict transform of", "$\\mathcal{F}|_{X \\times_B W}$", "(Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-strict-transform-local})", "and hence is the twice repeated strict transform of", "$\\mathcal{F}|_{X \\times_B W}$ by the two blowups displayed above", "(Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-strict-transform-composition-blowups}).", "After the first blowup our sheaf is already flat over", "the base and of finite presentation (by construction). Whence this holds ", "after the second strict transform as well (since this is a", "pullback by Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-strict-transform-flat}).", "Thus we see that the restriction of $\\mathcal{F}'$", "to an \\'etale cover of $B' \\times_B X$ has the desired properties", "and the theorem is proved." ], "refs": [ "spaces-divisors-lemma-strict-transform-local", "flat-theorem-flatten-module", "spaces-more-morphisms-lemma-push-ideal", "spaces-divisors-lemma-flat-base-change-blowing-up", "spaces-divisors-lemma-blowing-up-two-ideals", "spaces-divisors-lemma-strict-transform-local", "spaces-divisors-lemma-strict-transform-composition-blowups", "spaces-divisors-lemma-strict-transform-flat" ], "ref_ids": [ 13000, 5975, 189, 12990, 12997, 13000, 13005, 13002 ] } ], "ref_ids": [] }, { "id": 16, "type": "theorem", "label": "spaces-more-morphisms-theorem-grothendieck-existence", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-theorem-grothendieck-existence", "contents": [ "In Situation \\ref{situation-existence} the functor", "(\\ref{equation-completion-functor-proper-over-A})", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "We will use the equivalence of categories of", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-i-star-equivalence}", "without further mention in the proof of the theorem.", "By Lemma \\ref{lemma-fully-faithful} the functor is fully faithful.", "Thus we need to prove the functor is essentially surjective.", "\\medskip\\noindent", "Consider the collection $\\Xi$ of quasi-coherent sheaves of ideals", "$\\mathcal{K} \\subset \\mathcal{O}_X$ such that the statement holds", "for every object $(\\mathcal{F}_n)$ of", "$\\textit{Coh}_{\\text{support proper over }A}(X, \\mathcal{I})$", "annihilated by $\\mathcal{K}$. We want to show $(0)$ is in $\\Xi$.", "If not, then since $X$ is Noetherian there exists a maximal", "quasi-coherent sheaf of ideals $\\mathcal{K}$ not in $\\Xi$, see", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-acc-coherent}.", "After replacing $X$ by the closed subscheme of $X$", "corresponding to $\\mathcal{K}$ we may assume that every nonzero", "$\\mathcal{K}$ is in $\\Xi$. Let $(\\mathcal{F}_n)$ be an object of", "$\\textit{Coh}_{\\text{support proper over }A}(X, \\mathcal{I})$.", "We will show that this object is in the essential image, thereby", "completing the proof of the theorem.", "\\medskip\\noindent", "Apply Chow's lemma (Lemma \\ref{lemma-chow-noetherian-separated})", "to find a proper surjective morphism $f : Y \\to X$ which is an isomorphism", "over a dense open $U \\subset X$ such that $Y$ is H-quasi-projective", "over $A$. Note that $Y$ is a scheme and $f$ representable.", "Choose an open immersion $j : Y \\to Y'$ with $Y'$ projective over $A$, see", "Morphisms, Lemma \\ref{morphisms-lemma-H-quasi-projective-open-H-projective}.", "Let $T_n$ be the scheme theoretic support of $\\mathcal{F}_n$.", "Note that $|T_n| = |T_1|$, hence $T_n$ is proper over $A$ for all $n$", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-image-proper-is-proper}).", "Then $f^*\\mathcal{F}_n$ is supported on the closed subscheme", "$f^{-1}T_n$ which is proper over $A$ (by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-composition-proper}", "and properness of $f$).", "In particular, the composition $f^{-1}T_n \\to Y \\to Y'$ is closed", "(Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}).", "Let $T'_n \\subset Y'$ be the corresponding closed subscheme;", "it is contained in the open subscheme $Y$ and equal to $f^{-1}T_n$", "as a closed subscheme of $Y$. Let $\\mathcal{F}_n'$", "be the coherent $\\mathcal{O}_{Y'}$-module corresponding to", "$f^*\\mathcal{F}_n$ viewed as a coherent module on $Y'$ via", "the closed immersion $f^{-1}T_n = T'_n \\subset Y'$.", "Then $(\\mathcal{F}_n')$", "is an object of $\\textit{Coh}(Y', I\\mathcal{O}_{Y'})$.", "By the projective case of Grothendieck's existence theorem", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-existence-projective})", "there exists a coherent $\\mathcal{O}_{Y'}$-module", "$\\mathcal{F}'$ and an isomorphism", "$(\\mathcal{F}')^\\wedge \\cong (\\mathcal{F}'_n)$ in", "$\\textit{Coh}(Y', I\\mathcal{O}_{Y'})$.", "Let $Z' \\subset Y'$ be the scheme theoretic support of $\\mathcal{F}'$.", "Since $\\mathcal{F}'/I\\mathcal{F}' = \\mathcal{F}'_1$ we see", "that $Z' \\cap V(I\\mathcal{O}_{Y'}) = T'_1$ set-theoretically.", "The structure morphism $p' : Y' \\to \\Spec(A)$ is proper, hence", "$p'(Z' \\cap (Y' \\setminus Y))$ is closed in $\\Spec(A)$.", "If nonempty, then it would contain a point of $V(I)$", "as $I$ is contained in the Jacobson radical of $A$", "(Algebra, Lemma \\ref{algebra-lemma-radical-completion}).", "But we've seen above that $Z' \\cap (p')^{-1}V(I) = T'_1 \\subset Y$", "hence we conclude that $Z' \\subset Y$. Thus $\\mathcal{F}'|_Y$", "is supported on a closed subscheme of $Y$ proper over $A$.", "\\medskip\\noindent", "Let $\\mathcal{K}$ be the quasi-coherent sheaf of ideals cutting", "out the reduced complement $X \\setminus U$.", "By Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-pushforward-coherent}", "the $\\mathcal{O}_X$-module $\\mathcal{H} = f_*\\mathcal{F}'$ is coherent", "and by Lemma \\ref{lemma-inverse-systems-push-pull}", "there exists a morphism $\\alpha : (\\mathcal{F}_n) \\to \\mathcal{H}^\\wedge$", "in the category $\\textit{Coh}_{\\text{support proper over } A}(X, \\mathcal{I})$", "whose kernel and cokernel are annihilated by a power of $\\mathcal{K}$.", "Let $Z_0 \\subset X$ be the scheme theoretic support of $\\mathcal{H}$.", "It is clear that $|Z_0| \\subset f(|Z'|)$. Hence", "$Z_0 \\to \\Spec(A)$ is proper", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-image-proper-is-proper}).", "Thus $\\mathcal{H}$ is an object of ", "$\\textit{Coh}_{\\text{support proper over } A}(\\mathcal{O}_X)$.", "Since each of the sheaves of ideals $\\mathcal{K}^e$ is an element of", "$\\Xi$ we see that the assumptions of Lemma \\ref{lemma-existence-tricky}", "are satisfied and we conclude." ], "refs": [ "spaces-cohomology-lemma-i-star-equivalence", "spaces-more-morphisms-lemma-fully-faithful", "spaces-cohomology-lemma-acc-coherent", "spaces-more-morphisms-lemma-chow-noetherian-separated", "morphisms-lemma-H-quasi-projective-open-H-projective", "spaces-morphisms-lemma-image-proper-is-proper", "spaces-morphisms-lemma-composition-proper", "morphisms-lemma-image-proper-scheme-closed", "coherent-lemma-existence-projective", "algebra-lemma-radical-completion", "spaces-cohomology-lemma-proper-pushforward-coherent", "spaces-more-morphisms-lemma-inverse-systems-push-pull", "spaces-morphisms-lemma-image-proper-is-proper", "spaces-more-morphisms-lemma-existence-tricky" ], "ref_ids": [ 11303, 205, 11305, 198, 5428, 4921, 4918, 5411, 3383, 862, 11331, 208, 4921, 207 ] } ], "ref_ids": [] }, { "id": 17, "type": "theorem", "label": "spaces-more-morphisms-lemma-radicial-implies-universally-injective", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-radicial-implies-universally-injective", "contents": [ "A radicial morphism of algebraic spaces is universally injective." ], "refs": [], "proofs": [ { "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a radicial", "morphism of algebraic spaces over $S$.", "It is clear from the definition that given a morphism", "$\\Spec(K) \\to Y$ there is at most one lift of this morphism", "to a morphism into $X$. Hence we conclude that $f$ is universally", "injective by", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-universally-injective}." ], "refs": [ "spaces-morphisms-lemma-universally-injective" ], "ref_ids": [ 4793 ] } ], "ref_ids": [] }, { "id": 18, "type": "theorem", "label": "spaces-more-morphisms-lemma-when-universally-injective-radicial", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-when-universally-injective-radicial", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a universally injective", "morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item If $f$ is decent then $f$ is radicial.", "\\item If $f$ is quasi-separated then $f$ is radicial.", "\\item If $f$ is locally separated then $f$ is radicial.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces", "which is stable under base change and composition and holds for", "closed immersions. Assume $f : X \\to Y$ has $\\mathcal{P}$ and", "is universally injective. Then, in the situation of", "Definition \\ref{definition-radicial}", "the morphism $(\\Spec(K) \\times_Y X)_{red} \\to \\Spec(K)$", "is universally injective and has $\\mathcal{P}$. This reduces the", "problem of proving", "$$", "\\mathcal{P} + \\text{universally injective}", "\\Rightarrow", "\\text{radicial}", "$$", "to the problem of proving that any nonempty reduced algebraic space $X$", "over field whose structure morphism $X \\to \\Spec(K)$ is universally", "injective and $\\mathcal{P}$ is representable by the spectrum of a field.", "Namely, then $X \\to \\Spec(K)$ will be a morphism of schemes and", "we conclude by the equivalence of radicial and universally injective for", "morphisms of schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-universally-injective}.", "\\medskip\\noindent", "Let us prove (1). Assume $f$ is decent and universally injective. By", "Decent Spaces,", "Lemmas \\ref{decent-spaces-lemma-base-change-relative-conditions},", "\\ref{decent-spaces-lemma-composition-relative-conditions}, and", "\\ref{decent-spaces-lemma-properties-trivial-implications}", "(to see that an immersion is decent) we see that the discussion in", "the first paragraph applies.", "Let $X$ be a nonempty decent reduced algebraic space", "universally injective over a field $K$. In particular we see that $|X|$", "is a singleton. By", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-when-field}", "we conclude that $X \\cong \\Spec(L)$ for some extension", "$K \\subset L$ as desired.", "\\medskip\\noindent", "A quasi-separated morphism is decent, see", "Decent Spaces,", "Lemma \\ref{decent-spaces-lemma-properties-trivial-implications}.", "Hence (1) implies (2).", "\\medskip\\noindent", "Let us prove (3).", "Recall that the separation axioms are stable under base change", "and composition and that closed immersions are separated, see", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-base-change-separated},", "\\ref{spaces-morphisms-lemma-composition-separated}, and", "\\ref{spaces-morphisms-lemma-immersions-monomorphisms}.", "Thus the discussion in the first paragraph of the proof applies.", "Let $X$ be a reduced algebraic space universally injective and", "locally separated over a field $K$.", "In particular $|X|$ is a singleton hence $X$ is quasi-compact, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quasi-compact-space}.", "We can find a surjective \\'etale morphism $U \\to X$ with $U$ affine, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-quasi-compact-affine-cover}.", "Consider the morphism of schemes", "$$", "j :", "U \\times_X U", "\\longrightarrow", "U \\times_{\\Spec(K)} U", "$$", "As $X \\to \\Spec(K)$ is universally injective $j$ is surjective,", "and as $X \\to \\Spec(K)$ is locally separated $j$ is an immersion.", "A surjective immersion is a closed immersion, see", "Schemes, Lemma \\ref{schemes-lemma-immersion-when-closed}.", "Hence $R = U \\times_X U$ is affine as a closed subscheme of an affine scheme.", "In particular $R$ is quasi-compact.", "It follows that $X = U/R$ is quasi-separated, and the result follows from (2)." ], "refs": [ "spaces-more-morphisms-definition-radicial", "morphisms-lemma-universally-injective", "decent-spaces-lemma-base-change-relative-conditions", "decent-spaces-lemma-composition-relative-conditions", "decent-spaces-lemma-properties-trivial-implications", "decent-spaces-lemma-when-field", "decent-spaces-lemma-properties-trivial-implications", "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-composition-separated", "spaces-morphisms-lemma-immersions-monomorphisms", "spaces-properties-lemma-quasi-compact-space", "spaces-properties-lemma-quasi-compact-affine-cover", "schemes-lemma-immersion-when-closed" ], "ref_ids": [ 278, 5167, 9515, 9517, 9513, 9507, 9513, 4714, 4718, 4756, 11827, 11832, 7671 ] } ], "ref_ids": [] }, { "id": 19, "type": "theorem", "label": "spaces-more-morphisms-lemma-check-universally-injective", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-check-universally-injective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. Assume", "\\begin{enumerate}", "\\item $f$ is locally of finite type,", "\\item for every \\'etale morphism $V \\to Y$ the map $|X \\times_Y V| \\to |V|$", "is injective.", "\\end{enumerate}", "Then $f$ is universally injective." ], "refs": [], "proofs": [ { "contents": [ "The question is \\'etale local on $Y$ by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-injective-local}.", "Hence we may assume that $Y$ is a scheme.", "Then $Y$ is in particular decent and by Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-conditions-on-point-in-fibre-and-qf}", "we see that $f$ is locally quasi-finite.", "Let $y \\in Y$ be a point and let $X_y$ be the scheme theoretic", "fibre. Assume $X_y$ is not empty. By Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-locally-quasi-finite-over-field}", "we see that $X_y$ is a scheme which is locally quasi-finite over", "$\\kappa(y)$. Since $|X_y| \\subset |X|$ is the fibre of $|X| \\to |Y|$", "over $y$ we see that $X_y$ has a unique point $x$. The same is true", "for $X_y \\times_{\\Spec(\\kappa(y))} \\Spec(k)$ for any", "finite separable extension $\\kappa(y) \\subset k$", "because we can realize $k$ as the residue field at a point", "lying over $y$ in an \\'etale scheme over $Y$,", "see More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-realize-prescribed-residue-field-extension-etale}.", "Thus $X_y$ is geometrically connected, see", "Varieties, Lemma \\ref{varieties-lemma-characterize-geometrically-disconnected}.", "This implies that the finite extension $\\kappa(y) \\subset \\kappa(x)$", "is purely inseparable.", "\\medskip\\noindent", "We conclude (in the case that $Y$ is a scheme)", "that for every $y \\in Y$ either the fibre $X_y$ is empty,", "or $(X_y)_{red} = \\Spec(\\kappa(x))$ with", "$\\kappa(y) \\subset \\kappa(x)$ purely inseparable.", "Hence $f$ is radicial (some details omitted), whence universally injective by", "Lemma \\ref{lemma-radicial-implies-universally-injective}." ], "refs": [ "spaces-morphisms-lemma-universally-injective-local", "decent-spaces-lemma-conditions-on-point-in-fibre-and-qf", "spaces-over-fields-lemma-locally-quasi-finite-over-field", "more-morphisms-lemma-realize-prescribed-residue-field-extension-etale", "varieties-lemma-characterize-geometrically-disconnected", "spaces-more-morphisms-lemma-radicial-implies-universally-injective" ], "ref_ids": [ 4795, 9528, 12852, 13866, 10923, 17 ] } ], "ref_ids": [] }, { "id": 20, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-case", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-case", "contents": [ "A flat monomorphism of algebraic spaces is representable by schemes." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a flat monomorphism of algebraic spaces.", "To prove $f$ is representable, we have to show", "$X \\times_Y V$ is a scheme for every scheme $V$ mapping to $Y$.", "Since being a scheme is local (Properties of Spaces, ", "Lemma \\ref{spaces-properties-lemma-subscheme}), we may", "assume $V$ is affine. Thus we may assume $Y = \\Spec(B)$", "is an affine scheme. Next, we can assume that $X$ is quasi-compact", "by replacing $X$ by a quasi-compact open. The space $X$ is", "separated as $X \\to X \\times_{\\Spec(B)} X$ is an isomorphism.", "Applying Limits of Spaces, Lemma \\ref{spaces-limits-lemma-enough-local}", "we reduce to the case where $B$ is local, $X \\to \\Spec(B)$ is a", "flat monomorphism, and", "there exists a point $x \\in X$ mapping to the closed point of $\\Spec(B)$.", "Then $X \\to \\Spec(B)$ is surjective as generalizations", "lift along flat morphisms of separated algebraic spaces, see", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-generalizations-lift-flat}.", "Hence we see that $\\{X \\to \\Spec(B)\\}$ is an fpqc cover.", "Then $X \\to \\Spec(B)$ is a morphism which becomes an isomorphism", "after base change by $X \\to \\Spec(B)$. Hence it is an isomorphism by", "fpqc descent, see Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-isomorphism}." ], "refs": [ "spaces-properties-lemma-subscheme", "spaces-limits-lemma-enough-local", "decent-spaces-lemma-generalizations-lift-flat", "spaces-descent-lemma-descending-property-isomorphism" ], "ref_ids": [ 11848, 4632, 9474, 9395 ] } ], "ref_ids": [] }, { "id": 21, "type": "theorem", "label": "spaces-more-morphisms-lemma-ui-case", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-ui-case", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact monomorphism", "of algebraic spaces such that for every $T \\to Y$ the map", "$$", "\\mathcal{O}_T \\to f_{T,*}\\mathcal{O}_{X \\times_Y T}", "$$", "is injective. Then $f$ is an isomorphism (and hence representable by schemes)." ], "refs": [], "proofs": [ { "contents": [ "The question is \\'etale local on $Y$, hence we may assume $Y = \\Spec(A)$", "is affine. Then $X$ is quasi-compact and we may choose an affine scheme", "$U = \\Spec(B)$ and a surjective \\'etale morphism $U \\to X$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "Note that $U \\times_X U = \\Spec(B \\otimes_A B)$. Hence the category of", "quasi-coherent $\\mathcal{O}_X$-modules is equivalent to the", "category $DD_{B/A}$ of descent data on modules for $A \\to B$.", "See Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-quasi-coherent},", "Descent, Definition \\ref{descent-definition-descent-datum-modules}, and", "Descent, Subsection \\ref{descent-subsection-descent-modules-morphisms}.", "On the other hand,", "$$", "A \\to B", "$$", "is a universally injective ring map. Namely, given an", "$A$-module $M$ we see that $A \\oplus M \\to B \\otimes_A (A \\oplus M)$", "is injective by the assumption of the lemma. Hence", "$DD_{B/A}$ is equivalent to the category of $A$-modules by", "Descent, Theorem \\ref{descent-theorem-descent}. Thus pullback along", "$f : X \\to \\Spec(A)$ determines an equivalence of categories of", "quasi-coherent modules. In particular $f^*$ is exact on", "quasi-coherent modules and we see that $f$ is flat", "(small detail omitted). Moreover, it is clear that $f$ is surjective", "(for example because $\\Spec(B) \\to \\Spec(A)$ is surjective).", "Hence we see that $\\{X \\to \\Spec(A)\\}$ is an fpqc cover.", "Then $X \\to \\Spec(A)$ is a morphism which becomes an isomorphism", "after base change by $X \\to \\Spec(A)$. Hence it is an isomorphism by", "fpqc descent, see Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-isomorphism}." ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-properties-proposition-quasi-coherent", "descent-definition-descent-datum-modules", "descent-theorem-descent", "spaces-descent-lemma-descending-property-isomorphism" ], "ref_ids": [ 11832, 11920, 14759, 14590, 9395 ] } ], "ref_ids": [] }, { "id": 22, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-surjective-monomorphism", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-surjective-monomorphism", "contents": [ "A quasi-compact flat surjective monomorphism of algebraic spaces", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Such a morphism satisfies the assumptions of Lemma \\ref{lemma-ui-case}." ], "refs": [ "spaces-more-morphisms-lemma-ui-case" ], "ref_ids": [ 21 ] } ], "ref_ids": [] }, { "id": 23, "type": "theorem", "label": "spaces-more-morphisms-lemma-etale-conormal", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-etale-conormal", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion.", "Let $\\varphi : U \\to X$ be an \\'etale morphism where $U$ is a scheme.", "Set $Z_U = U \\times_X Z$ which is a locally closed subscheme of $U$.", "Then", "$$", "\\mathcal{C}_{Z/X}|_{Z_U} = \\mathcal{C}_{Z_U/U}", "$$", "canonically and functorially in $U$." ], "refs": [], "proofs": [ { "contents": [ "Let $T \\subset X$ be a closed subspace such that $i$ defines a closed", "immersion into $X \\setminus T$.", "Let $\\mathcal{I}$ be the quasi-coherent sheaf of ideals on", "$X \\setminus T$ defining $Z$. Then the lemma just states that", "$\\mathcal{I}|_{U \\setminus \\varphi^{-1}(T)}$ is the sheaf of ideals of", "the immersion $Z_U \\to U \\setminus \\varphi^{-1}(T)$.", "This is clear from the construction of $\\mathcal{I}$ in", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-closed-immersion-ideals}." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-ideals" ], "ref_ids": [ 4765 ] } ], "ref_ids": [] }, { "id": 24, "type": "theorem", "label": "spaces-more-morphisms-lemma-conormal-functorial", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-conormal-functorial", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\", "Z' \\ar[r]^{i'} & X'", "}", "$$", "be a commutative diagram of algebraic spaces over $S$.", "Assume $i$, $i'$ immersions. There is a canonical map", "of $\\mathcal{O}_Z$-modules", "$$", "f^*\\mathcal{C}_{Z'/X'}", "\\longrightarrow", "\\mathcal{C}_{Z/X}", "$$" ], "refs": [], "proofs": [ { "contents": [ "First find open subspaces $U' \\subset X'$ and $U \\subset X$ such that", "$g(U) \\subset U'$ and such that $i(Z) \\subset U$ and $i(Z') \\subset U'$", "are closed (proof existence omitted). Replacing $X$ by $U$ and $X'$ by", "$U'$ we may assume that $i$ and $i'$ are closed immersions.", "Let $\\mathcal{I}' \\subset \\mathcal{O}_{X'}$ and", "$\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent sheaves of", "ideals associated to $i'$ and $i$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-closed-immersion-ideals}.", "Consider the composition", "$$", "g^{-1}\\mathcal{I}' \\to g^{-1}\\mathcal{O}_{X'}", "\\xrightarrow{g^\\sharp} \\mathcal{O}_X \\to", "\\mathcal{O}_X/\\mathcal{I} = i_*\\mathcal{O}_Z", "$$", "Since $g(i(Z)) \\subset Z'$ we conclude this composition is zero (see", "statement on factorizations in", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-closed-immersion-ideals}).", "Thus we obtain a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{I} \\ar[r] &", "\\mathcal{O}_X \\ar[r] &", "i_*\\mathcal{O}_Z \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "g^{-1}\\mathcal{I}' \\ar[r] \\ar[u] &", "g^{-1}\\mathcal{O}_{X'} \\ar[r] \\ar[u] &", "g^{-1}i'_*\\mathcal{O}_{Z'} \\ar[r] \\ar[u] &", "0", "}", "$$", "The lower row is exact since $g^{-1}$ is an exact functor.", "By exactness we also see that", "$(g^{-1}\\mathcal{I}')^2 = g^{-1}((\\mathcal{I}')^2)$.", "Hence the diagram induces a map", "$g^{-1}(\\mathcal{I}'/(\\mathcal{I}')^2) \\to \\mathcal{I}/\\mathcal{I}^2$.", "Pulling back (using $i^{-1}$ for example) to $Z$ we obtain", "$i^{-1}g^{-1}(\\mathcal{I}'/(\\mathcal{I}')^2) \\to \\mathcal{C}_{Z/X}$.", "Since $i^{-1}g^{-1} = f^{-1}(i')^{-1}$ this gives a map", "$f^{-1}\\mathcal{C}_{Z'/X'} \\to \\mathcal{C}_{Z/X}$, which induces", "the desired map." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-ideals", "spaces-morphisms-lemma-closed-immersion-ideals" ], "ref_ids": [ 4765, 4765 ] } ], "ref_ids": [] }, { "id": 25, "type": "theorem", "label": "spaces-more-morphisms-lemma-conormal-functorial-more", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-conormal-functorial-more", "contents": [ "Let $S$ be a scheme. The conormal sheaf of", "Definition \\ref{definition-conormal-sheaf}, and its functoriality of", "Lemma \\ref{lemma-conormal-functorial}", "satisfy the following properties:", "\\begin{enumerate}", "\\item If $Z \\to X$ is an immersion of schemes over $S$, then the conormal", "sheaf agrees with the one from", "Morphisms, Definition \\ref{morphisms-definition-conormal-sheaf}.", "\\item If in", "Lemma \\ref{lemma-conormal-functorial}", "all the spaces are schemes, then the map", "$f^*\\mathcal{C}_{Z'/X'} \\to \\mathcal{C}_{Z/X}$ is the same", "as the one constructed in", "Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial}.", "\\item Given a commutative diagram", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\", "Z' \\ar[r]^{i'} \\ar[d]_{f'} & X' \\ar[d]^{g'} \\\\", "Z'' \\ar[r]^{i''} & X''", "}", "$$", "then the map $(f' \\circ f)^*\\mathcal{C}_{Z''/X''} \\to \\mathcal{C}_{Z/X}$", "is the same as the composition of", "$f^*\\mathcal{C}_{Z'/X'} \\to \\mathcal{C}_{Z/X}$", "with the pullback by $f$ of", "$(f')^*\\mathcal{C}_{Z''/X''} \\to \\mathcal{C}_{Z'/X'}$", "\\end{enumerate}" ], "refs": [ "spaces-more-morphisms-definition-conormal-sheaf", "spaces-more-morphisms-lemma-conormal-functorial", "morphisms-definition-conormal-sheaf", "spaces-more-morphisms-lemma-conormal-functorial", "morphisms-lemma-conormal-functorial" ], "proofs": [ { "contents": [ "Omitted. Note that Part (1) is a special case of", "Lemma \\ref{lemma-etale-conormal}." ], "refs": [ "spaces-more-morphisms-lemma-etale-conormal" ], "ref_ids": [ 23 ] } ], "ref_ids": [ 279, 24, 5562, 24, 5304 ] }, { "id": 26, "type": "theorem", "label": "spaces-more-morphisms-lemma-conormal-functorial-flat", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-conormal-functorial-flat", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\", "Z' \\ar[r]^{i'} & X'", "}", "$$", "be a fibre product diagram of algebraic spaces over $S$. Assume", "$i$, $i'$ immersions. Then the canonical map", "$f^*\\mathcal{C}_{Z'/X'} \\to \\mathcal{C}_{Z/X}$ of", "Lemma \\ref{lemma-conormal-functorial}", "is surjective. If $g$ is flat, then it is an isomorphism." ], "refs": [ "spaces-more-morphisms-lemma-conormal-functorial" ], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[r] \\ar[d] & X \\ar[d] \\\\", "U' \\ar[r] & X'", "}", "$$", "where $U$, $U'$ are schemes and the horizontal arrows are surjective", "and \\'etale, see", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}.", "Then using", "Lemmas \\ref{lemma-etale-conormal} and \\ref{lemma-conormal-functorial-more}", "we see that the question reduces to the case of a morphism of schemes.", "In the schemes case this is", "Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial-flat}." ], "refs": [ "spaces-lemma-lift-morphism-presentations", "spaces-more-morphisms-lemma-etale-conormal", "spaces-more-morphisms-lemma-conormal-functorial-more", "morphisms-lemma-conormal-functorial-flat" ], "ref_ids": [ 8159, 23, 25, 5305 ] } ], "ref_ids": [ 24 ] }, { "id": 27, "type": "theorem", "label": "spaces-more-morphisms-lemma-transitivity-conormal", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-transitivity-conormal", "contents": [ "Let $S$ be a scheme.", "Let $Z \\to Y \\to X$ be immersions of algebraic spaces.", "Then there is a canonical exact sequence", "$$", "i^*\\mathcal{C}_{Y/X} \\to", "\\mathcal{C}_{Z/X} \\to", "\\mathcal{C}_{Z/Y} \\to 0", "$$", "where the maps come from", "Lemma \\ref{lemma-conormal-functorial}", "and $i : Z \\to Y$ is the first morphism." ], "refs": [ "spaces-more-morphisms-lemma-conormal-functorial" ], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $U \\to X$ be a surjective \\'etale morphism. Via", "Lemmas \\ref{lemma-etale-conormal} and \\ref{lemma-conormal-functorial-more}", "the exactness of the sequence translates immediately into the", "exactness of the corresponding sequence for the immersions of schemes", "$Z \\times_X U \\to Y \\times_X U \\to U$. Hence the lemma follows from", "Morphisms, Lemma \\ref{morphisms-lemma-transitivity-conormal}." ], "refs": [ "spaces-more-morphisms-lemma-etale-conormal", "spaces-more-morphisms-lemma-conormal-functorial-more", "morphisms-lemma-transitivity-conormal" ], "ref_ids": [ 23, 25, 5306 ] } ], "ref_ids": [ 24 ] }, { "id": 28, "type": "theorem", "label": "spaces-more-morphisms-lemma-etale-conormal-algebra", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-etale-conormal-algebra", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic spaces", "over $S$. Let $\\varphi : U \\to X$ be an \\'etale morphism where $U$ is a", "scheme. Set $Z_U = U \\times_X Z$ which is a locally closed subscheme of $U$.", "Then", "$$", "\\mathcal{C}_{Z/X, *}|_{Z_U} = \\mathcal{C}_{Z_U/U, *}", "$$", "canonically and functorially in $U$." ], "refs": [], "proofs": [ { "contents": [ "Let $T \\subset X$ be a closed subspace such that $i$ defines a closed", "immersion into $X \\setminus T$. Let $\\mathcal{I}$ be the quasi-coherent", "sheaf of ideals on $X \\setminus T$ defining $Z$. Then the lemma follows", "from the fact that", "$\\mathcal{I}|_{U \\setminus \\varphi^{-1}(T)}$ is the sheaf of ideals of", "the immersion $Z_U \\to U \\setminus \\varphi^{-1}(T)$.", "This is clear from the construction of $\\mathcal{I}$ in", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-closed-immersion-ideals}." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-ideals" ], "ref_ids": [ 4765 ] } ], "ref_ids": [] }, { "id": 29, "type": "theorem", "label": "spaces-more-morphisms-lemma-conormal-algebra-functorial", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-conormal-algebra-functorial", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\", "Z' \\ar[r]^{i'} & X'", "}", "$$", "be a commutative diagram of algebraic spaces over $S$.", "Assume $i$, $i'$ immersions. There is a canonical map", "of graded $\\mathcal{O}_Z$-algebras", "$$", "f^*\\mathcal{C}_{Z'/X', *}", "\\longrightarrow", "\\mathcal{C}_{Z/X, *}", "$$" ], "refs": [], "proofs": [ { "contents": [ "First find open subspaces $U' \\subset X'$ and $U \\subset X$ such that", "$g(U) \\subset U'$ and such that $i(Z) \\subset U$ and $i(Z') \\subset U'$", "are closed (proof existence omitted). Replacing $X$ by $U$ and $X'$ by", "$U'$ we may assume that $i$ and $i'$ are closed immersions.", "Let $\\mathcal{I}' \\subset \\mathcal{O}_{X'}$ and", "$\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent sheaves of", "ideals associated to $i'$ and $i$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-closed-immersion-ideals}.", "Consider the composition", "$$", "g^{-1}\\mathcal{I}' \\to g^{-1}\\mathcal{O}_{X'}", "\\xrightarrow{g^\\sharp} \\mathcal{O}_X \\to", "\\mathcal{O}_X/\\mathcal{I} = i_*\\mathcal{O}_Z", "$$", "Since $g(i(Z)) \\subset Z'$ we conclude this composition is zero (see", "statement on factorizations in", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-closed-immersion-ideals}).", "Thus we obtain a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{I} \\ar[r] &", "\\mathcal{O}_X \\ar[r] &", "i_*\\mathcal{O}_Z \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "g^{-1}\\mathcal{I}' \\ar[r] \\ar[u] &", "g^{-1}\\mathcal{O}_{X'} \\ar[r] \\ar[u] &", "g^{-1}i'_*\\mathcal{O}_{Z'} \\ar[r] \\ar[u] &", "0", "}", "$$", "The lower row is exact since $g^{-1}$ is an exact functor.", "By exactness we also see that", "$(g^{-1}\\mathcal{I}')^n = g^{-1}((\\mathcal{I}')^n)$ for all $n \\geq 1$.", "Hence the diagram induces a map", "$g^{-1}((\\mathcal{I}')^n/(\\mathcal{I}')^{n + 1}) \\to", "\\mathcal{I}^n/\\mathcal{I}^{n + 1}$.", "Pulling back (using $i^{-1}$ for example) to $Z$ we obtain", "$i^{-1}g^{-1}((\\mathcal{I}')^n/(\\mathcal{I}')^{n + 1}) \\to", "\\mathcal{C}_{Z/X, n}$.", "Since $i^{-1}g^{-1} = f^{-1}(i')^{-1}$ this gives maps", "$f^{-1}\\mathcal{C}_{Z'/X', n} \\to \\mathcal{C}_{Z/X, n}$, which induce", "the desired map." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-ideals", "spaces-morphisms-lemma-closed-immersion-ideals" ], "ref_ids": [ 4765, 4765 ] } ], "ref_ids": [] }, { "id": 30, "type": "theorem", "label": "spaces-more-morphisms-lemma-conormal-algebra-functorial-flat", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-conormal-algebra-functorial-flat", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\", "Z' \\ar[r]^{i'} & X'", "}", "$$", "be a cartesian square of algebraic spaces over $S$ with", "$i$, $i'$ immersions. Then the canonical map", "$f^*\\mathcal{C}_{Z'/X', *} \\to \\mathcal{C}_{Z/X, *}$ of", "Lemma \\ref{lemma-conormal-algebra-functorial}", "is surjective. If $g$ is flat, then it is an isomorphism." ], "refs": [ "spaces-more-morphisms-lemma-conormal-algebra-functorial" ], "proofs": [ { "contents": [ "We may check the statement after \\'etale localizing $X'$.", "In this case we may assume $X' \\to X$ is a morphism of schemes,", "hence $Z$ and $Z'$ are schemes and the result follows from", "the case of schemes, see", "Divisors, Lemma \\ref{divisors-lemma-conormal-algebra-functorial-flat}." ], "refs": [ "divisors-lemma-conormal-algebra-functorial-flat" ], "ref_ids": [ 7981 ] } ], "ref_ids": [ 29 ] }, { "id": 31, "type": "theorem", "label": "spaces-more-morphisms-lemma-match-modules-differentials", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-match-modules-differentials", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let", "$f_{small} : X_\\etale \\to Y_\\etale$ be the associated", "morphism of small \\'etale sites, see", "Descent, Remark \\ref{descent-remark-change-topologies-ringed}.", "Then there is a canonical isomorphism", "$$", "(\\Omega_{X/Y})^a = \\Omega_{X_\\etale/Y_\\etale}", "$$", "compatible with universal derivations. Here the first module", "is the sheaf on $X_\\etale$ associated", "to the quasi-coherent $\\mathcal{O}_X$-module $\\Omega_{X/Y}$, see", "Morphisms, Definition \\ref{morphisms-definition-sheaf-differentials},", "and the second module is the one from", "Modules on Sites,", "Definition \\ref{sites-modules-definition-module-differentials}." ], "refs": [ "descent-remark-change-topologies-ringed", "morphisms-definition-sheaf-differentials", "sites-modules-definition-module-differentials" ], "proofs": [ { "contents": [ "Let $h : U \\to X$ be an \\'etale morphism. In this case the natural map", "$h^*\\Omega_{X/Y} \\to \\Omega_{U/Y}$ is an isomorphism, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-sheaf-differentials-etale-localization}.", "This means that there is a natural $\\mathcal{O}_{Y_\\etale}$-derivation", "$$", "\\text{d}^a : \\mathcal{O}_{X_\\etale} \\longrightarrow (\\Omega_{X/Y})^a", "$$", "since we have just seen that the value of $(\\Omega_{X/Y})^a$ on any object", "$U$ of $X_\\etale$ is canonically identified with", "$\\Gamma(U, \\Omega_{U/Y})$. By the universal property of", "$\\text{d}_{X/Y} :", "\\mathcal{O}_{X_\\etale}", "\\to", "\\Omega_{X_\\etale/Y_\\etale}$", "there is a unique $\\mathcal{O}_{X_\\etale}$-linear map", "$c : \\Omega_{X_\\etale/Y_\\etale} \\to (\\Omega_{X/Y})^a$", "such that", "$\\text{d}^a = c \\circ \\text{d}_{X/Y}$.", "\\medskip\\noindent", "Conversely, suppose that $\\mathcal{F}$ is an", "$\\mathcal{O}_{X_\\etale}$-module", "and $D : \\mathcal{O}_{X_\\etale} \\to \\mathcal{F}$ is a", "$\\mathcal{O}_{Y_\\etale}$-derivation. Then we can simply restrict", "$D$ to the small Zariski site $X_{Zar}$ of $X$. Since sheaves on $X_{Zar}$", "agree with sheaves on $X$, see", "Descent, Remark \\ref{descent-remark-Zariski-site-space},", "we see that $D|_{X_{Zar}} : \\mathcal{O}_X \\to \\mathcal{F}|_{X_{Zar}}$", "is just a ``usual'' $Y$-derivation. Hence we obtain a map", "$\\psi : \\Omega_{X/Y} \\longrightarrow \\mathcal{F}|_{X_{Zar}}$", "such that $D|_{X_{Zar}} = \\psi \\circ \\text{d}$. In particular, if we", "apply this with $\\mathcal{F} = \\Omega_{X_\\etale/Y_\\etale}$", "we obtain a map", "$$", "c' :", "\\Omega_{X/Y}", "\\longrightarrow", "\\Omega_{X_\\etale/Y_\\etale}|_{X_{Zar}}", "$$", "Consider the morphism of ringed sites", "$\\text{id}_{small, \\etale, Zar} : X_\\etale \\to X_{Zar}$", "discussed in", "Descent, Remark \\ref{descent-remark-change-topologies-ringed} and", "Lemma \\ref{descent-lemma-compare-sites}.", "Since the restriction functor $\\mathcal{F} \\mapsto \\mathcal{F}|_{X_{Zar}}$", "is equal to $\\text{id}_{small, \\etale, Zar, *}$, since", "$\\text{id}_{small, \\etale, Zar}^*$ is left adjoint to", "$\\text{id}_{small, \\etale, Zar, *}$ and since", "$(\\Omega_{X/Y})^a = \\text{id}_{small, \\etale, Zar}^*\\Omega_{X/Y}$", "we see that $c'$ is adjoint to a map", "$$", "c'' :", "(\\Omega_{X/Y})^a", "\\longrightarrow", "\\Omega_{X_\\etale/Y_\\etale}.", "$$", "We claim that $c''$ and $c'$ are mutually inverse.", "This claim finishes the proof of the lemma.", "To see this it is enough to show that $c''(\\text{d}(f)) = \\text{d}_{X/Y}(f)$", "and $c(\\text{d}_{X/Y}(f)) = \\text{d}(f)$ if $f$ is a local section of", "$\\mathcal{O}_X$ over an open of $X$. We omit the verification." ], "refs": [ "more-morphisms-lemma-sheaf-differentials-etale-localization", "descent-remark-Zariski-site-space", "descent-remark-change-topologies-ringed", "descent-lemma-compare-sites" ], "ref_ids": [ 13721, 14791, 14792, 14622 ] } ], "ref_ids": [ 14792, 5563, 14296 ] }, { "id": 32, "type": "theorem", "label": "spaces-more-morphisms-lemma-localize-differentials", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-localize-differentials", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Consider any commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_\\psi & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "where the vertical arrows are \\'etale morphisms of algebraic spaces. Then", "$$", "\\Omega_{X/Y}|_{U_\\etale} = \\Omega_{U/V}", "$$", "In particular, if $U$, $V$ are schemes, then this is equal to the usual", "sheaf of differentials of the morphism of schemes $U \\to V$." ], "refs": [], "proofs": [ { "contents": [ "By", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-morphism-topoi}", "and Equation (\\ref{spaces-properties-equation-restrict})", "we may think of the restriction of a sheaf on $X_\\etale$ to", "$U_\\etale$ as the pullback by $a_{small}$. Similarly for $b$. By", "Modules on Sites, Lemma \\ref{sites-modules-lemma-localize-differentials}", "we have", "$$", "\\Omega_{X/Y}|_{U_\\etale} =", "\\Omega_{\\mathcal{O}_{U_\\etale}/", "a_{small}^{-1}f_{small}^{-1}\\mathcal{O}_{Y_\\etale}}", "$$", "Since $a_{small}^{-1}f_{small}^{-1}\\mathcal{O}_{Y_\\etale}", "= \\psi_{small}^{-1}b_{small}^{-1}\\mathcal{O}_{Y_\\etale}", "= \\psi_{small}^{-1}\\mathcal{O}_{V_\\etale}$ we see that the lemma holds." ], "refs": [ "spaces-properties-lemma-etale-morphism-topoi", "sites-modules-lemma-localize-differentials" ], "ref_ids": [ 11866, 14231 ] } ], "ref_ids": [] }, { "id": 33, "type": "theorem", "label": "spaces-more-morphisms-lemma-module-differentials-quasi-coherent", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-module-differentials-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Then $\\Omega_{X/Y}$ is a quasi-coherent $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram as in", "Lemma \\ref{lemma-localize-differentials}", "with $a$ and $b$ surjective and $U$ and $V$ schemes.", "Then we see that $\\Omega_{X/Y}|_U = \\Omega_{U/V}$ which is", "quasi-coherent (for example by", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-diagonal}).", "Hence we conclude that $\\Omega_{X/Y}$ is quasi-coherent by", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-characterize-quasi-coherent}." ], "refs": [ "spaces-more-morphisms-lemma-localize-differentials", "morphisms-lemma-differentials-diagonal", "spaces-properties-lemma-characterize-quasi-coherent" ], "ref_ids": [ 32, 5311, 11911 ] } ], "ref_ids": [] }, { "id": 34, "type": "theorem", "label": "spaces-more-morphisms-lemma-functoriality-differentials", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-functoriality-differentials", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\", "Y' \\ar[r] & Y", "}", "$$", "be a commutative diagram of algebraic spaces. The map", "$f^\\sharp : \\mathcal{O}_X \\to f_*\\mathcal{O}_{X'}$ composed with the map", "$f_*\\text{d}_{X'/Y'} : f_*\\mathcal{O}_{X'} \\to f_*\\Omega_{X'/Y'}$ is a", "$Y$-derivation. Hence we obtain a canonical map of $\\mathcal{O}_X$-modules", "$\\Omega_{X/Y} \\to f_*\\Omega_{X'/Y'}$, and by", "adjointness of $f_*$ and $f^*$ a", "canonical $\\mathcal{O}_{X'}$-module homomorphism", "$$", "c_f : f^*\\Omega_{X/Y} \\longrightarrow \\Omega_{X'/Y'}.", "$$", "It is uniquely characterized by the property that", "$f^*\\text{d}_{X/Y}(t)$ mapsto $\\text{d}_{X'/Y'}(f^* t)$", "for any local section $t$ of $\\mathcal{O}_X$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-functoriality-differentials-ringed-topoi}." ], "refs": [ "sites-modules-lemma-functoriality-differentials-ringed-topoi" ], "ref_ids": [ 14235 ] } ], "ref_ids": [] }, { "id": 35, "type": "theorem", "label": "spaces-more-morphisms-lemma-check-functoriality-differentials", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-check-functoriality-differentials", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X'' \\ar[d] \\ar[r]_g & X' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\", "Y'' \\ar[r] & Y' \\ar[r] & Y", "}", "$$", "be a commutative diagram of algebraic spaces over $S$. Then we have", "$$", "c_{f \\circ g} = c_g \\circ g^* c_f", "$$", "as maps $(f \\circ g)^*\\Omega_{X/Y} \\to \\Omega_{X''/Y''}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use the characterization of $c_f, c_g, c_{f \\circ g}$", "in terms of the effect these maps have on local sections." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 36, "type": "theorem", "label": "spaces-more-morphisms-lemma-triangle-differentials", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-triangle-differentials", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$, $g : Y \\to B$ be morphisms of algebraic spaces over $S$.", "Then there is a canonical exact sequence", "$$", "f^*\\Omega_{Y/B} \\to \\Omega_{X/B} \\to \\Omega_{X/Y} \\to 0", "$$", "where the maps come from applications of", "Lemma \\ref{lemma-functoriality-differentials}." ], "refs": [ "spaces-more-morphisms-lemma-functoriality-differentials" ], "proofs": [ { "contents": [ "Follows from the schemes version, see", "Morphisms, Lemma \\ref{morphisms-lemma-triangle-differentials},", "of this result via \\'etale localization, see", "Lemma \\ref{lemma-localize-differentials}." ], "refs": [ "morphisms-lemma-triangle-differentials", "spaces-more-morphisms-lemma-localize-differentials" ], "ref_ids": [ 5313, 32 ] } ], "ref_ids": [ 34 ] }, { "id": 37, "type": "theorem", "label": "spaces-more-morphisms-lemma-immersion-differentials", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-immersion-differentials", "contents": [ "Let $S$ be a scheme. If $X \\to Y$ is an immersion", "of algebraic spaces over $S$ then $\\Omega_{X/S}$ is zero." ], "refs": [], "proofs": [ { "contents": [ "Follows from the schemes version, see", "Morphisms, Lemma \\ref{morphisms-lemma-immersion-differentials},", "of this result via \\'etale localization, see", "Lemma \\ref{lemma-localize-differentials}." ], "refs": [ "morphisms-lemma-immersion-differentials", "spaces-more-morphisms-lemma-localize-differentials" ], "ref_ids": [ 5318, 32 ] } ], "ref_ids": [] }, { "id": 38, "type": "theorem", "label": "spaces-more-morphisms-lemma-differentials-relative-immersion", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-differentials-relative-immersion", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $i : Z \\to X$ be an immersion of algebraic spaces over $B$.", "There is a canonical exact sequence", "$$", "\\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/B} \\to \\Omega_{Z/B} \\to 0", "$$", "where the first arrow is induced by $\\text{d}_{X/B}$", "and the second arrow comes from", "Lemma \\ref{lemma-functoriality-differentials}." ], "refs": [ "spaces-more-morphisms-lemma-functoriality-differentials" ], "proofs": [ { "contents": [ "This is the algebraic spaces version of", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-relative-immersion}", "and will be a consequence of that lemma by", "\\'etale localization, see", "Lemmas \\ref{lemma-localize-differentials} and", "\\ref{lemma-etale-conormal}.", "However, we should make sure we can define the first arrow globally.", "Hence we explain the meaning of ``induced by $\\text{d}_{X/B}$'' here.", "Namely, we may assume that $i$ is a closed immersion after replacing $X$", "by an open subspace. Let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be the quasi-coherent sheaf of ideals corresponding to $Z \\subset X$.", "Then $\\text{d}_{X/S} : \\mathcal{I} \\to \\Omega_{X/S}$", "maps the subsheaf $\\mathcal{I}^2 \\subset \\mathcal{I}$ to", "$\\mathcal{I}\\Omega_{X/S}$. Hence it induces a map", "$\\mathcal{I}/\\mathcal{I}^2 \\to \\Omega_{X/S}/\\mathcal{I}\\Omega_{X/S}$", "which is $\\mathcal{O}_X/\\mathcal{I}$-linear.", "By", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-i-star-equivalence}", "this corresponds to a map $\\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/S}$ as desired." ], "refs": [ "morphisms-lemma-differentials-relative-immersion", "spaces-more-morphisms-lemma-localize-differentials", "spaces-more-morphisms-lemma-etale-conormal", "spaces-morphisms-lemma-i-star-equivalence" ], "ref_ids": [ 5319, 32, 23, 4771 ] } ], "ref_ids": [ 34 ] }, { "id": 39, "type": "theorem", "label": "spaces-more-morphisms-lemma-differentials-relative-immersion-section", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-differentials-relative-immersion-section", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $i : Z \\to X$ be an immersion of algebraic spaces over $B$, and", "assume $i$ (\\'etale locally) has a left inverse. Then the canonical", "sequence", "$$", "0 \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/B} \\to \\Omega_{Z/B} \\to 0", "$$", "of", "Lemma \\ref{lemma-differentials-relative-immersion}", "is (\\'etale locally) split exact." ], "refs": [ "spaces-more-morphisms-lemma-differentials-relative-immersion" ], "proofs": [ { "contents": [ "Clarification: we claim that if $g : X \\to Z$ is a left inverse of $i$", "over $B$, then $i^*c_g$ is a right inverse of the map", "$i^*\\Omega_{X/B} \\to \\Omega_{Z/B}$.", "Having said this, the result follows from the corresponding result for", "morphisms of schemes by \\'etale localization, see", "Lemmas \\ref{lemma-localize-differentials} and", "\\ref{lemma-etale-conormal}." ], "refs": [ "spaces-more-morphisms-lemma-localize-differentials", "spaces-more-morphisms-lemma-etale-conormal" ], "ref_ids": [ 32, 23 ] } ], "ref_ids": [ 38 ] }, { "id": 40, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-differentials", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-differentials", "contents": [ "Let $S$ be a scheme.", "Let $X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $g : Y' \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $X' = X_{Y'}$ be the base change of $X$.", "Denote $g' : X' \\to X$ the projection.", "Then the map", "$$", "(g')^*\\Omega_{X/Y} \\to \\Omega_{X'/Y'}", "$$", "of", "Lemma \\ref{lemma-functoriality-differentials}", "is an isomorphism." ], "refs": [ "spaces-more-morphisms-lemma-functoriality-differentials" ], "proofs": [ { "contents": [ "Follows from the schemes version, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-differentials}", "and \\'etale localization, see", "Lemma \\ref{lemma-localize-differentials}." ], "refs": [ "morphisms-lemma-base-change-differentials", "spaces-more-morphisms-lemma-localize-differentials" ], "ref_ids": [ 5314, 32 ] } ], "ref_ids": [ 34 ] }, { "id": 41, "type": "theorem", "label": "spaces-more-morphisms-lemma-differential-product", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-differential-product", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to B$ and $g : Y \\to B$ be morphisms of algebraic spaces", "over $S$ with the same target.", "Let $p : X \\times_B Y \\to X$ and $q : X \\times_B Y \\to Y$ be the", "projection morphisms. The maps from", "Lemma \\ref{lemma-functoriality-differentials}", "$$", "p^*\\Omega_{X/B} \\oplus q^*\\Omega_{Y/B}", "\\longrightarrow", "\\Omega_{X \\times_B Y/B}", "$$", "give an isomorphism." ], "refs": [ "spaces-more-morphisms-lemma-functoriality-differentials" ], "proofs": [ { "contents": [ "Follows from the schemes version, see", "Morphisms, Lemma \\ref{morphisms-lemma-differential-product}", "and \\'etale localization, see", "Lemma \\ref{lemma-localize-differentials}." ], "refs": [ "morphisms-lemma-differential-product", "spaces-more-morphisms-lemma-localize-differentials" ], "ref_ids": [ 5315, 32 ] } ], "ref_ids": [ 34 ] }, { "id": 42, "type": "theorem", "label": "spaces-more-morphisms-lemma-finite-type-differentials", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-finite-type-differentials", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $f$ is locally of finite type, then $\\Omega_{X/Y}$ is", "a finite type $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "Follows from the schemes version, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-differentials}", "and \\'etale localization, see", "Lemma \\ref{lemma-localize-differentials}." ], "refs": [ "morphisms-lemma-finite-type-differentials", "spaces-more-morphisms-lemma-localize-differentials" ], "ref_ids": [ 5316, 32 ] } ], "ref_ids": [] }, { "id": 43, "type": "theorem", "label": "spaces-more-morphisms-lemma-finite-presentation-differentials", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-finite-presentation-differentials", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $f$ is locally of finite presentation, then $\\Omega_{X/Y}$ is", "an $\\mathcal{O}_X$-module of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Follows from the schemes version, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-differentials}", "and \\'etale localization, see", "Lemma \\ref{lemma-localize-differentials}." ], "refs": [ "morphisms-lemma-finite-presentation-differentials", "spaces-more-morphisms-lemma-localize-differentials" ], "ref_ids": [ 5317, 32 ] } ], "ref_ids": [] }, { "id": 44, "type": "theorem", "label": "spaces-more-morphisms-lemma-smooth-omega-finite-locally-free", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-smooth-omega-finite-locally-free", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a smooth morphism of algebraic spaces over $S$.", "Then the module of differentials $\\Omega_{X/Y}$", "is finite locally free." ], "refs": [], "proofs": [ { "contents": [ "The statement is \\'etale local on $X$ and $Y$ by", "Lemma \\ref{lemma-localize-differentials}.", "Hence this follows from the case of schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-omega-finite-locally-free}." ], "refs": [ "spaces-more-morphisms-lemma-localize-differentials", "morphisms-lemma-smooth-omega-finite-locally-free" ], "ref_ids": [ 32, 5334 ] } ], "ref_ids": [] }, { "id": 45, "type": "theorem", "label": "spaces-more-morphisms-lemma-topological-invariance", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-topological-invariance", "contents": [ "With assumption and notation as in", "Theorem \\ref{theorem-topological-invariance}", "the equivalence of categories", "$Y_{spaces, \\etale} \\to X_{spaces, \\etale}$", "restricts to equivalences of categories", "$Y_\\etale \\to X_\\etale$ and $Y_{affine, \\etale} \\to X_{affine, \\etale}$." ], "refs": [ "spaces-more-morphisms-theorem-topological-invariance" ], "proofs": [ { "contents": [ "This is just the statement that given an object", "$V \\in Y_{spaces, \\etale}$ we have $V$ is a(n affine) scheme if and", "only if $V \\times_Y X$ is a(n affine) scheme. Since $V \\times_Y X \\to V$", "is integral, universally injective, and surjective (as a base", "change of $X \\to Y$) this", "follows from Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-integral-universally-bijective-scheme} and", "Proposition \\ref{spaces-limits-proposition-affine}." ], "refs": [ "spaces-limits-lemma-integral-universally-bijective-scheme", "spaces-limits-proposition-affine" ], "ref_ids": [ 4628, 4658 ] } ], "ref_ids": [ 9 ] }, { "id": 46, "type": "theorem", "label": "spaces-more-morphisms-lemma-first-order-thickening-maps", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-first-order-thickening-maps", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $X \\subset X'$ and $Y \\subset Y'$ be thickenings", "of algebraic spaces over $B$. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $B$. Given any map of $\\mathcal{O}_B$-algebras", "$$", "\\alpha : f_{spaces, \\etale}^{-1}\\mathcal{O}_{Y'} \\to \\mathcal{O}_{X'}", "$$", "such that", "$$", "\\xymatrix{", "f_{spaces, \\etale}^{-1}\\mathcal{O}_Y \\ar[r]_-{f^\\sharp} \\ar[r] &", "\\mathcal{O}_X \\\\", "f_{spaces, \\etale}^{-1}\\mathcal{O}_{Y'} \\ar[r]^-\\alpha", "\\ar[u]^{i_Y^\\sharp} &", "\\mathcal{O}_{X'} \\ar[u]_{i_X^\\sharp}", "}", "$$", "commutes, there exists a unique morphism of $(f, f')$ of", "thickenings over $B$ such that $\\alpha = (f')^\\sharp$." ], "refs": [], "proofs": [ { "contents": [ "To find $f'$, by", "Properties of Spaces, Theorem \\ref{spaces-properties-theorem-fully-faithful},", "all we have to do is show that the morphism of ringed topoi", "$$", "(f_{spaces, \\etale}, \\alpha) :", "(\\Sh(X_{spaces, \\etale}), \\mathcal{O}_{X'})", "\\longrightarrow", "(\\Sh(Y_{spaces, \\etale}), \\mathcal{O}_{Y'})", "$$", "is a morphism of locally ringed topoi. This follows directly", "from the definition of morphisms of locally ringed topoi", "(Modules on Sites,", "Definition \\ref{sites-modules-definition-morphism-locally-ringed-topoi}),", "the fact that $(f, f^\\sharp)$ is a morphism of locally ringed topoi", "(Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-morphism-locally-ringed}),", "that $\\alpha$ fits into the given commutative diagram, and", "the fact that the kernels of $i_X^\\sharp$ and $i_Y^\\sharp$ are", "locally nilpotent. Finally, the fact that $f' \\circ i_X = i_Y \\circ f$", "follows from the commutativity of the diagram and another application of", "Properties of Spaces, Theorem \\ref{spaces-properties-theorem-fully-faithful}.", "We omit the verification that $f'$ is a morphism over $B$." ], "refs": [ "spaces-properties-theorem-fully-faithful", "sites-modules-definition-morphism-locally-ringed-topoi", "spaces-properties-lemma-morphism-locally-ringed", "spaces-properties-theorem-fully-faithful" ], "ref_ids": [ 11814, 14304, 11903, 11814 ] } ], "ref_ids": [] }, { "id": 47, "type": "theorem", "label": "spaces-more-morphisms-lemma-open-subspace-thickening", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-open-subspace-thickening", "contents": [ "Let $S$ be a scheme. Let $X \\subset X'$ be a thickening", "of algebraic spaces over $S$. For any open subspace $U \\subset X$ there", "exists a unique open subspace $U' \\subset X'$ such that", "$U = X \\times_{X'} U'$." ], "refs": [], "proofs": [ { "contents": [ "Let $U' \\to X'$ be the object of $X'_{spaces, \\etale}$", "corresponding to the object $U \\to X$ of $X_{spaces, \\etale}$", "via (\\ref{equation-equivalence-etale-spaces}). The morphism", "$U' \\to X'$ is \\'etale and universally injective, hence an open immersion, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-etale-universally-injective-open}." ], "refs": [ "spaces-morphisms-lemma-etale-universally-injective-open" ], "ref_ids": [ 4973 ] } ], "ref_ids": [] }, { "id": 48, "type": "theorem", "label": "spaces-more-morphisms-lemma-first-order-thickening-surjective", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-first-order-thickening-surjective", "contents": [ "Let $S$ be a scheme. Let $X \\subset X'$ be a thickening", "of algebraic spaces over $S$. Let $U$ be an affine object of", "$X_{spaces, \\etale}$. Then", "$$", "\\Gamma(U, \\mathcal{O}_{X'}) \\to \\Gamma(U, \\mathcal{O}_X)", "$$", "is surjective where we think of $\\mathcal{O}_{X'}$ as a sheaf on", "$X_{spaces, \\etale}$ via (\\ref{equation-fundamental-equivalence})." ], "refs": [], "proofs": [ { "contents": [ "Let $U' \\to X'$ be the \\'etale morphism of algebraic spaces such that", "$U = X \\times_{X'} U'$, see Theorem \\ref{theorem-topological-invariance}.", "By Limits of Spaces, Lemma \\ref{spaces-limits-lemma-affine} we see", "that $U'$ is an affine scheme. Hence", "$\\Gamma(U, \\mathcal{O}_{X'}) = \\Gamma(U', \\mathcal{O}_{U'}) \\to", "\\Gamma(U, \\mathcal{O}_U)$", "is surjective as $U \\to U'$ is a closed immersion of affine schemes.", "Below we give a direct proof for finite order thickenings", "which is the case most used in practice." ], "refs": [ "spaces-more-morphisms-theorem-topological-invariance", "spaces-limits-lemma-affine" ], "ref_ids": [ 9, 4626 ] } ], "ref_ids": [] }, { "id": 49, "type": "theorem", "label": "spaces-more-morphisms-lemma-thickening-scheme", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-thickening-scheme", "contents": [ "Let $S$ be a scheme. Let $X \\subset X'$ be a thickening of algebraic spaces", "over $S$. If $X$ is (representable by) a scheme, then so is $X'$." ], "refs": [], "proofs": [ { "contents": [ "Note that $X'_{red} = X_{red}$. Hence if $X$ is a scheme, then", "$X'_{red}$ is a scheme. Thus the result follows from", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-reduction-scheme}.", "Below we give a direct proof for finite order thickenings which is", "the case most often used in practice." ], "refs": [ "spaces-limits-lemma-reduction-scheme" ], "ref_ids": [ 4627 ] } ], "ref_ids": [] }, { "id": 50, "type": "theorem", "label": "spaces-more-morphisms-lemma-thickening-equivalence", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-thickening-equivalence", "contents": [ "Let $S$ be a scheme. Let $X \\subset X'$ be a thickening of algebraic spaces", "over $S$. The functor", "$$", "V' \\longmapsto V = X \\times_{X'} V'", "$$", "defines an equivalence of categories", "$X'_\\etale \\to X_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "The functor $V' \\mapsto V$ defines an equivalence of categories", "$X'_{spaces, \\etale} \\to X_{spaces, \\etale}$, see", "Theorem \\ref{theorem-topological-invariance}.", "Thus it suffices to show that $V$ is a scheme if and only if $V'$ is", "a scheme. This is the content of", "Lemma \\ref{lemma-thickening-scheme}." ], "refs": [ "spaces-more-morphisms-theorem-topological-invariance", "spaces-more-morphisms-lemma-thickening-scheme" ], "ref_ids": [ 9, 49 ] } ], "ref_ids": [] }, { "id": 51, "type": "theorem", "label": "spaces-more-morphisms-lemma-first-order-thickening", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-first-order-thickening", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to B$ be a morphism of algebraic spaces over $S$.", "Consider a short exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{A} \\to \\mathcal{O}_X \\to 0", "$$", "of sheaves on $X_\\etale$ where $\\mathcal{A}$ is a sheaf of", "$f^{-1}\\mathcal{O}_B$-algebras, $\\mathcal{A} \\to \\mathcal{O}_X$ is a surjection", "of sheaves of $f^{-1}\\mathcal{O}_B$-algebras, and $\\mathcal{I}$ is its kernel.", "If", "\\begin{enumerate}", "\\item $\\mathcal{I}$ is an ideal of square zero in $\\mathcal{A}$, and", "\\item $\\mathcal{I}$ is quasi-coherent as an $\\mathcal{O}_X$-module", "\\end{enumerate}", "then there exists a first order thickening", "$X \\subset X'$ over $B$ and an isomorphism", "$\\mathcal{O}_{X'} \\to \\mathcal{A}$ of $f^{-1}\\mathcal{O}_B$-algebras", "compatible with the surjections to $\\mathcal{O}_X$." ], "refs": [], "proofs": [ { "contents": [ "In this proof we redo some of the arguments used in the", "proofs of", "Lemmas \\ref{lemma-first-order-thickening-surjective} and", "\\ref{lemma-thickening-scheme}.", "We first handle the case $B = S = \\Spec(\\mathbf{Z})$.", "Let $U$ be an affine scheme, and let $U \\to X$ be \\'etale.", "Then", "$$", "0 \\to \\mathcal{I}(U) \\to \\mathcal{A}(U) \\to \\mathcal{O}_X(U) \\to 0", "$$", "is exact as $H^1(U_\\etale, \\mathcal{I}) = 0$ as", "$\\mathcal{I}$ is quasi-coherent, see", "Descent, Proposition \\ref{descent-proposition-same-cohomology-quasi-coherent}", "and Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "If $V \\to U$ is a morphism of affine objects of $X_{spaces, \\etale}$", "then", "$$", "\\mathcal{I}(V) = \\mathcal{I}(U) \\otimes_{\\mathcal{O}_X(U)} \\mathcal{O}_X(V)", "$$", "since $\\mathcal{I}$ is a quasi-coherent $\\mathcal{O}_X$-module, see", "Descent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent}.", "Hence $\\mathcal{A}(U) \\to \\mathcal{A}(V)$ is an", "\\'etale ring map, see", "Algebra, Lemma \\ref{algebra-lemma-lift-etale-infinitesimal}.", "Hence we see that", "$$", "U \\longmapsto U' = \\Spec(\\mathcal{A}(U))", "$$", "is a functor from $X_{affine, \\etale}$ to the category of affine", "schemes and \\'etale morphisms. In fact, we claim that this functor can", "be extended to a functor $U \\mapsto U'$ on all of $X_\\etale$.", "To see this, if $U$ is an object of $X_\\etale$, note that", "$$", "0 \\to \\mathcal{I}|_{U_{Zar}} \\to \\mathcal{A}|_{U_{Zar}} \\to", "\\mathcal{O}_X|_{U_{Zar}} \\to 0", "$$", "and $\\mathcal{I}|_{U_{Zar}}$ is a quasi-coherent sheaf on $U$, see", "Descent,", "Proposition \\ref{descent-proposition-equivalence-quasi-coherent-functorial}.", "Hence by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-first-order-thickening}", "we obtain a first order thickening $U \\subset U'$ of schemes such that", "$\\mathcal{O}_{U'}$ is isomorphic to $\\mathcal{A}|_{U_{Zar}}$. It is clear that", "this construction is compatible with the construction for affines above.", "\\medskip\\noindent", "Choose a presentation $X = U/R$, see", "Spaces, Definition \\ref{spaces-definition-presentation}", "so that $s, t : R \\to U$ define an \\'etale equivalence relation.", "Applying the functor above we obtain an \\'etale equivalence", "relation $s', t' : R' \\to U'$ in schemes. Consider the algebraic space", "$X' = U'/R'$ (see", "Spaces, Theorem \\ref{spaces-theorem-presentation}).", "The morphism $X = U/R \\to U'/R' = X'$ is a first order thickening.", "Consider $\\mathcal{O}_{X'}$ viewed as a sheaf on $X_\\etale$.", "By construction we have an isomorphism", "$$", "\\gamma :", "\\mathcal{O}_{X'}|_{U_\\etale}", "\\longrightarrow", "\\mathcal{A}|_{U_\\etale}", "$$", "such that $s^{-1}\\gamma$ agrees with $t^{-1}\\gamma$ on $R_\\etale$.", "Hence by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-descent-sheaf}", "this implies that $\\gamma$ comes from a unique isomorphism", "$\\mathcal{O}_{X'} \\to \\mathcal{A}$ as desired.", "\\medskip\\noindent", "To handle the case of a general base algebraic space $B$, we first", "construct $X'$ as an algebraic space over $\\mathbf{Z}$ as above.", "Then we use the isomorphism $\\mathcal{O}_{X'} \\to \\mathcal{A}$ to", "define $f^{-1}\\mathcal{O}_B \\to \\mathcal{O}_{X'}$. According to", "Lemma \\ref{lemma-first-order-thickening-maps}", "this defines a morphism $X' \\to B$ compatible with the given morphism", "$X \\to B$ and we are done." ], "refs": [ "spaces-more-morphisms-lemma-first-order-thickening-surjective", "spaces-more-morphisms-lemma-thickening-scheme", "descent-proposition-same-cohomology-quasi-coherent", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "descent-proposition-equivalence-quasi-coherent", "algebra-lemma-lift-etale-infinitesimal", "descent-proposition-equivalence-quasi-coherent-functorial", "more-morphisms-lemma-first-order-thickening", "spaces-definition-presentation", "spaces-theorem-presentation", "spaces-properties-lemma-descent-sheaf", "spaces-more-morphisms-lemma-first-order-thickening-maps" ], "ref_ids": [ 48, 49, 14754, 3282, 14755, 1239, 14756, 13677, 8177, 8124, 11869, 46 ] } ], "ref_ids": [] }, { "id": 52, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-thickening", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-thickening", "contents": [ "Let $S$ be a scheme. Let $Y \\subset Y'$ be a thickening of algebraic spaces", "over $S$. Let $X' \\to Y'$ be a morphism and set $X = Y \\times_{Y'} X'$.", "Then $(X \\subset X') \\to (Y \\subset Y')$", "is a morphism of thickenings. If $Y \\subset Y'$ is a first", "(resp.\\ finite order) thickening, then $X \\subset X'$ is a first", "(resp.\\ finite order) thickening." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 53, "type": "theorem", "label": "spaces-more-morphisms-lemma-composition-thickening", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-composition-thickening", "contents": [ "Let $S$ be a scheme. If $X \\subset X'$ and $X' \\subset X''$ are", "thickenings of algebraic spaces over $S$, then so is $X \\subset X''$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 54, "type": "theorem", "label": "spaces-more-morphisms-lemma-descending-property-thickening", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-descending-property-thickening", "contents": [ "The property of being a thickening is fpqc local.", "Similarly for first order thickenings." ], "refs": [], "proofs": [ { "contents": [ "The statement means the following: Let $S$ be a scheme and let", "$X \\to X'$ be a morphism of algebraic spaces over $S$.", "Let $\\{g_i : X'_i \\to X'\\}$ be an fpqc covering of algebraic spaces", "such that the base change $X_i \\to X'_i$ is a thickening for all $i$.", "Then $X \\to X'$ is a thickening. Since the morphisms $g_i$ are jointly", "surjective we conclude that $X \\to X'$ is surjective. By", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-closed-immersion}", "we conclude that $X \\to X'$ is a closed immersion.", "Thus $X \\to X'$ is a thickening. We omit the proof in the", "case of first order thickenings." ], "refs": [ "spaces-descent-lemma-descending-property-closed-immersion" ], "ref_ids": [ 9397 ] } ], "ref_ids": [] }, { "id": 55, "type": "theorem", "label": "spaces-more-morphisms-lemma-thicken-property-morphisms", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-thicken-property-morphisms", "contents": [ "Let $S$ be a scheme. Let $(f, f') : (X \\subset X') \\to (Y \\subset Y')$", "be a morphism of thickenings of algebraic spaces over $S$. Then", "\\begin{enumerate}", "\\item $f$ is an affine morphism if and only if $f'$ is an affine morphism,", "\\item $f$ is a surjective morphism if and only if $f'$ is a surjective morphism,", "\\item $f$ is quasi-compact if and only if $f'$ quasi-compact,", "\\item $f$ is universally closed if and only if $f'$ is universally closed,", "\\item $f$ is integral if and only if $f'$ is integral,", "\\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,", "\\item $f$ is universally injective if and only if $f'$ is universally injective,", "\\item $f$ is universally open if and only if $f'$ is universally open,", "\\item $f$ is representable if and only if $f'$ is representable, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $Y \\to Y'$ and $X \\to X'$ are integral and", "universal homeomorphisms. This immediately implies parts", "(2), (3), (4), (7), and (8).", "Part (1) follows from", "Limits of Spaces, Proposition \\ref{spaces-limits-proposition-affine}", "which tells us that there is a 1-to-1 correspondence between", "affine schemes \\'etale over $X$ and $X'$ and between affine schemes", "\\'etale over $Y$ and $Y'$.", "Part (5) follows from (1) and (4) by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-integral-universally-closed}.", "Finally, note that", "$$", "X \\times_Y X = X \\times_{Y'} X \\to X \\times_{Y'} X' \\to X' \\times_{Y'} X'", "$$", "is a thickening (the two arrows are thickenings by", "Lemma \\ref{lemma-base-change-thickening}).", "Hence applying (3) and (4) to the morphism", "$(X \\subset X') \\to (X \\times_Y X \\to X' \\times_{Y'} X')$", "we obtain (6). Finally, part (9) follows from the fact that an", "algebraic space thickening of a scheme is again a scheme, see", "Lemma \\ref{lemma-thickening-scheme}." ], "refs": [ "spaces-limits-proposition-affine", "spaces-morphisms-lemma-integral-universally-closed", "spaces-more-morphisms-lemma-base-change-thickening", "spaces-more-morphisms-lemma-thickening-scheme" ], "ref_ids": [ 4658, 4944, 52, 49 ] } ], "ref_ids": [] }, { "id": 56, "type": "theorem", "label": "spaces-more-morphisms-lemma-thicken-property-morphisms-cartesian", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-thicken-property-morphisms-cartesian", "contents": [ "Let $S$ be a scheme. Let $(f, f') : (X \\subset X') \\to (Y \\subset Y')$ be a", "morphism of thickenings of algebraic spaces over $S$ such that", "$X = Y \\times_{Y'} X'$. If $X \\subset X'$ is a finite order thickening, then", "\\begin{enumerate}", "\\item $f$ is a closed immersion if and only if $f'$ is a closed immersion,", "\\item $f$ is locally of finite type if and only if $f'$ is", "locally of finite type,", "\\item $f$ is locally quasi-finite if and only if $f'$ is locally", "quasi-finite,", "\\item $f$ is locally of finite type of relative dimension $d$ if and", "only if $f'$ is locally of finite type of relative dimension $d$,", "\\item $\\Omega_{X/Y} = 0$ if and only if $\\Omega_{X'/Y'} = 0$,", "\\item $f$ is unramified if and only if $f'$ is unramified,", "\\item $f$ is proper if and only if $f'$ is proper,", "\\item $f$ is a finite morphism if and only if $f'$ is an finite morphism,", "\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,", "\\item $f$ is an immersion if and only if $f'$ is an immersion, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V'$ and a surjective \\'etale morphism $V' \\to Y'$.", "Choose a scheme $U'$ and a surjective \\'etale morphism", "$U' \\to X' \\times_{Y'} V'$. Set $V = Y \\times_{Y'} V'$ and", "$U = X \\times_{X'} U'$. Then for \\'etale local properties of morphisms", "we can reduce to the morphism of thickenings of schemes", "$(U \\subset U') \\to (V \\subset V')$ and apply More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-thicken-property-morphisms-cartesian}.", "This proves (2), (3), (4), (5), and (6).", "\\medskip\\noindent", "The properties of morphisms in (1), (7), (8), (9), (10) are stable", "under base change, hence if $f'$ has property $\\mathcal{P}$, then so", "does $f$. See", "Spaces, Lemma \\ref{spaces-lemma-base-change-immersions},", "and", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-proper},", "\\ref{spaces-morphisms-lemma-base-change-integral}, and", "\\ref{spaces-morphisms-lemma-base-change-monomorphism}.", "\\medskip\\noindent", "The interesting direction in (1), (7), (8), (9), (10) is to assume", "that $f$ has the property and deduce that $f'$ has it too.", "By induction on the order of the thickening we may", "assume that $Y \\subset Y'$ is a first order thickening, see", "discussion on finite order thickenings above.", "\\medskip\\noindent", "Proof of (1). Choose a scheme $V'$ and a surjective \\'etale morphism", "$V' \\to Y'$. Set $V = Y \\times_{Y'} V'$, $U' = X' \\times_{Y'} V'$", "and $U = X \\times_Y V$. Then $U \\to V$ is a closed immersion, which", "implies that $U$ is a scheme, which in turn implies that $U'$ is", "a scheme (Lemma \\ref{lemma-thickening-scheme}). Thus we can apply", "the lemma in the case of schemes", "(More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-thicken-property-morphisms-cartesian})", "to $(U \\subset U') \\to (V \\subset V')$ to conclude.", "\\medskip\\noindent", "Proof of (7). Follows by combining (2) with", "results of Lemma \\ref{lemma-thicken-property-morphisms}", "and the fact that proper equals quasi-compact $+$", "separated $+$ locally of finite type $+$ universally closed.", "\\medskip\\noindent", "Proof of (8). Follows by combining (2) with", "results of Lemma \\ref{lemma-thicken-property-morphisms}", "and using the fact that finite equals integral $+$ locally", "of finite type (Morphisms, Lemma \\ref{morphisms-lemma-finite-integral}).", "\\medskip\\noindent", "Proof of (9). As $f$ is a monomorphism we have $X = X \\times_Y X$.", "We may apply the results proved so far to the morphism", "of thickenings $(X \\subset X') \\to (X \\times_Y X \\subset X' \\times_{Y'} X')$.", "We conclude $X' \\to X' \\times_{Y'} X'$ is a closed immersion by (1).", "In fact, it is a first order thickening as the ideal defining the", "closed immersion $X' \\to X' \\times_{Y'} X'$ is contained in the pullback", "of the ideal $\\mathcal{I} \\subset \\mathcal{O}_{Y'}$ cutting out $Y$ in $Y'$.", "Indeed, $X = X \\times_Y X = (X' \\times_{Y'} X') \\times_{Y'} Y$ is contained", "in $X'$. The conormal sheaf of the closed immersion", "$\\Delta : X' \\to X' \\times_{Y'} X'$ is equal to $\\Omega_{X'/Y'}$", "(this is the analogue of", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-diagonal}", "for algebraic spaces and follows either by \\'etale localization", "or by combining", "Lemmas \\ref{lemma-differentials-relative-immersion-section} and", "\\ref{lemma-differential-product}; some details omitted).", "Thus it suffices to show that $\\Omega_{X'/Y'} = 0$ which follows from (5)", "and the corresponding statement for $X/Y$.", "\\medskip\\noindent", "Proof of (10). If $f : X \\to Y$ is an immersion, then it factors as", "$X \\to V \\to Y$ where $V \\to Y$ is an open subspace and $X \\to V$ is a", "closed immersion, see", "Morphisms of Spaces, Remark \\ref{spaces-morphisms-remark-immersion}.", "Let $V' \\subset Y'$ be the open subspace whose", "underlying topological space $|V'|$ is the same as $|V| \\subset |Y| = |Y'|$.", "Then $X' \\to Y'$ factors through $V'$ and we conclude that $X' \\to V'$", "is a closed immersion by part (1). This finishes the proof." ], "refs": [ "more-morphisms-lemma-thicken-property-morphisms-cartesian", "spaces-lemma-base-change-immersions", "spaces-morphisms-lemma-base-change-proper", "spaces-morphisms-lemma-base-change-integral", "spaces-morphisms-lemma-base-change-monomorphism", "spaces-more-morphisms-lemma-thickening-scheme", "more-morphisms-lemma-thicken-property-morphisms-cartesian", "spaces-more-morphisms-lemma-thicken-property-morphisms", "spaces-more-morphisms-lemma-thicken-property-morphisms", "morphisms-lemma-finite-integral", "morphisms-lemma-differentials-diagonal", "spaces-more-morphisms-lemma-differential-product", "spaces-morphisms-remark-immersion" ], "ref_ids": [ 13684, 8161, 4917, 4942, 4754, 49, 13684, 55, 55, 5438, 5311, 41, 5029 ] } ], "ref_ids": [] }, { "id": 57, "type": "theorem", "label": "spaces-more-morphisms-lemma-properties-that-extend-over-thickenings", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-properties-that-extend-over-thickenings", "contents": [ "Let $S$ be a scheme. Let $(f, f') : (X \\subset X') \\to (Y \\to Y')$ be a", "morphism of thickenings of algebraic spaces over $S$. Assume $f$ and $f'$", "are locally of finite type and $X = Y \\times_{Y'} X'$. Then", "\\begin{enumerate}", "\\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,", "\\item $f$ is finite if and only if $f'$ is finite,", "\\item $f$ is a closed immersion if and only if $f'$ is a closed immersion,", "\\item $\\Omega_{X/Y} = 0$ if and only if $\\Omega_{X'/Y'} = 0$,", "\\item $f$ is unramified if and only if $f'$ is unramified,", "\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,", "\\item $f$ is an immersion if and only if $f'$ is an immersion,", "\\item $f$ is proper if and only if $f'$ is proper, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V'$ and a surjective \\'etale morphism $V' \\to Y'$.", "Choose a scheme $U'$ and a surjective \\'etale morphism", "$U' \\to X' \\times_{Y'} V'$. Set $V = Y \\times_{Y'} V'$ and", "$U = X \\times_{X'} U'$. Then for \\'etale local properties of morphisms", "we can reduce to the morphism of thickenings of schemes", "$(U \\subset U') \\to (V \\subset V')$ and apply", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-properties-that-extend-over-thickenings}.", "This proves (1), (4), and (5).", "\\medskip\\noindent", "The properties in (2), (3), (6), (7), and (8) are stable", "under base change, hence if $f'$ has property $\\mathcal{P}$, then so", "does $f$. See Spaces, Lemma \\ref{spaces-lemma-base-change-immersions},", "and", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-proper},", "\\ref{spaces-morphisms-lemma-base-change-integral}, and", "\\ref{spaces-morphisms-lemma-base-change-monomorphism}.", "Hence in each case we need only to prove that if $f$ has", "the desired property, so does $f'$.", "\\medskip\\noindent", "Case (2) follows from case (5) of Lemma \\ref{lemma-thicken-property-morphisms}", "and the fact that the finite morphisms are precisely", "the integral morphisms that are locally of finite type", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-integral}).", "\\medskip\\noindent", "Case (3). This follows immediately from", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-check-closed-infinitesimally}.", "\\medskip\\noindent", "Proof of (6). As $f$ is a monomorphism we have $X = X \\times_Y X$.", "We may apply the results proved so far to the morphism of thickenings", "$(X \\subset X') \\to (X \\times_Y X \\subset X' \\times_{Y'} X')$.", "We conclude $\\Delta_{X'/Y'} : X' \\to X' \\times_{Y'} X'$", "is a closed immersion by (3). In fact $\\Delta_{X'/Y'}$ induces a bijection", "$|X'| \\to |X' \\times_{Y'} X'|$, hence $\\Delta_{X'/Y'}$ is a thickening.", "On the other hand $\\Delta_{X'/Y'}$ is locally of finite presentation by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-diagonal-morphism-finite-type}.", "In other words, $\\Delta_{X'/Y'}(X')$ is cut out by", "a quasi-coherent sheaf of ideals", "$\\mathcal{J} \\subset \\mathcal{O}_{X' \\times_{Y'} X'}$ of finite type.", "Since $\\Omega_{X'/Y'} = 0$ by (5) we see that", "the conormal sheaf of $X' \\to X' \\times_{Y'} X'$ is zero.", "(The conormal sheaf of the closed immersion $\\Delta_{X'/Y'}$ is equal to", "$\\Omega_{X'/Y'}$; this is the analogue of", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-diagonal}", "for algebraic spaces and follows either by \\'etale localization", "or by combining", "Lemmas \\ref{lemma-differentials-relative-immersion-section} and", "\\ref{lemma-differential-product}; some details omitted.)", "In other words, $\\mathcal{J}/\\mathcal{J}^2 = 0$.", "This implies $\\Delta_{X'/Y'}$ is an isomorphism, for example", "by Algebra, Lemma \\ref{algebra-lemma-ideal-is-squared-union-connected}.", "\\medskip\\noindent", "Proof of (7). If $f : X \\to Y$ is an immersion, then it factors as", "$X \\to V \\to Y$ where $V \\to Y$ is an open subspace and $X \\to V$ is a", "closed immersion, see", "Morphisms of Spaces, Remark \\ref{spaces-morphisms-remark-immersion}.", "Let $V' \\subset Y'$ be the open subspace whose", "underlying topological space $|V'|$ is the same as $|V| \\subset |Y| = |Y'|$.", "Then $X' \\to Y'$ factors through $V'$ and we conclude that $X' \\to V'$", "is a closed immersion by part (3).", "\\medskip\\noindent", "Case (8) follows from Lemma \\ref{lemma-thicken-property-morphisms}", "and the definition of proper morphisms as being the quasi-compact,", "universally closed, and separated morphisms that are locally of finite type." ], "refs": [ "more-morphisms-lemma-properties-that-extend-over-thickenings", "spaces-lemma-base-change-immersions", "spaces-morphisms-lemma-base-change-proper", "spaces-morphisms-lemma-base-change-integral", "spaces-morphisms-lemma-base-change-monomorphism", "spaces-more-morphisms-lemma-thicken-property-morphisms", "spaces-morphisms-lemma-finite-integral", "spaces-limits-lemma-check-closed-infinitesimally", "spaces-morphisms-lemma-diagonal-morphism-finite-type", "morphisms-lemma-differentials-diagonal", "spaces-more-morphisms-lemma-differential-product", "algebra-lemma-ideal-is-squared-union-connected", "spaces-morphisms-remark-immersion", "spaces-more-morphisms-lemma-thicken-property-morphisms" ], "ref_ids": [ 13685, 8161, 4917, 4942, 4754, 55, 4943, 4629, 4847, 5311, 41, 407, 5029, 55 ] } ], "ref_ids": [] }, { "id": 58, "type": "theorem", "label": "spaces-more-morphisms-lemma-picard-group-first-order-thickening", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-picard-group-first-order-thickening", "contents": [ "Let $S$ be a scheme. Let $X \\subset X'$ be a first order thickening", "of algebraic spaces over $S$ with ideal sheaf $\\mathcal{I}$.", "Then there is a canonical exact sequence", "$$", "\\xymatrix{", "0 \\ar[r] &", "H^0(X, \\mathcal{I}) \\ar[r] &", "H^0(X', \\mathcal{O}_{X'}^*) \\ar[r] &", "H^0(X, \\mathcal{O}^*_X) \\ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\\\", "& H^1(X, \\mathcal{I}) \\ar[r] &", "\\Pic(X') \\ar[r] &", "\\Pic(X) \\ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\\\", "& H^2(X, \\mathcal{I}) \\ar[r] & \\ldots \\ar[r] & \\ldots", "}", "$$", "of abelian groups." ], "refs": [], "proofs": [ { "contents": [ "Recall that $X_\\etale = X'_\\etale$, see", "Lemma \\ref{lemma-thickening-equivalence} and more generally the", "discussion in Section \\ref{section-thickenings}.", "The sequence of the lemma is the long exact cohomology sequence", "associated to the short exact sequence of sheaves of abelian groups", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_{X'}^* \\to \\mathcal{O}_X^* \\to 0", "$$", "on $X_\\etale$ where the first map sends a local section $f$ of $\\mathcal{I}$", "to the invertible section $1 + f$ of $\\mathcal{O}_{X'}$.", "We also use the identification of the Picard group of a", "ringed site with the first cohomology group of the sheaf", "of invertible functions, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-h1-invertible}." ], "refs": [ "spaces-more-morphisms-lemma-thickening-equivalence", "sites-cohomology-lemma-h1-invertible" ], "ref_ids": [ 50, 4185 ] } ], "ref_ids": [] }, { "id": 59, "type": "theorem", "label": "spaces-more-morphisms-lemma-first-order-infinitesimal-neighbourhood", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-first-order-infinitesimal-neighbourhood", "contents": [ "Let $i : Z \\to X$ be an immersion of algebraic spaces.", "The first order infinitesimal neighbourhood $Z'$ of $Z$ in $X$", "has the following universal property:", "Given any commutative diagram", "$$", "\\xymatrix{", "Z \\ar[d]_i & T \\ar[l]^a \\ar[d] \\\\", "X & T' \\ar[l]_b", "}", "$$", "where $T \\subset T'$ is a first order thickening over $X$, there exists", "a unique morphism $(a', a) : (T \\subset T') \\to (Z \\subset Z')$ of", "thickenings over $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be the open subspace used in the construction of $Z'$,", "i.e., an open such that $Z$ is identified with a closed subspace of $U$", "cut out by the quasi-coherent sheaf of ideals $\\mathcal{I}$.", "Since $|T| = |T'|$ we see that $|b|(|T'|) \\subset |U|$. Hence we can", "think of $b$ as a morphism into $U$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-factor-through-open-subspace}.", "Let $\\mathcal{J} \\subset \\mathcal{O}_{T'}$", "be the square zero quasi-coherent sheaf of ideals cutting out $T$.", "By the commutativity of the diagram we have $b|_T = i \\circ a$ where", "$i : Z \\to U$ is the closed immersion. We conclude that", "$b^\\sharp(b^{-1}\\mathcal{I}) \\subset \\mathcal{J}$ by", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-closed-immersion-ideals}.", "As $T'$ is a first order thickening of $T$ we see that $\\mathcal{J}^2 = 0$", "hence $b^\\sharp(b^{-1}(\\mathcal{I}^2)) = 0$. By", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-closed-immersion-ideals}", "this implies that $b$ factors through $Z'$. Letting $a' : T' \\to Z'$", "be this factorization we win." ], "refs": [ "spaces-properties-lemma-factor-through-open-subspace", "spaces-morphisms-lemma-closed-immersion-ideals", "spaces-morphisms-lemma-closed-immersion-ideals" ], "ref_ids": [ 11824, 4765, 4765 ] } ], "ref_ids": [] }, { "id": 60, "type": "theorem", "label": "spaces-more-morphisms-lemma-infinitesimal-neighbourhood-conormal", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-infinitesimal-neighbourhood-conormal", "contents": [ "Let $i : Z \\to X$ be an immersion of algebraic spaces.", "Let $Z \\subset Z'$ be the first order infinitesimal neighbourhood", "of $Z$ in $X$. Then the diagram", "$$", "\\xymatrix{", "Z \\ar[r] \\ar[d] & Z' \\ar[d] \\\\", "Z \\ar[r] & X", "}", "$$", "induces a map of conormal sheaves", "$\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Z'}$ by", "Lemma \\ref{lemma-conormal-functorial}.", "This map is an isomorphism." ], "refs": [ "spaces-more-morphisms-lemma-conormal-functorial" ], "proofs": [ { "contents": [ "This is clear from the construction of $Z'$ above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 24 ] }, { "id": 61, "type": "theorem", "label": "spaces-more-morphisms-lemma-formally-etale-is-combination", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-formally-etale-is-combination", "contents": [ "Let $S$ be a scheme.", "Let $a : F \\to G$ be a transformation of functors", "$F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Then $a$ is formally \\'etale if and only if $a$ is both formally", "smooth and formally unramified." ], "refs": [], "proofs": [ { "contents": [ "Formal from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 62, "type": "theorem", "label": "spaces-more-morphisms-lemma-composition-formally-smooth-etale-unramified", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-composition-formally-smooth-etale-unramified", "contents": [ "Composition.", "\\begin{enumerate}", "\\item A composition of formally smooth transformations of functors is formally", "smooth.", "\\item A composition of formally \\'etale transformations of functors is formally", "\\'etale.", "\\item A composition of formally unramified transformations of functors is", "formally unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 63, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-formally-smooth-etale-unramified", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-formally-smooth-etale-unramified", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$, $b : H \\to G$ be transformations of functors.", "Consider the fibre product diagram", "$$", "\\xymatrix{", "H \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\", "H \\ar[r]^b & G", "}", "$$", "\\begin{enumerate}", "\\item If $a$ is formally smooth, then the base change $a'$ is", "formally smooth.", "\\item If $a$ is formally \\'etale, then the base change $a'$ is", "formally \\'etale.", "\\item If $a$ is formally unramified, then the base change $a'$ is", "formally unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 64, "type": "theorem", "label": "spaces-more-morphisms-lemma-representable-property-formally-property", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-representable-property-formally-property", "contents": [ "Let $S$ be a scheme.", "Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$ be a representable transformation of functors.", "\\begin{enumerate}", "\\item If $a$ is smooth then $a$ is formally smooth.", "\\item If $a$ is \\'etale, then $a$ is formally \\'etale.", "\\item If $a$ is unramified, then $a$ is formally unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider a solid commutative diagram", "$$", "\\xymatrix{", "F \\ar[d]_a & T \\ar[d]^i \\ar[l] \\\\", "G & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "as in", "Definition \\ref{definition-formally-smooth-etale-unramified}.", "Then $F \\times_G T'$ is a scheme smooth (resp.\\ \\'etale, resp.\\ unramified)", "over $T'$. Hence by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-smooth-formally-smooth}", "(resp.\\ Lemma \\ref{more-morphisms-lemma-etale-formally-etale},", "resp.\\ Lemma \\ref{more-morphisms-lemma-unramified-formally-unramified})", "we can fill in (resp.\\ uniquely fill in, resp.\\ fill in at most", "one way) the dotted arrow in the diagram", "$$", "\\xymatrix{", "F \\times_G T' \\ar[d] & T \\ar[d]^i \\ar[l] \\\\", "T' & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "an hence we also obtain the corresponding assertion in the first diagram." ], "refs": [ "spaces-more-morphisms-definition-formally-smooth-etale-unramified", "more-morphisms-lemma-smooth-formally-smooth", "more-morphisms-lemma-etale-formally-etale", "more-morphisms-lemma-unramified-formally-unramified" ], "ref_ids": [ 285, 13734, 13715, 13696 ] } ], "ref_ids": [] }, { "id": 65, "type": "theorem", "label": "spaces-more-morphisms-lemma-etale-on-top", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-etale-on-top", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$, $b : G \\to H$ be transformations of functors.", "Assume that $a$ is representable, surjective, and \\'etale.", "\\begin{enumerate}", "\\item If $b$ is formally smooth, then $b \\circ a$ is formally smooth.", "\\item If $b$ is formally \\'etale, then $b \\circ a$ is formally \\'etale.", "\\item If $b$ is formally unramified, then $b \\circ a$ is formally unramified.", "\\end{enumerate}", "Conversely, consider a solid commutative diagram", "$$", "\\xymatrix{", "G \\ar[d]_b & T \\ar[d]^i \\ar[l] \\\\", "H & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "with $T'$ an affine scheme over $S$", "and $i : T \\to T'$ a closed immersion defined by an ideal of square zero.", "\\begin{enumerate}", "\\item[(4)] If $b \\circ a$ is formally smooth, then for every $t \\in T$", "there exists an \\'etale morphism of affines $U' \\to T'$ and a morphism", "$U' \\to G$ such that", "$$", "\\xymatrix{", "G \\ar[d]_b & T \\ar[l] & T \\times_{T'} U' \\ar[d] \\ar[l]\\\\", "H & T' \\ar[l] & U' \\ar[llu] \\ar[l]", "}", "$$", "commutes and $t$ is in the image of $U' \\to T'$.", "\\item[(5)] If $b \\circ a$ is formally unramified, then there exists at most", "one dotted arrow in the diagram above, i.e., $b$ is formally unramified.", "\\item[(6)] If $b \\circ a$ is formally \\'etale, then there exists exactly one", "dotted arrow in the diagram above, i.e., $b$ is formally \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $b$ is formally smooth (resp.\\ formally \\'etale,", "resp.\\ formally unramified). Since an \\'etale morphism is both", "smooth and unramified we see that $a$ is representable and smooth", "(resp.\\ \\'etale, resp. unramified). Hence parts (1), (2) and (3)", "follow from a combination of", "Lemma \\ref{lemma-representable-property-formally-property}", "and", "Lemma \\ref{lemma-composition-formally-smooth-etale-unramified}.", "\\medskip\\noindent", "Assume that $b \\circ a$ is formally smooth. Consider a diagram", "as in the statement of the lemma. Let $W = F \\times_G T$.", "By assumption $W$ is a scheme surjective \\'etale over $T$. By", "\\'Etale Morphisms, Theorem \\ref{etale-theorem-remarkable-equivalence}", "there exists a scheme $W'$ \\'etale over $T'$ such that $W = T \\times_{T'} W'$.", "Choose an affine open subscheme $U' \\subset W'$ such that $t$ is in", "the image of $U' \\to T'$. Because $b \\circ a$ is formally", "smooth we see that the exist morphisms $U' \\to F$ such that", "$$", "\\xymatrix{", "F \\ar[d]_{b \\circ a} & W \\ar[l] & T \\times_{T'} U' \\ar[d] \\ar[l]\\\\", "H & T' \\ar[l] & U' \\ar[llu] \\ar[l]", "}", "$$", "commutes. Taking the composition $U' \\to F \\to G$ gives a", "map as in part (5) of the lemma.", "\\medskip\\noindent", "Assume that $f, g : T' \\to G$ are two dotted arrows fitting into the", "diagram of the lemma. Let $W = F \\times_G T$.", "By assumption $W$ is a scheme surjective \\'etale over $T$. By", "\\'Etale Morphisms, Theorem \\ref{etale-theorem-remarkable-equivalence}", "there exists a scheme $W'$ \\'etale over $T'$ such that $W = T \\times_{T'} W'$.", "Since $a$ is formally \\'etale the compositions", "$$", "W' \\to T' \\xrightarrow{f} G", "\\quad\\text{and}\\quad", "W' \\to T' \\xrightarrow{g} G", "$$", "lift to morphisms $f', g' : W' \\to F$ (lift on affine opens and glue by", "uniqueness). Now if $b \\circ a : F \\to H$ is formally unramified, then", "$f' = g'$ and hence $f = g$ as $W' \\to T'$ is an \\'etale covering. This proves", "part (6) of the lemma.", "\\medskip\\noindent", "Assume that $b \\circ a$ is formally \\'etale. Then by part (4) we", "can \\'etale locally on $T'$ find a dotted arrow fitting into the diagram", "and by part (5) this dotted arrow is unique. Hence we may glue the", "local solutions to get assertion (6). Some details omitted." ], "refs": [ "spaces-more-morphisms-lemma-representable-property-formally-property", "spaces-more-morphisms-lemma-composition-formally-smooth-etale-unramified", "etale-theorem-remarkable-equivalence", "etale-theorem-remarkable-equivalence" ], "ref_ids": [ 64, 62, 10694, 10694 ] } ], "ref_ids": [] }, { "id": 66, "type": "theorem", "label": "spaces-more-morphisms-lemma-formally-permanence", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-formally-permanence", "contents": [ "Let $S$ be a scheme.", "Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$, $b : G \\to H$ be transformations of functors.", "Assume $b$ is formally unramified.", "\\begin{enumerate}", "\\item If $b \\circ a$ is formally unramified then $a$ is formally unramified.", "\\item If $b \\circ a$ is formally \\'etale then $a$ is formally \\'etale.", "\\item If $b \\circ a$ is formally smooth then $a$ is formally smooth.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $T \\subset T'$ be a closed immersion of affine schemes defined by an ideal", "of square zero. Let $g' : T' \\to G$ and $f : T \\to F$ be given such that", "$g'|_T = a \\circ f$. Because $b$ is formally unramified, there is a one", "to one correspondence between", "$$", "\\{f' : T' \\to F \\mid f = f'|_T\\text{ and }a \\circ f' = g'\\}", "$$", "and", "$$", "\\{f' : T' \\to F \\mid f = f'|_T\\text{ and }b \\circ a \\circ f' = b \\circ g'\\}.", "$$", "From this the lemma follows formally." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 67, "type": "theorem", "label": "spaces-more-morphisms-lemma-formally-unramified", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-formally-unramified", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over", "$S$. The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is formally unramified,", "\\item for every diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$ and $V$ are schemes and the vertical arrows are \\'etale", "the morphism of schemes $\\psi$ is formally unramified (as in", "More on Morphisms,", "Definition \\ref{more-morphisms-definition-formally-unramified}), and", "\\item for one such diagram with surjective vertical arrows the morphism", "$\\psi$ is formally unramified.", "\\end{enumerate}" ], "refs": [ "more-morphisms-definition-formally-unramified" ], "proofs": [ { "contents": [ "Assume $f$ is formally unramified. By", "Lemma \\ref{lemma-representable-property-formally-property}", "the morphisms $U \\to X$ and $V \\to Y$ are formally unramified. Thus by", "Lemma \\ref{lemma-composition-formally-smooth-etale-unramified}", "the composition $U \\to Y$ is formally unramified. Then it follows from", "Lemma \\ref{lemma-formally-permanence}", "that $U \\to V$ is formally unramified. Thus (1) implies (2). And (2)", "implies (3) trivially", "\\medskip\\noindent", "Assume given a diagram as in (3). By", "Lemma \\ref{lemma-representable-property-formally-property}", "the morphism $V \\to Y$ is formally unramified. Thus by", "Lemma \\ref{lemma-composition-formally-smooth-etale-unramified}", "the composition $U \\to Y$ is formally unramified. Then it follows from", "Lemma \\ref{lemma-etale-on-top}", "that $X \\to Y$ is formally unramified, i.e., (1) holds." ], "refs": [ "spaces-more-morphisms-lemma-representable-property-formally-property", "spaces-more-morphisms-lemma-composition-formally-smooth-etale-unramified", "spaces-more-morphisms-lemma-formally-permanence", "spaces-more-morphisms-lemma-representable-property-formally-property", "spaces-more-morphisms-lemma-composition-formally-smooth-etale-unramified", "spaces-more-morphisms-lemma-etale-on-top" ], "ref_ids": [ 64, 62, 66, 64, 62, 65 ] } ], "ref_ids": [ 14108 ] }, { "id": 68, "type": "theorem", "label": "spaces-more-morphisms-lemma-formally-unramified-not-affine", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-formally-unramified-not-affine", "contents": [ "Let $S$ be a scheme.", "If $f : X \\to Y$ is a formally unramified morphism of algebraic spaces", "over $S$, then given any solid commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\", "S & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "where $T \\subset T'$ is a first order thickening of algebraic spaces", "over $S$ there exists at most one dotted arrow making the diagram commute.", "In other words, in", "Definition \\ref{definition-formally-unramified}", "the condition that $T$ be an affine scheme may be dropped." ], "refs": [ "spaces-more-morphisms-definition-formally-unramified" ], "proofs": [ { "contents": [ "This is true because there exists a surjective \\'etale morphism", "$U' \\to T'$ where $U'$ is a disjoint union of affine schemes (see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-cover-by-union-affines})", "and a morphism $T' \\to X$ is determined by its restriction to $U'$." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines" ], "ref_ids": [ 11830 ] } ], "ref_ids": [ 286 ] }, { "id": 69, "type": "theorem", "label": "spaces-more-morphisms-lemma-composition-formally-unramified", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-composition-formally-unramified", "contents": [ "A composition of formally unramified morphisms is formally unramified." ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 70, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-formally-unramified", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-formally-unramified", "contents": [ "A base change of a formally unramified morphism is formally unramified." ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 71, "type": "theorem", "label": "spaces-more-morphisms-lemma-characterize-formally-unramified", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-characterize-formally-unramified", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over", "$S$. The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is formally unramified, and", "\\item $\\Omega_{X/Y} = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a combination of", "Lemma \\ref{lemma-formally-unramified},", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-formally-unramified-differentials},", "and", "Lemma \\ref{lemma-localize-differentials}." ], "refs": [ "spaces-more-morphisms-lemma-formally-unramified", "more-morphisms-lemma-formally-unramified-differentials", "spaces-more-morphisms-lemma-localize-differentials" ], "ref_ids": [ 67, 13695, 32 ] } ], "ref_ids": [] }, { "id": 72, "type": "theorem", "label": "spaces-more-morphisms-lemma-unramified-formally-unramified", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-unramified-formally-unramified", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is unramified,", "\\item the morphism $f$ is locally of finite type and $\\Omega_{X/Y} = 0$, and", "\\item the morphism $f$ is locally of finite type and formally unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$ and $V$ are schemes and the vertical arrows are \\'etale and", "surjective. Then we see", "\\begin{align*}", "f\\text{ unramified}", "& \\Leftrightarrow", "\\psi\\text{ unramified} \\\\", "& \\Leftrightarrow", "\\psi\\text{ locally finite type and }\\Omega_{U/V} = 0 \\\\", "& \\Leftrightarrow", "f\\text{ locally finite type and }\\Omega_{X/Y} = 0 \\\\", "& \\Leftrightarrow", "f\\text{ locally finite type and formally unramified}", "\\end{align*}", "Here we have used", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-omega-zero} and", "Lemma \\ref{lemma-characterize-formally-unramified}." ], "refs": [ "morphisms-lemma-unramified-omega-zero", "spaces-more-morphisms-lemma-characterize-formally-unramified" ], "ref_ids": [ 5343, 71 ] } ], "ref_ids": [] }, { "id": 73, "type": "theorem", "label": "spaces-more-morphisms-lemma-universally-injective-unramified", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-universally-injective-unramified", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is unramified and a monomorphism,", "\\item $f$ is unramified and universally injective,", "\\item $f$ is locally of finite type and a monomorphism,", "\\item $f$ is universally injective, locally of finite type, and", "formally unramified.", "\\end{enumerate}", "Moreover, in this case $f$ is also representable, separated, and", "locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "We have seen in", "Lemma \\ref{lemma-unramified-formally-unramified}", "that being formally unramified and locally of finite type is the same thing", "as being unramified.", "Hence (4) is equivalent to (2).", "A monomorphism is certainly formally unramified hence (3) implies (4).", "It is clear that (1) implies (3). Finally, if (2) holds, then", "$\\Delta : X \\to X \\times_Y X$ is both an open immersion", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-diagonal-unramified-morphism})", "and surjective", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-injective})", "hence an isomorphism, i.e., $f$ is a monomorphism. In this way we see that", "(2) implies (1).", "Finally, we see that $f$ is representable, separated, and locally", "quasi-finite by", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite} and", "\\ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable}." ], "refs": [ "spaces-more-morphisms-lemma-unramified-formally-unramified", "spaces-morphisms-lemma-diagonal-unramified-morphism", "spaces-morphisms-lemma-universally-injective", "spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "spaces-morphisms-lemma-locally-quasi-finite-separated-representable" ], "ref_ids": [ 72, 4902, 4793, 4838, 4972 ] } ], "ref_ids": [] }, { "id": 74, "type": "theorem", "label": "spaces-more-morphisms-lemma-characterize-closed-immersion", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-characterize-closed-immersion", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is a closed immersion,", "\\item $f$ is universally closed, unramified, and a monomorphism,", "\\item $f$ is universally closed, unramified, and universally injective,", "\\item $f$ is universally closed, locally of finite type, and a monomorphism,", "\\item $f$ is universally closed, universally injective, locally of", "finite type, and formally unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (2) -- (5) follows immediately from", "Lemma \\ref{lemma-universally-injective-unramified}.", "Moreover, if (2) -- (5) are satisfied then $f$ is representable.", "Similarly, if (1) is satisfied then $f$ is representable.", "Hence the result follows from the case of schemes, see", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-characterize-closed-immersion}." ], "refs": [ "spaces-more-morphisms-lemma-universally-injective-unramified", "etale-lemma-characterize-closed-immersion" ], "ref_ids": [ 73, 10702 ] } ], "ref_ids": [] }, { "id": 75, "type": "theorem", "label": "spaces-more-morphisms-lemma-check-universal-first-order-thickening", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-check-universal-first-order-thickening", "contents": [ "Let $S$ be a scheme.", "Let $h : Z \\to X$ be a morphism of algebraic spaces over $S$.", "Let $Z \\subset Z'$ be a first order thickening over $X$.", "The following are equivalent", "\\begin{enumerate}", "\\item $Z \\subset Z'$ is a universal first order thickening,", "\\item for any diagram (\\ref{equation-universal-first-order-thickening})", "with $T'$ a scheme a unique dotted arrow exists making the diagram commute, and", "\\item for any diagram (\\ref{equation-universal-first-order-thickening})", "with $T'$ an affine scheme a unique dotted arrow exists making the", "diagram commute.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implications (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3) are formal.", "Assume (3) a assume given an arbitrary diagram", "(\\ref{equation-universal-first-order-thickening}).", "Choose a presentation $T' = U'/R'$, see", "Spaces, Definition \\ref{spaces-definition-presentation}.", "We may assume that $U' = \\coprod U'_i$ is a disjoint union", "of affines, so $R' = U' \\times_{T'} U' = \\coprod_{i, j} U'_i \\times_T' U'_j$.", "For each pair $(i, j)$ choose an affine open covering", "$U'_i \\times_T' U'_j = \\bigcup_k R'_{ijk}$. Denote $U_i, R_{ijk}$", "the fibre products with $T$ over $T'$. Then each", "$U_i \\subset U'_i$ and $R_{ijk} \\subset R'_{ijk}$", "is a first order thickening of affine schemes.", "Denote $a_i : U_i \\to Z$, resp.\\ $a_{ijk} : R_{ijk} \\to Z$", "the composition of $a : T \\to Z$ with the morphism", "$U_i \\to T$, resp.\\ $R_{ijk} \\to T$.", "By (3) applied to $a_i : U_i \\to Z$", "we obtain unique morphisms $a'_i : U'_i \\to Z'$.", "By (3) applied to $a_{ijk}$ we see that the two compositions", "$R'_{ijk} \\to R'_i \\to Z'$ and $R'_{ijk} \\to R'_j \\to Z'$", "are equal. Hence $a' = \\coprod a'_i : U' = \\coprod U'_i \\to Z'$", "descends to the quotient sheaf $T' = U'/R'$ and we win." ], "refs": [ "spaces-definition-presentation" ], "ref_ids": [ 8177 ] } ], "ref_ids": [] }, { "id": 76, "type": "theorem", "label": "spaces-more-morphisms-lemma-universal-thickening-over-formally-etale", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-universal-thickening-over-formally-etale", "contents": [ "Let $S$ be a scheme.", "Let $Z \\to Y \\to X$ be morphisms of algebraic spaces over $S$.", "If $Z \\subset Z'$ is a universal first order thickening of", "$Z$ over $Y$ and $Y \\to X$ is formally \\'etale, then $Z \\subset Z'$ is", "a universal first order thickening of $Z$ over $X$." ], "refs": [], "proofs": [ { "contents": [ "This is formal. Namely, by", "Lemma \\ref{lemma-check-universal-first-order-thickening}", "it suffices to consider solid commutative diagrams", "(\\ref{equation-universal-first-order-thickening})", "with $T'$ an affine scheme. The composition", "$T \\to Z \\to Y$ lifts uniquely to $T' \\to Y$ as $Y \\to X$ is", "assumed formally \\'etale. Hence the fact that", "$Z \\subset Z'$ is a universal first order thickening over $Y$", "produces the desired morphism $a' : T' \\to Z'$." ], "refs": [ "spaces-more-morphisms-lemma-check-universal-first-order-thickening" ], "ref_ids": [ 75 ] } ], "ref_ids": [] }, { "id": 77, "type": "theorem", "label": "spaces-more-morphisms-lemma-etale-morphism-of-universal-thickenings", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-etale-morphism-of-universal-thickenings", "contents": [ "Let $S$ be a scheme.", "Let $Z \\to Y \\to X$ be morphisms of algebraic spaces over $S$.", "Assume $Z \\to Y$ is \\'etale.", "\\begin{enumerate}", "\\item If $Y \\subset Y'$ is a universal first order thickening of", "$Y$ over $X$, then the unique \\'etale morphism $Z' \\to Y'$ such", "that $Z = Y \\times_{Y'} Z'$ (see", "Theorem \\ref{theorem-topological-invariance})", "is a universal first order thickening of $Z$ over $X$.", "\\item If $Z \\to Y$ is surjective and", "$(Z \\subset Z') \\to (Y \\subset Y')$ is an \\'etale morphism", "of first order thickenings over $X$ and $Z'$ is a universal first", "order thickening of $Z$ over $X$, then $Y'$ is a universal first", "order thickening of $Y$ over $X$.", "\\end{enumerate}" ], "refs": [ "spaces-more-morphisms-theorem-topological-invariance" ], "proofs": [ { "contents": [ "Proof of (1). By", "Lemma \\ref{lemma-check-universal-first-order-thickening}", "it suffices to consider solid commutative diagrams", "(\\ref{equation-universal-first-order-thickening})", "with $T'$ an affine scheme. The composition", "$T \\to Z \\to Y$ lifts uniquely to $T' \\to Y'$ as $Y'$ is", "the universal first order thickening. Then the fact that", "$Z' \\to Y'$ is \\'etale implies (see", "Lemma \\ref{lemma-representable-property-formally-property})", "that $T' \\to Y'$ lifts to the", "desired morphism $a' : T' \\to Z'$.", "\\medskip\\noindent", "Proof of (2). Let $T \\subset T'$ be a first order thickening over", "$X$ and let $a : T \\to Y$ be a morphism. Set $W = T \\times_Y Z$", "and denote $c : W \\to Z$ the projection", "Let $W' \\to T'$ be the unique \\'etale morphism such that", "$W = T \\times_{T'} W'$, see", "Theorem \\ref{theorem-topological-invariance}.", "Note that $W' \\to T'$ is surjective as $Z \\to Y$ is surjective.", "By assumption we obtain a unique morphism $c' : W' \\to Z'$", "over $X$ restricting to $c$ on $W$. By uniqueness the two restrictions", "of $c'$ to $W' \\times_{T'} W'$ are equal (as the two restrictions of", "$c$ to $W \\times_T W$ are equal). Hence $c'$ descends to a unique", "morphism $a' : T' \\to Y'$ and we win." ], "refs": [ "spaces-more-morphisms-lemma-check-universal-first-order-thickening", "spaces-more-morphisms-lemma-representable-property-formally-property", "spaces-more-morphisms-theorem-topological-invariance" ], "ref_ids": [ 75, 64, 9 ] } ], "ref_ids": [ 9 ] }, { "id": 78, "type": "theorem", "label": "spaces-more-morphisms-lemma-universal-thickening", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-universal-thickening", "contents": [ "Let $S$ be a scheme.", "Let $h : Z \\to X$ be a formally unramified morphism of algebraic", "spaces over $S$.", "There exists a universal first order thickening $Z \\subset Z'$ of", "$Z$ over $X$." ], "refs": [], "proofs": [ { "contents": [ "Choose any commutative diagram", "$$", "\\xymatrix{", "V \\ar[d] \\ar[r] & U \\ar[d] \\\\", "Z \\ar[r] & X", "}", "$$", "where $V$ and $U$ are schemes and the vertical arrows are \\'etale.", "Note that $V \\to U$ is a formally unramified morphism of schemes, see", "Lemma \\ref{lemma-formally-unramified}.", "Combining", "Lemma \\ref{lemma-check-universal-first-order-thickening}", "and", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-universal-thickening}", "we see that a universal first order thickening $V \\subset V'$", "of $V$ over $U$ exists. By", "Lemma \\ref{lemma-universal-thickening-over-formally-etale} part (1)", "$V'$ is a universal first order thickening of $V$ over $X$.", "\\medskip\\noindent", "Fix a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "The argument above shows that for any $V \\to Z$ \\'etale with $V$", "a scheme such that $V \\to Z \\to X$ factors through $U$ a", "universal first order thickening $V \\subset V'$ of $V$ over $X$", "exists (but does not depend on the chosen factorization of $V \\to X$", "through $U$). Now we may choose $V$ such that $V \\to Z$ is surjective", "\\'etale (see", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}).", "Then $R = V \\times_Z V$ a scheme \\'etale over $Z$ such that", "$R \\to X$ factors through $U$ also.", "Hence we obtain universal first order thickenings", "$V \\subset V'$ and $R \\subset R'$ over $X$.", "As $V \\subset V'$ is a universal first order thickening,", "the two projections $s, t : R \\to V$ lift to morphisms", "$s', t': R' \\to V'$. By", "Lemma \\ref{lemma-etale-morphism-of-universal-thickenings}", "as $R'$ is the universal first order thickening of $R$ over $X$", "these morphisms are \\'etale.", "Then $(t', s') : R' \\to V'$ is an \\'etale equivalence relation", "and we can set $Z' = V'/R'$. Since $V' \\to Z'$ is surjective \\'etale", "and $v'$ is the universal first order thickening of $V$ over $X$", "we conclude from", "Lemma \\ref{lemma-universal-thickening-over-formally-etale} part (2)", "that $Z'$ is a universal first order thickening of $Z$ over $X$." ], "refs": [ "spaces-more-morphisms-lemma-formally-unramified", "spaces-more-morphisms-lemma-check-universal-first-order-thickening", "more-morphisms-lemma-universal-thickening", "spaces-more-morphisms-lemma-universal-thickening-over-formally-etale", "spaces-lemma-lift-morphism-presentations", "spaces-more-morphisms-lemma-etale-morphism-of-universal-thickenings", "spaces-more-morphisms-lemma-universal-thickening-over-formally-etale" ], "ref_ids": [ 67, 75, 13697, 76, 8159, 77, 76 ] } ], "ref_ids": [] }, { "id": 79, "type": "theorem", "label": "spaces-more-morphisms-lemma-immersion-universal-thickening", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-immersion-universal-thickening", "contents": [ "Let $S$ be a scheme.", "Let $i : Z \\to X$ be an immersion of algebraic spaces over $S$. Then", "\\begin{enumerate}", "\\item $i$ is formally unramified,", "\\item the universal first order thickening of $Z$ over $X$ is the first order", "infinitesimal neighbourhood of $Z$ in $X$ of", "Definition \\ref{definition-first-order-infinitesimal-neighbourhood},", "\\item the conormal sheaf of $i$ in the sense of", "Definition \\ref{definition-conormal-sheaf}", "agrees with the conormal sheaf of $i$ in the sense of", "Definition \\ref{definition-universal-thickening}.", "\\end{enumerate}" ], "refs": [ "spaces-more-morphisms-definition-first-order-infinitesimal-neighbourhood", "spaces-more-morphisms-definition-conormal-sheaf", "spaces-more-morphisms-definition-universal-thickening" ], "proofs": [ { "contents": [ "An immersion of algebraic spaces is by definition a representable morphism.", "Hence by", "Morphisms, Lemmas \\ref{morphisms-lemma-open-immersion-unramified} and", "\\ref{morphisms-lemma-closed-immersion-unramified}", "an immersion is unramified (via the abstract principle of", "Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}).", "Hence it is formally unramified by", "Lemma \\ref{lemma-unramified-formally-unramified}.", "The other assertions follow by combining", "Lemmas \\ref{lemma-first-order-infinitesimal-neighbourhood} and", "\\ref{lemma-infinitesimal-neighbourhood-conormal}", "and the definitions." ], "refs": [ "morphisms-lemma-open-immersion-unramified", "morphisms-lemma-closed-immersion-unramified", "spaces-lemma-representable-transformations-property-implication", "spaces-more-morphisms-lemma-unramified-formally-unramified", "spaces-more-morphisms-lemma-first-order-infinitesimal-neighbourhood", "spaces-more-morphisms-lemma-infinitesimal-neighbourhood-conormal" ], "ref_ids": [ 5348, 5349, 8136, 72, 59, 60 ] } ], "ref_ids": [ 284, 279, 287 ] }, { "id": 80, "type": "theorem", "label": "spaces-more-morphisms-lemma-universal-thickening-unramified", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-universal-thickening-unramified", "contents": [ "Let $S$ be a scheme.", "Let $Z \\to X$ be a formally unramified morphism of algebraic spaces over $S$.", "Then the universal first order thickening $Z'$ is formally", "unramified over $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $T \\subset T'$ be a first order thickening of affine schemes over $X$.", "Let", "$$", "\\xymatrix{", "Z' \\ar[d] & T \\ar[l]^c \\ar[d] \\\\", "X & T' \\ar[l] \\ar[lu]^{a, b}", "}", "$$", "be a commutative diagram. Set $T_0 = c^{-1}(Z) \\subset T$ and", "$T'_a = a^{-1}(Z)$ (scheme theoretically).", "Since $Z'$ is a first order thickening of $Z$, we see that $T'$", "is a first order thickening of $T'_a$. Moreover, since $c = a|_T$ we see that", "$T_0 = T \\cap T'_a$ (scheme theoretically). As $T'$ is a first order", "thickening of $T$ it follows that $T'_a$", "is a first order thickening of $T_0$. Now $a|_{T'_a}$ and $b|_{T'_a}$", "are morphisms of $T'_a$ into $Z'$ over $X$ which agree on $T_0$ as", "morphisms into $Z$. Hence by the universal property of $Z'$ we conclude that", "$a|_{T'_a} = b|_{T'_a}$. Thus $a$ and $b$ are morphism from", "the first order thickening $T'$ of $T'_a$ whose restrictions to", "$T'_a$ agree as morphisms into $Z$. Thus using the universal property of", "$Z'$ once more we conclude that $a = b$. In other words, the defining", "property of a formally unramified morphism holds for $Z' \\to X$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 81, "type": "theorem", "label": "spaces-more-morphisms-lemma-universal-thickening-functorial", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-universal-thickening-functorial", "contents": [ "Let $S$ be a scheme", "Consider a commutative diagram of algebraic spaces over $S$", "$$", "\\xymatrix{", "Z \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\", "W \\ar[r]^{h'} & Y", "}", "$$", "with $h$ and $h'$ formally unramified. Let $Z \\subset Z'$ be the universal", "first order thickening of $Z$ over $X$. Let $W \\subset W'$ be the universal", "first order thickening of $W$ over $Y$. There exists a canonical morphism", "$(f, f') : (Z, Z') \\to (W, W')$ of thickenings over $Y$ which fits into", "the following commutative diagram", "$$", "\\xymatrix{", "& & & Z' \\ar[ld] \\ar[d]^{f'} \\\\", "Z \\ar[rr] \\ar[d]_f \\ar[rrru] & & X \\ar[d] & W' \\ar[ld] \\\\", "W \\ar[rrru]|!{[rr];[rruu]}\\hole \\ar[rr] & & Y", "}", "$$", "In particular the morphism $(f, f')$ of thickenings induces a morphism", "of conormal sheaves $f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$." ], "refs": [], "proofs": [ { "contents": [ "The first assertion is clear from the universal property of $W'$.", "The induced map on conormal sheaves is the map of", "Lemma \\ref{lemma-conormal-functorial}", "applied to $(Z \\subset Z') \\to (W \\subset W')$." ], "refs": [ "spaces-more-morphisms-lemma-conormal-functorial" ], "ref_ids": [ 24 ] } ], "ref_ids": [] }, { "id": 82, "type": "theorem", "label": "spaces-more-morphisms-lemma-universal-thickening-fibre-product", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-universal-thickening-fibre-product", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "Z \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\", "W \\ar[r]^{h'} & Y", "}", "$$", "be a fibre product diagram of algebraic spaces over $S$ with", "$h'$ formally unramified. Then $h$ is formally unramified and if", "$W \\subset W'$ is the universal first order thickening of $W$ over $Y$,", "then $Z = X \\times_Y W \\subset X \\times_Y W'$ is the universal", "first order thickening of $Z$ over $X$. In particular the canonical map", "$f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$ of", "Lemma \\ref{lemma-universal-thickening-functorial}", "is surjective." ], "refs": [ "spaces-more-morphisms-lemma-universal-thickening-functorial" ], "proofs": [ { "contents": [ "The morphism $h$ is formally unramified by", "Lemma \\ref{lemma-base-change-formally-unramified}.", "It is clear that $X \\times_Y W'$ is a first order thickening.", "It is straightforward to check that it has the universal property", "because $W'$ has the universal property (by mapping properties of", "fibre products). See", "Lemma \\ref{lemma-conormal-functorial-flat}", "for why this implies that the map of conormal sheaves is surjective." ], "refs": [ "spaces-more-morphisms-lemma-base-change-formally-unramified", "spaces-more-morphisms-lemma-conormal-functorial-flat" ], "ref_ids": [ 70, 26 ] } ], "ref_ids": [ 81 ] }, { "id": 83, "type": "theorem", "label": "spaces-more-morphisms-lemma-universal-thickening-fibre-product-flat", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-universal-thickening-fibre-product-flat", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "Z \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\", "W \\ar[r]^{h'} & Y", "}", "$$", "be a fibre product diagram of algebraic spaces over $S$ with", "$h'$ formally unramified and $g$ flat. In this case the corresponding", "map $Z' \\to W'$ of universal first order thickenings is flat, and", "$f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Flatness is preserved under base change, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-flat}.", "Hence the first statement follows from the description of $W'$ in", "Lemma \\ref{lemma-universal-thickening-fibre-product}.", "It is clear that $X \\times_Y W'$ is a first order thickening.", "It is straightforward to check that it has the universal property", "because $W'$ has the universal property (by mapping properties of", "fibre products). See", "Lemma \\ref{lemma-conormal-functorial-flat}", "for why this implies that the map of conormal sheaves is an isomorphism." ], "refs": [ "spaces-morphisms-lemma-base-change-flat", "spaces-more-morphisms-lemma-universal-thickening-fibre-product", "spaces-more-morphisms-lemma-conormal-functorial-flat" ], "ref_ids": [ 4853, 82, 26 ] } ], "ref_ids": [] }, { "id": 84, "type": "theorem", "label": "spaces-more-morphisms-lemma-universal-thickening-localize", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-universal-thickening-localize", "contents": [ "Taking the universal first order thickenings commutes with \\'etale", "localization. More precisely, let $h : Z \\to X$ be a formally unramified", "morphism of algebraic spaces over a base scheme $S$.", "Let", "$$", "\\xymatrix{", "V \\ar[d] \\ar[r] & U \\ar[d] \\\\", "Z \\ar[r] & X", "}", "$$", "be a commutative diagram with \\'etale vertical arrows.", "Let $Z'$ be the universal first order thickening of $Z$ over $X$.", "Then $V \\to U$ is formally unramified and the universal first", "order thickening $V'$ of $V$ over $U$ is \\'etale over $Z'$.", "In particular, $\\mathcal{C}_{Z/X}|_V = \\mathcal{C}_{V/U}$." ], "refs": [], "proofs": [ { "contents": [ "The first statement is", "Lemma \\ref{lemma-formally-unramified}.", "The compatibility of universal first order thickenings is", "a consequence of", "Lemmas \\ref{lemma-universal-thickening-over-formally-etale} and", "\\ref{lemma-etale-morphism-of-universal-thickenings}." ], "refs": [ "spaces-more-morphisms-lemma-formally-unramified", "spaces-more-morphisms-lemma-universal-thickening-over-formally-etale", "spaces-more-morphisms-lemma-etale-morphism-of-universal-thickenings" ], "ref_ids": [ 67, 76, 77 ] } ], "ref_ids": [] }, { "id": 85, "type": "theorem", "label": "spaces-more-morphisms-lemma-differentials-universally-unramified", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-differentials-universally-unramified", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $h : Z \\to X$ be a formally unramified morphism of algebraic spaces", "over $B$. Let $Z \\subset Z'$ be the universal first order thickening of $Z$", "over $X$ with structure morphism $h' : Z' \\to X$. The canonical map", "$$", "\\text{d}h' : (h')^*\\Omega_{X/B} \\to \\Omega_{Z'/B}", "$$", "induces an isomorphism", "$h^*\\Omega_{X/B} \\to \\Omega_{Z'/B} \\otimes \\mathcal{O}_Z$." ], "refs": [], "proofs": [ { "contents": [ "The map $c_{h'}$ is the map defined in", "Lemma \\ref{lemma-functoriality-differentials}.", "If $i : Z \\to Z'$ is the given closed immersion, then", "$i^*c_{h'}$ is a map", "$h^*\\Omega_{X/S} \\to \\Omega_{Z'/S} \\otimes \\mathcal{O}_Z$.", "Checking that it is an isomorphism reduces to the case of schemes", "by \\'etale localization, see", "Lemma \\ref{lemma-universal-thickening-localize}", "and", "Lemma \\ref{lemma-localize-differentials}.", "In this case the result is", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-differentials-universally-unramified}." ], "refs": [ "spaces-more-morphisms-lemma-functoriality-differentials", "spaces-more-morphisms-lemma-universal-thickening-localize", "spaces-more-morphisms-lemma-localize-differentials", "more-morphisms-lemma-differentials-universally-unramified" ], "ref_ids": [ 34, 84, 32, 13704 ] } ], "ref_ids": [] }, { "id": 86, "type": "theorem", "label": "spaces-more-morphisms-lemma-universally-unramified-differentials-sequence", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-universally-unramified-differentials-sequence", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $h : Z \\to X$ be a formally unramified morphism of algebraic", "spaces over $B$.", "There is a canonical exact sequence", "$$", "\\mathcal{C}_{Z/X} \\to h^*\\Omega_{X/B} \\to \\Omega_{Z/B} \\to 0.", "$$", "The first arrow is induced by $\\text{d}_{Z'/B}$ where", "$Z'$ is the universal first order neighbourhood of $Z$ over $X$." ], "refs": [], "proofs": [ { "contents": [ "We know that there is a canonical exact sequence", "$$", "\\mathcal{C}_{Z/Z'} \\to", "\\Omega_{Z'/S} \\otimes \\mathcal{O}_Z \\to", "\\Omega_{Z/S} \\to 0.", "$$", "see", "Lemma \\ref{lemma-differentials-relative-immersion}.", "Hence the result follows on applying", "Lemma \\ref{lemma-differentials-universally-unramified}." ], "refs": [ "spaces-more-morphisms-lemma-differentials-relative-immersion", "spaces-more-morphisms-lemma-differentials-universally-unramified" ], "ref_ids": [ 38, 85 ] } ], "ref_ids": [] }, { "id": 87, "type": "theorem", "label": "spaces-more-morphisms-lemma-two-unramified-morphisms", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-two-unramified-morphisms", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[rd]_j & X \\ar[d] \\\\", "& Y", "}", "$$", "be a commutative diagram of algebraic spaces over $S$", "where $i$ and $j$ are formally unramified. Then there is a", "canonical exact sequence", "$$", "\\mathcal{C}_{Z/Y} \\to", "\\mathcal{C}_{Z/X} \\to", "i^*\\Omega_{X/Y} \\to 0", "$$", "where the first arrow comes from", "Lemma \\ref{lemma-universal-thickening-functorial}", "and the second from", "Lemma \\ref{lemma-universally-unramified-differentials-sequence}." ], "refs": [ "spaces-more-morphisms-lemma-universal-thickening-functorial", "spaces-more-morphisms-lemma-universally-unramified-differentials-sequence" ], "proofs": [ { "contents": [ "Since the maps have been defined, checking the sequence is exact", "reduces to the case of schemes by \\'etale localization, see", "Lemma \\ref{lemma-universal-thickening-localize}", "and", "Lemma \\ref{lemma-localize-differentials}.", "In this case the result is", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-two-unramified-morphisms}." ], "refs": [ "spaces-more-morphisms-lemma-universal-thickening-localize", "spaces-more-morphisms-lemma-localize-differentials", "more-morphisms-lemma-two-unramified-morphisms" ], "ref_ids": [ 84, 32, 13706 ] } ], "ref_ids": [ 81, 86 ] }, { "id": 88, "type": "theorem", "label": "spaces-more-morphisms-lemma-transitivity-conormal-unramified", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-transitivity-conormal-unramified", "contents": [ "Let $S$ be a scheme.", "Let $Z \\to Y \\to X$ be formally unramified morphisms of", "algebraic spaces over $S$.", "\\begin{enumerate}", "\\item If $Z \\subset Z'$ is the universal first order thickening of $Z$", "over $X$ and $Y \\subset Y'$ is the universal first order thickening of $Y$", "over $X$, then there is a morphism $Z' \\to Y'$ and $Y \\times_{Y'} Z'$ is", "the universal first order thickening of $Z$ over $Y$.", "\\item There is a canonical exact sequence", "$$", "i^*\\mathcal{C}_{Y/X} \\to", "\\mathcal{C}_{Z/X} \\to", "\\mathcal{C}_{Z/Y} \\to 0", "$$", "where the maps come from", "Lemma \\ref{lemma-universal-thickening-functorial}", "and $i : Z \\to Y$ is the first morphism.", "\\end{enumerate}" ], "refs": [ "spaces-more-morphisms-lemma-universal-thickening-functorial" ], "proofs": [ { "contents": [ "The map $h : Z' \\to Y'$ in (1) comes from", "Lemma \\ref{lemma-universal-thickening-functorial}.", "The assertion that $Y \\times_{Y'} Z'$ is the universal first order", "thickening of $Z$ over $Y$ is clear from the universal properties", "of $Z'$ and $Y'$. By", "Lemma \\ref{lemma-transitivity-conormal}", "we have an exact sequence", "$$", "(i')^*\\mathcal{C}_{Y \\times_{Y'} Z'/Z'} \\to", "\\mathcal{C}_{Z/Z'} \\to", "\\mathcal{C}_{Z/Y \\times_{Y'} Z'} \\to 0", "$$", "where $i' : Z \\to Y \\times_{Y'} Z'$ is the given morphism. By", "Lemma \\ref{lemma-conormal-functorial-flat}", "there exists a surjection", "$h^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{Y \\times_{Y'} Z'/Z'}$.", "Combined with the equalities", "$\\mathcal{C}_{Y/Y'} = \\mathcal{C}_{Y/X}$,", "$\\mathcal{C}_{Z/Z'} = \\mathcal{C}_{Z/X}$, and", "$\\mathcal{C}_{Z/Y \\times_{Y'} Z'} = \\mathcal{C}_{Z/Y}$", "this proves the lemma." ], "refs": [ "spaces-more-morphisms-lemma-universal-thickening-functorial", "spaces-more-morphisms-lemma-transitivity-conormal", "spaces-more-morphisms-lemma-conormal-functorial-flat" ], "ref_ids": [ 81, 27, 26 ] } ], "ref_ids": [ 81 ] }, { "id": 89, "type": "theorem", "label": "spaces-more-morphisms-lemma-formally-etale", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-formally-etale", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over", "$S$. The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is formally \\'etale,", "\\item for every diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$ and $V$ are schemes and the vertical arrows are \\'etale", "the morphism of schemes $\\psi$ is formally \\'etale (as in", "More on Morphisms,", "Definition \\ref{more-morphisms-definition-formally-etale}), and", "\\item for one such diagram with surjective vertical arrows the morphism", "$\\psi$ is formally \\'etale.", "\\end{enumerate}" ], "refs": [ "more-morphisms-definition-formally-etale" ], "proofs": [ { "contents": [ "Assume $f$ is formally \\'etale. By", "Lemma \\ref{lemma-representable-property-formally-property}", "the morphisms $U \\to X$ and $V \\to Y$ are formally \\'etale. Thus by", "Lemma \\ref{lemma-composition-formally-smooth-etale-unramified}", "the composition $U \\to Y$ is formally \\'etale. Then it follows from", "Lemma \\ref{lemma-formally-permanence}", "that $U \\to V$ is formally \\'etale. Thus (1) implies (2). And (2)", "implies (3) trivially", "\\medskip\\noindent", "Assume given a diagram as in (3). By", "Lemma \\ref{lemma-representable-property-formally-property}", "the morphism $V \\to Y$ is formally \\'etale. Thus by", "Lemma \\ref{lemma-composition-formally-smooth-etale-unramified}", "the composition $U \\to Y$ is formally \\'etale. Then it follows from", "Lemma \\ref{lemma-etale-on-top}", "that $X \\to Y$ is formally \\'etale, i.e., (1) holds." ], "refs": [ "spaces-more-morphisms-lemma-representable-property-formally-property", "spaces-more-morphisms-lemma-composition-formally-smooth-etale-unramified", "spaces-more-morphisms-lemma-formally-permanence", "spaces-more-morphisms-lemma-representable-property-formally-property", "spaces-more-morphisms-lemma-composition-formally-smooth-etale-unramified", "spaces-more-morphisms-lemma-etale-on-top" ], "ref_ids": [ 64, 62, 66, 64, 62, 65 ] } ], "ref_ids": [ 14110 ] }, { "id": 90, "type": "theorem", "label": "spaces-more-morphisms-lemma-formally-etale-not-affine", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-formally-etale-not-affine", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a formally \\'etale morphism of algebraic spaces over $S$.", "Then given any solid commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & T \\ar[d]^i \\ar[l]_a \\\\", "Y & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "where $T \\subset T'$ is a first order thickening of algebraic spaces", "over $Y$ there exists exactly one dotted arrow making the diagram commute.", "In other words, in", "Definition \\ref{definition-formally-etale}", "the condition that $T$ be affine may be dropped." ], "refs": [ "spaces-more-morphisms-definition-formally-etale" ], "proofs": [ { "contents": [ "Let $U' \\to T'$ be a surjective \\'etale morphism where $U' = \\coprod U'_i$", "is a disjoint union of affine schemes. Let", "$U_i = T \\times_{T'} U'_i$. Then we get morphisms", "$a'_i : U'_i \\to X$ such that $a'_i|_{U_i}$ equals the composition", "$U_i \\to T \\to X$. By uniqueness (see", "Lemma \\ref{lemma-formally-unramified-not-affine})", "we see that $a'_i$ and $a'_j$ agree on the fibre product", "$U'_i \\times_{T'} U'_j$. Hence $\\coprod a'_i : U' \\to X$", "descends to give a unique morphism $a' : T' \\to X$." ], "refs": [ "spaces-more-morphisms-lemma-formally-unramified-not-affine" ], "ref_ids": [ 68 ] } ], "ref_ids": [ 288 ] }, { "id": 91, "type": "theorem", "label": "spaces-more-morphisms-lemma-composition-formally-etale", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-composition-formally-etale", "contents": [ "A composition of formally \\'etale morphisms is formally \\'etale." ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 92, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-formally-etale", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-formally-etale", "contents": [ "A base change of a formally \\'etale morphism is formally \\'etale." ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 93, "type": "theorem", "label": "spaces-more-morphisms-lemma-characterize-formally-etale", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-characterize-formally-etale", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is formally \\'etale,", "\\item $f$ is formally unramified and the universal first order thickening", "of $X$ over $Y$ is equal to $X$,", "\\item $f$ is formally unramified and $\\mathcal{C}_{X/Y} = 0$, and", "\\item $\\Omega_{X/Y} = 0$ and $\\mathcal{C}_{X/Y} = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Actually, the last assertion only make sense because $\\Omega_{X/Y} = 0$", "implies that $\\mathcal{C}_{X/Y}$ is defined via", "Lemma \\ref{lemma-characterize-formally-unramified}", "and", "Definition \\ref{definition-universal-thickening}.", "This also makes it clear that (3) and (4) are equivalent.", "\\medskip\\noindent", "Either of the assumptions (1), (2), and (3) imply that $f$ is formally", "unramified. Hence we may assume $f$ is formally unramified. The equivalence", "of (1), (2), and (3) follow from the universal property of the universal", "first order thickening $X'$ of $X$ over $S$ and the fact that", "$X = X' \\Leftrightarrow \\mathcal{C}_{X/Y} = 0$ since", "after all by definition $\\mathcal{C}_{X/Y} = \\mathcal{C}_{X/X'}$", "is the ideal sheaf of $X$ in $X'$." ], "refs": [ "spaces-more-morphisms-lemma-characterize-formally-unramified", "spaces-more-morphisms-definition-universal-thickening" ], "ref_ids": [ 71, 287 ] } ], "ref_ids": [] }, { "id": 94, "type": "theorem", "label": "spaces-more-morphisms-lemma-unramified-flat-formally-etale", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-unramified-flat-formally-etale", "contents": [ "An unramified flat morphism is formally \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Follows from the case of schemes, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-unramified-flat-formally-etale}", "and \\'etale localization, see", "Lemmas \\ref{lemma-formally-unramified} and \\ref{lemma-formally-etale}", "and", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-local}." ], "refs": [ "more-morphisms-lemma-unramified-flat-formally-etale", "spaces-more-morphisms-lemma-formally-unramified", "spaces-more-morphisms-lemma-formally-etale", "spaces-morphisms-lemma-flat-local" ], "ref_ids": [ 13713, 67, 89, 4854 ] } ], "ref_ids": [] }, { "id": 95, "type": "theorem", "label": "spaces-more-morphisms-lemma-etale-formally-etale", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-etale-formally-etale", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is \\'etale, and", "\\item the morphism $f$ is locally of finite presentation and", "formally \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows from the case of schemes, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-etale-formally-etale}", "and \\'etale localization, see", "Lemma \\ref{lemma-formally-etale}", "and", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-finite-presentation-local} and", "\\ref{spaces-morphisms-lemma-etale-local}." ], "refs": [ "more-morphisms-lemma-etale-formally-etale", "spaces-more-morphisms-lemma-formally-etale", "spaces-morphisms-lemma-finite-presentation-local", "spaces-morphisms-lemma-etale-local" ], "ref_ids": [ 13715, 89, 4841, 4905 ] } ], "ref_ids": [] }, { "id": 96, "type": "theorem", "label": "spaces-more-morphisms-lemma-difference-derivation", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-difference-derivation", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $X \\subset X'$ and $Y \\subset Y'$ be two first order thickenings", "of algebraic spaces over $B$.", "Let $(a, a'), (b, b') : (X \\subset X') \\to (Y \\subset Y')$", "be two morphisms of thickenings over $B$. Assume that", "\\begin{enumerate}", "\\item $a = b$, and", "\\item the two maps $a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$", "(Lemma \\ref{lemma-conormal-functorial})", "are equal.", "\\end{enumerate}", "Then the map $(a')^\\sharp - (b')^\\sharp$ factors as", "$$", "\\mathcal{O}_{Y'} \\to \\mathcal{O}_Y \\xrightarrow{D}", "a_*\\mathcal{C}_{X/X'} \\to a_*\\mathcal{O}_{X'}", "$$", "where $D$ is an $\\mathcal{O}_B$-derivation." ], "refs": [ "spaces-more-morphisms-lemma-conormal-functorial" ], "proofs": [ { "contents": [ "Instead of working on $Y$ we work on $X$. The advantage is that the pullback", "functor $a^{-1}$ is exact. Using (1) and (2) we obtain a commutative diagram", "with exact rows", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{C}_{X/X'} \\ar[r] &", "\\mathcal{O}_{X'} \\ar[r] &", "\\mathcal{O}_X \\ar[r] & 0 \\\\", "0 \\ar[r] &", "a^{-1}\\mathcal{C}_{Y/Y'} \\ar[r] \\ar[u] &", "a^{-1}\\mathcal{O}_{Y'}", "\\ar[r] \\ar@<1ex>[u]^{(a')^\\sharp} \\ar@<-1ex>[u]_{(b')^\\sharp} &", "a^{-1}\\mathcal{O}_Y \\ar[r] \\ar[u] & 0", "}", "$$", "Now it is a general fact that in such a situation the difference of the", "$\\mathcal{O}_B$-algebra maps $(a')^\\sharp$ and $(b')^\\sharp$ is an", "$\\mathcal{O}_B$-derivation from $a^{-1}\\mathcal{O}_Y$ to $\\mathcal{C}_{X/X'}$.", "By adjointness of the functors $a^{-1}$ and $a_*$ this is the same", "thing as an $\\mathcal{O}_B$-derivation from", "$\\mathcal{O}_Y$ into $a_*\\mathcal{C}_{X/X'}$. Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 24 ] }, { "id": 97, "type": "theorem", "label": "spaces-more-morphisms-lemma-action-by-derivations", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-action-by-derivations", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $(a, a') : (X \\subset X') \\to (Y \\subset Y')$", "be a morphism of first order thickenings over $B$.", "Let", "$$", "\\theta : a^*\\Omega_{Y/B} \\to \\mathcal{C}_{X/X'}", "$$", "be an $\\mathcal{O}_X$-linear map. Then there exists a unique morphism of pairs", "$(b, b') : (X \\subset X') \\to (Y \\subset Y')$ such that", "(1) and (2) of", "Lemma \\ref{lemma-difference-derivation}", "hold and the derivation $D$ and $\\theta$ are related by", "Equation (\\ref{equation-D})." ], "refs": [ "spaces-more-morphisms-lemma-difference-derivation" ], "proofs": [ { "contents": [ "Consider the map", "$$", "\\alpha = (a')^\\sharp + D : a^{-1}\\mathcal{O}_{Y'} \\to \\mathcal{O}_{X'}", "$$", "where $D$ is as in Equation (\\ref{equation-D}). As $D$ is an", "$\\mathcal{O}_B$-derivation it follows that $\\alpha$ is a map of", "sheaves of $\\mathcal{O}_B$-algebras. By construction we have", "$i_X^\\sharp \\circ \\alpha = a^\\sharp \\circ i_Y^\\sharp$ where", "$i_X : X \\to X'$ and $i_Y : Y \\to Y'$ are the given closed immersions. By", "Lemma \\ref{lemma-first-order-thickening-maps}", "we obtain a unique morphism", "$(a, b') : (X \\subset X') \\to (Y \\subset Y')$ of thickenings", "over $B$ such that $\\alpha = (b')^\\sharp$. Setting $b = a$", "we win." ], "refs": [ "spaces-more-morphisms-lemma-first-order-thickening-maps" ], "ref_ids": [ 46 ] } ], "ref_ids": [ 96 ] }, { "id": 98, "type": "theorem", "label": "spaces-more-morphisms-lemma-sheaf", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-sheaf", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $X \\subset X'$ and $Y \\subset Y'$ be first order thickenings", "over $B$. Assume given a morphism $a : X \\to Y$ and a map", "$A : a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$ of", "$\\mathcal{O}_X$-modules. For an object $U'$ of", "$(X')_{spaces, \\etale}$ with $U = X \\times_{X'} U'$", "consider morphisms $a' : U' \\to Y'$ such that", "\\begin{enumerate}", "\\item $a'$ is a morphism over $B$,", "\\item $a'|_U = a|_U$, and", "\\item the induced map", "$a^*\\mathcal{C}_{Y/Y'}|_U \\to \\mathcal{C}_{X/X'}|_U$", "is the restriction of $A$ to $U$.", "\\end{enumerate}", "Then the rule", "\\begin{equation}", "\\label{equation-sheaf}", "U' \\mapsto", "\\{a' : U' \\to Y'\\text{ such that (1), (2), (3) hold.}\\}", "\\end{equation}", "defines a sheaf of sets on $(X')_{spaces, \\etale}$." ], "refs": [], "proofs": [ { "contents": [ "Denote $\\mathcal{F}$ the rule of the lemma.", "The restriction mapping $\\mathcal{F}(U') \\to \\mathcal{F}(V')$ for", "$V' \\subset U' \\subset X'$", "of $\\mathcal{F}$ is really the restriction map $a' \\mapsto a'|_{V'}$.", "With this definition in place it is clear that $\\mathcal{F}$ is a", "sheaf since morphisms of algebraic spaces satisfy \\'etale descent, see", "Descent on Spaces,", "Lemma \\ref{spaces-descent-lemma-fpqc-universal-effective-epimorphisms}." ], "refs": [ "spaces-descent-lemma-fpqc-universal-effective-epimorphisms" ], "ref_ids": [ 9367 ] } ], "ref_ids": [] }, { "id": 99, "type": "theorem", "label": "spaces-more-morphisms-lemma-action-sheaf", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-action-sheaf", "contents": [ "Same notation and assumptions as in Lemma \\ref{lemma-sheaf}.", "We identify sheaves on $X$ and $X'$ via", "(\\ref{equation-equivalence-etale-spaces}).", "There is an action of the sheaf", "$$", "\\SheafHom_{\\mathcal{O}_X}(a^*\\Omega_{Y/B}, \\mathcal{C}_{X/X'})", "$$", "on the sheaf (\\ref{equation-sheaf}). Moreover, the action", "is simply transitive for any object $U'$ of $(X')_{spaces, \\etale}$", "over which the sheaf (\\ref{equation-sheaf}) has a section." ], "refs": [ "spaces-more-morphisms-lemma-sheaf" ], "proofs": [ { "contents": [ "This is a combination of", "Lemmas \\ref{lemma-difference-derivation},", "\\ref{lemma-action-by-derivations},", "and \\ref{lemma-sheaf}." ], "refs": [ "spaces-more-morphisms-lemma-difference-derivation", "spaces-more-morphisms-lemma-action-by-derivations", "spaces-more-morphisms-lemma-sheaf" ], "ref_ids": [ 96, 97, 98 ] } ], "ref_ids": [ 98 ] }, { "id": 100, "type": "theorem", "label": "spaces-more-morphisms-lemma-action-by-derivations-etale-localization", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-action-by-derivations-etale-localization", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram of first order", "thickenings", "$$", "\\vcenter{", "\\xymatrix{", "(T_2 \\subset T_2') \\ar[d]_{(h, h')} \\ar[rr]_{(a_2, a_2')} & &", "(X_2 \\subset X_2') \\ar[d]^{(f, f')} \\\\", "(T_1 \\subset T_1') \\ar[rr]^{(a_1, a_1')} & &", "(X_1 \\subset X_1')", "}", "}", "\\quad", "\\begin{matrix}", "\\text{and a commutative} \\\\", "\\text{diagram}", "\\end{matrix}", "\\quad", "\\vcenter{", "\\xymatrix{", "X_2' \\ar[r] \\ar[d] & B_2 \\ar[d] \\\\", "X_1' \\ar[r] & B_1", "}", "}", "$$", "of algebraic spaces over $S$", "with $X_2 \\to X_1$ and $B_2 \\to B_1$ \\'etale.", "For any $\\mathcal{O}_{T_1}$-linear map", "$\\theta_1 : a_1^*\\Omega_{X_1/B_1} \\to \\mathcal{C}_{T_1/T'_1}$ let", "$\\theta_2$ be the composition", "$$", "\\xymatrix{", "a_2^*\\Omega_{X_2/B_2} \\ar@{=}[r] &", "h^*a_1^*\\Omega_{X_1/B_1} \\ar[r]^-{h^*\\theta_1} &", "h^*\\mathcal{C}_{T_1/T'_1} \\ar[r] &", "\\mathcal{C}_{T_2/T'_2}", "}", "$$", "(equality sign is explained in the proof). Then the diagram", "$$", "\\xymatrix{", "T_2' \\ar[rr]_{\\theta_2 \\cdot a_2'} \\ar[d] & & X'_2 \\ar[d] \\\\", "T_1' \\ar[rr]^{\\theta_1 \\cdot a_1'} & & X'_1", "}", "$$", "commutes where the actions $\\theta_2 \\cdot a_2'$ and $\\theta_1 \\cdot a_1'$", "are as in Remark \\ref{remark-action-by-derivations}." ], "refs": [ "spaces-more-morphisms-remark-action-by-derivations" ], "proofs": [ { "contents": [ "The equality sign comes from the identification", "$f^*\\Omega_{X_1/S_1} = \\Omega_{X_2/S_2}$ we get", "as the construction of the sheaf of differentials is", "compatible with \\'etale localization (both on source and target), see", "Lemma \\ref{lemma-localize-differentials}.", "Namely, using this we have", "$a_2^*\\Omega_{X_2/S_2} = a_2^*f^*\\Omega_{X_1/S_1} =", "h^*a_1^*\\Omega_{X_1/S_1}$ because $f \\circ a_2 = a_1 \\circ h$.", "Having said this, the commutativity of the diagram may be checked", "on \\'etale locally. Thus we may assume $T'_i$, $X'_i$,", "$B_2$, and $B_1$ are schemes and in this case the lemma", "follows from", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-action-by-derivations-etale-localization}.", "Alternative proof: using Lemma \\ref{lemma-first-order-thickening-maps}", "it suffices to show a certain diagram of sheaves", "of rings on $X_1'$ is commutative; then argue exactly", "as in the proof of the aforementioned", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-action-by-derivations-etale-localization}", "to see that this is indeed the case." ], "refs": [ "spaces-more-morphisms-lemma-localize-differentials", "more-morphisms-lemma-action-by-derivations-etale-localization", "spaces-more-morphisms-lemma-first-order-thickening-maps", "more-morphisms-lemma-action-by-derivations-etale-localization" ], "ref_ids": [ 32, 13722, 46, 13722 ] } ], "ref_ids": [ 306 ] }, { "id": 101, "type": "theorem", "label": "spaces-more-morphisms-lemma-deform", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-deform", "contents": [ "Let $S$ be a scheme. Let $(f, f') : (X \\subset X') \\to (Y \\subset Y')$ be a", "morphism of first order thickenings of algebraic spaces over $S$. Assume that", "$f$ is flat. Then the following are equivalent", "\\begin{enumerate}", "\\item $f'$ is flat and $X = Y \\times_{Y'} X'$, and", "\\item the canonical map $f^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V'$ and a surjective \\'etale morphism $V' \\to Y'$.", "Choose a scheme $U'$ and a surjective \\'etale morphism", "$U' \\to X' \\times_{Y'} V'$. Set $U = X \\times_{X'} U'$ and", "$V = Y \\times_{Y'} V'$. According to our definition of a flat morphism", "of algebraic spaces we see that the induced map $g : U \\to V$ is a flat", "morphism of schemes and that $f'$ is flat if and only if the corresponding", "morphism $g' : U' \\to V'$ is flat. Also, $X = Y \\times_{Y'} X'$ if and only", "if $U = V \\times_{V'} V'$. Finally, the map", "$f^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$", "is an isomorphism if and only if", "$g^*\\mathcal{C}_{V/V'} \\to \\mathcal{C}_{U/U'}$ is an isomorphism.", "Hence the lemma follows from its analogue for morphisms of schemes, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-deform}." ], "refs": [ "more-morphisms-lemma-deform" ], "ref_ids": [ 13723 ] } ], "ref_ids": [] }, { "id": 102, "type": "theorem", "label": "spaces-more-morphisms-lemma-flatness-morphism-thickenings", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flatness-morphism-thickenings", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\", "& (B \\subset B')", "}", "$$", "of thickenings of algebraic spaces over $S$. Assume", "\\begin{enumerate}", "\\item $X'$ is flat over $B'$,", "\\item $f$ is flat,", "\\item $B \\subset B'$ is a finite order thickening, and", "\\item $X = B \\times_{B'} X'$ and $Y = B \\times_{B'} Y'$.", "\\end{enumerate}", "Then $f'$ is flat and $Y'$ is flat over $B'$ at all points in", "the image of $f'$." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U'$ and a surjective \\'etale morphism $U' \\to B'$.", "Choose a scheme $V'$ and a surjective \\'etale morphism", "$V' \\to U' \\times_{B'} Y'$.", "Choose a scheme $W'$ and a surjective \\'etale morphism", "$W' \\to V' \\times_{Y'} X'$. Let $U, V, W$ be the base change", "of $U', V', W'$ by $B \\to B'$. Then flatness of $f'$ is", "equivalent to flatness of $W' \\to V'$ and we are", "given that $W \\to V$ is flat. Hence we may apply the lemma", "in the case of schemes to the diagram", "$$", "\\xymatrix{", "(W \\subset W') \\ar[rr] \\ar[rd] & & (V \\subset V') \\ar[ld] \\\\", "& (U \\subset U')", "}", "$$", "of thickenings of schemes. See", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-flatness-morphism-thickenings}.", "The statement about flatness of $Y'/B'$ at points in the", "image of $f'$ follows in the same manner." ], "refs": [ "more-morphisms-lemma-flatness-morphism-thickenings" ], "ref_ids": [ 13724 ] } ], "ref_ids": [] }, { "id": 103, "type": "theorem", "label": "spaces-more-morphisms-lemma-deform-property", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-deform-property", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\", "& (B \\subset B')", "}", "$$", "of thickenings of algebraic spaces over $S$. Assume $B \\subset B'$", "is a finite order thickening, $X'$ flat over $B'$, $X = B \\times_{B'} X'$,", "and $Y = B \\times_{B'} Y'$. Then", "\\begin{enumerate}", "\\item $f$ is representable if and only if $f'$ is representable,", "\\label{item-representable}", "\\item $f$ is flat if and only if $f'$ is flat,", "\\label{item-flat}", "\\item $f$ is an isomorphism if and only if $f'$ is an isomorphism,", "\\label{item-isomorphism}", "\\item $f$ is an open immersion if and only if $f'$ is an open immersion,", "\\label{item-open-immersion}", "\\item $f$ is quasi-compact if and only if $f'$ is quasi-compact,", "\\label{item-quasi-compact}", "\\item $f$ is universally closed if and only if $f'$ is universally closed,", "\\label{item-universally-closed}", "\\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,", "\\label{item-separated}", "\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,", "\\label{item-monomorphism}", "\\item $f$ is surjective if and only if $f'$ is surjective,", "\\label{item-surjective}", "\\item $f$ is universally injective if and only if $f'$ is universally injective,", "\\label{item-universally-injective}", "\\item $f$ is affine if and only if $f'$ is affine,", "\\label{item-affine}", "\\item", "\\label{item-finite-type}", "$f$ is locally of finite type if and only if $f'$ is locally of finite type,", "\\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,", "\\label{item-quasi-finite}", "\\item", "\\label{item-finite-presentation}", "$f$ is locally of finite presentation if and only if $f'$ is locally of", "finite presentation,", "\\item", "\\label{item-relative-dimension-d}", "$f$ is locally of finite type of relative dimension $d$ if and only if", "$f'$ is locally of finite type of relative dimension $d$,", "\\item $f$ is universally open if and only if $f'$ is universally open,", "\\label{item-universally-open}", "\\item $f$ is syntomic if and only if $f'$ is syntomic,", "\\label{item-syntomic}", "\\item $f$ is smooth if and only if $f'$ is smooth,", "\\label{item-smooth}", "\\item $f$ is unramified if and only if $f'$ is unramified,", "\\label{item-unramified}", "\\item $f$ is \\'etale if and only if $f'$ is \\'etale,", "\\label{item-etale}", "\\item $f$ is proper if and only if $f'$ is proper,", "\\label{item-proper}", "\\item $f$ is integral if and only if $f'$ is integral,", "\\label{item-integral}", "\\item $f$ is finite if and only if $f'$ is finite,", "\\label{item-finite}", "\\item", "\\label{item-finite-locally-free}", "$f$ is finite locally free (of rank $d$) if and only if $f'$", "is finite locally free (of rank $d$), and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Case (\\ref{item-representable}) follows from", "Lemma \\ref{lemma-thicken-property-morphisms}.", "\\medskip\\noindent", "Choose a scheme $U'$ and a surjective \\'etale morphism $U' \\to B'$.", "Choose a scheme $V'$ and a surjective \\'etale morphism", "$V' \\to U' \\times_{B'} Y'$.", "Choose a scheme $W'$ and a surjective \\'etale morphism", "$W' \\to V' \\times_{Y'} X'$. Let $U, V, W$ be the base change", "of $U', V', W'$ by $B \\to B'$. Consider the diagram", "$$", "\\xymatrix{", "(W \\subset W') \\ar[rr] \\ar[rd] & & (V \\subset V') \\ar[ld] \\\\", "& (U \\subset U')", "}", "$$", "of thickenings of schemes. For any of the properties which are", "\\'etale local on the source-and-target the result follows immediately", "from the corresponding result for morphisms of thickenings of schemes", "applied to the diagram above. Thus cases", "(\\ref{item-flat}), (\\ref{item-finite-type}),", "(\\ref{item-quasi-finite}), (\\ref{item-finite-presentation}),", "(\\ref{item-relative-dimension-d}), (\\ref{item-syntomic}),", "(\\ref{item-smooth}), (\\ref{item-unramified}), (\\ref{item-etale})", "follow from the corresponding cases of", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-deform-property}.", "\\medskip\\noindent", "Since $X \\to X'$ and $Y \\to Y'$ are universal homeomorphisms", "we see that any question about the topology of the maps", "$X \\to Y$ and $X' \\to Y'$ has the same answer. Thus we see", "that cases (\\ref{item-quasi-compact}), (\\ref{item-universally-closed}),", "(\\ref{item-surjective}), (\\ref{item-universally-injective}), and", "(\\ref{item-universally-open}) hold.", "\\medskip\\noindent", "In each of the remaining cases we only prove the implication", "$f\\text{ has }P \\Rightarrow f'\\text{ has }P$ since the other", "implication follows from the fact that $P$ is stable under", "base change, see", "Spaces, Lemma \\ref{spaces-lemma-base-change-immersions} and", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-separated},", "\\ref{spaces-morphisms-lemma-base-change-monomorphism},", "\\ref{spaces-morphisms-lemma-base-change-affine},", "\\ref{spaces-morphisms-lemma-base-change-proper},", "\\ref{spaces-morphisms-lemma-base-change-integral}, and", "\\ref{spaces-morphisms-lemma-base-change-finite-locally-free}.", "\\medskip\\noindent", "The case (\\ref{item-open-immersion}). Assume $f$ is an open immersion.", "Then $f'$ is \\'etale by (\\ref{item-etale}) and universally injective", "by (\\ref{item-universally-injective})", "hence $f'$ is an open immersion, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-etale-universally-injective-open}.", "You can avoid using this lemma at the cost of first", "using (\\ref{item-representable}) to reduce to the case of schemes.", "\\medskip\\noindent", "The case (\\ref{item-isomorphism}). Follows from cases", "(\\ref{item-open-immersion}) and (\\ref{item-surjective}).", "\\medskip\\noindent", "The case (\\ref{item-separated}). See", "Lemma \\ref{lemma-thicken-property-morphisms}.", "\\medskip\\noindent", "The case (\\ref{item-monomorphism}). Assume $f$ is a monomorphism.", "Consider the diagonal morphism $\\Delta_{X'/Y'} : X' \\to X' \\times_{Y'} X'$.", "The base change of $\\Delta_{X'/Y'}$ by $B \\to B'$ is $\\Delta_{X/Y}$", "which is an isomorphism by assumption. By (\\ref{item-isomorphism})", "we conclude that $\\Delta_{X'/Y'}$ is an isomorphism and hence", "$f'$ is a monomorphism.", "\\medskip\\noindent", "The case (\\ref{item-affine}). See Lemma \\ref{lemma-thicken-property-morphisms}.", "\\medskip\\noindent", "The case (\\ref{item-proper}). See", "Lemma \\ref{lemma-thicken-property-morphisms-cartesian}.", "\\medskip\\noindent", "The case (\\ref{item-integral}). See", "Lemma \\ref{lemma-thicken-property-morphisms}.", "\\medskip\\noindent", "The case (\\ref{item-finite}). See", "Lemma \\ref{lemma-thicken-property-morphisms-cartesian}.", "\\medskip\\noindent", "The case (\\ref{item-finite-locally-free}). Assume $f$ finite locally free.", "By (\\ref{item-finite}) we see that $f'$ is finite.", "By (\\ref{item-flat}) we see that $f'$ is flat.", "By (\\ref{item-finite-presentation}) $f'$ is locally of finite", "presentation. Hence $f'$ is finite locally free by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-flat}." ], "refs": [ "spaces-more-morphisms-lemma-thicken-property-morphisms", "more-morphisms-lemma-deform-property", "spaces-lemma-base-change-immersions", "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-base-change-monomorphism", "spaces-morphisms-lemma-base-change-affine", "spaces-morphisms-lemma-base-change-proper", "spaces-morphisms-lemma-base-change-integral", "spaces-morphisms-lemma-base-change-finite-locally-free", "spaces-morphisms-lemma-etale-universally-injective-open", "spaces-more-morphisms-lemma-thicken-property-morphisms", "spaces-more-morphisms-lemma-thicken-property-morphisms", "spaces-more-morphisms-lemma-thicken-property-morphisms-cartesian", "spaces-more-morphisms-lemma-thicken-property-morphisms", "spaces-more-morphisms-lemma-thicken-property-morphisms-cartesian", "spaces-morphisms-lemma-finite-flat" ], "ref_ids": [ 55, 13725, 8161, 4714, 4754, 4800, 4917, 4942, 4953, 4973, 55, 55, 56, 55, 56, 4954 ] } ], "ref_ids": [] }, { "id": 104, "type": "theorem", "label": "spaces-more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\", "& (B \\subset B')", "}", "$$", "of thickenings of algebraic spaces over $S$. Assume", "\\begin{enumerate}", "\\item $Y' \\to B'$ is locally of finite type,", "\\item $X' \\to B'$ is flat and locally of finite presentation,", "\\item $f$ is flat, and", "\\item $X = B \\times_{B'} X'$ and $Y = B \\times_{B'} Y'$.", "\\end{enumerate}", "Then $f'$ is flat and for all $y' \\in |Y'|$ in the image of $|f'|$", "the morphism $Y' \\to B'$ is flat at $y'$." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U'$ and a surjective \\'etale morphism $U' \\to B'$.", "Choose a scheme $V'$ and a surjective \\'etale morphism", "$V' \\to U' \\times_{B'} Y'$.", "Choose a scheme $W'$ and a surjective \\'etale morphism", "$W' \\to V' \\times_{Y'} X'$. Let $U, V, W$ be the base change", "of $U', V', W'$ by $B \\to B'$. Then flatness of $f'$ is", "equivalent to flatness of $W' \\to V'$ and we are", "given that $W \\to V$ is flat. Hence we may apply the lemma", "in the case of schemes to the diagram", "$$", "\\xymatrix{", "(W \\subset W') \\ar[rr] \\ar[rd] & & (V \\subset V') \\ar[ld] \\\\", "& (U \\subset U')", "}", "$$", "of thickenings of schemes. See", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft}.", "The statement about flatness of $Y'/B'$ at points in the", "image of $f'$ follows in the same manner." ], "refs": [ "more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft" ], "ref_ids": [ 13726 ] } ], "ref_ids": [] }, { "id": 105, "type": "theorem", "label": "spaces-more-morphisms-lemma-deform-property-fp-over-ft", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-deform-property-fp-over-ft", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\", "& (B \\subset B')", "}", "$$", "of thickenings of algebraic spaces over $S$.", "Assume $Y' \\to B'$ locally of finite type,", "$X' \\to B'$ flat and locally of finite presentation,", "$X = B \\times_{B'} X'$, and $Y = B \\times_{B'} Y'$. Then", "\\begin{enumerate}", "\\item $f$ is representable if and only if $f'$ is representable,", "\\label{item-representable-fp-over-ft}", "\\item $f$ is flat if and only if $f'$ is flat,", "\\label{item-flat-fp-over-ft}", "\\item $f$ is an isomorphism if and only if $f'$ is an isomorphism,", "\\label{item-isomorphism-fp-over-ft}", "\\item $f$ is an open immersion if and only if $f'$ is an open immersion,", "\\label{item-open-immersion-fp-over-ft}", "\\item $f$ is quasi-compact if and only if $f'$ is quasi-compact,", "\\label{item-quasi-compact-fp-over-ft}", "\\item $f$ is universally closed if and only if $f'$ is universally closed,", "\\label{item-universally-closed-fp-over-ft}", "\\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,", "\\label{item-separated-fp-over-ft}", "\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,", "\\label{item-monomorphism-fp-over-ft}", "\\item $f$ is surjective if and only if $f'$ is surjective,", "\\label{item-surjective-fp-over-ft}", "\\item $f$ is universally injective if and only if $f'$ is universally injective,", "\\label{item-universally-injective-fp-over-ft}", "\\item $f$ is affine if and only if $f'$ is affine,", "\\label{item-affine-fp-over-ft}", "\\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,", "\\label{item-quasi-finite-fp-over-ft}", "\\item", "\\label{item-relative-dimension-d-fp-over-ft}", "$f$ is locally of finite type of relative dimension $d$ if and only if", "$f'$ is locally of finite type of relative dimension $d$,", "\\item $f$ is universally open if and only if $f'$ is universally open,", "\\label{item-universally-open-fp-over-ft}", "\\item $f$ is syntomic if and only if $f'$ is syntomic,", "\\label{item-syntomic-fp-over-ft}", "\\item $f$ is smooth if and only if $f'$ is smooth,", "\\label{item-smooth-fp-over-ft}", "\\item $f$ is unramified if and only if $f'$ is unramified,", "\\label{item-unramified-fp-over-ft}", "\\item $f$ is \\'etale if and only if $f'$ is \\'etale,", "\\label{item-etale-fp-over-ft}", "\\item $f$ is proper if and only if $f'$ is proper,", "\\label{item-proper-fp-over-ft}", "\\item $f$ is finite if and only if $f'$ is finite,", "\\label{item-finite-fp-over-ft}", "\\item", "\\label{item-finite-locally-free-fp-over-ft}", "$f$ is finite locally free (of rank $d$) if and only if $f'$", "is finite locally free (of rank $d$), and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Case (\\ref{item-representable-fp-over-ft}) follows from", "Lemma \\ref{lemma-thicken-property-morphisms}.", "\\medskip\\noindent", "Choose a scheme $U'$ and a surjective \\'etale morphism $U' \\to B'$.", "Choose a scheme $V'$ and a surjective \\'etale morphism", "$V' \\to U' \\times_{B'} Y'$.", "Choose a scheme $W'$ and a surjective \\'etale morphism", "$W' \\to V' \\times_{Y'} X'$. Let $U, V, W$ be the base change", "of $U', V', W'$ by $B \\to B'$. Consider the diagram", "$$", "\\xymatrix{", "(W \\subset W') \\ar[rr] \\ar[rd] & & (V \\subset V') \\ar[ld] \\\\", "& (U \\subset U')", "}", "$$", "of thickenings of schemes. For any of the properties which are", "\\'etale local on the source-and-target the result follows immediately", "from the corresponding result for morphisms of thickenings of schemes", "applied to the diagram above. Thus cases", "(\\ref{item-flat-fp-over-ft}),", "(\\ref{item-quasi-finite-fp-over-ft}),", "(\\ref{item-relative-dimension-d-fp-over-ft}),", "(\\ref{item-syntomic-fp-over-ft}),", "(\\ref{item-smooth-fp-over-ft}),", "(\\ref{item-unramified-fp-over-ft}),", "(\\ref{item-etale-fp-over-ft})", "follow from the corresponding cases of", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-deform-property-fp-over-ft}.", "\\medskip\\noindent", "Since $X \\to X'$ and $Y \\to Y'$ are universal homeomorphisms", "we see that any question about the topology of the maps", "$X \\to Y$ and $X' \\to Y'$ has the same answer. Thus we see", "that cases (\\ref{item-quasi-compact-fp-over-ft}),", "(\\ref{item-universally-closed-fp-over-ft}),", "(\\ref{item-surjective-fp-over-ft}),", "(\\ref{item-universally-injective-fp-over-ft}), and", "(\\ref{item-universally-open-fp-over-ft}) hold.", "\\medskip\\noindent", "In each of the remaining cases we only prove the implication", "$f\\text{ has }P \\Rightarrow f'\\text{ has }P$ since the other", "implication follows from the fact that $P$ is stable under", "base change, see", "Spaces, Lemma \\ref{spaces-lemma-base-change-immersions} and", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-separated},", "\\ref{spaces-morphisms-lemma-base-change-monomorphism},", "\\ref{spaces-morphisms-lemma-base-change-affine},", "\\ref{spaces-morphisms-lemma-base-change-proper},", "\\ref{spaces-morphisms-lemma-base-change-integral}, and", "\\ref{spaces-morphisms-lemma-base-change-finite-locally-free}.", "\\medskip\\noindent", "The case (\\ref{item-open-immersion-fp-over-ft}).", "Assume $f$ is an open immersion.", "Then $f'$ is \\'etale by (\\ref{item-etale-fp-over-ft}) and universally injective", "by (\\ref{item-universally-injective-fp-over-ft})", "hence $f'$ is an open immersion, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-etale-universally-injective-open}.", "You can avoid using this lemma at the cost of first", "using (\\ref{item-representable-fp-over-ft}) to reduce to the case of schemes.", "\\medskip\\noindent", "The case (\\ref{item-isomorphism-fp-over-ft}). Follows from cases", "(\\ref{item-open-immersion-fp-over-ft}) and (\\ref{item-surjective-fp-over-ft}).", "\\medskip\\noindent", "The case (\\ref{item-separated-fp-over-ft}). See", "Lemma \\ref{lemma-thicken-property-morphisms}.", "\\medskip\\noindent", "The case (\\ref{item-monomorphism-fp-over-ft}). Assume $f$ is a monomorphism.", "Consider the diagonal morphism $\\Delta_{X'/Y'} : X' \\to X' \\times_{Y'} X'$.", "The base change of $\\Delta_{X'/Y'}$ by $B \\to B'$ is $\\Delta_{X/Y}$", "which is an isomorphism by assumption. By (\\ref{item-isomorphism-fp-over-ft})", "we conclude that $\\Delta_{X'/Y'}$ is an isomorphism and hence", "$f'$ is a monomorphism.", "\\medskip\\noindent", "The case (\\ref{item-affine-fp-over-ft}).", "See Lemma \\ref{lemma-thicken-property-morphisms}.", "\\medskip\\noindent", "The case (\\ref{item-proper-fp-over-ft}). See", "Lemma \\ref{lemma-properties-that-extend-over-thickenings}.", "\\medskip\\noindent", "The case (\\ref{item-finite-fp-over-ft}). See", "Lemma \\ref{lemma-properties-that-extend-over-thickenings}.", "\\medskip\\noindent", "The case (\\ref{item-finite-locally-free-fp-over-ft}).", "Assume $f$ finite locally free.", "By (\\ref{item-finite-fp-over-ft}) we see that $f'$ is finite.", "By (\\ref{item-flat-fp-over-ft}) we see that $f'$ is flat.", "Also $f'$ is locally finite presentation by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-permanence}.", "Hence $f'$ is finite locally free by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-flat}." ], "refs": [ "spaces-more-morphisms-lemma-thicken-property-morphisms", "more-morphisms-lemma-deform-property-fp-over-ft", "spaces-lemma-base-change-immersions", "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-base-change-monomorphism", "spaces-morphisms-lemma-base-change-affine", "spaces-morphisms-lemma-base-change-proper", "spaces-morphisms-lemma-base-change-integral", "spaces-morphisms-lemma-base-change-finite-locally-free", "spaces-morphisms-lemma-etale-universally-injective-open", "spaces-more-morphisms-lemma-thicken-property-morphisms", "spaces-more-morphisms-lemma-thicken-property-morphisms", "spaces-more-morphisms-lemma-properties-that-extend-over-thickenings", "spaces-more-morphisms-lemma-properties-that-extend-over-thickenings", "spaces-morphisms-lemma-finite-presentation-permanence", "spaces-morphisms-lemma-finite-flat" ], "ref_ids": [ 55, 13727, 8161, 4714, 4754, 4800, 4917, 4942, 4953, 4973, 55, 55, 57, 57, 4846, 4954 ] } ], "ref_ids": [] }, { "id": 106, "type": "theorem", "label": "spaces-more-morphisms-lemma-composition-formally-smooth", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-composition-formally-smooth", "contents": [ "A composition of formally smooth morphisms is formally smooth." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 107, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-formally-smooth", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-formally-smooth", "contents": [ "A base change of a formally smooth morphism is formally smooth." ], "refs": [], "proofs": [ { "contents": [ "Omitted, but see", "Algebra, Lemma \\ref{algebra-lemma-base-change-fs}", "for the algebraic version." ], "refs": [ "algebra-lemma-base-change-fs" ], "ref_ids": [ 1204 ] } ], "ref_ids": [] }, { "id": 108, "type": "theorem", "label": "spaces-more-morphisms-lemma-formally-etale-unramified-smooth", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-formally-etale-unramified-smooth", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Then $f$ is formally \\'etale if and only if", "$f$ is formally smooth and formally unramified." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 109, "type": "theorem", "label": "spaces-more-morphisms-lemma-helper-formally-smooth", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-helper-formally-smooth", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "be a commutative diagram of morphisms of algebraic spaces over $S$.", "If the vertical arrows are \\'etale and $f$ is formally smooth, then", "$\\psi$ is formally smooth." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-representable-property-formally-property}", "the morphisms $U \\to X$ and $V \\to Y$ are formally \\'etale. By", "Lemma \\ref{lemma-composition-formally-smooth-etale-unramified}", "the composition $U \\to Y$ is formally smooth. By", "Lemma \\ref{lemma-formally-permanence}", "we see $\\psi : U \\to V$ is formally smooth." ], "refs": [ "spaces-more-morphisms-lemma-representable-property-formally-property", "spaces-more-morphisms-lemma-composition-formally-smooth-etale-unramified", "spaces-more-morphisms-lemma-formally-permanence" ], "ref_ids": [ 64, 62, 66 ] } ], "ref_ids": [] }, { "id": 110, "type": "theorem", "label": "spaces-more-morphisms-lemma-smooth-formally-smooth", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-smooth-formally-smooth", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is smooth.", "\\item The morphism $f$ is locally of finite presentation, and", "formally smooth.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f : X \\to S$ is locally of finite presentation and formally smooth.", "Consider a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$ and $V$ are schemes and the vertical arrows are \\'etale and", "surjective. By", "Lemma \\ref{lemma-helper-formally-smooth}", "we see $\\psi : U \\to V$ is formally smooth. By", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-local}", "the morphism $\\psi$ is locally of finite presentation.", "Hence by the case of schemes the morphism", "$\\psi$ is smooth, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-smooth-formally-smooth}.", "Hence $f$ is smooth, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-smooth-local}.", "\\medskip\\noindent", "Conversely, assume that $f : X \\to Y$ is smooth.", "Consider a solid commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & T \\ar[d]^i \\ar[l]^a \\\\", "Y & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "as in Definition \\ref{definition-formally-smooth}. We will show the", "dotted arrow exists thereby proving that $f$ is formally smooth.", "Let $\\mathcal{F}$ be the sheaf of sets on $(T')_{spaces, \\etale}$ of", "Lemma \\ref{lemma-sheaf} as in the special case discussed in", "Remark \\ref{remark-special-case}. Let", "$$", "\\mathcal{H} =", "\\SheafHom_{\\mathcal{O}_T}(a^*\\Omega_{X/Y}, \\mathcal{C}_{T/T'})", "$$", "be the sheaf of $\\mathcal{O}_T$-modules on $T_{spaces, \\etale}$", "with action $\\mathcal{H} \\times \\mathcal{F} \\to \\mathcal{F}$", "as in Lemma \\ref{lemma-action-sheaf}.", "The action $\\mathcal{H} \\times \\mathcal{F} \\to \\mathcal{F}$", "turns $\\mathcal{F}$ into a pseudo $\\mathcal{H}$-torsor, see", "Cohomology on Sites, Definition \\ref{sites-cohomology-definition-torsor}.", "Our goal is to show that $\\mathcal{F}$ is a trivial $\\mathcal{H}$-torsor.", "There are two steps: (I) To show that $\\mathcal{F}$ is a torsor", "we have to show that $\\mathcal{F}$ has \\'etale locally a", "section. (II) To show that $\\mathcal{F}$ is the trivial torsor", "it suffices to show that $H^1(T_\\etale, \\mathcal{H}) = 0$, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-torsors-h1}.", "\\medskip\\noindent", "First we prove (I). To see this choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$ and $V$ are schemes and the vertical arrows are \\'etale and", "surjective. As $f$ is assumed smooth we see that $\\psi$ is smooth and", "hence formally smooth by", "Lemma \\ref{lemma-representable-property-formally-property}.", "By the same lemma the morphism $V \\to Y$ is formally \\'etale. Thus by", "Lemma \\ref{lemma-composition-formally-smooth-etale-unramified}", "the composition $U \\to Y$ is formally smooth. Then (I) follows from", "Lemma \\ref{lemma-etale-on-top} part (4).", "\\medskip\\noindent", "Finally we prove (II). By", "Lemma \\ref{lemma-finite-presentation-differentials}", "we see that $\\Omega_{X/S}$ is of finite presentation.", "Hence $a^*\\Omega_{X/S}$ is of finite presentation (see", "Properties of Spaces,", "Section \\ref{spaces-properties-section-properties-modules}).", "Hence the sheaf", "$\\mathcal{H} =", "\\SheafHom_{\\mathcal{O}_T}(a^*\\Omega_{X/Y}, \\mathcal{C}_{T/T'})$", "is quasi-coherent by", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-properties-quasi-coherent}.", "Thus by", "Descent, Proposition \\ref{descent-proposition-same-cohomology-quasi-coherent}", "and Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}", "we have", "$$", "H^1(T_{spaces, \\etale}, \\mathcal{H}) =", "H^1(T_\\etale, \\mathcal{H}) =", "H^1(T, \\mathcal{H}) = 0", "$$", "as desired." ], "refs": [ "spaces-more-morphisms-lemma-helper-formally-smooth", "spaces-morphisms-lemma-finite-presentation-local", "more-morphisms-lemma-smooth-formally-smooth", "spaces-morphisms-lemma-smooth-local", "spaces-more-morphisms-definition-formally-smooth", "spaces-more-morphisms-lemma-sheaf", "spaces-more-morphisms-remark-special-case", "spaces-more-morphisms-lemma-action-sheaf", "sites-cohomology-definition-torsor", "sites-cohomology-lemma-torsors-h1", "spaces-more-morphisms-lemma-representable-property-formally-property", "spaces-more-morphisms-lemma-composition-formally-smooth-etale-unramified", "spaces-more-morphisms-lemma-etale-on-top", "spaces-more-morphisms-lemma-finite-presentation-differentials", "spaces-properties-lemma-properties-quasi-coherent", "descent-proposition-same-cohomology-quasi-coherent", "coherent-lemma-quasi-coherent-affine-cohomology-zero" ], "ref_ids": [ 109, 4841, 13734, 4888, 289, 98, 307, 99, 4411, 4182, 64, 62, 65, 43, 11912, 14754, 3282 ] } ], "ref_ids": [] }, { "id": 111, "type": "theorem", "label": "spaces-more-morphisms-lemma-smooth-strong-lift", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-smooth-strong-lift", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] & T \\ar[l] \\ar[d] \\\\", "Y & T' \\ar[l]", "}", "$$", "of algebraic spaces over $S$ where $X \\to Y$ is smooth", "and $T \\to T'$ is a thickening. Then there exists an", "\\'etale covering $\\{T'_i \\to T'\\}$ such that we", "can find the dotted arrow in", "$$", "\\xymatrix{", "X \\ar[d] & T \\ar[l] \\ar[d] & T \\times_{T'} T'_i \\ar[l] \\ar[d] \\\\", "Y & T' \\ar[l] & T'_i \\ar[l] \\ar@{..>}[llu]", "}", "$$", "making the diagram commute (for all $i$)." ], "refs": [], "proofs": [ { "contents": [ "Choose an \\'etale covering $\\{Y_i \\to Y\\}$ with each $Y_i$ affine.", "After replacing $T'$ by the induced \\'etale covering we may assume", "$Y$ is affine.", "\\medskip\\noindent", "Assume $Y$ is affine. Choose an \\'etale covering $\\{X_i \\to X\\}$.", "This gives rise to an \\'etale covering of $T$. This \\'etale covering of $T$", "comes from an \\'etale covering of $T'$", "(by Theorem \\ref{theorem-topological-invariance}, see", "discussion in Section \\ref{section-thickenings}).", "Hence we may assume $X$ is affine.", "\\medskip\\noindent", "Assume $X$ and $Y$ are affine. We can do one more \\'etale covering of", "$T'$ and assume $T'$ is affine. In this case the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-smooth-strong-lift}." ], "refs": [ "spaces-more-morphisms-theorem-topological-invariance", "algebra-lemma-smooth-strong-lift" ], "ref_ids": [ 9, 1216 ] } ], "ref_ids": [] }, { "id": 112, "type": "theorem", "label": "spaces-more-morphisms-lemma-formally-smooth-sheaf-differentials", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-formally-smooth-sheaf-differentials", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a formally smooth morphism of algebraic spaces over $S$.", "Then $\\Omega_{X/Y}$ is locally projective on $X$." ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$ and $V$ are affine(!) schemes and the vertical arrows are \\'etale.", "By", "Lemma \\ref{lemma-helper-formally-smooth}", "we see $\\psi : U \\to V$ is formally smooth. Hence", "$\\Gamma(V, \\mathcal{O}_V) \\to \\Gamma(U, \\mathcal{O}_U)$ is", "a formally smooth ring map, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-affine-formally-smooth}.", "Hence by", "Algebra, Lemma \\ref{algebra-lemma-characterize-formally-smooth-again}", "the $\\Gamma(U, \\mathcal{O}_U)$-module", "$\\Omega_{\\Gamma(U, \\mathcal{O}_U)/\\Gamma(V, \\mathcal{O}_V)}$", "is projective. Hence $\\Omega_{U/V}$ is locally projective, see", "Properties, Section \\ref{properties-section-locally-projective}.", "Since $\\Omega_{X/Y}|_U = \\Omega_{U/V}$ we see that $\\Omega_{X/Y}$ is", "locally projective too. (Because we can find an \\'etale covering of", "$X$ by the affine $U$'s fitting into diagrams as above -- details", "omitted.)" ], "refs": [ "spaces-more-morphisms-lemma-helper-formally-smooth", "more-morphisms-lemma-affine-formally-smooth", "algebra-lemma-characterize-formally-smooth-again" ], "ref_ids": [ 109, 13733, 1208 ] } ], "ref_ids": [] }, { "id": 113, "type": "theorem", "label": "spaces-more-morphisms-lemma-h1-is-zero", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-h1-is-zero", "contents": [ "Let $T$ be an affine scheme.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be quasi-coherent", "$\\mathcal{O}_T$-modules on $T_\\etale$.", "Consider the internal hom sheaf", "$\\mathcal{H} = \\SheafHom_{\\mathcal{O}_T}(\\mathcal{F}, \\mathcal{G})$", "on $T_\\etale$.", "If $\\mathcal{F}$ is locally projective, then", "$H^1(T_\\etale, \\mathcal{H}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "By the definition of a locally projective sheaf on an algebraic space (see", "Properties of Spaces,", "Definition \\ref{spaces-properties-definition-locally-projective})", "we see that $\\mathcal{F}_{Zar} = \\mathcal{F}|_{T_{Zar}}$ is a locally", "projective sheaf on the scheme $T$. Thus $\\mathcal{F}_{Zar}$ is a", "direct summand of a free $\\mathcal{O}_{T_{Zar}}$-module. Whereupon", "we conclude (as $\\mathcal{F} = (\\mathcal{F}_{Zar})^a$, see", "Descent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent})", "that $\\mathcal{F}$ is a direct summand of a free $\\mathcal{O}_T$-module", "on $T_\\etale$. Hence we may assume that", "$\\mathcal{F} = \\bigoplus_{i \\in I} \\mathcal{O}_T$ is a free module.", "In this case $\\mathcal{H} = \\prod_{i \\in I} \\mathcal{G}$ is", "a product of quasi-coherent modules. By", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-products}", "we conclude that $H^1 = 0$ because the cohomology of a quasi-coherent sheaf", "on an affine scheme is zero, see", "Descent, Proposition \\ref{descent-proposition-same-cohomology-quasi-coherent}", "and Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}." ], "refs": [ "spaces-properties-definition-locally-projective", "descent-proposition-equivalence-quasi-coherent", "sites-cohomology-lemma-cohomology-products", "descent-proposition-same-cohomology-quasi-coherent", "coherent-lemma-quasi-coherent-affine-cohomology-zero" ], "ref_ids": [ 11949, 14755, 4211, 14754, 3282 ] } ], "ref_ids": [] }, { "id": 114, "type": "theorem", "label": "spaces-more-morphisms-lemma-formally-smooth", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-formally-smooth", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over", "$S$. The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is formally smooth,", "\\item for every diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$ and $V$ are schemes and the vertical arrows are \\'etale", "the morphism of schemes $\\psi$ is formally smooth (as in", "More on Morphisms,", "Definition \\ref{more-morphisms-definition-formally-unramified}), and", "\\item for one such diagram with surjective vertical arrows the morphism", "$\\psi$ is formally smooth.", "\\end{enumerate}" ], "refs": [ "more-morphisms-definition-formally-unramified" ], "proofs": [ { "contents": [ "We have seen that (1) implies (2) and (3) in", "Lemma \\ref{lemma-helper-formally-smooth}.", "Assume (3).", "The proof that $f$ is formally smooth is entirely similar to", "the proof of (1) $\\Rightarrow$ (2) of", "Lemma \\ref{lemma-smooth-formally-smooth}.", "\\medskip\\noindent", "Consider a solid commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & T \\ar[d]^i \\ar[l]^a \\\\", "Y & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "as in Definition \\ref{definition-formally-smooth}.", "We will show the dotted arrow exists thereby", "proving that $f$ is formally smooth.", "Let $\\mathcal{F}$ be the sheaf of sets on $(T')_{spaces, \\etale}$ of", "Lemma \\ref{lemma-sheaf} as in the special case discussed in", "Remark \\ref{remark-special-case}.", "Let", "$$", "\\mathcal{H} =", "\\SheafHom_{\\mathcal{O}_T}(a^*\\Omega_{X/Y}, \\mathcal{C}_{T/T'})", "$$", "be the sheaf of $\\mathcal{O}_T$-modules on $T_{spaces, \\etale}$", "with action $\\mathcal{H} \\times \\mathcal{F} \\to \\mathcal{F}$ as in", "Lemma \\ref{lemma-action-sheaf}.", "The action $\\mathcal{H} \\times \\mathcal{F} \\to \\mathcal{F}$", "turns $\\mathcal{F}$ into a pseudo $\\mathcal{H}$-torsor, see", "Cohomology on Sites, Definition \\ref{sites-cohomology-definition-torsor}.", "Our goal is to show that $\\mathcal{F}$ is a trivial $\\mathcal{H}$-torsor.", "There are two steps: (I) To show that $\\mathcal{F}$ is a torsor", "we have to show that $\\mathcal{F}$ has \\'etale locally a", "section. (II) To show that $\\mathcal{F}$ is the trivial torsor", "it suffices to show that $H^1(T_\\etale, \\mathcal{H}) = 0$, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-torsors-h1}.", "\\medskip\\noindent", "First we prove (I). To see this consider a diagram", "(which exists because we are assuming (3))", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$ and $V$ are schemes, the vertical arrows are \\'etale and", "surjective, and $\\psi$ is formally smooth. By", "Lemma \\ref{lemma-representable-property-formally-property}", "the morphism $V \\to Y$ is formally \\'etale. Thus by", "Lemma \\ref{lemma-composition-formally-smooth-etale-unramified}", "the composition $U \\to Y$ is formally smooth. Then (I) follows from", "Lemma \\ref{lemma-etale-on-top} part (4).", "\\medskip\\noindent", "Finally we prove (II). By", "Lemma \\ref{lemma-formally-smooth-sheaf-differentials}", "we see that $\\Omega_{U/V}$ locally projective.", "Hence $\\Omega_{X/Y}$ is locally projective, see", "Descent on Spaces,", "Lemma \\ref{spaces-descent-lemma-locally-projective-descends}.", "Hence $a^*\\Omega_{X/Y}$ is locally projective, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-locally-projective-pullback}.", "Hence", "$$", "H^1(T_\\etale, \\mathcal{H}) =", "H^1(T_\\etale,", "\\SheafHom_{\\mathcal{O}_T}(a^*\\Omega_{X/Y}, \\mathcal{C}_{T/T'}) = 0", "$$", "by", "Lemma \\ref{lemma-h1-is-zero}", "as desired." ], "refs": [ "spaces-more-morphisms-lemma-helper-formally-smooth", "spaces-more-morphisms-lemma-smooth-formally-smooth", "spaces-more-morphisms-definition-formally-smooth", "spaces-more-morphisms-lemma-sheaf", "spaces-more-morphisms-remark-special-case", "spaces-more-morphisms-lemma-action-sheaf", "sites-cohomology-definition-torsor", "sites-cohomology-lemma-torsors-h1", "spaces-more-morphisms-lemma-representable-property-formally-property", "spaces-more-morphisms-lemma-composition-formally-smooth-etale-unramified", "spaces-more-morphisms-lemma-etale-on-top", "spaces-more-morphisms-lemma-formally-smooth-sheaf-differentials", "spaces-descent-lemma-locally-projective-descends", "spaces-properties-lemma-locally-projective-pullback", "spaces-more-morphisms-lemma-h1-is-zero" ], "ref_ids": [ 109, 110, 289, 98, 307, 99, 4411, 4182, 64, 62, 65, 112, 9363, 11914, 113 ] } ], "ref_ids": [ 14108 ] }, { "id": 115, "type": "theorem", "label": "spaces-more-morphisms-lemma-descending-property-formally-smooth", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-descending-property-formally-smooth", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is formally smooth''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a morphism of algebraic spaces over a scheme $S$.", "Choose an index set $I$ and diagrams", "$$", "\\xymatrix{", "U_i \\ar[d] \\ar[r]_{\\psi_i} & V_i \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "with \\'etale vertical arrows and $U_i$, $V_i$ affine schemes. Moreover,", "assume that $\\coprod U_i \\to X$ and $\\coprod V_i \\to Y$ are surjective, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-cover-by-union-affines}.", "By", "Lemma \\ref{lemma-formally-smooth}", "we see that $f$ is formally smooth if and only if each of the morphisms", "$\\psi_i$ are formally smooth. Hence we reduce to the case of a morphism", "of affine schemes. In this case the result follows from", "Algebra, Lemma \\ref{algebra-lemma-descent-formally-smooth}.", "Some details omitted." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-more-morphisms-lemma-formally-smooth", "algebra-lemma-descent-formally-smooth" ], "ref_ids": [ 11830, 114, 1215 ] } ], "ref_ids": [] }, { "id": 116, "type": "theorem", "label": "spaces-more-morphisms-lemma-triangle-differentials-formally-smooth", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-triangle-differentials-formally-smooth", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$, $g : Y \\to Z$ be morphisms of algebraic spaces over $S$.", "Assume $f$ is formally smooth. Then", "$$", "0 \\to f^*\\Omega_{Y/Z} \\to \\Omega_{X/Z} \\to \\Omega_{X/Y} \\to 0", "$$", "Lemma \\ref{lemma-triangle-differentials}", "is short exact." ], "refs": [ "spaces-more-morphisms-lemma-triangle-differentials" ], "proofs": [ { "contents": [ "Follows from the case of schemes, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-triangle-differentials-formally-smooth},", "by \\'etale localization, see", "Lemmas \\ref{lemma-formally-smooth} and \\ref{lemma-localize-differentials}." ], "refs": [ "more-morphisms-lemma-triangle-differentials-formally-smooth", "spaces-more-morphisms-lemma-formally-smooth", "spaces-more-morphisms-lemma-localize-differentials" ], "ref_ids": [ 13738, 114, 32 ] } ], "ref_ids": [ 36 ] }, { "id": 117, "type": "theorem", "label": "spaces-more-morphisms-lemma-differentials-formally-unramified-formally-smooth", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-differentials-formally-unramified-formally-smooth", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $h : Z \\to X$ be a formally unramified morphism of algebraic spaces", "over $B$.", "Assume that $Z$ is formally smooth over $B$. Then the", "canonical exact sequence", "$$", "0 \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/B} \\to \\Omega_{Z/B} \\to 0", "$$", "of", "Lemma \\ref{lemma-universally-unramified-differentials-sequence}", "is short exact." ], "refs": [ "spaces-more-morphisms-lemma-universally-unramified-differentials-sequence" ], "proofs": [ { "contents": [ "Let $Z \\to Z'$ be the universal first order thickening of $Z$ over $X$.", "From the proof of", "Lemma \\ref{lemma-universally-unramified-differentials-sequence}", "we see that our sequence is identified with the sequence", "$$", "\\mathcal{C}_{Z/Z'} \\to \\Omega_{Z'/B} \\otimes \\mathcal{O}_Z \\to", "\\Omega_{Z/B} \\to 0.", "$$", "Since $Z \\to S$ is formally smooth we can \\'etale locally on $Z'$ find", "a left inverse $Z' \\to Z$ over $B$ to the inclusion map $Z \\to Z'$.", "Thus the sequence is \\'etale locally split, see", "Lemma \\ref{lemma-differentials-relative-immersion-section}." ], "refs": [ "spaces-more-morphisms-lemma-universally-unramified-differentials-sequence" ], "ref_ids": [ 86 ] } ], "ref_ids": [ 86 ] }, { "id": 118, "type": "theorem", "label": "spaces-more-morphisms-lemma-two-unramified-morphisms-formally-smooth", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-two-unramified-morphisms-formally-smooth", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[rd]_j & X \\ar[d]^f \\\\", "& Y", "}", "$$", "be a commutative diagram of algebraic spaces over $S$", "where $i$ and $j$ are formally unramified and $f$ is formally smooth.", "Then the canonical exact sequence", "$$", "0 \\to", "\\mathcal{C}_{Z/Y} \\to", "\\mathcal{C}_{Z/X} \\to", "i^*\\Omega_{X/Y} \\to 0", "$$", "of", "Lemma \\ref{lemma-two-unramified-morphisms}", "is exact and locally split." ], "refs": [ "spaces-more-morphisms-lemma-two-unramified-morphisms" ], "proofs": [ { "contents": [ "Denote $Z \\to Z'$ the universal first order thickening of $Z$ over $X$.", "Denote $Z \\to Z''$ the universal first order thickening of $Z$ over $Y$.", "By", "Lemma \\ref{lemma-universally-unramified-differentials-sequence}", "here is a canonical morphism $Z' \\to Z''$ so that we have a commutative", "diagram", "$$", "\\xymatrix{", "Z \\ar[r]_{i'} \\ar[rd]_{j'} & Z' \\ar[r]_a \\ar[d]^k & X \\ar[d]^f \\\\", "& Z'' \\ar[r]^b & Y", "}", "$$", "The sequence above is identified with the sequence", "$$", "\\mathcal{C}_{Z/Z''} \\to", "\\mathcal{C}_{Z/Z'} \\to", "(i')^*\\Omega_{Z'/Z''} \\to 0", "$$", "via our definitions concerning conormal sheaves of formally unramified", "morphisms. Let $U'' \\to Z''$ be an \\'etale morphism with $U''$ affine.", "Denote $U \\to Z$ and $U' \\to Z'$ the corresponding affine", "schemes \\'etale over $Z$ and $Z'$.", "As $f$ is formally smooth there exists a morphism $h : U'' \\to X$", "which agrees with $i$ on $U$ and such that $f \\circ h$ equals $b|_{U''}$.", "Since $Z'$ is the universal first order thickening we obtain a unique", "morphism $g : U'' \\to Z'$ such that $g = a \\circ h$. The universal", "property of $Z''$ implies that $k \\circ g$ is the inclusion map", "$U'' \\to Z''$. Hence $g$ is a left inverse to $k$. Picture", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & Z' \\ar[d]^k \\\\", "U'' \\ar[r] \\ar[ru]^g & Z''", "}", "$$", "Thus $g$ induces a map $\\mathcal{C}_{Z/Z'}|_U \\to \\mathcal{C}_{Z/Z''}|_U$", "which is a left inverse to the map", "$\\mathcal{C}_{Z/Z''} \\to \\mathcal{C}_{Z/Z'}$ over $U$." ], "refs": [ "spaces-more-morphisms-lemma-universally-unramified-differentials-sequence" ], "ref_ids": [ 86 ] } ], "ref_ids": [ 87 ] }, { "id": 119, "type": "theorem", "label": "spaces-more-morphisms-lemma-lifting-along-artinian-at-point", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-lifting-along-artinian-at-point", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $x \\in |X|$.", "Assume that $Y$ is locally Noetherian and $f$ locally of finite type.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is smooth at $x$,", "\\item for every solid commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & \\Spec(B) \\ar[d]^i \\ar[l]^-\\alpha \\\\", "Y & \\Spec(B') \\ar[l]_-{\\beta} \\ar@{-->}[lu]", "}", "$$", "where $B' \\to B$ is a surjection of local rings with", "$\\Ker(B' \\to B)$ of square zero, and $\\alpha$ mapping the", "closed point of $\\Spec(B)$ to $x$ there exists", "a dotted arrow making the diagram commute, and", "\\item same as in (2) but with $B' \\to B$ ranging over small", "extensions (see Algebra, Definition \\ref{algebra-definition-small-extension}).", "\\end{enumerate}" ], "refs": [ "algebra-definition-small-extension" ], "proofs": [ { "contents": [ "Condition (1) means there is an open subspace $X' \\subset X$", "such that $X' \\to Y$ is smooth. Hence (1) implies conditions (2) and (3) by", "Lemma \\ref{lemma-smooth-formally-smooth}. Condition (2) implies", "condition (3) trivially. Assume (3). Choose a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] & U \\ar[l] \\ar[d] \\\\", "Y & V \\ar[l]", "}", "$$", "with $U$ and $V$ affine, horizontal arrows \\'etale and", "such that there is a point $u \\in U$ mapping to $x$. Next, consider", "a diagram", "$$", "\\xymatrix{", "X \\ar[d] & U \\ar[l] \\ar[d] & \\Spec(B) \\ar[d]^i \\ar[l]^-\\alpha \\\\", "Y & V \\ar[l] & \\Spec(B') \\ar[l]_-{\\beta}", "}", "$$", "as in (3) but for $u \\in U \\to V$. Let $\\gamma : \\Spec(B') \\to X$", "be the arrow we get from our assumption that (3) holds for $X$.", "Because $U \\to X$ is \\'etale and", "hence formally \\'etale (Lemma \\ref{lemma-etale-formally-etale})", "the morphism $\\gamma$", "has a unique lift to $U$ compatible with $\\alpha$. Then because", "$V \\to Y$ is \\'etale hence formally \\'etale this lift is compatible", "with $\\beta$. Hence (3) holds for $u \\in U \\to V$ and we conclude", "that $U \\to V$ is smooth at $u$ by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-lifting-along-artinian-at-point}.", "This proves that $X \\to Y$ is smooth at $x$, thereby", "finishing the proof." ], "refs": [ "spaces-more-morphisms-lemma-smooth-formally-smooth", "spaces-more-morphisms-lemma-etale-formally-etale", "more-morphisms-lemma-lifting-along-artinian-at-point" ], "ref_ids": [ 110, 95, 13741 ] } ], "ref_ids": [ 1538 ] }, { "id": 120, "type": "theorem", "label": "spaces-more-morphisms-lemma-lifting-along-artinian", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-lifting-along-artinian", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $Y$ is locally Noetherian and $f$ locally of finite type.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is smooth,", "\\item for every solid commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & \\Spec(B) \\ar[d]^i \\ar[l]^-\\alpha \\\\", "Y & \\Spec(B') \\ar[l]_-{\\beta} \\ar@{-->}[lu]", "}", "$$", "where $B' \\to B$ is a small extension of Artinian local rings", "and $\\beta$ of finite type (!) there exists a dotted arrow making", "the diagram commute.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $f$ is smooth, then the infinitesimal lifting criterion", "(Lemma \\ref{lemma-smooth-formally-smooth}) says", "$f$ is formally smooth and (2) holds.", "\\medskip\\noindent", "Assume $f$ is not smooth. The set of points $x \\in X$ where $f$ is not smooth", "forms a closed subset $T$ of $|X|$. By", "Morphisms of Spaces, Lemma ", "\\ref{spaces-morphisms-lemma-enough-finite-type-points}, there exists", "a point $x \\in T \\subset X$ with $x \\in X_{\\text{ft-pts}}$. Choose a", "commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] & U \\ar[l] \\ar[d] & u \\ar@{|->}[d] \\\\", "Y & V \\ar[l] & v", "}", "$$", "with $U$ and $V$ affine, horizontal arrows \\'etale and", "such that there is a point $u \\in U$ mapping to $x$. Then $u$", "is a finite type point of $U$. Since $U \\to V$ is not smooth at", "the point $u$, by", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-lifting-along-artinian-at-point}", "there is a diagram", "$$", "\\xymatrix{", "X \\ar[d] & U \\ar[l] \\ar[d] & \\Spec(B) \\ar[d]^i \\ar[l]^-\\alpha \\\\", "Y & V \\ar[l] & \\Spec(B') \\ar[l]_-{\\beta} \\ar@{-->}[lu]", "}", "$$", "with $B' \\to B$ a small extension of (Artinian) local rings", "such that the residue field of $B$ is equal to $\\kappa(v)$ and such", "that the dotted arrow does not exist. Since $U \\to V$ is", "of finite type, we see that $v$ is a finite type point of $V$. By", "Morphisms, Lemma \\ref{morphisms-lemma-artinian-finite-type}", "the morphism $\\beta$ is of finite type, hence the composition", "$\\Spec(B) \\to Y$ is of finite type also.", "Arguing exactly as in the proof of", "Lemma \\ref{lemma-lifting-along-artinian-at-point}", "(using that $U \\to X$ and $V \\to Y$ are \\'etale hence", "formally \\'etale)", "we see that there cannot be an arrow $\\Spec(B) \\to X$", "fitting into the outer rectangle of the last displayed diagram.", "In other words, (2) doesn't hold and the proof is complete." ], "refs": [ "spaces-more-morphisms-lemma-smooth-formally-smooth", "spaces-morphisms-lemma-enough-finite-type-points", "more-morphisms-lemma-lifting-along-artinian-at-point", "morphisms-lemma-artinian-finite-type", "spaces-more-morphisms-lemma-lifting-along-artinian-at-point" ], "ref_ids": [ 110, 4826, 13741, 5206, 119 ] } ], "ref_ids": [] }, { "id": 121, "type": "theorem", "label": "spaces-more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "algebraic spaces over $S$. Assume $f$ is", "locally of finite type and $Y$ locally Noetherian.", "Let $Z \\subset Y$ be a closed subspace with $n$th infinitesimal", "neighbourhood $Z_n \\subset Y$. Set $X_n = Z_n \\times_Y X$.", "\\begin{enumerate}", "\\item If $X_n \\to Z_n$ is smooth for all $n$, then $f$", "is smooth at every point of $f^{-1}(Z)$.", "\\item If $X_n \\to Z_n$ is \\'etale for all $n$, then $f$", "is \\'etale at every point of $f^{-1}(Z)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $X_n \\to Z_n$ is smooth for all $n$.", "Let $x \\in X$ be a point lying over a point of $Z$.", "Given a small extension $B' \\to B$ and morphisms $\\alpha$, $\\beta$ as in", "Lemma \\ref{lemma-lifting-along-artinian-at-point} part (3)", "the maximal ideal of $B'$ is nilpotent (as $B'$ is Artinian)", "and hence the morphism $\\beta$ factors through $Z_n$ and $\\alpha$ factors", "through $X_n$ for a suitable $n$. Thus the lifting property for", "$X_n \\to Z_n$ kicks in to get the desired dotted arrow in the diagram.", "This proves (1). Part (2) follows from (1) and the fact that a morphism", "is \\'etale if and only if it is smooth of relative dimension $0$." ], "refs": [ "spaces-more-morphisms-lemma-lifting-along-artinian-at-point" ], "ref_ids": [ 119 ] } ], "ref_ids": [] }, { "id": 122, "type": "theorem", "label": "spaces-more-morphisms-lemma-NL-etale-localization", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-NL-etale-localization", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_p \\ar[r]_g & V \\ar[d]^q \\\\", "X \\ar[r]^f & Y", "}", "$$", "of algebraic spaces over $S$ with $p$ and $q$ \\'etale.", "Then there is a canonical identification", "$\\NL_{X/Y}|_{U_\\etale} = \\NL_{U/V}$ in $D(\\mathcal{O}_U)$." ], "refs": [], "proofs": [ { "contents": [ "Formation of the naive cotangent complex commutes with pullback", "(Modules on Sites, Lemma \\ref{sites-modules-lemma-pullback-NL})", "and we have $p_{small}^{-1}\\mathcal{O}_X = \\mathcal{O}_U$ and", "$g_{small}^{-1}\\mathcal{O}_{V_\\etale} =", "p_{small}^{-1}f_{small}^{-1}\\mathcal{O}_{Y_\\etale}$", "because $q_{small}^{-1}\\mathcal{O}_{Y_\\etale} =", "\\mathcal{O}_{V_\\etale}$ by Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-etale-exact-pullback}.", "Tracing through the definitions we conclude that", "$\\NL_{X/Y}|_{U_\\etale} = \\NL_{U/V}$." ], "refs": [ "sites-modules-lemma-pullback-NL", "spaces-properties-lemma-etale-exact-pullback" ], "ref_ids": [ 14241, 11897 ] } ], "ref_ids": [] }, { "id": 123, "type": "theorem", "label": "spaces-more-morphisms-lemma-NL-compare-spaces-schemes", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-NL-compare-spaces-schemes", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $X$ and $Y$ representable by schemes $X_0$ and $Y_0$.", "Then there is a canonical identification", "$\\NL_{X/Y} = \\epsilon^*\\NL_{X_0/Y_0}$ in $D(\\mathcal{O}_X)$", "where $\\epsilon$ is as in Derived Categories of Spaces, Section", "\\ref{spaces-perfect-section-derived-quasi-coherent-etale}", "and $\\NL_{X_0/Y_0}$ is as in", "More on Morphisms, Definition", "\\ref{more-morphisms-definition-netherlander}." ], "refs": [ "more-morphisms-definition-netherlander" ], "proofs": [ { "contents": [ "Let $f_0 : X_0 \\to Y_0$ be the morphism of schemes corresponding to $f$.", "There is a canonical map", "$\\epsilon^{-1}f_0^{-1}\\mathcal{O}_{Y_0} \\to f_{small}^{-1}\\mathcal{O}_Y$", "compatible with", "$\\epsilon^\\sharp : \\epsilon^{-1}\\mathcal{O}_{X_0} \\to \\mathcal{O}_X$", "because there is a commutative diagram", "$$", "\\xymatrix{", "X_{0, Zar} \\ar[d]_{f_0} & X_\\etale \\ar[l]^\\epsilon \\ar[d]^f \\\\", "Y_{0, Zar} & Y_\\etale \\ar[l]_\\epsilon", "}", "$$", "see Derived Categories of Spaces, Remark", "\\ref{spaces-perfect-remark-match-total-direct-images}.", "Thus we obtain a canonical map", "$$", "\\epsilon^{-1}\\NL_{X_0/Y_0} =", "\\epsilon^{-1}\\NL_{\\mathcal{O}_{X_0}/f_0^{-1}\\mathcal{O}_{Y_0}} =", "\\NL_{\\epsilon^{-1}\\mathcal{O}_{X_0}/\\epsilon^{-1}f_0^{-1}\\mathcal{O}_{Y_0}}", "\\to", "\\NL_{\\mathcal{O}_X/f^{-1}_{small}\\mathcal{O}_Y} = \\NL_{X/Y}", "$$", "by functoriality of the naive cotangent complex.", "To see that the induced map $\\epsilon^*\\NL_{X_0/Y_0} \\to \\NL_{X/Y}$ is an", "isomorphism in $D(\\mathcal{O}_X)$ we may check on stalks at geometric points", "(Properties of Spaces, Theorem", "\\ref{spaces-properties-theorem-exactness-stalks}).", "Let $\\overline{x} : \\Spec(k) \\to X_0$", "be a geometric point lying over $x \\in X_0$, with", "$\\overline{y} = f \\circ \\overline{x}$ lying over $y \\in Y_0$. Then", "$$", "\\NL_{X/Y, \\overline{x}} =", "\\NL_{\\mathcal{O}_{X, \\overline{x}}/\\mathcal{O}_{Y, \\overline{y}}}", "$$", "This is true because taking stalks at $\\overline{x}$ is the same", "as taking inverse image via $\\overline{x} : \\Spec(k) \\to X$", "and we may apply Modules on Sites, Lemma \\ref{sites-modules-lemma-pullback-NL}.", "On the other hand we have", "$$", "(\\epsilon^*\\NL_{X_0/Y_0})_{\\overline{x}} =", "\\NL_{X_0/Y_0, x} \\otimes_{\\mathcal{O}_{X_0, x}}", "\\mathcal{O}_{X, \\overline{x}} =", "\\NL_{\\mathcal{O}_{X_0, x}/\\mathcal{O}_{Y_0, y}}", "\\otimes_{\\mathcal{O}_{X_0, x}} \\mathcal{O}_{X, \\overline{x}}", "$$", "Some details omitted (hint: use that the stalk of a pullback", "is the stalk at the image point, see", "Sites, Lemma \\ref{sites-lemma-point-morphism-sites},", "as well as the corresponding result for modules, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-pullback-stalk}).", "Observe that $\\mathcal{O}_{X, \\overline{x}}$ is the strict", "henselization of $\\mathcal{O}_{X_0, x}$ and similarly", "for $\\mathcal{O}_{Y, \\overline{y}}$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring}).", "Thus the result follows from", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-henselization-NL}." ], "refs": [ "spaces-perfect-remark-match-total-direct-images", "spaces-properties-theorem-exactness-stalks", "sites-modules-lemma-pullback-NL", "sites-lemma-point-morphism-sites", "sites-modules-lemma-pullback-stalk", "spaces-properties-lemma-describe-etale-local-ring", "more-algebra-lemma-henselization-NL" ], "ref_ids": [ 2768, 11813, 14241, 8603, 14245, 11884, 10005 ] } ], "ref_ids": [ 14112 ] }, { "id": 124, "type": "theorem", "label": "spaces-more-morphisms-lemma-netherlander-quasi-coherent", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-netherlander-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "algebraic spaces over $S$. The cohomology sheaves", "of the complex $\\NL_{X/Y}$ are quasi-coherent, zero outside", "degrees $-1$, $0$ and equal to $\\Omega_{X/Y}$ in degree $0$." ], "refs": [], "proofs": [ { "contents": [ "By construction of the naive cotangent complex in", "Modules on Sites, Section \\ref{sites-modules-section-netherlander}", "we have that $\\NL_{X/Y}$ is a complex sitting in degrees $-1$, $0$", "and that its cohomology in degree $0$ is $\\Omega_{X/Y}$ (by our", "construction of $\\Omega_{X/Y}$ in Section \\ref{section-sheaf-differentials}).", "The sheaf of differentials is quasi-coherent (by", "Lemma \\ref{lemma-module-differentials-quasi-coherent}).", "To finish the proof it suffices to show that $H^{-1}(\\NL_{X/Y})$", "is quasi-coherent. This follows by checking \\'etale locally", "(allowed by Lemma \\ref{lemma-NL-etale-localization} and", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-characterize-quasi-coherent})", "reducing to the case of schemes", "(Lemma \\ref{lemma-NL-compare-spaces-schemes})", "and finally using the result in the case of schemes", "(More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-netherlander-quasi-coherent})." ], "refs": [ "spaces-more-morphisms-lemma-module-differentials-quasi-coherent", "spaces-more-morphisms-lemma-NL-etale-localization", "spaces-properties-lemma-characterize-quasi-coherent", "spaces-more-morphisms-lemma-NL-compare-spaces-schemes", "more-morphisms-lemma-netherlander-quasi-coherent" ], "ref_ids": [ 33, 122, 11911, 123, 13746 ] } ], "ref_ids": [] }, { "id": 125, "type": "theorem", "label": "spaces-more-morphisms-lemma-netherlander-fp", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-netherlander-fp", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $f$ is locally of finite", "presentation, then $\\NL_{X/Y}$ is \\'etale locally on $X$", "quasi-isomorphic to a complex", "$$", "\\ldots \\to 0 \\to \\mathcal{F}^{-1} \\to \\mathcal{F}^0 \\to 0 \\to \\ldots", "$$", "of quasi-coherent $\\mathcal{O}_X$-modules", "with $\\mathcal{F}^0$ of finite presentation", "and $\\mathcal{F}^{-1}$ of finite type." ], "refs": [], "proofs": [ { "contents": [ "Formation of the naive cotangent complex commutes with \\'etale", "localization by Lemma \\ref{lemma-NL-etale-localization}.", "This reduces us to the case of schemes by", "Lemma \\ref{lemma-NL-compare-spaces-schemes}.", "The result in the case of schemes is", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-netherlander-fp}." ], "refs": [ "spaces-more-morphisms-lemma-NL-etale-localization", "spaces-more-morphisms-lemma-NL-compare-spaces-schemes", "more-morphisms-lemma-netherlander-fp" ], "ref_ids": [ 122, 123, 13747 ] } ], "ref_ids": [] }, { "id": 126, "type": "theorem", "label": "spaces-more-morphisms-lemma-NL-formally-smooth", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-NL-formally-smooth", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is formally smooth,", "\\item $H^{-1}(\\NL_{X/Y}) = 0$ and $H^0(\\NL_{X/Y}) = \\Omega_{X/Y}$", "is locally projective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-formally-smooth},", "Lemma \\ref{lemma-NL-etale-localization},", "Lemma \\ref{lemma-NL-compare-spaces-schemes}", "and the case of schemes which is", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-NL-formally-smooth}." ], "refs": [ "spaces-more-morphisms-lemma-formally-smooth", "spaces-more-morphisms-lemma-NL-etale-localization", "spaces-more-morphisms-lemma-NL-compare-spaces-schemes", "more-morphisms-lemma-NL-formally-smooth" ], "ref_ids": [ 114, 122, 123, 13748 ] } ], "ref_ids": [] }, { "id": 127, "type": "theorem", "label": "spaces-more-morphisms-lemma-NL-formally-etale", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-NL-formally-etale", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is formally \\'etale,", "\\item $H^{-1}(\\NL_{X/Y}) = H^0(\\NL_{X/Y}) = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). A formally \\'etale morphism is a formally smooth morphism.", "Thus $H^{-1}(\\NL_{X/Y}) = 0$ by Lemma \\ref{lemma-NL-formally-smooth}.", "On the other hand, a formally \\'etale morphism if formally unramified", "hence we have $\\Omega_{X/Y} = 0$ by", "Lemma \\ref{lemma-characterize-formally-unramified}.", "Conversely, if (2) holds, then $f$ is formally smooth by", "Lemma \\ref{lemma-NL-formally-smooth}", "and formally unramified by", "Lemma \\ref{lemma-characterize-formally-unramified}", "and hence formally \\'etale by", "Lemmas \\ref{lemma-formally-etale-unramified-smooth}." ], "refs": [ "spaces-more-morphisms-lemma-NL-formally-smooth", "spaces-more-morphisms-lemma-characterize-formally-unramified", "spaces-more-morphisms-lemma-NL-formally-smooth", "spaces-more-morphisms-lemma-characterize-formally-unramified", "spaces-more-morphisms-lemma-formally-etale-unramified-smooth" ], "ref_ids": [ 126, 71, 126, 71, 108 ] } ], "ref_ids": [] }, { "id": 128, "type": "theorem", "label": "spaces-more-morphisms-lemma-NL-smooth", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-NL-smooth", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is smooth, and", "\\item $f$ is locally of finite presentation,", "$H^{-1}(\\NL_{X/Y}) = 0$, and $H^0(\\NL_{X/Y}) = \\Omega_{X/Y}$", "is finite locally free.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-formally-smooth},", "Lemma \\ref{lemma-NL-etale-localization},", "Lemma \\ref{lemma-NL-compare-spaces-schemes}", "and the case of schemes which is", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-NL-smooth}." ], "refs": [ "spaces-more-morphisms-lemma-formally-smooth", "spaces-more-morphisms-lemma-NL-etale-localization", "spaces-more-morphisms-lemma-NL-compare-spaces-schemes", "more-morphisms-lemma-NL-smooth" ], "ref_ids": [ 114, 122, 123, 13750 ] } ], "ref_ids": [] }, { "id": 129, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-locus-base-change", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-locus-base-change", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "be a cartesian diagram of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume $g$ is flat, $f$ is locally of finite presentation,", "and $\\mathcal{F}$ is locally of finite presentation.", "Then", "$$", "\\{x' \\in |X'| : (g')^*\\mathcal{F}\\text{ is flat over }Y'\\text{ at }x'\\}", "$$", "is the inverse image of the open subset of", "Theorem \\ref{theorem-openness-flatness}", "under the continuous map $|g'| : |X'| \\to |X|$." ], "refs": [ "spaces-more-morphisms-theorem-openness-flatness" ], "proofs": [ { "contents": [ "This follows from", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-module-flat}." ], "refs": [ "spaces-morphisms-lemma-base-change-module-flat" ], "ref_ids": [ 4863 ] } ], "ref_ids": [ 10 ] }, { "id": 130, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-on-fibres-at-point", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-on-fibres-at-point", "contents": [ "In the situation above the following are equivalent", "\\begin{enumerate}", "\\item Pick a geometric point $\\overline{x}$ of $X$ lying over $x$.", "Set $\\overline{y} = f \\circ \\overline{x}$ and", "$\\overline{z} = g \\circ \\overline{x}$. Then the module", "$\\mathcal{F}_{\\overline{x}}/", "\\mathfrak m_{\\overline{z}}\\mathcal{F}_{\\overline{x}}$", "is flat over", "$\\mathcal{O}_{Y, \\overline{y}}/", "\\mathfrak m_{\\overline{z}}\\mathcal{O}_{Y, \\overline{y}}$.", "\\item Pick a morphism $x : \\Spec(K) \\to X$ in the equivalence class of", "$x$. Set $z = g \\circ x$, $X_z = \\Spec(K) \\times_{z, Z} X$,", "$Y_z = \\Spec(K) \\times_{z, Z} Y$, and $\\mathcal{F}_z$ the pullback", "of $\\mathcal{F}$ to $X_z$. Then $\\mathcal{F}_z$ is flat at $x$ over", "$Y_z$ (as defined in Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-flat-module}).", "\\item Pick a commutative diagram", "$$", "\\xymatrix{", "& & & U \\ar[llld]_a \\ar[rr] \\ar[dr] & & V \\ar[llld]_>>>>>>>b \\ar[dl] \\\\", "X \\ar[rr]_f \\ar[dr]_g & & Y \\ar[dl]^h & & W \\ar[llld]_c \\\\", "& Z", "}", "$$", "where $U, V, W$ are schemes, and $a, b, c$ are \\'etale,", "and a point $u \\in U$ mapping to $x$. Let $w \\in W$ be the image of", "$u$. Let $\\mathcal{F}_w$ be the pullback of $\\mathcal{F}$ to", "the fibre $U_w$ of $U \\to W$ at $w$. Then $\\mathcal{F}_w$", "is flat over $V_w$ at $u$.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-flat-module" ], "proofs": [ { "contents": [ "Note that in (2) the morphism $x : \\Spec(K) \\to X$ defines a", "$K$-rational point of $X_z$, hence the statement makes sense. Moreover,", "the condition in (2) is independent of the choice of $\\Spec(K) \\to X$", "in the equivalence class of $x$ (details omitted; this will also follow", "from the arguments below because the other conditions do not depend", "on this choice). Also note that we can always choose a diagram as in", "(3) by: first choosing", "a scheme $W$ and a surjective \\'etale morphism $W \\to Z$, then choosing", "a scheme $V$ and a surjective \\'etale morphism $V \\to W \\times_Z Y$, and", "finally choosing a scheme $U$ and a surjective \\'etale morphism", "$U \\to V \\times_Y X$. Having made these choices we set $U \\to W$ equal", "to the composition $U \\to V \\to W$ and we can pick a point $u \\in U$ mapping", "to $x$ because the morphism $U \\to X$ is surjective.", "\\medskip\\noindent", "Suppose given both a diagram as in (3) and a geometric point", "$\\overline{x} : \\Spec(k) \\to X$ as in (1). By", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-geometric-lift-to-usual}", "we can choose a geometric point $\\overline{u} : \\Spec(k) \\to U$", "lying over $u$ such that $\\overline{x} = a \\circ \\overline{u}$.", "Denote $\\overline{v} : \\Spec(k) \\to V$ and", "$\\overline{w} : \\Spec(k) \\to W$ the induced geometric points of", "$V$ and $W$. In this setting we know that", "$\\mathcal{O}_{X, \\overline{x}} = \\mathcal{O}_{U, u}^{sh}$", "and similarly for $Y$ and $Z$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-describe-etale-local-ring}.", "In the same vein we have", "$$", "\\mathcal{F}_{\\overline{x}} =", "(a^*\\mathcal{F})_u \\otimes_{\\mathcal{O}_{U, u}}", "\\mathcal{O}_{U, u}^{sh}", "$$", "see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-stalk-quasi-coherent}.", "Note that the stalk of $\\mathcal{F}_w$ at $u$ is given by", "$$", "(\\mathcal{F}_w)_u = (a^*\\mathcal{F})_u/\\mathfrak m_w(a^*\\mathcal{F})_u", "$$", "and the local ring of $V_w$ at $v$ is given by", "$$", "\\mathcal{O}_{V_w, v} = \\mathcal{O}_{V, v}/\\mathfrak m_w\\mathcal{O}_{V, v}.", "$$", "Since $\\mathfrak m_{\\overline{z}} =", "\\mathfrak m_w \\mathcal{O}_{Z, \\overline{z}} =", "\\mathfrak m_w \\mathcal{O}_{W, w}^{sh}$", "we see that", "\\begin{align*}", "\\mathcal{F}_{\\overline{x}}/", "\\mathfrak m_{\\overline{z}}\\mathcal{F}_{\\overline{x}} & =", "(a^*\\mathcal{F})_u \\otimes_{\\mathcal{O}_{U, u}}", "\\mathcal{O}_{X, \\overline{x}}/", "\\mathfrak m_{\\overline{z}}\\mathcal{O}_{X, \\overline{x}} \\\\", "& =", "(\\mathcal{F}_w)_u \\otimes_{\\mathcal{O}_{U_w, u}}", "\\mathcal{O}_{U, u}^{sh}/\\mathfrak m_w\\mathcal{O}_{U, u}^{sh} \\\\", "& = (\\mathcal{F}_w)_u \\otimes_{\\mathcal{O}_{U_w, u}}", "\\mathcal{O}_{U_w, \\overline{u}}^{sh} \\\\", "& = (\\mathcal{F}_w)_{\\overline{u}}", "\\end{align*}", "the penultimate equality by", "Algebra, Lemma \\ref{algebra-lemma-quotient-strict-henselization}", "and the last equality by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-stalk-quasi-coherent}.", "The same arguments applied to the structure sheaves of $V$ and $Y$", "show that", "$$", "\\mathcal{O}_{V_w, \\overline{v}}^{sh} =", "\\mathcal{O}_{V, v}^{sh}/\\mathfrak m_w \\mathcal{O}_{V, v}^{sh} =", "\\mathcal{O}_{Y, \\overline{y}}/", "\\mathfrak m_{\\overline{z}}\\mathcal{O}_{Y, \\overline{y}}.", "$$", "OK, and now we can use", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-at-point}", "to see that (1) is equivalent to (3).", "\\medskip\\noindent", "Finally we prove the equivalence of (2) and (3).", "To do this we pick a field extension $\\tilde K$ of $K$", "and a morphism $\\tilde x : \\Spec(\\tilde K) \\to U$ which", "lies over $u$ (this is possible because $u \\times_{X, x} \\Spec(K)$", "is a nonempty scheme). Set $\\tilde z : \\Spec(\\tilde K) \\to U \\to W$", "be the composition. We obtain a commutative diagram", "$$", "\\xymatrix{", "& & & U_w \\times_w \\tilde z \\ar[llld]_a \\ar[rr] \\ar[dr] & &", "V_w \\times_w \\tilde z \\ar[llld]_>>>>>>>b \\ar[dl] \\\\", "X_z \\ar[rr]_f \\ar[dr]_g & & Y_z \\ar[dl]^h & & \\tilde z \\ar[llld]_c \\\\", "& z", "}", "$$", "where $z = \\Spec(K)$ and $w = \\Spec(\\kappa(w))$. Now it", "is clear that $\\mathcal{F}_w$ and $\\mathcal{F}_z$ pull back to the", "same module on $U_w \\times_w \\tilde z$. This leads to a commutative", "diagram", "$$", "\\xymatrix{", "X_z \\ar[d] & U_w \\times_w \\tilde z \\ar[l] \\ar[d] \\ar[r] & U_w \\ar[d] \\\\", "Y_z & V_w \\times_w \\tilde z \\ar[l] \\ar[r] & V_w", "}", "$$", "both of whose squares are cartesian and whose bottom horizontal", "arrows are flat: the lower left horizontal arrow is the composition", "of the morphism $Y \\times_Z \\tilde z \\to Y \\times_Z z = Y_z$ (base change", "of a flat morphism), the \\'etale morphism", "$V \\times_Z \\tilde z \\to Y \\times_Z \\tilde z$, and", "the \\'etale morphism $V \\times_W \\tilde z \\to V \\times_Z \\tilde z$.", "Thus it follows from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-base-change-module-flat}", "that", "$$", "\\mathcal{F}_z\\text{ flat at }x\\text{ over }Y_z", "\\Leftrightarrow", "\\mathcal{F}|_{U_w \\times_w \\tilde z}", "\\text{ flat at }\\tilde x\\text{ over }V_w \\times_w \\tilde z", "\\Leftrightarrow", "\\mathcal{F}_w\\text{ flat at }u\\text{ over }V_w", "$$", "and we win." ], "refs": [ "spaces-properties-lemma-geometric-lift-to-usual", "spaces-properties-lemma-describe-etale-local-ring", "spaces-properties-lemma-stalk-quasi-coherent", "algebra-lemma-quotient-strict-henselization", "spaces-properties-lemma-stalk-quasi-coherent", "spaces-morphisms-lemma-flat-at-point", "spaces-morphisms-lemma-base-change-module-flat" ], "ref_ids": [ 11871, 11884, 11909, 1307, 11909, 4862, 4863 ] } ], "ref_ids": [ 5008 ] }, { "id": 131, "type": "theorem", "label": "spaces-more-morphisms-lemma-morphism-between-flat", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-morphism-between-flat", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ and $Y \\to Z$ be a morphism of algebraic spaces over $S$.", "Assume", "\\begin{enumerate}", "\\item $X$ is locally of finite presentation over $Z$,", "\\item $X$ is flat over $Z$,", "\\item for every $z \\in |Z|$ the fibre of $X$ over $z$", "is flat over the fibre of $Y$ over $z$, and", "\\item $Y$ is locally of finite type over $Z$.", "\\end{enumerate}", "Then $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $Z$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Theorem \\ref{theorem-criterion-flatness-fibre}." ], "refs": [ "spaces-more-morphisms-theorem-criterion-flatness-fibre" ], "ref_ids": [ 11 ] } ], "ref_ids": [] }, { "id": 132, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-criterion-flatness-fibre", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-criterion-flatness-fibre", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $Y \\to Z$ be morphisms of", "algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module.", "Assume", "\\begin{enumerate}", "\\item $X$ is locally of finite presentation over $Z$,", "\\item $\\mathcal{F}$ an $\\mathcal{O}_X$-module of finite presentation,", "\\item $\\mathcal{F}$ is flat over $Z$, and", "\\item $Y$ is locally of finite type over $Z$.", "\\end{enumerate}", "Then the set", "$$", "A = \\{x \\in |X| : \\mathcal{F} \\text{ flat at }x \\text{ over }Y\\}.", "$$", "is open in $|X|$ and its formation commutes with arbitrary base change:", "If $Z' \\to Z$ is a morphism of algebraic spaces, and $A'$ is the set of", "points of $X' = X \\times_Z Z'$ where $\\mathcal{F}' = \\mathcal{F} \\times_Z Z'$", "is flat over $Y' = Y \\times_Z Z'$, then $A'$ is the inverse image of", "$A$ under the continuous map $|X'| \\to |X|$." ], "refs": [], "proofs": [ { "contents": [ "One way to prove this is to translate the proof as given in", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-morphism-between-flat}", "into the category of algebraic spaces. Instead we will prove this", "by reducing to the case of schemes. Namely, choose a diagram as in", "Lemma \\ref{lemma-flat-on-fibres-at-point} part (3)", "such that $a$, $b$, and $c$ are surjective.", "It follows from the definitions that this reduces to the", "corresponding theorem for the morphisms of schemes", "$U \\to V \\to W$, the quasi-coherent sheaf $a^*\\mathcal{F}$,", "and the point $u \\in U$. The only minor point to make is that", "given a morphism of algebraic spaces $Z' \\to Z$ we choose a scheme", "$W'$ and a surjective \\'etale morphism $W' \\to W \\times_Z Z'$.", "Then we set $U' = W' \\times_W U$ and $V' = W' \\times_W V$.", "We write $a', b', c'$ for the morphisms from $U', V', W'$ to", "$X', Y', Z'$. In this case $A$, resp.\\ $A'$ are images of the open", "subsets of $U$, resp.\\ $U'$ associated to", "$a^*\\mathcal{F}$, resp.\\ $(a')^*\\mathcal{F}'$.", "This indeed does reduce the lemma to", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-morphism-between-flat}." ], "refs": [ "more-morphisms-lemma-morphism-between-flat", "spaces-more-morphisms-lemma-flat-on-fibres-at-point", "more-morphisms-lemma-morphism-between-flat" ], "ref_ids": [ 13768, 130, 13768 ] } ], "ref_ids": [] }, { "id": 133, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-flatness-fibres", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-flatness-fibres", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ and $Y \\to Z$ be a morphism of algebraic spaces over $S$.", "Assume", "\\begin{enumerate}", "\\item $X$ is locally of finite presentation over $Z$,", "\\item $X$ is flat over $Z$, and", "\\item $Y$ is locally of finite type over $Z$.", "\\end{enumerate}", "Then the set", "$$", "\\{x \\in |X| : X\\text{ flat at }x \\text{ over }Y\\}.", "$$", "is open in $|X|$ and its formation commutes with arbitrary base change", "$Z' \\to Z$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-base-change-criterion-flatness-fibre}." ], "refs": [ "spaces-more-morphisms-lemma-base-change-criterion-flatness-fibre" ], "ref_ids": [ 132 ] } ], "ref_ids": [] }, { "id": 134, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-and-free-at-point-fibre", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-and-free-at-point-fibre", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ which is locally of finite presentation.", "Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module.", "Let $x \\in |X|$ with image $y \\in |Y|$. If $\\mathcal{F}$ is flat at $x$", "over $Y$, then the following are equivalent", "\\begin{enumerate}", "\\item $(\\mathcal{F}_{\\overline{y}})_{\\overline{x}}$ is a flat", "$\\mathcal{O}_{X_{\\overline{y}}, \\overline{x}}$-module,", "\\item $(\\mathcal{F}_{\\overline{y}})_{\\overline{x}}$ is a free", "$\\mathcal{O}_{X_{\\overline{y}}, \\overline{x}}$-module,", "\\item $\\mathcal{F}_{\\overline{y}}$ is finite free in an", "\\'etale neighbourhood of $\\overline{x}$ in $X_{\\overline{y}}$, and", "\\item $\\mathcal{F}$ is finite free in an \\'etale neighbourhood of $x$ in $X$.", "\\end{enumerate}", "Here $\\overline{x}$ is a geometric point of $X$ lying over $x$", "and $\\overline{y} = f \\circ \\overline{x}$." ], "refs": [], "proofs": [ { "contents": [ "Pick a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$ and $V$ are schemes and the vertical arrows are \\'etale", "such that there is a point $u \\in U$ mapping to $x$. Let $v \\in V$", "be the image of $u$. Applying Lemma \\ref{lemma-flat-on-fibres-at-point}", "to $\\text{id} : X \\to X$ over $Y$ we see that (1) translates into", "the condition ``$\\mathcal{F}|_{U_v}$ is flat over $U_v$ at $u$''.", "In other words, (1) is equivalent to $(\\mathcal{F}|_{U_v})_u$", "being a flat $\\mathcal{O}_{U_v, u}$-module.", "By the case of schemes (More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-flat-and-free-at-point-fibre}),", "we find that this implies that", "$\\mathcal{F}|_U$ is finite free in an open neighbourhood", "of $u$. In this way we see that (1) implies (4).", "The implications (4) $\\Rightarrow$ (3) and", "(2) $\\Rightarrow$ (1) are immediate.", "For the implication (3) $\\Rightarrow$ (2) use", "the description of local rings and stalks in", "Properties of Spaces, Lemmas", "\\ref{spaces-properties-lemma-describe-etale-local-ring} and", "\\ref{spaces-properties-lemma-stalk-quasi-coherent}." ], "refs": [ "spaces-more-morphisms-lemma-flat-on-fibres-at-point", "more-morphisms-lemma-flat-and-free-at-point-fibre", "spaces-properties-lemma-describe-etale-local-ring", "spaces-properties-lemma-stalk-quasi-coherent" ], "ref_ids": [ 130, 13771, 11884, 11909 ] } ], "ref_ids": [] }, { "id": 135, "type": "theorem", "label": "spaces-more-morphisms-lemma-finite-free-open", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-finite-free-open", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ which is locally of finite presentation.", "Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module", "flat over $Y$. Then the set", "$$", "\\{x \\in |X| : \\mathcal{F}\\text{ free in an \\'etale neighbourhood of }x\\}", "$$", "is open in $|X|$ and its formation commutes with arbitrary base change", "$Y' \\to Y$." ], "refs": [], "proofs": [ { "contents": [ "Openness holds trivially. Let $Y' \\to Y$ be a morphism of algebraic spaces,", "set $X' = Y' \\times_Y X$, ", "and let $x' \\in |X'|$ be a point lying over $x \\in |X|$.", "By Lemma \\ref{lemma-flat-and-free-at-point-fibre}", "we see that $x$ is in our set if and only if", "$(\\mathcal{F}_{\\overline{y}})_{\\overline{x}}$ is a flat", "$\\mathcal{O}_{X_{\\overline{y}}, \\overline{x}}$-module.", "Simiarly, $x'$ is in the analogue of our set for the pullback", "$\\mathcal{F}'$ of $\\mathcal{F}$ to $X'$ if and only if", "$(\\mathcal{F}'_{\\overline{y}'})_{\\overline{x}'}$ is a flat", "$\\mathcal{O}_{X'_{\\overline{y}'}, \\overline{x}'}$-module", "(with obvious notation). These two assertions are equivalent", "by Lemma \\ref{lemma-flat-on-fibres-at-point} applied to", "the morphism $\\text{id} : X \\to X$ over $Y$.", "Thus the statement on base change holds." ], "refs": [ "spaces-more-morphisms-lemma-flat-and-free-at-point-fibre", "spaces-more-morphisms-lemma-flat-on-fibres-at-point" ], "ref_ids": [ 134, 130 ] } ], "ref_ids": [] }, { "id": 136, "type": "theorem", "label": "spaces-more-morphisms-lemma-morphism-between-flat-Noetherian", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-morphism-between-flat-Noetherian", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ and $Y \\to Z$ be a morphism of algebraic spaces over $S$.", "Assume", "\\begin{enumerate}", "\\item $X$, $Y$, $Z$ locally Noetherian,", "\\item $X$ is flat over $Z$,", "\\item for every $z \\in |Z|$ the fibre of $X$ over $z$", "is flat over the fibre of $Y$ over $z$.", "\\end{enumerate}", "Then $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $Z$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Theorem \\ref{theorem-criterion-flatness-fibre-Noetherian}." ], "refs": [ "spaces-more-morphisms-theorem-criterion-flatness-fibre-Noetherian" ], "ref_ids": [ 12 ] } ], "ref_ids": [] }, { "id": 137, "type": "theorem", "label": "spaces-more-morphisms-lemma-flatness-over-Noetherian-ring", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flatness-over-Noetherian-ring", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.", "Let $X$ be an algebraic space locally of finite presentation over", "$S = \\Spec(A)$. For $n \\geq 1$ set $S_n = \\Spec(A/I^n)$ and", "$X_n = S_n \\times_S X$. Let $\\mathcal{F}$ be coherent $\\mathcal{O}_X$-module.", "If for every $n \\geq 1$ the pullback $\\mathcal{F}_n$ of $\\mathcal{F}$ to $X$", "is flat over $S_n$, then the (open) locus where $\\mathcal{F}$", "is flat over $X$ contains the inverse image of $V(I)$ under $X \\to S$." ], "refs": [], "proofs": [ { "contents": [ "The locus where $\\mathcal{F}$ is flat over $S$ is open in $|X|$ by", "Theorem \\ref{theorem-openness-flatness}.", "The statement is insensitive to replacing $X$ by the members of an", "\\'etale covering, hence we may assume $X$ is an affine scheme.", "In this case the result follows immediately from", "Algebra, Lemma \\ref{algebra-lemma-flat-module-powers}.", "Some details omitted." ], "refs": [ "spaces-more-morphisms-theorem-openness-flatness", "algebra-lemma-flat-module-powers" ], "ref_ids": [ 10, 893 ] } ], "ref_ids": [] }, { "id": 138, "type": "theorem", "label": "spaces-more-morphisms-lemma-integral-closure-smooth-pullback", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-integral-closure-smooth-pullback", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a smooth morphism of", "algebraic spaces over $S$. Let $\\mathcal{A}$ be a quasi-coherent", "sheaf of $\\mathcal{O}_X$-algebras. The integral closure", "of $\\mathcal{O}_Y$ in $f^*\\mathcal{A}$ is equal to $f^*\\mathcal{A}'$", "where $\\mathcal{A}' \\subset \\mathcal{A}$ is the integral closure of", "$\\mathcal{O}_X$ in $\\mathcal{A}$." ], "refs": [], "proofs": [ { "contents": [ "By our construction of the integral closure, see", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-integral-closure},", "this reduces immediately to the case where $X$ and $Y$ are affine.", "In this case the result is", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-commutes-smooth}." ], "refs": [ "spaces-morphisms-definition-integral-closure", "algebra-lemma-integral-closure-commutes-smooth" ], "ref_ids": [ 5025, 1252 ] } ], "ref_ids": [] }, { "id": 139, "type": "theorem", "label": "spaces-more-morphisms-lemma-normalization-smooth-localization", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-normalization-smooth-localization", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "Y_2 \\ar[r] \\ar[d] & Y_1 \\ar[d]^f \\\\", "X_2 \\ar[r]^\\varphi & X_1", "}", "$$", "be a fibre square of algebraic spaces over $S$. Assume $f$ is quasi-compact", "and quasi-separated and $\\varphi$ is smooth.", "Let $Y_i \\to X_i' \\to X_i$ be the normalization of $X_i$ in $Y_i$.", "Then $X_2' \\cong X_2 \\times_{X_1} X_1'$." ], "refs": [], "proofs": [ { "contents": [ "The base change of the factorization $Y_1 \\to X_1' \\to X_1$ to $X_2$", "is a factorization $Y_2 \\to X_2 \\times_{X_1} X_1' \\to X_1$ and", "$X_2 \\times_{X_1} X_1' \\to X_1$ is integral", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-integral}).", "Hence we get a morphism", "$h : X_2' \\to X_2 \\times_{X_1} X_1'$ by the universal property of", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-characterize-normalization}.", "Observe that $X_2'$ is the relative spectrum of the integral closure", "of $\\mathcal{O}_{X_2}$ in $f_{2, *}\\mathcal{O}_{Y_2}$.", "If $\\mathcal{A}' \\subset f_{1, *}\\mathcal{O}_{Y_1}$ denotes the integral", "closure of $\\mathcal{O}_{X_2}$, then $X_2 \\times_{X_1} X_1'$ is the", "relative spectrum of $\\varphi^*\\mathcal{A}'$ as the construction of", "the relative spectrum commutes with arbitrary base change. By", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-flat-base-change-cohomology}", "we know that $f_{2, *}\\mathcal{O}_{Y_2} = \\varphi^*f_{1, *}\\mathcal{O}_{Y_1}$.", "Hence the result follows from", "Lemma \\ref{lemma-integral-closure-smooth-pullback}." ], "refs": [ "spaces-morphisms-lemma-base-change-integral", "spaces-morphisms-lemma-characterize-normalization", "spaces-cohomology-lemma-flat-base-change-cohomology", "spaces-more-morphisms-lemma-integral-closure-smooth-pullback" ], "ref_ids": [ 4942, 4959, 11296, 138 ] } ], "ref_ids": [] }, { "id": 140, "type": "theorem", "label": "spaces-more-morphisms-lemma-CM-local-ring-fibre", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-CM-local-ring-fibre", "contents": [ "The property of morphisms of germs of schemes", "\\begin{align*}", "& \\mathcal{P}((X, x) \\to (S, s)) = \\\\", "& \\text{the local ring }", "\\mathcal{O}_{X_s, x}", "\\text{ of the fibre is Noetherian and Cohen-Macaulay}", "\\end{align*}", "is \\'etale local on the source-and-target (Descent, Definition", "\\ref{descent-definition-local-source-target-at-point})." ], "refs": [ "descent-definition-local-source-target-at-point" ], "proofs": [ { "contents": [ "Given a diagram as in", "Descent, Definition \\ref{descent-definition-local-source-target-at-point}", "we obtain an \\'etale morphism of fibres", "$U'_{v'} \\to U_v$ mapping $u'$ to $u$, see", "Descent, Lemma \\ref{descent-lemma-etale-on-fiber}.", "Thus the strict henselizations of the local rings", "$\\mathcal{O}_{U'_{v'}, u'}$ and $\\mathcal{O}_{U_v, u}$", "are the same. We conclude by", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-CM}." ], "refs": [ "descent-definition-local-source-target-at-point", "descent-lemma-etale-on-fiber", "more-algebra-lemma-henselization-CM" ], "ref_ids": [ 14775, 14728, 10063 ] } ], "ref_ids": [ 14775 ] }, { "id": 141, "type": "theorem", "label": "spaces-more-morphisms-lemma-CM", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-CM", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume the fibres of $f$ are locally Noetherian.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is Cohen-Macaulay,", "\\item $f$ is flat and for some surjective \\'etale morphism $V \\to Y$", "where $V$ is a scheme, the fibres of $X_V \\to V$", "are Cohen-Macaulay algebraic spaces, and", "\\item $f$ is flat and for any \\'etale morphism $V \\to Y$", "where $V$ is a scheme, the fibres of $X_V \\to V$", "are Cohen-Macaulay algebraic spaces.", "\\end{enumerate}", "Given $x \\in |X|$ with image $y \\in |Y|$ the following are", "equivalent", "\\begin{enumerate}", "\\item[(a)] $f$ is Cohen-Macaulay at $x$, and", "\\item[(b)] $\\mathcal{O}_{Y, \\overline{y}} \\to \\mathcal{O}_{X, \\overline{x}}$", "is flat and", "$\\mathcal{O}_{X, \\overline{x}}/", "\\mathfrak m_{\\overline{y}}\\mathcal{O}_{X, \\overline{x}}$ is Cohen-Macaulay.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Given an \\'etale morphism $V \\to Y$ where $V$ is a scheme", "choose a scheme $U$ and a surjective \\'etale morphism $U \\to X \\times_Y V$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "Let $u \\in U$ with images $x \\in |X|$, $y \\in |Y|$, and $v \\in V$.", "Then $f$ is Cohen-Macaulay at $x$ if and only if $U \\to V$ is", "Cohen-Macaulay at $u$ (by definition). Moreover the morphism", "$U_v \\to X_v = (X_V)_v$ is surjective \\'etale. Hence the scheme $U_v$ is", "Cohen-Macaulay if and only if the algebraic space $X_v$ is Cohen-Macaulay.", "Thus the equivalence of (1), (2), and (3) follows from the", "corresponding equivalence for morphisms of", "schemes, see More on Morphisms, Lemma \\ref{more-morphisms-lemma-CM}", "by a formal argument.", "\\medskip\\noindent", "Proof of equivalence of (a) and (b). The corresponding equivalence", "for flatness is Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-flat-at-point-etale-local-rings}.", "Thus we may assume $f$ is flat at $x$ when proving the equivalence.", "Consider a diagram and $x, y, u, v$ as above. Then", "$\\mathcal{O}_{Y, \\overline{y}} \\to \\mathcal{O}_{X, \\overline{x}}$", "is equal to the map", "$\\mathcal{O}_{V, v}^{sh} \\to \\mathcal{O}_{U, u}^{sh}$", "on strict henselizations of local rings, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring}.", "Thus we have", "$$", "\\mathcal{O}_{X, \\overline{x}}/", "\\mathfrak m_{\\overline{y}}\\mathcal{O}_{X, \\overline{x}} =", "(\\mathcal{O}_{U, u}/\\mathfrak m_v \\mathcal{O}_{U, u})^{sh}", "$$", "by Algebra, Lemma \\ref{algebra-lemma-quotient-strict-henselization}.", "Thus we have to show that the Noetherian local ring", "$\\mathcal{O}_{U, u}/\\mathfrak m_v \\mathcal{O}_{U, u}$", "is Cohen-Macaulay if and only if its strict henselization is.", "This is More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-CM}." ], "refs": [ "more-morphisms-lemma-CM", "spaces-morphisms-lemma-flat-at-point-etale-local-rings", "spaces-properties-lemma-describe-etale-local-ring", "algebra-lemma-quotient-strict-henselization", "more-algebra-lemma-henselization-CM" ], "ref_ids": [ 13784, 4857, 11884, 1307, 10063 ] } ], "ref_ids": [] }, { "id": 142, "type": "theorem", "label": "spaces-more-morphisms-lemma-composition-CM", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-composition-CM", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces", "over $S$. Assume that the", "fibres of $f$, $g$, and $g \\circ f$ are locally Noetherian.", "Let $x \\in |X|$ with images $y \\in |Y|$ and $z \\in |Z|$.", "\\begin{enumerate}", "\\item If $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay", "at $f(x)$, then $g \\circ f$ is Cohen-Macaulay at $x$.", "\\item If $f$ and $g$ are Cohen-Macaulay, then $g \\circ f$ is Cohen-Macaulay.", "\\item If $g \\circ f$ is Cohen-Macaulay at $x$ and $f$ is flat at $x$,", "then $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$.", "\\item If $f \\circ g$ is Cohen-Macaulay and $f$ is flat, then", "$f$ is Cohen-Macaulay and $g$ is Cohen-Macaulay at every point in", "the image of $f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Working \\'etale locally this follows from the corresponding result for", "schemes, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-composition-CM}.", "Alternatively, we can use the equivalence of (a) and (b) in", "Lemma \\ref{lemma-CM}. Thus we consider the local homomorphism", "of Noetherian local rings", "$$", "\\mathcal{O}_{Y, \\overline{y}}/", "\\mathfrak m_{\\overline{z}}\\mathcal{O}_{Y, \\overline{y}}", "\\longrightarrow", "\\mathcal{O}_{X, \\overline{x}}/", "\\mathfrak m_{\\overline{z}}\\mathcal{O}_{X, \\overline{x}}", "$$", "whose fibre is", "$$", "\\mathcal{O}_{X, \\overline{x}}/", "\\mathfrak m_{\\overline{y}}\\mathcal{O}_{X, \\overline{x}}", "$$", "and we use Algebra, Lemma \\ref{algebra-lemma-CM-goes-up}." ], "refs": [ "more-morphisms-lemma-composition-CM", "spaces-more-morphisms-lemma-CM", "algebra-lemma-CM-goes-up" ], "ref_ids": [ 13786, 141, 1362 ] } ], "ref_ids": [] }, { "id": 143, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-morphism-from-CM", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-morphism-from-CM", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a flat morphism of locally Noetherian", "algebraic spaces over $S$.", "If $X$ is Cohen-Macaulay, then $f$ is Cohen-Macaulay and", "$\\mathcal{O}_{Y, f(\\overline{x})}$ is Cohen-Macaulay for all $x \\in |X|$." ], "refs": [], "proofs": [ { "contents": [ "After translating into algebra using Lemma \\ref{lemma-CM}", "(compare with the proof of", "Lemma \\ref{lemma-composition-CM}) this follows from", "Algebra, Lemma \\ref{algebra-lemma-CM-goes-up}." ], "refs": [ "spaces-more-morphisms-lemma-CM", "spaces-more-morphisms-lemma-composition-CM", "algebra-lemma-CM-goes-up" ], "ref_ids": [ 141, 142, 1362 ] } ], "ref_ids": [] }, { "id": 144, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-CM", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-CM", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume the fibres of $f$ are locally Noetherian.", "Let $Y' \\to Y$ be locally of finite type. Let $f' : X' \\to Y'$", "be the base change of $f$.", "Let $x' \\in |X'|$ be a point with image $x \\in |X|$.", "\\begin{enumerate}", "\\item If $f$ is Cohen-Macaulay at $x$, then", "$f' : X' \\to Y'$ is Cohen-Macaulay at $x'$.", "\\item If $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$, then $f$", "is Cohen-Macaulay at $x$.", "\\item If $Y' \\to Y$ is flat at $f'(x')$ and $f'$ is Cohen-Macaulay at", "$x'$, then $f$ is Cohen-Macaulay at $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $y \\in |Y|$ and $y' \\in |Y'|$ the image of $x'$.", "Choose a surjective \\'etale morphism $V \\to Y$ where $V$ is a scheme.", "Choose a surjective \\'etale morphism $U \\to X \\times_Y V$ where", "$U$ is a scheme.", "Choose a surjectiev \\'etale morphism $V' \\to Y' \\times_Y V$", "where $V'$ is a scheme.", "Then $U' = U \\times_V V'$ is a scheme which comes equipped", "with a surjective \\'etale morphism $U' \\to X'$.", "Choose $u' \\in U'$ mapping to $x'$. Denote $u \\in U$ the image of $u'$.", "Then the lemma follows from the lemma for", "$U \\to V$ and its base change $U' \\to V'$ and the", "points $u'$ and $u$ (this follows from the definitions).", "Thus the lemma follows from the case of schemes, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-base-change-CM}." ], "refs": [ "more-morphisms-lemma-base-change-CM" ], "ref_ids": [ 13788 ] } ], "ref_ids": [] }, { "id": 145, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-finite-presentation-CM-open", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-finite-presentation-CM-open", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "which is flat and locally of finite presentation. Let", "$$", "W = \\{x \\in |X| : f\\text{ is Cohen-Macaulay at }x\\}", "$$", "Then $W$ is open in $|X|$ and the formation of $W$", "commutes with arbitrary base change of $f$:", "For any morphism $g : Y' \\to Y$, consider", "the base change $f' : X' \\to Y'$ of $f$ and the", "projection $g' : X' \\to X$. Then the corresponding", "set $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with \\'etale vertical arrows and $U$ and $V$ schemes.", "Let $u \\in U$ with image $x \\in |X|$.", "Then $f$ is Cohen-Macaulay at $x$ if and only if $U \\to V$ is", "Cohen-Macaulay at $u$ (by definition).", "Thus we reduce to the case of the morphism $U \\to V$.", "See More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-flat-finite-presentation-CM-open}." ], "refs": [ "more-morphisms-lemma-flat-finite-presentation-CM-open" ], "ref_ids": [ 13789 ] } ], "ref_ids": [] }, { "id": 146, "type": "theorem", "label": "spaces-more-morphisms-lemma-lfp-CM-relative-dimension", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-lfp-CM-relative-dimension", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a", "morphism of algebraic spaces over $S$. Assume that", "$f$ is locally of finite presentation and Cohen-Macaulay.", "Then there exist open and closed subschemes $X_d \\subset X$", "such that $X = \\coprod_{d \\geq 0} X_d$ and $f|_{X_d} : X_d \\to Y$", "has relative dimension $d$." ], "refs": [], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with \\'etale vertical arrows and $U$ and $V$ schemes.", "Then $U \\to V$ is locally of finite presentation and Cohen-Macaulay", "(immediate from our definitions).", "Thus we have a decomposition $U = \\coprod_{d \\geq 0} U_d$", "into open and closed subschemes with $f|_{U_d} : U_d \\to V$", "of relative dimension $d$, see Morphisms, Lemma", "\\ref{morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension}.", "Let $u \\in U$ with image $x \\in |X|$. Then", "$f$ has relative dimension $d$ at $x$ if and only if", "$U \\to V$ has relative dimension $d$ at $u$", "(this follows from our definitions).", "In this way we see that $U_d$ is the inverse image", "of a subset $X_d \\subset |X|$ which is necessarily", "open and closed. Denoting $X_d$ the corresponding open", "and closed algebraic subspace of $X$ we see that the lemma is true." ], "refs": [ "morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension" ], "ref_ids": [ 5286 ] } ], "ref_ids": [] }, { "id": 147, "type": "theorem", "label": "spaces-more-morphisms-lemma-gorenstein-local-ring-fibre", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-gorenstein-local-ring-fibre", "contents": [ "The property of morphisms of germs of schemes", "\\begin{align*}", "& \\mathcal{P}((X, x) \\to (S, s)) = \\\\", "& \\text{the local ring }", "\\mathcal{O}_{X_s, x}", "\\text{ of the fibre is Noetherian and Gorenstein}", "\\end{align*}", "is \\'etale local on the source-and-target (Descent, Definition", "\\ref{descent-definition-local-source-target-at-point})." ], "refs": [ "descent-definition-local-source-target-at-point" ], "proofs": [ { "contents": [ "Given a diagram as in", "Descent, Definition \\ref{descent-definition-local-source-target-at-point}", "we obtain an \\'etale morphism of fibres", "$U'_{v'} \\to U_v$ mapping $u'$ to $u$, see", "Descent, Lemma \\ref{descent-lemma-etale-on-fiber}.", "Thus $\\mathcal{O}_{U_v, u} \\to \\mathcal{O}_{U'_{v'}, u'}$", "is the localization of an \\'etale ring map. Hence", "the first is Noetherian if and only if the second is Noetherian, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-Noetherian-etale-extension}.", "Then, since $\\mathcal{O}_{U'_{v'}, u'}/\\mathfrak m_u \\mathcal{O}_{U'_{v'}, u'}", "= \\kappa(u')$ (Algebra, Lemma \\ref{algebra-lemma-etale-at-prime})", "is a Gorenstein ring, we see that", "$\\mathcal{O}_{U_v, u}$ is Gorenstein if and only if", "$\\mathcal{O}_{U'_{v'}, u'}$ is Gorenstein by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-flat-under-gorenstein}." ], "refs": [ "descent-definition-local-source-target-at-point", "descent-lemma-etale-on-fiber", "more-algebra-lemma-Noetherian-etale-extension", "algebra-lemma-etale-at-prime", "dualizing-lemma-flat-under-gorenstein" ], "ref_ids": [ 14775, 14728, 10051, 1233, 2885 ] } ], "ref_ids": [ 14775 ] }, { "id": 148, "type": "theorem", "label": "spaces-more-morphisms-lemma-gorenstein", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-gorenstein", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume the fibres of $f$ are locally Noetherian.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is Gorenstein,", "\\item $f$ is flat and for some surjective \\'etale morphism $V \\to Y$", "where $V$ is a scheme, the fibres of $X_V \\to V$", "are Gorenstein algebraic spaces, and", "\\item $f$ is flat and for any \\'etale morphism $V \\to Y$", "where $V$ is a scheme, the fibres of $X_V \\to V$", "are Gorenstein algebraic spaces.", "\\end{enumerate}", "Given $x \\in |X|$ with image $y \\in |Y|$ the following are", "equivalent", "\\begin{enumerate}", "\\item[(a)] $f$ is Gorenstein at $x$, and", "\\item[(b)] $\\mathcal{O}_{Y, \\overline{y}} \\to \\mathcal{O}_{X, \\overline{x}}$", "is flat and", "$\\mathcal{O}_{X, \\overline{x}}/", "\\mathfrak m_{\\overline{y}}\\mathcal{O}_{X, \\overline{x}}$ is Gorenstein.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Given an \\'etale morphism $V \\to Y$ where $V$ is a scheme", "choose a scheme $U$ and a surjective \\'etale morphism $U \\to X \\times_Y V$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "Let $u \\in U$ with images $x \\in |X|$, $y \\in |Y|$, and $v \\in V$.", "Then $f$ is Gorenstein at $x$ if and only if $U \\to V$ is", "Gorenstein at $u$ (by definition). Moreover the morphism", "$U_v \\to X_v = (X_V)_v$ is surjective \\'etale. Hence the scheme $U_v$ is", "Gorenstein if and only if the algebraic space $X_v$ is Gorenstein.", "Thus the equivalence of (1), (2), and (3) follows from the", "corresponding equivalence for morphisms of", "schemes, see Duality for Schemes, Lemma \\ref{duality-lemma-gorenstein}", "by a formal argument.", "\\medskip\\noindent", "Proof of equivalence of (a) and (b). The corresponding equivalence", "for flatness is Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-flat-at-point-etale-local-rings}.", "Thus we may assume $f$ is flat at $x$ when proving the equivalence.", "Consider a diagram and $x, y, u, v$ as above. Then", "$\\mathcal{O}_{Y, \\overline{y}} \\to \\mathcal{O}_{X, \\overline{x}}$", "is equal to the map", "$\\mathcal{O}_{V, v}^{sh} \\to \\mathcal{O}_{U, u}^{sh}$", "on strict henselizations of local rings, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring}.", "Thus we have", "$$", "\\mathcal{O}_{X, \\overline{x}}/", "\\mathfrak m_{\\overline{y}}\\mathcal{O}_{X, \\overline{x}} =", "(\\mathcal{O}_{U, u}/\\mathfrak m_v \\mathcal{O}_{U, u})^{sh}", "$$", "by Algebra, Lemma \\ref{algebra-lemma-quotient-strict-henselization}.", "Thus we have to show that the Noetherian local ring", "$\\mathcal{O}_{U, u}/\\mathfrak m_v \\mathcal{O}_{U, u}$", "is Gorenstein if and only if its strict henselization is.", "This follows immediately from", "Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-completion-henselization-dualizing}", "and the definition of a Gorenstein local ring as a", "Noetherian local ring which is a dualizing complex over itself." ], "refs": [ "duality-lemma-gorenstein", "spaces-morphisms-lemma-flat-at-point-etale-local-rings", "spaces-properties-lemma-describe-etale-local-ring", "algebra-lemma-quotient-strict-henselization", "dualizing-lemma-completion-henselization-dualizing" ], "ref_ids": [ 13591, 4857, 11884, 1307, 2889 ] } ], "ref_ids": [] }, { "id": 149, "type": "theorem", "label": "spaces-more-morphisms-lemma-composition-gorenstein", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-composition-gorenstein", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces", "over $S$. Assume that the", "fibres of $f$, $g$, and $g \\circ f$ are locally Noetherian.", "Let $x \\in |X|$ with images $y \\in |Y|$ and $z \\in |Z|$.", "\\begin{enumerate}", "\\item If $f$ is Gorenstein at $x$ and $g$ is Gorenstein", "at $f(x)$, then $g \\circ f$ is Gorenstein at $x$.", "\\item If $f$ and $g$ are Gorenstein, then $g \\circ f$ is Gorenstein.", "\\item If $g \\circ f$ is Gorenstein at $x$ and $f$ is flat at $x$,", "then $f$ is Gorenstein at $x$ and $g$ is Gorenstein at $f(x)$.", "\\item If $f \\circ g$ is Gorenstein and $f$ is flat, then", "$f$ is Gorenstein and $g$ is Gorenstein at every point in", "the image of $f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Working \\'etale locally this follows from the corresponding result for", "schemes, see Duality for Schemes, Lemma", "\\ref{duality-lemma-composition-gorenstein}.", "Alternatively, we can use the equivalence of (a) and (b) in", "Lemma \\ref{lemma-gorenstein}. Thus we consider the local homomorphism", "of Noetherian local rings", "$$", "\\mathcal{O}_{Y, \\overline{y}}/", "\\mathfrak m_{\\overline{z}}\\mathcal{O}_{Y, \\overline{y}}", "\\longrightarrow", "\\mathcal{O}_{X, \\overline{x}}/", "\\mathfrak m_{\\overline{z}}\\mathcal{O}_{X, \\overline{x}}", "$$", "whose fibre is", "$$", "\\mathcal{O}_{X, \\overline{x}}/", "\\mathfrak m_{\\overline{y}}\\mathcal{O}_{X, \\overline{x}}", "$$", "and we use Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-flat-under-gorenstein}." ], "refs": [ "duality-lemma-composition-gorenstein", "spaces-more-morphisms-lemma-gorenstein", "dualizing-lemma-flat-under-gorenstein" ], "ref_ids": [ 13598, 148, 2885 ] } ], "ref_ids": [] }, { "id": 150, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-morphism-from-gorenstein", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-morphism-from-gorenstein", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a flat morphism of locally Noetherian", "algebraic spaces over $S$.", "If $X$ is Gorenstein, then $f$ is Gorenstein and", "$\\mathcal{O}_{Y, f(\\overline{x})}$ is Gorenstein for all $x \\in |X|$." ], "refs": [], "proofs": [ { "contents": [ "After translating into algebra using Lemma \\ref{lemma-gorenstein}", "(compare with the proof of", "Lemma \\ref{lemma-composition-gorenstein}) this follows from", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-flat-under-gorenstein}." ], "refs": [ "spaces-more-morphisms-lemma-gorenstein", "spaces-more-morphisms-lemma-composition-gorenstein", "dualizing-lemma-flat-under-gorenstein" ], "ref_ids": [ 148, 149, 2885 ] } ], "ref_ids": [] }, { "id": 151, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-gorenstein", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-gorenstein", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume the fibres of $f$ are locally Noetherian.", "Let $Y' \\to Y$ be locally of finite type. Let $f' : X' \\to Y'$", "be the base change of $f$.", "Let $x' \\in |X'|$ be a point with image $x \\in |X|$.", "\\begin{enumerate}", "\\item If $f$ is Gorenstein at $x$, then", "$f' : X' \\to Y'$ is Gorenstein at $x'$.", "\\item If $f$ is flat at $x$ and $f'$ is Gorenstein at $x'$, then $f$", "is Gorenstein at $x$.", "\\item If $Y' \\to Y$ is flat at $f'(x')$ and $f'$ is Gorenstein at", "$x'$, then $f$ is Gorenstein at $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $y \\in |Y|$ and $y' \\in |Y'|$ the image of $x'$.", "Choose a surjective \\'etale morphism $V \\to Y$ where $V$ is a scheme.", "Choose a surjective \\'etale morphism $U \\to X \\times_Y V$ where", "$U$ is a scheme.", "Choose a surjectiev \\'etale morphism $V' \\to Y' \\times_Y V$", "where $V'$ is a scheme.", "Then $U' = U \\times_V V'$ is a scheme which comes equipped", "with a surjective \\'etale morphism $U' \\to X'$.", "Choose $u' \\in U'$ mapping to $x'$. Denote $u \\in U$ the image of $u'$.", "Then the lemma follows from the lemma for", "$U \\to V$ and its base change $U' \\to V'$ and the", "points $u'$ and $u$ (this follows from the definitions).", "Thus the lemma follows from the case of schemes, see", "Duality for Schemes, Lemma \\ref{duality-lemma-base-change-gorenstein}." ], "refs": [ "duality-lemma-base-change-gorenstein" ], "ref_ids": [ 13600 ] } ], "ref_ids": [] }, { "id": 152, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-finite-presentation-gorenstein-open", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-finite-presentation-gorenstein-open", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "which is flat and locally of finite presentation. Let", "$$", "W = \\{x \\in |X| : f\\text{ is Gorenstein at }x\\}", "$$", "Then $W$ is open in $|X|$ and the formation of $W$", "commutes with arbitrary base change of $f$:", "For any morphism $g : Y' \\to Y$, consider", "the base change $f' : X' \\to Y'$ of $f$ and the", "projection $g' : X' \\to X$. Then the corresponding", "set $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "Let $u \\in U$ with image $x \\in |X|$.", "Then $f$ is Gorenstein at $x$ if and only if $U \\to V$ is", "Gorenstein at $u$ (by definition).", "Thus we reduce to the case of the morphism $U \\to V$", "of schemes. Openness is proven in", "Duality for Schemes, Lemma", "\\ref{duality-lemma-flat-finite-presentation-characterize-gorenstein}", "and compatibility with base change in", "Duality for Schemes, Lemma \\ref{duality-lemma-flat-lft-base-change-gorenstein}." ], "refs": [ "duality-lemma-flat-finite-presentation-characterize-gorenstein", "duality-lemma-flat-lft-base-change-gorenstein" ], "ref_ids": [ 13603, 13601 ] } ], "ref_ids": [] }, { "id": 153, "type": "theorem", "label": "spaces-more-morphisms-lemma-slice", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-slice", "contents": [ "Let $S$ be a scheme. Consider a cartesian diagram", "$$", "\\xymatrix{", "X \\ar[d] & F \\ar[l]^p \\ar[d] \\\\", "Y & \\Spec(k) \\ar[l]", "}", "$$", "where $X \\to Y$ is a morphism of algebraic spaces over $S$", "which is flat and locally of finite presentation, and where", "$k$ is a field over $S$. Let $f_1, \\ldots, f_r \\in \\Gamma(X, \\mathcal{O}_X)$", "and $z \\in |F|$ such that $f_1, \\ldots, f_r$ map to a regular sequence", "in the local ring $\\mathcal{O}_{F, \\overline{z}}$.", "Then, after replacing $X$ by an open subspace containing $p(z)$, the morphism", "$$", "V(f_1, \\ldots, f_r) \\longrightarrow Y", "$$", "is flat and locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Set $Z = V(f_1, \\ldots, f_r)$. It is clear that $Z \\to X$ is locally of", "finite presentation, hence the composition $Z \\to Y$ is locally of finite", "presentation, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-composition-finite-presentation}.", "Hence it suffices to show that $Z \\to Y$ is flat in a neighbourhood of $p(z)$.", "Let $k \\subset k'$ be an extension field. Then", "$F' = F \\times_{\\Spec(k)} \\Spec(k')$ is surjective and", "flat over $F$, hence we can find a point $z' \\in |F'|$ mapping to $z$", "and the local ring map", "$\\mathcal{O}_{F, \\overline{z}} \\to \\mathcal{O}_{F', \\overline{z}'}$ is", "flat, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-flat-at-point-etale-local-rings}.", "Hence the image of $f_1, \\ldots, f_r$ in", "$\\mathcal{O}_{F', \\overline{z}'}$ is a regular sequence too, see", "Algebra, Lemma \\ref{algebra-lemma-flat-increases-depth}.", "Thus, during the proof we may replace $k$ by an extension field.", "In particular, we may assume that $z \\in |F|$ comes from a section", "$z : \\Spec(k) \\to F$ of the structure morphism $F \\to \\Spec(k)$.", "\\medskip\\noindent", "Choose a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$. Choose a scheme $U$ and a surjective \\'etale morphism", "$U \\to X \\times_Y V$. After possibly enlarging $k$ once more we may", "assume that $\\Spec(k) \\to F \\to X$ factors through $U$ (as", "$U \\to X$ is surjective). Let", "$u : \\Spec(k) \\to U$ be such a factorization and denote $v \\in V$", "the image of $u$. Note that the morphisms", "$$", "U_v \\times_{\\Spec(\\kappa(v))} \\Spec(k) =", "U \\times_V \\Spec(k) \\to U \\times_Y \\Spec(k) \\to F", "$$", "are \\'etale (the first as the base change of $V \\to V \\times_Y V$ and", "the second as the base change of $U \\to X$). Moreover, by construction", "the point $u : \\Spec(k) \\to U$ gives a point of the left most", "space which maps to $z$ on the right. Hence the elements", "$f_1, \\ldots, f_r$ map to a regular sequence in the local ring", "on the right of the following map", "$$", "\\mathcal{O}_{U_v, u}", "\\longrightarrow", "\\mathcal{O}_{U_v \\times_{\\Spec(\\kappa(v)} \\Spec(k), \\overline{u}}", "=", "\\mathcal{O}_{U \\times_V \\Spec(k), \\overline{u}}.", "$$", "But since the displayed arrow is flat (combine", "More on Flatness, Lemma \\ref{flat-lemma-flat-up-down-henselization}", "and", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-flat-at-point-etale-local-rings})", "we see from", "Algebra, Lemma \\ref{algebra-lemma-flat-increases-depth}", "that $f_1, \\ldots, f_r$ maps to a regular sequence in", "$\\mathcal{O}_{U_v, u}$. By", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-slice-given-elements}", "we conclude that the morphism of schemes", "$$", "V(f_1, \\ldots, f_r) \\times_X U = V(f_1|_U, \\ldots, f_r|_U) \\to V", "$$", "is flat in an open neighbourhood $U'$ of $u$. Let $X' \\subset X$", "be the open subspace corresponding to the image of", "$|U'| \\to |X|$ (see", "Properties of Spaces, Lemmas", "\\ref{spaces-properties-lemma-topology-points} and", "\\ref{spaces-properties-lemma-open-subspaces}).", "We conclude that $V(f_1, \\ldots, f_r) \\cap X' \\to Y$ is flat", "(see", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-flat})", "as", "we have the commutative diagram", "$$", "\\xymatrix{", "V(f_1, \\ldots, f_r) \\times_X U' \\ar[d]_a \\ar[r] & V \\ar[d]^b \\\\", "V(f_1, \\ldots, f_r) \\cap X' \\ar[r] & Y", "}", "$$", "with $a, b$ \\'etale and $a$ surjective." ], "refs": [ "spaces-morphisms-lemma-composition-finite-presentation", "spaces-morphisms-lemma-flat-at-point-etale-local-rings", "algebra-lemma-flat-increases-depth", "flat-lemma-flat-up-down-henselization", "spaces-morphisms-lemma-flat-at-point-etale-local-rings", "algebra-lemma-flat-increases-depth", "more-morphisms-lemma-slice-given-elements", "spaces-properties-lemma-topology-points", "spaces-properties-lemma-open-subspaces", "spaces-morphisms-definition-flat" ], "ref_ids": [ 4839, 4857, 740, 5982, 4857, 740, 13795, 11822, 11823, 5007 ] } ], "ref_ids": [] }, { "id": 154, "type": "theorem", "label": "spaces-more-morphisms-lemma-geometrically-reduced-fibre", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-geometrically-reduced-fibre", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a", "morphism of algebraic spaces over $S$. Let $y \\in |Y|$.", "The following are equivalent", "\\begin{enumerate}", "\\item for some morphism $\\Spec(k) \\to Y$ in the equivalence class", "of $y$ the algebraic space $X_k$ is geometrically reduced over $k$,", "\\item for every morphism $\\Spec(k) \\to Y$ in the equivalence class", "of $y$ the algebraic space $X_k$ is geometrically reduced over $k$,", "\\item for every morphism $\\Spec(k) \\to Y$ in the equivalence class", "of $y$ the algebraic space $X_k$ is reduced.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-geometrically-reduced-upstairs}", "and the definition of the equivalence relation defining $|X|$", "given in", "Properties of Spaces, Section \\ref{spaces-properties-section-points}." ], "refs": [ "spaces-over-fields-lemma-geometrically-reduced-upstairs" ], "ref_ids": [ 12858 ] } ], "ref_ids": [] }, { "id": 155, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-fibres-geometrically-reduced", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-fibres-geometrically-reduced", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y' \\to Y$", "be morphisms of algebraic spaces over $S$. Denote", "$f' : X' \\to Y'$ the base change of $f$ by $g$. Then", "\\begin{align*}", "\\{y' \\in |Y'| :", "\\text{the fibre of }f' : X' \\to Y'\\text{ at }y'", "\\text{ is geometrically reduced}\\} \\\\", "= g^{-1}(\\{y \\in |Y| :", "\\text{the fibre of }f : X \\to Y\\text{ at }y", "\\text{ is geometrically reduced}\\}).", "\\end{align*}" ], "refs": [], "proofs": [ { "contents": [ "For $y' \\in |Y'|$ choose a morphism $\\Spec(k) \\to Y'$", "in the equivalence class of $y'$. Then $g(y')$ is", "represented by the composition $\\Spec(k) \\to Y' \\to Y$.", "Hence $X' \\times_{Y'} \\Spec(k) = X \\times_Y \\Spec(k)$", "and the result follows from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 156, "type": "theorem", "label": "spaces-more-morphisms-lemma-geometrically-reduced-constructible", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-geometrically-reduced-constructible", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ which is quasi-compact and", "locally of finite presentation. Then the set", "$$", "E = \\{y \\in |Y| : \\text{the fibre of }f : X \\to Y\\text{ at }y", "\\text{ is geometrically reduced}\\}", "$$", "is \\'etale locally constructible." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $V$ and an \\'etale morphism $V \\to Y$.", "The meaning of the statement is that the inverse image of $E$", "in $|V|$ is constructible. By", "Lemma \\ref{lemma-base-change-fibres-geometrically-reduced}", "we may replace $Y$ by $V$, i.e., we may assume that $Y$", "is an affine scheme. Then $X$ is quasi-compact. Choose an", "affine scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "For a morphism $\\Spec(k) \\to Y$ the morphism between fibres", "$U_k \\to X_k$ is surjective \\'etale. Hence $U_k$ is geometrically", "reduced over $k$ if and only if $X_k$ is geometrically reduced", "over $k$, see Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-geometrically-reduced-etale-local}.", "Thus the set $E$ for $X \\to Y$ is the same as the set $E$", "for $U \\to Y$.", "In this way we see that the lemma follows from the case", "of schemes, see More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-geometrically-reduced-constructible}." ], "refs": [ "spaces-more-morphisms-lemma-base-change-fibres-geometrically-reduced", "spaces-over-fields-lemma-geometrically-reduced-etale-local", "more-morphisms-lemma-geometrically-reduced-constructible" ], "ref_ids": [ 155, 12859, 13817 ] } ], "ref_ids": [] }, { "id": 157, "type": "theorem", "label": "spaces-more-morphisms-lemma-proper-flat-over-dvr-reduced-fibre", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-proper-flat-over-dvr-reduced-fibre", "contents": [ "Let $X$ be an algebraic space over a discrete valuation ring $R$", "whose structure morphism $X \\to \\Spec(R)$ is proper and flat.", "If the special fibre is reduced, then", "both $X$ and the generic fibre $X_\\eta$ are reduced." ], "refs": [], "proofs": [ { "contents": [ "Choose an \\'etale morphism $U \\to X$ where $U$ is an affine scheme.", "Then $U$ is of finite type over $R$. Let $u \\in U$ be in the special fibre.", "The local ring $A = \\mathcal{O}_{U, u}$ is essentially of finite", "type over $R$, hence Noetherian. Let $\\pi \\in R$ be a uniformizer.", "Since $X$ is flat over $R$, we see that $\\pi \\in \\mathfrak m_A$", "is a nonzerodivisor on $A$ and since the special fibre of $X$", "is reduced, we have that $A/\\pi A$ is reduced.", "If $a \\in A$, $a \\not = 0$ then there exists an $n \\geq 0$ and an element", "$a' \\in A$ such that $a = \\pi^n a'$ and $a' \\not \\in \\pi A$.", "This follows from Krull intersection theorem", "(Algebra, Lemma \\ref{algebra-lemma-intersect-powers-ideal-module-zero}).", "If $a$ is nilpotent, so is $a'$, because $\\pi$ is a nonzerodivisor.", "But $a'$ maps to a nonzero element of the reduced ring $A/\\pi A$", "so this is impossible. Hence $A$ is reduced. It follows that", "there exists an open neighbourhood of $u$ in $U$ which is reduced", "(small detail omitted; use that $U$ is Noetherian).", "Thus we can find an \\'etale morphism $U \\to X$ with $U$ a reduced", "scheme, such that every point of the special fibre of $X$ is", "in the image. Since $X$ is proper over $R$ it follows that", "$U \\to X$ is surjective. Hence $X$ is reduced. Since the generic fibre of", "$U \\to \\Spec(R)$ is reduced as well (on affine pieces", "it is computed by taking localizations), we conclude the same thing", "is true for the generic fibre." ], "refs": [ "algebra-lemma-intersect-powers-ideal-module-zero" ], "ref_ids": [ 627 ] } ], "ref_ids": [] }, { "id": 158, "type": "theorem", "label": "spaces-more-morphisms-lemma-geometrically-reduced-open", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-geometrically-reduced-open", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. If $f$ is flat, proper, and of finite presentation,", "then the set", "$$", "E = \\{y \\in |Y| : \\text{the fibre of }f : X \\to Y\\text{ at }y", "\\text{ is geometrically reduced}\\}", "$$", "is open in $|Y|$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-base-change-fibres-geometrically-reduced}", "formation of $E$ commutes with base change. To check a subset", "of $|Y|$ is open, we may replace $Y$ by the members of an", "\\'etale covering. Thus we may assume $Y$ is affine.", "Then $Y$ is a cofiltered limit of affine", "schemes of finite type over $\\mathbf{Z}$.", "Hence we can assume $X \\to Y$ is the", "base change of $X_0 \\to Y_0$ where $Y_0$ is the spectrum of a finite", "type $\\mathbf{Z}$-algebra and $X_0 \\to Y_0$ is flat and proper.", "See Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-finite-presentation},", "\\ref{spaces-limits-lemma-descend-flat}, and", "\\ref{spaces-limits-lemma-eventually-proper}. Since the formation of", "$E$ commutes with base change (see above),", "we may assume the base is Noetherian.", "\\medskip\\noindent", "Assume $Y$ is Noetherian. The set is constructible by", "Lemma \\ref{lemma-geometrically-reduced-constructible}.", "Hence it suffices to show the set is stable under generalization", "(Topology, Lemma \\ref{topology-lemma-characterize-closed-Noetherian}). By", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian-specialization-dvr}", "we reduce to the case where $Y = \\Spec(R)$, $R$ is a discrete", "valuation ring, and the closed fibre $X_y$ is geometrically", "reduced. To show: the generic fibre $X_\\eta$ is geometrically reduced.", "\\medskip\\noindent", "If not then there exists a finite extension $L$ of the fraction", "field of $R$ such that $X_L$ is not reduced, see", "Spaces over Fields, Lemmas", "\\ref{spaces-over-fields-lemma-perfect-reduced} (characteristic zero) and", "\\ref{spaces-over-fields-lemma-geometrically-reduced-positive-characteristic}", "(positive characteristic). There exists a discrete valuation ring", "$R' \\subset L$ with fraction field $L$ dominating $R$, see", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-Dedekind}.", "After replacing $R$ by $R'$ we reduce to", "Lemma \\ref{lemma-proper-flat-over-dvr-reduced-fibre}." ], "refs": [ "spaces-more-morphisms-lemma-base-change-fibres-geometrically-reduced", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-flat", "spaces-limits-lemma-eventually-proper", "spaces-more-morphisms-lemma-geometrically-reduced-constructible", "topology-lemma-characterize-closed-Noetherian", "properties-lemma-locally-Noetherian-specialization-dvr", "spaces-over-fields-lemma-perfect-reduced", "spaces-over-fields-lemma-geometrically-reduced-positive-characteristic", "algebra-lemma-integral-closure-Dedekind", "spaces-more-morphisms-lemma-proper-flat-over-dvr-reduced-fibre" ], "ref_ids": [ 155, 4598, 4595, 4596, 156, 8290, 2959, 12856, 12857, 1042, 157 ] } ], "ref_ids": [] }, { "id": 159, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let", "$$", "n_{X/Y} : |Y| \\to \\{0, 1, 2, 3, \\ldots, \\infty\\}", "$$", "be the function which associates to $y \\in Y$ the number of connected", "components of $X_k$ where $\\Spec(k) \\to Y$ is in the equivalence", "class of $y$ with $k$ algebraically closed.", "This is well defined and if $g : Y' \\to Y$ is a morphism then", "$$", "n_{X'/Y'} = n_{X/Y} \\circ g", "$$", "where $X' \\to Y'$ is the base change of $f$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $y' \\in Y'$ has image $y \\in Y$. Let $\\Spec(k') \\to Y'$", "be in the equivalence class of $y'$ with $k'$ algebraically closed.", "Then we can choose a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[rd] &", "\\Spec(k') \\ar[r] & Y' \\ar[d] \\\\", "& \\Spec(k) \\ar[r] & Y", "}", "$$", "where $K$ is an algebraically closed field.", "The result follows as the morphisms of schemes", "$$", "\\xymatrix{", "X'_{k'} & (X'_{k'})_K = (X_k)_K \\ar[l] \\ar[r] & X_k", "}", "$$", "induce bijections between connected components, see", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-separably-closed-field-connected-components}.", "To use this to prove the function is well defined take $Y' = Y$." ], "refs": [ "spaces-over-fields-lemma-separably-closed-field-connected-components" ], "ref_ids": [ 12862 ] } ], "ref_ids": [] }, { "id": 160, "type": "theorem", "label": "spaces-more-morphisms-lemma-dimension-fibre", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-dimension-fibre", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a finite type morphism of", "algebraic spaces over $S$. Let $y \\in |Y|$. The following quantities", "are the same", "\\begin{enumerate}", "\\item the minimal integer $d$ such that $f$ has relative dimension $\\leq d$", "at every $x \\in |X|$ mapping to $y$,", "\\item the dimension of the algebraic space $X_k = \\Spec(k) \\times_Y X$", "for any morphism $\\Spec(k) \\to Y$ in the equivalence class defining $y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To parse this one has to consult", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-dimension-fibre},", "Properties of Spaces,", "Definition \\ref{spaces-properties-definition-dimension},", "Properties of Spaces,", "Definition \\ref{spaces-properties-definition-dimension-at-point}.", "We will show that the numbers in (1) and (2) are equal for", "a fixed morphism $\\Spec(k) \\to Y$.", "Choose an \\'etale morphism $V \\to Y$ where $V$ is an affine", "scheme and a point $v \\in V$ mapping to $y$.", "Since $V \\times_Y \\Spec(k) \\to \\Spec(k)$ is surjective \\'etale", "(by Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian})", "we can find a finite separable extension $k'/k$", "(by Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field})", "and a commutative diagram", "$$", "\\xymatrix{", "\\Spec(k') \\ar[r] \\ar[d] & V \\ar[d] \\\\", "\\Spec(k) \\ar[r] & Y", "}", "$$", "We may replace $X \\to Y$ by $V \\times_Y X \\to V$ and", "$X_k$ by $X_{k'} = \\Spec(k') \\times_V (V \\times_Y X)$", "because this does not change the dimensions in question by", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-dimension-decent-invariant-under-etale}", "and Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}.", "Thus we may assume that $Y$ is an affine scheme.", "In this case we may assume that $k = \\kappa(y)$", "because the dimension of $X_{\\kappa(y)}$ and $X_k$", "are the same by the aforementioned Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}", "and the fact that for an algebraic space $Z$ over a field $K$ the", "relative dimension of $Z$ at a point $z \\in |Z|$", "is the same as $\\dim_z(Z)$ by definition.", "Assume $Y$ is affine and $k = \\kappa(y)$. Then", "$X$ is quasi-compact we can choose an affine scheme $U$ and", "an surjective \\'etale morphism $U \\to X$.", "Then $\\dim(X_k) = \\dim(U_k) = \\max \\dim_u(U_k)$", "is equal to the number given in (1) by definition." ], "refs": [ "spaces-morphisms-definition-dimension-fibre", "spaces-properties-definition-dimension", "spaces-properties-definition-dimension-at-point", "spaces-properties-lemma-points-cartesian", "morphisms-lemma-etale-over-field", "spaces-properties-lemma-dimension-decent-invariant-under-etale", "spaces-morphisms-lemma-dimension-fibre-after-base-change", "spaces-morphisms-lemma-dimension-fibre-after-base-change" ], "ref_ids": [ 5009, 11930, 11929, 11819, 5364, 11887, 4872, 4872 ] } ], "ref_ids": [] }, { "id": 161, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-dimension-fibres", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-dimension-fibres", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a finite type morphism of", "algebraic spaces over $S$. Let", "$$", "n_{X/Y} : |Y| \\to \\{0, 1, 2, 3, \\ldots, \\infty\\}", "$$", "be the function which associates to $y \\in |Y|$ the", "integer discussed in Lemma \\ref{lemma-dimension-fibre}.", "If $g : Y' \\to Y$ is a morphism then", "$$", "n_{X'/Y'} = n_{X/Y} \\circ |g|", "$$", "where $X' \\to Y'$ is the base change of $f$." ], "refs": [ "spaces-more-morphisms-lemma-dimension-fibre" ], "proofs": [ { "contents": [ "This follows immediately from", "Lemma \\ref{lemma-dimension-fibre}." ], "refs": [ "spaces-more-morphisms-lemma-dimension-fibre" ], "ref_ids": [ 160 ] } ], "ref_ids": [ 160 ] }, { "id": 162, "type": "theorem", "label": "spaces-more-morphisms-lemma-dimension-fibres-flat", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-dimension-fibres-flat", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a flat morphism of finite presentation of", "algebraic spaces over $S$. Let", "$n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$", "introduced in Lemma \\ref{lemma-base-change-dimension-fibres}.", "Then $n_{X/Y}$ is lower semi-continuous." ], "refs": [ "spaces-more-morphisms-lemma-base-change-dimension-fibres" ], "proofs": [ { "contents": [ "Let $V \\to Y$ be a surjective \\'etale morphism where $V$ is a scheme.", "If we can show that the composition $n_{X/Y} \\circ |g|$", "is lower semi-continuous, then the lemma follows as $|g|$ is open.", "Hence we may assume $Y$ is a scheme.", "Working locally we may assume $V$ is an affine scheme.", "Then we can choose an affine scheme $U$ and a surjective", "\\'etale morphism $U \\to X$. Then $n_{X/Y} = n_{U/Y}$.", "Hence we may assume $X$ and $Y$ are both schemes.", "In this case the lemma follows from", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-dimension-fibres-flat}." ], "refs": [ "more-morphisms-lemma-dimension-fibres-flat" ], "ref_ids": [ 13842 ] } ], "ref_ids": [ 161 ] }, { "id": 163, "type": "theorem", "label": "spaces-more-morphisms-lemma-dimension-fibres-proper", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-dimension-fibres-proper", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a proper morphism of algebraic spaces over $S$. Let", "$n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$", "introduced in Lemma \\ref{lemma-base-change-dimension-fibres}.", "Then $n_{X/Y}$ is upper semi-continuous." ], "refs": [ "spaces-more-morphisms-lemma-base-change-dimension-fibres" ], "proofs": [ { "contents": [ "Let $Z_d = \\{x \\in |X| :", "\\text{the fibre of }f\\text{ at }x\\text{ has dimension }> d\\}$.", "Then $Z_d$ is a closed subset of $|X|$ by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-openness-bounded-dimension-fibres}.", "Since $f$ is proper $f(Z_d)$ is closed in $|Y|$.", "Since $y \\in f(Z_d) \\Leftrightarrow n_{X/Y}(y) > d$", "we see that the lemma is true." ], "refs": [ "spaces-morphisms-lemma-openness-bounded-dimension-fibres" ], "ref_ids": [ 4873 ] } ], "ref_ids": [ 161 ] }, { "id": 164, "type": "theorem", "label": "spaces-more-morphisms-lemma-dimension-fibres-proper-flat", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-dimension-fibres-proper-flat", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper, flat, finitely presented", "morphism of algebraic spaces over $S$.", "Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$", "introduced in Lemma \\ref{lemma-base-change-dimension-fibres}.", "Then $n_{X/Y}$ is locally constant." ], "refs": [ "spaces-more-morphisms-lemma-base-change-dimension-fibres" ], "proofs": [ { "contents": [ "Immediate consequence of", "Lemmas \\ref{lemma-dimension-fibres-flat} and", "\\ref{lemma-dimension-fibres-proper}." ], "refs": [ "spaces-more-morphisms-lemma-dimension-fibres-flat", "spaces-more-morphisms-lemma-dimension-fibres-proper" ], "ref_ids": [ 162, 163 ] } ], "ref_ids": [ 161 ] }, { "id": 165, "type": "theorem", "label": "spaces-more-morphisms-lemma-construct-glueing", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-construct-glueing", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an integral morphism", "of algebraic spaces over $S$.", "Let $y \\in |Y|$ be a point which can be represented by a closed immersion", "$y : \\Spec(k) \\to Y$. Then there exists", "a factorization $X \\to X' \\to Y$ of $f$ such that", "\\begin{enumerate}", "\\item $X' \\to Y$ is integral,", "\\item $X \\to X'$ is an isomorphism over $X' \\setminus X'_y$,", "\\item $X'_y$ has a unique point $x'$ with $\\kappa(x') = k$.", "\\end{enumerate}", "Moreover, if $f$ is finite and $Y$ is locally Noetherian, then", "$X' \\to Y$ is finite." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-pushforward}", "the sheaves $f_*\\mathcal{O}_X$, $(X_y \\to Y)_*\\mathcal{O}_{X_y}$, and", "$y_*\\mathcal{O}_{\\Spec(k)}$ are quasi-coherent sheaves of", "$\\mathcal{O}_Y$-algebras. Consider the maps", "$$", "f_*\\mathcal{O}_Y \\longrightarrow", "(X_y \\to Y)_*\\mathcal{O}_{X_y} \\longleftarrow", "y_*\\mathcal{O}_{\\Spec(k)}", "$$", "The fibre product is a quasi-coherent sheaf of $\\mathcal{O}_Y$-algebras", "$\\mathcal{A}'$ and we can define $X' \\to Y$ as the relative spectrum", "of $\\mathcal{A}'$ over $Y$, see", "Morphisms, Lemma \\ref{morphisms-lemma-affine-equivalence-algebras}.", "This construction commutes with arbitrary change of base.", "In particular, it is clear that over the open subspace", "$|Y| \\setminus \\{y\\}$ the morphism $X \\to X'$ is an isomorphism", "and over $|Y| \\setminus \\{y\\}$ the morphism $X' \\to Y$ is integral.", "It remains to prove the statements in a small neighbourhood of $y$.", "Choose an affine scheme $V = \\Spec(R)$ and an \\'etale", "morphism $\\varphi : V \\to Y$ such that $y$ is in the image of", "$\\varphi$. Then $V_y$ is a closed subscheme of $V$", "\\'etale over $k$, whence consists of finitely many points", "each with residue field separable over $k$", "(see Decent Spaces, Remark \\ref{decent-spaces-remark-recall}).", "After shrinking $V$", "we may assume there is a unique closed point $v = \\Spec(l) \\to V$", "mapping to $y$ with $l/k$ finite separable.", "We may write $V \\times_Y X = \\Spec(C)$ with $R \\to C$", "an integral ring map. The stated compatibility with", "base change gives us that $U \\times_X Y' = \\Spec(C')$ where", "$$", "C' = C \\times_{C \\otimes_R l} l", "$$", "Since $R \\to l$ is surjective, also $C \\to C \\otimes_R l$", "is surjective and we see that this is a fibre product of the", "kind studied in More on Algebra, Situation", "\\ref{more-algebra-situation-module-over-fibre-product}", "(with $A, A', B, B'$ corresponding to $C \\otimes_R l, C, l, C'$).", "Observe that $C'$ is an $R$-subalgebra of $C$ and hence", "is integral over $R$; this proves (1).", "Finally, More on Algebra, Lemma", "\\ref{more-algebra-lemma-points-of-fibre-product}", "shows that $V \\times_X Y' = \\Spec(C')$", "has a unique point $y''$ lying over $v$ with residue $l$", "(this corresponds with the obvious surjective map $C' \\to l$).", "Thus $X_y \\times_{\\Spec(k)} \\Spec(l)$ has a unique point", "with residue field $l$. Since $l/k$ is finite separable,", "this implies $X'_y$ has a unique point with residue field $k$, i.e.,", "(3) holds.", "\\medskip\\noindent", "To prove the final statement, observe that if $Y$ is locally Noetherian,", "then $R$ is a Noetherian ring and if $f$ is finite, then $R \\to C$ is", "finite. Then $C'$ is a finite type $R$-algebra by More on Algebra,", "Lemma \\ref{more-algebra-lemma-fibre-product-finite-type}.", "This proves that $X' \\to Y$ is finite." ], "refs": [ "spaces-morphisms-lemma-pushforward", "morphisms-lemma-affine-equivalence-algebras", "decent-spaces-remark-recall", "more-algebra-lemma-points-of-fibre-product", "more-algebra-lemma-fibre-product-finite-type" ], "ref_ids": [ 4760, 5173, 9573, 9818, 9814 ] } ], "ref_ids": [] }, { "id": 166, "type": "theorem", "label": "spaces-more-morphisms-lemma-universally-catenary-dimension-function", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-universally-catenary-dimension-function", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $\\delta : |B| \\to \\mathbf{Z}$ be a function.", "Assume $B$ is decent, locally Noetherian, and", "universally catenary and $\\delta$ is a dimension function.", "If $X$ is a decent algebraic space over $B$ whose structure morphism", "$f : X \\to B$ is locally of finite type we define", "$\\delta_X : |X| \\to \\mathbf{Z}$ by the rule", "$$", "\\delta_X(x) = \\delta(f(x)) + \\text{transcendence degreeof }x/f(x)", "$$", "(Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-dimension-fibre}).", "Then $\\delta_X$ is a dimension function." ], "refs": [ "spaces-morphisms-definition-dimension-fibre" ], "proofs": [ { "contents": [ "The problem is local on $B$. Thus we may assume $B$ is quasi-compact.", "By Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-locally-Noetherian-decent-quasi-separated}", "we see $B$ is quasi-separated. By", "Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-there-is-a-scheme-finite-over}", "we can choose a finite surjective morphism $\\pi : Y \\to X$", "where $Y$ is a scheme.", "Claim: $\\delta_Y$ is a dimension function.", "\\medskip\\noindent", "The claim implies the lemma. With $X \\to B$ as in the lemma", "set $Z = Y \\times_B X$ with projections $p : Z \\to Y$ and $q : Z \\to X$.", "Then we have", "$$", "\\delta_Z(z) = \\delta_Y(p(z)) + \\text{transcendence degreeof }z/p(z)", "$$", "and $\\delta_Z(z) = \\delta_X(q(z))$. This follows from", "Morphisms of Spaces, Lemma ", "\\ref{spaces-morphisms-lemma-dimension-fibre-at-a-point-additive}", "and the fact that these transcendence degrees are zero", "for finite morphisms. By Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-scheme-with-dimension-function}", "and the claim we find that $\\delta_Z$ is a dimension function.", "Then we find that $\\delta_X$ is a dimension function by", "Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-check-dimension-function-finite-cover}.", "\\medskip\\noindent", "Proof of the claim. Consider a specialization $y \\leadsto y'$,", "$y \\not = y'$ of points of the Noetherian scheme $Y$.", "Then $\\delta_Y(y) > \\delta_Y(y')$ because there are", "no specializations between points in fibres of $Y$", "(see Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-conditions-on-fibre-and-qf}).", "Using this for a chain of specializations we find", "$$", "\\delta_Y(y) - \\delta_Y(y') \\geq", "\\text{codim}(\\overline{\\{y'\\}}, \\overline{\\{y\\}})", "$$", "Our task is to show equality. By", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian-closed-point}", "we can choose a specialization $y' \\leadsto y_0$.", "It suffices to show", "$\\delta_Y(y) - \\delta_Y(y_0) =", "\\text{codim}(\\overline{\\{y_0\\}}, \\overline{\\{y\\}})$", "because this will imply the equality for both", "$y \\leadsto y'$ and $y' \\leadsto y_0$.", "\\medskip\\noindent", "Choose a maximal chain", "$y = y_c \\leadsto y_{c - 1} \\leadsto \\ldots \\leadsto y_0$", "of specializations in $Y$.", "Set $b = \\pi(y)$ and $b_0 = \\pi(y_0)$.", "Choose a maximal chain", "$b = b_e \\leadsto b_{e - 1} \\leadsto \\ldots \\leadsto b_0$", "of specializations in $|B|$.", "We have to show $e = c$.", "Since $\\pi$ is closed", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-proper})", "we can find a sequence of specializations", "$y = y'_e \\leadsto y'_{e - 1} \\leadsto \\ldots \\leadsto y'_0$", "mapping to", "$b = b_e \\leadsto b_{e - 1} \\leadsto \\ldots \\leadsto b_0$.", "Observe that $y'_e \\leadsto y'_{e - 1} \\leadsto \\ldots \\leadsto y'_0$", "is a maximal chain as well.", "If $y_0 = y'_0$, then because $Y$ is catenary, we conclude", "that $e = c$ as desired. In the next paragraph we reduce to", "this case by sleight of hand and we conclude in the same manner.", "\\medskip\\noindent", "Since $\\pi$ is closed we see that $b_0$ is a closed point of $|B|$.", "By Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-space-closed-point}", "we can represent $b_0$ by a closed immersion $b_0 : \\Spec(k) \\to B$.", "By Lemma \\ref{lemma-construct-glueing}", "we can find a factorization", "$$", "Y \\to Y' \\to X", "$$", "with $\\pi' : Y' \\to X$ finite and $Y \\to Y'$ a morphism which", "map $y_0$ and $y'_0$ to the same point and is an isomorphism", "away from the inverse image of $b_0$.", "(Of course $Y'$ won't be a scheme but this doesn't matter for the", "argument that follows.)", "Clearly the maximal chains of specializations", "$y_c \\leadsto y_{c - 1} \\leadsto \\ldots \\leadsto y_0$ and", "$y'_e \\leadsto y'_{e - 1} \\leadsto \\ldots \\leadsto y'_0$", "map to maximal chains of specializations in $Y'$ having", "the same start and end.", "Since $B$ is universally catenary, we see that", "$|Y'|$ is catenary and we conclude as before." ], "refs": [ "decent-spaces-lemma-locally-Noetherian-decent-quasi-separated", "spaces-limits-proposition-there-is-a-scheme-finite-over", "spaces-morphisms-lemma-dimension-fibre-at-a-point-additive", "decent-spaces-lemma-scheme-with-dimension-function", "decent-spaces-lemma-check-dimension-function-finite-cover", "decent-spaces-lemma-conditions-on-fibre-and-qf", "properties-lemma-locally-Noetherian-closed-point", "spaces-morphisms-lemma-finite-proper", "decent-spaces-lemma-decent-space-closed-point", "spaces-more-morphisms-lemma-construct-glueing" ], "ref_ids": [ 9506, 4659, 4871, 9554, 9557, 9529, 2958, 4946, 9510, 165 ] } ], "ref_ids": [ 5009 ] }, { "id": 167, "type": "theorem", "label": "spaces-more-morphisms-lemma-etale-splits-off-quasi-finite-part", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-etale-splits-off-quasi-finite-part", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $y \\in |Y|$. Let $x_1, \\ldots, x_n \\in |X|$ mapping to $y$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite type,", "\\item $f$ is separated,", "\\item $f$ is quasi-finite at $x_1, \\ldots, x_n$, and", "\\item $f$ is quasi-compact or $Y$ is decent.", "\\end{enumerate}", "Then there exists an \\'etale morphism $(U, u) \\to (Y, y)$", "of pointed algebraic spaces and a decomposition", "$$", "U \\times_Y X = W \\amalg V", "$$", "into open and closed subspaces such that the morphism $V \\to U$ is finite,", "every point of the fibre of $|V| \\to |U|$ over $u$ maps to an $x_i$,", "and the fibre of $|W| \\to |U|$ over $u$ contains no point mapping to an", "$x_i$." ], "refs": [], "proofs": [ { "contents": [ "Let $(U, u) \\to (Y, y)$ be an \\'etale morphism of algebraic spaces", "and consider the set of $w \\in |U \\times_Y X|$ mapping to $u \\in |U|$", "and one of the $x_i \\in |X|$. By", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-qf-and-qc-finite-fibre}", "(if $f$ is of finite type) or", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-finite-fibre}", "(if $Y$ is decent) this set is finite.", "It follows that we may replace $f$ by the base change", "$U \\times_Y X \\to U$ and $x_1, \\ldots, x_n$ by the set of these $w$.", "In particular we may and do assume that $Y$ is an affine scheme,", "whence $X$ is a separated algebraic space.", "\\medskip\\noindent", "Choose an affine scheme $Z$ and an \\'etale morphism $Z \\to X$ such that", "$x_1, \\ldots, x_n$ are in the image of $|Z| \\to |X|$. The fibres of", "$|Z| \\to |X|$ are finite, see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-finite-fibres-presentation}", "(or the more general discussion in Decent Spaces, Section", "\\ref{decent-spaces-section-reasonable-decent}).", "Let $\\{z_1, \\ldots, z_m\\} \\subset |Z|$ be the preimage of", "$\\{x_1, \\ldots, x_n\\}$. By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical}", "there exists an \\'etale morphism $(U, u) \\to (Y, y)$ such", "that $U \\times_Y Z = Z_1 \\amalg Z_2$ with $Z_1 \\to U$ finite and", "$(Z_1)_y = \\{z_1, \\ldots, z_m\\}$. We may assume that $U$ is affine", "and hence $Z_1$ is affine too.", "\\medskip\\noindent", "Since $f$ is separated, the image $V$ of $Z_1 \\to X$ is both open and closed", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-permanence}).", "Set $W = X \\setminus V$ to get a decomposition as in the lemma.", "To finish the proof we have to show that $V \\to U$ is finite.", "As $Z_1 \\to V$ is surjective and \\'etale, $V$ is the quotient of", "$Z_1$ by the \\'etale equivalence relation $R = Z_1 \\times_V Z_1$, see", "Spaces, Lemma \\ref{spaces-lemma-space-presentation}.", "Since $f$ is separated, $V \\to U$ is separated and $R$ is closed in", "$Z_1 \\times_U Z_1$. Since $Z_1 \\to U$ is finite,", "the projections $s, t : R \\to Z_1$ are finite.", "Thus $V$ is an affine scheme by", "Groupoids, Proposition \\ref{groupoids-proposition-finite-flat-equivalence}.", "By Morphisms, Lemma \\ref{morphisms-lemma-image-proper-is-proper}", "we conclude that $V \\to U$ is proper and by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}", "we conclude that $V \\to U$ is finite,", "thereby finishing the proof." ], "refs": [ "decent-spaces-lemma-qf-and-qc-finite-fibre", "decent-spaces-lemma-decent-finite-fibre", "spaces-properties-lemma-finite-fibres-presentation", "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-lemma-space-presentation", "groupoids-proposition-finite-flat-equivalence", "morphisms-lemma-image-proper-is-proper", "morphisms-lemma-finite-proper" ], "ref_ids": [ 9523, 9524, 11836, 13895, 4920, 8149, 9669, 5413, 5445 ] } ], "ref_ids": [] }, { "id": 168, "type": "theorem", "label": "spaces-more-morphisms-lemma-etale-splits-off-just-one-quasi-finite-part", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-etale-splits-off-just-one-quasi-finite-part", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $x \\in |X|$ with image $y \\in |Y|$. Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite type,", "\\item $f$ is separated, and", "\\item $f$ is quasi-finite at $x$.", "\\end{enumerate}", "Then there exists an \\'etale morphism $(U, u) \\to (Y, y)$", "of pointed algebraic spaces and a decomposition", "$$", "U \\times_Y X = W \\amalg V", "$$", "into open and closed subspaces such that the morphism $V \\to U$ is finite", "and there exists a point $v \\in |V|$ which maps to $x$ in $|X|$", "and $u$ in $|U|$." ], "refs": [], "proofs": [ { "contents": [ "Pick a scheme $U$, a point $u \\in U$, and an \\'etale morphism", "$U \\to Y$ mapping $u$ to $y$. There exists a point $x' \\in |U \\times_Y X|$", "mapping to $x$ in $|X|$ and $u$ in $|U|$ (Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-points-cartesian}).", "To finish, apply Lemma \\ref{lemma-etale-splits-off-quasi-finite-part}", "to the morphism $U \\times_Y X \\to U$ and the point $x'$.", "It applies because $U$ is a scheme and hence $u$ comes from", "the monomorphism $\\Spec(\\kappa(u)) \\to U$." ], "refs": [ "spaces-properties-lemma-points-cartesian", "spaces-more-morphisms-lemma-etale-splits-off-quasi-finite-part" ], "ref_ids": [ 11819, 167 ] } ], "ref_ids": [] }, { "id": 169, "type": "theorem", "label": "spaces-more-morphisms-lemma-finite-type-separated", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-finite-type-separated", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$ which is of finite type and separated.", "Let $Y'$ be the normalization of $Y$ in $X$. Picture:", "$$", "\\xymatrix{", "X \\ar[rd]_f \\ar[rr]_{f'} & & Y' \\ar[ld]^\\nu \\\\", "& Y &", "}", "$$", "Then there exists an open subspace $U' \\subset Y'$ such that", "\\begin{enumerate}", "\\item $(f')^{-1}(U') \\to U'$ is an isomorphism, and", "\\item $(f')^{-1}(U') \\subset X$ is the set of points at which", "$f$ is quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part}", "there is an open subspace $U \\subset X$ corresponding to the points", "of $|X|$ where $f$ is quasi-finite. We have to prove", "\\begin{enumerate}", "\\item[(a)] the image of $|U| \\to |Y'|$ is $|U'|$ for some open subspace", "$U'$ of $Y'$,", "\\item[(b)] $U = f^{-1}(U')$, and", "\\item[(c)] $U \\to U'$ is an isomorphism.", "\\end{enumerate}", "Since formation of $U$ commutes with arbitrary base change", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part}),", "since formation of the normalization $Y'$ commutes with smooth", "base change (Lemma \\ref{lemma-normalization-smooth-localization}),", "since \\'etale morphisms are open, and", "since ``being an isomorphism'' is fpqc local on the base", "(Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-isomorphism}),", "it suffices to prove (a), (b), (c) \\'etale locally on $Y$", "(some details omitted). Thus", "we may assume $Y$ is an affine scheme. This implies that $Y'$ is", "an (affine) scheme as well.", "\\medskip\\noindent", "Let $x \\in |U|$. Claim: there exists an open", "neighbourhood $f'(x) \\in V \\subset Y'$ such that $(f')^{-1}V \\to V$ is an", "isomorphism. We first prove the claim implies the lemma.", "Namely, then $(f')^{-1}V \\cong V$ is a scheme (as an open", "of $Y'$), locally of finite type over $Y$ (as an open subspace of $X$),", "and for $v \\in V$ the residue field extension", "$\\kappa(v) \\supset \\kappa(\\nu(v))$ is algebraic (as", "$V \\subset Y'$ and $Y'$ is integral over $Y$). Hence the fibres", "of $V \\to Y$ are discrete (Morphisms, Lemma", "\\ref{morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre})", "and $(f')^{-1}V \\to Y$ is locally quasi-finite", "(Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-fibres}).", "This implies $(f')^{-1}V \\subset U$ and $V \\subset U'$. Since $x$ was", "arbitrary we see that (a), (b), and (c) are true.", "\\medskip\\noindent", "Let $y = f(x) \\in |Y|$. Let $(T, t) \\to (Y, y)$ be an \\'etale morphism", "of pointed schemes. Denote by a subscript ${}_T$ the base change to $T$.", "Let $z \\in X_T$ be a point in the fibre $X_t$ lying over $x$.", "Note that $U_T \\subset X_T$ is the set of points where $f_T$ is", "quasi-finite, see Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part}.", "Note that", "$$", "X_T \\xrightarrow{f'_T} Y'_T \\xrightarrow{\\nu_T} T", "$$", "is the normalization of $T$ in $X_T$, see", "Lemma \\ref{lemma-normalization-smooth-localization}.", "Suppose that the claim holds for $z \\in U_T \\subset X_T \\to Y'_T \\to T$, i.e.,", "suppose that we can find an open neighbourhood", "$f'_T(z) \\in V' \\subset Y'_T$ such that $(f'_T)^{-1}V' \\to V'$ is an", "isomorphism. The morphism $Y'_T \\to Y'$ is \\'etale hence the image", "$V \\subset Y'$ of $V'$ is open. Observe that $f'(x) \\in V$ as $f'_T(z) \\in V'$.", "Observe that", "$$", "\\xymatrix{", "(f'_T)^{-1}V' \\ar[r] \\ar[d] & (f')^{-1}(V) \\ar[d] \\\\", "V' \\ar[r] & V", "}", "$$", "is a fibre square (as $Y'_T \\times_{Y'} X = X_T$).", "Since the left vertical arrow is an isomorphism", "and $\\{V' \\to V\\}$ is a \\'etale covering, we conclude that the right vertical", "arrow is an isomorphism by", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-isomorphism}.", "In other words, the claim holds for $x \\in U \\subset X \\to Y' \\to Y$.", "\\medskip\\noindent", "By the result of the previous paragraph to prove the claim for", "$x \\in |U|$, we may replace $Y$ by an \\'etale neighbourhood $T$ of", "$y = f(x)$ and $x$ by any point lying over $x$ in $T \\times_Y X$.", "Thus we may assume there is a decomposition", "$$", "X = V \\amalg W", "$$", "into open and closed subspaces where $V \\to Y$ is finite and $x \\in V$,", "see Lemma \\ref{lemma-etale-splits-off-quasi-finite-part}.", "Since $X$ is a disjoint union of $V$ and $W$ over $Y$ and since", "$V \\to Y$ is finite we see that the", "normalization of $Y$ in $X$ is the morphism", "$$", "X = V \\amalg W \\longrightarrow V \\amalg W' \\longrightarrow S", "$$", "where $W'$ is the normalization of $Y$ in $W$, see", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-normalization-in-disjoint-union},", "\\ref{spaces-morphisms-lemma-finite-integral}, and", "\\ref{spaces-morphisms-lemma-normalization-in-integral}.", "The claim follows and we win." ], "refs": [ "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part", "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part", "spaces-more-morphisms-lemma-normalization-smooth-localization", "spaces-descent-lemma-descending-property-isomorphism", "morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre", "morphisms-lemma-locally-quasi-finite-fibres", "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part", "spaces-more-morphisms-lemma-normalization-smooth-localization", "spaces-descent-lemma-descending-property-isomorphism", "spaces-more-morphisms-lemma-etale-splits-off-quasi-finite-part", "spaces-morphisms-lemma-normalization-in-disjoint-union", "spaces-morphisms-lemma-finite-integral", "spaces-morphisms-lemma-normalization-in-integral" ], "ref_ids": [ 4876, 4876, 139, 9395, 5222, 5228, 4876, 139, 9395, 167, 4962, 4943, 4964 ] } ], "ref_ids": [] }, { "id": 170, "type": "theorem", "label": "spaces-more-morphisms-lemma-quasi-finite-separated-quasi-affine", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-quasi-finite-separated-quasi-affine", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume $f$ is quasi-finite and separated.", "Let $Y'$ be the normalization of $Y$ in $X$.", "Picture:", "$$", "\\xymatrix{", "X \\ar[rd]_f \\ar[rr]_{f'} & & Y' \\ar[ld]^\\nu \\\\", "& Y &", "}", "$$", "Then $f'$ is a quasi-compact open immersion and $\\nu$ is integral.", "In particular $f$ is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-finite-type-separated}. Namely, by", "that lemma there exists an open subspace $U' \\subset Y'$ such that", "$(f')^{-1}(U') = X$ (!) and $X \\to U'$ is an isomorphism! In other", "words, $f'$ is an open immersion. Note that $f'$ is quasi-compact as", "$f$ is quasi-compact and $\\nu : Y' \\to Y$ is separated", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-permanence}).", "Hence for every affine scheme $Z$ and morphism $Z \\to Y$ the", "fibre product $Z \\times_Y X$ is a quasi-compact open subscheme", "of the affine scheme $Z \\times_Y Y'$. Hence $f$ is quasi-affine by", "definition." ], "refs": [ "spaces-more-morphisms-lemma-finite-type-separated", "spaces-morphisms-lemma-quasi-compact-permanence" ], "ref_ids": [ 169, 4743 ] } ], "ref_ids": [] }, { "id": 171, "type": "theorem", "label": "spaces-more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume $f$ is quasi-finite and separated and assume that", "$Y$ is quasi-compact and quasi-separated. Then there exists", "a factorization", "$$", "\\xymatrix{", "X \\ar[rd]_f \\ar[rr]_j & & T \\ar[ld]^\\pi \\\\", "& Y &", "}", "$$", "where $j$ is a quasi-compact open immersion and $\\pi$ is finite." ], "refs": [], "proofs": [ { "contents": [ "Let $X \\to Y' \\to Y$ be as in the conclusion of", "Lemma \\ref{lemma-quasi-finite-separated-quasi-affine}.", "By", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-integral-algebra-directed-colimit-finite}", "we can write", "$\\nu_*\\mathcal{O}_{Y'} = \\colim_{i \\in I} \\mathcal{A}_i$ as a", "directed colimit of finite quasi-coherent $\\mathcal{O}_X$-algebras", "$\\mathcal{A}_i \\subset \\nu_*\\mathcal{O}_{Y'}$. Then", "$\\pi_i : T_i = \\underline{\\Spec}_Y(\\mathcal{A}_i) \\to Y$", "is a finite morphism for each $i$.", "Note that the transition morphisms $T_{i'} \\to T_i$ are affine", "and that $Y' = \\lim T_i$.", "\\medskip\\noindent", "By Limits of Spaces, Lemma \\ref{spaces-limits-lemma-descend-opens}", "there exists an $i$ and a quasi-compact open", "$U_i \\subset T_i$ whose inverse image in $Y'$ equals", "$f'(X)$. For $i' \\geq i$ let $U_{i'}$ be the inverse image", "of $U_i$ in $T_{i'}$. Then $X \\cong f'(X) = \\lim_{i' \\geq i} U_{i'}$, see", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-directed-inverse-system-has-limit}.", "By", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-finite-type-eventually-closed} we see that", "$X \\to U_{i'}$ is a closed immersion for some $i' \\geq i$.", "(In fact $X \\cong U_{i'}$ for sufficiently", "large $i'$ but we don't need this.) Hence $X \\to T_{i'}$ is an immersion. By", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-factor-the-other-way}", "we can factor this as $X \\to T \\to T_{i'}$ where the first arrow", "is an open immersion and the second a closed immersion. Thus we win." ], "refs": [ "spaces-more-morphisms-lemma-quasi-finite-separated-quasi-affine", "spaces-limits-lemma-integral-algebra-directed-colimit-finite", "spaces-limits-lemma-descend-opens", "spaces-limits-lemma-directed-inverse-system-has-limit", "spaces-limits-lemma-finite-type-eventually-closed", "spaces-morphisms-lemma-factor-the-other-way" ], "ref_ids": [ 170, 4607, 4575, 4565, 4580, 4764 ] } ], "ref_ids": [] }, { "id": 172, "type": "theorem", "label": "spaces-more-morphisms-lemma-quasi-finite-separated-pass-through-finite-addendum", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-quasi-finite-separated-pass-through-finite-addendum", "contents": [ "With notation and hypotheses as in", "Lemma \\ref{lemma-quasi-finite-separated-pass-through-finite}.", "Assume moreover that $f$ is locally of finite presentation. Then we can", "choose the factorization such that $T$ is finite and of", "finite presentation over $Y$." ], "refs": [ "spaces-more-morphisms-lemma-quasi-finite-separated-pass-through-finite" ], "proofs": [ { "contents": [ "By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-finite-in-finite-and-finite-presentation} we can write", "$T = \\lim T_i$ where all $T_i$ are finite and of finite presentation", "over $Y$ and the transition morphisms $T_{i'} \\to T_i$ are closed", "immersions. By", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-descend-opens}", "there exists an $i$ and an open subscheme $U_i \\subset T_i$ whose inverse", "image in $T$ is $X$. By", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-finite-type-eventually-closed}", "we see that $X \\cong U_i$ for large enough $i$.", "Replacing $T$ by $T_i$ finishes the proof." ], "refs": [ "spaces-limits-lemma-finite-in-finite-and-finite-presentation", "spaces-limits-lemma-descend-opens", "spaces-limits-lemma-finite-type-eventually-closed" ], "ref_ids": [ 4612, 4575, 4580 ] } ], "ref_ids": [ 171 ] }, { "id": 173, "type": "theorem", "label": "spaces-more-morphisms-lemma-characterize-finite", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-characterize-finite", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is finite,", "\\item $f$ is proper and locally quasi-finite,", "\\item $f$ is proper and $|X_k|$ is a discrete space for every morphism", "$\\Spec(k) \\to Y$ where $k$ is a field,", "\\item $f$ is universally closed, separated, locally of finite type", "and $|X_k|$ is a discrete space for every morphism $\\Spec(k) \\to Y$", "where $k$ is a field.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have (1) $\\Rightarrow$ (2) by", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-finite-proper},", "\\ref{spaces-morphisms-lemma-finite-quasi-finite}.", "We have (2) $\\Rightarrow$ (3) by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-quasi-finite}.", "By definition (3) implies (4).", "\\medskip\\noindent", "Assume (4). Since $f$ is universally closed it is quasi-compact", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-quasi-compact}).", "Pick a point $y$ of $|Y|$. We represent $y$ by a", "morphism $\\Spec(k) \\to Y$. Note that $|X_k|$ is finite discrete", "as a quasi-compact discrete space. The map $|X_k| \\to |X|$ surjects", "onto the fibre of $|X| \\to |Y|$ over $y$", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}).", "By", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-quasi-finite-at-point}", "we see that $X \\to Y$ is quasi-finite at all the points of the fibre", "of $|X| \\to |Y|$ over $y$.", "Choose an elementary \\'etale neighbourhood $(U, u) \\to (Y, y)$", "and decomposition $X_U = V \\amalg W$ as in", "Lemma \\ref{lemma-etale-splits-off-quasi-finite-part}", "adapted to all the points of $|X|$ lying over $y$.", "Note that $W_u = \\emptyset$ because we used all the points", "in the fibre of $|X| \\to |Y|$ over $y$.", "Since $f$ is universally closed we see that", "the image of $|W|$ in $|U|$ is a closed set not containing $u$.", "After shrinking $U$ we may assume that $W = \\emptyset$.", "In other words we see that $X_U = V$ is finite over $U$.", "Since $y \\in |Y|$ was arbitrary", "this means there exists a family $\\{U_i \\to Y\\}$", "of \\'etale morphisms whose images cover $Y$ such that", "the base changes $X_{U_i} \\to U_i$ are finite.", "We conclude that $f$ is finite by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-integral-local}." ], "refs": [ "spaces-morphisms-lemma-finite-proper", "spaces-morphisms-lemma-finite-quasi-finite", "spaces-morphisms-lemma-locally-quasi-finite", "spaces-morphisms-lemma-universally-closed-quasi-compact", "spaces-properties-lemma-points-cartesian", "spaces-morphisms-lemma-quasi-finite-at-point", "spaces-more-morphisms-lemma-etale-splits-off-quasi-finite-part", "spaces-morphisms-lemma-integral-local" ], "ref_ids": [ 4946, 4945, 4833, 4749, 11819, 4877, 167, 4940 ] } ], "ref_ids": [] }, { "id": 174, "type": "theorem", "label": "spaces-more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $y \\in |Y|$. Assume", "\\begin{enumerate}", "\\item $f$ is proper, and", "\\item $f$ is quasi-finite at all $x \\in |X|$ lying over $y$", "(Decent Spaces, Lemma \\ref{decent-spaces-lemma-conditions-on-fibre-and-qf}).", "\\end{enumerate}", "Then there exists an open neighbourhood $V \\subset Y$ of $y$", "such that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite." ], "refs": [ "decent-spaces-lemma-conditions-on-fibre-and-qf" ], "proofs": [ { "contents": [ "By", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part} the", "set of points at which $f$ is quasi-finite is an open $U \\subset X$.", "Let $Z = X \\setminus U$. Then $y \\not \\in f(Z)$. Since $f$ is proper", "the set $f(Z) \\subset Y$ is closed. Choose any open neighbourhood", "$V \\subset Y$ of $y$ with $Z \\cap V = \\emptyset$. Then", "$f^{-1}(V) \\to V$ is locally quasi-finite and proper.", "Hence $f^{-1}(V) \\to V$ is finite by Lemma \\ref{lemma-characterize-finite}." ], "refs": [ "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part", "spaces-more-morphisms-lemma-characterize-finite" ], "ref_ids": [ 4876, 173 ] } ], "ref_ids": [ 9529 ] }, { "id": 175, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-proper-family-cannot-collapse-fibre", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-proper-family-cannot-collapse-fibre", "contents": [ "\\begin{slogan}", "Collapsing a fibre of a proper family forces nearby ones to collapse too.", "\\end{slogan}", "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X \\ar[rr]_h \\ar[rd]_f & & Y \\ar[ld]^g \\\\", "& B", "}", "$$", "be a commutative diagram of morphism of algebraic spaces over $S$.", "Let $b \\in B$ and let $\\Spec(k) \\to B$ be a morphism in the equivalence", "class of $b$. Assume", "\\begin{enumerate}", "\\item $X \\to B$ is a proper morphism,", "\\item $Y \\to B$ is separated and locally of finite type,", "\\item one of the following is true", "\\begin{enumerate}", "\\item the image of $|X_k| \\to |Y_k|$ is finite,", "\\item the image of $|f|^{-1}(\\{b\\})$ in $|Y|$ is finite", "and $B$ is decent.", "\\end{enumerate}", "\\end{enumerate}", "Then there is an open", "subspace $B' \\subset B$ containing $b$ such that $X_{B'} \\to Y_{B'}$", "factors through a closed subspace $Z \\subset Y_{B'}$ finite over $B'$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset Y$ be the scheme theoretic image of $h$, see", "Morphisms of Spaces, Section", "\\ref{spaces-morphisms-section-scheme-theoretic-image}.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-scheme-theoretic-image-is-proper}", "the morphism $X \\to Z$ is surjective and $Z \\to B$ is proper.", "Thus", "$$", "\\{x \\in |X|\\text{ lying over }b\\} \\to", "\\{z \\in |Z|\\text{ lying over }b\\}", "$$", "and $|X_k| \\to |Z_k|$ are surjective. We see that either", "(3)(a) or (3)(b) imply that $Z \\to B$ is quasi-finite", "all points of $|Z|$ lying over $b$ by", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-conditions-on-fibre-and-qf}.", "Hence $Z \\to B$ is finite in an open neighbourhood of $b$ by", "Lemma \\ref{lemma-proper-finite-fibre-finite-in-neighbourhood}." ], "refs": [ "spaces-morphisms-lemma-scheme-theoretic-image-is-proper", "decent-spaces-lemma-conditions-on-fibre-and-qf", "spaces-more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood" ], "ref_ids": [ 4922, 9529, 174 ] } ], "ref_ids": [] }, { "id": 176, "type": "theorem", "label": "spaces-more-morphisms-lemma-stein-universally-closed", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-stein-universally-closed", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a universally closed and", "quasi-separated morphism of algebraic spaces over $S$.", "There exists a factorization", "$$", "\\xymatrix{", "X \\ar[rr]_{f'} \\ar[rd]_f & & Y' \\ar[dl]^\\pi \\\\", "& Y &", "}", "$$", "with the following properties:", "\\begin{enumerate}", "\\item the morphism $f'$ is universally closed, quasi-compact, quasi-separated,", "and surjective,", "\\item the morphism $\\pi : Y' \\to Y$ is integral,", "\\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{Y'}$,", "\\item we have $Y' = \\underline{\\Spec}_Y(f_*\\mathcal{O}_X)$, and", "\\item $Y'$ is the normalization of $Y$ in $X$ as defined in", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-normalization-X-in-Y}.", "\\end{enumerate}", "Formation of the factorization $f = \\pi \\circ f'$ commutes with flat", "base change." ], "refs": [ "spaces-morphisms-definition-normalization-X-in-Y" ], "proofs": [ { "contents": [ "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-quasi-compact}", "the morphism $f$ is quasi-compact.", "We just define $Y'$ as the normalization of $Y$ in $X$, so (5) and (2) hold", "automatically. By", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-normalization-in-universally-closed}", "we see that (4) holds. The morphism $f'$ is universally closed by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-permanence}.", "It is quasi-compact by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-permanence}", "and quasi-separated by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-compose-after-separated}.", "\\medskip\\noindent", "To show the remaining statements we may assume the base $Y$ is affine", "(as taking normalization commutes with \\'etale localization).", "Say $Y = \\Spec(R)$. Then $Y' = \\Spec(A)$ with", "$A = \\Gamma(X, \\mathcal{O}_X)$ an integral $R$-algebra.", "Thus it is clear that $f'_*\\mathcal{O}_X$", "is $\\mathcal{O}_{Y'}$ (because $f'_*\\mathcal{O}_X$ is quasi-coherent,", "by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-pushforward},", "and hence equal to $\\widetilde{A}$). This proves (3).", "\\medskip\\noindent", "Let us show that $f'$ is surjective. As $f'$ is universally closed (see above)", "the image of $f'$ is a closed subset", "$V(I) \\subset Y' = \\Spec(A)$. Pick $h \\in I$. Then", "$h|_X = f^\\sharp(h)$ is a global section of the structure sheaf of", "$X$ which vanishes at every point. As $X$ is quasi-compact this means", "that $h|_X$ is a nilpotent section, i.e., $h^n|X = 0$ for some $n > 0$.", "But $A = \\Gamma(X, \\mathcal{O}_X)$, hence $h^n = 0$.", "In other words $I$ is contained in the Jacobson radical of $A$ and we conclude", "that $V(I) = Y'$ as desired." ], "refs": [ "spaces-morphisms-lemma-universally-closed-quasi-compact", "spaces-morphisms-lemma-normalization-in-universally-closed", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-morphisms-lemma-quasi-compact-permanence", "spaces-morphisms-lemma-compose-after-separated", "spaces-morphisms-lemma-pushforward" ], "ref_ids": [ 4749, 4963, 4920, 4743, 4720, 4760 ] } ], "ref_ids": [ 5026 ] }, { "id": 177, "type": "theorem", "label": "spaces-more-morphisms-lemma-stein-universally-closed-residue-fields", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-stein-universally-closed-residue-fields", "contents": [ "In Lemma \\ref{lemma-stein-universally-closed} assume in addition that", "$f$ is locally of finite type and $Y$ affine. Then for $y \\in Y$ the fibre", "$\\pi^{-1}(\\{y\\}) = \\{y_1, \\ldots, y_n\\}$ is finite and the field extensions", "$\\kappa(y_i)/\\kappa(y)$ are finite." ], "refs": [ "spaces-more-morphisms-lemma-stein-universally-closed" ], "proofs": [ { "contents": [ "Recall that there are no specializations among the points of $\\pi^{-1}(\\{y\\})$,", "see Algebra, Lemma \\ref{algebra-lemma-integral-no-inclusion}.", "As $f'$ is surjective, we find that $|X_y| \\to \\pi^{-1}(\\{y\\})$ is surjective.", "Observe that $X_y$ is a quasi-separated algebraic space of finite type", "over a field (quasi-compactness was shown in the proof of the", "referenced lemma). Thus $|X_y|$ is a Noetherian topological space", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-noetherian}).", "A topological argument (omitted) now shows that $\\pi^{-1}(\\{y\\})$ is finite.", "For each $i$ we can pick a finite type point $x_i \\in |X_y|$", "mapping to $y_i$ (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-enough-finite-type-points}).", "We conclude that $\\kappa(y_i)/\\kappa(y)$ is finite:", "$x_i$ can be represented by a morphism $\\Spec(k_i) \\to X_y$", "of finite type (by our definition of finite type points)", "and hence $\\Spec(k_i) \\to y = \\Spec(\\kappa(y))$ is of finite type", "(as a composition of finite type morphisms),", "hence $k_i/\\kappa(y)$ is finite (Morphisms, Lemma", "\\ref{morphisms-lemma-point-finite-type})." ], "refs": [ "algebra-lemma-integral-no-inclusion", "spaces-morphisms-lemma-finite-presentation-noetherian", "spaces-morphisms-lemma-enough-finite-type-points", "morphisms-lemma-point-finite-type" ], "ref_ids": [ 498, 4843, 4826, 5205 ] } ], "ref_ids": [ 176 ] }, { "id": 178, "type": "theorem", "label": "spaces-more-morphisms-lemma-characterize-geometrically-connected-fibres", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-characterize-geometrically-connected-fibres", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $\\overline{y}$ be a geometric point of $Y$. Then", "$X_{\\overline{y}}$ is connected, if and only if for every \\'etale", "neighbourhood $(V, \\overline{v}) \\to (Y, \\overline{y})$ where $V$", "is a scheme the base change $X_V \\to V$ has connected fibre $X_v$." ], "refs": [], "proofs": [ { "contents": [ "Since the category of \\'etale neighbourhoods of $\\overline{y}$ is", "cofiltered and contains a cofinal collection of schemes", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-cofinal-etale})", "we may replace $Y$ by one of these neighbourhoods and assume that", "$Y$ is a scheme. Let $y \\in Y$ be the point corresponding to $\\overline{y}$.", "Then $X_y$ is geometrically connected over $\\kappa(y)$ if and only if", "$X_{\\overline{y}}$ is connected and if and only if $(X_y)_{k'}$", "is connected for every finite separable extension $k'$ of $\\kappa(y)$.", "See Spaces over Fields, Section", "\\ref{spaces-over-fields-section-geometrically-connected} and especially", "Lemma \\ref{spaces-over-fields-lemma-characterize-geometrically-disconnected}.", "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-realize-prescribed-residue-field-extension-etale}", "there exists an affine \\'etale neighbourhood $(V, v) \\to (Y, y)$ such that", "$\\kappa(s) \\subset \\kappa(u)$ is identified with $\\kappa(s) \\subset k'$", "any given finite separable extension. The lemma follows." ], "refs": [ "spaces-properties-lemma-cofinal-etale", "spaces-over-fields-lemma-characterize-geometrically-disconnected", "more-morphisms-lemma-realize-prescribed-residue-field-extension-etale" ], "ref_ids": [ 11870, 12866, 13866 ] } ], "ref_ids": [] }, { "id": 179, "type": "theorem", "label": "spaces-more-morphisms-lemma-geometrically-connected-fibres-towards-normal", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-geometrically-connected-fibres-towards-normal", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. Assume", "\\begin{enumerate}", "\\item $f$ is proper,", "\\item $Y$ is integral (Spaces over Fields, Definition", "\\ref{spaces-over-fields-definition-integral-algebraic-space})", "with generic point $\\xi$,", "\\item $Y$ is normal,", "\\item $X$ is reduced,", "\\item every generic point of an irreducible component of $|X|$ maps to $\\xi$,", "\\item we have $H^0(X_\\xi, \\mathcal{O}) = \\kappa(\\xi)$.", "\\end{enumerate}", "Then $f_*\\mathcal{O}_X = \\mathcal{O}_Y$ and $f$", "has geometrically connected fibres." ], "refs": [ "spaces-over-fields-definition-integral-algebraic-space" ], "proofs": [ { "contents": [ "Apply Theorem \\ref{theorem-stein-factorization-general} to get a", "factorization $X \\to Y' \\to Y$. It is enough to show that $Y' = Y$.", "It suffices to show that $Y' \\times_Y V \\to V$ is an isomorphism,", "where $V \\to Y$ is an \\'etale morphism and $V$ an affine integral scheme,", "see Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-normal-integral-cover-by-affines}.", "The formation of $Y'$ commutes with \\'etale base change, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-properties-normalization}.", "The generic points of $X \\times_Y V$ lie over the generic points of $X$", "(Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-generic-points})", "hence map to the generic point of $V$ by assumption (5). Moreover, condition", "(6) is preserved under the base change by $V \\to Y$, for example by", "flat base change (Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-flat-base-change-cohomology}).", "Thus it suffices to prove the lemma in case $Y$ is a normal", "integral affine scheme.", "\\medskip\\noindent", "Assume $Y$ is a normal integral affine scheme. We will show $Y' \\to Y$", "is an isomorphism by an application of Morphisms, Lemma", "\\ref{morphisms-lemma-finite-birational-over-normal}.", "Namely, $Y'$ is reduced because $X$ is reduced (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-normalization-in-reduced}).", "The morphism $Y' \\to Y$ is integral by the theorem cited above.", "Since $Y$ is decent and $X \\to Y$ is separated, we see that", "$X$ is decent too; to see this use", "Decent Spaces, Lemmas", "\\ref{decent-spaces-lemma-properties-trivial-implications} and", "\\ref{decent-spaces-lemma-property-over-property}. By assumption (5),", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-normalization-generic},", "and Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-generic-points}", "we see that every generic point of an irreducible component of $|Y'|$", "maps to $\\xi$. On the other hand, since $Y'$ is the relative", "spectrum of $f_*\\mathcal{O}_X$ we see that the scheme theoretic fibre", "$Y'_\\xi$ is the spectrum of $H^0(X_\\xi, \\mathcal{O})$ which is", "equal to $\\kappa(\\xi)$ by assumption. Hence $Y'$ is an integral", "scheme with function field equal to the function field of $Y$.", "This finishes the proof." ], "refs": [ "spaces-more-morphisms-theorem-stein-factorization-general", "spaces-over-fields-lemma-normal-integral-cover-by-affines", "spaces-morphisms-lemma-properties-normalization", "decent-spaces-lemma-decent-generic-points", "spaces-cohomology-lemma-flat-base-change-cohomology", "morphisms-lemma-finite-birational-over-normal", "spaces-morphisms-lemma-normalization-in-reduced", "decent-spaces-lemma-properties-trivial-implications", "decent-spaces-lemma-property-over-property", "spaces-morphisms-lemma-normalization-generic", "decent-spaces-lemma-decent-generic-points" ], "ref_ids": [ 14, 12826, 4958, 9531, 11296, 5518, 4960, 9513, 9516, 4961, 9531 ] } ], "ref_ids": [ 12883 ] }, { "id": 180, "type": "theorem", "label": "spaces-more-morphisms-lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous", "contents": [ "Let $S$ be a scheme.", "Let $X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $f$ is proper, flat, and of finite presentation, then the function", "$n_{X/Y} : |Y| \\to \\mathbf{Z}$ counting the number of geometric", "connected components of fibres of $f$", "(Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components})", "is lower semi-continuous." ], "refs": [ "spaces-more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components" ], "proofs": [ { "contents": [ "The question is \\'etale local on $Y$, hence we may and do assume $Y$", "is an affine scheme. Let $y \\in Y$. Set $n = n_{X/S}(y)$.", "Note that $n < \\infty$ as the geometric fibre of $X \\to Y$ at $y$", "is a proper algebraic space over a field, hence Noetherian, hence", "has a finite number of connected components.", "We have to find an open neighbourhood $V$ of $y$ such that $n_{X/S}|_V \\geq n$.", "Let $X \\to Y' \\to Y$ be the Stein factorization as in", "Theorem \\ref{theorem-stein-factorization-general}.", "By Lemma \\ref{lemma-stein-universally-closed-residue-fields}", "there are finitely many points", "$y'_1, \\ldots, y'_m \\in Y'$ lying over $y$", "and the extensions $\\kappa(y'_i)/\\kappa(y)$ are finite.", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-etale-makes-integral-split}", "tells us that after replacing $Y$ by an \\'etale neighbourhood", "of $y$ we may assume $Y' = V_1 \\amalg \\ldots \\amalg V_m$ as a scheme", "with $y'_i \\in V_i$ and $\\kappa(y'_i)/\\kappa(y)$ purely inseparable.", "Then the algebraic spaces $X_{y_i'}$ are geometrically connected over", "$\\kappa(y)$, hence $m = n$. The algebraic spaces", "$X_i = (f')^{-1}(V_i)$, $i = 1, \\ldots, n$", "are flat and of finite presentation over $Y$. Hence the image of $X_i \\to Y$", "is open (Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-fppf-open}).", "Thus in a neighbourhood of $y$ we see that $n_{X/Y}$ is", "at least $n$." ], "refs": [ "spaces-more-morphisms-theorem-stein-factorization-general", "spaces-more-morphisms-lemma-stein-universally-closed-residue-fields", "more-morphisms-lemma-etale-makes-integral-split", "spaces-morphisms-lemma-fppf-open" ], "ref_ids": [ 14, 177, 13898, 4855 ] } ], "ref_ids": [ 159 ] }, { "id": 181, "type": "theorem", "label": "spaces-more-morphisms-lemma-proper-flat-geom-red", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-proper-flat-geom-red", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume", "\\begin{enumerate}", "\\item $f$ is proper, flat, and of finite presentation, and", "\\item the geometric fibres of $f$ are reduced.", "\\end{enumerate}", "Then the function $n_{X/S} : |Y| \\to \\mathbf{Z}$", "counting the numbers of geometric connected components", "of fibres of $f$", "(Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components})", "is locally constant." ], "refs": [ "spaces-more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous}", "the function $n_{X/Y}$ is lower semincontinuous.", "Thus it suffices to show it is upper semi-continuous.", "To do this we may work \\'etale locally on $Y$, hence we", "may assume $Y$ is an affine scheme.", "For $y \\in Y$ consider the $\\kappa(y)$-algebra", "$$", "A = H^0(X_y, \\mathcal{O}_{X_y})", "$$", "By Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-proper-geometrically-reduced-global-sections}", "and the fact that $X_y$ is geometrically reduced", "$A$ is finite product of finite separable extensions of $\\kappa(y)$.", "Hence $A \\otimes_{\\kappa(y)} \\kappa(\\overline{y})$ is a product", "of $\\beta_0(y) = \\dim_{\\kappa(y)} A$ copies of $\\kappa(\\overline{y})$.", "Thus $X_{\\overline{y}}$ has $\\beta_0(y)$ connected components.", "In other words, we have $n_{X/S} = \\beta_0$ as functions on $Y$.", "Thus $n_{X/Y}$ is upper semi-continuous by", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-jump-loci-geometric}.", "This finishes the proof." ], "refs": [ "spaces-more-morphisms-lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous", "spaces-over-fields-lemma-proper-geometrically-reduced-global-sections", "spaces-perfect-lemma-jump-loci-geometric" ], "ref_ids": [ 180, 12868, 2744 ] } ], "ref_ids": [ 159 ] }, { "id": 182, "type": "theorem", "label": "spaces-more-morphisms-lemma-stein-factorization-etale", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-stein-factorization-etale", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a proper morphism of algebraic spaces over $S$.", "Let $X \\to Y' \\to Y$ be the Stein factorization of $f$", "(Theorem \\ref{theorem-stein-factorization-general}).", "If $f$ is of finite presentation, flat, with geometrically", "reduced fibres (Definition \\ref{definition-geometrically-reduced-fibre}),", "then $Y' \\to Y$ is finite \\'etale." ], "refs": [ "spaces-more-morphisms-theorem-stein-factorization-general", "spaces-more-morphisms-definition-geometrically-reduced-fibre" ], "proofs": [ { "contents": [ "Formation of the Stein factorization commutes with flat base change,", "see Lemma \\ref{lemma-stein-universally-closed}.", "Thus we may work \\'etale locally on $Y$ and we may assume $Y$", "is an affine scheme. Then $Y'$ is an affine scheme and $Y' \\to Y$", "is integral.", "\\medskip\\noindent", "Let $y \\in Y$. Set $n$ be the number of connected components of", "the geometric fibre $X_{\\overline{y}}$. Note that $n < \\infty$", "as the geometric fibre of $X \\to Y$ at $y$ is a proper", "algebraic space over a field, hence Noetherian,", "hence has a finite number of connected components.", "By Lemma \\ref{lemma-stein-universally-closed-residue-fields}", "there are finitely many points $y'_1, \\ldots, y'_m \\in Y'$ lying over $y$", "and for each $i$ we can pick a finite type point $x_i \\in |X_y|$", "mapping to $y'_i$ the extension $\\kappa(y'_i)/\\kappa(y)$ is finite.", "Thus More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-etale-makes-integral-split}", "tells us that after replacing $Y$ by an \\'etale neighbourhood", "of $y$ we may assume $Y' = V_1 \\amalg \\ldots \\amalg V_m$ as a scheme", "with $y'_i \\in V_i$ and $\\kappa(y'_i)/\\kappa(y)$ purely inseparable.", "In this case the algebraic spaces $X_{y_i'}$", "are geometrically connected over $\\kappa(y)$, hence $m = n$.", "The algebraic spaces $X_i = (f')^{-1}(V_i)$, $i = 1, \\ldots, n$", "are proper, flat, of finite presentation, with geometrically", "reduced fibres over $Y$. It suffices to prove the lemma", "for each of the morphisms $X_i \\to Y$. This reduces us to the case where", "$X_{\\overline{y}}$ is connected.", "\\medskip\\noindent", "Assume that $X_{\\overline{y}}$ is connected. By", "Lemma \\ref{lemma-proper-flat-geom-red}", "we see that $X \\to Y$ has geometrically connected", "fibres in a neighbourhood of $y$. Thus", "we may assume the fibres of $X \\to Y$ are geometrically connected.", "Then $f_*\\mathcal{O}_X = \\mathcal{O}_Y$ by", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-proper-flat-geom-red-connected}", "which finishes the proof." ], "refs": [ "spaces-more-morphisms-lemma-stein-universally-closed", "spaces-more-morphisms-lemma-stein-universally-closed-residue-fields", "more-morphisms-lemma-etale-makes-integral-split", "spaces-more-morphisms-lemma-proper-flat-geom-red", "spaces-perfect-lemma-proper-flat-geom-red-connected" ], "ref_ids": [ 176, 177, 13898, 181, 2750 ] } ], "ref_ids": [ 14, 294 ] }, { "id": 183, "type": "theorem", "label": "spaces-more-morphisms-lemma-split-off-proper-part-henselian", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-split-off-proper-part-henselian", "contents": [ "Let $(A, I)$ be a henselian pair. Let $X$ be an algebraic space", "separated and of finite type over $A$. Set", "$X_0 = X \\times_{\\Spec(A)} \\Spec(A/I)$.", "Let $Y \\subset X_0$ be an open and closed subspace such that", "$Y \\to \\Spec(A/I)$ is proper. Then there exists an open and closed", "subspace $W \\subset X$ which is proper over $A$ with", "$W \\times_{\\Spec(A)} \\Spec(A/I) = Y$." ], "refs": [], "proofs": [ { "contents": [ "We will denote $T \\mapsto T_0$ the base change by", "$\\Spec(A/I) \\to \\Spec(A)$.", "By a weak version of Chow's lemma (in the form of", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-weak-chow})", "there exists a surjective proper morphism $\\varphi : X' \\to X$ such", "that $X'$ admits an immersion into $\\mathbf{P}^n_A$.", "Set $Y' = \\varphi^{-1}(Y)$. This is an open and closed subscheme", "of $X'_0$. The lemma holds for $(X', Y')$ by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-split-off-proper-part-henselian}.", "Let $W' \\subset X'$ be the open and closed subscheme proper", "over $A$ such that $Y' = W'_0$.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-permanence}", "$Q_1 = \\varphi(|W'|) \\subset |X|$ and", "$Q_2 = \\varphi(|X' \\setminus W'|) \\subset |X|$", "are closed subsets and by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-image-proper-is-proper}", "any closed subspace structure on $Q_1$ is proper over $A$.", "The image of $Q_1 \\cap Q_2$ in $\\Spec(A)$ is closed.", "Since $(A, I)$ is henselian, if $Q_1 \\cap Q_2$ is nonempty, then we", "find that $Q_1 \\cap Q_2$ has a point lying over $\\Spec(A/I)$.", "This is impossible as $W'_0 = Y' = \\varphi^{-1}(Y)$.", "We conclude that $Q_1$ is open and closed in $|X|$.", "Let $W \\subset X$ be the corresponding open and closed", "subspace. Then $W$ is proper over $A$ with $W_0 = Y$." ], "refs": [ "spaces-cohomology-lemma-weak-chow", "more-morphisms-lemma-split-off-proper-part-henselian", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-morphisms-lemma-image-proper-is-proper" ], "ref_ids": [ 11327, 13948, 4920, 4921 ] } ], "ref_ids": [] }, { "id": 184, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-finite-type-finitely-presented-over-dense-open", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-finite-type-finitely-presented-over-dense-open", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $V \\subset Y$ be an open subspace. Assume", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $\\mathcal{F}$ is of finite type and flat over $Y$,", "\\item $V \\to Y$ is quasi-compact and scheme theoretically dense,", "\\item $\\mathcal{F}|_{f^{-1}V}$ is of finite presentation.", "\\end{enumerate}", "Then $\\mathcal{F}$ is of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove the pullback of $\\mathcal{F}$ to a scheme surjective", "and \\'etale over $X$ is of finite presentation. Hence we may assume $X$", "is a scheme. Similarly, we can replace $Y$ by a scheme surjective and", "\\'etale and over $Y$ (the inverse image of $V$ in this scheme is scheme", "theoretically dense, see", "Morphisms of Spaces, Section", "\\ref{spaces-morphisms-section-scheme-theoretic-closure}).", "Thus we reduce to the case of schemes which is", "More on Flatness, Lemma", "\\ref{flat-lemma-flat-finite-type-finitely-presented-over-dense-open}." ], "refs": [ "flat-lemma-flat-finite-type-finitely-presented-over-dense-open" ], "ref_ids": [ 6024 ] } ], "ref_ids": [] }, { "id": 185, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-finite-type-finitely-presented-over-dense-open-X", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-finite-type-finitely-presented-over-dense-open-X", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $V \\subset Y$ be an open subspace.", "Assume", "\\begin{enumerate}", "\\item $f$ is locally of finite type and flat,", "\\item $V \\to Y$ is quasi-compact and scheme theoretically dense,", "\\item $f|_{f^{-1}V} : f^{-1}V \\to V$ is locally of finite presentation.", "\\end{enumerate}", "Then $f$ is of locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Lemma \\ref{lemma-flat-finite-type-finitely-presented-over-dense-open}", "except one uses", "More on Flatness, Lemma", "\\ref{flat-lemma-flat-finite-type-finitely-presented-over-dense-open-X}." ], "refs": [ "spaces-more-morphisms-lemma-flat-finite-type-finitely-presented-over-dense-open", "flat-lemma-flat-finite-type-finitely-presented-over-dense-open-X" ], "ref_ids": [ 184, 6025 ] } ], "ref_ids": [] }, { "id": 186, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-fp-dimension-over-dense-open", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-fp-dimension-over-dense-open", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ which is flat and locally of finite type. Let $V \\subset Y$ be an", "open subspace such that $|V| \\subset |Y|$ is dense and such that $X_V \\to V$", "has relative dimension $\\leq d$. If also either", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation, or", "\\item $V \\to Y$ is quasi-compact,", "\\end{enumerate}", "then $f : X \\to Y$ has relative dimension $\\leq d$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $Y$ by its reduction, hence we may assume $Y$ is reduced.", "Then $V$ is scheme theoretically dense in $Y$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-immersion}.", "By definition the property of having relative dimension $\\leq d$ can", "be checked on an \\'etale covering, see", "Morphisms of Spaces, Sections \\ref{spaces-morphisms-section-relative-dimension}.", "Thus it suffices to prove $f$ has relative dimension $\\leq d$", "after replacing $X$ by a scheme surjective and \\'etale over $X$.", "Similarly, we can replace $Y$ by a scheme surjective and", "\\'etale and over $Y$. The inverse image of $V$ in this scheme is scheme", "theoretically dense, see", "Morphisms of Spaces, Section", "\\ref{spaces-morphisms-section-scheme-theoretic-closure}.", "Since a scheme theoretically dense open of a scheme is in particular", "dense, we reduce to the case of schemes which is", "More on Flatness, Lemma", "\\ref{flat-lemma-flat-finite-presentation-dimension-over-dense-open}." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-immersion", "flat-lemma-flat-finite-presentation-dimension-over-dense-open" ], "ref_ids": [ 4790, 6026 ] } ], "ref_ids": [] }, { "id": 187, "type": "theorem", "label": "spaces-more-morphisms-lemma-proper-flat-finite-over-dense-open", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-proper-flat-finite-over-dense-open", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ which is flat and proper. Let $V \\to Y$ be an open subspace", "with $|V| \\subset |Y|$ dense such that $X_V \\to V$ is finite. If also", "either $f$ is locally of finite presentation or $V \\to Y$ is quasi-compact,", "then $f$ is finite." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-flat-fp-dimension-over-dense-open}", "the fibres of $f$ have dimension zero.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0}", "this implies that $f$ is locally quasi-finite.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable}", "this implies that $f$ is representable.", "We can check whether $f$ is finite \\'etale locally on $Y$,", "hence we may assume $Y$ is a scheme. Since $f$ is representable,", "we reduce to the case of schemes which is", "More on Flatness, Lemma \\ref{flat-lemma-proper-flat-finite-over-dense-open}." ], "refs": [ "spaces-more-morphisms-lemma-flat-fp-dimension-over-dense-open", "spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0", "spaces-morphisms-lemma-locally-quasi-finite-separated-representable", "flat-lemma-proper-flat-finite-over-dense-open" ], "ref_ids": [ 186, 4875, 4972, 6027 ] } ], "ref_ids": [] }, { "id": 188, "type": "theorem", "label": "spaces-more-morphisms-lemma-zariski", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-zariski", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $V \\subset Y$ be an open subspace. If", "\\begin{enumerate}", "\\item $f$ is separated, locally of finite type, and flat,", "\\item $f^{-1}(V) \\to V$ is an isomorphism, and", "\\item $V \\to Y$ is quasi-compact and scheme theoretically dense,", "\\end{enumerate}", "then $f$ is an open immersion." ], "refs": [], "proofs": [ { "contents": [ "Applying", "Lemma \\ref{lemma-flat-finite-type-finitely-presented-over-dense-open-X}", "we see that $f$ is locally of finite presentation. Applying", "Lemma \\ref{lemma-flat-fp-dimension-over-dense-open}", "we see that $f$ has relative dimension $\\leq 0$.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0}", "this implies that $f$ is locally quasi-finite.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable}", "this implies that $f$ is representable.", "By Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-open-immersion}", "we can check whether $f$ is an open immersion \\'etale locally on $Y$.", "Hence we may assume that $Y$ is a scheme. Since $f$ is representable,", "we reduce to the case of schemes which is", "More on Flatness, Lemma \\ref{flat-lemma-zariski}." ], "refs": [ "spaces-more-morphisms-lemma-flat-finite-type-finitely-presented-over-dense-open-X", "spaces-more-morphisms-lemma-flat-fp-dimension-over-dense-open", "spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0", "spaces-morphisms-lemma-locally-quasi-finite-separated-representable", "spaces-descent-lemma-descending-property-open-immersion", "flat-lemma-zariski" ], "ref_ids": [ 185, 186, 4875, 4972, 9394, 6028 ] } ], "ref_ids": [] }, { "id": 189, "type": "theorem", "label": "spaces-more-morphisms-lemma-push-ideal", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-push-ideal", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $\\varphi : W \\to X$ be a quasi-compact", "separated \\'etale morphism. Let $U \\subset X$ be a quasi-compact open", "subspace. Let $\\mathcal{I} \\subset \\mathcal{O}_U$ be a finite type", "quasi-coherent sheaf of ideals such that", "$V(\\mathcal{I}) \\cap \\varphi^{-1}(U) = \\emptyset$.", "Then there exists a finite type quasi-coherent sheaf of ideals", "$\\mathcal{J} \\subset \\mathcal{O}_X$ such that", "\\begin{enumerate}", "\\item $V(\\mathcal{J}) \\cap U = \\emptyset$, and", "\\item $\\varphi^{-1}(\\mathcal{J})\\mathcal{O}_W = \\mathcal{I} \\mathcal{I}'$", "for some finite type quasi-coherent ideal", "$\\mathcal{I}' \\subset \\mathcal{O}_W$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a factorization $W \\to Y \\to X$ where $j : W \\to Y$ is a", "quasi-compact open immersion and $\\pi : Y \\to X$ is a finite", "morphism of finite presentation", "(Lemma \\ref{lemma-quasi-finite-separated-pass-through-finite-addendum}).", "Let $V = j(W) \\cup \\pi^{-1}(U) \\subset Y$. Note that", "$\\mathcal{I}$ on $W \\cong j(W)$ and $\\mathcal{O}_{\\pi^{-1}(U)}$", "glue to a finite type quasi-coherent sheaf of ideals", "$\\mathcal{I}_1 \\subset \\mathcal{O}_V$. By", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-extend}", "there exists a finite type quasi-coherent sheaf of ideals", "$\\mathcal{I}_2 \\subset \\mathcal{O}_Y$ such that", "$\\mathcal{I}_2|_V = \\mathcal{I}_1$. In other words,", "$\\mathcal{I}_2 \\subset \\mathcal{O}_Y$", "is a finite type quasi-coherent sheaf of ideals such that", "$V(\\mathcal{I}_2)$ is disjoint from $\\pi^{-1}(U)$ and", "$j^{-1}\\mathcal{I}_2 = \\mathcal{I}$. Denote $i : Z \\to Y$", "the corresponding closed immersion which is of finite presentation", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-closed-immersion-finite-presentation}).", "In particular the composition $\\tau = \\pi \\circ i : Z \\to X$ is finite", "and of finite presentation", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-composition-finite-presentation} and", "\\ref{spaces-morphisms-lemma-composition-integral}).", "\\medskip\\noindent", "Let $\\mathcal{F} = \\tau_*\\mathcal{O}_Z$ which we think of as", "a quasi-coherent $\\mathcal{O}_X$-module. By", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-finite-finitely-presented-module}", "we see that $\\mathcal{F}$ is a finitely presented $\\mathcal{O}_X$-module.", "Let $\\mathcal{J} = \\text{Fit}_0(\\mathcal{F})$. (Insert reference to", "fitting modules on ringed topoi here.) This is a finite type quasi-coherent", "sheaf of ideals on $X$ (as $\\mathcal{F}$ is of finite presentation, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-fitting-ideal-basics}).", "Part (1) of the lemma holds because $|\\tau|(|Z|) \\cap |U| = \\emptyset$", "by our choice of $\\mathcal{I}_2$ and because the $0$th Fitting ideal", "of the trivial module equals the structure sheaf. To prove (2) note that", "$\\varphi^{-1}(\\mathcal{J})\\mathcal{O}_W = \\text{Fit}_0(\\varphi^*\\mathcal{F})$", "because taking Fitting ideals commutes with base change.", "On the other hand, as $\\varphi : W \\to X$ is separated and \\'etale", "we see that $(1, j) : W \\to W \\times_X Y$ is an open and closed immersion.", "Hence $W \\times_Y Z = V(\\mathcal{I}) \\amalg Z'$ for some finite and", "finitely presented morphism of algebraic spaces $\\tau' : Z' \\to W$.", "Thus we see that", "\\begin{align*}", "\\text{Fit}_0(\\varphi^*\\mathcal{F}) & =", "\\text{Fit}_0((W \\times_Y Z \\to W)_*\\mathcal{O}_{W \\times_Y Z}) \\\\", "& =", "\\text{Fit}_0(\\mathcal{O}_W/\\mathcal{I}) \\cdot", "\\text{Fit}_0(\\tau'_*\\mathcal{O}_{Z'}) \\\\", "& =", "\\mathcal{I} \\cdot \\text{Fit}_0(\\tau'_*\\mathcal{O}_{Z'})", "\\end{align*}", "the second equality by", "More on Algebra, Lemma \\ref{more-algebra-lemma-fitting-ideal-basics}", "translated in sheaves on ringed topoi.", "Setting $\\mathcal{I}' = \\text{Fit}_0(\\tau'_*\\mathcal{O}_{Z'})$", "finishes the proof of the lemma." ], "refs": [ "spaces-more-morphisms-lemma-quasi-finite-separated-pass-through-finite-addendum", "spaces-limits-lemma-extend", "spaces-morphisms-lemma-closed-immersion-finite-presentation", "spaces-morphisms-lemma-composition-finite-presentation", "spaces-morphisms-lemma-composition-integral", "spaces-descent-lemma-finite-finitely-presented-module", "more-algebra-lemma-fitting-ideal-basics", "more-algebra-lemma-fitting-ideal-basics" ], "ref_ids": [ 172, 4608, 4849, 4839, 4941, 9365, 9834, 9834 ] } ], "ref_ids": [] }, { "id": 190, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-after-blowing-up", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-after-blowing-up", "contents": [ "Let $S$ be a scheme.", "Let $B$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $X$ be an algebraic space over $B$.", "Let $U \\subset B$ be a quasi-compact open subspace.", "Assume", "\\begin{enumerate}", "\\item $X \\to B$ is of finite type and quasi-separated, and", "\\item $X_U \\to U$ is flat and locally of finite presentation.", "\\end{enumerate}", "Then there exists a $U$-admissible blowup $B' \\to B$ such that", "the strict transform of $X$ is flat and of finite presentation", "over $B'$." ], "refs": [], "proofs": [ { "contents": [ "Let $B' \\to B$ be a $U$-admissible blowup. Note that the strict transform", "of $X$ is quasi-compact and quasi-separated over $B'$ as $X$ is quasi-compact", "and quasi-separated over $B$. Hence we only need to worry about finding", "a $U$-admissible blowup such that the strict transform becomes flat and", "locally of finite presentation. We cannot directly apply", "Theorem \\ref{theorem-flatten-module} because $X$ is not locally of finite", "presentation over $B$.", "\\medskip\\noindent", "Choose an affine scheme $V$ and a surjective \\'etale morphism $V \\to X$.", "(This is possible as $X$ is quasi-compact as a finite type space over", "the quasi-compact space $B$.) Then it suffices to show the result for", "the morphism $V \\to B$ (as strict transform commutes with \\'etale", "localization, see Divisors on Spaces,", "Lemma \\ref{spaces-divisors-lemma-strict-transform-local}).", "Hence we may assume that $X \\to B$ is separated as well as finite type.", "In this case we can find a closed immersion $i : X \\to Y$ with $Y \\to B$", "separated and of finite presentation, see", "Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-separated-closed-in-finite-presentation}.", "\\medskip\\noindent", "Apply Theorem \\ref{theorem-flatten-module} to $\\mathcal{F} = i_*\\mathcal{O}_X$", "on $Y/B$. We find a $U$-admissible blowup $B' \\to B$ such that strict", "transform of $\\mathcal{F}$ is flat over $B'$ and of finite presentation.", "Let $X'$ be the strict transform of $X$ under the blowup $B' \\to B$.", "Let $i' : X' \\to Y \\times_B B'$ be the induced morphism.", "Since taking strict transform commutes with pushforward along affine", "morphisms (Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-strict-transform-affine}),", "we see that $i'_*\\mathcal{O}_{X'}$ is flat over $B'$ and of", "finite presentation as a $\\mathcal{O}_{Y \\times_B B'}$-module.", "Thus $X' \\to B'$ is flat and locally of finite presentation.", "This implies the lemma by our earlier remarks." ], "refs": [ "spaces-more-morphisms-theorem-flatten-module", "spaces-divisors-lemma-strict-transform-local", "spaces-limits-proposition-separated-closed-in-finite-presentation", "spaces-more-morphisms-theorem-flatten-module", "spaces-divisors-lemma-strict-transform-affine" ], "ref_ids": [ 15, 13000, 4657, 15, 13003 ] } ], "ref_ids": [] }, { "id": 191, "type": "theorem", "label": "spaces-more-morphisms-lemma-finite-after-blowing-up", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-finite-after-blowing-up", "contents": [ "Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated", "algebraic space over $B$. Let $X$ be an algebraic space over $S$.", "Let $U \\subset B$ be a quasi-compact open subspace. Assume", "\\begin{enumerate}", "\\item $X \\to B$ is proper, and", "\\item $X_U \\to U$ is finite locally free.", "\\end{enumerate}", "Then there exists a $U$-admissible blowup $B' \\to B$ such that", "the strict transform of $X$ is finite locally free over $B'$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-flat-after-blowing-up} we may assume that", "$X \\to B$ is flat and of finite presentation. After replacing", "$B$ by a $U$-admissible blowup if necessary, we may assume", "that $U \\subset B$ is scheme theoretically dense. Then $f$ is", "finite by Lemma \\ref{lemma-proper-flat-finite-over-dense-open}.", "Hence $f$ is finite locally free by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-flat}." ], "refs": [ "spaces-more-morphisms-lemma-flat-after-blowing-up", "spaces-more-morphisms-lemma-proper-flat-finite-over-dense-open", "spaces-morphisms-lemma-finite-flat" ], "ref_ids": [ 190, 187, 4954 ] } ], "ref_ids": [] }, { "id": 192, "type": "theorem", "label": "spaces-more-morphisms-lemma-zariski-after-blowup", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-zariski-after-blowup", "contents": [ "Let $S$ be a scheme.", "Let $\\varphi : X \\to B$ be a morphism of algebraic spaces over $S$.", "Assume $\\varphi$ is of finite type with $B$ quasi-compact and quasi-separated.", "Let $U \\subset B$ be a quasi-compact open subspace such that", "$\\varphi^{-1}U \\to U$ is an isomorphism. Then there exists a $U$-admissible", "blowup $B' \\to B$ such that $U$ is scheme theoretically dense in $B'$", "and such that the strict transform $X'$ of $X$ is isomorphic", "to an open subspace of $B'$." ], "refs": [], "proofs": [ { "contents": [ "As the composition of $U$-admissible blowups is $U$-admissible", "(Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-composition-admissible-blowups})", "we can proceed in stages. Pick a finite type quasi-coherent sheaf", "of ideals $\\mathcal{I} \\subset \\mathcal{O}_B$ with", "$|B| \\setminus |U| = |V(\\mathcal{I})|$. Replace $B$ by the blowup", "of $B$ in $\\mathcal{I}$ and $X$ by the strict transform of $X$.", "After this replacement $B \\setminus U$ is the support of an effective", "Cartier divisor $D$ (Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-blowing-up-gives-effective-Cartier-divisor}).", "In particular $U$ is scheme theoretically dense in $B$", "(Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-complement-effective-Cartier-divisor}).", "Next, we do another $U$-admissible blowup to get to the situation where", "$X \\to B$ is flat and of finite presentation, see", "Lemma \\ref{lemma-flat-after-blowing-up}. Note that $U$ is still scheme", "theoretically dense in $B$. Hence $X \\to B$ is an open immersion by", "Lemma \\ref{lemma-zariski}." ], "refs": [ "spaces-divisors-lemma-composition-admissible-blowups", "spaces-divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "spaces-divisors-lemma-complement-effective-Cartier-divisor", "spaces-more-morphisms-lemma-flat-after-blowing-up", "spaces-more-morphisms-lemma-zariski" ], "ref_ids": [ 13009, 12991, 12937, 190, 188 ] } ], "ref_ids": [] }, { "id": 193, "type": "theorem", "label": "spaces-more-morphisms-lemma-dominate-modification-by-blowup", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-dominate-modification-by-blowup", "contents": [ "Let $S$ be a scheme.", "Let $\\varphi : X \\to B$ be a proper morphism of algebraic spaces over $S$.", "Assume $B$ quasi-compact and quasi-separated. Let $U \\subset B$ be a", "quasi-compact open subspace such that $\\varphi^{-1}U \\to U$ is an isomorphism.", "Then there exists a $U$-admissible blowup $B' \\to B$", "which dominates $X$, i.e., such that there exists a factorization", "$B' \\to X \\to B$ of the blowup morphism." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-zariski-after-blowup} we may find a $U$-admissible", "blowup $B' \\to B$ such that the strict transform $X'$ is an open subspace", "of $B'$ and $U$ is scheme theoretically dense in $B'$.", "Since $X' \\to B'$ is proper we see that $|X'|$ is closed in $|B'|$.", "As $U \\subset B'$ is dense $X' = B'$." ], "refs": [ "spaces-more-morphisms-lemma-zariski-after-blowup" ], "ref_ids": [ 192 ] } ], "ref_ids": [] }, { "id": 194, "type": "theorem", "label": "spaces-more-morphisms-lemma-get-section-after-blowup", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-get-section-after-blowup", "contents": [ "Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$.", "Let $U \\subset W \\subset Y$ be open subspaces.", "Let $f : X \\to W$ be a morphism and let $s : U \\to X$ be a", "morphism such that $f \\circ s = \\text{id}_U$. Assume", "\\begin{enumerate}", "\\item $f$ is proper,", "\\item $Y$ is quasi-compact and quasi-separated, and", "\\item $U$ and $W$ are quasi-compact.", "\\end{enumerate}", "Then there exists a $U$-admissible blowup $b : Y' \\to Y$ and a morphism", "$s' : b^{-1}(W) \\to X$ extending $s$ with $f \\circ s' = b|_{b^{-1}(W)}$." ], "refs": [], "proofs": [ { "contents": [ "We may and do replace $X$ by the scheme theoretic image of $s$.", "Then $X \\to W$ is an isomorphism over $U$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-scheme-theoretic-image-of-partial-section}.", "By Lemma \\ref{lemma-dominate-modification-by-blowup}", "there exists a $U$-admissible blowup $W' \\to W$ and an", "extension $W' \\to X$ of $s$.", "We finish the proof by applying", "Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-extend-admissible-blowups}", "to extend $W' \\to W$ to a $U$-admissible blowup of $Y$." ], "refs": [ "spaces-more-morphisms-lemma-dominate-modification-by-blowup", "spaces-divisors-lemma-extend-admissible-blowups" ], "ref_ids": [ 193, 13010 ] } ], "ref_ids": [] }, { "id": 195, "type": "theorem", "label": "spaces-more-morphisms-lemma-find-common-blowups", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-find-common-blowups", "contents": [ "Let $S$ be a scheme. Let $Y$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $U \\to X_1$ and $U \\to X_2$ be open immersions", "of algebraic spaces over $Y$ and assume $U$, $X_1$, $X_2$ of finite", "type and separated over $Y$. Then there exists a commutative diagram", "$$", "\\xymatrix{", "X_1' \\ar[d] \\ar[r] & X & X_2' \\ar[l] \\ar[d] \\\\", "X_1 & U \\ar[l] \\ar[lu] \\ar[u] \\ar[ru] \\ar[r] & X_2", "}", "$$", "of algebraic spaces over $Y$ where $X_i' \\to X_i$ is a $U$-admissible", "blowup, $X_i' \\to X$ is an open immersion, and $X$ is separated and finite", "type over $Y$." ], "refs": [], "proofs": [ { "contents": [ "Throughout the proof all the algebraic spaces will be separated of finite", "type over $Y$. This in particular implies these algebraic spaces are", "quasi-compact and quasi-separated and that the morphisms between them", "will be quasi-compact and separated. See", "Morphisms of Spaces, Sections", "\\ref{spaces-morphisms-section-separation-axioms} and", "\\ref{spaces-morphisms-section-quasi-compact}.", "We will use that if $U \\to W$ is an immersion of such spaces over $Y$,", "then the scheme theoretic image $Z$ of $U$ in $W$ is a closed subspace", "of $W$ and $U \\to Z$ is an open immersion, $U \\subset Z$ is scheme", "theoretically dense, and $|U| \\subset |Z|$ is dense. See", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-immersion}.", "\\medskip\\noindent", "Let $X_{12} \\subset X_1 \\times_Y X_2$ be the scheme theoretic image", "of $U \\to X_1 \\times_Y X_2$. The projections $p_i : X_{12} \\to X_i$", "induce isomorphisms $p_i^{-1}(U) \\to U$ by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-scheme-theoretic-image-of-partial-section}.", "Choose a $U$-admissible blowup $X_i^i \\to X_i$ such that", "the strict transform $X_{12}^i$ of $X_{12}$ is isomorphic to an", "open subspace of $X_i^i$, see", "Lemma \\ref{lemma-zariski-after-blowup}.", "Let $\\mathcal{I}_i \\subset \\mathcal{O}_{X_i}$ be the corresponding", "finite type quasi-coherent sheaf of ideals.", "Recall that $X_{12}^i \\to X_{12}$ is the blowup in", "$p_i^{-1}\\mathcal{I}_i \\mathcal{O}_{X_{12}}$, see", "Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-strict-transform}.", "Let $X_{12}'$ be the blowup of $X_{12}$ in", "$p_1^{-1}\\mathcal{I}_1 p_2^{-1}\\mathcal{I}_2 \\mathcal{O}_{X_{12}}$, see", "Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-blowing-up-two-ideals}", "for what this entails. We obtain a commutative diagram", "$$", "\\xymatrix{", "X_{12}' \\ar[d] \\ar[r] & X_{12}^2 \\ar[d] \\\\", "X_{12}^1 \\ar[r] & X_{12}", "}", "$$", "where all the morphisms are $U$-admissible blowing ups. Since", "$X_{12}^i \\subset X_i^i$ is an open we may choose a $U$-admissible blowup", "$X_i' \\to X_i^i$ restricting to $X_{12}' \\to X_{12}^i$, see", "Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-extend-admissible-blowups}.", "Then $X_{12}' \\subset X_i'$ is an open subspace and the diagram", "$$", "\\xymatrix{", "X_{12}' \\ar[d] \\ar[r] & X_i' \\ar[d] \\\\", "X_{12}^i \\ar[r] & X_i^i", "}", "$$", "is commutative with vertical arrows blowing ups and horizontal arrows", "open immersions. Note that $X'_{12} \\to X_1' \\times_Y X_2'$ is", "an immersion and proper (use that $X'_{12} \\to X_{12}$ is proper", "and $X_{12} \\to X_1 \\times_Y X_2$ is closed and $X_1' \\times_Y X_2' \\to", "X_1 \\times_Y X_2$ is separated and apply Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-permanence}).", "Thus $X'_{12} \\to X_1' \\times_Y X_2'$ is a closed immersion.", "If we define $X$ by glueing $X_1'$ and $X_2'$ along the common open", "subspace $X_{12}'$, then $X \\to Y$ is of finite type and", "separated\\footnote{Because we may check closedness of the diagonal", "$X \\to X \\times_Y X$ over the four open parts $X'_i \\times_Y X'_j$", "of $X \\times_Y X$ where it is clear.}. As compositions of", "$U$-admissible blowups are $U$-admissible blowups", "(Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-composition-admissible-blowups})", "the lemma is proved." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-immersion", "spaces-more-morphisms-lemma-zariski-after-blowup", "spaces-divisors-lemma-strict-transform", "spaces-divisors-lemma-blowing-up-two-ideals", "spaces-divisors-lemma-extend-admissible-blowups", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-divisors-lemma-composition-admissible-blowups" ], "ref_ids": [ 4790, 192, 13001, 12997, 13010, 4920, 13009 ] } ], "ref_ids": [] }, { "id": 196, "type": "theorem", "label": "spaces-more-morphisms-lemma-blowup-to-find-embedding", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-blowup-to-find-embedding", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $U \\subset X$ be an open subspace. Assume", "\\begin{enumerate}", "\\item $U$ is quasi-compact,", "\\item $Y$ is quasi-compact and quasi-separated,", "\\item there exists an immersion $U \\to \\mathbf{P}^n_Y$ over $Y$,", "\\item $f$ is of finite type and separated.", "\\end{enumerate}", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "& U \\ar[ld] \\ar[d] \\ar[rd] \\ar[rrd] \\\\", "X \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & Z' \\ar[ld] \\ar[r] & Z \\ar[ld] \\\\", "& Y & \\mathbf{P}^n_Y \\ar[l]", "}", "$$", "where", "the arrows with source $U$ are open immersions,", "$X' \\to X$ is a $U$-admissible blowup,", "$X' \\to Z'$ is an open immersion,", "$Z' \\to Y$ is a proper and representable morphism of algebraic spaces.", "More precisely, $Z' \\to Z$ is a $U$-admissible blowup", "and $Z \\to \\mathbf{P}^n_Y$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset \\mathbf{P}^n_Y$ be the scheme theoretic image of", "the immersion $U \\to \\mathbf{P}^n_Y$. Since $U \\to \\mathbf{P}^n_Y$", "is quasi-compact we see that $U \\subset Z$ is a", "(scheme theoretically) dense open subspace", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-immersion}).", "Apply Lemma \\ref{lemma-find-common-blowups} to find a diagram", "$$", "\\xymatrix{", "X' \\ar[d] \\ar[r] & \\overline{X}' & Z' \\ar[l] \\ar[d] \\\\", "X & U \\ar[l] \\ar[lu] \\ar[u] \\ar[ru] \\ar[r] & Z", "}", "$$", "with properties as listed in the statement of that lemma.", "As $X' \\to X$ and $Z' \\to Z$ are $U$-admissible blowups", "we find that $U$ is a scheme theoretically dense open of", "both $X'$ and $Z'$ (see Divisors on Spaces, Lemmas", "\\ref{spaces-divisors-lemma-blowing-up-gives-effective-Cartier-divisor} and", "\\ref{spaces-divisors-lemma-complement-effective-Cartier-divisor}).", "Since $Z' \\to Z \\to Y$ is proper we see that $Z' \\subset \\overline{X}'$", "is a closed subspace (see Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-permanence}).", "It follows that $X' \\subset Z'$ (scheme theoretically), hence $X'$", "is an open subspace of $Z'$ (small detail omitted) and the lemma is proved." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-immersion", "spaces-more-morphisms-lemma-find-common-blowups", "spaces-divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "spaces-divisors-lemma-complement-effective-Cartier-divisor", "spaces-morphisms-lemma-universally-closed-permanence" ], "ref_ids": [ 4790, 195, 12991, 12937, 4920 ] } ], "ref_ids": [] }, { "id": 197, "type": "theorem", "label": "spaces-more-morphisms-lemma-chow-noetherian", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-chow-noetherian", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ separated, of finite type, and $Y$ Noetherian.", "Then there exists a dense open subspace $U \\subset X$ and", "a commutative diagram", "$$", "\\xymatrix{", "& U \\ar[ld] \\ar[d] \\ar[rd] \\ar[rrd] \\\\", "X \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & Z' \\ar[ld] \\ar[r] & Z \\ar[ld] \\\\", "& Y & \\mathbf{P}^n_Y \\ar[l]", "}", "$$", "where", "the arrows with source $U$ are open immersions,", "$X' \\to X$ is a $U$-admissible blowup,", "$X' \\to Z'$ is an open immersion,", "$Z' \\to Y$ is a proper and representable morphism of algebraic spaces.", "More precisely, $Z' \\to Z$ is a $U$-admissible blowup", "and $Z \\to \\mathbf{P}^n_Y$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-embedding-into-affine-over-qs}", "there exists a dense open subspace $U \\subset X$ and an immersion", "$U \\to \\mathbf{A}^n_Y$ over $Y$. Composing with the open immersion", "$\\mathbf{A}^n_Y \\to \\mathbf{P}^n_Y$ we obtain a situation as in", "Lemma \\ref{lemma-blowup-to-find-embedding} and the result", "follows." ], "refs": [ "spaces-limits-lemma-embedding-into-affine-over-qs", "spaces-more-morphisms-lemma-blowup-to-find-embedding" ], "ref_ids": [ 4620, 196 ] } ], "ref_ids": [] }, { "id": 198, "type": "theorem", "label": "spaces-more-morphisms-lemma-chow-noetherian-separated", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-chow-noetherian-separated", "contents": [ "\\begin{reference}", "\\cite[IV Theorem 3.1]{Kn}", "\\end{reference}", "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ separated of finite type, and $Y$ separated and", "Noetherian. Then there exists a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & \\mathbf{P}^n_Y \\ar[ld] \\\\", "& Y", "}", "$$", "where $X' \\to X$ is a $U$-admissible blowup for some dense open", "$U \\subset X$ and the morphism $X' \\to \\mathbf{P}^n_Y$ is an immersion." ], "refs": [], "proofs": [ { "contents": [ "In this first paragraph of the proof we reduce the lemma to the case", "where $Y$ is of finite type over $\\Spec(\\mathbf{Z})$.", "We may and do replace the base scheme $S$ by $\\Spec(\\mathbf{Z})$.", "We can write $Y = \\lim Y_i$ as a directed limit of separated", "algebraic spaces of finite type over $\\Spec(\\mathbf{Z})$, see", "Limits of Spaces, Proposition \\ref{spaces-limits-proposition-approximate} and", "Lemma \\ref{spaces-limits-lemma-descend-separated}.", "For all $i$ sufficiently large we can find a separated finite type morphism", "$X_i \\to Y_i$ such that $X = Y \\times_{Y_i} X_i$, see", "Limits of Spaces, Lemmas", "\\ref{spaces-limits-lemma-descend-finite-presentation} and", "\\ref{spaces-limits-lemma-descend-separated-morphism}.", "Let $\\eta_1, \\ldots, \\eta_n$ be the generic points of the irreducible", "components of $|X|$ ($X$ is Noetherian as a finite type separated", "algebraic space over the Noetherian algebraic space $Y$ and therefore", "$|X|$ is a Noetherian topological space).", "By Limits of Spaces, Lemma \\ref{spaces-limits-lemma-topology-limit}", "we find that the images of $\\eta_1, \\ldots, \\eta_n$ in $|X_i|$", "are distinct for $i$ large enough. We may replace", "$X_i$ by the scheme theoretic image of the (quasi-compact, in fact affine)", "morphism $X \\to X_i$.", "After this replacement we see that the images", "of $\\eta_1, \\ldots, \\eta_n$ in $|X_i|$ are the generic points of the", "irreducible components of $|X_i|$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image}.", "Having said this, suppose we can find a diagram", "$$", "\\xymatrix{", "X_i \\ar[rd] & X_i' \\ar[l] \\ar[d] \\ar[r] & \\mathbf{P}^n_{Y_i} \\ar[ld] \\\\", "& Y", "}", "$$", "where $X_i' \\to X_i$ is a $U_i$-admissible blowup for some dense open", "$U_i \\subset X_i$ and the morphism $X_i' \\to \\mathbf{P}^n_{Y_i}$", "is an immersion. Then the strict transform $X' \\to X$ of $X$ relative", "to $X_i' \\to X_i$ is a $U$-admissible blowing up where $U \\subset X$", "is the inverse image of $U_i$ in $X$. Because of our carefuly chosen", "index $i$ it follows that $\\eta_1, \\ldots, \\eta_n \\in |U|$ and", "$U \\subset X$ is dense. Moreover, $X' \\to \\mathbf{P}^n_Y$ is an", "immersion as $X'$ is closed in", "$X_i' \\times_{X_i} X = X_i' \\times_{Y_i} Y$", "which comes with an immersion into $\\mathbf{P}^n_Y$. Thus we have reduced", "to the situation of the following paragraph.", "\\medskip\\noindent", "Assume that $Y$ is separated of finite type over $\\Spec(\\mathbf{Z})$.", "Then $X \\to \\Spec(\\mathbf{Z})$ is separated of finite type as well.", "We apply Lemma \\ref{lemma-chow-noetherian} to $X \\to \\Spec(\\mathbf{Z})$", "to find a dense open subspace $U \\subset X$ and a commutative diagram", "$$", "\\xymatrix{", "& U \\ar[ld] \\ar[d] \\ar[rd] \\ar[rrd] \\\\", "X \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & Z' \\ar[ld] \\ar[r] & Z \\ar[ld] \\\\", "& \\Spec(\\mathbf{Z}) & \\mathbf{P}^n_\\mathbf{Z} \\ar[l]", "}", "$$", "with all the properties listed in the lemma.", "Note that $Z$ has an ample invertible sheaf, namely", "$\\mathcal{O}_{\\mathbf{P}^n}(1)|_Z$. Hence $Z' \\to Z$", "is a H-projective morphism by Morphisms, Lemma", "\\ref{morphisms-lemma-projective-over-quasi-projective-is-H-projective}.", "It follows that $Z' \\to \\Spec(\\mathbf{Z})$ is H-projective", "by Morphisms, Lemma \\ref{morphisms-lemma-H-projective-composition}.", "Thus there exists a closed immersion", "$Z' \\to \\mathbf{P}^m_{\\Spec(\\mathbf{Z})}$ for some $m \\geq 0$.", "It follows that the diagonal morphism", "$$", "X' \\to Y \\times \\mathbf{P}^m_\\mathbf{Z} = \\mathbf{P}^m_Y", "$$", "is an immersion (because the composition with the projection", "to $\\mathbf{P}^m_\\mathbf{Z}$ is an immersion) and we win." ], "refs": [ "spaces-limits-proposition-approximate", "spaces-limits-lemma-descend-separated", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-separated-morphism", "spaces-limits-lemma-topology-limit", "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image", "spaces-more-morphisms-lemma-chow-noetherian", "morphisms-lemma-projective-over-quasi-projective-is-H-projective", "morphisms-lemma-H-projective-composition" ], "ref_ids": [ 4656, 4577, 4598, 4592, 4571, 4780, 197, 5433, 5424 ] } ], "ref_ids": [] }, { "id": 199, "type": "theorem", "label": "spaces-more-morphisms-lemma-chow-finite-type", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-chow-finite-type", "contents": [ "Let $S$ be a scheme. Let $Y$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $f : X \\to Y$ be a separated morphism of", "finite type. Then there exists a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & \\overline{X}' \\ar[ld] \\\\", "& Y", "}", "$$", "where $X' \\to X$ is proper surjective,", "$X' \\to \\overline{X}'$ is an open immersion, and", "$\\overline{X}' \\to Y$ is proper and representable morphism", "of algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "By", "Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-separated-closed-in-finite-presentation}", "we can find a closed immersion $X \\to X_1$ where $X_1$ is separated", "and of finite presentation over $Y$. Clearly, if we prove the assertion", "for $X_1 \\to Y$, then the result follows for $X$. Hence we may assume that", "$X$ is of finite presentation over $Y$.", "\\medskip\\noindent", "We may and do replace the base scheme $S$ by $\\Spec(\\mathbf{Z})$.", "Write $Y = \\lim_i Y_i$ as a directed limit of", "quasi-separated algebraic spaces of finite type over $\\Spec(\\mathbf{Z})$, see", "Limits of Spaces,", "Proposition \\ref{spaces-limits-proposition-approximate}.", "By", "Limits of Spaces,", "Lemma \\ref{spaces-limits-lemma-descend-finite-presentation}", "we can", "find an index $i \\in I$ and a scheme $X_i \\to Y_i$ of finite presentation", "so that $X = Y \\times_{Y_i} X_i$.", "By", "Limits of Spaces,", "Lemma \\ref{spaces-limits-lemma-descend-separated-morphism}", "we may assume that $X_i \\to Y_i$ is separated.", "Clearly, if we prove the assertion for", "$X_i$ over $Y_i$, then the assertion holds for $X$. The case", "$X_i \\to Y_i$ is treated by", "Lemma \\ref{lemma-chow-noetherian}." ], "refs": [ "spaces-limits-proposition-separated-closed-in-finite-presentation", "spaces-limits-proposition-approximate", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-separated-morphism", "spaces-more-morphisms-lemma-chow-noetherian" ], "ref_ids": [ 4657, 4656, 4598, 4592, 197 ] } ], "ref_ids": [] }, { "id": 200, "type": "theorem", "label": "spaces-more-morphisms-lemma-chow-finite-type-separated", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-chow-finite-type-separated", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ separated of finite type, and $Y$ separated and", "quasi-compact. Then there exists a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & \\mathbf{P}^n_Y \\ar[ld] \\\\", "& Y", "}", "$$", "where $X' \\to X$ is proper surjective morphism", "and the morphism $X' \\to \\mathbf{P}^n_Y$ is an immersion." ], "refs": [], "proofs": [ { "contents": [ "By", "Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-separated-closed-in-finite-presentation}", "we can find a closed immersion $X \\to X_1$ where $X_1$ is separated", "and of finite presentation over $Y$. Clearly, if we prove the assertion", "for $X_1 \\to Y$, then the result follows for $X$. Hence we may assume that", "$X$ is of finite presentation over $Y$.", "\\medskip\\noindent", "We may and do replace the base scheme $S$ by $\\Spec(\\mathbf{Z})$.", "Write $Y = \\lim_i Y_i$ as a directed limit of", "quasi-separated algebraic spaces of finite type over $\\Spec(\\mathbf{Z})$, see", "Limits of Spaces,", "Proposition \\ref{spaces-limits-proposition-approximate}.", "By Limits of Spaces, Lemma \\ref{spaces-limits-lemma-descend-separated}", "we may assume that $Y_i$ is separated for all $i$.", "By", "Limits of Spaces,", "Lemma \\ref{spaces-limits-lemma-descend-finite-presentation}", "we can", "find an index $i \\in I$ and a scheme $X_i \\to Y_i$ of finite presentation", "so that $X = Y \\times_{Y_i} X_i$.", "By", "Limits of Spaces,", "Lemma \\ref{spaces-limits-lemma-descend-separated-morphism}", "we may assume that $X_i \\to Y_i$ is separated.", "Clearly, if we prove the assertion for", "$X_i$ over $Y_i$, then the assertion holds for $X$. The case", "$X_i \\to Y_i$ is treated by", "Lemma \\ref{lemma-chow-noetherian-separated}." ], "refs": [ "spaces-limits-proposition-separated-closed-in-finite-presentation", "spaces-limits-proposition-approximate", "spaces-limits-lemma-descend-separated", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-separated-morphism", "spaces-more-morphisms-lemma-chow-noetherian-separated" ], "ref_ids": [ 4657, 4656, 4577, 4598, 4592, 198 ] } ], "ref_ids": [] }, { "id": 201, "type": "theorem", "label": "spaces-more-morphisms-lemma-inverse-systems-abelian", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-inverse-systems-abelian", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and", "let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals.", "\\begin{enumerate}", "\\item The category $\\textit{Coh}(X, \\mathcal{I})$ is abelian.", "\\item Exactness in $\\textit{Coh}(X, \\mathcal{I})$", "can be checked \\'etale locally.", "\\item For any flat morphism $f : X' \\to X$ of Noetherian algebraic spaces", "the functor $f^* : \\textit{Coh}(X, \\mathcal{I}) \\to", "\\textit{Coh}(X', f^{-1}\\mathcal{I}\\mathcal{O}_{X'})$ is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Choose an affine scheme $U_0$ and a surjective \\'etale morphism", "$U_0 \\to X$. Set $U_1 = U_0 \\times_X U_0$ and", "$U_2 = U_0 \\times_X U_0 \\times_X U_0$ as in our discussion of", "\\'etale descent above. The categories $\\textit{Coh}(U_i, \\mathcal{I}_i)$", "are abelian", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-abelian})", "and the pullback functors are exact functors", "$\\textit{Coh}(U_0, \\mathcal{I}_0) \\to \\textit{Coh}(U_1, \\mathcal{I}_1)$", "and", "$\\textit{Coh}(U_1, \\mathcal{I}_1) \\to \\textit{Coh}(U_2, \\mathcal{I}_2)$", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-pullback}).", "The lemma then follows formally from the description of", "$\\textit{Coh}(X, \\mathcal{I})$ as a category of descent data.", "Some details omitted; compare with the proof of", "Groupoids, Lemma \\ref{groupoids-lemma-abelian}.", "\\medskip\\noindent", "Part (2) follows immediately from the discussion in the previous paragraph.", "In the situation of (3) choose a commutative diagram", "$$", "\\xymatrix{", "U' \\ar[d] \\ar[r] & U \\ar[d] \\\\", "X' \\ar[r] & X", "}", "$$", "where $U'$ and $U$ are affine schemes and the vertical morphisms are", "surjective \\'etale. Then $U' \\to U$ is a flat morphism of Noetherian", "schemes (Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-local})", "whence the pullback functor", "$\\textit{Coh}(U, \\mathcal{I}\\mathcal{O}_U) \\to", "\\textit{Coh}(U', \\mathcal{I}\\mathcal{O}_{U'})$", "is exact by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-pullback}.", "Since we can check exactness in $\\textit{Coh}(X, \\mathcal{O}_X)$", "on $U$ and similarly for $X', U'$ the assertion follows." ], "refs": [ "coherent-lemma-inverse-systems-abelian", "coherent-lemma-inverse-systems-pullback", "groupoids-lemma-abelian", "spaces-morphisms-lemma-flat-local", "coherent-lemma-inverse-systems-pullback" ], "ref_ids": [ 3371, 3378, 9629, 4854, 3378 ] } ], "ref_ids": [] }, { "id": 202, "type": "theorem", "label": "spaces-more-morphisms-lemma-inverse-systems-surjective", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-inverse-systems-surjective", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$", "and let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf", "of ideals. A map $(\\mathcal{F}_n) \\to (\\mathcal{G}_n)$ is surjective in", "$\\textit{Coh}(X, \\mathcal{I})$ if and only if", "$\\mathcal{F}_1 \\to \\mathcal{G}_1$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "We can check on an affine \\'etale cover of $X$ by", "Lemma \\ref{lemma-inverse-systems-abelian}.", "Thus we reduce to the case of schemes which is", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-surjective}." ], "refs": [ "spaces-more-morphisms-lemma-inverse-systems-abelian", "coherent-lemma-inverse-systems-surjective" ], "ref_ids": [ 201, 3372 ] } ], "ref_ids": [] }, { "id": 203, "type": "theorem", "label": "spaces-more-morphisms-lemma-exact", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-exact", "contents": [ "The functor (\\ref{equation-completion-functor}) is exact." ], "refs": [], "proofs": [ { "contents": [ "It suffices to check this \\'etale locally on $X$, see", "Lemma \\ref{lemma-inverse-systems-abelian}.", "Thus we reduce to the case of schemes which is", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-exact}." ], "refs": [ "spaces-more-morphisms-lemma-inverse-systems-abelian", "coherent-lemma-exact" ], "ref_ids": [ 201, 3373 ] } ], "ref_ids": [] }, { "id": 204, "type": "theorem", "label": "spaces-more-morphisms-lemma-completion-internal-hom", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-completion-internal-hom", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and", "let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules. Set", "$\\mathcal{H} = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$.", "Then", "$$", "\\lim H^0(X, \\mathcal{H}/\\mathcal{I}^n\\mathcal{H}) =", "\\Mor_{\\textit{Coh}(X, \\mathcal{I})}", "(\\mathcal{F}^\\wedge, \\mathcal{G}^\\wedge).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathcal{H}$ is a sheaf on $X_\\etale$ and since", "we have \\'etale descent for objects of $\\textit{Coh}(X, \\mathcal{I})$", "it suffices to prove this \\'etale locally.", "Thus we reduce to the case of schemes which is", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-completion-internal-hom}." ], "refs": [ "coherent-lemma-completion-internal-hom" ], "ref_ids": [ 3374 ] } ], "ref_ids": [] }, { "id": 205, "type": "theorem", "label": "spaces-more-morphisms-lemma-fully-faithful", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-fully-faithful", "contents": [ "In Situation \\ref{situation-existence}.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.", "Assume that the intersection of the supports of", "$\\mathcal{F}$ and $\\mathcal{G}$ is proper over $\\Spec(A)$. Then the map", "$$", "\\Mor_{\\textit{Coh}(\\mathcal{O}_X)}(\\mathcal{F}, \\mathcal{G})", "\\longrightarrow", "\\Mor_{\\textit{Coh}(X, \\mathcal{I})}", "(\\mathcal{F}^\\wedge, \\mathcal{G}^\\wedge)", "$$", "coming from (\\ref{equation-completion-functor}) is a bijection.", "In particular, (\\ref{equation-completion-functor-proper-over-A})", "is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{H} = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{F})$.", "This is a coherent $\\mathcal{O}_X$-module because its restriction", "of schemes \\'etale over $X$ is coherent by", "Modules, Lemma \\ref{modules-lemma-internal-hom-locally-kernel-direct-sum}.", "By Lemma \\ref{lemma-completion-internal-hom} the map", "$$", "\\lim_n H^0(X, \\mathcal{H}/\\mathcal{I}^n\\mathcal{H})", "\\to", "\\Mor_{\\textit{Coh}(X, \\mathcal{I})}", "(\\mathcal{G}^\\wedge, \\mathcal{F}^\\wedge)", "$$", "is bijective. Let $i : Z \\to X$ be the scheme theoretic support of", "$\\mathcal{H}$. It is clear that $Z$ is a closed subspace such", "that $|Z|$ is contained in the intersection of the supports of $\\mathcal{F}$", "and $\\mathcal{G}$. Hence $Z \\to \\Spec(A)$ is proper by assumption", "(see Derived Categories of Spaces, Section", "\\ref{spaces-perfect-section-proper-over-base}).", "Write $\\mathcal{H} = i_*\\mathcal{H}'$ for some coherent", "$\\mathcal{O}_Z$-module $\\mathcal{H}'$. We have", "$i_*(\\mathcal{H}'/I^n\\mathcal{H}') = \\mathcal{H}/I^n\\mathcal{H}$.", "Hence we obtain", "\\begin{align*}", "\\lim_n H^0(X, \\mathcal{H}/\\mathcal{I}^n\\mathcal{H})", "& =", "\\lim_n H^0(Z, \\mathcal{H}'/\\mathcal{I}^n\\mathcal{H}') \\\\", "& =", "H^0(Z, \\mathcal{H}') \\\\", "& =", "H^0(X, \\mathcal{H}) \\\\", "& = ", "\\Mor_{\\textit{Coh}(\\mathcal{O}_X)}(\\mathcal{F}, \\mathcal{G})", "\\end{align*}", "the second equality by the theorem on formal functions", "(Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-spell-out-theorem-formal-functions}).", "This proves the lemma." ], "refs": [ "modules-lemma-internal-hom-locally-kernel-direct-sum", "spaces-more-morphisms-lemma-completion-internal-hom", "spaces-cohomology-lemma-spell-out-theorem-formal-functions" ], "ref_ids": [ 13298, 204, 11339 ] } ], "ref_ids": [] }, { "id": 206, "type": "theorem", "label": "spaces-more-morphisms-lemma-existence-easy", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-existence-easy", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let", "$\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals.", "Let $\\mathcal{G}$ be a coherent $\\mathcal{O}_X$-module, $(\\mathcal{F}_n)$", "an object of $\\textit{Coh}(X, \\mathcal{I})$, and", "$\\alpha : (\\mathcal{F}_n) \\to \\mathcal{G}^\\wedge$", "a map whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$.", "Then there exists a unique (up to unique isomorphism) triple", "$(\\mathcal{F}, a, \\beta)$ where", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module,", "\\item $a : \\mathcal{F} \\to \\mathcal{G}$ is an $\\mathcal{O}_X$-module map", "whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$,", "\\item $\\beta : (\\mathcal{F}_n) \\to \\mathcal{F}^\\wedge$ is an isomorphism, and", "\\item $\\alpha = a^\\wedge \\circ \\beta$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The uniqueness and \\'etale descent for objects of", "$\\textit{Coh}(X, \\mathcal{I})$ and $\\textit{Coh}(\\mathcal{O}_X)$", "implies it suffices to construct $(\\mathcal{F}, a, \\beta)$ \\'etale", "locally on $X$. Thus we reduce to the case of schemes which is", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-existence-easy}." ], "refs": [ "coherent-lemma-existence-easy" ], "ref_ids": [ 3375 ] } ], "ref_ids": [] }, { "id": 207, "type": "theorem", "label": "spaces-more-morphisms-lemma-existence-tricky", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-existence-tricky", "contents": [ "In Situation \\ref{situation-existence}. Let $\\mathcal{K} \\subset \\mathcal{O}_X$", "be a quasi-coherent sheaf of ideals. Let $X_e \\subset X$ be the closed subspace", "cut out by $\\mathcal{K}^e$. Let $\\mathcal{I}_e = \\mathcal{I}\\mathcal{O}_{X_e}$.", "Let $(\\mathcal{F}_n)$ be an object of", "$\\textit{Coh}_{\\text{support proper over } A}(X, \\mathcal{I})$.", "Assume", "\\begin{enumerate}", "\\item the functor", "$\\textit{Coh}_{\\text{support proper over } A}(\\mathcal{O}_{X_e})", "\\to \\textit{Coh}_{\\text{support proper over } A}(X_e, \\mathcal{I}_e)$", "is an equivalence for all $e \\geq 1$, and", "\\item there exists an object $\\mathcal{H}$ of", "$\\textit{Coh}_{\\text{support proper over } A}(\\mathcal{O}_X)$ and a map", "$\\alpha : (\\mathcal{F}_n) \\to \\mathcal{H}^\\wedge$ whose", "kernel and cokernel are annihilated by a power of $\\mathcal{K}$.", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ is in the essential image of", "(\\ref{equation-completion-functor-proper-over-A})." ], "refs": [], "proofs": [ { "contents": [ "During this proof we will use without further mention that for a closed", "immersion $i : Z \\to X$ the functor $i_*$ gives an equivalence between the", "category of coherent modules on $Z$ and coherent modules on $X$ annihilated", "by the ideal sheaf of $Z$, see", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-i-star-equivalence}.", "In particular we think of", "$$", "\\textit{Coh}_{\\text{support proper over } A}(\\mathcal{O}_{X_e})", "\\subset", "\\textit{Coh}_{\\text{support proper over } A}(\\mathcal{O}_X)", "$$", "as the full subcategory of consisting of modules annihilated by", "$\\mathcal{K}^e$ and", "$$", "\\textit{Coh}_{\\text{support proper over } A}(X_e, \\mathcal{I}_e)", "\\subset", "\\textit{Coh}_{\\text{support proper over } A}(X, \\mathcal{I})", "$$", "as the full subcategory of objects annihilated by $\\mathcal{K}^e$.", "Moreover (1) tells us these two categories are equivalent under the", "completion functor (\\ref{equation-completion-functor-proper-over-A}).", "\\medskip\\noindent", "Applying this equivalence we get a coherent $\\mathcal{O}_X$-module", "$\\mathcal{G}_e$ annihilated by $\\mathcal{K}^e$ corresponding to the system", "$(\\mathcal{F}_n/\\mathcal{K}^e\\mathcal{F}_n)$ of", "$\\textit{Coh}_{\\text{support proper over } A}(X, \\mathcal{I})$. The maps", "$\\mathcal{F}_n/\\mathcal{K}^{e + 1}\\mathcal{F}_n \\to", "\\mathcal{F}_n/\\mathcal{K}^e\\mathcal{F}_n$ correspond to canonical maps", "$\\mathcal{G}_{e + 1} \\to \\mathcal{G}_e$ which induce isomorphisms", "$\\mathcal{G}_{e + 1}/\\mathcal{K}^e\\mathcal{G}_{e + 1} \\to \\mathcal{G}_e$.", "We obtain an object $(\\mathcal{G}_e)$ of the category", "$\\textit{Coh}_{\\text{support proper over } A}(X, \\mathcal{K})$.", "The map $\\alpha$ induces a system of maps", "$$", "\\mathcal{F}_n/\\mathcal{K}^e\\mathcal{F}_n", "\\longrightarrow", "\\mathcal{H}/(\\mathcal{I}^n + \\mathcal{K}^e)\\mathcal{H}", "$$", "whence maps $\\mathcal{G}_e \\to \\mathcal{H}/\\mathcal{K}^e\\mathcal{H}$", "(by the equivalence of categories again).", "Let $t \\geq 1$ be an integer, which exists by assumption (2),", "such that $\\mathcal{K}^t$ annihilates the kernel and cokernel of all the maps", "$\\mathcal{F}_n \\to \\mathcal{H}/\\mathcal{I}^n\\mathcal{H}$.", "Then $\\mathcal{K}^{2t}$ annihilates the kernel and cokernel of the maps", "$\\mathcal{F}_n/\\mathcal{K}^e\\mathcal{F}_n \\to", "\\mathcal{H}/(\\mathcal{I}^n + \\mathcal{K}^e)\\mathcal{H}$", "(details omitted; see Cohomology of Schemes,", "Remark \\ref{coherent-remark-inverse-systems-kernel-cokernel-annihilated-by}).", "Whereupon we conclude that $\\mathcal{K}^{4t}$ annihilates the kernel and", "the cokernel of the maps", "$$", "\\mathcal{G}_e", "\\longrightarrow", "\\mathcal{H}/\\mathcal{K}^e\\mathcal{H},", "$$", "(details omitted; see Cohomology of Schemes,", "Remark \\ref{coherent-remark-inverse-systems-kernel-cokernel-annihilated-by}).", "We apply Lemma \\ref{lemma-existence-easy} to obtain a coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$, a map", "$a : \\mathcal{F} \\to \\mathcal{H}$ and an isomorphism", "$\\beta : (\\mathcal{G}_e) \\to (\\mathcal{F}/\\mathcal{K}^e\\mathcal{F})$", "in $\\textit{Coh}(X, \\mathcal{K})$. Working backwards, for a given $n$", "the triple", "$(\\mathcal{F}/\\mathcal{I}^n\\mathcal{F}, a \\bmod \\mathcal{I}^n, \\beta", "\\bmod \\mathcal{I}^n)$ is a triple as in the lemma for the morphism", "$\\alpha_n \\bmod \\mathcal{K}^e :", "(\\mathcal{F}_n/\\mathcal{K}^e\\mathcal{F}_n) \\to", "(\\mathcal{H}/(\\mathcal{I}^n + \\mathcal{K}^e)\\mathcal{H})$", "of $\\textit{Coh}(X, \\mathcal{K})$. Thus the uniqueness in", "Lemma \\ref{lemma-existence-easy}", "gives a canonical isomorphism", "$\\mathcal{F}/\\mathcal{I}^n\\mathcal{F} \\to \\mathcal{F}_n$", "compatible with all the morphisms in sight.", "\\medskip\\noindent", "To finish the proof of the lemma we still have to show that the", "support of $\\mathcal{F}$ is proper over $A$.", "By construction the kernel of $a : \\mathcal{F} \\to \\mathcal{H}$", "is annihilated by a power of $\\mathcal{K}$. Hence the support of", "this kernel is contained in the support of $\\mathcal{G}_1$. Since", "$\\mathcal{G}_1$ is an object of", "$\\textit{Coh}_{\\text{support proper over } A}(\\mathcal{O}_{X_1})$", "we see this is proper over $A$. Combined with the fact that the", "support of $\\mathcal{H}$ is proper over $A$ we conclude that the", "support of $\\mathcal{F}$ is proper over $A$ by", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-union-closed-proper-over-base}." ], "refs": [ "spaces-cohomology-lemma-i-star-equivalence", "coherent-remark-inverse-systems-kernel-cokernel-annihilated-by", "coherent-remark-inverse-systems-kernel-cokernel-annihilated-by", "spaces-more-morphisms-lemma-existence-easy", "spaces-more-morphisms-lemma-existence-easy", "spaces-perfect-lemma-union-closed-proper-over-base" ], "ref_ids": [ 11303, 3408, 3408, 206, 206, 2660 ] } ], "ref_ids": [] }, { "id": 208, "type": "theorem", "label": "spaces-more-morphisms-lemma-inverse-systems-push-pull", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-inverse-systems-push-pull", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable", "proper morphism of Noetherian algebraic spaces over $S$. Let", "$\\mathcal{J}, \\mathcal{K} \\subset \\mathcal{O}_Y$", "be quasi-coherent sheaves of ideals.", "Assume $f$ is an isomorphism over $V = Y \\setminus V(\\mathcal{K})$.", "Set $\\mathcal{I} = f^{-1}\\mathcal{J} \\mathcal{O}_X$.", "Let $(\\mathcal{G}_n)$ be an object of $\\textit{Coh}(Y, \\mathcal{J})$,", "let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module, and let", "$\\beta : (f^*\\mathcal{G}_n) \\to \\mathcal{F}^\\wedge$ be an isomorphism in", "$\\textit{Coh}(X, \\mathcal{I})$. Then there exists a map", "$$", "\\alpha :", "(\\mathcal{G}_n)", "\\longrightarrow", "(f_*\\mathcal{F})^\\wedge", "$$", "in $\\textit{Coh}(Y, \\mathcal{J})$ whose kernel and cokernel", "are annihilated by a power of $\\mathcal{K}$." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is a proper morphism we see that $f_*\\mathcal{F}$ is a coherent", "$\\mathcal{O}_Y$-module (Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-pushforward-coherent}).", "Thus the statement of the lemma makes sense. Consider the compositions", "$$", "\\gamma_n : \\mathcal{G}_n \\to", "f_*f^*\\mathcal{G}_n \\to", "f_*(\\mathcal{F}/\\mathcal{I}^n\\mathcal{F}).", "$$", "Here the first map is the adjunction map and the second is $f_*\\beta_n$.", "We claim that there exists a unique $\\alpha$ as in the lemma", "such that the compositions", "$$", "\\mathcal{G}_n \\xrightarrow{\\alpha_n}", "f_*\\mathcal{F}/\\mathcal{J}^nf_*\\mathcal{F} \\to", "f_*(\\mathcal{F}/\\mathcal{I}^n\\mathcal{F})", "$$", "equal $\\gamma_n$ for all $n$. Because of the uniqueness and \\'etale", "descent for $\\textit{Coh}(Y, \\mathcal{J})$ it suffices", "to prove this \\'etale locally on $Y$. Thus we may assume $Y$", "is the spectrum of a Noetherian ring. As $f$ is representable", "we see that $X$ is a scheme as well. Thus we reduce to the case of", "schemes, see proof of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-push-pull}." ], "refs": [ "spaces-cohomology-lemma-proper-pushforward-coherent", "coherent-lemma-inverse-systems-push-pull" ], "ref_ids": [ 11331, 3385 ] } ], "ref_ids": [] }, { "id": 209, "type": "theorem", "label": "spaces-more-morphisms-lemma-algebraize-formal-closed-subscheme", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-algebraize-formal-closed-subscheme", "contents": [ "Let $A$ be a Noetherian ring complete with respect to an ideal $I$.", "Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$.", "Let $X \\to S$ be a morphism of algebraic spaces that is separated", "and of finite type.", "For $n \\geq 1$ we set $X_n = X \\times_S S_n$.", "Suppose given a commutative diagram", "$$", "\\xymatrix{", "Z_1 \\ar[r] \\ar[d] & Z_2 \\ar[r] \\ar[d] & Z_3 \\ar[r] \\ar[d] & \\ldots \\\\", "X_1 \\ar[r]^{i_1} & X_2 \\ar[r]^{i_2} & X_3 \\ar[r] & \\ldots", "}", "$$", "of algebraic spaces with cartesian squares. Assume that", "\\begin{enumerate}", "\\item $Z_1 \\to X_1$ is a closed immersion, and", "\\item $Z_1 \\to S_1$ is proper.", "\\end{enumerate}", "Then there exists a closed immersion of algebraic spaces $Z \\to X$ such that", "$Z_n = Z \\times_S S_n$ for all $n \\geq 1$. Moreover, $Z$ is proper over $S$." ], "refs": [], "proofs": [ { "contents": [ "Let's write $j_n : Z_n \\to X_n$ for the vertical morphisms.", "As the squares in the statement are cartesian", "we see that the base change of $j_n$ to $X_1$ is $j_1$.", "Thus Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-check-closed-infinitesimally}", "shows that $j_n$ is a closed immersion.", "Set $\\mathcal{F}_n = j_{n, *}\\mathcal{O}_{Z_n}$, so that", "$j_n^\\sharp$ is a surjection $\\mathcal{O}_{X_n} \\to \\mathcal{F}_n$.", "Again using that the squares are cartesian we see that", "the pullback of $\\mathcal{F}_{n + 1}$ to $X_n$ is $\\mathcal{F}_n$.", "Hence Grothendieck's existence theorem, as reformulated in", "Remark \\ref{remark-reformulate-existence-theorem},", "tells us there exists a map", "$\\mathcal{O}_X \\to \\mathcal{F}$", "of coherent $\\mathcal{O}_X$-modules whose restriction to", "$X_n$ recovers $\\mathcal{O}_{X_n} \\to \\mathcal{F}_n$.", "Moreover, the support of $\\mathcal{F}$ is proper over $S$.", "As the completion functor is exact (Lemma \\ref{lemma-exact})", "we see that $\\mathcal{O}_X \\to \\mathcal{F}$", "is surjective. Thus $\\mathcal{F} = \\mathcal{O}_X/\\mathcal{J}$", "for some quasi-coherent sheaf of ideals $\\mathcal{J}$.", "Setting $Z = V(\\mathcal{J})$ finishes the proof." ], "refs": [ "spaces-limits-lemma-check-closed-infinitesimally", "spaces-more-morphisms-remark-reformulate-existence-theorem", "spaces-more-morphisms-lemma-exact" ], "ref_ids": [ 4629, 311, 203 ] } ], "ref_ids": [] }, { "id": 210, "type": "theorem", "label": "spaces-more-morphisms-lemma-algebraize-formal-algebraic-space-finite-over-proper", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-algebraize-formal-algebraic-space-finite-over-proper", "contents": [ "Let $A$ be a Noetherian ring complete with respect to an ideal $I$.", "Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$.", "Let $X \\to S$ be a morphism of algebraic spaces that is separated", "and of finite type.", "For $n \\geq 1$ we set $X_n = X \\times_S S_n$.", "Suppose given a commutative diagram", "$$", "\\xymatrix{", "Y_1 \\ar[r] \\ar[d] & Y_2 \\ar[r] \\ar[d] & Y_3 \\ar[r] \\ar[d] & \\ldots \\\\", "X_1 \\ar[r]^{i_1} & X_2 \\ar[r]^{i_2} & X_3 \\ar[r] & \\ldots", "}", "$$", "of algebraic spaces with cartesian squares. Assume that", "\\begin{enumerate}", "\\item $Y_1 \\to X_1$ is a finite morphism, and", "\\item $Y_1 \\to S_1$ is proper.", "\\end{enumerate}", "Then there exists a finite morphism of algebraic spaces $Y \\to X$ such that", "$Y_n = Y \\times_S S_n$ for all $n \\geq 1$. Moreover, $Y$ is proper over $S$." ], "refs": [], "proofs": [ { "contents": [ "Let's write $f_n : Y_n \\to X_n$ for the vertical morphisms.", "As the squares in the statement are cartesian", "we see that the base change of $f_n$ to $X_1$ is $f_1$.", "Thus Lemma \\ref{lemma-thicken-property-morphisms-cartesian}", "shows that $f_n$ is a finite morphism.", "Set $\\mathcal{F}_n = f_{n, *}\\mathcal{O}_{Y_n}$.", "Using that the squares are cartesian we see that", "the pullback of $\\mathcal{F}_{n + 1}$ to $X_n$ is $\\mathcal{F}_n$.", "Hence Grothendieck's existence theorem, as reformulated in", "Remark \\ref{remark-reformulate-existence-theorem},", "tells us there exists a coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "whose restriction to $X_n$ recovers $\\mathcal{F}_n$.", "Moreover, the support of $\\mathcal{F}$ is proper over $S$.", "As the completion functor is fully faithful", "(Theorem \\ref{theorem-grothendieck-existence})", "we see that the multiplication maps", "$\\mathcal{F}_n \\otimes_{\\mathcal{O}_{X_n}} \\mathcal{F}_n \\to", "\\mathcal{F}_n$ fit together to give an algebra structure on $\\mathcal{F}$.", "Setting $Y = \\underline{\\Spec}_X(\\mathcal{F})$ finishes the proof." ], "refs": [ "spaces-more-morphisms-lemma-thicken-property-morphisms-cartesian", "spaces-more-morphisms-remark-reformulate-existence-theorem", "spaces-more-morphisms-theorem-grothendieck-existence" ], "ref_ids": [ 56, 311, 16 ] } ], "ref_ids": [] }, { "id": 211, "type": "theorem", "label": "spaces-more-morphisms-lemma-algebraize-morphism", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-algebraize-morphism", "contents": [ "Let $A$ be a Noetherian ring complete with respect to an ideal $I$.", "Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Let $X$, $Y$ be algebraic", "spaces over $S$. For $n \\geq 1$ we set $X_n = X \\times_S S_n$ and", "$Y_n = Y \\times_S S_n$. Suppose given a compatible system of", "commutative diagrams", "$$", "\\xymatrix{", "& & X_{n + 1} \\ar[rd] \\ar[rr]_{g_{n + 1}} & & Y_{n + 1} \\ar[ld] \\\\", "X_n \\ar[rru] \\ar[rd] \\ar[rr]_{g_n} & & Y_n \\ar[rru] \\ar[ld] & S_{n + 1} \\\\", "& S_n \\ar[rru]", "}", "$$", "Assume that", "\\begin{enumerate}", "\\item $X \\to S$ is proper, and", "\\item $Y \\to S$ is separated of finite type.", "\\end{enumerate}", "Then there exists a unique morphism of algebraic spaces $g : X \\to Y$", "over $S$ such that $g_n$ is the base change of $g$ to $S_n$." ], "refs": [], "proofs": [ { "contents": [ "The morphisms $(1, g_n) : X_n \\to X_n \\times_S Y_n$ are closed immersions", "because $Y_n \\to S_n$ is separated", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-section-immersion}).", "Thus by Lemma \\ref{lemma-algebraize-formal-closed-subscheme}", "there exists a closed subspace $Z \\subset X \\times_S Y$", "proper over $S$ whose base change to $S_n$ recovers", "$X_n \\subset X_n \\times_S Y_n$. The first projection $p : Z \\to X$", "is a proper morphism (as $Z$ is proper over $S$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-permanence})", "whose base change to $S_n$ is an isomorphism for all $n$.", "In particular, $p : Z \\to X$ is quasi-finite on an open subspace", "of $Z$ containing every point of $Z_0$ for example by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part}.", "As $Z$ is proper over $S$ this open neighbourhood is all of $Z$.", "We conclude that $p : Z \\to X$ is finite by Zariski's main theorem", "(for example apply", "Lemma \\ref{lemma-quasi-finite-separated-pass-through-finite}", "and use properness of $Z$ over $X$ to see that the immersion is", "a closed immersion). Applying the equivalence of", "Theorem \\ref{theorem-grothendieck-existence}", "we see that $p_*\\mathcal{O}_Z = \\mathcal{O}_X$ as this is true", "modulo $I^n$ for all $n$. Hence $p$ is an isomorphism and we obtain", "the morphism $g$ as the composition $X \\cong Z \\to Y$.", "We omit the proof of uniqueness." ], "refs": [ "spaces-more-morphisms-lemma-algebraize-formal-closed-subscheme", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part", "spaces-more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "spaces-more-morphisms-theorem-grothendieck-existence" ], "ref_ids": [ 209, 4920, 4876, 171, 16 ] } ], "ref_ids": [] }, { "id": 212, "type": "theorem", "label": "spaces-more-morphisms-lemma-formal-algebraic-space-proper-reldim-1", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-formal-algebraic-space-proper-reldim-1", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a complete local Noetherian ring.", "Set $S = \\Spec(A)$ and $S_n = \\Spec(A/\\mathfrak m^n)$.", "Consider a commutative diagram", "$$", "\\xymatrix{", "X_1 \\ar[r]_{i_1} \\ar[d] & X_2 \\ar[r]_{i_2} \\ar[d] & X_3 \\ar[r] \\ar[d] &", "\\ldots \\\\", "S_1 \\ar[r] & S_2 \\ar[r] & S_3 \\ar[r] & \\ldots", "}", "$$", "of algebraic spaces with cartesian squares. If $\\dim(X_1) \\leq 1$,", "then there exists a projective morphism of schemes $X \\to S$", "and isomorphisms $X_n \\cong X \\times_S S_n$ compatible with $i_n$." ], "refs": [], "proofs": [ { "contents": [ "By Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-codim-1-point-in-schematic-locus}", "the algebraic space $X_1$ is a scheme. Hence $X_1$", "is a proper scheme of dimension $\\leq 1$ over $\\kappa$.", "By Varieties, Lemma \\ref{varieties-lemma-dim-1-proper-projective}", "we see that $X_1$ is H-projective over $\\kappa$.", "Let $\\mathcal{L}_1$ be an ample invertible sheaf on $X_1$.", "\\medskip\\noindent", "We are going to show that $\\mathcal{L}_1$ lifts to a compatible system", "$\\{\\mathcal{L}_n\\}$ of invertible sheaves on $\\{X_n\\}$.", "Observe that $X_n$ is a scheme too by Lemma \\ref{lemma-thickening-scheme}.", "Recall that $X_1 \\to X_n$ induces homeomorphisms of underlying", "topological spaces. In the rest of the proof we do not distinguish", "between sheaves on $X_n$ and sheaves on $X_1$.", "Suppose, given a lift $\\mathcal{L}_n$ to $X_n$. We consider", "the exact sequence", "$$", "1 \\to", "(1 + \\mathfrak m^n\\mathcal{O}_{X_{n + 1}})^* \\to", "\\mathcal{O}_{X_{n + 1}}^* \\to \\mathcal{O}_{X_n}^* \\to 1", "$$", "of sheaves on $X_{n + 1}$. The class of $\\mathcal{L}_n$ in", "$H^1(X_n, \\mathcal{O}_{X_n}^*)$ (see", "Cohomology, Lemma \\ref{cohomology-lemma-h1-invertible})", "can be lifted to an element of $H^1(X_{n + 1}, \\mathcal{O}_{X_{n + 1}}^*)$", "if and only if the obstruction in", "$H^2(X_{n + 1}, (1 + \\mathfrak m^n\\mathcal{O}_{X_{n + 1}})^*)$", "is zero. As $X_1$ is a Noetherian scheme of dimension $\\leq 1$", "this cohomology group vanishes (Cohomology, Proposition", "\\ref{cohomology-proposition-vanishing-Noetherian}).", "\\medskip\\noindent", "By Grothendieck's algebraization theorem", "(Cohomology of Schemes, Theorem \\ref{coherent-theorem-algebraization})", "we find a projective morphism of schemes $X \\to S = \\Spec(A)$", "and a compatible system of isomorphisms $X_n = S_n \\times_S X$." ], "refs": [ "spaces-over-fields-lemma-codim-1-point-in-schematic-locus", "varieties-lemma-dim-1-proper-projective", "spaces-more-morphisms-lemma-thickening-scheme", "cohomology-lemma-h1-invertible", "cohomology-proposition-vanishing-Noetherian", "coherent-theorem-algebraization" ], "ref_ids": [ 12845, 11099, 49, 2036, 2246, 3280 ] } ], "ref_ids": [] }, { "id": 213, "type": "theorem", "label": "spaces-more-morphisms-lemma-projective-over-complete", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-projective-over-complete", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a complete Noetherian local ring.", "Let $X$ be an algebraic space over $\\Spec(A)$.", "If $X \\to \\Spec(A)$ is proper and $\\dim(X_\\kappa) \\leq 1$, then", "$X$ is a scheme projective over $A$." ], "refs": [], "proofs": [ { "contents": [ "Set $X_n = X \\times_{\\Spec(A)} \\Spec(A/\\mathfrak m^n)$.", "By Lemma \\ref{lemma-formal-algebraic-space-proper-reldim-1}", "there exists a projective morphism $Y \\to \\Spec(A)$", "and compatible isomorphisms", "$Y \\times_{\\Spec(A)} \\Spec(A/\\mathfrak m^n) \\cong", "X \\times_{\\Spec(A)} \\Spec(A/\\mathfrak m^n)$.", "By Lemma \\ref{lemma-algebraize-morphism}", "we see that $X \\cong Y$ and the proof is complete." ], "refs": [ "spaces-more-morphisms-lemma-formal-algebraic-space-proper-reldim-1", "spaces-more-morphisms-lemma-algebraize-morphism" ], "ref_ids": [ 212, 211 ] } ], "ref_ids": [] }, { "id": 214, "type": "theorem", "label": "spaces-more-morphisms-lemma-representable-etale-local-target", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-representable-etale-local-target", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes which is \\'etale", "local on the target. Let $S$ be a scheme.", "Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$.", "Consider commutative diagrams", "$$", "\\xymatrix{", "X \\times_Y V \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $V$ is a scheme and $V \\to Y$ is \\'etale.", "The following are equivalent", "\\begin{enumerate}", "\\item for any diagram as above the projection $X \\times_Y V \\to V$", "has property $\\mathcal{P}$, and", "\\item for some diagram as above with $V \\to Y$ surjective", "the projection $X \\times_Y V \\to V$ has property $\\mathcal{P}$.", "\\end{enumerate}", "If $X$ and $Y$ are representable, then this is also equivalent to", "$f$ (as a morphism of schemes) having property $\\mathcal{P}$." ], "refs": [], "proofs": [ { "contents": [ "Let us prove the equivalence of (1) and (2).", "The implication (1) $\\Rightarrow$ (2) is immediate.", "Assume", "$$", "\\xymatrix{", "X \\times_Y V \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "\\quad\\quad", "\\xymatrix{", "X \\times_Y V' \\ar[d] \\ar[r] & V' \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "are two diagrams as in the lemma. Assume $V \\to Y$ is", "surjective and $X \\times_Y V \\to V$ has property $\\mathcal{P}$.", "To show that (2) implies (1) we have to prove that", "$X \\times_Y V' \\to V'$ has $\\mathcal{P}$. To do", "this consider the diagram", "$$", "\\xymatrix{", "X \\times_Y V \\ar[d] &", "(X \\times_Y V) \\times_X (X \\times_Y V') \\ar[l] \\ar[d] \\ar[r] &", "X \\times_Y V' \\ar[d] \\\\", "V &", "V \\times_Y V' \\ar[l] \\ar[r] &", "V'", "}", "$$", "By our assumption that $\\mathcal{P}$ is \\'etale local on the source,", "we see that $\\mathcal{P}$ is preserved under \\'etale base change, see", "Descent, Lemma \\ref{descent-lemma-pullback-property-local-target}.", "Hence if the left vertical arrow has $\\mathcal{P}$ the so does", "the middle vertical arrow. Since $U \\times_X U' \\to U'$ is surjective", "and \\'etale (hence defines an \\'etale covering of $U'$)", "this implies (as $\\mathcal{P}$ is assumed local for the \\'etale topology", "on the target) that the left vertical arrow has $\\mathcal{P}$.", "\\medskip\\noindent", "If $X$ and $Y$ are representable, then we can take", "$\\text{id}_Y : Y \\to Y$ as our \\'etale covering to see the", "final statement of the lemma is true." ], "refs": [ "descent-lemma-pullback-property-local-target" ], "ref_ids": [ 14663 ] } ], "ref_ids": [] }, { "id": 215, "type": "theorem", "label": "spaces-more-morphisms-lemma-regular-quasi-regular-immersion", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-regular-quasi-regular-immersion", "contents": [ "Let $S$ be a scheme.", "Let $i : Z \\to X$ be an immersion of algebraic spaces over $S$.", "We have the following implications:", "$i$ is Koszul-regular $\\Rightarrow$", "$i$ is $H_1$-regular $\\Rightarrow$", "$i$ is quasi-regular." ], "refs": [], "proofs": [ { "contents": [ "Via the definition this lemma immediately reduces to", "Divisors, Lemma \\ref{divisors-lemma-regular-quasi-regular-immersion}." ], "refs": [ "divisors-lemma-regular-quasi-regular-immersion" ], "ref_ids": [ 7989 ] } ], "ref_ids": [] }, { "id": 216, "type": "theorem", "label": "spaces-more-morphisms-lemma-regular-immersion-noetherian", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-regular-immersion-noetherian", "contents": [ "Let $S$ be a scheme.", "Let $i : Z \\to X$ be an immersion of algebraic spaces over $S$.", "Assume $X$ is locally Noetherian. Then", "$i$ is Koszul-regular $\\Leftrightarrow$", "$i$ is $H_1$-regular $\\Leftrightarrow$", "$i$ is quasi-regular." ], "refs": [], "proofs": [ { "contents": [ "Via Definition \\ref{definition-regular-immersion}", "(and the definition of a locally Noetherian algebraic space", "in Properties of Spaces, Section", "\\ref{spaces-properties-section-types-properties})", "this immediately translates to the case of schemes which is", "Divisors, Lemma \\ref{divisors-lemma-regular-immersion-noetherian}." ], "refs": [ "spaces-more-morphisms-definition-regular-immersion", "divisors-lemma-regular-immersion-noetherian" ], "ref_ids": [ 295, 7990 ] } ], "ref_ids": [] }, { "id": 217, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-base-change-regular-immersion", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-base-change-regular-immersion", "contents": [ "\\begin{slogan}", "Regular immersions are stable under flat base change.", "\\end{slogan}", "Let $S$ be a scheme. Let $i : Z \\to X$ be a Koszul-regular,", "$H_1$-regular, or quasi-regular immersion of algebraic spaces over $S$.", "Let $X' \\to X$ be a flat morphism of algebraic spaces over $S$.", "Then the base change $i' : Z \\times_X X' \\to X'$ is a Koszul-regular,", "$H_1$-regular, or quasi-regular immersion." ], "refs": [], "proofs": [ { "contents": [ "Via Definition \\ref{definition-regular-immersion}", "(and the definition of a flat morphism of algebraic spaces", "in Morphisms of Spaces, Section", "\\ref{spaces-morphisms-section-flat})", "this lemma reduces to the case of schemes, see", "Divisors, Lemma \\ref{divisors-lemma-flat-base-change-regular-immersion}." ], "refs": [ "spaces-more-morphisms-definition-regular-immersion", "divisors-lemma-flat-base-change-regular-immersion" ], "ref_ids": [ 295, 7991 ] } ], "ref_ids": [] }, { "id": 218, "type": "theorem", "label": "spaces-more-morphisms-lemma-quasi-regular-immersion", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-quasi-regular-immersion", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic spaces", "over $S$. Then $i$ is a quasi-regular immersion if and only if the following", "conditions are satisfied", "\\begin{enumerate}", "\\item $i$ is locally of finite presentation,", "\\item the conormal sheaf $\\mathcal{C}_{Z/X}$ is finite locally free, and", "\\item the map (\\ref{equation-conormal-algebra-quotient}) is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows from the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-quasi-regular-immersion})", "via \\'etale localization (use Definition \\ref{definition-regular-immersion}", "and", "Lemma \\ref{lemma-etale-conormal-algebra})." ], "refs": [ "divisors-lemma-quasi-regular-immersion", "spaces-more-morphisms-definition-regular-immersion", "spaces-more-morphisms-lemma-etale-conormal-algebra" ], "ref_ids": [ 7992, 295, 28 ] } ], "ref_ids": [] }, { "id": 219, "type": "theorem", "label": "spaces-more-morphisms-lemma-transitivity-conormal-quasi-regular", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-transitivity-conormal-quasi-regular", "contents": [ "Let $S$ be a scheme. Let $Z \\to Y \\to X$ be immersions of algebraic spaces", "over $S$. Assume that $Z \\to Y$ is $H_1$-regular. Then the canonical", "sequence of Lemma \\ref{lemma-transitivity-conormal}", "$$", "0 \\to i^*\\mathcal{C}_{Y/X} \\to", "\\mathcal{C}_{Z/X} \\to", "\\mathcal{C}_{Z/Y} \\to 0", "$$", "is exact and (\\'etale) locally split." ], "refs": [ "spaces-more-morphisms-lemma-transitivity-conormal" ], "proofs": [ { "contents": [ "Since $\\mathcal{C}_{Z/Y}$ is finite locally free (see", "Lemma \\ref{lemma-quasi-regular-immersion}", "and", "Lemma \\ref{lemma-regular-quasi-regular-immersion})", "it suffices to prove that the sequence is exact.", "It suffices to show that the first map is injective", "as the sequence is already right exact in general.", "After \\'etale localization on $X$ this reduces to the case", "of schemes, see", "Divisors, Lemma \\ref{divisors-lemma-transitivity-conormal-quasi-regular}." ], "refs": [ "spaces-more-morphisms-lemma-quasi-regular-immersion", "spaces-more-morphisms-lemma-regular-quasi-regular-immersion", "divisors-lemma-transitivity-conormal-quasi-regular" ], "ref_ids": [ 218, 215, 7993 ] } ], "ref_ids": [ 27 ] }, { "id": 220, "type": "theorem", "label": "spaces-more-morphisms-lemma-composition-regular-immersion", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-composition-regular-immersion", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of", "algebraic spaces over $S$.", "\\begin{enumerate}", "\\item If $i$ and $j$ are Koszul-regular immersions, so is $j \\circ i$.", "\\item If $i$ and $j$ are $H_1$-regular immersions, so is $j \\circ i$.", "\\item If $i$ is an $H_1$-regular immersion and $j$ is a quasi-regular", "immersion, then $j \\circ i$ is a quasi-regular immersion.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Immediate from the case of schemes, see", "Divisors, Lemma \\ref{divisors-lemma-composition-regular-immersion}." ], "refs": [ "divisors-lemma-composition-regular-immersion" ], "ref_ids": [ 7994 ] } ], "ref_ids": [] }, { "id": 221, "type": "theorem", "label": "spaces-more-morphisms-lemma-permanence-regular-immersion", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-permanence-regular-immersion", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of", "algebraic spaces over $S$. Assume that the sequence", "$$", "0 \\to i^*\\mathcal{C}_{Y/X} \\to", "\\mathcal{C}_{Z/X} \\to", "\\mathcal{C}_{Z/Y} \\to 0", "$$", "of Lemma \\ref{lemma-transitivity-conormal} is exact and locally split.", "\\begin{enumerate}", "\\item If $j \\circ i$ is a quasi-regular immersion, so is $i$.", "\\item If $j \\circ i$ is a $H_1$-regular immersion, so is $i$.", "\\item If both $j$ and $j \\circ i$ are Koszul-regular immersions, so is $i$.", "\\end{enumerate}" ], "refs": [ "spaces-more-morphisms-lemma-transitivity-conormal" ], "proofs": [ { "contents": [ "Immediate from the case of schemes, see", "Divisors, Lemma \\ref{divisors-lemma-permanence-regular-immersion}." ], "refs": [ "divisors-lemma-permanence-regular-immersion" ], "ref_ids": [ 7995 ] } ], "ref_ids": [ 27 ] }, { "id": 222, "type": "theorem", "label": "spaces-more-morphisms-lemma-extra-permanence-regular-immersion-noetherian", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-extra-permanence-regular-immersion-noetherian", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of", "algebraic spaces over $S$. Assume $X$ is locally Noetherian.", "The following are equivalent", "\\begin{enumerate}", "\\item $i$ and $j$ are Koszul regular immersions,", "\\item $i$ and $j \\circ i$ are Koszul regular immersions,", "\\item $j \\circ i$ is a Koszul regular immersion and the conormal sequence", "$$", "0 \\to i^*\\mathcal{C}_{Y/X} \\to", "\\mathcal{C}_{Z/X} \\to", "\\mathcal{C}_{Z/Y} \\to 0", "$$", "is exact and locally split.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Immediate from the case of schemes, see Divisors, Lemma", "\\ref{divisors-lemma-extra-permanence-regular-immersion-noetherian}." ], "refs": [ "divisors-lemma-extra-permanence-regular-immersion-noetherian" ], "ref_ids": [ 7996 ] } ], "ref_ids": [] }, { "id": 223, "type": "theorem", "label": "spaces-more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ which is locally of finite type. Let $m \\in \\mathbf{Z}$.", "Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. With notation as explained in", "Remark \\ref{remark-match-relative-pseudo-coherence}", "the following are equivalent:", "\\begin{enumerate}", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes and the vertical arrows are \\'etale, the complex", "$E|_U$ is $m$-pseudo-coherent relative to $V$,", "\\item for some commutative diagram as in (1) with $U \\to X$", "surjective, the complex $E|_U$ is $m$-pseudo-coherent relative to $V$,", "\\item for every commutative diagram as in (1) with $U$ and $V$", "affine the complex $R\\Gamma(U, E)$ of $\\mathcal{O}_X(U)$-modules", "is $m$-pseudo-coherent relative to $\\mathcal{O}_Y(V)$.", "\\end{enumerate}" ], "refs": [ "spaces-more-morphisms-remark-match-relative-pseudo-coherence" ], "proofs": [ { "contents": [ "Part (1) implies (3) by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize}.", "\\medskip\\noindent", "Assume (3). Pick any commutative diagram as in (1) with $U \\to X$ surjective.", "Choose an affine open covering $V = \\bigcup V_j$ and affine open coverings", "$(U \\to V)^{-1}(V_j) = \\bigcup U_{ij}$. By (3) and More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize}", "we see that $E|_U$ is $m$-pseudo-coherent relative to $V$.", "Thus (3) implies (2).", "\\medskip\\noindent", "Assume (2). Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes, the vertical arrows are \\'etale, the", "morphism $U \\to X$ is surjective, and $E|_U$ is $m$-pseudo-coherent", "relative to $V$. Next, suppose given a second commutative diagram", "$$", "\\xymatrix{", "U' \\ar[d] \\ar[r] & V' \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with \\'etale vertical arrows and $U', V'$ schemes. We want to show", "that $E|_{U'}$ is $m$-pseudo-coherent relative to $V'$.", "The morphism $U'' = U \\times_X U' \\to U'$ is surjective \\'etale", "and $U'' \\to V'$ factors through $V'' = V' \\times_Y V$ which", "is \\'etale over $V'$. Hence it suffices to show that $E|_{U''}$", "is $m$-pseudo-coherent relative to $V''$, see", "More on Morphisms, Lemmas", "\\ref{more-morphisms-lemma-relative-pseudo-coherent-descends-fppf} and", "\\ref{more-morphisms-lemma-relative-pseudo-coherent-post-compose}.", "Using the second lemma once more it suffices to show that", "$E|_{U''}$ is $m$-pseudo-coherent relative to $V$.", "This is true by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-pull-relative-pseudo-coherent}", "and the fact that an \\'etale morphism of schemes is pseudo-coherent by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-flat-finite-presentation-pseudo-coherent}." ], "refs": [ "more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize", "more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize", "more-morphisms-lemma-relative-pseudo-coherent-descends-fppf", "more-morphisms-lemma-relative-pseudo-coherent-post-compose", "more-morphisms-lemma-pull-relative-pseudo-coherent", "more-morphisms-lemma-flat-finite-presentation-pseudo-coherent" ], "ref_ids": [ 13963, 13963, 14063, 14064, 13972, 13979 ] } ], "ref_ids": [ 313 ] }, { "id": 224, "type": "theorem", "label": "spaces-more-morphisms-lemma-relative-pseudo-coherent-is-moot", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-relative-pseudo-coherent-is-moot", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "algebraic spaces over $S$. Let $E$ in $D_\\QCoh(\\mathcal{O}_X)$.", "If $f$ is flat and locally of finite presentation, then", "the following are equivalent", "\\begin{enumerate}", "\\item $E$ is pseudo-coherent relative to $Y$, and", "\\item $E$ is pseudo-coherent on $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By \\'etale localization and the definitions we may assume", "$X$ and $Y$ are schemes. For the case of schemes this follows", "from More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-check-relative-pseudo-coherence-on-charts}." ], "refs": [ "more-morphisms-lemma-check-relative-pseudo-coherence-on-charts" ], "ref_ids": [ 13974 ] } ], "ref_ids": [] }, { "id": 225, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-base-change-pseudo-coherent", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-base-change-pseudo-coherent", "contents": [ "A flat base change of a pseudo-coherent morphism is pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use the schemes version of this lemma, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-flat-base-change-pseudo-coherent}." ], "refs": [ "more-morphisms-lemma-flat-base-change-pseudo-coherent" ], "ref_ids": [ 13976 ] } ], "ref_ids": [] }, { "id": 226, "type": "theorem", "label": "spaces-more-morphisms-lemma-composition-pseudo-coherent", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-composition-pseudo-coherent", "contents": [ "A composition of pseudo-coherent morphisms is pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use the schemes version of this lemma, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-composition-pseudo-coherent}." ], "refs": [ "more-morphisms-lemma-composition-pseudo-coherent" ], "ref_ids": [ 13977 ] } ], "ref_ids": [] }, { "id": 227, "type": "theorem", "label": "spaces-more-morphisms-lemma-pseudo-coherent-finite-presentation", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-pseudo-coherent-finite-presentation", "contents": [ "A pseudo-coherent morphism is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 228, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-finite-presentation-pseudo-coherent", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-finite-presentation-pseudo-coherent", "contents": [ "A flat morphism which is locally of finite presentation is pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use the schemes version of this lemma, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-flat-finite-presentation-pseudo-coherent}." ], "refs": [ "more-morphisms-lemma-flat-finite-presentation-pseudo-coherent" ], "ref_ids": [ 13979 ] } ], "ref_ids": [] }, { "id": 229, "type": "theorem", "label": "spaces-more-morphisms-lemma-permanence-pseudo-coherent", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-permanence-pseudo-coherent", "contents": [ "Let $f : X \\to Y$ be a morphism of algebraic spaces pseudo-coherent", "over a base algebraic space $B$. Then $f$ is pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use the schemes version of this lemma, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-permanence-pseudo-coherent}." ], "refs": [ "more-morphisms-lemma-permanence-pseudo-coherent" ], "ref_ids": [ 13980 ] } ], "ref_ids": [] }, { "id": 230, "type": "theorem", "label": "spaces-more-morphisms-lemma-Noetherian-pseudo-coherent", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-Noetherian-pseudo-coherent", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. If $Y$ is locally Noetherian, then $f$ is pseudo-coherent if", "and only if $f$ is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use the schemes version of this lemma, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-Noetherian-pseudo-coherent}." ], "refs": [ "more-morphisms-lemma-Noetherian-pseudo-coherent" ], "ref_ids": [ 13982 ] } ], "ref_ids": [] }, { "id": 231, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-base-change-perfect", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-base-change-perfect", "contents": [ "A flat base change of a perfect morphism is perfect." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use the schemes version of this lemma, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-flat-base-change-perfect}." ], "refs": [ "more-morphisms-lemma-flat-base-change-perfect" ], "ref_ids": [ 13988 ] } ], "ref_ids": [] }, { "id": 232, "type": "theorem", "label": "spaces-more-morphisms-lemma-composition-perfect", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-composition-perfect", "contents": [ "A composition of perfect morphisms is perfect." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use the schemes version of this lemma, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-composition-perfect}." ], "refs": [ "more-morphisms-lemma-composition-perfect" ], "ref_ids": [ 13989 ] } ], "ref_ids": [] }, { "id": 233, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-finite-presentation-perfect", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-finite-presentation-perfect", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over", "$S$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is flat and perfect, and", "\\item $f$ is flat and locally of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use the schemes version of this lemma, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-flat-finite-presentation-perfect}." ], "refs": [ "more-morphisms-lemma-flat-finite-presentation-perfect" ], "ref_ids": [ 13990 ] } ], "ref_ids": [] }, { "id": 234, "type": "theorem", "label": "spaces-more-morphisms-lemma-perfect-proper-perfect-direct-image", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-perfect-proper-perfect-direct-image", "contents": [ "Let $S$ be a scheme. Let $Y$ be a Noetherian algebraic space over $S$.", "Let $f : X \\to Y$ be a perfect proper morphism of algebraic spaces.", "Let $E \\in D(\\mathcal{O}_X)$ be perfect. Then", "$Rf_*E$ is a perfect object of $D(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "We claim that Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-perfect-direct-image} applies.", "Conditions (1) and (2) are immediate. Condition (3) is local", "on $X$. Thus we may assume $X$ and $Y$ affine and $E$", "represented by a strictly perfect complex of $\\mathcal{O}_X$-modules.", "Thus it suffices to show that $\\mathcal{O}_X$ has finite", "tor dimension as a sheaf of $f^{-1}\\mathcal{O}_Y$-modules", "on the \\'etale site. By Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-tor-dimension-rel} it suffices to", "check this on the Zariski site. This is equivalent to being perfect", "for finite type morphisms of schemes by More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-check-perfect-stalks}." ], "refs": [ "spaces-perfect-lemma-perfect-direct-image", "spaces-perfect-lemma-tor-dimension-rel", "more-morphisms-lemma-check-perfect-stalks" ], "ref_ids": [ 2728, 2694, 13995 ] } ], "ref_ids": [] }, { "id": 235, "type": "theorem", "label": "spaces-more-morphisms-lemma-lci", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-lci", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a local complete intersection morphism", "of algebraic spaces over $S$.", "Let $P$ be an algebraic space smooth over $Y$.", "Let $U \\to X$ be an \\'etale morphism of algebraic spaces", "and let $i : U \\to P$ an immersion of algebraic spaces over $Y$.", "Picture:", "$$", "\\xymatrix{", "X \\ar[rd] & U \\ar[l] \\ar[d] \\ar[r]_i & P \\ar[ld] \\\\", "& Y", "}", "$$", "Then $i$ is a Koszul-regular immersion of algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Choose a scheme $W$ and a surjective \\'etale morphism $W \\to P \\times_Y V$.", "Set $U' = U \\times_P W$, which is a scheme \\'etale over $U$.", "We have to show that $U' \\to W$ is a Koszul-regular immersion of", "schemes, see", "Definition \\ref{definition-regular-immersion}.", "By", "Definition \\ref{definition-lci}", "above the morphism of schemes $U' \\to V$ is a local complete intersection", "morphism. Hence the result follows from", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-lci}." ], "refs": [ "spaces-more-morphisms-definition-regular-immersion", "spaces-more-morphisms-definition-lci", "more-morphisms-lemma-lci" ], "ref_ids": [ 295, 299, 14001 ] } ], "ref_ids": [] }, { "id": 236, "type": "theorem", "label": "spaces-more-morphisms-lemma-lci-properties", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-lci-properties", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a local complete intersection", "morphism of algebraic spaces over $S$. Then", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $f$ is pseudo-coherent, and", "\\item $f$ is perfect.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use the schemes version of this lemma, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-lci-properties}." ], "refs": [ "more-morphisms-lemma-lci-properties" ], "ref_ids": [ 14002 ] } ], "ref_ids": [] }, { "id": 237, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-base-change-lci", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-base-change-lci", "contents": [ "A flat base change of a local complete intersection morphism is a", "local complete intersection morphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use the schemes version of this lemma, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-flat-base-change-lci}." ], "refs": [ "more-morphisms-lemma-flat-base-change-lci" ], "ref_ids": [ 14004 ] } ], "ref_ids": [] }, { "id": 238, "type": "theorem", "label": "spaces-more-morphisms-lemma-composition-lci", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-composition-lci", "contents": [ "A composition of local complete intersection morphisms is a", "local complete intersection morphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use the schemes version of this lemma, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-composition-lci}." ], "refs": [ "more-morphisms-lemma-composition-lci" ], "ref_ids": [ 14005 ] } ], "ref_ids": [] }, { "id": 239, "type": "theorem", "label": "spaces-more-morphisms-lemma-flat-lci", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-flat-lci", "contents": [ "\\begin{slogan}", "Syntomic equals flat plus lci (for algebraic spaces).", "\\end{slogan}", "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is flat and a local complete intersection morphism, and", "\\item $f$ is syntomic.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use the schemes version of this lemma, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-flat-lci}." ], "refs": [ "more-morphisms-lemma-flat-lci" ], "ref_ids": [ 14006 ] } ], "ref_ids": [] }, { "id": 240, "type": "theorem", "label": "spaces-more-morphisms-lemma-regular-immersion-lci", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-regular-immersion-lci", "contents": [ "Let $S$ be a scheme. A Koszul-regular immersion of algebraic spaces", "over $S$ is a local complete intersection morphism." ], "refs": [], "proofs": [ { "contents": [ "Let $i : X \\to Y$ be a Koszul-regular immersion of algebraic spaces", "over $S$. By definition there exists a surjective \\'etale morphism", "$V \\to Y$ where $V$ is a scheme such that $X \\times_Y V$ is a scheme", "and the base change $X \\times_Y V \\to V$ is a Koszul-regular immersion of", "schemes. By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-regular-immersion-lci} we see that", "$X \\times_Y V \\to V$ is a local complete intersection morphism.", "From Definition \\ref{definition-lci} we conclude that $i$ is a", "local complete intersection morphism of algebraic spaces." ], "refs": [ "more-morphisms-lemma-regular-immersion-lci", "spaces-more-morphisms-definition-lci" ], "ref_ids": [ 14007, 299 ] } ], "ref_ids": [] }, { "id": 241, "type": "theorem", "label": "spaces-more-morphisms-lemma-lci-permanence", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-lci-permanence", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\", "& Z", "}", "$$", "be a commutative diagram of morphisms of algebraic spaces over $S$.", "Assume $Y \\to Z$ is smooth and $X \\to Z$ is a", "local complete intersection morphism.", "Then $f : X \\to Y$ is a local complete intersection morphism." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $W$ and a surjective \\'etale morphism $W \\to Z$.", "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to W \\times_Z Y$.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to V \\times_Y X$.", "Then $U \\to W$ is a local complete intersection morphism of schemes and", "$V \\to W$ is a smooth morphism of schemes. By the result for schemes", "(More on Morphisms, Lemma \\ref{more-morphisms-lemma-lci-permanence})", "we conclude that $U \\to V$ is a local complete intersection morphism.", "By definition this means that $f$ is a local complete intersection morphism." ], "refs": [ "more-morphisms-lemma-lci-permanence" ], "ref_ids": [ 14008 ] } ], "ref_ids": [] }, { "id": 242, "type": "theorem", "label": "spaces-more-morphisms-lemma-descending-property-lci", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-descending-property-lci", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is a local complete intersection", "morphism'' is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a", "morphism of algebraic spaces over $S$.", "Let $\\{Y_i \\to Y\\}$ be an fpqc covering", "(Topologies on Spaces, Definition", "\\ref{spaces-topologies-definition-fpqc-covering}).", "Let $f_i : X_i \\to Y_i$ be the base change of $f$ by $Y_i \\to Y$.", "If $f$ is a local complete intersection morphism,", "then each $f_i$ is a local complete intersection morphism", "by Lemma \\ref{lemma-flat-base-change-lci}.", "\\medskip\\noindent", "Conversely, assume each $f_i$ is a local complete intersection morphism.", "We may replace the covering by a refinement (again because", "flat base change preserves the property of being a", "local complete intersection morphism). Hence we may assume", "$Y_i$ is a scheme for each $i$, see", "Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-refine-fpqc-schemes}.", "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Choose a scheme $U$ and a surjective \\'etale morphism", "$U \\to V \\times_Y X$. We have to show that $U \\to V$ is a", "local complete intersection morphism of schemes.", "By Topologies on Spaces, Lemma", "\\ref{spaces-topologies-lemma-recognize-fpqc-covering}", "we have that $\\{Y_i \\times_Y V \\to V\\}$ is an fpqc covering", "of schemes. By the case of schemes", "(More on Morphisms, Lemma \\ref{more-morphisms-lemma-descending-property-lci})", "it suffices to prove the base change", "$$", "U \\times_Y Y_i = U \\times_V (V \\times_Y Y_i) \\longrightarrow V", "$$", "of $U \\to V$ by $V \\times_Y Y_i \\to V$ is a", "local complete intersection morphism. We can write this as the", "composition", "$$", "U \\times_Y Y_i \\longrightarrow", "(V \\times_Y X) \\times_Y Y_i = V \\times_Y X_i \\longrightarrow", "V \\times_Y Y_i", "$$", "The first arrow is an \\'etale morphism of schemes (as a base change of", "$U \\to V \\times_Y X$) and the second arrow is a", "local complete intersection morphism of schemes as a flat base change of $f_i$.", "The result follows as being a local complete intersection morphism", "is syntomic local on the source and since \\'etale morphisms", "are syntomic (More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-lci-syntomic-local-source}", "and Morphisms, Lemma \\ref{morphisms-lemma-etale-syntomic})." ], "refs": [ "spaces-topologies-definition-fpqc-covering", "spaces-more-morphisms-lemma-flat-base-change-lci", "spaces-topologies-lemma-refine-fpqc-schemes", "spaces-topologies-lemma-recognize-fpqc-covering", "more-morphisms-lemma-descending-property-lci", "more-morphisms-lemma-lci-syntomic-local-source", "morphisms-lemma-etale-syntomic" ], "ref_ids": [ 3694, 237, 3680, 3679, 14017, 14018, 5367 ] } ], "ref_ids": [] }, { "id": 243, "type": "theorem", "label": "spaces-more-morphisms-lemma-lci-syntomic-local-source", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-lci-syntomic-local-source", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is a local complete intersection", "morphism'' is syntomic local on the source." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-transfer-from-schemes} and", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-lci-syntomic-local-source}." ], "refs": [ "spaces-descent-lemma-transfer-from-schemes", "more-morphisms-lemma-lci-syntomic-local-source" ], "ref_ids": [ 9416, 14018 ] } ], "ref_ids": [] }, { "id": 244, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-lci-fibres", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-lci-fibres", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\", "& Z", "}", "$$", "of algebraic spaces over $S$. Assume that both $p$ and $q$", "are flat and locally of finite presentation.", "Then there exists an open subspace $U(f) \\subset X$", "such that $|U(f)| \\subset |X|$ is the set of points where $f$ is Koszul.", "Moreover, for any morphism of algebraic spaces $Z' \\to Z$, if", "$f' : X' \\to Y'$ is the base change of $f$ by $Z' \\to Z$, then", "$U(f')$ is the inverse image of $U(f)$ under the projection $X' \\to X$." ], "refs": [], "proofs": [ { "contents": [ "This lemma is the analogue of", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-base-change-lci-fibres}", "and in fact we will deduce the lemma from it. By", "Definition \\ref{definition-lci}", "the set $\\{x \\in |X| : f \\text{ is Koszul at }x\\}$ is", "open in $|X|$ hence by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-open-subspaces}", "it corresponds to an open subspace $U(f)$ of $X$. Hence we only need to", "prove the final statement.", "\\medskip\\noindent", "Choose a scheme $W$ and a surjective \\'etale morphism $W \\to Z$.", "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to W \\times_Z Y$.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to V \\times_Y X$.", "Finally, choose a scheme $W'$ and a surjective \\'etale morphism", "$W' \\to W \\times_Z Z'$.", "Set $V' = W' \\times_W V$ and $U' = W' \\times_W U$, so that we obtain", "surjective \\'etale morphisms $V' \\to Y'$ and $U' \\to X'$.", "We will use without further mention an \\'etale morphism of algebraic spaces", "induces an open map of associated topological spaces (see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-etale-open}).", "Note that by definition $U(f)$ is the image in $|X|$ of the set $T$", "of points in $U$ where the morphism of schemes $U \\to V$ is Koszul.", "Similarly, $U(f')$ is the image in $|X'|$ of the set $T'$ of points in", "$U'$ where the morphism of schemes $U' \\to V'$ is Koszul. Now, by construction", "the diagram", "$$", "\\xymatrix{", "U' \\ar[r] \\ar[d] & U \\ar[d] \\\\", "V' \\ar[r] & V", "}", "$$", "is cartesian (in the category of schemes). Hence the aforementioned", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-base-change-lci-fibres}", "applies to show that $T'$ is the inverse image of $T$. Since", "$|U'| \\to |X'|$ is surjective this implies the lemma." ], "refs": [ "more-morphisms-lemma-base-change-lci-fibres", "spaces-more-morphisms-definition-lci", "spaces-properties-lemma-open-subspaces", "spaces-properties-lemma-etale-open", "more-morphisms-lemma-base-change-lci-fibres" ], "ref_ids": [ 14019, 299, 11823, 11860, 14019 ] } ], "ref_ids": [] }, { "id": 245, "type": "theorem", "label": "spaces-more-morphisms-lemma-unramified-lci", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-unramified-lci", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a local complete intersection", "morphism of algebraic spaces over $S$. Then $f$ is unramified if and only", "if $f$ is formally unramified and in this case the conormal sheaf", "$\\mathcal{C}_{X/Y}$ is finite locally free on $X$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the corresponding result for morphisms of schemes, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-unramified-lci},", "by \\'etale localization, see", "Lemma \\ref{lemma-universal-thickening-localize}.", "(Note that in the situation of this lemma the morphism $V \\to U$", "is unramified and a local complete intersection morphism by definition.)" ], "refs": [ "more-morphisms-lemma-unramified-lci", "spaces-more-morphisms-lemma-universal-thickening-localize" ], "ref_ids": [ 14020, 84 ] } ], "ref_ids": [] }, { "id": 246, "type": "theorem", "label": "spaces-more-morphisms-lemma-transitivity-conormal-lci", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-transitivity-conormal-lci", "contents": [ "Let $S$ be a scheme. Let $Z \\to Y \\to X$ be formally unramified morphisms", "of algebraic spaces over $S$. Assume that $Z \\to Y$ is a local complete", "intersection morphism. The exact sequence", "$$", "0 \\to i^*\\mathcal{C}_{Y/X} \\to", "\\mathcal{C}_{Z/X} \\to", "\\mathcal{C}_{Z/Y} \\to 0", "$$", "of", "Lemma \\ref{lemma-transitivity-conormal}", "is short exact." ], "refs": [ "spaces-more-morphisms-lemma-transitivity-conormal" ], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to U \\times_X Y$.", "Choose a scheme $W$ and a surjective \\'etale morphism $W \\to V \\times_Y Z$.", "By", "Lemma \\ref{lemma-universal-thickening-localize}", "the morphisms $W \\to V$ and $V \\to U$ are formally unramified.", "Moreover the sequence", "$i^*\\mathcal{C}_{Y/X} \\to \\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Y} \\to 0$", "restricts to the corresponding sequence", "$i^*\\mathcal{C}_{V/U} \\to \\mathcal{C}_{W/U} \\to \\mathcal{C}_{W/V} \\to 0$", "for $W \\to V \\to U$. Hence the result follows from the result for schemes", "(More on Morphisms, Lemma \\ref{more-morphisms-lemma-transitivity-conormal-lci})", "as by definition the morphism $W \\to V$ is a local complete intersection", "morphism." ], "refs": [ "spaces-more-morphisms-lemma-universal-thickening-localize", "more-morphisms-lemma-transitivity-conormal-lci" ], "ref_ids": [ 84, 14021 ] } ], "ref_ids": [ 27 ] }, { "id": 247, "type": "theorem", "label": "spaces-more-morphisms-lemma-where-unramified", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-where-unramified", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\", "& Z", "}", "$$", "of algebraic spaces. Assume that $p$ is locally of finite type and closed.", "Then there exists an open subspace $W \\subset Z$", "such that a morphism $Z' \\to Z$ factors through $W$ if and only if the", "base change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is unramified." ], "refs": [], "proofs": [ { "contents": [ "By", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-where-unramified}", "there exists an open subspace $U(f) \\subset X$ which is the set of", "points where $f$ is unramified. Moreover, formation of $U(f)$ commutes", "with arbitrary base change. Let $W \\subset Z$ be the open subspace", "(see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-open-subspaces})", "with underlying set of points", "$$", "|W| = |Z| \\setminus |p|\\left(|X| \\setminus |U(f)|\\right)", "$$", "i.e., $z \\in |Z|$ is a point of $W$ if and only if $f$ is unramified", "at every point of $X$ above $z$. Note that this is open because we", "assumed that $p$ is closed. Since the formation of $U(f)$", "commutes with arbitrary base change we immediately see (using", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-factor-through-open-subspace})", "that $W$ has the desired universal property." ], "refs": [ "spaces-morphisms-lemma-where-unramified", "spaces-properties-lemma-open-subspaces", "spaces-properties-lemma-factor-through-open-subspace" ], "ref_ids": [ 4903, 11823, 11824 ] } ], "ref_ids": [] }, { "id": 248, "type": "theorem", "label": "spaces-more-morphisms-lemma-where-unramified-universally-injective", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-where-unramified-universally-injective", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\", "& Z", "}", "$$", "of algebraic spaces. Assume that", "\\begin{enumerate}", "\\item $p$ is locally of finite type,", "\\item $p$ is closed, and", "\\item $p_2 : X \\times_Y X \\to Z$ is closed.", "\\end{enumerate}", "Then there exists an open subspace $W \\subset Z$", "such that a morphism $Z' \\to Z$ factors through $W$ if and only if the", "base change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is unramified and universally", "injective." ], "refs": [], "proofs": [ { "contents": [ "After replacing $Z$ by the open subspace found in", "Lemma \\ref{lemma-where-unramified}", "we may assume that $f$ is already unramified; note that this does not", "destroy assumption (2) or (3). By", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-diagonal-unramified-morphism}", "we see that $\\Delta_{X/Y} : X \\to X \\times_Y X$ is an open immersion.", "This remains true after any base change. Hence by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-injective}", "we see that $f_{Z'}$ is universally injective if and only if", "the base change of the diagonal $X_{Z'} \\to (X \\times_Y X)_{Z'}$", "is an isomorphism. Let $W \\subset Z$ be the open subspace", "(see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-open-subspaces})", "with underlying set of points", "$$", "|W| = |Z| \\setminus", "|p_2|\\left(|X \\times_Y X| \\setminus \\Im(|\\Delta_{X/Y}|)\\right)", "$$", "i.e., $z \\in |Z|$ is a point of $W$ if and only if the fibre of", "$|X \\times_Y X| \\to |Z|$ over $z$ is in the image of", "$|X| \\to |X \\times_Y X|$. Then it is clear from the discussion above", "that the restriction $p^{-1}(W) \\to q^{-1}(W)$ of $f$ is", "unramified and universally injective.", "\\medskip\\noindent", "Conversely, suppose that $f_{Z'}$ is unramified and universally injective.", "In order to show that $Z' \\to Z$ factors through $W$ it suffices to show", "that $|Z'| \\to |Z|$ has image contained in $|W|$, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-factor-through-open-subspace}.", "Hence it suffices to prove the result when $Z'$ is the spectrum of a field.", "Denote $z \\in |Z|$ the image of $|Z'| \\to |Z|$. The discussion above shows", "that", "$$", "|X_{Z'}| \\longrightarrow |(X \\times_Y X)_{Z'}|", "$$", "is surjective. By", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-points-cartesian}", "in the commutative diagram", "$$", "\\xymatrix{", "|X_{Z'}| \\ar[d] \\ar[r] &", "|(X \\times_Y X)_{Z'}| \\ar[d] \\\\", "|p|^{-1}(\\{z\\}) \\ar[r] &", "|p_2|^{-1}(\\{z\\})", "}", "$$", "the vertical arrows are surjective. It follows that $z \\in |W|$ as desired." ], "refs": [ "spaces-more-morphisms-lemma-where-unramified", "spaces-morphisms-lemma-diagonal-unramified-morphism", "spaces-morphisms-lemma-universally-injective", "spaces-properties-lemma-open-subspaces", "spaces-properties-lemma-factor-through-open-subspace", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 247, 4902, 4793, 11823, 11824, 11819 ] } ], "ref_ids": [] }, { "id": 249, "type": "theorem", "label": "spaces-more-morphisms-lemma-where-closed-immersion", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-where-closed-immersion", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\", "& Z", "}", "$$", "of algebraic spaces. Assume that", "\\begin{enumerate}", "\\item $p$ is locally of finite type,", "\\item $p$ is universally closed, and", "\\item $q : Y \\to Z$ is separated.", "\\end{enumerate}", "Then there exists an open subspace $W \\subset Z$", "such that a morphism $Z' \\to Z$ factors through $W$ if and only if the", "base change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "We will use the characterization of closed immersions as", "universally closed, unramified, and universally injective morphisms, see", "Lemma \\ref{lemma-characterize-closed-immersion}.", "First, note that since $p$ is universally closed and $q$ is", "separated, we see that $f$ is universally closed, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-permanence}.", "It follows that any base change of $f$ is universally closed, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-universally-closed}.", "Thus to finish the proof of the lemma it suffices to prove that", "the assumptions of", "Lemma \\ref{lemma-where-unramified-universally-injective}", "are satisfied. The projection $\\text{pr}_0 : X \\times_Y X \\to X$", "is universally closed as a base change of $f$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-universally-closed}.", "Hence $X \\times_Y X \\to Z$ is universally closed as", "a composition of universally closed morphisms (see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-composition-universally-closed}).", "This finishes the proof of the lemma." ], "refs": [ "spaces-more-morphisms-lemma-characterize-closed-immersion", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-morphisms-lemma-base-change-universally-closed", "spaces-more-morphisms-lemma-where-unramified-universally-injective", "spaces-morphisms-lemma-base-change-universally-closed", "spaces-morphisms-lemma-composition-universally-closed" ], "ref_ids": [ 74, 4920, 4746, 248, 4746, 4747 ] } ], "ref_ids": [] }, { "id": 250, "type": "theorem", "label": "spaces-more-morphisms-lemma-where-flat", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-where-flat", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\", "& Z", "}", "$$", "of algebraic spaces. Assume that", "\\begin{enumerate}", "\\item $p$ is locally of finite presentation,", "\\item $p$ is flat,", "\\item $p$ is closed, and", "\\item $q$ is locally of finite type.", "\\end{enumerate}", "Then there exists an open subspace $W \\subset Z$", "such that a morphism $Z' \\to Z$ factors through $W$ if and only if the", "base change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is flat." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-base-change-flatness-fibres}", "the set", "$$", "A = \\{x \\in |X| : X\\text{ flat at }x \\text{ over }Y\\}.", "$$", "is open in $|X|$ and its formation commutes with arbitrary base", "change. Let $W \\subset Z$ be the open subspace", "(see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-open-subspaces})", "with underlying set of points", "$$", "|W| = |Z| \\setminus |p|\\left(|X| \\setminus A\\right)", "$$", "i.e., $z \\in |Z|$ is a point of $W$ if and only if the whole fibre", "of $|X| \\to |Z|$ over $z$ is contained in $A$. This is open because", "$p$ is closed. Since the formation of $A$ commutes with arbitrary", "base change it follows that $W$ works." ], "refs": [ "spaces-more-morphisms-lemma-base-change-flatness-fibres", "spaces-properties-lemma-open-subspaces" ], "ref_ids": [ 133, 11823 ] } ], "ref_ids": [] }, { "id": 251, "type": "theorem", "label": "spaces-more-morphisms-lemma-where-surjective-flat", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-where-surjective-flat", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\", "& Z", "}", "$$", "of algebraic spaces. Assume that", "\\begin{enumerate}", "\\item $p$ is locally of finite presentation,", "\\item $p$ is flat,", "\\item $p$ is closed,", "\\item $q$ is locally of finite type, and", "\\item $q$ is closed.", "\\end{enumerate}", "Then there exists an open subspace $W \\subset Z$", "such that a morphism $Z' \\to Z$ factors through $W$ if and only if the", "base change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is surjective and flat." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-where-flat}", "we may assume that $f$ is flat.", "Note that $f$ is locally of finite presentation by", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-finite-presentation-permanence}.", "Hence $f$ is open, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-fppf-open}.", "Let $W \\subset Z$ be the open subspace (see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-open-subspaces})", "with underlying set of points", "$$", "|W| = |Z| \\setminus |q|\\left(|Y| \\setminus |f|(|X|)\\right).", "$$", "in other words for $z \\in |Z|$ we have $z \\in |W|$ if and only", "if the whole fibre of $|Y| \\to |Z|$ over $z$ is in the image of", "$|X| \\to |Y|$. Since $q$ is closed this set is open in $|Z|$.", "The morphism $X_W \\to Y_W$ is surjective by construction.", "Finally, suppose that $X_{Z'} \\to Y_{Z'}$ is surjective.", "In order to show that $Z' \\to Z$ factors through $W$ it suffices to show", "that $|Z'| \\to |Z|$ has image contained in $|W|$, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-factor-through-open-subspace}.", "Hence it suffices to prove the result when $Z'$ is the spectrum of a field.", "Denote $z \\in |Z|$ the image of $|Z'| \\to |Z|$. By", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-points-cartesian}", "in the commutative diagram", "$$", "\\xymatrix{", "|X_{Z'}| \\ar[d] \\ar[r] &", "|Y_{Z'}| \\ar[d] \\\\", "|p|^{-1}(\\{z\\}) \\ar[r] &", "|q|^{-1}(\\{z\\})", "}", "$$", "the vertical arrows are surjective. It follows that $z \\in |W|$ as desired." ], "refs": [ "spaces-more-morphisms-lemma-where-flat", "spaces-morphisms-lemma-finite-presentation-permanence", "spaces-morphisms-lemma-fppf-open", "spaces-properties-lemma-open-subspaces", "spaces-properties-lemma-factor-through-open-subspace", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 250, 4846, 4855, 11823, 11824, 11819 ] } ], "ref_ids": [] }, { "id": 252, "type": "theorem", "label": "spaces-more-morphisms-lemma-where-isomorphism", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-where-isomorphism", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\", "& Z", "}", "$$", "of algebraic spaces. Assume that", "\\begin{enumerate}", "\\item $p$ is locally of finite presentation,", "\\item $p$ is flat,", "\\item $p$ is universally closed,", "\\item $q$ is locally of finite type,", "\\item $q$ is closed, and", "\\item $q$ is separated.", "\\end{enumerate}", "Then there exists an open subspace $W \\subset Z$", "such that a morphism $Z' \\to Z$ factors through $W$ if and only if the", "base change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-where-surjective-flat}", "there exists an open subspace $W_1 \\subset Z$ such that", "$f_{Z'}$ is surjective and flat if and only if $Z' \\to Z$", "factors through $W_1$. By", "Lemma \\ref{lemma-where-closed-immersion}", "there exists an open subspace $W_2 \\subset Z$ such that", "$f_{Z'}$ is a closed immersion if and only if $Z' \\to Z$", "factors through $W_2$. We claim that $W = W_1 \\cap W_2$ works.", "Certainly, if $f_{Z'}$ is an isomorphism, then $Z' \\to Z$", "factors through $W$. Hence it suffices to show that", "$f_W$ is an isomorphism. By construction $f_W$ is a", "surjective flat closed immersion. In particular $f_W$ is", "representable. Since a surjective flat closed immersion of", "schemes is an isomorphism (see", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-flat-closed-immersions})", "we win. (Note that actually $f_W$ is locally of finite presentation,", "whence open, so you can avoid the use of this lemma if you like.)" ], "refs": [ "spaces-more-morphisms-lemma-where-surjective-flat", "spaces-more-morphisms-lemma-where-closed-immersion", "morphisms-lemma-characterize-flat-closed-immersions" ], "ref_ids": [ 251, 249, 5274 ] } ], "ref_ids": [] }, { "id": 253, "type": "theorem", "label": "spaces-more-morphisms-lemma-where-lci", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-where-lci", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\", "& Z", "}", "$$", "of algebraic spaces. Assume that", "\\begin{enumerate}", "\\item $p$ is flat and locally of finite presentation,", "\\item $p$ is closed, and", "\\item $q$ is flat and locally of finite presentation,", "\\end{enumerate}", "Then there exists an open subspace $W \\subset Z$", "such that a morphism $Z' \\to Z$ factors through $W$ if and only if the", "base change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is a local complete intersection", "morphism." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-base-change-lci-fibres}", "there exists an open subspace $U(f) \\subset X$ which is the set of", "points where $f$ is Koszul. Moreover, formation of $U(f)$ commutes", "with arbitrary base change. Let $W \\subset Z$ be the open subspace", "(see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-open-subspaces})", "with underlying set of points", "$$", "|W| = |Z| \\setminus |p|\\left(|X| \\setminus |U(f)|\\right)", "$$", "i.e., $z \\in |Z|$ is a point of $W$ if and only if $f$ is Koszul", "at every point of $X$ above $z$. Note that this is open because we", "assumed that $p$ is closed. Since the formation of $U(f)$", "commutes with arbitrary base change we immediately see (using", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-factor-through-open-subspace})", "that $W$ has the desired universal property." ], "refs": [ "spaces-more-morphisms-lemma-base-change-lci-fibres", "spaces-properties-lemma-open-subspaces", "spaces-properties-lemma-factor-through-open-subspace" ], "ref_ids": [ 244, 11823, 11824 ] } ], "ref_ids": [] }, { "id": 254, "type": "theorem", "label": "spaces-more-morphisms-lemma-case-of-tor-independence", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-case-of-tor-independence", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram of algebraic spaces", "$$", "\\xymatrix{", "Z' \\ar[d] \\ar[r] & Y' \\ar[d] \\\\", "X' \\ar[r] & B'", "}", "$$", "over $S$.", "Let $B \\to B'$ be a morphism. Denote by $X$ and $Y$ the base", "changes of $X'$ and $Y'$ to $B$.", "Assume $Y' \\to B'$ and $Z' \\to X'$ are flat.", "Then $X \\times_B Y$ and $Z'$ are Tor independent over $X' \\times_{B'} Y'$." ], "refs": [], "proofs": [ { "contents": [ "By Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-tor-independent}", "we may check tor independence \\'etale locally on $X \\times_B Y$", "and $Z'$. This\\footnote{Here is the argument in more detail.", "Choose a surjective \\'etale morphism $W' \\to B'$", "with $W'$ a scheme. Choose a surjective \\'etale morphism", "$W \\to B \\times_{B'} W'$ with $W$ a scheme. Choose a", "surjective \\'etale morphism", "$U' \\to X' \\times_{B'} W'$ with $U'$ a scheme. Choose a", "surjective \\'etale morphism $V' \\to Y' \\times_{B'} W'$ with $V'$ a scheme.", "Observe that $U' \\times_{W'} V' \\to X' \\times_{B'} Y'$ is surjective", "\\'etale. Choose a surjective \\'etale morphism", "$T' \\to Z' \\times_{X' \\times_{B'} Y'} U' \\times_{W'} V'$", "with $T'$ a scheme. Denote $U$ and $V$ the base changes of $U'$ and $V'$", "to $W$. Then the lemma says that $X \\times_B Y$ and $Z'$", "are Tor independent over $X' \\times_{B'} Y'$ as algebraic spaces", "if and only if $U \\times_W V$ and $T'$ are Tor independent over", "$U' \\times_{W'} V'$ as schemes. Thus", "it suffices to prove the lemma for", "the square with corners $T', U', V', W'$ and base change by $W \\to W'$.", "The flatness of $Y' \\to B'$ and $Z' \\to X'$ implies flatness", "of $V' \\to W'$ and $T' \\to U'$.}", "reduces the lemma to the case of schemes", "which is More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-case-of-tor-independence}." ], "refs": [ "spaces-perfect-lemma-tor-independent", "more-morphisms-lemma-case-of-tor-independence" ], "ref_ids": [ 2719, 14057 ] } ], "ref_ids": [] }, { "id": 255, "type": "theorem", "label": "spaces-more-morphisms-lemma-derived-chow", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-derived-chow", "contents": [ "Let $A$ be a ring. Let $X$ be a separated algebraic space", "of finite presentation over $A$. Let $x \\in |X|$. Then there exist", "an $n \\geq 0$,", "a closed subspace $Z \\subset X \\times_A \\mathbf{P}^n_A$,", "a point $z \\in |Z|$,", "an open $V \\subset \\mathbf{P}^n_A$, and", "an object $E$ in $D(\\mathcal{O}_{X \\times_A \\mathbf{P}^n_A})$ such that", "\\begin{enumerate}", "\\item $Z \\to X \\times_A \\mathbf{P}^n_A$ is of finite presentation,", "\\item $c : Z \\to \\mathbf{P}^n_A$ is a closed immersion over $V$,", "set $W = c^{-1}(V)$,", "\\item the restriction of $b : Z \\to X$ to $W$ is \\'etale,", "$z \\in W$, and $b(z) = x$,", "\\item $E|_{X \\times_A V} \\cong", "(b, c)_*\\mathcal{O}_Z|_{X \\times_A V}$,", "\\item $E$ is pseudo-coherent and supported on $Z$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We can find a finite type $\\mathbf{Z}$-subalgebra $A' \\subset A$", "and an algebraic space $X'$ separated and of finite presentation over $A'$", "whose base change to $A$ is $X$. See", "Limits of Spaces, Lemmas", "\\ref{spaces-limits-lemma-descend-finite-presentation} and", "\\ref{spaces-limits-lemma-descend-separated-morphism}.", "Let $x' \\in |X'|$ be the image of $x$.", "If we can prove the lemma for $(X'/A', x')$, then", "the lemma follows for $(X/A, x)$.", "Namely, if $n', Z', z', V', E'$ provide the solution", "for $(X'/A', x')$, then we can let", "$n = n'$,", "let $Z \\subset X \\times \\mathbf{P}^n$ be the inverse image of $Z'$,", "let $z \\in Z$ be the unique point mapping to $x$,", "let $V \\subset \\mathbf{P}^n_A$ be the inverse image of $V'$, and", "let $E$ be the derived pullback of $E'$.", "Observe that $E$ is pseudo-coherent by", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-pseudo-coherent-pullback}.", "It only remains to check (5). To see this", "set $W = c^{-1}(V)$ and $W' = (c')^{-1}(V')$", "and consider the cartesian square", "$$", "\\xymatrix{", "W \\ar[d]_{(b, c)} \\ar[r] & W' \\ar[d]^{(b', c')} \\\\", "X \\times_A V \\ar[r] & X' \\times_{A'} V'", "}", "$$", "By Lemma \\ref{lemma-case-of-tor-independence} $X \\times_A V$ and $W'$", "are tor-independent over $X' \\times_{A'} V'$.", "Thus the derived pullback of", "$(b', c')_*\\mathcal{O}_{W'}$ to $X \\times_A V$", "is $(b, c)_*\\mathcal{O}_W$ by", "Derived Categories of Spaces,", "Lemma \\ref{spaces-perfect-lemma-compare-base-change}.", "This also uses that $R(b', c')_*\\mathcal{O}_{Z'} = (b', c')_*\\mathcal{O}_{Z'}$", "because $(b', c')$ is a closed immersion and simiarly for", "$(b, c)_*\\mathcal{O}_Z$.", "Since $E'|_{U' \\times_{A'} V'} =", "(b', c')_*\\mathcal{O}_{W'}$ we obtain", "$E|_{U \\times_A V} = (b, c)_*\\mathcal{O}_W$", "and (5) holds.", "This reduces us to the situation described in the next", "paragraph.", "\\medskip\\noindent", "Assume $A$ is of finite type over $\\mathbf{Z}$.", "Choose an \\'etale morphism $U \\to X$ where $U$ is an affine scheme", "and a point $u \\in U$ mapping to $x$. Then $U$ is of finite type over $A$.", "Choose a closed immersion $U \\to \\mathbf{A}^n_A$ and denote", "$j : U \\to \\mathbf{P}^n_A$ the immersion we get by composing", "with the open immersion $\\mathbf{A}^n_A \\to \\mathbf{P}^n_A$.", "Let $Z$ be the scheme theoretic closure of", "$$", "(\\text{id}_U, j) : U \\longrightarrow X \\times_A \\mathbf{P}^n_A", "$$", "Let $z \\in Z$ be the image of $u$.", "Let $Y \\subset \\mathbf{P}^n_A$ be the scheme theoretic", "closure of $j$. Then it is clear that $Z \\subset X \\times_A Y$", "is the scheme theoretic closure of", "$(\\text{id}_U, j) : U \\to X \\times_A Y$.", "As $X$ is separated, the morphism", "$X \\times_A Y \\to Y$ is separated as well.", "Hence we see that $Z \\to Y$ is an isomorphism over", "the open subscheme $j(U) \\subset Y$ by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-scheme-theoretic-image-of-partial-section}.", "Choose $V \\subset \\mathbf{P}^n_A$ open with $V \\cap Y = j(U)$.", "Then we see that (2) holds, that $W = (\\text{id}_U, j)(U)$, and hence", "that (3) holds. Part (1) holds because $A$ is Noetherian.", "\\medskip\\noindent", "Because $A$ is Noetherian we see that $X$ and $X \\times_A \\mathbf{P}^n_A$", "are Noetherian algebraic spaces. Hence we can take $E = (b, c)_*\\mathcal{O}_Z$", "in this case: (4) is clear and for (5) see", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-identify-pseudo-coherent-noetherian}.", "This finishes the proof." ], "refs": [ "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-separated-morphism", "sites-cohomology-lemma-pseudo-coherent-pullback", "spaces-more-morphisms-lemma-case-of-tor-independence", "spaces-perfect-lemma-compare-base-change", "spaces-perfect-lemma-identify-pseudo-coherent-noetherian" ], "ref_ids": [ 4598, 4592, 4367, 254, 2720, 2697 ] } ], "ref_ids": [] }, { "id": 256, "type": "theorem", "label": "spaces-more-morphisms-lemma-compute-Fourier-Mukai-for-derived-chow", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-compute-Fourier-Mukai-for-derived-chow", "contents": [ "Let $X/A$, $x \\in |X|$, and", "$n, Z, z, V, E$ be as in Lemma \\ref{lemma-derived-chow}.", "For any $K \\in D_\\QCoh(\\mathcal{O}_X)$ we have", "$$", "Rq_*(Lp^*K \\otimes^\\mathbf{L} E)|_V = R(W \\to V)_*K|_W", "$$", "where $p : X \\times_A \\mathbf{P}^n_A \\to X$ and", "$q : X \\times_A \\mathbf{P}^n_A \\to \\mathbf{P}^n_A$ are", "the projections and where the morphism $W \\to V$ is", "the finitely presented closed immersion $c|_W : W \\to V$." ], "refs": [ "spaces-more-morphisms-lemma-derived-chow" ], "proofs": [ { "contents": [ "Since $W = c^{-1}(V)$ and since $c$ is a closed immersion", "over $V$, we see that $c|_W$ is a closed immersion.", "It is of finite presentation because $W$ and $V$ are of finite", "presentation over $A$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-permanence}.", "First we have", "$$", "Rq_*(Lp^*K \\otimes^\\mathbf{L} E)|_V =", "Rq'_*\\left((Lp^*K \\otimes^\\mathbf{L} E)|_{X \\times_A V}\\right)", "$$", "where $q' : X \\times_A V \\to V$ is the projection because", "formation of total direct image commutes with localization.", "Denote $i = (b, c)|_W : W \\to X \\times_A V$ the given closed immersion.", "Then", "$$", "Rq'_*\\left((Lp^*K \\otimes^\\mathbf{L} E)|_{X \\times_A V}\\right) =", "Rq'_*(Lp^*K|_{X \\times_A V} \\otimes^\\mathbf{L} i_*\\mathcal{O}_W)", "$$", "by property (5). Since $i$ is a closed immersion we have", "$i_*\\mathcal{O}_W = Ri_*\\mathcal{O}_W$.", "Using", "Derived Categories of Spaces,", "Lemma \\ref{spaces-perfect-lemma-cohomology-base-change}", "we can rewrite this as", "$$", "Rq'_* Ri_* Li^* Lp^*K|_{X \\times_A V} =", "R(q' \\circ i)_* Lb^*K|_W =", "R(W \\to V)_* K|_W", "$$", "which is what we want. (Note that restricting to $W$", "and derived pulling back via $W \\to X$ is the same thing", "as $W$ is \\'etale over $X$.)" ], "refs": [ "spaces-morphisms-lemma-finite-presentation-permanence", "spaces-perfect-lemma-cohomology-base-change" ], "ref_ids": [ 4846, 2718 ] } ], "ref_ids": [ 255 ] }, { "id": 257, "type": "theorem", "label": "spaces-more-morphisms-lemma-characterize-pseudo-coherent", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-characterize-pseudo-coherent", "contents": [ "Let $A$ be a ring. Let $X$ be an algebraic space separated and", "of finite presentation over $A$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$.", "If $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent", "in $D(A)$ for every pseudo-coherent $E$ in $D(\\mathcal{O}_X)$,", "then $K$ is pseudo-coherent relative to $A$", "(Definition \\ref{definition-relative-pseudo-coherence})." ], "refs": [ "spaces-more-morphisms-definition-relative-pseudo-coherence" ], "proofs": [ { "contents": [ "Assume $K \\in D_\\QCoh(\\mathcal{O}_X)$ and", "$R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent", "in $D(A)$ for every pseudo-coherent $E$ in $D(\\mathcal{O}_X)$.", "Let $x \\in |X|$. We will show that $K$ is pseudo-coherent relative to $A$", "in an \\'etale neighbourhood of $x$. This will prove the lemma", "by our definition of relative pseudo-coherence.", "\\medskip\\noindent", "Choose $n, Z, z, V, E$ as in Lemma \\ref{lemma-derived-chow}.", "Denote $p : X \\times \\mathbf{P}^n \\to X$ and", "$q : X \\times \\mathbf{P}^n \\to \\mathbf{P}^n_A$", "the projections.", "Then for any $i \\in \\mathbf{Z}$ we have", "\\begin{align*}", "& R\\Gamma(\\mathbf{P}^n_A,", "Rq_*(Lp^*K \\otimes^\\mathbf{L} E)", "\\otimes^\\mathbf{L}", "\\mathcal{O}_{\\mathbf{P}^n_A}(i)) \\\\", "& =", "R\\Gamma(X \\times \\mathbf{P}^n,", "Lp^*K \\otimes^\\mathbf{L} E \\otimes^\\mathbf{L}", "Lq^*\\mathcal{O}_{\\mathbf{P}^n_A}(i)) \\\\", "& =", "R\\Gamma(X,", "K \\otimes^\\mathbf{L} Rq_*(E \\otimes^\\mathbf{L}", "Lq^*\\mathcal{O}_{\\mathbf{P}^n_A}(i)))", "\\end{align*}", "by", "Derived Categories of Spaces,", "Lemma \\ref{spaces-perfect-lemma-cohomology-base-change}.", "By", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-flat-proper-pseudo-coherent-direct-image-general}", "the complex $Rq_*(E \\otimes^\\mathbf{L} Lq^*\\mathcal{O}_{\\mathbf{P}^n_A}(i))$", "is pseudo-coherent on $X$. Hence the assumption tells us the expression", "in the displayed formula is a pseudo-coherent object of $D(A)$.", "By", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-pseudo-coherent-on-projective-space}", "we conclude that $Rq_*(Lp^*K \\otimes^\\mathbf{L} E)$ is", "pseudo-coherent on $\\mathbf{P}^n_A$.", "By Lemma \\ref{lemma-compute-Fourier-Mukai-for-derived-chow}", "we have", "$$", "Rq_*(Lp^*K \\otimes^\\mathbf{L} E)|_{X \\times_A V} =", "R(W \\to V)_*K|_W", "$$", "Since $W \\to V$ is a closed immersion into an open subscheme of", "$\\mathbf{P}^n_A$ this means $K|_W$ is pseudo-coherent relative to $A$", "for example by", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-check-relative-pseudo-coherence-on-charts}." ], "refs": [ "spaces-more-morphisms-lemma-derived-chow", "spaces-perfect-lemma-cohomology-base-change", "spaces-perfect-lemma-flat-proper-pseudo-coherent-direct-image-general", "perfect-lemma-pseudo-coherent-on-projective-space", "spaces-more-morphisms-lemma-compute-Fourier-Mukai-for-derived-chow", "more-morphisms-lemma-check-relative-pseudo-coherence-on-charts" ], "ref_ids": [ 255, 2718, 2739, 7075, 256, 13974 ] } ], "ref_ids": [ 296 ] }, { "id": 258, "type": "theorem", "label": "spaces-more-morphisms-lemma-characterize-pseudo-coh-improved", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-characterize-pseudo-coh-improved", "contents": [ "Let $A$ be a ring. Let $X$ be an algebraic space separated and", "of finite presentation over $A$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$. If ", "$R \\Gamma (X, E \\otimes ^{\\mathbf{L}} K)$ is", "pseudo-coherent in $D(A)$ for every perfect ", "$E \\in D(\\mathcal{O}_X)$, then $K$ is pseudo-coherent", "relative to $A$." ], "refs": [], "proofs": [ { "contents": [ "In view of Lemma \\ref{lemma-characterize-pseudo-coherent}, it suffices", "to show $R \\Gamma (X, E \\otimes ^{\\mathbf{L}} K)$ is", "pseudo-coherent in $D(A)$ for every pseudo-coherent ", "$E \\in D(\\mathcal{O}_X)$. By Derived Categories of Spaces,", "Proposition \\ref{spaces-perfect-proposition-detecting-bounded-above}", "it follows that $K \\in D^-_\\QCoh (\\mathcal{O}_X)$. Now the", "result follows by Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-perfect-enough}." ], "refs": [ "spaces-more-morphisms-lemma-characterize-pseudo-coherent", "spaces-perfect-proposition-detecting-bounded-above", "spaces-perfect-lemma-perfect-enough" ], "ref_ids": [ 257, 2759, 2741 ] } ], "ref_ids": [] }, { "id": 259, "type": "theorem", "label": "spaces-more-morphisms-lemma-affine-locally-rel-perfect", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-affine-locally-rel-perfect", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ which is flat and locally of finite presentation.", "Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. The following are equivalent:", "\\begin{enumerate}", "\\item $E$ is $Y$-perfect,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_g & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$, $V$ are schemes and the vertical arrows are \\'etale, the complex", "$E|_U$ is $V$-perfect in the sense of Derived Categories of Schemes,", "Definition \\ref{perfect-definition-relatively-perfect},", "\\item for some commutative diagram as in (2) with $U \\to X$", "surjective, the complex $E|_U$ is $V$-perfect in the sense of", "Derived Categories of Schemes,", "Definition \\ref{perfect-definition-relatively-perfect},", "\\item for every commutative diagram as in (2) with $U$ and $V$", "affine the complex $R\\Gamma(U, E)$ is $\\mathcal{O}_Y(V)$-perfect.", "\\end{enumerate}" ], "refs": [ "perfect-definition-relatively-perfect", "perfect-definition-relatively-perfect" ], "proofs": [ { "contents": [ "To make sense of parts (2), (3), (4) of the lemma, observe that", "the object $E|_U$ of $D_\\QCoh(\\mathcal{O}_U)$ corresponds to", "an object $E_0$ of $D_\\QCoh(\\mathcal{O}_{U_0})$ where $U_0$", "denotes the scheme underlying $U$, see Derived Categories of Spaces,", "Lemma \\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}.", "Moreover, in this case $E_0$ is pseudo-coherent if and only if", "$E|_U$ is pseudo-coherent, see Derived Categories of Spaces,", "Lemma \\ref{spaces-perfect-lemma-descend-pseudo-coherent}.", "Also, $E|_U$ locally has finite tor dimension over", "$f^{-1}\\mathcal{O}_Y|_U = g^{-1}\\mathcal{O}_V$ if and only if", "$E_0$ locally has finite tor dimension over $g_0^{-1}\\mathcal{O}_{V_0}$", "by Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-tor-dimension-rel}.", "Here $g_0 : U_0 \\to V_0$ is the morphism of schemes representing", "$g : U \\to V$ (notation as in Derived Categories of Spaces,", "Remark \\ref{spaces-perfect-remark-match-total-direct-images}).", "Finally, observe that ``being pseudo-coherent'' is \\'etale local and", "of course ``having locally finite tor dimension'' is \\'etale local.", "Thus we see that it suffices to check $Y$-perfectness \\'etale locally", "and by the above discussion we see that (1) implies (2) and", "(3) implies (1). Since part (4) is equivalent", "to (2) and (3) by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-locally-rel-perfect}", "the proof is complete." ], "refs": [ "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-descend-pseudo-coherent", "spaces-perfect-lemma-tor-dimension-rel", "spaces-perfect-remark-match-total-direct-images", "perfect-lemma-affine-locally-rel-perfect" ], "ref_ids": [ 2644, 2692, 2694, 2768, 7077 ] } ], "ref_ids": [ 7119, 7119 ] }, { "id": 260, "type": "theorem", "label": "spaces-more-morphisms-lemma-triangulated", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-triangulated", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "algebraic spaces over $S$ which is flat and locally of finite presentation.", "The full subcategory of $D(\\mathcal{O}_X)$ consisting of $Y$-perfect objects is", "a saturated\\footnote{Derived Categories, Definition", "\\ref{derived-definition-saturated}.} triangulated subcategory." ], "refs": [ "derived-definition-saturated" ], "proofs": [ { "contents": [ "This follows from Cohomology on Sites, Lemmas", "\\ref{sites-cohomology-lemma-cone-pseudo-coherent},", "\\ref{sites-cohomology-lemma-summands-pseudo-coherent},", "\\ref{sites-cohomology-lemma-cone-tor-amplitude}, and", "\\ref{sites-cohomology-lemma-summands-tor-amplitude}." ], "refs": [ "sites-cohomology-lemma-cone-pseudo-coherent", "sites-cohomology-lemma-summands-pseudo-coherent", "sites-cohomology-lemma-cone-tor-amplitude", "sites-cohomology-lemma-summands-tor-amplitude" ], "ref_ids": [ 4368, 4370, 4376, 4378 ] } ], "ref_ids": [ 1974 ] }, { "id": 261, "type": "theorem", "label": "spaces-more-morphisms-lemma-perfect-relatively-perfect", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-perfect-relatively-perfect", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "which is flat and locally of finite presentation.", "A perfect object of $D(\\mathcal{O}_X)$ is $Y$-perfect.", "If $K, M \\in D(\\mathcal{O}_X)$, then $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M$", "is $Y$-perfect if $K$ is perfect and $M$ is $Y$-perfect." ], "refs": [], "proofs": [ { "contents": [ "Reduce to the case of schemes using", "Lemma \\ref{lemma-affine-locally-rel-perfect}", "and then apply", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-perfect-relatively-perfect}." ], "refs": [ "spaces-more-morphisms-lemma-affine-locally-rel-perfect", "perfect-lemma-perfect-relatively-perfect" ], "ref_ids": [ 259, 7079 ] } ], "ref_ids": [] }, { "id": 262, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-relatively-perfect", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-relatively-perfect", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "which is flat and locally of finite presentation.", "Let $g : Y' \\to Y$ be a morphism of algebraic spaces over $S$.", "Set $X' = Y' \\times_Y X$ and denote $g' : X' \\to X$ the projection.", "If $K \\in D(\\mathcal{O}_X)$ is $Y$-perfect, then $L(g')^*K$", "is $Y'$-perfect." ], "refs": [], "proofs": [ { "contents": [ "Reduce to the case of schemes using", "Lemma \\ref{lemma-affine-locally-rel-perfect}", "and then apply", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-base-change-relatively-perfect}." ], "refs": [ "spaces-more-morphisms-lemma-affine-locally-rel-perfect", "perfect-lemma-base-change-relatively-perfect" ], "ref_ids": [ 259, 7080 ] } ], "ref_ids": [] }, { "id": 263, "type": "theorem", "label": "spaces-more-morphisms-lemma-relative-descend-homomorphisms", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-relative-descend-homomorphisms", "contents": [ "In Situation \\ref{situation-relative-descent}.", "Let $K_0$ and $L_0$ be objects of $D(\\mathcal{O}_{X_0})$.", "Set $K_i = Lf_{i0}^*K_0$ and $L_i = Lf_{i0}^*L_0$ for $i \\geq 0$", "and set $K = Lf_0^*K_0$ and $L = Lf_0^*L_0$. Then the map", "$$", "\\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{X_i})}(K_i, L_i)", "\\longrightarrow", "\\Hom_{D(\\mathcal{O}_X)}(K, L)", "$$", "is an isomorphism if $K_0$ is pseudo-coherent and", "$L_0 \\in D_\\QCoh(\\mathcal{O}_{X_0})$ has (locally)", "finite tor dimension as an object of", "$D((X_0 \\to Y_0)^{-1}\\mathcal{O}_{Y_0})$" ], "refs": [], "proofs": [ { "contents": [ "For every quasi-compact and quasi-separated object $U_0$ of", "$(X_0)_{spaces, \\etale}$ consider the condition $P$ that", "$$", "\\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{U_i})}(K_i|_{U_i}, L_i|_{U_i})", "\\longrightarrow", "\\Hom_{D(\\mathcal{O}_U)}(K|_U, L|_U)", "$$", "is an isomorphism where $U = X \\times_{X_0} U_0$ and", "$U_i = X_i \\times_{X_0} U_0$. We will prove $P$ holds for each $U_0$.", "\\medskip\\noindent", "Suppose that $(U_0 \\subset W_0, V_0 \\to W_0)$ is an elementary", "distinguished square in $(X_0)_{spaces, \\etale}$", "and $P$ holds for $U_0, V_0, U_0 \\times_{W_0} V_0$.", "Then $P$ holds for $W_0$ by Mayer-Vietoris", "for hom in the derived category, see Derived Categories of Spaces,", "Lemma \\ref{spaces-perfect-lemma-mayer-vietoris-hom}.", "\\medskip\\noindent", "We first consider $U_0 = W_0 \\times_{Y_0} X_0$ with $W_0$ a", "quasi-compact and quasi-separated object of $(Y_0)_{spaces, \\etale}$.", "By the induction principle of Derived Categories of Spaces,", "Lemma \\ref{spaces-perfect-lemma-induction-principle}", "applied to these $W_0$ and the previous paragraph,", "we find that it is enough to prove", "$P$ for $U_0 = W_0 \\times_{Y_0} X_0$ with $W_0$ affine.", "In other words, we have reduced to the case where $Y_0$ is affine.", "Next, we apply the induction principle again, this time to all", "quasi-compact and quasi-separated opens of $X_0$, to reduce to the", "case where $X_0$ is affine as well.", "\\medskip\\noindent", "If $X_0$ and $Y_0$ are affine, then we are back in the case", "of schemes which is proved in", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-relative-descend-homomorphisms}.", "The reader may use", "Derived Categories of Spaces, Lemmas", "\\ref{spaces-perfect-lemma-pseudo-coherent},", "\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site},", "\\ref{spaces-perfect-lemma-descend-pseudo-coherent}, and", "\\ref{spaces-perfect-lemma-tor-dimension-rel}", "to accomplish the translation of the statement into a statement", "involving only schemes and derived categories of modules on schemes." ], "refs": [ "spaces-perfect-lemma-mayer-vietoris-hom", "spaces-perfect-lemma-induction-principle", "perfect-lemma-relative-descend-homomorphisms", "spaces-perfect-lemma-pseudo-coherent", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-descend-pseudo-coherent", "spaces-perfect-lemma-tor-dimension-rel" ], "ref_ids": [ 2676, 2670, 7081, 2696, 2644, 2692, 2694 ] } ], "ref_ids": [] }, { "id": 264, "type": "theorem", "label": "spaces-more-morphisms-lemma-descend-relatively-perfect", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-descend-relatively-perfect", "contents": [ "In Situation \\ref{situation-relative-descent} the category of", "$Y$-perfect objects of $D(\\mathcal{O}_X)$ is the colimit of the categories", "of $Y_i$-perfect objects of $D(\\mathcal{O}_{X_i})$." ], "refs": [], "proofs": [ { "contents": [ "For every quasi-compact and quasi-separated object $U_0$ of", "$(X_0)_{spaces, \\etale}$ consider the condition $P$", "that the functor", "$$", "\\colim_{i \\geq 0} D_{Y_i\\text{-perfect}}(\\mathcal{O}_{U_i})", "\\longrightarrow", "D_{Y\\text{-perfect}}(\\mathcal{O}_U)", "$$", "is an equivalence where $U = X \\times_{X_0} U_0$ and", "$U_i = X_i \\times_{X_0} U_0$.", "We observe that we already know this functor is fully faithful", "by Lemma \\ref{lemma-relative-descend-homomorphisms}. Thus it suffices to prove", "essential surjectivity.", "\\medskip\\noindent", "Suppose that $(U_0 \\subset W_0, V_0 \\to W_0)$ is an elementary", "distinguished square in $(X_0)_{spaces, \\etale}$", "and $P$ holds for $U_0, V_0, U_0 \\times_{W_0} V_0$.", "We claim that $P$ holds for $W_0$. We will use the notation", "$U_i = X_i \\times_{X_0} U_0$, $U = X \\times_{X_0} U_0$,", "and similarly for $V_0$ and $W_0$. We will abusively use the symbol", "$f_i$ for all the morphisms $U \\to U_i$, $V \\to V_i$,", "$U \\times_W V \\to U_i \\times_{W_i} V_i$, and $W \\to W_i$.", "Suppose $E$ is an $Y$-perfect object of $D(\\mathcal{O}_W)$.", "Goal: show $E$ is in the essential image of the functor.", "By assumption,", "we can find $i \\geq 0$, an $Y_i$-perfect object $E_{U, i}$ on $U_i$,", "an $Y_i$-perfect object $E_{V, i}$ on $V_i$, and", "isomorphisms $Lf_i^*E_{U, i} \\to E|_U$ and $Lf_i^*E_{V, i} \\to E|_V$.", "Let", "$$", "a : E_{U, i} \\to (Rf_{i, *}E)|_{U_i}", "\\quad\\text{and}\\quad", "b : E_{V, i} \\to (Rf_{i, *}E)|_{V_i}", "$$", "the maps adjoint to the isomorphisms $Lf_i^*E_{U, i} \\to E|_U$", "and $Lf_i^*E_{V, i} \\to E|_V$.", "By fully faithfulness, after increasing $i$,", "we can find an isomorphism", "$c : E_{U, i}|_{U_i \\times_{W_i} V_i} \\to E_{V, i}|_{U_i \\times_{W_i} V_i}$", "which pulls back to the identifications ", "$$", "Lf_i^*E_{U, i}|_{U \\times_W V} \\to E|_{U \\times_W V} \\to", "Lf_i^*E_{V, i}|_{U \\times_W V}.", "$$", "Apply Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-glue}", "to get an object $E_i$ on $W_i$ and a map $d : E_i \\to Rf_{i, *}E$", "which restricts to the maps $a$ and $b$ over $U_i$ and $V_i$.", "Then it is clear that $E_i$ is $Y_i$-perfect (because being", "relatively perfect is an \\'etale local property) and that", "$d$ is adjoint to an isomorphism $Lf_i^*E_i \\to E$.", "\\medskip\\noindent", "By exactly the same argument as used in", "the proof of Lemma \\ref{lemma-relative-descend-homomorphisms}", "using the induction principle", "(Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-induction-principle})", "we reduce to the case where both $X_0$ and $Y_0$", "are affine: first work with quasi-compact and quasi-separated objects", "in $(Y_0)_{spaces, \\etale}$ to reduce to", "$Y_0$ affine, then work with quasi-compact and quasi-separated object", "in $(X_0)_{spaces, \\etale}$ to reduce to $X_0$ affine.", "In the affine case the result follows from the case of schemes which is", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-descend-relatively-perfect}.", "The translation into the case for schemes is done by", "Lemma \\ref{lemma-affine-locally-rel-perfect}." ], "refs": [ "spaces-more-morphisms-lemma-relative-descend-homomorphisms", "spaces-perfect-lemma-glue", "spaces-more-morphisms-lemma-relative-descend-homomorphisms", "spaces-perfect-lemma-induction-principle", "perfect-lemma-descend-relatively-perfect", "spaces-more-morphisms-lemma-affine-locally-rel-perfect" ], "ref_ids": [ 263, 2680, 263, 2670, 7082, 259 ] } ], "ref_ids": [] }, { "id": 265, "type": "theorem", "label": "spaces-more-morphisms-lemma-derived-pushforward-rel-perfect", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-derived-pushforward-rel-perfect", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "which is flat, proper, and", "of finite presentation. Let $E \\in D(\\mathcal{O}_X)$ be $Y$-perfect.", "Then $Rf_*E$ is a perfect object of $D(\\mathcal{O}_Y)$", "and its formation commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "The statement on base change is Derived Categories of Spaces,", "Lemma \\ref{spaces-perfect-lemma-base-change-tensor} (with", "$\\mathcal{G}^\\bullet$ equal to $\\mathcal{O}_X$ in degree $0$).", "Thus it suffices to show that $Rf_*E$ is a perfect object. We will reduce", "to the case where $Y$ is Noetherian affine by a limit argument.", "\\medskip\\noindent", "The question is \\'etale local on $Y$, hence we may assume $Y$ is affine.", "Say $Y = \\Spec(R)$. We write $R = \\colim R_i$ as a filtered colimit", "of Noetherian rings $R_i$. By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-finite-presentation}", "there exists an $i$ and an algebraic space", "$X_i$ of finite presentation over $R_i$", "whose base change to $R$ is $X$. By", "Limits of Spaces, Lemmas \\ref{spaces-limits-lemma-eventually-proper} and", "\\ref{spaces-limits-lemma-descend-flat}", "we may assume $X_i$ is proper and flat over $R_i$.", "By Lemma \\ref{lemma-descend-relatively-perfect}", "we may assume there exists a $R_i$-perfect object $E_i$ of", "$D(\\mathcal{O}_{X_i})$ whose pullback to $X$ is $E$.", "Applying Derived Categories of Spaces,", "Lemma \\ref{spaces-perfect-lemma-perfect-direct-image}", "to $X_i \\to \\Spec(R_i)$ and $E_i$ and using the", "base change property already shown we obtain the result." ], "refs": [ "spaces-perfect-lemma-base-change-tensor", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-eventually-proper", "spaces-limits-lemma-descend-flat", "spaces-more-morphisms-lemma-descend-relatively-perfect", "spaces-perfect-lemma-perfect-direct-image" ], "ref_ids": [ 2726, 4598, 4596, 4595, 264, 2728 ] } ], "ref_ids": [] }, { "id": 266, "type": "theorem", "label": "spaces-more-morphisms-lemma-compute-ext-rel-perfect", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-compute-ext-rel-perfect", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $E, K \\in D(\\mathcal{O}_X)$.", "Assume", "\\begin{enumerate}", "\\item $Y$ is quasi-compact and quasi-separated,", "\\item $f$ is proper, flat, and of finite presentation,", "\\item $E$ is $Y$-perfect,", "\\item $K$ is pseudo-coherent.", "\\end{enumerate}", "Then there exists a pseudo-coherent $L \\in D(\\mathcal{O}_Y)$ such that", "$$", "Rf_*R\\SheafHom(K, E) = R\\SheafHom(L, \\mathcal{O}_Y)", "$$", "and the same is true after arbitrary base change: given", "$$", "\\vcenter{", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "Y' \\ar[r]^g &", "Y", "}", "}", "\\quad\\quad", "\\begin{matrix}", "\\text{cartesian, then we have } \\\\", "Rf'_*R\\SheafHom(L(g')^*K, L(g')^*E) \\\\", "= R\\SheafHom(Lg^*L, \\mathcal{O}_{Y'})", "\\end{matrix}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since $Y$ is quasi-compact and quasi-separated, the same is true for $X$.", "By Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-pseudo-coherent-hocolim} we can write", "$K = \\text{hocolim} K_n$ with $K_n$ perfect and $K_n \\to K$ inducing", "an isomorphism on truncations $\\tau_{\\geq -n}$. Let $K_n^\\vee$", "be the dual perfect complex", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-dual-perfect-complex}).", "We obtain an inverse system $\\ldots \\to K_3^\\vee \\to K_2^\\vee \\to K_1^\\vee$", "of perfect objects. By Lemma \\ref{lemma-perfect-relatively-perfect}", "we see that $K_n^\\vee \\otimes_{\\mathcal{O}_X} E$ is $Y$-perfect.", "Thus we may apply Lemma \\ref{lemma-derived-pushforward-rel-perfect}", "to $K_n^\\vee \\otimes_{\\mathcal{O}_X} E$ and we obtain an inverse system", "$$", "\\ldots \\to M_3 \\to M_2 \\to M_1", "$$", "of perfect complexes on $Y$ with", "$$", "M_n = Rf_*(K_n^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) =", "Rf_*R\\SheafHom(K_n, E)", "$$", "Moreover, the formation of these complexes commutes with any", "base change, namely $Lg^*M_n =", "Rf'_*((L(g')^*K_n)^\\vee \\otimes_{\\mathcal{O}_{X'}}^\\mathbf{L} L(g')^*E) =", "Rf'_*R\\SheafHom(L(g')^*K_n, L(g')^*E)$.", "\\medskip\\noindent", "As $K_n \\to K$ induces an isomorphism on $\\tau_{\\geq -n}$, we see that", "$K_n \\to K_{n + 1}$ induces an isomorphism on $\\tau_{\\geq -n}$.", "It follows that $K_{n + 1}^\\vee \\to K_n^\\vee$", "induces an isomorphism on $\\tau_{\\leq n}$ as", "$K_n^\\vee = R\\SheafHom(K_n, \\mathcal{O}_X)$.", "Suppose that $E$ has tor amplitude in $[a, b]$ as a complex", "of $f^{-1}\\mathcal{O}_Y$-modules. Then the same is true after", "any base change, see", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-tor-independence-and-tor-amplitude}.", "We find that", "$K_{n + 1}^\\vee \\otimes_{\\mathcal{O}_X} E \\to", "K_n^\\vee \\otimes_{\\mathcal{O}_X} E$", "induces an isomorphism on $\\tau_{\\leq n + a}$", "and the same is true after any base change.", "Applying the right derived functor $Rf_*$", "we conclude the maps $M_{n + 1} \\to M_n$", "induce isomorphisms on $\\tau_{\\leq n + a}$", "and the same is true after any base change.", "Choose a distinguished triangle", "$$", "M_{n + 1} \\to M_n \\to C_n \\to M_{n + 1}[1]", "$$", "Pick $y \\in |Y|$. Choose an elementary \\'etale neighbourhood", "$(U, u) \\to (Y, y)$; this is possible by", "Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-decent-space-elementary-etale-neighbourhood}.", "Take $Y'$ equal to the spectrum of the residue field at $u$.", "Pull back to see that $C_n|_U \\otimes_{\\mathcal{O}_U}^\\mathbf{L} \\kappa(u)$", "has nonzero cohomology only in degrees $\\geq n + a$. By", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-perfect-from-residue-field}", "we see that the perfect complex $C_n|_U$ has tor amplitude in", "$[n + a, m_n]$ for some integer $m_n$ and after possibly shrinking $U$.", "Thus $C_n$ has tor amplitude in $[n + a, m_n]$ for some integer $m_n$", "(because $Y$ is quasi-compact).", "In particular, the dual perfect complex $C_n^\\vee$ has tor amplitude in", "$[-m_n, -n - a]$.", "\\medskip\\noindent", "Let $L_n = M_n^\\vee$ be the dual perfect complex. The", "conclusion from the discussion in the previous paragraph is that", "$L_n \\to L_{n + 1}$ induces isomorphisms on $\\tau_{\\geq -n - a}$.", "Thus $L = \\text{hocolim} L_n$ is pseudo-coherent, see", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-pseudo-coherent-hocolim}.", "Since we have", "$$", "R\\SheafHom(K, E) = R\\SheafHom(\\text{hocolim} K_n, E) =", "R\\lim R\\SheafHom(K_n, E) = R\\lim K_n^\\vee \\otimes_{\\mathcal{O}_X} E", "$$", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-colim-and-lim-of-duals})", "and since $R\\lim$ commutes with $Rf_*$ we find that", "$$", "Rf_*R\\SheafHom(K, E) = R\\lim M_n = R\\lim R\\SheafHom(L_n, \\mathcal{O}_Y) =", "R\\SheafHom(L, \\mathcal{O}_Y)", "$$", "This proves the formula over $Y$. Since the construction of $M_n$ is", "compatible with base chance, the formula continues to hold after", "any base change." ], "refs": [ "spaces-perfect-lemma-pseudo-coherent-hocolim", "sites-cohomology-lemma-dual-perfect-complex", "spaces-more-morphisms-lemma-perfect-relatively-perfect", "spaces-more-morphisms-lemma-derived-pushforward-rel-perfect", "spaces-perfect-lemma-tor-independence-and-tor-amplitude", "decent-spaces-lemma-decent-space-elementary-etale-neighbourhood", "more-algebra-lemma-lift-perfect-from-residue-field", "spaces-perfect-lemma-pseudo-coherent-hocolim", "sites-cohomology-lemma-colim-and-lim-of-duals" ], "ref_ids": [ 2713, 4390, 261, 265, 2722, 9488, 10232, 2713, 4393 ] } ], "ref_ids": [] }, { "id": 267, "type": "theorem", "label": "spaces-more-morphisms-lemma-bounded-on-fibres", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-bounded-on-fibres", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ such that", "the structure morphism $f : X \\to S$ is flat and", "locally of finite presentation. Let $E$ be a pseudo-coherent", "object of $D(\\mathcal{O}_X)$. The following are equivalent", "\\begin{enumerate}", "\\item $E$ is $S$-perfect, and", "\\item $E$ is locally bounded below and for every point $s \\in S$", "the object $L(X_s \\to X)^*E$ of $D(\\mathcal{O}_{X_s})$", "is locally bounded below.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since everything is local we immediately reduce to the", "case that $X$ and $S$ are affine, see Lemma", "\\ref{lemma-affine-locally-rel-perfect}.", "This case is handled by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-bounded-on-fibres}." ], "refs": [ "spaces-more-morphisms-lemma-affine-locally-rel-perfect", "perfect-lemma-bounded-on-fibres" ], "ref_ids": [ 259, 7085 ] } ], "ref_ids": [] }, { "id": 268, "type": "theorem", "label": "spaces-more-morphisms-lemma-characterize-relatively-perfect", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-characterize-relatively-perfect", "contents": [ "Let $A$ be a ring. Let $X$ be an algebraic space separated, of", "finite presentation, and flat over $A$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$.", "If $R \\Gamma (X, E \\otimes^\\mathbf{L} K)$ is perfect in", "$D(A)$ for every perfect $E \\in D(\\mathcal{O}_X)$, then $K$ is", "$\\Spec(A)$-perfect." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-characterize-pseudo-coh-improved},", "$K$ is pseudo-coherent relative to $A$. By Lemma", "\\ref{lemma-relative-pseudo-coherent-is-moot},", "$K$ is pseudo-coherent in $D(\\mathcal{O}_X)$.", "By Derived Categories of Spaces, Proposition", "\\ref{spaces-perfect-proposition-detecting-bounded-below}", "we see that $K$ is in $D^-(\\mathcal{O}_X)$.", "Let $\\mathfrak{p}$ be a prime ideal of $A$ and denote", "$i : Y \\to X$ the inclusion of the scheme theoretic", "fibre over $\\mathfrak{p}$, i.e., $Y$ is a scheme over $\\kappa(\\mathfrak p)$.", "By Lemma \\ref{lemma-bounded-on-fibres},", "we will be done if we can show $Li^*(K)$ is bounded below. ", "Let $G \\in D_{perf} (\\mathcal{O}_X)$ be a perfect", "complex which generates $D_\\QCoh (\\mathcal{O}_X)$,", "see Derived Categories of Spaces, Theorem", "\\ref{spaces-perfect-theorem-bondal-van-den-Bergh}.", "We have", "\\begin{align*}", "R\\Hom _{\\mathcal{O}_Y}(Li^*(G), Li^*(K))", "& =", "R\\Gamma(Y, Li^*(G ^\\vee \\otimes ^\\mathbf{L} K)) \\\\", "& =", "R\\Gamma(X, G^\\vee \\otimes ^{\\mathbf{L}} K)", "\\otimes^\\mathbf{L}_A \\kappa(\\mathfrak{p})", "\\end{align*}", "The first equality uses that $Li^*$ preserves perfect objects and duals", "and Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-dual-perfect-complex}; we omit", "some details. The second equality follows from", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-compare-base-change}", "as $X$ is flat over $A$. It follows from our hypothesis that this is a", "perfect object of $D(\\kappa(\\mathfrak{p}))$. The object", "$Li^*(G) \\in D_{perf}(\\mathcal{O}_Y)$ generates $D_\\QCoh(\\mathcal{O}_Y)$ by", "Derived Categories of Spaces, Remark", "\\ref{spaces-perfect-remark-pullback-generator}.", "Hence Derived Categories of Spaces, Proposition", "\\ref{spaces-perfect-proposition-detecting-bounded-below}", "now implies that $Li^*(K)$ is bounded below and we win." ], "refs": [ "spaces-more-morphisms-lemma-characterize-pseudo-coh-improved", "spaces-more-morphisms-lemma-relative-pseudo-coherent-is-moot", "spaces-perfect-proposition-detecting-bounded-below", "spaces-more-morphisms-lemma-bounded-on-fibres", "spaces-perfect-theorem-bondal-van-den-Bergh", "sites-cohomology-lemma-dual-perfect-complex", "spaces-perfect-lemma-compare-base-change", "spaces-perfect-remark-pullback-generator", "spaces-perfect-proposition-detecting-bounded-below" ], "ref_ids": [ 258, 224, 2760, 267, 2640, 4390, 2720, 2771, 2760 ] } ], "ref_ids": [] }, { "id": 269, "type": "theorem", "label": "spaces-more-morphisms-lemma-diagonal-picard-flat-proper", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-diagonal-picard-flat-proper", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a flat, proper morphism of finite presentation", "of algebraic spaces over $S$.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module.", "For a morphism $g : Y' \\to Y$ consider the base change diagram", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "Assume $\\mathcal{O}_{Y'} \\to f'_*\\mathcal{O}_{X'}$ is an", "isomorphism for all $g : Y' \\to Y$.", "Then there exists an immersion $j : Z \\to Y$ of finite presentation", "such that a morphism $g : Y' \\to Y$ factors through $Z$ if and only if", "there exists a finite locally free $\\mathcal{O}_{Y'}$-module $\\mathcal{N}$", "with $(f')^*\\mathcal{N} \\cong (g')^*\\mathcal{L}$." ], "refs": [], "proofs": [ { "contents": [ "Let $y : \\Spec(k) \\to Y$ be a field valued point.", "Then the fibre $X_y$ of $f$ at $y$ is connected by our assumption", "that $H^0(X_y, \\mathcal{O}_{X_y}) = k$. Thus the rank of", "$\\mathcal{E}$ is constant on the fibres. Since $f$ is open", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-fppf-open})", "and closed we conclude that there is a decomposition $Y = \\coprod Y_r$", "of $Y$ into open and closed subspaces such that $\\mathcal{E}$", "has constant rank $r$ on the inverse image of $Y_r$.", "Thus we may assume $\\mathcal{E}$ has constant rank $r$.", "We will denote $\\mathcal{E}^\\vee = \\SheafHom(\\mathcal{E}, \\mathcal{O}_X)$", "the dual rank $r$ module.", "\\medskip\\noindent", "By cohomology and base change (more precisely by", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-flat-proper-perfect-direct-image-general})", "we see that $E = Rf_*\\mathcal{E}$ is a perfect object of the", "derived category of $Y$ and that its formation commutes with", "arbitrary change of base. Similarly for $E' = Rf_*\\mathcal{E}^\\vee$.", "Since there is never any cohomology in degrees $< 0$, we see that", "$E$ and $E'$ have (locally) tor-amplitude in $[0, b]$ for some $b$.", "Observe that for any $g : Y' \\to Y$ we have", "$f'_*((g')^*\\mathcal{E}) = H^0(Lg^*E)$ and", "$f'_*((g')^*\\mathcal{E}^\\vee) = H^0(Lg^*E')$.", "Let $j : Z \\to Y$ and $j' : Z' \\to Y$ be the locally closed", "immersions constructed in Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-locally-closed-where-H0-locally-free}", "for $E$ and $E'$ with $a = 0$ and $r = r$; these are characterized", "by the property that $H^0(Lj^*E)$ and $H^0((j')^*E')$", "are locally free modules of rank $r$ compatible with pullback.", "\\medskip\\noindent", "Let $g : Y' \\to Y$ be a morphism. If there exists an $\\mathcal{N}$", "as in the lemma, then, using the projection formula", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-projection-formula},", "we see that the modules", "$$", "f'_*((g')^*\\mathcal{E}) \\cong", "f'_*((f')^*\\mathcal{N}) \\cong", "\\mathcal{N} \\otimes_{\\mathcal{O}_{Y'}} f'_*\\mathcal{O}_{X'} \\cong", "\\mathcal{N}\\quad\\text{and similarly }\\quad", "f'_*((g')^*\\mathcal{E}^\\vee) \\cong \\mathcal{N}^\\vee", "$$", "are locally free of rank $r$ and remain locally free of rank $r$", "after any further base change $Y'' \\to Y'$.", "Hence in this case $g : Y' \\to Y$ factors through $j$ and through $j'$.", "Thus we may replace $Y$ by $Z \\times_Y Z'$ and assume that", "$f_*\\mathcal{E}$ and $f_*\\mathcal{E}^\\vee$ are locally free", "$\\mathcal{O}_Y$-modules of rank $r$", "whose formation commutes with arbitrary change of base.", "\\medskip\\noindent", "In this sitation if $g : Y' \\to Y$ is a morphism and there exists an", "$\\mathcal{N}$ as in the lemma, then the map (cup product in degree $0$)", "$$", "f'_*((g')^*\\mathcal{E})", "\\otimes_{\\mathcal{O}_{Y'}}", "f'_*((g')^*\\mathcal{E}^\\vee)", "\\longrightarrow \\mathcal{O}_{Y'}", "$$", "is a perfect pairing. Conversely, if this cup product map is a", "perfect pairing, then we see that locally on $Y'$ we have", "a basis of sections", "$\\sigma_1, \\ldots, \\sigma_r$ in $f'_*((g')^*\\mathcal{L})$ and", "$\\tau_1, \\ldots, \\tau_r$ in", "$f'_*((g')^*\\mathcal{E}^\\vee)$ whose products satisfy", "$\\sigma_i \\tau_j = \\delta_{ij}$. Thinking of $\\sigma_i$", "as a section of $(g')^*\\mathcal{L}$ on $X'$", "and $\\tau_j$ as a section of $(g')^*\\mathcal{L}^\\vee$ on $X'$,", "we conclude that", "$$", "\\sigma_1, \\ldots, \\sigma_r :", "\\mathcal{O}_{X'}^{\\oplus r}", "\\longrightarrow", "(g')^*\\mathcal{E}", "$$", "is an isomorphism with inverse given by", "$$", "\\tau_1, \\ldots, \\tau_r :", "(g')^*\\mathcal{E}", "\\longrightarrow", "\\mathcal{O}_{X'}^{\\oplus r}", "$$", "In other words, we see that", "$(f')^*f'_*(g')^*\\mathcal{E} \\cong (g')^*\\mathcal{E}$.", "But the condition that the cupproduct is nondegenerate", "picks out a retrocompact open subscheme (namely, the locus where a suitable", "determinant is nonzero) and the proof is complete." ], "refs": [ "spaces-morphisms-lemma-fppf-open", "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "spaces-perfect-lemma-locally-closed-where-H0-locally-free", "sites-cohomology-lemma-projection-formula" ], "ref_ids": [ 4855, 2738, 2748, 4396 ] } ], "ref_ids": [] }, { "id": 270, "type": "theorem", "label": "spaces-more-morphisms-lemma-pseudo-coherent-descends-fpqc", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-pseudo-coherent-descends-fpqc", "contents": [ "Let $S$ be a scheme. Let $\\{f_i : X_i \\to X\\}$ be an fpqc covering of", "algebraic spaces over $S$. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$.", "Let $m \\in \\mathbf{Z}$. Then $E$ is $m$-pseudo-coherent if and only if each", "$Lf_i^*E$ is $m$-pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "Pullback always preserves $m$-pseudo-coherence, see", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-pseudo-coherent-pullback}.", "Thus it suffices to assume $Lf_i^*E$ is $m$-pseudo-coherent", "and to prove that $E$ is $m$-pseudo-coherent.", "Then first we may assume $X_i$ is a scheme for all $i$, see", "Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-refine-fpqc-schemes}.", "Next, choose a surjective \\'etale morphism $U \\to X$ where $U$ is a scheme.", "Then $U_i = U \\times_X X_i$ is a scheme and we obtain an fpqc covering", "$\\{U_i \\to U\\}$ of schemes, see", "Topologies on Spaces, Lemma", "\\ref{spaces-topologies-lemma-recognize-fpqc-covering}.", "We know the result is true for", "$\\{U_i \\to U\\}_{i \\in I}$ by the case for schemes, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-pseudo-coherent-descends-fpqc}.", "On the other hand, the restriction $E|_U$ comes from", "an object of $D_\\QCoh(\\mathcal{O}_U)$ (defined using the Zariski", "topology and the ``usual'' structure sheaf of $U$), see", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}.", "The lemma follows as the two notions of pseudo-coherent", "(\\'etale and Zariski) agree by", "Derived Categories of Spaces,", "Lemma \\ref{spaces-perfect-lemma-descend-pseudo-coherent}." ], "refs": [ "sites-cohomology-lemma-pseudo-coherent-pullback", "spaces-topologies-lemma-refine-fpqc-schemes", "spaces-topologies-lemma-recognize-fpqc-covering", "perfect-lemma-pseudo-coherent-descends-fpqc", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-descend-pseudo-coherent" ], "ref_ids": [ 4367, 3680, 3679, 6991, 2644, 2692 ] } ], "ref_ids": [] }, { "id": 271, "type": "theorem", "label": "spaces-more-morphisms-lemma-tor-amplitude-descends-fppf", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-tor-amplitude-descends-fppf", "contents": [ "Let $S$ be a scheme. Let $\\{g_i : Y_i \\to Y\\}$ be an fpqc covering of", "algebraic spaces over $S$. Let $f : X \\to Y$ be a morphism of algebraic", "spaces and set $X_i = Y_i \\times_Y X$ with projections $f_i : X_i \\to Y_i$", "and $g'_i : X_i \\to X$. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$.", "Let $a, b \\in \\mathbf{Z}$. Then the following are equivalent", "\\begin{enumerate}", "\\item $E$ has tor amplitude in $[a, b]$ as an object of", "$D(f^{-1}\\mathcal{O}_Y)$, and", "\\item $L(g'_i)^*E$ has tor amplitude in $[a, b]$ as a object of", "$D(f_i^{-1}\\mathcal{O}_{Y_i})$ for all $i$.", "\\end{enumerate}", "Also true if ``tor amplitude in $[a, b]$'' is replaced by", "``locally finite tor dimension''." ], "refs": [], "proofs": [ { "contents": [ "Pullback preserves ``tor amplitude in $[a, b]$'' by", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-tor-independence-and-tor-amplitude}", "Observe that $Y_i$ and $X$ are tor independent over $Y$", "as $Y_i \\to Y$ is flat. Let us assume (2) and prove (1).", "We can compute tor dimension at stalks, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-tor-amplitude-stalk}", "and Properties of Spaces, Theorem", "\\ref{spaces-properties-theorem-exactness-stalks}.", "Let $\\overline{x}$ be a geometric point of $X$. Choose", "an $i$ and a geometric point $\\overline{x}_i$ in $X_i$ with image", "$\\overline{x}$ in $X$. Then", "$$", "(L(g_i')^*E)_{\\overline{x}_i} =", "E_{\\overline{x}}", "\\otimes_{\\mathcal{O}_{X, \\overline{x}}}^\\mathbf{L}", "\\mathcal{O}_{X_, \\overline{x}_i}", "$$", "Let $\\overline{y}_i$ in $Y_i$ and $\\overline{y}$ in $Y$", "be the image of $\\overline{x}_i$ and $\\overline{x}$.", "Since $X$ and $Y_i$ are tor independent over $Y$, we can apply", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-comparison}", "to see that the right hand side of the displayed formula is equal to", "$E_{\\overline{x}}", "\\otimes_{\\mathcal{O}_{Y, \\overline{y}}}^\\mathbf{L}", "\\mathcal{O}_{Y_i, \\overline{y}_i}$", "in $D(\\mathcal{O}_{Y_i, \\overline{y}_i})$.", "Since we have assume the tor amplitude of this is in", "$[a, b]$, we conclude that the tor amplitude of", "$E_{\\overline{x}}$ in $D(\\mathcal{O}_{Y, \\overline{y}})$", "is in $[a, b]$ by More on Algebra, Lemma", "\\ref{more-algebra-lemma-flat-descent-tor-amplitude}.", "Thus (1) follows.", "\\medskip\\noindent", "Using some elementary topology the case", "``locally finite tor dimension'' follows too." ], "refs": [ "spaces-perfect-lemma-tor-independence-and-tor-amplitude", "sites-cohomology-lemma-tor-amplitude-stalk", "spaces-properties-theorem-exactness-stalks", "more-algebra-lemma-base-change-comparison", "more-algebra-lemma-flat-descent-tor-amplitude" ], "ref_ids": [ 2722, 4380, 11813, 10139, 10184 ] } ], "ref_ids": [] }, { "id": 272, "type": "theorem", "label": "spaces-more-morphisms-lemma-thickening-pseudo-coherent", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-thickening-pseudo-coherent", "contents": [ "Let $S$ be a scheme. Let $i : X \\to X'$ be a finite order thickening of", "algebraic spaces. Let $K' \\in D(\\mathcal{O}_{X'})$ be an object such that", "$K = Li^*K'$ is pseudo-coherent. Then $K'$ is pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "We first prove $K'$ has quasi-coherent cohomology sheaves; we urge", "the reader to skip this part.", "To do this, we may reduce to the case of a first order thickening, see", "Section \\ref{section-thickenings}. Let $\\mathcal{I} \\subset \\mathcal{O}_{X'}$", "be the quasi-coherent sheaf of ideals cutting out $X$.", "Tensoring the short exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_{X'} \\to i_*\\mathcal{O}_X \\to 0", "$$", "with $K'$ we obtain a distinguished triangle", "$$", "K' \\otimes_{\\mathcal{O}_{X'}}^\\mathbf{L} \\mathcal{I}", "\\to K' \\to", "K' \\otimes_{\\mathcal{O}_{X'}}^\\mathbf{L} i_*\\mathcal{O}_X", "\\to", "(K' \\otimes_{\\mathcal{O}_{X'}}^\\mathbf{L} \\mathcal{I})[1]", "$$", "Since $i_* = Ri_*$ and since we may view $\\mathcal{I}$", "as a quasi-coherent $\\mathcal{O}_X$-module (as we have a first", "order thickening) we may rewrite this as", "$$", "i_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{I})", "\\to K' \\to", "i_*K \\to", "i_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{I})[1]", "$$", "Please use Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-projection-formula-finite}", "to identify the terms. Since $K$ is in", "$D_\\QCoh(\\mathcal{O}_X)$ we conclude that", "$K'$ is in $D_\\QCoh(\\mathcal{O}_{X'})$; this uses", "Derived Categories of Spaces, Lemmas", "\\ref{spaces-perfect-lemma-pseudo-coherent},", "\\ref{spaces-perfect-lemma-quasi-coherence-tensor-product}, and", "\\ref{spaces-perfect-lemma-quasi-coherence-direct-image}.", "\\medskip\\noindent", "Assume $K'$ is in $D_\\QCoh(\\mathcal{O}_{X'})$.", "The question is \\'etale local on $X'$", "hence we may assume $X'$ is affine.", "In this case the result follows from the case of schemes", "(More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-thickening-pseudo-coherent}).", "The translation into the language of schemes uses", "Derived Categories of Spaces, Lemmas", "\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site} and", "\\ref{spaces-perfect-lemma-descend-pseudo-coherent} and", "Remark \\ref{spaces-perfect-remark-match-total-direct-images}." ], "refs": [ "spaces-cohomology-lemma-projection-formula-finite", "spaces-perfect-lemma-pseudo-coherent", "spaces-perfect-lemma-quasi-coherence-tensor-product", "spaces-perfect-lemma-quasi-coherence-direct-image", "more-morphisms-lemma-thickening-pseudo-coherent", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-descend-pseudo-coherent", "spaces-perfect-remark-match-total-direct-images" ], "ref_ids": [ 11276, 2696, 2649, 2652, 14066, 2644, 2692, 2768 ] } ], "ref_ids": [] }, { "id": 273, "type": "theorem", "label": "spaces-more-morphisms-lemma-thickening-relatively-perfect", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-thickening-relatively-perfect", "contents": [ "Let $S$ be a scheme. Consider a cartesian diagram", "$$", "\\xymatrix{", "X \\ar[r]_i \\ar[d]_f & X' \\ar[d]^{f'} \\\\", "Y \\ar[r]^j & Y'", "}", "$$", "of algebraic spaces over $S$. Assume $X' \\to Y'$ is flat and locally", "of finite presentation and $Y \\to Y'$ is a finite order thickening.", "Let $E' \\in D(\\mathcal{O}_{X'})$. If $E = Li^*(E')$ is $Y$-perfect,", "then $E'$ is $Y'$-perfect." ], "refs": [], "proofs": [ { "contents": [ "Recall that being $Y$-perfect for $E$ means $E$ is", "pseudo-coherent and locally has finite tor dimension as a complex", "of $f^{-1}\\mathcal{O}_Y$-modules", "(Definition \\ref{definition-relatively-perfect}).", "By Lemma \\ref{lemma-thickening-pseudo-coherent}", "we find that $E'$ is pseudo-coherent.", "In particular, $E'$ is in $D_\\QCoh(\\mathcal{O}_{X'})$, see", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-pseudo-coherent}.", "By Lemma \\ref{lemma-affine-locally-rel-perfect}", "this reduces us to the case of schemes.", "The case of schemes is", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-thickening-relatively-perfect}." ], "refs": [ "spaces-more-morphisms-definition-relatively-perfect", "spaces-more-morphisms-lemma-thickening-pseudo-coherent", "spaces-perfect-lemma-pseudo-coherent", "spaces-more-morphisms-lemma-affine-locally-rel-perfect", "more-morphisms-lemma-thickening-relatively-perfect" ], "ref_ids": [ 300, 272, 2696, 259, 14067 ] } ], "ref_ids": [] }, { "id": 274, "type": "theorem", "label": "spaces-more-morphisms-lemma-henselian-relatively-perfect", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-henselian-relatively-perfect", "contents": [ "Let $(R, I)$ be a pair consisting of a ring and an ideal $I$", "contained in the Jacobson radical. Set $S = \\Spec(R)$ and $S_0 = \\Spec(R/I)$.", "Let $X$ be an algebraic space over $R$ whose structure morphism", "$f : X \\to S$ is proper, flat, and of finite presentation.", "Denote $X_0 = S_0 \\times_S X$. Let $E \\in D(\\mathcal{O}_X)$", "be pseudo-coherent. If the derived restriction $E_0$ of $E$", "to $X_0$ is $S_0$-perfect, then $E$ is $S$-perfect." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjective \\'etale morphism $U \\to X$ with $U$ affine.", "Choose a closed immersion $U \\to \\mathbf{A}^d_S$.", "Set $U_0 = S_0 \\times_S U$.", "The complex $E_0|_{U_0}$ has tor amplitude", "in $[a, b]$ for some $a, b \\in \\mathbf{Z}$.", "Let $\\overline{x}$ be a geometric point of $X$.", "We will show that the tor amplitude of", "$E_{\\overline{x}}$ over $R$ is in $[a - d, b]$.", "This will finish the proof as the tor amplitude can be", "read off from the stalks by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-tor-amplitude-stalk}", "and Properties of Spaces, Theorem", "\\ref{spaces-properties-theorem-exactness-stalks}.", "\\medskip\\noindent", "Let $x \\in |X|$ be the point determined by $\\overline{x}$.", "Recall that $|X| \\to |S|$ is closed (by definition of proper morphisms).", "Since $I$ is contained in the Jacobson radical, any nonempty closed", "subset of $S$ contains a point of the closed subscheme $S_0$.", "Hence we can find a specialization $x \\leadsto x_0$ in $|X|$", "with $x_0 \\in |X_0|$. Choose $u_0 \\in U_0$ mapping to $x_0$.", "By Decent Spaces, Lemma \\ref{decent-spaces-lemma-generalizations-lift-flat}", "(or by Decent Spaces, Lemma \\ref{decent-spaces-lemma-specialization}", "which applies directly to \\'etale morphisms)", "we find a specialization $u \\leadsto u_0$ in $U$", "such that $u$ maps to $x$. We may lift $\\overline{x}$", "to a geometric point $\\overline{u}$ of $U$ lying over $u$.", "Then we have $E_{\\overline{x}} = (E|_U)_{\\overline{u}}$.", "\\medskip\\noindent", "Write $U = \\Spec(A)$. Then $A$ is a flat, finitely presented", "$R$-algebra which is a quotient of a polynomial $R$-algebra in", "$d$-variables. The restriction $E|_U$ corresponds", "(by Derived Categories of Spaces, Lemmas", "\\ref{spaces-perfect-lemma-pseudo-coherent},", "\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}, and", "\\ref{spaces-perfect-lemma-descend-pseudo-coherent}", "and", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded} and", "\\ref{perfect-lemma-pseudo-coherent-affine})", "to a pseudo-coherent object $K$ of $D(A)$.", "Observe that $E_0$ corresponds to $K \\otimes_A^\\mathbf{L} A/IA$.", "Let $\\mathfrak q \\subset \\mathfrak q_0 \\subset A$ be the prime", "ideals corresponding to $u \\leadsto u_0$.", "Then", "$$", "E_{\\overline{x}} =", "(E|_U)_{\\overline{u}} =", "E_u \\otimes_{\\mathcal{O}_{U, u}}^\\mathbf{L} \\mathcal{O}_{U, \\overline{u}} =", "K_{\\mathfrak q} \\otimes_{A_\\mathfrak q}^\\mathbf{L} A_{\\mathfrak q}^{sh}", "$$", "(some details omitted). Since $A_\\mathfrak q \\to A_\\mathfrak q^{sh}$", "is flat, the tor amplitude of this as an $R$-module is the same as", "the tor amplitude of $K_\\mathfrak q$ as an $R$-module", "(More on Algebra, Lemma \\ref{more-algebra-lemma-no-change-tor-amplitude}).", "Also, $K_{\\mathfrak q}$ is a localization of $K_{\\mathfrak q_0}$.", "Hence it suffices to show that $K_{\\mathfrak q_0}$ has tor amplitude in", "$[a - d, b]$ as a complex of $R$-modules.", "\\medskip\\noindent", "Let $I \\subset \\mathfrak p_0 \\subset R$ be the prime", "ideal corresponding to $f(x_0)$. Then we have", "\\begin{align*}", "K \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p_0)", "& =", "(K \\otimes_R^\\mathbf{L} R/I) \\otimes_{R/I}^\\mathbf{L}", "\\kappa(\\mathfrak p_0) \\\\", "& =", "(K \\otimes_A^\\mathbf{L} A/IA) \\otimes_{R/I}^\\mathbf{L} \\kappa(\\mathfrak p_0)", "\\end{align*}", "the second equality because $R \\to A$ is flat.", "By our choice of $a, b$ this complex has cohomology", "only in degrees in the interval $[a, b]$.", "Thus we may finally apply", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-from-fibre-relatively-perfect}", "to $R \\to A$, $\\mathfrak q_0$, $\\mathfrak p_0$ and $K$", "to conclude." ], "refs": [ "sites-cohomology-lemma-tor-amplitude-stalk", "spaces-properties-theorem-exactness-stalks", "decent-spaces-lemma-generalizations-lift-flat", "decent-spaces-lemma-specialization", "spaces-perfect-lemma-pseudo-coherent", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-descend-pseudo-coherent", "perfect-lemma-affine-compare-bounded", "perfect-lemma-pseudo-coherent-affine", "more-algebra-lemma-no-change-tor-amplitude", "more-algebra-lemma-lift-from-fibre-relatively-perfect" ], "ref_ids": [ 4380, 11813, 9474, 9473, 2696, 2644, 2692, 6941, 6975, 10185, 10295 ] } ], "ref_ids": [] }, { "id": 275, "type": "theorem", "label": "spaces-more-morphisms-lemma-base-change-nodal", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-base-change-nodal", "contents": [ "The property of being at-worst-nodal of relative dimension $1$", "is preserved under base change." ], "refs": [], "proofs": [ { "contents": [ "See", "Morphisms of Spaces, Remark \\ref{spaces-morphisms-remark-base-change-P}", "and", "Algebraic Curves, Lemma \\ref{curves-lemma-base-change-nodal-family}." ], "refs": [ "spaces-morphisms-remark-base-change-P", "curves-lemma-base-change-nodal-family" ], "ref_ids": [ 5034, 6319 ] } ], "ref_ids": [] }, { "id": 276, "type": "theorem", "label": "spaces-more-morphisms-lemma-nodal-local", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-nodal-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is at-worst-nodal of relative dimension $1$,", "\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism", "$Z \\times_Y X \\to Z$ is at-worst-nodal of relative dimension $1$,", "\\item for every affine scheme $Z$ and any morphism", "$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is", "at-worst-nodal of relative dimension $1$,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is", "at-worst-nodal of relative dimension $1$,", "\\item there exists a scheme $U$ and a surjective \\'etale morphism", "$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$", "is at-worst-nodal of relative dimension $1$,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes and the vertical arrows are \\'etale", "the top horizontal arrow is at-worst-nodal of relative dimension $1$,", "\\item there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes, the vertical arrows are \\'etale, and", "$U \\to X$ is surjective such that the top horizontal arrow is", "at-worst-nodal of relative dimension $1$, and", "\\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$,", "and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that", "each morphism $X_{ij} \\to Y_i$ is", "at-worst-nodal of relative dimension $1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 277, "type": "theorem", "label": "spaces-more-morphisms-lemma-locus-where-nodal", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-lemma-locus-where-nodal", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "which is flat and locally of finite presentation. Then there", "is a maximal open subspace $X' \\subset X$ such that $f|_{X'} : X' \\to Y$", "is at-worst-nodal of relative dimension $1$. Moreover, formation", "of $X'$ commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_h & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$, $V$ are schemes, the vertical arrows are \\'etale, and", "$U \\to X$ is surjective. By the lemma for the case of schemes", "(Algebraic Curves, Lemma \\ref{curves-lemma-locus-where-nodal})", "we find a maximal open subscheme $U' \\subset U$", "such that $h|_{U'} : U' \\to V$ is at-worst-nodal of relative dimension $1$", "and such that formation of $U'$ commutes with base change.", "Let $X' \\subset X$ be the open subspace whose points correspond", "to the open subset $\\Im(|U'| \\to |X|)$.", "By Lemma \\ref{lemma-nodal-local} we see that $X' \\to Y$ is", "at-worst-nodal of relative dimension $1$ and that $X'$ is the", "largest open subspace with this property (this also implies", "that $U'$ is the inverse image of $X'$ in $U$, but we do", "not need this). Since the same is true after base change", "the proof is complete." ], "refs": [ "curves-lemma-locus-where-nodal", "spaces-more-morphisms-lemma-nodal-local" ], "ref_ids": [ 6320, 276 ] } ], "ref_ids": [] }, { "id": 315, "type": "theorem", "label": "algebra-theorem-chevalley", "categories": [ "algebra" ], "title": "algebra-theorem-chevalley", "contents": [ "Suppose that $R \\to S$ is of finite presentation.", "The image of a constructible subset of", "$\\Spec(S)$ in $\\Spec(R)$ is constructible." ], "refs": [], "proofs": [ { "contents": [ "Write $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$.", "We may factor $R \\to S$ as $R \\to R[x_1] \\to R[x_1, x_2]", "\\to \\ldots \\to R[x_1, \\ldots, x_{n-1}] \\to S$. Hence", "we may assume that $S = R[x]/(f_1, \\ldots, f_m)$.", "In this case we factor the map as $R \\to R[x] \\to S$,", "and by Lemma \\ref{lemma-closed-fp} we reduce to", "the case $S = R[x]$. By Lemma \\ref{lemma-qc-open} suffices", "to show that if", "$T = (\\bigcup_{i = 1\\ldots n} D(f_i)) \\cap V(g_1, \\ldots, g_m)$", "for $f_i , g_j \\in R[x]$ then the image in $\\Spec(R)$ is", "constructible. Since finite unions of constructible sets", "are constructible, it suffices to deal with the case $n = 1$,", "i.e., when $T = D(f) \\cap V(g_1, \\ldots, g_m)$.", "\\medskip\\noindent", "Note that if $c \\in R$, then we have", "$$", "\\Spec(R) =", "V(c) \\amalg D(c) =", "\\Spec(R/(c)) \\amalg \\Spec(R_c),", "$$", "and correspondingly $\\Spec(R[x]) =", "V(c) \\amalg D(c) = \\Spec(R/(c)[x]) \\amalg", "\\Spec(R_c[x])$. The intersection of $T = D(f) \\cap V(g_1, \\ldots, g_m)$", "with each part still has the same shape, with $f$, $g_i$ replaced", "by their images in $R/(c)[x]$, respectively $R_c[x]$.", "Note that the image of $T$", "in $\\Spec(R)$ is the union of the image of", "$T \\cap V(c)$ and $T \\cap D(c)$. Using Lemmas \\ref{lemma-open-fp}", "and \\ref{lemma-closed-fp} it suffices to prove the images of both", "parts are constructible in $\\Spec(R/(c))$, respectively", "$\\Spec(R_c)$.", "\\medskip\\noindent", "Let us assume we have $T = D(f) \\cap V(g_1, \\ldots, g_m)$", "as above, with $\\deg(g_1) \\leq \\deg(g_2) \\leq \\ldots \\leq \\deg(g_m)$.", "We are going to use induction on $m$, and on the", "degrees of the $g_i$. Let $d = \\deg(g_1)$, i.e., $g_1 = c x^{d_1} + l.o.t$", "with $c \\in R$ not zero. Cutting $R$ up into the pieces", "$R/(c)$ and $R_c$ we either lower the degree of $g_1$ (and this", "is covered by induction)", "or we reduce to the case where $c$ is invertible.", "If $c$ is invertible, and $m > 1$, then write", "$g_2 = c' x^{d_2} + l.o.t$. In this case consider", "$g_2' = g_2 - (c'/c) x^{d_2 - d_1} g_1$. Since the ideals", "$(g_1, g_2, \\ldots, g_m)$ and $(g_1, g_2', g_3, \\ldots, g_m)$", "are equal we see that $T = D(f) \\cap V(g_1, g_2', g_3\\ldots, g_m)$.", "But here the degree of $g_2'$ is strictly less than the degree", "of $g_2$ and hence this case is covered by induction.", "\\medskip\\noindent", "The bases case for the induction above are the cases", "(a) $T = D(f) \\cap V(g)$ where the leading coefficient", "of $g$ is invertible, and (b) $T = D(f)$. These two cases", "are dealt with in Lemmas \\ref{lemma-affineline-special}", "and \\ref{lemma-affineline-open}." ], "refs": [ "algebra-lemma-closed-fp", "algebra-lemma-qc-open", "algebra-lemma-open-fp", "algebra-lemma-closed-fp", "algebra-lemma-affineline-special", "algebra-lemma-affineline-open" ], "ref_ids": [ 437, 432, 436, 437, 440, 438 ] } ], "ref_ids": [] }, { "id": 316, "type": "theorem", "label": "algebra-theorem-nullstellensatz", "categories": [ "algebra" ], "title": "algebra-theorem-nullstellensatz", "contents": [ "Let $k$ be a field.", "\\begin{enumerate}", "\\item", "\\label{item-finite-kappa}", "For any maximal ideal $\\mathfrak m \\subset k[x_1, \\ldots, x_n]$", "the field extension $k \\subset \\kappa(\\mathfrak m)$ is finite.", "\\item", "\\label{item-polynomial-ring-Jacobson}", "Any radical ideal $I \\subset k[x_1, \\ldots, x_n]$", "is the intersection of maximal ideals containing it.", "\\end{enumerate}", "The same is true in any finite type $k$-algebra." ], "refs": [], "proofs": [ { "contents": [ "It is enough to prove part (\\ref{item-finite-kappa}) of", "the theorem for the case of a polynomial", "algebra $k[x_1, \\ldots, x_n]$, because any finitely generated", "$k$-algebra is a quotient of such a polynomial algebra.", "We prove this by induction on $n$. The case $n = 0$ is clear.", "Suppose that $\\mathfrak m$ is a maximal ideal in $k[x_1, \\ldots, x_n]$.", "Let $\\mathfrak p \\subset k[x_n]$ be the intersection", "of $\\mathfrak m$ with $k[x_n]$.", "\\medskip\\noindent", "If $\\mathfrak p \\not = (0)$,", "then $\\mathfrak p$ is maximal and generated by an irreducible", "monic polynomial $P$ (because of the Euclidean algorithm", "in $k[x_n]$). Then", "$k' = k[x_n]/\\mathfrak p$ is a finite field extension of $k$", "and contained in $\\kappa(\\mathfrak m)$. In this case", "we get a surjection", "$$", "k'[x_1, \\ldots, x_{n-1}]", "\\to", "k'[x_1, \\ldots, x_n] =", "k' \\otimes_k k[x_1, \\ldots, x_n]", "\\longrightarrow", "\\kappa(\\mathfrak m)", "$$", "and hence we see that $\\kappa(\\mathfrak m)$ is a finite", "extension of $k'$ by induction hypothesis. Thus $\\kappa(\\mathfrak m)$", "is finite over $k$ as well.", "\\medskip\\noindent", "If $\\mathfrak p = (0)$ we consider the ring", "extension $k[x_n] \\subset k[x_1, \\ldots, x_n]/\\mathfrak m$.", "This is a finitely generated ring extension, hence", "of finite presentation by", "Lemmas \\ref{lemma-obvious-Noetherian} and", "\\ref{lemma-Noetherian-finite-type-is-finite-presentation}.", "Thus the image of $\\Spec(k[x_1, \\ldots, x_n]/\\mathfrak m)$", "in $\\Spec(k[x_n])$ is constructible by", "Theorem \\ref{theorem-chevalley}. Since the image", "contains $(0)$ we conclude that it contains a standard", "open $D(f)$ for some $f\\in k[x_n]$ nonzero. Since clearly", "$D(f)$ is infinite we get a contradiction with the", "assumption that $k[x_1, \\ldots, x_n]/\\mathfrak m$ is", "a field (and hence has a spectrum consisting of one point).", "\\medskip\\noindent", "To prove part (\\ref{item-polynomial-ring-Jacobson}) let", "$I \\subset R$ be a radical ideal, with $R$ of finite type over $k$.", "Let $f \\in R$, $f \\not \\in I$. Pick a maximal ideal $\\mathfrak m'$", "in the nonzero ring $R_f/IR_f = (R/I)_f$. Let $\\mathfrak m \\subset R$", "be the inverse image of $\\mathfrak m'$ in $R$. We see that", "$I \\subset \\mathfrak m$", "and $f \\not \\in \\mathfrak m$. If we show that $\\mathfrak m$ is a maximal", "ideal of $R$, then we are done. We clearly have", "$$", "k \\subset R/\\mathfrak m \\subset \\kappa(\\mathfrak m').", "$$", "By part (\\ref{item-finite-kappa}) the field extension", "$k \\subset \\kappa(\\mathfrak m')$ is finite. Hence", "$R/\\mathfrak m$ is a field by Fields, Lemma", "\\ref{fields-lemma-subalgebra-algebraic-extension-field}.", "Thus $\\mathfrak m$ is maximal and the proof is complete." ], "refs": [ "algebra-lemma-obvious-Noetherian", "algebra-lemma-Noetherian-finite-type-is-finite-presentation", "algebra-theorem-chevalley", "fields-lemma-subalgebra-algebraic-extension-field" ], "ref_ids": [ 450, 451, 315, 4457 ] } ], "ref_ids": [] }, { "id": 317, "type": "theorem", "label": "algebra-theorem-uncountable-nullstellensatz", "categories": [ "algebra" ], "title": "algebra-theorem-uncountable-nullstellensatz", "contents": [ "Let $k$ be a field. Let $S$ be a $k$-algebra generated over $k$", "by the elements $\\{x_i\\}_{i \\in I}$. Assume the cardinality of $I$", "is smaller than the cardinality of $k$. Then", "\\begin{enumerate}", "\\item for all maximal ideals $\\mathfrak m \\subset S$ the field", "extension $k \\subset \\kappa(\\mathfrak m)$", "is algebraic, and", "\\item $S$ is a Jacobson ring.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $I$ is finite then the result follows from the Hilbert Nullstellensatz,", "Theorem \\ref{theorem-nullstellensatz}. In the rest of the proof we assume", "$I$ is infinite. It suffices to prove the result for", "$\\mathfrak m \\subset k[\\{x_i\\}_{i \\in I}]$ maximal in the polynomial", "ring on variables $x_i$, since $S$ is a quotient of this.", "As $I$ is infinite the set", "of monomials $x_{i_1}^{e_1} \\ldots x_{i_r}^{e_r}$, $i_1, \\ldots, i_r \\in I$", "and $e_1, \\ldots, e_r \\geq 0$ has cardinality at most equal to the", "cardinality of $I$. Because the cardinality of $I \\times \\ldots \\times I$", "is the cardinality of $I$, and also the cardinality of", "$\\bigcup_{n \\geq 0} I^n$ has the same cardinality.", "(If $I$ is finite, then this is not true and", "in that case this proof only works if $k$ is uncountable.)", "\\medskip\\noindent", "To arrive at a contradiction pick $T \\in \\kappa(\\mathfrak m)$ transcendental", "over $k$. Note that the $k$-linear map", "$T : \\kappa(\\mathfrak m) \\to \\kappa(\\mathfrak m)$", "given by multiplication by $T$ has the property that $P(T)$ is invertible", "for all monic polynomials $P(t) \\in k[t]$.", "Also, $\\kappa(\\mathfrak m)$ has dimension at most the cardinality of $I$", "over $k$ since it is a quotient of the vector space", "$k[\\{x_i\\}_{i \\in I}]$ over $k$ (whose dimension is $\\# I$ as we saw above).", "This is impossible by Lemma \\ref{lemma-dimension}.", "\\medskip\\noindent", "To show that $S$ is Jacobson we argue as follows. If not then", "there exists a prime $\\mathfrak q \\subset S$ and an element $f \\in S$,", "$f \\not \\in \\mathfrak q$ such that $\\mathfrak q$ is not maximal", "and $(S/\\mathfrak q)_f$ is a field, see", "Lemma \\ref{lemma-characterize-jacobson}.", "But note that $(S/\\mathfrak q)_f$ is generated by at most $\\# I + 1$ elements.", "Hence the field extension $k \\subset (S/\\mathfrak q)_f$ is algebraic", "(by the first part of the proof).", "This implies that $\\kappa(\\mathfrak q)$ is an algebraic extension of $k$", "hence $\\mathfrak q$ is maximal by", "Lemma \\ref{lemma-finite-residue-extension-closed}. This contradiction", "finishes the proof." ], "refs": [ "algebra-theorem-nullstellensatz", "algebra-lemma-dimension", "algebra-lemma-characterize-jacobson", "algebra-lemma-finite-residue-extension-closed" ], "ref_ids": [ 316, 473, 470, 472 ] } ], "ref_ids": [] }, { "id": 318, "type": "theorem", "label": "algebra-theorem-lazard", "categories": [ "algebra" ], "title": "algebra-theorem-lazard", "contents": [ "Let $M$ be an $R$-module. Then $M$ is flat if and only if it is the colimit of", "a directed system of free finite $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "A colimit of a directed system of flat modules is flat, as taking directed", "colimits is exact and commutes with tensor product. Hence if $M$ is the colimit", "of a directed system of free finite modules then $M$ is flat.", "\\medskip\\noindent", "For the converse, first recall that any module $M$ can be written as the", "colimit of a directed system of finitely presented modules, in the following", "way. Choose a surjection $f: R^I \\to M$ for some set $I$, and let $K$", "be the kernel. Let $E$ be the set of ordered pairs $(J, N)$ where $J$ is a", "finite subset of $I$ and $N$ is a finitely generated submodule of $R^J \\cap K$.", "Then $E$ is made into a directed partially ordered set by defining $(J, N) \\leq", "(J', N')$ if and only if $J \\subset J'$ and $N \\subset N'$. Define $M_e =", "R^J/N$ for $e = (J, N)$, and define $f_{ee'}: M_e \\to M_{e'}$ to be", "the natural map for $e \\leq e'$. Then $(M_e, f_{ee'})$ is a directed system and", "the natural maps $f_e: M_e \\to M$ induce an isomorphism", "$\\colim_{e", "\\in E} M_e \\xrightarrow{\\cong} M$.", "\\medskip\\noindent", "Now suppose $M$ is flat. Let $I = M \\times \\mathbf{Z}$, write $(x_i)$ for", "the canonical basis of $R^{I}$, and take in the above discussion $f: R^I", "\\to M$ to be the map sending $x_i$ to the projection of $i$ onto $M$.", "To prove the theorem it suffices to show that the $e \\in E$ such that $M_e$", "is free form a cofinal subset of $E$. So let $e = (J, N) \\in E$ be arbitrary.", "By Lemma \\ref{lemma-flat-factors-fp} there is a free finite module $F$ and maps", "$h: R^J/N \\to F$ and $g: F \\to M$ such that the natural map", "$f_e: R^J/N \\to M$ factors as $R^J/N \\xrightarrow{h} F", "\\xrightarrow{g} M$. We are going to realize $F$ as $M_{e'}$ for some $e' \\geq", "e$.", "\\medskip\\noindent", "Let $\\{ b_1, \\ldots, b_n \\}$ be a finite basis of $F$. Choose $n$ distinct", "elements $i_1, \\ldots, i_n \\in I$ such that $i_{\\ell} \\notin J$ for all $\\ell$,", "and such that the image of $x_{i_{\\ell}}$ under $f: R^I \\to M$ equals", "the image of $b_{\\ell}$ under $g: F \\to M$. This is possible since", "every element of $M$ can be written as $f(x_i)$ for infinitely many", "distinct $i \\in I$ (by our choice of $I$). Now let", "$J' = J \\cup \\{i_1, \\ldots , i_n \\}$, and define $R^{J'}", "\\to F$ by $x_i \\mapsto h(x_i)$ for $i \\in J$ and $x_{i_{\\ell}} \\mapsto", "b_{\\ell}$ for $\\ell = 1, \\ldots, n$. Let $N' = \\Ker(R^{J'} \\to F)$.", "Observe:", "\\begin{enumerate}", "\\item The square", "$$", "\\xymatrix{", "R^{J'} \\ar[r] \\ar@{^{(}->}[d] & F \\ar[d]^{g} \\\\", "R^{I} \\ar[r]_{f} & M", "}", "$$", "is commutative,", "%$R^{J'} \\to F$ factors $f: R^I \\to M$,", "hence $N' \\subset K = \\Ker(f)$;", "\\item $R^{J'} \\to F$ is a surjection onto a free finite module, hence", "it splits and so $N'$ is finitely generated;", "\\item $J \\subset J'$ and $N \\subset N'$.", "\\end{enumerate}", "By (1) and (2) $e' = (J', N')$ is in $E$, by (3) $e' \\geq e$, and by", "construction $M_{e'} = R^{J'}/N' \\cong F$ is free." ], "refs": [ "algebra-lemma-flat-factors-fp" ], "ref_ids": [ 806 ] } ], "ref_ids": [] }, { "id": 319, "type": "theorem", "label": "algebra-theorem-universally-exact-criteria", "categories": [ "algebra" ], "title": "algebra-theorem-universally-exact-criteria", "contents": [ "Let", "$$", "0 \\to M_1 \\xrightarrow{f_1} M_2 \\xrightarrow{f_2} M_3 \\to 0", "$$", "be an exact sequence of $R$-modules. The following are equivalent:", "\\begin{enumerate}", "\\item The sequence $0 \\to M_1 \\to M_2 \\to M_3", "\\to 0$ is universally exact.", "\\item For every finitely presented $R$-module $Q$, the sequence", "$$", "0 \\to M_1 \\otimes_R Q \\to M_2 \\otimes_R Q \\to", "M_3 \\otimes_R Q \\to 0", "$$", "is exact.", "\\item Given elements $x_i \\in M_1$ $(i = 1, \\ldots, n)$, $y_j \\in M_2$ $(j = 1,", "\\ldots, m)$, and $a_{ij} \\in R$ $(i = 1, \\ldots, n, j = 1, \\ldots, m)$ such that", "for all $i$", "$$", "f_1(x_i) = \\sum\\nolimits_j a_{ij} y_j,", "$$", "there exists $z_j \\in M_1$ $(j =1, \\ldots, m)$ such that for all $i$,", "$$", "x_i = \\sum\\nolimits_j a_{ij} z_j .", "$$", "\\item Given a commutative diagram of $R$-module maps", "$$", "\\xymatrix{", "R^n \\ar[r] \\ar[d] & R^m \\ar[d] \\\\", "M_1 \\ar[r]^{f_1} & M_2", "}", "$$", "where $m$ and $n$ are integers, there exists a map $R^m \\to M_1$ making", "the top triangle commute.", "\\item For every finitely presented $R$-module $P$, the $R$-module", "map $\\Hom_R(P, M_2) \\to \\Hom_R(P, M_3)$ is surjective.", "\\item The sequence $0 \\to M_1 \\to M_2 \\to M_3", "\\to 0$ is the colimit of a directed system of split exact sequences of", "the form", "$$", "0 \\to M_{1} \\to M_{2, i} \\to M_{3, i} \\to 0", "$$", "where the $M_{3, i}$ are finitely presented.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Obviously (1) implies (2).", "\\medskip\\noindent", "Next we show (2) implies (3). Let $f_1(x_i) = \\sum_j a_{ij} y_j$ be relations", "as in (3). Let $(d_j)$ be a basis for $R^m$, $(e_i)$ a basis for $R^n$, and", "$R^m \\to R^n$ the map given by $d_j \\mapsto \\sum_i a_{ij} e_i$.", "Let $Q$ be the cokernel of $R^m \\to R^n$. Then tensoring", "$R^m \\to R^n \\to Q \\to 0$ by the map $f_1: M_1 \\to M_2$, we get a", "commutative diagram", "$$", "\\xymatrix{", "M_1^{\\oplus m} \\ar[r] \\ar[d] & M_1^{\\oplus n} \\ar[r] \\ar[d] & M_1 \\otimes_R Q", "\\ar[r] \\ar[d] & 0 \\\\", "M_2^{\\oplus m} \\ar[r] & M_2^{\\oplus n} \\ar[r] & M_2 \\otimes_R Q \\ar[r] & 0", "}", "$$", "where $M_1^{\\oplus m} \\to M_1^{\\oplus n}$ is given by", "$$", "(z_1, \\ldots, z_m) \\mapsto", "(\\sum\\nolimits_j a_{1j} z_j, \\ldots, \\sum\\nolimits_j a_{nj} z_j),", "$$", "and $M_2^{\\oplus m} \\to M_2^{\\oplus n}$ is given similarly. We want to", "show $x = (x_1, \\ldots, x_n) \\in M_1^{\\oplus n}$ is in the image of $M_1^{\\oplus", "m} \\to M_1^{\\oplus n}$. By (2) the map $M_1 \\otimes Q \\to M_2", "\\otimes Q$ is injective, hence by exactness of the top row it is enough to show", "$x$ maps to $0$ in $M_2 \\otimes Q$, and so by exactness of the bottom row it is", "enough to show the image of $x$ in $M_2^{\\oplus n}$ is in the image of", "$M_2^{\\oplus m} \\to M_2^{\\oplus n}$. This is true by assumption.", "\\medskip\\noindent", "Condition (4) is just a translation of (3) into diagram form.", "\\medskip\\noindent", "Next we show (4) implies (5). Let $\\varphi : P \\to M_3$ be a map from a", "finitely presented $R$-module $P$. We must show that $\\varphi$ lifts to a map", "$P \\to M_2$. Choose a presentation of $P$,", "$$", "R^n \\xrightarrow{g_1} R^m \\xrightarrow{g_2} P \\to 0.", "$$", "Using freeness of $R^n$ and $R^m$, we can construct $h_2: R^m \\to M_2$", "and then $h_1: R^n \\to M_1$ such that the following diagram commutes", "$$", "\\xymatrix{", " & R^n \\ar[r]^{g_1} \\ar[d]^{h_1} & R^m \\ar[r]^{g_2} \\ar[d]^{h_2} & P", "\\ar[r] \\ar[d]^{\\varphi} & 0 \\\\", "0 \\ar[r] & M_1 \\ar[r]^{f_1} & M_2 \\ar[r]^{f_2} & M_3 \\ar[r] & 0 .", "}", "$$", "By (4) there is a map $k_1: R^m \\to M_1$ such that $k_1 \\circ g_1 =", "h_1$. Now define $h'_2: R^m \\to M_2$ by $h_2' = h_2 - f_1 \\circ k_1$.", "Then", "$$", "h'_2 \\circ g_1 = h_2 \\circ g_1 - f_1 \\circ k_1 \\circ g_1 =", "h_2 \\circ g_1 - f_1 \\circ h_1 = 0 .", "$$", "Hence by passing to the quotient $h'_2$ defines a map $\\varphi': P \\to", "M_2$ such that $\\varphi' \\circ g_2 = h_2'$. In a diagram, we have", "$$", "\\xymatrix{", "R^m \\ar[r]^{g_2} \\ar[d]_{h'_2} & P \\ar[d]^{\\varphi} \\ar[dl]_{\\varphi'} \\\\", "M_2 \\ar[r]^{f_2} & M_3.", "}", "$$", "where the top triangle commutes. We claim that $\\varphi'$ is the desired lift,", "i.e.\\ that $f_2 \\circ \\varphi' = \\varphi$. From the definitions we have", "$$", "f_2 \\circ \\varphi' \\circ g_2 = f_2 \\circ h'_2 =", "f_2 \\circ h_2 - f_2 \\circ f_1 \\circ k_1 = f_2 \\circ h_2 =", "\\varphi \\circ g_2.", "$$", "Since $g_2$ is surjective, this finishes the proof.", "\\medskip\\noindent", "Now we show (5) implies (6). Write $M_{3}$ as the colimit of a directed system", "of finitely presented modules $M_{3, i}$, see", "Lemma \\ref{lemma-module-colimit-fp}. Let $M_{2, i}$ be the fiber product", "of $M_{3, i}$ and $M_{2}$ over $M_{3}$---by definition this is the submodule of", "$M_2 \\times M_{3, i}$ consisting of elements whose two projections onto $M_3$", "are equal. Let $M_{1, i}$ be the kernel of the projection", "$M_{2, i} \\to M_{3, i}$. Then we have a directed system of exact sequences", "$$", "0 \\to M_{1, i} \\to M_{2, i} \\to M_{3, i} \\to 0,", "$$", "and for each $i$ a map of exact sequences", "$$", "\\xymatrix{", "0 \\ar[r] & M_{1, i} \\ar[d] \\ar[r] & M_{2, i} \\ar[r] \\ar[d] & M_{3, i} \\ar[d]", "\\ar[r] & 0 \\\\", "0 \\ar[r] & M_{1} \\ar[r] & M_{2} \\ar[r] & M_{3} \\ar[r] & 0", "}", "$$", "compatible with the directed system. From the definition of the fiber product", "$M_{2, i}$, it follows that the map $M_{1, i} \\to M_1$ is an isomorphism.", " By (5) there is a map $M_{3, i} \\to M_{2}$ lifting $M_{3, i} \\to", "M_3$, and by the universal property of the fiber product this gives rise to a", "section of $M_{2, i} \\to M_{3, i}$. Hence the sequences", "$$", "0 \\to M_{1, i} \\to M_{2, i} \\to M_{3, i} \\to 0", "$$", "split. Passing to the colimit, we have a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] & \\colim M_{1, i} \\ar[d]^{\\cong} \\ar[r] & \\colim M_{2, i}", " \\ar[r]", "\\ar[d] & \\colim M_{3, i} \\ar[d]^{\\cong} \\ar[r] & 0 \\\\", "0 \\ar[r] & M_{1} \\ar[r] & M_{2} \\ar[r] & M_{3} \\ar[r] & 0", "}", "$$", "with exact rows and outer vertical maps isomorphisms. Hence $\\colim", "M_{2, i}", "\\to M_2$ is also an isomorphism and (6) holds.", "\\medskip\\noindent", "Condition (6) implies (1) by", "Example \\ref{example-universally-exact} (2)." ], "refs": [ "algebra-lemma-module-colimit-fp" ], "ref_ids": [ 355 ] } ], "ref_ids": [] }, { "id": 320, "type": "theorem", "label": "algebra-theorem-kaplansky-direct-sum", "categories": [ "algebra" ], "title": "algebra-theorem-kaplansky-direct-sum", "contents": [ "Suppose $M$ is a direct sum of countably generated $R$-modules. If $P$ is a", "direct summand of $M$, then $P$ is also a direct sum of countably generated", "$R$-modules." ], "refs": [], "proofs": [ { "contents": [ "Write $M = P \\oplus Q$. We are going to construct a Kaplansky d\\'evissage", "$(M_{\\alpha})_{\\alpha \\in S}$ of $M$ which, in addition to the defining", "properties (0)-(4), satisfies:", "\\begin{enumerate}", "\\item[(5)] Each $M_{\\alpha}$ is a direct summand of $M$;", "\\item[(6)] $M_{\\alpha} = P_{\\alpha} \\oplus Q_{\\alpha}$, where $P_{\\alpha} =P", "\\cap M_{\\alpha}$ and $Q = Q \\cap M_{\\alpha}$.", "\\end{enumerate}", "(Note: if properties (0)-(2) hold, then in fact property (3) is equivalent to", "property (5).)", "\\medskip\\noindent", "To see how this implies the theorem, it is enough to show that", "$(P_{\\alpha})_{\\alpha \\in S}$ forms a Kaplansky d\\'evissage of $P$. Properties", "(0), (1), and (2) are clear. By (5) and (6) for $(M_{\\alpha})$, each", "$P_{\\alpha}$ is a direct summand of $M$. Since $P_{\\alpha} \\subset P_{\\alpha +", "1}$, this implies $P_{\\alpha}$ is a direct summand of $P_{\\alpha + 1}$; hence", "(3) holds for $(P_{\\alpha})$. For (4), note that", "$$", "M_{\\alpha + 1}/M_{\\alpha} \\cong P_{\\alpha + 1}/P_{\\alpha} \\oplus", "Q_{\\alpha + 1}/Q_{\\alpha},", "$$", "so $P_{\\alpha + 1}/P_{\\alpha}$ is countably generated because this is true of", "$M_{\\alpha + 1}/M_{\\alpha}$.", "\\medskip\\noindent", "It remains to construct the $M_{\\alpha}$. Write $M = \\bigoplus_{i \\in I} N_i$", "where each $N_i$ is a countably generated $R$-module. Choose a well-ordering", "of $I$. By transfinite induction we are going to define an increasing family", "of submodules $M_{\\alpha}$ of $M$, one for each ordinal $\\alpha$, such that", "$M_{\\alpha}$ is a direct sum of some subset of the $N_i$.", "\\medskip\\noindent", "For $\\alpha = 0$ let $M_{0} = 0$. If $\\alpha$ is a limit ordinal and", "$M_{\\beta}$ has been defined for all $\\beta < \\alpha$, then define $M_{\\alpha}", "= \\bigcup_{\\beta < \\alpha} M_{\\beta}$. Since each $M_{\\beta}$ for $\\beta <", "\\alpha$ is a direct sum of a subset of the $N_i$, the same will be true of", "$M_{\\alpha}$. If $\\alpha + 1$ is a successor ordinal and $M_{\\alpha}$ has been", "defined, then define $M_{\\alpha + 1}$ as follows. If $M_{\\alpha} = M$, then let", "$M_{\\alpha + 1} = M$. If not, choose the smallest $j \\in I$ such that $N_j$ is", "not contained in $M_{\\alpha}$. We will construct an infinite matrix $(x_{mn}),", "m, n = 1, 2, 3, \\ldots$ such that:", "\\begin{enumerate}", "\\item $N_j$ is contained in the submodule of $M$ generated by the entries", "$x_{mn}$;", "\\item if we write any entry $x_{k\\ell}$ in terms of its $P$- and", "$Q$-components, $x_{k\\ell} = y_{k\\ell} + z_{k\\ell}$, then the matrix $(x_{mn})$", "contains a set of generators for each $N_i$ for which $y_{k\\ell}$ or", "$z_{k\\ell}$ has nonzero component.", "\\end{enumerate}", "Then we define $M_{\\alpha + 1}$ to be the submodule of $M$ generated by", "$M_{\\alpha}$ and all $x_{mn}$; by property (2) of the matrix $(x_{mn})$,", "$M_{\\alpha + 1}$ will be a direct sum of some subset of the $N_i$.", "To construct the matrix $(x_{mn})$, let $x_{11}, x_{12}, x_{13}, \\ldots$", "be a countable set of generators for $N_j$. Then if", "$x_{11} = y_{11} + z_{11}$ is the decomposition into $P$- and", "$Q$-components, let $x_{21}, x_{22}, x_{23}, \\ldots$ be a countable", "set of generators for the sum of the $N_i$ for which $y_{11}$ or $z_{11}$ have", "nonzero component. Repeat this process on $x_{12}$ to get elements $x_{31},", "x_{32}, \\ldots$, the third row of our matrix. Repeat on $x_{21}$ to get the", "fourth row, on $x_{13}$ to get the fifth, and so on, going down along", "successive anti-diagonals as indicated below:", "$$", "\\left(", "\\vcenter{", "\\xymatrix@R=2mm@C=2mm{", "x_{11} & x_{12} \\ar[dl] & x_{13} \\ar[dl] & x_{14} \\ar[dl] & \\ldots \\\\", "x_{21} & x_{22} \\ar[dl] & x_{23} \\ar[dl] & \\ldots \\\\", "x_{31} & x_{32} \\ar[dl] & \\ldots \\\\", "x_{41} & \\ldots \\\\", "\\ldots", "}", "}", "\\right).", "$$", "\\medskip\\noindent", "Transfinite induction on $I$ (using the fact that we constructed", "$M_{\\alpha + 1}$", "to contain $N_j$ for the smallest $j$ such that $N_j$ is not contained in", "$M_{\\alpha}$) shows that for each $i \\in I$, $N_i$ is contained in some", "$M_{\\alpha}$. Thus, there is some large enough ordinal $S$ satisfying: for", "each $i \\in I$ there is $\\alpha \\in S$ such that $N_i$ is contained in", "$M_{\\alpha}$. This means $(M_{\\alpha})_{\\alpha \\in S}$ satisfies property (1)", "of a Kaplansky d\\'evissage of $M$. The family $(M_{\\alpha})_{\\alpha \\in S}$", "moreover satisfies the other defining properties, and also (5) and (6) above:", "properties (0), (2), (4), and (6) are clear by construction; property (5) is", "true because each $M_{\\alpha}$ is by construction a direct sum of some $N_i$;", "and (3) is implied by (5) and the fact that $M_{\\alpha} \\subset M_{\\alpha + 1}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 321, "type": "theorem", "label": "algebra-theorem-projective-direct-sum", "categories": [ "algebra" ], "title": "algebra-theorem-projective-direct-sum", "contents": [ "\\begin{slogan}", "Any projective module is a direct sum of countably generated", "projective modules.", "\\end{slogan}", "If $P$ is a projective $R$-module, then $P$ is a direct sum of countably", "generated projective $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "A module is projective if and only if it is a direct summand of a free module,", "so this follows from Theorem \\ref{theorem-kaplansky-direct-sum}." ], "refs": [ "algebra-theorem-kaplansky-direct-sum" ], "ref_ids": [ 320 ] } ], "ref_ids": [] }, { "id": 322, "type": "theorem", "label": "algebra-theorem-projective-free-over-local-ring", "categories": [ "algebra" ], "title": "algebra-theorem-projective-free-over-local-ring", "contents": [ "\\begin{slogan}", "Projective modules over local rings are free.", "\\end{slogan}", "If $P$ is a projective module over a local ring $R$, then $P$ is free." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemmas \\ref{lemma-projective-free}, \\ref{lemma-freeness-criteria},", "and \\ref{lemma-projective-freeness-criteria}." ], "refs": [ "algebra-lemma-projective-free", "algebra-lemma-freeness-criteria", "algebra-lemma-projective-freeness-criteria" ], "ref_ids": [ 822, 823, 824 ] } ], "ref_ids": [] }, { "id": 323, "type": "theorem", "label": "algebra-theorem-projectivity-characterization", "categories": [ "algebra" ], "title": "algebra-theorem-projectivity-characterization", "contents": [ "Let $M$ be an $R$-module. Then $M$ is projective if and only it satisfies:", "\\begin{enumerate}", "\\item $M$ is flat,", "\\item $M$ is Mittag-Leffler,", "\\item $M$ is a direct sum of countably generated $R$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First suppose $M$ is projective. Then $M$ is a direct summand of a free", "module, so $M$ is flat and Mittag-Leffler since these properties pass to direct", "summands. By Kaplansky's theorem (Theorem \\ref{theorem-projective-direct-sum}),", "$M$ satisfies (3).", "\\medskip\\noindent", "Conversely, suppose $M$ satisfies (1)-(3). Since being flat and Mittag-Leffler", "passes to direct summands, $M$ is a direct sum of flat, Mittag-Leffler,", "countably generated $R$-modules.", "Lemma \\ref{lemma-countgen-projective}", "implies $M$ is a direct sum of projective modules. Hence $M$ is projective." ], "refs": [ "algebra-theorem-projective-direct-sum", "algebra-lemma-countgen-projective" ], "ref_ids": [ 321, 850 ] } ], "ref_ids": [] }, { "id": 324, "type": "theorem", "label": "algebra-theorem-ffdescent-projectivity", "categories": [ "algebra" ], "title": "algebra-theorem-ffdescent-projectivity", "contents": [ "Let $R \\to S$ be a faithfully flat ring map. Let $M$ be an $R$-module.", " If the $S$-module $M \\otimes_R S$ is projective, then $M$ is projective." ], "refs": [], "proofs": [ { "contents": [ "We are going to construct a Kaplansky d\\'evissage of $M$ to show that it is a", "direct sum of projective modules and hence projective. By Theorem", "\\ref{theorem-projective-direct-sum} we can write $M \\otimes_R S =", "\\bigoplus_{i \\in I} Q_i$ as a direct sum of countably generated $S$-modules", "$Q_i$. Choose a well-ordering on $M$. By transfinite induction we are going", "to define an increasing family of submodules $M_{\\alpha}$ of $M$, one for each", "ordinal $\\alpha$, such that $M_{\\alpha} \\otimes_R S$ is a direct sum of some", "subset of the $Q_i$.", "\\medskip\\noindent", "For $\\alpha = 0$ let $M_0 = 0$. If $\\alpha$ is a limit ordinal and $M_{\\beta}$", "has been defined for all $\\beta < \\alpha$, then define $M_{\\beta} =", "\\bigcup_{\\beta < \\alpha} M_{\\beta}$. Since each $M_{\\beta} \\otimes_R S$ for", "$\\beta < \\alpha$ is a direct sum of a subset of the $Q_i$, the same will be", "true of $M_{\\alpha} \\otimes_R S$. If $\\alpha + 1$ is a successor ordinal and", "$M_{\\alpha}$ has been defined, then define $M_{\\alpha + 1}$ as follows. If", "$M_{\\alpha} = M$, then let $M_{\\alpha +1} = M$. Otherwise choose the smallest", "$x \\in M$ (with respect to the fixed well-ordering) such that $x \\notin", "M_{\\alpha}$. Since $S$ is flat over $R$, $(M/M_{\\alpha}) \\otimes_R S = M", "\\otimes_R S/M_{\\alpha} \\otimes_R S$, so since $M_{\\alpha} \\otimes_R S$ is", "a direct sum of some $Q_i$, the same is true of $(M/M_{\\alpha}) \\otimes_R", "S$. By Lemma \\ref{lemma-adapted-submodule}, we can find a countably generated", "$R$-submodule $P$ of $M/M_{\\alpha}$ containing the image of $x$ in", "$M/M_{\\alpha}$ and such that $P \\otimes_R S$ (which equals $\\Im(P", "\\otimes_R S \\to M \\otimes_R S)$ since $S$ is flat over $R$) is a", "direct sum of some $Q_i$. Since $M \\otimes_R S = \\bigoplus_{i \\in I} Q_i$", "is projective and projectivity passes to direct summands, $P \\otimes_R S$ is", "also projective. Thus by Lemma \\ref{lemma-ffdescent-countable-projectivity},", "$P$ is projective. Finally we define $M_{\\alpha + 1}$ to be the preimage of $P$", "in $M$, so that $M_{\\alpha + 1}/M_{\\alpha} = P$ is countably generated and", "projective. In particular $M_{\\alpha}$ is a direct summand of $M_{\\alpha + 1}$", "since projectivity of $M_{\\alpha + 1}/M_{\\alpha}$ implies the sequence $0", "\\to M_{\\alpha} \\to M_{\\alpha + 1} \\to", "M_{\\alpha + 1}/M_{\\alpha} \\to 0$ splits.", "\\medskip\\noindent", "Transfinite induction on $M$ (using the fact that we constructed", "$M_{\\alpha + 1}$ to contain the smallest $x \\in M$ not contained in", "$M_{\\alpha}$) shows that", "each $x \\in M$ is contained in some $M_{\\alpha}$. Thus, there is some large", "enough ordinal $S$ satisfying: for each $x \\in M$ there is $\\alpha \\in S$ such", "that $x \\in M_{\\alpha}$. This means $(M_{\\alpha})_{\\alpha \\in S}$ satisfies", "property (1) of a Kaplansky d\\'evissage of $M$. The other properties are clear", "by construction. We conclude", "$M = \\bigoplus_{\\alpha + 1 \\in S} M_{\\alpha + 1}/M_{\\alpha}$.", "Since each $M_{\\alpha + 1}/M_{\\alpha}$ is projective", "by construction, $M$ is projective." ], "refs": [ "algebra-theorem-projective-direct-sum", "algebra-lemma-adapted-submodule", "algebra-lemma-ffdescent-countable-projectivity" ], "ref_ids": [ 321, 857, 855 ] } ], "ref_ids": [] }, { "id": 325, "type": "theorem", "label": "algebra-theorem-main-theorem", "categories": [ "algebra" ], "title": "algebra-theorem-main-theorem", "contents": [ "Let $R$ be a ring. Let $R \\to S$ be a finite type $R$-algebra.", "Let $S' \\subset S$ be the integral closure of $R$ in $S$.", "Let $\\mathfrak q \\subset S$ be a prime of $S$.", "If $R \\to S$ is quasi-finite at $\\mathfrak q$ then", "there exists a $g \\in S'$, $g \\not \\in \\mathfrak q$", "such that $S'_g \\cong S_g$." ], "refs": [], "proofs": [ { "contents": [ "There exist finitely many elements", "$x_1, \\ldots, x_n \\in S$ such that $S$ is finite", "over the $R$-sub algebra generated by $x_1, \\ldots, x_n$. (For", "example generators of $S$ over $R$.) We prove the proposition", "by induction on the minimal such number $n$.", "\\medskip\\noindent", "The case $n = 0$ is trivial, because in this case $S' = S$,", "see Lemma \\ref{lemma-finite-is-integral}.", "\\medskip\\noindent", "The case $n = 1$. We may replace $R$ by its integral closure in $S$", "(Lemma \\ref{lemma-quasi-finite-permanence} guarantees that $R \\to S$", "is still quasi-finite at $\\mathfrak q$). Thus we may assume", "$R \\subset S$ is integrally closed in $S$, in other words $R = S'$.", "Consider the map $\\varphi : R[x] \\to S$, $x \\mapsto x_1$.", "(We will see that $\\varphi$ is not injective below.)", "By assumption $\\varphi$ is finite. Hence we are in Situation", "\\ref{situation-one-transcendental-element}.", "Let $J \\subset S$ be the ``conductor ideal'' defined", "in Situation \\ref{situation-one-transcendental-element}.", "Consider the diagram", "$$", "\\xymatrix{", "R[x] \\ar[r] & S \\ar[r] & S/\\sqrt{J} & R/(R \\cap \\sqrt{J})[x] \\ar[l]", "\\\\", "& R \\ar[lu] \\ar[r] \\ar[u] & R/(R \\cap \\sqrt{J}) \\ar[u] \\ar[ru] &", "}", "$$", "According to Lemma \\ref{lemma-all-coefficients-in-J}", "the image of $x$ in the quotient $S/\\sqrt{J}$", "is strongly transcendental over $R/ (R \\cap \\sqrt{J})$.", "Hence by Lemma \\ref{lemma-reduced-strongly-transcendental-not-quasi-finite}", "the ring map $R/ (R \\cap \\sqrt{J}) \\to S/\\sqrt{J}$", "is not quasi-finite at any prime of $S/\\sqrt{J}$.", "By Lemma \\ref{lemma-four-rings} we deduce that $\\mathfrak q$", "does not lie in $V(J) \\subset \\Spec(S)$.", "Thus there exists an element $s \\in J$,", "$s \\not\\in \\mathfrak q$. By definition of $J$ we may write", "$s = \\varphi(f)$ for some polynomial $f \\in R[x]$.", "Let $I = \\Ker(\\varphi : R[x] \\to S)$. Since $\\varphi(f) \\in J$", "we get $(R[x]/I)_f \\cong S_{\\varphi(f)}$. Also $s \\not \\in \\mathfrak q$", "means that $f \\not \\in \\varphi^{-1}(\\mathfrak q)$. Thus", "$\\varphi^{-1}(\\mathfrak q)$ is a prime of $R[x]/I$", "at which $R \\to R[x]/I$ is quasi-finite, see", "Lemma \\ref{lemma-quasi-finite-local}. Note that $R$ is integrally closed", "in $R[x]/I$ since $R$ is integrally closed in $S$. By", "Lemma \\ref{lemma-quasi-finite-monogenic}", "there exists an element $h \\in R$, $h \\not \\in R \\cap \\mathfrak q$", "such that $R_h \\cong (R[x]/I)_h$. Thus", "$(R[x]/I)_{fh} = S_{\\varphi(fh)}$ is isomorphic to a principal", "localization $R_{h'}$ of $R$ for some", "$h' \\in R$, $h' \\not \\in \\mathfrak q$.", "\\medskip\\noindent", "The case $n > 1$. Consider the subring $R' \\subset S$", "which is the integral closure of $R[x_1, \\ldots, x_{n-1}]$", "in $S$. By Lemma \\ref{lemma-quasi-finite-permanence} the extension", "$S/R'$ is quasi-finite at $\\mathfrak q$. Also, note", "that $S$ is finite over $R'[x_n]$.", "By the case $n = 1$ above, there exists a $g' \\in R'$,", "$g' \\not \\in \\mathfrak q$ such that", "$(R')_{g'} \\cong S_{g'}$. At this point we cannot", "apply induction to $R \\to R'$ since $R'$ may not be finite type over $R$.", "Since $S$ is finitely generated over $R$ we deduce in particular", "that $(R')_{g'}$ is finitely generated over $R$. Say", "the elements $g'$, and $y_1/(g')^{n_1}, \\ldots, y_N/(g')^{n_N}$", "with $y_i \\in R'$", "generate $(R')_{g'}$ over $R$. Let $R''$ be the $R$-sub algebra", "of $R'$ generated by $x_1, \\ldots, x_{n-1}, y_1, \\ldots, y_N, g'$.", "This has the property $(R'')_{g'} \\cong S_{g'}$. Surjectivity", "because of how we chose $y_i$, injectivity because", "$R'' \\subset R'$, and localization is exact. Note that", "$R''$ is finite over $R[x_1, \\ldots, x_{n-1}]$ because", "of our choice of $R'$, see Lemma \\ref{lemma-characterize-integral}.", "Let $\\mathfrak q'' = R'' \\cap \\mathfrak q$. Since", "$(R'')_{\\mathfrak q''} = S_{\\mathfrak q}$ we see that", "$R \\to R''$ is quasi-finite at $\\mathfrak q''$, see", "Lemma \\ref{lemma-isolated-point-fibre}.", "We apply our induction hypothesis to $R \\to R''$, $\\mathfrak q''$", "and $x_1, \\ldots, x_{n-1} \\in R''$ and we find a subring", "$R''' \\subset R''$ which is integral over $R$ and an", "element $g'' \\in R'''$, $g'' \\not \\in \\mathfrak q''$", "such that $(R''')_{g''} \\cong (R'')_{g''}$. Write the image of $g'$ in", "$(R'')_{g''}$ as $g'''/(g'')^n$ for some $g''' \\in R'''$.", "Set $g = g''g''' \\in R'''$. Then it is clear that $g \\not\\in", "\\mathfrak q$ and $(R''')_g \\cong S_g$. Since by construction", "we have $R''' \\subset S'$ we also have $S'_g \\cong S_g$ as desired." ], "refs": [ "algebra-lemma-finite-is-integral", "algebra-lemma-quasi-finite-permanence", "algebra-lemma-all-coefficients-in-J", "algebra-lemma-reduced-strongly-transcendental-not-quasi-finite", "algebra-lemma-four-rings", "algebra-lemma-quasi-finite-local", "algebra-lemma-quasi-finite-monogenic", "algebra-lemma-quasi-finite-permanence", "algebra-lemma-characterize-integral", "algebra-lemma-isolated-point-fibre" ], "ref_ids": [ 482, 1055, 1061, 1064, 1052, 1051, 1065, 1055, 483, 1049 ] } ], "ref_ids": [] }, { "id": 326, "type": "theorem", "label": "algebra-theorem-openness-flatness", "categories": [ "algebra" ], "title": "algebra-theorem-openness-flatness", "contents": [ "Let $R$ be a ring. Let $R \\to S$ be a ring map of finite", "presentation. Let $M$ be a finitely presented $S$-module.", "The set", "$$", "\\{ \\mathfrak q \\in \\Spec(S) \\mid", "M_{\\mathfrak q}\\text{ is flat over }R\\}", "$$", "is open in $\\Spec(S)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q \\in \\Spec(S)$ be a prime.", "Let $\\mathfrak p \\subset R$ be the inverse image of $\\mathfrak q$ in $R$.", "Note that $M_{\\mathfrak q}$ is flat over $R$ if and only if", "it is flat over $R_{\\mathfrak p}$.", "Let us assume that $M_{\\mathfrak q}$ is flat over $R$.", "We claim that there exists a $g \\in S$, $g \\not \\in \\mathfrak q$", "such that $M_g$ is flat over $R$.", "\\medskip\\noindent", "We first reduce to the case where $R$ and $S$ are", "of finite type over $\\mathbf{Z}$.", "Choose a directed set $\\Lambda$ and", "a system $(R_\\lambda \\to S_\\lambda, M_\\lambda)$", "as in Lemma \\ref{lemma-limit-module-finite-presentation}.", "Set $\\mathfrak p_\\lambda$ equal to the inverse image of", "$\\mathfrak p$ in $R_\\lambda$.", "Set $\\mathfrak q_\\lambda$ equal to the inverse image of", "$\\mathfrak q$ in $S_\\lambda$.", "Then the system", "$$", "((R_\\lambda)_{\\mathfrak p_\\lambda},", "(S_\\lambda)_{\\mathfrak q_\\lambda},", "(M_\\lambda)_{\\mathfrak q_{\\lambda}})", "$$", "is a system as in", "Lemma \\ref{lemma-limit-module-essentially-finite-presentation}.", "Hence by Lemma \\ref{lemma-colimit-eventually-flat}", "we see that for some $\\lambda$ the module", "$M_\\lambda$ is flat over $R_\\lambda$ at the prime", "$\\mathfrak q_{\\lambda}$. Suppose we", "can prove our claim for the system", "$(R_\\lambda \\to S_\\lambda, M_\\lambda, \\mathfrak q_{\\lambda})$.", "In other words, suppose that we can find a $g \\in S_\\lambda$,", "$g \\not\\in \\mathfrak q_\\lambda$ such that $(M_\\lambda)_g$", "is flat over $R_\\lambda$. By Lemma \\ref{lemma-limit-module-finite-presentation}", "we have $M = M_\\lambda \\otimes_{R_\\lambda} R$ and hence", "also $M_g = (M_\\lambda)_g \\otimes_{R_\\lambda} R$. Thus by", "Lemma \\ref{lemma-flat-base-change} we deduce the claim", "for the system $(R \\to S, M, \\mathfrak q)$.", "\\medskip\\noindent", "At this point we may assume that $R$ and $S$ are of finite type", "over $\\mathbf{Z}$. We may write $S$ as a quotient of a", "polynomial ring $R[x_1, \\ldots, x_n]$. Of course, we may replace", "$S$ by $R[x_1, \\ldots, x_n]$ and assume that $S$ is a polynomial", "ring over $R$. In particular we see that $R \\to S$ is flat", "and all fibres rings $S \\otimes_R \\kappa(\\mathfrak p)$", "have global dimension $n$.", "\\medskip\\noindent", "Choose a resolution $F_\\bullet$ of $M$ over $S$ with each", "$F_i$ finite free, see Lemma \\ref{lemma-resolution-by-finite-free}.", "Let $K_n = \\Ker(F_{n-1} \\to F_{n-2})$. Note that", "$(K_n)_{\\mathfrak q}$ is flat over $R$, since each $F_i$", "is flat over $R$ and by assumption on $M$, see Lemma", "\\ref{lemma-flat-ses}. In addition, the sequence", "$$", "0 \\to", "K_n/\\mathfrak p K_n \\to", "F_{n-1}/ \\mathfrak p F_{n-1} \\to", "\\ldots \\to", "F_0 / \\mathfrak p F_0 \\to", "M/\\mathfrak p M \\to", "0", "$$", "is exact upon localizing at $\\mathfrak q$, because of vanishing", "of $\\text{Tor}_i^{R_\\mathfrak p}(\\kappa(\\mathfrak p), M_{\\mathfrak q})$.", "Since the global dimension of $S_\\mathfrak q/\\mathfrak p S_{\\mathfrak q}$", "is $n$ we conclude that $K_n / \\mathfrak p K_n$ localized", "at $\\mathfrak q$ is a finite free module over", "$S_\\mathfrak q/\\mathfrak p S_{\\mathfrak q}$. By", "Lemma \\ref{lemma-free-fibre-flat-free} $(K_n)_{\\mathfrak q}$", "is free over $S_{\\mathfrak q}$. In particular, there exists a", "$g \\in S$, $g \\not \\in \\mathfrak q$ such that $(K_n)_g$", "is finite free over $S_g$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-exact-on-fibres-open}", "there exists a further localization $S_g$ such that", "the complex", "$$", "0 \\to K_n \\to F_{n-1} \\to \\ldots \\to F_0", "$$", "is exact on {\\it all fibres} of $R \\to S$. By", "Lemma \\ref{lemma-complex-exact-mod}", "this implies that the cokernel of $F_1 \\to F_0$ is", "flat. This proves the theorem in the Noetherian case." ], "refs": [ "algebra-lemma-limit-module-finite-presentation", "algebra-lemma-limit-module-essentially-finite-presentation", "algebra-lemma-colimit-eventually-flat", "algebra-lemma-limit-module-finite-presentation", "algebra-lemma-flat-base-change", "algebra-lemma-resolution-by-finite-free", "algebra-lemma-flat-ses", "algebra-lemma-free-fibre-flat-free", "algebra-lemma-exact-on-fibres-open", "algebra-lemma-complex-exact-mod" ], "ref_ids": [ 1106, 1101, 1109, 1106, 527, 761, 533, 886, 1119, 887 ] } ], "ref_ids": [] }, { "id": 327, "type": "theorem", "label": "algebra-theorem-cohen-structure-theorem", "categories": [ "algebra" ], "title": "algebra-theorem-cohen-structure-theorem", "contents": [ "Let $(R, \\mathfrak m)$ be a complete local ring.", "\\begin{enumerate}", "\\item $R$ has a coefficient ring (see", "Definition \\ref{definition-coefficient-ring}),", "\\item if $\\mathfrak m$ is a finitely generated ideal, then", "$R$ is isomorphic to a quotient", "$$", "\\Lambda[[x_1, \\ldots, x_n]]/I", "$$", "where $\\Lambda$ is either a field or a Cohen ring.", "\\end{enumerate}" ], "refs": [ "algebra-definition-coefficient-ring" ], "proofs": [ { "contents": [ "Let us prove a coefficient ring exists.", "First we prove this in case the characteristic of the residue field $\\kappa$", "is zero. Namely, in this case we will prove by induction", "on $n > 0$ that there exists a section", "$$", "\\varphi_n : \\kappa \\longrightarrow R/\\mathfrak m^n", "$$", "to the canonical map $R/\\mathfrak m^n \\to \\kappa = R/\\mathfrak m$.", "This is trivial for $n = 1$. If $n > 1$, let $\\varphi_{n - 1}$ be given.", "The field extension $\\mathbf{Q} \\subset \\kappa$ is formally smooth by", "Proposition \\ref{proposition-characterize-separable-field-extensions}.", "Hence we can find the dotted arrow", "in the following diagram", "$$", "\\xymatrix{", "R/\\mathfrak m^{n - 1} &", "R/\\mathfrak m^n \\ar[l] \\\\", "\\kappa \\ar[u]^{\\varphi_{n - 1}} \\ar@{..>}[ru] & \\mathbf{Q} \\ar[l] \\ar[u]", "}", "$$", "This proves the induction step. Putting these maps together", "$$", "\\lim_n\\ \\varphi_n : \\kappa \\longrightarrow", "R = \\lim_n\\ R/\\mathfrak m^n", "$$", "gives a map whose image is the desired coefficient ring.", "\\medskip\\noindent", "Next, we prove the existence of a coefficient ring in the case", "where the characteristic of the residue field $\\kappa$ is $p > 0$.", "Namely, choose a Cohen ring $\\Lambda$ with $\\kappa = \\Lambda/p\\Lambda$,", "see Lemma \\ref{lemma-cohen-rings-exist}. In this case we will prove by", "induction on $n > 0$ that there exists a map", "$$", "\\varphi_n :", "\\Lambda/p^n\\Lambda", "\\longrightarrow", "R/\\mathfrak m^n", "$$", "whose composition with the reduction map $R/\\mathfrak m^n \\to \\kappa$", "produces the given isomorphism $\\Lambda/p\\Lambda = \\kappa$. This is trivial", "for $n = 1$. If $n > 1$, let $\\varphi_{n - 1}$ be given.", "The ring map $\\mathbf{Z}/p^n\\mathbf{Z} \\to \\Lambda/p^n\\Lambda$", "is formally smooth by Lemma \\ref{lemma-cohen-ring-formally-smooth}.", "Hence we can find the dotted arrow", "in the following diagram", "$$", "\\xymatrix{", "R/\\mathfrak m^{n - 1} &", "R/\\mathfrak m^n \\ar[l] \\\\", "\\Lambda/p^n\\Lambda \\ar[u]^{\\varphi_{n - 1}} \\ar@{..>}[ru] &", "\\mathbf{Z}/p^n\\mathbf{Z} \\ar[l] \\ar[u]", "}", "$$", "This proves the induction step. Putting these maps together", "$$", "\\lim_n\\ \\varphi_n :", "\\Lambda = \\lim_n\\ \\Lambda/p^n\\Lambda", "\\longrightarrow", "R = \\lim_n\\ R/\\mathfrak m^n", "$$", "gives a map whose image is the desired coefficient ring.", "\\medskip\\noindent", "The final statement of the theorem follows readily. Namely, if", "$y_1, \\ldots, y_n$ are generators of the ideal $\\mathfrak m$,", "then we can use the map $\\Lambda \\to R$ just constructed", "to get a map", "$$", "\\Lambda[[x_1, \\ldots, x_n]] \\longrightarrow R,", "\\quad x_i \\longmapsto y_i.", "$$", "Since both sides are $(x_1, \\ldots, x_n)$-adically complete", "this map is surjective by Lemma \\ref{lemma-completion-generalities}", "as it is surjective modulo $(x_1, \\ldots, x_n)$ by", "construction." ], "refs": [ "algebra-proposition-characterize-separable-field-extensions", "algebra-lemma-cohen-rings-exist", "algebra-lemma-cohen-ring-formally-smooth", "algebra-lemma-completion-generalities" ], "ref_ids": [ 1429, 1329, 1330, 858 ] } ], "ref_ids": [ 1549 ] }, { "id": 328, "type": "theorem", "label": "algebra-lemma-snake", "categories": [ "algebra" ], "title": "algebra-lemma-snake", "contents": [ "\\begin{reference}", "\\cite[III, Lemma 3.3]{Cartan-Eilenberg}", "\\end{reference}", "Suppose given a commutative diagram", "$$", "\\xymatrix{", "& X \\ar[r] \\ar[d]^\\alpha &", "Y \\ar[r] \\ar[d]^\\beta &", "Z \\ar[r] \\ar[d]^\\gamma &", "0 \\\\", "0 \\ar[r] & U \\ar[r] & V \\ar[r] & W", "}", "$$", "of abelian groups with exact rows, then there is a canonical exact sequence", "$$", "\\Ker(\\alpha) \\to \\Ker(\\beta) \\to \\Ker(\\gamma)", "\\to", "\\Coker(\\alpha) \\to \\Coker(\\beta) \\to \\Coker(\\gamma)", "$$", "Moreover, if $X \\to Y$ is injective, then the first map is", "injective, and if $V \\to W$ is surjective, then the last", "map is surjective." ], "refs": [], "proofs": [ { "contents": [ "The map $\\partial : \\Ker(\\gamma) \\to \\Coker(\\alpha)$ is defined", "as follows. Take $z \\in \\Ker(\\gamma)$. Choose $y \\in Y$ mapping to $z$.", "Then $\\beta(y) \\in V$ maps to zero in $W$. Hence $\\beta(y)$ is the image of", "some $u \\in U$. Set $\\partial z = \\overline{u}$ the class of $u$ in the", "cokernel of $\\alpha$. Proof of exactness is omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 329, "type": "theorem", "label": "algebra-lemma-lift-map", "categories": [ "algebra" ], "title": "algebra-lemma-lift-map", "contents": [ "Let $R$ be a ring. Let $\\alpha : R^{\\oplus n} \\to M$ and $\\beta : N \\to M$ be", "module maps. If $\\Im(\\alpha) \\subset \\Im(\\beta)$, then there", "exists an $R$-module map $\\gamma : R^{\\oplus n} \\to N$ such that", "$\\alpha = \\beta \\circ \\gamma$." ], "refs": [], "proofs": [ { "contents": [ "Let $e_i = (0, \\ldots, 0, 1, 0, \\ldots, 0)$ be the $i$th basis vector", "of $R^{\\oplus n}$. Let $x_i \\in N$ be an element with", "$\\alpha(e_i) = \\beta(x_i)$ which exists by assumption. Set", "$\\gamma(a_1, \\ldots, a_n) = \\sum a_i x_i$. By construction", "$\\alpha = \\beta \\circ \\gamma$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 330, "type": "theorem", "label": "algebra-lemma-extension", "categories": [ "algebra" ], "title": "algebra-lemma-extension", "contents": [ "Let $R$ be a ring.", "Let", "$$", "0 \\to M_1 \\to M_2 \\to M_3 \\to 0", "$$", "be a short exact sequence of $R$-modules.", "\\begin{enumerate}", "\\item If $M_1$ and $M_3$ are finite $R$-modules, then $M_2$ is a finite", "$R$-module.", "\\item If $M_1$ and $M_3$ are finitely presented $R$-modules, then $M_2$", "is a finitely presented $R$-module.", "\\item If $M_2$ is a finite $R$-module, then $M_3$ is a finite $R$-module.", "\\item If $M_2$ is a finitely presented $R$-module and $M_1$ is a", "finite $R$-module, then $M_3$ is a finitely presented $R$-module.", "\\item If $M_3$ is a finitely presented $R$-module and $M_2$ is a finite", "$R$-module, then $M_1$ is a finite $R$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). If $x_1, \\ldots, x_n$ are generators of $M_1$ and", "$y_1, \\ldots, y_m \\in M_2$ are elements whose images in $M_3$ are", "generators of $M_3$, then $x_1, \\ldots, x_n, y_1, \\ldots, y_m$", "generate $M_2$.", "\\medskip\\noindent", "Part (3) is immediate from the definition.", "\\medskip\\noindent", "Proof of (5). Assume $M_3$ is finitely presented and $M_2$ finite.", "Choose a presentation", "$$", "R^{\\oplus m} \\to R^{\\oplus n} \\to M_3 \\to 0", "$$", "By Lemma \\ref{lemma-lift-map} there exists a map", "$R^{\\oplus n} \\to M_2$ such that", "the solid diagram", "$$", "\\xymatrix{", "& R^{\\oplus m} \\ar[r] \\ar@{..>}[d] & R^{\\oplus n} \\ar[r] \\ar[d] &", "M_3 \\ar[r] \\ar[d]^{\\text{id}} & 0 \\\\", "0 \\ar[r] & M_1 \\ar[r] & M_2 \\ar[r] & M_3 \\ar[r] & 0", "}", "$$", "commutes. This produces the dotted arrow. By the snake lemma", "(Lemma \\ref{lemma-snake}) we see that we get an isomorphism", "$$", "\\Coker(R^{\\oplus m} \\to M_1)", "\\cong", "\\Coker(R^{\\oplus n} \\to M_2)", "$$", "In particular we conclude that $\\Coker(R^{\\oplus m} \\to M_1)$", "is a finite $R$-module. Since $\\Im(R^{\\oplus m} \\to M_1)$", "is finite by (3), we see that $M_1$ is finite by part (1).", "\\medskip\\noindent", "Proof of (4). Assume $M_2$ is finitely presented and $M_1$ is finite.", "Choose a presentation $R^{\\oplus m} \\to R^{\\oplus n} \\to M_2 \\to 0$.", "Choose a surjection $R^{\\oplus k} \\to M_1$. By Lemma \\ref{lemma-lift-map}", "there exists a factorization $R^{\\oplus k} \\to R^{\\oplus n} \\to M_2$", "of the composition $R^{\\oplus k} \\to M_1 \\to M_2$. Then", "$R^{\\oplus k + m} \\to R^{\\oplus n} \\to M_3 \\to 0$", "is a presentation.", "\\medskip\\noindent", "Proof of (2). Assume that $M_1$ and $M_3$ are finitely presented.", "The argument in the proof of part (1) produces a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] & R^{\\oplus n} \\ar[d] \\ar[r] & R^{\\oplus n + m} \\ar[d] \\ar[r] &", "R^{\\oplus m} \\ar[d] \\ar[r] & 0 \\\\", "0 \\ar[r] & M_1 \\ar[r] & M_2 \\ar[r] & M_3 \\ar[r] & 0", "}", "$$", "with surjective vertical arrows. By the snake lemma we obtain a short", "exact sequence", "$$", "0 \\to \\Ker(R^{\\oplus n} \\to M_1) \\to", "\\Ker(R^{\\oplus n + m} \\to M_2) \\to", "\\Ker(R^{\\oplus m} \\to M_3) \\to 0", "$$", "By part (5) we see that the outer two modules are finite. Hence the", "middle one is finite too. By (4) we see that $M_2$ is of finite presentation." ], "refs": [ "algebra-lemma-lift-map", "algebra-lemma-snake", "algebra-lemma-lift-map" ], "ref_ids": [ 329, 328, 329 ] } ], "ref_ids": [] }, { "id": 331, "type": "theorem", "label": "algebra-lemma-trivial-filter-finite-module", "categories": [ "algebra" ], "title": "algebra-lemma-trivial-filter-finite-module", "contents": [ "\\begin{slogan}", "Finite modules have filtrations such that successive quotients are", "cyclic modules.", "\\end{slogan}", "Let $R$ be a ring, and let $M$ be a finite $R$-module.", "There exists a filtration by $R$-submodules", "$$", "0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M", "$$", "such that each quotient $M_i/M_{i-1}$ is isomorphic", "to $R/I_i$ for some ideal $I_i$ of $R$." ], "refs": [], "proofs": [ { "contents": [ "By induction on the number of generators of $M$. Let", "$x_1, \\ldots, x_r \\in M$ be a minimal number of generators.", "Let $M' = Rx_1 \\subset M$. Then $M/M'$ has $r - 1$ generators", "and the induction hypothesis applies. And clearly $M' \\cong R/I_1$", "with $I_1 = \\{f \\in R \\mid fx_1 = 0\\}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 332, "type": "theorem", "label": "algebra-lemma-finite-over-subring", "categories": [ "algebra" ], "title": "algebra-lemma-finite-over-subring", "contents": [ "Let $R \\to S$ be a ring map.", "Let $M$ be an $S$-module.", "If $M$ is finite as an $R$-module, then $M$ is finite as an $S$-module." ], "refs": [], "proofs": [ { "contents": [ "In fact, any $R$-generating set of $M$ is also an $S$-generating set of", "$M$, since the $R$-module structure is induced by the image of $R$ in $S$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 333, "type": "theorem", "label": "algebra-lemma-compose-finite-type", "categories": [ "algebra" ], "title": "algebra-lemma-compose-finite-type", "contents": [ "The notions finite type and finite presentation have the following", "permanence properties.", "\\begin{enumerate}", "\\item A composition of ring maps of finite type is of finite type.", "\\item A composition of ring maps of finite presentation is of finite", "presentation.", "\\item Given $R \\to S' \\to S$ with $R \\to S$ of finite type,", "then $S' \\to S$ is of finite type.", "\\item Given $R \\to S' \\to S$, with $R \\to S$ of finite presentation,", "and $R \\to S'$ of finite type, then $S' \\to S$ is of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We only prove the last assertion.", "Write $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$", "and $S' = R[y_1, \\ldots, y_a]/I$. Say that the class", "$\\bar y_i$ of $y_i$ maps", "to $h_i \\bmod (f_1, \\ldots, f_m)$ in $S$.", "Then it is clear that", "$S = S'[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m,", "h_1 - \\bar y_1, \\ldots, h_a - \\bar y_a)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 334, "type": "theorem", "label": "algebra-lemma-finite-presentation-independent", "categories": [ "algebra" ], "title": "algebra-lemma-finite-presentation-independent", "contents": [ "Let $R \\to S$ be a ring map of finite presentation.", "For any surjection $\\alpha : R[x_1, \\ldots, x_n] \\to S$ the", "kernel of $\\alpha$ is a finitely generated ideal in $R[x_1, \\ldots, x_n]$." ], "refs": [], "proofs": [ { "contents": [ "Write $S = R[y_1, \\ldots, y_m]/(f_1, \\ldots, f_k)$.", "Choose $g_i \\in R[y_1, \\ldots, y_m]$ which are lifts", "of $\\alpha(x_i)$. Then we see that $S = R[x_i, y_j]/(f_l, x_i - g_i)$.", "Choose $h_j \\in R[x_1, \\ldots, x_n]$ such that $\\alpha(h_j)$", "corresponds to $y_j \\bmod (f_1, \\ldots, f_k)$. Consider", "the map $\\psi : R[x_i, y_j] \\to R[x_i]$, $x_i \\mapsto x_i$,", "$y_j \\mapsto h_j$. Then the kernel of $\\alpha$", "is the image of $(f_l, x_i - g_i)$ under $\\psi$ and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 335, "type": "theorem", "label": "algebra-lemma-finitely-presented-over-subring", "categories": [ "algebra" ], "title": "algebra-lemma-finitely-presented-over-subring", "contents": [ "Let $R \\to S$ be a ring map.", "Let $M$ be an $S$-module.", "Assume $R \\to S$ is of finite type and", "$M$ is finitely presented as an $R$-module.", "Then $M$ is finitely presented as an $S$-module." ], "refs": [], "proofs": [ { "contents": [ "This is similar to the proof of part (4) of", "Lemma \\ref{lemma-compose-finite-type}.", "We may assume $S = R[x_1, \\ldots, x_n]/J$.", "Choose $y_1, \\ldots, y_m \\in M$ which generate $M$ as an $R$-module", "and choose relations $\\sum a_{ij} y_j = 0$, $i = 1, \\ldots, t$ which", "generate the kernel of $R^{\\oplus m} \\to M$. For any", "$i = 1, \\ldots, n$ and $j = 1, \\ldots, m$ write", "$$", "x_i y_j = \\sum a_{ijk} y_k", "$$", "for some $a_{ijk} \\in R$. Consider the $S$-module $N$ generated by", "$y_1, \\ldots, y_m$ subject to the relations", "$\\sum a_{ij} y_j = 0$, $i = 1, \\ldots, t$ and", "$x_i y_j = \\sum a_{ijk} y_k$, $i = 1, \\ldots, n$ and $j = 1, \\ldots, m$.", "Then $N$ has a presentation", "$$", "S^{\\oplus nm + t} \\longrightarrow S^{\\oplus m} \\longrightarrow N", "\\longrightarrow 0", "$$", "By construction there is a surjective map $\\varphi : N \\to M$.", "To finish the proof we show $\\varphi$ is injective.", "Suppose $z = \\sum b_j y_j \\in N$ for some $b_j \\in S$.", "We may think of $b_j$ as a polynomial in $x_1, \\ldots, x_n$", "with coefficients in $R$.", "By applying the relations of the form $x_i y_j = \\sum a_{ijk} y_k$", "we can inductively lower the degree of the polynomials.", "Hence we see that $z = \\sum c_j y_j$ for some $c_j \\in R$.", "Hence if $\\varphi(z) = 0$ then the vector $(c_1, \\ldots, c_m)$", "is an $R$-linear combination of the vectors $(a_{i1}, \\ldots, a_{im})$", "and we conclude that $z = 0$ as desired." ], "refs": [ "algebra-lemma-compose-finite-type" ], "ref_ids": [ 333 ] } ], "ref_ids": [] }, { "id": 336, "type": "theorem", "label": "algebra-lemma-finite-module-over-finite-extension", "categories": [ "algebra" ], "title": "algebra-lemma-finite-module-over-finite-extension", "contents": [ "Let $R \\to S$ be a finite ring map.", "Let $M$ be an $S$-module.", "Then $M$ is finite as an $R$-module if and only if $M$ is finite", "as an $S$-module." ], "refs": [], "proofs": [ { "contents": [ "One of the implications follows from", "Lemma \\ref{lemma-finite-over-subring}.", "To see the other assume that $M$ is finite as an $S$-module.", "Pick $x_1, \\ldots, x_n \\in S$ which generate $S$ as an $R$-module.", "Pick $y_1, \\ldots, y_m \\in M$ which generate $M$ as an $S$-module.", "Then $x_i y_j$ generate $M$ as an $R$-module." ], "refs": [ "algebra-lemma-finite-over-subring" ], "ref_ids": [ 332 ] } ], "ref_ids": [] }, { "id": 337, "type": "theorem", "label": "algebra-lemma-finite-transitive", "categories": [ "algebra" ], "title": "algebra-lemma-finite-transitive", "contents": [ "Suppose that $R \\to S$ and $S \\to T$ are finite ring maps.", "Then $R \\to T$ is finite." ], "refs": [], "proofs": [ { "contents": [ "If $t_i$ generate $T$ as an $S$-module and $s_j$ generate $S$ as an", "$R$-module, then $t_i s_j$ generate $T$ as an $R$-module.", "(Also follows from", "Lemma \\ref{lemma-finite-module-over-finite-extension}.)" ], "refs": [ "algebra-lemma-finite-module-over-finite-extension" ], "ref_ids": [ 336 ] } ], "ref_ids": [] }, { "id": 338, "type": "theorem", "label": "algebra-lemma-finite-finite-type", "categories": [ "algebra" ], "title": "algebra-lemma-finite-finite-type", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "\\begin{enumerate}", "\\item If $\\varphi$ is finite, then $\\varphi$ is of finite type.", "\\item If $S$ is of finite presentation as an $R$-module, then", "$\\varphi$ is of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For (1) if $x_1, \\ldots, x_n \\in S$ generate $S$ as an $R$-module,", "then $x_1, \\ldots, x_n$ generate $S$ as an $R$-algebra. For (2),", "suppose that $\\sum r_j^ix_i = 0$, $j = 1, \\ldots, m$ is a set", "of generators of the relations among the $x_i$ when viewed as", "$R$-module generators of $S$. Furthermore, write", "$1 = \\sum r_ix_i$ for some $r_i \\in R$ and", "$x_ix_j = \\sum r_{ij}^k x_k$ for some $r_{ij}^k \\in R$.", "Then", "$$", "S = R[t_1, \\ldots, t_n]/", "(\\sum r_j^it_i,\\ 1 - \\sum r_it_i,\\ t_it_j - \\sum r_{ij}^k t_k)", "$$", "as an $R$-algebra which proves (2)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 339, "type": "theorem", "label": "algebra-lemma-colimit", "categories": [ "algebra" ], "title": "algebra-lemma-colimit", "contents": [ "Let $(M_i, \\mu_{ij})$ be a system of $R$-modules over the preordered set $I$.", "The colimit of the system $(M_i, \\mu_{ij})$ is the quotient $R$-module", "$(\\bigoplus_{i\\in I} M_i) /Q$ where $Q$ is the", "$R$-submodule generated by all elements", "$$", "\\iota_i(x_i) - \\iota_j(\\mu_{ij}(x_i))", "$$", "where $\\iota_i : M_i \\to \\bigoplus_{i\\in I} M_i$", "is the natural inclusion. We denote the colimit", "$M = \\colim_i M_i$. We denote", "$\\pi : \\bigoplus_{i\\in I} M_i \\to M$ the", "projection map and", "$\\phi_i = \\pi \\circ \\iota_i : M_i \\to M$." ], "refs": [], "proofs": [ { "contents": [ "This lemma is a special case of", "Categories, Lemma \\ref{categories-lemma-colimits-coproducts-coequalizers}", "but we will also prove it directly in this case.", "Namely, note that $\\phi_i = \\phi_j\\circ \\mu_{ij}$ in the above", "construction. To show the pair $(M, \\phi_i)$ is the colimit we have", "to show it satisfies the universal property: for any other such pair", "$(Y, \\psi_i)$ with $\\psi_i : M_i \\to", "Y$, $\\psi_i = \\psi_j\\circ \\mu_{ij}$, there is a unique $R$-module", "homomorphism $g : M \\to Y$ such that the", "following diagram commutes:", "$$", "\\xymatrix{", "M_i \\ar[rr]^{\\mu_{ij}} \\ar[dr]^{\\phi_i} \\ar[ddr]_{\\psi_i} & &", "M_j\\ar[dl]_{\\phi_j} \\ar[ddl]^{\\psi_j} \\\\", "& M \\ar[d]^{g}\\\\", "& Y", "}", "$$", "And this is clear because we can define $g$ by taking the", "map $\\psi_i$ on the summand $M_i$ in the direct sum", "$\\bigoplus M_i$." ], "refs": [ "categories-lemma-colimits-coproducts-coequalizers" ], "ref_ids": [ 12214 ] } ], "ref_ids": [] }, { "id": 340, "type": "theorem", "label": "algebra-lemma-directed-colimit", "categories": [ "algebra" ], "title": "algebra-lemma-directed-colimit", "contents": [ "Let $(M_i, \\mu_{ij})$ be a system of $R$-modules over the", "preordered set $I$. Assume that $I$ is directed.", "The colimit of the system $(M_i, \\mu_{ij})$ is canonically", "isomorphic to the module $M$ defined as follows:", "\\begin{enumerate}", "\\item as a set let", "$$", "M = \\left(\\coprod\\nolimits_{i \\in I} M_i\\right)/\\sim", "$$", "where for $m \\in M_i$ and $m' \\in M_{i'}$ we have", "$$", "m \\sim m' \\Leftrightarrow", "\\mu_{ij}(m) = \\mu_{i'j}(m')\\text{ for some }j \\geq i, i'", "$$", "\\item as an abelian group for $m \\in M_i$ and $m' \\in M_{i'}$", "we define the sum of the classes of $m$ and $m'$ in $M$", "to be the class of $\\mu_{ij}(m) + \\mu_{i'j}(m')$ where", "$j \\in I$ is any index with $i \\leq j$ and $i' \\leq j$, and", "\\item as an $R$-module define for $m \\in M_i$ and $x \\in R$", "the product of $x$ and the class of $m$ in $M$ to be the", "class of $xm$ in $M$.", "\\end{enumerate}", "The canonical maps $\\phi_i : M_i \\to M$ are induced by the canonical", "maps $M_i \\to \\coprod_{i \\in I} M_i$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Compare with", "Categories, Section \\ref{categories-section-directed-colimits}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 341, "type": "theorem", "label": "algebra-lemma-zero-directed-limit", "categories": [ "algebra" ], "title": "algebra-lemma-zero-directed-limit", "contents": [ "Let $(M_i, \\mu_{ij})$ be a directed system.", "Let $M = \\colim M_i$ with $\\mu_i : M_i \\to M$.", "Then, $\\mu_i(x_i) = 0$ for $x_i \\in M_i$ if and only if", "there exists $j \\geq i$ such that $\\mu_{ij}(x_i) = 0$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the description of the directed colimit", "in Lemma \\ref{lemma-directed-colimit}." ], "refs": [ "algebra-lemma-directed-colimit" ], "ref_ids": [ 340 ] } ], "ref_ids": [] }, { "id": 342, "type": "theorem", "label": "algebra-lemma-homomorphism-limit", "categories": [ "algebra" ], "title": "algebra-lemma-homomorphism-limit", "contents": [ "Let $(M_i, \\mu_{ij})$, $(N_i, \\nu_{ij})$ be", "systems of $R$-modules over the same preordered set.", "A morphism of systems $\\Phi = (\\phi_i)$ from $(M_i, \\mu_{ij})$ to", "$(N_i, \\nu_{ij})$ induces a unique homomorphism", "$$", "\\colim \\phi_i : \\colim M_i \\longrightarrow \\colim N_i", "$$", "such that", "$$", "\\xymatrix{", "M_i \\ar[r] \\ar[d]_{\\phi_i} & \\colim M_i \\ar[d]^{\\colim \\phi_i} \\\\", "N_i \\ar[r] & \\colim N_i", "}", "$$", "commutes for all $i \\in I$." ], "refs": [], "proofs": [ { "contents": [ "Write $M = \\colim M_i$ and $N = \\colim N_i$ and $\\phi = \\colim \\phi_i$", "(as yet to be constructed). We will use the explicit description of $M$", "and $N$ in Lemma \\ref{lemma-colimit} without further mention.", "The condition of the lemma is equivalent to the condition that", "$$", "\\xymatrix{", "\\bigoplus_{i\\in I} M_i \\ar[r] \\ar[d]_{\\bigoplus\\phi_i} & M \\ar[d]^\\phi \\\\", "\\bigoplus_{i\\in I} N_i \\ar[r] & N", "}", "$$", "commutes. Hence it is clear that if $\\phi$ exists, then it is unique.", "To see that $\\phi$ exists, it suffices to show that the kernel of the", "upper horizontal arrow is mapped by $\\bigoplus \\phi_i$ to the kernel", "of the lower horizontal arrow. To see this, let $j \\leq k$ and", "$x_j \\in M_j$. Then", "$$", "(\\bigoplus \\phi_i)(x_j - \\mu_{jk}(x_j))", "=", "\\phi_j(x_j) - \\phi_k(\\mu_{jk}(x_j))", "=", "\\phi_j(x_j) - \\nu_{jk}(\\phi_j(x_j))", "$$", "which is in the kernel of the lower horizontal arrow as required." ], "refs": [ "algebra-lemma-colimit" ], "ref_ids": [ 339 ] } ], "ref_ids": [] }, { "id": 343, "type": "theorem", "label": "algebra-lemma-directed-colimit-exact", "categories": [ "algebra" ], "title": "algebra-lemma-directed-colimit-exact", "contents": [ "\\begin{slogan}", "Filtered colimits are exact. Directed colimits are exact.", "\\end{slogan}", "Let $I$ be a directed set.", "Let $(L_i, \\lambda_{ij})$, $(M_i, \\mu_{ij})$, and", "$(N_i, \\nu_{ij})$ be systems of $R$-modules over $I$.", "Let $\\varphi_i : L_i \\to M_i$ and $\\psi_i : M_i \\to N_i$ be", "morphisms of systems over $I$. Assume that for all $i \\in I$ the", "sequence of $R$-modules", "$$", "\\xymatrix{", "L_i \\ar[r]^{\\varphi_i} &", "M_i \\ar[r]^{\\psi_i} &", "N_i", "}", "$$", "is a complex with homology $H_i$.", "Then the $R$-modules $H_i$ form a system over $I$,", "the sequence of $R$-modules", "$$", "\\xymatrix{", "\\colim_i L_i \\ar[r]^\\varphi &", "\\colim_i M_i \\ar[r]^\\psi &", "\\colim_i N_i", "}", "$$", "is a complex as well, and denoting $H$ its homology we have", "$$", "H = \\colim_i H_i.", "$$" ], "refs": [], "proofs": [ { "contents": [ "It is clear that", "$", "\\xymatrix{", "\\colim_i L_i \\ar[r]^\\varphi &", "\\colim_i M_i \\ar[r]^\\psi &", "\\colim_i N_i", "}", "$", "is a complex. For each $i \\in I$, there is a canonical", "$R$-module morphism $H_i \\to H$ (sending each", "$[m] \\in H_i = \\Ker(\\psi_i) / \\Im(\\varphi_i)$ to the", "residue class in $H = \\Ker(\\psi) / \\Im(\\varphi)$", "of the image of $m$ in $\\colim_i M_i$). These give rise", "to a morphism $\\colim_i H_i \\to H$. It remains to", "show that this morphism is surjective and injective.", "\\medskip\\noindent", "We are going to repeatedly use the description of colimits over $I$", "as in Lemma \\ref{lemma-directed-colimit} without further mention.", "Let $h \\in H$.", "Since $H = \\Ker(\\psi)/\\Im(\\varphi)$ we see that", "$h$ is the class mod $\\Im(\\varphi)$ of an element $[m]$", "in $\\Ker(\\psi) \\subset \\colim_i M_i$. Choose an", "$i$ such that $[m]$ comes from an element $m \\in M_i$. Choose", "a $j \\geq i$ such that $\\nu_{ij}(\\psi_i(m)) = 0$ which is possible", "since $[m] \\in \\Ker(\\psi)$. After replacing $i$ by $j$ and", "$m$ by $\\mu_{ij}(m)$ we see that we may assume $m \\in \\Ker(\\psi_i)$.", "This shows that the map $\\colim_i H_i \\to H$ is surjective.", "\\medskip\\noindent", "Suppose that $h_i \\in H_i$ has image zero on $H$. Since", "$H_i = \\Ker(\\psi_i)/\\Im(\\varphi_i)$ we may represent", "$h_i$ by an element $m \\in \\Ker(\\psi_i) \\subset M_i$.", "The assumption on the vanishing of $h_i$ in $H$ means that", "the class of $m$ in $\\colim_i M_i$ lies in the image", "of $\\varphi$. Hence there exists a $j \\geq i$ and an $l \\in L_j$", "such that $\\varphi_j(l) = \\mu_{ij}(m)$. Clearly this shows that", "the image of $h_i$ in $H_j$ is zero. This proves the", "injectivity of $\\colim_i H_i \\to H$." ], "refs": [ "algebra-lemma-directed-colimit" ], "ref_ids": [ 340 ] } ], "ref_ids": [] }, { "id": 344, "type": "theorem", "label": "algebra-lemma-almost-directed-colimit-exact", "categories": [ "algebra" ], "title": "algebra-lemma-almost-directed-colimit-exact", "contents": [ "Let $\\mathcal{I}$ be an index category satisfying the assumptions of", "Categories, Lemma \\ref{categories-lemma-split-into-directed}.", "Then taking colimits of diagrams of abelian groups over $\\mathcal{I}$", "is exact (i.e., the analogue of", "Lemma \\ref{lemma-directed-colimit-exact}", "holds in this situation)." ], "refs": [ "categories-lemma-split-into-directed", "algebra-lemma-directed-colimit-exact" ], "proofs": [ { "contents": [ "By", "Categories, Lemma \\ref{categories-lemma-split-into-directed}", "we may write $\\mathcal{I} = \\coprod_{j \\in J} \\mathcal{I}_j$ with each", "$\\mathcal{I}_j$ a filtered category, and $J$ possibly empty. By", "Categories, Lemma \\ref{categories-lemma-directed-category-system}", "taking colimits over the index categories $\\mathcal{I}_j$ is", "the same as taking the colimit over some directed set. Hence", "Lemma \\ref{lemma-directed-colimit-exact}", "applies to these colimits. This reduces the problem to showing that", "coproducts in the category of $R$-modules over the set $J$ are exact.", "In other words, exact sequences", "$L_j \\to M_j \\to N_j$ of $R$ modules we have to show that", "$$", "\\bigoplus\\nolimits_{j \\in J} L_j", "\\longrightarrow", "\\bigoplus\\nolimits_{j \\in J} M_j", "\\longrightarrow", "\\bigoplus\\nolimits_{j \\in J} N_j", "$$", "is exact. This can be verified by hand, and holds even if $J$ is empty." ], "refs": [ "categories-lemma-split-into-directed", "categories-lemma-directed-category-system", "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 12234, 12236, 343 ] } ], "ref_ids": [ 12234, 343 ] }, { "id": 345, "type": "theorem", "label": "algebra-lemma-localization-zero", "categories": [ "algebra" ], "title": "algebra-lemma-localization-zero", "contents": [ "The localization $S^{-1}A$ is the zero ring if and only if $0\\in S$." ], "refs": [], "proofs": [ { "contents": [ "If $0\\in S$, any pair $(a, s)\\sim (0, 1)$ by definition.", "If $0\\not \\in S$, then clearly $1/1 \\neq 0/1$ in $S^{-1}A$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 346, "type": "theorem", "label": "algebra-lemma-localization-and-modules", "categories": [ "algebra" ], "title": "algebra-lemma-localization-and-modules", "contents": [ "Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset.", "The category of $S^{-1}R$-modules is equivalent to the category", "of $R$-modules $N$ with the property that every $s \\in S$ acts as", "an automorphism on $N$." ], "refs": [], "proofs": [ { "contents": [ "The functor which defines the equivalence associates to an $S^{-1}R$-module", "$M$ the same module but now viewed as an $R$-module via the localization", "map $R \\to S^{-1}R$. Conversely, if $N$ is an $R$-module, such that every", "$s \\in S$ acts via an automorphism $s_N$, then we can think of $N$ as an", "$S^{-1}R$-module by letting $x/s$ act via $x_N \\circ s_N^{-1}$.", "We omit the verification that these two functors are quasi-inverse to", "each other." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 347, "type": "theorem", "label": "algebra-lemma-universal-property-localization-module", "categories": [ "algebra" ], "title": "algebra-lemma-universal-property-localization-module", "contents": [ "Let $R$ be a ring. Let $S \\subset R$ a multiplicative subset. Let $M$, $N$", "be $R$-modules. Assume all the elements of $S$ act as automorphisms on $N$.", "Then the canonical map", "$$", "\\Hom_R(S^{-1}M, N) \\longrightarrow \\Hom_R(M, N)", "$$", "induced by the localization map, is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "It is clear that the map is well-defined and R-linear.", "Injectivity: Let $\\alpha \\in \\Hom_R(S^{-1}M, N)$ and take an arbitrary", "element $m/s \\in S^{-1}M$. Then, since $s \\cdot \\alpha(m/s) = \\alpha(m/1)$,", "we have $ \\alpha(m/s) =s^{-1}(\\alpha (m/1))$, so $\\alpha$ is completely", "determined by what it does on the image of $M$ in $S^{-1}M$.", "Surjectivity: Let $\\beta : M \\rightarrow N$ be a given R-linear map.", "We need to show that it can be \"extended\" to $S^{-1}M$. Define a map of", "sets", "$$", "M \\times S \\rightarrow N,\\quad", "(m,s) \\mapsto s^{-1}\\beta(m)", "$$", "Clearly, this map respects the equivalence relation from above, so it", "descends to a well-defined map $\\alpha : S^{-1}M \\rightarrow N$.", "It remains to show that this map is $R$-linear, so take", "$r, r^\\prime \\in R$ as well as $s, s^\\prime \\in S$ and", "$m, m^\\prime \\in M$. Then", "\\begin{align*}", "\\alpha(r \\cdot m/s + r^\\prime \\cdot m^\\prime /s^\\prime)", "& = \\alpha ((r \\cdot s\\prime \\cdot m + r\\prime \\cdot s \\cdot m^\\prime)", "/(ss^\\prime)) \\\\", "& =", "(ss^\\prime)^{-1}(\\beta(r \\cdot s\\prime \\cdot m +", "r\\prime \\cdot s \\cdot m^\\prime) \\\\", "& =", "(ss^\\prime)^{-1} (r \\cdot s^\\prime \\beta (m) +", "r^\\prime \\cdot s \\beta (m^\\prime) \\\\", "& =", "r \\alpha (m/s) + r^\\prime \\alpha (m^\\prime /s^\\prime)", "\\end{align*}", "and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 348, "type": "theorem", "label": "algebra-lemma-localization-colimit", "categories": [ "algebra" ], "title": "algebra-lemma-localization-colimit", "contents": [ "Let $R$ be a ring.", "Let $S \\subset R$ be a multiplicative subset.", "Let $M$ be an $R$-module.", "Then", "$$", "S^{-1}M = \\colim_{f \\in S} M_f", "$$", "where the preorder on $S$ is given by", "$f \\geq f' \\Leftrightarrow f = f'f''$ for some $f'' \\in R$", "in which case the map $M_{f'} \\to M_f$ is given", "by $m/(f')^e \\mapsto m(f'')^e/f^e$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use the universal property of", "Lemma \\ref{lemma-universal-property-localization-module}." ], "refs": [ "algebra-lemma-universal-property-localization-module" ], "ref_ids": [ 347 ] } ], "ref_ids": [] }, { "id": 349, "type": "theorem", "label": "algebra-lemma-localize-quotient-modules", "categories": [ "algebra" ], "title": "algebra-lemma-localize-quotient-modules", "contents": [ "Localization respects quotients, i.e. if $N$ is a submodule of", "$M$, then $S^{-1}(M/N)\\simeq (S^{-1}M)/(S^{-1}N)$." ], "refs": [], "proofs": [ { "contents": [ "From the exact sequence", "$$", "0 \\longrightarrow N \\longrightarrow M \\longrightarrow M/N \\longrightarrow 0", "$$", "we have", "$$", "0 \\longrightarrow S^{-1}N \\longrightarrow S^{-1}M", "\\longrightarrow S^{-1}(M/N) \\longrightarrow 0", "$$", "The corollary then follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 350, "type": "theorem", "label": "algebra-lemma-submodule-localization", "categories": [ "algebra" ], "title": "algebra-lemma-submodule-localization", "contents": [ "Any submodule $N'$ of $S^{-1}M$ is of the form $S^{-1}N$ for some", "$N\\subset M$. Indeed one can take $N$ to be the inverse image of", "$N'$ in $M$." ], "refs": [], "proofs": [ { "contents": [ "Let $N$ be the inverse image of $N'$ in $M$. Then one can see that", "$S^{-1}N\\supset N'$. To show they are equal, take $x/s$ in", "$S^{-1}N$, where $s\\in S$ and $x\\in N$. This yields that $x/1\\in", "N'$. Since $N'$ is an $S^{-1}R$-submodule we have", "$x/s = x/1\\cdot 1/s\\in N'$. This finishes the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 351, "type": "theorem", "label": "algebra-lemma-ideal-in-localization", "categories": [ "algebra" ], "title": "algebra-lemma-ideal-in-localization", "contents": [ "\\begin{slogan}", "Ideals in the localization of a ring are localizations of ideals.", "\\end{slogan}", "Each ideal $I'$ of $S^{-1}A$ takes the form $S^{-1}I$, where one can", "take $I$ to be the inverse image of $I'$ in $A$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-submodule-localization}." ], "refs": [ "algebra-lemma-submodule-localization" ], "ref_ids": [ 350 ] } ], "ref_ids": [] }, { "id": 352, "type": "theorem", "label": "algebra-lemma-hom-exact", "categories": [ "algebra" ], "title": "algebra-lemma-hom-exact", "contents": [ "Exactness and $\\Hom_R$. Let $R$ be a ring.", "\\begin{enumerate}", "\\item Let $M_1 \\to M_2 \\to M_3 \\to 0$ be a complex of $R$-modules.", "Then $M_1 \\to M_2 \\to M_3 \\to 0$ is exact if and only if", "$0 \\to \\Hom_R(M_3, N) \\to \\Hom_R(M_2, N) \\to \\Hom_R(M_1, N)$", "is exact for all $R$-modules $N$.", "\\item Let $0 \\to M_1 \\to M_2 \\to M_3$ be a complex of $R$-modules.", "Then $0 \\to M_1 \\to M_2 \\to M_3$ is exact if and only if", "$0 \\to \\Hom_R(N, M_1) \\to \\Hom_R(N, M_2) \\to \\Hom_R(N, M_3)$", "is exact for all $R$-modules $N$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 353, "type": "theorem", "label": "algebra-lemma-hom-from-finitely-presented", "categories": [ "algebra" ], "title": "algebra-lemma-hom-from-finitely-presented", "contents": [ "Let $R$ be a ring. Let $M$ be a finitely presented $R$-module.", "Let $N$ be an $R$-module.", "\\begin{enumerate}", "\\item For $f \\in R$ we have", "$\\Hom_R(M, N)_f = \\Hom_{R_f}(M_f, N_f) = \\Hom_R(M_f, N_f)$,", "\\item for a multiplicative subset $S$ of $R$ we have", "$$", "S^{-1}\\Hom_R(M, N) = \\Hom_{S^{-1}R}(S^{-1}M, S^{-1}N) =", "\\Hom_R(S^{-1}M, S^{-1}N).", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is a special case of part (2).", "The second equality in (2) follows from", "Lemma \\ref{lemma-universal-property-localization-module}.", "Choose a presentation", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} R", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} R", "\\to M \\to 0.", "$$", "By", "Lemma \\ref{lemma-hom-exact}", "this gives an exact sequence", "$$", "0 \\to", "\\Hom_R(M, N) \\to", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} N", "\\longrightarrow", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} N.", "$$", "Inverting $S$ and using Proposition \\ref{proposition-localization-exact}", "we get an exact sequence", "$$", "0 \\to", "S^{-1}\\Hom_R(M, N) \\to", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} S^{-1}N", "\\longrightarrow", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} S^{-1}N", "$$", "and the result follows since $S^{-1}M$ sits in", "an exact sequence", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} S^{-1}R", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} S^{-1}R \\to S^{-1}M \\to 0", "$$", "which induces (by Lemma \\ref{lemma-hom-exact})", "the exact sequence", "$$", "0 \\to", "\\Hom_{S^{-1}R}(S^{-1}M, S^{-1}N) \\to", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} S^{-1}N", "\\longrightarrow", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} S^{-1}N", "$$", "which is the same as the one above." ], "refs": [ "algebra-lemma-universal-property-localization-module", "algebra-lemma-hom-exact", "algebra-proposition-localization-exact", "algebra-lemma-hom-exact" ], "ref_ids": [ 347, 352, 1402, 352 ] } ], "ref_ids": [] }, { "id": 354, "type": "theorem", "label": "algebra-lemma-characterize-finite-module-hom", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-finite-module-hom", "contents": [ "Let $R$ be a ring. Let $N$ be an $R$-module. The following are equivalent", "\\begin{enumerate}", "\\item $N$ is a finite $R$-module,", "\\item for any filtered colimit $M = \\colim M_i$ of $R$-modules the map", "$\\colim \\Hom_R(N, M_i) \\to \\Hom_R(K, M)$ is injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and choose generators $x_1, \\ldots, x_m$ for $N$.", "If $N \\to M_i$ is a module map and the composition", "$N \\to M_i \\to M$ is zero, then because $M = \\colim_{i' \\geq i} M_{i'}$", "for each $j \\in \\{1, \\ldots, m\\}$ we can find a $i' \\geq i$ such that", "$x_j$ maps to zero in $M_{i'}$. Since there are finitely many", "$x_j$ we can find a single $i'$ which works for all of them.", "Then the composition $N \\to M_i \\to M_{i'}$ is zero and we conclude", "the map is injective, i.e., part (2) holds.", "\\medskip\\noindent", "Assume (2). For a finite subset $E \\subset N$ denote $N_E \\subset N$", "the $R$-submodule generated by the elements of $E$. Then", "$0 = \\colim N/N_E$ is a filtered colimit. Hence we see that", "$\\text{id} : N \\to N$ maps into $N_E$ for some $E$, i.e., $N$", "is finitely generated." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 355, "type": "theorem", "label": "algebra-lemma-module-colimit-fp", "categories": [ "algebra" ], "title": "algebra-lemma-module-colimit-fp", "contents": [ "Let $R$ be a ring and let $M$ be an $R$-module.", "Then $M$ is the colimit of a directed system", "$(M_i, \\mu_{ij})$ of $R$-modules", "with all $M_i$ finitely presented $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "Consider any finite subset $S \\subset M$ and any finite", "collection of relations $E$ among the elements", "of $S$. So each $s \\in S$ corresponds to $x_s \\in M$ and", "each $e \\in E$ consists of a vector", "of elements $f_{e, s} \\in R$ such that $\\sum f_{e, s} x_s = 0$.", "Let $M_{S, E}$ be the cokernel of the map", "$$", "R^{\\# E} \\longrightarrow R^{\\# S}, \\quad", "(g_e)_{e\\in E} \\longmapsto (\\sum g_e f_{e, s})_{s\\in S}.", "$$", "There are canonical maps $M_{S, E} \\to M$.", "If $S \\subset S'$ and if the elements of", "$E$ correspond, via this map, to relations", "in $E'$, then there is an obvious map", "$M_{S, E} \\to M_{S', E'}$ commuting with the", "maps to $M$. Let $I$ be the set of pairs", "$(S, E)$ with ordering by inclusion as above.", "It is clear that the colimit of this directed system is $M$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 356, "type": "theorem", "label": "algebra-lemma-characterize-finitely-presented-module-hom", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-finitely-presented-module-hom", "contents": [ "Let $R$ be a ring. Let $N$ be an $R$-module. The following are equivalent", "\\begin{enumerate}", "\\item $N$ is a finitely presented $R$-module,", "\\item for any filtered colimit $M = \\colim M_i$ of $R$-modules the map", "$\\colim \\Hom_R(N, M_i) \\to \\Hom_R(K, M)$ is bijective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and choose an exact sequence $F_{-1} \\to F_0 \\to N \\to 0$", "with $F_i$ finite free. Then we have an exact sequence", "$$", "0 \\to \\Hom_R(N, M) \\to \\Hom_R(F_0, M) \\to \\Hom_R(F_{-1}, M)", "$$", "functorial in the $R$-module $M$. The functors $\\Hom_R(F_i, M)$ commute", "with filtered colimits as $\\Hom_R(R^{\\oplus n}, M) = M^{\\oplus n}$.", "Since filtered colimits are exact", "(Lemma \\ref{lemma-directed-colimit-exact})", "we see that (2) holds.", "\\medskip\\noindent", "Assume (2). By Lemma \\ref{lemma-module-colimit-fp}", "we can write $M = \\colim M_i$ as a filtered", "colimit such that $M_i$ is of finite presentation for all $i$.", "Thus $\\text{id}_M$ factors through $M_i$ for some $i$.", "This means that $M$ is a direct summand of a finitely", "presented $R$-module (namely $M_i$) and hence finitely presented." ], "refs": [ "algebra-lemma-directed-colimit-exact", "algebra-lemma-module-colimit-fp" ], "ref_ids": [ 343, 355 ] } ], "ref_ids": [] }, { "id": 357, "type": "theorem", "label": "algebra-lemma-tensor-product", "categories": [ "algebra" ], "title": "algebra-lemma-tensor-product", "contents": [ "Let $M, N$ be $R$-modules. Then there exists a pair $(T, g)$", "where $T$ is an $R$-module, and", "$g : M \\times N \\to T$ an $R$-bilinear", "mapping, with the following universal property:", "For any $R$-module $P$ and any $R$-bilinear mapping", "$f : M \\times N \\to P$, there", "exists a unique $R$-linear", "mapping $\\tilde{f} : T \\to P$ such that $f = \\tilde{f} \\circ g$.", "In other words, the following diagram commutes:", "$$", "\\xymatrix{", "M \\times N \\ar[rr]^f \\ar[dr]_g & & P\\\\", "& T \\ar[ur]_{\\tilde f}", "}", "$$", "Moreover, if $(T, g)$ and $(T', g')$", "are two pairs with this property, then there", "exists a unique isomorphism", "$j : T \\to T'$ such that $j\\circ g = g'$." ], "refs": [], "proofs": [ { "contents": [ "We first prove the existence of such $R$-module $T$.", "Let $M, N$ be $R$-modules.", "Let $T$ be the quotient module", "$P/Q$, where $P$ is the free $R$-module $R^{(M \\times N)}$ and $Q$ is the", "$R$-module generated by all elements of", "the following types: ($x\\in M, y\\in N$)", "\\begin{align*}", "(x + x', y) - (x, y) - (x', y), \\\\", "(x, y + y') - (x, y) - (x, y'), \\\\", "(ax, y) - a(x, y), \\\\", "(x, ay) - a(x, y)", "\\end{align*}", "Let $\\pi : M \\times N \\to T$ denote the natural map.", "This map is $R$-bilinear, as", "implied by the above relations", "when we check the bilinearity conditions. Denote the image", "$\\pi(x, y) = x \\otimes", "y$, then these elements generate", "$T$. Now let $f : M \\times N \\to P$ be an $R$-bilinear map,", "then we can define", "$f' : T \\to P$ by extending the mapping", "$f'(x \\otimes y) = f(x, y)$. Clearly $f = f'\\circ \\pi$. Moreover, $f'$ is", "uniquely determined by the value on the", "generating sets $\\{x \\otimes y : x\\in M, y\\in N\\}$.", "Suppose there is another pair $(T', g')$ satisfying the same properties.", "Then there is a unique $j : T \\to T'$ and", "also $j' : T' \\to T$ such that $g' = j\\circ g$, $g = j'\\circ g'$.", "But then both the maps $(j\\circ j') \\circ g$ and $g$", "satisfies the universal properties, so by uniqueness they are equal,", "and hence $j'\\circ j$ is identity on $T$.", "Similarly $(j'\\circ j) \\circ g' = g'$ and $j\\circ j'$ is identity on $T'$.", "So $j$ is an isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 358, "type": "theorem", "label": "algebra-lemma-flip-tensor-product", "categories": [ "algebra" ], "title": "algebra-lemma-flip-tensor-product", "contents": [ "Let $M, N, P$ be $R$-modules, then the bilinear maps", "\\begin{align*}", "(x, y) & \\mapsto y \\otimes x\\\\", "(x + y, z) & \\mapsto x \\otimes z + y \\otimes z\\\\", "(r, x) & \\mapsto rx", "\\end{align*}", "induce unique isomorphisms", "\\begin{align*}", "M \\otimes_R N & \\to N \\otimes_R M, \\\\", "(M\\oplus N)\\otimes_R P & \\to (M \\otimes_R P)\\oplus(N \\otimes_R P), \\\\", "R \\otimes_R M & \\to M", "\\end{align*}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 359, "type": "theorem", "label": "algebra-lemma-multilinear", "categories": [ "algebra" ], "title": "algebra-lemma-multilinear", "contents": [ "Let $M_1, \\ldots, M_r$ be $R$-modules. Then there exists a pair $(T, g)$", "consisting of an $R$-module T and an $R$-multilinear mapping", "$g : M_1\\times \\ldots \\times M_r \\to T$ with the universal", "property: For any $R$-multilinear mapping", "$f : M_1\\times \\ldots \\times M_r \\to P$ there exists a unique $R$-module", "homomorphism $f' : T \\to P$ such that $f'\\circ g = f$.", "Such a module $T$ is unique up to unique isomorphism. We denote it", "$M_1\\otimes_R \\ldots \\otimes_R M_r$ and we denote the universal", "multilinear map $(m_1, \\ldots, m_r) \\mapsto m_1 \\otimes \\ldots \\otimes m_r$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 360, "type": "theorem", "label": "algebra-lemma-transitive", "categories": [ "algebra" ], "title": "algebra-lemma-transitive", "contents": [ "The homomorphisms", "$$", "(M \\otimes_R N)\\otimes_R P \\to", "M \\otimes_R N \\otimes_R P \\to", "M \\otimes_R (N \\otimes_R P)", "$$", "such that", "$f((x \\otimes y)\\otimes z) = x \\otimes y \\otimes z$", "and $g(x \\otimes y \\otimes z) = x \\otimes (y \\otimes z)$,", "$x\\in M, y\\in N, z\\in P$ are well-defined and are isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "We shall prove $f$ is well-defined and is an isomorphism, and this proof", "carries analogously to $g$. Fix any", "$z\\in P$, then the mapping $(x, y)\\mapsto x \\otimes y \\otimes z$,", "$x\\in M, y\\in N$, is $R$-bilinear in $x$ and $y$,", "and hence induces homomorphism $f_z : M \\otimes N \\to M \\otimes N \\otimes P$", "which sends", "$f_z(x \\otimes y) = x \\otimes y \\otimes z$.", "Then consider $(M \\otimes N)\\times P \\to M \\otimes N \\otimes P$ given by", "$(w, z)\\mapsto f_z(w)$. The map is", "$R$-bilinear and thus induces", "$f : (M \\otimes_R N)\\otimes_R P \\to M \\otimes_R N \\otimes_R P$", "and $f((x \\otimes y)\\otimes z) = x \\otimes y \\otimes z$.", "To construct the inverse, we note that the map", "$\\pi : M \\times N \\times P \\to (M \\otimes N)\\otimes P$ is", "$R$-trilinear.", "Therefore, it induces an $R$-linear map", "$h : M \\otimes N \\otimes P \\to (M \\otimes N)\\otimes P$ which", "agrees with the universal property. Here we see that", "$h(x \\otimes y \\otimes z) = (x \\otimes y)\\otimes z$.", "From the explicit expression of $f$ and $h$, $f\\circ h$ and $h\\circ f$ are", "identity maps of $M \\otimes N \\otimes", "P$ and $(M \\otimes N)\\otimes P$ respectively, hence $f$ is our desired", "isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 361, "type": "theorem", "label": "algebra-lemma-tensor-with-bimodule", "categories": [ "algebra" ], "title": "algebra-lemma-tensor-with-bimodule", "contents": [ "For $A$-module $M$, $B$-module $P$ and $(A, B)$-bimodule $N$, the modules", "$(M \\otimes_A N)\\otimes_B P$ and $M \\otimes_A(N \\otimes_B P)$ can both be", "given $(A, B)$-bimodule structure,", "and moreover", "$$", "(M \\otimes_A N)\\otimes_B P \\cong M \\otimes_A(N \\otimes_B P).", "$$" ], "refs": [], "proofs": [ { "contents": [ "A priori $M \\otimes_A N$ is an $A$-module, but we can give it a", "$B$-module structure by letting", "$$", "(x \\otimes y)b = x \\otimes yb, \\quad x\\in M, y\\in N, b\\in B", "$$", "Thus $M \\otimes_A N$ becomes an $(A, B)$-bimodule. Similarly for", "$N \\otimes_B P$, and thus for", "$(M \\otimes_A N)\\otimes_B P$ and $M \\otimes_A(N \\otimes_B P)$. By", "Lemma \\ref{lemma-transitive}, these two", "modules are isomorphic as both as $A$-module and $B$-module via the same", "mapping." ], "refs": [ "algebra-lemma-transitive" ], "ref_ids": [ 360 ] } ], "ref_ids": [] }, { "id": 362, "type": "theorem", "label": "algebra-lemma-hom-from-tensor-product", "categories": [ "algebra" ], "title": "algebra-lemma-hom-from-tensor-product", "contents": [ "For any three $R$-modules $M, N, P$,", "$$", "\\Hom_R(M \\otimes_R N, P) \\cong \\Hom_R(M, \\Hom_R(N, P))", "$$" ], "refs": [], "proofs": [ { "contents": [ "An $R$-linear map $\\hat{f}\\in \\Hom_R(M \\otimes_R N, P)$ corresponds to an", "$R$-bilinear map $f : M \\times N \\to P$. For", "each $x\\in M$ the mapping $y\\mapsto f(x, y)$ is $R$-linear by the universal", "property. Thus $f$ corresponds to a", "map $\\phi_f : M \\to \\Hom_R(N, P)$. This map is $R$-linear since", "$$", "\\phi_f(ax + y)(z) =", "f(ax + y, z) = af(x, z)+f(y, z) =", "(a\\phi_f(x)+\\phi_f(y))(z),", "$$", "for all $a \\in R$, $x \\in M$, $y \\in M$ and", "$z \\in N$. Conversely, any", "$f \\in \\Hom_R(M, \\Hom_R(N, P))$ defines an $R$-bilinear", "map $M \\times N \\to P$, namely $(x, y)\\mapsto f(x)(y)$.", "So this is a natural one-to-one correspondence between the", "two modules", "$\\Hom_R(M \\otimes_R N, P)$ and $\\Hom_R(M, \\Hom_R(N, P))$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 363, "type": "theorem", "label": "algebra-lemma-tensor-products-commute-with-limits", "categories": [ "algebra" ], "title": "algebra-lemma-tensor-products-commute-with-limits", "contents": [ "Let $(M_i, \\mu_{ij})$ be a system over the preordered set $I$.", "Let $N$ be an $R$-module. Then", "$$", "\\colim (M_i \\otimes N) \\cong (\\colim M_i)\\otimes N.", "$$", "Moreover, the isomorphism is induced by the homomorphisms", "$\\mu_i \\otimes 1: M_i \\otimes N \\to M \\otimes N$", "where $M = \\colim_i M_i$ with natural maps $\\mu_i : M_i \\to M$." ], "refs": [], "proofs": [ { "contents": [ "First proof. The functor $M' \\mapsto M' \\otimes_R N$ is left adjoint", "to the functor $N' \\mapsto \\Hom_R(N, N')$ by", "Lemma \\ref{lemma-hom-from-tensor-product}. Thus $M' \\mapsto M' \\otimes_R N$", "commutes with all colimits, see", "Categories, Lemma \\ref{categories-lemma-adjoint-exact}.", "\\medskip\\noindent", "Second direct proof. Let $P = \\colim (M_i \\otimes N)$, $M = \\colim M_i$.", "Then for all $i\\leq j$, the following diagram commutes:", "$$", "\\xymatrix{", "M_i \\otimes N \\ar[r]_{\\mu_i \\otimes 1} \\ar[d]_{\\mu_{ij} \\otimes 1} &", "M \\otimes N \\ar[d]^{\\text{id}} \\\\", "M_j \\otimes N \\ar[r]^{\\mu_j \\otimes 1} &", "M \\otimes N", "}", "$$", "By Lemma \\ref{lemma-homomorphism-limit},", "these maps induce a unique homomorphism", "$\\psi : P \\to M \\otimes N$, with $\\lambda_i : M_i \\otimes N \\to P$ given by", "$\\lambda_i = \\pi \\circ (\\iota_i \\otimes 1)$.", "\\medskip\\noindent", "To construct the inverse map, for each $i\\in I$, there is the canonical", "$R$-bilinear mapping $g_i : M_i \\times N \\to", "M_i \\otimes N$. This induces a unique mapping", "$\\widehat{\\phi} : M \\times N \\to P$", "such that $\\widehat{\\phi} \\circ (\\mu_i \\times 1) = \\lambda_i \\circ g_i$.", "It is $R$-bilinear. Thus it induces an", "$R$-linear mapping $\\phi : M \\otimes N \\to P$.", "From the commutative diagram below:", "$$", "\\xymatrix{", "M_i \\times N \\ar[r]^{g_i} \\ar[d]^{\\mu_i \\times \\text{id}} &", "M_i \\otimes N\\ar[r]_{\\text{id}} \\ar[d]_{\\lambda_i} &", "M_i \\otimes N \\ar[d]_{\\mu_i \\otimes \\text{id}} \\ar[rd]^{\\lambda_i} \\\\", "M \\times N \\ar[r]^{\\widehat{\\phi}} &", "P \\ar[r]^{\\psi} & M \\otimes N \\ar[r]^{\\phi} & P", "}", "$$", "we see that $\\psi\\circ\\widehat{\\phi} = g$, the canonical $R$-bilinear mapping", "$g : M \\times N \\to M \\otimes N$. So", "$\\psi\\circ\\phi$ is identity on $M \\otimes N$. From the right-hand square and", "triangle, $\\phi\\circ\\psi$ is also", "identity on $P$." ], "refs": [ "algebra-lemma-hom-from-tensor-product", "categories-lemma-adjoint-exact", "algebra-lemma-homomorphism-limit" ], "ref_ids": [ 362, 12249, 342 ] } ], "ref_ids": [] }, { "id": 364, "type": "theorem", "label": "algebra-lemma-tensor-product-exact", "categories": [ "algebra" ], "title": "algebra-lemma-tensor-product-exact", "contents": [ "Let", "\\begin{align*}", "M_1\\xrightarrow{f} M_2\\xrightarrow{g} M_3 \\to 0", "\\end{align*}", "be an exact sequence of $R$-modules and homomorphisms, and let $N$ be any", "$R$-module. Then the sequence", "\\begin{equation}", "\\label{equation-2ndex}", "M_1\\otimes N\\xrightarrow{f \\otimes 1} M_2\\otimes N \\xrightarrow{g \\otimes 1}", "M_3\\otimes N \\to 0", "\\end{equation}", "is exact. In other words, the functor $- \\otimes_R N$ is", "{\\it right exact}, in the sense that tensoring", "each term in the original right exact sequence preserves the exactness." ], "refs": [], "proofs": [ { "contents": [ "We apply the functor $\\Hom(-, \\Hom(N, P))$ to the first exact", "sequence. We obtain", "$$", "0 \\to", "\\Hom(M_3, \\Hom(N, P)) \\to", "\\Hom(M_2, \\Hom(N, P)) \\to", "\\Hom(M_1, \\Hom(N, P))", "$$", "By Lemma \\ref{lemma-hom-from-tensor-product}, we have", "$$", "0 \\to \\Hom(M_3 \\otimes N, P) \\to", "\\Hom(M_2 \\otimes N, P) \\to \\Hom(M_1 \\otimes N, P)", "$$", "Using the pullback property again, we arrive at the desired exact sequence." ], "refs": [ "algebra-lemma-hom-from-tensor-product" ], "ref_ids": [ 362 ] } ], "ref_ids": [] }, { "id": 365, "type": "theorem", "label": "algebra-lemma-tensor-finiteness", "categories": [ "algebra" ], "title": "algebra-lemma-tensor-finiteness", "contents": [ "Let $R$ be a ring. Let $M$ and $N$ be $R$-modules.", "\\begin{enumerate}", "\\item If $N$ and $M$ are finite, then so is $M \\otimes_R N$.", "\\item If $N$ and $M$ are finitely presented, then so is $M \\otimes_R N$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose $M$ is finite. Then choose a presentation", "$0 \\to K \\to R^{\\oplus n} \\to M \\to 0$. This gives an exact sequence", "$K \\otimes_R N \\to N^{\\oplus n} \\to M \\otimes_R N \\to 0$ by", "Lemma \\ref{lemma-tensor-product-exact}.", "We conclude that if $N$ is finite too then $M \\otimes_R N$", "is a quotient of a finite module, hence finite, see", "Lemma \\ref{lemma-extension}.", "Similarly, if both $N$ and $M$ are finitely presented, then", "we see that $K$ is finite and that $M \\otimes_R N$", "is a quotient of the finitely presented module $N^{\\oplus n}$ by", "a finite module, namely $K \\otimes_R N$, and hence finitely presented, see", "Lemma \\ref{lemma-extension}." ], "refs": [ "algebra-lemma-tensor-product-exact", "algebra-lemma-extension", "algebra-lemma-extension" ], "ref_ids": [ 364, 330, 330 ] } ], "ref_ids": [] }, { "id": 366, "type": "theorem", "label": "algebra-lemma-tensor-localization", "categories": [ "algebra" ], "title": "algebra-lemma-tensor-localization", "contents": [ "Let $M$ be an $R$-module. Then the $S^{-1}R$-modules $S^{-1}M$", "and $S^{-1}R \\otimes_R M$ are canonically isomorphic, and the", "canonical isomorphism $f : S^{-1}R \\otimes_R M \\to S^{-1}M$", "is given by", "$$", "f((a/s) \\otimes m) = am/s, \\forall a \\in R, m \\in M, s \\in S", "$$" ], "refs": [], "proofs": [ { "contents": [ "Obviously, the map", "$f' : S^{-1}R \\times M \\to S^{-1}M$ given by $f((a/s, m)) = am/s$ is", "bilinear, and thus by the", "universal property, this map induces a unique $S^{-1}R$-module homomorphism", "$f : S^{-1}R \\otimes_R M \\to S^{-1}M$ as in the statement of the lemma.", "Actually every element in $S^{-1}M$ is of the form $m/s$, $m\\in M, s\\in S$ and", "every element in", "$S^{-1}R \\otimes_R M$ is of the form $1/s \\otimes m$. To see the latter fact,", "write an element in", "$S^{-1}R \\otimes_R M$ as", "$$", "\\sum_k \\frac{a_k}{s_k} \\otimes m_k =", "\\sum_k \\frac{a_k t_k}{s} \\otimes m_k =", "\\frac{1}{s} \\otimes \\sum_k {a_k t_k}m_k = \\frac{1}{s} \\otimes m", "$$", "Where $m = \\sum_k {a_k t_k}m_k$. Then it is obvious that $f$ is surjective,", "and if $f(\\frac{1}{s} \\otimes m) = m/s = 0$ then there exists $t'\\in S$ with", "$tm = 0$ in $M$. Then we have", "$$", "\\frac{1}{s} \\otimes m = \\frac{1}{st} \\otimes tm = \\frac{1}{st} \\otimes 0 = 0", "$$", "Therefore $f$ is injective." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 367, "type": "theorem", "label": "algebra-lemma-tensor-product-localization", "categories": [ "algebra" ], "title": "algebra-lemma-tensor-product-localization", "contents": [ "Let $M, N$ be $R$-modules, then there is a canonical", "$S^{-1}R$-module isomorphism", "$f : S^{-1}M \\otimes_{S^{-1}R}S^{-1}N \\to S^{-1}(M \\otimes_R N)$,", "given by", "$$", "f((m/s)\\otimes(n/t)) = (m \\otimes n)/st", "$$" ], "refs": [], "proofs": [ { "contents": [ "We may use Lemma \\ref{lemma-tensor-with-bimodule}", "and Lemma \\ref{lemma-tensor-localization} repeatedly to", "see that these two", "$S^{-1}R$-modules are isomorphic, noting that $S^{-1}R$ is an", "$(R, S^{-1}R)$-bimodule:", "\\begin{align*}", "S^{-1}(M \\otimes_R N) & \\cong S^{-1}R \\otimes_R (M \\otimes_R N)\\\\", " & \\cong S^{-1}M \\otimes_R N\\\\", " & \\cong (S^{-1}M \\otimes_{S^{-1}R}S^{-1}R)\\otimes_R N\\\\", " & \\cong S^{-1}M \\otimes_{S^{-1}R}(S^{-1}R \\otimes_R N)\\\\", " & \\cong S^{-1}M \\otimes_{S^{-1}R}S^{-1}N", "\\end{align*}", "This isomorphism is easily seen to be the one stated in the lemma." ], "refs": [ "algebra-lemma-tensor-with-bimodule", "algebra-lemma-tensor-localization" ], "ref_ids": [ 361, 366 ] } ], "ref_ids": [] }, { "id": 368, "type": "theorem", "label": "algebra-lemma-free-tensor-algebra", "categories": [ "algebra" ], "title": "algebra-lemma-free-tensor-algebra", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "If $M$ is a free $R$-module, so is each symmetric and exterior power." ], "refs": [], "proofs": [ { "contents": [ "Omitted, but see above for the finite free case." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 369, "type": "theorem", "label": "algebra-lemma-presentation-sym-exterior", "categories": [ "algebra" ], "title": "algebra-lemma-presentation-sym-exterior", "contents": [ "Let $R$ be a ring.", "Let $M_2 \\to M_1 \\to M \\to 0$ be an exact sequence of $R$-modules.", "There are exact sequences", "$$", "M_2 \\otimes_R \\text{Sym}^{n - 1}(M_1)", "\\to", "\\text{Sym}^n(M_1)", "\\to", "\\text{Sym}^n(M)", "\\to", "0", "$$", "and similarly", "$$", "M_2 \\otimes_R \\wedge^{n - 1}(M_1)", "\\to", "\\wedge^n(M_1)", "\\to", "\\wedge^n(M)", "\\to", "0", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 370, "type": "theorem", "label": "algebra-lemma-present-sym-wedge", "categories": [ "algebra" ], "title": "algebra-lemma-present-sym-wedge", "contents": [ "Let $R$ be a ring.", "Let $M$ be an $R$-module.", "Let $x_i$, $i \\in I$ be a given system of generators of", "$M$ as an $R$-module. Let $n \\geq 2$.", "There exists a canonical exact sequence", "$$", "\\bigoplus_{1 \\leq j_1 < j_2 \\leq n}", "\\bigoplus_{i_1, i_2 \\in I}", "\\text{T}^{n - 2}(M)", "\\oplus", "\\bigoplus_{1 \\leq j_1 < j_2 \\leq n}", "\\bigoplus_{i \\in I}", "\\text{T}^{n - 2}(M)", "\\to", "\\text{T}^n(M)", "\\to", "\\wedge^n(M)", "\\to", "0", "$$", "where the pure tensor $m_1 \\otimes \\ldots \\otimes m_{n - 2}$ in the first", "summand maps to", "\\begin{align*}", "\\underbrace{", "m_1 \\otimes \\ldots \\otimes x_{i_1} \\otimes \\ldots", "\\otimes x_{i_2} \\otimes \\ldots \\otimes m_{n - 2}", "}_{\\text{with } x_{i_1} \\text{ and } x_{i_2}", "\\text{ occupying slots } j_1 \\text{ and } j_2", "\\text{ in the tensor}} \\\\", "+", "\\underbrace{", "m_1 \\otimes \\ldots \\otimes x_{i_2} \\otimes \\ldots", "\\otimes x_{i_1} \\otimes \\ldots \\otimes m_{n - 2}", "}_{\\text{with } x_{i_2} \\text{ and } x_{i_1}", "\\text{ occupying slots } j_1 \\text{ and } j_2", "\\text{ in the tensor}}", "\\end{align*}", "and $m_1 \\otimes \\ldots \\otimes m_{n - 2}$ in the second", "summand maps to", "$$", "\\underbrace{", "m_1 \\otimes \\ldots \\otimes x_i \\otimes \\ldots", "\\otimes x_i \\otimes \\ldots \\otimes m_{n - 2}", "}_{\\text{with } x_{i} \\text{ and } x_{i}", "\\text{ occupying slots } j_1 \\text{ and } j_2", "\\text{ in the tensor}}", "$$", "There is also a canonical exact sequence", "$$", "\\bigoplus_{1 \\leq j_1 < j_2 \\leq n}", "\\bigoplus_{i_1, i_2 \\in I}", "\\text{T}^{n - 2}(M)", "\\to", "\\text{T}^n(M)", "\\to", "\\text{Sym}^n(M)", "\\to", "0", "$$", "where the pure tensor $m_1 \\otimes \\ldots \\otimes m_{n - 2}$ maps to", "\\begin{align*}", "\\underbrace{", "m_1 \\otimes \\ldots \\otimes x_{i_1} \\otimes \\ldots", "\\otimes x_{i_2} \\otimes \\ldots \\otimes m_{n - 2}", "}_{\\text{with } x_{i_1} \\text{ and } x_{i_2}", "\\text{ occupying slots } j_1 \\text{ and } j_2", "\\text{ in the tensor}} \\\\", "-", "\\underbrace{", "m_1 \\otimes \\ldots \\otimes x_{i_2} \\otimes \\ldots", "\\otimes x_{i_1} \\otimes \\ldots \\otimes m_{n - 2}", "}_{\\text{with } x_{i_2} \\text{ and } x_{i_1}", "\\text{ occupying slots } j_1 \\text{ and } j_2", "\\text{ in the tensor}}", "\\end{align*}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 371, "type": "theorem", "label": "algebra-lemma-colimit-tensor-algebra", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-tensor-algebra", "contents": [ "\\begin{slogan}", "Taking tensor algebras commutes with filtered colimits.", "\\end{slogan}", "Let $R$ be a ring. Let $M_i$ be a directed system of", "$R$-modules. Then", "$\\colim_i \\text{T}(M_i) = \\text{T}(\\colim_i M_i)$", "and similarly for the symmetric and exterior algebras." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Apply Lemma \\ref{lemma-tensor-products-commute-with-limits}." ], "refs": [ "algebra-lemma-tensor-products-commute-with-limits" ], "ref_ids": [ 363 ] } ], "ref_ids": [] }, { "id": 372, "type": "theorem", "label": "algebra-lemma-tensor-algebra-localization", "categories": [ "algebra" ], "title": "algebra-lemma-tensor-algebra-localization", "contents": [ "Let $R$ be a ring and let $S \\subset R$ be a multiplicative subset.", "Then $S^{-1}T_R(M) = T_{S^{-1}R}(S^{-1}M)$ for any $R$-module $M$.", "Similar for symmetric and exterior algebras." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Apply Lemma \\ref{lemma-tensor-product-localization}." ], "refs": [ "algebra-lemma-tensor-product-localization" ], "ref_ids": [ 367 ] } ], "ref_ids": [] }, { "id": 373, "type": "theorem", "label": "algebra-lemma-base-change-finiteness", "categories": [ "algebra" ], "title": "algebra-lemma-base-change-finiteness", "contents": [ "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module.", "Let $R \\to R'$ be a ring map and let $S' = S \\otimes_R R'$ and", "$M' = M \\otimes_R R'$ be the base changes.", "\\begin{enumerate}", "\\item If $M$ is a finite $S$-module, then the base change", "$M'$ is a finite $S'$-module.", "\\item If $M$ is an $S$-module finite presentation, then", "the base change $M'$ is an $S'$-module of finite presentation.", "\\item If $R \\to S$ is of finite type, then the base change", "$R' \\to S'$ is of finite type.", "\\item If $R \\to S$ is of finite presentation, then", "the base change $R' \\to S'$ is of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Take a surjective, $S$-linear map", "$S^{\\oplus n} \\to M \\to 0$.", "By Lemma \\ref{lemma-flip-tensor-product} and \\ref{lemma-tensor-product-exact}", "the result after tensoring with $R^\\prime$ is a surjection", "${S^\\prime}^{\\oplus n} \\to M^\\prime \\rightarrow 0$,", "so $M^\\prime$ is a finitely generated $S^\\prime$-module.", "Proof of (2). Take a presentation", "$S^{\\oplus m} \\to S^{\\oplus n} \\to M \\to 0$.", "By Lemma \\ref{lemma-flip-tensor-product} and \\ref{lemma-tensor-product-exact}", "the result after tensoring with $R^\\prime$ gives a finite presentation", "${S^\\prime}^{\\oplus m} \\to {S^\\prime}^{\\oplus n} \\to M^\\prime \\to 0$, of", "the $S^\\prime$-module $M^\\prime$. Proof of (3). This follows by the remark", "preceding the lemma as we can take $I$ to be finite by assumption.", "Proof of (4). This follows by the remark preceding the lemma", "as we can take $I$ and $J$ to be finite by assumption." ], "refs": [ "algebra-lemma-flip-tensor-product", "algebra-lemma-tensor-product-exact", "algebra-lemma-flip-tensor-product", "algebra-lemma-tensor-product-exact" ], "ref_ids": [ 358, 364, 358, 364 ] } ], "ref_ids": [] }, { "id": 374, "type": "theorem", "label": "algebra-lemma-adjoint-tensor-restrict", "categories": [ "algebra" ], "title": "algebra-lemma-adjoint-tensor-restrict", "contents": [ "Let $R \\to S$ be a ring map. The functors", "$\\text{Mod}_S \\to \\text{Mod}_R$, $N \\mapsto N_R$ (restriction)", "and $\\text{Mod}_R \\to \\text{Mod}_S$, $M \\mapsto M \\otimes_R S$", "(base change) are adjoint functors. In a formula", "$$", "\\Hom_R(M, N_R) = \\Hom_S(M \\otimes_R S, N)", "$$" ], "refs": [], "proofs": [ { "contents": [ "If $\\alpha : M \\to N_R$ is an $R$-module map, then we define", "$\\alpha' : M \\otimes_R S \\to N$ by the rule", "$\\alpha'(m \\otimes s) = s\\alpha(m)$. If $\\beta : M \\otimes_R S \\to N$", "is an $S$-module map, we define $\\beta' : M \\to N_R$ by the rule", "$\\beta'(m) = \\beta(m \\otimes 1)$.", "We omit the verification that these constructions are mutually inverse." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 375, "type": "theorem", "label": "algebra-lemma-adjoint-hom-restrict", "categories": [ "algebra" ], "title": "algebra-lemma-adjoint-hom-restrict", "contents": [ "Let $R \\to S$ be a ring map. The functors", "$\\text{Mod}_S \\to \\text{Mod}_R$, $N \\mapsto N_R$ (restriction)", "and $\\text{Mod}_R \\to \\text{Mod}_S$, $M \\mapsto \\Hom_R(S, M)$", "are adjoint functors. In a formula", "$$", "\\Hom_R(N_R, M) = \\Hom_S(N, \\Hom_R(S, M))", "$$" ], "refs": [], "proofs": [ { "contents": [ "If $\\alpha : N_R \\to M$ is an $R$-module map, then we define", "$\\alpha' : N \\to \\Hom_R(S, M)$ by the rule", "$\\alpha'(n) = (s \\mapsto \\alpha(sn))$. If $\\beta : N \\to \\Hom_R(S, M)$", "is an $S$-module map, we define $\\beta' : N_R \\to M$ by the rule", "$\\beta'(n) = \\beta(n)(1)$.", "We omit the verification that these constructions are mutually inverse." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 376, "type": "theorem", "label": "algebra-lemma-hom-from-tensor-product-variant", "categories": [ "algebra" ], "title": "algebra-lemma-hom-from-tensor-product-variant", "contents": [ "Let $R \\to S$ be a ring map. Given $S$-modules $M, N$ and an $R$-module $P$", "we have", "$$", "\\Hom_R(M \\otimes_S N, P) = \\Hom_S(M, \\Hom_R(N, P))", "$$" ], "refs": [], "proofs": [ { "contents": [ "This can be proved directly, but it is also a consequence of", "Lemmas \\ref{lemma-adjoint-hom-restrict} and \\ref{lemma-hom-from-tensor-product}.", "Namely, we have", "\\begin{align*}", "\\Hom_R(M \\otimes_S N, P)", "& =", "\\Hom_S(M \\otimes_S N, \\Hom_R(S, P)) \\\\", "& =", "\\Hom_S(M, \\Hom_S(N, \\Hom_R(S, P))) \\\\", "& =", "\\Hom_S(M, \\Hom_R(N, P))", "\\end{align*}", "as desired." ], "refs": [ "algebra-lemma-adjoint-hom-restrict", "algebra-lemma-hom-from-tensor-product" ], "ref_ids": [ 375, 362 ] } ], "ref_ids": [] }, { "id": 377, "type": "theorem", "label": "algebra-lemma-product-ideals-in-prime", "categories": [ "algebra" ], "title": "algebra-lemma-product-ideals-in-prime", "contents": [ "Let $R$ be a ring, $I$ and $J$ two ideals and $\\mathfrak p$ a prime ideal", "containing the product $IJ$. Then $\\mathfrak{p}$ contains $I$ or $J$." ], "refs": [], "proofs": [ { "contents": [ "Assume the contrary and take $x \\in I \\setminus \\mathfrak p$ and", "$y \\in J \\setminus \\mathfrak p$. Their product is an element of", "$IJ \\subset \\mathfrak p$, which contradicts the assumption that", "$\\mathfrak p$ was prime." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 378, "type": "theorem", "label": "algebra-lemma-silly", "categories": [ "algebra" ], "title": "algebra-lemma-silly", "contents": [ "\\begin{slogan}", "1. In an affine scheme if a finite number of points are contained in an", "open subset then they are contained in a smaller principal open subset.", "2. Affine opens are cofinal among the neighborhoods of a given finite set", "of an affine scheme", "\\end{slogan}", "Let $R$ be a ring. Let $I_i \\subset R$, $i = 1, \\ldots, r$,", "and $J \\subset R$ be ideals. Assume", "\\begin{enumerate}", "\\item $J \\not\\subset I_i$ for $i = 1, \\ldots, r$, and", "\\item all but two of $I_i$ are prime ideals.", "\\end{enumerate}", "Then there exists an $x \\in J$, $x\\not\\in I_i$ for all $i$." ], "refs": [], "proofs": [ { "contents": [ "The result is true for $r = 1$. If $r = 2$, then let $x, y \\in J$ with", "$x \\not \\in I_1$ and $y \\not \\in I_2$. We are done unless $x \\in I_2$", "and $y \\in I_1$. Then the element $x + y$ cannot be in $I_1$ (since that", "would mean $x + y - y \\in I_1$) and it also cannot be in $I_2$.", "\\medskip\\noindent", "For $r \\geq 3$, assume the result holds for $r - 1$. After renumbering", "we may assume that $I_r$ is prime. We may also assume there are no", "inclusions among the $I_i$. Pick $x \\in J$, $x \\not \\in I_i$ for all", "$i = 1, \\ldots, r - 1$. If $x \\not\\in I_r$ we are done. So assume", "$x \\in I_r$. If $J I_1 \\ldots I_{r - 1} \\subset I_r$ then", "$J \\subset I_r$ (by Lemma \\ref{lemma-product-ideals-in-prime}) a contradiction.", "Pick $y \\in J I_1 \\ldots I_{r - 1}$, $y \\not \\in I_r$. Then $x + y$ works." ], "refs": [ "algebra-lemma-product-ideals-in-prime" ], "ref_ids": [ 377 ] } ], "ref_ids": [] }, { "id": 379, "type": "theorem", "label": "algebra-lemma-silly-silly", "categories": [ "algebra" ], "title": "algebra-lemma-silly-silly", "contents": [ "Let $R$ be a ring. Let $x \\in R$, $I \\subset R$ an ideal, and", "$\\mathfrak p_i$, $i = 1, \\ldots, r$ be prime ideals.", "Suppose that $x + I \\not \\subset \\mathfrak p_i$ for", "$i = 1, \\ldots, r$. Then there exists an $y \\in I$", "such that $x + y \\not \\in \\mathfrak p_i$ for all $i$." ], "refs": [], "proofs": [ { "contents": [ "We may assume there are no inclusions among the $\\mathfrak p_i$.", "After reordering we may assume $x \\not \\in \\mathfrak p_i$ for $i < s$", "and $x \\in \\mathfrak p_i$ for $i \\geq s$. If $s = r + 1$ then we are done.", "If not, then we can find $y \\in I$ with $y \\not \\in \\mathfrak p_s$.", "Choose $f \\in \\bigcap_{i < s} \\mathfrak p_i$ with $f \\not \\in \\mathfrak p_s$.", "Then $x + fy$ is not contained in $\\mathfrak p_1, \\ldots, \\mathfrak p_s$.", "Thus we win by induction on $s$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 380, "type": "theorem", "label": "algebra-lemma-chinese-remainder", "categories": [ "algebra" ], "title": "algebra-lemma-chinese-remainder", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item If $I_1, \\ldots, I_r$ are ideals such that $I_a + I_b = R$", "when $a \\not = b$, then $I_1 \\cap \\ldots \\cap I_r =", "I_1I_2\\ldots I_r$ and $R/(I_1I_2\\ldots I_r)", "\\cong R/I_1 \\times \\ldots \\times R/I_r$.", "\\item If $\\mathfrak m_1, \\ldots, \\mathfrak m_r$ are pairwise distinct maximal", "ideals then $\\mathfrak m_a + \\mathfrak m_b = R$ for $a \\not = b$ and the", "above applies.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us first prove $I_1 \\cap \\ldots \\cap I_r = I_1 \\ldots I_r$", "as this will also imply the injectivity of the induced ring", "homomorphism $R/(I_1 \\ldots I_r) \\rightarrow R/I_1 \\times \\ldots \\times R/I_r$.", "The inclusion $I_1 \\cap \\ldots \\cap I_r \\supset I_1 \\ldots I_r$ is always", "fulfilled since ideals are closed under multiplication with arbitrary ring", "elements. To prove the other inclusion, we claim that the ideals", "$$", "I_1 \\ldots \\hat I_i \\ldots I_r,\\quad i = 1, \\ldots, r", "$$", "generate the ring $R$. We prove this by induction on $r$. It holds when", "$r = 2$. If $r > 2$, then we see that $R$ is the sum of the ideals", "$I_1 \\ldots \\hat I_i \\ldots I_{r - 1}$, $i = 1, \\ldots, r - 1$.", "Hence $I_r$ is the sum of the ideals", "$I_1 \\ldots \\hat I_i \\ldots I_r$, $i = 1, \\ldots, r - 1$.", "Applying the same argument with the reverse ordering on the ideals", "we see that $I_1$ is the sum of the ideals", "$I_1 \\ldots \\hat I_i \\ldots I_r$, $i = 2, \\ldots, r$.", "Since $R = I_1 + I_r$ by assumption we see that $R$ is the sum of the", "ideals displayed above. Therefore we can find elements", "$a_i \\in I_1 \\ldots \\hat I_i \\ldots I_r$", "such that their sum is one. Multiplying this equation by an element", "of $I_1 \\cap \\ldots \\cap I_r$ gives the other inclusion.", "It remains to show that the canonical map", "$R/(I_1 \\ldots I_r) \\rightarrow R/I_1 \\times \\ldots \\times R/I_r$", "is surjective. For this, consider its action on the equation", "$1 = \\sum_{i=1}^r a_i$ we derived above. On the one hand, a", "ring morphism sends 1 to 1 and on the other hand, the image of any", "$a_i$ is zero in $R/I_j$ for $j \\neq i$. Therefore, the image of $a_i$", "in $R/I_i$ is the identity. So given any element", "$(\\bar{b_1}, \\ldots, \\bar{b_r}) \\in R/I_1 \\times \\ldots \\times R/I_r$,", "the element $\\sum_{i=1}^r a_i \\cdot b_i$ is an inverse image in $R$.", "\\medskip\\noindent", "To see (2), by the very definition of being distinct maximal ideals, we have", "$\\mathfrak{m}_a + \\mathfrak{m}_b = R$ for $a \\neq b$ and so the above applies." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 381, "type": "theorem", "label": "algebra-lemma-matrix-left-inverse", "categories": [ "algebra" ], "title": "algebra-lemma-matrix-left-inverse", "contents": [ "Let $R$ be a ring. Let $n \\geq m$. Let $A$ be an", "$n \\times m$ matrix with coefficients in $R$. Let $J \\subset R$", "be the ideal generated by the $m \\times m$ minors of $A$.", "\\begin{enumerate}", "\\item For any $f \\in J$ there exists a $m \\times n$ matrix $B$", "such that $BA = f 1_{m \\times m}$.", "\\item If $f \\in R$ and $BA = f 1_{m \\times m}$ for some $m \\times n$ matrix", "$B$, then $f^m \\in J$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For $I \\subset \\{1, \\ldots, n\\}$ with $|I| = m$, we denote", "by $E_I$ the $m \\times n$ matrix of the projection", "$$", "R^{\\oplus n} = \\bigoplus\\nolimits_{i \\in \\{1, \\ldots, n\\}} R", "\\longrightarrow \\bigoplus\\nolimits_{i \\in I} R", "$$", "and set $A_I = E_I A$, i.e., $A_I$ is the $m \\times m$ matrix", "whose rows are the rows of $A$ with indices in $I$.", "Let $B_I$ be the adjugate (transpose of", "cofactor) matrix to $A_I$, i.e., such that", "$A_I B_I = B_I A_I = \\det(A_I) 1_{m \\times m}$.", "The $m \\times m$ minors of $A$ are the determinants $\\det A_I$", "for all the $I \\subset \\{1, \\ldots, n\\}$ with $|I| = m$.", "If $f \\in J$ then we can write $f = \\sum c_I \\det(A_I)$ for some", "$c_I \\in R$. Set $B = \\sum c_I B_I E_I$ to see that (1) holds.", "\\medskip\\noindent", "If $f 1_{m \\times m} = BA$ then by the", "Cauchy-Binet formula (\\ref{item-cauchy-binet}) we", "have $f^m = \\sum b_I \\det(A_I)$ where $b_I$ is the determinant", "of the $m \\times m$ matrix whose columns are the columns of $B$ with", "indices in $I$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 382, "type": "theorem", "label": "algebra-lemma-matrix-right-inverse", "categories": [ "algebra" ], "title": "algebra-lemma-matrix-right-inverse", "contents": [ "Let $R$ be a ring. Let $n \\geq m$. Let $A = (a_{ij})$ be an", "$n \\times m$ matrix with coefficients in $R$, written in block form", "as", "$$", "A =", "\\left(", "\\begin{matrix}", "A_1 \\\\", "A_2", "\\end{matrix}", "\\right)", "$$", "where $A_1$ has size $m \\times m$. Let $B$ be the adjugate (transpose of", "cofactor) matrix to $A_1$. Then", "$$", "AB = ", "\\left(", "\\begin{matrix}", "f 1_{m \\times m} \\\\", "C", "\\end{matrix}", "\\right)", "$$", "where $f = \\det(A_1)$ and $c_{ij}$ is (up to sign) the determinant of the", "$m \\times m$ minor of $A$ corresponding to the rows", "$1, \\ldots, \\hat j, \\ldots, m, i$." ], "refs": [], "proofs": [ { "contents": [ "Since the adjugate has the property $A_1B = B A_1 = f$ the first block", "of the expression for $AB$ is correct. Note that", "$$", "c_{ij} = \\sum\\nolimits_k a_{ik}b_{kj} = \\sum (-1)^{j + k}a_{ik} \\det(A_1^{jk})", "$$", "where $A_1^{ij}$ means $A_1$ with the $j$th row and $k$th column removed.", "This last expression is the row expansion of the determinant of the matrix", "in the statement of the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 383, "type": "theorem", "label": "algebra-lemma-map-cannot-be-injective", "categories": [ "algebra" ], "title": "algebra-lemma-map-cannot-be-injective", "contents": [ "\\begin{slogan}", "A map of finite free modules cannot be injective if the source has", "rank bigger than the target.", "\\end{slogan}", "Let $R$ be a nonzero ring. Let $n \\geq 1$. Let $M$ be an $R$-module generated", "by $< n$ elements. Then any $R$-module map $f : R^{\\oplus n} \\to M$ has a", "nonzero kernel." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjection $R^{\\oplus n - 1} \\to M$.", "We may lift the map $f$ to a map $f' : R^{\\oplus n} \\to R^{\\oplus n - 1}$", "(Lemma \\ref{lemma-lift-map}).", "It suffices to prove $f'$ has a nonzero kernel.", "The map $f' : R^{\\oplus n} \\to R^{\\oplus n - 1}$ is given by a", "matrix $A = (a_{ij})$. If one of the $a_{ij}$ is not nilpotent, say", "$a = a_{ij}$ is not, then we can replace $R$ by the localization $R_a$", "and we may assume $a_{ij}$ is a unit. Since if we find a nonzero kernel", "after localization then there was a nonzero kernel to start with as", "localization is exact, see Proposition \\ref{proposition-localization-exact}.", "In this case we can do a base change on both $R^{\\oplus n}$", "and $R^{\\oplus n - 1}$ and reduce to the case where", "$$", "A =", "\\left(", "\\begin{matrix}", "1 & 0 & 0 & \\ldots \\\\", "0 & a_{22} & a_{23} & \\ldots \\\\", "0 & a_{32} & \\ldots \\\\", "\\ldots & \\ldots", "\\end{matrix}", "\\right)", "$$", "Hence in this case we win by induction on $n$. If not then each", "$a_{ij}$ is nilpotent. Set $I = (a_{ij}) \\subset R$. Note that", "$I^{m + 1} = 0$ for some $m \\geq 0$. Let $m$ be the largest integer", "such that $I^m \\not = 0$. Then we see that $(I^m)^{\\oplus n}$ is", "contained in the kernel of the map and we win." ], "refs": [ "algebra-lemma-lift-map", "algebra-proposition-localization-exact" ], "ref_ids": [ 329, 1402 ] } ], "ref_ids": [] }, { "id": 384, "type": "theorem", "label": "algebra-lemma-rank", "categories": [ "algebra" ], "title": "algebra-lemma-rank", "contents": [ "\\begin{slogan}", "The rank of a finite free module is well defined.", "\\end{slogan}", "Let $R$ be a nonzero ring. Let $n, m \\geq 0$ be integers.", "If $R^{\\oplus n}$ is isomorphic to $R^{\\oplus m}$ as", "$R$-modules, then $n = m$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-map-cannot-be-injective}." ], "refs": [ "algebra-lemma-map-cannot-be-injective" ], "ref_ids": [ 383 ] } ], "ref_ids": [] }, { "id": 385, "type": "theorem", "label": "algebra-lemma-charpoly", "categories": [ "algebra" ], "title": "algebra-lemma-charpoly", "contents": [ "Let $R$ be a ring. Let $A = (a_{ij})$ be an $n \\times n$", "matrix with coefficients in $R$. Let $P(x) \\in R[x]$", "be the characteristic polynomial of $A$ (defined", "as $\\det(x\\text{id}_{n \\times n} - A)$).", "Then $P(A) = 0$ in $\\text{Mat}(n \\times n, R)$." ], "refs": [], "proofs": [ { "contents": [ "We reduce the question to the well-known Cayley-Hamilton", "theorem from linear algebra in several steps:", "\\begin{enumerate}", "\\item If $\\phi :S \\rightarrow R$ is a ring morphism and $b_{ij}$", "are inverse images of the $a_{ij}$ under this map, then it suffices", "to show the statement for $S$ and $(b_{ij})$ since $\\phi$ is a ring morphism.", "\\item If $\\psi :R \\hookrightarrow S$ is an injective ring morphism, it", "clearly suffices to show the result for $S$ and the $a_{ij}$ considered as", "elements of $S$. ", "\\item Thus we may first reduce to the case $R = \\mathbf{Z}[X_{ij}]$,", "$a_{ij} = X_{ij}$ of a polynomial ring and then further to", "the case $R = \\mathbf{Q}(X_{ij})$ where we may finally apply Cayley-Hamilton.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 386, "type": "theorem", "label": "algebra-lemma-charpoly-module", "categories": [ "algebra" ], "title": "algebra-lemma-charpoly-module", "contents": [ "Let $R$ be a ring.", "Let $M$ be a finite $R$-module.", "Let $\\varphi : M \\to M$ be an endomorphism.", "Then there exists a monic polynomial $P \\in R[T]$ such that", "$P(\\varphi) = 0$ as an endomorphism of $M$." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjective $R$-module map $R^{\\oplus n} \\to M$, given by", "$(a_1, \\ldots, a_n) \\mapsto \\sum a_ix_i$ for some generators $x_i \\in M$.", "Choose $(a_{i1}, \\ldots, a_{in}) \\in R^{\\oplus n}$ such that", "$\\varphi(x_i) = \\sum a_{ij} x_j$. In other words the diagram", "$$", "\\xymatrix{", "R^{\\oplus n} \\ar[d]_A \\ar[r] & M \\ar[d]^\\varphi \\\\", "R^{\\oplus n} \\ar[r] & M", "}", "$$", "is commutative where $A = (a_{ij})$. By", "Lemma \\ref{lemma-charpoly}", "there exists a monic polynomial $P$ such that $P(A) = 0$.", "Then it follows that $P(\\varphi) = 0$." ], "refs": [ "algebra-lemma-charpoly" ], "ref_ids": [ 385 ] } ], "ref_ids": [] }, { "id": 387, "type": "theorem", "label": "algebra-lemma-charpoly-module-ideal", "categories": [ "algebra" ], "title": "algebra-lemma-charpoly-module-ideal", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal.", "Let $M$ be a finite $R$-module.", "Let $\\varphi : M \\to M$ be an endomorphism such", "that $\\varphi(M) \\subset IM$.", "Then there exists a monic polynomial", "$P = t^n + a_1 t^{n - 1} + \\ldots + a_n \\in R[T]$", "such that $a_j \\in I^j$ and $P(\\varphi) = 0$ as an endomorphism of $M$." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjective $R$-module map $R^{\\oplus n} \\to M$, given by", "$(a_1, \\ldots, a_n) \\mapsto \\sum a_ix_i$ for some generators $x_i \\in M$.", "Choose $(a_{i1}, \\ldots, a_{in}) \\in I^{\\oplus n}$ such that", "$\\varphi(x_i) = \\sum a_{ij} x_j$. In other words the diagram", "$$", "\\xymatrix{", "R^{\\oplus n} \\ar[d]_A \\ar[r] & M \\ar[d]^\\varphi \\\\", "I^{\\oplus n} \\ar[r] & M", "}", "$$", "is commutative where $A = (a_{ij})$. By", "Lemma \\ref{lemma-charpoly}", "the polynomial", "$P(t) = \\det(t\\text{id}_{n \\times n} - A)$", "has all the desired properties." ], "refs": [ "algebra-lemma-charpoly" ], "ref_ids": [ 385 ] } ], "ref_ids": [] }, { "id": 388, "type": "theorem", "label": "algebra-lemma-fun", "categories": [ "algebra" ], "title": "algebra-lemma-fun", "contents": [ "Let $R$ be a ring.", "Let $M$ be a finite $R$-module.", "Let $\\varphi : M \\to M$ be a surjective $R$-module map.", "Then $\\varphi$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "Write $R' = R[x]$ and think of $M$ as a finite $R'$-module with", "$x$ acting via $\\varphi$. Set $I = (x) \\subset R'$. By our assumption that", "$\\varphi$ is surjective we have $IM = M$. Hence we may apply", "Lemma \\ref{lemma-charpoly-module-ideal}", "to $M$ as an $R'$-module, the ideal $I$ and the endomorphism $\\text{id}_M$.", "We conclude that", "$(1 + a_1 + \\ldots + a_n)\\text{id}_M = 0$ with $a_j \\in I$.", "Write $a_j = b_j(x)x$ for some $b_j(x) \\in R[x]$.", "Translating back into $\\varphi$ we see that", "$\\text{id}_M = -(\\sum_{j = 1, \\ldots, n} b_j(\\varphi)) \\varphi$, and hence", "$\\varphi$ is invertible." ], "refs": [ "algebra-lemma-charpoly-module-ideal" ], "ref_ids": [ 387 ] } ], "ref_ids": [] }, { "id": 389, "type": "theorem", "label": "algebra-lemma-Zariski-topology", "categories": [ "algebra" ], "title": "algebra-lemma-Zariski-topology", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item The spectrum of a ring $R$ is empty if and only if $R$", "is the zero ring.", "\\item Every nonzero ring has a maximal ideal.", "\\item Every nonzero ring has a minimal prime ideal.", "\\item Given an ideal $I \\subset R$ and a prime ideal", "$I \\subset \\mathfrak p$ there exists a prime", "$I \\subset \\mathfrak q \\subset \\mathfrak p$ such", "that $\\mathfrak q$ is minimal over $I$.", "\\item If $T \\subset R$, and if $(T)$ is the ideal generated by", "$T$ in $R$, then $V((T)) = V(T)$.", "\\item If $I$ is an ideal and $\\sqrt{I}$ is its radical,", "see basic notion (\\ref{item-radical-ideal}), then $V(I) = V(\\sqrt{I})$.", "\\item Given an ideal $I$ of $R$ we have $\\sqrt{I} =", "\\bigcap_{I \\subset \\mathfrak p} \\mathfrak p$.", "\\item If $I$ is an ideal then $V(I) = \\emptyset$ if and only", "if $I$ is the unit ideal.", "\\item If $I$, $J$ are ideals of $R$ then $V(I) \\cup V(J) =", "V(I \\cap J)$.", "\\item If $(I_a)_{a\\in A}$ is a set of ideals of $R$ then", "$\\cap_{a\\in A} V(I_a) = V(\\cup_{a\\in A} I_a)$.", "\\item If $f \\in R$, then $D(f) \\amalg V(f) = \\Spec(R)$.", "\\item If $f \\in R$ then $D(f) = \\emptyset$ if and only if $f$", "is nilpotent.", "\\item If $f = u f'$ for some unit $u \\in R$, then $D(f) = D(f')$.", "\\item If $I \\subset R$ is an ideal, and $\\mathfrak p$ is a prime of", "$R$ with $\\mathfrak p \\not\\in V(I)$, then there exists an $f \\in R$", "such that $\\mathfrak p \\in D(f)$, and $D(f) \\cap V(I) = \\emptyset$.", "\\item If $f, g \\in R$, then $D(fg) = D(f) \\cap D(g)$.", "\\item If $f_i \\in R$ for $i \\in I$, then", "$\\bigcup_{i\\in I} D(f_i)$ is the complement of $V(\\{f_i \\}_{i\\in I})$", "in $\\Spec(R)$.", "\\item If $f \\in R$ and $D(f) = \\Spec(R)$, then $f$ is a unit.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We address each part in the corresponding item below.", "\\begin{enumerate}", "\\item This is a direct consequence of (2) or (3).", "\\item Let $\\mathfrak{A}$ be the set of all proper ideals of $R$. This set is", "ordered by inclusion and is non-empty, since $(0) \\in \\mathfrak{A}$ is a proper", "ideal. Let $A$ be a totally ordered subset of $\\mathfrak A$.", "Then $\\bigcup_{I \\in A} I$ is in", "fact an ideal. Since 1 $\\notin I$ for all $I \\in A$, the union does not contain", "1 and thus is proper. Hence $\\bigcup_{I \\in A} I$ is in $\\mathfrak{A}$ and is", "an upper bound for the set $A$. Thus by Zorn's lemma $\\mathfrak{A}$ has a", "maximal element, which is the sought-after maximal ideal.", "\\item Since $R$ is nonzero, it contains a maximal ideal which is a prime ideal.", "Thus the set $\\mathfrak{A}$ of all prime ideals of $R$ is nonempty.", "$\\mathfrak{A}$ is ordered by reverse-inclusion. Let $A$ be a totally ordered", "subset of $\\mathfrak{A}$. It's pretty clear that $J = \\bigcap_{I \\in A} I$ is", "in fact an ideal. Not so clear, however, is that it is prime. Let $xy \\in J$.", "Then $xy \\in I$ for all $I \\in A$. Now let $B = \\{I \\in A | y \\in I\\}$. Let $K", "= \\bigcap_{I \\in B} I$. Since $A$ is totally ordered, either $K = J$ (and we're", "done, since then $y \\in J$) or $K \\supset J$ and for all $I \\in A$ such that", "$I$ is properly contained in $K$, we have $y \\notin I$. But that means that for", "all those $I, x \\in I$, since they are prime. Hence $x \\in J$. In either case,", "$J$ is prime as desired. Hence by Zorn's lemma we get a maximal element which", "in this case is a minimal prime ideal.", "\\item This is the same exact argument as (3) except you only consider prime", "ideals contained in $\\mathfrak{p}$ and containing $I$.", "\\item $(T)$ is the smallest ideal containing $T$. Hence if $T \\subset I$, some", "ideal, then $(T) \\subset I$ as well. Hence if $I \\in V(T)$, then $I \\in V((T))$", "as well. The other inclusion is obvious.", "\\item Since $I \\subset \\sqrt{I}, V(\\sqrt{I}) \\subset V(I)$. Now let", "$\\mathfrak{p} \\in V(I)$. Let $x \\in \\sqrt{I}$. Then $x^n \\in I$ for some $n$.", "Hence $x^n \\in \\mathfrak{p}$. But since $\\mathfrak{p}$ is prime, a boring", "induction argument gets you that $x \\in \\mathfrak{p}$. Hence $\\sqrt{I} \\subset", "\\mathfrak{p}$ and $\\mathfrak{p} \\in V(\\sqrt{I})$.", "\\item Let $f \\in R \\setminus \\sqrt{I}$. Then $f^n \\notin I$ for all $n$. Hence", "$S = \\{1, f, f^2, \\ldots\\}$ is a multiplicative subset, not containing $0$.", "Take a", "prime ideal $\\bar{\\mathfrak{p}} \\subset S^{-1}R$ containing $S^{-1}I$. Then the", "pull-back $\\mathfrak{p}$ in $R$ of $\\bar{\\mathfrak{p}}$ is a prime ideal", "containing $I$ that does not intersect $S$. This shows that $\\bigcap_{I \\subset", "\\mathfrak p} \\mathfrak p \\subset \\sqrt{I}$. Now if $a \\in \\sqrt{I}$, then $a^n", "\\in I$ for some $n$. Hence if $I \\subset \\mathfrak{p}$, then $a^n \\in", "\\mathfrak{p}$. But since $\\mathfrak{p}$ is prime, we have $a \\in \\mathfrak{p}$.", "Thus the equality is shown.", "\\item $I$ is not the unit ideal if and only if $I$", "is contained in some maximal ideal (to", "see this, apply (2) to the ring $R/I$) which is therefore prime.", "\\item If $\\mathfrak{p} \\in V(I) \\cup V(J)$, then $I \\subset \\mathfrak{p}$ or $J", "\\subset \\mathfrak{p}$ which means that $I \\cap J \\subset \\mathfrak{p}$. Now if", "$I \\cap J \\subset \\mathfrak{p}$, then $IJ \\subset \\mathfrak{p}$ and hence", "either $I$ of $J$ is in $\\mathfrak{p}$, since $\\mathfrak{p}$ is prime.", "\\item $\\mathfrak{p} \\in \\bigcap_{a \\in A} V(I_a) \\Leftrightarrow I_a \\subset", "\\mathfrak{p}, \\forall a \\in A \\Leftrightarrow \\mathfrak{p} \\in V(\\cup_{a\\in A}", "I_a)$", "\\item If $\\mathfrak{p}$ is a prime ideal and $f \\in R$, then either $f \\in", "\\mathfrak{p}$ or $f \\notin \\mathfrak{p}$ (strictly) which is what the disjoint", "union says.", "\\item If $a \\in R$ is nilpotent, then $a^n = 0$ for some $n$. Hence $a^n \\in", "\\mathfrak{p}$ for any prime ideal. Thus $a \\in \\mathfrak{p}$ as can be shown by", "induction and $D(f) = \\emptyset$. Now, as shown in (7), if $a \\in R$ is not", "nilpotent, then there is a prime ideal that does not contain it.", "\\item $f \\in \\mathfrak{p} \\Leftrightarrow uf \\in \\mathfrak{p}$, since $u$ is", "invertible.", "\\item If $\\mathfrak{p} \\notin V(I)$, then $\\exists f \\in I \\setminus", "\\mathfrak{p}$. Then $f \\notin \\mathfrak{p}$ so $\\mathfrak{p} \\in D(f)$. Also if", "$\\mathfrak{q} \\in D(f)$, then $f \\notin \\mathfrak{q}$ and thus $I$ is not", "contained in $\\mathfrak{q}$. Thus $D(f) \\cap V(I) = \\emptyset$.", "\\item If $fg \\in \\mathfrak{p}$, then $f \\in \\mathfrak{p}$ or $g \\in", "\\mathfrak{p}$. Hence if $f \\notin \\mathfrak{p}$ and $g \\notin \\mathfrak{p}$,", "then $fg \\notin \\mathfrak{p}$. Since $\\mathfrak{p}$ is an ideal, if $fg \\notin", "\\mathfrak{p}$, then $f \\notin \\mathfrak{p}$ and $g \\notin \\mathfrak{p}$.", "\\item $\\mathfrak{p} \\in \\bigcup_{i \\in I} D(f_i) \\Leftrightarrow \\exists i \\in", "I, f_i \\notin \\mathfrak{p} \\Leftrightarrow \\mathfrak{p} \\in \\Spec(R)", "\\setminus V(\\{f_i\\}_{i \\in I})$", "\\item If $D(f) = \\Spec(R)$, then $V(f) = \\emptyset$ and", "hence $fR = R$, so $f$ is a unit.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 390, "type": "theorem", "label": "algebra-lemma-spec-functorial", "categories": [ "algebra" ], "title": "algebra-lemma-spec-functorial", "contents": [ "\\begin{slogan}", "Functoriality of the spectrum", "\\end{slogan}", "Suppose that $\\varphi : R \\to R'$ is a ring homomorphism.", "The induced map", "$$", "\\Spec(\\varphi) : \\Spec(R') \\longrightarrow \\Spec(R),", "\\quad", "\\mathfrak p' \\longmapsto \\varphi^{-1}(\\mathfrak p')", "$$", "is continuous for the Zariski topologies. In fact, for any", "element $f \\in R$ we have", "$\\Spec(\\varphi)^{-1}(D(f)) = D(\\varphi(f))$." ], "refs": [], "proofs": [ { "contents": [ "It is basic notion (\\ref{item-inverse-image-prime}) that", "$\\mathfrak p := \\varphi^{-1}(\\mathfrak p')$", "is indeed a prime ideal of $R$. The last assertion", "of the lemma follows directly from the definitions,", "and implies the first." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 391, "type": "theorem", "label": "algebra-lemma-spec-localization", "categories": [ "algebra" ], "title": "algebra-lemma-spec-localization", "contents": [ "Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset.", "The map $R \\to S^{-1}R$ induces via the functoriality of $\\Spec$", "a homeomorphism", "$$", "\\Spec(S^{-1}R)", "\\longrightarrow", "\\{\\mathfrak p \\in \\Spec(R) \\mid S \\cap \\mathfrak p = \\emptyset \\}", "$$", "where the topology on the right hand side is that induced from the", "Zariski topology on $\\Spec(R)$. The inverse map is given", "by $\\mathfrak p \\mapsto S^{-1}\\mathfrak p$." ], "refs": [], "proofs": [ { "contents": [ "Denote the right hand side of the arrow of the lemma by $D$.", "Choose a prime $\\mathfrak p' \\subset S^{-1}R$ and let $\\mathfrak p$", "the inverse image of $\\mathfrak p'$ in $R$. Since $\\mathfrak p'$", "does not contain $1$ we see that $\\mathfrak p$ does not contain", "any element of $S$. Hence $\\mathfrak p \\in D$ and we see that", "the image is contained in $D$. Let $\\mathfrak p \\in D$.", "By assumption the image $\\overline{S}$ does not contain $0$.", "By basic notion (\\ref{item-localization-zero})", "$\\overline{S}^{-1}(R/\\mathfrak p)$ is not the zero ring.", "By basic notion (\\ref{item-localize-ideal}) we see", "$S^{-1}R / S^{-1}\\mathfrak p = \\overline{S}^{-1}(R/\\mathfrak p)$", "is a domain, and hence $S^{-1}\\mathfrak p$ is a prime.", "The equality of rings also shows that the inverse image of", "$S^{-1}\\mathfrak p$ in $R$ is equal to $\\mathfrak p$,", "because $R/\\mathfrak p \\to \\overline{S}^{-1}(R/\\mathfrak p)$", "is injective by basic notion (\\ref{item-localize-nonzerodivisors}).", "This proves that the map $\\Spec(S^{-1}R) \\to \\Spec(R)$", "is bijective onto $D$ with inverse as given.", "It is continuous by Lemma \\ref{lemma-spec-functorial}.", "Finally, let $D(g) \\subset \\Spec(S^{-1}R)$ be a standard", "open. Write $g = h/s$ for some $h\\in R$ and $s\\in S$.", "Since $g$ and $h/1$ differ by a unit we have $D(g) =", "D(h/1)$ in $\\Spec(S^{-1}R)$.", "Hence by Lemma \\ref{lemma-spec-functorial} and the bijectivity", "above the image of $D(g) = D(h/1)$ is $D \\cap D(h)$.", "This proves the map is open as well." ], "refs": [ "algebra-lemma-spec-functorial", "algebra-lemma-spec-functorial" ], "ref_ids": [ 390, 390 ] } ], "ref_ids": [] }, { "id": 392, "type": "theorem", "label": "algebra-lemma-standard-open", "categories": [ "algebra" ], "title": "algebra-lemma-standard-open", "contents": [ "Let $R$ be a ring. Let $f \\in R$.", "The map $R \\to R_f$ induces via the functoriality of", "$\\Spec$ a homeomorphism", "$$", "\\Spec(R_f) \\longrightarrow D(f) \\subset \\Spec(R).", "$$", "The inverse is given by $\\mathfrak p \\mapsto \\mathfrak p \\cdot R_f$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-spec-localization}." ], "refs": [ "algebra-lemma-spec-localization" ], "ref_ids": [ 391 ] } ], "ref_ids": [] }, { "id": 393, "type": "theorem", "label": "algebra-lemma-spec-closed", "categories": [ "algebra" ], "title": "algebra-lemma-spec-closed", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal.", "The map $R \\to R/I$ induces via the functoriality of", "$\\Spec$ a homeomorphism", "$$", "\\Spec(R/I) \\longrightarrow V(I) \\subset \\Spec(R).", "$$", "The inverse is given by $\\mathfrak p \\mapsto \\mathfrak p / I$." ], "refs": [], "proofs": [ { "contents": [ "It is immediate that the image is contained in $V(I)$.", "On the other hand, if $\\mathfrak p \\in V(I)$", "then $\\mathfrak p \\supset I$ and we may consider", "the ideal $\\mathfrak p /I \\subset R/I$. Using", "basic notion (\\ref{item-isomorphism-theorem}) we see that", "$(R/I)/(\\mathfrak p/I) = R/\\mathfrak p$ is a domain", "and hence $\\mathfrak p/I$ is a prime ideal. From this", "it is immediately clear that the image of $D(f + I)$", "is $D(f) \\cap V(I)$, and hence the map is a homeomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 394, "type": "theorem", "label": "algebra-lemma-in-image", "categories": [ "algebra" ], "title": "algebra-lemma-in-image", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Let $\\mathfrak p$", "be a prime of $R$. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathfrak p$ is in the image of", "$\\Spec(S) \\to \\Spec(R)$,", "\\item $S \\otimes_R \\kappa(\\mathfrak p) \\not = 0$,", "\\item $S_{\\mathfrak p}/\\mathfrak p S_{\\mathfrak p} \\not = 0$,", "\\item $(S/\\mathfrak pS)_{\\mathfrak p} \\not = 0$, and", "\\item $\\mathfrak p = \\varphi^{-1}(\\mathfrak pS)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have already seen the equivalence of the first two", "in Remark \\ref{remark-fundamental-diagram}. The others", "are just reformulations of this." ], "refs": [ "algebra-remark-fundamental-diagram" ], "ref_ids": [ 1558 ] } ], "ref_ids": [] }, { "id": 395, "type": "theorem", "label": "algebra-lemma-quasi-compact", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-compact", "contents": [ "\\begin{slogan}", "The spectrum of a ring is quasi-compact", "\\end{slogan}", "Let $R$ be a ring. The space $\\Spec(R)$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove that any covering of $\\Spec(R)$", "by standard opens can be refined by a finite covering.", "Thus suppose that $\\Spec(R) = \\cup D(f_i)$", "for a set of elements $\\{f_i\\}_{i\\in I}$ of $R$. This means that", "$\\cap V(f_i) = \\emptyset$. According to Lemma", "\\ref{lemma-Zariski-topology} this means that", "$V(\\{f_i \\}) = \\emptyset$. According to the", "same lemma this means that the ideal generated", "by the $f_i$ is the unit ideal of $R$. This means", "that we can write $1$ as a {\\it finite} sum:", "$1 = \\sum_{i \\in J} r_i f_i$ with $J \\subset I$ finite.", "And then it follows that $\\Spec(R)", "= \\cup_{i \\in J} D(f_i)$." ], "refs": [ "algebra-lemma-Zariski-topology" ], "ref_ids": [ 389 ] } ], "ref_ids": [] }, { "id": 396, "type": "theorem", "label": "algebra-lemma-topology-spec", "categories": [ "algebra" ], "title": "algebra-lemma-topology-spec", "contents": [ "Let $R$ be a ring.", "The topology on $X = \\Spec(R)$ has the following properties:", "\\begin{enumerate}", "\\item $X$ is quasi-compact,", "\\item $X$ has a basis for the topology consisting of quasi-compact opens, and", "\\item the intersection of any two quasi-compact opens is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The spectrum of a ring is quasi-compact, see", "Lemma \\ref{lemma-quasi-compact}.", "It has a basis for the topology consisting of the standard opens", "$D(f) = \\Spec(R_f)$", "(Lemma \\ref{lemma-standard-open})", "which are quasi-compact by the first remark.", "The intersection of two standard opens is quasi-compact", "as $D(f) \\cap D(g) = D(fg)$. Given any two quasi-compact opens", "$U, V \\subset X$ we may write $U = D(f_1) \\cup \\ldots \\cup D(f_n)$", "and $V = D(g_1) \\cup \\ldots \\cup D(g_m)$. Then", "$U \\cap V = \\bigcup D(f_ig_j)$ which is quasi-compact." ], "refs": [ "algebra-lemma-quasi-compact", "algebra-lemma-standard-open" ], "ref_ids": [ 395, 392 ] } ], "ref_ids": [] }, { "id": 397, "type": "theorem", "label": "algebra-lemma-characterize-local-ring", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-local-ring", "contents": [ "Let $R$ be a ring. The following are equivalent:", "\\begin{enumerate}", "\\item $R$ is a local ring,", "\\item $\\Spec(R)$ has exactly one closed point,", "\\item $R$ has a maximal ideal $\\mathfrak m$", "and every element of $R \\setminus \\mathfrak m$", "is a unit, and", "\\item $R$ is not the zero ring and for every $x \\in R$ either $x$", "or $1 - x$ is invertible or both.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a ring, and $\\mathfrak m$ a maximal ideal.", "If $x \\in R \\setminus \\mathfrak m$, and $x$ is not a unit", "then there is a maximal ideal $\\mathfrak m'$ containing $x$.", "Hence $R$ has at least two maximal ideals. Conversely,", "if $\\mathfrak m'$ is another maximal ideal, then choose", "$x \\in \\mathfrak m'$, $x \\not \\in \\mathfrak m$. Clearly", "$x$ is not a unit. This proves the equivalence of (1) and (3).", "The equivalence (1) and (2) is tautological.", "If $R$ is local then (4) holds since $x$ is either in $\\mathfrak m$", "or not. If (4) holds, and $\\mathfrak m$, $\\mathfrak m'$ are distinct", "maximal ideals then we may choose $x \\in R$ such that", "$x \\bmod \\mathfrak m' = 0$ and $x \\bmod \\mathfrak m = 1$", "by the Chinese remainder theorem", "(Lemma \\ref{lemma-chinese-remainder}).", "This element $x$ is not invertible and neither is $1 - x$ which is", "a contradiction. Thus (4) and (1) are equivalent." ], "refs": [ "algebra-lemma-chinese-remainder" ], "ref_ids": [ 380 ] } ], "ref_ids": [] }, { "id": 398, "type": "theorem", "label": "algebra-lemma-characterize-local-ring-map", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-local-ring-map", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Assume $R$ and $S$ are local rings.", "The following are equivalent:", "\\begin{enumerate}", "\\item $\\varphi$ is a local ring map,", "\\item $\\varphi(\\mathfrak m_R) \\subset \\mathfrak m_S$, and", "\\item $\\varphi^{-1}(\\mathfrak m_S) = \\mathfrak m_R$.", "\\item For any $x \\in R$, if $\\varphi(x)$ is invertible in $S$, then $x$", "is invertible in $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Conditions (1) and (2) are equivalent by definition.", "If (3) holds then (2) holds. Conversely, if (2) holds, then", "$\\varphi^{-1}(\\mathfrak m_S)$ is a prime ideal containing", "the maximal ideal $\\mathfrak m_R$, hence", "$\\varphi^{-1}(\\mathfrak m_S) = \\mathfrak m_R$. Finally, (4) is the", "contrapositive of (2) by Lemma \\ref{lemma-characterize-local-ring}." ], "refs": [ "algebra-lemma-characterize-local-ring" ], "ref_ids": [ 397 ] } ], "ref_ids": [] }, { "id": 399, "type": "theorem", "label": "algebra-lemma-contained-in-radical", "categories": [ "algebra" ], "title": "algebra-lemma-contained-in-radical", "contents": [ "Let $R$ be a ring with Jacobson radical $\\text{rad}(R)$.", "Let $I \\subset R$ be an ideal. The following are", "equivalent", "\\begin{enumerate}", "\\item $I \\subset \\text{rad}(R)$, and", "\\item every element of $1 + I$ is a unit in $R$.", "\\end{enumerate}", "In this case every element of $R$ which maps to a unit of $R/I$ is a unit." ], "refs": [], "proofs": [ { "contents": [ "If $f \\in \\text{rad}(R)$, then $f \\in \\mathfrak m$ for all", "maximal ideals $\\mathfrak m$ of $R$. Hence $1 + f \\not \\in \\mathfrak m$", "for all maximal ideals $\\mathfrak m$ of $R$. Thus the closed", "subset $V(1 + f)$ of $\\Spec(R)$ is empty. This implies", "that $1 + f$ is a unit, see Lemma \\ref{lemma-Zariski-topology}.", "\\medskip\\noindent", "Conversely, assume that $1 + f$ is a unit for all $f \\in I$.", "If $\\mathfrak m$ is a maximal ideal and $I \\not \\subset \\mathfrak m$,", "then $I + \\mathfrak m = R$. Hence $1 = f + g$ for some $g \\in \\mathfrak m$", "and $f \\in I$. Then $g = 1 + (-f)$ is not a unit, contradiction.", "\\medskip\\noindent", "For the final statement let $f \\in R$ map to a unit in $R/I$.", "Then we can find $g \\in R$ mapping to the multiplicative inverse", "of $f \\bmod I$. Then $fg = 1 \\bmod I$. Hence $fg$ is a unit of $R$", "by (2) which implies that $f$ is a unit." ], "refs": [ "algebra-lemma-Zariski-topology" ], "ref_ids": [ 389 ] } ], "ref_ids": [] }, { "id": 400, "type": "theorem", "label": "algebra-lemma-surjective-on-spec-units", "categories": [ "algebra" ], "title": "algebra-lemma-surjective-on-spec-units", "contents": [ "Let $\\varphi : R \\to S$ be a ring map such that the induced map", "$\\Spec(S) \\to \\Spec(R)$ is surjective. Then an element $x \\in R$", "is a unit if and only if $\\varphi(x) \\in S$ is a unit." ], "refs": [], "proofs": [ { "contents": [ "If $x$ is a unit, then so is $\\varphi(x)$. Conversely, if $\\varphi(x)$", "is a unit, then $\\varphi(x) \\not \\in \\mathfrak q$ for all", "$\\mathfrak q \\in \\Spec(S)$. Hence", "$x \\not \\in \\varphi^{-1}(\\mathfrak q) = \\Spec(\\varphi)(\\mathfrak q)$", "for all $\\mathfrak q \\in \\Spec(S)$. Since $\\Spec(\\varphi)$ is surjective", "we conclude that $x$ is a unit by", "part (17) of Lemma \\ref{lemma-Zariski-topology}." ], "refs": [ "algebra-lemma-Zariski-topology" ], "ref_ids": [ 389 ] } ], "ref_ids": [] }, { "id": 401, "type": "theorem", "label": "algebra-lemma-NAK", "categories": [ "algebra" ], "title": "algebra-lemma-NAK", "contents": [ "\\begin{reference}", "\\cite[1.M Lemma (NAK) page 11]{MatCA}", "\\end{reference}", "\\begin{history}", "We quote from \\cite{MatCA}: ``This simple but", "important lemma is due to T.~Nakayama, G.~Azumaya and W.~Krull. Priority", "is obscure, and although it is usually called the Lemma of Nakayama, late", "Prof.~Nakayama did not like the name.''", "\\end{history}", "Let $R$ be a ring with Jacobson radical $\\text{rad}(R)$.", "Let $M$ be an $R$-module. Let $I \\subset R$", "be an ideal.", "\\begin{enumerate}", "\\item", "\\label{item-nakayama}", "If $IM = M$ and $M$ is finite, then there exists a $f \\in 1 + I$ such that", "$fM = 0$.", "\\item If $IM = M$, $M$ is finite, and $I \\subset \\text{rad}(R)$, then $M = 0$.", "\\item If $N, N' \\subset M$, $M = N + IN'$, and $N'$ is finite,", "then there exists a $f \\in 1 + I$ such that $fM \\subset N$ and $M_f = N_f$.", "\\item If $N, N' \\subset M$, $M = N + IN'$, $N'$ is finite, and", "$I \\subset \\text{rad}(R)$, then $M = N$.", "\\item If $N \\to M$ is a module map, $N/IN \\to M/IM$ is", "surjective, and $M$ is finite, then there exists a $f \\in 1 + I$", "such that $N_f \\to M_f$ is surjective.", "\\item If $N \\to M$ is a module map, $N/IN \\to M/IM$ is", "surjective, $M$ is finite, and $I \\subset \\text{rad}(R)$,", "then $N \\to M$ is surjective.", "\\item If $x_1, \\ldots, x_n \\in M$ generate $M/IM$ and $M$ is finite,", "then there exists an $f \\in 1 + I$ such that $x_1, \\ldots, x_n$", "generate $M_f$ over $R_f$.", "\\item If $x_1, \\ldots, x_n \\in M$ generate $M/IM$, $M$ is finite, and", "$I \\subset \\text{rad}(R)$, then $M$ is generated by $x_1, \\ldots, x_n$.", "\\item If $IM = M$, $I$ is nilpotent, then $M = 0$.", "\\item If $N, N' \\subset M$, $M = N + IN'$, and $I$ is nilpotent then $M = N$.", "\\item If $N \\to M$ is a module map, $I$ is nilpotent, and $N/IN \\to M/IM$", "is surjective, then $N \\to M$ is surjective.", "\\item If $\\{x_\\alpha\\}_{\\alpha \\in A}$ is a set of elements of $M$", "which generate $M/IM$ and $I$ is nilpotent, then $M$ is generated", "by the $x_\\alpha$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (\\ref{item-nakayama}). Choose generators $y_1, \\ldots, y_m$ of $M$", "over $R$. For each $i$ we can write $y_i = \\sum z_{ij} y_j$ with", "$z_{ij} \\in I$ (since $M = IM$).", "In other words $\\sum_j (\\delta_{ij} - z_{ij})y_j = 0$.", "Let $f$ be the determinant of the $m \\times m$ matrix", "$A = (\\delta_{ij} - z_{ij})$. Note that $f \\in 1 + I$", "(since the matrix $A$ is entrywise congruent to the", "$m \\times m$ identity matrix modulo $I$).", "By Lemma \\ref{lemma-matrix-left-inverse} (1),", "there exists an $m \\times m$", "matrix $B$ such that $BA = f 1_{m \\times m}$. Writing out we see that", "$\\sum_{i} b_{hi} a_{ij} = f \\delta_{hj}$ for all", "$h$ and $j$; hence, $\\sum_{i, j} b_{hi} a_{ij} y_j", "= \\sum_{j} f \\delta_{hj} y_j = f y_h$ for every $h$.", "In other words, $0 = f y_h$ for every $h$ (since each", "$i$ satisfies $\\sum_j a_{ij} y_j = 0$).", "This implies that $f$ annihilates $M$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-contained-in-radical} an element of $1 + \\text{rad}(R)$ is", "invertible element of $R$. Hence we see that (\\ref{item-nakayama}) implies", "(2). We obtain (3) by applying (1) to $M/N$ which is finite as $N'$ is finite.", "We obtain (4) by applying (2) to $M/N$ which is finite as $N'$ is finite.", "We obtain (5) by applying (3) to $M$ and the submodules $\\Im(N \\to M)$", "and $M$. We obtain (6) by applying (4) to $M$ and the submodules", "$\\Im(N \\to M)$ and $M$.", "We obtain (7) by applying (5) to the map $R^{\\oplus n} \\to M$,", "$(a_1, \\ldots, a_n) \\mapsto a_1x_1 + \\ldots + a_nx_n$.", "We obtain (8) by applying (6) to the map $R^{\\oplus n} \\to M$,", "$(a_1, \\ldots, a_n) \\mapsto a_1x_1 + \\ldots + a_nx_n$.", "\\medskip\\noindent", "Part (9) holds because if $M = IM$ then $M = I^nM$ for all $n \\geq 0$", "and $I$ being nilpotent means $I^n = 0$ for some $n \\gg 0$. Parts", "(10), (11), and (12) follow from (9) by the arguments used above." ], "refs": [ "algebra-lemma-matrix-left-inverse", "algebra-lemma-contained-in-radical" ], "ref_ids": [ 381, 399 ] } ], "ref_ids": [] }, { "id": 402, "type": "theorem", "label": "algebra-lemma-when-surjective-local", "categories": [ "algebra" ], "title": "algebra-lemma-when-surjective-local", "contents": [ "Let $A \\to B$ be a local homomorphism of local rings.", "Assume", "\\begin{enumerate}", "\\item $B$ is finite as an $A$-module,", "\\item $\\mathfrak m_B$ is a finitely generated ideal,", "\\item $A \\to B$ induces an isomorphism on residue fields, and", "\\item $\\mathfrak m_A/\\mathfrak m_A^2 \\to \\mathfrak m_B/\\mathfrak m_B^2$", "is surjective.", "\\end{enumerate}", "Then $A \\to B$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "To show that $A \\to B$ is surjective, we view it as a map of $A$-modules", "and apply Lemma \\ref{lemma-NAK} (6). We conclude it suffices", "to show that $A/\\mathfrak m_A \\to B/\\mathfrak m_AB$ is surjective.", "As $A/\\mathfrak m_A = B/\\mathfrak m_B$ it suffices to show that", "$\\mathfrak m_AB \\to \\mathfrak m_B$ is surjective. View", "$\\mathfrak m_AB \\to \\mathfrak m_B$ as a map of $B$-modules and apply", "Lemma \\ref{lemma-NAK} (6). We conclude it suffices to see that", "$\\mathfrak m_AB/\\mathfrak m_A\\mathfrak m_B \\to \\mathfrak m_B/\\mathfrak m_B^2$", "is surjective. This follows from assumption (4)." ], "refs": [ "algebra-lemma-NAK", "algebra-lemma-NAK" ], "ref_ids": [ 401, 401 ] } ], "ref_ids": [] }, { "id": 403, "type": "theorem", "label": "algebra-lemma-idempotent-spec", "categories": [ "algebra" ], "title": "algebra-lemma-idempotent-spec", "contents": [ "Let $R$ be a ring. Let $e \\in R$ be an idempotent.", "In this case", "$$", "\\Spec(R) = D(e) \\amalg D(1-e).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Note that an idempotent $e$ of a domain is either $1$ or $0$.", "Hence we see that", "\\begin{eqnarray*}", "D(e)", "& = &", "\\{ \\mathfrak p \\in \\Spec(R)", "\\mid", "e \\not\\in \\mathfrak p \\} \\\\", "& = &", "\\{ \\mathfrak p \\in \\Spec(R)", "\\mid", "e \\not = 0\\text{ in }\\kappa(\\mathfrak p) \\} \\\\", "& = &", "\\{ \\mathfrak p \\in \\Spec(R)", "\\mid", "e = 1\\text{ in }\\kappa(\\mathfrak p) \\}", "\\end{eqnarray*}", "Similarly we have", "\\begin{eqnarray*}", "D(1-e)", "& = &", "\\{ \\mathfrak p \\in \\Spec(R)", "\\mid", "1 - e \\not\\in \\mathfrak p \\} \\\\", "& = &", "\\{ \\mathfrak p \\in \\Spec(R)", "\\mid", "e \\not = 1\\text{ in }\\kappa(\\mathfrak p) \\} \\\\", "& = &", "\\{ \\mathfrak p \\in \\Spec(R)", "\\mid", "e = 0\\text{ in }\\kappa(\\mathfrak p) \\}", "\\end{eqnarray*}", "Since the image of $e$ in any residue field is either $1$ or $0$", "we deduce that $D(e)$ and $D(1-e)$ cover all of $\\Spec(R)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 404, "type": "theorem", "label": "algebra-lemma-spec-product", "categories": [ "algebra" ], "title": "algebra-lemma-spec-product", "contents": [ "Let $R_1$ and $R_2$ be rings.", "Let $R = R_1 \\times R_2$.", "The maps $R \\to R_1$, $(x, y) \\mapsto x$ and $R \\to R_2$,", "$(x, y) \\mapsto y$", "induce continuous maps $\\Spec(R_1) \\to \\Spec(R)$ and", "$\\Spec(R_2) \\to \\Spec(R)$.", "The induced map", "$$", "\\Spec(R_1) \\amalg \\Spec(R_2)", "\\longrightarrow", "\\Spec(R)", "$$", "is a homeomorphism. In other words,", "the spectrum of $R = R_1\\times R_2$ is the", "disjoint union of the spectrum of $R_1$ and the", "spectrum of $R_2$." ], "refs": [], "proofs": [ { "contents": [ "Write $1 = e_1 + e_2$ with $e_1 = (1, 0)$ and $e_2 = (0, 1)$.", "Note that $e_1$ and $e_2 = 1 - e_1$ are idempotents.", "We leave it to the reader to show that", "$R_1 = R_{e_1}$ is the localization of $R$ at $e_1$.", "Similarly for $e_2$.", "Thus the statement of the lemma follows from Lemma", "\\ref{lemma-idempotent-spec} combined with Lemma", "\\ref{lemma-standard-open}." ], "refs": [ "algebra-lemma-idempotent-spec", "algebra-lemma-standard-open" ], "ref_ids": [ 403, 392 ] } ], "ref_ids": [] }, { "id": 405, "type": "theorem", "label": "algebra-lemma-disjoint-decomposition", "categories": [ "algebra" ], "title": "algebra-lemma-disjoint-decomposition", "contents": [ "Let $R$ be a ring. For each $U \\subset \\Spec(R)$", "which is open and closed", "there exists a unique idempotent $e \\in R$ such that", "$U = D(e)$. This induces a 1-1 correspondence between", "open and closed subsets $U \\subset \\Spec(R)$ and", "idempotents $e \\in R$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset \\Spec(R)$ be open and closed.", "Since $U$ is closed it is quasi-compact by", "Lemma \\ref{lemma-quasi-compact}, and similarly for", "its complement.", "Write $U = \\bigcup_{i = 1}^n D(f_i)$ as a finite union of standard opens.", "Similarly, write $\\Spec(R) \\setminus U = \\bigcup_{j = 1}^m D(g_j)$", "as a finite union of standard opens. Since $\\emptyset =", "D(f_i) \\cap D(g_j) = D(f_i g_j)$ we see that $f_i g_j$ is", "nilpotent by Lemma \\ref{lemma-Zariski-topology}.", "Let $I = (f_1, \\ldots, f_n) \\subset R$ and let", "$J = (g_1, \\ldots, g_m) \\subset R$.", "Note that $V(J)$ equals $U$, that $V(I)$", "equals the complement of $U$, so $\\Spec(R) = V(I) \\amalg V(J)$.", "By the remark on nilpotency above,", "we see that $(IJ)^N = (0)$ for some sufficiently large integer $N$.", "Since $\\bigcup D(f_i) \\cup \\bigcup D(g_j) = \\Spec(R)$", "we see that $I + J = R$, see Lemma \\ref{lemma-Zariski-topology}.", "By raising this equation to the $2N$th power we conclude that", "$I^N + J^N = R$. Write $1 = x + y$ with $x \\in I^N$ and $y \\in J^N$.", "Then $0 = xy = x(1 - x)$ as $I^N J^N = (0)$. Thus $x = x^2$", "is idempotent and contained", "in $I^N \\subset I$. The idempotent $y = 1 - x$ is contained in $J^N \\subset J$. ", "This shows that the idempotent $x$ maps to $1$ in every residue field", "$\\kappa(\\mathfrak p)$ for $\\mathfrak p \\in V(J)$ and that $x$ maps to $0$", "in $\\kappa(\\mathfrak p)$ for every $\\mathfrak p \\in V(I)$.", "\\medskip\\noindent", "To see uniqueness suppose that $e_1, e_2$ are", "distinct idempotents in $R$. We have to show there", "exists a prime $\\mathfrak p$ such that $e_1 \\in \\mathfrak p$", "and $e_2 \\not \\in \\mathfrak p$, or conversely.", "Write $e_i' = 1 - e_i$. If $e_1 \\not = e_2$, then", "$0 \\not = e_1 - e_2 = e_1(e_2 + e_2') - (e_1 + e_1')e_2", "= e_1 e_2' - e_1' e_2$. Hence either the idempotent", "$e_1 e_2' \\not = 0$ or $e_1' e_2 \\not = 0$. An idempotent", "is not nilpotent, and hence we find a prime", "$\\mathfrak p$ such that either $e_1e_2' \\not \\in \\mathfrak p$", "or $e_1'e_2 \\not \\in \\mathfrak p$, by Lemma \\ref{lemma-Zariski-topology}.", "It is easy to see this gives the desired prime." ], "refs": [ "algebra-lemma-quasi-compact", "algebra-lemma-Zariski-topology", "algebra-lemma-Zariski-topology", "algebra-lemma-Zariski-topology" ], "ref_ids": [ 395, 389, 389, 389 ] } ], "ref_ids": [] }, { "id": 406, "type": "theorem", "label": "algebra-lemma-characterize-spec-connected", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-spec-connected", "contents": [ "Let $R$ be a nonzero ring. Then $\\Spec(R)$ is", "connected if and only if $R$ has no nontrivial", "idempotents." ], "refs": [], "proofs": [ { "contents": [ "Obvious from Lemma \\ref{lemma-disjoint-decomposition}", "and the definition of a connected topological space." ], "refs": [ "algebra-lemma-disjoint-decomposition" ], "ref_ids": [ 405 ] } ], "ref_ids": [] }, { "id": 407, "type": "theorem", "label": "algebra-lemma-ideal-is-squared-union-connected", "categories": [ "algebra" ], "title": "algebra-lemma-ideal-is-squared-union-connected", "contents": [ "Let $R$ be a ring.", "Let $I$ be a finitely generated ideal.", "Assume that $I = I^2$.", "Then $V(I)$ is open and closed in $\\Spec(R)$,", "and $R/I \\cong R_e$ for some idempotent $e \\in R$." ], "refs": [], "proofs": [ { "contents": [ "By Nakayama's Lemma \\ref{lemma-NAK} there exists an element", "$f = 1 + i$, $i \\in I$ in $R$ such that $fI = 0$.", "It follows that $V(I) = D(f)$ by a simple argument.", "Also, $0 = fi = i + i^2$, and hence", "$f^2 = 1 + i + i + i^2 = 1 + i = f$, so $f$ is an idempotent.", "Consider the canonical map $R \\to R_f$. It is surjective", "since $x/f^n = x/f = xf/f^2 = xf/f = x/1$ in $R_f$.", "Any element of $I$ is in the kernel since $fI = 0$.", "If $x \\mapsto 0$ in $R_f$, then $f^nx = 0$ for some $n > 0$", "and hence $(1 + i)x = 0$ hence $x \\in I$." ], "refs": [ "algebra-lemma-NAK" ], "ref_ids": [ 401 ] } ], "ref_ids": [] }, { "id": 408, "type": "theorem", "label": "algebra-lemma-closed-union-connected-components", "categories": [ "algebra" ], "title": "algebra-lemma-closed-union-connected-components", "contents": [ "Let $R$ be a ring. Let $T \\subset \\Spec(R)$ be a subset of the spectrum.", "The following are equivalent", "\\begin{enumerate}", "\\item $T$ is closed and is a union of connected components of", "$\\Spec(R)$,", "\\item $T$ is an intersection of open and closed subsets of", "$\\Spec(R)$, and", "\\item $T = V(I)$ where $I \\subset R$ is an ideal generated by idempotents.", "\\end{enumerate}", "Moreover, the ideal in (3) if it exists is unique." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-topology-spec}", "and", "Topology, Lemma \\ref{topology-lemma-closed-union-connected-components}", "we see that (1) and (2) are equivalent.", "Assume (2) and write $T = \\bigcap U_\\alpha$ with", "$U_\\alpha \\subset \\Spec(R)$ open and closed.", "Then $U_\\alpha = D(e_\\alpha)$ for some idempotent $e_\\alpha \\in R$ by", "Lemma \\ref{lemma-disjoint-decomposition}.", "Then setting $I = (1 - e_\\alpha)$ we see that $T = V(I)$, i.e., (3) holds.", "Finally, assume (3). Write $T = V(I)$ and $I = (e_\\alpha)$ for some", "collection of idempotents $e_\\alpha$. Then it is clear that", "$T = \\bigcap V(e_\\alpha) = \\bigcap D(1 - e_\\alpha)$.", "\\medskip\\noindent", "Suppose that $I$ is an ideal generated by idempotents.", "Let $e \\in R$ be an idempotent such that $V(I) \\subset V(e)$. Then by", "Lemma \\ref{lemma-Zariski-topology}", "we see that $e^n \\in I$ for some $n \\geq 1$. As $e$ is an idempotent", "this means that $e \\in I$. Hence we see that $I$ is generated by", "exactly those idempotents $e$ such that $T \\subset V(e)$.", "In other words, the ideal $I$ is completely determined by the", "closed subset $T$ which proves uniqueness." ], "refs": [ "algebra-lemma-topology-spec", "topology-lemma-closed-union-connected-components", "algebra-lemma-disjoint-decomposition", "algebra-lemma-Zariski-topology" ], "ref_ids": [ 396, 8238, 405, 389 ] } ], "ref_ids": [] }, { "id": 409, "type": "theorem", "label": "algebra-lemma-connected-component", "categories": [ "algebra" ], "title": "algebra-lemma-connected-component", "contents": [ "Let $R$ be a ring.", "A connected component of", "$\\Spec(R)$ is of the form $V(I)$,", "where $I$ is an ideal generated by idempotents", "such that every idempotent of $R$ either maps to", "$0$ or $1$ in $R/I$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p$ be a prime of $R$. By", "Lemma \\ref{lemma-topology-spec}", "we have see that the hypotheses of", "Topology, Lemma \\ref{topology-lemma-connected-component-intersection}", "are satisfied for the topological space $\\Spec(R)$.", "Hence the connected component of $\\mathfrak p$ in $\\Spec(R)$", "is the intersection of open and closed subsets of $\\Spec(R)$", "containing $\\mathfrak p$. Hence it equals $V(I)$ where", "$I$ is generated by the idempotents $e \\in R$ such that $e$ maps to $0$", "in $\\kappa(\\mathfrak p)$, see", "Lemma \\ref{lemma-disjoint-decomposition}.", "Any idempotent $e$ which is not in this collection clearly maps to $1$", "in $R/I$." ], "refs": [ "algebra-lemma-topology-spec", "topology-lemma-connected-component-intersection", "algebra-lemma-disjoint-decomposition" ], "ref_ids": [ 396, 8236, 405 ] } ], "ref_ids": [] }, { "id": 410, "type": "theorem", "label": "algebra-lemma-characterize-zero-local", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-zero-local", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item For an element $x$ of an $R$-module $M$ the following are equivalent", "\\begin{enumerate}", "\\item $x = 0$,", "\\item $x$ maps to zero in $M_\\mathfrak p$ for all $\\mathfrak p \\in \\Spec(R)$,", "\\item $x$ maps to zero in $M_{\\mathfrak m}$ for all maximal ideals", "$\\mathfrak m$ of $R$.", "\\end{enumerate}", "In other words, the map $M \\to \\prod_{\\mathfrak m} M_{\\mathfrak m}$", "is injective.", "\\item Given an $R$-module $M$ the following are equivalent", "\\begin{enumerate}", "\\item $M$ is zero,", "\\item $M_{\\mathfrak p}$ is zero for all $\\mathfrak p \\in \\Spec(R)$,", "\\item $M_{\\mathfrak m}$ is zero for all maximal ideals $\\mathfrak m$ of $R$.", "\\end{enumerate}", "\\item Given a complex $M_1 \\to M_2 \\to M_3$", "of $R$-modules the following are equivalent", "\\begin{enumerate}", "\\item $M_1 \\to M_2 \\to M_3$ is exact,", "\\item for every prime $\\mathfrak p$ of $R$ the localization", "$M_{1, \\mathfrak p} \\to M_{2, \\mathfrak p} \\to M_{3, \\mathfrak p}$", "is exact,", "\\item for every maximal ideal $\\mathfrak m$ of $R$ the localization", "$M_{1, \\mathfrak m} \\to M_{2, \\mathfrak m} \\to M_{3, \\mathfrak m}$", "is exact.", "\\end{enumerate}", "\\item Given a map $f : M \\to M'$ of $R$-modules the following are equivalent", "\\begin{enumerate}", "\\item $f$ is injective,", "\\item $f_{\\mathfrak p} : M_\\mathfrak p \\to M'_\\mathfrak p$ is injective", "for all primes $\\mathfrak p$ of $R$,", "\\item $f_{\\mathfrak m} : M_\\mathfrak m \\to M'_\\mathfrak m$ is injective", "for all maximal ideals $\\mathfrak m$ of $R$.", "\\end{enumerate}", "\\item Given a map $f : M \\to M'$ of $R$-modules the following are equivalent", "\\begin{enumerate}", "\\item $f$ is surjective,", "\\item $f_{\\mathfrak p} : M_\\mathfrak p \\to M'_\\mathfrak p$ is surjective", "for all primes $\\mathfrak p$ of $R$,", "\\item $f_{\\mathfrak m} : M_\\mathfrak m \\to M'_\\mathfrak m$ is surjective", "for all maximal ideals $\\mathfrak m$ of $R$.", "\\end{enumerate}", "\\item Given a map $f : M \\to M'$ of $R$-modules the following are equivalent", "\\begin{enumerate}", "\\item $f$ is bijective,", "\\item $f_{\\mathfrak p} : M_\\mathfrak p \\to M'_\\mathfrak p$ is bijective", "for all primes $\\mathfrak p$ of $R$,", "\\item $f_{\\mathfrak m} : M_\\mathfrak m \\to M'_\\mathfrak m$ is bijective", "for all maximal ideals $\\mathfrak m$ of $R$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in M$ as in (1). Let $I = \\{f \\in R \\mid fx = 0\\}$.", "It is easy to see that $I$ is an ideal (it is the", "annihilator of $x$). Condition (1)(c) means that for", "all maximal ideals $\\mathfrak m$ there exists an", "$f \\in R \\setminus \\mathfrak m$ such that $fx =0$.", "In other words, $V(I)$ does not contain a closed point.", "By Lemma \\ref{lemma-Zariski-topology} we see $I$ is the unit ideal.", "Hence $x$ is zero, i.e., (1)(a) holds. This proves (1).", "\\medskip\\noindent", "Part (2) follows by applying (1) to all elements of $M$ simultaneously.", "\\medskip\\noindent", "Proof of (3). Let $H$ be the homology of the sequence, i.e.,", "$H = \\Ker(M_2 \\to M_3)/\\Im(M_1 \\to M_2)$. By", "Proposition \\ref{proposition-localization-exact}", "we have that $H_\\mathfrak p$ is the homology of the sequence", "$M_{1, \\mathfrak p} \\to M_{2, \\mathfrak p} \\to M_{3, \\mathfrak p}$.", "Hence (3) is a consequence of (2).", "\\medskip\\noindent", "Parts (4) and (5) are special cases of (3). Part (6) follows", "formally on combining (4) and (5)." ], "refs": [ "algebra-lemma-Zariski-topology", "algebra-proposition-localization-exact" ], "ref_ids": [ 389, 1402 ] } ], "ref_ids": [] }, { "id": 411, "type": "theorem", "label": "algebra-lemma-cover", "categories": [ "algebra" ], "title": "algebra-lemma-cover", "contents": [ "\\begin{slogan}", "Zariski-local properties of modules and algebras", "\\end{slogan}", "Let $R$ be a ring. Let $M$ be an $R$-module. Let $S$ be an $R$-algebra.", "Suppose that $f_1, \\ldots, f_n$ is a finite list of", "elements of $R$ such that $\\bigcup D(f_i) = \\Spec(R)$,", "in other words $(f_1, \\ldots, f_n) = R$.", "\\begin{enumerate}", "\\item If each $M_{f_i} = 0$ then $M = 0$.", "\\item If each $M_{f_i}$ is a finite $R_{f_i}$-module,", "then $M$ is a finite $R$-module.", "\\item If each $M_{f_i}$ is a finitely presented $R_{f_i}$-module,", "then $M$ is a finitely presented $R$-module.", "\\item Let $M \\to N$ be a map of $R$-modules. If $M_{f_i} \\to N_{f_i}$", "is an isomorphism for each $i$ then $M \\to N$ is an isomorphism.", "\\item Let $0 \\to M'' \\to M \\to M' \\to 0$ be a complex of $R$-modules.", "If $0 \\to M''_{f_i} \\to M_{f_i} \\to M'_{f_i} \\to 0$ is exact for each $i$,", "then $0 \\to M'' \\to M \\to M' \\to 0$ is exact.", "\\item If each $R_{f_i}$ is Noetherian, then $R$ is Noetherian.", "\\item If each $S_{f_i}$ is a finite type $R$-algebra, so is $S$.", "\\item If each $S_{f_i}$ is of finite presentation over $R$, so is $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We prove each of the parts in turn.", "\\begin{enumerate}", "\\item By Proposition \\ref{proposition-localize-twice}", "this implies $M_\\mathfrak p = 0$ for all $\\mathfrak p \\in \\Spec(R)$,", "so we conclude by Lemma \\ref{lemma-characterize-zero-local}.", "\\item For each $i$ take a finite generating set $X_i$ of $M_{f_i}$.", "Without loss of generality, we may assume that the elements of $X_i$ are", "in the image of the localization map $M \\rightarrow M_{f_i}$, so we take", "a finite set $Y_i$ of preimages of the elements of $X_i$ in $M$. Let $Y$", "be the union of these sets. This is still a finite set.", "Consider the obvious $R$-linear map $R^Y \\rightarrow M$ sending the basis", "element $e_y$ to $y$. By assumption this map is surjective after localizing", "at an arbitrary prime ideal $\\mathfrak p$ of $R$, so it surjective by", "Lemma \\ref{lemma-characterize-zero-local}", "and $M$ is finitely generated.", "\\item By (2) we have a short exact sequence", "$$", "0 \\rightarrow K \\rightarrow R^n \\rightarrow M \\rightarrow 0", "$$", "Since localization is an exact functor and $M_{f_i}$ is finitely", "presented we see that $K_{f_i}$ is finitely generated for all", "$1 \\leq i \\leq n$ by Lemma \\ref{lemma-extension}.", "By (2) this implies that $K$ is a finite $R$-module and therefore", "$M$ is finitely presented.", "\\item By Proposition \\ref{proposition-localize-twice}", "the assumption implies that the induced morphism", "on localizations at all prime ideals is an isomorphism, so we conclude", "by Lemma \\ref{lemma-characterize-zero-local}.", "\\item By Proposition \\ref{proposition-localize-twice} the assumption", "implies that the induced", "sequence of localizations at all prime ideals is short exact, so we", "conclude by Lemma \\ref{lemma-characterize-zero-local}.", "\\item We will show that every ideal of $R$ has a finite generating set:", "For this, let $I \\subset R$ be an arbitrary ideal. By", "Proposition \\ref{proposition-localization-exact}", "each $I_{f_i} \\subset R_{f_i}$ is an ideal. These are all", "finitely generated by assumption, so we conclude by (2).", "\\item For each $i$ take a finite generating set $X_i$ of $S_{f_i}$.", "Without loss of generality, we may assume that the elements of $X_i$", "are in the image of the localization map $S \\rightarrow S_{f_i}$, so", "we take a finite set $Y_i$ of preimages of the elements of $X_i$ in", "$S$. Let $Y$ be the union of these sets. This is still a finite set.", "Consider the algebra homomorphism $R[X_y]_{y \\in Y} \\rightarrow S$", "induced by $Y$. Since it is an algebra homomorphism, the image $T$", "is an $R$-submodule of the $R$-module $S$, so we can consider the", "quotient module $S/T$. By assumption, this is zero if we localize", "at the $f_i$, so it is zero by (1) and therefore $S$ is an", "$R$-algebra of finite type.", "\\item By the previous item, there exists a surjective $R$-algebra", "homomorphism $R[X_1, \\ldots, X_n] \\rightarrow S$. Let $K$ be the kernel", "of this map. This is an ideal in $R[X_1, \\ldots, X_n]$, finitely generated", "in each localization at $f_i$. Since the $f_i$ generate the unit ideal", "in $R$, they also generate the unit ideal in $R[X_1, \\ldots, X_n]$, so an", "application of (2) finishes the proof.", "\\end{enumerate}" ], "refs": [ "algebra-proposition-localize-twice", "algebra-lemma-characterize-zero-local", "algebra-lemma-characterize-zero-local", "algebra-lemma-extension", "algebra-proposition-localize-twice", "algebra-lemma-characterize-zero-local", "algebra-proposition-localize-twice", "algebra-lemma-characterize-zero-local", "algebra-proposition-localization-exact" ], "ref_ids": [ 1400, 410, 410, 330, 1400, 410, 1400, 410, 1402 ] } ], "ref_ids": [] }, { "id": 412, "type": "theorem", "label": "algebra-lemma-cover-upstairs", "categories": [ "algebra" ], "title": "algebra-lemma-cover-upstairs", "contents": [ "Let $R \\to S$ be a ring map.", "Suppose that $g_1, \\ldots, g_n$ is a finite list of", "elements of $S$ such that $\\bigcup D(g_i) = \\Spec(S)$", "in other words $(g_1, \\ldots, g_n) = S$.", "\\begin{enumerate}", "\\item If each $S_{g_i}$ is of finite type over $R$, then $S$ is", "of finite type over $R$.", "\\item If each $S_{g_i}$ is of finite presentation over $R$,", "then $S$ is of finite presentation over $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose $h_1, \\ldots, h_n \\in S$ such that $\\sum h_i g_i = 1$.", "\\medskip\\noindent", "Proof of (1). For each $i$ choose a finite list of elements", "$x_{i, j} \\in S_{g_i}$, $j = 1, \\ldots, m_i$", "which generate $S_{g_i}$ as an $R$-algebra.", "Write $x_{i, j} = y_{i, j}/g_i^{n_{i, j}}$ for some $y_{i, j} \\in S$ and", "some $n_{i, j} \\ge 0$. Consider the $R$-subalgebra $S' \\subset S$", "generated by $g_1, \\ldots, g_n$, $h_1, \\ldots, h_n$ and", "$y_{i, j}$, $i = 1, \\ldots, n$, $j = 1, \\ldots, m_i$.", "Since localization is exact (Proposition \\ref{proposition-localization-exact}),", "we see that $S'_{g_i} \\to S_{g_i}$ is injective.", "On the other hand, it is surjective by our choice of $y_{i, j}$.", "The elements $g_1, \\ldots, g_n$ generate the unit ideal in $S'$", "as $h_1, \\ldots, h_n \\in S'$.", "Thus $S' \\to S$ viewed as an $S'$-module map is an isomorphism", "by Lemma \\ref{lemma-cover}.", "\\medskip\\noindent", "Proof of (2). We already know that $S$ is of finite type.", "Write $S = R[x_1, \\ldots, x_m]/J$ for some ideal $J$.", "For each $i$ choose a lift $g'_i \\in R[x_1, \\ldots, x_m]$ of $g_i$", "and we choose a lift $h'_i \\in R[x_1, \\ldots, x_m]$ of $h_i$.", "Then we see that", "$$", "S_{g_i} = R[x_1, \\ldots, x_m, y_i]/J_i + (1 - y_ig'_i)", "$$", "where $J_i$ is the ideal of $R[x_1, \\ldots, x_m, y_i]$", "generated by $J$. Small detail omitted. By", "Lemma \\ref{lemma-finite-presentation-independent}", "we may choose a finite list of elements", "$f_{i, j} \\in J$, $j = 1, \\ldots, m_i$", "such that the images of $f_{i, j}$ in $J_i$ and $1 - y_ig'_i$", "generate the ideal $J_i + (1 - y_ig'_i)$.", "Set", "$$", "S' = R[x_1, \\ldots, x_m]/(\\sum h'_ig'_i - 1, f_{i, j}; ", "i = 1, \\ldots, n, j = 1, \\ldots, m_i)", "$$", "There is a surjective $R$-algebra map $S' \\to S$.", "The classes of the elements $g'_1, \\ldots, g'_n$ in $S'$", "generate the unit ideal and by construction the maps", "$S'_{g'_i} \\to S_{g_i}$ are injective.", "Thus we conclude as in part (1)." ], "refs": [ "algebra-proposition-localization-exact", "algebra-lemma-cover", "algebra-lemma-finite-presentation-independent" ], "ref_ids": [ 1402, 411, 334 ] } ], "ref_ids": [] }, { "id": 413, "type": "theorem", "label": "algebra-lemma-cover-module", "categories": [ "algebra" ], "title": "algebra-lemma-cover-module", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_n$ be elements of $R$", "generating the unit ideal. Let $M$ be an $R$-module.", "The sequence", "$$", "0 \\to", "M \\xrightarrow{\\alpha}", "\\bigoplus\\nolimits_{i = 1}^n M_{f_i} \\xrightarrow{\\beta}", "\\bigoplus\\nolimits_{i, j = 1}^n M_{f_i f_j}", "$$", "is exact, where $\\alpha(m) = (m/1, \\ldots, m/1)$", "and $\\beta(m_1/f_1^{e_1}, \\ldots, m_n/f_n^{e_n})", "= (m_i/f_i^{e_i} - m_j/f_j^{e_j})_{(i, j)}$." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that the localization of the sequence at", "any maximal ideal $\\mathfrak m$ is exact, see", "Lemma \\ref{lemma-characterize-zero-local}.", "Since $f_1, \\ldots, f_n$ generate the unit ideal,", "there is an $i$ such that $f_i \\not \\in \\mathfrak m$.", "After renumbering we may assume $i = 1$.", "Note that $(M_{f_i})_\\mathfrak m = (M_\\mathfrak m)_{f_i}$", "and $(M_{f_if_j})_\\mathfrak m = (M_\\mathfrak m)_{f_if_j}$, see", "Proposition \\ref{proposition-localize-twice-module}.", "In particular $(M_{f_1})_\\mathfrak m = M_\\mathfrak m$ and", "$(M_{f_1 f_i})_\\mathfrak m = (M_\\mathfrak m)_{f_i}$, because", "$f_1$ is a unit.", "Note that the maps in the sequence are the canonical ones", "coming from", "Lemma \\ref{lemma-universal-property-localization-module}", "and the identity map on $M$.", "Having said all of this, after replacing $R$ by $R_\\mathfrak m$,", "$M$ by $M_\\mathfrak m$, and $f_i$ by their image in $R_\\mathfrak m$,", "and $f_1$ by $1 \\in R_\\mathfrak m$,", "we reduce to the case where $f_1 = 1$.", "\\medskip\\noindent", "Assume $f_1 = 1$. Injectivity of $\\alpha$ is now trivial. Let", "$m = (m_i) \\in \\bigoplus_{i = 1}^n M_{f_i}$ be in the kernel of $\\beta$.", "Then $m_1 \\in M_{f_1} = M$. Moreover, $\\beta(m) = 0$", "implies that $m_1$ and $m_i$ map to the same element of", "$M_{f_1f_i} = M_{f_i}$. Thus $\\alpha(m_1) = m$ and the", "proof is complete." ], "refs": [ "algebra-lemma-characterize-zero-local", "algebra-proposition-localize-twice-module", "algebra-lemma-universal-property-localization-module" ], "ref_ids": [ 410, 1401, 347 ] } ], "ref_ids": [] }, { "id": 414, "type": "theorem", "label": "algebra-lemma-standard-covering", "categories": [ "algebra" ], "title": "algebra-lemma-standard-covering", "contents": [ "Let $R$ be a ring, and let $f_1, f_2, \\ldots f_n\\in R$ generate", "the unit ideal in $R$.", "Then the following sequence is exact:", "$$", "0 \\longrightarrow", "R \\longrightarrow", "\\bigoplus\\nolimits_i R_{f_i} \\longrightarrow", "\\bigoplus\\nolimits_{i, j}R_{f_if_j}", "$$", "where the maps $\\alpha : R \\longrightarrow \\bigoplus_i R_{f_i}$", "and $\\beta : \\bigoplus_i R_{f_i} \\longrightarrow \\bigoplus_{i, j} R_{f_if_j}$", "are defined as", "$$", "\\alpha(x) = \\left(\\frac{x}{1}, \\ldots, \\frac{x}{1}\\right)", "\\text{ and }", "\\beta\\left(\\frac{x_1}{f_1^{r_1}}, \\ldots, \\frac{x_n}{f_n^{r_n}}\\right)", "=", "\\left(\\frac{x_i}{f_i^{r_i}}-\\frac{x_j}{f_j^{r_j}}~\\text{in}~R_{f_if_j}\\right).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-cover-module}." ], "refs": [ "algebra-lemma-cover-module" ], "ref_ids": [ 413 ] } ], "ref_ids": [] }, { "id": 415, "type": "theorem", "label": "algebra-lemma-disjoint-implies-product", "categories": [ "algebra" ], "title": "algebra-lemma-disjoint-implies-product", "contents": [ "Let $R$ be a ring.", "If $\\Spec(R) = U \\amalg V$ with both $U$ and $V$ open", "then $R \\cong R_1 \\times R_2$ with $U \\cong \\Spec(R_1)$", "and $V \\cong \\Spec(R_2)$ via the maps in Lemma \\ref{lemma-spec-product}.", "Moreover, both $R_1$ and $R_2$ are localizations as well as quotients", "of the ring $R$." ], "refs": [ "algebra-lemma-spec-product" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-disjoint-decomposition} we have", "$U = D(e)$ and $V = D(1-e)$ for some idempotent $e$.", "By Lemma \\ref{lemma-standard-covering} we see that", "$R \\cong R_e \\times R_{1 - e}$ (since clearly $R_{e(1-e)} = 0$", "so the glueing condition is trivial; of course it is", "trivial to prove the product decomposition directly in this", "case). The lemma follows." ], "refs": [ "algebra-lemma-disjoint-decomposition", "algebra-lemma-standard-covering" ], "ref_ids": [ 405, 414 ] } ], "ref_ids": [ 404 ] }, { "id": 416, "type": "theorem", "label": "algebra-lemma-when-injective-covering", "categories": [ "algebra" ], "title": "algebra-lemma-when-injective-covering", "contents": [ "Let $R$ be a ring.", "Let $f_1, \\ldots, f_n \\in R$.", "Let $M$ be an $R$-module.", "Then $M \\to \\bigoplus M_{f_i}$ is injective if and only if", "$$", "M \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} M, \\quad", "m \\longmapsto (f_1m, \\ldots, f_nm)", "$$", "is injective." ], "refs": [], "proofs": [ { "contents": [ "The map $M \\to \\bigoplus M_{f_i}$ is injective if and only if", "for all $m \\in M$ and $e_1, \\ldots, e_n \\geq 1$ such that", "$f_i^{e_i}m = 0$, $i = 1, \\ldots, n$ we have $m = 0$.", "This clearly implies the displayed map is injective.", "Conversely, suppose the displayed map is injective and", "$m \\in M$ and $e_1, \\ldots, e_n \\geq 1$ are such that", "$f_i^{e_i}m = 0$, $i = 1, \\ldots, n$. If $e_i = 1$ for all $i$,", "then we immediately conclude that $m = 0$ from the injectivity of", "the displayed map. Next, we prove this holds for any such data", "by induction on $e = \\sum e_i$. The base case is $e = n$, and we have", "just dealt with this. If some $e_i > 1$, then set $m' = f_im$.", "By induction we see that $m' = 0$. Hence we see that $f_i m = 0$,", "i.e., we may take $e_i = 1$ which decreases $e$ and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 417, "type": "theorem", "label": "algebra-lemma-glue-modules", "categories": [ "algebra" ], "title": "algebra-lemma-glue-modules", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_n \\in R$. Suppose we are given", "the following data:", "\\begin{enumerate}", "\\item For each $i$ an $R_{f_i}$-module $M_i$.", "\\item For each pair $i, j$ an $R_{f_if_j}$-module isomorphism", "$\\psi_{ij} : (M_i)_{f_j} \\to (M_j)_{f_i}$.", "\\end{enumerate}", "which satisfy the ``cocycle condition'' that all the diagrams", "$$", "\\xymatrix{", "(M_i)_{f_jf_k}", "\\ar[rd]_{\\psi_{ij}}", "\\ar[rr]^{\\psi_{ik}}", "& &", "(M_k)_{f_if_j} \\\\", "&", "(M_j)_{f_if_k} \\ar[ru]_{\\psi_{jk}}", "}", "$$", "commute (for all triples $i, j, k$). Given this data define", "$$", "M = \\Ker\\left(", "\\bigoplus\\nolimits_{1 \\leq i \\leq n} M_i", "\\longrightarrow", "\\bigoplus\\nolimits_{1 \\leq i, j \\leq n} (M_i)_{f_j}", "\\right)", "$$", "where $(m_1, \\ldots, m_n)$ maps to the element whose", "$(i, j)$th entry is $m_i/1 - \\psi_{ji}(m_j/1)$.", "Then the natural map $M \\to M_i$ induces an isorphism", "$M_{f_i} \\to M_i$. Moreover $\\psi_{ij}(m/1) = m/1$", "for all $m \\in M$ (with obvious notation)." ], "refs": [], "proofs": [ { "contents": [ "To show that $M_{f_1} \\to M_1$ is an isomorphism, it suffices", "to show that its localization at every prime $\\mathfrak p'$", "of $R_{f_1}$ is an isomorphism, see", "Lemma \\ref{lemma-characterize-zero-local}.", "Write $\\mathfrak p' = \\mathfrak p R_{f_1}$", "for some prime $\\mathfrak p \\subset R$, $f_1 \\not \\in \\mathfrak p$, see", "Lemma \\ref{lemma-standard-open}.", "Since localization is exact", "(Proposition \\ref{proposition-localization-exact}),", "we see that", "\\begin{align*}", "(M_{f_1})_{\\mathfrak p'} & =", "M_\\mathfrak p \\\\", "& =", "\\Ker\\left(", "\\bigoplus\\nolimits_{1 \\leq i \\leq n} M_{i, \\mathfrak p}", "\\longrightarrow", "\\bigoplus\\nolimits_{1 \\leq i, j \\leq n} ((M_i)_{f_j})_\\mathfrak p", "\\right) \\\\", "& =", "\\Ker\\left(", "\\bigoplus\\nolimits_{1 \\leq i \\leq n} M_{i, \\mathfrak p}", "\\longrightarrow", "\\bigoplus\\nolimits_{1 \\leq i, j \\leq n} (M_{i, \\mathfrak p})_{f_j}", "\\right)", "\\end{align*}", "Here we also used Proposition \\ref{proposition-localize-twice-module}.", "Since $f_1$ is a unit in $R_\\mathfrak p$, this reduces us to the case", "where $f_1 = 1$ by replacing $R$ by $R_\\mathfrak p$, $f_i$ by the", "image of $f_i$ in $R_\\mathfrak p$, $M$ by $M_\\mathfrak p$, and", "$f_1$ by $1$.", "\\medskip\\noindent", "Assume $f_1 = 1$. Then $\\psi_{1j} : (M_1)_{f_j} \\to M_j$", "is an isomorphism for $j = 2, \\ldots, n$. If we use these", "isomorphisms to identify $M_j = (M_1)_{f_j}$, then we see", "that $\\psi_{ij} : (M_1)_{f_if_j} \\to (M_1)_{f_if_j}$ is", "the canonical identification. Thus the complex", "$$", "0 \\to M_1 \\to", "\\bigoplus\\nolimits_{1 \\leq i \\leq n} (M_1)_{f_i}", "\\longrightarrow", "\\bigoplus\\nolimits_{1 \\leq i, j \\leq n}", "(M_1)_{f_if_j}", "$$", "is exact by Lemma \\ref{lemma-cover-module}.", "Thus the first map identifies $M_1$ with $M$ in this case", "and everything is clear." ], "refs": [ "algebra-lemma-characterize-zero-local", "algebra-lemma-standard-open", "algebra-proposition-localization-exact", "algebra-proposition-localize-twice-module", "algebra-lemma-cover-module" ], "ref_ids": [ 410, 392, 1402, 1401, 413 ] } ], "ref_ids": [] }, { "id": 418, "type": "theorem", "label": "algebra-lemma-minimal-prime-reduced-ring", "categories": [ "algebra" ], "title": "algebra-lemma-minimal-prime-reduced-ring", "contents": [ "Let $\\mathfrak p$ be a minimal prime of a ring $R$.", "Every element of the maximal ideal of $R_{\\mathfrak p}$", "is nilpotent. If $R$ is reduced then $R_{\\mathfrak p}$", "is a field." ], "refs": [], "proofs": [ { "contents": [ "If some element $x$ of ${\\mathfrak p}R_{\\mathfrak p}$", "is not nilpotent, then $D(x) \\not = \\emptyset$, see", "Lemma \\ref{lemma-Zariski-topology}. This contradicts", "the minimality of $\\mathfrak p$. If $R$ is reduced,", "then ${\\mathfrak p}R_{\\mathfrak p} = 0$ and", "hence it is a field." ], "refs": [ "algebra-lemma-Zariski-topology" ], "ref_ids": [ 389 ] } ], "ref_ids": [] }, { "id": 419, "type": "theorem", "label": "algebra-lemma-reduced-ring-sub-product-fields", "categories": [ "algebra" ], "title": "algebra-lemma-reduced-ring-sub-product-fields", "contents": [ "Let $R$ be a reduced ring. Then", "\\begin{enumerate}", "\\item $R$ is a subring of a product of fields,", "\\item $R \\to \\prod_{\\mathfrak p\\text{ minimal}} R_{\\mathfrak p}$", "is an embedding into a product of fields,", "\\item $\\bigcup_{\\mathfrak p\\text{ minimal}} \\mathfrak p$ is the set", "of zerodivisors of $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-minimal-prime-reduced-ring} each of the rings", "$R_\\mathfrak p$ is a field. In particular, the kernel of the ring", "map $R \\to R_\\mathfrak p$ is $\\mathfrak p$.", "By Lemma \\ref{lemma-Zariski-topology}", "we have $\\bigcap_{\\mathfrak p} \\mathfrak p = (0)$.", "Hence (2) and (1) are true. If $x y = 0$ and $y \\not = 0$, then", "$y \\not \\in \\mathfrak p$ for some minimal prime $\\mathfrak p$.", "Hence $x \\in \\mathfrak p$. Thus every zerodivisor of $R$ is contained", "in $\\bigcup_{\\mathfrak p\\text{ minimal}} \\mathfrak p$.", "Conversely, suppose that $x \\in \\mathfrak p$ for some minimal", "prime $\\mathfrak p$. Then $x$ maps to zero in $R_\\mathfrak p$,", "hence there exists $y \\in R$, $y \\not \\in \\mathfrak p$ such that", "$xy = 0$. In other words, $x$ is a zerodivisor. This finishes the", "proof of (3) and the lemma." ], "refs": [ "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-Zariski-topology" ], "ref_ids": [ 418, 389 ] } ], "ref_ids": [] }, { "id": 420, "type": "theorem", "label": "algebra-lemma-total-ring-fractions", "categories": [ "algebra" ], "title": "algebra-lemma-total-ring-fractions", "contents": [ "Let $R$ be a ring.", "Let $S \\subset R$ be a multiplicative subset consisting of nonzerodivisors.", "Then $Q(R) \\cong Q(S^{-1}R)$.", "In particular $Q(R) \\cong Q(Q(R))$." ], "refs": [], "proofs": [ { "contents": [ "If $x \\in S^{-1}R$ is a nonzerodivisor, and", "$x = r/f$ for some $r \\in R$, $f \\in S$, then", "$r$ is a nonzerodivisor in $R$. Whence the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 421, "type": "theorem", "label": "algebra-lemma-total-ring-fractions-no-embedded-points", "categories": [ "algebra" ], "title": "algebra-lemma-total-ring-fractions-no-embedded-points", "contents": [ "Let $R$ be a ring.", "Assume that $R$ has finitely many minimal primes", "$\\mathfrak q_1, \\ldots, \\mathfrak q_t$, and that", "$\\mathfrak q_1 \\cup \\ldots \\cup \\mathfrak q_t$ is the set", "of zerodivisors of $R$.", "Then the total ring of fractions $Q(R)$ is equal to", "$R_{\\mathfrak q_1} \\times \\ldots \\times R_{\\mathfrak q_t}$." ], "refs": [], "proofs": [ { "contents": [ "There are natural maps $Q(R) \\to R_{\\mathfrak q_i}$ since", "any nonzerodivisor is contained in $R \\setminus \\mathfrak q_i$.", "Hence a natural map", "$Q(R) \\to R_{\\mathfrak q_1} \\times \\ldots \\times R_{\\mathfrak q_t}$.", "For any nonminimal prime $\\mathfrak p \\subset R$ we see that", "$\\mathfrak p \\not \\subset \\mathfrak q_1 \\cup \\ldots \\cup \\mathfrak q_t$", "by Lemma \\ref{lemma-silly}. Hence", "$\\Spec(Q(R)) = \\{\\mathfrak q_1, \\ldots, \\mathfrak q_t\\}$", "(as subsets of $\\Spec(R)$, see Lemma \\ref{lemma-spec-localization}).", "Therefore $\\Spec(Q(R))$ is a finite discrete set and", "it follows that $Q(R) = A_1 \\times \\ldots \\times A_t$", "with $\\Spec(A_i) = \\{q_i\\}$, see", "Lemma \\ref{lemma-disjoint-implies-product}.", "Moreover $A_i$ is a local ring, which is a localization", "of $R$. Hence $A_i \\cong R_{\\mathfrak q_i}$." ], "refs": [ "algebra-lemma-silly", "algebra-lemma-spec-localization", "algebra-lemma-disjoint-implies-product" ], "ref_ids": [ 378, 391, 415 ] } ], "ref_ids": [] }, { "id": 422, "type": "theorem", "label": "algebra-lemma-irreducible", "categories": [ "algebra" ], "title": "algebra-lemma-irreducible", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item For a prime $\\mathfrak p \\subset R$ the closure", "of $\\{\\mathfrak p\\}$ in the Zariski topology is $V(\\mathfrak p)$.", "In a formula $\\overline{\\{\\mathfrak p\\}} = V(\\mathfrak p)$.", "\\item The irreducible closed subsets of $\\Spec(R)$ are", "exactly the subsets $V(\\mathfrak p)$, with $\\mathfrak p \\subset R$", "a prime.", "\\item The irreducible components (see Topology,", "Definition \\ref{topology-definition-irreducible-components})", "of $\\Spec(R)$ are exactly the subsets $V(\\mathfrak p)$,", "with $\\mathfrak p \\subset R$ a minimal prime.", "\\end{enumerate}" ], "refs": [ "topology-definition-irreducible-components" ], "proofs": [ { "contents": [ "Note that if $ \\mathfrak p \\in V(I)$, then", "$I \\subset \\mathfrak p$. Hence,", "clearly $\\overline{\\{\\mathfrak p\\}} = V(\\mathfrak p)$.", "In particular $V(\\mathfrak p)$ is the closure of", "a singleton and hence irreducible.", "The second assertion implies the third.", "To show the second, let", "$V(I) \\subset \\Spec(R)$ with $I$ a radical ideal.", "If $I$ is not prime, then choose $a, b\\in R$, $a, b\\not \\in I$", "with $ab\\in I$. In this case $V(I, a) \\cup V(I, b) = V(I)$,", "but neither $V(I, b) = V(I)$ nor $V(I, a) = V(I)$, by", "Lemma \\ref{lemma-Zariski-topology}. Hence $V(I)$ is not", "irreducible." ], "refs": [ "algebra-lemma-Zariski-topology" ], "ref_ids": [ 389 ] } ], "ref_ids": [ 8353 ] }, { "id": 423, "type": "theorem", "label": "algebra-lemma-spec-spectral", "categories": [ "algebra" ], "title": "algebra-lemma-spec-spectral", "contents": [ "The spectrum of a ring is a spectral space, see Topology, Definition", "\\ref{topology-definition-spectral-space}." ], "refs": [ "topology-definition-spectral-space" ], "proofs": [ { "contents": [ "Formally this follows from Lemma \\ref{lemma-irreducible} and", "Lemma \\ref{lemma-topology-spec}. See also discussion above." ], "refs": [ "algebra-lemma-irreducible", "algebra-lemma-topology-spec" ], "ref_ids": [ 422, 396 ] } ], "ref_ids": [ 8370 ] }, { "id": 424, "type": "theorem", "label": "algebra-lemma-irreducible-components-containing-x", "categories": [ "algebra" ], "title": "algebra-lemma-irreducible-components-containing-x", "contents": [ "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime.", "\\begin{enumerate}", "\\item the set of irreducible closed subsets of $\\Spec(R)$", "passing through $\\mathfrak p$ is in one-to-one correspondence with", "primes $\\mathfrak q \\subset R_{\\mathfrak p}$.", "\\item The set of irreducible components of $\\Spec(R)$ passing through", "$\\mathfrak p$ is in one-to-one correspondence with minimal", "primes $\\mathfrak q \\subset R_{\\mathfrak p}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-irreducible}", "and the description of $\\Spec(R_\\mathfrak p)$ in", "Lemma \\ref{lemma-spec-localization} which shows that", "$\\Spec(R_\\mathfrak p)$ corresponds to primes $\\mathfrak q$ in $R$", "with $\\mathfrak q \\subset \\mathfrak p$." ], "refs": [ "algebra-lemma-irreducible", "algebra-lemma-spec-localization" ], "ref_ids": [ 422, 391 ] } ], "ref_ids": [] }, { "id": 425, "type": "theorem", "label": "algebra-lemma-standard-open-containing-maximal-point", "categories": [ "algebra" ], "title": "algebra-lemma-standard-open-containing-maximal-point", "contents": [ "Let $R$ be a ring.", "Let $\\mathfrak p$ be a minimal prime of $R$.", "Let $W \\subset \\Spec(R)$ be a quasi-compact open", "not containing the point $\\mathfrak p$. Then there", "exists an $f \\in R$, $f \\not \\in \\mathfrak p$ such", "that $D(f) \\cap W = \\emptyset$." ], "refs": [], "proofs": [ { "contents": [ "Since $W$ is quasi-compact we may write it as a finite union", "of standard affine opens $D(g_i)$, $i = 1, \\ldots, n$.", "Since $\\mathfrak p \\not \\in W$ we have $g_i \\in \\mathfrak p$ for", "all $i$. By Lemma \\ref{lemma-minimal-prime-reduced-ring}", "each $g_i$ is nilpotent in $R_{\\mathfrak p}$. Hence we can find", "an $f \\in R$, $f \\not \\in \\mathfrak p$ such that for all $i$ we have", "$f g_i^{n_i} = 0$ for some $n_i > 0$. Then $D(f)$ works." ], "refs": [ "algebra-lemma-minimal-prime-reduced-ring" ], "ref_ids": [ 418 ] } ], "ref_ids": [] }, { "id": 426, "type": "theorem", "label": "algebra-lemma-ring-with-only-minimal-primes", "categories": [ "algebra" ], "title": "algebra-lemma-ring-with-only-minimal-primes", "contents": [ "Let $R$ be a ring. Let $X = \\Spec(R)$ as a topological space.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is profinite,", "\\item $X$ is Hausdorff,", "\\item $X$ is totally disconnected.", "\\item every quasi-compact open of $X$ is closed,", "\\item there are no nontrivial inclusions between its prime ideals,", "\\item every prime ideal is a maximal ideal,", "\\item every prime ideal is minimal,", "\\item every standard open $D(f) \\subset X$ is closed, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First proof. It is clear that (5), (6), and (7) are equivalent.", "It is clear that (4) and (8) are equivalent as every quasi-compact", "open is a finite union of standard opens.", "The implication (7) $\\Rightarrow$ (4) follows from", "Lemma \\ref{lemma-standard-open-containing-maximal-point}.", "Assume (4) holds. Let $\\mathfrak p, \\mathfrak p'$ be distinct", "primes of $R$. Choose an $f \\in \\mathfrak p'$, $f \\not \\in \\mathfrak p$", "(if needed switch $\\mathfrak p$ with $\\mathfrak p'$).", "Then $\\mathfrak p' \\not \\in D(f)$ and $\\mathfrak p \\in D(f)$.", "By (4) the open $D(f)$ is also closed.", "Hence $\\mathfrak p$ and $\\mathfrak p'$ are in disjoint open", "neighbourhoods whose union is $X$. Thus $X$ is Hausdorff and totally", "disconnected. Thus (4) $\\Rightarrow$ (2) and (3).", "If (3) holds then there cannot be any specializations", "between points of $\\Spec(R)$ and we see that (5) holds.", "If $X$ is Hausdorff then every point is closed, so (2) implies (6).", "Thus (2), (3), (4), (5), (6), (7) and (8) are equivalent.", "Any profinite space is Hausdorff, so (1) implies (2).", "If $X$ satisfies (2) and (3), then $X$ (being quasi-compact by", "Lemma \\ref{lemma-quasi-compact}) is profinite by", "Topology, Lemma \\ref{topology-lemma-profinite}.", "\\medskip\\noindent", "Second proof. Besides the equivalence of (4) and (8) this follows", "from Lemma \\ref{lemma-spec-spectral} and purely topological facts, see", "Topology, Lemma \\ref{topology-lemma-characterize-profinite-spectral}." ], "refs": [ "algebra-lemma-standard-open-containing-maximal-point", "algebra-lemma-quasi-compact", "topology-lemma-profinite", "algebra-lemma-spec-spectral", "topology-lemma-characterize-profinite-spectral" ], "ref_ids": [ 425, 395, 8299, 423, 8309 ] } ], "ref_ids": [] }, { "id": 427, "type": "theorem", "label": "algebra-lemma-colon", "categories": [ "algebra" ], "title": "algebra-lemma-colon", "contents": [ "Let $R$ be a ring. For a principal ideal $J \\subset R$, and for any ideal", "$I \\subset J$ we have $I = J (I : J)$." ], "refs": [], "proofs": [ { "contents": [ "Say $J = (a)$. Then $(I : J) = (I : a)$.", "Since $I \\subset J$ we see that any $y \\in I$ is of the form", "$y = xa$ for some $x \\in (I : a)$. Hence $I \\subset J (I : J)$.", "Conversely, if $x \\in (I : a)$, then $xJ = (xa) \\subset I$, which", "proves the other inclusion." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 428, "type": "theorem", "label": "algebra-lemma-simple", "categories": [ "algebra" ], "title": "algebra-lemma-simple", "contents": [ "Let $R$ be a ring. Let $S$ be a multiplicative subset of $R$.", "An ideal $I \\subset R$ which is maximal with respect to the property", "that $I \\cap S = \\emptyset$ is prime." ], "refs": [], "proofs": [ { "contents": [ "This is the example discussed in the introduction to this section.", "For an alternative proof, combine", "Example \\ref{example-oka-family-not-meet-multiplicative-set}", "with", "Proposition \\ref{proposition-oka}." ], "refs": [ "algebra-proposition-oka" ], "ref_ids": [ 1404 ] } ], "ref_ids": [] }, { "id": 429, "type": "theorem", "label": "algebra-lemma-cohen", "categories": [ "algebra" ], "title": "algebra-lemma-cohen", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item An ideal $I \\subset R$ maximal with respect to not being", "finitely generated is prime.", "\\item If every prime ideal of $R$ is", "finitely generated, then", "every ideal of $R$ is finitely generated\\footnote{Later we will say", "that $R$ is Noetherian.}.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first assertion is an immediate consequence of", "Example \\ref{example-oka-family-finitely-generated} and", "Proposition \\ref{proposition-oka}. For the second,", "suppose that there exists an ideal $I \\subset R$ which is not finitely", "generated. The union of a totally ordered chain $\\left\\{I_\\alpha\\right\\}$", "of ideals that are not finitely generated is not finitely generated;", "indeed, if $I = \\bigcup I_\\alpha$ were generated by", "$a_1, \\ldots, a_n$, then all the generators would belong to some", "$I_\\alpha $ and would consequently generate it.", "By Zorn's lemma, there is an ideal maximal with respect to being not finitely", "generated. By the first part this ideal is prime." ], "refs": [ "algebra-proposition-oka" ], "ref_ids": [ 1404 ] } ], "ref_ids": [] }, { "id": 430, "type": "theorem", "label": "algebra-lemma-primes-principal", "categories": [ "algebra" ], "title": "algebra-lemma-primes-principal", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item An ideal $I \\subset R$ maximal with respect to not being", "principal is prime.", "\\item If every prime ideal of $R$ is principal, then", "every ideal of $R$ is principal.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first part follows from", "Example \\ref{example-oka-family-principal} and", "Proposition \\ref{proposition-oka}.", "For the second, suppose that there exists an ideal $I \\subset R$", "which is not principal. The union of a totally ordered chain", "$\\left\\{I_\\alpha\\right\\}$ of ideals that not principal is not principal;", "indeed, if $I = \\bigcup I_\\alpha$ were generated by", "$a$, then $a$ would belong to some $I_\\alpha $ and $a$ would generate it.", "By Zorn's lemma, there is an ideal maximal with respect to not being", "principal. This ideal is necessarily prime by the first part." ], "refs": [ "algebra-proposition-oka" ], "ref_ids": [ 1404 ] } ], "ref_ids": [] }, { "id": 431, "type": "theorem", "label": "algebra-lemma-characterize-domain", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-domain", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item An ideal maximal among the ideals which do not contain a", "nonzerodivisor is prime.", "\\item If $R$ is nonzero and every nonzero prime ideal in $R$", "contains a nonzerodivisor, then $R$ is a domain.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the set $S$ of nonzerodivisors. It is a multiplicative", "subset of $R$. Hence any ideal maximal with respect to not intersecting", "$S$ is prime, see", "Lemma \\ref{lemma-simple}.", "Thus, if every nonzero prime ideal contains a nonzerodivisor, then", "$(0)$ is prime, i.e., $R$ is a domain." ], "refs": [ "algebra-lemma-simple" ], "ref_ids": [ 428 ] } ], "ref_ids": [] }, { "id": 432, "type": "theorem", "label": "algebra-lemma-qc-open", "categories": [ "algebra" ], "title": "algebra-lemma-qc-open", "contents": [ "Let $U \\subset \\Spec(R)$ be open. The following", "are equivalent:", "\\begin{enumerate}", "\\item $U$ is retrocompact in $\\Spec(R)$,", "\\item $U$ is quasi-compact,", "\\item $U$ is a finite union of standard opens, and", "\\item there exists a finitely generated ideal $I \\subset R$ such", "that $X \\setminus V(I) = U$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have (1) $\\Rightarrow$ (2) because $\\Spec(R)$ is quasi-compact, see", "Lemma \\ref{lemma-quasi-compact}. We have (2) $\\Rightarrow$ (3) because", "standard opens form a basis for the topology. Proof of (3) $\\Rightarrow$ (1).", "Let $U = \\bigcup_{i = 1\\ldots n} D(f_i)$. To show that $U$ is retrocompact", "in $\\Spec(R)$ it suffices to show that $U \\cap V$ is quasi-compact for any", "quasi-compact open $V$ of $\\Spec(R)$. Write", "$V = \\bigcup_{j = 1\\ldots m} D(g_j)$ which is possible by (2) $\\Rightarrow$", "(3). Each standard open is homeomorphic to the spectrum of a ring and hence", "quasi-compact, see Lemmas \\ref{lemma-standard-open} and", "\\ref{lemma-quasi-compact}. Thus", "$U \\cap V =", "(\\bigcup_{i = 1\\ldots n} D(f_i)) \\cap (\\bigcup_{j = 1\\ldots m} D(g_j))", "= \\bigcup_{i, j} D(f_i g_j)$ is a finite union of quasi-compact opens", "hence quasi-compact. To finish the proof note", "that (4) is equivalent to (3) by ", "Lemma \\ref{lemma-Zariski-topology}." ], "refs": [ "algebra-lemma-quasi-compact", "algebra-lemma-standard-open", "algebra-lemma-quasi-compact", "algebra-lemma-Zariski-topology" ], "ref_ids": [ 395, 392, 395, 389 ] } ], "ref_ids": [] }, { "id": 433, "type": "theorem", "label": "algebra-lemma-affine-map-quasi-compact", "categories": [ "algebra" ], "title": "algebra-lemma-affine-map-quasi-compact", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "The induced continuous map $f : \\Spec(S) \\to \\Spec(R)$", "is quasi-compact. For any constructible set $E \\subset \\Spec(R)$", "the inverse image $f^{-1}(E)$ is constructible in $\\Spec(S)$." ], "refs": [], "proofs": [ { "contents": [ "We first show that the inverse image of any quasi-compact", "open $U \\subset \\Spec(R)$ is quasi-compact. By", "Lemma \\ref{lemma-qc-open} we may write $U$ as a finite", "open of standard opens. Thus by Lemma \\ref{lemma-spec-functorial}", "we see that $f^{-1}(U)$ is a finite union of standard opens.", "Hence $f^{-1}(U)$ is quasi-compact by Lemma \\ref{lemma-qc-open} again.", "The second assertion now follows from Topology, Lemma", "\\ref{topology-lemma-inverse-images-constructibles}." ], "refs": [ "algebra-lemma-qc-open", "algebra-lemma-spec-functorial", "algebra-lemma-qc-open", "topology-lemma-inverse-images-constructibles" ], "ref_ids": [ 432, 390, 432, 8254 ] } ], "ref_ids": [] }, { "id": 434, "type": "theorem", "label": "algebra-lemma-constructible", "categories": [ "algebra" ], "title": "algebra-lemma-constructible", "contents": [ "Let $R$ be a ring. A subset of $\\Spec(R)$ is constructible if and only", "if it can be written as a finite union of subsets of the form", "$D(f) \\cap V(g_1, \\ldots, g_m)$ for $f, g_1, \\ldots, g_m \\in R$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-qc-open} the subset $D(f)$ and the complement of", "$V(g_1, \\ldots, g_m)$ are retro-compact open. Hence", "$D(f) \\cap V(g_1, \\ldots, g_m)$ is a constructible subset and so is", "any finite union of such. Conversely, let $T \\subset \\Spec(R)$ be", "constructible. By Topology, Definition \\ref{topology-definition-constructible},", "we may assume that $T = U \\cap V^c$, where $U, V \\subset \\Spec(R)$", "are retrocompact open. By Lemma \\ref{lemma-qc-open} we may write", "$U = \\bigcup_{i = 1, \\ldots, n} D(f_i)$ and", "$V = \\bigcup_{j = 1, \\ldots, m} D(g_j)$. Then", "$T = \\bigcup_{i = 1, \\ldots, n} \\big(D(f_i) \\cap V(g_1, \\ldots, g_m)\\big)$." ], "refs": [ "algebra-lemma-qc-open", "topology-definition-constructible", "algebra-lemma-qc-open" ], "ref_ids": [ 432, 8362, 432 ] } ], "ref_ids": [] }, { "id": 435, "type": "theorem", "label": "algebra-lemma-constructible-is-image", "categories": [ "algebra" ], "title": "algebra-lemma-constructible-is-image", "contents": [ "Let $R$ be a ring and let $T \\subset \\Spec(R)$", "be constructible. Then there exists a ring map $R \\to S$ of", "finite presentation such that $T$ is the image of", "$\\Spec(S)$ in $\\Spec(R)$." ], "refs": [], "proofs": [ { "contents": [ "The spectrum of a finite product of rings", "is the disjoint union of the spectra, see", "Lemma \\ref{lemma-spec-product}. Hence if $T = T_1 \\cup T_2$", "and the result holds for $T_1$ and $T_2$, then the", "result holds for $T$.", "By Lemma \\ref{lemma-constructible} we may assume", "that $T = D(f) \\cap V(g_1, \\ldots, g_m)$.", "In this case $T$ is the image of the map", "$\\Spec((R/(g_1, \\ldots, g_m))_f) \\to \\Spec(R)$, see Lemmas", "\\ref{lemma-standard-open} and \\ref{lemma-spec-closed}." ], "refs": [ "algebra-lemma-spec-product", "algebra-lemma-constructible", "algebra-lemma-standard-open", "algebra-lemma-spec-closed" ], "ref_ids": [ 404, 434, 392, 393 ] } ], "ref_ids": [] }, { "id": 436, "type": "theorem", "label": "algebra-lemma-open-fp", "categories": [ "algebra" ], "title": "algebra-lemma-open-fp", "contents": [ "Let $R$ be a ring.", "Let $f$ be an element of $R$.", "Let $S = R_f$.", "Then the image of a constructible subset of $\\Spec(S)$", "is constructible in $\\Spec(R)$." ], "refs": [], "proofs": [ { "contents": [ "We repeatedly use Lemma \\ref{lemma-qc-open} without mention.", "Let $U, V$ be quasi-compact open in $\\Spec(S)$.", "We will show that the image of $U \\cap V^c$ is constructible.", "Under the identification", "$\\Spec(S) = D(f)$ of Lemma \\ref{lemma-standard-open}", "the sets $U, V$ correspond to quasi-compact opens", "$U', V'$ of $\\Spec(R)$.", "Hence it suffices to show that $U' \\cap (V')^c$", "is constructible in $\\Spec(R)$ which is clear." ], "refs": [ "algebra-lemma-qc-open", "algebra-lemma-standard-open" ], "ref_ids": [ 432, 392 ] } ], "ref_ids": [] }, { "id": 437, "type": "theorem", "label": "algebra-lemma-closed-fp", "categories": [ "algebra" ], "title": "algebra-lemma-closed-fp", "contents": [ "Let $R$ be a ring.", "Let $I$ be a finitely generated ideal of $R$.", "Let $S = R/I$.", "Then the image of a constructible of $\\Spec(S)$", "is constructible in $\\Spec(R)$." ], "refs": [], "proofs": [ { "contents": [ "If $I = (f_1, \\ldots, f_m)$, then we see that", "$V(I)$ is the complement of $\\bigcup D(f_i)$,", "see Lemma \\ref{lemma-Zariski-topology}.", "Hence it is constructible, by Lemma \\ref{lemma-qc-open}.", "Denote the map $R \\to S$ by $f \\mapsto \\overline{f}$.", "We have to show that if $\\overline{U}, \\overline{V}$", "are retrocompact opens of $\\Spec(S)$, then the", "image of $\\overline{U} \\cap \\overline{V}^c$", "in $\\Spec(R)$ is constructible.", "By Lemma \\ref{lemma-qc-open} we may write", "$\\overline{U} = \\bigcup D(\\overline{g_i})$.", "Setting $U = \\bigcup D({g_i})$ we see $\\overline{U}$", "has image $U \\cap V(I)$ which is constructible in", "$\\Spec(R)$. Similarly the image of $\\overline{V}$ equals", "$V \\cap V(I)$ for some retrocompact open $V$ of $\\Spec(R)$.", "Hence the image of $\\overline{U} \\cap \\overline{V}^c$", "equals $U \\cap V(I) \\cap V^c$ as desired." ], "refs": [ "algebra-lemma-Zariski-topology", "algebra-lemma-qc-open", "algebra-lemma-qc-open" ], "ref_ids": [ 389, 432, 432 ] } ], "ref_ids": [] }, { "id": 438, "type": "theorem", "label": "algebra-lemma-affineline-open", "categories": [ "algebra" ], "title": "algebra-lemma-affineline-open", "contents": [ "Let $R$ be a ring. The map $\\Spec(R[x]) \\to \\Spec(R)$", "is open, and the image of any standard open is a quasi-compact", "open." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that the image of a standard open", "$D(f)$, $f\\in R[x]$ is quasi-compact open.", "The image of $D(f)$ is the image of", "$\\Spec(R[x]_f) \\to \\Spec(R)$.", "Let $\\mathfrak p \\subset R$ be a prime ideal.", "Let $\\overline{f}$ be the image of $f$ in", "$\\kappa(\\mathfrak p)[x]$.", "Recall, see Lemma \\ref{lemma-in-image},", "that $\\mathfrak p$ is in the image", "if and only if $R[x]_f \\otimes_R \\kappa(\\mathfrak p) =", "\\kappa(\\mathfrak p)[x]_{\\overline{f}}$ is not the", "zero ring. This is exactly the condition that $f$ does not map", "to zero in $\\kappa(\\mathfrak p)[x]$, in other words, that", "some coefficient of $f$ is not in $\\mathfrak p$.", "Hence we see: if $f = a_d x^d + \\ldots + a_0$, then", "the image of $D(f)$ is $D(a_d) \\cup \\ldots \\cup D(a_0)$." ], "refs": [ "algebra-lemma-in-image" ], "ref_ids": [ 394 ] } ], "ref_ids": [] }, { "id": 439, "type": "theorem", "label": "algebra-lemma-characteristic-polynomial-prime", "categories": [ "algebra" ], "title": "algebra-lemma-characteristic-polynomial-prime", "contents": [ "Let $R \\to A$ be a ring homomorphism.", "Assume $A \\cong R^{\\oplus n}$ as an $R$-module.", "Let $f \\in A$. The multiplication map $m_f: A", "\\to A$ is $R$-linear and hence", "has a characteristic polynomial", "$P(T) = T^n + r_{n-1}T^{n-1} + \\ldots + r_0 \\in R[T]$.", "For any prime", "$\\mathfrak{p} \\in \\Spec(R)$, $f$ acts nilpotently on $A", "\\otimes_R \\kappa(\\mathfrak{p})$ if and only if $\\mathfrak p \\in", "V(r_0, \\ldots, r_{n-1})$." ], "refs": [], "proofs": [ { "contents": [ "This follows quite easily once we prove that the characteristic", "polynomial $\\bar P(T) \\in \\kappa(\\mathfrak p)[T]$ of the", "multiplication map $m_{\\bar f}: A \\otimes_R \\kappa(\\mathfrak p) \\to", "A \\otimes_R \\kappa(\\mathfrak p)$ which multiplies elements of $A", "\\otimes_R \\kappa(\\mathfrak p)$ by $\\bar f$, the image of $f$ viewed in", "$\\kappa(\\mathfrak p)$, is just the image of $P(T)$ in", "$\\kappa(\\mathfrak p)[T]$. Let $(a_{ij})$ be the matrix of the map", "$m_f$ with entries in $R$, using a basis $e_1, \\ldots, e_n$", "of $A$ as an $R$-module.", "Then, $A \\otimes_R \\kappa(\\mathfrak p) \\cong (R \\otimes_R", "\\kappa(\\mathfrak p))^{\\oplus n} = \\kappa(\\mathfrak p)^n$, which is", "an $n$-dimensional vector space over $\\kappa(\\mathfrak p)$ with", "basis $e_1 \\otimes 1, \\ldots, e_n \\otimes 1$. The image $\\bar f = f", "\\otimes 1$, and so the multiplication map $m_{\\bar f}$ has matrix", "$(a_{ij} \\otimes 1)$. Thus, the characteristic polynomial is", "precisely the image of $P(T)$.", "\\medskip\\noindent", "From linear algebra, we know that a linear transformation acts", "nilpotently on an $n$-dimensional vector space if and only if the", "characteristic polynomial is $T^n$ (since the characteristic", "polynomial divides some power of the minimal polynomial). Hence,", "$f$ acts nilpotently on $A \\otimes_R \\kappa(\\mathfrak p)$ if and", "only if $\\bar P(T) = T^n$. This occurs if and only if $r_i \\in", "\\mathfrak p$ for all $0 \\leq i \\leq n - 1$, that is when $\\mathfrak p \\in", "V(r_0, \\ldots, r_{n - 1}).$" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 440, "type": "theorem", "label": "algebra-lemma-affineline-special", "categories": [ "algebra" ], "title": "algebra-lemma-affineline-special", "contents": [ "Let $R$ be a ring. Let $f, g \\in R[x]$ be polynomials.", "Assume the leading coefficient of $g$ is a unit of $R$.", "There exists elements $r_i\\in R$, $i = 1\\ldots, n$ such that", "the image of $D(f) \\cap V(g)$ in $\\Spec(R)$ is", "$\\bigcup_{i = 1, \\ldots, n} D(r_i)$." ], "refs": [], "proofs": [ { "contents": [ "Write $g = ux^d + a_{d-1}x^{d-1} + \\ldots + a_0$, where", "$d$ is the degree of $g$, and hence $u \\in R^*$.", "Consider the ring $A = R[x]/(g)$.", "It is, as an $R$-module, finite free with basis the images", "of $1, x, \\ldots, x^{d-1}$. Consider multiplication", "by (the image of) $f$ on $A$. This is an $R$-module map.", "Hence we can let $P(T) \\in R[T]$ be the characteristic polynomial", "of this map. Write $P(T) = T^d + r_{d-1} T^{d-1} + \\ldots + r_0$.", "We claim that $r_0, \\ldots, r_{d-1}$ have the desired property.", "We will use below the property of characteristic polynomials", "that", "$$", "\\mathfrak p \\in V(r_0, \\ldots, r_{d-1})", "\\Leftrightarrow", "\\text{multiplication by }f\\text{ is nilpotent on }", "A \\otimes_R \\kappa(\\mathfrak p).", "$$", "This was proved in Lemma \\ref{lemma-characteristic-polynomial-prime}.", "\\medskip\\noindent", "Suppose $\\mathfrak q\\in D(f) \\cap V(g)$, and let", "$\\mathfrak p = \\mathfrak q \\cap R$. Then there is a nonzero map", "$A \\otimes_R \\kappa(\\mathfrak p) \\to \\kappa(\\mathfrak q)$ which", "is compatible with multiplication by $f$.", "And $f$ acts as a unit on $\\kappa(\\mathfrak q)$.", "Thus we conclude $\\mathfrak p \\not \\in V(r_0, \\ldots, r_{d-1})$.", "\\medskip\\noindent", "On the other hand, suppose that $r_i \\not\\in \\mathfrak p$ for some", "prime $\\mathfrak p$ of $R$ and some $0 \\leq i \\leq d - 1$.", "Then multiplication by $f$ is not nilpotent on the algebra", "$A \\otimes_R \\kappa(\\mathfrak p)$.", "Hence there exists a prime ideal $\\overline{\\mathfrak q} \\subset", "A \\otimes_R \\kappa(\\mathfrak p)$ not containing the image of $f$.", "The inverse image of $\\overline{\\mathfrak q}$ in $R[x]$", "is an element of $D(f) \\cap V(g)$ mapping to $\\mathfrak p$." ], "refs": [ "algebra-lemma-characteristic-polynomial-prime" ], "ref_ids": [ 439 ] } ], "ref_ids": [] }, { "id": 441, "type": "theorem", "label": "algebra-lemma-generic-finite-presentation", "categories": [ "algebra" ], "title": "algebra-lemma-generic-finite-presentation", "contents": [ "Let $R \\subset S$ be an inclusion of domains.", "Assume that $R \\to S$ is of finite type.", "There exists a nonzero $f \\in R$, and a nonzero $g \\in S$", "such that $R_f \\to S_{fg}$ is of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "By induction on the number of generators of $S$ over $R$.", "During the proof we may replace $R$ by $R_f$ and $S$ by $S_f$", "for some nonzero $f \\in R$.", "\\medskip\\noindent", "Suppose that $S$ is generated by a single element", "over $R$. Then $S = R[x]/\\mathfrak q$ for some", "prime ideal $\\mathfrak q \\subset R[x]$. If $\\mathfrak q = (0)$", "there is nothing to prove. If $\\mathfrak q \\not = (0)$,", "then let $h \\in \\mathfrak q$ be a nonzero element with minimal", "degree in $x$. Write $h = f x^d + a_{d - 1} x^{d - 1} + \\ldots + a_0$", "with $a_i \\in R$ and $f \\not = 0$. After inverting $f$", "in $R$ and $S$ we may assume that $h$ is monic. We obtain", "a surjective $R$-algebra map $R[x]/(h) \\to S$.", "We have $R[x]/(h) = R \\oplus Rx \\oplus \\ldots \\oplus Rx^{d - 1}$", "as an $R$-module and by minimality of $d$ we see that", "$R[x]/(h)$ maps injectively into $S$. Thus $R[x]/(h) \\cong S$", "is finitely presented over $R$.", "\\medskip\\noindent", "Suppose that $S$ is generated by $n > 1$ elements over $R$.", "Say $x_1, \\ldots, x_n \\in S$ generate $S$. Denote $S' \\subset S$", "the subring generated by $x_1, \\ldots, x_{n-1}$. By induction", "hypothesis we see that there exist $f\\in R$ and $g \\in S'$", "nonzero such that $R_f \\to S'_{fg}$ is of finite presentation.", "Next we apply the induction hypothesis to $S'_{fg} \\to S_{fg}$", "to see that there exist $f' \\in S'_{fg}$ and", "$g' \\in S_{fg}$ such that $S'_{fgf'} \\to S_{fgf'g'}$", "is of finite presentation. We leave it to the reader to conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 442, "type": "theorem", "label": "algebra-lemma-characterize-image-finite-type", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-image-finite-type", "contents": [ "Let $R \\to S$ be a finite type ring map.", "Denote $X = \\Spec(R)$ and $Y = \\Spec(S)$.", "Write $f : Y \\to X$ the induced", "map of spectra. Let $E \\subset Y = \\Spec(S)$ be a", "constructible set.", "If a point $\\xi \\in X$ is in $f(E)$, then", "$\\overline{\\{\\xi\\}} \\cap f(E)$ contains an open", "dense subset of $\\overline{\\{\\xi\\}}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\xi \\in X$ be a point of $f(E)$. Choose a point $\\eta \\in E$", "mapping to $\\xi$. Let $\\mathfrak p \\subset R$ be the prime", "corresponding to $\\xi$ and let $\\mathfrak q \\subset S$ be the", "prime corresponding to $\\eta$. Consider the diagram", "$$", "\\xymatrix{", "\\eta \\ar[r] \\ar@{|->}[d] & E \\cap Y' \\ar[r] \\ar[d] &", "Y' = \\Spec(S/\\mathfrak q) \\ar[r] \\ar[d] &", "Y \\ar[d] \\\\", "\\xi \\ar[r] & f(E) \\cap X' \\ar[r] &", "X' = \\Spec(R/\\mathfrak p) \\ar[r] &", "X", "}", "$$", "By Lemma \\ref{lemma-affine-map-quasi-compact} the set $E \\cap Y'$", "is constructible in $Y'$.", "It follows that we may replace $X$ by $X'$ and", "$Y$ by $Y'$. Hence we may assume that $R \\subset S$ is an", "inclusion of domains, $\\xi$ is the generic", "point of $X$, and $\\eta$ is the generic point of $Y$.", "By Lemma \\ref{lemma-generic-finite-presentation}", "combined with Chevalley's theorem", "(Theorem \\ref{theorem-chevalley})", "we see that there exist dense opens $U \\subset X$,", "$V \\subset Y$ such that $f(V) \\subset U$ and", "such that $f : V \\to U$ maps constructible sets", "to constructible sets. Note that $E \\cap V$ is", "constructible in $V$, see Topology,", "Lemma \\ref{topology-lemma-open-immersion-constructible-inverse-image}.", "Hence $f(E \\cap V)$ is constructible in $U$ and contains $\\xi$.", "By Topology, Lemma \\ref{topology-lemma-generic-point-in-constructible}", "we see that $f(E \\cap V)$ contains a dense open $U' \\subset U$." ], "refs": [ "algebra-lemma-affine-map-quasi-compact", "algebra-lemma-generic-finite-presentation", "algebra-theorem-chevalley", "topology-lemma-open-immersion-constructible-inverse-image", "topology-lemma-generic-point-in-constructible" ], "ref_ids": [ 433, 441, 315, 8255, 8266 ] } ], "ref_ids": [] }, { "id": 443, "type": "theorem", "label": "algebra-lemma-surjective-spec-radical-ideal", "categories": [ "algebra" ], "title": "algebra-lemma-surjective-spec-radical-ideal", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "The following are equivalent:", "\\begin{enumerate}", "\\item The map $\\Spec(S) \\to \\Spec(R)$ is surjective.", "\\item For any ideal $I \\subset R$", "the inverse image of $\\sqrt{IS}$ in $R$ is equal to $\\sqrt{I}$.", "\\item For any radical ideal $I \\subset R$ the inverse image", "of $IS$ in $R$ is equal to $I$.", "\\item For every prime $\\mathfrak p$ of $R$ the inverse", "image of $\\mathfrak p S$ in $R$ is $\\mathfrak p$.", "\\end{enumerate}", "In this case the same is true after any base change: Given a ring map", "$R \\to R'$ the ring map $R' \\to R' \\otimes_R S$ has the equivalent", "properties (1), (2), (3) as well." ], "refs": [], "proofs": [ { "contents": [ "If $J \\subset S$ is an ideal, then", "$\\sqrt{\\varphi^{-1}(J)} = \\varphi^{-1}(\\sqrt{J})$. This shows that (2)", "and (3) are equivalent.", "The implication (3) $\\Rightarrow$ (4) is immediate.", "If $I \\subset R$ is a radical ideal, then", "Lemma \\ref{lemma-Zariski-topology}", "guarantees that $I = \\bigcap_{I \\subset \\mathfrak p} \\mathfrak p$.", "Hence (4) $\\Rightarrow$ (2). By", "Lemma \\ref{lemma-in-image}", "we have $\\mathfrak p = \\varphi^{-1}(\\mathfrak p S)$ if and only if", "$\\mathfrak p$ is in the image. Hence (1) $\\Leftrightarrow$ (4).", "Thus (1), (2), (3), and (4) are equivalent.", "\\medskip\\noindent", "Assume (1) holds. Let $R \\to R'$ be a ring map. Let", "$\\mathfrak p' \\subset R'$ be a prime ideal lying over the prime", "$\\mathfrak p$ of $R$. To see that $\\mathfrak p'$ is in the image", "of $\\Spec(R' \\otimes_R S) \\to \\Spec(R')$ we have to show", "that $(R' \\otimes_R S) \\otimes_{R'} \\kappa(\\mathfrak p')$ is not zero, see", "Lemma \\ref{lemma-in-image}.", "But we have", "$$", "(R' \\otimes_R S) \\otimes_{R'} \\kappa(\\mathfrak p') =", "S \\otimes_R \\kappa(\\mathfrak p)", "\\otimes_{\\kappa(\\mathfrak p)} \\kappa(\\mathfrak p')", "$$", "which is not zero as $S \\otimes_R \\kappa(\\mathfrak p)$ is not zero", "by assumption and $\\kappa(\\mathfrak p) \\to \\kappa(\\mathfrak p')$ is", "an extension of fields." ], "refs": [ "algebra-lemma-Zariski-topology", "algebra-lemma-in-image", "algebra-lemma-in-image" ], "ref_ids": [ 389, 394, 394 ] } ], "ref_ids": [] }, { "id": 444, "type": "theorem", "label": "algebra-lemma-domain-image-dense-set-points-generic-point", "categories": [ "algebra" ], "title": "algebra-lemma-domain-image-dense-set-points-generic-point", "contents": [ "Let $R$ be a domain. Let $\\varphi : R \\to S$ be a ring map.", "The following are equivalent:", "\\begin{enumerate}", "\\item The ring map $R \\to S$ is injective.", "\\item The image $\\Spec(S) \\to \\Spec(R)$", "contains a dense set of points.", "\\item There exists a prime ideal $\\mathfrak q \\subset S$", "whose inverse image in $R$ is $(0)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $K$ be the field of fractions of the domain $R$.", "Assume that $R \\to S$ is injective. Since localization", "is exact we see that $K \\to S \\otimes_R K$ is injective.", "Hence there is a prime mapping to $(0)$ by", "Lemma \\ref{lemma-in-image}.", "\\medskip\\noindent", "Note that $(0)$ is dense in $\\Spec(R)$, so that the", "last condition implies the second.", "\\medskip\\noindent", "Suppose the second condition holds. Let $f \\in R$,", "$f \\not = 0$. As $R$ is a domain we see that $V(f)$", "is a proper closed subset of $R$. By assumption", "there exists a prime $\\mathfrak q$", "of $S$ such that $\\varphi(f) \\not \\in \\mathfrak q$.", "Hence $\\varphi(f) \\not = 0$.", "Hence $R \\to S$ is injective." ], "refs": [ "algebra-lemma-in-image" ], "ref_ids": [ 394 ] } ], "ref_ids": [] }, { "id": 445, "type": "theorem", "label": "algebra-lemma-injective-minimal-primes-in-image", "categories": [ "algebra" ], "title": "algebra-lemma-injective-minimal-primes-in-image", "contents": [ "Let $R \\subset S$ be an injective ring map.", "Then $\\Spec(S) \\to \\Spec(R)$", "hits all the minimal primes." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p \\subset R$ be a minimal prime.", "In this case $R_{\\mathfrak p}$ has a unique prime ideal.", "Hence it suffices to show that $S_{\\mathfrak p}$ is not zero.", "And this follows from the fact that localization is exact,", "see Proposition \\ref{proposition-localization-exact}." ], "refs": [ "algebra-proposition-localization-exact" ], "ref_ids": [ 1402 ] } ], "ref_ids": [] }, { "id": 446, "type": "theorem", "label": "algebra-lemma-image-dense-generic-points", "categories": [ "algebra" ], "title": "algebra-lemma-image-dense-generic-points", "contents": [ "Let $R \\to S$ be a ring map. The following are equivalent:", "\\begin{enumerate}", "\\item The kernel of $R \\to S$ consists of nilpotent elements.", "\\item The minimal primes of $R$ are in the image of", "$\\Spec(S) \\to \\Spec(R)$.", "\\item The image of $\\Spec(S) \\to \\Spec(R)$ is dense", "in $\\Spec(R)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $I = \\Ker(R \\to S)$. Note that", "$\\sqrt{(0)} = \\bigcap_{\\mathfrak q \\subset S} \\mathfrak q$, see", "Lemma \\ref{lemma-Zariski-topology}.", "Hence $\\sqrt{I} = \\bigcap_{\\mathfrak q \\subset S} R \\cap \\mathfrak q$.", "Thus $V(I) = V(\\sqrt{I})$ is the closure of the image of", "$\\Spec(S) \\to \\Spec(R)$.", "This shows that (1) is equivalent to (3). It is clear that", "(2) implies (3). Finally, assume (1). We may replace", "$R$ by $R/I$ and $S$ by $S/IS$ without affecting the topology", "of the spectra and the map. Hence the implication (1) $\\Rightarrow$ (2)", "follows from Lemma \\ref{lemma-injective-minimal-primes-in-image}." ], "refs": [ "algebra-lemma-Zariski-topology", "algebra-lemma-injective-minimal-primes-in-image" ], "ref_ids": [ 389, 445 ] } ], "ref_ids": [] }, { "id": 447, "type": "theorem", "label": "algebra-lemma-minimal-prime-image-minimal-prime", "categories": [ "algebra" ], "title": "algebra-lemma-minimal-prime-image-minimal-prime", "contents": [ "Let $R \\to S$ be a ring map. If a minimal prime $\\mathfrak p \\subset R$", "is in the image of $\\Spec(S) \\to \\Spec(R)$, then it is the image", "of a minimal prime." ], "refs": [], "proofs": [ { "contents": [ "Say $\\mathfrak p = \\mathfrak q \\cap R$. Then choose a minimal", "prime $\\mathfrak r \\subset S$ with $\\mathfrak r \\subset \\mathfrak q$, see", "Lemma \\ref{lemma-Zariski-topology}.", "By minimality of $\\mathfrak p$ we see that", "$\\mathfrak p = \\mathfrak r \\cap R$." ], "refs": [ "algebra-lemma-Zariski-topology" ], "ref_ids": [ 389 ] } ], "ref_ids": [] }, { "id": 448, "type": "theorem", "label": "algebra-lemma-Noetherian-permanence", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-permanence", "contents": [ "\\begin{slogan}", "Noetherian property is stable by passage to finite type extension", "and localization.", "\\end{slogan}", "Any finitely generated ring over a Noetherian ring", "is Noetherian. Any localization of a Noetherian ring", "is Noetherian." ], "refs": [], "proofs": [ { "contents": [ "The statement on localizations follows from the fact", "that any ideal $J \\subset S^{-1}R$ is of the form", "$I \\cdot S^{-1}R$. Any quotient $R/I$ of a Noetherian", "ring $R$ is Noetherian because any ideal $\\overline{J} \\subset R/I$", "is of the form $J/I$ for some ideal $I \\subset J \\subset R$.", "Thus it suffices to show that if $R$ is Noetherian so", "is $R[X]$. Suppose $J_1 \\subset J_2 \\subset \\ldots$ is an", "ascending chain of ideals in $R[X]$. Consider the ideals $I_{i, d}$", "defined as the ideal of elements of $R$ which occur as leading", "coefficients of degree $d$ polynomials in $J_i$.", "Clearly $I_{i, d} \\subset I_{i', d'}$ whenever", "$i \\leq i'$ and $d \\leq d'$. By the ascending chain condition", "in $R$ there are at most finitely many distinct ideals among all of", "the $I_{i, d}$.", "(Hint: Any infinite set of elements of", "$\\mathbf{N} \\times \\mathbf{N}$ contains an increasing", "infinite sequence.)", "Take $i_0$ so large that $I_{i, d} = I_{i_0, d}$", "for all $i \\geq i_0$ and all $d$. Suppose $f \\in J_i$ for some $i \\geq i_0$.", "By induction on the degree $d = \\deg(f)$ we show that $f \\in J_{i_0}$.", "Namely, there exists a $g\\in J_{i_0}$ whose degree is $d$ and which", "has the same leading coefficient as $f$. By induction", "$f - g \\in J_{i_0}$ and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 449, "type": "theorem", "label": "algebra-lemma-Noetherian-power-series", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-power-series", "contents": [ "If $R$ is a Noetherian ring, then so is the formal power", "series ring $R[[x_1, \\ldots, x_n]]$." ], "refs": [], "proofs": [ { "contents": [ "Since $R[[x_1, \\ldots, x_{n + 1}]] \\cong R[[x_1, \\ldots, x_n]][[x_{n + 1}]]$", "it suffices to prove the statement that $R[[x]]$ is Noetherian if", "$R$ is Noetherian. Let $I \\subset R[[x]]$ be a ideal.", "We have to show that $I$ is a finitely generated ideal.", "For each integer", "$d$ denote $I_d = \\{a \\in R \\mid ax^d + \\text{h.o.t.} \\in I\\}$.", "Then we see that $I_0 \\subset I_1 \\subset \\ldots$ stabilizes as $R$", "is Noetherian. Choose $d_0$ such that $I_{d_0} = I_{d_0 + 1} = \\ldots$.", "For each $d \\leq d_0$ choose elements $f_{d, j} \\in I \\cap (x^d)$,", "$j = 1, \\ldots, n_d$ such that if we write", "$f_{d, j} = a_{d, j}x^d + \\text{h.o.t}$ then $I_d = (a_{d, j})$.", "Denote $I' = (\\{f_{d, j}\\}_{d = 0, \\ldots, d_0, j = 1, \\ldots, n_d})$.", "Then it is clear that $I' \\subset I$. Pick $f \\in I$.", "First we may choose $c_{d, i} \\in R$ such that", "$$", "f - \\sum c_{d, i} f_{d, i} \\in (x^{d_0 + 1}) \\cap I.", "$$", "Next, we can choose $c_{i, 1} \\in R$, $i = 1, \\ldots, n_{d_0}$ such that", "$$", "f - \\sum c_{d, i} f_{d, i} - \\sum c_{i, 1}xf_{d_0, i} \\in (x^{d_0 + 2}) \\cap I.", "$$", "Next, we can choose $c_{i, 2} \\in R$, $i = 1, \\ldots, n_{d_0}$ such that", "$$", "f - \\sum c_{d, i} f_{d, i} - \\sum c_{i, 1}xf_{d_0, i}", "- \\sum c_{i, 2}x^2f_{d_0, i}", "\\in (x^{d_0 + 3}) \\cap I.", "$$", "And so on. In the end we see that", "$$", "f = \\sum c_{d, i} f_{d, i} +", "\\sum\\nolimits_i (\\sum\\nolimits_e c_{i, e} x^e)f_{d_0, i}", "$$", "is contained in $I'$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 450, "type": "theorem", "label": "algebra-lemma-obvious-Noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-obvious-Noetherian", "contents": [ "Any finite type algebra over a field is Noetherian.", "Any finite type algebra over $\\mathbf{Z}$ is Noetherian." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from Lemma \\ref{lemma-Noetherian-permanence}", "and the fact that fields are Noetherian rings and that", "$\\mathbf{Z}$ is Noetherian ring (because it is a", "principal ideal domain)." ], "refs": [ "algebra-lemma-Noetherian-permanence" ], "ref_ids": [ 448 ] } ], "ref_ids": [] }, { "id": 451, "type": "theorem", "label": "algebra-lemma-Noetherian-finite-type-is-finite-presentation", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-finite-type-is-finite-presentation", "contents": [ "Let $R$ be a Noetherian ring.", "\\begin{enumerate}", "\\item Any finite $R$-module is of finite presentation.", "\\item Any finite type $R$-algebra is of finite presentation over $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be a finite $R$-module. By", "Lemma \\ref{lemma-trivial-filter-finite-module}", "we can find a finite filtration of $M$ whose successive quotients are", "of the form $R/I$. Since any ideal is finitely generated, each of", "the quotients $R/I$ is finitely presented. Hence $M$ is finitely", "presented by", "Lemma \\ref{lemma-extension}.", "This proves (1).", "To see (2) note that any ideal of", "$R[x_1, \\ldots, x_n]$ is finitely generated by", "Lemma \\ref{lemma-Noetherian-permanence}." ], "refs": [ "algebra-lemma-trivial-filter-finite-module", "algebra-lemma-extension", "algebra-lemma-Noetherian-permanence" ], "ref_ids": [ 331, 330, 448 ] } ], "ref_ids": [] }, { "id": 452, "type": "theorem", "label": "algebra-lemma-Noetherian-topology", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-topology", "contents": [ "If $R$ is a Noetherian ring then $\\Spec(R)$", "is a Noetherian topological space, see Topology,", "Definition \\ref{topology-definition-noetherian}." ], "refs": [ "topology-definition-noetherian" ], "proofs": [ { "contents": [ "This is because any closed subset of $\\Spec(R)$", "is uniquely of the form $V(I)$ with $I$ a radical ideal,", "see Lemma \\ref{lemma-Zariski-topology}.", "And this correspondence is inclusion reversing.", "Thus the result follows from the definitions." ], "refs": [ "algebra-lemma-Zariski-topology" ], "ref_ids": [ 389 ] } ], "ref_ids": [ 8355 ] }, { "id": 453, "type": "theorem", "label": "algebra-lemma-Noetherian-irreducible-components", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-irreducible-components", "contents": [ "\\begin{slogan}", "A Noetherian affine scheme has finitely many generic points.", "\\end{slogan}", "If $R$ is a Noetherian ring then $\\Spec(R)$", "has finitely many irreducible components. In other words", "$R$ has finitely many minimal primes." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-Noetherian-topology} and", "Topology, Lemma \\ref{topology-lemma-Noetherian}", "we see there are finitely many irreducible components.", "By Lemma \\ref{lemma-irreducible} these correspond to", "minimal primes of $R$." ], "refs": [ "algebra-lemma-Noetherian-topology", "topology-lemma-Noetherian", "algebra-lemma-irreducible" ], "ref_ids": [ 452, 8220, 422 ] } ], "ref_ids": [] }, { "id": 454, "type": "theorem", "label": "algebra-lemma-Noetherian-base-change-finite-type", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-base-change-finite-type", "contents": [ "Let $R \\to S$ be a ring map. Let $R \\to R'$ be of finite type.", "If $S$ is Noetherian, then the base change $S' = R' \\otimes_R S$", "is Noetherian." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-base-change-finiteness} finite type is stable under", "base change. Thus $S \\to S'$ is of finite type. Since $S$ is Noetherian we", "can apply Lemma \\ref{lemma-Noetherian-permanence}." ], "refs": [ "algebra-lemma-base-change-finiteness", "algebra-lemma-Noetherian-permanence" ], "ref_ids": [ 373, 448 ] } ], "ref_ids": [] }, { "id": 455, "type": "theorem", "label": "algebra-lemma-Noetherian-field-extension", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-field-extension", "contents": [ "Let $k$ be a field and let $R$ be a Noetherian $k$-algebra.", "If $K/k$ is a finitely generated field extension then", "$K \\otimes_k R$ is Noetherian." ], "refs": [], "proofs": [ { "contents": [ "Since $K/k$ is a finitely generated field extension, there exists", "a finitely generated $k$-algebra $B \\subset K$ such that $K$ is", "the fraction field of $B$. In other words, $K = S^{-1}B$", "with $S = B \\setminus \\{0\\}$. Then $K \\otimes_k R = S^{-1}(B \\otimes_k R)$.", "Then $B \\otimes_k R$ is Noetherian by", "Lemma \\ref{lemma-Noetherian-base-change-finite-type}.", "Finally, $K \\otimes_k R = S^{-1}(B \\otimes_k R)$ is Noetherian by", "Lemma \\ref{lemma-Noetherian-permanence}." ], "refs": [ "algebra-lemma-Noetherian-base-change-finite-type", "algebra-lemma-Noetherian-permanence" ], "ref_ids": [ 454, 448 ] } ], "ref_ids": [] }, { "id": 456, "type": "theorem", "label": "algebra-lemma-subring-of-local-ring", "categories": [ "algebra" ], "title": "algebra-lemma-subring-of-local-ring", "contents": [ "Let $R$ be a ring and $\\mathfrak p \\subset R$ be a prime.", "There exists an $f \\in R$, $f \\not \\in \\mathfrak p$ such", "that $R_f \\to R_\\mathfrak p$ is injective in each of the", "following cases", "\\begin{enumerate}", "\\item $R$ is a domain,", "\\item $R$ is Noetherian, or", "\\item $R$ is reduced and has finitely many minimal primes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $R$ is a domain, then $R \\subset R_\\mathfrak p$, hence $f = 1$ works.", "If $R$ is Noetherian, then the kernel $I$ of $R \\to R_\\mathfrak p$", "is a finitely generated ideal and we can find", "$f \\in R$, $f \\not \\in \\mathfrak p$ such that $IR_f = 0$.", "For this $f$ the map $R_f \\to R_\\mathfrak p$ is injective", "and $f$ works. If $R$ is reduced with finitely", "many minimal primes $\\mathfrak p_1, \\ldots, \\mathfrak p_n$,", "then we can choose", "$f \\in \\bigcap_{\\mathfrak p_i \\not \\subset \\mathfrak p} \\mathfrak p_i$,", "$f \\not \\in \\mathfrak p$. Indeed, if $\\mathfrak{p}_i\\not\\subset", "\\mathfrak{p}$ then there exist $f_i \\in \\mathfrak{p}_i$,", "$f_i \\not\\in \\mathfrak{p}$ and $f = \\prod f_i$ works.", "For this $f$ we have $R_f \\subset R_\\mathfrak p$ because the minimal", "primes of $R_f$ correspond to minimal primes of $R_\\mathfrak p$", "and we can apply Lemma \\ref{lemma-reduced-ring-sub-product-fields}", "(some details omitted)." ], "refs": [ "algebra-lemma-reduced-ring-sub-product-fields" ], "ref_ids": [ 419 ] } ], "ref_ids": [] }, { "id": 457, "type": "theorem", "label": "algebra-lemma-surjective-endo-noetherian-ring-is-iso", "categories": [ "algebra" ], "title": "algebra-lemma-surjective-endo-noetherian-ring-is-iso", "contents": [ "Any surjective endomorphism of a Noetherian ring is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "If $f : R \\to R$ were such an endomorphism but not injective, then", "$$", "\\Ker(f) \\subset \\Ker(f \\circ f) \\subset", "\\Ker(f \\circ f \\circ f) \\subset \\ldots", "$$", "would be a strictly increasing chain of ideals." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 458, "type": "theorem", "label": "algebra-lemma-locally-nilpotent", "categories": [ "algebra" ], "title": "algebra-lemma-locally-nilpotent", "contents": [ "Let $R \\to R'$ be a ring map and let $I \\subset R$ be a locally nilpotent", "ideal. Then $IR'$ is a locally nilpotent ideal of $R'$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that if $x, y \\in R'$ are nilpotent, then", "$x + y$ is nilpotent too. Namely, if $x^n = 0$ and $y^m = 0$, then", "$(x + y)^{n + m - 1} = 0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 459, "type": "theorem", "label": "algebra-lemma-locally-nilpotent-unit", "categories": [ "algebra" ], "title": "algebra-lemma-locally-nilpotent-unit", "contents": [ "Let $R$ be a ring and let $I \\subset R$ be a locally nilpotent", "ideal.", "An element $x$ of $R$ is a unit if and only if the image of $x$", "in $R/I$ is a unit." ], "refs": [], "proofs": [ { "contents": [ "If $x$ is a unit in $R$, then its image is clearly a unit in $R/I$.", "It remains to prove the converse.", "Assume the image of $y \\in R$ in $R/I$ is the inverse of the image of $x$.", "Then $xy = 1 - z$ for some $z \\in I$.", "This means that $1\\equiv z$ modulo $xR$.", "Since $z$ lies in the locally nilpotent ideal", "$I$, we have $z^N = 0$ for some sufficiently large $N$.", "It follows that $1 = 1^N \\equiv z^N = 0$ modulo $xR$.", "In other words, $x$ divides $1$ and is hence a unit." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 460, "type": "theorem", "label": "algebra-lemma-Noetherian-power", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-power", "contents": [ "\\begin{slogan}", "An ideal in a Noetherian ring is nilpotent if each element", "of the ideal is nilpotent.", "\\end{slogan}", "Let $R$ be a Noetherian ring. Let $I, J$ be ideals of $R$.", "Suppose $J \\subset \\sqrt{I}$. Then $J^n \\subset I$ for some $n$.", "In particular, in a Noetherian ring the notions of", "``locally nilpotent ideal''", "and ``nilpotent ideal'' coincide." ], "refs": [], "proofs": [ { "contents": [ "Say $J = (f_1, \\ldots, f_s)$.", "By assumption $f_i^{d_i} \\in I$.", "Take $n = d_1 + d_2 + \\ldots + d_s + 1$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 461, "type": "theorem", "label": "algebra-lemma-lift-idempotents", "categories": [ "algebra" ], "title": "algebra-lemma-lift-idempotents", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be a locally nilpotent ideal.", "Then $R \\to R/I$ induces a bijection on idempotents." ], "refs": [], "proofs": [ { "contents": [ "[First proof of Lemma \\ref{lemma-lift-idempotents}]", "As $I$ is locally nilpotent it is contained in every prime ideal.", "Hence $\\Spec(R/I) = V(I) = \\Spec(R)$. Hence the", "lemma follows from Lemma \\ref{lemma-disjoint-decomposition}." ], "refs": [ "algebra-lemma-lift-idempotents", "algebra-lemma-disjoint-decomposition" ], "ref_ids": [ 461, 405 ] } ], "ref_ids": [] }, { "id": 462, "type": "theorem", "label": "algebra-lemma-lift-idempotents-noncommutative", "categories": [ "algebra" ], "title": "algebra-lemma-lift-idempotents-noncommutative", "contents": [ "Let $A$ be a possibly noncommutative algebra.", "Let $e \\in A$ be an element such that $x = e^2 - e$ is nilpotent.", "Then there exists an idempotent of the form", "$e' = e + x(\\sum a_{i, j}e^ix^j) \\in A$", "with $a_{i, j} \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the ring $R_n = \\mathbf{Z}[e]/((e^2 - e)^n)$. It is clear that", "if we can prove the result for each $R_n$ then the lemma follows.", "In $R_n$ consider the ideal $I = (e^2 - e)$ and apply", "Lemma \\ref{lemma-lift-idempotents}." ], "refs": [ "algebra-lemma-lift-idempotents" ], "ref_ids": [ 461 ] } ], "ref_ids": [] }, { "id": 463, "type": "theorem", "label": "algebra-lemma-lift-nth-roots", "categories": [ "algebra" ], "title": "algebra-lemma-lift-nth-roots", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be a locally nilpotent ideal.", "Let $n \\geq 1$ be an integer which is invertible in $R/I$. Then", "\\begin{enumerate}", "\\item the $n$th power map $1 + I \\to 1 + I$, $1 + x \\mapsto (1 + x)^n$", "is a bijection,", "\\item a unit of $R$ is a $n$th power if and only if its image in $R/I$", "is an $n$th power.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $a \\in R$ be a unit whose image in $R/I$ is the same as the image", "of $b^n$ with $b \\in R$. Then $b$ is a unit", "(Lemma \\ref{lemma-locally-nilpotent-unit}) and", "$ab^{-n} = 1 + x$ for some $x \\in I$. Hence $ab^{-n} = c^n$ by", "part (1). Thus (2) follows from (1).", "\\medskip\\noindent", "Proof of (1). This is true because there is an inverse to the", "map $1 + x \\mapsto (1 + x)^n$. Namely, we can consider the map", "which sends $1 + x$ to", "\\begin{align*}", "(1 + x)^{1/n}", "& =", "1 + {1/n \\choose 1}x +", "{1/n \\choose 2}x^2 +", "{1/n \\choose 3}x^3 + \\ldots \\\\", "& =", "1 + \\frac{1}{n} x + \\frac{1 - n}{2n^2}x^2 +", "\\frac{(1 - n)(1 - 2n)}{6n^3}x^3 + \\ldots", "\\end{align*}", "as in elementary calculus. This makes sense because the series is finite", "as $x^k = 0$ for all $k \\gg 0$ and each coefficient", "${1/n \\choose k} \\in \\mathbf{Z}[1/n]$ (details omitted; observe that", "$n$ is invertible in $R$ by Lemma \\ref{lemma-locally-nilpotent-unit})." ], "refs": [ "algebra-lemma-locally-nilpotent-unit", "algebra-lemma-locally-nilpotent-unit" ], "ref_ids": [ 459, 459 ] } ], "ref_ids": [] }, { "id": 464, "type": "theorem", "label": "algebra-lemma-invert-closed-quotient", "categories": [ "algebra" ], "title": "algebra-lemma-invert-closed-quotient", "contents": [ "Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset.", "Assume the image of the map $\\Spec(S^{-1}R) \\to \\Spec(R)$", "is closed. Then $S^{-1}R \\cong R/I$ for some ideal $I \\subset R$." ], "refs": [], "proofs": [ { "contents": [ "Let $I = \\Ker(R \\to S^{-1}R)$ so that $V(I)$ contains the image.", "Say the image is the closed subset $V(I') \\subset \\Spec(R)$ for", "some ideal $I' \\subset R$. So $V(I') \\subset V(I)$.", "For $f \\in I'$ we see that $f/1 \\in S^{-1}R$", "is contained in every prime ideal. Hence $f^n$ maps to zero in $S^{-1}R$", "for some $n \\geq 1$ (Lemma \\ref{lemma-Zariski-topology}).", "Hence $V(I') = V(I)$.", "Then this implies every $g \\in S$ is invertible mod $I$.", "Hence we get ring maps $R/I \\to S^{-1}R$ and $S^{-1}R \\to R/I$.", "The first map is injective by choice of $I$.", "The second is the map $S^{-1}R \\to S^{-1}(R/I) = R/I$ which", "has kernel $S^{-1}I$ because localization is exact.", "Since $S^{-1}I = 0$ we see also the second map is injective.", "Hence $S^{-1}R \\cong R/I$." ], "refs": [ "algebra-lemma-Zariski-topology" ], "ref_ids": [ 389 ] } ], "ref_ids": [] }, { "id": 465, "type": "theorem", "label": "algebra-lemma-invert-closed-split", "categories": [ "algebra" ], "title": "algebra-lemma-invert-closed-split", "contents": [ "Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset.", "Assume the image of the map $\\Spec(S^{-1}R) \\to \\Spec(R)$", "is closed. If $R$ is Noetherian, or $\\Spec(R)$ is a", "Noetherian topological space, or $S$ is finitely generated as a monoid,", "then $R \\cong S^{-1}R \\times R'$ for some ring $R'$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-invert-closed-quotient} we have $S^{-1}R \\cong R/I$", "for some ideal $I \\subset R$. By Lemma \\ref{lemma-disjoint-implies-product}", "it suffices to show that $V(I)$ is open.", "If $R$ is Noetherian then $\\Spec(R)$ is a Noetherian", "topological space, see Lemma \\ref{lemma-Noetherian-topology}.", "If $\\Spec(R)$ is a Noetherian topological space,", "then the complement $\\Spec(R) \\setminus V(I)$ is quasi-compact, see", "Topology, Lemma \\ref{topology-lemma-Noetherian-quasi-compact}.", "Hence there exist finitely many $f_1, \\ldots, f_n \\in I$ such", "that $V(I) = V(f_1, \\ldots, f_n)$.", "Since each $f_i$ maps to zero in $S^{-1}R$", "there exists a $g \\in S$ such that $gf_i = 0$ for", "$i = 1, \\ldots, n$. Hence $D(g) = V(I)$ as desired.", "In case $S$ is finitely generated as a monoid, say $S$ is generated", "by $g_1, \\ldots, g_m$, then $S^{-1}R \\cong R_{g_1 \\ldots g_m}$", "and we conclude that $V(I) = D(g_1 \\ldots g_m)$." ], "refs": [ "algebra-lemma-invert-closed-quotient", "algebra-lemma-disjoint-implies-product", "algebra-lemma-Noetherian-topology", "topology-lemma-Noetherian-quasi-compact" ], "ref_ids": [ 464, 415, 452, 8239 ] } ], "ref_ids": [] }, { "id": 466, "type": "theorem", "label": "algebra-lemma-field-finite-type-over-domain", "categories": [ "algebra" ], "title": "algebra-lemma-field-finite-type-over-domain", "contents": [ "Let $R$ be a ring. Let $K$ be a field.", "If $R \\subset K$ and $K$ is of finite type over $R$,", "then there exists an $f \\in R$ such that $R_f$ is a field,", "and $R_f \\subset K$ is a finite field extension." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-characterize-image-finite-type} there", "exist a nonempty open $U \\subset \\Spec(R)$", "contained in the image $\\{(0)\\}$ of $\\Spec(K) \\to \\Spec(R)$.", "Choose $f \\in R$, $f \\not = 0$ such that $D(f) \\subset U$, i.e.,", "$D(f) = \\{(0)\\}$. Then $R_f$ is a domain whose spectrum has exactly one", "point and $R_f$ is a field. Then $K$ is a finitely generated algebra", "over the field $R_f$ and hence a finite field extension of", "$R_f$ by the Hilbert Nullstellensatz (Theorem \\ref{theorem-nullstellensatz})." ], "refs": [ "algebra-lemma-characterize-image-finite-type", "algebra-theorem-nullstellensatz" ], "ref_ids": [ 442, 316 ] } ], "ref_ids": [] }, { "id": 467, "type": "theorem", "label": "algebra-lemma-finite-type-field-Jacobson", "categories": [ "algebra" ], "title": "algebra-lemma-finite-type-field-Jacobson", "contents": [ "Any algebra of finite type over a field is Jacobson." ], "refs": [], "proofs": [ { "contents": [ "This follows from Theorem \\ref{theorem-nullstellensatz}", "and Definition \\ref{definition-ring-jacobson}." ], "refs": [ "algebra-theorem-nullstellensatz", "algebra-definition-ring-jacobson" ], "ref_ids": [ 316, 1449 ] } ], "ref_ids": [] }, { "id": 468, "type": "theorem", "label": "algebra-lemma-jacobson-prime", "categories": [ "algebra" ], "title": "algebra-lemma-jacobson-prime", "contents": [ "Let $R$ be a ring. If every prime ideal of $R$ is the", "intersection of the maximal ideals containing it,", "then $R$ is Jacobson." ], "refs": [], "proofs": [ { "contents": [ "This is immediately clear from the fact that", "every radical ideal $I \\subset R$ is the", "intersection of the primes containing it.", "See Lemma \\ref{lemma-Zariski-topology}." ], "refs": [ "algebra-lemma-Zariski-topology" ], "ref_ids": [ 389 ] } ], "ref_ids": [] }, { "id": 469, "type": "theorem", "label": "algebra-lemma-jacobson", "categories": [ "algebra" ], "title": "algebra-lemma-jacobson", "contents": [ "A ring $R$ is Jacobson if and only if $\\Spec(R)$", "is Jacobson, see Topology,", "Definition \\ref{topology-definition-space-jacobson}." ], "refs": [ "topology-definition-space-jacobson" ], "proofs": [ { "contents": [ "Suppose $R$ is Jacobson. Let $Z \\subset \\Spec(R)$", "be a closed subset. We have to show that the set of closed", "points in $Z$ is dense in $Z$. Let $U \\subset \\Spec(R)$", "be an open such that $U \\cap Z$ is nonempty.", "We have to show $Z \\cap U$ contains a closed point", "of $\\Spec(R)$. We may", "assume $U = D(f)$ as standard opens form a basis for the", "topology on $\\Spec(R)$. According to", "Lemma \\ref{lemma-Zariski-topology} we may assume that", "$Z = V(I)$, where $I$ is a radical ideal. We see also", "that $f \\not \\in I$. By assumption, there exists a", "maximal ideal $\\mathfrak m \\subset R$ such that", "$I \\subset \\mathfrak m$ but $f \\not\\in \\mathfrak m$.", "Hence $\\mathfrak m \\in D(f) \\cap V(I) = U \\cap Z$ as desired.", "\\medskip\\noindent", "Conversely, suppose that $\\Spec(R)$ is Jacobson.", "Let $I \\subset R$ be a radical ideal. Let", "$J = \\cap_{I \\subset \\mathfrak m} \\mathfrak m$", "be the intersection of the maximal ideals containing $I$.", "Clearly $J$ is a radical ideal, $V(J) \\subset V(I)$, and", "$V(J)$ is the smallest closed subset of $V(I)$ containing", "all the closed points of $V(I)$. By assumption we see that", "$V(J) = V(I)$. But Lemma \\ref{lemma-Zariski-topology}", "shows there is a bijection between Zariski closed", "sets and radical ideals, hence $I = J$ as desired." ], "refs": [ "algebra-lemma-Zariski-topology", "algebra-lemma-Zariski-topology" ], "ref_ids": [ 389, 389 ] } ], "ref_ids": [ 8364 ] }, { "id": 470, "type": "theorem", "label": "algebra-lemma-characterize-jacobson", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-jacobson", "contents": [ "Let $R$ be a ring. If $R$ is not Jacobson there exist", "a prime $\\mathfrak p \\subset R$, an element $f \\in R$", "such that the following hold", "\\begin{enumerate}", "\\item $\\mathfrak p$ is not a maximal ideal,", "\\item $f \\not \\in \\mathfrak p$,", "\\item $V(\\mathfrak p) \\cap D(f) = \\{\\mathfrak p\\}$, and", "\\item $(R/\\mathfrak p)_f$ is a field.", "\\end{enumerate}", "On the other hand, if $R$ is Jacobson, then for any pair $(\\mathfrak p, f)$", "such that (1) and (2) hold the set $V(\\mathfrak p) \\cap D(f)$ is", "infinite." ], "refs": [], "proofs": [ { "contents": [ "Assume $R$ is not Jacobson.", "By Lemma \\ref{lemma-jacobson} this means there exists an", "closed subset $T \\subset \\Spec(R)$", "whose set $T_0 \\subset T$ of closed points is not dense in $T$.", "Choose an $f \\in R$ such that $T_0 \\subset V(f)$ but", "$T \\not \\subset V(f)$. Note that $T \\cap D(f)$", "is homeomorphic to $\\Spec((R/I)_f)$ if $T = V(I)$, see", "Lemmas \\ref{lemma-spec-closed} and \\ref{lemma-standard-open}.", "As any ring has a maximal ideal", "(Lemma \\ref{lemma-Zariski-topology}) we can choose a closed point $t$ of", "space $T \\cap D(f)$. Then $t$ corresponds to a prime ideal", "$\\mathfrak p \\subset R$ which is not maximal (as $t \\not \\in T_0$).", "Thus (1) holds. By construction $f \\not \\in \\mathfrak p$, hence (2).", "As $t$ is a closed point of $T \\cap D(f)$ we see that", "$V(\\mathfrak p) \\cap D(f) = \\{\\mathfrak p\\}$, i.e., (3) holds. Hence we", "conclude that $(R/\\mathfrak p)_f$ is a domain whose", "spectrum has one point, hence (4) holds", "(for example combine Lemmas \\ref{lemma-characterize-local-ring} and", "\\ref{lemma-minimal-prime-reduced-ring}).", "\\medskip\\noindent", "Conversely, suppose that $R$ is Jacobson and $(\\mathfrak p, f)$", "satisfy (1) and (2). If", "$V(\\mathfrak p) \\cap D(f) =", "\\{\\mathfrak p, \\mathfrak q_1, \\ldots, \\mathfrak q_t\\}$", "then $\\mathfrak p \\not = \\mathfrak q_i$", "implies there exists an element $g \\in R$ such that $g \\not \\in \\mathfrak p$", "but $g \\in \\mathfrak q_i$ for all $i$. Hence", "$V(\\mathfrak p) \\cap D(fg) = \\{\\mathfrak p\\}$ which", "is impossible since each locally closed subset of $\\Spec(R)$", "contains at least one closed point as $\\Spec(R)$ is", "a Jacobson topological space." ], "refs": [ "algebra-lemma-jacobson", "algebra-lemma-spec-closed", "algebra-lemma-standard-open", "algebra-lemma-Zariski-topology", "algebra-lemma-characterize-local-ring", "algebra-lemma-minimal-prime-reduced-ring" ], "ref_ids": [ 469, 393, 392, 389, 397, 418 ] } ], "ref_ids": [] }, { "id": 471, "type": "theorem", "label": "algebra-lemma-pid-jacobson", "categories": [ "algebra" ], "title": "algebra-lemma-pid-jacobson", "contents": [ "The ring $\\mathbf{Z}$ is a Jacobson ring.", "More generally, let $R$ be a ring such that", "\\begin{enumerate}", "\\item $R$ is a domain,", "\\item $R$ is Noetherian,", "\\item any nonzero prime ideal is a maximal ideal, and", "\\item $R$ has infinitely many maximal ideals.", "\\end{enumerate}", "Then $R$ is a Jacobson ring." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ satisfy (1), (2), (3) and (4). The statement", "means that $(0) = \\bigcap_{\\mathfrak m \\subset R} \\mathfrak m$.", "Since $R$ has infinitely many maximal ideals it suffices to", "show that any nonzero $x \\in R$ is contained in at most", "finitely many maximal ideals, in other words that $V(x)$ is finite.", "By Lemma \\ref{lemma-spec-closed}", "we see that $V(x)$ is homeomorphic to $\\Spec(R/xR)$.", "By assumption (3) every prime of $R/xR$ is minimal and hence", "corresponds to an irreducible component of $\\Spec(R)$", "(Lemma \\ref{lemma-irreducible}).", "As $R/xR$ is Noetherian, the topological space $\\Spec(R/xR)$", "is Noetherian (Lemma \\ref{lemma-Noetherian-topology})", "and has finitely many irreducible components", "(Topology, Lemma \\ref{topology-lemma-Noetherian}).", "Thus $V(x)$ is finite as desired." ], "refs": [ "algebra-lemma-spec-closed", "algebra-lemma-irreducible", "algebra-lemma-Noetherian-topology", "topology-lemma-Noetherian" ], "ref_ids": [ 393, 422, 452, 8220 ] } ], "ref_ids": [] }, { "id": 472, "type": "theorem", "label": "algebra-lemma-finite-residue-extension-closed", "categories": [ "algebra" ], "title": "algebra-lemma-finite-residue-extension-closed", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak m \\subset R$ be a maximal ideal.", "Let $\\mathfrak q \\subset S$ be a prime ideal", "lying over $\\mathfrak m$ such that $\\kappa(\\mathfrak m)", "\\subset \\kappa(\\mathfrak q)$ is an algebraic field extension.", "Then $\\mathfrak q$ is a maximal ideal of $S$." ], "refs": [], "proofs": [ { "contents": [ "Consider the diagram", "$$", "\\xymatrix{", "S \\ar[r] & S/\\mathfrak q \\ar[r] & \\kappa(\\mathfrak q) \\\\", "R \\ar[r] \\ar[u] & R/\\mathfrak m \\ar[u]", "}", "$$", "We see that $\\kappa(\\mathfrak m) \\subset S/\\mathfrak q \\subset", "\\kappa(\\mathfrak q)$. Because the field extension", "$\\kappa(\\mathfrak m) \\subset \\kappa(\\mathfrak q)$", "is algebraic, any ring between $\\kappa(\\mathfrak m)$", "and $\\kappa(\\mathfrak q)$ is a field", "(Fields, Lemma \\ref{fields-lemma-subalgebra-algebraic-extension-field}).", "Thus $S/\\mathfrak q$ is a field, and a posteriori equal", "to $\\kappa(\\mathfrak q)$." ], "refs": [ "fields-lemma-subalgebra-algebraic-extension-field" ], "ref_ids": [ 4457 ] } ], "ref_ids": [] }, { "id": 473, "type": "theorem", "label": "algebra-lemma-dimension", "categories": [ "algebra" ], "title": "algebra-lemma-dimension", "contents": [ "Suppose that $k$ is a field and suppose that $V$ is a nonzero vector", "space over $k$. Assume the dimension of $V$ (which is a cardinal number)", "is smaller than the cardinality of $k$. Then for any linear operator", "$T : V \\to V$ there exists some monic polynomial $P(t) \\in k[t]$ such that", "$P(T)$ is not invertible." ], "refs": [], "proofs": [ { "contents": [ "If not then $V$ inherits the structure of a vector space over", "the field $k(t)$. But the dimension of $k(t)$ over $k$ is", "at least the cardinality of $k$ for example due to the fact that the elements", "$\\frac{1}{t - \\lambda}$ are $k$-linearly independent." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 474, "type": "theorem", "label": "algebra-lemma-base-change-Jacobson", "categories": [ "algebra" ], "title": "algebra-lemma-base-change-Jacobson", "contents": [ "Let $k$ be a field. Let $S$ be a $k$-algebra.", "For any field extension $k \\subset K$ whose cardinality is larger", "than the cardinality of $S$ we have", "\\begin{enumerate}", "\\item for every maximal ideal $\\mathfrak m$ of $S_K$ the field", "$\\kappa(\\mathfrak m)$ is algebraic over $K$, and", "\\item $S_K$ is a Jacobson ring.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose $k \\subset K$ such that the cardinality of $K$ is greater", "than the cardinality of $S$. Since the elements of $S$ generate", "the $K$-algebra $S_K$ we see that", "Theorem \\ref{theorem-uncountable-nullstellensatz}", "applies." ], "refs": [ "algebra-theorem-uncountable-nullstellensatz" ], "ref_ids": [ 317 ] } ], "ref_ids": [] }, { "id": 475, "type": "theorem", "label": "algebra-lemma-Jacobson-invert-element", "categories": [ "algebra" ], "title": "algebra-lemma-Jacobson-invert-element", "contents": [ "Let $R$ be a Jacobson ring. Let $f \\in R$. The ring $R_f$ is Jacobson and", "maximal ideals of $R_f$ correspond to maximal ideals of $R$ not containing $f$." ], "refs": [], "proofs": [ { "contents": [ "By Topology, Lemma \\ref{topology-lemma-jacobson-inherited}", "we see that $D(f) = \\Spec(R_f)$ is Jacobson and", "that closed points of $D(f)$", "correspond to closed points in $\\Spec(R)$", "which happen to lie in $D(f)$. Thus $R_f$ is Jacobson by", "Lemma \\ref{lemma-jacobson}." ], "refs": [ "topology-lemma-jacobson-inherited", "algebra-lemma-jacobson" ], "ref_ids": [ 8279, 469 ] } ], "ref_ids": [] }, { "id": 476, "type": "theorem", "label": "algebra-lemma-Jacobson-mod-ideal", "categories": [ "algebra" ], "title": "algebra-lemma-Jacobson-mod-ideal", "contents": [ "Let $R$ be a Jacobson ring. Let $I \\subset R$ be an ideal.", "The ring $R/I$ is Jacobson and maximal ideals", "of $R/I$ correspond to maximal ideals of $R$ containing $I$." ], "refs": [], "proofs": [ { "contents": [ "The proof is the same as the proof of", "Lemma \\ref{lemma-Jacobson-invert-element}." ], "refs": [ "algebra-lemma-Jacobson-invert-element" ], "ref_ids": [ 475 ] } ], "ref_ids": [] }, { "id": 477, "type": "theorem", "label": "algebra-lemma-silly-jacobson", "categories": [ "algebra" ], "title": "algebra-lemma-silly-jacobson", "contents": [ "Let $R$ be a Jacobson ring. Let $K$ be a field. Let $R \\subset K$ and", "$K$ is of finite type over $R$. Then $R$ is a field and $K/R$", "is a finite field extension." ], "refs": [], "proofs": [ { "contents": [ "First note that $R$ is a domain.", "By Lemma \\ref{lemma-field-finite-type-over-domain}", "we see that $R_f$ is a field and $K/R_f$ is a finite field extension", "for some nonzero $f \\in R$. Hence $(0)$ is a maximal ideal of $R_f$", "and by", "Lemma \\ref{lemma-Jacobson-invert-element}", "we conclude $(0)$ is a maximal ideal of $R$." ], "refs": [ "algebra-lemma-field-finite-type-over-domain", "algebra-lemma-Jacobson-invert-element" ], "ref_ids": [ 466, 475 ] } ], "ref_ids": [] }, { "id": 478, "type": "theorem", "label": "algebra-lemma-corollary-jacobson", "categories": [ "algebra" ], "title": "algebra-lemma-corollary-jacobson", "contents": [ "Any finite type algebra over $\\mathbf{Z}$ is Jacobson." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-pid-jacobson} and", "Proposition \\ref{proposition-Jacobson-permanence}." ], "refs": [ "algebra-lemma-pid-jacobson", "algebra-proposition-Jacobson-permanence" ], "ref_ids": [ 471, 1405 ] } ], "ref_ids": [] }, { "id": 479, "type": "theorem", "label": "algebra-lemma-image-finite-type-map-Jacobson-rings", "categories": [ "algebra" ], "title": "algebra-lemma-image-finite-type-map-Jacobson-rings", "contents": [ "Let $R \\to S$ be a finite type ring map of Jacobson rings.", "Denote $X = \\Spec(R)$ and $Y = \\Spec(S)$.", "Write $f : Y \\to X$ the induced", "map of spectra. Let $E \\subset Y = \\Spec(S)$ be a", "constructible set. Denote with a subscript ${}_0$ the set", "of closed points of a topological space.", "\\begin{enumerate}", "\\item We have $f(E)_0 = f(E_0) = X_0 \\cap f(E)$.", "\\item A point $\\xi \\in X$ is in $f(E)$ if and only if", "$\\overline{\\{\\xi\\}} \\cap f(E_0)$ is dense in $\\overline{\\{\\xi\\}}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have a commutative diagram of continuous maps", "$$", "\\xymatrix{", "E \\ar[r] \\ar[d] & Y \\ar[d] \\\\", "f(E) \\ar[r] & X", "}", "$$", "Suppose $x \\in f(E)$ is closed in $f(E)$. Then $f^{-1}(\\{x\\})\\cap E$", "is nonempty and closed in $E$. Applying", "Topology, Lemma \\ref{topology-lemma-jacobson-inherited}", "to both inclusions", "$$", "f^{-1}(\\{x\\}) \\cap E \\subset E \\subset Y", "$$", "we find there exists a point $y \\in f^{-1}(\\{x\\}) \\cap E$ which is", "closed in $Y$. In other words, there exists $y \\in Y_0$ and $y \\in E_0$", "mapping to $x$. Hence $x \\in f(E_0)$.", "This proves that $f(E)_0 \\subset f(E_0)$.", "Proposition \\ref{proposition-Jacobson-permanence} implies that", "$f(E_0) \\subset X_0 \\cap f(E)$. The inclusion", "$X_0 \\cap f(E) \\subset f(E)_0$ is trivial. This proves the", "first assertion.", "\\medskip\\noindent", "Suppose that $\\xi \\in f(E)$. According to", "Lemma \\ref{lemma-characterize-image-finite-type}", "the set $f(E) \\cap \\overline{\\{\\xi\\}}$ contains a dense", "open subset of $\\overline{\\{\\xi\\}}$. Since $X$ is Jacobson", "we conclude that $f(E) \\cap \\overline{\\{\\xi\\}}$ contains a", "dense set of closed points, see Topology,", "Lemma \\ref{topology-lemma-jacobson-inherited}.", "We conclude by part (1) of the lemma.", "\\medskip\\noindent", "On the other hand, suppose that $\\overline{\\{\\xi\\}} \\cap f(E_0)$", "is dense in $\\overline{\\{\\xi\\}}$. By", "Lemma \\ref{lemma-constructible-is-image}", "there exists a ring map $S \\to S'$ of finite presentation", "such that $E$ is the image of $Y' := \\Spec(S') \\to Y$.", "Then $E_0$ is the image of $Y'_0$ by the first part of the", "lemma applied to the ring map $S \\to S'$. Thus we may assume that", "$E = Y$ by replacing $S$ by $S'$. Suppose $\\xi$ corresponds", "to $\\mathfrak p \\subset R$. Consider the diagram", "$$", "\\xymatrix{", "S \\ar[r] & S/\\mathfrak p S \\\\", "R \\ar[r] \\ar[u] & R/\\mathfrak p \\ar[u]", "}", "$$", "This diagram and the density of $f(Y_0) \\cap V(\\mathfrak p)$", "in $V(\\mathfrak p)$", "shows that the morphism $R/\\mathfrak p \\to S/\\mathfrak p S$", "satisfies condition (2) of", "Lemma \\ref{lemma-domain-image-dense-set-points-generic-point}.", "Hence we conclude", "there exists a prime $\\overline{\\mathfrak q} \\subset S/\\mathfrak pS$", "mapping to $(0)$. In other words the inverse image $\\mathfrak q$", "of $\\overline{\\mathfrak q}$ in $S$ maps to $\\mathfrak p$ as desired." ], "refs": [ "topology-lemma-jacobson-inherited", "algebra-proposition-Jacobson-permanence", "algebra-lemma-characterize-image-finite-type", "topology-lemma-jacobson-inherited", "algebra-lemma-constructible-is-image", "algebra-lemma-domain-image-dense-set-points-generic-point" ], "ref_ids": [ 8279, 1405, 442, 8279, 435, 444 ] } ], "ref_ids": [] }, { "id": 480, "type": "theorem", "label": "algebra-lemma-conclude-jacobson-Noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-conclude-jacobson-Noetherian", "contents": [ "With notation as above. Assume that $R$ is a Noetherian Jacobson ring.", "Further assume $R \\to S$ is of finite type.", "There is a commutative diagram", "$$", "\\xymatrix{", "\\text{Constr}(Y) \\ar[r]^{E \\mapsto E_0} \\ar[d]^{E \\mapsto f(E)} &", "\\text{Constr}(Y_0) \\ar[d]^{E \\mapsto f(E)} \\\\", "\\text{Constr}(X) \\ar[r]^{E \\mapsto E_0} &", "\\text{Constr}(X_0)", "}", "$$", "where the horizontal arrows are the bijections from", "Topology, Lemma \\ref{topology-lemma-jacobson-equivalent-constructible}." ], "refs": [ "topology-lemma-jacobson-equivalent-constructible" ], "proofs": [ { "contents": [ "Since $R \\to S$ is of finite type, it is of finite presentation,", "see Lemma \\ref{lemma-Noetherian-finite-type-is-finite-presentation}.", "Thus the image of a constructible set in $X$ is constructible", "in $Y$ by Chevalley's theorem", "(Theorem \\ref{theorem-chevalley}).", "Combined with", "Lemma \\ref{lemma-image-finite-type-map-Jacobson-rings}", "the lemma follows." ], "refs": [ "algebra-lemma-Noetherian-finite-type-is-finite-presentation", "algebra-theorem-chevalley", "algebra-lemma-image-finite-type-map-Jacobson-rings" ], "ref_ids": [ 451, 315, 479 ] } ], "ref_ids": [ 8282 ] }, { "id": 481, "type": "theorem", "label": "algebra-lemma-characterize-integral-element", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-integral-element", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Let $y \\in S$. If there exists a", "finite $R$-submodule $M$ of $S$ such that $1 \\in M$ and $yM \\subset M$,", "then $y$ is integral over $R$." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1 = 1 \\in M$ and $x_i \\in M$, $i = 2, \\ldots, n$ be a finite set of", "elements generating $M$ as an $R$-module.", "Write $yx_i = \\sum \\varphi(a_{ij}) x_j$", "for some $a_{ij} \\in R$. Let $P(T) \\in R[T]$ be", "the characteristic polynomial of the $n \\times n$ matrix", "$A = (a_{ij})$. By", "Lemma \\ref{lemma-charpoly}", "we see $P(A) = 0$. By construction the map $\\pi : R^n \\to M$,", "$(a_1, \\ldots, a_n) \\mapsto \\sum \\varphi(a_i) x_i$", "commutes with $A : R^n \\to R^n$ and multiplication by $y$. In a formula", "$\\pi(Av) = y\\pi(v)$. Thus $P(y) = P(y) \\cdot 1", "= P(y) \\cdot x_1 = P(y) \\cdot \\pi((1, 0, \\ldots, 0))", "= \\pi(P(A)(1, 0, \\ldots, 0)) = 0$." ], "refs": [ "algebra-lemma-charpoly" ], "ref_ids": [ 385 ] } ], "ref_ids": [] }, { "id": 482, "type": "theorem", "label": "algebra-lemma-finite-is-integral", "categories": [ "algebra" ], "title": "algebra-lemma-finite-is-integral", "contents": [ "A finite ring extension is integral." ], "refs": [], "proofs": [ { "contents": [ "Let $R \\to S$ be finite. Let $y \\in S$. Apply", "Lemma \\ref{lemma-characterize-integral-element}", "to $M = S$ to see that $y$ is integral over $R$." ], "refs": [ "algebra-lemma-characterize-integral-element" ], "ref_ids": [ 481 ] } ], "ref_ids": [] }, { "id": 483, "type": "theorem", "label": "algebra-lemma-characterize-integral", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-integral", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Let $s_1, \\ldots, s_n$", "be a finite set of elements of $S$.", "In this case $s_i$ is integral over $R$ for all $i = 1, \\ldots, n$", "if and only if", "there exists an $R$-subalgebra $S' \\subset S$ finite over $R$", "containing all of the $s_i$." ], "refs": [], "proofs": [ { "contents": [ "If each $s_i$ is integral, then the subalgebra", "generated by $\\varphi(R)$ and the $s_i$ is finite", "over $R$. Namely, if $s_i$ satisfies a monic equation", "of degree $d_i$ over $R$, then this subalgebra is generated as an", "$R$-module by the elements $s_1^{e_1} \\ldots s_n^{e_n}$", "with $0 \\leq e_i \\leq d_i - 1$.", "Conversely, suppose given a finite $R$-subalgebra", "$S'$ containing all the $s_i$. Then all of the", "$s_i$ are integral by Lemma \\ref{lemma-finite-is-integral}." ], "refs": [ "algebra-lemma-finite-is-integral" ], "ref_ids": [ 482 ] } ], "ref_ids": [] }, { "id": 484, "type": "theorem", "label": "algebra-lemma-characterize-finite-in-terms-of-integral", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-finite-in-terms-of-integral", "contents": [ "Let $R \\to S$ be a ring map. The following are equivalent", "\\begin{enumerate}", "\\item $R \\to S$ is finite,", "\\item $R \\to S$ is integral and of finite type, and", "\\item there exist $x_1, \\ldots, x_n \\in S$ which generate $S$ as an", "algebra over $R$ such that each $x_i$ is integral over $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Clear from Lemma \\ref{lemma-characterize-integral}." ], "refs": [ "algebra-lemma-characterize-integral" ], "ref_ids": [ 483 ] } ], "ref_ids": [] }, { "id": 485, "type": "theorem", "label": "algebra-lemma-integral-transitive", "categories": [ "algebra" ], "title": "algebra-lemma-integral-transitive", "contents": [ "\\begin{slogan}", "A composition of integral ring maps is integral", "\\end{slogan}", "Suppose that $R \\to S$ and $S \\to T$ are integral", "ring maps. Then $R \\to T$ is integral." ], "refs": [], "proofs": [ { "contents": [ "Let $t \\in T$. Let $P(x) \\in S[x]$ be a", "monic polynomial such that $P(t) = 0$.", "Apply Lemma \\ref{lemma-characterize-integral}", "to the finite set of coefficients of $P$.", "Hence $t$ is integral over some subalgebra", "$S' \\subset S$ finite over $R$. Apply Lemma", "\\ref{lemma-characterize-integral} again to find", "a subalgebra $T' \\subset T$ finite over $S'$ and", "containing $t$. Lemma \\ref{lemma-finite-transitive}", "applied to $R \\to S' \\to T'$ shows that $T'$ is finite", "over $R$. The integrality of $t$ over $R$", "now follows from Lemma \\ref{lemma-finite-is-integral}." ], "refs": [ "algebra-lemma-characterize-integral", "algebra-lemma-characterize-integral", "algebra-lemma-finite-transitive", "algebra-lemma-finite-is-integral" ], "ref_ids": [ 483, 483, 337, 482 ] } ], "ref_ids": [] }, { "id": 486, "type": "theorem", "label": "algebra-lemma-integral-closure-is-ring", "categories": [ "algebra" ], "title": "algebra-lemma-integral-closure-is-ring", "contents": [ "Let $R \\to S$ be a ring homomorphism.", "The set", "$$", "S' = \\{s \\in S \\mid s\\text{ is integral over }R\\}", "$$", "is an $R$-subalgebra of $S$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from Lemmas \\ref{lemma-characterize-integral}", "and \\ref{lemma-finite-is-integral}." ], "refs": [ "algebra-lemma-characterize-integral", "algebra-lemma-finite-is-integral" ], "ref_ids": [ 483, 482 ] } ], "ref_ids": [] }, { "id": 487, "type": "theorem", "label": "algebra-lemma-finite-product-integral", "categories": [ "algebra" ], "title": "algebra-lemma-finite-product-integral", "contents": [ "Let $R_i\\to S_i$ be ring maps $i = 1, \\ldots, n$.", "Let $R$ and $S$ denote the product of the $R_i$ and $S_i$ respectively.", "Then an element $s = (s_1, \\ldots, s_n) \\in S$ is integral over $R$", "if and only if each $s_i$ is integral over $R_i$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 488, "type": "theorem", "label": "algebra-lemma-finite-product-integral-closure", "categories": [ "algebra" ], "title": "algebra-lemma-finite-product-integral-closure", "contents": [ "Let $R_i\\to S_i$ be ring maps $i = 1, \\ldots, n$.", "Denote the integral closure of $R_i$ in $S_i$ by $S'_i$.", "Further let $R$ and $S$ denote the product of the $R_i$ and $S_i$ respectively.", "Then the integral closure of $R$ in $S$", "is the product of the $S'_i$. In particular $R \\to S$ is", "integrally closed if and only if each $R_i \\to S_i$ is integrally closed." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from Lemma \\ref{lemma-finite-product-integral}." ], "refs": [ "algebra-lemma-finite-product-integral" ], "ref_ids": [ 487 ] } ], "ref_ids": [] }, { "id": 489, "type": "theorem", "label": "algebra-lemma-integral-closure-localize", "categories": [ "algebra" ], "title": "algebra-lemma-integral-closure-localize", "contents": [ "Integral closure commutes with localization: If $A \\to B$ is a ring", "map, and $S \\subset A$ is a multiplicative subset, then the integral", "closure of $S^{-1}A$ in $S^{-1}B$ is $S^{-1}B'$, where $B' \\subset B$", "is the integral closure of $A$ in $B$." ], "refs": [], "proofs": [ { "contents": [ "Since localization is exact we see that $S^{-1}B' \\subset S^{-1}B$.", "Suppose $x \\in B'$ and $f \\in S$. Then", "$x^d + \\sum_{i = 1, \\ldots, d} a_i x^{d - i} = 0$", "in $B$ for some $a_i \\in A$. Hence also", "$$", "(x/f)^d + \\sum\\nolimits_{i = 1, \\ldots, d} a_i/f^i (x/f)^{d - i} = 0", "$$", "in $S^{-1}B$. In this way we see that $S^{-1}B'$ is contained in", "the integral closure of $S^{-1}A$ in $S^{-1}B$. Conversely, suppose", "that $x/f \\in S^{-1}B$ is integral over $S^{-1}A$. Then we have", "$$", "(x/f)^d + \\sum\\nolimits_{i = 1, \\ldots, d} (a_i/f_i) (x/f)^{d - i} = 0", "$$", "in $S^{-1}B$ for some $a_i \\in A$ and $f_i \\in S$. This means that", "$$", "(f'f_1 \\ldots f_d x)^d +", "\\sum\\nolimits_{i = 1, \\ldots, d}", "f^i(f')^if_1^i \\ldots f_i^{i - 1} \\ldots f_d^i a_i", "(f'f_1 \\ldots f_dx)^{d - i} = 0", "$$", "for a suitable $f' \\in S$. Hence $f'f_1\\ldots f_dx \\in B'$ and thus", "$x/f \\in S^{-1}B'$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 490, "type": "theorem", "label": "algebra-lemma-integral-closure-stalks", "categories": [ "algebra" ], "title": "algebra-lemma-integral-closure-stalks", "contents": [ "\\begin{slogan}", "An element of an algebra over a ring is integral over the ring", "if and only if it is locally integral at every prime ideal of the ring.", "\\end{slogan}", "Let $\\varphi : R \\to S$ be a ring map.", "Let $x \\in S$. The following are equivalent:", "\\begin{enumerate}", "\\item $x$ is integral over $R$, and", "\\item for every prime ideal $\\mathfrak p \\subset R$ the element", "$x \\in S_{\\mathfrak p}$ is integral over $R_{\\mathfrak p}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2). Assume (2). Consider the $R$-algebra", "$S' \\subset S$ generated by $\\varphi(R)$ and $x$. Let $\\mathfrak p$ be", "a prime ideal of $R$. Then we know that", "$x^d + \\sum_{i = 1, \\ldots, d} \\varphi(a_i) x^{d - i} = 0$", "in $S_{\\mathfrak p}$ for some $a_i \\in R_{\\mathfrak p}$. Hence we see,", "by looking at which denominators occur, that", "for some $f \\in R$, $f \\not \\in \\mathfrak p$ we have", "$a_i \\in R_f$ and", "$x^d + \\sum_{i = 1, \\ldots, d} \\varphi(a_i) x^{d - i} = 0$", "in $S_f$. This implies that $S'_f$ is finite over $R_f$.", "Since $\\mathfrak p$ was arbitrary and $\\Spec(R)$ is quasi-compact", "(Lemma \\ref{lemma-quasi-compact}) we can find finitely many elements", "$f_1, \\ldots, f_n \\in R$", "which generate the unit ideal of $R$ such that $S'_{f_i}$ is finite", "over $R_{f_i}$. Hence we conclude from Lemma \\ref{lemma-cover} that", "$S'$ is finite over $R$. Hence $x$ is integral over $R$ by", "Lemma \\ref{lemma-characterize-integral}." ], "refs": [ "algebra-lemma-quasi-compact", "algebra-lemma-cover", "algebra-lemma-characterize-integral" ], "ref_ids": [ 395, 411, 483 ] } ], "ref_ids": [] }, { "id": 491, "type": "theorem", "label": "algebra-lemma-base-change-integral", "categories": [ "algebra" ], "title": "algebra-lemma-base-change-integral", "contents": [ "\\begin{slogan}", "Integrality and finiteness are preserved under base change.", "\\end{slogan}", "Let $R \\to S$ and $R \\to R'$ be ring maps.", "Set $S' = R' \\otimes_R S$.", "\\begin{enumerate}", "\\item If $R \\to S$ is integral so is $R' \\to S'$.", "\\item If $R \\to S$ is finite so is $R' \\to S'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We prove (1).", "Let $s_i \\in S$ be generators for $S$ over $R$.", "Each of these satisfies a monic polynomial equation $P_i$", "over $R$. Hence the elements $1 \\otimes s_i \\in S'$ generate", "$S'$ over $R'$ and satisfy the corresponding polynomial", "$P_i'$ over $R'$. Since these elements generate $S'$ over $R'$", "we see that $S'$ is integral over $R'$.", "Proof of (2) omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 492, "type": "theorem", "label": "algebra-lemma-integral-local", "categories": [ "algebra" ], "title": "algebra-lemma-integral-local", "contents": [ "Let $R \\to S$ be a ring map.", "Let $f_1, \\ldots, f_n \\in R$ generate the unit ideal.", "\\begin{enumerate}", "\\item If each $R_{f_i} \\to S_{f_i}$ is integral, so is $R \\to S$.", "\\item If each $R_{f_i} \\to S_{f_i}$ is finite, so is $R \\to S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1).", "Let $s \\in S$. Consider the ideal $I \\subset R[x]$ of", "polynomials $P$ such that $P(s) = 0$. Let $J \\subset R$", "denote the ideal (!) of leading coefficients of elements of $I$.", "By assumption and clearing denominators", "we see that $f_i^{n_i} \\in J$ for all $i$", "and certain $n_i \\geq 0$. Hence $J$ contains $1$ and we see", "$s$ is integral over $R$. Proof of (2) omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 493, "type": "theorem", "label": "algebra-lemma-integral-permanence", "categories": [ "algebra" ], "title": "algebra-lemma-integral-permanence", "contents": [ "Let $A \\to B \\to C$ be ring maps.", "\\begin{enumerate}", "\\item If $A \\to C$ is integral so is $B \\to C$.", "\\item If $A \\to C$ is finite so is $B \\to C$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 494, "type": "theorem", "label": "algebra-lemma-integral-closure-transitive", "categories": [ "algebra" ], "title": "algebra-lemma-integral-closure-transitive", "contents": [ "Let $A \\to B \\to C$ be ring maps.", "Let $B'$ be the integral closure of $A$ in $B$,", "let $C'$ be the integral closure of $B'$ in $C$. Then", "$C'$ is the integral closure of $A$ in $C$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 495, "type": "theorem", "label": "algebra-lemma-integral-overring-surjective", "categories": [ "algebra" ], "title": "algebra-lemma-integral-overring-surjective", "contents": [ "Suppose that $R \\to S$ is an integral", "ring extension with $R \\subset S$.", "Then $\\varphi : \\Spec(S) \\to \\Spec(R)$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p \\subset R$ be a prime ideal.", "We have to show $\\mathfrak pS_{\\mathfrak p} \\not = S_{\\mathfrak p}$, see", "Lemma \\ref{lemma-in-image}.", "The localization $R_{\\mathfrak p} \\to S_{\\mathfrak p}$ is injective", "(as localization is exact) and integral by", "Lemma \\ref{lemma-integral-closure-localize} or", "\\ref{lemma-base-change-integral}.", "Hence we may replace $R$, $S$ by $R_{\\mathfrak p}$, $S_{\\mathfrak p}$ and", "we may assume $R$ is local with maximal ideal $\\mathfrak m$ and", "it suffices to show that $\\mathfrak mS \\not = S$.", "Suppose $1 = \\sum f_i s_i$ with $f_i \\in \\mathfrak m$", "and $s_i \\in S$ in order to get a contradiction.", "Let $R \\subset S' \\subset S$", "be such that $R \\to S'$ is finite and $s_i \\in S'$, see", "Lemma \\ref{lemma-characterize-integral}.", "The equation $1 = \\sum f_i s_i$ implies that", "the finite $R$-module $S'$ satisfies $S' = \\mathfrak m S'$. Hence by", "Nakayama's Lemma \\ref{lemma-NAK}", "we see $S' = 0$. Contradiction." ], "refs": [ "algebra-lemma-in-image", "algebra-lemma-integral-closure-localize", "algebra-lemma-base-change-integral", "algebra-lemma-characterize-integral", "algebra-lemma-NAK" ], "ref_ids": [ 394, 489, 491, 483, 401 ] } ], "ref_ids": [] }, { "id": 496, "type": "theorem", "label": "algebra-lemma-integral-under-field", "categories": [ "algebra" ], "title": "algebra-lemma-integral-under-field", "contents": [ "Let $R$ be a ring. Let $K$ be a field.", "If $R \\subset K$ and $K$ is integral over $R$,", "then $R$ is a field and $K$ is an algebraic extension.", "If $R \\subset K$ and $K$ is finite over $R$,", "then $R$ is a field and $K$ is a finite algebraic extension." ], "refs": [], "proofs": [ { "contents": [ "Assume that $R \\subset K$ is integral.", "By Lemma \\ref{lemma-integral-overring-surjective} we see that", "$\\Spec(R)$ has $1$ point. Since clearly $R$ is a domain we see", "that $R = R_{(0)}$ is a field (Lemma \\ref{lemma-minimal-prime-reduced-ring}).", "The other assertions are immediate from this." ], "refs": [ "algebra-lemma-integral-overring-surjective", "algebra-lemma-minimal-prime-reduced-ring" ], "ref_ids": [ 495, 418 ] } ], "ref_ids": [] }, { "id": 497, "type": "theorem", "label": "algebra-lemma-integral-over-field", "categories": [ "algebra" ], "title": "algebra-lemma-integral-over-field", "contents": [ "Let $k$ be a field. Let $S$ be a $k$-algebra over $k$.", "\\begin{enumerate}", "\\item If $S$ is a domain and finite dimensional over $k$,", "then $S$ is a field.", "\\item If $S$ is integral over $k$ and a domain,", "then $S$ is a field.", "\\item If $S$ is integral over $k$ then every prime of", "$S$ is a maximal ideal (see", "Lemma \\ref{lemma-ring-with-only-minimal-primes}", "for more consequences).", "\\end{enumerate}" ], "refs": [ "algebra-lemma-ring-with-only-minimal-primes" ], "proofs": [ { "contents": [ "The statement on primes follows from the statement", "``integral $+$ domain $\\Rightarrow$ field''.", "Let $S$ integral over $k$ and assume $S$ is a domain,", "Take $s \\in S$. By Lemma", "\\ref{lemma-characterize-integral} we may find a", "finite dimensional $k$-subalgebra $k \\subset S' \\subset S$", "containing $s$. Hence $S$ is a field if we can prove the", "first statement. Assume $S$ finite dimensional", "over $k$ and a domain. Pick $s\\in S$.", "Since $S$ is a domain the multiplication", "map $s : S \\to S$ is surjective by dimension", "reasons. Hence there exists an element $s_1 \\in S$", "such that $ss_1 = 1$. So $S$ is a field." ], "refs": [ "algebra-lemma-characterize-integral" ], "ref_ids": [ 483 ] } ], "ref_ids": [ 426 ] }, { "id": 498, "type": "theorem", "label": "algebra-lemma-integral-no-inclusion", "categories": [ "algebra" ], "title": "algebra-lemma-integral-no-inclusion", "contents": [ "Suppose $R \\to S$ is integral.", "Let $\\mathfrak q, \\mathfrak q' \\in \\Spec(S)$", "be distinct primes", "having the same image in $\\Spec(R)$.", "Then neither $\\mathfrak q \\subset \\mathfrak q'$", "nor $\\mathfrak q' \\subset \\mathfrak q$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p \\subset R$ be the image.", "By Remark \\ref{remark-fundamental-diagram}", "the primes $\\mathfrak q, \\mathfrak q'$", "correspond to ideals in", "$S \\otimes_R \\kappa(\\mathfrak p)$.", "Thus the lemma follows from Lemma \\ref{lemma-integral-over-field}." ], "refs": [ "algebra-remark-fundamental-diagram", "algebra-lemma-integral-over-field" ], "ref_ids": [ 1558, 497 ] } ], "ref_ids": [] }, { "id": 499, "type": "theorem", "label": "algebra-lemma-finite-finite-fibres", "categories": [ "algebra" ], "title": "algebra-lemma-finite-finite-fibres", "contents": [ "Suppose $R \\to S$ is finite.", "Then the fibres of $\\Spec(S) \\to \\Spec(R)$ are finite." ], "refs": [], "proofs": [ { "contents": [ "By the discussion in", "Remark \\ref{remark-fundamental-diagram}", "the fibres are the spectra of the rings $S \\otimes_R \\kappa(\\mathfrak p)$.", "As $R \\to S$ is finite, these fibre rings are finite over", "$\\kappa(\\mathfrak p)$ hence Noetherian by", "Lemma \\ref{lemma-Noetherian-permanence}.", "By", "Lemma \\ref{lemma-integral-no-inclusion}", "every prime of $S \\otimes_R \\kappa(\\mathfrak p)$ is a minimal", "prime. Hence by", "Lemma \\ref{lemma-Noetherian-irreducible-components}", "there are at most finitely many." ], "refs": [ "algebra-remark-fundamental-diagram", "algebra-lemma-Noetherian-permanence", "algebra-lemma-integral-no-inclusion", "algebra-lemma-Noetherian-irreducible-components" ], "ref_ids": [ 1558, 448, 498, 453 ] } ], "ref_ids": [] }, { "id": 500, "type": "theorem", "label": "algebra-lemma-integral-going-up", "categories": [ "algebra" ], "title": "algebra-lemma-integral-going-up", "contents": [ "Let $R \\to S$ be a ring map such that", "$S$ is integral over $R$.", "Let $\\mathfrak p \\subset \\mathfrak p' \\subset R$", "be primes. Let $\\mathfrak q$ be a prime of $S$ mapping", "to $\\mathfrak p$. Then there exists a prime $\\mathfrak q'$", "with $\\mathfrak q \\subset \\mathfrak q'$", "mapping to $\\mathfrak p'$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $R$ by $R/\\mathfrak p$ and $S$ by $S/\\mathfrak q$.", "This reduces us to the situation of having an integral", "extension of domains $R \\subset S$ and a prime $\\mathfrak p' \\subset R$.", "By Lemma \\ref{lemma-integral-overring-surjective} we win." ], "refs": [ "algebra-lemma-integral-overring-surjective" ], "ref_ids": [ 495 ] } ], "ref_ids": [] }, { "id": 501, "type": "theorem", "label": "algebra-lemma-finite-finitely-presented-extension", "categories": [ "algebra" ], "title": "algebra-lemma-finite-finitely-presented-extension", "contents": [ "Let $R \\to S$ be a finite and finitely presented ring map.", "Let $M$ be an $S$-module.", "Then $M$ is finitely presented as an $R$-module if and only if", "$M$ is finitely presented as an $S$-module." ], "refs": [], "proofs": [ { "contents": [ "One of the implications follows from", "Lemma \\ref{lemma-finitely-presented-over-subring}.", "To see the other assume that $M$ is finitely presented as an $S$-module.", "Pick a presentation", "$$", "S^{\\oplus m} \\longrightarrow", "S^{\\oplus n} \\longrightarrow", "M \\longrightarrow 0", "$$", "As $S$ is finite as an $R$-module, the kernel of", "$S^{\\oplus n} \\to M$ is a finite $R$-module. Thus from", "Lemma \\ref{lemma-extension}", "we see that it suffices to prove that $S$ is finitely presented as an", "$R$-module.", "\\medskip\\noindent", "Pick $y_1, \\ldots, y_n \\in S$ such that $y_1, \\ldots, y_n$ generate $S$", "as an $R$-module. By Lemma \\ref{lemma-characterize-integral-element}", "each $y_i$ is integral over $R$. Choose monic polynomials", "$P_i(x) \\in R[x]$ with $P_i(y_i) = 0$. Consider the ring", "$$", "S' = R[x_1, \\ldots, x_n]/(P_1(x_1), \\ldots, P_n(x_n))", "$$", "Then we see that $S$ is of finite presentation as an $S'$-algebra", "by Lemma \\ref{lemma-compose-finite-type}. Since $S' \\to S$ is surjective,", "the kernel $J = \\Ker(S' \\to S)$ is finitely generated as an ideal by", "Lemma \\ref{lemma-finite-presentation-independent}. Hence $J$ is a finite", "$S'$-module (immediate from the definitions).", "Thus $S = \\Coker(J \\to S')$ is of finite presentation as an $S'$-module", "by Lemma \\ref{lemma-extension}.", "Hence, arguing as in the first paragraph, it suffices to show that $S'$ is", "of finite presentation as an $R$-module. Actually, $S'$ is free as an", "$R$-module with basis the monomials $x_1^{e_1} \\ldots x_n^{e_n}$", "for $0 \\leq e_i < \\deg(P_i)$. Namely, write $R \\to S'$ as the composition", "$$", "R \\to R[x_1]/(P_1(x_1)) \\to R[x_1, x_2]/(P_1(x_1), P_2(x_2)) \\to", "\\ldots \\to S'", "$$", "This shows that the $i$th ring in this sequence is free as a module over the", "$(i - 1)$st one with basis $1, x_i, \\ldots, x_i^{\\deg(P_i) - 1}$. The result", "follows easily from this by induction. Some details omitted." ], "refs": [ "algebra-lemma-finitely-presented-over-subring", "algebra-lemma-extension", "algebra-lemma-characterize-integral-element", "algebra-lemma-compose-finite-type", "algebra-lemma-finite-presentation-independent", "algebra-lemma-extension" ], "ref_ids": [ 335, 330, 481, 333, 334, 330 ] } ], "ref_ids": [] }, { "id": 502, "type": "theorem", "label": "algebra-lemma-silly-normal", "categories": [ "algebra" ], "title": "algebra-lemma-silly-normal", "contents": [ "Let $R$ be a ring. Let $x, y \\in R$ be nonzerodivisors.", "Let $R[x/y] \\subset R_{xy}$ be the $R$-subalgebra generated", "by $x/y$, and similarly for the subalgebras $R[y/x]$ and $R[x/y, y/x]$.", "If $R$ is integrally closed in $R_x$ or $R_y$, then the sequence", "$$", "0 \\to R \\xrightarrow{(-1, 1)} R[x/y] \\oplus R[y/x] \\xrightarrow{(1, 1)}", "R[x/y, y/x] \\to 0", "$$", "is a short exact sequence of $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "Since $x/y \\cdot y/x = 1$ it is clear that the map", "$R[x/y] \\oplus R[y/x] \\to R[x/y, y/x]$ is surjective.", "Let $\\alpha \\in R[x/y] \\cap R[y/x]$. To show exactness in the middle", "we have to prove that $\\alpha \\in R$. By assumption we may write", "$$", "\\alpha = a_0 + a_1 x/y + \\ldots + a_n (x/y)^n =", "b_0 + b_1 y/x + \\ldots + b_m(y/x)^m", "$$", "for some $n, m \\geq 0$ and $a_i, b_j \\in R$.", "Pick some $N > \\max(n, m)$.", "Consider the finite $R$-submodule $M$ of $R_{xy}$ generated by the elements", "$$", "(x/y)^N, (x/y)^{N - 1}, \\ldots, x/y, 1, y/x, \\ldots, (y/x)^{N - 1}, (y/x)^N", "$$", "We claim that $\\alpha M \\subset M$. Namely, it is clear that", "$(x/y)^i (b_0 + b_1 y/x + \\ldots + b_m(y/x)^m) \\in M$ for", "$0 \\leq i \\leq N$ and that", "$(y/x)^i (a_0 + a_1 x/y + \\ldots + a_n(x/y)^n) \\in M$ for", "$0 \\leq i \\leq N$. Hence $\\alpha$ is integral over $R$ by", "Lemma \\ref{lemma-characterize-integral-element}. Note that", "$\\alpha \\in R_x$, so if $R$ is integrally closed in $R_x$", "then $\\alpha \\in R$ as desired." ], "refs": [ "algebra-lemma-characterize-integral-element" ], "ref_ids": [ 481 ] } ], "ref_ids": [] }, { "id": 503, "type": "theorem", "label": "algebra-lemma-integral-closure-in-normal", "categories": [ "algebra" ], "title": "algebra-lemma-integral-closure-in-normal", "contents": [ "Let $R \\to S$ be a ring map.", "If $S$ is a normal domain, then the integral closure of $R$", "in $S$ is a normal domain." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 504, "type": "theorem", "label": "algebra-lemma-almost-integral", "categories": [ "algebra" ], "title": "algebra-lemma-almost-integral", "contents": [ "Let $R$ be a domain with fraction field $K$.", "If $u, v \\in K$ are almost integral over $R$, then so are", "$u + v$ and $uv$. Any element $g \\in K$ which is integral over $R$", "is almost integral over $R$. If $R$ is Noetherian", "then the converse holds as well." ], "refs": [], "proofs": [ { "contents": [ "If $ru^n \\in R$ for all $n \\geq 0$ and", "$v^nr' \\in R$ for all $n \\geq 0$, then", "$(uv)^nrr'$ and $(u + v)^nrr'$ are in $R$ for", "all $n \\geq 0$. Hence the first assertion.", "Suppose $g \\in K$ is integral over $R$.", "In this case there exists an $d > 0$ such that", "the ring $R[g]$ is generated by $1, g, \\ldots, g^d$ as an $R$-module.", "Let $r \\in R$ be a common denominator of the elements", "$1, g, \\ldots, g^d \\in K$. It is follows that $rR[g] \\subset R$,", "and hence $g$ is almost integral over $R$.", "\\medskip\\noindent", "Suppose $R$ is Noetherian and $g \\in K$ is almost integral over $R$.", "Let $r \\in R$, $r\\not = 0$ be as in the definition.", "Then $R[g] \\subset \\frac{1}{r}R$ as an $R$-module.", "Since $R$ is Noetherian this implies that $R[g]$ is", "finite over $R$. Hence $g$ is integral over $R$, see", "Lemma \\ref{lemma-finite-is-integral}." ], "refs": [ "algebra-lemma-finite-is-integral" ], "ref_ids": [ 482 ] } ], "ref_ids": [] }, { "id": 505, "type": "theorem", "label": "algebra-lemma-localize-normal-domain", "categories": [ "algebra" ], "title": "algebra-lemma-localize-normal-domain", "contents": [ "Any localization of a normal domain is normal." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a normal domain, and let $S \\subset R$ be", "a multiplicative subset. Suppose $g$ is an element", "of the fraction field of $R$ which is integral over $S^{-1}R$.", "Let $P = x^d + \\sum_{j < d} a_j x^j$ be a polynomial", "with $a_i \\in S^{-1}R$ such that $P(g) = 0$.", "Choose $s \\in S$ such that $sa_i \\in R$ for all $i$.", "Then $sg$ satisfies the monic polynomial", "$x^d + \\sum_{j < d} s^{d-j}a_j x^j$ which has coefficients", "$s^{d-j}a_j$ in $R$. Hence $sg \\in R$ because $R$ is normal.", "Hence $g \\in S^{-1}R$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 506, "type": "theorem", "label": "algebra-lemma-PID-normal", "categories": [ "algebra" ], "title": "algebra-lemma-PID-normal", "contents": [ "A principal ideal domain is normal." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a principal ideal domain.", "Let $g = a/b$ be an element of the fraction field", "of $R$ integral over $R$. Because $R$ is a principal ideal domain", "we may divide out a common factor of $a$ and $b$", "and assume $(a, b) = R$. In this case, any equation", "$(a/b)^n + r_{n-1} (a/b)^{n-1} + \\ldots + r_0 = 0$", "with $r_i \\in R$ would imply $a^n \\in (b)$. This", "contradicts $(a, b) = R$ unless $b$ is a unit in $R$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 507, "type": "theorem", "label": "algebra-lemma-prepare-polynomial-ring-normal", "categories": [ "algebra" ], "title": "algebra-lemma-prepare-polynomial-ring-normal", "contents": [ "Let $R$ be a domain with fraction field $K$.", "Suppose $f = \\sum \\alpha_i x^i$ is an", "element of $K[x]$.", "\\begin{enumerate}", "\\item If $f$ is integral over $R[x]$", "then all $\\alpha_i$ are integral over $R$, and", "\\item If $f$ is almost integral over $R[x]$", "then all $\\alpha_i$ are almost integral over $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We first prove the second statement.", "Write $f = \\alpha_0 + \\alpha_1 x + \\ldots + \\alpha_r x^r$", "with $\\alpha_r \\not = 0$.", "By assumption there exists $h = b_0 + b_1 x + \\ldots + b_s x^s \\in R[x]$,", "$b_s \\not = 0$ such that $f^n h \\in R[x]$ for all", "$n \\geq 0$. This implies that $b_s \\alpha_r^n \\in R$", "for all $n \\geq 0$. Hence $\\alpha_r$ is almost", "integral over $R$. Since the set of almost integral", "elements form a subring (Lemma \\ref{lemma-almost-integral}) we deduce that", "$f - \\alpha_r x^r = \\alpha_0 + \\alpha_1 x + \\ldots + \\alpha_{r - 1} x^{r - 1}$", "is almost integral over $R[x]$. By induction on $r$ we win.", "\\medskip\\noindent", "In order to prove the first statement we will use absolute Noetherian", "reduction. Namely, write $\\alpha_i = a_i / b_i$ and", "let $P(t) = t^d + \\sum_{j < d} f_j t^j$ be a polynomial", "with coefficients $f_j \\in R[x]$ such that $P(f) = 0$.", "Let $f_j = \\sum f_{ji}x^i$. Consider the subring", "$R_0 \\subset R$ generated by the finite list of elements", "$a_i, b_i, f_{ji}$ of $R$. It is a domain; let", "$K_0$ be its field of fractions. Since $R_0$ is a finite type", "$\\mathbf{Z}$-algebra it is Noetherian, see", "Lemma \\ref{lemma-obvious-Noetherian}. It is still", "the case that $f \\in K_0[x]$ is integral over $R_0[x]$,", "because all the identities in $R$", "among the elements $a_i, b_i, f_{ji}$ also hold in $R_0$.", "By Lemma \\ref{lemma-almost-integral} the element", "$f$ is almost integral over $R_0[x]$. By the second statement of", "the lemma, the elements $\\alpha_i$ are almost integral", "over $R_0$. And since $R_0$ is Noetherian, they are", "integral over $R_0$, see Lemma \\ref{lemma-almost-integral}.", "Of course, then they are integral over $R$." ], "refs": [ "algebra-lemma-almost-integral", "algebra-lemma-obvious-Noetherian", "algebra-lemma-almost-integral", "algebra-lemma-almost-integral" ], "ref_ids": [ 504, 450, 504, 504 ] } ], "ref_ids": [] }, { "id": 508, "type": "theorem", "label": "algebra-lemma-polynomial-domain-normal", "categories": [ "algebra" ], "title": "algebra-lemma-polynomial-domain-normal", "contents": [ "Let $R$ be a normal domain.", "Then $R[x]$ is a normal domain." ], "refs": [], "proofs": [ { "contents": [ "The result is true if $R$ is a field $K$ because", "$K[x]$ is a euclidean domain and hence a principal ideal", "domain and hence normal by Lemma \\ref{lemma-PID-normal}.", "Let $g$ be an element of the fraction field of", "$R[x]$ which is integral over $R[x]$. Because $g$", "is integral over $K[x]$ where $K$ is the fraction", "field of $R$ we may write $g = \\alpha_d x^d + \\alpha_{d-1}x^{d-1} +", "\\ldots + \\alpha_0$ with $\\alpha_i \\in K$.", "By Lemma \\ref{lemma-prepare-polynomial-ring-normal}", "the elements $\\alpha_i$ are integral over $R$ and", "hence are in $R$." ], "refs": [ "algebra-lemma-PID-normal", "algebra-lemma-prepare-polynomial-ring-normal" ], "ref_ids": [ 506, 507 ] } ], "ref_ids": [] }, { "id": 509, "type": "theorem", "label": "algebra-lemma-power-series-over-Noetherian-normal-domain", "categories": [ "algebra" ], "title": "algebra-lemma-power-series-over-Noetherian-normal-domain", "contents": [ "Let $R$ be a Noetherian normal domain. Then $R[[x]]$ is", "a Noetherian normal domain." ], "refs": [], "proofs": [ { "contents": [ "The power series ring is Noetherian by", "Lemma \\ref{lemma-Noetherian-power-series}.", "Let $f, g \\in R[[x]]$ be nonzero elements such that", "$w = f/g$ is integral over $R[[x]]$.", "Let $K$ be the fraction field of $R$. Since the ring of Laurent series", "$K((x)) = K[[x]][1/x]$ is a field, we can write", "$w = a_n x^n + a_{n + 1} x^{n + 1} + \\ldots)$", "for some $n \\in \\mathbf{Z}$, $a_i \\in K$, and $a_n \\not = 0$.", "By Lemma \\ref{lemma-almost-integral} we see there exists a", "nonzero element $h = b_m x^m + b_{m + 1} x^{m + 1} + \\ldots$", "in $R[[x]]$ with $b_m \\not = 0$ such that", "$w^e h \\in R[[x]]$ for all $e \\geq 1$. We conclude that $n \\geq 0$ and that", "$b_m a_n^e \\in R$ for all $e \\geq 1$.", "Since $R$ is Noetherian this implies that $a_n \\in R$ by", "the same lemma. Now, if $a_n, a_{n + 1}, \\ldots, a_{N - 1} \\in R$,", "then we can apply the same argument to", "$w - a_n x^n - \\ldots - a_{N - 1} x^{N - 1} = a_N x^N + \\ldots$.", "In this way we see that all $a_i \\in R$ and the lemma is proved." ], "refs": [ "algebra-lemma-Noetherian-power-series", "algebra-lemma-almost-integral" ], "ref_ids": [ 449, 504 ] } ], "ref_ids": [] }, { "id": 510, "type": "theorem", "label": "algebra-lemma-normality-is-local", "categories": [ "algebra" ], "title": "algebra-lemma-normality-is-local", "contents": [ "Let $R$ be a domain. The following are equivalent:", "\\begin{enumerate}", "\\item The domain $R$ is a normal domain,", "\\item for every prime $\\mathfrak p \\subset R$ the local ring", "$R_{\\mathfrak p}$ is a normal domain, and", "\\item for every maximal ideal $\\mathfrak m$ the ring $R_{\\mathfrak m}$", "is a normal domain.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows easily from the fact that for any domain $R$ we have", "$$", "R = \\bigcap\\nolimits_{\\mathfrak m} R_{\\mathfrak m}", "$$", "inside the fraction field of $R$. Namely, if $g$ is an element of", "the right hand side then the ideal $I = \\{x \\in R \\mid xg \\in R\\}$", "is not contained in any maximal ideal $\\mathfrak m$, whence $I = R$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 511, "type": "theorem", "label": "algebra-lemma-normal-ring-integrally-closed", "categories": [ "algebra" ], "title": "algebra-lemma-normal-ring-integrally-closed", "contents": [ "A normal ring is integrally closed in its total ring of fractions." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a normal ring. Let $x \\in Q(R)$ be an element of the total ring", "of fractions of $R$ integral over $R$. Set $I = \\{f \\in R, fx \\in R\\}$. Let", "$\\mathfrak p \\subset R$ be a prime. As $R \\subset R_{\\mathfrak p}$ is", "flat we see that $R_{\\mathfrak p} \\subset Q(R) \\otimes_R R_{\\mathfrak p}$. As", "$R_{\\mathfrak p}$ is a normal domain we see that $x \\otimes 1$ is an element of", "$R_{\\mathfrak p}$. Hence we can find $a, f \\in R$, $f \\not \\in \\mathfrak p$", "such that $x \\otimes 1 = a \\otimes 1/f$. This means that $fx - a$ maps to", "zero in $Q(R) \\otimes_R R_{\\mathfrak p} = Q(R)_{\\mathfrak p}$, which", "in turn means that there exists an $f' \\in R$, $f' \\not \\in \\mathfrak p$", "such that $f'fx = f'a$ in $R$. In other words, $ff' \\in I$. Thus $I$", "is an ideal which isn't contained in any of the prime ideals of $R$, i.e.,", "$I = R$ and $x \\in R$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 512, "type": "theorem", "label": "algebra-lemma-localization-normal-ring", "categories": [ "algebra" ], "title": "algebra-lemma-localization-normal-ring", "contents": [ "A localization of a normal ring is a normal ring." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 513, "type": "theorem", "label": "algebra-lemma-polynomial-ring-normal", "categories": [ "algebra" ], "title": "algebra-lemma-polynomial-ring-normal", "contents": [ "Let $R$ be a normal ring. Then $R[x]$ is a normal ring." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q$ be a prime of $R[x]$. Set $\\mathfrak p = R \\cap \\mathfrak q$.", "Then we see that $R_{\\mathfrak p}[x]$ is a normal domain by", "Lemma \\ref{lemma-polynomial-domain-normal}.", "Hence $(R[x])_{\\mathfrak q}$ is a normal domain by", "Lemma \\ref{lemma-localize-normal-domain}." ], "refs": [ "algebra-lemma-polynomial-domain-normal", "algebra-lemma-localize-normal-domain" ], "ref_ids": [ 508, 505 ] } ], "ref_ids": [] }, { "id": 514, "type": "theorem", "label": "algebra-lemma-finite-product-normal", "categories": [ "algebra" ], "title": "algebra-lemma-finite-product-normal", "contents": [ "A finite product of normal rings is normal." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that the product of two normal rings, say $R$ and $S$, is", "normal. By Lemma \\ref{lemma-disjoint-decomposition} the prime ideals of", "$R\\times S$ are of the form $\\mathfrak{p}\\times S$ and $R\\times", "\\mathfrak{q}$, where $\\mathfrak{p}$ and $\\mathfrak{q}$ are primes of $R$", "and $S$ respectively. Localization yields ", "$(R\\times S)_{\\mathfrak{p}\\times S}=R_{\\mathfrak{p}}$ which is a normal domain", "by assumption. Similarly for $S$." ], "refs": [ "algebra-lemma-disjoint-decomposition" ], "ref_ids": [ 405 ] } ], "ref_ids": [] }, { "id": 515, "type": "theorem", "label": "algebra-lemma-characterize-reduced-ring-normal", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-reduced-ring-normal", "contents": [ "Let $R$ be a ring. Assume $R$ is reduced and has finitely many", "minimal primes. Then the following are equivalent:", "\\begin{enumerate}", "\\item $R$ is a normal ring,", "\\item $R$ is integrally closed in its total ring of fractions, and", "\\item $R$ is a finite product of normal domains.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implications (1) $\\Rightarrow$ (2) and", "(3) $\\Rightarrow$ (1) hold in general,", "see Lemmas \\ref{lemma-normal-ring-integrally-closed} and", "\\ref{lemma-finite-product-normal}.", "\\medskip\\noindent", "Let $\\mathfrak p_1, \\ldots, \\mathfrak p_n$ be the minimal primes of $R$.", "By Lemmas \\ref{lemma-reduced-ring-sub-product-fields} and", "\\ref{lemma-total-ring-fractions-no-embedded-points} we have", "$Q(R) = R_{\\mathfrak p_1} \\times \\ldots \\times R_{\\mathfrak p_n}$, and", "by Lemma \\ref{lemma-minimal-prime-reduced-ring} each factor is a field.", "Denote $e_i = (0, \\ldots, 0, 1, 0, \\ldots, 0)$ the $i$th idempotent", "of $Q(R)$.", "\\medskip\\noindent", "If $R$ is integrally closed in $Q(R)$, then it contains in particular", "the idempotents $e_i$, and we see that $R$ is a product of $n$", "domains (see Sections \\ref{section-connected-components} and", "\\ref{section-tilde-module-sheaf}). Each factor is of the form", "$R/\\mathfrak p_i$ with field of fractions $R_{\\mathfrak p_i}$. ", "By Lemma \\ref{lemma-finite-product-integral-closure} each map", "$R/\\mathfrak p_i \\to R_{\\mathfrak p_i}$ is integrally closed. ", "Hence $R$ is a finite product of normal domains." ], "refs": [ "algebra-lemma-normal-ring-integrally-closed", "algebra-lemma-finite-product-normal", "algebra-lemma-reduced-ring-sub-product-fields", "algebra-lemma-total-ring-fractions-no-embedded-points", "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-finite-product-integral-closure" ], "ref_ids": [ 511, 514, 419, 421, 418, 488 ] } ], "ref_ids": [] }, { "id": 516, "type": "theorem", "label": "algebra-lemma-colimit-normal-ring", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-normal-ring", "contents": [ "Let $(R_i, \\varphi_{ii'})$ be a directed system", "(Categories, Definition \\ref{definition-directed-system})", "of rings. If each $R_i$ is a normal ring so is", "$R = \\colim_i R_i$." ], "refs": [ "algebra-definition-directed-system" ], "proofs": [ { "contents": [ "Let $\\mathfrak p \\subset R$ be a prime ideal.", "Set $\\mathfrak p_i = R_i \\cap \\mathfrak p$ (usual abuse of notation).", "Then we see that", "$R_{\\mathfrak p} = \\colim_i (R_i)_{\\mathfrak p_i}$.", "Since each $(R_i)_{\\mathfrak p_i}$ is a normal domain we", "reduce to proving the statement of the lemma for normal", "domains. If $a, b \\in R$ and $a/b$ satisfies a monic polynomial", "$P(T) \\in R[T]$, then we can find a (sufficiently large) $i \\in I$", "such that $a, b$ come from objects $a_i, b_i$ over $R_i$, $P$ comes from a", "monic polynomial $P_i\\in R_i[T]$ and $P_i(a_i/b_i)=0$. Since $R_i$ is normal we", "see $a_i/b_i \\in R_i$ and hence also $a/b \\in R$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1435 ] }, { "id": 517, "type": "theorem", "label": "algebra-lemma-characterize-integral-ideal", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-integral-ideal", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "Let $I \\subset R$ be an ideal.", "Let $A = \\sum I^nt^n \\subset R[t]$ be the", "subring of the polynomial ring", "generated by $R \\oplus It \\subset R[t]$.", "An element $s \\in S$ is integral over $I$ if", "and only if the element $st \\in S[t]$", "is integral over $A$." ], "refs": [], "proofs": [ { "contents": [ "Suppose $st$ is integral over $A$.", "Let $P = x^d + \\sum_{j < d} a_j x^j$", "be a monic polynomial with coefficients in $A$", "such that $P^\\varphi(st) = 0$. Let $a_j' \\in A$", "be the degree $d-j$ part of $a_i$, in other", "words $a_j' = a_j'' t^{d-j}$ with $a_j'' \\in I^{d-j}$.", "For degree reasons we still have", "$(st)^d + \\sum_{j < d} \\varphi(a_j'') t^{d-j} (st)^j = 0$.", "Hence we see that $s$ is integral over $I$.", "\\medskip\\noindent", "Suppose that $s$ is integral over $I$.", "Say $P = x^d + \\sum_{j < d} a_j x^j$", "with $a_j \\in I^{d-j}$. Then we immediately find a", "polynomial $Q = x^d + \\sum_{j < d} (a_j t^{d-j}) x^j$", "with coefficients in $A$ which proves that", "$st$ is integral over $A$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 518, "type": "theorem", "label": "algebra-lemma-integral-over-ideal-is-submodule", "categories": [ "algebra" ], "title": "algebra-lemma-integral-over-ideal-is-submodule", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "Let $I \\subset R$ be an ideal.", "The set of elements of $S$ which are integral", "over $I$ form a $R$-submodule of $S$.", "Furthermore, if $s \\in S$ is integral over", "$R$, and $s'$ is integral over $I$, then", "$ss'$ is integral over $I$." ], "refs": [], "proofs": [ { "contents": [ "Closure under addition is clear from the", "characterization of Lemma \\ref{lemma-characterize-integral-ideal}.", "Any element $s \\in S$ which is integral over", "$R$ corresponds to the degree $0$ element $s$ of $S[x]$", "which is integral over $A$ (because $R \\subset A$).", "Hence we see that multiplication by $s$ on $S[x]$", "preserves the property of being integral over $A$,", "by Lemma \\ref{lemma-integral-closure-is-ring}." ], "refs": [ "algebra-lemma-characterize-integral-ideal", "algebra-lemma-integral-closure-is-ring" ], "ref_ids": [ 517, 486 ] } ], "ref_ids": [] }, { "id": 519, "type": "theorem", "label": "algebra-lemma-integral-integral-over-ideal", "categories": [ "algebra" ], "title": "algebra-lemma-integral-integral-over-ideal", "contents": [ "Suppose $\\varphi : R \\to S$ is integral.", "Suppose $I \\subset R$ is an ideal.", "Then every element of $IS$ is integral over $I$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-integral-over-ideal-is-submodule}." ], "refs": [ "algebra-lemma-integral-over-ideal-is-submodule" ], "ref_ids": [ 518 ] } ], "ref_ids": [] }, { "id": 520, "type": "theorem", "label": "algebra-lemma-polynomials-divide", "categories": [ "algebra" ], "title": "algebra-lemma-polynomials-divide", "contents": [ "Let $K$ be a field. Let $n, m \\in \\mathbf{N}$ and", "$a_0, \\ldots, a_{n - 1}, b_0, \\ldots, b_{m - 1} \\in K$.", "If the polynomial $x^n + a_{n - 1}x^{n - 1} + \\ldots + a_0$", "divides the polynomial $x^m + b_{m - 1} x^{m - 1} + \\ldots + b_0$", "in $K[x]$ then", "\\begin{enumerate}", "\\item $a_0, \\ldots, a_{n - 1}$ are integral over any subring", "$R_0$ of $K$ containing the elements $b_0, \\ldots, b_{m - 1}$, and", "\\item each $a_i$ lies in $\\sqrt{(b_0, \\ldots, b_{m-1})R}$", "for any subring $R \\subset K$ containing the elements", "$a_0, \\ldots, a_{n - 1}, b_0, \\ldots, b_{m - 1}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $L/K$ be a field extension such that we can write", "$x^m + b_{m - 1} x^{m - 1} + \\ldots + b_0 =", "\\prod_{i = 1}^m (x - \\beta_i)$ with $\\beta_i \\in L$.", "See Fields, Section \\ref{fields-section-splitting-fieds}.", "Each $\\beta_i$ is integral over $R_0$.", "Since each $a_i$ is a homogeneous polynomial in $\\beta_1, \\ldots, \\beta_m$", "we deduce the same for the $a_i$", "(use Lemma \\ref{lemma-integral-closure-is-ring}).", "\\medskip\\noindent", "Choose $c_0, \\ldots, c_{m - n - 1} \\in K$ such that", "$$", "\\begin{matrix}", "x^m + b_{m - 1} x^{m - 1} + \\ldots + b_0 = \\\\", "(x^n + a_{n - 1}x^{n - 1} + \\ldots + a_0)", "(x^{m - n} + c_{m - n - 1}x^{m - n - 1}+ \\ldots + c_0).", "\\end{matrix}", "$$", "By part (1) the elements $c_i$ are integral over $R$. Consider", "the integral extension", "$$", "R \\subset R' = R[c_0, \\ldots, c_{m - n - 1}] \\subset K", "$$", "By Lemmas \\ref{lemma-integral-overring-surjective}", "and \\ref{lemma-surjective-spec-radical-ideal}", "we see that $R \\cap \\sqrt{(b_0, \\ldots, b_{m - 1})R'}", "= \\sqrt{(b_0, \\ldots, b_{m - 1})R}$. Thus we may replace", "$R$ by $R'$ and assume $c_i \\in R$.", "Dividing out the radical $\\sqrt{(b_0, \\ldots, b_{m - 1})}$", "we get a reduced ring $\\overline{R}$.", "We have to show that the images $\\overline{a}_i \\in \\overline{R}$", "are zero. And in", "$\\overline{R}[x]$ we have the relation", "$$", "\\begin{matrix}", "x^m = x^m + \\overline{b}_{m - 1} x^{m - 1} + \\ldots + \\overline{b}_0 = \\\\", "(x^n + \\overline{a}_{n - 1}x^{n - 1} + \\ldots + \\overline{a}_0)", "(x^{m - n} + \\overline{c}_{m - n - 1}x^{m - n - 1}+ \\ldots + \\overline{c}_0).", "\\end{matrix}", "$$", "It is easy to see that this implies $\\overline{a}_i = 0$ for all $i$. Indeed", "by Lemma \\ref{lemma-minimal-prime-reduced-ring} the localization of", "$\\overline{R}$ at a minimal prime $\\mathfrak{p}$ is a field and", "$\\overline{R}_{\\mathfrak p}[x]$ a UFD. Thus", "$f = x^n + \\sum \\overline{a}_i x^i$", "is associated to $x^n$ and since $f$ is monic $f = x^n$", "in $\\overline{R}_{\\mathfrak p}[x]$.", "Then there exists an $s \\in \\overline{R}$, $s \\not\\in \\mathfrak p$", "such that $s(f - x^n) = 0$. Therefore all $\\overline{a}_i$ lie", "in $\\mathfrak p$ and we conclude by", "Lemma \\ref{lemma-reduced-ring-sub-product-fields}." ], "refs": [ "algebra-lemma-integral-closure-is-ring", "algebra-lemma-integral-overring-surjective", "algebra-lemma-surjective-spec-radical-ideal", "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-reduced-ring-sub-product-fields" ], "ref_ids": [ 486, 495, 443, 418, 419 ] } ], "ref_ids": [] }, { "id": 521, "type": "theorem", "label": "algebra-lemma-minimal-polynomial-normal-domain", "categories": [ "algebra" ], "title": "algebra-lemma-minimal-polynomial-normal-domain", "contents": [ "Let $R \\subset S$ be an inclusion of domains.", "Assume $R$ is normal. Let $g \\in S$ be integral", "over $R$. Then the minimal polynomial of $g$", "has coefficients in $R$." ], "refs": [], "proofs": [ { "contents": [ "Let $P = x^m + b_{m-1} x^{m-1} + \\ldots + b_0$", "be a polynomial with coefficients in $R$", "such that $P(g) = 0$. Let $Q = x^n + a_{n-1}x^{n-1} + \\ldots + a_0$", "be the minimal polynomial for $g$ over the fraction field", "$K$ of $R$. Then $Q$ divides $P$ in $K[x]$. By Lemma", "\\ref{lemma-polynomials-divide} we see the $a_i$ are", "integral over $R$. Since $R$ is normal this", "means they are in $R$." ], "refs": [ "algebra-lemma-polynomials-divide" ], "ref_ids": [ 520 ] } ], "ref_ids": [] }, { "id": 522, "type": "theorem", "label": "algebra-lemma-flat-intersect-ideals", "categories": [ "algebra" ], "title": "algebra-lemma-flat-intersect-ideals", "contents": [ "Let $R$ be a ring. Let $I, J \\subset R$ be ideals. Let $M$ be a flat", "$R$-module. Then $IM \\cap JM = (I \\cap J)M$." ], "refs": [], "proofs": [ { "contents": [ "Consider the exact sequence $0 \\to I \\cap J \\to R \\to R/I \\oplus R/J$.", "Tensoring with the flat module $M$ we obtain an exact sequence", "$$", "0 \\to (I \\cap J) \\otimes_R M \\to M \\to M/IM \\oplus M/JM", "$$", "Since the kernel of $M \\to M/IM \\oplus M/JM$ is equal to", "$IM \\cap JM$ we conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 523, "type": "theorem", "label": "algebra-lemma-colimit-flat", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-flat", "contents": [ "Let $R$ be a ring. Let $\\{M_i, \\varphi_{ii'}\\}$ be a directed system of", "flat $R$-modules. Then $\\colim_i M_i$ is a flat $R$-module." ], "refs": [], "proofs": [ { "contents": [ "This follows as $\\otimes$ commutes with colimits and because", "directed colimits are exact, see", "Lemma \\ref{lemma-directed-colimit-exact}." ], "refs": [ "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 343 ] } ], "ref_ids": [] }, { "id": 524, "type": "theorem", "label": "algebra-lemma-composition-flat", "categories": [ "algebra" ], "title": "algebra-lemma-composition-flat", "contents": [ "A composition of (faithfully) flat ring maps is", "(faithfully) flat.", "If $R \\to R'$ is (faithfully) flat, and $M'$ is a", "(faithfully) flat $R'$-module, then $M'$ is a", "(faithfully) flat $R$-module." ], "refs": [], "proofs": [ { "contents": [ "The first statement of the lemma is a particular case of the", "second, so it is clearly enough to prove the latter. Let", "$R \\to R'$ be a flat ring map, and $M'$ a flat $R'$-module.", "We need to prove that $M'$ is a flat $R$-module. Let", "$N_1 \\to N_2 \\to N_3$ be an exact complex of $R$-modules.", "Then, the complex $R' \\otimes_R N_1 \\to", "R' \\otimes_R N_2 \\to R' \\otimes_R N_3$ is exact (since $R'$", "is flat as an $R$-module), and so the complex", "$M' \\otimes_{R'} \\left(R' \\otimes_R N_1\\right)", "\\to M' \\otimes_{R'} \\left(R' \\otimes_R N_2\\right)", "\\to M' \\otimes_{R'} \\left(R' \\otimes_R N_3\\right)$ is", "exact (since $M'$ is a flat $R'$-module). Since", "$M' \\otimes_{R'} \\left(R' \\otimes_R N\\right)", "\\cong \\left(M' \\otimes_{R'} R'\\right) \\otimes_R N", "\\cong M' \\otimes_R N$ for any $R$-module $N$ functorially", "(by Lemmas \\ref{lemma-tensor-with-bimodule} and", "\\ref{lemma-flip-tensor-product}), this complex is isomorphic", "to the complex", "$M' \\otimes_R N_1 \\to M' \\otimes_R N_2 \\to M' \\otimes_R N_3$,", "which is therefore also exact. This shows that $M'$ is a flat", "$R$-module. Tracing this argument backwards, we can show", "that if $R \\to R'$ is faithfully flat, and if $M'$ is", "faithfully flat as an $R'$-module, then $M'$ is faithfully", "flat as an $R$-module." ], "refs": [ "algebra-lemma-tensor-with-bimodule", "algebra-lemma-flip-tensor-product" ], "ref_ids": [ 361, 358 ] } ], "ref_ids": [] }, { "id": 525, "type": "theorem", "label": "algebra-lemma-flat", "categories": [ "algebra" ], "title": "algebra-lemma-flat", "contents": [ "Let $M$ be an $R$-module. The following are equivalent:", "\\begin{enumerate}", "\\item", "\\label{item-flat}", "$M$ is flat over $R$.", "\\item", "\\label{item-injective}", "for every injection of $R$-modules $N \\subset N'$", "the map $N \\otimes_R M \\to N'\\otimes_R M$ is injective.", "\\item", "\\label{item-f-ideal}", "for every ideal $I \\subset R$ the map", "$I \\otimes_R M \\to R \\otimes_R M = M$ is injective.", "\\item", "\\label{item-ffg-ideal}", "for every finitely generated ideal $I \\subset R$", "the map $I \\otimes_R M \\to R \\otimes_R M = M$ is injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implications (\\ref{item-flat}) implies (\\ref{item-injective})", "implies (\\ref{item-f-ideal}) implies (\\ref{item-ffg-ideal}) are all", "trivial. Thus we prove (\\ref{item-ffg-ideal}) implies (\\ref{item-flat}).", "Suppose that $N_1 \\to N_2 \\to N_3$ is exact.", "Let $K = \\Ker(N_2 \\to N_3)$ and $Q = \\Im(N_2 \\to N_3)$.", "Then we get maps", "$$", "N_1 \\otimes_R M \\to", "K \\otimes_R M \\to", "N_2 \\otimes_R M \\to", "Q \\otimes_R M \\to", "N_3 \\otimes_R M", "$$", "Observe that the first and third arrows are surjective. Thus if we show", "that the second and fourth arrows are injective, then we are", "done\\footnote{Here is the argument in more detail:", "Assume that we know that the second and fourth arrows are", "injective. Lemma \\ref{lemma-tensor-product-exact} (applied", "to the exact sequence $K \\to N_2 \\to Q \\to 0$) yields that", "the sequence $K \\otimes_R M \\to N_2 \\otimes_R M \\to", "Q \\otimes_R M \\to 0$ is exact. Hence,", "$\\Ker \\left(N_2 \\otimes_R M \\to Q \\otimes_R M\\right)", "= \\Im \\left(K \\otimes_R M \\to N_2 \\otimes_R M\\right)$.", "Since", "$\\Im \\left(K \\otimes_R M \\to N_2 \\otimes_R M\\right)", "= \\Im \\left(N_1 \\otimes_R M \\to N_2 \\otimes_R M\\right)$", "(due to the surjectivity of $N_1 \\otimes_R M \\to", "K \\otimes_R M$) and", "$\\Ker \\left(N_2 \\otimes_R M \\to Q \\otimes_R M\\right)", "= \\Ker \\left(N_2 \\otimes_R M \\to N_3 \\otimes_R M\\right)$", "(due to the injectivity of $Q \\otimes_R M \\to", "N_3 \\otimes_R M$), this becomes", "$\\Ker \\left(N_2 \\otimes_R M \\to N_3 \\otimes_R M\\right)", "= \\Im \\left(N_1 \\otimes_R M \\to N_2 \\otimes_R M\\right)$,", "which shows that the functor $- \\otimes_R M$ is exact,", "whence $M$ is flat.}.", "Hence it suffices to show that $- \\otimes_R M$ transforms", "injective $R$-module maps into injective $R$-module maps.", "\\medskip\\noindent", "Assume $K \\to N$ is an injective $R$-module map and", "let $x \\in \\Ker(K \\otimes_R M \\to N \\otimes_R M)$.", "We have to show that $x$ is zero.", "The $R$-module $K$ is the union of its finite", "$R$-submodules; hence, $K \\otimes_R M$ is", "the colimit of $R$-modules of the form", "$K_i \\otimes_R M$ where $K_i$ runs over all finite", "$R$-submodules of $K$", "(because tensor product commutes with colimits).", "Thus, for some $i$ our $x$ comes from an element", "$x_i \\in K_i \\otimes_R M$. Thus we may assume that $K$", "is a finite $R$-module. Assume this. We regard the", "injection $K \\to N$ as an inclusion, so that", "$K \\subset N$.", "\\medskip\\noindent", "The $R$-module $N$ is the union of its finite", "$R$-submodules that contain $K$. Hence, $N \\otimes_R M$", "is the colimit of $R$-modules of the form", "$N_i \\otimes_R M$ where $N_i$ runs over all finite", "$R$-submodules of $N$ that contain $K$", "(again since tensor product commutes with colimits).", "Notice that this is a colimit over a directed system", "(since the sum of two finite submodules of $N$ is", "again finite).", "Hence, (by Lemma \\ref{lemma-zero-directed-limit})", "the element $x \\in K \\otimes_R M$ maps to", "zero in at least one of these $R$-modules", "$N_i \\otimes_R M$ (since $x$ maps to zero", "in $N \\otimes_R M$).", "Thus we may assume $N$ is a finite $R$-module.", "\\medskip\\noindent", "Assume $N$ is a finite $R$-module. Write $N = R^{\\oplus n}/L$ and $K = L'/L$", "for some $L \\subset L' \\subset R^{\\oplus n}$.", "For any $R$-submodule $G \\subset R^{\\oplus n}$,", "we have a canonical map $G \\otimes_R M \\to M^{\\oplus n}$", "obtained by composing", "$G \\otimes_R M \\to R^n \\otimes_R M = M^{\\oplus n}$.", "It suffices to prove that $L \\otimes_R M \\to M^{\\oplus n}$", "and $L' \\otimes_R M \\to M^{\\oplus n}$ are injective.", "Namely, if so, then we see that", "$K \\otimes_R M = L' \\otimes_R M/L \\otimes_R M \\to M^{\\oplus n}/L \\otimes_R M$", "is injective too\\footnote{This becomes obvious if we", "identify $L' \\otimes_R M$ and $L \\otimes_R M$ with", "submodules of $M^{\\oplus n}$ (which is legitimate since", "the maps $L \\otimes_R M \\to M^{\\oplus n}$", "and $L' \\otimes_R M \\to M^{\\oplus n}$ are injective and", "commute with the obvious map $L' \\otimes_R M \\to L \\otimes_R M$).}.", "\\medskip\\noindent", "Thus it suffices to show that $L \\otimes_R M \\to M^{\\oplus n}$", "is injective when $L \\subset R^{\\oplus n}$ is an $R$-submodule.", "We do this by induction on $n$. The base case $n = 1$ we handle below.", "For the induction step assume $n > 1$ and set", "$L' = L \\cap R \\oplus 0^{\\oplus n - 1}$. Then $L'' = L/L'$ is a submodule", "of $R^{\\oplus n - 1}$. We obtain a diagram", "$$", "\\xymatrix{", "&", "L' \\otimes_R M \\ar[r] \\ar[d] &", "L \\otimes_R M \\ar[r] \\ar[d] &", "L'' \\otimes_R M \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "M \\ar[r] &", "M^{\\oplus n} \\ar[r] &", "M^{\\oplus n - 1} \\ar[r] & 0", "}", "$$", "By induction hypothesis and the base case the left and right vertical", "arrows are injective. The rows are exact. It follows that the middle vertical", "arrow is injective too.", "\\medskip\\noindent", "The base case of the induction above is when $L \\subset R$ is an ideal.", "In other words, we have to show that $I \\otimes_R M \\to M$ is injective", "for any ideal $I$ of $R$. We know this is true when $I$ is finitely", "generated. However, $I = \\bigcup I_\\alpha$ is the union of the", "finitely generated ideals $I_\\alpha$ contained in it. In other words,", "$I = \\colim I_\\alpha$. Since $\\otimes$ commutes with colimits we see that", "$I \\otimes_R M = \\colim I_\\alpha \\otimes_R M$ and since all", "the morphisms $I_\\alpha \\otimes_R M \\to M$ are injective by", "assumption, the same is true for $I \\otimes_R M \\to M$." ], "refs": [ "algebra-lemma-tensor-product-exact", "algebra-lemma-zero-directed-limit" ], "ref_ids": [ 364, 341 ] } ], "ref_ids": [] }, { "id": 526, "type": "theorem", "label": "algebra-lemma-colimit-rings-flat", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-rings-flat", "contents": [ "Let $\\{R_i, \\varphi_{ii'}\\}$ be a system of rings over the directed set $I$.", "Let $R = \\colim_i R_i$. Let $M$ be an $R$-module such that", "$M$ is flat as an $R_i$-module for all $i$. Then $M$ is flat as", "an $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak a \\subset R$ be a finitely generated ideal. By", "Lemma \\ref{lemma-flat}", "it suffices to show that $\\mathfrak a \\otimes_R M \\to M$ is", "injective. We can find an $i \\in I$ and a finitely generated ideal", "$\\mathfrak a' \\subset R_i$ such that $\\mathfrak a = \\mathfrak a'R$.", "Then $\\mathfrak a = \\colim_{i' \\geq i} \\mathfrak a'R_{i'}$.", "Hence the map $\\mathfrak a \\otimes_R M \\to M$ is the colimit of the", "maps", "$$", "\\mathfrak a'R_{i'} \\otimes_{R_{i'}} M \\longrightarrow M", "$$", "which are all injective by assumption. Since $\\otimes$ commutes with", "colimits and since colimits over $I$ are exact by", "Lemma \\ref{lemma-directed-colimit-exact}", "we win." ], "refs": [ "algebra-lemma-flat", "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 525, 343 ] } ], "ref_ids": [] }, { "id": 527, "type": "theorem", "label": "algebra-lemma-flat-base-change", "categories": [ "algebra" ], "title": "algebra-lemma-flat-base-change", "contents": [ "Suppose that $M$ is (faithfully) flat over $R$, and that $R \\to R'$", "is a ring map. Then $M \\otimes_R R'$ is (faithfully) flat over $R'$." ], "refs": [], "proofs": [ { "contents": [ "For any $R'$-module $N$ we have a canonical", "isomorphism $N \\otimes_{R'} (R'\\otimes_R M)", "= N \\otimes_R M$. Hence the desired exactness properties of the functor", "$-\\otimes_{R'}(R'\\otimes_R M)$ follow from", "the corresponding exactness properties of the functor $-\\otimes_R M$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 528, "type": "theorem", "label": "algebra-lemma-flatness-descends", "categories": [ "algebra" ], "title": "algebra-lemma-flatness-descends", "contents": [ "Let $R \\to R'$ be a faithfully flat ring map.", "Let $M$ be a module over $R$, and set $M' = R' \\otimes_R M$.", "Then $M$ is flat over $R$ if and only if $M'$ is flat over $R'$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-flat-base-change} we see that if $M$ is flat", "then $M'$ is flat. For the converse, suppose that $M'$ is flat.", "Let $N_1 \\to N_2 \\to N_3$ be an exact sequence of $R$-modules.", "We want to show that $N_1 \\otimes_R M \\to N_2 \\otimes_R M \\to N_3 \\otimes_R M$", "is exact. We know that", "$N_1 \\otimes_R R' \\to N_2 \\otimes_R R' \\to N_3 \\otimes_R R'$ is", "exact, because $R \\to R'$ is flat. Flatness of $M'$ implies that", "$N_1 \\otimes_R R' \\otimes_{R'} M'", "\\to N_2 \\otimes_R R' \\otimes_{R'} M'", "\\to N_3 \\otimes_R R' \\otimes_{R'} M'$ is exact.", "We may write this as", "$N_1 \\otimes_R M \\otimes_R R'", "\\to N_2 \\otimes_R M \\otimes_R R'", "\\to N_3 \\otimes_R M \\otimes_R R'$.", "Finally, faithful flatness implies that", "$N_1 \\otimes_R M \\to N_2 \\otimes_R M \\to N_3 \\otimes_R M$", "is exact." ], "refs": [ "algebra-lemma-flat-base-change" ], "ref_ids": [ 527 ] } ], "ref_ids": [] }, { "id": 529, "type": "theorem", "label": "algebra-lemma-flatness-descends-more-general", "categories": [ "algebra" ], "title": "algebra-lemma-flatness-descends-more-general", "contents": [ "Let $R$ be a ring. Let $S \\to S'$ be a flat map of $R$-algebras.", "Let $M$ be a module over $S$, and set $M' = S' \\otimes_S M$.", "\\begin{enumerate}", "\\item If $M$ is flat over $R$, then $M'$ is flat over $R$.", "\\item If $S \\to S'$ is faithfully flat, then $M$ is flat", "over $R$ if and only if $M'$ is flat over $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $N \\to N'$ be an injection of $R$-modules. By the flatness", "of $S \\to S'$ we have", "$$", "\\Ker(N \\otimes_R M \\to N' \\otimes_R M) \\otimes_S S'", "=", "\\Ker(N \\otimes_R M' \\to N' \\otimes_R M')", "$$", "If $M$ is flat over $R$, then the left hand side is zero and", "we find that $M'$ is flat over $R$ by the second characterization", "of flatness in Lemma \\ref{lemma-flat}.", "If $M'$ is flat over $R$ then we have the vanishing of the right hand side", "and if in addition $S \\to S'$ is faithfully flat, this implies that", "$\\Ker(N \\otimes_R M \\to N' \\otimes_R M)$ is zero which in turn", "shows that $M$ is flat over $R$." ], "refs": [ "algebra-lemma-flat" ], "ref_ids": [ 525 ] } ], "ref_ids": [] }, { "id": 530, "type": "theorem", "label": "algebra-lemma-flat-permanence", "categories": [ "algebra" ], "title": "algebra-lemma-flat-permanence", "contents": [ "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module.", "If $M$ is flat as an $R$-module and faithfully flat as an $S$-module,", "then $R \\to S$ is flat." ], "refs": [], "proofs": [ { "contents": [ "Let $N_1 \\to N_2 \\to N_3$ be an exact sequence of $R$-modules.", "By assumption $N_1 \\otimes_R M \\to N_2 \\otimes_R M \\to N_3 \\otimes_R M$", "is exact. We may write this as", "$$", "N_1 \\otimes_R S \\otimes_S M", "\\to", "N_2 \\otimes_R S \\otimes_S M", "\\to", "N_3 \\otimes_R S \\otimes_S M.", "$$", "By faithful flatness of $M$ over $S$ we conclude that", "$N_1 \\otimes_R S \\to N_2 \\otimes_R S \\to N_3 \\otimes_R S$ is exact.", "Hence $R \\to S$ is flat." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 531, "type": "theorem", "label": "algebra-lemma-flat-eq", "categories": [ "algebra" ], "title": "algebra-lemma-flat-eq", "contents": [ "A module $M$ over $R$ is flat if and only if", "every relation in $M$ is trivial." ], "refs": [], "proofs": [ { "contents": [ "Assume $M$ is flat and let $\\sum f_i x_i = 0$ be a relation in $M$.", "Let $I = (f_1, \\ldots, f_n)$, and let", "$K = \\Ker(R^n \\to I, (a_1, \\ldots, a_n) \\mapsto \\sum_i a_i f_i)$.", "So we have the short exact sequence", "$0 \\to K \\to R^n \\to I \\to 0$. Then $\\sum f_i \\otimes x_i$", "is an element of $I \\otimes_R M$ which maps", "to zero in $R \\otimes_R M = M$. By flatness", "$\\sum f_i \\otimes x_i$ is zero in $I \\otimes_R M$.", "Thus there exists an element of $K \\otimes_R M$ mapping", "to $\\sum e_i \\otimes x_i \\in R^n \\otimes_R M$ where $e_i$", "is the $i$th basis element of $R^n$.", "Write this element as $\\sum k_j \\otimes y_j$", "and then write the image of $k_j$ in $R^n$ as", "$\\sum a_{ij} e_i$ to get the result.", "\\medskip\\noindent", "Assume every relation is trivial, let $I$", "be a finitely generated ideal, and let $x = \\sum f_i \\otimes x_i$", "be an element of $I \\otimes_R M$ mapping to zero in $R \\otimes_R M = M$.", "This just means exactly that $\\sum f_i x_i$ is a relation in", "$M$. And the fact that it is trivial implies easily that", "$x$ is zero, because", "$$", "x", "=", "\\sum f_i \\otimes x_i", "=", "\\sum f_i \\otimes \\left(\\sum a_{ij}y_j\\right)", "=", "\\sum \\left(\\sum f_i a_{ij}\\right) \\otimes y_j", "=", "0", "$$" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 532, "type": "theorem", "label": "algebra-lemma-flat-tor-zero", "categories": [ "algebra" ], "title": "algebra-lemma-flat-tor-zero", "contents": [ "Suppose that $R$ is a ring, $0 \\to M'' \\to M' \\to M \\to 0$", "a short exact sequence, and $N$ an $R$-module. If $M$ is flat", "then $N \\otimes_R M'' \\to N \\otimes_R M'$ is injective, i.e., the", "sequence", "$$", "0 \\to N \\otimes_R M'' \\to N \\otimes_R M' \\to N \\otimes_R M \\to 0", "$$", "is a short exact sequence." ], "refs": [], "proofs": [ { "contents": [ "Let $R^{(I)} \\to N$ be a surjection from a free module", "onto $N$ with kernel $K$. The result follows", "from the snake lemma applied to the following diagram", "$$", "\\begin{matrix}", " & & 0 & & 0 & & 0 & & \\\\", " & & \\uparrow & & \\uparrow & & \\uparrow & & \\\\", " & & M''\\otimes_R N & \\to & M' \\otimes_R N & \\to & M \\otimes_R N & \\to & 0 \\\\", " & & \\uparrow & & \\uparrow & & \\uparrow & & \\\\", "0 & \\to & (M'')^{(I)} & \\to & (M')^{(I)} & \\to & M^{(I)} & \\to & 0 \\\\", " & & \\uparrow & & \\uparrow & & \\uparrow & & \\\\", " & & M''\\otimes_R K & \\to & M' \\otimes_R K & \\to & M \\otimes_R K & \\to & 0 \\\\", " & & & & & & \\uparrow & & \\\\", " & & & & & & 0 & &", "\\end{matrix}", "$$", "with exact rows and columns. The middle row is exact because tensoring", "with the free module $R^{(I)}$ is exact." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 533, "type": "theorem", "label": "algebra-lemma-flat-ses", "categories": [ "algebra" ], "title": "algebra-lemma-flat-ses", "contents": [ "Suppose that $0 \\to M' \\to M \\to M'' \\to 0$ is", "a short exact sequence of $R$-modules.", "If $M'$ and $M''$ are flat so is $M$.", "If $M$ and $M''$ are flat so is $M'$." ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion that a module $N$ is flat if for", "every ideal $I \\subset R$ the map $N \\otimes_R I \\to N$ is injective,", "see Lemma \\ref{lemma-flat}.", "Consider an ideal $I \\subset R$.", "Consider the diagram", "$$", "\\begin{matrix}", "0 & \\to & M' & \\to & M & \\to & M'' & \\to & 0 \\\\", "& & \\uparrow & & \\uparrow & & \\uparrow & & \\\\", "& & M'\\otimes_R I & \\to & M \\otimes_R I & \\to & M''\\otimes_R I & \\to & 0", "\\end{matrix}", "$$", "with exact rows. This immediately proves the first assertion.", "The second follows because if $M''$ is flat then the lower left", "horizontal arrow is injective by Lemma \\ref{lemma-flat-tor-zero}." ], "refs": [ "algebra-lemma-flat", "algebra-lemma-flat-tor-zero" ], "ref_ids": [ 525, 532 ] } ], "ref_ids": [] }, { "id": 534, "type": "theorem", "label": "algebra-lemma-easy-ff", "categories": [ "algebra" ], "title": "algebra-lemma-easy-ff", "contents": [ "Let $R$ be a ring.", "Let $M$ be an $R$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $M$ is faithfully flat, and", "\\item $M$ is flat and for all $R$-module homomorphisms $\\alpha : N \\to N'$", "we have $\\alpha = 0$ if and only if $\\alpha \\otimes \\text{id}_M = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $M$ is faithfully flat, then", "$0 \\to \\Ker(\\alpha) \\to N \\to N'$ is exact if and only if the same holds", "after tensoring with $M$. This proves (1) implies (2).", "For the other, assume (2). Let $N_1 \\to N_2 \\to N_3$", "be a complex, and assume the complex", "$N_1 \\otimes_R M \\to N_2 \\otimes_R M \\to N_3\\otimes_R M$", "is exact. Take $x \\in \\Ker(N_2 \\to N_3)$,", "and consider the map $\\alpha : R \\to N_2/\\Im(N_1)$,", "$r \\mapsto rx + \\Im(N_1)$. By the exactness", "of the complex $-\\otimes_R M$ we see that $\\alpha \\otimes", "\\text{id}_M$ is zero. By assumption we get that $\\alpha$ is", "zero. Hence $x $ is in the image of $N_1 \\to N_2$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 535, "type": "theorem", "label": "algebra-lemma-ff", "categories": [ "algebra" ], "title": "algebra-lemma-ff", "contents": [ "\\begin{slogan}", "A flat module is faithfully flat if and only if it has nonzero fibers.", "\\end{slogan}", "Let $M$ be a flat $R$-module.", "The following are equivalent:", "\\begin{enumerate}", "\\item $M$ is faithfully flat,", "\\item for every nonzero $R$-module $N$, then tensor product $M \\otimes_R N$", "is nonzero,", "\\item for all $\\mathfrak p \\in \\Spec(R)$", "the tensor product $M \\otimes_R \\kappa(\\mathfrak p)$ is nonzero, and", "\\item for all maximal ideals $\\mathfrak m$ of $R$", "the tensor product $M \\otimes_R \\kappa(\\mathfrak m) = M/{\\mathfrak m}M$", "is nonzero.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $M$ faithfully flat and $N \\not = 0$. By Lemma \\ref{lemma-easy-ff}", "the nonzero map $1 : N \\to N$ induces a nonzero map", "$M \\otimes_R N \\to M \\otimes_R N$, so $M \\otimes_R N \\not = 0$.", "Thus (1) implies (2). The imlpications (2) $\\Rightarrow$ (3) $\\Rightarrow$ (4)", "are immediate.", "\\medskip\\noindent", "Assume (4). Suppose that $N_1 \\to N_2 \\to N_3$ is a complex and", "suppose that $N_1 \\otimes_R M \\to N_2\\otimes_R M \\to", "N_3\\otimes_R M$ is exact. Let $H$ be the cohomology of the complex,", "so $H = \\Ker(N_2 \\to N_3)/\\Im(N_1 \\to N_2)$. To finish the proof", "we will show $H = 0$. By flatness we see that $H \\otimes_R M = 0$.", "Take $x \\in H$ and let $I = \\{f \\in R \\mid fx = 0 \\}$", "be its annihilator. Since $R/I \\subset H$ we get", "$M/IM \\subset H \\otimes_R M = 0$ by flatness of $M$.", "If $I \\not = R$ we may choose", "a maximal ideal $I \\subset \\mathfrak m \\subset R$.", "This immediately gives a contradiction." ], "refs": [ "algebra-lemma-easy-ff" ], "ref_ids": [ 534 ] } ], "ref_ids": [] }, { "id": 536, "type": "theorem", "label": "algebra-lemma-ff-rings", "categories": [ "algebra" ], "title": "algebra-lemma-ff-rings", "contents": [ "Let $R \\to S$ be a flat ring map.", "The following are equivalent:", "\\begin{enumerate}", "\\item $R \\to S$ is faithfully flat,", "\\item the induced map on $\\Spec$ is surjective, and", "\\item any closed point $x \\in \\Spec(R)$ is", "in the image of the map $\\Spec(S) \\to \\Spec(R)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows quickly from Lemma \\ref{lemma-ff}, because we", "saw in Remark \\ref{remark-fundamental-diagram}", "that $\\mathfrak p$ is in the image", "if and only if the ring $S \\otimes_R \\kappa(\\mathfrak p)$", "is nonzero." ], "refs": [ "algebra-lemma-ff", "algebra-remark-fundamental-diagram" ], "ref_ids": [ 535, 1558 ] } ], "ref_ids": [] }, { "id": 537, "type": "theorem", "label": "algebra-lemma-local-flat-ff", "categories": [ "algebra" ], "title": "algebra-lemma-local-flat-ff", "contents": [ "A flat local ring homomorphism of local rings is faithfully flat." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-ff-rings}." ], "refs": [ "algebra-lemma-ff-rings" ], "ref_ids": [ 536 ] } ], "ref_ids": [] }, { "id": 538, "type": "theorem", "label": "algebra-lemma-flat-localization", "categories": [ "algebra" ], "title": "algebra-lemma-flat-localization", "contents": [ "Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset.", "\\begin{enumerate}", "\\item The localization $S^{-1}R$ is a flat $R$-algebra.", "\\item If $M$ is an $S^{-1}R$-module, then $M$ is a flat $R$-module", "if and only if $M$ is a flat $S^{-1}R$-module.", "\\item Suppose $M$ is an $R$-module. Then", "$M$ is a flat $R$-module if and only if $M_{\\mathfrak p}$ is a flat", "$R_{\\mathfrak p}$-module for all primes $\\mathfrak p$ of $R$.", "\\item Suppose $M$ is an $R$-module. Then $M$ is a flat $R$-module if", "and only if $M_{\\mathfrak m}$ is a flat", "$R_{\\mathfrak m}$-module for all maximal ideals $\\mathfrak m$ of $R$.", "\\item Suppose $R \\to A$ is a ring map, $M$ is an $A$-module,", "and $g_1, \\ldots, g_m \\in A$ are elements generating the unit", "ideal of $A$. Then $M$ is flat over $R$ if and only if each localization", "$M_{g_i}$ is flat over $R$.", "\\item Suppose $R \\to A$ is a ring map, and $M$ is an $A$-module.", "Then $M$ is a flat $R$-module if and only if the localization", "$M_{\\mathfrak q}$ is a flat $R_{\\mathfrak p}$-module", "(with $\\mathfrak p$ the prime of $R$ lying under $\\mathfrak q$)", "for all primes $\\mathfrak q$ of $A$.", "\\item Suppose $R \\to A$ is a ring map, and $M$ is an $A$-module.", "Then $M$ is a flat $R$-module if and only if the localization", "$M_{\\mathfrak m}$ is a flat $R_{\\mathfrak p}$-module", "(with $\\mathfrak p = R \\cap \\mathfrak m$)", "for all maximal ideals $\\mathfrak m$ of $A$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us prove the last statement of the lemma.", "In the proof we will use repeatedly that localization is exact", "and commutes with tensor product, see Sections \\ref{section-localization}", "and \\ref{section-tensor-product}.", "\\medskip\\noindent", "Suppose $R \\to A$ is a ring map, and $M$ is an $A$-module.", "Assume that $M_{\\mathfrak m}$ is a flat $R_{\\mathfrak p}$-module", "for all maximal ideals $\\mathfrak m$ of $A$ (with", "$\\mathfrak p = R \\cap \\mathfrak m$). Let $I \\subset R$ be an ideal.", "We have to show the map $I \\otimes_R M \\to M$ is injective.", "We can think of this as a map of $A$-modules.", "By assumption the localization", "$(I \\otimes_R M)_{\\mathfrak m} \\to M_{\\mathfrak m}$ is injective", "because", "$(I \\otimes_R M)_{\\mathfrak m} =", "I_{\\mathfrak p} \\otimes_{R_{\\mathfrak p}} M_{\\mathfrak m}$.", "Hence the kernel of $I \\otimes_R M \\to M$ is zero by", "Lemma \\ref{lemma-characterize-zero-local}.", "Hence $M$ is flat over $R$.", "\\medskip\\noindent", "Conversely, assume $M$ is flat over $R$. Pick a prime $\\mathfrak q$", "of $A$ lying over the prime $\\mathfrak p$ of $R$. Suppose that", "$I \\subset R_{\\mathfrak p}$ is an ideal. We have to show that", "$I \\otimes_{R_{\\mathfrak p}} M_{\\mathfrak q} \\to M_{\\mathfrak q}$", "is injective. We can write $I = J_{\\mathfrak p}$ for some", "ideal $J \\subset R$. Then the map", "$I \\otimes_{R_{\\mathfrak p}} M_{\\mathfrak q} \\to M_{\\mathfrak q}$", "is just the localization (at $\\mathfrak q$) of the map", "$J \\otimes_R M \\to M$ which is injective. Since localization is exact", "we see that $M_{\\mathfrak q}$ is a flat $R_{\\mathfrak p}$-module.", "\\medskip\\noindent", "This proves (7) and (6). The other statements follow in a straightforward", "way from the last statement (proofs omitted)." ], "refs": [ "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 410 ] } ], "ref_ids": [] }, { "id": 539, "type": "theorem", "label": "algebra-lemma-flat-going-down", "categories": [ "algebra" ], "title": "algebra-lemma-flat-going-down", "contents": [ "Let $R \\to S$ be flat. Let $\\mathfrak p \\subset \\mathfrak p'$", "be primes of $R$. Let $\\mathfrak q' \\subset S$ be a prime of $S$", "mapping to $\\mathfrak p'$. Then there exists a prime", "$\\mathfrak q \\subset \\mathfrak q'$ mapping to $\\mathfrak p$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-flat-localization} the local ring map", "$R_{\\mathfrak p'} \\to S_{\\mathfrak q'}$ is flat.", "By Lemma \\ref{lemma-local-flat-ff} this local ring map is faithfully", "flat. By Lemma \\ref{lemma-ff-rings} there is a prime mapping to", "$\\mathfrak p R_{\\mathfrak p'}$. The inverse image of this", "prime in $S$ does the job." ], "refs": [ "algebra-lemma-flat-localization", "algebra-lemma-local-flat-ff", "algebra-lemma-ff-rings" ], "ref_ids": [ 538, 537, 536 ] } ], "ref_ids": [] }, { "id": 540, "type": "theorem", "label": "algebra-lemma-colimit-faithfully-flat", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-faithfully-flat", "contents": [ "Let $R$ be a ring. Let $\\{S_i, \\varphi_{ii'}\\}$ be a directed system of", "faithfully flat $R$-algebras. Then $S = \\colim_i S_i$ is a faithfully flat", "$R$-algebra." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-colimit-flat} we see that $S$ is flat.", "Let $\\mathfrak m \\subset R$ be a maximal ideal. By", "Lemma \\ref{lemma-ff-rings}", "none of the rings $S_i/\\mathfrak m S_i$ is zero.", "Hence $S/\\mathfrak mS = \\colim S_i/\\mathfrak mS_i$ is nonzero", "as well because $1$ is not equal to zero. Thus the image of", "$\\Spec(S) \\to \\Spec(R)$ contains $\\mathfrak m$ and we see that $R \\to S$", "is faithfully flat by Lemma \\ref{lemma-ff-rings}." ], "refs": [ "algebra-lemma-colimit-flat", "algebra-lemma-ff-rings", "algebra-lemma-ff-rings" ], "ref_ids": [ 523, 536, 536 ] } ], "ref_ids": [] }, { "id": 541, "type": "theorem", "label": "algebra-lemma-support-zero", "categories": [ "algebra" ], "title": "algebra-lemma-support-zero", "contents": [ "\\begin{slogan}", "A module over a ring has empty support if and only if it is the trivial module.", "\\end{slogan}", "Let $R$ be a ring. Let $M$ be an $R$-module. Then", "$$", "M = (0) \\Leftrightarrow \\text{Supp}(M) = \\emptyset.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Actually,", "Lemma \\ref{lemma-characterize-zero-local}", "even shows that $\\text{Supp}(M)$ always contains a maximal ideal", "if $M$ is not zero." ], "refs": [ "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 410 ] } ], "ref_ids": [] }, { "id": 542, "type": "theorem", "label": "algebra-lemma-annihilator-flat-base-change", "categories": [ "algebra" ], "title": "algebra-lemma-annihilator-flat-base-change", "contents": [ "Let $R \\to S$ be a flat ring map. Let $M$ be an $R$-module and", "$m \\in M$. Then $\\text{Ann}_R(m) S = \\text{Ann}_S(m \\otimes 1)$.", "If $M$ is a finite $R$-module, then", "$\\text{Ann}_R(M) S = \\text{Ann}_S(M \\otimes_R S)$." ], "refs": [], "proofs": [ { "contents": [ "Set $I = \\text{Ann}_R(m)$. By definition there is an exact sequence", "$0 \\to I \\to R \\to M$ where the map $R \\to M$ sends $f$ to $fm$. Using", "flatness we obtain an exact sequence", "$0 \\to I \\otimes_R S \\to S \\to M \\otimes_R S$ which proves the first", "assertion. If $m_1, \\ldots, m_n$ is a set of generators of $M$", "then $\\text{Ann}_R(M) = \\bigcap \\text{Ann}_R(m_i)$. Similarly", "$\\text{Ann}_S(M \\otimes_R S) = \\bigcap \\text{Ann}_S(m_i \\otimes 1)$.", "Set $I_i = \\text{Ann}_R(m_i)$. Then it suffices to show that", "$\\bigcap_{i = 1, \\ldots, n} (I_i S) = (\\bigcap_{i = 1, \\ldots, n} I_i)S$.", "This is Lemma \\ref{lemma-flat-intersect-ideals}." ], "refs": [ "algebra-lemma-flat-intersect-ideals" ], "ref_ids": [ 522 ] } ], "ref_ids": [] }, { "id": 543, "type": "theorem", "label": "algebra-lemma-support-closed", "categories": [ "algebra" ], "title": "algebra-lemma-support-closed", "contents": [ "Let $R$ be a ring and let $M$ be an $R$-module.", "If $M$ is finite, then $\\text{Supp}(M)$ is closed.", "More precisely, if $I = \\text{Ann}(M)$ is the annihilator of $M$, then", "$V(I) = \\text{Supp}(M)$." ], "refs": [], "proofs": [ { "contents": [ "We will show that $V(I) = \\text{Supp}(M)$.", "\\medskip\\noindent", "Suppose $\\mathfrak p \\in \\text{Supp}(M)$. Then $M_{\\mathfrak p} \\not = 0$.", "Choose an element $m \\in M$ whose image in $M_\\mathfrak p$ is nonzero.", "Then the annihilator of $m$ is contained in $\\mathfrak p$ by construction", "of the localization $M_\\mathfrak p$. Hence", "a fortiori $I = \\text{Ann}(M)$ must be contained in $\\mathfrak p$.", "\\medskip\\noindent", "Conversely, suppose that $\\mathfrak p \\not \\in \\text{Supp}(M)$.", "Then $M_{\\mathfrak p} = 0$.", "Let $x_1, \\ldots, x_r \\in M$ be generators.", "By Lemma \\ref{lemma-localization-colimit} there exists", "an $f \\in R$, $f\\not\\in \\mathfrak p$ such that", "$x_i/1 = 0$ in $M_f$. Hence $f^{n_i} x_i = 0$ for some $n_i \\geq 1$.", "Hence $f^nM = 0$ for $n = \\max\\{n_i\\}$ as desired." ], "refs": [ "algebra-lemma-localization-colimit" ], "ref_ids": [ 348 ] } ], "ref_ids": [] }, { "id": 544, "type": "theorem", "label": "algebra-lemma-support-base-change", "categories": [ "algebra" ], "title": "algebra-lemma-support-base-change", "contents": [ "Let $R \\to R'$ be a ring map and let $M$ be a finite $R$-module.", "Then $\\text{Supp}(M \\otimes_R R')$ is the inverse image of", "$\\text{Supp}(M)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p \\in \\text{Supp}(M)$. By Nakayama's lemma", "(Lemma \\ref{lemma-NAK}) we see that", "$$", "M \\otimes_R \\kappa(\\mathfrak p) = M_\\mathfrak p/\\mathfrak p M_\\mathfrak p", "$$", "is a nonzero $\\kappa(\\mathfrak p)$ vector space.", "Hence for every prime $\\mathfrak p' \\subset R'$ lying", "over $\\mathfrak p$ we see that", "$$", "(M \\otimes_R R')_{\\mathfrak p'}/\\mathfrak p' (M \\otimes_R R')_{\\mathfrak p'} =", "(M \\otimes_R R') \\otimes_{R'} \\kappa(\\mathfrak p') =", "M \\otimes_R \\kappa(\\mathfrak p) \\otimes_{\\kappa(\\mathfrak p)}", "\\kappa(\\mathfrak p')", "$$", "is nonzero. This implies $\\mathfrak p' \\in \\text{Supp}(M \\otimes_R R')$.", "For the converse, if $\\mathfrak p' \\subset R'$ is a prime lying", "over an arbitrary prime $\\mathfrak p \\subset R$, then", "$$", "(M \\otimes_R R')_{\\mathfrak p'} =", "M_\\mathfrak p \\otimes_{R_\\mathfrak p} R'_{\\mathfrak p'}.", "$$", "Hence if $\\mathfrak p' \\in \\text{Supp}(M \\otimes_R R')$", "lies over the prime $\\mathfrak p \\subset R$, then", "$\\mathfrak p \\in \\text{Supp}(M)$." ], "refs": [ "algebra-lemma-NAK" ], "ref_ids": [ 401 ] } ], "ref_ids": [] }, { "id": 545, "type": "theorem", "label": "algebra-lemma-support-element", "categories": [ "algebra" ], "title": "algebra-lemma-support-element", "contents": [ "Let $R$ be a ring, let $M$ be an $R$-module, and let $m \\in M$.", "Then $\\mathfrak p \\in V(\\text{Ann}(m))$ if and only if", "$m$ does not map to zero in $M_\\mathfrak p$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $M$ by $Rm \\subset M$. Then (1) $\\text{Ann}(m) = \\text{Ann}(M)$", "and (2) $x$ does not map to zero in $M_\\mathfrak p$ if and only if", "$\\mathfrak p \\in \\text{Supp}(M)$.", "The result now follows from Lemma \\ref{lemma-support-closed}." ], "refs": [ "algebra-lemma-support-closed" ], "ref_ids": [ 543 ] } ], "ref_ids": [] }, { "id": 546, "type": "theorem", "label": "algebra-lemma-support-finite-presentation-constructible", "categories": [ "algebra" ], "title": "algebra-lemma-support-finite-presentation-constructible", "contents": [ "Let $R$ be a ring and let $M$ be an $R$-module.", "If $M$ is a finitely presented $R$-module, then $\\text{Supp}(M)$ is a", "closed subset of $\\Spec(R)$ whose complement is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation", "$$", "R^{\\oplus m} \\longrightarrow R^{\\oplus n} \\longrightarrow M \\to 0", "$$", "Let $A \\in \\text{Mat}(n \\times m, R)$ be the matrix of the first", "map. By Nakayama's Lemma \\ref{lemma-NAK} we see that", "$$", "M_{\\mathfrak p} \\not = 0 \\Leftrightarrow", "M \\otimes \\kappa(\\mathfrak p) \\not = 0 \\Leftrightarrow", "\\text{rank}(A \\bmod \\mathfrak p) < n.", "$$", "Hence, if $I$ is the ideal of $R$ generated by the $n \\times n$ minors", "of $A$, then $\\text{Supp}(M) = V(I)$. Since $I$", "is finitely generated, say $I = (f_1, \\ldots, f_t)$,", "we see that $\\Spec(R) \\setminus V(I)$ is", "a finite union of the standard opens $D(f_i)$, hence quasi-compact." ], "refs": [ "algebra-lemma-NAK" ], "ref_ids": [ 401 ] } ], "ref_ids": [] }, { "id": 547, "type": "theorem", "label": "algebra-lemma-support-quotient", "categories": [ "algebra" ], "title": "algebra-lemma-support-quotient", "contents": [ "Let $R$ be a ring and let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item If $M$ is finite then the support", "of $M/IM$ is $\\text{Supp}(M) \\cap V(I)$.", "\\item If $N \\subset M$, then $\\text{Supp}(N) \\subset", "\\text{Supp}(M)$.", "\\item If $Q$ is a quotient module of $M$ then $\\text{Supp}(Q) \\subset", "\\text{Supp}(M)$.", "\\item If $0 \\to N \\to M \\to Q \\to 0$ is a short exact sequence", "then $\\text{Supp}(M) = \\text{Supp}(Q) \\cup \\text{Supp}(N)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The functors $M \\mapsto M_{\\mathfrak p}$ are exact. This immediately", "implies all but the first assertion. For the first assertion", "we need to show that $M_\\mathfrak p \\not = 0$ and", "$I \\subset \\mathfrak p$ implies $(M/IM)_{\\mathfrak p}", "= M_\\mathfrak p/IM_\\mathfrak p \\not = 0$. This follows", "from Nakayama's Lemma \\ref{lemma-NAK}." ], "refs": [ "algebra-lemma-NAK" ], "ref_ids": [ 401 ] } ], "ref_ids": [] }, { "id": 548, "type": "theorem", "label": "algebra-lemma-open-going-down", "categories": [ "algebra" ], "title": "algebra-lemma-open-going-down", "contents": [ "Let $R \\to S$ be a ring map. If the induced map", "$\\varphi : \\Spec(S) \\to \\Spec(R)$ is open, then", "$R \\to S$ satisfies going down." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $\\mathfrak p \\subset \\mathfrak p' \\subset R$ and", "$\\mathfrak q' \\subset S$ lies over $\\mathfrak p'$. As $\\varphi$ is open,", "for every $g \\in S$, $g \\not \\in \\mathfrak q'$ we see that $\\mathfrak p$", "is in the image of $D(g) \\subset \\Spec(S)$. In other words", "$S_g \\otimes_R \\kappa(\\mathfrak p)$ is not zero. Since $S_{\\mathfrak q'}$", "is the directed colimit of these $S_g$ this implies", "that $S_{\\mathfrak q'} \\otimes_R \\kappa(\\mathfrak p)$ is not", "zero, see", "Lemmas \\ref{lemma-localization-colimit} and", "\\ref{lemma-tensor-products-commute-with-limits}.", "Hence $\\mathfrak p$ is in the image of", "$\\Spec(S_{\\mathfrak q'}) \\to \\Spec(R)$ as desired." ], "refs": [ "algebra-lemma-localization-colimit", "algebra-lemma-tensor-products-commute-with-limits" ], "ref_ids": [ 348, 363 ] } ], "ref_ids": [] }, { "id": 549, "type": "theorem", "label": "algebra-lemma-going-up-down-specialization", "categories": [ "algebra" ], "title": "algebra-lemma-going-up-down-specialization", "contents": [ "Let $R \\to S$ be a ring map.", "\\begin{enumerate}", "\\item $R \\to S$ satisfies going down if and only if", "generalizations lift along the map $\\Spec(S) \\to \\Spec(R)$,", "see Topology, Definition \\ref{topology-definition-lift-specializations}.", "\\item $R \\to S$ satisfies going up if and only if", "specializations lift along the map $\\Spec(S) \\to \\Spec(R)$,", "see Topology, Definition \\ref{topology-definition-lift-specializations}.", "\\end{enumerate}" ], "refs": [ "topology-definition-lift-specializations", "topology-definition-lift-specializations" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8366, 8366 ] }, { "id": 550, "type": "theorem", "label": "algebra-lemma-going-up-down-composition", "categories": [ "algebra" ], "title": "algebra-lemma-going-up-down-composition", "contents": [ "Suppose $R \\to S$ and $S \\to T$ are ring maps satisfying", "going down. Then so does $R \\to T$. Similarly for going up." ], "refs": [], "proofs": [ { "contents": [ "According to Lemma \\ref{lemma-going-up-down-specialization}", "this follows from", "Topology, Lemma \\ref{topology-lemma-lift-specialization-composition}" ], "refs": [ "algebra-lemma-going-up-down-specialization", "topology-lemma-lift-specialization-composition" ], "ref_ids": [ 549, 8285 ] } ], "ref_ids": [] }, { "id": 551, "type": "theorem", "label": "algebra-lemma-image-stable-specialization-closed", "categories": [ "algebra" ], "title": "algebra-lemma-image-stable-specialization-closed", "contents": [ "Let $R \\to S$ be a ring map. Let $T \\subset \\Spec(R)$", "be the image of $\\Spec(S)$. If $T$ is stable under specialization,", "then $T$ is closed." ], "refs": [], "proofs": [ { "contents": [ "We give two proofs.", "\\medskip\\noindent", "First proof. Let $\\mathfrak p \\subset R$ be a prime ideal such that", "the corresponding point of $\\Spec(R)$ is in the closure", "of $T$. This means that for every $f \\in R$, $f \\not \\in \\mathfrak p$", "we have $D(f) \\cap T \\not = \\emptyset$. Note that $D(f) \\cap T$", "is the image of $\\Spec(S_f)$ in $\\Spec(R)$. Hence", "we conclude that $S_f \\not = 0$. In other words, $1 \\not = 0$ in", "the ring $S_f$. Since $S_{\\mathfrak p}$ is the directed colimit", "of the rings $S_f$ we conclude that $1 \\not = 0$ in", "$S_{\\mathfrak p}$. In other words, $S_{\\mathfrak p} \\not = 0$ and", "considering the image of $\\Spec(S_{\\mathfrak p})", "\\to \\Spec(S) \\to \\Spec(R)$ we see there exists", "a $\\mathfrak p' \\in T$ with $\\mathfrak p' \\subset \\mathfrak p$.", "As we assumed $T$ closed under specialization we conclude $\\mathfrak p$", "is a point of $T$ as desired.", "\\medskip\\noindent", "Second proof. Let $I = \\Ker(R \\to S)$. We may replace $R$ by $R/I$.", "In this case the ring map $R \\to S$ is injective.", "By Lemma \\ref{lemma-injective-minimal-primes-in-image}", "all the minimal primes of $R$ are contained in the image $T$. Hence", "if $T$ is stable under specialization then it contains all primes." ], "refs": [ "algebra-lemma-injective-minimal-primes-in-image" ], "ref_ids": [ 445 ] } ], "ref_ids": [] }, { "id": 552, "type": "theorem", "label": "algebra-lemma-going-up-closed", "categories": [ "algebra" ], "title": "algebra-lemma-going-up-closed", "contents": [ "Let $R \\to S$ be a ring map. The following are equivalent:", "\\begin{enumerate}", "\\item Going up holds for $R \\to S$, and", "\\item the map $\\Spec(S) \\to \\Spec(R)$ is closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is a general fact that specializations lift along a", "closed map of topological spaces, see", "Topology, Lemma \\ref{topology-lemma-closed-open-map-specialization}.", "Hence the second condition implies the first.", "\\medskip\\noindent", "Assume that going up holds for $R \\to S$.", "Let $V(I) \\subset \\Spec(S)$ be a closed set.", "We want to show that the image of $V(I)$ in $\\Spec(R)$ is closed.", "The ring map $S \\to S/I$ obviously satisfies going up.", "Hence $R \\to S \\to S/I$ satisfies going up,", "by Lemma \\ref{lemma-going-up-down-composition}.", "Replacing $S$ by $S/I$ it suffices to show the image $T$", "of $\\Spec(S)$ in $\\Spec(R)$ is closed.", "By Topology, Lemmas \\ref{topology-lemma-open-closed-specialization}", "and \\ref{topology-lemma-lift-specializations-images} this", "image is stable under specialization. Thus the result follows", "from Lemma \\ref{lemma-image-stable-specialization-closed}." ], "refs": [ "topology-lemma-closed-open-map-specialization", "algebra-lemma-going-up-down-composition", "topology-lemma-open-closed-specialization", "topology-lemma-lift-specializations-images", "algebra-lemma-image-stable-specialization-closed" ], "ref_ids": [ 8287, 550, 8283, 8286, 551 ] } ], "ref_ids": [] }, { "id": 553, "type": "theorem", "label": "algebra-lemma-constructible-stable-specialization-closed", "categories": [ "algebra" ], "title": "algebra-lemma-constructible-stable-specialization-closed", "contents": [ "Let $R$ be a ring. Let $E \\subset \\Spec(R)$ be a constructible subset.", "\\begin{enumerate}", "\\item If $E$ is stable under specialization, then $E$ is closed.", "\\item If $E$ is stable under generalization, then $E$ is open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First proof. The first assertion", "follows from Lemma \\ref{lemma-image-stable-specialization-closed}", "combined with Lemma \\ref{lemma-constructible-is-image}.", "The second follows because the complement of a constructible", "set is constructible", "(see Topology, Lemma \\ref{topology-lemma-constructible}),", "the first part of the lemma and Topology,", "Lemma \\ref{topology-lemma-open-closed-specialization}.", "\\medskip\\noindent", "Second proof. Since $\\Spec(R)$ is a spectral space by", "Lemma \\ref{lemma-spec-spectral} this is a special case of", "Topology, Lemma", "\\ref{topology-lemma-constructible-stable-specialization-closed}." ], "refs": [ "algebra-lemma-image-stable-specialization-closed", "algebra-lemma-constructible-is-image", "topology-lemma-constructible", "topology-lemma-open-closed-specialization", "algebra-lemma-spec-spectral", "topology-lemma-constructible-stable-specialization-closed" ], "ref_ids": [ 551, 435, 8253, 8283, 423, 8307 ] } ], "ref_ids": [] }, { "id": 554, "type": "theorem", "label": "algebra-lemma-same-image", "categories": [ "algebra" ], "title": "algebra-lemma-same-image", "contents": [ "Let $k$ be a field, and let $R$, $S$ be $k$-algebras.", "Let $S' \\subset S$ be a sub $k$-algebra, and let $f \\in S' \\otimes_k R$.", "In the commutative diagram", "$$", "\\xymatrix{", "\\Spec((S \\otimes_k R)_f) \\ar[rd] \\ar[rr] & &", "\\Spec((S' \\otimes_k R)_f) \\ar[ld] \\\\", "& \\Spec(R) &", "}", "$$", "the images of the diagonal arrows are the same." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p \\subset R$ be in the image of the south-west", "arrow. This means (Lemma \\ref{lemma-in-image}) that", "$$", "(S' \\otimes_k R)_f \\otimes_R \\kappa(\\mathfrak p)", "=", "(S' \\otimes_k \\kappa(\\mathfrak p))_f", "$$", "is not the zero ring, i.e., $S' \\otimes_k \\kappa(\\mathfrak p)$", "is not the zero ring and the image of $f$ in it is not nilpotent.", "The ring map", "$S' \\otimes_k \\kappa(\\mathfrak p) \\to S \\otimes_k \\kappa(\\mathfrak p)$", "is injective. Hence also $S \\otimes_k \\kappa(\\mathfrak p)$", "is not the zero ring and the image of $f$ in it is not nilpotent.", "Hence $(S \\otimes_k R)_f \\otimes_R \\kappa(\\mathfrak p)$", "is not the zero ring. Thus (Lemma \\ref{lemma-in-image})", "we see that $\\mathfrak p$ is in the image of the south-east arrow", "as desired." ], "refs": [ "algebra-lemma-in-image", "algebra-lemma-in-image" ], "ref_ids": [ 394, 394 ] } ], "ref_ids": [] }, { "id": 555, "type": "theorem", "label": "algebra-lemma-map-into-tensor-algebra-open", "categories": [ "algebra" ], "title": "algebra-lemma-map-into-tensor-algebra-open", "contents": [ "Let $k$ be a field.", "Let $R$ and $S$ be $k$-algebras.", "The map $\\Spec(S \\otimes_k R) \\to \\Spec(R)$", "is open." ], "refs": [], "proofs": [ { "contents": [ "Let $f \\in S \\otimes_k R$.", "It suffices to prove that the image of the standard open $D(f)$ is open.", "Let $S' \\subset S$ be a finite type $k$-subalgebra such that", "$f \\in S' \\otimes_k R$. The map $R \\to S' \\otimes_k R$ is flat", "and of finite presentation, hence the image $U$ of", "$\\Spec((S' \\otimes_k R)_f) \\to \\Spec(R)$ is open", "by Proposition \\ref{proposition-fppf-open}.", "By Lemma \\ref{lemma-same-image} this is also the image of $D(f)$ and we win." ], "refs": [ "algebra-proposition-fppf-open", "algebra-lemma-same-image" ], "ref_ids": [ 1407, 554 ] } ], "ref_ids": [] }, { "id": 556, "type": "theorem", "label": "algebra-lemma-unique-prime-over-localize-below", "categories": [ "algebra" ], "title": "algebra-lemma-unique-prime-over-localize-below", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak p \\subset R$ be a prime.", "Assume that", "\\begin{enumerate}", "\\item there exists a unique prime $\\mathfrak q \\subset S$ lying over", "$\\mathfrak p$, and", "\\item either", "\\begin{enumerate}", "\\item going up holds for $R \\to S$, or", "\\item going down holds for $R \\to S$ and there is at most one prime", "of $S$ above every prime of $R$.", "\\end{enumerate}", "\\end{enumerate}", "Then $S_{\\mathfrak p} = S_{\\mathfrak q}$." ], "refs": [], "proofs": [ { "contents": [ "Consider any prime $\\mathfrak q' \\subset S$ which corresponds to", "a point of $\\Spec(S_{\\mathfrak p})$. This means that", "$\\mathfrak p' = R \\cap \\mathfrak q'$ is contained in $\\mathfrak p$.", "Here is a picture", "$$", "\\xymatrix{", "\\mathfrak q' \\ar@{-}[d] \\ar@{-}[r] & ? \\ar@{-}[r] \\ar@{-}[d] & S \\ar@{-}[d] \\\\", "\\mathfrak p' \\ar@{-}[r] & \\mathfrak p \\ar@{-}[r] & R", "}", "$$", "Assume (1) and (2)(a).", "By going up there exists a prime $\\mathfrak q'' \\subset S$", "with $\\mathfrak q' \\subset \\mathfrak q''$ and $\\mathfrak q''$", "lying over $\\mathfrak p$. By the uniqueness of $\\mathfrak q$ we", "conclude that $\\mathfrak q'' = \\mathfrak q$. In other words", "$\\mathfrak q'$ defines a point of $\\Spec(S_{\\mathfrak q})$.", "\\medskip\\noindent", "Assume (1) and (2)(b).", "By going down there exists a prime $\\mathfrak q'' \\subset \\mathfrak q$", "lying over $\\mathfrak p'$. By the uniqueness of primes lying over", "$\\mathfrak p'$ we see that $\\mathfrak q' = \\mathfrak q''$. In other words", "$\\mathfrak q'$ defines a point of $\\Spec(S_{\\mathfrak q})$.", "\\medskip\\noindent", "In both cases we conclude that the map", "$\\Spec(S_{\\mathfrak q}) \\to \\Spec(S_{\\mathfrak p})$", "is bijective. Clearly this means all the elements of $S - \\mathfrak q$", "are all invertible in $S_{\\mathfrak p}$, in other words", "$S_{\\mathfrak p} = S_{\\mathfrak q}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 557, "type": "theorem", "label": "algebra-lemma-going-down-flat-module", "categories": [ "algebra" ], "title": "algebra-lemma-going-down-flat-module", "contents": [ "Let $R \\to S$ be a ring map. Let $N$ be a finite $S$-module flat over $R$.", "Endow $\\text{Supp}(N) \\subset \\Spec(S)$ with the induced topology.", "Then generalizations lift along $\\text{Supp}(N) \\to \\Spec(R)$." ], "refs": [], "proofs": [ { "contents": [ "The meaning of the statement is as follows. Let", "$\\mathfrak p \\subset \\mathfrak p' \\subset R$ be primes. Let", "$\\mathfrak q' \\subset S$ be a prime $\\mathfrak q' \\in \\text{Supp}(N)$", "Then there exists a prime $\\mathfrak q \\subset \\mathfrak q'$,", "$\\mathfrak q \\in \\text{Supp}(N)$ lying over $\\mathfrak p$.", "As $N$ is flat over $R$ we see that $N_{\\mathfrak q'}$ is flat", "over $R_{\\mathfrak p'}$, see Lemma \\ref{lemma-flat-localization}.", "As $N_{\\mathfrak q'}$ is finite over $S_{\\mathfrak q'}$", "and not zero since $\\mathfrak q' \\in \\text{Supp}(N)$ we see", "that $N_{\\mathfrak q'} \\otimes_{S_{\\mathfrak q'}} \\kappa(\\mathfrak q')$", "is nonzero by Nakayama's Lemma \\ref{lemma-NAK}.", "Thus $N_{\\mathfrak q'} \\otimes_{R_{\\mathfrak p'}} \\kappa(\\mathfrak p')$", "is also not zero. We conclude from Lemma \\ref{lemma-ff}", "that $N_{\\mathfrak q'} \\otimes_{R_{\\mathfrak p'}} \\kappa(\\mathfrak p)$", "is nonzero. Let", "$J \\subset S_{\\mathfrak q'} \\otimes_{R_{\\mathfrak p'}} \\kappa(\\mathfrak p)$", "be the annihilator of the finite nonzero module", "$N_{\\mathfrak q'} \\otimes_{R_{\\mathfrak p'}} \\kappa(\\mathfrak p)$.", "Since $J$ is a proper ideal we can choose a prime $\\mathfrak q \\subset S$", "which corresponds to a prime of", "$S_{\\mathfrak q'} \\otimes_{R_{\\mathfrak p'}} \\kappa(\\mathfrak p)/J$.", "This prime is in the support of $N$, lies over $\\mathfrak p$, and", "is contained in $\\mathfrak q'$ as desired." ], "refs": [ "algebra-lemma-flat-localization", "algebra-lemma-NAK", "algebra-lemma-ff" ], "ref_ids": [ 538, 401, 535 ] } ], "ref_ids": [] }, { "id": 558, "type": "theorem", "label": "algebra-lemma-subextensions-are-separable", "categories": [ "algebra" ], "title": "algebra-lemma-subextensions-are-separable", "contents": [ "Let $k \\subset K$ be a separable field extension.", "For any subextension $k \\subset K' \\subset K$ the field", "extension $k \\subset K'$ is separable." ], "refs": [], "proofs": [ { "contents": [ "This is direct from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 559, "type": "theorem", "label": "algebra-lemma-generating-finitely-generated-separable-field-extensions", "categories": [ "algebra" ], "title": "algebra-lemma-generating-finitely-generated-separable-field-extensions", "contents": [ "Let $k \\subset K$ be a separably generated, and finitely generated", "field extension.", "Set $r = \\text{trdeg}_k(K)$. Then there exist elements", "$x_1, \\ldots, x_{r + 1}$ of $K$ such that", "\\begin{enumerate}", "\\item $x_1, \\ldots, x_r$ is a transcendence basis of $K$ over $k$,", "\\item $K = k(x_1, \\ldots, x_{r + 1})$, and", "\\item $x_{r + 1}$ is separable over $k(x_1, \\ldots, x_r)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Combine the definition with Fields, Lemma \\ref{fields-lemma-primitive-element}." ], "refs": [ "fields-lemma-primitive-element" ], "ref_ids": [ 4498 ] } ], "ref_ids": [] }, { "id": 560, "type": "theorem", "label": "algebra-lemma-make-separably-generated", "categories": [ "algebra" ], "title": "algebra-lemma-make-separably-generated", "contents": [ "Let $k \\subset K$ be a finitely generated field extension.", "There exists a diagram", "$$", "\\xymatrix{", "K \\ar[r] & K' \\\\", "k \\ar[u] \\ar[r] & k' \\ar[u]", "}", "$$", "where $k \\subset k'$, $K \\subset K'$ are finite purely inseparable field", "extensions such that $k' \\subset K'$ is a separably generated field extension." ], "refs": [], "proofs": [ { "contents": [ "This lemma is only interesting when the characteristic of $k$ is $p > 0$.", "Choose $x_1, \\ldots, x_r$ a transcendence basis of $K$ over $k$.", "As $K$ is finitely generated over $k$ the extension", "$k(x_1, \\ldots, x_r) \\subset K$ is finite.", "Let $k(x_1, \\ldots, x_r) \\subset K_{sep} \\subset K$ be the subextension", "found in", "Fields, Lemma \\ref{fields-lemma-separable-first}.", "If $K = K_{sep}$ then we are done.", "We will use induction on $d = [K : K_{sep}]$.", "\\medskip\\noindent", "Assume that $d > 1$. Choose a $\\beta \\in K$ with", "$\\alpha = \\beta^p \\in K_{sep}$ and $\\beta \\not \\in K_{sep}$.", "Let $P = T^d + a_1T^{d - 1} + \\ldots + a_d$", "be the minimal polynomial of $\\alpha$ over $k(x_1, \\ldots, x_r)$.", "Let $k \\subset k'$ be a finite purely inseparable extension", "obtained by adjoining $p$th roots such that each $a_i$ is a", "$p$th power in $k'(x_1^{1/p}, \\ldots, x_r^{1/p})$.", "Such an extension exists; details omitted.", "Let $L$ be a field fitting into the diagram", "$$", "\\xymatrix{", "K \\ar[r] & L \\\\", "k(x_1, \\ldots, x_r) \\ar[u] \\ar[r] & k'(x_1^{1/p}, \\ldots, x_r^{1/p}) \\ar[u]", "}", "$$", "We may and do assume $L$ is the compositum of $K$ and", "$k'(x_1^{1/p}, \\ldots, x_r^{1/p})$. Let", "$k'(x_1^{1/p}, \\ldots, x_r^{1/p}) \\subset L_{sep} \\subset L$", "be the subextension found in", "Fields, Lemma \\ref{fields-lemma-separable-first}.", "Then $L_{sep}$ is the compositum of", "$K_{sep}$ and $k'(x_1^{1/p}, \\ldots, x_r^{1/p})$.", "The element $\\alpha \\in L_{sep}$ is a zero of the polynomial", "$P$ all of whose coefficients are $p$th powers in", "$k'(x_1^{1/p}, \\ldots, x_r^{1/p})$ and whose roots are", "pairwise distinct. By", "Fields, Lemma \\ref{fields-lemma-pth-root}", "we see that $\\alpha = (\\alpha')^p$ for some $\\alpha' \\in L_{sep}$.", "Clearly, this means that $\\beta$ maps to $\\alpha' \\in L_{sep}$.", "In other words, we get the tower of fields", "$$", "\\xymatrix{", "K \\ar[r] & L \\\\", "K_{sep}(\\beta) \\ar[r] \\ar[u] & L_{sep} \\ar[u] \\\\", "K_{sep} \\ar[r] \\ar[u] & L_{sep} \\ar@{=}[u] \\\\", "k(x_1, \\ldots, x_r) \\ar[u] \\ar[r] & k'(x_1^{1/p}, \\ldots, x_r^{1/p}) \\ar[u] \\\\", "k \\ar[r] \\ar[u] & k' \\ar[u]", "}", "$$", "Thus this construction leads to a new situation with", "$[L : L_{sep}] < [K : K_{sep}]$. By induction we can find", "$k' \\subset k''$ and $L \\subset L'$ as in the lemma for the", "extension $k' \\subset L$. Then the extensions $k \\subset k''$ and", "$K \\subset L'$ work for the extension $k \\subset K$.", "This proves the lemma." ], "refs": [ "fields-lemma-separable-first", "fields-lemma-separable-first", "fields-lemma-pth-root" ], "ref_ids": [ 4482, 4482, 4523 ] } ], "ref_ids": [] }, { "id": 561, "type": "theorem", "label": "algebra-lemma-subalgebra-separable", "categories": [ "algebra" ], "title": "algebra-lemma-subalgebra-separable", "contents": [ "Elementary properties of geometrically reduced algebras.", "Let $k$ be a field. Let $S$ be a $k$-algebra.", "\\begin{enumerate}", "\\item If $S$ is geometrically reduced over $k$ so is every", "$k$-subalgebra.", "\\item If all finitely generated $k$-subalgebras of $S$ are", "geometrically reduced, then $S$ is geometrically reduced.", "\\item A directed colimit of geometrically reduced $k$-algebras", "is geometrically reduced.", "\\item If $S$ is geometrically reduced over $k$, then any localization", "of $S$ is geometrically reduced over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. The second and third property follow from the fact that", "tensor product commutes with colimits." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 562, "type": "theorem", "label": "algebra-lemma-geometrically-reduced-permanence", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-reduced-permanence", "contents": [ "Let $k$ be a field.", "If $R$ is geometrically reduced over $k$,", "and $S \\subset R$ is a multiplicative subset, then the localization", "$S^{-1}R$ is geometrically reduced over $k$.", "If $R$ is geometrically reduced over $k$, then $R[x]$ is geometrically", "reduced over $k$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints: A localization of a reduced ring is reduced, and", "localization commutes with tensor products." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 563, "type": "theorem", "label": "algebra-lemma-limit-argument", "categories": [ "algebra" ], "title": "algebra-lemma-limit-argument", "contents": [ "Let $k$ be a field. Let $R$, $S$ be $k$-algebras.", "\\begin{enumerate}", "\\item If $R \\otimes_k S$ is nonreduced, then there exist", "finitely generated subalgebras $R' \\subset R$,", "$S' \\subset S$ such that $R' \\otimes_k S'$ is not reduced.", "\\item If $R \\otimes_k S$ contains a nonzero zerodivisor, then there exist", "finitely generated subalgebras $R' \\subset R$,", "$S' \\subset S$ such that $R' \\otimes_k S'$ contains a nonzero zerodivisor.", "\\item If $R \\otimes_k S$ contains a nontrivial idempotent, then there exist", "finitely generated subalgebras $R' \\subset R$,", "$S' \\subset S$ such that $R' \\otimes_k S'$ contains a nontrivial idempotent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose $z \\in R \\otimes_k S$ is nilpotent. We may write", "$z = \\sum_{i = 1, \\ldots, n} x_i \\otimes y_i$.", "Thus we may take $R'$ the $k$-subalgebra generated by", "the $x_i$ and $S'$ the $k$-subalgebra generated by the $y_i$.", "The second and third statements are proved in the same way." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 564, "type": "theorem", "label": "algebra-lemma-geometrically-reduced-any-reduced-base-change", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-reduced-any-reduced-base-change", "contents": [ "Let $k$ be a field.", "Let $S$ be a geometrically reduced $k$-algebra.", "Let $R$ be any reduced $k$-algebra.", "Then $R \\otimes_k S$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-limit-argument}", "we may assume that $R$ is of finite type over $k$.", "Then $R$, as a reduced Noetherian ring, embeds into a finite", "product of fields", "(see Lemmas \\ref{lemma-total-ring-fractions-no-embedded-points},", "\\ref{lemma-Noetherian-irreducible-components}, and", "\\ref{lemma-minimal-prime-reduced-ring}).", "Hence we may assume $R$ is a finite product of", "fields. In this case it follows from", "Definition \\ref{definition-geometrically-reduced}", "that $R \\otimes_k S$ is reduced." ], "refs": [ "algebra-lemma-limit-argument", "algebra-lemma-total-ring-fractions-no-embedded-points", "algebra-lemma-Noetherian-irreducible-components", "algebra-lemma-minimal-prime-reduced-ring", "algebra-definition-geometrically-reduced" ], "ref_ids": [ 563, 421, 453, 418, 1461 ] } ], "ref_ids": [] }, { "id": 565, "type": "theorem", "label": "algebra-lemma-separable-extension-preserves-reducedness", "categories": [ "algebra" ], "title": "algebra-lemma-separable-extension-preserves-reducedness", "contents": [ "Let $k$ be a field.", "Let $S$ be a reduced $k$-algebra.", "Let $k \\subset K$ be either a separable field extension,", "or a separably generated field extension.", "Then $K \\otimes_k S$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "Assume $k \\subset K$ is separable.", "By Lemma \\ref{lemma-limit-argument}", "we may assume that $S$ is of finite type over $k$", "and $K$ is finitely generated over $k$.", "Then $S$ embeds into a finite product of fields,", "namely its total ring of fractions (see", "Lemmas \\ref{lemma-minimal-prime-reduced-ring} and", "\\ref{lemma-total-ring-fractions-no-embedded-points}).", "Hence we may actually assume that $S$ is a domain.", "We choose $x_1, \\ldots, x_{r + 1} \\in K$ as in", "Lemma \\ref{lemma-generating-finitely-generated-separable-field-extensions}.", "Let $P \\in k(x_1, \\ldots, x_r)[T]$", "be the minimal polynomial of $x_{r + 1}$. It is a separable polynomial.", "It is easy to see that", "$k[x_1, \\ldots, x_r] \\otimes_k S = S[x_1, \\ldots, x_r]$ is a domain.", "This implies $k(x_1, \\ldots, x_r) \\otimes_k S$ is a domain", "as it is a localization of $S[x_1, \\ldots, x_r]$.", "The ring extension $k(x_1, \\ldots, x_r) \\otimes_k S \\subset K \\otimes_k S$", "is generated by a single element $x_{r + 1}$ with a single", "equation, namely $P$. Hence $K \\otimes_k S$ embeds into", "$F[T]/(P)$ where $F$ is the fraction field of $k(x_1, \\ldots, x_r) \\otimes_k S$.", "Since $P$ is separable this is a finite product of fields and we win.", "\\medskip\\noindent", "At this point we do not yet know that a separably generated field", "extension is separable, so we have to prove the lemma in this case also.", "To do this suppose that $\\{x_i\\}_{i \\in I}$ is a separating", "transcendence basis for $K$ over $k$. For any finite set of elements", "$\\lambda_j \\in K$ there exists a finite subset $T \\subset I$ such", "that $k(\\{x_i\\}_{i\\in T}) \\subset k(\\{x_i\\}_{i \\in T} \\cup \\{\\lambda_j\\})$", "is finite separable. Hence we see that $K$ is a directed colimit of", "finitely generated and separably generated extensions of $k$. Thus", "the argument of the preceding paragraph applies to this case as well." ], "refs": [ "algebra-lemma-limit-argument", "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-total-ring-fractions-no-embedded-points", "algebra-lemma-generating-finitely-generated-separable-field-extensions" ], "ref_ids": [ 563, 418, 421, 559 ] } ], "ref_ids": [] }, { "id": 566, "type": "theorem", "label": "algebra-lemma-generic-points-geometrically-reduced", "categories": [ "algebra" ], "title": "algebra-lemma-generic-points-geometrically-reduced", "contents": [ "Let $k$ be a field and let $S$ be a $k$-algebra. Assume that", "$S$ is reduced and that $S_{\\mathfrak p}$ is geometrically", "reduced for every minimal prime $\\mathfrak p$ of $S$.", "Then $S$ is geometrically reduced." ], "refs": [], "proofs": [ { "contents": [ "Since $S$ is reduced the map", "$S \\to \\prod_{\\mathfrak p\\text{ minimal}} S_{\\mathfrak p}$", "is injective, see", "Lemma \\ref{lemma-reduced-ring-sub-product-fields}.", "If $k \\subset K$ is a field extension, then the maps", "$$", "S \\otimes_k K \\to (\\prod S_\\mathfrak p) \\otimes_k K \\to", "\\prod S_\\mathfrak p \\otimes_k K", "$$", "are injective: the first as $k \\to K$ is flat and the second by inspection", "because $K$ is a free $k$-module. As $S_\\mathfrak p$ is geometrically", "reduced the ring on the right is reduced. Thus we see that $S \\otimes_k K$", "is reduced as a subring of a reduced ring." ], "refs": [ "algebra-lemma-reduced-ring-sub-product-fields" ], "ref_ids": [ 419 ] } ], "ref_ids": [] }, { "id": 567, "type": "theorem", "label": "algebra-lemma-separable-algebraic-diagonal", "categories": [ "algebra" ], "title": "algebra-lemma-separable-algebraic-diagonal", "contents": [ "Let $k'/k$ be a separable algebraic extension.", "Then there exists a multiplicative subset $S \\subset k' \\otimes_k k'$", "such that the multiplication map $k' \\otimes_k k' \\to k'$", "is identified with $k' \\otimes_k k' \\to S^{-1}(k' \\otimes_k k')$." ], "refs": [], "proofs": [ { "contents": [ "First assume $k'/k$ is finite separable. Then $k' = k(\\alpha)$,", "see Fields, Lemma \\ref{fields-lemma-primitive-element}.", "Let $P \\in k[x]$ be the minimal polynomial of $\\alpha$ over $k$.", "Then $P$ is an irreducible, separable, monic polynomial, see", "Fields, Section \\ref{fields-section-separable-extensions}.", "Then $k'[x]/(P) \\to k' \\otimes_k k'$,", "$\\sum \\alpha_i x^i \\mapsto \\alpha_i \\otimes \\alpha^i$ is an isomorphism.", "We can factor $P = (x - \\alpha) Q$ in $k'[x]$ and since $P$", "is separable we see that $Q(\\alpha) \\not = 0$.", "Then it is clear that the multiplicative set $S'$ generated by", "$Q$ in $k'[x]/(P)$ works, i.e., that $k' = (S')^{-1}(k'[x]/(P))$.", "By transport of structure the image $S$ of $S'$ in $k' \\otimes_k k'$", "works.", "\\medskip\\noindent", "In the general case we write $k' = \\bigcup k_i$ as the union", "of its finite subfield extensions over $k$. For each $i$ there", "is a multiplicative subset $S_i \\subset k_i \\otimes_k k_i$", "such that $k_i = S_i^{-1}(k_i \\otimes_k k_i)$. Then", "$S = \\bigcup S_i \\subset k' \\otimes_k k'$ works." ], "refs": [ "fields-lemma-primitive-element" ], "ref_ids": [ 4498 ] } ], "ref_ids": [] }, { "id": 568, "type": "theorem", "label": "algebra-lemma-geometrically-reduced-over-separable-algebraic", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-reduced-over-separable-algebraic", "contents": [ "Let $k \\subset k'$ be a separable algebraic field extension.", "Let $A$ be an algebra over $k'$. Then $A$ is geometrically", "reduced over $k$ if and only if it is geometrically reduced over $k'$." ], "refs": [], "proofs": [ { "contents": [ "Assume $A$ is geometrically reduced over $k'$.", "Let $K/k$ be a field extension. Then $K \\otimes_k k'$ is", "a reduced ring by", "Lemma \\ref{lemma-separable-extension-preserves-reducedness}.", "Hence by Lemma \\ref{lemma-geometrically-reduced-any-reduced-base-change}", "we find that $K \\otimes_k A = (K \\otimes_k k') \\otimes_{k'} A$ is reduced.", "\\medskip\\noindent", "Assume $A$ is geometrically reduced over $k$. Let $K/k'$ be a field", "extension. Then", "$$", "K \\otimes_{k'} A = (K \\otimes_k A) \\otimes_{(k' \\otimes_k k')} k'", "$$", "Since $k' \\otimes_k k' \\to k'$ is a localization by", "Lemma \\ref{lemma-separable-algebraic-diagonal},", "we see that $K \\otimes_{k'} A$", "is a localization of a reduced algebra, hence reduced." ], "refs": [ "algebra-lemma-separable-extension-preserves-reducedness", "algebra-lemma-geometrically-reduced-any-reduced-base-change", "algebra-lemma-separable-algebraic-diagonal" ], "ref_ids": [ 565, 564, 567 ] } ], "ref_ids": [] }, { "id": 569, "type": "theorem", "label": "algebra-lemma-characterize-separable-field-extensions", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-separable-field-extensions", "contents": [ "Let $k$ be a field of characteristic $p > 0$.", "Let $k \\subset K$ be a field extension.", "The following are equivalent:", "\\begin{enumerate}", "\\item $K$ is separable over $k$,", "\\item the ring $K \\otimes_k k^{1/p}$ is reduced, and", "\\item $K$ is geometrically reduced over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (3) follows from", "Lemma \\ref{lemma-separable-extension-preserves-reducedness}.", "The implication (3) $\\Rightarrow$ (2) is immediate.", "\\medskip\\noindent", "Assume (2). Let $k \\subset L \\subset K$ be a subextension such that", "$L$ is a finitely generated field extension of $k$.", "We have to show that we can find a separating transcendence basis of $L$.", "The assumption implies that $L \\otimes_k k^{1/p}$ is reduced.", "Let $x_1, \\ldots, x_r$ be a transcendence basis of $L$ over $k$ such", "that the degree of inseparability of the finite extension", "$k(x_1, \\ldots, x_r) \\subset L$ is minimal.", "If $L$ is separable over $k(x_1, \\ldots, x_r)$ then we win.", "Assume this is not the case to get a contradiction.", "Then there exists an element $\\alpha \\in L$ which is not", "separable over $k(x_1, \\ldots, x_r)$. Let $P(T) \\in k(x_1, \\ldots, x_r)[T]$", "be the minimal polynomial of $\\alpha$ over $k(x_1, \\ldots, x_r)$.", "After replacing $\\alpha$ by $f \\alpha$ for some nonzero", "$f \\in k[x_1, \\ldots, x_r]$", "we may and do assume that $P$ lies in $k[x_1, \\ldots, x_r, T]$.", "Because $\\alpha$ is not separable $P$ is a polynomial in $T^p$, see", "Fields, Lemma \\ref{fields-lemma-irreducible-polynomials}.", "Let $dp$ be the degree of $P$ as a polynomial in $T$.", "Since $P$ is the minimal polynomial of $\\alpha$ the monomials", "$$", "x_1^{e_1} \\ldots x_r^{e_r} \\alpha^e", "$$", "for $e < dp$ are linearly independent over $k$ in $L$. We claim that", "the element $\\partial P/\\partial x_i \\in k[x_1, \\ldots, x_r, T]$ is not zero", "for at least one $i$.", "Namely, if this was not the case, then $P$ is actually a polynomial in", "$x_1^p, \\ldots, x_r^p, T^p$. In that case we can consider", "$P^{1/p} \\in k^{1/p}[x_1, \\ldots, x_r, T]$. This would map to", "$P^{1/p}(x_1, \\ldots, x_r, \\alpha)$ which is a nilpotent element of", "$k^{1/p} \\otimes_k L$ and hence zero. On the other hand,", "$P^{1/p}(x_1, \\ldots, x_r, \\alpha)$ is a $k^{1/p}$-linear combination", "the monomials listed above, hence nonzero in $k^{1/p} \\otimes_k L$.", "This is a contradiction which proves our claim.", "\\medskip\\noindent", "Thus, after renumbering, we may assume that $\\partial P/\\partial x_1$", "is not zero. As $P$ is an irreducible polynomial in $T$ over", "$k(x_1, \\ldots, x_r)$ it is irreducible as a polynomial in", "$x_1, \\ldots, x_r, T$, hence by Gauss's lemma it is irreducible", "as a polynomial in $x_1$ over $k(x_2, \\ldots, x_r, T)$.", "Since the transcendence degree of $L$ is $r$ we see that", "$x_2, \\ldots, x_r, \\alpha$ are algebraically independent.", "Hence $P(X, x_2, \\ldots, x_r, \\alpha) \\in k(x_2, \\ldots, x_r, \\alpha)[X]$", "is irreducible. It follows that $x_1$ is separably algebraic over", "$k(x_2, \\ldots, x_r, \\alpha)$. This means that", "the degree of inseparability of the finite extension", "$k(x_2, \\ldots, x_r, \\alpha) \\subset L$ is less than the", "degree of inseparability of the finite extension", "$k(x_1, \\ldots, x_r) \\subset L$, which is a contradiction." ], "refs": [ "algebra-lemma-separable-extension-preserves-reducedness", "fields-lemma-irreducible-polynomials" ], "ref_ids": [ 565, 4464 ] } ], "ref_ids": [] }, { "id": 570, "type": "theorem", "label": "algebra-lemma-separably-generated-separable", "categories": [ "algebra" ], "title": "algebra-lemma-separably-generated-separable", "contents": [ "A separably generated field extension is separable." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-separable-extension-preserves-reducedness}", "with Lemma \\ref{lemma-characterize-separable-field-extensions}." ], "refs": [ "algebra-lemma-separable-extension-preserves-reducedness", "algebra-lemma-characterize-separable-field-extensions" ], "ref_ids": [ 565, 569 ] } ], "ref_ids": [] }, { "id": 571, "type": "theorem", "label": "algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension", "contents": [ "Let $k$ be a field. Let $S$ be a $k$-algebra.", "The following are equivalent:", "\\begin{enumerate}", "\\item $k' \\otimes_k S$ is reduced for every finite", "purely inseparable extension $k'$ of $k$,", "\\item $k^{1/p} \\otimes_k S$ is reduced,", "\\item $k^{perf} \\otimes_k S$ is reduced, where $k^{perf}$ is the", "perfect closure of $k$,", "\\item $\\overline{k} \\otimes_k S$ is reduced, where $\\overline{k}$ is the", "algebraic closure of $k$, and", "\\item $S$ is geometrically reduced over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that any finite purely inseparable extension $k \\subset k'$ embeds", "in $k^{perf}$. Moreover, $k^{1/p}$ embeds into $k^{perf}$ which embeds", "into $\\overline{k}$. Thus it is", "clear that (5) $\\Rightarrow$ (4) $\\Rightarrow$ (3) $\\Rightarrow$ (2)", "and that (3) $\\Rightarrow$ (1).", "\\medskip\\noindent", "We prove that (1) $\\Rightarrow$ (5).", "Assume $k' \\otimes_k S$ is reduced for every finite", "purely inseparable extension $k'$ of $k$. Let $k \\subset K$ be", "an extension of fields. We have to show that $K \\otimes_k S$", "is reduced. By Lemma \\ref{lemma-limit-argument} we reduce to the case where", "$k \\subset K$ is a finitely generated field extension. Choose a diagram", "$$", "\\xymatrix{", "K \\ar[r] & K' \\\\", "k \\ar[u] \\ar[r] & k' \\ar[u]", "}", "$$", "as in Lemma \\ref{lemma-make-separably-generated}.", "By assumption $k' \\otimes_k S$ is reduced.", "By Lemma \\ref{lemma-separable-extension-preserves-reducedness}", "it follows that $K' \\otimes_k S$ is reduced.", "Hence we conclude that $K \\otimes_k S$ is reduced as desired.", "\\medskip\\noindent", "Finally we prove that (2) $\\Rightarrow$ (5).", "Assume $k^{1/p} \\otimes_k S$ is reduced. Then $S$ is reduced.", "Moreover, for each localization $S_{\\mathfrak p}$ at a minimal", "prime $\\mathfrak p$, the ring $k^{1/p}\\otimes_k S_{\\mathfrak p}$", "is a localization of $k^{1/p} \\otimes_k S$ hence is reduced.", "But $S_{\\mathfrak p}$ is a field by", "Lemma \\ref{lemma-minimal-prime-reduced-ring},", "hence $S_{\\mathfrak p}$ is geometrically reduced by", "Lemma \\ref{lemma-characterize-separable-field-extensions}.", "It follows from Lemma \\ref{lemma-generic-points-geometrically-reduced}", "that $S$ is geometrically reduced." ], "refs": [ "algebra-lemma-limit-argument", "algebra-lemma-make-separably-generated", "algebra-lemma-separable-extension-preserves-reducedness", "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-characterize-separable-field-extensions", "algebra-lemma-generic-points-geometrically-reduced" ], "ref_ids": [ 563, 560, 565, 418, 569, 566 ] } ], "ref_ids": [] }, { "id": 572, "type": "theorem", "label": "algebra-lemma-perfect", "categories": [ "algebra" ], "title": "algebra-lemma-perfect", "contents": [ "A field $k$ is perfect if and only if it is a field of characteristic $0$", "or a field of characteristic $p > 0$ such that every element has a $p$th", "root." ], "refs": [], "proofs": [ { "contents": [ "The characteristic zero case is clear.", "Assume the characteristic of $k$ is $p > 0$.", "If $k$ is perfect, then all the field extensions where we adjoin", "a $p$th root of an element of $k$ have to be trivial, hence every", "element of $k$ has a $p$th root. Conversely if every element has a $p$th", "root, then $k = k^{1/p}$ and every field extension of $k$ is", "separable by", "Lemma \\ref{lemma-characterize-separable-field-extensions}." ], "refs": [ "algebra-lemma-characterize-separable-field-extensions" ], "ref_ids": [ 569 ] } ], "ref_ids": [] }, { "id": 573, "type": "theorem", "label": "algebra-lemma-make-separable", "categories": [ "algebra" ], "title": "algebra-lemma-make-separable", "contents": [ "Let $k \\subset K$ be a finitely generated field extension.", "There exists a diagram", "$$", "\\xymatrix{", "K \\ar[r] & K' \\\\", "k \\ar[u] \\ar[r] & k' \\ar[u]", "}", "$$", "where $k \\subset k'$, $K \\subset K'$ are finite purely inseparable field", "extensions such that $k' \\subset K'$ is a separable field extension.", "In this situation we can assume that $K' = k'K$ is the compositum,", "and also that $K' = (k' \\otimes_k K)_{red}$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-make-separably-generated}", "we can find such a diagram with $k' \\subset K'$ separably generated.", "By", "Lemma \\ref{lemma-separably-generated-separable}", "this implies that $K'$ is separable over $k'$.", "The compositum $k'K$ is a subextension of $k' \\subset K'$ and hence", "$k' \\subset k'K$ is separable by", "Lemma \\ref{lemma-subextensions-are-separable}.", "The ring $(k' \\otimes_k K)_{red}$ is a domain as for some", "$n \\gg 0$ the map $x \\mapsto x^{p^n}$ maps it into $K$.", "Hence it is a field by", "Lemma \\ref{lemma-integral-over-field}.", "Thus $(k' \\otimes_k K)_{red} \\to K'$ maps it isomorphically onto $k'K$." ], "refs": [ "algebra-lemma-make-separably-generated", "algebra-lemma-separably-generated-separable", "algebra-lemma-subextensions-are-separable", "algebra-lemma-integral-over-field" ], "ref_ids": [ 560, 570, 558, 497 ] } ], "ref_ids": [] }, { "id": 574, "type": "theorem", "label": "algebra-lemma-perfection", "categories": [ "algebra" ], "title": "algebra-lemma-perfection", "contents": [ "\\begin{slogan}", "Every field has a unique perfect closure.", "\\end{slogan}", "For every field $k$ there exists a purely inseparable extension", "$k \\subset k'$ such that $k'$ is perfect. The field extension", "$k \\subset k'$ is unique up to unique isomorphism." ], "refs": [], "proofs": [ { "contents": [ "If the characteristic of $k$ is zero, then $k' = k$ is the", "unique choice. Assume the characteristic of $k$ is $p > 0$.", "For every $n > 0$ there exists a unique algebraic extension", "$k \\subset k^{1/p^n}$ such that (a) every element $\\lambda \\in k$", "has a $p^n$th root in $k^{1/p^n}$ and (b) for every element", "$\\mu \\in k^{1/p^n}$ we have $\\mu^{p^n} \\in k$.", "Namely, consider the ring map $k \\to k^{1/p^n} = k$, $x \\mapsto x^{p^n}$.", "This is injective and satisfies (a) and (b). It is clear that", "$k^{1/p^n} \\subset k^{1/p^{n + 1}}$ as extensions of $k$ via", "the map $y \\mapsto y^p$. Then we can take $k' = \\bigcup k^{1/p^n}$.", "Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 575, "type": "theorem", "label": "algebra-lemma-perfect-reduced", "categories": [ "algebra" ], "title": "algebra-lemma-perfect-reduced", "contents": [ "Let $k$ be a perfect field.", "Any reduced $k$ algebra is geometrically reduced over $k$.", "Let $R$, $S$ be $k$-algebras.", "Assume both $R$ and $S$ are reduced.", "Then the $k$-algebra $R \\otimes_k S$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from", "Lemma \\ref{lemma-geometrically-reduced-finite-purely-inseparable-extension}.", "For the second statement use the first statement and", "Lemma \\ref{lemma-geometrically-reduced-any-reduced-base-change}." ], "refs": [ "algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension", "algebra-lemma-geometrically-reduced-any-reduced-base-change" ], "ref_ids": [ 571, 564 ] } ], "ref_ids": [] }, { "id": 576, "type": "theorem", "label": "algebra-lemma-surjective-locally-nilpotent-kernel", "categories": [ "algebra" ], "title": "algebra-lemma-surjective-locally-nilpotent-kernel", "contents": [ "Let $\\varphi : R \\to S$ be a surjective map with locally nilpotent kernel.", "Then $\\varphi$ induces a homeomorphism of spectra and isomorphisms", "on residue fields. For any ring map $R \\to R'$ the ring map", "$R' \\to R' \\otimes_R S$ is surjective with locally nilpotent kernel." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-spec-closed} the map $\\Spec(S) \\to \\Spec(R)$ is", "a homeomorphism onto the closed subset $V(\\Ker(\\varphi))$. Of course", "$V(\\Ker(\\varphi)) = \\Spec(R)$ because every prime ideal of $R$ contains", "every nilpotent element of $R$. This also implies the statement on", "residue fields. By right exactness of tensor product we see that", "$\\Ker(\\varphi)R'$ is the kernel of the surjective map $R' \\to R' \\otimes_R S$.", "Hence the final statement by Lemma \\ref{lemma-locally-nilpotent}." ], "refs": [ "algebra-lemma-spec-closed", "algebra-lemma-locally-nilpotent" ], "ref_ids": [ 393, 458 ] } ], "ref_ids": [] }, { "id": 577, "type": "theorem", "label": "algebra-lemma-powers-field", "categories": [ "algebra" ], "title": "algebra-lemma-powers-field", "contents": [ "\\begin{reference}", "\\cite[Lemma 3.1.6]{Alper-adequate}", "\\end{reference}", "Let $k \\subset k'$ be a field extension. The following are equivalent", "\\begin{enumerate}", "\\item for each $x \\in k'$ there exists an $n > 0$ such that $x^n \\in k$, and", "\\item $k' = k$ or $k$ and $k'$ have characteristic $p > 0$ and", "either $k'/k$ is a purely inseparable extension or", "$k$ and $k'$ are algebraic extensions of $\\mathbf{F}_p$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that each of the possibilities listed in (2) satisfies (1).", "Thus we assume $k'/k$ satisfies (1) and we prove that we are in", "one of the cases of (2). Discarding the case $k = k'$ we may assume", "$k' \\not = k$. It is clear that $k'/k$ is algebraic.", "Hence we may assume that $k'/k$ is a nontrivial finite extension.", "Let $k \\subset k'_{sep} \\subset k'$ be the separable subextension", "found in Fields, Lemma \\ref{fields-lemma-separable-first}.", "We have to show that $k = k'_{sep}$ or that $k$ is an algebraic over", "$\\mathbf{F}_p$. Thus we may assume that $k'/k$", "is a nontrivial finite separable extension and we have to show", "$k$ is algebraic over $\\mathbf{F}_p$.", "\\medskip\\noindent", "Pick $x \\in k'$, $x \\not \\in k$. Pick $n, m > 0$ such that", "$x^n \\in k$ and $(x + 1)^m \\in k$. Let $\\overline{k}$ be an", "algebraic closure of $k$. We can choose embeddings", "$\\sigma, \\tau : k' \\to \\overline{k}$ with $\\sigma(x) \\not = \\tau(x)$.", "This follows from the discussion in", "Fields, Section \\ref{fields-section-separable-extensions}", "(more precisely, after replacing $k'$ by the $k$-extension", "generated by $x$ it follows from", "Fields, Lemma \\ref{fields-lemma-count-embeddings}).", "Then we see that $\\sigma(x) = \\zeta \\tau(x)$ for some", "$n$th root of unity $\\zeta$ in $\\overline{k}$.", "Similarly, we see that $\\sigma(x + 1) = \\zeta' \\tau(x + 1)$", "for some $m$th root of unity $\\zeta' \\in \\overline{k}$.", "Since $\\sigma(x + 1) \\not = \\tau(x + 1)$ we see $\\zeta' \\not = 1$.", "Then", "$$", "\\zeta' (\\tau(x) + 1) =", "\\zeta' \\tau(x + 1) =", "\\sigma(x + 1) =", "\\sigma(x) + 1 =", "\\zeta \\tau(x) + 1", "$$", "implies that", "$$", "\\tau(x) (\\zeta' - \\zeta) = 1 - \\zeta'", "$$", "hence $\\zeta' \\not = \\zeta$ and", "$$", "\\tau(x) = (1 - \\zeta')/(\\zeta' - \\zeta)", "$$", "Hence every element of $k'$ which is not in $k$ is algebraic over the prime", "subfield. Since $k'$ is generated over the prime subfield by the elements", "of $k'$ which are not in $k$, we conclude that $k'$ (and hence $k$)", "is algebraic over the prime subfield.", "\\medskip\\noindent", "Finally, if the characteristic of $k$ is $0$, the above leads to a", "contradiction as follows (we encourage the reader to find their own proof).", "For every rational number $y$ we similarly get a root of unity", "$\\zeta_y$ such that $\\sigma(x + y) = \\zeta_y\\tau(x + y)$.", "Then we find", "$$", "\\zeta \\tau(x) + y = \\zeta_y(\\tau(x) + y)", "$$", "and by our formula for $\\tau(x)$ above we conclude", "$\\zeta_y \\in \\mathbf{Q}(\\zeta, \\zeta')$. Since the number field", "$\\mathbf{Q}(\\zeta, \\zeta')$ contains only a finite number of roots of", "unity we find two distinct rational numbers $y, y'$ with", "$\\zeta_y = \\zeta_{y'}$. Then we conclude that", "$$", "y - y' =", "\\sigma(x + y) - \\sigma(x + y') =", "\\zeta_y(\\tau(x + y)) - \\zeta_{y'}\\tau(x + y') = \\zeta_y(y - y')", "$$", "which implies $\\zeta_y = 1$ a contradiction." ], "refs": [ "fields-lemma-separable-first", "fields-lemma-count-embeddings" ], "ref_ids": [ 4482, 4468 ] } ], "ref_ids": [] }, { "id": 578, "type": "theorem", "label": "algebra-lemma-powers", "categories": [ "algebra" ], "title": "algebra-lemma-powers", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. If", "\\begin{enumerate}", "\\item for any $x \\in S$ there exists $n > 0$ such that", "$x^n$ is in the image of $\\varphi$, and", "\\item $\\Ker(\\varphi)$ is locally nilpotent,", "\\end{enumerate}", "then $\\varphi$ induces a homeomorphism on spectra and induces residue", "field extensions satisfying the equivalent conditions of", "Lemma \\ref{lemma-powers-field}." ], "refs": [ "algebra-lemma-powers-field" ], "proofs": [ { "contents": [ "Assume (1) and (2). Let $\\mathfrak q, \\mathfrak q'$ be primes of $S$", "lying over the same prime ideal $\\mathfrak p$ of $R$. Suppose $x \\in S$ with", "$x \\in \\mathfrak q$, $x \\not \\in \\mathfrak q'$. Then $x^n \\in \\mathfrak q$", "and $x^n \\not \\in \\mathfrak q'$ for all $n > 0$. If $x^n = \\varphi(y)$ with", "$y \\in R$ for some $n > 0$ then", "$$", "x^n \\in \\mathfrak q \\Rightarrow y \\in \\mathfrak p \\Rightarrow", "x^n \\in \\mathfrak q'", "$$", "which is a contradiction. Hence there does not exist an $x$ as above and", "we conclude that $\\mathfrak q = \\mathfrak q'$, i.e., the map on spectra", "is injective. By assumption (2) the kernel $I = \\Ker(\\varphi)$ is", "contained in every prime, hence $\\Spec(R) = \\Spec(R/I)$ as", "topological spaces. As the induced map $R/I \\to S$ is integral by", "assumption (1)", "Lemma \\ref{lemma-integral-overring-surjective}", "shows that $\\Spec(S) \\to \\Spec(R/I)$ is surjective. Combining", "the above we see that $\\Spec(S) \\to \\Spec(R)$ is bijective.", "If $x \\in S$ is arbitrary, and we pick $y \\in R$ such that", "$\\varphi(y) = x^n$ for some $n > 0$, then we see that the open", "$D(x) \\subset \\Spec(S)$ corresponds to the open", "$D(y) \\subset \\Spec(R)$ via the bijection above. Hence we see that", "the map $\\Spec(S) \\to \\Spec(R)$ is a homeomorphism.", "\\medskip\\noindent", "To see the statement on residue fields, let $\\mathfrak q \\subset S$", "be a prime lying over a prime ideal $\\mathfrak p \\subset R$. Let", "$x \\in \\kappa(\\mathfrak q)$. If we think of $\\kappa(\\mathfrak q)$", "as the residue field of the local ring $S_\\mathfrak q$, then we", "see that $x$ is the image of some $y/z \\in S_\\mathfrak q$", "with $y \\in S$, $z \\in S$, $z \\not \\in \\mathfrak q$.", "Choose $n, m > 0$ such that $y^n, z^m$ are in the image of $\\varphi$.", "Then $x^{nm}$ is the residue of $(y/z)^{nm} = (y^n)^m/(z^m)^n$", "which is in the image of $R_\\mathfrak p \\to S_\\mathfrak q$.", "Hence $x^{nm}$ is in the image of", "$\\kappa(\\mathfrak p) \\to \\kappa(\\mathfrak q)$." ], "refs": [ "algebra-lemma-integral-overring-surjective" ], "ref_ids": [ 495 ] } ], "ref_ids": [ 577 ] }, { "id": 579, "type": "theorem", "label": "algebra-lemma-2-3-ring-map", "categories": [ "algebra" ], "title": "algebra-lemma-2-3-ring-map", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Assume", "\\begin{enumerate}", "\\item[(a)] $S$ is generated as an $R$-algebra by elements $x$ such", "that $x^2, x^3 \\in \\varphi(R)$, and", "\\item[(b)] $\\Ker(\\varphi)$ is locally nilpotent,", "\\end{enumerate}", "Then $\\varphi$ induces isomorphisms on residue fields and", "a homeomorphism of spectra. For any ring map $R \\to R'$", "the ring map $R' \\to R' \\otimes_R S$ also satisfies (a) and (b)." ], "refs": [], "proofs": [ { "contents": [ "Assume (a) and (b). The map on spectra is closed as $S$ is integral", "over $R$, see Lemmas \\ref{lemma-going-up-closed} and", "\\ref{lemma-integral-going-up}. The image is dense by", "Lemma \\ref{lemma-image-dense-generic-points}. Thus $\\Spec(S) \\to \\Spec(R)$", "is surjective. If $\\mathfrak q \\subset S$ is a prime lying over", "$\\mathfrak p \\subset R$ then the field extension", "$\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$ is generated by elements", "$\\alpha \\in \\kappa(\\mathfrak q)$ whose square and cube are", "in $\\kappa(\\mathfrak p)$. Thus clearly $\\alpha \\in \\kappa(\\mathfrak p)$", "and we find that $\\kappa(\\mathfrak q) = \\kappa(\\mathfrak p)$.", "If $\\mathfrak q, \\mathfrak q'$ were two distinct primes lying over", "$\\mathfrak p$, then at least one of the generators $x$ of $S$ as", "in (a) would have distinct images in", "$\\kappa(\\mathfrak q) = \\kappa(\\mathfrak p)$ and", "$\\kappa(\\mathfrak q') = \\kappa(\\mathfrak p)$.", "This would contradict the fact that both $x^2$ and $x^3$", "do have the same image. This proves that $\\Spec(S) \\to \\Spec(R)$", "is injective hence a homeomorphism (by what was already shown).", "\\medskip\\noindent", "Since $\\varphi$ induces a homeomorphism on spectra, it is in particular", "surjective on spectra which is a property preserved under any base change, see", "Lemma \\ref{lemma-surjective-spec-radical-ideal}.", "Therefore for any $R \\to R'$ the kernel of the ring map", "$R' \\to R' \\otimes_R S$ consists of nilpotent elements, see", "Lemma \\ref{lemma-image-dense-generic-points},", "in other words (b) holds for $R' \\to R' \\otimes_R S$.", "It is clear that (a) is preserved under base change." ], "refs": [ "algebra-lemma-going-up-closed", "algebra-lemma-integral-going-up", "algebra-lemma-image-dense-generic-points", "algebra-lemma-surjective-spec-radical-ideal", "algebra-lemma-image-dense-generic-points" ], "ref_ids": [ 552, 500, 446, 443, 446 ] } ], "ref_ids": [] }, { "id": 580, "type": "theorem", "label": "algebra-lemma-help-with-powers", "categories": [ "algebra" ], "title": "algebra-lemma-help-with-powers", "contents": [ "Let $p$ be a prime number. Let $n, m > 0$ be two integers. There exists", "an integer $a$ such that", "$(x + y)^{p^a}, p^a(x + y) \\in \\mathbf{Z}[x^{p^n}, p^nx, y^{p^m}, p^my]$." ], "refs": [], "proofs": [ { "contents": [ "This is clear for $p^a(x + y)$ as soon as $a \\geq n, m$.", "In fact, pick $a \\gg n, m$. Write", "$$", "(x + y)^{p^a} = \\sum\\nolimits_{i, j \\geq 0, i + j = p^a}", "{p^a \\choose i, j} x^iy^j", "$$", "For every $i, j \\geq 0$ with $i + j = p^a$ write", "$i = q p^n + r$ with $r \\in \\{0, \\ldots, p^n - 1\\}$ and", "$j = q' p^m + r'$ with $r' \\in \\{0, \\ldots, p^m - 1\\}$.", "The condition $(x + y)^{p^a} \\in \\mathbf{Z}[x^{p^n}, p^nx, y^{p^m}, p^my]$", "holds if", "$$", "p^{nr + mr'} \\text{ divides } {p^a \\choose i, j}", "$$", "If $r = r' = 0$ then the divisibility holds. If $r \\not = 0$, then", "we write", "$$", "{p^a \\choose i, j} = \\frac{p^a}{i} {p^a - 1 \\choose i - 1, j}", "$$", "Since $r \\not = 0$ the rational number $p^a/i$ has $p$-adic", "valuation at least $a - (n - 1)$ (because $i$ is not divisible by $p^n$).", "Thus ${p^a \\choose i, j}$ is divisible by $p^{a - n + 1}$ in this case.", "Similarly, we see that if $r' \\not = 0$, then ${p^a \\choose i, j}$ is", "divisible by $p^{a - m + 1}$. Picking $a = np^n + mp^m + n + m$ will work." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 581, "type": "theorem", "label": "algebra-lemma-p-ring-map-field", "categories": [ "algebra" ], "title": "algebra-lemma-p-ring-map-field", "contents": [ "Let $k \\subset k'$ be a field extension. Let $p$ be a prime number.", "The following are equivalent", "\\begin{enumerate}", "\\item $k'$ is generated as a field extension of $k$ by elements", "$x$ such that there exists an $n > 0$ with $x^{p^n} \\in k$ and", "$p^nx \\in k$, and", "\\item $k = k'$ or the characteristic of $k$", "and $k'$ is $p$ and $k'/k$ is purely inseparable.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in k'$. If there exists an $n > 0$ with $x^{p^n} \\in k$ and", "$p^nx \\in k$ and if the characteristic is not $p$, then $x \\in k$.", "If the characteristic is $p$, then we find $x^{p^n} \\in k$", "and hence $x$ is purely inseparable over $k$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 582, "type": "theorem", "label": "algebra-lemma-p-ring-map", "categories": [ "algebra" ], "title": "algebra-lemma-p-ring-map", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Let $p$ be a prime number. Assume", "\\begin{enumerate}", "\\item[(a)] $S$ is generated as an $R$-algebra by elements $x$ such", "that there exists an $n > 0$ with $x^{p^n} \\in \\varphi(R)$ and", "$p^nx \\in \\varphi(R)$, and", "\\item[(b)] $\\Ker(\\varphi)$ is locally nilpotent,", "\\end{enumerate}", "Then $\\varphi$ induces a homeomorphism of spectra and induces", "residue field extensions satisfying the equivalent conditions", "of Lemma \\ref{lemma-p-ring-map-field}. For any ring map $R \\to R'$", "the ring map $R' \\to R' \\otimes_R S$ also satisfies (a) and (b)." ], "refs": [ "algebra-lemma-p-ring-map-field" ], "proofs": [ { "contents": [ "Assume (a) and (b). Note that (b) is equivalent to condition (2)", "of Lemma \\ref{lemma-powers}. Let $T \\subset S$ be the set of", "elements $x \\in S$ such that there exists an", "integer $n > 0$ such that $x^{p^n} , p^n x \\in \\varphi(R)$.", "We claim that $T = S$. This will prove that condition (1) of", "Lemma \\ref{lemma-powers} holds and hence $\\varphi$ induces", "a homeomorphism on spectra.", "By assumption (a) it suffices to show that $T \\subset S$ is an $R$-sub algebra.", "If $x \\in T$ and $y \\in R$, then it is clear that $yx \\in T$.", "Suppose $x, y \\in T$ and $n, m > 0$ such that", "$x^{p^n}, y^{p^m}, p^n x, p^m y \\in \\varphi(R)$.", "Then $(xy)^{p^{n + m}}, p^{n + m}xy \\in \\varphi(R)$", "hence $xy \\in T$. We have $x + y \\in T$ by Lemma \\ref{lemma-help-with-powers}", "and the claim is proved.", "\\medskip\\noindent", "Since $\\varphi$ induces a homeomorphism on spectra, it is in particular", "surjective on spectra which is a property preserved under any base change, see", "Lemma \\ref{lemma-surjective-spec-radical-ideal}.", "Therefore for any $R \\to R'$ the kernel of the ring map", "$R' \\to R' \\otimes_R S$ consists of nilpotent elements, see", "Lemma \\ref{lemma-image-dense-generic-points},", "in other words (b) holds for $R' \\to R' \\otimes_R S$.", "It is clear that (a) is preserved under base change.", "Finally, the condition on residue fields follows from (a)", "as generators for $S$ as an $R$-algebra map to generators for", "the residue field extensions." ], "refs": [ "algebra-lemma-powers", "algebra-lemma-powers", "algebra-lemma-help-with-powers", "algebra-lemma-surjective-spec-radical-ideal", "algebra-lemma-image-dense-generic-points" ], "ref_ids": [ 578, 578, 580, 443, 446 ] } ], "ref_ids": [ 581 ] }, { "id": 583, "type": "theorem", "label": "algebra-lemma-radicial", "categories": [ "algebra" ], "title": "algebra-lemma-radicial", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Assume", "\\begin{enumerate}", "\\item $\\varphi$ induces an injective map of spectra,", "\\item $\\varphi$ induces purely inseparable residue field extensions.", "\\end{enumerate}", "Then for any ring map $R \\to R'$ properties (1) and (2) are true for", "$R' \\to R' \\otimes_R S$." ], "refs": [], "proofs": [ { "contents": [ "Set $S' = R' \\otimes_R S$ so that we have a commutative diagram", "of continuous maps of spectra of rings", "$$", "\\xymatrix{", "\\Spec(S') \\ar[r] \\ar[d] & \\Spec(S) \\ar[d] \\\\", "\\Spec(R') \\ar[r] & \\Spec(R)", "}", "$$", "Let $\\mathfrak p' \\subset R'$ be a prime ideal lying over", "$\\mathfrak p \\subset R$. If there is no prime ideal of $S$", "lying over $\\mathfrak p$, then there is no prime ideal of", "$S'$ lying over $\\mathfrak p'$. Otherwise, by", "Remark \\ref{remark-fundamental-diagram} there is a unique", "prime ideal $\\mathfrak r$ of $F = S \\otimes_R \\kappa(\\mathfrak p)$", "whose residue field is purely inseparable over $\\kappa(\\mathfrak p)$.", "Consider the ring maps", "$$", "\\kappa(\\mathfrak p) \\to F \\to \\kappa(\\mathfrak r)", "$$", "By Lemma \\ref{lemma-minimal-prime-reduced-ring} the ideal", "$\\mathfrak r \\subset F$ is locally nilpotent, hence", "we may apply Lemma \\ref{lemma-surjective-locally-nilpotent-kernel}", "to the ring map $F \\to \\kappa(\\mathfrak r)$.", "We may apply Lemma \\ref{lemma-p-ring-map}", "to the ring map $\\kappa(\\mathfrak p) \\to \\kappa(\\mathfrak r)$.", "Hence the composition and the second arrow in the maps", "$$", "\\kappa(\\mathfrak p') \\to", "\\kappa(\\mathfrak p') \\otimes_{\\kappa(\\mathfrak p)} F \\to", "\\kappa(\\mathfrak p') \\otimes_{\\kappa(\\mathfrak p)} \\kappa(\\mathfrak r)", "$$", "induces bijections on spectra and purely inseparable residue", "field extensions. This implies the same thing for the first", "map. Since", "$$", "\\kappa(\\mathfrak p') \\otimes_{\\kappa(\\mathfrak p)} F =", "\\kappa(\\mathfrak p') \\otimes_{\\kappa(\\mathfrak p)}", "\\kappa(\\mathfrak p) \\otimes_R S =", "\\kappa(\\mathfrak p') \\otimes_R S =", "\\kappa(\\mathfrak p') \\otimes_{R'} R' \\otimes_R S", "$$", "we conclude by the discussion in Remark \\ref{remark-fundamental-diagram}." ], "refs": [ "algebra-remark-fundamental-diagram", "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-surjective-locally-nilpotent-kernel", "algebra-lemma-p-ring-map", "algebra-remark-fundamental-diagram" ], "ref_ids": [ 1558, 418, 576, 582, 1558 ] } ], "ref_ids": [] }, { "id": 584, "type": "theorem", "label": "algebra-lemma-radicial-integral", "categories": [ "algebra" ], "title": "algebra-lemma-radicial-integral", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Assume", "\\begin{enumerate}", "\\item $\\varphi$ is integral,", "\\item $\\varphi$ induces an injective map of spectra,", "\\item $\\varphi$ induces purely inseparable residue field extensions.", "\\end{enumerate}", "Then $\\varphi$ induces a homeomorphism from $\\Spec(S)$ onto a closed", "subset of $\\Spec(R)$ and for any ring map", "$R \\to R'$ properties (1), (2), (3) are true for $R' \\to R' \\otimes_R S$." ], "refs": [], "proofs": [ { "contents": [ "The map on spectra is closed by", "Lemmas \\ref{lemma-going-up-closed} and \\ref{lemma-integral-going-up}.", "The properties are preserved under base change by", "Lemmas \\ref{lemma-radicial} and \\ref{lemma-base-change-integral}." ], "refs": [ "algebra-lemma-going-up-closed", "algebra-lemma-integral-going-up", "algebra-lemma-radicial", "algebra-lemma-base-change-integral" ], "ref_ids": [ 552, 500, 583, 491 ] } ], "ref_ids": [] }, { "id": 585, "type": "theorem", "label": "algebra-lemma-radicial-integral-bijective", "categories": [ "algebra" ], "title": "algebra-lemma-radicial-integral-bijective", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Assume", "\\begin{enumerate}", "\\item $\\varphi$ is integral,", "\\item $\\varphi$ induces an bijective map of spectra,", "\\item $\\varphi$ induces purely inseparable residue field extensions.", "\\end{enumerate}", "Then $\\varphi$ induces a homeomorphism on spectra and for any ring map", "$R \\to R'$ properties (1), (2), (3) are true for $R' \\to R' \\otimes_R S$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemmas \\ref{lemma-radicial-integral} and", "\\ref{lemma-surjective-spec-radical-ideal}." ], "refs": [ "algebra-lemma-radicial-integral", "algebra-lemma-surjective-spec-radical-ideal" ], "ref_ids": [ 584, 443 ] } ], "ref_ids": [] }, { "id": 586, "type": "theorem", "label": "algebra-lemma-universally-bijective", "categories": [ "algebra" ], "title": "algebra-lemma-universally-bijective", "contents": [ "Let $\\varphi : R \\to S$ be a ring map such that", "\\begin{enumerate}", "\\item the kernel of $\\varphi$ is locally nilpotent, and", "\\item $S$ is generated as an $R$-algebra by elements $x$", "such that there exist $n > 0$ and a polynomial $P(T) \\in R[T]$", "whose image in $S[T]$ is $(T - x)^n$.", "\\end{enumerate}", "Then $\\Spec(S) \\to \\Spec(R)$ is a homeomorphism and $R \\to S$", "induces purely inseparable extensions of residue fields.", "Moreover, conditions (1) and (2) remain true on arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "We may replace $R$ by $R/\\Ker(\\varphi)$, see", "Lemma \\ref{lemma-surjective-locally-nilpotent-kernel}.", "Assumption (2) implies $S$ is generated over $R$ by", "elements which are integral over $R$.", "Hence $R \\subset S$ is integral", "(Lemma \\ref{lemma-integral-closure-is-ring}).", "In particular $\\Spec(S) \\to \\Spec(R)$ is surjective and closed", "(Lemmas \\ref{lemma-integral-overring-surjective},", "\\ref{lemma-going-up-closed}, and", "\\ref{lemma-integral-going-up}).", "\\medskip\\noindent", "Let $x \\in S$ be one of the generators in (2), i.e., there exists an", "$n > 0$ be such that $(T - x)^n \\in R[T]$.", "Let $\\mathfrak p \\subset R$ be a prime.", "The $\\kappa(\\mathfrak p) \\otimes_R S$ ring is nonzero by", "the above and Lemma \\ref{lemma-in-image}.", "If the characteristic of $\\kappa(\\mathfrak p)$ is zero", "then we see that $nx \\in R$ implies $1 \\otimes x$ is in the image", "of $\\kappa(\\mathfrak p) \\to \\kappa(\\mathfrak p) \\otimes_R S$.", "Hence $\\kappa(\\mathfrak p) \\to \\kappa(\\mathfrak p) \\otimes_R S$", "is an isomorphism.", "If the characteristic of $\\kappa(\\mathfrak p)$ is $p > 0$,", "then write $n = p^k m$ with $m$ prime to $p$.", "In $\\kappa(\\mathfrak p) \\otimes_R S[T]$ we have", "$$", "(T - 1 \\otimes x)^n = ((T - 1 \\otimes x)^{p^k})^m =", "(T^{p^k} - 1 \\otimes x^{p^k})^m", "$$", "and we see that $mx^{p^k} \\in R$. This implies that", "$1 \\otimes x^{p^k}$ is in the image of", "$\\kappa(\\mathfrak p) \\to \\kappa(\\mathfrak p) \\otimes_R S$.", "Hence Lemma \\ref{lemma-p-ring-map} applies to", "$\\kappa(\\mathfrak p) \\to \\kappa(\\mathfrak p) \\otimes_R S$.", "In both cases we conclude that $\\kappa(\\mathfrak p) \\otimes_R S$", "has a unique prime ideal with residue field purely inseparable", "over $\\kappa(\\mathfrak p)$. By Remark \\ref{remark-fundamental-diagram}", "we conclude that $\\varphi$ is bijective on spectra.", "\\medskip\\noindent", "The statement on base change is immediate." ], "refs": [ "algebra-lemma-surjective-locally-nilpotent-kernel", "algebra-lemma-integral-closure-is-ring", "algebra-lemma-integral-overring-surjective", "algebra-lemma-going-up-closed", "algebra-lemma-integral-going-up", "algebra-lemma-in-image", "algebra-lemma-p-ring-map", "algebra-remark-fundamental-diagram" ], "ref_ids": [ 576, 486, 495, 552, 500, 394, 582, 1558 ] } ], "ref_ids": [] }, { "id": 587, "type": "theorem", "label": "algebra-lemma-flat-fibres-irreducible", "categories": [ "algebra" ], "title": "algebra-lemma-flat-fibres-irreducible", "contents": [ "Let $R \\to S$ be a ring map. Assume", "\\begin{enumerate}", "\\item[(a)] $\\Spec(R)$ is irreducible,", "\\item[(b)] $R \\to S$ is flat,", "\\item[(c)] $R \\to S$ is of finite presentation,", "\\item[(d)] the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$", "have irreducible spectra for a dense collection of primes $\\mathfrak p$ of $R$.", "\\end{enumerate}", "Then $\\Spec(S)$ is irreducible.", "This is true more generally with (b) $+$ (c)", "replaced by ``the map $\\Spec(S) \\to \\Spec(R)$ is open''." ], "refs": [], "proofs": [ { "contents": [ "The assumptions (b) and (c) imply that the map on spectra is open,", "see", "Proposition \\ref{proposition-fppf-open}.", "Hence the lemma follows from", "Topology, Lemma \\ref{topology-lemma-irreducible-on-top}." ], "refs": [ "algebra-proposition-fppf-open", "topology-lemma-irreducible-on-top" ], "ref_ids": [ 1407, 8217 ] } ], "ref_ids": [] }, { "id": 588, "type": "theorem", "label": "algebra-lemma-separably-closed-irreducible", "categories": [ "algebra" ], "title": "algebra-lemma-separably-closed-irreducible", "contents": [ "Let $k$ be a separably closed field.", "Let $R$, $S$ be $k$-algebras. If $R$, $S$ have a unique", "minimal prime, so does $R \\otimes_k S$." ], "refs": [], "proofs": [ { "contents": [ "Let $k \\subset \\overline{k}$ be a perfect closure, see", "Definition \\ref{definition-perfection}.", "By assumption $\\overline{k}$ is algebraically closed.", "The ring maps $R \\to R \\otimes_k \\overline{k}$ and", "$S \\to S \\otimes_k \\overline{k}$ and", "$R \\otimes_k S \\to (R \\otimes_k S) \\otimes_k \\overline{k}", "= (R \\otimes_k \\overline{k}) \\otimes_{\\overline{k}} (S \\otimes_k \\overline{k})$", "satisfy the assumptions of Lemma \\ref{lemma-p-ring-map}.", "Hence we may assume $k$ is algebraically closed.", "\\medskip\\noindent", "We may replace $R$ and $S$ by their reductions.", "Hence we may assume that $R$ and $S$ are domains.", "By Lemma \\ref{lemma-perfect-reduced} we see that $R \\otimes_k S$ is", "reduced. Hence its spectrum is reducible if and only if it contains a nonzero", "zerodivisor. By Lemma \\ref{lemma-limit-argument} we reduce to the case where", "$R$ and $S$ are domains of finite type over $k$ algebraically closed.", "\\medskip\\noindent", "Note that the ring map $R \\to R \\otimes_k S$ is of finite", "presentation and flat. Moreover, for every maximal ideal", "$\\mathfrak m$ of $R$ we have", "$(R \\otimes_k S) \\otimes_R R/\\mathfrak m \\cong S$ because", "$k \\cong R/\\mathfrak m$ by the Hilbert Nullstellensatz Theorem", "\\ref{theorem-nullstellensatz}. Moreover, the set of", "maximal ideals is dense in the spectrum of $R$ since", "$\\Spec(R)$ is Jacobson, see Lemma \\ref{lemma-finite-type-field-Jacobson}.", "Hence we see that Lemma \\ref{lemma-flat-fibres-irreducible} applies", "to the ring map $R \\to R \\otimes_k S$ and we conclude that", "the spectrum of $R \\otimes_k S$ is irreducible as desired." ], "refs": [ "algebra-definition-perfection", "algebra-lemma-p-ring-map", "algebra-lemma-perfect-reduced", "algebra-lemma-limit-argument", "algebra-theorem-nullstellensatz", "algebra-lemma-finite-type-field-Jacobson", "algebra-lemma-flat-fibres-irreducible" ], "ref_ids": [ 1463, 582, 575, 563, 316, 467, 587 ] } ], "ref_ids": [] }, { "id": 589, "type": "theorem", "label": "algebra-lemma-geometrically-irreducible", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-irreducible", "contents": [ "Let $k$ be a field.", "Let $R$ be a $k$-algebra.", "The following are equivalent", "\\begin{enumerate}", "\\item for every field extension $k \\subset k'$ the", "spectrum of $R \\otimes_k k'$ is irreducible,", "\\item for every finite separable field extension $k \\subset k'$ the", "spectrum of $R \\otimes_k k'$ is irreducible,", "\\item the spectrum of $R \\otimes_k \\overline{k}$ is irreducible", "where $\\overline{k}$ is the separable algebraic closure of $k$, and", "\\item the spectrum of $R \\otimes_k \\overline{k}$ is irreducible", "where $\\overline{k}$ is the algebraic closure of $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2).", "\\medskip\\noindent", "Assume (2) and let $\\overline{k}$ is the separable algebraic closure of $k$.", "Suppose $\\mathfrak q_i \\subset R \\otimes_k \\overline{k}$, $i = 1, 2$", "are two minimal prime ideals. For every finite subextension", "$\\overline{k}/k'/k$ the extension $k'/k$ is separable and", "the ring map $R \\otimes_k k' \\to R \\otimes_k \\overline{k}$", "is flat. Hence $\\mathfrak p_i = (R \\otimes_k k') \\cap \\mathfrak q_i$", "are minimal prime ideals (as we have going down for flat ring maps", "by Lemma \\ref{lemma-flat-going-down}). Thus we see that", "$\\mathfrak p_1 = \\mathfrak p_2$", "by assumption (2). Since $\\overline{k} = \\bigcup k'$ we conclude", "$\\mathfrak q_1 = \\mathfrak q_2$. Hence $\\Spec(R \\otimes_k \\overline{k})$", "is irreducible.", "\\medskip\\noindent", "Assume (3) and let $\\overline{k}$ be the algebraic closure of $k$.", "Let $\\overline{k}/\\overline{k}'/k$ be the corresponding", "separable algebraic closure of $k$. Then $\\overline{k}/\\overline{k}'$", "is purely inseparable (in positive characteristic) or trivial.", "Hence $R \\otimes_k \\overline{k}' \\to R \\otimes_k \\overline{k}$", "induces a homeomorphism on spectra, for example by", "Lemma \\ref{lemma-p-ring-map}. Thus we have (4).", "\\medskip\\noindent", "Assume (4). Let $k'/k$ be an arbitrary field extension and let", "$\\overline{k}$ be the algebraic closure of $k$. We may choose a", "field $F$ such that both $k'$ and $\\overline{k}$ are isomorphic", "to subfields of $F$. Then", "$$", "R \\otimes_k F = (R \\otimes_k \\overline{k}) \\otimes_{\\overline{k}} F", "$$", "and hence we see from Lemma \\ref{lemma-separably-closed-irreducible}", "that $R \\otimes_k F$ has a unique minimal prime. Finally, the", "ring map $R \\otimes_k k' \\to R \\otimes_k F$ is flat and injective", "and hence any minimal prime of $R \\otimes_k k'$ is the image of", "a minimal prime of $R \\otimes_k F$ (by", "Lemma \\ref{lemma-injective-minimal-primes-in-image}", "and going down). We conclude that there is only one", "such minimal prime and the proof is complete." ], "refs": [ "algebra-lemma-flat-going-down", "algebra-lemma-p-ring-map", "algebra-lemma-separably-closed-irreducible", "algebra-lemma-injective-minimal-primes-in-image" ], "ref_ids": [ 539, 582, 588, 445 ] } ], "ref_ids": [] }, { "id": 590, "type": "theorem", "label": "algebra-lemma-separably-closed-irreducible-implies-geometric", "categories": [ "algebra" ], "title": "algebra-lemma-separably-closed-irreducible-implies-geometric", "contents": [ "Let $k$ be a field.", "Let $R$ be a $k$-algebra.", "If $k$ is separably algebraically closed then $R$ is", "geometrically irreducible over $k$ if and only if the", "spectrum of $R$ is irreducible." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the remark following", "Definition \\ref{definition-geometrically-irreducible}." ], "refs": [ "algebra-definition-geometrically-irreducible" ], "ref_ids": [ 1464 ] } ], "ref_ids": [] }, { "id": 591, "type": "theorem", "label": "algebra-lemma-subalgebra-geometrically-irreducible", "categories": [ "algebra" ], "title": "algebra-lemma-subalgebra-geometrically-irreducible", "contents": [ "Let $k$ be a field. Let $S$ be a $k$-algebra.", "\\begin{enumerate}", "\\item If $S$ is geometrically irreducible over $k$ so is every", "$k$-subalgebra.", "\\item If all finitely generated $k$-subalgebras of $S$ are", "geometrically irreducible, then $S$ is geometrically irreducible.", "\\item A directed colimit of geometrically irreducible $k$-algebras", "is geometrically irreducible.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $S' \\subset S$ be a subalgebra. Then for any extension $k \\subset k'$", "the ring map $S' \\otimes_k k' \\to S \\otimes_k k'$ is injective also.", "Hence (1) follows from Lemma \\ref{lemma-injective-minimal-primes-in-image}", "(and the fact that the image of an irreducible space under a continuous", "map is irreducible). The second and third property follow from the fact", "that tensor product commutes with colimits." ], "refs": [ "algebra-lemma-injective-minimal-primes-in-image" ], "ref_ids": [ 445 ] } ], "ref_ids": [] }, { "id": 592, "type": "theorem", "label": "algebra-lemma-geometrically-irreducible-any-base-change", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-irreducible-any-base-change", "contents": [ "Let $k$ be a field.", "Let $S$ be a geometrically irreducible $k$-algebra.", "Let $R$ be any $k$-algebra.", "The map", "$$", "\\Spec(R \\otimes_k S) \\longrightarrow \\Spec(R)", "$$", "induces a bijection on irreducible components." ], "refs": [], "proofs": [ { "contents": [ "Recall that irreducible components correspond to minimal primes", "(Lemma \\ref{lemma-irreducible}).", "As $R \\to R \\otimes_k S$ is flat we see by going down", "(Lemma \\ref{lemma-flat-going-down}) that", "any minimal prime of $R \\otimes_k S$ lies over a minimal prime of $R$.", "Conversely, if $\\mathfrak p \\subset R$ is a (minimal) prime then", "$$", "R \\otimes_k S/\\mathfrak p(R \\otimes_k S)", "=", "(R/\\mathfrak p) \\otimes_k S", "\\subset", "\\kappa(\\mathfrak p) \\otimes_k S", "$$", "by flatness of $R \\to R \\otimes_k S$. The ring", "$\\kappa(\\mathfrak p) \\otimes_k S$ has irreducible spectrum", "by assumption. It follows that", "$R \\otimes_k S/\\mathfrak p(R \\otimes_k S)$ has a single minimal", "prime (Lemma \\ref{lemma-injective-minimal-primes-in-image}).", "In other words, the inverse image of the irreducible set", "$V(\\mathfrak p)$ is irreducible.", "Hence the lemma follows." ], "refs": [ "algebra-lemma-irreducible", "algebra-lemma-flat-going-down", "algebra-lemma-injective-minimal-primes-in-image" ], "ref_ids": [ 422, 539, 445 ] } ], "ref_ids": [] }, { "id": 593, "type": "theorem", "label": "algebra-lemma-field-extension-geometrically-irreducible", "categories": [ "algebra" ], "title": "algebra-lemma-field-extension-geometrically-irreducible", "contents": [ "Let $K/k$ be a field extension. If $k$ is algebraically closed in $K$, then", "$K$ is geometrically irreducible over $k$." ], "refs": [], "proofs": [ { "contents": [ "Assume $k$ is algebraically closed in $K$. By", "Definition \\ref{definition-geometrically-irreducible}", "and Lemma \\ref{lemma-geometrically-irreducible} it suffices to show", "that the spectrum of $K \\otimes_k k'$ is irreducible for every", "finite separable extension $k'/k$. Say $k'$ is generated by $\\alpha \\in k'$", "over $k$, see", "Fields, Lemma \\ref{fields-lemma-primitive-element}. Let", "$P = T^d + a_1 T^{d - 1} + \\ldots + a_d \\in k[T]$ be the minimal", "polynomial of $\\alpha$. Then $K \\otimes_k k' \\cong K[T]/(P)$.", "The only way the spectrum of $K[T]/(P)$ can be reducible", "is if $P$ is reducible in $K[T]$. Assume $P = P_1 P_2$ is a nontrivial", "factorization in $K[T]$ to get a contradiction.", "By Lemma \\ref{lemma-polynomials-divide} we see that", "the coefficients of $P_1$ and $P_2$ are algebraic over $k$.", "Our assumption implies the coefficients of $P_1$ and $P_2$", "are in $k$ which contradicts the fact that $P$ is irreducible", "over $k$." ], "refs": [ "algebra-definition-geometrically-irreducible", "algebra-lemma-geometrically-irreducible", "fields-lemma-primitive-element", "algebra-lemma-polynomials-divide" ], "ref_ids": [ 1464, 589, 4498, 520 ] } ], "ref_ids": [] }, { "id": 594, "type": "theorem", "label": "algebra-lemma-geometrically-irreducible-transitive", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-irreducible-transitive", "contents": [ "Let $K/k$ be a geometrically irreducible field extension.", "Let $S$ be a geometrically irreducible $K$-algebra.", "Then $S$ is geometrically irreducible over $k$." ], "refs": [], "proofs": [ { "contents": [ "By Definition \\ref{definition-geometrically-irreducible}", "and Lemma \\ref{lemma-geometrically-irreducible} it suffices to show", "that the spectrum of $S \\otimes_k k'$ is irreducible for every", "finite separable extension $k'/k$. Since $K$ is geometrically irreducible", "over $k$ we see that $K' = K \\otimes_k k'$", "is a finite, separable field extension of $K$.", "Hence the spectrum of $S \\otimes_k k' = S \\otimes_K K'$ is", "irreducible as $S$ is assumed geometrically irreducible over $K$." ], "refs": [ "algebra-definition-geometrically-irreducible", "algebra-lemma-geometrically-irreducible" ], "ref_ids": [ 1464, 589 ] } ], "ref_ids": [] }, { "id": 595, "type": "theorem", "label": "algebra-lemma-geometrically-irreducible-base-change-transcendental", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-irreducible-base-change-transcendental", "contents": [ "Let $K/k$ be a field extension. The following are equivalent", "\\begin{enumerate}", "\\item $K$ is geometrically irreducible over $k$, and", "\\item the induced extension $K(t)/k(t)$ of purely transcendental extensions", "is geometrically irreducible.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Denote $\\Omega$ an algebraic closure of $k(t)$.", "By Definition \\ref{definition-geometrically-irreducible}", "we find that the spectrum of", "$$", "K \\otimes_k \\Omega = K \\otimes_k k(t) \\otimes_{k(t)} \\Omega", "$$", "is irreducible. Since $K(t)$ is a localization of $K \\otimes_k k(T)$", "we conclude that the spectrum of $K(t) \\otimes_{k(t)} \\Omega$", "is irreducible. Thus by Lemma \\ref{lemma-geometrically-irreducible}", "we find that $K(t)/k(t)$ is geometrically irreducible.", "\\medskip\\noindent", "Assume (2). Let $k'/k$ be a field extension.", "We have to show that $K \\otimes_k k'$ has a unique minimal prime.", "We know that the spectrum of", "$$", "K(t) \\otimes_{k(t)} k'(t)", "$$", "is irreducible, i.e., has a unique minimal prime.", "Since there is an injective map", "$K \\otimes_k k' \\to K(t) \\otimes_{k(t)} k'(t)$ (details omitted)", "we conclude by", "Lemmas \\ref{lemma-injective-minimal-primes-in-image} and", "\\ref{lemma-minimal-prime-image-minimal-prime}." ], "refs": [ "algebra-definition-geometrically-irreducible", "algebra-lemma-geometrically-irreducible", "algebra-lemma-injective-minimal-primes-in-image", "algebra-lemma-minimal-prime-image-minimal-prime" ], "ref_ids": [ 1464, 589, 445, 447 ] } ], "ref_ids": [] }, { "id": 596, "type": "theorem", "label": "algebra-lemma-geometrically-irreducible-add-transcendental", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-irreducible-add-transcendental", "contents": [ "Let $K/L/M$ be a tower of fields with $L/M$ geometrically irreducible.", "Let $x \\in K$ be transcendental over $L$. Then $L(x)/M(x)$ is geometrically", "irreducible." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-geometrically-irreducible-base-change-transcendental}", "because the fields $L(x)$ and $M(x)$ are purely transcendental", "extensions of $L$ and $M$." ], "refs": [ "algebra-lemma-geometrically-irreducible-base-change-transcendental" ], "ref_ids": [ 595 ] } ], "ref_ids": [] }, { "id": 597, "type": "theorem", "label": "algebra-lemma-geometrically-irreducible-separable-elements", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-irreducible-separable-elements", "contents": [ "Let $K/k$ be a field extension. The following are equivalent", "\\begin{enumerate}", "\\item $K/k$ is geometrically irreducible, and", "\\item every element $\\alpha \\in K$ separably algebraic over $k$ is", "in $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and let $\\alpha \\in K$ be separably algebraic over $k$.", "Then $k' = k(\\alpha)$ is a finite separable extension of $k$ contained", "in $k$. By Lemma \\ref{lemma-subalgebra-geometrically-irreducible}", "the extension $k'/k$ is geometrically irreducible.", "In particular, we see that the spectrum of $k' \\otimes_k \\overline{k}$", "is irreducible (and hence if it is a product of fields, then there", "is exactly one factor).", "By Fields, Lemma \\ref{fields-lemma-finite-separable-tensor-alg-closed}", "it follows that $\\Hom_k(k', \\overline{k})$ has one element which in turn", "implies that $k' = k$ by", "Fields, Lemma \\ref{fields-lemma-separable-equality}.", "Thus (2) holds.", "\\medskip\\noindent", "Assume (2). Let $k' \\subset K$ be the subfield consisting of elements", "algebraic over $k$. By", "Lemma \\ref{lemma-field-extension-geometrically-irreducible}", "the extension $K/k'$ is geometrically irreducible.", "By assumption $k'/k$ is a purely inseparable extension.", "By Lemma \\ref{lemma-p-ring-map} the extension", "$k'/k$ is geometrically irreducible. Hence by", "Lemma \\ref{lemma-geometrically-irreducible-transitive}", "we see that $K/k$ is geometrically irreducible." ], "refs": [ "algebra-lemma-subalgebra-geometrically-irreducible", "fields-lemma-finite-separable-tensor-alg-closed", "fields-lemma-separable-equality", "algebra-lemma-field-extension-geometrically-irreducible", "algebra-lemma-p-ring-map", "algebra-lemma-geometrically-irreducible-transitive" ], "ref_ids": [ 591, 4477, 4471, 593, 582, 594 ] } ], "ref_ids": [] }, { "id": 598, "type": "theorem", "label": "algebra-lemma-make-geometrically-irreducible", "categories": [ "algebra" ], "title": "algebra-lemma-make-geometrically-irreducible", "contents": [ "Let $K/k$ be a field extension. Consider the subextension $K/k'/k$ consisting", "of elements separably algebraic over $k$. Then $K$ is geometrically irreducible", "over $k'$. If $K/k$ is a finitely generated field extension, then", "$[k' : k] < \\infty$." ], "refs": [], "proofs": [ { "contents": [ "The first statement is immediate from", "Lemma \\ref{lemma-geometrically-irreducible-separable-elements}", "and the fact that elements separably algebraic over $k'$", "are in $k'$ by the transitivity of separable algebraic", "extensions, see Fields, Lemma \\ref{fields-lemma-separable-permanence}.", "If $k \\subset K$ is finitely generated, then $k'$ is finite over $k$ by", "Fields, Lemma \\ref{fields-lemma-algebraic-closure-in-finitely-generated}." ], "refs": [ "algebra-lemma-geometrically-irreducible-separable-elements", "fields-lemma-separable-permanence", "fields-lemma-algebraic-closure-in-finitely-generated" ], "ref_ids": [ 597, 4472, 4521 ] } ], "ref_ids": [] }, { "id": 599, "type": "theorem", "label": "algebra-lemma-Galois-orbit", "categories": [ "algebra" ], "title": "algebra-lemma-Galois-orbit", "contents": [ "Let $k \\subset K$ be an extension of fields.", "Let $k \\subset \\overline{k}$ be a separable algebraic closure.", "Then $\\text{Gal}(\\overline{k}/k)$ acts transitively on the", "primes of $\\overline{k} \\otimes_k K$." ], "refs": [], "proofs": [ { "contents": [ "Let $k \\subset k' \\subset K$ be the subextension found in", "Lemma \\ref{lemma-make-geometrically-irreducible}.", "Note that as $k \\subset \\overline{k}$ is integral all the prime ideals", "of $\\overline{k} \\otimes_k K$ and $\\overline{k} \\otimes_k k'$ are maximal, see", "Lemma \\ref{lemma-integral-no-inclusion}.", "By Lemma \\ref{lemma-geometrically-irreducible-any-base-change}", "the map", "$$", "\\Spec(\\overline{k} \\otimes_k K) \\to \\Spec(\\overline{k} \\otimes_k k')", "$$", "is bijective because (1) all primes are minimal primes, (2)", "$\\overline{k} \\otimes_k K = (\\overline{k} \\otimes_k k') \\otimes_{k'} K$,", "and (3) $K$ is geometrically irreducible over $k'$.", "Hence it suffices to prove the lemma for the action of", "$\\text{Gal}(\\overline{k}/k)$ on the primes of $\\overline{k} \\otimes_k k'$.", "\\medskip\\noindent", "As every prime of $\\overline{k} \\otimes_k k'$ is maximal, the residue fields", "are isomorphic to $\\overline{k}$. Hence the prime ideals of", "$\\overline{k} \\otimes_k k'$ correspond one to one to elements of", "$\\Hom_k(k', \\overline{k})$ with $\\sigma \\in \\Hom_k(k', \\overline{k})$", "corresponding to the kernel $\\mathfrak p_\\sigma$ of", "$1 \\otimes \\sigma : \\overline{k} \\otimes_k k' \\to \\overline{k}$.", "In particular $\\text{Gal}(\\overline{k}/k)$ acts transitively on", "this set as desired." ], "refs": [ "algebra-lemma-make-geometrically-irreducible", "algebra-lemma-integral-no-inclusion", "algebra-lemma-geometrically-irreducible-any-base-change" ], "ref_ids": [ 598, 498, 592 ] } ], "ref_ids": [] }, { "id": 600, "type": "theorem", "label": "algebra-lemma-separably-closed-connected", "categories": [ "algebra" ], "title": "algebra-lemma-separably-closed-connected", "contents": [ "Let $k$ be a separably algebraically closed field.", "Let $R$, $S$ be $k$-algebras. If $\\Spec(R)$, and", "$\\Spec(S)$ are connected, then so is", "$\\Spec(R \\otimes_k S)$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\Spec(R)$ is connected if and only if", "$R$ has no nontrivial idempotents, see", "Lemma \\ref{lemma-characterize-spec-connected}.", "Hence, by Lemma \\ref{lemma-limit-argument} we may assume $R$ and $S$ are of", "finite type over $k$.", "In this case $R$ and $S$ are Noetherian,", "and have finitely many minimal primes, see", "Lemma \\ref{lemma-Noetherian-irreducible-components}.", "Thus we may argue by induction on $n + m$ where $n$, resp.\\ $m$", "is the number of irreducible components of $\\Spec(R)$,", "resp.\\ $\\Spec(S)$. Of course the case where either $n$ or", "$m$ is zero is trivial. If $n = m = 1$, i.e.,", "$\\Spec(R)$ and $\\Spec(S)$ both have one irreducible component,", "then the result holds by Lemma \\ref{lemma-separably-closed-irreducible}.", "Suppose that $n > 1$. Let $\\mathfrak p \\subset R$ be a minimal prime", "corresponding to the irreducible closed subset $T \\subset \\Spec(R)$.", "Let $I \\subset R$ be such that $T' = V(I) \\subset \\Spec(R)$", "is the closure of the complement of $T$. Note that this means", "that $T' = \\Spec(R/I)$ (Lemma \\ref{lemma-spec-closed}) has $n - 1$", "irreducible components. Then $T \\cup T' = \\Spec(R)$,", "and $T \\cap T' = V(\\mathfrak p + I) = \\Spec(R/(\\mathfrak p + I))$", "is not empty as $\\Spec(R)$ is assumed connected.", "The inverse image of $T$ in $\\Spec(R \\otimes_k S)$", "is $\\Spec(R/\\mathfrak p \\otimes_k S)$, and the inverse", "of $T'$ in $\\Spec(R \\otimes_k S)$", "is $\\Spec(R/I \\otimes_k S)$. By induction these are both", "connected. The inverse image of $T \\cap T'$ is", "$\\Spec(R/(\\mathfrak p + I) \\otimes_k S)$ which is nonempty.", "Hence $\\Spec(R \\otimes_k S)$ is connected." ], "refs": [ "algebra-lemma-characterize-spec-connected", "algebra-lemma-limit-argument", "algebra-lemma-Noetherian-irreducible-components", "algebra-lemma-separably-closed-irreducible", "algebra-lemma-spec-closed" ], "ref_ids": [ 406, 563, 453, 588, 393 ] } ], "ref_ids": [] }, { "id": 601, "type": "theorem", "label": "algebra-lemma-geometrically-connected", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-connected", "contents": [ "Let $k$ be a field.", "Let $R$ be a $k$-algebra.", "The following are equivalent", "\\begin{enumerate}", "\\item for every field extension $k \\subset k'$ the", "spectrum of $R \\otimes_k k'$ is connected, and", "\\item for every finite separable field extension $k \\subset k'$ the", "spectrum of $R \\otimes_k k'$ is connected.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For any extension of fields $k \\subset k'$ the connectivity", "of the spectrum of $R \\otimes_k k'$ is equivalent to $R \\otimes_k k'$", "having no nontrivial idempotents, see", "Lemma \\ref{lemma-characterize-spec-connected}. Assume (2).", "Let $k \\subset \\overline{k}$ be a separable algebraic closure of $k$.", "Using Lemma \\ref{lemma-limit-argument}", "we see that (2) is equivalent to $R \\otimes_k \\overline{k}$", "having no nontrivial idempotents.", "For any field extension $k \\subset k'$, there exists a field", "extension $\\overline{k} \\subset \\overline{k}'$ with", "$k' \\subset \\overline{k}'$. By Lemma \\ref{lemma-separably-closed-connected}", "we see that $R \\otimes_k \\overline{k}'$ has no nontrivial idempotents.", "If $R \\otimes_k k'$ has a nontrivial idempotent,", "then also $R \\otimes_k \\overline{k}'$, contradiction." ], "refs": [ "algebra-lemma-characterize-spec-connected", "algebra-lemma-limit-argument", "algebra-lemma-separably-closed-connected" ], "ref_ids": [ 406, 563, 600 ] } ], "ref_ids": [] }, { "id": 602, "type": "theorem", "label": "algebra-lemma-separably-closed-connected-implies-geometric", "categories": [ "algebra" ], "title": "algebra-lemma-separably-closed-connected-implies-geometric", "contents": [ "Let $k$ be a field.", "Let $R$ be a $k$-algebra.", "If $k$ is separably algebraically closed then $R$ is", "geometrically connected over $k$ if and only if the", "spectrum of $R$ is connected." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the remark following", "Definition \\ref{definition-geometrically-connected}." ], "refs": [ "algebra-definition-geometrically-connected" ], "ref_ids": [ 1465 ] } ], "ref_ids": [] }, { "id": 603, "type": "theorem", "label": "algebra-lemma-subalgebra-geometrically-connected", "categories": [ "algebra" ], "title": "algebra-lemma-subalgebra-geometrically-connected", "contents": [ "Let $k$ be a field. Let $S$ be a $k$-algebra.", "\\begin{enumerate}", "\\item If $S$ is geometrically connected over $k$ so is every", "$k$-subalgebra.", "\\item If all finitely generated $k$-subalgebras of $S$ are", "geometrically connected, then $S$ is geometrically connected.", "\\item A directed colimit of geometrically connected $k$-algebras", "is geometrically connected.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from the characterization of connectedness in terms of the", "nonexistence of nontrivial idempotents. The second and third property follow", "from the fact that tensor product commutes with colimits." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 604, "type": "theorem", "label": "algebra-lemma-geometrically-connected-any-base-change", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-connected-any-base-change", "contents": [ "Let $k$ be a field.", "Let $S$ be a geometrically connected $k$-algebra.", "Let $R$ be any $k$-algebra.", "The map", "$$", "R \\longrightarrow R \\otimes_k S", "$$", "induces a bijection on idempotents, and the map", "$$", "\\Spec(R \\otimes_k S) \\longrightarrow \\Spec(R)", "$$", "induces a bijection on connected components." ], "refs": [], "proofs": [ { "contents": [ "The second assertion follows from the first combined with", "Lemma \\ref{lemma-connected-component}.", "By Lemmas \\ref{lemma-subalgebra-geometrically-connected}", "and \\ref{lemma-limit-argument} we may assume that $R$ and $S$", "are of finite type over $k$. Then we see that also", "$R \\otimes_k S$ is of finite type over $k$. Note that in this", "case all the rings are Noetherian and hence their spectra", "have finitely many connected components (since they have", "finitely many irreducible components, see", "Lemma \\ref{lemma-Noetherian-irreducible-components}).", "In particular, all connected components in question are open!", "Hence via Lemma \\ref{lemma-disjoint-implies-product}", "we see that the first statement of the", "lemma in this case is equivalent to the second. Let's prove this.", "As the algebra $S$ is geometrically connected", "and nonzero we see that all fibres of $X = \\Spec(R \\otimes_k S)", "\\to \\Spec(R) = Y$ are connected and nonempty. Also, as", "$R \\to R \\otimes_k S$ is flat of finite presentation the map", "$X \\to Y$ is open", "(Proposition \\ref{proposition-fppf-open}).", "Topology, Lemma \\ref{topology-lemma-connected-fibres-connected-components}", "shows that $X \\to Y$ induces bijection on connected components." ], "refs": [ "algebra-lemma-connected-component", "algebra-lemma-subalgebra-geometrically-connected", "algebra-lemma-limit-argument", "algebra-lemma-Noetherian-irreducible-components", "algebra-lemma-disjoint-implies-product", "algebra-proposition-fppf-open", "topology-lemma-connected-fibres-connected-components" ], "ref_ids": [ 409, 603, 563, 453, 415, 1407, 8208 ] } ], "ref_ids": [] }, { "id": 605, "type": "theorem", "label": "algebra-lemma-geometrically-integral", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-integral", "contents": [ "Let $k$ be a field.", "Let $S$ be a $k$-algebra.", "In this case $S$ is geometrically integral over $k$ if and only if", "$S$ is geometrically irreducible as well as geometrically reduced over $k$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 606, "type": "theorem", "label": "algebra-lemma-characterize-geometrically-integral", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-geometrically-integral", "contents": [ "Let $k$ be a field. Let $S$ be a $k$-algebra.", "The following are equivalent", "\\begin{enumerate}", "\\item $S$ is geometrically integral over $k$,", "\\item for every finite extension $k'/k$ of fields", "the ring $S \\otimes_k k'$ is a domain,", "\\item $S \\otimes_k \\overline{k}$ is a domain", "where $\\overline{k}$ is the algebraic closure of $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemmas \\ref{lemma-geometrically-integral},", "\\ref{lemma-geometrically-reduced-finite-purely-inseparable-extension}, and", "\\ref{lemma-geometrically-irreducible}." ], "refs": [ "algebra-lemma-geometrically-integral", "algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension", "algebra-lemma-geometrically-irreducible" ], "ref_ids": [ 605, 571, 589 ] } ], "ref_ids": [] }, { "id": 607, "type": "theorem", "label": "algebra-lemma-geometrically-integral-any-integral-base-change", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-integral-any-integral-base-change", "contents": [ "Let $k$ be a field. Let $S$ be a geometrically integral $k$-algebra.", "Let $R$ be a $k$-algebra and an integral domain. Then $R \\otimes_k S$", "is an integral domain." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-geometrically-reduced-any-reduced-base-change}", "the ring $R \\otimes_k S$ is reduced and by", "Lemma \\ref{lemma-geometrically-irreducible-any-base-change}", "the ring $R \\otimes_k S$ is irreducible (the spectrum", "has just one irreducible component), so $R \\otimes_k S$ is", "an integral domain." ], "refs": [ "algebra-lemma-geometrically-reduced-any-reduced-base-change", "algebra-lemma-geometrically-irreducible-any-base-change" ], "ref_ids": [ 564, 592 ] } ], "ref_ids": [] }, { "id": 608, "type": "theorem", "label": "algebra-lemma-dominate", "categories": [ "algebra" ], "title": "algebra-lemma-dominate", "contents": [ "Let $K$ be a field. Let $A \\subset K$ be a local subring.", "Then there exists a valuation ring with fraction field $K$", "dominating $A$." ], "refs": [], "proofs": [ { "contents": [ "We consider the collection of local subrings", "of $K$ as a partially ordered set using the relation of domination.", "Suppose that $\\{A_i\\}_{i \\in I}$ is a totally ordered", "collection of local subrings of $K$. Then $B = \\bigcup A_i$", "is a local subring which dominates all of the $A_i$.", "Hence by Zorn's Lemma, it suffices to show that if $A \\subset K$", "is a local ring whose fraction field is not $K$, then there", "exists a local ring $B \\subset K$, $B \\not = A$ dominating $A$.", "\\medskip\\noindent", "Pick $t \\in K$ which is not in the fraction field of $A$.", "If $t$ is transcendental over $A$, then $A[t] \\subset K$", "and hence $A[t]_{(t, \\mathfrak m)} \\subset K$ is a local ring", "distinct from $A$ dominating $A$. Suppose $t$ is algebraic over $A$.", "Then for some $a \\in A$ the element $at$ is integral over $A$.", "In this case the subring $A' \\subset K$ generated by $A$ and", "$ta$ is finite over $A$.", "By Lemma \\ref{lemma-integral-overring-surjective} there exists", "a prime ideal $\\mathfrak m' \\subset A'$ lying over", "$\\mathfrak m$. Then $A'_{\\mathfrak m'}$ dominates", "$A$. If $A = A'_{\\mathfrak m'}$, then $t$", "is in the fraction field of $A$ which we assumed not to be the case.", "Thus $A \\not = A'_{\\mathfrak m'}$ as desired." ], "refs": [ "algebra-lemma-integral-overring-surjective" ], "ref_ids": [ 495 ] } ], "ref_ids": [] }, { "id": 609, "type": "theorem", "label": "algebra-lemma-valuation-ring-x-or-x-inverse", "categories": [ "algebra" ], "title": "algebra-lemma-valuation-ring-x-or-x-inverse", "contents": [ "Let $A$ be a valuation ring with maximal ideal $\\mathfrak m$ and", "fraction field $K$.", "Let $x \\in K$. Then either $x \\in A$ or $x^{-1} \\in A$ or both." ], "refs": [], "proofs": [ { "contents": [ "Assume that $x$ is not in $A$.", "Let $A'$ denote the subring of $K$ generated by $A$ and $x$.", "Since $A$ is a valuation ring we see that there is no prime", "of $A'$ lying over $\\mathfrak m$. Since $\\mathfrak m$ is maximal", "we see that $V(\\mathfrak m A') = \\emptyset$. Then $\\mathfrak m A' = A'$", "by Lemma \\ref{lemma-Zariski-topology}.", "Hence we can write", "$1 = \\sum_{i = 0}^d t_i x^i$ with $t_i \\in \\mathfrak m$.", "This implies that $(1 - t_0) (x^{-1})^d - \\sum t_i (x^{-1})^{d - i} = 0$.", "In particular we see that $x^{-1}$ is integral over $A$.", "Thus the subring $A''$ of $K$ generated by $A$ and $x^{-1}$ is", "finite over $A$ and we see there exists a prime ideal", "$\\mathfrak m'' \\subset A''$ lying over $\\mathfrak m$ by", "Lemma \\ref{lemma-integral-overring-surjective}. Since $A$", "is a valuation ring we conclude that $A = (A'')_{\\mathfrak m''}$", "and hence $x^{-1} \\in A$." ], "refs": [ "algebra-lemma-Zariski-topology", "algebra-lemma-integral-overring-surjective" ], "ref_ids": [ 389, 495 ] } ], "ref_ids": [] }, { "id": 610, "type": "theorem", "label": "algebra-lemma-x-or-x-inverse-valuation-ring", "categories": [ "algebra" ], "title": "algebra-lemma-x-or-x-inverse-valuation-ring", "contents": [ "Let $A \\subset K$ be a subring of a field $K$ such that for all", "$x \\in K$ either $x \\in A$ or $x^{-1} \\in A$ or both.", "Then $A$ is a valuation ring with fraction field $K$." ], "refs": [], "proofs": [ { "contents": [ "If $A$ is not $K$, then $A$ is not a field and there is a nonzero", "maximal ideal $\\mathfrak m$.", "If $\\mathfrak m'$ is a second maximal ideal, then choose $x, y \\in A$", "with $x \\in \\mathfrak m$, $y \\not \\in \\mathfrak m$,", "$x \\not \\in \\mathfrak m'$, and $y \\in \\mathfrak m'$ (see", "Lemma \\ref{lemma-silly}). Then neither $x/y \\in A$ nor $y/x \\in A$", "contradicting the assumption of the lemma. Thus we see that $A$ is", "a local ring. Suppose that $A'$ is a local ring contained in $K$ which", "dominates $A$. Let $x \\in A'$. We have to show that $x \\in A$. If not, then", "$x^{-1} \\in A$, and of course $x^{-1} \\in \\mathfrak m_A$. But then", "$x^{-1} \\in \\mathfrak m_{A'}$ which contradicts $x \\in A'$." ], "refs": [ "algebra-lemma-silly" ], "ref_ids": [ 378 ] } ], "ref_ids": [] }, { "id": 611, "type": "theorem", "label": "algebra-lemma-colimit-valuation-rings", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-valuation-rings", "contents": [ "\\begin{slogan}", "Valuation rings are stable under filtered direct limits", "\\end{slogan}", "Let $I$ be a directed set. Let $(A_i, \\varphi_{ij})$", "be a system of valuation rings over $I$.", "Then $A = \\colim A_i$ is a valuation ring." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $A$ is a domain. Let $a, b \\in A$.", "Lemma \\ref{lemma-x-or-x-inverse-valuation-ring} tells us we have", "to show that either $a | b$ or $b | a$ in $A$. Choose $i$", "so large that there exist $a_i, b_i \\in A_i$ mapping to $a, b$.", "Then Lemma \\ref{lemma-valuation-ring-x-or-x-inverse}", "applied to $a_i, b_i$ in $A_i$ implies the result for $a, b$ in $A$." ], "refs": [ "algebra-lemma-x-or-x-inverse-valuation-ring", "algebra-lemma-valuation-ring-x-or-x-inverse" ], "ref_ids": [ 610, 609 ] } ], "ref_ids": [] }, { "id": 612, "type": "theorem", "label": "algebra-lemma-valuation-ring-cap-field", "categories": [ "algebra" ], "title": "algebra-lemma-valuation-ring-cap-field", "contents": [ "Let $K \\subset L$ be an extension of fields. If $B \\subset L$", "is a valuation ring, then $A = K \\cap B$ is a valuation ring." ], "refs": [], "proofs": [ { "contents": [ "We can replace $L$ by the fraction field $F$ of $B$ and $K$ by", "$K \\cap F$. Then the lemma follows from a combination of", "Lemmas \\ref{lemma-valuation-ring-x-or-x-inverse} and", "\\ref{lemma-x-or-x-inverse-valuation-ring}." ], "refs": [ "algebra-lemma-valuation-ring-x-or-x-inverse", "algebra-lemma-x-or-x-inverse-valuation-ring" ], "ref_ids": [ 609, 610 ] } ], "ref_ids": [] }, { "id": 613, "type": "theorem", "label": "algebra-lemma-valuation-ring-cap-field-finite", "categories": [ "algebra" ], "title": "algebra-lemma-valuation-ring-cap-field-finite", "contents": [ "Let $K \\subset L$ be an algebraic extension of fields. If $B \\subset L$", "is a valuation ring with fraction field $L$ and not a field, then", "$A = K \\cap B$ is a valuation ring and not a field." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-valuation-ring-cap-field} the ring $A$ is a valuation", "ring. If $A$ is a field, then $A = K$. Then $A = K \\subset B$ is an integral", "extension, hence there are no proper inclusions among the primes of $B$", "(Lemma \\ref{lemma-integral-no-inclusion}).", "This contradicts the assumption that $B$ is a local domain and not a field." ], "refs": [ "algebra-lemma-valuation-ring-cap-field", "algebra-lemma-integral-no-inclusion" ], "ref_ids": [ 612, 498 ] } ], "ref_ids": [] }, { "id": 614, "type": "theorem", "label": "algebra-lemma-make-valuation-rings", "categories": [ "algebra" ], "title": "algebra-lemma-make-valuation-rings", "contents": [ "Let $A$ be a valuation ring. For any prime ideal $\\mathfrak p \\subset A$ the", "quotient $A/\\mathfrak p$ is a valuation ring. The same is true for the", "localization $A_\\mathfrak p$ and in fact any localization of $A$." ], "refs": [], "proofs": [ { "contents": [ "Use the characterization of valuation rings given", "in Lemma \\ref{lemma-x-or-x-inverse-valuation-ring}." ], "refs": [ "algebra-lemma-x-or-x-inverse-valuation-ring" ], "ref_ids": [ 610 ] } ], "ref_ids": [] }, { "id": 615, "type": "theorem", "label": "algebra-lemma-stack-valuation-rings", "categories": [ "algebra" ], "title": "algebra-lemma-stack-valuation-rings", "contents": [ "Let $A'$ be a valuation ring with residue field $K$.", "Let $A$ be a valuation ring with fraction field $K$.", "Then", "$C = \\{\\lambda \\in A' \\mid \\lambda \\bmod \\mathfrak m_{A'} \\in A\\}$", "is a valuation ring." ], "refs": [], "proofs": [ { "contents": [ "Note that $\\mathfrak m_{A'} \\subset C$ and $C/\\mathfrak m_{A'} = A$.", "In particular, the fraction field of $C$ is equal to the fraction field", "of $A'$. We will use the criterion of", "Lemma \\ref{lemma-x-or-x-inverse-valuation-ring} to prove the lemma.", "Let $x$ be an element of the fraction field of $C$.", "By the lemma we may assume $x \\in A'$. If $x \\in \\mathfrak m_{A'}$,", "then we see $x \\in C$. If not, then $x$ is a unit of $A'$ and we", "also have $x^{-1} \\in A'$. Hence either $x$ or $x^{-1}$ maps to", "an element of $A$ by the lemma again." ], "refs": [ "algebra-lemma-x-or-x-inverse-valuation-ring" ], "ref_ids": [ 610 ] } ], "ref_ids": [] }, { "id": 616, "type": "theorem", "label": "algebra-lemma-valuation-ring-normal", "categories": [ "algebra" ], "title": "algebra-lemma-valuation-ring-normal", "contents": [ "Let $A$ be a valuation ring.", "Then $A$ is a normal domain." ], "refs": [], "proofs": [ { "contents": [ "Suppose $x$ is in the field of fractions of $A$ and integral over $A$,", "say $x^{d + 1} + \\sum_{i \\leq d} a_i x^i = 0$. By", "Lemma \\ref{lemma-x-or-x-inverse-valuation-ring}", "either $x \\in A$ (and we're done) or $x^{-1} \\in A$. In the second case", "we see that $x = - \\sum a_i x^{i - d} \\in A$ as well." ], "refs": [ "algebra-lemma-x-or-x-inverse-valuation-ring" ], "ref_ids": [ 610 ] } ], "ref_ids": [] }, { "id": 617, "type": "theorem", "label": "algebra-lemma-find-valuation-rings", "categories": [ "algebra" ], "title": "algebra-lemma-find-valuation-rings", "contents": [ "Let $A$ be a normal domain with fraction field $K$.", "\\begin{enumerate}", "\\item For every $x \\in K$, $x \\not \\in A$ there exists a valuation ring", "$A \\subset V \\subset K$ with fraction field $K$ such that $x \\not \\in V$.", "\\item If $A$ is local, we can moreover choose $V$ which dominates $A$.", "\\end{enumerate}", "In other words, $A$ is the intersection of all valuation rings in $K$", "containing $A$ and if $A$ is local, then $A$ is the intersection of", "all valuation rings in $K$ dominating $A$." ], "refs": [], "proofs": [ { "contents": [ "Suppose $x \\in K$, $x \\not \\in A$. Consider $B = A[x^{-1}]$.", "Then $x \\not \\in B$. Namely, if $x = a_0 + a_1x^{-1} + \\ldots + a_d x^{-d}$", "then $x^{d + 1} - a_0x^d - \\ldots - a_d = 0$ and $x$ is integral", "over $A$ in contradiction with the fact that $A$ is normal.", "Thus $x^{-1}$ is not a unit in $B$. Thus $V(x^{-1}) \\subset \\Spec(B)$", "is not empty (Lemma \\ref{lemma-Zariski-topology}), and we can choose a prime", "$\\mathfrak p \\subset B$ with $x^{-1} \\in \\mathfrak p$.", "Choose a valuation ring $V \\subset K$ dominating $B_\\mathfrak p$", "(Lemma \\ref{lemma-dominate}).", "Then $x \\not \\in V$ as $x^{-1} \\in \\mathfrak m_V$.", "\\medskip\\noindent", "If $A$ is local, then we claim that $x^{-1} B + \\mathfrak m_A B \\not = B$.", "Namely, if $1 = (a_0 + a_1x^{-1} + \\ldots + a_d x^{-d})x^{-1} +", "a'_0 + \\ldots + a'_d x^{-d}$ with $a_i \\in A$ and $a'_i \\in \\mathfrak m_A$,", "then we'd get", "$$", "(1 - a'_0) x^{d + 1} - (a_0 + a'_1) x^d - \\ldots - a_d = 0", "$$", "Since $a'_0 \\in \\mathfrak m_A$ we see that $1 - a'_0$ is a unit in $A$", "and we conclude that $x$ would be integral over $A$, a contradiction as", "before. Then choose the prime $\\mathfrak p \\supset x^{-1} B + \\mathfrak m_A B$", "we find $V$ dominating $A$." ], "refs": [ "algebra-lemma-Zariski-topology", "algebra-lemma-dominate" ], "ref_ids": [ 389, 608 ] } ], "ref_ids": [] }, { "id": 618, "type": "theorem", "label": "algebra-lemma-valuation-group", "categories": [ "algebra" ], "title": "algebra-lemma-valuation-group", "contents": [ "Let $A$ be a valuation ring with field of fractions $K$.", "Set $\\Gamma = K^*/A^*$ (with group law written additively).", "For $\\gamma, \\gamma' \\in \\Gamma$", "define $\\gamma \\geq \\gamma'$ if and only if", "$\\gamma - \\gamma'$ is in the image of $A - \\{0\\} \\to \\Gamma$.", "Then $(\\Gamma, \\geq)$ is a totally ordered abelian group." ], "refs": [], "proofs": [ { "contents": [ "Omitted, but follows easily from", "Lemma \\ref{lemma-valuation-ring-x-or-x-inverse}.", "Note that in case $A = K$ we obtain the zero group $\\Gamma = \\{0\\}$", "endowed with its unique total ordering." ], "refs": [ "algebra-lemma-valuation-ring-x-or-x-inverse" ], "ref_ids": [ 609 ] } ], "ref_ids": [] }, { "id": 619, "type": "theorem", "label": "algebra-lemma-properties-valuation", "categories": [ "algebra" ], "title": "algebra-lemma-properties-valuation", "contents": [ "Let $A$ be a valuation ring. The valuation $v : A -\\{0\\} \\to \\Gamma_{\\geq 0}$", "has the following properties:", "\\begin{enumerate}", "\\item $v(a) = 0 \\Leftrightarrow a \\in A^*$,", "\\item $v(ab) = v(a) + v(b)$,", "\\item $v(a + b) \\geq \\min(v(a), v(b))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 620, "type": "theorem", "label": "algebra-lemma-characterize-valuation-ring", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-valuation-ring", "contents": [ "Let $A$ be a ring. The following are equivalent", "\\begin{enumerate}", "\\item $A$ is a valuation ring,", "\\item $A$ is a local domain and every finitely generated", "ideal of $A$ is principal.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $A$ is a valuation ring and let $f_1, \\ldots, f_n \\in A$.", "Choose $i$ such that $v(f_i)$ is minimal among $v(f_j)$.", "Then $(f_i) = (f_1, \\ldots, f_n)$. Conversely, assume $A$ is", "a local domain and every finitely generated ideal of $A$ is principal.", "Pick $f, g \\in A$ and write $(f, g) = (h)$. Then $f = ah$ and $g = bh$", "and $h = cf + dg$ for some $a, b, c, d \\in A$. Thus $ac + bd = 1$", "and we see that either $a$ or $b$ is a unit, i.e., either", "$g/f$ or $f/g$ is an element of $A$. This shows $A$ is a valuation ring", "by Lemma \\ref{lemma-x-or-x-inverse-valuation-ring}." ], "refs": [ "algebra-lemma-x-or-x-inverse-valuation-ring" ], "ref_ids": [ 610 ] } ], "ref_ids": [] }, { "id": 621, "type": "theorem", "label": "algebra-lemma-valuation-valuation-ring", "categories": [ "algebra" ], "title": "algebra-lemma-valuation-valuation-ring", "contents": [ "Let $(\\Gamma, \\geq)$ be a totally ordered abelian group.", "Let $K$ be a field. Let $v : K^* \\to \\Gamma$ be a homomorphism", "of abelian groups such that $v(a + b) \\geq \\min(v(a), v(b))$ for", "$a, b \\in K$ with $a, b, a + b$ not zero. Then", "$$", "A =", "\\{", "x \\in K \\mid x = 0 \\text{ or } v(x) \\geq 0", "\\}", "$$", "is a valuation ring with value group $\\Im(v) \\subset \\Gamma$,", "with maximal ideal", "$$", "\\mathfrak m =", "\\{", "x \\in K \\mid x = 0 \\text{ or } v(x) > 0", "\\}", "$$", "and with group of units", "$$", "A^* =", "\\{", "x \\in K^* \\mid v(x) = 0", "\\}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 622, "type": "theorem", "label": "algebra-lemma-ideals-valuation-ring", "categories": [ "algebra" ], "title": "algebra-lemma-ideals-valuation-ring", "contents": [ "Let $A$ be a valuation ring.", "Ideals in $A$ correspond $1 - 1$ with ideals of $\\Gamma$.", "This bijection is inclusion preserving, and maps prime", "ideals to prime ideals." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 623, "type": "theorem", "label": "algebra-lemma-valuation-ring-Noetherian-discrete", "categories": [ "algebra" ], "title": "algebra-lemma-valuation-ring-Noetherian-discrete", "contents": [ "A valuation ring is Noetherian if and only if it is", "a discrete valuation ring or a field." ], "refs": [], "proofs": [ { "contents": [ "Suppose $A$ is a discrete valuation ring", "with valuation $v : A \\setminus \\{0\\} \\to \\mathbf{Z}$", "normalized so that $\\Im(v) = \\mathbf{Z}_{\\geq 0}$.", "By Lemma \\ref{lemma-ideals-valuation-ring} the ideals of $A$ are the subsets", "$I_n = \\{0\\} \\cup v^{-1}(\\mathbf{Z}_{\\geq n})$. It is clear", "that any element $x \\in A$ with $v(x) = n$ generates $I_n$.", "Hence $A$ is a PID so certainly Noetherian.", "\\medskip\\noindent", "Suppose $A$ is a Noetherian valuation ring with value group $\\Gamma$.", "By Lemma \\ref{lemma-ideals-valuation-ring} we see the ascending chain", "condition holds for ideals in $\\Gamma$.", "We may assume $A$ is not a field, i.e., there is a $\\gamma \\in \\Gamma$", "with $\\gamma > 0$. Applying the ascending chain condition to the subsets", "$\\gamma + \\Gamma_{\\geq 0}$ with $\\gamma > 0$ we see", "there exists a smallest element $\\gamma_0$ which is bigger than $0$.", "Let $\\gamma \\in \\Gamma$ be an element $\\gamma > 0$. Consider the sequence", "of elements $\\gamma$, $\\gamma - \\gamma_0$, $\\gamma - 2\\gamma_0$,", "etc. By the ascending chain condition these cannot all be $> 0$.", "Let $\\gamma - n \\gamma_0$ be the last one $\\geq 0$. By minimality", "of $\\gamma_0$ we see that $0 = \\gamma - n \\gamma_0$. Hence $\\Gamma$", "is a cyclic group as desired." ], "refs": [ "algebra-lemma-ideals-valuation-ring", "algebra-lemma-ideals-valuation-ring" ], "ref_ids": [ 622, 622 ] } ], "ref_ids": [] }, { "id": 624, "type": "theorem", "label": "algebra-lemma-Noetherian-basic", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-basic", "contents": [ "Let $R$ be a Noetherian ring.", "Any finite $R$-module is of finite presentation.", "Any submodule of a finite $R$-module is finite.", "The ascending chain condition holds for $R$-submodules", "of a finite $R$-module." ], "refs": [], "proofs": [ { "contents": [ "We first show that any submodule $N$ of a finite $R$-module", "$M$ is finite. We do this by induction on the number of", "generators of $M$. If this number is $1$, then $N = J/I \\subset", "M = R/I$ for some ideals $I \\subset J \\subset R$. Thus the definition", "of Noetherian implies the result. If the number of generators of", "$M$ is greater than $1$, then we can find a short exact sequence", "$0 \\to M' \\to M \\to M'' \\to 0$ where $M'$ and $M''$ have fewer", "generators. Note that setting $N' = M' \\cap N$ and $N'' = \\Im(N \\to", "M'')$ gives a similar short exact sequence for $N$. Hence the result", "follows from the induction hypothesis", "since the number of generators of $N$ is at most the number of", "generators of $N'$ plus the number of generators of $N''$.", "\\medskip\\noindent", "To show that $M$ is finitely presented just apply the previous result", "to the kernel of a presentation $R^n \\to M$.", "\\medskip\\noindent", "It is well known and easy to prove that the ascending chain condition for", "$R$-submodules of $M$ is equivalent to the condition that every submodule", "of $M$ is a finite $R$-module. We omit the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 625, "type": "theorem", "label": "algebra-lemma-Artin-Rees", "categories": [ "algebra" ], "title": "algebra-lemma-Artin-Rees", "contents": [ "Suppose that $R$ is Noetherian, $I \\subset R$ an ideal.", "Let $N \\subset M$ be finite $R$-modules.", "There exists a constant $c > 0$ such that", "$I^n M \\cap N = I^{n-c}(I^cM \\cap N)$ for all $n \\geq c$." ], "refs": [], "proofs": [ { "contents": [ "Consider the ring $S = R \\oplus I \\oplus I^2 \\oplus \\ldots", "= \\bigoplus_{n \\geq 0} I^n$. Convention: $I^0 = R$.", "Multiplication maps $I^n \\times I^m$", "into $I^{n + m}$ by multiplication in $R$.", "Note that if $I = (f_1, \\ldots, f_t)$", "then $S$ is a quotient of the Noetherian ring $R[X_1, \\ldots, X_t]$.", "The map just sends the monomial $X_1^{e_1}\\ldots X_t^{e_t}$", "to $f_1^{e_1}\\ldots f_t^{e_t}$. Thus $S$ is Noetherian.", "Similarly, consider the module $M \\oplus IM \\oplus I^2M \\oplus \\ldots", "= \\bigoplus_{n \\geq 0} I^nM$. This is a finitely generated $S$-module.", "Namely, if $x_1, \\ldots, x_r$ generate $M$ over $R$, then they also generate", "$\\bigoplus_{n \\geq 0} I^nM$ over $S$. Next, consider the", "submodule $\\bigoplus_{n \\geq 0} I^nM \\cap N$.", "This is an $S$-submodule, as is easily verified. By", "Lemma \\ref{lemma-Noetherian-basic} it is finitely generated as", "an $S$-module,", "say by $\\xi_j \\in \\bigoplus_{n \\geq 0} I^nM \\cap N$, $j = 1, \\ldots, s$.", "We may assume by decomposing each $\\xi_j$ into its homogeneous", "pieces that each $\\xi_j \\in I^{d_j}M \\cap N$ for some $d_j$.", "Set $c = \\max\\{d_j\\}$. Then for all $n \\geq c$ every element", "in $I^nM \\cap N$ is of the form $\\sum h_j \\xi_j$ with", "$h_j \\in I^{n - d_j}$. The lemma now follows from this and the trivial", "observation that $I^{n-d_j}(I^{d_j}M \\cap N) \\subset I^{n-c}(I^cM \\cap N)$." ], "refs": [ "algebra-lemma-Noetherian-basic" ], "ref_ids": [ 624 ] } ], "ref_ids": [] }, { "id": 626, "type": "theorem", "label": "algebra-lemma-map-AR", "categories": [ "algebra" ], "title": "algebra-lemma-map-AR", "contents": [ "Suppose that $0 \\to K \\to M \\xrightarrow{f} N$ is an", "exact sequence of finitely generated modules", "over a Noetherian ring $R$. Let $I \\subset R$ be an ideal.", "Then there exists a $c$ such that", "$$", "f^{-1}(I^nN) = K + I^{n-c}f^{-1}(I^cN)", "\\quad\\text{and}\\quad", "f(M) \\cap I^nN \\subset f(I^{n - c}M)", "$$", "for all $n \\geq c$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-Artin-Rees} to", "$\\Im(f) \\subset N$ and note that", "$f : I^{n-c}M \\to I^{n-c}f(M)$ is surjective." ], "refs": [ "algebra-lemma-Artin-Rees" ], "ref_ids": [ 625 ] } ], "ref_ids": [] }, { "id": 627, "type": "theorem", "label": "algebra-lemma-intersect-powers-ideal-module-zero", "categories": [ "algebra" ], "title": "algebra-lemma-intersect-powers-ideal-module-zero", "contents": [ "Let $R$ be a Noetherian local ring. Let $I \\subset R$ be", "a proper ideal. Let $M$ be a finite $R$-module.", "Then $\\bigcap_{n \\geq 0} I^nM = 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $N = \\bigcap_{n \\geq 0} I^nM$.", "Then $N = I^nM \\cap N$ for all $n \\geq 0$.", "By the Artin-Rees Lemma \\ref{lemma-Artin-Rees}", "we see that $N = I^nM \\cap N \\subset IN$ for", "some suitably large $n$. By Nakayama's Lemma \\ref{lemma-NAK}", "we see that $N = 0$." ], "refs": [ "algebra-lemma-Artin-Rees", "algebra-lemma-NAK" ], "ref_ids": [ 625, 401 ] } ], "ref_ids": [] }, { "id": 628, "type": "theorem", "label": "algebra-lemma-intersection-powers-ideal-module", "categories": [ "algebra" ], "title": "algebra-lemma-intersection-powers-ideal-module", "contents": [ "Let $R$ be a Noetherian ring. Let $I \\subset R$ be an ideal.", "Let $M$ be a finite $R$-module. Let $N = \\bigcap_n I^n M$.", "\\begin{enumerate}", "\\item For every prime $\\mathfrak p$, $I \\subset \\mathfrak p$ there", "exists a $f \\in R$, $f \\not \\in \\mathfrak p$ such that $N_f = 0$.", "\\item If $I$ is contained in the Jacobson radical", "of $R$, then $N = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $x_1, \\ldots, x_n$ be generators for the module $N$,", "see Lemma \\ref{lemma-Noetherian-basic}. For every prime", "$\\mathfrak p$, $I \\subset \\mathfrak p$ we see that", "the image of $N$ in the localization $M_{\\mathfrak p}$", "is zero, by Lemma \\ref{lemma-intersect-powers-ideal-module-zero}.", "Hence we can find $g_i \\in R$, $g_i \\not \\in \\mathfrak p$", "such that $x_i$ maps to zero in $N_{g_i}$. Thus", "$N_{g_1g_2\\ldots g_n} = 0$.", "\\medskip\\noindent", "Part (2) follows from (1) and Lemma \\ref{lemma-characterize-zero-local}." ], "refs": [ "algebra-lemma-Noetherian-basic", "algebra-lemma-intersect-powers-ideal-module-zero", "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 624, 627, 410 ] } ], "ref_ids": [] }, { "id": 629, "type": "theorem", "label": "algebra-lemma-Artin-Tate", "categories": [ "algebra" ], "title": "algebra-lemma-Artin-Tate", "contents": [ "Let $R$ be a Noetherian ring. Let $S$ be a finitely", "generated $R$-algebra. If $T \\subset S$ is an $R$-subalgebra such", "that $S$ is finitely generated as a $T$-module, then $T$ is of", "finite type over $R$." ], "refs": [], "proofs": [ { "contents": [ "Choose elements $x_1, \\ldots, x_n \\in S$ which generate $S$ as an $R$-algebra.", "Choose $y_1, \\ldots, y_m$ in $S$ which generate $S$ as a $T$-module.", "Thus there exist $a_{ij} \\in T$ such that", "$x_i = \\sum a_{ij} y_j$. There also exist $b_{ijk} \\in T$ such", "that $y_i y_j = \\sum b_{ijk} y_k$. Let $T' \\subset T$ be the", "sub $R$-algebra generated by $a_{ij}$ and $b_{ijk}$. This is a finitely", "generated $R$-algebra, hence Noetherian. Consider the algebra", "$$", "S' = T'[Y_1, \\ldots, Y_m]/(Y_i Y_j - \\sum b_{ijk} Y_k).", "$$", "Note that $S'$ is finite over $T'$, namely as a $T'$-module it is", "generated by the classes of $1, Y_1, \\ldots, Y_m$.", "Consider the $T'$-algebra homomorphism $S' \\to S$ which maps", "$Y_i$ to $y_i$. Because $a_{ij} \\in T'$ we see that $x_j$ is", "in the image of this map. Thus $S' \\to S$ is surjective.", "Therefore $S$ is finite over $T'$ as well. Since $T'$ is Noetherian", "and we conclude that $T \\subset S$ is finite over $T'$ and", "we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 630, "type": "theorem", "label": "algebra-lemma-finite-length-finite", "categories": [ "algebra" ], "title": "algebra-lemma-finite-length-finite", "contents": [ "\\begin{slogan}", "Modules of finite length are finite.", "\\end{slogan}", "Let $R$ be a ring.", "Let $M$ be an $R$-module.", "If $\\text{length}_R(M) < \\infty$ then $M$ is a finite $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 631, "type": "theorem", "label": "algebra-lemma-length-additive", "categories": [ "algebra" ], "title": "algebra-lemma-length-additive", "contents": [ "\\begin{slogan}", "Length is additive in short exact sequences.", "\\end{slogan}", "If $0 \\to M' \\to M \\to M'' \\to 0$", "is a short exact sequence of modules over $R$ then", "the length of $M$ is the sum of the", "lengths of $M'$ and $M''$." ], "refs": [], "proofs": [ { "contents": [ "Given filtrations of $M'$ and $M''$ of lengths $n', n''$", "it is easy to make a corresponding filtration of $M$", "of length $n' + n''$. Thus we see that $\\text{length}_R M", "\\geq \\text{length}_R M' + \\text{length}_R M''$.", "Conversely, given a filtration", "$M_0 \\subset M_1 \\subset \\ldots \\subset M_n$ of", "$M$ consider the induced filtrations", "$M_i' = M_i \\cap M'$ and $M_i'' = \\Im(M_i \\to M'')$.", "Let $n'$ (resp.\\ $n''$) be the number of steps in the filtration", "$\\{M'_i\\}$ (resp.\\ $\\{M''_i\\}$).", "If $M_i' = M_{i + 1}'$ and $M_i'' = M_{i + 1}''$ then", "$M_i = M_{i + 1}$. Hence we conclude that $n' + n'' \\geq n$.", "Combined with the earlier result we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 632, "type": "theorem", "label": "algebra-lemma-length-infinite", "categories": [ "algebra" ], "title": "algebra-lemma-length-infinite", "contents": [ "Let $R$ be a local ring with maximal ideal $\\mathfrak m$.", "Let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item If $M$ is a finite module and", "$\\mathfrak m^n M \\not = 0$ for all $n\\geq 0$,", "then $\\text{length}_R(M) = \\infty$.", "\\item If $M$ has finite length then $\\mathfrak m^nM = 0$", "for some $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathfrak m^n M \\not = 0$ for all $n\\geq 0$.", "Choose $x \\in M$ and $f_1, \\ldots, f_n \\in \\mathfrak m$", "such that $f_1f_2 \\ldots f_n x \\not = 0$.", "By Nakayama's Lemma \\ref{lemma-NAK} the first $n$ steps in the filtration", "$$", "0 \\subset R f_1 \\ldots f_n x \\subset R f_1 \\ldots f_{n - 1} x", "\\subset \\ldots \\subset R x \\subset M", "$$", "are distinct. This can also be seen directly. For example, if", "$R f_1 x = R f_1 f_2 x$ , then $f_1 x = g f_1 f_2 x$ for some $g$,", "hence $(1 - gf_2) f_1 x = 0$ hence $f_1 x = 0$ as $1 - gf_2$ is a unit", "which is a contradiction with the choice of $x$ and $f_1, \\ldots, f_n$.", "Hence the length is infinite, i.e., (1) holds.", "Combine (1) and Lemma \\ref{lemma-finite-length-finite} to see (2)." ], "refs": [ "algebra-lemma-NAK", "algebra-lemma-finite-length-finite" ], "ref_ids": [ 401, 630 ] } ], "ref_ids": [] }, { "id": 633, "type": "theorem", "label": "algebra-lemma-length-independent", "categories": [ "algebra" ], "title": "algebra-lemma-length-independent", "contents": [ "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module.", "We always have $\\text{length}_R(M) \\geq \\text{length}_S(M)$.", "If $R \\to S$ is surjective then equality holds." ], "refs": [], "proofs": [ { "contents": [ "A filtration of $M$ by $S$-submodules gives rise a filtration", "of $M$ by $R$-submodules. This proves the inequality.", "And if $R \\to S$ is surjective, then any $R$-submodule", "of $M$ is automatically an $S$-submodule. Hence equality", "in this case." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 634, "type": "theorem", "label": "algebra-lemma-dimension-is-length", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-is-length", "contents": [ "Let $R$ be a ring with maximal ideal $\\mathfrak m$.", "Suppose that $M$ is an $R$-module with", "$\\mathfrak m M = 0$. Then the length of $M$ as", "an $R$-module agrees with the dimension of $M$ as", "a $R/\\mathfrak m$ vector space.", "The length is finite if and only if $M$ is a finite $R$-module." ], "refs": [], "proofs": [ { "contents": [ "The first part is a special case of Lemma \\ref{lemma-length-independent}.", "Thus the length is finite if and only if $M$ has a finite basis", "as a $R/\\mathfrak m$-vector space if and only if $M$ has a finite", "set of generators as an $R$-module." ], "refs": [ "algebra-lemma-length-independent" ], "ref_ids": [ 633 ] } ], "ref_ids": [] }, { "id": 635, "type": "theorem", "label": "algebra-lemma-length-localize", "categories": [ "algebra" ], "title": "algebra-lemma-length-localize", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \\subset R$ be", "a multiplicative subset. Then", "$\\text{length}_R(M) \\geq \\text{length}_{S^{-1}R}(S^{-1}M)$." ], "refs": [], "proofs": [ { "contents": [ "Any submodule $N' \\subset S^{-1}M$ is of the form", "$S^{-1}N$ for some $R$-submodule $N \\subset M$, by Lemma", "\\ref{lemma-submodule-localization}. The lemma follows." ], "refs": [ "algebra-lemma-submodule-localization" ], "ref_ids": [ 350 ] } ], "ref_ids": [] }, { "id": 636, "type": "theorem", "label": "algebra-lemma-length-finite", "categories": [ "algebra" ], "title": "algebra-lemma-length-finite", "contents": [ "Let $R$ be a ring with finitely generated", "maximal ideal $\\mathfrak m$. (For example $R$ Noetherian.)", "Suppose that $M$ is a finite $R$-module with", "$\\mathfrak m^n M = 0$ for some $n$.", "Then $\\text{length}_R(M) < \\infty$." ], "refs": [], "proofs": [ { "contents": [ "Consider the filtration", "$0 = \\mathfrak m^n M \\subset", "\\mathfrak m^{n-1} M \\subset", "\\ldots \\subset \\mathfrak m M \\subset M$.", "All of the subquotients are finitely generated $R$-modules", "to which Lemma \\ref{lemma-dimension-is-length} applies. We conclude", "by additivity, see Lemma \\ref{lemma-length-additive}." ], "refs": [ "algebra-lemma-dimension-is-length", "algebra-lemma-length-additive" ], "ref_ids": [ 634, 631 ] } ], "ref_ids": [] }, { "id": 637, "type": "theorem", "label": "algebra-lemma-characterize-length-1", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-length-1", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "The following are equivalent:", "\\begin{enumerate}", "\\item $M$ is simple,", "\\item $\\text{length}_R(M) = 1$, and", "\\item $M \\cong R/\\mathfrak m$ for some maximal ideal", "$\\mathfrak m \\subset R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak m$ be a maximal ideal of $R$.", "By Lemma \\ref{lemma-dimension-is-length} the module", "$R/\\mathfrak m$ has length $1$. The equivalence of", "the first two assertions is tautological.", "Suppose that $M$ is simple. Choose $x \\in M$, $x \\not = 0$.", "As $M$ is simple we have $M = R \\cdot x$.", "Let $I \\subset R$ be the annihilator of $x$, i.e.,", "$I = \\{f \\in R \\mid fx = 0\\}$. The map $R/I \\to M$,", "$f \\bmod I \\mapsto fx$ is an isomorphism, hence", "$R/I$ is a simple $R$-module. Since $R/I \\not = 0$ we see $I \\not = R$.", "Let $I \\subset \\mathfrak m$ be a maximal ideal containing $I$.", "If $I \\not = \\mathfrak m$, then $\\mathfrak m /I \\subset R/I$", "is a nontrivial submodule contradicting the simplicity", "of $R/I$. Hence we see $I = \\mathfrak m$ as desired." ], "refs": [ "algebra-lemma-dimension-is-length" ], "ref_ids": [ 634 ] } ], "ref_ids": [] }, { "id": 638, "type": "theorem", "label": "algebra-lemma-simple-pieces", "categories": [ "algebra" ], "title": "algebra-lemma-simple-pieces", "contents": [ "Let $R$ be a ring. Let $M$ be a finite length $R$-module.", "Choose any maximal chain of submodules", "$$", "0 = M_0 \\subset M_1 \\subset M_2 \\subset \\ldots \\subset M_n = M", "$$", "with $M_i \\not = M_{i-1}$, $i = 1, \\ldots, n$. Then", "\\begin{enumerate}", "\\item $n = \\text{length}_R(M)$,", "\\item each $M_i/M_{i-1}$ is simple,", "\\item each $M_i/M_{i-1}$ is of the form", "$R/\\mathfrak m_i$ for some maximal ideal $\\mathfrak m_i$,", "\\item given a maximal ideal $\\mathfrak m \\subset R$", "we have", "$$", "\\# \\{i \\mid \\mathfrak m_i = \\mathfrak m\\}", "=", "\\text{length}_{R_{\\mathfrak m}} (M_{\\mathfrak m}).", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $M_i/M_{i-1}$ is not simple then we can refine the filtration", "and the filtration is not maximal. Thus we see that $M_i/M_{i-1}$", "is simple. By Lemma \\ref{lemma-characterize-length-1} the modules", "$M_i/M_{i-1}$ have length $1$ and are of the form $R/\\mathfrak m_i$", "for some maximal ideals $\\mathfrak m_i$. By additivity of length,", "Lemma \\ref{lemma-length-additive}, we see $n = \\text{length}_R(M)$.", "Since localization is exact, we see that", "$$", "0 = (M_0)_{\\mathfrak m}", "\\subset (M_1)_{\\mathfrak m}", "\\subset (M_2)_{\\mathfrak m}", "\\subset \\ldots", "\\subset (M_n)_{\\mathfrak m} = M_{\\mathfrak m}", "$$", "is a filtration of $M_{\\mathfrak m}$ with successive quotients", "$(M_i/M_{i-1})_{\\mathfrak m}$. Thus the last statement follows", "directly from the fact that given maximal ideals $\\mathfrak m$,", "$\\mathfrak m'$ of $R$ we have", "$$", "(R/\\mathfrak m')_{\\mathfrak m}", "\\cong", "\\left\\{", "\\begin{matrix}", "0 &", "\\text{if } \\mathfrak m \\not = \\mathfrak m', \\\\", "R_{\\mathfrak m}/\\mathfrak m R_{\\mathfrak m} &", "\\text{if } \\mathfrak m = \\mathfrak m'", "\\end{matrix}", "\\right.", "$$", "This we leave to the reader." ], "refs": [ "algebra-lemma-characterize-length-1", "algebra-lemma-length-additive" ], "ref_ids": [ 637, 631 ] } ], "ref_ids": [] }, { "id": 639, "type": "theorem", "label": "algebra-lemma-pushdown-module", "categories": [ "algebra" ], "title": "algebra-lemma-pushdown-module", "contents": [ "Let $A$ be a local ring with maximal ideal $\\mathfrak m$.", "Let $B$ be a semi-local ring with maximal ideals $\\mathfrak m_i$,", "$i = 1, \\ldots, n$.", "Suppose that $A \\to B$ is a homomorphism such that each $\\mathfrak m_i$", "lies over $\\mathfrak m$ and such that", "$$", "[\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m)] < \\infty.", "$$", "Let $M$ be a $B$-module of finite length.", "Then", "$$", "\\text{length}_A(M) = \\sum\\nolimits_{i = 1, \\ldots, n}", "[\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m)]", "\\text{length}_{B_{\\mathfrak m_i}}(M_{\\mathfrak m_i}),", "$$", "in particular $\\text{length}_A(M) < \\infty$." ], "refs": [], "proofs": [ { "contents": [ "Choose a maximal chain", "$$", "0 = M_0", "\\subset M_1", "\\subset M_2", "\\subset \\ldots", "\\subset M_m = M", "$$", "by $B$-submodules as in Lemma \\ref{lemma-simple-pieces}.", "Then each quotient $M_j/M_{j - 1}$ is isomorphic to", "$\\kappa(\\mathfrak m_{i(j)})$ for some $i(j) \\in \\{1, \\ldots, n\\}$.", "Moreover", "$\\text{length}_A(\\kappa(\\mathfrak m_i)) =", "[\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m)]$ by", "Lemma \\ref{lemma-dimension-is-length}. The lemma follows", "by additivity of lengths (Lemma \\ref{lemma-length-additive})." ], "refs": [ "algebra-lemma-simple-pieces", "algebra-lemma-dimension-is-length", "algebra-lemma-length-additive" ], "ref_ids": [ 638, 634, 631 ] } ], "ref_ids": [] }, { "id": 640, "type": "theorem", "label": "algebra-lemma-pullback-module", "categories": [ "algebra" ], "title": "algebra-lemma-pullback-module", "contents": [ "Let $A \\to B$ be a flat local homomorphism of local rings.", "Then for any $A$-module $M$ we have", "$$", "\\text{length}_A(M) \\text{length}_B(B/\\mathfrak m_AB)", "=", "\\text{length}_B(M \\otimes_A B).", "$$", "In particular, if $\\text{length}_B(B/\\mathfrak m_AB) < \\infty$", "then $M$ has finite length if and only if $M \\otimes_A B$ has finite length." ], "refs": [], "proofs": [ { "contents": [ "The ring map $A \\to B$ is faithfully flat by", "Lemma \\ref{lemma-local-flat-ff}.", "Hence if $0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M$", "is a chain of length $n$ in $M$, then the corresponding chain", "$0 = M_0 \\otimes_A B \\subset M_1 \\otimes_A B \\subset", "\\ldots \\subset M_n \\otimes_A B = M \\otimes_A B$ has length $n$", "also. This proves", "$\\text{length}_A(M) = \\infty \\Rightarrow", "\\text{length}_B(M \\otimes_A B) = \\infty$.", "Next, assume $\\text{length}_A(M) < \\infty$. In this case we see", "that $M$ has a filtration of length $\\ell = \\text{length}_A(M)$", "whose quotients are $A/\\mathfrak m_A$. Arguing as above", "we see that $M \\otimes_A B$ has a filtration of length $\\ell$", "whose quotients are isomorphic to", "$B \\otimes_A A/\\mathfrak m_A = B/\\mathfrak m_AB$.", "Thus the lemma follows." ], "refs": [ "algebra-lemma-local-flat-ff" ], "ref_ids": [ 537 ] } ], "ref_ids": [] }, { "id": 641, "type": "theorem", "label": "algebra-lemma-pullback-transitive", "categories": [ "algebra" ], "title": "algebra-lemma-pullback-transitive", "contents": [ "Let $A \\to B \\to C$ be flat local homomorphisms of local rings. Then", "$$", "\\text{length}_B(B/\\mathfrak m_A B)", "\\text{length}_C(C/\\mathfrak m_B C)", "=", "\\text{length}_C(C/\\mathfrak m_A C)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-pullback-module} applied to the ring map", "$B \\to C$ and the $B$-module $M = B/\\mathfrak m_A B$" ], "refs": [ "algebra-lemma-pullback-module" ], "ref_ids": [ 640 ] } ], "ref_ids": [] }, { "id": 642, "type": "theorem", "label": "algebra-lemma-finite-dimensional-algebra", "categories": [ "algebra" ], "title": "algebra-lemma-finite-dimensional-algebra", "contents": [ "Suppose $R$ is a finite dimensional algebra over a field.", "Then $R$ is Artinian." ], "refs": [], "proofs": [ { "contents": [ "The descending chain condition for ideals obviously holds." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 643, "type": "theorem", "label": "algebra-lemma-artinian-finite-nr-max", "categories": [ "algebra" ], "title": "algebra-lemma-artinian-finite-nr-max", "contents": [ "If $R$ is Artinian then $R$ has only finitely many maximal ideals." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $\\mathfrak m_i$, $i = 1, 2, 3, \\ldots$ are", "pairwise distinct maximal ideals.", "Then $\\mathfrak m_1 \\supset \\mathfrak m_1\\cap \\mathfrak m_2", "\\supset \\mathfrak m_1 \\cap \\mathfrak m_2 \\cap \\mathfrak m_3 \\supset \\ldots$", "is an infinite descending sequence (because by the Chinese", "remainder theorem all the maps $R \\to \\oplus_{i = 1}^n R/\\mathfrak m_i$", "are surjective)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 644, "type": "theorem", "label": "algebra-lemma-artinian-radical-nilpotent", "categories": [ "algebra" ], "title": "algebra-lemma-artinian-radical-nilpotent", "contents": [ "Let $R$ be Artinian. The Jacobson radical of $R$ is a nilpotent ideal." ], "refs": [], "proofs": [ { "contents": [ "Let $I \\subset R$ be the Jacobson radical.", "Note that $I \\supset I^2 \\supset I^3 \\supset \\ldots$ is a descending", "sequence. Thus $I^n = I^{n + 1}$ for some $n$.", "Set $J = \\{ x\\in R \\mid xI^n = 0\\}$. We have to show $J = R$.", "If not, choose an ideal $J' \\not = J$, $J \\subset J'$ minimal (possible", "by the Artinian property). Then $J' = J + Rx$ for some $x \\in R$.", "By NAK, Lemma \\ref{lemma-NAK}, we have $IJ' \\subset J$.", "Hence $xI^{n + 1} \\subset xI \\cdot I^n \\subset J \\cdot I^n = 0$.", "Since $I^{n + 1} = I^n$ we conclude $x\\in J$. Contradiction." ], "refs": [ "algebra-lemma-NAK" ], "ref_ids": [ 401 ] } ], "ref_ids": [] }, { "id": 645, "type": "theorem", "label": "algebra-lemma-product-local", "categories": [ "algebra" ], "title": "algebra-lemma-product-local", "contents": [ "Any ring with finitely many maximal ideals and", "locally nilpotent Jacobson radical is the product of its localizations", "at its maximal ideals. Also, all primes are maximal." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a ring with finitely many maximal ideals", "$\\mathfrak m_1, \\ldots, \\mathfrak m_n$.", "Let $I = \\bigcap_{i = 1}^n \\mathfrak m_i$", "be the Jacobson radical of $R$. Assume $I$ is locally nilpotent.", "Let $\\mathfrak p$ be a prime ideal of $R$.", "Since every prime contains every nilpotent", "element of $R$ we see", "$ \\mathfrak p \\supset \\mathfrak m_1 \\cap \\ldots \\cap \\mathfrak m_n$.", "Since $\\mathfrak m_1 \\cap \\ldots \\cap \\mathfrak m_n \\supset", "\\mathfrak m_1 \\ldots \\mathfrak m_n$", "we conclude $\\mathfrak p \\supset \\mathfrak m_1 \\ldots \\mathfrak m_n$.", "Hence $\\mathfrak p \\supset \\mathfrak m_i$ for some $i$, and so", "$\\mathfrak p = \\mathfrak m_i$. By the Chinese remainder theorem", "(Lemma \\ref{lemma-chinese-remainder})", "we have $R/I \\cong \\bigoplus R/\\mathfrak m_i$", "which is a product of fields.", "Hence by Lemma \\ref{lemma-lift-idempotents}", "there are idempotents $e_i$, $i = 1, \\ldots, n$", "with $e_i \\bmod \\mathfrak m_j = \\delta_{ij}$.", "Hence $R = \\prod Re_i$, and each $Re_i$ is a", "ring with exactly one maximal ideal." ], "refs": [ "algebra-lemma-chinese-remainder", "algebra-lemma-lift-idempotents" ], "ref_ids": [ 380, 461 ] } ], "ref_ids": [] }, { "id": 646, "type": "theorem", "label": "algebra-lemma-artinian-finite-length", "categories": [ "algebra" ], "title": "algebra-lemma-artinian-finite-length", "contents": [ "A ring $R$ is Artinian if and only if it has finite length", "as a module over itself. Any such ring $R$ is both Artinian and", "Noetherian, any prime ideal of $R$ is a maximal ideal, and $R$ is equal", "to the (finite) product of its localizations at its maximal ideals." ], "refs": [], "proofs": [ { "contents": [ "If $R$ has finite length over itself then it satisfies both", "the ascending chain condition and the descending chain", "condition for ideals. Hence it is both Noetherian and Artinian.", "Any Artinian ring is equal to product of its localizations", "at maximal ideals by Lemmas \\ref{lemma-artinian-finite-nr-max},", "\\ref{lemma-artinian-radical-nilpotent}, and \\ref{lemma-product-local}.", "\\medskip\\noindent", "Suppose that $R$ is Artinian. We will show $R$ has finite", "length over itself. It suffices to exhibit a chain of", "submodules whose successive quotients have finite length.", "By what we said above", "we may assume that $R$ is local, with maximal ideal $\\mathfrak m$.", "By Lemma \\ref{lemma-artinian-radical-nilpotent} we have", "$\\mathfrak m^n =0$ for some $n$.", "Consider the sequence", "$0 = \\mathfrak m^n \\subset \\mathfrak m^{n-1} \\subset", "\\ldots \\subset \\mathfrak m \\subset R$. By Lemma", "\\ref{lemma-dimension-is-length} the length of each subquotient", "$\\mathfrak m^j/\\mathfrak m^{j + 1}$ is the dimension of this", "as a vector space over $\\kappa(\\mathfrak m)$. This has to be", "finite since otherwise we would have an infinite descending", "chain of sub vector spaces which would correspond to an", "infinite descending chain of ideals in $R$." ], "refs": [ "algebra-lemma-artinian-finite-nr-max", "algebra-lemma-artinian-radical-nilpotent", "algebra-lemma-product-local", "algebra-lemma-artinian-radical-nilpotent", "algebra-lemma-dimension-is-length" ], "ref_ids": [ 643, 644, 645, 644, 634 ] } ], "ref_ids": [] }, { "id": 647, "type": "theorem", "label": "algebra-lemma-composition-essentially-of-finite-type", "categories": [ "algebra" ], "title": "algebra-lemma-composition-essentially-of-finite-type", "contents": [ "The class of ring maps which are essentially of finite type is", "preserved under composition. Similarly for essentially of finite", "presentation." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 648, "type": "theorem", "label": "algebra-lemma-base-change-essentially-of-finite-type", "categories": [ "algebra" ], "title": "algebra-lemma-base-change-essentially-of-finite-type", "contents": [ "The class of ring maps which are essentially of finite type is", "preserved by base change. Similarly for essentially of finite", "presentation." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 649, "type": "theorem", "label": "algebra-lemma-essentially-of-finite-type-into-artinian-local", "categories": [ "algebra" ], "title": "algebra-lemma-essentially-of-finite-type-into-artinian-local", "contents": [ "Let $R \\to S$ be a ring map. Assume $S$ is an Artinian local ring with", "maximal ideal $\\mathfrak m$. Then", "\\begin{enumerate}", "\\item $R \\to S$ is finite if and only if $R \\to S/\\mathfrak m$ is finite,", "\\item $R \\to S$ is of finite type if and only if $R \\to S/\\mathfrak m$", "is of finite type.", "\\item $R \\to S$ is essentially of finite type if and", "only if the composition $R \\to S/\\mathfrak m$ is essentially", "of finite type.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $R \\to S$ is finite, then $R \\to S/\\mathfrak m$", "is finite by Lemma \\ref{lemma-finite-transitive}.", "Conversely, assume $R \\to S/\\mathfrak m$ is finite.", "As $S$ has finite length over itself", "(Lemma \\ref{lemma-artinian-finite-length})", "we can choose a filtration", "$$", "0 \\subset I_1 \\subset \\ldots \\subset I_n = S", "$$", "by ideals such that $I_i/I_{i - 1} \\cong S/\\mathfrak m$ as $S$-modules.", "Thus $S$ has a filtration by $R$-submodules $I_i$ such that each", "successive quotient is a finite $R$-module. Thus $S$ is a finite", "$R$-module by Lemma \\ref{lemma-extension}.", "\\medskip\\noindent", "If $R \\to S$ is of finite type, then $R \\to S/\\mathfrak m$", "is of finite type by Lemma \\ref{lemma-compose-finite-type}.", "Conversely, assume that $R \\to S/\\mathfrak m$ is of finite type.", "Choose $f_1, \\ldots, f_n \\in S$ which map to generators of $S/\\mathfrak m$.", "Then $A = R[x_1, \\ldots, x_n] \\to S$, $x_i \\mapsto f_i$ is a ring map such", "that $A \\to S/\\mathfrak m$ is surjective (in particular finite).", "Hence $A \\to S$ is finite by part (1) and we see that $R \\to S$", "is of finite type by Lemma \\ref{lemma-compose-finite-type}.", "\\medskip\\noindent", "If $R \\to S$ is essentially of finite type, then $R \\to S/\\mathfrak m$", "is essentially of finite type by", "Lemma \\ref{lemma-composition-essentially-of-finite-type}.", "Conversely, assume that $R \\to S/\\mathfrak m$ is essentially", "of finite type. Suppose $S/\\mathfrak m$ is the localization", "of $R[x_1, \\ldots, x_n]/I$. Choose $f_1, \\ldots, f_n \\in S$", "whose congruence classes modulo $\\mathfrak m$ correspond to", "the congruence classes of $x_1, \\ldots, x_n$ modulo $I$.", "Consider the map $R[x_1, \\ldots, x_n] \\to S$, $x_i \\mapsto f_i$", "with kernel $J$. Set $A = R[x_1, \\ldots, x_n]/J \\subset S$", "and $\\mathfrak p = A \\cap \\mathfrak m$. Note that", "$A/\\mathfrak p \\subset S/\\mathfrak m$ is equal to the image", "of $R[x_1, \\ldots, x_n]/I$ in $S/\\mathfrak m$. Hence", "$\\kappa(\\mathfrak p) = S/\\mathfrak m$. Thus $A_\\mathfrak p \\to S$", "is finite by part (1). We conclude that $S$ is essentially of finite", "type by Lemma \\ref{lemma-composition-essentially-of-finite-type}." ], "refs": [ "algebra-lemma-finite-transitive", "algebra-lemma-artinian-finite-length", "algebra-lemma-extension", "algebra-lemma-compose-finite-type", "algebra-lemma-compose-finite-type", "algebra-lemma-composition-essentially-of-finite-type", "algebra-lemma-composition-essentially-of-finite-type" ], "ref_ids": [ 337, 646, 330, 333, 333, 647, 647 ] } ], "ref_ids": [] }, { "id": 650, "type": "theorem", "label": "algebra-lemma-localization-at-closed-point-special-fibre", "categories": [ "algebra" ], "title": "algebra-lemma-localization-at-closed-point-special-fibre", "contents": [ "Let $\\varphi : R \\to S$ be essentially of finite type with $R$ and $S$", "local (but not necessarily $\\varphi$ local). Then there exists", "an $n$ and a maximal ideal $\\mathfrak m \\subset R[x_1, \\ldots, x_n]$", "lying over $\\mathfrak m_R$ such that $S$ is a localization of a", "quotient of $R[x_1, \\ldots, x_n]_\\mathfrak m$." ], "refs": [], "proofs": [ { "contents": [ "We can write $S$ as a localization of a quotient of $R[x_1, \\ldots, x_n]$.", "Hence it suffices to prove the lemma in case", "$S = R[x_1, \\ldots, x_n]_\\mathfrak q$ for some prime", "$\\mathfrak q \\subset R[x_1, \\ldots, x_n]$.", "If $\\mathfrak q + \\mathfrak m_R R[x_1, \\ldots, x_n] \\not = R[x_1, \\ldots, x_n]$", "then we can find a maximal ideal $\\mathfrak m$ as in the statement", "of the lemma with $\\mathfrak q \\subset \\mathfrak m$ and the result is clear.", "\\medskip\\noindent", "Choose a valuation ring $A \\subset \\kappa(\\mathfrak q)$", "which dominates the image of $R \\to \\kappa(\\mathfrak q)$", "(Lemma \\ref{lemma-dominate}). If the image $\\lambda_i \\in \\kappa(\\mathfrak q)$", "of $x_i$ is contained in $A$, then $\\mathfrak q$ is contained in the inverse", "image of $\\mathfrak m_A$ via $R[x_1, \\ldots, x_n] \\to A$", "which means we are back in the preceding case.", "Hence there exists an $i$ such that $\\lambda_i^{-1} \\in A$", "and such that $\\lambda_j/\\lambda_i \\in A$ for all $j = 1, \\ldots, n$", "(because the value group of $A$ is totally ordered, see", "Lemma \\ref{lemma-valuation-group}). Then we consider the map", "$$", "R[y_0, y_1, \\ldots, \\hat{y_i}, \\ldots, y_n]", "\\to R[x_1, \\ldots, x_n]_\\mathfrak q,\\quad", "y_0 \\mapsto 1/x_i,\\quad y_j \\mapsto x_j/x_i", "$$", "Let $\\mathfrak q' \\subset R[y_0, \\ldots, \\hat{y_i}, \\ldots, y_n]$", "be the inverse image", "of $\\mathfrak q$. Since $y_0 \\not \\in \\mathfrak q'$ it is easy to see", "that the displayed arrow defines an isomorphism on localizations.", "On the other hand, the result of the first paragraph applies to", "$R[y_0, \\ldots, \\hat{y_i}, \\ldots, y_n]$", "because $y_j$ maps to an element of $A$.", "This finishes the proof." ], "refs": [ "algebra-lemma-dominate", "algebra-lemma-valuation-group" ], "ref_ids": [ 608, 618 ] } ], "ref_ids": [] }, { "id": 651, "type": "theorem", "label": "algebra-lemma-length-K0", "categories": [ "algebra" ], "title": "algebra-lemma-length-K0", "contents": [ "If $R$ is an Artinian local ring then the length function", "defines a natural abelian group homomorphism", "$\\text{length}_R : K'_0(R) \\to \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "The length of any finite $R$-module is finite,", "because it is the quotient of $R^n$ which has finite length by", "Lemma \\ref{lemma-artinian-finite-length}. And the length function", "is additive, see Lemma \\ref{lemma-length-additive}." ], "refs": [ "algebra-lemma-artinian-finite-length", "algebra-lemma-length-additive" ], "ref_ids": [ 646, 631 ] } ], "ref_ids": [] }, { "id": 652, "type": "theorem", "label": "algebra-lemma-K0-product", "categories": [ "algebra" ], "title": "algebra-lemma-K0-product", "contents": [ "Let $R = R_1 \\times R_2$. Then $K_0(R) = K_0(R_1) \\times K_0(R_2)$", "and $K'_0(R) = K'_0(R_1) \\times K'_0(R_2)$" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 653, "type": "theorem", "label": "algebra-lemma-K0prime-Artinian", "categories": [ "algebra" ], "title": "algebra-lemma-K0prime-Artinian", "contents": [ "Let $R$ be an Artinian local ring.", "The map $\\text{length}_R : K'_0(R) \\to \\mathbf{Z}$", "of Lemma \\ref{lemma-length-K0} is an isomorphism." ], "refs": [ "algebra-lemma-length-K0" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 651 ] }, { "id": 654, "type": "theorem", "label": "algebra-lemma-K0-local", "categories": [ "algebra" ], "title": "algebra-lemma-K0-local", "contents": [ "Let $(R, \\mathfrak m)$ be a local ring. Every finite projective $R$-module", "is finite free. The map $\\text{rank}_R : K_0(R) \\to \\mathbf{Z}$", "defined by $[M] \\to \\text{rank}_R(M)$ is well defined", "and an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Let $P$ be a finite projective $R$-module. Choose elements", "$x_1, \\ldots, x_n \\in P$ which map to a basis of $P/\\mathfrak m P$.", "By Nakayama's Lemma \\ref{lemma-NAK} these elements generate $P$.", "The corresponding surjection $u : R^{\\oplus n} \\to P$ has a splitting", "as $P$ is projective. Hence $R^{\\oplus n} = P \\oplus Q$ with $Q = \\Ker(u)$.", "It follows that $Q/\\mathfrak m Q = 0$, hence $Q$ is zero by", "Nakayama's lemma. In this way we see that every finite projective", "$R$-module is finite free.", "A finite free module has a well defined rank by Lemma \\ref{lemma-rank}.", "Given a short exact sequence of finite free $R$-modules", "$$", "0 \\to M' \\to M \\to M'' \\to 0", "$$", "we have $\\text{rank}(M) = \\text{rank}(M') + \\text{rank}(M'')$", "because we have $M \\cong M' \\oplus M'$ in this case (for example", "we have a splitting by Lemma \\ref{lemma-lift-map}).", "We conclude $K_0(R) = \\mathbf{Z}$." ], "refs": [ "algebra-lemma-NAK", "algebra-lemma-rank", "algebra-lemma-lift-map" ], "ref_ids": [ 401, 384, 329 ] } ], "ref_ids": [] }, { "id": 655, "type": "theorem", "label": "algebra-lemma-K0-and-K0prime-Artinian-local", "categories": [ "algebra" ], "title": "algebra-lemma-K0-and-K0prime-Artinian-local", "contents": [ "Let $R$ be a local Artinian ring. There is a commutative", "diagram", "$$", "\\xymatrix{", "K_0(R) \\ar[rr] \\ar[d]_{\\text{rank}_R} & &", "K'_0(R) \\ar[d]^{\\text{length}_R} \\\\", "\\mathbf{Z} \\ar[rr]^{\\text{length}_R(R)} & &", "\\mathbf{Z}", "}", "$$", "where the vertical maps are isomorphisms by", "Lemmas \\ref{lemma-K0prime-Artinian} and \\ref{lemma-K0-local}." ], "refs": [ "algebra-lemma-K0prime-Artinian", "algebra-lemma-K0-local" ], "proofs": [ { "contents": [ "Let $P$ be a finite projective $R$-module. We have to show that", "$\\text{length}_R(P) = \\text{rank}_R(P) \\text{length}_R(R)$.", "By Lemma \\ref{lemma-K0-local} the module $P$ is finite free. So", "$P \\cong R^{\\oplus n}$ for some $n \\geq 0$. Then $\\text{rank}_R(P) = n$ and", "$\\text{length}_R(R^{\\oplus n}) = n \\text{length}_R(R)$", "by additivity of lenghts (Lemma \\ref{lemma-length-additive}).", "Thus the result holds." ], "refs": [ "algebra-lemma-K0-local", "algebra-lemma-length-additive" ], "ref_ids": [ 654, 631 ] } ], "ref_ids": [ 653, 654 ] }, { "id": 656, "type": "theorem", "label": "algebra-lemma-graded-NAK", "categories": [ "algebra" ], "title": "algebra-lemma-graded-NAK", "contents": [ "Let $S$ be a graded ring. Let $M$ be a graded $S$-module.", "\\begin{enumerate}", "\\item If $S_+M = M$ and $M$ is finite, then $M = 0$.", "\\item If $N, N' \\subset M$ are graded submodules,", "$M = N + S_+N'$, and $N'$ is finite, then $M = N$.", "\\item If $N \\to M$ is a map of graded modules, $N/S_+N \\to M/S_+M$", "is surjective, and $M$ is finite, then $N \\to M$ is surjective.", "\\item If $x_1, \\ldots, x_n \\in M$ are homogeneous and generate $M/S_+M$", "and $M$ is finite, then $x_1, \\ldots, x_n$ generate $M$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Choose generators $y_1, \\ldots, y_r$ of $M$ over $S$.", "We may assume that $y_i$ is homogeneous of degree $d_i$. After", "renumbering we may assume $d_r = \\min(d_i)$. Then the condition that", "$S_+M = M$ implies $y_r = 0$. Hence $M = 0$ by induction on $r$.", "Part (2) follows by applying (1) to $M/N$. Part (3) follows by", "applying (2) to the submodules $\\Im(N \\to M)$ and $M$.", "Part (4) follows by applying (3) to the module map", "$\\bigoplus S(-d_i) \\to M$, $(s_1, \\ldots, s_n) \\mapsto \\sum s_i x_i$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 657, "type": "theorem", "label": "algebra-lemma-uple-generated-degree-1", "categories": [ "algebra" ], "title": "algebra-lemma-uple-generated-degree-1", "contents": [ "Let $S$ be a graded ring, which is finitely generated over $S_0$.", "Then for all sufficiently divisible $d$ the algebra", "$S^{(d)}$ is generated in degree $1$ over $S_0$." ], "refs": [], "proofs": [ { "contents": [ "Say $S$ is generated by $f_1, \\ldots, f_r \\in S$ over $S_0$.", "After replacing $f_i$ by their homogeneous parts, we may assume", "$f_i$ is homogeneous of degree $d_i > 0$. Then any element of", "$S_n$ is a linear combination with coefficients in $S_0$ of monomials", "$f_1^{e_1} \\ldots f_r^{e_r}$ with $\\sum e_i d_i = n$.", "Let $m$ be a multiple of $\\text{lcm}(d_i)$. For any $N \\geq r$ if", "$$", "\\sum e_i d_i = N m", "$$", "then for some $i$ we have $e_i \\geq m/d_i$ by an elementary argument.", "Hence every monomial of degree $N m$ is a product of a monomial", "of degree $m$, namely $f_i^{m/d_i}$, and a monomial of degree $(N - 1)m$.", "It follows that any monomial of degree $nrm$ with $n \\geq 2$", "is a product of monomials of degree $rm$. Thus $S^{(rm)}$ is generated", "in degree $1$ over $S_0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 658, "type": "theorem", "label": "algebra-lemma-integral-closure-graded", "categories": [ "algebra" ], "title": "algebra-lemma-integral-closure-graded", "contents": [ "Let $R \\to S$ be a homomorphism of graded rings.", "Let $S' \\subset S$ be the integral closure of $R$ in $S$.", "Then", "$$", "S' = \\bigoplus\\nolimits_{d \\geq 0} S' \\cap S_d,", "$$", "i.e., $S'$ is a graded $R$-subalgebra of $S$." ], "refs": [], "proofs": [ { "contents": [ "We have to show the following: If", "$s = s_n + s_{n + 1} + \\ldots + s_m \\in S'$, then each homogeneous", "part $s_j \\in S'$. We will prove this by induction on $m - n$ over", "all homomorphisms $R \\to S$ of graded rings. First note that it", "is immediate that $s_0$ is integral over $R_0$ (hence over $R$) as", "there is a ring map $S \\to S_0$ compatible with the ring map $R \\to R_0$.", "Thus, after replacing $s$ by $s - s_0$, we may assume $n > 0$. Consider the", "extension of graded rings $R[t, t^{-1}] \\to S[t, t^{-1}]$ where", "$t$ has degree $0$. There is a commutative diagram", "$$", "\\xymatrix{", "S[t, t^{-1}] \\ar[rr]_{s \\mapsto t^{\\deg(s)}s} & & S[t, t^{-1}] \\\\", "R[t, t^{-1}] \\ar[u] \\ar[rr]^{r \\mapsto t^{\\deg(r)}r} & & R[t, t^{-1}] \\ar[u]", "}", "$$", "where the horizontal maps are ring automorphisms. Hence the integral", "closure $C$ of $S[t, t^{-1}]$ over $R[t, t^{-1}]$ maps into itself.", "Thus we see that", "$$", "t^m(s_n + s_{n + 1} + \\ldots + s_m) -", "(t^ns_n + t^{n + 1}s_{n + 1} + \\ldots + t^ms_m) \\in C", "$$", "which implies by induction hypothesis that each $(t^m - t^i)s_i \\in C$", "for $i = n, \\ldots, m - 1$. Note that for any ring $A$ and $m > i \\geq n > 0$", "we have $A[t, t^{-1}]/(t^m - t^i - 1) \\cong A[t]/(t^m - t^i - 1) \\supset A$", "because $t(t^{m - 1} - t^{i - 1}) = 1$ in $A[t]/(t^m - t^i - 1)$.", "Since $t^m - t^i$ maps to $1$ we see the image of $s_i$ in the ring", "$S[t]/(t^m - t^i - 1)$ is integral over $R[t]/(t^m - t^i - 1)$ for", "$i = n, \\ldots, m - 1$. Since $R \\to R[t]/(t^m - t^i - 1)$ is finite", "we see that $s_i$ is integral over $R$ by transitivity, see", "Lemma \\ref{lemma-integral-transitive}.", "Finally, we also conclude that $s_m = s - \\sum_{i = n, \\ldots, m - 1} s_i$", "is integral over $R$." ], "refs": [ "algebra-lemma-integral-transitive" ], "ref_ids": [ 485 ] } ], "ref_ids": [] }, { "id": 659, "type": "theorem", "label": "algebra-lemma-Z-graded", "categories": [ "algebra" ], "title": "algebra-lemma-Z-graded", "contents": [ "Let $S$ be a $\\mathbf{Z}$-graded ring containing a homogeneous", "invertible element of positive degree. Then the set", "$G \\subset \\Spec(S)$ of $\\mathbf{Z}$-graded primes of $S$", "(with induced topology) maps homeomorphically to $\\Spec(S_0)$." ], "refs": [], "proofs": [ { "contents": [ "First we show that the map is a bijection by constructing an inverse.", "Let $f \\in S_d$, $d > 0$ be invertible in $S$.", "If $\\mathfrak p_0$ is a prime of $S_0$, then $\\mathfrak p_0S$", "is a $\\mathbf{Z}$-graded ideal of $S$ such that", "$\\mathfrak p_0S \\cap S_0 = \\mathfrak p_0$. And if $ab \\in \\mathfrak p_0S$", "with $a$, $b$ homogeneous, then", "$a^db^d/f^{\\deg(a) + \\deg(b)} \\in \\mathfrak p_0$.", "Thus either $a^d/f^{\\deg(a)} \\in \\mathfrak p_0$ or", "$b^d/f^{\\deg(b)} \\in \\mathfrak p_0$, in other words either", "$a^d \\in \\mathfrak p_0S$ or $b^d \\in \\mathfrak p_0S$.", "It follows that $\\sqrt{\\mathfrak p_0S}$ is a $\\mathbf{Z}$-graded", "prime ideal of $S$ whose intersection with $S_0$ is $\\mathfrak p_0$.", "\\medskip\\noindent", "To show that the map is a homeomorphism we show that", "the image of $G \\cap D(g)$ is open. If $g = \\sum g_i$", "with $g_i \\in S_i$, then by the above $G \\cap D(g)$", "maps onto the set $\\bigcup D(g_i^d/f^i)$ which is open." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 660, "type": "theorem", "label": "algebra-lemma-topology-proj", "categories": [ "algebra" ], "title": "algebra-lemma-topology-proj", "contents": [ "Let $S = \\oplus_{d \\geq 0} S_d$ be a graded ring.", "\\begin{enumerate}", "\\item The sets $D_{+}(f)$ are open in $\\text{Proj}(S)$.", "\\item We have $D_{+}(ff') = D_{+}(f) \\cap D_{+}(f')$.", "\\item Let $g = g_0 + \\ldots + g_m$ be an element", "of $S$ with $g_i \\in S_i$. Then", "$$", "D(g) \\cap \\text{Proj}(S) =", "(D(g_0) \\cap \\text{Proj}(S))", "\\cup", "\\bigcup\\nolimits_{i \\geq 1} D_{+}(g_i).", "$$", "\\item", "Let $g_0\\in S_0$ be a homogeneous element of degree $0$. Then", "$$", "D(g_0) \\cap \\text{Proj}(S)", "=", "\\bigcup\\nolimits_{f \\in S_d, \\ d\\geq 1} D_{+}(g_0 f).", "$$", "\\item The open sets $D_{+}(f)$ form a", "basis for the topology of $\\text{Proj}(S)$.", "\\item Let $f \\in S$ be homogeneous of positive degree.", "The ring $S_f$ has a natural $\\mathbf{Z}$-grading.", "The ring maps $S \\to S_f \\leftarrow S_{(f)}$ induce", "homeomorphisms", "$$", "D_{+}(f)", "\\leftarrow", "\\{\\mathbf{Z}\\text{-graded primes of }S_f\\}", "\\to", "\\Spec(S_{(f)}).", "$$", "\\item There exists an $S$ such that $\\text{Proj}(S)$ is not", "quasi-compact.", "\\item The sets $V_{+}(I)$ are closed.", "\\item Any closed subset $T \\subset \\text{Proj}(S)$ is of", "the form $V_{+}(I)$ for some homogeneous ideal $I \\subset S$.", "\\item For any graded ideal $I \\subset S$ we have", "$V_{+}(I) = \\emptyset$ if and only if $S_{+} \\subset \\sqrt{I}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $D_{+}(f) = \\text{Proj}(S) \\cap D(f)$, these sets are open.", "This proves (1). Also (2) follows as $D(ff') = D(f) \\cap D(f')$.", "Similarly the sets $V_{+}(I) = \\text{Proj}(S) \\cap V(I)$", "are closed. This proves (8).", "\\medskip\\noindent", "Suppose that $T \\subset \\text{Proj}(S)$ is closed.", "Then we can write $T = \\text{Proj}(S) \\cap V(J)$ for some", "ideal $J \\subset S$. By definition of a homogeneous ideal", "if $g \\in J$, $g = g_0 + \\ldots + g_m$", "with $g_d \\in S_d$ then $g_d \\in \\mathfrak p$ for all", "$\\mathfrak p \\in T$. Thus, letting $I \\subset S$", "be the ideal generated by the homogeneous parts of the elements", "of $J$ we have $T = V_{+}(I)$. This proves (9).", "\\medskip\\noindent", "The formula for $\\text{Proj}(S) \\cap D(g)$, with $g \\in S$ is direct", "from the definitions. This proves (3).", "Consider the formula for $\\text{Proj}(S) \\cap D(g_0)$.", "The inclusion of the right hand side in the left hand side is", "obvious. For the other inclusion, suppose $g_0 \\not \\in \\mathfrak p$", "with $\\mathfrak p \\in \\text{Proj}(S)$. If all $g_0f \\in \\mathfrak p$", "for all homogeneous $f$ of positive degree, then we see that", "$S_{+} \\subset \\mathfrak p$ which is a contradiction. This gives", "the other inclusion. This proves (4).", "\\medskip\\noindent", "The collection of opens $D(g) \\cap \\text{Proj}(S)$", "forms a basis for the topology since the standard opens", "$D(g) \\subset \\Spec(S)$ form a basis for the topology on", "$\\Spec(S)$. By the formulas above we can express", "$D(g) \\cap \\text{Proj}(S)$ as a union of opens $D_{+}(f)$.", "Hence the collection of opens $D_{+}(f)$ forms a basis for the topology", "also. This proves (5).", "\\medskip\\noindent", "Proof of (6). First we note that $D_{+}(f)$ may be identified", "with a subset (with induced topology) of $D(f) = \\Spec(S_f)$", "via Lemma \\ref{lemma-standard-open}. Note that the ring", "$S_f$ has a $\\mathbf{Z}$-grading. The homogeneous elements are", "of the form $r/f^n$ with $r \\in S$ homogeneous and have", "degree $\\deg(r/f^n) = \\deg(r) - n\\deg(f)$. The subset", "$D_{+}(f)$ corresponds exactly to those prime ideals", "$\\mathfrak p \\subset S_f$ which are $\\mathbf{Z}$-graded ideals", "(i.e., generated by homogeneous elements). Hence we have to show that", "the set of $\\mathbf{Z}$-graded prime ideals of $S_f$ maps homeomorphically", "to $\\Spec(S_{(f)})$. This follows from Lemma \\ref{lemma-Z-graded}.", "\\medskip\\noindent", "Let $S = \\mathbf{Z}[X_1, X_2, X_3, \\ldots]$ with grading such that", "each $X_i$ has degree $1$. Then it is easy to see that", "$$", "\\text{Proj}(S) = \\bigcup\\nolimits_{i = 1}^\\infty D_{+}(X_i)", "$$", "does not have a finite refinement. This proves (7).", "\\medskip\\noindent", "Let $I \\subset S$ be a graded ideal.", "If $\\sqrt{I} \\supset S_{+}$ then $V_{+}(I) = \\emptyset$ since", "every prime $\\mathfrak p \\in \\text{Proj}(S)$ does not contain", "$S_{+}$ by definition. Conversely, suppose that", "$S_{+} \\not \\subset \\sqrt{I}$. Then we can find an element", "$f \\in S_{+}$ such that $f$ is not nilpotent modulo $I$.", "Clearly this means that one of the homogeneous parts of $f$", "is not nilpotent modulo $I$, in other words we may (and do)", "assume that $f$ is homogeneous. This implies that", "$I S_f \\not = S_f$, in other words that $(S/I)_f$ is not", "zero. Hence $(S/I)_{(f)} \\not = 0$ since it is a ring", "which maps into $(S/I)_f$. Pick a prime", "$\\mathfrak q \\subset (S/I)_{(f)}$. This corresponds to", "a graded prime of $S/I$, not containing the irrelevant ideal", "$(S/I)_{+}$. And this in turn corresponds to a graded prime", "ideal $\\mathfrak p$ of $S$, containing $I$ but not containing $S_{+}$", "as desired. This proves (10) and finishes the proof." ], "refs": [ "algebra-lemma-standard-open", "algebra-lemma-Z-graded" ], "ref_ids": [ 392, 659 ] } ], "ref_ids": [] }, { "id": 661, "type": "theorem", "label": "algebra-lemma-proj-prime", "categories": [ "algebra" ], "title": "algebra-lemma-proj-prime", "contents": [ "Let $S$ be a graded ring. Let $M$ be a graded $S$-module.", "Let $\\mathfrak p$ be an element of $\\text{Proj}(S)$.", "Let $f \\in S$ be a homogeneous element of positive degree", "such that $f \\not \\in \\mathfrak p$, i.e., $\\mathfrak p \\in D_{+}(f)$.", "Let $\\mathfrak p' \\subset S_{(f)}$ be the element of", "$\\Spec(S_{(f)})$ corresponding to $\\mathfrak p$ as in", "Lemma \\ref{lemma-topology-proj}. Then", "$S_{(\\mathfrak p)} = (S_{(f)})_{\\mathfrak p'}$", "and compatibly", "$M_{(\\mathfrak p)} = (M_{(f)})_{\\mathfrak p'}$." ], "refs": [ "algebra-lemma-topology-proj" ], "proofs": [ { "contents": [ "We define a map $\\psi : M_{(\\mathfrak p)} \\to (M_{(f)})_{\\mathfrak p'}$.", "Let $x/g \\in M_{(\\mathfrak p)}$. We set", "$$", "\\psi(x/g) = (x g^{\\deg(f) - 1}/f^{\\deg(x)})/(g^{\\deg(f)}/f^{\\deg(g)}).", "$$", "This makes sense since $\\deg(x) = \\deg(g)$ and since", "$g^{\\deg(f)}/f^{\\deg(g)} \\not \\in \\mathfrak p'$.", "We omit the verification that $\\psi$ is well defined, a module map", "and an isomorphism. Hint: the inverse sends $(x/f^n)/(g/f^m)$ to", "$(xf^m)/(g f^n)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 660 ] }, { "id": 662, "type": "theorem", "label": "algebra-lemma-graded-silly", "categories": [ "algebra" ], "title": "algebra-lemma-graded-silly", "contents": [ "Suppose $S$ is a graded ring, $\\mathfrak p_i$, $i = 1, \\ldots, r$", "homogeneous prime ideals and $I \\subset S_{+}$ a graded ideal.", "Assume $I \\not\\subset \\mathfrak p_i$ for all $i$. Then there", "exists a homogeneous element $x\\in I$ of positive degree such", "that $x\\not\\in \\mathfrak p_i$ for all $i$." ], "refs": [], "proofs": [ { "contents": [ "We may assume there are no inclusions among the $\\mathfrak p_i$.", "The result is true for $r = 1$. Suppose the result holds for $r - 1$.", "Pick $x \\in I$ homogeneous of positive degree such that", "$x \\not \\in \\mathfrak p_i$ for all $i = 1, \\ldots, r - 1$.", "If $x \\not\\in \\mathfrak p_r$ we are done. So assume $x \\in \\mathfrak p_r$.", "If $I \\mathfrak p_1 \\ldots \\mathfrak p_{r-1} \\subset \\mathfrak p_r$", "then $I \\subset \\mathfrak p_r$ a contradiction.", "Pick $y \\in I\\mathfrak p_1 \\ldots \\mathfrak p_{r-1}$ homogeneous", "and $y \\not \\in \\mathfrak p_r$. Then $x^{\\deg(y)} + y^{\\deg(x)}$ works." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 663, "type": "theorem", "label": "algebra-lemma-smear-out", "categories": [ "algebra" ], "title": "algebra-lemma-smear-out", "contents": [ "Let $S$ be a graded ring.", "Let $\\mathfrak p \\subset S$ be a prime.", "Let $\\mathfrak q$ be the homogeneous ideal of $S$ generated by the", "homogeneous elements of $\\mathfrak p$. Then $\\mathfrak q$ is a", "prime ideal of $S$." ], "refs": [], "proofs": [ { "contents": [ "Suppose $f, g \\in S$ are such that $fg \\in \\mathfrak q$.", "Let $f_d$ (resp.\\ $g_e$) be the homogeneous part of", "$f$ (resp.\\ $g$) of degree $d$ (resp.\\ $e$). Assume $d, e$ are", "maxima such that $f_d \\not = 0$ and $g_e \\not = 0$.", "By assumption we can write $fg = \\sum a_i f_i$ with", "$f_i \\in \\mathfrak p$ homogeneous. Say $\\deg(f_i) = d_i$.", "Then $f_d g_e = \\sum a_i' f_i$ with $a_i'$ to homogeneous", "par of degree $d + e - d_i$ of $a_i$ (or $0$ if $d + e -d_i < 0$).", "Hence $f_d \\in \\mathfrak p$ or $g_e \\in \\mathfrak p$. Hence", "$f_d \\in \\mathfrak q$ or $g_e \\in \\mathfrak q$. In the first", "case replace $f$ by $f - f_d$, in the second case replace", "$g$ by $g - g_e$. Then still $fg \\in \\mathfrak q$ but the discrete", "invariant $d + e$ has been decreased. Thus we may continue in this", "fashion until either $f$ or $g$ is zero. This clearly shows that", "$fg \\in \\mathfrak q$ implies either $f \\in \\mathfrak q$ or $g \\in \\mathfrak q$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 664, "type": "theorem", "label": "algebra-lemma-graded-ring-minimal-prime", "categories": [ "algebra" ], "title": "algebra-lemma-graded-ring-minimal-prime", "contents": [ "Let $S$ be a graded ring.", "\\begin{enumerate}", "\\item Any minimal prime of $S$ is a homogeneous ideal of $S$.", "\\item Given a homogeneous ideal $I \\subset S$ any minimal", "prime over $I$ is homogeneous.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first assertion holds because the prime $\\mathfrak q$ constructed in", "Lemma \\ref{lemma-smear-out} satisfies $\\mathfrak q \\subset \\mathfrak p$.", "The second because we may consider $S/I$ and apply the first part." ], "refs": [ "algebra-lemma-smear-out" ], "ref_ids": [ 663 ] } ], "ref_ids": [] }, { "id": 665, "type": "theorem", "label": "algebra-lemma-dehomogenize-finite-type", "categories": [ "algebra" ], "title": "algebra-lemma-dehomogenize-finite-type", "contents": [ "Let $R$ be a ring. Let $S$ be a graded $R$-algebra. Let $f \\in S_{+}$", "be homogeneous. Assume that $S$ is of finite type over $R$. Then", "\\begin{enumerate}", "\\item the ring $S_{(f)}$ is of finite type over $R$, and", "\\item for any finite graded $S$-module $M$ the module $M_{(f)}$", "is a finite $S_{(f)}$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose $f_1, \\ldots, f_n \\in S$ which generate $S$ as an $R$-algebra.", "We may assume that each $f_i$ is homogeneous (by decomposing each $f_i$", "into its homogeneous components). An element of $S_{(f)}$ is a sum", "of the form", "$$", "\\sum\\nolimits_{e\\deg(f) =", "\\sum e_i\\deg(f_i)} \\lambda_{e_1 \\ldots e_n} f_1^{e_1} \\ldots f_n^{e_n}/f^e", "$$", "with $\\lambda_{e_1 \\ldots e_n} \\in R$. Thus $S_{(f)}$ is generated", "as an $R$-algebra by the $f_1^{e_1} \\ldots f_n^{e_n} /f^e$ with the", "property that $e\\deg(f) = \\sum e_i\\deg(f_i)$. If $e_i \\geq \\deg(f)$", "then we can write this as", "$$", "f_1^{e_1} \\ldots f_n^{e_n}/f^e =", "f_i^{\\deg(f)}/f^{\\deg(f_i)} \\cdot", "f_1^{e_1} \\ldots f_i^{e_i - \\deg(f)} \\ldots f_n^{e_n}/f^{e - \\deg(f_i)}", "$$", "Thus we only need the elements $f_i^{\\deg(f)}/f^{\\deg(f_i)}$ as well", "as the elements $f_1^{e_1} \\ldots f_n^{e_n} /f^e$ with", "$e \\deg(f) = \\sum e_i \\deg(f_i)$ and $e_i < \\deg(f)$.", "This is a finite list and we see that (1) is true.", "\\medskip\\noindent", "To see (2) suppose that $M$ is generated by homogeneous elements", "$x_1, \\ldots, x_m$. Then arguing", "as above we find that $M_{(f)}$ is generated as an $S_{(f)}$-module", "by the finite list of elements of the form", "$f_1^{e_1} \\ldots f_n^{e_n} x_j /f^e$", "with $e \\deg(f) = \\sum e_i \\deg(f_i) + \\deg(x_j)$ and", "$e_i < \\deg(f)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 666, "type": "theorem", "label": "algebra-lemma-homogenize", "categories": [ "algebra" ], "title": "algebra-lemma-homogenize", "contents": [ "Let $R$ be a ring.", "Let $R'$ be a finite type $R$-algebra, and let $M$ be a finite $R'$-module.", "There exists a graded $R$-algebra $S$, a graded $S$-module $N$ and", "an element $f \\in S$ homogeneous of degree $1$ such that", "\\begin{enumerate}", "\\item $R' \\cong S_{(f)}$ and $M \\cong N_{(f)}$ (as modules),", "\\item $S_0 = R$ and $S$ is generated by finitely many elements", "of degree $1$ over $R$, and", "\\item $N$ is a finite $S$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may write $R' = R[x_1, \\ldots, x_n]/I$ for some ideal $I$.", "For an element $g \\in R[x_1, \\ldots, x_n]$ denote", "$\\tilde g \\in R[X_0, \\ldots, X_n]$ the element homogeneous of minimal", "degree such that $g = \\tilde g(1, x_1, \\ldots, x_n)$.", "Let $\\tilde I \\subset R[X_0, \\ldots, X_n]$ generated by all", "elements $\\tilde g$, $g \\in I$.", "Set $S = R[X_0, \\ldots, X_n]/\\tilde I$ and denote $f$ the image", "of $X_0$ in $S$. By construction we have an isomorphism", "$$", "S_{(f)} \\longrightarrow R', \\quad", "X_i/X_0 \\longmapsto x_i.", "$$", "To do the same thing with the module $M$ we choose a presentation", "$$", "M = (R')^{\\oplus r}/\\sum\\nolimits_{j \\in J} R'k_j", "$$", "with $k_j = (k_{1j}, \\ldots, k_{rj})$. Let $d_{ij} = \\deg(\\tilde k_{ij})$.", "Set $d_j = \\max\\{d_{ij}\\}$. Set $K_{ij} = X_0^{d_j - d_{ij}}\\tilde k_{ij}$", "which is homogeneous of degree $d_j$. With this notation we set", "$$", "N = \\Coker\\Big(", "\\bigoplus\\nolimits_{j \\in J} S(-d_j) \\xrightarrow{(K_{ij})} S^{\\oplus r}", "\\Big)", "$$", "which works. Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 667, "type": "theorem", "label": "algebra-lemma-S-plus-generated", "categories": [ "algebra" ], "title": "algebra-lemma-S-plus-generated", "contents": [ "Let $S$ be a graded ring. A set of homogeneous elements", "$f_i \\in S_{+}$ generates $S$ as an algebra over $S_0$ if", "and only if they generate $S_{+}$ as an ideal of $S$." ], "refs": [], "proofs": [ { "contents": [ "If the $f_i$ generate $S$ as an algebra over $S_0$ then every element", "in $S_{+}$ is a polynomial without constant term in the $f_i$ and hence", "$S_{+}$ is generated by the $f_i$ as an ideal. Conversely, suppose that", "$S_{+} = \\sum Sf_i$. We will prove that any element $f$ of $S$ can be written", "as a polynomial in the $f_i$ with coefficients in $S_0$. It suffices", "to do this for homogeneous elements. Say $f$ has degree $d$. Then we may", "perform induction on $d$. The case $d = 0$ is immediate. If $d > 0$", "then $f \\in S_{+}$ hence we can write $f = \\sum g_i f_i$", "for some $g_i \\in S$. As $S$ is graded we can replace $g_i$ by its", "homogeneous component of degree $d - \\deg(f_i)$. By induction we", "see that each $g_i$ is a polynomial in the $f_i$ and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 668, "type": "theorem", "label": "algebra-lemma-graded-Noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-graded-Noetherian", "contents": [ "A graded ring $S$ is Noetherian if and only if $S_0$ is", "Noetherian and $S_{+}$ is finitely generated as an ideal of $S$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that if $S$ is Noetherian then $S_0 = S/S_{+}$ is Noetherian", "and $S_{+}$ is finitely generated. Conversely, assume $S_0$ is Noetherian", "and $S_{+}$ finitely generated as an ideal of $S$. Pick generators", "$S_{+} = (f_1, \\ldots, f_n)$. By decomposing the $f_i$ into homogeneous", "pieces we may assume each $f_i$ is homogeneous. By", "Lemma \\ref{lemma-S-plus-generated}", "we see that $S_0[X_1, \\ldots X_n] \\to S$ sending $X_i$ to $f_i$", "is surjective. Thus $S$ is Noetherian by", "Lemma \\ref{lemma-Noetherian-permanence}." ], "refs": [ "algebra-lemma-S-plus-generated", "algebra-lemma-Noetherian-permanence" ], "ref_ids": [ 667, 448 ] } ], "ref_ids": [] }, { "id": 669, "type": "theorem", "label": "algebra-lemma-numerical-polynomial-functorial", "categories": [ "algebra" ], "title": "algebra-lemma-numerical-polynomial-functorial", "contents": [ "If $A \\to A'$ is a homomorphism of abelian groups and if", "$f : n \\mapsto f(n) \\in A$ is a numerical polynomial,", "then so is the composition." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 670, "type": "theorem", "label": "algebra-lemma-numerical-polynomial", "categories": [ "algebra" ], "title": "algebra-lemma-numerical-polynomial", "contents": [ "Suppose that $f: n \\mapsto f(n) \\in A$", "is defined for all $n$ sufficiently large", "and suppose that $n \\mapsto f(n) - f(n-1)$", "is a numerical polynomial. Then $f$ is a", "numerical polynomial." ], "refs": [], "proofs": [ { "contents": [ "Let $f(n) - f(n-1) = \\sum\\nolimits_{i = 0}^r \\binom{n}{i} a_i$", "for all $n \\gg 0$. Set", "$g(n) = f(n) - \\sum\\nolimits_{i = 0}^r \\binom{n + 1}{i + 1} a_i$.", "Then $g(n) - g(n-1) = 0$ for all $n \\gg 0$. Hence $g$ is", "eventually constant, say equal to $a_{-1}$. We leave it", "to the reader to show that", "$a_{-1} + \\sum\\nolimits_{i = 0}^r \\binom{n + 1}{i + 1} a_i$", "has the required shape (see remark above the lemma)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 671, "type": "theorem", "label": "algebra-lemma-graded-module-fg", "categories": [ "algebra" ], "title": "algebra-lemma-graded-module-fg", "contents": [ "If $M$ is a finitely generated graded $S$-module,", "and if $S$ is finitely generated over $S_0$, then", "each $M_n$ is a finite $S_0$-module." ], "refs": [], "proofs": [ { "contents": [ "Suppose the generators of $M$ are $m_i$ and the generators", "of $S$ are $f_i$. By taking homogeneous components we may", "assume that the $m_i$ and the $f_i$ are homogeneous", "and we may assume $f_i \\in S_{+}$. In this case it is", "clear that each $M_n$ is generated over $S_0$", "by the ``monomials'' $\\prod f_i^{e_i} m_j$ whose", "degree is $n$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 672, "type": "theorem", "label": "algebra-lemma-quotient-smaller-d", "categories": [ "algebra" ], "title": "algebra-lemma-quotient-smaller-d", "contents": [ "Let $k$ be a field. Suppose that $I \\subset k[X_1, \\ldots, X_d]$", "is a nonzero graded ideal. Let $M = k[X_1, \\ldots, X_d]/I$.", "Then the numerical polynomial $n \\mapsto \\dim_k(M_n)$ (see", "Example \\ref{example-hilbert-function})", "has degree $ < d - 1$ (or is zero if $d = 1$)." ], "refs": [], "proofs": [ { "contents": [ "The numerical polynomial associated to the graded module", "$k[X_1, \\ldots, X_d]$ is $n \\mapsto \\binom{n - 1 + d}{d - 1}$.", "For any nonzero homogeneous $f \\in I$ of degree $e$", "and any degree $n >> e$ we have $I_n \\supset f \\cdot k[X_1, \\ldots, X_d]_{n-e}$", "and hence $\\dim_k(I_n) \\geq \\binom{n - e - 1 + d}{d - 1}$. Hence", "$\\dim_k(M_n) \\leq \\binom{n - 1 + d}{d - 1} - \\binom{n - e - 1 + d}{d - 1}$.", "We win because the last expression", "has degree $ < d - 1$ (or is zero if $d = 1$)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 673, "type": "theorem", "label": "algebra-lemma-differ-finite", "categories": [ "algebra" ], "title": "algebra-lemma-differ-finite", "contents": [ "Suppose that $M' \\subset M$ are finite $R$-modules", "with finite length quotient. Then there exists a", "constants $c_1, c_2$ such that for all $n \\geq c_2$ we have", "$$", "c_1 + \\chi_{I, M'}(n - c_2) \\leq \\chi_{I, M}(n) \\leq", "c_1 + \\chi_{I, M'}(n)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since $M/M'$ has finite length there is a $c_2 \\geq 0$ such that", "$I^{c_2}M \\subset M'$. Let $c_1 = \\text{length}_R(M/M')$.", "For $n \\geq c_2$ we have", "\\begin{eqnarray*}", "\\chi_{I, M}(n)", "& = &", "\\text{length}_R(M/I^{n + 1}M) \\\\", "& = &", "c_1 + \\text{length}_R(M'/I^{n + 1}M) \\\\", "& \\leq &", "c_1 + \\text{length}_R(M'/I^{n + 1}M') \\\\", "& = &", "c_1 + \\chi_{I, M'}(n)", "\\end{eqnarray*}", "On the other hand, since $I^{c_2}M \\subset M'$,", "we have $I^nM \\subset I^{n - c_2}M'$ for $n \\geq c_2$.", "Thus for $n \\geq c_2$ we get", "\\begin{eqnarray*}", "\\chi_{I, M}(n)", "& = &", "\\text{length}_R(M/I^{n + 1}M) \\\\", "& = &", "c_1 + \\text{length}_R(M'/I^{n + 1}M) \\\\", "& \\geq &", "c_1 + \\text{length}_R(M'/I^{n + 1 - c_2}M') \\\\", "& = &", "c_1 + \\chi_{I, M'}(n - c_2)", "\\end{eqnarray*}", "which finishes the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 674, "type": "theorem", "label": "algebra-lemma-hilbert-ses", "categories": [ "algebra" ], "title": "algebra-lemma-hilbert-ses", "contents": [ "Suppose that $0 \\to M' \\to M \\to M'' \\to 0$", "is a short exact sequence of finite $R$-modules.", "Then there exists a submodule $N \\subset M'$ with", "finite colength $l$ and $c \\geq 0$ such that", "$$", "\\chi_{I, M}(n) = \\chi_{I, M''}(n) + \\chi_{I, N}(n - c) + l", "$$", "and", "$$", "\\varphi_{I, M}(n) = \\varphi_{I, M''}(n) + \\varphi_{I, N}(n - c)", "$$", "for all $n \\geq c$." ], "refs": [], "proofs": [ { "contents": [ "Note that $M/I^nM \\to M''/I^nM''$ is surjective", "with kernel $M' / M' \\cap I^nM$. By the Artin-Rees", "Lemma \\ref{lemma-Artin-Rees} there exists a", "constant $c$ such that $M' \\cap I^nM =", "I^{n - c}(M' \\cap I^cM)$. Denote $N = M' \\cap I^cM$.", "Note that $I^c M' \\subset N \\subset M'$.", "Hence $\\text{length}_R(M' / M' \\cap I^nM)", "= \\text{length}_R(M'/N) + \\text{length}_R(N/I^{n - c}N)$ for $n \\geq c$.", "From the short exact sequence", "$$", "0 \\to M' / M' \\cap I^nM \\to M/I^nM \\to M''/I^nM'' \\to 0", "$$", "and additivity of lengths (Lemma \\ref{lemma-length-additive})", "we obtain the equality", "$$", "\\chi_{I, M}(n - 1)", "=", "\\chi_{I, M''}(n - 1)", "+", "\\chi_{I, N}(n - c - 1)", "+", "\\text{length}_R(M'/N)", "$$", "for $n \\geq c$. We have", "$\\varphi_{I, M}(n) = \\chi_{I, M}(n) - \\chi_{I, M}(n - 1)$", "and similarly for the modules $M''$ and $N$. Hence", "we get $\\varphi_{I, M}(n) = \\varphi_{I, M''}(n) + \\varphi_{I, N}(n-c)$ for", "$n \\geq c$." ], "refs": [ "algebra-lemma-Artin-Rees", "algebra-lemma-length-additive" ], "ref_ids": [ 625, 631 ] } ], "ref_ids": [] }, { "id": 675, "type": "theorem", "label": "algebra-lemma-hilbert-change-I", "categories": [ "algebra" ], "title": "algebra-lemma-hilbert-change-I", "contents": [ "Suppose that $I$, $I'$ are two ideals of definition", "for the Noetherian local ring $R$. Let $M$ be a", "finite $R$-module. There exists a constant $a$ such that", "$\\chi_{I, M}(n) \\leq \\chi_{I', M}(an)$ for $n \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "There exists an integer $c \\geq 1$ such that $(I')^c \\subset I$.", "Hence we get a surjection $M/(I')^{c(n + 1)}M \\to M/I^{n + 1}M$.", "Whence the result with $a = 2c - 1$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 676, "type": "theorem", "label": "algebra-lemma-d-independent", "categories": [ "algebra" ], "title": "algebra-lemma-d-independent", "contents": [ "Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module.", "\\begin{enumerate}", "\\item The degree of the numerical polynomial $\\varphi_{I, M}$ is independent", "of the ideal of definition $I$.", "\\item The degree of the numerical polynomial $\\chi_{I, M}$ is independent", "of the ideal of definition $I$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) follows immediately from Lemma \\ref{lemma-hilbert-change-I}.", "Part (1) follows from (2) because", "$\\varphi_{I, M}(n) = \\chi_{I, M}(n) - \\chi_{I, M}(n - 1)$", "for $n \\geq 1$." ], "refs": [ "algebra-lemma-hilbert-change-I" ], "ref_ids": [ 675 ] } ], "ref_ids": [] }, { "id": 677, "type": "theorem", "label": "algebra-lemma-differ-finite-chi", "categories": [ "algebra" ], "title": "algebra-lemma-differ-finite-chi", "contents": [ "Let $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal", "of definition. Let $M$ be a finite $R$-module", "which does not have finite length. If $M' \\subset M$ is a submodule", "with finite colength, then $\\chi_{I, M} - \\chi_{I, M'}$", "is a polynomial of degree $<$ degree of either polynomial." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-differ-finite} by elementary calculus." ], "refs": [ "algebra-lemma-differ-finite" ], "ref_ids": [ 673 ] } ], "ref_ids": [] }, { "id": 678, "type": "theorem", "label": "algebra-lemma-hilbert-ses-chi", "categories": [ "algebra" ], "title": "algebra-lemma-hilbert-ses-chi", "contents": [ "Let $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal of", "definition. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence", "of finite $R$-modules. Then", "\\begin{enumerate}", "\\item if $M'$ does not have finite length, then", "$\\chi_{I, M} - \\chi_{I, M''} - \\chi_{I, M'}$", "is a numerical polynomial of degree $<$ the degree of", "$\\chi_{I, M'}$,", "\\item $\\max\\{ \\deg(\\chi_{I, M'}), \\deg(\\chi_{I, M''}) \\} = \\deg(\\chi_{I, M})$,", "and", "\\item $\\max\\{d(M'), d(M'')\\} = d(M)$,", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We first prove (1). Let $N \\subset M'$ be as in Lemma \\ref{lemma-hilbert-ses}.", "By Lemma \\ref{lemma-differ-finite-chi} the numerical polynomial", "$\\chi_{I, M'} - \\chi_{I, N}$ has degree $<$ the common degree of", "$\\chi_{I, M'}$ and $\\chi_{I, N}$. By Lemma \\ref{lemma-hilbert-ses}", "the difference", "$$", "\\chi_{I, M}(n) - \\chi_{I, M''}(n) - \\chi_{I, N}(n - c)", "$$", "is constant for $n \\gg 0$. By elementary calculus the difference", "$\\chi_{I, N}(n) - \\chi_{I, N}(n - c)$ has degree $<$ the degree of", "$\\chi_{I, N}$ which is bigger than zero (see above). Putting everything", "together we obtain (1).", "\\medskip\\noindent", "Note that the leading coefficients of $\\chi_{I, M'}$ and $\\chi_{I, M''}$ are", "nonnegative. Thus the degree of $\\chi_{I, M'} + \\chi_{I, M''}$ is equal", "to the maximum of the degrees. Thus if $M'$ does not have finite", "length, then (2) follows from (1). If $M'$ does have finite length, then", "$I^nM \\to I^nM''$ is an isomorphism for all $n \\gg 0$ by Artin-Rees", "(Lemma \\ref{lemma-Artin-Rees}). Thus $M/I^nM \\to M''/I^nM''$ is a", "surjection with kernel $M'$ for $n \\gg 0$ and we see that", "$\\chi_{I, M}(n) - \\chi_{I, M''}(n) = \\text{length}(M')$", "for all $n \\gg 0$. Thus (2) holds in this case also.", "\\medskip\\noindent", "Proof of (3). This follows from (2) except if one of $M$, $M'$, or $M''$", "is zero. We omit the proof in these special cases." ], "refs": [ "algebra-lemma-hilbert-ses", "algebra-lemma-differ-finite-chi", "algebra-lemma-hilbert-ses", "algebra-lemma-Artin-Rees" ], "ref_ids": [ 674, 677, 674, 625 ] } ], "ref_ids": [] }, { "id": 679, "type": "theorem", "label": "algebra-lemma-dimension-height", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-height", "contents": [ "The Krull dimension of $R$ is the supremum of the", "heights of its (maximal) primes." ], "refs": [], "proofs": [ { "contents": [ "This is so because we can always add a maximal ideal at the end of a chain", "of prime ideals." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 680, "type": "theorem", "label": "algebra-lemma-Noetherian-dimension-0", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-dimension-0", "contents": [ "A Noetherian ring of dimension $0$ is Artinian.", "Conversely, any Artinian ring is Noetherian of dimension zero." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-Noetherian-topology} the space $\\Spec(R)$", "is Noetherian. By Topology, Lemma \\ref{topology-lemma-Noetherian} we see", "that $\\Spec(R)$ has finitely many irreducible", "components, say $\\Spec(R) = Z_1 \\cup \\ldots Z_r$.", "According to Lemma \\ref{lemma-irreducible}, each $Z_i = V(\\mathfrak p_i)$", "with $\\mathfrak p_i$ a minimal ideal. Since the dimension is $0$", "these $\\mathfrak p_i$ are also maximal. Thus $\\Spec(R)$", "is the discrete topological space with elements $\\mathfrak p_i$.", "All elements $f$ of the Jacobson radical $I = \\cap \\mathfrak p_i$", "are nilpotent since otherwise $R_f$ would not be the zero ring", "and we would have another prime. Since $I$ is finitely generated", "we conclude that $I$ is nilpotent, Lemma \\ref{lemma-Noetherian-power}.", "By Lemma \\ref{lemma-product-local} $R$ is the product of its", "local rings. By Lemma \\ref{lemma-length-finite} each of these", "has finite length over $R$. Hence we conclude that $R$", "is Artinian by Lemma \\ref{lemma-artinian-finite-length}.", "\\medskip\\noindent", "If $R$ is Artinian then by Lemma \\ref{lemma-artinian-finite-length}", "it is Noetherian. All of its primes are maximal by a combination", "of Lemmas \\ref{lemma-artinian-finite-nr-max},", "\\ref{lemma-artinian-radical-nilpotent} and \\ref{lemma-product-local}." ], "refs": [ "algebra-lemma-Noetherian-topology", "topology-lemma-Noetherian", "algebra-lemma-irreducible", "algebra-lemma-Noetherian-power", "algebra-lemma-product-local", "algebra-lemma-length-finite", "algebra-lemma-artinian-finite-length", "algebra-lemma-artinian-finite-length", "algebra-lemma-artinian-finite-nr-max", "algebra-lemma-artinian-radical-nilpotent", "algebra-lemma-product-local" ], "ref_ids": [ 452, 8220, 422, 460, 645, 636, 646, 646, 643, 644, 645 ] } ], "ref_ids": [] }, { "id": 681, "type": "theorem", "label": "algebra-lemma-dimension-0-d-0", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-0-d-0", "contents": [ "Let $R$ be a Noetherian local ring.", "Then $\\dim(R) = 0 \\Leftrightarrow d(R) = 0$." ], "refs": [], "proofs": [ { "contents": [ "This is because $d(R) = 0$ if and only if $R$ has finite", "length as an $R$-module. See Lemma \\ref{lemma-artinian-finite-length}." ], "refs": [ "algebra-lemma-artinian-finite-length" ], "ref_ids": [ 646 ] } ], "ref_ids": [] }, { "id": 682, "type": "theorem", "label": "algebra-lemma-height-1", "categories": [ "algebra" ], "title": "algebra-lemma-height-1", "contents": [ "Let $R$ be a local Noetherian ring.", "The following are equivalent:", "\\begin{enumerate}", "\\item", "\\label{item-dim-1}", "$\\dim(R) = 1$,", "\\item", "\\label{item-d-1}", "$d(R) = 1$,", "\\item", "\\label{item-Vx}", "there exists an $x \\in \\mathfrak m$, $x$ not nilpotent", "such that $V(x) = \\{\\mathfrak m\\}$,", "\\item", "\\label{item-x}", "there exists an $x \\in \\mathfrak m$, $x$ not nilpotent", "such that $\\mathfrak m = \\sqrt{(x)}$, and", "\\item", "\\label{item-ideal-1}", "there exists an ideal of definition generated by $1$ element,", "and no ideal of definition is generated by $0$ elements.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First, assume that $\\dim(R) = 1$.", "Let $\\mathfrak p_i$ be the minimal primes of $R$.", "Because the dimension is $1$ the only other prime of $R$", "is $\\mathfrak m$.", "According to Lemma \\ref{lemma-Noetherian-irreducible-components}", "there are finitely many. Hence we can find $x \\in \\mathfrak m$,", "$x \\not \\in \\mathfrak p_i$, see Lemma \\ref{lemma-silly}.", "Thus the only prime containing $x$ is $\\mathfrak m$ and", "hence (\\ref{item-Vx}).", "\\medskip\\noindent", "If (\\ref{item-Vx}) then $\\mathfrak m = \\sqrt{(x)}$ by", "Lemma \\ref{lemma-Zariski-topology}, and hence (\\ref{item-x}).", "The converse is clear as well.", "The equivalence of (\\ref{item-x}) and (\\ref{item-ideal-1}) follows", "from directly the definitions.", "\\medskip\\noindent", "Assume (\\ref{item-ideal-1}).", "Let $I = (x)$ be an ideal of definition.", "Note that $I^n/I^{n + 1}$ is a quotient of $R/I$ via multiplication", "by $x^n$ and hence $\\text{length}_R(I^n/I^{n + 1})$ is bounded.", "Thus $d(R) = 0$ or $d(R) = 1$, but $d(R) = 0$ is excluded", "by the assumption that $0$ is not an ideal of definition.", "\\medskip\\noindent", "Assume (\\ref{item-d-1}). To get a contradiction, assume there", "exist primes $\\mathfrak p \\subset \\mathfrak q \\subset \\mathfrak m$,", "with both inclusions strict. Pick some ideal of definition $I \\subset R$.", "We will repeatedly use", "Lemma \\ref{lemma-hilbert-ses-chi}. First of all", "it implies, via the exact sequence", "$0 \\to \\mathfrak p \\to R \\to R/\\mathfrak p \\to 0$,", "that $d(R/\\mathfrak p) \\leq 1$. But it clearly cannot", "be zero. Pick $x\\in \\mathfrak q$, $x\\not \\in \\mathfrak p$.", "Consider the short exact sequence", "$$", "0 \\to R/\\mathfrak p \\to R/\\mathfrak p \\to R/(xR + \\mathfrak p) \\to 0.", "$$", "This implies that $\\chi_{I, R/\\mathfrak p} - \\chi_{I, R/\\mathfrak p}", "- \\chi_{I, R/(xR + \\mathfrak p)} = - \\chi_{I, R/(xR + \\mathfrak p)}$", "has degree $ < 1$. In other words, $d(R/(xR + \\mathfrak p) = 0$,", "and hence $\\dim(R/(xR + \\mathfrak p)) = 0$, by", "Lemma \\ref{lemma-dimension-0-d-0}. But $R/(xR + \\mathfrak p)$", "has the distinct primes $\\mathfrak q/(xR + \\mathfrak p)$ and", "$\\mathfrak m/(xR + \\mathfrak p)$ which gives the desired contradiction." ], "refs": [ "algebra-lemma-Noetherian-irreducible-components", "algebra-lemma-silly", "algebra-lemma-Zariski-topology", "algebra-lemma-hilbert-ses-chi", "algebra-lemma-dimension-0-d-0" ], "ref_ids": [ 453, 378, 389, 678, 681 ] } ], "ref_ids": [] }, { "id": 683, "type": "theorem", "label": "algebra-lemma-minimal-over-1", "categories": [ "algebra" ], "title": "algebra-lemma-minimal-over-1", "contents": [ "Let $R$ be a Noetherian ring. Let $x \\in R$.", "\\begin{enumerate}", "\\item If $\\mathfrak p$ is minimal over $(x)$", "then the height of $\\mathfrak p$ is $0$ or $1$.", "\\item If $\\mathfrak p, \\mathfrak q \\in \\Spec(R)$ and $\\mathfrak q$", "is minimal over $(\\mathfrak p, x)$, then there is no prime strictly", "between $\\mathfrak p$ and $\\mathfrak q$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). If $\\mathfrak p$ is minimal over $x$, then the only", "prime ideal of $R_\\mathfrak p$ containing $x$ is the maximal ideal", "$\\mathfrak p R_\\mathfrak p$. This is true because the primes of", "$R_\\mathfrak p$ correspond $1$-to-$1$ with the primes of $R$ contained", "in $\\mathfrak p$, see Lemma \\ref{lemma-spec-localization}.", "Hence Lemma \\ref{lemma-height-1} shows $\\dim(R_\\mathfrak p) = 1$", "if $x$ is not nilpotent in $R_\\mathfrak p$. Of course, if", "$x$ is nilpotent in $R_\\mathfrak p$ the argument gives that", "$\\mathfrak pR_\\mathfrak p$ is the only prime ideal and we see", "that the height is $0$.", "\\medskip\\noindent", "Proof of (2). By part (1) we see that $\\mathfrak q/\\mathfrak p$", "is a prime of height $1$ or $0$ in $R/\\mathfrak p$. This immediately", "implies there cannot be a prime strictly between $\\mathfrak p$", "and $\\mathfrak q$." ], "refs": [ "algebra-lemma-spec-localization", "algebra-lemma-height-1" ], "ref_ids": [ 391, 682 ] } ], "ref_ids": [] }, { "id": 684, "type": "theorem", "label": "algebra-lemma-minimal-over-r", "categories": [ "algebra" ], "title": "algebra-lemma-minimal-over-r", "contents": [ "Let $R$ be a Noetherian ring. Let $f_1, \\ldots, f_r \\in R$.", "\\begin{enumerate}", "\\item If $\\mathfrak p$ is minimal over $(f_1, \\ldots, f_r)$", "then the height of $\\mathfrak p$ is $\\leq r$.", "\\item If $\\mathfrak p, \\mathfrak q \\in \\Spec(R)$ and", "$\\mathfrak q$ is minimal over $(\\mathfrak p, f_1, \\ldots, f_r)$,", "then every chain of primes between $\\mathfrak p$ and $\\mathfrak q$", "has length at most $r$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). If $\\mathfrak p$ is minimal over $f_1, \\ldots, f_r$,", "then the only prime ideal of $R_\\mathfrak p$ containing $f_1, \\ldots, f_r$", "is the maximal ideal $\\mathfrak p R_\\mathfrak p$. This is true because", "the primes of $R_\\mathfrak p$ correspond $1$-to-$1$ with the primes of", "$R$ contained in $\\mathfrak p$, see Lemma \\ref{lemma-spec-localization}.", "Hence Proposition \\ref{proposition-dimension} shows", "$\\dim(R_\\mathfrak p) \\leq r$.", "\\medskip\\noindent", "Proof of (2). By part (1) we see that $\\mathfrak q/\\mathfrak p$", "is a prime of height $\\leq r$. This immediately", "implies the statement about chains of primes between $\\mathfrak p$", "and $\\mathfrak q$." ], "refs": [ "algebra-lemma-spec-localization", "algebra-proposition-dimension" ], "ref_ids": [ 391, 1411 ] } ], "ref_ids": [] }, { "id": 685, "type": "theorem", "label": "algebra-lemma-one-equation", "categories": [ "algebra" ], "title": "algebra-lemma-one-equation", "contents": [ "Suppose that $R$ is a Noetherian local ring and $x\\in \\mathfrak m$ an", "element of its maximal ideal. Then $\\dim R \\leq \\dim R/xR + 1$.", "If $x$ is not contained in any of the minimal primes of $R$", "then equality holds. (For example if $x$ is a nonzerodivisor.)" ], "refs": [], "proofs": [ { "contents": [ "If $x_1, \\ldots, x_{\\dim R/xR} \\in R$ map to elements of $R/xR$ which", "generate an ideal of definition for $R/xR$, then $x, x_1, \\ldots,", "x_{\\dim R/xR}$ generate an ideal of definition for $R$. Hence", "the inequality by Proposition \\ref{proposition-dimension}.", "On the other hand, if $x$ is not contained in any minimal", "prime of $R$, then the chains of primes in $R/xR$ all give", "rise to chains in $R$ which are at least one step away", "from being maximal." ], "refs": [ "algebra-proposition-dimension" ], "ref_ids": [ 1411 ] } ], "ref_ids": [] }, { "id": 686, "type": "theorem", "label": "algebra-lemma-elements-generate-ideal-definition", "categories": [ "algebra" ], "title": "algebra-lemma-elements-generate-ideal-definition", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring.", "Suppose $x_1, \\ldots, x_d \\in \\mathfrak m$ generate an", "ideal of definition and $d = \\dim(R)$. Then", "$\\dim(R/(x_1, \\ldots, x_i)) = d - i$ for all $i = 1, \\ldots, d$." ], "refs": [], "proofs": [ { "contents": [ "Follows either from the proof of Proposition \\ref{proposition-dimension},", "or by using induction on $d$ and Lemma \\ref{lemma-one-equation}." ], "refs": [ "algebra-proposition-dimension" ], "ref_ids": [ 1411 ] } ], "ref_ids": [] }, { "id": 687, "type": "theorem", "label": "algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens", "contents": [ "Let $R$ be a Noetherian local domain of dimension $\\geq 2$.", "A nonempty open subset $U \\subset \\Spec(R)$ is", "infinite." ], "refs": [], "proofs": [ { "contents": [ "To get a contradiction, assume that $U \\subset \\Spec(R)$ is finite.", "In this case $(0) \\in U$ and $\\{(0)\\}$ is an open subset of $U$ (because", "the complement of $\\{(0)\\}$ is the union of the closures of the other points).", "Thus we may assume $U = \\{(0)\\}$.", "Let $\\mathfrak m \\subset R$ be the maximal ideal.", "We can find an $x \\in \\mathfrak m$, $x \\not = 0$ such that", "$V(x) \\cup U = \\Spec(R)$. In other words we see that", "$D(x) = \\{(0)\\}$. In particular we see", "that $\\dim(R/xR) = \\dim(R) - 1 \\geq 1$, see Lemma \\ref{lemma-one-equation}.", "Let $\\overline{y}_2, \\ldots, \\overline{y}_{\\dim(R)} \\in R/xR$ generate", "an ideal of definition of $R/xR$, see Proposition \\ref{proposition-dimension}.", "Choose lifts $y_2, \\ldots, y_{\\dim(R)} \\in R$, so that", "$x, y_2, \\ldots, y_{\\dim(R)}$ generate an ideal of definition in $R$.", "This implies that $\\dim(R/(y_2)) = \\dim(R) - 1$ and", "$\\dim(R/(y_2, x)) = \\dim(R) - 2$, see", "Lemma \\ref{lemma-elements-generate-ideal-definition}.", "Hence there exists a prime", "$\\mathfrak p$ containing $y_2$ but not $x$. This contradicts", "the fact that $D(x) = \\{(0)\\}$." ], "refs": [ "algebra-proposition-dimension", "algebra-lemma-elements-generate-ideal-definition" ], "ref_ids": [ 1411, 686 ] } ], "ref_ids": [] }, { "id": 688, "type": "theorem", "label": "algebra-lemma-Noetherian-finite-nr-primes", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-finite-nr-primes", "contents": [ "A Noetherian ring with finitely many primes has dimension $\\leq 1$." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a Noetherian ring with finitely many primes.", "If $R$ is a local domain, then the lemma follows from", "Lemma \\ref{lemma-Noetherian-local-domain-dim-2-infinite-opens}.", "If $R$ is a domain, then $R_\\mathfrak m$ has dimension $\\leq 1$", "for all maximal ideals $\\mathfrak m$ by the local case.", "Hence $\\dim(R) \\leq 1$ by Lemma \\ref{lemma-dimension-height}.", "If $R$ is general, then $\\dim(R/\\mathfrak q) \\leq 1$", "for every minimal prime $\\mathfrak q$ of $R$.", "Since every prime contains a minimal prime", "(Lemma \\ref{lemma-Zariski-topology}), this implies $\\dim(R) \\leq 1$." ], "refs": [ "algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens", "algebra-lemma-dimension-height", "algebra-lemma-Zariski-topology" ], "ref_ids": [ 687, 679, 389 ] } ], "ref_ids": [] }, { "id": 689, "type": "theorem", "label": "algebra-lemma-finite-type-algebra-finite-nr-primes", "categories": [ "algebra" ], "title": "algebra-lemma-finite-type-algebra-finite-nr-primes", "contents": [ "Let $S$ be a nonzero finite type algebra over a field $k$.", "Then $\\dim(S) = 0$ if and only if $S$ has", "finitely many primes." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\Spec(S)$ is sober, Noetherian, and Jacobson, see", "Lemmas \\ref{lemma-spec-spectral}, \\ref{lemma-Noetherian-topology},", "\\ref{lemma-finite-type-field-Jacobson}, and \\ref{lemma-jacobson}.", "If it has dimension $0$, then every point defines an", "irreducible component and there are only a finite number", "of irreducible components (Topology, Lemma \\ref{topology-lemma-Noetherian}).", "Conversely, if $\\Spec(S)$ is finite, then it is discrete", "by Topology, Lemma \\ref{topology-lemma-finite-jacobson}", "and hence the dimension is $0$." ], "refs": [ "algebra-lemma-spec-spectral", "algebra-lemma-Noetherian-topology", "algebra-lemma-finite-type-field-Jacobson", "algebra-lemma-jacobson", "topology-lemma-Noetherian", "topology-lemma-finite-jacobson" ], "ref_ids": [ 423, 452, 467, 469, 8220, 8280 ] } ], "ref_ids": [] }, { "id": 690, "type": "theorem", "label": "algebra-lemma-noetherian-dim-1-Jacobson", "categories": [ "algebra" ], "title": "algebra-lemma-noetherian-dim-1-Jacobson", "contents": [ "Noetherian Jacobson rings.", "\\begin{enumerate}", "\\item Any Noetherian domain $R$ of dimension $1$", "with infinitely many primes is Jacobson.", "\\item Any Noetherian ring such that every prime", "$\\mathfrak p$ is either maximal or contained in", "infinitely many prime ideals is Jacobson.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is a reformulation of Lemma \\ref{lemma-pid-jacobson}.", "\\medskip\\noindent", "Let $R$ be a Noetherian ring such that", "every non-maximal prime $\\mathfrak p$ is contained", "in infinitely many prime ideals.", "Assume $\\Spec(R)$ is not Jacobson to get", "a contradiction.", "By Lemmas \\ref{lemma-irreducible}", "and \\ref{lemma-Noetherian-topology}", "we see that $\\Spec(R)$ is a sober, Noetherian topological space.", "By Topology, Lemma \\ref{topology-lemma-non-jacobson-Noetherian-characterize}", "we see that there exists a non-maximal ideal $\\mathfrak p \\subset R$", "such that $\\{\\mathfrak p\\}$ is a locally closed subset of", "$\\Spec(R)$. In other words, $\\mathfrak p$ is not maximal", "and $\\{\\mathfrak p\\}$ is an open subset of $V(\\mathfrak p)$.", "Consider a prime $\\mathfrak q \\subset R$ with", "$\\mathfrak p \\subset \\mathfrak q$. Recall that the topology on the spectrum of", "$(R/\\mathfrak p)_{\\mathfrak q} = R_{\\mathfrak q}/\\mathfrak pR_{\\mathfrak q}$", "is induced from that of $\\Spec(R)$, see Lemmas", "\\ref{lemma-spec-localization} and \\ref{lemma-spec-closed}.", "Hence we see that $\\{(0)\\}$ is a locally closed subset of", "$\\Spec((R/\\mathfrak p)_{\\mathfrak q})$. By", "Lemma \\ref{lemma-Noetherian-local-domain-dim-2-infinite-opens}", "we conclude that $\\dim((R/\\mathfrak p)_{\\mathfrak q}) = 1$.", "Since this holds for every $\\mathfrak q \\supset \\mathfrak p$", "we conclude that $\\dim(R/\\mathfrak p) = 1$. At this point we use", "the assumption that $\\mathfrak p$ is contained in infinitely many", "primes to see that $\\Spec(R/\\mathfrak p)$ is infinite.", "Hence by part (1) of the lemma we see that", "$V(\\mathfrak p) \\cong \\Spec(R/\\mathfrak p)$", "is the closure of its closed points.", "This is the desired contradiction since it means that", "$\\{\\mathfrak p\\} \\subset V(\\mathfrak p)$ cannot be open." ], "refs": [ "algebra-lemma-pid-jacobson", "algebra-lemma-irreducible", "algebra-lemma-Noetherian-topology", "topology-lemma-non-jacobson-Noetherian-characterize", "algebra-lemma-spec-localization", "algebra-lemma-spec-closed", "algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens" ], "ref_ids": [ 471, 422, 452, 8277, 391, 393, 687 ] } ], "ref_ids": [] }, { "id": 691, "type": "theorem", "label": "algebra-lemma-filter-Noetherian-module", "categories": [ "algebra" ], "title": "algebra-lemma-filter-Noetherian-module", "contents": [ "Let $R$ be a Noetherian ring, and let $M$ be a finite $R$-module.", "There exists a filtration by $R$-submodules", "$$", "0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M", "$$", "such that each quotient $M_i/M_{i-1}$ is isomorphic", "to $R/\\mathfrak p_i$ for some prime ideal $\\mathfrak p_i$", "of $R$." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "By Lemma \\ref{lemma-trivial-filter-finite-module}", "it suffices to do the case $M = R/I$ for some ideal $I$.", "Consider the set $S$ of ideals $J$ such that the lemma", "does not hold for the module $R/J$, and order it by", "inclusion. To arrive at a", "contradiction, assume that $S$ is not empty. Because", "$R$ is Noetherian, $S$ has a maximal element $J$.", "By definition of $S$, the ideal $J$ cannot be prime.", "Pick $a, b\\in R$ such that $ab \\in J$, but neither", "$a \\in J$ nor $b\\in J$. Consider the filtration", "$0 \\subset aR/(J \\cap aR) \\subset R/J$.", "Note that $aR/(J \\cap aR)$ is a quotient of $R/(J + bR)$", "and the second quotient equals $R/(aR + J)$. Hence by", "maximality of $J$, each of these has a filtration as", "above and hence so does $R/J$. Contradiction." ], "refs": [ "algebra-lemma-trivial-filter-finite-module" ], "ref_ids": [ 331 ] } ], "ref_ids": [] }, { "id": 692, "type": "theorem", "label": "algebra-lemma-filter-primes-in-support", "categories": [ "algebra" ], "title": "algebra-lemma-filter-primes-in-support", "contents": [ "Let $R$, $M$, $M_i$, $\\mathfrak p_i$ as in", "Lemma \\ref{lemma-filter-Noetherian-module}.", "Then $\\text{Supp}(M) = \\bigcup V(\\mathfrak p_i)$", "and in particular $\\mathfrak p_i \\in \\text{Supp}(M)$." ], "refs": [ "algebra-lemma-filter-Noetherian-module" ], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-support-closed} and", "\\ref{lemma-support-quotient}." ], "refs": [ "algebra-lemma-support-closed", "algebra-lemma-support-quotient" ], "ref_ids": [ 543, 547 ] } ], "ref_ids": [ 691 ] }, { "id": 693, "type": "theorem", "label": "algebra-lemma-support-point", "categories": [ "algebra" ], "title": "algebra-lemma-support-point", "contents": [ "Suppose that $R$ is a Noetherian local ring with", "maximal ideal $\\mathfrak m$. Let $M$ be a nonzero finite", "$R$-module. Then $\\text{Supp}(M) = \\{ \\mathfrak m\\}$", "if and only if $M$ has finite length over $R$." ], "refs": [], "proofs": [ { "contents": [ "Assume that $\\text{Supp}(M) = \\{ \\mathfrak m\\}$.", "It suffices to show that all the primes $\\mathfrak p_i$", "in the filtration of Lemma \\ref{lemma-filter-Noetherian-module}", "are the maximal ideal. This is clear by", "Lemma \\ref{lemma-filter-primes-in-support}.", "\\medskip\\noindent", "Suppose that $M$ has finite length over $R$.", "Then $\\mathfrak m^n M = 0$ by Lemma \\ref{lemma-length-infinite}.", "Since some element of $\\mathfrak m$ maps to a unit", "in $R_{\\mathfrak p}$ for any prime", "$\\mathfrak p \\not = \\mathfrak m$ in $R$ we see $M_{\\mathfrak p} = 0$." ], "refs": [ "algebra-lemma-filter-Noetherian-module", "algebra-lemma-filter-primes-in-support", "algebra-lemma-length-infinite" ], "ref_ids": [ 691, 692, 632 ] } ], "ref_ids": [] }, { "id": 694, "type": "theorem", "label": "algebra-lemma-Noetherian-power-ideal-kills-module", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-power-ideal-kills-module", "contents": [ "Let $R$ be a Noetherian ring.", "Let $I \\subset R$ be an ideal.", "Let $M$ be a finite $R$-module.", "Then $I^nM = 0$ for some $n \\geq 0$ if and only if", "$\\text{Supp}(M) \\subset V(I)$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $I^nM = 0$ for some $n \\geq 0$ implies", "$\\text{Supp}(M) \\subset V(I)$. Suppose that $\\text{Supp}(M) \\subset V(I)$.", "Choose a filtration $0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M$", "as in Lemma \\ref{lemma-filter-Noetherian-module}. Each of the primes", "$\\mathfrak p_i$ is contained in $V(I)$ by", "Lemma \\ref{lemma-filter-primes-in-support}.", "Hence $I \\subset \\mathfrak p_i$ and $I$ annihilates $M_i/M_{i - 1}$.", "Hence $I^n$ annihilates $M$." ], "refs": [ "algebra-lemma-filter-Noetherian-module", "algebra-lemma-filter-primes-in-support" ], "ref_ids": [ 691, 692 ] } ], "ref_ids": [] }, { "id": 695, "type": "theorem", "label": "algebra-lemma-filter-minimal-primes-in-support", "categories": [ "algebra" ], "title": "algebra-lemma-filter-minimal-primes-in-support", "contents": [ "Let $R$, $M$, $M_i$, $\\mathfrak p_i$ as in", "Lemma \\ref{lemma-filter-Noetherian-module}.", "The minimal elements of the set $\\{\\mathfrak p_i\\}$", "are the minimal elements of $\\text{Supp}(M)$.", "The number of times a minimal prime $\\mathfrak p$", "occurs is", "$$", "\\#\\{i \\mid \\mathfrak p_i = \\mathfrak p\\}", "=", "\\text{length}_{R_\\mathfrak p} M_{\\mathfrak p}.", "$$" ], "refs": [ "algebra-lemma-filter-Noetherian-module" ], "proofs": [ { "contents": [ "The first statement follows because", "$\\text{Supp}(M) = \\bigcup V(\\mathfrak p_i)$, see", "Lemma \\ref{lemma-filter-primes-in-support}.", "Let $\\mathfrak p \\in \\text{Supp}(M)$ be minimal.", "The support of $M_{\\mathfrak p}$ is the set", "consisting of the maximal ideal $\\mathfrak p R_{\\mathfrak p}$.", "Hence by Lemma \\ref{lemma-support-point} the length", "of $M_{\\mathfrak p}$ is finite and $> 0$. Next we", "note that $M_{\\mathfrak p}$ has a filtration with subquotients", "$", "(R/\\mathfrak p_i)_{\\mathfrak p}", "=", "R_{\\mathfrak p}/{\\mathfrak p_i}R_{\\mathfrak p}", "$.", "These are zero if $\\mathfrak p_i \\not \\subset \\mathfrak p$", "and equal to $\\kappa(\\mathfrak p)$ if $\\mathfrak p_i \\subset", "\\mathfrak p$ because by minimality of $\\mathfrak p$", "we have $\\mathfrak p_i = \\mathfrak p$ in this case.", "The result follows since $\\kappa(\\mathfrak p)$ has length $1$." ], "refs": [ "algebra-lemma-filter-primes-in-support", "algebra-lemma-support-point" ], "ref_ids": [ 692, 693 ] } ], "ref_ids": [ 691 ] }, { "id": 696, "type": "theorem", "label": "algebra-lemma-support-dimension-d", "categories": [ "algebra" ], "title": "algebra-lemma-support-dimension-d", "contents": [ "Let $R$ be a Noetherian local ring.", "Let $M$ be a finite $R$-module.", "Then $d(M) = \\dim(\\text{Supp}(M))$." ], "refs": [], "proofs": [ { "contents": [ "Let $M_i, \\mathfrak p_i$ be as in Lemma \\ref{lemma-filter-Noetherian-module}.", "By Lemma \\ref{lemma-hilbert-ses-chi} we obtain the equality", "$d(M) = \\max \\{ d(R/\\mathfrak p_i) \\}$. By", "Proposition \\ref{proposition-dimension} we have", "$d(R/\\mathfrak p_i) = \\dim(R/\\mathfrak p_i)$.", "Trivially $\\dim(R/\\mathfrak p_i) = \\dim V(\\mathfrak p_i)$.", "Since all minimal primes of $\\text{Supp}(M)$ occur among", "the $\\mathfrak p_i$ (Lemma \\ref{lemma-filter-minimal-primes-in-support}) we win." ], "refs": [ "algebra-lemma-filter-Noetherian-module", "algebra-lemma-hilbert-ses-chi", "algebra-proposition-dimension", "algebra-lemma-filter-minimal-primes-in-support" ], "ref_ids": [ 691, 678, 1411, 695 ] } ], "ref_ids": [] }, { "id": 697, "type": "theorem", "label": "algebra-lemma-ses-dimension", "categories": [ "algebra" ], "title": "algebra-lemma-ses-dimension", "contents": [ "Let $R$ be a Noetherian ring. Let $0 \\to M' \\to M \\to M'' \\to 0$", "be a short exact sequence of finite $R$-modules. Then", "$\\max\\{\\dim(\\text{Supp}(M')), \\dim(\\text{Supp}(M''))\\} =", "\\dim(\\text{Supp}(M))$." ], "refs": [], "proofs": [ { "contents": [ "If $R$ is local, this follows immediately from", "Lemmas \\ref{lemma-support-dimension-d} and \\ref{lemma-hilbert-ses-chi}.", "A more elementary argument, which works also if $R$ is not local,", "is to use that $\\text{Supp}(M')$, $\\text{Supp}(M'')$, and", "$\\text{Supp}(M)$ are closed (Lemma \\ref{lemma-support-closed})", "and that $\\text{Supp}(M) = \\text{Supp}(M') \\cup \\text{Supp}(M'')$", "(Lemma \\ref{lemma-support-quotient})." ], "refs": [ "algebra-lemma-support-dimension-d", "algebra-lemma-hilbert-ses-chi", "algebra-lemma-support-closed", "algebra-lemma-support-quotient" ], "ref_ids": [ 696, 678, 543, 547 ] } ], "ref_ids": [] }, { "id": 698, "type": "theorem", "label": "algebra-lemma-ass-support", "categories": [ "algebra" ], "title": "algebra-lemma-ass-support", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Then $\\text{Ass}(M) \\subset \\text{Supp}(M)$." ], "refs": [], "proofs": [ { "contents": [ "If $m \\in M$ has annihilator $\\mathfrak p$, then in particular", "no element of $R \\setminus \\mathfrak p$ annihilates $m$.", "Hence $m$ is a nonzero element of $M_{\\mathfrak p}$, i.e.,", "$\\mathfrak p \\in \\text{Supp}(M)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 699, "type": "theorem", "label": "algebra-lemma-ass", "categories": [ "algebra" ], "title": "algebra-lemma-ass", "contents": [ "Let $R$ be a ring. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence", "of $R$-modules. Then $\\text{Ass}(M') \\subset \\text{Ass}(M)$ and", "$\\text{Ass}(M) \\subset \\text{Ass}(M') \\cup \\text{Ass}(M'')$.", "Also $\\text{Ass}(M' \\oplus M'') = \\text{Ass}(M') \\cup \\text{Ass}(M'')$." ], "refs": [], "proofs": [ { "contents": [ "If $m' \\in M'$, then the annihilator of $m'$ viewed as an element of $M'$", "is the same as the annihilator of $m'$ viewed as an element of $M$. Hence", "the inclusion $\\text{Ass}(M') \\subset \\text{Ass}(M)$. Let $m \\in M$", "be an element whose annihilator is a prime ideal $\\mathfrak p$. If there", "exists a $g \\in R$, $g \\not \\in \\mathfrak p$ such that $m' = gm \\in M'$", "then the annihilator of $m'$ is $\\mathfrak p$. If there does not", "exist a $g \\in R$, $g \\not \\in \\mathfrak p$ such that $gm \\in M'$,", "then the annilator of the image $m'' \\in M''$ of $m$ is $\\mathfrak p$.", "This proves the inclusion", "$\\text{Ass}(M) \\subset \\text{Ass}(M') \\cup \\text{Ass}(M'')$.", "We omit the proof of the final statement." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 700, "type": "theorem", "label": "algebra-lemma-ass-filter", "categories": [ "algebra" ], "title": "algebra-lemma-ass-filter", "contents": [ "Let $R$ be a ring, and $M$ an $R$-module.", "Suppose there exists a filtration by $R$-submodules", "$$", "0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M", "$$", "such that each quotient $M_i/M_{i-1}$ is isomorphic to $R/\\mathfrak p_i$", "for some prime ideal $\\mathfrak p_i$ of $R$.", "Then $\\text{Ass}(M) \\subset \\{\\mathfrak p_1, \\ldots, \\mathfrak p_n\\}$." ], "refs": [], "proofs": [ { "contents": [ "By induction on the length $n$ of the filtration $\\{ M_i \\}$.", "Pick $m \\in M$ whose annihilator is a prime $\\mathfrak p$.", "If $m \\in M_{n-1}$ we are done by induction. If not,", "then $m$ maps to a nonzero element of $M/M_{n-1} \\cong", "R/\\mathfrak p_n$. Hence we have $\\mathfrak p \\subset \\mathfrak p_n$.", "If equality does not hold, then we can find $f \\in \\mathfrak p_n$,", "$f \\not\\in \\mathfrak p$. In this case the annihilator of $fm$ is still", "$\\mathfrak p$ and $fm \\in M_{n-1}$. Thus we win by induction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 701, "type": "theorem", "label": "algebra-lemma-finite-ass", "categories": [ "algebra" ], "title": "algebra-lemma-finite-ass", "contents": [ "Let $R$ be a Noetherian ring.", "Let $M$ be a finite $R$-module.", "Then $\\text{Ass}(M)$ is finite." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-ass-filter} and", "Lemma \\ref{lemma-filter-Noetherian-module}." ], "refs": [ "algebra-lemma-ass-filter", "algebra-lemma-filter-Noetherian-module" ], "ref_ids": [ 700, 691 ] } ], "ref_ids": [] }, { "id": 702, "type": "theorem", "label": "algebra-lemma-ass-zero", "categories": [ "algebra" ], "title": "algebra-lemma-ass-zero", "contents": [ "\\begin{slogan}", "Over a Noetherian ring each nonzero module has an associated prime.", "\\end{slogan}", "Let $R$ be a Noetherian ring. Let $M$ be an $R$-module.", "Then", "$$", "M = (0) \\Leftrightarrow \\text{Ass}(M) = \\emptyset.", "$$" ], "refs": [], "proofs": [ { "contents": [ "If $M = (0)$, then $\\text{Ass}(M) = \\emptyset$ by definition.", "If $M \\not = 0$, pick any nonzero finitely generated submodule", "$M' \\subset M$, for example a submodule generated by a single nonzero", "element. By", "Lemma \\ref{lemma-support-zero}", "we see that $\\text{Supp}(M')$ is nonempty. By", "Proposition \\ref{proposition-minimal-primes-associated-primes}", "this implies that $\\text{Ass}(M')$ is nonempty.", "By", "Lemma \\ref{lemma-ass}", "this implies $\\text{Ass}(M) \\not = \\emptyset$." ], "refs": [ "algebra-lemma-support-zero", "algebra-proposition-minimal-primes-associated-primes", "algebra-lemma-ass" ], "ref_ids": [ 541, 1412, 699 ] } ], "ref_ids": [] }, { "id": 703, "type": "theorem", "label": "algebra-lemma-ass-minimal-prime-support", "categories": [ "algebra" ], "title": "algebra-lemma-ass-minimal-prime-support", "contents": [ "Let $R$ be a Noetherian ring.", "Let $M$ be an $R$-module.", "Any $\\mathfrak p \\in \\text{Supp}(M)$ which is minimal among the elements", "of $\\text{Supp}(M)$ is an element of $\\text{Ass}(M)$." ], "refs": [], "proofs": [ { "contents": [ "If $M$ is a finite $R$-module, then this is a consequence of", "Proposition \\ref{proposition-minimal-primes-associated-primes}.", "In general write $M = \\bigcup M_\\lambda$ as the union of its", "finite submodules, and use that", "$\\text{Supp}(M) = \\bigcup \\text{Supp}(M_\\lambda)$", "and", "$\\text{Ass}(M) = \\bigcup \\text{Ass}(M_\\lambda)$." ], "refs": [ "algebra-proposition-minimal-primes-associated-primes" ], "ref_ids": [ 1412 ] } ], "ref_ids": [] }, { "id": 704, "type": "theorem", "label": "algebra-lemma-ass-zero-divisors", "categories": [ "algebra" ], "title": "algebra-lemma-ass-zero-divisors", "contents": [ "Let $R$ be a Noetherian ring.", "Let $M$ be an $R$-module.", "The union $\\bigcup_{\\mathfrak q \\in \\text{Ass}(M)} \\mathfrak q$", "is the set of elements of $R$ which are zerodivisors on $M$." ], "refs": [], "proofs": [ { "contents": [ "Any element in any associated prime clearly is a zerodivisor on $M$.", "Conversely, suppose $x \\in R$ is a zerodivisor on $M$.", "Consider the submodule $N = \\{m \\in M \\mid xm = 0\\}$.", "Since $N$ is not zero it has an associated prime $\\mathfrak q$ by", "Lemma \\ref{lemma-ass-zero}.", "Then $x \\in \\mathfrak q$ and $\\mathfrak q$", "is an associated prime of $M$ by", "Lemma \\ref{lemma-ass}." ], "refs": [ "algebra-lemma-ass-zero", "algebra-lemma-ass" ], "ref_ids": [ 702, 699 ] } ], "ref_ids": [] }, { "id": 705, "type": "theorem", "label": "algebra-lemma-one-equation-module", "categories": [ "algebra" ], "title": "algebra-lemma-one-equation-module", "contents": [ "Let $R$ is a Noetherian local ring, $M$ a finite $R$-module, and", "$f \\in \\mathfrak m$ an element of the maximal ideal of $R$. Then", "$$", "\\dim(\\text{Supp}(M/fM)) \\leq", "\\dim(\\text{Supp}(M)) \\leq", "\\dim(\\text{Supp}(M/fM)) + 1", "$$", "If $f$ is not in any of the minimal primes of the support of $M$", "(for example if $f$ is a nonzerodivisor on $M$), then equality", "holds for the right inequality." ], "refs": [], "proofs": [ { "contents": [ "(The parenthetical statement follows from Lemma \\ref{lemma-ass-zero-divisors}.)", "The first inequality follows from $\\text{Supp}(M/fM) \\subset \\text{Supp}(M)$,", "see Lemma \\ref{lemma-support-quotient}. For the second inequality, note", "that $\\text{Supp}(M/fM) = \\text{Supp}(M) \\cap V(f)$, see", "Lemma \\ref{lemma-support-quotient}. It follows, for example by", "Lemma \\ref{lemma-filter-primes-in-support} and elementary properties", "of dimension, that it suffices to show", "$\\dim V(\\mathfrak p) \\leq \\dim (V(\\mathfrak p) \\cap V(f)) + 1$", "for primes $\\mathfrak p$ of $R$. This is a consequence of", "Lemma \\ref{lemma-one-equation}.", "Finally, if $f$ is not contained in any minimal", "prime of the support of $M$, then the chains of primes in", "$\\text{Supp}(M/fM)$ all give", "rise to chains in $\\text{Supp}(M)$ which are at least one step away", "from being maximal." ], "refs": [ "algebra-lemma-ass-zero-divisors", "algebra-lemma-support-quotient", "algebra-lemma-support-quotient", "algebra-lemma-filter-primes-in-support" ], "ref_ids": [ 704, 547, 547, 692 ] } ], "ref_ids": [] }, { "id": 706, "type": "theorem", "label": "algebra-lemma-ass-functorial", "categories": [ "algebra" ], "title": "algebra-lemma-ass-functorial", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "Let $M$ be an $S$-module.", "Then $\\Spec(\\varphi)(\\text{Ass}_S(M)) \\subset \\text{Ass}_R(M)$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathfrak q \\in \\text{Ass}_S(M)$, then there exists an $m$ in $M$", "such that the annihilator of $m$ in $S$ is $\\mathfrak q$. Then the annihilator", "of $m$ in $R$ is $\\mathfrak q \\cap R$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 707, "type": "theorem", "label": "algebra-lemma-ass-functorial-Noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-ass-functorial-Noetherian", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "Let $M$ be an $S$-module. If $S$ is Noetherian, then", "$\\Spec(\\varphi)(\\text{Ass}_S(M)) = \\text{Ass}_R(M)$." ], "refs": [], "proofs": [ { "contents": [ "We have already seen in", "Lemma \\ref{lemma-ass-functorial}", "that", "$\\Spec(\\varphi)(\\text{Ass}_S(M)) \\subset \\text{Ass}_R(M)$.", "For the converse, choose a prime $\\mathfrak p \\in \\text{Ass}_R(M)$.", "Let $m \\in M$ be an element such that the annihilator of $m$ in $R$", "is $\\mathfrak p$. Let $I = \\{g \\in S \\mid gm = 0\\}$ be the annihilator", "of $m$ in $S$. Then $R/\\mathfrak p \\subset S/I$ is injective.", "Combining Lemmas \\ref{lemma-injective-minimal-primes-in-image} and", "\\ref{lemma-minimal-prime-image-minimal-prime} we see that", "there is a prime $\\mathfrak q \\subset S$ minimal over $I$", "mapping to $\\mathfrak p$. By", "Proposition \\ref{proposition-minimal-primes-associated-primes}", "we see that $\\mathfrak q$ is an associated prime of $S/I$, hence", "$\\mathfrak q$ is an associated prime of $M$ by", "Lemma \\ref{lemma-ass}", "and we win." ], "refs": [ "algebra-lemma-ass-functorial", "algebra-lemma-injective-minimal-primes-in-image", "algebra-lemma-minimal-prime-image-minimal-prime", "algebra-proposition-minimal-primes-associated-primes", "algebra-lemma-ass" ], "ref_ids": [ 706, 445, 447, 1412, 699 ] } ], "ref_ids": [] }, { "id": 708, "type": "theorem", "label": "algebra-lemma-ass-quotient-ring", "categories": [ "algebra" ], "title": "algebra-lemma-ass-quotient-ring", "contents": [ "Let $R$ be a ring.", "Let $I$ be an ideal.", "Let $M$ be an $R/I$-module.", "Via the canonical injection", "$\\Spec(R/I) \\to \\Spec(R)$", "we have $\\text{Ass}_{R/I}(M) = \\text{Ass}_R(M)$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 709, "type": "theorem", "label": "algebra-lemma-associated-primes-localize", "categories": [ "algebra" ], "title": "algebra-lemma-associated-primes-localize", "contents": [ "Let $R$ be a ring.", "Let $M$ be an $R$-module.", "Let $\\mathfrak p \\subset R$ be a prime.", "\\begin{enumerate}", "\\item If $\\mathfrak p \\in \\text{Ass}(M)$ then", "$\\mathfrak pR_{\\mathfrak p} \\in \\text{Ass}(M_{\\mathfrak p})$.", "\\item If $\\mathfrak p$ is finitely generated then the converse holds", "as well.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $\\mathfrak p \\in \\text{Ass}(M)$ there exists an element $m \\in M$", "whose annihilator is $\\mathfrak p$. As localization is exact", "(Proposition \\ref{proposition-localization-exact})", "we see that the annihilator of $m/1$ in", "$M_{\\mathfrak p}$ is $\\mathfrak pR_{\\mathfrak p}$ hence (1) holds.", "Assume $\\mathfrak pR_{\\mathfrak p} \\in \\text{Ass}(M_{\\mathfrak p})$", "and $\\mathfrak p = (f_1, \\ldots, f_n)$. Let $m/g$ be an element of", "$M_{\\mathfrak p}$ whose annihilator is $\\mathfrak pR_{\\mathfrak p}$.", "This implies that the annihilator of $m$ is contained in $\\mathfrak p$.", "As $f_i m/g = 0$ in $M_{\\mathfrak p}$ we see there exists a", "$g_i \\in R$, $g_i \\not \\in \\mathfrak p$ such that $g_i f_i m = 0$ in $M$.", "Combined we see the annihilator of $g_1\\ldots g_nm$ is $\\mathfrak p$. Hence", "$\\mathfrak p \\in \\text{Ass}(M)$." ], "refs": [ "algebra-proposition-localization-exact" ], "ref_ids": [ 1402 ] } ], "ref_ids": [] }, { "id": 710, "type": "theorem", "label": "algebra-lemma-localize-ass", "categories": [ "algebra" ], "title": "algebra-lemma-localize-ass", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Let $S \\subset R$ be a multiplicative subset.", "Via the canonical injection $\\Spec(S^{-1}R) \\to \\Spec(R)$", "we have", "\\begin{enumerate}", "\\item $\\text{Ass}_R(S^{-1}M) = \\text{Ass}_{S^{-1}R}(S^{-1}M)$,", "\\item", "$\\text{Ass}_R(M) \\cap \\Spec(S^{-1}R) \\subset \\text{Ass}_R(S^{-1}M)$, and", "\\item if $R$ is Noetherian this inclusion is an equality.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first equality follows, since if $m \\in S^{-1}M$, then the annihilator", "of $m$ in $R$ is the intersection of the annihilator of $m$ in $S^{-1}R$", "with $R$.", "The displayed inclusion and equality in the Noetherian case follows from", "Lemma \\ref{lemma-associated-primes-localize}", "since for $\\mathfrak p \\in R$, $S \\cap \\mathfrak p = \\emptyset$ we have", "$M_{\\mathfrak p} = (S^{-1}M)_{S^{-1}\\mathfrak p}$." ], "refs": [ "algebra-lemma-associated-primes-localize" ], "ref_ids": [ 709 ] } ], "ref_ids": [] }, { "id": 711, "type": "theorem", "label": "algebra-lemma-localize-ass-nonzero-divisors", "categories": [ "algebra" ], "title": "algebra-lemma-localize-ass-nonzero-divisors", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Let $S \\subset R$ be a multiplicative subset.", "Assume that every $s \\in S$ is a nonzerodivisor on $M$.", "Then", "$$", "\\text{Ass}_R(M) = \\text{Ass}_R(S^{-1}M).", "$$" ], "refs": [], "proofs": [ { "contents": [ "As $M \\subset S^{-1}M$ by assumption we get the inclusion", "$\\text{Ass}(M) \\subset \\text{Ass}(S^{-1}M)$ from", "Lemma \\ref{lemma-ass}.", "Conversely, suppose that $n/s \\in S^{-1}M$ is an element whose", "annihilator is a prime ideal $\\mathfrak p$. Then the annihilator", "of $n \\in M$ is also $\\mathfrak p$." ], "refs": [ "algebra-lemma-ass" ], "ref_ids": [ 699 ] } ], "ref_ids": [] }, { "id": 712, "type": "theorem", "label": "algebra-lemma-ideal-nonzerodivisor", "categories": [ "algebra" ], "title": "algebra-lemma-ideal-nonzerodivisor", "contents": [ "Let $R$ be a Noetherian local ring with", "maximal ideal $\\mathfrak m$. Let $I \\subset \\mathfrak m$", "be an ideal. Let $M$ be a finite $R$-module.", "The following are equivalent:", "\\begin{enumerate}", "\\item There exists an $x \\in I$ which is not a zerodivisor on $M$.", "\\item We have $I \\not \\subset \\mathfrak q$ for all", "$\\mathfrak q \\in \\text{Ass}(M)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If there exists a nonzerodivisor $x$ in $I$,", "then $x$ clearly cannot be in any associated", "prime of $M$. Conversely, suppose $I \\not \\subset \\mathfrak q$", "for all $\\mathfrak q \\in \\text{Ass}(M)$. In this case we can", "choose $x \\in I$, $x \\not \\in \\mathfrak q$ for all", "$\\mathfrak q \\in \\text{Ass}(M)$ by Lemmas", "\\ref{lemma-finite-ass} and \\ref{lemma-silly}.", "By Lemma \\ref{lemma-ass-zero-divisors} the element $x$", "is not a zerodivisor on $M$." ], "refs": [ "algebra-lemma-finite-ass", "algebra-lemma-silly", "algebra-lemma-ass-zero-divisors" ], "ref_ids": [ 701, 378, 704 ] } ], "ref_ids": [] }, { "id": 713, "type": "theorem", "label": "algebra-lemma-zero-at-ass-zero", "categories": [ "algebra" ], "title": "algebra-lemma-zero-at-ass-zero", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module. If $R$ is Noetherian", "the map", "$$", "M", "\\longrightarrow", "\\prod\\nolimits_{\\mathfrak p \\in \\text{Ass}(M)} M_{\\mathfrak p}", "$$", "is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in M$ be an element of the kernel of the map.", "Then if $\\mathfrak p$ is an associated prime of $Rx \\subset M$", "we see on the one hand that $\\mathfrak p \\in \\text{Ass}(M)$", "(Lemma \\ref{lemma-ass}) and", "on the other hand that $(Rx)_{\\mathfrak p} \\subset M_{\\mathfrak p}$", "is not zero. This contradiction shows that $\\text{Ass}(Rx) = \\emptyset$.", "Hence $Rx = 0$ by", "Lemma \\ref{lemma-ass-zero}." ], "refs": [ "algebra-lemma-ass", "algebra-lemma-ass-zero" ], "ref_ids": [ 699, 702 ] } ], "ref_ids": [] }, { "id": 714, "type": "theorem", "label": "algebra-lemma-symbolic-power-associated", "categories": [ "algebra" ], "title": "algebra-lemma-symbolic-power-associated", "contents": [ "Let $R$ be a Noetherian ring.", "Let $\\mathfrak p$ be a prime ideal.", "Let $n > 0$. Then $\\text{Ass}(R/\\mathfrak p^{(n)}) = \\{\\mathfrak p\\}$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathfrak q$ is an associated prime of $R/\\mathfrak p^{(n)}$", "then clearly $\\mathfrak p \\subset \\mathfrak q$.", "On the other hand, any element $x \\in R$, $x \\not \\in \\mathfrak p$", "is a nonzerodivisor on $R/\\mathfrak p^{(n)}$.", "Namely, if $y \\in R$ and", "$xy \\in \\mathfrak p^{(n)} = R \\cap \\mathfrak p^nR_{\\mathfrak p}$", "then $y \\in \\mathfrak p^nR_{\\mathfrak p}$, hence $y \\in \\mathfrak p^{(n)}$.", "Hence the lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 715, "type": "theorem", "label": "algebra-lemma-symbolic-power-flat-extension", "categories": [ "algebra" ], "title": "algebra-lemma-symbolic-power-flat-extension", "contents": [ "Let $R \\to S$ be flat ring map. Let $\\mathfrak p \\subset R$ be a prime", "such that $\\mathfrak q = \\mathfrak p S$ is a prime of $S$.", "Then $\\mathfrak p^{(n)} S = \\mathfrak q^{(n)}$." ], "refs": [], "proofs": [ { "contents": [ "Since", "$\\mathfrak p^{(n)} = \\Ker(R \\to R_\\mathfrak p/\\mathfrak p^nR_\\mathfrak p)$", "we see using flatness that $\\mathfrak p^{(n)} S$ is the kernel of the map", "$S \\to S_\\mathfrak p/\\mathfrak p^nS_\\mathfrak p$. On the other hand", "$\\mathfrak q^{(n)}$ is the kernel of the map", "$S \\to S_\\mathfrak q/\\mathfrak q^nS_\\mathfrak q =", "S_\\mathfrak q/\\mathfrak p^nS_\\mathfrak q$. Hence it suffices", "to show that", "$$", "S_\\mathfrak p/\\mathfrak p^nS_\\mathfrak p", "\\longrightarrow", "S_\\mathfrak q/\\mathfrak p^nS_\\mathfrak q", "$$", "is injective. Observe that the right hand module is the localization", "of the left hand module by elements $f \\in S$, $f \\not \\in \\mathfrak q$.", "Thus it suffices to show these elements are nonzerodivisors on", "$S_\\mathfrak p/\\mathfrak p^nS_\\mathfrak p$. By flatness, the module", "$S_\\mathfrak p/\\mathfrak p^nS_\\mathfrak p$ has a finite filtration whose", "subquotients are", "$$", "\\mathfrak p^iS_\\mathfrak p/\\mathfrak p^{i + 1}S_\\mathfrak p", "\\cong \\mathfrak p^iR_\\mathfrak p/\\mathfrak p^{i + 1}R_\\mathfrak p", "\\otimes_{R_\\mathfrak p} S_\\mathfrak p \\cong", "V \\otimes_{\\kappa(\\mathfrak p)} (S/\\mathfrak q)_\\mathfrak p", "$$", "where $V$ is a $\\kappa(\\mathfrak p)$ vector space. Thus $f$", "acts invertibly as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 716, "type": "theorem", "label": "algebra-lemma-compare-relative-assassins", "categories": [ "algebra" ], "title": "algebra-lemma-compare-relative-assassins", "contents": [ "Let $R \\to S$ be a ring map. Let $N$ be an $S$-module.", "Let $A$, $A'$, $A_{fin}$, $B$, and $B_{fin}$ be the subsets of", "$\\Spec(S)$ introduced above.", "\\begin{enumerate}", "\\item We always have $A = A'$.", "\\item We always have $A_{fin} \\subset A$,", "$B_{fin} \\subset B$, $A_{fin} \\subset A'_{fin} \\subset B_{fin}$", "and $A \\subset B$.", "\\item If $S$ is Noetherian, then $A = A_{fin}$ and $B = B_{fin}$.", "\\item If $N$ is flat over $R$, then $A = A_{fin} = A'_{fin}$ and $B = B_{fin}$.", "\\item If $R$ is Noetherian and $N$ is flat over $R$, then all of the sets", "are equal, i.e., $A = A' = A_{fin} = A'_{fin} = B = B_{fin}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Some of the arguments in the proof will be repeated in the proofs of", "later lemmas which are more precise than this one (because they deal", "with a given module $M$ or a given prime $\\mathfrak p$ and not with", "the collection of all of them).", "\\medskip\\noindent", "Proof of (1). Let $\\mathfrak p$ be a prime of $R$. Then we have", "$$", "\\text{Ass}_S(N \\otimes_R \\kappa(\\mathfrak p)) =", "\\text{Ass}_{S/\\mathfrak pS}(N \\otimes_R \\kappa(\\mathfrak p)) =", "\\text{Ass}_{S \\otimes_R \\kappa(\\mathfrak p)}(N \\otimes_R \\kappa(\\mathfrak p))", "$$", "the first equality by", "Lemma \\ref{lemma-ass-quotient-ring}", "and the second by", "Lemma \\ref{lemma-localize-ass} part (1). This prove that $A = A'$.", "The inclusion $A_{fin} \\subset A'_{fin}$ is clear.", "\\medskip\\noindent", "Proof of (2). Each of the inclusions is immediate from the definitions", "except perhaps $A_{fin} \\subset A$ which follows from", "Lemma \\ref{lemma-localize-ass}", "and the fact that we require $\\mathfrak p = R \\cap \\mathfrak q$ in", "the formulation of $A_{fin}$.", "\\medskip\\noindent", "Proof of (3). The equality $A = A_{fin}$ follows from", "Lemma \\ref{lemma-localize-ass} part (3)", "if $S$ is Noetherian. Let $\\mathfrak q = (g_1, \\ldots, g_m)$ be a finitely", "generated prime ideal of $S$.", "Say $z \\in N \\otimes_R M$ is an element whose annihilator is $\\mathfrak q$.", "We may pick a finite submodule $M' \\subset M$ such that $z$ is the", "image of $z' \\in N \\otimes_R M'$. Then", "$\\text{Ann}_S(z') \\subset \\mathfrak q = \\text{Ann}_S(z)$.", "Since $N \\otimes_R -$ commutes with colimits and since $M$ is the", "directed colimit of finite $R$-modules we can find $M' \\subset M'' \\subset M$", "such that the image $z'' \\in N \\otimes_R M''$ is annihilated by", "$g_1, \\ldots, g_m$. Hence $\\text{Ann}_S(z'') = \\mathfrak q$. This proves", "that $B = B_{fin}$ if $S$ is Noetherian.", "\\medskip\\noindent", "Proof of (4). If $N$ is flat, then the functor $N \\otimes_R -$ is exact.", "In particular, if $M' \\subset M$, then $N \\otimes_R M' \\subset N \\otimes_R M$.", "Hence if $z \\in N \\otimes_R M$ is an element whose annihilator", "$\\mathfrak q = \\text{Ann}_S(z)$ is a prime, then we can pick any", "finite $R$-submodule $M' \\subset M$ such that $z \\in N \\otimes_R M'$", "and we see that the annihilator of $z$ as an element of $N \\otimes_R M'$", "is equal to $\\mathfrak q$. Hence $B = B_{fin}$. Let $\\mathfrak p'$ be a", "prime of $R$ and let $\\mathfrak q$ be a prime of $S$ which is", "an associated prime of $N/\\mathfrak p'N$. This implies that", "$\\mathfrak p'S \\subset \\mathfrak q$. As $N$ is flat over $R$ we", "see that $N/\\mathfrak p'N$ is flat over the integral domain $R/\\mathfrak p'$.", "Hence every nonzero element of $R/\\mathfrak p'$ is a nonzerodivisor on", "$N/\\mathfrak p'$. Hence none of these elements can map to an element of", "$\\mathfrak q$ and we conclude that $\\mathfrak p' = R \\cap \\mathfrak q$.", "Hence $A_{fin} = A'_{fin}$. Finally, by", "Lemma \\ref{lemma-localize-ass-nonzero-divisors}", "we see that", "$\\text{Ass}_S(N/\\mathfrak p'N) =", "\\text{Ass}_S(N \\otimes_R \\kappa(\\mathfrak p'))$, i.e., $A'_{fin} = A$.", "\\medskip\\noindent", "Proof of (5). We only need to prove $A'_{fin} = B_{fin}$ as the other", "equalities have been proved in (4). To see this let $M$ be a finite", "$R$-module. By", "Lemma \\ref{lemma-filter-Noetherian-module}", "there exists a filtration by $R$-submodules", "$$", "0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M", "$$", "such that each quotient $M_i/M_{i-1}$ is isomorphic", "to $R/\\mathfrak p_i$ for some prime ideal $\\mathfrak p_i$", "of $R$. Since $N$ is flat we obtain a filtration by $S$-submodules", "$$", "0 = N \\otimes_R M_0 \\subset N \\otimes_R M_1 \\subset \\ldots \\subset", "N \\otimes_R M_n = N \\otimes_R M", "$$", "such that each subquotient is isomorphic to $N/\\mathfrak p_iN$. By", "Lemma \\ref{lemma-ass}", "we conclude that", "$\\text{Ass}_S(N \\otimes_R M) \\subset \\bigcup \\text{Ass}_S(N/\\mathfrak p_iN)$.", "Hence we see that $B_{fin} \\subset A'_{fin}$. Since the other inclusion is part", "of (2) we win." ], "refs": [ "algebra-lemma-ass-quotient-ring", "algebra-lemma-localize-ass", "algebra-lemma-localize-ass", "algebra-lemma-localize-ass", "algebra-lemma-localize-ass-nonzero-divisors", "algebra-lemma-filter-Noetherian-module", "algebra-lemma-ass" ], "ref_ids": [ 708, 710, 710, 710, 711, 691, 699 ] } ], "ref_ids": [] }, { "id": 717, "type": "theorem", "label": "algebra-lemma-bourbaki", "categories": [ "algebra" ], "title": "algebra-lemma-bourbaki", "contents": [ "Let $R \\to S$ be a ring map.", "Let $M$ be an $R$-module, and let $N$ be an $S$-module.", "If $N$ is flat as $R$-module, then", "$$", "\\text{Ass}_S(M \\otimes_R N)", "\\supset", "\\bigcup\\nolimits_{\\mathfrak p \\in \\text{Ass}_R(M)} \\text{Ass}_S(N/\\mathfrak pN)", "$$", "and if $R$ is Noetherian then we have equality." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathfrak p \\in \\text{Ass}_R(M)$ then there exists an injection", "$R/\\mathfrak p \\to M$. As $N$ is flat over $R$ we obtain an injection", "$R/\\mathfrak p \\otimes_R N \\to M \\otimes_R N$. Since", "$R/\\mathfrak p \\otimes_R N = N/\\mathfrak pN$ we conclude that", "$\\text{Ass}_S(N/\\mathfrak pN) \\subset \\text{Ass}_S(M \\otimes_R N)$, see", "Lemma \\ref{lemma-ass}. Hence the right hand side is", "contained in the left hand side.", "\\medskip\\noindent", "Write $M = \\bigcup M_\\lambda$ as the union of its finitely generated", "$R$-submodules. Then also $N \\otimes_R M = \\bigcup N \\otimes_R M_\\lambda$", "(as $N$ is $R$-flat). By definition of associated primes we see that", "$\\text{Ass}_S(N \\otimes_R M) = \\bigcup \\text{Ass}_S(N \\otimes_R M_\\lambda)$", "and $\\text{Ass}_R(M) = \\bigcup \\text{Ass}(M_\\lambda)$. Hence we may assume", "$M$ is finitely generated.", "\\medskip\\noindent", "Let $\\mathfrak q \\in \\text{Ass}_S(M \\otimes_R N)$, and assume $R$ is", "Noetherian and $M$", "is a finite $R$-module. To finish the proof we have to show that", "$\\mathfrak q$ is an element of the right hand side. First we observe that", "$\\mathfrak qS_{\\mathfrak q} \\in", "\\text{Ass}_{S_{\\mathfrak q}}((M \\otimes_R N)_{\\mathfrak q})$,", "see Lemma \\ref{lemma-associated-primes-localize}.", "Let $\\mathfrak p$ be the corresponding prime of $R$.", "Note that", "$$", "(M \\otimes_R N)_{\\mathfrak q} = M \\otimes_R N_{\\mathfrak q}", "= M_{\\mathfrak p} \\otimes_{R_{\\mathfrak p}} N_{\\mathfrak q}", "$$", "If", "$\\mathfrak pR_{\\mathfrak p} \\not \\in", "\\text{Ass}_{R_{\\mathfrak p}}(M_{\\mathfrak p})$", "then there exists an element $x \\in \\mathfrak pR_{\\mathfrak p}$ which", "is a nonzerodivisor in $M_{\\mathfrak p}$ (see", "Lemma \\ref{lemma-ideal-nonzerodivisor}). Since", "$N_{\\mathfrak q}$ is flat over $R_{\\mathfrak p}$ we see that", "the image of $x$ in $\\mathfrak qS_{\\mathfrak q}$ is a nonzerodivisor on", "$(M \\otimes_R N)_{\\mathfrak q}$. This is a contradiction", "with the assumption that", "$\\mathfrak qS_{\\mathfrak q} \\in \\text{Ass}_S((M \\otimes_R N)_{\\mathfrak q})$.", "Hence we conclude that $\\mathfrak p$ is one of the associated", "primes of $M$.", "\\medskip\\noindent", "Continuing the argument we choose a filtration", "$$", "0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M", "$$", "such that each quotient $M_i/M_{i-1}$ is isomorphic", "to $R/\\mathfrak p_i$ for some prime ideal $\\mathfrak p_i$", "of $R$, see Lemma \\ref{lemma-filter-Noetherian-module}.", "(By Lemma \\ref{lemma-ass-filter} we have $\\mathfrak p_i = \\mathfrak p$ for", "at least one $i$.) This gives a filtration", "$$", "0 = M_0 \\otimes_R N \\subset M_1 \\otimes_R N \\subset \\ldots", "\\subset M_n \\otimes_R N = M \\otimes_R N", "$$", "with subquotients isomorphic to $N/\\mathfrak p_iN$. If", "$\\mathfrak p_i \\not = \\mathfrak p$ then $\\mathfrak q$ cannot be", "associated to the module $N/\\mathfrak p_iN$ by the result of the", "preceding paragraph (as $\\text{Ass}_R(R/\\mathfrak p_i) = \\{\\mathfrak p_i\\}$).", "Hence we conclude that $\\mathfrak q$ is associated to", "$N/\\mathfrak pN$ as desired." ], "refs": [ "algebra-lemma-ass", "algebra-lemma-associated-primes-localize", "algebra-lemma-ideal-nonzerodivisor", "algebra-lemma-filter-Noetherian-module", "algebra-lemma-ass-filter" ], "ref_ids": [ 699, 709, 712, 691, 700 ] } ], "ref_ids": [] }, { "id": 718, "type": "theorem", "label": "algebra-lemma-post-bourbaki", "categories": [ "algebra" ], "title": "algebra-lemma-post-bourbaki", "contents": [ "Let $R \\to S$ be a ring map.", "Let $N$ be an $S$-module.", "Assume $N$ is flat as an $R$-module and", "$R$ is a domain with fraction field $K$.", "Then", "$$", "\\text{Ass}_S(N) =", "\\text{Ass}_S(N \\otimes_R K) =", "\\text{Ass}_{S \\otimes_R K}(N \\otimes_R K)", "$$", "via the canonical inclusion", "$\\Spec(S \\otimes_R K) \\subset \\Spec(S)$." ], "refs": [], "proofs": [ { "contents": [ "Note that $S \\otimes_R K = (R \\setminus \\{0\\})^{-1}S$ and", "$N \\otimes_R K = (R \\setminus \\{0\\})^{-1}N$.", "For any nonzero $x \\in R$ multiplication by $x$ on $N$ is injective as", "$N$ is flat over $R$. Hence the lemma follows from", "Lemma \\ref{lemma-localize-ass-nonzero-divisors}", "combined with", "Lemma \\ref{lemma-localize-ass} part (1)." ], "refs": [ "algebra-lemma-localize-ass-nonzero-divisors", "algebra-lemma-localize-ass" ], "ref_ids": [ 711, 710 ] } ], "ref_ids": [] }, { "id": 719, "type": "theorem", "label": "algebra-lemma-bourbaki-fibres", "categories": [ "algebra" ], "title": "algebra-lemma-bourbaki-fibres", "contents": [ "Let $R \\to S$ be a ring map.", "Let $M$ be an $R$-module, and let $N$ be an $S$-module.", "Assume $N$ is flat as $R$-module. Then", "$$", "\\text{Ass}_S(M \\otimes_R N)", "\\supset", "\\bigcup\\nolimits_{\\mathfrak p \\in \\text{Ass}_R(M)}", "\\text{Ass}_{S \\otimes_R \\kappa(\\mathfrak p)}(N \\otimes_R \\kappa(\\mathfrak p))", "$$", "where we use", "Remark \\ref{remark-fundamental-diagram}", "to think of the spectra of fibre rings as subsets of $\\Spec(S)$.", "If $R$ is Noetherian then this inclusion is an equality." ], "refs": [ "algebra-remark-fundamental-diagram" ], "proofs": [ { "contents": [ "This is equivalent to", "Lemma \\ref{lemma-bourbaki}", "by", "Lemmas \\ref{lemma-ass-quotient-ring},", "\\ref{lemma-flat-base-change}, and", "\\ref{lemma-post-bourbaki}." ], "refs": [ "algebra-lemma-bourbaki", "algebra-lemma-ass-quotient-ring", "algebra-lemma-flat-base-change", "algebra-lemma-post-bourbaki" ], "ref_ids": [ 717, 708, 527, 718 ] } ], "ref_ids": [ 1558 ] }, { "id": 720, "type": "theorem", "label": "algebra-lemma-weakly-ass-local", "categories": [ "algebra" ], "title": "algebra-lemma-weakly-ass-local", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Let $\\mathfrak p$ be a prime of $R$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $\\mathfrak p$ is weakly associated to $M$,", "\\item $\\mathfrak pR_{\\mathfrak p}$ is weakly associated to $M_{\\mathfrak p}$,", "and", "\\item $M_{\\mathfrak p}$ contains an element whose", "annihilator has radical equal to $\\mathfrak pR_{\\mathfrak p}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Then there exists an element $m \\in M$ such that", "$\\mathfrak p$ is minimal among the primes containing the annihilator", "$I = \\{x \\in R \\mid xm = 0\\}$ of $m$. As localization is exact, the", "annihilator of $m$ in $M_{\\mathfrak p}$ is $I_{\\mathfrak p}$.", "Hence $\\mathfrak pR_{\\mathfrak p}$ is a minimal prime of", "$R_{\\mathfrak p}$ containing the annihilator $I_{\\mathfrak p}$", "of $m$ in $M_{\\mathfrak p}$. This implies (2) holds, and also (3)", "as it implies that $\\sqrt{I_{\\mathfrak p}} = \\mathfrak pR_{\\mathfrak p}$.", "\\medskip\\noindent", "Applying the implication (1) $\\Rightarrow$ (3) to $M_{\\mathfrak p}$", "over $R_{\\mathfrak p}$ we see that (2) $\\Rightarrow$ (3).", "\\medskip\\noindent", "Finally, assume (3). This means there exists an element", "$m/f \\in M_{\\mathfrak p}$ whose annihilator has radical equal", "to $\\mathfrak pR_{\\mathfrak p}$. Then the annihilator", "$I = \\{x \\in R \\mid xm = 0\\}$ of $m$ in $M$ is such that", "$\\sqrt{I_{\\mathfrak p}} = \\mathfrak pR_{\\mathfrak p}$. Clearly", "this means that $\\mathfrak p$ contains $I$ and is minimal among the", "primes containing $I$, i.e., (1) holds." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 721, "type": "theorem", "label": "algebra-lemma-reduced-weakly-ass-minimal", "categories": [ "algebra" ], "title": "algebra-lemma-reduced-weakly-ass-minimal", "contents": [ "For a reduced ring the weakly associated primes of the ring are", "the minimal primes." ], "refs": [], "proofs": [ { "contents": [ "Let $(R, \\mathfrak m)$ be a reduced local ring.", "Suppose $x \\in R$ is an element whose annihilator", "has radical $\\mathfrak m$. If $\\mathfrak m \\not = 0$, then $x$", "cannot be a unit, so $x \\in \\mathfrak m$. Then in particular $x^{1 + n} = 0$", "for some $n \\geq 0$. Hence $x = 0$. Which contradicts the assumption", "that the annihilator of $x$ is contained in $\\mathfrak m$.", "Thus we see that $\\mathfrak m = 0$, i.e., $R$ is a field.", "By Lemma \\ref{lemma-weakly-ass-local} this", "implies the statement of the lemma." ], "refs": [ "algebra-lemma-weakly-ass-local" ], "ref_ids": [ 720 ] } ], "ref_ids": [] }, { "id": 722, "type": "theorem", "label": "algebra-lemma-weakly-ass", "categories": [ "algebra" ], "title": "algebra-lemma-weakly-ass", "contents": [ "Let $R$ be a ring.", "Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence", "of $R$-modules.", "Then $\\text{WeakAss}(M') \\subset \\text{WeakAss}(M)$ and", "$\\text{WeakAss}(M) \\subset \\text{WeakAss}(M') \\cup \\text{WeakAss}(M'')$." ], "refs": [], "proofs": [ { "contents": [ "We will use the characterization of weakly associated primes of", "Lemma \\ref{lemma-weakly-ass-local}.", "Let $\\mathfrak p$ be a prime of $R$. As localization is exact we obtain", "the short exact sequence", "$0 \\to M'_{\\mathfrak p} \\to M_{\\mathfrak p} \\to M''_{\\mathfrak p} \\to 0$.", "Suppose that $m \\in M_{\\mathfrak p}$ is an element whose annihilator", "has radical $\\mathfrak pR_{\\mathfrak p}$. Then either the image $\\overline{m}$", "of $m$ in $M''_{\\mathfrak p}$ is zero and $m \\in M'_{\\mathfrak p}$, or the", "radical of the annihilator of $\\overline{m}$ is $\\mathfrak pR_{\\mathfrak p}$.", "This proves that", "$\\text{WeakAss}(M) \\subset \\text{WeakAss}(M') \\cup \\text{WeakAss}(M'')$.", "The inclusion $\\text{WeakAss}(M') \\subset \\text{WeakAss}(M)$ is immediate", "from the definitions." ], "refs": [ "algebra-lemma-weakly-ass-local" ], "ref_ids": [ 720 ] } ], "ref_ids": [] }, { "id": 723, "type": "theorem", "label": "algebra-lemma-weakly-ass-zero", "categories": [ "algebra" ], "title": "algebra-lemma-weakly-ass-zero", "contents": [ "\\begin{slogan}", "Every nonzero module has a weakly associated prime.", "\\end{slogan}", "Let $R$ be a ring. Let $M$ be an $R$-module. Then", "$$", "M = (0) \\Leftrightarrow \\text{WeakAss}(M) = \\emptyset", "$$" ], "refs": [], "proofs": [ { "contents": [ "If $M = (0)$ then $\\text{WeakAss}(M) = \\emptyset$ by definition.", "Conversely, suppose that $M \\not = 0$. Pick a nonzero element $m \\in M$.", "Write $I = \\{x \\in R \\mid xm = 0\\}$ the annihilator of $m$.", "Then $R/I \\subset M$. Hence $\\text{WeakAss}(R/I) \\subset \\text{WeakAss}(M)$ by", "Lemma \\ref{lemma-weakly-ass}.", "But as $I \\not = R$ we have $V(I) = \\Spec(R/I)$ contains a minimal", "prime, see", "Lemmas \\ref{lemma-Zariski-topology} and", "\\ref{lemma-spec-closed},", "and we win." ], "refs": [ "algebra-lemma-weakly-ass", "algebra-lemma-Zariski-topology", "algebra-lemma-spec-closed" ], "ref_ids": [ 722, 389, 393 ] } ], "ref_ids": [] }, { "id": 724, "type": "theorem", "label": "algebra-lemma-weakly-ass-support", "categories": [ "algebra" ], "title": "algebra-lemma-weakly-ass-support", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module. Then", "$$", "\\text{Ass}(M) \\subset \\text{WeakAss}(M) \\subset \\text{Supp}(M).", "$$" ], "refs": [], "proofs": [ { "contents": [ "The first inclusion is immediate from the definitions.", "If $\\mathfrak p \\in \\text{WeakAss}(M)$, then by", "Lemma \\ref{lemma-weakly-ass-local}", "we have $M_{\\mathfrak p} \\not = 0$, hence $\\mathfrak p \\in \\text{Supp}(M)$." ], "refs": [ "algebra-lemma-weakly-ass-local" ], "ref_ids": [ 720 ] } ], "ref_ids": [] }, { "id": 725, "type": "theorem", "label": "algebra-lemma-weakly-ass-zero-divisors", "categories": [ "algebra" ], "title": "algebra-lemma-weakly-ass-zero-divisors", "contents": [ "Let $R$ be a ring.", "Let $M$ be an $R$-module.", "The union $\\bigcup_{\\mathfrak q \\in \\text{WeakAss}(M)} \\mathfrak q$", "is the set elements of $R$ which are zerodivisors on $M$." ], "refs": [], "proofs": [ { "contents": [ "Suppose $f \\in \\mathfrak q \\in \\text{WeakAss}(M)$.", "Then there exists an element $m \\in M$ such that", "$\\mathfrak q$ is minimal over $I = \\{x \\in R \\mid xm = 0\\}$.", "Hence there exists a $g \\in R$, $g \\not \\in \\mathfrak q$ and $n > 0$", "such that $f^ngm = 0$. Note that $gm \\not = 0$ as $g \\not \\in I$.", "If we take $n$ minimal as above, then $f (f^{n - 1}gm) = 0$", "and $f^{n - 1}gm \\not = 0$, so $f$ is a zerodivisor on $M$.", "Conversely, suppose $f \\in R$ is a zerodivisor on $M$.", "Consider the submodule $N = \\{m \\in M \\mid fm = 0\\}$.", "Since $N$ is not zero it has a weakly associated prime $\\mathfrak q$ by", "Lemma \\ref{lemma-weakly-ass-zero}.", "Clearly $f \\in \\mathfrak q$ and by", "Lemma \\ref{lemma-weakly-ass}", "$\\mathfrak q$ is a weakly associated prime of $M$." ], "refs": [ "algebra-lemma-weakly-ass-zero", "algebra-lemma-weakly-ass" ], "ref_ids": [ 723, 722 ] } ], "ref_ids": [] }, { "id": 726, "type": "theorem", "label": "algebra-lemma-weakly-ass-minimal-prime-support", "categories": [ "algebra" ], "title": "algebra-lemma-weakly-ass-minimal-prime-support", "contents": [ "Let $R$ be a ring.", "Let $M$ be an $R$-module.", "Any $\\mathfrak p \\in \\text{Supp}(M)$ which is minimal among the elements", "of $\\text{Supp}(M)$ is an element of $\\text{WeakAss}(M)$." ], "refs": [], "proofs": [ { "contents": [ "Note that $\\text{Supp}(M_{\\mathfrak p}) = \\{\\mathfrak pR_{\\mathfrak p}\\}$", "in $\\Spec(R_{\\mathfrak p})$. In particular $M_{\\mathfrak p}$", "is nonzero, and hence $\\text{WeakAss}(M_{\\mathfrak p}) \\not = \\emptyset$ by", "Lemma \\ref{lemma-weakly-ass-zero}.", "Since $\\text{WeakAss}(M_{\\mathfrak p}) \\subset \\text{Supp}(M_{\\mathfrak p})$", "by", "Lemma \\ref{lemma-weakly-ass-support}", "we conclude that", "$\\text{WeakAss}(M_{\\mathfrak p}) = \\{\\mathfrak pR_{\\mathfrak p}\\}$,", "whence $\\mathfrak p \\in \\text{WeakAss}(M)$ by", "Lemma \\ref{lemma-weakly-ass-local}." ], "refs": [ "algebra-lemma-weakly-ass-zero", "algebra-lemma-weakly-ass-support", "algebra-lemma-weakly-ass-local" ], "ref_ids": [ 723, 724, 720 ] } ], "ref_ids": [] }, { "id": 727, "type": "theorem", "label": "algebra-lemma-ass-weakly-ass", "categories": [ "algebra" ], "title": "algebra-lemma-ass-weakly-ass", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Let $\\mathfrak p$ be a prime ideal of $R$ which is finitely generated.", "Then", "$$", "\\mathfrak p \\in \\text{Ass}(M) \\Leftrightarrow", "\\mathfrak p \\in \\text{WeakAss}(M).", "$$", "In particular, if $R$ is Noetherian, then $\\text{Ass}(M) = \\text{WeakAss}(M)$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\mathfrak p = (g_1, \\ldots, g_n)$ for some $g_i \\in R$.", "It is enough the prove the implication ``$\\Leftarrow$'' as the other", "implication holds in general, see", "Lemma \\ref{lemma-weakly-ass-support}.", "Assume $\\mathfrak p \\in \\text{WeakAss}(M)$.", "By", "Lemma \\ref{lemma-weakly-ass-local}", "there exists an element $m \\in M_{\\mathfrak p}$ such that", "$I = \\{x \\in R_{\\mathfrak p} \\mid xm = 0\\}$ has radical", "$\\mathfrak pR_{\\mathfrak p}$. Hence for each $i$ there exists", "a smallest $e_i > 0$ such that $g_i^{e_i}m = 0$ in $M_{\\mathfrak p}$.", "If $e_i > 1$ for some $i$, then we can replace $m$ by", "$g_i^{e_i - 1} m \\not = 0$ and decrease $\\sum e_i$.", "Hence we may assume that the annihilator of $m \\in M_{\\mathfrak p}$ is", "$(g_1, \\ldots, g_n)R_{\\mathfrak p} = \\mathfrak p R_{\\mathfrak p}$. By", "Lemma \\ref{lemma-associated-primes-localize}", "we see that $\\mathfrak p \\in \\text{Ass}(M)$." ], "refs": [ "algebra-lemma-weakly-ass-support", "algebra-lemma-weakly-ass-local", "algebra-lemma-associated-primes-localize" ], "ref_ids": [ 724, 720, 709 ] } ], "ref_ids": [] }, { "id": 728, "type": "theorem", "label": "algebra-lemma-weakly-ass-reverse-functorial", "categories": [ "algebra" ], "title": "algebra-lemma-weakly-ass-reverse-functorial", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module.", "Then we have", "$\\Spec(\\varphi)(\\text{WeakAss}_S(M)) \\supset \\text{WeakAss}_R(M)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p$ be an element of $\\text{WeakAss}_R(M)$.", "Then there exists an $m \\in M_{\\mathfrak p}$ whose annihilator", "$I = \\{x \\in R_{\\mathfrak p} \\mid xm = 0\\}$ has radical", "$\\mathfrak pR_{\\mathfrak p}$. Consider the annihilator", "$J = \\{x \\in S_{\\mathfrak p} \\mid xm = 0 \\}$ of $m$ in $S_{\\mathfrak p}$.", "As $IS_{\\mathfrak p} \\subset J$ we see that any minimal prime", "$\\mathfrak q \\subset S_{\\mathfrak p}$ over $J$ lies over $\\mathfrak p$.", "Moreover such a $\\mathfrak q$ corresponds to a weakly associated prime", "of $M$ for example by", "Lemma \\ref{lemma-weakly-ass-local}." ], "refs": [ "algebra-lemma-weakly-ass-local" ], "ref_ids": [ 720 ] } ], "ref_ids": [] }, { "id": 729, "type": "theorem", "label": "algebra-lemma-weakly-ass-finite-ring-map", "categories": [ "algebra" ], "title": "algebra-lemma-weakly-ass-finite-ring-map", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module.", "Denote $f : \\Spec(S) \\to \\Spec(R)$ the associated map on spectra.", "If $\\varphi$ is a finite ring map, then", "$$", "\\text{WeakAss}_R(M) = f(\\text{WeakAss}_S(M)).", "$$" ], "refs": [], "proofs": [ { "contents": [ "One of the inclusions has already been proved, see", "Remark \\ref{remark-ass-functorial}.", "To prove the other assume $\\mathfrak q \\in \\text{WeakAss}_S(M)$", "and let $\\mathfrak p$ be the corresponding prime of $R$. Let $m \\in M$", "be an element such that $\\mathfrak q$ is a minimal prime over", "$J = \\{g \\in S \\mid gm = 0\\}$. Thus the radical of", "$JS_{\\mathfrak q}$ is $\\mathfrak qS_{\\mathfrak q}$.", "As $R \\to S$ is finite there are", "finitely many primes", "$\\mathfrak q = \\mathfrak q_1, \\mathfrak q_2, \\ldots, \\mathfrak q_l$", "over $\\mathfrak p$, see", "Lemma \\ref{lemma-finite-finite-fibres}.", "Pick $x \\in \\mathfrak q$ with $x \\not \\in \\mathfrak q_i$ for $i > 1$, see", "Lemma \\ref{lemma-silly}.", "By the above there exists an element $y \\in S$, $y \\not \\in \\mathfrak q$", "and an integer $t > 0$ such that $y x^t m = 0$. Thus the element", "$ym \\in M$ is annihilated by $x^t$, hence $ym$ maps to zero in", "$M_{\\mathfrak q_i}$, $i = 2, \\ldots, l$. To be sure, $ym$ does not", "map to zero in $S_{\\mathfrak q}$.", "\\medskip\\noindent", "The ring $S_{\\mathfrak p}$ is semi-local with maximal ideals", "$\\mathfrak q_i S_{\\mathfrak p}$ by going up for finite ring maps, see", "Lemma \\ref{lemma-integral-going-up}.", "If $f \\in \\mathfrak pR_{\\mathfrak p}$ then some power of $f$ ends", "up in $JS_{\\mathfrak q}$ hence for some $n > 0$ we see that", "$f^t ym$ maps to zero in $M_{\\mathfrak q}$. As $ym$ vanishes at the", "other maximal ideals of $S_{\\mathfrak p}$ we conclude that $f^t ym$ is zero", "in $M_{\\mathfrak p}$, see", "Lemma \\ref{lemma-characterize-zero-local}.", "In this way we see that $\\mathfrak p$ is a minimal prime over", "the annihilator of $ym$ in $R$ and we win." ], "refs": [ "algebra-remark-ass-functorial", "algebra-lemma-finite-finite-fibres", "algebra-lemma-silly", "algebra-lemma-integral-going-up", "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 1565, 499, 378, 500, 410 ] } ], "ref_ids": [] }, { "id": 730, "type": "theorem", "label": "algebra-lemma-weakly-ass-quotient-ring", "categories": [ "algebra" ], "title": "algebra-lemma-weakly-ass-quotient-ring", "contents": [ "Let $R$ be a ring.", "Let $I$ be an ideal.", "Let $M$ be an $R/I$-module.", "Via the canonical injection", "$\\Spec(R/I) \\to \\Spec(R)$", "we have $\\text{WeakAss}_{R/I}(M) = \\text{WeakAss}_R(M)$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-weakly-ass-finite-ring-map}." ], "refs": [ "algebra-lemma-weakly-ass-finite-ring-map" ], "ref_ids": [ 729 ] } ], "ref_ids": [] }, { "id": 731, "type": "theorem", "label": "algebra-lemma-localize-weakly-ass", "categories": [ "algebra" ], "title": "algebra-lemma-localize-weakly-ass", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Let $S \\subset R$ be a multiplicative subset.", "Via the canonical injection $\\Spec(S^{-1}R) \\to \\Spec(R)$", "we have $\\text{WeakAss}_R(S^{-1}M) = \\text{WeakAss}_{S^{-1}R}(S^{-1}M)$", "and", "$$", "\\text{WeakAss}(M) \\cap \\Spec(S^{-1}R) = \\text{WeakAss}(S^{-1}M).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $m \\in S^{-1}M$. Let $I = \\{x \\in R \\mid xm = 0\\}$", "and $I' = \\{x' \\in S^{-1}R \\mid x'm = 0\\}$. Then $I' = S^{-1}I$", "and $I \\cap S = \\emptyset$ unless $I = R$ (verifications omitted).", "Thus primes in $S^{-1}R$ minimal over $I'$ correspond bijectively", "to primes in $R$ minimal over $I$ and avoiding $S$. This proves the", "equality $\\text{WeakAss}_R(S^{-1}M) = \\text{WeakAss}_{S^{-1}R}(S^{-1}M)$.", "The second equality follows from", "Lemma \\ref{lemma-weakly-ass-local}", "since for $\\mathfrak p \\in R$, $S \\cap \\mathfrak p = \\emptyset$ we have", "$M_{\\mathfrak p} = (S^{-1}M)_{S^{-1}\\mathfrak p}$." ], "refs": [ "algebra-lemma-weakly-ass-local" ], "ref_ids": [ 720 ] } ], "ref_ids": [] }, { "id": 732, "type": "theorem", "label": "algebra-lemma-localize-weakly-ass-nonzero-divisors", "categories": [ "algebra" ], "title": "algebra-lemma-localize-weakly-ass-nonzero-divisors", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Let $S \\subset R$ be a multiplicative subset.", "Assume that every $s \\in S$ is a nonzerodivisor on $M$.", "Then", "$$", "\\text{WeakAss}(M) = \\text{WeakAss}(S^{-1}M).", "$$" ], "refs": [], "proofs": [ { "contents": [ "As $M \\subset S^{-1}M$ by assumption we obtain", "$\\text{WeakAss}(M) \\subset \\text{WeakAss}(S^{-1}M)$ from", "Lemma \\ref{lemma-weakly-ass}.", "Conversely, suppose that $n/s \\in S^{-1}M$ is an element with annihilator", "$I$ and $\\mathfrak p$ a prime which is minimal over $I$.", "Then the annihilator of $n \\in M$ is $I$ and $\\mathfrak p$ is a prime", "minimal over $I$." ], "refs": [ "algebra-lemma-weakly-ass" ], "ref_ids": [ 722 ] } ], "ref_ids": [] }, { "id": 733, "type": "theorem", "label": "algebra-lemma-zero-at-weakly-ass-zero", "categories": [ "algebra" ], "title": "algebra-lemma-zero-at-weakly-ass-zero", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module. The map", "$$", "M", "\\longrightarrow", "\\prod\\nolimits_{\\mathfrak p \\in \\text{WeakAss}(M)} M_{\\mathfrak p}", "$$", "is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in M$ be an element of the kernel of the map. Set", "$N = Rx \\subset M$. If $\\mathfrak p$ is a weakly associated prime of $N$", "we see on the one hand that $\\mathfrak p \\in \\text{WeakAss}(M)$", "(Lemma \\ref{lemma-weakly-ass})", "and on the other hand that $N_{\\mathfrak p} \\subset M_{\\mathfrak p}$", "is not zero. This contradiction shows that $\\text{WeakAss}(N) = \\emptyset$.", "Hence $N = 0$, i.e., $x = 0$ by", "Lemma \\ref{lemma-weakly-ass-zero}." ], "refs": [ "algebra-lemma-weakly-ass", "algebra-lemma-weakly-ass-zero" ], "ref_ids": [ 722, 723 ] } ], "ref_ids": [] }, { "id": 734, "type": "theorem", "label": "algebra-lemma-weak-post-bourbaki", "categories": [ "algebra" ], "title": "algebra-lemma-weak-post-bourbaki", "contents": [ "Let $R \\to S$ be a ring map.", "Let $N$ be an $S$-module.", "Assume $N$ is flat as an $R$-module and", "$R$ is a domain with fraction field $K$.", "Then", "$$", "\\text{WeakAss}_S(N) = \\text{WeakAss}_{S \\otimes_R K}(N \\otimes_R K)", "$$", "via the canonical inclusion", "$\\Spec(S \\otimes_R K) \\subset \\Spec(S)$." ], "refs": [], "proofs": [ { "contents": [ "Note that $S \\otimes_R K = (R \\setminus \\{0\\})^{-1}S$ and", "$N \\otimes_R K = (R \\setminus \\{0\\})^{-1}N$.", "For any nonzero $x \\in R$ multiplication by $x$ on $N$ is injective as", "$N$ is flat over $R$. Hence the lemma follows from", "Lemma \\ref{lemma-localize-weakly-ass-nonzero-divisors}." ], "refs": [ "algebra-lemma-localize-weakly-ass-nonzero-divisors" ], "ref_ids": [ 732 ] } ], "ref_ids": [] }, { "id": 735, "type": "theorem", "label": "algebra-lemma-weakly-ass-change-fields", "categories": [ "algebra" ], "title": "algebra-lemma-weakly-ass-change-fields", "contents": [ "Let $K/k$ be a field extension. Let $R$ be a $k$-algebra.", "Let $M$ be an $R$-module. Let $\\mathfrak q \\subset R \\otimes_k K$", "be a prime lying over $\\mathfrak p \\subset R$. If", "$\\mathfrak q$ is weakly associated to $M \\otimes_k K$,", "then $\\mathfrak p$ is weakly associated to $M$." ], "refs": [], "proofs": [ { "contents": [ "Let $z \\in M \\otimes_k K$ be an element such that $\\mathfrak q$", "is minimal over the annihilator $J \\subset R \\otimes_k K$ of $z$.", "Choose a finitely generated subextension $K/L/k$ such that", "$z \\in M \\otimes_k L$. Since $R \\otimes_k L \\to R \\otimes_k K$", "is flat we see that $J = I(R \\otimes_k K)$ where $I \\subset R \\otimes_k L$", "is the annihilator of $z$ in the smaller ring", "(Lemma \\ref{lemma-annihilator-flat-base-change}).", "Thus $\\mathfrak q \\cap (R \\otimes_k L)$ is minimal over $I$ by", "going down (Lemma \\ref{lemma-flat-going-down}).", "In this way we reduce to the case described in the next paragraph.", "\\medskip\\noindent", "Assume $K/k$ is a finitely generated field extension.", "Let $x_1, \\ldots, x_r \\in K$ be a transcendence basis", "of $K$ over $k$, see Fields, Section \\ref{fields-section-transcendence}.", "Set $L = k(x_1, \\ldots, x_r)$. Say $[K : L] = n$. Then", "$R \\otimes_k L \\to R \\otimes_k K$ is a finite ring map.", "Hence $\\mathfrak q \\cap (R \\otimes_k L)$", "is a weakly associated prime of $M \\otimes_k K$", "viewed as a $R \\otimes_k L$-module by", "Lemma \\ref{lemma-weakly-ass-finite-ring-map}.", "Since $M \\otimes_k K \\cong (M \\otimes_k L)^{\\oplus n}$", "as a $R \\otimes_k L$-module, we see that", "$\\mathfrak q \\cap (R \\otimes_k L)$", "is a weakly associated prime of $M \\otimes_k L$", "(for example by using Lemma \\ref{lemma-weakly-ass} and induction).", "In this way we reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $K = k(x_1, \\ldots, x_r)$ is a purely transcendental field extension.", "We may replace $R$ by $R_\\mathfrak p$, $M$ by $M_\\mathfrak p$", "and $\\mathfrak q$ by $\\mathfrak q(R_\\mathfrak p \\otimes_k K)$.", "See Lemma \\ref{lemma-localize-weakly-ass}.", "In this way we reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $K = k(x_1, \\ldots, x_r)$ is a purely transcendental field extension", "and $R$ is local with maximal ideal $\\mathfrak p$. We claim that any", "$f \\in R \\otimes_k K$, $f \\not \\in \\mathfrak p(R \\otimes_k K)$", "is a nonzerodivisor on $M \\otimes_k K$. Namely, let", "$z \\in M \\otimes_k K$ be an element.", "There is a finite $R$-submodule $M' \\subset M$ such that", "$z \\in M' \\otimes_k K$ and such that $M'$ is minimal with", "this property: choose a basis $\\{t_\\alpha\\}$ of $K$ as a", "$k$-vector space, write $z = \\sum m_\\alpha \\otimes t_\\alpha$ and let", "$M'$ be the $R$-submodule generated by the $m_\\alpha$.", "If $z \\in \\mathfrak p(M' \\otimes_k K) = \\mathfrak p M' \\otimes_k K$,", "then $\\mathfrak pM' = M'$ and $M' = 0$ by Lemma \\ref{lemma-NAK}", "a contradiction.", "Thus $z$ has nonzero image $\\overline{z}$ in $M'/\\mathfrak p M' \\otimes_k K$", "But $R/\\mathfrak p \\otimes_k K$ is a domain as a localization", "of $\\kappa(\\mathfrak p)[x_1, \\ldots, x_n]$ and", "$M'/\\mathfrak p M' \\otimes_k K$ is a free module, hence", "$f\\overline{z} \\not = 0$. This proves the claim.", "\\medskip\\noindent", "Finally, pick $z \\in M \\otimes_k K$ such that $\\mathfrak q$", "is minimal over the annihilator $J \\subset R \\otimes_k K$ of $z$.", "For $f \\in \\mathfrak p$ there exists an $n \\geq 1$ and a", "$g \\in R \\otimes_k K$, $g \\not \\in \\mathfrak q$ such that", "$g f^n z \\in J$, i.e., $g f^n z = 0$.", "(This holds because $\\mathfrak q$ lies over $\\mathfrak p$", "and $\\mathfrak q$ is minimal over $J$.)", "Above we have seen that $g$ is a nonzerodivisor hence $f^n z = 0$.", "This means that $\\mathfrak p$ is a weakly associated prime", "of $M \\otimes_k K$ viewed as an $R$-module.", "Since $M \\otimes_k K$ is a direct sum of copies of $M$", "we conclude that $\\mathfrak p$ is a weakly associated", "prime of $M$ as before." ], "refs": [ "algebra-lemma-annihilator-flat-base-change", "algebra-lemma-flat-going-down", "algebra-lemma-weakly-ass-finite-ring-map", "algebra-lemma-weakly-ass", "algebra-lemma-localize-weakly-ass", "algebra-lemma-NAK" ], "ref_ids": [ 542, 539, 729, 722, 731, 401 ] } ], "ref_ids": [] }, { "id": 736, "type": "theorem", "label": "algebra-lemma-remove-embedded-primes", "categories": [ "algebra" ], "title": "algebra-lemma-remove-embedded-primes", "contents": [ "Let $R$ be a Noetherian ring.", "Let $M$ be a finite $R$-module.", "Consider the set of $R$-submodules", "$$", "\\{", "K \\subset M", "\\mid", "\\text{Supp}(K)", "\\text{ nowhere dense in }", "\\text{Supp}(M)", "\\}.", "$$", "This set has a maximal element $K$ and the quotient", "$M' = M/K$ has the following properties", "\\begin{enumerate}", "\\item $\\text{Supp}(M) = \\text{Supp}(M')$,", "\\item $M'$ has no embedded associated primes,", "\\item for any $f \\in R$ which is contained in all", "embedded associated primes of $M$ we have $M_f \\cong M'_f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ denote the", "minimal primes in the support of $M$. Let", "$\\mathfrak p_1, \\ldots, \\mathfrak p_s$ denote the", "embedded associated primes of $M$. Then", "$\\text{Ass}(M) = \\{\\mathfrak q_j, \\mathfrak p_i\\}$.", "There are finitely many of these,", "see Lemma \\ref{lemma-finite-ass}.", "Set $I = \\prod_{i = 1, \\ldots, s} \\mathfrak p_i$.", "Then $I \\not \\subset \\mathfrak q_j$ for", "any $j$. Hence by Lemma \\ref{lemma-silly} we can find an", "$f \\in I$ such that $f \\not \\in \\mathfrak q_j$ for", "all $j = 1, \\ldots, t$. Set $M' = \\Im(M \\to M_f)$.", "This implies that $M_f \\cong M'_f$. Since", "$M' \\subset M_f$ we see that", "$\\text{Ass}(M') \\subset \\text{Ass}(M_f) = \\{\\mathfrak q_j\\}$.", "Thus $M'$ has no embedded associated primes.", "\\medskip\\noindent", "Moreover, the support of $K = \\Ker(M \\to M')$", "is contained in $V(\\mathfrak p_1) \\cup \\ldots \\cup V(\\mathfrak p_s)$,", "because $\\text{Ass}(K) \\subset \\text{Ass}(M)$", "(see Lemma \\ref{lemma-ass}) and $\\text{Ass}(K)$ contains none", "of the $\\mathfrak q_i$ by construction.", "Clearly, $K$ is in fact the largest submodule of $M$ whose support is", "contained in $V(\\mathfrak p_1) \\cup \\ldots \\cup V(\\mathfrak p_t)$.", "This implies that $K$ is the maximal element of the set displayed", "in the lemma." ], "refs": [ "algebra-lemma-finite-ass", "algebra-lemma-silly", "algebra-lemma-ass" ], "ref_ids": [ 701, 378, 699 ] } ], "ref_ids": [] }, { "id": 737, "type": "theorem", "label": "algebra-lemma-remove-embedded-primes-localize", "categories": [ "algebra" ], "title": "algebra-lemma-remove-embedded-primes-localize", "contents": [ "Let $R$ be a Noetherian ring.", "Let $M$ be a finite $R$-module.", "For any $f \\in R$ we have $(M')_f = (M_f)'$ where", "$M \\to M'$ and $M_f \\to (M_f)'$ are the quotients", "constructed in Lemma \\ref{lemma-remove-embedded-primes}." ], "refs": [ "algebra-lemma-remove-embedded-primes" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 736 ] }, { "id": 738, "type": "theorem", "label": "algebra-lemma-no-embedded-primes-endos", "categories": [ "algebra" ], "title": "algebra-lemma-no-embedded-primes-endos", "contents": [ "Let $R$ be a Noetherian ring.", "Let $M$ be a finite $R$-module without embedded associated primes.", "Let $I = \\{x \\in R \\mid xM = 0\\}$. Then the ring $R/I$ has no", "embedded primes." ], "refs": [], "proofs": [ { "contents": [ "We may replace $R$ by $R/I$.", "Hence we may assume every nonzero element", "of $R$ acts nontrivially on $M$.", "By Lemma \\ref{lemma-support-closed} this implies that", "$\\Spec(R)$ equals the support of $M$.", "Suppose that $\\mathfrak p$ is an embedded prime of $R$.", "Let $x \\in R$ be an element whose annihilator is $\\mathfrak p$.", "Consider the nonzero module $N = xM \\subset M$. It is annihilated", "by $\\mathfrak p$. Hence any associated prime $\\mathfrak q$ of $N$", "contains $\\mathfrak p$ and is also an associated prime of $M$.", "Then $\\mathfrak q$ would be an embedded associated prime of", "$M$ which contradicts the assumption of the lemma." ], "refs": [ "algebra-lemma-support-closed" ], "ref_ids": [ 543 ] } ], "ref_ids": [] }, { "id": 739, "type": "theorem", "label": "algebra-lemma-permute-xi", "categories": [ "algebra" ], "title": "algebra-lemma-permute-xi", "contents": [ "Let $R$ be a local Noetherian ring.", "Let $M$ be a finite $R$-module.", "Let $x_1, \\ldots, x_c$ be an $M$-regular sequence.", "Then any permutation of the $x_i$ is a regular", "sequence as well." ], "refs": [], "proofs": [ { "contents": [ "First we do the case $c = 2$.", "Consider $K \\subset M$ the kernel of $x_2 : M \\to M$.", "For any $z \\in K$ we know that $z = x_1 z'$", "for some $z' \\in M$ because", "$x_2$ is a nonzerodivisor on $M/x_1M$.", "Because $x_1$ is a nonzerodivisor on $M$ we see that $x_2 z' = 0$", "as well. Hence $x_1 : K \\to K$ is surjective.", "Thus $K = 0$ by Nakayama's Lemma \\ref{lemma-NAK}.", "Next, consider multiplication by $x_1$ on $M/x_2M$.", "If $z \\in M$ maps to an element $\\overline{z} \\in M/x_2M$", "in the kernel of this map, then $x_1 z = x_2 y$ for some $y \\in M$.", "But then since $x_1, x_2$ is a regular sequence we see that", "$y = x_1 y'$ for some $y' \\in M$. Hence $x_1 ( z - x_2 y' ) =0$", "and hence $z = x_2 y'$ and hence $\\overline{z} = 0$ as desired.", "\\medskip\\noindent", "For the general case, observe that any permutation is", "a composition of transpositions of adjacent indices.", "Hence it suffices to prove that", "$$", "x_1, \\ldots, x_{i-2}, x_i, x_{i-1}, x_{i + 1}, \\ldots, x_c", "$$", "is an $M$-regular sequence. This follows from the case we", "just did applied to the module $M/(x_1, \\ldots, x_{i-2})$", "and the length $2$ regular sequence $x_{i-1}, x_i$." ], "refs": [ "algebra-lemma-NAK" ], "ref_ids": [ 401 ] } ], "ref_ids": [] }, { "id": 740, "type": "theorem", "label": "algebra-lemma-flat-increases-depth", "categories": [ "algebra" ], "title": "algebra-lemma-flat-increases-depth", "contents": [ "Let $R, S$ be local rings. Let $R \\to S$ be a flat local ring homomorphism.", "Let $x_1, \\ldots, x_r$ be a sequence in $R$. Let $M$ be an $R$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $x_1, \\ldots, x_r$ is an $M$-regular sequence in $R$, and", "\\item the images of $x_1, \\ldots, x_r$ in $S$ form a $M \\otimes_R S$-regular", "sequence.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is so because $R \\to S$ is faithfully flat", "by Lemma \\ref{lemma-local-flat-ff}." ], "refs": [ "algebra-lemma-local-flat-ff" ], "ref_ids": [ 537 ] } ], "ref_ids": [] }, { "id": 741, "type": "theorem", "label": "algebra-lemma-regular-sequence-in-neighbourhood", "categories": [ "algebra" ], "title": "algebra-lemma-regular-sequence-in-neighbourhood", "contents": [ "Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module.", "Let $\\mathfrak p$ be a prime. Let $x_1, \\ldots, x_r$ be a sequence", "in $R$ whose image in $R_{\\mathfrak p}$ forms an $M_{\\mathfrak p}$-regular", "sequence. Then there exists a $g \\in R$, $g \\not \\in \\mathfrak p$", "such that the image of $x_1, \\ldots, x_r$ in $R_g$ forms", "an $M_g$-regular sequence." ], "refs": [], "proofs": [ { "contents": [ "Set", "$$", "K_i = \\Ker\\left(x_i : M/(x_1, \\ldots, x_{i - 1})M \\to", "M/(x_1, \\ldots, x_{i - 1})M\\right).", "$$", "This is a finite $R$-module whose localization at $\\mathfrak p$ is", "zero by assumption. Hence there exists a $g \\in R$, $g \\not \\in \\mathfrak p$", "such that $(K_i)_g = 0$ for all $i = 1, \\ldots, r$. This $g$ works." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 742, "type": "theorem", "label": "algebra-lemma-join-regular-sequences", "categories": [ "algebra" ], "title": "algebra-lemma-join-regular-sequences", "contents": [ "Let $A$ be a ring. Let $I$ be an ideal generated by a regular", "sequence $f_1, \\ldots, f_n$ in $A$. Let $g_1, \\ldots, g_m \\in A$ be", "elements whose images $\\overline{g}_1, \\ldots, \\overline{g}_m$ form a", "regular sequence in $A/I$. Then $f_1, \\ldots, f_n, g_1, \\ldots, g_m$", "is a regular sequence in $A$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 743, "type": "theorem", "label": "algebra-lemma-regular-sequence-short-exact-sequence", "categories": [ "algebra" ], "title": "algebra-lemma-regular-sequence-short-exact-sequence", "contents": [ "Let $R$ be a ring. Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$", "be a short exact sequence of $R$-modules. Let $f_1, \\ldots, f_r \\in R$.", "If $f_1, \\ldots, f_r$ is $M_1$-regular and $M_3$-regular, then", "$f_1, \\ldots, f_r$ is $M_2$-regular." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-snake}, if $f_1 : M_1 \\to M_1$ and", "$f_1 : M_3 \\to M_3$ are injective, then so is $f_1 : M_2 \\to M_2$", "and we obtain a short exact sequence", "$$", "0 \\to M_1/f_1M_1 \\to M_2/f_1M_2 \\to M_3/f_1M_3 \\to 0", "$$", "The lemma follows from this and induction on $r$. Some details omitted." ], "refs": [ "algebra-lemma-snake" ], "ref_ids": [ 328 ] } ], "ref_ids": [] }, { "id": 744, "type": "theorem", "label": "algebra-lemma-regular-sequence-powers", "categories": [ "algebra" ], "title": "algebra-lemma-regular-sequence-powers", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Let $f_1, \\ldots, f_r \\in R$ and $e_1, \\ldots, e_r > 0$ integers.", "Then $f_1, \\ldots, f_r$ is an $M$-regular sequence", "if and only if $f_1^{e_1}, \\ldots, f_r^{e_r}$", "is an $M$-regular sequence." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by induction on $r$. If $r = 1$ this follows from the", "following two easy facts: (a) a power of a nonzerodivisor on $M$", "is a nonzerodivisor on $M$ and (b) a divisor of a nonzerodivisor on $M$", "is a nonzerodivisor on $M$.", "If $r > 1$, then by induction applied to $M/f_1M$ we have that", "$f_1, f_2, \\ldots, f_r$ is an $M$-regular sequence if and only if", "$f_1, f_2^{e_2}, \\ldots, f_r^{e_r}$ is an $M$-regular sequence.", "Thus it suffices to show, given $e > 0$, that $f_1^e, f_2, \\ldots, f_r$", "is an $M$-regular sequence if and only if $f_1, \\ldots, f_r$", "is an $M$-regular sequence. We will prove this", "by induction on $e$. The case $e = 1$ is trivial. Since $f_1$ is a", "nonzerodivisor under both assumptions (by the case $r = 1$)", "we have a short exact sequence", "$$", "0 \\to M/f_1M \\xrightarrow{f_1^{e - 1}} M/f_1^eM \\to M/f_1^{e - 1}M \\to 0", "$$", "Suppose that $f_1, f_2, \\ldots, f_r$ is an $M$-regular sequence.", "Then by induction the elements $f_2, \\ldots, f_r$ are $M/f_1M$ and", "$M/f_1^{e - 1}M$-regular sequences. By", "Lemma \\ref{lemma-regular-sequence-short-exact-sequence}", "$f_2, \\ldots, f_r$ is $M/f_1^eM$-regular. Hence $f_1^e, f_2, \\ldots, f_r$", "is $M$-regular. Conversely, suppose", "that $f_1^e, f_2, \\ldots, f_r$ is an $M$-regular sequence. Then", "$f_2 : M/f_1^eM \\to M/f_1^eM$ is injective, hence", "$f_2 : M/f_1M \\to M/f_1M$ is injective, hence by induction(!)", "$f_2 : M/f_1^{e - 1}M \\to M/f_1^{e - 1}M$ is injective, hence", "$$", "0 \\to", "M/(f_1, f_2)M \\xrightarrow{f_1^{e - 1}}", "M/(f_1^e, f_2)M \\to", "M/(f_1^{e - 1}, f_2)M \\to 0", "$$", "is a short exact sequence by Lemma \\ref{lemma-snake}. This proves the", "converse for $r = 2$. If $r > 2$, then we have", "$f_3 : M/(f_1^e, f_2)M \\to M/(f_1^e, f_2)M$ is injective, hence", "$f_3 : M/(f_1, f_2)M \\to M/(f_1, f_2)M$ is injective, and so on.", "Some details omitted." ], "refs": [ "algebra-lemma-regular-sequence-short-exact-sequence", "algebra-lemma-snake" ], "ref_ids": [ 743, 328 ] } ], "ref_ids": [] }, { "id": 745, "type": "theorem", "label": "algebra-lemma-regular-sequence-in-polynomial-ring", "categories": [ "algebra" ], "title": "algebra-lemma-regular-sequence-in-polynomial-ring", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ which do not generate", "the unit ideal. The following are equivalent:", "\\begin{enumerate}", "\\item any permutation of $f_1, \\ldots, f_r$ is a regular sequence,", "\\item any subsequence of $f_1, \\ldots, f_r$ (in the given order) is", "a regular sequence, and", "\\item $f_1x_1, \\ldots, f_rx_r$ is a regular sequence in the polynomial", "ring $R[x_1, \\ldots, x_r]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2). We prove (2) implies (1) by induction", "on $r$. The case $r = 1$ is trivial. The case $r = 2$ says that if", "$a, b \\in R$ are a regular sequence and $b$ is a nonzerodivisor, then", "$b, a$ is a regular sequence. This is clear because the kernel of", "$a : R/(b) \\to R/(b)$ is isomorphic to the kernel of $b : R/(a) \\to R/(a)$", "if both $a$ and $b$ are nonzerodivisors. The case $r > 2$. Assume", "(2) holds and say we want to prove $f_{\\sigma(1)}, \\ldots, f_{\\sigma(r)}$", "is a regular sequence for some permutation $\\sigma$. We already know", "that $f_{\\sigma(1)}, \\ldots, f_{\\sigma(r - 1)}$ is a regular sequence", "by induction. Hence it suffices to show that $f_s$ where $s = \\sigma(r)$", "is a nonzerodivisor modulo $f_1, \\ldots, \\hat f_s, \\ldots, f_r$.", "If $s = r$ we are done. If $s < r$, then note that $f_s$ and $f_r$", "are both nonzerodivisors in the ring", "$R/(f_1, \\ldots, \\hat f_s, \\ldots, f_{r - 1})$", "(by induction hypothesis again). Since we know $f_s, f_r$ is a", "regular sequence in that ring we conclude by the case of sequence of length", "$2$ that $f_r, f_s$ is too.", "\\medskip\\noindent", "Note that $R[x_1, \\ldots, x_r]/(f_1x_1, \\ldots, f_ix_i)$ as an $R$-module", "is a direct sum of the modules", "$$", "R/I_E \\cdot x_1^{e_1} \\ldots x_r^{e_r}", "$$", "indexed by multi-indices $E = (e_1, \\ldots, e_r)$ where", "$I_E$ is the ideal generated by $f_j$ for $1 \\leq j \\leq i$", "with $e_j > 0$. Hence $f_{i + 1}x_i$ is a nonzerodivisor on this if", "and only if $f_{i + 1}$ is a nonzerodivisor on $R/I_E$ for all $E$.", "Taking $E$ with all positive entries, we see that $f_{i + 1}$", "is a nonzerodivisor on $R/(f_1, \\ldots, f_i)$. Thus (3) implies (2).", "Conversely, if (2) holds, then any subsequence of", "$f_1, \\ldots, f_i, f_{i + 1}$ is a regular sequence", "in particular $f_{i + 1}$ is a nonzerodivisor on all $R/I_E$.", "In this way we see that (2) implies (3)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 746, "type": "theorem", "label": "algebra-lemma-regular-quasi-regular", "categories": [ "algebra" ], "title": "algebra-lemma-regular-quasi-regular", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item A regular sequence $f_1, \\ldots, f_c$ of $R$ is a quasi-regular", "sequence.", "\\item Suppose that $M$ is an $R$-module and that $f_1, \\ldots, f_c$", "is an $M$-regular sequence. Then $f_1, \\ldots, f_c$ is an", "$M$-quasi-regular sequence.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Set $J = (f_1, \\ldots, f_c)$.", "We prove the first assertion by induction on $c$.", "We have to show that given any relation", "$\\sum_{|I| = n} a_I f^I \\in J^{n + 1}$ with $a_I \\in R$ we", "actually have $a_I \\in J$ for all multi-indices $I$. Since", "any element of $J^{n + 1}$ is of the form $\\sum_{|I| = n} b_I f^I$", "with $b_I \\in J$ we may assume, after replacing $a_I$ by $a_I - b_I$,", "the relation reads $\\sum_{|I| = n} a_I f^I = 0$. We can rewrite", "this as", "$$", "\\sum\\nolimits_{e = 0}^n", "\\left(", "\\sum\\nolimits_{|I'| = n - e}", "a_{I', e} f^{I'}", "\\right)", "f_c^e", "=", "0", "$$", "Here and below the ``primed'' multi-indices $I'$ are required to be of the form", "$I' = (i_1, \\ldots, i_{c - 1}, 0)$. We will show by descending", "induction on $l \\in \\{0, \\ldots, n\\}$", "that if we have a relation", "$$", "\\sum\\nolimits_{e = 0}^l", "\\left(", "\\sum\\nolimits_{|I'| = n - e}", "a_{I', e} f^{I'}", "\\right)", "f_c^e", "=", "0", "$$", "then $a_{I', e} \\in J$ for all $I', e$.", "Namely, set $J' = (f_1, \\ldots, f_{c-1})$.", "Observe that $\\sum\\nolimits_{|I'| = n - l} a_{I', l} f^{I'}$", "is mapped into $(J')^{n - l + 1}$ by $f_c^{l}$.", "By induction hypothesis (for the induction on $c$)", "we see that $f_c^l a_{I', l} \\in J'$.", "Because $f_c$ is not a zerodivisor on $R/J'$ (as $f_1, \\ldots, f_c$", "is a regular sequence) we conclude that $a_{I', l} \\in J'$.", "This allows us to rewrite the term", "$(\\sum\\nolimits_{|I'| = n - l} a_{I', l} f^{I'})f_c^l$", "in the form $(\\sum\\nolimits_{|I'| = n - l + 1} f_c b_{I', l - 1}", "f^{I'})f_c^{l-1}$. This gives a new relation of the form", "$$", "\\left(\\sum\\nolimits_{|I'| = n - l + 1}", "(a_{I', l-1} + f_c b_{I', l - 1}) f^{I'}\\right)f_c^{l-1}", "+", "\\sum\\nolimits_{e = 0}^{l - 2}", "\\left(", "\\sum\\nolimits_{|I'| = n - e}", "a_{I', e} f^{I'}", "\\right)", "f_c^e", "=", "0", "$$", "Now by the induction hypothesis (on $l$ this time) we see that", "all $a_{I', l-1} + f_c b_{I', l - 1} \\in J$ and", "all $a_{I', e} \\in J$ for $e \\leq l - 2$. This, combined with", "$a_{I', l} \\in J' \\subset J$ seen above, finishes the proof of the", "induction step.", "\\medskip\\noindent", "The second assertion means that given any formal expression", "$F = \\sum_{|I| = n} m_I X^I$, $m_I \\in M$ with $\\sum m_I f^I", "\\in J^{n + 1}M$, then all the coefficients $m_I$ are in $J$.", "This is proved in exactly the same way as we prove the corresponding", "result for the first assertion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 747, "type": "theorem", "label": "algebra-lemma-flat-base-change-quasi-regular", "categories": [ "algebra" ], "title": "algebra-lemma-flat-base-change-quasi-regular", "contents": [ "Let $R \\to R'$ be a flat ring map. Let $M$ be an $R$-module.", "Suppose that $f_1, \\ldots, f_r \\in R$ form an $M$-quasi-regular sequence.", "Then the images of $f_1, \\ldots, f_r$ in", "$R'$ form a $M \\otimes_R R'$-quasi-regular sequence." ], "refs": [], "proofs": [ { "contents": [ "Set $J = (f_1, \\ldots, f_r)$, $J' = JR'$ and $M' = M \\otimes_R R'$.", "We have to show the canonical map", "$\\mu : R'/J'[X_1, \\ldots X_r] \\otimes_{R'/J'} M'/J'M' \\to", "\\bigoplus (J')^nM'/(J')^{n + 1}M'$ is an isomorphism.", "Because $R \\to R'$ is flat the sequences", "$0 \\to J^nM \\to M$ and", "$0 \\to J^{n + 1}M \\to J^nM \\to J^nM/J^{n + 1}M \\to 0$", "remain exact on tensoring with $R'$. This first implies that", "$J^nM \\otimes_R R' = (J')^nM'$ and then that", "$(J')^nM'/(J')^{n + 1}M' = J^nM/J^{n + 1}M \\otimes_R R'$.", "Thus $\\mu$ is the tensor product of (\\ref{equation-quasi-regular}),", "which is an isomorphism by assumption,", "with $\\text{id}_{R'}$ and we conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 748, "type": "theorem", "label": "algebra-lemma-quasi-regular-sequence-in-neighbourhood", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-regular-sequence-in-neighbourhood", "contents": [ "Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module.", "Let $\\mathfrak p$ be a prime. Let $x_1, \\ldots, x_c$ be a sequence", "in $R$ whose image in $R_{\\mathfrak p}$ forms an", "$M_{\\mathfrak p}$-quasi-regular sequence. Then there exists a", "$g \\in R$, $g \\not \\in \\mathfrak p$", "such that the image of $x_1, \\ldots, x_c$ in $R_g$ forms", "an $M_g$-quasi-regular sequence." ], "refs": [], "proofs": [ { "contents": [ "Consider the kernel $K$ of the map (\\ref{equation-quasi-regular}).", "As $M/JM \\otimes_{R/J} R/J[X_1, \\ldots, X_c]$ is a finite", "$R/J[X_1, \\ldots, X_c]$-module and as $R/J[X_1, \\ldots, X_c]$ is", "Noetherian, we see that $K$ is also a finite $R/J[X_1, \\ldots, X_c]$-module.", "Pick homogeneous generators $k_1, \\ldots, k_t \\in K$.", "By assumption for each $i = 1, \\ldots, t$ there exists a $g_i \\in R$,", "$g_i \\not \\in \\mathfrak p$ such that $g_i k_i = 0$.", "Hence $g = g_1 \\ldots g_t$ works." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 749, "type": "theorem", "label": "algebra-lemma-truncate-quasi-regular", "categories": [ "algebra" ], "title": "algebra-lemma-truncate-quasi-regular", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Let $f_1, \\ldots, f_c \\in R$ be an $M$-quasi-regular sequence.", "For any $i$ the sequence", "$\\overline{f}_{i + 1}, \\ldots, \\overline{f}_c$", "of $\\overline{R} = R/(f_1, \\ldots, f_i)$ is an", "$\\overline{M} = M/(f_1, \\ldots, f_i)M$-quasi-regular sequence." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove this for $i = 1$. Set", "$\\overline{J} = (\\overline{f}_2, \\ldots, \\overline{f}_c) \\subset \\overline{R}$.", "Then", "\\begin{align*}", "\\overline{J}^n\\overline{M}/\\overline{J}^{n + 1}\\overline{M}", "& =", "(J^nM + f_1M)/(J^{n + 1}M + f_1M) \\\\", "& = J^nM / (J^{n + 1}M + J^nM \\cap f_1M).", "\\end{align*}", "Thus, in order to prove the lemma it suffices to show that", "$J^{n + 1}M + J^nM \\cap f_1M = J^{n + 1}M + f_1J^{n - 1}M$", "because that will show that", "$\\bigoplus_{n \\geq 0}", "\\overline{J}^n\\overline{M}/\\overline{J}^{n + 1}\\overline{M}$", "is the quotient of", "$\\bigoplus_{n \\geq 0} J^nM/J^{n + 1} \\cong M/JM[X_1, \\ldots, X_c]$", "by $X_1$. Actually, we have $J^nM \\cap f_1M = f_1J^{n - 1}M$.", "Namely, if $m \\not \\in J^{n - 1}M$, then $f_1m \\not \\in J^nM$", "because $\\bigoplus J^nM/J^{n + 1}M$ is the polynomial algebra", "$M/J[X_1, \\ldots, X_c]$ by assumption." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 750, "type": "theorem", "label": "algebra-lemma-quasi-regular-regular", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-regular-regular", "contents": [ "Let $(R, \\mathfrak m)$ be a local Noetherian ring.", "Let $M$ be a nonzero finite $R$-module.", "Let $f_1, \\ldots, f_c \\in \\mathfrak m$ be an $M$-quasi-regular sequence.", "Then $f_1, \\ldots, f_c$ is an $M$-regular sequence." ], "refs": [], "proofs": [ { "contents": [ "Set $J = (f_1, \\ldots, f_c)$.", "Let us show that $f_1$ is a nonzerodivisor on $M$.", "Suppose $x \\in M$ is not zero.", "By Krull's intersection theorem there exists an integer $r$", "such that $x \\in J^rM$ but $x \\not \\in J^{r + 1}M$, see", "Lemma \\ref{lemma-intersect-powers-ideal-module-zero}.", "Then $f_1 x \\in J^{r + 1}M$ is an element whose class in", "$J^{r + 1}M/J^{r + 2}M$ is nonzero by the assumed structure of", "$\\bigoplus J^nM/J^{n + 1}M$. Whence $f_1x \\not = 0$.", "\\medskip\\noindent", "Now we can finish the proof by induction on $c$ using", "Lemma \\ref{lemma-truncate-quasi-regular}." ], "refs": [ "algebra-lemma-intersect-powers-ideal-module-zero", "algebra-lemma-truncate-quasi-regular" ], "ref_ids": [ 627, 749 ] } ], "ref_ids": [] }, { "id": 751, "type": "theorem", "label": "algebra-lemma-quasi-regular-on-quotient", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-regular-on-quotient", "contents": [ "Let $R$ be a ring. Let $J = (f_1, \\ldots, f_r)$ be an ideal of $R$.", "Let $M$ be an $R$-module. Set $\\overline{R} = R/\\bigcap_{n \\geq 0} J^n$,", "$\\overline{M} = M/\\bigcap_{n \\geq 0} J^nM$, and denote", "$\\overline{f}_i$ the image of $f_i$ in $\\overline{R}$.", "Then $f_1, \\ldots, f_r$ is $M$-quasi-regular if and only if", "$\\overline{f}_1, \\ldots, \\overline{f}_r$ is $\\overline{M}$-quasi-regular." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$J^nM/J^{n + 1}M \\cong", "\\overline{J}^n\\overline{M}/\\overline{J}^{n + 1}\\overline{M}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 752, "type": "theorem", "label": "algebra-lemma-affine-blowup", "categories": [ "algebra" ], "title": "algebra-lemma-affine-blowup", "contents": [ "Let $R$ be a ring, $I \\subset R$ an ideal, and $a \\in I$.", "Let $R' = R[\\frac{I}{a}]$ be the affine blowup algebra. Then", "\\begin{enumerate}", "\\item the image of $a$ in $R'$ is a nonzerodivisor,", "\\item $IR' = aR'$, and", "\\item $(R')_a = R_a$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Immediate from the description of $R[\\frac{I}{a}]$ above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 753, "type": "theorem", "label": "algebra-lemma-blowup-base-change", "categories": [ "algebra" ], "title": "algebra-lemma-blowup-base-change", "contents": [ "Let $R \\to S$ be a ring map. Let $I \\subset R$ be an ideal", "and $a \\in I$. Set $J = IS$ and let $b \\in J$ be the image of $a$.", "Then $S[\\frac{J}{b}]$ is the quotient of $S \\otimes_R R[\\frac{I}{a}]$", "by the ideal of elements annihilated by some power of $b$." ], "refs": [], "proofs": [ { "contents": [ "Let $S'$ be the quotient of $S \\otimes_R R[\\frac{I}{a}]$ by its", "$b$-power torsion elements. The ring map", "$$", "S \\otimes_R R[\\textstyle{\\frac{I}{a}}]", "\\longrightarrow", "S[\\textstyle{\\frac{J}{b}}]", "$$", "is surjective and annihilates $a$-power torsion as $b$ is a nonzerodivisor", "in $S[\\frac{J}{b}]$. Hence we obtain a surjective map $S' \\to S[\\frac{J}{b}]$.", "To see that the kernel is trivial, we construct an inverse map. Namely, let", "$z = y/b^n$ be an element of $S[\\frac{J}{b}]$, i.e., $y \\in J^n$.", "Write $y = \\sum x_is_i$ with $x_i \\in I^n$ and $s_i \\in S$.", "We map $z$ to the class of $\\sum s_i \\otimes x_i/a^n$ in", "$S'$. This is well defined because an element of the kernel of the map", "$S \\otimes_R I^n \\to J^n$ is annihilated by $a^n$, hence maps to zero in $S'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 754, "type": "theorem", "label": "algebra-lemma-affine-blowup-quotient-description", "categories": [ "algebra" ], "title": "algebra-lemma-affine-blowup-quotient-description", "contents": [ "Let $R$ be a ring. Let $I = (a_1, \\ldots, a_n)$ be an ideal of $R$.", "Let $a = a_1$. Then there is a surjection", "$$", "R[x_2, \\ldots, x_n]/(a x_2 - a_2, \\ldots, a x_n - a_n)", "\\longrightarrow", "\\textstyle{R[\\frac{I}{a}]}", "$$", "whose kernel is the $a$-power torsion in the source." ], "refs": [], "proofs": [ { "contents": [ "Consider the ring map $P = \\mathbf{Z}[t_1, \\ldots, t_n] \\to R$", "sending $t_i$ to $a_i$. Set $J = (t_1, \\ldots, t_n)$.", "By Example \\ref{example-affine-blowup-algebra-polynomial} we have", "$P[\\frac{J}{t_1}] =", "P[x_2, \\ldots, x_n]/(t_1 x_2 - t_2, \\ldots, t_1 x_n - t_n)$.", "Apply Lemma \\ref{lemma-blowup-base-change} to the map $P \\to A$ to conclude." ], "refs": [ "algebra-lemma-blowup-base-change" ], "ref_ids": [ 753 ] } ], "ref_ids": [] }, { "id": 755, "type": "theorem", "label": "algebra-lemma-blowup-in-principal", "categories": [ "algebra" ], "title": "algebra-lemma-blowup-in-principal", "contents": [ "Let $R$ be a ring, $I \\subset R$ an ideal, and $a \\in I$.", "Set $R' = R[\\frac{I}{a}]$. If $f \\in R$ is such that $V(f) = V(I)$,", "then $f$ maps to a nonzerodivisor in $R'$ and $R'_f = R'_a = R_a$." ], "refs": [], "proofs": [ { "contents": [ "We will use the results of Lemma \\ref{lemma-affine-blowup}", "without further mention.", "The assumption $V(f) = V(I)$ implies $V(fR') = V(IR') = V(aR')$.", "Hence $a^n = fb$ and $f^m = ac$ for some $b, c \\in R'$.", "The lemma follows." ], "refs": [ "algebra-lemma-affine-blowup" ], "ref_ids": [ 752 ] } ], "ref_ids": [] }, { "id": 756, "type": "theorem", "label": "algebra-lemma-blowup-add-principal", "categories": [ "algebra" ], "title": "algebra-lemma-blowup-add-principal", "contents": [ "Let $R$ be a ring, $I \\subset R$ an ideal, $a \\in I$, and $f \\in R$.", "Set $R' = R[\\frac{I}{a}]$ and $R'' = R[\\frac{fI}{fa}]$. Then", "there is a surjective $R$-algebra map $R' \\to R''$ whose kernel", "is the set of $f$-power torsion elements of $R'$." ], "refs": [], "proofs": [ { "contents": [ "The map is given by sending $x/a^n$ for $x \\in I^n$ to $f^nx/(fa)^n$.", "It is straightforward to check this map is well defined and surjective.", "Since $af$ is a nonzero divisor in $R''$", "(Lemma \\ref{lemma-affine-blowup}) we see that the set of $f$-power", "torsion elements are mapped to zero. Conversely, if $x \\in R'$", "and $f^n x \\not = 0$ for all $n > 0$, then $(af)^n x \\not = 0$", "for all $n$ as $a$ is a nonzero divisor in $R'$. It follows", "that the image of $x$ in $R''$ is not zero by the description of", "$R''$ following Definition \\ref{definition-blow-up}." ], "refs": [ "algebra-lemma-affine-blowup", "algebra-definition-blow-up" ], "ref_ids": [ 752, 1488 ] } ], "ref_ids": [] }, { "id": 757, "type": "theorem", "label": "algebra-lemma-blowup-reduced", "categories": [ "algebra" ], "title": "algebra-lemma-blowup-reduced", "contents": [ "\\begin{slogan}", "Being reduced is invariant under blowup", "\\end{slogan}", "If $R$ is reduced then every (affine) blowup algebra of $R$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "Let $I \\subset R$ be an ideal and $a \\in I$. Suppose $x/a^n$ with", "$x \\in I^n$ is a nilpotent element of $R[\\frac{I}{a}]$. Then", "$(x/a^n)^m = 0$. Hence $a^N x^m = 0$ in $R$ for some $N \\geq 0$.", "After increasing $N$ if necessary we may assume $N = me$ for some", "$e \\geq 0$. Then $(a^e x)^m = 0$ and since $R$ is reduced we find", "$a^e x = 0$. This means that $x/a^n = 0$ in $R[\\frac{I}{a}]$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 758, "type": "theorem", "label": "algebra-lemma-blowup-domain", "categories": [ "algebra" ], "title": "algebra-lemma-blowup-domain", "contents": [ "Let $R$ be a domain, $I \\subset R$ an ideal, and $a \\in I$ a nonzero", "element. Then the affine blowup algebra $R[\\frac{I}{a}]$ is a domain." ], "refs": [], "proofs": [ { "contents": [ "Suppose $x/a^n$, $y/a^m$ with $x \\in I^n$, $y \\in I^m$", "are elements of $R[\\frac{I}{a}]$ whose product is zero.", "Then $a^N x y = 0$ in $R$. Since $R$ is a domain we conclude", "that either $x = 0$ or $y = 0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 759, "type": "theorem", "label": "algebra-lemma-blowup-dominant", "categories": [ "algebra" ], "title": "algebra-lemma-blowup-dominant", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $a \\in I$.", "If $a$ is not contained in any minimal prime of $R$, then", "$\\Spec(R[\\frac{I}{a}]) \\to \\Spec(R)$ has dense image." ], "refs": [], "proofs": [ { "contents": [ "If $a^k x = 0$ for $x \\in R$, then $x$ is contained in all the", "minimal primes of $R$ and hence nilpotent, see", "Lemma \\ref{lemma-Zariski-topology}.", "Thus the kernel of $R \\to R[\\frac{I}{a}]$ consists of nilpotent", "elements. Hence the result follows from", "Lemma \\ref{lemma-image-dense-generic-points}." ], "refs": [ "algebra-lemma-Zariski-topology", "algebra-lemma-image-dense-generic-points" ], "ref_ids": [ 389, 446 ] } ], "ref_ids": [] }, { "id": 760, "type": "theorem", "label": "algebra-lemma-valuation-ring-colimit-affine-blowups", "categories": [ "algebra" ], "title": "algebra-lemma-valuation-ring-colimit-affine-blowups", "contents": [ "Let $(R, \\mathfrak m)$ be a local domain with fraction field $K$.", "Let $R \\subset A \\subset K$ be a valuation ring which dominates $R$.", "Then", "$$", "A = \\colim R[\\textstyle{\\frac{I}{a}}]", "$$", "is a directed colimit of affine blowups $R \\to R[\\frac{I}{a}]$ with", "the following properties", "\\begin{enumerate}", "\\item $a \\in I \\subset \\mathfrak m$,", "\\item $I$ is finitely generated, and", "\\item the fibre ring of $R \\to R[\\frac{I}{a}]$ at $\\mathfrak m$", "is not zero.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Any blowup algebra $R[\\frac{I}{a}]$ is a domain contained in $K$ see", "Lemma \\ref{lemma-blowup-domain}. The lemma simply says that $A$ is the", "directed union of the ones where $a \\in I$ have properties (1), (2), (3).", "If $R[\\frac{I}{a}] \\subset A$ and $R[\\frac{J}{b}] \\subset A$, then", "we have", "$$", "R[\\textstyle{\\frac{I}{a}}] \\cup R[\\textstyle{\\frac{J}{b}}] \\subset", "R[\\textstyle{\\frac{IJ}{ab}}] \\subset A", "$$", "The first inclusion because $x/a^n = b^nx/(ab)^n$ and the second one", "because if $z \\in (IJ)^n$, then $z = \\sum x_iy_i$ with $x_i \\in I^n$", "and $y_i \\in J^n$ and hence $z/(ab)^n = \\sum (x_i/a^n)(y_i/b^n)$", "is contained in $A$.", "\\medskip\\noindent", "Consider a finite subset $E \\subset A$. Say $E = \\{e_1, \\ldots, e_n\\}$.", "Choose a nonzero $a \\in R$ such that we can write $e_i = f_i/a$ for", "all $i = 1, \\ldots, n$. Set $I = (f_1, \\ldots, f_n, a)$.", "We claim that $R[\\frac{I}{a}] \\subset A$. This is clear as an element", "of $R[\\frac{I}{a}]$ can be represented as a polynomial in the elements", "$e_i$. The lemma follows immediately from this observation." ], "refs": [ "algebra-lemma-blowup-domain" ], "ref_ids": [ 758 ] } ], "ref_ids": [] }, { "id": 761, "type": "theorem", "label": "algebra-lemma-resolution-by-finite-free", "categories": [ "algebra" ], "title": "algebra-lemma-resolution-by-finite-free", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item There exists an exact complex", "$$", "\\ldots \\to F_2 \\to F_1 \\to F_0 \\to M \\to 0.", "$$", "with $F_i$ free $R$-modules.", "\\item If $R$ is Noetherian and $M$ finite over $R$, then we", "can choose the complex such that $F_i$ is finite free.", "In other words, we can find an exact complex", "$$", "\\ldots \\to R^{\\oplus n_2} \\to R^{\\oplus n_1} \\to R^{\\oplus n_0} \\to M \\to 0.", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us explain only the Noetherian case.", "As a first step choose a surjection $R^{n_0} \\to M$.", "Then having constructed an exact complex of length", "$e$ we simply choose a surjection $R^{n_{e + 1}} \\to", "\\Ker(R^{n_e} \\to R^{n_{e-1}})$ which is possible", "because $R$ is Noetherian." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 762, "type": "theorem", "label": "algebra-lemma-homotopic-equal-homology", "categories": [ "algebra" ], "title": "algebra-lemma-homotopic-equal-homology", "contents": [ "Any two homotopic maps of complexes induce the same maps on", "(co)homology groups." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 763, "type": "theorem", "label": "algebra-lemma-compare-resolutions", "categories": [ "algebra" ], "title": "algebra-lemma-compare-resolutions", "contents": [ "Let $R$ be a ring. Let $M \\to N$ be a map of $R$-modules.", "Let $N_\\bullet \\to N$ be an arbitrary resolution.", "Let", "$$", "\\ldots \\to F_2 \\to F_1 \\to F_0 \\to M", "$$", "be a complex of $R$-modules where each $F_i$ is a free $R$-module. Then", "\\begin{enumerate}", "\\item there exists a map of complexes $F_\\bullet \\to N_\\bullet$ such that", "$$", "\\xymatrix{", "F_0 \\ar[r] \\ar[d] & M \\ar[d] \\\\", "N_0 \\ar[r] & N", "}", "$$", "is commutative, and", "\\item any two maps $\\alpha, \\beta : F_\\bullet \\to N_\\bullet$ as in (1)", "are homotopic.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Because $F_0$ is free we can find a map $F_0 \\to N_0$", "lifting the map $F_0 \\to M \\to N$. We obtain an induced", "map $F_1 \\to F_0 \\to N_0$ which ends up in the image", "of $N_1 \\to N_0$. Since $F_1$ is free we may lift this", "to a map $F_1 \\to N_1$. This in turn induces a map", "$F_2 \\to F_1 \\to N_1$ which maps to zero into", "$N_0$. Since $N_\\bullet$ is exact we see that", "the image of this map is contained in the image", "of $N_2 \\to N_1$. Hence we may lift to get a map", "$F_2 \\to N_2$. Repeat.", "\\medskip\\noindent", "Proof of (2). To show that $\\alpha, \\beta$ are homotopic it suffices", "to show the difference $\\gamma = \\alpha - \\beta$ is homotopic", "to zero. Note that the image of $\\gamma_0 : F_0 \\to N_0$", "is contained in the image of $N_1 \\to N_0$. Hence we may lift", "$\\gamma_0$ to a map $h_0 : F_0 \\to N_1$. Consider the map", "$\\gamma_1' = \\gamma_1 - h_0 \\circ d_{F, 1}$. By our choice of $h_0$", "we see that the image of $\\gamma_1'$ is contained in", "the kernel of $N_1 \\to N_0$. Since $N_\\bullet$ is exact", "we may lift $\\gamma_1'$ to a map $h_1 : F_1 \\to N_2$.", "At this point we have $\\gamma_1 = h_0 \\circ d_{F, 1}", "+ d_{N, 2} \\circ h_1$. Repeat." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 764, "type": "theorem", "label": "algebra-lemma-ext-welldefined", "categories": [ "algebra" ], "title": "algebra-lemma-ext-welldefined", "contents": [ "Let $R$ be a ring. Let $M_1, M_2, N$ be $R$-modules.", "Suppose that $F_{\\bullet}$ is a free resolution of the module $M_1$,", "and $G_{\\bullet}$ is a free resolution of the module $M_2$.", "Let $\\varphi : M_1 \\to M_2$ be a module map.", "Let $\\alpha : F_{\\bullet} \\to G_{\\bullet}$ be", "a map of complexes inducing $\\varphi$ on", "$M_1 = \\Coker(d_{F, 1}) \\to M_2 = \\Coker(d_{G, 1})$,", "see Lemma \\ref{lemma-compare-resolutions}.", "Then the induced maps", "$$", "H^i(\\alpha) :", "H^i(\\Hom_R(F_{\\bullet}, N))", "\\longrightarrow", "H^i(\\Hom_R(G_{\\bullet}, N))", "$$", "are independent of the choice of $\\alpha$.", "If $\\varphi$ is an isomorphism, so are all the maps", "$H^i(\\alpha)$. If $M_1 = M_2$, $F_\\bullet = G_\\bullet$, and", "$\\varphi$ is the identity, so are all the maps $H_i(\\alpha)$." ], "refs": [ "algebra-lemma-compare-resolutions" ], "proofs": [ { "contents": [ "Another map $\\beta : F_{\\bullet} \\to G_{\\bullet}$", "inducing $\\varphi$ is homotopic to $\\alpha$ by", "Lemma \\ref{lemma-compare-resolutions}. Hence the", "maps $\\Hom_R(F_\\bullet, N) \\to", "\\Hom_R(G_\\bullet, N)$ are homotopic.", "Hence the independence result follows from", "Lemma \\ref{lemma-homotopic-equal-homology}.", "\\medskip\\noindent", "Suppose that $\\varphi$ is an isomorphism.", "Let $\\psi : M_2 \\to M_1$ be an inverse.", "Choose $\\beta : G_{\\bullet} \\to F_{\\bullet}$", "be a map inducing $\\psi :", "M_2 = \\Coker(d_{G, 1}) \\to M_1 = \\Coker(d_{F, 1})$,", "see Lemma \\ref{lemma-compare-resolutions}.", "OK, and now consider the map", "$H^i(\\alpha) \\circ H^i(\\beta) =", "H^i(\\alpha \\circ \\beta)$. By the above the", "map $H^i(\\alpha \\circ \\beta)$ is the {\\it same}", "as the map $H^i(\\text{id}_{G_{\\bullet}}) = \\text{id}$.", "Similarly for the composition $H^i(\\beta) \\circ H^i(\\alpha)$.", "Hence $H^i(\\alpha)$ and $H^i(\\beta)$ are inverses of each other." ], "refs": [ "algebra-lemma-compare-resolutions", "algebra-lemma-homotopic-equal-homology", "algebra-lemma-compare-resolutions" ], "ref_ids": [ 763, 762, 763 ] } ], "ref_ids": [ 763 ] }, { "id": 765, "type": "theorem", "label": "algebra-lemma-long-exact-seq-ext", "categories": [ "algebra" ], "title": "algebra-lemma-long-exact-seq-ext", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Let $0 \\to N' \\to N \\to N'' \\to 0$ be a", "short exact sequence. Then we get a long exact", "sequence", "$$", "\\begin{matrix}", "0", "\\to \\Hom_R(M, N')", "\\to \\Hom_R(M, N)", "\\to \\Hom_R(M, N'')", "\\\\", "\\phantom{0\\ }", "\\to \\Ext^1_R(M, N')", "\\to \\Ext^1_R(M, N)", "\\to \\Ext^1_R(M, N'')", "\\to \\ldots", "\\end{matrix}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Pick a free resolution $F_{\\bullet} \\to M$.", "Since each of the $F_i$ are free we see that", "we get a short exact sequence of complexes", "$$", "0 \\to", "\\Hom_R(F_{\\bullet}, N') \\to", "\\Hom_R(F_{\\bullet}, N) \\to", "\\Hom_R(F_{\\bullet}, N'') \\to", "0", "$$", "Thus we get the long exact sequence from", "the snake lemma applied to this." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 766, "type": "theorem", "label": "algebra-lemma-reverse-long-exact-seq-ext", "categories": [ "algebra" ], "title": "algebra-lemma-reverse-long-exact-seq-ext", "contents": [ "Let $R$ be a ring. Let $N$ be an $R$-module.", "Let $0 \\to M' \\to M \\to M'' \\to 0$ be a", "short exact sequence. Then we get a long exact", "sequence", "$$", "\\begin{matrix}", "0", "\\to \\Hom_R(M'', N)", "\\to \\Hom_R(M, N)", "\\to \\Hom_R(M', N)", "\\\\", "\\phantom{0\\ }", "\\to \\Ext^1_R(M'', N)", "\\to \\Ext^1_R(M, N)", "\\to \\Ext^1_R(M', N)", "\\to \\ldots", "\\end{matrix}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Pick sets of generators $\\{m'_{i'}\\}_{i' \\in I'}$ and", "$\\{m''_{i''}\\}_{i'' \\in I''}$ of $M'$ and $M''$.", "For each $i'' \\in I''$ choose a lift $\\tilde m''_{i''} \\in M$", "of the element $m''_{i''} \\in M''$. Set $F' = \\bigoplus_{i' \\in I'} R$,", "$F'' = \\bigoplus_{i'' \\in I''} R$ and $F = F' \\oplus F''$.", "Mapping the generators of these free modules to the corresponding", "chosen generators gives surjective $R$-module maps $F' \\to M'$,", "$F'' \\to M''$, and $F \\to M$. We obtain a map of short exact sequences", "$$", "\\begin{matrix}", "0 & \\to & M' & \\to & M & \\to & M'' & \\to & 0 \\\\", "& & \\uparrow & & \\uparrow & & \\uparrow \\\\", "0 & \\to & F' & \\to & F & \\to & F'' & \\to & 0 \\\\", "\\end{matrix}", "$$", "By the snake lemma we see that the sequence of kernels", "$0 \\to K' \\to K \\to K'' \\to 0$ is short exact sequence of $R$-modules.", "Hence we can continue this process indefinitely. In other words", "we obtain a short exact sequence of resolutions fitting into the diagram", "$$", "\\begin{matrix}", "0 & \\to & M' & \\to & M & \\to & M'' & \\to & 0 \\\\", "& & \\uparrow & & \\uparrow & & \\uparrow \\\\", "0 & \\to & F_\\bullet' & \\to & F_\\bullet & \\to & F_\\bullet'' & \\to & 0 \\\\", "\\end{matrix}", "$$", "Because each of the sequences $0 \\to F'_n \\to F_n \\to F''_n \\to 0$", "is split exact (by construction) we obtain a short exact sequence of", "complexes", "$$", "0 \\to", "\\Hom_R(F''_{\\bullet}, N) \\to", "\\Hom_R(F_{\\bullet}, N) \\to", "\\Hom_R(F'_{\\bullet}, N) \\to", "0", "$$", "by applying the $\\Hom_R(-, N)$ functor.", "Thus we get the long exact sequence from", "the snake lemma applied to this." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 767, "type": "theorem", "label": "algebra-lemma-annihilate-ext", "categories": [ "algebra" ], "title": "algebra-lemma-annihilate-ext", "contents": [ "Let $R$ be a ring. Let $M$, $N$ be $R$-modules.", "Any $x\\in R$ such that either $xN = 0$, or $xM = 0$", "annihilates each of the modules $\\Ext^i_R(M, N)$." ], "refs": [], "proofs": [ { "contents": [ "Pick a free resolution $F_{\\bullet}$ of $M$.", "Since $\\Ext^i_R(M, N)$", "is defined as the cohomology of the complex", "$\\Hom_R(F_{\\bullet}, N)$ the lemma is", "clear when $xN = 0$. If $xM = 0$, then", "we see that multiplication by $x$ on $F_{\\bullet}$", "lifts the zero map on $M$. Hence by Lemma", "\\ref{lemma-ext-welldefined} we see that it", "induces the same map on Ext groups as the", "zero map." ], "refs": [ "algebra-lemma-ext-welldefined" ], "ref_ids": [ 764 ] } ], "ref_ids": [] }, { "id": 768, "type": "theorem", "label": "algebra-lemma-ext-noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-ext-noetherian", "contents": [ "Let $R$ be a Noetherian ring. Let $M$, $N$ be finite $R$-modules.", "Then $\\Ext^i_R(M, N)$ is a finite $R$-module for all $i$." ], "refs": [], "proofs": [ { "contents": [ "This holds because $\\Ext^i_R(M, N)$ is computed as the", "cohomology groups of a complex $\\Hom_R(F_\\bullet, N)$", "with each $F_n$ a finite free $R$-module, see", "Lemma \\ref{lemma-resolution-by-finite-free}." ], "refs": [ "algebra-lemma-resolution-by-finite-free" ], "ref_ids": [ 761 ] } ], "ref_ids": [] }, { "id": 769, "type": "theorem", "label": "algebra-lemma-depth-weak-sequence", "categories": [ "algebra" ], "title": "algebra-lemma-depth-weak-sequence", "contents": [ "Let $R$ be a ring, $I \\subset R$ an ideal, and $M$ a finite $R$-module.", "Then $\\text{depth}_I(M)$ is equal to the supremum of the lengths of", "sequences $f_1, \\ldots, f_r \\in I$ such that $f_i$ is a nonzerodivisor", "on $M/(f_1, \\ldots, f_{i - 1})M$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $IM = M$. Then Lemma \\ref{lemma-NAK} shows there exists", "an $f \\in I$ such that $f : M \\to M$ is $\\text{id}_M$. Hence", "$f, 0, 0, 0, \\ldots$ is an infinite sequence of successive", "nonzerodivisors and we see agreement holds in this case.", "If $IM \\not = M$, then we see that a sequence as in the lemma", "is an $M$-regular sequence and we conclude that agreement holds as well." ], "refs": [ "algebra-lemma-NAK" ], "ref_ids": [ 401 ] } ], "ref_ids": [] }, { "id": 770, "type": "theorem", "label": "algebra-lemma-bound-depth", "categories": [ "algebra" ], "title": "algebra-lemma-bound-depth", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring.", "Let $M$ be a nonzero finite $R$-module.", "Then $\\dim(\\text{Supp}(M)) \\geq \\text{depth}(M)$." ], "refs": [], "proofs": [ { "contents": [ "The proof is by induction on $\\dim(\\text{Supp}(M))$.", "If $\\dim(\\text{Supp}(M)) = 0$, then", "$\\text{Supp}(M) = \\{\\mathfrak m\\}$, whence $\\text{Ass}(M) = \\{\\mathfrak m\\}$", "(by Lemmas \\ref{lemma-ass-support} and \\ref{lemma-ass-zero}), and hence", "the depth of $M$ is zero for example by", "Lemma \\ref{lemma-ideal-nonzerodivisor}.", "For the induction step we assume $\\dim(\\text{Supp}(M)) > 0$.", "Let $f_1, \\ldots, f_d$ be a sequence of elements of $\\mathfrak m$", "such that $f_i$ is a nonzerodivisor on $M/(f_1, \\ldots, f_{i - 1})M$.", "According to Lemma \\ref{lemma-depth-weak-sequence} it suffices to prove", "$\\dim(\\text{Supp}(M)) \\geq d$. We may assume", "$d > 0$ otherwise the lemma holds. By", "Lemma \\ref{lemma-one-equation-module}", "we have $\\dim(\\text{Supp}(M/f_1M)) = \\dim(\\text{Supp}(M)) - 1$.", "By induction we conclude $\\dim(\\text{Supp}(M/f_1M)) \\geq d - 1$", "as desired." ], "refs": [ "algebra-lemma-ass-support", "algebra-lemma-ass-zero", "algebra-lemma-ideal-nonzerodivisor", "algebra-lemma-depth-weak-sequence" ], "ref_ids": [ 698, 702, 712, 769 ] } ], "ref_ids": [] }, { "id": 771, "type": "theorem", "label": "algebra-lemma-depth-finite-noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-depth-finite-noetherian", "contents": [ "Let $R$ be a Noetherian ring, $I \\subset R$ an ideal, and $M$ a", "finite nonzero $R$-module such that $IM \\not = M$. Then", "$\\text{depth}_I(M) < \\infty$." ], "refs": [], "proofs": [ { "contents": [ "Since $M/IM$ is nonzero we can choose $\\mathfrak p \\in \\text{Supp}(M/IM)$", "by Lemma \\ref{lemma-support-zero}. Then $(M/IM)_\\mathfrak p \\not = 0$", "which implies $I \\subset \\mathfrak p$ and moreover implies", "$M_\\mathfrak p \\not = IM_\\mathfrak p$ as localization is exact.", "Let $f_1, \\ldots, f_r \\in I$ be an $M$-regular sequence.", "Then $M_\\mathfrak p/(f_1, \\ldots, f_r)M_\\mathfrak p$ is", "nonzero as $(f_1, \\ldots, f_r) \\subset I$. As localization is", "flat we see that the images of $f_1, \\ldots, f_r$ form a", "$M_\\mathfrak p$-regular sequence in $I_\\mathfrak p$. Since this", "works for every $M$-regular sequence in $I$ we conclude that", "$\\text{depth}_I(M) \\leq \\text{depth}_{I_\\mathfrak p}(M_\\mathfrak p)$.", "The latter is $\\leq \\text{depth}(M_\\mathfrak p)$ which is", "$< \\infty$ by Lemma \\ref{lemma-bound-depth}." ], "refs": [ "algebra-lemma-support-zero", "algebra-lemma-bound-depth" ], "ref_ids": [ 541, 770 ] } ], "ref_ids": [] }, { "id": 772, "type": "theorem", "label": "algebra-lemma-depth-ext", "categories": [ "algebra" ], "title": "algebra-lemma-depth-ext", "contents": [ "Let $R$ be a Noetherian local ring with maximal ideal $\\mathfrak m$.", "Let $M$ be a nonzero finite $R$-module. Then $\\text{depth}(M)$", "is equal to the smallest integer $i$ such that", "$\\Ext^i_R(R/\\mathfrak m, M)$ is nonzero." ], "refs": [], "proofs": [ { "contents": [ "Let $\\delta(M)$ denote the depth of $M$ and let $i(M)$ denote", "the smallest integer $i$ such that $\\Ext^i_R(R/\\mathfrak m, M)$", "is nonzero. We will see in a moment that $i(M) < \\infty$.", "By Lemma \\ref{lemma-ideal-nonzerodivisor} we have", "$\\delta(M) = 0$ if and only if $i(M) = 0$, because", "$\\mathfrak m \\in \\text{Ass}(M)$ exactly means", "that $i(M) = 0$. Hence if $\\delta(M)$ or $i(M)$ is $> 0$, then we may", "choose $x \\in \\mathfrak m$ such that (a) $x$ is a nonzerodivisor", "on $M$, and (b) $\\text{depth}(M/xM) = \\delta(M) - 1$.", "Consider the long exact sequence", "of Ext-groups associated to the short exact sequence", "$0 \\to M \\to M \\to M/xM \\to 0$ by Lemma \\ref{lemma-long-exact-seq-ext}:", "$$", "\\begin{matrix}", "0", "\\to \\Hom_R(\\kappa, M)", "\\to \\Hom_R(\\kappa, M)", "\\to \\Hom_R(\\kappa, M/xM)", "\\\\", "\\phantom{0\\ }", "\\to \\Ext^1_R(\\kappa, M)", "\\to \\Ext^1_R(\\kappa, M)", "\\to \\Ext^1_R(\\kappa, M/xM)", "\\to \\ldots", "\\end{matrix}", "$$", "Since $x \\in \\mathfrak m$ all the maps $\\Ext^i_R(\\kappa, M)", "\\to \\Ext^i_R(\\kappa, M)$ are zero, see", "Lemma \\ref{lemma-annihilate-ext}.", "Thus it is clear that $i(M/xM) = i(M) - 1$. Induction on", "$\\delta(M)$ finishes the proof." ], "refs": [ "algebra-lemma-ideal-nonzerodivisor", "algebra-lemma-long-exact-seq-ext", "algebra-lemma-annihilate-ext" ], "ref_ids": [ 712, 765, 767 ] } ], "ref_ids": [] }, { "id": 773, "type": "theorem", "label": "algebra-lemma-depth-in-ses", "categories": [ "algebra" ], "title": "algebra-lemma-depth-in-ses", "contents": [ "Let $R$ be a local Noetherian ring. Let $0 \\to N' \\to N \\to N'' \\to 0$", "be a short exact sequence of nonzero finite $R$-modules.", "\\begin{enumerate}", "\\item", "$\\text{depth}(N) \\geq \\min\\{\\text{depth}(N'), \\text{depth}(N'')\\}$", "\\item", "$\\text{depth}(N'') \\geq \\min\\{\\text{depth}(N), \\text{depth}(N') - 1\\}$", "\\item", "$\\text{depth}(N') \\geq \\min\\{\\text{depth}(N), \\text{depth}(N'') + 1\\}$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Use the characterization of depth using the Ext groups", "$\\Ext^i(\\kappa, N)$, see Lemma \\ref{lemma-depth-ext},", "and use the long exact cohomology sequence", "$$", "\\begin{matrix}", "0", "\\to \\Hom_R(\\kappa, N')", "\\to \\Hom_R(\\kappa, N)", "\\to \\Hom_R(\\kappa, N'')", "\\\\", "\\phantom{0\\ }", "\\to \\Ext^1_R(\\kappa, N')", "\\to \\Ext^1_R(\\kappa, N)", "\\to \\Ext^1_R(\\kappa, N'')", "\\to \\ldots", "\\end{matrix}", "$$", "from Lemma \\ref{lemma-long-exact-seq-ext}." ], "refs": [ "algebra-lemma-depth-ext", "algebra-lemma-long-exact-seq-ext" ], "ref_ids": [ 772, 765 ] } ], "ref_ids": [] }, { "id": 774, "type": "theorem", "label": "algebra-lemma-depth-drops-by-one", "categories": [ "algebra" ], "title": "algebra-lemma-depth-drops-by-one", "contents": [ "Let $R$ be a local Noetherian ring and $M$ a nonzero finite $R$-module.", "\\begin{enumerate}", "\\item If $x \\in \\mathfrak m$ is a nonzerodivisor on $M$, then", "$\\text{depth}(M/xM) = \\text{depth}(M) - 1$.", "\\item Any $M$-regular sequence $x_1, \\ldots, x_r$ can be extended to an", "$M$-regular sequence of length $\\text{depth}(M)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) is a formal consequence of part (1). Let $x \\in R$ be as in (1).", "By the short exact sequence $0 \\to M \\to M \\to M/xM \\to 0$", "and Lemma \\ref{lemma-depth-in-ses} we see that the depth drops by at most 1.", "On the other hand, if $x_1, \\ldots, x_r \\in \\mathfrak m$", "is a regular sequence for $M/xM$, then $x, x_1, \\ldots, x_r$", "is a regular sequence for $M$. Hence we see that the depth drops by", "at least 1." ], "refs": [ "algebra-lemma-depth-in-ses" ], "ref_ids": [ 773 ] } ], "ref_ids": [] }, { "id": 775, "type": "theorem", "label": "algebra-lemma-inherit-minimal-primes", "categories": [ "algebra" ], "title": "algebra-lemma-inherit-minimal-primes", "contents": [ "Let $(R, \\mathfrak m)$ be a local Noetherian ring and $M$ a finite $R$-module.", "Let $x \\in \\mathfrak m$, $\\mathfrak p \\in \\text{Ass}(M)$, and $\\mathfrak q$", "minimal over $\\mathfrak p + (x)$. Then $\\mathfrak q \\in \\text{Ass}(M/x^nM)$", "for some $n \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "Pick a submodule $N \\subset M$ with $N \\cong R/\\mathfrak p$.", "By the Artin-Rees lemma (Lemma \\ref{lemma-Artin-Rees})", "we can pick $n > 0$ such that $N \\cap x^nM \\subset xN$.", "Let $\\overline{N} \\subset M/x^nM$ be the image of $N \\to M \\to M/x^nM$.", "By Lemma \\ref{lemma-ass} it suffices to show", "$\\mathfrak q \\in \\text{Ass}(\\overline{N})$.", "By our choice of $n$ there is a surjection", "$\\overline{N} \\to N/xN = R/\\mathfrak p + (x)$", "and hence $\\mathfrak q$ is in the support of $\\overline{N}$.", "Since $\\overline{N}$ is annihilated by $x^n$ and $\\mathfrak p$ we see that", "$\\mathfrak q$ is minimal among the primes in the support of $\\overline{N}$.", "Thus $\\mathfrak q$ is an associated prime of $\\overline{N}$ by", "Lemma \\ref{lemma-ass-minimal-prime-support}." ], "refs": [ "algebra-lemma-Artin-Rees", "algebra-lemma-ass", "algebra-lemma-ass-minimal-prime-support" ], "ref_ids": [ 625, 699, 703 ] } ], "ref_ids": [] }, { "id": 776, "type": "theorem", "label": "algebra-lemma-depth-dim-associated-primes", "categories": [ "algebra" ], "title": "algebra-lemma-depth-dim-associated-primes", "contents": [ "Let $(R, \\mathfrak m)$ be a local Noetherian ring and $M$ a finite $R$-module.", "For $\\mathfrak p \\in \\text{Ass}(M)$ we have", "$\\dim(R/\\mathfrak p) \\geq \\text{depth}(M)$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathfrak m \\in \\text{Ass}(M)$ then there is a nonzero element", "$x \\in M$ which is annihilated by all elements of $\\mathfrak m$.", "Thus $\\text{depth}(M) = 0$. In particular the lemma holds in this case.", "\\medskip\\noindent", "If $\\text{depth}(M) = 1$, then by the first paragraph", "we find that $\\mathfrak m \\not \\in \\text{Ass}(M)$.", "Hence $\\dim(R/\\mathfrak p) \\geq 1$ for all $\\mathfrak p \\in \\text{Ass}(M)$", "and the lemma is true in this case as well.", "\\medskip\\noindent", "We will prove the lemma in general by induction on $\\text{depth}(M)$", "which we may and do assume to be $> 1$. Pick $x \\in \\mathfrak m$ which", "is a nonzerodivisor on $M$. Note $x \\not \\in \\mathfrak p$", "(Lemma \\ref{lemma-ass-zero-divisors}).", "By Lemma \\ref{lemma-one-equation} we have", "$\\dim(R/\\mathfrak p + (x)) = \\dim(R/\\mathfrak p) - 1$.", "Thus there exists a prime $\\mathfrak q$ minimal over $\\mathfrak p + (x)$ with", "$\\dim(R/\\mathfrak q) = \\dim(R/\\mathfrak p) - 1$ (small argument omitted;", "hint: the dimension of a Noetherian local ring $A$ is the maximum", "of the dimensions of $A/\\mathfrak r$ taken over the minimal", "primes $\\mathfrak r$ of $A$). Pick $n$ as in", "Lemma \\ref{lemma-inherit-minimal-primes} so that", "$\\mathfrak q$ is an associated prime of $M/x^nM$.", "We may apply induction hypothesis to $M/x^nM$ and $\\mathfrak q$", "because $\\text{depth}(M/x^nM) = \\text{depth}(M) - 1$ by", "Lemma \\ref{lemma-depth-drops-by-one}. We find", "$\\dim(R/\\mathfrak q) \\geq \\text{depth}(M/x^nM)$ and we win." ], "refs": [ "algebra-lemma-ass-zero-divisors", "algebra-lemma-inherit-minimal-primes", "algebra-lemma-depth-drops-by-one" ], "ref_ids": [ 704, 775, 774 ] } ], "ref_ids": [] }, { "id": 777, "type": "theorem", "label": "algebra-lemma-depth-localization", "categories": [ "algebra" ], "title": "algebra-lemma-depth-localization", "contents": [ "Let $R$ be a local Noetherian ring and $M$ a finite $R$-module.", "For a prime ideal $\\mathfrak p \\subset R$ we have", "$\\text{depth}(M_\\mathfrak p) + \\dim(R/\\mathfrak p) \\geq \\text{depth}(M)$." ], "refs": [], "proofs": [ { "contents": [ "If $M_\\mathfrak p = 0$, then $\\text{depth}(M_\\mathfrak p) = \\infty$ and", "the lemma holds.", "If $\\text{depth}(M) \\leq \\dim(R/\\mathfrak p)$, then the lemma is true.", "If $\\text{depth}(M) > \\dim(R/\\mathfrak p)$, then $\\mathfrak p$ is not", "contained in any associated prime $\\mathfrak q$ of $M$ by", "Lemma \\ref{lemma-depth-dim-associated-primes}.", "Hence we can find an $x \\in \\mathfrak p$ not contained in any", "associated prime of $M$ by Lemma \\ref{lemma-silly} and", "Lemma \\ref{lemma-finite-ass}. Then $x$ is a nonzerodivisor", "on $M$, see Lemma \\ref{lemma-ass-zero-divisors}.", "Hence $\\text{depth}(M/xM) = \\text{depth}(M) - 1$ and", "$\\text{depth}(M_\\mathfrak p / x M_\\mathfrak p) =", "\\text{depth}(M_\\mathfrak p) - 1$ provided $M_\\mathfrak p$ is nonzero,", "see Lemma \\ref{lemma-depth-drops-by-one}.", "Thus we conclude by induction on $\\text{depth}(M)$." ], "refs": [ "algebra-lemma-depth-dim-associated-primes", "algebra-lemma-silly", "algebra-lemma-finite-ass", "algebra-lemma-ass-zero-divisors", "algebra-lemma-depth-drops-by-one" ], "ref_ids": [ 776, 378, 701, 704, 774 ] } ], "ref_ids": [] }, { "id": 778, "type": "theorem", "label": "algebra-lemma-depth-goes-down-finite", "categories": [ "algebra" ], "title": "algebra-lemma-depth-goes-down-finite", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring. Let $R \\to S$", "be a finite ring map. Let $\\mathfrak m_1, \\ldots, \\mathfrak m_n$", "be the maximal ideals of $S$. Let $N$ be a finite $S$-module.", "Then", "$$", "\\min\\nolimits_{i = 1, \\ldots, n} \\text{depth}(N_{\\mathfrak m_i}) =", "\\text{depth}_\\mathfrak m(N)", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-integral-no-inclusion}, \\ref{lemma-integral-going-up},", "and Lemma \\ref{lemma-finite-finite-fibres} the maximal ideals of", "$S$ are exactly the primes of $S$ lying over $\\mathfrak m$ and", "there are finitely many of them. Hence the statement of the lemma", "makes sense. We will prove the lemma by induction on", "$k = \\min\\nolimits_{i = 1, \\ldots, n} \\text{depth}(N_{\\mathfrak m_i})$.", "If $k = 0$, then $\\text{depth}(N_{\\mathfrak m_i}) = 0$ for some $i$.", "By Lemma \\ref{lemma-depth-ext} this means", "$\\mathfrak m_i S_{\\mathfrak m_i}$ is an associated prime", "of $N_{\\mathfrak m_i}$ and hence $\\mathfrak m_i$ is an", "associated prime of $N$ (Lemma \\ref{lemma-localize-ass}).", "By Lemma \\ref{lemma-ass-functorial-Noetherian} we see that", "$\\mathfrak m$ is an associated prime of $N$ as an $R$-module.", "Whence $\\text{depth}_\\mathfrak m(N) = 0$. This proves the base case.", "If $k > 0$, then we see that $\\mathfrak m_i \\not \\in \\text{Ass}_S(N)$.", "Hence $\\mathfrak m \\not \\in \\text{Ass}_R(N)$, again by", "Lemma \\ref{lemma-ass-functorial-Noetherian}.", "Thus we can find $f \\in \\mathfrak m$ which is not a zerodivisor on", "$N$, see Lemma \\ref{lemma-ideal-nonzerodivisor}. By", "Lemma \\ref{lemma-depth-drops-by-one}", "all the depths drop exactly by $1$ when passing from $N$ to", "$N/fN$ and the induction hypothesis does the rest." ], "refs": [ "algebra-lemma-integral-no-inclusion", "algebra-lemma-integral-going-up", "algebra-lemma-finite-finite-fibres", "algebra-lemma-depth-ext", "algebra-lemma-localize-ass", "algebra-lemma-ass-functorial-Noetherian", "algebra-lemma-ass-functorial-Noetherian", "algebra-lemma-ideal-nonzerodivisor", "algebra-lemma-depth-drops-by-one" ], "ref_ids": [ 498, 500, 499, 772, 710, 707, 707, 712, 774 ] } ], "ref_ids": [] }, { "id": 779, "type": "theorem", "label": "algebra-lemma-flat-base-change-ext", "categories": [ "algebra" ], "title": "algebra-lemma-flat-base-change-ext", "contents": [ "Given a flat ring map $R \\to R'$, an $R$-module $M$, and an", "$R'$-module $N'$ the natural map", "$$", "\\Ext^i_{R'}(M \\otimes_R R', N') \\to \\text{Ext}^i_R(M, N')", "$$", "is an isomorphism for $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Choose a free resolution $F_\\bullet$ of $M$.", "Since $R \\to R'$ is flat we see that $F_\\bullet \\otimes_R R'$ is", "a free resolution of $M \\otimes_R R'$ over $R'$.", "The statement is that the map", "$$", "\\Hom_{R'}(F_\\bullet \\otimes_R R', N') \\to", "\\Hom_R(F_\\bullet, N')", "$$", "induces an isomorphism on homology groups, which is true because", "it is an isomorphism of complexes by", "Lemma \\ref{lemma-adjoint-tensor-restrict}." ], "refs": [ "algebra-lemma-adjoint-tensor-restrict" ], "ref_ids": [ 374 ] } ], "ref_ids": [] }, { "id": 780, "type": "theorem", "label": "algebra-lemma-split-injection-after-completion", "categories": [ "algebra" ], "title": "algebra-lemma-split-injection-after-completion", "contents": [ "Let $R$ be a Noetherian ring. Let $I \\subset R$ be an ideal", "contained in the Jacobson radical of $R$.", "Let $N \\to M$ be a homomorphism of finite $R$-modules.", "Suppose that there exists arbitrarily large $n$ such that", "$N/I^nN \\to M/I^nM$ is a split injection.", "Then $N \\to M$ is a split injection." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\varphi : N \\to M$ satisfies the assumptions of the lemma.", "Note that this implies that $\\Ker(\\varphi) \\subset I^nN$", "for arbitrarily large $n$. Hence by", "Lemma \\ref{lemma-intersection-powers-ideal-module} we see that $\\varphi$", "is injection. Let $Q = M/N$ so that we have a short exact sequence", "$$", "0 \\to N \\to M \\to Q \\to 0.", "$$", "Let", "$$", "F_2 \\xrightarrow{d_2} F_1 \\xrightarrow{d_1} F_0 \\to Q \\to 0", "$$", "be a finite free resolution of $Q$. We can choose a map", "$\\alpha : F_0 \\to M$ lifting the map $F_0 \\to Q$. This induces a map", "$\\beta : F_1 \\to N$ such that $\\beta \\circ d_2 = 0$. The extension", "above is split if and only if there exists a map $\\gamma : F_0 \\to N$", "such that $\\beta = \\gamma \\circ d_1$. In other words, the class of", "$\\beta$ in $\\Ext^1_R(Q, N)$ is the obstruction to splitting", "the short exact sequence above.", "\\medskip\\noindent", "Suppose $n$ is a large integer such that $N/I^nN \\to M/I^nM$ is a", "split injection. This implies", "$$", "0 \\to N/I^nN \\to M/I^nM \\to Q/I^nQ \\to 0.", "$$", "is still short exact. Also, the sequence", "$$", "F_1/I^nF_1 \\xrightarrow{d_1} F_0/I^nF_0 \\to Q/I^nQ \\to 0", "$$", "is still exact. Arguing as above we see that the map", "$\\overline{\\beta} : F_1/I^nF_1 \\to N/I^nN$", "induced by $\\beta$ is equal to $\\gamma_n \\circ d_1$ for some", "map $\\overline{\\gamma_n} : F_0/I^nF_0 \\to N/I^n$.", "Since $F_0$ is free we can lift $\\overline{\\gamma_n}$ to a map", "$\\gamma_n : F_0 \\to N$ and then we see that", "$\\beta - \\gamma_n \\circ d_1$ is a map from $F_1$ into $I^nN$.", "In other words we conclude that", "$$", "\\beta \\in", "\\Im\\Big(\\Hom_R(F_0, N) \\to \\Hom_R(F_1, N)\\Big) + I^n\\Hom_R(F_1, N).", "$$", "for this $n$.", "\\medskip\\noindent", "Since we have this property for arbitrarily large $n$ by assumption", "we conclude that the image of $\\beta$ in the cokernel of", "$\\Hom_R(F_0, N) \\to \\Hom_R(F_1, N)$ is zero by ", "Lemma \\ref{lemma-intersection-powers-ideal-module}. Hence", "$\\beta$ is in the image of the map $\\Hom_R(F_0, N) \\to \\Hom_R(F_1, N)$ as", "desired." ], "refs": [ "algebra-lemma-intersection-powers-ideal-module", "algebra-lemma-intersection-powers-ideal-module" ], "ref_ids": [ 628, 628 ] } ], "ref_ids": [] }, { "id": 781, "type": "theorem", "label": "algebra-lemma-tor-welldefined", "categories": [ "algebra" ], "title": "algebra-lemma-tor-welldefined", "contents": [ "Let $R$ be a ring. Let $M_1, M_2, N$ be $R$-modules.", "Suppose that $F_\\bullet$ is a free resolution of", "the module $M_1$ and that $G_\\bullet$ is a free", "resolution of the module $M_2$. Let $\\varphi : M_1 \\to M_2$", "be a module map. Let $\\alpha : F_\\bullet \\to G_\\bullet$", "be a map of complexes inducing $\\varphi$ on", "$M_1 = \\Coker(d_{F, 1}) \\to M_2 = \\Coker(d_{G, 1})$,", "see Lemma \\ref{lemma-compare-resolutions}.", "Then the induced maps", "$$", "H_i(\\alpha) :", "H_i(F_\\bullet \\otimes_R N)", "\\longrightarrow", "H_i(G_\\bullet \\otimes_R N)", "$$", "are independent of the choice of $\\alpha$. If $\\varphi$", "is an isomorphism, so are all the maps $H_i(\\alpha)$.", "If $M_1 = M_2$, $F_\\bullet = G_\\bullet$, and", "$\\varphi$ is the identity, so are all the maps $H_i(\\alpha)$." ], "refs": [ "algebra-lemma-compare-resolutions" ], "proofs": [ { "contents": [ "The proof of this lemma is identical to the proof of Lemma", "\\ref{lemma-ext-welldefined}." ], "refs": [ "algebra-lemma-ext-welldefined" ], "ref_ids": [ 764 ] } ], "ref_ids": [ 763 ] }, { "id": 782, "type": "theorem", "label": "algebra-lemma-long-exact-sequence-tor", "categories": [ "algebra" ], "title": "algebra-lemma-long-exact-sequence-tor", "contents": [ "Let $R$ be a ring and let $M$ be an $R$-module.", "Suppose that $0 \\to N' \\to N \\to N'' \\to 0$ is a short", "exact sequence of $R$-modules. There exists a long", "exact sequence", "$$", "\\text{Tor}_1^R(M, N')", "\\to \\text{Tor}_1^R(M, N)", "\\to \\text{Tor}_1^R(M, N'')", "\\to", "M \\otimes_R N'", "\\to M \\otimes_R N", "\\to M \\otimes_R N''", "\\to 0", "$$" ], "refs": [], "proofs": [ { "contents": [ "The proof of this is the same as the proof of", "Lemma \\ref{lemma-long-exact-seq-ext}." ], "refs": [ "algebra-lemma-long-exact-seq-ext" ], "ref_ids": [ 765 ] } ], "ref_ids": [] }, { "id": 783, "type": "theorem", "label": "algebra-lemma-no-spectral-sequence", "categories": [ "algebra" ], "title": "algebra-lemma-no-spectral-sequence", "contents": [ "Let $(A_{\\bullet, \\bullet}, d, \\delta)$ be a double complex such", "that", "\\begin{enumerate}", "\\item Each row $A_{\\bullet, j}$ is a resolution of $R(A)_j$.", "\\item Each column $A_{i, \\bullet}$ is a resolution of $U(A)_i$.", "\\end{enumerate}", "Then there are canonical isomorphisms", "$$", "H_i(R(A)_\\bullet)", "\\cong", "H_i(U(A)_\\bullet).", "$$", "The isomorphisms are functorial with respect to morphisms", "of double complexes with the properties above." ], "refs": [], "proofs": [ { "contents": [ "We will show that $H_i(R(A)_\\bullet))$", "and $H_i(U(A)_\\bullet)$ are canonically", "isomorphic to a third group. Namely", "$$", "\\mathbf{H}_i(A) :=", "\\frac{", "\\{", "(a_{i, 0}, a_{i-1, 1}, \\ldots, a_{0, i})", "\\mid", "d(a_{i, 0}) = \\delta(a_{i-1, 1}), \\ldots,", "d(a_{1, i-1}) = \\delta(a_{0, i})", "\\}}", "{", "\\{", "d(a_{i + 1, 0}) + \\delta(a_{i, 1}),", "d(a_{i, 1}) + \\delta(a_{i-1, 2}),", "\\ldots,", "d(a_{1, i}) + \\delta(a_{0, i + 1})", "\\}", "}", "$$", "Here we use the notational convention that $a_{i, j}$ denotes", "an element of $A_{i, j}$. In other words, an element of $\\mathbf{H}_i$", "is represented by a zig-zag, represented as follows for $i = 2$", "$$", "\\xymatrix{", "a_{2, 0} \\ar@{|->}[r] & d(a_{2, 0}) = \\delta(a_{1, 1}) & \\\\", "& a_{1, 1} \\ar@{|->}[u] \\ar@{|->}[r] & d(a_{1, 1}) = \\delta(a_{0, 2}) \\\\", "& & a_{0, 2} \\ar@{|->}[u] \\\\", "}", "$$", "Naturally, we divide out by ``trivial'' zig-zags, namely the submodule", "generated by elements of the form $(0, \\ldots, 0, -\\delta(a_{t + 1, t-i}),", "d(a_{t + 1, t-i}), 0, \\ldots, 0)$. Note that there are canonical", "homomorphisms", "$$", "\\mathbf{H}_i(A) \\to H_i(R(A)_\\bullet), \\quad", "(a_{i, 0}, a_{i-1, 1}, \\ldots, a_{0, i}) \\mapsto", "\\text{class of image of }a_{0, i}", "$$", "and", "$$", "\\mathbf{H}_i(A) \\to H_i(U(A)_\\bullet), \\quad", "(a_{i, 0}, a_{i-1, 1}, \\ldots, a_{0, i}) \\mapsto", "\\text{class of image of }a_{i, 0}", "$$", "\\medskip\\noindent", "First we show that these maps are surjective.", "Suppose that $\\overline{r} \\in H_i(R(A)_\\bullet)$.", "Let $r \\in R(A)_i$ be a cocycle representing the", "class of $\\overline{r}$.", "Let $a_{0, i} \\in A_{0, i}$ be an element which", "maps to $r$. Because $\\delta(r) = 0$,", "we see that $\\delta(a_{0, i})$ is in the", "image of $d$. Hence there exists an element", "$a_{1, i-1} \\in A_{1, i-1}$ such that", "$d(a_{1, i-1}) = \\delta(a_{0, i})$. This in turn", "implies that $\\delta(a_{1, i-1})$ is in the kernel", "of $d$ (because $d(\\delta(a_{1, i-1})) = \\delta(d(a_{1, i-1}))", "= \\delta(\\delta(a_{0, i})) = 0$. By exactness of the", "rows we find an element $a_{2, i-2}$ such that", "$d(a_{2, i-2}) = \\delta(a_{1, i-1})$. And so on", "until a full zig-zag is found. Of course surjectivity", "of $\\mathbf{H}_i \\to H_i(U(A))$ is shown similarly.", "\\medskip\\noindent", "To prove injectivity we argue in exactly the same way.", "Namely, suppose we are given a zig-zag", "$(a_{i, 0}, a_{i-1, 1}, \\ldots, a_{0, i})$", "which maps to zero in $H_i(R(A)_\\bullet)$.", "This means that $a_{0, i}$ maps to an element", "of $\\Coker(A_{i, 1} \\to A_{i, 0})$", "which is in the image of", "$\\delta : \\Coker(A_{i + 1, 1} \\to A_{i + 1, 0}) \\to", "\\Coker(A_{i, 1} \\to A_{i, 0})$.", "In other words, $a_{0, i}$ is in the image of", "$\\delta \\oplus d : A_{0, i + 1} \\oplus A_{1, i} \\to A_{0, i}$.", "From the definition of trivial zig-zags we see that", "we may modify our zig-zag by a trivial one and", "assume that $a_{0, i} = 0$. This immediately", "implies that $d(a_{1, i-1}) = 0$. As the rows", "are exact this implies that $a_{1, i-1}$ is", "in the image of $d : A_{2, i-1} \\to A_{1, i-1}$.", "Thus we may modify our zig-zag once again by a", "trivial zig-zag and assume that our zig-zag looks", "like $(a_{i, 0}, a_{i-1, 1}, \\ldots, a_{2, i-2}, 0, 0)$.", "Continuing like this we obtain the desired injectivity.", "\\medskip\\noindent", "If $\\Phi : (A_{\\bullet, \\bullet}, d, \\delta)", "\\to (B_{\\bullet, \\bullet}, d, \\delta)$ is a morphism", "of double complexes both of which satisfy the conditions", "of the lemma, then we clearly obtain a commutative", "diagram", "$$", "\\xymatrix{", "H_i(U(A)_\\bullet) \\ar[d] &", "\\mathbf{H}_i(A) \\ar[r] \\ar[l] \\ar[d] &", "H_i(R(A)_\\bullet) \\ar[d] \\\\", "H_i(U(B)_\\bullet) &", "\\mathbf{H}_i(B) \\ar[r] \\ar[l] &", "H_i(R(B)_\\bullet) \\\\", "}", "$$", "This proves the functoriality." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 784, "type": "theorem", "label": "algebra-lemma-tor-left-right", "categories": [ "algebra" ], "title": "algebra-lemma-tor-left-right", "contents": [ "Let $R$ be a ring. For any $i \\geq 0$ the functors", "$\\text{Mod}_R \\times \\text{Mod}_R \\to \\text{Mod}_R$,", "$(M, N) \\mapsto \\text{Tor}_i^R(M, N)$ and", "$(M, N) \\mapsto \\text{Tor}_i^R(N, M)$ are", "canonically isomorphic." ], "refs": [], "proofs": [ { "contents": [ "Let $F_\\bullet$ be a free resolution of the module $M$ and", "let $G_\\bullet$ be a free resolution of the module $N$.", "Consider the double complex $(A_{i, j}, d, \\delta)$ defined", "as follows:", "\\begin{enumerate}", "\\item set $A_{i, j} = F_i \\otimes_R G_j$,", "\\item set $d_{i, j} : F_i \\otimes_R G_j \\to F_{i-1} \\otimes G_j$", "equal to $d_{F, i} \\otimes \\text{id}$, and", "\\item set $\\delta_{i, j} : F_i \\otimes_R G_j \\to F_i \\otimes G_{j-1}$", "equal to $\\text{id} \\otimes d_{G, j}$.", "\\end{enumerate}", "This double complex is usually simply denoted $F_\\bullet \\otimes_R G_\\bullet$.", "\\medskip\\noindent", "Since each $G_j$ is free, and hence flat we see that each", "row of the double complex is exact except in homological", "degree $0$. Since each $F_i$ is free and hence flat we see that each", "column of the double complex is exact except in homological", "degree $0$. Hence the double complex satisfies the conditions", "of Lemma \\ref{lemma-no-spectral-sequence}.", "\\medskip\\noindent", "To see what the lemma says we compute $R(A)_\\bullet$ and $U(A)_\\bullet$.", "Namely,", "\\begin{eqnarray*}", "R(A)_i & = & \\Coker(A_{1, i} \\to A_{0, i}) \\\\", "& = & \\Coker(F_1 \\otimes_R G_i \\to F_0 \\otimes_R G_i) \\\\", "& = & \\Coker(F_1 \\to F_0) \\otimes_R G_i \\\\", "& = & M \\otimes_R G_i", "\\end{eqnarray*}", "In fact these isomorphisms are compatible with the differentials", "$\\delta$ and we see that $R(A)_\\bullet = M \\otimes_R G_\\bullet$", "as homological complexes. In exactly the same way we see that", "$U(A)_\\bullet = F_\\bullet \\otimes_R N$. We get", "\\begin{eqnarray*}", "\\text{Tor}_i^R(M, N)", "& = & H_i(F_\\bullet \\otimes_R N) \\\\", "& = & H_i(U(A)_\\bullet) \\\\", "& = & H_i(R(A)_\\bullet) \\\\", "& = & H_i(M \\otimes_R G_\\bullet) \\\\", "& = & H_i(G_\\bullet \\otimes_R M) \\\\", "& = & \\text{Tor}_i^R(N, M)", "\\end{eqnarray*}", "Here the third equality is Lemma \\ref{lemma-no-spectral-sequence}, and", "the fifth equality uses the isomorphism $V \\otimes W = W \\otimes V$", "of the tensor product.", "\\medskip\\noindent", "Functoriality. Suppose that we have $R$-modules $M_\\nu$, $N_\\nu$,", "$\\nu = 1, 2$. Let $\\varphi : M_1 \\to M_2$ and $\\psi : N_1 \\to N_2$", "be morphisms of $R$-modules.", "Suppose that we have free resolutions $F_{\\nu, \\bullet}$", "for $M_\\nu$ and free resolutions $G_{\\nu, \\bullet}$ for $N_\\nu$.", "By Lemma \\ref{lemma-compare-resolutions} we may choose", "maps of complexes $\\alpha : F_{1, \\bullet} \\to F_{2, \\bullet}$", "and $\\beta : G_{1, \\bullet} \\to G_{2, \\bullet}$ compatible", "with $\\varphi$ and $\\psi$. We claim that", "the pair $(\\alpha, \\beta)$ induces a morphism of double", "complexes", "$$", "\\alpha \\otimes \\beta :", "F_{1, \\bullet} \\otimes_R G_{1, \\bullet}", "\\longrightarrow", "F_{2, \\bullet} \\otimes_R G_{2, \\bullet}", "$$", "This is really a very straightforward check using the rule", "that $F_{1, i} \\otimes_R G_{1, j} \\to F_{2, i} \\otimes_R G_{2, j}$", "is given by $\\alpha_i \\otimes \\beta_j$ where $\\alpha_i$,", "resp.\\ $\\beta_j$ is the degree $i$, resp.\\ $j$ component of $\\alpha$,", "resp.\\ $\\beta$. The reader also readily verifies that the", "induced maps $R(F_{1, \\bullet} \\otimes_R G_{1, \\bullet})_\\bullet", "\\to R(F_{2, \\bullet} \\otimes_R G_{2, \\bullet})_\\bullet$", "agrees with the map $M_1 \\otimes_R G_{1, \\bullet}", "\\to M_2 \\otimes_R G_{2, \\bullet}$ induced by $\\varphi \\otimes \\beta$.", "Similarly for the map induced on the $U(-)_\\bullet$ complexes.", "Thus the statement on functoriality follows from the statement", "on functoriality in Lemma \\ref{lemma-no-spectral-sequence}." ], "refs": [ "algebra-lemma-no-spectral-sequence", "algebra-lemma-no-spectral-sequence", "algebra-lemma-compare-resolutions", "algebra-lemma-no-spectral-sequence" ], "ref_ids": [ 783, 783, 763, 783 ] } ], "ref_ids": [] }, { "id": 785, "type": "theorem", "label": "algebra-lemma-tor-noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-tor-noetherian", "contents": [ "Let $R$ be a Noetherian ring. Let $M$, $N$ be finite $R$-modules.", "Then $\\text{Tor}_p^R(M, N)$ is a finite $R$-module for all $p$." ], "refs": [], "proofs": [ { "contents": [ "This holds because $\\text{Tor}_p^R(M, N)$ is computed as the", "cohomology groups of a complex $F_\\bullet \\otimes_R N$", "with each $F_n$ a finite free $R$-module, see", "Lemma \\ref{lemma-resolution-by-finite-free}." ], "refs": [ "algebra-lemma-resolution-by-finite-free" ], "ref_ids": [ 761 ] } ], "ref_ids": [] }, { "id": 786, "type": "theorem", "label": "algebra-lemma-characterize-flat", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-flat", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "The following are equivalent:", "\\begin{enumerate}", "\\item The module $M$ is flat over $R$.", "\\item For all $i > 0$ the functor $\\text{Tor}_i^R(M, -)$ is zero.", "\\item The functor $\\text{Tor}_1^R(M, -)$ is zero.", "\\item For all ideals $I \\subset R$ we have $\\text{Tor}_1^R(M, R/I) = 0$.", "\\item For all finitely generated ideals $I \\subset R$ we have", "$\\text{Tor}_1^R(M, R/I) = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose $M$ is flat. Let $N$ be an $R$-module.", "Let $F_\\bullet$ be a free resolution of $N$.", "Then $F_\\bullet \\otimes_R M$ is a resolution of $N \\otimes_R M$,", "by flatness of $M$. Hence all higher Tor groups vanish.", "\\medskip\\noindent", "It now suffices to show that the last condition implies that", "$M$ is flat. Let $I \\subset R$ be an ideal.", "Consider the short exact sequence", "$0 \\to I \\to R \\to R/I \\to 0$. Apply", "Lemma \\ref{lemma-long-exact-sequence-tor}. We get an", "exact sequence", "$$", "\\text{Tor}_1^R(M, R/I) \\to", "M \\otimes_R I \\to", "M \\otimes_R R \\to", "M \\otimes_R R/I \\to", "0", "$$", "Since obviously $M \\otimes_R R = M$ we conclude that the", "last hypothesis implies that $M \\otimes_R I \\to M$ is", "injective for every finitely generated ideal $I$.", "Thus $M$ is flat by Lemma \\ref{lemma-flat}." ], "refs": [ "algebra-lemma-long-exact-sequence-tor", "algebra-lemma-flat" ], "ref_ids": [ 782, 525 ] } ], "ref_ids": [] }, { "id": 787, "type": "theorem", "label": "algebra-lemma-flat-base-change-tor", "categories": [ "algebra" ], "title": "algebra-lemma-flat-base-change-tor", "contents": [ "Given a flat ring map $R \\to R'$ and $R$-modules", "$M$, $N$ the natural $R$-module map", "$\\text{Tor}_i^R(M, N)\\otimes_R R'", "\\to \\text{Tor}_i^{R'}(M \\otimes_R R', N \\otimes_R R')$", "is an isomorphism for all $i$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. This is true because a free resolution $F_\\bullet$ of $M$ over", "$R$ stays exact when tensoring with $R'$ over $R$ and hence", "$(F_\\bullet \\otimes_R N)\\otimes_R R'$ computes the Tor groups", "over $R'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 788, "type": "theorem", "label": "algebra-lemma-tor-commutes-filtered-colimits", "categories": [ "algebra" ], "title": "algebra-lemma-tor-commutes-filtered-colimits", "contents": [ "Let $R$ be a ring. Let $M = \\colim M_i$ be a filtered colimit of", "$R$-modules. Let $N$ be an $R$-module. Then", "$\\text{Tor}_n^R(M, N) = \\colim \\text{Tor}_n^R(M_i, N)$ for all $n$." ], "refs": [], "proofs": [ { "contents": [ "Choose a free resolution $F_\\bullet$ of $N$. Then", "$F_\\bullet \\otimes_R M = \\colim F_\\bullet \\otimes_R M_i$", "as complexes by Lemma \\ref{lemma-tensor-products-commute-with-limits}.", "Thus the result by Lemma \\ref{lemma-directed-colimit-exact}." ], "refs": [ "algebra-lemma-tensor-products-commute-with-limits", "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 363, 343 ] } ], "ref_ids": [] }, { "id": 789, "type": "theorem", "label": "algebra-lemma-characterize-projective", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-projective", "contents": [ "Let $R$ be a ring. Let $P$ be an $R$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $P$ is projective,", "\\item $P$ is a direct summand of a free $R$-module, and", "\\item $\\Ext^1_R(P, M) = 0$ for every $R$-module $M$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $P$ is projective. Choose a surjection $\\pi : F \\to P$ where $F$", "is a free $R$-module. As $P$ is projective there exists a", "$i \\in \\Hom_R(P, F)$ such that $\\pi \\circ i = \\text{id}_P$.", "In other words $F \\cong \\Ker(\\pi) \\oplus i(P)$ and we see", "that $P$ is a direct summand of $F$.", "\\medskip\\noindent", "Conversely, assume that $P \\oplus Q = F$ is a free $R$-module.", "Note that the free module $F = \\bigoplus_{i \\in I} R$ is projective", "as $\\Hom_R(F, M) = \\prod_{i \\in I} M$ and the functor", "$M \\mapsto \\prod_{i \\in I} M$ is exact.", "Then $\\Hom_R(F, -) = \\Hom_R(P, -) \\times \\Hom_R(Q, -)$", "as functors, hence both $P$ and $Q$ are projective.", "\\medskip\\noindent", "Assume $P \\oplus Q = F$ is a free $R$-module. Then we have a", "free resolution $F_\\bullet$ of the form", "$$", "\\ldots F \\xrightarrow{a} F \\xrightarrow{b} F \\to P \\to 0", "$$", "where the maps $a, b$ alternate and are equal to the projector onto", "$P$ and $Q$. Hence the complex $\\Hom_R(F_\\bullet, M)$ is split", "exact in degrees $\\geq 1$, whence we see the vanishing in (3).", "\\medskip\\noindent", "Assume $\\Ext^1_R(P, M) = 0$ for every $R$-module $M$.", "Pick a free resolution $F_\\bullet \\to P$. Set", "$M = \\Im(F_1 \\to F_0) = \\Ker(F_0 \\to P)$.", "Consider the element $\\xi \\in \\Ext^1_R(P, M)$ given by", "the class of the quotient map $\\pi : F_1 \\to M$. Since $\\xi$ is zero", "there exists a map $s : F_0 \\to M$ such that $\\pi = s \\circ (F_1 \\to F_0)$.", "Clearly, this means that", "$$", "F_0 = \\Ker(s) \\oplus \\Ker(F_0 \\to P) =", "P \\oplus \\Ker(F_0 \\to P)", "$$", "and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 790, "type": "theorem", "label": "algebra-lemma-characterize-finite-projective-noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-finite-projective-noetherian", "contents": [ "Let $R$ be a Noetherian ring. Let $P$ be a finite $R$-module.", "If $\\Ext^1_R(P, M) = 0$ for every finite $R$-module $M$, then", "$P$ is projective." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjection $R^{\\oplus n} \\to P$ with kernel $M$.", "Since $\\Ext^1_R(P, M) = 0$ this surjection is split and", "we conclude by Lemma \\ref{lemma-characterize-projective}." ], "refs": [ "algebra-lemma-characterize-projective" ], "ref_ids": [ 789 ] } ], "ref_ids": [] }, { "id": 791, "type": "theorem", "label": "algebra-lemma-direct-sum-projective", "categories": [ "algebra" ], "title": "algebra-lemma-direct-sum-projective", "contents": [ "A direct sum of projective modules is projective." ], "refs": [], "proofs": [ { "contents": [ "This is true by the characterization of projectives as direct", "summands of free modules in", "Lemma \\ref{lemma-characterize-projective}." ], "refs": [ "algebra-lemma-characterize-projective" ], "ref_ids": [ 789 ] } ], "ref_ids": [] }, { "id": 792, "type": "theorem", "label": "algebra-lemma-lift-projective-module", "categories": [ "algebra" ], "title": "algebra-lemma-lift-projective-module", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be a nilpotent ideal. Let", "$\\overline{P}$ be a projective $R/I$-module. Then there exists a", "projective $R$-module $P$ such that $P/IP \\cong \\overline{P}$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-characterize-projective}", "we can choose a set $A$ and a direct sum decomposition", "$\\bigoplus_{\\alpha \\in A} R/I = \\overline{P} \\oplus \\overline{K}$", "for some $R/I$-module $\\overline{K}$. Write $F = \\bigoplus_{\\alpha \\in A} R$", "for the free $R$-module on $A$. Choose a lift", "$p : F \\to F$ of the projector $\\overline{p}$ associated to", "the direct summand $\\overline{P}$ of $\\bigoplus_{\\alpha \\in A} R/I$.", "Note that $p^2 - p \\in \\text{End}_R(F)$ is a nilpotent", "endomorphism of $F$ (as $I$ is nilpotent and the matrix entries of", "$p^2 - p$ are in $I$; more precisely, if $I^n = 0$, then $(p^2 - p)^n = 0$).", "Hence by Lemma \\ref{lemma-lift-idempotents-noncommutative}", "we can modify our choice of $p$ and assume that $p$ is a projector.", "Set $P = \\Im(p)$." ], "refs": [ "algebra-lemma-characterize-projective", "algebra-lemma-lift-idempotents-noncommutative" ], "ref_ids": [ 789, 462 ] } ], "ref_ids": [] }, { "id": 793, "type": "theorem", "label": "algebra-lemma-lift-finite-projective-module", "categories": [ "algebra" ], "title": "algebra-lemma-lift-finite-projective-module", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be a locally nilpotent ideal. Let", "$\\overline{P}$ be a finite projective $R/I$-module. Then there exists a", "finite projective $R$-module $P$ such that $P/IP \\cong \\overline{P}$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\overline{P}$ is a direct summand of a free $R/I$-module", "$\\bigoplus_{\\alpha \\in A} R/I$ by Lemma \\ref{lemma-characterize-projective}.", "As $\\overline{P}$ is finite, it follows that $\\overline{P}$ is contained", "in $\\bigoplus_{\\alpha \\in A'} R/I$ for some $A' \\subset A$ finite.", "Hence we may assume we have a direct sum decomposition", "$(R/I)^{\\oplus n} = \\overline{P} \\oplus \\overline{K}$", "for some $n$ and some $R/I$-module $\\overline{K}$. Choose a lift", "$p \\in \\text{Mat}(n \\times n, R)$ of the projector $\\overline{p}$", "associated to the direct summand $\\overline{P}$ of $(R/I)^{\\oplus n}$.", "Note that $p^2 - p \\in \\text{Mat}(n \\times n, R)$ is nilpotent:", "as $I$ is locally nilpotent and the matrix entries $c_{ij}$ of", "$p^2 - p$ are in $I$ we have $c_{ij}^t = 0$ for some $t > 0$ and", "then $(p^2 - p)^{tn^2} = 0$ (by looking at the matrix coefficients).", "Hence by Lemma \\ref{lemma-lift-idempotents-noncommutative}", "we can modify our choice of $p$ and assume that $p$ is a projector.", "Set $P = \\Im(p)$." ], "refs": [ "algebra-lemma-characterize-projective", "algebra-lemma-lift-idempotents-noncommutative" ], "ref_ids": [ 789, 462 ] } ], "ref_ids": [] }, { "id": 794, "type": "theorem", "label": "algebra-lemma-lift-projective", "categories": [ "algebra" ], "title": "algebra-lemma-lift-projective", "contents": [ "Let $R$ be a ring.", "Let $I \\subset R$ be an ideal.", "Let $M$ be an $R$-module.", "Assume", "\\begin{enumerate}", "\\item $I$ is nilpotent,", "\\item $M/IM$ is a projective $R/I$-module,", "\\item $M$ is a flat $R$-module.", "\\end{enumerate}", "Then $M$ is a projective $R$-module." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-lift-projective-module} we can find a projective", "$R$-module $P$ and an isomorphism $P/IP \\to M/IM$. We are going to show", "that $M$ is isomorphic to $P$ which will finish the proof. Because $P$", "is projective we can lift the map $P \\to P/IP \\to M/IM$ to an $R$-module", "map $P \\to M$ which is an isomorphism modulo $I$. Since $I^n = 0$", "for some $n$, we can use the filtrations", "\\begin{align*}", "0 = I^nM \\subset I^{n - 1}M \\subset \\ldots \\subset IM \\subset M \\\\", "0 = I^nP \\subset I^{n - 1}P \\subset \\ldots \\subset IP \\subset P", "\\end{align*}", "to see that it suffices to show that the induced maps", "$I^aP/I^{a + 1}P \\to I^aM/I^{a + 1}M$ are bijective. Since both $P$", "and $M$ are flat $R$-modules we can identify this with the map", "$$", "I^a/I^{a + 1} \\otimes_{R/I} P/IP", "\\longrightarrow", "I^a/I^{a + 1} \\otimes_{R/I} M/IM", "$$", "induced by $P \\to M$. Since we chose $P \\to M$ such that the induced", "map $P/IP \\to M/IM$ is an isomorphism, we win." ], "refs": [ "algebra-lemma-lift-projective-module" ], "ref_ids": [ 792 ] } ], "ref_ids": [] }, { "id": 795, "type": "theorem", "label": "algebra-lemma-finite-projective", "categories": [ "algebra" ], "title": "algebra-lemma-finite-projective", "contents": [ "Let $R$ be a ring and let $M$ be an $R$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $M$ is finitely presented and $R$-flat,", "\\item $M$ is finite projective,", "\\item $M$ is a direct summand of a finite free $R$-module,", "\\item $M$ is finitely presented and", "for all $\\mathfrak p \\in \\Spec(R)$ the", "localization $M_{\\mathfrak p}$ is free,", "\\item $M$ is finitely presented and", "for all maximal ideals $\\mathfrak m \\subset R$ the", "localization $M_{\\mathfrak m}$ is free,", "\\item $M$ is finite and locally free,", "\\item $M$ is finite locally free, and", "\\item $M$ is finite, for every prime $\\mathfrak p$ the module", "$M_{\\mathfrak p}$ is free, and the function", "$$", "\\rho_M : \\Spec(R) \\to \\mathbf{Z}, \\quad", "\\mathfrak p", "\\longmapsto", "\\dim_{\\kappa(\\mathfrak p)} M \\otimes_R \\kappa(\\mathfrak p)", "$$", "is locally constant in the Zariski topology.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First suppose $M$ is finite projective, i.e., (2) holds.", "Take a surjection $R^n \\to M$ and let $K$ be the kernel.", "Since $M$ is projective,", "$0 \\to K \\to R^n \\to M \\to 0$ splits.", "Hence (2) $\\Rightarrow$ (3).", "The implication (3) $\\Rightarrow$ (2) follows from the fact that", "a direct summand of a projective is projective, see", "Lemma \\ref{lemma-characterize-projective}.", "\\medskip\\noindent", "Assume (3), so we can write $K \\oplus M \\cong R^{\\oplus n}$.", "So $K$ is a direct summand of $R^n$ and thus finitely generated.", "This shows $M = R^{\\oplus n}/K$ is finitely presented.", "In other words, (3) $\\Rightarrow$ (1).", "\\medskip\\noindent", "Assume $M$ is finitely presented and flat, i.e., (1) holds.", "We will prove that (7) holds. Pick any prime $\\mathfrak p$ and", "$x_1, \\ldots, x_r \\in M$ which map to a basis of", "$M \\otimes_R \\kappa(\\mathfrak p)$. By", "Nakayama's Lemma \\ref{lemma-NAK}", "these elements generate $M_g$ for some $g \\in R$, $g \\not \\in \\mathfrak p$.", "The corresponding surjection $\\varphi : R_g^{\\oplus r} \\to M_g$", "has the following two properties: (a) $\\Ker(\\varphi)$ is a finite", "$R_g$-module (see Lemma \\ref{lemma-extension})", "and (b) $\\Ker(\\varphi) \\otimes \\kappa(\\mathfrak p) = 0$", "by flatness of $M_g$ over $R_g$ (see", "Lemma \\ref{lemma-flat-tor-zero}).", "Hence by Nakayama's lemma again there exists a $g' \\in R_g$ such that", "$\\Ker(\\varphi)_{g'} = 0$. In other words, $M_{gg'}$ is free.", "\\medskip\\noindent", "A finite locally free module is a finite module, see", "Lemma \\ref{lemma-cover},", "hence (7) $\\Rightarrow$ (6).", "It is clear that (6) $\\Rightarrow$ (7) and that (7) $\\Rightarrow$ (8).", "\\medskip\\noindent", "A finite locally free module is a finitely presented module, see", "Lemma \\ref{lemma-cover},", "hence (7) $\\Rightarrow$ (4).", "Of course (4) implies (5).", "Since we may check flatness locally (see", "Lemma \\ref{lemma-flat-localization})", "we conclude that (5) implies (1).", "At this point we have", "$$", "\\xymatrix{", "(2) \\ar@{<=>}[r] & (3) \\ar@{=>}[r] & (1) \\ar@{=>}[r] &", "(7) \\ar@{<=>}[r] \\ar@{=>}[rd] \\ar@{=>}[d] & (6) \\\\", "& & (5) \\ar@{=>}[u] & (4) \\ar@{=>}[l] & (8)", "}", "$$", "\\medskip\\noindent", "Suppose that $M$ satisfies (1), (4), (5), (6), and (7).", "We will prove that (3) holds. It suffices", "to show that $M$ is projective. We have to show that $\\Hom_R(M, -)$", "is exact. Let $0 \\to N'' \\to N \\to N'\\to 0$ be a short exact sequence of", "$R$-module. We have to show that", "$0 \\to \\Hom_R(M, N'') \\to \\Hom_R(M, N) \\to", "\\Hom_R(M, N') \\to 0$ is exact.", "As $M$ is finite locally free there exist a covering", "$\\Spec(R) = \\bigcup D(f_i)$ such that $M_{f_i}$ is finite free.", "By", "Lemma \\ref{lemma-hom-from-finitely-presented}", "we see that", "$$", "0 \\to \\Hom_R(M, N'')_{f_i} \\to \\Hom_R(M, N)_{f_i} \\to", "\\Hom_R(M, N')_{f_i} \\to 0", "$$", "is equal to", "$0 \\to \\Hom_{R_{f_i}}(M_{f_i}, N''_{f_i}) \\to", "\\Hom_{R_{f_i}}(M_{f_i}, N_{f_i}) \\to", "\\Hom_{R_{f_i}}(M_{f_i}, N'_{f_i}) \\to 0$", "which is exact as $M_{f_i}$ is free and as the localization", "$0 \\to N''_{f_i} \\to N_{f_i} \\to N'_{f_i} \\to 0$", "is exact (as localization is exact). Whence we see that", "$0 \\to \\Hom_R(M, N'') \\to \\Hom_R(M, N) \\to", "\\Hom_R(M, N') \\to 0$ is exact by", "Lemma \\ref{lemma-cover}.", "\\medskip\\noindent", "Finally, assume that (8) holds. Pick a maximal ideal $\\mathfrak m \\subset R$.", "Pick $x_1, \\ldots, x_r \\in M$ which map to a $\\kappa(\\mathfrak m)$-basis of", "$M \\otimes_R \\kappa(\\mathfrak m) = M/\\mathfrak mM$. In particular", "$\\rho_M(\\mathfrak m) = r$. By", "Nakayama's Lemma \\ref{lemma-NAK}", "there exists an $f \\in R$, $f \\not \\in \\mathfrak m$ such that", "$x_1, \\ldots, x_r$ generate $M_f$ over $R_f$. By the assumption that", "$\\rho_M$ is locally constant there exists a $g \\in R$, $g \\not \\in \\mathfrak m$", "such that $\\rho_M$ is constant equal to $r$ on $D(g)$. We claim that", "$$", "\\Psi : R_{fg}^{\\oplus r} \\longrightarrow M_{fg}, \\quad", "(a_1, \\ldots, a_r) \\longmapsto \\sum a_i x_i", "$$", "is an isomorphism. This claim will show that $M$ is finite locally", "free, i.e., that (7) holds. To see the claim", "it suffices to show that the induced map on localizations", "$\\Psi_{\\mathfrak p} : R_{\\mathfrak p}^{\\oplus r} \\to M_{\\mathfrak p}$", "is an isomorphism for all $\\mathfrak p \\in D(fg)$, see", "Lemma \\ref{lemma-characterize-zero-local}.", "By our choice of $f$ the map $\\Psi_{\\mathfrak p}$", "is surjective. By assumption (8) we have", "$M_{\\mathfrak p} \\cong R_{\\mathfrak p}^{\\oplus \\rho_M(\\mathfrak p)}$", "and by our choice of $g$ we have $\\rho_M(\\mathfrak p) = r$.", "Hence $\\Psi_{\\mathfrak p}$ determines a surjection", "$R_{\\mathfrak p}^{\\oplus r} \\to", "M_{\\mathfrak p} \\cong R_{\\mathfrak p}^{\\oplus r}$", "whence is an isomorphism by", "Lemma \\ref{lemma-fun}.", "(Of course this last fact follows from a simple matrix argument also.)" ], "refs": [ "algebra-lemma-characterize-projective", "algebra-lemma-NAK", "algebra-lemma-extension", "algebra-lemma-flat-tor-zero", "algebra-lemma-cover", "algebra-lemma-cover", "algebra-lemma-flat-localization", "algebra-lemma-hom-from-finitely-presented", "algebra-lemma-cover", "algebra-lemma-NAK", "algebra-lemma-characterize-zero-local", "algebra-lemma-fun" ], "ref_ids": [ 789, 401, 330, 532, 411, 411, 538, 353, 411, 401, 410, 388 ] } ], "ref_ids": [] }, { "id": 796, "type": "theorem", "label": "algebra-lemma-finite-projective-reduced", "categories": [ "algebra" ], "title": "algebra-lemma-finite-projective-reduced", "contents": [ "Let $R$ be a reduced ring and let $M$ be an $R$-module. Then the", "equivalent conditions of Lemma \\ref{lemma-finite-projective}", "are also equivalent to", "\\begin{enumerate}", "\\item[(9)] $M$ is finite and the function", "$\\rho_M : \\Spec(R) \\to \\mathbf{Z}$, $\\mathfrak p \\mapsto", "\\dim_{\\kappa(\\mathfrak p)} M \\otimes_R \\kappa(\\mathfrak p)$", "is locally constant in the Zariski topology.", "\\end{enumerate}" ], "refs": [ "algebra-lemma-finite-projective" ], "proofs": [ { "contents": [ "Pick a maximal ideal $\\mathfrak m \\subset R$.", "Pick $x_1, \\ldots, x_r \\in M$ which map to a $\\kappa(\\mathfrak m)$-basis of", "$M \\otimes_R \\kappa(\\mathfrak m) = M/\\mathfrak mM$. In particular", "$\\rho_M(\\mathfrak m) = r$. By", "Nakayama's Lemma \\ref{lemma-NAK}", "there exists an $f \\in R$, $f \\not \\in \\mathfrak m$ such that", "$x_1, \\ldots, x_r$ generate $M_f$ over $R_f$. By the assumption that", "$\\rho_M$ is locally constant there exists a $g \\in R$, $g \\not \\in \\mathfrak m$", "such that $\\rho_M$ is constant equal to $r$ on $D(g)$. We claim that", "$$", "\\Psi : R_{fg}^{\\oplus r} \\longrightarrow M_{fg}, \\quad", "(a_1, \\ldots, a_r) \\longmapsto \\sum a_i x_i", "$$", "is an isomorphism. This claim will show that $M$ is finite locally", "free, i.e., that (7) holds. Since $\\Psi$ is surjective, it suffices to", "show that $\\Psi$ is injective. Since $R_{fg}$ is reduced,", "it suffices to show that $\\Psi$ is injective after localization", "at all minimal primes $\\mathfrak p$ of $R_{fg}$, see", "Lemma \\ref{lemma-reduced-ring-sub-product-fields}.", "However, we know that $R_\\mathfrak p = \\kappa(\\mathfrak p)$", "by Lemma \\ref{lemma-minimal-prime-reduced-ring} and", "$\\rho_M(\\mathfrak p) = r$ hence $\\Psi_\\mathfrak p :", "R_\\mathfrak p^{\\oplus r} \\to M \\otimes_R \\kappa(\\mathfrak p)$", "is an isomorphism as a surjective map of finite dimensional", "vector spaces of the same dimension." ], "refs": [ "algebra-lemma-NAK", "algebra-lemma-reduced-ring-sub-product-fields", "algebra-lemma-minimal-prime-reduced-ring" ], "ref_ids": [ 401, 419, 418 ] } ], "ref_ids": [ 795 ] }, { "id": 797, "type": "theorem", "label": "algebra-lemma-finite-flat-local", "categories": [ "algebra" ], "title": "algebra-lemma-finite-flat-local", "contents": [ "(Warning: see Remark \\ref{remark-warning}.)", "Suppose $R$ is a local ring, and $M$ is a finite", "flat $R$-module. Then $M$ is finite free." ], "refs": [ "algebra-remark-warning" ], "proofs": [ { "contents": [ "Follows from the equational criterion of flatness, see", "Lemma \\ref{lemma-flat-eq}. Namely, suppose that", "$x_1, \\ldots, x_r \\in M$ map to a basis of", "$M/\\mathfrak mM$. By Nakayama's Lemma \\ref{lemma-NAK}", "these elements generate $M$. We want to show there", "is no relation among the $x_i$. Instead, we will show", "by induction on $n$ that if $x_1, \\ldots, x_n \\in M$", "are linearly independent in the vector space", "$M/\\mathfrak mM$ then they are independent over $R$.", "\\medskip\\noindent", "The base case of the induction is where we have", "$x \\in M$, $x \\not\\in \\mathfrak mM$ and a relation", "$fx = 0$. By the equational criterion there", "exist $y_j \\in M$ and $a_j \\in R$ such that", "$x = \\sum a_j y_j$ and $fa_j = 0$ for all $j$.", "Since $x \\not\\in \\mathfrak mM$ we see that", "at least one $a_j$ is a unit and hence $f = 0 $.", "\\medskip\\noindent", "Suppose that $\\sum f_i x_i$ is a relation among $x_1, \\ldots, x_n$.", "By our choice of $x_i$ we have $f_i \\in \\mathfrak m$.", "According to the equational criterion of flatness there exist", "$a_{ij} \\in R$ and $y_j \\in M$ such that", "$x_i = \\sum a_{ij} y_j$ and $\\sum f_i a_{ij} = 0$.", "Since $x_n \\not \\in \\mathfrak mM$ we see that", "$a_{nj}\\not\\in \\mathfrak m$ for at least one $j$.", "Since $\\sum f_i a_{ij} = 0$ we get", "$f_n = \\sum_{i = 1}^{n-1} (-a_{ij}/a_{nj}) f_i$.", "The relation $\\sum f_i x_i = 0$ now can be rewritten", "as $\\sum_{i = 1}^{n-1} f_i( x_i + (-a_{ij}/a_{nj}) x_n) = 0$.", "Note that the elements $x_i + (-a_{ij}/a_{nj}) x_n$ map", "to $n-1$ linearly independent elements of $M/\\mathfrak mM$.", "By induction assumption we get that all the $f_i$, $i \\leq n-1$", "have to be zero, and also $f_n = \\sum_{i = 1}^{n-1} (-a_{ij}/a_{nj}) f_i$.", "This proves the induction step." ], "refs": [ "algebra-lemma-flat-eq", "algebra-lemma-NAK" ], "ref_ids": [ 531, 401 ] } ], "ref_ids": [ 1571 ] }, { "id": 798, "type": "theorem", "label": "algebra-lemma-finite-projective-descends", "categories": [ "algebra" ], "title": "algebra-lemma-finite-projective-descends", "contents": [ "Let $R \\to S$ be a flat local homomorphism of local rings.", "Let $M$ be a finite $R$-module. Then $M$ is finite projective", "over $R$ if and only if $M \\otimes_R S$ is finite projective", "over $S$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-projective} being finite projective", "over a local ring is the same thing as being finite free.", "Suppose that $M \\otimes_R S$ is a finite free $S$-module.", "Pick $x_1, \\ldots, x_r \\in M$ whose images in $M/\\mathfrak m_RM$", "form a basis over $\\kappa(\\mathfrak m)$. Then", "we see that $x_1 \\otimes 1, \\ldots, x_r \\otimes 1$", "are a basis for $M \\otimes_R S$. This implies that", "the map $R^{\\oplus r} \\to M, (a_i) \\mapsto \\sum a_i x_i$", "becomes an isomorphism after tensoring with $S$.", "By faithful flatness of $R \\to S$, see Lemma \\ref{lemma-local-flat-ff}", "we see that it is an isomorphism." ], "refs": [ "algebra-lemma-finite-projective", "algebra-lemma-local-flat-ff" ], "ref_ids": [ 795, 537 ] } ], "ref_ids": [] }, { "id": 799, "type": "theorem", "label": "algebra-lemma-locally-free-semi-local-free", "categories": [ "algebra" ], "title": "algebra-lemma-locally-free-semi-local-free", "contents": [ "Let $R$ be a semi-local ring.", "Let $M$ be a finite locally free module.", "If $M$ has constant rank, then $M$ is free. In particular, if $R$ has", "connected spectrum, then $M$ is free." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints: First show that $M/\\mathfrak m_iM$ has the", "same dimension $d$ for all maximal ideal $\\mathfrak m_1, \\ldots, \\mathfrak m_n$", "of $R$ using the rank is constant.", "Next, show that there exist elements $x_1, \\ldots, x_d \\in M$", "which form a basis for each $M/\\mathfrak m_iM$ by the Chinese", "remainder theorem. Finally show that $x_1, \\ldots, x_d$ is a basis for $M$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 800, "type": "theorem", "label": "algebra-lemma-semi-local-module-basis-in-submodule", "categories": [ "algebra" ], "title": "algebra-lemma-semi-local-module-basis-in-submodule", "contents": [ "Let $R$ be a local ring with maximal ideal $\\mathfrak m$ and", "infinite residue field.", "Let $R \\to S$ be a ring map.", "Let $M$ be an $S$-module and let $N \\subset M$ be an $R$-submodule.", "Assume", "\\begin{enumerate}", "\\item $S$ is semi-local and $\\mathfrak mS$ is contained in the", "Jacobson radical of $S$,", "\\item $M$ is a finite free $S$-module, and", "\\item $N$ generates $M$ as an $S$-module.", "\\end{enumerate}", "Then $N$ contains an $S$-basis of $M$." ], "refs": [], "proofs": [ { "contents": [ "Assume $M$ is free of rank $n$. Let $I \\subset S$ be the Jacobson radical.", "By Nakayama's Lemma \\ref{lemma-NAK} a sequence of elements", "$m_1, \\ldots, m_n$ is a basis for $M$ if and only if", "$\\overline{m}_i \\in M/IM$ generate $M/IM$. Hence we may replace", "$M$ by $M/IM$, $N$ by $N/(N \\cap IM)$, $R$ by $R/\\mathfrak m$,", "and $S$ by $S/IS$. In this case we see that $S$ is a finite product", "of fields $S = k_1 \\times \\ldots \\times k_r$ and", "$M = k_1^{\\oplus n} \\times \\ldots \\times k_r^{\\oplus n}$.", "The fact that $N \\subset M$ generates $M$ as an $S$-module", "means that there exist $x_j \\in N$ such that a linear combination", "$\\sum a_j x_j$ with $a_j \\in S$ has a nonzero component in each", "factor $k_i^{\\oplus n}$.", "Because $R = k$ is an infinite field, this means that also", "some linear combination $y = \\sum c_j x_j$ with $c_j \\in k$ has a", "nonzero component in each factor. Hence $y \\in N$ generates a", "free direct summand $Sy \\subset M$. By induction on $n$ the result", "holds for $M/Sy$ and the submodule $\\overline{N} = N/(N \\cap Sy)$.", "In other words there exist $\\overline{y}_2, \\ldots, \\overline{y}_n$", "in $\\overline{N}$ which (freely) generate $M/Sy$. Then", "$y, y_2, \\ldots, y_n$ (freely) generate $M$ and we win." ], "refs": [ "algebra-lemma-NAK" ], "ref_ids": [ 401 ] } ], "ref_ids": [] }, { "id": 801, "type": "theorem", "label": "algebra-lemma-evaluation-map-iso-finite-projective", "categories": [ "algebra" ], "title": "algebra-lemma-evaluation-map-iso-finite-projective", "contents": [ "Let $R$ be ring. Let $L$, $M$, $N$ be $R$-modules.", "The canonical map", "$$", "\\Hom_R(M, N) \\otimes_R L \\to \\Hom_R(M, N \\otimes_R L)", "$$", "is an isomorphism if $M$ is finite projective." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-projective} we see that $M$", "is finitely presented as well as finite locally free.", "By Lemmas \\ref{lemma-hom-from-finitely-presented} and", "\\ref{lemma-tensor-product-localization} formation of", "the left and right hand side of the arrow commutes with", "localization. We may check that our map is an isomorphism", "after localization, see Lemma \\ref{lemma-cover}.", "Thus we may assume $M$ is finite free. In this case", "the lemma is immediate." ], "refs": [ "algebra-lemma-finite-projective", "algebra-lemma-hom-from-finitely-presented", "algebra-lemma-tensor-product-localization", "algebra-lemma-cover" ], "ref_ids": [ 795, 353, 367, 411 ] } ], "ref_ids": [] }, { "id": 802, "type": "theorem", "label": "algebra-lemma-map-between-finite", "categories": [ "algebra" ], "title": "algebra-lemma-map-between-finite", "contents": [ "Let $R$ be a ring. Let $\\varphi : M \\to N$ be a map of $R$-modules", "with $N$ a finite $R$-module. Then we have the equality", "\\begin{align*}", "U & = \\{\\mathfrak p \\subset R \\mid", "\\varphi_{\\mathfrak p} : M_{\\mathfrak p} \\to N_{\\mathfrak p}", "\\text{ is surjective}\\} \\\\", "& = \\{\\mathfrak p \\subset R \\mid", "\\varphi \\otimes \\kappa(\\mathfrak p) :", "M \\otimes \\kappa(\\mathfrak p) \\to N \\otimes \\kappa(\\mathfrak p)", "\\text{ is surjective}\\}", "\\end{align*}", "and $U$ is an open subset of $\\Spec(R)$. Moreover, for any $f \\in R$", "such that $D(f) \\subset U$ the map $M_f \\to N_f$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "The equality in the displayed formula follows from Nakayama's lemma.", "Nakayama's lemma also implies that $U$ is open. See", "Lemma \\ref{lemma-NAK} especially part (3). If $D(f) \\subset U$, then", "$M_f \\to N_f$ is surjective on all localizations at primes of", "$R_f$, and hence it is surjective by Lemma \\ref{lemma-characterize-zero-local}." ], "refs": [ "algebra-lemma-NAK", "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 401, 410 ] } ], "ref_ids": [] }, { "id": 803, "type": "theorem", "label": "algebra-lemma-map-between-finitely-presented", "categories": [ "algebra" ], "title": "algebra-lemma-map-between-finitely-presented", "contents": [ "Let $R$ be a ring. Let $\\varphi : M \\to N$ be a map of $R$-modules", "with $M$ finite and $N$ finitely presented. Then", "$$", "U = \\{\\mathfrak p \\subset R \\mid", "\\varphi_{\\mathfrak p} : M_{\\mathfrak p} \\to N_{\\mathfrak p}", "\\text{ is an isomorphism}\\}", "$$", "is an open subset of $\\Spec(R)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p \\in U$. Pick a presentation", "$N = R^{\\oplus n}/\\sum_{j = 1, \\ldots, m} R k_j$.", "Denote $e_i$ the image in $N$ of the $i$th basis vector of $R^{\\oplus n}$.", "For each $i \\in \\{1, \\ldots, n\\}$ choose an element", "$m_i \\in M_{\\mathfrak p}$ such that $\\varphi(m_i) = f_i e_i$", "for some $f_i \\in R$, $f_i \\not \\in \\mathfrak p$. This is possible", "as $\\varphi_{\\mathfrak p}$ is an isomorphism. Set $f = f_1 \\ldots f_n$", "and let $\\psi : R_f^{\\oplus n} \\to M_f$ be the map which maps", "the $i$th basis vector to $m_i/f_i$. Note that", "$\\varphi_f \\circ \\psi$ is the localization", "at $f$ of the given map $R^{\\oplus n} \\to N$.", "As $\\varphi_{\\mathfrak p}$ is an isomorphism we", "see that $\\psi(k_j)$ is an element of $M$ which maps to zero", "in $M_{\\mathfrak p}$. Hence we see that there exist", "$g_j \\in R$, $g_j \\not \\in \\mathfrak p$ such that $g_j \\psi(k_j) = 0$.", "Setting $g = g_1 \\ldots g_m$, we see that $\\psi_g$ factors through", "$N_{fg}$ to give a map $\\chi : N_{fg} \\to M_{fg}$. By construction", "$\\chi$ is a right inverse to $\\varphi_{fg}$. It follows that", "$\\chi_\\mathfrak p$ is an isomorphism. By Lemma \\ref{lemma-map-between-finite}", "there is an $h \\in R$, $h \\not \\in \\mathfrak p$ such that", "$\\chi_h : N_{fgh} \\to M_{fgh}$ is surjective.", "Hence $\\varphi_{fgh}$ and $\\chi_h$ are mutually inverse maps,", "which implies that $D(fgh) \\subset U$ as desired." ], "refs": [ "algebra-lemma-map-between-finite" ], "ref_ids": [ 802 ] } ], "ref_ids": [] }, { "id": 804, "type": "theorem", "label": "algebra-lemma-cokernel-flat", "categories": [ "algebra" ], "title": "algebra-lemma-cokernel-flat", "contents": [ "Let $R$ be a ring. Let $\\varphi : P_1 \\to P_2$ be a map of", "finite projective modules. Then", "\\begin{enumerate}", "\\item The set $U$ of primes $\\mathfrak p \\in \\Spec(R)$ such that", "$\\varphi \\otimes \\kappa(\\mathfrak p)$ is injective is open and", "for any $f\\in R$ such that $D(f) \\subset U$ we have", "\\begin{enumerate}", "\\item $P_{1, f} \\to P_{2, f}$ is injective, and", "\\item the module $\\Coker(\\varphi)_f$ is finite projective over $R_f$.", "\\end{enumerate}", "\\item The set $W$ of primes $\\mathfrak p \\in \\Spec(R)$ such that", "$\\varphi \\otimes \\kappa(\\mathfrak p)$ is surjective is open and", "for any $f\\in R$ such that $D(f) \\subset W$ we have", "\\begin{enumerate}", "\\item $P_{1, f} \\to P_{2, f}$ is surjective, and", "\\item the module $\\Ker(\\varphi)_f$ is finite projective over $R_f$.", "\\end{enumerate}", "\\item The set $V$ of primes $\\mathfrak p \\in \\Spec(R)$ such that", "$\\varphi \\otimes \\kappa(\\mathfrak p)$ is an isomorphism is open and", "for any $f\\in R$ such that $D(f) \\subset V$ the map", "$\\varphi : P_{1, f} \\to P_{2, f}$ is an isomorphism of modules over $R_f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove the set $U$ is open we may work locally on", "$\\Spec(R)$. Thus we may replace $R$ by a suitable localization", "and assume that $P_1 = R^{n_1}$ and $P_2 = R^{n_2}$, see Lemma", "\\ref{lemma-finite-projective}. In this case injectivity of", "$\\varphi \\otimes \\kappa(\\mathfrak p)$ is equivalent to $n_1 \\leq n_2$", "and some $n_1 \\times n_1$ minor $f$ of the matrix of $\\varphi$ being", "invertible in $\\kappa(\\mathfrak p)$. Thus $D(f) \\subset U$.", "This argument also shows that $P_{1, \\mathfrak p} \\to P_{2, \\mathfrak p}$", "is injective for $\\mathfrak p \\in U$.", "\\medskip\\noindent", "Now suppose $D(f) \\subset U$. By the remark in the previous paragraph", "and Lemma \\ref{lemma-characterize-zero-local} we see that", "$P_{1, f} \\to P_{2, f}$ is injective, i.e., (1)(a) holds.", "By Lemma \\ref{lemma-finite-projective} to prove (1)(b)", "it suffices to prove that $\\Coker(\\varphi)$ is finite projective", "locally on $D(f)$. Thus, as we saw above, we may", "assume that $P_1 = R^{n_1}$ and $P_2 = R^{n_2}$", "and that some minor of the matrix of $\\varphi$ is invertible in $R$.", "If the minor in question corresponds to the first $n_1$", "basis vectors of $R^{n_2}$, then using the last $n_2 - n_1$ basis", "vectors we get a map $R^{n_2 - n_1} \\to R^{n_2} \\to \\Coker(\\varphi)$", "which is easily seen to be an isomorphism.", "\\medskip\\noindent", "Openness of $W$ and (2)(a) for $D(f) \\subset W$ follow from", "Lemma \\ref{lemma-map-between-finite}. Since $P_{2, f}$ is projective", "over $R_f$ we see that $\\varphi_f : P_{1, f} \\to P_{2, f}$ has a section", "and it follows that $\\Ker(\\varphi)_f$ is a direct summand of $P_{2, f}$.", "Therefore $\\Ker(\\varphi)_f$ is finite projective. Thus (2)(b) holds as well.", "\\medskip\\noindent", "It is clear that $V = U \\cap W$ is open and the other statement in (3)", "follows from (1)(a) and (2)(a)." ], "refs": [ "algebra-lemma-finite-projective", "algebra-lemma-characterize-zero-local", "algebra-lemma-finite-projective", "algebra-lemma-map-between-finite" ], "ref_ids": [ 795, 410, 795, 802 ] } ], "ref_ids": [] }, { "id": 805, "type": "theorem", "label": "algebra-lemma-flat-factors-free", "categories": [ "algebra" ], "title": "algebra-lemma-flat-factors-free", "contents": [ "Let $M$ be an $R$-module. The following are equivalent:", "\\begin{enumerate}", "\\item $M$ is flat.", "\\item If $f: R^n \\to M$ is a module map and $x \\in \\Ker(f)$, then there", "are module maps $h: R^n \\to R^m$ and $g: R^m \\to M$ such that", "$f = g \\circ h$ and $x \\in \\Ker(h)$.", "\\item Suppose $f: R^n \\to M$ is a module map, $N \\subset \\Ker(f)$ any", "submodule, and $h: R^n \\to R^{m}$ a map such that $N \\subset \\Ker(h)$", "and $f$ factors through $h$. Then given any $x \\in \\Ker(f)$ we can find a map", "$h': R^n \\to R^{m'}$ such that $N + Rx \\subset \\Ker(h')$ and $f$", "factors through $h'$.", "\\item If $f: R^n \\to M$ is a module map and $N \\subset \\Ker(f)$ is a", "finitely generated submodule, then there are module maps $h: R^n \\to", "R^m$ and $g: R^m \\to M$ such that $f = g \\circ h$ and $N \\subset", "\\Ker(h)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "That (1) is equivalent to (2) is just a reformulation of the equational", "criterion for flatness\\footnote{In fact, a module map $f : R^n \\to M$", "corresponds to a choice of elements $x_1, x_2, \\ldots, x_n$ of $M$", "(namely, the images of the standard basis elements $e_1, e_2, \\ldots,", "e_n$); furthermore, an element $x \\in \\Ker(f)$ corresponds to a", "relation between these $x_1, x_2, \\ldots, x_n$ (namely, the relation", "$\\sum_i f_i x_i = 0$, where the $f_i$ are the coordinates of $x$).", "The module map $h$ (represented as an $m \\times n$-matrix)", "corresponds to the matrix $(a_{ij})$ from Lemma \\ref{lemma-flat-eq},", "and the $y_j$ of Lemma \\ref{lemma-flat-eq} are the images of the", "standard basis vectors of $R^m$ under $g$.}. To show", "(2) implies (3), let $g: R^m \\to M$", "be the map such that $f$ factors as $f = g \\circ h$. By (2) find $h'': R^m", "\\to R^{m'}$ such that $h''$ kills $h(x)$ and $g: R^m \\to M$", "factors through $h''$. Then taking $h' = h'' \\circ h$ works. (3) implies (4)", "by induction on the number of generators of $N \\subset \\Ker(f)$ in (4).", "Clearly (4) implies (2)." ], "refs": [ "algebra-lemma-flat-eq", "algebra-lemma-flat-eq" ], "ref_ids": [ 531, 531 ] } ], "ref_ids": [] }, { "id": 806, "type": "theorem", "label": "algebra-lemma-flat-factors-fp", "categories": [ "algebra" ], "title": "algebra-lemma-flat-factors-fp", "contents": [ "Let $M$ be an $R$-module. Then $M$ is flat if and only if the following", "condition holds: if $P$ is a finitely presented $R$-module and $f: P", "\\to M$ a module map, then there is a free finite $R$-module $F$ and", "module maps $h: P \\to F$ and $g: F \\to M$ such that $f = g", "\\circ h$." ], "refs": [], "proofs": [ { "contents": [ "This is just a reformulation of condition (4) from", "Lemma \\ref{lemma-flat-factors-free}." ], "refs": [ "algebra-lemma-flat-factors-free" ], "ref_ids": [ 805 ] } ], "ref_ids": [] }, { "id": 807, "type": "theorem", "label": "algebra-lemma-flat-surjective-hom", "categories": [ "algebra" ], "title": "algebra-lemma-flat-surjective-hom", "contents": [ "Let $M$ be an $R$-module. Then $M$ is flat if and only if the following", "condition holds: for every finitely presented $R$-module $P$, if $N \\to", "M$ is a surjective $R$-module map, then the induced map $\\Hom_R(P, N)", "\\to \\Hom_R(P, M)$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "First suppose $M$ is flat. We must show that if $P$ is finitely presented,", "then given a map $f: P \\to M$, it factors through the map $N", "\\to M$. By", "Lemma \\ref{lemma-flat-factors-fp}", "the map $f$ factors through a map $F \\to M$ where $F$ is free and finite.", "Since $F$ is free, this map factors through", "$N \\to M$. Thus $f$ factors through $N \\to M$.", "\\medskip\\noindent", "Conversely, suppose the condition of the lemma holds. Let $f: P \\to M$", "be a map from a finitely presented module $P$. Choose a free module $N$ with a", "surjection $N \\to M$ onto $M$. Then $f$ factors through $N \\to", "M$, and since $P$ is finitely generated, $f$ factors through a free finite", "submodule of $N$. Thus $M$ satisfies the condition of Lemma", "\\ref{lemma-flat-factors-fp}, hence is flat." ], "refs": [ "algebra-lemma-flat-factors-fp", "algebra-lemma-flat-factors-fp" ], "ref_ids": [ 806, 806 ] } ], "ref_ids": [] }, { "id": 808, "type": "theorem", "label": "algebra-lemma-universally-exact-split", "categories": [ "algebra" ], "title": "algebra-lemma-universally-exact-split", "contents": [ "Let", "$$", "0 \\to M_1 \\to M_2 \\to M_3 \\to 0", "$$", "be an exact sequence of $R$-modules. Suppose $M_3$ is of finite presentation.", "Then", "$$", "0 \\to M_1 \\to M_2 \\to M_3 \\to 0", "$$", "is universally exact if and only if it is split." ], "refs": [], "proofs": [ { "contents": [ "A split short exact sequence is always universally exact, see", "Example \\ref{example-universally-exact}.", "Conversely, if the sequence is universally exact, then by", "Theorem \\ref{theorem-universally-exact-criteria} (5)", "applied to $P = M_3$, the map $M_2 \\to M_3$ admits a section." ], "refs": [ "algebra-theorem-universally-exact-criteria" ], "ref_ids": [ 319 ] } ], "ref_ids": [] }, { "id": 809, "type": "theorem", "label": "algebra-lemma-flat-universally-injective", "categories": [ "algebra" ], "title": "algebra-lemma-flat-universally-injective", "contents": [ "Let $M$ be an $R$-module. Then $M$ is flat if and only if any exact sequence", "of $R$-modules", "$$", "0 \\to M_1 \\to M_2 \\to M \\to 0", "$$", "is universally exact." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-flat-surjective-hom} and", "Theorem \\ref{theorem-universally-exact-criteria} (5)." ], "refs": [ "algebra-lemma-flat-surjective-hom", "algebra-theorem-universally-exact-criteria" ], "ref_ids": [ 807, 319 ] } ], "ref_ids": [] }, { "id": 810, "type": "theorem", "label": "algebra-lemma-ui-flat-domain", "categories": [ "algebra" ], "title": "algebra-lemma-ui-flat-domain", "contents": [ "Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a universally exact sequence", "of $R$-modules, and suppose $M_2$ is flat.", "Then $M_1$ and $M_3$ are flat." ], "refs": [], "proofs": [ { "contents": [ "Let $0 \\to N \\to N' \\to N'' \\to 0$ be a short exact sequence of", "$R$-modules. Consider the commutative diagram", "$$", "\\xymatrix{", "M_1 \\otimes_R N \\ar[r] \\ar[d] &", "M_2 \\otimes_R N \\ar[r] \\ar[d] &", "M_3 \\otimes_R N \\ar[d] \\\\", "M_1 \\otimes_R N' \\ar[r] \\ar[d] &", "M_2 \\otimes_R N' \\ar[r] \\ar[d] &", "M_3 \\otimes_R N' \\ar[d] \\\\", "M_1 \\otimes_R N'' \\ar[r] &", "M_2 \\otimes_R N'' \\ar[r] &", "M_3 \\otimes_R N''", "}", "$$", "(we have dropped the $0$'s on the boundary).", "By assumption the rows give short exact sequences and the arrow", "$M_2 \\otimes N \\to M_2 \\otimes N'$ is injective. Clearly this implies", "that $M_1 \\otimes N \\to M_1 \\otimes N'$ is injective and we see that $M_1$", "is flat. In particular the left and middle columns give rise to short", "exact sequences. It follows from a diagram chase that the arrow", "$M_3 \\otimes N \\to M_3 \\otimes N'$ is injective. Hence $M_3$ is flat." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 811, "type": "theorem", "label": "algebra-lemma-universally-injective-tensor", "categories": [ "algebra" ], "title": "algebra-lemma-universally-injective-tensor", "contents": [ "Let $R$ be a ring.", "Let $M \\to M'$ be a universally injective $R$-module map.", "Then for any $R$-module $N$ the map $M \\otimes_R N \\to M' \\otimes_R N$", "is universally injective." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 812, "type": "theorem", "label": "algebra-lemma-composition-universally-injective", "categories": [ "algebra" ], "title": "algebra-lemma-composition-universally-injective", "contents": [ "Let $R$ be a ring. A composition of universally injective", "$R$-module maps is universally injective." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 813, "type": "theorem", "label": "algebra-lemma-universally-injective-permanence", "categories": [ "algebra" ], "title": "algebra-lemma-universally-injective-permanence", "contents": [ "Let $R$ be a ring. Let $M \\to M'$ and $M' \\to M''$ be $R$-module maps.", "If their composition $M \\to M''$ is universally injective, then", "$M \\to M'$ is universally injective." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 814, "type": "theorem", "label": "algebra-lemma-faithfully-flat-universally-injective", "categories": [ "algebra" ], "title": "algebra-lemma-faithfully-flat-universally-injective", "contents": [ "Let $R \\to S$ be a faithfully flat ring map.", "Then $R \\to S$ is universally injective as a map of $R$-modules.", "In particular $R \\cap IS = I$ for any ideal $I \\subset R$." ], "refs": [], "proofs": [ { "contents": [ "Let $N$ be an $R$-module. We have to show that $N \\to N \\otimes_R S$ is", "injective. As $S$ is faithfully flat as an $R$-module, it suffices to prove", "this after tensoring with $S$. Hence it suffices to show that", "$N \\otimes_R S \\to N \\otimes_R S \\otimes_R S$,", "$n \\otimes s \\mapsto n \\otimes 1 \\otimes s$ is injective. This is true", "because there is a retraction, namely,", "$n \\otimes s \\otimes s' \\mapsto n \\otimes ss'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 815, "type": "theorem", "label": "algebra-lemma-universally-injective-check-stalks", "categories": [ "algebra" ], "title": "algebra-lemma-universally-injective-check-stalks", "contents": [ "Let $R \\to S$ be a ring map.", "Let $M \\to M'$ be a map of $S$-modules.", "The following are equivalent", "\\begin{enumerate}", "\\item $M \\to M'$ is universally injective as a map of $R$-modules,", "\\item for each prime $\\mathfrak q$ of $S$ the map", "$M_{\\mathfrak q} \\to M'_{\\mathfrak q}$ is universally injective", "as a map of $R$-modules,", "\\item for each maximal ideal $\\mathfrak m$ of $S$ the map", "$M_{\\mathfrak m} \\to M'_{\\mathfrak m}$ is universally injective", "as a map of $R$-modules,", "\\item for each prime $\\mathfrak q$ of $S$ the map", "$M_{\\mathfrak q} \\to M'_{\\mathfrak q}$ is universally injective", "as a map of $R_{\\mathfrak p}$-modules, where $\\mathfrak p$ is the", "inverse image of $\\mathfrak q$ in $R$, and", "\\item for each maximal ideal $\\mathfrak m$ of $S$ the map", "$M_{\\mathfrak m} \\to M'_{\\mathfrak m}$ is universally injective", "as a map of $R_{\\mathfrak p}$-modules, where $\\mathfrak p$ is the", "inverse image of $\\mathfrak m$ in $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $N$ be an $R$-module. Let $\\mathfrak q$ be a prime of $S$ lying over", "the prime $\\mathfrak p$ of $R$. Then we have", "$$", "(M \\otimes_R N)_{\\mathfrak q} =", "M_{\\mathfrak q} \\otimes_R N =", "M_{\\mathfrak q} \\otimes_{R_{\\mathfrak p}} N_{\\mathfrak p}.", "$$", "Moreover, the same thing holds for $M'$ and localization is exact.", "Also, if $N$ is an $R_{\\mathfrak p}$-module, then $N_{\\mathfrak p} = N$.", "Using this the equivalences can be proved in a straightforward manner.", "\\medskip\\noindent", "For example, suppose that (5) holds. Let", "$K = \\Ker(M \\otimes_R N \\to M' \\otimes_R N)$. By the remarks", "above we see that $K_{\\mathfrak m} = 0$ for each maximal ideal $\\mathfrak m$", "of $S$. Hence $K = 0$ by", "Lemma \\ref{lemma-characterize-zero-local}.", "Thus (1) holds. Conversely, suppose that (1) holds. Take any", "$\\mathfrak q \\subset S$ lying over $\\mathfrak p \\subset R$.", "Take any module $N$ over $R_{\\mathfrak p}$. Then", "by assumption $\\Ker(M \\otimes_R N \\to M' \\otimes_R N) = 0$.", "Hence by the formulae above and the fact that $N = N_{\\mathfrak p}$", "we see that", "$\\Ker(M_{\\mathfrak q} \\otimes_{R_{\\mathfrak p}} N \\to", "M'_{\\mathfrak q} \\otimes_{R_{\\mathfrak p}} N) = 0$. In other words", "(4) holds. Of course (4) $\\Rightarrow$ (5) is immediate. Hence", "(1), (4) and (5) are all equivalent.", "We omit the proof of the other equivalences." ], "refs": [ "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 410 ] } ], "ref_ids": [] }, { "id": 816, "type": "theorem", "label": "algebra-lemma-universally-injective-localize", "categories": [ "algebra" ], "title": "algebra-lemma-universally-injective-localize", "contents": [ "Let $\\varphi : A \\to B$ be a ring map. Let $S \\subset A$ and", "$S' \\subset B$ be multiplicative subsets such that $\\varphi(S) \\subset S'$.", "Let $M \\to M'$ be a map of $B$-modules.", "\\begin{enumerate}", "\\item If $M \\to M'$ is universally injective as a map of $A$-modules,", "then $(S')^{-1}M \\to (S')^{-1}M'$ is universally injective as a map of", "$A$-modules and as a map of $S^{-1}A$-modules.", "\\item If $M$ and $M'$ are $(S')^{-1}B$-modules, then $M \\to M'$", "is universally injective as a map of $A$-modules if and only if", "it is universally injective as a map of $S^{-1}A$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "You can prove this using", "Lemma \\ref{lemma-universally-injective-check-stalks}", "but you can also prove it directly as follows.", "Assume $M \\to M'$ is $A$-universally injective.", "Let $Q$ be an $A$-module. Then $Q \\otimes_A M \\to Q \\otimes_A M'$", "is injective. Since localization is exact we see that", "$(S')^{-1}(Q \\otimes_A M) \\to (S')^{-1}(Q \\otimes_A M')$ is injective.", "As $(S')^{-1}(Q \\otimes_A M) = Q \\otimes_A (S')^{-1}M$ and similarly for $M'$", "we see that", "$Q \\otimes_A (S')^{-1}M \\to Q \\otimes_A (S')^{-1}M'$ is injective, hence", "$(S')^{-1}M \\to (S')^{-1}M'$ is universally injective as a map of", "$A$-modules. This proves the first part of (1).", "To see (2) we can use the following two facts: (a) if $Q$ is an", "$S^{-1}A$-module, then $Q \\otimes_A S^{-1}A = Q$, i.e., tensoring", "with $Q$ over $A$ is the same thing as tensoring with $Q$ over $S^{-1}A$,", "(b) if $M$ is any $A$-module on which the elements of $S$ are invertible,", "then $M \\otimes_A Q = M \\otimes_{S^{-1}A} S^{-1}Q$.", "Part (2) follows from this immediately." ], "refs": [ "algebra-lemma-universally-injective-check-stalks" ], "ref_ids": [ 815 ] } ], "ref_ids": [] }, { "id": 817, "type": "theorem", "label": "algebra-lemma-check-universally-injective-into-flat", "categories": [ "algebra" ], "title": "algebra-lemma-check-universally-injective-into-flat", "contents": [ "Let $R$ be a ring and let $M \\to M'$ be a map of $R$-modules.", "If $M'$ is flat, then $M \\to M'$ is universally injective if", "and only if $M/IM \\to M'/IM'$ is injective for every finitely", "generated ideal $I$ of $R$." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that $M \\otimes_R Q \\to M' \\otimes_R Q$ is", "injective for every finite $R$-module $Q$, see", "Theorem \\ref{theorem-universally-exact-criteria}.", "Then $Q$ has a finite filtration", "$0 = Q_0 \\subset Q_1 \\subset \\ldots \\subset Q_n = Q$", "by submodules whose subquotients", "are isomorphic to cyclic modules $R/I_i$, see", "Lemma \\ref{lemma-trivial-filter-finite-module}.", "Since $M'$ is flat, we obtain a filtration", "$$", "\\xymatrix{", "M \\otimes Q_1 \\ar[r] \\ar[d] &", "M \\otimes Q_2 \\ar[r] \\ar[d] &", "\\ldots \\ar[r] &", "M \\otimes Q \\ar[d] \\\\", "M' \\otimes Q_1 \\ar@{^{(}->}[r] &", "M' \\otimes Q_2 \\ar@{^{(}->}[r] &", "\\ldots \\ar@{^{(}->}[r] &", "M' \\otimes Q", "}", "$$", "of $M' \\otimes_R Q$ by submodules $M' \\otimes_R Q_i$ whose successive", "quotients are $M' \\otimes_R R/I_i = M'/I_iM'$. A simple induction argument", "shows that it suffices to check $M/I_i M \\to M'/I_i M'$ is injective.", "Note that the collection of finitely generated ideals $I'_i \\subset I_i$", "is a directed set. Thus $M/I_iM = \\colim M/I'_iM$ is a filtered", "colimit, similarly for $M'$, the maps $M/I'_iM \\to M'/I'_i M'$ are", "injective by assumption, and since filtered colimits are exact", "(Lemma \\ref{lemma-directed-colimit-exact}) we conclude." ], "refs": [ "algebra-theorem-universally-exact-criteria", "algebra-lemma-trivial-filter-finite-module", "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 319, 331, 343 ] } ], "ref_ids": [] }, { "id": 818, "type": "theorem", "label": "algebra-lemma-finite-projective-again", "categories": [ "algebra" ], "title": "algebra-lemma-finite-projective-again", "contents": [ "Let $M$ be an $R$-module. Then $M$ is finite projective if and only if $M$ is", "finitely presented and flat." ], "refs": [], "proofs": [ { "contents": [ "This is part of", "Lemma \\ref{lemma-finite-projective}.", "However, at this point we can give a more elegant proof of the implication", "(1) $\\Rightarrow$ (2) of that lemma as follows.", "If $M$ is finitely presented and flat, then take a surjection", "$R^n \\to M$. By", "Lemma \\ref{lemma-flat-surjective-hom}", "applied to $P = M$, the map $R^n \\to M$ admits a section.", "So $M$ is a direct summand of a free module and hence projective." ], "refs": [ "algebra-lemma-finite-projective", "algebra-lemma-flat-surjective-hom" ], "ref_ids": [ 795, 807 ] } ], "ref_ids": [] }, { "id": 819, "type": "theorem", "label": "algebra-lemma-descend-properties-modules", "categories": [ "algebra" ], "title": "algebra-lemma-descend-properties-modules", "contents": [ "Let $R \\to S$ be a faithfully flat ring map.", "Let $M$ be an $R$-module. Then", "\\begin{enumerate}", "\\item if the $S$-module $M \\otimes_R S$ is of finite type, then", "$M$ is of finite type,", "\\item if the $S$-module $M \\otimes_R S$ is of finite presentation, then", "$M$ is of finite presentation,", "\\item if the $S$-module $M \\otimes_R S$ is flat, then", "$M$ is flat, and", "\\item add more here as needed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $M \\otimes_R S$ is of finite type. Let $y_1, \\ldots, y_m$ be generators", "of $M \\otimes_R S$ over $S$. Write $y_j = \\sum x_i \\otimes f_i$ for some", "$x_1, \\ldots, x_n \\in M$. Then we see that the map", "$\\varphi : R^{\\oplus n} \\to M$", "has the property that", "$\\varphi \\otimes \\text{id}_S : S^{\\oplus n} \\to M \\otimes_R S$", "is surjective. Since $R \\to S$ is faithfully flat we see that", "$\\varphi$ is surjective, and $M$ is finitely generated.", "\\medskip\\noindent", "Assume $M \\otimes_R S$ is of finite presentation. By (1) we see that", "$M$ is of finite type. Choose a surjection $R^{\\oplus n} \\to M$ and", "denote $K$ the kernel. As $R \\to S$ is flat we see that $K \\otimes_R S$", "is the kernel of the base change $S^{\\oplus n} \\to M \\otimes_R S$.", "As $M \\otimes_R S$ is of finite presentation we conclude that $K \\otimes_R S$", "is of finite type. Hence by (1) we see that $K$ is of finite type", "and hence $M$ is of finite presentation.", "\\medskip\\noindent", "Part (3) is", "Lemma \\ref{lemma-flatness-descends}." ], "refs": [ "algebra-lemma-flatness-descends" ], "ref_ids": [ 528 ] } ], "ref_ids": [] }, { "id": 820, "type": "theorem", "label": "algebra-lemma-direct-sum-devissage", "categories": [ "algebra" ], "title": "algebra-lemma-direct-sum-devissage", "contents": [ "Let $M$ be an $R$-module. If $(M_{\\alpha})_{\\alpha \\in S}$ is a direct sum", "d\\'evissage of $M$, then", "$M \\cong \\bigoplus_{\\alpha + 1 \\in S} M_{\\alpha + 1}/M_{\\alpha}$." ], "refs": [], "proofs": [ { "contents": [ "By property (3) of a direct sum d\\'evissage, there is an inclusion", "$M_{\\alpha + 1}/M_{\\alpha} \\to M$ for each $\\alpha \\in S$. Consider the", "map", "$$", "f : \\bigoplus\\nolimits_{\\alpha + 1\\in S} M_{\\alpha + 1}/M_{\\alpha} \\to M", "$$", "given by the sum of these inclusions.", "Further consider the restrictions", "$$", "f_{\\beta} :", "\\bigoplus\\nolimits_{\\alpha + 1 \\leq \\beta} M_{\\alpha + 1}/M_{\\alpha}", "\\longrightarrow", "M", "$$", "for $\\beta\\in S$. Transfinite induction on $S$ shows that the image of", "$f_{\\beta}$ is $M_{\\beta}$. For $\\beta=0$ this is true by $(0)$. If $\\beta+1$", "is a successor ordinal and it is true for $\\beta$, then it is true for ", "$\\beta + 1$ by (3). And if $\\beta$ is a limit ordinal and it is true for", "$\\alpha < \\beta$, then it is true for $\\beta$ by (2). Hence $f$ is surjective", "by (1). ", "\\medskip\\noindent", "Transfinite induction on $S$ also shows that the restrictions $f_{\\beta}$", "are injective. For $\\beta = 0$ it is true. If $\\beta+1$ is a", "successor ordinal and $f_{\\beta}$ is injective, then let $x$ be in the kernel", "and write $x = (x_{\\alpha + 1})_{\\alpha + 1 \\leq \\beta + 1}$ in terms of its", "components $x_{\\alpha + 1} \\in M_{\\alpha + 1}/M_{\\alpha}$. By property (3) and", "the fact that the image of $f_{\\beta}$ is $M_{\\beta}$ both", "$(x_{\\alpha + 1})_{\\alpha + 1 \\leq \\beta}$ and $x_{\\beta + 1}$ map to $0$.", "Hence $x_{\\beta+1} = 0$ and, by the assumption that the restriction", "$f_{\\beta}$ is injective also $x_{\\alpha + 1} = 0$", "for every $\\alpha + 1 \\leq \\beta$. So $x = 0$ and $f_{\\beta+1}$ is injective.", "If $\\beta$ is a limit ordinal consider an element $x$ of the kernel. Then $x$", "is already contained in the domain of $f_{\\alpha}$ for some $\\alpha < \\beta$.", "Thus $x = 0$ which finishes the induction.", "We conclude that $f$ is injective since $f_{\\beta}$ is for each $\\beta \\in S$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 821, "type": "theorem", "label": "algebra-lemma-Kaplansky-devissage", "categories": [ "algebra" ], "title": "algebra-lemma-Kaplansky-devissage", "contents": [ "Let $M$ be an $R$-module. Then $M$ is a direct sum of countably generated", "$R$-modules if and only if it admits a Kaplansky d\\'evissage." ], "refs": [], "proofs": [ { "contents": [ "The lemma takes care of the ``if'' direction. Conversely, suppose $M =", "\\bigoplus_{i \\in I} N_i$ where each $N_i$ is a countably generated $R$-module.", "Well-order $I$ so that we can think of it as an ordinal. Then setting $M_i =", "\\bigoplus_{j < i} N_j$ gives a Kaplansky d\\'evissage $(M_i)_{i \\in I}$ of", "$M$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 822, "type": "theorem", "label": "algebra-lemma-projective-free", "categories": [ "algebra" ], "title": "algebra-lemma-projective-free", "contents": [ "Let $R$ be a ring. Then every projective $R$-module is free if and only if", "every countably generated projective $R$-module is free." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Theorem \\ref{theorem-projective-direct-sum}." ], "refs": [ "algebra-theorem-projective-direct-sum" ], "ref_ids": [ 321 ] } ], "ref_ids": [] }, { "id": 823, "type": "theorem", "label": "algebra-lemma-freeness-criteria", "categories": [ "algebra" ], "title": "algebra-lemma-freeness-criteria", "contents": [ "Let $M$ be a countably generated $R$-module. Suppose any direct summand $N$ of", "$M$ satisfies: any element of $N$ is contained in a free direct summand of $N$.", " Then $M$ is free." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1, x_2, \\ldots$ be a countable set of generators for $M$. By the", "assumption on $M$, we can construct by induction free $R$-modules $F_1, F_2,", "\\ldots$ such that for every positive integer $n$, $\\bigoplus_{i=1}^{n} F_i$ is a", "direct summand of $M$ and contains $x_1, \\ldots, x_n$. Then $M = \\bigoplus_{i =", "1}^{\\infty} F_i$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 824, "type": "theorem", "label": "algebra-lemma-projective-freeness-criteria", "categories": [ "algebra" ], "title": "algebra-lemma-projective-freeness-criteria", "contents": [ "Let $P$ be a projective module over a local ring $R$. Then any element of $P$", "is contained in a free direct summand of $P$." ], "refs": [], "proofs": [ { "contents": [ "Since $P$ is projective it is a direct summand of some free $R$-module $F$, say", "$F = P \\oplus Q$. Let $x \\in P$ be the element that we wish to show is", "contained in a free direct summand of $P$. Let $B$ be a basis of $F$ such that", "the number of basis elements needed in the expression of $x$ is minimal, say $x", "= \\sum_{i=1}^n a_i e_i$ for some $e_i \\in B$ and $a_i \\in R$. Then no $a_j$", "can be expressed as a linear combination of the other $a_i$; for if $a_j =", "\\sum_{i \\neq j} a_i b_i$ for some $b_i \\in R$, then replacing $e_i$ by $e_i +", "b_ie_j$ for $i \\neq j$ and leaving unchanged the other elements of $B$, we get", "a new basis for $F$ in terms of which $x$ has a shorter expression.", "\\medskip\\noindent", "Let $e_i = y_i + z_i, y_i \\in P, z_i \\in Q$ be the decomposition of $e_i$ into", "its $P$- and $Q$-components. Write $y_i = \\sum_{j=1}^{n} b_{ij} e_j + t_i$,", "where $t_i$ is a linear combination of elements in $B$ other than $e_1, \\ldots,", "e_n$. To finish the proof it suffices to show that the matrix $(b_{ij})$ is", "invertible. For then the map $F \\to F$ sending $e_i \\mapsto y_i$ for", "$i=1, \\ldots, n$ and fixing $B \\setminus \\{e_1, \\ldots, e_n\\}$ is an", "isomorphism,", "so that $y_1, \\ldots, y_n$ together with $B \\setminus \\{e_1, \\ldots, e_n\\}$", "form a basis for $F$. Then the submodule $N$ spanned by $y_1, \\ldots, y_n$", "is a free submodule of $P$; $N$ is a direct summand of $P$ since $N \\subset P$", "and both $N$ and $P$ are direct summands of $F$; and $x \\in N$ since $x \\in P$", "implies $x = \\sum_{i=1}^n a_i e_i = \\sum_{i=1}^n a_i y_i$.", "\\medskip\\noindent", "Now we prove that $(b_{ij})$ is invertible. Plugging $y_i = \\sum_{j=1}^{n}", "b_{ij} e_j + t_i$ into $\\sum_{i=1}^n a_i e_i = \\sum_{i=1}^n a_i y_i$ and", "equating the coefficients of $e_j$ gives $a_j = \\sum_{i=1}^n a_i b_{ij}$. But", "as noted above, our choice of $B$ guarantees that no $a_j$ can be written as a", "linear combination of the other $a_i$. Thus $b_{ij}$ is a non-unit for $i \\neq", "j$, and $1-b_{ii}$ is a non-unit---so in particular $b_{ii}$ is a unit---for", "all $i$. But a matrix over a local ring having units along the diagonal and", "non-units elsewhere is invertible, as its determinant is a unit." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 825, "type": "theorem", "label": "algebra-lemma-ML-limit-nonempty", "categories": [ "algebra" ], "title": "algebra-lemma-ML-limit-nonempty", "contents": [ "Let $(A_i, \\varphi_{ji})$ be a directed inverse system over $I$. Suppose $I$", "is countable. If $(A_i, \\varphi_{ji})$ is Mittag-Leffler and the $A_i$ are", "nonempty, then $\\lim A_i$ is nonempty." ], "refs": [], "proofs": [ { "contents": [ "Let $i_1, i_2, i_3, \\ldots$ be an enumeration of the elements of $I$. Define", "inductively a sequence of elements $j_n \\in I$ for $n = 1, 2, 3, \\ldots$ by the", "conditions: $j_1 = i_1$, and $j_n \\geq i_n$ and $j_n \\geq j_m$ for $m < n$.", " Then the sequence $j_n$ is increasing and forms a cofinal subset of $I$.", "Hence we may assume $I =\\{1, 2, 3, \\ldots \\}$. So by Example", "\\ref{example-ML-surjective-maps} we are reduced to showing that the limit of an", "inverse system of nonempty sets with surjective maps indexed by the positive", "integers is nonempty. This is obvious." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 826, "type": "theorem", "label": "algebra-lemma-ML-exact-sequence", "categories": [ "algebra" ], "title": "algebra-lemma-ML-exact-sequence", "contents": [ "Let", "$$", "0 \\to A_i \\xrightarrow{f_i} B_i \\xrightarrow{g_i} C_i \\to 0", "$$", "be an exact sequence of directed inverse systems of abelian groups over $I$.", "Suppose $I$ is countable. If $(A_i)$ is Mittag-Leffler, then", "$$", "0 \\to \\lim A_i \\to \\lim B_i \\to \\lim C_i\\to 0", "$$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "Taking limits of directed inverse systems is left exact, hence we only need to", "prove surjectivity of $\\lim B_i \\to \\lim C_i$. So let $(c_i) \\in \\lim", "C_i$. For each $i \\in I$, let $E_i = g_i^{-1}(c_i)$, which is nonempty since", "$g_i: B_i \\to C_i$ is surjective. The system of maps $\\varphi_{ji}: B_j", "\\to B_i$ for $(B_i)$ restrict to maps $E_j \\to E_i$ which", "make $(E_i)$ into an inverse system of nonempty sets. It is enough to show", "that $(E_i)$ is Mittag-Leffler. For then Lemma \\ref{lemma-ML-limit-nonempty}", "would show $\\lim E_i$ is nonempty, and taking any element of $\\lim E_i$ would", "give an element of $\\lim B_i$ mapping to $(c_i)$.", "\\medskip\\noindent", "By the injection $f_i: A_i \\to B_i$ we will regard $A_i$ as a subset of", "$B_i$. Since $(A_i)$ is Mittag-Leffler, if $i \\in I$ then there exists $j \\geq", "i$ such that $\\varphi_{ki}(A_k) = \\varphi_{ji}(A_j)$ for $k \\geq j$. We claim", "that also $\\varphi_{ki}(E_k) = \\varphi_{ji}(E_j)$ for $k \\geq j$. Always", "$\\varphi_{ki}(E_k) \\subset \\varphi_{ji}(E_j)$ for $k \\geq j$. For the reverse", "inclusion let $e_j \\in E_j$, and we need to find $x_k \\in E_k$ such that", "$\\varphi_{ki}(x_k) = \\varphi_{ji}(e_j)$. Let $e'_k \\in E_k$ be any element,", "and set $e'_j = \\varphi_{kj}(e'_k)$. Then $g_j(e_j - e'_j) = c_j - c_j = 0$,", "hence $e_j - e'_j = a_j \\in A_j$. Since $\\varphi_{ki}(A_k) =", "\\varphi_{ji}(A_j)$, there exists $a_k \\in A_k$ such that $\\varphi_{ki}(a_k) =", "\\varphi_{ji}(a_j)$. Hence", "$$", "\\varphi_{ki}(e'_k + a_k) = \\varphi_{ji}(e'_j) + \\varphi_{ji}(a_j) =", "\\varphi_{ji}(e_j),", "$$", "so we can take $x_k = e'_k + a_k$." ], "refs": [ "algebra-lemma-ML-limit-nonempty" ], "ref_ids": [ 825 ] } ], "ref_ids": [] }, { "id": 827, "type": "theorem", "label": "algebra-lemma-Mittag-Leffler", "categories": [ "algebra" ], "title": "algebra-lemma-Mittag-Leffler", "contents": [ "Let $R$ be a ring.", "Let $0 \\to K_i \\to L_i \\to M_i \\to 0$ be short exact sequences of", "$R$-modules, $i \\geq 1$ which fit into maps of short exact sequences", "$$", "\\xymatrix{", "0 \\ar[r] &", "K_i \\ar[r] &", "L_i \\ar[r] &", "M_i \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "K_{i + 1} \\ar[r] \\ar[u] &", "L_{i + 1} \\ar[r] \\ar[u] &", "M_{i + 1} \\ar[r] \\ar[u] &", "0}", "$$", "If for every $i$ there exists a $c = c(i) \\geq i$ such that", "$\\Im(K_c \\to K_i) = \\Im(K_j \\to K_i)$", "for all $j \\geq c$, then the sequence", "$$", "0 \\to \\lim K_i \\to \\lim L_i \\to \\lim M_i \\to 0", "$$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of the more general", "Lemma \\ref{lemma-ML-exact-sequence}." ], "refs": [ "algebra-lemma-ML-exact-sequence" ], "ref_ids": [ 826 ] } ], "ref_ids": [] }, { "id": 828, "type": "theorem", "label": "algebra-lemma-domination-fp", "categories": [ "algebra" ], "title": "algebra-lemma-domination-fp", "contents": [ "Let $f: M \\to N$ and $g: M \\to M'$ be maps of $R$-modules.", "Then $g$ dominates $f$ if and only if for any finitely presented $R$-module", "$Q$, we have $\\Ker(f \\otimes_R \\text{id}_Q) \\subset \\Ker(g \\otimes_R", "\\text{id}_Q)$." ], "refs": [], "proofs": [ { "contents": [ "Suppose $\\Ker(f \\otimes_R \\text{id}_Q) \\subset \\Ker(g \\otimes_R", "\\text{id}_Q)$ for all finitely presented modules $Q$. If $Q$ is an", "arbitrary module, write $Q = \\colim_{i \\in I} Q_i$ as a colimit of a", "directed", "system of finitely presented modules $Q_i$. Then $\\Ker(f \\otimes_R", "\\text{id}_{Q_i}) \\subset \\Ker(g \\otimes_R \\text{id}_{Q_i})$ for", "all $i$. Since taking directed colimits is exact and commutes with tensor", "product, it follows that $\\Ker(f \\otimes_R \\text{id}_Q) \\subset \\Ker(g", "\\otimes_R \\text{id}_Q)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 829, "type": "theorem", "label": "algebra-lemma-domination-universally-injective", "categories": [ "algebra" ], "title": "algebra-lemma-domination-universally-injective", "contents": [ "Let $f : M \\to N$ and $g : M \\to M'$ be maps of $R$-modules.", "Consider the pushout of $f$ and $g$,", "$$", "\\xymatrix{", "M \\ar[r]_f \\ar[d]_g & N \\ar[d]^{g'} \\\\", "M' \\ar[r]^{f'} & N'", "}", "$$", "Then $g$ dominates $f$ if and only if $f'$ is universally injective." ], "refs": [], "proofs": [ { "contents": [ "Recall that $N'$ is $M' \\oplus N$ modulo the submodule consisting of elements", "$(g(x), -f(x))$ for $x \\in M$.", "From the construction of $N'$ we have a short exact sequence", "$$", "0 \\to \\Ker(f) \\cap \\Ker(g) \\to \\Ker(f) \\to \\Ker(f')", "\\to 0.", "$$", "Since tensoring commutes with taking pushouts, we have such a short exact", "sequence", "$$", "0 \\to \\Ker(f \\otimes \\text{id}_Q ) \\cap \\Ker(g \\otimes", "\\text{id}_Q) \\to \\Ker(f \\otimes \\text{id}_Q)", "\\to \\Ker(f' \\otimes \\text{id}_Q) \\to 0", "$$", "for every $R$-module $Q$. So $f'$ is universally injective if and only if", "$\\Ker(f \\otimes \\text{id}_Q ) \\subset \\Ker(g \\otimes", "\\text{id}_Q)$ for every $Q$, if and only if $g$ dominates $f$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 830, "type": "theorem", "label": "algebra-lemma-domination", "categories": [ "algebra" ], "title": "algebra-lemma-domination", "contents": [ "Let $f: M \\to N$ and $g: M \\to M'$ be maps of $R$-modules.", "Suppose $\\Coker(f)$ is of finite presentation. Then $g$ dominates $f$ if", "and", "only if $g$ factors through $f$, i.e.\\ there exists a module map $h: N", "\\to M'$ such that $g = h \\circ f$." ], "refs": [], "proofs": [ { "contents": [ "Consider the pushout of $f$ and $g$ as in the statement of", "Lemma \\ref{lemma-domination-universally-injective}.", "From the construction of the pushout it follows that", "$\\Coker(f') = \\Coker(f)$, so $\\Coker(f')$ is of finite", "presentation. Then by", "Lemma \\ref{lemma-universally-exact-split}, $f'$ is universally injective if and", "only if", "$$", "0 \\to M' \\xrightarrow{f'} N' \\to \\Coker(f') \\to 0", "$$", "splits. This is the case if and only if there is a map $h' : N' \\to M'$", "such that $h' \\circ f' = \\text{id}_{M'}$. From the universal", "property of the pushout, the existence of such an $h'$ is equivalent to $g$", "factoring through $f$." ], "refs": [ "algebra-lemma-domination-universally-injective", "algebra-lemma-universally-exact-split" ], "ref_ids": [ 829, 808 ] } ], "ref_ids": [] }, { "id": 831, "type": "theorem", "label": "algebra-lemma-tensor-ML-modules", "categories": [ "algebra" ], "title": "algebra-lemma-tensor-ML-modules", "contents": [ "If $R$ is a ring and $M$, $N$ are Mittag-Leffler modules over $R$,", "then $M \\otimes_R N$ is a Mittag-Leffler module." ], "refs": [], "proofs": [ { "contents": [ "Write $M = \\colim_{i \\in I} M_i$ and $N = \\colim_{j \\in J} N_j$", "as directed colimits of finitely presented $R$-modules.", "Denote $f_{ii'} : M_i \\to M_{i'}$ and $g_{jj'} : N_j \\to N_{j'}$ the", "transition maps. Then $M_i \\otimes_R N_j$ is a finitely presented", "$R$-module (see", "Lemma \\ref{lemma-tensor-finiteness}),", "and $M \\otimes_R N = \\colim_{(i, j) \\in I \\times J} M_i \\otimes_R M_j$.", "Pick $(i, j) \\in I \\times J$. By the definition of a Mittag-Leffler module", "we have", "Proposition \\ref{proposition-ML-characterization} (3)", "for both systems. In other words there exist $i' \\geq i$ and $j' \\geq j$", "such that for every choice of $i'' \\geq i$ and $j'' \\geq j$ there exist", "maps $a : M_{i''} \\to M_{i'}$ and $b : M_{j''} \\to M_{j'}$ such that", "$f_{ii'} = a \\circ f_{ii''}$ and $g_{jj'} = b \\circ g_{jj''}$.", "Then it is clear that", "$a \\otimes b : M_{i''} \\otimes_R N_{j''} \\to M_{i'} \\otimes_R N_{j'}$", "serves the same purpose for the system", "$(M_i \\otimes_R N_j, f_{ii'} \\otimes g_{jj'})$.", "Thus by the characterization", "Proposition \\ref{proposition-ML-characterization} (3)", "we conclude that $M \\otimes_R N$ is Mittag-Leffler." ], "refs": [ "algebra-lemma-tensor-finiteness", "algebra-proposition-ML-characterization", "algebra-proposition-ML-characterization" ], "ref_ids": [ 365, 1414, 1414 ] } ], "ref_ids": [] }, { "id": 832, "type": "theorem", "label": "algebra-lemma-ML-also", "categories": [ "algebra" ], "title": "algebra-lemma-ML-also", "contents": [ "Let $R$ be a ring and $M$ an $R$-module. Then $M$ is Mittag-Leffler if and", "only if for every finite free $R$-module $F$ and module map", "$f: F \\to M$, there exists a finitely presented $R$-module $Q$", "and a module map $g : F \\to Q$ such that $g$ and $f$ dominate each other, i.e.,", "$\\Ker(f \\otimes_R \\text{id}_N) = \\Ker(g \\otimes_R \\text{id}_N)$", "for every $R$-module $N$." ], "refs": [], "proofs": [ { "contents": [ "Since the condition is clear weaker than condition (1) of", "Proposition \\ref{proposition-ML-characterization}", "we see that a Mittag-Leffler module satisfies the condition.", "Conversely, suppose that $M$ satisfies the condition and that", "$f : P \\to M$ is an $R$-module map from a finitely presented", "$R$-module $P$ into $M$. Choose a surjection $F \\to P$ where", "$F$ is a finite free $R$-module. By assumption we can find a map", "$F \\to Q$ where $Q$ is a finitely presented $R$-module such that", "$F \\to Q$ and $F \\to M$ dominate each other. In particular, the kernel", "of $F \\to Q$ contains the kernel of $F \\to P$, hence we obtain an", "$R$-module map $g : P \\to Q$ such that $F \\to Q$ is equal to", "the composition $F \\to P \\to Q$. Let $N$ be any $R$-module and", "consider the commutative diagram", "$$", "\\xymatrix{", "F \\otimes_R N \\ar[d] \\ar[r] & Q \\otimes_R N \\\\", "P \\otimes_R N \\ar[ru] \\ar[r] & M \\otimes_R N", "}", "$$", "By assumption the kernels of $F \\otimes_R N \\to Q \\otimes_R N$", "and $F \\otimes_R N \\to M \\otimes_R N$ are equal. Hence, as", "$F \\otimes_R N \\to P \\otimes_R N$ is surjective, also the kernels", "of $P \\otimes_R N \\to Q \\otimes_R N$", "and $P \\otimes_R N \\to M \\otimes_R N$ are equal." ], "refs": [ "algebra-proposition-ML-characterization" ], "ref_ids": [ 1414 ] } ], "ref_ids": [] }, { "id": 833, "type": "theorem", "label": "algebra-lemma-restrict-ML-modules", "categories": [ "algebra" ], "title": "algebra-lemma-restrict-ML-modules", "contents": [ "Let $R \\to S$ be a finite and finitely presented ring map.", "Let $M$ be an $S$-module.", "If $M$ is a Mittag-Leffler module over $S$ then", "$M$ is a Mittag-Leffler module over $R$." ], "refs": [], "proofs": [ { "contents": [ "Assume $M$ is a Mittag-Leffler module over $S$.", "Write $M = \\colim M_i$ as a directed colimit of finitely presented", "$S$-modules $M_i$. As $M$ is Mittag-Leffler over $S$ there exists for each", "$i$ an index $j \\geq i$ such that for all $k \\geq j$ there is a factorization", "$f_{ij} = h \\circ f_{ik}$ (where $h$ depends on $i$, the choice of $j$ and", "$k$). Note that by", "Lemma \\ref{lemma-finite-finitely-presented-extension}", "the modules $M_i$ are also finitely presented as $R$-modules. Moreover, all", "the maps $f_{ij}, f_{ik}, h$ are maps of $R$-modules. Thus we see that the", "system $(M_i, f_{ij})$ satisfies the same condition when viewed as a system", "of $R$-modules. Thus $M$ is Mittag-Leffler as an $R$-module." ], "refs": [ "algebra-lemma-finite-finitely-presented-extension" ], "ref_ids": [ 501 ] } ], "ref_ids": [] }, { "id": 834, "type": "theorem", "label": "algebra-lemma-mod-ideal-ML-modules", "categories": [ "algebra" ], "title": "algebra-lemma-mod-ideal-ML-modules", "contents": [ "Let $R$ be a ring.", "Let $S = R/I$ for some finitely generated ideal $I$.", "Let $M$ be an $S$-module.", "Then $M$ is a Mittag-Leffler module over $R$ if and only if", "$M$ is a Mittag-Leffler module over $S$." ], "refs": [], "proofs": [ { "contents": [ "One implication follows from", "Lemma \\ref{lemma-restrict-ML-modules}.", "To prove the other, assume $M$ is Mittag-Leffler as an $R$-module.", "Write $M = \\colim M_i$ as a directed colimit of finitely presented", "$S$-modules. As $I$ is finitely generated, the ring $S$ is finite and finitely", "presented as an $R$-algebra, hence the modules $M_i$ are finitely", "presented as $R$-modules, see", "Lemma \\ref{lemma-finite-finitely-presented-extension}.", "Next, let $N$ be any $S$-module. Note that for each $i$ we have", "$\\Hom_R(M_i, N) = \\Hom_S(M_i, N)$ as $R \\to S$ is surjective.", "Hence the condition that the inverse system", "$(\\Hom_R(M_i, N))_i$ satisfies Mittag-Leffler, implies that the system", "$(\\Hom_S(M_i, N))_i$ satisfies Mittag-Leffler. Thus $M$ is", "Mittag-Leffler over $S$ by definition." ], "refs": [ "algebra-lemma-restrict-ML-modules", "algebra-lemma-finite-finitely-presented-extension" ], "ref_ids": [ 833, 501 ] } ], "ref_ids": [] }, { "id": 835, "type": "theorem", "label": "algebra-lemma-kernel-tensored-fp", "categories": [ "algebra" ], "title": "algebra-lemma-kernel-tensored-fp", "contents": [ "Let $M$ be an $R$-module, $P$ a finitely presented $R$-module, and $f: P", "\\to M$ a map. Let $Q$ be an $R$-module and suppose $x \\in \\Ker(P", "\\otimes Q \\to M \\otimes Q)$. Then there exists a finitely presented", "$R$-module $P'$ and a map $f': P \\to P'$ such that $f$ factors through", "$f'$ and $x \\in \\Ker(P \\otimes Q \\to P' \\otimes Q)$." ], "refs": [], "proofs": [ { "contents": [ "Write $M$ as a colimit $M = \\colim_{i \\in I} M_i$ of a directed system of", "finitely presented modules $M_i$. Since $P$ is finitely presented, the map $f:", "P \\to M$ factors through $M_j \\to M$ for some $j \\in I$. Upon", "tensoring by $Q$ we have a commutative diagram", "$$", "\\xymatrix{", "& M_j \\otimes Q \\ar[dr] & \\\\", "P \\otimes Q \\ar[ur] \\ar[rr] & & M \\otimes Q .", "}", "$$", "The image $y$ of $x$ in $M_j \\otimes Q$ is in the kernel of $M_j \\otimes Q", "\\to M \\otimes Q$. Since $M \\otimes Q = \\colim_{i \\in I} (M_i", "\\otimes", "Q)$, this means $y$ maps to $0$ in $M_{j'} \\otimes Q$ for some $j' \\geq j$.", "Thus we may take $P' = M_{j'}$ and $f'$ to be the composite $P \\to M_j", "\\to M_{j'}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 836, "type": "theorem", "label": "algebra-lemma-minimal-contains", "categories": [ "algebra" ], "title": "algebra-lemma-minimal-contains", "contents": [ "Let $M$ be a flat Mittag-Leffler module over $R$. Let $F$ be an $R$-module", "and let $x \\in F \\otimes_R M$. Then there exists a smallest submodule", "$F' \\subset F$ such that $x \\in F' \\otimes_R M$." ], "refs": [], "proofs": [ { "contents": [ "Since $M$ is flat we have $F' \\otimes_R M \\subset F \\otimes_R M$", "if $F' \\subset F$ is a submodule, hence the statement makes sense.", "Let $I = \\{F' \\subset F \\mid x \\in F' \\otimes_R M\\}$ and for", "$i \\in I$ denote $F_i \\subset F$ the corresponding submodule.", "Then $x$ maps to zero under the map", "$$", "F \\otimes_R M \\longrightarrow \\prod (F/F_i \\otimes_R M)", "$$", "whence by Proposition \\ref{proposition-ML-tensor}", "$x$ maps to zero under the map", "$$", "F \\otimes_R M \\longrightarrow \\left(\\prod F/F_i\\right) \\otimes_R M", "$$", "Since $M$ is flat the kernel of this arrow is", "$(\\bigcap F_i) \\otimes_R M$ which proves the lemma." ], "refs": [ "algebra-proposition-ML-tensor" ], "ref_ids": [ 1417 ] } ], "ref_ids": [] }, { "id": 837, "type": "theorem", "label": "algebra-lemma-pure-submodule-ML", "categories": [ "algebra" ], "title": "algebra-lemma-pure-submodule-ML", "contents": [ "Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a", "universally exact sequence of $R$-modules. Then:", "\\begin{enumerate}", "\\item If $M_2$ is Mittag-Leffler, then $M_1$ is Mittag-Leffler.", "\\item If $M_1$ and $M_3$ are Mittag-Leffler, then $M_2$ is Mittag-Leffler.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For any family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules we have a", "commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] & M_1 \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r] \\ar[d] & M_2", "\\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r] \\ar[d] & M_3 \\otimes_R", "(\\prod_{\\alpha} Q_{\\alpha}) \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] & \\prod_{\\alpha}(M_1 \\otimes Q_{\\alpha}) \\ar[r] & \\prod_{\\alpha}(M_2", "\\otimes Q_{\\alpha}) \\ar[r] & \\prod_{\\alpha}(M_3 \\otimes Q_{\\alpha})\\ar[r] & 0", "}", "$$", "with exact rows. Thus (1) and (2) follow from", "Proposition \\ref{proposition-ML-tensor}." ], "refs": [ "algebra-proposition-ML-tensor" ], "ref_ids": [ 1417 ] } ], "ref_ids": [] }, { "id": 838, "type": "theorem", "label": "algebra-lemma-quotient-module-ML", "categories": [ "algebra" ], "title": "algebra-lemma-quotient-module-ML", "contents": [ "Let $M_1 \\to M_2 \\to M_3 \\to 0$ be an exact sequence of $R$-modules.", "If $M_1$ is finitely generated and $M_2$ is Mittag-Leffler, then $M_3$", "is Mittag-Leffler." ], "refs": [], "proofs": [ { "contents": [ "For any family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules,", "since tensor product is right exact, we have a commutative diagram", "$$", "\\xymatrix{", "M_1 \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r] \\ar[d] & M_2", "\\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r] \\ar[d] & M_3 \\otimes_R", "(\\prod_{\\alpha} Q_{\\alpha}) \\ar[r] \\ar[d] & 0 \\\\", "\\prod_{\\alpha}(M_1 \\otimes Q_{\\alpha}) \\ar[r] & \\prod_{\\alpha}(M_2", "\\otimes Q_{\\alpha}) \\ar[r] & \\prod_{\\alpha}(M_3 \\otimes Q_{\\alpha})\\ar[r] & 0", "}", "$$", "with exact rows. By Proposition \\ref{proposition-fg-tensor}", "the left vertical arrow is surjective. By", "Proposition \\ref{proposition-ML-tensor} the middle vertical arrow", "is injective. A diagram chase shows the right vertical arrow", "is injective. Hence $M_3$ is Mittag-Leffler by", "Proposition \\ref{proposition-ML-tensor}." ], "refs": [ "algebra-proposition-fg-tensor", "algebra-proposition-ML-tensor", "algebra-proposition-ML-tensor" ], "ref_ids": [ 1415, 1417, 1417 ] } ], "ref_ids": [] }, { "id": 839, "type": "theorem", "label": "algebra-lemma-colimit-universally-injective-ML", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-universally-injective-ML", "contents": [ "If $M = \\colim M_i$ is the colimit of a directed system of Mittag-Leffler", "$R$-modules $M_i$ with universally injective transition maps, then $M$ is", "Mittag-Leffler." ], "refs": [], "proofs": [ { "contents": [ "Let $(Q_{\\alpha})_{\\alpha \\in A}$ be a family of $R$-modules. We have to", "show that $M \\otimes_R (\\prod Q_\\alpha) \\to \\prod M \\otimes_R Q_\\alpha$", "is injective and we know that", "$M_i \\otimes_R (\\prod Q_\\alpha) \\to \\prod M_i \\otimes_R Q_\\alpha$", "is injective for each $i$, see Proposition \\ref{proposition-ML-tensor}.", "Since $\\otimes$ commutes with filtered colimits, it suffices to show that", "$\\prod M_i \\otimes_R Q_\\alpha \\to \\prod M \\otimes_R Q_\\alpha$", "is injective. This is clear as each of the maps", "$M_i \\otimes_R Q_\\alpha \\to M \\otimes_R Q_\\alpha$ is injective", "by our assumption that the transition maps are universally injective." ], "refs": [ "algebra-proposition-ML-tensor" ], "ref_ids": [ 1417 ] } ], "ref_ids": [] }, { "id": 840, "type": "theorem", "label": "algebra-lemma-direct-sum-ML", "categories": [ "algebra" ], "title": "algebra-lemma-direct-sum-ML", "contents": [ "If $M = \\bigoplus_{i \\in I} M_i$ is a direct sum of $R$-modules, then $M$ is", "Mittag-Leffler if and only if each $M_i$ is Mittag-Leffler." ], "refs": [], "proofs": [ { "contents": [ "The ``only if'' direction follows from Lemma \\ref{lemma-pure-submodule-ML} (1)", "and the fact that a split short exact sequence is universally exact. The", "converse follows from Lemma \\ref{lemma-colimit-universally-injective-ML}", "but we can also argue it directly as follows. First note that if $I$ is", "finite then this follows from Lemma", "\\ref{lemma-pure-submodule-ML} (2). For general $I$, if all $M_i$ are", "Mittag-Leffler then we prove the same of $M$ by verifying condition (1) of", "Proposition \\ref{proposition-ML-characterization}.", "Let $f: P \\to M$ be a map from a finitely presented module $P$.", "Then $f$ factors as", "$P \\xrightarrow{f'} \\bigoplus_{i' \\in I'} M_{i'} \\hookrightarrow", "\\bigoplus_{i \\in I} M_i$", "for some finite subset $I'$ of $I$. By the finite case", "$\\bigoplus_{i' \\in I'} M_{i'}$ is Mittag-Leffler and hence there exists", "a finitely presented module $Q$ and a map $g: P \\to Q$ such that $g$", "and $f'$ dominate each other. Then also $g$ and $f$ dominate each other." ], "refs": [ "algebra-lemma-pure-submodule-ML", "algebra-lemma-colimit-universally-injective-ML", "algebra-lemma-pure-submodule-ML", "algebra-proposition-ML-characterization" ], "ref_ids": [ 837, 839, 837, 1414 ] } ], "ref_ids": [] }, { "id": 841, "type": "theorem", "label": "algebra-lemma-flat-ML-over-ML-ring", "categories": [ "algebra" ], "title": "algebra-lemma-flat-ML-over-ML-ring", "contents": [ "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module.", "If $S$ is Mittag-Leffler as an $R$-module, and $M$ is flat and Mittag-Leffler", "as an $S$-module, then $M$ is Mittag-Leffler as an $R$-module." ], "refs": [], "proofs": [ { "contents": [ "We deduce this from the characterization of", "Proposition \\ref{proposition-ML-tensor}.", "Namely, suppose that $Q_\\alpha$ is a family of $R$-modules.", "Consider the composition", "$$", "\\xymatrix{", "M \\otimes_R \\prod_\\alpha Q_\\alpha =", "M \\otimes_S S \\otimes_R \\prod_\\alpha Q_\\alpha \\ar[d] \\\\", "M \\otimes_S \\prod_\\alpha (S \\otimes_R Q_\\alpha) \\ar[d] \\\\", "\\prod_\\alpha (M \\otimes_S \\otimes_R Q_\\alpha) =", "\\prod_\\alpha (M \\otimes_R Q_\\alpha)", "}", "$$", "The first arrows is injective as $M$ is flat over $S$ and $S$ is", "Mittag-Leffler over $R$ and the second arrow is injective as", "$M$ is Mittag-Leffler over $S$. Hence $M$ is Mittag-Leffler over $R$." ], "refs": [ "algebra-proposition-ML-tensor" ], "ref_ids": [ 1417 ] } ], "ref_ids": [] }, { "id": 842, "type": "theorem", "label": "algebra-lemma-coherent", "categories": [ "algebra" ], "title": "algebra-lemma-coherent", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item A finite submodule of a coherent module is coherent.", "\\item Let $\\varphi : N \\to M$ be a homomorphism from a finite", "module to a coherent module. Then $\\Ker(\\varphi)$ is finite.", "\\item Let $\\varphi : N \\to M$ be a homomorphism of coherent modules.", "Then $\\Ker(\\varphi)$ and $\\Coker(\\varphi)$ are coherent", "modules.", "\\item Given a short exact sequence of $R$-modules", "$0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ if two out of three are coherent", "so is the third.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first statement is immediate from the definition.", "During the rest of the proof we will use the results of", "Lemma \\ref{lemma-extension}", "without further mention.", "\\medskip\\noindent", "Let $\\varphi : N \\to M$ satisfy the assumptions of (2).", "Suppose that $N$ is generated by $x_1, \\ldots, x_n$. By", "Definition \\ref{definition-coherent}", "the kernel $K$ of the induced map $R^{\\oplus n} \\to M$,", "$e_i \\mapsto \\varphi(x_i)$ is of finite type.", "Hence $\\Ker(\\varphi)$ which is the image of the", "composition $K \\to R^{\\oplus n} \\to N$", "is of finite type. This proves (2).", "\\medskip\\noindent", "Let $\\varphi : N \\to M$ satisfy the assumptions of (3).", "By (2) the kernel of $\\varphi$ is of finite type and", "hence by (1) it is coherent.", "\\medskip\\noindent", "With the same hypotheses", "let us show that $\\Coker(\\varphi)$ is coherent.", "Since $M$ is finite so is $\\Coker(\\varphi)$.", "Let $\\overline{x}_i \\in \\Coker(\\varphi)$.", "We have to show that the kernel of the associated morphism", "$\\overline{\\Psi} : R^{\\oplus n} \\to \\Coker(\\varphi)$", "is finite. Choose $x_i \\in M$ lifting $\\overline{x}_i$.", "Choose additionally generators $y_1, \\ldots, y_m$ of $\\Im(\\varphi)$.", "Let $\\Phi : R^{\\oplus m} \\to \\Im(\\varphi)$ using $y_j$ and", "$\\Psi : R^{\\oplus m} \\oplus R^{\\oplus n} \\to M$ using $y_j$ and $x_i$", "be the corresponding maps.", "Consider the following commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "R^{\\oplus m} \\ar[d]_\\Phi \\ar[r] &", "R^{\\oplus m} \\oplus R^{\\oplus n} \\ar[d]_\\Psi \\ar[r] &", "R^{\\oplus n} \\ar[d]_{\\overline{\\Psi}} \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "\\Im(\\varphi) \\ar[r] &", "M \\ar[r] &", "\\Coker(\\varphi) \\ar[r] &", "0", "}", "$$", "with exact rows. By Lemma \\ref{lemma-snake} we get an exact sequence", "$\\Ker(\\Psi) \\to \\Ker(\\overline{\\Psi}) \\to 0$.", "Since $\\Ker(\\Psi)$ is a finite $R$-module,", "we see that $\\Ker(\\overline{\\Psi})$ is finite.", "\\medskip\\noindent", "Statement (4) follows from (3).", "\\medskip\\noindent", "Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$", "be a short exact sequence of $R$-modules. It suffices", "to prove that if $M_1$ and $M_3$ are coherent", "so is $M_2$. By", "Lemma \\ref{lemma-extension}", "we see that $M_2$ is finite. Let $x_1, \\ldots, x_n$ be finitely many", "elements of $M_2$.", "We have to show that the module of relations $K$", "between them is finite.", "Consider the following commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "0 \\ar[r] \\ar[d] &", "\\bigoplus_{i = 1}^{n} R \\ar[r] \\ar[d] &", "\\bigoplus_{i = 1}^{n} R \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "M_1 \\ar[r] &", "M_2 \\ar[r] &", "M_3 \\ar[r] &", "0", "}", "$$", "with obvious notation. By the snake lemma we get an exact sequence", "$0 \\to K \\to K_3 \\to M_1$", "where $K_3$ is the module of relations among", "the images of the $x_i$ in $M_3$.", "Since $M_3$ is coherent we see that", "$K_3$ is a finite module. Since $M_1$", "is coherent we see that the image $I$", "of $K_3 \\to M_1$", "is coherent. Hence $K$", "is the kernel of the map $K_3 \\to I$", "between a finite module and a coherent module and hence", "finite by (2)." ], "refs": [ "algebra-lemma-extension", "algebra-definition-coherent", "algebra-lemma-snake", "algebra-lemma-extension" ], "ref_ids": [ 330, 1499, 328, 330 ] } ], "ref_ids": [] }, { "id": 843, "type": "theorem", "label": "algebra-lemma-coherent-ring", "categories": [ "algebra" ], "title": "algebra-lemma-coherent-ring", "contents": [ "Let $R$ be a ring. If $R$ is coherent, then a module is coherent", "if and only if it is finitely presented." ], "refs": [], "proofs": [ { "contents": [ "It is clear that a coherent module is finitely presented (over any ring).", "Conversely, if $R$ is coherent, then $R^{\\oplus n}$ is coherent and so is", "the cokernel of any map $R^{\\oplus m} \\to R^{\\oplus n}$, see", "Lemma \\ref{lemma-coherent}." ], "refs": [ "algebra-lemma-coherent" ], "ref_ids": [ 842 ] } ], "ref_ids": [] }, { "id": 844, "type": "theorem", "label": "algebra-lemma-Noetherian-coherent", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-coherent", "contents": [ "A Noetherian ring is a coherent ring." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-Noetherian-finite-type-is-finite-presentation}", "any finite $R$-module is finitely presented. In particular any ideal of $R$", "is finitely presented." ], "refs": [ "algebra-lemma-Noetherian-finite-type-is-finite-presentation" ], "ref_ids": [ 451 ] } ], "ref_ids": [] }, { "id": 845, "type": "theorem", "label": "algebra-lemma-flat-ML-criterion", "categories": [ "algebra" ], "title": "algebra-lemma-flat-ML-criterion", "contents": [ "Let $M$ be a flat $R$-module. The following are equivalent", "\\begin{enumerate}", "\\item $M$ is Mittag-Leffler, and", "\\item if $F$ is a finite free $R$-module and", "$x \\in F \\otimes_R M$, then there exists a smallest submodule $F'$ of $F$", "such that $x \\in F' \\otimes_R M$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) is a special case of", "Lemma \\ref{lemma-minimal-contains}. Assume (2).", "By Theorem \\ref{theorem-lazard} we can write $M$ as the colimit", "$M = \\colim_{i \\in I} M_i$ of a directed system $(M_i, f_{ij})$ of", "finite free $R$-modules.", "By Remark \\ref{remark-flat-ML}, it suffices to show that the inverse system", "$(\\Hom_R(M_i, R), \\Hom_R(f_{ij}, R))$ is Mittag-Leffler. In", "other words,", "fix $i \\in I$ and for $j \\geq i$ let $Q_j$ be the image of", "$\\Hom_R(M_j, R)", "\\to \\Hom_R(M_i, R)$; we must show that the $Q_j$ stabilize.", "\\medskip\\noindent", "Since $M_i$ is free and finite, we can make the identification", "$\\Hom_R(M_i, M_j) = \\Hom_R(M_i, R) \\otimes_R M_j$ for all $j$.", " Using the", "fact that the $M_j$ are free, it follows that for $j \\geq i$, $Q_j$ is the", "smallest submodule of $\\Hom_R(M_i, R)$ such that $f_{ij} \\in Q_j", "\\otimes_R", "M_j$. Under the identification $\\Hom_R(M_i, M) = \\Hom_R(M_i, R)", "\\otimes_R", "M$, the canonical map $f_i: M_i \\to M$ is in $\\Hom_R(M_i, R)", "\\otimes_R M$. By the assumption on $M$, there exists a smallest submodule", "$Q$ of $\\Hom_R(M_i, R)$ such that $f_i \\in Q \\otimes_R M$. We are", "going to", "show that the $Q_j$ stabilize to $Q$.", "\\medskip\\noindent", "For $j \\geq i$ we have a commutative diagram", "$$", "\\xymatrix{", "Q_j \\otimes_R M_j \\ar[r] \\ar[d] & \\Hom_R(M_i, R) \\otimes_R M_j", "\\ar[d] \\\\", "Q_j \\otimes_R M \\ar[r] & \\Hom_R(M_i, R) \\otimes_R M.", "}", "$$", "Since $f_{ij} \\in Q_j \\otimes_R M_j$ maps to $f_i \\in \\Hom_R(M_i, R)", "\\otimes_R M$, it follows that $f_i \\in Q_j \\otimes_R M$. Hence, by the", "choice of $Q$, we have $Q \\subset Q_j$ for all $j \\geq i$.", "\\medskip\\noindent", "Since the $Q_j$ are decreasing and $Q \\subset Q_j$ for all $j \\geq i$, to show", "that the $Q_j$ stabilize to $Q$ it suffices to find a $j \\geq i$ such that $Q_j", "\\subset Q$. As an element of", "$$", "\\Hom_R(M_i, R) \\otimes_R M = \\colim_{j \\in J}", "(\\Hom_R(M_i, R) \\otimes_R", "M_j),", "$$", "$f_i$ is the colimit of $f_{ij}$ for $j \\geq i$, and $f_i$ also lies in the", "submodule", "$$", "\\colim_{j \\in J} (Q \\otimes_R M_j) \\subset \\colim_{j \\in J}", "(\\Hom_R(M_i, R)", "\\otimes_R M_j).", "$$", "It follows that for some $j \\geq i$, $f_{ij}$ lies in $Q \\otimes_R M_j$.", "Since $Q_j$ is the smallest submodule of $\\Hom_R(M_i, R)$ with $f_{ij}", "\\in", "Q_j \\otimes_R M_j$, we conclude $Q_j\\subset Q$." ], "refs": [ "algebra-lemma-minimal-contains", "algebra-theorem-lazard", "algebra-remark-flat-ML" ], "ref_ids": [ 836, 318, 1572 ] } ], "ref_ids": [] }, { "id": 846, "type": "theorem", "label": "algebra-lemma-product-over-Noetherian-ring", "categories": [ "algebra" ], "title": "algebra-lemma-product-over-Noetherian-ring", "contents": [ "Let $R$ be a Noetherian ring and $A$ a set.", "Then $M = R^A$ is a flat and Mittag-Leffler $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Combining", "Lemma \\ref{lemma-Noetherian-coherent}", "and", "Proposition \\ref{proposition-characterize-coherent}", "we see that $M$ is flat over $R$. We show that $M$ satisfies the condition of", "Lemma \\ref{lemma-flat-ML-criterion}.", "Let $F$ be a free finite $R$-module. If $F'$ is any submodule of $F$ then it", "is finitely presented since $R$ is Noetherian. So by", "Proposition \\ref{proposition-fp-tensor}", "we have a commutative diagram", "$$", "\\xymatrix{", "F' \\otimes_R M \\ar[r] \\ar[d]^{\\cong} & F \\otimes_R M \\ar[d]^{\\cong} \\\\", "(F')^A \\ar[r] & F^A", "}", "$$", "by which we can identify the map $F' \\otimes_R M \\to F \\otimes_R M$", "with $(F')^A \\to F^A$. Hence if $x \\in F \\otimes_R M$ corresponds to", "$(x_\\alpha) \\in F^A$, then the submodule of $F'$ of $F$ generated by the", "$x_\\alpha$ is the smallest submodule of $F$ such that $x \\in F' \\otimes_R M$." ], "refs": [ "algebra-lemma-Noetherian-coherent", "algebra-proposition-characterize-coherent", "algebra-lemma-flat-ML-criterion", "algebra-proposition-fp-tensor" ], "ref_ids": [ 844, 1418, 845, 1416 ] } ], "ref_ids": [] }, { "id": 847, "type": "theorem", "label": "algebra-lemma-power-series-ML", "categories": [ "algebra" ], "title": "algebra-lemma-power-series-ML", "contents": [ "Let $R$ be a Noetherian ring and $n$ a positive integer. Then the $R$-module", "$M = R[[t_1, \\ldots, t_n]]$ is flat and Mittag-Leffler." ], "refs": [], "proofs": [ { "contents": [ "As an $R$-module, we have $M = R^A$ for a (countable) set $A$.", "Hence this lemma is a special case of", "Lemma \\ref{lemma-product-over-Noetherian-ring}." ], "refs": [ "algebra-lemma-product-over-Noetherian-ring" ], "ref_ids": [ 846 ] } ], "ref_ids": [] }, { "id": 848, "type": "theorem", "label": "algebra-lemma-ML-countable-colimit", "categories": [ "algebra" ], "title": "algebra-lemma-ML-countable-colimit", "contents": [ "Let $M$ be an $R$-module. Write $M = \\colim_{i \\in I} M_i$ where $(M_i,", "f_{ij})$ is a directed system of finitely presented $R$-modules. If $M$ is", "Mittag-Leffler and countably generated, then there is a directed countable", "subset $I' \\subset I$ such that $M \\cong \\colim_{i \\in I'} M_i$." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1, x_2, \\ldots$ be a countable set of generators for $M$. For each $x_n$", "choose $i \\in I$ such that $x_n$ is in the image of the canonical map $f_i:", "M_i \\to M$; let $I'_{0} \\subset I$ be the set of all these $i$. Now", "since $M$ is Mittag-Leffler, for each $i \\in I'_{0}$ we can choose $j \\in I$", "such that $j \\geq i$ and $f_{ij}: M_i \\to M_j$ factors through", "$f_{ik}: M_i \\to M_k$ for all $k \\geq i$ (condition (3) of Proposition", "\\ref{proposition-ML-characterization}); let $I'_1$ be the union of $I'_0$ with", "all of these $j$. Since $I'_1$ is a countable, we can enlarge it to a", "countable directed set $I'_{2} \\subset I$. Now we can apply the same procedure", "to $I'_{2}$ as we did to $I'_{0}$ to get a new countable set $I'_{3} \\subset", "I$. Then we enlarge $I'_{3}$ to a countable directed set $I'_{4}$. Continuing", "in this way---adding in a $j$ as in Proposition", "\\ref{proposition-ML-characterization} (3) for each $ i \\in I'_{\\ell}$ if $\\ell$", "is odd and enlarging $I'_{\\ell}$ to a directed set if $\\ell$ is even---we get a", "sequence of subsets $I'_{\\ell} \\subset I$ for $\\ell \\geq 0$. The union $I' =", "\\bigcup I'_{\\ell}$ satisfies:", "\\begin{enumerate}", "\\item $I'$ is countable and directed;", "\\item each $x_n$ is in the image of $f_i: M_i \\to M$ for some $i", "\\in I'$;", "\\item if $i \\in I'$, then there is $j \\in I'$ such that $j \\geq i$ and $f_{ij}:", "M_i \\to M_j$ factors through $f_{ik}: M_i \\to M_k$ for all", "$k \\in I$ with $k \\geq i$. In particular $\\Ker(f_{ik}) \\subset \\Ker(f_{ij})$", "for $k \\geq i$.", "\\end{enumerate}", "We claim that the canonical map $\\colim_{i \\in I'} M_i \\to", "\\colim_{i", "\\in I} M_i = M$ is an isomorphism. By (2) it is surjective. For injectivity,", "suppose $x \\in \\colim_{i \\in I'} M_i$ maps to $0$ in $\\colim_{i \\in", "I} M_i$.", "Representing $x$ by an element $\\tilde{x} \\in M_i$ for some $i \\in I'$, this", "means that $f_{ik}(\\tilde{x}) = 0$ for some $k \\in I, k \\geq i$. But then by", "(3) there is $j \\in I', j \\geq i,$ such that $f_{ij}(\\tilde{x}) = 0$. Hence $x", "= 0$ in $\\colim_{i \\in I'} M_i$." ], "refs": [ "algebra-proposition-ML-characterization", "algebra-proposition-ML-characterization" ], "ref_ids": [ 1414, 1414 ] } ], "ref_ids": [] }, { "id": 849, "type": "theorem", "label": "algebra-lemma-ML-countable", "categories": [ "algebra" ], "title": "algebra-lemma-ML-countable", "contents": [ "Let $R$ be a ring.", "Let $M$ be an $R$-module.", "Assume $M$ is Mittag-Leffler and countably generated.", "For any $R$-module map $f : P \\to M$ with $P$ finitely generated there", "exists an endomorphism $\\alpha : M \\to M$ such that", "\\begin{enumerate}", "\\item $\\alpha : M \\to M$ factors through a finitely presented $R$-module, and", "\\item $\\alpha \\circ f = f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $M = \\colim_{i \\in I} M_i$ as a directed colimit of finitely", "presented $R$-modules with $I$ countable, see", "Lemma \\ref{lemma-ML-countable-colimit}.", "The transition maps are denoted $f_{ij}$ and we use $f_i : M_i \\to M$", "to denote the canonical maps into $M$. Set $N = \\prod_{s \\in I} M_s$. Denote", "$$", "M_i^* = \\Hom_R(M_i, N) = \\prod\\nolimits_{s \\in I} \\Hom_R(M_i, M_s)", "$$", "so that $(M_i^*)$ is an inverse system of $R$-modules over $I$.", "Note that $\\Hom_R(M, N) = \\lim M_i^*$.", "As $M$ is Mittag-Leffler, we find for every", "$i \\in I$ an index $k(i) \\geq i$ such that", "$$", "E_i := \\bigcap\\nolimits_{i' \\geq i} \\Im(M_{i'}^* \\to M_i^*)", "=", "\\Im(M_{k(i)}^* \\to M_i^*)", "$$", "Choose and fix $j \\in I$ such that $\\Im(P \\to M) \\subset \\Im(M_j \\to M)$.", "This is possible as $P$ is finitely generated. Set $k = k(j)$.", "Let", "$x = (0, \\ldots, 0, \\text{id}_{M_k}, 0, \\ldots, 0) \\in M_k^*$ and", "note that this maps to $y = (0, \\ldots, 0, f_{jk}, 0, \\ldots, 0) \\in M_j^*$.", "By our choice of $k$ we see that $y \\in E_j$. By", "Example \\ref{example-ML-surjective-maps}", "the transition maps $E_i \\to E_j$ are surjective for each $i \\geq j$", "and $\\lim E_i = \\lim M_i^* = \\Hom_R(M, N)$. Hence", "Lemma \\ref{lemma-ML-limit-nonempty}", "guarantees there exists an element $z \\in \\Hom_R(M, N)$", "which maps to $y$ in $E_j \\subset M_j^*$. Let $z_k$ be the $k$th component", "of $z$. Then $z_k : M \\to M_k$ is a homomorphism such that", "$$", "\\xymatrix{", "M \\ar[r]_{z_k} & M_k \\\\", "M_j \\ar[ru]_{f_{jk}} \\ar[u]^{f_j}", "}", "$$", "commutes. Let $\\alpha : M \\to M$ be the composition", "$f_k \\circ z_k : M \\to M_k \\to M$.", "Then $\\alpha$ factors through a finitely presented module by construction and", "$\\alpha \\circ f_j = f_j$. Since the image of $f$ is contained in the image of", "$f_j$ this also implies that $\\alpha \\circ f = f$." ], "refs": [ "algebra-lemma-ML-countable-colimit", "algebra-lemma-ML-limit-nonempty" ], "ref_ids": [ 848, 825 ] } ], "ref_ids": [] }, { "id": 850, "type": "theorem", "label": "algebra-lemma-countgen-projective", "categories": [ "algebra" ], "title": "algebra-lemma-countgen-projective", "contents": [ "Let $M$ be an $R$-module. If $M$ is flat, Mittag-Leffler, and countably", "generated, then $M$ is projective." ], "refs": [], "proofs": [ { "contents": [ "By Lazard's theorem (Theorem \\ref{theorem-lazard}), we can write $M =", "\\colim_{i", "\\in I} M_i$ for a directed system of finite free $R$-modules $(M_i, f_{ij})$", "indexed by a set $I$. By Lemma \\ref{lemma-ML-countable-colimit}, we may assume", "$I$ is countable. Now let", "$$", "0 \\to N_1 \\to N_2 \\to N_3 \\to 0", "$$", "be an exact sequence of $R$-modules. We must show that applying", "$\\Hom_R(M, -)$", "preserves exactness. Since $M_i$ is finite free,", "$$", "0 \\to \\Hom_R(M_i, N_1) \\to \\Hom_R(M_i, N_2) \\to", "\\Hom_R(M_i, N_3) \\to 0", "$$", "is exact for each $i$. Since $M$ is Mittag-Leffler, $(\\Hom_R(M_i,", "N_{1}))$ is", "a Mittag-Leffler inverse system. So by Lemma \\ref{lemma-ML-exact-sequence},", "$$", "0 \\to \\lim_{i \\in I} \\Hom_R(M_i, N_1) \\to", "\\lim_{i \\in I} \\Hom_R(M_i, N_2) \\to", "\\lim_{i \\in I} \\Hom_R(M_i, N_3) \\to 0", "$$", "is exact. But for any $R$-module $N$ there is a functorial isomorphism", "$\\Hom_R(M, N) \\cong \\lim_{i \\in I} \\Hom_R(M_i, N)$, so", "$$", "0 \\to \\Hom_R(M, N_1) \\to \\Hom_R(M, N_2) \\to", "\\Hom_R(M, N_3) \\to 0", "$$", "is exact." ], "refs": [ "algebra-theorem-lazard", "algebra-lemma-ML-countable-colimit", "algebra-lemma-ML-exact-sequence" ], "ref_ids": [ 318, 848, 826 ] } ], "ref_ids": [] }, { "id": 851, "type": "theorem", "label": "algebra-lemma-ML-ui-descent", "categories": [ "algebra" ], "title": "algebra-lemma-ML-ui-descent", "contents": [ "Let $f: M \\to N$ be universally injective map of $R$-modules. Suppose", "$M$ is a direct sum of countably generated $R$-modules, and suppose $N$ is flat", "and Mittag-Leffler. Then $M$ is projective." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemmas \\ref{lemma-ui-flat-domain} and", "\\ref{lemma-pure-submodule-ML},", "$M$ is flat and Mittag-Leffler, so the conclusion follows from Theorem", "\\ref{theorem-projectivity-characterization}." ], "refs": [ "algebra-lemma-ui-flat-domain", "algebra-lemma-pure-submodule-ML", "algebra-theorem-projectivity-characterization" ], "ref_ids": [ 810, 837, 323 ] } ], "ref_ids": [] }, { "id": 852, "type": "theorem", "label": "algebra-lemma-universally-injective-submodule-powerseries", "categories": [ "algebra" ], "title": "algebra-lemma-universally-injective-submodule-powerseries", "contents": [ "Let $R$ be a Noetherian ring and let $M$ be a $R$-module. Suppose $M$ is a", "direct sum of countably generated $R$-modules, and suppose there is a", "universally injective map $M \\to R[[t_1, \\ldots, t_n]]$ for some $n$.", "Then $M$ is projective." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemmas \\ref{lemma-ML-ui-descent} and", "\\ref{lemma-power-series-ML}." ], "refs": [ "algebra-lemma-ML-ui-descent", "algebra-lemma-power-series-ML" ], "ref_ids": [ 851, 847 ] } ], "ref_ids": [] }, { "id": 853, "type": "theorem", "label": "algebra-lemma-ascend-properties-modules", "categories": [ "algebra" ], "title": "algebra-lemma-ascend-properties-modules", "contents": [ "Let $R \\to S$ be a ring map. Let $M$ be an $R$-module. Then:", "\\begin{enumerate}", "\\item If $M$ is flat, then the $S$-module $M \\otimes_R S$ is flat.", "\\item If $M$ is Mittag-Leffler, then the $S$-module $M \\otimes_R S$ is", "Mittag-Leffler.", "\\item If $M$ is a direct sum of countably generated $R$-modules, then the", "$S$-module $M \\otimes_R S$ is a direct sum of countably generated $S$-modules.", "\\item If $M$ is projective, then the $S$-module $M \\otimes_R S$ is projective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "All are obvious except (2). For this, use formulation (3) of being", "Mittag-Leffler from Proposition \\ref{proposition-ML-characterization} and the", "fact that tensoring commutes with taking colimits." ], "refs": [ "algebra-proposition-ML-characterization" ], "ref_ids": [ 1414 ] } ], "ref_ids": [] }, { "id": 854, "type": "theorem", "label": "algebra-lemma-ffdescent-ML", "categories": [ "algebra" ], "title": "algebra-lemma-ffdescent-ML", "contents": [ "\\begin{reference}", "Email from Juan Pablo Acosta Lopez dated 12/20/14.", "\\end{reference}", "Let $R \\to S$ be a faithfully flat ring map. Let $M$ be an $R$-module. If the", "$S$-module $M \\otimes_R S$ is Mittag-Leffler, then $M$ is Mittag-Leffler." ], "refs": [], "proofs": [ { "contents": [ "Write $M = \\colim_{i\\in I} M_i$ as a directed colimit", "of finitely presented $R$-modules $M_i$.", "Using Proposition \\ref{proposition-ML-characterization}, we see that we", "have to prove that for each $i \\in I$ there exists $i \\leq j$, $j\\in I$", "such that $M_i\\rightarrow M_j$ dominates $M_i\\rightarrow M$.", "\\medskip\\noindent", "Take $N$ the pushout", "$$", "\\xymatrix{", "M_i \\ar[r] \\ar[d] & M_j \\ar[d] \\\\", "M \\ar[r] & N", "}", "$$", "Then the lemma is equivalent to the existence of $j$ such that", "$M_j\\rightarrow N$ is universally injective, see", "Lemma \\ref{lemma-domination-universally-injective}.", "Observe that the tensorization by $S$", "$$", "\\xymatrix{", "M_i\\otimes_R S \\ar[r] \\ar[d] & M_j\\otimes_R S \\ar[d] \\\\", "M\\otimes_R S \\ar[r] & N\\otimes_R S", "}", "$$", "Is a pushout diagram. So because", "$M \\otimes_R S = \\colim_{i\\in I} M_i \\otimes_R S$", "expresses $M\\otimes_R S$ as a colimit of $S$-modules of", "finite presentation, and $M\\otimes_R S$ is Mittag-Leffler,", "there exists $j \\geq i$ such that $M_j\\otimes_R S\\rightarrow N\\otimes_R S$", "is universally injective. So using that $R\\rightarrow S$ is faithfully flat", "we conclude that $M_j\\rightarrow N$ is universally injective too." ], "refs": [ "algebra-proposition-ML-characterization", "algebra-lemma-domination-universally-injective" ], "ref_ids": [ 1414, 829 ] } ], "ref_ids": [] }, { "id": 855, "type": "theorem", "label": "algebra-lemma-ffdescent-countable-projectivity", "categories": [ "algebra" ], "title": "algebra-lemma-ffdescent-countable-projectivity", "contents": [ "Let $R \\to S$ be a faithfully flat ring map. Let $M$ be an $R$-module.", " If the $S$-module $M \\otimes_R S$ is countably generated and projective,", "then $M$ is countably generated and projective." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-descend-properties-modules},", "Lemma \\ref{lemma-ffdescent-ML}, the fact that countable", "generation descends, and", "Theorem \\ref{theorem-projectivity-characterization}." ], "refs": [ "algebra-lemma-descend-properties-modules", "algebra-lemma-ffdescent-ML", "algebra-theorem-projectivity-characterization" ], "ref_ids": [ 819, 854, 323 ] } ], "ref_ids": [] }, { "id": 856, "type": "theorem", "label": "algebra-lemma-lift-countably-generated-submodule", "categories": [ "algebra" ], "title": "algebra-lemma-lift-countably-generated-submodule", "contents": [ "Let $R \\to S$ be a ring map, let $M$ be an $R$-module, and let $Q$ be a", "countably generated $S$-submodule of $M \\otimes_R S$. Then there exists a", "countably generated $R$-submodule $P$ of $M$ such that", "$\\Im(P \\otimes_R S \\to M \\otimes_R S)$ contains $Q$." ], "refs": [], "proofs": [ { "contents": [ "Let $y_1, y_2, \\ldots$ be generators for $Q$ and write $y_j = \\sum_k x_{jk}", "\\otimes s_{jk}$ for some $x_{jk} \\in M$ and $s_{jk} \\in S$. Then take $P$ be", "the submodule of $M$ generated by the $x_{jk}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 857, "type": "theorem", "label": "algebra-lemma-adapted-submodule", "categories": [ "algebra" ], "title": "algebra-lemma-adapted-submodule", "contents": [ "Let $R \\to S$ be a ring map, and let $M$ be an $R$-module. Suppose $M", "\\otimes_R S = \\bigoplus_{i \\in I} Q_i$ is a direct sum of countably generated", "$S$-modules $Q_i$. If $N$ is a countably generated submodule of $M$, then", "there is a countably generated submodule $N'$ of $M$ such that $N' \\supset N$", "and $\\Im(N' \\otimes_R S \\to M \\otimes_R S) =", "\\bigoplus_{i \\in I'} Q_i$ for some subset $I' \\subset I$." ], "refs": [], "proofs": [ { "contents": [ "Let $N'_0 = N$. We construct by induction an increasing sequence of countably", "generated submodules $N'_{\\ell} \\subset M$ for $\\ell = 0, 1, 2, \\ldots$", "such that: if $I'_{\\ell}$ is the set of $i \\in I$ such that the projection of", "$\\Im(N'_{\\ell} \\otimes_R S \\to M \\otimes_R S)$ onto $Q_i$ is", "nonzero, then $\\Im(N'_{\\ell + 1} \\otimes_R S \\to M \\otimes_R", "S)$ contains $Q_i$ for all $i \\in I'_{\\ell}$. To construct $N'_{\\ell + 1}$", "from $N'_\\ell$, let $Q$ be the sum of (the countably many) $Q_i$ for", "$i \\in I'_{\\ell}$, choose $P$ as in Lemma", "\\ref{lemma-lift-countably-generated-submodule}, and then let $N'_{\\ell + 1} =", "N'_{\\ell} + P$. Having constructed the $N'_{\\ell}$, just take $N' =", "\\bigcup_{\\ell} N'_{\\ell}$ and $I' = \\bigcup_{\\ell} I'_{\\ell}$." ], "refs": [ "algebra-lemma-lift-countably-generated-submodule" ], "ref_ids": [ 856 ] } ], "ref_ids": [] }, { "id": 858, "type": "theorem", "label": "algebra-lemma-completion-generalities", "categories": [ "algebra" ], "title": "algebra-lemma-completion-generalities", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal.", "Let $\\varphi : M \\to N$ be a map of $R$-modules.", "\\begin{enumerate}", "\\item If $M/IM \\to N/IN$ is surjective, then $M^\\wedge \\to N^\\wedge$", "is surjective.", "\\item If $M \\to N$ is surjective, then $M^\\wedge \\to N^\\wedge$ is surjective.", "\\item If $0 \\to K \\to M \\to N \\to 0$ is a short exact sequence of", "$R$-modules and $N$ is flat, then", "$0 \\to K^\\wedge \\to M^\\wedge \\to N^\\wedge \\to 0$ is a short exact sequence.", "\\item The map $M \\otimes_R R^\\wedge \\to M^\\wedge$ is", "surjective for any finite $R$-module $M$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $M/IM \\to N/IN$ is surjective. Then the map $M/I^nM \\to N/I^nN$", "is surjective for each $n \\geq 1$ by Nakayama's lemma. More precisely,", "apply Lemma \\ref{lemma-NAK} part (11) to the", "map $M/I^nM \\to N/I^nN$ over the ring $R/I^n$ and the nilpotent", "ideal $I/I^n$ to see this. Set $K_n = \\{x \\in M \\mid \\varphi(x) \\in I^nN\\}$.", "Thus we get short exact sequences", "$$", "0 \\to K_n/I^nM \\to M/I^nM \\to N/I^nN \\to 0", "$$", "We claim that the canonical map $K_{n + 1}/I^{n + 1}M \\to K_n/I^nM$", "is surjective. Namely, if $x \\in K_n$ write", "$\\varphi(x) = \\sum z_j n_j$ with $z_j \\in I^n$, $n_j \\in N$.", "By assumption we can write $n_j = \\varphi(m_j) + \\sum z_{jk}n_{jk}$", "with $m_j \\in M$, $z_{jk} \\in I$ and $n_{jk} \\in N$. Hence", "$$", "\\varphi(x - \\sum z_j m_j) = \\sum z_jz_{jk} n_{jk}.", "$$", "This means that $x' = x - \\sum z_j m_j \\in K_{n + 1}$ maps", "to $x$ which proves the claim. Now we may apply", "Lemma \\ref{lemma-Mittag-Leffler}", "to the inverse system of short exact sequences above to see (1).", "Part (2) is a special case of (1).", "If the assumptions of (3) hold, then for each $n$ the sequence", "$$", "0 \\to K/I^nK \\to M/I^nM \\to N/I^nN \\to 0", "$$", "is short exact by", "Lemma \\ref{lemma-flat-tor-zero}.", "Hence we can directly apply", "Lemma \\ref{lemma-Mittag-Leffler}", "to conclude (3) is true.", "To see (4) choose generators $x_i \\in M$, $i = 1, \\ldots, n$.", "Then the map $R^{\\oplus n} \\to M$, $(a_1, \\ldots, a_n) \\mapsto \\sum a_ix_i$", "is surjective. Hence by (2) we see", "$(R^\\wedge)^{\\oplus n} \\to M^\\wedge$, $(a_1, \\ldots, a_n) \\mapsto \\sum a_ix_i$", "is surjective. Assertion (4) follows from this." ], "refs": [ "algebra-lemma-NAK", "algebra-lemma-Mittag-Leffler", "algebra-lemma-flat-tor-zero", "algebra-lemma-Mittag-Leffler" ], "ref_ids": [ 401, 827, 532, 827 ] } ], "ref_ids": [] }, { "id": 859, "type": "theorem", "label": "algebra-lemma-hathat-finitely-generated", "categories": [ "algebra" ], "title": "algebra-lemma-hathat-finitely-generated", "contents": [ "\\begin{reference}", "\\cite[Theorem 15]{Matlis}. The slick proof given here is from", "an email of Bjorn Poonen dated Nov 5, 2016.", "\\end{reference}", "Let $R$ be a ring. Let $I$ be a finitely generated ideal of $R$.", "Let $M$ be an $R$-module. Then", "\\begin{enumerate}", "\\item the completion $M^\\wedge$ is $I$-adically complete, and", "\\item $I^nM^\\wedge = \\Ker(M^\\wedge \\to M/I^nM) = (I^nM)^\\wedge$ for all", "$n \\geq 1$.", "\\end{enumerate}", "In particular $R^\\wedge$ is $I$-adically complete,", "$I^nR^\\wedge = (I^n)^\\wedge$, and", "$R^\\wedge/I^nR^\\wedge = R/I^n$." ], "refs": [], "proofs": [ { "contents": [ "Since $I$ is finitely generated,", "$I^n$ is finitely generated, say by $f_1, \\ldots, f_r$.", "Applying Lemma \\ref{lemma-completion-generalities} part (2)", "to the surjection $(f_1, \\ldots, f_r) : M^{\\oplus r} \\to I^n M$", "yields a surjection", "$$", "(M^\\wedge)^{\\oplus r} \\xrightarrow{(f_1, \\ldots, f_r)} (I^n M)^\\wedge =", "\\lim_{m \\geq n} I^n M/I^m M = \\Ker(M^\\wedge \\to M/I^n M).", "$$", "On the other hand, the image of", "$(f_1, \\ldots, f_r) : (M^\\wedge)^{\\oplus r} \\to M^\\wedge$", "is $I^n M^\\wedge$.", "Thus $M^\\wedge / I^n M^\\wedge \\simeq M/I^n M$.", "Taking inverse limits yields $(M^\\wedge)^\\wedge \\simeq M^\\wedge$;", "that is, $M^\\wedge$ is $I$-adically complete." ], "refs": [ "algebra-lemma-completion-generalities" ], "ref_ids": [ 858 ] } ], "ref_ids": [] }, { "id": 860, "type": "theorem", "label": "algebra-lemma-completion-differ-by-torsion", "categories": [ "algebra" ], "title": "algebra-lemma-completion-differ-by-torsion", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let", "$0 \\to M \\to N \\to Q \\to 0$ be an exact sequence of", "$R$-modules such that $Q$ is annihilated by a power of $I$.", "Then completion produces an exact sequence", "$0 \\to M^\\wedge \\to N^\\wedge \\to Q \\to 0$." ], "refs": [], "proofs": [ { "contents": [ "Say $I^c Q = 0$. Then $Q/I^nQ = Q$ for $n \\geq c$.", "On the other hand, it is clear that", "$I^nM \\subset M \\cap I^nN \\subset I^{n - c}M$ for $n \\geq c$.", "Thus $M^\\wedge = \\lim M/(M \\cap I^n N)$. Apply Lemma \\ref{lemma-Mittag-Leffler}", "to the system of exact sequences", "$$", "0 \\to M/(M \\cap I^n N) \\to N/I^n N \\to Q \\to 0", "$$", "for $n \\geq c$ to conclude." ], "refs": [ "algebra-lemma-Mittag-Leffler" ], "ref_ids": [ 827 ] } ], "ref_ids": [] }, { "id": 861, "type": "theorem", "label": "algebra-lemma-hathat", "categories": [ "algebra" ], "title": "algebra-lemma-hathat", "contents": [ "\\begin{reference}", "Taken from an unpublished note of Lenstra and de Smit.", "\\end{reference}", "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module.", "Denote $K_n = \\Ker(M^\\wedge \\to M/I^nM)$. Then $M^\\wedge$ is $I$-adically", "complete if and only if $K_n$ is equal to $I^nM^\\wedge$ for all $n \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "The module $I^n M^\\wedge$ is contained in $K_n$.", "Thus for each $n \\geq 1$ there is a canonical exact sequence", "$$", "0 \\to K_n/I^nM^\\wedge \\to M^\\wedge/I^nM^\\wedge \\to M/I^nM \\to 0.", "$$", "As $I^nM^\\wedge$ maps onto $I^nM/I^{n + 1}M$ we see that", "$K_{n + 1} + I^n M^\\wedge = K_n$. Thus the inverse system", "$\\{K_n/I^n M^\\wedge\\}_{n \\geq 1}$ has surjective transition maps.", "By", "Lemma \\ref{lemma-Mittag-Leffler}", "we see that there is a short exact sequence", "$$", "0 \\to", "\\lim_n K_n/I^n M^\\wedge \\to", "(M^\\wedge)^\\wedge \\to", "M^\\wedge \\to 0", "$$", "Hence $M^\\wedge$ is complete if and only if $K_n/I^n M^\\wedge = 0$", "for all $n \\geq 1$." ], "refs": [ "algebra-lemma-Mittag-Leffler" ], "ref_ids": [ 827 ] } ], "ref_ids": [] }, { "id": 862, "type": "theorem", "label": "algebra-lemma-radical-completion", "categories": [ "algebra" ], "title": "algebra-lemma-radical-completion", "contents": [ "Let $R$ be a ring, let $I \\subset R$ be an ideal, and let", "$R^\\wedge = \\lim R/I^n$.", "\\begin{enumerate}", "\\item any element of $R^\\wedge$ which maps to a unit of $R/I$ is a unit,", "\\item any element of $1 + I$ maps to an invertible element of $R^\\wedge$,", "\\item any element of $1 + IR^\\wedge$ is invertible in $R^\\wedge$, and", "\\item the ideals $IR^\\wedge$ and $\\Ker(R^\\wedge \\to R/I)$ are contained", "in the Jacobson radical of $R^\\wedge$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in R^\\wedge$ map to a unit $x_1$ in $R/I$.", "Then $x$ maps to a unit $x_n$ in $R/I^n$ for every $n$ by", "Lemma \\ref{lemma-locally-nilpotent-unit}.", "Hence $y = (x_n^{-1}) \\in \\lim R/I^n = R^\\wedge$ is an inverse to $x$.", "Parts (2) and (3) follow immediately from (1).", "Part (4) follows from (1) and Lemma \\ref{lemma-contained-in-radical}." ], "refs": [ "algebra-lemma-locally-nilpotent-unit", "algebra-lemma-contained-in-radical" ], "ref_ids": [ 459, 399 ] } ], "ref_ids": [] }, { "id": 863, "type": "theorem", "label": "algebra-lemma-when-surjective-to-completion", "categories": [ "algebra" ], "title": "algebra-lemma-when-surjective-to-completion", "contents": [ "Let $A$ be a ring. Let $I = (f_1, \\ldots, f_r)$ be a finitely", "generated ideal. If $M \\to \\lim M/f_i^nM$ is surjective for", "each $i$, then $M \\to \\lim M/I^nM$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Note that $\\lim M/I^nM = \\lim M/(f_1^n, \\ldots, f_r^n)M$ as", "$I^n \\supset (f_1^n, \\ldots, f_r^n) \\supset I^{rn}$.", "An element $\\xi$ of $\\lim M/(f_1^n, \\ldots, f_r^n)M$ can be symbolically", "written as", "$$", "\\xi = \\sum\\nolimits_{n \\geq 0} \\sum\\nolimits_i f_i^n x_{n, i}", "$$", "with $x_{n, i} \\in M$. If $M \\to \\lim M/f_i^nM$ is surjective, then there is", "an $x_i \\in M$ mapping to $\\sum x_{n, i} f_i^n$ in $\\lim M/f_i^nM$.", "Then $x = \\sum x_i$ maps to $\\xi$ in $\\lim M/I^nM$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 864, "type": "theorem", "label": "algebra-lemma-complete-by-sub", "categories": [ "algebra" ], "title": "algebra-lemma-complete-by-sub", "contents": [ "Let $A$ be a ring. Let $I \\subset J \\subset A$ be ideals.", "If $M$ is $J$-adically complete and $I$ is finitely generated, then", "$M$ is $I$-adically complete." ], "refs": [], "proofs": [ { "contents": [ "Assume $M$ is $J$-adically complete and $I$ is finitely generated.", "We have $\\bigcap I^nM = 0$ because $\\bigcap J^nM = 0$. By", "Lemma \\ref{lemma-when-surjective-to-completion}", "it suffices to prove the surjectivity of $M \\to \\lim M/I^nM$ in case", "$I$ is generated by a single element. Say $I = (f)$.", "Let $x_n \\in M$ with $x_{n + 1} - x_n \\in f^nM$. We have to show there exists", "an $x \\in M$ such that $x_n - x \\in f^nM$ for all $n$.", "As $x_{n + 1} - x_n \\in J^nM$ and as $M$ is $J$-adically complete,", "there exists an element $x \\in M$ such that $x_n - x \\in J^nM$.", "Replacing $x_n$ by $x_n - x$ we may assume that $x_n \\in J^nM$.", "To finish the proof we will show that this implies $x_n \\in I^nM$.", "Namely, write $x_n - x_{n + 1} = f^nz_n$.", "Then", "$$", "x_n = f^n(z_n + fz_{n + 1} + f^2z_{n + 2} + \\ldots)", "$$", "The sum $z_n + fz_{n + 1} + f^2z_{n + 2} + \\ldots$ converges in $M$", "as $f^c \\in J^c$. The sum $f^n(z_n + fz_{n + 1} + f^2z_{n + 2} + \\ldots)$", "converges in $M$ to $x_n$ because", "the partial sums equal $x_n - x_{n + c}$ and $x_{n + c} \\in J^{n + c}M$." ], "refs": [ "algebra-lemma-when-surjective-to-completion" ], "ref_ids": [ 863 ] } ], "ref_ids": [] }, { "id": 865, "type": "theorem", "label": "algebra-lemma-change-ideal-completion", "categories": [ "algebra" ], "title": "algebra-lemma-change-ideal-completion", "contents": [ "Let $R$ be a ring.", "Let $I$, $J$ be ideals of $R$.", "Assume there exist integers $c, d > 0$ such that", "$I^c \\subset J$ and $J^d \\subset I$.", "Then completion with respect to $I$ agrees with completion", "with respect to $J$ for any $R$-module.", "In particular an $R$-module $M$ is $I$-adically complete", "if and only if it is $J$-adically complete." ], "refs": [], "proofs": [ { "contents": [ "Consider the system of maps", "$M/I^nM \\to M/J^{\\lfloor n/d \\rfloor}M$ and", "the system of maps $M/J^mM \\to M/I^{\\lfloor m/c \\rfloor}M$", "to get mutually inverse maps between the completions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 866, "type": "theorem", "label": "algebra-lemma-quotient-complete", "categories": [ "algebra" ], "title": "algebra-lemma-quotient-complete", "contents": [ "Let $R$ be a ring. Let $I$ be an ideal of $R$.", "Let $M$ be an $I$-adically complete $R$-module,", "and let $K \\subset M$ be an $R$-submodule.", "The following are equivalent", "\\begin{enumerate}", "\\item $K = \\bigcap (K + I^nM)$ and", "\\item $M/K$ is $I$-adically complete.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Set $N = M/K$. By", "Lemma \\ref{lemma-completion-generalities}", "the map $M = M^\\wedge \\to N^\\wedge$ is surjective.", "Hence $N \\to N^\\wedge$ is surjective. It is easy to see that the", "kernel of $N \\to N^\\wedge$ is the module $\\bigcap (K + I^nM) / K$." ], "refs": [ "algebra-lemma-completion-generalities" ], "ref_ids": [ 858 ] } ], "ref_ids": [] }, { "id": 867, "type": "theorem", "label": "algebra-lemma-when-finite-module-complete-over-complete-ring", "categories": [ "algebra" ], "title": "algebra-lemma-when-finite-module-complete-over-complete-ring", "contents": [ "Let $R$ be a ring. Let $I$ be an ideal of $R$.", "Let $M$ be an $R$-module.", "If (a) $R$ is $I$-adically complete, (b) $M$ is a finite $R$-module,", "and (c) $\\bigcap I^nM = (0)$, then $M$ is $I$-adically complete." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-completion-generalities}", "the map $M = M \\otimes_R R = M \\otimes_R R^\\wedge \\to M^\\wedge$", "is surjective. The kernel of this map is $\\bigcap I^nM$ hence zero", "by assumption. Hence $M \\cong M^\\wedge$ and $M$ is complete." ], "refs": [ "algebra-lemma-completion-generalities" ], "ref_ids": [ 858 ] } ], "ref_ids": [] }, { "id": 868, "type": "theorem", "label": "algebra-lemma-finite-over-complete-ring", "categories": [ "algebra" ], "title": "algebra-lemma-finite-over-complete-ring", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module.", "Assume", "\\begin{enumerate}", "\\item $R$ is $I$-adically complete,", "\\item $\\bigcap_{n \\geq 1} I^nM = (0)$, and", "\\item $M/IM$ is a finite $R/I$-module.", "\\end{enumerate}", "Then $M$ is a finite $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1, \\ldots, x_n \\in M$ be elements whose images in $M/IM$ generate", "$M/IM$ as a $R/I$-module. Denote $M' \\subset M$ the $R$-submodule", "generated by $x_1, \\ldots, x_n$. By Lemma \\ref{lemma-completion-generalities}", "the map $(M')^\\wedge \\to M^\\wedge$ is surjective.", "Since $\\bigcap I^nM = 0$ we see in particular that $\\bigcap I^nM' = (0)$.", "Hence by Lemma \\ref{lemma-when-finite-module-complete-over-complete-ring}", "we see that $M'$ is complete, and we conclude that $M' \\to M^\\wedge$", "is surjective. Finally, the kernel of $M \\to M^\\wedge$ is", "zero since it is equal to $\\bigcap I^nM = (0)$.", "Hence we conclude that $M \\cong M' \\cong M^\\wedge$", "is finitely generated." ], "refs": [ "algebra-lemma-completion-generalities", "algebra-lemma-when-finite-module-complete-over-complete-ring" ], "ref_ids": [ 858, 867 ] } ], "ref_ids": [] }, { "id": 869, "type": "theorem", "label": "algebra-lemma-completion-tensor", "categories": [ "algebra" ], "title": "algebra-lemma-completion-tensor", "contents": [ "Let $I$ be an ideal of a Noetherian ring $R$.", "Denote ${}^\\wedge$ completion with respect to $I$.", "\\begin{enumerate}", "\\item If $K \\to N$ is an injective map of finite $R$-modules,", "then the map on completions $K^\\wedge \\to N^\\wedge$ is injective.", "\\item If $0 \\to K \\to N \\to M \\to 0$ is a short exact sequence", "of finite $R$-modules, then $0 \\to K^\\wedge \\to N^\\wedge \\to M^\\wedge \\to 0$", "is a short exact sequence.", "\\item If $M$ is a finite $R$-module, then $M^\\wedge = M \\otimes_R R^\\wedge$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Setting $M = N/K$ we find that part (1) follows from part (2).", "Let $0 \\to K \\to N \\to M \\to 0$ be as in (2).", "For each $n$ we get the short exact sequence", "$$", "0 \\to K/(I^nN \\cap K) \\to N/I^nN \\to M/I^nM \\to 0.", "$$", "By Lemma \\ref{lemma-Mittag-Leffler}", "we obtain the exact sequence", "$$", "0 \\to \\lim K/(I^nN \\cap K) \\to N^\\wedge \\to M^\\wedge \\to 0.", "$$", "By the Artin-Rees Lemma \\ref{lemma-Artin-Rees} we may choose $c$ such that", "$I^nK \\subset I^n N \\cap K \\subset I^{n-c} K$ for $n \\geq c$.", "Hence $K^\\wedge = \\lim K/I^nK = \\lim K/(I^nN \\cap K)$", "and we conclude that (2) is true.", "\\medskip\\noindent", "Let $M$ be as in (3) and let $0 \\to K \\to R^{\\oplus t} \\to M \\to 0$", "be a presentation of $M$. We get a commutative diagram", "$$", "\\xymatrix{", "&", "K \\otimes_R R^\\wedge \\ar[r] \\ar[d] &", "R^{\\oplus t} \\otimes_R R^\\wedge \\ar[r] \\ar[d] &", "M \\otimes_R R^\\wedge \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "K^\\wedge \\ar[r] &", "(R^{\\oplus t})^\\wedge \\ar[r] &", "M^\\wedge \\ar[r] & 0", "}", "$$", "The top row is exact, see Section \\ref{section-flat}.", "The bottom row is exact by part (2).", "By Lemma \\ref{lemma-completion-generalities}", "the vertical arrows are surjective.", "The middle vertical arrow is an isomorphism.", "We conclude (3) holds by the Snake Lemma \\ref{lemma-snake}." ], "refs": [ "algebra-lemma-Mittag-Leffler", "algebra-lemma-Artin-Rees", "algebra-lemma-completion-generalities", "algebra-lemma-snake" ], "ref_ids": [ 827, 625, 858, 328 ] } ], "ref_ids": [] }, { "id": 870, "type": "theorem", "label": "algebra-lemma-completion-flat", "categories": [ "algebra" ], "title": "algebra-lemma-completion-flat", "contents": [ "Let $I$ be a ideal of a Noetherian ring $R$.", "Denote ${}^\\wedge$ completion with respect to $I$.", "\\begin{enumerate}", "\\item The ring map $R \\to R^\\wedge$ is flat.", "\\item The functor $M \\mapsto M^\\wedge$ is exact on the category of", "finitely generated $R$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider $J \\otimes_R R^\\wedge \\to R \\otimes_R R^\\wedge = R^\\wedge$", "where $J$ is an arbitrary ideal of $R$.", "According to Lemma \\ref{lemma-completion-tensor} this", "is identified with $J^\\wedge \\to R^\\wedge$ and $J^\\wedge \\to R^\\wedge$", "is injective. Part (1) follows from Lemma \\ref{lemma-flat}.", "Part (2) is a reformulation of", "Lemma \\ref{lemma-completion-tensor} part (2)." ], "refs": [ "algebra-lemma-completion-tensor", "algebra-lemma-flat", "algebra-lemma-completion-tensor" ], "ref_ids": [ 869, 525, 869 ] } ], "ref_ids": [] }, { "id": 871, "type": "theorem", "label": "algebra-lemma-completion-faithfully-flat", "categories": [ "algebra" ], "title": "algebra-lemma-completion-faithfully-flat", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring.", "Let $I \\subset \\mathfrak m$ be an ideal. Denote $R^\\wedge$", "the completion of $R$ with respect to $I$.", "The ring map $R \\to R^\\wedge$ is faithfully flat.", "In particular the completion with respect to $\\mathfrak m$,", "namely $\\lim_n R/\\mathfrak m^n$ is faithfully flat." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-completion-flat} it is flat.", "The composition $R \\to R^\\wedge \\to R/\\mathfrak m$ where", "the last map is the projection map $R^\\wedge \\to R/I$", "combined with $R/I \\to R/\\mathfrak m$ shows that", "$\\mathfrak m$ is in the image of $\\Spec(R^\\wedge)", "\\to \\Spec(R)$. Hence the map is faithfully", "flat by Lemma \\ref{lemma-ff}." ], "refs": [ "algebra-lemma-completion-flat", "algebra-lemma-ff" ], "ref_ids": [ 870, 535 ] } ], "ref_ids": [] }, { "id": 872, "type": "theorem", "label": "algebra-lemma-completion-complete", "categories": [ "algebra" ], "title": "algebra-lemma-completion-complete", "contents": [ "Let $R$ be a Noetherian ring.", "Let $I$ be an ideal of $R$.", "Let $M$ be an $R$-module.", "Then the completion $M^\\wedge$", "of $M$ with respect to $I$ is $I$-adically complete,", "$I^n M^\\wedge = (I^nM)^\\wedge$, and $M^\\wedge/I^nM^\\wedge = M/I^nM$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-hathat-finitely-generated}", "because $I$ is a finitely generated ideal." ], "refs": [ "algebra-lemma-hathat-finitely-generated" ], "ref_ids": [ 859 ] } ], "ref_ids": [] }, { "id": 873, "type": "theorem", "label": "algebra-lemma-completion-Noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-completion-Noetherian", "contents": [ "Let $I$ be an ideal of a ring $R$. Assume", "\\begin{enumerate}", "\\item $R/I$ is a Noetherian ring,", "\\item $I$ is finitely generated.", "\\end{enumerate}", "Then the completion $R^\\wedge$ of $R$ with respect to $I$", "is a Noetherian ring complete with respect to $IR^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-hathat-finitely-generated}", "we see that $R^\\wedge$ is $I$-adically complete. Hence it is also", "$IR^\\wedge$-adically complete. Since $R^\\wedge/IR^\\wedge = R/I$ is", "Noetherian we see that after replacing $R$ by $R^\\wedge$ we may in", "addition to assumptions (1) and (2) assume that also $R$ is $I$-adically", "complete.", "\\medskip\\noindent", "Let $f_1, \\ldots, f_t$ be generators of $I$.", "Then there is a surjection of rings", "$R/I[T_1, \\ldots, T_t] \\to \\bigoplus I^n/I^{n + 1}$", "mapping $T_i$ to the element $\\overline{f}_i \\in I/I^2$.", "Hence $\\bigoplus I^n/I^{n + 1}$ is a Noetherian ring.", "Let $J \\subset R$ be an ideal. Consider the ideal", "$$", "\\bigoplus J \\cap I^n/J \\cap I^{n + 1} \\subset \\bigoplus I^n/I^{n + 1}.", "$$", "Let $\\overline{g}_1, \\ldots, \\overline{g}_m$ be generators of this", "ideal. We may choose $\\overline{g}_j$ to be a homogeneous element", "of degree $d_j$ and we may pick $g_j \\in J \\cap I^{d_j}$ mapping to", "$\\overline{g}_j \\in J \\cap I^{d_j}/J \\cap I^{d_j + 1}$. We claim", "that $g_1, \\ldots, g_m$ generate $J$.", "\\medskip\\noindent", "Let $x \\in J \\cap I^n$. There exist $a_j \\in I^{\\max(0, n - d_j)}$ such that", "$x - \\sum a_j g_j \\in J \\cap I^{n + 1}$.", "The reason is that $J \\cap I^n/J \\cap I^{n + 1}$ is equal to", "$\\sum \\overline{g}_j I^{n - d_j}/I^{n - d_j + 1}$ by our choice", "of $g_1, \\ldots, g_m$. Hence starting with $x \\in J$ we can find", "a sequence of vectors $(a_{1, n}, \\ldots, a_{m, n})_{n \\geq 0}$", "with $a_{j, n} \\in I^{\\max(0, n - d_j)}$ such that", "$$", "x =", "\\sum\\nolimits_{n = 0, \\ldots, N}", "\\sum\\nolimits_{j = 1, \\ldots, m} a_{j, n} g_j \\bmod I^{N + 1}", "$$", "Setting $A_j = \\sum_{n \\geq 0} a_{j, n}$ we see that", "$x = \\sum A_j g_j$ as $R$ is complete. Hence $J$ is finitely generated and", "we win." ], "refs": [ "algebra-lemma-hathat-finitely-generated" ], "ref_ids": [ 859 ] } ], "ref_ids": [] }, { "id": 874, "type": "theorem", "label": "algebra-lemma-completion-Noetherian-Noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-completion-Noetherian-Noetherian", "contents": [ "Let $R$ be a Noetherian ring.", "Let $I$ be an ideal of $R$.", "The completion $R^\\wedge$ of $R$ with respect to $I$ is", "Noetherian." ], "refs": [], "proofs": [ { "contents": [ "This is a consequence of", "Lemma \\ref{lemma-completion-Noetherian}.", "It can also be seen directly as follows.", "Choose generators $f_1, \\ldots, f_n$ of $I$.", "Consider the map", "$$", "R[[x_1, \\ldots, x_n]] \\longrightarrow R^\\wedge,", "\\quad", "x_i \\longmapsto f_i.", "$$", "This is a well defined and surjective ring map", "(details omitted).", "Since $R[[x_1, \\ldots, x_n]]$ is Noetherian (see", "Lemma \\ref{lemma-Noetherian-power-series}) we win." ], "refs": [ "algebra-lemma-completion-Noetherian", "algebra-lemma-Noetherian-power-series" ], "ref_ids": [ 873, 449 ] } ], "ref_ids": [] }, { "id": 875, "type": "theorem", "label": "algebra-lemma-finite-after-completion", "categories": [ "algebra" ], "title": "algebra-lemma-finite-after-completion", "contents": [ "Let $R \\to S$ be a local homomorphism of local rings $(R, \\mathfrak m)$", "and $(S, \\mathfrak n)$. Let $R^\\wedge$, resp.\\ $S^\\wedge$ be the completion", "of $R$, resp.\\ $S$ with respect to $\\mathfrak m$, resp.\\ $\\mathfrak n$.", "If $\\mathfrak m$ and $\\mathfrak n$ are finitely generated and", "$\\dim_{\\kappa(\\mathfrak m)} S/\\mathfrak mS < \\infty$, then", "\\begin{enumerate}", "\\item $S^\\wedge$ is equal to the $\\mathfrak m$-adic completion of $S$, and", "\\item $S^\\wedge$ is a finite $R^\\wedge$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have $\\mathfrak mS \\subset \\mathfrak n$ because $R \\to S$ is a local", "ring map.", "The assumption $\\dim_{\\kappa(\\mathfrak m)} S/\\mathfrak mS < \\infty$", "implies that $S/\\mathfrak mS$ is an Artinian ring, see", "Lemma \\ref{lemma-finite-dimensional-algebra}.", "Hence has dimension $0$, see", "Lemma \\ref{lemma-Noetherian-dimension-0},", "hence $\\mathfrak n = \\sqrt{\\mathfrak mS}$.", "This and the fact that $\\mathfrak n$ is finitely generated", "implies that $\\mathfrak n^t \\subset \\mathfrak mS$ for", "some $t \\geq 1$. By", "Lemma \\ref{lemma-change-ideal-completion}", "we see that $S^\\wedge$ can be identified with the $\\mathfrak m$-adic", "completion of $S$. As $\\mathfrak m$ is finitely generated we see from", "Lemma \\ref{lemma-hathat-finitely-generated}", "that $S^\\wedge$ and $R^\\wedge$ are $\\mathfrak m$-adically complete.", "At this point we may apply", "Lemma \\ref{lemma-finite-over-complete-ring}", "to $S^\\wedge$ as an $R^\\wedge$-module to conclude." ], "refs": [ "algebra-lemma-finite-dimensional-algebra", "algebra-lemma-Noetherian-dimension-0", "algebra-lemma-change-ideal-completion", "algebra-lemma-hathat-finitely-generated", "algebra-lemma-finite-over-complete-ring" ], "ref_ids": [ 642, 680, 865, 859, 868 ] } ], "ref_ids": [] }, { "id": 876, "type": "theorem", "label": "algebra-lemma-completion-finite-extension", "categories": [ "algebra" ], "title": "algebra-lemma-completion-finite-extension", "contents": [ "Let $R$ be a Noetherian ring. Let $R \\to S$ be a finite ring map.", "Let $\\mathfrak p \\subset R$ be a prime and let", "$\\mathfrak q_1, \\ldots, \\mathfrak q_m$ be the primes of $S$", "lying over $\\mathfrak p$", "(Lemma \\ref{lemma-finite-finite-fibres}).", "Then", "$$", "R_\\mathfrak p^\\wedge \\otimes_R S =", "(S_\\mathfrak p)^\\wedge =", "S_{\\mathfrak q_1}^\\wedge \\times \\ldots \\times S_{\\mathfrak q_m}^\\wedge", "$$", "where the $(S_\\mathfrak p)^\\wedge$ is the completion with respect to", "$\\mathfrak p$ and the local rings $R_\\mathfrak p$ and", "$S_{\\mathfrak q_i}$ are completed with respect to their maximal ideals." ], "refs": [ "algebra-lemma-finite-finite-fibres" ], "proofs": [ { "contents": [ "The first equality follows from Lemma \\ref{lemma-completion-tensor}.", "We may replace $R$ by the localization $R_\\mathfrak p$ and", "$S$ by $S_\\mathfrak p = S \\otimes_R R_\\mathfrak p$.", "Hence we may assume that $R$ is a local Noetherian ring and", "that $\\mathfrak p = \\mathfrak m$ is its maximal ideal.", "The $\\mathfrak q_iS_{\\mathfrak q_i}$-adic completion", "$S_{\\mathfrak q_i}^\\wedge$ is equal to the $\\mathfrak m$-adic", "completion by Lemma \\ref{lemma-finite-after-completion}.", "For every $n \\geq 1$ prime ideals of $S/\\mathfrak m^nS$ are in 1-to-1", "correspondence with the maximal ideals", "$\\mathfrak q_1, \\ldots, \\mathfrak q_m$ of $S$", "(by going up for $S$ over $R$, see", "Lemma \\ref{lemma-integral-going-up}).", "Hence", "$S/\\mathfrak m^nS = \\prod S_{\\mathfrak q_i}/\\mathfrak m^nS_{\\mathfrak q_i}$", "by Lemma \\ref{lemma-artinian-finite-length}", "(using for example", "Proposition \\ref{proposition-dimension-zero-ring}", "to see that $S/\\mathfrak m^nS$ is Artinian).", "Hence the $\\mathfrak m$-adic completion $S^\\wedge$ of $S$ is equal to", "$\\prod S_{\\mathfrak q_i}^\\wedge$. Finally, we have", "$R^\\wedge \\otimes_R S = S^\\wedge$ by Lemma \\ref{lemma-completion-tensor}." ], "refs": [ "algebra-lemma-completion-tensor", "algebra-lemma-finite-after-completion", "algebra-lemma-integral-going-up", "algebra-lemma-artinian-finite-length", "algebra-proposition-dimension-zero-ring", "algebra-lemma-completion-tensor" ], "ref_ids": [ 869, 875, 500, 646, 1410, 869 ] } ], "ref_ids": [ 499 ] }, { "id": 877, "type": "theorem", "label": "algebra-lemma-split-completed-sequence", "categories": [ "algebra" ], "title": "algebra-lemma-split-completed-sequence", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal.", "Let $0 \\to K \\to P \\to M \\to 0$ be a short exact sequence of", "$R$-modules. If $M$ is flat over $R$ and $M/IM$ is a projective", "$R/I$-module, then the sequence of $I$-adic completions", "$$", "0 \\to K^\\wedge \\to P^\\wedge \\to M^\\wedge \\to 0", "$$", "is a split exact sequence." ], "refs": [], "proofs": [ { "contents": [ "As $M$ is flat, each of the sequences", "$$", "0 \\to K/I^nK \\to P/I^nP \\to M/I^nM \\to 0", "$$", "is short exact, see", "Lemma \\ref{lemma-flat-tor-zero}", "and the sequence $0 \\to K^\\wedge \\to P^\\wedge \\to M^\\wedge \\to 0$", "is a short exact sequence, see", "Lemma \\ref{lemma-completion-generalities}.", "It suffices to show that we can find splittings", "$s_n : M/I^nM \\to P/I^nP$ such that $s_{n + 1} \\bmod I^n = s_n$.", "We will construct these $s_n$ by induction on $n$.", "Pick any splitting $s_1$, which exists as $M/IM$ is a projective $R/I$-module.", "Assume given $s_n$ for some $n > 0$. Set", "$P_{n + 1} = \\{x \\in P \\mid x \\bmod I^nP \\in \\Im(s_n)\\}$.", "The map $\\pi : P_{n + 1}/I^{n + 1}P_{n + 1} \\to M/I^{n + 1}M$ is surjective", "(details omitted). As $M/I^{n + 1}M$ is projective as a $R/I^{n + 1}$-module", "by", "Lemma \\ref{lemma-lift-projective}", "we may choose a section", "$t : M/I^{n + 1}M \\to P_{n + 1}/I^{n + 1}P_{n + 1}$", "of $\\pi$. Setting $s_{n + 1}$ equal to the composition of", "$t$ with the canonical map $P_{n + 1}/I^{n + 1}P_{n + 1} \\to P/I^{n + 1}P$", "works." ], "refs": [ "algebra-lemma-flat-tor-zero", "algebra-lemma-completion-generalities", "algebra-lemma-lift-projective" ], "ref_ids": [ 532, 858, 794 ] } ], "ref_ids": [] }, { "id": 878, "type": "theorem", "label": "algebra-lemma-complete-modulo-nilpotent", "categories": [ "algebra" ], "title": "algebra-lemma-complete-modulo-nilpotent", "contents": [ "Let $A$ be a Noetherian ring. Let $I, J \\subset A$ be ideals.", "If $A$ is $I$-adically complete and $A/I$ is $J$-adically complete,", "then $A$ is $J$-adically complete." ], "refs": [], "proofs": [ { "contents": [ "Let $B$ be the $(I + J)$-adic completion of $A$. By", "Lemma \\ref{lemma-completion-flat} $B/IB$ is the $J$-adic completion of $A/I$", "hence isomorphic to $A/I$ by assumption. Moreover $B$ is $I$-adically", "complete by Lemma \\ref{lemma-complete-by-sub}. Hence $B$ is a finite", "$A$-module by Lemma \\ref{lemma-finite-over-complete-ring}.", "By Nakayama's lemma (Lemma \\ref{lemma-NAK} using $I$ is in the", "Jacobson radical of $A$", "by Lemma \\ref{lemma-radical-completion}) we find that $A \\to B$ is surjective.", "The map $A \\to B$ is flat by Lemma \\ref{lemma-completion-flat}.", "The image of $\\Spec(B) \\to \\Spec(A)$ contains $V(I)$ and as $I$", "is contained in the Jacobson radical of $A$ we find $A \\to B$ is faithfully flat", "(Lemma \\ref{lemma-ff-rings}). Thus $A \\to B$ is injective. Thus $A$ is", "complete with respect to $I + J$, hence a fortiori complete", "with respect to $J$." ], "refs": [ "algebra-lemma-completion-flat", "algebra-lemma-complete-by-sub", "algebra-lemma-finite-over-complete-ring", "algebra-lemma-NAK", "algebra-lemma-radical-completion", "algebra-lemma-completion-flat", "algebra-lemma-ff-rings" ], "ref_ids": [ 870, 864, 868, 401, 862, 870, 536 ] } ], "ref_ids": [] }, { "id": 879, "type": "theorem", "label": "algebra-lemma-limit-complete-pre", "categories": [ "algebra" ], "title": "algebra-lemma-limit-complete-pre", "contents": [ "Let $I \\subset A$ be a finitely generated ideal of a ring.", "Let $(M_n)$ be an inverse system of $A$-modules with $I^n M_n = 0$.", "Then $M = \\lim M_n$ is $I$-adically complete." ], "refs": [], "proofs": [ { "contents": [ "We have $M \\to M/I^nM \\to M_n$. Taking the limit we get", "$M \\to M^\\wedge \\to M$. Hence $M$ is a direct summand of $M^\\wedge$.", "Since $M^\\wedge$ is $I$-adically complete by", "Lemma \\ref{lemma-hathat-finitely-generated}, so is $M$." ], "refs": [ "algebra-lemma-hathat-finitely-generated" ], "ref_ids": [ 859 ] } ], "ref_ids": [] }, { "id": 880, "type": "theorem", "label": "algebra-lemma-limit-complete", "categories": [ "algebra" ], "title": "algebra-lemma-limit-complete", "contents": [ "Let $I \\subset A$ be a finitely generated ideal of a ring.", "Let $(M_n)$ be an inverse system of $A$-modules with", "$M_n = M_{n + 1}/I^nM_{n + 1}$. Then $M/I^nM = M_n$ and $M$ is", "$I$-adically complete." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-limit-complete-pre} we see that $M$ is $I$-adically", "complete. Since the transition maps are surjective, the maps $M \\to M_n$", "are surjective. Consider the inverse system of short exact sequences", "$$", "0 \\to N_n \\to M \\to M_n \\to 0", "$$", "defining $N_n$. Since $M_n = M_{n + 1}/I^nM_{n + 1}$ the map", "$N_{n + 1} + I^nM \\to N_n$ is surjective. Hence", "$N_{n + 1}/(N_{n + 1} \\cap I^{n + 1}M) \\to N_n/(N_n \\cap I^nM)$", "is surjective. Taking the inverse limit of the short exact sequences", "$$", "0 \\to N_n/(N_n \\cap I^nM) \\to M/I^nM \\to M_n \\to 0", "$$", "we obtain an exact sequence", "$$", "0 \\to \\lim N_n/(N_n \\cap I^nM) \\to M^\\wedge \\to M", "$$", "Since $M$ is $I$-adically complete we conclude that", "$\\lim N_n/(N_n \\cap I^nM) = 0$ and hence by the surjectivity", "of the transition maps we get $N_n/(N_n \\cap I^nM) = 0$ for all $n$.", "Thus $M_n = M/I^nM$ as desired." ], "refs": [ "algebra-lemma-limit-complete-pre" ], "ref_ids": [ 879 ] } ], "ref_ids": [] }, { "id": 881, "type": "theorem", "label": "algebra-lemma-finiteness-graded", "categories": [ "algebra" ], "title": "algebra-lemma-finiteness-graded", "contents": [ "Let $A$ be a Noetherian graded ring. Let $I \\subset A_+$ be a homogeneous", "ideal. Let $(N_n)$ be an inverse system of finite graded $A$-modules with", "$N_n = N_{n + 1}/I^n N_{n + 1}$. Then there is a finite graded $A$-module", "$N$ such that $N_n = N/I^nN$ as graded modules for all $n$." ], "refs": [], "proofs": [ { "contents": [ "Pick $r$ and homogeneous elements $x_{1, 1}, \\ldots, x_{1, r} \\in N_1$ of", "degrees $d_1, \\ldots, d_r$ generating $N_1$. Since the transition maps", "are surjective, we can pick a compatible system of homogeneous elements", "$x_{n, i} \\in N_n$ lifting $x_{1, i}$. By the graded Nakayama lemma", "(Lemma \\ref{lemma-graded-NAK}) we see that", "$N_n$ is generated by the elements $x_{n, 1}, \\ldots, x_{n, r}$", "sitting in degrees $d_1, \\ldots, d_r$.", "Thus for $m \\leq n$ we see that $N_n \\to N_n/I^m N_n$", "is an isomorphism in degrees $< \\min(d_i) + m$ (as $I^mN_n$ is zero", "in those degrees). Thus the inverse system of degree $d$ parts", "$$", "\\ldots", "= N_{2 + d - \\min(d_i), d}", "= N_{1 + d - \\min(d_i), d}", "= N_{d - \\min(d_i), d} \\to N_{-1 + d - \\min(d_i), d} \\to \\ldots", "$$", "stabilizes as indicated. Let $N$ be the graded $A$-module whose", "$d$th graded part is this stabilization. In particular, we have the", "elements $x_i = \\lim x_{n, i}$ in $N$. We claim the $x_i$ generate $N$:", "any $x \\in N_d$ is a linear combination of $x_1, \\ldots, x_r$", "because we can check this in $N_{d - \\min(d_i), d}$ where it holds", "as $x_{d - \\min(d_i), i}$ generate $N_{d - \\min(d_i)}$.", "Finally, the reader checks that the surjective map", "$N/I^nN \\to N_n$ is an isomorphism by checking to see", "what happens in each degree as before. Details omitted." ], "refs": [ "algebra-lemma-graded-NAK" ], "ref_ids": [ 656 ] } ], "ref_ids": [] }, { "id": 882, "type": "theorem", "label": "algebra-lemma-daniel-litt", "categories": [ "algebra" ], "title": "algebra-lemma-daniel-litt", "contents": [ "Let $A$ be a graded ring. Let $I \\subset A_+$ be a homogeneous ideal.", "Denote $A' = \\lim A/I^n$. Let $(G_n)$ be an inverse system", "of graded $A$-modules with $G_n$ annihilated by $I^n$.", "Let $M$ be a graded $A$-module and let", "$\\varphi_n : M \\to G_n$ be a compatible system of graded", "$A$-module maps. If the induced map", "$$", "\\varphi : M \\otimes_A A' \\longrightarrow \\lim G_n", "$$", "is an isomorphism, then $M_d \\to \\lim G_{n, d}$", "is an isomorphism for all $d \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "By convention graded rings are in degrees $\\geq 0$ and graded modules", "may have nonzero parts of any degree, see Section \\ref{section-graded}.", "The map $\\varphi$ exists because $\\lim G_n$ is", "a module over $A'$ as $G_n$ is annihilated by $I^n$.", "Another useful thing to keep in mind is that we have", "$$", "\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\lim G_{n, d} \\subset", "\\lim G_n \\subset", "\\prod\\nolimits_{d \\in \\mathbf{Z}} \\lim G_{n, d}", "$$", "where a subscript ${\\ }_d$ indicates the $d$th graded part.", "\\medskip\\noindent", "Injective. Let $x \\in M_d$. If $x \\mapsto 0$ in $\\lim G_{n, d}$", "then $x \\otimes 1 = 0$ in $M \\otimes_A A'$. Then we can find", "a finitely generated submodule $M' \\subset M$ with $x \\in M'$", "such that $x \\otimes 1$ is zero in $M' \\otimes_A A'$.", "Say $M'$ is generated by homogeneous elements sitting", "in degrees $d_1, \\ldots, d_r$. Let $n = d - \\min(d_i) + 1$.", "Since $A'$ has a map to $A/I^n$ and since", "$A \\to A/I^n$ is an isomorphism in degrees $\\leq n - 1$", "we see that $M' \\to M' \\otimes_A A'$ is injective in", "degrees $\\leq n - 1$. Thus $x = 0$ as desired.", "\\medskip\\noindent", "Surjective. Let $y \\in \\lim G_{n, d}$. Choose a finite", "sum $\\sum x_i \\otimes f'_i$ in $M \\otimes_A A'$ mapping to $y$.", "We may assume $x_i$ is homogeneous, say of degree $d_i$.", "Observe that although $A'$ is not a graded ring, it is a", "limit of the graded rings $A/I^nA$ and moreover, in any", "given degree the transition maps eventually become isomorphisms", "(see above). This gives", "$$", "A = \\bigoplus\\nolimits_{d \\geq 0} A_d \\subset A' \\subset", "\\prod\\nolimits_{d \\geq 0} A_d", "$$", "Thus we can write", "$$", "f'_i = \\sum\\nolimits_{j = 0, \\ldots, d - d_i - 1} f_{i, j} + f_i + g'_i", "$$", "with $f_{i, j} \\in A_j$, $f_i \\in A_{d - d_i}$, and", "$g'_i \\in A'$ mapping to zero in $\\prod_{j \\leq d - d_i} A_j$.", "Now if we compute $\\varphi_n(\\sum_{i, j} f_{i, j}x_i) \\in G_n$,", "then we get a sum of homogeneous elements of degree $< d$.", "Hence $\\varphi(\\sum x_i \\otimes f_{i, j})$ maps to zero in", "$\\lim G_{n, d}$.", "Similarly, a computation shows the element $\\varphi(\\sum x_i \\otimes g'_i)$", "maps to zero in $\\prod_{d' \\leq d} \\lim G_{n, d'}$.", "Since we know that $\\varphi(\\sum x_i \\otimes f'_i)$", "is $y$, we conclude that $\\sum f_ix_i \\in M_d$ maps to $y$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 883, "type": "theorem", "label": "algebra-lemma-mod-injective", "categories": [ "algebra" ], "title": "algebra-lemma-mod-injective", "contents": [ "Suppose that $R \\to S$ is a local homomorphism of Noetherian local rings.", "Denote $\\mathfrak m$ the maximal ideal of $R$. Let $M$ be a flat $R$-module", "and $N$ a finite $S$-module. Let $u : N \\to M$ be a map of $R$-modules.", "If $\\overline{u} : N/\\mathfrak m N \\to M/\\mathfrak m M$", "is injective then $u$ is injective.", "In this case $M/u(N)$ is flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "First we claim that $u_n : N/{\\mathfrak m}^nN \\to M/{\\mathfrak m}^nM$", "is injective for all $n \\geq 1$. We proceed by induction, the base", "case is that $\\overline{u} = u_1$ is injective. By our assumption that $M$", "is flat over $R$ we have a short exact sequence", "$0 \\to M \\otimes_R {\\mathfrak m}^n/{\\mathfrak m}^{n + 1}", "\\to M/{\\mathfrak m}^{n + 1}M \\to M/{\\mathfrak m}^n M \\to 0$.", "Also, $M \\otimes_R {\\mathfrak m}^n/{\\mathfrak m}^{n + 1}", "= M/{\\mathfrak m}M \\otimes_{R/{\\mathfrak m}}", "{\\mathfrak m}^n/{\\mathfrak m}^{n + 1}$. We have", "a similar exact sequence $N \\otimes_R {\\mathfrak m}^n/{\\mathfrak m}^{n + 1}", "\\to N/{\\mathfrak m}^{n + 1}N \\to N/{\\mathfrak m}^n N \\to 0$", "for $N$ except we do not have the zero on the left. We also", "have $N \\otimes_R {\\mathfrak m}^n/{\\mathfrak m}^{n + 1}", "= N/{\\mathfrak m}N \\otimes_{R/{\\mathfrak m}}", "{\\mathfrak m}^n/{\\mathfrak m}^{n + 1}$. Thus the map $u_{n + 1}$ is", "injective as both $u_n$ and the map", "$\\overline{u} \\otimes \\text{id}_{{\\mathfrak m}^n/{\\mathfrak m}^{n + 1}}$ are.", "\\medskip\\noindent", "By Krull's intersection theorem", "(Lemma \\ref{lemma-intersect-powers-ideal-module-zero})", "applied to $N$ over the ring $S$ and the ideal $\\mathfrak mS$", "we have $\\bigcap \\mathfrak m^nN = 0$. Thus the injectivity", "of $u_n$ for all $n$ implies $u$ is injective.", "\\medskip\\noindent", "To show that $M/u(N)$ is flat over $R$, it suffices to show that", "$I \\otimes_R M/u(N) \\to M/u(N)$ is injective for every ideal $I \\subset R$,", "see Lemma \\ref{lemma-flat}. Consider the diagram", "$$", "\\begin{matrix}", "& & 0 & & 0 & & 0 & & \\\\", "& & \\uparrow & & \\uparrow & & \\uparrow & & \\\\", "& & N/IN & \\to & M/IM & \\to & M/(IM + u(N)) & \\to & 0 \\\\", "& & \\uparrow & & \\uparrow & & \\uparrow & & \\\\", "0 & \\to & N & \\to & M & \\to & M/u(N) & \\to & 0 \\\\", "& & \\uparrow & & \\uparrow & & \\uparrow & & \\\\", "& & N \\otimes_R I & \\to & M \\otimes_R I & \\to & M/u(N)\\otimes_R I & \\to & 0", "\\end{matrix}", "$$", "The arrow $M \\otimes_R I \\to M$ is injective. By the snake lemma", "(Lemma \\ref{lemma-snake}) we see that it suffices to prove that", "$N/IN$ injects into $M/IM$. Note that $R/I \\to S/IS$ is a local", "homomorphism of Noetherian local rings, $N/IN \\to M/IM$ is a map", "of $R/I$-modules, $N/IN$ is finite over $S/IS$, and $M/IM$ is flat over", "$R/I$ and $u \\bmod I : N/IN \\to M/IM$ is injective modulo $\\mathfrak m$.", "Thus we may apply the first part of the proof to $u \\bmod I$ and we conclude." ], "refs": [ "algebra-lemma-intersect-powers-ideal-module-zero", "algebra-lemma-flat", "algebra-lemma-snake" ], "ref_ids": [ 627, 525, 328 ] } ], "ref_ids": [] }, { "id": 884, "type": "theorem", "label": "algebra-lemma-grothendieck", "categories": [ "algebra" ], "title": "algebra-lemma-grothendieck", "contents": [ "Suppose that $R \\to S$ is a flat and local ring homomorphism of Noetherian", "local rings. Denote $\\mathfrak m$ the maximal ideal of $R$.", "Suppose $f \\in S$ is a nonzerodivisor in $S/{\\mathfrak m}S$.", "Then $S/fS$ is flat over $R$, and $f$ is a nonzerodivisor in $S$." ], "refs": [], "proofs": [ { "contents": [ "Follows directly from Lemma \\ref{lemma-mod-injective}." ], "refs": [ "algebra-lemma-mod-injective" ], "ref_ids": [ 883 ] } ], "ref_ids": [] }, { "id": 885, "type": "theorem", "label": "algebra-lemma-grothendieck-regular-sequence", "categories": [ "algebra" ], "title": "algebra-lemma-grothendieck-regular-sequence", "contents": [ "Suppose that $R \\to S$ is a flat and local ring homomorphism of Noetherian", "local rings. Denote $\\mathfrak m$ the maximal ideal of $R$.", "Suppose $f_1, \\ldots, f_c$ is a sequence of elements of", "$S$ such that the images $\\overline{f}_1, \\ldots, \\overline{f}_c$", "form a regular sequence in $S/{\\mathfrak m}S$.", "Then $f_1, \\ldots, f_c$ is a regular sequence in $S$ and each", "of the quotients $S/(f_1, \\ldots, f_i)$ is flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "Induction and Lemma \\ref{lemma-grothendieck}." ], "refs": [ "algebra-lemma-grothendieck" ], "ref_ids": [ 884 ] } ], "ref_ids": [] }, { "id": 886, "type": "theorem", "label": "algebra-lemma-free-fibre-flat-free", "categories": [ "algebra" ], "title": "algebra-lemma-free-fibre-flat-free", "contents": [ "Let $R \\to S$ be a local homomorphism of Noetherian", "local rings. Let $\\mathfrak m$ be the maximal", "ideal of $R$. Let $M$ be a finite $S$-module.", "Suppose that (a) $M/\\mathfrak mM$", "is a free $S/\\mathfrak mS$-module, and (b) $M$ is flat over $R$.", "Then $M$ is free and $S$ is flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\overline{x}_1, \\ldots, \\overline{x}_n$ be a basis", "for the free module $M/\\mathfrak mM$. Choose", "$x_1, \\ldots, x_n \\in M$ with $x_i$ mapping to $\\overline{x}_i$. Let", "$u : S^{\\oplus n} \\to M$ be the map which maps the $i$th", "standard basis vector to $x_i$. By Lemma \\ref{lemma-mod-injective}", "we see that $u$ is injective. On the other hand, by", "Nakayama's Lemma \\ref{lemma-NAK} the map is surjective. The", "lemma follows." ], "refs": [ "algebra-lemma-mod-injective", "algebra-lemma-NAK" ], "ref_ids": [ 883, 401 ] } ], "ref_ids": [] }, { "id": 887, "type": "theorem", "label": "algebra-lemma-complex-exact-mod", "categories": [ "algebra" ], "title": "algebra-lemma-complex-exact-mod", "contents": [ "Let $R \\to S$ be a local homomorphism of local Noetherian", "rings. Let $\\mathfrak m$ be the maximal ideal of $R$.", "Let $0 \\to F_e \\to F_{e-1} \\to \\ldots \\to F_0$", "be a finite complex of finite $S$-modules. Assume that", "each $F_i$ is $R$-flat, and that the complex", "$0 \\to F_e/\\mathfrak m F_e \\to F_{e-1}/\\mathfrak m F_{e-1}", "\\to \\ldots \\to F_0 / \\mathfrak m F_0$ is exact.", "Then $0 \\to F_e \\to F_{e-1} \\to \\ldots \\to F_0$", "is exact, and moreover the module", "$\\Coker(F_1 \\to F_0)$ is $R$-flat." ], "refs": [], "proofs": [ { "contents": [ "By induction on $e$. If $e = 1$, then this is exactly", "Lemma \\ref{lemma-mod-injective}. If $e > 1$, we see", "by Lemma \\ref{lemma-mod-injective} that $F_e \\to F_{e-1}$", "is injective and that $C = \\Coker(F_e \\to F_{e-1})$", "is a finite $S$-module flat over $R$. Hence we can", "apply the induction hypothesis to the complex", "$0 \\to C \\to F_{e-2} \\to \\ldots \\to F_0$.", "We deduce that $C \\to F_{e-2}$ is injective", "and the exactness of the complex follows, as well", "as the flatness of the cokernel of $F_1 \\to F_0$." ], "refs": [ "algebra-lemma-mod-injective", "algebra-lemma-mod-injective" ], "ref_ids": [ 883, 883 ] } ], "ref_ids": [] }, { "id": 888, "type": "theorem", "label": "algebra-lemma-prepare-local-criterion-flatness", "categories": [ "algebra" ], "title": "algebra-lemma-prepare-local-criterion-flatness", "contents": [ "Let $R$ be a local ring with maximal ideal $\\mathfrak m$", "and residue field $\\kappa = R/\\mathfrak m$.", "Let $M$ be an $R$-module. If $\\text{Tor}_1^R(\\kappa, M) = 0$,", "then for every finite length $R$-module $N$ we have", "$\\text{Tor}_1^R(N, M) = 0$." ], "refs": [], "proofs": [ { "contents": [ "By descending induction on the length of $N$.", "If the length of $N$ is $1$, then $N \\cong \\kappa$", "and we are done. If the length of $N$ is more than", "$1$, then we can fit $N$ into a short exact sequence", "$0 \\to N' \\to N \\to N'' \\to 0$ where $N'$, $N''$ are", "finite length $R$-modules of smaller length.", "The vanishing of $\\text{Tor}_1^R(N, M)$ follows", "from the vanishing of $\\text{Tor}_1^R(N', M)$", "and $\\text{Tor}_1^R(N'', M)$ (induction hypothesis)", "and the long exact sequence of Tor groups, see Lemma", "\\ref{lemma-long-exact-sequence-tor}." ], "refs": [ "algebra-lemma-long-exact-sequence-tor" ], "ref_ids": [ 782 ] } ], "ref_ids": [] }, { "id": 889, "type": "theorem", "label": "algebra-lemma-local-criterion-flatness", "categories": [ "algebra" ], "title": "algebra-lemma-local-criterion-flatness", "contents": [ "Let $R \\to S$ be a local homomorphism of local Noetherian", "rings. Let $\\mathfrak m$ be the maximal ideal of $R$,", "and let $\\kappa = R/\\mathfrak m$.", "Let $M$ be a finite $S$-module. If $\\text{Tor}_1^R(\\kappa, M) = 0$,", "then $M$ is flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "Let $I \\subset R$ be an ideal. By Lemma \\ref{lemma-flat} it suffices", "to show that $I \\otimes_R M \\to M$ is injective. By Remark", "\\ref{remark-Tor-ring-mod-ideal} we see that this kernel is", "equal to $\\text{Tor}_1^R(M, R/I)$. By", "Lemma \\ref{lemma-prepare-local-criterion-flatness}", "we see that $J \\otimes_R M \\to M$ is injective for all ideals", "of finite colength.", "\\medskip\\noindent", "Choose $n >> 0$ and consider the following short exact", "sequence", "$$", "0", "\\to I \\cap \\mathfrak m^n", "\\to I \\oplus \\mathfrak m^n", "\\to I + \\mathfrak m^n", "\\to 0", "$$", "This is a sub sequence of the short exact sequence", "$0 \\to R \\to R^{\\oplus 2} \\to R \\to 0$. Thus we get the diagram", "$$", "\\xymatrix{", "(I\\cap \\mathfrak m^n) \\otimes_R M \\ar[r] \\ar[d] &", "I \\otimes_R M \\oplus \\mathfrak m^n \\otimes_R M \\ar[r] \\ar[d] &", "(I + \\mathfrak m^n) \\otimes_R M \\ar[d] \\\\", "M \\ar[r] &", "M \\oplus M \\ar[r] &", "M", "}", "$$", "Note that $I + \\mathfrak m^n$ and $\\mathfrak m^n$", "are ideals of finite colength.", "Thus a diagram chase shows that", "$\\Ker((I \\cap \\mathfrak m^n)\\otimes_R M \\to M)", "\\to \\Ker(I \\otimes_R M \\to M)$", "is surjective. We conclude in particular that", "$K = \\Ker(I \\otimes_R M \\to M)$ is contained", "in the image of $(I \\cap \\mathfrak m^n) \\otimes_R M$", "in $I \\otimes_R M$. By Artin-Rees, Lemma \\ref{lemma-Artin-Rees}", "we see that $K$ is contained", "in $\\mathfrak m^{n-c}(I \\otimes_R M)$ for some $c > 0$", "and all $n >> 0$. Since $I \\otimes_R M$ is a finite", "$S$-module (!) and since $S$ is Noetherian, we see", "that this implies $K = 0$. Namely, the above implies", "$K$ maps to zero in the $\\mathfrak mS$-adic completion", "of $I \\otimes_R M$. But the map from $S$", "to its $\\mathfrak mS$-adic completion is faithfully flat", "by Lemma \\ref{lemma-completion-faithfully-flat}.", "Hence $K = 0$, as desired." ], "refs": [ "algebra-lemma-flat", "algebra-remark-Tor-ring-mod-ideal", "algebra-lemma-prepare-local-criterion-flatness", "algebra-lemma-Artin-Rees", "algebra-lemma-completion-faithfully-flat" ], "ref_ids": [ 525, 1570, 888, 625, 871 ] } ], "ref_ids": [] }, { "id": 890, "type": "theorem", "label": "algebra-lemma-what-does-it-mean", "categories": [ "algebra" ], "title": "algebra-lemma-what-does-it-mean", "contents": [ "Let $R$ be a ring.", "Let $I \\subset R$ be an ideal.", "Let $M$ be an $R$-module.", "If $M/IM$ is flat over $R/I$ and $\\text{Tor}_1^R(R/I, M) = 0$ then", "\\begin{enumerate}", "\\item $M/I^nM$ is flat over $R/I^n$ for all $n \\geq 1$, and", "\\item for any module $N$ which is annihilated by $I^m$ for some $m \\geq 0$", "we have $\\text{Tor}_1^R(N, M) = 0$.", "\\end{enumerate}", "In particular, if $I$ is nilpotent, then $M$ is flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "Assume $M/IM$ is flat over $R/I$ and $\\text{Tor}_1^R(R/I, M) = 0$.", "Let $N$ be an $R/I$-module. Choose a short exact sequence", "$$", "0 \\to K \\to \\bigoplus\\nolimits_{i \\in I} R/I \\to N \\to 0", "$$", "By the long exact sequence of $\\text{Tor}$ and the vanishing of", "$\\text{Tor}_1^R(R/I, M)$ we get", "$$", "0 \\to \\text{Tor}_1^R(N, M) \\to K \\otimes_R M \\to", "(\\bigoplus\\nolimits_{i \\in I} R/I) \\otimes_R M \\to N \\otimes_R M \\to 0", "$$", "But since $K$, $\\bigoplus_{i \\in I} R/I$, and $N$ are all annihilated", "by $I$ we see that", "\\begin{align*}", "K \\otimes_R M & = K \\otimes_{R/I} M/IM, \\\\", "(\\bigoplus\\nolimits_{i \\in I} R/I) \\otimes_R M & =", "(\\bigoplus\\nolimits_{i \\in I} R/I) \\otimes_{R/I} M/IM, \\\\", "N \\otimes_R M & = N \\otimes_{R/I} M/IM.", "\\end{align*}", "As $M/IM$ is flat over $R/I$ we conclude that", "$$", "0 \\to K \\otimes_{R/I} M/IM \\to", "(\\bigoplus\\nolimits_{i \\in I} R/I) \\otimes_{R/I} M/IM \\to", "N \\otimes_{R/} M/IM \\to 0", "$$", "is exact. Combining this with the above we conclude that", "$\\text{Tor}_1^R(N, M) = 0$ for any $R$-module $N$ annihilated by $I$.", "\\medskip\\noindent", "In particular, if we apply this to the", "module $I/I^2$, then we conclude that the sequence", "$$", "0 \\to I^2 \\otimes_R M \\to I \\otimes_R M \\to I/I^2 \\otimes_R M \\to 0", "$$", "is short exact. This implies that $I^2 \\otimes_R M \\to M$ is injective", "and it implies that $I/I^2 \\otimes_{R/I} M/IM = IM/I^2M$.", "\\medskip\\noindent", "Let us prove that $M/I^2M$ is flat over $R/I^2$. Let $I^2 \\subset J$", "be an ideal. We have to show that", "$J/I^2 \\otimes_{R/I^2} M/I^2M \\to M/I^2M$ is injective, see", "Lemma \\ref{lemma-flat}.", "As $M/IM$ is flat over $R/I$ we know that the map", "$(I + J)/I \\otimes_{R/I} M/IM \\to M/IM$ is injective.", "The sequence", "$$", "(I \\cap J)/I^2 \\otimes_{R/I^2} M/I^2M \\to", "J/I^2 \\otimes_{R/I^2} M/I^2M \\to", "(I + J)/I \\otimes_{R/I} M/IM \\to 0", "$$", "is exact, as you get it by tensoring the exact sequence", "$0 \\to (I \\cap J) \\to J \\to (I + J)/I \\to 0$ by $M/I^2M$.", "Hence suffices to prove the injectivity of the map", "$(I \\cap J)/I^2 \\otimes_{R/I} M/IM \\to IM/I^2M$. However, the map", "$(I \\cap J)/I^2 \\to I/I^2$ is injective and as $M/IM$", "is flat over $R/I$ the map", "$(I \\cap J)/I^2 \\otimes_{R/I} M/IM \\to I/I^2 \\otimes_{R/I} M/IM$", "is injective. Since we have previously seen that", "$I/I^2 \\otimes_{R/I} M/IM = IM/I^2M$ we obtain the desired injectivity.", "\\medskip\\noindent", "Hence we have proven that the assumptions imply:", "(a) $\\text{Tor}_1^R(N, M) = 0$ for all $N$ annihilated by $I$,", "(b) $I^2 \\otimes_R M \\to M$ is injective, and (c) $M/I^2M$ is flat", "over $R/I^2$. Thus we can continue by induction to get the", "same results for $I^n$ for all $n \\geq 1$." ], "refs": [ "algebra-lemma-flat" ], "ref_ids": [ 525 ] } ], "ref_ids": [] }, { "id": 891, "type": "theorem", "label": "algebra-lemma-what-does-it-mean-again", "categories": [ "algebra" ], "title": "algebra-lemma-what-does-it-mean-again", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal.", "Let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item If $M/IM$ is flat over $R/I$ and $M \\otimes_R I/I^2 \\to IM/I^2M$", "is injective, then $M/I^2M$ is flat over $R/I^2$.", "\\item If $M/IM$ is flat over $R/I$ and $M \\otimes_R I^n/I^{n + 1}", "\\to I^nM/I^{n + 1}M$ is injective for $n = 1, \\ldots, k$,", "then $M/I^{k + 1}M$ is flat over $R/I^{k + 1}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first statement is a consequence of", "Lemma \\ref{lemma-what-does-it-mean} applied with $R$ replaced by $R/I^2$", "and $M$ replaced by $M/I^2M$ using that", "$$", "\\text{Tor}_1^{R/I^2}(M/I^2M, R/I) =", "\\Ker(M \\otimes_R I/I^2 \\to IM/I^2M),", "$$", "see Remark \\ref{remark-Tor-ring-mod-ideal}.", "The second statement follows in the same manner using induction", "on $n$ to show that $M/I^{n + 1}M$ is flat over $R/I^{n + 1}$ for", "$n = 1, \\ldots, k$. Here we use that", "$$", "\\text{Tor}_1^{R/I^{n + 1}}(M/I^{n + 1}M, R/I) =", "\\Ker(M \\otimes_R I^n/I^{n + 1} \\to I^nM/I^{n + 1}M)", "$$", "for every $n$." ], "refs": [ "algebra-lemma-what-does-it-mean", "algebra-remark-Tor-ring-mod-ideal" ], "ref_ids": [ 890, 1570 ] } ], "ref_ids": [] }, { "id": 892, "type": "theorem", "label": "algebra-lemma-variant-local-criterion-flatness", "categories": [ "algebra" ], "title": "algebra-lemma-variant-local-criterion-flatness", "contents": [ "Let $R \\to S$ be a local homomorphism of Noetherian", "local rings. Let $I \\not = R$ be an ideal in $R$.", "Let $M$ be a finite $S$-module. If $\\text{Tor}_1^R(M, R/I) = 0$", "and $M/IM$ is flat over $R/I$, then $M$ is flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "First proof: By", "Lemma \\ref{lemma-what-does-it-mean}", "we see that $\\text{Tor}_1^R(\\kappa, M)$ is zero where $\\kappa$", "is the residue field of $R$. Hence we see that $M$", "is flat over $R$ by", "Lemma \\ref{lemma-local-criterion-flatness}.", "\\medskip\\noindent", "Second proof: Let $\\mathfrak m$ be the maximal ideal of $R$.", "We will show that $\\mathfrak m \\otimes_R M \\to M$ is injective,", "and then apply", "Lemma \\ref{lemma-local-criterion-flatness}.", "Suppose that $\\sum f_i \\otimes x_i \\in \\mathfrak m \\otimes_R M$", "and that $\\sum f_i x_i = 0$ in $M$. By the equational criterion", "for flatness Lemma \\ref{lemma-flat-eq} applied to $M/IM$", "over $R/I$ we see there exist $\\overline{a}_{ij} \\in R/I$", "and $\\overline{y}_j \\in M/IM$ such that", "$x_i \\bmod IM = \\sum_j \\overline{a}_{ij} \\overline{y}_j $", "and $0 = \\sum_i (f_i \\bmod I) \\overline{a}_{ij}$.", "Let $a_{ij} \\in R$ be a lift of $\\overline{a}_{ij}$ and", "similarly let $y_j \\in M$ be a lift of $\\overline{y}_j$.", "Then we see that", "\\begin{eqnarray*}", "\\sum f_i \\otimes x_i", "& = &", "\\sum f_i \\otimes x_i +", "\\sum f_ia_{ij} \\otimes y_j -", "\\sum f_i \\otimes a_{ij} y_j", "\\\\", "& = &", "\\sum f_i \\otimes (x_i - \\sum a_{ij} y_j) +", "\\sum (\\sum f_i a_{ij}) \\otimes y_j", "\\end{eqnarray*}", "Since $x_i - \\sum a_{ij} y_j \\in IM$ and", "$\\sum f_i a_{ij} \\in I$ we see that there exists", "an element in $I \\otimes_R M$ which maps to our given", "element $\\sum f_i \\otimes x_i$ in $\\mathfrak m \\otimes_R M$.", "But $I \\otimes_R M \\to M$ is injective by assumption (see", "Remark \\ref{remark-Tor-ring-mod-ideal}) and we win." ], "refs": [ "algebra-lemma-what-does-it-mean", "algebra-lemma-local-criterion-flatness", "algebra-lemma-local-criterion-flatness", "algebra-lemma-flat-eq", "algebra-remark-Tor-ring-mod-ideal" ], "ref_ids": [ 890, 889, 889, 531, 1570 ] } ], "ref_ids": [] }, { "id": 893, "type": "theorem", "label": "algebra-lemma-flat-module-powers", "categories": [ "algebra" ], "title": "algebra-lemma-flat-module-powers", "contents": [ "Let $R \\to S$ be a ring map. Let $I \\subset R$ be an ideal.", "Let $M$ be an $S$-module. Assume", "\\begin{enumerate}", "\\item $R$ is a Noetherian ring,", "\\item $S$ is a Noetherian ring,", "\\item $M$ is a finite $S$-module, and", "\\item for each $n \\geq 1$ the module $M/I^n M$ is flat over", "$R/I^n$.", "\\end{enumerate}", "Then for every $\\mathfrak q \\in V(IS)$", "the localization $M_{\\mathfrak q}$ is flat over $R$.", "In particular, if $S$ is local and $IS$ is contained", "in its maximal ideal, then $M$ is flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "We are going to use", "Lemma \\ref{lemma-variant-local-criterion-flatness}.", "By assumption $M/IM$ is flat over $R/I$. Hence it suffices to check", "that $\\text{Tor}_1^R(M, R/I)$ is zero on localization at $\\mathfrak q$. By", "Remark \\ref{remark-Tor-ring-mod-ideal}", "this Tor group is equal to $K = \\Ker(I \\otimes_R M \\to M)$.", "We know for each $n \\geq 1$ that the kernel", "$\\Ker(I/I^n \\otimes_{R/I^n} M/I^nM \\to M/I^nM)$ is zero.", "Since there is a module map", "$I/I^n \\otimes_{R/I^n} M/I^nM \\to (I \\otimes_R M)/I^{n - 1}(I \\otimes_R M)$", "we conclude that $K \\subset I^{n - 1}(I \\otimes_R M)$ for each $n$.", "By the Artin-Rees lemma, and more precisely", "Lemma \\ref{lemma-intersection-powers-ideal-module}", "we conclude that $K_{\\mathfrak q} = 0$, as desired." ], "refs": [ "algebra-lemma-variant-local-criterion-flatness", "algebra-remark-Tor-ring-mod-ideal", "algebra-lemma-intersection-powers-ideal-module" ], "ref_ids": [ 892, 1570, 628 ] } ], "ref_ids": [] }, { "id": 894, "type": "theorem", "label": "algebra-lemma-surjective-on-tor-one", "categories": [ "algebra" ], "title": "algebra-lemma-surjective-on-tor-one", "contents": [ "Let $R \\to R' \\to R''$ be ring maps.", "Let $M$ be an $R$-module. Suppose that $M \\otimes_R R'$", "is flat over $R'$. Then the natural map", "$\\text{Tor}_1^R(M, R') \\otimes_{R'} R'' \\to", "\\text{Tor}_1^R(M, R'')$ is onto." ], "refs": [], "proofs": [ { "contents": [ "Let $F_\\bullet$ be a free resolution of $M$ over $R$.", "The complex $F_2 \\otimes_R R' \\to F_1\\otimes_R R' \\to F_0 \\otimes_R R'$", "computes $\\text{Tor}_1^R(M, R')$.", "The complex $F_2 \\otimes_R R'' \\to F_1\\otimes_R R'' \\to F_0 \\otimes_R R''$", "computes $\\text{Tor}_1^R(M, R'')$. Note that", "$F_i \\otimes_R R' \\otimes_{R'} R'' = F_i \\otimes_R R''$. Let", "$K' = \\Ker(F_1\\otimes_R R' \\to F_0 \\otimes_R R')$ and", "similarly $K'' = \\Ker(F_1\\otimes_R R'' \\to F_0 \\otimes_R R'')$.", "Thus we have an exact sequence", "$$", "0 \\to K' \\to F_1\\otimes_R R' \\to F_0 \\otimes_R R' \\to M \\otimes_R R' \\to 0.", "$$", "By the assumption that $M \\otimes_R R'$ is flat over $R'$,", "the sequence", "$$", "K' \\otimes_{R'} R'' \\to", "F_1 \\otimes_R R'' \\to", "F_0 \\otimes_R R'' \\to", "M \\otimes_R R'' \\to 0", "$$", "is still exact. This means that $K' \\otimes_{R'} R'' \\to K''$", "is surjective. Since $\\text{Tor}_1^R(M, R')$ is a quotient of $K'$ and", "$\\text{Tor}_1^R(M, R'')$ is a quotient of $K''$ we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 895, "type": "theorem", "label": "algebra-lemma-surjective-on-tor-one-trivial", "categories": [ "algebra" ], "title": "algebra-lemma-surjective-on-tor-one-trivial", "contents": [ "Let $R \\to R'$ be a ring map. Let $I \\subset R$ be", "an ideal and $I' = IR'$. Let $M$ be an $R$-module", "and set $M' = M \\otimes_R R'$. The natural map", "$\\text{Tor}_1^R(R'/I', M) \\to \\text{Tor}_1^{R'}(R'/I', M')$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $F_2 \\to F_1 \\to F_0 \\to M \\to 0$ be a free resolution of", "$M$ over $R$. Set $F_i' = F_i \\otimes_R R'$. The sequence", "$F_2' \\to F_1' \\to F_0' \\to M' \\to 0$ may no longer be exact", "at $F_1'$. A free resolution of $M'$ over $R'$ therefore looks", "like", "$$", "F_2' \\oplus F_2'' \\to F_1' \\to F_0' \\to M' \\to 0", "$$", "for a suitable free module $F_2''$ over $R'$. Next, note that", "$F_i \\otimes_R R'/I' = F_i' / IF_i' = F_i'/I'F_i'$.", "So the complex $F_2'/I'F_2' \\to F_1'/I'F_1' \\to F_0'/I'F_0'$", "computes $\\text{Tor}_1^R(M, R'/I')$. On the other hand", "$F_i' \\otimes_{R'} R'/I' = F_i'/I'F_i'$ and similarly", "for $F_2''$. Thus the complex", "$F_2'/I'F_2' \\oplus F_2''/I'F_2'' \\to F_1'/I'F_1' \\to F_0'/I'F_0'$", "computes $\\text{Tor}_1^{R'}(M', R'/I')$. Since the vertical", "map on complexes", "$$", "\\xymatrix{", "F_2'/I'F_2' \\ar[r] \\ar[d] &", "F_1'/I'F_1' \\ar[r] \\ar[d] &", "F_0'/I'F_0' \\ar[d] \\\\", "F_2'/I'F_2' \\oplus F_2''/I'F_2'' \\ar[r] &", "F_1'/I'F_1' \\ar[r] &", "F_0'/I'F_0'", "}", "$$", "clearly induces a surjection on cohomology we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 896, "type": "theorem", "label": "algebra-lemma-another-variant-local-criterion-flatness", "categories": [ "algebra" ], "title": "algebra-lemma-another-variant-local-criterion-flatness", "contents": [ "Let", "$$", "\\xymatrix{", "S \\ar[r] & S' \\\\", "R \\ar[r] \\ar[u] & R' \\ar[u]", "}", "$$", "be a commutative diagram of local homomorphisms of local Noetherian rings.", "Let $I \\subset R$ be a proper ideal.", "Let $M$ be a finite $S$-module.", "Denote $I' = IR'$ and $M' = M \\otimes_S S'$.", "Assume that", "\\begin{enumerate}", "\\item $S'$ is a localization of the tensor product", "$S \\otimes_R R'$,", "\\item $M/IM$ is flat over $R/I$,", "\\item $\\text{Tor}_1^R(M, R/I) \\to \\text{Tor}_1^{R'}(M', R'/I')$", "is zero.", "\\end{enumerate}", "Then $M'$ is flat over $R'$." ], "refs": [], "proofs": [ { "contents": [ "Since $S'$ is a localization of $S \\otimes_R R'$ we see that", "$M'$ is a localization of $M \\otimes_R R'$. Note that", "by Lemma \\ref{lemma-flat-base-change} the module $M/IM \\otimes_{R/I} R'/I'", "= M \\otimes_R R' /I'(M \\otimes_R R')$ is flat over $R'/I'$. Hence also", "$M'/I'M'$ is flat over $R'/I'$ as the localization of a flat module", "is flat. By Lemma \\ref{lemma-variant-local-criterion-flatness}", "it suffices to show that $\\text{Tor}_1^{R'}(M', R'/I')$ is zero.", "Since $M'$ is a localization of $M \\otimes_R R'$, the last assumption", "implies that it suffices to show that", "$\\text{Tor}_1^R(M, R/I) \\otimes_R R'", "\\to", "\\text{Tor}_1^{R'}(M \\otimes_R R', R'/I')$", "is surjective.", "\\medskip\\noindent", "By Lemma \\ref{lemma-surjective-on-tor-one-trivial} we see that", "$\\text{Tor}_1^R(M, R'/I') \\to \\text{Tor}_1^{R'}(M \\otimes_R R', R'/I')$", "is surjective. So now it suffices to show that", "$\\text{Tor}_1^R(M, R/I) \\otimes_R R'", "\\to", "\\text{Tor}_1^R(M, R'/I')$", "is surjective. This follows from Lemma \\ref{lemma-surjective-on-tor-one}", "by looking at the ring maps $R \\to R/I \\to R'/I'$ and the module $M$." ], "refs": [ "algebra-lemma-flat-base-change", "algebra-lemma-variant-local-criterion-flatness", "algebra-lemma-surjective-on-tor-one-trivial", "algebra-lemma-surjective-on-tor-one" ], "ref_ids": [ 527, 892, 895, 894 ] } ], "ref_ids": [] }, { "id": 897, "type": "theorem", "label": "algebra-lemma-criterion-flatness-fibre-Noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-criterion-flatness-fibre-Noetherian", "contents": [ "Let $R$, $S$, $S'$ be Noetherian local rings and let $R \\to S \\to S'$", "be local ring homomorphisms. Let $\\mathfrak m \\subset R$ be the", "maximal ideal. Let $M$ be an $S'$-module. Assume", "\\begin{enumerate}", "\\item The module $M$ is finite over $S'$.", "\\item The module $M$ is not zero.", "\\item The module $M/\\mathfrak m M$", "is a flat $S/\\mathfrak m S$-module.", "\\item The module $M$ is a flat $R$-module.", "\\end{enumerate}", "Then $S$ is flat over $R$ and $M$ is a flat $S$-module." ], "refs": [], "proofs": [ { "contents": [ "Set $I = \\mathfrak mS \\subset S$. Then we see that $M/IM$ is a flat", "$S/I$-module because of (3). Since", "$\\mathfrak m \\otimes_R S' \\to I \\otimes_S S'$ is surjective we see", "that also $\\mathfrak m \\otimes_R M \\to I \\otimes_S M$ is surjective.", "Consider", "$$", "\\mathfrak m \\otimes_R M \\to I \\otimes_S M \\to M.", "$$", "As $M$ is flat over $R$ the composition is injective", "and so both arrows are injective.", "In particular $\\text{Tor}_1^S(S/I, M) = 0$ see", "Remark \\ref{remark-Tor-ring-mod-ideal}. By", "Lemma \\ref{lemma-variant-local-criterion-flatness} we conclude", "that $M$ is flat over $S$. Note that since $M/\\mathfrak m_{S'}M$", "is not zero by Nakayama's Lemma \\ref{lemma-NAK}", "we see that actually $M$ is faithfully flat over $S$ by", "Lemma \\ref{lemma-ff} (since it forces $M/\\mathfrak m_SM \\not = 0$).", "\\medskip\\noindent", "Consider the exact sequence", "$0 \\to \\mathfrak m \\to R \\to \\kappa \\to 0$.", "This gives an exact sequence", "$0 \\to \\text{Tor}_1^R(\\kappa, S) \\to \\mathfrak m \\otimes_R S \\to I \\to 0$.", "Since $M$ is flat over $S$ this gives an exact sequence", "$0 \\to \\text{Tor}_1^R(\\kappa, S)\\otimes_S M \\to", "\\mathfrak m \\otimes_R M \\to I \\otimes_S M \\to 0$.", "By the above this implies that $\\text{Tor}_1^R(\\kappa, S)\\otimes_S M = 0$.", "Since $M$ is faithfully flat over $S$ this implies that", "$\\text{Tor}_1^R(\\kappa, S) = 0$ and we conclude that", "$S$ is flat over $R$ by Lemma \\ref{lemma-local-criterion-flatness}." ], "refs": [ "algebra-remark-Tor-ring-mod-ideal", "algebra-lemma-variant-local-criterion-flatness", "algebra-lemma-NAK", "algebra-lemma-ff", "algebra-lemma-local-criterion-flatness" ], "ref_ids": [ 1570, 892, 401, 535, 889 ] } ], "ref_ids": [] }, { "id": 898, "type": "theorem", "label": "algebra-lemma-base-change-flat-up-down", "categories": [ "algebra" ], "title": "algebra-lemma-base-change-flat-up-down", "contents": [ "Let", "$$", "\\xymatrix{", "S \\ar[r] & S' \\\\", "R \\ar[r] \\ar[u] & R' \\ar[u]", "}", "$$", "be a commutative diagram of local homomorphisms of local rings.", "Assume that $S'$ is a localization of the tensor product $S \\otimes_R R'$.", "Let $M$ be an $S$-module and set $M' = S' \\otimes_S M$.", "\\begin{enumerate}", "\\item If $M$ is flat over $R$ then $M'$ is flat over $R'$.", "\\item If $M'$ is flat over $R'$ and $R \\to R'$ is flat then", "$M$ is flat over $R$.", "\\end{enumerate}", "In particular we have", "\\begin{enumerate}", "\\item[(3)] If $S$ is flat over $R$ then $S'$ is flat over $R'$.", "\\item[(4)] If $R' \\to S'$ and $R \\to R'$ are flat then $S$ is flat over $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). If $M$ is flat over $R$, then $M \\otimes_R R'$", "is flat over $R'$ by", "Lemma \\ref{lemma-flat-base-change}.", "If $W \\subset S \\otimes_R R'$ is the multiplicative subset such that", "$W^{-1}(S \\otimes_R R') = S'$ then $M' = W^{-1}(M \\otimes_R R')$.", "Hence $M'$ is flat over $R'$ as the localization of a flat module, see", "Lemma \\ref{lemma-flat-localization} part (5). This proves (1) and in", "particular, we see that (3) holds.", "\\medskip\\noindent", "Proof of (2). Suppose that $M'$ is flat over $R'$ and $R \\to R'$ is flat.", "By (3) applied to the diagram reflected in the northwest diagonal", "we see that $S \\to S'$ is flat. Thus $S \\to S'$ is faithfully flat by", "Lemma \\ref{lemma-local-flat-ff}.", "We are going to use the criterion of", "Lemma \\ref{lemma-flat} (\\ref{item-f-ideal})", "to show that $M$ is flat.", "Let $I \\subset R$ be an ideal. If $I \\otimes_R M \\to M$ has a kernel,", "so does $(I \\otimes_R M) \\otimes_S S' \\to M \\otimes_S S' = M'$.", "Note that $I \\otimes_R R' = IR'$ as $R \\to R'$ is flat, and that", "$$", "(I \\otimes_R M) \\otimes_S S' =", "(I \\otimes_R R') \\otimes_{R'} (M \\otimes_S S') =", "IR' \\otimes_{R'} M'.", "$$", "From flatness of $M'$ over $R'$", "we conclude that this maps injectively into $M'$.", "This concludes the proof of (2), and hence (4) is true as well." ], "refs": [ "algebra-lemma-flat-base-change", "algebra-lemma-flat-localization", "algebra-lemma-local-flat-ff", "algebra-lemma-flat" ], "ref_ids": [ 527, 538, 537, 525 ] } ], "ref_ids": [] }, { "id": 899, "type": "theorem", "label": "algebra-lemma-yet-another-variant-local-criterion-flatness", "categories": [ "algebra" ], "title": "algebra-lemma-yet-another-variant-local-criterion-flatness", "contents": [ "Consider a commutative diagram of local rings and local homomorphisms", "$$", "\\xymatrix{", "S \\ar[r] & S' \\\\", "R \\ar[r] \\ar[u] & R' \\ar[u]", "}", "$$", "Let $M$ be a finite $S$-module. Assume that", "\\begin{enumerate}", "\\item the horizontal arrows are flat ring maps", "\\item $M$ is flat over $R$,", "\\item $\\mathfrak m_R R' = \\mathfrak m_{R'}$,", "\\item $R'$ and $S'$ are Noetherian.", "\\end{enumerate}", "Then $M' = M \\otimes_R S'$ is flat over $R'$." ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathfrak m_R \\subset R$ and $R \\to R'$ is flat, we get", "$\\mathfrak m_R \\otimes_R R' = \\mathfrak m_R R' = \\mathfrak m_{R'}$", "by assumption (3). Observe that $M'$ is a finite $S'$-module", "which is flat over $R$ by Lemma \\ref{lemma-flatness-descends-more-general}.", "Thus $\\mathfrak m_R \\otimes_R M' \\to M'$ is injective.", "Then we get", "$$", "\\mathfrak m_R \\otimes_R M' =", "\\mathfrak m_R \\otimes_R R' \\otimes_{R'} M' =", "\\mathfrak m_{R'} \\otimes_{R'} M'", "$$", "Thus $\\mathfrak m_{R'} \\otimes_{R'} M' \\to M'$ is injective.", "This shows that $\\text{Tor}_1^{R'}(\\kappa_{R'}, M') = 0$", "(Remark \\ref{remark-Tor-ring-mod-ideal}).", "Thus $M'$ is flat over $R'$ by", "Lemma \\ref{lemma-local-criterion-flatness}." ], "refs": [ "algebra-lemma-flatness-descends-more-general", "algebra-remark-Tor-ring-mod-ideal", "algebra-lemma-local-criterion-flatness" ], "ref_ids": [ 529, 1570, 889 ] } ], "ref_ids": [] }, { "id": 900, "type": "theorem", "label": "algebra-lemma-local-artinian-basis-when-flat", "categories": [ "algebra" ], "title": "algebra-lemma-local-artinian-basis-when-flat", "contents": [ "Let $(R, \\mathfrak m)$ be a local ring with nilpotent maximal ideal", "$\\mathfrak m$. Let $M$ be a flat $R$-module.", "If $A$ is a set and $x_\\alpha \\in M$, $\\alpha \\in A$ is a collection", "of elements of $M$, then the following are equivalent:", "\\begin{enumerate}", "\\item $\\{\\overline{x}_\\alpha\\}_{\\alpha \\in A}$ forms a basis", "for the vector space $M/\\mathfrak mM$ over $R/\\mathfrak m$, and", "\\item $\\{x_\\alpha\\}_{\\alpha \\in A}$ forms a basis for $M$ over $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is immediate.", "Assume (1). By Nakayama's Lemma \\ref{lemma-NAK}", "the elements $x_\\alpha$ generate $M$. Then one gets a short exact", "sequence", "$$", "0 \\to K \\to \\bigoplus\\nolimits_{\\alpha \\in A} R \\to M \\to 0", "$$", "Tensoring with $R/\\mathfrak m$ and using Lemma \\ref{lemma-flat-tor-zero}", "we obtain $K/\\mathfrak mK = 0$. By Nakayama's Lemma \\ref{lemma-NAK}", "we conclude $K = 0$." ], "refs": [ "algebra-lemma-NAK", "algebra-lemma-flat-tor-zero", "algebra-lemma-NAK" ], "ref_ids": [ 401, 532, 401 ] } ], "ref_ids": [] }, { "id": 901, "type": "theorem", "label": "algebra-lemma-local-artinian-characterize-flat", "categories": [ "algebra" ], "title": "algebra-lemma-local-artinian-characterize-flat", "contents": [ "Let $R$ be a local ring with nilpotent maximal ideal. Let $M$ be an $R$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $M$ is flat over $R$,", "\\item $M$ is a free $R$-module, and", "\\item $M$ is a projective $R$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since any projective module is flat (as a direct summand of a free module)", "and every free module is projective, it suffices to prove that a flat module", "is free. Let $M$ be a flat module. Let $A$ be a set and let $x_\\alpha \\in M$,", "$\\alpha \\in A$ be elements such that", "$\\overline{x_\\alpha} \\in M/\\mathfrak m M$ forms a basis over the residue", "field of $R$. By", "Lemma \\ref{lemma-local-artinian-basis-when-flat}", "the $x_\\alpha$ are a basis for $M$ over $R$ and we win." ], "refs": [ "algebra-lemma-local-artinian-basis-when-flat" ], "ref_ids": [ 900 ] } ], "ref_ids": [] }, { "id": 902, "type": "theorem", "label": "algebra-lemma-lift-basis", "categories": [ "algebra" ], "title": "algebra-lemma-lift-basis", "contents": [ "Let $R$ be a ring.", "Let $I \\subset R$ be an ideal.", "Let $M$ be an $R$-module.", "Let $A$ be a set and let $x_\\alpha \\in M$, $\\alpha \\in A$ be a collection", "of elements of $M$.", "Assume", "\\begin{enumerate}", "\\item $I$ is nilpotent,", "\\item $\\{\\overline{x}_\\alpha\\}_{\\alpha \\in A}$ forms a basis for $M/IM$ over", "$R/I$, and", "\\item $\\text{Tor}_1^R(R/I, M) = 0$.", "\\end{enumerate}", "Then $M$ is free on $\\{x_\\alpha\\}_{\\alpha \\in A}$ over $R$." ], "refs": [], "proofs": [ { "contents": [ "Let $R$, $I$, $M$, $\\{x_\\alpha\\}_{\\alpha \\in A}$ be as in the lemma", "and satisfy assumptions (1), (2), and (3). By", "Nakayama's Lemma \\ref{lemma-NAK}", "the elements $x_\\alpha$ generate $M$ over $R$.", "The assumption $\\text{Tor}_1^R(R/I, M) = 0$ implies that we have a short", "exact sequence", "$$", "0 \\to I \\otimes_R M \\to M \\to M/IM \\to 0.", "$$", "Let $\\sum f_\\alpha x_\\alpha = 0$ be a relation in $M$.", "By choice of $x_\\alpha$ we see that $f_\\alpha \\in I$.", "Hence we conclude that $\\sum f_\\alpha \\otimes x_\\alpha = 0$ in", "$I \\otimes_R M$. The map $I \\otimes_R M \\to I/I^2 \\otimes_{R/I} M/IM$", "and the fact that $\\{x_\\alpha\\}_{\\alpha \\in A}$ forms a basis", "for $M/IM$ implies that $f_\\alpha \\in I^2$! Hence we conclude that", "there are no relations among the images of the $x_\\alpha$ in", "$M/I^2M$. In other words, we see that $M/I^2M$ is free with basis", "the images of the $x_\\alpha$. Using the map", "$I \\otimes_R M \\to I/I^3 \\otimes_{R/I^2} M/I^2M$", "we then conclude that $f_\\alpha \\in I^3$!", "And so on. Since $I^n = 0$ for some $n$ by assumption (1) we win." ], "refs": [ "algebra-lemma-NAK" ], "ref_ids": [ 401 ] } ], "ref_ids": [] }, { "id": 903, "type": "theorem", "label": "algebra-lemma-prepare-lift-flatness", "categories": [ "algebra" ], "title": "algebra-lemma-prepare-lift-flatness", "contents": [ "Let $\\varphi : R \\to R'$ be a ring map.", "Let $I \\subset R$ be an ideal.", "Let $M$ be an $R$-module.", "Assume", "\\begin{enumerate}", "\\item $M/IM$ is flat over $R/I$, and", "\\item $R' \\otimes_R M$ is flat over $R'$.", "\\end{enumerate}", "Set $I_2 = \\varphi^{-1}(\\varphi(I^2)R')$.", "Then $M/I_2M$ is flat over $R/I_2$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $R$, $M$, and $R'$ by $R/I_2$, $M/I_2M$, and", "$R'/\\varphi(I)^2R'$. Then $I^2 = 0$ and $\\varphi$ is injective. By", "Lemma \\ref{lemma-what-does-it-mean}", "and the fact that $I^2 = 0$ it suffices to prove that", "$\\text{Tor}^R_1(R/I, M) = K = \\Ker(I \\otimes_R M \\to M)$ is zero.", "Set $M' = M \\otimes_R R'$ and $I' = IR'$.", "By assumption the map $I' \\otimes_{R'} M' \\to M'$ is injective.", "Hence $K$ maps to zero in", "$$", "I' \\otimes_{R'} M' = I' \\otimes_R M = I' \\otimes_{R/I} M/IM.", "$$", "Then $I \\to I'$ is an injective map of $R/I$-modules.", "Since $M/IM$ is flat over $R/I$ the map", "$$", "I \\otimes_{R/I} M/IM \\longrightarrow I' \\otimes_{R/I} M/IM", "$$", "is injective. This implies that $K$ is zero in", "$I \\otimes_R M = I \\otimes_{R/I} M/IM$ as desired." ], "refs": [ "algebra-lemma-what-does-it-mean" ], "ref_ids": [ 890 ] } ], "ref_ids": [] }, { "id": 904, "type": "theorem", "label": "algebra-lemma-lift-flatness", "categories": [ "algebra" ], "title": "algebra-lemma-lift-flatness", "contents": [ "Let $\\varphi : R \\to R'$ be a ring map.", "Let $I \\subset R$ be an ideal.", "Let $M$ be an $R$-module.", "Assume", "\\begin{enumerate}", "\\item $I$ is nilpotent,", "\\item $R \\to R'$ is injective,", "\\item $M/IM$ is flat over $R/I$, and", "\\item $R' \\otimes_R M$ is flat over $R'$.", "\\end{enumerate}", "Then $M$ is flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "Define inductively $I_1 = I$ and $I_{n + 1} = \\varphi^{-1}(\\varphi(I_n)^2R')$", "for $n \\geq 1$. Note that by", "Lemma \\ref{lemma-prepare-lift-flatness}", "we find that $M/I_nM$ is flat over $R/I_n$ for each $n \\geq 1$.", "It is clear that $\\varphi(I_n) \\subset \\varphi(I)^{2^n}R'$. Since", "$I$ is nilpotent we see that $\\varphi(I_n) = 0$ for some $n$. As", "$\\varphi$ is injective we conclude that $I_n = 0$ for some $n$ and", "we win." ], "refs": [ "algebra-lemma-prepare-lift-flatness" ], "ref_ids": [ 903 ] } ], "ref_ids": [] }, { "id": 905, "type": "theorem", "label": "algebra-lemma-artinian-variant-local-criterion-flatness", "categories": [ "algebra" ], "title": "algebra-lemma-artinian-variant-local-criterion-flatness", "contents": [ "Let $R$ be an Artinian local ring. Let $M$ be an $R$-module.", "Let $I \\subset R$ be a proper ideal. The following are", "equivalent", "\\begin{enumerate}", "\\item $M$ is flat over $R$, and", "\\item $M/IM$ is flat over $R/I$ and $\\text{Tor}_1^R(R/I, M) = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) follows immediately from the", "definitions. Assume $M/IM$ is flat over $R/I$ and", "$\\text{Tor}_1^R(R/I, M) = 0$. By", "Lemma \\ref{lemma-local-artinian-characterize-flat}", "this implies that $M/IM$ is free over $R/I$. Pick a set $A$", "and elements $x_\\alpha \\in M$ such that the images in $M/IM$ form", "a basis. By", "Lemma \\ref{lemma-lift-basis}", "we conclude that $M$ is free and in particular flat." ], "refs": [ "algebra-lemma-local-artinian-characterize-flat", "algebra-lemma-lift-basis" ], "ref_ids": [ 901, 902 ] } ], "ref_ids": [] }, { "id": 906, "type": "theorem", "label": "algebra-lemma-descent-flatness-injective-map-artinian-rings", "categories": [ "algebra" ], "title": "algebra-lemma-descent-flatness-injective-map-artinian-rings", "contents": [ "Let $R \\to S$ be a ring map. Let $M$ be an $R$-module.", "Assume", "\\begin{enumerate}", "\\item $R$ is Artinian", "\\item $R \\to S$ is injective, and", "\\item $M \\otimes_R S$ is a flat $S$-module.", "\\end{enumerate}", "Then $M$ is a flat $R$-module." ], "refs": [], "proofs": [ { "contents": [ "First proof: Let $I \\subset R$ be the Jacobson radical of $R$.", "Then $I$ is nilpotent and $M/IM$ is flat over $R/I$ as $R/I$", "is a product of fields, see", "Section \\ref{section-artinian}.", "Hence $M$ is flat by an application of", "Lemma \\ref{lemma-lift-flatness}.", "\\medskip\\noindent", "Second proof: By", "Lemma \\ref{lemma-artinian-finite-length}", "we may write $R = \\prod R_i$ as a finite product of local Artinian", "rings. This induces similar product decompositions for both $R$ and $S$.", "Hence we reduce to the case where $R$ is local Artinian (details omitted).", "\\medskip\\noindent", "Assume that $R \\to S$, $M$ are as in the lemma satisfying (1), (2), and (3)", "and in addition that $R$ is local with maximal ideal $\\mathfrak m$.", "Let $A$ be a set and $x_\\alpha \\in A$ be elements such that", "$\\overline{x}_\\alpha$ forms a basis for $M/\\mathfrak mM$", "over $R/\\mathfrak m$. By", "Nakayama's Lemma \\ref{lemma-NAK}", "we see that the elements $x_\\alpha$ generate $M$ as an $R$-module.", "Set $N = S \\otimes_R M$ and $I = \\mathfrak mS$.", "Then $\\{1 \\otimes x_\\alpha\\}_{\\alpha \\in A}$ is a family of elements", "of $N$ which form a basis for $N/IN$. Moreover, since $N$ is flat over", "$S$ we have $\\text{Tor}_1^S(S/I, N) = 0$. Thus we conclude from", "Lemma \\ref{lemma-lift-basis}", "that $N$ is free on $\\{1 \\otimes x_\\alpha\\}_{\\alpha \\in A}$.", "The injectivity of $R \\to S$ then guarantees that there cannot be a", "nontrivial relation among the $x_\\alpha$ with coefficients in $R$." ], "refs": [ "algebra-lemma-lift-flatness", "algebra-lemma-artinian-finite-length", "algebra-lemma-NAK", "algebra-lemma-lift-basis" ], "ref_ids": [ 904, 646, 401, 902 ] } ], "ref_ids": [] }, { "id": 907, "type": "theorem", "label": "algebra-lemma-criterion-flatness-fibre-nilpotent", "categories": [ "algebra" ], "title": "algebra-lemma-criterion-flatness-fibre-nilpotent", "contents": [ "Let", "$$", "\\xymatrix{", "S \\ar[rr] & & S' \\\\", "& R \\ar[lu] \\ar[ru]", "}", "$$", "be a commutative diagram in the category of rings.", "Let $I \\subset R$ be a nilpotent ideal and $M$ an $S'$-module. Assume", "\\begin{enumerate}", "\\item The module $M/IM$ is a flat $S/IS$-module.", "\\item The module $M$ is a flat $R$-module.", "\\end{enumerate}", "Then $M$ is a flat $S$-module and $S_{\\mathfrak q}$ is flat over $R$", "for every $\\mathfrak q \\subset S$ such that $M \\otimes_S \\kappa(\\mathfrak q)$", "is nonzero." ], "refs": [], "proofs": [ { "contents": [ "As $M$ is flat over $R$ tensoring with the short exact", "sequence $0 \\to I \\to R \\to R/I \\to 0$ gives a short exact sequence", "$$", "0 \\to I \\otimes_R M \\to M \\to M/IM \\to 0.", "$$", "Note that $I \\otimes_R M \\to IS \\otimes_S M$ is surjective. Combined with", "the above this means both maps in", "$$", "I \\otimes_R M \\to IS \\otimes_S M \\to M", "$$", "are injective. Hence $\\text{Tor}_1^S(IS, M) = 0$ (see", "Remark \\ref{remark-Tor-ring-mod-ideal})", "and we conclude that $M$ is a flat $S$-module by", "Lemma \\ref{lemma-what-does-it-mean}.", "To finish we need to show that $S_{\\mathfrak q}$ is flat over", "$R$ for any prime $\\mathfrak q \\subset S$ such that", "$M \\otimes_S \\kappa(\\mathfrak q)$ is nonzero. This follows from", "Lemma \\ref{lemma-ff} and \\ref{lemma-flat-permanence}." ], "refs": [ "algebra-remark-Tor-ring-mod-ideal", "algebra-lemma-what-does-it-mean", "algebra-lemma-ff", "algebra-lemma-flat-permanence" ], "ref_ids": [ 1570, 890, 535, 530 ] } ], "ref_ids": [] }, { "id": 908, "type": "theorem", "label": "algebra-lemma-add-trivial-complex", "categories": [ "algebra" ], "title": "algebra-lemma-add-trivial-complex", "contents": [ "Suppose $R$ is a ring. Let", "$$", "\\ldots", "\\xrightarrow{\\varphi_{i + 1}}", "R^{n_i}", "\\xrightarrow{\\varphi_i}", "R^{n_{i-1}}", "\\xrightarrow{\\varphi_{i-1}}", "\\ldots", "$$", "be a complex of finite free $R$-modules. Suppose that for some $i$", "some matrix coefficient of the map $\\varphi_i$ is invertible.", "Then the displayed complex is isomorphic to the direct sum of a complex", "$$", "\\ldots \\to", "R^{n_{i + 2}} \\xrightarrow{\\varphi_{i + 2}}", "R^{n_{i + 1}} \\to", "R^{n_i - 1} \\to", "R^{n_{i - 1} - 1} \\to", "R^{n_{i - 2}} \\xrightarrow{\\varphi_{i - 2}}", "R^{n_{i - 3}} \\to", "\\ldots", "$$", "and the complex $\\ldots \\to 0 \\to R \\to R \\to 0 \\to \\ldots$", "where the map $R \\to R$ is the identity map." ], "refs": [], "proofs": [ { "contents": [ "The assumption means, after a change of basis of", "$R^{n_i}$ and $R^{n_{i-1}}$ that the first basis", "vector of $R^{n_i}$ is mapped via $\\varphi_i$ to the first basis", "vector of $R^{n_{i-1}}$. Let $e_j$ denote the", "$j$th basis vector of $R^{n_i}$ and $f_k$ the $k$th", "basis vector of $R^{n_{i-1}}$. Write $\\varphi_i(e_j)", "= \\sum a_{jk} f_k$. So $a_{1k} = 0$ unless $k = 1$", "and $a_{11} = 1$. Change basis on $R^{n_i}$ again", "by setting $e'_j = e_j - a_{j1} e_1$ for $j > 1$.", "After this change of coordinates we have $a_{j1} = 0$", "for $j > 1$. Note the image", "of $R^{n_{i + 1}} \\to R^{n_i}$ is contained in the", "subspace spanned by $e_j$, $j > 1$. Note also", "that $R^{n_{i-1}} \\to R^{n_{i-2}}$ has to annihilate", "$f_1$ since it is in the image. These conditions", "and the shape of the matrix $(a_{jk})$ for $\\varphi_i$", "imply the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 909, "type": "theorem", "label": "algebra-lemma-exact-depth-zero-local", "categories": [ "algebra" ], "title": "algebra-lemma-exact-depth-zero-local", "contents": [ "In Situation \\ref{situation-complex}. Suppose $R$ is", "a local Noetherian ring with maximal ideal $\\mathfrak m$.", "Assume $\\mathfrak m \\in \\text{Ass}(R)$, in other words", "$R$ has depth $0$. Suppose that", "$0 \\to R^{n_e} \\to R^{n_{e-1}} \\to \\ldots \\to R^{n_0}$", "is exact at $R^{n_e}, \\ldots, R^{n_1}$.", "Then the complex is isomorphic to a direct sum of trivial", "complexes." ], "refs": [], "proofs": [ { "contents": [ "Pick $x \\in R$, $x \\not = 0$, with $\\mathfrak m x = 0$.", "Let $i$ be the biggest index such that $n_i > 0$.", "If $i = 0$, then the statement is true. If", "$i > 0$ denote $f_1$ the first basis vector of $R^{n_i}$.", "Since $xf_1$ is not mapped to zero by", "exactness of the complex we deduce that some matrix", "coefficient of the map $R^{n_i} \\to R^{n_{i - 1}}$", "is not in $\\mathfrak m$.", "Lemma \\ref{lemma-add-trivial-complex} then allows", "us to decrease $n_e + \\ldots + n_1$. Induction finishes the proof." ], "refs": [ "algebra-lemma-add-trivial-complex" ], "ref_ids": [ 908 ] } ], "ref_ids": [] }, { "id": 910, "type": "theorem", "label": "algebra-lemma-exact-artinian-local", "categories": [ "algebra" ], "title": "algebra-lemma-exact-artinian-local", "contents": [ "In Situation \\ref{situation-complex}. Let $R$ be a Artinian local ring.", "Suppose that $0 \\to R^{n_e} \\to R^{n_{e-1}} \\to \\ldots \\to R^{n_0}$", "is exact at $R^{n_e}, \\ldots, R^{n_1}$. Then the complex is isomorphic", "to a direct sum of trivial complexes." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-exact-depth-zero-local}", "because an Artinian local ring has depth $0$." ], "refs": [ "algebra-lemma-exact-depth-zero-local" ], "ref_ids": [ 909 ] } ], "ref_ids": [] }, { "id": 911, "type": "theorem", "label": "algebra-lemma-trivial-case-exact", "categories": [ "algebra" ], "title": "algebra-lemma-trivial-case-exact", "contents": [ "In Situation \\ref{situation-complex}, suppose the complex is", "isomorphic to a direct sum of trivial complexes. Then", "we have", "\\begin{enumerate}", "\\item the maps $\\varphi_i$ have rank", "$r_i = n_i - n_{i + 1} + \\ldots + (-1)^{e-i-1} n_{e-1} + (-1)^{e-i} n_e$,", "\\item for all $i$, $1 \\leq i \\leq e - 1$ we have", "$\\text{rank}(\\varphi_{i + 1}) + \\text{rank}(\\varphi_i) = n_i$,", "\\item each $I(\\varphi_i) = R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may assume the complex is the direct sum of trivial", "complexes. Then for each $i$ we can split the standard basis", "elements of $R^{n_i}$ into those that map to a basis element", "of $R^{n_{i-1}}$ and those that are mapped to zero (and these", "are mapped onto by basis elements of $R^{n_{i + 1}}$ if $i > 0$).", "Using descending", "induction starting with $i = e$ it is easy to prove that there", "are $r_{i + 1}$-basis elements of $R^{n_i}$ which are mapped", "to zero and $r_i$ which are mapped to basis elements of", "$R^{n_{i-1}}$. From this the result follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 912, "type": "theorem", "label": "algebra-lemma-div-x-exact-one-less", "categories": [ "algebra" ], "title": "algebra-lemma-div-x-exact-one-less", "contents": [ "In Situation \\ref{situation-complex}. Suppose $R$ is", "a local ring with maximal ideal $\\mathfrak m$.", "Suppose that $0 \\to R^{n_e} \\to R^{n_{e-1}} \\to \\ldots \\to R^{n_0}$", "is exact at $R^{n_e}, \\ldots, R^{n_1}$.", "Let $x \\in \\mathfrak m$ be a nonzerodivisor. The complex", "$0 \\to (R/xR)^{n_e} \\to \\ldots \\to (R/xR)^{n_1}$", "is exact at $(R/xR)^{n_e}, \\ldots, (R/xR)^{n_2}$." ], "refs": [], "proofs": [ { "contents": [ "Denote $F_\\bullet$ the complex with terms $F_i = R^{n_i}$", "and differential given by $\\varphi_i$. Then we have a short", "exact sequence of complexes", "$$", "0 \\to F_\\bullet \\xrightarrow{x} F_\\bullet \\to F_\\bullet/xF_\\bullet \\to 0", "$$", "Applying the snake lemma we get a long exact sequence", "$$", "H_i(F_\\bullet) \\xrightarrow{x} H_i(F_\\bullet) \\to", "H_i(F_\\bullet/xF_\\bullet) \\to H_{i - 1}(F_\\bullet)", "\\xrightarrow{x} H_{i - 1}(F_\\bullet)", "$$", "The lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 913, "type": "theorem", "label": "algebra-lemma-acyclic", "categories": [ "algebra" ], "title": "algebra-lemma-acyclic", "contents": [ "\\begin{reference}", "\\cite[Lemma 1.8]{Peskine-Szpiro}", "\\end{reference}", "Let $R$ be a local Noetherian ring.", "Let $0 \\to M_e \\to M_{e-1} \\to \\ldots \\to M_0$", "be a complex of finite $R$-modules.", "Assume $\\text{depth}(M_i) \\geq i$.", "Let $i$ be the largest index such that the complex is", "not exact at $M_i$. If $i > 0$ then", "$\\Ker(M_i \\to M_{i-1})/\\Im(M_{i + 1} \\to M_i)$", "has depth $\\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "Let $H = \\Ker(M_i \\to M_{i-1})/\\Im(M_{i + 1} \\to M_i)$ be the", "cohomology group in question.", "We may break the complex into short exact sequences", "$0 \\to M_e \\to M_{e-1} \\to K_{e-2} \\to 0$,", "$0 \\to K_j \\to M_j \\to K_{j-1} \\to 0$, for $i + 2 \\leq j \\leq e-2 $,", "$0 \\to K_{i + 1} \\to M_{i + 1} \\to B_i \\to 0$,", "$0 \\to K_i \\to M_i \\to M_{i-1}$, and", "$0 \\to B_i \\to K_i \\to H \\to 0$.", "We proceed up through these complexes to", "prove the statements about depths, repeatedly using", "Lemma \\ref{lemma-depth-in-ses}.", "First of all, since $\\text{depth}(M_e) \\geq e$,", "and $\\text{depth}(M_{e-1}) \\geq e-1$ we deduce", "that $\\text{depth}(K_{e-2}) \\geq e - 1$. At this point the", "sequences $0 \\to K_j \\to M_j \\to K_{j-1} \\to 0$ for $i + 2 \\leq j \\leq e-2 $", "imply similarly that $\\text{depth}(K_{j-1}) \\geq j$ for", "$i + 2 \\leq j \\leq e-2$. The sequence", "$0 \\to K_{i + 1} \\to M_{i + 1} \\to B_i \\to 0$", "then shows that $\\text{depth}(B_i) \\geq i + 1$. The sequence", "$0 \\to K_i \\to M_i \\to M_{i-1}$ shows that $\\text{depth}(K_i) \\geq 1$", "since $M_i$ has depth $\\geq i \\geq 1$ by assumption.", "The sequence $0 \\to B_i \\to K_i \\to H \\to 0$ then", "implies the result." ], "refs": [ "algebra-lemma-depth-in-ses" ], "ref_ids": [ 773 ] } ], "ref_ids": [] }, { "id": 914, "type": "theorem", "label": "algebra-lemma-good-element", "categories": [ "algebra" ], "title": "algebra-lemma-good-element", "contents": [ "Notation and assumptions as above. If $g$ is good with respect to", "$(M, f_1, \\ldots, f_d)$, then (a) $g$ is a nonzerodivisor on $M$,", "and (b) $M/gM$ is Cohen-Macaulay with maximal regular", "sequence $f_1, \\ldots, f_{d - 1}$." ], "refs": [], "proofs": [ { "contents": [ "We prove the lemma by induction on $d$.", "If $d = 0$, then $M$ is finite and there is no case", "to which the lemma applies.", "If $d = 1$, then we have to show that $g : M \\to M$ is", "injective. The kernel $K$ has support $\\{\\mathfrak m\\}$", "because by assumption $\\dim \\text{Supp}(M) \\cap V(g) = 0$.", "Hence $K$ has finite length. Hence $f_1 : K \\to K$ injective", "implies the length of the image is the length of $K$, and hence", "$f_1 K = K$, which by Nakayama's Lemma \\ref{lemma-NAK} implies $K = 0$.", "Also, $\\dim \\text{Supp}(M/gM) = 0$ and so $M/gM$ is Cohen-Macaulay", "of depth $0$.", "\\medskip\\noindent", "Assume $d > 1$. Observe that $g$ is good for $(M/f_1M, f_2, \\ldots, f_d)$,", "as is easily seen from the definition. By induction, we have that", "(a) $g$ is a nonzerodivisor on $M/f_1M$ and", "(b) $M/(g, f_1)M$ is Cohen-Macaulay with maximal regular sequence", "$f_2, \\ldots, f_{d - 1}$. By", "Lemma \\ref{lemma-permute-xi}", "we see that $g, f_1$ is an $M$-regular sequence.", "Hence $g$ is a nonzerodivisor on $M$ and", "$f_1, \\ldots, f_{d - 1}$ is an $M/gM$-regular sequence." ], "refs": [ "algebra-lemma-NAK", "algebra-lemma-permute-xi" ], "ref_ids": [ 401, 739 ] } ], "ref_ids": [] }, { "id": 915, "type": "theorem", "label": "algebra-lemma-CM-one-g", "categories": [ "algebra" ], "title": "algebra-lemma-CM-one-g", "contents": [ "Let $R$ be a Noetherian local ring.", "Let $M$ be a Cohen-Macaulay module over $R$.", "Suppose $g \\in \\mathfrak m$ is such that $\\dim(\\text{Supp}(M) \\cap V(g))", "= \\dim(\\text{Supp}(M)) - 1$. Then (a) $g$ is a nonzerodivisor on $M$,", "and (b) $M/gM$ is Cohen-Macaulay of depth one less." ], "refs": [], "proofs": [ { "contents": [ "Choose a $M$-regular sequence $f_1, \\ldots, f_d$ with", "$d = \\dim(\\text{Supp}(M))$. If $g$ is good with respect to", "$(M, f_1, \\ldots, f_d)$ we win by Lemma \\ref{lemma-good-element}.", "In particular the lemma holds if $d = 1$. (The case $d = 0$ does", "not occur.) Assume $d > 1$. Choose an element $h \\in R$ such that", "(\\romannumeral1) $h$ is good with respect to $(M, f_1, \\ldots, f_d)$,", "and (\\romannumeral2) $\\dim(\\text{Supp}(M) \\cap V(h, g)) = d - 2$.", "To see $h$ exists, let $\\{\\mathfrak q_j\\}$ be the (finite) set of", "minimal primes of the closed sets $\\text{Supp}(M)$,", "$\\text{Supp}(M)\\cap V(f_1, \\ldots, f_i)$, $i = 1, \\ldots, d - 1$,", "and $\\text{Supp}(M) \\cap V(g)$. None of these $\\mathfrak q_j$", "is equal to $\\mathfrak m$ and hence we may find $h \\in \\mathfrak m$,", "$h \\not \\in \\mathfrak q_j$ by Lemma \\ref{lemma-silly}. It is clear", "that $h$ satisfies (\\romannumeral1) and (\\romannumeral2). From", "Lemma \\ref{lemma-good-element} we conclude that", "$M/hM$ is Cohen-Macaulay. By (\\romannumeral2) we see that the pair", "$(M/hM, g)$ satisfies the induction hypothesis. Hence", "$M/(h, g)M$ is Cohen-Macaulay and $g : M/hM \\to M/hM$", "is injective. By Lemma \\ref{lemma-permute-xi} we see", "that $g : M \\to M$ and $h : M/gM \\to M/gM$", "are injective. Combined with the fact that $M/(g, h)M$", "is Cohen-Macaulay this finishes the proof." ], "refs": [ "algebra-lemma-good-element", "algebra-lemma-silly", "algebra-lemma-good-element", "algebra-lemma-permute-xi" ], "ref_ids": [ 914, 378, 914, 739 ] } ], "ref_ids": [] }, { "id": 916, "type": "theorem", "label": "algebra-lemma-nonzerodivisor-on-CM", "categories": [ "algebra" ], "title": "algebra-lemma-nonzerodivisor-on-CM", "contents": [ "Let $R$ be a Noetherian local ring with maximal ideal $\\mathfrak m$.", "Let $M$ be a finite $R$-module. Let $x \\in \\mathfrak m$ be a", "nonzerodivisor on $M$. Then $M$ is Cohen-Macaulay if and only", "if $M/xM$ is Cohen-Macaulay." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-depth-drops-by-one} we have", "$\\text{depth}(M/xM) = \\text{depth}(M)-1$.", "By Lemma \\ref{lemma-one-equation-module}", "we have $\\dim(\\text{Supp}(M/xM)) = \\dim(\\text{Supp}(M)) - 1$." ], "refs": [ "algebra-lemma-depth-drops-by-one" ], "ref_ids": [ 774 ] } ], "ref_ids": [] }, { "id": 917, "type": "theorem", "label": "algebra-lemma-CM-over-quotient", "categories": [ "algebra" ], "title": "algebra-lemma-CM-over-quotient", "contents": [ "Let $R \\to S$ be a surjective homomorphism of Noetherian local rings.", "Let $N$ be a finite $S$-module. Then $N$ is Cohen-Macaulay as an $S$-module", "if and only if $N$ is Cohen-Macaulay as an $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 918, "type": "theorem", "label": "algebra-lemma-CM-ass-minimal-support", "categories": [ "algebra" ], "title": "algebra-lemma-CM-ass-minimal-support", "contents": [ "\\begin{reference}", "\\cite[Chapter 0, Proposition 16.5.4]{EGA}", "\\end{reference}", "Let $R$ be a Noetherian local ring. Let $M$ be a finite Cohen-Macaulay", "$R$-module. If $\\mathfrak p \\in \\text{Ass}(M)$, then", "$\\dim(R/\\mathfrak p) = \\dim(\\text{Supp}(M))$ and $\\mathfrak p$", "is a minimal prime in the support of $M$.", "In particular, $M$ has no embedded associated primes." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-depth-dim-associated-primes} we have", "$\\text{depth}(M) \\leq \\dim(R/\\mathfrak p)$.", "Of course $\\dim(R/\\mathfrak p) \\leq \\dim(\\text{Supp}(M))$", "as $\\mathfrak p \\in \\text{Supp}(M)$ (Lemma \\ref{lemma-ass-support}).", "Thus we have equality in both inequalities as $M$ is Cohen-Macaulay.", "Then $\\mathfrak p$ must be minimal in $\\text{Supp}(M)$ otherwise", "we would have $\\dim(R/\\mathfrak p) < \\dim(\\text{Supp}(M))$.", "Finally, minimal primes in the support of $M$ are equal to", "the minimal elements of $\\text{Ass}(M)$", "(Proposition \\ref{proposition-minimal-primes-associated-primes})", "hence $M$ has no embedded associated primes", "(Definition \\ref{definition-embedded-primes})." ], "refs": [ "algebra-lemma-depth-dim-associated-primes", "algebra-lemma-ass-support", "algebra-proposition-minimal-primes-associated-primes", "algebra-definition-embedded-primes" ], "ref_ids": [ 776, 698, 1412, 1485 ] } ], "ref_ids": [] }, { "id": 919, "type": "theorem", "label": "algebra-lemma-maximal-chain-maximal-CM", "categories": [ "algebra" ], "title": "algebra-lemma-maximal-chain-maximal-CM", "contents": [ "\\begin{slogan}", "In a local Cohen-Macaulay ring, any maximal chain of prime ideals has", "length equal to the dimension.", "\\end{slogan}", "Let $R$ be a Noetherian local ring. Assume there exists a", "Cohen-Macaulay module $M$ with $\\Spec(R) = \\text{Supp}(M)$.", "Then any maximal chain of ideals $\\mathfrak p_0 \\subset", "\\mathfrak p_1 \\subset \\ldots \\subset \\mathfrak p_n$", "has length $n = \\dim(R)$." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by induction on $\\dim(R)$. If $\\dim(R) = 0$,", "then the statement is clear. Assume $\\dim(R) > 0$. Then $n > 0$.", "Choose an element $x \\in \\mathfrak p_1$, with $x$ not in", "any of the minimal primes of $R$, and in particular", "$x \\not \\in \\mathfrak p_0$. (See Lemma \\ref{lemma-silly}.)", "Then $\\dim(R/xR) = \\dim(R) - 1$ by Lemma \\ref{lemma-one-equation}.", "The module $M/xM$ is Cohen-Macaulay over $R/xR$ by", "Proposition \\ref{proposition-CM-module} and", "Lemma \\ref{lemma-CM-over-quotient}.", "The support of $M/xM$ is $\\Spec(R/xR)$ by", "Lemma \\ref{lemma-support-quotient}.", "After replacing $x$ by $x^n$ for some $n$,", "we may assume that $\\mathfrak p_1$ is an associated prime of $M/xM$, see", "Lemma \\ref{lemma-inherit-minimal-primes}.", "By Lemma \\ref{lemma-CM-ass-minimal-support}", "we conclude that $\\mathfrak p_1/(x)$ is a minimal prime of $R/xR$.", "It follows that the chain", "$\\mathfrak p_1/(x) \\subset \\ldots \\subset \\mathfrak p_n/(x)$", "is a maximal chain of primes in $R/xR$.", "By induction we find that this chain", "has length $\\dim(R/xR) = \\dim(R) - 1$ as desired." ], "refs": [ "algebra-lemma-silly", "algebra-proposition-CM-module", "algebra-lemma-CM-over-quotient", "algebra-lemma-support-quotient", "algebra-lemma-inherit-minimal-primes", "algebra-lemma-CM-ass-minimal-support" ], "ref_ids": [ 378, 1420, 917, 547, 775, 918 ] } ], "ref_ids": [] }, { "id": 920, "type": "theorem", "label": "algebra-lemma-dim-formula-maximal-CM", "categories": [ "algebra" ], "title": "algebra-lemma-dim-formula-maximal-CM", "contents": [ "Suppose $R$ is a Noetherian local ring. Assume there exists a", "Cohen-Macaulay module $M$ with $\\Spec(R) = \\text{Supp}(M)$. Then for", "a prime $\\mathfrak p \\subset R$ we have", "$$", "\\dim(R) = \\dim(R_{\\mathfrak p}) + \\dim(R/\\mathfrak p).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-maximal-chain-maximal-CM}." ], "refs": [ "algebra-lemma-maximal-chain-maximal-CM" ], "ref_ids": [ 919 ] } ], "ref_ids": [] }, { "id": 921, "type": "theorem", "label": "algebra-lemma-localize-CM-module", "categories": [ "algebra" ], "title": "algebra-lemma-localize-CM-module", "contents": [ "Suppose $R$ is a Noetherian local ring. Let $M$ be a Cohen-Macaulay", "module over $R$. For any prime $\\mathfrak p \\subset R$ the", "module $M_{\\mathfrak p}$ is Cohen-Macaulay over $R_\\mathfrak p$." ], "refs": [], "proofs": [ { "contents": [ "We may and do assume $\\mathfrak p \\not = \\mathfrak m$ and $M$ not zero.", "Choose a maximal chain of primes $\\mathfrak p = \\mathfrak p_c \\subset", "\\mathfrak p_{c - 1} \\subset \\ldots \\subset \\mathfrak p_1 \\subset \\mathfrak m$.", "If we prove the result for $M_{\\mathfrak p_1}$ over $R_{\\mathfrak p_1}$,", "then the lemma will follow by induction on $c$. Thus we may assume that", "there is no prime strictly between $\\mathfrak p$ and $\\mathfrak m$.", "Note that", "$\\dim(\\text{Supp}(M_\\mathfrak p)) \\leq \\dim(\\text{Supp}(M)) - 1$", "because any chain of primes in the support of $M_\\mathfrak p$", "can be extended by one more prime (namely $\\mathfrak m$) in the", "support of $M$. On the other hand, we have", "$\\text{depth}(M_\\mathfrak p) \\geq \\text{depth}(M) - \\dim(R/\\mathfrak p) =", "\\text{depth}(M) - 1$ by Lemma \\ref{lemma-depth-localization} and our", "choice of $\\mathfrak p$.", "Thus $\\text{depth}(M_\\mathfrak p) \\geq \\dim(\\text{Supp}(M_\\mathfrak p)$", "as desired (the other inequality is Lemma \\ref{lemma-bound-depth})." ], "refs": [ "algebra-lemma-depth-localization", "algebra-lemma-bound-depth" ], "ref_ids": [ 777, 770 ] } ], "ref_ids": [] }, { "id": 922, "type": "theorem", "label": "algebra-lemma-maximal-CM-polynomial-algebra", "categories": [ "algebra" ], "title": "algebra-lemma-maximal-CM-polynomial-algebra", "contents": [ "Let $R$ be a Noetherian ring. Let $M$ be a Cohen-Macaulay module", "over $R$. Then $M \\otimes_R R[x_1, \\ldots, x_n]$ is a", "Cohen-Macaulay module over $R[x_1, \\ldots, x_n]$." ], "refs": [], "proofs": [ { "contents": [ "By induction on the number of variables it suffices to prove this for", "$M[x] = M \\otimes_R R[x]$ over $R[x]$. Let $\\mathfrak m \\subset R[x]$", "be a maximal ideal, and let $\\mathfrak p = R \\cap \\mathfrak m$.", "Let $f_1, \\ldots, f_d$ be a $M_\\mathfrak p$-regular sequence in the maximal", "ideal of $R_{\\mathfrak p}$ of length $d = \\dim(\\text{Supp}(M_{\\mathfrak p}))$.", "Note that since $R[x]$ is flat over $R$ the localization", "$R[x]_{\\mathfrak m}$ is flat over $R_{\\mathfrak p}$.", "Hence, by Lemma \\ref{lemma-flat-increases-depth}, the sequence", "$f_1, \\ldots, f_d$ is a $M[x]_{\\mathfrak m}$-regular sequence of length $d$", "in $R[x]_{\\mathfrak m}$. The quotient", "$$", "Q = M[x]_{\\mathfrak m}/(f_1, \\ldots, f_d)M[x]_{\\mathfrak m} =", "M_{\\mathfrak p}/(f_1, \\ldots, f_d)M_{\\mathfrak p}", "\\otimes_{R_\\mathfrak p} R[x]_{\\mathfrak m}", "$$", "has support equal to the primes lying over $\\mathfrak p$", "because $R_\\mathfrak p \\to R[x]_\\mathfrak m$ is flat and", "the support of $M_{\\mathfrak p}/(f_1, \\ldots, f_d)M_{\\mathfrak p}$", "is equal to $\\{\\mathfrak p\\}$ (details omitted; hint: follows from", "Lemmas \\ref{lemma-annihilator-flat-base-change} and", "\\ref{lemma-support-closed}). Hence the dimension is $1$.", "To finish the proof it suffices to find an $f \\in \\mathfrak m$", "which is a nonzerodivisor on $Q$. Since $\\mathfrak m$ is", "a maximal ideal, the field extension", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak m)$", "is finite (Theorem \\ref{theorem-nullstellensatz}).", "Hence we can find $f \\in \\mathfrak m$ which", "viewed as a polynomial in $x$ has leading", "coefficient not in $\\mathfrak p$. Such an $f$ acts as", "a nonzerodivisor on", "$$", "M_{\\mathfrak p}/(f_1, \\ldots, f_d)M_{\\mathfrak p} \\otimes_R R[x] =", "\\bigoplus\\nolimits_{n \\geq 0}", "M_{\\mathfrak p}/(f_1, \\ldots, f_d)M_{\\mathfrak p} \\cdot x^n", "$$", "and hence acts as a nonzerodivisor on $Q$." ], "refs": [ "algebra-lemma-flat-increases-depth", "algebra-lemma-annihilator-flat-base-change", "algebra-lemma-support-closed", "algebra-theorem-nullstellensatz" ], "ref_ids": [ 740, 542, 543, 316 ] } ], "ref_ids": [] }, { "id": 923, "type": "theorem", "label": "algebra-lemma-reformulate-CM", "categories": [ "algebra" ], "title": "algebra-lemma-reformulate-CM", "contents": [ "\\begin{slogan}", "Regular sequences in Cohen-Macaulay local rings are characterized", "by cutting out something of the correct dimension.", "\\end{slogan}", "Let $R$ be a Noetherian local Cohen-Macaulay ring with maximal", "ideal $\\mathfrak m $. Let $x_1, \\ldots, x_c \\in \\mathfrak m$ be", "elements. Then", "$$", "x_1, \\ldots, x_c", "\\text{ is a regular sequence }", "\\Leftrightarrow", "\\dim(R/(x_1, \\ldots, x_c)) = \\dim(R) - c", "$$", "If so", "$x_1, \\ldots, x_c$ can be extended to", "a regular sequence of length $\\dim(R)$ and each quotient", "$R/(x_1, \\ldots, x_i)$ is a Cohen-Macaulay ring of dimension", "$\\dim(R) - i$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Proposition \\ref{proposition-CM-module}." ], "refs": [ "algebra-proposition-CM-module" ], "ref_ids": [ 1420 ] } ], "ref_ids": [] }, { "id": 924, "type": "theorem", "label": "algebra-lemma-maximal-chain-CM", "categories": [ "algebra" ], "title": "algebra-lemma-maximal-chain-CM", "contents": [ "Let $R$ be Noetherian local.", "Suppose $R$ is Cohen-Macaulay of dimension $d$.", "Any maximal chain of ideals $\\mathfrak p_0 \\subset", "\\mathfrak p_1 \\subset \\ldots \\subset \\mathfrak p_n$", "has length $n = d$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-maximal-chain-maximal-CM}." ], "refs": [ "algebra-lemma-maximal-chain-maximal-CM" ], "ref_ids": [ 919 ] } ], "ref_ids": [] }, { "id": 925, "type": "theorem", "label": "algebra-lemma-CM-dim-formula", "categories": [ "algebra" ], "title": "algebra-lemma-CM-dim-formula", "contents": [ "Suppose $R$ is a Noetherian local Cohen-Macaulay ring of dimension $d$.", "For any prime $\\mathfrak p \\subset R$ we have", "$$", "\\dim(R) = \\dim(R_{\\mathfrak p}) + \\dim(R/\\mathfrak p).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-maximal-chain-CM}.", "(Also, this is a special case of Lemma \\ref{lemma-dim-formula-maximal-CM}.)" ], "refs": [ "algebra-lemma-maximal-chain-CM", "algebra-lemma-dim-formula-maximal-CM" ], "ref_ids": [ 924, 920 ] } ], "ref_ids": [] }, { "id": 926, "type": "theorem", "label": "algebra-lemma-localize-CM", "categories": [ "algebra" ], "title": "algebra-lemma-localize-CM", "contents": [ "Suppose $R$ is a Cohen-Macaulay local ring.", "For any prime $\\mathfrak p \\subset R$ the", "ring $R_{\\mathfrak p}$ is Cohen-Macaulay as well." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-localize-CM-module}." ], "refs": [ "algebra-lemma-localize-CM-module" ], "ref_ids": [ 921 ] } ], "ref_ids": [] }, { "id": 927, "type": "theorem", "label": "algebra-lemma-CM-polynomial-algebra", "categories": [ "algebra" ], "title": "algebra-lemma-CM-polynomial-algebra", "contents": [ "Suppose $R$ is a Noetherian Cohen-Macaulay ring.", "Any polynomial algebra over $R$ is Cohen-Macaulay." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-maximal-CM-polynomial-algebra}." ], "refs": [ "algebra-lemma-maximal-CM-polynomial-algebra" ], "ref_ids": [ 922 ] } ], "ref_ids": [] }, { "id": 928, "type": "theorem", "label": "algebra-lemma-dimension-shift", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-shift", "contents": [ "Let $R$ be a Noetherian local Cohen-Macaulay ring of dimension $d$.", "Let $0 \\to K \\to R^{\\oplus n} \\to M \\to 0$ be an exact sequence of", "$R$-modules. Then either $M = 0$, or $\\text{depth}(K) > \\text{depth}(M)$, or", "$\\text{depth}(K) = \\text{depth}(M) = d$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-depth-in-ses}." ], "refs": [ "algebra-lemma-depth-in-ses" ], "ref_ids": [ 773 ] } ], "ref_ids": [] }, { "id": 929, "type": "theorem", "label": "algebra-lemma-mcm-resolution", "categories": [ "algebra" ], "title": "algebra-lemma-mcm-resolution", "contents": [ "Let $R$ be a local Noetherian Cohen-Macaulay ring of dimension $d$.", "Let $M$ be a finite $R$ module of depth $e$.", "There exists an exact complex", "$$", "0 \\to K \\to F_{d-e-1} \\to \\ldots \\to F_0 \\to M \\to 0", "$$", "with each $F_i$ finite free and $K$ maximal Cohen-Macaulay." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definition and Lemma \\ref{lemma-dimension-shift}." ], "refs": [ "algebra-lemma-dimension-shift" ], "ref_ids": [ 928 ] } ], "ref_ids": [] }, { "id": 930, "type": "theorem", "label": "algebra-lemma-find-sequence-image-regular", "categories": [ "algebra" ], "title": "algebra-lemma-find-sequence-image-regular", "contents": [ "Let $\\varphi : A \\to B$ be a map of local rings.", "Assume that $B$ is Noetherian and Cohen-Macaulay and that", "$\\mathfrak m_B = \\sqrt{\\varphi(\\mathfrak m_A) B}$. Then there exists", "a sequence of elements $f_1, \\ldots, f_{\\dim(B)}$ in $A$", "such that $\\varphi(f_1), \\ldots, \\varphi(f_{\\dim(B)})$ is a", "regular sequence in $B$." ], "refs": [], "proofs": [ { "contents": [ "By induction on $\\dim(B)$ it suffices to prove: If $\\dim(B) \\geq 1$, then we", "can find an element $f$ of $A$ which maps to a nonzerodivisor in $B$.", "By", "Lemma \\ref{lemma-reformulate-CM}", "it suffices to find $f \\in A$ whose image in $B$ is not contained in any", "of the finitely many minimal primes $\\mathfrak q_1, \\ldots, \\mathfrak q_r$", "of $B$. By the assumption that", "$\\mathfrak m_B = \\sqrt{\\varphi(\\mathfrak m_A) B}$", "we see that $\\mathfrak m_A \\not \\subset \\varphi^{-1}(\\mathfrak q_i)$.", "Hence we can find $f$ by", "Lemma \\ref{lemma-silly}." ], "refs": [ "algebra-lemma-reformulate-CM", "algebra-lemma-silly" ], "ref_ids": [ 923, 378 ] } ], "ref_ids": [] }, { "id": 931, "type": "theorem", "label": "algebra-lemma-catenary", "categories": [ "algebra" ], "title": "algebra-lemma-catenary", "contents": [ "A ring $R$ is catenary if and only if the topological space", "$\\Spec(R)$ is catenary (see", "Topology, Definition \\ref{topology-definition-catenary})." ], "refs": [ "topology-definition-catenary" ], "proofs": [ { "contents": [ "Immediate from the definition and the characterization of", "irreducible closed subsets in Lemma \\ref{lemma-irreducible}." ], "refs": [ "algebra-lemma-irreducible" ], "ref_ids": [ 422 ] } ], "ref_ids": [ 8359 ] }, { "id": 932, "type": "theorem", "label": "algebra-lemma-localization-catenary", "categories": [ "algebra" ], "title": "algebra-lemma-localization-catenary", "contents": [ "Any localization of a catenary ring is catenary.", "Any localization of a Noetherian universally catenary", "ring is universally catenary." ], "refs": [], "proofs": [ { "contents": [ "Let $A$ be a ring and let $S \\subset A$ be a multiplicative subset.", "The description of $\\Spec(S^{-1}A)$ in Lemma \\ref{lemma-spec-localization}", "shows that if $A$ is catenary, then so is $S^{-1}A$. If $S^{-1}A \\to C$", "is of finite type, then $C = S^{-1}B$ for some finite type ring map", "$A \\to B$. Hence if $A$ is Noetherian and universally catenary, then", "$B$ is catenary and we see that $C$ is catenary too. This proves the lemma." ], "refs": [ "algebra-lemma-spec-localization" ], "ref_ids": [ 391 ] } ], "ref_ids": [] }, { "id": 933, "type": "theorem", "label": "algebra-lemma-universally-catenary", "categories": [ "algebra" ], "title": "algebra-lemma-universally-catenary", "contents": [ "Let $A$ be a Noetherian universally catenary ring.", "Any $A$-algebra essentially of finite type over $A$", "is universally catenary." ], "refs": [], "proofs": [ { "contents": [ "If $B$ is a finite type $A$-algebra, then $B$ is Noetherian", "by Lemma \\ref{lemma-Noetherian-permanence}. Any finite type", "$B$-algebra is a finite type $A$-algebra and hence catenary", "by our assumption that $A$ is universally catenary. Thus $B$", "is universally catenary. Any localization of $B$ is universally", "catenary by Lemma \\ref{lemma-localization-catenary} and this", "finishes the proof." ], "refs": [ "algebra-lemma-Noetherian-permanence", "algebra-lemma-localization-catenary" ], "ref_ids": [ 448, 932 ] } ], "ref_ids": [] }, { "id": 934, "type": "theorem", "label": "algebra-lemma-catenary-check-local", "categories": [ "algebra" ], "title": "algebra-lemma-catenary-check-local", "contents": [ "Let $R$ be a ring. The following are equivalent", "\\begin{enumerate}", "\\item $R$ is catenary,", "\\item $R_\\mathfrak p$ is catenary for all prime ideals $\\mathfrak p$,", "\\item $R_\\mathfrak m$ is catenary for all maximal ideals $\\mathfrak m$.", "\\end{enumerate}", "Assume $R$ is Noetherian. The following are equivalent", "\\begin{enumerate}", "\\item $R$ is universally catenary,", "\\item $R_\\mathfrak p$ is universally catenary for all prime ideals", "$\\mathfrak p$,", "\\item $R_\\mathfrak m$ is universally catenary for all maximal ideals", "$\\mathfrak m$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) follows from", "Lemma \\ref{lemma-localization-catenary} in both cases.", "The implication (2) $\\Rightarrow$ (3) is immediate in both cases.", "Assume $R_\\mathfrak m$ is catenary for all maximal ideals", "$\\mathfrak m$ of $R$. If $\\mathfrak p \\subset \\mathfrak q$ are primes", "in $R$, then choose a maximal ideal $\\mathfrak q \\subset \\mathfrak m$.", "Chains of primes ideals between $\\mathfrak p$ and $\\mathfrak q$ are", "in 1-to-1 correspondence with chains of prime ideals between", "$\\mathfrak pR_\\mathfrak m$ and $\\mathfrak qR_\\mathfrak m$ hence we", "see $R$ is catenary. Assume $R$ is Noetherian and $R_\\mathfrak m$ is", "universally catenary for all maximal ideals $\\mathfrak m$ of $R$.", "Let $R \\to S$ be a finite type ring map. Let $\\mathfrak q$ be a prime", "ideal of $S$ lying over the prime $\\mathfrak p \\subset R$.", "Choose a maximal ideal $\\mathfrak p \\subset \\mathfrak m$ in $R$.", "Then $R_\\mathfrak p$ is a localization of $R_\\mathfrak m$ hence", "universally catenary by Lemma \\ref{lemma-localization-catenary}.", "Then $S_\\mathfrak p$ is catenary as a finite type ring over $R_\\mathfrak p$.", "Hence $S_\\mathfrak q$ is catenary as a localization. Thus $S$ is catenary", "by the first case treated above." ], "refs": [ "algebra-lemma-localization-catenary", "algebra-lemma-localization-catenary" ], "ref_ids": [ 932, 932 ] } ], "ref_ids": [] }, { "id": 935, "type": "theorem", "label": "algebra-lemma-quotient-catenary", "categories": [ "algebra" ], "title": "algebra-lemma-quotient-catenary", "contents": [ "Any quotient of a catenary ring is catenary.", "Any quotient of a Noetherian universally catenary ring is", "universally catenary." ], "refs": [], "proofs": [ { "contents": [ "Let $A$ be a ring and let $I \\subset A$ be an ideal.", "The description of $\\Spec(A/I)$ in Lemma \\ref{lemma-spec-closed}", "shows that if $A$ is catenary, then so is $A/I$.", "The second statement is a special case of", "Lemma \\ref{lemma-universally-catenary}." ], "refs": [ "algebra-lemma-spec-closed", "algebra-lemma-universally-catenary" ], "ref_ids": [ 393, 933 ] } ], "ref_ids": [] }, { "id": 936, "type": "theorem", "label": "algebra-lemma-catenary-check-irreducible", "categories": [ "algebra" ], "title": "algebra-lemma-catenary-check-irreducible", "contents": [ "Let $R$ be a Noetherian ring.", "\\begin{enumerate}", "\\item $R$ is catenary if and only if $R/\\mathfrak p$ is catenary", "for every minimal prime $\\mathfrak p$.", "\\item $R$ is universally catenary if and only if $R/\\mathfrak p$ is", "universally catenary for every minimal prime $\\mathfrak p$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $\\mathfrak a \\subset \\mathfrak b$ is an inclusion of primes of $R$,", "then we can find a minimal prime $\\mathfrak p \\subset \\mathfrak a$", "and the first assertion is clear. We omit the proof of the second." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 937, "type": "theorem", "label": "algebra-lemma-CM-ring-catenary", "categories": [ "algebra" ], "title": "algebra-lemma-CM-ring-catenary", "contents": [ "A Noetherian Cohen-Macaulay ring is universally catenary.", "More generally, if $R$ is a Noetherian ring and $M$ is", "a Cohen-Macaulay $R$-module with $\\text{Supp}(M) = \\Spec(R)$,", "then $R$ is universally catenary." ], "refs": [], "proofs": [ { "contents": [ "Since a polynomial algebra over $R$ is Cohen-Macaulay,", "by Lemma \\ref{lemma-CM-polynomial-algebra},", "it suffices to show that a Cohen-Macaulay ring is", "catenary.", "Let $R$ be Cohen-Macaulay and $\\mathfrak p \\subset \\mathfrak q$", "primes of $R$. By definition $R_{\\mathfrak q}$ and", "$R_{\\mathfrak p}$ are Cohen-Macaulay.", "Take a maximal chain of primes", "$\\mathfrak p = \\mathfrak p_0 \\subset \\mathfrak p_1 \\subset", "\\ldots \\subset \\mathfrak p_n = \\mathfrak q$.", "Next choose a maximal chain of primes", "$\\mathfrak q_0 \\subset \\mathfrak q_1 \\subset \\ldots \\subset", "\\mathfrak q_m = \\mathfrak p$. By", "Lemma \\ref{lemma-maximal-chain-CM}", "we have $n + m = \\dim(R_{\\mathfrak q})$. And we have", "$m = \\dim(R_{\\mathfrak p})$ by the same lemma.", "Hence $n = \\dim(R_{\\mathfrak q}) - \\dim(R_{\\mathfrak p})$", "is independent of choices.", "\\medskip\\noindent", "To prove the more general statement, argue exactly as above but", "using Lemmas \\ref{lemma-maximal-CM-polynomial-algebra}", "and \\ref{lemma-maximal-chain-maximal-CM}." ], "refs": [ "algebra-lemma-CM-polynomial-algebra", "algebra-lemma-maximal-chain-CM", "algebra-lemma-maximal-CM-polynomial-algebra", "algebra-lemma-maximal-chain-maximal-CM" ], "ref_ids": [ 927, 924, 922, 919 ] } ], "ref_ids": [] }, { "id": 938, "type": "theorem", "label": "algebra-lemma-catenary-Noetherian-local", "categories": [ "algebra" ], "title": "algebra-lemma-catenary-Noetherian-local", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring. The following are equivalent", "\\begin{enumerate}", "\\item $A$ is catenary, and", "\\item $\\mathfrak p \\mapsto \\dim(A/\\mathfrak p)$ is a dimension function", "on $\\Spec(A)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $A$ is catenary, then $\\Spec(A)$ has a dimension function $\\delta$ by", "Topology, Lemma \\ref{topology-lemma-locally-dimension-function}", "(and Lemma \\ref{lemma-catenary}). We may assume $\\delta(\\mathfrak m) = 0$.", "Then we see that", "$$", "\\delta(\\mathfrak p) = \\text{codim}(V(\\mathfrak m), V(\\mathfrak p)) =", "\\dim(A/\\mathfrak p)", "$$", "by Topology, Lemma \\ref{topology-lemma-dimension-function-catenary}.", "In this way we see that (1) implies (2). The reverse implication", "follows from", "Topology, Lemma \\ref{topology-lemma-dimension-function-catenary}", "as well." ], "refs": [ "topology-lemma-locally-dimension-function", "algebra-lemma-catenary", "topology-lemma-dimension-function-catenary", "topology-lemma-dimension-function-catenary" ], "ref_ids": [ 8293, 931, 8291, 8291 ] } ], "ref_ids": [] }, { "id": 939, "type": "theorem", "label": "algebra-lemma-regular-graded", "categories": [ "algebra" ], "title": "algebra-lemma-regular-graded", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $d$.", "The graded ring $\\bigoplus \\mathfrak m^n / \\mathfrak m^{n + 1}$", "is isomorphic to the graded polynomial algebra", "$\\kappa[X_1, \\ldots, X_d]$." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1, \\ldots, x_d$ be a minimal set of generators", "for the maximal ideal $\\mathfrak m$, see", "Definition \\ref{definition-regular-local}.", "There is a surjection $\\kappa[X_1, \\ldots, X_d]", "\\to \\bigoplus \\mathfrak m^n/\\mathfrak m^{n + 1}$,", "which maps $X_i$ to the class of $x_i$ in $\\mathfrak m/\\mathfrak m^2$.", "Since $d(R) = d$ by", "Proposition \\ref{proposition-dimension}", "we know that the numerical", "polynomial $n \\mapsto \\dim_\\kappa \\mathfrak m^n/\\mathfrak m^{n + 1}$", "has degree $d - 1$. By Lemma \\ref{lemma-quotient-smaller-d} we", "conclude that the surjection $\\kappa[X_1, \\ldots, X_d]", "\\to \\bigoplus \\mathfrak m^n/\\mathfrak m^{n + 1}$ is an isomorphism." ], "refs": [ "algebra-definition-regular-local", "algebra-proposition-dimension", "algebra-lemma-quotient-smaller-d" ], "ref_ids": [ 1480, 1411, 672 ] } ], "ref_ids": [] }, { "id": 940, "type": "theorem", "label": "algebra-lemma-regular-domain", "categories": [ "algebra" ], "title": "algebra-lemma-regular-domain", "contents": [ "Any regular local ring is a domain." ], "refs": [], "proofs": [ { "contents": [ "We will use that $\\bigcap \\mathfrak m^n = 0$", "by Lemma \\ref{lemma-intersect-powers-ideal-module-zero}.", "Let $f, g \\in R$ such that $fg = 0$.", "Suppose that $f \\in \\mathfrak m^a$ and", "$g \\in \\mathfrak m^b$, with $a, b$ maximal.", "Since $fg = 0 \\in \\mathfrak m^{a + b + 1}$", "we see from the result of Lemma \\ref{lemma-regular-graded}", "that either $f \\in \\mathfrak m^{a + 1}$ or", "$g \\in \\mathfrak m^{b + 1}$. Contradiction." ], "refs": [ "algebra-lemma-intersect-powers-ideal-module-zero", "algebra-lemma-regular-graded" ], "ref_ids": [ 627, 939 ] } ], "ref_ids": [] }, { "id": 941, "type": "theorem", "label": "algebra-lemma-regular-ring-CM", "categories": [ "algebra" ], "title": "algebra-lemma-regular-ring-CM", "contents": [ "Let $R$ be a regular local ring and let", "$x_1, \\ldots, x_d$ be a minimal set of generators", "for the maximal ideal $\\mathfrak m$. Then", "$x_1, \\ldots, x_d$ is a regular sequence, and", "each $R/(x_1, \\ldots, x_c)$ is a regular local ring", "of dimension $d - c$. In particular $R$ is Cohen-Macaulay." ], "refs": [], "proofs": [ { "contents": [ "Note that $R/x_1R$ is a Noetherian local ring of dimension $\\geq d - 1$", "by Lemma \\ref{lemma-one-equation} with $x_2, \\ldots, x_d$", "generating the maximal ideal. Hence it is a regular local ring by definition.", "Since $R$ is a domain by Lemma \\ref{lemma-regular-domain}", "$x_1$ is a nonzerodivisor." ], "refs": [ "algebra-lemma-regular-domain" ], "ref_ids": [ 940 ] } ], "ref_ids": [] }, { "id": 942, "type": "theorem", "label": "algebra-lemma-regular-quotient-regular", "categories": [ "algebra" ], "title": "algebra-lemma-regular-quotient-regular", "contents": [ "Let $R$ be a regular local ring. Let $I \\subset R$ be an ideal", "such that $R/I$ is a regular local ring as well. Then", "there exists a minimal set of generators $x_1, \\ldots, x_d$", "for the maximal ideal $\\mathfrak m$ of $R$ such that", "$I = (x_1, \\ldots, x_c)$ for some $0 \\leq c \\leq d$." ], "refs": [], "proofs": [ { "contents": [ "Say $\\dim(R) = d$ and $\\dim(R/I) = d - c$.", "Denote $\\overline{\\mathfrak m} = \\mathfrak m/I$ the", "maximal ideal of $R/I$. Let $\\kappa = R/\\mathfrak m$. We have", "$$", "\\dim_\\kappa((I + \\mathfrak m^2)/\\mathfrak m^2) =", "\\dim_\\kappa(\\mathfrak m/\\mathfrak m^2)", "- \\dim(\\overline{\\mathfrak m}/\\overline{\\mathfrak m}^2) = d - (d - c) = c", "$$", "by the definition of a regular local ring. Hence we can choose", "$x_1, \\ldots, x_c \\in I$ whose images in $\\mathfrak m/\\mathfrak m^2$", "are linearly independent and supplement with", "$x_{c + 1}, \\ldots, x_d$ to get a minimal system of generators of", "$\\mathfrak m$. The induced map $R/(x_1, \\ldots, x_c) \\to R/I$ is a", "surjection between regular local rings of the same dimension", "(Lemma \\ref{lemma-regular-ring-CM}). It follows that", "the kernel is zero, i.e., $I = (x_1, \\ldots, x_c)$. Namely, if not", "then we would have $\\dim(R/I) < \\dim(R/(x_1, \\ldots, x_c))$ by", "Lemmas \\ref{lemma-regular-domain} and \\ref{lemma-one-equation}." ], "refs": [ "algebra-lemma-regular-ring-CM", "algebra-lemma-regular-domain" ], "ref_ids": [ 941, 940 ] } ], "ref_ids": [] }, { "id": 943, "type": "theorem", "label": "algebra-lemma-free-mod-x", "categories": [ "algebra" ], "title": "algebra-lemma-free-mod-x", "contents": [ "Let $R$ be a Noetherian local ring.", "Let $x \\in \\mathfrak m$.", "Let $M$ be a finite $R$-module such that", "$x$ is a nonzerodivisor on $M$ and", "$M/xM$ is free over $R/xR$.", "Then $M$ is free over $R$." ], "refs": [], "proofs": [ { "contents": [ "Let $m_1, \\ldots, m_r$ be elements of $M$ which map to", "a $R/xR$-basis of $M/xM$. By Nakayama's Lemma \\ref{lemma-NAK}", "$m_1, \\ldots, m_r$ generate $M$. If $\\sum a_i m_i = 0$", "is a relation, then $a_i \\in xR$ for all $i$. Hence", "$a_i = b_i x$ for some $b_i \\in R$. Hence", "the kernel $K$ of $R^r \\to M$ satisfies $xK = K$", "and hence is zero by Nakayama's lemma." ], "refs": [ "algebra-lemma-NAK" ], "ref_ids": [ 401 ] } ], "ref_ids": [] }, { "id": 944, "type": "theorem", "label": "algebra-lemma-regular-mcm-free", "categories": [ "algebra" ], "title": "algebra-lemma-regular-mcm-free", "contents": [ "Let $R$ be a regular local ring.", "Any maximal Cohen-Macaulay module over $R$ is free." ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be a maximal Cohen-Macaulay module over $R$.", "Let $x \\in \\mathfrak m$ be part of a regular sequence", "generating $\\mathfrak m$. Then $x$ is a nonzerodivisor", "on $M$ by Proposition \\ref{proposition-CM-module}, and", "$M/xM$ is a maximal Cohen-Macaulay module over $R/xR$.", "By induction on $\\dim(R)$ we see that $M/xM$ is free.", "We win by Lemma \\ref{lemma-free-mod-x}." ], "refs": [ "algebra-proposition-CM-module", "algebra-lemma-free-mod-x" ], "ref_ids": [ 1420, 943 ] } ], "ref_ids": [] }, { "id": 945, "type": "theorem", "label": "algebra-lemma-regular-mod-x", "categories": [ "algebra" ], "title": "algebra-lemma-regular-mod-x", "contents": [ "Suppose $R$ is a Noetherian local ring.", "Let $x \\in \\mathfrak m$ be a nonzerodivisor", "such that $R/xR$ is a regular local ring. Then $R$ is a regular local ring.", "More generally, if $x_1, \\ldots, x_r$ is a regular sequence in $R$", "such that $R/(x_1, \\ldots, x_r)$ is a regular local ring, then", "$R$ is a regular local ring." ], "refs": [], "proofs": [ { "contents": [ "This is true because $x$ together with the lifts of a system", "of minimal generators of the maximal ideal of $R/xR$ will give", "$\\dim(R)$ generators of $\\mathfrak m$.", "Use Lemma \\ref{lemma-one-equation}.", "The last statement follows from the first and induction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 946, "type": "theorem", "label": "algebra-lemma-colimit-regular", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-regular", "contents": [ "Let $(R_i, \\varphi_{ii'})$ be a directed system of local rings whose", "transition maps are local ring maps. If each $R_i$ is a regular local ring and", "$R = \\colim R_i$ is Noetherian, then $R$ is a regular local ring." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak m \\subset R$ be the maximal ideal; it is the colimit", "of the maximal ideal $\\mathfrak m_i \\subset R_i$.", "We prove the lemma by induction on $d = \\dim \\mathfrak m/\\mathfrak m^2$.", "If $d = 0$, then $R = R/\\mathfrak m$ is a field and $R$ is a regular local ring.", "If $d > 0$ pick an $x \\in \\mathfrak m$, $x \\not \\in \\mathfrak m^2$.", "For some $i$ we can find an $x_i \\in \\mathfrak m_i$ mapping to $x$.", "Note that $R/xR = \\colim_{i' \\geq i} R_{i'}/x_iR_{i'}$ is a Noetherian", "local ring. By", "Lemma \\ref{lemma-regular-ring-CM}", "we see that $R_{i'}/x_iR_{i'}$ is a regular local ring.", "Hence by induction we see", "that $R/xR$ is a regular local ring. Since each $R_i$ is a domain", "(Lemma \\ref{lemma-regular-graded}) we see that $R$ is a domain.", "Hence $x$ is a nonzerodivisor and we conclude that $R$ is", "a regular local ring by Lemma \\ref{lemma-regular-mod-x}." ], "refs": [ "algebra-lemma-regular-ring-CM", "algebra-lemma-regular-graded", "algebra-lemma-regular-mod-x" ], "ref_ids": [ 941, 939, 945 ] } ], "ref_ids": [] }, { "id": 947, "type": "theorem", "label": "algebra-lemma-epimorphism", "categories": [ "algebra" ], "title": "algebra-lemma-epimorphism", "contents": [ "Let $R \\to S$ be a ring map. The following are equivalent", "\\begin{enumerate}", "\\item $R \\to S$ is an epimorphism,", "\\item the two ring maps $S \\to S \\otimes_R S$ are equal,", "\\item either of the ring maps $S \\to S \\otimes_R S$ is an isomorphism, and", "\\item the ring map $S \\otimes_R S \\to S$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 948, "type": "theorem", "label": "algebra-lemma-composition-epimorphism", "categories": [ "algebra" ], "title": "algebra-lemma-composition-epimorphism", "contents": [ "The composition of two epimorphisms of rings is an epimorphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is true in any category." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 949, "type": "theorem", "label": "algebra-lemma-base-change-epimorphism", "categories": [ "algebra" ], "title": "algebra-lemma-base-change-epimorphism", "contents": [ "If $R \\to S$ is an epimorphism of rings and $R \\to R'$ is any ring map,", "then $R' \\to R' \\otimes_R S$ is an epimorphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: True in any category with pushouts." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 950, "type": "theorem", "label": "algebra-lemma-permanence-epimorphism", "categories": [ "algebra" ], "title": "algebra-lemma-permanence-epimorphism", "contents": [ "If $A \\to B \\to C$ are ring maps and $A \\to C$ is an epimorphism, so is", "$B \\to C$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is true in any category." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 951, "type": "theorem", "label": "algebra-lemma-epimorphism-local", "categories": [ "algebra" ], "title": "algebra-lemma-epimorphism-local", "contents": [ "Let $R \\to S$ be a ring map. The following are equivalent:", "\\begin{enumerate}", "\\item $R \\to S$ is an epimorphism, and", "\\item $R_{\\mathfrak p} \\to S_{\\mathfrak p}$ is an epimorphism for", "each prime $\\mathfrak p$ of $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $S_{\\mathfrak p} = R_{\\mathfrak p} \\otimes_R S$ (see", "Lemma \\ref{lemma-tensor-localization})", "we see that (1) implies (2) by", "Lemma \\ref{lemma-base-change-epimorphism}.", "Conversely, assume that (2) holds. Let $a, b : S \\to A$ be two ring maps", "from $S$ to a ring $A$ equalizing the map $R \\to S$. By assumption we see", "that for every prime $\\mathfrak p$ of $R$ the induced maps", "$a_{\\mathfrak p}, b_{\\mathfrak p} : S_{\\mathfrak p} \\to A_{\\mathfrak p}$ are", "the same. Hence $a = b$ as $A \\subset \\prod_{\\mathfrak p} A_{\\mathfrak p}$, see", "Lemma \\ref{lemma-characterize-zero-local}." ], "refs": [ "algebra-lemma-tensor-localization", "algebra-lemma-base-change-epimorphism", "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 366, 949, 410 ] } ], "ref_ids": [] }, { "id": 952, "type": "theorem", "label": "algebra-lemma-finite-epimorphism-surjective", "categories": [ "algebra" ], "title": "algebra-lemma-finite-epimorphism-surjective", "contents": [ "\\begin{slogan}", "A ring map is surjective if and only if it is a finite epimorphism.", "\\end{slogan}", "Let $R \\to S$ be a ring map. The following are equivalent", "\\begin{enumerate}", "\\item $R \\to S$ is an epimorphism and finite, and", "\\item $R \\to S$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "(This lemma seems to have been reproved many times in the literature, and", "has many different proofs.)", "It is clear that a surjective ring map is an epimorphism.", "Suppose that $R \\to S$ is a finite ring map such that", "$S \\otimes_R S \\to S$ is an isomorphism. Our goal is to show that", "$R \\to S$ is surjective. Assume $S/R$ is not zero.", "The exact sequence $R \\to S \\to S/R \\to 0$", "leads to an exact sequence", "$$", "R \\otimes_R S \\to S \\otimes_R S \\to S/R \\otimes_R S \\to 0.", "$$", "Our assumption implies that the first arrow is an isomorphism, hence", "we conclude that $S/R \\otimes_R S = 0$. Hence also $S/R \\otimes_R S/R = 0$. By", "Lemma \\ref{lemma-trivial-filter-finite-module}", "there exists a surjection of $R$-modules $S/R \\to R/I$ for some proper", "ideal $I \\subset R$. Hence there exists a", "surjection $S/R \\otimes_R S/R \\to R/I \\otimes_R R/I = R/I \\not = 0$,", "contradiction." ], "refs": [ "algebra-lemma-trivial-filter-finite-module" ], "ref_ids": [ 331 ] } ], "ref_ids": [] }, { "id": 953, "type": "theorem", "label": "algebra-lemma-faithfully-flat-epimorphism", "categories": [ "algebra" ], "title": "algebra-lemma-faithfully-flat-epimorphism", "contents": [ "A faithfully flat epimorphism is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is clear from", "Lemma \\ref{lemma-epimorphism} part (3)", "as the map $S \\to S \\otimes_R S$ is the map $R \\to S$ tensored with $S$." ], "refs": [ "algebra-lemma-epimorphism" ], "ref_ids": [ 947 ] } ], "ref_ids": [] }, { "id": 954, "type": "theorem", "label": "algebra-lemma-epimorphism-over-field", "categories": [ "algebra" ], "title": "algebra-lemma-epimorphism-over-field", "contents": [ "If $k \\to S$ is an epimorphism and $k$ is a field, then $S = k$ or $S = 0$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the result of", "Lemma \\ref{lemma-faithfully-flat-epimorphism}", "(as any nonzero algebra over $k$ is faithfully flat), or", "by arguing directly that $R \\to R \\otimes_k R$ cannot be", "surjective unless $\\dim_k(R) \\leq 1$." ], "refs": [ "algebra-lemma-faithfully-flat-epimorphism" ], "ref_ids": [ 953 ] } ], "ref_ids": [] }, { "id": 955, "type": "theorem", "label": "algebra-lemma-epimorphism-injective-spec", "categories": [ "algebra" ], "title": "algebra-lemma-epimorphism-injective-spec", "contents": [ "Let $R \\to S$ be an epimorphism of rings. Then", "\\begin{enumerate}", "\\item $\\Spec(S) \\to \\Spec(R)$ is injective, and", "\\item for $\\mathfrak q \\subset S$ lying over $\\mathfrak p \\subset R$", "we have $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak q)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p$ be a prime of $R$. The fibre of the map is the spectrum", "of the fibre ring $S \\otimes_R \\kappa(\\mathfrak p)$. By", "Lemma \\ref{lemma-base-change-epimorphism}", "the map $\\kappa(\\mathfrak p) \\to S \\otimes_R \\kappa(\\mathfrak p)$", "is an epimorphism, and hence by", "Lemma \\ref{lemma-epimorphism-over-field}", "we have either $S \\otimes_R \\kappa(\\mathfrak p) = 0$", "or $S \\otimes_R \\kappa(\\mathfrak p) = \\kappa(\\mathfrak p)$", "which proves (1) and (2)." ], "refs": [ "algebra-lemma-base-change-epimorphism", "algebra-lemma-epimorphism-over-field" ], "ref_ids": [ 949, 954 ] } ], "ref_ids": [] }, { "id": 956, "type": "theorem", "label": "algebra-lemma-relations", "categories": [ "algebra" ], "title": "algebra-lemma-relations", "contents": [ "Let $R$ be a ring.", "Let $M$, $N$ be $R$-modules.", "Let $\\{x_i\\}_{i \\in I}$ be a set of generators of $M$.", "Let $\\{y_j\\}_{j \\in J}$ be a set of generators of $N$.", "Let $\\{m_j\\}_{j \\in J}$ be a family of elements of $M$ with $m_j = 0$", "for all but finitely many $j$.", "Then", "$$", "\\sum\\nolimits_{j \\in J} m_j \\otimes y_j = 0 \\text{ in } M \\otimes_R N", "$$", "is equivalent to the following:", "There exist $a_{i, j} \\in R$ with $a_{i, j} = 0$ for all but finitely many", "pairs $(i, j)$ such that", "\\begin{align*}", "m_j & = \\sum\\nolimits_{i \\in I} a_{i, j} x_i \\quad\\text{for all } j \\in J, \\\\", "0 & = \\sum\\nolimits_{j \\in J} a_{i, j} y_j \\quad\\text{for all } i \\in I.", "\\end{align*}" ], "refs": [], "proofs": [ { "contents": [ "The sufficiency is immediate. Suppose that", "$\\sum_{j \\in J} m_j \\otimes y_j = 0$.", "Consider the short exact sequence", "$$", "0 \\to K \\to \\bigoplus\\nolimits_{j \\in J} R \\to N \\to 0", "$$", "where the $j$th basis vector of $\\bigoplus\\nolimits_{j \\in J} R$ maps", "to $y_j$. Tensor this with $M$ to get the exact sequence", "$$", "K \\otimes_R M \\to \\bigoplus\\nolimits_{j \\in J} M \\to N \\otimes_R M \\to 0.", "$$", "The assumption implies that there exist elements $k_i \\in K$ such that", "$\\sum k_i \\otimes x_i$ maps to the element $(m_j)_{j \\in J}$ of the middle.", "Writing $k_i = (a_{i, j})_{j \\in J}$ and we obtain what we want." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 957, "type": "theorem", "label": "algebra-lemma-kernel-difference-projections", "categories": [ "algebra" ], "title": "algebra-lemma-kernel-difference-projections", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "Let $g \\in S$. The following are equivalent:", "\\begin{enumerate}", "\\item $g \\otimes 1 = 1 \\otimes g$ in $S \\otimes_R S$, and", "\\item there exist $n \\geq 0$ and elements $y_i, z_j \\in S$", "and $x_{i, j} \\in R$ for $1 \\leq i, j \\leq n$ such that", "\\begin{enumerate}", "\\item $g = \\sum_{i, j \\leq n} x_{i, j} y_i z_j$,", "\\item for each $j$ we have $\\sum x_{i, j}y_i \\in \\varphi(R)$, and", "\\item for each $i$ we have $\\sum x_{i, j}z_j \\in \\varphi(R)$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (2) implies (1). Conversely, suppose that", "$g \\otimes 1 = 1 \\otimes g$. Choose generators $\\{s_i\\}_{i \\in I}$", "of $S$ as an $R$-module with $0, 1 \\in I$ and $s_0 = 1$ and $s_1 = g$.", "Apply", "Lemma \\ref{lemma-relations}", "to the relation $g \\otimes s_0 + (-1) \\otimes s_1 = 0$.", "We see that there exist $a_{i, j} \\in R$ such that", "$g = \\sum_i a_{i, 0} s_i$, $-1 = \\sum_i a_{i, 1} s_i$, and for", "$j \\not = 0, 1$ we have $0 = \\sum_i a_{i, j} s_i$, and moreover", "for all $i$ we have $\\sum_j a_{i, j}s_j = 0$.", "Then we have", "$$", "\\sum\\nolimits_{i, j \\not = 0} a_{i, j} s_i s_j = -g + a_{0, 0}", "$$", "and for each $j \\not = 0$ we have", "$\\sum_{i \\not = 0} a_{i, j}s_i \\in R$. This proves that $-g + a_{0, 0}$", "can be written as in (2). It follows that $g$ can be written as", "in (2). Details omitted.", "Hint: Show that the set of elements of $S$ which have an", "expression as in (2) form an $R$-subalgebra of $S$." ], "refs": [ "algebra-lemma-relations" ], "ref_ids": [ 956 ] } ], "ref_ids": [] }, { "id": 958, "type": "theorem", "label": "algebra-lemma-epimorphism-cardinality", "categories": [ "algebra" ], "title": "algebra-lemma-epimorphism-cardinality", "contents": [ "Let $R \\to S$ be an epimorphism of rings.", "Then the cardinality of $S$ is at most the cardinality of $R$.", "In a formula: $|S| \\leq |R|$." ], "refs": [], "proofs": [ { "contents": [ "The condition that $R \\to S$ is an epimorphism means that each $g \\in S$", "satisfies $g \\otimes 1 = 1 \\otimes g$, see", "Lemma \\ref{lemma-epimorphism}.", "We are going to use the notation introduced in", "Remark \\ref{remark-matrices-associated-to-elements-epicenter}.", "Suppose that $g, g' \\in S$ and suppose that $(P, U, V)$ is an $n$-triple", "which is associated to both $g$ and $g'$. Then we claim that", "$g = g'$. Namely, write $(P, U, V) = (X, YX, XZ)$ for a matrix", "factorization $(g) = YXZ$ of $g$ and write $(P, U, V) = (X', Y'X', X'Z')$", "for a matrix factorization $(g') = Y'X'Z'$ of $g'$.", "Then we see that", "$$", "(g) = YXZ = UZ = Y'X'Z = Y'PZ = Y'XZ = Y'V = Y'X'Z' = (g')", "$$", "and hence $g = g'$. This implies that the cardinality of $S$ is bounded", "by the number of possible triples, which has cardinality at most", "$\\sup_{n \\in \\mathbf{N}} |R|^n$. If $R$ is infinite then this is", "at most $|R|$, see \\cite[Ch. I, 10.13]{Kunen}.", "\\medskip\\noindent", "If $R$ is a finite ring then the argument above only proves that $S$", "is at worst countable. In fact in this case $R$ is Artinian and the", "map $R \\to S$ is surjective. We omit the proof of this case." ], "refs": [ "algebra-lemma-epimorphism", "algebra-remark-matrices-associated-to-elements-epicenter" ], "ref_ids": [ 947, 1575 ] } ], "ref_ids": [] }, { "id": 959, "type": "theorem", "label": "algebra-lemma-epimorphism-modules", "categories": [ "algebra" ], "title": "algebra-lemma-epimorphism-modules", "contents": [ "Let $R \\to S$ be an epimorphism of rings. Let $N_1, N_2$ be $S$-modules.", "Then $\\Hom_S(N_1, N_2) = \\Hom_R(N_1, N_2)$. In other words, the", "restriction functor $\\text{Mod}_S \\to \\text{Mod}_R$ is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi : N_1 \\to N_2$ be an $R$-linear map. For any $x \\in N_1$", "consider the map $S \\otimes_R S \\to N_2$ defined by the rule", "$g \\otimes g' \\mapsto g\\varphi(g'x)$. Since both maps $S \\to S \\otimes_R S$", "are isomorphisms (Lemma \\ref{lemma-epimorphism}), we conclude that", "$g \\varphi(g'x) = gg'\\varphi(x) = \\varphi(gg' x)$. Thus $\\varphi$", "is $S$-linear." ], "refs": [ "algebra-lemma-epimorphism" ], "ref_ids": [ 947 ] } ], "ref_ids": [] }, { "id": 960, "type": "theorem", "label": "algebra-lemma-pure", "categories": [ "algebra" ], "title": "algebra-lemma-pure", "contents": [ "Let $R$ be a ring.", "Let $I \\subset R$ be an ideal.", "The following are equivalent:", "\\begin{enumerate}", "\\item $I$ is pure,", "\\item for every ideal $J \\subset R$ we have $J \\cap I = IJ$,", "\\item for every finitely generated ideal $J \\subset R$ we have", "$J \\cap I = JI$,", "\\item for every $x \\in R$ we have $(x) \\cap I = xI$,", "\\item for every $x \\in I$ we have $x = yx$ for some $y \\in I$,", "\\item for every $x_1, \\ldots, x_n \\in I$ there exists a", "$y \\in I$ such that $x_i = yx_i$ for all $i = 1, \\ldots, n$,", "\\item for every prime $\\mathfrak p$ of $R$ we have", "$IR_{\\mathfrak p} = 0$ or $IR_{\\mathfrak p} = R_{\\mathfrak p}$,", "\\item $\\text{Supp}(I) = \\Spec(R) \\setminus V(I)$,", "\\item $I$ is the kernel of the map $R \\to (1 + I)^{-1}R$,", "\\item $R/I \\cong S^{-1}R$ as $R$-algebras for some multiplicative", "subset $S$ of $R$, and", "\\item $R/I \\cong (1 + I)^{-1}R$ as $R$-algebras.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For any ideal $J$ of $R$ we have the short exact sequence", "$0 \\to J \\to R \\to R/J \\to 0$. Tensoring with $R/I$ we get", "an exact sequence $J \\otimes_R R/I \\to R/I \\to R/I + J \\to 0$", "and $J \\otimes_R R/I = J/JI$. Thus the", "equivalence of (1), (2), and (3) follows from", "Lemma \\ref{lemma-flat}. Moreover, these imply (4).", "\\medskip\\noindent", "The implication (4) $\\Rightarrow$ (5) is trivial.", "Assume (5) and let $x_1, \\ldots, x_n \\in I$.", "Choose $y_i \\in I$ such that $x_i = y_ix_i$.", "Let $y \\in I$ be the element such that", "$1 - y = \\prod_{i = 1, \\ldots, n} (1 - y_i)$.", "Then $x_i = yx_i$ for all $i = 1, \\ldots, n$.", "Hence (6) holds, and it follows that (5) $\\Leftrightarrow$ (6).", "\\medskip\\noindent", "Assume (5). Let $x \\in I$. Then $x = yx$ for some $y \\in I$.", "Hence $x(1 - y) = 0$, which shows that $x$ maps to zero in", "$(1 + I)^{-1}R$. Of course the kernel of the map", "$R \\to (1 + I)^{-1}R$ is always contained in $I$. Hence we", "see that (5) implies (9). Assume (9). Then for any $x \\in I$", "we see that $x(1 - y) = 0$ for some $y \\in I$.", "In other words, $x = yx$. We conclude that (5) is equivalent to (9).", "\\medskip\\noindent", "Assume (5). Let $\\mathfrak p$ be a prime of $R$.", "If $\\mathfrak p \\not \\in V(I)$, then $IR_{\\mathfrak p} = R_{\\mathfrak p}$.", "If $\\mathfrak p \\in V(I)$, in other words, if $I \\subset \\mathfrak p$,", "then $x \\in I$ implies $x(1 - y) = 0$ for some $y \\in I$, implies", "$x$ maps to zero in $R_{\\mathfrak p}$, i.e., $IR_{\\mathfrak p} = 0$.", "Thus we see that (7) holds.", "\\medskip\\noindent", "Assume (7). Then $(R/I)_{\\mathfrak p}$ is either $0$ or $R_{\\mathfrak p}$", "for any prime $\\mathfrak p$ of $R$. Hence by", "Lemma \\ref{lemma-flat-localization}", "we see that (1) holds. At this point we see that all of", "(1) -- (7) and (9) are equivalent.", "\\medskip\\noindent", "As $IR_{\\mathfrak p} = I_{\\mathfrak p}$ we see that (7) implies (8).", "Finally, if (8) holds, then this means exactly that $I_{\\mathfrak p}$", "is the zero module if and only if $\\mathfrak p \\in V(I)$, which", "is clearly saying that (7) holds. Now (1) -- (9) are equivalent.", "\\medskip\\noindent", "Assume (1) -- (9) hold. Then $R/I \\subset (1 + I)^{-1}R$ by (9) and", "the map $R/I \\to (1 + I)^{-1}R$ is also surjective by the description", "of localizations at primes afforded by (7). Hence (11) holds.", "\\medskip\\noindent", "The implication (11) $\\Rightarrow$ (10) is trivial.", "And (10) implies that (1) holds because a localization of", "$R$ is flat over $R$, see", "Lemma \\ref{lemma-flat-localization}." ], "refs": [ "algebra-lemma-flat", "algebra-lemma-flat-localization", "algebra-lemma-flat-localization" ], "ref_ids": [ 525, 538, 538 ] } ], "ref_ids": [] }, { "id": 961, "type": "theorem", "label": "algebra-lemma-pure-ideal-determined-by-zero-set", "categories": [ "algebra" ], "title": "algebra-lemma-pure-ideal-determined-by-zero-set", "contents": [ "\\begin{slogan}", "Pure ideals are determined by their vanishing locus.", "\\end{slogan}", "Let $R$ be a ring.", "If $I, J \\subset R$ are pure ideals, then $V(I) = V(J)$", "implies $I = J$." ], "refs": [], "proofs": [ { "contents": [ "For example, by property (7) of", "Lemma \\ref{lemma-pure}", "we see that", "$I = \\Ker(R \\to \\prod_{\\mathfrak p \\in V(I)} R_{\\mathfrak p})$", "can be recovered from the closed subset associated to it." ], "refs": [ "algebra-lemma-pure" ], "ref_ids": [ 960 ] } ], "ref_ids": [] }, { "id": 962, "type": "theorem", "label": "algebra-lemma-pure-open-closed-specializations", "categories": [ "algebra" ], "title": "algebra-lemma-pure-open-closed-specializations", "contents": [ "Let $R$ be a ring. The rule", "$I \\mapsto V(I)$", "determines a bijection", "$$", "\\{I \\subset R \\text{ pure}\\}", "\\leftrightarrow", "\\{Z \\subset \\Spec(R)\\text{ closed and closed under generalizations}\\}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $I$ be a pure ideal. Then since $R \\to R/I$ is flat, by going up", "generalizations lift along the map $\\Spec(R/I) \\to \\Spec(R)$.", "Hence $V(I)$ is closed under generalizations. This shows that the map", "is well defined. By", "Lemma \\ref{lemma-pure-ideal-determined-by-zero-set}", "the map is injective. Suppose that", "$Z \\subset \\Spec(R)$ is closed and closed under generalizations.", "Let $J \\subset R$ be the radical ideal such that $Z = V(J)$.", "Let $I = \\{x \\in R : x \\in xJ\\}$. Note that $I$ is an ideal:", "if $x, y \\in I$ then there exist $f, g \\in J$ such that", "$x = xf$ and $y = yg$. Then", "$$", "x + y = (x + y)(f + g - fg)", "$$", "Verification left to the reader.", "We claim that $I$ is pure and that $V(I) = V(J)$.", "If the claim is true then the map of the lemma is surjective and", "the lemma holds.", "\\medskip\\noindent", "Note that $I \\subset J$, so that $V(J) \\subset V(I)$.", "Let $I \\subset \\mathfrak p$ be a prime. Consider the multiplicative", "subset $S = (R \\setminus \\mathfrak p)(1 + J)$. By definition of", "$I$ and $I \\subset \\mathfrak p$ we see that $0 \\not \\in S$.", "Hence we can find a prime $\\mathfrak q$ of $R$ which is disjoint", "from $S$, see", "Lemmas \\ref{lemma-localization-zero} and", "\\ref{lemma-spec-localization}.", "Hence $\\mathfrak q \\subset \\mathfrak p$ and", "$\\mathfrak q \\cap (1 + J) = \\emptyset$.", "This implies that $\\mathfrak q + J$ is a proper ideal of $R$.", "Let $\\mathfrak m$ be a maximal ideal containing $\\mathfrak q + J$.", "Then we get", "$\\mathfrak m \\in V(J)$ and hence $\\mathfrak q \\in V(J) = Z$", "as $Z$ was assumed to be closed under generalization.", "This in turn implies $\\mathfrak p \\in V(J)$ as", "$\\mathfrak q \\subset \\mathfrak p$. Thus we see that $V(I) = V(J)$.", "\\medskip\\noindent", "Finally, since $V(I) = V(J)$ (and $J$ radical) we see that $J = \\sqrt{I}$.", "Pick $x \\in I$, so that $x = xy$ for some $y \\in J$ by definition.", "Then $x = xy = xy^2 = \\ldots = xy^n$. Since $y^n \\in I$ for some $n > 0$", "we conclude that property (5) of", "Lemma \\ref{lemma-pure}", "holds and we see that $I$ is indeed pure." ], "refs": [ "algebra-lemma-pure-ideal-determined-by-zero-set", "algebra-lemma-localization-zero", "algebra-lemma-spec-localization", "algebra-lemma-pure" ], "ref_ids": [ 961, 345, 391, 960 ] } ], "ref_ids": [] }, { "id": 963, "type": "theorem", "label": "algebra-lemma-finitely-generated-pure-ideal", "categories": [ "algebra" ], "title": "algebra-lemma-finitely-generated-pure-ideal", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal.", "The following are equivalent", "\\begin{enumerate}", "\\item $I$ is pure and finitely generated,", "\\item $I$ is generated by an idempotent,", "\\item $I$ is pure and $V(I)$ is open, and", "\\item $R/I$ is a projective $R$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (1) holds, then $I = I \\cap I = I^2$ by", "Lemma \\ref{lemma-pure}.", "Hence $I$ is generated by an idempotent by", "Lemma \\ref{lemma-ideal-is-squared-union-connected}.", "Thus (1) $\\Rightarrow$ (2).", "If (2) holds, then $I = (e)$ and $R = (1 - e) \\oplus (e)$ as", "an $R$-module hence $R/I$ is flat and $I$ is pure and $V(I) = D(1 - e)$", "is open. Thus (2) $\\Rightarrow$ (1) $+$ (3).", "Finally, assume (3). Then $V(I)$ is open and closed, hence", "$V(I) = D(1 - e)$ for some idempotent $e$ of $R$, see", "Lemma \\ref{lemma-disjoint-decomposition}.", "The ideal $J = (e)$ is a pure ideal such that $V(J) = V(I)$ hence", "$I = J$ by", "Lemma \\ref{lemma-pure-ideal-determined-by-zero-set}.", "In this way we see that (3) $\\Rightarrow$ (2). By", "Lemma \\ref{lemma-finite-projective}", "we see that (4) is equivalent to the assertion that $I$ is pure", "and $R/I$ finitely presented. Moreover, $R/I$ is finitely presented", "if and only if $I$ is finitely generated, see Lemma \\ref{lemma-extension}.", "Hence (4) is equivalent to (1)." ], "refs": [ "algebra-lemma-pure", "algebra-lemma-ideal-is-squared-union-connected", "algebra-lemma-disjoint-decomposition", "algebra-lemma-pure-ideal-determined-by-zero-set", "algebra-lemma-finite-projective", "algebra-lemma-extension" ], "ref_ids": [ 960, 407, 405, 961, 795, 330 ] } ], "ref_ids": [] }, { "id": 964, "type": "theorem", "label": "algebra-lemma-finite-flat-module-finitely-presented", "categories": [ "algebra" ], "title": "algebra-lemma-finite-flat-module-finitely-presented", "contents": [ "Let $R$ be a ring. The following are equivalent:", "\\begin{enumerate}", "\\item every $Z \\subset \\Spec(R)$ which is closed and closed under", "generalizations is also open, and", "\\item any finite flat $R$-module is finite locally free.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If any finite flat $R$-module is finite locally free then the support", "of $R/I$ where $I$ is a pure ideal is open. Hence the implication", "(2) $\\Rightarrow$ (1) follows from", "Lemma \\ref{lemma-pure-ideal-determined-by-zero-set}.", "\\medskip\\noindent", "For the converse assume that $R$ satisfies (1).", "Let $M$ be a finite flat $R$-module.", "The support $Z = \\text{Supp}(M)$ of $M$ is closed, see", "Lemma \\ref{lemma-support-closed}.", "On the other hand, if $\\mathfrak p \\subset \\mathfrak p'$, then by", "Lemma \\ref{lemma-finite-flat-local}", "the module $M_{\\mathfrak p'}$ is free, and", "$M_{\\mathfrak p} = M_{\\mathfrak p'} \\otimes_{R_{\\mathfrak p'}} R_{\\mathfrak p}$", "Hence", "$\\mathfrak p' \\in \\text{Supp}(M) \\Rightarrow \\mathfrak p \\in \\text{Supp}(M)$,", "in other words, the support is closed under generalization.", "As $R$ satisfies (1) we see that the support of $M$ is open and closed.", "Suppose that $M$ is generated by $r$ elements $m_1, \\ldots, m_r$.", "The modules $\\wedge^i(M)$, $i = 1, \\ldots, r$ are finite flat $R$-modules", "also, because $\\wedge^i(M)_{\\mathfrak p} = \\wedge^i(M_{\\mathfrak p})$", "is free over $R_{\\mathfrak p}$. Note that", "$\\text{Supp}(\\wedge^{i + 1}(M)) \\subset \\text{Supp}(\\wedge^i(M))$.", "Thus we see that there exists a decomposition", "$$", "\\Spec(R) = U_0 \\amalg U_1 \\amalg \\ldots \\amalg U_r", "$$", "by open and closed subsets such that the support of", "$\\wedge^i(M)$ is $U_r \\cup \\ldots \\cup U_i$ for all $i = 0, \\ldots, r$.", "Let $\\mathfrak p$ be a prime of $R$, and say $\\mathfrak p \\in U_i$.", "Note that", "$\\wedge^i(M) \\otimes_R \\kappa(\\mathfrak p) =", "\\wedge^i(M \\otimes_R \\kappa(\\mathfrak p))$.", "Hence, after possibly renumbering $m_1, \\ldots, m_r$ we may assume that", "$m_1, \\ldots, m_i$ generate $M \\otimes_R \\kappa(\\mathfrak p)$. By", "Nakayama's Lemma \\ref{lemma-NAK}", "we get a surjection", "$$", "R_f^{\\oplus i} \\longrightarrow M_f, \\quad", "(a_1, \\ldots, a_i) \\longmapsto \\sum a_im_i", "$$", "for some $f \\in R$, $f \\not \\in \\mathfrak p$. We may also assume that", "$D(f) \\subset U_i$. This means that $\\wedge^i(M_f) = \\wedge^i(M)_f$", "is a flat $R_f$ module whose support is all of $\\Spec(R_f)$.", "By the above it is generated by a single element, namely", "$m_1 \\wedge \\ldots \\wedge m_i$. Hence $\\wedge^i(M)_f \\cong R_f/J$", "for some pure ideal $J \\subset R_f$ with $V(J) = \\Spec(R_f)$.", "Clearly this means that $J = (0)$, see", "Lemma \\ref{lemma-pure-ideal-determined-by-zero-set}.", "Thus $m_1 \\wedge \\ldots \\wedge m_i$ is a basis for", "$\\wedge^i(M_f)$ and it follows that the displayed map is injective", "as well as surjective. This proves that $M$ is finite locally free", "as desired." ], "refs": [ "algebra-lemma-pure-ideal-determined-by-zero-set", "algebra-lemma-support-closed", "algebra-lemma-finite-flat-local", "algebra-lemma-NAK", "algebra-lemma-pure-ideal-determined-by-zero-set" ], "ref_ids": [ 961, 543, 797, 401, 961 ] } ], "ref_ids": [] }, { "id": 965, "type": "theorem", "label": "algebra-lemma-Schanuel", "categories": [ "algebra" ], "title": "algebra-lemma-Schanuel", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Suppose that", "$$", "0 \\to K \\xrightarrow{c_1} P_1 \\xrightarrow{p_1} M \\to 0", "\\quad\\text{and}\\quad", "0 \\to L \\xrightarrow{c_2} P_2 \\xrightarrow{p_2} M \\to 0", "$$", "are two short exact sequences, with $P_i$ projective.", "Then $K \\oplus P_2 \\cong L \\oplus P_1$. More precisely,", "there exist a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "K \\oplus P_2 \\ar[r]_{(c_1, \\text{id})} \\ar[d] &", "P_1 \\oplus P_2 \\ar[r]_{(p_1, 0)} \\ar[d] &", "M \\ar[r] \\ar@{=}[d] &", "0 \\\\", "0 \\ar[r] &", "P_1 \\oplus L \\ar[r]^{(\\text{id}, c_2)} &", "P_1 \\oplus P_2 \\ar[r]^{(0, p_2)} &", "M \\ar[r] &", "0", "}", "$$", "whose vertical arrows are isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "Consider the module $N$ defined by the short exact sequence", "$0 \\to N \\to P_1 \\oplus P_2 \\to M \\to 0$,", "where the last map is the sum of the two maps", "$P_i \\to M$. It is easy to see that the projection", "$N \\to P_1$ is surjective with kernel $L$, and that", "$N \\to P_2$ is surjective with kernel $K$.", "Since $P_i$ are projective we have $N \\cong K \\oplus P_2", "\\cong L \\oplus P_1$. This proves the first statement.", "\\medskip\\noindent", "To prove the second statement (and to reprove the first), choose", "$a : P_1 \\to P_2$ and $b : P_2 \\to P_1$ such that", "$p_1 = p_2 \\circ a$ and $p_2 = p_1 \\circ b$. This is possible", "because $P_1$ and $P_2$ are projective. Then we get a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "K \\oplus P_2 \\ar[r]_{(c_1, \\text{id})} &", "P_1 \\oplus P_2 \\ar[r]_{(p_1, 0)} &", "M \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "N \\ar[r] \\ar[d] \\ar[u] &", "P_1 \\oplus P_2 \\ar[r]_{(p_1, p_2)}", "\\ar[d]_S \\ar[u]^T &", "M \\ar[r] \\ar@{=}[d] \\ar@{=}[u] &", "0 \\\\", "0 \\ar[r] &", "P_1 \\oplus L \\ar[r]^{(\\text{id}, c_2)} &", "P_1 \\oplus P_2 \\ar[r]^{(0, p_2)} &", "M \\ar[r] &", "0", "}", "$$", "with $T$ and $S$ given by the matrices", "$$", "S = \\left(", "\\begin{matrix}", "\\text{id} & 0 \\\\", "a & \\text{id}", "\\end{matrix}", "\\right)", "\\quad\\text{and}\\quad", "T = \\left(", "\\begin{matrix}", "\\text{id} & b \\\\", "0 & \\text{id}", "\\end{matrix}", "\\right)", "$$", "Then $S$, $T$ and the maps $N \\to P_1 \\oplus L$", "and $N \\to K \\oplus P_2$ are isomorphisms as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 966, "type": "theorem", "label": "algebra-lemma-independent-resolution", "categories": [ "algebra" ], "title": "algebra-lemma-independent-resolution", "contents": [ "Let $R$ be a ring. Suppose that $M$ is an $R$-module of projective", "dimension $d$. Suppose that $F_e \\to F_{e-1} \\to \\ldots \\to F_0 \\to M \\to 0$", "is exact with $F_i$ projective and $e \\geq d - 1$.", "Then the kernel of $F_e \\to F_{e-1}$ is projective", "(or the kernel of $F_0 \\to M$ is projective in case", "$e = 0$)." ], "refs": [], "proofs": [ { "contents": [ "We prove this by induction on $d$. If $d = 0$, then", "$M$ is projective. In this case there is a splitting", "$F_0 = \\Ker(F_0 \\to M) \\oplus M$, and hence", "$\\Ker(F_0 \\to M)$ is projective. This finishes", "the proof if $e = 0$, and if $e > 0$, then replacing", "$M$ by $\\Ker(F_0 \\to M)$ we decrease $e$.", "\\medskip\\noindent", "Next assume $d > 0$.", "Let $0 \\to P_d \\to P_{d-1} \\to \\ldots \\to P_0 \\to M \\to 0$", "be a minimal length finite resolution with $P_i$ projective.", "According to", "Schanuel's Lemma \\ref{lemma-Schanuel}", "we have", "$P_0 \\oplus \\Ker(F_0 \\to M) \\cong F_0 \\oplus \\Ker(P_0 \\to M)$.", "This proves the case $d = 1$, $e = 0$, because then the right", "hand side is $F_0 \\oplus P_1$ which is projective. Hence now we may", "assume $e > 0$. The module $F_0 \\oplus \\Ker(P_0 \\to M)$ has the", "finite projective resolution", "$$", "0 \\to P_d \\to P_{d-1} \\to \\ldots \\to", "P_2 \\to P_1 \\oplus F_0 \\to \\Ker(P_0 \\to M) \\oplus F_0 \\to 0", "$$", "of length $d - 1$. By induction applied to the exact sequence", "$$", "F_e \\to F_{e-1} \\to \\ldots \\to F_2 \\to P_0 \\oplus F_1 \\to", "P_0 \\oplus \\Ker(F_0 \\to M) \\to 0", "$$", "of length $e - 1$ we conclude $\\Ker(F_e \\to F_{e - 1})$", "is projective (if $e \\geq 2$)", "or that $\\Ker(F_1 \\oplus P_0 \\to F_0 \\oplus P_0)$ is projective.", "This implies the lemma." ], "refs": [ "algebra-lemma-Schanuel" ], "ref_ids": [ 965 ] } ], "ref_ids": [] }, { "id": 967, "type": "theorem", "label": "algebra-lemma-what-kind-of-resolutions", "categories": [ "algebra" ], "title": "algebra-lemma-what-kind-of-resolutions", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module. Let $d \\geq 0$.", "The following are equivalent", "\\begin{enumerate}", "\\item $M$ has projective dimension $\\leq d$,", "\\item there exists a resolution", "$0 \\to P_d \\to P_{d - 1} \\to \\ldots \\to P_0 \\to M \\to 0$", "with $P_i$ projective,", "\\item for some resolution", "$\\ldots \\to P_2 \\to P_1 \\to P_0 \\to M \\to 0$ with", "$P_i$ projective we have $\\Ker(P_{d - 1} \\to P_{d - 2})$", "is projective if $d \\geq 2$, or $\\Ker(P_0 \\to M)$ is projective if", "$d = 1$, or $M$ is projective if $d = 0$,", "\\item for any resolution", "$\\ldots \\to P_2 \\to P_1 \\to P_0 \\to M \\to 0$ with", "$P_i$ projective we have $\\Ker(P_{d - 1} \\to P_{d - 2})$", "is projective if $d \\geq 2$, or $\\Ker(P_0 \\to M)$ is projective if", "$d = 1$, or $M$ is projective if $d = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is the definition of projective", "dimension, see Definition \\ref{definition-finite-proj-dim}.", "We have (2) $\\Rightarrow$ (4) by Lemma \\ref{lemma-independent-resolution}.", "The implications (4) $\\Rightarrow$ (3) and (3) $\\Rightarrow$ (2) are", "immediate." ], "refs": [ "algebra-definition-finite-proj-dim", "algebra-lemma-independent-resolution" ], "ref_ids": [ 1510, 966 ] } ], "ref_ids": [] }, { "id": 968, "type": "theorem", "label": "algebra-lemma-what-kind-of-resolutions-local", "categories": [ "algebra" ], "title": "algebra-lemma-what-kind-of-resolutions-local", "contents": [ "Let $R$ be a local ring. Let $M$ be an $R$-module. Let $d \\geq 0$.", "The equivalent conditions (1) -- (4) of", "Lemma \\ref{lemma-what-kind-of-resolutions}", "are also equivalent to", "\\begin{enumerate}", "\\item[(5)] there exists a resolution", "$0 \\to P_d \\to P_{d - 1} \\to \\ldots \\to P_0 \\to M \\to 0$", "with $P_i$ free.", "\\end{enumerate}" ], "refs": [ "algebra-lemma-what-kind-of-resolutions" ], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-what-kind-of-resolutions} and", "Theorem \\ref{theorem-projective-free-over-local-ring}." ], "refs": [ "algebra-lemma-what-kind-of-resolutions", "algebra-theorem-projective-free-over-local-ring" ], "ref_ids": [ 967, 322 ] } ], "ref_ids": [ 967 ] }, { "id": 969, "type": "theorem", "label": "algebra-lemma-what-kind-of-resolutions-Noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-what-kind-of-resolutions-Noetherian", "contents": [ "Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module.", "Let $d \\geq 0$. The equivalent conditions (1) -- (4) of", "Lemma \\ref{lemma-what-kind-of-resolutions}", "are also equivalent to", "\\begin{enumerate}", "\\item[(6)] there exists a resolution", "$0 \\to P_d \\to P_{d - 1} \\to \\ldots \\to P_0 \\to M \\to 0$", "with $P_i$ finite projective.", "\\end{enumerate}" ], "refs": [ "algebra-lemma-what-kind-of-resolutions" ], "proofs": [ { "contents": [ "Choose a resolution $\\ldots \\to F_2 \\to F_1 \\to F_0 \\to M \\to 0$", "with $F_i$ finite free (Lemma \\ref{lemma-resolution-by-finite-free}).", "By Lemma \\ref{lemma-what-kind-of-resolutions} we see that", "$P_d = \\Ker(F_{d - 1} \\to F_{d - 2})$ is projective at least if $d \\geq 2$.", "Then $P_d$ is a finite $R$-module as $R$ is Noetherian and", "$P_d \\subset F_{d - 1}$ which is finite free.", "Whence $0 \\to P_d \\to F_{d - 1} \\to \\ldots \\to F_1 \\to F_0 \\to M \\to 0$", "is the desired resolution." ], "refs": [ "algebra-lemma-resolution-by-finite-free", "algebra-lemma-what-kind-of-resolutions" ], "ref_ids": [ 761, 967 ] } ], "ref_ids": [ 967 ] }, { "id": 970, "type": "theorem", "label": "algebra-lemma-what-kind-of-resolutions-Noetherian-local", "categories": [ "algebra" ], "title": "algebra-lemma-what-kind-of-resolutions-Noetherian-local", "contents": [ "Let $R$ be a local Noetherian ring. Let $M$ be a finite $R$-module.", "Let $d \\geq 0$. The equivalent conditions (1) -- (4) of", "Lemma \\ref{lemma-what-kind-of-resolutions},", "condition (5) of Lemma \\ref{lemma-what-kind-of-resolutions-local},", "and condition (6) of Lemma \\ref{lemma-what-kind-of-resolutions-Noetherian}", "are also equivalent to", "\\begin{enumerate}", "\\item[(7)] there exists a resolution", "$0 \\to F_d \\to F_{d - 1} \\to \\ldots \\to F_0 \\to M \\to 0$", "with $F_i$ finite free.", "\\end{enumerate}" ], "refs": [ "algebra-lemma-what-kind-of-resolutions", "algebra-lemma-what-kind-of-resolutions-local", "algebra-lemma-what-kind-of-resolutions-Noetherian" ], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-what-kind-of-resolutions},", "\\ref{lemma-what-kind-of-resolutions-local}, and", "\\ref{lemma-what-kind-of-resolutions-Noetherian}", "and because a finite projective module over a local ring", "is finite free, see Lemma \\ref{lemma-finite-projective}." ], "refs": [ "algebra-lemma-what-kind-of-resolutions", "algebra-lemma-what-kind-of-resolutions-local", "algebra-lemma-what-kind-of-resolutions-Noetherian", "algebra-lemma-finite-projective" ], "ref_ids": [ 967, 968, 969, 795 ] } ], "ref_ids": [ 967, 968, 969 ] }, { "id": 971, "type": "theorem", "label": "algebra-lemma-projective-dimension-ext", "categories": [ "algebra" ], "title": "algebra-lemma-projective-dimension-ext", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module. Let $n \\geq 0$.", "The following are equivalent", "\\begin{enumerate}", "\\item $M$ has projective dimension $\\leq n$,", "\\item $\\Ext^i_R(M, N) = 0$ for all $R$-modules $N$ and all", "$i \\geq n + 1$, and", "\\item $\\Ext^{n + 1}_R(M, N) = 0$ for all $R$-modules $N$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Choose a free resolution $F_\\bullet \\to M$ of $M$. Denote", "$d_e : F_e \\to F_{e - 1}$. By", "Lemma \\ref{lemma-independent-resolution}", "we see that $P_e = \\Ker(d_e)$ is projective for $e \\geq n - 1$.", "This implies that $F_e \\cong P_e \\oplus P_{e - 1}$ for $e \\geq n$", "where $d_e$ maps the summand $P_{e - 1}$ isomorphically to $P_{e - 1}$", "in $F_{e - 1}$. Hence, for any $R$-module $N$ the complex", "$\\Hom_R(F_\\bullet, N)$ is split exact in degrees $\\geq n + 1$.", "Whence (2) holds. The implication (2) $\\Rightarrow$ (3) is trivial.", "\\medskip\\noindent", "Assume (3) holds. If $n = 0$ then $M$ is projective by", "Lemma \\ref{lemma-characterize-projective}", "and we see that (1) holds. If $n > 0$ choose a free $R$-module $F$", "and a surjection $F \\to M$ with kernel $K$. By", "Lemma \\ref{lemma-reverse-long-exact-seq-ext}", "and the vanishing of $\\Ext_R^i(F, N)$ for all $i > 0$ by part (1)", "we see that $\\Ext_R^n(K, N) = 0$ for all $R$-modules $N$.", "Hence by induction we see that $K$ has projective dimension $\\leq n - 1$.", "Then $M$ has projective dimension $\\leq n$ as any finite projective", "resolution of $K$ gives a projective resolution of length one more", "for $M$ by adding $F$ to the front." ], "refs": [ "algebra-lemma-independent-resolution", "algebra-lemma-characterize-projective", "algebra-lemma-reverse-long-exact-seq-ext" ], "ref_ids": [ 966, 789, 766 ] } ], "ref_ids": [] }, { "id": 972, "type": "theorem", "label": "algebra-lemma-exact-sequence-projective-dimension", "categories": [ "algebra" ], "title": "algebra-lemma-exact-sequence-projective-dimension", "contents": [ "Let $R$ be a ring. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short", "exact sequence of $R$-modules.", "\\begin{enumerate}", "\\item If $M$ has projective dimension $\\leq n$ and $M''$", "has projective dimension $\\leq n + 1$, then $M'$ has projective", "dimension $\\leq n$.", "\\item If $M'$ and $M''$ have projective dimension", "$\\leq n$ then $M$ has projective dimension $\\leq n$.", "\\item If $M'$ has projective dimension $\\leq n$ and", "$M$ has projective dimension $\\leq n + 1$ then", "$M''$ has projective dimension $\\leq n + 1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Combine the characterization of projective dimension in", "Lemma \\ref{lemma-projective-dimension-ext}", "with the long exact sequence of ext groups in", "Lemma \\ref{lemma-reverse-long-exact-seq-ext}." ], "refs": [ "algebra-lemma-projective-dimension-ext", "algebra-lemma-reverse-long-exact-seq-ext" ], "ref_ids": [ 971, 766 ] } ], "ref_ids": [] }, { "id": 973, "type": "theorem", "label": "algebra-lemma-colimit-projective-dimension", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-projective-dimension", "contents": [ "Let $R$ be a ring. Suppose we have a module $M = \\bigcup_{e \\in E} M_e$", "where the $M_e$ are submodules well-ordered by inclusion. Assume the quotients", "$M_e/\\bigcup\\nolimits_{e' < e} M_{e'}$ have projective dimension $\\leq n$.", "Then $M$ has projective dimension $\\leq n$." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by induction on $n$.", "\\medskip\\noindent", "Base case: $n = 0$. Then $P_e = M_e/\\bigcup_{e' < e} M_{e'}$ is projective.", "Thus we may choose a section $P_e \\to M_e$ of the projection $M_e \\to P_e$.", "We claim that the induced map $\\psi : \\bigoplus_{e \\in E} P_e \\to M$ is an", "isomorphism. Namely, if $x = \\sum x_e \\in \\bigoplus P_e$ is nonzero,", "then we let $e_{max}$ be maximal such that $x_{e_{max}}$ is nonzero", "and we conclude that $y = \\psi(x) = \\psi(\\sum x_e)$ is nonzero because", "$y \\in M_{e_{max}}$ has nonzero image $x_{e_{max}}$ in $P_{e_{max}}$.", "On the other hand, let $y \\in M$. Then $y \\in M_e$ for some $e$.", "We show that $y \\in \\Im(\\psi)$ by transfinite induction on $e$.", "Let $x_e \\in P_e$ be the image of $y$. Then", "$y - \\psi(x_e) \\in \\bigcup_{e' < e} M_{e'}$.", "By induction hypothesis we conclude that $y - \\psi(x_e) \\in \\Im(\\psi)$", "hence $y \\in \\Im(\\psi)$. Thus the claim is true and", "$\\psi$ is an isomorphism. We conclude that $M$ is projective as", "a direct sum of projectives, see", "Lemma \\ref{lemma-direct-sum-projective}.", "\\medskip\\noindent", "If $n > 0$, then for $e \\in E$ we denote $F_e$ the free $R$-module", "on the set of elements of $M_e$. Then we have a system of", "short exact sequences", "$$", "0 \\to K_e \\to F_e \\to M_e \\to 0", "$$", "over the well-ordered set $E$. Note that the transition maps", "$F_{e'} \\to F_e$ and $K_{e'} \\to K_e$ are injective too.", "Set $F = \\bigcup F_e$ and $K = \\bigcup K_e$. Then", "$$", "0 \\to", "K_e/\\bigcup\\nolimits_{e' < e} K_{e'} \\to", "F_e/\\bigcup\\nolimits_{e' < e} F_{e'} \\to", "M_e/\\bigcup\\nolimits_{e' < e} M_{e'} \\to 0", "$$", "is a short exact sequence of $R$-modules too and", "$F_e/\\bigcup_{e' < e} F_{e'}$ is the free $R$-module on the", "set of elements in $M_e$ which are not contained in $\\bigcup_{e' < e} M_{e'}$.", "Hence by", "Lemma \\ref{lemma-exact-sequence-projective-dimension}", "we see that the projective dimension of $K_e/\\bigcup_{e' < e} K_{e'}$", "is at most $n - 1$. By induction we conclude that $K$ has projective", "dimension at most $n - 1$. Whence $M$ has projective dimension at most", "$n$ and we win." ], "refs": [ "algebra-lemma-direct-sum-projective", "algebra-lemma-exact-sequence-projective-dimension" ], "ref_ids": [ 791, 972 ] } ], "ref_ids": [] }, { "id": 974, "type": "theorem", "label": "algebra-lemma-finite-gl-dim", "categories": [ "algebra" ], "title": "algebra-lemma-finite-gl-dim", "contents": [ "Let $R$ be a ring. The following are equivalent", "\\begin{enumerate}", "\\item $R$ has finite global dimension $\\leq n$,", "\\item every finite $R$-module has projective dimension $\\leq n$, and", "\\item every cyclic $R$-module $R/I$ has projective dimension $\\leq n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) $\\Rightarrow$ (2) and (2) $\\Rightarrow$ (3).", "Assume (3). Choose a set $E \\subset M$ of generators of $M$.", "Choose a well ordering on $E$. For $e \\in E$ denote", "$M_e$ the submodule of $M$ generated by the elements $e' \\in E$", "with $e' \\leq e$. Then $M = \\bigcup_{e \\in E} M_e$.", "Note that for each $e \\in E$ the quotient", "$$", "M_e/\\bigcup\\nolimits_{e' < e} M_{e'}", "$$", "is either zero or generated by one element, hence has projective", "dimension $\\leq n$ by (3). By Lemma \\ref{lemma-colimit-projective-dimension}", "this means that $M$ has projective dimension $\\leq n$." ], "refs": [ "algebra-lemma-colimit-projective-dimension" ], "ref_ids": [ 973 ] } ], "ref_ids": [] }, { "id": 975, "type": "theorem", "label": "algebra-lemma-localize-finite-gl-dim", "categories": [ "algebra" ], "title": "algebra-lemma-localize-finite-gl-dim", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Let $S \\subset R$ be a multiplicative subset.", "\\begin{enumerate}", "\\item If $M$ has projective dimension $\\leq n$, then $S^{-1}M$ has", "projective dimension $\\leq n$ over $S^{-1}R$.", "\\item If $R$ has finite global dimension $\\leq n$, then", "$S^{-1}R$ has finite global dimension $\\leq n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $0 \\to P_n \\to P_{n - 1} \\to \\ldots \\to P_0 \\to M \\to 0$", "be a projective resolution. As localization is exact, see", "Proposition \\ref{proposition-localization-exact},", "and as each $S^{-1}P_i$ is a projective $S^{-1}R$-module, see", "Lemma \\ref{lemma-ascend-properties-modules},", "we see that $0 \\to S^{-1}P_n \\to \\ldots \\to S^{-1}P_0 \\to S^{-1}M \\to 0$", "is a projective resolution of $S^{-1}M$. This proves (1).", "Let $M'$ be an $S^{-1}R$-module.", "Note that $M' = S^{-1}M'$.", "Hence we see that (2) follows from (1)." ], "refs": [ "algebra-proposition-localization-exact", "algebra-lemma-ascend-properties-modules" ], "ref_ids": [ 1402, 853 ] } ], "ref_ids": [] }, { "id": 976, "type": "theorem", "label": "algebra-lemma-finite-gl-dim-primes", "categories": [ "algebra" ], "title": "algebra-lemma-finite-gl-dim-primes", "contents": [ "Let $R$ be a Noetherian ring.", "Then $R$ has finite global dimension if and", "only if there exists an integer $n$ such that", "for all maximal ideals $\\mathfrak m$ of $R$", "the ring $R_{\\mathfrak m}$ has global dimension", "$\\leq n$." ], "refs": [], "proofs": [ { "contents": [ "We saw, Lemma \\ref{lemma-localize-finite-gl-dim}", "that if $R$ has finite global dimension $n$,", "then all the localizations $R_{\\mathfrak m}$", "have finite global dimension at most $n$.", "Conversely, suppose that all the $R_{\\mathfrak m}$", "have global dimension $\\leq n$. Let $M$ be a finite", "$R$-module. Let", "$0 \\to K_n \\to F_{n-1} \\to \\ldots \\to F_0 \\to M \\to 0$", "be a resolution with $F_i$ finite free.", "Then $K_n$ is a finite $R$-module.", "According to", "Lemma \\ref{lemma-independent-resolution}", "and the assumption all the modules $K_n \\otimes_R R_{\\mathfrak m}$", "are projective. Hence by", "Lemma \\ref{lemma-finite-projective}", "the module $K_n$ is finite projective." ], "refs": [ "algebra-lemma-localize-finite-gl-dim", "algebra-lemma-independent-resolution", "algebra-lemma-finite-projective" ], "ref_ids": [ 975, 966, 795 ] } ], "ref_ids": [] }, { "id": 977, "type": "theorem", "label": "algebra-lemma-length-resolution-residue-field", "categories": [ "algebra" ], "title": "algebra-lemma-length-resolution-residue-field", "contents": [ "Suppose that $R$ is a Noetherian local ring", "with maximal ideal $\\mathfrak m$ and", "residue field $\\kappa$. In this case", "the projective dimension of $\\kappa$ is", "$\\geq \\dim_\\kappa \\mathfrak m / \\mathfrak m^2$." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1 , \\ldots, x_n$ be elements of $\\mathfrak m$", "whose images in $\\mathfrak m / \\mathfrak m^2$ form a basis.", "Consider the {\\it Koszul complex} on $x_1, \\ldots, x_n$.", "This is the complex", "$$", "0 \\to \\wedge^n R^n \\to \\wedge^{n-1} R^n \\to \\wedge^{n-2} R^n \\to", "\\ldots \\to \\wedge^i R^n \\to \\ldots \\to R^n \\to R", "$$", "with maps given by", "$$", "e_{j_1} \\wedge \\ldots \\wedge e_{j_i}", "\\longmapsto", "\\sum_{a = 1}^i (-1)^{i + 1} x_{j_a} e_{j_1} \\wedge \\ldots", "\\wedge \\hat e_{j_a} \\wedge \\ldots \\wedge e_{j_i}", "$$", "It is easy to see that this is a complex $K_{\\bullet}(R, x_{\\bullet})$.", "Note that the cokernel of the last map of $K_{\\bullet}(R, x_{\\bullet})$", "is $\\kappa$ by Lemma \\ref{lemma-NAK} part (8).", "\\medskip\\noindent", "If $\\kappa$ has finite projective dimension $d$, then we can find", "a resolution $F_{\\bullet} \\to \\kappa$ by finite free $R$-modules", "of length $d$", "(Lemma \\ref{lemma-what-kind-of-resolutions-Noetherian-local}).", "By Lemma \\ref{lemma-add-trivial-complex}", "we may assume all the maps in the complex $F_{\\bullet}$", "have the property that $\\Im(F_i \\to F_{i-1})", "\\subset \\mathfrak m F_{i-1}$, because removing a trivial", "summand from the resolution can at worst shorten the resolution.", "By Lemma \\ref{lemma-compare-resolutions} we can find a map", "of complexes $\\alpha : K_{\\bullet}(R, x_{\\bullet}) \\to F_{\\bullet}$", "inducing the identity on $\\kappa$. We will prove by induction", "that the maps $\\alpha_i : \\wedge^i R^n = K_i(R, x_{\\bullet}) \\to F_i$", "have the property that", "$\\alpha_i \\otimes \\kappa : \\wedge^i \\kappa^n \\to F_i \\otimes \\kappa$", "are injective. This shows that $F_n \\not = 0$ and hence $d \\geq n$", "as desired.", "\\medskip\\noindent", "The result is clear for $i = 0$ because the composition", "$R \\xrightarrow{\\alpha_0} F_0 \\to \\kappa$ is nonzero.", "Note that $F_0$ must have rank $1$ since", "otherwise the map $F_1 \\to F_0$ whose cokernel is a single", "copy of $\\kappa$ cannot have image contained in $\\mathfrak m F_0$.", "\\medskip\\noindent", "Next we check the case $i = 1$ as we feel that it is instructive;", "the reader can skip this as the induction step will deduce the $i = 1$", "case from the case $i = 0$. We saw above that", "$F_0 = R$ and $F_1 \\to F_0 = R$ has image $\\mathfrak m$.", "We have a commutative diagram", "$$", "\\begin{matrix}", "R^n & = & K_1(R, x_{\\bullet}) & \\to & K_0(R, x_{\\bullet}) & = & R \\\\", "& & \\downarrow & & \\downarrow & & \\downarrow \\\\", "& & F_1 & \\to & F_0 & = & R", "\\end{matrix}", "$$", "where the rightmost vertical arrow is given by multiplication", "by a unit. Hence we see that the image of the composition", "$R^n \\to F_1 \\to F_0 = R$ is also equal to $\\mathfrak m$.", "Thus the map $R^n \\otimes \\kappa \\to F_1 \\otimes \\kappa$", "has to be injective since $\\dim_\\kappa (\\mathfrak m / \\mathfrak m^2) = n$.", "\\medskip\\noindent", "Let $i \\geq 1$ and assume injectivity of $\\alpha_j \\otimes \\kappa$ has been", "proved for all $j \\leq i - 1$. Consider the commutative diagram", "$$", "\\begin{matrix}", "\\wedge^i R^n & = & K_i(R, x_{\\bullet}) & \\to & K_{i-1}(R, x_{\\bullet})", "& = & \\wedge^{i-1} R^n \\\\", "& & \\downarrow & & \\downarrow & & \\\\", "& & F_i & \\to & F_{i-1} & &", "\\end{matrix}", "$$", "We know that $\\wedge^{i-1} \\kappa^n \\to F_{i-1} \\otimes \\kappa$", "is injective. This proves that", "$\\wedge^{i-1} \\kappa^n \\otimes_{\\kappa} \\mathfrak m/\\mathfrak m^2", "\\to F_{i-1} \\otimes \\mathfrak m/\\mathfrak m^2$ is injective.", "Also, by our choice of the complex, $F_i$ maps into", "$\\mathfrak mF_{i-1}$, and similarly for the Koszul complex.", "Hence we get a commutative diagram", "$$", "\\begin{matrix}", "\\wedge^i \\kappa^n & \\to &", "\\wedge^{i-1} \\kappa^n \\otimes \\mathfrak m/\\mathfrak m^2 \\\\", "\\downarrow & & \\downarrow \\\\", "F_i \\otimes \\kappa & \\to & F_{i-1} \\otimes \\mathfrak m/\\mathfrak m^2", "\\end{matrix}", "$$", "At this point it suffices to verify the map", "$\\wedge^i \\kappa^n \\to", "\\wedge^{i-1} \\kappa^n \\otimes \\mathfrak m/\\mathfrak m^2$", "is injective, which can be done by hand." ], "refs": [ "algebra-lemma-NAK", "algebra-lemma-what-kind-of-resolutions-Noetherian-local", "algebra-lemma-add-trivial-complex", "algebra-lemma-compare-resolutions" ], "ref_ids": [ 401, 970, 908, 763 ] } ], "ref_ids": [] }, { "id": 978, "type": "theorem", "label": "algebra-lemma-dim-gl-dim", "categories": [ "algebra" ], "title": "algebra-lemma-dim-gl-dim", "contents": [ "Let $R$ be a Noetherian local ring.", "Suppose that the residue field $\\kappa$ has finite", "projective dimension $n$ over $R$.", "In this case $\\dim(R) \\geq n$." ], "refs": [], "proofs": [ { "contents": [ "Let $F_{\\bullet}$ be a finite resolution of $\\kappa$ by finite free", "$R$-modules (Lemma \\ref{lemma-what-kind-of-resolutions-Noetherian-local}).", "By Lemma \\ref{lemma-add-trivial-complex}", "we may assume all the maps in the complex $F_{\\bullet}$", "have to property that $\\Im(F_i \\to F_{i-1})", "\\subset \\mathfrak m F_{i-1}$, because removing a trivial", "summand from the resolution can at worst shorten the resolution.", "Say $F_n \\not = 0$ and $F_i = 0$ for $i > n$, so that", "the projective dimension of $\\kappa$ is $n$.", "By Proposition \\ref{proposition-what-exact} we see that", "$\\text{depth}_{I(\\varphi_n)}(R) \\geq n$ since $I(\\varphi_n)$", "cannot equal $R$ by our choice of the complex.", "Thus by Lemma \\ref{lemma-bound-depth} also $\\dim(R) \\geq n$." ], "refs": [ "algebra-lemma-what-kind-of-resolutions-Noetherian-local", "algebra-lemma-add-trivial-complex", "algebra-proposition-what-exact", "algebra-lemma-bound-depth" ], "ref_ids": [ 970, 908, 1419, 770 ] } ], "ref_ids": [] }, { "id": 979, "type": "theorem", "label": "algebra-lemma-localization-of-regular-local-is-regular", "categories": [ "algebra" ], "title": "algebra-lemma-localization-of-regular-local-is-regular", "contents": [ "A Noetherian local ring $R$ is a regular local ring if and only if", "it has finite global dimension. In this case", "$R_{\\mathfrak p}$ is a regular local ring for all primes $\\mathfrak p$." ], "refs": [], "proofs": [ { "contents": [ "By Propositions \\ref{proposition-finite-gl-dim-regular} and", "\\ref{proposition-regular-finite-gl-dim}", "we see that a Noetherian local ring is a regular local ring if and only if", "it has finite global dimension. Furthermore, any localization", "$R_{\\mathfrak p}$ has finite global dimension,", "see Lemma \\ref{lemma-localize-finite-gl-dim},", "and hence is a regular local ring." ], "refs": [ "algebra-proposition-finite-gl-dim-regular", "algebra-proposition-regular-finite-gl-dim", "algebra-lemma-localize-finite-gl-dim" ], "ref_ids": [ 1422, 1421, 975 ] } ], "ref_ids": [] }, { "id": 980, "type": "theorem", "label": "algebra-lemma-finite-gl-dim-finite-dim-regular", "categories": [ "algebra" ], "title": "algebra-lemma-finite-gl-dim-finite-dim-regular", "contents": [ "Let $R$ be a Noetherian ring.", "The following are equivalent:", "\\begin{enumerate}", "\\item $R$ has finite global dimension $n$,", "\\item there exists an integer $n$ such that", "all the localizations $R_{\\mathfrak m}$ at maximal ideals", "are regular of dimension $\\leq n$ with equality for at least", "one $\\mathfrak m$, and", "\\item there exists an integer $n$ such that", "all the localizations $R_{\\mathfrak p}$ at prime ideals", "are regular of dimension $\\leq n$ with equality for at least", "one $\\mathfrak p$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of Lemma \\ref{lemma-finite-gl-dim-primes}", "in view of the discussion surrounding Definition \\ref{definition-regular}.", "See especially Propositions", "\\ref{proposition-regular-finite-gl-dim} and", "\\ref{proposition-finite-gl-dim-regular}." ], "refs": [ "algebra-lemma-finite-gl-dim-primes", "algebra-definition-regular", "algebra-proposition-regular-finite-gl-dim", "algebra-proposition-finite-gl-dim-regular" ], "ref_ids": [ 976, 1512, 1421, 1422 ] } ], "ref_ids": [] }, { "id": 981, "type": "theorem", "label": "algebra-lemma-flat-under-regular", "categories": [ "algebra" ], "title": "algebra-lemma-flat-under-regular", "contents": [ "Let $R \\to S$ be a local homomorphism of local Noetherian rings.", "Assume that $R \\to S$ is flat and that $S$ is regular.", "Then $R$ is regular." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak m \\subset R$ be the maximal ideal", "and let $\\kappa = R/\\mathfrak m$ be the residue field.", "Let $d = \\dim S$.", "Choose any resolution $F_\\bullet \\to \\kappa$", "with each $F_i$ a finite free $R$-module. Set", "$K_d = \\Ker(F_{d - 1} \\to F_{d - 2})$.", "By flatness of $R \\to S$ the complex", "$0 \\to K_d \\otimes_R S \\to F_{d - 1} \\otimes_R S \\to \\ldots", "\\to F_0 \\otimes_R S \\to \\kappa \\otimes_R S \\to 0$", "is still exact. Because the global dimension of $S$", "is $d$, see Proposition \\ref{proposition-regular-finite-gl-dim},", "we see that $K_d \\otimes_R S$ is a finite free $S$-module", "(see also Lemma \\ref{lemma-independent-resolution}).", "By Lemma \\ref{lemma-finite-projective-descends} we see", "that $K_d$ is a finite free $R$-module.", "Hence $\\kappa$ has finite projective dimension and $R$ is regular by", "Proposition \\ref{proposition-finite-gl-dim-regular}." ], "refs": [ "algebra-proposition-regular-finite-gl-dim", "algebra-lemma-independent-resolution", "algebra-lemma-finite-projective-descends", "algebra-proposition-finite-gl-dim-regular" ], "ref_ids": [ 1421, 966, 798, 1422 ] } ], "ref_ids": [] }, { "id": 982, "type": "theorem", "label": "algebra-lemma-dimension-going-up", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-going-up", "contents": [ "Suppose $R \\to S$ is a ring map satisfying either going up, see", "Definition \\ref{definition-going-up-down}, or going down", "see Definition \\ref{definition-going-up-down}.", "Assume in addition that $\\Spec(S) \\to \\Spec(R)$", "is surjective. Then $\\dim(R) \\leq \\dim(S)$." ], "refs": [ "algebra-definition-going-up-down", "algebra-definition-going-up-down" ], "proofs": [ { "contents": [ "Assume going up.", "Take any chain $\\mathfrak p_0 \\subset \\mathfrak p_1 \\subset \\ldots", "\\subset \\mathfrak p_e$ of prime ideals in $R$.", "By surjectivity we may choose a prime $\\mathfrak q_0$ mapping", "to $\\mathfrak p_0$. By going up we may extend this to a chain", "of length $e$ of primes $\\mathfrak q_i$ lying over", "$\\mathfrak p_i$. Thus $\\dim(S) \\geq \\dim(R)$.", "The case of going down is exactly the same.", "See also Topology, Lemma \\ref{topology-lemma-dimension-specializations-lift}", "for a purely topological version." ], "refs": [ "topology-lemma-dimension-specializations-lift" ], "ref_ids": [ 8289 ] } ], "ref_ids": [ 1459, 1459 ] }, { "id": 983, "type": "theorem", "label": "algebra-lemma-going-up-maximal-on-top", "categories": [ "algebra" ], "title": "algebra-lemma-going-up-maximal-on-top", "contents": [ "Suppose that $R \\to S$ is a ring map with the going up property,", "see Definition \\ref{definition-going-up-down}. If", "$\\mathfrak q \\subset S$ is a maximal ideal.", "Then the inverse image of $\\mathfrak q$ in $R$", "is a maximal ideal too." ], "refs": [ "algebra-definition-going-up-down" ], "proofs": [ { "contents": [ "Trivial." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1459 ] }, { "id": 984, "type": "theorem", "label": "algebra-lemma-integral-dim-up", "categories": [ "algebra" ], "title": "algebra-lemma-integral-dim-up", "contents": [ "Suppose that $R \\to S$ is a ring map such that $S$ is integral over $R$.", "Then $\\dim (R) \\geq \\dim(S)$, and every closed point of $\\Spec(S)$", "maps to a closed point of $\\Spec(R)$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemmas \\ref{lemma-integral-no-inclusion} and", "\\ref{lemma-going-up-maximal-on-top}", "and the definitions." ], "refs": [ "algebra-lemma-integral-no-inclusion", "algebra-lemma-going-up-maximal-on-top" ], "ref_ids": [ 498, 983 ] } ], "ref_ids": [] }, { "id": 985, "type": "theorem", "label": "algebra-lemma-integral-sub-dim-equal", "categories": [ "algebra" ], "title": "algebra-lemma-integral-sub-dim-equal", "contents": [ "Suppose $R \\subset S$ and $S$ integral over $R$.", "Then $\\dim(R) = \\dim(S)$." ], "refs": [], "proofs": [ { "contents": [ "This is a combination of Lemmas", "\\ref{lemma-integral-going-up},", "\\ref{lemma-integral-overring-surjective},", "\\ref{lemma-dimension-going-up}, and", "\\ref{lemma-integral-dim-up}." ], "refs": [ "algebra-lemma-integral-going-up", "algebra-lemma-integral-overring-surjective", "algebra-lemma-dimension-going-up", "algebra-lemma-integral-dim-up" ], "ref_ids": [ 500, 495, 982, 984 ] } ], "ref_ids": [] }, { "id": 986, "type": "theorem", "label": "algebra-lemma-dimension-base-fibre-total", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-base-fibre-total", "contents": [ "Let $R \\to S$ be a homomorphism of Noetherian rings.", "Let $\\mathfrak q \\subset S$ be a prime lying", "over the prime $\\mathfrak p$. Then", "$$", "\\dim(S_{\\mathfrak q})", "\\leq", "\\dim(R_{\\mathfrak p})", "+", "\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "We use the characterization of dimension of", "Proposition \\ref{proposition-dimension}.", "Let $x_1, \\ldots, x_d$ be elements of $\\mathfrak p$", "generating an ideal of definition of $R_{\\mathfrak p}$ with", "$d = \\dim(R_{\\mathfrak p})$.", "Let $y_1, \\ldots, y_e$ be elements of $\\mathfrak q$", "generating an ideal of definition of", "$S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}$", "with $e = \\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q})$.", "It is clear that $S_{\\mathfrak q}/(x_1, \\ldots, x_d, y_1, \\ldots, y_e)$", "has a nilpotent maximal ideal. Hence", "$x_1, \\ldots, x_d, y_1, \\ldots, y_e$ generate an ideal of definition", "of $S_{\\mathfrak q}$." ], "refs": [ "algebra-proposition-dimension" ], "ref_ids": [ 1411 ] } ], "ref_ids": [] }, { "id": 987, "type": "theorem", "label": "algebra-lemma-dimension-base-fibre-equals-total", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-base-fibre-equals-total", "contents": [ "Let $R \\to S$ be a homomorphism of Noetherian rings.", "Let $\\mathfrak q \\subset S$ be a prime lying", "over the prime $\\mathfrak p$. Assume the going down property holds", "for $R \\to S$ (for example if $R \\to S$ is flat, see", "Lemma \\ref{lemma-flat-going-down}). Then", "$$", "\\dim(S_{\\mathfrak q})", "=", "\\dim(R_{\\mathfrak p})", "+", "\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}).", "$$" ], "refs": [ "algebra-lemma-flat-going-down" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-dimension-base-fibre-total}", "we have an inequality", "$\\dim(S_{\\mathfrak q}) \\leq", "\\dim(R_{\\mathfrak p}) + \\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q})$.", "To get equality, choose a chain of primes", "$\\mathfrak pS \\subset \\mathfrak q_0 \\subset \\mathfrak q_1 \\subset \\ldots", "\\subset \\mathfrak q_d = \\mathfrak q$ with", "$d = \\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q})$.", "On the other hand, choose a chain of primes", "$\\mathfrak p_0 \\subset \\mathfrak p_1 \\subset \\ldots \\subset \\mathfrak p_e", "= \\mathfrak p$ with $e = \\dim(R_{\\mathfrak p})$.", "By the going down theorem we may choose", "$\\mathfrak q_{-1} \\subset \\mathfrak q_0$ lying over", "$\\mathfrak p_{e-1}$. And then we may choose", "$\\mathfrak q_{-2} \\subset \\mathfrak q_{e-1}$ lying over", "$\\mathfrak p_{e-2}$. Inductively we keep going until we", "get a chain", "$\\mathfrak q_{-e} \\subset \\ldots \\subset \\mathfrak q_d$", "of length $e + d$." ], "refs": [ "algebra-lemma-dimension-base-fibre-total" ], "ref_ids": [ 986 ] } ], "ref_ids": [ 539 ] }, { "id": 988, "type": "theorem", "label": "algebra-lemma-flat-over-regular-with-regular-fibre", "categories": [ "algebra" ], "title": "algebra-lemma-flat-over-regular-with-regular-fibre", "contents": [ "Let $R \\to S$ be a local homomorphism of local Noetherian rings.", "Assume", "\\begin{enumerate}", "\\item $R$ is regular,", "\\item $S/\\mathfrak m_RS$ is regular, and", "\\item $R \\to S$ is flat.", "\\end{enumerate}", "Then $S$ is regular." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-dimension-base-fibre-equals-total}", "we have", "$\\dim(S) = \\dim(R) + \\dim(S/\\mathfrak m_RS)$.", "Pick generators $x_1, \\ldots, x_d \\in \\mathfrak m_R$ with $d = \\dim(R)$,", "and pick $y_1, \\ldots, y_e \\in \\mathfrak m_S$", "which generate the maximal ideal of $S/\\mathfrak m_RS$ with", "$e = \\dim(S/\\mathfrak m_RS)$. Then we see that", "$x_1, \\ldots, x_d, y_1, \\ldots, y_e$ are elements which generate", "the maximal ideal of $S$ and $e + d = \\dim(S)$." ], "refs": [ "algebra-lemma-dimension-base-fibre-equals-total" ], "ref_ids": [ 987 ] } ], "ref_ids": [] }, { "id": 989, "type": "theorem", "label": "algebra-lemma-finite-flat-over-regular-CM", "categories": [ "algebra" ], "title": "algebra-lemma-finite-flat-over-regular-CM", "contents": [ "Let $R \\to S$ be a local homomorphism of Noetherian local rings.", "Assume $R$ Cohen-Macaulay.", "If $S$ is finite flat over $R$, or if $S$ is flat over $R$ and", "$\\dim(S) \\leq \\dim(R)$, then $S$ is Cohen-Macaulay and $\\dim(R) = \\dim(S)$." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1, \\ldots, x_d \\in \\mathfrak m_R$ be a regular sequence", "of length $d = \\dim(R)$. By Lemma \\ref{lemma-flat-increases-depth}", "this maps to a regular sequence in $S$.", "Hence $S$ is Cohen-Macaulay if $\\dim(S) \\leq d$. This is true", "if $S$ is finite flat over $R$ by Lemma \\ref{lemma-integral-sub-dim-equal}.", "And in the second case we assumed it." ], "refs": [ "algebra-lemma-flat-increases-depth", "algebra-lemma-integral-sub-dim-equal" ], "ref_ids": [ 740, 985 ] } ], "ref_ids": [] }, { "id": 990, "type": "theorem", "label": "algebra-lemma-dimension-formula", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-formula", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak q$ be a prime of $S$ lying over the prime $\\mathfrak p$ of $R$.", "Assume that", "\\begin{enumerate}", "\\item $R$ is Noetherian,", "\\item $R \\to S$ is of finite type,", "\\item $R$, $S$ are domains, and", "\\item $R \\subset S$.", "\\end{enumerate}", "Then we have", "$$", "\\text{height}(\\mathfrak q)", "\\leq", "\\text{height}(\\mathfrak p) + \\text{trdeg}_R(S)", "- \\text{trdeg}_{\\kappa(\\mathfrak p)} \\kappa(\\mathfrak q)", "$$", "with equality if $R$ is universally catenary." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $R \\subset S' \\subset S$ is a finitely generated $R$-subalgebra", "of $S$. In this case set $\\mathfrak q' = S' \\cap \\mathfrak q$.", "The lemma for the ring maps $R \\to S'$ and $S' \\to S$ implies the", "lemma for $R \\to S$ by additivity of transcendence degree in towers", "of fields (Fields, Lemma \\ref{fields-lemma-transcendence-degree-tower}).", "Hence we can use induction on the number of generators", "of $S$ over $R$ and reduce to the case where $S$ is generated by", "one element over $R$.", "\\medskip\\noindent", "Case I: $S = R[x]$ is a polynomial algebra over $R$.", "In this case we have $\\text{trdeg}_R(S) = 1$.", "Also $R \\to S$ is flat and hence", "$$", "\\dim(S_{\\mathfrak q}) =", "\\dim(R_{\\mathfrak p}) + \\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q})", "$$", "see Lemma \\ref{lemma-dimension-base-fibre-equals-total}.", "Let $\\mathfrak r = \\mathfrak pS$. Then", "$\\text{trdeg}_{\\kappa(\\mathfrak p)} \\kappa(\\mathfrak q) = 1$", "is equivalent to $\\mathfrak q = \\mathfrak r$, and implies that", "$\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) = 0$.", "In the same vein $\\text{trdeg}_{\\kappa(\\mathfrak p)} \\kappa(\\mathfrak q) = 0$", "is equivalent to having a strict inclusion", "$\\mathfrak r \\subset \\mathfrak q$, which implies that", "$\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) = 1$.", "Thus we are done with case I with equality in every instance.", "\\medskip\\noindent", "Case II: $S = R[x]/\\mathfrak n$ with $\\mathfrak n \\not = 0$.", "In this case we have $\\text{trdeg}_R(S) = 0$.", "Denote $\\mathfrak q' \\subset R[x]$ the prime corresponding to $\\mathfrak q$.", "Thus we have", "$$", "S_{\\mathfrak q} = (R[x])_{\\mathfrak q'}/\\mathfrak n(R[x])_{\\mathfrak q'}", "$$", "By the previous case we have", "$\\dim((R[x])_{\\mathfrak q'}) =", "\\dim(R_{\\mathfrak p}) + 1", "- \\text{trdeg}_{\\kappa(\\mathfrak p)} \\kappa(\\mathfrak q)$.", "Since $\\mathfrak n \\not = 0$ we see that the dimension of", "$S_{\\mathfrak q}$ decreases by at least one, see", "Lemma \\ref{lemma-one-equation},", "which proves the inequality of the lemma.", "To see the equality in case $R$ is universally catenary note that", "$\\mathfrak n \\subset R[x]$ is a height one prime as it corresponds", "to a nonzero prime in $F[x]$ where $F$ is the fraction field of $R$.", "Hence any maximal chain of primes in", "$S_\\mathfrak q = R[x]_{\\mathfrak q'}/\\mathfrak nR[x]_{\\mathfrak q'}$", "corresponds to a maximal chain of primes", "with length 1 greater between $\\mathfrak q'$ and $(0)$ in $R[x]$.", "If $R$ is universally catenary these all have the same length equal to", "the height of $\\mathfrak q'$. This proves that", "$\\dim(S_\\mathfrak q) = \\dim(R[x]_{\\mathfrak q'}) - 1$", "and this implies equality holds as desired." ], "refs": [ "fields-lemma-transcendence-degree-tower", "algebra-lemma-dimension-base-fibre-equals-total" ], "ref_ids": [ 4519, 987 ] } ], "ref_ids": [] }, { "id": 991, "type": "theorem", "label": "algebra-lemma-finite-in-codim-1", "categories": [ "algebra" ], "title": "algebra-lemma-finite-in-codim-1", "contents": [ "Let $A \\to B$ be a ring map.", "Assume", "\\begin{enumerate}", "\\item $A \\subset B$ is an extension of domains,", "\\item the induced extension of fraction fields is finite,", "\\item $A$ is Noetherian, and", "\\item $A \\to B$ is of finite type.", "\\end{enumerate}", "Let $\\mathfrak p \\subset A$ be a prime of height $1$.", "Then there are at most finitely many primes of $B$", "lying over $\\mathfrak p$ and they all have height $1$." ], "refs": [], "proofs": [ { "contents": [ "By the dimension formula (Lemma \\ref{lemma-dimension-formula})", "for any prime $\\mathfrak q$ lying over $\\mathfrak p$ we have", "$$", "\\dim(B_{\\mathfrak q}) \\leq", "\\dim(A_{\\mathfrak p}) - \\text{trdeg}_{\\kappa(\\mathfrak p)} \\kappa(\\mathfrak q).", "$$", "As the domain $B_\\mathfrak q$ has at least $2$ prime ideals we see that", "$\\dim(B_{\\mathfrak q}) \\geq 1$. We conclude that", "$\\dim(B_{\\mathfrak q}) = 1$ and that the extension", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$ is algebraic.", "Hence $\\mathfrak q$ defines a closed point of its fibre", "$\\Spec(B \\otimes_A \\kappa(\\mathfrak p))$, see", "Lemma \\ref{lemma-finite-residue-extension-closed}.", "Since $B \\otimes_A \\kappa(\\mathfrak p)$ is a Noetherian ring", "the fibre $\\Spec(B \\otimes_A \\kappa(\\mathfrak p))$", "is a Noetherian topological space, see", "Lemma \\ref{lemma-Noetherian-topology}.", "A Noetherian topological space consisting of closed points", "is finite, see for example", "Topology, Lemma \\ref{topology-lemma-Noetherian}." ], "refs": [ "algebra-lemma-dimension-formula", "algebra-lemma-finite-residue-extension-closed", "algebra-lemma-Noetherian-topology", "topology-lemma-Noetherian" ], "ref_ids": [ 990, 472, 452, 8220 ] } ], "ref_ids": [] }, { "id": 992, "type": "theorem", "label": "algebra-lemma-dim-affine-space", "categories": [ "algebra" ], "title": "algebra-lemma-dim-affine-space", "contents": [ "Let $\\mathfrak m$ be a maximal ideal in $k[x_1, \\ldots, x_n]$.", "The ideal $\\mathfrak m$ is generated by $n$ elements.", "The dimension of $k[x_1, \\ldots, x_n]_{\\mathfrak m}$ is $n$.", "Hence $k[x_1, \\ldots, x_n]_{\\mathfrak m}$ is a regular local", "ring of dimension $n$." ], "refs": [], "proofs": [ { "contents": [ "By the Hilbert Nullstellensatz", "(Theorem \\ref{theorem-nullstellensatz})", "we know the residue field $\\kappa = \\kappa(\\mathfrak m)$ is", "a finite extension of $k$. Denote $\\alpha_i \\in \\kappa$ the", "image of $x_i$. Denote $\\kappa_i = k(\\alpha_1, \\ldots, \\alpha_i)", "\\subset \\kappa$, $i = 1, \\ldots, n$ and $\\kappa_0 = k$.", "Note that $\\kappa_i = k[\\alpha_1, \\ldots, \\alpha_i]$", "by field theory. Define inductively elements", "$f_i \\in \\mathfrak m \\cap k[x_1, \\ldots, x_i]$", "as follows: Let $P_i(T) \\in \\kappa_{i-1}[T]$", "be the monic minimal polynomial of $\\alpha_i $ over $\\kappa_{i-1}$.", "Let $Q_i(T) \\in k[x_1, \\ldots, x_{i-1}][T]$ be a monic lift of $P_i(T)$", "(of the same degree). Set $f_i = Q_i(x_i)$.", "Note that if $d_i = \\deg_T(P_i) = \\deg_T(Q_i) = \\deg_{x_i}(f_i)$", "then $d_1d_2\\ldots d_i = [\\kappa_i : k]$ by", "Fields, Lemmas \\ref{fields-lemma-multiplicativity-degrees} and", "\\ref{fields-lemma-degree-minimal-polynomial}.", "\\medskip\\noindent", "We claim that for all $i = 0, 1, \\ldots, n$ there is an", "isomorphism", "$$", "\\psi_i : k[x_1, \\ldots, x_i] /(f_1, \\ldots, f_i) \\cong \\kappa_i.", "$$", "By construction the composition", "$k[x_1, \\ldots, x_i] \\to k[x_1, \\ldots, x_n] \\to \\kappa$", "is surjective onto $\\kappa_i$ and $f_1, \\ldots, f_i$ are", "in the kernel. This gives a surjective homomorphism.", "We prove $\\psi_i$ is injective by induction. It is clear for $i = 0$.", "Given the statement for $i$ we prove it for $i + 1$.", "The ring extension $k[x_1, \\ldots, x_i]/(f_1, \\ldots, f_i) \\to", "k[x_1, \\ldots, x_{i + 1}]/(f_1, \\ldots, f_{i + 1})$", "is generated by $1$ element over a field and one", "irreducible equation. By elementary field theory", "$k[x_1, \\ldots, x_{i + 1}]/(f_1, \\ldots, f_{i + 1})$", "is a field, and hence $\\psi_i$ is injective.", "\\medskip\\noindent", "This implies that $\\mathfrak m = (f_1, \\ldots, f_n)$.", "Moreover, we also conclude that", "$$", "k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_i)", "\\cong", "\\kappa_i[x_{i + 1}, \\ldots, x_n].", "$$", "Hence $(f_1, \\ldots, f_i)$ is a prime ideal. Thus", "$$", "(0) \\subset (f_1) \\subset (f_1, f_2) \\subset \\ldots \\subset", "(f_1, \\ldots, f_n) = \\mathfrak m", "$$", "is a chain of primes of length $n$. The lemma follows." ], "refs": [ "algebra-theorem-nullstellensatz", "fields-lemma-multiplicativity-degrees", "fields-lemma-degree-minimal-polynomial" ], "ref_ids": [ 316, 4450, 4459 ] } ], "ref_ids": [] }, { "id": 993, "type": "theorem", "label": "algebra-lemma-dimension-height-polynomial-ring", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-height-polynomial-ring", "contents": [ "Let $k$ be a field.", "Let $\\mathfrak p \\subset \\mathfrak q \\subset k[x_1, \\ldots, x_n]$", "be a pair of primes.", "Any maximal chain of primes between $\\mathfrak p$ and $\\mathfrak q$", "has length $\\text{height}(\\mathfrak q) - \\text{height}(\\mathfrak p)$." ], "refs": [], "proofs": [ { "contents": [ "By", "Proposition \\ref{proposition-finite-gl-dim-polynomial-ring}", "any local ring of $k[x_1, \\ldots, x_n]$ is regular.", "Hence all local rings are Cohen-Macaulay, see", "Lemma \\ref{lemma-regular-ring-CM}.", "The local rings at maximal ideals have dimension $n$ hence", "every maximal chain of primes in $k[x_1, \\ldots, x_n]$", "has length $n$, see", "Lemma \\ref{lemma-maximal-chain-CM}.", "Hence every maximal chain of primes between $(0)$ and $\\mathfrak p$", "has length $\\text{height}(\\mathfrak p)$, see", "Lemma \\ref{lemma-CM-dim-formula}", "for example.", "Putting these together leads to the assertion of the lemma." ], "refs": [ "algebra-proposition-finite-gl-dim-polynomial-ring", "algebra-lemma-regular-ring-CM", "algebra-lemma-maximal-chain-CM", "algebra-lemma-CM-dim-formula" ], "ref_ids": [ 1424, 941, 924, 925 ] } ], "ref_ids": [] }, { "id": 994, "type": "theorem", "label": "algebra-lemma-dimension-spell-it-out", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-spell-it-out", "contents": [ "Let $k$ be a field.", "Let $S$ be a finite type $k$-algebra which is an integral domain.", "Then $\\dim(S) = \\dim(S_{\\mathfrak m})$ for any maximal", "ideal $\\mathfrak m$ of $S$. In words: every maximal chain", "of primes has length equal to the dimension of $S$." ], "refs": [], "proofs": [ { "contents": [ "Write $S = k[x_1, \\ldots, x_n]/\\mathfrak p$.", "By Proposition \\ref{proposition-finite-gl-dim-polynomial-ring} and", "Lemma \\ref{lemma-dimension-height-polynomial-ring}", "all the maximal chains of primes in $S$ (which necessarily end", "with a maximal ideal) have length $n - \\text{height}(\\mathfrak p)$.", "Thus this number is the dimension of $S$ and of $S_{\\mathfrak m}$", "for any maximal ideal $\\mathfrak m$ of $S$." ], "refs": [ "algebra-proposition-finite-gl-dim-polynomial-ring", "algebra-lemma-dimension-height-polynomial-ring" ], "ref_ids": [ 1424, 993 ] } ], "ref_ids": [] }, { "id": 995, "type": "theorem", "label": "algebra-lemma-dimension-at-a-point-finite-type-over-field", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-at-a-point-finite-type-over-field", "contents": [ "Let $k$ be a field.", "Let $S$ be a finite type $k$-algebra.", "Let $X = \\Spec(S)$.", "Let $\\mathfrak p \\subset S$ be a prime ideal and let", "$x \\in X$ be the corresponding point.", "The following numbers are equal", "\\begin{enumerate}", "\\item $\\dim_x(X)$,", "\\item $\\max \\dim(Z)$ where the maximum is over those", "irreducible components $Z$ of $X$ passing through $x$, and", "\\item $\\min \\dim(S_{\\mathfrak m})$ where the minimum", "is over maximal ideals $\\mathfrak m$ with", "$\\mathfrak p \\subset \\mathfrak m$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X = \\bigcup_{i \\in I} Z_i$ be the decomposition of $X$ into", "its irreducible components. There are finitely many of", "them (see", "Lemmas \\ref{lemma-obvious-Noetherian} and \\ref{lemma-Noetherian-topology}).", "Let $I' = \\{i \\mid x \\in Z_i\\}$, and let", "$T = \\bigcup_{i \\not \\in I'} Z_i$. Then $U = X \\setminus T$", "is an open subset of $X$ containing the point $x$.", "The number (2) is $\\max_{i \\in I'} \\dim(Z_i)$.", "For any open $W \\subset U$ with $x \\in W$", "the irreducible components of $W$ are the irreducible sets", "$W_i = Z_i \\cap W$ for $i \\in I'$ and $x$ is contained", "in each of these.", "Note that each $W_i$, $i \\in I'$ contains a closed point because", "$X$ is Jacobson, see Section \\ref{section-ring-jacobson}.", "Since $W_i \\subset Z_i$ we have $\\dim(W_i) \\leq \\dim(Z_i)$.", "The existence of a closed point implies, via Lemma", "\\ref{lemma-dimension-spell-it-out}, that there is a chain of", "irreducible closed subsets of length equal to $\\dim(Z_i)$ in the open $W_i$.", "Thus $\\dim(W_i) = \\dim(Z_i)$ for any $i \\in I'$. Hence $\\dim(W)$", "is equal to the number (2). This proves that (1) $ = $ (2).", "\\medskip\\noindent", "Let $\\mathfrak m \\supset \\mathfrak p$ be any maximal ideal", "containing $\\mathfrak p$. Let $x_0 \\in X$ be the corresponding", "point. First of all, $x_0$ is contained in all the", "irreducible components $Z_i$, $i \\in I'$. Let $\\mathfrak q_i$", "denote the minimal primes of $S$ corresponding to the", "irreducible components $Z_i$. For each $i$ such that", "$x_0 \\in Z_i$ (which is equivalent to $\\mathfrak m \\supset \\mathfrak q_i$)", "we have a surjection", "$$", "S_{\\mathfrak m} \\longrightarrow", "S_\\mathfrak m/\\mathfrak q_i S_\\mathfrak m =(S/\\mathfrak q_i)_{\\mathfrak m}", "$$", "Moreover, the primes $\\mathfrak q_i S_\\mathfrak m$ so obtained", "exhaust the minimal", "primes of the Noetherian local ring $S_{\\mathfrak m}$, see", "Lemma \\ref{lemma-irreducible-components-containing-x}.", "We conclude, using Lemma \\ref{lemma-dimension-spell-it-out},", "that the dimension of $S_{\\mathfrak m}$ is the", "maximum of the dimensions of the $Z_i$ passing through $x_0$.", "To finish the proof of the lemma it suffices to show that", "we can choose $x_0$ such that $x_0 \\in Z_i \\Rightarrow i \\in I'$.", "Because $S$ is Jacobson (as we saw above)", "it is enough to show that $V(\\mathfrak p) \\setminus T$", "(with $T$ as above) is nonempty. And this is clear since it", "contains the point $x$ (i.e. $\\mathfrak p$)." ], "refs": [ "algebra-lemma-obvious-Noetherian", "algebra-lemma-Noetherian-topology", "algebra-lemma-dimension-spell-it-out", "algebra-lemma-irreducible-components-containing-x", "algebra-lemma-dimension-spell-it-out" ], "ref_ids": [ 450, 452, 994, 424, 994 ] } ], "ref_ids": [] }, { "id": 996, "type": "theorem", "label": "algebra-lemma-dimension-closed-point-finite-type-field", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-closed-point-finite-type-field", "contents": [ "Let $k$ be a field.", "Let $S$ be a finite type $k$-algebra.", "Let $X = \\Spec(S)$.", "Let $\\mathfrak m \\subset S$ be a maximal ideal and let", "$x \\in X$ be the associated closed point.", "Then $\\dim_x(X) = \\dim(S_{\\mathfrak m})$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-dimension-at-a-point-finite-type-over-field}." ], "refs": [ "algebra-lemma-dimension-at-a-point-finite-type-over-field" ], "ref_ids": [ 995 ] } ], "ref_ids": [] }, { "id": 997, "type": "theorem", "label": "algebra-lemma-disjoint-decomposition-CM-algebra", "categories": [ "algebra" ], "title": "algebra-lemma-disjoint-decomposition-CM-algebra", "contents": [ "Let $k$ be a field.", "Let $S$ be a finite type $k$ algebra.", "Assume that $S$ is Cohen-Macaulay.", "Then $\\Spec(S) = \\coprod T_d$ is a finite disjoint union of", "open and closed subsets $T_d$ with $T_d$ equidimensional", "(see Topology, Definition \\ref{topology-definition-equidimensional})", "of dimension $d$. Equivalently, $S$ is a product of rings", "$S_d$, $d = 0, \\ldots, \\dim(S)$ such that every maximal ideal", "$\\mathfrak m$ of $S_d$ has height $d$." ], "refs": [ "topology-definition-equidimensional" ], "proofs": [ { "contents": [ "The equivalence of the two statements follows from", "Lemma \\ref{lemma-disjoint-implies-product}.", "Let $\\mathfrak m \\subset S$ be a maximal ideal.", "Every maximal chain of primes in $S_{\\mathfrak m}$ has", "the same length equal to $\\dim(S_{\\mathfrak m})$, see", "Lemma \\ref{lemma-maximal-chain-CM}. Hence, the dimension of the irreducible", "components passing through the point corresponding to $\\mathfrak m$", "all have dimension equal to $\\dim(S_{\\mathfrak m})$, see", "Lemma \\ref{lemma-dimension-spell-it-out}.", "Since $\\Spec(S)$ is a Jacobson topological space", "the intersection", "of any two irreducible components of it contains a closed point if nonempty,", "see", "Lemmas \\ref{lemma-finite-type-field-Jacobson} and", "\\ref{lemma-jacobson}.", "Thus we have shown that any two irreducible components", "that meet have the same dimension. The lemma follows", "easily from this, and the fact that $\\Spec(S)$", "has a finite number of irreducible components (see", "Lemmas \\ref{lemma-obvious-Noetherian} and \\ref{lemma-Noetherian-topology})." ], "refs": [ "algebra-lemma-disjoint-implies-product", "algebra-lemma-maximal-chain-CM", "algebra-lemma-dimension-spell-it-out", "algebra-lemma-finite-type-field-Jacobson", "algebra-lemma-jacobson", "algebra-lemma-obvious-Noetherian", "algebra-lemma-Noetherian-topology" ], "ref_ids": [ 415, 924, 994, 467, 469, 450, 452 ] } ], "ref_ids": [ 8357 ] }, { "id": 998, "type": "theorem", "label": "algebra-lemma-helper", "categories": [ "algebra" ], "title": "algebra-lemma-helper", "contents": [ "Let $n \\in \\mathbf{N}$.", "Let $N$ be a finite nonempty", "set of multi-indices $\\nu = (\\nu_1, \\ldots, \\nu_n)$.", "Given $e = (e_1, \\ldots, e_n)$ we set $e \\cdot \\nu = \\sum e_i\\nu_i$.", "Then for $e_1 \\gg e_2 \\gg \\ldots \\gg e_{n-1} \\gg e_n$ we have:", "If $\\nu, \\nu' \\in N$ then", "$$", "(e \\cdot \\nu = e \\cdot \\nu')", "\\Leftrightarrow", "(\\nu = \\nu')", "$$" ], "refs": [], "proofs": [ { "contents": [ "Say $N = \\{\\nu_j\\}$ with $\\nu_j = (\\nu_{j1}, \\ldots, \\nu_{jn})$.", "Let $A_i = \\max_j \\nu_{ji} - \\min_j \\nu_{ji}$. If for each $i$ we have", "$e_{i - 1} > A_ie_i + A_{i + 1}e_{i + 1} + \\ldots + A_ne_n$ then", "the lemma holds. For suppose that $e \\cdot (\\nu - \\nu') = 0$. Then for", "$n \\ge 2$,", "$$", "e_1(\\nu_1 - \\nu'_1) = \\sum\\nolimits_{i = 2}^n e_i(\\nu'_i - \\nu_i).", "$$", "We may assume that $(\\nu_1 - \\nu'_1) \\ge 0$. If $(\\nu_1 - \\nu'_1) > 0$, then", "$$", "e_1(\\nu_1 - \\nu'_1) \\ge e_1 >", "A_2e_2 + \\ldots + A_ne_n \\ge", "\\sum\\nolimits_{i = 2}^n e_i|\\nu'_i - \\nu_i| \\ge", "\\sum\\nolimits_{i = 2}^n e_i(\\nu'_i - \\nu_i).", "$$", "This contradiction implies that $\\nu'_1 = \\nu_1$. By", "induction, $\\nu'_i = \\nu_i$ for $2 \\le i \\le n$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 999, "type": "theorem", "label": "algebra-lemma-helper-polynomial", "categories": [ "algebra" ], "title": "algebra-lemma-helper-polynomial", "contents": [ "Let $R$ be a ring. Let $g \\in R[x_1, \\ldots, x_n]$ be an element", "which is nonconstant, i.e., $g \\not \\in R$.", "For $e_1 \\gg e_2 \\gg \\ldots \\gg e_{n-1} \\gg e_n = 1$ the polynomial", "$$", "g(x_1 + x_n^{e_1}, x_2 + x_n^{e_2}, \\ldots, x_{n - 1} + x_n^{e_{n - 1}}, x_n)", "=", "ax_n^d + \\text{lower order terms in }x_n", "$$", "where $d > 0$ and $a \\in R$ is one of the nonzero coefficients of $g$." ], "refs": [], "proofs": [ { "contents": [ "Write $g = \\sum_{\\nu \\in N} a_\\nu x^\\nu$ with $a_\\nu \\in R$ not zero.", "Here $N$ is a finite set of multi-indices as in", "Lemma \\ref{lemma-helper}", "and $x^\\nu = x_1^{\\nu_1} \\ldots x_n^{\\nu_n}$.", "Note that the leading term in", "$$", "(x_1 + x_n^{e_1})^{\\nu_1} \\ldots (x_{n-1} + x_n^{e_{n-1}})^{\\nu_{n-1}}", "x_n^{\\nu_n}", "\\quad\\text{is}\\quad", "x_n^{e_1\\nu_1 + \\ldots + e_{n-1}\\nu_{n-1} + \\nu_n}.", "$$", "Hence the lemma follows from", "Lemma \\ref{lemma-helper}", "which guarantees that there is exactly one nonzero term $a_\\nu x^\\nu$ of $g$", "which gives rise to the leading term of", "$g(x_1 + x_n^{e_1}, x_2 + x_n^{e_2}, \\ldots, x_{n - 1} + x_n^{e_{n - 1}},", "x_n)$, i.e., $a = a_\\nu$ for the unique $\\nu \\in N$ such that", "$e \\cdot \\nu$ is maximal." ], "refs": [ "algebra-lemma-helper", "algebra-lemma-helper" ], "ref_ids": [ 998, 998 ] } ], "ref_ids": [] }, { "id": 1000, "type": "theorem", "label": "algebra-lemma-one-relation", "categories": [ "algebra" ], "title": "algebra-lemma-one-relation", "contents": [ "Let $k$ be a field.", "Let $S = k[x_1, \\ldots, x_n]/I$ for some proper ideal $I$.", "If $I \\not = 0$, then there exist $y_1, \\ldots, y_{n-1} \\in k[x_1, \\ldots, x_n]$", "such that $S$ is finite over $k[y_1, \\ldots, y_{n-1}]$. Moreover we may", "choose $y_i$ to be in the $\\mathbf{Z}$-subalgebra of $k[x_1, \\ldots, x_n]$", "generated by $x_1, \\ldots, x_n$." ], "refs": [], "proofs": [ { "contents": [ "Pick $f \\in I$, $f\\not = 0$. It suffices to show the lemma", "for $k[x_1, \\ldots, x_n]/(f)$ since $S$ is a quotient of that ring.", "We will take $y_i = x_i - x_n^{e_i}$, $i = 1, \\ldots, n-1$", "for suitable integers $e_i$. When does this work? It suffices", "to show that $\\overline{x_n} \\in k[x_1, \\ldots, x_n]/(f)$", "is integral over the ring $k[y_1, \\ldots, y_{n-1}]$. The", "equation for $\\overline{x_n}$ over this ring is", "$$", "f(y_1 + x_n^{e_1}, \\ldots, y_{n-1} + x_n^{e_{n-1}}, x_n) = 0.", "$$", "Hence we are done if we can show there exists integers $e_i$ such", "that the leading coefficient with respect to $x_n$ of the equation", "above is a nonzero element of $k$. This can be achieved for example", "by choosing $e_1 \\gg e_2 \\gg \\ldots \\gg e_{n-1}$, see", "Lemma \\ref{lemma-helper-polynomial}." ], "refs": [ "algebra-lemma-helper-polynomial" ], "ref_ids": [ 999 ] } ], "ref_ids": [] }, { "id": 1001, "type": "theorem", "label": "algebra-lemma-Noether-normalization", "categories": [ "algebra" ], "title": "algebra-lemma-Noether-normalization", "contents": [ "Let $k$ be a field. Let $S = k[x_1, \\ldots, x_n]/I$ for some ideal $I$.", "If $I \\neq (1)$, there exist $r\\geq 0$, and", "$y_1, \\ldots, y_r \\in k[x_1, \\ldots, x_n]$", "such that (a) the map $k[y_1, \\ldots, y_r] \\to S$ is injective,", "and (b) the map $k[y_1, \\ldots, y_r] \\to S$ is finite.", "In this case the integer $r$ is the dimension of $S$.", "Moreover we may choose $y_i$ to be in the", "$\\mathbf{Z}$-subalgebra of $k[x_1, \\ldots, x_n]$", "generated by $x_1, \\ldots, x_n$." ], "refs": [], "proofs": [ { "contents": [ "By induction on $n$, with $n = 0$ being trivial.", "If $I = 0$, then take $r = n$ and $y_i = x_i$.", "If $I \\not = 0$, then choose $y_1, \\ldots, y_{n-1}$", "as in Lemma \\ref{lemma-one-relation}. Let", "$S' \\subset S$ be the subring generated by", "the images of the $y_i$. By induction we can", "choose $r$ and $z_1, \\ldots, z_r \\in k[y_1, \\ldots, y_{n-1}]$", "such that (a), (b) hold for $k[z_1, \\ldots, z_r]", "\\to S'$. Since $S' \\to S$ is injective and finite", "we see (a), (b) hold for $k[z_1, \\ldots, z_r]", "\\to S$. The last assertion follows from Lemma", "\\ref{lemma-integral-sub-dim-equal}." ], "refs": [ "algebra-lemma-one-relation", "algebra-lemma-integral-sub-dim-equal" ], "ref_ids": [ 1000, 985 ] } ], "ref_ids": [] }, { "id": 1002, "type": "theorem", "label": "algebra-lemma-Noether-normalization-at-point", "categories": [ "algebra" ], "title": "algebra-lemma-Noether-normalization-at-point", "contents": [ "Let $k$ be a field.", "Let $S$ be a finite type $k$ algebra and denote $X = \\Spec(S)$.", "Let $\\mathfrak q$ be a prime of $S$, and let $x \\in X$ be the", "corresponding point. There exists a $g \\in S$, $g \\not \\in \\mathfrak q$", "such that $\\dim(S_g) = \\dim_x(X) =: d$ and such that", "there exists a finite injective map $k[y_1, \\ldots, y_d] \\to S_g$." ], "refs": [], "proofs": [ { "contents": [ "Note that by definition $\\dim_x(X)$ is the minimum", "of the dimensions of $S_g$ for $g \\in S$, $g \\not \\in \\mathfrak q$, i.e.,", "the minimum is attained. Thus the lemma follows from", "Lemma \\ref{lemma-Noether-normalization}." ], "refs": [ "algebra-lemma-Noether-normalization" ], "ref_ids": [ 1001 ] } ], "ref_ids": [] }, { "id": 1003, "type": "theorem", "label": "algebra-lemma-refined-Noether-normalization", "categories": [ "algebra" ], "title": "algebra-lemma-refined-Noether-normalization", "contents": [ "Let $k$ be a field. Let $\\mathfrak q \\subset k[x_1, \\ldots, x_n]$", "be a prime ideal. Set $r = \\text{trdeg}_k\\ \\kappa(\\mathfrak q)$.", "Then there exists a finite ring map", "$\\varphi : k[y_1, \\ldots, y_n] \\to k[x_1, \\ldots, x_n]$ such", "that $\\varphi^{-1}(\\mathfrak q) = (y_{r + 1}, \\ldots, y_n)$." ], "refs": [], "proofs": [ { "contents": [ "By induction on $n$. The case $n = 0$ is clear. Assume $n > 0$.", "If $r = n$, then $\\mathfrak q = (0)$ and the result is clear.", "Choose a nonzero $f \\in \\mathfrak q$. Of course $f$ is nonconstant.", "After applying an automorphism of the form", "$$", "k[x_1, \\ldots, x_n] \\longrightarrow k[x_1, \\ldots, x_n],", "\\quad", "x_n \\mapsto x_n,", "\\quad", "x_i \\mapsto x_i + x_n^{e_i}\\ (i < n)", "$$", "we may assume that $f$ is monic in $x_n$ over $k[x_1, \\ldots, x_n]$, see", "Lemma \\ref{lemma-helper-polynomial}. Hence the ring map", "$$", "k[y_1, \\ldots, y_n] \\longrightarrow k[x_1, \\ldots, x_n],", "\\quad", "y_n \\mapsto f,", "\\quad", "y_i \\mapsto x_i\\ (i < n)", "$$", "is finite. Moreover $y_n \\in \\mathfrak q \\cap k[y_1, \\ldots, y_n]$ by", "construction. Thus", "$\\mathfrak q \\cap k[y_1, \\ldots, y_n] = \\mathfrak pk[y_1, \\ldots, y_n] + (y_n)$", "where $\\mathfrak p \\subset k[y_1, \\ldots, y_{n - 1}]$ is a prime ideal.", "Note that $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$ is finite, and", "hence $r = \\text{trdeg}_k\\ \\kappa(\\mathfrak p)$.", "Apply the induction hypothesis to the pair", "$(k[y_1, \\ldots, y_{n - 1}], \\mathfrak p)$ and we obtain a finite ring map", "$k[z_1, \\ldots, z_{n - 1}] \\to k[y_1, \\ldots, y_{n - 1}]$ such that", "$\\mathfrak p \\cap k[z_1, \\ldots, z_{n - 1}] = (z_{r + 1}, \\ldots, z_{n - 1})$.", "We extend the ring map", "$k[z_1, \\ldots, z_{n - 1}] \\to k[y_1, \\ldots, y_{n - 1}]$", "to a ring map", "$k[z_1, \\ldots, z_n] \\to k[y_1, \\ldots, y_n]$", "by mapping $z_n$ to $y_n$.", "The composition of the ring maps", "$$", "k[z_1, \\ldots, z_n] \\to k[y_1, \\ldots, y_n] \\to k[x_1, \\ldots, x_n]", "$$", "solves the problem." ], "refs": [ "algebra-lemma-helper-polynomial" ], "ref_ids": [ 999 ] } ], "ref_ids": [] }, { "id": 1004, "type": "theorem", "label": "algebra-lemma-Noether-normalization-over-a-domain", "categories": [ "algebra" ], "title": "algebra-lemma-Noether-normalization-over-a-domain", "contents": [ "Let $R \\to S$ be an injective finite type ring map. Assume $R$ is a domain.", "Then there exists an integer $d$ and a factorization", "$$", "R \\to R[y_1, \\ldots, y_d] \\to S' \\to S", "$$", "by injective maps such that $S'$ is finite over $R[y_1, \\ldots, y_d]$", "and such that $S'_f \\cong S_f$ for some nonzero $f \\in R$." ], "refs": [], "proofs": [ { "contents": [ "Pick $x_1, \\ldots, x_n \\in S$ which generate $S$ over $R$.", "Let $K$ be the fraction field of $R$ and $S_K = S \\otimes_R K$.", "By Lemma \\ref{lemma-Noether-normalization}", "we can find $y_1, \\ldots, y_d \\in S$ such that $K[y_1, \\ldots, y_d] \\to S_K$", "is a finite injective map. Note that $y_i \\in S$ because we may pick the", "$y_j$ in the $\\mathbf{Z}$-algebra generated by $x_1, \\ldots, x_n$.", "As a finite ring map is integral (see", "Lemma \\ref{lemma-finite-is-integral})", "we can find monic $P_i \\in K[y_1, \\ldots, y_d][T]$ such that", "$P_i(x_i) = 0$ in $S_K$. Let $f \\in R$ be a nonzero element such that", "$fP_i \\in R[y_1, \\ldots, y_d][T]$ for all $i$. Then $fP_i(x_i)$", "maps to zero in $S_K$. Hence after replacing $f$ by another", "nonzero element of $R$ we may also assume $fP_i(x_i)$ is zero in $S$.", "Set $x_i' = fx_i$ and let $S' \\subset S$ be the $R$-subalgebra generated by", "$y_1, \\ldots, y_d$ and $x'_1, \\ldots, x'_n$. Note that $x'_i$ is integral over", "$R[y_1, \\ldots, y_d]$ as we have $Q_i(x_i') = 0$ where", "$Q_i = f^{\\deg_T(P_i)}P_i(T/f)$ which is a monic polynomial in", "$T$ with coefficients in $R[y_1, \\ldots, y_d]$ by our choice of $f$.", "Hence $R[y_1, \\ldots, y_d] \\subset S'$ is finite by", "Lemma \\ref{lemma-characterize-finite-in-terms-of-integral}.", "Since $S' \\subset S$ we have $S'_f \\subset S_f$ (localization is exact).", "On the other hand, the elements $x_i = x'_i/f$ in $S'_f$ generate $S_f$", "over $R_f$ and hence $S'_f \\to S_f$ is surjective. Whence", "$S'_f \\cong S_f$ and we win." ], "refs": [ "algebra-lemma-Noether-normalization", "algebra-lemma-finite-is-integral", "algebra-lemma-characterize-finite-in-terms-of-integral" ], "ref_ids": [ 1001, 482, 484 ] } ], "ref_ids": [] }, { "id": 1005, "type": "theorem", "label": "algebra-lemma-dimension-prime-polynomial-ring", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-prime-polynomial-ring", "contents": [ "Let $k$ be a field.", "Let $S$ be a finite type $k$ algebra which is an integral domain.", "Let $K$ be the field of fractions of $S$.", "Let $r = \\text{trdeg}(K/k)$ be the transcendence degree of $K$ over $k$.", "Then $\\dim(S) = r$. Moreover, the local ring of $S$ at every maximal", "ideal has dimension $r$." ], "refs": [], "proofs": [ { "contents": [ "We may write $S = k[x_1, \\ldots, x_n]/\\mathfrak p$.", "By Lemma \\ref{lemma-dimension-height-polynomial-ring}", "all local rings of $S$ at maximal ideals have the same dimension.", "Apply Lemma \\ref{lemma-Noether-normalization}.", "We get a finite injective ring map", "$$", "k[y_1, \\ldots, y_d] \\to S", "$$", "with $d = \\dim(S)$. Clearly, $k(y_1, \\ldots, y_d) \\subset K$", "is a finite extension and we win." ], "refs": [ "algebra-lemma-dimension-height-polynomial-ring", "algebra-lemma-Noether-normalization" ], "ref_ids": [ 993, 1001 ] } ], "ref_ids": [] }, { "id": 1006, "type": "theorem", "label": "algebra-lemma-tr-deg-specialization", "categories": [ "algebra" ], "title": "algebra-lemma-tr-deg-specialization", "contents": [ "Let $k$ be a field. Let $S$ be a finite type $k$-algebra.", "Let $\\mathfrak q \\subset \\mathfrak q' \\subset S$ be distinct", "prime ideals. Then", "$\\text{trdeg}_k\\ \\kappa(\\mathfrak q') < \\text{trdeg}_k\\ \\kappa(\\mathfrak q)$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-dimension-prime-polynomial-ring}", "we have $\\dim V(\\mathfrak q) = \\text{trdeg}_k\\ \\kappa(\\mathfrak q)$", "and similarly for $\\mathfrak q'$. Hence the result follows", "as the strict inclusion $V(\\mathfrak q') \\subset V(\\mathfrak q)$", "implies a strict inequality of dimensions." ], "refs": [ "algebra-lemma-dimension-prime-polynomial-ring" ], "ref_ids": [ 1005 ] } ], "ref_ids": [] }, { "id": 1007, "type": "theorem", "label": "algebra-lemma-dimension-at-a-point-finite-type-field", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-at-a-point-finite-type-field", "contents": [ "Let $k$ be a field.", "Let $S$ be a finite type $k$ algebra.", "Let $X = \\Spec(S)$.", "Let $\\mathfrak p \\subset S$ be a prime ideal,", "and let $x \\in X$ be the corresponding point.", "Then we have", "$$", "\\dim_x(X) = \\dim(S_{\\mathfrak p}) + \\text{trdeg}_k\\ \\kappa(\\mathfrak p).", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-dimension-prime-polynomial-ring} we know that", "$r = \\text{trdeg}_k\\ \\kappa(\\mathfrak p)$ is equal to the", "dimension of $V(\\mathfrak p)$.", "Pick any maximal chain of primes", "$\\mathfrak p \\subset \\mathfrak p_1 \\subset \\ldots \\subset \\mathfrak p_r$", "starting with $\\mathfrak p$ in $S$.", "This has length $r$ by Lemma \\ref{lemma-dimension-spell-it-out}.", "Let $\\mathfrak q_j$, $j \\in J$ be the minimal", "primes of $S$ which are contained in $\\mathfrak p$.", "These correspond $1-1$ to minimal primes in $S_{\\mathfrak p}$", "via the rule $\\mathfrak q_j \\mapsto \\mathfrak q_jS_{\\mathfrak p}$.", "By Lemma \\ref{lemma-dimension-at-a-point-finite-type-over-field}", "we know that $\\dim_x(X)$ is equal", "to the maximum of the dimensions of the rings $S/\\mathfrak q_j$.", "For each $j$ pick a maximal chain of primes", "$\\mathfrak q_j \\subset \\mathfrak p'_1 \\subset \\ldots \\subset \\mathfrak p'_{s(j)}", "= \\mathfrak p$.", "Then $\\dim(S_{\\mathfrak p}) = \\max_{j \\in J} s(j)$.", "Now, each chain", "$$", "\\mathfrak q_i \\subset \\mathfrak p'_1 \\subset \\ldots \\subset", "\\mathfrak p'_{s(j)} = \\mathfrak p \\subset", "\\mathfrak p_1 \\subset \\ldots \\subset \\mathfrak p_r", "$$", "is a maximal chain in $S/\\mathfrak q_j$, and by what was said", "before we have", "$\\dim_x(X) = \\max_{j \\in J} r + s(j)$.", "The lemma follows." ], "refs": [ "algebra-lemma-dimension-prime-polynomial-ring", "algebra-lemma-dimension-spell-it-out", "algebra-lemma-dimension-at-a-point-finite-type-over-field" ], "ref_ids": [ 1005, 994, 995 ] } ], "ref_ids": [] }, { "id": 1008, "type": "theorem", "label": "algebra-lemma-codimension", "categories": [ "algebra" ], "title": "algebra-lemma-codimension", "contents": [ "Let $k$ be a field.", "Let $S' \\to S$ be a surjection of finite type $k$ algebras.", "Let $\\mathfrak p \\subset S$ be a prime ideal,", "and let $\\mathfrak p'$ be the corresponding prime ideal of $S'$.", "Let $X = \\Spec(S)$, resp.\\ $X' = \\Spec(S')$,", "and let $x \\in X$, resp. $x'\\in X'$ be the point corresponding", "to $\\mathfrak p$, resp.\\ $\\mathfrak p'$.", "Then", "$$", "\\dim_{x'} X' - \\dim_x X =", "\\text{height}(\\mathfrak p') - \\text{height}(\\mathfrak p).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-dimension-at-a-point-finite-type-field}." ], "refs": [ "algebra-lemma-dimension-at-a-point-finite-type-field" ], "ref_ids": [ 1007 ] } ], "ref_ids": [] }, { "id": 1009, "type": "theorem", "label": "algebra-lemma-dimension-preserved-field-extension", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-preserved-field-extension", "contents": [ "Let $k$ be a field.", "Let $S$ be a finite type $k$-algebra.", "Let $k \\subset K$ be a field extension.", "Then $\\dim(S) = \\dim(K \\otimes_k S)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-Noether-normalization}", "there exists a finite injective map", "$k[y_1, \\ldots, y_d] \\to S$ with $d = \\dim(S)$.", "Since $K$ is flat over $k$ we also get a finite injective", "map $K[y_1, \\ldots, y_d] \\to K \\otimes_k S$.", "The result follows from Lemma \\ref{lemma-integral-sub-dim-equal}." ], "refs": [ "algebra-lemma-Noether-normalization", "algebra-lemma-integral-sub-dim-equal" ], "ref_ids": [ 1001, 985 ] } ], "ref_ids": [] }, { "id": 1010, "type": "theorem", "label": "algebra-lemma-dimension-at-a-point-preserved-field-extension", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-at-a-point-preserved-field-extension", "contents": [ "Let $k$ be a field.", "Let $S$ be a finite type $k$-algebra.", "Set $X = \\Spec(S)$.", "Let $k \\subset K$ be a field extension.", "Set $S_K = K \\otimes_k S$, and $X_K = \\Spec(S_K)$.", "Let $\\mathfrak q \\subset S$ be a prime corresponding to $x \\in X$", "and let $\\mathfrak q_K \\subset S_K$ be a prime corresponding", "to $x_K \\in X_K$ lying over $\\mathfrak q$.", "Then $\\dim_x X = \\dim_{x_K} X_K$." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $S = k[x_1, \\ldots, x_n]/I$.", "This gives a presentation", "$K \\otimes_k S = K[x_1, \\ldots, x_n]/(K \\otimes_k I)$.", "Let $\\mathfrak q_K' \\subset K[x_1, \\ldots, x_n]$,", "resp.\\ $\\mathfrak q' \\subset k[x_1, \\ldots, x_n]$ be", "the corresponding primes. Consider the following", "commutative diagram of Noetherian local rings", "$$", "\\xymatrix{", "K[x_1, \\ldots, x_n]_{\\mathfrak q_K'} \\ar[r] &", "(K \\otimes_k S)_{\\mathfrak q_K} \\\\", "k[x_1, \\ldots, x_n]_{\\mathfrak q'} \\ar[r] \\ar[u] &", "S_{\\mathfrak q} \\ar[u]", "}", "$$", "Both vertical arrows are flat because they are localizations of", "the flat ring maps $S \\to S_K$ and", "$k[x_1, \\ldots, x_n] \\to K[x_1, \\ldots, x_n]$.", "Moreover, the vertical arrows have the same fibre rings.", "Hence, we see from", "Lemma \\ref{lemma-dimension-base-fibre-equals-total} that", "$\\text{height}(\\mathfrak q') - \\text{height}(\\mathfrak q)", "= \\text{height}(\\mathfrak q_K') - \\text{height}(\\mathfrak q_K)$.", "Denote $x' \\in X' = \\Spec(k[x_1, \\ldots, x_n])$", "and $x'_K \\in X'_K = \\Spec(K[x_1, \\ldots, x_n])$", "the points corresponding to $\\mathfrak q'$ and", "$\\mathfrak q_K'$. By Lemma \\ref{lemma-codimension} and what we showed", "above we have", "\\begin{eqnarray*}", "n - \\dim_x X & = & \\dim_{x'} X' - \\dim_x X \\\\", "& = & \\text{height}(\\mathfrak q') - \\text{height}(\\mathfrak q) \\\\", "& = & \\text{height}(\\mathfrak q_K') - \\text{height}(\\mathfrak q_K) \\\\", "& = & \\dim_{x'_K} X'_K - \\dim_{x_K} X_K \\\\", "& = & n - \\dim_{x_K} X_K", "\\end{eqnarray*}", "and the lemma follows." ], "refs": [ "algebra-lemma-dimension-base-fibre-equals-total", "algebra-lemma-codimension" ], "ref_ids": [ 987, 1008 ] } ], "ref_ids": [] }, { "id": 1011, "type": "theorem", "label": "algebra-lemma-inequalities-under-field-extension", "categories": [ "algebra" ], "title": "algebra-lemma-inequalities-under-field-extension", "contents": [ "Let $k$ be a field. Let $S$ be a finite type $k$-algebra.", "Let $k \\subset K$ be a field extension. Set $S_K = K \\otimes_k S$.", "Let $\\mathfrak q \\subset S$ be a prime and let $\\mathfrak q_K \\subset S_K$", "be a prime lying over $\\mathfrak q$. Then", "$$", "\\dim (S_K \\otimes_S \\kappa(\\mathfrak q))_{\\mathfrak q_K} =", "\\dim (S_K)_{\\mathfrak q_K} - \\dim S_\\mathfrak q =", "\\text{trdeg}_k \\kappa(\\mathfrak q) - \\text{trdeg}_K \\kappa(\\mathfrak q_K)", "$$", "Moreover, given $\\mathfrak q$ we can always choose $\\mathfrak q_K$ such", "that the number above is zero." ], "refs": [], "proofs": [ { "contents": [ "Observe that $S_\\mathfrak q \\to (S_K)_{\\mathfrak q_K}$ is a flat", "local homomorphism of local Noetherian rings with special fibre", "$(S_K \\otimes_S \\kappa(\\mathfrak q))_{\\mathfrak q_K}$. Hence the first", "equality by Lemma \\ref{lemma-dimension-base-fibre-equals-total}.", "The second equality follows from the fact that we have", "$\\dim_x X = \\dim_{x_K} X_K$ with notation as in", "Lemma \\ref{lemma-dimension-at-a-point-preserved-field-extension}", "and we have", "$\\dim_x X = \\dim S_\\mathfrak q + \\text{trdeg}_k \\kappa(\\mathfrak q)$", "by Lemma \\ref{lemma-dimension-at-a-point-finite-type-field}", "and similarly for $\\dim_{x_K} X_K$.", "If we choose $\\mathfrak q_K$ minimal over $\\mathfrak q S_K$, then", "the dimension of the fibre ring will be zero." ], "refs": [ "algebra-lemma-dimension-base-fibre-equals-total", "algebra-lemma-dimension-at-a-point-preserved-field-extension", "algebra-lemma-dimension-at-a-point-finite-type-field" ], "ref_ids": [ 987, 1010, 1007 ] } ], "ref_ids": [] }, { "id": 1012, "type": "theorem", "label": "algebra-lemma-dimension-graded", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-graded", "contents": [ "Let $k$ be a field.", "Let $S$ be a finitely generated graded algebra over $k$.", "Assume $S_0 = k$. Let $P(T) \\in \\mathbf{Q}[T]$ be the polynomial", "such that $\\dim(S_d) = P(d)$ for all $d \\gg 0$. See", "Proposition \\ref{proposition-graded-hilbert-polynomial}.", "Then", "\\begin{enumerate}", "\\item The irrelevant ideal $S_{+}$ is a maximal ideal $\\mathfrak m$.", "\\item Any minimal prime of $S$ is a homogeneous ideal and is contained", "in $S_{+} = \\mathfrak m$.", "\\item We have $\\dim(S) = \\deg(P) + 1 = \\dim_x\\Spec(S)$", "(with the convention that $\\deg(0) = -1$)", "where $x$ is the point corresponding to the maximal ideal", "$S_{+} = \\mathfrak m$.", "\\item The Hilbert function of the local ring $R = S_{\\mathfrak m}$", "is equal to the Hilbert function of $S$.", "\\end{enumerate}" ], "refs": [ "algebra-proposition-graded-hilbert-polynomial" ], "proofs": [ { "contents": [ "The first statement is obvious.", "The second follows from Lemma \\ref{lemma-graded-ring-minimal-prime}.", "The equality $\\dim(S) = \\dim_x\\Spec(S)$ follows from the", "fact that every irreducible component passes through $x$ according", "to (2). Hence we may compute this dimension as the dimension of", "the local ring $R = S_{\\mathfrak m}$ with $\\mathfrak m = S_{+}$ by", "Lemma \\ref{lemma-dimension-closed-point-finite-type-field}.", "Since", "$\\mathfrak m^d/\\mathfrak m^{d + 1} \\cong \\mathfrak m^dR/\\mathfrak m^{d + 1}R$", "we see that the Hilbert function of the local ring $R$ is equal to the", "Hilbert function of $S$, which is (4). We conclude the last equality", "of (3) by Proposition \\ref{proposition-dimension}." ], "refs": [ "algebra-lemma-graded-ring-minimal-prime", "algebra-lemma-dimension-closed-point-finite-type-field", "algebra-proposition-dimension" ], "ref_ids": [ 664, 996, 1411 ] } ], "ref_ids": [ 1408 ] }, { "id": 1013, "type": "theorem", "label": "algebra-lemma-generic-flatness-Noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-generic-flatness-Noetherian", "contents": [ "Let $R \\to S$ be a ring map.", "Let $M$ be an $S$-module.", "Assume", "\\begin{enumerate}", "\\item $R$ is Noetherian,", "\\item $R$ is a domain,", "\\item $R \\to S$ is of finite type, and", "\\item $M$ is a finite type $S$-module.", "\\end{enumerate}", "Then there exists a nonzero $f \\in R$ such that", "$M_f$ is a free $R_f$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $K$ be the fraction field of $R$. Set $S_K = K \\otimes_R S$. This", "is an algebra of finite type over $K$. We will argue by induction on", "$d = \\dim(S_K)$ (which is finite for example by Noether normalization, see", "Section \\ref{section-Noether-normalization}).", "Fix $d \\geq 0$.", "Assume we know that the lemma holds in all cases where $\\dim(S_K) < d$.", "\\medskip\\noindent", "Suppose given $R \\to S$ and $M$ as in the lemma with $\\dim(S_K) = d$. By", "Lemma \\ref{lemma-filter-Noetherian-module}", "there exists a filtration", "$0 \\subset M_1 \\subset M_2 \\subset \\ldots \\subset M_n = M$", "so that $M_i/M_{i - 1}$ is isomorphic to $S/\\mathfrak q$", "for some prime $\\mathfrak q$ of $S$. Note that", "$\\dim((S/\\mathfrak q)_K) \\leq \\dim(S_K)$. Also, note that an extension of", "free modules is free (see basic notion \\ref{item-extension-free}).", "Thus we may assume $M = S$ and that $S$ is a domain of finite type over $R$.", "\\medskip\\noindent", "If $R \\to S$ has a nontrivial kernel, then take a nonzero $f \\in R$ in", "this kernel. In this case $S_f = 0$ and the lemma holds. (This is really", "the case $d = -1$ and the start of the induction.) Hence we", "may assume that $R \\to S$ is a finite type extension of Noetherian domains.", "\\medskip\\noindent", "Apply Lemma \\ref{lemma-Noether-normalization-over-a-domain}", "and replace $R$ by $R_f$ (with $f$ as in the lemma) to get a", "factorization", "$$", "R \\subset R[y_1, \\ldots, y_d] \\subset S", "$$", "where the second extension is finite. Choose $z_1, \\ldots, z_r \\in S$ which", "form a basis for the fraction field of $S$ over the fraction field of", "$R[y_1, \\ldots, y_d]$. This gives a short exact sequence", "$$", "0 \\to", "R[y_1, \\ldots, y_d]^{\\oplus r} \\xrightarrow{(z_1, \\ldots, z_r)}", "S \\to N \\to 0", "$$", "By construction $N$ is a finite $R[y_1, \\ldots, y_d]$-module whose", "support does not contain the generic point $(0)$ of", "$\\Spec(R[y_1, \\ldots, y_d])$. By", "Lemma \\ref{lemma-support-closed}", "there exists a nonzero $g \\in R[y_1, \\ldots, y_d]$ such that", "$g$ annihilates $N$, so we may view $N$ as a finite module over", "$S' = R[y_1, \\ldots, y_d]/(g)$. Since $\\dim(S'_K) < d$ by induction", "there exists a nonzero $f \\in R$ such that $N_f$ is a free", "$R_f$-module. Since", "$(R[y_1, \\ldots, y_d])_f \\cong R_f[y_1, \\ldots, y_d]$ is free", "also we conclude by the already mentioned fact that an extension", "of free modules is free." ], "refs": [ "algebra-lemma-filter-Noetherian-module", "algebra-lemma-Noether-normalization-over-a-domain", "algebra-lemma-support-closed" ], "ref_ids": [ 691, 1004, 543 ] } ], "ref_ids": [] }, { "id": 1014, "type": "theorem", "label": "algebra-lemma-generic-flatness-finitely-presented", "categories": [ "algebra" ], "title": "algebra-lemma-generic-flatness-finitely-presented", "contents": [ "\\begin{slogan}", "Generic freeness.", "\\end{slogan}", "Let $R \\to S$ be a ring map.", "Let $M$ be an $S$-module.", "Assume", "\\begin{enumerate}", "\\item $R$ is a domain,", "\\item $R \\to S$ is of finite presentation, and", "\\item $M$ is an $S$-module of finite presentation.", "\\end{enumerate}", "Then there exists a nonzero $f \\in R$ such that", "$M_f$ is a free $R_f$-module." ], "refs": [], "proofs": [ { "contents": [ "Write $S = R[x_1, \\ldots, x_n]/(g_1, \\ldots, g_m)$.", "For $g \\in R[x_1, \\ldots, x_n]$ denote $\\overline{g}$ its image in $S$.", "We may write $M = S^{\\oplus t}/\\sum Sn_i$ for some $n_i \\in S^{\\oplus t}$.", "Write $n_i = (\\overline{g}_{i1}, \\ldots, \\overline{g}_{it})$ for some", "$g_{ij} \\in R[x_1, \\ldots, x_n]$. Let $R_0 \\subset R$ be the subring", "generated by all the coefficients of all the elements", "$g_i, g_{ij} \\in R[x_1, \\ldots, x_n]$. Define", "$S_0 = R_0[x_1, \\ldots, x_n]/(g_1, \\ldots, g_m)$.", "Define $M_0 = S_0^{\\oplus t}/\\sum S_0n_i$.", "Then $R_0$ is a domain of finite type over $\\mathbf{Z}$ and hence", "Noetherian (see", "Lemma \\ref{lemma-Noetherian-permanence}).", "Moreover via the injection $R_0 \\to R$ we have $S \\cong R \\otimes_{R_0} S_0$", "and $M \\cong R \\otimes_{R_0} M_0$. Applying", "Lemma \\ref{lemma-generic-flatness-Noetherian}", "we obtain a nonzero $f \\in R_0$ such that $(M_0)_f$ is a free", "$(R_0)_f$-module. Hence $M_f = R_f \\otimes_{(R_0)_f} (M_0)_f$", "is a free $R_f$-module." ], "refs": [ "algebra-lemma-Noetherian-permanence", "algebra-lemma-generic-flatness-Noetherian" ], "ref_ids": [ 448, 1013 ] } ], "ref_ids": [] }, { "id": 1015, "type": "theorem", "label": "algebra-lemma-generic-flatness", "categories": [ "algebra" ], "title": "algebra-lemma-generic-flatness", "contents": [ "Let $R \\to S$ be a ring map.", "Let $M$ be an $S$-module.", "Assume", "\\begin{enumerate}", "\\item $R$ is a domain,", "\\item $R \\to S$ is of finite type, and", "\\item $M$ is a finite type $S$-module.", "\\end{enumerate}", "Then there exists a nonzero $f \\in R$ such that", "\\begin{enumerate}", "\\item[(a)] $M_f$ and $S_f$ are free as $R_f$-modules, and", "\\item[(b)] $S_f$ is a finitely presented $R_f$-algebra and $M_f$ is a", "finitely presented $S_f$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We first prove the lemma for $S = R[x_1, \\ldots, x_n]$, and then", "we deduce the result in general.", "\\medskip\\noindent", "Assume $S = R[x_1, \\ldots, x_n]$.", "Choose elements $m_1, \\ldots, m_t$ which generate $M$. This gives", "a short exact sequence", "$$", "0 \\to N \\to S^{\\oplus t} \\xrightarrow{(m_1, \\ldots, m_t)} M \\to 0.", "$$", "Denote $K$ the fraction field of $R$. Denote", "$S_K = K \\otimes_R S = K[x_1, \\ldots, x_n]$, and similarly", "$N_K = K \\otimes_R N$, $M_K = K \\otimes_R M$.", "As $R \\to K$ is flat the sequence remains exact after tensoring with $K$.", "As $S_K = K[x_1, \\ldots, x_n]$ is a Noetherian ring (see", "Lemma \\ref{lemma-Noetherian-permanence})", "we can find finitely many elements $n'_1, \\ldots, n'_s \\in N_K$", "which generate it. Choose $n_1, \\ldots, n_r \\in N$ such that", "$n'_i = \\sum a_{ij}n_j$ for some $a_{ij} \\in K$. Set", "$$", "M' = S^{\\oplus t}/\\sum\\nolimits_{i = 1, \\ldots, r} Sn_i", "$$", "By construction $M'$ is a finitely presented $S$-module, and", "there is a surjection $M' \\to M$ which induces an isomorphism", "$M'_K \\cong M_K$. We may apply", "Lemma \\ref{lemma-generic-flatness-finitely-presented}", "to $R \\to S$ and $M'$ and we find an $f \\in R$ such that $M'_f$", "is a free $R_f$-module. Thus $M'_f \\to M_f$ is a surjection of", "modules over the domain $R_f$ where the source is a free module", "and which becomes an isomorphism upon tensoring with $K$.", "Thus it is injective as $M'_f \\subset M'_K$ as it is free over", "the domain $R_f$. Hence $M'_f \\to M_f$ is an isomorphism and the", "result is proved.", "\\medskip\\noindent", "For the general case, choose a surjection $R[x_1, \\ldots, x_n] \\to S$.", "Think of both $S$ and $M$ as finite modules over $R[x_1, \\ldots, x_n]$.", "By the special case proved above there exists a nonzero $f \\in R$", "such that both $S_f$ and $M_f$ are free as $R_f$-modules and finitely", "presented as $R_f[x_1, \\ldots, x_n]$-modules. Clearly this implies that", "$S_f$ is a finitely presented $R_f$-algebra and that $M_f$ is a", "finitely presented $S_f$-module." ], "refs": [ "algebra-lemma-Noetherian-permanence", "algebra-lemma-generic-flatness-finitely-presented" ], "ref_ids": [ 448, 1014 ] } ], "ref_ids": [] }, { "id": 1016, "type": "theorem", "label": "algebra-lemma-generic-flatness-locus-extension", "categories": [ "algebra" ], "title": "algebra-lemma-generic-flatness-locus-extension", "contents": [ "Let $R \\to S$ be a ring map.", "Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a short exact sequence", "of $S$-modules.", "Then", "$$", "U(R \\to S, M_1) \\cap U(R \\to S, M_3) \\subset U(R \\to S, M_2).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $u \\in U(R \\to S, M_1) \\cap U(R \\to S, M_3)$. Choose", "$f_1, f_3 \\in R$ such that $u \\in D(f_1)$, $u \\in D(f_3)$ and", "such that (\\ref{equation-flat-and-finitely-presented}) holds for", "$f_1$ and $M_1$ and for $f_3$ and $M_3$. Then set $f = f_1f_3$.", "Then $u \\in D(f)$ and (\\ref{equation-flat-and-finitely-presented})", "holds for $f$ and both $M_1$ and $M_3$. An extension of free modules", "is free, and an extension of finitely presented modules is finitely presented", "(Lemma \\ref{lemma-extension}). Hence we see that", "(\\ref{equation-flat-and-finitely-presented}) holds for $f$ and $M_2$.", "Thus $u \\in U(R \\to S, M_2)$ and we win." ], "refs": [ "algebra-lemma-extension" ], "ref_ids": [ 330 ] } ], "ref_ids": [] }, { "id": 1017, "type": "theorem", "label": "algebra-lemma-generic-flatness-locus-localize", "categories": [ "algebra" ], "title": "algebra-lemma-generic-flatness-locus-localize", "contents": [ "Let $R \\to S$ be a ring map.", "Let $M$ be an $S$-module.", "Let $f \\in R$.", "Using the identification $\\Spec(R_f) = D(f)$ we have", "$U(R_f \\to S_f, M_f) = D(f) \\cap U(R \\to S, M)$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $u \\in U(R_f \\to S_f, M_f)$. Then there exists an", "element $g \\in R_f$ such that $u \\in D(g)$ and such that", "(\\ref{equation-flat-and-finitely-presented})", "holds for the pair $((R_f)_g \\to (S_f)_g, (M_f)_g)$.", "Write $g = a/f^n$ for some $a \\in R$. Set $h = af$.", "Then $R_h = (R_f)_g$, $S_h = (S_f)_g$, and $M_h = (M_f)_g$.", "Moreover $u \\in D(h)$. Hence $u \\in U(R \\to S, M)$.", "Conversely, suppose that $u \\in D(f) \\cap U(R \\to S, M)$.", "Then there exists an element $g \\in R$ such that $u \\in D(g)$ and such that", "(\\ref{equation-flat-and-finitely-presented})", "holds for the pair $(R_g \\to S_g, M_g)$.", "Then it is clear that (\\ref{equation-flat-and-finitely-presented})", "also holds for the pair", "$(R_{fg} \\to S_{fg}, M_{fg}) = ((R_f)_g \\to (S_f)_g, (M_f)_g)$.", "Hence $u \\in U(R_f \\to S_f, M_f)$ and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1018, "type": "theorem", "label": "algebra-lemma-generic-flatness-locus-reduce", "categories": [ "algebra" ], "title": "algebra-lemma-generic-flatness-locus-reduce", "contents": [ "Let $R \\to S$ be a ring map.", "Let $M$ be an $S$-module.", "Let $U \\subset \\Spec(R)$ be a dense open.", "Assume there is a covering $U = \\bigcup_{i \\in I} D(f_i)$ of", "opens such that $U(R_{f_i} \\to S_{f_i}, M_{f_i})$ is dense in", "$D(f_i)$ for each $i \\in I$. Then $U(R \\to S, M)$ is dense in", "$\\Spec(R)$." ], "refs": [], "proofs": [ { "contents": [ "In view of", "Lemma \\ref{lemma-generic-flatness-locus-localize}", "this is a purely topological statement. Namely, by that lemma", "we see that $U(R \\to S, M) \\cap D(f_i)$ is dense in $D(f_i)$", "for each $i \\in I$. By", "Topology, Lemma \\ref{topology-lemma-nowhere-dense-local}", "we see that $U(R \\to S, M) \\cap U$ is dense in $U$.", "Since $U$ is dense in $\\Spec(R)$ we conclude that $U(R \\to S, M)$", "is dense in $\\Spec(R)$." ], "refs": [ "algebra-lemma-generic-flatness-locus-localize", "topology-lemma-nowhere-dense-local" ], "ref_ids": [ 1017, 8296 ] } ], "ref_ids": [] }, { "id": 1019, "type": "theorem", "label": "algebra-lemma-generic-flatness-reduced", "categories": [ "algebra" ], "title": "algebra-lemma-generic-flatness-reduced", "contents": [ "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module.", "Assume", "\\begin{enumerate}", "\\item $R \\to S$ is of finite type,", "\\item $M$ is a finite $S$-module, and", "\\item $R$ is reduced.", "\\end{enumerate}", "Then there exists a subset $U \\subset \\Spec(R)$ such that", "\\begin{enumerate}", "\\item $U$ is open and dense in $\\Spec(R)$,", "\\item for every $u \\in U$ there exists an $f \\in R$ such", "that $u \\in D(f) \\subset U$ and such that we have", "\\begin{enumerate}", "\\item $M_f$ and $S_f$ are free over $R_f$,", "\\item $S_f$ is a finitely presented $R_f$-algebra, and", "\\item $M_f$ is a finitely presented $S_f$-module.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that the lemma is equivalent to the statement that the open", "$U(R \\to S, M)$, see Equation (\\ref{equation-good-locus}), is dense", "in $\\Spec(R)$. We first prove the lemma for", "$S = R[x_1, \\ldots, x_n]$, and then", "we deduce the result in general.", "\\medskip\\noindent", "Proof of the case $S = R[x_1, \\ldots, x_n]$ and $M$ any finite module", "over $S$. Note that in this case $S_f = R_f[x_1, \\ldots, x_n]$", "is free and of finite presentation over $R_f$, so we do not have", "to worry about the conditions regarding $S$,", "only those that concern $M$. We will use induction on $n$.", "\\medskip\\noindent", "There exists a finite filtration", "$$", "0 \\subset M_1 \\subset M_2 \\subset \\ldots \\subset M_t = M", "$$", "such that $M_i/M_{i - 1} \\cong S/J_i$ for some ideal $J_i \\subset S$, see", "Lemma \\ref{lemma-trivial-filter-finite-module}. Since", "a finite intersection of dense opens is dense open,", "we see from", "Lemma \\ref{lemma-generic-flatness-locus-extension}", "that it suffices to prove the lemma for each of the modules $R/J_i$.", "Hence we may assume that $M = S/J$ for some ideal $J$ of", "$S = R[x_1, \\ldots, x_n]$.", "\\medskip\\noindent", "Let $I \\subset R$ be the ideal generated by the coefficients of", "elements of $J$. Let $U_1 = \\Spec(R) \\setminus V(I)$ and", "let", "$$", "U_2 = \\Spec(R) \\setminus \\overline{U_1}.", "$$", "Then it is clear that $U = U_1 \\cup U_2$ is dense in $\\Spec(R)$.", "Let $f \\in R$ be an element such that either (a) $D(f) \\subset U_1$ or", "(b) $D(f) \\subset U_2$. If for any such $f$", "the lemma holds for the pair $(R_f \\to R_f[x_1, \\ldots, x_n], M_f)$", "then by", "Lemma \\ref{lemma-generic-flatness-locus-reduce}", "we see that $U(R \\to S, M)$ is dense in $\\Spec(R)$.", "Hence we may assume either (a) $I = R$, or (b) $V(I) = \\Spec(R)$.", "\\medskip\\noindent", "In case (b) we actually have $I = 0$ as $R$ is reduced! Hence $J = 0$", "and $M = S$ and the lemma holds in this case.", "\\medskip\\noindent", "In case (a) we have to do a little bit more work. Note that every element", "of $I$ is actually the coefficient of a monomial of an element of $J$, because", "the set of coefficients of elements of $J$ forms an ideal (details omitted).", "Hence we find an element", "$$", "g = \\sum\\nolimits_{K \\in E} a_K x^K \\in J", "$$", "where $E$ is a finite set of multi-indices $K = (k_1, \\ldots, k_n)$", "with at least one coefficient $a_{K_0}$ a unit in $R$. Actually we can", "find one which has a coefficient equal to $1$ as $1 \\in I$ in case (a).", "Let $m = \\#\\{K \\in E \\mid a_K \\text{ is not a unit}\\}$.", "Note that $0 \\leq m \\leq \\# E - 1$.", "We will argue by induction on $m$.", "\\medskip\\noindent", "The case $m = 0$. In this case all the coefficients $a_K$, $K \\in E$", "of $g$ are units and $E \\not = \\emptyset$.", "If $E = \\{K_0\\}$ is a singleton and $K_0 = (0, \\ldots, 0)$, then $g$", "is a unit and $J = S$ so the result holds for sure. (This happens in", "particular when $n = 0$ and it provides the base case of the induction", "on $n$.) If not $E = \\{(0, \\ldots, 0)\\}$, then at least one $K$ is not", "equal to $(0, \\ldots, 0)$, i.e., $g \\not \\in R$. At this point we employ", "the usual trick of Noether normalization. Namely, we consider", "$$", "G(y_1, \\ldots, y_n) =", "g(y_1 + y_n^{e_1}, y_2 + y_n^{e_2}, \\ldots, y_{n - 1} + y_n^{e_{n - 1}}, y_n)", "$$", "with $0 \\ll e_{n -1} \\ll e_{n - 2} \\ll \\ldots \\ll e_1$. By", "Lemma \\ref{lemma-helper-polynomial}", "it follows that $G(y_1, \\ldots, y_n)$ as a polynomial in $y_n$", "looks like", "$$", "a_K y_n^{k_n + \\sum_{i = 1, \\ldots, n - 1} e_i k_i} +", "\\text{lower order terms in }y_n", "$$", "As $a_K$ is a unit we conclude that $M = R[x_1, \\ldots, x_n]/J$ is", "finite over $R[y_1, \\ldots, y_{n - 1}]$. Hence", "$U(R \\to R[x_1, \\ldots, x_n], M) = U(R \\to R[y_1, \\ldots, y_{n - 1}], M)$", "and we win by induction on $n$.", "\\medskip\\noindent", "The case $m > 0$. Pick a multi-index $K \\in E$ such that $a_K$ is", "not a unit. As before set", "$U_1 = \\Spec(R_{a_K}) = \\Spec(R) \\setminus V(a_K)$", "and set", "$$", "U_2 = \\Spec(R) \\setminus \\overline{U_1}.", "$$", "Then it is clear that $U = U_1 \\cup U_2$ is dense in $\\Spec(R)$.", "Let $f \\in R$ be an element such that either (a) $D(f) \\subset U_1$ or", "(b) $D(f) \\subset U_2$. If for any such $f$", "the lemma holds for the pair $(R_f \\to R_f[x_1, \\ldots, x_n], M_f)$", "then by", "Lemma \\ref{lemma-generic-flatness-locus-reduce}", "we see that $U(R \\to S, M)$ is dense in $\\Spec(R)$.", "Hence we may assume either (a) $a_KR = R$, or (b) $V(a_K) = \\Spec(R)$.", "In case (a) the number $m$ drops, as $a_K$ has turned into a unit.", "In case (b), since $R$ is reduced, we conclude that $a_K = 0$.", "Hence the set $E$ decreases so the number $m$ drops as well.", "In both cases we win by induction on $m$.", "\\medskip\\noindent", "At this point we have proven the lemma in case $S = R[x_1, \\ldots, x_n]$.", "Assume that $(R \\to S, M)$ is an arbitrary pair satisfying the conditions of", "the lemma. Choose a surjection $R[x_1, \\ldots, x_n] \\to S$. Observe that,", "with the notation introduced in (\\ref{equation-good-locus}), we have", "$$", "U(R \\to S, M) =", "U(R \\to R[x_1, \\ldots, x_n], S)", "\\cap", "U(R \\to R[x_1, \\ldots, x_n], M)", "$$", "Hence as we've just finished proving the right two opens are dense also", "the open on the left is dense." ], "refs": [ "algebra-lemma-trivial-filter-finite-module", "algebra-lemma-generic-flatness-locus-extension", "algebra-lemma-generic-flatness-locus-reduce", "algebra-lemma-helper-polynomial", "algebra-lemma-generic-flatness-locus-reduce" ], "ref_ids": [ 331, 1016, 1018, 999, 1018 ] } ], "ref_ids": [] }, { "id": 1020, "type": "theorem", "label": "algebra-lemma-dominate-by-dimension-1", "categories": [ "algebra" ], "title": "algebra-lemma-dominate-by-dimension-1", "contents": [ "Let $R$ be a local Noetherian domain with fraction field $K$.", "Assume $R$ is not a field.", "Then there exist $R \\subset R' \\subset K$ with", "\\begin{enumerate}", "\\item $R'$ local Noetherian of dimension $1$,", "\\item $R \\to R'$ a local ring map, i.e., $R'$ dominates $R$, and", "\\item $R \\to R'$ essentially of finite type.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose any valuation ring $A \\subset K$ dominating $R$ (which", "exist by Lemma \\ref{lemma-dominate}).", "Denote $v$ the corresponding valuation.", "Let $x_1, \\ldots, x_r$ be a minimal set of generators of the", "maximal ideal $\\mathfrak m$ of $R$. We may and do assume that", "$v(x_r) = \\min\\{v(x_1), \\ldots, v(x_r)\\}$. Consider the ring", "$$", "S = R[x_1/x_r, x_2/x_r, \\ldots, x_{r - 1}/x_r] \\subset K.", "$$", "Note that $\\mathfrak mS = x_rS$ is a principal ideal.", "Note that $S \\subset A$ and that $v(x_r) > 0$, hence we see", "that $x_rS \\not = S$. Choose a minimal prime $\\mathfrak q$", "over $x_rS$. Then $\\text{height}(\\mathfrak q) = 1$ by", "Lemma \\ref{lemma-minimal-over-1}", "and $\\mathfrak q$ lies over $\\mathfrak m$. Hence we", "see that $R' = S_{\\mathfrak q}$ is a solution." ], "refs": [ "algebra-lemma-dominate", "algebra-lemma-minimal-over-1" ], "ref_ids": [ 608, 683 ] } ], "ref_ids": [] }, { "id": 1021, "type": "theorem", "label": "algebra-lemma-hart-serre-loc-thm", "categories": [ "algebra" ], "title": "algebra-lemma-hart-serre-loc-thm", "contents": [ "\\begin{reference}", "This is taken from a forthcoming paper by", "J\\'anos Koll\\'ar entitled ``Variants of normality for Noetherian schemes''.", "\\end{reference}", "Let $(R, \\mathfrak m)$ be a local Noetherian ring.", "Then exactly one of the following holds:", "\\begin{enumerate}", "\\item $(R, \\mathfrak m)$ is Artinian,", "\\item $(R, \\mathfrak m)$ is regular of dimension $1$,", "\\item $\\text{depth}(R) \\geq 2$, or", "\\item there exists a finite ring map $R \\to R'$ which is not", "an isomorphism whose kernel and cokernel are annihilated by a power", "of $\\mathfrak m$ such that $\\mathfrak m$ is not an associated", "prime of $R'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $(R, \\mathfrak m)$ is not Artinian if and only if", "$V(\\mathfrak m) \\subset \\Spec(R)$ is nowhere dense. See", "Proposition \\ref{proposition-dimension-zero-ring}. We assume this from now on.", "\\medskip\\noindent", "Let $J \\subset R$ be the largest ideal killed by a power of $\\mathfrak m$.", "If $J \\not = 0$ then $R \\to R/J$ shows that $(R, \\mathfrak m)$", "is as in (4).", "\\medskip\\noindent", "Otherwise $J = 0$. In particular $\\mathfrak m$ is not an associated prime", "of $R$ and we see that there is a nonzerodivisor $x \\in \\mathfrak m$ by", "Lemma \\ref{lemma-ideal-nonzerodivisor}. If $\\mathfrak m$", "is not an associated prime of $R/xR$ then $\\text{depth}(R) \\geq 2$", "by the same lemma. Thus we are left with the case when there is an", "$y \\in R$, $y \\not \\in xR$ such that $y \\mathfrak m \\subset xR$.", "\\medskip\\noindent", "If $y \\mathfrak m \\subset x \\mathfrak m$ then we can consider the", "map $\\varphi : \\mathfrak m \\to \\mathfrak m$, $f \\mapsto yf/x$", "(well defined as $x$ is a nonzerodivisor). By the determinantal trick", "of Lemma \\ref{lemma-charpoly-module} there exists a monic", "polynomial $P$ with coefficients in $R$ such that $P(\\varphi) = 0$.", "We conclude that $P(y/x) = 0$ in $R_x$.", "Let $R' \\subset R_x$ be the ring generated by", "$R$ and $y/x$. Then $R \\subset R'$ and $R'/R$ is a finite $R$-module", "annihilated by a power of $\\mathfrak m$. Thus $R$ is as in (4).", "\\medskip\\noindent", "Otherwise there is a $t \\in \\mathfrak m$ such that $y t = u x$", "for some unit $u$ of $R$. After replacing $t$ by $u^{-1}t$", "we get $yt = x$. In particular $y$ is a nonzerodivisor.", "For any $t' \\in \\mathfrak m$ we have $y t' = x s$ for some $s \\in R$.", "Thus $y (t' - s t ) = x s - x s = 0$.", "Since $y$ is not a zero-divisor this implies that $t' = ts$ and so", "$\\mathfrak m = (t)$. Thus $(R, \\mathfrak m)$ is regular of dimension 1." ], "refs": [ "algebra-proposition-dimension-zero-ring", "algebra-lemma-ideal-nonzerodivisor", "algebra-lemma-charpoly-module" ], "ref_ids": [ 1410, 712, 386 ] } ], "ref_ids": [] }, { "id": 1022, "type": "theorem", "label": "algebra-lemma-nonregular-dimension-one", "categories": [ "algebra" ], "title": "algebra-lemma-nonregular-dimension-one", "contents": [ "Let $R$ be a local ring with maximal ideal $\\mathfrak m$.", "Assume $R$ is Noetherian, has dimension $1$, and that", "$\\dim(\\mathfrak m/\\mathfrak m^2) > 1$. Then there exists", "a ring map $R \\to R'$ such that", "\\begin{enumerate}", "\\item $R \\to R'$ is finite,", "\\item $R \\to R'$ is not an isomorphism,", "\\item the kernel and cokernel of $R \\to R'$ are annihilated by", "a power of $\\mathfrak m$, and", "\\item $\\mathfrak m$ is not an associated prime of $R'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-hart-serre-loc-thm}", "and the fact that $R$ is not Artinian, not regular, and", "does not have depth $\\geq 2$ (the last part because the", "depth does not exceed the dimension by", "Lemma \\ref{lemma-bound-depth})." ], "refs": [ "algebra-lemma-hart-serre-loc-thm", "algebra-lemma-bound-depth" ], "ref_ids": [ 1021, 770 ] } ], "ref_ids": [] }, { "id": 1023, "type": "theorem", "label": "algebra-lemma-characterize-dvr", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-dvr", "contents": [ "Let $A$ be a ring. The following are equivalent.", "\\begin{enumerate}", "\\item The ring $A$ is a discrete valuation ring.", "\\item The ring $A$ is a valuation ring and Noetherian but not a field.", "\\item The ring $A$ is a regular local ring of dimension $1$.", "\\item The ring $A$ is a Noetherian local domain with maximal ideal", "$\\mathfrak m$ generated by a single nonzero element.", "\\item The ring $A$ is a Noetherian local normal domain of dimension $1$.", "\\end{enumerate}", "In this case if $\\pi$ is a generator of the maximal ideal of", "$A$, then every element of $A$ can be uniquely written as", "$u\\pi^n$, where $u \\in A$ is a unit." ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is", "Lemma \\ref{lemma-valuation-ring-Noetherian-discrete}.", "Moreover, in the proof of Lemma \\ref{lemma-valuation-ring-Noetherian-discrete}", "we saw that if $A$ is a discrete valuation ring, then $A$ is a PID, hence (3).", "Note that a regular local ring is a domain (see", "Lemma \\ref{lemma-regular-domain}). Using this the equivalence of (3) and (4)", "follows from dimension theory, see Section \\ref{section-dimension}.", "\\medskip\\noindent", "Assume (3) and let $\\pi$ be a generator of the maximal ideal $\\mathfrak m$.", "For all $n \\geq 0$ we have", "$\\dim_{A/\\mathfrak m} \\mathfrak m^n/\\mathfrak m^{n + 1} = 1$", "because it is generated by $\\pi^n$ (and it cannot be zero).", "In particular $\\mathfrak m^n = (\\pi^n)$ and", "the graded ring $\\bigoplus \\mathfrak m^n/\\mathfrak m^{n + 1}$", "is isomorphic to the polynomial ring $A/\\mathfrak m[T]$.", "For $x \\in A \\setminus \\{0\\}$ define", "$v(x) = \\max\\{n \\mid x \\in \\mathfrak m^n\\}$.", "In other words $x = u \\pi^{v(x)}$ with $u \\in A^*$.", "By the remarks above we have $v(xy) = v(x) + v(y)$", "for all $x, y \\in A \\setminus \\{0\\}$. We extend this to the field of fractions", "$K$ of $A$ by setting $v(a/b) = v(a) - v(b)$ (well defined by multiplicativity", "shown above). Then it is clear that $A$ is the set of elements of", "$K$ which have valuation $\\geq 0$. Hence we see that $A$ is a", "valuation ring by Lemma \\ref{lemma-valuation-valuation-ring}.", "\\medskip\\noindent", "A valuation ring is a normal domain by Lemma \\ref{lemma-valuation-ring-normal}.", "Hence we see that the equivalent conditions (1) -- (3) imply", "(5). Assume (5). Suppose that $\\mathfrak m$ cannot be generated", "by $1$ element to get a contradiction.", "Then Lemma \\ref{lemma-nonregular-dimension-one} implies there is a finite", "ring map $A \\to A'$ which is an isomorphism after inverting", "any nonzero element of $\\mathfrak m$ but not an isomorphism.", "In particular we may identify $A'$ with a subset of the fraction field of $A$.", "Since $A \\to A'$ is finite it is integral (see", "Lemma \\ref{lemma-finite-is-integral}).", "Since $A$ is normal we get $A = A'$ a contradiction." ], "refs": [ "algebra-lemma-valuation-ring-Noetherian-discrete", "algebra-lemma-valuation-ring-Noetherian-discrete", "algebra-lemma-regular-domain", "algebra-lemma-valuation-valuation-ring", "algebra-lemma-valuation-ring-normal", "algebra-lemma-nonregular-dimension-one", "algebra-lemma-finite-is-integral" ], "ref_ids": [ 623, 623, 940, 621, 616, 1022, 482 ] } ], "ref_ids": [] }, { "id": 1024, "type": "theorem", "label": "algebra-lemma-finite-length", "categories": [ "algebra" ], "title": "algebra-lemma-finite-length", "contents": [ "Let $R$ be a domain with fraction field $K$.", "Let $M$ be an $R$-submodule of $K^{\\oplus r}$.", "Assume $R$ is local Noetherian of dimension $1$.", "For any nonzero $x \\in R$ we have $\\text{length}_R(R/xR) < \\infty$", "and", "$$", "\\text{length}_R(M/xM) \\leq r \\cdot \\text{length}_R(R/xR).", "$$" ], "refs": [], "proofs": [ { "contents": [ "If $x$ is a unit then the result is true. Hence we may assume", "$x \\in \\mathfrak m$ the maximal ideal of $R$. Since $x$ is not", "zero and $R$ is a domain we have $\\dim(R/xR) = 0$, and hence", "$R/xR$ has finite length. Consider $M \\subset K^{\\oplus r}$ as", "in the lemma. We may assume that the elements of $M$ generate", "$K^{\\oplus r}$ as a $K$-vector space after replacing $K^{\\oplus r}$", "by a smaller subspace if necessary.", "\\medskip\\noindent", "Suppose first that $M$ is a finite $R$-module. In that case we can clear", "denominators and assume $M \\subset R^{\\oplus r}$. Since", "$M$ generates $K^{\\oplus r}$ as a vectors space we see that", "$R^{\\oplus r}/M$ has finite length. In particular there exists", "an integer $c \\geq 0$ such that $x^cR^{\\oplus r} \\subset M$.", "Note that $M \\supset xM \\supset x^2M \\supset \\ldots$ is a sequence of", "modules with successive quotients each isomorphic to $M/xM$. Hence", "we see that", "$$", "n \\text{length}_R(M/xM) = \\text{length}_R(M/x^nM).", "$$", "The same argument for $M = R^{\\oplus r}$ shows that", "$$", "n \\text{length}_R(R^{\\oplus r}/xR^{\\oplus r}) =", "\\text{length}_R(R^{\\oplus r}/x^nR^{\\oplus r}).", "$$", "By our choice of $c$ above we see that $x^nM$ is sandwiched between", "$x^n R^{\\oplus r}$ and $x^{n + c}R^{\\oplus r}$. This easily gives that", "$$", "r(n + c) \\text{length}_R(R/xR)", "\\geq", "n \\text{length}_R(M/xM)", "\\geq", "r (n - c) \\text{length}_R(R/xR)", "$$", "Hence in the finite case we actually get the result of the lemma with", "equality.", "\\medskip\\noindent", "Suppose now that $M$ is not finite. Suppose that the length of $M/xM$ is", "$\\geq k$ for some natural number $k$. Then we can find", "$$", "0 \\subset N_0 \\subset N_1 \\subset N_2 \\subset \\ldots N_k \\subset M/xM", "$$", "with $N_i \\not = N_{i + 1}$ for $i = 0, \\ldots k - 1$.", "Choose an element $m_i \\in M$ whose congruence class mod $xM$ falls", "into $N_i$ but not into $N_{i - 1}$ for $i = 1, \\ldots, k$.", "Consider the finite $R$-module $M' = Rm_1 + \\ldots + Rm_k \\subset M$.", "Let $N'_i \\subset M'/xM'$ be the inverse image of $N_i$.", "It is clear that $N'_i \\not =N'_{i + 1}$ by our choice of $m_i$.", "Hence we see that $\\text{length}_R(M'/xM') \\geq k$. By the", "finite case we conclude $k \\leq r\\text{length}_R(R/xR)$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1025, "type": "theorem", "label": "algebra-lemma-finite-extension-residue-fields-dimension-1", "categories": [ "algebra" ], "title": "algebra-lemma-finite-extension-residue-fields-dimension-1", "contents": [ "Let $R \\to S$ be a homomorphism of domains inducing an", "injection of fraction fields $K \\subset L$. If $R$ is Noetherian", "local of dimension $1$ and $[L : K] < \\infty$ then", "\\begin{enumerate}", "\\item each prime ideal $\\mathfrak n_i$ of $S$ lying over", "the maximal ideal $\\mathfrak m$ of $R$ is maximal,", "\\item there are finitely many of these, and", "\\item $[\\kappa(\\mathfrak n_i) : \\kappa(\\mathfrak m)] < \\infty$ for each $i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Pick $x \\in \\mathfrak m$ nonzero. Apply Lemma \\ref{lemma-finite-length}", "to the submodule $S \\subset L \\cong K^{\\oplus n}$ where $n = [L : K]$.", "Thus the ring $S/xS$ has finite length over $R$. It follows that", "$S/\\mathfrak m S$ has finite length over $\\kappa(\\mathfrak m)$.", "In other words, $\\dim_{\\kappa(\\mathfrak m)} S/\\mathfrak m S$", "is finite (Lemma \\ref{lemma-dimension-is-length}). Thus $S/\\mathfrak mS$", "is Artinian (Lemma \\ref{lemma-finite-dimensional-algebra}). The", "structural results on Artinian rings implies parts (1) and (2), see", "for example Lemma \\ref{lemma-artinian-finite-length}.", "Part (3) is implied by the finiteness established above." ], "refs": [ "algebra-lemma-finite-length", "algebra-lemma-dimension-is-length", "algebra-lemma-finite-dimensional-algebra", "algebra-lemma-artinian-finite-length" ], "ref_ids": [ 1024, 634, 642, 646 ] } ], "ref_ids": [] }, { "id": 1026, "type": "theorem", "label": "algebra-lemma-finite-length-global", "categories": [ "algebra" ], "title": "algebra-lemma-finite-length-global", "contents": [ "Let $R$ be a domain with fraction field $K$.", "Let $M$ be an $R$-submodule of $K^{\\oplus r}$.", "Assume $R$ is Noetherian of dimension $1$.", "For any nonzero $x \\in R$ we have", "$\\text{length}_R(M/xM) < \\infty$." ], "refs": [], "proofs": [ { "contents": [ "Since $R$ has dimension $1$ we see that $x$ is contained in finitely many", "primes $\\mathfrak m_i$, $i = 1, \\ldots, n$, each maximal. Since $R$ is", "Noetherian we see that $R/xR$ is Artinian and", "$R/xR = \\prod_{i = 1, \\ldots, n} (R/xR)_{\\mathfrak m_i}$ by", "Proposition \\ref{proposition-dimension-zero-ring} and", "Lemma \\ref{lemma-artinian-finite-length}. Hence $M/xM$ similarly", "decomposes as the product $M/xM = \\prod (M/xM)_{\\mathfrak m_i}$", "of its localizations at the $\\mathfrak m_i$. By Lemma \\ref{lemma-finite-length}", "applied to $M_{\\mathfrak m_i}$ over $R_{\\mathfrak m_i}$ we see each", "$M_{\\mathfrak m_i}/xM_{\\mathfrak m_i} = (M/xM)_{\\mathfrak m_i}$", "has finite length over $R_{\\mathfrak m_i}$. Thus $M/xM$", "has finite length over $R$ as the above implies $M/xM$", "has a finite filtration by $R$-submodules whose successive", "quotients are isomorphic to the residue fields $\\kappa(\\mathfrak m_i)$." ], "refs": [ "algebra-proposition-dimension-zero-ring", "algebra-lemma-artinian-finite-length", "algebra-lemma-finite-length" ], "ref_ids": [ 1410, 646, 1024 ] } ], "ref_ids": [] }, { "id": 1027, "type": "theorem", "label": "algebra-lemma-krull-akizuki", "categories": [ "algebra" ], "title": "algebra-lemma-krull-akizuki", "contents": [ "Let $R$ be a domain with fraction field $K$.", "Let $K \\subset L$ be a finite extension of fields.", "Assume $R$ is Noetherian and $\\dim(R) = 1$.", "In this case any ring $A$ with $R \\subset A \\subset L$ is", "Noetherian." ], "refs": [], "proofs": [ { "contents": [ "To begin we may assume that $L$ is the fraction field of $A$", "by replacing $L$ by the fraction field of $A$ if necessary.", "Let $I \\subset A$ be a nonzero ideal. Clearly $I$ generates $L$ as", "a $K$-vector space. Hence we see that $I \\cap R \\not = (0)$.", "Pick any nonzero $x \\in I \\cap R$. Then we get", "$I/xA \\subset A/xA$. By Lemma \\ref{lemma-finite-length-global}", "the $R$-module $A/xA$ has finite length as an $R$-module. Hence", "$I/xA$ has finite length as an $R$-module. Hence $I$ is finitely", "generated as an ideal in $A$." ], "refs": [ "algebra-lemma-finite-length-global" ], "ref_ids": [ 1026 ] } ], "ref_ids": [] }, { "id": 1028, "type": "theorem", "label": "algebra-lemma-exists-dvr", "categories": [ "algebra" ], "title": "algebra-lemma-exists-dvr", "contents": [ "Let $R$ be a Noetherian local domain with fraction field $K$.", "Assume that $R$ is not a field.", "Let $K \\subset L$ be a finitely generated field extension.", "Then there exists discrete valuation ring $A$ with fraction field", "$L$ which dominates $R$." ], "refs": [], "proofs": [ { "contents": [ "If $L$ is not finite over $K$ choose a transcendence basis", "$x_1, \\ldots, x_r$ of $L$ over $K$ and replace $R$ by", "$R[x_1, \\ldots, x_r]$ localized at the maximal ideal", "generated by $\\mathfrak m_R$ and $x_1, \\ldots, x_r$.", "Thus we may assume $K \\subset L$ finite.", "\\medskip\\noindent", "By Lemma \\ref{lemma-dominate-by-dimension-1} we may assume $\\dim(R) = 1$.", "\\medskip\\noindent", "Let $A \\subset L$ be the integral closure of $R$ in $L$.", "By Lemma \\ref{lemma-krull-akizuki} this is Noetherian.", "By Lemma \\ref{lemma-integral-overring-surjective} there", "is a prime ideal $\\mathfrak q \\subset A$ lying", "over the maximal ideal of $R$.", "By Lemma \\ref{lemma-characterize-dvr} the ring $A_{\\mathfrak q}$ is a discrete", "valuation ring dominating $R$ as desired." ], "refs": [ "algebra-lemma-dominate-by-dimension-1", "algebra-lemma-krull-akizuki", "algebra-lemma-integral-overring-surjective", "algebra-lemma-characterize-dvr" ], "ref_ids": [ 1020, 1027, 495, 1023 ] } ], "ref_ids": [] }, { "id": 1029, "type": "theorem", "label": "algebra-lemma-easy-divisibility", "categories": [ "algebra" ], "title": "algebra-lemma-easy-divisibility", "contents": [ "Let $R$ be a domain. Let $x, y \\in R$.", "Then $x$, $y$ are associates if and only if $(x) = (y)$." ], "refs": [], "proofs": [ { "contents": [ "If $x = uy$ for some unit $u \\in R$, then $(x) \\subset (y)$ and", "$y = u^{-1}x$ so also $(y) \\subset (x)$. Conversely, suppose that", "$(x) = (y)$. Then $x = fy$ and $y = gx$ for some $f, g \\in A$.", "Then $x = fg x$ and since $R$ is a domain $fg = 1$. Thus", "$x$ and $y$ are associates." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1030, "type": "theorem", "label": "algebra-lemma-factorization-exists", "categories": [ "algebra" ], "title": "algebra-lemma-factorization-exists", "contents": [ "Let $R$ be a domain. Consider the following conditions:", "\\begin{enumerate}", "\\item The ring $R$ satisfies the ascending chain condition for", "principal ideals.", "\\item Every nonzero, nonunit element $a \\in R$", "has a factorization $a = b_1 \\ldots b_k$", "with each $b_i$ an irreducible element of $R$.", "\\end{enumerate}", "Then (1) implies (2)." ], "refs": [], "proofs": [ { "contents": [ "Let $x$ be a nonzero element, not a unit, which does not have a", "factorization into", "irreducibles. Set $x_1 = x$. We can write $x = yz$ where neither", "$y$ nor $z$ is irreducible or a unit.", "Then either $y$ does not have a factorization", "into irreducibles, in which case we set $x_2 = y$, or $z$ does not", "have a factorization into irreducibles, in which case we set $x_2 = z$.", "Continuing in this fashion we find a sequence", "$$", "x_1 | x_2 | x_3 | \\ldots", "$$", "of elements of $R$ with $x_n/x_{n + 1}$ not a unit.", "This gives a strictly increasing sequence of principal ideals", "$(x_1) \\subset (x_2) \\subset (x_3) \\subset \\ldots$ thereby finishing the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1031, "type": "theorem", "label": "algebra-lemma-characterize-UFD", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-UFD", "contents": [ "Let $R$ be a domain. Assume every nonzero, nonunit factors into", "irreducibles. Then $R$ is a UFD if and only if every irreducible", "element is prime." ], "refs": [], "proofs": [ { "contents": [ "Assume $R$ is a UFD and let $x \\in R$ be an irreducible element.", "Say $ab \\in (x)$, i.e., $ab = cx$. Choose factorizations", "$a = a_1 \\ldots a_n$, $b = b_1 \\ldots b_m$, and $c = c_1 \\ldots c_r$.", "By uniqueness of the factorization", "$$", "a_1 \\ldots a_n b_1 \\ldots b_m = c_1 \\ldots c_r x", "$$", "we find that $x$ is an associate of one of the elements", "$a_1, \\ldots, b_m$. In other words, either $a \\in (x)$ or $b \\in (x)$", "and we conclude that $x$ is prime.", "\\medskip\\noindent", "Assume every irreducible element is prime. We have to prove that", "factorization into irreducibles is unique up to permutation and", "taking associates. Say $a_1 \\ldots a_m = b_1 \\ldots b_n$ with $a_i$", "and $b_j$ irreducible. Since $a_1$ is prime, we see that", "$b_j \\in (a_1)$ for some $j$. After renumbering we may assume", "$b_1 \\in (a_1)$. Then $b_1 = a_1 u$ and since $b_1$ is irreducible", "we see that $u$ is a unit. Hence $a_1$ and $b_1$ are associates", "and $a_2 \\ldots a_n = ub_2\\ldots b_m$. By induction on $n + m$", "we see that $n = m$ and $a_i$ associate to $b_{\\sigma(i)}$ for", "$i = 2, \\ldots, n$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1032, "type": "theorem", "label": "algebra-lemma-characterize-UFD-height-1", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-UFD-height-1", "contents": [ "Let $R$ be a Noetherian domain. Then $R$ is a UFD if and only if every", "height $1$ prime ideal is principal." ], "refs": [], "proofs": [ { "contents": [ "Assume $R$ is a UFD and let $\\mathfrak p$ be a height 1 prime ideal.", "Take $x \\in \\mathfrak p$ nonzero and let $x = a_1 \\ldots a_n$", "be a factorization into irreducibles.", "Since $\\mathfrak p$ is prime we see that $a_i \\in \\mathfrak p$", "for some $i$. By Lemma \\ref{lemma-characterize-UFD} the ideal", "$(a_i)$ is prime. Since $\\mathfrak p$ has height $1$ we conclude", "that $(a_i) = \\mathfrak p$.", "\\medskip\\noindent", "Assume every height $1$ prime is principal. Since $R$ is Noetherian", "every nonzero nonunit element $x$ has a factorization into irreducibles,", "see Lemma \\ref{lemma-factorization-exists}. It suffices to prove that", "an irreducible element $x$ is prime, see Lemma \\ref{lemma-characterize-UFD}.", "Let $(x) \\subset \\mathfrak p$ be a prime minimal over $(x)$. Then", "$\\mathfrak p$ has height $1$ by Lemma \\ref{lemma-minimal-over-1}.", "By assumption $\\mathfrak p = (y)$. Hence $x = yz$ and $z$ is a unit", "as $x$ is irreducible. Thus $(x) = (y)$ and we see that $x$ is prime." ], "refs": [ "algebra-lemma-characterize-UFD", "algebra-lemma-factorization-exists", "algebra-lemma-characterize-UFD", "algebra-lemma-minimal-over-1" ], "ref_ids": [ 1031, 1030, 1031, 683 ] } ], "ref_ids": [] }, { "id": 1033, "type": "theorem", "label": "algebra-lemma-invert-prime-elements", "categories": [ "algebra" ], "title": "algebra-lemma-invert-prime-elements", "contents": [ "\\begin{reference}", "\\cite[Lemma 2]{Nagata-UFD}", "\\end{reference}", "Let $A$ be a domain. Let $S \\subset A$ be a multiplicative subset", "generated by prime elements. Let $x \\in A$ be irreducible. Then", "\\begin{enumerate}", "\\item the image of $x$ in $S^{-1}A$ is irreducible or a unit, and", "\\item $x$ is prime if and only if the image of $x$ in $S^{-1}A$ is", "a unit or a prime element in $S^{-1}A$.", "\\end{enumerate}", "Moreover, then $A$ is a UFD if and only if every element of $A$ has a", "factorization into irreducibles and $S^{-1}A$ is a UFD." ], "refs": [], "proofs": [ { "contents": [ "Say $x = \\alpha \\beta$ for $\\alpha, \\beta \\in S^{-1}A$. Then", "$\\alpha = a/s$ and $\\beta = b/s'$ for $a, b \\in A$, $s, s' \\in S$.", "Thus we get $ss'x = ab$. By assumption we can write", "$ss' = p_1 \\ldots p_r$ for some prime elements $p_i$.", "For each $i$ the element $p_i$ divides either $a$ or $b$.", "Dividing we find a factorization $x = a' b'$ and", "$a = s'' a'$, $b = s''' b'$ for some $s'', s''' \\in S$.", "As $x$ is irreducible, either $a'$ or $b'$ is a unit.", "Tracing back we find that either $\\alpha$ or $\\beta$ is a unit.", "This proves (1).", "\\medskip\\noindent", "Suppose $x$ is prime. Then $A/(x)$ is a domain. Hence", "$S^{-1}A/xS^{-1}A = S^{-1}(A/(x))$ is a domain or zero.", "Thus $x$ maps to a prime element or a unit.", "\\medskip\\noindent", "Suppose that the image of $x$ in $S^{-1}A$ is a unit.", "Then $y x = s$ for some $s \\in S$ and $y \\in A$. By assumption", "$s = p_1 \\ldots p_r$ with $p_i$ a prime element. For each $i$ either", "$p_i$ divides $y$ or $p_i$ divides $x$. In the second case", "$p_i$ and $x$ are associates (as $x$ is irreducible) and we are done.", "But if the first case happens for all $i = 1, \\ldots, r$, then", "$x$ is a unit which is a contradiction.", "\\medskip\\noindent", "Suppose that the image of $x$ in $S^{-1}A$ is a prime element.", "Assume $a, b \\in A$ and $ab \\in (x)$. Then $sa = xy$ or", "$sb = xy$ for some $s \\in S$ and $y \\in A$. Say the first", "case happens. By assumption", "$s = p_1 \\ldots p_r$ with $p_i$ a prime element. For each $i$ either", "$p_i$ divides $y$ or $p_i$ divides $x$. In the second case", "$p_i$ and $x$ are associates (as $x$ is irreducible) and we are done.", "If the first case happens for all $i = 1, \\ldots, r$, then", "$a \\in (x)$ as desired. This completes the proof of (2).", "\\medskip\\noindent", "The final statement of the lemma follows from (1) and (2)", "and Lemma \\ref{lemma-characterize-UFD}." ], "refs": [ "algebra-lemma-characterize-UFD" ], "ref_ids": [ 1031 ] } ], "ref_ids": [] }, { "id": 1034, "type": "theorem", "label": "algebra-lemma-UFD-ascending-chain-condition-principal-ideals", "categories": [ "algebra" ], "title": "algebra-lemma-UFD-ascending-chain-condition-principal-ideals", "contents": [ "A UFD satisfies the ascending chain condition for principal", "ideals." ], "refs": [], "proofs": [ { "contents": [ "Consider an ascending chain $(a_1) \\subset (a_2) \\subset (a_3) \\subset \\ldots$", "of principal ideals in $R$. Write $a_1 = p_1^{e_1} \\ldots p_r^{e_r}$", "with $p_i$ prime. Then we see that $a_n$ is an associate of", "$p_1^{c_1} \\ldots p_r^{c_r}$ for some $0 \\leq c_i \\leq e_i$.", "Since there are only finitely many possibilities we conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1035, "type": "theorem", "label": "algebra-lemma-factoring-in-polynomial", "categories": [ "algebra" ], "title": "algebra-lemma-factoring-in-polynomial", "contents": [ "Let $R$ be a domain. Assume $R$ has the ascending chain condition", "for principal ideals. Then the same property holds for a polynomial", "ring over $R$." ], "refs": [], "proofs": [ { "contents": [ "Consider an ascending chain $(f_1) \\subset (f_2) \\subset (f_3) \\subset \\ldots$", "of principal ideals in $R[x]$. Since $f_{n + 1}$ divides $f_n$ we see", "that the degrees decrease in the sequence. Thus $f_n$", "has fixed degree $d \\geq 0$ for all $n \\gg 0$. Let $a_n$ be the", "leading coefficient of $f_n$. The condition $f_n \\in (f_{n + 1})$", "implies that $a_{n + 1}$ divides $a_n$ for all $n$.", "By our assumption on $R$ we see that $a_{n + 1}$ and $a_n$", "are associates for all $n$ large enough (Lemma \\ref{lemma-easy-divisibility}).", "Thus for large $n$ we see that $f_n = u f_{n + 1}$ where", "$u \\in R$ (for reasons of degree) is a unit (as $a_n$ and $a_{n + 1}$", "are associates)." ], "refs": [ "algebra-lemma-easy-divisibility" ], "ref_ids": [ 1029 ] } ], "ref_ids": [] }, { "id": 1036, "type": "theorem", "label": "algebra-lemma-polynomial-ring-UFD", "categories": [ "algebra" ], "title": "algebra-lemma-polynomial-ring-UFD", "contents": [ "A polynomial ring over a UFD is a UFD. In particular, if $k$ is a field,", "then $k[x_1, \\ldots, x_n]$ is a UFD." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a UFD. Then $R$ satisfies the ascending chain condition for", "principal ideals", "(Lemma \\ref{lemma-UFD-ascending-chain-condition-principal-ideals}),", "hence $R[x]$ satisfies the ascending chain condition for principal", "ideals (Lemma \\ref{lemma-factoring-in-polynomial}), and hence every", "element of $R[x]$ has a factorization into irreducibles", "(Lemma \\ref{lemma-factorization-exists}).", "Let $S \\subset R$ be the multiplicative subset", "generated by prime elements. Since every nonunit of $R$ is a product", "of prime elements we see that $K = S^{-1}R$ is the fraction field", "of $R$. Observe that every prime element of $R$ maps to a prime element", "of $R[x]$ and that $S^{-1}(R[x]) = S^{-1}R[x] = K[x]$ is a", "UFD (and even a PID). Thus we may apply", "Lemma \\ref{lemma-invert-prime-elements} to conclude." ], "refs": [ "algebra-lemma-UFD-ascending-chain-condition-principal-ideals", "algebra-lemma-factoring-in-polynomial", "algebra-lemma-factorization-exists", "algebra-lemma-invert-prime-elements" ], "ref_ids": [ 1034, 1035, 1030, 1033 ] } ], "ref_ids": [] }, { "id": 1037, "type": "theorem", "label": "algebra-lemma-UFD-normal", "categories": [ "algebra" ], "title": "algebra-lemma-UFD-normal", "contents": [ "A unique factorization domain is normal." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a UFD. Let $x$ be an element of the fraction field of", "$R$ which is integral over $R$. Say $x^d - a_1 x^{d - 1} - \\ldots - a_d = 0$", "with $a_i \\in R$. We can write", "$x = u p_1^{e_1} \\ldots p_r^{e_r}$ with $u$ a unit, $e_i \\in \\mathbf{Z}$,", "and $p_1, \\ldots, p_r$ irreducible elements which are not associates.", "To prove the lemma we have to show $e_i \\geq 0$. If not, say $e_1 < 0$,", "then for $N \\gg 0$ we get", "$$", "u^d p_2^{de_2 + N} \\ldots p_r^{de_r + N} =", "p_1^{-de_1}p_2^N \\ldots p_r^N(", "\\sum\\nolimits_{i = 1, \\ldots, d} a_i x^{d - i} ) \\in (p_1)", "$$", "which contradicts uniqueness of factorization in $R$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1038, "type": "theorem", "label": "algebra-lemma-PID-UFD", "categories": [ "algebra" ], "title": "algebra-lemma-PID-UFD", "contents": [ "A principal ideal domain is a unique factorization domain." ], "refs": [], "proofs": [ { "contents": [ "As a PID is Noetherian this follows from", "Lemma \\ref{lemma-characterize-UFD-height-1}." ], "refs": [ "algebra-lemma-characterize-UFD-height-1" ], "ref_ids": [ 1032 ] } ], "ref_ids": [] }, { "id": 1039, "type": "theorem", "label": "algebra-lemma-PID-dedekind", "categories": [ "algebra" ], "title": "algebra-lemma-PID-dedekind", "contents": [ "A PID is a Dedekind domain." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a PID. Since every nonzero ideal of $R$ is principal,", "and $R$ is a UFD (Lemma \\ref{lemma-PID-UFD}), this follows from", "the fact that every irreducible element in $R$ is prime", "(Lemma \\ref{lemma-characterize-UFD})", "so that factorizations of elements turn into factorizations into primes." ], "refs": [ "algebra-lemma-PID-UFD", "algebra-lemma-characterize-UFD" ], "ref_ids": [ 1038, 1031 ] } ], "ref_ids": [] }, { "id": 1040, "type": "theorem", "label": "algebra-lemma-product-ideals-principal", "categories": [ "algebra" ], "title": "algebra-lemma-product-ideals-principal", "contents": [ "\\begin{slogan}", "A product of ideals is an invertible module iff both factors are.", "\\end{slogan}", "Let $A$ be a ring. Let $I$ and $J$ be nonzero ideals of $A$", "such that $IJ = (f)$ for some nonzerodivisor $f \\in A$. Then $I$ and $J$ are", "finitely generated ideals and finitely locally free of rank $1$ as $A$-modules." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that $I$ and $J$ are finite locally free $A$-modules", "of rank $1$, see Lemma \\ref{lemma-finite-projective}.", "To do this, write $f = \\sum_{i = 1, \\ldots, n} x_i y_i$ with $x_i \\in I$", "and $y_i \\in J$. We can", "also write $x_i y_i = a_i f$ for some $a_i \\in A$.", "Since $f$ is a nonzerodivisor we see that $\\sum a_i = 1$.", "Thus it suffices to show that each $I_{a_i}$ and", "$J_{a_i}$ is free of rank $1$ over $A_{a_i}$. After replacing $A$ by", "$A_{a_i}$ we conclude that $f = xy$ for some $x \\in I$ and $y \\in J$.", "Note that both $x$ and $y$ are nonzerodivisors. We claim that", "$I = (x)$ and $J = (y)$ which finishes the proof. Namely, if $x' \\in I$,", "then $x'y = af = axy$ for some $a \\in A$. Hence $x' = ax$ and we win." ], "refs": [ "algebra-lemma-finite-projective" ], "ref_ids": [ 795 ] } ], "ref_ids": [] }, { "id": 1041, "type": "theorem", "label": "algebra-lemma-characterize-Dedekind", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-Dedekind", "contents": [ "Let $R$ be a ring. The following are equivalent", "\\begin{enumerate}", "\\item $R$ is a Dedekind domain,", "\\item $R$ is a Noetherian domain, and for every maximal ideal $\\mathfrak m$", "the local ring $R_{\\mathfrak m}$ is a discrete valuation ring, and", "\\item $R$ is a Noetherian, normal domain, and $\\dim(R) \\leq 1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). The argument is nontrivial because we did not assume that $R$", "was Noetherian in our definition of a Dedekind domain. Let", "$\\mathfrak p \\subset R$ be a prime ideal. Observe that", "$\\mathfrak p \\not = \\mathfrak p^2$ by uniqueness", "of the factorizations in the definition. Pick $x \\in \\mathfrak p$", "with $x \\not \\in \\mathfrak p^2$. Let $y \\in \\mathfrak p$ be a", "second element (for example $y = 0$).", "Write $(x, y) = \\mathfrak p_1 \\ldots \\mathfrak p_r$.", "Since $(x, y) \\subset \\mathfrak p$ at least one of the primes", "$\\mathfrak p_i$ is contained in $\\mathfrak p$.", "But as $x \\not \\in \\mathfrak p^2$ there is at most one.", "Thus exactly one of $\\mathfrak p_1, \\ldots, \\mathfrak p_r$ is contained", "in $\\mathfrak p$, say $\\mathfrak p_1 \\subset \\mathfrak p$.", "We conclude that $(x, y)R_\\mathfrak p = \\mathfrak p_1R_\\mathfrak p$", "is prime for every choice of $y$. We claim that", "$(x)R_\\mathfrak p = \\mathfrak pR_\\mathfrak p$. Namely,", "pick $y \\in \\mathfrak p$. By the above applied with $y^2$ we see", "that $(x, y^2)R_\\mathfrak p$ is prime.", "Hence $y \\in (x, y^2)R_\\mathfrak p$, i.e., $y = ax + by^2$ in", "$R_\\mathfrak p$. Thus $(1 - by)y = ax \\in (x)R_\\mathfrak p$, i.e.,", "$y \\in (x)R_\\mathfrak p$ as desired.", "\\medskip\\noindent", "Writing $(x) = \\mathfrak p_1 \\ldots \\mathfrak p_r$ anew with", "$\\mathfrak p_1 \\subset \\mathfrak p$ we conclude that", "$\\mathfrak p_1 R_\\mathfrak p = \\mathfrak p R_\\mathfrak p$, i.e.,", "$\\mathfrak p_1 = \\mathfrak p$. Moreover, $\\mathfrak p_1 = \\mathfrak p$ is", "a finitely generated ideal of $R$ by", "Lemma \\ref{lemma-product-ideals-principal}.", "We conclude that $R$ is Noetherian by Lemma \\ref{lemma-cohen}.", "Moreover, it follows that $R_\\mathfrak m$ is a discrete", "valuation ring for every prime ideal $\\mathfrak p$, see", "Lemma \\ref{lemma-characterize-dvr}.", "\\medskip\\noindent", "The equivalence of (2) and (3) follows from", "Lemmas \\ref{lemma-normality-is-local} and", "\\ref{lemma-characterize-dvr}. Assume (2) and (3) are satisfied.", "Let $I \\subset R$ be an ideal. We will construct a factorization", "of $I$. If $I$ is prime, then there is nothing to prove.", "If not, pick $I \\subset \\mathfrak p$ with $\\mathfrak p \\subset R$", "maximal. Let $J = \\{x \\in R \\mid x \\mathfrak p \\subset I\\}$.", "We claim $J \\mathfrak p = I$. It suffices to check this after localization", "at the maximal ideals $\\mathfrak m$ of $R$ (the formation of $J$ commutes with", "localization and we use Lemma \\ref{lemma-characterize-zero-local}).", "Then either $\\mathfrak p R_\\mathfrak m = R_\\mathfrak m$ and the result", "is clear, or $\\mathfrak p R_\\mathfrak m = \\mathfrak m R_\\mathfrak m$.", "In the last case $\\mathfrak p R_\\mathfrak m = (\\pi)$ and the case where", "$\\mathfrak p$ is principal is immediate. By Noetherian induction the", "ideal $J$ has a factorization and we obtain the desired factorization", "of $I$. We omit the proof of uniqueness of the factorization." ], "refs": [ "algebra-lemma-product-ideals-principal", "algebra-lemma-cohen", "algebra-lemma-characterize-dvr", "algebra-lemma-normality-is-local", "algebra-lemma-characterize-dvr", "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 1040, 429, 1023, 510, 1023, 410 ] } ], "ref_ids": [] }, { "id": 1042, "type": "theorem", "label": "algebra-lemma-integral-closure-Dedekind", "categories": [ "algebra" ], "title": "algebra-lemma-integral-closure-Dedekind", "contents": [ "Let $A$ be a Noetherian domain of dimension $1$ with fraction field $K$.", "Let $K \\subset L$ be a finite extension. Let $B$ be the", "integral closure of $A$ in $L$. Then $B$ is a Dedekind domain and", "$\\Spec(B) \\to \\Spec(A)$ is surjective, has finite fibres, and", "induces finite residue field extensions." ], "refs": [], "proofs": [ { "contents": [ "By Krull-Akizuki (Lemma \\ref{lemma-krull-akizuki})", "the ring $B$ is Noetherian. By Lemma \\ref{lemma-integral-sub-dim-equal}", "$\\dim(B) = 1$. Thus $B$ is a Dedekind domain by", "Lemma \\ref{lemma-characterize-Dedekind}.", "Surjectivity of the map on spectra follows from", "Lemma \\ref{lemma-integral-overring-surjective}.", "The last two statements follow from", "Lemma \\ref{lemma-finite-extension-residue-fields-dimension-1}." ], "refs": [ "algebra-lemma-krull-akizuki", "algebra-lemma-integral-sub-dim-equal", "algebra-lemma-characterize-Dedekind", "algebra-lemma-integral-overring-surjective", "algebra-lemma-finite-extension-residue-fields-dimension-1" ], "ref_ids": [ 1027, 985, 1041, 495, 1025 ] } ], "ref_ids": [] }, { "id": 1043, "type": "theorem", "label": "algebra-lemma-ord-additive", "categories": [ "algebra" ], "title": "algebra-lemma-ord-additive", "contents": [ "Let $R$ be a semi-local Noetherian ring of dimension $1$.", "If $a, b \\in R$ are nonzerodivisors then", "$$", "\\text{length}_R(R/(ab)) =", "\\text{length}_R(R/(a)) +", "\\text{length}_R(R/(b))", "$$", "and these lengths are finite." ], "refs": [], "proofs": [ { "contents": [ "We saw the finiteness in Lemma \\ref{lemma-finite-length-global}.", "Additivity holds since there is a short exact sequence", "$0 \\to R/(a) \\to R/(ab) \\to R/(b) \\to 0$ where the first map", "is given by multiplication by $b$. (Use length is additive,", "see Lemma \\ref{lemma-length-additive}.)" ], "refs": [ "algebra-lemma-finite-length-global", "algebra-lemma-length-additive" ], "ref_ids": [ 1026, 631 ] } ], "ref_ids": [] }, { "id": 1044, "type": "theorem", "label": "algebra-lemma-compare-lattices", "categories": [ "algebra" ], "title": "algebra-lemma-compare-lattices", "contents": [ "Let $R$ be a Noetherian local domain of dimension $1$ with", "fraction field $K$. Let $V$ be a finite dimensional $K$-vector space.", "\\begin{enumerate}", "\\item If $M$ is a lattice in $V$ and $M \\subset M' \\subset V$", "is an $R$-submodule of $V$ containing $M$", "then the following are equivalent", "\\begin{enumerate}", "\\item $M'$ is a lattice,", "\\item $\\text{length}_R(M'/M)$ is finite, and", "\\item $M'$ is finitely generated.", "\\end{enumerate}", "\\item If $M$ is a lattice in $V$ and $M' \\subset M$ is an $R$-submodule", "of $M$ then $M'$ is a lattice if and only if", "$\\text{length}_R(M/M')$ is finite.", "\\item If $M$, $M'$ are lattices in $V$, then so are", "$M \\cap M'$ and $M + M'$.", "\\item If $M \\subset M' \\subset M'' \\subset V$ are lattices in $V$", "then", "$$", "\\text{length}_R(M''/M) =", "\\text{length}_R(M'/M) +", "\\text{length}_R(M''/M').", "$$", "\\item If $M$, $M'$, $N$, $N'$ are lattices in $V$ and", "$N \\subset M \\cap M'$, $M + M' \\subset N'$, then we have", "\\begin{eqnarray*}", "& & \\text{length}_R(M/M \\cap M') - \\text{length}_R(M'/M \\cap M')\\\\", "& = &", "\\text{length}_R(M/N) - \\text{length}_R(M'/N) \\\\", "& = &", "\\text{length}_R(M + M' / M') - \\text{length}_R(M + M'/M) \\\\", "& = &", "\\text{length}_R(N' / M') - \\text{length}_R(N'/M)", "\\end{eqnarray*}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Assume (1)(a). Say $y_1, \\ldots, y_m$ generate $M'$.", "Then each $y_i = x_i/f_i$ for some $x_i \\in M$ and", "nonzero $f_i \\in R$.", "Hence we see that $f_1 \\ldots f_m M' \\subset M$.", "Since $R$ is Noetherian local of dimension $1$", "we see that $\\mathfrak m^n \\subset (f_1 \\ldots f_m)$", "for some $n$ (for example combine", "Lemmas \\ref{lemma-one-equation} and", "Proposition \\ref{proposition-dimension-zero-ring} or combine", "Lemmas \\ref{lemma-finite-length} and \\ref{lemma-length-infinite}).", "In other words $\\mathfrak m^nM' \\subset M$ for some $n$", "Hence", "$\\text{length}(M'/M) < \\infty$ by Lemma \\ref{lemma-length-finite},", "in other words (1)(b) holds.", "Assume (1)(b). Then $M'/M$ is a finite $R$-module", "(see Lemma \\ref{lemma-finite-length-finite}).", "Hence $M'$ is a finite $R$-module as an extension of finite $R$-modules.", "Hence (1)(c). The implication", "(1)(c) $\\Rightarrow$ (1)(a) follows from the remark following", "Definition \\ref{definition-lattice}.", "\\medskip\\noindent", "Proof of (2). Suppose", "$M$ is a lattice in $V$ and $M' \\subset M$ is an $R$-submodule.", "We have seen in (1) that if $M'$ is a lattice, then", "$\\text{length}_R(M/M') < \\infty$. Conversely, assume that", "$\\text{length}_R(M/M') < \\infty$. Then $M'$ is finitely generated", "as $R$ is Noetherian and for some $n$ we have", "$\\mathfrak m^n M \\subset M'$ (Lemma \\ref{lemma-length-infinite}).", "Hence it follows", "that $M'$ contains a basis for $V$, and $M'$ is a lattice.", "\\medskip\\noindent", "Proof of (3). Assume $M$, $M'$ are lattices in $V$.", "Since $R$ is Noetherian the submodule $M \\cap M'$ of $M$ is finite.", "As $M$ is a lattice we can find", "$x_1, \\ldots, x_n \\in M$ which form a $K$-basis for", "$V$. Because $M'$ is a lattice we can write $x_i = y_i/f_i$ with", "$y_i \\in M'$ and $f_i \\in R$. Hence $f_ix_i \\in M \\cap M'$. Hence", "$M \\cap M'$ is a lattice also.", "The fact that $M + M'$ is a lattice follows from part (1).", "\\medskip\\noindent", "Part (4) follows from additivity of lengths", "(Lemma \\ref{lemma-length-additive})", "and the exact sequence", "$$", "0 \\to M'/M \\to M''/M \\to M''/M' \\to 0", "$$", "Part (5) follows from repeatedly applying part (4)." ], "refs": [ "algebra-proposition-dimension-zero-ring", "algebra-lemma-finite-length", "algebra-lemma-length-infinite", "algebra-lemma-length-finite", "algebra-lemma-finite-length-finite", "algebra-definition-lattice", "algebra-lemma-length-infinite", "algebra-lemma-length-additive" ], "ref_ids": [ 1410, 1024, 632, 636, 630, 1520, 632, 631 ] } ], "ref_ids": [] }, { "id": 1045, "type": "theorem", "label": "algebra-lemma-properties-distance-function", "categories": [ "algebra" ], "title": "algebra-lemma-properties-distance-function", "contents": [ "Let $R$ be a Noetherian local domain of dimension $1$ with", "fraction field $K$. Let $V$ be a finite dimensional $K$-vector space.", "This distance function has the property that", "$$", "d(M, M'') = d(M, M') + d(M', M'')", "$$", "whenever given three lattices $M$, $M'$, $M''$ of $V$.", "In particular we have $d(M, M') = - d(M', M)$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1046, "type": "theorem", "label": "algebra-lemma-order-vanishing-determinant", "categories": [ "algebra" ], "title": "algebra-lemma-order-vanishing-determinant", "contents": [ "Let $R$ be a Noetherian local domain of dimension $1$ with", "fraction field $K$. Let $V$ be a finite dimensional $K$-vector space.", "Let $\\varphi : V \\to V$ be a $K$-linear isomorphism.", "For any lattice $M \\subset V$ we have", "$$", "d(M, \\varphi(M)) = \\text{ord}_R(\\det(\\varphi))", "$$" ], "refs": [], "proofs": [ { "contents": [ "We can see that the integer $d(M, \\varphi(M))$ does not depend", "on the lattice $M$ as follows. Suppose that $M'$ is a second such", "lattice. Then we see that", "\\begin{eqnarray*}", "d(M, \\varphi(M)) & = & d(M, M') + d(M', \\varphi(M)) \\\\", "& = & d(M, M') + d(\\varphi(M'), \\varphi(M)) + d(M', \\varphi(M'))", "\\end{eqnarray*}", "Since $\\varphi$ is an isomorphism we see that", "$d(\\varphi(M'), \\varphi(M)) = d(M', M) = -d(M, M')$, and hence", "$d(M, \\varphi(M)) = d(M', \\varphi(M'))$. Moreover, both sides of the", "equation (of the lemma) are additive in $\\varphi$, i.e.,", "$$", "\\text{ord}_R(\\det(\\varphi \\circ \\psi))", "=", "\\text{ord}_R(\\det(\\varphi))", "+", "\\text{ord}_R(\\det(\\psi))", "$$", "and also", "\\begin{eqnarray*}", "d(M, \\varphi(\\psi((M))) & = &", "d(M, \\psi(M)) + d(\\psi(M), \\varphi(\\psi(M))) \\\\", "& = & d(M, \\psi(M)) + d(M, \\varphi(M))", "\\end{eqnarray*}", "by the independence shown above. Hence it suffices to prove the lemma", "for generators of $\\text{GL}(V)$. Choose an isomorphism", "$K^{\\oplus n} \\cong V$. Then $\\text{GL}(V) = \\text{GL}_n(K)$ is", "generated by elementary matrices $E$.", "The result is clear for $E$ equal to the identity matrix.", "If $E = E_{ij}(\\lambda)$ with $i \\not = j$, $\\lambda \\in K$,", "$\\lambda \\not = 0$, for example", "$$", "E_{12}(\\lambda) =", "\\left(", "\\begin{matrix}", "1 & \\lambda & \\ldots \\\\", "0 & 1 & \\ldots \\\\", "\\ldots & \\ldots & \\ldots", "\\end{matrix}", "\\right)", "$$", "then with respect to a different basis we get $E_{12}(1)$.", "The result is clear for $E = E_{12}(1)$ by taking as lattice", "$R^{\\oplus n} \\subset K^{\\oplus n}$. Finally, if $E = E_i(a)$,", "with $a \\in K^*$ for example", "$$", "E_1(a) =", "\\left(", "\\begin{matrix}", "a & 0 & \\ldots \\\\", "0 & 1 & \\ldots \\\\", "\\ldots & \\ldots & \\ldots", "\\end{matrix}", "\\right)", "$$", "then $E_1(a)(R^{\\oplus b}) = aR \\oplus R^{\\oplus n - 1}$ and", "it is clear that $d(R^{\\oplus n}, aR \\oplus R^{\\oplus n - 1})", "= \\text{ord}_R(a)$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1047, "type": "theorem", "label": "algebra-lemma-finite-extension-dim-1", "categories": [ "algebra" ], "title": "algebra-lemma-finite-extension-dim-1", "contents": [ "Let $A \\to B$ be a ring map. Assume", "\\begin{enumerate}", "\\item $A$ is a Noetherian local domain of dimension $1$,", "\\item $A \\subset B$ is a finite extension of domains.", "\\end{enumerate}", "Let $L/K$ be the corresponding finite extension of fraction fields.", "Let $y \\in L^*$ and $x = \\text{Nm}_{L/K}(y)$.", "In this situation $B$ is semi-local.", "Let $\\mathfrak m_i$, $i = 1, \\ldots, n$ be the maximal ideals of $B$.", "Then", "$$", "\\text{ord}_A(x) =", "\\sum\\nolimits_i", "[\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m_A)]", "\\text{ord}_{B_{\\mathfrak m_i}}(y)", "$$", "where $\\text{ord}$ is defined as in Definition \\ref{definition-ord}." ], "refs": [ "algebra-definition-ord" ], "proofs": [ { "contents": [ "The ring $B$ is semi-local by Lemma \\ref{lemma-finite-in-codim-1}.", "Write $y = b/b'$ for some $b, b' \\in B$.", "By the additivity of $\\text{ord}$ and multiplicativity of", "$\\text{Nm}$ it suffices to prove the lemma for", "$y = b$ or $y = b'$. In other words we may assume $y \\in B$.", "In this case the right hand side of the formula is", "$$", "\\sum [\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m_A)]", "\\text{length}_{B_{\\mathfrak m_i}}((B/yB)_{\\mathfrak m_i})", "$$", "By Lemma \\ref{lemma-pushdown-module} this is equal to", "$\\text{length}_A(B/yB)$. By Lemma \\ref{lemma-order-vanishing-determinant}", "we have", "$$", "\\text{length}_A(B/yB) = d(B, yB) =", "\\text{ord}_A(\\det\\nolimits_K(L \\xrightarrow{y} L)).", "$$", "Since $x = \\text{Nm}_{L/K}(y) = \\det\\nolimits_K(L \\xrightarrow{y} L)$", "by definition the lemma is proved." ], "refs": [ "algebra-lemma-finite-in-codim-1", "algebra-lemma-pushdown-module", "algebra-lemma-order-vanishing-determinant" ], "ref_ids": [ 991, 639, 1046 ] } ], "ref_ids": [ 1519 ] }, { "id": 1048, "type": "theorem", "label": "algebra-lemma-isolated-point", "categories": [ "algebra" ], "title": "algebra-lemma-isolated-point", "contents": [ "Let $k$ be a field.", "Let $S$ be a finite type $k$ algebra.", "Let $\\mathfrak q$ be a prime of $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $\\mathfrak q$ is an isolated point of $\\Spec(S)$,", "\\item $S_{\\mathfrak q}$ is finite over $k$,", "\\item there exists a $g \\in S$, $g \\not\\in \\mathfrak q$ such that", "$D(g) = \\{ \\mathfrak q \\}$,", "\\item $\\dim_{\\mathfrak q} \\Spec(S) = 0$,", "\\item $\\mathfrak q$ is a closed point of $\\Spec(S)$ and", "$\\dim(S_{\\mathfrak q}) = 0$, and", "\\item the field extension $k \\subset \\kappa(\\mathfrak q)$ is finite", "and $\\dim(S_{\\mathfrak q}) = 0$.", "\\end{enumerate}", "In this case $S = S_{\\mathfrak q} \\times S'$ for some", "finite type $k$-algebra $S'$. Also, the element $g$", "as in (3) has the property $S_{\\mathfrak q} = S_g$." ], "refs": [], "proofs": [ { "contents": [ "Suppose $\\mathfrak q$ is an isolated point of $\\Spec(S)$, i.e.,", "$\\{\\mathfrak q\\}$ is open in $\\Spec(S)$.", "Because $\\Spec(S)$ is a Jacobson space (see", "Lemmas \\ref{lemma-finite-type-field-Jacobson} and", "\\ref{lemma-jacobson})", "we see that $\\mathfrak q$ is a closed point. Hence", "$\\{\\mathfrak q\\}$ is open and closed in $\\Spec(S)$.", "By", "Lemmas \\ref{lemma-disjoint-decomposition} and", "\\ref{lemma-disjoint-implies-product} we may", "write $S = S_1 \\times S_2$ with $\\mathfrak q$", "corresponding to the only point $\\Spec(S_1)$.", "Hence $S_1 = S_{\\mathfrak q}$ is a zero dimensional", "ring of finite type over $k$. Hence it is finite over $k$", "for example by Lemma \\ref{lemma-Noether-normalization}.", "We have proved (1) implies (2).", "\\medskip\\noindent", "Suppose $S_{\\mathfrak q}$ is finite over $k$.", "Then $S_{\\mathfrak q}$ is Artinian local, see", "Lemma \\ref{lemma-finite-dimensional-algebra}. So", "$\\Spec(S_{\\mathfrak q}) = \\{\\mathfrak qS_{\\mathfrak q}\\}$ by", "Lemma \\ref{lemma-artinian-finite-length}.", "Consider the exact sequence $0 \\to K \\to S \\to S_{\\mathfrak q}", "\\to Q \\to 0$. It is clear that $K_{\\mathfrak q} = Q_{\\mathfrak q} = 0$.", "Also, $K$ is a finite $S$-module as $S$ is Noetherian and", "$Q$ is a finite $S$-module since $S_{\\mathfrak q}$ is finite over $k$.", "Hence there exists $g \\in S$, $g \\not \\in \\mathfrak q$ such that", "$K_g = Q_g = 0$. Thus $S_{\\mathfrak q} = S_g$ and", "$D(g) = \\{ \\mathfrak q \\}$. We have proved that (2) implies (3).", "\\medskip\\noindent", "Suppose $D(g) = \\{ \\mathfrak q \\}$. Since $D(g)$ is open by", "construction of the topology on $\\Spec(S)$ we see that", "$\\mathfrak q$ is an isolated point of $\\Spec(S)$.", "We have proved that (3) implies (1).", "In other words (1), (2) and (3) are equivalent.", "\\medskip\\noindent", "Assume $\\dim_{\\mathfrak q} \\Spec(S) = 0$. This means that", "there is some open neighbourhood of $\\mathfrak q$ in $\\Spec(S)$", "which has dimension zero. Then there is an open neighbourhood of the", "form $D(g)$ which has dimension zero. Since $S_g$ is Noetherian", "we conclude that $S_g$ is Artinian and", "$D(g) = \\Spec(S_g)$ is a finite discrete set, see", "Proposition \\ref{proposition-dimension-zero-ring}.", "Thus $\\mathfrak q$ is an isolated point of $D(g)$ and,", "by the equivalence of (1) and (2) above applied to", "$\\mathfrak qS_g \\subset S_g$, we see that", "$S_{\\mathfrak q} = (S_g)_{\\mathfrak qS_g}$ is finite over $k$.", "Hence (4) implies (2). It is clear that (1) implies (4).", "Thus (1) -- (4) are all equivalent.", "\\medskip\\noindent", "Lemma \\ref{lemma-dimension-closed-point-finite-type-field}", "gives the implication (5) $\\Rightarrow$ (4).", "The implication (4) $\\Rightarrow$ (6) follows from", "Lemma \\ref{lemma-dimension-at-a-point-finite-type-field}.", "The implication (6) $\\Rightarrow$ (5) follows from", "Lemma \\ref{lemma-finite-residue-extension-closed}.", "At this point we know (1) -- (6) are equivalent.", "\\medskip\\noindent", "The two statements at the end of the lemma we saw during the", "course of the proof of the equivalence of (1), (2) and (3) above." ], "refs": [ "algebra-lemma-finite-type-field-Jacobson", "algebra-lemma-jacobson", "algebra-lemma-disjoint-decomposition", "algebra-lemma-disjoint-implies-product", "algebra-lemma-Noether-normalization", "algebra-lemma-finite-dimensional-algebra", "algebra-lemma-artinian-finite-length", "algebra-proposition-dimension-zero-ring", "algebra-lemma-dimension-closed-point-finite-type-field", "algebra-lemma-dimension-at-a-point-finite-type-field", "algebra-lemma-finite-residue-extension-closed" ], "ref_ids": [ 467, 469, 405, 415, 1001, 642, 646, 1410, 996, 1007, 472 ] } ], "ref_ids": [] }, { "id": 1049, "type": "theorem", "label": "algebra-lemma-isolated-point-fibre", "categories": [ "algebra" ], "title": "algebra-lemma-isolated-point-fibre", "contents": [ "\\begin{slogan}", "Equivalent conditions for isolated points in fibres", "\\end{slogan}", "Let $R \\to S$ be a ring map of finite type.", "Let $\\mathfrak q \\subset S$ be a prime lying over", "$\\mathfrak p \\subset R$. Let $F = \\Spec(S \\otimes_R \\kappa(\\mathfrak p))$", "be the fibre of $\\Spec(S) \\to \\Spec(R)$, see", "Remark \\ref{remark-fundamental-diagram}.", "Denote $\\overline{\\mathfrak q} \\in F$ the point corresponding to", "$\\mathfrak q$. The following are equivalent", "\\begin{enumerate}", "\\item $\\overline{\\mathfrak q}$ is an isolated point of $F$,", "\\item $S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}$ is finite over", "$\\kappa(\\mathfrak p)$,", "\\item there exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that", "the only prime of $D(g)$ mapping to $\\mathfrak p$ is $\\mathfrak q$,", "\\item $\\dim_{\\overline{\\mathfrak q}}(F) = 0$,", "\\item $\\overline{\\mathfrak q}$ is a closed point of $F$ and", "$\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) = 0$, and", "\\item the field extension $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$", "is finite and $\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) = 0$.", "\\end{enumerate}" ], "refs": [ "algebra-remark-fundamental-diagram" ], "proofs": [ { "contents": [ "Note that $S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q} =", "(S \\otimes_R \\kappa(\\mathfrak p))_{\\overline{\\mathfrak q}}$.", "Moreover $S \\otimes_R \\kappa(\\mathfrak p)$ is of finite type over", "$\\kappa(\\mathfrak p)$.", "The conditions correspond exactly to the conditions of", "Lemma \\ref{lemma-isolated-point}", "for the $\\kappa(\\mathfrak p)$-algebra $S \\otimes_R \\kappa(\\mathfrak p)$", "and the prime $\\overline{\\mathfrak q}$, hence they are equivalent." ], "refs": [ "algebra-lemma-isolated-point" ], "ref_ids": [ 1048 ] } ], "ref_ids": [ 1558 ] }, { "id": 1050, "type": "theorem", "label": "algebra-lemma-quasi-finite", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-finite", "contents": [ "Let $R \\to S$ be a finite type ring map.", "Then $R \\to S$ is quasi-finite if and only if for all", "primes $\\mathfrak p \\subset R$", "the fibre $S \\otimes_R \\kappa(\\mathfrak p)$ is finite", "over $\\kappa(\\mathfrak p)$." ], "refs": [], "proofs": [ { "contents": [ "If the fibres are finite then the map is clearly quasi-finite.", "For the converse, note that $S \\otimes_R \\kappa(\\mathfrak p)$", "is a $\\kappa(\\mathfrak p)$-algebra of finite type and of dimension $0$.", "Hence it is finite over $\\kappa(\\mathfrak p)$ for example", "by Lemma \\ref{lemma-Noether-normalization}." ], "refs": [ "algebra-lemma-Noether-normalization" ], "ref_ids": [ 1001 ] } ], "ref_ids": [] }, { "id": 1051, "type": "theorem", "label": "algebra-lemma-quasi-finite-local", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-finite-local", "contents": [ "Let $R \\to S$ be a finite type ring map. Let $\\mathfrak q \\subset S$", "be a prime lying over $\\mathfrak p \\subset R$. Let", "$f \\in R$, $f \\not \\in \\mathfrak p$ and $g \\in S$, $g \\not \\in \\mathfrak q$.", "Then $R \\to S$ is quasi-finite at $\\mathfrak q$ if and only if", "$R_f \\to S_{fg}$ is quasi-finite at $\\mathfrak qS_{fg}$." ], "refs": [], "proofs": [ { "contents": [ "The fibre of $\\Spec(S_{fg}) \\to \\Spec(R_f)$ is homeomorphic", "to an open subset of the fibre of $\\Spec(S) \\to \\Spec(R)$.", "Hence the lemma follows from part (1) of the equivalent conditions of", "Lemma \\ref{lemma-isolated-point-fibre}." ], "refs": [ "algebra-lemma-isolated-point-fibre" ], "ref_ids": [ 1049 ] } ], "ref_ids": [] }, { "id": 1052, "type": "theorem", "label": "algebra-lemma-four-rings", "categories": [ "algebra" ], "title": "algebra-lemma-four-rings", "contents": [ "Let", "$$", "\\xymatrix{", "S \\ar[r] & S' & &", "\\mathfrak q \\ar@{-}[r] & \\mathfrak q' \\\\", "R \\ar[u] \\ar[r] & R' \\ar[u] & &", "\\mathfrak p \\ar@{-}[r] \\ar@{-}[u] & \\mathfrak p' \\ar@{-}[u]", "}", "$$", "be a commutative diagram of rings with primes as indicated.", "Assume $R \\to S$ of finite type, and $S \\otimes_R R' \\to S'$ surjective.", "If $R \\to S$ is quasi-finite at $\\mathfrak q$, then", "$R' \\to S'$ is quasi-finite at $\\mathfrak q'$." ], "refs": [], "proofs": [ { "contents": [ "Write $S \\otimes_R \\kappa(\\mathfrak p) = S_1 \\times S_2$", "with $S_1$ finite over $\\kappa(\\mathfrak p)$ and such that", "$\\mathfrak q$ corresponds to a point of $S_1$ as in", "Lemma \\ref{lemma-isolated-point}. This product decomposition", "induces a corresponding product decomposition for any", "$S \\otimes_R \\kappa(\\mathfrak p)$-algebra. In particular,", "we obtain $S' \\otimes_{R'} \\kappa(\\mathfrak p') = S'_1 \\times S'_2$.", "Because $S \\otimes_R R' \\to S'$ is surjective the canonical map", "$(S \\otimes_R \\kappa(\\mathfrak p)) \\otimes_{\\kappa(\\mathfrak p)}", "\\kappa(\\mathfrak p') \\to S' \\otimes_{R'} \\kappa(\\mathfrak p')$", "is surjective and hence", "$S_i \\otimes_{\\kappa(\\mathfrak p)} \\kappa(\\mathfrak p') \\to S'_i$", "is surjective. It follows that $S'_1$ is finite over $\\kappa(\\mathfrak p')$.", "The map $S' \\otimes_{R'} \\kappa(\\mathfrak p') \\to", "\\kappa(\\mathfrak q')$ factors through $S_1'$", "(i.e.\\ it annihilates the factor $S_2'$)", "because the map $S \\otimes_R \\kappa(\\mathfrak p) \\to", "\\kappa(\\mathfrak q)$ factors through $S_1$", "(i.e.\\ it annihilates the factor $S_2$). Thus", "$\\mathfrak q'$ corresponds to a point of", "$\\Spec(S_1')$ in the disjoint union decomposition", "of the fibre: $\\Spec(S' \\otimes_{R'} \\kappa(\\mathfrak p'))", "= \\Spec(S_1') \\amalg \\Spec(S_2')$, see", "Lemma \\ref{lemma-spec-product}.", "Since $S_1'$ is finite over a field, it is Artinian ring,", "and hence $\\Spec(S_1')$ is a finite discrete set.", "(See Proposition \\ref{proposition-dimension-zero-ring}.)", "We conclude $\\mathfrak q'$ is isolated in its fibre as", "desired." ], "refs": [ "algebra-lemma-isolated-point", "algebra-lemma-spec-product", "algebra-proposition-dimension-zero-ring" ], "ref_ids": [ 1048, 404, 1410 ] } ], "ref_ids": [] }, { "id": 1053, "type": "theorem", "label": "algebra-lemma-quasi-finite-composition", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-finite-composition", "contents": [ "A composition of quasi-finite ring maps is quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Suppose $A \\to B$ and $B \\to C$ are quasi-finite ring maps. By", "Lemma \\ref{lemma-compose-finite-type}", "we see that $A \\to C$ is of finite type.", "Let $\\mathfrak r \\subset C$ be a prime of $C$ lying over", "$\\mathfrak q \\subset B$ and $\\mathfrak p \\subset A$. Since $A \\to B$ and", "$B \\to C$ are quasi-finite at $\\mathfrak q$ and $\\mathfrak r$ respectively,", "then there exist $b \\in B$ and $c \\in C$ such that $\\mathfrak q$ is", "the only prime of $D(b)$ which maps to $\\mathfrak p$ and similarly", "$\\mathfrak r$ is the only prime of $D(c)$ which maps to $\\mathfrak q$.", "If $c' \\in C$ is the image of $b \\in B$, then $\\mathfrak r$ is the only", "prime of $D(cc')$ which maps to $\\mathfrak p$.", "Therefore $A \\to C$ is quasi-finite at $\\mathfrak r$." ], "refs": [ "algebra-lemma-compose-finite-type" ], "ref_ids": [ 333 ] } ], "ref_ids": [] }, { "id": 1054, "type": "theorem", "label": "algebra-lemma-quasi-finite-base-change", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-finite-base-change", "contents": [ "Let $R \\to S$ be a ring map of finite type.", "Let $R \\to R'$ be any ring map. Set $S' = R' \\otimes_R S$.", "\\begin{enumerate}", "\\item The set", "$\\{\\mathfrak q' \\mid R' \\to S' \\text{ quasi-finite at }\\mathfrak q'\\}$", "is the inverse image of the corresponding set of $\\Spec(S)$", "under the canonical map $\\Spec(S') \\to \\Spec(S)$.", "\\item If $\\Spec(R') \\to \\Spec(R)$ is surjective,", "then $R \\to S$ is quasi-finite if and only if $R' \\to S'$ is quasi-finite.", "\\item Any base change of a quasi-finite ring map is quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p' \\subset R'$ be a prime lying over $\\mathfrak p \\subset R$.", "Then the fibre ring $S' \\otimes_{R'} \\kappa(\\mathfrak p')$ is the", "base change of the fibre ring $S \\otimes_R \\kappa(\\mathfrak p)$", "by the field extension $\\kappa(\\mathfrak p) \\to \\kappa(\\mathfrak p')$.", "Hence the first assertion follows from the invariance of dimension", "under field extension", "(Lemma \\ref{lemma-dimension-at-a-point-preserved-field-extension})", "and Lemma \\ref{lemma-isolated-point}.", "The stability of quasi-finite maps under base change follows from", "this and the stability of finite type property under base change.", "The second assertion follows", "since the assumption implies that given a prime $\\mathfrak q \\subset S$ we can", "find a prime $\\mathfrak q' \\subset S'$ lying over it." ], "refs": [ "algebra-lemma-dimension-at-a-point-preserved-field-extension", "algebra-lemma-isolated-point" ], "ref_ids": [ 1010, 1048 ] } ], "ref_ids": [] }, { "id": 1055, "type": "theorem", "label": "algebra-lemma-quasi-finite-permanence", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-finite-permanence", "contents": [ "Let $A \\to B$ and $B \\to C$ be ring homomorphisms such that $A \\to C$", "is of finite type. Let $\\mathfrak r$ be a prime of $C$ lying over", "$\\mathfrak q \\subset B$ and $\\mathfrak p \\subset A$.", "If $A \\to C$ is quasi-finite at $\\mathfrak r$, then", "$B \\to C$ is quasi-finite at $\\mathfrak r$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $B \\to C$ is of finite type", "(Lemma \\ref{lemma-compose-finite-type})", "so that the statement makes sense.", "Let us use characterization (3) of Lemma \\ref{lemma-isolated-point-fibre}.", "If $A \\to C$ is quasi-finite at $\\mathfrak r$, then", "there exists some $c \\in C$ such that", "$$", "\\{\\mathfrak r' \\subset C \\text{ lying over }\\mathfrak p\\} \\cap D(c) =", "\\{\\mathfrak{r}\\}.", "$$", "Since the primes $\\mathfrak r' \\subset C$ lying over $\\mathfrak q$", "form a subset of the primes $\\mathfrak r' \\subset C$ lying over", "$\\mathfrak p$ we conclude $B \\to C$ is quasi-finite at $\\mathfrak r$." ], "refs": [ "algebra-lemma-compose-finite-type", "algebra-lemma-isolated-point-fibre" ], "ref_ids": [ 333, 1049 ] } ], "ref_ids": [] }, { "id": 1056, "type": "theorem", "label": "algebra-lemma-generically-finite", "categories": [ "algebra" ], "title": "algebra-lemma-generically-finite", "contents": [ "Let $R \\to S$ be a ring map of finite type.", "Let $\\mathfrak p \\subset R$ be a minimal prime.", "Assume that there are at most finitely many primes of $S$", "lying over $\\mathfrak p$. Then there exists a", "$g \\in R$, $g \\not \\in \\mathfrak p$ such that the", "ring map $R_g \\to S_g$ is finite." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1, \\ldots, x_n$ be generators of $S$ over $R$.", "Since $\\mathfrak p$ is a minimal prime we have that", "$\\mathfrak pR_{\\mathfrak p}$ is a locally nilpotent ideal, see", "Lemma \\ref{lemma-minimal-prime-reduced-ring}.", "Hence $\\mathfrak pS_{\\mathfrak p}$ is a locally nilpotent ideal, see", "Lemma \\ref{lemma-locally-nilpotent}.", "By assumption the finite type $\\kappa(\\mathfrak p)$-algebra", "$S_{\\mathfrak p}/\\mathfrak pS_{\\mathfrak p}$ has finitely many", "primes. Hence (for example by", "Lemmas \\ref{lemma-finite-type-algebra-finite-nr-primes} and", "\\ref{lemma-Noether-normalization})", "$\\kappa(\\mathfrak p) \\to S_{\\mathfrak p}/\\mathfrak pS_{\\mathfrak p}$", "is a finite ring map. Thus we may find monic polynomials", "$P_i \\in R_{\\mathfrak p}[X]$ such that $P_i(x_i)$ maps to zero", "in $S_{\\mathfrak p}/\\mathfrak pS_{\\mathfrak p}$. By what we said", "above there exist $e_i \\geq 1$ such that $P(x_i)^{e_i} = 0$", "in $S_{\\mathfrak p}$. Let $g_1 \\in R$, $g_1 \\not \\in \\mathfrak p$", "be an element such that $P_i$ has coefficients in $R[1/g_1]$ for all $i$.", "Next, let $g_2 \\in R$, $g_2 \\not \\in \\mathfrak p$ be an element", "such that $P(x_i)^{e_i} = 0$ in $S_{g_1g_2}$. Setting $g = g_1g_2$", "we win." ], "refs": [ "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-locally-nilpotent", "algebra-lemma-finite-type-algebra-finite-nr-primes", "algebra-lemma-Noether-normalization" ], "ref_ids": [ 418, 458, 689, 1001 ] } ], "ref_ids": [] }, { "id": 1057, "type": "theorem", "label": "algebra-lemma-make-integral-trivial", "categories": [ "algebra" ], "title": "algebra-lemma-make-integral-trivial", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "Suppose $t \\in S$ satisfies the", "relation $\\varphi(a_0) + \\varphi(a_1)t + \\ldots + \\varphi(a_n) t^n = 0$.", "Then $\\varphi(a_n)t$ is integral over $R$." ], "refs": [], "proofs": [ { "contents": [ "Namely, multiply the equation", "$\\varphi(a_0) + \\varphi(a_1)t + \\ldots + \\varphi(a_n) t^n = 0$", "with $\\varphi(a_n)^{n-1}$ and write it as", "$\\varphi(a_0 a_n^{n-1}) +", "\\varphi(a_1 a_n^{n-2}) (\\varphi(a_n)t) +", "\\ldots +", "(\\varphi(a_n) t)^n = 0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1058, "type": "theorem", "label": "algebra-lemma-make-integral-trick", "categories": [ "algebra" ], "title": "algebra-lemma-make-integral-trick", "contents": [ "Let $R$ be a ring. Let $\\varphi : R[x] \\to S$ be", "a ring map. Let $t \\in S$.", "Assume that (a) $t$ is integral over $R[x]$,", "and (b) there exists a monic $p \\in R[x]$ such that", "$t \\varphi(p) \\in \\Im(\\varphi)$. Then there", "exists a $q \\in R[x]$ such that $t - \\varphi(q)$", "is integral over $R$." ], "refs": [], "proofs": [ { "contents": [ "Write $t \\varphi(p) = \\varphi(r)$ for some $r \\in R[x]$.", "Using euclidean division, write $r = qp + r'$ with", "$q, r' \\in R[x]$ and $\\deg(r') < \\deg(p)$. We may replace", "$t$ by $t - \\varphi(q)$ which is still integral over", "$R[x]$, so that we obtain $t \\varphi(p) = \\varphi(r')$.", "In the ring $S_t$ we may write this as", "$\\varphi(p) - (1/t) \\varphi(r') = 0$.", "This implies that $\\varphi(x)$ gives an element of the", "localization $S_t$ which is integral over", "$\\varphi(R)[1/t] \\subset S_t$. On the other hand,", "$t$ is integral over the subring $\\varphi(R)[\\varphi(x)] \\subset S$.", "Combined we conclude that $t$ is integral over", "the subring $\\varphi(R)[1/t] \\subset S_t$, see Lemma", "\\ref{lemma-integral-transitive}. In other words", "there exists an equation of the form", "$$", "t^d + \\sum\\nolimits_{i < d}", "\\left(\\sum\\nolimits_{j = 0, \\ldots, n_i} \\varphi(r_{i, j})/t^j\\right) t^i = 0", "$$", "in $S_t$ with $r_{i, j} \\in R$. This means that", "$t^{d + N} +", "\\sum_{i < d} \\sum_{j = 0, \\ldots, n_i} \\varphi(r_{i, j}) t^{i + N - j} = 0$", "in $S$ for some $N$ large enough. In other words", "$t$ is integral over $R$." ], "refs": [ "algebra-lemma-integral-transitive" ], "ref_ids": [ 485 ] } ], "ref_ids": [] }, { "id": 1059, "type": "theorem", "label": "algebra-lemma-combine-lemmas", "categories": [ "algebra" ], "title": "algebra-lemma-combine-lemmas", "contents": [ "Let $R$ be a ring. Let $\\varphi : R[x] \\to S$ be", "a ring map. Let $t \\in S$. Assume $t$ is integral", "over $R[x]$. Let $p \\in R[x]$, $p = a_0 + a_1x + \\ldots +", "a_k x^k$ such that $t \\varphi(p) \\in \\Im(\\varphi)$.", "Then there exists a $q \\in R[x]$ and $n \\geq 0$", "such that $\\varphi(a_k)^n t - \\varphi(q)$ is integral", "over $R$." ], "refs": [], "proofs": [ { "contents": [ "Let $R'$ and $S'$ be the localization of $R$ and $S$ at the element $a_k$.", "Let $\\varphi' : R'[x] \\to S'$ be the localization of $\\varphi$.", "Let $t' \\in S'$ be the image of $t$. Set $p' = p/a_k \\in R'[x]$.", "Then $t' \\varphi'(p') \\in \\Im(\\varphi')$ since $t \\varphi(p) \\in \\Im(\\varphi)$.", "As $p'$ is monic, by", "Lemma \\ref{lemma-make-integral-trick} there exists a $q' \\in R'[x]$", "such that $t' - \\varphi'(q')$ is integral over $R'$.", "We may choose an $n \\geq 0$ and an element $q \\in R[x]$", "such that $a_k^n q'$ is the image of $q$.", "Then $\\varphi(a_k)^n t - \\varphi(q)$ is an element of $S$ whose", "image in $S'$ is integral over $R'$.", "By Lemma \\ref{lemma-integral-closure-localize}", "there exists an $m \\geq 0$ such that", "$\\varphi(a_k)^m(\\varphi(a_k)^n t - \\varphi(q))$ is integral over $R$.", "Thus $\\varphi(a_k)^{m + n}t - \\varphi(a_k^m q)$", "is integral over $R$ as desired." ], "refs": [ "algebra-lemma-make-integral-trick", "algebra-lemma-integral-closure-localize" ], "ref_ids": [ 1058, 489 ] } ], "ref_ids": [] }, { "id": 1060, "type": "theorem", "label": "algebra-lemma-leading-coefficient-in-J", "categories": [ "algebra" ], "title": "algebra-lemma-leading-coefficient-in-J", "contents": [ "In Situation \\ref{situation-one-transcendental-element}.", "Suppose $u \\in S$, $a_0, \\ldots, a_k \\in R$,", "$u \\varphi(a_0 + a_1x + \\ldots + a_k x^k) \\in J$.", "Then there exists an $m \\geq 0$ such that", "$u \\varphi(a_k)^m \\in J$." ], "refs": [], "proofs": [ { "contents": [ "Assume that $S$ is generated by $t_1, \\ldots, t_n$", "as an $R[x]$-module. In this case", "$J = \\{ g \\in S \\mid gt_i \\in \\Im(\\varphi)\\text{ for all }i\\}$.", "Note that each element $u t_i$ is integral over", "$R[x]$, see Lemma \\ref{lemma-finite-is-integral}.", "We have $\\varphi(a_0 + a_1x + \\ldots + a_k x^k) u t_i \\in", "\\Im(\\varphi)$. By Lemma \\ref{lemma-combine-lemmas}, for", "each $i$ there exists an integer $n_i$ and an element", "$q_i \\in R[x]$ such that $\\varphi(a_k^{n_i}) u t_i - \\varphi(q_i)$", "is integral over $R$. By assumption this element is in $\\varphi(R)$", "and hence $\\varphi(a_k^{n_i}) u t_i \\in \\Im(\\varphi)$.", "It follows that $m = \\max\\{n_1, \\ldots, n_n\\}$ works." ], "refs": [ "algebra-lemma-finite-is-integral", "algebra-lemma-combine-lemmas" ], "ref_ids": [ 482, 1059 ] } ], "ref_ids": [] }, { "id": 1061, "type": "theorem", "label": "algebra-lemma-all-coefficients-in-J", "categories": [ "algebra" ], "title": "algebra-lemma-all-coefficients-in-J", "contents": [ "In Situation \\ref{situation-one-transcendental-element}.", "Suppose $u \\in S$, $a_0, \\ldots, a_k \\in R$,", "$u \\varphi(a_0 + a_1x + \\ldots + a_k x^k) \\in \\sqrt{J}$.", "Then $u \\varphi(a_i) \\in \\sqrt{J}$ for all $i$." ], "refs": [], "proofs": [ { "contents": [ "Under the assumptions of the lemma we have", "$u^n \\varphi(a_0 + a_1x + \\ldots + a_k x^k)^n \\in J$ for", "some $n \\geq 1$. By Lemma \\ref{lemma-leading-coefficient-in-J}", "we deduce $u^n \\varphi(a_k^{nm}) \\in J$ for some $m \\geq 1$.", "Thus $u \\varphi(a_k) \\in \\sqrt{J}$, and so", "$u \\varphi(a_0 + a_1x + \\ldots + a_k x^k) - u \\varphi(a_k x^k) =", "u \\varphi(a_0 + a_1x + \\ldots + a_{k-1} x^{k-1}) \\in \\sqrt{J}$.", "We win by induction on $k$." ], "refs": [ "algebra-lemma-leading-coefficient-in-J" ], "ref_ids": [ 1060 ] } ], "ref_ids": [] }, { "id": 1062, "type": "theorem", "label": "algebra-lemma-reduced-strongly-transcendental-minimal-prime", "categories": [ "algebra" ], "title": "algebra-lemma-reduced-strongly-transcendental-minimal-prime", "contents": [ "Suppose $R \\subset S$ is an inclusion of reduced rings", "and suppose that $x \\in S$ is strongly transcendental over $R$.", "Let $\\mathfrak q \\subset S$ be a minimal prime", "and let $\\mathfrak p = R \\cap \\mathfrak q$.", "Then the image of $x$ in $S/\\mathfrak q$ is strongly", "transcendental over the subring $R/\\mathfrak p$." ], "refs": [], "proofs": [ { "contents": [ "Suppose $u(a_0 + a_1x + \\ldots + a_k x^k) \\in \\mathfrak q$.", "By Lemma \\ref{lemma-minimal-prime-reduced-ring}", "the local ring $S_{\\mathfrak q}$ is a field,", "and hence $u(a_0 + a_1x + \\ldots + a_k x^k) $ is zero", "in $S_{\\mathfrak q}$. Thus $uu'(a_0 + a_1x + \\ldots + a_k x^k) = 0$", "for some $u' \\in S$, $u' \\not\\in \\mathfrak q$.", "Since $x$ is strongly transcendental over $R$ we get", "$uu'a_i = 0$ for all $i$. This in turn implies", "that $ua_i \\in \\mathfrak q$." ], "refs": [ "algebra-lemma-minimal-prime-reduced-ring" ], "ref_ids": [ 418 ] } ], "ref_ids": [] }, { "id": 1063, "type": "theorem", "label": "algebra-lemma-domains-transcendental-not-quasi-finite", "categories": [ "algebra" ], "title": "algebra-lemma-domains-transcendental-not-quasi-finite", "contents": [ "Suppose $R\\subset S$ is an inclusion of domains and", "let $x \\in S$. Assume $x$ is (strongly) transcendental over $R$", "and that $S$ is finite over $R[x]$. Then $R \\to S$ is not", "quasi-finite at any prime of $S$." ], "refs": [], "proofs": [ { "contents": [ "As a first case, assume that $R$ is normal, see", "Definition \\ref{definition-ring-normal}.", "By Lemma \\ref{lemma-polynomial-ring-normal}", "we see that $R[x]$ is normal.", "Take a prime $\\mathfrak q \\subset S$,", "and set $\\mathfrak p = R \\cap \\mathfrak q$.", "Assume that the extension $\\kappa(\\mathfrak p)", "\\subset \\kappa(\\mathfrak q)$ is finite.", "This would be the case if $R \\to S$ is", "quasi-finite at $\\mathfrak q$.", "Let $\\mathfrak r = R[x] \\cap \\mathfrak q$.", "Then since $\\kappa(\\mathfrak p)", "\\subset \\kappa(\\mathfrak r) \\subset \\kappa(\\mathfrak q)$", "we see that the extension $\\kappa(\\mathfrak p)", "\\subset \\kappa(\\mathfrak r)$ is finite too.", "Thus the inclusion $\\mathfrak r \\supset \\mathfrak p R[x]$", "is strict. By going down for $R[x] \\subset S$,", "see Proposition \\ref{proposition-going-down-normal-integral},", "we find a prime $\\mathfrak q' \\subset \\mathfrak q$,", "lying over the prime $\\mathfrak pR[x]$. Hence", "the fibre $\\Spec(S \\otimes_R \\kappa(\\mathfrak p))$", "contains a point not equal to $\\mathfrak q$,", "namely $\\mathfrak q'$, whose closure contains $\\mathfrak q$ and hence", "$\\mathfrak q$ is not isolated in its fibre.", "\\medskip\\noindent", "If $R$ is not normal, let $R \\subset R' \\subset K$ be", "the integral closure $R'$ of $R$ in its field of fractions", "$K$. Let $S \\subset S' \\subset L$ be the subring $S'$ of", "the field of fractions $L$ of $S$ generated by $R'$ and", "$S$. Note that by construction the map $S \\otimes_R R'", "\\to S'$ is surjective. This implies that $R'[x] \\subset S'$", "is finite. Also, the map $S \\subset S'$", "induces a surjection on $\\Spec$, see", "Lemma \\ref{lemma-integral-overring-surjective}.", "We conclude by Lemma \\ref{lemma-four-rings} and the normal case", "we just discussed." ], "refs": [ "algebra-definition-ring-normal", "algebra-lemma-polynomial-ring-normal", "algebra-proposition-going-down-normal-integral", "algebra-lemma-integral-overring-surjective", "algebra-lemma-four-rings" ], "ref_ids": [ 1454, 513, 1406, 495, 1052 ] } ], "ref_ids": [] }, { "id": 1064, "type": "theorem", "label": "algebra-lemma-reduced-strongly-transcendental-not-quasi-finite", "categories": [ "algebra" ], "title": "algebra-lemma-reduced-strongly-transcendental-not-quasi-finite", "contents": [ "Suppose $R \\subset S$ is an inclusion of reduced rings.", "Assume $x \\in S$ be strongly transcendental over $R$,", "and $S$ finite over $R[x]$. Then $R \\to S$ is not", "quasi-finite at any prime of $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q \\subset S$ be any prime.", "Choose a minimal prime $\\mathfrak q' \\subset \\mathfrak q$.", "According to Lemmas", "\\ref{lemma-reduced-strongly-transcendental-minimal-prime} and", "\\ref{lemma-domains-transcendental-not-quasi-finite}", "the extension $R/(R \\cap \\mathfrak q') \\subset", "S/\\mathfrak q'$ is not quasi-finite at the prime corresponding", "to $\\mathfrak q$. By Lemma \\ref{lemma-four-rings}", "the extension $R \\to S$ is not quasi-finite", "at $\\mathfrak q$." ], "refs": [ "algebra-lemma-reduced-strongly-transcendental-minimal-prime", "algebra-lemma-domains-transcendental-not-quasi-finite", "algebra-lemma-four-rings" ], "ref_ids": [ 1062, 1063, 1052 ] } ], "ref_ids": [] }, { "id": 1065, "type": "theorem", "label": "algebra-lemma-quasi-finite-monogenic", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-finite-monogenic", "contents": [ "Let $R$ be a ring. Let $S = R[x]/I$.", "Let $\\mathfrak q \\subset S$ be a prime.", "Assume $R \\to S$ is quasi-finite at $\\mathfrak q$.", "Let $S' \\subset S$ be the integral closure of $R$ in $S$.", "Then there exists an element", "$g \\in S'$, $g \\not\\in \\mathfrak q$ such that", "$S'_g \\cong S_g$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p$ be the image of $\\mathfrak q$ in $\\Spec(R)$.", "There exists an $f \\in I$, $f = a_nx^n + \\ldots + a_0$ such that", "$a_i \\not \\in \\mathfrak p$ for some $i$. Namely, otherwise the fibre ring", "$S \\otimes_R \\kappa(\\mathfrak p)$ would be $\\kappa(\\mathfrak p)[x]$", "and the map would not be quasi-finite at any prime lying", "over $\\mathfrak p$. We conclude there exists a relation", "$b_m x^m + \\ldots + b_0 = 0$ with $b_j \\in S'$, $j = 0, \\ldots, m$", "and $b_j \\not \\in \\mathfrak q \\cap S'$ for some $j$.", "We prove the lemma by induction on $m$. The base case is", "$m = 0$ is vacuous (because the statements $b_0 = 0$ and", "$b_0 \\not \\in \\mathfrak q$ are contradictory).", "\\medskip\\noindent", "The case $b_m \\not \\in \\mathfrak q$. In this case $x$ is integral", "over $S'_{b_m}$, in fact $b_mx \\in S'$ by", "Lemma \\ref{lemma-make-integral-trivial}.", "Hence the injective map $S'_{b_m} \\to S_{b_m}$ is also surjective, i.e.,", "an isomorphism as desired.", "\\medskip\\noindent", "The case $b_m \\in \\mathfrak q$. In this case we have $b_mx \\in S'$ by", "Lemma \\ref{lemma-make-integral-trivial}.", "Set $b'_{m - 1} = b_mx + b_{m - 1}$. Then", "$$", "b'_{m - 1}x^{m - 1} + b_{m - 2}x^{m - 2} + \\ldots + b_0 = 0", "$$", "Since $b'_{m - 1}$ is congruent to $b_{m - 1}$ modulo $S' \\cap \\mathfrak q$", "we see that it is still the case that one of", "$b'_{m - 1}, b_{m - 2}, \\ldots, b_0$ is not in $S' \\cap \\mathfrak q$.", "Thus we win by induction on $m$." ], "refs": [ "algebra-lemma-make-integral-trivial", "algebra-lemma-make-integral-trivial" ], "ref_ids": [ 1057, 1057 ] } ], "ref_ids": [] }, { "id": 1066, "type": "theorem", "label": "algebra-lemma-quasi-finite-open", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-finite-open", "contents": [ "Let $R \\to S$ be a finite type ring map.", "The set of points $\\mathfrak q$ of $\\Spec(S)$ at which", "$S/R$ is quasi-finite is open in $\\Spec(S)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q \\subset S$ be a point at which the ring map", "is quasi-finite. By Theorem \\ref{theorem-main-theorem}", "there exists an integral ring extension $R \\to S'$, $S' \\subset S$", "and an element $g \\in S'$, $g\\not \\in \\mathfrak q$ such that", "$S'_g \\cong S_g$. Since $S$ and hence $S_g$ are of finite type", "over $R$ we may find finitely many elements", "$y_1, \\ldots, y_N$ of $S'$ such that $S''_g \\cong S_g$", "where $S'' \\subset S'$ is the sub $R$-algebra generated", "by $g, y_1, \\ldots, y_N$. Since $S''$ is finite over $R$", "(see Lemma \\ref{lemma-characterize-integral}) we see that", "$S''$ is quasi-finite over $R$ (see Lemma \\ref{lemma-quasi-finite}).", "It is easy to see that this implies that $S''_g$ is quasi-finite over $R$,", "for example because the property of being quasi-finite at a prime depends", "only on the local ring at the prime. Thus we see that $S_g$ is quasi-finite", "over $R$. By the same token this implies that $R \\to S$ is quasi-finite", "at every prime of $S$ which lies in $D(g)$." ], "refs": [ "algebra-theorem-main-theorem", "algebra-lemma-characterize-integral", "algebra-lemma-quasi-finite" ], "ref_ids": [ 325, 483, 1050 ] } ], "ref_ids": [] }, { "id": 1067, "type": "theorem", "label": "algebra-lemma-quasi-finite-open-integral-closure", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-finite-open-integral-closure", "contents": [ "Let $R \\to S$ be a finite type ring map.", "Suppose that $S$ is quasi-finite over $R$.", "Let $S' \\subset S$ be the integral closure of $R$ in $S$. Then", "\\begin{enumerate}", "\\item $\\Spec(S) \\to \\Spec(S')$ is a homeomorphism", "onto an open subset,", "\\item if $g \\in S'$ and $D(g)$ is contained in the image", "of the map, then $S'_g \\cong S_g$, and", "\\item there exists a finite $R$-algebra $S'' \\subset S'$", "such that (1) and (2) hold for the ring map", "$S'' \\to S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Because $S/R$ is quasi-finite we may apply", "Theorem \\ref{theorem-main-theorem} to", "each point $\\mathfrak q$ of $\\Spec(S)$.", "Since $\\Spec(S)$ is quasi-compact, see", "Lemma \\ref{lemma-quasi-compact}, we may choose", "a finite number of $g_i \\in S'$, $i = 1, \\ldots, n$", "such that $S'_{g_i} = S_{g_i}$, and such that", "$g_1, \\ldots, g_n$ generate the unit ideal in $S$", "(in other words the standard opens of $\\Spec(S)$ associated", "to $g_1, \\ldots, g_n$ cover all of $\\Spec(S)$).", "\\medskip\\noindent", "Suppose that $D(g) \\subset \\Spec(S')$", "is contained in the image. Then $D(g) \\subset \\bigcup D(g_i)$.", "In other words, $g_1, \\ldots, g_n$ generate the unit ideal of", "$S'_g$. Note that $S'_{gg_i} \\cong S_{gg_i}$ by our choice", "of $g_i$. Hence $S'_g \\cong S_g$ by Lemma \\ref{lemma-cover}.", "\\medskip\\noindent", "We construct a finite algebra $S'' \\subset S'$ as", "in (3). To do this note that each $S'_{g_i} \\cong S_{g_i}$", "is a finite type $R$-algebra. For each $i$ pick", "some elements $y_{ij} \\in S'$ such that each", "$S'_{g_i}$ is generated as $R$-algebra by $1/g_i$", "and the elements $y_{ij}$. Then set $S''$", "equal to the sub $R$-algebra of $S'$ generated by all $g_i$", "and all the $y_{ij}$. Details omitted." ], "refs": [ "algebra-theorem-main-theorem", "algebra-lemma-quasi-compact", "algebra-lemma-cover" ], "ref_ids": [ 325, 395, 411 ] } ], "ref_ids": [] }, { "id": 1068, "type": "theorem", "label": "algebra-lemma-quasi-finite-extension-dim-1", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-finite-extension-dim-1", "contents": [ "Let $A \\subset B$ be an extension of domains. Assume", "\\begin{enumerate}", "\\item $A$ is a local Noetherian ring of dimension $1$,", "\\item $A \\to B$ is of finite type, and", "\\item the induced extension $L/K$ of fraction fields is finite.", "\\end{enumerate}", "Then $B$ is semi-local.", "Let $x \\in \\mathfrak m_A$, $x \\not = 0$.", "Let $\\mathfrak m_i$, $i = 1, \\ldots, n$", "be the maximal ideals of $B$.", "Then", "$$", "[L : K]\\text{ord}_A(x)", "\\geq", "\\sum\\nolimits_i", "[\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m_A)]", "\\text{ord}_{B_{\\mathfrak m_i}}(x)", "$$", "where $\\text{ord}$ is defined as in Definition \\ref{definition-ord}.", "We have equality if and only if $A \\to B$ is finite." ], "refs": [ "algebra-definition-ord" ], "proofs": [ { "contents": [ "The ring $B$ is semi-local by Lemma \\ref{lemma-finite-in-codim-1}.", "Let $B'$ be the integral closure of $A$ in $B$. By", "Lemma \\ref{lemma-quasi-finite-open-integral-closure}", "we can find a finite $A$-subalgebra $C \\subset B'$ such that", "on setting $\\mathfrak n_i = C \\cap \\mathfrak m_i$ we have", "$C_{\\mathfrak n_i} \\cong B_{\\mathfrak m_i}$ and the primes", "$\\mathfrak n_1, \\ldots, \\mathfrak n_n$ are pairwise distinct.", "The ring $C$ is semi-local by Lemma \\ref{lemma-finite-in-codim-1}.", "Let $\\mathfrak p_j$, $j = 1, \\ldots, m$ be the other maximal", "ideals of $C$ (the ``missing points''). By", "Lemma \\ref{lemma-finite-extension-dim-1} we have", "$$", "\\text{ord}_A(x^{[L : K]}) =", "\\sum\\nolimits_i", "[\\kappa(\\mathfrak n_i) : \\kappa(\\mathfrak m_A)]", "\\text{ord}_{C_{\\mathfrak n_i}}(x)", "+", "\\sum\\nolimits_j", "[\\kappa(\\mathfrak p_j) : \\kappa(\\mathfrak m_A)]", "\\text{ord}_{C_{\\mathfrak p_j}}(x)", "$$", "hence the inequality follows. In case of equality we conclude that", "$m = 0$ (no ``missing points''). Hence $C \\subset B$ is an inclusion", "of semi-local rings inducing a bijection on maximal ideals and", "an isomorphism on all localizations at maximal ideals. So if $b \\in B$,", "then $I = \\{x \\in C \\mid xb \\in C\\}$ is an ideal of $C$ which is not", "contained in any of the maximal ideals of $C$, and hence $I = C$,", "hence $b \\in C$. Thus $B = C$ and $B$ is finite over $A$." ], "refs": [ "algebra-lemma-finite-in-codim-1", "algebra-lemma-quasi-finite-open-integral-closure", "algebra-lemma-finite-in-codim-1", "algebra-lemma-finite-extension-dim-1" ], "ref_ids": [ 991, 1067, 991, 1047 ] } ], "ref_ids": [ 1519 ] }, { "id": 1069, "type": "theorem", "label": "algebra-lemma-essentially-finite-type-fibre-dim-zero", "categories": [ "algebra" ], "title": "algebra-lemma-essentially-finite-type-fibre-dim-zero", "contents": [ "Let $(R, \\mathfrak m_R) \\to (S, \\mathfrak m_S)$ be a local homomorphism", "of local rings. Assume", "\\begin{enumerate}", "\\item $R \\to S$ is essentially of finite type,", "\\item $\\kappa(\\mathfrak m_R) \\subset \\kappa(\\mathfrak m_S)$ is finite, and", "\\item $\\dim(S/\\mathfrak m_RS) = 0$.", "\\end{enumerate}", "Then $S$ is the localization of a finite $R$-algebra." ], "refs": [], "proofs": [ { "contents": [ "Let $S'$ be a finite type $R$-algebra such that $S = S'_{\\mathfrak q'}$", "for some prime $\\mathfrak q'$ of $S'$. By", "Definition \\ref{definition-quasi-finite}", "we see that $R \\to S'$ is quasi-finite at $\\mathfrak q'$.", "After replacing $S'$ by $S'_{g'}$ for some", "$g' \\in S'$, $g' \\not \\in \\mathfrak q'$ we may assume that $R \\to S'$ is", "quasi-finite, see", "Lemma \\ref{lemma-quasi-finite-open}.", "Then by", "Lemma \\ref{lemma-quasi-finite-open-integral-closure}", "there exists a finite $R$-algebra $S''$ and elements", "$g' \\in S'$, $g' \\not \\in \\mathfrak q'$ and $g'' \\in S''$", "such that $S'_{g'} \\cong S''_{g''}$ as $R$-algebras.", "This proves the lemma." ], "refs": [ "algebra-definition-quasi-finite", "algebra-lemma-quasi-finite-open", "algebra-lemma-quasi-finite-open-integral-closure" ], "ref_ids": [ 1522, 1066, 1067 ] } ], "ref_ids": [] }, { "id": 1070, "type": "theorem", "label": "algebra-lemma-completion-at-quasi-finite-prime", "categories": [ "algebra" ], "title": "algebra-lemma-completion-at-quasi-finite-prime", "contents": [ "Let $R \\to S$ be a ring map, $\\mathfrak q$ a prime of $S$", "lying over $\\mathfrak p$ in $R$. If", "\\begin{enumerate}", "\\item $R$ is Noetherian,", "\\item $R \\to S$ is of finite type, and", "\\item $R \\to S$ is quasi-finite at $\\mathfrak q$,", "\\end{enumerate}", "then $R_\\mathfrak p^\\wedge \\otimes_R S = S_\\mathfrak q^\\wedge \\times B$", "for some $R_\\mathfrak p^\\wedge$-algebra $B$." ], "refs": [], "proofs": [ { "contents": [ "There exists a finite $R$-algebra $S' \\subset S$ and an element", "$g \\in S'$, $g \\not \\in \\mathfrak q' = S' \\cap \\mathfrak q$", "such that $S'_g = S_g$ and in particular", "$S'_{\\mathfrak q'} = S_\\mathfrak q$, see", "Lemma \\ref{lemma-quasi-finite-open-integral-closure}.", "We have", "$$", "R_\\mathfrak p^\\wedge \\otimes_R S' = (S'_{\\mathfrak q'})^\\wedge \\times B'", "$$", "by Lemma \\ref{lemma-completion-finite-extension}. Observe that under this", "product decomposition $g$ maps to a pair $(u, b')$ with", "$u \\in (S'_{\\mathfrak q'})^\\wedge$ a unit because $g \\not \\in \\mathfrak q'$.", "The product decomposition for $R_\\mathfrak p^\\wedge \\otimes_R S'$", "induces a product decomposition", "$$", "R_\\mathfrak p^\\wedge \\otimes_R S = A \\times B", "$$", "Since $S'_g = S_g$ we also have", "$(R_\\mathfrak p^\\wedge \\otimes_R S')_g = (R_\\mathfrak p^\\wedge \\otimes_R S)_g$", "and since $g \\mapsto (u, b')$ where $u$ is a unit we see that", "$(S'_{\\mathfrak q'})^\\wedge = A$. Since the isomorphism", "$S'_{\\mathfrak q'} = S_\\mathfrak q$ determines an isomorphism on", "completions this also tells us that $A = S_\\mathfrak q^\\wedge$.", "This finishes the proof, except that we should perform the sanity check", "that the induced map", "$\\phi : R_\\mathfrak p^\\wedge \\otimes_R S \\to A = S_\\mathfrak q^\\wedge$", "is the natural one. For elements of the form $x \\otimes 1$", "with $x \\in R_\\mathfrak p^\\wedge$ this is clear as the natural", "map $R_\\mathfrak p^\\wedge \\to S_\\mathfrak q^\\wedge$ factors through", "$(S'_{\\mathfrak q'})^\\wedge$. For elements of the form $1 \\otimes y$", "with $y \\in S$ we can argue that for some $n \\geq 1$ the element", "$g^ny$ is the image of some $y' \\in S'$. Thus $\\phi(1 \\otimes g^ny)$", "is the image of $y'$ by the composition", "$S' \\to (S'_{\\mathfrak q'})^\\wedge \\to S_\\mathfrak q^\\wedge$ which is", "equal to the image of $g^ny$ by the map $S \\to S_\\mathfrak q^\\wedge$.", "Since $g$ maps to a unit this also", "implies that $\\phi(1 \\otimes y)$ has the correct value, i.e., the", "image of $y$ by $S \\to S_\\mathfrak q^\\wedge$." ], "refs": [ "algebra-lemma-quasi-finite-open-integral-closure", "algebra-lemma-completion-finite-extension" ], "ref_ids": [ 1067, 876 ] } ], "ref_ids": [] }, { "id": 1071, "type": "theorem", "label": "algebra-lemma-quasi-finite-over-polynomial-algebra", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-finite-over-polynomial-algebra", "contents": [ "Let $R \\to S$ be a finite type ring map.", "Let $\\mathfrak q \\subset S$ be a prime.", "Suppose that $\\dim_{\\mathfrak q}(S/R) = n$.", "There exists a $g \\in S$, $g \\not\\in \\mathfrak q$", "such that $S_g$ is quasi-finite over a", "polynomial algebra $R[t_1, \\ldots, t_n]$." ], "refs": [], "proofs": [ { "contents": [ "The ring $\\overline{S} = S \\otimes_R \\kappa(\\mathfrak p)$ is", "of finite type over $\\kappa(\\mathfrak p)$.", "Let $\\overline{\\mathfrak q}$ be the prime of $\\overline{S}$", "corresponding to $\\mathfrak q$.", "By definition of", "the dimension of a topological space at a point there exists", "an open $U \\subset \\Spec(\\overline{S})$ with", "$\\overline{q} \\in U$ and $\\dim(U) = n$.", "Since the topology on $\\Spec(\\overline{S})$ is", "induced from the topology on $\\Spec(S)$ (see", "Remark \\ref{remark-fundamental-diagram}), we can find", "a $g \\in S$, $g \\not \\in \\mathfrak q$ with image", "$\\overline{g} \\in \\overline{S}$ such that", "$D(\\overline{g}) \\subset U$.", "Thus after replacing $S$ by $S_g$ we see that", "$\\dim(\\overline{S}) = n$.", "\\medskip\\noindent", "Next, choose generators $x_1, \\ldots, x_N$ for $S$ as an $R$-algebra. By", "Lemma \\ref{lemma-Noether-normalization}", "there exist elements $y_1, \\ldots, y_n$ in the $\\mathbf{Z}$-subalgebra of $S$", "generated by $x_1, \\ldots, x_N$ such that the map", "$R[t_1, \\ldots, t_n] \\to S$, $t_i \\mapsto y_i$ has the property", "that $\\kappa(\\mathfrak p)[t_1\\ldots, t_n] \\to \\overline{S}$", "is finite. In particular, $S$ is quasi-finite over $R[t_1, \\ldots, t_n]$", "at $\\mathfrak q$. Hence, by Lemma \\ref{lemma-quasi-finite-open}", "we may replace $S$ by $S_g$ for some $g\\in S$, $g \\not \\in \\mathfrak q$", "such that $R[t_1, \\ldots, t_n] \\to S$ is quasi-finite." ], "refs": [ "algebra-remark-fundamental-diagram", "algebra-lemma-Noether-normalization", "algebra-lemma-quasi-finite-open" ], "ref_ids": [ 1558, 1001, 1066 ] } ], "ref_ids": [] }, { "id": 1072, "type": "theorem", "label": "algebra-lemma-refined-quasi-finite-over-polynomial-algebra", "categories": [ "algebra" ], "title": "algebra-lemma-refined-quasi-finite-over-polynomial-algebra", "contents": [ "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$", "be a prime lying over the prime $\\mathfrak p$ of $R$.", "Assume", "\\begin{enumerate}", "\\item $R \\to S$ is of finite type,", "\\item $\\dim_{\\mathfrak q}(S/R) = n$, and", "\\item $\\text{trdeg}_{\\kappa(\\mathfrak p)}\\kappa(\\mathfrak q) = r$.", "\\end{enumerate}", "Then there exist $f \\in R$, $f \\not \\in \\mathfrak p$,", "$g \\in S$, $g \\not\\in \\mathfrak q$ and a quasi-finite ring map", "$$", "\\varphi : R_f[x_1, \\ldots, x_n] \\longrightarrow S_g", "$$", "such that $\\varphi^{-1}(\\mathfrak qS_g) =", "(\\mathfrak p, x_{r + 1}, \\ldots, x_n)R_f[x_{r + 1}, \\ldots, x_n]$" ], "refs": [], "proofs": [ { "contents": [ "After replacing $S$ by a principal localization we may assume there", "exists a quasi-finite ring map $\\varphi : R[t_1, \\ldots, t_n] \\to S$, see", "Lemma \\ref{lemma-quasi-finite-over-polynomial-algebra}.", "Set $\\mathfrak q' = \\varphi^{-1}(\\mathfrak q)$.", "Let $\\overline{\\mathfrak q}' \\subset \\kappa(\\mathfrak p)[t_1, \\ldots, t_n]$", "be the prime corresponding to $\\mathfrak q'$. By", "Lemma \\ref{lemma-refined-Noether-normalization}", "there exists a finite ring map", "$\\kappa(\\mathfrak p)[x_1, \\ldots, x_n] \\to", "\\kappa(\\mathfrak p)[t_1, \\ldots, t_n]$", "such that the inverse image of $\\overline{\\mathfrak q}'$ is", "$(x_{r + 1}, \\ldots, x_n)$. Let", "$\\overline{h}_i \\in \\kappa(\\mathfrak p)[t_1, \\ldots, t_n]$", "be the image of $x_i$. We can find an element", "$f \\in R$, $f \\not \\in \\mathfrak p$", "and $h_i \\in R_f[t_1, \\ldots, t_n]$ which map to $\\overline{h}_i$", "in $\\kappa(\\mathfrak p)[t_1, \\ldots, t_n]$. Then the ring map", "$$", "R_f[x_1, \\ldots, x_n] \\longrightarrow R_f[t_1, \\ldots, t_n]", "$$", "becomes finite after tensoring with $\\kappa(\\mathfrak p)$.", "In particular, $R_f[t_1, \\ldots, t_n]$ is quasi-finite over", "$R_f[x_1, \\ldots, x_n]$ at the prime $\\mathfrak q'R_f[t_1, \\ldots, t_n]$.", "Hence, by", "Lemma \\ref{lemma-quasi-finite-open}", "there exists a $g \\in R_f[t_1, \\ldots, t_n]$,", "$g \\not \\in \\mathfrak q'R_f[t_1, \\ldots, t_n]$", "such that $R_f[x_1, \\ldots, x_n] \\to R_f[t_1, \\ldots, t_n, 1/g]$", "is quasi-finite. Thus we see that the composition", "$$", "R_f[x_1, \\ldots, x_n] \\longrightarrow", "R_f[t_1, \\ldots, t_n, 1/g] \\longrightarrow S_{\\varphi(g)}", "$$", "is quasi-finite and we win." ], "refs": [ "algebra-lemma-quasi-finite-over-polynomial-algebra", "algebra-lemma-refined-Noether-normalization", "algebra-lemma-quasi-finite-open" ], "ref_ids": [ 1071, 1003, 1066 ] } ], "ref_ids": [] }, { "id": 1073, "type": "theorem", "label": "algebra-lemma-dimension-inequality-quasi-finite", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-inequality-quasi-finite", "contents": [ "Let $R \\to S$ be a finite type ring map.", "Let $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p \\subset R$.", "If $R \\to S$ is quasi-finite at $\\mathfrak q$, then", "$\\dim(S_{\\mathfrak q}) \\leq \\dim(R_{\\mathfrak p})$." ], "refs": [], "proofs": [ { "contents": [ "If $R_{\\mathfrak p}$ is Noetherian", "(and hence $S_{\\mathfrak q}$ Noetherian since it is essentially of", "finite type over $R_{\\mathfrak p}$)", "then this follows immediately from", "Lemma \\ref{lemma-dimension-base-fibre-total} and the", "definitions. In the general case, let $S'$ be the integral", "closure of $R_\\mathfrak p$ in $S_\\mathfrak p$.", "By Zariski's Main Theorem \\ref{theorem-main-theorem}", "we have $S_{\\mathfrak q} = S'_{\\mathfrak q'}$ for some", "$\\mathfrak q' \\subset S'$ lying over $\\mathfrak q$.", "By Lemma \\ref{lemma-integral-dim-up} we have", "$\\dim(S') \\leq \\dim(R_\\mathfrak p)$ and hence a fortiori", "$\\dim(S_\\mathfrak q) = \\dim(S'_{\\mathfrak q'}) \\leq \\dim(R_\\mathfrak p)$." ], "refs": [ "algebra-lemma-dimension-base-fibre-total", "algebra-theorem-main-theorem", "algebra-lemma-integral-dim-up" ], "ref_ids": [ 986, 325, 984 ] } ], "ref_ids": [] }, { "id": 1074, "type": "theorem", "label": "algebra-lemma-dimension-quasi-finite-over-polynomial-algebra", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-quasi-finite-over-polynomial-algebra", "contents": [ "\\begin{slogan}", "A quasi-finite cover of affine n-space has dimension at most n.", "\\end{slogan}", "Let $k$ be a field. Let $S$ be a finite type $k$-algebra.", "Suppose there is a quasi-finite $k$-algebra map", "$k[t_1, \\ldots, t_n] \\subset S$. Then $\\dim(S) \\leq n$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-dim-affine-space} the dimension of", "any local ring of $k[t_1, \\ldots, t_n]$ is at most $n$.", "Thus the result follows from", "Lemma \\ref{lemma-dimension-inequality-quasi-finite}." ], "refs": [ "algebra-lemma-dim-affine-space", "algebra-lemma-dimension-inequality-quasi-finite" ], "ref_ids": [ 992, 1073 ] } ], "ref_ids": [] }, { "id": 1075, "type": "theorem", "label": "algebra-lemma-dimension-fibres-bounded-open-upstairs", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-fibres-bounded-open-upstairs", "contents": [ "Let $R \\to S$ be a finite type ring map.", "Let $\\mathfrak q \\subset S$ be a prime.", "Suppose that $\\dim_{\\mathfrak q}(S/R) = n$.", "There exists an open neighbourhood $V$ of $\\mathfrak q$", "in $\\Spec(S)$ such that", "$\\dim_{\\mathfrak q'}(S/R) \\leq n$ for all $\\mathfrak q' \\in V$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-quasi-finite-over-polynomial-algebra}", "we see that we may assume that $S$ is quasi-finite over", "a polynomial algebra $R[t_1, \\ldots, t_n]$. Considering", "the fibres, we reduce to", "Lemma \\ref{lemma-dimension-quasi-finite-over-polynomial-algebra}." ], "refs": [ "algebra-lemma-quasi-finite-over-polynomial-algebra", "algebra-lemma-dimension-quasi-finite-over-polynomial-algebra" ], "ref_ids": [ 1071, 1074 ] } ], "ref_ids": [] }, { "id": 1076, "type": "theorem", "label": "algebra-lemma-dimension-fibres-bounded-open-upstairs-base-change", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-fibres-bounded-open-upstairs-base-change", "contents": [ "Let $R \\to S$ be a finite type ring map.", "Let $R \\to R'$ be any ring map.", "Set $S' = R' \\otimes_R S$ and denote $f : \\Spec(S') \\to \\Spec(S)$", "the associated map on spectra.", "Let $n \\geq 0$.", "The inverse image", "$f^{-1}(\\{\\mathfrak q \\in \\Spec(S) \\mid", "\\dim_{\\mathfrak q}(S/R) \\leq n\\})$", "is equal to", "$\\{\\mathfrak q' \\in \\Spec(S') \\mid", "\\dim_{\\mathfrak q'}(S'/R') \\leq n\\}$." ], "refs": [], "proofs": [ { "contents": [ "The condition is formulated in terms of dimensions", "of fibre rings which are of finite type over a field.", "Combined with", "Lemma \\ref{lemma-dimension-at-a-point-preserved-field-extension}", "this yields the lemma." ], "refs": [ "algebra-lemma-dimension-at-a-point-preserved-field-extension" ], "ref_ids": [ 1010 ] } ], "ref_ids": [] }, { "id": 1077, "type": "theorem", "label": "algebra-lemma-dimension-fibres-bounded-quasi-compact-open-upstairs", "categories": [ "algebra" ], "title": "algebra-lemma-dimension-fibres-bounded-quasi-compact-open-upstairs", "contents": [ "Let $R \\to S$ be a ring homomorphism of finite presentation.", "Let $n \\geq 0$. The set", "$$", "V_n = \\{\\mathfrak q \\in \\Spec(S) \\mid \\dim_{\\mathfrak q}(S/R) \\leq n\\}", "$$", "is a quasi-compact open subset of $\\Spec(S)$." ], "refs": [], "proofs": [ { "contents": [ "It is open by Lemma \\ref{lemma-dimension-fibres-bounded-open-upstairs}.", "Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$ be a presentation of", "$S$. Let $R_0$ be the $\\mathbf{Z}$-subalgebra of $R$ generated by the", "coefficients of the polynomials $f_i$.", "Let $S_0 = R_0[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$.", "Then $S = R \\otimes_{R_0} S_0$. By", "Lemma \\ref{lemma-dimension-fibres-bounded-open-upstairs-base-change}", "$V_n$ is the inverse image of an open $V_{0, n}$ under the quasi-compact", "continuous map $\\Spec(S) \\to \\Spec(S_0)$. Since", "$S_0$ is Noetherian we see that $V_{0, n}$ is quasi-compact." ], "refs": [ "algebra-lemma-dimension-fibres-bounded-open-upstairs", "algebra-lemma-dimension-fibres-bounded-open-upstairs-base-change" ], "ref_ids": [ 1075, 1076 ] } ], "ref_ids": [] }, { "id": 1078, "type": "theorem", "label": "algebra-lemma-finite-type-domain-over-valuation-ring-dim-fibres", "categories": [ "algebra" ], "title": "algebra-lemma-finite-type-domain-over-valuation-ring-dim-fibres", "contents": [ "Let $R$ be a valuation ring with residue field $k$ and field", "of fractions $K$. Let $S$ be a domain containing $R$ such that", "$S$ is of finite type over $R$. If $S \\otimes_R k$ is not the", "zero ring then", "$$", "\\dim(S \\otimes_R k) = \\dim(S \\otimes_R K)", "$$", "In fact, $\\Spec(S \\otimes_R k)$ is equidimensional." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that $\\dim_{\\mathfrak q}(S/k)$ is equal", "to $\\dim(S \\otimes_R K)$ for every prime $\\mathfrak q$ of", "$S$ containing $\\mathfrak m_RS$. Pick such a prime. By", "Lemma \\ref{lemma-dimension-fibres-bounded-open-upstairs}", "the inequality $\\dim_{\\mathfrak q}(S/k) \\geq \\dim(S \\otimes_R K)$", "holds. Set $n = \\dim_{\\mathfrak q}(S/k)$. By", "Lemma \\ref{lemma-quasi-finite-over-polynomial-algebra}", "after replacing $S$ by $S_g$ for some $g \\in S$, $g \\not \\in \\mathfrak q$", "there exists a quasi-finite ring map", "$R[t_1, \\ldots, t_n] \\to S$. If $\\dim(S \\otimes_R K) < n$,", "then $K[t_1, \\ldots, t_n] \\to S \\otimes_R K$ has a nonzero kernel.", "Say $f = \\sum a_I t_1^{i_1}\\ldots t_n^{i_n}$. After dividing", "$f$ by a nonzero coefficient of $f$ with minimal valuation, we may", "assume $f\\in R[t_1, \\ldots, t_n]$ and some $a_I$ does not map", "to zero in $k$. Hence the ring map $k[t_1, \\ldots, t_n] \\to S \\otimes_R k$", "has a nonzero kernel which implies that $\\dim(S \\otimes_R k) < n$.", "Contradiction." ], "refs": [ "algebra-lemma-dimension-fibres-bounded-open-upstairs", "algebra-lemma-quasi-finite-over-polynomial-algebra" ], "ref_ids": [ 1075, 1071 ] } ], "ref_ids": [] }, { "id": 1079, "type": "theorem", "label": "algebra-lemma-finite-type-descends", "categories": [ "algebra" ], "title": "algebra-lemma-finite-type-descends", "contents": [ "Let $R \\to S$ be a ring map.", "Let $R \\to R'$ be a faithfully flat ring map.", "Set $S' = R'\\otimes_R S$.", "Then $R \\to S$ is of finite type if and only if $R' \\to S'$", "is of finite type." ], "refs": [], "proofs": [ { "contents": [ "It is clear that if $R \\to S$ is of finite type then $R' \\to S'$", "is of finite type. Assume that $R' \\to S'$ is of finite type.", "Say $y_1, \\ldots, y_m$ generate $S'$ over $R'$.", "Write $y_j = \\sum_i a_{ij} \\otimes x_{ji}$ for some", "$a_{ij} \\in R'$ and $x_{ji} \\in S$. Let $A \\subset S$", "be the $R$-subalgebra generated by the $x_{ij}$.", "By flatness we have $A' := R' \\otimes_R A \\subset S'$, and", "by construction $y_j \\in A'$. Hence $A' = S'$.", "By faithful flatness $A = S$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1080, "type": "theorem", "label": "algebra-lemma-finite-presentation-descends", "categories": [ "algebra" ], "title": "algebra-lemma-finite-presentation-descends", "contents": [ "Let $R \\to S$ be a ring map.", "Let $R \\to R'$ be a faithfully flat ring map.", "Set $S' = R'\\otimes_R S$.", "Then $R \\to S$ is of finite presentation if and only if $R' \\to S'$", "is of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "It is clear that if $R \\to S$ is of finite presentation then $R' \\to S'$", "is of finite presentation. Assume that $R' \\to S'$ is of finite presentation.", "By Lemma \\ref{lemma-finite-type-descends} we see", "that $R \\to S$ is of finite type. Write $S = R[x_1, \\ldots, x_n]/I$.", "By flatness $S' = R'[x_1, \\ldots, x_n]/R'\\otimes I$.", "Say $g_1, \\ldots, g_m$ generate $R'\\otimes I$ over $R'[x_1, \\ldots, x_n]$.", "Write $g_j = \\sum_i a_{ij} \\otimes f_{ji}$ for some", "$a_{ij} \\in R'$ and $f_{ji} \\in I$. Let $J \\subset I$", "be the ideal generated by the $f_{ij}$.", "By flatness we have $R' \\otimes_R J \\subset R'\\otimes_R I$, and", "both are ideals over $R'[x_1, \\ldots, x_n]$.", "By construction $g_j \\in R' \\otimes_R J$. Hence", "$R' \\otimes_R J = R'\\otimes_R I$.", "By faithful flatness $J = I$." ], "refs": [ "algebra-lemma-finite-type-descends" ], "ref_ids": [ 1079 ] } ], "ref_ids": [] }, { "id": 1081, "type": "theorem", "label": "algebra-lemma-construct-fp-module", "categories": [ "algebra" ], "title": "algebra-lemma-construct-fp-module", "contents": [ "Let $R$ be a ring.", "Let $I \\subset R$ be an ideal.", "Let $S \\subset R$ be a multiplicative subset.", "Set $R' = S^{-1}(R/I) = S^{-1}R/S^{-1}I$.", "\\begin{enumerate}", "\\item For any finite $R'$-module $M'$ there exists a", "finite $R$-module $M$ such that $S^{-1}(M/IM) \\cong M'$.", "\\item For any finitely presented $R'$-module $M'$ there exists a", "finitely presented $R$-module $M$ such that $S^{-1}(M/IM) \\cong M'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Choose a short exact sequence", "$0 \\to K' \\to (R')^{\\oplus n} \\to M' \\to 0$.", "Let $K \\subset R^{\\oplus n}$ be the inverse image of", "$K'$ under the map $R^{\\oplus n} \\to (R')^{\\oplus n}$.", "Then $M = R^{\\oplus n}/K$ works.", "\\medskip\\noindent", "Proof of (2).", "Choose a presentation $(R')^{\\oplus m} \\to (R')^{\\oplus n} \\to M' \\to 0$.", "Suppose that the first map is given by the matrix", "$A' = (a'_{ij})$ and the second map is determined by generators", "$x'_i \\in M'$, $i = 1, \\ldots, n$. As $R' = S^{-1}(R/I)$ we can choose", "$s \\in S$ and a matrix $A = (a_{ij})$ with coefficients in $R$", "such that $a'_{ij} = a_{ij} / s \\bmod S^{-1}I$. Let $M$ be the", "finitely presented $R$-module with presentation", "$R^{\\oplus m} \\to R^{\\oplus n} \\to M \\to 0$", "where the first map is given by the matrix $A$ and the second map is", "determined by generators $x_i \\in M$, $i = 1, \\ldots, n$.", "Then the map $M \\to M'$, $x_i \\mapsto x'_i$ induces an isomorphism", "$S^{-1}(M/IM) \\cong M'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1082, "type": "theorem", "label": "algebra-lemma-construct-fp-module-from-localization", "categories": [ "algebra" ], "title": "algebra-lemma-construct-fp-module-from-localization", "contents": [ "Let $R$ be a ring.", "Let $S \\subset R$ be a multiplicative subset.", "Let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item If $S^{-1}M$ is a finite $S^{-1}R$-module then there", "exists a finite $R$-module $M'$ and a map $M' \\to M$ which induces an", "isomorphism $S^{-1}M' \\to S^{-1}M$.", "\\item If $S^{-1}M$ is a finitely presented $S^{-1}R$-module", "then there exists an $R$-module $M'$ of finite presentation", "and a map $M' \\to M$ which induces an isomorphism", "$S^{-1}M' \\to S^{-1}M$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $x_1, \\ldots, x_n \\in M$ be elements which generate", "$S^{-1}M$ as an $S^{-1}R$-module. Let $M'$ be the", "$R$-submodule of $M$ generated by $x_1, \\ldots, x_n$.", "\\medskip\\noindent", "Proof of (2). Let $x_1, \\ldots, x_n \\in M$ be elements which generate", "$S^{-1}M$ as an $S^{-1}R$-module. Let", "$K = \\Ker(R^{\\oplus n} \\to M)$ where the map is given by", "the rule $(a_1, \\ldots, a_n) \\mapsto \\sum a_i x_i$. By", "Lemma \\ref{lemma-extension}", "we see that $S^{-1}K$ is a finite $S^{-1}R$-module.", "By (1) we can find a finite submodule $K' \\subset K$", "with $S^{-1}K' = S^{-1}K$. Take", "$M' = \\Coker(K' \\to R^{\\oplus n})$." ], "refs": [ "algebra-lemma-extension" ], "ref_ids": [ 330 ] } ], "ref_ids": [] }, { "id": 1083, "type": "theorem", "label": "algebra-lemma-construct-fp-module-from-stalk", "categories": [ "algebra" ], "title": "algebra-lemma-construct-fp-module-from-stalk", "contents": [ "Let $R$ be a ring.", "Let $\\mathfrak p \\subset R$ be a prime ideal.", "Let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item If $M_{\\mathfrak p}$ is a finite $R_{\\mathfrak p}$-module then there", "exists a finite $R$-module $M'$ and a map $M' \\to M$ which induces an", "isomorphism $M'_{\\mathfrak p} \\to M_{\\mathfrak p}$.", "\\item If $M_{\\mathfrak p}$ is a finitely presented $R_{\\mathfrak p}$-module", "then there exists an $R$-module $M'$ of finite presentation", "and a map $M' \\to M$ which induces an isomorphism", "$M'_{\\mathfrak p} \\to M_{\\mathfrak p}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-construct-fp-module-from-localization}" ], "refs": [ "algebra-lemma-construct-fp-module-from-localization" ], "ref_ids": [ 1082 ] } ], "ref_ids": [] }, { "id": 1084, "type": "theorem", "label": "algebra-lemma-local-isomorphism", "categories": [ "algebra" ], "title": "algebra-lemma-local-isomorphism", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$", "be a prime lying over $\\mathfrak p \\subset R$. Assume", "\\begin{enumerate}", "\\item $S$ is of finite presentation over $R$,", "\\item $\\varphi$ induces an isomorphism $R_\\mathfrak p \\cong S_\\mathfrak q$.", "\\end{enumerate}", "Then there exist $f \\in R$, $f \\not \\in \\mathfrak p$ and an", "$R_f$-algebra $C$ such that $S_f \\cong R_f \\times C$ as $R_f$-algebras." ], "refs": [], "proofs": [ { "contents": [ "Write $S = R[x_1, \\ldots, x_n]/(g_1, \\ldots, g_m)$. Let $a_i \\in R_\\mathfrak p$", "be an element mapping to the image of $x_i$ in $S_\\mathfrak q$.", "Write $a_i = b_i/f$ for some $f \\in R$, $f \\not \\in \\mathfrak p$.", "After replacing $R$ by $R_f$ and $x_i$ by $x_i - a_i$ we may", "assume that $S = R[x_1, \\ldots, x_n]/(g_1, \\ldots, g_m)$ such", "that $x_i$ maps to zero in $S_\\mathfrak q$. Then if $c_j$ denotes", "the constant term of $g_j$ we conclude that $c_j$ maps to zero", "in $R_\\mathfrak p$. After another replacement of $R$ we may", "assume that the constant coefficients $c_j$ of the $g_j$ are zero.", "Thus we obtain an $R$-algebra map $S \\to R$, $x_i \\mapsto 0$ whose", "kernel is the ideal $(x_1, \\ldots, x_n)$.", "\\medskip\\noindent", "Note that $\\mathfrak q = \\mathfrak pS + (x_1, \\ldots, x_n)$.", "Write $g_j = \\sum a_{ji}x_i + h.o.t.$. Since $S_\\mathfrak q = R_\\mathfrak p$", "we have $\\mathfrak p \\otimes \\kappa(\\mathfrak p) =", "\\mathfrak q \\otimes \\kappa(\\mathfrak q)$. It follows that", "$m \\times n$ matrix $A = (a_{ji})$ defines a surjective", "map $\\kappa(\\mathfrak p)^{\\oplus m} \\to \\kappa(\\mathfrak p)^{\\oplus n}$.", "Thus after inverting some element of $R$ not in $\\mathfrak p$ we may", "assume there are $b_{ij} \\in R$ such that $\\sum b_{ij} g_j = x_i + h.o.t.$.", "We conclude that $(x_1, \\ldots, x_n) = (x_1, \\ldots, x_n)^2$ in $S$.", "It follows from Lemma \\ref{lemma-ideal-is-squared-union-connected}", "that $(x_1, \\ldots, x_n)$ is generated by an idempotent $e$.", "Setting $C = eS$ finishes the proof." ], "refs": [ "algebra-lemma-ideal-is-squared-union-connected" ], "ref_ids": [ 407 ] } ], "ref_ids": [] }, { "id": 1085, "type": "theorem", "label": "algebra-lemma-isomorphic-local-rings", "categories": [ "algebra" ], "title": "algebra-lemma-isomorphic-local-rings", "contents": [ "Let $R$ be a ring.", "Let $S$, $S'$ be of finite presentation over $R$.", "Let $\\mathfrak q \\subset S$ and $\\mathfrak q' \\subset S'$", "be primes. If $S_{\\mathfrak q} \\cong S'_{\\mathfrak q'}$ as", "$R$-algebras, then there exist $g \\in S$, $g \\not \\in \\mathfrak q$", "and $g' \\in S'$, $g' \\not \\in \\mathfrak q'$ such that", "$S_g \\cong S'_{g'}$ as $R$-algebras." ], "refs": [], "proofs": [ { "contents": [ "Let $\\psi : S_{\\mathfrak q} \\to S'_{\\mathfrak q'}$ be the isomorphism", "of the hypothesis of the lemma.", "Write $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_r)$ and", "$S' = R[y_1, \\ldots, y_m]/J$.", "For each $i = 1, \\ldots, n$ choose a fraction", "$h_i/g_i$ with $h_i, g_i \\in R[y_1, \\ldots, y_m]$", "and $g_i \\bmod J$ not in $\\mathfrak q'$ which represents", "the image of $x_i$ under $\\psi$. After replacing", "$S'$ by $S'_{g_1 \\ldots g_n}$ and", "$R[y_1, \\ldots, y_m, y_{m + 1}]$ (mapping $y_{m + 1}$ to $1/(g_1\\ldots g_n)$)", "we may assume that $\\psi(x_i)$ is the image of some", "$h_i \\in R[y_1, \\ldots, y_m]$. Consider the elements", "$f_j(h_1, \\ldots, h_n) \\in R[y_1, \\ldots, y_m]$.", "Since $\\psi$ kills each $f_j$ we see that", "there exists a $g \\in R[y_1, \\ldots, y_m]$, $g \\bmod J \\not \\in \\mathfrak q'$", "such that $g f_j(h_1, \\ldots, h_n) \\in J$ for each $j = 1, \\ldots, r$.", "After replacing $S'$ by $S'_g$ and", "$R[y_1, \\ldots, y_m, y_{m + 1}]$ as before we may assume that", "$f_j(h_1, \\ldots, h_n) \\in J$. Thus we obtain a ring map", "$S \\to S'$, $x_i \\mapsto h_i$ which induces $\\psi$ on local rings.", "By Lemma \\ref{lemma-compose-finite-type}", "the map $S \\to S'$ is of finite presentation.", "By Lemma \\ref{lemma-local-isomorphism}", "we may assume that $S' = S \\times C$. Thus localizing $S'$ at the", "idempotent corresponding to the factor $C$ we obtain the result." ], "refs": [ "algebra-lemma-compose-finite-type", "algebra-lemma-local-isomorphism" ], "ref_ids": [ 333, 1084 ] } ], "ref_ids": [] }, { "id": 1086, "type": "theorem", "label": "algebra-lemma-finite-type-mod-nilpotent", "categories": [ "algebra" ], "title": "algebra-lemma-finite-type-mod-nilpotent", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be a nilpotent ideal.", "Let $S$ be an $R$-algebra such that $R/I \\to S/IS$", "is of finite type. Then $R \\to S$ is of finite type." ], "refs": [], "proofs": [ { "contents": [ "Choose $s_1, \\ldots, s_n \\in S$ whose images in $S/IS$ generate", "$S/IS$ as an algebra over $R/I$. By Lemma \\ref{lemma-NAK} part (11)", "we see that the $R$-algebra map $R[x_1, \\ldots, x_n \\to S$, $x_i \\mapsto s_i$", "is surjective and we conclude." ], "refs": [ "algebra-lemma-NAK" ], "ref_ids": [ 401 ] } ], "ref_ids": [] }, { "id": 1087, "type": "theorem", "label": "algebra-lemma-surjective-mod-locally-nilpotent", "categories": [ "algebra" ], "title": "algebra-lemma-surjective-mod-locally-nilpotent", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be a locally nilpotent ideal.", "Let $S \\to S'$ be an $R$-algebra map such that $S \\to S'/IS'$ is surjective", "and such that $S'$ is of finite type over $R$. Then $S \\to S'$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Write $S' = R[x_1, \\ldots, x_m]/K$ for some ideal $K$. By assumption there", "exist $g_j = x_j + \\sum \\delta_{j, J} x^J \\in R[x_1, \\ldots, x_n]$ with", "$\\delta_{j, J} \\in I$ and with $g_j \\bmod K \\in \\Im(S \\to S')$.", "Hence it suffices to show that $g_1, \\ldots, g_m$ generate", "$R[x_1, \\ldots, x_n]$. Let $R_0 \\subset R$ be a finitely generated", "$\\mathbf{Z}$-subalgebra of $R$ containing at least the $\\delta_{j, J}$.", "Then $R_0 \\cap I$ is a nilpotent ideal (by", "Lemma \\ref{lemma-Noetherian-power}). It follows that", "$R_0[x_1, \\ldots, x_n]$ is generated by $g_1, \\ldots, g_m$ (because", "$x_j \\mapsto g_j$ defines an automorphism of $R_0[x_1, \\ldots, x_m]$;", "details omitted). Since $R$ is the union of the subrings $R_0$ we win." ], "refs": [ "algebra-lemma-Noetherian-power" ], "ref_ids": [ 460 ] } ], "ref_ids": [] }, { "id": 1088, "type": "theorem", "label": "algebra-lemma-isomorphism-modulo-ideal", "categories": [ "algebra" ], "title": "algebra-lemma-isomorphism-modulo-ideal", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $S \\to S'$", "be an $R$-algebra map. Let $IS \\subset \\mathfrak q \\subset S$", "be a prime", "ideal. Assume that", "\\begin{enumerate}", "\\item $S \\to S'$ is surjective,", "\\item $S_\\mathfrak q/IS_\\mathfrak q \\to S'_\\mathfrak q/IS'_\\mathfrak q$", "is an isomorphism,", "\\item $S$ is of finite type over $R$,", "\\item $S'$ of finite presentation over $R$, and", "\\item $S'_\\mathfrak q$ is flat over $R$.", "\\end{enumerate}", "Then $S_g \\to S'_g$ is an isomorphism for some", "$g \\in S$, $g \\not \\in \\mathfrak q$." ], "refs": [], "proofs": [ { "contents": [ "Let $J = \\Ker(S \\to S')$. By", "Lemma \\ref{lemma-compose-finite-type}", "$J$ is a finitely generated ideal. Since $S'_\\mathfrak q$ is flat", "over $R$ we see that", "$J_\\mathfrak q/IJ_\\mathfrak q \\subset S_\\mathfrak q/IS_{\\mathfrak q}$", "(apply Lemma \\ref{lemma-flat-tor-zero} to $0 \\to J \\to S \\to S' \\to 0$).", "By assumption (2) we see that $J_\\mathfrak q/IJ_\\mathfrak q$ is zero.", "By Nakayama's lemma (Lemma \\ref{lemma-NAK}) we see that", "there exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such", "that $J_g = 0$. Hence $S_g \\cong S'_g$ as desired." ], "refs": [ "algebra-lemma-compose-finite-type", "algebra-lemma-flat-tor-zero", "algebra-lemma-NAK" ], "ref_ids": [ 333, 532, 401 ] } ], "ref_ids": [] }, { "id": 1089, "type": "theorem", "label": "algebra-lemma-isomorphism-modulo-locally-nilpotent", "categories": [ "algebra" ], "title": "algebra-lemma-isomorphism-modulo-locally-nilpotent", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $S \\to S'$", "be an $R$-algebra map. Assume that", "\\begin{enumerate}", "\\item $I$ is locally nilpotent,", "\\item $S/IS \\to S'/IS'$ is an isomorphism,", "\\item $S$ is of finite type over $R$,", "\\item $S'$ of finite presentation over $R$, and", "\\item $S'$ is flat over $R$.", "\\end{enumerate}", "Then $S \\to S'$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-surjective-mod-locally-nilpotent} the map", "$S \\to S'$ is surjective. As $I$ is locally nilpotent, so are the", "ideals $IS$ and $IS'$ (Lemma \\ref{lemma-locally-nilpotent}). Hence", "every prime ideal $\\mathfrak q$ of $S$ contains $IS$ and (trivially)", "$S_\\mathfrak q/IS_\\mathfrak q \\cong S'_\\mathfrak q/IS'_\\mathfrak q$.", "Thus Lemma \\ref{lemma-isomorphism-modulo-ideal} applies", "and we see that $S_\\mathfrak q \\to S'_\\mathfrak q$ is an", "isomorphism for every prime $\\mathfrak q \\subset S$.", "It follows that $S \\to S'$ is injective for example by", "Lemma \\ref{lemma-characterize-zero-local}." ], "refs": [ "algebra-lemma-surjective-mod-locally-nilpotent", "algebra-lemma-locally-nilpotent", "algebra-lemma-isomorphism-modulo-ideal", "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 1087, 458, 1088, 410 ] } ], "ref_ids": [] }, { "id": 1090, "type": "theorem", "label": "algebra-lemma-ring-colimit-fp-category", "categories": [ "algebra" ], "title": "algebra-lemma-ring-colimit-fp-category", "contents": [ "Let $R \\to A$ be a ring map. Consider the category $\\mathcal{I}$ of all", "diagrams of $R$-algebra maps $A' \\to A$ with $A'$ finitely presented over", "$R$. Then $\\mathcal{I}$ is filtered, and the colimit of the $A'$ over", "$\\mathcal{I}$ is isomorphic to $A$." ], "refs": [], "proofs": [ { "contents": [ "The category\\footnote{To avoid set theoretical difficulties we", "consider only $A' \\to A$ such that $A'$ is a quotient of", "$R[x_1, x_2, x_3, \\ldots]$.}", "$\\mathcal{I}$ is nonempty as $R \\to R$ is an object of it.", "Consider a pair of objects $A' \\to A$, $A'' \\to A$ of $\\mathcal{I}$.", "Then $A' \\otimes_R A'' \\to A$ is in", "$\\mathcal{I}$ (use Lemmas \\ref{lemma-compose-finite-type} and", "\\ref{lemma-base-change-finiteness}). The ring maps", "$A' \\to A' \\otimes_R A''$ and $A'' \\to A' \\otimes_R A''$", "define arrows in $\\mathcal{I}$ thereby proving the second defining", "property of a filtered category, see", "Categories, Definition \\ref{categories-definition-directed}.", "Finally, suppose that we have two morphisms $\\sigma, \\tau : A' \\to A''$", "in $\\mathcal{I}$. If $x_1, \\ldots, x_r \\in A'$ are generators of", "$A'$ as an $R$-algebra, then we can consider", "$A''' = A''/(\\sigma(x_i) - \\tau(x_i))$.", "This is a finitely presented $R$-algebra and the given $R$-algebra map", "$A'' \\to A$ factors through the surjection $\\nu : A'' \\to A'''$.", "Thus $\\nu$ is a morphism in $\\mathcal{I}$ equalizing $\\sigma$ and $\\tau$", "as desired.", "\\medskip\\noindent", "The fact that our index category is cofiltered means that we may", "compute the value of $B = \\colim_{A' \\to A} A'$ in the category of sets", "(some details omitted; compare with the discussion in", "Categories, Section \\ref{categories-section-directed-colimits}).", "To see that $B \\to A$ is surjective, for", "every $a \\in A$ we can use $R[x] \\to A$, $x \\mapsto a$ to see that", "$a$ is in the image of $B \\to A$. Conversely, if $b \\in B$ is mapped", "to zero in $A$, then we can find $A' \\to A$ in $\\mathcal{I}$ and", "$a' \\in A'$ which maps to $b$. Then $A'/(a') \\to A$ is in $\\mathcal{I}$", "as well and the map $A' \\to B$ factors as $A' \\to A'/(a') \\to B$", "which shows that $b = 0$ as desired." ], "refs": [ "algebra-lemma-compose-finite-type", "algebra-lemma-base-change-finiteness", "categories-definition-directed" ], "ref_ids": [ 333, 373, 12363 ] } ], "ref_ids": [] }, { "id": 1091, "type": "theorem", "label": "algebra-lemma-ring-colimit-fp", "categories": [ "algebra" ], "title": "algebra-lemma-ring-colimit-fp", "contents": [ "Let $R \\to A$ be a ring map. There exists a directed system $A_\\lambda$ of", "$R$-algebras of finite presentation such that $A = \\colim_\\lambda A_\\lambda$.", "If $A$ is of finite type over $R$ we may arrange it so that all the", "transition maps in the system of $A_\\lambda$ are surjective." ], "refs": [], "proofs": [ { "contents": [ "The first proof is that this follows from", "Lemma \\ref{lemma-ring-colimit-fp-category} and", "Categories, Lemma \\ref{categories-lemma-directed-category-system}.", "\\medskip\\noindent", "Second proof.", "Compare with the proof of Lemma \\ref{lemma-module-colimit-fp}.", "Consider any finite subset $S \\subset A$, and any finite", "collection of polynomial relations $E$ among the elements of $S$.", "So each $s \\in S$ corresponds to $x_s \\in A$ and", "each $e \\in E$ consists of a polynomial", "$f_e \\in R[X_s; s\\in S]$ such that $f_e(x_s) = 0$.", "Let $A_{S, E} = R[X_s; s\\in S]/(f_e; e\\in E)$", "which is a finitely presented $R$-algebra.", "There are canonical maps $A_{S, E} \\to A$.", "If $S \\subset S'$ and if the elements of", "$E$ correspond, via the map $R[X_s; s \\in S] \\to R[X_s; s\\in S']$,", "to a subset of $E'$, then there is an obvious map", "$A_{S, E} \\to A_{S', E'}$ commuting with the", "maps to $A$. Thus, setting $\\Lambda$ equal the set of pairs", "$(S, E)$ with ordering by inclusion as above, we get a", "directed partially ordered set.", "It is clear that the colimit of this directed system is $A$.", "\\medskip\\noindent", "For the last statement, suppose $A = R[x_1, \\ldots, x_n]/I$.", "In this case, consider the subset $\\Lambda' \\subset \\Lambda$", "consisting of those systems $(S, E)$ above", "with $S = \\{x_1, \\ldots, x_n\\}$. It is easy to see that", "still $A = \\colim_{\\lambda' \\in \\Lambda'} A_{\\lambda'}$.", "Moreover, the transition maps are clearly surjective." ], "refs": [ "algebra-lemma-ring-colimit-fp-category", "categories-lemma-directed-category-system", "algebra-lemma-module-colimit-fp" ], "ref_ids": [ 1090, 12236, 355 ] } ], "ref_ids": [] }, { "id": 1092, "type": "theorem", "label": "algebra-lemma-characterize-finite-presentation", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-finite-presentation", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. The following are equivalent", "\\begin{enumerate}", "\\item $\\varphi$ is of finite presentation,", "\\item for every directed system $A_\\lambda$ of $R$-algebras", "the map", "$$", "\\colim_\\lambda \\Hom_R(S, A_\\lambda) \\longrightarrow", "\\Hom_R(S, \\colim_\\lambda A_\\lambda)", "$$", "is bijective, and", "\\item for every directed system $A_\\lambda$ of $R$-algebras", "the map", "$$", "\\colim_\\lambda \\Hom_R(S, A_\\lambda) \\longrightarrow", "\\Hom_R(S, \\colim_\\lambda A_\\lambda)", "$$", "is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and write $S = R[x_1, \\ldots, x_n] / (f_1, \\ldots, f_m)$.", "Let $A = \\colim A_\\lambda$. Observe that an $R$-algebra homomorphism", "$S \\to A$ or $S \\to A_\\lambda$ is determined by the images of", "$x_1, \\ldots, x_n$. Hence it is clear that", "$\\colim_\\lambda \\Hom_R(S, A_\\lambda) \\to \\Hom_R(S, A)$", "is injective. To see that it is surjective, let $\\chi : S \\to A$", "be an $R$-algebra homomorphism. Then each", "$x_i$ maps to some element in the image of some $A_{\\lambda_i}$.", "We may pick $\\mu \\geq \\lambda_i$, $i = 1, \\ldots, n$ and", "assume $\\chi(x_i)$ is the image of $y_i \\in A_\\mu$ for", "$i = 1, \\ldots, n$. Consider $z_j = f_j(y_1, \\ldots, y_n) \\in A_\\mu$.", "Since $\\chi$ is a homomorphism the image of $z_j$ in", "$A = \\colim_\\lambda A_\\lambda$ is zero. Hence there exists a", "$\\mu_j \\geq \\mu$ such that $z_j$ maps to zero in $A_{\\mu_j}$.", "Pick $\\nu \\geq \\mu_j$, $j = 1, \\ldots, m$. Then the", "images of $z_1, \\ldots, z_m$ are zero in $A_\\nu$. This", "exactly means that the $y_i$ map to elements", "$y'_i \\in A_\\nu$ which satisfy the relations $f_j(y'_1, \\ldots, y'_n) = 0$.", "Thus we obtain a ring map $S \\to A_\\nu$. This shows that", "(1) implies (2).", "\\medskip\\noindent", "It is clear that (2) implies (3). Assume (3).", "By Lemma \\ref{lemma-ring-colimit-fp} we may write", "$S = \\colim_\\lambda S_\\lambda$ with $S_\\lambda$", "of finite presentation over $R$. Then the identity map", "factors as", "$$", "S \\to S_\\lambda \\to S", "$$", "for some $\\lambda$. This implies that $S$", "is finitely presented over $S_\\lambda$ by", "Lemma \\ref{lemma-compose-finite-type} part (4)", "applied to $S \\to S_\\lambda \\to S$. Applying part (2) of the same", "lemma to $R \\to S_\\lambda \\to S$ we conclude that $S$ is of finite", "presentation over $R$." ], "refs": [ "algebra-lemma-ring-colimit-fp", "algebra-lemma-compose-finite-type" ], "ref_ids": [ 1091, 333 ] } ], "ref_ids": [] }, { "id": 1093, "type": "theorem", "label": "algebra-lemma-when-colimit", "categories": [ "algebra" ], "title": "algebra-lemma-when-colimit", "contents": [ "Let $R \\to \\Lambda$ be a ring map. Let $\\mathcal{E}$ be a set of $R$-algebras", "such that each $A \\in \\mathcal{E}$ is of finite presentation over $R$.", "Then the following two statements are equivalent", "\\begin{enumerate}", "\\item $\\Lambda$ is a filtered colimit of elements of $\\mathcal{E}$, and", "\\item for any $R$ algebra map $A \\to \\Lambda$ with $A$ of finite", "presentation over $R$ we can find a factorization $A \\to B \\to \\Lambda$", "with $B \\in \\mathcal{E}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $\\mathcal{I} \\to \\mathcal{E}$, $i \\mapsto A_i$", "is a filtered diagram such that $\\Lambda = \\colim_i A_i$.", "Let $A \\to \\Lambda$ be an $R$-algebra map with $A$ of finite", "presentation over $R$. Then we get a factorization $A \\to A_i \\to \\Lambda$", "by applying Lemma \\ref{lemma-characterize-finite-presentation}.", "Thus (1) implies (2).", "\\medskip\\noindent", "Consider the category", "$\\mathcal{I}$ of Lemma \\ref{lemma-ring-colimit-fp-category}.", "By Categories, Lemma \\ref{categories-lemma-cofinal-in-filtered}", "the full subcategory $\\mathcal{J}$ consisting of those", "$A \\to \\Lambda$ with $A \\in \\mathcal{E}$ is cofinal in $\\mathcal{I}$ and", "is a filtered category. Then $\\Lambda$ is also the colimit", "over $\\mathcal{J}$ by Categories, Lemma \\ref{categories-lemma-cofinal}." ], "refs": [ "algebra-lemma-characterize-finite-presentation", "algebra-lemma-ring-colimit-fp-category", "categories-lemma-cofinal-in-filtered", "categories-lemma-cofinal" ], "ref_ids": [ 1092, 1090, 12229, 12217 ] } ], "ref_ids": [] }, { "id": 1094, "type": "theorem", "label": "algebra-lemma-module-map-property-in-colimit", "categories": [ "algebra" ], "title": "algebra-lemma-module-map-property-in-colimit", "contents": [ "Let $A$ be a ring and let $M, N$ be $A$-modules.", "Suppose that $R = \\colim_{i \\in I} R_i$ is a directed colimit", "of $A$-algebras.", "\\begin{enumerate}", "\\item If $M$ is a finite $A$-module, and $u, u' : M \\to N$ are", "$A$-module maps such that", "$u \\otimes 1 = u' \\otimes 1 : M \\otimes_A R \\to N \\otimes_A R$", "then for some $i$ we have", "$u \\otimes 1 = u' \\otimes 1 : M \\otimes_A R_i \\to N \\otimes_A R_i$.", "\\item If $N$ is a finite $A$-module and $u : M \\to N$ is an $A$-module", "map such that $u \\otimes 1 : M \\otimes_A R \\to N \\otimes_A R$ is surjective,", "then for some $i$ the map $u \\otimes 1 : M \\otimes_A R_i \\to N \\otimes_A R_i$", "is surjective.", "\\item If $N$ is a finitely presented $A$-module, and", "$v : N \\otimes_A R \\to M \\otimes_A R$ is an $R$-module", "map, then there exists an $i$ and an $R_i$-module map", "$v_i : N \\otimes_A R_i \\to M \\otimes_A R_i$ such that $v = v_i \\otimes 1$.", "\\item If $M$ is a finite $A$-module, $N$ is a finitely presented $A$-module,", "and $u : M \\to N$ is an $A$-module map such that", "$u \\otimes 1 : M \\otimes_A R \\to N \\otimes_A R$ is an isomorphism, then", "for some $i$ the map $u \\otimes 1 : M \\otimes_A R_i \\to N \\otimes_A R_i$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove (1) assume $u$ is as in (1) and", "let $x_1, \\ldots, x_m \\in M$ be generators. Since", "$N \\otimes_A R = \\colim_i N \\otimes_A R_i$", "we may pick an $i \\in I$ such that $u(x_j) \\otimes 1 = u'(x_j) \\otimes 1$", "in $M \\otimes_A R_i$, $j = 1, \\ldots, m$.", "For such an $i$ we have", "$u \\otimes 1 = u' \\otimes 1 : M \\otimes_A R_i \\to N \\otimes_A R_i$.", "\\medskip\\noindent", "To prove (2) assume $u \\otimes 1$ surjective and", "let $y_1, \\ldots, y_m \\in N$ be generators. Since", "$N \\otimes_A R = \\colim_i N \\otimes_A R_i$", "we may pick an $i \\in I$ and $z_j \\in M \\otimes_A R_i$, $j = 1, \\ldots, m$", "whose images in $N \\otimes_A R$ equal $y_j \\otimes 1$.", "For such an $i$ the map $u \\otimes 1 : M \\otimes_A R_i \\to N \\otimes_A R_i$", "is surjective.", "\\medskip\\noindent", "To prove (3) let $y_1, \\ldots, y_m \\in N$ be generators. Let", "$K = \\Ker(A^{\\oplus m} \\to N)$ where the map is given by", "the rule $(a_1, \\ldots, a_m) \\mapsto \\sum a_j x_j$. Let $k_1, \\ldots, k_t$", "be generators for $K$. Say $k_s = (k_{s1}, \\ldots, k_{sm})$.", "Since $M \\otimes_A R = \\colim_i M \\otimes_A R_i$", "we may pick an $i \\in I$ and $z_j \\in M \\otimes_A R_i$, $j = 1, \\ldots, m$", "whose images in $M \\otimes_A R$ equal $v(y_j \\otimes 1)$.", "We want to use the $z_j$ to define the map", "$v_i : N \\otimes_A R_i \\to M \\otimes_A R_i$.", "Since $K \\otimes_A R_i \\to R_i^{\\oplus m} \\to N \\otimes_A R_i \\to 0$", "is a presentation, it suffices to check that $\\xi_s = \\sum_j k_{sj}z_j$ is", "zero in $M \\otimes_A R_i$ for each $s = 1, \\ldots, t$. This may not", "be the case, but since the image of $\\xi_s$ in $M \\otimes_A R$ is zero", "we see that it will be the case after increasing $i$ a bit.", "\\medskip\\noindent", "To prove (4) assume $u \\otimes 1$ is an isomorphism, that", "$M$ is finite, and that $N$ is finitely presented.", "Let $v : N \\otimes_A R \\to M \\otimes_A R$ be an inverse to", "$u \\otimes 1$. Apply part (3) to get a map", "$v_i : N \\otimes_A R_i \\to M \\otimes_A R_i$ for some $i$.", "Apply part (1) to see that, after increasing $i$ we have", "$v_i \\circ (u \\otimes 1) = \\text{id}_{M \\otimes_R R_i}$ and", "$(u \\otimes 1) \\circ v_i = \\text{id}_{N \\otimes_R R_i}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1095, "type": "theorem", "label": "algebra-lemma-colimit-category-fp-modules", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-category-fp-modules", "contents": [ "Suppose that $R = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$ is a directed colimit", "of rings. Then the category of finitely presented $R$-modules is", "the colimit of the categories of finitely presented $R_\\lambda$-modules.", "More precisely", "\\begin{enumerate}", "\\item Given a finitely presented $R$-module $M$ there exists a", "$\\lambda \\in \\Lambda$ and a finitely presented $R_\\lambda$-module", "$M_\\lambda$ such that $M \\cong M_\\lambda \\otimes_{R_\\lambda} R$.", "\\item Given a $\\lambda \\in \\Lambda$, finitely presented", "$R_\\lambda$-modules $M_\\lambda, N_\\lambda$, and an $R$-module map", "$\\varphi : M_\\lambda \\otimes_{R_\\lambda} R \\to N_\\lambda \\otimes_{R_\\lambda} R$,", "then there exists a $\\mu \\geq \\lambda$ and an $R_\\mu$-module map", "$\\varphi_\\mu : M_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to", "N_\\lambda \\otimes_{R_\\lambda} R_\\mu$", "such that $\\varphi = \\varphi_\\mu \\otimes 1_R$.", "\\item Given a $\\lambda \\in \\Lambda$, finitely presented", "$R_\\lambda$-modules $M_\\lambda, N_\\lambda$, and $R$-module maps", "$\\varphi_\\lambda, \\psi_\\lambda : M_\\lambda \\to N_\\lambda$", "such that $\\varphi \\otimes 1_R = \\psi \\otimes 1_R$, then", "$\\varphi \\otimes 1_{R_\\mu} = \\psi \\otimes 1_{R_\\mu}$ for some", "$\\mu \\geq \\lambda$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove (1) choose a presentation", "$R^{\\oplus m} \\to R^{\\oplus n} \\to M \\to 0$.", "Suppose that the first map is given by the matrix $A = (a_{ij})$.", "We can choose a $\\lambda \\in \\Lambda$ and a matrix", "$A_\\lambda = (a_{\\lambda, ij})$ with coefficients in $R_\\lambda$", "which maps to $A$ in $R$.", "Then we simply let $M_\\lambda$ be the $R_\\lambda$-module with presentation", "$R_\\lambda^{\\oplus m} \\to R_\\lambda^{\\oplus n} \\to M_\\lambda \\to 0$", "where the first arrow is given by $A_\\lambda$.", "\\medskip\\noindent", "Parts (2) and (3) follow from", "Lemma \\ref{lemma-module-map-property-in-colimit}." ], "refs": [ "algebra-lemma-module-map-property-in-colimit" ], "ref_ids": [ 1094 ] } ], "ref_ids": [] }, { "id": 1096, "type": "theorem", "label": "algebra-lemma-algebra-map-property-in-colimit", "categories": [ "algebra" ], "title": "algebra-lemma-algebra-map-property-in-colimit", "contents": [ "Let $A$ be a ring and let $B, C$ be $A$-algebras.", "Suppose that $R = \\colim_{i \\in I} R_i$ is a directed colimit", "of $A$-algebras.", "\\begin{enumerate}", "\\item If $B$ is a finite type $A$-algebra, and $u, u' : B \\to C$ are", "$A$-algebra maps such that", "$u \\otimes 1 = u' \\otimes 1 : B \\otimes_A R \\to C \\otimes_A R$", "then for some $i$ we have", "$u \\otimes 1 = u' \\otimes 1 : B \\otimes_A R_i \\to C \\otimes_A R_i$.", "\\item If $C$ is a finite type $A$-algebra and $u : B \\to C$ is an", "$A$-algebra map such that", "$u \\otimes 1 : B \\otimes_A R \\to C \\otimes_A R$ is surjective, then", "for some $i$ the map $u \\otimes 1 : B \\otimes_A R_i \\to C \\otimes_A R_i$", "is surjective.", "\\item If $C$ is of finite presentation over $A$ and", "$v : C \\otimes_A R \\to B \\otimes_A R$ is an $R$-algebra map, then there", "exists an $i$ and an $R_i$-algebra map", "$v_i : C \\otimes_A R_i \\to B \\otimes_A R_i$ such that", "$v = v_i \\otimes 1$.", "\\item If $B$ is a finite type $A$-algebra, $C$ is a finitely presented", "$A$-algebra, and", "$u \\otimes 1 : B \\otimes_A R \\to C \\otimes_A R$ is an isomorphism, then", "for some $i$ the map $u \\otimes 1 : B \\otimes_A R_i \\to C \\otimes_A R_i$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove (1) assume $u$ is as in (1) and", "let $x_1, \\ldots, x_m \\in B$ be generators. Since", "$B \\otimes_A R = \\colim_i B \\otimes_A R_i$", "we may pick an $i \\in I$ such that $u(x_j) \\otimes 1 = u'(x_j) \\otimes 1$", "in $B \\otimes_A R_i$, $j = 1, \\ldots, m$.", "For such an $i$ we have", "$u \\otimes 1 = u' \\otimes 1 : B \\otimes_A R_i \\to C \\otimes_A R_i$.", "\\medskip\\noindent", "To prove (2) assume $u \\otimes 1$ surjective and", "let $y_1, \\ldots, y_m \\in C$ be generators. Since", "$B \\otimes_A R = \\colim_i B \\otimes_A R_i$", "we may pick an $i \\in I$ and $z_j \\in B \\otimes_A R_i$, $j = 1, \\ldots, m$", "whose images in $C \\otimes_A R$ equal $y_j \\otimes 1$.", "For such an $i$ the map $u \\otimes 1 : B \\otimes_A R_i \\to C \\otimes_A R_i$", "is surjective.", "\\medskip\\noindent", "To prove (3) let $c_1, \\ldots, c_m \\in C$ be generators. Let", "$K = \\Ker(A[x_1, \\ldots, x_m] \\to N)$ where the map is given by", "the rule $x_j \\mapsto \\sum c_j$. Let $f_1, \\ldots, f_t$", "be generators for $K$ as an ideal in $A[x_1, \\ldots, x_m]$.", "We think of $f_j = f_j(x_1, \\ldots, x_m)$ as a polynomial.", "Since $B \\otimes_A R = \\colim_i B \\otimes_A R_i$", "we may pick an $i \\in I$ and $z_j \\in B \\otimes_A R_i$, $j = 1, \\ldots, m$", "whose images in $B \\otimes_A R$ equal $v(c_j \\otimes 1)$.", "We want to use the $z_j$ to define a map", "$v_i : C \\otimes_A R_i \\to B \\otimes_A R_i$.", "Since $K \\otimes_A R_i \\to R_i[x_1, \\ldots, x_m] \\to C \\otimes_A R_i \\to 0$", "is a presentation, it suffices to check that", "$\\xi_s = f_j(z_1, \\ldots, z_m)$ is", "zero in $B \\otimes_A R_i$ for each $s = 1, \\ldots, t$. This may not", "be the case, but since the image of $\\xi_s$ in $B \\otimes_A R$ is zero", "we see that it will be the case after increasing $i$ a bit.", "\\medskip\\noindent", "To prove (4) assume $u \\otimes 1$ is an isomorphism, that", "$B$ is a finite type $A$-algebra, and that $C$ is a finitely presented", "$A$-algebra. Let $v : B \\otimes_A R \\to C \\otimes_A R$ be an inverse to", "$u \\otimes 1$. Let $v_i : C \\otimes_A R_i \\to B \\otimes_A R_i$ be as", "in part (3). Apply part (1) to see that, after increasing $i$ we have", "$v_i \\circ (u \\otimes 1) = \\text{id}_{B \\otimes_R R_i}$ and", "$(u \\otimes 1) \\circ v_i = \\text{id}_{C \\otimes_R R_i}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1097, "type": "theorem", "label": "algebra-lemma-colimit-category-fp-algebras", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-category-fp-algebras", "contents": [ "Suppose that $R = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$ is a directed colimit", "of rings. Then the category of finitely presented $R$-algebras is", "the colimit of the categories of finitely presented $R_\\lambda$-algebras.", "More precisely", "\\begin{enumerate}", "\\item Given a finitely presented $R$-algebra $A$ there exists a", "$\\lambda \\in \\Lambda$ and a finitely presented $R_\\lambda$-algebra", "$A_\\lambda$ such that $A \\cong A_\\lambda \\otimes_{R_\\lambda} R$.", "\\item Given a $\\lambda \\in \\Lambda$, finitely presented", "$R_\\lambda$-algebras $A_\\lambda, B_\\lambda$, and an $R$-algebra map", "$\\varphi : A_\\lambda \\otimes_{R_\\lambda} R \\to B_\\lambda \\otimes_{R_\\lambda} R$,", "then there exists a $\\mu \\geq \\lambda$ and an $R_\\mu$-algebra map", "$\\varphi_\\mu : A_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to", "B_\\lambda \\otimes_{R_\\lambda} R_\\mu$", "such that $\\varphi = \\varphi_\\mu \\otimes 1_R$.", "\\item Given a $\\lambda \\in \\Lambda$, finitely presented", "$R_\\lambda$-algebras $A_\\lambda, B_\\lambda$, and $R_\\lambda$-algebra maps", "$\\varphi_\\lambda, \\psi_\\lambda : A_\\lambda \\to B_\\lambda$", "such that $\\varphi \\otimes 1_R = \\psi \\otimes 1_R$, then", "$\\varphi \\otimes 1_{R_\\mu} = \\psi \\otimes 1_{R_\\mu}$ for some", "$\\mu \\geq \\lambda$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove (1) choose a presentation", "$A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$.", "We can choose a $\\lambda \\in \\Lambda$ and elements", "$f_{\\lambda, j} \\in R_\\lambda[x_1, \\ldots, x_n]$ mapping to", "$f_j \\in R[x_1, \\ldots, x_n]$.", "Then we simply let", "$A_\\lambda =", "R_\\lambda[x_1, \\ldots, x_n]/(f_{\\lambda, 1}, \\ldots, f_{\\lambda, m})$.", "\\medskip\\noindent", "Parts (2) and (3) follow from", "Lemma \\ref{lemma-algebra-map-property-in-colimit}." ], "refs": [ "algebra-lemma-algebra-map-property-in-colimit" ], "ref_ids": [ 1096 ] } ], "ref_ids": [] }, { "id": 1098, "type": "theorem", "label": "algebra-lemma-limit-no-condition-local", "categories": [ "algebra" ], "title": "algebra-lemma-limit-no-condition-local", "contents": [ "Suppose $R \\to S$ is a local homomorphism of local rings.", "There exists a directed set $(\\Lambda, \\leq)$, and", "a system of local homomorphisms $R_\\lambda \\to S_\\lambda$", "of local rings such that", "\\begin{enumerate}", "\\item The colimit of the system $R_\\lambda \\to S_\\lambda$", "is equal to $R \\to S$.", "\\item Each $R_\\lambda$ is essentially of finite type", "over $\\mathbf{Z}$.", "\\item Each $S_\\lambda$ is essentially of finite type", "over $R_\\lambda$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $\\varphi : R \\to S$ the ring map.", "Let $\\mathfrak m \\subset R$ be the maximal ideal", "of $R$ and let $\\mathfrak n \\subset S$ be the maximal", "ideal of $S$. Let", "$$", "\\Lambda = \\{", "(A, B)", "\\mid", "A \\subset R, B \\subset S, \\# A < \\infty, \\# B < \\infty, \\varphi(A) \\subset B", "\\}.", "$$", "As partial ordering we take the inclusion relation. For each", "$\\lambda = (A, B) \\in \\Lambda$ we let $R'_\\lambda$ be", "the sub $\\mathbf{Z}$-algebra generated by", "$a \\in A$, and we let $S'_\\lambda$ be the sub", "$\\mathbf{Z}$-algebra generated by $b$, $b \\in B$.", "Let $R_\\lambda$ be the localization of $R'_\\lambda$", "at the prime ideal $R'_\\lambda \\cap \\mathfrak m$ and let", "$S_\\lambda$ be the localization of $S'_\\lambda$ at", "the prime ideal $S'_\\lambda \\cap \\mathfrak n$.", "In a picture", "$$", "\\xymatrix{", "B \\ar[r] &", "S'_\\lambda \\ar[r] &", "S_\\lambda \\ar[r] &", "S \\\\", "A \\ar[r] \\ar[u] &", "R'_\\lambda \\ar[r] \\ar[u] &", "R_\\lambda \\ar[r] \\ar[u] &", "R \\ar[u]", "}.", "$$", "The transition maps are clear. We leave the proofs of the other", "assertions to the reader." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1099, "type": "theorem", "label": "algebra-lemma-limit-essentially-finite-type", "categories": [ "algebra" ], "title": "algebra-lemma-limit-essentially-finite-type", "contents": [ "Suppose $R \\to S$ is a local homomorphism of local rings.", "Assume that $S$ is essentially of finite type over $R$.", "Then there exists a directed set $(\\Lambda, \\leq)$, and", "a system of local homomorphisms $R_\\lambda \\to S_\\lambda$", "of local rings such that", "\\begin{enumerate}", "\\item The colimit of the system $R_\\lambda \\to S_\\lambda$", "is equal to $R \\to S$.", "\\item Each $R_\\lambda$ is essentially of finite type", "over $\\mathbf{Z}$.", "\\item Each $S_\\lambda$ is essentially of finite type", "over $R_\\lambda$.", "\\item For each $\\lambda \\leq \\mu$ the map", "$S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$", "presents $S_\\mu$ as the localization of a quotient", "of $S_\\lambda \\otimes_{R_\\lambda} R_\\mu$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $\\varphi : R \\to S$ the ring map.", "Let $\\mathfrak m \\subset R$ be the maximal ideal", "of $R$ and let $\\mathfrak n \\subset S$ be the maximal", "ideal of $S$. Let $x_1, \\ldots, x_n \\in S$ be elements such that", "$S$ is a localization of the sub $R$-algebra of $S$", "generated by $x_1, \\ldots, x_n$. In other words, $S$", "is a quotient of a localization of the polynomial ring", "$R[x_1, \\ldots, x_n]$.", "\\medskip\\noindent", "Let $\\Lambda = \\{ A \\subset R \\mid \\# A < \\infty\\}$", "be the set of finite subsets of $R$.", "As partial ordering we take the inclusion relation. For each", "$\\lambda = A \\in \\Lambda$ we let $R'_\\lambda$ be", "the sub $\\mathbf{Z}$-algebra generated by", "$a \\in A$, and we let $S'_\\lambda$ be the sub", "$\\mathbf{Z}$-algebra generated by $\\varphi(a)$, $a \\in A$", "and the elements $x_1, \\ldots, x_n$. Let $R_\\lambda$ be", "the localization of $R'_\\lambda$ at the prime ideal", "$R'_\\lambda \\cap \\mathfrak m$ and let", "$S_\\lambda$ be the localization of $S'_\\lambda$ at", "the prime ideal $S'_\\lambda \\cap \\mathfrak n$.", "In a picture", "$$", "\\xymatrix{", "\\varphi(A) \\amalg \\{x_i\\} \\ar[r] &", "S'_\\lambda \\ar[r] &", "S_\\lambda \\ar[r] &", "S \\\\", "A \\ar[r] \\ar[u] &", "R'_\\lambda \\ar[r] \\ar[u] &", "R_\\lambda \\ar[r] \\ar[u] &", "R \\ar[u]", "}", "$$", "It is clear that if $A \\subset B$ corresponds to", "$\\lambda \\leq \\mu$ in $\\Lambda$, then there are", "canonical maps $R_\\lambda \\to R_\\mu$, and $S_\\lambda \\to S_\\mu$", "and we obtain a system over the directed set $\\Lambda$.", "\\medskip\\noindent", "The assertion that $R = \\colim R_\\lambda$ is clear", "because all the maps $R_\\lambda \\to R$ are injective and", "any element of $R$ eventually is in the image. The same", "argument works for $S = \\colim S_\\lambda$.", "Assertions (2), (3) are true by construction.", "The final assertion holds because clearly", "the maps $S'_\\lambda \\otimes_{R'_\\lambda} R'_\\mu", "\\to S'_\\mu$ are surjective." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1100, "type": "theorem", "label": "algebra-lemma-limit-essentially-finite-presentation", "categories": [ "algebra" ], "title": "algebra-lemma-limit-essentially-finite-presentation", "contents": [ "Suppose $R \\to S$ is a local homomorphism of local rings.", "Assume that $S$ is essentially of finite presentation over $R$.", "Then there exists a directed set $(\\Lambda, \\leq)$, and", "a system of local homomorphism $R_\\lambda \\to S_\\lambda$", "of local rings such that", "\\begin{enumerate}", "\\item The colimit of the system $R_\\lambda \\to S_\\lambda$", "is equal to $R \\to S$.", "\\item Each $R_\\lambda$ is essentially of finite type", "over $\\mathbf{Z}$.", "\\item Each $S_\\lambda$ is essentially of finite type", "over $R_\\lambda$.", "\\item For each $\\lambda \\leq \\mu$ the map", "$S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$", "presents $S_\\mu$ as the localization of", "$S_\\lambda \\otimes_{R_\\lambda} R_\\mu$", "at a prime ideal.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By assumption we may choose an isomorphism", "$\\Phi : (R[x_1, \\ldots, x_n]/I)_{\\mathfrak q} \\to S$", "where $I \\subset R[x_1, \\ldots, x_n]$ is a finitely generated ideal,", "and $\\mathfrak q \\subset R[x_1, \\ldots, x_n]/I$ is a prime.", "(Note that $R \\cap \\mathfrak q$", "is equal to the maximal ideal $\\mathfrak m$ of $R$.)", "We also choose generators $f_1, \\ldots, f_m \\in I$ for the ideal $I$.", "Write $R$ in any way as a colimit $R = \\colim R_\\lambda$", "over a directed set $(\\Lambda, \\leq )$, with each $R_\\lambda$", "local and essentially of finite type over $\\mathbf{Z}$.", "There exists some $\\lambda_0 \\in \\Lambda$ such that $f_j$ is the image", "of some $f_{j, \\lambda_0} \\in R_{\\lambda_0}[x_1, \\ldots, x_n]$.", "For all $\\lambda \\geq \\lambda_0$ denote", "$f_{j, \\lambda} \\in R_{\\lambda}[x_1, \\ldots, x_n]$ the image", "of $f_{j, \\lambda_0}$. Thus we obtain a system of ring maps", "$$", "R_\\lambda[x_1, \\ldots, x_n]/(f_{1, \\lambda}, \\ldots, f_{n, \\lambda})", "\\to", "R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n) \\to S", "$$", "Set $\\mathfrak q_\\lambda$ the inverse image of $\\mathfrak q$.", "Set $S_\\lambda = (R_\\lambda[x_1, \\ldots, x_n]/", "(f_{1, \\lambda}, \\ldots, f_{n, \\lambda}))_{\\mathfrak q_\\lambda}$.", "We leave it to the reader to see that this works." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1101, "type": "theorem", "label": "algebra-lemma-limit-module-essentially-finite-presentation", "categories": [ "algebra" ], "title": "algebra-lemma-limit-module-essentially-finite-presentation", "contents": [ "Suppose $R \\to S$ is a local homomorphism of local rings.", "Assume that $S$ is essentially of finite presentation over $R$.", "Let $M$ be a finitely presented $S$-module.", "Then there exists a directed set $(\\Lambda, \\leq)$, and", "a system of local homomorphisms $R_\\lambda \\to S_\\lambda$", "of local rings together with $S_\\lambda$-modules $M_\\lambda$,", "such that", "\\begin{enumerate}", "\\item The colimit of the system $R_\\lambda \\to S_\\lambda$", "is equal to $R \\to S$. The colimit of the system $M_\\lambda$", "is $M$.", "\\item Each $R_\\lambda$ is essentially of finite type", "over $\\mathbf{Z}$.", "\\item Each $S_\\lambda$ is essentially of finite type", "over $R_\\lambda$.", "\\item Each $M_\\lambda$ is finite over $S_\\lambda$.", "\\item For each $\\lambda \\leq \\mu$ the map", "$S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$", "presents $S_\\mu$ as the localization of", "$S_\\lambda \\otimes_{R_\\lambda} R_\\mu$", "at a prime ideal.", "\\item For each $\\lambda \\leq \\mu$ the map", "$M_\\lambda \\otimes_{S_\\lambda} S_\\mu \\to M_\\mu$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As in the proof of Lemma \\ref{lemma-limit-essentially-finite-presentation}", "we may first write $R = \\colim R_\\lambda$ as a directed colimit", "of local $\\mathbf{Z}$-algebras which are essentially of finite type.", "Next, we may assume that for some $\\lambda_1 \\in \\Lambda$ there", "exist $f_{j, \\lambda_1} \\in R_{\\lambda_1}[x_1, \\ldots, x_n]$", "such that", "$$", "S =", "\\colim_{\\lambda \\geq \\lambda_1} S_\\lambda, \\text{ with }", "S_\\lambda =", "(R_\\lambda[x_1, \\ldots, x_n]/", "(f_{1, \\lambda}, \\ldots, f_{m, \\lambda}))_{\\mathfrak q_\\lambda}", "$$", "Choose a presentation", "$$", "S^{\\oplus s} \\to S^{\\oplus t} \\to M \\to 0", "$$", "of $M$ over $S$. Let $A \\in \\text{Mat}(t \\times s, S)$ be", "the matrix of the presentation. For some $\\lambda_2 \\in \\Lambda$,", "$\\lambda_2 \\geq \\lambda_1$", "we can find a matrix $A_{\\lambda_2} \\in \\text{Mat}(t \\times s, S_{\\lambda_2})$", "which maps to $A$. For all $\\lambda \\geq \\lambda_2$ we let", "$M_\\lambda = \\Coker(S_\\lambda^{\\oplus s} \\xrightarrow{A_\\lambda}", "S_\\lambda^{\\oplus t})$. We leave it to the reader to see that", "this works." ], "refs": [ "algebra-lemma-limit-essentially-finite-presentation" ], "ref_ids": [ 1100 ] } ], "ref_ids": [] }, { "id": 1102, "type": "theorem", "label": "algebra-lemma-limit-no-condition", "categories": [ "algebra" ], "title": "algebra-lemma-limit-no-condition", "contents": [ "Suppose $R \\to S$ is a ring map.", "Then there exists a directed set $(\\Lambda, \\leq)$, and", "a system of ring maps $R_\\lambda \\to S_\\lambda$", "such that", "\\begin{enumerate}", "\\item The colimit of the system $R_\\lambda \\to S_\\lambda$", "is equal to $R \\to S$.", "\\item Each $R_\\lambda$ is of finite type", "over $\\mathbf{Z}$.", "\\item Each $S_\\lambda$ is of finite type", "over $R_\\lambda$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is the non-local version of", "Lemma \\ref{lemma-limit-no-condition-local}.", "Proof is similar and left to the reader." ], "refs": [ "algebra-lemma-limit-no-condition-local" ], "ref_ids": [ 1098 ] } ], "ref_ids": [] }, { "id": 1103, "type": "theorem", "label": "algebra-lemma-limit-integral", "categories": [ "algebra" ], "title": "algebra-lemma-limit-integral", "contents": [ "Suppose $R \\to S$ is a ring map.", "Assume that $S$ is integral over $R$.", "Then there exists a directed set $(\\Lambda, \\leq)$, and", "a system of ring maps $R_\\lambda \\to S_\\lambda$", "such that", "\\begin{enumerate}", "\\item The colimit of the system $R_\\lambda \\to S_\\lambda$", "is equal to $R \\to S$.", "\\item Each $R_\\lambda$ is of finite type", "over $\\mathbf{Z}$.", "\\item Each $S_\\lambda$ is of finite over $R_\\lambda$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the set $\\Lambda$ of pairs $(E, F)$ where $E \\subset R$", "is a finite subset, $F \\subset S$ is a finite subset, and", "every element $f \\in F$ is the root of a monic $P(X) \\in R[X]$", "whose coefficients are in $E$. Say $(E, F) \\leq (E', F')$", "if $E \\subset E'$ and $F \\subset F'$.", "Given $\\lambda = (E, F) \\in \\Lambda$ set $R_\\lambda \\subset R$ equal", "to the $\\mathbf{Z}$-subalgebra of $R$ generated by $E$ and", "$S_\\lambda \\subset S$ equal to the $\\mathbf{Z}$-subalgebra generated by", "$F$ and the image of $E$ in $S$. It is clear that $R = \\colim R_\\lambda$.", "We have $S = \\colim S_\\lambda$ as every element of $S$ is integral", "over $S$. The ring maps $R_\\lambda \\to S_\\lambda$ are finite by", "Lemma \\ref{lemma-characterize-finite-in-terms-of-integral} and the fact that", "$S_\\lambda$ is generated over $R_\\lambda$ by the elements of", "$F$ which are integral over $R_\\lambda$ by our condition on the", "pairs $(E, F)$. The lemma follows." ], "refs": [ "algebra-lemma-characterize-finite-in-terms-of-integral" ], "ref_ids": [ 484 ] } ], "ref_ids": [] }, { "id": 1104, "type": "theorem", "label": "algebra-lemma-limit-finite-type", "categories": [ "algebra" ], "title": "algebra-lemma-limit-finite-type", "contents": [ "Suppose $R \\to S$ is a ring map.", "Assume that $S$ is of finite type over $R$.", "Then there exists a directed set $(\\Lambda, \\leq)$, and", "a system of ring maps $R_\\lambda \\to S_\\lambda$", "such that", "\\begin{enumerate}", "\\item The colimit of the system $R_\\lambda \\to S_\\lambda$", "is equal to $R \\to S$.", "\\item Each $R_\\lambda$ is of finite type", "over $\\mathbf{Z}$.", "\\item Each $S_\\lambda$ is of finite type", "over $R_\\lambda$.", "\\item For each $\\lambda \\leq \\mu$ the map", "$S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$", "presents $S_\\mu$ as a quotient", "of $S_\\lambda \\otimes_{R_\\lambda} R_\\mu$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is the non-local version of", "Lemma \\ref{lemma-limit-essentially-finite-type}.", "Proof is similar and left to the reader." ], "refs": [ "algebra-lemma-limit-essentially-finite-type" ], "ref_ids": [ 1099 ] } ], "ref_ids": [] }, { "id": 1105, "type": "theorem", "label": "algebra-lemma-limit-finite-presentation", "categories": [ "algebra" ], "title": "algebra-lemma-limit-finite-presentation", "contents": [ "Suppose $R \\to S$ is a ring map.", "Assume that $S$ is of finite presentation over $R$.", "Then there exists a directed set $(\\Lambda, \\leq)$, and", "a system of ring maps $R_\\lambda \\to S_\\lambda$", "such that", "\\begin{enumerate}", "\\item The colimit of the system $R_\\lambda \\to S_\\lambda$", "is equal to $R \\to S$.", "\\item Each $R_\\lambda$ is of finite type", "over $\\mathbf{Z}$.", "\\item Each $S_\\lambda$ is of finite type", "over $R_\\lambda$.", "\\item For each $\\lambda \\leq \\mu$ the map", "$S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is the non-local version of", "Lemma \\ref{lemma-limit-essentially-finite-presentation}.", "Proof is similar and left to the reader." ], "refs": [ "algebra-lemma-limit-essentially-finite-presentation" ], "ref_ids": [ 1100 ] } ], "ref_ids": [] }, { "id": 1106, "type": "theorem", "label": "algebra-lemma-limit-module-finite-presentation", "categories": [ "algebra" ], "title": "algebra-lemma-limit-module-finite-presentation", "contents": [ "Suppose $R \\to S$ is a ring map.", "Assume that $S$ is of finite presentation over $R$.", "Let $M$ be a finitely presented $S$-module.", "Then there exists a directed set $(\\Lambda, \\leq)$, and", "a system of ring maps $R_\\lambda \\to S_\\lambda$", "together with $S_\\lambda$-modules $M_\\lambda$,", "such that", "\\begin{enumerate}", "\\item The colimit of the system $R_\\lambda \\to S_\\lambda$", "is equal to $R \\to S$. The colimit of the system $M_\\lambda$", "is $M$.", "\\item Each $R_\\lambda$ is of finite type", "over $\\mathbf{Z}$.", "\\item Each $S_\\lambda$ is of finite type", "over $R_\\lambda$.", "\\item Each $M_\\lambda$ is finite over $S_\\lambda$.", "\\item For each $\\lambda \\leq \\mu$ the map", "$S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$", "is an isomorphism.", "\\item For each $\\lambda \\leq \\mu$ the map", "$M_\\lambda \\otimes_{S_\\lambda} S_\\mu \\to M_\\mu$", "is an isomorphism.", "\\end{enumerate}", "In particular, for every $\\lambda \\in \\Lambda$ we have", "$$", "M = M_\\lambda \\otimes_{S_\\lambda} S", "= M_\\lambda \\otimes_{R_\\lambda} R.", "$$" ], "refs": [], "proofs": [ { "contents": [ "This is the non-local version of", "Lemma \\ref{lemma-limit-module-essentially-finite-presentation}.", "Proof is similar and left to the reader." ], "refs": [ "algebra-lemma-limit-module-essentially-finite-presentation" ], "ref_ids": [ 1101 ] } ], "ref_ids": [] }, { "id": 1107, "type": "theorem", "label": "algebra-lemma-CM-over-regular-flat", "categories": [ "algebra" ], "title": "algebra-lemma-CM-over-regular-flat", "contents": [ "\\begin{slogan}", "Miracle flatness", "\\end{slogan}", "Let $R \\to S$ be a local homomorphism of Noetherian local", "rings. Assume", "\\begin{enumerate}", "\\item $R$ is regular,", "\\item $S$ Cohen-Macaulay,", "\\item $\\dim(S) = \\dim(R) + \\dim(S/\\mathfrak m_R S)$.", "\\end{enumerate}", "Then $R \\to S$ is flat." ], "refs": [], "proofs": [ { "contents": [ "By induction on $\\dim(R)$. The case $\\dim(R) = 0$ is trivial, because", "then $R$ is a field. Assume $\\dim(R) > 0$. By (3) this implies that", "$\\dim(S) > 0$. Let $\\mathfrak q_1, \\ldots, \\mathfrak q_r$ be the minimal", "primes of $S$. Note that $\\mathfrak q_i \\not \\supset \\mathfrak m_R S$ since", "$$", "\\dim(S/\\mathfrak q_i) = \\dim(S) > \\dim(S/\\mathfrak m_R S)", "$$", "the first equality by Lemma \\ref{lemma-maximal-chain-CM} and the", "inequality by (3). Thus", "$\\mathfrak p_i = R \\cap \\mathfrak q_i$ is not equal to $\\mathfrak m_R$.", "Pick $x \\in \\mathfrak m$, $x \\not \\in \\mathfrak m^2$, and", "$x \\not \\in \\mathfrak p_i$, see", "Lemma \\ref{lemma-silly}.", "Hence we see that $x$ is not contained in any of the minimal", "primes of $S$. Hence $x$ is a nonzerodivisor on $S$ by (2), see", "Lemma \\ref{lemma-reformulate-CM} and", "$S/xS$ is Cohen-Macaulay with $\\dim(S/xS) = \\dim(S) - 1$.", "By (1) and Lemma \\ref{lemma-regular-ring-CM} the ring $R/xR$ is regular", "with $\\dim(R/xR) = \\dim(R) - 1$.", "By induction we see that $R/xR \\to S/xS$ is flat. Hence we", "conclude by Lemma \\ref{lemma-variant-local-criterion-flatness}", "and the remark following it." ], "refs": [ "algebra-lemma-maximal-chain-CM", "algebra-lemma-silly", "algebra-lemma-reformulate-CM", "algebra-lemma-regular-ring-CM", "algebra-lemma-variant-local-criterion-flatness" ], "ref_ids": [ 924, 378, 923, 941, 892 ] } ], "ref_ids": [] }, { "id": 1108, "type": "theorem", "label": "algebra-lemma-flat-over-regular", "categories": [ "algebra" ], "title": "algebra-lemma-flat-over-regular", "contents": [ "Let $R \\to S$ be a homomorphism of Noetherian local rings.", "Assume that $R$ is a regular local ring and that a regular system", "of parameters maps to a regular sequence in $S$. Then $R \\to S$", "is flat." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $x_1, \\ldots, x_d$ are a system of parameters of $R$", "which map to a regular sequence in $S$. Note that", "$S/(x_1, \\ldots, x_d)S$ is flat over $R/(x_1, \\ldots, x_d)$", "as the latter is a field. Then $x_d$ is a nonzerodivisor in", "$S/(x_1, \\ldots, x_{d - 1})S$ hence $S/(x_1, \\ldots, x_{d - 1})S$", "is flat over $R/(x_1, \\ldots, x_{d - 1})$ by the local criterion", "of flatness (see Lemma \\ref{lemma-variant-local-criterion-flatness}", "and remarks following). Then $x_{d - 1}$ is a nonzerodivisor in", "$S/(x_1, \\ldots, x_{d - 2})S$ hence $S/(x_1, \\ldots, x_{d - 2})S$", "is flat over $R/(x_1, \\ldots, x_{d - 2})$ by the local criterion", "of flatness (see Lemma \\ref{lemma-variant-local-criterion-flatness}", "and remarks following). Continue till one reaches the conclusion", "that $S$ is flat over $R$." ], "refs": [ "algebra-lemma-variant-local-criterion-flatness", "algebra-lemma-variant-local-criterion-flatness" ], "ref_ids": [ 892, 892 ] } ], "ref_ids": [] }, { "id": 1109, "type": "theorem", "label": "algebra-lemma-colimit-eventually-flat", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-eventually-flat", "contents": [ "Let $R \\to S$, $M$, $\\Lambda$, $R_\\lambda \\to S_\\lambda$, $M_\\lambda$", "be as in Lemma \\ref{lemma-limit-module-essentially-finite-presentation}.", "Assume that $M$ is flat over $R$.", "Then for some $\\lambda \\in \\Lambda$ the module", "$M_\\lambda$ is flat over $R_\\lambda$." ], "refs": [ "algebra-lemma-limit-module-essentially-finite-presentation" ], "proofs": [ { "contents": [ "Pick some $\\lambda \\in \\Lambda$ and consider", "$$", "\\text{Tor}_1^{R_\\lambda}(M_\\lambda, R_\\lambda/\\mathfrak m_\\lambda)", "=", "\\Ker(\\mathfrak m_\\lambda \\otimes_{R_\\lambda} M_\\lambda", "\\to M_\\lambda).", "$$", "See Remark \\ref{remark-Tor-ring-mod-ideal}. The right hand side", "shows that this is a finitely generated $S_\\lambda$-module (because", "$S_\\lambda$ is Noetherian and the modules in question are finite).", "Let $\\xi_1, \\ldots, \\xi_n$ be generators.", "Because $M$ is flat over $R$ we", "have that $0 = \\Ker(\\mathfrak m_\\lambda R \\otimes_R M \\to M)$.", "Since $\\otimes$ commutes with colimits we see there exists", "a $\\lambda' \\geq \\lambda$ such that each $\\xi_i$ maps to", "zero in", "$\\mathfrak m_{\\lambda}R_{\\lambda'} \\otimes_{R_{\\lambda'}} M_{\\lambda'}$.", "Hence we see that", "$$", "\\text{Tor}_1^{R_\\lambda}(M_\\lambda, R_\\lambda/\\mathfrak m_\\lambda)", "\\longrightarrow", "\\text{Tor}_1^{R_{\\lambda'}}(M_{\\lambda'},", "R_{\\lambda'}/\\mathfrak m_{\\lambda}R_{\\lambda'})", "$$", "is zero. Note that", "$M_\\lambda \\otimes_{R_\\lambda} R_\\lambda/\\mathfrak m_\\lambda$", "is flat over $R_\\lambda/\\mathfrak m_\\lambda$ because this last", "ring is a field. Hence we may apply Lemma", "\\ref{lemma-another-variant-local-criterion-flatness}", "to get that $M_{\\lambda'}$ is flat over $R_{\\lambda'}$." ], "refs": [ "algebra-remark-Tor-ring-mod-ideal", "algebra-lemma-another-variant-local-criterion-flatness" ], "ref_ids": [ 1570, 896 ] } ], "ref_ids": [ 1101 ] }, { "id": 1110, "type": "theorem", "label": "algebra-lemma-mod-injective-general", "categories": [ "algebra" ], "title": "algebra-lemma-mod-injective-general", "contents": [ "Suppose that $R \\to S$ is a local homomorphism of local rings.", "Denote $\\mathfrak m$ the maximal ideal of $R$.", "Let $u : M \\to N$ be a map of $S$-modules.", "Assume", "\\begin{enumerate}", "\\item $S$ is essentially of finite presentation over $R$,", "\\item $M$, $N$ are finitely presented over $S$,", "\\item $N$ is flat over $R$, and", "\\item $\\overline{u} : M/\\mathfrak mM \\to N/\\mathfrak mN$ is injective.", "\\end{enumerate}", "Then $u$ is injective, and $N/u(M)$ is flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-limit-module-essentially-finite-presentation}", "and its proof we can find a system $R_\\lambda \\to S_\\lambda$ of", "local ring maps together with maps of $S_\\lambda$-modules", "$u_\\lambda : M_\\lambda \\to N_\\lambda$ satisfying the conclusions", "(1) -- (6) for both $N$ and $M$ of that lemma and such that the", "colimit of the maps $u_\\lambda$ is $u$. By", "Lemma \\ref{lemma-colimit-eventually-flat}", "we may assume that $N_\\lambda$ is flat over $R_\\lambda$ for all", "sufficiently large $\\lambda$. Denote $\\mathfrak m_\\lambda \\subset R_\\lambda$", "the maximal ideal and $\\kappa_\\lambda = R_\\lambda / \\mathfrak m_\\lambda$,", "resp.\\ $\\kappa = R/\\mathfrak m$ the residue fields.", "\\medskip\\noindent", "Consider the map", "$$", "\\Psi_\\lambda :", "M_\\lambda/\\mathfrak m_\\lambda M_\\lambda \\otimes_{\\kappa_\\lambda} \\kappa", "\\longrightarrow", "M/\\mathfrak m M.", "$$", "Since $S_\\lambda/\\mathfrak m_\\lambda S_\\lambda$ is essentially of finite type", "over the field $\\kappa_\\lambda$ we see that the tensor product", "$S_\\lambda/\\mathfrak m_\\lambda S_\\lambda \\otimes_{\\kappa_\\lambda} \\kappa$", "is essentially of finite type over $\\kappa$. Hence it is a Noetherian", "ring and we conclude the kernel of $\\Psi_\\lambda$ is finitely generated.", "Since $M/\\mathfrak m M$ is the colimit of the system", "$M_\\lambda/\\mathfrak m_\\lambda M_\\lambda$ and $\\kappa$ is the colimit of", "the fields $\\kappa_\\lambda$ there exists a $\\lambda' > \\lambda$ such that", "the kernel of $\\Psi_\\lambda$ is generated by the kernel of", "$$", "\\Psi_{\\lambda, \\lambda'} :", "M_\\lambda/\\mathfrak m_\\lambda M_\\lambda", "\\otimes_{\\kappa_\\lambda}", "\\kappa_{\\lambda'}", "\\longrightarrow", "M_{\\lambda'}/\\mathfrak m_{\\lambda'} M_{\\lambda'}.", "$$", "By construction there exists a multiplicative subset", "$W \\subset S_\\lambda \\otimes_{R_\\lambda} R_{\\lambda'}$ such that", "$S_{\\lambda'} = W^{-1}(S_\\lambda \\otimes_{R_\\lambda} R_{\\lambda'})$ and", "$$", "W^{-1}(M_\\lambda/\\mathfrak m_\\lambda M_\\lambda", "\\otimes_{\\kappa_\\lambda}", "\\kappa_{\\lambda'})", "=", "M_{\\lambda'}/\\mathfrak m_{\\lambda'} M_{\\lambda'}.", "$$", "Now suppose that $x$ is an element of the kernel of", "$$", "\\Psi_{\\lambda'} :", "M_{\\lambda'}/\\mathfrak m_{\\lambda'} M_{\\lambda'}", "\\otimes_{\\kappa_{\\lambda'}} \\kappa", "\\longrightarrow", "M/\\mathfrak m M.", "$$", "Then for some $w \\in W$ we have", "$wx \\in M_\\lambda/\\mathfrak m_\\lambda M_\\lambda \\otimes \\kappa$.", "Hence $wx \\in \\Ker(\\Psi_\\lambda)$. Hence $wx$ is a linear", "combination of elements in the kernel of $\\Psi_{\\lambda, \\lambda'}$.", "Hence $wx = 0$ in $M_{\\lambda'}/\\mathfrak m_{\\lambda'} M_{\\lambda'}", "\\otimes_{\\kappa_{\\lambda'}} \\kappa$, hence $x = 0$ because $w$ is invertible", "in $S_{\\lambda'}$.", "We conclude that the kernel of $\\Psi_{\\lambda'}$ is zero for all sufficiently", "large $\\lambda'$!", "\\medskip\\noindent", "By the result of the preceding paragraph we may assume that", "the kernel of $\\Psi_\\lambda$ is zero for all $\\lambda$ sufficiently large,", "which implies that the map", "$M_\\lambda/\\mathfrak m_\\lambda M_\\lambda \\to M/\\mathfrak m M$", "is injective. Combined with $\\overline{u}$ being injective this", "formally implies that also", "$\\overline{u_\\lambda} : M_\\lambda/\\mathfrak m_\\lambda M_\\lambda", "\\to N_\\lambda/\\mathfrak m_\\lambda N_\\lambda$ is injective.", "By", "Lemma \\ref{lemma-mod-injective}", "we conclude that (for all sufficiently large $\\lambda$) the map", "$u_\\lambda$ is injective and that $N_\\lambda/u_\\lambda(M_\\lambda)$ is flat", "over $R_\\lambda$.", "The lemma follows." ], "refs": [ "algebra-lemma-limit-module-essentially-finite-presentation", "algebra-lemma-colimit-eventually-flat", "algebra-lemma-mod-injective" ], "ref_ids": [ 1101, 1109, 883 ] } ], "ref_ids": [] }, { "id": 1111, "type": "theorem", "label": "algebra-lemma-grothendieck-general", "categories": [ "algebra" ], "title": "algebra-lemma-grothendieck-general", "contents": [ "Suppose that $R \\to S$ is a local ring homomorphism of local rings.", "Denote $\\mathfrak m$ the maximal ideal of $R$.", "Suppose", "\\begin{enumerate}", "\\item $S$ is essentially of finite presentation over $R$,", "\\item $S$ is flat over $R$, and", "\\item $f \\in S$ is a nonzerodivisor in $S/{\\mathfrak m}S$.", "\\end{enumerate}", "Then $S/fS$ is flat over $R$, and $f$ is a nonzerodivisor in $S$." ], "refs": [], "proofs": [ { "contents": [ "Follows directly from Lemma \\ref{lemma-mod-injective-general}." ], "refs": [ "algebra-lemma-mod-injective-general" ], "ref_ids": [ 1110 ] } ], "ref_ids": [] }, { "id": 1112, "type": "theorem", "label": "algebra-lemma-grothendieck-regular-sequence-general", "categories": [ "algebra" ], "title": "algebra-lemma-grothendieck-regular-sequence-general", "contents": [ "Suppose that $R \\to S$ is a local ring homomorphism of local rings.", "Denote $\\mathfrak m$ the maximal ideal of $R$.", "Suppose", "\\begin{enumerate}", "\\item $R \\to S$ is essentially of finite presentation,", "\\item $R \\to S$ is flat, and", "\\item $f_1, \\ldots, f_c$ is a sequence of elements of", "$S$ such that the images $\\overline{f}_1, \\ldots, \\overline{f}_c$", "form a regular sequence in $S/{\\mathfrak m}S$.", "\\end{enumerate}", "Then $f_1, \\ldots, f_c$ is a regular sequence in $S$ and each", "of the quotients $S/(f_1, \\ldots, f_i)$ is flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "Induction and Lemma \\ref{lemma-grothendieck-general}." ], "refs": [ "algebra-lemma-grothendieck-general" ], "ref_ids": [ 1111 ] } ], "ref_ids": [] }, { "id": 1113, "type": "theorem", "label": "algebra-lemma-variant-local-criterion-flatness-general", "categories": [ "algebra" ], "title": "algebra-lemma-variant-local-criterion-flatness-general", "contents": [ "Let $R \\to S$ be a local homomorphism of local rings.", "Let $I \\not = R$ be an ideal in $R$. Let $M$ be an $S$-module. Assume", "\\begin{enumerate}", "\\item $S$ is essentially of finite presentation over $R$,", "\\item $M$ is of finite presentation over $S$,", "\\item $\\text{Tor}_1^R(M, R/I) = 0$, and", "\\item $M/IM$ is flat over $R/I$.", "\\end{enumerate}", "Then $M$ is flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\Lambda$, $R_\\lambda \\to S_\\lambda$, $M_\\lambda$ be as in", "Lemma \\ref{lemma-limit-module-essentially-finite-presentation}.", "Denote $I_\\lambda \\subset R_\\lambda$ the inverse image of $I$.", "In this case the system", "$R/I \\to S/IS$, $M/IM$, $R_\\lambda \\to S_\\lambda/I_\\lambda S_\\lambda$,", "and $M_\\lambda/I_\\lambda M_\\lambda$ satisfies the conclusions of", "Lemma \\ref{lemma-limit-module-essentially-finite-presentation}", "as well. Hence by", "Lemma \\ref{lemma-colimit-eventually-flat}", "we may assume (after shrinking the index set $\\Lambda$)", "that $M_\\lambda/I_\\lambda M_\\lambda$ is flat for all $\\lambda$.", "Pick some $\\lambda$ and consider", "$$", "\\text{Tor}_1^{R_\\lambda}(M_\\lambda, R_\\lambda/I_\\lambda)", "=", "\\Ker(I_\\lambda \\otimes_{R_\\lambda} M_\\lambda", "\\to M_\\lambda).", "$$", "See Remark \\ref{remark-Tor-ring-mod-ideal}. The right hand side", "shows that this is a finitely generated $S_\\lambda$-module (because", "$S_\\lambda$ is Noetherian and the modules in question are finite).", "Let $\\xi_1, \\ldots, \\xi_n$ be generators.", "Because $\\text{Tor}_1^R(M, R/I) = 0$ and since $\\otimes$ commutes", "with colimits we see there exists", "a $\\lambda' \\geq \\lambda$ such that each $\\xi_i$ maps to", "zero in", "$\\text{Tor}_1^{R_{\\lambda'}}(M_{\\lambda'}, R_{\\lambda'}/I_{\\lambda'})$.", "The composition of the maps", "$$", "\\xymatrix{", "R_{\\lambda'} \\otimes_{R_\\lambda}", "\\text{Tor}_1^{R_\\lambda}(M_\\lambda, R_\\lambda/I_\\lambda)", "\\ar[d]^{\\text{surjective by Lemma \\ref{lemma-surjective-on-tor-one}}} \\\\", "\\text{Tor}_1^{R_\\lambda}(M_\\lambda, R_{\\lambda'}/I_\\lambda R_{\\lambda'})", "\\ar[d]^{\\text{surjective up to localization by", "Lemma \\ref{lemma-surjective-on-tor-one-trivial}}} \\\\", "\\text{Tor}_1^{R_{\\lambda'}}(M_{\\lambda'}, R_{\\lambda'}/I_\\lambda R_{\\lambda'})", "\\ar[d]^{\\text{surjective by Lemma \\ref{lemma-surjective-on-tor-one}}} \\\\", "\\text{Tor}_1^{R_{\\lambda'}}(M_{\\lambda'}, R_{\\lambda'}/I_{\\lambda'}).", "}", "$$", "is surjective up to a localization by the reasons indicated.", "The localization is necessary since $M_{\\lambda'}$ is not equal", "to $M_\\lambda \\otimes_{R_\\lambda} R_{\\lambda'}$. Namely, it is equal", "to $M_\\lambda \\otimes_{S_\\lambda} S_{\\lambda'}$ and $S_{\\lambda'}$", "is the localization of $S_{\\lambda} \\otimes_{R_\\lambda} R_{\\lambda'}$ whence", "the statement up to a localization (or tensoring with $S_{\\lambda'}$).", "Note that", "Lemma \\ref{lemma-surjective-on-tor-one}", "applies to the first and third arrows because", "$M_\\lambda/I_\\lambda M_\\lambda$ is flat over", "$R_\\lambda/I_\\lambda$ and because $M_{\\lambda'}/I_\\lambda M_{\\lambda'}$", "is flat over $R_{\\lambda'}/I_\\lambda R_{\\lambda'}$ as it is a base", "change of the flat module $M_\\lambda/I_\\lambda M_\\lambda$.", "The composition maps the generators $\\xi_i$ to zero as we explained above.", "We finally conclude that", "$\\text{Tor}_1^{R_{\\lambda'}}(M_{\\lambda'}, R_{\\lambda'}/I_{\\lambda'})$", "is zero. This implies that $M_{\\lambda'}$ is flat over $R_{\\lambda'}$ by", "Lemma \\ref{lemma-variant-local-criterion-flatness}." ], "refs": [ "algebra-lemma-limit-module-essentially-finite-presentation", "algebra-lemma-limit-module-essentially-finite-presentation", "algebra-lemma-colimit-eventually-flat", "algebra-remark-Tor-ring-mod-ideal", "algebra-lemma-surjective-on-tor-one", "algebra-lemma-surjective-on-tor-one-trivial", "algebra-lemma-surjective-on-tor-one", "algebra-lemma-surjective-on-tor-one", "algebra-lemma-variant-local-criterion-flatness" ], "ref_ids": [ 1101, 1101, 1109, 1570, 894, 895, 894, 894, 892 ] } ], "ref_ids": [] }, { "id": 1114, "type": "theorem", "label": "algebra-lemma-criterion-flatness-fibre", "categories": [ "algebra" ], "title": "algebra-lemma-criterion-flatness-fibre", "contents": [ "Let $R$, $S$, $S'$ be local rings and let $R \\to S \\to S'$ be local ring", "homomorphisms. Let $M$ be an $S'$-module. Let $\\mathfrak m \\subset R$", "be the maximal ideal. Assume", "\\begin{enumerate}", "\\item The ring maps $R \\to S$ and $R \\to S'$ are essentially", "of finite presentation.", "\\item The module $M$ is of finite presentation over $S'$.", "\\item The module $M$ is not zero.", "\\item The module $M/\\mathfrak mM$ is a flat $S/\\mathfrak mS$-module.", "\\item The module $M$ is a flat $R$-module.", "\\end{enumerate}", "Then $S$ is flat over $R$ and $M$ is a flat $S$-module." ], "refs": [], "proofs": [ { "contents": [ "As in the proof of Lemma \\ref{lemma-limit-essentially-finite-presentation}", "we may first write $R = \\colim R_\\lambda$ as a directed colimit", "of local $\\mathbf{Z}$-algebras which are essentially of finite type.", "Denote $\\mathfrak p_\\lambda$ the maximal ideal of $R_\\lambda$.", "Next, we may assume that for some $\\lambda_1 \\in \\Lambda$ there", "exist $f_{j, \\lambda_1} \\in R_{\\lambda_1}[x_1, \\ldots, x_n]$", "such that", "$$", "S =", "\\colim_{\\lambda \\geq \\lambda_1} S_\\lambda, \\text{ with }", "S_\\lambda =", "(R_\\lambda[x_1, \\ldots, x_n]/", "(f_{1, \\lambda}, \\ldots, f_{u, \\lambda}))_{\\mathfrak q_\\lambda}", "$$", "For some $\\lambda_2 \\in \\Lambda$,", "$\\lambda_2 \\geq \\lambda_1$ there exist", "$g_{j, \\lambda_2} \\in R_{\\lambda_2}[x_1, \\ldots, x_n, y_1, \\ldots, y_m]$", "with images", "$\\overline{g}_{j, \\lambda_2} \\in S_{\\lambda_2}[y_1, \\ldots, y_m]$", "such that", "$$", "S' =", "\\colim_{\\lambda \\geq \\lambda_2} S'_\\lambda, \\text{ with }", "S'_\\lambda =", "(S_\\lambda[y_1, \\ldots, y_m]/", "(\\overline{g}_{1, \\lambda}, \\ldots,", "\\overline{g}_{v, \\lambda}))_{\\overline{\\mathfrak q}'_\\lambda}", "$$", "Note that this also implies that", "$$", "S'_\\lambda =", "(R_\\lambda[x_1, \\ldots, x_n, y_1, \\ldots, y_m]/", "(g_{1, \\lambda}, \\ldots, g_{v, \\lambda}))_{\\mathfrak q'_\\lambda}", "$$", "Choose a presentation", "$$", "(S')^{\\oplus s} \\to (S')^{\\oplus t} \\to M \\to 0", "$$", "of $M$ over $S'$. Let $A \\in \\text{Mat}(t \\times s, S')$ be", "the matrix of the presentation. For some $\\lambda_3 \\in \\Lambda$,", "$\\lambda_3 \\geq \\lambda_2$", "we can find a matrix $A_{\\lambda_3} \\in \\text{Mat}(t \\times s, S_{\\lambda_3})$", "which maps to $A$. For all $\\lambda \\geq \\lambda_3$ we let", "$M_\\lambda = \\Coker((S'_\\lambda)^{\\oplus s} \\xrightarrow{A_\\lambda}", "(S'_\\lambda)^{\\oplus t})$.", "\\medskip\\noindent", "With these choices, we have for each $\\lambda_3 \\leq \\lambda \\leq \\mu$", "that $S_\\lambda \\otimes_{R_{\\lambda}} R_\\mu \\to S_\\mu$ is a localization,", "$S'_\\lambda \\otimes_{S_{\\lambda}} S_\\mu \\to S'_\\mu$ is a localization, and", "the map $M_\\lambda \\otimes_{S'_\\lambda} S'_\\mu \\to M_\\mu$ is an", "isomorphism. This also implies that", "$S'_\\lambda \\otimes_{R_{\\lambda}} R_\\mu \\to S'_\\mu$ is a localization.", "Thus, since $M$ is flat over $R$ we see by", "Lemma \\ref{lemma-colimit-eventually-flat} that", "for all $\\lambda$ big enough the module $M_\\lambda$ is", "flat over $R_\\lambda$.", "Moreover, note that", "$", "\\mathfrak m = \\colim \\mathfrak p_\\lambda", "$,", "$", "S/\\mathfrak mS = \\colim S_\\lambda/\\mathfrak p_\\lambda S_\\lambda", "$,", "$", "S'/\\mathfrak mS' = \\colim S'_\\lambda/\\mathfrak p_\\lambda S'_\\lambda", "$,", "and", "$", "M/\\mathfrak mM = \\colim M_\\lambda/\\mathfrak p_\\lambda M_\\lambda", "$. Also, for each $\\lambda_3 \\leq \\lambda \\leq \\mu$ we see (from the", "properties listed above) that", "$$", "S'_\\lambda/\\mathfrak p_\\lambda S'_\\lambda", "\\otimes_{S_{\\lambda}/\\mathfrak p_\\lambda S_\\lambda}", "S_\\mu/\\mathfrak p_\\mu S_\\mu", "\\longrightarrow", "S'_\\mu/\\mathfrak p_\\mu S'_\\mu", "$$", "is a localization, and the map", "$$", "M_\\lambda / \\mathfrak p_\\lambda M_\\lambda", "\\otimes_{S'_\\lambda/\\mathfrak p_\\lambda S'_\\lambda}", "S'_\\mu /\\mathfrak p_\\mu S'_\\mu", "\\longrightarrow", "M_\\mu/\\mathfrak p_\\mu M_\\mu", "$$", "is an isomorphism. Hence the system", "$(S_\\lambda/\\mathfrak p_\\lambda S_\\lambda \\to", "S'_\\lambda/\\mathfrak p_\\lambda S'_\\lambda,", "M_\\lambda/\\mathfrak p_\\lambda M_\\lambda)$", "is a system as in", "Lemma \\ref{lemma-limit-module-essentially-finite-presentation} as well.", "We may apply Lemma \\ref{lemma-colimit-eventually-flat} again because", "$M/\\mathfrak m M$ is assumed flat over $S/\\mathfrak mS$ and we see that", "$M_\\lambda/\\mathfrak p_\\lambda M_\\lambda$ is flat over", "$S_\\lambda/\\mathfrak p_\\lambda S_\\lambda$ for all $\\lambda$ big enough.", "Thus for $\\lambda$ big enough the data", "$R_\\lambda \\to S_\\lambda \\to S'_\\lambda, M_\\lambda$ satisfies", "the hypotheses of Lemma \\ref{lemma-criterion-flatness-fibre-Noetherian}.", "Pick such a $\\lambda$. Then $S = S_\\lambda \\otimes_{R_\\lambda} R$", "is flat over $R$, and $M = M_\\lambda \\otimes_{S_\\lambda} S$", "is flat over $S$ (since the base change of a flat module is flat)." ], "refs": [ "algebra-lemma-limit-essentially-finite-presentation", "algebra-lemma-colimit-eventually-flat", "algebra-lemma-limit-module-essentially-finite-presentation", "algebra-lemma-colimit-eventually-flat", "algebra-lemma-criterion-flatness-fibre-Noetherian" ], "ref_ids": [ 1100, 1109, 1101, 1109, 897 ] } ], "ref_ids": [] }, { "id": 1115, "type": "theorem", "label": "algebra-lemma-criterion-flatness-fibre-fp-over-ft", "categories": [ "algebra" ], "title": "algebra-lemma-criterion-flatness-fibre-fp-over-ft", "contents": [ "Let $R$, $S$, $S'$ be local rings and let $R \\to S \\to S'$ be local ring", "homomorphisms. Let $M$ be an $S'$-module. Let $\\mathfrak m \\subset R$", "be the maximal ideal. Assume", "\\begin{enumerate}", "\\item $R \\to S'$ is essentially of finite presentation,", "\\item $R \\to S$ is essentially of finite type,", "\\item $M$ is of finite presentation over $S'$,", "\\item $M$ is not zero,", "\\item $M/\\mathfrak mM$ is a flat $S/\\mathfrak mS$-module, and", "\\item $M$ is a flat $R$-module.", "\\end{enumerate}", "Then $S$ is essentially of finite presentation and flat over $R$", "and $M$ is a flat $S$-module." ], "refs": [], "proofs": [ { "contents": [ "As $S$ is essentially of finite presentation over $R$ we can write", "$S = C_{\\overline{\\mathfrak q}}$ for some finite type $R$-algebra $C$.", "Write $C = R[x_1, \\ldots, x_n]/I$. Denote", "$\\mathfrak q \\subset R[x_1, \\ldots, x_n]$ be the prime ideal corresponding", "to $\\overline{\\mathfrak q}$. Then we see that $S = B/J$ where", "$B = R[x_1, \\ldots, x_n]_{\\mathfrak q}$ is essentially of finite presentation", "over $R$ and $J = IB$. We can find $f_1, \\ldots, f_k \\in J$ such that", "the images $\\overline{f}_i \\in B/\\mathfrak mB$", "generate the image $\\overline{J}$ of $J$ in the Noetherian ring", "$B/\\mathfrak mB$. Hence there exist finitely generated ideals", "$J' \\subset J$ such that $B/J' \\to B/J$ induces an isomorphism", "$$", "(B/J') \\otimes_R R/\\mathfrak m \\longrightarrow", "B/J \\otimes_R R/\\mathfrak m = S/\\mathfrak mS.", "$$", "For any $J'$ as above we see that", "Lemma \\ref{lemma-criterion-flatness-fibre}", "applies to the ring maps", "$$", "R \\longrightarrow B/J' \\longrightarrow S'", "$$", "and the module $M$. Hence we conclude that $B/J'$ is flat over $R$", "for any choice $J'$ as above. Now, if $J' \\subset J' \\subset J$ are", "two finitely generated ideals as above, then we conclude that", "$B/J' \\to B/J''$ is a surjective map between flat $R$-algebras", "which are essentially of finite presentation which is an isomorphism", "modulo $\\mathfrak m$. Hence", "Lemma \\ref{lemma-mod-injective-general}", "implies that $B/J' = B/J''$, i.e., $J' = J''$. Clearly this means that", "$J$ is finitely generated, i.e., $S$ is essentially of finite presentation", "over $R$. Thus we may apply", "Lemma \\ref{lemma-criterion-flatness-fibre}", "to $R \\to S \\to S'$ and we win." ], "refs": [ "algebra-lemma-criterion-flatness-fibre", "algebra-lemma-mod-injective-general", "algebra-lemma-criterion-flatness-fibre" ], "ref_ids": [ 1114, 1110, 1114 ] } ], "ref_ids": [] }, { "id": 1116, "type": "theorem", "label": "algebra-lemma-criterion-flatness-fibre-locally-nilpotent", "categories": [ "algebra" ], "title": "algebra-lemma-criterion-flatness-fibre-locally-nilpotent", "contents": [ "Let", "$$", "\\xymatrix{", "S \\ar[rr] & & S' \\\\", "& R \\ar[lu] \\ar[ru]", "}", "$$", "be a commutative diagram in the category of rings.", "Let $I \\subset R$ be a locally nilpotent ideal and", "$M$ an $S'$-module. Assume", "\\begin{enumerate}", "\\item $R \\to S$ is of finite type,", "\\item $R \\to S'$ is of finite presentation,", "\\item $M$ is a finitely presented $S'$-module,", "\\item $M/IM$ is flat as a $S/IS$-module, and", "\\item $M$ is flat as an $R$-module.", "\\end{enumerate}", "Then $M$ is a flat $S$-module and $S_\\mathfrak q$ is flat", "and essentially of finite presentation over $R$", "for every $\\mathfrak q \\subset S$ such that", "$M \\otimes_S \\kappa(\\mathfrak q)$ is nonzero." ], "refs": [], "proofs": [ { "contents": [ "If $M \\otimes_S \\kappa(\\mathfrak q)$ is nonzero, then", "$S' \\otimes_S \\kappa(\\mathfrak q)$ is nonzero and hence", "there exists a prime $\\mathfrak q' \\subset S'$ lying over", "$\\mathfrak q$ (Lemma \\ref{lemma-in-image}). Let", "$\\mathfrak p \\subset R$ be the image of $\\mathfrak q$ in $\\Spec(R)$.", "Then $I \\subset \\mathfrak p$ as $I$ is locally nilpotent", "hence $M/\\mathfrak p M$ is flat over $S/\\mathfrak pS$.", "Hence we may apply Lemma \\ref{lemma-criterion-flatness-fibre-fp-over-ft}", "to $R_\\mathfrak p \\to S_\\mathfrak q \\to S'_{\\mathfrak q'}$", "and $M_{\\mathfrak q'}$. We conclude that $M_{\\mathfrak q'}$", "is flat over $S$ and $S_\\mathfrak q$ is flat and essentially", "of finite presentation over $R$.", "Since $\\mathfrak q'$ was an arbitrary prime of $S'$ we also", "see that $M$ is flat over $S$ (Lemma \\ref{lemma-flat-localization})." ], "refs": [ "algebra-lemma-in-image", "algebra-lemma-criterion-flatness-fibre-fp-over-ft", "algebra-lemma-flat-localization" ], "ref_ids": [ 394, 1115, 538 ] } ], "ref_ids": [] }, { "id": 1117, "type": "theorem", "label": "algebra-lemma-CM-dim-finite-type", "categories": [ "algebra" ], "title": "algebra-lemma-CM-dim-finite-type", "contents": [ "Let $k$ be a field. Let $S$ be a finite type", "$k$-algebra. Let $f_1, \\ldots, f_i$ be elements", "of $S$. Assume that $S$ is Cohen-Macaulay and", "equidimensional of dimension $d$, and that", "$\\dim V(f_1, \\ldots, f_i) \\leq d - i$. Then equality", "holds and $f_1, \\ldots, f_i$ forms a regular", "sequence in $S_{\\mathfrak q}$ for every prime $\\mathfrak q$", "of $V(f_1, \\ldots, f_i)$." ], "refs": [], "proofs": [ { "contents": [ "If $S$ is Cohen-Macaulay and equidimensional of dimension", "$d$, then we have $\\dim(S_{\\mathfrak m}) = d$ for all maximal", "ideals $\\mathfrak m$ of $S$, see", "Lemma \\ref{lemma-disjoint-decomposition-CM-algebra}.", "By Proposition \\ref{proposition-CM-module} we see that", "for all maximal ideals $\\mathfrak m \\in V(f_1, \\ldots, f_i)$", "the sequence is a regular sequence in $S_{\\mathfrak m}$ and", "the local ring $S_{\\mathfrak m}/(f_1, \\ldots, f_i)$ is", "Cohen-Macaulay of dimension $d - i$. This actually", "means that $S/(f_1, \\ldots, f_i)$ is Cohen-Macaulay", "and equidimensional of dimension $d - i$." ], "refs": [ "algebra-lemma-disjoint-decomposition-CM-algebra", "algebra-proposition-CM-module" ], "ref_ids": [ 997, 1420 ] } ], "ref_ids": [] }, { "id": 1118, "type": "theorem", "label": "algebra-lemma-open-regular-sequence", "categories": [ "algebra" ], "title": "algebra-lemma-open-regular-sequence", "contents": [ "Suppose that $R \\to S$ is a ring map which is", "finite type, flat. Let $d$ be an integer", "such that all fibres", "$S \\otimes_R \\kappa(\\mathfrak p)$ are", "Cohen-Macaulay and equidimensional", "of dimension $d$. Let $f_1, \\ldots, f_i$", "be elements of $S$. The set", "$$", "\\{ \\mathfrak q \\in V(f_1, \\ldots, f_i)", "\\mid f_1, \\ldots, f_i", "\\text{ are a regular sequence in }", "S_{\\mathfrak q}/\\mathfrak p S_{\\mathfrak q}", "\\text{ where }\\mathfrak p = R \\cap \\mathfrak q", "\\}", "$$", "is open in $V(f_1, \\ldots, f_i)$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\overline{S} = S/(f_1, \\ldots, f_i)$.", "Suppose $\\mathfrak q$ is an element of the set defined in the", "lemma, and $\\mathfrak p$ is the corresponding prime of $R$.", "We will use relative dimension as defined in", "Definition \\ref{definition-relative-dimension}.", "First, note that $d = \\dim_{\\mathfrak q}(S/R) =", "\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) +", "\\text{trdeg}_{\\kappa(\\mathfrak p)}\\ \\kappa(\\mathfrak q)$", "by Lemma \\ref{lemma-dimension-at-a-point-finite-type-field}.", "Since $f_1, \\ldots, f_i$ form a regular sequence in the", "Noetherian local ring $S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}$", "general dimension theory tells us that", "$\\dim(\\overline{S}_{\\mathfrak q}/\\mathfrak p\\overline{S}_{\\mathfrak q})", "= \\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) - i$.", "By the same Lemma \\ref{lemma-dimension-at-a-point-finite-type-field}", "we then conclude that $\\dim_{\\mathfrak q}(\\overline{S}/R)", "= \\dim(\\overline{S}_{\\mathfrak q}/\\mathfrak p\\overline{S}_{\\mathfrak q}) +", "\\text{trdeg}_{\\kappa(\\mathfrak p)}\\ \\kappa(\\mathfrak q)", "= d - i$. By Lemma", "\\ref{lemma-dimension-fibres-bounded-open-upstairs}", "we have $\\dim_{\\mathfrak q'}(\\overline{S}/R) \\leq d - i$", "for all $\\mathfrak q' \\in V(f_1, \\ldots, f_i) = \\Spec(\\overline{S})$", "in a neighbourhood of $\\mathfrak q$. Thus after replacing", "$S$ by $S_g$ for some $g \\in S$, $g \\not \\in \\mathfrak q$", "we may assume that the inequality holds for all", "$\\mathfrak q'$. The result follows from Lemma", "\\ref{lemma-CM-dim-finite-type}." ], "refs": [ "algebra-definition-relative-dimension", "algebra-lemma-dimension-at-a-point-finite-type-field", "algebra-lemma-dimension-at-a-point-finite-type-field", "algebra-lemma-dimension-fibres-bounded-open-upstairs", "algebra-lemma-CM-dim-finite-type" ], "ref_ids": [ 1524, 1007, 1007, 1075, 1117 ] } ], "ref_ids": [] }, { "id": 1119, "type": "theorem", "label": "algebra-lemma-exact-on-fibres-open", "categories": [ "algebra" ], "title": "algebra-lemma-exact-on-fibres-open", "contents": [ "Let $R \\to S$ is a ring map.", "Consider a finite homological complex of", "finite free $S$-modules:", "$$", "F_{\\bullet} :", "0", "\\to", "S^{n_e}", "\\xrightarrow{\\varphi_e}", "S^{n_{e-1}}", "\\xrightarrow{\\varphi_{e-1}}", "\\ldots", "\\xrightarrow{\\varphi_{i + 1}}", "S^{n_i}", "\\xrightarrow{\\varphi_i}", "S^{n_{i-1}}", "\\xrightarrow{\\varphi_{i-1}}", "\\ldots", "\\xrightarrow{\\varphi_1}", "S^{n_0}", "$$", "For every prime $\\mathfrak q$ of $S$ consider the", "complex $\\overline{F}_{\\bullet, \\mathfrak q} =", "F_{\\bullet, \\mathfrak q} \\otimes_R \\kappa(\\mathfrak p)$", "where $\\mathfrak p$ is inverse image of $\\mathfrak q$ in $R$.", "Assume there exists an integer $d$ such", "that $R \\to S$ is finite type, flat", "with fibres $S \\otimes_R \\kappa(\\mathfrak p)$", "Cohen-Macaulay of dimension $d$.", "The set", "$$", "\\{\\mathfrak q \\in \\Spec(S) \\mid", "\\overline{F}_{\\bullet, \\mathfrak q}\\text{ is exact}\\}", "$$", "is open in $\\Spec(S)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q$ be an element of the set defined in the lemma.", "We are going to use Proposition \\ref{proposition-what-exact}", "to show there exists a $g \\in S$, $g \\not \\in \\mathfrak q$", "such that $D(g)$ is contained in the set defined in the lemma.", "In other words, we are going to show that after replacing $S$", "by $S_g$, the set of the lemma is all of $\\Spec(S)$.", "Thus during the proof we will, finitely often, replace", "$S$ by such a localization.", "Recall that Proposition \\ref{proposition-what-exact}", "characterizes exactness of complexes", "in terms of ranks of the maps $\\varphi_i$ and the ideals", "$I(\\varphi_i)$, in case the ring is local. We first address", "the rank condition. Set", "$r_i = n_i - n_{i + 1} + \\ldots + (-1)^{e - i} n_e$.", "Note that $r_i + r_{i + 1} = n_i$ and note that", "$r_i$ is the expected rank of $\\varphi_i$ (in the", "exact case).", "\\medskip\\noindent", "By Lemma \\ref{lemma-complex-exact-mod} we see that if", "$\\overline{F}_{\\bullet, \\mathfrak q}$ is exact, then", "the localization $F_{\\bullet, \\mathfrak q}$ is exact.", "In particular the complex $F_\\bullet$ becomes", "exact after localizing by an element", "$g \\in S$, $g \\not \\in \\mathfrak q$. In this case", "Proposition \\ref{proposition-what-exact} applied", "to all localizations of $S$ at prime ideals", "implies that all $(r_i + 1) \\times (r_i + 1)$-minors", "of $\\varphi_i$ are zero. Thus we see that the rank", "of $\\varphi_i$ is at most $r_i$.", "\\medskip\\noindent", "Let $I_i \\subset S$ denote the ideal generated", "by the $r_i \\times r_i$-minors of the matrix", "of $\\varphi_i$. By Proposition \\ref{proposition-what-exact}", "the complex $\\overline{F}_{\\bullet, \\mathfrak q}$ is exact", "if and only if for every $1 \\leq i \\leq e$ we have", "either $(I_i)_{\\mathfrak q} = S_{\\mathfrak q}$ or", "$(I_i)_{\\mathfrak q}$ contains a $S_{\\mathfrak q}/\\mathfrak p", "S_{\\mathfrak q}$-regular sequence of length $i$.", "Namely, by our choice of $r_i$ above and by the", "bound on the ranks of the $\\varphi_i$ this is the", "only way the conditions of Proposition \\ref{proposition-what-exact}", "can be satisfied.", "\\medskip\\noindent", "If $(I_i)_{\\mathfrak q} = S_{\\mathfrak q}$, then after localizing $S$ at", "some element $g \\not\\in \\mathfrak q$ we may assume that", "$I_i = S$. Clearly, this is an open condition.", "\\medskip\\noindent", "If $(I_i)_{\\mathfrak q} \\not = S_{\\mathfrak q}$, then we have", "a sequence $f_1, \\ldots, f_i \\in (I_i)_{\\mathfrak q}$ which", "form a regular sequence in $S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}$.", "Note that for any prime $\\mathfrak q' \\subset S$ such that", "$(f_1, \\ldots, f_i) \\not \\subset \\mathfrak q'$ we have", "$(I_i)_{\\mathfrak q'} = S_{\\mathfrak q'}$.", "Thus the result follows from Lemma \\ref{lemma-open-regular-sequence}." ], "refs": [ "algebra-proposition-what-exact", "algebra-proposition-what-exact", "algebra-lemma-complex-exact-mod", "algebra-proposition-what-exact", "algebra-proposition-what-exact", "algebra-proposition-what-exact", "algebra-lemma-open-regular-sequence" ], "ref_ids": [ 1419, 1419, 887, 1419, 1419, 1419, 1118 ] } ], "ref_ids": [] }, { "id": 1120, "type": "theorem", "label": "algebra-lemma-where-CM", "categories": [ "algebra" ], "title": "algebra-lemma-where-CM", "contents": [ "Let $S$ be a finite type algebra over a field $k$.", "Let $\\varphi : k[y_1, \\ldots, y_d] \\to S$ be a quasi-finite ring map.", "As subsets of $\\Spec(S)$ we have", "$$", "\\{ \\mathfrak q \\mid", "S_{\\mathfrak q} \\text{ flat over }k[y_1, \\ldots, y_d]\\}", "=", "\\{ \\mathfrak q \\mid", "S_{\\mathfrak q} \\text{ CM and }\\dim_{\\mathfrak q}(S/k) = d\\}", "$$", "For notation see Definition \\ref{definition-relative-dimension}." ], "refs": [ "algebra-definition-relative-dimension" ], "proofs": [ { "contents": [ "Let $\\mathfrak q \\subset S$ be a prime. Denote", "$\\mathfrak p = k[y_1, \\ldots, y_d] \\cap \\mathfrak q$.", "Note that always", "$\\dim(S_{\\mathfrak q}) \\leq \\dim(k[y_1, \\ldots, y_d]_{\\mathfrak p})$", "by Lemma \\ref{lemma-dimension-inequality-quasi-finite} for example.", "Moreover, the field extension $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$", "is finite and hence", "$\\text{trdeg}_k(\\kappa(\\mathfrak p)) = \\text{trdeg}_k(\\kappa(\\mathfrak q))$.", "\\medskip\\noindent", "Let $\\mathfrak q$ be an element of the left hand side.", "Then Lemma \\ref{lemma-finite-flat-over-regular-CM} applies", "and we conclude that $S_{\\mathfrak q}$ is Cohen-Macaulay", "and $\\dim(S_{\\mathfrak q}) = \\dim(k[y_1, \\ldots, y_d]_{\\mathfrak p})$.", "Combined with the equality of transcendence degrees above and", "Lemma \\ref{lemma-dimension-at-a-point-finite-type-field} this", "implies that $\\dim_{\\mathfrak q}(S/k) = d$. Hence $\\mathfrak q$", "is an element of the right hand side.", "\\medskip\\noindent", "Let $\\mathfrak q$ be an element of the right hand side.", "By the equality of transcendence degrees above, the assumption", "that $\\dim_{\\mathfrak q}(S/k) = d$ and", "Lemma \\ref{lemma-dimension-at-a-point-finite-type-field}", "we conclude that", "$\\dim(S_{\\mathfrak q}) = \\dim(k[y_1, \\ldots, y_d]_{\\mathfrak p})$.", "Hence Lemma \\ref{lemma-CM-over-regular-flat}", "applies and we see that $\\mathfrak q$ is an", "element of the left hand side." ], "refs": [ "algebra-lemma-dimension-inequality-quasi-finite", "algebra-lemma-finite-flat-over-regular-CM", "algebra-lemma-dimension-at-a-point-finite-type-field", "algebra-lemma-dimension-at-a-point-finite-type-field", "algebra-lemma-CM-over-regular-flat" ], "ref_ids": [ 1073, 989, 1007, 1007, 1107 ] } ], "ref_ids": [ 1524 ] }, { "id": 1121, "type": "theorem", "label": "algebra-lemma-finite-type-over-field-CM-open", "categories": [ "algebra" ], "title": "algebra-lemma-finite-type-over-field-CM-open", "contents": [ "Let $S$ be a finite type algebra over a field $k$.", "The set of primes $\\mathfrak q$ such that $S_{\\mathfrak q}$ is", "Cohen-Macaulay is open in $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q \\subset S$ be a prime such that $S_{\\mathfrak q}$ is", "Cohen-Macaulay. We have to show there exists a", "$g \\in S$, $g \\not \\in \\mathfrak q$ such that the ring", "$S_g$ is Cohen-Macaulay. For any $g \\in S$, $g \\not \\in \\mathfrak q$", "we may replace $S$ by $S_g$ and $\\mathfrak q$ by $\\mathfrak qS_g$.", "Combining this with", "Lemmas \\ref{lemma-Noether-normalization-at-point} and", "\\ref{lemma-dimension-at-a-point-finite-type-field}", "we may assume that there exists a finite injective", "ring map $k[y_1, \\ldots, y_d] \\to S$ with", "$d = \\dim(S_{\\mathfrak q}) + \\text{trdeg}_k(\\kappa(\\mathfrak q))$.", "Set $\\mathfrak p = k[y_1, \\ldots, y_d] \\cap \\mathfrak q$.", "By construction we see that $\\mathfrak q$ is an element of", "the right hand side of the displayed equality of", "Lemma \\ref{lemma-where-CM}. Hence it is also an element of", "the left hand side.", "\\medskip\\noindent", "By Theorem \\ref{theorem-openness-flatness} we see that for some $g \\in S$,", "$g \\not \\in \\mathfrak q$ the ring $S_g$ is flat over $k[y_1, \\ldots, y_d]$.", "Hence by the equality of Lemma \\ref{lemma-where-CM} again we conclude that", "all local rings of $S_g$ are Cohen-Macaulay as desired." ], "refs": [ "algebra-lemma-Noether-normalization-at-point", "algebra-lemma-dimension-at-a-point-finite-type-field", "algebra-lemma-where-CM", "algebra-theorem-openness-flatness", "algebra-lemma-where-CM" ], "ref_ids": [ 1002, 1007, 1120, 326, 1120 ] } ], "ref_ids": [] }, { "id": 1122, "type": "theorem", "label": "algebra-lemma-generic-CM", "categories": [ "algebra" ], "title": "algebra-lemma-generic-CM", "contents": [ "Let $k$ be a field. Let $S$ be a finite type $k$ algebra.", "The set of Cohen-Macaulay primes forms a dense open", "$U \\subset \\Spec(S)$." ], "refs": [], "proofs": [ { "contents": [ "The set is open by Lemma \\ref{lemma-finite-type-over-field-CM-open}.", "It contains all minimal primes $\\mathfrak q \\subset S$", "since the local ring at a minimal prime $S_{\\mathfrak q}$", "has dimension zero and hence is Cohen-Macaulay." ], "refs": [ "algebra-lemma-finite-type-over-field-CM-open" ], "ref_ids": [ 1121 ] } ], "ref_ids": [] }, { "id": 1123, "type": "theorem", "label": "algebra-lemma-dim-not-zero-exists-nonzerodivisor-nonunit", "categories": [ "algebra" ], "title": "algebra-lemma-dim-not-zero-exists-nonzerodivisor-nonunit", "contents": [ "Let $k$ be a field. Let $S$ be a finite type $k$ algebra.", "If $\\dim(S) > 0$, then there exists an element $f \\in S$", "which is a nonzerodivisor and a nonunit." ], "refs": [], "proofs": [ { "contents": [ "Let $I \\subset S$ be the radical ideal such that $V(I) \\subset \\Spec(S)$", "is the set of primes $\\mathfrak q \\subset S$ with $S_\\mathfrak q$", "not Cohen-Macaulay. See Lemma \\ref{lemma-generic-CM} which also", "tells us that $V(I)$ is nowhere dense in $\\Spec(S)$.", "Let $\\mathfrak m \\subset S$ be a maximal ideal such that", "$\\dim(S_\\mathfrak m) > 0$ and $\\mathfrak m \\not \\in V(I)$.", "Such a maximal ideal exists as $\\dim(S) > 0$ using the", "Hilbert Nullstellensatz (Theorem \\ref{theorem-nullstellensatz}) and", "Lemma \\ref{lemma-dimension-at-a-point-finite-type-over-field}", "which implies that any dense open of", "$\\Spec(S)$ has the same dimension as $\\Spec(S)$.", "Finally, let $\\mathfrak q_1, \\ldots, \\mathfrak q_m$ be the minimal", "primes of $S$. Choose $f \\in S$ with", "$$", "f \\equiv 1 \\bmod I,\\quad", "f \\in \\mathfrak m,\\quad", "f \\not \\in \\bigcup \\mathfrak q_i", "$$", "This is possible by Lemma \\ref{lemma-silly-silly}. Namely, we have", "$S/(I \\cap \\mathfrak m) = S/I \\times S/\\mathfrak m$ by", "Lemma \\ref{lemma-chinese-remainder}. Thus we can first choose", "$g \\in S$ such that $g \\equiv 1 \\bmod I$ and $g \\in \\mathfrak m$.", "Then $g + (I \\cap \\mathfrak m) \\not \\subset \\mathfrak q_i$", "since $V(I \\cap \\mathfrak m) \\not \\supset V(\\mathfrak q_i)$.", "Hence the lemma applies. Clearly $f$ is not a unit.", "To show that $f$ is a nonzerodivisor, it suffices to", "prove that $f : S_\\mathfrak q \\to S_\\mathfrak q$ is", "injective for every prime ideal $\\mathfrak q \\subset S$.", "If $S_\\mathfrak q$ is not Cohen-Macaulay, then $\\mathfrak q \\in V(I)$", "and $f$ maps to a unit of $S_\\mathfrak q$. On the other hand, if", "$S_\\mathfrak q$ is Cohen-Macaulay, then we use that", "$\\dim(S_\\mathfrak q/fS_\\mathfrak q) < \\dim(S_\\mathfrak q)$", "by the requirement $f \\not \\in \\mathfrak q_i$ and we conclude", "that $f$ is a nonzerodivisor in $S_\\mathfrak q$ by", "Lemma \\ref{lemma-reformulate-CM}." ], "refs": [ "algebra-lemma-generic-CM", "algebra-theorem-nullstellensatz", "algebra-lemma-dimension-at-a-point-finite-type-over-field", "algebra-lemma-silly-silly", "algebra-lemma-chinese-remainder", "algebra-lemma-reformulate-CM" ], "ref_ids": [ 1122, 316, 995, 379, 380, 923 ] } ], "ref_ids": [] }, { "id": 1124, "type": "theorem", "label": "algebra-lemma-finite-presentation-flat-CM-locus-open", "categories": [ "algebra" ], "title": "algebra-lemma-finite-presentation-flat-CM-locus-open", "contents": [ "Let $R$ be a ring. Let $R \\to S$ be of finite presentation", "and flat. For any $d \\geq 0$ the set", "$$", "\\left\\{", "\\begin{matrix}", "\\mathfrak q \\in \\Spec(S)", "\\text{ such that setting }\\mathfrak p = R \\cap \\mathfrak q", "\\text{ the fibre ring}\\\\", "S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}", "\\text{ is Cohen-Macaulay}", "\\text{ and } \\dim_{\\mathfrak q}(S/R) = d", "\\end{matrix}", "\\right\\}", "$$", "is open in $\\Spec(S)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q$ be an element of the set indicated, with", "$\\mathfrak p$ the corresponding prime of $R$.", "We have to find a $g \\in S$, $g \\not \\in \\mathfrak q$ such that", "all fibre rings of $R \\to S_g$ are Cohen-Macaulay.", "During the course of the proof we may (finitely many times)", "replace $S$ by $S_g$ for a $g \\in S$, $g \\not \\in \\mathfrak q$.", "Thus by Lemma \\ref{lemma-quasi-finite-over-polynomial-algebra}", "we may assume there is a quasi-finite ring map", "$R[t_1, \\ldots, t_d] \\to S$ with $d = \\dim_{\\mathfrak q}(S/R)$.", "Let $\\mathfrak q' = R[t_1, \\ldots, t_d] \\cap \\mathfrak q$.", "By Lemma \\ref{lemma-where-CM} we see that the ring map", "$$", "R[t_1, \\ldots, t_d]_{\\mathfrak q'} /", "\\mathfrak p R[t_1, \\ldots, t_d]_{\\mathfrak q'}", "\\longrightarrow", "S_{\\mathfrak q}/\\mathfrak p S_{\\mathfrak q}", "$$", "is flat. Hence by the crit\\`ere de platitude par fibres", "Lemma \\ref{lemma-criterion-flatness-fibre} we see that", "$R[t_1, \\ldots, t_d]_{\\mathfrak q'} \\to S_{\\mathfrak q}$ is flat.", "Hence by Theorem \\ref{theorem-openness-flatness} we see that", "for some $g \\in S$, $g \\not \\in \\mathfrak q$ the ring map", "$R[t_1, \\ldots, t_d] \\to S_g$ is flat. Replacing $S$ by $S_g$", "we see that for every prime $\\mathfrak r \\subset S$,", "setting $\\mathfrak r' = R[t_1, \\ldots, t_d] \\cap \\mathfrak r$", "and $\\mathfrak p' = R \\cap \\mathfrak r$", "the local ring map", "$R[t_1, \\ldots, t_d]_{\\mathfrak r'} \\to S_{\\mathfrak r}$ is flat.", "Hence also the base change", "$$", "R[t_1, \\ldots, t_d]_{\\mathfrak r'} /", "\\mathfrak p' R[t_1, \\ldots, t_d]_{\\mathfrak r'}", "\\longrightarrow", "S_{\\mathfrak r}/\\mathfrak p' S_{\\mathfrak r}", "$$", "is flat. Hence by Lemma \\ref{lemma-where-CM} applied with", "$k = \\kappa(\\mathfrak p')$ we see", "$\\mathfrak r$ is in the set of the lemma", "as desired." ], "refs": [ "algebra-lemma-quasi-finite-over-polynomial-algebra", "algebra-lemma-where-CM", "algebra-lemma-criterion-flatness-fibre", "algebra-theorem-openness-flatness", "algebra-lemma-where-CM" ], "ref_ids": [ 1071, 1120, 1114, 326, 1120 ] } ], "ref_ids": [] }, { "id": 1125, "type": "theorem", "label": "algebra-lemma-generic-CM-flat-finite-presentation", "categories": [ "algebra" ], "title": "algebra-lemma-generic-CM-flat-finite-presentation", "contents": [ "Let $R$ be a ring. Let $R \\to S$ be flat of finite presentation.", "The set of primes $\\mathfrak q$ such that the fibre ring", "$S_{\\mathfrak q} \\otimes_R \\kappa(\\mathfrak p)$,", "with $\\mathfrak p = R \\cap \\mathfrak q$ is Cohen-Macaulay", "is open and dense in every fibre of $\\Spec(S) \\to \\Spec(R)$." ], "refs": [], "proofs": [ { "contents": [ "The set, call it $W$, is open by", "Lemma \\ref{lemma-finite-presentation-flat-CM-locus-open}.", "It is dense in the fibres because the intersection of $W$", "with a fibre is the corresponding set of the fibre", "to which Lemma \\ref{lemma-generic-CM} applies." ], "refs": [ "algebra-lemma-finite-presentation-flat-CM-locus-open", "algebra-lemma-generic-CM" ], "ref_ids": [ 1124, 1122 ] } ], "ref_ids": [] }, { "id": 1126, "type": "theorem", "label": "algebra-lemma-extend-field-CM-locus", "categories": [ "algebra" ], "title": "algebra-lemma-extend-field-CM-locus", "contents": [ "Let $k$ be a field. Let $S$ be a finite type $k$-algebra.", "Let $k \\subset K$ be a field extension, and set $S_K = K \\otimes_k S$.", "Let $\\mathfrak q \\subset S$ be a prime of $S$.", "Let $\\mathfrak q_K \\subset S_K$ be a prime of $S_K$ lying", "over $\\mathfrak q$. Then $S_{\\mathfrak q}$ is Cohen-Macaulay", "if and only if $(S_K)_{\\mathfrak q_K}$ is Cohen-Macaulay." ], "refs": [], "proofs": [ { "contents": [ "During the course of the proof we may (finitely many times) replace", "$S$ by $S_g$ for any $g \\in S$, $g \\not \\in \\mathfrak q$. Hence", "using Lemma \\ref{lemma-Noether-normalization-at-point} we may", "assume that $\\dim(S) = \\dim_{\\mathfrak q}(S/k) =: d$ and", "find a finite injective map $k[x_1, \\ldots, x_d] \\to S$.", "Note that this also induces a finite injective map", "$K[x_1, \\ldots, x_d] \\to S_K$ by base change.", "By Lemma \\ref{lemma-dimension-at-a-point-preserved-field-extension}", "we have $\\dim_{\\mathfrak q_K}(S_K/K) = d$.", "Set $\\mathfrak p = k[x_1, \\ldots, x_d] \\cap \\mathfrak q$", "and $\\mathfrak p_K = K[x_1, \\ldots, x_d] \\cap \\mathfrak q_K$.", "Consider the following commutative diagram of Noetherian local", "rings", "$$", "\\xymatrix{", "S_{\\mathfrak q} \\ar[r] &", "(S_K)_{\\mathfrak q_K} \\\\", "k[x_1, \\ldots, x_d]_{\\mathfrak p} \\ar[r] \\ar[u] &", "K[x_1, \\ldots, x_d]_{\\mathfrak p_K} \\ar[u]", "}", "$$", "By Lemma \\ref{lemma-where-CM} we have to show that", "the left vertical arrow is flat if and only if the right", "vertical arrow is flat. Because the bottom arrow is flat", "this equivalence holds by Lemma \\ref{lemma-base-change-flat-up-down}." ], "refs": [ "algebra-lemma-Noether-normalization-at-point", "algebra-lemma-dimension-at-a-point-preserved-field-extension", "algebra-lemma-where-CM", "algebra-lemma-base-change-flat-up-down" ], "ref_ids": [ 1002, 1010, 1120, 898 ] } ], "ref_ids": [] }, { "id": 1127, "type": "theorem", "label": "algebra-lemma-CM-locus-commutes-base-change", "categories": [ "algebra" ], "title": "algebra-lemma-CM-locus-commutes-base-change", "contents": [ "Let $R$ be a ring. Let $R \\to S$ be of finite type.", "Let $R \\to R'$ be any ring map. Set $S' = R' \\otimes_R S$.", "Denote $f : \\Spec(S') \\to \\Spec(S)$ the map", "associated to the ring map $S \\to S'$.", "Set $W$ equal to the", "set of primes $\\mathfrak q$ such that the fibre ring", "$S_{\\mathfrak q} \\otimes_R \\kappa(\\mathfrak p)$,", "$\\mathfrak p = R \\cap \\mathfrak q$ is Cohen-Macaulay,", "and let $W'$ denote the analogue for $S'/R'$. Then", "$W' = f^{-1}(W)$." ], "refs": [], "proofs": [ { "contents": [ "Trivial from Lemma \\ref{lemma-extend-field-CM-locus} and the definitions." ], "refs": [ "algebra-lemma-extend-field-CM-locus" ], "ref_ids": [ 1126 ] } ], "ref_ids": [] }, { "id": 1128, "type": "theorem", "label": "algebra-lemma-relative-dimension-CM", "categories": [ "algebra" ], "title": "algebra-lemma-relative-dimension-CM", "contents": [ "Let $R$ be a ring. Let $R \\to S$ be a ring map which is (a) flat,", "(b) of finite presentation, (c) has Cohen-Macaulay fibres. Then we can write", "$S = S_0 \\times \\ldots \\times S_n$ as a product of $R$-algebras $S_d$", "such that each $S_d$ satisfies", "(a), (b), (c) and has all fibres equidimensional of dimension $d$." ], "refs": [], "proofs": [ { "contents": [ "For each integer $d$ denote $W_d \\subset \\Spec(S)$ the set", "defined in Lemma \\ref{lemma-finite-presentation-flat-CM-locus-open}.", "Clearly we have $\\Spec(S) = \\coprod W_d$, and each $W_d$", "is open by the lemma we just quoted. Hence the result follows", "from Lemma \\ref{lemma-disjoint-implies-product}." ], "refs": [ "algebra-lemma-finite-presentation-flat-CM-locus-open", "algebra-lemma-disjoint-implies-product" ], "ref_ids": [ 1124, 415 ] } ], "ref_ids": [] }, { "id": 1129, "type": "theorem", "label": "algebra-lemma-universal-omega", "categories": [ "algebra" ], "title": "algebra-lemma-universal-omega", "contents": [ "\\begin{slogan}", "Maps out of the module of differentials are the same as derivations.", "\\end{slogan}", "The module of differentials of $S$ over $R$ has the following", "universal property. The map", "$$", "\\Hom_S(\\Omega_{S/R}, M)", "\\longrightarrow", "\\text{Der}_R(S, M), \\quad", "\\alpha", "\\longmapsto", "\\alpha \\circ \\text{d}", "$$", "is an isomorphism of functors." ], "refs": [], "proofs": [ { "contents": [ "By definition an $R$-derivation is a rule which associates", "to each $a \\in S$ an element $D(a) \\in M$. Thus $D$ gives", "rise to a map $[D] : \\bigoplus S[a] \\to M$. However, the conditions", "of being an $R$-derivation exactly mean that $[D]$ annihilates", "the image of the map in the displayed presentation of", "$\\Omega_{S/R}$ above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1130, "type": "theorem", "label": "algebra-lemma-colimit-differentials", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-differentials", "contents": [ "Let $I$ be a directed set.", "Let $(R_i \\to S_i, \\varphi_{ii'})$ be a system of", "ring maps over $I$, see", "Categories, Section \\ref{categories-section-posets-limits}.", "Then we have", "$$", "\\Omega_{S/R} =", "\\colim_i \\Omega_{S_i/R_i}.", "$$", "where $R \\to S = \\colim (R_i \\to S_i)$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the presentation of $\\Omega_{S/R}$ given above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1131, "type": "theorem", "label": "algebra-lemma-trivial-differential-surjective", "categories": [ "algebra" ], "title": "algebra-lemma-trivial-differential-surjective", "contents": [ "Suppose that $R \\to S$ is surjective.", "Then $\\Omega_{S/R} = 0$." ], "refs": [], "proofs": [ { "contents": [ "You can see this either because all $R$-derivations", "clearly have to be zero, or because", "the map in the presentation of $\\Omega_{S/R}$ is surjective." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1132, "type": "theorem", "label": "algebra-lemma-differential-surjective", "categories": [ "algebra" ], "title": "algebra-lemma-differential-surjective", "contents": [ "In diagram (\\ref{equation-functorial-omega}), suppose", "that $S \\to S'$ is surjective with kernel $I \\subset S$.", "Then $\\Omega_{S/R} \\to \\Omega_{S'/R'}$ is surjective with", "kernel generated as an $S$-module by the elements", "$\\text{d}a$, where $a \\in S$ is such that $\\varphi(a) \\in \\beta(R')$.", "(This includes in particular the elements $\\text{d}(i)$, $i \\in I$.)" ], "refs": [], "proofs": [ { "contents": [ "Consider the map of presentations above. Clearly the right vertical", "map of free modules is surjective. Thus the map is surjective.", "A diagram chase shows that the following elements generate", "the kernel as an $S$-module for sure: $i\\text{d}a, i\\in I, a \\in S$,", "and $\\text{d}a$, with $a \\in S$ such that", "$\\varphi(a) = \\beta(r')$ for some $r' \\in R'$.", "Note that $\\varphi(i) = \\varphi(ia) = 0 = \\beta(0)$, and that", "$\\text{d}(ia) = i\\text{d}a + a \\text{d}i$.", "Hence $i\\text{d}a = \\text{d}(ia) - a \\text{d}i$ is", "an $S$-linear combination of elements of the second kind." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1133, "type": "theorem", "label": "algebra-lemma-exact-sequence-differentials", "categories": [ "algebra" ], "title": "algebra-lemma-exact-sequence-differentials", "contents": [ "Let $A \\to B \\to C$ be ring maps.", "Then there is a canonical exact sequence", "$$", "C \\otimes_B \\Omega_{B/A} \\to", "\\Omega_{C/A} \\to", "\\Omega_{C/B} \\to 0", "$$", "of $C$-modules." ], "refs": [], "proofs": [ { "contents": [ "We get a diagram (\\ref{equation-functorial-omega}) by putting", "$R = A$, $S = C$, $R' = B$, and $S' = C$.", "By Lemma \\ref{lemma-differential-surjective} the map", "$\\Omega_{C/A} \\to \\Omega_{C/B}$ is surjective, and the kernel", "is generated by the elements $\\text{d}(c)$, where $c \\in C$", "is in the image of $B \\to C$. The lemma follows." ], "refs": [ "algebra-lemma-differential-surjective" ], "ref_ids": [ 1132 ] } ], "ref_ids": [] }, { "id": 1134, "type": "theorem", "label": "algebra-lemma-differentials-localize", "categories": [ "algebra" ], "title": "algebra-lemma-differentials-localize", "contents": [ "Let $\\varphi : A \\to B$ be a ring map.", "\\begin{enumerate}", "\\item If $S \\subset A$ is a multiplicative subset mapping to", "invertible elements of $B$, then $\\Omega_{B/A} = \\Omega_{B/S^{-1}A}$.", "\\item If $S \\subset B$ is a multiplicative subset then", "$S^{-1}\\Omega_{B/A} = \\Omega_{S^{-1}B/A}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To show the equality of (1) it is enough to show that any", "$A$-derivation $D : B \\to M$ annihilates the elements $\\varphi(s)^{-1}$.", "This is clear from the Leibniz rule applied to", "$1 = \\varphi(s) \\varphi(s)^{-1}$.", "To show (2) note that there is an obvious map", "$S^{-1}\\Omega_{B/A} \\to \\Omega_{S^{-1}B/A}$.", "To show it is an isomorphism it is enough to show that", "there is a $A$-derivation $\\text{d}'$ of $S^{-1}B$ into $S^{-1}\\Omega_{B/A}$.", "To define it we simply set", "$\\text{d}'(b/s) = (1/s)\\text{d}b - (1/s^2)b\\text{d}s$.", "Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1135, "type": "theorem", "label": "algebra-lemma-differential-seq", "categories": [ "algebra" ], "title": "algebra-lemma-differential-seq", "contents": [ "In diagram (\\ref{equation-functorial-omega}),", "suppose that $S \\to S'$ is surjective with kernel $I \\subset S$,", "and assume that $R' = R$.", "Then there is a canonical exact sequence of $S'$-modules", "$$", "I/I^2", "\\longrightarrow", "\\Omega_{S/R} \\otimes_S S'", "\\longrightarrow", "\\Omega_{S'/R}", "\\longrightarrow", "0", "$$", "The leftmost map is characterized by the rule that", "$f \\in I$ maps to $\\text{d}f \\otimes 1$." ], "refs": [], "proofs": [ { "contents": [ "The middle term is $\\Omega_{S/R} \\otimes_S S/I$.", "For $f \\in I$ denote $\\overline{f}$ the image of $f$ in $I/I^2$.", "To show that the map $\\overline{f} \\mapsto \\text{d}f \\otimes 1$", "is well defined we just have to check that", "$\\text{d} f_1f_2 \\otimes 1 = 0$ if $f_1, f_2 \\in I$.", "And this is clear from the Leibniz rule", "$\\text{d} f_1f_2 \\otimes 1 = (f_1 \\text{d}f_2 + f_2 \\text{d} f_1 )\\otimes 1 =", "\\text{d}f_2 \\otimes f_1 + \\text{d}f_1 \\otimes f_2 = 0$.", "A similar computation show this map is $S' = S/I$-linear.", "\\medskip\\noindent", "The map $\\Omega_{S/R} \\otimes_S S' \\to \\Omega_{S'/R}$", "is the canonical $S'$-linear map associated to the", "$S$-linear map $\\Omega_{S/R} \\to \\Omega_{S'/R}$.", "It is surjective because $\\Omega_{S/R} \\to \\Omega_{S'/R}$", "is surjective by Lemma \\ref{lemma-differential-surjective}.", "\\medskip\\noindent", "The composite of the two maps is zero because", "$\\text{d}f$ maps to zero in $\\Omega_{S'/R}$", "for $f \\in I$. Note that exactness just says that", "the kernel of $\\Omega_{S/R} \\to \\Omega_{S'/R}$", "is generated as an $S$-submodule by the submodule $I\\Omega_{S/R}$ together", "with the elements $\\text{d}f$, with $f \\in I$. We know by", "Lemma \\ref{lemma-differential-surjective}", "that this kernel is generated by the elements $\\text{d}(a)$", "where $\\varphi(a) = \\beta(r)$ for some $r \\in R$.", "But then $a = \\alpha(r) + a - \\alpha(r)$, so", "$\\text{d}(a) = \\text{d}(a - \\alpha(r))$. And", "$a - \\alpha(r) \\in I$ since $\\varphi(a - \\alpha(r)) =", "\\varphi(a) - \\varphi(\\alpha(r)) = \\beta(r) - \\beta(r) = 0$.", "We conclude the elements $\\text{d}f$ with $f \\in I$ already", "generate the kernel as an $S$-module, as desired." ], "refs": [ "algebra-lemma-differential-surjective", "algebra-lemma-differential-surjective" ], "ref_ids": [ 1132, 1132 ] } ], "ref_ids": [] }, { "id": 1136, "type": "theorem", "label": "algebra-lemma-differential-seq-split", "categories": [ "algebra" ], "title": "algebra-lemma-differential-seq-split", "contents": [ "In diagram (\\ref{equation-functorial-omega}),", "suppose that $S \\to S'$ is surjective with kernel $I \\subset S$,", "and assume that $R' = R$. Moreover, assume that there exists", "an $R$-algebra map $S' \\to S$ which is a right inverse to", "$S \\to S'$. Then the exact sequence of $S'$-modules", "of Lemma \\ref{lemma-differential-seq} turns into a short exact sequence", "$$", "0 \\longrightarrow", "I/I^2", "\\longrightarrow", "\\Omega_{S/R} \\otimes_S S'", "\\longrightarrow", "\\Omega_{S'/R}", "\\longrightarrow", "0", "$$", "which is even a split short exact sequence." ], "refs": [ "algebra-lemma-differential-seq" ], "proofs": [ { "contents": [ "Let $\\beta : S' \\to S$ be the right inverse to the surjection", "$\\alpha : S \\to S'$, so $S = I \\oplus \\beta(S')$.", "Clearly we can use $\\beta : \\Omega_{S'/R} \\to \\Omega_{S/R}$,", "to get a right inverse to the map $\\Omega_{S/R} \\otimes_S S' \\to \\Omega_{S'/R}$.", "On the other hand, consider the map", "$$", "D : S \\longrightarrow I/I^2,", "\\quad", "x \\longmapsto x - \\beta(\\alpha(x))", "$$", "It is easy to show that $D$ is an $R$-derivation (omitted).", "Moreover $x D(s) = 0$ if $x \\in I, s \\in S$. Hence, by the universal property", "$D$ induces a map $\\tau : \\Omega_{S/R} \\otimes_S S' \\to I/I^2$.", "We omit the verification that it is a left inverse to", "$\\text{d} : I/I^2 \\to \\Omega_{S/R} \\otimes_S S'$. Hence we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1135 ] }, { "id": 1137, "type": "theorem", "label": "algebra-lemma-differential-mod-power-ideal", "categories": [ "algebra" ], "title": "algebra-lemma-differential-mod-power-ideal", "contents": [ "Let $R \\to S$ be a ring map. Let $I \\subset S$ be an ideal.", "Let $n \\geq 1$ be an integer. Set $S' = S/I^{n + 1}$.", "The map $\\Omega_{S/R} \\to \\Omega_{S'/R}$ induces an", "isomorphism", "$$", "\\Omega_{S/R} \\otimes_S S/I^n", "\\longrightarrow", "\\Omega_{S'/R} \\otimes_{S'} S/I^n.", "$$" ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-differential-seq} and the fact that", "$\\text{d}(I^{n + 1}) \\subset I^n\\Omega_{S/R}$ by the", "Leibniz rule for $\\text{d}$." ], "refs": [ "algebra-lemma-differential-seq" ], "ref_ids": [ 1135 ] } ], "ref_ids": [] }, { "id": 1138, "type": "theorem", "label": "algebra-lemma-differentials-base-change", "categories": [ "algebra" ], "title": "algebra-lemma-differentials-base-change", "contents": [ "Suppose that we have ring maps $R \\to R'$ and $R \\to S$.", "Set $S' = S \\otimes_R R'$, so that we obtain a diagram", "(\\ref{equation-functorial-omega}). Then the canonical map defined above", "induces an isomorphism $\\Omega_{S/R} \\otimes_R R' = \\Omega_{S'/R'}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\text{d}' : S' = S \\otimes_R R' \\to \\Omega_{S/R} \\otimes_R R'$ denote the", "map $\\text{d}'( \\sum a_i \\otimes x_i ) = \\text{d}(a_i) \\otimes x_i$.", "It exists because the map $S \\times R' \\to \\Omega_{S/R} \\otimes_R R'$,", "$(a, x)\\mapsto \\text{d}a \\otimes_R x$ is $R$-bilinear.", "This is an $R'$-derivation, as can be verified by a simple computation.", "We will show that $(\\Omega_{S/R} \\otimes_R R', \\text{d}')$ satisfies", "the universal property. Let $D : S' \\to M'$ be an $R'$ derivation", "into an $S'$-module. The composition $S \\to S' \\to M'$ is an $R$-derivation,", "hence we get an $S$-linear map $\\varphi_D : \\Omega_{S/R} \\to M'$. We may", "tensor this with $R'$ and get the map $\\varphi'_D :", "\\Omega_{S/R} \\otimes_R R' \\to M'$, $\\varphi'_D(\\eta \\otimes x) =", "x\\varphi_D(\\eta)$. It is clear that $D = \\varphi'_D \\circ \\text{d}'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1139, "type": "theorem", "label": "algebra-lemma-differentials-diagonal", "categories": [ "algebra" ], "title": "algebra-lemma-differentials-diagonal", "contents": [ "Let $R \\to S$ be a ring map. Let $J = \\Ker(S \\otimes_R S \\to S)$", "be the kernel of the multiplication map. There is a canonical", "isomorphism of $S$-modules $\\Omega_{S/R} \\to J/J^2$,", "$a \\text{d} b \\mapsto a \\otimes b - ab \\otimes 1$." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "Apply Lemma \\ref{lemma-differential-seq-split} to the commutative diagram", "$$", "\\xymatrix{", "S \\otimes_R S \\ar[r] & S \\\\", "S \\ar[r] \\ar[u] & S \\ar[u]", "}", "$$", "where the left vertical arrow is $a \\mapsto a \\otimes 1$. We get the", "exact sequence", "$0 \\to J/J^2 \\to", "\\Omega_{S \\otimes_R S/S} \\otimes_{S \\otimes_R S} S \\to \\Omega_{S/S} \\to 0$.", "By Lemma \\ref{lemma-trivial-differential-surjective}", "the term $\\Omega_{S/S}$ is $0$, and we obtain an", "isomorphism between the other two terms. We have", "$\\Omega_{S \\otimes_R S/S} = \\Omega_{S/R} \\otimes_S (S \\otimes_R S)$", "by Lemma \\ref{lemma-differentials-base-change} as $S \\to S \\otimes_R S$", "is the base change of $R \\to S$ and hence", "$$", "\\Omega_{S \\otimes_R S/S} \\otimes_{S \\otimes_R S} S =", "\\Omega_{S/R} \\otimes_S (S \\otimes_R S) \\otimes_{S \\otimes_R S} S =", "\\Omega_{S/R}", "$$", "We omit the verification that the map is given by the rule of the lemma." ], "refs": [ "algebra-lemma-differential-seq-split", "algebra-lemma-trivial-differential-surjective", "algebra-lemma-differentials-base-change" ], "ref_ids": [ 1136, 1131, 1138 ] } ], "ref_ids": [] }, { "id": 1140, "type": "theorem", "label": "algebra-lemma-differentials-polynomial-ring", "categories": [ "algebra" ], "title": "algebra-lemma-differentials-polynomial-ring", "contents": [ "If $S = R[x_1, \\ldots, x_n]$, then", "$\\Omega_{S/R}$ is a finite free $S$-module with", "basis $\\text{d}x_1, \\ldots, \\text{d}x_n$." ], "refs": [], "proofs": [ { "contents": [ "We first show that $\\text{d}x_1, \\ldots, \\text{d}x_n$", "generate $\\Omega_{S/R}$ as an $S$-module. To prove this", "we show that $\\text{d}g$ can be expressed as a", "sum $\\sum g_i \\text{d}x_i$ for any $g \\in R[x_1, \\ldots, x_n]$.", "We do this by induction on the (total) degree of $g$.", "It is clear if the degree of $g$ is $0$, because then", "$\\text{d}g = 0$. If the degree of $g$ is $> 0$, then", "we may write $g$ as $c + \\sum g_i x_i$ with $c\\in R$", "and $\\deg(g_i) < \\deg(g)$. By the Leibniz rule we have", "$\\text{d}g = \\sum g_i \\text{d} x_i + \\sum x_i \\text{d}g_i$,", "and hence we win by induction.", "\\medskip\\noindent", "Consider the $R$-derivation $\\partial / \\partial x_i :", "R[x_1, \\ldots, x_n] \\to R[x_1, \\ldots, x_n]$. (We leave it to", "the reader to define this; the defining property", "being that $\\partial / \\partial x_i (x_j) = \\delta_{ij}$.)", "By the universal property this corresponds to an $S$-module map $l_i :", "\\Omega_{S/R} \\to R[x_1, \\ldots, x_n]$ which maps $\\text{d}x_i$", "to $1$ and $\\text{d}x_j$ to $0$ for $j \\not = i$.", "Thus it is clear that there are no $S$-linear relations", "among the elements $\\text{d}x_1, \\ldots, \\text{d}x_n$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1141, "type": "theorem", "label": "algebra-lemma-differentials-finitely-presented", "categories": [ "algebra" ], "title": "algebra-lemma-differentials-finitely-presented", "contents": [ "Suppose $R \\to S$ is of finite presentation.", "Then $\\Omega_{S/R}$ is a finitely presented", "$S$-module." ], "refs": [], "proofs": [ { "contents": [ "Write $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$.", "Write $I = (f_1, \\ldots, f_m)$. According", "to Lemma \\ref{lemma-differential-seq} there is an exact sequence", "of $S$-modules", "$$", "I/I^2", "\\to", "\\Omega_{R[x_1, \\ldots, x_n]/R} \\otimes_{R[x_1, \\ldots, x_n]} S", "\\to", "\\Omega_{S/R}", "\\to", "0", "$$", "The result follows from the fact that $I/I^2$ is a finite", "$S$-module (generated by the images of the $f_i$), and that", "the middle term is finite free by", "Lemma \\ref{lemma-differentials-polynomial-ring}." ], "refs": [ "algebra-lemma-differential-seq", "algebra-lemma-differentials-polynomial-ring" ], "ref_ids": [ 1135, 1140 ] } ], "ref_ids": [] }, { "id": 1142, "type": "theorem", "label": "algebra-lemma-differentials-finitely-generated", "categories": [ "algebra" ], "title": "algebra-lemma-differentials-finitely-generated", "contents": [ "Suppose $R \\to S$ is of finite type.", "Then $\\Omega_{S/R}$ is finitely generated", "$S$-module." ], "refs": [], "proofs": [ { "contents": [ "This is very similar to, but easier than the proof", "of Lemma \\ref{lemma-differentials-finitely-presented}." ], "refs": [ "algebra-lemma-differentials-finitely-presented" ], "ref_ids": [ 1141 ] } ], "ref_ids": [] }, { "id": 1143, "type": "theorem", "label": "algebra-lemma-de-rham-complex", "categories": [ "algebra" ], "title": "algebra-lemma-de-rham-complex", "contents": [ "Let $A \\to B$ be a ring map. Let $\\pi : \\Omega_{B/A} \\to \\Omega$", "be a surjective $B$-module map. Denote $\\text{d} : B \\to \\Omega$", "the composition of $\\pi$ with the universal derivation", "$\\text{d}_{B/A} : B \\to \\Omega_{B/A}$. Set $\\Omega^i = \\wedge_B^i(\\Omega)$.", "Assume that the kernel of $\\pi$ is generated, as a $B$-module,", "by elements $\\omega \\in \\Omega_{B/A}$ such that", "$\\text{d}_{B/A}(\\omega) \\in \\Omega_{B/A}^2$ maps to zero in $\\Omega^2$.", "Then there is a de Rham complex", "$$", "\\Omega^0 \\to \\Omega^1 \\to \\Omega^2 \\to \\ldots", "$$", "whose differential is defined by the rule", "$$", "\\text{d} : \\Omega^p \\to \\Omega^{p + 1},\\quad", "\\text{d}\\left(f_0\\text{d}f_1 \\wedge \\ldots \\wedge \\text{d}f_p\\right) =", "\\text{d}f_0 \\wedge \\text{d}f_1 \\wedge \\ldots \\wedge \\text{d}f_p", "$$" ], "refs": [], "proofs": [ { "contents": [ "We will show that there exists a commutative diagram", "$$", "\\xymatrix{", "\\Omega_{B/A}^0 \\ar[d] \\ar[r]_{\\text{d}_{B/A}} &", "\\Omega_{B/A}^1 \\ar[d]_\\pi \\ar[r]_{\\text{d}_{B/A}} &", "\\Omega_{B/A}^2 \\ar[d]_{\\wedge^2\\pi} \\ar[r]_{\\text{d}_{B/A}} &", "\\ldots \\\\", "\\Omega^0 \\ar[r]^{\\text{d}} &", "\\Omega^1 \\ar[r]^{\\text{d}} &", "\\Omega^2 \\ar[r]^{\\text{d}} &", "\\ldots", "}", "$$", "the description of the map $\\text{d}$ will follow from the construction", "of the differentials", "$\\text{d}_{B/A} : \\Omega^p_{B/A} \\to \\Omega^{p + 1}_{B/A}$ of the", "de Rham complex of $B$ over $A$ given above.", "Since the left most vertical arrow is an isomorphism we have", "the first square. Because $\\pi$ is surjective, to get the second", "square it suffices to show that $\\text{d}_{B/A}$ maps the kernel", "of $\\pi$ into the kernel of $\\wedge^2\\pi$. We are given that any element", "of the kernel of $\\pi$ is of the form", "$\\sum b_i\\omega_i$ with $\\pi(\\omega_i) = 0$ and", "$\\wedge^2\\pi(\\text{d}_{B/A}(\\omega_i)) = 0$.", "By the Leibniz rule for $\\text{d}_{B/A}$ we have", "$\\text{d}_{B/A}(\\sum b_i\\omega_i) = \\sum b_i \\text{d}_{B/A}(\\omega_i) +", "\\sum \\text{d}_{B/A}(b_i) \\wedge \\omega_i$. Hence this maps to zero", "under $\\wedge^2\\pi$.", "\\medskip\\noindent", "For $i > 1$ we note that $\\wedge^i \\pi$ is surjective with", "kernel the image of $\\Ker(\\pi) \\wedge \\Omega^{i - 1}_{B/A}", "\\to \\Omega_{B/A}^i$. For $\\omega_1 \\in \\Ker(\\pi)$ and", "$\\omega_2 \\in \\Omega^{i - 1}_{B/A}$ we have", "$$", "\\text{d}_{B/A}(\\omega_1 \\wedge \\omega_2) =", "\\text{d}_{B/A}(\\omega_1) \\wedge \\omega_2 -", "\\omega_1 \\wedge \\text{d}_{B/A}(\\omega_2)", "$$", "which is in the kernel of $\\wedge^{i + 1}\\pi$ by what we just proved above.", "Hence we get the $(i + 1)$st square in the diagram above.", "This concludes the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1144, "type": "theorem", "label": "algebra-lemma-composition-differential-operators", "categories": [ "algebra" ], "title": "algebra-lemma-composition-differential-operators", "contents": [ "Let $R \\to S$ be a ring map. Let $L, M, N$ be $S$-modules.", "If $D : L \\to M$ and $D' : M \\to N$ are differential", "operators of order $k$ and $k'$, then $D' \\circ D$ is a", "differential operator of order $k + k'$." ], "refs": [], "proofs": [ { "contents": [ "Let $g \\in S$. Then the map which sends $x \\in L$ to", "$$", "D'(D(gx)) - gD'(D(x)) = D'(D(gx)) - D'(gD(x)) + D'(gD(x)) - gD'(D(x))", "$$", "is a sum of two compositions of differential operators of lower order.", "Hence the lemma follows by induction on $k + k'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1145, "type": "theorem", "label": "algebra-lemma-module-principal-parts", "categories": [ "algebra" ], "title": "algebra-lemma-module-principal-parts", "contents": [ "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module.", "Let $k \\geq 0$. There exists an $S$-module $P^k_{S/R}(M)$", "and a canonical isomorphism", "$$", "\\text{Diff}^k_{S/R}(M, N) = \\Hom_S(P^k_{S/R}(M), N)", "$$", "functorial in the $S$-module $N$." ], "refs": [], "proofs": [ { "contents": [ "The existence of $P^k_{S/R}(M)$ follows from general category theoretic", "arguments (insert future reference here), but we will also give a", "construction. Set $F = \\bigoplus_{m \\in M} S[m]$ where $[m]$ is a", "symbol indicating the basis element in the summand corresponding to $m$.", "Given any differential operator $D : M \\to N$ we obtain an $S$-linear", "map $L_D : F \\to N$ sending $[m]$ to $D(m)$. If $D$ has order $0$,", "then $L_D$ annihilates the elements", "$$", "[m + m'] - [m] - [m'],\\quad", "g_0[m] - [g_0m]", "$$", "where $g_0 \\in S$ and $m, m' \\in M$.", "If $D$ has order $1$, then $L_D$ annihilates the elements", "$$", "[m + m'] - [m] - [m'],\\quad", "f[m] - [fm], \\quad", "g_0g_1[m] - g_0[g_1m] - g_1[g_0m] + [g_1g_0m]", "$$", "where", "$f \\in R$, $g_0, g_1 \\in S$, and $m \\in M$.", "If $D$ has order $k$, then $L_D$ annihilates the elements", "$[m + m'] - [m] - [m']$, $f[m] - [fm]$, and the elements", "$$", "g_0g_1\\ldots g_k[m] - \\sum g_0 \\ldots \\hat g_i \\ldots g_k[g_im] + \\ldots", "+(-1)^{k + 1}[g_0\\ldots g_km]", "$$", "Conversely, if $L : F \\to N$ is an", "$S$-linear map annihilating all the elements listed in the previous", "sentence, then $m \\mapsto L([m])$ is a differential operator", "of order $k$. Thus we see that $P^k_{S/R}(M)$ is the quotient", "of $F$ by the submodule generated by these elements." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1146, "type": "theorem", "label": "algebra-lemma-sequence-of-principal-parts", "categories": [ "algebra" ], "title": "algebra-lemma-sequence-of-principal-parts", "contents": [ "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. There is a", "canonical short exact sequence", "$$", "0 \\to \\Omega_{S/R} \\otimes_S M \\to P^1_{S/R}(M) \\to M \\to 0", "$$", "functorial in $M$ called the {\\it sequence of principal parts}." ], "refs": [], "proofs": [ { "contents": [ "The map $P^1_{S/R}(M) \\to M$ is given above.", "Let $N$ be an $S$-module and let $D : M \\to N$ be a", "differential operator of order $1$. For $m \\in M$ the map", "$$", "g \\longmapsto D(gm) - gD(m)", "$$", "is an $R$-derivation $S \\to N$ by the axioms for differential operators", "of order $1$. Thus it corresponds to a linear map $D_m : \\Omega_{S/R} \\to N$", "determined by the rule $a\\text{d}b \\mapsto aD(bm) - abD(m)$", "(see Lemma \\ref{lemma-universal-omega}). The map", "$$", "\\Omega_{S/R} \\times M \\longrightarrow N,\\quad", "(\\eta, m) \\longmapsto D_m(\\eta)", "$$", "is $S$-bilinear (details omitted) and hence determines an $S$-linear", "map", "$$", "\\sigma_D : \\Omega_{S/R} \\otimes_S M \\to N", "$$", "In this way we obtain a map", "$\\text{Diff}^1(M, N) \\to \\Hom_S(\\Omega_{S/R} \\otimes_S M, N)$,", "$D \\mapsto \\sigma_D$ functorial in $N$. By the Yoneda lemma this corresponds", "a map $\\Omega_{S/R} \\otimes_S M \\to P^1_{S/R}(M)$. It is immediate", "from the construction that this map is functorial in $M$. The sequence", "$$", "\\Omega_{S/R} \\otimes_S M \\to P^1_{S/R}(M) \\to M \\to 0", "$$", "is exact because for every module $N$ the sequence", "$$", "0 \\to \\Hom_S(M, N) \\to", "\\text{Diff}^1(M, N) \\to", "\\Hom_S(\\Omega_{S/R} \\otimes_S M, N)", "$$", "is exact by inspection.", "\\medskip\\noindent", "To see that $\\Omega_{S/R} \\otimes_S M \\to P^1_{S/R}(M)$ is injective", "we argue as follows. Choose an exact sequence", "$$", "0 \\to M' \\to F \\to M \\to 0", "$$", "with $F$ a free $S$-module. This induces an exact sequence", "$$", "0 \\to \\text{Diff}^1(M, N) \\to \\text{Diff}^1(F, N) \\to \\text{Diff}^1(M', N)", "$$", "for all $N$. This proves that in the commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Omega_{S/R} \\otimes_S M' \\ar[r] \\ar[d] &", "P^1_{S/R}(M') \\ar[r] \\ar[d] &", "M' \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\Omega_{S/R} \\otimes_S F \\ar[r] \\ar[d] &", "P^1_{S/R}(F) \\ar[r] \\ar[d] &", "F \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\Omega_{S/R} \\otimes_S M \\ar[r] \\ar[d] &", "P^1_{S/R}(M) \\ar[r] \\ar[d] &", "M \\ar[r] \\ar[d] &", "0 \\\\", "& 0 & 0 & 0", "}", "$$", "the middle column is exact. The left column is exact by", "right exactness of $\\Omega_{S/R} \\otimes_S -$. By the snake lemma", "(see Section \\ref{section-snake}) it suffices to prove exactness", "on the left for the free module $F$.", "Using that $P^1_{S/R}(-)$ commutes with direct sums we reduce to the case", "$M = S$. This case is a consequence of the discussion in", "Example \\ref{example-derivations-and-differential-operators}." ], "refs": [ "algebra-lemma-universal-omega" ], "ref_ids": [ 1129 ] } ], "ref_ids": [] }, { "id": 1147, "type": "theorem", "label": "algebra-lemma-differentials-de-rham-complex-order-1", "categories": [ "algebra" ], "title": "algebra-lemma-differentials-de-rham-complex-order-1", "contents": [ "Let $A \\to B$ be a ring map. The differentials", "$\\text{d} : \\Omega^i_{B/A} \\to \\Omega^{i + 1}_{B/A}$", "are differential operators of order $1$." ], "refs": [], "proofs": [ { "contents": [ "Given $b \\in B$ we have to show that $\\text{d} \\circ b - b \\circ \\text{d}$", "is a linear operator. Thus we have to show that", "$$", "\\text{d} \\circ b \\circ b' - b \\circ \\text{d} \\circ b' -", "b' \\circ \\text{d} \\circ b + b' \\circ b \\circ \\text{d} = 0", "$$", "To see this it suffices to check this on additive generators for", "$\\Omega^i_{B/A}$. Thus it suffices to show that", "$$", "\\text{d}(bb'b_0\\text{d}b_1 \\wedge \\ldots \\wedge \\text{d}b_i) -", "b\\text{d}(b'b_0\\text{d}b_1 \\wedge \\ldots \\wedge \\text{d}b_i) -", "b'\\text{d}(bb_0\\text{d}b_1 \\wedge \\ldots \\wedge \\text{d}b_i) +", "bb'\\text{d}(b_0\\text{d}b_1 \\wedge \\ldots \\wedge \\text{d}b_i)", "$$", "is zero. This is a pleasant calculation using the Leibniz rule", "which is left to the reader." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1148, "type": "theorem", "label": "algebra-lemma-check-differential-operators", "categories": [ "algebra" ], "title": "algebra-lemma-check-differential-operators", "contents": [ "Let $A \\to B$ be a ring map. Let $g_i \\in B$, $i \\in I$ be a set of generators", "for $B$ as an $A$-algebra. Let $M, N$ be $B$-modules.", "Let $D : M \\to N$ be an $A$-linear map. In order to show", "that $D$ is a differential operator of order $k$ it suffices", "to show that $D \\circ g_i - g_i \\circ D$ is a differential", "operator of order $k - 1$ for $i \\in I$." ], "refs": [], "proofs": [ { "contents": [ "Namely, we claim that the set of elements $g \\in B$ such that", "$D \\circ g - g \\circ D$ is a differential operator of order $k - 1$", "is an $A$-subalgebra of $B$. This follows from the relations", "$$", "D \\circ (g + g') - (g + g') \\circ D =", "(D \\circ g - g \\circ D) + (D \\circ g' - g' \\circ D)", "$$", "and", "$$", "D \\circ gg' - gg' \\circ D =", "(D \\circ g - g \\circ D) \\circ g' + g \\circ (D \\circ g' - g' \\circ D)", "$$", "Strictly speaking, to conclude for products we also use", "Lemma \\ref{lemma-composition-differential-operators}." ], "refs": [ "algebra-lemma-composition-differential-operators" ], "ref_ids": [ 1144 ] } ], "ref_ids": [] }, { "id": 1149, "type": "theorem", "label": "algebra-lemma-invert-system-differential-operators", "categories": [ "algebra" ], "title": "algebra-lemma-invert-system-differential-operators", "contents": [ "Let $A \\to B$ be a ring map. Let $M, N$ be $B$-modules.", "Let $S \\subset B$ be a multiplicative subset. Any differential operator", "$D : M \\to N$ of order $k$ extends uniquely to a differential operator", "$E : S^{-1}M \\to S^{-1}N$ of order $k$." ], "refs": [], "proofs": [ { "contents": [ "By induction on $k$. If $k = 0$, then $D$ is $B$-linear and hence we", "get the extension by the functoriality of localization. Given $b \\in B$", "the operator $L_b : m \\mapsto D(bm) - bD(m)$ has order $k - 1$. Hence", "it has a unique extension to a differential operator", "$E_b : S^{-1}M \\to S^{-1}N$ of order $k - 1$ by induction.", "Moreover, a computation shows that $L_{b'b} = L_{b'} \\circ b + b' \\circ L_b$", "hence by uniqueness we obtain $E_{b'b} = E_{b'} \\circ b + b' \\circ E_b$.", "Similarly, we obtain $E_{b'} \\circ b - b \\circ E_{b'} =", "E_b \\circ b' - b' \\circ E_b$.", "Now for $m \\in M$ and $g \\in S$ we set", "$$", "E(m/g) = (1/g)(D(m) - E_g(m/g))", "$$", "To show that this is well defined it suffices to show that", "for $g' \\in S$ if we use the representative $g'm/g'g$ we get the", "same result. We compute", "\\begin{align*}", "(1/g'g)(D(g'm) - E_{g'g}(g'm/gg'))", "& =", "(1/gg')(g'D(m) + E_{g'}(m) - E_{g'g}(g'm/gg')) \\\\", "& =", "(1/g'g)(g'D(m) - g' E_g(m/g))", "\\end{align*}", "which is the same as before. It is clear that $E$ is $R$-linear", "as $D$ and $E_g$ are $R$-linear. Taking $g = 1$ and using that $E_1 = 0$", "we see that $E$ extends $D$. By Lemma \\ref{lemma-check-differential-operators}", "it now suffices to show that $E \\circ b - b \\circ E$ for $b \\in B$ and", "$E \\circ 1/g' - 1/g' \\circ E$ for $g' \\in S$ are differential operators of", "order $k - 1$ in order to show that $E$ is a differential operator of", "order $k$. For the first, choose an element $m/g$ in $S^{-1}M$ and", "observe that", "\\begin{align*}", "E(b m/g) - bE(m/g)", "& =", "(1/g)(D(bm) - bD(m) - E_g(bm/g) + bE_g(m/g)) \\\\", "& =", "(1/g)(L_b(m) - E_b(m) + gE_b(m/g)) \\\\", "& =", "E_b(m/g)", "\\end{align*}", "which is a differential operator of order $k - 1$. Finally, we have", "\\begin{align*}", "E(m/g'g) - (1/g')E(m/g)", "& =", "(1/g'g)(D(m) - E_{g'g}(m/g'g)) -", "(1/g'g)(D(m) - E_g(m/g)) \\\\", "& =", "-(1/g')E_{g'}(m/g'g)", "\\end{align*}", "which also is a differential operator of order $k - 1$ as the composition", "of linear maps (multiplication by $1/g'$ and signs) and $E_{g'}$.", "We omit the proof of uniqueness." ], "refs": [ "algebra-lemma-check-differential-operators" ], "ref_ids": [ 1148 ] } ], "ref_ids": [] }, { "id": 1150, "type": "theorem", "label": "algebra-lemma-base-change-differential-operators", "categories": [ "algebra" ], "title": "algebra-lemma-base-change-differential-operators", "contents": [ "Let $R \\to A$ and $R \\to B$ be ring maps. Let $M$ and $M'$ be $A$-modules.", "Let $D : M \\to M'$ be a differential operator of order $k$ with respect to", "$R \\to A$. Let $N$ be any $B$-module. Then the map", "$$", "D \\otimes \\text{id}_N : M \\otimes_R N \\to M' \\otimes_R N", "$$", "is a differential operator of order $k$ with respect to $B \\to A \\otimes_R B$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $D' = D \\otimes \\text{id}_N$ is $B$-linear.", "By Lemma \\ref{lemma-check-differential-operators} it suffices", "to show that", "$$", "D' \\circ a \\otimes 1 - a \\otimes 1 \\circ D' =", "(D \\circ a - a \\circ D) \\otimes \\text{id}_N", "$$", "is a differential operator of order $k - 1$ which follows", "by induction on $k$." ], "refs": [ "algebra-lemma-check-differential-operators" ], "ref_ids": [ 1148 ] } ], "ref_ids": [] }, { "id": 1151, "type": "theorem", "label": "algebra-lemma-NL-homotopy", "categories": [ "algebra" ], "title": "algebra-lemma-NL-homotopy", "contents": [ "Suppose given a diagram (\\ref{equation-functoriality-NL}).", "Let $\\alpha : P \\to S$ and $\\alpha' : P' \\to S'$ be presentations.", "\\begin{enumerate}", "\\item There exists a morphism of presentations from $\\alpha$ to $\\alpha'$.", "\\item Any two morphisms of presentations induce homotopic", "morphisms of complexes $\\NL(\\alpha) \\to \\NL(\\alpha')$.", "\\item The construction is compatible with compositions of morphisms", "of presentations (see proof for exact statement).", "\\item If $R \\to R'$ and $S \\to S'$ are isomorphisms, then", "for any map $\\varphi$ of presentations from $\\alpha$ to $\\alpha'$", "the induced map $\\NL(\\alpha) \\to \\NL(\\alpha')$ is a homotopy equivalence", "and a quasi-isomorphism.", "\\end{enumerate}", "In particular, comparing $\\alpha$ to the canonical presentation", "(\\ref{equation-canonical-presentation}) we conclude there is a", "quasi-isomorphism $\\NL(\\alpha) \\to \\NL_{S/R}$ well defined", "up to homotopy and compatible with all functorialities (up to homotopy)." ], "refs": [], "proofs": [ { "contents": [ "Since $P$ is a polynomial algebra over $R$ we can write", "$P = R[x_a, a \\in A]$ for some set $A$.", "As $\\alpha'$ is surjective, we can choose", "for every $a \\in A$ an element $f_a \\in P'$", "such that $\\alpha'(f_a) = \\phi(\\alpha(x_a))$. Let", "$\\varphi : P = R[x_a, a \\in A] \\to P'$ be the", "unique $R$-algebra map such that $\\varphi(x_a) = f_a$.", "This gives the morphism in (1).", "\\medskip\\noindent", "Let $\\varphi$ and $\\varphi'$ morphisms of presentations from $\\alpha$", "to $\\alpha'$. Let $I = \\Ker(\\alpha)$ and $I' = \\Ker(\\alpha')$.", "We have to construct the diagonal map $h$ in the diagram", "$$", "\\xymatrix{", "I/I^2 \\ar[r]^-{\\text{d}}", "\\ar@<1ex>[d]^{\\varphi'_1} \\ar@<-1ex>[d]_{\\varphi_1}", "&", "\\Omega_{P/R} \\otimes_P S", "\\ar@<1ex>[d]^{\\varphi'_0} \\ar@<-1ex>[d]_{\\varphi_0}", "\\ar[ld]_h", "\\\\", "I'/(I')^2 \\ar[r]^-{\\text{d}}", "&", "\\Omega_{P'/R'} \\otimes_{P'} S'", "}", "$$", "where the vertical maps are induced by $\\varphi$, $\\varphi'$ such that", "$$", "\\varphi_1 - \\varphi'_1 = h \\circ \\text{d}", "\\quad\\text{and}\\quad", "\\varphi_0 - \\varphi'_0 = \\text{d} \\circ h", "$$", "Consider the map $\\varphi - \\varphi' : P \\to P'$. Since both $\\varphi$", "and $\\varphi'$ are compatible with $\\alpha$ and $\\alpha'$ we obtain", "$\\varphi - \\varphi' : P \\to I'$. This implies that", "$\\varphi, \\varphi' : P \\to P'$ induce the same $P$-module structure", "on $I'/(I')^2$, since", "$\\varphi(p)i' - \\varphi'(p)i' = (\\varphi - \\varphi')(p)i' \\in (I')^2$.", "Also $\\varphi - \\varphi'$ is $R$-linear and", "$$", "(\\varphi - \\varphi')(fg) =", "\\varphi(f)(\\varphi - \\varphi')(g) + (\\varphi - \\varphi')(f)\\varphi'(g)", "$$", "Hence the induced map $D : P \\to I'/(I')^2$ is a $R$-derivation.", "Thus we obtain a canonical map $h : \\Omega_{P/R} \\otimes_P S \\to I'/(I')^2$", "such that $D = h \\circ \\text{d}$. ", "A calculation (omitted) shows that $h$ is the desired homotopy.", "\\medskip\\noindent", "Suppose that we have a commutative diagram", "$$", "\\xymatrix{", "S \\ar[r]_{\\phi} & S' \\ar[r]_{\\phi'} & S'' \\\\", "R \\ar[r] \\ar[u] & R' \\ar[u] \\ar[r] & R'' \\ar[u]", "}", "$$", "and that", "\\begin{enumerate}", "\\item $\\alpha : P \\to S$,", "\\item $\\alpha' : P' \\to S'$, and", "\\item $\\alpha'' : P'' \\to S''$", "\\end{enumerate}", "are presentations. Suppose that", "\\begin{enumerate}", "\\item $\\varphi : P \\to P$ is a morphism of presentations from", "$\\alpha$ to $\\alpha'$ and", "\\item $\\varphi' : P' \\to P''$", "is a morphism of presentations from $\\alpha'$ to $\\alpha''$.", "\\end{enumerate}", "Then it is immediate that", "$\\varphi' \\circ \\varphi : P \\to P''$", "is a morphism of presentations from $\\alpha$ to $\\alpha''$ and that", "the induced map $\\NL(\\alpha) \\to \\NL(\\alpha'')$ of naive cotangent complexes", "is the composition of the maps $\\NL(\\alpha) \\to \\NL(\\alpha')$ and", "$\\NL(\\alpha') \\to \\NL(\\alpha'')$ induced by $\\varphi$ and $\\varphi'$.", "\\medskip\\noindent", "In the simple case of complexes with 2 terms a quasi-isomorphism", "is just a map that induces an isomorphism on both the cokernel", "and the kernel of the maps between the terms. Note that homotopic", "maps of 2 term complexes (as explained above) define the same maps on", "kernel and cokernel. Hence if $\\varphi$ is a map from a presentation", "$\\alpha$ of $S$ over $R$ to itself, then the induced map", "$\\NL(\\alpha) \\to \\NL(\\alpha)$ is a quasi-isomorphism being homotopic", "to the identity by part (2). To prove (4) in full generality, consider", "a morphism $\\varphi'$ from $\\alpha'$ to $\\alpha$ which exists by (1).", "The compositions $\\NL(\\alpha) \\to \\NL(\\alpha') \\to \\NL(\\alpha)$ and", "$\\NL(\\alpha') \\to \\NL(\\alpha) \\to \\NL(\\alpha')$ are homotopic to the identity", "maps by (3), hence these maps are homotopy equivalences by definition.", "It follows formally that both maps", "$\\NL(\\alpha) \\to \\NL(\\alpha')$ and $\\NL(\\alpha') \\to \\NL(\\alpha)$ are", "quasi-isomorphisms. Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1152, "type": "theorem", "label": "algebra-lemma-NL-polynomial-algebra", "categories": [ "algebra" ], "title": "algebra-lemma-NL-polynomial-algebra", "contents": [ "Let $A \\to B$ be a polynomial algebra. Then $\\NL_{B/A}$ is homotopy equivalent", "to the chain complex $(0 \\to \\Omega_{B/A})$ with $\\Omega_{B/A}$", "in degree $0$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-NL-homotopy}", "and the fact that $\\text{id}_B : B \\to B$ is a presentation of $B$ over $A$", "with zero kernel." ], "refs": [ "algebra-lemma-NL-homotopy" ], "ref_ids": [ 1151 ] } ], "ref_ids": [] }, { "id": 1153, "type": "theorem", "label": "algebra-lemma-exact-sequence-NL", "categories": [ "algebra" ], "title": "algebra-lemma-exact-sequence-NL", "contents": [ "Let $A \\to B \\to C$ be ring maps. Choose a presentation", "$\\alpha : A[x_s, s \\in S] \\to B$ with kernel $I$. Choose a presentation", "$\\beta : B[y_t, t \\in T] \\to C$ with kernel $J$. Let", "$\\gamma : A[x_s, y_t] \\to C$ be the induced presentation of $C$ with kernel", "$K$. Then we get a canonical commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Omega_{A[x_s]/A} \\otimes C \\ar[r] &", "\\Omega_{A[x_s, y_t]/A} \\otimes C \\ar[r] &", "\\Omega_{B[y_t]/B} \\otimes C \\ar[r] &", "0 \\\\", "&", "I/I^2 \\otimes C \\ar[r] \\ar[u] &", "K/K^2 \\ar[r] \\ar[u] &", "J/J^2 \\ar[r] \\ar[u] &", "0", "}", "$$", "with exact rows. We get the following exact sequence", "of homology groups", "$$", "H_1(\\NL_{B/A} \\otimes_B C) \\to", "H_1(L_{C/A}) \\to", "H_1(L_{C/B}) \\to", "C \\otimes_B \\Omega_{B/A} \\to", "\\Omega_{C/A} \\to", "\\Omega_{C/B} \\to 0", "$$", "of $C$-modules extending the sequence of", "Lemma \\ref{lemma-exact-sequence-differentials}.", "If $\\text{Tor}_1^B(\\Omega_{B/A}, C) = 0$, then", "$H_1(\\NL_{B/A} \\otimes_B C) = H_1(L_{B/A}) \\otimes_B C$." ], "refs": [ "algebra-lemma-exact-sequence-differentials" ], "proofs": [ { "contents": [ "The precise definition of the maps is omitted.", "The exactness of the top row follows as the $\\text{d}x_s$,", "$\\text{d}y_t$ form a basis for the middle module.", "The map $\\gamma$ factors", "$$", "A[x_s, y_t] \\to B[y_t] \\to C", "$$", "with surjective first arrow and second arrow equal to $\\beta$.", "Thus we see that $K \\to J$ is surjective.", "Moreover, the kernel of the first displayed arrow is", "$IA[x_s, y_t]$. Hence $I/I^2 \\otimes C$ surjects onto the", "kernel of $K/K^2 \\to J/J^2$. Finally, we can use", "Lemma \\ref{lemma-NL-homotopy}", "to identify the terms as homology groups of the naive", "cotangent complexes.", "The final assertion follows as the degree $0$ term of the complex", "$\\NL_{B/A}$ is a free $B$-module." ], "refs": [ "algebra-lemma-NL-homotopy" ], "ref_ids": [ 1151 ] } ], "ref_ids": [ 1133 ] }, { "id": 1154, "type": "theorem", "label": "algebra-lemma-NL-surjection", "categories": [ "algebra" ], "title": "algebra-lemma-NL-surjection", "contents": [ "Let $A \\to B$ be a surjective ring map with kernel $I$.", "Then $\\NL_{B/A}$ is homotopy equivalent to the chain complex", "$(I/I^2 \\to 0)$ with $I/I^2$ in degree $1$. In particular", "$H_1(L_{B/A}) = I/I^2$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-NL-homotopy}", "and the fact that $A \\to B$ is a presentation of $B$ over $A$." ], "refs": [ "algebra-lemma-NL-homotopy" ], "ref_ids": [ 1151 ] } ], "ref_ids": [] }, { "id": 1155, "type": "theorem", "label": "algebra-lemma-application-NL", "categories": [ "algebra" ], "title": "algebra-lemma-application-NL", "contents": [ "Let $A \\to B \\to C$ be ring maps. Assume $A \\to C$ is surjective (so", "also $B \\to C$ is). Denote $I = \\Ker(A \\to C)$ and", "$J = \\Ker(B \\to C)$. Then the sequence", "$$", "I/I^2 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B B/J \\to 0", "$$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-exact-sequence-NL}", "and the description of the naive cotangent complexes", "$\\NL_{C/B}$ and $\\NL_{C/A}$ in Lemma \\ref{lemma-NL-surjection}." ], "refs": [ "algebra-lemma-exact-sequence-NL", "algebra-lemma-NL-surjection" ], "ref_ids": [ 1153, 1154 ] } ], "ref_ids": [] }, { "id": 1156, "type": "theorem", "label": "algebra-lemma-change-base-NL", "categories": [ "algebra" ], "title": "algebra-lemma-change-base-NL", "contents": [ "Let $R \\to S$ be a ring map. Let $\\alpha : P \\to S$ be a presentation.", "Let $R \\to R'$ be a flat ring map.", "Let $\\alpha' : P \\otimes_R R' \\to S' = S \\otimes_R R'$", "be the induced presentation.", "Then $\\NL(\\alpha) \\otimes_R R' = \\NL(\\alpha) \\otimes_S S' = \\NL(\\alpha')$.", "In particular, the canonical map", "$$", "\\NL_{S/R} \\otimes_S S' \\longrightarrow \\NL_{S \\otimes_R R'/R'}", "$$", "is a homotopy equivalence if $R \\to R'$ is flat." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$\\Ker(\\alpha') = R' \\otimes_R \\Ker(\\alpha)$", "since $R \\to R'$ is flat." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1157, "type": "theorem", "label": "algebra-lemma-colimits-NL", "categories": [ "algebra" ], "title": "algebra-lemma-colimits-NL", "contents": [ "Let $R_i \\to S_i$ be a system of ring maps over the directed set $I$.", "Set $R = \\colim R_i$ and $S = \\colim S_i$.", "Then $\\NL_{S/R} = \\colim \\NL_{S_i/R_i}$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\NL_{S/R}$ is the complex", "$I/I^2 \\to \\bigoplus_{s \\in S} S\\text{d}[s]$ where $I \\subset R[S]$", "is the kernel of the canonical presentation $R[S] \\to S$.", "Now it is clear that $R[S] = \\colim R_i[S_i]$ and similarly", "that $I = \\colim I_i$ where $I_i = \\Ker(R_i[S_i] \\to S_i)$.", "Hence the lemma is clear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1158, "type": "theorem", "label": "algebra-lemma-NL-of-localization", "categories": [ "algebra" ], "title": "algebra-lemma-NL-of-localization", "contents": [ "If $S \\subset A$ is a multiplicative subset of $A$, then", "$\\NL_{S^{-1}A/A}$ is homotopy equivalent to the zero complex." ], "refs": [], "proofs": [ { "contents": [ "Since $A \\to S^{-1}A$ is flat we see that", "$\\NL_{S^{-1}A/A} \\otimes_A S^{-1}A \\to \\NL_{S^{-1}A/S^{-1}A}$", "is a homotopy equivalence by flat base change", "(Lemma \\ref{lemma-change-base-NL}). Since the source of the arrow", "is isomorphic to $\\NL_{S^{-1}A/A}$ and the target of the arrow is", "zero (by Lemma \\ref{lemma-NL-surjection}) we win." ], "refs": [ "algebra-lemma-change-base-NL", "algebra-lemma-NL-surjection" ], "ref_ids": [ 1156, 1154 ] } ], "ref_ids": [] }, { "id": 1159, "type": "theorem", "label": "algebra-lemma-NL-localize-bottom", "categories": [ "algebra" ], "title": "algebra-lemma-NL-localize-bottom", "contents": [ "Let $S \\subset A$ is a multiplicative subset of $A$.", "Let $S^{-1}A \\to B$ be a ring map.", "Then $\\NL_{B/A} \\to \\NL_{B/S^{-1}A}$ is a homotopy equivalence." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $\\alpha : P \\to B$ of $B$ over $A$.", "Then $\\beta : S^{-1}P \\to B$ is a presentation of $B$ over $S^{-1}A$.", "A direct computation shows that we have $\\NL(\\alpha) = \\NL(\\beta)$", "which proves the lemma as the naive cotangent complex is well defined", "up to homotopy by Lemma \\ref{lemma-NL-homotopy}." ], "refs": [ "algebra-lemma-NL-homotopy" ], "ref_ids": [ 1151 ] } ], "ref_ids": [] }, { "id": 1160, "type": "theorem", "label": "algebra-lemma-principal-localization-NL", "categories": [ "algebra" ], "title": "algebra-lemma-principal-localization-NL", "contents": [ "\\begin{slogan}", "The formation of the naive cotangent complex commutes with localization", "at an element.", "\\end{slogan}", "Let $A \\to B$ be a ring map. Let $g \\in B$. Suppose $\\alpha : P \\to B$", "is a presentation with kernel $I$. Then a presentation of $B_g$ over $A$ is", "the map", "$$", "\\beta : P[x] \\longrightarrow B_g", "$$", "extending $\\alpha$ and sending $x$ to $1/g$.", "The kernel $J$ of $\\beta$ is generated by $I$ and the element $f x - 1$", "where $f \\in P$ is an element mapped to $g \\in B$ by $\\alpha$. In this", "situation we have", "\\begin{enumerate}", "\\item $J/J^2 = (I/I^2)_g \\oplus B_g (f x - 1)$,", "\\item $\\Omega_{P[x]/A} \\otimes_{P[x]} B_g =", "\\Omega_{P/A} \\otimes_P B_g \\oplus B_g \\text{d}x$,", "\\item $\\NL(\\beta) \\cong", "\\NL(\\alpha) \\otimes_B B_g \\oplus (B_g \\xrightarrow{g} B_g)$", "\\end{enumerate}", "Hence the canonical map $\\NL_{B/A} \\otimes_B B_g \\to \\NL_{B_g/A}$", "is a homotopy equivalence." ], "refs": [], "proofs": [ { "contents": [ "Since $P[x]/(I, fx - 1) = B[x]/(gx - 1) = B_g$ we get the statement about", "$I$ and $fx - 1$ generating $J$. Consider the commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Omega_{P/A} \\otimes B_g \\ar[r] &", "\\Omega_{P[x]/A} \\otimes B_g \\ar[r] &", "\\Omega_{B[x]/B} \\otimes B_g \\ar[r] &", "0 \\\\", "&", "(I/I^2)_g \\ar[r] \\ar[u] &", "J/J^2 \\ar[r] \\ar[u] &", "(gx - 1)/(gx - 1)^2 \\ar[r] \\ar[u] &", "0", "}", "$$", "with exact rows of Lemma \\ref{lemma-exact-sequence-NL}.", "The $B_g$-module $\\Omega_{B[x]/B} \\otimes B_g$ is free of", "rank $1$ on $\\text{d}x$. The element $\\text{d}x$ in the", "$B_g$-module $\\Omega_{P[x]/A} \\otimes B_g$ provides", "a splitting for the top row. The element $gx - 1 \\in (gx - 1)/(gx - 1)^2$", "is mapped to $g\\text{d}x$ in $\\Omega_{B[x]/B} \\otimes B_g$", "and hence $(gx - 1)/(gx - 1)^2$ is free of rank $1$ over $B_g$.", "(This can also be seen by arguing that $gx - 1$ is a nonzerodivisor", "in $B[x]$ because it is a polynomial with invertible constant term", "and any nonzerodivisor gives a quasi-regular sequence of length $1$", "by Lemma \\ref{lemma-regular-quasi-regular}.)", "\\medskip\\noindent", "Let us prove $(I/I^2)_g \\to J/J^2$ injective. Consider the $P$-algebra map", "$$", "\\pi : P[x] \\to (P/I^2)_f = P_f/I_f^2", "$$", "sending $x$ to $1/f$. Since $J$ is generated by $I$ and $fx - 1$", "we see that $\\pi(J) \\subset (I/I^2)_f = (I/I^2)_g$. Since this", "is an ideal of square zero we see that $\\pi(J^2) = 0$.", "If $a \\in I$ maps to an element of $J^2$ in $J$, then", "$\\pi(a) = 0$, which implies that $a$ maps to zero in $I_f/I_f^2$.", "This proves the desired injectivity.", "\\medskip\\noindent", "Thus we have a short exact sequence of two term complexes", "$$", "0 \\to \\NL(\\alpha) \\otimes_B B_g \\to \\NL(\\beta)", "\\to (B_g \\xrightarrow{g} B_g) \\to 0", "$$", "Such a short exact sequence can always be split in the category of", "complexes. In our particular case we can take as splittings", "$$", "J/J^2 = (I/I^2)_g \\oplus B_g (fx - 1)\\quad\\text{and}\\quad", "\\Omega_{P[x]/A} \\otimes B_g = \\Omega_{P/A} \\otimes B_g \\oplus", "B_g (g^{-2}\\text{d}f + \\text{d}x)", "$$", "This works because", "$\\text{d}(fx - 1) = x\\text{d}f + f \\text{d}x =", "g(g^{-2}\\text{d}f + \\text{d}x)$", "in $\\Omega_{P[x]/A} \\otimes B_g$." ], "refs": [ "algebra-lemma-exact-sequence-NL", "algebra-lemma-regular-quasi-regular" ], "ref_ids": [ 1153, 746 ] } ], "ref_ids": [] }, { "id": 1161, "type": "theorem", "label": "algebra-lemma-localize-NL", "categories": [ "algebra" ], "title": "algebra-lemma-localize-NL", "contents": [ "Let $A \\to B$ be a ring map. Let $S \\subset B$ be a multiplicative subset.", "The canonical map $\\NL_{B/A} \\otimes_B S^{-1}B \\to \\NL_{S^{-1}B/A}$", "is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "We have $S^{-1}B = \\colim_{g \\in S} B_g$ where we think of $S$", "as a directed set (ordering by divisibility), see", "Lemma \\ref{lemma-localization-colimit}.", "By Lemma \\ref{lemma-principal-localization-NL} each of the maps", "$\\NL_{B/A} \\otimes_B B_g \\to \\NL_{B_g/A}$", "are quasi-isomorphisms.", "The lemma follows from Lemma \\ref{lemma-colimits-NL}." ], "refs": [ "algebra-lemma-localization-colimit", "algebra-lemma-principal-localization-NL", "algebra-lemma-colimits-NL" ], "ref_ids": [ 348, 1160, 1157 ] } ], "ref_ids": [] }, { "id": 1162, "type": "theorem", "label": "algebra-lemma-sum-two-terms", "categories": [ "algebra" ], "title": "algebra-lemma-sum-two-terms", "contents": [ "Let $R$ be a ring.", "Let $A_1 \\to A_0$, and $B_1 \\to B_0$ be", "two term complexes. Suppose that there exist", "morphisms of complexes $\\varphi : A_\\bullet \\to B_\\bullet$", "and $\\psi : B_\\bullet \\to A_\\bullet$ such that", "$\\varphi \\circ \\psi$ and $\\psi \\circ \\varphi$ are", "homotopic to the identity maps.", "Then $A_1 \\oplus B_0 \\cong B_1 \\oplus A_0$ as", "$R$-modules." ], "refs": [], "proofs": [ { "contents": [ "Choose a map $h : A_0 \\to A_1$ such that", "$$", "\\text{id}_{A_1} - \\psi_1 \\circ \\varphi_1 = h \\circ d_A", "\\text{ and }", "\\text{id}_{A_0} - \\psi_0 \\circ \\varphi_0 = d_A \\circ h.", "$$", "Similarly, choose a map $h' : B_0 \\to B_1$ such that", "$$", "\\text{id}_{B_1} - \\varphi_1 \\circ \\psi_1 = h' \\circ d_B", "\\text{ and }", "\\text{id}_{B_0} - \\varphi_0 \\circ \\psi_0 = d_B \\circ h'.", "$$", "A trivial computation shows that", "$$", "\\left(", "\\begin{matrix}", "\\text{id}_{A_1} & -h' \\circ \\psi_1 + h \\circ \\psi_0 \\\\", "0 & \\text{id}_{B_0}", "\\end{matrix}", "\\right)", "=", "\\left(", "\\begin{matrix}", "\\psi_1 & h \\\\", "-d_B & \\varphi_0", "\\end{matrix}", "\\right)", "\\left(", "\\begin{matrix}", "\\varphi_1 & - h' \\\\", "d_A & \\psi_0", "\\end{matrix}", "\\right)", "$$", "This shows that both matrices on the right hand side", "are invertible and proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1163, "type": "theorem", "label": "algebra-lemma-conormal-module", "categories": [ "algebra" ], "title": "algebra-lemma-conormal-module", "contents": [ "Let $R \\to S$ be a ring map of finite type.", "For any presentations $\\alpha : R[x_1, \\ldots, x_n] \\to S$, and", "$\\beta : R[y_1, \\ldots, y_m] \\to S$ we have", "$$", "I/I^2 \\oplus S^{\\oplus m} \\cong J/J^2 \\oplus S^{\\oplus n}", "$$", "as $S$-modules where $I = \\Ker(\\alpha)$ and $J = \\Ker(\\beta)$." ], "refs": [], "proofs": [ { "contents": [ "See Lemmas \\ref{lemma-NL-homotopy} and \\ref{lemma-sum-two-terms}." ], "refs": [ "algebra-lemma-NL-homotopy", "algebra-lemma-sum-two-terms" ], "ref_ids": [ 1151, 1162 ] } ], "ref_ids": [] }, { "id": 1164, "type": "theorem", "label": "algebra-lemma-conormal-module-localize", "categories": [ "algebra" ], "title": "algebra-lemma-conormal-module-localize", "contents": [ "Let $R \\to S$ be a ring map of finite type.", "Let $g \\in S$. For any presentations", "$\\alpha : R[x_1, \\ldots, x_n] \\to S$, and", "$\\beta : R[y_1, \\ldots, y_m] \\to S_g$ we have", "$$", "(I/I^2)_g \\oplus S^{\\oplus m}_g \\cong J/J^2 \\oplus S_g^{\\oplus n}", "$$", "as $S_g$-modules where", "$I = \\Ker(\\alpha)$ and $J = \\Ker(\\beta)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-conormal-module}, we see that it suffices to", "prove this for a single choice of $\\alpha$ and $\\beta$. Thus we may take", "$\\beta$ the presentation of Lemma \\ref{lemma-principal-localization-NL}", "and the result is clear." ], "refs": [ "algebra-lemma-conormal-module", "algebra-lemma-principal-localization-NL" ], "ref_ids": [ 1163, 1160 ] } ], "ref_ids": [] }, { "id": 1165, "type": "theorem", "label": "algebra-lemma-localize-lci", "categories": [ "algebra" ], "title": "algebra-lemma-localize-lci", "contents": [ "Let $k$ be a field.", "Let $S$ be a finite type $k$-algebra.", "Let $g \\in S$.", "\\begin{enumerate}", "\\item If $S$ is a global complete intersection so is $S_g$.", "\\item If $S$ is a local complete intersection so is $S_g$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The second statement follows immediately from the first.", "Proof of the first statement. If $S_g$ is the zero ring,", "then it is true. Assume $S_g$ is nonzero.", "Write $S = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "with $n - c = \\dim(S)$ as in Definition \\ref{definition-lci-field}.", "By the remarks following the definition $S$ is equidimensional", "of dimension $n - c$, so $\\dim(S_g) = n - c$ as well. Let", "$g' \\in k[x_1, \\ldots, x_n]$ be an element whose residue class", "corresponds to $g$. Then", "$S_g = k[x_1, \\ldots, x_n, x_{n + 1}]/(f_1, \\ldots, f_c, x_{n + 1}g' - 1)$", "as desired." ], "refs": [ "algebra-definition-lci-field" ], "ref_ids": [ 1530 ] } ], "ref_ids": [] }, { "id": 1166, "type": "theorem", "label": "algebra-lemma-lci-CM", "categories": [ "algebra" ], "title": "algebra-lemma-lci-CM", "contents": [ "Let $k$ be a field. Let $S$ be a finite type $k$-algebra.", "If $S$ is a local complete intersection, then", "$S$ is a Cohen-Macaulay ring." ], "refs": [], "proofs": [ { "contents": [ "Choose a maximal prime $\\mathfrak m$ of $S$.", "We have to show that $S_\\mathfrak m$ is Cohen-Macaulay.", "By assumption we may assume $S = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "with $\\dim(S) = n - c$. Let $\\mathfrak m' \\subset k[x_1, \\ldots, x_n]$", "be the maximal ideal corresponding to $\\mathfrak m$.", "According to Proposition \\ref{proposition-finite-gl-dim-polynomial-ring}", "the local ring", "$k[x_1, \\ldots, x_n]_{\\mathfrak m'}$ is regular local of", "dimension $n$. In particular it is Cohen-Macaulay by", "Lemma \\ref{lemma-regular-ring-CM}.", "By Lemma \\ref{lemma-one-equation} applied $c$ times the local ring", "$S_{\\mathfrak m} = k[x_1, \\ldots, x_n]_{\\mathfrak m'}/(f_1, \\ldots, f_c)$", "has dimension $\\geq n - c$. By assumption $\\dim(S_{\\mathfrak m}) \\leq n - c$.", "Thus we get equality. This implies that $f_1, \\ldots, f_c$ is a regular", "sequence in $k[x_1, \\ldots, x_n]_{\\mathfrak m'}$ and that", "$S_{\\mathfrak m}$ is Cohen-Macaulay, see Proposition", "\\ref{proposition-CM-module}." ], "refs": [ "algebra-proposition-finite-gl-dim-polynomial-ring", "algebra-lemma-regular-ring-CM", "algebra-proposition-CM-module" ], "ref_ids": [ 1424, 941, 1420 ] } ], "ref_ids": [] }, { "id": 1167, "type": "theorem", "label": "algebra-lemma-lci", "categories": [ "algebra" ], "title": "algebra-lemma-lci", "contents": [ "Let $k$ be a field.", "Let $S$ be a finite type $k$-algebra.", "Let $\\mathfrak q$ be a prime of $S$.", "Choose any presentation $S = k[x_1, \\ldots, x_n]/I$.", "Let $\\mathfrak q'$ be the prime of $k[x_1, \\ldots, x_n]$ corresponding", "to $\\mathfrak q$. Set", "$c = \\text{height}(\\mathfrak q') - \\text{height}(\\mathfrak q)$,", "in other words $\\dim_{\\mathfrak q}(S) = n - c$", "(see Lemma \\ref{lemma-codimension}). The following are equivalent", "\\begin{enumerate}", "\\item There exists a $g \\in S$, $g \\not \\in \\mathfrak q$", "such that $S_g$ is a global complete intersection over $k$.", "\\item The ideal $I_{\\mathfrak q'} \\subset k[x_1, \\ldots, x_n]_{\\mathfrak q'}$", "can be generated by $c$ elements.", "\\item The conormal module $(I/I^2)_{\\mathfrak q}$ can be generated by", "$c$ elements over $S_{\\mathfrak q}$.", "\\item The conormal module $(I/I^2)_{\\mathfrak q}$ is a free", "$S_{\\mathfrak q}$-module of rank $c$.", "\\item The ideal $I_{\\mathfrak q'}$ can be generated by a regular sequence", "in the regular local ring $k[x_1, \\ldots, x_n]_{\\mathfrak q'}$.", "\\end{enumerate}", "In this case any $c$ elements of $I_{\\mathfrak q'}$", "which generate $I_{\\mathfrak q'}/\\mathfrak q'I_{\\mathfrak q'}$", "form a regular sequence in the local", "ring $k[x_1, \\ldots, x_n]_{\\mathfrak q'}$." ], "refs": [ "algebra-lemma-codimension" ], "proofs": [ { "contents": [ "Set $R = k[x_1, \\ldots, x_n]_{\\mathfrak q'}$. This is a", "Cohen-Macaulay local", "ring of dimension $\\text{height}(\\mathfrak q')$, see for example", "Lemma \\ref{lemma-lci-CM}. Moreover,", "$\\overline{R} = R/IR = R/I_{\\mathfrak q'} = S_{\\mathfrak q}$", "is a quotient of dimension $\\text{height}(\\mathfrak q)$.", "Let $f_1, \\ldots, f_c \\in I_{\\mathfrak q'}$ be elements", "which generate $(I/I^2)_{\\mathfrak q}$. By Lemma \\ref{lemma-NAK}", "we see that $f_1, \\ldots, f_c$ generate $I_{\\mathfrak q'}$.", "Since the dimensions work out, we conclude", "by Proposition \\ref{proposition-CM-module} that", "$f_1, \\ldots, f_c$ is a regular sequence in $R$.", "By Lemma \\ref{lemma-regular-quasi-regular} we see that", "$(I/I^2)_{\\mathfrak q}$ is free.", "These arguments show that (2), (3), (4) are equivalent and", "that they imply the last statement of the lemma, and therefore", "they imply (5).", "\\medskip\\noindent", "If (5) holds, say $I_{\\mathfrak q'}$ is generated by a regular", "sequence of length $e$, then", "$\\text{height}(\\mathfrak q) = \\dim(S_{\\mathfrak q}) =", "\\dim(k[x_1, \\ldots, x_n]_{\\mathfrak q'}) - e =", "\\text{height}(\\mathfrak q') - e$ by dimension theory,", "see Section \\ref{section-dimension}. We conclude that $e = c$.", "Thus (5) implies (2).", "\\medskip\\noindent", "We continue with the notation introduced in the first paragraph.", "For each $f_i$ we may find $d_i \\in k[x_1, \\ldots, x_n]$,", "$d_i \\not \\in \\mathfrak q'$ such that", "$f_i' = d_i f_i \\in k[x_1, \\ldots, x_n]$.", "Then it is still true that $I_{\\mathfrak q'} = (f_1', \\ldots, f_c')R$.", "Hence there exists a $g' \\in k[x_1, \\ldots, x_n]$, $g' \\not \\in \\mathfrak q'$", "such that $I_{g'} = (f_1', \\ldots, f_c')$.", "Moreover, pick $g'' \\in k[x_1, \\ldots, x_n]$, $g'' \\not \\in \\mathfrak q'$", "such that $\\dim(S_{g''}) = \\dim_{\\mathfrak q} \\Spec(S)$.", "By Lemma \\ref{lemma-codimension} this dimension is equal to $n - c$.", "Finally, set $g$ equal to the image of $g'g''$ in $S$.", "Then we see that", "$$", "S_g \\cong k[x_1, \\ldots, x_n, x_{n + 1}]", "/", "(f_1', \\ldots, f_c', x_{n + 1}g'g'' - 1)", "$$", "and by our choice of $g''$ this ring has dimension $n - c$.", "Therefore it is a global complete intersection.", "Thus each of (2), (3), and (4) implies (1).", "\\medskip\\noindent", "Assume (1). Let $S_g \\cong k[y_1, \\ldots, y_m]/(f_1, \\ldots, f_t)$", "be a presentation of $S_g$ as a global complete intersection.", "Write $J = (f_1, \\ldots, f_t)$. Let $\\mathfrak q'' \\subset k[y_1, \\ldots, y_m]$", "be the prime corresponding to $\\mathfrak qS_g$. Note that", "$t = m - \\dim(S_g) =", "\\text{height}(\\mathfrak q'') - \\text{height}(\\mathfrak q)$,", "see Lemma \\ref{lemma-codimension} for the last equality.", "As seen in the proof of Lemma \\ref{lemma-lci-CM} (and also above) the elements", "$f_1, \\ldots, f_t$ form a regular sequence in the local ring", "$k[y_1, \\ldots, y_m]_{\\mathfrak q''}$.", "By Lemma \\ref{lemma-regular-quasi-regular} we see that", "$(J/J^2)_{\\mathfrak q}$ is free of rank $t$.", "By Lemma \\ref{lemma-conormal-module-localize} we have", "$$", "J/J^2 \\oplus S_g^n \\cong (I/I^2)_g \\oplus S_g^m", "$$", "Thus $(I/I^2)_{\\mathfrak q}$ is free of rank", "$t + n - m = m - \\dim(S_g) + n - m = n - \\dim(S_g) =", "\\text{height}(\\mathfrak q') - \\text{height}(\\mathfrak q) = c$.", "Thus we obtain (4)." ], "refs": [ "algebra-lemma-lci-CM", "algebra-lemma-NAK", "algebra-proposition-CM-module", "algebra-lemma-regular-quasi-regular", "algebra-lemma-codimension", "algebra-lemma-codimension", "algebra-lemma-lci-CM", "algebra-lemma-regular-quasi-regular", "algebra-lemma-conormal-module-localize" ], "ref_ids": [ 1166, 401, 1420, 746, 1008, 1008, 1166, 746, 1164 ] } ], "ref_ids": [ 1008 ] }, { "id": 1168, "type": "theorem", "label": "algebra-lemma-ci-well-defined", "categories": [ "algebra" ], "title": "algebra-lemma-ci-well-defined", "contents": [ "Let $A \\to B \\to C$ be surjective local ring homomorphisms.", "Assume $A$ and $B$ are regular local rings. The following are equivalent", "\\begin{enumerate}", "\\item $\\Ker(A \\to C)$ is generated by a regular sequence,", "\\item $\\Ker(A \\to C)$ is generated by $\\dim(A) - \\dim(C)$ elements,", "\\item $\\Ker(B \\to C)$ is generated by a regular sequence, and", "\\item $\\Ker(B \\to C)$ is generated by $\\dim(B) - \\dim(C)$ elements.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "A regular local ring is Cohen-Macaulay, see Lemma \\ref{lemma-regular-ring-CM}.", "Hence the equivalences (1) $\\Leftrightarrow$ (2) and", "(3) $\\Leftrightarrow$ (4), see Proposition \\ref{proposition-CM-module}.", "By Lemma \\ref{lemma-regular-quotient-regular}", "the ideal $\\Ker(A \\to B)$ can be generated", "by $\\dim(A) - \\dim(B)$ elements.", "Hence we see that (4) implies (2).", "\\medskip\\noindent", "It remains to show that (1) implies (4). We do this by induction on", "$\\dim(A) - \\dim(B)$. The case $\\dim(A) - \\dim(B) = 0$ is trivial.", "Assume $\\dim(A) > \\dim (B)$.", "Write $I = \\Ker(A \\to C)$ and $J = \\Ker(A \\to B)$.", "Note that $J \\subset I$. Our assumption is that the minimal number", "of generators of $I$ is $\\dim(A) - \\dim(C)$.", "Let $\\mathfrak m \\subset A$ be the maximal", "ideal. Consider the maps", "$$", "J/ \\mathfrak m J \\to I / \\mathfrak m I \\to \\mathfrak m /\\mathfrak m^2", "$$", "By Lemma \\ref{lemma-regular-quotient-regular} and its proof the", "composition is injective. Take any element $x \\in J$ which is", "not zero in $J /\\mathfrak mJ$. By the above and Nakayama's lemma", "$x$ is an element of a minimal set of generators of $I$.", "Hence we may replace $A$ by $A/xA$ and $I$ by $I/xA$ which", "decreases both $\\dim(A)$ and the minimal number of generators of $I$", "by $1$. Thus we win." ], "refs": [ "algebra-lemma-regular-ring-CM", "algebra-proposition-CM-module", "algebra-lemma-regular-quotient-regular", "algebra-lemma-regular-quotient-regular" ], "ref_ids": [ 941, 1420, 942, 942 ] } ], "ref_ids": [] }, { "id": 1169, "type": "theorem", "label": "algebra-lemma-lci-local", "categories": [ "algebra" ], "title": "algebra-lemma-lci-local", "contents": [ "Let $k$ be a field. Let $S$ be a local $k$-algebra essentially of finite", "type over $k$. The following are equivalent:", "\\begin{enumerate}", "\\item $S$ is a complete intersection over $k$,", "\\item for any surjection $R \\to S$ with $R$ a regular local ring", "essentially of finite presentation over $k$ the ideal", "$\\Ker(R \\to S)$ can be generated by a regular sequence,", "\\item for some surjection $R \\to S$ with $R$ a regular local ring", "essentially of finite presentation over $k$ the ideal", "$\\Ker(R \\to S)$ can be generated by", "$\\dim(R) - \\dim(S)$ elements,", "\\item there exists a global complete intersection", "$A$ over $k$ and a prime $\\mathfrak a$ of $A$ such", "that $S \\cong A_{\\mathfrak a}$, and", "\\item there exists a local complete intersection", "$A$ over $k$ and a prime $\\mathfrak a$ of $A$ such", "that $S \\cong A_{\\mathfrak a}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (2) implies (1) and (1) implies (3).", "It is also clear that (4) implies (5). Let us show that (3) implies", "(4). Thus we assume there exists a surjection", "$R \\to S$ with $R$ a regular local ring", "essentially of finite presentation over $k$ such that the ideal", "$\\Ker(R \\to S)$ can be generated by $\\dim(R) - \\dim(S)$ elements.", "We may write $R = (k[x_1, \\ldots, x_n]/J)_{\\mathfrak q}$", "for some $J \\subset k[x_1, \\ldots, x_n]$ and", "some prime $\\mathfrak q \\subset k[x_1, \\ldots, x_n]$ with", "$J \\subset \\mathfrak q$. Let $I \\subset k[x_1, \\ldots, x_n]$", "be the kernel of the map $k[x_1, \\ldots, x_n] \\to S$ so that", "$S \\cong (k[x_1, \\ldots, x_n]/I)_{\\mathfrak q}$.", "By assumption $(I/J)_{\\mathfrak q}$ is generated by", "$\\dim(R) - \\dim(S)$ elements. We conclude that", "$I_{\\mathfrak q}$ can be generated by", "$\\dim(k[x_1, \\ldots, x_n]_{\\mathfrak q}) - \\dim(S)$ elements", "by Lemma \\ref{lemma-ci-well-defined}.", "From Lemma \\ref{lemma-lci} we see that for some", "$g \\in k[x_1, \\ldots, x_n]$, $g \\not \\in \\mathfrak q$", "the algebra $(k[x_1, \\ldots, x_n]/I)_g$ is a global", "complete intersection and $S$ is isomorphic to", "a local ring of it.", "\\medskip\\noindent", "To finish the proof of the lemma we have to show that (5) implies (2).", "Assume (5) and let $\\pi : R \\to S$ be a surjection with $R$ a regular local", "$k$-algebra essentially of finite type over $k$.", "By assumption we have $S = A_{\\mathfrak a}$ for some local", "complete intersection $A$ over $k$.", "Choose a presentation $R = (k[y_1, \\ldots, y_m]/J)_{\\mathfrak q}$", "with $J \\subset \\mathfrak q \\subset k[y_1, \\ldots, y_m]$.", "We may and do assume that $J$ is the kernel of the map", "$k[y_1, \\ldots, y_m] \\to R$. Let $I \\subset k[y_1, \\ldots, y_m]$", "be the kernel of the map $k[y_1, \\ldots, y_m] \\to S = A_{\\mathfrak a}$.", "Then $J \\subset I$ and $(I/J)_{\\mathfrak q}$ is the kernel of", "the surjection $\\pi : R \\to S$. So", "$S = (k[y_1, \\ldots, y_m]/I)_{\\mathfrak q}$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-isomorphic-local-rings} we see that there exist", "$g \\in A$, $g \\not \\in \\mathfrak a$ and", "$g' \\in k[y_1, \\ldots, y_m]$, $g' \\not \\in \\mathfrak q$", "such that $A_g \\cong (k[y_1, \\ldots, y_m]/I)_{g'}$.", "After replacing $A$ by $A_g$ and $k[y_1, \\ldots, y_m]$ by", "$k[y_1, \\ldots, y_{m + 1}]$ we may assume that", "$A \\cong k[y_1, \\ldots, y_m]/I$. Consider the surjective", "maps of local rings", "$$", "k[y_1, \\ldots, y_m]_{\\mathfrak q} \\to R \\to S.", "$$", "We have to show that the kernel of $R \\to S$ is generated by", "a regular sequence. By Lemma \\ref{lemma-lci} we know that", "$k[y_1, \\ldots, y_m]_{\\mathfrak q} \\to A_{\\mathfrak a} = S$", "has this property (as $A$ is a local complete intersection over $k$).", "We win by Lemma \\ref{lemma-ci-well-defined}." ], "refs": [ "algebra-lemma-ci-well-defined", "algebra-lemma-lci", "algebra-lemma-isomorphic-local-rings", "algebra-lemma-lci", "algebra-lemma-ci-well-defined" ], "ref_ids": [ 1168, 1167, 1085, 1167, 1168 ] } ], "ref_ids": [] }, { "id": 1170, "type": "theorem", "label": "algebra-lemma-lci-at-prime", "categories": [ "algebra" ], "title": "algebra-lemma-lci-at-prime", "contents": [ "Let $k$ be a field. Let $S$ be a finite type $k$-algebra.", "Let $\\mathfrak q$ be a prime of $S$. The following are", "equivalent:", "\\begin{enumerate}", "\\item The local ring $S_{\\mathfrak q}$ is a complete intersection", "ring (Definition \\ref{definition-lci-local-ring}).", "\\item There exists a $g \\in S$, $g \\not \\in \\mathfrak q$", "such that $S_g$ is a local complete intersection over $k$.", "\\item There exists a $g \\in S$, $g \\not \\in \\mathfrak q$", "such that $S_g$ is a global complete intersection over $k$.", "\\item For any presentation $S = k[x_1, \\ldots, x_n]/I$ with", "$\\mathfrak q' \\subset k[x_1, \\ldots, x_n]$ corresponding to $\\mathfrak q$", "any of the equivalent conditions (1) -- (5) of Lemma \\ref{lemma-lci} hold.", "\\end{enumerate}" ], "refs": [ "algebra-definition-lci-local-ring", "algebra-lemma-lci" ], "proofs": [ { "contents": [ "This is a combination of Lemmas \\ref{lemma-lci} and \\ref{lemma-lci-local}", "and the definitions." ], "refs": [ "algebra-lemma-lci", "algebra-lemma-lci-local" ], "ref_ids": [ 1167, 1169 ] } ], "ref_ids": [ 1531, 1167 ] }, { "id": 1171, "type": "theorem", "label": "algebra-lemma-lci-global", "categories": [ "algebra" ], "title": "algebra-lemma-lci-global", "contents": [ "Let $k$ be a field. Let $S$ be a finite type $k$-algebra.", "The following are equivalent:", "\\begin{enumerate}", "\\item The ring $S$ is a local complete intersection over $k$.", "\\item All local rings of $S$ are complete intersection rings over $k$.", "\\item All localizations of $S$", "at maximal ideals are complete intersection rings over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-lci-at-prime},", "the fact that $\\Spec(S)$ is quasi-compact and the definitions." ], "refs": [ "algebra-lemma-lci-at-prime" ], "ref_ids": [ 1170 ] } ], "ref_ids": [] }, { "id": 1172, "type": "theorem", "label": "algebra-lemma-lci-field-change-local", "categories": [ "algebra" ], "title": "algebra-lemma-lci-field-change-local", "contents": [ "Let $k \\subset K$ be a field extension.", "Let $S$ be a finite type algebra over $k$.", "Let $\\mathfrak q_K$ be a prime of $S_K = K \\otimes_k S$", "and let $\\mathfrak q$ be the corresponding prime of $S$.", "Then $S_{\\mathfrak q}$ is a complete intersection", "over $k$ (Definition \\ref{definition-lci-local-ring})", "if and only if $(S_K)_{\\mathfrak q_K}$ is a complete", "intersection over $K$." ], "refs": [ "algebra-definition-lci-local-ring" ], "proofs": [ { "contents": [ "Choose a presentation $S = k[x_1, \\ldots, x_n]/I$.", "This gives a presentation", "$S_K = K[x_1, \\ldots, x_n]/I_K$ where $I_K = K \\otimes_k I$.", "Let $\\mathfrak q_K' \\subset K[x_1, \\ldots, x_n]$,", "resp.\\ $\\mathfrak q' \\subset k[x_1, \\ldots, x_n]$ be", "the corresponding prime. We will show that the equivalent conditions", "of Lemma \\ref{lemma-lci}", "hold for the pair $(S = k[x_1, \\ldots, x_n]/I, \\mathfrak q)$", "if and only if they hold for the pair", "$(S_K = K[x_1, \\ldots, x_n]/I_K, \\mathfrak q_K)$.", "The lemma will follow from this (see Lemma \\ref{lemma-lci-at-prime}).", "\\medskip\\noindent", "By Lemma \\ref{lemma-dimension-at-a-point-preserved-field-extension} we have", "$\\dim_{\\mathfrak q} S = \\dim_{\\mathfrak q_K} S_K$.", "Hence the integer $c$ occurring in Lemma \\ref{lemma-lci}", "is the same for the pair $(S = k[x_1, \\ldots, x_n]/I, \\mathfrak q)$", "as for the pair $(S_K = K[x_1, \\ldots, x_n]/I_K, \\mathfrak q_K)$.", "On the other hand we have", "\\begin{eqnarray*}", "I \\otimes_{k[x_1, \\ldots, x_n]} \\kappa(\\mathfrak q')", "\\otimes_{\\kappa(\\mathfrak q')} \\kappa(\\mathfrak q_K')", "& = &", "I \\otimes_{k[x_1, \\ldots, x_n]} \\kappa(\\mathfrak q_K') \\\\", "& = &", "I \\otimes_{k[x_1, \\ldots, x_n]} K[x_1, \\ldots, x_n]", "\\otimes_{K[x_1, \\ldots, x_n]} \\kappa(\\mathfrak q_K') \\\\", "& = &", "(K \\otimes_k I) \\otimes_{K[x_1, \\ldots, x_n]} \\kappa(\\mathfrak q_K') \\\\", "& = &", "I_K \\otimes_{K[x_1, \\ldots, x_n]} \\kappa(\\mathfrak q'_K).", "\\end{eqnarray*}", "Therefore,", "$\\dim_{\\kappa(\\mathfrak q')}", "I \\otimes_{k[x_1, \\ldots, x_n]} \\kappa(\\mathfrak q')", "=", "\\dim_{\\kappa(\\mathfrak q'_K)}", "I_K \\otimes_{K[x_1, \\ldots, x_n]} \\kappa(\\mathfrak q_K')$.", "Thus it follows from", "Nakayama's Lemma \\ref{lemma-NAK} that the minimal number", "of generators of $I_{\\mathfrak q'}$ is the same as the minimal", "number of generators of $(I_K)_{\\mathfrak q'_K}$.", "Thus the lemma follows from characterization (2) of Lemma \\ref{lemma-lci}." ], "refs": [ "algebra-lemma-lci", "algebra-lemma-lci-at-prime", "algebra-lemma-dimension-at-a-point-preserved-field-extension", "algebra-lemma-lci", "algebra-lemma-NAK", "algebra-lemma-lci" ], "ref_ids": [ 1167, 1170, 1010, 1167, 401, 1167 ] } ], "ref_ids": [ 1531 ] }, { "id": 1173, "type": "theorem", "label": "algebra-lemma-lci-field-change", "categories": [ "algebra" ], "title": "algebra-lemma-lci-field-change", "contents": [ "Let $k \\to K$ be a field extension.", "Let $S$ be a finite type $k$-algebra.", "Then $S$ is a local complete intersection over $k$ if and", "only if $S \\otimes_k K$ is a local complete intersection over $K$." ], "refs": [], "proofs": [ { "contents": [ "This follows from a combination of Lemmas", "\\ref{lemma-lci-global} and \\ref{lemma-lci-field-change-local}.", "But we also give a different", "proof here (based on the same principles).", "\\medskip\\noindent", "Set $S' = S \\otimes_k K$. Let $\\alpha : k[x_1, \\ldots, x_n] \\to S$ be a", "presentation with kernel $I$. Let $\\alpha' : K[x_1, \\ldots, x_n] \\to S'$", "be the induced presentation with kernel $I'$.", "\\medskip\\noindent", "Suppose that $S$ is a local complete intersection.", "Pick a prime $\\mathfrak q \\subset S'$. Denote", "$\\mathfrak q'$ the corresponding prime of $K[x_1, \\ldots, x_n]$,", "$\\mathfrak p$ the corresponding prime of $S$, and", "$\\mathfrak p'$ the corresponding prime of $k[x_1, \\ldots, x_n]$.", "Consider the following diagram of Noetherian local rings", "$$", "\\xymatrix{", "S'_{\\mathfrak q} & K[x_1, \\ldots, x_n]_{\\mathfrak q'} \\ar[l] \\\\", "S_{\\mathfrak p}\\ar[u] & k[x_1, \\ldots, x_n]_{\\mathfrak p'} \\ar[u] \\ar[l]", "}", "$$", "By Lemma \\ref{lemma-lci} we know that $S_{\\mathfrak p}$", "is cut out by some regular sequence $f_1, \\ldots, f_c$ in", "$k[x_1, \\ldots, x_n]_{\\mathfrak p'}$. Since the right vertical", "arrow is flat we see that the images of $f_1, \\ldots, f_c$", "form a regular sequence in $K[x_1, \\ldots, x_n]_{\\mathfrak q'}$.", "Because tensoring with $K$ over $k$ is an exact functor we have", "$S'_{\\mathfrak q} = K[x_1, \\ldots, x_n]_{\\mathfrak q'}/(f_1, \\ldots, f_c)$.", "Hence by Lemma \\ref{lemma-lci} again we see that $S'$ is a local", "complete intersection in a neighbourhood of $\\mathfrak q$. Since", "$\\mathfrak q$ was arbitrary we see that $S'$ is a local complete", "intersection over $K$.", "\\medskip\\noindent", "Suppose that $S'$ is a local complete intersection.", "Pick a maximal ideal $\\mathfrak m$ of $S$. Let $\\mathfrak m'$", "denote the corresponding maximal ideal of $k[x_1, \\ldots, x_n]$.", "Denote $\\kappa = \\kappa(\\mathfrak m)$ the residue field.", "By Remark \\ref{remark-fundamental-diagram} the primes of", "$S'$ lying over $\\mathfrak m$ correspond to primes", "in $K \\otimes_k \\kappa$. By the Hilbert-Nullstellensatz", "Theorem \\ref{theorem-nullstellensatz} we have $[\\kappa : k] < \\infty$.", "Hence $K \\otimes_k \\kappa$ is finite nonzero over $K$.", "Hence $K \\otimes_k \\kappa$ has a finite number $> 0$ of primes", "which are all maximal, each of which has a residue field", "finite over $K$ (see Section \\ref{section-artinian}).", "Hence there are finitely many $> 0$ prime ideals", "$\\mathfrak n \\subset S'$ lying over $\\mathfrak m$,", "each of which is maximal and has a residue field", "which is finite over $K$. Pick one, say $\\mathfrak n \\subset S'$,", "and let $\\mathfrak n' \\subset K[x_1, \\ldots, x_n]$ denote the corresponding", "prime ideal of $K[x_1, \\ldots, x_n]$.", "Note that since $V(\\mathfrak mS')$ is finite, we see that", "$\\mathfrak n$ is an isolated closed point of it, and we", "deduce that $\\mathfrak mS'_{\\mathfrak n}$ is an ideal of definition", "of $S'_{\\mathfrak n}$. This implies that", "$\\dim(S_{\\mathfrak m}) = \\dim(S'_{\\mathfrak n})$ for example by", "Lemma \\ref{lemma-dimension-base-fibre-equals-total}.", "(This can also be seen using", "Lemma \\ref{lemma-dimension-at-a-point-preserved-field-extension}.)", "Consider the corresponding diagram of Noetherian local rings", "$$", "\\xymatrix{", "S'_{\\mathfrak n} & K[x_1, \\ldots, x_n]_{\\mathfrak n'} \\ar[l] \\\\", "S_{\\mathfrak m}\\ar[u] & k[x_1, \\ldots, x_n]_{\\mathfrak m'} \\ar[u] \\ar[l]", "}", "$$", "According to Lemma \\ref{lemma-change-base-NL} we have", "$\\NL(\\alpha) \\otimes_S S' = \\NL(\\alpha')$, in particular", "$I'/(I')^2 = I/I^2 \\otimes_S S'$. Thus", "$(I/I^2)_{\\mathfrak m} \\otimes_{S_{\\mathfrak m}} \\kappa$", "and", "$(I'/(I')^2)_{\\mathfrak n} \\otimes_{S'_{\\mathfrak n}} \\kappa(\\mathfrak n)$", "have the same dimension. Since $(I'/(I')^2)_{\\mathfrak n}$", "is free of rank $n - \\dim S'_{\\mathfrak n}$ we deduce that", "$(I/I^2)_{\\mathfrak m}$ can be generated by", "$n - \\dim S'_{\\mathfrak n} = n - \\dim S_{\\mathfrak m}$ elements.", "By Lemma \\ref{lemma-lci} we see that $S$ is a local", "complete intersection in a neighbourhood of $\\mathfrak m$.", "Since $\\mathfrak m$ was any maximal ideal we conclude that", "$S$ is a local complete intersection." ], "refs": [ "algebra-lemma-lci-global", "algebra-lemma-lci-field-change-local", "algebra-lemma-lci", "algebra-lemma-lci", "algebra-remark-fundamental-diagram", "algebra-theorem-nullstellensatz", "algebra-lemma-dimension-base-fibre-equals-total", "algebra-lemma-dimension-at-a-point-preserved-field-extension", "algebra-lemma-change-base-NL", "algebra-lemma-lci" ], "ref_ids": [ 1171, 1172, 1167, 1167, 1558, 316, 987, 1010, 1156, 1167 ] } ], "ref_ids": [] }, { "id": 1174, "type": "theorem", "label": "algebra-lemma-lci-permanence-initial", "categories": [ "algebra" ], "title": "algebra-lemma-lci-permanence-initial", "contents": [ "Let", "$$", "\\xymatrix{", "B & S \\ar[l] \\\\", "A \\ar[u] & R \\ar[l] \\ar[u]", "}", "$$", "be a commutative square of local rings. Assume", "\\begin{enumerate}", "\\item $R$ and $\\overline{S} = S/\\mathfrak m_R S$ are regular local rings,", "\\item $A = R/I$ and $B = S/J$ for some ideals $I$, $J$,", "\\item $J \\subset S$ and", "$\\overline{J} = J/\\mathfrak m_R \\cap J \\subset \\overline{S}$", "are generated by regular sequences, and", "\\item $A \\to B$ and $R \\to S$ are flat.", "\\end{enumerate}", "Then $I$ is generated by a regular sequence." ], "refs": [], "proofs": [ { "contents": [ "Set $\\overline{B} = B/\\mathfrak m_RB = B/\\mathfrak m_AB$ so that", "$\\overline{B} = \\overline{S}/\\overline{J}$.", "Let $f_1, \\ldots, f_{\\overline{c}} \\in J$ be elements such that", "$\\overline{f}_1, \\ldots, \\overline{f}_{\\overline{c}} \\in \\overline{J}$", "form a regular sequence generating $\\overline{J}$.", "Note that $\\overline{c} = \\dim(\\overline{S}) - \\dim(\\overline{B})$,", "see Lemma \\ref{lemma-ci-well-defined}.", "By Lemma \\ref{lemma-grothendieck-regular-sequence}", "the ring $S/(f_1, \\ldots, f_{\\overline{c}})$ is flat", "over $R$. Hence $S/(f_1, \\ldots, f_{\\overline{c}}) + IS$ is flat over $A$.", "The map $S/(f_1, \\ldots, f_{\\overline{c}}) + IS \\to B$ is therefore a", "surjection of finite $S/IS$-modules flat over $A$ which", "is an isomorphism modulo $\\mathfrak m_A$, and hence an", "isomorphism by Lemma \\ref{lemma-mod-injective}. In other words,", "$J = (f_1, \\ldots, f_{\\overline{c}}) + IS$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-ci-well-defined} again the ideal $J$ is", "generated by a regular sequence of $c = \\dim(S) - \\dim(B)$ elements. Hence", "$J/\\mathfrak m_SJ$ is a vector space of dimension $c$.", "By the description of $J$ above there exist", "$g_1, \\ldots, g_{c - \\overline{c}} \\in I$ such that", "$J$ is generated by", "$f_1, \\ldots, f_{\\overline{c}}, g_1, \\ldots, g_{c - \\overline{c}}$", "(use Nakayama's Lemma \\ref{lemma-NAK}). Consider the ring", "$A' = R/(g_1, \\ldots, g_{c - \\overline{c}})$ and the surjection", "$A' \\to A$. We see from the above that", "$B = S/(f_1, \\ldots, f_{\\overline{c}}, g_1, \\ldots, g_{c - \\overline{c}})$", "is flat over $A'$ (as $S/(f_1, \\ldots, f_{\\overline{c}})$ is flat", "over $R$). Hence $A' \\to B$ is injective (as it is faithfully flat,", "see Lemma \\ref{lemma-local-flat-ff}).", "Since this map factors through $A$ we get $A' = A$.", "Note that $\\dim(B) = \\dim(A) + \\dim(\\overline{B})$, and", "$\\dim(S) = \\dim(R) + \\dim(\\overline{S})$, see", "Lemma \\ref{lemma-dimension-base-fibre-equals-total}.", "Hence $c - \\overline{c} = \\dim(R) -\\dim(A)$ by elementary algebra.", "Thus $I = (g_1, \\ldots, g_{c - \\overline{c}})$ is generated", "by a regular sequence according to Lemma \\ref{lemma-ci-well-defined}." ], "refs": [ "algebra-lemma-ci-well-defined", "algebra-lemma-grothendieck-regular-sequence", "algebra-lemma-mod-injective", "algebra-lemma-ci-well-defined", "algebra-lemma-NAK", "algebra-lemma-local-flat-ff", "algebra-lemma-dimension-base-fibre-equals-total", "algebra-lemma-ci-well-defined" ], "ref_ids": [ 1168, 885, 883, 1168, 401, 537, 987, 1168 ] } ], "ref_ids": [] }, { "id": 1175, "type": "theorem", "label": "algebra-lemma-syntomic-descends", "categories": [ "algebra" ], "title": "algebra-lemma-syntomic-descends", "contents": [ "\\begin{slogan}", "Being syntomic is fpqc local on the base.", "\\end{slogan}", "Let $R \\to S$ be a ring map.", "Let $R \\to R'$ be a faithfully flat ring map.", "Set $S' = R'\\otimes_R S$.", "Then $R \\to S$ is syntomic if and only if $R' \\to S'$ is syntomic." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-presentation-descends} and", "Lemma \\ref{lemma-flatness-descends} this holds for the property", "of being flat and for the property of being of finite presentation.", "The map $\\Spec(R') \\to \\Spec(R)$ is surjective,", "see Lemma \\ref{lemma-ff-rings}. Thus it suffices to show", "given primes $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p \\subset R$", "that $S \\otimes_R \\kappa(\\mathfrak p)$ is a local complete", "intersection if and only if $S' \\otimes_{R'} \\kappa(\\mathfrak p')$", "is a local complete intersection. Note that", "$S' \\otimes_{R'} \\kappa(\\mathfrak p') =", "S \\otimes_R \\kappa(\\mathfrak p)", "\\otimes_{\\kappa(\\mathfrak p)} \\kappa(\\mathfrak p')$.", "Thus Lemma \\ref{lemma-lci-field-change} applies." ], "refs": [ "algebra-lemma-finite-presentation-descends", "algebra-lemma-flatness-descends", "algebra-lemma-ff-rings", "algebra-lemma-lci-field-change" ], "ref_ids": [ 1080, 528, 536, 1173 ] } ], "ref_ids": [] }, { "id": 1176, "type": "theorem", "label": "algebra-lemma-base-change-syntomic", "categories": [ "algebra" ], "title": "algebra-lemma-base-change-syntomic", "contents": [ "Any base change of a syntomic map is syntomic." ], "refs": [], "proofs": [ { "contents": [ "This is true for being flat, for being of finite presentation,", "and for having local complete intersections as fibres by", "Lemmas \\ref{lemma-flat-base-change}, \\ref{lemma-compose-finite-type} and", "\\ref{lemma-lci-field-change}." ], "refs": [ "algebra-lemma-flat-base-change", "algebra-lemma-compose-finite-type", "algebra-lemma-lci-field-change" ], "ref_ids": [ 527, 333, 1173 ] } ], "ref_ids": [] }, { "id": 1177, "type": "theorem", "label": "algebra-lemma-local-syntomic", "categories": [ "algebra" ], "title": "algebra-lemma-local-syntomic", "contents": [ "Let $R \\to S$ be a ring map.", "Suppose we have $g_1, \\ldots g_m \\in S$ which generate the", "unit ideal such that each $R \\to S_{g_i}$ is syntomic.", "Then $R \\to S$ is syntomic." ], "refs": [], "proofs": [ { "contents": [ "This is true for being flat and for being of finite presentation by", "Lemmas \\ref{lemma-flat-localization} and \\ref{lemma-cover-upstairs}.", "The property of having fibre rings which are local complete intersections", "is local on $S$ by its very definition, see", "Definition \\ref{definition-lci-field}." ], "refs": [ "algebra-lemma-flat-localization", "algebra-lemma-cover-upstairs", "algebra-definition-lci-field" ], "ref_ids": [ 538, 412, 1530 ] } ], "ref_ids": [] }, { "id": 1178, "type": "theorem", "label": "algebra-lemma-huber", "categories": [ "algebra" ], "title": "algebra-lemma-huber", "contents": [ "Let $S$ be a finitely presented $R$-algebra which has a presentation", "$S = R[x_1, \\ldots, x_n]/I$ such that $I/I^2$ is free over $S$. Then", "$S$ has a presentation $S = R[y_1, \\ldots, y_m]/(f_1, \\ldots, f_c)$", "such that $(f_1, \\ldots, f_c)/(f_1, \\ldots, f_c)^2$ is free with", "basis given by the classes of $f_1, \\ldots, f_c$." ], "refs": [], "proofs": [ { "contents": [ "Note that $I$ is a finitely generated ideal by", "Lemma \\ref{lemma-finite-presentation-independent}.", "Let $f_1, \\ldots, f_c \\in I$ be elements which map to a basis of $I/I^2$.", "By Nakayama's lemma (Lemma \\ref{lemma-NAK})", "there exists a $g \\in 1 + I$ such that", "$$", "g \\cdot I \\subset (f_1, \\ldots, f_c)", "$$", "and $I_g \\cong (f_1, \\ldots, f_c)_g$. Hence we see that", "$$", "S \\cong R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)[1/g]", "\\cong R[x_1, \\ldots, x_n, x_{n + 1}]/(f_1, \\ldots, f_c, gx_{n + 1} - 1)", "$$", "as desired. It follows that $f_1, \\ldots, f_c,gx_{n + 1} - 1$", "form a basis for", "$(f_1, \\ldots, f_c, gx_{n + 1} - 1)/(f_1, \\ldots, f_c, gx_{n + 1} - 1)^2$", "for example by applying Lemma \\ref{lemma-principal-localization-NL}." ], "refs": [ "algebra-lemma-finite-presentation-independent", "algebra-lemma-NAK", "algebra-lemma-principal-localization-NL" ], "ref_ids": [ 334, 401, 1160 ] } ], "ref_ids": [] }, { "id": 1179, "type": "theorem", "label": "algebra-lemma-adjoin-roots", "categories": [ "algebra" ], "title": "algebra-lemma-adjoin-roots", "contents": [ "Suppose that $A$ is a ring, and", "$P(x) = x^n + b_1 x^{n-1} + \\ldots + b_n \\in A[x]$ is", "a monic polynomial over $A$. Then there exists a", "syntomic, finite locally free, faithfully flat ring extension", "$A \\subset A'$ such that $P(x) = \\prod_{i = 1, \\ldots, n} (x - \\beta_i)$", "for certain $\\beta_i \\in A'$." ], "refs": [], "proofs": [ { "contents": [ "Take $A' = A \\otimes_R S$, where $R$ and $S$ are as in", "Example \\ref{example-roots-universal-polynomial},", "where $R \\to A$ maps $a_i$ to $b_i$, and let", "$\\beta_i = -1 \\otimes \\alpha_i$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1180, "type": "theorem", "label": "algebra-lemma-base-change-relative-global-complete-intersection", "categories": [ "algebra" ], "title": "algebra-lemma-base-change-relative-global-complete-intersection", "contents": [ "Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ be a", "relative global complete intersection", "(Definition \\ref{definition-relative-global-complete-intersection})", "\\begin{enumerate}", "\\item For any $R \\to R'$ the base change", "$R' \\otimes_R S = R'[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ is a relative", "global complete intersection.", "\\item For any $g \\in S$ which is the image of $h \\in R[x_1, \\ldots, x_n]$", "the ring", "$S_g = R[x_1, \\ldots, x_n, x_{n + 1}]/(f_1, \\ldots, f_c, hx_{n + 1} - 1)$", "is a relative global complete intersection.", "\\item If $R \\to S$ factors as $R \\to R_f \\to S$ for some $f \\in R$.", "Then the ring $S = R_f[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "is a relative global complete intersection over $R_f$.", "\\end{enumerate}" ], "refs": [ "algebra-definition-relative-global-complete-intersection" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-dimension-preserved-field-extension}", "the fibres of a base change have the same dimension as the", "fibres of the original map. Moreover", "$R' \\otimes_R R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)", "= R'[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$. Thus (1) follows.", "The proof of (2) is that", "the localization at one element can be described as", "$S_g \\cong S[x_{n + 1}]/(gx_{n + 1} - 1)$.", "Assertion (3) follows from (1) since under the assumptions of (3) we have", "$R_f \\otimes_R S \\cong S$." ], "refs": [ "algebra-lemma-dimension-preserved-field-extension" ], "ref_ids": [ 1009 ] } ], "ref_ids": [ 1533 ] }, { "id": 1181, "type": "theorem", "label": "algebra-lemma-localize-relative-complete-intersection", "categories": [ "algebra" ], "title": "algebra-lemma-localize-relative-complete-intersection", "contents": [ "Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$.", "We will find $h \\in R[x_1, \\ldots, x_n]$ which maps to $g \\in S$", "such that", "$$", "S_g = R[x_1, \\ldots, x_n, x_{n + 1}]/(f_1, \\ldots, f_c, hx_{n + 1} - 1)", "$$", "is a relative global complete intersection with a presentation as in", "Definition \\ref{definition-relative-global-complete-intersection}", "in each of the following cases:", "\\begin{enumerate}", "\\item Let $I \\subset R$ be an ideal. If the fibres of", "$\\Spec(S/IS) \\to \\Spec(R/I)$ have dimension $n - c$, then we can", "find $(h, g)$ as above such that $g$ maps to $1 \\in S/IS$.", "\\item Let $\\mathfrak p \\subset R$ be a prime. If", "$\\dim(S \\otimes_R \\kappa(\\mathfrak p)) = n - c$, then we can", "find $(h, g)$ as above such that $g$ maps to a unit of", "$S \\otimes_R \\kappa(\\mathfrak p)$.", "\\item Let $\\mathfrak q \\subset S$ be a prime lying over", "$\\mathfrak p \\subset R$. If $\\dim_{\\mathfrak q}(S/R) = n - c$, then we can", "find $(h, g)$ as above such that $g \\not \\in \\mathfrak q$.", "\\end{enumerate}" ], "refs": [ "algebra-definition-relative-global-complete-intersection" ], "proofs": [ { "contents": [ "Ad (1). By Lemma \\ref{lemma-dimension-fibres-bounded-open-upstairs}", "there exists an open subset $W \\subset \\Spec(S)$ containing $V(IS)$", "such that all fibres of $W \\to \\Spec(R)$ have dimension $\\leq n - c$.", "Say $W = \\Spec(S) \\setminus V(J)$. Then $V(J) \\cap V(IS) = \\emptyset$", "hence we can find a $g \\in J$ which maps to $1 \\in S/IS$.", "Let $h \\in R[x_1, \\ldots, x_n]$ be any preimage of $g$.", "\\medskip\\noindent", "Ad (2). By Lemma \\ref{lemma-dimension-fibres-bounded-open-upstairs}", "there exists an open subset $W \\subset \\Spec(S)$ containing", "$\\Spec(S \\otimes_R \\kappa(\\mathfrak p))$", "such that all fibres of $W \\to \\Spec(R)$ have dimension $\\leq n - c$.", "Say $W = \\Spec(S) \\setminus V(J)$. Then", "$V(J \\cdot S \\otimes_R \\kappa(\\mathfrak p)) = \\emptyset$.", "Hence we can find a $g \\in J$ which maps to a unit in", "$S \\otimes_R \\kappa(\\mathfrak p)$ (details omitted).", "Let $h \\in R[x_1, \\ldots, x_n]$ be any preimage of $g$.", "\\medskip\\noindent", "Ad (3). By Lemma \\ref{lemma-dimension-fibres-bounded-open-upstairs}", "there exists a $g \\in S$, $g \\not \\in \\mathfrak q$", "such that all nonempty fibres of $R \\to S_g$", "have dimension $\\leq n - c$. Let $h \\in R[x_1, \\ldots, x_n]$", "be any element that maps to $g$." ], "refs": [ "algebra-lemma-dimension-fibres-bounded-open-upstairs", "algebra-lemma-dimension-fibres-bounded-open-upstairs", "algebra-lemma-dimension-fibres-bounded-open-upstairs" ], "ref_ids": [ 1075, 1075, 1075 ] } ], "ref_ids": [ 1533 ] }, { "id": 1182, "type": "theorem", "label": "algebra-lemma-relative-global-complete-intersection-Noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-relative-global-complete-intersection-Noetherian", "contents": [ "Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "be a relative global complete intersection", "(Definition \\ref{definition-relative-global-complete-intersection}).", "There exist a finite type $\\mathbf{Z}$-subalgebra $R_0 \\subset R$", "such that $f_i \\in R_0[x_1, \\ldots, x_n]$ and such that", "$$", "S_0 = R_0[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)", "$$", "is a relative global complete intersection." ], "refs": [ "algebra-definition-relative-global-complete-intersection" ], "proofs": [ { "contents": [ "Let $R_0 \\subset R$ be the $\\mathbf{Z}$-algebra of $R$ generated by all the", "coefficients of the polynomials $f_1, \\ldots, f_c$. Let", "$S_0 = R_0[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$.", "Clearly, $S = R \\otimes_{R_0} S_0$.", "Pick a prime $\\mathfrak q \\subset S$ and denote", "$\\mathfrak p \\subset R$, $\\mathfrak q_0 \\subset S_0$, and", "$\\mathfrak p_0 \\subset R_0$ the primes it lies over.", "Because $\\dim (S \\otimes_R \\kappa(\\mathfrak p) ) = n - c$", "we also have $\\dim (S_0 \\otimes_{R_0} \\kappa(\\mathfrak p_0)) = n - c$,", "see Lemma \\ref{lemma-dimension-preserved-field-extension}.", "By Lemma \\ref{lemma-dimension-fibres-bounded-open-upstairs}", "there exists a $g \\in S_0$, $g \\not \\in \\mathfrak q_0$", "such that all nonempty fibres of $R_0 \\to (S_0)_g$", "have dimension $\\leq n - c$. As $\\mathfrak q$ was arbitrary and", "$\\Spec(S)$ quasi-compact, we can find finitely many", "$g_1, \\ldots, g_m \\in S_0$ such that (a) for $j = 1, \\ldots, m$", "the nonempty fibres of", "$R_0 \\to (S_0)_{g_j}$ have dimension $\\leq n - c$ and (b) the image of", "$\\Spec(S) \\to \\Spec(S_0)$ is contained in $D(g_1) \\cup \\ldots \\cup D(g_m)$.", "In other words, the images of $g_1, \\ldots, g_m$ in $S = R \\otimes_{R_0} S_0$", "generate the unit ideal. After increasing $R_0$ we may assume", "that $g_1, \\ldots, g_m$ generate the unit ideal in $S_0$. By (a)", "the nonempty fibres of $R_0 \\to S_0$ all have dimension $\\leq n - c$", "and we conclude." ], "refs": [ "algebra-lemma-dimension-preserved-field-extension", "algebra-lemma-dimension-fibres-bounded-open-upstairs" ], "ref_ids": [ 1009, 1075 ] } ], "ref_ids": [ 1533 ] }, { "id": 1183, "type": "theorem", "label": "algebra-lemma-relative-global-complete-intersection-conormal", "categories": [ "algebra" ], "title": "algebra-lemma-relative-global-complete-intersection-conormal", "contents": [ "Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "be a relative global complete intersection (Definition", "\\ref{definition-relative-global-complete-intersection}). For every prime", "$\\mathfrak q$ of $S$, let $\\mathfrak q'$ denote the corresponding", "prime of $R[x_1, \\ldots, x_n]$. Then", "\\begin{enumerate}", "\\item $f_1, \\ldots, f_c$ is a regular sequence in the local ring", "$R[x_1, \\ldots, x_n]_{\\mathfrak q'}$,", "\\item each of the rings", "$R[x_1, \\ldots, x_n]_{\\mathfrak q'}/(f_1, \\ldots, f_i)$ is flat over $R$, and", "\\item the $S$-module $(f_1, \\ldots, f_c)/(f_1, \\ldots, f_c)^2$", "is free with basis given by the elements $f_i \\bmod (f_1, \\ldots, f_c)^2$.", "\\end{enumerate}" ], "refs": [ "algebra-definition-relative-global-complete-intersection" ], "proofs": [ { "contents": [ "First, by Lemma \\ref{lemma-regular-quasi-regular}, part (3) follows", "from part (1). Parts (1) and (2) immediately reduce to the Noetherian case", "by Lemma \\ref{lemma-relative-global-complete-intersection-Noetherian}", "(some minor details omitted). Assume $R$ is Noetherian. Let", "$\\mathfrak p = R \\cap \\mathfrak q'$.", "By Lemma \\ref{lemma-lci} for example we see", "that $f_1, \\ldots, f_c$ form a regular sequence in the local ring", "$R[x_1, \\ldots, x_n]_{\\mathfrak q'} \\otimes_R \\kappa(\\mathfrak p)$.", "Moreover, the local ring $R[x_1, \\ldots, x_n]_{\\mathfrak q'}$", "is flat over $R_{\\mathfrak p}$. Since $R$, and hence", "$R[x_1, \\ldots, x_n]_{\\mathfrak q'}$ is Noetherian we", "may apply Lemma \\ref{lemma-grothendieck-regular-sequence}", "to conclude." ], "refs": [ "algebra-lemma-regular-quasi-regular", "algebra-lemma-relative-global-complete-intersection-Noetherian", "algebra-lemma-lci", "algebra-lemma-grothendieck-regular-sequence" ], "ref_ids": [ 746, 1182, 1167, 885 ] } ], "ref_ids": [ 1533 ] }, { "id": 1184, "type": "theorem", "label": "algebra-lemma-relative-global-complete-intersection", "categories": [ "algebra" ], "title": "algebra-lemma-relative-global-complete-intersection", "contents": [ "A relative global complete intersection is syntomic, i.e., flat." ], "refs": [], "proofs": [ { "contents": [ "Let $R \\to S$ be a relative global complete intersection.", "The fibres are global complete intersections, and", "$S$ is of finite presentation over $R$.", "Thus the only thing to prove is that $R \\to S$ is flat.", "This is true by (2) of", "Lemma \\ref{lemma-relative-global-complete-intersection-conormal}." ], "refs": [ "algebra-lemma-relative-global-complete-intersection-conormal" ], "ref_ids": [ 1183 ] } ], "ref_ids": [] }, { "id": 1185, "type": "theorem", "label": "algebra-lemma-syntomic", "categories": [ "algebra" ], "title": "algebra-lemma-syntomic", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak q \\subset S$ be a prime lying over", "the prime $\\mathfrak p$ of $R$.", "The following are equivalent:", "\\begin{enumerate}", "\\item There exists an element $g \\in S$, $g \\not \\in \\mathfrak q$ such that", "$R \\to S_g$ is syntomic.", "\\item There exists an element $g \\in S$, $g \\not \\in \\mathfrak q$", "such that $S_g$ is a relative global complete intersection over $R$.", "\\item There exists an element $g \\in S$, $g \\not \\in \\mathfrak q$,", "such that $R \\to S_g$ is of finite presentation,", "the local ring map $R_{\\mathfrak p} \\to S_{\\mathfrak q}$ is flat, and", "the local ring $S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}$ is", "a complete intersection ring over $\\kappa(\\mathfrak p)$ (see", "Definition \\ref{definition-lci-local-ring}).", "\\end{enumerate}" ], "refs": [ "algebra-definition-lci-local-ring" ], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (3) is Lemma \\ref{lemma-lci-at-prime}.", "The implication (2) $\\Rightarrow$ (1) is", "Lemma \\ref{lemma-relative-global-complete-intersection}.", "It remains to show that (3) implies (2).", "\\medskip\\noindent", "Assume (3). After replacing $S$ by $S_g$ for some $g \\in S$,", "$g\\not\\in \\mathfrak q$ we may assume $S$ is finitely presented over $R$.", "Choose a presentation $S = R[x_1, \\ldots, x_n]/I$. Let", "$\\mathfrak q' \\subset R[x_1, \\ldots, x_n]$ be the prime corresponding", "to $\\mathfrak q$. Write $\\kappa(\\mathfrak p) = k$.", "Note that $S \\otimes_R k = k[x_1, \\ldots, x_n]/\\overline{I}$ where", "$\\overline{I} \\subset k[x_1, \\ldots, x_n]$ is the ideal generated", "by the image of $I$. Let $\\overline{\\mathfrak q}' \\subset k[x_1, \\ldots, x_n]$", "be the prime ideal generated by the image of $\\mathfrak q'$.", "By Lemma \\ref{lemma-lci-at-prime} the equivalent conditions of", "Lemma \\ref{lemma-lci} hold for $\\overline{I}$ and $\\overline{\\mathfrak q}'$.", "Say the dimension of", "$\\overline{I}_{\\overline{\\mathfrak q}'}/", "\\overline{\\mathfrak q}'\\overline{I}_{\\overline{\\mathfrak q}'}$", "over $\\kappa(\\overline{\\mathfrak q}')$ is $c$.", "Pick $f_1, \\ldots, f_c \\in I$ mapping to a basis of this vector space.", "The images $\\overline{f}_j \\in \\overline{I}$ generate", "$\\overline{I}_{\\overline{\\mathfrak q}'}$ (by Lemma \\ref{lemma-lci}).", "Set $S' = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$. Let $J$ be the", "kernel of the surjection $S' \\to S$. Since $S$ is of finite presentation", "$J$ is a finitely generated ideal", "(Lemma \\ref{lemma-compose-finite-type}). Consider the short exact sequence", "$$", "0 \\to J \\to S' \\to S \\to 0", "$$", "As $S_\\mathfrak q$ is flat over $R$ we see that", "$J_{\\mathfrak q'} \\otimes_R k \\to S'_{\\mathfrak q'} \\otimes_R k$", "is injective (Lemma \\ref{lemma-flat-tor-zero}).", "However, by construction $S'_{\\mathfrak q'} \\otimes_R k$", "maps isomorphically to $S_\\mathfrak q \\otimes_R k$. Hence we", "conclude that $J_{\\mathfrak q'} \\otimes_R k =", "J_{\\mathfrak q'}/\\mathfrak pJ_{\\mathfrak q'} = 0$. By Nakayama's", "lemma (Lemma \\ref{lemma-NAK}) we conclude that there exists a", "$g \\in R[x_1, \\ldots, x_n]$, $g \\not \\in \\mathfrak q'$ such that", "$J_g = 0$. In other words $S'_g \\cong S_g$. After further localizing", "we see that $S'$ (and hence $S$) becomes a relative global complete", "intersection by", "Lemma \\ref{lemma-localize-relative-complete-intersection}", "as desired." ], "refs": [ "algebra-lemma-lci-at-prime", "algebra-lemma-relative-global-complete-intersection", "algebra-lemma-lci-at-prime", "algebra-lemma-lci", "algebra-lemma-lci", "algebra-lemma-compose-finite-type", "algebra-lemma-flat-tor-zero", "algebra-lemma-NAK", "algebra-lemma-localize-relative-complete-intersection" ], "ref_ids": [ 1170, 1184, 1170, 1167, 1167, 333, 532, 401, 1181 ] } ], "ref_ids": [ 1531 ] }, { "id": 1186, "type": "theorem", "label": "algebra-lemma-syntomic-presentation-ideal-mod-squares", "categories": [ "algebra" ], "title": "algebra-lemma-syntomic-presentation-ideal-mod-squares", "contents": [ "Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/I$ for some", "finitely generated ideal $I$. If $g \\in S$ is such that", "$S_g$ is syntomic over $R$, then $(I/I^2)_g$ is a finite projective", "$S_g$-module." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-syntomic} there exist finitely many elements", "$g_1, \\ldots, g_m \\in S$ which generate the unit ideal in $S_g$", "such that each $S_{gg_j}$ is a relative global complete intersection", "over $R$. Since it suffices to prove that $(I/I^2)_{gg_j}$ is", "finite projective, see", "Lemma \\ref{lemma-finite-projective},", "we may assume that $S_g$ is a relative global complete intersection.", "In this case the result follows from", "Lemmas \\ref{lemma-conormal-module-localize} and", "\\ref{lemma-relative-global-complete-intersection-conormal}." ], "refs": [ "algebra-lemma-syntomic", "algebra-lemma-finite-projective", "algebra-lemma-conormal-module-localize", "algebra-lemma-relative-global-complete-intersection-conormal" ], "ref_ids": [ 1185, 795, 1164, 1183 ] } ], "ref_ids": [] }, { "id": 1187, "type": "theorem", "label": "algebra-lemma-composition-syntomic", "categories": [ "algebra" ], "title": "algebra-lemma-composition-syntomic", "contents": [ "Let $R \\to S$, $S \\to S'$ be ring maps.", "\\begin{enumerate}", "\\item If $R \\to S$ and $S \\to S'$ are syntomic, then $R \\to S'$", "is syntomic.", "\\item If $R \\to S$ and $S \\to S'$ are relative global complete intersections,", "then $R \\to S'$ is a relative global complete intersection.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $R \\to S$ and $S \\to S'$ are syntomic.", "This implies that $R \\to S'$ is flat by", "Lemma \\ref{lemma-composition-flat}.", "It also implies that $R \\to S'$ is of finite presentation by", "Lemma \\ref{lemma-compose-finite-type}.", "Thus it suffices to show that the fibres of $R \\to S'$ are", "local complete intersections.", "Choose a prime $\\mathfrak p \\subset R$.", "We have a factorization", "$$", "\\kappa(\\mathfrak p) \\to", "S \\otimes_R \\kappa(\\mathfrak p) \\to", "S' \\otimes_R \\kappa(\\mathfrak p).", "$$", "By assumption $S \\otimes_R \\kappa(\\mathfrak p)$ is", "a local complete intersection, and by Lemma \\ref{lemma-base-change-syntomic}", "we see that $S \\otimes_R \\kappa(\\mathfrak p)$ is syntomic over", "$S \\otimes_R \\kappa(\\mathfrak p)$.", "After replacing $S$ by $S \\otimes_R \\kappa(\\mathfrak p)$", "and $S'$ by $S' \\otimes_R \\kappa(\\mathfrak p)$ we may assume", "that $R$ is a field. Say $R = k$.", "\\medskip\\noindent", "Choose a prime $\\mathfrak q' \\subset S'$ lying over the prime", "$\\mathfrak q$ of $S$. Our goal is to find a $g' \\in S'$,", "$g' \\not \\in \\mathfrak q'$ such that $S'_{g'}$ is a global complete", "intersection over $k$. Choose a $g \\in S$, $g \\not \\in \\mathfrak q$", "such that $S_g = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ is", "a global complete intersection over $k$.", "Since $S_g \\to S'_g$ is still syntomic also, and $g \\not \\in \\mathfrak q'$", "we may replace $S$ by $S_g$ and $S'$ by $S'_g$ and assume that", "$S = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ is", "a global complete intersection over $k$. Next we choose a $g' \\in S'$,", "$g' \\not \\in \\mathfrak q'$ such that", "$S' = S[y_1, \\ldots, y_m]/(h_1, \\ldots, h_d)$", "is a relative global complete intersection over $S$.", "Hence we have reduced to part (2) of the lemma.", "\\medskip\\noindent", "Suppose that $R \\to S$ and $S \\to S'$ are", "relative global complete intersections. Say", "$S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "and", "$S' = S[y_1, \\ldots, y_m]/(h_1, \\ldots, h_d)$.", "Then", "$$", "S' \\cong", "R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]/(f_1, \\ldots, f_c, h'_1, \\ldots, h'_d)", "$$", "for some lifts $h_j' \\in R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]$ of the $h_j$.", "Hence it suffices to bound the dimensions of the fibres.", "Thus we may yet again assume $R = k$ is a field.", "In this case we see that we have a ring, namely $S$, which is of finite", "type over $k$ and equidimensional of dimension $n - c$, and a", "finite type ring map $S \\to S'$ all of whose nonempty fibre", "rings are equidimensional of dimension $m - d$. Then, by", "Lemma \\ref{lemma-dimension-base-fibre-total} for example applied", "to localizations at maximal ideals of $S'$, we see that", "$\\dim(S') \\leq n - c + m - d$ as desired." ], "refs": [ "algebra-lemma-composition-flat", "algebra-lemma-compose-finite-type", "algebra-lemma-base-change-syntomic", "algebra-lemma-dimension-base-fibre-total" ], "ref_ids": [ 524, 333, 1176, 986 ] } ], "ref_ids": [] }, { "id": 1188, "type": "theorem", "label": "algebra-lemma-lift-syntomic", "categories": [ "algebra" ], "title": "algebra-lemma-lift-syntomic", "contents": [ "Let $R$ be a ring and let $I \\subset R$ be an ideal.", "Let $R/I \\to \\overline{S}$ be a syntomic map.", "Then there exists elements $\\overline{g}_i \\in \\overline{S}$", "which generate the unit ideal of $\\overline{S}$", "such that each $\\overline{S}_{g_i} \\cong S_i/IS_i$", "for some relative global complete intersection $S_i$", "over $R$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-syntomic} we find a collection of elements", "$\\overline{g}_i \\in \\overline{S}$", "which generate the unit ideal of $\\overline{S}$", "such that each $\\overline{S}_{g_i}$ is a relative", "global complete intersection over $R/I$.", "Hence we may assume that $\\overline{S}$ is a", "relative global complete intersection.", "Write", "$\\overline{S} =", "(R/I)[x_1, \\ldots, x_n]/(\\overline{f}_1, \\ldots, \\overline{f}_c)$", "as in Definition \\ref{definition-relative-global-complete-intersection}.", "Choose $f_1, \\ldots, f_c \\in R[x_1, \\ldots, x_n]$", "lifting $\\overline{f}_1, \\ldots, \\overline{f}_c$.", "Set $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$.", "Note that $S/IS \\cong \\overline{S}$.", "By Lemma \\ref{lemma-localize-relative-complete-intersection}", "we can find $g \\in S$ mapping to $1$ in $\\overline{S}$ such", "that $S_g$ is a relative global complete intersection over $R$.", "Since $\\overline{S} \\cong S_g/IS_g$ this finishes the proof." ], "refs": [ "algebra-lemma-syntomic", "algebra-definition-relative-global-complete-intersection", "algebra-lemma-localize-relative-complete-intersection" ], "ref_ids": [ 1185, 1533, 1181 ] } ], "ref_ids": [] }, { "id": 1189, "type": "theorem", "label": "algebra-lemma-smooth-independent-presentation", "categories": [ "algebra" ], "title": "algebra-lemma-smooth-independent-presentation", "contents": [ "Let $R \\to S$ be a ring map of finite presentation.", "If for some presentation $\\alpha$ of $S$ over $R$ the", "naive cotangent complex $\\NL(\\alpha)$ is quasi-isomorphic", "to a finite projective $S$-module placed in degree $0$, then", "this holds for any presentation." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-NL-homotopy}." ], "refs": [ "algebra-lemma-NL-homotopy" ], "ref_ids": [ 1151 ] } ], "ref_ids": [] }, { "id": 1190, "type": "theorem", "label": "algebra-lemma-localize-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-localize-smooth", "contents": [ "Let $R \\to S$ be a smooth ring map.", "Any localization $S_g$ is smooth over $R$.", "If $f \\in R$ maps to an invertible element of $S$,", "then $R_f \\to S$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-localize-NL} the naive cotangent", "complex for $S_g$ over $R$ is the base change of the naive cotangent", "complex of $S$ over $R$. The assumption is that the naive cotangent", "complex of $S/R$ is $\\Omega_{S/R}$ and that this is a finite projective", "$S$-module. Hence so is its base change. Thus $S_g$ is smooth over $R$.", "\\medskip\\noindent", "The second assertion follows in the same way from", "Lemma \\ref{lemma-NL-localize-bottom}." ], "refs": [ "algebra-lemma-localize-NL", "algebra-lemma-NL-localize-bottom" ], "ref_ids": [ 1161, 1159 ] } ], "ref_ids": [] }, { "id": 1191, "type": "theorem", "label": "algebra-lemma-base-change-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-base-change-smooth", "contents": [ "\\begin{slogan}", "Smoothness is preserved under base change", "\\end{slogan}", "Let $R \\to S$ be a smooth ring map.", "Let $R \\to R'$ be any ring map.", "Then the base change $R' \\to S' = R' \\otimes_R S$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha : R[x_1, \\ldots, x_n] \\to S$ be a presentation", "with kernel $I$. Let $\\alpha' : R'[x_1, \\ldots, x_n] \\to R' \\otimes_R S$", "be the induced presentation. Let $I' = \\Ker(\\alpha')$.", "Since $0 \\to I \\to R[x_1, \\ldots, x_n] \\to S \\to 0$", "is exact, the sequence", "$R' \\otimes_R I \\to R'[x_1, \\ldots, x_n] \\to R' \\otimes_R S \\to 0$", "is exact. Thus $R' \\otimes_R I \\to I'$ is surjective.", "By Definition \\ref{definition-smooth} there is a short exact sequence", "$$", "0 \\to I/I^2 \\to", "\\Omega_{R[x_1, \\ldots, x_n]/R} \\otimes_{R[x_1, \\ldots, x_n]} S \\to", "\\Omega_{S/R} \\to", "0", "$$", "and the $S$-module $\\Omega_{S/R}$ is finite projective.", "In particular $I/I^2$ is a direct summand of", "$\\Omega_{R[x_1, \\ldots, x_n]/R} \\otimes_{R[x_1, \\ldots, x_n]} S$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "R' \\otimes_R (I/I^2) \\ar[r] \\ar[d] &", "R' \\otimes_R (\\Omega_{R[x_1, \\ldots, x_n]/R} \\otimes_{R[x_1, \\ldots, x_n]} S)", "\\ar[d] \\\\", "I'/(I')^2 \\ar[r] &", "\\Omega_{R'[x_1, \\ldots, x_n]/R'}", "\\otimes_{R'[x_1, \\ldots, x_n]} (R' \\otimes_R S)", "}", "$$", "Since the right vertical map is an isomorphism we see that", "the left vertical map is injective and surjective by what was", "said above. Thus we conclude that $\\NL(\\alpha')$ is quasi-isomorphic", "to $\\Omega_{S'/R'} \\cong S' \\otimes_S \\Omega_{S/R}$.", "And this is finite projective since it is the base change", "of a finite projective module." ], "refs": [ "algebra-definition-smooth" ], "ref_ids": [ 1534 ] } ], "ref_ids": [] }, { "id": 1192, "type": "theorem", "label": "algebra-lemma-smooth-over-field", "categories": [ "algebra" ], "title": "algebra-lemma-smooth-over-field", "contents": [ "Let $k$ be a field.", "Let $S$ be a smooth $k$-algebra.", "Then $S$ is a local complete intersection." ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-base-change-smooth} and", "\\ref{lemma-lci-field-change} it suffices to prove this when", "$k$ is algebraically closed. Choose a presentation", "$\\alpha : k[x_1, \\ldots, x_n] \\to S$ with kernel $I$. Let $\\mathfrak m$", "be a maximal ideal of $S$, and let $\\mathfrak m' \\supset I$ be the", "corresponding maximal ideal of $k[x_1, \\ldots, x_n]$.", "We will show that condition (5) of", "Lemma \\ref{lemma-lci}", "holds (with $\\mathfrak m$ instead of $\\mathfrak q$).", "We may write $\\mathfrak m' = (x_1 - a_1, \\ldots, x_n - a_n)$", "for some $a_i \\in k$, because $k$ is algebraically closed, see", "Theorem \\ref{theorem-nullstellensatz}.", "By our assumption that $k \\to S$ is smooth the $S$-module map", "$\\text{d} : I/I^2 \\to \\bigoplus_{i = 1}^n S \\text{d}x_i$", "is a split injection. Hence the corresponding map", "$I/\\mathfrak m' I \\to \\bigoplus \\kappa(\\mathfrak m') \\text{d}x_i$", "is injective. Say $\\dim_{\\kappa(\\mathfrak m')}(I/\\mathfrak m' I) = c$", "and pick $f_1, \\ldots, f_c \\in I$ which map to a $\\kappa(\\mathfrak m')$-basis", "of $I/\\mathfrak m' I$. By", "Nakayama's Lemma \\ref{lemma-NAK}", "we see that $f_1, \\ldots, f_c$ generate $I_{\\mathfrak m'}$ over", "$k[x_1, \\ldots, x_n]_{\\mathfrak m'}$. Consider the commutative diagram", "$$", "\\xymatrix{", "I \\ar[r] \\ar[d] & I/I^2 \\ar[rr] \\ar[d] & &", "I/\\mathfrak m'I \\ar[d] \\\\", "\\Omega_{k[x_1, \\ldots, x_n]/k} \\ar[r] &", "\\bigoplus S\\text{d}x_i \\ar[rr]^{\\text{d}x_i \\mapsto x_i - a_i} & &", "\\mathfrak m'/(\\mathfrak m')^2", "}", "$$", "(proof commutativity omitted). The middle vertical map is the one defining", "the naive cotangent complex of $\\alpha$. Note that the right lower", "horizontal arrow induces an isomorphism", "$\\bigoplus \\kappa(\\mathfrak m') \\text{d}x_i \\to \\mathfrak m'/(\\mathfrak m')^2$.", "Hence our generators $f_1, \\ldots, f_c$ of $I_{\\mathfrak m'}$ map to a", "collection of elements in $k[x_1, \\ldots, x_n]_{\\mathfrak m'}$ whose", "classes in $\\mathfrak m'/(\\mathfrak m')^2$ are linearly independent", "over $\\kappa(\\mathfrak m')$. Therefore they form a regular sequence", "in the ring $k[x_1, \\ldots, x_n]_{\\mathfrak m'}$ by", "Lemma \\ref{lemma-regular-ring-CM}.", "This verifies condition (5) of", "Lemma \\ref{lemma-lci}", "hence $S_g$ is a global complete intersection over $k$ for some", "$g \\in S$, $g \\not \\in \\mathfrak m$. As this works for any maximal", "ideal of $S$ we conclude that $S$ is a local complete intersection over $k$." ], "refs": [ "algebra-lemma-base-change-smooth", "algebra-lemma-lci-field-change", "algebra-lemma-lci", "algebra-theorem-nullstellensatz", "algebra-lemma-NAK", "algebra-lemma-regular-ring-CM", "algebra-lemma-lci" ], "ref_ids": [ 1191, 1173, 1167, 316, 401, 941, 1167 ] } ], "ref_ids": [] }, { "id": 1193, "type": "theorem", "label": "algebra-lemma-standard-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-standard-smooth", "contents": [ "Let", "$S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c) = R[x_1, \\ldots, x_n]/I$", "be a standard smooth algebra. Then", "\\begin{enumerate}", "\\item the ring map $R \\to S$ is smooth,", "\\item the $S$-module $\\Omega_{S/R}$ is free on", "$\\text{d}x_{c + 1}, \\ldots, \\text{d}x_n$,", "\\item the $S$-module $I/I^2$ is free on the classes of $f_1, \\ldots, f_c$,", "\\item for any $g \\in S$ the ring map $R \\to S_g$ is standard smooth,", "\\item for any ring map $R \\to R'$ the base change", "$R' \\to R'\\otimes_R S$ is standard smooth,", "\\item if $f \\in R$ maps to an invertible element in $S$, then", "$R_f \\to S$ is standard smooth, and", "\\item the ring $S$ is a relative global complete intersection over $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the naive cotangent complex of the given presentation", "$$", "(f_1, \\ldots, f_c)/(f_1, \\ldots, f_c)^2", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1}^n S \\text{d}x_i", "$$", "Let us compose this map with the projection onto the first $c$ direct summands", "of the direct sum. According to the definition of a standard smooth", "algebra the classes $f_i \\bmod (f_1, \\ldots, f_c)^2$ map to a basis of", "$\\bigoplus_{i = 1}^c S\\text{d}x_i$. We conclude that", "$(f_1, \\ldots, f_c)/(f_1, \\ldots, f_c)^2$ is free of rank $c$ with", "a basis given by the elements $f_i \\bmod (f_1, \\ldots, f_c)^2$, and", "that the homology in degree $0$, i.e., $\\Omega_{S/R}$,", "of the naive cotangent complex is a free $S$-module with basis the images of", "$\\text{d}x_{c + j}$, $j = 1, \\ldots, n - c$.", "In particular, this proves $R \\to S$ is smooth.", "\\medskip\\noindent", "The proofs of (4) and (6) are omitted. But see the example below and", "the proof of", "Lemma \\ref{lemma-base-change-relative-global-complete-intersection}.", "\\medskip\\noindent", "Let $\\varphi : R \\to R'$ be any ring map.", "Denote $S' = R'[x_1, \\ldots, x_n]/(f_1^\\varphi, \\ldots, f_c^\\varphi)$", "where $f^\\varphi$ is the polynomial obtained from $f \\in R[x_1, \\ldots, x_n]$", "by applying $\\varphi$ to all the coefficients. Then $S' \\cong R' \\otimes_R S$.", "Moreover, the determinant of Definition \\ref{definition-standard-smooth}", "for $S'/R'$ is equal to $g^\\varphi$. Its image in $S'$ is therefore", "the image of $g$ via $R[x_1, \\ldots, x_n] \\to S \\to S'$", "and hence invertible. This proves (5).", "\\medskip\\noindent", "To prove (7) it suffices to show that", "$S \\otimes_R \\kappa(\\mathfrak p)$ has dimension $n - c$", "for every prime $\\mathfrak p \\subset R$.", "By (5) it suffices to prove that any standard smooth", "algebra $k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "over a field $k$ has dimension $n - c$. We already", "know that $k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ is a local", "complete intersection by Lemma \\ref{lemma-smooth-over-field}.", "Hence, since $I/I^2$ is free of rank $c$ we see that", "$k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ has dimension", "$n - c$, by Lemma \\ref{lemma-lci} for example." ], "refs": [ "algebra-lemma-base-change-relative-global-complete-intersection", "algebra-definition-standard-smooth", "algebra-lemma-smooth-over-field", "algebra-lemma-lci" ], "ref_ids": [ 1180, 1535, 1192, 1167 ] } ], "ref_ids": [] }, { "id": 1194, "type": "theorem", "label": "algebra-lemma-compose-standard-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-compose-standard-smooth", "contents": [ "A composition of standard smooth ring maps is standard smooth." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $R \\to S$ and $S \\to S'$ are standard smooth. We choose", "presentations", "$S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "and", "$S' = S[y_1, \\ldots, y_m]/(g_1, \\ldots, g_d)$.", "Choose elements $g_j' \\in R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]$ mapping", "to the $g_j$. In this way we see", "$S' = R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]/", "(f_1, \\ldots, f_c, g'_1, \\ldots, g'_d)$.", "To show that $S'$ is standard smooth it suffices to verify", "that the determinant", "$$", "\\det", "\\left(", "\\begin{matrix}", "\\partial f_1/\\partial x_1 &", "\\ldots &", "\\partial f_c/\\partial x_1 &", "\\partial g_1/\\partial x_1 &", "\\ldots &", "\\partial g_d/\\partial x_1 \\\\", "\\ldots &", "\\ldots &", "\\ldots &", "\\ldots &", "\\ldots &", "\\ldots \\\\", "\\partial f_1/\\partial x_c &", "\\ldots &", "\\partial f_c/\\partial x_c &", "\\partial g_1/\\partial x_c &", "\\ldots &", "\\partial g_d/\\partial x_c \\\\", "0 &", "\\ldots &", "0 &", "\\partial g_1/\\partial y_1 &", "\\ldots &", "\\partial g_d/\\partial y_1 \\\\", "\\ldots &", "\\ldots &", "\\ldots &", "\\ldots &", "\\ldots &", "\\ldots \\\\", "0 &", "\\ldots &", "0 &", "\\partial g_1/\\partial y_d &", "\\ldots &", "\\partial g_d/\\partial y_d", "\\end{matrix}", "\\right)", "$$", "is invertible in $S'$. This is clear since it is the product", "of the two determinants which were assumed to be invertible", "by hypothesis." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1195, "type": "theorem", "label": "algebra-lemma-smooth-syntomic", "categories": [ "algebra" ], "title": "algebra-lemma-smooth-syntomic", "contents": [ "Let $R \\to S$ be a smooth ring map.", "There exists an open covering of $\\Spec(S)$ by", "standard opens $D(g)$ such that each $S_g$ is standard smooth", "over $R$. In particular $R \\to S$ is syntomic." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $\\alpha : R[x_1, \\ldots, x_n] \\to S$", "with kernel $I = (f_1, \\ldots, f_m)$. For every subset", "$E \\subset \\{1, \\ldots, m\\}$ consider the open", "subset $U_E$ where the classes $f_e, e\\in E$ freely generate", "the finite projective $S$-module $I/I^2$, see Lemma \\ref{lemma-cokernel-flat}.", "We may cover $\\Spec(S)$ by standard opens $D(g)$ each", "completely contained in one of the opens $U_E$. For such a $g$", "we look at the presentation", "$$", "\\beta : R[x_1, \\ldots, x_n, x_{n + 1}] \\longrightarrow S_g", "$$", "mapping $x_{n + 1}$ to $1/g$. Setting $J = \\Ker(\\beta)$ we", "use Lemma \\ref{lemma-principal-localization-NL} to see that", "$J/J^2 \\cong (I/I^2)_g \\oplus S_g$ is free.", "We may and do replace $S$ by $S_g$. Then using", "Lemma \\ref{lemma-huber} we may assume we have a presentation", "$\\alpha : R[x_1, \\ldots, x_n] \\to S$ with kernel $I = (f_1, \\ldots, f_c)$", "such that $I/I^2$ is free on the classes of $f_1, \\ldots, f_c$.", "\\medskip\\noindent", "Using the presentation $\\alpha$ obtained at the end of the previous", "paragraph, we more or less repeat this argument with", "the basis elements $\\text{d}x_1, \\ldots, \\text{d}x_n$", "of $\\Omega_{R[x_1, \\ldots, x_n]/R}$.", "Namely, for any subset $E \\subset \\{1, \\ldots, n\\}$ of cardinality $c$", "we may consider the open subset $U_E$ of $\\Spec(S)$ where", "the differential of $\\NL(\\alpha)$ composed with the projection", "$$", "S^{\\oplus c} \\cong I/I^2", "\\longrightarrow", "\\Omega_{R[x_1, \\ldots, x_n]/R} \\otimes_{R[x_1, \\ldots, x_n]} S", "\\longrightarrow", "\\bigoplus\\nolimits_{i \\in E} S\\text{d}x_i", "$$", "is an isomorphism. Again we may find a covering of $\\Spec(S)$", "by (finitely many) standard opens $D(g)$ such that each $D(g)$", "is completely contained in one of the opens $U_E$.", "By renumbering, we may assume $E = \\{1, \\ldots, c\\}$.", "For a $g$ with $D(g) \\subset U_E$ we look at the presentation", "$$", "\\beta : R[x_1, \\ldots, x_n, x_{n + 1}] \\to S_g", "$$", "mapping $x_{n + 1}$ to $1/g$. Setting $J = \\Ker(\\beta)$", "we conclude from Lemma \\ref{lemma-principal-localization-NL}", "that $J = (f_1, \\ldots, f_c, fx_{n + 1} - 1)$ where $\\alpha(f) = g$", "and that the composition", "$$", "J/J^2 \\longrightarrow", "\\Omega_{R[x_1, \\ldots, x_{n + 1}]/R} \\otimes_{R[x_1, \\ldots, x_{n + 1}]} S_g", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1}^c S_g\\text{d}x_i \\oplus S_g \\text{d}x_{n + 1}", "$$", "is an isomorphism. Reordering the coordinates as", "$x_1, \\ldots, x_c, x_{n + 1}, x_{c + 1}, \\ldots, x_n$", "we conclude that $S_g$ is standard smooth over $R$ as desired.", "\\medskip\\noindent", "This finishes the proof as standard smooth algebras are syntomic", "(Lemmas \\ref{lemma-standard-smooth} and", "\\ref{lemma-relative-global-complete-intersection})", "and being syntomic over $R$ is local on $S$", "(Lemma \\ref{lemma-local-syntomic})." ], "refs": [ "algebra-lemma-cokernel-flat", "algebra-lemma-principal-localization-NL", "algebra-lemma-huber", "algebra-lemma-principal-localization-NL", "algebra-lemma-standard-smooth", "algebra-lemma-relative-global-complete-intersection", "algebra-lemma-local-syntomic" ], "ref_ids": [ 804, 1160, 1178, 1160, 1193, 1184, 1177 ] } ], "ref_ids": [] }, { "id": 1196, "type": "theorem", "label": "algebra-lemma-smooth-at-point", "categories": [ "algebra" ], "title": "algebra-lemma-smooth-at-point", "contents": [ "Let $R \\to S$ be of finite presentation. Let $\\mathfrak q$ be a", "prime of $S$. The following are equivalent", "\\begin{enumerate}", "\\item $R \\to S$ is smooth at $\\mathfrak q$,", "\\item $H_1(L_{S/R})_\\mathfrak q = 0$ and", "$\\Omega_{S/R, \\mathfrak q}$ is a finite free $S_\\mathfrak q$-module,", "\\item $H_1(L_{S/R})_\\mathfrak q = 0$ and", "$\\Omega_{S/R, \\mathfrak q}$ is a projective $S_\\mathfrak q$-module, and", "\\item $H_1(L_{S/R})_\\mathfrak q = 0$ and", "$\\Omega_{S/R, \\mathfrak q}$ is a flat $S_\\mathfrak q$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use without further mention that formation of the", "naive cotangent complex commutes with localization, see", "Section \\ref{section-netherlander}, especially", "Lemma \\ref{lemma-localize-NL}.", "Note that $\\Omega_{S/R}$ is a finitely presented $S$-module, see", "Lemma \\ref{lemma-differentials-finitely-presented}. Hence", "(2), (3), and (4) are equivalent by Lemma \\ref{lemma-finite-projective}.", "It is clear that (1) implies the equivalent conditions (2), (3), and (4).", "Assume (2) holds. Writing $S_\\mathfrak q$ as", "the colimit of principal localizations we see from", "Lemma \\ref{lemma-colimit-category-fp-modules}", "that we can find a $g \\in S$, $g \\not \\in \\mathfrak q$ such that", "$(\\Omega_{S/R})_g$ is finite free. Choose a presentation", "$\\alpha : R[x_1, \\ldots, x_n] \\to S$ with kernel $I$. We may work", "with $\\NL(\\alpha)$ instead of $\\NL_{S/R}$, see", "Lemma \\ref{lemma-NL-homotopy}. The surjection", "$$", "\\Omega_{R[x_1, \\ldots, x_n]/R} \\otimes_R S \\to \\Omega_{S/R} \\to 0", "$$", "has a right inverse after inverting $g$ because $(\\Omega_{S/R})_g$ is", "projective. Hence the image of", "$\\text{d} : (I/I^2)_g \\to \\Omega_{R[x_1, \\ldots, x_n]/R} \\otimes_R S_g$", "is a direct summand and this map has a right inverse too.", "We conclude that $H_1(L_{S/R})_g$ is a quotient of $(I/I^2)_g$.", "In particular $H_1(L_{S/R})_g$ is a finite $S_g$-module.", "Thus the vanishing of $H_1(L_{S/R})_{\\mathfrak q}$", "implies the vanishing of $H_1(L_{S/R})_{gg'}$ for some $g' \\in S$,", "$g' \\not \\in \\mathfrak q$. Then $R \\to S_{gg'}$ is smooth by", "definition." ], "refs": [ "algebra-lemma-localize-NL", "algebra-lemma-differentials-finitely-presented", "algebra-lemma-finite-projective", "algebra-lemma-colimit-category-fp-modules", "algebra-lemma-NL-homotopy" ], "ref_ids": [ 1161, 1141, 795, 1095, 1151 ] } ], "ref_ids": [] }, { "id": 1197, "type": "theorem", "label": "algebra-lemma-locally-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-locally-smooth", "contents": [ "\\begin{slogan}", "A ring map is smooth if and only if it is smooth at all primes of the target", "\\end{slogan}", "Let $R \\to S$ be a ring map.", "Then $R \\to S$ is smooth if and only if $R \\to S$ is smooth", "at every prime $\\mathfrak q$ of $S$." ], "refs": [], "proofs": [ { "contents": [ "The direct implication is trivial. Suppose that $R \\to S$ is smooth", "at every prime $\\mathfrak q$ of $S$. Since $\\Spec(S)$ is", "quasi-compact, see Lemma \\ref{lemma-quasi-compact},", "there exists a finite covering", "$\\Spec(S) = \\bigcup D(g_i)$ such that each $S_{g_i}$ is", "smooth. By Lemma \\ref{lemma-cover-upstairs} this implies that", "$S$ is of finite presentation over $R$. According to", "Lemma \\ref{lemma-localize-NL} we see that", "$\\NL_{S/R} \\otimes_S S_{g_i}$ is quasi-isomorphic to a finite projective", "$S_{g_i}$-module. By Lemma \\ref{lemma-finite-projective}", "this implies that $\\NL_{S/R}$ is quasi-isomorphic to a finite", "projective $S$-module." ], "refs": [ "algebra-lemma-quasi-compact", "algebra-lemma-cover-upstairs", "algebra-lemma-localize-NL", "algebra-lemma-finite-projective" ], "ref_ids": [ 395, 412, 1161, 795 ] } ], "ref_ids": [] }, { "id": 1198, "type": "theorem", "label": "algebra-lemma-compose-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-compose-smooth", "contents": [ "A composition of smooth ring maps is smooth." ], "refs": [], "proofs": [ { "contents": [ "You can prove this in many different ways. One way is to use", "the snake lemma (Lemma \\ref{lemma-snake}), the Jacobi-Zariski sequence", "(Lemma \\ref{lemma-exact-sequence-NL}), combined with the", "characterization of projective modules as being", "direct summands of free modules (Lemma \\ref{lemma-characterize-projective}).", "Another proof can be obtained by combining", "Lemmas \\ref{lemma-smooth-syntomic}, \\ref{lemma-compose-standard-smooth}", "and \\ref{lemma-locally-smooth}." ], "refs": [ "algebra-lemma-snake", "algebra-lemma-exact-sequence-NL", "algebra-lemma-characterize-projective", "algebra-lemma-smooth-syntomic", "algebra-lemma-compose-standard-smooth", "algebra-lemma-locally-smooth" ], "ref_ids": [ 328, 1153, 789, 1195, 1194, 1197 ] } ], "ref_ids": [] }, { "id": 1199, "type": "theorem", "label": "algebra-lemma-relative-global-complete-intersection-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-relative-global-complete-intersection-smooth", "contents": [ "Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "be a relative global complete intersection.", "Let $\\mathfrak q \\subset S$ be a prime. Then $R \\to S$", "is smooth at $\\mathfrak q$ if and only if there exists a", "subset $I \\subset \\{1, \\ldots, n\\}$ of cardinality $c$", "such that the polynomial", "$$", "g_I = \\det (\\partial f_j/\\partial x_i)_{j = 1, \\ldots, c, \\ i \\in I}.", "$$", "does not map to an element of $\\mathfrak q$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-relative-global-complete-intersection-conormal}", "we see that the naive cotangent complex", "associated to the given presentation of $S$ is the complex", "$$", "\\bigoplus\\nolimits_{j = 1}^c S \\cdot f_j", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1}^n S \\cdot \\text{d}x_i, \\quad", "f_j \\longmapsto \\sum \\frac{\\partial f_j}{\\partial x_i} \\text{d}x_i.", "$$", "The maximal minors of the matrix giving the map are exactly", "the polynomials $g_I$.", "\\medskip\\noindent", "Assume $g_I$ maps to $g \\in S$, with $g \\not \\in \\mathfrak q$.", "Then the algebra $S_g$ is smooth over $R$. Namely, its naive", "cotangent complex is quasi-isomorphic to the complex above", "localized at $g$, see Lemma \\ref{lemma-localize-NL}. And by", "construction it is quasi-isomorphic to a free rank $n - c$", "module in degree $0$.", "\\medskip\\noindent", "Conversely, suppose that all $g_I$ end up in $\\mathfrak q$.", "In this case the complex above tensored with $\\kappa(\\mathfrak q)$", "does not have maximal rank, and hence there is no localization", "by an element $g \\in S$, $g \\not \\in \\mathfrak q$", "where this map becomes a split injection. By Lemma \\ref{lemma-localize-NL}", "again there is no such localization which is smooth over $R$." ], "refs": [ "algebra-lemma-relative-global-complete-intersection-conormal", "algebra-lemma-localize-NL", "algebra-lemma-localize-NL" ], "ref_ids": [ 1183, 1161, 1161 ] } ], "ref_ids": [] }, { "id": 1200, "type": "theorem", "label": "algebra-lemma-flat-fibre-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-flat-fibre-smooth", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak q \\subset S$ be a prime lying over the", "prime $\\mathfrak p$ of $R$. Assume", "\\begin{enumerate}", "\\item there exists a $g \\in S$, $g \\not\\in \\mathfrak q$", "such that $R \\to S_g$ is of finite presentation,", "\\item the local ring homomorphism", "$R_{\\mathfrak p} \\to S_{\\mathfrak q}$ is flat,", "\\item the fibre $S \\otimes_R \\kappa(\\mathfrak p)$ is smooth", "over $\\kappa(\\mathfrak p)$ at the prime corresponding", "to $\\mathfrak q$.", "\\end{enumerate}", "Then $R \\to S$ is smooth at $\\mathfrak q$." ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-syntomic} and \\ref{lemma-smooth-over-field}", "we see that there exists a $g \\in S$ such that $S_g$ is a", "relative global complete intersection. Replacing $S$ by $S_g$ we may assume", "$S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ is a relative", "global complete intersection.", "For any subset $I \\subset \\{1, \\ldots, n\\}$ of cardinality", "$c$ consider the polynomial", "$g_I = \\det (\\partial f_j/\\partial x_i)_{j = 1, \\ldots, c, i \\in I}$", "of Lemma \\ref{lemma-relative-global-complete-intersection-smooth}.", "Note that the image $\\overline{g}_I$ of $g_I$ in the polynomial ring", "$\\kappa(\\mathfrak p)[x_1, \\ldots, x_n]$ is the determinant", "of the partial derivatives of the images $\\overline{f}_j$ of the $f_j$", "in the ring $\\kappa(\\mathfrak p)[x_1, \\ldots, x_n]$. Thus the lemma follows", "by applying Lemma \\ref{lemma-relative-global-complete-intersection-smooth}", "both to $R \\to S$ and to", "$\\kappa(\\mathfrak p) \\to S \\otimes_R \\kappa(\\mathfrak p)$." ], "refs": [ "algebra-lemma-syntomic", "algebra-lemma-smooth-over-field", "algebra-lemma-relative-global-complete-intersection-smooth", "algebra-lemma-relative-global-complete-intersection-smooth" ], "ref_ids": [ 1185, 1192, 1199, 1199 ] } ], "ref_ids": [] }, { "id": 1201, "type": "theorem", "label": "algebra-lemma-flat-base-change-locus-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-flat-base-change-locus-smooth", "contents": [ "Let $R \\to S$ be a ring map of finite presentation.", "Let $R \\to R'$ be a flat ring map.", "Denote $S' = R' \\otimes_R S$ the base change.", "Let $U \\subset \\Spec(S)$ be the set of primes at", "which $R \\to S$ is smooth.", "Let $V \\subset \\Spec(S')$ the set of primes at", "which $R' \\to S'$ is smooth.", "Then $V$ is the inverse image of $U$ under the", "map $f : \\Spec(S') \\to \\Spec(S)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-change-base-NL} we see that", "$\\NL_{S/R} \\otimes_S S'$ is homotopy equivalent to $\\NL_{S'/R'}$.", "This already implies that $f^{-1}(U) \\subset V$.", "\\medskip\\noindent", "Let $\\mathfrak q' \\subset S'$ be a prime lying over", "$\\mathfrak q \\subset S$. Assume $\\mathfrak q' \\in V$.", "We have to show that $\\mathfrak q \\in U$.", "Since $S \\to S'$ is flat, we see that $S_{\\mathfrak q} \\to S'_{\\mathfrak q'}$", "is faithfully flat (Lemma \\ref{lemma-local-flat-ff}). Thus the vanishing of", "$H_1(L_{S'/R'})_{\\mathfrak q'}$ implies the", "vanishing of $H_1(L_{S/R})_{\\mathfrak q}$.", "By Lemma \\ref{lemma-finite-projective-descends}", "applied to the $S_{\\mathfrak q}$-module $(\\Omega_{S/R})_{\\mathfrak q}$", "and the map $S_{\\mathfrak q} \\to S'_{\\mathfrak q'}$ we see that", "$(\\Omega_{S/R})_{\\mathfrak q}$ is projective. Hence", "$R \\to S$ is smooth at $\\mathfrak q$ by", "Lemma \\ref{lemma-smooth-at-point}." ], "refs": [ "algebra-lemma-change-base-NL", "algebra-lemma-local-flat-ff", "algebra-lemma-finite-projective-descends", "algebra-lemma-smooth-at-point" ], "ref_ids": [ 1156, 537, 798, 1196 ] } ], "ref_ids": [] }, { "id": 1202, "type": "theorem", "label": "algebra-lemma-smooth-field-change-local", "categories": [ "algebra" ], "title": "algebra-lemma-smooth-field-change-local", "contents": [ "Let $k \\subset K$ be a field extension.", "Let $S$ be a finite type algebra over $k$.", "Let $\\mathfrak q_K$ be a prime of $S_K = K \\otimes_k S$", "and let $\\mathfrak q$ be the corresponding prime of $S$.", "Then $S$ is smooth over $k$ at $\\mathfrak q$ if and only if", "$S_K$ is smooth at $\\mathfrak q_K$ over $K$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-flat-base-change-locus-smooth}." ], "refs": [ "algebra-lemma-flat-base-change-locus-smooth" ], "ref_ids": [ 1201 ] } ], "ref_ids": [] }, { "id": 1203, "type": "theorem", "label": "algebra-lemma-lift-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-lift-smooth", "contents": [ "Let $R$ be a ring and let $I \\subset R$ be an ideal.", "Let $R/I \\to \\overline{S}$ be a smooth ring map.", "Then there exists elements $\\overline{g}_i \\in \\overline{S}$", "which generate the unit ideal of $\\overline{S}$", "such that each $\\overline{S}_{g_i} \\cong S_i/IS_i$", "for some (standard) smooth ring $S_i$ over $R$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-smooth-syntomic} we find a collection of elements", "$\\overline{g}_i \\in \\overline{S}$", "which generate the unit ideal of $\\overline{S}$", "such that each $\\overline{S}_{g_i}$ is standard smooth over $R/I$.", "Hence we may assume that $\\overline{S}$ is standard smooth", "over $R/I$. Write", "$\\overline{S} =", "(R/I)[x_1, \\ldots, x_n]/(\\overline{f}_1, \\ldots, \\overline{f}_c)$", "as in Definition \\ref{definition-standard-smooth}.", "Choose $f_1, \\ldots, f_c \\in R[x_1, \\ldots, x_n]$", "lifting $\\overline{f}_1, \\ldots, \\overline{f}_c$. Set", "$S = R[x_1, \\ldots, x_n, x_{n + 1}]/(f_1, \\ldots, f_c, x_{n + 1}\\Delta - 1)$", "where $\\Delta = \\det(\\frac{\\partial f_j}{\\partial x_i})_{i, j = 1, \\ldots, c}$", "as in Example \\ref{example-make-standard-smooth}.", "This proves the lemma." ], "refs": [ "algebra-lemma-smooth-syntomic", "algebra-definition-standard-smooth" ], "ref_ids": [ 1195, 1535 ] } ], "ref_ids": [] }, { "id": 1204, "type": "theorem", "label": "algebra-lemma-base-change-fs", "categories": [ "algebra" ], "title": "algebra-lemma-base-change-fs", "contents": [ "Let $R \\to S$ be a formally smooth ring map.", "Let $R \\to R'$ be any ring map.", "Then the base change $S' = R' \\otimes_R S$ is formally smooth over $R'$." ], "refs": [], "proofs": [ { "contents": [ "Let a solid diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar@{-->}[rrd] & R' \\otimes_R S \\ar[r] \\ar@{-->}[rd] & A/I \\\\", "R \\ar[u] \\ar[r] & R' \\ar[r] \\ar[u] & A \\ar[u]", "}", "$$", "as in Definition \\ref{definition-formally-smooth} be given.", "By assumption the longer dotted arrow exists. By the universal", "property of tensor product we obtain the shorter dotted arrow." ], "refs": [ "algebra-definition-formally-smooth" ], "ref_ids": [ 1537 ] } ], "ref_ids": [] }, { "id": 1205, "type": "theorem", "label": "algebra-lemma-compose-formally-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-compose-formally-smooth", "contents": [ "A composition of formally smooth ring maps is formally smooth." ], "refs": [], "proofs": [ { "contents": [ "Omitted. (Hint: This is completely formal, and follows from considering", "a suitable diagram.)" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1206, "type": "theorem", "label": "algebra-lemma-polynomial-ring-formally-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-polynomial-ring-formally-smooth", "contents": [ "A polynomial ring over $R$ is formally smooth over $R$." ], "refs": [], "proofs": [ { "contents": [ "Suppose we have a diagram as in Definition \\ref{definition-formally-smooth}", "with $S = R[x_j; j \\in J]$. Then there exists a dotted arrow", "simply by choosing lifts $a_j \\in A$ of the elements in $A/I$", "to which the elements $x_j$ map to under the top horizontal arrow." ], "refs": [ "algebra-definition-formally-smooth" ], "ref_ids": [ 1537 ] } ], "ref_ids": [] }, { "id": 1207, "type": "theorem", "label": "algebra-lemma-characterize-formally-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-formally-smooth", "contents": [ "Let $R \\to S$ be a ring map.", "Let $P \\to S$ be a surjective $R$-algebra map from a", "polynomial ring $P$ onto $S$. Denote $J \\subset P$ the", "kernel. Then $R \\to S$ is formally smooth if and only", "if there exists an $R$-algebra map $\\sigma : S \\to P/J^2$", "which is a right inverse to the surjection", "$P/J^2 \\to S$." ], "refs": [], "proofs": [ { "contents": [ "Assume $R \\to S$ is formally smooth.", "Consider the commutative diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar@{-->}[rd] & P/J \\\\", "R \\ar[r] \\ar[u] & P/J^2\\ar[u]", "}", "$$", "By assumption the dotted arrow exists. This proves that", "$\\sigma$ exists.", "\\medskip\\noindent", "Conversely, suppose we have a $\\sigma$ as in the lemma.", "Let a solid diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar@{-->}[rd] & A/I \\\\", "R \\ar[r] \\ar[u] & A \\ar[u]", "}", "$$", "as in Definition \\ref{definition-formally-smooth} be given.", "Because $P$ is formally smooth by", "Lemma \\ref{lemma-polynomial-ring-formally-smooth},", "there exists an $R$-algebra homomorphism", "$\\psi : P \\to A$ which lifts the map $P \\to S \\to A/I$.", "Clearly $\\psi(J) \\subset I$ and since $I^2 = 0$ we conclude that", "$\\psi(J^2) = 0$. Hence $\\psi$ factors as", "$\\overline{\\psi} : P/J^2 \\to A$. The desired dotted arrow", "is the composition $\\overline{\\psi} \\circ \\sigma : S \\to A$." ], "refs": [ "algebra-definition-formally-smooth", "algebra-lemma-polynomial-ring-formally-smooth" ], "ref_ids": [ 1537, 1206 ] } ], "ref_ids": [] }, { "id": 1208, "type": "theorem", "label": "algebra-lemma-characterize-formally-smooth-again", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-formally-smooth-again", "contents": [ "Let $R \\to S$ be a ring map.", "Let $P \\to S$ be a surjective $R$-algebra map from a", "polynomial ring $P$ onto $S$. Denote $J \\subset P$ the", "kernel. Then $R \\to S$ is formally smooth if and only", "if the sequence", "$$", "0 \\to J/J^2 \\to \\Omega_{P/R} \\otimes_P S \\to \\Omega_{S/R} \\to 0", "$$", "of Lemma \\ref{lemma-differential-seq} is a split exact sequence." ], "refs": [ "algebra-lemma-differential-seq" ], "proofs": [ { "contents": [ "Assume $S$ is formally smooth over $R$. By", "Lemma \\ref{lemma-characterize-formally-smooth}", "this means there exists an $R$-algebra map", "$S \\to P/J^2$ which is a right inverse to the", "canonical map $P/J^2 \\to S$. By Lemma \\ref{lemma-differential-seq-split}", "we see that the sequence is split.", "\\medskip\\noindent", "Assume the exact sequence of the lemma is split exact.", "Choose a splitting $\\sigma : \\Omega_{S/R} \\to \\Omega_{P/R} \\otimes_R S$.", "For each $\\lambda \\in S$ choose $x_\\lambda \\in P$", "which maps to $\\lambda$. Next, for each $\\lambda \\in S$ choose", "$f_\\lambda \\in J$ such that", "$$", "\\text{d}f_\\lambda = \\text{d}x_\\lambda - \\sigma(\\text{d}\\lambda)", "$$", "in the middle term of the exact sequence.", "We claim that $s : \\lambda \\mapsto x_\\lambda - f_\\lambda \\mod J^2$", "is an $R$-algebra homomorphism $s : S \\to P/J^2$.", "To prove this we will repeatedly use that if $h \\in J$ and", "$\\text{d}h = 0$ in $\\Omega_{P/R} \\otimes_R S$, then $h \\in J^2$.", "Let $\\lambda, \\mu \\in S$.", "Then $\\sigma(\\text{d}\\lambda + \\text{d}\\mu - \\text{d}(\\lambda + \\mu)) = 0$.", "This implies", "$$", "\\text{d}(x_\\lambda + x_\\mu - x_{\\lambda + \\mu}", "- f_\\lambda - f_\\mu + f_{\\lambda + \\mu}) = 0", "$$", "which means that $x_\\lambda + x_\\mu - x_{\\lambda + \\mu}", "- f_\\lambda - f_\\mu + f_{\\lambda + \\mu} \\in J^2$, which in turn", "means that $s(\\lambda) + s(\\mu) = s(\\lambda + \\mu)$.", "Similarly, we have", "$\\sigma(\\lambda \\text{d}\\mu + \\mu \\text{d}\\lambda - \\text{d}\\lambda \\mu) = 0$", "which implies that", "$$", "\\mu(\\text{d}x_\\lambda - \\text{d}f_\\lambda) +", "\\lambda(\\text{d}x_\\mu - \\text{d}f_\\mu) -", "\\text{d}x_{\\lambda\\mu} + \\text{d}f_{\\lambda\\mu} = 0", "$$", "in the middle term of the exact sequence.", "Moreover we have", "$$", "\\text{d}(x_\\lambda x_\\mu) =", "x_\\lambda \\text{d}x_\\mu + x_\\mu \\text{d}x_\\lambda =", "\\lambda \\text{d}x_\\mu + \\mu \\text{d} x_\\lambda", "$$", "in the middle term again. Combined these equations mean that", "$x_\\lambda x_\\mu - x_{\\lambda\\mu}", "- \\mu f_\\lambda - \\lambda f_\\mu + f_{\\lambda\\mu} \\in J^2$,", "hence $(x_\\lambda - f_\\lambda)(x_\\mu - f_\\mu) -", "(x_{\\lambda\\mu} - f_{\\lambda\\mu}) \\in J^2$ as $f_\\lambda f_\\mu \\in J^2$,", "which means that $s(\\lambda)s(\\mu) = s(\\lambda\\mu)$.", "If $\\lambda \\in R$, then $\\text{d}\\lambda = 0$ and we see", "that $\\text{d}f_\\lambda = \\text{d}x_\\lambda$, hence", "$\\lambda - x_\\lambda + f_\\lambda \\in J^2$ and hence", "$s(\\lambda) = \\lambda$ as desired. At this point we can", "apply Lemma \\ref{lemma-characterize-formally-smooth}", "to conclude that $S/R$ is formally smooth." ], "refs": [ "algebra-lemma-characterize-formally-smooth", "algebra-lemma-differential-seq-split", "algebra-lemma-characterize-formally-smooth" ], "ref_ids": [ 1207, 1136, 1207 ] } ], "ref_ids": [ 1135 ] }, { "id": 1209, "type": "theorem", "label": "algebra-lemma-ses-formally-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-ses-formally-smooth", "contents": [ "Let $A \\to B \\to C$ be ring maps. Assume $B \\to C$ is formally smooth.", "Then the sequence", "$$", "0 \\to \\Omega_{B/A} \\otimes_B C \\to \\Omega_{C/A} \\to \\Omega_{C/B} \\to 0", "$$", "of", "Lemma \\ref{lemma-exact-sequence-differentials}", "is a split short exact sequence." ], "refs": [ "algebra-lemma-exact-sequence-differentials" ], "proofs": [ { "contents": [ "Follows from", "Proposition \\ref{proposition-characterize-formally-smooth}", "and", "Lemma \\ref{lemma-exact-sequence-NL}." ], "refs": [ "algebra-proposition-characterize-formally-smooth", "algebra-lemma-exact-sequence-NL" ], "ref_ids": [ 1425, 1153 ] } ], "ref_ids": [ 1133 ] }, { "id": 1210, "type": "theorem", "label": "algebra-lemma-differential-seq-formally-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-differential-seq-formally-smooth", "contents": [ "Let $A \\to B \\to C$ be ring maps with $A \\to C$ formally smooth", "and $B \\to C$ surjective with kernel $J \\subset B$.", "Then the exact sequence", "$$", "0 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B C \\to \\Omega_{C/A} \\to 0", "$$", "of", "Lemma \\ref{lemma-differential-seq}", "is split exact." ], "refs": [ "algebra-lemma-differential-seq" ], "proofs": [ { "contents": [ "Follows from", "Proposition \\ref{proposition-characterize-formally-smooth},", "Lemma \\ref{lemma-exact-sequence-NL}, and", "Lemma \\ref{lemma-differential-seq}." ], "refs": [ "algebra-proposition-characterize-formally-smooth", "algebra-lemma-exact-sequence-NL", "algebra-lemma-differential-seq" ], "ref_ids": [ 1425, 1153, 1135 ] } ], "ref_ids": [ 1135 ] }, { "id": 1211, "type": "theorem", "label": "algebra-lemma-application-NL-formally-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-application-NL-formally-smooth", "contents": [ "Let $A \\to B \\to C$ be ring maps. Assume $A \\to C$ is surjective (so", "also $B \\to C$ is) and $A \\to B$ formally smooth.", "Denote $I = \\Ker(A \\to C)$ and $J = \\Ker(B \\to C)$.", "Then the sequence", "$$", "0 \\to I/I^2 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B B/J \\to 0", "$$", "of", "Lemma \\ref{lemma-application-NL}", "is split exact." ], "refs": [ "algebra-lemma-application-NL" ], "proofs": [ { "contents": [ "Since $A \\to B$ is formally smooth there exists a ring map", "$\\sigma : B \\to A/I^2$ whose composition with $A \\to B$ equals", "the quotient map $A \\to A/I^2$. Then $\\sigma$ induces a map", "$J/J^2 \\to I/I^2$ which is inverse to the map $I/I^2 \\to J/J^2$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1155 ] }, { "id": 1212, "type": "theorem", "label": "algebra-lemma-lift-formal-smoothness", "categories": [ "algebra" ], "title": "algebra-lemma-lift-formal-smoothness", "contents": [ "Let $R \\to S$ be a ring map.", "Let $I \\subset R$ be an ideal. Assume", "\\begin{enumerate}", "\\item $I^2 = 0$,", "\\item $R \\to S$ is flat, and", "\\item $R/I \\to S/IS$ is formally smooth.", "\\end{enumerate}", "Then $R \\to S$ is formally smooth." ], "refs": [], "proofs": [ { "contents": [ "Assume (1), (2) and (3).", "Let $P = R[\\{x_t\\}_{t \\in T}] \\to S$ be a surjection of $R$-algebras", "with kernel $J$. Thus $0 \\to J \\to P \\to S \\to 0$ is a", "short exact sequence of flat $R$-modules. This implies that", "$I \\otimes_R S = IS$, $I \\otimes_R P = IP$ and $I \\otimes_R J = IJ$", "as well as $J \\cap IP = IJ$.", "We will use throughout the proof that", "$$", "\\Omega_{(S/IS)/(R/I)} = \\Omega_{S/R} \\otimes_S (S/IS)", "= \\Omega_{S/R} \\otimes_R R/I = \\Omega_{S/R} / I\\Omega_{S/R}", "$$", "and similarly for $P$ (see Lemma \\ref{lemma-differentials-base-change}).", "By Lemma \\ref{lemma-characterize-formally-smooth-again} the sequence", "\\begin{equation}", "\\label{equation-split}", "0 \\to J/(IJ + J^2) \\to", "\\Omega_{P/R} \\otimes_P S/IS \\to", "\\Omega_{S/R} \\otimes_S S/IS \\to 0", "\\end{equation}", "is split exact. Of course the middle term is", "$\\bigoplus_{t \\in T} S/IS \\text{d}x_t$. Choose a splitting", "$\\sigma : \\Omega_{P/R} \\otimes_P S/IS \\to J/(IJ + J^2)$.", "For each $t \\in T$ choose an element $f_t \\in J$ which maps", "to $\\sigma(\\text{d}x_t)$ in $J/(IJ + J^2)$. This determines a", "unique $S$-module map", "$$", "\\tilde \\sigma : \\Omega_{P/R} \\otimes_R S", "= \\bigoplus S\\text{d}x_t \\longrightarrow J/J^2", "$$", "with the property that $\\tilde\\sigma(\\text{d}x_t) = f_t$.", "As $\\sigma$ is a section to $\\text{d}$ the difference", "$$", "\\Delta = \\text{id}_{J/J^2} - \\tilde \\sigma \\circ \\text{d}", "$$", "is a self map $J/J^2 \\to J/J^2$ whose image is contained in", "$(IJ + J^2)/J^2$. In particular $\\Delta((IJ + J^2)/J^2) = 0$", "because $I^2 = 0$. This means that $\\Delta$ factors as", "$$", "J/J^2 \\to J/(IJ + J^2) \\xrightarrow{\\overline{\\Delta}}", "(IJ + J^2)/J^2 \\to J/J^2", "$$", "where $\\overline{\\Delta}$ is a $S/IS$-module map.", "Using again that the sequence (\\ref{equation-split})", "is split, we can find a $S/IS$-module map", "$\\overline{\\delta} : \\Omega_{P/R} \\otimes_P S/IS \\to (IJ + J^2)/J^2$", "such that $\\overline{\\delta} \\circ d$ is equal to $\\overline{\\Delta}$.", "In the same manner as above the map $\\overline{\\delta}$ determines", "an $S$-module map", "$\\delta : \\Omega_{P/R} \\otimes_P S \\to J/J^2$.", "After replacing $\\tilde \\sigma$ by $\\tilde \\sigma + \\delta$", "a simple computation shows that $\\Delta = 0$. In other words $\\tilde \\sigma$", "is a section of $J/J^2 \\to \\Omega_{P/R} \\otimes_P S$.", "By Lemma \\ref{lemma-characterize-formally-smooth-again}", "we conclude that $R \\to S$ is formally smooth." ], "refs": [ "algebra-lemma-differentials-base-change", "algebra-lemma-characterize-formally-smooth-again", "algebra-lemma-characterize-formally-smooth-again" ], "ref_ids": [ 1138, 1208, 1208 ] } ], "ref_ids": [] }, { "id": 1213, "type": "theorem", "label": "algebra-lemma-finite-presentation-fs-Noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-finite-presentation-fs-Noetherian", "contents": [ "Let $R \\to S$ be a smooth ring map. Then there exists a subring", "$R_0 \\subset R$ of finite type over $\\mathbf{Z}$ and a smooth", "ring map $R_0 \\to S_0$ such that $S \\cong R \\otimes_{R_0} S_0$." ], "refs": [], "proofs": [ { "contents": [ "We are going to use that smooth is equivalent to finite presentation", "and formally smooth, see Proposition \\ref{proposition-smooth-formally-smooth}.", "Write $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$", "and denote $I = (f_1, \\ldots, f_m)$.", "Choose a right inverse", "$\\sigma : S \\to R[x_1, \\ldots, x_n]/I^2$", "to the projection to $S$ as in", "Lemma \\ref{lemma-characterize-formally-smooth}.", "Choose $h_i \\in R[x_1, \\ldots, x_n]$ such that", "$\\sigma(x_i \\bmod I) = h_i \\bmod I^2$.", "The fact that $\\sigma$ is an $R$-algebra homomorphism", "$R[x_1, \\ldots, x_n]/I \\to R[x_1, \\ldots, x_n]/I^2$", "is equivalent to the condition that", "$$", "f_j(h_1, \\ldots, h_n) = \\sum\\nolimits_{j_1 j_2} a_{j_1 j_2} f_{j_1} f_{j_2}", "$$", "for certain $a_{kl} \\in R[x_1, \\ldots, x_n]$.", "Let $R_0 \\subset R$ be the subring generated over $\\mathbf{Z}$", "by all the coefficients of the polynomials $f_j, h_i, a_{kl}$.", "Set $S_0 = R_0[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$,", "with $I_0 = (f_1, \\ldots, f_m)$.", "Let $\\sigma_0 : S_0 \\to R_0[x_1, \\ldots, x_n]/I_0^2$ defined by", "the rule $x_i \\mapsto h_i \\bmod I_0^2$; this works since the", "$a_{lk}$ are defined over $R_0$ and satisfy the same relations.", "Thus by Lemma \\ref{lemma-characterize-formally-smooth}", "the ring $S_0$ is formally smooth over $R_0$." ], "refs": [ "algebra-proposition-smooth-formally-smooth", "algebra-lemma-characterize-formally-smooth", "algebra-lemma-characterize-formally-smooth" ], "ref_ids": [ 1426, 1207, 1207 ] } ], "ref_ids": [] }, { "id": 1214, "type": "theorem", "label": "algebra-lemma-smooth-descends-through-colimit", "categories": [ "algebra" ], "title": "algebra-lemma-smooth-descends-through-colimit", "contents": [ "Let $A = \\colim A_i$ be a filtered colimit of rings. Let", "$A \\to B$ be a smooth ring map. There exists an $i$ and", "a smooth ring map $A_i \\to B_i$ such that $B = B_i \\otimes_{A_i} A$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-finite-presentation-fs-Noetherian}", "since $R_0 \\to A$ will factor through $A_i$ for some $i$ by", "Lemma \\ref{lemma-characterize-finite-presentation}." ], "refs": [ "algebra-lemma-finite-presentation-fs-Noetherian", "algebra-lemma-characterize-finite-presentation" ], "ref_ids": [ 1213, 1092 ] } ], "ref_ids": [] }, { "id": 1215, "type": "theorem", "label": "algebra-lemma-descent-formally-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-descent-formally-smooth", "contents": [ "Let $R \\to S$ be a ring map. Let $R \\to R'$ be a faithfully flat ring map.", "Set $S' = S \\otimes_R R'$. Then $R \\to S$ is formally smooth if and only", "if $R' \\to S'$ is formally smooth." ], "refs": [], "proofs": [ { "contents": [ "If $R \\to S$ is formally smooth, then $R' \\to S'$ is formally smooth by", "Lemma \\ref{lemma-base-change-fs}.", "To prove the converse, assume $R' \\to S'$ is formally smooth.", "Note that $N \\otimes_R R' = N \\otimes_S S'$ for any $S$-module $N$. In", "particular $S \\to S'$ is faithfully flat also.", "Choose a polynomial ring $P = R[\\{x_i\\}_{i \\in I}]$ and a surjection", "of $R$-algebras $P \\to S$ with kernel $J$. Note that $P' = P \\otimes_R R'$", "is a polynomial algebra over $R'$. Since $R \\to R'$ is flat the kernel", "$J'$ of the surjection $P' \\to S'$ is $J \\otimes_R R'$. Hence the", "split exact sequence (see", "Lemma \\ref{lemma-characterize-formally-smooth-again})", "$$", "0 \\to J'/(J')^2 \\to \\Omega_{P'/R'} \\otimes_{P'} S' \\to \\Omega_{S'/R'} \\to 0", "$$", "is the base change via $S \\to S'$ of the corresponding sequence", "$$", "J/J^2 \\to \\Omega_{P/R} \\otimes_P S \\to \\Omega_{S/R} \\to 0", "$$", "see", "Lemma \\ref{lemma-differential-seq}.", "As $S \\to S'$ is faithfully flat we conclude two things:", "(1) this sequence (without ${}'$) is exact too, and (2)", "$\\Omega_{S/R}$ is a projective $S$-module. Namely, $\\Omega_{S'/R'}$", "is projective as a direct sum of the free module", "$\\Omega_{P'/R'} \\otimes_{P'} S'$ and", "$\\Omega_{S/R} \\otimes_S {S'} = \\Omega_{S'/R'}$ by what we said above.", "Thus (2) follows by descent of projectivity", "through faithfully flat ring maps, see", "Theorem \\ref{theorem-ffdescent-projectivity}.", "Hence the sequence", "$0 \\to J/J^2 \\to \\Omega_{P/R} \\otimes_P S \\to \\Omega_{S/R} \\to 0$", "is exact also and we win by applying", "Lemma \\ref{lemma-characterize-formally-smooth-again}", "once more." ], "refs": [ "algebra-lemma-base-change-fs", "algebra-lemma-characterize-formally-smooth-again", "algebra-lemma-differential-seq", "algebra-theorem-ffdescent-projectivity", "algebra-lemma-characterize-formally-smooth-again" ], "ref_ids": [ 1204, 1208, 1135, 324, 1208 ] } ], "ref_ids": [] }, { "id": 1216, "type": "theorem", "label": "algebra-lemma-smooth-strong-lift", "categories": [ "algebra" ], "title": "algebra-lemma-smooth-strong-lift", "contents": [ "Let $R \\to S$ be a smooth ring map. Given a commutative solid diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar@{-->}[rd] & A/I \\\\", "R \\ar[r] \\ar[u] & A \\ar[u]", "}", "$$", "where $I \\subset A$ is a locally nilpotent ideal, a dotted", "arrow exists which makes the diagram commute." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-presentation-fs-Noetherian} we can extend the", "diagram to a commutative diagram", "$$", "\\xymatrix{", "S_0 \\ar[r] & S \\ar[r] \\ar@{-->}[rd] & A/I \\\\", "R_0 \\ar[r] \\ar[u] & R \\ar[r] \\ar[u] & A \\ar[u]", "}", "$$", "with $R_0 \\to S_0$ smooth, $R_0$ of finite type over $\\mathbf{Z}$, and", "$S = S_0 \\otimes_{R_0} R$. Let $x_1, \\ldots, x_n \\in S_0$ be generators of", "$S_0$ over $R_0$. Let $a_1, \\ldots, a_n$ be elements of $A$ which", "map to the same elements in $A/I$ as the elements $x_1, \\ldots, x_n$.", "Denote $A_0 \\subset A$ the subring generated by the image of $R_0$", "and the elements $a_1, \\ldots, a_n$. Set $I_0 = A_0 \\cap I$. Then", "$A_0/I_0 \\subset A/I$ and $S_0 \\to A/I$ maps into $A_0/I_0$.", "Thus it suffices to find the dotted arrow in the diagram", "$$", "\\xymatrix{", "S_0 \\ar[r] \\ar@{-->}[rd] & A_0/I_0 \\\\", "R_0 \\ar[r] \\ar[u] & A_0 \\ar[u]", "}", "$$", "The ring $A_0$ is of finite type over $\\mathbf{Z}$ by construction.", "Hence $A_0$ is Noetherian, whence $I_0$ is nilpotent, see", "Lemma \\ref{lemma-Noetherian-power}.", "Say $I_0^n = 0$. By Proposition \\ref{proposition-smooth-formally-smooth}", "we can successively lift the $R_0$-algebra map $S_0 \\to A_0/I_0$ to", "$S_0 \\to A_0/I_0^2$, $S_0 \\to A_0/I_0^3$, $\\ldots$,", "and finally $S_0 \\to A_0/I_0^n = A_0$." ], "refs": [ "algebra-lemma-finite-presentation-fs-Noetherian", "algebra-lemma-Noetherian-power", "algebra-proposition-smooth-formally-smooth" ], "ref_ids": [ 1213, 460, 1426 ] } ], "ref_ids": [] }, { "id": 1217, "type": "theorem", "label": "algebra-lemma-triangle-differentials-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-triangle-differentials-smooth", "contents": [ "Given ring maps $A \\to B \\to C$ with $B \\to C$ smooth, then the sequence", "$$", "0 \\to C \\otimes_B \\Omega_{B/A} \\to \\Omega_{C/A} \\to \\Omega_{C/B} \\to 0", "$$", "of Lemma \\ref{lemma-exact-sequence-differentials} is exact." ], "refs": [ "algebra-lemma-exact-sequence-differentials" ], "proofs": [ { "contents": [ "This follows from the more general", "Lemma \\ref{lemma-ses-formally-smooth}", "because a smooth ring map is formally smooth, see", "Proposition \\ref{proposition-smooth-formally-smooth}.", "But it also follows directly from", "Lemma \\ref{lemma-exact-sequence-NL}", "since $H_1(L_{C/B}) = 0$ is part of the definition of smoothness of $B \\to C$." ], "refs": [ "algebra-lemma-ses-formally-smooth", "algebra-proposition-smooth-formally-smooth", "algebra-lemma-exact-sequence-NL" ], "ref_ids": [ 1209, 1426, 1153 ] } ], "ref_ids": [ 1133 ] }, { "id": 1218, "type": "theorem", "label": "algebra-lemma-differential-seq-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-differential-seq-smooth", "contents": [ "Let $A \\to B \\to C$ be ring maps with $A \\to C$ smooth", "and $B \\to C$ surjective with kernel $J \\subset B$.", "Then the exact sequence", "$$", "0 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B C \\to \\Omega_{C/A} \\to 0", "$$", "of", "Lemma \\ref{lemma-differential-seq}", "is split exact." ], "refs": [ "algebra-lemma-differential-seq" ], "proofs": [ { "contents": [ "This follows from the more general", "Lemma \\ref{lemma-differential-seq-formally-smooth}", "because a smooth ring map is formally smooth, see", "Proposition \\ref{proposition-smooth-formally-smooth}." ], "refs": [ "algebra-lemma-differential-seq-formally-smooth", "algebra-proposition-smooth-formally-smooth" ], "ref_ids": [ 1210, 1426 ] } ], "ref_ids": [ 1135 ] }, { "id": 1219, "type": "theorem", "label": "algebra-lemma-application-NL-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-application-NL-smooth", "contents": [ "Let $A \\to B \\to C$ be ring maps. Assume $A \\to C$ is surjective (so", "also $B \\to C$ is) and $A \\to B$ smooth.", "Denote $I = \\Ker(A \\to C)$ and $J = \\Ker(B \\to C)$.", "Then the sequence", "$$", "0 \\to I/I^2 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B B/J \\to 0", "$$", "of", "Lemma \\ref{lemma-application-NL}", "is exact." ], "refs": [ "algebra-lemma-application-NL" ], "proofs": [ { "contents": [ "This follows from the more general", "Lemma \\ref{lemma-application-NL-formally-smooth}", "because a smooth ring map is formally smooth, see", "Proposition \\ref{proposition-smooth-formally-smooth}." ], "refs": [ "algebra-lemma-application-NL-formally-smooth", "algebra-proposition-smooth-formally-smooth" ], "ref_ids": [ 1211, 1426 ] } ], "ref_ids": [ 1155 ] }, { "id": 1220, "type": "theorem", "label": "algebra-lemma-section-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-section-smooth", "contents": [ "\\begin{slogan}", "If $R$ is a summand of $S$ and $S$ is smooth over $R$, then the", "$I$-adic completion of $S$ is often a power series over $R$", "where $I$ is the kernel of the projection map from $S$ to $R$.", "\\end{slogan}", "Let $\\varphi : R \\to S$ be a smooth ring map.", "Let $\\sigma : S \\to R$ be a left inverse to $\\varphi$.", "Set $I = \\Ker(\\sigma)$. Then", "\\begin{enumerate}", "\\item $I/I^2$ is a finite locally free $R$-module, and", "\\item if $I/I^2$ is free, then $S^\\wedge \\cong R[[t_1, \\ldots, t_d]]$", "as $R$-algebras, where $S^\\wedge$ is the $I$-adic completion of $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-differential-seq-split}", "applied to $R \\to S \\to R$ we see that", "$I/I^2 = \\Omega_{S/R} \\otimes_{S, \\sigma} R$.", "Since by definition of a smooth morphism the module $\\Omega_{S/R}$ is", "finite locally free over $S$ we deduce that (1) holds.", "If $I/I^2$ is free, then choose $f_1, \\ldots, f_d \\in I$ whose images", "in $I/I^2$ form an $R$-basis. Consider the $R$-algebra map defined by", "$$", "\\Psi : R[[x_1, \\ldots, x_d]] \\longrightarrow S^\\wedge, \\quad", "x_i \\longmapsto f_i.", "$$", "Denote $P = R[[x_1, \\ldots, x_d]]$ and $J = (x_1, \\ldots, x_d) \\subset P$.", "We write $\\Psi_n : P/J^n \\to S/I^n$ for the induced map of quotient rings.", "Note that $S/I^2 = \\varphi(R) \\oplus I/I^2$. Thus $\\Psi_2$ is an", "isomorphism. Denote $\\sigma_2 : S/I^2 \\to P/J^2$ the inverse of $\\Psi_2$.", "We will prove by induction on $n$ that for all $n > 2$ there exists an inverse", "$\\sigma_n : S/I^n \\to P/J^n$ of $\\Psi_n$. Namely, as $S$ is formally", "smooth over $R$ (by", "Proposition \\ref{proposition-smooth-formally-smooth})", "we see that in the solid diagram", "$$", "\\xymatrix{", "S \\ar@{..>}[r] \\ar[rd]_{\\sigma_{n - 1}} & P/J^n \\ar[d] \\\\", " & P/J^{n - 1}", "}", "$$", "of $R$-algebras we can fill in the dotted arrow by some $R$-algebra", "map $\\tau : S \\to P/J^n$ making the diagram commute. This induces an", "$R$-algebra map $\\overline{\\tau} : S/I^n \\to P/J^n$ which is equal to", "$\\sigma_{n - 1}$ modulo $J^n$. By construction the map $\\Psi_n$ is surjective", "and now $\\overline{\\tau} \\circ \\Psi_n$ is an $R$-algebra endomorphism", "of $P/J^n$ which maps $x_i$ to $x_i + \\delta_{i, n}$ with", "$\\delta_{i, n} \\in J^{n -1}/J^n$. It follows that $\\Psi_n$ is an", "isomorphism and hence it has an inverse $\\sigma_n$.", "This proves the lemma." ], "refs": [ "algebra-lemma-differential-seq-split", "algebra-proposition-smooth-formally-smooth" ], "ref_ids": [ 1136, 1426 ] } ], "ref_ids": [] }, { "id": 1221, "type": "theorem", "label": "algebra-lemma-rank-omega", "categories": [ "algebra" ], "title": "algebra-lemma-rank-omega", "contents": [ "Let $k$ be an algebraically closed field.", "Let $S$ be a finite type $k$-algebra.", "Let $\\mathfrak m \\subset S$ be a maximal ideal.", "Then", "$$", "\\dim_{\\kappa(\\mathfrak m)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak m)", "=", "\\dim_{\\kappa(\\mathfrak m)} \\mathfrak m/\\mathfrak m^2.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Consider the exact sequence", "$$", "\\mathfrak m/\\mathfrak m^2 \\to", "\\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak m) \\to", "\\Omega_{\\kappa(\\mathfrak m)/k} \\to 0", "$$", "of Lemma \\ref{lemma-differential-seq}. We would like to show that the", "first map is an isomorphism. Since $k$ is algebraically closed the", "composition $k \\to \\kappa(\\mathfrak m)$ is an isomorphism by", "Theorem \\ref{theorem-nullstellensatz}.", "So the surjection $S \\to \\kappa(\\mathfrak m)$ splits as a map of", "$k$-algebras, and Lemma \\ref{lemma-differential-seq-split} shows", "that the sequence above is exact", "on the left. Since $\\Omega_{\\kappa(\\mathfrak m)/k} = 0$, we win." ], "refs": [ "algebra-lemma-differential-seq", "algebra-theorem-nullstellensatz", "algebra-lemma-differential-seq-split" ], "ref_ids": [ 1135, 316, 1136 ] } ], "ref_ids": [] }, { "id": 1222, "type": "theorem", "label": "algebra-lemma-characterize-smooth-kbar", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-smooth-kbar", "contents": [ "Let $k$ be an algebraically closed field.", "Let $S$ be a finite type $k$-algebra.", "Let $\\mathfrak m \\subset S$ be a maximal ideal.", "The following are equivalent:", "\\begin{enumerate}", "\\item The ring $S_{\\mathfrak m}$ is a regular local ring.", "\\item We have", "$\\dim_{\\kappa(\\mathfrak m)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak m)", "\\leq \\dim(S_{\\mathfrak m})$.", "\\item We have", "$\\dim_{\\kappa(\\mathfrak m)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak m)", "= \\dim(S_{\\mathfrak m})$.", "\\item There exists a $g \\in S$, $g \\not \\in \\mathfrak m$", "such that $S_g$ is smooth over $k$. In other words $S/k$", "is smooth at $\\mathfrak m$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that (1), (2) and (3) are equivalent by Lemma \\ref{lemma-rank-omega}", "and Definition \\ref{definition-regular}.", "\\medskip\\noindent", "Assume that $S$ is smooth at $\\mathfrak m$.", "By Lemma \\ref{lemma-smooth-syntomic} we see that", "$S_g$ is standard smooth over $k$", "for a suitable $g \\in S$, $g \\not \\in \\mathfrak m$.", "Hence by Lemma \\ref{lemma-standard-smooth}", "we see that $\\Omega_{S_g/k}$ is free of rank $\\dim(S_g)$.", "Hence by Lemma \\ref{lemma-rank-omega}", "we see that $\\dim(S_m) = \\dim (\\mathfrak m/\\mathfrak m^2)$", "in other words $S_\\mathfrak m$ is regular.", "\\medskip\\noindent", "Conversely, suppose that $S_{\\mathfrak m}$ is regular.", "Let $d = \\dim(S_{\\mathfrak m}) = \\dim \\mathfrak m/\\mathfrak m^2$.", "Choose a presentation $S = k[x_1, \\ldots, x_n]/I$", "such that $x_i$ maps to an element of $\\mathfrak m$ for", "all $i$. In other words, $\\mathfrak m'' = (x_1, \\ldots, x_n)$", "is the corresponding maximal ideal of $k[x_1, \\ldots, x_n]$.", "Note that we have a short exact sequence", "$$", "I/\\mathfrak m''I \\to \\mathfrak m''/(\\mathfrak m'')^2", "\\to \\mathfrak m/(\\mathfrak m)^2 \\to 0", "$$", "Pick $c = n - d$ elements $f_1, \\ldots, f_c \\in I$ such that", "their images in $\\mathfrak m''/(\\mathfrak m'')^2$ span the", "kernel of the map to $\\mathfrak m/\\mathfrak m^2$. This is clearly", "possible. Denote $J = (f_1, \\ldots, f_c)$. So $J \\subset I$.", "Denote $S' = k[x_1, \\ldots, x_n]/J$ so there is a surjection", "$S' \\to S$. Denote $\\mathfrak m' = \\mathfrak m''S'$ the corresponding", "maximal ideal of $S'$. Hence we have", "$$", "\\xymatrix{", "k[x_1, \\ldots, x_n] \\ar[r] & S' \\ar[r] & S \\\\", "\\mathfrak m'' \\ar[u] \\ar[r] & \\mathfrak m' \\ar[r] \\ar[u] &", "\\mathfrak m \\ar[u]", "}", "$$", "By our choice of $J$ the exact sequence", "$$", "J/\\mathfrak m''J \\to \\mathfrak m''/(\\mathfrak m'')^2", "\\to \\mathfrak m'/(\\mathfrak m')^2 \\to 0", "$$", "shows that $\\dim( \\mathfrak m'/(\\mathfrak m')^2 ) = d$.", "Since $S'_{\\mathfrak m'}$ surjects onto $S_{\\mathfrak m}$", "we see that $\\dim(S_{\\mathfrak m'}) \\geq d$. Hence by", "the discussion preceding Definition \\ref{definition-regular-local}", "we conclude that $S'_{\\mathfrak m'}$ is", "regular of dimension $d$ as well. Because $S'$ was cut out", "by $c = n - d$ equations we", "conclude that there exists a $g' \\in S'$, $g' \\not \\in \\mathfrak m'$", "such that $S'_{g'}$ is a global complete intersection over $k$,", "see Lemma \\ref{lemma-lci}.", "Also the map $S'_{\\mathfrak m'} \\to S_{\\mathfrak m}$", "is a surjection of Noetherian local domains of the same", "dimension and hence an isomorphism. Hence $S' \\to S$ is surjective", "with finitely generated kernel and becomes an isomorphism", "after localizing at $\\mathfrak m'$. Thus we can find $g' \\in S'$,", "$g \\not \\in \\mathfrak m'$ such that $S'_{g'} \\to S_{g'}$", "is an isomorphism. All in all we conclude that", "after replacing $S$ by a principal localization we may", "assume that $S$ is a global complete intersection.", "\\medskip\\noindent", "At this point we may write $S = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "with $\\dim S = n - c$. Recall that the naive cotangent complex", "of this algebra is given by", "$$", "\\bigoplus S \\cdot f_j", "\\to", "\\bigoplus S \\cdot \\text{d}x_i", "$$", "see Lemma \\ref{lemma-relative-global-complete-intersection-conormal}.", "By Lemma \\ref{lemma-relative-global-complete-intersection-smooth}", "in order to show that $S$ is smooth at", "$\\mathfrak m$ we have to show that one of the $c \\times c$", "minors $g_I$ of the matrix ``$A$'' giving the map above", "does not vanish at $\\mathfrak m$. By Lemma \\ref{lemma-rank-omega}", "the matrix $A \\bmod \\mathfrak m$ has rank $c$. Thus we win." ], "refs": [ "algebra-lemma-rank-omega", "algebra-definition-regular", "algebra-lemma-smooth-syntomic", "algebra-lemma-standard-smooth", "algebra-lemma-rank-omega", "algebra-definition-regular-local", "algebra-lemma-lci", "algebra-lemma-relative-global-complete-intersection-conormal", "algebra-lemma-relative-global-complete-intersection-smooth", "algebra-lemma-rank-omega" ], "ref_ids": [ 1221, 1512, 1195, 1193, 1221, 1480, 1167, 1183, 1199, 1221 ] } ], "ref_ids": [] }, { "id": 1223, "type": "theorem", "label": "algebra-lemma-characterize-smooth-over-field", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-smooth-over-field", "contents": [ "Let $k$ be any field.", "Let $S$ be a finite type $k$-algebra.", "Let $X = \\Spec(S)$.", "Let $\\mathfrak q \\subset S$ be a prime", "corresponding to $x \\in X$.", "The following are equivalent:", "\\begin{enumerate}", "\\item The $k$-algebra $S$ is smooth at $\\mathfrak q$ over $k$.", "\\item We have", "$\\dim_{\\kappa(\\mathfrak q)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak q)", "\\leq \\dim_x X$.", "\\item We have", "$\\dim_{\\kappa(\\mathfrak q)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak q)", "= \\dim_x X$.", "\\end{enumerate}", "Moreover, in this case the local ring $S_{\\mathfrak q}$ is regular." ], "refs": [], "proofs": [ { "contents": [ "If $S$ is smooth at $\\mathfrak q$ over $k$, then there exists", "a $g \\in S$, $g \\not \\in \\mathfrak q$ such that $S_g$ is", "standard smooth over $k$, see Lemma \\ref{lemma-smooth-syntomic}.", "A standard smooth algebra over $k$ has a module of differentials", "which is free of rank equal to the dimension, see", "Lemma \\ref{lemma-standard-smooth} (use that a relative global", "complete intersection over a field has dimension equal to the", "number of variables minus the number of equations). Thus we see that", "(1) implies (3). To finish the proof of the lemma it", "suffices to show that (2) implies (1) and that it implies", "that $S_{\\mathfrak q}$ is regular.", "\\medskip\\noindent", "Assume (2). By Nakayama's Lemma \\ref{lemma-NAK} we see that", "$\\Omega_{S/k, \\mathfrak q}$ can be generated by $\\leq \\dim_x X$ elements.", "We may replace $S$ by $S_g$ for some $g \\in S$, $g \\not \\in \\mathfrak q$", "such that $\\Omega_{S/k}$ is generated by at most", "$\\dim_x X$ elements.", "Let $K \\supset k$ be an algebraically closed field extension", "such that there exists a $k$-algebra map $\\psi : \\kappa(\\mathfrak q) \\to K$.", "Consider $S_K = K \\otimes_k S$. Let $\\mathfrak m \\subset S_K$", "be the maximal ideal corresponding to the surjection", "$$", "\\xymatrix{", "S_K = K \\otimes_k S \\ar[r] &", "K \\otimes_k \\kappa(\\mathfrak q)", "\\ar[r]^-{\\text{id}_K \\otimes \\psi} &", "K.", "}", "$$", "Note that $\\mathfrak m \\cap S = \\mathfrak q$, in other words", "$\\mathfrak m$ lies over $\\mathfrak q$.", "By Lemma \\ref{lemma-dimension-at-a-point-preserved-field-extension}", "the dimension of $X_K = \\Spec(S_K)$ at the point corresponding", "to $\\mathfrak m$ is $\\dim_x X$. By", "Lemma \\ref{lemma-dimension-closed-point-finite-type-field}", "this is equal to $\\dim((S_K)_{\\mathfrak m})$.", "By Lemma \\ref{lemma-differentials-base-change}", "the module of differentials of $S_K$ over $K$ is", "the base change of $\\Omega_{S/k}$, hence also", "generated by at most $\\dim_x X = \\dim((S_K)_{\\mathfrak m})$", "elements. By Lemma \\ref{lemma-characterize-smooth-kbar}", "we see that $S_K$ is smooth at $\\mathfrak m$ over $K$.", "By Lemma \\ref{lemma-flat-base-change-locus-smooth} this", "implies that $S$ is smooth at $\\mathfrak q$ over $k$.", "This proves (1). Moreover, we know by", "Lemma \\ref{lemma-characterize-smooth-kbar}", "that the local ring $(S_K)_{\\mathfrak m}$ is regular.", "Since $S_{\\mathfrak q} \\to (S_K)_{\\mathfrak m}$ is flat we", "conclude from Lemma \\ref{lemma-flat-under-regular}", "that $S_{\\mathfrak q}$ is regular." ], "refs": [ "algebra-lemma-smooth-syntomic", "algebra-lemma-standard-smooth", "algebra-lemma-NAK", "algebra-lemma-dimension-at-a-point-preserved-field-extension", "algebra-lemma-dimension-closed-point-finite-type-field", "algebra-lemma-differentials-base-change", "algebra-lemma-characterize-smooth-kbar", "algebra-lemma-flat-base-change-locus-smooth", "algebra-lemma-characterize-smooth-kbar", "algebra-lemma-flat-under-regular" ], "ref_ids": [ 1195, 1193, 401, 1010, 996, 1138, 1222, 1201, 1222, 981 ] } ], "ref_ids": [] }, { "id": 1224, "type": "theorem", "label": "algebra-lemma-computation-differential", "categories": [ "algebra" ], "title": "algebra-lemma-computation-differential", "contents": [ "Let $k$ be a field.", "Let $R$ be a Noetherian local ring containing $k$.", "Assume that the residue field $\\kappa = R/\\mathfrak m$", "is a finitely generated separable extension of $k$.", "Then the map", "$$", "\\text{d} :", "\\mathfrak m/\\mathfrak m^2", "\\longrightarrow", "\\Omega_{R/k} \\otimes_R \\kappa(\\mathfrak m)", "$$", "is injective." ], "refs": [], "proofs": [ { "contents": [ "We may replace $R$ by $R/\\mathfrak m^2$. Hence we may assume that", "$\\mathfrak m^2 = 0$. By assumption we may write", "$\\kappa = k(\\overline{x}_1, \\ldots, \\overline{x}_r, \\overline{y})$", "where $\\overline{x}_1, \\ldots, \\overline{x}_r$ is a transcendence basis", "of $\\kappa$ over $k$ and $\\overline{y}$ is separable algebraic over", "$k(\\overline{x}_1, \\ldots, \\overline{x}_r)$. Say its minimal", "equation is $P(\\overline{y}) = 0$ with $P(T) = T^d + \\sum_{i < d} a_iT^i$,", "with $a_i \\in k(\\overline{x}_1, \\ldots, \\overline{x}_r)$ and", "$P'(\\overline{y}) \\not = 0$. Choose any lifts", "$x_i \\in R$ of the elements $\\overline{x}_i \\in \\kappa$.", "This gives a commutative diagram", "$$", "\\xymatrix{", "R \\ar[r] & \\kappa \\\\", "& k(\\overline{x}_1, \\ldots, \\overline{x}_r) \\ar[lu]^\\varphi \\ar[u]", "}", "$$", "of $k$-algebras. We want to extend the left upwards arrow", "$\\varphi$ to a $k$-algebra", "map from $\\kappa$ to $R$. To do this choose any $y \\in R$ lifting", "$\\overline{y}$. To see that it defines a $k$-algebra map", "defined on $\\kappa \\cong k(\\overline{x}_1, \\ldots, \\overline{x}_r)[T]/(P)$", "all we have to show is that we may choose $y$ such that $P^\\varphi(y) = 0$.", "If not then we compute for $\\delta \\in \\mathfrak m$ that", "$$", "P(y + \\delta) = P(y) + P'(y)\\delta", "$$", "because $\\mathfrak m^2 = 0$. Since $P'(y)\\delta = P'(\\overline{y})\\delta$", "we see that we can adjust our choice as desired.", "This shows that $R \\cong \\kappa \\oplus \\mathfrak m$ as", "$k$-algebras! From a direct computation of", "$\\Omega_{\\kappa \\oplus \\mathfrak m/k}$ the lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1225, "type": "theorem", "label": "algebra-lemma-separable-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-separable-smooth", "contents": [ "Let $k$ be a field.", "Let $S$ be a finite type $k$-algebra.", "Let $\\mathfrak q \\subset S$ be a prime.", "Assume $\\kappa(\\mathfrak q)$ is separable over $k$.", "The following are equivalent:", "\\begin{enumerate}", "\\item The algebra $S$ is smooth at $\\mathfrak q$ over $k$.", "\\item The ring $S_{\\mathfrak q}$ is regular.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $R = S_{\\mathfrak q}$ and denote its maximal", "by $\\mathfrak m$ and its residue field $\\kappa$.", "By Lemma \\ref{lemma-computation-differential} and", "\\ref{lemma-differential-seq} we see that there is a short exact", "sequence", "$$", "0 \\to \\mathfrak m/\\mathfrak m^2 \\to", "\\Omega_{R/k} \\otimes_R \\kappa \\to", "\\Omega_{\\kappa/k} \\to 0", "$$", "Note that $\\Omega_{R/k} = \\Omega_{S/k, \\mathfrak q}$, see", "Lemma \\ref{lemma-differentials-localize}.", "Moreover, since $\\kappa$ is separable over $k$", "we have $\\dim_{\\kappa} \\Omega_{\\kappa/k} = \\text{trdeg}_k(\\kappa)$.", "Hence we get", "$$", "\\dim_{\\kappa} \\Omega_{R/k} \\otimes_R \\kappa", "=", "\\dim_\\kappa \\mathfrak m/\\mathfrak m^2 + \\text{trdeg}_k (\\kappa)", "\\geq", "\\dim R + \\text{trdeg}_k (\\kappa)", "=", "\\dim_{\\mathfrak q} S", "$$", "(see Lemma \\ref{lemma-dimension-at-a-point-finite-type-field} for", "the last equality)", "with equality if and only if $R$ is regular.", "Thus we win by applying Lemma \\ref{lemma-characterize-smooth-over-field}." ], "refs": [ "algebra-lemma-computation-differential", "algebra-lemma-differential-seq", "algebra-lemma-differentials-localize", "algebra-lemma-dimension-at-a-point-finite-type-field", "algebra-lemma-characterize-smooth-over-field" ], "ref_ids": [ 1224, 1135, 1134, 1007, 1223 ] } ], "ref_ids": [] }, { "id": 1226, "type": "theorem", "label": "algebra-lemma-characteristic-zero", "categories": [ "algebra" ], "title": "algebra-lemma-characteristic-zero", "contents": [ "Let $R \\to S$ be a $\\mathbf{Q}$-algebra map.", "Let $f \\in S$ be such that $\\Omega_{S/R} = S \\text{d}f \\oplus C$", "for some $S$-submodule $C$. Then", "\\begin{enumerate}", "\\item $f$ is not nilpotent, and", "\\item if $S$ is a Noetherian local ring, then $f$ is a nonzerodivisor in $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For $a \\in S$ write $\\text{d}(a) = \\theta(a)\\text{d}f + c(a)$ for some", "$\\theta(a) \\in S$ and $c(a) \\in C$.", "Consider the $R$-derivation $S \\to S$, $a \\mapsto \\theta(a)$.", "Note that $\\theta(f) = 1$.", "\\medskip\\noindent", "If $f^n = 0$ with $n > 1$ minimal, then $0 = \\theta(f^n) = n f^{n - 1}$", "contradicting the minimality of $n$. We conclude that $f$ is not nilpotent.", "\\medskip\\noindent", "Suppose $fa = 0$. If $f$ is a unit then $a = 0$ and we win. Assume", "$f$ is not a unit. Then", "$0 = \\theta(fa) = f\\theta(a) + a$ by the Leibniz rule and hence $a \\in (f)$.", "By induction suppose we have shown $fa = 0 \\Rightarrow a \\in (f^n)$.", "Then writing $a = f^nb$ we get", "$0 = \\theta(f^{n + 1}b) = (n + 1)f^nb + f^{n + 1}\\theta(b)$.", "Hence $a = f^n b = -f^{n + 1}\\theta(b)/(n + 1) \\in (f^{n + 1})$.", "Since in the Noetherian local ring $S$ we have $\\bigcap (f^n) = 0$, see", "Lemma \\ref{lemma-intersect-powers-ideal-module-zero}", "we win." ], "refs": [ "algebra-lemma-intersect-powers-ideal-module-zero" ], "ref_ids": [ 627 ] } ], "ref_ids": [] }, { "id": 1227, "type": "theorem", "label": "algebra-lemma-characteristic-zero-local-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-characteristic-zero-local-smooth", "contents": [ "Let $k$ be a field of characteristic $0$.", "Let $S$ be a finite type $k$-algebra.", "Let $\\mathfrak q \\subset S$ be a prime.", "The following are equivalent:", "\\begin{enumerate}", "\\item The algebra $S$ is smooth at $\\mathfrak q$ over $k$.", "\\item The $S_{\\mathfrak q}$-module $\\Omega_{S/k, \\mathfrak q}$", "is (finite) free.", "\\item The ring $S_{\\mathfrak q}$ is regular.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In characteristic zero any field extension is separable and hence the", "equivalence of (1) and (3) follows from Lemma \\ref{lemma-separable-smooth}.", "Also (1) implies (2) by definition of smooth algebras.", "Assume that $\\Omega_{S/k, \\mathfrak q}$ is free over $S_{\\mathfrak q}$.", "We are going to use the notation and observations made in the", "proof of Lemma \\ref{lemma-separable-smooth}. So $R = S_{\\mathfrak q}$", "with maximal ideal $\\mathfrak m$ and residue field $\\kappa$.", "Our goal is to prove $R$ is regular.", "\\medskip\\noindent", "If $\\mathfrak m/\\mathfrak m^2 = 0$, then $\\mathfrak m = 0$", "and $R \\cong \\kappa$. Hence $R$ is regular and we win.", "\\medskip\\noindent", "If $\\mathfrak m/ \\mathfrak m^2 \\not = 0$, then choose any", "$f \\in \\mathfrak m$ whose image in $\\mathfrak m/ \\mathfrak m^2$", "is not zero. By Lemma \\ref{lemma-computation-differential}", "we see that $\\text{d}f$ has nonzero image in", "$\\Omega_{R/k}/\\mathfrak m\\Omega_{R/k}$. By assumption", "$\\Omega_{R/k} = \\Omega_{S/k, \\mathfrak q}$ is finite free and", "hence by Nakayama's Lemma \\ref{lemma-NAK} we see that", "$\\text{d}f$ generates a direct summand. We apply", "Lemma \\ref{lemma-characteristic-zero}", "to deduce that $f$ is a nonzerodivisor in $R$.", "Furthermore, by Lemma \\ref{lemma-differential-seq} we get an exact sequence", "$$", "(f)/(f^2) \\to \\Omega_{R/k} \\otimes_R R/fR \\to \\Omega_{(R/fR)/k} \\to 0", "$$", "This implies that $\\Omega_{(R/fR)/k}$ is finite free as well.", "Hence by induction we see that $R/fR$ is a regular local ring.", "Since $f \\in \\mathfrak m$ was a nonzerodivisor we", "conclude that $R$ is regular, see Lemma \\ref{lemma-regular-mod-x}." ], "refs": [ "algebra-lemma-separable-smooth", "algebra-lemma-separable-smooth", "algebra-lemma-computation-differential", "algebra-lemma-NAK", "algebra-lemma-characteristic-zero", "algebra-lemma-differential-seq", "algebra-lemma-regular-mod-x" ], "ref_ids": [ 1225, 1225, 1224, 401, 1226, 1135, 945 ] } ], "ref_ids": [] }, { "id": 1228, "type": "theorem", "label": "algebra-lemma-smooth-at-generic-point", "categories": [ "algebra" ], "title": "algebra-lemma-smooth-at-generic-point", "contents": [ "Let $R \\to S$ be an injective finite type ring map with $R$ and $S$ domains.", "Then $R \\to S$ is smooth at $\\mathfrak q = (0)$ if and only if", "the induced extension $L/K$ of fraction fields is separable." ], "refs": [], "proofs": [ { "contents": [ "Assume $R \\to S$ is smooth at $(0)$. We may replace $S$ by $S_g$", "for some nonzero $g \\in S$ and assume that $R \\to S$ is smooth.", "Then $K \\to S \\otimes_R K$ is smooth", "(Lemma \\ref{lemma-base-change-smooth}). Moreover, for any", "field extension $K \\subset K'$ the ring map $K' \\to S \\otimes_R K'$", "is smooth as well. Hence $S \\otimes_R K'$ is a regular ring", "by Lemma \\ref{lemma-characterize-smooth-over-field}, in particular reduced.", "It follows that $S \\otimes_R K$ is a geometrically reduced over $K$.", "Hence $L$ is geometrically reduced over $K$, see", "Lemma \\ref{lemma-geometrically-reduced-permanence}.", "Hence $L/K$ is separable by", "Lemma \\ref{lemma-characterize-separable-field-extensions}.", "\\medskip\\noindent", "Conversely, assume that $L/K$ is separable.", "We may assume $R \\to S$ is of finite presentation, see", "Lemma \\ref{lemma-generic-finite-presentation}.", "It suffices to prove that $K \\to S \\otimes_R K$ is smooth", "at $(0)$, see", "Lemma \\ref{lemma-flat-base-change-locus-smooth}.", "This follows from Lemma \\ref{lemma-separable-smooth}, the", "fact that a field is a regular ring,", "and the assumption that $L/K$ is separable." ], "refs": [ "algebra-lemma-base-change-smooth", "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-geometrically-reduced-permanence", "algebra-lemma-characterize-separable-field-extensions", "algebra-lemma-generic-finite-presentation", "algebra-lemma-flat-base-change-locus-smooth", "algebra-lemma-separable-smooth" ], "ref_ids": [ 1191, 1223, 562, 569, 441, 1201, 1225 ] } ], "ref_ids": [] }, { "id": 1229, "type": "theorem", "label": "algebra-lemma-smooth-test-artinian", "categories": [ "algebra" ], "title": "algebra-lemma-smooth-test-artinian", "contents": [ "Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime ideal of", "$S$ lying over $\\mathfrak p \\subset R$. Assume $R$ is Noetherian", "and $R \\to S$ of finite type.", "The following are equivalent:", "\\begin{enumerate}", "\\item $R \\to S$ is smooth at $\\mathfrak q$,", "\\item for every surjection of local $R$-algebras", "$(B', \\mathfrak m') \\to (B, \\mathfrak m)$", "with $\\Ker(B' \\to B)$ having square zero", "and every solid commutative diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar@{-->}[rd] & B \\\\", "R \\ar[r] \\ar[u] & B' \\ar[u]", "}", "$$", "such that $\\mathfrak q = S \\cap \\mathfrak m$ there exists a dotted", "arrow making the diagram commute,", "\\item same as in (2) but with $B' \\to B$ ranging over small extensions, and", "\\item same as in (2) but with $B' \\to B$ ranging over small extensions", "such that in addition $S \\to B$ induces an isomorphism", "$\\kappa(\\mathfrak q) \\cong \\kappa(\\mathfrak m)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). This means there exists a $g \\in S$, $g \\not \\in \\mathfrak q$", "such that $R \\to S_g$ is smooth. By", "Proposition \\ref{proposition-smooth-formally-smooth}", "we know that $R \\to S_g$ is formally smooth. Note that given any diagram", "as in (2) the map $S \\to B$ factors automatically through $S_{\\mathfrak q}$", "and a fortiori through $S_g$. The formal smoothness of $S_g$ over $R$", "gives us a morphism $S_g \\to B'$ fitting into a similar diagram with $S_g$ at", "the upper left corner. Composing with $S \\to S_g$ gives the desired arrow.", "In other words, we have shown that (1) implies (2).", "\\medskip\\noindent", "Clearly (2) implies (3) and (3) implies (4).", "\\medskip\\noindent", "Assume (4). We are going to show that (1) holds, thereby finishing the", "proof of the lemma. Choose a presentation", "$S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$.", "This is possible as $S$ is of finite type over $R$ and therefore of finite", "presentation (see", "Lemma \\ref{lemma-Noetherian-finite-type-is-finite-presentation}).", "Set $I = (f_1, \\ldots, f_m)$.", "Consider the naive cotangent complex", "$$", "\\text{d} : I/I^2", "\\longrightarrow", "\\bigoplus\\nolimits_{j = 1}^m S\\text{d}x_j", "$$", "of this presentation (see Section \\ref{section-netherlander}).", "It suffices to show that when we localize this complex at $\\mathfrak q$", "then the map becomes a split injection, see Lemma \\ref{lemma-smooth-at-point}.", "Denote $S' = R[x_1, \\ldots, x_n]/I^2$.", "By Lemma \\ref{lemma-differential-mod-power-ideal} we have", "$$", "S \\otimes_{S'} \\Omega_{S'/R} =", "S \\otimes_{R[x_1, \\ldots, x_n]} \\Omega_{R[x_1, \\ldots, x_n]/R} =", "\\bigoplus\\nolimits_{j = 1}^m S\\text{d}x_j.", "$$", "Thus the map", "$$", "\\text{d} :", "I/I^2", "\\longrightarrow", "S \\otimes_{S'} \\Omega_{S'/R}", "$$", "is the same as the map in the naive cotangent complex above. In particular", "the truth of the assertion we are trying to prove", "depends only on the three rings $R \\to S' \\to S$.", "Let $\\mathfrak q' \\subset R[x_1, \\ldots, x_n]$ be the prime ideal", "corresponding to $\\mathfrak q$. Since", "localization commutes with taking modules of differentials", "(Lemma \\ref{lemma-differentials-localize}) we see that it suffices to show", "that the map", "\\begin{equation}", "\\label{equation-target-map}", "\\text{d} :", "I_{\\mathfrak q'}/I_{\\mathfrak q'}^2", "\\longrightarrow", "S_{\\mathfrak q} \\otimes_{S'_{\\mathfrak q'}} \\Omega_{S'_{\\mathfrak q'}/R}", "\\end{equation}", "coming from $R \\to S'_{\\mathfrak q'} \\to S_{\\mathfrak q}$", "is a split injection.", "\\medskip\\noindent", "Let $N \\in \\mathbf{N}$ be an integer.", "Consider the ring", "$$", "B'_N = S'_{\\mathfrak q'} / (\\mathfrak q')^N S'_{\\mathfrak q'}", "= (S'/(\\mathfrak q')^N S')_{\\mathfrak q'}", "$$", "and its quotient $B_N = B'_N/IB'_N$. Note that", "$B_N \\cong S_{\\mathfrak q}/\\mathfrak q^NS_{\\mathfrak q}$.", "Observe that $B'_N$ is an Artinian local ring since it is the", "quotient of a local Noetherian ring by a power of its maximal ideal.", "Consider a filtration of the kernel $I_N$ of $B'_N \\to B_N$", "by $B'_N$-submodules", "$$", "0 \\subset J_{N, 1} \\subset J_{N, 2} \\subset \\ldots \\subset J_{N, n(N)} = I_N", "$$", "such that each successive quotient $J_{N, i}/J_{N, i - 1}$ has length $1$.", "(As $B'_N$ is Artinian such a filtration exists.)", "This gives a sequence of small extensions", "$$", "B'_N \\to B'_N/J_{N, 1} \\to B'_N/J_{N, 2} \\to \\ldots \\to", "B'_N/J_{N, n(N)} = B'_N/I_N", "= B_N = S_{\\mathfrak q}/\\mathfrak q^NS_{\\mathfrak q}", "$$", "Applying condition (4) successively to these small extensions", "starting with the map $S \\to B_N$ we see there", "exists a commutative diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar[rd] & B_N \\\\", "R \\ar[r] \\ar[u] & B'_N \\ar[u]", "}", "$$", "Clearly the ring map $S \\to B'_N$ factors as $S \\to S_{\\mathfrak q} \\to B'_N$", "where $S_{\\mathfrak q} \\to B'_N$ is a local homomorphism of local rings.", "Moreover, since the maximal ideal of $B'_N$ to the $N$th power is zero", "we conclude that $S_{\\mathfrak q} \\to B'_N$ factors through", "$S_{\\mathfrak q}/(\\mathfrak q)^NS_{\\mathfrak q} = B_N$. In other words", "we have shown that for all $N \\in \\mathbf{N}$ the surjection of", "$R$-algebras $B'_N \\to B_N$ has a splitting.", "\\medskip\\noindent", "Consider the presentation", "$$", "I_N \\to B_N \\otimes_{B'_N} \\Omega_{B'_N/R} \\to \\Omega_{B_N/R} \\to 0", "$$", "coming from the surjection $B'_N \\to B_N$ with kernel $I_N$ (see", "Lemma \\ref{lemma-differential-seq}). By the above the $R$-algebra map", "$B'_N \\to B_N$ has a right inverse. Hence by", "Lemma \\ref{lemma-differential-seq-split} we see that the sequence above", "is split exact! Thus for every $N$ the map", "$$", "I_N \\longrightarrow B_N \\otimes_{B'_N} \\Omega_{B'_N/R}", "$$", "is a split injection. The rest of the proof is gotten by unwinding what", "this means exactly. Note that", "$$", "I_N = I_{\\mathfrak q'}/", "(I_{\\mathfrak q'}^2 + (\\mathfrak q')^N \\cap I_{\\mathfrak q'})", "$$", "By Artin-Rees (Lemma \\ref{lemma-Artin-Rees}) we find a $c \\geq 0$", "such that", "$$", "S_{\\mathfrak q}/\\mathfrak q^{N - c}S_{\\mathfrak q}", "\\otimes_{S_{\\mathfrak q}} I_N =", "S_{\\mathfrak q}/\\mathfrak q^{N - c}S_{\\mathfrak q}", "\\otimes_{S_{\\mathfrak q}}", "I_{\\mathfrak q'}/I_{\\mathfrak q'}^2", "$$", "for all $N \\geq c$", "(these tensor product are just a fancy way of dividing by", "$\\mathfrak q^{N - c}$). We may of course assume $c \\geq 1$.", "By Lemma \\ref{lemma-differential-mod-power-ideal} we see that", "$$", "S'_{\\mathfrak q'}/(\\mathfrak q')^{N - c}S'_{\\mathfrak q'}", "\\otimes_{S'_{\\mathfrak q'}} \\Omega_{B'_N/R} =", "S'_{\\mathfrak q'}/(\\mathfrak q')^{N - c}S'_{\\mathfrak q'}", "\\otimes_{S'_{\\mathfrak q'}} \\Omega_{S'_{\\mathfrak q'}/R}", "$$", "we can further tensor this by $B_N = S_{\\mathfrak q}/\\mathfrak q^N$", "to see that", "$$", "S_{\\mathfrak q}/\\mathfrak q^{N - c}S_{\\mathfrak q}", "\\otimes_{S'_{\\mathfrak q'}} \\Omega_{B'_N/R} =", "S_{\\mathfrak q}/\\mathfrak q^{N - c}S_{\\mathfrak q}", "\\otimes_{S'_{\\mathfrak q'}} \\Omega_{S'_{\\mathfrak q'}/R}.", "$$", "Since a split injection remains a split injection after tensoring", "with anything we see that", "$$", "S_{\\mathfrak q}/\\mathfrak q^{N - c}S_{\\mathfrak q}", "\\otimes_{S_{\\mathfrak q}}", "(\\ref{equation-target-map}) =", "S_{\\mathfrak q}/\\mathfrak q^{N - c}S_{\\mathfrak q}", "\\otimes_{S_{\\mathfrak q}/\\mathfrak q^N S_{\\mathfrak q}}", "(I_N \\longrightarrow B_N \\otimes_{B'_N} \\Omega_{B'_N/R})", "$$", "is a split injection for all $N \\geq c$. By", "Lemma \\ref{lemma-split-injection-after-completion} we see that", "(\\ref{equation-target-map}) is a split injection. This finishes the proof." ], "refs": [ "algebra-proposition-smooth-formally-smooth", "algebra-lemma-Noetherian-finite-type-is-finite-presentation", "algebra-lemma-smooth-at-point", "algebra-lemma-differential-mod-power-ideal", "algebra-lemma-differentials-localize", "algebra-lemma-differential-seq", "algebra-lemma-differential-seq-split", "algebra-lemma-Artin-Rees", "algebra-lemma-differential-mod-power-ideal", "algebra-lemma-split-injection-after-completion" ], "ref_ids": [ 1426, 451, 1196, 1137, 1134, 1135, 1136, 625, 1137, 780 ] } ], "ref_ids": [] }, { "id": 1230, "type": "theorem", "label": "algebra-lemma-etale-standard-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-etale-standard-smooth", "contents": [ "Any \\'etale ring map is standard smooth. More precisely, if", "$R \\to S$ is \\'etale, then there exists a presentation", "$S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$ such that", "the image of $\\det(\\partial f_j/\\partial x_i)$ is invertible in $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $R \\to S$ be \\'etale. Choose a presentation $S = R[x_1, \\ldots, x_n]/I$.", "As $R \\to S$ is \\'etale we know that", "$$", "\\text{d} :", "I/I^2", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} S\\text{d}x_i", "$$", "is an isomorphism, in particular $I/I^2$ is a free $S$-module.", "Thus by Lemma \\ref{lemma-huber} we may assume (after possibly changing", "the presentation), that $I = (f_1, \\ldots, f_c)$ such that the classes", "$f_i \\bmod I^2$ form a basis of $I/I^2$. It follows immediately from", "the fact that the displayed map above is an isomorphism that $c = n$ and", "that $\\det(\\partial f_j/\\partial x_i)$ is invertible in $S$." ], "refs": [ "algebra-lemma-huber" ], "ref_ids": [ 1178 ] } ], "ref_ids": [] }, { "id": 1231, "type": "theorem", "label": "algebra-lemma-etale", "categories": [ "algebra" ], "title": "algebra-lemma-etale", "contents": [ "Results on \\'etale ring maps.", "\\begin{enumerate}", "\\item The ring map $R \\to R_f$ is \\'etale for any ring $R$ and any $f \\in R$.", "\\item Compositions of \\'etale ring maps are \\'etale.", "\\item A base change of an \\'etale ring map is \\'etale.", "\\item The property of being \\'etale is local: Given a ring map", "$R \\to S$ and elements $g_1, \\ldots, g_m \\in S$ which generate the unit ideal", "such that $R \\to S_{g_j}$ is \\'etale for $j = 1, \\ldots, m$ then", "$R \\to S$ is \\'etale.", "\\item Given $R \\to S$ of finite presentation, and a flat ring map", "$R \\to R'$, set $S' = R' \\otimes_R S$. The set of primes where $R' \\to S'$", "is \\'etale is the inverse image via $\\Spec(S') \\to \\Spec(S)$", "of the set of primes where $R \\to S$ is \\'etale.", "\\item An \\'etale ring map is syntomic, in particular flat.", "\\item If $S$ is finite type over a field $k$, then $S$ is \\'etale over", "$k$ if and only if $\\Omega_{S/k} = 0$.", "\\item Any \\'etale ring map $R \\to S$ is the base change of an \\'etale", "ring map $R_0 \\to S_0$ with $R_0$ of finite type over $\\mathbf{Z}$.", "\\item Let $A = \\colim A_i$ be a filtered colimit of rings.", "Let $A \\to B$ be an \\'etale ring map. Then there exists an \\'etale ring", "map $A_i \\to B_i$ for some $i$ such that $B \\cong A \\otimes_{A_i} B_i$.", "\\item Let $A$ be a ring. Let $S$ be a multiplicative subset of $A$.", "Let $S^{-1}A \\to B'$ be \\'etale. Then there exists an \\'etale ring map", "$A \\to B$ such that $B' \\cong S^{-1}B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In each case we use the corresponding result for smooth ring maps with", "a small argument added to show that $\\Omega_{S/R}$ is zero.", "\\medskip\\noindent", "Proof of (1). The ring map $R \\to R_f$ is smooth and $\\Omega_{R_f/R} = 0$.", "\\medskip\\noindent", "Proof of (2). The composition $A \\to C$ of smooth maps $A \\to B$ and", "$B \\to C$ is smooth, see Lemma \\ref{lemma-compose-smooth}. By", "Lemma \\ref{lemma-exact-sequence-differentials} we see that", "$\\Omega_{C/A}$ is zero as both $\\Omega_{C/B}$ and $\\Omega_{B/A}$ are zero.", "\\medskip\\noindent", "Proof of (3). Let $R \\to S$ be \\'etale and $R \\to R'$ be arbitrary.", "Then $R' \\to S' = R' \\otimes_R S$ is smooth, see", "Lemma \\ref{lemma-base-change-smooth}. Since", "$\\Omega_{S'/R'} = S' \\otimes_S \\Omega_{S/R}$ by", "Lemma \\ref{lemma-differentials-base-change}", "we conclude that $\\Omega_{S'/R'} = 0$. Hence $R' \\to S'$ is \\'etale.", "\\medskip\\noindent", "Proof of (4). Assume the hypotheses of (4). By", "Lemma \\ref{lemma-locally-smooth} we see that $R \\to S$ is smooth.", "We are also given that $\\Omega_{S_{g_i}/R} = (\\Omega_{S/R})_{g_i} = 0$", "for all $i$. Then $\\Omega_{S/R} = 0$, see Lemma \\ref{lemma-cover}.", "\\medskip\\noindent", "Proof of (5). The result for smooth maps is", "Lemma \\ref{lemma-flat-base-change-locus-smooth}.", "In the proof of that lemma we used that $\\NL_{S/R} \\otimes_S S'$", "is homotopy equivalent to $\\NL_{S'/R'}$.", "This reduces us to showing that if $M$ is a finitely presented", "$S$-module the set of primes $\\mathfrak q'$ of $S'$", "such that $(M \\otimes_S S')_{\\mathfrak q'} = 0$ is the inverse", "image of the set of primes $\\mathfrak q$ of $S$ such that", "$M_{\\mathfrak q} = 0$. This follows from Lemma \\ref{lemma-support-base-change}.", "\\medskip\\noindent", "Proof of (6). Follows directly from the corresponding result for", "smooth ring maps (Lemma \\ref{lemma-smooth-syntomic}).", "\\medskip\\noindent", "Proof of (7). Follows from Lemma \\ref{lemma-characterize-smooth-over-field}", "and the definitions.", "\\medskip\\noindent", "Proof of (8). Lemma \\ref{lemma-finite-presentation-fs-Noetherian}", "gives the result for smooth ring maps. The resulting smooth ring map", "$R_0 \\to S_0$ satisfies the", "hypotheses of Lemma \\ref{lemma-relative-dimension-CM}, and hence we may", "replace $S_0$ by the factor of relative dimension $0$ over $R_0$.", "\\medskip\\noindent", "Proof of (9). Follows from (8) since $R_0 \\to A$ will factor through", "$A_i$ for some $i$ by Lemma \\ref{lemma-characterize-finite-presentation}.", "\\medskip\\noindent", "Proof of (10). Follows from (9), (1), and (2) since $S^{-1}A$ is a", "filtered colimit of principal localizations of $A$." ], "refs": [ "algebra-lemma-compose-smooth", "algebra-lemma-exact-sequence-differentials", "algebra-lemma-base-change-smooth", "algebra-lemma-differentials-base-change", "algebra-lemma-locally-smooth", "algebra-lemma-cover", "algebra-lemma-flat-base-change-locus-smooth", "algebra-lemma-support-base-change", "algebra-lemma-smooth-syntomic", "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-finite-presentation-fs-Noetherian", "algebra-lemma-relative-dimension-CM", "algebra-lemma-characterize-finite-presentation" ], "ref_ids": [ 1198, 1133, 1191, 1138, 1197, 411, 1201, 544, 1195, 1223, 1213, 1128, 1092 ] } ], "ref_ids": [] }, { "id": 1232, "type": "theorem", "label": "algebra-lemma-etale-over-field", "categories": [ "algebra" ], "title": "algebra-lemma-etale-over-field", "contents": [ "Let $k$ be a field. A ring map $k \\to S$ is \\'etale if and only if $S$", "is isomorphic as a $k$-algebra to a finite product of finite separable", "extensions of $k$." ], "refs": [], "proofs": [ { "contents": [ "If $k \\to k'$ is a finite separable field extension then we can", "write $k' = k(\\alpha) \\cong k[x]/(f)$. Here $f$ is the minimal", "polynomial of the element $\\alpha$. Since $k'$ is separable over $k$", "we have $\\gcd(f, f') = 1$. This", "implies that $\\text{d} : k'\\cdot f \\to k' \\cdot \\text{d}x$", "is an isomorphism. Hence $k \\to k'$ is \\'etale.", "\\medskip\\noindent", "Conversely, suppose that $k \\to S$ is \\'etale. Let $\\overline{k}$", "be an algebraic closure of $k$. Then $S \\otimes_k \\overline{k}$", "is \\'etale over $\\overline{k}$. Suppose we have the result over $\\overline{k}$.", "Then $S \\otimes_k \\overline{k}$ is reduced and hence $S$ is reduced.", "Also, $S \\otimes_k \\overline{k}$ is finite over $\\overline{k}$", "and hence $S$ is finite over $k$. Hence $S$ is a finite product", "$S = \\prod k_i$", "of fields, see", "Lemma \\ref{lemma-finite-dimensional-algebra}", "and", "Proposition \\ref{proposition-dimension-zero-ring}.", "The result over $\\overline{k}$ means $S \\otimes_k \\overline{k}$", "is isomorphic to a finite product of copies of $\\overline{k}$, which", "implies that each $k \\subset k_i$ is finite separable, see for example", "Lemmas \\ref{lemma-characterize-separable-field-extensions} and", "\\ref{lemma-geometrically-reduced-finite-purely-inseparable-extension}.", "Thus we have reduced to the case $k = \\overline{k}$.", "In this case", "Lemma \\ref{lemma-characterize-smooth-kbar}", "(combined with $\\Omega_{S/k} = 0$) we see that $S_{\\mathfrak m} \\cong k$", "for all maximal ideals $\\mathfrak m \\subset S$. This implies the result", "because $S$ is the product of the localizations at its maximal ideals by", "Lemma \\ref{lemma-finite-dimensional-algebra}", "and", "Proposition \\ref{proposition-dimension-zero-ring}", "again." ], "refs": [ "algebra-lemma-finite-dimensional-algebra", "algebra-proposition-dimension-zero-ring", "algebra-lemma-characterize-separable-field-extensions", "algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension", "algebra-lemma-characterize-smooth-kbar", "algebra-lemma-finite-dimensional-algebra", "algebra-proposition-dimension-zero-ring" ], "ref_ids": [ 642, 1410, 569, 571, 1222, 642, 1410 ] } ], "ref_ids": [] }, { "id": 1233, "type": "theorem", "label": "algebra-lemma-etale-at-prime", "categories": [ "algebra" ], "title": "algebra-lemma-etale-at-prime", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p$ in $R$.", "If $S/R$ is \\'etale at $\\mathfrak q$ then", "\\begin{enumerate}", "\\item we have $\\mathfrak p S_{\\mathfrak q} = \\mathfrak qS_{\\mathfrak q}$", "is the maximal ideal of the local ring $S_{\\mathfrak q}$, and", "\\item the field extension $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$", "is finite separable.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First we may replace $S$ by $S_g$ for some $g \\in S$, $g \\not \\in \\mathfrak q$", "and assume that $R \\to S$ is \\'etale. Then the lemma follows from", "Lemma \\ref{lemma-etale-over-field} by unwinding the", "fact that $S \\otimes_R \\kappa(\\mathfrak p)$ is \\'etale over", "$\\kappa(\\mathfrak p)$." ], "refs": [ "algebra-lemma-etale-over-field" ], "ref_ids": [ 1232 ] } ], "ref_ids": [] }, { "id": 1234, "type": "theorem", "label": "algebra-lemma-etale-quasi-finite", "categories": [ "algebra" ], "title": "algebra-lemma-etale-quasi-finite", "contents": [ "An \\'etale ring map is quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Let $R \\to S$ be an \\'etale ring map. By definition $R \\to S$ is of finite type.", "For any prime $\\mathfrak p \\subset R$ the fibre ring", "$S \\otimes_R \\kappa(\\mathfrak p)$ is \\'etale over $\\kappa(\\mathfrak p)$", "and hence a finite products of fields finite separable over", "$\\kappa(\\mathfrak p)$, in particular finite over $\\kappa(\\mathfrak p)$.", "Thus $R \\to S$ is quasi-finite by Lemma \\ref{lemma-quasi-finite}." ], "refs": [ "algebra-lemma-quasi-finite" ], "ref_ids": [ 1050 ] } ], "ref_ids": [] }, { "id": 1235, "type": "theorem", "label": "algebra-lemma-characterize-etale", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-etale", "contents": [ "Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime of $S$", "lying over a prime $\\mathfrak p$ of $R$. If", "\\begin{enumerate}", "\\item $R \\to S$ is of finite presentation,", "\\item $R_{\\mathfrak p} \\to S_{\\mathfrak q}$ is flat", "\\item $\\mathfrak p S_{\\mathfrak q}$ is the maximal ideal", "of the local ring $S_{\\mathfrak q}$, and", "\\item the field extension $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$", "is finite separable,", "\\end{enumerate}", "then $R \\to S$ is \\'etale at $\\mathfrak q$." ], "refs": [], "proofs": [ { "contents": [ "Apply", "Lemma \\ref{lemma-isolated-point-fibre}", "to find a $g \\in S$, $g \\not \\in \\mathfrak q$ such that", "$\\mathfrak q$ is the only prime of $S_g$ lying over $\\mathfrak p$.", "We may and do replace $S$ by $S_g$. Then", "$S \\otimes_R \\kappa(\\mathfrak p)$ has a unique prime, hence is a", "local ring, hence is equal to", "$S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}", "\\cong \\kappa(\\mathfrak q)$.", "By Lemma \\ref{lemma-flat-fibre-smooth}", "there exists a $g \\in S$, $g \\not \\in \\mathfrak q$", "such that $R \\to S_g$ is smooth. Replace $S$ by $S_g$ again we may", "assume that $R \\to S$ is smooth. By", "Lemma \\ref{lemma-smooth-syntomic} we may even assume that", "$R \\to S$ is standard smooth, say $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$.", "Since $S \\otimes_R \\kappa(\\mathfrak p) = \\kappa(\\mathfrak q)$", "has dimension $0$ we conclude that $n = c$, i.e., $R \\to S$ is \\'etale." ], "refs": [ "algebra-lemma-isolated-point-fibre", "algebra-lemma-flat-fibre-smooth", "algebra-lemma-smooth-syntomic" ], "ref_ids": [ 1049, 1200, 1195 ] } ], "ref_ids": [] }, { "id": 1236, "type": "theorem", "label": "algebra-lemma-map-between-etale", "categories": [ "algebra" ], "title": "algebra-lemma-map-between-etale", "contents": [ "Let $R \\to S$ and $R \\to S'$ be \\'etale.", "Then any $R$-algebra map $S' \\to S$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "First of all we note that $S' \\to S$ is of finite presentation by", "Lemma \\ref{lemma-compose-finite-type}.", "Let $\\mathfrak q \\subset S$ be a prime ideal lying over the primes", "$\\mathfrak q' \\subset S'$ and $\\mathfrak p \\subset R$.", "By Lemma \\ref{lemma-etale-at-prime} the ring map", "$S'_{\\mathfrak q'}/\\mathfrak p S'_{\\mathfrak q'} \\to", "S_{\\mathfrak q}/\\mathfrak p S_{\\mathfrak q}$", "is a map finite separable extensions of $\\kappa(\\mathfrak p)$.", "In particular it is flat. Hence by", "Lemma \\ref{lemma-criterion-flatness-fibre} we see that", "$S'_{\\mathfrak q'} \\to S_{\\mathfrak q}$ is flat. Thus $S' \\to S$", "is flat. Moreover, the above also shows that $\\mathfrak q'S_{\\mathfrak q}$", "is the maximal ideal of $S_{\\mathfrak q}$ and that the residue", "field extension of $S'_{\\mathfrak q'} \\to S_{\\mathfrak q}$ is", "finite separable. Hence from Lemma \\ref{lemma-characterize-etale}", "we conclude that $S' \\to S$ is \\'etale at $\\mathfrak q$. Since", "being \\'etale is local (see Lemma \\ref{lemma-etale}) we win." ], "refs": [ "algebra-lemma-compose-finite-type", "algebra-lemma-etale-at-prime", "algebra-lemma-criterion-flatness-fibre", "algebra-lemma-characterize-etale", "algebra-lemma-etale" ], "ref_ids": [ 333, 1233, 1114, 1235, 1231 ] } ], "ref_ids": [] }, { "id": 1237, "type": "theorem", "label": "algebra-lemma-surjective-flat-finitely-presented", "categories": [ "algebra" ], "title": "algebra-lemma-surjective-flat-finitely-presented", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. If $R \\to S$ is surjective, flat and", "finitely presented then there exist an idempotent $e \\in R$ such that", "$S = R_e$." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "Let $I$ be the kernel of $\\varphi$.", "We have that $I$ is finitely generated by", "Lemma \\ref{lemma-finite-presentation-independent}", "since $\\varphi$ is of finite presentation.", "Moreover, since $S$ is flat over $R$, tensoring the exact sequence", "$0 \\to I \\to R \\to S \\to 0$ over $R$ with $S$", "gives $I/I^2 = 0$. Now we conclude by", "Lemma \\ref{lemma-ideal-is-squared-union-connected}." ], "refs": [ "algebra-lemma-finite-presentation-independent", "algebra-lemma-ideal-is-squared-union-connected" ], "ref_ids": [ 334, 407 ] } ], "ref_ids": [] }, { "id": 1238, "type": "theorem", "label": "algebra-lemma-lift-etale", "categories": [ "algebra" ], "title": "algebra-lemma-lift-etale", "contents": [ "\\begin{slogan}", "\\'Etale ring maps lift along surjections of rings", "\\end{slogan}", "Let $R$ be a ring and let $I \\subset R$ be an ideal.", "Let $R/I \\to \\overline{S}$ be an \\'etale ring map.", "Then there exists an \\'etale ring map", "$R \\to S$ such that $\\overline{S} \\cong S/IS$ as $R/I$-algebras." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-etale-standard-smooth} we can write", "$\\overline{S} =", "(R/I)[x_1, \\ldots, x_n]/(\\overline{f}_1, \\ldots, \\overline{f}_n)$", "as in Definition \\ref{definition-standard-smooth} with", "$\\overline{\\Delta} =", "\\det(\\frac{\\partial \\overline{f}_i}{\\partial x_j})_{i, j = 1, \\ldots, n}$", "invertible in $\\overline{S}$. Just take some lifts $f_i$ and set", "$S = R[x_1, \\ldots, x_n, x_{n+1}]/(f_1, \\ldots, f_n, x_{n + 1}\\Delta - 1)$", "where $\\Delta = \\det(\\frac{\\partial f_i}{\\partial x_j})_{i, j = 1, \\ldots, n}$", "as in Example \\ref{example-make-standard-smooth}.", "This proves the lemma." ], "refs": [ "algebra-lemma-etale-standard-smooth", "algebra-definition-standard-smooth" ], "ref_ids": [ 1230, 1535 ] } ], "ref_ids": [] }, { "id": 1239, "type": "theorem", "label": "algebra-lemma-lift-etale-infinitesimal", "categories": [ "algebra" ], "title": "algebra-lemma-lift-etale-infinitesimal", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "J \\ar[r] &", "B' \\ar[r] &", "B \\ar[r] & 0 \\\\", "0 \\ar[r] &", "I \\ar[r] \\ar[u] &", "A' \\ar[r] \\ar[u] &", "A \\ar[r] \\ar[u] & 0", "}", "$$", "with exact rows where $B' \\to B$ and $A' \\to A$ are surjective ring maps", "whose kernels are ideals of square zero. If $A \\to B$ is \\'etale,", "and $J = I \\otimes_A B$, then $A' \\to B'$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-lift-etale}", "there exists an \\'etale ring map $A' \\to C$ such that $C/IC = B$.", "Then $A' \\to C$ is formally smooth (by", "Proposition \\ref{proposition-smooth-formally-smooth})", "hence we get an $A'$-algebra map $\\varphi : C \\to B'$.", "Since $A' \\to C$ is flat we have $I \\otimes_A B = I \\otimes_A C/IC = IC$.", "Hence the assumption that $J = I \\otimes_A B$ implies that", "$\\varphi$ induces an isomorphism $IC \\to J$ and an isomorphism", "$C/IC \\to B'/IB'$, whence $\\varphi$ is an isomorphism." ], "refs": [ "algebra-lemma-lift-etale", "algebra-proposition-smooth-formally-smooth" ], "ref_ids": [ 1238, 1426 ] } ], "ref_ids": [] }, { "id": 1240, "type": "theorem", "label": "algebra-lemma-factor-mod-lift-etale", "categories": [ "algebra" ], "title": "algebra-lemma-factor-mod-lift-etale", "contents": [ "Let $R$ be a ring. Let $f \\in R[x]$ be a monic polynomial. Let $\\mathfrak p$", "be a prime of $R$. Let $f \\bmod \\mathfrak p = \\overline{g} \\overline{h}$", "be a factorization of the image of $f$ in $\\kappa(\\mathfrak p)[x]$.", "If $\\gcd(\\overline{g}, \\overline{h}) = 1$, then there exist", "\\begin{enumerate}", "\\item an \\'etale ring map $R \\to R'$,", "\\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$, and", "\\item a factorization $f = g h$ in $R'[x]$", "\\end{enumerate}", "such that", "\\begin{enumerate}", "\\item $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$,", "\\item $\\overline{g} = g \\bmod \\mathfrak p'$,", "$\\overline{h} = h \\bmod \\mathfrak p'$, and", "\\item the polynomials $g, h$ generate the unit ideal in $R'[x]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose", "$\\overline{g} = \\overline{b}_0 x^n + \\overline{b}_1 x^{n - 1} + \\ldots", "+ \\overline{b}_n$, and", "$\\overline{h} = \\overline{c}_0 x^m + \\overline{c}_1 x^{m - 1} + \\ldots", "+ \\overline{c}_m$ with $\\overline{b}_0, \\overline{c}_0 \\in \\kappa(\\mathfrak p)$", "nonzero. After localizing $R$ at some element of $R$ not contained in", "$\\mathfrak p$ we may assume $\\overline{b}_0$ is the", "image of an invertible element $b_0 \\in R$. Replacing", "$\\overline{g}$ by $\\overline{g}/b_0$ and", "$\\overline{h}$ by $b_0\\overline{h}$ we reduce to the case where", "$\\overline{g}$, $\\overline{h}$ are monic (verification omitted).", "Say $\\overline{g} = x^n + \\overline{b}_1 x^{n - 1} + \\ldots + \\overline{b}_n$,", "and $\\overline{h} = x^m + \\overline{c}_1 x^{m - 1} + \\ldots + \\overline{c}_m$.", "Write $f = x^{n + m} + a_1 x^{n - 1} + \\ldots + a_{n + m}$.", "Consider the fibre product", "$$", "R' = R \\otimes_{\\mathbf{Z}[a_1, \\ldots, a_{n + m}]}", "\\mathbf{Z}[b_1, \\ldots, b_n, c_1, \\ldots, c_m]", "$$", "where the map $\\mathbf{Z}[a_k] \\to \\mathbf{Z}[b_i, c_j]$", "is as in Examples \\ref{example-factor-polynomials} and", "\\ref{example-factor-polynomials-etale}. By construction there", "is an $R$-algebra map", "$$", "R' = R \\otimes_{\\mathbf{Z}[a_1, \\ldots, a_{n + m}]}", "\\mathbf{Z}[b_1, \\ldots, b_n, c_1, \\ldots, c_m]", "\\longrightarrow", "\\kappa(\\mathfrak p)", "$$", "which maps $b_i$ to $\\overline{b}_i$ and $c_j$ to $\\overline{c}_j$.", "Denote $\\mathfrak p' \\subset R'$ the kernel of this map.", "Since by assumption the polynomials $\\overline{g}, \\overline{h}$", "are relatively prime we see that the element", "$\\Delta = \\text{Res}_x(g, h) \\in \\mathbf{Z}[b_i, c_j]$", "(see Example \\ref{example-factor-polynomials-etale})", "does not map to zero in $\\kappa(\\mathfrak p)$ under the displayed map.", "We conclude that $R \\to R'$ is \\'etale at $\\mathfrak p'$.", "In fact a solution to the problem posed in the lemma is", "the ring map $R \\to R'[1/\\Delta]$ and the prime", "$\\mathfrak p' R'[1/\\Delta]$. Because $\\text{Res}_x(f, g)$ is", "invertible in this ring the Sylvester matrix is invertible over", "$R'[1/\\Delta]$ and hence $1 = a g + b h$ for some $a, b \\in R'[1/\\Delta][x]$", "see Example \\ref{example-factor-polynomials-etale}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1241, "type": "theorem", "label": "algebra-lemma-standard-etale", "categories": [ "algebra" ], "title": "algebra-lemma-standard-etale", "contents": [ "Let $R \\to R[x]_g/(f)$ be standard \\'etale.", "\\begin{enumerate}", "\\item The ring map $R \\to R[x]_g/(f)$ is \\'etale.", "\\item For any ring map $R \\to R'$ the base change $R' \\to R'[x]_g/(f)$", "of the standard \\'etale ring map $R \\to R[x]_g/(f)$ is standard \\'etale.", "\\item Any principal localization of $R[x]_g/(f)$ is standard \\'etale over $R$.", "\\item A composition of standard \\'etale maps is {\\bf not} standard \\'etale", "in general.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Here is an example for (4).", "The ring map $\\mathbf{F}_2 \\to \\mathbf{F}_{2^2}$ is standard \\'etale.", "The ring map", "$\\mathbf{F}_{2^2} \\to \\mathbf{F}_{2^2} \\times \\mathbf{F}_{2^2}", "\\times \\mathbf{F}_{2^2} \\times \\mathbf{F}_{2^2}$ is standard \\'etale.", "But the ring map", "$\\mathbf{F}_2 \\to \\mathbf{F}_{2^2} \\times \\mathbf{F}_{2^2}", "\\times \\mathbf{F}_{2^2} \\times \\mathbf{F}_{2^2}$ is not standard \\'etale." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1242, "type": "theorem", "label": "algebra-lemma-make-etale-map-prescribed-residue-field", "categories": [ "algebra" ], "title": "algebra-lemma-make-etale-map-prescribed-residue-field", "contents": [ "Let $R$ be a ring.", "Let $\\mathfrak p$ be a prime of $R$.", "Let $\\kappa(\\mathfrak p) \\subset L$ be a finite separable field extension.", "There exists an \\'etale ring map $R \\to R'$ together with a prime $\\mathfrak p'$", "lying over $\\mathfrak p$ such that the field extension", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak p')$ is isomorphic", "to $\\kappa(\\mathfrak p) \\subset L$." ], "refs": [], "proofs": [ { "contents": [ "By the theorem of the primitive element we may write", "$L = \\kappa(\\mathfrak p)[\\alpha]$. Let", "$\\overline{f} \\in \\kappa(\\mathfrak p)[x]$", "denote the minimal polynomial for $\\alpha$ (in particular this is monic).", "After replacing $\\alpha$ by $c\\alpha$ for some $c \\in R$,", "$c\\not \\in \\mathfrak p$ we may assume all the coefficients", "of $\\overline{f}$ are in the image of $R \\to \\kappa(\\mathfrak p)$", "(verification omitted). Thus we can find a monic polynomial", "$f \\in R[x]$ which maps to $\\overline{f}$ in $\\kappa(\\mathfrak p)[x]$.", "Since $\\kappa(\\mathfrak p) \\subset L$ is separable, we see", "that $\\gcd(\\overline{f}, \\overline{f}') = 1$.", "Hence there is an element $\\gamma \\in L$ such that", "$\\overline{f}'(\\alpha) \\gamma = 1$. Thus we get a $R$-algebra map", "\\begin{eqnarray*}", "R[x, 1/f']/(f) & \\longrightarrow & L \\\\", "x & \\longmapsto & \\alpha \\\\", "1/f' & \\longmapsto & \\gamma", "\\end{eqnarray*}", "The left hand side is a standard \\'etale algebra $R'$ over $R$", "and the kernel of the ring map gives the desired prime." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1243, "type": "theorem", "label": "algebra-lemma-standard-etale-finite-flat-Zariski", "categories": [ "algebra" ], "title": "algebra-lemma-standard-etale-finite-flat-Zariski", "contents": [ "Let $R \\to S$ be a standard \\'etale morphism.", "There exists a ring map $R \\to S'$ with the following properties", "\\begin{enumerate}", "\\item $R \\to S'$ is finite, finitely presented, and flat", "(in other words $S'$ is finite projective as an $R$-module),", "\\item $\\Spec(S') \\to \\Spec(R)$ is surjective,", "\\item for every prime $\\mathfrak q \\subset S$, lying over", "$\\mathfrak p \\subset R$ and every prime", "$\\mathfrak q' \\subset S'$ lying over $\\mathfrak p$ there exists", "a $g' \\in S'$, $g' \\not \\in \\mathfrak q'$", "such that the ring map $R \\to S'_{g'}$ factors", "through a map $\\varphi : S \\to S'_{g'}$ with", "$\\varphi^{-1}(\\mathfrak q'S'_{g'}) = \\mathfrak q$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $S = R[x]_g/(f)$ be a presentation of $S$ as in", "Definition \\ref{definition-standard-etale}.", "Write $f = x^n + a_1 x^{n - 1} + \\ldots + a_n$ with $a_i \\in R$.", "By Lemma \\ref{lemma-adjoin-roots} there exists a finite locally free", "and faithfully flat ring map $R \\to S'$ such that $f = \\prod (x - \\alpha_i)$", "for certain $\\alpha_i \\in S'$. Hence $R \\to S'$ satisfies conditions (1), (2).", "Let $\\mathfrak q \\subset R[x]/(f)$ be a prime ideal with", "$g \\not \\in \\mathfrak q$ (i.e., it corresponds to a prime of $S$).", "Let $\\mathfrak p = R \\cap \\mathfrak q$ and let", "$\\mathfrak q' \\subset S'$ be a prime lying over $\\mathfrak p$.", "Note that there are", "$n$ maps of $R$-algebras", "\\begin{eqnarray*}", "\\varphi_i : R[x]/(f) & \\longrightarrow & S' \\\\", "x & \\longmapsto & \\alpha_i", "\\end{eqnarray*}", "To finish the proof we have to show that for some $i$ we have", "(a) the image of $\\varphi_i(g)$ in $\\kappa(\\mathfrak q')$ is not zero,", "and (b) $\\varphi_i^{-1}(\\mathfrak q') = \\mathfrak q$.", "Because then we can just take $g' = \\varphi_i(g)$, and", "$\\varphi = \\varphi_i$ for that $i$.", "\\medskip\\noindent", "Let $\\overline{f}$ denote the image of $f$ in $\\kappa(\\mathfrak p)[x]$.", "Note that as a point of $\\Spec(\\kappa(\\mathfrak p)[x]/(\\overline{f}))$", "the prime $\\mathfrak q$ corresponds to an irreducible factor", "$f_1$ of $\\overline{f}$. Moreover, $g \\not \\in \\mathfrak q$ means", "that $f_1$ does not divide the image $\\overline{g}$ of $g$ in", "$\\kappa(\\mathfrak p)[x]$.", "Denote $\\overline{\\alpha}_1, \\ldots, \\overline{\\alpha}_n$ the images", "of $\\alpha_1, \\ldots, \\alpha_n$ in $\\kappa(\\mathfrak q')$.", "Note that the polynomial $\\overline{f}$ splits completely", "in $\\kappa(\\mathfrak q')[x]$, namely", "$$", "\\overline{f} = \\prod\\nolimits_i (x - \\overline{\\alpha}_i)", "$$", "Moreover $\\varphi_i(g)$ reduces to $\\overline{g}(\\overline{\\alpha}_i)$.", "It follows we may pick $i$ such that $f_1(\\overline{\\alpha}_i) = 0$ and", "$\\overline{g}(\\overline{\\alpha}_i) \\not = 0$.", "For this $i$ properties (a) and (b) hold. Some details omitted." ], "refs": [ "algebra-definition-standard-etale", "algebra-lemma-adjoin-roots" ], "ref_ids": [ 1540, 1179 ] } ], "ref_ids": [] }, { "id": 1244, "type": "theorem", "label": "algebra-lemma-etale-finite-flat-zariski", "categories": [ "algebra" ], "title": "algebra-lemma-etale-finite-flat-zariski", "contents": [ "Let $R \\to S$ be a ring map.", "Assume that", "\\begin{enumerate}", "\\item $R \\to S$ is \\'etale, and", "\\item $\\Spec(S) \\to \\Spec(R)$ is surjective.", "\\end{enumerate}", "Then there exists a ring map $R \\to S'$ such that", "\\begin{enumerate}", "\\item $R \\to S'$ is finite, finitely presented, and flat", "(in other words it is finite projective as an $R$-module),", "\\item $\\Spec(S') \\to \\Spec(R)$ is surjective,", "\\item for every prime $\\mathfrak q' \\subset S'$ there exists a", "$g' \\in S'$, $g' \\not \\in \\mathfrak q'$ such that", "the ring map $R \\to S'_{g'}$ factors as $R \\to S \\to S'_{g'}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Proposition \\ref{proposition-etale-locally-standard} and", "the quasi-compactness of $\\Spec(S)$ (see Lemma \\ref{lemma-quasi-compact})", "we can find $g_1, \\ldots, g_n \\in S$ generating the unit ideal", "of $S$ such that each $R \\to S_{g_i}$ is standard \\'etale.", "If we prove the lemma for the ring map $R \\to \\prod_{i = 1, \\ldots, n} S_{g_i}$", "then the lemma follows for the ring map $R \\to S$.", "Hence we may assume that $S = \\prod_{i = 1, \\ldots, n} S_i$", "is a finite product of standard \\'etale morphisms.", "\\medskip\\noindent", "For each $i$ choose a ring map $R \\to S_i'$ as in", "Lemma \\ref{lemma-standard-etale-finite-flat-Zariski}", "adapted to the standard \\'etale morphism $R \\to S_i$.", "Set $S' = S_1' \\otimes_R \\ldots \\otimes_R S_n'$; we will use", "the $R$-algebra maps $S_i' \\to S'$ without further mention below.", "We claim this works. Properties (1) and (2) are immediate.", "For property (3) suppose that $\\mathfrak q' \\subset S'$ is a prime.", "Denote $\\mathfrak p$ its image in $\\Spec(R)$.", "Choose $i \\in \\{1, \\ldots, n\\}$ such that $\\mathfrak p$", "is in the image of $\\Spec(S_i) \\to \\Spec(R)$; this is", "possible by assumption. Set $\\mathfrak q_i' \\subset S_i'$", "the image of $\\mathfrak q'$ in the spectrum of $S_i'$.", "By construction of $S'_i$ there exists a $g'_i \\in S_i'$", "such that $R \\to (S_i')_{g_i'}$ factors as", "$R \\to S_i \\to (S_i')_{g_i'}$. Hence also", "$R \\to S'_{g_i'}$ factors as", "$$", "R \\to S_i \\to (S_i')_{g_i'} \\to S'_{g_i'}", "$$", "as desired." ], "refs": [ "algebra-proposition-etale-locally-standard", "algebra-lemma-quasi-compact", "algebra-lemma-standard-etale-finite-flat-Zariski" ], "ref_ids": [ 1427, 395, 1243 ] } ], "ref_ids": [] }, { "id": 1245, "type": "theorem", "label": "algebra-lemma-produce-finite", "categories": [ "algebra" ], "title": "algebra-lemma-produce-finite", "contents": [ "Let $R \\to S' \\to S$ be ring maps.", "Let $\\mathfrak p \\subset R$ be a prime.", "Let $g \\in S'$ be an element.", "Assume", "\\begin{enumerate}", "\\item $R \\to S'$ is integral,", "\\item $R \\to S$ is finite type,", "\\item $S'_g \\cong S_g$, and", "\\item $g$ invertible in $S' \\otimes_R \\kappa(\\mathfrak p)$.", "\\end{enumerate}", "Then there exists a $f \\in R$, $f \\not \\in \\mathfrak p$ such", "that $R_f \\to S_f$ is finite." ], "refs": [], "proofs": [ { "contents": [ "By assumption the image $T$ of $V(g) \\subset \\Spec(S')$ under", "the morphism $\\Spec(S') \\to \\Spec(R)$ does not", "contain $\\mathfrak p$. By Section \\ref{section-going-up}", "especially, Lemma \\ref{lemma-going-up-closed} we see $T$ is closed.", "Pick $f \\in R$, $f \\not \\in \\mathfrak p$ such that", "$T \\cap D(f) = \\emptyset$. Then we see that $g$ becomes invertible", "in $S'_f$. Hence $S'_f \\cong S_f$. Thus $S_f$ is both of finite type", "and integral over $R_f$, hence finite." ], "refs": [ "algebra-lemma-going-up-closed" ], "ref_ids": [ 552 ] } ], "ref_ids": [] }, { "id": 1246, "type": "theorem", "label": "algebra-lemma-etale-makes-quasi-finite-finite-one-prime", "categories": [ "algebra" ], "title": "algebra-lemma-etale-makes-quasi-finite-finite-one-prime", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak q \\subset S$ be a prime lying over", "the prime $\\mathfrak p \\subset R$.", "Assume $R \\to S$ finite type and quasi-finite at $\\mathfrak q$.", "Then there exists", "\\begin{enumerate}", "\\item an \\'etale ring map $R \\to R'$,", "\\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$,", "\\item a product decomposition", "$$", "R' \\otimes_R S = A \\times B", "$$", "\\end{enumerate}", "with the following properties", "\\begin{enumerate}", "\\item $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$,", "\\item $R' \\to A$ is finite,", "\\item $A$ has exactly one prime $\\mathfrak r$ lying over $\\mathfrak p'$, and", "\\item $\\mathfrak r$ lies over $\\mathfrak q$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $S' \\subset S$ be the integral closure of $R$ in $S$.", "Let $\\mathfrak q' = S' \\cap \\mathfrak q$.", "By Zariski's Main Theorem \\ref{theorem-main-theorem}", "there exists a $g \\in S'$, $g \\not \\in \\mathfrak q'$ such", "that $S'_g \\cong S_g$. Consider the fibre rings", "$F = S \\otimes_R \\kappa(\\mathfrak p)$ and", "$F' = S' \\otimes_R \\kappa(\\mathfrak p)$. Denote $\\overline{\\mathfrak q}'$", "the prime of $F'$ corresponding to $\\mathfrak q'$. Since", "$F'$ is integral over $\\kappa(\\mathfrak p)$ we see", "that $\\overline{\\mathfrak q}'$ is a closed point of", "$\\Spec(F')$, see Lemma \\ref{lemma-integral-over-field}.", "Note that $\\mathfrak q$ defines an isolated closed point", "$\\overline{\\mathfrak q}$ of", "$\\Spec(F)$ (see Definition \\ref{definition-quasi-finite}).", "Since $S'_g \\cong S_g$ we have $F'_g \\cong F_g$,", "so $\\overline{\\mathfrak q}$ and $\\overline{\\mathfrak q}'$", "have isomorphic open neighbourhoods in $\\Spec(F)$", "and $\\Spec(F')$. We conclude the set", "$\\{\\overline{\\mathfrak q}'\\} \\subset \\Spec(F')$ is", "open. Combined with $\\mathfrak q'$ being closed (shown above)", "we conclude that $\\overline{\\mathfrak q}'$ defines", "an isolated closed point of $\\Spec(F')$ as well.", "\\medskip\\noindent", "An additional small remark is that under the map", "$\\Spec(F) \\to \\Spec(F')$ the point $\\overline{\\mathfrak q}$", "is the only point mapping to $\\overline{\\mathfrak q}'$. This follows", "from the discussion above.", "\\medskip\\noindent", "By Lemma \\ref{lemma-disjoint-implies-product} we may write", "$F' = F'_1 \\times F'_2$ with", "$\\Spec(F'_1) = \\{\\overline{\\mathfrak q}'\\}$.", "Since $F' = S' \\otimes_R \\kappa(\\mathfrak p)$, there", "exists an $s' \\in S'$ which maps to the element", "$(r, 0) \\in F'_1 \\times F'_2 = F'$ for some $r \\in R$, $r \\not \\in \\mathfrak p$.", "In fact, what we will use about $s'$ is that it is an element of $S'$,", "not contained in $\\mathfrak q'$, and contained in any other prime", "lying over $\\mathfrak p$.", "\\medskip\\noindent", "Let $f(x) \\in R[x]$ be a monic polynomial such that $f(s') = 0$.", "Denote $\\overline{f} \\in \\kappa(\\mathfrak p)[x]$ the image.", "We can factor it as $\\overline{f} = x^e \\overline{h}$ where", "$\\overline{h}(0) \\not = 0$. By Lemma \\ref{lemma-factor-mod-lift-etale}", "we can find an \\'etale ring extension $R \\to R'$,", "a prime $\\mathfrak p'$ lying over $\\mathfrak p$, and", "a factorization $f = h i$ in $R'[x]$ such that", "$\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$,", "$\\overline{h} = h \\bmod \\mathfrak p'$,", "$x^e = i \\bmod \\mathfrak p'$, and", "we can write $a h + b i = 1$ in $R'[x]$ (for suitable $a, b$).", "\\medskip\\noindent", "Consider the elements $h(s'), i(s') \\in R' \\otimes_R S'$.", "By construction we have $h(s')i(s') = f(s') = 0$. On the other", "hand they generate the unit ideal since $a(s')h(s') + b(s')i(s') = 1$.", "Thus we see that $R' \\otimes_R S'$ is the product of the", "localizations at these elements:", "$$", "R' \\otimes_R S'", "=", "(R' \\otimes_R S')_{i(s')}", "\\times", "(R' \\otimes_R S')_{h(s')}", "=", "S'_1 \\times S'_2", "$$", "Moreover this product decomposition is compatible with the product", "decomposition we found for the fibre ring $F'$; this comes from our", "choices of $s', i, h$ which guarantee that $\\overline{\\mathfrak q}'$", "is the only prime of $F'$ which does not contain the image of $i(s')$", "in $F'$. Here we use that the fibre ring of $R'\\otimes_R S'$ over $R'$ at", "$\\mathfrak p'$ is the same as $F'$ due to the fact that", "$\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$.", "It follows that $S'_1$ has exactly", "one prime, say $\\mathfrak r'$,", "lying over $\\mathfrak p'$ and", "that this prime lies over $\\mathfrak q$.", "Hence the element $g \\in S'$ maps to an element of $S'_1$ not contained", "in $\\mathfrak r'$.", "\\medskip\\noindent", "The base change $R'\\otimes_R S$ inherits a similar product decomposition", "$$", "R' \\otimes_R S", "=", "(R' \\otimes_R S)_{i(s')}", "\\times", "(R' \\otimes_R S)_{h(s')}", "=", "S_1 \\times S_2", "$$", "It follows from the above that $S_1$ has exactly", "one prime, say $\\mathfrak r$,", "lying over $\\mathfrak p'$ (consider the fibre ring as above),", "and that this prime lies over $\\mathfrak q$.", "\\medskip\\noindent", "Now we may apply Lemma \\ref{lemma-produce-finite} to the ring maps", "$R' \\to S'_1 \\to S_1$, the prime $\\mathfrak p'$ and", "the element $g$ to see that after replacing $R'$ by", "a principal localization we can assume that $S_1$ is", "finite over $R'$ as desired." ], "refs": [ "algebra-theorem-main-theorem", "algebra-lemma-integral-over-field", "algebra-definition-quasi-finite", "algebra-lemma-disjoint-implies-product", "algebra-lemma-factor-mod-lift-etale", "algebra-lemma-produce-finite" ], "ref_ids": [ 325, 497, 1522, 415, 1240, 1245 ] } ], "ref_ids": [] }, { "id": 1247, "type": "theorem", "label": "algebra-lemma-etale-makes-quasi-finite-finite", "categories": [ "algebra" ], "title": "algebra-lemma-etale-makes-quasi-finite-finite", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak p \\subset R$ be a prime.", "Assume $R \\to S$ finite type.", "Then there exists", "\\begin{enumerate}", "\\item an \\'etale ring map $R \\to R'$,", "\\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$,", "\\item a product decomposition", "$$", "R' \\otimes_R S = A_1 \\times \\ldots \\times A_n \\times B", "$$", "\\end{enumerate}", "with the following properties", "\\begin{enumerate}", "\\item we have $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$,", "\\item each $A_i$ is finite over $R'$,", "\\item each $A_i$ has exactly one prime $\\mathfrak r_i$ lying over", "$\\mathfrak p'$, and", "\\item $R' \\to B$ not quasi-finite at any prime lying over $\\mathfrak p'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $F = S \\otimes_R \\kappa(\\mathfrak p)$ the fibre ring of $S/R$", "at the prime $\\mathfrak p$. As $F$ is of finite type over $\\kappa(\\mathfrak p)$", "it is Noetherian and hence $\\Spec(F)$ has finitely many isolated closed", "points. If there are no isolated closed points,", "i.e., no primes $\\mathfrak q$ of $S$ over $\\mathfrak p$ such that", "$S/R$ is quasi-finite at $\\mathfrak q$, then the lemma holds.", "If there exists at least one such prime $\\mathfrak q$, then", "we may apply Lemma \\ref{lemma-etale-makes-quasi-finite-finite-one-prime}.", "This gives a diagram", "$$", "\\xymatrix{", "S \\ar[r] & R'\\otimes_R S \\ar@{=}[r] & A_1 \\times B' \\\\", "R \\ar[r] \\ar[u] & R' \\ar[u] \\ar[ru]", "}", "$$", "as in said lemma. Since the residue fields at $\\mathfrak p$ and $\\mathfrak p'$", "are the same, the fibre rings of $S/R$ and $(A_1 \\times B')/R'$", "are the same. Hence, by induction on the number of isolated closed points", "of the fibre we may assume that the lemma holds for", "$R' \\to B'$ and $\\mathfrak p'$. Thus we get an \\'etale ring", "map $R' \\to R''$, a prime $\\mathfrak p'' \\subset R''$ and", "a decomposition", "$$", "R'' \\otimes_{R'} B' = A_2 \\times \\ldots \\times A_n \\times B", "$$", "We omit the verification that the ring map $R \\to R''$, the", "prime $\\mathfrak p''$ and the resulting decomposition", "$$", "R'' \\otimes_R S = (R'' \\otimes_{R'} A_1) \\times", "A_2 \\times \\ldots \\times A_n \\times B", "$$", "is a solution to the problem posed in the lemma." ], "refs": [ "algebra-lemma-etale-makes-quasi-finite-finite-one-prime" ], "ref_ids": [ 1246 ] } ], "ref_ids": [] }, { "id": 1248, "type": "theorem", "label": "algebra-lemma-etale-makes-quasi-finite-finite-variant", "categories": [ "algebra" ], "title": "algebra-lemma-etale-makes-quasi-finite-finite-variant", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak p \\subset R$ be a prime.", "Assume $R \\to S$ finite type.", "Then there exists", "\\begin{enumerate}", "\\item an \\'etale ring map $R \\to R'$,", "\\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$,", "\\item a product decomposition", "$$", "R' \\otimes_R S = A_1 \\times \\ldots \\times A_n \\times B", "$$", "\\end{enumerate}", "with the following properties", "\\begin{enumerate}", "\\item each $A_i$ is finite over $R'$,", "\\item each $A_i$ has exactly one prime $\\mathfrak r_i$ lying over", "$\\mathfrak p'$,", "\\item the finite field extensions", "$\\kappa(\\mathfrak p') \\subset \\kappa(\\mathfrak r_i)$", "are purely inseparable, and", "\\item $R' \\to B$ not quasi-finite at any prime lying over $\\mathfrak p'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The strategy of the proof is to make two \\'etale ring", "extensions: first we control the residue fields, then we", "apply Lemma \\ref{lemma-etale-makes-quasi-finite-finite}.", "\\medskip\\noindent", "Denote $F = S \\otimes_R \\kappa(\\mathfrak p)$ the fibre ring of $S/R$", "at the prime $\\mathfrak p$.", "As in the proof of Lemma \\ref{lemma-etale-makes-quasi-finite-finite}", "there are finitely may primes, say", "$\\mathfrak q_1, \\ldots, \\mathfrak q_n$ of $S$ lying over", "$R$ at which the ring map $R \\to S$ is quasi-finite.", "Let $\\kappa(\\mathfrak p) \\subset L_i \\subset \\kappa(\\mathfrak q_i)$", "be the subfield such that $\\kappa(\\mathfrak p) \\subset L_i$", "is separable, and the field extension $L_i \\subset \\kappa(\\mathfrak q_i)$", "is purely inseparable. Let $\\kappa(\\mathfrak p) \\subset L$", "be a finite Galois extension into which $L_i$ embeds for $i = 1, \\ldots, n$.", "By Lemma \\ref{lemma-make-etale-map-prescribed-residue-field}", "we can find an \\'etale ring extension", "$R \\to R'$ together with a prime $\\mathfrak p'$ lying over $\\mathfrak p$", "such that the field extension", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak p')$ is isomorphic", "to $\\kappa(\\mathfrak p) \\subset L$.", "Thus the fibre ring of $R' \\otimes_R S$ at $\\mathfrak p'$ is", "isomorphic to $F \\otimes_{\\kappa(\\mathfrak p)} L$.", "The primes lying over $\\mathfrak q_i$ correspond to primes", "of $\\kappa(\\mathfrak q_i) \\otimes_{\\kappa(\\mathfrak p)} L$", "which is a product of fields purely inseparable over", "$L$ by our choice of $L$ and elementary field theory.", "These are also the only primes over $\\mathfrak p'$", "at which $R' \\to R' \\otimes_R S$ is quasi-finite, by", "Lemma \\ref{lemma-quasi-finite-base-change}.", "Hence after replacing $R$ by $R'$, $\\mathfrak p$ by $\\mathfrak p'$,", "and $S$ by $R' \\otimes_R S$ we may assume that for all", "primes $\\mathfrak q$ lying over $\\mathfrak p$", "for which $S/R$ is quasi-finite the field extensions", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$", "are purely inseparable.", "\\medskip\\noindent", "Next apply Lemma \\ref{lemma-etale-makes-quasi-finite-finite}.", "The result is what we want since the field extensions do not", "change under this \\'etale ring extension." ], "refs": [ "algebra-lemma-etale-makes-quasi-finite-finite", "algebra-lemma-etale-makes-quasi-finite-finite", "algebra-lemma-make-etale-map-prescribed-residue-field", "algebra-lemma-quasi-finite-base-change", "algebra-lemma-etale-makes-quasi-finite-finite" ], "ref_ids": [ 1247, 1247, 1242, 1054, 1247 ] } ], "ref_ids": [] }, { "id": 1249, "type": "theorem", "label": "algebra-lemma-etale-under-finite-flat", "categories": [ "algebra" ], "title": "algebra-lemma-etale-under-finite-flat", "contents": [ "Let $(R, \\mathfrak m_R) \\to (S, \\mathfrak m_S)$ be a local homomorphism", "of local rings. Assume $S$ is the localization of an \\'etale ring extension", "of $R$. Then there exists a finite, finitely presented, faithfully flat", "ring map $R \\to S'$ such that for every maximal ideal $\\mathfrak m'$ of $S'$", "there is a factorization", "$$", "R \\to S \\to S'_{\\mathfrak m'}.", "$$", "of the ring map $R \\to S'_{\\mathfrak m'}$." ], "refs": [], "proofs": [ { "contents": [ "Write $S = T_{\\mathfrak q}$ for some \\'etale $R$-algebra $T$. By", "Proposition \\ref{proposition-etale-locally-standard}", "we may assume $T$ is standard \\'etale.", "Apply", "Lemma \\ref{lemma-standard-etale-finite-flat-Zariski}", "to the ring map $R \\to T$ to get $R \\to S'$. Then in particular", "for every maximal ideal $\\mathfrak m'$ of $S'$ we get a factorization", "$\\varphi : T \\to S'_{g'}$ for some $g' \\not \\in \\mathfrak m'$ such", "that $\\mathfrak q = \\varphi^{-1}(\\mathfrak m'S'_{g'})$. Thus $\\varphi$", "induces the desired local ring map $S \\to S'_{\\mathfrak m'}$." ], "refs": [ "algebra-proposition-etale-locally-standard", "algebra-lemma-standard-etale-finite-flat-Zariski" ], "ref_ids": [ 1427, 1243 ] } ], "ref_ids": [] }, { "id": 1250, "type": "theorem", "label": "algebra-lemma-trick", "categories": [ "algebra" ], "title": "algebra-lemma-trick", "contents": [ "Let $R$ be a ring.", "Let $f \\in R[x]$ be a monic polynomial.", "Let $R \\to B$ be a ring map.", "If $h \\in B[x]/(f)$ is integral over $R$, then the element", "$f' h$ can be written as $f'h = \\sum_i b_i x^i$ with $b_i \\in B$", "integral over $R$." ], "refs": [], "proofs": [ { "contents": [ "Say $h^e + r_1 h^{e - 1} + \\ldots + r_e = 0$ in the ring $B[x]/(f)$", "with $r_i \\in R$.", "There exists a finite free ring extension $B \\subset B'$ such that", "$f = (x - \\alpha_1) \\ldots (x - \\alpha_d)$ for some $\\alpha_i \\in B'$,", "see Lemma \\ref{lemma-adjoin-roots}.", "Note that each $\\alpha_i$ is integral over $R$.", "We may represent $h = h_0 + h_1 x + \\ldots + h_{d - 1} x^{d - 1}$", "with $h_i \\in B$. Then it is a universal fact that", "$$", "f' h", "\\equiv", "\\sum\\nolimits_{i = 1, \\ldots, d}", "h(\\alpha_i)", "(x - \\alpha_1) \\ldots \\widehat{(x - \\alpha_i)} \\ldots (x - \\alpha_d)", "$$", "as elements of $B[x]/(f)$. You prove this by", "evaluating both sides at the points $\\alpha_i$ over the ring", "$B_{univ} = \\mathbf{Z}[\\alpha_i, h_j]$ (some details omitted).", "By our assumption that $h$ satisfies", "$h^e + r_1 h^{e - 1} + \\ldots + r_e = 0$ in the ring $B[x]/(f)$", "we see that", "$$", "h(\\alpha_i)^e + r_1 h(\\alpha_i)^{e - 1} + \\ldots + r_e = 0", "$$", "in $B'$. Hence $h(\\alpha_i)$ is integral over $R$. Using the formula", "above we see that $f'h \\equiv \\sum_{j = 0, \\ldots, d - 1} b'_j x^j$", "in $B'[x]/(f)$ with $b'_j \\in B'$ integral over $R$. However,", "since $f' h \\in B[x]/(f)$ and since $1, x, \\ldots, x^{d - 1}$ is a", "$B'$-basis for $B'[x]/(f)$ we see that $b'_j \\in B$ as desired." ], "refs": [ "algebra-lemma-adjoin-roots" ], "ref_ids": [ 1179 ] } ], "ref_ids": [] }, { "id": 1251, "type": "theorem", "label": "algebra-lemma-integral-closure-commutes-etale", "categories": [ "algebra" ], "title": "algebra-lemma-integral-closure-commutes-etale", "contents": [ "Let $R \\to S$ be an \\'etale ring map.", "Let $R \\to B$ be any ring map.", "Let $A \\subset B$ be the integral closure of $R$ in $B$.", "Let $A' \\subset S \\otimes_R B$ be the integral closure of $S$ in", "$S \\otimes_R B$. Then the canonical map $S \\otimes_R A \\to A'$ is", "an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The map $S \\otimes_R A \\to A'$ is injective because $A \\subset B$ and", "$R \\to S$ is flat. We are going to use repeatedly that taking integral", "closure commutes with localization, see", "Lemma \\ref{lemma-integral-closure-localize}.", "Hence we may localize on $S$, by Lemma \\ref{lemma-cover} (the criterion", "for checking whether an $S$-module map is an isomorphism).", "Thus we may assume that $S = R[x]_g/(f) = (R[x]/(f))_g$", "is standard \\'etale over $R$,", "see Proposition \\ref{proposition-etale-locally-standard}.", "Applying localization one more time we see that", "$A'$ is $(A'')_g$ where $A''$ is the integral closure of", "$R[x]/(f)$ in $B[x]/(f)$. Suppose that $a \\in A''$. It suffices", "to show that $a$ is in $S \\otimes_R A$. By", "Lemma \\ref{lemma-trick} we see that $f' a = \\sum a_i x^i$ with $a_i \\in A$.", "Since $f'$ is invertible in $B[x]_g/(f)$ (by definition of a standard", "\\'etale ring map) we conclude that $a \\in S \\otimes_R A$ as desired." ], "refs": [ "algebra-lemma-integral-closure-localize", "algebra-lemma-cover", "algebra-proposition-etale-locally-standard", "algebra-lemma-trick" ], "ref_ids": [ 489, 411, 1427, 1250 ] } ], "ref_ids": [] }, { "id": 1252, "type": "theorem", "label": "algebra-lemma-integral-closure-commutes-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-integral-closure-commutes-smooth", "contents": [ "Let $R \\to S$ be a smooth ring map.", "Let $R \\to B$ be any ring map.", "Let $A \\subset B$ be the integral closure of $R$ in $B$.", "Let $A' \\subset S \\otimes_R B$ be the integral closure of $S$ in", "$S \\otimes_R B$. Then the canonical map $S \\otimes_R A \\to A'$ is", "an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Arguing as in the proof of Lemma \\ref{lemma-integral-closure-commutes-etale}", "we may localize on $S$. Hence we may assume that $R \\to S$ is a standard", "smooth ring map, see Lemma \\ref{lemma-smooth-syntomic}. By definition of", "a standard smooth ring map we see that $S$ is \\'etale over a polynomial", "ring $R[x_1, \\ldots, x_n]$. Since we have seen the result in the case of", "an \\'etale ring extension (Lemma \\ref{lemma-integral-closure-commutes-etale})", "this reduces us to the case where $S = R[x]$. Thus we have to show", "$$", "f = \\sum b_i x^i", "\\text{ integral over }R[x]", "\\Leftrightarrow", "\\text{each }b_i\\text{ integral over }R.", "$$", "The implication from right to left holds because the set of elements", "in $B[x]$ integral over $R[x]$ is a ring", "(Lemma \\ref{lemma-integral-closure-is-ring}) and contains", "$x$.", "\\medskip\\noindent", "Suppose that $f \\in B[x]$ is integral over $R[x]$, and assume that", "$f = \\sum_{i < d} b_i x^i$ has degree $< d$. Since integral closure", "and localization commute, it suffices to show there exist", "distinct primes $p, q$ such that each $b_i$ is", "integral both over $R[1/p]$ and over $R[1/q]$. Hence, we can find a finite", "free ring extension $R \\subset R'$ such that $R'$ contains", "$\\alpha_1, \\ldots, \\alpha_d$ with the property that", "$\\prod_{i < j} (\\alpha_i - \\alpha_j)$ is a unit in $R'$, see", "Example \\ref{example-fourier}.", "In this case we have the universal equality", "$$", "f", "=", "\\sum_i", "f(\\alpha_i)", "\\frac{(x - \\alpha_1) \\ldots \\widehat{(x - \\alpha_i)} \\ldots (x - \\alpha_d)}", "{(\\alpha_i - \\alpha_1) \\ldots \\widehat{(\\alpha_i - \\alpha_i)} \\ldots", "(\\alpha_i - \\alpha_d)}.", "$$", "OK, and the elements $f(\\alpha_i)$ are integral over $R'$ since", "$(R' \\otimes_R B)[x] \\to R' \\otimes_R B$, $h \\mapsto h(\\alpha_i)$", "is a ring map. Hence we see that the coefficients of $f$", "in $(R' \\otimes_R B)[x]$ are integral over $R'$. Since $R'$ is finite", "over $R$ (hence integral over $R$) we see that they are integral", "over $R$ also, as desired." ], "refs": [ "algebra-lemma-integral-closure-commutes-etale", "algebra-lemma-smooth-syntomic", "algebra-lemma-integral-closure-commutes-etale", "algebra-lemma-integral-closure-is-ring" ], "ref_ids": [ 1251, 1195, 1251, 486 ] } ], "ref_ids": [] }, { "id": 1253, "type": "theorem", "label": "algebra-lemma-integral-closure-commutes-colim-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-integral-closure-commutes-colim-smooth", "contents": [ "Let $R \\to S$ and $R \\to B$ be ring maps.", "Let $A \\subset B$ be the integral closure of $R$ in $B$.", "Let $A' \\subset S \\otimes_R B$ be the integral closure of $S$ in", "$S \\otimes_R B$. If $S$ is a filtered colimit of smooth $R$-algebras,", "then the canonical map $S \\otimes_R A \\to A'$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This follows from the straightforward fact that taking", "tensor products and taking integral closures", "commutes with filtered colimits and", "Lemma \\ref{lemma-integral-closure-commutes-smooth}." ], "refs": [ "algebra-lemma-integral-closure-commutes-smooth" ], "ref_ids": [ 1252 ] } ], "ref_ids": [] }, { "id": 1254, "type": "theorem", "label": "algebra-lemma-characterize-formally-unramified", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-formally-unramified", "contents": [ "Let $R \\to S$ be a ring map.", "The following are equivalent:", "\\begin{enumerate}", "\\item $R \\to S$ is formally unramified,", "\\item the module of differentials $\\Omega_{S/R}$ is zero.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $J = \\Ker(S \\otimes_R S \\to S)$ be the kernel of", "the multiplication map. Let $A_{univ} = S \\otimes_R S/J^2$. Recall", "that $I_{univ} = J/J^2$ is isomorphic to $\\Omega_{S/R}$, see", "Lemma \\ref{lemma-differentials-diagonal}. Moreover, the two $R$-algebra maps", "$\\sigma_1, \\sigma_2 : S \\to A_{univ}$, $\\sigma_1(s) = s \\otimes 1 \\bmod J^2$,", "and $\\sigma_2(s) = 1 \\otimes s \\bmod J^2$ differ by the", "universal derivation $\\text{d} : S \\to \\Omega_{S/R} = I_{univ}$.", "\\medskip\\noindent", "Assume $R \\to S$ formally unramified.", "Then we see that $\\sigma_1 = \\sigma_2$.", "Hence $\\text{d}(s) = 0$ for all $s \\in S$.", "Hence $\\Omega_{S/R} = 0$.", "\\medskip\\noindent", "Assume that $\\Omega_{S/R} = 0$. Let $A, I, R \\to A, S \\to A/I$", "be a solid diagram as in Definition \\ref{definition-formally-unramified}.", "Let $\\tau_1, \\tau_2 : S \\to A$ be two dotted arrows making the", "diagram commute. Consider the $R$-algebra map $A_{univ} \\to A$", "defined by the rule $s_1 \\otimes s_2 \\mapsto \\tau_1(s_1)\\tau_2(s_2)$.", "We omit the verification that this is well defined. Since $A_{univ} \\cong S$", "as $I_{univ} = \\Omega_{S/R} = 0$ we conclude that $\\tau_1 = \\tau_2$." ], "refs": [ "algebra-lemma-differentials-diagonal", "algebra-definition-formally-unramified" ], "ref_ids": [ 1139, 1541 ] } ], "ref_ids": [] }, { "id": 1255, "type": "theorem", "label": "algebra-lemma-formally-unramified-local", "categories": [ "algebra" ], "title": "algebra-lemma-formally-unramified-local", "contents": [ "Let $R \\to S$ be a ring map.", "The following are equivalent:", "\\begin{enumerate}", "\\item $R \\to S$ is formally unramified,", "\\item $R \\to S_{\\mathfrak q}$ is formally unramified for all", "primes $\\mathfrak q$ of $S$, and", "\\item $R_{\\mathfrak p} \\to S_{\\mathfrak q}$ is formally unramified", "for all primes $\\mathfrak q$ of $S$ with $\\mathfrak p = R \\cap \\mathfrak q$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have seen in", "Lemma \\ref{lemma-characterize-formally-unramified}", "that (1) is equivalent to", "$\\Omega_{S/R} = 0$. Similarly, by", "Lemma \\ref{lemma-differentials-localize}", "we see that (2) and (3)", "are equivalent to $(\\Omega_{S/R})_{\\mathfrak q} = 0$ for all", "$\\mathfrak q$. Hence the equivalence follows from", "Lemma \\ref{lemma-characterize-zero-local}." ], "refs": [ "algebra-lemma-characterize-formally-unramified", "algebra-lemma-differentials-localize", "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 1254, 1134, 410 ] } ], "ref_ids": [] }, { "id": 1256, "type": "theorem", "label": "algebra-lemma-formally-unramified-localize", "categories": [ "algebra" ], "title": "algebra-lemma-formally-unramified-localize", "contents": [ "Let $A \\to B$ be a formally unramified ring map.", "\\begin{enumerate}", "\\item For $S \\subset A$ a multiplicative subset,", "$S^{-1}A \\to S^{-1}B$ is formally unramified.", "\\item For $S \\subset B$ a multiplicative subset,", "$A \\to S^{-1}B$ is formally unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-formally-unramified-local}.", "(You can also deduce it from", "Lemma \\ref{lemma-characterize-formally-unramified}", "combined with", "Lemma \\ref{lemma-differentials-localize}.)" ], "refs": [ "algebra-lemma-formally-unramified-local", "algebra-lemma-characterize-formally-unramified", "algebra-lemma-differentials-localize" ], "ref_ids": [ 1255, 1254, 1134 ] } ], "ref_ids": [] }, { "id": 1257, "type": "theorem", "label": "algebra-lemma-colimit-formally-unramified", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-formally-unramified", "contents": [ "Let $R$ be a ring. Let $I$ be a directed set.", "Let $(S_i, \\varphi_{ii'})$ be a system of $R$-algebras", "over $I$. If each $R \\to S_i$ is formally unramified, then", "$S = \\colim_{i \\in I} S_i$ is formally unramified over $R$" ], "refs": [], "proofs": [ { "contents": [ "Consider a diagram as in Definition \\ref{definition-formally-unramified}.", "By assumption there exists at most one $R$-algebra map $S_i \\to A$ lifting", "the compositions $S_i \\to S \\to A/I$. Since every element of $S$", "is in the image of one of the maps $S_i \\to S$ we see that there", "is at most one map $S \\to A$ fitting into the diagram." ], "refs": [ "algebra-definition-formally-unramified" ], "ref_ids": [ 1541 ] } ], "ref_ids": [] }, { "id": 1258, "type": "theorem", "label": "algebra-lemma-universal-thickening", "categories": [ "algebra" ], "title": "algebra-lemma-universal-thickening", "contents": [ "Let $R \\to S$ be a formally unramified ring map. There exists a surjection of", "$R$-algebras $S' \\to S$ whose kernel is an ideal of square zero with the", "following universal property: Given any commutative diagram", "$$", "\\xymatrix{", "S \\ar[r]_a & A/I \\\\", "R \\ar[r]^b \\ar[u] & A \\ar[u]", "}", "$$", "where $I \\subset A$ is an ideal of square zero, there is a unique $R$-algebra", "map $a' : S' \\to A$ such that $S' \\to A \\to A/I$ is equal to $S' \\to S \\to A/I$." ], "refs": [], "proofs": [ { "contents": [ "Choose a set of generators $z_i \\in S$, $i \\in I$ for $S$ as an $R$-algebra.", "Let $P = R[\\{x_i\\}_{i \\in I}]$ denote the polynomial ring on generators", "$x_i$, $i \\in I$. Consider the $R$-algebra map $P \\to S$ which maps", "$x_i$ to $z_i$. Let $J = \\Ker(P \\to S)$. Consider the map", "$$", "\\text{d} : J/J^2 \\longrightarrow \\Omega_{P/R} \\otimes_P S", "$$", "see", "Lemma \\ref{lemma-differential-seq}.", "This is surjective since $\\Omega_{S/R} = 0$ by assumption, see", "Lemma \\ref{lemma-characterize-formally-unramified}.", "Note that $\\Omega_{P/R}$ is free on $\\text{d}x_i$, and hence the module", "$\\Omega_{P/R} \\otimes_P S$ is free over $S$. Thus we may choose a splitting", "of the surjection above and write", "$$", "J/J^2 = K \\oplus \\Omega_{P/R} \\otimes_P S", "$$", "Let $J^2 \\subset J' \\subset J$ be the ideal of $P$ such that", "$J'/J^2$ is the second summand in the decomposition above.", "Set $S' = P/J'$. We obtain a short exact sequence", "$$", "0 \\to J/J' \\to S' \\to S \\to 0", "$$", "and we see that $J/J' \\cong K$ is a square zero ideal in $S'$. Hence", "$$", "\\xymatrix{", "S \\ar[r]_1 & S \\\\", "R \\ar[r] \\ar[u] & S' \\ar[u]", "}", "$$", "is a diagram as above. In fact we claim that this is an initial object in", "the category of diagrams. Namely, let $(I \\subset A, a, b)$ be an arbitrary", "diagram. We may choose an $R$-algebra map $\\beta : P \\to A$ such that", "$$", "\\xymatrix{", "S \\ar[r]_1 & S \\ar[r]_a & A/I \\\\", "R \\ar[r] \\ar@/_/[rr]_b \\ar[u] & P \\ar[u] \\ar[r]^\\beta & A \\ar[u]", "}", "$$", "is commutative. Now it may not be the case that $\\beta(J') = 0$, in other", "words it may not be true that $\\beta$ factors through $S' = P/J'$.", "But what is clear is that $\\beta(J') \\subset I$ and", "since $\\beta(J) \\subset I$ and $I^2 = 0$ we have $\\beta(J^2) = 0$.", "Thus the ``obstruction'' to finding a morphism from", "$(J/J' \\subset S', 1, R \\to S')$ to $(I \\subset A, a, b)$ is", "the corresponding $S$-linear map $\\overline{\\beta} : J'/J^2 \\to I$.", "The choice in picking $\\beta$ lies in the choice of $\\beta(x_i)$.", "A different choice of $\\beta$, say $\\beta'$, is gotten by taking", "$\\beta'(x_i) = \\beta(x_i) + \\delta_i$ with $\\delta_i \\in I$.", "In this case, for $g \\in J'$, we obtain", "$$", "\\beta'(g) =", "\\beta(g) + \\sum\\nolimits_i \\delta_i \\frac{\\partial g}{\\partial x_i}.", "$$", "Since the map $\\text{d}|_{J'/J^2} : J'/J^2 \\to \\Omega_{P/R} \\otimes_P S$", "given by $g \\mapsto \\frac{\\partial g}{\\partial x_i}\\text{d}x_i$", "is an isomorphism by construction, we see that there is a unique choice", "of $\\delta_i \\in I$ such that $\\beta'(g) = 0$ for all $g \\in J'$.", "(Namely, $\\delta_i$ is $-\\overline{\\beta}(g)$ where $g \\in J'/J^2$", "is the unique element with $\\frac{\\partial g}{\\partial x_j} = 1$ if", "$i = j$ and $0$ else.) The uniqueness of the solution implies the", "uniqueness required in the lemma." ], "refs": [ "algebra-lemma-differential-seq", "algebra-lemma-characterize-formally-unramified" ], "ref_ids": [ 1135, 1254 ] } ], "ref_ids": [] }, { "id": 1259, "type": "theorem", "label": "algebra-lemma-universal-thickening-quotient", "categories": [ "algebra" ], "title": "algebra-lemma-universal-thickening-quotient", "contents": [ "Let $I \\subset R$ be an ideal of a ring.", "The universal first order thickening of $R/I$ over $R$", "is the surjection $R/I^2 \\to R/I$. The conormal module", "of $R/I$ over $R$ is $C_{(R/I)/R} = I/I^2$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1260, "type": "theorem", "label": "algebra-lemma-universal-thickening-localize", "categories": [ "algebra" ], "title": "algebra-lemma-universal-thickening-localize", "contents": [ "Let $A \\to B$ be a formally unramified ring map.", "Let $\\varphi : B' \\to B$ be the universal first order thickening of", "$B$ over $A$.", "\\begin{enumerate}", "\\item Let $S \\subset A$ be a multiplicative subset.", "Then $S^{-1}B' \\to S^{-1}B$ is the universal first order thickening of", "$S^{-1}B$ over $S^{-1}A$. In particular $S^{-1}C_{B/A} = C_{S^{-1}B/S^{-1}A}$.", "\\item Let $S \\subset B$ be a multiplicative subset.", "Then $S' = \\varphi^{-1}(S)$ is a multiplicative subset in $B'$", "and $(S')^{-1}B' \\to S^{-1}B$ is the universal first order thickening", "of $S^{-1}B$ over $A$. In particular $S^{-1}C_{B/A} = C_{S^{-1}B/A}$.", "\\end{enumerate}", "Note that the lemma makes sense by", "Lemma \\ref{lemma-formally-unramified-localize}." ], "refs": [ "algebra-lemma-formally-unramified-localize" ], "proofs": [ { "contents": [ "With notation and assumptions as in (1). Let $(S^{-1}B)' \\to S^{-1}B$", "be the universal first order thickening of $S^{-1}B$ over $S^{-1}A$.", "Note that $S^{-1}B' \\to S^{-1}B$ is a surjection of $S^{-1}A$-algebras", "whose kernel has square zero. Hence by definition we obtain a map", "$(S^{-1}B)' \\to S^{-1}B'$ compatible with the maps towards $S^{-1}B$.", "Consider any commutative diagram", "$$", "\\xymatrix{", "B \\ar[r] & S^{-1}B \\ar[r] & D/I \\\\", "A \\ar[r] \\ar[u] & S^{-1}A \\ar[r] \\ar[u] & D \\ar[u]", "}", "$$", "where $I \\subset D$ is an ideal of square zero. Since $B'$ is the universal", "first order thickening of $B$ over $A$ we obtain an $A$-algebra map", "$B' \\to D$. But it is clear that the image of $S$ in $D$ is mapped to", "invertible elements of $D$, and hence we obtain a compatible map", "$S^{-1}B' \\to D$. Applying this to $D = (S^{-1}B)'$ we see that we get", "a map $S^{-1}B' \\to (S^{-1}B)'$. We omit the verification that this map", "is inverse to the map described above.", "\\medskip\\noindent", "With notation and assumptions as in (2). Let $(S^{-1}B)' \\to S^{-1}B$", "be the universal first order thickening of $S^{-1}B$ over $A$.", "Note that $(S')^{-1}B' \\to S^{-1}B$ is a surjection of $A$-algebras", "whose kernel has square zero. Hence by definition we obtain a map", "$(S^{-1}B)' \\to (S')^{-1}B'$ compatible with the maps towards $S^{-1}B$.", "Consider any commutative diagram", "$$", "\\xymatrix{", "B \\ar[r] & S^{-1}B \\ar[r] & D/I \\\\", "A \\ar[r] \\ar[u] & A \\ar[r] \\ar[u] & D \\ar[u]", "}", "$$", "where $I \\subset D$ is an ideal of square zero. Since $B'$ is the universal", "first order thickening of $B$ over $A$ we obtain an $A$-algebra map", "$B' \\to D$. But it is clear that the image of $S'$ in $D$ is mapped to", "invertible elements of $D$, and hence we obtain a compatible map", "$(S')^{-1}B' \\to D$. Applying this to $D = (S^{-1}B)'$ we see that we get", "a map $(S')^{-1}B' \\to (S^{-1}B)'$. We omit the verification that this map", "is inverse to the map described above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1256 ] }, { "id": 1261, "type": "theorem", "label": "algebra-lemma-differentials-universal-thickening", "categories": [ "algebra" ], "title": "algebra-lemma-differentials-universal-thickening", "contents": [ "Let $R \\to A \\to B$ be ring maps. Assume $A \\to B$ formally unramified.", "Let $B' \\to B$ be the universal first order thickening of $B$ over $A$.", "Then $B'$ is formally unramified over $A$, and the canonical map", "$\\Omega_{A/R} \\otimes_A B \\to \\Omega_{B'/R} \\otimes_{B'} B$ is an", "isomorphism." ], "refs": [], "proofs": [ { "contents": [ "We are going to use the construction of $B'$ from the proof of", "Lemma \\ref{lemma-universal-thickening}", "although in principle it should be possible to deduce these results", "formally from the definition. Namely, we choose a presentation", "$B = P/J$, where $P = A[x_i]$ is a polynomial ring over $A$.", "Next, we choose elements $f_i \\in J$ such that", "$\\text{d}f_i = \\text{d}x_i \\otimes 1$ in $\\Omega_{P/A} \\otimes_P B$.", "Having made these choices we have", "$B' = P/J'$ with $J' = (f_i) + J^2$, see proof of", "Lemma \\ref{lemma-universal-thickening}.", "\\medskip\\noindent", "Consider the canonical exact sequence", "$$", "J'/(J')^2 \\to \\Omega_{P/A} \\otimes_P B' \\to \\Omega_{B'/A} \\to 0", "$$", "see", "Lemma \\ref{lemma-differential-seq}.", "By construction the classes of the $f_i \\in J'$ map to elements of", "the module $\\Omega_{P/A} \\otimes_P B'$ which generate it modulo", "$J'/J^2$ by construction. Since $J'/J^2$ is a nilpotent ideal, we see", "that these elements generate the module altogether (by", "Nakayama's Lemma \\ref{lemma-NAK}). This proves that $\\Omega_{B'/A} = 0$", "and hence that $B'$ is formally unramified over $A$, see", "Lemma \\ref{lemma-characterize-formally-unramified}.", "\\medskip\\noindent", "Since $P$ is a polynomial ring over $A$ we have", "$\\Omega_{P/R} = \\Omega_{A/R} \\otimes_A P \\oplus \\bigoplus P\\text{d}x_i$.", "We are going to use this decomposition.", "Consider the following exact sequence", "$$", "J'/(J')^2 \\to", "\\Omega_{P/R} \\otimes_P B' \\to", "\\Omega_{B'/R} \\to 0", "$$", "see", "Lemma \\ref{lemma-differential-seq}.", "We may tensor this with $B$ and obtain the exact sequence", "$$", "J'/(J')^2 \\otimes_{B'} B \\to", "\\Omega_{P/R} \\otimes_P B \\to", "\\Omega_{B'/R} \\otimes_{B'} B \\to 0", "$$", "If we remember that $J' = (f_i) + J^2$", "then we see that the first arrow annihilates the submodule $J^2/(J')^2$.", "In terms of the direct sum decomposition", "$\\Omega_{P/R} \\otimes_P B =", "\\Omega_{A/R} \\otimes_A B \\oplus \\bigoplus B\\text{d}x_i $ given", "we see that the submodule $(f_i)/(J')^2 \\otimes_{B'} B$ maps", "isomorphically onto the summand $\\bigoplus B\\text{d}x_i$. Hence what is", "left of this exact sequence is an isomorphism", "$\\Omega_{A/R} \\otimes_A B \\to \\Omega_{B'/R} \\otimes_{B'} B$", "as desired." ], "refs": [ "algebra-lemma-universal-thickening", "algebra-lemma-universal-thickening", "algebra-lemma-differential-seq", "algebra-lemma-NAK", "algebra-lemma-characterize-formally-unramified", "algebra-lemma-differential-seq" ], "ref_ids": [ 1258, 1258, 1135, 401, 1254, 1135 ] } ], "ref_ids": [] }, { "id": 1262, "type": "theorem", "label": "algebra-lemma-formally-etale-etale", "categories": [ "algebra" ], "title": "algebra-lemma-formally-etale-etale", "contents": [ "Let $R \\to S$ be a ring map of finite presentation.", "The following are equivalent:", "\\begin{enumerate}", "\\item $R \\to S$ is formally \\'etale,", "\\item $R \\to S$ is \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume that $R \\to S$ is formally \\'etale.", "Then $R \\to S$ is smooth by", "Proposition \\ref{proposition-smooth-formally-smooth}.", "By Lemma \\ref{lemma-characterize-formally-unramified}", "we have $\\Omega_{S/R} = 0$.", "Hence $R \\to S$ is \\'etale by definition.", "\\medskip\\noindent", "Assume that $R \\to S$ is \\'etale.", "Then $R \\to S$ is formally smooth by", "Proposition \\ref{proposition-smooth-formally-smooth}.", "By Lemma \\ref{lemma-characterize-formally-unramified}", "it is formally unramified. Hence $R \\to S$ is formally \\'etale." ], "refs": [ "algebra-proposition-smooth-formally-smooth", "algebra-lemma-characterize-formally-unramified", "algebra-proposition-smooth-formally-smooth", "algebra-lemma-characterize-formally-unramified" ], "ref_ids": [ 1426, 1254, 1426, 1254 ] } ], "ref_ids": [] }, { "id": 1263, "type": "theorem", "label": "algebra-lemma-colimit-formally-etale", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-formally-etale", "contents": [ "Let $R$ be a ring. Let $I$ be a directed set.", "Let $(S_i, \\varphi_{ii'})$ be a system of $R$-algebras", "over $I$. If each $R \\to S_i$ is formally \\'etale, then", "$S = \\colim_{i \\in I} S_i$ is formally \\'etale over $R$" ], "refs": [], "proofs": [ { "contents": [ "Consider a diagram as in Definition \\ref{definition-formally-etale}.", "By assumption we get unique $R$-algebra maps $S_i \\to A$ lifting", "the compositions $S_i \\to S \\to A/I$. Hence these are compatible", "with the transition maps $\\varphi_{ii'}$ and define a lift", "$S \\to A$. This proves existence.", "The uniqueness is clear by restricting to each $S_i$." ], "refs": [ "algebra-definition-formally-etale" ], "ref_ids": [ 1543 ] } ], "ref_ids": [] }, { "id": 1264, "type": "theorem", "label": "algebra-lemma-localization-formally-etale", "categories": [ "algebra" ], "title": "algebra-lemma-localization-formally-etale", "contents": [ "Let $R$ be a ring. Let $S \\subset R$ be any multiplicative subset.", "Then the ring map $R \\to S^{-1}R$ is formally \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Let $I \\subset A$ be an ideal of square zero. What we are saying", "here is that given a ring map $\\varphi : R \\to A$ such that", "$\\varphi(f) \\mod I$ is invertible for all $f \\in S$ we have also that", "$\\varphi(f)$ is invertible in $A$ for all $f \\in S$. This is true because", "$A^*$ is the inverse image of $(A/I)^*$ under the canonical map", "$A \\to A/I$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1265, "type": "theorem", "label": "algebra-lemma-formally-unramified-unramified", "categories": [ "algebra" ], "title": "algebra-lemma-formally-unramified-unramified", "contents": [ "Let $R \\to S$ be a ring map. The following are equivalent", "\\begin{enumerate}", "\\item $R \\to S$ is formally unramified and of finite type, and", "\\item $R \\to S$ is unramified.", "\\end{enumerate}", "Moreover, also the following are equivalent", "\\begin{enumerate}", "\\item $R \\to S$ is formally unramified and of finite presentation, and", "\\item $R \\to S$ is G-unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-characterize-formally-unramified}", "and the definitions." ], "refs": [ "algebra-lemma-characterize-formally-unramified" ], "ref_ids": [ 1254 ] } ], "ref_ids": [] }, { "id": 1266, "type": "theorem", "label": "algebra-lemma-unramified", "categories": [ "algebra" ], "title": "algebra-lemma-unramified", "contents": [ "Properties of unramified and G-unramified ring maps.", "\\begin{enumerate}", "\\item The base change of an unramified ring map is unramified.", "The base change of a G-unramified ring map is G-unramified.", "\\item The composition of unramified ring maps is unramified.", "The composition of G-unramified ring maps is G-unramified.", "\\item Any principal localization $R \\to R_f$ is G-unramified and", "unramified.", "\\item If $I \\subset R$ is an ideal, then $R \\to R/I$ is unramified.", "If $I \\subset R$ is a finitely generated ideal, then $R \\to R/I$ is", "G-unramified.", "\\item An \\'etale ring map is G-unramified and unramified.", "\\item If $R \\to S$ is of finite type (resp.\\ finite presentation),", "$\\mathfrak q \\subset S$ is a prime and $(\\Omega_{S/R})_{\\mathfrak q} = 0$,", "then $R \\to S$ is unramified (resp.\\ G-unramified) at $\\mathfrak q$.", "\\item If $R \\to S$ is of finite type (resp.\\ finite presentation),", "$\\mathfrak q \\subset S$ is a prime and", "$\\Omega_{S/R} \\otimes_S \\kappa(\\mathfrak q) = 0$, then", "$R \\to S$ is unramified (resp.\\ G-unramified) at $\\mathfrak q$.", "\\item If $R \\to S$ is of finite type (resp.\\ finite presentation),", "$\\mathfrak q \\subset S$ is a prime lying over $\\mathfrak p \\subset R$ and", "$(\\Omega_{S \\otimes_R \\kappa(\\mathfrak p)/\\kappa(\\mathfrak p)})_{\\mathfrak q}", "= 0$, then $R \\to S$ is unramified (resp.\\ G-unramified) at $\\mathfrak q$.", "\\item If $R \\to S$ is of finite type (resp.\\ presentation),", "$\\mathfrak q \\subset S$ is a prime lying over $\\mathfrak p \\subset R$ and", "$(\\Omega_{S \\otimes_R \\kappa(\\mathfrak p)/\\kappa(\\mathfrak p)})", "\\otimes_{S \\otimes_R \\kappa(\\mathfrak p)} \\kappa(\\mathfrak q) = 0$,", "then $R \\to S$ is unramified (resp.\\ G-unramified) at $\\mathfrak q$.", "\\item If $R \\to S$ is a ring map, $g_1, \\ldots, g_m \\in S$ generate", "the unit ideal and $R \\to S_{g_j}$ is unramified (resp.\\ G-unramified) for", "$j = 1, \\ldots, m$, then $R \\to S$ is unramified (resp.\\ G-unramified).", "\\item If $R \\to S$ is a ring map which is unramified (resp.\\ G-unramified)", "at every prime of $S$, then $R \\to S$ is unramified (resp.\\ G-unramified).", "\\item If $R \\to S$ is G-unramified, then there exists a finite type", "$\\mathbf{Z}$-algebra $R_0$ and a G-unramified ring map $R_0 \\to S_0$", "and a ring map $R_0 \\to R$ such that $S = R \\otimes_{R_0} S_0$.", "\\item If $R \\to S$ is unramified, then there exists a finite type", "$\\mathbf{Z}$-algebra $R_0$ and an unramified ring map $R_0 \\to S_0$", "and a ring map $R_0 \\to R$ such that $S$ is a quotient of", "$R \\otimes_{R_0} S_0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We prove each point, in order.", "\\medskip\\noindent", "Ad (1). Follows from Lemmas \\ref{lemma-differentials-base-change}", "and \\ref{lemma-base-change-finiteness}.", "\\medskip\\noindent", "Ad (2). Follows from Lemmas \\ref{lemma-exact-sequence-differentials}", "and \\ref{lemma-base-change-finiteness}.", "\\medskip\\noindent", "Ad (3). Follows by direct computation of $\\Omega_{R_f/R}$ which we omit.", "\\medskip\\noindent", "Ad (4). We have $\\Omega_{(R/I)/R} = 0$, see", "Lemma \\ref{lemma-trivial-differential-surjective},", "and the ring map $R \\to R/I$", "is of finite type. If $I$ is a finitely generated ideal then $R \\to R/I$", "is of finite presentation.", "\\medskip\\noindent", "Ad (5). See discussion following Definition \\ref{definition-etale}.", "\\medskip\\noindent", "Ad (6). In this case $\\Omega_{S/R}$ is a finite $S$-module (see", "Lemma \\ref{lemma-differentials-finitely-generated}) and hence there", "exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that", "$(\\Omega_{S/R})_g = 0$. By Lemma \\ref{lemma-differentials-localize}", "this means that $\\Omega_{S_g/R} = 0$ and hence $R \\to S_g$ is", "unramified as desired.", "\\medskip\\noindent", "Ad (7). Use Nakayama's lemma (Lemma \\ref{lemma-NAK}) to see that", "the condition is equivalent to the condition of (6).", "\\medskip\\noindent", "Ad (8) and (9). These are equivalent in the same manner that (6) and (7)", "are equivalent. Moreover", "$\\Omega_{S \\otimes_R \\kappa(\\mathfrak p)/\\kappa(\\mathfrak p)} =", "\\Omega_{S/R} \\otimes_S (S \\otimes_R \\kappa(\\mathfrak p))$ by", "Lemma \\ref{lemma-differentials-base-change}.", "Hence we see that (9) is equivalent to (7) since", "the $\\kappa(\\mathfrak q)$ vector spaces in both are canonically", "isomorphic.", "\\medskip\\noindent", "Ad (10). Follows from Lemmas \\ref{lemma-cover}", "and \\ref{lemma-differentials-localize}.", "\\medskip\\noindent", "Ad (11). Follows from (6) and (7) and the fact that the spectrum of $S$", "is quasi-compact.", "\\medskip\\noindent", "Ad (12). Write $S = R[x_1, \\ldots, x_n]/(g_1, \\ldots, g_m)$.", "As $\\Omega_{S/R} = 0$ we can write", "$$", "\\text{d}x_i = \\sum h_{ij}\\text{d}g_j + \\sum a_{ijk}g_j\\text{d}x_k", "$$", "in $\\Omega_{R[x_1, \\ldots, x_n]/R}$", "for some $h_{ij}, a_{ijk} \\in R[x_1, \\ldots, x_n]$.", "Choose a finitely generated", "$\\mathbf{Z}$-subalgebra $R_0 \\subset R$ containing all the coefficients of the", "polynomials $g_i, h_{ij}, a_{ijk}$. Set", "$S_0 = R_0[x_1, \\ldots, x_n]/(g_1, \\ldots, g_m)$. This works.", "\\medskip\\noindent", "Ad (13). Write $S = R[x_1, \\ldots, x_n]/I$.", "As $\\Omega_{S/R} = 0$ we can write", "$$", "\\text{d}x_i = \\sum h_{ij}\\text{d}g_{ij} + \\sum g'_{ik}\\text{d}x_k", "$$", "in $\\Omega_{R[x_1, \\ldots, x_n]/R}$", "for some $h_{ij} \\in R[x_1, \\ldots, x_n]$ and $g_{ij}, g'_{ik} \\in I$.", "Choose a finitely generated $\\mathbf{Z}$-subalgebra $R_0 \\subset R$", "containing all the coefficients of the", "polynomials $g_{ij}, h_{ij}, g'_{ik}$. Set", "$S_0 = R_0[x_1, \\ldots, x_n]/(g_{ij}, g'_{ik})$. This works." ], "refs": [ "algebra-lemma-differentials-base-change", "algebra-lemma-base-change-finiteness", "algebra-lemma-exact-sequence-differentials", "algebra-lemma-base-change-finiteness", "algebra-lemma-trivial-differential-surjective", "algebra-definition-etale", "algebra-lemma-differentials-finitely-generated", "algebra-lemma-differentials-localize", "algebra-lemma-NAK", "algebra-lemma-differentials-base-change", "algebra-lemma-cover", "algebra-lemma-differentials-localize" ], "ref_ids": [ 1138, 373, 1133, 373, 1131, 1539, 1142, 1134, 401, 1138, 411, 1134 ] } ], "ref_ids": [] }, { "id": 1267, "type": "theorem", "label": "algebra-lemma-diagonal-unramified", "categories": [ "algebra" ], "title": "algebra-lemma-diagonal-unramified", "contents": [ "Let $R \\to S$ be a ring map.", "If $R \\to S$ is unramified, then there exists an idempotent", "$e \\in S \\otimes_R S$ such that $S \\otimes_R S \\to S$ is isomorphic", "to $S \\otimes_R S \\to (S \\otimes_R S)_e$." ], "refs": [], "proofs": [ { "contents": [ "Let $J = \\Ker(S \\otimes_R S \\to S)$. By assumption", "$J/J^2 = 0$, see", "Lemma \\ref{lemma-differentials-diagonal}.", "Since $S$ is of finite type over $R$ we", "see that $J$ is finitely generated, namely by", "$x_i \\otimes 1 - 1 \\otimes x_i$, where $x_i$ generate $S$ over $R$.", "We win by Lemma \\ref{lemma-ideal-is-squared-union-connected}." ], "refs": [ "algebra-lemma-differentials-diagonal", "algebra-lemma-ideal-is-squared-union-connected" ], "ref_ids": [ 1139, 407 ] } ], "ref_ids": [] }, { "id": 1268, "type": "theorem", "label": "algebra-lemma-unramified-at-prime", "categories": [ "algebra" ], "title": "algebra-lemma-unramified-at-prime", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak q \\subset S$ be", "a prime lying over $\\mathfrak p$ in $R$.", "If $S/R$ is unramified at $\\mathfrak q$ then", "\\begin{enumerate}", "\\item we have $\\mathfrak p S_{\\mathfrak q} = \\mathfrak qS_{\\mathfrak q}$", "is the maximal ideal of the local ring $S_{\\mathfrak q}$, and", "\\item the field extension $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$", "is finite separable.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may first replace $S$ by $S_g$ for some $g \\in S$, $g \\not \\in \\mathfrak q$", "and assume that $R \\to S$ is unramified.", "The base change $S \\otimes_R \\kappa(\\mathfrak p)$", "is unramified over $\\kappa(\\mathfrak p)$ by", "Lemma \\ref{lemma-unramified}.", "By", "Lemma \\ref{lemma-characterize-smooth-over-field}", "it is smooth hence \\'etale over $\\kappa(\\mathfrak p)$.", "Hence we see that", "$S \\otimes_R \\kappa(\\mathfrak p) =", "(R \\setminus \\mathfrak p)^{-1} S/\\mathfrak pS$", "is a product of finite separable field extensions of", "$\\kappa(\\mathfrak p)$ by Lemma \\ref{lemma-etale-over-field}.", "This implies the lemma." ], "refs": [ "algebra-lemma-unramified", "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-etale-over-field" ], "ref_ids": [ 1266, 1223, 1232 ] } ], "ref_ids": [] }, { "id": 1269, "type": "theorem", "label": "algebra-lemma-unramified-quasi-finite", "categories": [ "algebra" ], "title": "algebra-lemma-unramified-quasi-finite", "contents": [ "Let $R \\to S$ be a finite type ring map.", "Let $\\mathfrak q$ be a prime of $S$.", "If $R \\to S$ is unramified at $\\mathfrak q$ then", "$R \\to S$ is quasi-finite at $\\mathfrak q$.", "In particular, an unramified ring map is quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "An unramified ring map is of finite type.", "Thus it is clear that the second statement follows from the first.", "To see the first statement apply the characterization of", "Lemma \\ref{lemma-isolated-point-fibre} part (2) using", "Lemma \\ref{lemma-unramified-at-prime}." ], "refs": [ "algebra-lemma-isolated-point-fibre", "algebra-lemma-unramified-at-prime" ], "ref_ids": [ 1049, 1268 ] } ], "ref_ids": [] }, { "id": 1270, "type": "theorem", "label": "algebra-lemma-characterize-unramified", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-unramified", "contents": [ "Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime of $S$", "lying over a prime $\\mathfrak p$ of $R$. If", "\\begin{enumerate}", "\\item $R \\to S$ is of finite type,", "\\item $\\mathfrak p S_{\\mathfrak q}$ is the maximal ideal", "of the local ring $S_{\\mathfrak q}$, and", "\\item the field extension $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$", "is finite separable,", "\\end{enumerate}", "then $R \\to S$ is unramified at $\\mathfrak q$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-unramified} (8) it suffices to show that", "$\\Omega_{S \\otimes_R \\kappa(\\mathfrak p) / \\kappa(\\mathfrak p)}$", "is zero when localized at $\\mathfrak q$. Hence we may replace $S$", "by $S \\otimes_R \\kappa(\\mathfrak p)$ and $R$ by $\\kappa(\\mathfrak p)$.", "In other words, we may assume that $R = k$ is a field and $S$", "is a finite type $k$-algebra.", "In this case the hypotheses imply that", "$S_{\\mathfrak q} \\cong \\kappa(\\mathfrak q)$.", "Thus $(\\Omega_{S/k})_{\\mathfrak q} = \\Omega_{S_\\mathfrak q/k} =", "\\Omega_{\\kappa(\\mathfrak q)/k}$ is zero as desired (the", "first equality is Lemma \\ref{lemma-differentials-localize})." ], "refs": [ "algebra-lemma-unramified", "algebra-lemma-differentials-localize" ], "ref_ids": [ 1266, 1134 ] } ], "ref_ids": [] }, { "id": 1271, "type": "theorem", "label": "algebra-lemma-etale-flat-unramified-finite-presentation", "categories": [ "algebra" ], "title": "algebra-lemma-etale-flat-unramified-finite-presentation", "contents": [ "Let $R \\to S$ be a ring map. The following are equivalent", "\\begin{enumerate}", "\\item $R \\to S$ is \\'etale,", "\\item $R \\to S$ is flat and G-unramified, and", "\\item $R \\to S$ is flat, unramified, and of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (2) and (3) are equivalent by definition.", "The implication (1) $\\Rightarrow$ (3) follows from", "the fact that \\'etale ring maps are of finite presentation,", "Lemma \\ref{lemma-etale} (flatness of \\'etale maps), and", "Lemma \\ref{lemma-unramified} (\\'etale maps are unramified).", "Conversely, the characterization of \\'etale ring maps in", "Lemma \\ref{lemma-characterize-etale}", "and the structure of unramified ring maps in", "Lemma \\ref{lemma-unramified-at-prime}", "shows that (3) implies (1). (This uses that $R \\to S$", "is \\'etale if $R \\to S$ is \\'etale at every prime $\\mathfrak q \\subset S$,", "see Lemma \\ref{lemma-etale}.)" ], "refs": [ "algebra-lemma-etale", "algebra-lemma-unramified", "algebra-lemma-characterize-etale", "algebra-lemma-unramified-at-prime", "algebra-lemma-etale" ], "ref_ids": [ 1231, 1266, 1235, 1268, 1231 ] } ], "ref_ids": [] }, { "id": 1272, "type": "theorem", "label": "algebra-lemma-characterize-etale-over-polynomial-ring", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-etale-over-polynomial-ring", "contents": [ "Let $k$ be a field. Let", "$$", "\\varphi : k[x_1, \\ldots, x_n] \\to A, \\quad x_i \\longmapsto a_i", "$$", "be a finite type ring map. Then $\\varphi$ is \\'etale if and only if we", "have the following two conditions: (a) the local rings of $A$ at maximal ideals", "have dimension $n$, and (b) the elements $\\text{d}(a_1), \\ldots, \\text{d}(a_n)$", "generate $\\Omega_{A/k}$ as an $A$-module." ], "refs": [], "proofs": [ { "contents": [ "Assume (a) and (b). Condition (b) implies that", "$\\Omega_{A/k[x_1, \\ldots, x_n]} = 0$ and hence $\\varphi$ is unramified.", "Thus it suffices to prove that $\\varphi$ is flat, see", "Lemma \\ref{lemma-etale-flat-unramified-finite-presentation}.", "Let $\\mathfrak m \\subset A$ be a maximal ideal.", "Set $X = \\Spec(A)$ and denote $x \\in X$ the closed point corresponding", "to $\\mathfrak m$. Then $\\dim(A_\\mathfrak m)$ is $\\dim_x X$, see", "Lemma \\ref{lemma-dimension-closed-point-finite-type-field}.", "Thus by Lemma \\ref{lemma-characterize-smooth-over-field}", "we see that if (a) and (b) hold, then $A_\\mathfrak m$ is", "a regular local ring for every maximal ideal $\\mathfrak m$. Then", "$k[x_1, \\ldots, x_n]_{\\varphi^{-1}(\\mathfrak m)} \\to A_\\mathfrak m$", "is flat by Lemma \\ref{lemma-CM-over-regular-flat}", "(and the fact that a regular local ring is CM, see", "Lemma \\ref{lemma-regular-ring-CM}).", "Thus $\\varphi$ is flat by Lemma \\ref{lemma-flat-localization}.", "\\medskip\\noindent", "Assume $\\varphi$ is \\'etale. Then $\\Omega_{A/k[x_1, \\ldots, x_n]} = 0$", "and hence (b) holds. On the other hand, \\'etale ring maps are flat", "(Lemma \\ref{lemma-etale}) and quasi-finite", "(Lemma \\ref{lemma-etale-quasi-finite}).", "Hence for every maximal ideal $\\mathfrak m$ of $A$ we my apply", "Lemma \\ref{lemma-dimension-base-fibre-equals-total} to", "$k[x_1, \\ldots, x_n]_{\\varphi^{-1}(\\mathfrak m)} \\to A_\\mathfrak m$", "to see that $\\dim(A_\\mathfrak m) = n$ and hence (a) holds." ], "refs": [ "algebra-lemma-etale-flat-unramified-finite-presentation", "algebra-lemma-dimension-closed-point-finite-type-field", "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-CM-over-regular-flat", "algebra-lemma-regular-ring-CM", "algebra-lemma-flat-localization", "algebra-lemma-etale", "algebra-lemma-etale-quasi-finite", "algebra-lemma-dimension-base-fibre-equals-total" ], "ref_ids": [ 1271, 996, 1223, 1107, 941, 538, 1231, 1234, 987 ] } ], "ref_ids": [] }, { "id": 1273, "type": "theorem", "label": "algebra-lemma-etale-makes-unramified-closed-at-prime", "categories": [ "algebra" ], "title": "algebra-lemma-etale-makes-unramified-closed-at-prime", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak p \\subset R$.", "Assume that $R \\to S$ is of finite type and unramified at $\\mathfrak q$.", "Then there exist", "\\begin{enumerate}", "\\item an \\'etale ring map $R \\to R'$,", "\\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$.", "\\item a product decomposition", "$$", "R' \\otimes_R S = A \\times B", "$$", "\\end{enumerate}", "with the following properties", "\\begin{enumerate}", "\\item $R' \\to A$ is surjective, and", "\\item $\\mathfrak p'A$ is a prime of $A$ lying over $\\mathfrak p'$ and", "over $\\mathfrak q$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may replace $(R \\to S, \\mathfrak p, \\mathfrak q)$", "with any base change $(R' \\to R'\\otimes_R S, \\mathfrak p', \\mathfrak q')$", "by an \\'etale ring map $R \\to R'$ with a prime $\\mathfrak p'$", "lying over $\\mathfrak p$, and a choice of $\\mathfrak q'$ lying over", "both $\\mathfrak q$ and $\\mathfrak p'$. Note also that given", "$R \\to R'$ and $\\mathfrak p'$ a suitable $\\mathfrak q'$ can always", "be found.", "\\medskip\\noindent", "The assumption that $R \\to S$ is of finite type means that we may apply", "Lemma \\ref{lemma-etale-makes-quasi-finite-finite-variant}. Thus we may", "assume that $S = A_1 \\times \\ldots \\times A_n \\times B$, that", "each $R \\to A_i$ is finite with exactly one prime $\\mathfrak r_i$", "lying over $\\mathfrak p$ such that", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak r_i)$ is purely inseparable", "and that $R \\to B$ is not quasi-finite at any prime lying over $\\mathfrak p$.", "Then clearly $\\mathfrak q = \\mathfrak r_i$ for some $i$, since", "an unramified morphism is quasi-finite", "(see Lemma \\ref{lemma-unramified-quasi-finite}).", "Say $\\mathfrak q = \\mathfrak r_1$.", "By Lemma \\ref{lemma-unramified-at-prime} we see that", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak r_1)$", "is separable hence the trivial field extension, and that", "$\\mathfrak p(A_1)_{\\mathfrak r_1}$ is the maximal ideal.", "Also, by Lemma \\ref{lemma-unique-prime-over-localize-below}", "(which applies to $R \\to A_1$ because a finite ring map satisfies going up by", "Lemma \\ref{lemma-integral-going-up})", "we have $(A_1)_{\\mathfrak r_1} = (A_1)_{\\mathfrak p}$.", "It follows from Nakayama's Lemma \\ref{lemma-NAK}", "that the map of local rings", "$R_{\\mathfrak p} \\to (A_1)_{\\mathfrak p} = (A_1)_{\\mathfrak r_1}$", "is surjective. Since $A_1$ is finite over $R$ we see that there", "exists a $f \\in R$, $f \\not \\in \\mathfrak p$ such that", "$R_f \\to (A_1)_f$ is surjective. After replacing $R$ by $R_f$ we win." ], "refs": [ "algebra-lemma-etale-makes-quasi-finite-finite-variant", "algebra-lemma-unramified-quasi-finite", "algebra-lemma-unramified-at-prime", "algebra-lemma-unique-prime-over-localize-below", "algebra-lemma-integral-going-up", "algebra-lemma-NAK" ], "ref_ids": [ 1248, 1269, 1268, 556, 500, 401 ] } ], "ref_ids": [] }, { "id": 1274, "type": "theorem", "label": "algebra-lemma-etale-makes-unramified-closed", "categories": [ "algebra" ], "title": "algebra-lemma-etale-makes-unramified-closed", "contents": [ "\\begin{slogan}", "In an unramified ring map, one can separate the points in a fiber", "by passing to an \\'etale neighbourhood.", "\\end{slogan}", "Let $R \\to S$ be a ring map.", "Let $\\mathfrak p$ be a prime of $R$.", "If $R \\to S$ is unramified then there exist", "\\begin{enumerate}", "\\item an \\'etale ring map $R \\to R'$,", "\\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$.", "\\item a product decomposition", "$$", "R' \\otimes_R S = A_1 \\times \\ldots \\times A_n \\times B", "$$", "\\end{enumerate}", "with the following properties", "\\begin{enumerate}", "\\item $R' \\to A_i$ is surjective,", "\\item $\\mathfrak p'A_i$ is a prime of $A_i$ lying over $\\mathfrak p'$, and", "\\item there is no prime of $B$ lying over $\\mathfrak p'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may apply Lemma \\ref{lemma-etale-makes-quasi-finite-finite-variant}.", "Thus, after an \\'etale base change,", "we may assume that $S = A_1 \\times \\ldots \\times A_n \\times B$,", "that each $R \\to A_i$ is finite with exactly one prime $\\mathfrak r_i$", "lying over $\\mathfrak p$ such that", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak r_i)$ is purely inseparable,", "and that $R \\to B$ is not quasi-finite at any prime lying over $\\mathfrak p$.", "Since $R \\to S$ is quasi-finite (see", "Lemma \\ref{lemma-unramified-quasi-finite})", "we see there is no prime of $B$ lying over $\\mathfrak p$.", "By Lemma \\ref{lemma-unramified-at-prime} we see that", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak r_i)$", "is separable hence the trivial field extension, and that", "$\\mathfrak p(A_i)_{\\mathfrak r_i}$ is the maximal ideal.", "Also, by Lemma \\ref{lemma-unique-prime-over-localize-below}", "(which applies to $R \\to A_i$ because a finite ring map satisfies going up by", "Lemma \\ref{lemma-integral-going-up})", "we have $(A_i)_{\\mathfrak r_i} = (A_i)_{\\mathfrak p}$.", "It follows from Nakayama's Lemma \\ref{lemma-NAK}", "that the map of local rings", "$R_{\\mathfrak p} \\to (A_i)_{\\mathfrak p} = (A_i)_{\\mathfrak r_i}$", "is surjective. Since $A_i$ is finite over $R$ we see that there", "exists a $f \\in R$, $f \\not \\in \\mathfrak p$ such that", "$R_f \\to (A_i)_f$ is surjective. After replacing $R$ by $R_f$ we win." ], "refs": [ "algebra-lemma-etale-makes-quasi-finite-finite-variant", "algebra-lemma-unramified-quasi-finite", "algebra-lemma-unramified-at-prime", "algebra-lemma-unique-prime-over-localize-below", "algebra-lemma-integral-going-up", "algebra-lemma-NAK" ], "ref_ids": [ 1248, 1269, 1268, 556, 500, 401 ] } ], "ref_ids": [] }, { "id": 1275, "type": "theorem", "label": "algebra-lemma-uniqueness", "categories": [ "algebra" ], "title": "algebra-lemma-uniqueness", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring.", "Let $f \\in R[T]$. Let $a, b \\in R$ such that $f(a) = f(b) = 0$,", "$a = b \\bmod \\mathfrak m$, and $f'(a) \\not \\in \\mathfrak m$.", "Then $a = b$." ], "refs": [], "proofs": [ { "contents": [ "Write $f(x + y) - f(x) = f'(x)y + g(x, y) y^2$ in $R[x, y]$ (this is possible", "as one sees by expanding $f(x + y)$; details omitted).", "Then we see that $0 = f(b) - f(a) = f(a + (b - a)) - f(a) =", "f'(a)(b - a) + c (b - a)^2$ for some $c \\in R$. By assumption", "$f'(a)$ is a unit in $R$. Hence $(b - a)(1 + f'(a)^{-1}c(b - a)) = 0$.", "By assumption $b - a \\in \\mathfrak m$, hence $1 + f'(a)^{-1}c(b - a)$", "is a unit in $R$. Hence $b - a = 0$ in $R$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1276, "type": "theorem", "label": "algebra-lemma-characterize-henselian", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-henselian", "contents": [ "\\begin{slogan}", "Characterizations of henselian local rings", "\\end{slogan}", "Let $(R, \\mathfrak m, \\kappa)$ be a local ring.", "The following are equivalent", "\\begin{enumerate}", "\\item $R$ is henselian,", "\\item for every $f \\in R[T]$ and every root $a_0 \\in \\kappa$", "of $\\overline{f}$ such that $\\overline{f'}(a_0) \\not = 0$", "there exists an $a \\in R$ such that $f(a) = 0$ and", "$a_0 = \\overline{a}$,", "\\item for any monic $f \\in R[T]$ and any factorization", "$\\overline{f} = g_0 h_0$ with $\\gcd(g_0, h_0) = 1$ there", "exists a factorization $f = gh$ in $R[T]$ such that", "$g_0 = \\overline{g}$ and $h_0 = \\overline{h}$,", "\\item for any monic $f \\in R[T]$ and any factorization", "$\\overline{f} = g_0 h_0$ with $\\gcd(g_0, h_0) = 1$ there", "exists a factorization $f = gh$ in $R[T]$ such that", "$g_0 = \\overline{g}$ and $h_0 = \\overline{h}$ and moreover", "$\\deg_T(g) = \\deg_T(g_0)$,", "\\item for any $f \\in R[T]$ and any factorization", "$\\overline{f} = g_0 h_0$ with $\\gcd(g_0, h_0) = 1$ there", "exists a factorization $f = gh$ in $R[T]$ such that", "$g_0 = \\overline{g}$ and $h_0 = \\overline{h}$,", "\\item for any $f \\in R[T]$ and any factorization", "$\\overline{f} = g_0 h_0$ with $\\gcd(g_0, h_0) = 1$ there", "exists a factorization $f = gh$ in $R[T]$ such that", "$g_0 = \\overline{g}$ and $h_0 = \\overline{h}$ and", "moreover $\\deg_T(g) = \\deg_T(g_0)$,", "\\item for any \\'etale ring map $R \\to S$ and prime $\\mathfrak q$ of $S$", "lying over $\\mathfrak m$ with $\\kappa = \\kappa(\\mathfrak q)$", "there exists a section $\\tau : S \\to R$ of $R \\to S$,", "\\item for any \\'etale ring map $R \\to S$ and prime $\\mathfrak q$ of $S$", "lying over $\\mathfrak m$ with $\\kappa = \\kappa(\\mathfrak q)$", "there exists a section $\\tau : S \\to R$ of $R \\to S$ with", "$\\mathfrak q = \\tau^{-1}(\\mathfrak m)$,", "\\item any finite $R$-algebra is a product of local rings,", "\\item any finite $R$-algebra is a finite product of local rings,", "\\item any finite type $R$-algebra $S$ can be written as", "$A \\times B$ with $R \\to A$ finite", "and $R \\to B$ not quasi-finite at any prime lying over $\\mathfrak m$,", "\\item any finite type $R$-algebra $S$ can be written as", "$A \\times B$ with $R \\to A$ finite", "such that each irreducible component of $\\Spec(B \\otimes_R \\kappa)$", "has dimension $\\geq 1$, and", "\\item any quasi-finite $R$-algebra $S$ can be written as", "$S = A \\times B$ with $R \\to A$ finite such that $B \\otimes_R \\kappa = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Here is a list of the easier implications:", "\\begin{enumerate}", "\\item 2$\\Rightarrow$1 because in (2) we consider all polynomials and", "in (1) only monic ones,", "\\item 5$\\Rightarrow$3 because in (5) we consider all polynomials and", "in (3) only monic ones,", "\\item 6$\\Rightarrow$4 because in (6) we consider all polynomials and", "in (4) only monic ones,", "\\item 4$\\Rightarrow$3 is obvious,", "\\item 6$\\Rightarrow$5 is obvious,", "\\item 8$\\Rightarrow$7 is obvious,", "\\item 10$\\Rightarrow$9 is obvious,", "\\item 11$\\Leftrightarrow$12 by definition of being quasi-finite at a prime,", "\\item 11$\\Rightarrow$13 by definition of being quasi-finite,", "\\end{enumerate}", "\\noindent", "Proof of 1$\\Rightarrow$8. Assume (1).", "Let $R \\to S$ be \\'etale, and let $\\mathfrak q \\subset S$", "be a prime ideal such that $\\kappa(\\mathfrak q) \\cong \\kappa$. By", "Proposition \\ref{proposition-etale-locally-standard}", "we can find a $g \\in S$, $g \\not \\in \\mathfrak q$ such that", "$R \\to S_g$ is standard \\'etale. After replacing $S$ by $S_g$ we may assume", "that $S = R[t]_g/(f)$ is standard \\'etale. Since the prime $\\mathfrak q$", "has residue field $\\kappa$ it corresponds to a root $a_0$ of", "$\\overline{f}$ which is not a root of $\\overline{g}$. By definition", "of a standard \\'etale algebra this also means that", "$\\overline{f'}(a_0) \\not = 0$.", "Since also $f$ is monic by definition of a standard \\'etale algebra again we", "may use that $R$ is henselian to conclude that there exists an $a \\in R$", "with $a_0 = \\overline{a}$ such that $f(a) = 0$. This implies that", "$g(a)$ is a unit of $R$ and we obtain the desired map", "$\\tau : S = R[t]_g/(f) \\to R$ by the rule $t \\mapsto a$. By construction", "$\\tau^{-1}(\\mathfrak q) = \\mathfrak m$. This proves (8) holds.", "\\medskip\\noindent", "Proof of 7$\\Rightarrow$8. (This is really unimportant and should be", "skipped.) Assume (7) holds and assume $R \\to S$ is \\'etale.", "Let $\\mathfrak q_1, \\ldots, \\mathfrak q_r$ be", "the other primes of $S$ lying over $\\mathfrak m$.", "Then we can find a $g \\in S$, $g \\not \\in \\mathfrak q$ and", "$g \\in \\mathfrak q_i$ for $i = 1, \\ldots, r$.", "Namely, we can argue that", "$\\bigcap_{i=1}^{r} \\mathfrak{q}_{i} \\not\\subset \\mathfrak{q}$", "since otherwise", "$\\mathfrak{q}_{i} \\subset \\mathfrak{q}$", "for some $i$, but this cannot happen as the fiber of an", "\\'etale morphism is discrete (use Lemma \\ref{lemma-etale-over-field}", "for example).", "Apply (7) to the \\'etale ring map", "$R \\to S_g$ and the prime $\\mathfrak qS_g$. This gives a section", "$\\tau_g : S_g \\to R$ such that the composition $\\tau : S \\to S_g \\to R$", "has the property $\\tau^{-1}(\\mathfrak m) = \\mathfrak q$.", "Minor details omitted.", "\\medskip\\noindent", "Proof of 8$\\Rightarrow$11. Assume (8) and let $R \\to S$ be a finite type", "ring map. Apply", "Lemma \\ref{lemma-etale-makes-quasi-finite-finite}.", "We find an \\'etale ring map $R \\to R'$ and a prime $\\mathfrak m' \\subset R'$", "lying over $\\mathfrak m$ with $\\kappa = \\kappa(\\mathfrak m')$", "such that $R' \\otimes_R S = A' \\times B'$ with $A'$ finite over $R'$", "and $B'$ not quasi-finite over $R'$ at any prime lying over $\\mathfrak m'$.", "Apply (8) to get a section $\\tau : R' \\to R$ with", "$\\mathfrak m = \\tau^{-1}(\\mathfrak m')$. Then use that", "$$", "S = (S \\otimes_R R') \\otimes_{R', \\tau} R", "= (A' \\times B') \\otimes_{R', \\tau} R", "= (A' \\otimes_{R', \\tau} R) \\times (B' \\otimes_{R', \\tau} R)", "$$", "which gives a decomposition as in (11).", "\\medskip\\noindent", "Proof of 8$\\Rightarrow$10. Assume (8) and let $R \\to S$ be a finite", "ring map. Apply", "Lemma \\ref{lemma-etale-makes-quasi-finite-finite}.", "We find an \\'etale ring map $R \\to R'$ and a prime $\\mathfrak m' \\subset R'$", "lying over $\\mathfrak m$ with $\\kappa = \\kappa(\\mathfrak m')$", "such that $R' \\otimes_R S = A'_1 \\times \\ldots \\times A'_n \\times B'$", "with $A'_i$ finite over $R'$ having exactly one prime over $\\mathfrak m'$", "and $B'$ not quasi-finite over $R'$ at any prime lying over $\\mathfrak m'$.", "Apply (8) to get a section $\\tau : R' \\to R$ with", "$\\mathfrak m = \\tau^{-1}(\\mathfrak m')$. Then we obtain", "\\begin{align*}", "S & = (S \\otimes_R R') \\otimes_{R', \\tau} R \\\\", "& = (A'_1 \\times \\ldots \\times A'_n \\times B') \\otimes_{R', \\tau} R \\\\", "& = (A'_1 \\otimes_{R', \\tau} R) \\times", "\\ldots \\times (A'_1 \\otimes_{R', \\tau} R) \\times", "(B' \\otimes_{R', \\tau} R) \\\\", "& = A_1 \\times \\ldots \\times A_n \\times B", "\\end{align*}", "The factor $B$ is finite over $R$ but $R \\to B$", "is not quasi-finite at any prime lying over $\\mathfrak m$. Hence", "$B = 0$. The factors $A_i$ are finite $R$-algebras having exactly", "one prime lying over $\\mathfrak m$, hence they are local rings.", "This proves that $S$ is a finite product of local rings.", "\\medskip\\noindent", "Proof of 9$\\Rightarrow$10. This holds because if $S$ is finite over the local", "ring $R$, then it has at most finitely many maximal ideals. Namely, by", "going up for $R \\to S$ the maximal ideals of $S$ all lie over $\\mathfrak m$,", "and $S/\\mathfrak mS$ is Artinian hence has finitely many primes.", "\\medskip\\noindent", "Proof of 10$\\Rightarrow$1. Assume (10). Let $f \\in R[T]$ be a monic", "polynomial and $a_0 \\in \\kappa$ a simple root of $\\overline{f}$.", "Then $S = R[T]/(f)$ is a finite $R$-algebra. Applying (10)", "we get $S = A_1 \\times \\ldots \\times A_r$ is a finite product of", "local $R$-algebras. In particular we see that", "$S/\\mathfrak mS = \\prod A_i/\\mathfrak mA_i$ is the decomposition", "of $\\kappa[T]/(\\overline{f})$ as a product of local rings.", "This means that one of the factors, say $A_1/\\mathfrak mA_1$", "is the quotient $\\kappa[T]/(\\overline{f}) \\to \\kappa[T]/(T - a_0)$.", "Since $A_1$ is a summand of the finite free $R$-module $S$ it", "is a finite free $R$-module itself. As $A_1/\\mathfrak mA_1$ is a", "$\\kappa$-vector space of dimension 1 we see that $A_1 \\cong R$ as an", "$R$-module. Clearly this means that $R \\to A_1$ is an isomorphism.", "Let $a \\in R$ be the image of $T$ under the map", "$R[T] \\to S \\to A_1 \\to R$. Then $f(a) = 0$ and $\\overline{a} = a_0$", "as desired.", "\\medskip\\noindent", "Proof of 13$\\Rightarrow$1. Assume (13). Let $f \\in R[T]$ be a monic", "polynomial and $a_0 \\in \\kappa$ a simple root of $\\overline{f}$.", "Then $S_1 = R[T]/(f)$ is a finite $R$-algebra. Let $g \\in R[T]$", "be any element such that $\\overline{g} = \\overline{f}/(T - a_0)$.", "Then $S = (S_1)_g$ is a quasi-finite $R$-algebra such that", "$S \\otimes_R \\kappa \\cong \\kappa[T]_{\\overline{g}}/(\\overline{f})", "\\cong \\kappa[T]/(T - a_0) \\cong \\kappa$.", "Applying (13) to $S$ we get $S = A \\times B$ with $A$ finite over $R$ and", "$B \\otimes_R \\kappa = 0$. In particular we see that", "$\\kappa \\cong S/\\mathfrak mS = A/\\mathfrak mA$.", "Since $A$ is a summand of the flat $R$-algebra $S$ we see", "that it is finite flat, hence free over $R$.", "As $A/\\mathfrak mA$ is a", "$\\kappa$-vector space of dimension 1 we see that $A \\cong R$ as an", "$R$-module. Clearly this means that $R \\to A$ is an isomorphism.", "Let $a \\in R$ be the image of $T$ under the map", "$R[T] \\to S \\to A \\to R$. Then $f(a) = 0$ and $\\overline{a} = a_0$", "as desired.", "\\medskip\\noindent", "Proof of 8$\\Rightarrow$2. Assume (8). Let $f \\in R[T]$ be any", "polynomial and let $a_0 \\in \\kappa$ be a simple root. Then", "the algebra $S = R[T]_{f'}/(f)$ is \\'etale over $R$.", "Let $\\mathfrak q \\subset S$ be the prime", "generated by $\\mathfrak m$ and $T - b$ where $b \\in R$ is any", "element such that $\\overline{b} = a_0$. Apply (8) to $S$ and $\\mathfrak q$", "to get $\\tau : S \\to R$.", "Then the image $\\tau(T) = a \\in R$ works in (2).", "\\medskip\\noindent", "At this point we see that (1), (2), (7), (8), (9), (10), (11), (12), (13) are", "all equivalent. The weakest assertion of (3), (4), (5) and (6)", "is (3) and the strongest is (6). Hence we still have to prove that", "(3) implies (1) and (1) implies (6).", "\\medskip\\noindent", "Proof of 3$\\Rightarrow$1. Assume (3). Let $f \\in R[T]$ be monic and", "let $a_0 \\in \\kappa$ be a simple root of $\\overline{f}$. This gives", "a factorization $\\overline{f} = (T - a_0)h_0$ with $h_0(a_0) \\not = 0$,", "so $\\gcd(T - a_0, h_0) = 1$. Apply (3) to get a factorization", "$f = gh$ with $\\overline{g} = T - a_0$ and $\\overline{h} = h_0$.", "Set $S = R[T]/(f)$ which is a finite free $R$-algebra. We will write", "$g$, $h$ also for the images of $g$ and $h$ in $S$. Then", "$gS + hS = S$ by", "Nakayama's Lemma \\ref{lemma-NAK}", "as the equality holds modulo $\\mathfrak m$. Since $gh = f = 0$ in $S$", "this also implies that $gS \\cap hS = 0$. Hence by the Chinese Remainder", "theorem we obtain $S = S/(g) \\times S/(h)$. This implies that", "$A = S/(g)$ is a summand of a finite free $R$-module, hence finite", "free. Moreover, the rank of $A$ is $1$ as", "$A/\\mathfrak mA = \\kappa[T]/(T - a_0)$. Thus the map $R \\to A$", "is an isomorphism. Setting $a \\in R$ equal to the image of $T$", "under the maps $R[T] \\to S \\to A \\to R$ gives an element of $R$", "with $f(a) = 0$ and $\\overline{a} = a_0$.", "\\medskip\\noindent", "Proof of 1$\\Rightarrow$6. Assume (1) or equivalently all of", "(1), (2), (7), (8), (9), (10), (11), (12), (13).", "Let $f \\in R[T]$ be a polynomial.", "Suppose that $\\overline{f} = g_0h_0$ is a factorization with", "$\\gcd(g_0, h_0) = 1$. We may and do assume that $g_0$ is monic.", "Consider $S = R[T]/(f)$. Because we", "have the factorization we see that the coefficients of", "$f$ generate the unit ideal in $R$.", "This implies that $S$ has finite fibres over $R$, hence is", "quasi-finite over $R$. It also implies that $S$ is flat over $R$ by", "Lemma \\ref{lemma-grothendieck-general}.", "Combining (13) and (10) we may write", "$S = A_1 \\times \\ldots \\times A_n \\times B$", "where each $A_i$ is local and finite over $R$, and", "$B \\otimes_R \\kappa = 0$. After reordering the factors $A_1, \\ldots, A_n$", "we may assume that", "$$", "\\kappa[T]/(g_0) =", "A_1/\\mathfrak m A_1 \\times \\ldots \\times A_r/\\mathfrak mA_r,", "\\ \\kappa[T]/(h_0) =", "A_{r + 1}/\\mathfrak mA_{r + 1} \\times \\ldots \\times A_n/\\mathfrak mA_n", "$$", "as quotients of $\\kappa[T]$. The finite flat $R$-algebra", "$A = A_1 \\times \\ldots \\times A_r$ is free as an $R$-module, see", "Lemma \\ref{lemma-finite-flat-local}.", "Its rank is $\\deg_T(g_0)$. Let $g \\in R[T]$ be the characteristic polynomial", "of the $R$-linear operator $T : A \\to A$. Then $g$ is a monic polynomial", "of degree $\\deg_T(g) = \\deg_T(g_0)$ and moreover $\\overline{g} = g_0$.", "By Cayley-Hamilton", "(Lemma \\ref{lemma-charpoly})", "we see that $g(T_A) = 0$ where $T_A$ indicates", "the image of $T$ in $A$. Hence we obtain a well defined surjective map", "$R[T]/(g) \\to A$ which is an isomorphism by", "Nakayama's Lemma \\ref{lemma-NAK}. The map $R[T] \\to A$ factors", "through $R[T]/(f)$ by construction hence we may write $f = gh$ for", "some $h$. This finishes the proof." ], "refs": [ "algebra-proposition-etale-locally-standard", "algebra-lemma-etale-over-field", "algebra-lemma-etale-makes-quasi-finite-finite", "algebra-lemma-etale-makes-quasi-finite-finite", "algebra-lemma-NAK", "algebra-lemma-grothendieck-general", "algebra-lemma-finite-flat-local", "algebra-lemma-charpoly", "algebra-lemma-NAK" ], "ref_ids": [ 1427, 1232, 1247, 1247, 401, 1111, 797, 385, 401 ] } ], "ref_ids": [] }, { "id": 1277, "type": "theorem", "label": "algebra-lemma-finite-over-henselian", "categories": [ "algebra" ], "title": "algebra-lemma-finite-over-henselian", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a henselian local ring.", "\\begin{enumerate}", "\\item If $R \\subset S$ is a finite ring extension then $S$ is", "a finite product of henselian local rings.", "\\item If $R \\subset S$ is a finite local homomorphism of local rings,", "then $S$ is a henselian local ring.", "\\item If $R \\to S$ is a finite type ring map, and $\\mathfrak q$ is", "a prime of $S$ lying over $\\mathfrak m$", "at which $R \\to S$ is quasi-finite, then", "$S_{\\mathfrak q}$ is henselian.", "\\item If $R \\to S$ is quasi-finite then $S_{\\mathfrak q}$ is henselian", "for every prime $\\mathfrak q$ lying over $\\mathfrak m$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) implies part (1) since $S$ as in part (1) is a finite product", "of its localizations at the primes lying over $\\mathfrak m$.", "Part (2) follows from", "Lemma \\ref{lemma-characterize-henselian} part (10)", "since any finite $S$-algebra is also a finite $R$-algebra.", "If $R \\to S$ and $\\mathfrak q$ are as in (3), then", "$S_{\\mathfrak q}$ is a local ring of a finite $R$-algebra by", "Lemma \\ref{lemma-characterize-henselian} part (11).", "Hence (3) follows from (1).", "Part (4) follows from part (3)." ], "refs": [ "algebra-lemma-characterize-henselian", "algebra-lemma-characterize-henselian" ], "ref_ids": [ 1276, 1276 ] } ], "ref_ids": [] }, { "id": 1278, "type": "theorem", "label": "algebra-lemma-mop-up", "categories": [ "algebra" ], "title": "algebra-lemma-mop-up", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a henselian local ring.", "Any finite type $R$-algebra $S$ can be written as", "$S = A_1 \\times \\ldots \\times A_n \\times B$ with $A_i$ local", "and finite over $R$ and $R \\to B$ not quasi-finite at any", "prime of $B$ lying over $\\mathfrak m$." ], "refs": [], "proofs": [ { "contents": [ "This is a combination of parts (11) and (10) of", "Lemma \\ref{lemma-characterize-henselian}." ], "refs": [ "algebra-lemma-characterize-henselian" ], "ref_ids": [ 1276 ] } ], "ref_ids": [] }, { "id": 1279, "type": "theorem", "label": "algebra-lemma-mop-up-strictly-henselian", "categories": [ "algebra" ], "title": "algebra-lemma-mop-up-strictly-henselian", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a strictly henselian local ring.", "Any finite type $R$-algebra $S$ can be written as", "$S = A_1 \\times \\ldots \\times A_n \\times B$ with $A_i$ local", "and finite over $R$ and $\\kappa \\subset \\kappa(\\mathfrak m_{A_i})$", "finite purely inseparable and $R \\to B$ not quasi-finite", "at any prime of $B$ lying over $\\mathfrak m$." ], "refs": [], "proofs": [ { "contents": [ "First write $S = A_1 \\times \\ldots \\times A_n \\times B$ as in", "Lemma \\ref{lemma-mop-up}.", "The field extension $\\kappa \\subset \\kappa(\\mathfrak m_{A_i})$", "is finite and $\\kappa$ is separably algebraically closed, hence", "it is finite purely inseparable." ], "refs": [ "algebra-lemma-mop-up" ], "ref_ids": [ 1278 ] } ], "ref_ids": [] }, { "id": 1280, "type": "theorem", "label": "algebra-lemma-henselian-cat-finite-etale", "categories": [ "algebra" ], "title": "algebra-lemma-henselian-cat-finite-etale", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a henselian local ring.", "The category of finite \\'etale ring extensions $R \\to S$ is", "equivalent to the category of finite \\'etale algebras", "$\\kappa \\to \\overline{S}$ via the functor $S \\mapsto S/\\mathfrak mS$." ], "refs": [], "proofs": [ { "contents": [ "Denote $\\mathcal{C} \\to \\mathcal{D}$ the functor of categories", "of the statement.", "Suppose that $R \\to S$ is finite \\'etale. Then we may write", "$$", "S = A_1 \\times \\ldots \\times A_n", "$$", "with $A_i$ local and finite \\'etale over $S$, use either", "Lemma \\ref{lemma-mop-up}", "or", "Lemma \\ref{lemma-characterize-henselian} part (10).", "In particular $A_i/\\mathfrak mA_i$ is a finite separable field", "extension of $\\kappa$, see", "Lemma \\ref{lemma-etale-at-prime}.", "Thus we see that every object of $\\mathcal{C}$ and", "$\\mathcal{D}$ decomposes canonically into irreducible pieces", "which correspond via the given functor.", "Next, suppose that $S_1$, $S_2$ are finite \\'etale over $R$ such that", "$\\kappa_1 = S_1/\\mathfrak mS_1$ and $\\kappa_2 = S_2/\\mathfrak mS_2$", "are fields (finite separable over $\\kappa$). Then $S_1 \\otimes_R S_2$", "is finite \\'etale over $R$ and we may write", "$$", "S_1 \\otimes_R S_2 = A_1 \\times \\ldots \\times A_n", "$$", "as before. Then we see that $\\Hom_R(S_1, S_2)$ is identified", "with the set of indices $i \\in \\{1, \\ldots, n\\}$ such that", "$S_2 \\to A_i$ is an isomorphism. To see this use that given any $R$-algebra", "map $\\varphi : S_1 \\to S_2$ the map", "$\\varphi \\times 1 : S_1 \\otimes_R S_2 \\to S_2$", "is surjective, and hence is equal to projection onto one of the factors $A_i$.", "But in exactly the same way we see that", "$\\Hom_\\kappa(\\kappa_1, \\kappa_2)$ is identified with", "the set of indices $i \\in \\{1, \\ldots, n\\}$ such that", "$\\kappa_2 \\to A_i/\\mathfrak mA_i$ is an isomorphism.", "By the discussion above these sets of indices match, and we conclude", "that our functor is fully faithful.", "Finally, let $\\kappa \\subset \\kappa'$ be a finite", "separable field extension. By", "Lemma \\ref{lemma-make-etale-map-prescribed-residue-field}", "there exists an \\'etale ring map $R \\to S$ and a prime $\\mathfrak q$", "of $S$ lying over $\\mathfrak m$ such that $\\kappa \\subset \\kappa(\\mathfrak q)$", "is isomorphic to the given extension. By part (1)", "we may write $S = A_1 \\times \\ldots \\times A_n \\times B$.", "Since $R \\to S$ is quasi-finite we see that there exists no", "prime of $B$ over $\\mathfrak m$. Hence $S_{\\mathfrak q}$ is", "equal to $A_i$ for some $i$. Hence $R \\to A_i$ is finite \\'etale", "and produces the given residue field extension. Thus the functor", "is essentially surjective and we win." ], "refs": [ "algebra-lemma-mop-up", "algebra-lemma-characterize-henselian", "algebra-lemma-etale-at-prime", "algebra-lemma-make-etale-map-prescribed-residue-field" ], "ref_ids": [ 1278, 1276, 1233, 1242 ] } ], "ref_ids": [] }, { "id": 1281, "type": "theorem", "label": "algebra-lemma-unramified-over-strictly-henselian", "categories": [ "algebra" ], "title": "algebra-lemma-unramified-over-strictly-henselian", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a strictly henselian local ring.", "Let $R \\to S$ be an unramified ring map. Then", "$$", "S = A_1 \\times \\ldots \\times A_n \\times B", "$$", "with each $R \\to A_i$ surjective and no prime of $B$ lying", "over $\\mathfrak m$." ], "refs": [], "proofs": [ { "contents": [ "First write $S = A_1 \\times \\ldots \\times A_n \\times B$ as in", "Lemma \\ref{lemma-mop-up}.", "Now we see that $R \\to A_i$ is finite unramified and $A_i$ local.", "Hence the maximal ideal of $A_i$ is $\\mathfrak mA_i$ and its", "residue field $A_i / \\mathfrak m A_i$ is a finite", "separable extension of $\\kappa$, see", "Lemma \\ref{lemma-unramified-at-prime}.", "However, the condition that $R$ is strictly henselian means that", "$\\kappa$ is separably algebraically closed, so", "$\\kappa = A_i / \\mathfrak m A_i$. By", "Nakayama's Lemma \\ref{lemma-NAK}", "we conclude that $R \\to A_i$ is surjective as desired." ], "refs": [ "algebra-lemma-mop-up", "algebra-lemma-unramified-at-prime", "algebra-lemma-NAK" ], "ref_ids": [ 1278, 1268, 401 ] } ], "ref_ids": [] }, { "id": 1282, "type": "theorem", "label": "algebra-lemma-complete-henselian", "categories": [ "algebra" ], "title": "algebra-lemma-complete-henselian", "contents": [ "\\begin{slogan}", "Complete local rings are Henselian by Newton's method", "\\end{slogan}", "Let $(R, \\mathfrak m, \\kappa)$ be a complete local ring, see", "Definition \\ref{definition-complete-local-ring}.", "Then $R$ is henselian." ], "refs": [ "algebra-definition-complete-local-ring" ], "proofs": [ { "contents": [ "Let $f \\in R[T]$ be monic.", "Denote $f_n \\in R/\\mathfrak m^{n + 1}[T]$ the image.", "Denote $f'_n$ the derivative of $f_n$ with respect to $T$.", "Let $a_0 \\in \\kappa$ be a simple root of $f_0$. We lift this", "to a solution of $f$ over $R$ inductively as follows:", "Suppose given $a_n \\in R/\\mathfrak m^{n + 1}$ such that", "$a_n \\bmod \\mathfrak m = a_0$ and $f_n(a_n) = 0$. Pick any", "element $b \\in R/\\mathfrak m^{n + 2}$ such that", "$a_n = b \\bmod \\mathfrak m^{n + 1}$. Then", "$f_{n + 1}(b) \\in \\mathfrak m^{n + 1}/\\mathfrak m^{n + 2}$.", "Set", "$$", "a_{n + 1} = b - f_{n + 1}(b)/f'_{n + 1}(b)", "$$", "(Newton's method). This makes sense as", "$f'_{n + 1}(b) \\in R/\\mathfrak m^{n + 1}$", "is invertible by the condition on $a_0$. Then we compute", "$f_{n + 1}(a_{n + 1}) = f_{n + 1}(b) - f_{n + 1}(b) = 0$", "in $R/\\mathfrak m^{n + 2}$. Since the system of elements", "$a_n \\in R/\\mathfrak m^{n + 1}$ so constructed is compatible", "we get an element", "$a \\in \\lim R/\\mathfrak m^n = R$ (here we use that $R$ is complete).", "Moreover, $f(a) = 0$ since it maps to zero in each $R/\\mathfrak m^n$.", "Finally $\\overline{a} = a_0$ and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1548 ] }, { "id": 1283, "type": "theorem", "label": "algebra-lemma-local-dimension-zero-henselian", "categories": [ "algebra" ], "title": "algebra-lemma-local-dimension-zero-henselian", "contents": [ "\\begin{slogan}", "Local rings of dimension zero are henselian.", "\\end{slogan}", "Let $(R, \\mathfrak m)$ be a local ring of dimension $0$.", "Then $R$ is henselian." ], "refs": [], "proofs": [ { "contents": [ "Let $R \\to S$ be a finite ring map. By", "Lemma \\ref{lemma-characterize-henselian}", "it suffices to show that $S$ is a product of local rings. By", "Lemma \\ref{lemma-finite-finite-fibres}", "$S$ has finitely many primes $\\mathfrak m_1, \\ldots, \\mathfrak m_r$", "which all lie over $\\mathfrak m$. There are no inclusions among these", "primes, see", "Lemma \\ref{lemma-integral-no-inclusion},", "hence they are all maximal. Every element of", "$\\mathfrak m_1 \\cap \\ldots \\cap \\mathfrak m_r$ is nilpotent by", "Lemma \\ref{lemma-Zariski-topology}.", "It follows $S$ is the product of the localizations of $S$ at the primes", "$\\mathfrak m_i$ by", "Lemma \\ref{lemma-product-local}." ], "refs": [ "algebra-lemma-characterize-henselian", "algebra-lemma-finite-finite-fibres", "algebra-lemma-integral-no-inclusion", "algebra-lemma-Zariski-topology", "algebra-lemma-product-local" ], "ref_ids": [ 1276, 499, 498, 389, 645 ] } ], "ref_ids": [] }, { "id": 1284, "type": "theorem", "label": "algebra-lemma-map-into-henselian", "categories": [ "algebra" ], "title": "algebra-lemma-map-into-henselian", "contents": [ "Let $R \\to S$ be a ring map with $S$ henselian local.", "Given", "\\begin{enumerate}", "\\item an \\'etale ring map $R \\to A$,", "\\item a prime $\\mathfrak q$ of $A$ lying over", "$\\mathfrak p = R \\cap \\mathfrak m_S$,", "\\item a $\\kappa(\\mathfrak p)$-algebra map", "$\\tau : \\kappa(\\mathfrak q) \\to S/\\mathfrak m_S$,", "\\end{enumerate}", "then there exists a unique homomorphism of $R$-algebras $f : A \\to S$", "such that $\\mathfrak q = f^{-1}(\\mathfrak m_S)$ and", "$f \\bmod \\mathfrak q = \\tau$." ], "refs": [], "proofs": [ { "contents": [ "Consider $A \\otimes_R S$. This is an \\'etale algebra over $S$, see", "Lemma \\ref{lemma-etale}. Moreover, the kernel", "$$", "\\mathfrak q' = \\Ker(A \\otimes_R S \\to", "\\kappa(\\mathfrak q) \\otimes_{\\kappa(\\mathfrak p)} \\kappa(\\mathfrak m_S) \\to", "\\kappa(\\mathfrak m_S))", "$$", "of the map using the map given in (3) is a prime ideal lying over", "$\\mathfrak m_S$ with residue field equal to the residue field of $S$.", "Hence by Lemma \\ref{lemma-characterize-henselian}", "there exists a unique splitting $\\tau : A \\otimes_R S \\to S$", "with $\\tau^{-1}(\\mathfrak m_S) = \\mathfrak q'$.", "Set $f$ equal to the composition $A \\to A \\otimes_R S \\to S$." ], "refs": [ "algebra-lemma-etale", "algebra-lemma-characterize-henselian" ], "ref_ids": [ 1231, 1276 ] } ], "ref_ids": [] }, { "id": 1285, "type": "theorem", "label": "algebra-lemma-strictly-henselian-solutions", "categories": [ "algebra" ], "title": "algebra-lemma-strictly-henselian-solutions", "contents": [ "Let $\\varphi : R \\to S$ be a local homomorphism", "of strictly henselian local rings.", "Let $P_1, \\ldots, P_n \\in R[x_1, \\ldots, x_n]$ be polynomials such that", "$R[x_1, \\ldots, x_n]/(P_1, \\ldots, P_n)$ is \\'etale over $R$.", "Then the map", "$$", "R^n \\longrightarrow S^n, \\quad", "(h_1, \\ldots, h_n) \\longmapsto (\\varphi(h_1), \\ldots, \\varphi(h_n))", "$$", "induces a bijection between", "$$", "\\{", "(r_1, \\ldots, r_n) \\in R^n", "\\mid", "P_i(r_1, \\ldots, r_n) = 0, \\ i = 1, \\ldots, n", "\\}", "$$", "and", "$$", "\\{", "(s_1, \\ldots, s_n) \\in S^n", "\\mid", "P'_i(s_1, \\ldots, s_n) = 0, \\ i = 1, \\ldots, n", "\\}", "$$", "where $P'_i \\in S[x_1, \\ldots, x_n]$ are the images of the $P_i$", "under $\\varphi$." ], "refs": [], "proofs": [ { "contents": [ "The first solution set is canonically isomorphic to the set", "$$", "\\Hom_R(R[x_1, \\ldots, x_n]/(P_1, \\ldots, P_n), R).", "$$", "As $R$ is henselian the map $R \\to R/\\mathfrak m_R$ induces a bijection", "between this set and the set of solutions in the", "residue field $R/\\mathfrak m_R$, see", "Lemma \\ref{lemma-characterize-henselian}.", "The same is true for $S$.", "Now since $R[x_1, \\ldots, x_n]/(P_1, \\ldots, P_n)$ is \\'etale over $R$", "and $R/\\mathfrak m_R$ is separably algebraically closed we see that", "$R/\\mathfrak m_R[x_1, \\ldots, x_n]/(\\overline{P_1}, \\ldots, \\overline{P_n})$", "is a finite product of copies of $R/\\mathfrak m_R$. Hence the", "tensor product", "$$", "R/\\mathfrak m_R[x_1, \\ldots, x_n]/(\\overline{P_1}, \\ldots, \\overline{P_n})", "\\otimes_{R/\\mathfrak m_R} S/\\mathfrak m_S", "=", "S/\\mathfrak m_S[x_1, \\ldots, x_n]/(\\overline{P_1'}, \\ldots, \\overline{P_n'})", "$$", "is also a finite product of copies of $S/\\mathfrak m_S$ with the same", "index set. This proves the lemma." ], "refs": [ "algebra-lemma-characterize-henselian" ], "ref_ids": [ 1276 ] } ], "ref_ids": [] }, { "id": 1286, "type": "theorem", "label": "algebra-lemma-split-ML-henselian", "categories": [ "algebra" ], "title": "algebra-lemma-split-ML-henselian", "contents": [ "Let $R$ be a henselian local ring.", "Any countably generated Mittag-Leffler module over $R$ is a direct", "sum of finitely presented $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be a countably generated and Mittag-Leffler $R$-module.", "We claim that for any element $x \\in M$ there exists a direct", "sum decomposition $M = N \\oplus K$ with $x \\in N$, the module", "$N$ finitely presented, and $K$ Mittag-Leffler.", "\\medskip\\noindent", "Suppose the claim is true. Choose generators $x_1, x_2, x_3, \\ldots$", "of $M$. By the claim we can inductively find direct sum decompositions", "$$", "M = N_1 \\oplus N_2 \\oplus \\ldots \\oplus N_n \\oplus K_n", "$$", "with $N_i$ finitely presented,", "$x_1, \\ldots, x_n \\in N_1 \\oplus \\ldots \\oplus N_n$, and $K_n$ Mittag-Leffler.", "Repeating ad infinitum we see that $M = \\bigoplus N_i$.", "\\medskip\\noindent", "We still have to prove the claim. Let $x \\in M$. By", "Lemma \\ref{lemma-ML-countable}", "there exists an endomorphism $\\alpha : M \\to M$", "such that $\\alpha$ factors through a finitely presented module, and", "$\\alpha (x) = x$. Say $\\alpha$ factors as", "$$", "\\xymatrix{", "M \\ar[r]^\\pi & P \\ar[r]^i & M", "}", "$$", "Set $a = \\pi \\circ \\alpha \\circ i : P \\to P$, so", "$i \\circ a \\circ \\pi = \\alpha^3$. By", "Lemma \\ref{lemma-charpoly-module}", "there exists a monic polynomial $P \\in R[T]$ such that $P(a) = 0$.", "Note that this implies formally that $\\alpha^2 P(\\alpha) = 0$.", "Hence we may think of $M$ as a module over $R[T]/(T^2P)$.", "Assume that $x \\not = 0$. Then $\\alpha(x) = x$ implies that", "$0 = \\alpha^2P(\\alpha)x = P(1)x$ hence $P(1) = 0$ in $R/I$ where", "$I = \\{r \\in R \\mid rx = 0\\}$ is the annihilator of $x$.", "As $x \\not = 0$ we see $I \\subset \\mathfrak m_R$, hence", "$1$ is a root of $\\overline{P} = P \\bmod \\mathfrak m_R \\in R/\\mathfrak m_R[T]$.", "As $R$ is henselian we can find a factorization", "$$", "T^2P = (T^2 Q_1) Q_2", "$$", "for some $Q_1, Q_2 \\in R[T]$ with", "$Q_2 = (T - 1)^e \\bmod \\mathfrak m_R R[T]$ and", "$Q_1(1) \\not = 0 \\bmod \\mathfrak m_R$, see", "Lemma \\ref{lemma-characterize-henselian}.", "Let $N = \\Im(\\alpha^2Q_1(\\alpha) : M \\to M)$ and", "$K = \\Im(Q_2(\\alpha) : M \\to M)$. As $T^2Q_1$ and", "$Q_2$ generate the unit ideal of $R[T]$ we get a direct sum", "decomposition $M = N \\oplus K$. Moreover, $Q_2$ acts as zero on $N$ and", "$T^2Q_1$ acts as zero on $K$. Note that $N$ is a quotient of $P$", "hence is finitely generated. Also $x \\in N$ because", "$\\alpha^2Q_1(\\alpha)x = Q_1(1)x$ and $Q_1(1)$ is a unit in $R$. By", "Lemma \\ref{lemma-direct-sum-ML}", "the modules $N$ and $K$ are Mittag-Leffler. Finally, the finitely generated", "module $N$ is finitely presented as a finitely generated Mittag-Leffler", "module is finitely presented, see", "Example \\ref{example-ML} part (1)." ], "refs": [ "algebra-lemma-ML-countable", "algebra-lemma-charpoly-module", "algebra-lemma-characterize-henselian", "algebra-lemma-direct-sum-ML" ], "ref_ids": [ 849, 386, 1276, 840 ] } ], "ref_ids": [] }, { "id": 1287, "type": "theorem", "label": "algebra-lemma-base-change-colimit-etale", "categories": [ "algebra" ], "title": "algebra-lemma-base-change-colimit-etale", "contents": [ "Let $R \\to A$ and $R \\to R'$ be ring maps. If $A$ is a filtered", "colimit of \\'etale ring maps, then so is $R' \\to R' \\otimes_R A$." ], "refs": [], "proofs": [ { "contents": [ "This is true because colimits commute with tensor products", "and \\'etale ring maps are preserved under base change", "(Lemma \\ref{lemma-etale})." ], "refs": [ "algebra-lemma-etale" ], "ref_ids": [ 1231 ] } ], "ref_ids": [] }, { "id": 1288, "type": "theorem", "label": "algebra-lemma-composition-colimit-etale", "categories": [ "algebra" ], "title": "algebra-lemma-composition-colimit-etale", "contents": [ "Let $A \\to B \\to C$ be ring maps. If $A \\to B$ is a filtered", "colimit of \\'etale ring maps and $B \\to C$ is a filtered colimit", "of \\'etale ring maps, then $A \\to C$ is a filtered colimit of", "\\'etale ring maps." ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion of Lemma \\ref{lemma-when-colimit}.", "Let $A \\to P \\to C$ be a factorization of $A \\to C$", "with $P$ of finite presentation over $A$.", "Write $B = \\colim_{i \\in I} B_i$ where $I$ is a directed set and", "where $B_i$ is an \\'etale $A$-algebra. ", "Write $C = \\colim_{j \\in J} C_j$ where $J$ is a directed set and", "where $C_j$ is an \\'etale $B$-algebra.", "We can factor $P \\to C$ as $P \\to C_j \\to C$ for", "some $j$ by Lemma \\ref{lemma-characterize-finite-presentation}.", "By Lemma \\ref{lemma-etale} we can find an", "$i \\in I$ and an \\'etale ring map $B_i \\to C'_j$", "such that $C_j = B \\otimes_{B_i} C'_j$.", "Then $C_j = \\colim_{i' \\geq i} B_{i'} \\otimes_{B_i} C'_j$", "and again we see that $P \\to C_j$ factors as", "$P \\to B_{i'} \\otimes_{B_i} C'_j \\to C$.", "As $A \\to C' = B_{i'} \\otimes_{B_i} C'_j$ is \\'etale as", "compositions and tensor products of \\'etale ring maps", "are \\'etale. Hence we have factored $P \\to C$ as", "$P \\to C' \\to C$ with $C'$ \\'etale over $A$ and the criterion", "of Lemma \\ref{lemma-when-colimit} applies." ], "refs": [ "algebra-lemma-when-colimit", "algebra-lemma-characterize-finite-presentation", "algebra-lemma-etale", "algebra-lemma-when-colimit" ], "ref_ids": [ 1093, 1092, 1231, 1093 ] } ], "ref_ids": [] }, { "id": 1289, "type": "theorem", "label": "algebra-lemma-colimit-colimit-etale", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-colimit-etale", "contents": [ "Let $R$ be a ring. Let $A = \\colim A_i$ be a filtered colimit", "of $R$-algebras such that each $A_i$ is a filtered colimit of", "\\'etale $R$-algebras. Then $A$ is a filtered colimit of \\'etale", "$R$-algebras." ], "refs": [], "proofs": [ { "contents": [ "Write $A_i = \\colim_{j \\in J_i} A_j$ where $J_i$ is a directed set and", "$A_j$ is an \\'etale $R$-algebra.", "For each $i \\leq i'$ and $j \\in J_i$ there exists an", "$j' \\in J_{i'}$ and an $R$-algebra map $\\varphi_{jj'} : A_j \\to A_{j'}$", "making the diagram", "$$", "\\xymatrix{", "A_i \\ar[r] & A_{i'} \\\\", "A_j \\ar[u] \\ar[r]^{\\varphi_{jj'}} & A_{j'} \\ar[u]", "}", "$$", "commute. This is true because $R \\to A_j$ is of finite presentation", "so that Lemma \\ref{lemma-characterize-finite-presentation} applies.", "Let $\\mathcal{J}$ be the category with objects $\\coprod_{i \\in I} J_i$", "and morphisms triples $(j, j', \\varphi_{jj'})$ as above (and obvious", "composition law). Then $\\mathcal{J}$ is a filtered category and", "$A = \\colim_\\mathcal{J} A_j$. Details omitted." ], "refs": [ "algebra-lemma-characterize-finite-presentation" ], "ref_ids": [ 1092 ] } ], "ref_ids": [] }, { "id": 1290, "type": "theorem", "label": "algebra-lemma-colimits-of-etale", "categories": [ "algebra" ], "title": "algebra-lemma-colimits-of-etale", "contents": [ "Let $R$ be a ring. Let $A \\to B$ be an $R$-algebra homomorphism.", "If $A$ and $B$ are filtered colimits of \\'etale $R$-algebras, then", "$B$ is a filtered colimit of \\'etale $A$-algebras." ], "refs": [], "proofs": [ { "contents": [ "Write $A = \\colim A_i$ and $B = \\colim B_j$ as filtered colimits with $A_i$", "and $B_j$ \\'etale over $R$. For each $i$ we can find a $j$ such that", "$A_i \\to B$ factors through $B_j$, see", "Lemma \\ref{lemma-characterize-finite-presentation}.", "The factorization $A_i \\to B_j$ is \\'etale by", "Lemma \\ref{lemma-map-between-etale}.", "Since $A \\to A \\otimes_{A_i} B_j$ is \\'etale (Lemma \\ref{lemma-etale})", "it suffices to prove that $B = \\colim A \\otimes_{A_i} B_j$ where the", "colimit is over pairs $(i, j)$ and factorizations $A_i \\to B_j \\to B$", "of $A_i \\to B$ (this is a directed system; details omitted).", "This is clear because colimits commute with tensor products", "and hence $\\colim A \\otimes_{A_i} B_j = A \\otimes_A B = B$." ], "refs": [ "algebra-lemma-characterize-finite-presentation", "algebra-lemma-map-between-etale", "algebra-lemma-etale" ], "ref_ids": [ 1092, 1236, 1231 ] } ], "ref_ids": [] }, { "id": 1291, "type": "theorem", "label": "algebra-lemma-map-into-henselian-colimit", "categories": [ "algebra" ], "title": "algebra-lemma-map-into-henselian-colimit", "contents": [ "Let $R \\to S$ be a ring map with $S$ henselian local. Given", "\\begin{enumerate}", "\\item an $R$-algebra $A$ which is a filtered colimit of \\'etale $R$-algebras,", "\\item a prime $\\mathfrak q$ of $A$ lying over", "$\\mathfrak p = R \\cap \\mathfrak m_S$,", "\\item a $\\kappa(\\mathfrak p)$-algebra map", "$\\tau : \\kappa(\\mathfrak q) \\to S/\\mathfrak m_S$,", "\\end{enumerate}", "then there exists a unique homomorphism of $R$-algebras $f : A \\to S$", "such that $\\mathfrak q = f^{-1}(\\mathfrak m_S)$ and", "$f \\bmod \\mathfrak q = \\tau$." ], "refs": [], "proofs": [ { "contents": [ "Write $A = \\colim A_i$ as a filtered colimit of \\'etale $R$-algebras.", "Set $\\mathfrak q_i = A_i \\cap \\mathfrak q$. We obtain $f_i : A_i \\to S$", "by applying Lemma \\ref{lemma-map-into-henselian}. Set $f = \\colim f_i$." ], "refs": [ "algebra-lemma-map-into-henselian" ], "ref_ids": [ 1284 ] } ], "ref_ids": [] }, { "id": 1292, "type": "theorem", "label": "algebra-lemma-uniqueness-henselian", "categories": [ "algebra" ], "title": "algebra-lemma-uniqueness-henselian", "contents": [ "Let $R$ be a ring. Given a commutative diagram of ring maps", "$$", "\\xymatrix{", "S \\ar[r] & K \\\\", "R \\ar[u] \\ar[r] & S' \\ar[u]", "}", "$$", "where $S$, $S'$ are henselian local, $S$, $S'$ are filtered colimits", "of \\'etale $R$-algebras, $K$ is a field and the arrows $S \\to K$ and ", "$S' \\to K$ identify $K$ with the residue field of both $S$ and $S'$.", "Then there exists an unique $R$-algebra isomorphism $S \\to S'$", "compatible with the maps to $K$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-map-into-henselian-colimit}." ], "refs": [ "algebra-lemma-map-into-henselian-colimit" ], "ref_ids": [ 1291 ] } ], "ref_ids": [] }, { "id": 1293, "type": "theorem", "label": "algebra-lemma-colimit-henselian", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-henselian", "contents": [ "A filtered colimit of henselian local rings along local homomorphisms", "is henselian." ], "refs": [], "proofs": [ { "contents": [ "Categories, Lemma \\ref{categories-lemma-directed-category-system}", "says that this is really just a question about a colimit of", "henselian local rings over a directed set.", "Let $(R_i, \\varphi_{ii'})$ be such a system with each $\\varphi_{ii'}$", "local. Then $R = \\colim_i R_i$ is local, and", "its residue field $\\kappa$ is $\\colim \\kappa_i$", "(argument omitted).", "Suppose that $f \\in R[T]$ is monic and that $a_0 \\in \\kappa$ is", "a simple root of $\\overline{f}$. Then for some large enough $i$", "there exists an $f_i \\in R_i[T]$ mapping to $f$ and an", "$a_{0, i} \\in \\kappa_i$ mapping to $a_0$. Since", "$\\overline{f_i}(a_{0, i}) \\in \\kappa_i$,", "resp.\\ $\\overline{f_i'}(a_{0, i}) \\in \\kappa_i$ maps to", "$0 = \\overline{f}(a_0) \\in \\kappa$,", "resp.\\ $0 \\not = \\overline{f'}(a_0) \\in \\kappa$", "we conclude that $a_{0, i}$ is a simple root of $\\overline{f_i}$.", "As $R_i$ is henselian we can find $a_i \\in R_i$ such that", "$f_i(a_i) = 0$ and $a_{0, i} = \\overline{a_i}$.", "Then the image $a \\in R$ of $a_i$ is the desired solution.", "Thus $R$ is henselian." ], "refs": [ "categories-lemma-directed-category-system" ], "ref_ids": [ 12236 ] } ], "ref_ids": [] }, { "id": 1294, "type": "theorem", "label": "algebra-lemma-henselization", "categories": [ "algebra" ], "title": "algebra-lemma-henselization", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. There exists a", "local ring map $R \\to R^h$ with the following properties", "\\begin{enumerate}", "\\item $R^h$ is henselian,", "\\item $R^h$ is a filtered colimit of \\'etale $R$-algebras,", "\\item $\\mathfrak m R^h$ is the", "maximal ideal of $R^h$, and", "\\item $\\kappa = R^h/\\mathfrak m R^h$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the category of pairs $(S, \\mathfrak q)$ where $R \\to S$ is an", "\\'etale ring map, and $\\mathfrak q$ is a prime of $S$ lying over", "$\\mathfrak m$ with $\\kappa = \\kappa(\\mathfrak q)$. A morphism of pairs", "$(S, \\mathfrak q) \\to (S', \\mathfrak q')$ is given by an $R$-algebra", "map $\\varphi : S \\to S'$ such that $\\varphi^{-1}(\\mathfrak q') = \\mathfrak q$.", "We set", "$$", "R^h = \\colim_{(S, \\mathfrak q)} S.", "$$", "Let us show that the category of pairs is filtered, see", "Categories, Definition \\ref{categories-definition-directed}.", "The category contains the pair $(R, \\mathfrak m)$ and hence is not empty,", "which proves part (1) of", "Categories, Definition \\ref{categories-definition-directed}.", "For any pair $(S, \\mathfrak q)$ the prime ideal $\\mathfrak q$", "is maximal with residue field $\\kappa$ since the composition", "$\\kappa \\to S/\\mathfrak q \\to \\kappa(\\mathfrak q)$ is an isomorphism.", "Suppose that $(S, \\mathfrak q)$ and $(S', \\mathfrak q')$ are two objects. Set", "$S'' = S \\otimes_R S'$ and $\\mathfrak q'' = \\mathfrak qS'' + \\mathfrak q'S''$.", "Then $S''/\\mathfrak q'' = S/\\mathfrak q \\otimes_R S'/\\mathfrak q' = \\kappa$", "by what we said above. Moreover, $R \\to S''$ is \\'etale by", "Lemma \\ref{lemma-etale}.", "This proves part (2) of", "Categories, Definition \\ref{categories-definition-directed}.", "Next, suppose that", "$\\varphi, \\psi : (S, \\mathfrak q) \\to (S', \\mathfrak q')$", "are two morphisms of pairs. Then $\\varphi$, $\\psi$, and", "$S' \\otimes_R S' \\to S'$ are \\'etale ring maps by", "Lemma \\ref{lemma-map-between-etale}.", "Consider", "$$", "S'' = (S' \\otimes_{\\varphi, S, \\psi} S')", "\\otimes_{S' \\otimes_R S'} S'", "$$", "with prime ideal", "$$", "\\mathfrak q'' =", "(\\mathfrak q' \\otimes S' + S' \\otimes \\mathfrak q') \\otimes S'", "+", "(S' \\otimes_{\\varphi, S, \\psi} S') \\otimes \\mathfrak q'", "$$", "Arguing as above (base change of \\'etale maps is \\'etale, composition of", "\\'etale maps is \\'etale) we see that $S''$ is \\'etale over $R$. Moreover,", "the canonical map $S' \\to S''$ (using the right most factor for example)", "equalizes $\\varphi$ and $\\psi$. This proves part (3) of", "Categories, Definition \\ref{categories-definition-directed}.", "Hence we conclude that $R^h$ consists of triples $(S, \\mathfrak q, f)$", "with $f \\in S$, and two such triples", "$(S, \\mathfrak q, f)$, $(S', \\mathfrak q', f')$", "define the same element of $R^h$ if and only if there exists", "a pair $(S'', \\mathfrak q'')$ and morphisms of pairs", "$\\varphi : (S, \\mathfrak q) \\to (S'', \\mathfrak q'')$", "and", "$\\varphi' : (S', \\mathfrak q') \\to (S'', \\mathfrak q'')$", "such that $\\varphi(f) = \\varphi'(f')$.", "\\medskip\\noindent", "Suppose that $x \\in R^h$. Represent $x$ by a triple $(S, \\mathfrak q, f)$.", "Let $\\mathfrak q_1, \\ldots, \\mathfrak q_r$ be the other primes of $S$", "lying over $\\mathfrak m$. Then $\\mathfrak q \\not \\subset \\mathfrak q_i$", "as we have seen above that $\\mathfrak q$ is maximal.", "Thus, since $\\mathfrak q$ is a prime ideal,", "we can find a $g \\in S$, $g \\not \\in \\mathfrak q$ and", "$g \\in \\mathfrak q_i$ for $i = 1, \\ldots, r$. Consider the morphism of", "pairs $(S, \\mathfrak q) \\to (S_g, \\mathfrak qS_g)$.", "In this way we see that we may always assume that $x$", "is given by a triple $(S, \\mathfrak q, f)$ where", "$\\mathfrak q$ is the only prime of $S$ lying over $\\mathfrak m$,", "i.e., $\\sqrt{\\mathfrak mS} = \\mathfrak q$. But since", "$R \\to S$ is \\'etale, we have", "$\\mathfrak mS_{\\mathfrak q} = \\mathfrak qS_{\\mathfrak q}$, see", "Lemma \\ref{lemma-etale-at-prime}.", "Hence we actually get that $\\mathfrak mS = \\mathfrak q$.", "\\medskip\\noindent", "Suppose that $x \\not \\in \\mathfrak mR^h$.", "Represent $x$ by a triple $(S, \\mathfrak q, f)$ with", "$\\mathfrak mS = \\mathfrak q$.", "Then $f \\not \\in \\mathfrak mS$, i.e., $f \\not \\in \\mathfrak q$.", "Hence $(S, \\mathfrak q) \\to (S_f, \\mathfrak qS_f)$ is a morphism", "of pairs such that the image of $f$ becomes invertible.", "Hence $x$ is invertible with inverse represented by the triple", "$(S_f, \\mathfrak qS_f, 1/f)$. We conclude that $R^h$ is a local", "ring with maximal ideal $\\mathfrak mR^h$. The residue field is", "$\\kappa$ since we can define $R^h/\\mathfrak mR^h \\to \\kappa$", "by mapping a triple $(S, \\mathfrak q, f)$ to the residue", "class of $f$ modulo $\\mathfrak q$.", "\\medskip\\noindent", "We still have to show that $R^h$ is henselian.", "Namely, suppose that $P \\in R^h[T]$ is a monic", "polynomial and $a_0 \\in \\kappa$ is a simple root of", "the reduction $\\overline{P} \\in \\kappa[T]$.", "Then we can find a pair $(S, \\mathfrak q)$ such that", "$P$ is the image of a monic polynomial $Q \\in S[T]$.", "Since $S \\to R^h$ induces an isomorphism of residue", "fields we see that $S' = S[T]/(Q)$ has a prime ideal", "$\\mathfrak q' = (\\mathfrak q, T - a_0)$ at which", "$S \\to S'$ is standard \\'etale. Moreover, $\\kappa = \\kappa(\\mathfrak q')$.", "Pick $g \\in S'$, $g \\not \\in \\mathfrak q'$ such that", "$S'' = S'_g$ is \\'etale over $S$. Then", "$(S, \\mathfrak q) \\to (S'', \\mathfrak q'S'')$ is a morphism", "of pairs. Now that triple $(S'', \\mathfrak q'S'', \\text{class of }T)$", "determines an element $a \\in R^h$ with the properties $P(a) = 0$,", "and $\\overline{a} = a_0$ as desired." ], "refs": [ "categories-definition-directed", "categories-definition-directed", "algebra-lemma-etale", "categories-definition-directed", "algebra-lemma-map-between-etale", "categories-definition-directed", "algebra-lemma-etale-at-prime" ], "ref_ids": [ 12363, 12363, 1231, 12363, 1236, 12363, 1233 ] } ], "ref_ids": [] }, { "id": 1295, "type": "theorem", "label": "algebra-lemma-strict-henselization", "categories": [ "algebra" ], "title": "algebra-lemma-strict-henselization", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring.", "Let $\\kappa \\subset \\kappa^{sep}$ be a separable algebraic closure.", "There exists a commutative diagram", "$$", "\\xymatrix{", "\\kappa \\ar[r] & \\kappa \\ar[r] & \\kappa^{sep} \\\\", "R \\ar[r] \\ar[u] & R^h \\ar[r] \\ar[u] & R^{sh} \\ar[u]", "}", "$$", "with the following properties", "\\begin{enumerate}", "\\item the map $R^h \\to R^{sh}$ is local", "\\item $R^{sh}$ is strictly henselian,", "\\item $R^{sh}$ is a filtered colimit of \\'etale $R$-algebras,", "\\item $\\mathfrak m R^{sh}$ is the", "maximal ideal of $R^{sh}$, and", "\\item $\\kappa^{sep} = R^{sh}/\\mathfrak m R^{sh}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is proved by exactly the same proof as used for", "Lemma \\ref{lemma-henselization}.", "The only difference is that, instead of pairs, one uses triples", "$(S, \\mathfrak q, \\alpha)$ where $R \\to S$ \\'etale,", "$\\mathfrak q$ is a prime of $S$ lying over $\\mathfrak m$, and", "$\\alpha : \\kappa(\\mathfrak q) \\to \\kappa^{sep}$ is an embedding", "of extensions of $\\kappa$." ], "refs": [ "algebra-lemma-henselization" ], "ref_ids": [ 1294 ] } ], "ref_ids": [] }, { "id": 1296, "type": "theorem", "label": "algebra-lemma-henselian-functorial-prepare", "categories": [ "algebra" ], "title": "algebra-lemma-henselian-functorial-prepare", "contents": [ "Let $R \\to S$ be a local map of local rings.", "Let $S \\to S^h$ be the henselization.", "Let $R \\to A$ be an \\'etale ring map and let $\\mathfrak q$", "be a prime of $A$ lying over $\\mathfrak m_R$", "such that $R/\\mathfrak m_R \\cong \\kappa(\\mathfrak q)$.", "Then there exists a unique morphism of rings", "$f : A \\to S^h$ fitting into the commutative diagram", "$$", "\\xymatrix{", "A \\ar[r]_f & S^h \\\\", "R \\ar[u] \\ar[r] & S \\ar[u]", "}", "$$", "such that $f^{-1}(\\mathfrak m_{S^h}) = \\mathfrak q$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-map-into-henselian}." ], "refs": [ "algebra-lemma-map-into-henselian" ], "ref_ids": [ 1284 ] } ], "ref_ids": [] }, { "id": 1297, "type": "theorem", "label": "algebra-lemma-henselian-functorial", "categories": [ "algebra" ], "title": "algebra-lemma-henselian-functorial", "contents": [ "Let $R \\to S$ be a local map of local rings.", "Let $R \\to R^h$ and $S \\to S^h$ be the henselizations.", "There exists a unique local ring map $R^h \\to S^h$ fitting", "into the commutative diagram", "$$", "\\xymatrix{", "R^h \\ar[r]_f & S^h \\\\", "R \\ar[u] \\ar[r] & S \\ar[u]", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-map-into-henselian-colimit}." ], "refs": [ "algebra-lemma-map-into-henselian-colimit" ], "ref_ids": [ 1291 ] } ], "ref_ids": [] }, { "id": 1298, "type": "theorem", "label": "algebra-lemma-henselization-different", "categories": [ "algebra" ], "title": "algebra-lemma-henselization-different", "contents": [ "Let $R$ be a ring.", "Let $\\mathfrak p \\subset R$ be a prime ideal.", "Consider the category of pairs $(S, \\mathfrak q)$ where", "$R \\to S$ is \\'etale and $\\mathfrak q$ is a prime lying over $\\mathfrak p$", "such that $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak q)$.", "This category is filtered and", "$$", "(R_{\\mathfrak p})^h = \\colim_{(S, \\mathfrak q)} S", "= \\colim_{(S, \\mathfrak q)} S_{\\mathfrak q}", "$$", "canonically." ], "refs": [], "proofs": [ { "contents": [ "A morphism of pairs $(S, \\mathfrak q) \\to (S', \\mathfrak q')$", "is given by an $R$-algebra map $\\varphi : S \\to S'$ such that", "$\\varphi^{-1}(\\mathfrak q') = \\mathfrak q$.", "Let us show that the category of pairs is filtered, see", "Categories, Definition \\ref{categories-definition-directed}.", "The category contains the pair $(R, \\mathfrak p)$ and hence is not empty,", "which proves part (1) of", "Categories, Definition \\ref{categories-definition-directed}.", "Suppose that $(S, \\mathfrak q)$ and $(S', \\mathfrak q')$ are two pairs.", "Note that $\\mathfrak q$, resp.\\ $\\mathfrak q'$ correspond to primes", "of the fibre rings $S \\otimes \\kappa(\\mathfrak p)$,", "resp.\\ $S' \\otimes \\kappa(\\mathfrak p)$ with residue fields", "$\\kappa(\\mathfrak p)$, hence they correspond to maximal ideals of", "$S \\otimes \\kappa(\\mathfrak p)$, resp.\\ $S' \\otimes \\kappa(\\mathfrak p)$.", "Set $S'' = S \\otimes_R S'$. By the above there exists a unique", "prime $\\mathfrak q'' \\subset S''$ lying over $\\mathfrak q$ and over", "$\\mathfrak q'$ whose residue field is $\\kappa(\\mathfrak p)$.", "The ring map $R \\to S''$ is \\'etale by", "Lemma \\ref{lemma-etale}.", "This proves part (2) of", "Categories, Definition \\ref{categories-definition-directed}.", "Next, suppose that", "$\\varphi, \\psi : (S, \\mathfrak q) \\to (S', \\mathfrak q')$", "are two morphisms of pairs. Then $\\varphi$, $\\psi$, and", "$S' \\otimes_R S' \\to S'$ are \\'etale ring maps by", "Lemma \\ref{lemma-map-between-etale}. Consider", "$$", "S'' = (S' \\otimes_{\\varphi, S, \\psi} S')", "\\otimes_{S' \\otimes_R S'} S'", "$$", "Arguing as above (base change of \\'etale maps is \\'etale, composition of", "\\'etale maps is \\'etale) we see that $S''$ is \\'etale over $R$. The fibre", "ring of $S''$ over $\\mathfrak p$ is", "$$", "F'' = (F' \\otimes_{\\varphi, F, \\psi} F')", "\\otimes_{F' \\otimes_{\\kappa(\\mathfrak p)} F'} F'", "$$", "where $F', F$ are the fibre rings of $S'$ and $S$. Since $\\varphi$ and", "$\\psi$ are morphisms of pairs the map $F' \\to \\kappa(\\mathfrak p)$", "corresponding to $\\mathfrak p'$ extends to a map $F'' \\to \\kappa(\\mathfrak p)$", "and in turn corresponds to a prime ideal $\\mathfrak q'' \\subset S''$", "whose residue field is $\\kappa(\\mathfrak p)$.", "The canonical map $S' \\to S''$ (using the right most factor for example)", "is a morphism of pairs $(S', \\mathfrak q') \\to (S'', \\mathfrak q'')$", "which equalizes $\\varphi$ and $\\psi$. This proves part (3) of", "Categories, Definition \\ref{categories-definition-directed}.", "Hence we conclude that the category is filtered.", "\\medskip\\noindent", "Recall that in the proof of", "Lemma \\ref{lemma-henselization}", "we constructed $(R_{\\mathfrak p})^h$ as the corresponding colimit", "but starting with $R_{\\mathfrak p}$ and its maximal ideal", "$\\mathfrak pR_{\\mathfrak p}$. Now, given any pair $(S, \\mathfrak q)$", "for $(R, \\mathfrak p)$ we obtain a pair", "$(S_{\\mathfrak p}, \\mathfrak qS_{\\mathfrak p})$ for", "$(R_{\\mathfrak p}, \\mathfrak pR_{\\mathfrak p})$.", "Moreover, in this situation", "$$", "S_{\\mathfrak p} = \\colim_{f \\in R, f \\not \\in \\mathfrak p} S_f.", "$$", "Hence in order to show the equalities", "of the lemma, it suffices to show that any pair $(S_{loc}, \\mathfrak q_{loc})$", "for $(R_{\\mathfrak p}, \\mathfrak pR_{\\mathfrak p})$ is of the form", "$(S_{\\mathfrak p}, \\mathfrak qS_{\\mathfrak p})$ for some pair", "$(S, \\mathfrak q)$ over $(R, \\mathfrak p)$ (some details omitted).", "This follows from", "Lemma \\ref{lemma-etale}." ], "refs": [ "categories-definition-directed", "categories-definition-directed", "algebra-lemma-etale", "categories-definition-directed", "algebra-lemma-map-between-etale", "categories-definition-directed", "algebra-lemma-henselization", "algebra-lemma-etale" ], "ref_ids": [ 12363, 12363, 1231, 12363, 1236, 12363, 1294, 1231 ] } ], "ref_ids": [] }, { "id": 1299, "type": "theorem", "label": "algebra-lemma-henselian-functorial-improve", "categories": [ "algebra" ], "title": "algebra-lemma-henselian-functorial-improve", "contents": [ "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime lying", "over $\\mathfrak p \\subset R$. Let $R \\to R^h$ and $S \\to S^h$ be the", "henselizations of $R_\\mathfrak p$ and $S_\\mathfrak q$. The local ring map", "$R^h \\to S^h$ of Lemma \\ref{lemma-henselian-functorial} identifies $S^h$", "with the henselization of $R^h \\otimes_R S$ at the unique prime", "lying over $\\mathfrak m^h$ and $\\mathfrak q$." ], "refs": [ "algebra-lemma-henselian-functorial" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-henselization-different} we see that $R^h$, resp.\\ $S^h$", "are filtered colimits of \\'etale $R$, resp.\\ $S$-algebras.", "Hence we see that $R^h \\otimes_R S$ is a filtered colimit of", "\\'etale $S$-algebras $A_i$ (Lemma \\ref{lemma-etale}). By", "Lemma \\ref{lemma-colimits-of-etale} we see that $S^h$ is a", "filtered colimit of \\'etale $R^h \\otimes_R S$-algebras.", "Since moreover $S^h$ is a henselian local ring with residue field", "equal to $\\kappa(\\mathfrak q)$, the statement follows from the uniqueness", "result of Lemma \\ref{lemma-uniqueness-henselian}." ], "refs": [ "algebra-lemma-henselization-different", "algebra-lemma-etale", "algebra-lemma-colimits-of-etale", "algebra-lemma-uniqueness-henselian" ], "ref_ids": [ 1298, 1231, 1290, 1292 ] } ], "ref_ids": [ 1297 ] }, { "id": 1300, "type": "theorem", "label": "algebra-lemma-quasi-finite-henselization", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-finite-henselization", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak p$ in $R$.", "Assume $R \\to S$ is quasi-finite at $\\mathfrak q$.", "The commutative diagram", "$$", "\\xymatrix{", "R_{\\mathfrak p}^h \\ar[r] & S_{\\mathfrak q}^h \\\\", "R_{\\mathfrak p} \\ar[u] \\ar[r] & S_{\\mathfrak q} \\ar[u]", "}", "$$", "of", "Lemma \\ref{lemma-henselian-functorial}", "identifies $S_{\\mathfrak q}^h$ with the localization of", "$R_{\\mathfrak p}^h \\otimes_{R_{\\mathfrak p}} S_{\\mathfrak q}$", "at the prime generated by $\\mathfrak q$." ], "refs": [ "algebra-lemma-henselian-functorial" ], "proofs": [ { "contents": [ "Note that $R_{\\mathfrak p}^h \\otimes_R S$ is quasi-finite over", "$R_{\\mathfrak p}^h$ at the prime ideal corresponding to $\\mathfrak q$, see", "Lemma \\ref{lemma-four-rings}. Hence the localization $S'$ of", "$R_{\\mathfrak p}^h \\otimes_{R_{\\mathfrak p}} S_{\\mathfrak q}$ is henselian, see", "Lemma \\ref{lemma-finite-over-henselian}. As a localization $S'$ is a filtered", "colimit of \\'etale", "$R_{\\mathfrak p}^h \\otimes_{R_{\\mathfrak p}} S_{\\mathfrak q}$-algebras.", "By Lemma \\ref{lemma-henselian-functorial-improve} we see that", "$S_\\mathfrak q^h$ is the henselization of", "$R_{\\mathfrak p}^h \\otimes_{R_{\\mathfrak p}} S_{\\mathfrak q}$.", "Thus $S' = S_\\mathfrak q^h$ by the uniqueness", "result of Lemma \\ref{lemma-uniqueness-henselian}." ], "refs": [ "algebra-lemma-four-rings", "algebra-lemma-finite-over-henselian", "algebra-lemma-henselian-functorial-improve", "algebra-lemma-uniqueness-henselian" ], "ref_ids": [ 1052, 1277, 1299, 1292 ] } ], "ref_ids": [ 1297 ] }, { "id": 1301, "type": "theorem", "label": "algebra-lemma-quotient-henselization", "categories": [ "algebra" ], "title": "algebra-lemma-quotient-henselization", "contents": [ "\\begin{slogan}", "Henselization is compatible with quotients.", "\\end{slogan}", "Let $R$ be a local ring with henselization $R^h$.", "Let $I \\subset \\mathfrak m_R$.", "Then $R^h/IR^h$ is the henselization of $R/I$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-quasi-finite-henselization}." ], "refs": [ "algebra-lemma-quasi-finite-henselization" ], "ref_ids": [ 1300 ] } ], "ref_ids": [] }, { "id": 1302, "type": "theorem", "label": "algebra-lemma-strictly-henselian-functorial-prepare", "categories": [ "algebra" ], "title": "algebra-lemma-strictly-henselian-functorial-prepare", "contents": [ "Let $\\varphi : R \\to S$ be a local map of local rings.", "Let $S/\\mathfrak m_S \\subset \\kappa^{sep}$ be a separable algebraic closure.", "Let $S \\to S^{sh}$ be the strict henselization of $S$", "with respect to $S/\\mathfrak m_S \\subset \\kappa^{sep}$.", "Let $R \\to A$ be an \\'etale ring map and let $\\mathfrak q$", "be a prime of $A$ lying over $\\mathfrak m_R$.", "Given any commutative diagram", "$$", "\\xymatrix{", "\\kappa(\\mathfrak q) \\ar[r]_{\\phi} & \\kappa^{sep} \\\\", "R/\\mathfrak m_R \\ar[r]^{\\varphi} \\ar[u] & S/\\mathfrak m_S \\ar[u]", "}", "$$", "there exists a unique morphism of rings", "$f : A \\to S^{sh}$ fitting into the commutative diagram", "$$", "\\xymatrix{", "A \\ar[r]_f & S^{sh} \\\\", "R \\ar[u] \\ar[r]^{\\varphi} & S \\ar[u]", "}", "$$", "such that $f^{-1}(\\mathfrak m_{S^h}) = \\mathfrak q$ and the induced", "map $\\kappa(\\mathfrak q) \\to \\kappa^{sep}$ is the given one." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-map-into-henselian}." ], "refs": [ "algebra-lemma-map-into-henselian" ], "ref_ids": [ 1284 ] } ], "ref_ids": [] }, { "id": 1303, "type": "theorem", "label": "algebra-lemma-strictly-henselian-functorial", "categories": [ "algebra" ], "title": "algebra-lemma-strictly-henselian-functorial", "contents": [ "Let $R \\to S$ be a local map of local rings.", "Choose separable algebraic closures", "$R/\\mathfrak m_R \\subset \\kappa_1^{sep}$", "and", "$S/\\mathfrak m_S \\subset \\kappa_2^{sep}$.", "Let $R \\to R^{sh}$ and $S \\to S^{sh}$ be the corresponding strict", "henselizations. Given any commutative diagram", "$$", "\\xymatrix{", "\\kappa_1^{sep} \\ar[r]_{\\phi} & \\kappa_2^{sep} \\\\", "R/\\mathfrak m_R \\ar[r]^{\\varphi} \\ar[u] & S/\\mathfrak m_S \\ar[u]", "}", "$$", "There exists a unique local ring map $R^{sh} \\to S^{sh}$ fitting", "into the commutative diagram", "$$", "\\xymatrix{", "R^{sh} \\ar[r]_f & S^{sh} \\\\", "R \\ar[u] \\ar[r] & S \\ar[u]", "}", "$$", "and inducing $\\phi$ on the residue fields of", "$R^{sh}$ and $S^{sh}$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-map-into-henselian-colimit}." ], "refs": [ "algebra-lemma-map-into-henselian-colimit" ], "ref_ids": [ 1291 ] } ], "ref_ids": [] }, { "id": 1304, "type": "theorem", "label": "algebra-lemma-strict-henselization-different", "categories": [ "algebra" ], "title": "algebra-lemma-strict-henselization-different", "contents": [ "Let $R$ be a ring.", "Let $\\mathfrak p \\subset R$ be a prime ideal.", "Let $\\kappa(\\mathfrak p) \\subset \\kappa^{sep}$ be a", "separable algebraic closure.", "Consider the category of triples $(S, \\mathfrak q, \\phi)$", "where $R \\to S$ is \\'etale, $\\mathfrak q$ is a prime lying over $\\mathfrak p$,", "and $\\phi : \\kappa(\\mathfrak q) \\to \\kappa^{sep}$ is a", "$\\kappa(\\mathfrak p)$-algebra map. This category is filtered and", "$$", "(R_{\\mathfrak p})^{sh} =", "\\colim_{(S, \\mathfrak q, \\phi)} S =", "\\colim_{(S, \\mathfrak q, \\phi)} S_{\\mathfrak q}", "$$", "canonically." ], "refs": [], "proofs": [ { "contents": [ "A morphism of triples $(S, \\mathfrak q, \\phi) \\to (S', \\mathfrak q', \\phi')$", "is given by an $R$-algebra map $\\varphi : S \\to S'$ such that", "$\\varphi^{-1}(\\mathfrak q') = \\mathfrak q$ and such that", "$\\phi' \\circ \\varphi = \\phi$.", "Let us show that the category of pairs is filtered, see", "Categories, Definition \\ref{categories-definition-directed}.", "The category contains the triple", "$(R, \\mathfrak p, \\kappa(\\mathfrak p) \\subset \\kappa^{sep})$", "and hence is not empty, which proves part (1) of", "Categories, Definition \\ref{categories-definition-directed}.", "Suppose that $(S, \\mathfrak q, \\phi)$ and $(S', \\mathfrak q', \\phi')$", "are two triples.", "Note that $\\mathfrak q$, resp.\\ $\\mathfrak q'$ correspond to primes", "of the fibre rings $S \\otimes \\kappa(\\mathfrak p)$,", "resp.\\ $S' \\otimes \\kappa(\\mathfrak p)$ with residue fields", "finite separable over $\\kappa(\\mathfrak p)$ and $\\phi$, resp.\\ $\\phi'$", "correspond to maps into $\\kappa^{sep}$. Hence this data corresponds to", "$\\kappa(\\mathfrak p)$-algebra maps", "$$", "\\phi : S \\otimes_R \\kappa(\\mathfrak p) \\longrightarrow \\kappa^{sep},", "\\quad", "\\phi' : S' \\otimes_R \\kappa(\\mathfrak p) \\longrightarrow \\kappa^{sep}.", "$$", "Set $S'' = S \\otimes_R S'$. Combining the maps the above we get a unique", "$\\kappa(\\mathfrak p)$-algebra map", "$$", "\\phi'' = \\phi \\otimes \\phi' :", "S'' \\otimes_R \\kappa(\\mathfrak p)", "\\longrightarrow", "\\kappa^{sep}", "$$", "whose kernel corresponds to a prime $\\mathfrak q'' \\subset S''$", "lying over $\\mathfrak q$ and over $\\mathfrak q'$, and whose residue field", "maps via $\\phi''$ to the compositum of", "$\\phi(\\kappa(\\mathfrak q))$ and $\\phi'(\\kappa(\\mathfrak q'))$ in", "$\\kappa^{sep}$. The ring map $R \\to S''$ is \\'etale by", "Lemma \\ref{lemma-etale}.", "Hence $(S'', \\mathfrak q'', \\phi'')$ is a triple dominating both", "$(S, \\mathfrak q, \\phi)$ and $(S', \\mathfrak q', \\phi')$.", "This proves part (2) of", "Categories, Definition \\ref{categories-definition-directed}.", "Next, suppose that", "$\\varphi, \\psi : (S, \\mathfrak q, \\phi) \\to (S', \\mathfrak q', \\phi')$", "are two morphisms of pairs. Then $\\varphi$, $\\psi$, and", "$S' \\otimes_R S' \\to S'$ are \\'etale ring maps by", "Lemma \\ref{lemma-map-between-etale}.", "Consider", "$$", "S'' = (S' \\otimes_{\\varphi, S, \\psi} S')", "\\otimes_{S' \\otimes_R S'} S'", "$$", "Arguing as above (base change of \\'etale maps is \\'etale, composition of", "\\'etale maps is \\'etale) we see that $S''$ is \\'etale over $R$. The fibre", "ring of $S''$ over $\\mathfrak p$ is", "$$", "F'' = (F' \\otimes_{\\varphi, F, \\psi} F')", "\\otimes_{F' \\otimes_{\\kappa(\\mathfrak p)} F'} F'", "$$", "where $F', F$ are the fibre rings of $S'$ and $S$. Since $\\varphi$ and", "$\\psi$ are morphisms of triples the map $\\phi' : F' \\to \\kappa^{sep}$", "extends to a map $\\phi'' : F'' \\to \\kappa^{sep}$", "which in turn corresponds to a prime ideal $\\mathfrak q'' \\subset S''$.", "The canonical map $S' \\to S''$ (using the right most factor for example)", "is a morphism of triples", "$(S', \\mathfrak q', \\phi') \\to (S'', \\mathfrak q'', \\phi'')$", "which equalizes $\\varphi$ and $\\psi$. This proves part (3) of", "Categories, Definition \\ref{categories-definition-directed}.", "Hence we conclude that the category is filtered.", "\\medskip\\noindent", "We still have to show that the colimit $R_{colim}$ of the system", "is equal to the strict henselization", "of $R_{\\mathfrak p}$ with respect to $\\kappa^{sep}$. To see this note that", "the system of triples $(S, \\mathfrak q, \\phi)$ contains as a subsystem", "the pairs $(S, \\mathfrak q)$ of", "Lemma \\ref{lemma-henselization-different}.", "Hence $R_{colim}$ contains $R_{\\mathfrak p}^h$ by the result of that lemma.", "Moreover, it is clear that $R_{\\mathfrak p}^h \\subset R_{colim}$", "is a directed colimit of \\'etale ring extensions.", "It follows that $R_{colim}$ is henselian by", "Lemmas \\ref{lemma-finite-over-henselian} and", "\\ref{lemma-colimit-henselian}.", "Finally, by", "Lemma \\ref{lemma-make-etale-map-prescribed-residue-field}", "we see that the residue field of $R_{colim}$ is equal to", "$\\kappa^{sep}$. Hence we conclude that $R_{colim}$ is strictly henselian", "and hence equals the strict henselization of $R_{\\mathfrak p}$ as desired.", "Some details omitted." ], "refs": [ "categories-definition-directed", "categories-definition-directed", "algebra-lemma-etale", "categories-definition-directed", "algebra-lemma-map-between-etale", "categories-definition-directed", "algebra-lemma-henselization-different", "algebra-lemma-finite-over-henselian", "algebra-lemma-colimit-henselian", "algebra-lemma-make-etale-map-prescribed-residue-field" ], "ref_ids": [ 12363, 12363, 1231, 12363, 1236, 12363, 1298, 1277, 1293, 1242 ] } ], "ref_ids": [] }, { "id": 1305, "type": "theorem", "label": "algebra-lemma-strictly-henselian-functorial-improve", "categories": [ "algebra" ], "title": "algebra-lemma-strictly-henselian-functorial-improve", "contents": [ "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime lying", "over $\\mathfrak p \\subset R$. Choose separable algebraic closures", "$\\kappa(\\mathfrak p) \\subset \\kappa_1^{sep}$", "and", "$\\kappa(\\mathfrak q) \\subset \\kappa_2^{sep}$.", "Let $R^{sh}$ and $S^{sh}$ be the corresponding strict", "henselizations of $R_\\mathfrak p$ and $S_\\mathfrak q$.", "Given any commutative diagram", "$$", "\\xymatrix{", "\\kappa_1^{sep} \\ar[r]_{\\phi} & \\kappa_2^{sep} \\\\", "\\kappa(\\mathfrak p) \\ar[r]^{\\varphi} \\ar[u] & \\kappa(\\mathfrak q) \\ar[u]", "}", "$$", "The local ring map $R^{sh} \\to S^{sh}$ of", "Lemma \\ref{lemma-strictly-henselian-functorial} identifies $S^{sh}$", "with the strict henselization of $R^{sh} \\otimes_R S$ at a prime", "lying over $\\mathfrak q$ and the maximal ideal", "$\\mathfrak m^{sh} \\subset R^{sh}$." ], "refs": [ "algebra-lemma-strictly-henselian-functorial" ], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Lemma \\ref{lemma-henselian-functorial-improve}", "except that it uses", "Lemma \\ref{lemma-strict-henselization-different}", "instead of", "Lemma \\ref{lemma-henselization-different}." ], "refs": [ "algebra-lemma-henselian-functorial-improve", "algebra-lemma-strict-henselization-different", "algebra-lemma-henselization-different" ], "ref_ids": [ 1299, 1304, 1298 ] } ], "ref_ids": [ 1303 ] }, { "id": 1306, "type": "theorem", "label": "algebra-lemma-quasi-finite-strict-henselization", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-finite-strict-henselization", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak p$ in $R$.", "Let $\\kappa(\\mathfrak q) \\subset \\kappa^{sep}$ be a separable", "algebraic closure. Assume $R \\to S$ is quasi-finite at $\\mathfrak q$.", "The commutative diagram", "$$", "\\xymatrix{", "R_{\\mathfrak p}^{sh} \\ar[r] & S_{\\mathfrak q}^{sh} \\\\", "R_{\\mathfrak p} \\ar[u] \\ar[r] & S_{\\mathfrak q} \\ar[u]", "}", "$$", "of", "Lemma \\ref{lemma-strictly-henselian-functorial}", "identifies $S_{\\mathfrak q}^{sh}$ with a localization of", "$R_{\\mathfrak p}^{sh} \\otimes_{R_{\\mathfrak p}} S_{\\mathfrak q}$." ], "refs": [ "algebra-lemma-strictly-henselian-functorial" ], "proofs": [ { "contents": [ "The residue field of $R_{\\mathfrak p}^{sh}$ is the separable", "algebraic closure of $\\kappa(\\mathfrak p)$ in $\\kappa^{sep}$.", "Note that $R_{\\mathfrak p}^{sh} \\otimes_R S$ is quasi-finite over", "$R_{\\mathfrak p}^{sh}$ at the prime ideal corresponding to $\\mathfrak q$, see", "Lemma \\ref{lemma-four-rings}. Hence the localization $S'$ of", "$R_{\\mathfrak p}^{sh} \\otimes_{R_{\\mathfrak p}} S_{\\mathfrak q}$", "is henselian, see", "Lemma \\ref{lemma-finite-over-henselian}.", "Note that the residue field of $S'$ is $\\kappa^{sep}$ since it", "contains both the separable algebraic closure of", "$\\kappa(\\mathfrak p)$ and $\\kappa(\\mathfrak q)$.", "Furthermore, as a localization $S'$ is a filtered colimit of \\'etale", "$R_{\\mathfrak p}^{sh} \\otimes_{R_{\\mathfrak p}} S_{\\mathfrak q}$-algebras.", "By Lemma \\ref{lemma-strictly-henselian-functorial-improve}", "we see that $S_{\\mathfrak q}^{sh}$ is a strict henselization of", "$R_{\\mathfrak p}^{sh} \\otimes_{R_{\\mathfrak p}} S_{\\mathfrak q}$.", "Thus $S' = S_\\mathfrak q^{sh}$ by the uniqueness", "result of Lemma \\ref{lemma-uniqueness-henselian}." ], "refs": [ "algebra-lemma-four-rings", "algebra-lemma-finite-over-henselian", "algebra-lemma-strictly-henselian-functorial-improve", "algebra-lemma-uniqueness-henselian" ], "ref_ids": [ 1052, 1277, 1305, 1292 ] } ], "ref_ids": [ 1303 ] }, { "id": 1307, "type": "theorem", "label": "algebra-lemma-quotient-strict-henselization", "categories": [ "algebra" ], "title": "algebra-lemma-quotient-strict-henselization", "contents": [ "Let $R$ be a local ring with strict henselization $R^{sh}$.", "Let $I \\subset \\mathfrak m_R$.", "Then $R^{sh}/IR^{sh}$ is a strict henselization of $R/I$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-quasi-finite-strict-henselization}." ], "refs": [ "algebra-lemma-quasi-finite-strict-henselization" ], "ref_ids": [ 1306 ] } ], "ref_ids": [] }, { "id": 1308, "type": "theorem", "label": "algebra-lemma-sh-from-h-map", "categories": [ "algebra" ], "title": "algebra-lemma-sh-from-h-map", "contents": [ "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime", "lying over $\\mathfrak p \\subset R$ such that", "$\\kappa(\\mathfrak p) \\to \\kappa(\\mathfrak q)$ is an isomorphism.", "Choose a separable algebraic closure $\\kappa^{sep}$ of", "$\\kappa(\\mathfrak p) = \\kappa(\\mathfrak q)$.", "Then", "$$", "(S_\\mathfrak q)^{sh} =", "(S_\\mathfrak q)^h \\otimes_{(R_\\mathfrak p)^h} (R_\\mathfrak p)^{sh}", "$$" ], "refs": [], "proofs": [ { "contents": [ "This follows from the alternative construction of the strict henselization", "of a local ring in Remark \\ref{remark-construct-sh-from-h} and the", "fact that the residue fields are equal. Some details omitted." ], "refs": [ "algebra-remark-construct-sh-from-h" ], "ref_ids": [ 1581 ] } ], "ref_ids": [] }, { "id": 1309, "type": "theorem", "label": "algebra-lemma-criterion-no-embedded-primes", "categories": [ "algebra" ], "title": "algebra-lemma-criterion-no-embedded-primes", "contents": [ "Let $R$ be a Noetherian ring.", "Let $M$ be a finite $R$-module.", "The following are equivalent:", "\\begin{enumerate}", "\\item $M$ has no embedded associated prime, and", "\\item $M$ has property $(S_1)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p$ be an embedded associated prime of $M$.", "Then there exists another associated prime $\\mathfrak q$ of $M$", "such that $\\mathfrak p \\supset \\mathfrak q$. In particular this", "implies that $\\dim(\\text{Supp}(M_{\\mathfrak p})) \\geq 1$ (since $\\mathfrak q$", "is in the support as well). On the other hand $\\mathfrak pR_{\\mathfrak p}$", "is associated to $M_{\\mathfrak p}$", "(Lemma \\ref{lemma-associated-primes-localize}) and hence", "$\\text{depth}(M_{\\mathfrak p}) = 0$", "(see Lemma \\ref{lemma-ideal-nonzerodivisor}).", "In other words $(S_1)$ does not hold.", "Conversely, if $(S_1)$ does not hold then there exists a prime", "$\\mathfrak p$ such that $\\dim(\\text{Supp}(M_{\\mathfrak p})) \\geq 1$", "and $\\text{depth}(M_{\\mathfrak p}) = 0$. Then we see", "(arguing backwards using the lemmas cited above) that $\\mathfrak p$", "is an embedded associated prime." ], "refs": [ "algebra-lemma-associated-primes-localize", "algebra-lemma-ideal-nonzerodivisor" ], "ref_ids": [ 709, 712 ] } ], "ref_ids": [] }, { "id": 1310, "type": "theorem", "label": "algebra-lemma-criterion-reduced", "categories": [ "algebra" ], "title": "algebra-lemma-criterion-reduced", "contents": [ "\\begin{slogan}", "Reduced equals R0 plus S1.", "\\end{slogan}", "Let $R$ be a Noetherian ring.", "The following are equivalent:", "\\begin{enumerate}", "\\item $R$ is reduced, and", "\\item $R$ has properties $(R_0)$ and $(S_1)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $R$ is reduced. Then $R_{\\mathfrak p}$ is a field for", "every minimal prime $\\mathfrak p$ of $R$, according to", "Lemma \\ref{lemma-minimal-prime-reduced-ring}. Hence we have $(R_0)$.", "Let $\\mathfrak p$ be a prime of height $\\geq 1$. Then $A = R_{\\mathfrak p}$", "is a reduced local ring of dimension $\\geq 1$. Hence its maximal", "ideal $\\mathfrak m$ is not an associated prime", "since this would mean there exists a $x \\in \\mathfrak m$", "with annihilator $\\mathfrak m$ so $x^2 = 0$. Hence the depth of", "$A = R_{\\mathfrak p}$ is at least one, by Lemma \\ref{lemma-ass-zero-divisors}.", "This shows that $(S_1)$ holds.", "\\medskip\\noindent", "Conversely, assume that $R$ satisfies $(R_0)$ and $(S_1)$.", "If $\\mathfrak p$ is a minimal prime of $R$, then", "$R_{\\mathfrak p}$ is a field by $(R_0)$, and hence is reduced.", "If $\\mathfrak p$ is not minimal, then we see that $R_{\\mathfrak p}$", "has depth $\\geq 1$ by $(S_1)$ and we conclude there exists an element", "$t \\in \\mathfrak pR_{\\mathfrak p}$ such that", "$R_{\\mathfrak p} \\to R_{\\mathfrak p}[1/t]$ is injective.", "This implies that $R_{\\mathfrak p}$ is a subring of localizations", "of $R$ at primes of smaller height. Thus by induction on the height we", "conclude that $R$ is reduced." ], "refs": [ "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-ass-zero-divisors" ], "ref_ids": [ 418, 704 ] } ], "ref_ids": [] }, { "id": 1311, "type": "theorem", "label": "algebra-lemma-criterion-normal", "categories": [ "algebra" ], "title": "algebra-lemma-criterion-normal", "contents": [ "\\begin{reference}", "\\cite[IV, Theorem 5.8.6]{EGA}", "\\end{reference}", "\\begin{slogan}", "Normal equals R1 plus S2.", "\\end{slogan}", "Let $R$ be a Noetherian ring.", "The following are equivalent:", "\\begin{enumerate}", "\\item $R$ is a normal ring, and", "\\item $R$ has properties $(R_1)$ and $(S_2)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1) $\\Rightarrow$ (2). Assume $R$ is normal, i.e., all", "localizations $R_{\\mathfrak p}$ at primes are normal domains.", "In particular we see that $R$ has $(R_0)$ and $(S_1)$ by", "Lemma \\ref{lemma-criterion-reduced}. Hence it suffices to show", "that a local Noetherian normal domain $R$ of dimension $d$ has", "depth $\\geq \\min(2, d)$ and is regular if $d = 1$. The assertion", "if $d = 1$ follows from Lemma \\ref{lemma-characterize-dvr}.", "\\medskip\\noindent", "Let $R$ be a local Noetherian normal domain with maximal ideal", "$\\mathfrak m$ and dimension $d \\geq 2$. Apply", "Lemma \\ref{lemma-hart-serre-loc-thm} to $R$.", "It is clear that $R$ does not fall into cases (1) or (2)", "of the lemma.", "Let $R \\to R'$ as in (4) of the lemma.", "Since $R$ is a domain we have $R \\subset R'$. Since $\\mathfrak m$", "is not an associated prime of $R'$ there exists an $x \\in \\mathfrak m$", "which is a nonzerodivisor on $R'$. Then $R_x = R'_x$ so", "$R$ and $R'$ are domains with the same fraction field. But", "finiteness of $R \\subset R'$ implies every element of $R'$ is integral", "over $R$ (Lemma \\ref{lemma-finite-is-integral})", "and we conclude that $R = R'$ as $R$ is normal.", "This means (4) does not happen. Thus we get the remaining possibility", "(3), i.e., $\\text{depth}(R) \\geq 2$ as desired.", "\\medskip\\noindent", "Proof of (2) $\\Rightarrow$ (1). Assume $R$ satisfies $(R_1)$ and $(S_2)$.", "By Lemma \\ref{lemma-criterion-reduced} we conclude that $R$ is", "reduced. Hence it suffices to show that if $R$ is a reduced local", "Noetherian ring of dimension $d$ satisfying $(S_2)$ and $(R_1)$", "then $R$ is a normal domain. If $d = 0$, the result is clear.", "If $d = 1$, then the result follows from Lemma \\ref{lemma-characterize-dvr}.", "\\medskip\\noindent", "Let $R$ be a reduced local Noetherian ring with maximal ideal", "$\\mathfrak m$ and dimension $d \\geq 2$ which satisfies $(R_1)$ and", "$(S_2)$. By Lemma \\ref{lemma-characterize-reduced-ring-normal}", "it suffices to show that $R$ is integrally closed in its", "total ring of fractions $Q(R)$. Pick $x \\in Q(R)$ which is integral", "over $R$. Then $R' = R[x]$ is a finite ring extension of $R$", "(Lemma \\ref{lemma-characterize-finite-in-terms-of-integral}).", "Because $\\dim(R_\\mathfrak p) < d$ for", "every nonmaximal prime $\\mathfrak p \\subset R$", "we have $R_\\mathfrak p = R'_\\mathfrak p$ by induction.", "Hence the support of $R'/R$ is $\\{\\mathfrak m\\}$.", "It follows that $R'/R$ is annihilated by a power of $\\mathfrak m$", "(Lemma \\ref{lemma-Noetherian-power-ideal-kills-module}).", "By Lemma \\ref{lemma-hart-serre-loc-thm} this", "contradicts the assumption that the depth of $R$ is $\\geq 2 = \\min(2, d)$", "and the proof is complete." ], "refs": [ "algebra-lemma-criterion-reduced", "algebra-lemma-characterize-dvr", "algebra-lemma-hart-serre-loc-thm", "algebra-lemma-finite-is-integral", "algebra-lemma-criterion-reduced", "algebra-lemma-characterize-dvr", "algebra-lemma-characterize-reduced-ring-normal", "algebra-lemma-characterize-finite-in-terms-of-integral", "algebra-lemma-Noetherian-power-ideal-kills-module", "algebra-lemma-hart-serre-loc-thm" ], "ref_ids": [ 1310, 1023, 1021, 482, 1310, 1023, 515, 484, 694, 1021 ] } ], "ref_ids": [] }, { "id": 1312, "type": "theorem", "label": "algebra-lemma-regular-normal", "categories": [ "algebra" ], "title": "algebra-lemma-regular-normal", "contents": [ "A regular ring is normal." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a regular ring. By", "Lemma \\ref{lemma-criterion-normal}", "it suffices to prove that $R$ is $(R_1)$ and $(S_2)$.", "As a regular local ring is Cohen-Macaulay, see", "Lemma \\ref{lemma-regular-ring-CM},", "it is clear that $R$ is $(S_2)$.", "Property $(R_1)$ is immediate." ], "refs": [ "algebra-lemma-criterion-normal", "algebra-lemma-regular-ring-CM" ], "ref_ids": [ 1311, 941 ] } ], "ref_ids": [] }, { "id": 1313, "type": "theorem", "label": "algebra-lemma-normal-domain-intersection-localizations-height-1", "categories": [ "algebra" ], "title": "algebra-lemma-normal-domain-intersection-localizations-height-1", "contents": [ "Let $R$ be a Noetherian normal domain with fraction field $K$. Then", "\\begin{enumerate}", "\\item for any nonzero $a \\in R$ the quotient $R/aR$ has no embedded primes,", "and all its associated primes have height $1$", "\\item", "$$", "R = \\bigcap\\nolimits_{\\text{height}(\\mathfrak p) = 1} R_{\\mathfrak p}", "$$", "\\item For any nonzero $x \\in K$ the quotient $R/(R \\cap xR)$", "has no embedded primes, and all its associates primes have height $1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-criterion-normal} we see that $R$ has $(S_2)$.", "Hence for any nonzero element $a \\in R$ we see that $R/aR$ has $(S_1)$", "(use Lemma \\ref{lemma-depth-in-ses} for example)", "Hence $R/aR$ has no embedded primes", "(Lemma \\ref{lemma-criterion-no-embedded-primes}).", "We conclude the associated primes of $R/aR$ are exactly", "the minimal primes $\\mathfrak p$ over $(a)$, which have height $1$", "as $a$ is not zero (Lemma \\ref{lemma-minimal-over-1}). This proves (1).", "\\medskip\\noindent", "Thus, given $b \\in R$ we have $b \\in aR$ if and only if", "$b \\in aR_{\\mathfrak p}$ for every minimal prime $\\mathfrak p$", "over $(a)$ (see Lemma \\ref{lemma-zero-at-ass-zero}).", "These primes all have height $1$ as seen above so", "$b/a \\in R$ if and only if $b/a \\in R_{\\mathfrak p}$ for all", "height 1 primes. Hence (2) holds.", "\\medskip\\noindent", "For (3) write $x = a/b$. Let $\\mathfrak p_1, \\ldots, \\mathfrak p_r$", "be the minimal primes over $(ab)$. These all have height 1 by the above.", "Then we see that", "$R \\cap xR = \\bigcap_{i = 1, \\ldots, r} (R \\cap xR_{\\mathfrak p_i})$", "by part (2) of the lemma. Hence $R/(R \\cap xR)$ is a submodule of", "$\\bigoplus R/(R \\cap xR_{\\mathfrak p_i})$.", "As $R_{\\mathfrak p_i}$ is a discrete valuation ring (by property $(R_1)$", "for the Noetherian normal domain $R$, see Lemma \\ref{lemma-criterion-normal})", "we have $xR_{\\mathfrak p_i} = \\mathfrak p_i^{e_i}R_{\\mathfrak p_i}$", "for some $e_i \\in \\mathbf{Z}$. Hence the direct sum is equal", "to $\\bigoplus_{e_i > 0} R/\\mathfrak p_i^{(e_i)}$, see", "Definition \\ref{definition-symbolic-power}.", "By Lemma \\ref{lemma-symbolic-power-associated}", "the only associated prime of the module", "$R/\\mathfrak p^{(n)}$ is $\\mathfrak p$. Hence the set of associate primes", "of $R/(R \\cap xR)$ is a subset of $\\{\\mathfrak p_i\\}$ and there are", "no inclusion relations among them. This proves (3)." ], "refs": [ "algebra-lemma-criterion-normal", "algebra-lemma-depth-in-ses", "algebra-lemma-criterion-no-embedded-primes", "algebra-lemma-minimal-over-1", "algebra-lemma-zero-at-ass-zero", "algebra-lemma-criterion-normal", "algebra-definition-symbolic-power", "algebra-lemma-symbolic-power-associated" ], "ref_ids": [ 1311, 773, 1309, 683, 713, 1311, 1482, 714 ] } ], "ref_ids": [] }, { "id": 1314, "type": "theorem", "label": "algebra-lemma-characterize-separable-algebraic-field-extensions", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-separable-algebraic-field-extensions", "contents": [ "Let $k \\subset K$ be a finitely generated field extension.", "The following are equivalent", "\\begin{enumerate}", "\\item $K$ is a finite separable field extension of $k$,", "\\item $\\Omega_{K/k} = 0$,", "\\item $K$ is formally unramified over $k$,", "\\item $K$ is unramified over $k$,", "\\item $K$ is formally \\'etale over $k$,", "\\item $K$ is \\'etale over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (2) and (3) is", "Lemma \\ref{lemma-characterize-formally-unramified}.", "By Lemma \\ref{lemma-etale-over-field}", "we see that (1) is equivalent to (6).", "Property (6) implies (5) and (4) which both in turn imply (3)", "(Lemmas \\ref{lemma-formally-etale-etale}, \\ref{lemma-unramified},", "and \\ref{lemma-formally-unramified-unramified}).", "Thus it suffices to show that (2) implies (1).", "Choose a finitely generated $k$-subalgebra $A \\subset K$", "such that $K$ is the fraction field of the domain $A$.", "Set $S = A \\setminus \\{0\\}$.", "Since $0 = \\Omega_{K/k} = S^{-1}\\Omega_{A/k}$", "(Lemma \\ref{lemma-differentials-localize})", "and since $\\Omega_{A/k}$ is finitely generated", "(Lemma \\ref{lemma-differentials-finitely-generated}),", "we can replace $A$ by a localization $A_f$ to reduce to the case", "that $\\Omega_{A/k} = 0$ (details omitted).", "Then $A$ is unramified over $k$, hence", "$K/k$ is finite separable for example by", "Lemma \\ref{lemma-unramified-at-prime} applied with $\\mathfrak q = (0)$." ], "refs": [ "algebra-lemma-characterize-formally-unramified", "algebra-lemma-etale-over-field", "algebra-lemma-formally-etale-etale", "algebra-lemma-unramified", "algebra-lemma-formally-unramified-unramified", "algebra-lemma-differentials-localize", "algebra-lemma-differentials-finitely-generated", "algebra-lemma-unramified-at-prime" ], "ref_ids": [ 1254, 1232, 1262, 1266, 1265, 1134, 1142, 1268 ] } ], "ref_ids": [] }, { "id": 1315, "type": "theorem", "label": "algebra-lemma-derivative-zero-pth-power", "categories": [ "algebra" ], "title": "algebra-lemma-derivative-zero-pth-power", "contents": [ "Let $k$ be a perfect field of characteristic $p > 0$.", "Let $K/k$ be an extension.", "Let $a \\in K$. Then $\\text{d}a = 0$ in $\\Omega_{K/k}$", "if and only if $a$ is a $p$th power." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-colimit-differentials} we see that there exists a subfield", "$k \\subset L \\subset K$ such that $k \\subset L$", "is a finitely generated field extension and such that", "$\\text{d}a$ is zero in $\\Omega_{L/k}$.", "Hence we may assume that $K$ is a finitely generated field extension", "of $k$.", "\\medskip\\noindent", "Choose a transcendence basis $x_1, \\ldots, x_r \\in K$", "such that $K$ is finite separable over $k(x_1, \\ldots, x_r)$.", "This is possible by the definitions, see", "Definitions \\ref{definition-perfect} and", "\\ref{definition-separable-field-extension}.", "We remark that the result holds for the purely transcendental", "subfield $k(x_1, \\ldots, x_r) \\subset K$.", "Namely,", "$$", "\\Omega_{k(x_1, \\ldots, x_r)/k} =", "\\bigoplus\\nolimits_{i = 1}^r k(x_1, \\ldots, x_r) \\text{d}x_i", "$$", "and any rational function all of whose partial derivatives are zero", "is a $p$th power. Moreover, we also have", "$$", "\\Omega_{K/k} =", "\\bigoplus\\nolimits_{i = 1}^r K\\text{d}x_i", "$$", "since $k(x_1, \\ldots, x_r) \\subset K$ is finite separable", "(computation omitted). Suppose $a \\in K$ is an element such that", "$\\text{d}a = 0$ in the module of differentials. By our choice of $x_i$ we", "see that the minimal polynomial $P(T) \\in k(x_1, \\ldots, x_r)[T]$", "of $a$ is separable. Write", "$$", "P(T) = T^d + \\sum\\nolimits_{i = 1}^d a_i T^{d - i}", "$$", "and hence", "$$", "0 = \\text{d}P(a) = \\sum\\nolimits_{i = 1}^d a^{d - i}\\text{d}a_i", "$$", "in $\\Omega_{K/k}$. By the description of", "$\\Omega_{K/k}$ above and the fact that $P$ was the minimal", "polynomial of $a$, we see that this implies $\\text{d}a_i = 0$.", "Hence $a_i = b_i^p$ for each $i$. Therefore by", "Fields, Lemma \\ref{fields-lemma-pth-root}", "we see that $a$ is a $p$th power." ], "refs": [ "algebra-lemma-colimit-differentials", "algebra-definition-perfect", "algebra-definition-separable-field-extension", "fields-lemma-pth-root" ], "ref_ids": [ 1130, 1462, 1460, 4523 ] } ], "ref_ids": [] }, { "id": 1316, "type": "theorem", "label": "algebra-lemma-size-extension-pth-roots", "categories": [ "algebra" ], "title": "algebra-lemma-size-extension-pth-roots", "contents": [ "Let $k$ be a field of characteristic $p > 0$.", "Let $a_1, \\ldots, a_n \\in k$ be elements such that", "$\\text{d}a_1, \\ldots, \\text{d}a_n$ are linearly independent in", "$\\Omega_{k/\\mathbf{F}_p}$. Then the field extension", "$k(a_1^{1/p}, \\ldots, a_n^{1/p})$ has degree $p^n$ over $k$." ], "refs": [], "proofs": [ { "contents": [ "By induction on $n$. If $n = 1$ the result is", "Lemma \\ref{lemma-derivative-zero-pth-power}.", "For the induction step, suppose that $k(a_1^{1/p}, \\ldots, a_{n - 1}^{1/p})$", "has degree $p^{n - 1}$ over $k$. We have to show that $a_n$ does not", "map to a $p$th power in $k(a_1^{1/p}, \\ldots, a_{n - 1}^{1/p})$.", "If it does then we can write", "\\begin{align*}", "a_n & =", "\\left(\\sum\\nolimits_{I = (i_1, \\ldots, i_{n - 1}),\\ 0 \\leq i_j \\leq p - 1}", "\\lambda_I a_1^{i_1/p} \\ldots a_{n - 1}^{i_{n - 1}/p}\\right)^p \\\\", "& = \\sum\\nolimits_{I = (i_1, \\ldots, i_{n - 1}),\\ 0 \\leq i_j \\leq p - 1}", "\\lambda_I^p a_1^{i_1} \\ldots a_{n - 1}^{i_{n - 1}}", "\\end{align*}", "Applying $\\text{d}$ we see that $\\text{d}a_n$ is linearly dependent on", "$\\text{d}a_i$, $i < n$. This is a contradiction." ], "refs": [ "algebra-lemma-derivative-zero-pth-power" ], "ref_ids": [ 1315 ] } ], "ref_ids": [] }, { "id": 1317, "type": "theorem", "label": "algebra-lemma-separable-differentials", "categories": [ "algebra" ], "title": "algebra-lemma-separable-differentials", "contents": [ "Let $k$ be a field of characteristic $p > 0$.", "The following are equivalent:", "\\begin{enumerate}", "\\item the field extension $K/k$ is separable", "(see Definition \\ref{definition-separable-field-extension}), and", "\\item the map", "$K \\otimes_k \\Omega_{k/\\mathbf{F}_p} \\to \\Omega_{K/\\mathbf{F}_p}$", "is injective.", "\\end{enumerate}" ], "refs": [ "algebra-definition-separable-field-extension" ], "proofs": [ { "contents": [ "Write $K$ as a directed colimit $K = \\colim_i K_i$ of finitely generated", "field extensions $k \\subset K_i$. By definition $K$ is separable if and only", "if each $K_i$ is separable over $k$, and by", "Lemma \\ref{lemma-colimit-differentials} we see that", "$K \\otimes_k \\Omega_{k/\\mathbf{F}_p} \\to \\Omega_{K/\\mathbf{F}_p}$", "is injective if and only if each", "$K_i \\otimes_k \\Omega_{k/\\mathbf{F}_p} \\to \\Omega_{K_i/\\mathbf{F}_p}$", "is injective. Hence we may assume that $K/k$ is a finitely generated field", "extension.", "\\medskip\\noindent", "Assume $k \\subset K$ is a finitely generated field extension which is", "separable. Choose $x_1, \\ldots, x_{r + 1} \\in K$ as in", "Lemma \\ref{lemma-generating-finitely-generated-separable-field-extensions}.", "In this case there exists an irreducible polynomial", "$G(X_1, \\ldots, X_{r + 1}) \\in k[X_1, \\ldots, X_{r + 1}]$", "such that $G(x_1, \\ldots, x_{r + 1}) = 0$ and such that", "$\\partial G/\\partial X_{r + 1}$ is not identically zero.", "Moreover $K$ is the field of fractions of the domain.", "$S = K[X_1, \\ldots, X_{r + 1}]/(G)$.", "Write", "$$", "G = \\sum a_I X^I, \\quad X^I = X_1^{i_1}\\ldots X_{r + 1}^{i_{r + 1}}.", "$$", "Using the presentation of $S$ above we see that", "$$", "\\Omega_{S/\\mathbf{F}_p}", "=", "\\frac{", "S \\otimes_k \\Omega_k \\oplus", "\\bigoplus\\nolimits_{i = 1, \\ldots, r + 1} S\\text{d}X_i", "}{", "\\langle", "\\sum X^I \\text{d}a_I + \\sum \\partial G/\\partial X_i \\text{d}X_i", "\\rangle", "}", "$$", "Since $\\Omega_{K/\\mathbf{F}_p}$ is the localization", "of the $S$-module $\\Omega_{S/\\mathbf{F}_p}$ (see", "Lemma \\ref{lemma-differentials-localize}) we conclude", "that", "$$", "\\Omega_{K/\\mathbf{F}_p}", "=", "\\frac{", "K \\otimes_k \\Omega_k \\oplus", "\\bigoplus\\nolimits_{i = 1, \\ldots, r + 1} K\\text{d}X_i", "}{", "\\langle", "\\sum X^I \\text{d}a_I + \\sum \\partial G/\\partial X_i \\text{d}X_i", "\\rangle", "}", "$$", "Now, since the polynomial $\\partial G/\\partial X_{r + 1}$ is not identically", "zero we conclude that the map", "$K \\otimes_k \\Omega_{k/\\mathbf{F}_p} \\to \\Omega_{S/\\mathbf{F}_p}$", "is injective as desired.", "\\medskip\\noindent", "Assume $k \\subset K$ is a finitely generated field extension", "and that", "$K \\otimes_k \\Omega_{k/\\mathbf{F}_p} \\to \\Omega_{K/\\mathbf{F}_p}$", "is injective.", "(This part of the proof is the same as the argument proving", "Lemma \\ref{lemma-characterize-separable-field-extensions}.)", "Let $x_1, \\ldots, x_r$ be a transcendence basis of $K$ over $k$ such", "that the degree of inseparability of the finite extension", "$k(x_1, \\ldots, x_r) \\subset K$ is minimal.", "If $K$ is separable over $k(x_1, \\ldots, x_r)$ then we win.", "Assume this is not the case to get a contradiction.", "Then there exists an element $\\alpha \\in K$ which is not", "separable over $k(x_1, \\ldots, x_r)$. Let $P(T) \\in k(x_1, \\ldots, x_r)[T]$", "be its minimal polynomial. Because $\\alpha$ is not separable", "actually $P$ is a polynomial in $T^p$. Clear denominators", "to get an irreducible polynomial", "$$", "G(X_1, \\ldots, X_r, T) = \\sum a_{I, i} X^I T^i \\in k[X_1, \\ldots, X_r, T]", "$$", "such that $G(x_1, \\ldots, x_r, \\alpha) = 0$ in $L$.", "Note that this means $k[X_1, \\ldots, X_r, T]/(G) \\subset L$.", "We may assume that for some pair $(I_0, i_0)$ the coefficient", "$a_{I_0, i_0} = 1$.", "We claim that $\\text{d}G/\\text{d}X_i$ is not identically zero", "for at least one $i$. Namely, if this is not the case, then", "$G$ is actually a polynomial in $X_1^p, \\ldots, X_r^p, T^p$.", "Then this means that", "$$", "\\sum\\nolimits_{(I, i) \\not = (I_0, i_0)} x^I\\alpha^i \\text{d}a_{I, i}", "$$", "is zero in $\\Omega_{K/\\mathbf{F}_p}$. Note that there is no", "$k$-linear relation among the elements", "$$", "\\{x^I\\alpha^i \\mid a_{I, i} \\not = 0 \\text{ and } (I, i) \\not = (I_0, i_0)\\}", "$$", "of $K$. Hence the assumption", "that $K \\otimes_k \\Omega_{k/\\mathbf{F}_p} \\to \\Omega_{K/\\mathbf{F}_p}$", "is injective this implies that $\\text{d}a_{I, i} = 0$", "in $\\Omega_{k/\\mathbf{F}_p}$ for all $(I, i)$.", "By Lemma \\ref{lemma-derivative-zero-pth-power}", "we see that each $a_{I, i}$ is a $p$th power, which", "implies that $G$ is a $p$th power contradicting the irreducibility of", "$G$. Thus,", "after renumbering, we may assume that $\\text{d}G/\\text{d}X_1$ is not zero.", "Then we see that $x_1$ is separably algebraic over", "$k(x_2, \\ldots, x_r, \\alpha)$, and that $x_2, \\ldots, x_r, \\alpha$", "is a transcendence basis of $L$ over $k$. This means that", "the degree of inseparability of the finite extension", "$k(x_2, \\ldots, x_r, \\alpha) \\subset L$ is less than the", "degree of inseparability of the finite extension", "$k(x_1, \\ldots, x_r) \\subset L$, which is a contradiction." ], "refs": [ "algebra-lemma-colimit-differentials", "algebra-lemma-generating-finitely-generated-separable-field-extensions", "algebra-lemma-differentials-localize", "algebra-lemma-characterize-separable-field-extensions", "algebra-lemma-derivative-zero-pth-power" ], "ref_ids": [ 1130, 559, 1134, 569, 1315 ] } ], "ref_ids": [ 1460 ] }, { "id": 1318, "type": "theorem", "label": "algebra-lemma-formally-smooth-implies-separable", "categories": [ "algebra" ], "title": "algebra-lemma-formally-smooth-implies-separable", "contents": [ "Let $k \\subset K$ be an extension of fields.", "If $K$ is formally smooth over $k$, then $K$ is", "a separable extension of $k$." ], "refs": [], "proofs": [ { "contents": [ "Assume $K$ is formally smooth over $k$.", "By Lemma \\ref{lemma-ses-formally-smooth} we see that", "$K \\otimes_k \\Omega_{k/\\mathbf{Z}} \\to \\Omega_{K/\\mathbf{Z}}$", "is injective. Hence $K$ is separable over $k$ by", "Lemma \\ref{lemma-separable-differentials}." ], "refs": [ "algebra-lemma-ses-formally-smooth", "algebra-lemma-separable-differentials" ], "ref_ids": [ 1209, 1317 ] } ], "ref_ids": [] }, { "id": 1319, "type": "theorem", "label": "algebra-lemma-characterize-formally-smooth-field-extension", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-formally-smooth-field-extension", "contents": [ "Let $k \\subset K$ be an extension of fields.", "Then $K$ is formally smooth over $k$ if and only if", "$H_1(L_{K/k}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Proposition \\ref{proposition-characterize-formally-smooth}", "and the fact that a vector spaces is free (hence projective)." ], "refs": [ "algebra-proposition-characterize-formally-smooth" ], "ref_ids": [ 1425 ] } ], "ref_ids": [] }, { "id": 1320, "type": "theorem", "label": "algebra-lemma-formally-smooth-extensions-easy", "categories": [ "algebra" ], "title": "algebra-lemma-formally-smooth-extensions-easy", "contents": [ "Let $k \\subset K$ be an extension of fields.", "\\begin{enumerate}", "\\item If $K$ is purely transcendental over $k$, then", "$K$ is formally smooth over $k$.", "\\item If $K$ is separable algebraic over $k$, then $K$ is", "formally smooth over $k$.", "\\item If $K$ is separable over $k$, then $K$ is formally smooth", "over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For (1) write $K = k(x_j; j \\in J)$. Suppose that", "$A$ is a $k$-algebra, and $I \\subset A$ is an ideal of", "square zero. Let $\\varphi : K \\to A/I$ be a $k$-algebra map.", "Let $a_j \\in A$ be an element such that $a_j \\mod I = \\varphi(x_j)$.", "Then it is easy to see that there is a unique $k$-algebra", "map $K \\to A$ which maps $x_j$ to $a_j$ and which reduces", "to $\\varphi$ mod $I$. Hence $k \\subset K$ is formally smooth.", "\\medskip\\noindent", "In case (2) we see that $k \\subset K$ is a colimit of", "\\'etale ring extensions. An \\'etale ring map is formally \\'etale", "(Lemma \\ref{lemma-formally-etale-etale}). Hence this case follows from", "Lemma \\ref{lemma-colimit-formally-etale} and the trivial observation", "that a formally \\'etale ring map is formally smooth.", "\\medskip\\noindent", "In case (3), write $K = \\colim K_i$ as the filtered colimit of its", "finitely generated sub $k$-extensions. By", "Definition \\ref{definition-separable-field-extension}", "each $K_i$ is separable algebraic over a purely transcendental", "extension of $k$. Hence $K_i/k$ is formally smooth by cases (1) and (2) and", "Lemma \\ref{lemma-compose-formally-smooth}. Thus", "$H_1(L_{K_i/k}) = 0$ by", "Lemma \\ref{lemma-characterize-formally-smooth-field-extension}.", "Hence $H_1(L_{K/k}) = 0$ by Lemma \\ref{lemma-colimits-NL}.", "Hence $K/k$ is formally smooth by", "Lemma \\ref{lemma-characterize-formally-smooth-field-extension} again." ], "refs": [ "algebra-lemma-formally-etale-etale", "algebra-lemma-colimit-formally-etale", "algebra-definition-separable-field-extension", "algebra-lemma-compose-formally-smooth", "algebra-lemma-characterize-formally-smooth-field-extension", "algebra-lemma-colimits-NL", "algebra-lemma-characterize-formally-smooth-field-extension" ], "ref_ids": [ 1262, 1263, 1460, 1205, 1319, 1157, 1319 ] } ], "ref_ids": [] }, { "id": 1321, "type": "theorem", "label": "algebra-lemma-fields-are-formally-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-fields-are-formally-smooth", "contents": [ "\\begin{slogan}", "Formally smooth equals separable for field extensions.", "\\end{slogan}", "Let $k$ be a field.", "\\begin{enumerate}", "\\item If the characteristic of $k$ is zero, then any extension field", "of $k$ is formally smooth over $k$.", "\\item If the characteristic of $k$ is $p > 0$, then $k \\subset K$ is", "formally smooth if and only if it is a separable field extension.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-formally-smooth-implies-separable} and", "\\ref{lemma-formally-smooth-extensions-easy}." ], "refs": [ "algebra-lemma-formally-smooth-implies-separable", "algebra-lemma-formally-smooth-extensions-easy" ], "ref_ids": [ 1318, 1320 ] } ], "ref_ids": [] }, { "id": 1322, "type": "theorem", "label": "algebra-lemma-localization-smooth-separable", "categories": [ "algebra" ], "title": "algebra-lemma-localization-smooth-separable", "contents": [ "Let $k \\subset K$ be a finitely generated field extension.", "Then $K$ is separable over $k$ if and only if $K$ is", "the localization of a smooth $k$-algebra." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite type $k$-algebra $R$ which is a domain whose", "fraction field is $K$. Lemma \\ref{lemma-smooth-at-generic-point}", "says that $k \\to R$ is smooth", "at $(0)$ if and only if $K/k$ is separable.", "This proves the lemma." ], "refs": [ "algebra-lemma-smooth-at-generic-point" ], "ref_ids": [ 1228 ] } ], "ref_ids": [] }, { "id": 1323, "type": "theorem", "label": "algebra-lemma-colimit-syntomic", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-syntomic", "contents": [ "Let $k \\subset K$ be a field extension.", "Then $K$ is a filtered colimit of global complete intersection", "algebras over $k$. If $K/k$ is separable, then $K$ is a filtered", "colimit of smooth algebras over $k$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $E \\subset K$ is a finite subset. It suffices to show that", "there exists a $k$ subalgebra $A \\subset K$ which contains $E$", "and which is a global complete intersection (resp.\\ smooth) over $k$.", "The separable/smooth case follows from", "Lemma \\ref{lemma-localization-smooth-separable}.", "In general let $L \\subset K$ be the subfield generated by $E$.", "Pick a transcendence basis $x_1, \\ldots, x_d \\in L$ over $k$.", "The extension $k(x_1, \\ldots, x_d) \\subset L$ is finite.", "Say $L = k(x_1, \\ldots, x_d)[y_1, \\ldots, y_r]$.", "Pick inductively polynomials $P_i \\in k(x_1, \\ldots, x_d)[Y_1, \\ldots, Y_r]$", "such that $P_i = P_i(Y_1, \\ldots, Y_i)$ is monic in $Y_i$ over", "$k(x_1, \\ldots, x_d)[Y_1, \\ldots, Y_{i - 1}]$ and maps to the", "minimum polynomial of $y_i$ in", "$k(x_1, \\ldots, x_d)[y_1, \\ldots, y_{i - 1}][Y_i]$.", "Then it is clear that $P_1, \\ldots, P_r$ is a regular sequence", "in $k(x_1, \\ldots, x_r)[Y_1, \\ldots, Y_r]$ and that", "$L = k(x_1, \\ldots, x_r)[Y_1, \\ldots, Y_r]/(P_1, \\ldots, P_r)$.", "If $h \\in k[x_1, \\ldots, x_d]$ is a polynomial such that", "$P_i \\in k[x_1, \\ldots, x_d, 1/h, Y_1, \\ldots, Y_r]$, then", "we see that $P_1, \\ldots, P_r$ is a regular sequence in", "$k[x_1, \\ldots, x_d, 1/h, Y_1, \\ldots, Y_r]$ and", "$A = k[x_1, \\ldots, x_d, 1/h, Y_1, \\ldots, Y_r]/(P_1, \\ldots, P_r)$", "is a global complete intersection. After adjusting our choice of $h$", "we may assume $E \\subset A$ and we win." ], "refs": [ "algebra-lemma-localization-smooth-separable" ], "ref_ids": [ 1322 ] } ], "ref_ids": [] }, { "id": 1324, "type": "theorem", "label": "algebra-lemma-flat-local-given-residue-field", "categories": [ "algebra" ], "title": "algebra-lemma-flat-local-given-residue-field", "contents": [ "Let $(R, \\mathfrak m, k)$ be a local ring. Let $k \\subset K$ be a field", "extension. There exists a local ring $(R', \\mathfrak m', k')$, a flat local", "ring map $R \\to R'$ such that $\\mathfrak m' = \\mathfrak mR'$ and such that", "$k \\subset k'$ is isomorphic to $k \\subset K$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $k \\subset k' = k(\\alpha)$ is a monogenic extension of fields.", "Then $k'$ is the residue field of a flat local extension $R \\subset R'$", "as in the lemma. Namely, if $\\alpha$ is transcendental over $k$, then we let", "$R'$ be the localization of $R[x]$ at the prime $\\mathfrak mR[x]$.", "If $\\alpha$ is algebraic with minimal polynomial", "$T^d + \\sum \\overline{\\lambda}_iT^{d - i}$, then we let", "$R' = R[T]/(T^d + \\sum \\lambda_i T^{d - i})$.", "\\medskip\\noindent", "Consider the collection of triples $(k', R \\to R', \\phi)$, where", "$k \\subset k' \\subset K$ is a subfield,", "$R \\to R'$ is a local ring map as in the lemma, and", "$\\phi : R' \\to k'$ induces an isomorphism $R'/\\mathfrak mR' \\cong k'$", "of $k$-extensions. These form a ``big'' category $\\mathcal{C}$ with morphisms", "$(k_1, R_1, \\phi_1) \\to (k_2, R_2, \\phi_2)$", "given by ring maps $\\psi : R_1 \\to R_2$ such that", "$$", "\\xymatrix{", "R_1 \\ar[d]_\\psi \\ar[r]_{\\phi_1} & k_1 \\ar[r] & K \\ar@{=}[d] \\\\", "R_2 \\ar[r]^{\\phi_2} & k_2 \\ar[r] & K", "}", "$$", "commutes. This implies that $k_1 \\subset k_2$.", "\\medskip\\noindent", "Suppose that $I$ is a directed set, and", "$((R_i, k_i, \\phi_i), \\psi_{ii'})$ is a system over $I$, see", "Categories, Section \\ref{categories-section-posets-limits}.", "In this case we can consider", "$$", "R' = \\colim_{i \\in I} R_i", "$$", "This is a local ring with maximal ideal $\\mathfrak mR'$, and", "residue field $k' = \\bigcup_{i \\in I} k_i$. Moreover, the ring", "map $R \\to R'$ is flat as it is a colimit of flat maps (and tensor", "products commute with directed colimits).", "Hence we see that $(R', k', \\phi')$ is an ``upper bound'' for the system.", "\\medskip\\noindent", "An almost trivial application of Zorn's Lemma would finish the proof", "if $\\mathcal{C}$ was a set, but it isn't.", "(Actually, you can make this work by finding a reasonable bound on the", "cardinals of the local rings occurring.)", "To get around this problem we choose a well ordering on $K$.", "For $x \\in K$ we let $K(x)$ be the subfield of $K$ generated", "by all elements of $K$ which are $\\leq x$.", "By transfinite induction on $x \\in K$ we will produce ring maps", "$R \\subset R(x)$ as in the lemma with residue field extension", "$k \\subset K(x)$. Moreover, by construction we will have that", "$R(x)$ will contain $R(y)$ for all $y \\leq x$.", "Namely, if $x$ has a predecessor $x'$, then $K(x) = K(x')[x]$", "and hence we can let $R(x') \\subset R(x)$ be the local ring extension", "constructed in the first paragraph of the proof. If $x$ does not", "have a predecessor, then we first set", "$R'(x) = \\colim_{x' < x} R(x')$ as in the third paragraph", "of the proof. The residue field of $R'(x)$ is $K'(x) = \\bigcup_{x' < x} K(x')$.", "Since $K(x) = K'(x)[x]$ we see that we can use the construction of the", "first paragraph of the proof to produce $R'(x) \\subset R(x)$.", "This finishes the proof of the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1325, "type": "theorem", "label": "algebra-lemma-colimit-finite-etale-given-residue-field", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-finite-etale-given-residue-field", "contents": [ "Let $(R, \\mathfrak m, k)$ be a local ring. If $k \\subset K$ is a", "separable algebraic extension, then there exists a directed set $I$ and", "a system of finite \\'etale extensions $R \\subset R_i$, $i \\in I$", "of local rings such that $R' = \\colim R_i$ has residue field", "$K$ (as extension of $k$)." ], "refs": [], "proofs": [ { "contents": [ "Let $R \\subset R'$ be the extension constructed in the proof of", "Lemma \\ref{lemma-flat-local-given-residue-field}. By construction", "$R' = \\colim_{\\alpha \\in A} R_\\alpha$ where $A$ is a well-ordered", "set and the transition maps $R_\\alpha \\to R_{\\alpha + 1}$", "are finite \\'etale and $R_\\alpha = \\colim_{\\beta < \\alpha} R_\\beta$", "if $\\alpha$ is not a successor. We will prove the result by transfinite", "induction.", "\\medskip\\noindent", "Suppose the result holds for $R_\\alpha$, i.e., $R_\\alpha = \\colim R_i$", "with $R_i$ finite \\'etale over $R$. Since", "$R_\\alpha \\to R_{\\alpha + 1}$ is finite \\'etale", "there exists an $i$ and a finite \\'etale extension $R_i \\to R_{i, 1}$", "such that $R_{\\alpha + 1} = R_\\alpha \\otimes_{R_i} R_{i, 1}$.", "Thus $R_{\\alpha + 1} = \\colim_{i' \\geq i} R_{i'} \\otimes_{R_i} R_{i, 1}$", "and the result holds for $\\alpha + 1$. Suppose $\\alpha$ is not a successor", "and the result holds for $R_\\beta$ for all $\\beta < \\alpha$.", "Since every finite subset $E \\subset R_\\alpha$ is contained in $R_\\beta$", "for some $\\beta < \\alpha$ and we see that $E$ is contained in a finite \\'etale", "subextension by assumption. Thus the result holds for $R_\\alpha$." ], "refs": [ "algebra-lemma-flat-local-given-residue-field" ], "ref_ids": [ 1324 ] } ], "ref_ids": [] }, { "id": 1326, "type": "theorem", "label": "algebra-lemma-finite-free-given-residue-field-extension", "categories": [ "algebra" ], "title": "algebra-lemma-finite-free-given-residue-field-extension", "contents": [ "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime and", "let $\\kappa(\\mathfrak p) \\subset L$ be a finite extension of fields.", "Then there exists a finite free ring map $R \\to S$ such that", "$\\mathfrak q = \\mathfrak pS$ is prime and", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$ is isomorphic to the given", "extension $\\kappa(\\mathfrak p) \\subset L$." ], "refs": [], "proofs": [ { "contents": [ "By induction of the degree of $\\kappa(\\mathfrak p) \\subset L$.", "If the degree is $1$, then we take $R = S$.", "In general, if there exists a sub extension", "$\\kappa(\\mathfrak p) \\subset L' \\subset L$ then we win by induction", "on the degree (by first constructing $R \\subset S'$ corresponding", "to $L'/\\kappa(\\mathfrak p)$ and then construction $S' \\subset S$", "corresponding to $L/L'$). Thus we may assume that", "$L \\supset \\kappa(\\mathfrak p)$ is generated by a single element", "$\\alpha \\in L$. Let $X^d + \\sum_{i < d} a_iX^i$ be the minimal polynomial", "of $\\alpha$ over $\\kappa(\\mathfrak p)$, so $a_i \\in \\kappa(\\mathfrak p)$.", "We may write $a_i$ as the image", "of $f_i/g$ for some $f_i, g \\in R$ and $g \\not \\in \\mathfrak p$.", "After replacing $\\alpha$ by $g\\alpha$ (and correspondingly", "replacing $a_i$ by $g^{d - i}a_i$) we may assume that $a_i$ is", "the image of some $f_i \\in R$.", "Then we simply take $S = R[x]/(x^d + \\sum f_ix^i)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1327, "type": "theorem", "label": "algebra-lemma-quotient-complete-local", "categories": [ "algebra" ], "title": "algebra-lemma-quotient-complete-local", "contents": [ "Let $R$ be a Noetherian complete local ring.", "Any quotient of $R$ is also a Noetherian complete local ring.", "Given a finite ring map $R \\to S$, then $S$ is a product of", "Noetherian complete local rings." ], "refs": [], "proofs": [ { "contents": [ "The ring $S$ is Noetherian by Lemma \\ref{lemma-Noetherian-permanence}.", "As an $R$-module $S$ is complete by Lemma \\ref{lemma-completion-tensor}.", "Hence $S$ is the product of the completions at its maximal ideals", "by Lemma \\ref{lemma-completion-finite-extension}." ], "refs": [ "algebra-lemma-Noetherian-permanence", "algebra-lemma-completion-tensor", "algebra-lemma-completion-finite-extension" ], "ref_ids": [ 448, 869, 876 ] } ], "ref_ids": [] }, { "id": 1328, "type": "theorem", "label": "algebra-lemma-complete-local-ring-Noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-complete-local-ring-Noetherian", "contents": [ "Let $(R, \\mathfrak m)$ be a complete local ring.", "If $\\mathfrak m$ is a finitely generated ideal then", "$R$ is Noetherian." ], "refs": [], "proofs": [ { "contents": [ "See Lemma \\ref{lemma-completion-Noetherian}." ], "refs": [ "algebra-lemma-completion-Noetherian" ], "ref_ids": [ 873 ] } ], "ref_ids": [] }, { "id": 1329, "type": "theorem", "label": "algebra-lemma-cohen-rings-exist", "categories": [ "algebra" ], "title": "algebra-lemma-cohen-rings-exist", "contents": [ "Let $p$ be a prime number.", "Let $k$ be a field of characteristic $p$.", "There exists a Cohen ring $\\Lambda$ with $\\Lambda/p\\Lambda \\cong k$." ], "refs": [], "proofs": [ { "contents": [ "First note that the $p$-adic integers $\\mathbf{Z}_p$ form a Cohen ring", "for $\\mathbf{F}_p$. Let $k$ be an arbitrary field of characteristic $p$.", "Let $\\mathbf{Z}_p \\to R$ be a flat local ring map such that", "$\\mathfrak m_R = pR$ and $R/pR = k$, see", "Lemma \\ref{lemma-flat-local-given-residue-field}.", "Then clearly $R$ is a discrete valuation ring. Hence its", "completion is a Cohen ring for $k$." ], "refs": [ "algebra-lemma-flat-local-given-residue-field" ], "ref_ids": [ 1324 ] } ], "ref_ids": [] }, { "id": 1330, "type": "theorem", "label": "algebra-lemma-cohen-ring-formally-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-cohen-ring-formally-smooth", "contents": [ "Let $p > 0$ be a prime.", "Let $\\Lambda$ be a Cohen ring with residue field of characteristic $p$.", "For every $n \\geq 1$ the ring map", "$$", "\\mathbf{Z}/p^n\\mathbf{Z} \\to \\Lambda/p^n\\Lambda", "$$", "is formally smooth." ], "refs": [], "proofs": [ { "contents": [ "If $n = 1$, this follows from", "Proposition \\ref{proposition-characterize-separable-field-extensions}.", "For general $n$ we argue by induction on $n$.", "Namely, if $\\mathbf{Z}/p^n\\mathbf{Z} \\to \\Lambda/p^n\\Lambda$ is", "formally smooth, then we can apply Lemma \\ref{lemma-lift-formal-smoothness}", "to the ring map", "$\\mathbf{Z}/p^{n + 1}\\mathbf{Z} \\to \\Lambda/p^{n + 1}\\Lambda$", "and the ideal $I = (p^n) \\subset \\mathbf{Z}/p^{n + 1}\\mathbf{Z}$." ], "refs": [ "algebra-proposition-characterize-separable-field-extensions", "algebra-lemma-lift-formal-smoothness" ], "ref_ids": [ 1429, 1212 ] } ], "ref_ids": [] }, { "id": 1331, "type": "theorem", "label": "algebra-lemma-regular-complete-containing-coefficient-field", "categories": [ "algebra" ], "title": "algebra-lemma-regular-complete-containing-coefficient-field", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian complete local ring.", "Assume $R$ is regular.", "\\begin{enumerate}", "\\item If $R$ contains either $\\mathbf{F}_p$ or $\\mathbf{Q}$, then $R$", "is isomorphic to a power series ring over its residue field.", "\\item If $k$ is a field and $k \\to R$ is a ring map inducing", "an isomorphism $k \\to R/\\mathfrak m$, then $R$ is isomorphic", "as a $k$-algebra to a power series ring over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In case (1), by the Cohen structure theorem", "(Theorem \\ref{theorem-cohen-structure-theorem})", "there exists a coefficient ring which must be a field", "mapping isomorphically to the residue field. Thus", "it suffices to prove (2). In case (2) we pick", "$f_1, \\ldots, f_d \\in \\mathfrak m$ which", "map to a basis of $\\mathfrak m/\\mathfrak m^2$ and we consider", "the continuous $k$-algebra map $k[[x_1, \\ldots, x_d]] \\to R$", "sending $x_i$ to $f_i$. As both source and target are", "$(x_1, \\ldots, x_d)$-adically complete, this map is surjective by", "Lemma \\ref{lemma-completion-generalities}. On the other hand, it", "has to be injective because otherwise the dimension of", "$R$ would be $< d$ by Lemma \\ref{lemma-one-equation}." ], "refs": [ "algebra-theorem-cohen-structure-theorem", "algebra-lemma-completion-generalities" ], "ref_ids": [ 327, 858 ] } ], "ref_ids": [] }, { "id": 1332, "type": "theorem", "label": "algebra-lemma-complete-local-Noetherian-domain-finite-over-regular", "categories": [ "algebra" ], "title": "algebra-lemma-complete-local-Noetherian-domain-finite-over-regular", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian complete local domain.", "Then there exists a $R_0 \\subset R$ with the following properties", "\\begin{enumerate}", "\\item $R_0$ is a regular complete local ring,", "\\item $R_0 \\subset R$ is finite and induces an isomorphism on", "residue fields,", "\\item $R_0$ is either isomorphic to $k[[X_1, \\ldots, X_d]]$ where $k$", "is a field or $\\Lambda[[X_1, \\ldots, X_d]]$ where $\\Lambda$ is a Cohen ring.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\Lambda$ be a coefficient ring of $R$.", "Since $R$ is a domain we see that either $\\Lambda$ is a field", "or $\\Lambda$ is a Cohen ring.", "\\medskip\\noindent", "Case I: $\\Lambda = k$ is a field. Let $d = \\dim(R)$.", "Choose $x_1, \\ldots, x_d \\in \\mathfrak m$", "which generate an ideal of definition $I \\subset R$.", "(See Section \\ref{section-dimension}.)", "By Lemma \\ref{lemma-change-ideal-completion} we see that $R$", "is $I$-adically complete as well.", "Consider the map $R_0 = k[[X_1, \\ldots, X_d]] \\to R$", "which maps $X_i$ to $x_i$.", "Note that $R_0$ is complete with respect to the ideal", "$I_0 = (X_1, \\ldots, X_d)$,", "and that $R/I_0R \\cong R/IR$ is finite over $k = R_0/I_0$", "(because $\\dim(R/I) = 0$, see Section \\ref{section-dimension}.)", "Hence we conclude that $R_0 \\to R$ is finite by", "Lemma \\ref{lemma-finite-over-complete-ring}.", "Since $\\dim(R) = \\dim(R_0)$ this implies that", "$R_0 \\to R$ is injective (see Lemma \\ref{lemma-integral-dim-up}),", "and the lemma is proved.", "\\medskip\\noindent", "Case II: $\\Lambda$ is a Cohen ring. Let $d + 1 = \\dim(R)$.", "Let $p > 0$ be the characteristic of the residue field $k$.", "As $R$ is a domain we see that $p$ is a nonzerodivisor in $R$.", "Hence $\\dim(R/pR) = d$, see Lemma \\ref{lemma-one-equation}.", "Choose $x_1, \\ldots, x_d \\in R$", "which generate an ideal of definition in $R/pR$.", "Then $I = (p, x_1, \\ldots, x_d)$ is an ideal of definition of $R$.", "By Lemma \\ref{lemma-change-ideal-completion} we see that $R$", "is $I$-adically complete as well.", "Consider the map $R_0 = \\Lambda[[X_1, \\ldots, X_d]] \\to R$", "which maps $X_i$ to $x_i$.", "Note that $R_0$ is complete with respect to the ideal", "$I_0 = (p, X_1, \\ldots, X_d)$,", "and that $R/I_0R \\cong R/IR$ is finite over $k = R_0/I_0$", "(because $\\dim(R/I) = 0$, see Section \\ref{section-dimension}.)", "Hence we conclude that $R_0 \\to R$ is finite by", "Lemma \\ref{lemma-finite-over-complete-ring}.", "Since $\\dim(R) = \\dim(R_0)$ this implies that", "$R_0 \\to R$ is injective (see Lemma \\ref{lemma-integral-dim-up}),", "and the lemma is proved." ], "refs": [ "algebra-lemma-change-ideal-completion", "algebra-lemma-finite-over-complete-ring", "algebra-lemma-integral-dim-up", "algebra-lemma-change-ideal-completion", "algebra-lemma-finite-over-complete-ring", "algebra-lemma-integral-dim-up" ], "ref_ids": [ 865, 868, 984, 865, 868, 984 ] } ], "ref_ids": [] }, { "id": 1333, "type": "theorem", "label": "algebra-lemma-localize-N", "categories": [ "algebra" ], "title": "algebra-lemma-localize-N", "contents": [ "Let $R$ be a domain.", "If $R$ is N-1 then so is any localization of $R$.", "Same for N-2." ], "refs": [], "proofs": [ { "contents": [ "These statements hold because taking integral closure commutes", "with localization, see Lemma \\ref{lemma-integral-closure-localize}." ], "refs": [ "algebra-lemma-integral-closure-localize" ], "ref_ids": [ 489 ] } ], "ref_ids": [] }, { "id": 1334, "type": "theorem", "label": "algebra-lemma-Japanese-local", "categories": [ "algebra" ], "title": "algebra-lemma-Japanese-local", "contents": [ "Let $R$ be a domain. Let $f_1, \\ldots, f_n \\in R$ generate the", "unit ideal. If each domain $R_{f_i}$ is N-1 then so is $R$.", "Same for N-2." ], "refs": [], "proofs": [ { "contents": [ "Assume $R_{f_i}$ is N-2 (or N-1).", "Let $L$ be a finite extension of the fraction field of $R$ (equal to", "the fraction field in the N-1 case). Let $S$ be the integral", "closure of $R$ in $L$. By Lemma \\ref{lemma-integral-closure-localize}", "we see that $S_{f_i}$ is the integral closure of $R_{f_i}$ in $L$.", "Hence $S_{f_i}$ is finite over $R_{f_i}$ by assumption.", "Thus $S$ is finite over $R$ by Lemma \\ref{lemma-cover}." ], "refs": [ "algebra-lemma-integral-closure-localize", "algebra-lemma-cover" ], "ref_ids": [ 489, 411 ] } ], "ref_ids": [] }, { "id": 1335, "type": "theorem", "label": "algebra-lemma-quasi-finite-over-Noetherian-japanese", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-finite-over-Noetherian-japanese", "contents": [ "Let $R$ be a domain. Let $R \\subset S$ be a quasi-finite extension of domains", "(for example finite). Assume $R$ is N-2 and Noetherian. Then $S$ is N-2." ], "refs": [], "proofs": [ { "contents": [ "Let $L/K$ be the induced extension of fraction fields.", "Note that this is a finite field extension (for example by", "Lemma \\ref{lemma-isolated-point-fibre} (2)", "applied to the fibre $S \\otimes_R K$, and the definition of a", "quasi-finite ring map).", "Let $S'$ be the integral closure of $R$ in $S$.", "Then $S'$ is contained in the integral closure of $R$ in $L$", "which is finite over $R$ by assumption. As $R$ is Noetherian this", "implies $S'$ is finite over $R$.", "By Lemma \\ref{lemma-quasi-finite-open-integral-closure}", "there exist elements $g_1, \\ldots, g_n \\in S'$", "such that $S'_{g_i} \\cong S_{g_i}$ and such that $g_1, \\ldots, g_n$", "generate the unit ideal in $S$. Hence it suffices to show that", "$S'$ is N-2 by Lemmas \\ref{lemma-localize-N} and \\ref{lemma-Japanese-local}.", "Thus we have reduced to the case where $S$ is finite over $R$.", "\\medskip\\noindent", "Assume $R \\subset S$ with hypotheses as in the lemma and moreover", "that $S$ is finite over $R$. Let $M$ be a finite field extension", "of the fraction field of $S$. Then $M$ is also a finite field extension", "of $K$ and we conclude that the integral closure $T$ of $R$ in", "$M$ is finite over $R$. By Lemma \\ref{lemma-integral-closure-transitive}", "we see that $T$ is also the integral closure of $S$ in $M$ and we win by", "Lemma \\ref{lemma-integral-permanence}." ], "refs": [ "algebra-lemma-isolated-point-fibre", "algebra-lemma-quasi-finite-open-integral-closure", "algebra-lemma-localize-N", "algebra-lemma-Japanese-local", "algebra-lemma-integral-closure-transitive", "algebra-lemma-integral-permanence" ], "ref_ids": [ 1049, 1067, 1333, 1334, 494, 493 ] } ], "ref_ids": [] }, { "id": 1336, "type": "theorem", "label": "algebra-lemma-Laurent-ring-N-1", "categories": [ "algebra" ], "title": "algebra-lemma-Laurent-ring-N-1", "contents": [ "Let $R$ be a Noetherian domain.", "If $R[z, z^{-1}]$ is N-1, then so is $R$." ], "refs": [], "proofs": [ { "contents": [ "Let $R'$ be the integral closure of $R$ in its field of fractions $K$.", "Let $S'$ be the integral closure of $R[z, z^{-1}]$ in its field of fractions.", "Clearly $R' \\subset S'$.", "Since $K[z, z^{-1}]$ is a normal domain we see that $S' \\subset K[z, z^{-1}]$.", "Suppose that $f_1, \\ldots, f_n \\in S'$ generate $S'$ as $R[z, z^{-1}]$-module.", "Say $f_i = \\sum a_{ij}z^j$ (finite sum), with $a_{ij} \\in K$.", "For any $x \\in R'$ we can write", "$$", "x = \\sum h_i f_i", "$$", "with $h_i \\in R[z, z^{-1}]$. Thus we see that $R'$ is contained in the", "finite $R$-submodule $\\sum Ra_{ij} \\subset K$. Since $R$ is Noetherian", "we conclude that $R'$ is a finite $R$-module." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1337, "type": "theorem", "label": "algebra-lemma-finite-extension-N-2", "categories": [ "algebra" ], "title": "algebra-lemma-finite-extension-N-2", "contents": [ "Let $R$ be a Noetherian domain, and let $R \\subset S$ be a", "finite extension of domains. If $S$ is N-1, then so is $R$.", "If $S$ is N-2, then so is $R$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. (Hint: Integral closures of $R$ in extension fields", "are contained in integral closures of $S$ in extension fields.)" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1338, "type": "theorem", "label": "algebra-lemma-Noetherian-normal-domain-finite-separable-extension", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-normal-domain-finite-separable-extension", "contents": [ "Let $R$ be a Noetherian normal domain with fraction field $K$.", "Let $K \\subset L$ be a finite separable field extension.", "Then the integral closure of $R$ in $L$ is finite over $R$." ], "refs": [], "proofs": [ { "contents": [ "Consider the trace pairing", "(Fields, Definition \\ref{fields-definition-trace-pairing})", "$$", "L \\times L \\longrightarrow K,", "\\quad (x, y) \\longmapsto \\langle x, y\\rangle := \\text{Trace}_{L/K}(xy).", "$$", "Since $L/K$ is separable this is nondegenerate", "(Fields, Lemma \\ref{fields-lemma-separable-trace-pairing}).", "Moreover, if $x \\in L$ is integral over $R$, then", "$\\text{Trace}_{L/K}(x)$ is in $R$. This is true because the", "minimal polynomial of $x$ over $K$ has coefficients in $R$", "(Lemma \\ref{lemma-minimal-polynomial-normal-domain})", "and because $\\text{Trace}_{L/K}(x)$ is an", "integer multiple of one of these coefficients", "(Fields, Lemma \\ref{fields-lemma-trace-and-norm-from-minimal-polynomial}).", "Pick $x_1, \\ldots, x_n \\in L$ which are integral over $R$", "and which form a $K$-basis of $L$. Then the integral closure", "$S \\subset L$ is contained in the $R$-module", "$$", "M = \\{y \\in L \\mid \\langle x_i, y\\rangle \\in R, \\ i = 1, \\ldots, n\\}", "$$", "By linear algebra we see that $M \\cong R^{\\oplus n}$ as an $R$-module.", "Hence $S \\subset R^{\\oplus n}$ is a finitely generated $R$-module", "as $R$ is Noetherian." ], "refs": [ "fields-definition-trace-pairing", "fields-lemma-separable-trace-pairing", "algebra-lemma-minimal-polynomial-normal-domain", "fields-lemma-trace-and-norm-from-minimal-polynomial" ], "ref_ids": [ 4546, 4503, 521, 4500 ] } ], "ref_ids": [] }, { "id": 1339, "type": "theorem", "label": "algebra-lemma-Noetherian-normal-domain-insep-extension", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-normal-domain-insep-extension", "contents": [ "Let $R$ be a Noetherian normal domain with fraction field $K$", "of characteristic $p > 0$.", "Let $a \\in K$ be an element such that there exists a derivation", "$D : R \\to R$ with $D(a) \\not = 0$. Then the integral closure", "of $R$ in $L = K[x]/(x^p - a)$ is finite over $R$." ], "refs": [], "proofs": [ { "contents": [ "After replacing $x$ by $fx$ and $a$ by $f^pa$ for some $f \\in R$", "we may assume $a \\in R$. Hence also $D(a) \\in R$. We will show", "by induction on $i \\leq p - 1$ that if", "$$", "y = a_0 + a_1x + \\ldots + a_i x^i,\\quad a_j \\in K", "$$", "is integral over $R$, then $D(a)^i a_j \\in R$. Thus the integral", "closure is contained in the finite $R$-module with basis", "$D(a)^{-p + 1}x^j$, $j = 0, \\ldots, p - 1$. Since $R$ is Noetherian", "this proves the lemma.", "\\medskip\\noindent", "If $i = 0$, then $y = a_0$ is integral over $R$ if and only if $a_0 \\in R$", "and the statement is true. Suppose the statement holds for some $i < p - 1$", "and suppose that", "$$", "y = a_0 + a_1x + \\ldots + a_{i + 1} x^{i + 1},\\quad a_j \\in K", "$$", "is integral over $R$. Then", "$$", "y^p = a_0^p + a_1^p a + \\ldots + a_{i + 1}^pa^{i + 1}", "$$", "is an element of $R$ (as it is in $K$ and integral over $R$). Applying", "$D$ we obtain", "$$", "(a_1^p + 2a_2^p a + \\ldots + (i + 1)a_{i + 1}^p a^i)D(a)", "$$", "is in $R$. Hence it follows that", "$$", "D(a)a_1 + 2D(a) a_2 x + \\ldots + (i + 1)D(a) a_{i + 1} x^i", "$$", "is integral over $R$. By induction we find $D(a)^{i + 1}a_j \\in R$", "for $j = 1, \\ldots, i + 1$. (Here we use that $1, \\ldots, i + 1$", "are invertible.) Hence $D(a)^{i + 1}a_0$ is also in $R$ because it", "is the difference of $y$ and $\\sum_{j > 0} D(a)^{i + 1}a_jx^j$ which", "are integral over $R$ (since $x$ is integral over $R$ as $a \\in R$)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1340, "type": "theorem", "label": "algebra-lemma-domain-char-zero-N-1-2", "categories": [ "algebra" ], "title": "algebra-lemma-domain-char-zero-N-1-2", "contents": [ "A Noetherian domain whose fraction field has characteristic zero is N-1", "if and only if it is N-2 (i.e., Japanese)." ], "refs": [], "proofs": [ { "contents": [ "This is clear from", "Lemma \\ref{lemma-Noetherian-normal-domain-finite-separable-extension}", "since every field extension in characteristic zero is separable." ], "refs": [ "algebra-lemma-Noetherian-normal-domain-finite-separable-extension" ], "ref_ids": [ 1338 ] } ], "ref_ids": [] }, { "id": 1341, "type": "theorem", "label": "algebra-lemma-domain-char-p-N-1-2", "categories": [ "algebra" ], "title": "algebra-lemma-domain-char-p-N-1-2", "contents": [ "Let $R$ be a Noetherian domain with fraction field $K$ of", "characteristic $p > 0$. Then $R$ is N-2 if and only if", "for every finite purely inseparable extension $K \\subset L$ the integral", "closure of $R$ in $L$ is finite over $R$." ], "refs": [], "proofs": [ { "contents": [ "Assume the integral closure of $R$ in every finite purely inseparable", "field extension of $K$ is finite.", "Let $K \\subset L$ be any finite extension. We have to show the", "integral closure of $R$ in $L$ is finite over $R$.", "Choose a finite normal field extension $K \\subset M$", "containing $L$. As $R$ is Noetherian it suffices to show that", "the integral closure of $R$ in $M$ is finite over $R$.", "By Fields, Lemma \\ref{fields-lemma-normal-case}", "there exists a subextension $K \\subset M_{insep} \\subset M$", "such that $M_{insep}/K$ is purely inseparable, and $M/M_{insep}$", "is separable. By assumption the integral closure $R'$ of $R$ in", "$M_{insep}$ is finite over $R$. By", "Lemma \\ref{lemma-Noetherian-normal-domain-finite-separable-extension}", "the integral", "closure $R''$ of $R'$ in $M$ is finite over $R'$. Then $R''$ is finite", "over $R$ by Lemma \\ref{lemma-finite-transitive}.", "Since $R''$ is also the integral closure", "of $R$ in $M$ (see Lemma \\ref{lemma-integral-closure-transitive}) we win." ], "refs": [ "fields-lemma-normal-case", "algebra-lemma-Noetherian-normal-domain-finite-separable-extension", "algebra-lemma-finite-transitive", "algebra-lemma-integral-closure-transitive" ], "ref_ids": [ 4522, 1338, 337, 494 ] } ], "ref_ids": [] }, { "id": 1342, "type": "theorem", "label": "algebra-lemma-polynomial-ring-N-2", "categories": [ "algebra" ], "title": "algebra-lemma-polynomial-ring-N-2", "contents": [ "Let $R$ be a Noetherian domain.", "If $R$ is N-1 then $R[x]$ is N-1.", "If $R$ is N-2 then $R[x]$ is N-2." ], "refs": [], "proofs": [ { "contents": [ "Assume $R$ is N-1. Let $R'$ be the integral closure of $R$", "which is finite over $R$. Hence also $R'[x]$ is finite over", "$R[x]$. The ring $R'[x]$ is normal (see", "Lemma \\ref{lemma-polynomial-domain-normal}), hence N-1.", "This proves the first assertion.", "\\medskip\\noindent", "For the second assertion, by Lemma \\ref{lemma-finite-extension-N-2}", "it suffices to show that $R'[x]$ is N-2. In other words we may", "and do assume that $R$ is a normal N-2 domain. In characteristic zero", "we are done by Lemma \\ref{lemma-domain-char-zero-N-1-2}.", "In characteristic $p > 0$ we have to show that the integral", "closure of $R[x]$ is finite in any finite purely inseparable extension", "of $L/K(x)$ where $K$ is the fraction field of $R$. There", "exists a finite purely inseparable field extension $L'/K$", "and $q = p^e$ such that $L \\subset L'(x^{1/q})$; some details omitted.", "As $R[x]$ is Noetherian it suffices to show that the integral closure of $R[x]$", "in $L'(x^{1/q})$ is finite over $R[x]$. And this integral closure", "is equal to $R'[x^{1/q}]$ with $R \\subset R' \\subset L'$ the integral", "closure of $R$ in $L'$.", "Since $R$ is N-2 we see that $R'$ is finite over $R$ and hence", "$R'[x^{1/q}]$ is finite over $R[x]$." ], "refs": [ "algebra-lemma-polynomial-domain-normal", "algebra-lemma-finite-extension-N-2", "algebra-lemma-domain-char-zero-N-1-2" ], "ref_ids": [ 508, 1337, 1340 ] } ], "ref_ids": [] }, { "id": 1343, "type": "theorem", "label": "algebra-lemma-openness-normal-locus", "categories": [ "algebra" ], "title": "algebra-lemma-openness-normal-locus", "contents": [ "Let $R$ be a Noetherian domain.", "If there exists an $f \\in R$ such that $R_f$ is normal", "then", "$$", "U = \\{\\mathfrak p \\in \\Spec(R) \\mid R_{\\mathfrak p} \\text{ is normal}\\}", "$$", "is open in $\\Spec(R)$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that the standard open $D(f)$ is contained in $U$.", "By Serre's criterion Lemma \\ref{lemma-criterion-normal} we see that", "$\\mathfrak p \\not \\in U$ implies that for some", "$\\mathfrak q \\subset \\mathfrak p$ we have", "either", "\\begin{enumerate}", "\\item Case I: $\\text{depth}(R_{\\mathfrak q}) < 2$", "and $\\dim(R_{\\mathfrak q}) \\geq 2$, and", "\\item Case II: $R_{\\mathfrak q}$ is not regular", "and $\\dim(R_{\\mathfrak q}) = 1$.", "\\end{enumerate}", "This in particular also means that $R_{\\mathfrak q}$ is not", "normal, and hence $f \\in \\mathfrak q$. In case I we see that", "$\\text{depth}(R_{\\mathfrak q}) =", "\\text{depth}(R_{\\mathfrak q}/fR_{\\mathfrak q}) + 1$.", "Hence such a prime $\\mathfrak q$ is the same thing as an embedded", "associated prime of $R/fR$. In case II $\\mathfrak q$ is an associated", "prime of $R/fR$ of height 1. Thus there is a finite set $E$", "of such primes $\\mathfrak q$ (see Lemma \\ref{lemma-finite-ass}) and", "$$", "\\Spec(R) \\setminus U", "=", "\\bigcup\\nolimits_{\\mathfrak q \\in E} V(\\mathfrak q)", "$$", "as desired." ], "refs": [ "algebra-lemma-criterion-normal", "algebra-lemma-finite-ass" ], "ref_ids": [ 1311, 701 ] } ], "ref_ids": [] }, { "id": 1344, "type": "theorem", "label": "algebra-lemma-characterize-N-1", "categories": [ "algebra" ], "title": "algebra-lemma-characterize-N-1", "contents": [ "Let $R$ be a Noetherian domain. Then $R$ is N-1 if and only if the following", "two conditions hold", "\\begin{enumerate}", "\\item there exists a nonzero $f \\in R$ such that $R_f$ is normal, and", "\\item for every maximal ideal $\\mathfrak m \\subset R$", "the local ring $R_{\\mathfrak m}$ is N-1.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First assume $R$ is N-1. Let $R'$ be the integral closure of $R$ in its", "field of fractions $K$. By assumption we can find $x_1, \\ldots, x_n$ in $R'$", "which generate $R'$ as an $R$-module. Since $R' \\subset K$ we can find", "$f_i \\in R$ nonzero such that $f_i x_i \\in R$. Then $R_f \\cong R'_f$", "where $f = f_1 \\ldots f_n$. Hence $R_f$ is normal and we have (1).", "Part (2) follows from Lemma \\ref{lemma-localize-N}.", "\\medskip\\noindent", "Assume (1) and (2). Let $K$ be the fraction field of $R$.", "Suppose that $R \\subset R' \\subset K$ is a finite", "extension of $R$ contained in $K$. Note that $R_f = R'_f$ since", "$R_f$ is already normal. Hence by Lemma \\ref{lemma-openness-normal-locus}", "the set of primes", "$\\mathfrak p' \\in \\Spec(R')$ with $R'_{\\mathfrak p'}$ non-normal", "is closed in $\\Spec(R')$. Since $\\Spec(R') \\to \\Spec(R)$", "is closed the image of this set is closed in $\\Spec(R)$.", "For such a ring $R'$ denote $Z_{R'} \\subset \\Spec(R)$ this image.", "\\medskip\\noindent", "Pick a maximal ideal $\\mathfrak m \\subset R$.", "Let $R_{\\mathfrak m} \\subset R_{\\mathfrak m}'$ be the integral", "closure of the local ring in $K$. By assumption this is", "a finite ring extension. By Lemma \\ref{lemma-integral-closure-localize}", "we can find finitely", "many elements $r_1, \\ldots, r_n \\in K$ integral over $R$ such that", "$R_{\\mathfrak m}'$ is generated by $r_1, \\ldots, r_n$ over $R_{\\mathfrak m}$.", "Let $R' = R[x_1, \\ldots, x_n] \\subset K$. With this choice it is clear", "that $\\mathfrak m \\not \\in Z_{R'}$.", "\\medskip\\noindent", "As $\\Spec(R)$ is quasi-compact, the above shows that we can", "find a finite collection $R \\subset R'_i \\subset K$ such that", "$\\bigcap Z_{R'_i} = \\emptyset$. Let $R'$ be the subring of $K$", "generated by all of these. It is finite over $R$. Also $Z_{R'} = \\emptyset$.", "Namely, every prime $\\mathfrak p'$ lies over a prime $\\mathfrak p'_i$", "such that $(R'_i)_{\\mathfrak p'_i}$ is normal. This implies", "that $R'_{\\mathfrak p'} = (R'_i)_{\\mathfrak p'_i}$ is normal too.", "Hence $R'$ is normal, in other words", "$R'$ is the integral closure of $R$ in $K$." ], "refs": [ "algebra-lemma-localize-N", "algebra-lemma-openness-normal-locus", "algebra-lemma-integral-closure-localize" ], "ref_ids": [ 1333, 1343, 489 ] } ], "ref_ids": [] }, { "id": 1345, "type": "theorem", "label": "algebra-lemma-tate-japanese", "categories": [ "algebra" ], "title": "algebra-lemma-tate-japanese", "contents": [ "\\begin{reference}", "\\cite[Theorem 23.1.3]{EGA}", "\\end{reference}", "Let $R$ be a ring.", "Let $x \\in R$.", "Assume", "\\begin{enumerate}", "\\item $R$ is a normal Noetherian domain,", "\\item $R/xR$ is a domain and N-2,", "\\item $R \\cong \\lim_n R/x^nR$ is complete with respect to $x$.", "\\end{enumerate}", "Then $R$ is N-2." ], "refs": [], "proofs": [ { "contents": [ "We may assume $x \\not = 0$ since otherwise the lemma is trivial.", "Let $K$ be the fraction field of $R$. If the characteristic of $K$", "is zero the lemma follows from (1), see", "Lemma \\ref{lemma-domain-char-zero-N-1-2}. Hence we may assume", "that the characteristic of $K$ is $p > 0$, and we may apply", "Lemma \\ref{lemma-domain-char-p-N-1-2}. Thus given $K \\subset L$", "be a finite purely inseparable field extension we have to show", "that the integral closure $S$ of $R$ in $L$ is finite over $R$.", "\\medskip\\noindent", "Let $q$ be a power of $p$ such that $L^q \\subset K$.", "By enlarging $L$ if necessary we may assume there exists", "an element $y \\in L$ such that $y^q = x$. Since $R \\to S$", "induces a homeomorphism of spectra (see Lemma \\ref{lemma-p-ring-map})", "there is a unique prime ideal $\\mathfrak q \\subset S$ lying", "over the prime ideal $\\mathfrak p = xR$. It is clear that", "$$", "\\mathfrak q = \\{f \\in S \\mid f^q \\in \\mathfrak p\\} = yS", "$$", "since $y^q = x$. Observe that $R_{\\mathfrak p}$ is a discrete", "valuation ring by Lemma \\ref{lemma-characterize-dvr}. Then", "$S_{\\mathfrak q}$ is Noetherian by Krull-Akizuki", "(Lemma \\ref{lemma-krull-akizuki}). Whereupon we conclude", "$S_{\\mathfrak q}$ is a discrete valuation ring by", "Lemma \\ref{lemma-characterize-dvr} once again.", "By Lemma \\ref{lemma-finite-extension-residue-fields-dimension-1} we", "see that $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$ is", "a finite field extension. Hence the integral closure", "$S' \\subset \\kappa(\\mathfrak q)$ of $R/xR$ is finite over", "$R/xR$ by assumption (2). Since $S/yS \\subset S'$ this implies", "that $S/yS$ is finite over $R$. Note that $S/y^nS$ has a finite", "filtration whose subquotients are the modules", "$y^iS/y^{i + 1}S \\cong S/yS$. Hence we see that each $S/y^nS$", "is finite over $R$. In particular $S/xS$ is finite over $R$.", "Also, it is clear that $\\bigcap x^nS = (0)$ since an element", "in the intersection has $q$th power contained in $\\bigcap x^nR = (0)$", "(Lemma \\ref{lemma-intersect-powers-ideal-module-zero}).", "Thus we may apply Lemma \\ref{lemma-finite-over-complete-ring} to conclude", "that $S$ is finite over $R$, and we win." ], "refs": [ "algebra-lemma-domain-char-zero-N-1-2", "algebra-lemma-domain-char-p-N-1-2", "algebra-lemma-p-ring-map", "algebra-lemma-characterize-dvr", "algebra-lemma-krull-akizuki", "algebra-lemma-characterize-dvr", "algebra-lemma-finite-extension-residue-fields-dimension-1", "algebra-lemma-intersect-powers-ideal-module-zero", "algebra-lemma-finite-over-complete-ring" ], "ref_ids": [ 1340, 1341, 582, 1023, 1027, 1023, 1025, 627, 868 ] } ], "ref_ids": [] }, { "id": 1346, "type": "theorem", "label": "algebra-lemma-power-series-over-N-2", "categories": [ "algebra" ], "title": "algebra-lemma-power-series-over-N-2", "contents": [ "Let $R$ be a ring.", "If $R$ is Noetherian, a domain, and N-2, then so is $R[[x]]$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $R[[x]]$ is Noetherian by", "Lemma \\ref{lemma-Noetherian-power-series}.", "Let $R' \\supset R$ be the integral closure of $R$ in its fraction", "field. Because $R$ is N-2 this is finite over $R$. Hence $R'[[x]]$", "is finite over $R[[x]]$. By", "Lemma \\ref{lemma-power-series-over-Noetherian-normal-domain}", "we see that $R'[[x]]$ is a normal domain.", "Apply Lemma \\ref{lemma-tate-japanese} to the", "element $x \\in R'[[x]]$ to see that $R'[[x]]$ is N-2. Then", "Lemma \\ref{lemma-finite-extension-N-2} shows that $R[[x]]$ is N-2." ], "refs": [ "algebra-lemma-Noetherian-power-series", "algebra-lemma-power-series-over-Noetherian-normal-domain", "algebra-lemma-tate-japanese", "algebra-lemma-finite-extension-N-2" ], "ref_ids": [ 449, 509, 1345, 1337 ] } ], "ref_ids": [] }, { "id": 1347, "type": "theorem", "label": "algebra-lemma-nagata-in-reduced-finite-type-finite-integral-closure", "categories": [ "algebra" ], "title": "algebra-lemma-nagata-in-reduced-finite-type-finite-integral-closure", "contents": [ "Let $R$ be a Nagata ring.", "Let $R \\to S$ be essentially of finite type with $S$ reduced.", "Then the integral closure of $R$ in $S$ is finite over $R$." ], "refs": [], "proofs": [ { "contents": [ "As $S$ is essentially of finite type over $R$ it is Noetherian and", "has finitely many minimal primes $\\mathfrak q_1, \\ldots, \\mathfrak q_m$,", "see Lemma \\ref{lemma-Noetherian-irreducible-components}.", "Since $S$ is reduced we have $S \\subset \\prod S_{\\mathfrak q_i}$", "and each $S_{\\mathfrak q_i} = K_i$ is a field, see", "Lemmas \\ref{lemma-total-ring-fractions-no-embedded-points}", "and \\ref{lemma-minimal-prime-reduced-ring}.", "It suffices to show that the integral closure", "$A_i'$ of $R$ in each $K_i$ is finite over $R$.", "This is true because $R$ is Noetherian and $A \\subset \\prod A_i'$.", "Let $\\mathfrak p_i \\subset R$ be the prime of $R$", "corresponding to $\\mathfrak q_i$.", "As $S$ is essentially of finite type over $R$ we see that", "$K_i = S_{\\mathfrak q_i} = \\kappa(\\mathfrak q_i)$ is a finitely", "generated field extension of $\\kappa(\\mathfrak p_i)$. Hence the algebraic", "closure $L_i$ of $\\kappa(\\mathfrak p_i)$ in $\\subset K_i$", "is finite over $\\kappa(\\mathfrak p_i)$, see", "Fields, Lemma \\ref{fields-lemma-algebraic-closure-in-finitely-generated}.", "It is clear that $A_i'$ is the integral closure of $R/\\mathfrak p_i$", "in $L_i$, and hence we win by definition of a Nagata ring." ], "refs": [ "algebra-lemma-Noetherian-irreducible-components", "algebra-lemma-total-ring-fractions-no-embedded-points", "algebra-lemma-minimal-prime-reduced-ring", "fields-lemma-algebraic-closure-in-finitely-generated" ], "ref_ids": [ 453, 421, 418, 4521 ] } ], "ref_ids": [] }, { "id": 1348, "type": "theorem", "label": "algebra-lemma-check-universally-japanese", "categories": [ "algebra" ], "title": "algebra-lemma-check-universally-japanese", "contents": [ "Let $R$ be a ring.", "To check that $R$ is universally Japanese it suffices to show:", "If $R \\to S$ is of finite type, and $S$ a domain then $S$ is N-1." ], "refs": [], "proofs": [ { "contents": [ "Namely, assume the condition of the lemma.", "Let $R \\to S$ be a finite type ring map with $S$ a domain.", "Let $L$ be a finite extension of the fraction field of $S$.", "Then there exists a finite ring extension $S \\subset S' \\subset L$", "such that $L$ is the fraction field of $S'$.", "By assumption $S'$ is N-1, and hence the integral", "closure $S''$ of $S'$ in $L$ is finite over $S'$. Thus $S''$ is finite", "over $S$ (Lemma \\ref{lemma-finite-transitive})", "and $S''$ is the integral closure of $S$ in $L$", "(Lemma \\ref{lemma-integral-closure-transitive}).", "We conclude that $R$ is universally Japanese." ], "refs": [ "algebra-lemma-finite-transitive", "algebra-lemma-integral-closure-transitive" ], "ref_ids": [ 337, 494 ] } ], "ref_ids": [] }, { "id": 1349, "type": "theorem", "label": "algebra-lemma-universally-japanese", "categories": [ "algebra" ], "title": "algebra-lemma-universally-japanese", "contents": [ "If $R$ is universally Japanese then any algebra essentially of finite type", "over $R$ is universally Japanese." ], "refs": [], "proofs": [ { "contents": [ "The case of an algebra of finite type over $R$ is immediate from", "the definition. The general case follows on applying", "Lemma \\ref{lemma-localize-N}." ], "refs": [ "algebra-lemma-localize-N" ], "ref_ids": [ 1333 ] } ], "ref_ids": [] }, { "id": 1350, "type": "theorem", "label": "algebra-lemma-quasi-finite-over-nagata", "categories": [ "algebra" ], "title": "algebra-lemma-quasi-finite-over-nagata", "contents": [ "Let $R$ be a Nagata ring.", "If $R \\to S$ is a quasi-finite ring map (for example finite)", "then $S$ is a Nagata ring also." ], "refs": [], "proofs": [ { "contents": [ "First note that $S$ is Noetherian as $R$ is Noetherian and a quasi-finite", "ring map is of finite type.", "Let $\\mathfrak q \\subset S$ be a prime ideal, and set", "$\\mathfrak p = R \\cap \\mathfrak q$. Then", "$R/\\mathfrak p \\subset S/\\mathfrak q$ is quasi-finite and", "hence we conclude that $S/\\mathfrak q$ is N-2 by", "Lemma \\ref{lemma-quasi-finite-over-Noetherian-japanese}", "as desired." ], "refs": [ "algebra-lemma-quasi-finite-over-Noetherian-japanese" ], "ref_ids": [ 1335 ] } ], "ref_ids": [] }, { "id": 1351, "type": "theorem", "label": "algebra-lemma-nagata-localize", "categories": [ "algebra" ], "title": "algebra-lemma-nagata-localize", "contents": [ "A localization of a Nagata ring is a Nagata ring." ], "refs": [], "proofs": [ { "contents": [ "Clear from Lemma \\ref{lemma-localize-N}." ], "refs": [ "algebra-lemma-localize-N" ], "ref_ids": [ 1333 ] } ], "ref_ids": [] }, { "id": 1352, "type": "theorem", "label": "algebra-lemma-nagata-local", "categories": [ "algebra" ], "title": "algebra-lemma-nagata-local", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_n \\in R$ generate the", "unit ideal.", "\\begin{enumerate}", "\\item If each $R_{f_i}$ is universally Japanese then so is $R$.", "\\item If each $R_{f_i}$ is Nagata then so is $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi : R \\to S$ be a finite type ring map so that $S$ is a domain.", "Then $\\varphi(f_1), \\ldots, \\varphi(f_n)$ generate the unit ideal", "in $S$. Hence if each $S_{f_i} = S_{\\varphi(f_i)}$ is N-1 then so is", "$S$, see Lemma \\ref{lemma-Japanese-local}. This proves (1).", "\\medskip\\noindent", "If each $R_{f_i}$ is Nagata, then each $R_{f_i}$ is Noetherian and", "hence $R$ is Noetherian, see Lemma \\ref{lemma-cover}. And if", "$\\mathfrak p \\subset R$ is a prime, then we see each", "$R_{f_i}/\\mathfrak pR_{f_i} = (R/\\mathfrak p)_{f_i}$ is N-2", "and hence we conclude $R/\\mathfrak p$ is N-2 by", "Lemma \\ref{lemma-Japanese-local}. This proves (2)." ], "refs": [ "algebra-lemma-Japanese-local", "algebra-lemma-cover", "algebra-lemma-Japanese-local" ], "ref_ids": [ 1334, 411, 1334 ] } ], "ref_ids": [] }, { "id": 1353, "type": "theorem", "label": "algebra-lemma-Noetherian-complete-local-Nagata", "categories": [ "algebra" ], "title": "algebra-lemma-Noetherian-complete-local-Nagata", "contents": [ "A Noetherian complete local ring is a Nagata ring." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a complete local Noetherian ring.", "Let $\\mathfrak p \\subset R$ be a prime.", "Then $R/\\mathfrak p$ is also a complete local Noetherian ring,", "see Lemma \\ref{lemma-quotient-complete-local}.", "Hence it suffices to show that a Noetherian complete local", "domain $R$ is N-2. By", "Lemmas \\ref{lemma-quasi-finite-over-Noetherian-japanese}", "and \\ref{lemma-complete-local-Noetherian-domain-finite-over-regular}", "we reduce to the case $R = k[[X_1, \\ldots, X_d]]$ where $k$ is a field or", "$R = \\Lambda[[X_1, \\ldots, X_d]]$ where $\\Lambda$ is a Cohen ring.", "\\medskip\\noindent", "In the case $k[[X_1, \\ldots, X_d]]$ we reduce to the statement that a", "field is N-2 by Lemma \\ref{lemma-power-series-over-N-2}. This is clear.", "In the case $\\Lambda[[X_1, \\ldots, X_d]]$ we reduce to the statement", "that a Cohen ring $\\Lambda$ is N-2. Applying Lemma \\ref{lemma-tate-japanese}", "once more with $x = p \\in \\Lambda$ we reduce yet again to the case", "of a field. Thus we win." ], "refs": [ "algebra-lemma-quotient-complete-local", "algebra-lemma-quasi-finite-over-Noetherian-japanese", "algebra-lemma-complete-local-Noetherian-domain-finite-over-regular", "algebra-lemma-power-series-over-N-2", "algebra-lemma-tate-japanese" ], "ref_ids": [ 1327, 1335, 1332, 1346, 1345 ] } ], "ref_ids": [] }, { "id": 1354, "type": "theorem", "label": "algebra-lemma-analytically-unramified-easy", "categories": [ "algebra" ], "title": "algebra-lemma-analytically-unramified-easy", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring.", "\\begin{enumerate}", "\\item If $R$ is analytically unramified, then $R$ is reduced.", "\\item If $R$ is analytically unramified, then each minimal prime of", "$R$ is analytically unramified.", "\\item If $R$ is reduced with minimal primes", "$\\mathfrak q_1, \\ldots, \\mathfrak q_t$, and each $\\mathfrak q_i$", "is analytically unramified, then $R$ is analytically unramified.", "\\item If $R$ is analytically unramified, then the integral closure", "of $R$ in its total ring of fractions $Q(R)$ is finite over $R$.", "\\item If $R$ is a domain and analytically unramified, then $R$ is N-1.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In this proof we will use the remarks immediately following", "Definition \\ref{definition-analytically-unramified}.", "As $R \\to R^\\wedge$ is a faithfully flat local ring homomorphism", "it is injective and (1) follows.", "\\medskip\\noindent", "Let $\\mathfrak q$ be a minimal prime of $R$, and assume $R$ is", "analytically unramified.", "Then $\\mathfrak q$ is an associated", "prime of $R$ (see", "Proposition \\ref{proposition-minimal-primes-associated-primes}).", "Hence there exists an $f \\in R$", "such that $\\{x \\in R \\mid fx = 0\\} = \\mathfrak q$.", "Note that $(R/\\mathfrak q)^\\wedge = R^\\wedge/\\mathfrak q^\\wedge$,", "and that $\\{x \\in R^\\wedge \\mid fx = 0\\} = \\mathfrak q^\\wedge$,", "because completion is exact (Lemma \\ref{lemma-completion-flat}).", "If $x \\in R^\\wedge$ is such", "that $x^2 \\in \\mathfrak q^\\wedge$, then $fx^2 = 0$ hence", "$(fx)^2 = 0$ hence $fx = 0$ hence $x \\in \\mathfrak q^\\wedge$.", "Thus $\\mathfrak q$ is analytically unramified and (2) holds.", "\\medskip\\noindent", "Assume $R$ is reduced with minimal primes", "$\\mathfrak q_1, \\ldots, \\mathfrak q_t$, and each $\\mathfrak q_i$", "is analytically unramified. Then", "$R \\to R/\\mathfrak q_1 \\times \\ldots \\times R/\\mathfrak q_t$ is", "injective. Since completion is exact (see Lemma \\ref{lemma-completion-flat})", "we see that", "$R^\\wedge \\subset (R/\\mathfrak q_1)^\\wedge \\times \\ldots \\times", "(R/\\mathfrak q_t)^\\wedge$. Hence (3) is clear.", "\\medskip\\noindent", "Assume $R$ is analytically unramified.", "Let $\\mathfrak p_1, \\ldots, \\mathfrak p_s$ be the minimal primes", "of $R^\\wedge$. Then we see that", "$$", "Q(R^\\wedge) =", "R^\\wedge_{\\mathfrak p_1} \\times \\ldots \\times R^\\wedge_{\\mathfrak p_s}", "$$", "with each $R^\\wedge_{\\mathfrak p_i}$ a field", "as $R^\\wedge$ is reduced (see", "Lemma \\ref{lemma-total-ring-fractions-no-embedded-points}).", "Hence the integral closure $S$ of $R^\\wedge$", "in $Q(R^\\wedge)$ is equal to $S = S_1 \\times \\ldots \\times S_s$ with", "$S_i$ the integral closure of $R^\\wedge/\\mathfrak p_i$ in its fraction", "field. In particular $S$ is finite over $R^\\wedge$.", "Denote $R'$ the integral closure of $R$ in $Q(R)$.", "As $R \\to R^\\wedge$ is flat we see that", "$R' \\otimes_R R^\\wedge \\subset Q(R) \\otimes_R R^\\wedge \\subset Q(R^\\wedge)$.", "Moreover $R' \\otimes_R R^\\wedge$ is integral over $R^\\wedge$", "(Lemma \\ref{lemma-base-change-integral}).", "Hence $R' \\otimes_R R^\\wedge \\subset S$ is a $R^\\wedge$-submodule.", "As $R^\\wedge$ is Noetherian it is a finite $R^\\wedge$-module.", "Thus we may find $f_1, \\ldots, f_n \\in R'$ such that", "$R' \\otimes_R R^\\wedge$ is generated by the elements $f_i \\otimes 1$", "as a $R^\\wedge$-module.", "By faithful flatness we see that $R'$ is generated by $f_1, \\ldots, f_n$", "as an $R$-module. This proves (4).", "\\medskip\\noindent", "Part (5) is a special case of part (4)." ], "refs": [ "algebra-definition-analytically-unramified", "algebra-proposition-minimal-primes-associated-primes", "algebra-lemma-completion-flat", "algebra-lemma-completion-flat", "algebra-lemma-total-ring-fractions-no-embedded-points", "algebra-lemma-base-change-integral" ], "ref_ids": [ 1553, 1412, 870, 870, 421, 491 ] } ], "ref_ids": [] }, { "id": 1355, "type": "theorem", "label": "algebra-lemma-codimension-1-analytically-unramified", "categories": [ "algebra" ], "title": "algebra-lemma-codimension-1-analytically-unramified", "contents": [ "Let $R$ be a Noetherian local ring.", "Let $\\mathfrak p \\subset R$ be a prime.", "Assume", "\\begin{enumerate}", "\\item $R_{\\mathfrak p}$ is a discrete valuation ring, and", "\\item $\\mathfrak p$ is analytically unramified.", "\\end{enumerate}", "Then for any associated prime $\\mathfrak q$ of $R^\\wedge/\\mathfrak pR^\\wedge$", "the local ring $(R^\\wedge)_{\\mathfrak q}$ is a discrete valuation ring." ], "refs": [], "proofs": [ { "contents": [ "Assumption (2) says that $R^\\wedge/\\mathfrak pR^\\wedge$ is a reduced ring.", "Hence an associated prime $\\mathfrak q \\subset R^\\wedge$", "of $R^\\wedge/\\mathfrak pR^\\wedge$", "is the same thing as a minimal prime over $\\mathfrak pR^\\wedge$.", "In particular we see that the maximal ideal of $(R^\\wedge)_{\\mathfrak q}$", "is $\\mathfrak p(R^\\wedge)_{\\mathfrak q}$.", "Choose $x \\in R$ such that $xR_{\\mathfrak p} = \\mathfrak pR_{\\mathfrak p}$.", "By the above we see that $x \\in (R^\\wedge)_{\\mathfrak q}$ generates", "the maximal ideal. As $R \\to R^\\wedge$ is faithfully flat we see that", "$x$ is a nonzerodivisor in $(R^\\wedge)_{\\mathfrak q}$.", "Hence we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1356, "type": "theorem", "label": "algebra-lemma-criterion-analytically-unramified", "categories": [ "algebra" ], "title": "algebra-lemma-criterion-analytically-unramified", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local domain.", "Let $x \\in \\mathfrak m$. Assume", "\\begin{enumerate}", "\\item $x \\not = 0$,", "\\item $R/xR$ has no embedded primes, and", "\\item for each associated prime $\\mathfrak p \\subset R$", "of $R/xR$ we have", "\\begin{enumerate}", "\\item the local ring $R_{\\mathfrak p}$ is regular, and", "\\item $\\mathfrak p$ is analytically unramified.", "\\end{enumerate}", "\\end{enumerate}", "Then $R$ is analytically unramified." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p_1, \\ldots, \\mathfrak p_t$ be the associated primes", "of the $R$-module $R/xR$. Since $R/xR$ has no embedded primes we", "see that each $\\mathfrak p_i$ has height $1$, and is a minimal", "prime over $(x)$.", "For each $i$, let $\\mathfrak q_{i1}, \\ldots, \\mathfrak q_{is_i}$", "be the associated primes of the $R^\\wedge$-module", "$R^\\wedge/\\mathfrak p_iR^\\wedge$.", "By Lemma \\ref{lemma-codimension-1-analytically-unramified}", "we see that $(R^\\wedge)_{\\mathfrak q_{ij}}$ is regular.", "By Lemma \\ref{lemma-bourbaki} we see that", "$$", "\\text{Ass}_{R^\\wedge}(R^\\wedge/xR^\\wedge)", "=", "\\bigcup\\nolimits_{\\mathfrak p \\in \\text{Ass}_R(R/xR)}", "\\text{Ass}_{R^\\wedge}(R^\\wedge/\\mathfrak pR^\\wedge)", "=", "\\{\\mathfrak q_{ij}\\}.", "$$", "Let $y \\in R^\\wedge$ with $y^2 = 0$.", "As $(R^\\wedge)_{\\mathfrak q_{ij}}$ is regular, and hence a domain", "(Lemma \\ref{lemma-regular-domain})", "we see that $y$ maps to zero in $(R^\\wedge)_{\\mathfrak q_{ij}}$.", "Hence $y$ maps to zero in $R^\\wedge/xR^\\wedge$ by", "Lemma \\ref{lemma-zero-at-ass-zero}.", "Hence $y = xy'$. Since $x$ is a nonzerodivisor (as $R \\to R^\\wedge$ is flat)", "we see that $(y')^2 = 0$. Hence we conclude that", "$y \\in \\bigcap x^nR^\\wedge = (0)$", "(Lemma \\ref{lemma-intersect-powers-ideal-module-zero})." ], "refs": [ "algebra-lemma-codimension-1-analytically-unramified", "algebra-lemma-bourbaki", "algebra-lemma-regular-domain", "algebra-lemma-zero-at-ass-zero", "algebra-lemma-intersect-powers-ideal-module-zero" ], "ref_ids": [ 1355, 717, 940, 713, 627 ] } ], "ref_ids": [] }, { "id": 1357, "type": "theorem", "label": "algebra-lemma-local-nagata-domain-analytically-unramified", "categories": [ "algebra" ], "title": "algebra-lemma-local-nagata-domain-analytically-unramified", "contents": [ "Let $(R, \\mathfrak m)$ be a local ring.", "If $R$ is Noetherian, a domain, and Nagata, then $R$ is", "analytically unramified." ], "refs": [], "proofs": [ { "contents": [ "By induction on $\\dim(R)$.", "The case $\\dim(R) = 0$ is trivial. Hence we assume $\\dim(R) = d$ and that", "the lemma holds for all Noetherian Nagata domains of dimension $< d$.", "\\medskip\\noindent", "Let $R \\subset S$ be the integral closure", "of $R$ in the field of fractions of $R$. By assumption $S$ is a finite", "$R$-module. By Lemma \\ref{lemma-quasi-finite-over-nagata} we see that", "$S$ is Nagata. By Lemma \\ref{lemma-integral-sub-dim-equal} we see", "$\\dim(R) = \\dim(S)$.", "Let $\\mathfrak m_1, \\ldots, \\mathfrak m_t$ be the maximal", "ideals of $S$. Each of these lies over the maximal ideal $\\mathfrak m$", "of $R$. Moreover", "$$", "(\\mathfrak m_1 \\cap \\ldots \\cap \\mathfrak m_t)^n \\subset \\mathfrak mS", "$$", "for sufficiently large $n$ as $S/\\mathfrak mS$ is Artinian.", "By Lemma \\ref{lemma-completion-flat} $R^\\wedge \\to S^\\wedge$", "is an injective map, and by the Chinese Remainder", "Lemma \\ref{lemma-chinese-remainder} combined with", "Lemma \\ref{lemma-change-ideal-completion} we have", "$S^\\wedge = \\prod S^\\wedge_i$ where $S^\\wedge_i$", "is the completion of $S$ with respect to the maximal ideal $\\mathfrak m_i$.", "Hence it suffices to show that $S_{\\mathfrak m_i}$ is analytically unramified.", "In other words, we have reduced to the case where $R$ is a Noetherian", "normal Nagata domain.", "\\medskip\\noindent", "Assume $R$ is a Noetherian, normal, local Nagata domain.", "Pick a nonzero $x \\in \\mathfrak m$ in the maximal ideal.", "We are going to apply Lemma \\ref{lemma-criterion-analytically-unramified}.", "We have to check properties (1), (2), (3)(a) and (3)(b).", "Property (1) is clear.", "We have that $R/xR$ has no embedded primes by", "Lemma \\ref{lemma-normal-domain-intersection-localizations-height-1}.", "Thus property (2) holds. The same lemma also tells us each associated", "prime $\\mathfrak p$ of $R/xR$ has height $1$.", "Hence $R_{\\mathfrak p}$ is a $1$-dimensional normal domain", "hence regular (Lemma \\ref{lemma-characterize-dvr}). Thus (3)(a) holds.", "Finally (3)(b) holds by induction hypothesis, since", "$R/\\mathfrak p$ is Nagata (by Lemma \\ref{lemma-quasi-finite-over-nagata}", "or directly from the definition).", "Thus we conclude $R$ is analytically unramified." ], "refs": [ "algebra-lemma-quasi-finite-over-nagata", "algebra-lemma-integral-sub-dim-equal", "algebra-lemma-completion-flat", "algebra-lemma-chinese-remainder", "algebra-lemma-change-ideal-completion", "algebra-lemma-criterion-analytically-unramified", "algebra-lemma-normal-domain-intersection-localizations-height-1", "algebra-lemma-characterize-dvr", "algebra-lemma-quasi-finite-over-nagata" ], "ref_ids": [ 1350, 985, 870, 380, 865, 1356, 1313, 1023, 1350 ] } ], "ref_ids": [] }, { "id": 1358, "type": "theorem", "label": "algebra-lemma-local-nagata-and-analytically-unramified", "categories": [ "algebra" ], "title": "algebra-lemma-local-nagata-and-analytically-unramified", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring. The following", "are equivalent", "\\begin{enumerate}", "\\item $R$ is Nagata,", "\\item for $R \\to S$ finite with $S$ a domain and $\\mathfrak m' \\subset S$", "maximal the local ring $S_{\\mathfrak m'}$ is analytically unramified,", "\\item for $(R, \\mathfrak m) \\to (S, \\mathfrak m')$ finite", "local homomorphism with $S$ a domain, then $S$ is analytically", "unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $R$ is Nagata and let $R \\to S$ and $\\mathfrak m' \\subset S$", "be as in (2). Then $S$ is Nagata by Lemma \\ref{lemma-quasi-finite-over-nagata}.", "Hence the local ring $S_{\\mathfrak m'}$ is Nagata", "(Lemma \\ref{lemma-nagata-localize}). Thus it is analytically", "unramified by Lemma \\ref{lemma-local-nagata-domain-analytically-unramified}.", "It is clear that (2) implies (3).", "\\medskip\\noindent", "Assume (3) holds. Let $\\mathfrak p \\subset R$ be a prime ideal and", "let $L/\\kappa(\\mathfrak p)$ be a finite extension of fields.", "To prove (1) we have to show that the integral closure of $R/\\mathfrak p$", "is finite over $R/\\mathfrak p$. Choose $x_1, \\ldots, x_n \\in L$", "which generate $L$ over $\\kappa(\\mathfrak p)$. For each $i$ let", "$P_i(T) = T^{d_i} + a_{i, 1} T^{d_i - 1} + \\ldots + a_{i, d_i}$", "be the minimal polynomial for $x_i$ over $\\kappa(\\mathfrak p)$.", "After replacing $x_i$ by $f_i x_i$ for a suitable", "$f_i \\in R$, $f_i \\not \\in \\mathfrak p$ we may assume", "$a_{i, j} \\in R/\\mathfrak p$. In fact, after further multiplying", "by elements of $\\mathfrak m$, we may assume", "$a_{i, j} \\in \\mathfrak m/\\mathfrak p \\subset R/\\mathfrak p$ for all $i, j$.", "Having done this let $S = R/\\mathfrak p[x_1, \\ldots, x_n] \\subset L$.", "Then $S$ is finite over $R$, a domain, and $S/\\mathfrak m S$ is a quotient", "of $R/\\mathfrak m[T_1, \\ldots, T_n]/(T_1^{d_1}, \\ldots, T_n^{d_n})$.", "Hence $S$ is local. By (3) $S$ is analytically unramified and by", "Lemma \\ref{lemma-analytically-unramified-easy}", "we find that its integral closure $S'$ in $L$ is finite over $S$.", "Since $S'$ is also the integral closure of $R/\\mathfrak p$ in", "$L$ we win." ], "refs": [ "algebra-lemma-quasi-finite-over-nagata", "algebra-lemma-nagata-localize", "algebra-lemma-local-nagata-domain-analytically-unramified", "algebra-lemma-analytically-unramified-easy" ], "ref_ids": [ 1350, 1351, 1357, 1354 ] } ], "ref_ids": [] }, { "id": 1359, "type": "theorem", "label": "algebra-lemma-nagata-pth-roots", "categories": [ "algebra" ], "title": "algebra-lemma-nagata-pth-roots", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local domain which is Nagata", "and has fraction field of characteristic $p$. If $a \\in A$ has a", "$p$th root in $A^\\wedge$, then $a$ has a $p$th root in $A$." ], "refs": [], "proofs": [ { "contents": [ "Consider the ring extension $A \\subset B = A[x]/(x^p - a)$.", "If $a$ does not have a $p$th root in $A$, then $B$ is a domain", "whose completion isn't reduced. This contradicts our earlier", "results, as $B$ is a Nagata ring", "(Proposition \\ref{proposition-nagata-universally-japanese})", "and hence analytically unramified by", "Lemma \\ref{lemma-local-nagata-domain-analytically-unramified}." ], "refs": [ "algebra-proposition-nagata-universally-japanese", "algebra-lemma-local-nagata-domain-analytically-unramified" ], "ref_ids": [ 1430, 1357 ] } ], "ref_ids": [] }, { "id": 1360, "type": "theorem", "label": "algebra-lemma-apply-grothendieck-module", "categories": [ "algebra" ], "title": "algebra-lemma-apply-grothendieck-module", "contents": [ "\\begin{reference}", "\\cite[IV, Proposition 6.3.1]{EGA}", "\\end{reference}", "We have", "$$", "\\text{depth}(M \\otimes_R N)", "=", "\\text{depth}(M) + \\text{depth}(N/\\mathfrak m_RN)", "$$", "where $R \\to S$ is a local homomorphism of local Noetherian rings,", "$M$ is a finite $R$-module, and $N$ is a finite $S$-module flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "In the statement and in the proof below, we take the depth of $M$", "as an $R$-module, the depth of $M \\otimes_R N$ as an $S$-module, and", "the depth of $N/\\mathfrak m_RN$ as an $S/\\mathfrak m_RS$-module.", "Denote $n$ the right hand side. First assume that $n$ is zero.", "Then both $\\text{depth}(M) = 0$ and", "$\\text{depth}(N/\\mathfrak m_RN) = 0$.", "This means there is a $z \\in M$ whose annihilator is $\\mathfrak m_R$", "and a $\\overline{y} \\in N/\\mathfrak m_RN$", "whose annihilator is $\\mathfrak m_S/\\mathfrak m_RS$.", "Let $y \\in N$ be a lift of $\\overline{y}$.", "Since $N$ is flat over $R$ the map $z : R/\\mathfrak m_R \\to M$", "produces an injective map $N/\\mathfrak m_RN \\to M \\otimes_R N$.", "Hence the annihilator of $z \\otimes y$ is $\\mathfrak m_S$.", "Thus $\\text{depth}(M \\otimes_R N) = 0$ as well.", "\\medskip\\noindent", "Assume $n > 0$. If $\\text{depth}(N/\\mathfrak m_RN) > 0$, then we may choose", "$f \\in \\mathfrak m_S$ mapping to $\\overline{f} \\in S/\\mathfrak m_RS$ which", "is a nonzerodivisor on $N/\\mathfrak m_RN$.", "Then $\\text{depth}(N/\\mathfrak m_RN) =", "\\text{depth}(N/(f, \\mathfrak m_R)N) + 1$", "by Lemma \\ref{lemma-depth-drops-by-one}.", "According to Lemma \\ref{lemma-mod-injective} the element $f \\in S$ is a", "nonzerodivisor on $N$ and $N/fN$ is flat over $R$.", "Hence by induction on $n$ we have", "$$", "\\text{depth}(M \\otimes_R N/fN) =", "\\text{depth}(M) + \\text{depth}(N/(f, \\mathfrak m_R)N).", "$$", "Because $N/fN$ is flat over $R$ the sequence", "$$", "0 \\to M \\otimes_R N \\to M \\otimes_R N \\to M \\otimes_R N/fN \\to 0", "$$", "is exact where the first map is multiplication by $f$", "(Lemma \\ref{lemma-flat-tor-zero}). Hence by", "Lemma \\ref{lemma-depth-drops-by-one} we find that", "$\\text{depth}(M \\otimes_R N) = \\text{depth}(M \\otimes_R N/fN) + 1$", "and we conclude that equality holds in the formula of the lemma.", "\\medskip\\noindent", "If $n > 0$, but $\\text{depth}(N/\\mathfrak m_RN) = 0$,", "then we can choose $f \\in \\mathfrak m_R$ which is a nonzerodivisor on $M$.", "As $N$ is flat over $R$ it is also the case that $f$ is a nonzerodivisor on", "$M \\otimes_R N$. By induction on $n$ again we have", "$$", "\\text{depth}(M/fM \\otimes_R N) =", "\\text{depth}(M/fM) + \\text{depth}(N/\\mathfrak m_RN).", "$$", "In this case", "$\\text{depth}(M \\otimes_R N) = \\text{depth}(M/fM \\otimes_R N) + 1$", "and $\\text{depth}(M) = \\text{depth}(M/fM) + 1$", "by Lemma \\ref{lemma-depth-drops-by-one} and", "we conclude that equality holds in the formula of the lemma." ], "refs": [ "algebra-lemma-depth-drops-by-one", "algebra-lemma-mod-injective", "algebra-lemma-flat-tor-zero", "algebra-lemma-depth-drops-by-one", "algebra-lemma-depth-drops-by-one" ], "ref_ids": [ 774, 883, 532, 774, 774 ] } ], "ref_ids": [] }, { "id": 1361, "type": "theorem", "label": "algebra-lemma-apply-grothendieck", "categories": [ "algebra" ], "title": "algebra-lemma-apply-grothendieck", "contents": [ "Suppose that $R \\to S$ is a flat and local ring homomorphism of Noetherian", "local rings. Then", "$$", "\\text{depth}(S) = \\text{depth}(R) + \\text{depth}(S/\\mathfrak m_RS).", "$$" ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-apply-grothendieck-module}." ], "refs": [ "algebra-lemma-apply-grothendieck-module" ], "ref_ids": [ 1360 ] } ], "ref_ids": [] }, { "id": 1362, "type": "theorem", "label": "algebra-lemma-CM-goes-up", "categories": [ "algebra" ], "title": "algebra-lemma-CM-goes-up", "contents": [ "Let $R \\to S$ be a flat local homomorphism of local Noetherian rings.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $S$ is Cohen-Macaulay, and", "\\item $R$ and $S/\\mathfrak m_RS$ are Cohen-Macaulay.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows from the definitions and", "Lemmas \\ref{lemma-apply-grothendieck} and", "\\ref{lemma-dimension-base-fibre-equals-total}." ], "refs": [ "algebra-lemma-apply-grothendieck", "algebra-lemma-dimension-base-fibre-equals-total" ], "ref_ids": [ 1361, 987 ] } ], "ref_ids": [] }, { "id": 1363, "type": "theorem", "label": "algebra-lemma-Sk-goes-up", "categories": [ "algebra" ], "title": "algebra-lemma-Sk-goes-up", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Assume", "\\begin{enumerate}", "\\item $R$ is Noetherian,", "\\item $S$ is Noetherian,", "\\item $\\varphi$ is flat,", "\\item the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$ are $(S_k)$, and", "\\item $R$ has property $(S_k)$.", "\\end{enumerate}", "Then $S$ has property $(S_k)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q$ be a prime of $S$", "lying over a prime $\\mathfrak p$ of $R$. By", "Lemma \\ref{lemma-apply-grothendieck} we have", "$$", "\\text{depth}(S_{\\mathfrak q}) =", "\\text{depth}(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) +", "\\text{depth}(R_{\\mathfrak p}).", "$$", "On the other hand, we have", "$$", "\\dim(R_{\\mathfrak p})", "+", "\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q})", "\\geq", "\\dim(S_{\\mathfrak q})", "$$", "by Lemma \\ref{lemma-dimension-base-fibre-total}.", "(Actually equality holds, by", "Lemma \\ref{lemma-dimension-base-fibre-equals-total}", "but strictly speaking we do not need this.)", "Finally, as the fibre rings of the map", "are assumed $(S_k)$ we see that", "$\\text{depth}(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q})", "\\geq \\min(k, \\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}))$.", "Thus the lemma follows by the following string of inequalities", "\\begin{eqnarray*}", "\\text{depth}(S_{\\mathfrak q}) & = &", "\\text{depth}(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) +", "\\text{depth}(R_{\\mathfrak p}) \\\\", "& \\geq &", "\\min(k, \\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q})) +", "\\min(k, \\dim(R_{\\mathfrak p})) \\\\", "& = &", "\\min(2k, \\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) + k,", "k + \\dim(R_\\mathfrak p),", "\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) +", "\\dim(R_{\\mathfrak p})) \\\\", "& \\geq &", "\\min(k, \\dim(S_{\\mathfrak q}))", "\\end{eqnarray*}", "as desired." ], "refs": [ "algebra-lemma-apply-grothendieck", "algebra-lemma-dimension-base-fibre-total", "algebra-lemma-dimension-base-fibre-equals-total" ], "ref_ids": [ 1361, 986, 987 ] } ], "ref_ids": [] }, { "id": 1364, "type": "theorem", "label": "algebra-lemma-Rk-goes-up", "categories": [ "algebra" ], "title": "algebra-lemma-Rk-goes-up", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Assume", "\\begin{enumerate}", "\\item $R$ is Noetherian,", "\\item $S$ is Noetherian", "\\item $\\varphi$ is flat,", "\\item the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$", "have property $(R_k)$, and", "\\item $R$ has property $(R_k)$.", "\\end{enumerate}", "Then $S$ has property $(R_k)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q$ be a prime of $S$", "lying over a prime $\\mathfrak p$ of $R$.", "Assume that $\\dim(S_{\\mathfrak q}) \\leq k$.", "Since $\\dim(S_{\\mathfrak q}) = \\dim(R_{\\mathfrak p})", "+ \\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q})$ by", "Lemma \\ref{lemma-dimension-base-fibre-equals-total}", "we see that $\\dim(R_{\\mathfrak p}) \\leq k$ and", "$\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) \\leq k$.", "Hence $R_{\\mathfrak p}$ and $S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}$", "are regular by assumption.", "It follows that $S_{\\mathfrak q}$ is regular by", "Lemma \\ref{lemma-flat-over-regular-with-regular-fibre}." ], "refs": [ "algebra-lemma-dimension-base-fibre-equals-total", "algebra-lemma-flat-over-regular-with-regular-fibre" ], "ref_ids": [ 987, 988 ] } ], "ref_ids": [] }, { "id": 1365, "type": "theorem", "label": "algebra-lemma-reduced-goes-up-noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-reduced-goes-up-noetherian", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Assume", "\\begin{enumerate}", "\\item $R$ is Noetherian,", "\\item $S$ is Noetherian", "\\item $\\varphi$ is flat,", "\\item the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$ are reduced,", "\\item $R$ is reduced.", "\\end{enumerate}", "Then $S$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "For Noetherian rings reduced is the same as having properties", "$(S_1)$ and $(R_0)$, see Lemma \\ref{lemma-criterion-reduced}.", "Thus we know $R$ and the fibre rings have these properties.", "Hence we may apply Lemmas \\ref{lemma-Sk-goes-up} and \\ref{lemma-Rk-goes-up}", "and we see that $S$ is $(S_1)$ and $(R_0)$, in other words reduced", "by Lemma \\ref{lemma-criterion-reduced} again." ], "refs": [ "algebra-lemma-criterion-reduced", "algebra-lemma-Sk-goes-up", "algebra-lemma-Rk-goes-up", "algebra-lemma-criterion-reduced" ], "ref_ids": [ 1310, 1363, 1364, 1310 ] } ], "ref_ids": [] }, { "id": 1366, "type": "theorem", "label": "algebra-lemma-reduced-goes-up", "categories": [ "algebra" ], "title": "algebra-lemma-reduced-goes-up", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Assume", "\\begin{enumerate}", "\\item $\\varphi$ is smooth,", "\\item $R$ is reduced.", "\\end{enumerate}", "Then $S$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "Observe that $R \\to S$ is flat with regular fibres (see the list of", "results on smooth ring maps in Section \\ref{section-smooth-overview}).", "In particular, the fibres are reduced.", "Thus if $R$ is Noetherian, then $S$ is Noetherian and we get", "the result from Lemma \\ref{lemma-reduced-goes-up-noetherian}.", "\\medskip\\noindent", "In the general case we may find a finitely generated", "$\\mathbf{Z}$-subalgebra $R_0 \\subset R$ and a smooth ring", "map $R_0 \\to S_0$ such that $S \\cong R \\otimes_{R_0} S_0$, see", "remark (10) in Section \\ref{section-smooth-overview}.", "Now, if $x \\in S$ is an element with $x^2 = 0$,", "then we can enlarge $R_0$ and assume that $x$ comes", "from an element $x_0 \\in S_0$. After enlarging", "$R_0$ once more we may assume that $x_0^2 = 0$ in $S_0$.", "However, since $R_0 \\subset R$ is reduced we see that", "$S_0$ is reduced and hence $x_0 = 0$ as desired." ], "refs": [ "algebra-lemma-reduced-goes-up-noetherian" ], "ref_ids": [ 1365 ] } ], "ref_ids": [] }, { "id": 1367, "type": "theorem", "label": "algebra-lemma-normal-goes-up-noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-normal-goes-up-noetherian", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Assume", "\\begin{enumerate}", "\\item $R$ is Noetherian,", "\\item $S$ is Noetherian,", "\\item $\\varphi$ is flat,", "\\item the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$ are normal, and", "\\item $R$ is normal.", "\\end{enumerate}", "Then $S$ is normal." ], "refs": [], "proofs": [ { "contents": [ "For a Noetherian ring being normal is the same as having properties", "$(S_2)$ and $(R_1)$, see Lemma \\ref{lemma-criterion-normal}.", "Thus we know $R$ and the fibre rings have these properties.", "Hence we may apply Lemmas \\ref{lemma-Sk-goes-up} and \\ref{lemma-Rk-goes-up}", "and we see that $S$ is $(S_2)$ and $(R_1)$, in other words normal", "by Lemma \\ref{lemma-criterion-normal} again." ], "refs": [ "algebra-lemma-criterion-normal", "algebra-lemma-Sk-goes-up", "algebra-lemma-Rk-goes-up", "algebra-lemma-criterion-normal" ], "ref_ids": [ 1311, 1363, 1364, 1311 ] } ], "ref_ids": [] }, { "id": 1368, "type": "theorem", "label": "algebra-lemma-normal-goes-up", "categories": [ "algebra" ], "title": "algebra-lemma-normal-goes-up", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Assume", "\\begin{enumerate}", "\\item $\\varphi$ is smooth,", "\\item $R$ is normal.", "\\end{enumerate}", "Then $S$ is normal." ], "refs": [], "proofs": [ { "contents": [ "Observe that $R \\to S$ is flat with regular fibres (see the list of", "results on smooth ring maps in Section \\ref{section-smooth-overview}).", "In particular, the fibres are normal. Thus if $R$ is Noetherian,", "then $S$ is Noetherian and we get the result from", "Lemma \\ref{lemma-normal-goes-up-noetherian}.", "\\medskip\\noindent", "The general case. First note that $R$ is reduced and hence", "$S$ is reduced by Lemma \\ref{lemma-reduced-goes-up}.", "Let $\\mathfrak q$ be a prime of $S$ and let $\\mathfrak p$ be", "the corresponding prime of $R$. Note that $R_{\\mathfrak p}$", "is a normal domain. We have to show that $S_{\\mathfrak q}$ is", "a normal domain. To do this we may replace $R$ by $R_{\\mathfrak p}$", "and $S$ by $S_{\\mathfrak p}$. Hence we may assume that $R$ is", "a normal domain.", "\\medskip\\noindent", "Assume $R \\to S$ smooth, and $R$ a normal domain.", "We may find a finitely generated $\\mathbf{Z}$-subalgebra", "$R_0 \\subset R$ and a smooth ring map $R_0 \\to S_0$ such", "that $S \\cong R \\otimes_{R_0} S_0$, see", "remark (10) in Section \\ref{section-smooth-overview}.", "As $R_0$ is a Nagata domain (see Proposition \\ref{proposition-ubiquity-nagata})", "we see that its integral closure $R_0'$ is finite over $R_0$.", "Moreover, as $R$ is a normal domain it is clear that $R_0' \\subset R$.", "Hence we may replace $R_0$ by $R_0'$ and $S_0$ by", "$R_0' \\otimes_{R_0} S_0$ and assume that $R_0$ is a normal", "Noetherian domain. By the first paragraph of the proof we conclude", "that $S_0$ is a normal ring (it need not be a domain of course).", "In this way we see that $R = \\bigcup R_\\lambda$", "is the union of normal Noetherian domains and correspondingly", "$S = \\colim R_\\lambda \\otimes_{R_0} S_0$ is the colimit", "of normal rings. This implies that $S$ is a normal ring.", "Some details omitted." ], "refs": [ "algebra-lemma-normal-goes-up-noetherian", "algebra-lemma-reduced-goes-up", "algebra-proposition-ubiquity-nagata" ], "ref_ids": [ 1367, 1366, 1431 ] } ], "ref_ids": [] }, { "id": 1369, "type": "theorem", "label": "algebra-lemma-regular-goes-up", "categories": [ "algebra" ], "title": "algebra-lemma-regular-goes-up", "contents": [ "\\begin{slogan}", "Regularity ascends along smooth maps of rings.", "\\end{slogan}", "Let $\\varphi : R \\to S$ be a ring map. Assume", "\\begin{enumerate}", "\\item $\\varphi$ is smooth,", "\\item $R$ is a regular ring.", "\\end{enumerate}", "Then $S$ is regular." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-Rk-goes-up} applied for all $(R_k)$", "using Lemma \\ref{lemma-characterize-smooth-over-field} to see that the", "hypotheses are satisfied." ], "refs": [ "algebra-lemma-Rk-goes-up", "algebra-lemma-characterize-smooth-over-field" ], "ref_ids": [ 1364, 1223 ] } ], "ref_ids": [] }, { "id": 1370, "type": "theorem", "label": "algebra-lemma-descent-Noetherian", "categories": [ "algebra" ], "title": "algebra-lemma-descent-Noetherian", "contents": [ "Let $R \\to S$ be a ring map.", "Assume that", "\\begin{enumerate}", "\\item $R \\to S$ is faithfully flat, and", "\\item $S$ is Noetherian.", "\\end{enumerate}", "Then $R$ is Noetherian." ], "refs": [], "proofs": [ { "contents": [ "Let $I_0 \\subset I_1 \\subset I_2 \\subset \\ldots$ be a", "growing sequence of ideals of $R$. By assumption we have", "$I_nS = I_{n +1}S = I_{n + 2}S = \\ldots$ for some $n$.", "Since $R \\to S$ is flat we have $I_kS = I_k \\otimes_R S$.", "Hence, as $R \\to S$ is faithfully flat we see that", "$I_nS = I_{n +1}S = I_{n + 2}S = \\ldots$ implies that", "$I_n = I_{n +1} = I_{n + 2} = \\ldots$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1371, "type": "theorem", "label": "algebra-lemma-descent-reduced", "categories": [ "algebra" ], "title": "algebra-lemma-descent-reduced", "contents": [ "Let $R \\to S$ be a ring map.", "Assume that", "\\begin{enumerate}", "\\item $R \\to S$ is faithfully flat, and", "\\item $S$ is reduced.", "\\end{enumerate}", "Then $R$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "This is clear as $R \\to S$ is injective." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1372, "type": "theorem", "label": "algebra-lemma-descent-normal", "categories": [ "algebra" ], "title": "algebra-lemma-descent-normal", "contents": [ "Let $R \\to S$ be a ring map.", "Assume that", "\\begin{enumerate}", "\\item $R \\to S$ is faithfully flat, and", "\\item $S$ is a normal ring.", "\\end{enumerate}", "Then $R$ is a normal ring." ], "refs": [], "proofs": [ { "contents": [ "Since $S$ is reduced it follows that $R$ is reduced.", "Let $\\mathfrak p$ be a prime of $R$. We have to show that", "$R_{\\mathfrak p}$ is a normal domain. Since $S_{\\mathfrak p}$", "is faithfully over $R_{\\mathfrak p}$ too we may assume that", "$R$ is local with maximal ideal $\\mathfrak m$.", "Let $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak m$.", "Then we see that $R \\to S_{\\mathfrak q}$ is faithfully flat", "(Lemma \\ref{lemma-local-flat-ff}).", "Hence we may assume $S$ is local as well.", "In particular $S$ is a normal domain.", "Since $R \\to S$ is faithfully flat", "and $S$ is a normal domain we see that $R$ is a domain.", "Next, suppose that $a/b$ is integral over $R$ with $a, b \\in R$.", "Then $a/b \\in S$ as $S$ is normal. Hence $a \\in bS$.", "This means that $a : R \\to R/bR$ becomes the zero map", "after base change to $S$. By faithful flatness we see that", "$a \\in bR$, so $a/b \\in R$. Hence $R$ is normal." ], "refs": [ "algebra-lemma-local-flat-ff" ], "ref_ids": [ 537 ] } ], "ref_ids": [] }, { "id": 1373, "type": "theorem", "label": "algebra-lemma-descent-regular", "categories": [ "algebra" ], "title": "algebra-lemma-descent-regular", "contents": [ "Let $R \\to S$ be a ring map.", "Assume that", "\\begin{enumerate}", "\\item $R \\to S$ is faithfully flat, and", "\\item $S$ is a regular ring.", "\\end{enumerate}", "Then $R$ is a regular ring." ], "refs": [], "proofs": [ { "contents": [ "We see that $R$ is Noetherian by Lemma \\ref{lemma-descent-Noetherian}.", "Let $\\mathfrak p \\subset R$ be a prime. Choose a prime $\\mathfrak q \\subset S$", "lying over $\\mathfrak p$. Then Lemma \\ref{lemma-flat-under-regular}", "applies to $R_\\mathfrak p \\to S_\\mathfrak q$ and we conclude that", "$R_\\mathfrak p$ is regular. Since $\\mathfrak p$ was arbitrary we see", "$R$ is regular." ], "refs": [ "algebra-lemma-descent-Noetherian", "algebra-lemma-flat-under-regular" ], "ref_ids": [ 1370, 981 ] } ], "ref_ids": [] }, { "id": 1374, "type": "theorem", "label": "algebra-lemma-descent-Sk", "categories": [ "algebra" ], "title": "algebra-lemma-descent-Sk", "contents": [ "Let $R \\to S$ be a ring map.", "Assume that", "\\begin{enumerate}", "\\item $R \\to S$ is faithfully flat, and", "\\item $S$ is Noetherian and has property $(S_k)$.", "\\end{enumerate}", "Then $R$ is Noetherian and has property $(S_k)$." ], "refs": [], "proofs": [ { "contents": [ "We have already seen that (1) and (2) imply that $R$ is Noetherian,", "see Lemma \\ref{lemma-descent-Noetherian}.", "Let $\\mathfrak p \\subset R$ be a prime ideal.", "Choose a prime $\\mathfrak q \\subset S$ lying over $\\mathfrak p$", "which corresponds to a minimal prime of the fibre ring", "$S \\otimes_R \\kappa(\\mathfrak p)$. Then", "$A = R_{\\mathfrak p} \\to S_{\\mathfrak q} = B$ is a flat local ring", "homomorphism of Noetherian local rings with $\\mathfrak m_AB$ an", "ideal of definition of $B$. Hence", "$\\dim(A) = \\dim(B)$ (Lemma \\ref{lemma-dimension-base-fibre-equals-total}) and", "$\\text{depth}(A) = \\text{depth}(B)$ (Lemma \\ref{lemma-apply-grothendieck}).", "Hence since $B$ has $(S_k)$ we", "see that $A$ has $(S_k)$." ], "refs": [ "algebra-lemma-descent-Noetherian", "algebra-lemma-dimension-base-fibre-equals-total", "algebra-lemma-apply-grothendieck" ], "ref_ids": [ 1370, 987, 1361 ] } ], "ref_ids": [] }, { "id": 1375, "type": "theorem", "label": "algebra-lemma-descent-Rk", "categories": [ "algebra" ], "title": "algebra-lemma-descent-Rk", "contents": [ "Let $R \\to S$ be a ring map. Assume that", "\\begin{enumerate}", "\\item $R \\to S$ is faithfully flat, and", "\\item $S$ is Noetherian and has property $(R_k)$.", "\\end{enumerate}", "Then $R$ is Noetherian and has property $(R_k)$." ], "refs": [], "proofs": [ { "contents": [ "We have already seen that (1) and (2) imply that $R$ is Noetherian,", "see Lemma \\ref{lemma-descent-Noetherian}.", "Let $\\mathfrak p \\subset R$ be a prime ideal and assume", "$\\dim(R_{\\mathfrak p}) \\leq k$.", "Choose a prime $\\mathfrak q \\subset S$ lying over $\\mathfrak p$", "which corresponds to a minimal prime of the fibre ring", "$S \\otimes_R \\kappa(\\mathfrak p)$. Then", "$A = R_{\\mathfrak p} \\to S_{\\mathfrak q} = B$ is a flat local ring", "homomorphism of Noetherian local rings with $\\mathfrak m_AB$ an", "ideal of definition of $B$. Hence", "$\\dim(A) = \\dim(B)$ (Lemma \\ref{lemma-dimension-base-fibre-equals-total}).", "As $S$ has $(R_k)$ we conclude that $B$ is a regular local ring.", "By Lemma \\ref{lemma-flat-under-regular} we conclude that $A$ is regular." ], "refs": [ "algebra-lemma-descent-Noetherian", "algebra-lemma-dimension-base-fibre-equals-total", "algebra-lemma-flat-under-regular" ], "ref_ids": [ 1370, 987, 981 ] } ], "ref_ids": [] }, { "id": 1376, "type": "theorem", "label": "algebra-lemma-descent-nagata", "categories": [ "algebra" ], "title": "algebra-lemma-descent-nagata", "contents": [ "Let $R \\to S$ be a ring map. Assume that", "\\begin{enumerate}", "\\item $R \\to S$ is smooth and surjective on spectra, and", "\\item $S$ is a Nagata ring.", "\\end{enumerate}", "Then $R$ is a Nagata ring." ], "refs": [], "proofs": [ { "contents": [ "Recall that a Nagata ring is the same thing as a Noetherian", "universally Japanese ring", "(Proposition \\ref{proposition-nagata-universally-japanese}).", "We have already seen that $R$ is Noetherian in", "Lemma \\ref{lemma-descent-Noetherian}.", "Let $R \\to A$ be a finite type ring map into a domain.", "According to Lemma \\ref{lemma-check-universally-japanese}", "it suffices to check that $A$ is N-1.", "It is clear that $B = A \\otimes_R S$ is a finite type $S$-algebra", "and hence Nagata (Proposition \\ref{proposition-nagata-universally-japanese}).", "Since $A \\to B$ is smooth (Lemma \\ref{lemma-base-change-smooth})", "we see that $B$ is reduced (Lemma \\ref{lemma-reduced-goes-up}).", "Since $B$ is Noetherian it has only a finite number of minimal", "primes $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ (see", "Lemma \\ref{lemma-Noetherian-irreducible-components}).", "As $A \\to B$ is flat each of these lies over $(0) \\subset A$", "(by going down, see Lemma \\ref{lemma-flat-going-down})", "The total ring of fractions $Q(B)$ is the product of the", "$L_i = \\kappa(\\mathfrak q_i)$ (Lemmas", "\\ref{lemma-total-ring-fractions-no-embedded-points} and", "\\ref{lemma-minimal-prime-reduced-ring}).", "Moreover, the integral closure $B'$ of $B$ in $Q(B)$ is", "the product of the integral closures $B_i'$ of the $B/\\mathfrak q_i$", "in the factors $L_i$ (compare with", "Lemma \\ref{lemma-characterize-reduced-ring-normal}).", "Since $B$ is universally Japanese the", "ring extensions $B/\\mathfrak q_i \\subset B_i'$ are finite", "and we conclude that $B' = \\prod B_i'$ is finite over $B$.", "Since $A \\to B$ is flat we see that any", "nonzerodivisor on $A$ maps to a nonzerodivisor on $B$.", "The corresponding map", "$$", "Q(A) \\otimes_A B = (A \\setminus \\{0\\})^{-1}A \\otimes_A B", "= (A \\setminus \\{0\\})^{-1}B \\to Q(B)", "$$", "is injective (we used Lemma \\ref{lemma-tensor-localization}).", "Via this map $A'$ maps into $B'$. This induces a map", "$$", "A' \\otimes_A B \\longrightarrow B'", "$$", "which is injective (by the above and the flatness of $A \\to B$).", "Since $B'$ is a finite $B$-module", "and $B$ is Noetherian we see that $A' \\otimes_A B$ is a finite $B$-module.", "Hence there exist finitely many elements $x_i \\in A'$ such that", "the elements $x_i \\otimes 1$ generate $A' \\otimes_A B$ as a $B$-module.", "Finally, by faithful flatness of $A \\to B$ we conclude that", "the $x_i$ also generated $A'$ as an $A$-module, and we win." ], "refs": [ "algebra-proposition-nagata-universally-japanese", "algebra-lemma-descent-Noetherian", "algebra-lemma-check-universally-japanese", "algebra-proposition-nagata-universally-japanese", "algebra-lemma-base-change-smooth", "algebra-lemma-reduced-goes-up", "algebra-lemma-Noetherian-irreducible-components", "algebra-lemma-flat-going-down", "algebra-lemma-total-ring-fractions-no-embedded-points", "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-characterize-reduced-ring-normal", "algebra-lemma-tensor-localization" ], "ref_ids": [ 1430, 1370, 1348, 1430, 1191, 1366, 453, 539, 421, 418, 515, 366 ] } ], "ref_ids": [] }, { "id": 1377, "type": "theorem", "label": "algebra-lemma-geometrically-normal", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-normal", "contents": [ "Let $k$ be a field. Let $A$ be a $k$-algebra.", "The following properties of $A$ are equivalent:", "\\begin{enumerate}", "\\item $k' \\otimes_k A$ is a normal ring", "for every field extension $k'/k$,", "\\item $k' \\otimes_k A$ is a normal ring", "for every finitely generated field extension $k'/k$,", "\\item $k' \\otimes_k A$ is a normal ring", "for every finite purely inseparable extension $k'/k$,", "\\item $k^{perf} \\otimes_k A$ is a normal ring.", "\\end{enumerate}", "Here normal ring is defined in Definition \\ref{definition-ring-normal}." ], "refs": [ "algebra-definition-ring-normal" ], "proofs": [ { "contents": [ "It is clear that (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3)", "and (1) $\\Rightarrow$ (4).", "\\medskip\\noindent", "If $k'/k$ is a finite purely inseparable extension, then", "there is an embedding $k' \\to k^{perf}$ of $k$-extensions.", "The ring map $k' \\otimes_k A \\to k^{perf} \\otimes_k A$", "is faithfully flat, hence $k' \\otimes_k A$ is normal if", "$k^{perf} \\otimes_k A$ is normal by", "Lemma \\ref{lemma-descent-normal}. In this way we see that", "(4) $\\Rightarrow$ (3).", "\\medskip\\noindent", "Assume (2) and let $k \\subset k'$ be any field extension.", "Then we can write $k' = \\colim_i k_i$ as a directed", "colimit of finitely generated field extensions. Hence we", "see that $k' \\otimes_k A = \\colim_i k_i \\otimes_k A$", "is a directed colimit of normal rings. Thus we see", "that $k' \\otimes_k A$ is a normal ring by", "Lemma \\ref{lemma-colimit-normal-ring}.", "Hence (1) holds.", "\\medskip\\noindent", "Assume (3) and let $k \\subset K$ be a finitely generated field extension.", "By Lemma \\ref{lemma-make-separable} we can find a diagram", "$$", "\\xymatrix{", "K \\ar[r] & K' \\\\", "k \\ar[u] \\ar[r] & k' \\ar[u]", "}", "$$", "where $k \\subset k'$, $K \\subset K'$ are finite purely inseparable field", "extensions such that $k' \\subset K'$ is separable. By", "Lemma \\ref{lemma-localization-smooth-separable}", "there exists a smooth $k'$-algebra $B$ such that $K'$ is the", "fraction field of $B$. Now we can argue as follows:", "Step 1: $k' \\otimes_k A$ is a normal ring because we assumed (3).", "Step 2: $B \\otimes_{k'} k' \\otimes_k A$ is a normal ring as", "$k' \\otimes_k A \\to B \\otimes_{k'} k' \\otimes_k A$ is smooth", "(Lemma \\ref{lemma-base-change-smooth})", "and ascent of normality along smooth maps", "(Lemma \\ref{lemma-normal-goes-up}).", "Step 3. $K' \\otimes_{k'} k' \\otimes_k A = K' \\otimes_k A$ is", "a normal ring as it is a localization of a normal ring", "(Lemma \\ref{lemma-localization-normal-ring}).", "Step 4. Finally $K \\otimes_k A$ is a normal ring by descent of", "normality along the faithfully flat ring map", "$K \\otimes_k A \\to K' \\otimes_k A$ (Lemma \\ref{lemma-descent-normal}).", "This proves the lemma." ], "refs": [ "algebra-lemma-descent-normal", "algebra-lemma-colimit-normal-ring", "algebra-lemma-make-separable", "algebra-lemma-localization-smooth-separable", "algebra-lemma-base-change-smooth", "algebra-lemma-normal-goes-up", "algebra-lemma-localization-normal-ring", "algebra-lemma-descent-normal" ], "ref_ids": [ 1372, 516, 573, 1322, 1191, 1368, 512, 1372 ] } ], "ref_ids": [ 1454 ] }, { "id": 1378, "type": "theorem", "label": "algebra-lemma-localization-geometrically-normal-algebra", "categories": [ "algebra" ], "title": "algebra-lemma-localization-geometrically-normal-algebra", "contents": [ "\\begin{slogan}", "Localization preserves geometric normality.", "\\end{slogan}", "Let $k$ be a field. A localization of a geometrically normal $k$-algebra", "is geometrically normal." ], "refs": [], "proofs": [ { "contents": [ "This is clear as being a normal ring is checked at the localizations at", "prime ideals." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1379, "type": "theorem", "label": "algebra-lemma-separable-field-extension-geometrically-normal", "categories": [ "algebra" ], "title": "algebra-lemma-separable-field-extension-geometrically-normal", "contents": [ "Let $k$ be a field. Let $K/k$ be a separable field extension.", "Then $K$ is geometrically normal over $k$." ], "refs": [], "proofs": [ { "contents": [ "This is true because $k^{perf} \\otimes_k K$ is a field.", "Namely, it is reduced for example by", "Lemma \\ref{lemma-characterize-separable-field-extensions}", "and it has a unique prime ideal because $K \\subset k^{perf} \\otimes_k K$", "is a universal homeomorphism." ], "refs": [ "algebra-lemma-characterize-separable-field-extensions" ], "ref_ids": [ 569 ] } ], "ref_ids": [] }, { "id": 1380, "type": "theorem", "label": "algebra-lemma-geometrically-normal-tensor-normal", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-normal-tensor-normal", "contents": [ "Let $k$ be a field. Let $A, B$ be $k$-algebras. Assume $A$ is geometrically", "normal over $k$ and $B$ is a normal ring. Then $A \\otimes_k B$ is a normal", "ring." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak r$ be a prime ideal of $A \\otimes_k B$. Denote", "$\\mathfrak p$, resp.\\ $\\mathfrak q$ the corresponding prime of $A$,", "resp.\\ $B$. Then $(A \\otimes_k B)_{\\mathfrak r}$ is a localization of", "$A_{\\mathfrak p} \\otimes_k B_{\\mathfrak q}$. Hence it suffices to prove the", "result for the ring $A_{\\mathfrak p} \\otimes_k B_{\\mathfrak q}$, see", "Lemma \\ref{lemma-localization-normal-ring}", "and", "Lemma \\ref{lemma-localization-geometrically-normal-algebra}.", "Thus we may assume $A$ and $B$ are domains.", "\\medskip\\noindent", "Assume that $A$ and $B$ are domains with fractions fields $K$ and $L$.", "Note that $B$ is the filtered colimit of its finite type normal", "$k$-sub algebras (as $k$ is a Nagata ring, see", "Proposition \\ref{proposition-ubiquity-nagata},", "and hence the integral closure of a finite type $k$-sub algebra is still", "a finite type $k$-sub algebra by", "Proposition \\ref{proposition-nagata-universally-japanese}).", "By", "Lemma \\ref{lemma-colimit-normal-ring}", "we reduce to the case that $B$ is of finite type over $k$.", "\\medskip\\noindent", "Assume that $A$ and $B$ are domains with fractions fields $K$ and $L$", "and $B$ of finite type over $k$. In this case the ring $K \\otimes_k B$", "is of finite type over $K$, hence Noetherian", "(Lemma \\ref{lemma-Noetherian-permanence}).", "In particular $K \\otimes_k B$ has finitely many minimal primes", "(Lemma \\ref{lemma-Noetherian-irreducible-components}).", "Since $A \\to A \\otimes_k B$ is flat, this implies that $A \\otimes_k B$", "has finitely many minimal primes (by going down for flat ring maps --", "Lemma \\ref{lemma-flat-going-down}", "-- these primes all lie over $(0) \\subset A$). Thus it suffices to prove", "that $A \\otimes_k B$ is integrally closed in its total ring of fractions", "(Lemma \\ref{lemma-characterize-reduced-ring-normal}).", "\\medskip\\noindent", "We claim that $K \\otimes_k B$ and $A \\otimes_k L$ are both normal rings.", "If this is true then any element $x$ of $Q(A \\otimes_k B)$ which is", "integral over $A \\otimes_k B$ is (by", "Lemma \\ref{lemma-normal-ring-integrally-closed})", "contained in $K \\otimes_k B \\cap A \\otimes_k L = A \\otimes_k B$ and we're done.", "Since $A \\otimes_K L$ is a normal ring by assumption, it suffices to", "prove that $K \\otimes_k B$ is normal.", "\\medskip\\noindent", "As $A$ is geometrically normal over $k$ we see $K$ is geometrically normal", "over $k$", "(Lemma \\ref{lemma-localization-geometrically-normal-algebra})", "hence $K$ is geometrically reduced over $k$.", "Hence $K = \\bigcup K_i$ is the union of finitely generated field extensions", "of $k$ which are geometrically reduced", "(Lemma \\ref{lemma-subalgebra-separable}).", "Each $K_i$ is the localization of a smooth $k$-algebra", "(Lemma \\ref{lemma-localization-smooth-separable}).", "So $K_i \\otimes_k B$ is the localization of a smooth $B$-algebra hence normal", "(Lemma \\ref{lemma-normal-goes-up}).", "Thus $K \\otimes_k B$ is a normal ring", "(Lemma \\ref{lemma-colimit-normal-ring})", "and we win." ], "refs": [ "algebra-lemma-localization-normal-ring", "algebra-lemma-localization-geometrically-normal-algebra", "algebra-proposition-ubiquity-nagata", "algebra-proposition-nagata-universally-japanese", "algebra-lemma-colimit-normal-ring", "algebra-lemma-Noetherian-permanence", "algebra-lemma-Noetherian-irreducible-components", "algebra-lemma-flat-going-down", "algebra-lemma-characterize-reduced-ring-normal", "algebra-lemma-normal-ring-integrally-closed", "algebra-lemma-localization-geometrically-normal-algebra", "algebra-lemma-subalgebra-separable", "algebra-lemma-localization-smooth-separable", "algebra-lemma-normal-goes-up", "algebra-lemma-colimit-normal-ring" ], "ref_ids": [ 512, 1378, 1431, 1430, 516, 448, 453, 539, 515, 511, 1378, 561, 1322, 1368, 516 ] } ], "ref_ids": [] }, { "id": 1381, "type": "theorem", "label": "algebra-lemma-geometrically-normal-over-separable-algebraic", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-normal-over-separable-algebraic", "contents": [ "Let $k \\subset k'$ be a separable algebraic field extension.", "Let $A$ be an algebra over $k'$. Then $A$ is geometrically normal", "over $k$ if and only if it is geometrically normal over $k'$." ], "refs": [], "proofs": [ { "contents": [ "Let $k \\subset L$ be a finite purely inseparable field extension.", "Then $L' = k' \\otimes_k L$ is a field (see material in", "Fields, Section \\ref{fields-section-algebraic})", "and $A \\otimes_k L = A \\otimes_{k'} L'$. Hence if", "$A$ is geometrically normal over $k'$, then $A$ is geometrically", "normal over $k$.", "\\medskip\\noindent", "Assume $A$ is geometrically normal over $k$. Let $K/k'$ be a field", "extension. Then", "$$", "K \\otimes_{k'} A = (K \\otimes_k A) \\otimes_{(k' \\otimes_k k')} k'", "$$", "Since $k' \\otimes_k k' \\to k'$ is a localization by", "Lemma \\ref{lemma-separable-algebraic-diagonal},", "we see that $K \\otimes_{k'} A$", "is a localization of a normal ring, hence normal." ], "refs": [ "algebra-lemma-separable-algebraic-diagonal" ], "ref_ids": [ 567 ] } ], "ref_ids": [] }, { "id": 1382, "type": "theorem", "label": "algebra-lemma-geometrically-regular", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-regular", "contents": [ "Let $k$ be a field. Let $A$ be a $k$-algebra.", "Assume $A$ is Noetherian.", "The following properties of $A$ are equivalent:", "\\begin{enumerate}", "\\item $k' \\otimes_k A$ is regular for every finitely generated field", "extension $k \\subset k'$, and", "\\item $k' \\otimes_k A$ is regular for every finite purely inseparable", "extension $k \\subset k'$.", "\\end{enumerate}", "Here regular ring is as in Definition \\ref{definition-regular}." ], "refs": [ "algebra-definition-regular" ], "proofs": [ { "contents": [ "The lemma makes sense by the remarks preceding the lemma.", "It is clear that (1) $\\Rightarrow$ (2).", "\\medskip\\noindent", "Assume (2) and let $k \\subset K$ be a finitely generated field extension.", "By Lemma \\ref{lemma-make-separable} we can find a diagram", "$$", "\\xymatrix{", "K \\ar[r] & K' \\\\", "k \\ar[u] \\ar[r] & k' \\ar[u]", "}", "$$", "where $k \\subset k'$, $K \\subset K'$ are finite purely inseparable field", "extensions such that $k' \\subset K'$ is separable. By", "Lemma \\ref{lemma-localization-smooth-separable}", "there exists a smooth $k'$-algebra $B$ such that $K'$ is the", "fraction field of $B$. Now we can argue as follows:", "Step 1: $k' \\otimes_k A$ is a regular ring because we assumed (2).", "Step 2: $B \\otimes_{k'} k' \\otimes_k A$ is a regular ring as", "$k' \\otimes_k A \\to B \\otimes_{k'} k' \\otimes_k A$ is smooth", "(Lemma \\ref{lemma-base-change-smooth})", "and ascent of regularity along smooth maps", "(Lemma \\ref{lemma-regular-goes-up}).", "Step 3. $K' \\otimes_{k'} k' \\otimes_k A = K' \\otimes_k A$ is", "a regular ring as it is a localization of a regular ring", "(immediate from the definition).", "Step 4. Finally $K \\otimes_k A$ is a regular ring by descent of", "regularity along the faithfully flat ring map", "$K \\otimes_k A \\to K' \\otimes_k A$ (Lemma \\ref{lemma-descent-regular}).", "This proves the lemma." ], "refs": [ "algebra-lemma-make-separable", "algebra-lemma-localization-smooth-separable", "algebra-lemma-base-change-smooth", "algebra-lemma-regular-goes-up", "algebra-lemma-descent-regular" ], "ref_ids": [ 573, 1322, 1191, 1369, 1373 ] } ], "ref_ids": [ 1512 ] }, { "id": 1383, "type": "theorem", "label": "algebra-lemma-geometrically-regular-descent", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-regular-descent", "contents": [ "\\begin{slogan}", "Geometric regularity descends through faithfully flat maps of algebras", "\\end{slogan}", "Let $k$ be a field. Let $A \\to B$ be a faithfully flat $k$-algebra", "map. If $B$ is geometrically regular over $k$, so is $A$." ], "refs": [], "proofs": [ { "contents": [ "Assume $B$ is geometrically regular over $k$.", "Let $k \\subset k'$ be a finite, purely inseparable extension.", "Then $A \\otimes_k k' \\to B \\otimes_k k'$ is faithfully flat as a", "base change of $A \\to B$ (by", "Lemmas \\ref{lemma-surjective-spec-radical-ideal} and", "\\ref{lemma-flat-base-change})", "and $B \\otimes_k k'$ is regular by our", "assumption on $B$ over $k$. Then $A \\otimes_k k'$ is regular by", "Lemma \\ref{lemma-descent-regular}." ], "refs": [ "algebra-lemma-surjective-spec-radical-ideal", "algebra-lemma-flat-base-change", "algebra-lemma-descent-regular" ], "ref_ids": [ 443, 527, 1373 ] } ], "ref_ids": [] }, { "id": 1384, "type": "theorem", "label": "algebra-lemma-geometrically-regular-goes-up", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-regular-goes-up", "contents": [ "Let $k$ be a field. Let $A \\to B$ be a smooth ring map", "of $k$-algebras. If $A$ is geometrically regular over $k$,", "then $B$ is geometrically regular over $k$." ], "refs": [], "proofs": [ { "contents": [ "Let $k \\subset k'$ be a finitely generated field extension.", "Then $A \\otimes_k k' \\to B \\otimes_k k'$ is a smooth ring map", "(Lemma \\ref{lemma-base-change-smooth}) and $A \\otimes_k k'$", "is regular. Hence $B \\otimes_k k'$ is regular by", "Lemma \\ref{lemma-regular-goes-up}." ], "refs": [ "algebra-lemma-base-change-smooth", "algebra-lemma-regular-goes-up" ], "ref_ids": [ 1191, 1369 ] } ], "ref_ids": [] }, { "id": 1385, "type": "theorem", "label": "algebra-lemma-geometrically-regular-over-subfields", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-regular-over-subfields", "contents": [ "Let $k$ be a field. Let $A$ be an algebra over $k$.", "Let $k = \\colim k_i$ be a directed colimit of subfields.", "If $A$ is geometrically regular over each $k_i$, then", "$A$ is geometrically regular over $k$." ], "refs": [], "proofs": [ { "contents": [ "Let $k \\subset k'$ be a finite purely inseparable field extension.", "We can get $k'$ by adjoining finitely many variables to $k$ and", "imposing finitely many polynomial relations. Hence we see that", "there exists an $i$ and a finite purely inseparable field extension", "$k_i \\subset k_i'$ such that $k_i = k \\otimes_{k_i} k_i'$.", "Thus $A \\otimes_k k' = A \\otimes_{k_i} k_i'$ and the lemma is clear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1386, "type": "theorem", "label": "algebra-lemma-geometrically-regular-over-separable-algebraic", "categories": [ "algebra" ], "title": "algebra-lemma-geometrically-regular-over-separable-algebraic", "contents": [ "Let $k \\subset k'$ be a separable algebraic field extension.", "Let $A$ be an algebra over $k'$. Then $A$ is geometrically", "regular over $k$ if and only if it is geometrically regular over $k'$." ], "refs": [], "proofs": [ { "contents": [ "Let $k \\subset L$ be a finite purely inseparable field extension.", "Then $L' = k' \\otimes_k L$ is a field (see material in", "Fields, Section \\ref{fields-section-algebraic})", "and $A \\otimes_k L = A \\otimes_{k'} L'$. Hence if", "$A$ is geometrically regular over $k'$, then $A$ is geometrically", "regular over $k$.", "\\medskip\\noindent", "Assume $A$ is geometrically regular over $k$. Since $k'$", "is the filtered colimit of finite extensions of $k$ we may", "assume by Lemma \\ref{lemma-geometrically-regular-over-subfields}", "that $k'/k$ is finite separable. Consider the ring maps", "$$", "k' \\to A \\otimes_k k' \\to A.", "$$", "Note that $A \\otimes_k k'$ is geometrically regular over $k'$", "as a base change of $A$ to $k'$. Note that $A \\otimes_k k' \\to A$", "is the base change of $k' \\otimes_k k' \\to k'$ by the map", "$k' \\to A$. Since $k'/k$ is an \\'etale extension of rings, we", "see that $k' \\otimes_k k' \\to k'$ is \\'etale", "(Lemma \\ref{lemma-etale}). Hence $A$ is", "geometrically regular over $k'$ by", "Lemma \\ref{lemma-geometrically-regular-goes-up}." ], "refs": [ "algebra-lemma-geometrically-regular-over-subfields", "algebra-lemma-etale", "algebra-lemma-geometrically-regular-goes-up" ], "ref_ids": [ 1385, 1231, 1384 ] } ], "ref_ids": [] }, { "id": 1387, "type": "theorem", "label": "algebra-lemma-tensor-fields-CM", "categories": [ "algebra" ], "title": "algebra-lemma-tensor-fields-CM", "contents": [ "Let $k$ be a field and let $k \\subset K$ and $k \\subset L$ be", "two field extensions such that one of them is a field extension of finite type.", "Then $K \\otimes_k L$ is a Noetherian Cohen-Macaulay ring." ], "refs": [], "proofs": [ { "contents": [ "The ring $K \\otimes_k L$ is Noetherian by", "Lemma \\ref{lemma-Noetherian-field-extension}.", "Say $K$ is a finite extension of the purely transcendental extension", "$k(t_1, \\ldots, t_r)$. Then", "$k(t_1, \\ldots, t_r) \\otimes_k L \\to K \\otimes_k L$", "is a finite free ring map. By", "Lemma \\ref{lemma-finite-flat-over-regular-CM}", "it suffices to show that $k(t_1, \\ldots, t_r) \\otimes_k L$ is Cohen-Macaulay.", "This is clear because it is a localization of the polynomial", "ring $L[t_1, \\ldots, t_r]$. (See for example", "Lemma \\ref{lemma-CM-polynomial-algebra}", "for the fact that a polynomial ring is Cohen-Macaulay.)" ], "refs": [ "algebra-lemma-Noetherian-field-extension", "algebra-lemma-finite-flat-over-regular-CM", "algebra-lemma-CM-polynomial-algebra" ], "ref_ids": [ 455, 989, 927 ] } ], "ref_ids": [] }, { "id": 1388, "type": "theorem", "label": "algebra-lemma-CM-geometrically-CM", "categories": [ "algebra" ], "title": "algebra-lemma-CM-geometrically-CM", "contents": [ "Let $k$ be a field. Let $S$ be a Noetherian $k$-algebra.", "Let $k \\subset K$ be a finitely generated field extension,", "and set $S_K = K \\otimes_k S$. Let $\\mathfrak q \\subset S$", "be a prime of $S$. Let $\\mathfrak q_K \\subset S_K$ be a prime", "of $S_K$ lying over $\\mathfrak q$. Then $S_{\\mathfrak q}$ is Cohen-Macaulay", "if and only if $(S_K)_{\\mathfrak q_K}$ is Cohen-Macaulay." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-Noetherian-field-extension}", "the ring $S_K$ is Noetherian. Hence", "$S_{\\mathfrak q} \\to (S_K)_{\\mathfrak q_K}$ is a flat local homomorphism", "of Noetherian local rings. Note that the fibre", "$$", "(S_K)_{\\mathfrak q_K} / \\mathfrak q (S_K)_{\\mathfrak q_K}", "\\cong (\\kappa(\\mathfrak q) \\otimes_k K)_{\\mathfrak q'}", "$$", "is the localization of the Cohen-Macaulay (Lemma \\ref{lemma-tensor-fields-CM})", "ring $\\kappa(\\mathfrak q) \\otimes_k K$ at a suitable prime ideal", "$\\mathfrak q'$. Hence the lemma follows from Lemma \\ref{lemma-CM-goes-up}." ], "refs": [ "algebra-lemma-Noetherian-field-extension", "algebra-lemma-tensor-fields-CM", "algebra-lemma-CM-goes-up" ], "ref_ids": [ 455, 1387, 1362 ] } ], "ref_ids": [] }, { "id": 1389, "type": "theorem", "label": "algebra-lemma-flat-finite-presentation-limit-flat", "categories": [ "algebra" ], "title": "algebra-lemma-flat-finite-presentation-limit-flat", "contents": [ "Let $R \\to S$ be a ring map.", "Let $M$ be an $S$-module.", "Assume that", "\\begin{enumerate}", "\\item $R \\to S$ is of finite presentation,", "\\item $M$ is a finitely presented $S$-module, and", "\\item $M$ is flat over $R$.", "\\end{enumerate}", "In this case we have the following:", "\\begin{enumerate}", "\\item There exists a finite type $\\mathbf{Z}$-algebra $R_0$ and", "a finite type ring map $R_0 \\to S_0$ and a finite $S_0$-module $M_0$", "such that $M_0$ is flat over $R_0$, together with a ring maps", "$R_0 \\to R$ and $S_0 \\to S$ and an $S_0$-module map $M_0 \\to M$", "such that $S \\cong R \\otimes_{R_0} S_0$ and $M = S \\otimes_{S_0} M_0$.", "\\item If $R = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$ is written", "as a directed colimit, then there exists a $\\lambda$ and a ring map", "$R_\\lambda \\to S_\\lambda$ of finite presentation, and an $S_\\lambda$-module", "$M_\\lambda$ of finite presentation such that $M_\\lambda$ is flat over", "$R_\\lambda$ and such that $S = R \\otimes_{R_\\lambda} S_\\lambda$ and", "$M = S \\otimes_{S_{\\lambda}} M_\\lambda$.", "\\item If", "$$", "(R \\to S, M) =", "\\colim_{\\lambda \\in \\Lambda}", "(R_\\lambda \\to S_\\lambda, M_\\lambda)", "$$", "is written as a directed colimit such that", "\\begin{enumerate}", "\\item $R_\\mu \\otimes_{R_\\lambda} S_\\lambda \\to S_\\mu$ and", "$S_\\mu \\otimes_{S_\\lambda} M_\\lambda \\to M_\\mu$ are isomorphisms", "for $\\mu \\geq \\lambda$,", "\\item $R_\\lambda \\to S_\\lambda$ is of finite presentation,", "\\item $M_\\lambda$ is a finitely presented $S_\\lambda$-module,", "\\end{enumerate}", "then for all sufficiently large $\\lambda$ the module $M_\\lambda$", "is flat over $R_\\lambda$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We first write $(R \\to S, M)$ as the directed colimit of a system", "$(R_\\lambda \\to S_\\lambda, M_\\lambda)$ as in", "as in Lemma \\ref{lemma-limit-module-finite-presentation}.", "Let $\\mathfrak q \\subset S$ be a prime.", "Let $\\mathfrak p \\subset R$, $\\mathfrak q_\\lambda \\subset S_\\lambda$,", "and $\\mathfrak p_\\lambda \\subset R_\\lambda$ the corresponding primes.", "As seen in the proof of Theorem \\ref{theorem-openness-flatness}", "$$", "((R_\\lambda)_{\\mathfrak p_\\lambda},", "(S_\\lambda)_{\\mathfrak q_\\lambda},", "(M_\\lambda)_{\\mathfrak q_{\\lambda}})", "$$", "is a system as in", "Lemma \\ref{lemma-limit-module-essentially-finite-presentation}, and", "hence by Lemma \\ref{lemma-colimit-eventually-flat}", "we see that for some $\\lambda_{\\mathfrak q} \\in \\Lambda$", "for all $\\lambda \\geq \\lambda_{\\mathfrak q}$", "the module $M_\\lambda$ is flat over", "$R_\\lambda$ at the prime $\\mathfrak q_{\\lambda}$.", "\\medskip\\noindent", "By Theorem \\ref{theorem-openness-flatness} we get an open subset", "$U_\\lambda \\subset \\Spec(S_\\lambda)$ such that $M_\\lambda$", "flat over $R_\\lambda$ at all the primes of $U_\\lambda$.", "Denote $V_\\lambda \\subset \\Spec(S)$ the inverse image of", "$U_\\lambda$ under the map $\\Spec(S) \\to \\Spec(S_\\lambda)$.", "The argument above shows that for every $\\mathfrak q \\in \\Spec(S)$", "there exists a $\\lambda_{\\mathfrak q}$ such that", "$\\mathfrak q \\in V_\\lambda$ for all $\\lambda \\geq \\lambda_{\\mathfrak q}$.", "Since $\\Spec(S)$ is quasi-compact we see this implies there", "exists a single $\\lambda_0 \\in \\Lambda$ such that", "$V_{\\lambda_0} = \\Spec(S)$.", "\\medskip\\noindent", "The complement $\\Spec(S_{\\lambda_0}) \\setminus U_{\\lambda_0}$", "is $V(I)$ for some ideal $I \\subset S_{\\lambda_0}$. As", "$V_{\\lambda_0} = \\Spec(S)$ we see that $IS = S$.", "Choose $f_1, \\ldots, f_r \\in I$ and $s_1, \\ldots, s_n \\in S$ such", "that $\\sum f_i s_i = 1$. Since $\\colim S_\\lambda = S$, after", "increasing $\\lambda_0$ we may assume there exist", "$s_{i, \\lambda_0} \\in S_{\\lambda_0}$ such that", "$\\sum f_i s_{i, \\lambda_0} = 1$.", "Hence for this $\\lambda_0$ we have", "$U_{\\lambda_0} = \\Spec(S_{\\lambda_0})$.", "This proves (1).", "\\medskip\\noindent", "Proof of (2). Let $(R_0 \\to S_0, M_0)$ be as in (1) and suppose that", "$R = \\colim R_\\lambda$. Since $R_0$ is a finite type $\\mathbf{Z}$", "algebra, there exists a $\\lambda$ and a map $R_0 \\to R_\\lambda$ such", "that $R_0 \\to R_\\lambda \\to R$ is the given map $R_0 \\to R$ (see", "Lemma \\ref{lemma-characterize-finite-presentation}).", "Then, part (2) follows by taking $S_\\lambda = R_\\lambda \\otimes_{R_0} S_0$", "and $M_\\lambda = S_\\lambda \\otimes_{S_0} M_0$.", "\\medskip\\noindent", "Finally, we come to the proof of (3). Let", "$(R_\\lambda \\to S_\\lambda, M_\\lambda)$ be as in (3). Choose", "$(R_0 \\to S_0, M_0)$ and $R_0 \\to R$ as in (1).", "As in the proof of (2), there exists a $\\lambda_0$ and a ring map", "$R_0 \\to R_{\\lambda_0}$ such that $R_0 \\to R_{\\lambda_0} \\to R$ is the given", "map $R_0 \\to R$. Since $S_0$ is of finite presentation over $R_0$ and since", "$S = \\colim S_\\lambda$ we see that for some $\\lambda_1 \\geq \\lambda_0$", "we get an $R_0$-algebra map $S_0 \\to S_{\\lambda_1}$ such that the", "composition $S_0 \\to S_{\\lambda_1} \\to S$ is the given map $S_0 \\to S$", "(see Lemma \\ref{lemma-characterize-finite-presentation}).", "For all $\\lambda \\geq \\lambda_1$ this gives maps", "$$", "\\Psi_{\\lambda} :", "R_\\lambda \\otimes_{R_0} S_0", "\\longrightarrow", "R_\\lambda \\otimes_{R_{\\lambda_1}} S_{\\lambda_1}", "\\cong", "S_\\lambda", "$$", "the last isomorphism by assumption. By construction", "$\\colim_\\lambda \\Psi_\\lambda$ is an isomorphism. Hence $\\Psi_\\lambda$", "is an isomorphism for all $\\lambda$ large enough by", "Lemma \\ref{lemma-colimit-category-fp-algebras}.", "In the same vein, there exists a $\\lambda_2 \\geq \\lambda_1$", "and an $S_0$-module map $M_0 \\to M_{\\lambda_2}$ such that", "$M_0 \\to M_{\\lambda_2} \\to M$ is the given", "map $M_0 \\to M$ (see Lemma \\ref{lemma-module-map-property-in-colimit}).", "For $\\lambda \\geq \\lambda_2$ there is an induced map", "$$", "S_\\lambda \\otimes_{S_0} M_0", "\\longrightarrow", "S_\\lambda \\otimes_{S_{\\lambda_2}} M_{\\lambda_2}", "\\cong", "M_\\lambda", "$$", "and for $\\lambda$ large enough this map is an isomorphism by", "Lemma \\ref{lemma-colimit-category-fp-modules}.", "This implies (3) because $M_0$ is flat over $R_0$." ], "refs": [ "algebra-lemma-limit-module-finite-presentation", "algebra-theorem-openness-flatness", "algebra-lemma-limit-module-essentially-finite-presentation", "algebra-lemma-colimit-eventually-flat", "algebra-theorem-openness-flatness", "algebra-lemma-characterize-finite-presentation", "algebra-lemma-characterize-finite-presentation", "algebra-lemma-colimit-category-fp-algebras", "algebra-lemma-module-map-property-in-colimit", "algebra-lemma-colimit-category-fp-modules" ], "ref_ids": [ 1106, 326, 1101, 1109, 326, 1092, 1092, 1097, 1094, 1095 ] } ], "ref_ids": [] }, { "id": 1390, "type": "theorem", "label": "algebra-lemma-descend-faithfully-flat-finite-presentation", "categories": [ "algebra" ], "title": "algebra-lemma-descend-faithfully-flat-finite-presentation", "contents": [ "Let $R \\to A \\to B$ be ring maps.", "Assume $A \\to B$ faithfully flat of finite presentation.", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "R \\ar[r] \\ar@{=}[d] &", "A_0 \\ar[d] \\ar[r] &", "B_0 \\ar[d] \\\\", "R \\ar[r] & A \\ar[r] & B", "}", "$$", "with $R \\to A_0$ of finite presentation,", "$A_0 \\to B_0$ faithfully flat of finite presentation", "and $B = A \\otimes_{A_0} B_0$." ], "refs": [], "proofs": [ { "contents": [ "We first prove the lemma with $R$ replaced $\\mathbf{Z}$.", "By Lemma \\ref{lemma-flat-finite-presentation-limit-flat}", "there exists a diagram", "$$", "\\xymatrix{", "A_0 \\ar[r] & A \\\\", "B_0 \\ar[u] \\ar[r] & B \\ar[u]", "}", "$$", "where $A_0$ is of finite type over $\\mathbf{Z}$, $B_0$ is flat of finite", "presentation over $A_0$ such that $B = A \\otimes_{A_0} B_0$.", "As $A_0 \\to B_0$ is flat of finite presentation we see that the image of", "$\\Spec(B_0) \\to \\Spec(A_0)$ is open, see", "Proposition \\ref{proposition-fppf-open}. Hence the complement of the image", "is $V(I_0)$ for some ideal $I_0 \\subset A_0$.", "As $A \\to B$ is faithfully", "flat the map $\\Spec(B) \\to \\Spec(A)$ is surjective, see", "Lemma \\ref{lemma-ff-rings}.", "Now we use that", "the base change of the image is the image of the base change.", "Hence $I_0A = A$. Pick a relation", "$\\sum f_i r_i = 1$, with $r_i \\in A$, $f_i \\in I_0$. Then after", "enlarging $A_0$ to contain the elements $r_i$ (and correspondingly", "enlarging $B_0$) we see that $A_0 \\to B_0$ is surjective on spectra", "also, i.e., faithfully flat.", "\\medskip\\noindent", "Thus the lemma holds in case $R = \\mathbf{Z}$.", "In the general case, take the solution $A_0' \\to B_0'$", "just obtained and set $A_0 = A_0' \\otimes_{\\mathbf{Z}} R$,", "$B_0 = B_0' \\otimes_{\\mathbf{Z}} R$." ], "refs": [ "algebra-lemma-flat-finite-presentation-limit-flat", "algebra-proposition-fppf-open", "algebra-lemma-ff-rings" ], "ref_ids": [ 1389, 1407, 536 ] } ], "ref_ids": [] }, { "id": 1391, "type": "theorem", "label": "algebra-lemma-colimit-finite", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-finite", "contents": [ "Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings.", "Let $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras.", "Assume", "\\begin{enumerate}", "\\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is finite,", "\\item $C_0$ is of finite type over $B_0$.", "\\end{enumerate}", "Then there exists an $i \\geq 0$ such that the map", "$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$", "is finite." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1, \\ldots, x_m$ be generators for $C_0$ over $B_0$.", "Pick monic polynomials $P_j \\in A \\otimes_{A_0} B_0[T]$ such", "that $P_j(1 \\otimes x_j) = 0$ in $A \\otimes_{A_0} C_0$. For some", "$i \\geq 0$ we can find $P_{j, i} \\in A_i \\otimes_{A_0} B_0[T]$", "mapping to $P_j$. Since $\\otimes$", "commutes with colimits we see that $P_{j, i}(1 \\otimes x_j)$ is zero", "in $A_i \\otimes_{A_0} C_0$ after possibly increasing $i$.", "Then this $i$ works." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1392, "type": "theorem", "label": "algebra-lemma-colimit-surjective", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-surjective", "contents": [ "Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings.", "Let $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras.", "Assume", "\\begin{enumerate}", "\\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is surjective,", "\\item $C_0$ is of finite type over $B_0$.", "\\end{enumerate}", "Then for some $i \\geq 0$ the map", "$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1, \\ldots, x_m$ be generators for $C_0$ over $B_0$.", "Pick $b_j \\in A \\otimes_{A_0} B_0$ mapping to $1 \\otimes x_j$ in", "$A \\otimes_{A_0} C_0$. For some $i \\geq 0$ we can find", "$b_{j, i} \\in A_i \\otimes_{A_0} B_0$ mapping to $b_j$.", "Then this $i$ works." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1393, "type": "theorem", "label": "algebra-lemma-colimit-unramified", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-unramified", "contents": [ "Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings.", "Let $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras.", "Assume", "\\begin{enumerate}", "\\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is unramified,", "\\item $C_0$ is of finite type over $B_0$.", "\\end{enumerate}", "Then for some $i \\geq 0$ the map", "$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$", "is unramified." ], "refs": [], "proofs": [ { "contents": [ "Set $B_i = A_i \\otimes_{A_0} B_0$, $C_i = A_i \\otimes_{A_0} C_0$,", "$B = A \\otimes_{A_0} B_0$, and $C = A \\otimes_{A_0} C_0$.", "Let $x_1, \\ldots, x_m$ be generators for $C_0$ over $B_0$.", "Then $\\text{d}x_1, \\ldots, \\text{d}x_m$ generate $\\Omega_{C_0/B_0}$", "over $C_0$ and their images generate $\\Omega_{C_i/B_i}$ over $C_i$", "(Lemmas \\ref{lemma-differentials-polynomial-ring} and", "\\ref{lemma-differential-seq}).", "Observe that $0 = \\Omega_{C/B} = \\colim \\Omega_{C_i/B_i}$", "(Lemma \\ref{lemma-colimit-differentials}).", "Thus there is an $i$ such that $\\text{d}x_1, \\ldots, \\text{d}x_m$", "map to zero and hence $\\Omega_{C_i/B_i} = 0$ as desired." ], "refs": [ "algebra-lemma-differentials-polynomial-ring", "algebra-lemma-differential-seq", "algebra-lemma-colimit-differentials" ], "ref_ids": [ 1140, 1135, 1130 ] } ], "ref_ids": [] }, { "id": 1394, "type": "theorem", "label": "algebra-lemma-colimit-isomorphism", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-isomorphism", "contents": [ "Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings.", "Let $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras.", "Assume", "\\begin{enumerate}", "\\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is", "an isomorphism,", "\\item $B_0 \\to C_0$ is of finite presentation.", "\\end{enumerate}", "Then for some $i \\geq 0$ the map", "$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$ is", "an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-colimit-surjective} there exists an $i$ such that", "$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$ is", "surjective. Since the map is of finite presentation", "the kernel is a finitely generated ideal. Let", "$g_1, \\ldots, g_r \\in A_i \\otimes_{A_0} B_0$ generate the kernel.", "Then we may pick $i' \\geq i$ such that $g_j$ map to zero", "in $A_{i'} \\otimes_{A_0} B_0$. Then", "$A_{i'} \\otimes_{A_0} B_0 \\to A_{i'} \\otimes_{A_0} C_0$ is", "an isomorphism." ], "refs": [ "algebra-lemma-colimit-surjective" ], "ref_ids": [ 1392 ] } ], "ref_ids": [] }, { "id": 1395, "type": "theorem", "label": "algebra-lemma-colimit-etale", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-etale", "contents": [ "Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings.", "Let $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras.", "Assume", "\\begin{enumerate}", "\\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is \\'etale,", "\\item $B_0 \\to C_0$ is of finite presentation.", "\\end{enumerate}", "Then for some $i \\geq 0$ the map", "$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$", "is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Write $C_0 = B_0[x_1, \\ldots, x_n]/(f_{1, 0}, \\ldots, f_{m, 0})$.", "Write $B_i = A_i \\otimes_{A_0} B_0$ and $C_i = A_i \\otimes_{A_0} C_0$.", "Note that $C_i = B_i[x_1, \\ldots, x_n]/(f_{1, i}, \\ldots, f_{m, i})$", "where $f_{j, i}$ is the image of $f_{j, 0}$ in the polynomial ring", "over $B_i$. Write $B = A \\otimes_{A_0} B_0$ and $C = A \\otimes_{A_0} C_0$.", "Note that $C = B[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$", "where $f_j$ is the image of $f_{j, 0}$ in the polynomial ring", "over $B$. The assumption is that the map", "$$", "\\text{d} :", "(f_1, \\ldots, f_m)/(f_1, \\ldots, f_m)^2", "\\longrightarrow", "\\bigoplus C \\text{d}x_k", "$$", "is an isomorphism. Thus for sufficiently large $i$ we can find elements", "$$", "\\xi_{k, i} \\in (f_{1, i}, \\ldots, f_{m, i})/(f_{1, i}, \\ldots, f_{m, i})^2", "$$", "with $\\text{d}\\xi_{k, i} = \\text{d}x_k$ in $\\bigoplus C_i \\text{d}x_k$.", "Moreover, on increasing $i$ if necessary, we see that", "$\\sum (\\partial f_{j, i}/\\partial x_k) \\xi_{k, i} =", "f_{j, i} \\bmod (f_{1, i}, \\ldots, f_{m, i})^2$", "since this is true in the limit. Then this $i$ works." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1396, "type": "theorem", "label": "algebra-lemma-colimit-smooth", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-smooth", "contents": [ "Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings.", "Let $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras.", "Assume", "\\begin{enumerate}", "\\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is smooth,", "\\item $B_0 \\to C_0$ is of finite presentation.", "\\end{enumerate}", "Then for some $i \\geq 0$ the map", "$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "Write $C_0 = B_0[x_1, \\ldots, x_n]/(f_{1, 0}, \\ldots, f_{m, 0})$.", "Write $B_i = A_i \\otimes_{A_0} B_0$ and $C_i = A_i \\otimes_{A_0} C_0$.", "Note that $C_i = B_i[x_1, \\ldots, x_n]/(f_{1, i}, \\ldots, f_{m, i})$", "where $f_{j, i}$ is the image of $f_{j, 0}$ in the polynomial ring", "over $B_i$. Write $B = A \\otimes_{A_0} B_0$ and $C = A \\otimes_{A_0} C_0$.", "Note that $C = B[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$", "where $f_j$ is the image of $f_{j, 0}$ in the polynomial ring", "over $B$. The assumption is that the map", "$$", "\\text{d} :", "(f_1, \\ldots, f_m)/(f_1, \\ldots, f_m)^2", "\\longrightarrow", "\\bigoplus C \\text{d}x_k", "$$", "is a split injection. Let $\\xi_k \\in (f_1, \\ldots, f_m)/(f_1, \\ldots, f_m)^2$", "be elements such that $\\sum (\\partial f_j/\\partial x_k) \\xi_k =", "f_j \\bmod (f_1, \\ldots, f_m)^2$. Then for sufficiently large $i$ we can", "find elements", "$$", "\\xi_{k, i} \\in (f_{1, i}, \\ldots, f_{m, i})/(f_{1, i}, \\ldots, f_{m, i})^2", "$$", "with $\\sum (\\partial f_{j, i}/\\partial x_k) \\xi_{k, i} =", "f_{j, i} \\bmod (f_{1, i}, \\ldots, f_{m, i})^2$", "since this is true in the limit. Then this $i$ works." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1397, "type": "theorem", "label": "algebra-lemma-colimit-lci", "categories": [ "algebra" ], "title": "algebra-lemma-colimit-lci", "contents": [ "Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings.", "Let $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras.", "Assume", "\\begin{enumerate}", "\\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is", "syntomic (resp.\\ a relative global complete intersection),", "\\item $C_0$ is of finite presentation over $B_0$.", "\\end{enumerate}", "Then there exists an $i \\geq 0$ such that the map", "$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$", "is syntomic (resp.\\ a relative global complete intersection)." ], "refs": [], "proofs": [ { "contents": [ "Assume $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is a relative", "global complete intersection.", "By Lemma \\ref{lemma-relative-global-complete-intersection-Noetherian}", "there exists a finite type $\\mathbf{Z}$-algebra $R$,", "a ring map $R \\to A \\otimes_{A_0} B_0$, a relative", "global complete intersection $R \\to S$, and an isomorphism", "$$", "(A \\otimes_{A_0} B_0) \\otimes_R S", "\\longrightarrow", "A \\otimes_{A_0} C_0", "$$", "Because $R$ is of finite type (and hence finite presentation)", "over $\\mathbf{Z}$, there exists an $i$ and a map", "$R \\to A_i \\otimes_{A_0} B_0$ lifting the map $R \\to A \\otimes_{A_0} B_0$,", "see Lemma \\ref{lemma-characterize-finite-presentation}.", "Using the same lemma, there exists an $i' \\geq i$ such that", "$(A_i \\otimes_{A_0} B_0) \\otimes_R S \\to A \\otimes_{A_0} C_0$", "comes from a map", "$(A_i \\otimes_{A_0} B_0) \\otimes_R S \\to A_{i'} \\otimes_{A_0} C_0$.", "Thus we may assume, after replacing $i$ by $i'$,", "that the displayed map comes from an $A_i \\otimes_{A_0} B_0$-algebra map", "$$", "(A_i \\otimes_{A_0} B_0) \\otimes_R S", "\\longrightarrow", "A_i \\otimes_{A_0} C_0", "$$", "By Lemma \\ref{lemma-colimit-isomorphism} after increasing $i$ this", "map is an isomorphism. This finishes the proof in this case because the base", "change of a relative global complete intersection is a relative", "global complete intersection by", "Lemma \\ref{lemma-base-change-relative-global-complete-intersection}.", "\\medskip\\noindent", "Assume $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is syntomic.", "Then there exist elements $g_1, \\ldots, g_m$ in", "$A \\otimes_{A_0} C_0$ generating the unit ideal such that", "$A \\otimes_{A_0} B_0 \\to (A \\otimes_{A_0} C_0)_{g_j}$ is a", "relative global complete intersection, see Lemma \\ref{lemma-syntomic}.", "We can find an $i$ and elements $g_{i, j} \\in A_i \\otimes_{A_0} C_0$", "mapping to $g_j$. After increasing $i$ we may assume", "$g_{i, 1}, \\ldots, g_{i, m}$ generate the unit ideal", "of $A_i \\otimes_{A_0} C_0$. The result of the previous paragraph", "implies that, after increasing $i$, we may assume the maps", "$A_i \\otimes_{A_0} B_0 \\to (A_i \\otimes_{A_0} C_0)_{g_{i, j}}$", "are relative global complete intersections.", "Then $A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$", "is syntomic by Lemma \\ref{lemma-local-syntomic}", "(and the already used Lemma \\ref{lemma-syntomic})." ], "refs": [ "algebra-lemma-relative-global-complete-intersection-Noetherian", "algebra-lemma-characterize-finite-presentation", "algebra-lemma-colimit-isomorphism", "algebra-lemma-base-change-relative-global-complete-intersection", "algebra-lemma-syntomic", "algebra-lemma-local-syntomic", "algebra-lemma-syntomic" ], "ref_ids": [ 1182, 1092, 1394, 1180, 1185, 1177, 1185 ] } ], "ref_ids": [] }, { "id": 1398, "type": "theorem", "label": "algebra-lemma-fppf-fpqf", "categories": [ "algebra" ], "title": "algebra-lemma-fppf-fpqf", "contents": [ "Let $R \\to S$ be a faithfully flat ring map of finite presentation.", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "S \\ar[rr] & & S' \\\\", "& R \\ar[lu] \\ar[ru]", "}", "$$", "where $R \\to S'$ is quasi-finite, faithfully flat and of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "As a first step we reduce this lemma to the case where $R$ is of finite", "type over $\\mathbf{Z}$.", "By Lemma \\ref{lemma-descend-faithfully-flat-finite-presentation}", "there exists a diagram", "$$", "\\xymatrix{", "S_0 \\ar[r] & S \\\\", "R_0 \\ar[u] \\ar[r] & R \\ar[u]", "}", "$$", "where $R_0$ is of finite type over $\\mathbf{Z}$,", "and $S_0$ is faithfully flat of finite presentation over $R_0$", "such that $S = R \\otimes_{R_0} S_0$.", "If we prove the lemma for the ring map $R_0 \\to S_0$, then the lemma", "follows for $R \\to S$ by base change, as the base change of", "a quasi-finite ring map is quasi-finite, see", "Lemma \\ref{lemma-quasi-finite-base-change}. (Of course we", "also use that base changes of flat maps are flat and", "base changes of maps of finite presentation are of finite presentation.)", "\\medskip\\noindent", "Assume $R \\to S$ is a faithfully flat ring map of finite presentation", "and that $R$ is Noetherian (which we may assume by the preceding", "paragraph). Let $W \\subset \\Spec(S)$ be the open set of", "Lemma \\ref{lemma-finite-presentation-flat-CM-locus-open}.", "As $R \\to S$ is faithfully flat the map $\\Spec(S) \\to \\Spec(R)$", "is surjective, see Lemma \\ref{lemma-ff-rings}.", "By Lemma \\ref{lemma-generic-CM-flat-finite-presentation}", "the map $W \\to \\Spec(R)$ is also surjective.", "Hence by replacing $S$ with a product $S_{g_1} \\times \\ldots \\times S_{g_m}$", "we may assume $W = \\Spec(S)$; here we use that $\\Spec(R)$", "is quasi-compact (Lemma \\ref{lemma-quasi-compact}), and that the map", "$\\Spec(S) \\to \\Spec(R)$ is open", "(Proposition \\ref{proposition-fppf-open}).", "Suppose that $\\mathfrak p \\subset R$ is a prime. Choose a prime", "$\\mathfrak q \\subset S$ lying over $\\mathfrak p$ which corresponds", "to a maximal ideal of the fibre ring $S \\otimes_R \\kappa(\\mathfrak p)$.", "The Noetherian local ring", "$\\overline{S}_{\\mathfrak q} = S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}$", "is Cohen-Macaulay, say of dimension $d$. We may choose $f_1, \\ldots, f_d$", "in the maximal ideal of $S_{\\mathfrak q}$ which map to a regular sequence", "in $\\overline{S}_{\\mathfrak q}$. Choose a common denominator", "$g \\in S$, $g \\not \\in \\mathfrak q$ of $f_1, \\ldots, f_d$, and consider", "the $R$-algebra", "$$", "S' = S_g/(f_1, \\ldots, f_d).", "$$", "By construction there is a prime ideal $\\mathfrak q' \\subset S'$", "lying over $\\mathfrak p$ and corresponding to $\\mathfrak q$ (via", "$S_g \\to S'_g$). Also by construction the ring map $R \\to S'$ is", "quasi-finite at $\\mathfrak q$ as the local ring", "$$", "S'_{\\mathfrak q'}/\\mathfrak pS'_{\\mathfrak q'} =", "S_{\\mathfrak q}/(f_1, \\ldots, f_d) + \\mathfrak pS_{\\mathfrak q} =", "\\overline{S}_{\\mathfrak q}/(\\overline{f}_1, \\ldots, \\overline{f}_d)", "$$", "has dimension zero, see Lemma \\ref{lemma-isolated-point-fibre}.", "Also by construction $R \\to S'$ is of finite presentation.", "Finally, by Lemma \\ref{lemma-grothendieck-regular-sequence} the local ring map", "$R_{\\mathfrak p} \\to S'_{\\mathfrak q'}$ is flat (this is where we", "use that $R$ is Noetherian). Hence, by openness of flatness", "(Theorem \\ref{theorem-openness-flatness}), and openness of quasi-finiteness", "(Lemma \\ref{lemma-quasi-finite-open})", "we may after replacing", "$g$ by $gg'$ for a suitable $g' \\in S$, $g' \\not \\in \\mathfrak q$", "assume that $R \\to S'$ is flat and quasi-finite.", "The image $\\Spec(S') \\to \\Spec(R)$ is open and", "contains $\\mathfrak p$. In other words we have shown", "a ring $S'$ as in the statement of the lemma exists (except possibly", "the faithfulness part) whose image contains any given prime.", "Using one more time the quasi-compactness of $\\Spec(R)$", "we see that a finite product of such rings does the job." ], "refs": [ "algebra-lemma-descend-faithfully-flat-finite-presentation", "algebra-lemma-quasi-finite-base-change", "algebra-lemma-finite-presentation-flat-CM-locus-open", "algebra-lemma-ff-rings", "algebra-lemma-generic-CM-flat-finite-presentation", "algebra-lemma-quasi-compact", "algebra-proposition-fppf-open", "algebra-lemma-isolated-point-fibre", "algebra-lemma-grothendieck-regular-sequence", "algebra-theorem-openness-flatness", "algebra-lemma-quasi-finite-open" ], "ref_ids": [ 1390, 1054, 1124, 536, 1125, 395, 1407, 1049, 885, 326, 1066 ] } ], "ref_ids": [] }, { "id": 1399, "type": "theorem", "label": "algebra-proposition-universal-property-localization", "categories": [ "algebra" ], "title": "algebra-proposition-universal-property-localization", "contents": [ "Let $f : A \\to B$ be a ring map that sends every element in $S$ to a unit", "of $B$. Then there is a unique homomorphism $g : S^{-1}A \\to B$ such", "that the following diagram commutes.", "$$", "\\xymatrix{", "A \\ar[rr]^{f} \\ar[dr] & & B \\\\", "& S^{-1}A \\ar[ur]_g", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Existence. We define a map $g$ as follows. For $x/s\\in S^{-1}A$, let", "$g(x/s) = f(x)f(s)^{-1}\\in B$. It is easily checked from the definition", "that this is a well-defined ring map. And it is also clear that", "this makes the diagram commutative.", "\\medskip\\noindent", "Uniqueness. We now show that if $g' : S^{-1}A \\to B$", "satisfies $g'(x/1) = f(x)$, then $g = g'$. Hence $f(s) = g'(s/1)$ for", "$s \\in S$ by the commutativity of the diagram. But then $g'(1/s)f(s) = 1$", "in $B$, which implies that $g'(1/s) = f(s)^{-1}$ and hence", "$g'(x/s) = g'(x/1)g'(1/s) = f(x)f(s)^{-1} = g(x/s)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1400, "type": "theorem", "label": "algebra-proposition-localize-twice", "categories": [ "algebra" ], "title": "algebra-proposition-localize-twice", "contents": [ "Let $\\overline{S}$ be the image of $S$ in $S'^{-1}A$, then", "$(SS')^{-1}A$ is isomorphic to $\\overline{S}^{-1}(S'^{-1}A)$." ], "refs": [], "proofs": [ { "contents": [ "The map sending $x\\in A$ to $x/1\\in (SS')^{-1}A$ induces a map", "sending $x/s\\in S'^{-1}A$ to $x/s \\in (SS')^{-1}A$, by universal", "property. The image of the elements in $\\overline{S}$ are invertible", "in $(SS')^{-1}A$. By the universal property we get a map", "$f : \\overline{S}^{-1}(S'^{-1}A) \\to (SS')^{-1}A$ which maps", "$(x/t')/(s/s')$ to $(x/t')\\cdot(s/s')^{-1}$.", "\\medskip\\noindent", "On the other hand, the map from $A$ to $\\overline{S}^{-1}(S'^{-1}A)$", "sending $x\\in A$ to $(x/1)/(1/1)$ also induces a map", "$g : (SS')^{-1}A \\to \\overline{S}^{-1}(S'^{-1}A)$ which sends $x/ss'$", "to $(x/s')/(s/1)$, by the universal property again. It is", "immediately checked that $f$ and $g$ are inverse to each other,", "hence they are both isomorphisms." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1401, "type": "theorem", "label": "algebra-proposition-localize-twice-module", "categories": [ "algebra" ], "title": "algebra-proposition-localize-twice-module", "contents": [ "View $S'^{-1}M$ as an $A$-module, then $S^{-1}(S'^{-1}M)$ is", "isomorphic to $(SS')^{-1}M$." ], "refs": [], "proofs": [ { "contents": [ "Note that given a $A$-module M, we have not proved any", "universal property for $S^{-1}M$. Hence we cannot reason", "as in the preceding proof; we have to construct the isomorphism explicitly.", "\\medskip\\noindent", "We define the maps as follows", "\\begin{align*}", "& f : S^{-1}(S'^{-1}M) \\longrightarrow (SS')^{-1}M, \\quad \\frac{x/s'}{s}\\mapsto", "x/ss'\\\\", "& g : (SS')^{-1}M \\longrightarrow S^{-1}(S'^{-1}M), \\quad x/t\\mapsto", "\\frac{x/s'}{s}\\ \\text{for some }s\\in S, s'\\in S', \\text{ and }", "t = ss'", "\\end{align*}", "We have to check that these homomorphisms are well-defined, that is,", "independent the choice of the fraction. This is easily checked and it is also", "straightforward to show that they are inverse to each other." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1402, "type": "theorem", "label": "algebra-proposition-localization-exact", "categories": [ "algebra" ], "title": "algebra-proposition-localization-exact", "contents": [ "\\begin{slogan}", "Localization is exact.", "\\end{slogan}", "Let $L\\xrightarrow{u} M\\xrightarrow{v} N$ be an exact sequence", "of $R$-modules. Then", "$S^{-1}L \\to S^{-1}M \\to S^{-1}N$ is also exact." ], "refs": [], "proofs": [ { "contents": [ "First it is clear that $S^{-1}L \\to S^{-1}M \\to S^{-1}N$ is a complex", "since localization is a functor. Next suppose that $x/s$ maps to zero", "in $S^{-1}N$ for some $x/s \\in S^{-1}M$. Then by definition there is a", "$t\\in S$ such that $v(xt) = v(x)t = 0$ in $M$, which means", "$xt \\in \\Ker(v)$. By the exactness of $L \\to M \\to N$ we have", "$xt = u(y)$ for some $y$ in $L$. Then $x/s$ is the image of $y/st$.", "This proves the exactness." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1403, "type": "theorem", "label": "algebra-proposition-localize-quotient", "categories": [ "algebra" ], "title": "algebra-proposition-localize-quotient", "contents": [ "Let $I$ be an ideal of $A$, $S$ a multiplicative set of $A$. Then", "$S^{-1}I$ is an ideal of $S^{-1}A$ and $\\overline{S}^{-1}(A/I)$ is", "isomorphic to $S^{-1}A/S^{-1}I$, where $\\overline{S}$ is", "the image of $S$ in $A/I$." ], "refs": [], "proofs": [ { "contents": [ "The fact that $S^{-1}I$ is an ideal is clear since $I$ itself is an", "ideal. Define", "$$", "f : S^{-1}A\\longrightarrow \\overline{S}^{-1}(A/I), \\quad x/s\\mapsto", "\\overline{x}/\\overline{s}", "$$", "where $\\overline{x}$ and $\\overline{s}$ are the images of $x$ and", "$s$ in $A/I$. We shall keep similar notations in this proof.", "This map is well-defined by the universal property of", "$S^{-1}A$, and $S^{-1}I$ is contained in the kernel of it,", "therefore it induces a map", "$$", "\\overline{f} : S^{-1}A/S^{-1}I \\longrightarrow \\overline{S}^{-1}(A/I), \\quad", "\\overline{x/s}\\mapsto \\overline{x}/\\overline{s}", "$$", "\\medskip\\noindent", "On the other hand, the map $A \\to S^{-1}A/S^{-1}I$ sending $x$ to", "$\\overline{x/1}$ induces a map $A/I \\to S^{-1}A/S^{-1}I$ sending", "$\\overline{x}$ to $\\overline{x/1}$. The image of $\\overline{S}$ is", "invertible in $S^{-1}A/S^{-1}I$, thus induces a map", "$$", "g : \\overline{S}^{-1}(A/I) \\longrightarrow S^{-1}A/S^{-1}I, \\quad", "\\frac{\\overline{x}}{\\overline{s}}\\mapsto \\overline{x/s}", "$$", "by the universal property. It is then clear that $\\overline{f}$ and $g$", "are inverse to each other, hence are both isomorphisms." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1404, "type": "theorem", "label": "algebra-proposition-oka", "categories": [ "algebra" ], "title": "algebra-proposition-oka", "contents": [ "If $\\mathcal{F}$ is an Oka family of ideals, then any maximal element of", "the complement of $\\mathcal{F}$ is prime." ], "refs": [], "proofs": [ { "contents": [ "Suppose $I \\not \\in \\mathcal{F}$ is maximal with respect to not being in", "$\\mathcal{F}$ but $I$ is not prime. Note that $I \\not = R$ because", "$R \\in \\mathcal{F}$. Since $I$ is not prime we can find $a, b \\in R - I$", "with $ab \\in I$. It follows that $(I, a) \\neq I$ and $(I : a)$ contains", "$b \\not \\in I$ so also $(I : a) \\neq I$. Thus $(I : a), (I, a)$ both", "strictly contain $I$, so they must belong to $\\mathcal{F}$.", "By the Oka condition, we have $I \\in \\mathcal{F}$, a contradiction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1405, "type": "theorem", "label": "algebra-proposition-Jacobson-permanence", "categories": [ "algebra" ], "title": "algebra-proposition-Jacobson-permanence", "contents": [ "Let $R$ be a Jacobson ring. Let $R \\to S$ be a", "ring map of finite type. Then", "\\begin{enumerate}", "\\item The ring $S$ is Jacobson.", "\\item The map $\\Spec(S) \\to \\Spec(R)$ transforms", "closed points to closed points.", "\\item For $\\mathfrak m' \\subset S$ maximal lying over $\\mathfrak m \\subset R$", "the field extension $\\kappa(\\mathfrak m')/\\kappa(\\mathfrak m)$", "is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak m' \\subset S$ be a maximal ideal and", "$R \\cap \\mathfrak m' = \\mathfrak m$.", "Then $R/\\mathfrak m \\to S/\\mathfrak m'$ satisfies", "the conditions of Lemma \\ref{lemma-silly-jacobson}", "by Lemma \\ref{lemma-Jacobson-mod-ideal}.", "Hence $R/\\mathfrak m$ is a field and", "$\\mathfrak m$ a maximal ideal and the induced", "residue field extension is finite. This proves (2) and (3). ", "\\medskip\\noindent", "If $S$ is not Jacobson, then by Lemma \\ref{lemma-characterize-jacobson} there", "exists a non-maximal prime ideal $\\mathfrak q$ of $S$ and an", "$g \\in S$, $g \\not\\in \\mathfrak q$ such that $(S/\\mathfrak q)_g$ is a field.", "To arrive at a contradiction we show that $\\mathfrak q$ is a maximal ideal.", "Let $\\mathfrak p = \\mathfrak q \\cap R$. Then", "$R/\\mathfrak p \\to (S/\\mathfrak q)_g$ satisfies the conditions of", "Lemma \\ref{lemma-silly-jacobson} by", "Lemma \\ref{lemma-Jacobson-mod-ideal}.", "Hence $R/\\mathfrak p$ is a field and the field extension", "$\\kappa(\\mathfrak p) \\to (S/\\mathfrak q)_g = \\kappa(\\mathfrak q)$ is", "finite, thus algebraic. Then $\\mathfrak q$ is a maximal ideal of $S$ by", "Lemma \\ref{lemma-finite-residue-extension-closed}. Contradiction." ], "refs": [ "algebra-lemma-silly-jacobson", "algebra-lemma-Jacobson-mod-ideal", "algebra-lemma-characterize-jacobson", "algebra-lemma-silly-jacobson", "algebra-lemma-Jacobson-mod-ideal", "algebra-lemma-finite-residue-extension-closed" ], "ref_ids": [ 477, 476, 470, 477, 476, 472 ] } ], "ref_ids": [] }, { "id": 1406, "type": "theorem", "label": "algebra-proposition-going-down-normal-integral", "categories": [ "algebra" ], "title": "algebra-proposition-going-down-normal-integral", "contents": [ "Let $R \\subset S$ be an inclusion of domains.", "Assume $R$ is normal and $S$ integral over $R$.", "Let $\\mathfrak p \\subset \\mathfrak p' \\subset R$", "be primes. Let $\\mathfrak q'$ be a prime of $S$", "with $\\mathfrak p' = R \\cap \\mathfrak q'$.", "Then there exists a prime $\\mathfrak q$", "with $\\mathfrak q \\subset \\mathfrak q'$", "such that $\\mathfrak p = R \\cap \\mathfrak q$. In other words:", "the going down property holds for $R \\to S$, see", "Definition \\ref{definition-going-up-down}." ], "refs": [ "algebra-definition-going-up-down" ], "proofs": [ { "contents": [ "Let $\\mathfrak p$, $\\mathfrak p'$ and $\\mathfrak q'$", "be as in the statement. We have to show there is a prime", "$\\mathfrak q$, with $\\mathfrak q \\subset \\mathfrak q'$ and", "$R \\cap \\mathfrak q = \\mathfrak p$. This is the same", "as finding a prime of", "$S_{\\mathfrak q'}$ mapping to $\\mathfrak p$.", "According to Lemma \\ref{lemma-in-image} we have to show", "that $\\mathfrak p S_{\\mathfrak q'} \\cap R", "= \\mathfrak p$. Pick $z \\in \\mathfrak p S_{\\mathfrak q'} \\cap R$.", "We may write $z = y/g$ with $y \\in \\mathfrak pS$ and", "$g \\in S$, $g \\not\\in \\mathfrak q'$. Written", "differently we have $zg = y$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-integral-integral-over-ideal}", "there exists a monic polynomial", "$P = x^m + b_{m-1} x^{m-1} + \\ldots + b_0$", "with $b_i \\in \\mathfrak p$ such that $P(y) = 0$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-minimal-polynomial-normal-domain}", "the minimal polynomial of $g$ over $K$ has coefficients", "in $R$. Write it as $Q = x^n + a_{n-1} x^{n-1} + \\ldots", "+ a_0$. Note that not all $a_i$, $i = n-1, \\ldots, 0$", "are in $\\mathfrak p$ since that would imply", "$g^n = \\sum_{j < n} a_j g^j \\in \\mathfrak pS", "\\subset \\mathfrak p'S", "\\subset \\mathfrak q'$", "which is a contradiction.", "\\medskip\\noindent", "Since $y = zg$ we see immediately from the above", "that $Q' = x^n + za_{n-1} x^{n-1} + \\ldots + z^{n}a_0$", "is the minimal polynomial for $y$. Hence", "$Q'$ divides $P$ and by Lemma \\ref{lemma-polynomials-divide}", "we see that $z^ja_{n - j} \\in \\sqrt{(b_0, \\ldots, b_{m-1})}", "\\subset \\mathfrak p$, $j = 1, \\ldots, n$.", "Because not all $a_i$, $i = n-1, \\ldots, 0$", "are in $\\mathfrak p$ we conclude $z \\in \\mathfrak p$", "as desired." ], "refs": [ "algebra-lemma-in-image", "algebra-lemma-integral-integral-over-ideal", "algebra-lemma-minimal-polynomial-normal-domain", "algebra-lemma-polynomials-divide" ], "ref_ids": [ 394, 519, 521, 520 ] } ], "ref_ids": [ 1459 ] }, { "id": 1407, "type": "theorem", "label": "algebra-proposition-fppf-open", "categories": [ "algebra" ], "title": "algebra-proposition-fppf-open", "contents": [ "Let $R \\to S$ be flat and of finite presentation.", "Then $\\Spec(S) \\to \\Spec(R)$ is open.", "More generally this holds for any ring map $R \\to S$ of", "finite presentation which satisfies going down." ], "refs": [], "proofs": [ { "contents": [ "Assume that $R \\to S$ has finite presentation and satisfies", "going down.", "It suffices to prove that the image of a standard open $D(f)$ is open.", "Since $S \\to S_f$ satisfies going down as well, we see that", "$R \\to S_f$ satisfies going down. Thus after replacing", "$S$ by $S_f$ we see it suffices to prove the image is", "open. By Chevalley's theorem", "(Theorem \\ref{theorem-chevalley})", "the image is a constructible set $E$. And $E$ is stable", "under generalization because $R \\to S$ satisfies going down,", "see Topology, Lemmas \\ref{topology-lemma-open-closed-specialization}", "and \\ref{topology-lemma-lift-specializations-images}.", "Hence $E$ is open by", "Lemma \\ref{lemma-constructible-stable-specialization-closed}." ], "refs": [ "algebra-theorem-chevalley", "topology-lemma-open-closed-specialization", "topology-lemma-lift-specializations-images", "algebra-lemma-constructible-stable-specialization-closed" ], "ref_ids": [ 315, 8283, 8286, 553 ] } ], "ref_ids": [] }, { "id": 1408, "type": "theorem", "label": "algebra-proposition-graded-hilbert-polynomial", "categories": [ "algebra" ], "title": "algebra-proposition-graded-hilbert-polynomial", "contents": [ "Suppose that $S$ is a Noetherian graded ring", "and $M$ a finite graded $S$-module. Consider the", "function", "$$", "\\mathbf{Z} \\longrightarrow K'_0(S_0), \\quad", "n \\longmapsto [M_n]", "$$", "see Lemma \\ref{lemma-graded-module-fg}.", "If $S_{+}$ is generated by elements of degree $1$,", "then this function is a numerical polynomial." ], "refs": [ "algebra-lemma-graded-module-fg" ], "proofs": [ { "contents": [ "We prove this by induction on the minimal number of", "generators of $S_1$. If this number is $0$, then", "$M_n = 0$ for all $n \\gg 0$ and the result holds.", "To prove the induction step, let $x\\in S_1$", "be one of a minimal set of generators, such that", "the induction hypothesis applies to the", "graded ring $S/(x)$.", "\\medskip\\noindent", "First we show the result holds if $x$ is nilpotent on $M$.", "This we do by induction on the minimal integer $r$ such that", "$x^r M = 0$. If $r = 1$, then $M$ is a module over $S/xS$", "and the result holds (by the other induction hypothesis).", "If $r > 1$, then we can find a short exact sequence", "$0 \\to M' \\to M \\to M'' \\to 0$ such that the integers", "$r', r''$ are strictly smaller than $r$. Thus we know", "the result for $M''$ and $M'$. Hence", "we get the result for $M$ because of the relation", "$", "[M_d] = [M'_d] + [M''_d]", "$", "in $K'_0(S_0)$.", "\\medskip\\noindent", "If $x$ is not nilpotent on $M$, let $M' \\subset M$ be", "the largest submodule on which $x$ is nilpotent.", "Consider the exact sequence $0 \\to M' \\to M \\to M/M' \\to 0$", "we see again it suffices to prove the result for $M/M'$. In other", "words we may assume that multiplication by $x$ is injective.", "\\medskip\\noindent", "Let $\\overline{M} = M/xM$. Note that the map $x : M \\to M$", "is {\\it not} a map of graded $S$-modules, since it does", "not map $M_d$ into $M_d$. Namely, for each $d$ we have the", "following short exact sequence", "$$", "0 \\to M_d \\xrightarrow{x} M_{d + 1} \\to \\overline{M}_{d + 1} \\to 0", "$$", "This proves that $[M_{d + 1}] - [M_d] = [\\overline{M}_{d + 1}]$.", "Hence we win by Lemma \\ref{lemma-numerical-polynomial}." ], "refs": [ "algebra-lemma-numerical-polynomial" ], "ref_ids": [ 670 ] } ], "ref_ids": [ 671 ] }, { "id": 1409, "type": "theorem", "label": "algebra-proposition-hilbert-function-polynomial", "categories": [ "algebra" ], "title": "algebra-proposition-hilbert-function-polynomial", "contents": [ "Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module.", "Let $I \\subset R$ be an ideal of definition.", "The Hilbert function $\\varphi_{I, M}$ and the function", "$\\chi_{I, M}$ are numerical polynomials." ], "refs": [], "proofs": [ { "contents": [ "Consider the graded ring $S = R/I \\oplus I/I^2 \\oplus I^2/I^3 \\oplus", "\\ldots = \\bigoplus_{d \\geq 0} I^d/I^{d + 1}$. Consider the graded", "$S$-module $N = M/IM \\oplus IM/I^2M \\oplus \\ldots =", "\\bigoplus_{d \\geq 0} I^dM/I^{d + 1}M$. This pair $(S, N)$ satisfies", "the hypotheses of Proposition \\ref{proposition-graded-hilbert-polynomial}.", "Hence the result for $\\varphi_{I, M}$ follows from that proposition and", "Lemma \\ref{lemma-length-K0}. The result for $\\chi_{I, M}$ follows", "from this and Lemma \\ref{lemma-numerical-polynomial}." ], "refs": [ "algebra-proposition-graded-hilbert-polynomial", "algebra-lemma-length-K0", "algebra-lemma-numerical-polynomial" ], "ref_ids": [ 1408, 651, 670 ] } ], "ref_ids": [] }, { "id": 1410, "type": "theorem", "label": "algebra-proposition-dimension-zero-ring", "categories": [ "algebra" ], "title": "algebra-proposition-dimension-zero-ring", "contents": [ "Let $R$ be a ring. The following are equivalent:", "\\begin{enumerate}", "\\item $R$ is Artinian,", "\\item $R$ is Noetherian and $\\dim(R) = 0$,", "\\item $R$ has finite length as a module over itself,", "\\item $R$ is a finite product of Artinian local rings,", "\\item $R$ is Noetherian and $\\Spec(R)$ is a", "finite discrete topological space,", "\\item $R$ is a finite product of Noetherian local rings", "of dimension $0$,", "\\item $R$ is a finite product of Noetherian local rings", "$R_i$ with $d(R_i) = 0$,", "\\item $R$ is a finite product of Noetherian local rings", "$R_i$ whose maximal ideals are nilpotent,", "\\item $R$ is Noetherian, has finitely many maximal", "ideals and its Jacobson radical ideal is nilpotent, and", "\\item $R$ is Noetherian and there are no strict inclusions", "among its primes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a combination of Lemmas", "\\ref{lemma-product-local},", "\\ref{lemma-artinian-finite-length},", "\\ref{lemma-Noetherian-dimension-0}, and", "\\ref{lemma-dimension-0-d-0}." ], "refs": [ "algebra-lemma-product-local", "algebra-lemma-artinian-finite-length", "algebra-lemma-Noetherian-dimension-0", "algebra-lemma-dimension-0-d-0" ], "ref_ids": [ 645, 646, 680, 681 ] } ], "ref_ids": [] }, { "id": 1411, "type": "theorem", "label": "algebra-proposition-dimension", "categories": [ "algebra" ], "title": "algebra-proposition-dimension", "contents": [ "Let $R$ be a local Noetherian ring. Let $d \\geq 0$ be an integer.", "The following are equivalent:", "\\begin{enumerate}", "\\item", "\\label{item-dim-d}", "$\\dim(R) = d$,", "\\item", "\\label{item-d-d}", "$d(R) = d$,", "\\item", "\\label{item-ideal-d}", "there exists an ideal of definition generated by $d$ elements,", "and no ideal of definition is generated by fewer than $d$ elements.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This proof is really just the same as the proof of Lemma", "\\ref{lemma-height-1}. We will prove the proposition by induction", "on $d$. By Lemmas \\ref{lemma-dimension-0-d-0} and \\ref{lemma-height-1}", "we may assume that $d > 1$. Denote the minimal number of", "generators for an ideal of definition of $R$ by $d'(R)$.", "We will prove the inequalities", "$\\dim(R) \\geq d'(R) \\geq d(R) \\geq \\dim(R)$,", "and hence they are all equal.", "\\medskip\\noindent", "First, assume that $\\dim(R) = d$.", "Let $\\mathfrak p_i$ be the minimal primes of $R$.", "According to Lemma \\ref{lemma-Noetherian-irreducible-components}", "there are finitely many. Hence we can find $x \\in \\mathfrak m$,", "$x \\not \\in \\mathfrak p_i$, see Lemma \\ref{lemma-silly}.", "Note that every maximal chain of primes starts with some $\\mathfrak p_i$,", "hence the dimension of $R/xR$ is at most $d-1$. By induction", "there are $x_2, \\ldots, x_d$ which generate an ideal of definition", "in $R/xR$. Hence $R$ has an ideal of definition generated", "by (at most) $d$ elements.", "\\medskip\\noindent", "Assume $d'(R) = d$. Let $I = (x_1, \\ldots, x_d)$ be an ideal", "of definition. Note that $I^n/I^{n + 1}$ is a quotient of a direct", "sum of $\\binom{d + n - 1}{d - 1}$ copies $R/I$ via multiplication", "by all degree $n$ monomials in $x_1, \\ldots, x_n$.", "Hence $\\text{length}_R(I^n/I^{n + 1})$ is bounded by a polynomial", "of degree $d-1$. Thus $d(R) \\leq d$.", "\\medskip\\noindent", "Assume $d(R) = d$. Consider a chain of primes", "$\\mathfrak p \\subset \\mathfrak q \\subset", "\\mathfrak q_2 \\subset \\ldots \\subset \\mathfrak q_e = \\mathfrak m$,", "with all inclusions strict, and $e \\geq 2$.", "Pick some ideal of definition $I \\subset R$.", "We will repeatedly use", "Lemma \\ref{lemma-hilbert-ses-chi}. First of all", "it implies, via the exact sequence", "$0 \\to \\mathfrak p \\to R \\to R/\\mathfrak p \\to 0$,", "that $d(R/\\mathfrak p) \\leq d$. But it clearly cannot", "be zero. Pick $x\\in \\mathfrak q$, $x\\not \\in \\mathfrak p$.", "Consider the short exact sequence", "$$", "0 \\to R/\\mathfrak p \\to R/\\mathfrak p \\to R/(xR + \\mathfrak p) \\to 0.", "$$", "This implies that $\\chi_{I, R/\\mathfrak p} - \\chi_{I, R/\\mathfrak p}", "- \\chi_{I, R/(xR + \\mathfrak p)} = - \\chi_{I, R/(xR + \\mathfrak p)}$", "has degree $ < d$. In other words, $d(R/(xR + \\mathfrak p)) \\leq d - 1$,", "and hence $\\dim(R/(xR + \\mathfrak p)) \\leq d - 1$, by", "induction. Now $R/(xR + \\mathfrak p)$ has the chain of prime ideals", "$\\mathfrak q/(xR + \\mathfrak p) \\subset \\mathfrak q_2/(xR + \\mathfrak p)", "\\subset \\ldots \\subset \\mathfrak q_e/(xR + \\mathfrak p)$ which gives", "$e - 1 \\leq d - 1$. Since we started with an arbitrary chain of", "primes this proves that $\\dim(R) \\leq d(R)$.", "\\medskip\\noindent", "Reading back the reader will see we proved the circular", "inequalities as desired." ], "refs": [ "algebra-lemma-height-1", "algebra-lemma-dimension-0-d-0", "algebra-lemma-height-1", "algebra-lemma-Noetherian-irreducible-components", "algebra-lemma-silly", "algebra-lemma-hilbert-ses-chi" ], "ref_ids": [ 682, 681, 682, 453, 378, 678 ] } ], "ref_ids": [] }, { "id": 1412, "type": "theorem", "label": "algebra-proposition-minimal-primes-associated-primes", "categories": [ "algebra" ], "title": "algebra-proposition-minimal-primes-associated-primes", "contents": [ "Let $R$ be a Noetherian ring.", "Let $M$ be a finite $R$-module.", "The following sets of primes are the same:", "\\begin{enumerate}", "\\item The minimal primes in the support of $M$.", "\\item The minimal primes in $\\text{Ass}(M)$.", "\\item For any filtration $0 = M_0 \\subset M_1 \\subset \\ldots", "\\subset M_{n-1} \\subset M_n = M$ with $M_i/M_{i-1} \\cong R/\\mathfrak p_i$", "the minimal primes of the set $\\{\\mathfrak p_i\\}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a filtration as in (3).", "In Lemma \\ref{lemma-filter-minimal-primes-in-support}", "we have seen that the sets in (1) and (3) are equal.", "\\medskip\\noindent", "Let $\\mathfrak p$ be a minimal element of the set $\\{\\mathfrak p_i\\}$.", "Let $i$ be minimal such that $\\mathfrak p = \\mathfrak p_i$.", "Pick $m \\in M_i$, $m \\not \\in M_{i-1}$. The annihilator of $m$", "is contained in $\\mathfrak p_i = \\mathfrak p$ and contains", "$\\mathfrak p_1 \\mathfrak p_2 \\ldots \\mathfrak p_i$. By our choice of", "$i$ and $\\mathfrak p$ we have $\\mathfrak p_j \\not \\subset \\mathfrak p$", "for $j < i$ and hence we have", "$\\mathfrak p_1 \\mathfrak p_2 \\ldots \\mathfrak p_{i - 1}", "\\not \\subset \\mathfrak p_i$. Pick", "$f \\in \\mathfrak p_1 \\mathfrak p_2 \\ldots \\mathfrak p_{i - 1}$,", "$f \\not \\in \\mathfrak p$. Then $fm$ has annihilator $\\mathfrak p$.", "In this way we see that $\\mathfrak p$ is an associated prime of $M$.", "By Lemma \\ref{lemma-ass-support} we have $\\text{Ass}(M) \\subset \\text{Supp}(M)$", "and hence $\\mathfrak p$ is minimal in $\\text{Ass}(M)$.", "Thus the set of primes in (1) is contained in the set of primes of (2).", "\\medskip\\noindent", "Let $\\mathfrak p$ be a minimal element of $\\text{Ass}(M)$.", "Since $\\text{Ass}(M) \\subset \\text{Supp}(M)$ there is a minimal", "element $\\mathfrak q$ of $\\text{Supp}(M)$ with", "$\\mathfrak q \\subset \\mathfrak p$. We have just shown that", "$\\mathfrak q \\in \\text{Ass}(M)$. Hence $\\mathfrak q = \\mathfrak p$", "by minimality of $\\mathfrak p$. Thus the set of primes in (2) is", "contained in the set of primes of (1)." ], "refs": [ "algebra-lemma-filter-minimal-primes-in-support", "algebra-lemma-ass-support" ], "ref_ids": [ 695, 698 ] } ], "ref_ids": [] }, { "id": 1413, "type": "theorem", "label": "algebra-proposition-ffdescent-finite-projectivity", "categories": [ "algebra" ], "title": "algebra-proposition-ffdescent-finite-projectivity", "contents": [ "Let $R \\to S$ be a faithfully flat ring map. Let $M$ be an $R$-module.", "If the $S$-module $M \\otimes_R S$ is finite projective, then $M$ is finite", "projective." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemmas \\ref{lemma-finite-projective-again} and", "\\ref{lemma-descend-properties-modules}." ], "refs": [ "algebra-lemma-finite-projective-again", "algebra-lemma-descend-properties-modules" ], "ref_ids": [ 818, 819 ] } ], "ref_ids": [] }, { "id": 1414, "type": "theorem", "label": "algebra-proposition-ML-characterization", "categories": [ "algebra" ], "title": "algebra-proposition-ML-characterization", "contents": [ "Let $M$ be an $R$-module. Let $(M_i, f_{ij})$ be a directed system of finitely", "presented $R$-modules, indexed by $I$, such that $M = \\colim M_i$. Let", "$f_i:", "M_i \\to M$ be the canonical map. The following are equivalent:", "\\begin{enumerate}", "\\item For every finitely presented $R$-module $P$ and module map $f: P", "\\to M$, there exists a finitely presented $R$-module $Q$ and a module", "map $g: P \\to Q$ such that $g$ and $f$ dominate each other, i.e.,", "$\\Ker(f \\otimes_R \\text{id}_N) = \\Ker(g \\otimes_R \\text{id}_N)$", "for every $R$-module $N$.", "\\item For each $i \\in I$, there exists $j \\geq i$ such that $f_{ij}: M_i", "\\to M_j$ dominates $f_i: M_i \\to M$.", "\\item For each $i \\in I$, there exists $j \\geq i$ such that $f_{ij}: M_i", "\\to M_j$ factors through $f_{ik}: M_i \\to M_k$ for all $k \\geq", "i$.", "\\item For every $R$-module $N$, the inverse system", "$(\\Hom_R(M_i, N), \\Hom_R(f_{ij}, N))$ is Mittag-Leffler.", "\\item For $N = \\prod_{s \\in I} M_s$, the inverse system", "$(\\Hom_R(M_i, N), \\Hom_R(f_{ij}, N))$ is Mittag-Leffler.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First we prove the equivalence of (1) and (2). Suppose (1) holds and let $i", "\\in I$. Corresponding to the map $f_i: M_i \\to M$, we can choose $g:", "M_i \\to Q$ as in (1). Since $M_i$ and $Q$ are of finite presentation,", "so is $\\Coker(g)$. Then by Lemma \\ref{lemma-domination}, $f_i : M_i", "\\to M$ factors through $g: M_i \\to Q$, say $f_i = h \\circ g$", "for some $h: Q \\to M$. Then since $Q$ is finitely presented, $h$", "factors through $M_j \\to M$ for some $j \\geq i$, say $h = f_j \\circ h'$", "for some $h': Q \\to M_j$. In total we have a commutative diagram", "$$", "\\xymatrix{", " & M & \\\\", "M_i \\ar[dr]_g \\ar[ur]^{f_i} \\ar[rr]^{f_{ij}} &", "& M_j \\ar[ul]_{f_j} \\\\", " & Q \\ar[ur]_{h'} &", "}", "$$", "Thus $f_{ij}$ dominates $g$. But $g$ dominates $f_i$, so $f_{ij}$ dominates", "$f_i$.", "\\medskip\\noindent", "Conversely, suppose (2) holds. Let $P$ be of finite presentation and $f: P", "\\to M$ a module map. Then $f$ factors through $f_i: M_i \\to M$", "for some $i \\in I$, say $f = f_i \\circ g'$ for some $g': P \\to M_i$.", "Choose by (2) a $j \\geq i$ such that $f_{ij}$ dominates $f_i$. We have a", "commutative diagram", "$$", "\\xymatrix{", "P \\ar[d]_{g'} \\ar[r]^{f} & M \\\\", "M_i \\ar[ur]^{f_i} \\ar[r]_{f_{ij}} & M_j \\ar[u]_{f_j}", "}", "$$", "From the diagram and the fact that $f_{ij}$ dominates $f_i$, we find that $f$", "and $f_{ij} \\circ g'$ dominate each other. Hence taking $g = f_{ij} \\circ g' :", "P \\to M_j$ works.", "\\medskip\\noindent", "Next we prove (2) is equivalent to (3). Let $i \\in I$. It is always true that", "$f_i$ dominates $f_{ik}$ for $k \\geq i$, since $f_i$ factors through", "$f_{ik}$. If (2) holds, choose $j \\geq i$ such that $f_{ij}$ dominates", "$f_i$. Then since domination is a transitive relation, $f_{ij}$ dominates", "$f_{ik}$ for $k \\geq i$. All $M_i$ are of finite presentation, so", "$\\Coker(f_{ik})$ is of finite presentation for $k \\geq i$. By Lemma", "\\ref{lemma-domination}, $f_{ij}$ factors through $f_{ik}$ for all $k \\geq i$.", "Thus (2) implies (3). On the other hand, if (3) holds then for any $R$-module", "$N$, $f_{ij} \\otimes_R \\text{id}_N$ factors through $f_{ik}", "\\otimes_R \\text{id}_N$ for $k \\geq i$. So $\\Ker(f_{ik} \\otimes_R", "\\text{id}_N) \\subset \\Ker(f_{ij} \\otimes_R \\text{id}_N)$ for $k", "\\geq i$. But $\\Ker(f_i \\otimes_R \\text{id}_N: M_i \\otimes_R N", "\\to M \\otimes_R N)$ is the union of $\\Ker(f_{ik} \\otimes_R", "\\text{id}_N)$ for $k \\geq i$. Thus $\\Ker(f_i \\otimes_R", "\\text{id}_N) \\subset \\Ker(f_{ij} \\otimes_R \\text{id}_N)$ for", "any $R$-module $N$, which by definition means $f_{ij}$ dominates $f_i$.", "\\medskip\\noindent", "It is trivial that (3) implies (4) implies (5). We show (5) implies (3). Let", "$N = \\prod_{s \\in I} M_s$. If (5) holds, then given $i \\in I$ choose $j \\geq i$", "such that", "$$", "\\Im( \\Hom(M_j, N) \\to \\Hom(M_i, N)) =", "\\Im( \\Hom(M_k, N) \\to \\Hom(M_i, N))", "$$", "for all $k \\geq j$. Passing the product over $s \\in I$ outside of the", "$\\Hom$'s", "and looking at the maps on each component of the product, this says", "$$", "\\Im( \\Hom(M_j, M_s) \\to \\Hom(M_i, M_s)) =", "\\Im( \\Hom(M_k, M_s) \\to \\Hom(M_i, M_s))", "$$", "for all $k \\geq j$ and $s \\in I$. Taking $s = j$ we have", "$$", "\\Im( \\Hom(M_j, M_j) \\to \\Hom(M_i, M_j)) =", "\\Im( \\Hom(M_k, M_j) \\to \\Hom(M_i, M_j))", "$$", "for all $k \\geq j$. Since $f_{ij}$ is the image of", "$\\text{id} \\in \\Hom(M_j, M_j)$ under", "$\\Hom(M_j, M_j) \\to \\Hom(M_i, M_j)$,", "this shows that for any $k \\geq j$ there is $h \\in \\Hom(M_k, M_j)$", "such that $f_{ij} = h \\circ f_{ik}$. If $j \\geq k$ then we can take", "$h = f_{kj}$. Hence (3) holds." ], "refs": [ "algebra-lemma-domination", "algebra-lemma-domination" ], "ref_ids": [ 830, 830 ] } ], "ref_ids": [] }, { "id": 1415, "type": "theorem", "label": "algebra-proposition-fg-tensor", "categories": [ "algebra" ], "title": "algebra-proposition-fg-tensor", "contents": [ "Let $M$ be an $R$-module. The following are equivalent:", "\\begin{enumerate}", "\\item $M$ is finitely generated.", "\\item For every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules, the", "canonical map $M \\otimes_R \\left( \\prod_{\\alpha} Q_{\\alpha} \\right)", "\\to \\prod_{\\alpha} (M \\otimes_R Q_{\\alpha})$ is surjective.", "\\item For every $R$-module $Q$ and every set $A$, the canonical map $M", "\\otimes_R Q^{A} \\to (M \\otimes_R Q)^{A}$ is surjective.", "\\item For every set $A$, the canonical map $M \\otimes_R R^{A} \\to", "M^{A}$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First we prove (1) implies (2). Choose a surjection $R^n \\to M$ and", "consider the commutative diagram", "$$", "\\xymatrix{", "R^n \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r]^{\\cong} \\ar[d] &", "\\prod_{\\alpha} (R^n \\otimes_R Q_{\\alpha}) \\ar[d] \\\\", "M \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r] & \\prod_{\\alpha} ( M", "\\otimes_R Q_{\\alpha}).", "}", "$$", "The top arrow is an isomorphism and the vertical arrows are surjections. We", "conclude that the bottom arrow is a surjection.", "\\medskip\\noindent", "Obviously (2) implies (3) implies (4), so it remains to prove (4) implies (1).", "In fact for (1) to hold it suffices that the element $d = (x)_{x \\in M}$ of", "$M^M$ is in the image of the map $f: M \\otimes_R R^{M} \\to M^M$. In", "this case $d = \\sum_{i = 1}^{n} f(x_i \\otimes a_i)$ for some $x_i \\in M$ and", "$a_i \\in R^M$. If for $x \\in M$ we write $p_x: M^M \\to M$ for the", "projection onto the $x$-th factor, then", "$$", "x = p_x(d) = \\sum\\nolimits_{i = 1}^{n} p_x(f(x_i \\otimes a_i)) =", "\\sum\\nolimits_{i=1}^{n} p_x(a_i) x_i.", "$$", "Thus $x_1, \\ldots, x_n$ generate $M$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1416, "type": "theorem", "label": "algebra-proposition-fp-tensor", "categories": [ "algebra" ], "title": "algebra-proposition-fp-tensor", "contents": [ "Let $M$ be an $R$-module. The following are equivalent:", "\\begin{enumerate}", "\\item $M$ is finitely presented.", "\\item For every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules, the", "canonical map $M \\otimes_R \\left( \\prod_{\\alpha} Q_{\\alpha} \\right)", "\\to \\prod_{\\alpha} (M \\otimes_R Q_{\\alpha})$ is bijective.", "\\item For every $R$-module $Q$ and every set $A$, the canonical map $M", "\\otimes_R Q^{A} \\to (M \\otimes_R Q)^{A}$ is bijective.", "\\item For every set $A$, the canonical map $M \\otimes_R R^{A} \\to", "M^{A}$ is bijective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First we prove (1) implies (2). Choose a presentation $R^m \\to R^n", "\\to M$ and consider the commutative diagram", "$$", "\\xymatrix{", "R^m \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r] \\ar[d]^{\\cong} & R^m", "\\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r] \\ar[d]^{\\cong} & M \\otimes_R", "(\\prod_{\\alpha} Q_{\\alpha}) \\ar[r] \\ar[d] & 0 \\\\", "\\prod_{\\alpha} (R^m \\otimes_R Q_{\\alpha}) \\ar[r] & \\prod_{\\alpha} (R^n", "\\otimes_R Q_{\\alpha}) \\ar[r] & \\prod_{\\alpha} (M \\otimes_R Q_{\\alpha})", "\\ar[r] & 0.", "}", "$$", "The first two vertical arrows are isomorphisms and the rows are exact. This", "implies that the map", "$M \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\to", "\\prod_{\\alpha} ( M \\otimes_R Q_{\\alpha})$", "is surjective and, by a diagram chase, also injective. Hence (2) holds.", "\\medskip\\noindent", "Obviously (2) implies (3) implies (4), so it remains to prove (4) implies (1).", "From Proposition \\ref{proposition-fg-tensor}, if (4) holds we already know that", "$M$ is finitely generated. So we can choose a surjection $F \\to M$", "where $F$ is free and finite. Let $K$ be the kernel. We must show $K$ is", "finitely generated. For any set $A$, we have a commutative diagram", "$$", "\\xymatrix{", "& K \\otimes_R R^A \\ar[r] \\ar[d]_{f_3} & F \\otimes_R R^A \\ar[r]", "\\ar[d]_{f_2}^{\\cong} & M \\otimes_R R^A \\ar[r] \\ar[d]_{f_1}^{\\cong} & 0 \\\\", "0 \\ar[r] & K^A \\ar[r] & F^A \\ar[r] & M^A \\ar[r] & 0 .", "}", "$$", "The map $f_1$ is an isomorphism by assumption, the map $f_2$ is a isomorphism", "since $F$ is free and finite, and the rows are exact. A diagram chase shows", "that $f_3$ is surjective, hence by Proposition \\ref{proposition-fg-tensor} we", "get that $K$ is finitely generated." ], "refs": [ "algebra-proposition-fg-tensor", "algebra-proposition-fg-tensor" ], "ref_ids": [ 1415, 1415 ] } ], "ref_ids": [] }, { "id": 1417, "type": "theorem", "label": "algebra-proposition-ML-tensor", "categories": [ "algebra" ], "title": "algebra-proposition-ML-tensor", "contents": [ "Let $M$ be an $R$-module. The following are equivalent:", "\\begin{enumerate}", "\\item $M$ is Mittag-Leffler.", "\\item For every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules, the", "canonical map $M \\otimes_R \\left( \\prod_{\\alpha} Q_{\\alpha} \\right)", "\\to \\prod_{\\alpha} (M \\otimes_R Q_{\\alpha})$ is injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First we prove (1) implies (2). Suppose $M$ is Mittag-Leffler and let $x$ be", "in the kernel of $M \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\to", "\\prod_{\\alpha} (M \\otimes_R Q_{\\alpha})$. Write $M$ as a colimit $M =", "\\colim_{i \\in I} M_i$ of a directed system of finitely presented modules", "$M_i$.", " Then $M \\otimes_R (\\prod_{\\alpha} Q_{\\alpha})$ is the colimit of $M_i", "\\otimes_R (\\prod_{\\alpha} Q_{\\alpha})$. So $x$ is the image of an element", "$x_i \\in M_i \\otimes_R (\\prod_{\\alpha} Q_{\\alpha})$. We must show that $x_i$", "maps to $0$ in $M_j \\otimes_R (\\prod_{\\alpha} Q_{\\alpha})$ for some $j \\geq", "i$. Since $M$ is Mittag-Leffler, we may choose $j \\geq i$ such that $M_i", "\\to M_j$ and $M_i \\to M$ dominate each other. Then consider", "the commutative diagram", "$$", "\\xymatrix{", "M \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r] & \\prod_{\\alpha} (M", "\\otimes_R Q_{\\alpha}) \\\\", "M_i \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r]^{\\cong} \\ar[d] \\ar[u] &", "\\prod_{\\alpha} (M_i \\otimes_R Q_{\\alpha}) \\ar[d] \\ar[u] \\\\", "M_j \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r]^{\\cong} & \\prod_{\\alpha}", "(M_j \\otimes_R Q_{\\alpha})", "}", "$$", "whose bottom two horizontal maps are isomorphisms, according to Proposition", "\\ref{proposition-fp-tensor}. Since $x_i$ maps to $0$ in $\\prod_{\\alpha} (M", "\\otimes_R Q_{\\alpha})$, its image in $\\prod_{\\alpha} (M_i \\otimes_R", "Q_{\\alpha})$ is in the kernel of the map $\\prod_{\\alpha} (M_i \\otimes_R", "Q_{\\alpha}) \\to \\prod_{\\alpha} (M \\otimes_R Q_{\\alpha})$. But this", "kernel equals the kernel of $\\prod_{\\alpha} (M_i \\otimes_R Q_{\\alpha})", "\\to \\prod_{\\alpha} (M_j \\otimes_R Q_{\\alpha})$ according to the", "choice of $j$. Thus $x_i$ maps to $0$ in $\\prod_{\\alpha} (M_j \\otimes_R", "Q_{\\alpha})$ and hence to $0$ in $M_j \\otimes_R (\\prod_{\\alpha} Q_{\\alpha})$.", "\\medskip\\noindent", "Now suppose (2) holds. We prove $M$ satisfies formulation (1) of being", "Mittag-Leffler from Proposition \\ref{proposition-ML-characterization}. Let $f:", "P \\to M$ be a map from a finitely presented module $P$ to $M$. Choose", "a set $B$ of representatives of the isomorphism classes of finitely presented", "$R$-modules. Let $A$ be the set of pairs $(Q, x)$ where $Q \\in B$ and $x \\in", "\\Ker(P \\otimes Q \\to M \\otimes Q)$. For $\\alpha = (Q, x) \\in A$, we", "write $Q_{\\alpha}$ for $Q$ and $x_{\\alpha}$ for $x$. Consider the commutative", "diagram", "$$", "\\xymatrix{", "M \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r] &", "\\prod_{\\alpha} (M \\otimes_R Q_{\\alpha}) \\\\", "P \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r]^{\\cong} \\ar[u] &", "\\prod_{\\alpha} (P \\otimes_R Q_{\\alpha}) \\ar[u] .", "}", "$$", "The top arrow is an injection by assumption, and the bottom arrow is an", "isomorphism by Proposition \\ref{proposition-fp-tensor}. Let $x \\in P", "\\otimes_R (\\prod_{\\alpha} Q_{\\alpha})$ be the element corresponding to", "$(x_{\\alpha}) \\in \\prod_{\\alpha} (P \\otimes_R Q_{\\alpha})$ under this", "isomorphism. Then $x \\in \\Ker( P \\otimes_R (\\prod_{\\alpha} Q_{\\alpha})", "\\to M \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}))$ since the top arrow in", "the diagram is injective. By Lemma \\ref{lemma-kernel-tensored-fp}, we get a", "finitely presented module $P'$ and a map $f': P \\to P'$ such that $f: P", "\\to M$ factors through $f'$ and $x \\in \\Ker(P \\otimes_R", "(\\prod_{\\alpha} Q_{\\alpha}) \\to P' \\otimes_R (\\prod_{\\alpha}", "Q_{\\alpha}))$. We have a commutative diagram", "$$", "\\xymatrix{", "P' \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r]^{\\cong} &", "\\prod_{\\alpha} (P' \\otimes_R Q_{\\alpha}) \\\\", "P \\otimes_R (\\prod_{\\alpha} Q_{\\alpha}) \\ar[r]^{\\cong} \\ar[u] &", "\\prod_{\\alpha} (P \\otimes_R Q_{\\alpha}) \\ar[u] .", "}", "$$", "where both the top and bottom arrows are isomorphisms by Proposition", "\\ref{proposition-fp-tensor}. Thus since $x$ is in the kernel of the left", "vertical map, $(x_{\\alpha})$ is in the kernel of the right vertical map. This", "means $x_{\\alpha} \\in \\Ker(P \\otimes_R Q_{\\alpha} \\to P' \\otimes_R", "Q_{\\alpha})$ for every $\\alpha \\in A$. By the definition of $A$ this means", "$\\Ker(P \\otimes_R Q \\to P' \\otimes_R Q) \\supset \\Ker(P \\otimes_R", "Q \\to M \\otimes_R Q)$ for all finitely presented $Q$ and, since $f: P", "\\to M$ factors through $f': P \\to P'$, actually equality holds.", " By Lemma \\ref{lemma-domination-fp}, $f$ and $f'$ dominate each other." ], "refs": [ "algebra-proposition-fp-tensor", "algebra-proposition-ML-characterization", "algebra-proposition-fp-tensor", "algebra-lemma-kernel-tensored-fp", "algebra-proposition-fp-tensor", "algebra-lemma-domination-fp" ], "ref_ids": [ 1416, 1414, 1416, 835, 1416, 828 ] } ], "ref_ids": [] }, { "id": 1418, "type": "theorem", "label": "algebra-proposition-characterize-coherent", "categories": [ "algebra" ], "title": "algebra-proposition-characterize-coherent", "contents": [ "\\begin{reference}", "This is \\cite[Theorem 2.1]{Chase}.", "\\end{reference}", "Let $R$ be a ring. The following are equivalent", "\\begin{enumerate}", "\\item $R$ is coherent,", "\\item any product of flat $R$-modules is flat, and", "\\item for every set $A$ the module $R^A$ is flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $R$ coherent, and let $Q_\\alpha$, $\\alpha \\in A$ be a set of flat", "$R$-modules. We have to show that", "$I \\otimes_R \\prod_\\alpha Q_\\alpha \\to \\prod Q_\\alpha$ is injective", "for every finitely generated ideal $I$ of $R$, see", "Lemma \\ref{lemma-flat}.", "Since $R$ is coherent $I$ is an $R$-module of finite presentation.", "Hence $I \\otimes_R \\prod_\\alpha Q_\\alpha = \\prod I \\otimes_R Q_\\alpha$ by", "Proposition \\ref{proposition-fp-tensor}.", "The desired injectivity follows as $I \\otimes_R Q_\\alpha \\to Q_\\alpha$", "is injective by flatness of $Q_\\alpha$.", "\\medskip\\noindent", "The implication (2) $\\Rightarrow$ (3) is trivial.", "\\medskip\\noindent", "Assume that the $R$-module $R^A$ is flat for every set $A$. Let $I$", "be a finitely generated ideal in $R$. Then $I \\otimes_R R^A \\to R^A$", "is injective by assumption. By", "Proposition \\ref{proposition-fg-tensor}", "and the finiteness of $I$ the image is equal to $I^A$. Hence", "$I \\otimes_R R^A = I^A$ for every set $A$ and we conclude that $I$", "is finitely presented by", "Proposition \\ref{proposition-fp-tensor}." ], "refs": [ "algebra-lemma-flat", "algebra-proposition-fp-tensor", "algebra-proposition-fg-tensor", "algebra-proposition-fp-tensor" ], "ref_ids": [ 525, 1416, 1415, 1416 ] } ], "ref_ids": [] }, { "id": 1419, "type": "theorem", "label": "algebra-proposition-what-exact", "categories": [ "algebra" ], "title": "algebra-proposition-what-exact", "contents": [ "\\begin{reference}", "\\cite[Corollary 1]{WhatExact}", "\\end{reference}", "In Situation \\ref{situation-complex}, suppose $R$ is", "a local Noetherian ring. The following are equivalent", "\\begin{enumerate}", "\\item $0 \\to R^{n_e} \\to R^{n_{e-1}} \\to \\ldots \\to R^{n_0}$", "is exact at $R^{n_e}, \\ldots, R^{n_1}$, and", "\\item for all $i$, $1 \\leq i \\leq e$", "the following two conditions are satisfied:", "\\begin{enumerate}", "\\item $\\text{rank}(\\varphi_i) = r_i$ where", "$r_i = n_i - n_{i + 1} + \\ldots + (-1)^{e-i-1} n_{e-1} + (-1)^{e-i} n_e$,", "\\item $I(\\varphi_i) = R$, or $I(\\varphi_i)$ contains a", "regular sequence of length $i$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If for some $i$ some matrix coefficient of $\\varphi_i$", "is not in $\\mathfrak m$, then we apply Lemma \\ref{lemma-add-trivial-complex}.", "It is easy to see that the proposition for a complex and", "for the same complex with a trivial complex added to it", "are equivalent. Thus we may assume that all matrix entries", "of each $\\varphi_i$ are elements of the maximal ideal.", "We may also assume that $e \\geq 1$.", "\\medskip\\noindent", "Assume the complex is exact at $R^{n_e}, \\ldots, R^{n_1}$.", "Let $\\mathfrak q \\in \\text{Ass}(R)$.", "Note that the ring $R_{\\mathfrak q}$ has depth $0$", "and that the complex remains exact after localization at $\\mathfrak q$.", "We apply Lemmas \\ref{lemma-exact-depth-zero-local} and", "\\ref{lemma-trivial-case-exact} to the localized complex", "over $R_{\\mathfrak q}$. We conclude that", "$\\varphi_{i, \\mathfrak q}$ has rank $r_i$ for all $i$.", "Since $R \\to \\bigoplus_{\\mathfrak q \\in \\text{Ass}(R)} R_\\mathfrak q$", "is injective (Lemma \\ref{lemma-zero-at-ass-zero}), we conclude that", "$\\varphi_i$ has rank $r_i$ over $R$ by the definition of rank as given", "in Definition \\ref{definition-rank}. Therefore we see that", "$I(\\varphi_i)_\\mathfrak q = I(\\varphi_{i, \\mathfrak q})$", "as the ranks do not change. Since all of the ideals", "$I(\\varphi_i)_{\\mathfrak q}$, $e \\geq i \\geq 1$", "are equal to $R_{\\mathfrak q}$ (by the lemmas referenced above)", "we conclude none of the ideals $I(\\varphi_i)$ is contained in $\\mathfrak q$.", "This implies that $I(\\varphi_e)I(\\varphi_{e-1})\\ldots I(\\varphi_1)$", "is not contained in any of the associated primes", "of $R$. By Lemma \\ref{lemma-silly} we may choose", "$x \\in I(\\varphi_e)I(\\varphi_{e - 1})\\ldots I(\\varphi_1)$,", "$x \\not \\in \\mathfrak q$ for all $\\mathfrak q \\in \\text{Ass}(R)$.", "Observe that $x$ is a nonzerodivisor (Lemma \\ref{lemma-ass-zero-divisors}).", "According to Lemma \\ref{lemma-div-x-exact-one-less}", "the complex $0 \\to (R/xR)^{n_e} \\to \\ldots \\to (R/xR)^{n_1}$ is exact", "at $(R/xR)^{n_e}, \\ldots, (R/xR)^{n_2}$. By induction", "on $e$ all the ideals $I(\\varphi_i)/xR$ have a regular", "sequence of length $i - 1$. This proves that $I(\\varphi_i)$", "contains a regular sequence of length $i$.", "\\medskip\\noindent", "Assume (2)(a) and (2)(b) hold. We claim that for any prime", "$\\mathfrak p \\subset R$ conditions (2)(a) and (2)(b)", "hold for the complex", "$0 \\to R_\\mathfrak p^{n_e} \\to R_\\mathfrak p^{n_{e - 1}} \\to \\ldots \\to", "R_\\mathfrak p^{n_0}$ with maps $\\varphi_{i, \\mathfrak p}$", "over $R_\\mathfrak p$. Namely, since $I(\\varphi_i)$ contains a", "nonzero divisor, the image of $I(\\varphi_i)$ in $R_\\mathfrak p$", "is nonzero. This implies that the rank of $\\varphi_{i, \\mathfrak p}$", "is the same as the rank of $\\varphi_i$: the rank as defined above", "of a matrix $\\varphi$ over a ring $R$ can only drop when passing", "to an $R$-algebra $R'$ and this happens if and only if $I(\\varphi)$", "maps to zero in $R'$. Thus (2)(a) holds. Having said this", "we know that $I(\\varphi_{i, \\mathfrak p}) = I(\\varphi_i)_\\mathfrak p$", "and we see that (2)(b) is preserved under localization as well.", "By induction on the dimension of $R$ we may assume the complex", "is exact when localized at any nonmaximal prime $\\mathfrak p$ of $R$.", "Thus $\\Ker(\\varphi_i)/\\Im(\\varphi_{i + 1})$ has support contained in", "$\\{\\mathfrak m\\}$ and hence if nonzero has depth $0$.", "As $I(\\varphi_i) \\subset \\mathfrak m$ for all $i$ because", "of what was said in the first paragraph of the proof, we", "see that (2)(b) implies $\\text{depth}(R) \\geq e$.", "By Lemma \\ref{lemma-acyclic} we see", "that the complex is exact at $R^{n_e}, \\ldots, R^{n_1}$", "concluding the proof." ], "refs": [ "algebra-lemma-add-trivial-complex", "algebra-lemma-exact-depth-zero-local", "algebra-lemma-trivial-case-exact", "algebra-lemma-zero-at-ass-zero", "algebra-definition-rank", "algebra-lemma-silly", "algebra-lemma-ass-zero-divisors", "algebra-lemma-div-x-exact-one-less", "algebra-lemma-acyclic" ], "ref_ids": [ 908, 909, 911, 713, 1501, 378, 704, 912, 913 ] } ], "ref_ids": [] }, { "id": 1420, "type": "theorem", "label": "algebra-proposition-CM-module", "categories": [ "algebra" ], "title": "algebra-proposition-CM-module", "contents": [ "Let $R$ be a Noetherian local ring, with maximal ideal $\\mathfrak m$.", "Let $M$ be a Cohen-Macaulay module over $R$ whose support has dimension $d$.", "Suppose that $g_1, \\ldots, g_c$ are elements of", "$\\mathfrak m$ such that $\\dim(\\text{Supp}(M/(g_1, \\ldots, g_c)M))", "= d - c$. Then $g_1, \\ldots, g_c$ is an $M$-regular sequence,", "and can be extended to a maximal $M$-regular sequence." ], "refs": [], "proofs": [ { "contents": [ "Let $Z = \\text{Supp}(M) \\subset \\Spec(R)$.", "By Lemma \\ref{lemma-one-equation} in the chain", "$Z \\supset Z \\cap V(g_1) \\supset \\ldots \\supset Z \\cap V(g_1, \\ldots, g_c)$", "each step decreases the dimension at most by $1$. Hence by assumption", "each step decreases the dimension by exactly $1$ each time. Thus we", "may successively apply Lemma \\ref{lemma-CM-one-g} to the modules", "$M/(g_1, \\ldots, g_i)$ and the element $g_{i + 1}$.", "\\medskip\\noindent", "To extend $g_1, \\ldots, g_c$ by one element if $c < d$ we simply", "choose an element $g_{c + 1} \\in \\mathfrak m$ which is not", "in any of the finitely many minimal primes of $Z \\cap V(g_1, \\ldots, g_c)$,", "using Lemma \\ref{lemma-silly}." ], "refs": [ "algebra-lemma-CM-one-g", "algebra-lemma-silly" ], "ref_ids": [ 915, 378 ] } ], "ref_ids": [] }, { "id": 1421, "type": "theorem", "label": "algebra-proposition-regular-finite-gl-dim", "categories": [ "algebra" ], "title": "algebra-proposition-regular-finite-gl-dim", "contents": [ "Let $R$ be a regular local ring of dimension $d$.", "Every finite $R$-module $M$ of depth $e$ has a finite free", "resolution", "$$", "0 \\to F_{d-e} \\to \\ldots \\to F_0 \\to M \\to 0.", "$$", "In particular a regular local ring has global dimension $\\leq d$." ], "refs": [], "proofs": [ { "contents": [ "The first part holds in view of Lemma \\ref{lemma-regular-mcm-free}", "and Lemma \\ref{lemma-mcm-resolution}. The last part follows from this", "and Lemma \\ref{lemma-finite-gl-dim}." ], "refs": [ "algebra-lemma-regular-mcm-free", "algebra-lemma-mcm-resolution", "algebra-lemma-finite-gl-dim" ], "ref_ids": [ 944, 929, 974 ] } ], "ref_ids": [] }, { "id": 1422, "type": "theorem", "label": "algebra-proposition-finite-gl-dim-regular", "categories": [ "algebra" ], "title": "algebra-proposition-finite-gl-dim-regular", "contents": [ "A Noetherian local ring whose residue field", "has finite projective dimension is a regular local ring.", "In particular a Noetherian local ring of", "finite global dimension is a regular local ring." ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-length-resolution-residue-field}", "and \\ref{lemma-dim-gl-dim} we see that", "$\\dim(R) \\geq \\dim_\\kappa(\\mathfrak m /\\mathfrak m^2)$.", "Thus the result follows immediately from Definition", "\\ref{definition-regular-local}." ], "refs": [ "algebra-lemma-length-resolution-residue-field", "algebra-lemma-dim-gl-dim", "algebra-definition-regular-local" ], "ref_ids": [ 977, 978, 1480 ] } ], "ref_ids": [] }, { "id": 1423, "type": "theorem", "label": "algebra-proposition-Auslander-Buchsbaum", "categories": [ "algebra" ], "title": "algebra-proposition-Auslander-Buchsbaum", "contents": [ "Let $R$ be a Noetherian local ring. Let $M$ be a nonzero finite $R$-module", "which has finite projective dimension $\\text{pd}_R(M)$. Then we have", "$$", "\\text{depth}(R) = \\text{pd}_R(M) + \\text{depth}(M)", "$$" ], "refs": [], "proofs": [ { "contents": [ "We prove this by induction on $\\text{depth}(M)$. The most interesting", "case is the case $\\text{depth}(M) = 0$. In this case, let", "$$", "0 \\to R^{n_e} \\to R^{n_{e-1}} \\to \\ldots \\to R^{n_0} \\to M \\to 0", "$$", "be a minimal finite free resolution, so $e = \\text{pd}_R(M)$.", "By Lemma \\ref{lemma-add-trivial-complex} we may assume all matrix", "coefficients of the maps in the complex are contained in the maximal", "ideal of $R$. Then on the one hand, by", "Proposition \\ref{proposition-what-exact} we see that", "$\\text{depth}(R) \\geq e$. On the other hand, breaking the long", "exact sequence into short exact sequences", "\\begin{align*}", "0 \\to R^{n_e} \\to R^{n_{e - 1}} \\to K_{e - 2} \\to 0,\\\\", "0 \\to K_{e - 2} \\to R^{n_{e - 2}} \\to K_{e - 3} \\to 0,\\\\", "\\ldots,\\\\", "0 \\to K_0 \\to R^{n_0} \\to M \\to 0", "\\end{align*}", "we see, using Lemma \\ref{lemma-depth-in-ses}, that", "\\begin{align*}", "\\text{depth}(K_{e - 2}) \\geq \\text{depth}(R) - 1,\\\\", "\\text{depth}(K_{e - 3}) \\geq \\text{depth}(R) - 2,\\\\", "\\ldots,\\\\", "\\text{depth}(K_0) \\geq \\text{depth}(R) - (e - 1),\\\\", "\\text{depth}(M) \\geq \\text{depth}(R) - e", "\\end{align*}", "and since $\\text{depth}(M) = 0$ we conclude $\\text{depth}(R) \\leq e$.", "This finishes the proof of the case $\\text{depth}(M) = 0$.", "\\medskip\\noindent", "Induction step. If $\\text{depth}(M) > 0$, then we pick $x \\in \\mathfrak m$", "which is a nonzerodivisor on both $M$ and $R$. This is possible, because", "either $\\text{pd}_R(M) > 0$ and $\\text{depth}(R) > 0$ by the aforementioned", "Proposition \\ref{proposition-what-exact} or $\\text{pd}_R(M) = 0$ in which", "case $M$ is finite free hence also $\\text{depth}(R) = \\text{depth}(M) > 0$.", "Thus $\\text{depth}(R \\oplus M) > 0$ by Lemma \\ref{lemma-depth-in-ses}", "(for example) and we can find an $x \\in \\mathfrak m$ which is a nonzerodivisor", "on both $R$ and $M$. Let", "$$", "0 \\to R^{n_e} \\to R^{n_{e-1}} \\to \\ldots \\to R^{n_0} \\to M \\to 0", "$$", "be a minimal resolution as above. An application of the snake lemma", "shows that", "$$", "0 \\to (R/xR)^{n_e} \\to (R/xR)^{n_{e-1}} \\to \\ldots \\to (R/xR)^{n_0} \\to", "M/xM \\to 0", "$$", "is a minimal resolution too. Thus $\\text{pd}_R(M) = \\text{pd}_{R/xR}(M/xM)$.", "By Lemma \\ref{lemma-depth-drops-by-one} we have", "$\\text{depth}(R/xR) = \\text{depth}(R) - 1$ and", "$\\text{depth}(M/xM) = \\text{depth}(M) - 1$.", "Till now depths have all been depths as $R$ modules, but we observe that", "$\\text{depth}_R(M/xM) = \\text{depth}_{R/xR}(M/xM)$ and similarly for $R/xR$.", "By induction hypothesis we see that the", "Auslander-Buchsbaum formula holds for $M/xM$ over $R/xR$. Since the", "depths of both $R/xR$ and $M/xM$ have decreased by one and the projective", "dimension has not changed we conclude." ], "refs": [ "algebra-lemma-add-trivial-complex", "algebra-proposition-what-exact", "algebra-lemma-depth-in-ses", "algebra-proposition-what-exact", "algebra-lemma-depth-in-ses", "algebra-lemma-depth-drops-by-one" ], "ref_ids": [ 908, 1419, 773, 1419, 773, 774 ] } ], "ref_ids": [] }, { "id": 1424, "type": "theorem", "label": "algebra-proposition-finite-gl-dim-polynomial-ring", "categories": [ "algebra" ], "title": "algebra-proposition-finite-gl-dim-polynomial-ring", "contents": [ "A polynomial algebra in $n$ variables over a field is a regular ring.", "It has global dimension $n$. All localizations at maximal ideals", "are regular local rings of dimension $n$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-dim-affine-space}", "all localizations $k[x_1, \\ldots, x_n]_{\\mathfrak m}$", "at maximal ideals are regular local rings of dimension $n$. Hence", "we conclude by Lemma \\ref{lemma-finite-gl-dim-finite-dim-regular}." ], "refs": [ "algebra-lemma-dim-affine-space", "algebra-lemma-finite-gl-dim-finite-dim-regular" ], "ref_ids": [ 992, 980 ] } ], "ref_ids": [] }, { "id": 1425, "type": "theorem", "label": "algebra-proposition-characterize-formally-smooth", "categories": [ "algebra" ], "title": "algebra-proposition-characterize-formally-smooth", "contents": [ "Let $R \\to S$ be a ring map. Consider a formally smooth $R$-algebra $P$ and", "a surjection $P \\to S$ with kernel $J$. The following are equivalent", "\\begin{enumerate}", "\\item $S$ is formally smooth over $R$,", "\\item for some $P \\to S$ as above there exists a", "section to $P/J^2 \\to S$,", "\\item for all $P \\to S$ as above there exists a", "section to $P/J^2 \\to S$,", "\\item for some $P \\to S$ as above the sequence", "$0 \\to J/J^2 \\to \\Omega_{P/R} \\otimes S \\to \\Omega_{S/R} \\to 0$ is split exact,", "\\item for all $P \\to S$ as above the sequence", "$0 \\to J/J^2 \\to \\Omega_{P/R} \\otimes S \\to \\Omega_{S/R} \\to 0$ is split exact,", "and", "\\item the naive cotangent complex $\\NL_{S/R}$ is quasi-isomorphic to a", "projective $S$-module placed in degree $0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (3) implies (2), see first part of the proof of", "Lemma \\ref{lemma-characterize-formally-smooth}.", "It is also true that (3) implies (5) implies (4) and that (2) implies (4), see", "first part of the proof of", "Lemma \\ref{lemma-characterize-formally-smooth-again}.", "Finally, Lemma \\ref{lemma-characterize-formally-smooth-again}", "applied to the canonical surjection $R[S] \\to S$", "(\\ref{equation-canonical-presentation}) shows that (1) implies (6).", "\\medskip\\noindent", "Assume (4) and let's prove (6). Consider the sequence of", "Lemma \\ref{lemma-exact-sequence-NL}", "associated to the ring maps $R \\to P \\to S$. By the implication", "(1) $\\Rightarrow$ (6) proved above we see that $\\NL_{P/R} \\otimes_R S$", "is quasi-isomorphic to $\\Omega_{P/R} \\otimes_P S$ placed in degree $0$.", "Hence $H_1(\\NL_{P/R} \\otimes_P S) = 0$. Since $P \\to S$ is surjective we", "see that $\\NL_{S/P}$ is homotopy equivalent to $J/J^2$ placed in degree $1$", "(Lemma \\ref{lemma-NL-surjection}). Thus we obtain the exact sequence", "$0 \\to H_1(L_{S/R}) \\to J/J^2 \\to \\Omega_{P/R} \\otimes_P S \\to", "\\Omega_{S/R} \\to 0$.", "By assumption we see that $H_1(L_{S/R}) = 0$ and that $\\Omega_{S/R}$", "is a projective $S$-module. Thus (6) follows.", "\\medskip\\noindent", "Finally, let's prove that (6) implies (1). The assumption means that", "the complex $J/J^2 \\to \\Omega_{P/R} \\otimes S$ where $P = R[S]$ and", "$P \\to S$ is the canonical surjection (\\ref{equation-canonical-presentation}).", "Hence Lemma \\ref{lemma-characterize-formally-smooth-again} shows that $S$", "is formally smooth over $R$." ], "refs": [ "algebra-lemma-characterize-formally-smooth", "algebra-lemma-characterize-formally-smooth-again", "algebra-lemma-characterize-formally-smooth-again", "algebra-lemma-exact-sequence-NL", "algebra-lemma-NL-surjection", "algebra-lemma-characterize-formally-smooth-again" ], "ref_ids": [ 1207, 1208, 1208, 1153, 1154, 1208 ] } ], "ref_ids": [] }, { "id": 1426, "type": "theorem", "label": "algebra-proposition-smooth-formally-smooth", "categories": [ "algebra" ], "title": "algebra-proposition-smooth-formally-smooth", "contents": [ "Let $R \\to S$ be a ring map. The following are equivalent", "\\begin{enumerate}", "\\item $R \\to S$ is of finite presentation and formally smooth,", "\\item $R \\to S$ is smooth.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Proposition \\ref{proposition-characterize-formally-smooth}", "and Definition \\ref{definition-smooth}.", "(Note that $\\Omega_{S/R}$ is a finitely presented $S$-module if $R \\to S$ is", "of finite presentation, see", "Lemma \\ref{lemma-differentials-finitely-presented}.)" ], "refs": [ "algebra-proposition-characterize-formally-smooth", "algebra-definition-smooth", "algebra-lemma-differentials-finitely-presented" ], "ref_ids": [ 1425, 1534, 1141 ] } ], "ref_ids": [] }, { "id": 1427, "type": "theorem", "label": "algebra-proposition-etale-locally-standard", "categories": [ "algebra" ], "title": "algebra-proposition-etale-locally-standard", "contents": [ "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime.", "If $R \\to S$ is \\'etale at $\\mathfrak q$, then there exists", "a $g \\in S$, $g \\not \\in \\mathfrak q$ such that $R \\to S_g$", "is standard \\'etale." ], "refs": [], "proofs": [ { "contents": [ "The following proof is a little roundabout and there may be ways to", "shorten it.", "\\medskip\\noindent", "Step 1. By Definition \\ref{definition-etale}", "there exists a $g \\in S$, $g \\not \\in \\mathfrak q$", "such that $R \\to S_g$ is \\'etale. Thus we may assume that $S$ is \\'etale", "over $R$.", "\\medskip\\noindent", "Step 2. By Lemma \\ref{lemma-etale} there exists an \\'etale ring map", "$R_0 \\to S_0$ with $R_0$ of finite type over $\\mathbf{Z}$, and a ring map", "$R_0 \\to R$ such that $R = R \\otimes_{R_0} S_0$. Denote", "$\\mathfrak q_0$ the prime of $S_0$ corresponding to $\\mathfrak q$.", "If we show the result for $(R_0 \\to S_0, \\mathfrak q_0)$ then the", "result follows for $(R \\to S, \\mathfrak q)$ by base change. Hence", "we may assume that $R$ is Noetherian.", "\\medskip\\noindent", "Step 3.", "Note that $R \\to S$ is quasi-finite by Lemma \\ref{lemma-etale-quasi-finite}.", "By Lemma \\ref{lemma-quasi-finite-open-integral-closure}", "there exists a finite ring map $R \\to S'$, an $R$-algebra map", "$S' \\to S$, an element $g' \\in S'$ such that", "$g' \\not \\in \\mathfrak q$ such that $S' \\to S$ induces", "an isomorphism $S'_{g'} \\cong S_{g'}$.", "(Note that of course $S'$ is not \\'etale over $R$ in general.)", "Thus we may assume that (a) $R$ is Noetherian, (b) $R \\to S$ is finite", "and (c) $R \\to S$ is \\'etale at $\\mathfrak q$", "(but no longer necessarily \\'etale at all primes).", "\\medskip\\noindent", "Step 4. Let $\\mathfrak p \\subset R$ be the prime corresponding", "to $\\mathfrak q$. Consider the fibre ring", "$S \\otimes_R \\kappa(\\mathfrak p)$. This is a finite algebra over", "$\\kappa(\\mathfrak p)$. Hence it is Artinian", "(see Lemma \\ref{lemma-finite-dimensional-algebra}) and", "so a finite product of local rings", "$$", "S \\otimes_R \\kappa(\\mathfrak p) = \\prod\\nolimits_{i = 1}^n A_i", "$$", "see Proposition \\ref{proposition-dimension-zero-ring}. One of the factors,", "say $A_1$, is the local ring $S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}$", "which is isomorphic to $\\kappa(\\mathfrak q)$,", "see Lemma \\ref{lemma-etale-at-prime}. The other factors correspond to", "the other primes, say $\\mathfrak q_2, \\ldots, \\mathfrak q_n$ of", "$S$ lying over $\\mathfrak p$.", "\\medskip\\noindent", "Step 5. We may choose a nonzero element $\\alpha \\in \\kappa(\\mathfrak q)$ which", "generates the finite separable field extension", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$ (so even if the", "field extension is trivial we do not allow $\\alpha = 0$).", "Note that for any $\\lambda \\in \\kappa(\\mathfrak p)^*$ the", "element $\\lambda \\alpha$ also generates $\\kappa(\\mathfrak q)$", "over $\\kappa(\\mathfrak p)$. Consider the element", "$$", "\\overline{t} =", "(\\alpha, 0, \\ldots, 0) \\in", "\\prod\\nolimits_{i = 1}^n A_i =", "S \\otimes_R \\kappa(\\mathfrak p).", "$$", "After possibly replacing $\\alpha$ by $\\lambda \\alpha$ as above", "we may assume that $\\overline{t}$ is the image of $t \\in S$.", "Let $I \\subset R[x]$ be the kernel of the $R$-algebra", "map $R[x] \\to S$ which maps $x$ to $t$. Set $S' = R[x]/I$,", "so $S' \\subset S$. Here is a diagram", "$$", "\\xymatrix{", "R[x] \\ar[r] & S' \\ar[r] & S \\\\", "R \\ar[u] \\ar[ru] \\ar[rru] & &", "}", "$$", "By construction the primes $\\mathfrak q_j$, $j \\geq 2$ of $S$ all", "lie over the prime $(\\mathfrak p, x)$ of $R[x]$, whereas", "the prime $\\mathfrak q$ lies over a different prime of $R[x]$", "because $\\alpha \\not = 0$.", "\\medskip\\noindent", "Step 6. Denote $\\mathfrak q' \\subset S'$ the prime of $S'$", "corresponding to $\\mathfrak q$. By the above $\\mathfrak q$ is", "the only prime of $S$ lying over $\\mathfrak q'$. Thus we see that", "$S_{\\mathfrak q} = S_{\\mathfrak q'}$, see", "Lemma \\ref{lemma-unique-prime-over-localize-below} (we have", "going up for $S' \\to S$ by Lemma \\ref{lemma-integral-going-up}", "since $S' \\to S$ is finite as $R \\to S$ is finite).", "It follows that $S'_{\\mathfrak q'} \\to S_{\\mathfrak q}$ is finite", "and injective as the localization of the finite injective ring map", "$S' \\to S$. Consider the maps of local rings", "$$", "R_{\\mathfrak p} \\to S'_{\\mathfrak q'} \\to S_{\\mathfrak q}", "$$", "The second map is finite and injective. We have", "$S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q} = \\kappa(\\mathfrak q)$,", "see Lemma \\ref{lemma-etale-at-prime}.", "Hence a fortiori", "$S_{\\mathfrak q}/\\mathfrak q'S_{\\mathfrak q} = \\kappa(\\mathfrak q)$.", "Since", "$$", "\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q') \\subset \\kappa(\\mathfrak q)", "$$", "and since $\\alpha$ is in the image of $\\kappa(\\mathfrak q')$ in", "$\\kappa(\\mathfrak q)$", "we conclude that $\\kappa(\\mathfrak q') = \\kappa(\\mathfrak q)$.", "Hence by Nakayama's Lemma \\ref{lemma-NAK} applied to the", "$S'_{\\mathfrak q'}$-module map $S'_{\\mathfrak q'} \\to S_{\\mathfrak q}$,", "the map $S'_{\\mathfrak q'} \\to S_{\\mathfrak q}$ is surjective.", "In other words,", "$S'_{\\mathfrak q'} \\cong S_{\\mathfrak q}$.", "\\medskip\\noindent", "Step 7. By Lemma \\ref{lemma-isomorphic-local-rings} there exist", "$g \\in S$, $g \\not \\in \\mathfrak q$ and $g' \\in S'$, $g' \\not \\in \\mathfrak q'$", "such that $S'_{g'} \\cong S_g$. As $R$ is Noetherian the ring $S'$ is finite", "over $R$ because it is an $R$-submodule", "of the finite $R$-module $S$. Hence after replacing $S$ by $S'$ we may", "assume that (a) $R$ is Noetherian, (b) $S$ finite over $R$, (c)", "$S$ is \\'etale over $R$ at $\\mathfrak q$, and (d) $S = R[x]/I$.", "\\medskip\\noindent", "Step 8. Consider the ring", "$S \\otimes_R \\kappa(\\mathfrak p) = \\kappa(\\mathfrak p)[x]/\\overline{I}$", "where $\\overline{I} = I \\cdot \\kappa(\\mathfrak p)[x]$ is the ideal generated", "by $I$ in $\\kappa(\\mathfrak p)[x]$. As $\\kappa(\\mathfrak p)[x]$ is a PID", "we know that $\\overline{I} = (\\overline{h})$ for some monic", "$\\overline{h} \\in \\kappa(\\mathfrak p)[x]$. After replacing $\\overline{h}$", "by $\\lambda \\cdot \\overline{h}$ for some $\\lambda \\in \\kappa(\\mathfrak p)$", "we may assume that $\\overline{h}$ is the image of some $h \\in I \\subset R[x]$.", "(The problem is that we do not know if we may choose $h$ monic.)", "Also, as in Step 4 we know that", "$S \\otimes_R \\kappa(\\mathfrak p) = A_1 \\times \\ldots \\times A_n$ with", "$A_1 = \\kappa(\\mathfrak q)$ a finite separable extension of", "$\\kappa(\\mathfrak p)$ and $A_2, \\ldots, A_n$ local. This implies", "that", "$$", "\\overline{h} = \\overline{h}_1 \\overline{h}_2^{e_2} \\ldots \\overline{h}_n^{e_n}", "$$", "for certain pairwise coprime irreducible monic polynomials", "$\\overline{h}_i \\in \\kappa(\\mathfrak p)[x]$ and certain", "$e_2, \\ldots, e_n \\geq 1$. Here the numbering is chosen so that", "$A_i = \\kappa(\\mathfrak p)[x]/(\\overline{h}_i^{e_i})$ as", "$\\kappa(\\mathfrak p)[x]$-algebras. Note that $\\overline{h}_1$ is", "the minimal polynomial of $\\alpha \\in \\kappa(\\mathfrak q)$ and hence", "is a separable polynomial (its derivative is prime to itself).", "\\medskip\\noindent", "Step 9. Let $m \\in I$ be a monic element; such an element exists", "because the ring extension $R \\to R[x]/I$ is finite hence integral.", "Denote $\\overline{m}$ the image in $\\kappa(\\mathfrak p)[x]$.", "We may factor", "$$", "\\overline{m} = \\overline{k}", "\\overline{h}_1^{d_1} \\overline{h}_2^{d_2} \\ldots \\overline{h}_n^{d_n}", "$$", "for some $d_1 \\geq 1$, $d_j \\geq e_j$, $j = 2, \\ldots, n$ and", "$\\overline{k} \\in \\kappa(\\mathfrak p)[x]$ prime to all the $\\overline{h}_i$.", "Set $f = m^l + h$ where $l \\deg(m) > \\deg(h)$, and $l \\geq 2$.", "Then $f$ is monic as a polynomial over $R$. Also, the image $\\overline{f}$", "of $f$ in $\\kappa(\\mathfrak p)[x]$ factors as", "$$", "\\overline{f} =", "\\overline{h}_1 \\overline{h}_2^{e_2} \\ldots \\overline{h}_n^{e_n}", "+", "\\overline{k}^l \\overline{h}_1^{ld_1} \\overline{h}_2^{ld_2}", "\\ldots \\overline{h}_n^{ld_n}", "=", "\\overline{h}_1(\\overline{h}_2^{e_2} \\ldots \\overline{h}_n^{e_n}", "+", "\\overline{k}^l", "\\overline{h}_1^{ld_1 - 1} \\overline{h}_2^{ld_2} \\ldots \\overline{h}_n^{ld_n})", "= \\overline{h}_1 \\overline{w}", "$$", "with $\\overline{w}$ a polynomial relatively prime to $\\overline{h}_1$.", "Set $g = f'$ (the derivative with respect to $x$).", "\\medskip\\noindent", "Step 10. The ring map $R[x] \\to S = R[x]/I$ has the properties:", "(1) it maps $f$ to zero, and", "(2) it maps $g$ to an element of $S \\setminus \\mathfrak q$.", "The first assertion is clear since $f$ is an element of $I$.", "For the second assertion we just have to show that $g$ does", "not map to zero in", "$\\kappa(\\mathfrak q) = \\kappa(\\mathfrak p)[x]/(\\overline{h}_1)$.", "The image of $g$ in $\\kappa(\\mathfrak p)[x]$ is the derivative", "of $\\overline{f}$. Thus (2) is clear because", "$$", "\\overline{g} =", "\\frac{\\text{d}\\overline{f}}{\\text{d}x} =", "\\overline{w}\\frac{\\text{d}\\overline{h}_1}{\\text{d}x} +", "\\overline{h}_1\\frac{\\text{d}\\overline{w}}{\\text{d}x},", "$$", "$\\overline{w}$ is prime to $\\overline{h}_1$ and", "$\\overline{h}_1$ is separable.", "\\medskip\\noindent", "Step 11.", "We conclude that $\\varphi : R[x]/(f) \\to S$ is a surjective ring map,", "$R[x]_g/(f)$ is \\'etale over $R$ (because it is standard \\'etale,", "see Lemma \\ref{lemma-standard-etale}) and $\\varphi(g) \\not \\in \\mathfrak q$.", "Pick an element $g' \\in R[x]/(f)$ such that", "also $\\varphi(g') \\not \\in \\mathfrak q$ and $S_{\\varphi(g')}$", "is \\'etale over $R$ (which exists since $S$ is \\'etale over $R$ at", "$\\mathfrak q$). Then the ring map", "$R[x]_{gg'}/(f) \\to S_{\\varphi(gg')}$ is a surjective map of \\'etale", "algebras over $R$. Hence it is \\'etale by Lemma \\ref{lemma-map-between-etale}.", "Hence it is a localization by", "Lemma \\ref{lemma-surjective-flat-finitely-presented}.", "Thus a localization of $S$ at an element not in $\\mathfrak q$ is", "isomorphic to a localization of a standard \\'etale algebra over $R$", "which is what we wanted to show." ], "refs": [ "algebra-definition-etale", "algebra-lemma-etale", "algebra-lemma-etale-quasi-finite", "algebra-lemma-quasi-finite-open-integral-closure", "algebra-lemma-finite-dimensional-algebra", "algebra-proposition-dimension-zero-ring", "algebra-lemma-etale-at-prime", "algebra-lemma-unique-prime-over-localize-below", "algebra-lemma-integral-going-up", "algebra-lemma-etale-at-prime", "algebra-lemma-NAK", "algebra-lemma-isomorphic-local-rings", "algebra-lemma-standard-etale", "algebra-lemma-map-between-etale", "algebra-lemma-surjective-flat-finitely-presented" ], "ref_ids": [ 1539, 1231, 1234, 1067, 642, 1410, 1233, 556, 500, 1233, 401, 1085, 1241, 1236, 1237 ] } ], "ref_ids": [] }, { "id": 1428, "type": "theorem", "label": "algebra-proposition-unramified-locally-standard", "categories": [ "algebra" ], "title": "algebra-proposition-unramified-locally-standard", "contents": [ "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime.", "If $R \\to S$ is unramified at $\\mathfrak q$, then there exist", "\\begin{enumerate}", "\\item a $g \\in S$, $g \\not \\in \\mathfrak q$,", "\\item a standard \\'etale ring map $R \\to S'$, and", "\\item a surjective $R$-algebra map $S' \\to S_g$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This proof is the ``same'' as the proof of", "Proposition \\ref{proposition-etale-locally-standard}.", "The proof is a little roundabout and there may be ways to", "shorten it.", "\\medskip\\noindent", "Step 1. By Definition \\ref{definition-unramified}", "there exists a $g \\in S$, $g \\not \\in \\mathfrak q$", "such that $R \\to S_g$ is unramified. Thus we may assume that $S$ is", "unramified over $R$.", "\\medskip\\noindent", "Step 2. By Lemma \\ref{lemma-unramified}", "there exists an unramified ring map $R_0 \\to S_0$", "with $R_0$ of finite type over $\\mathbf{Z}$, and a ring map", "$R_0 \\to R$ such that $S$ is a quotient of $R \\otimes_{R_0} S_0$. Denote", "$\\mathfrak q_0$ the prime of $S_0$ corresponding to $\\mathfrak q$.", "If we show the result for $(R_0 \\to S_0, \\mathfrak q_0)$ then the", "result follows for $(R \\to S, \\mathfrak q)$ by base change. Hence", "we may assume that $R$ is Noetherian.", "\\medskip\\noindent", "Step 3.", "Note that $R \\to S$ is quasi-finite by", "Lemma \\ref{lemma-unramified-quasi-finite}.", "By Lemma \\ref{lemma-quasi-finite-open-integral-closure}", "there exists a finite ring map $R \\to S'$, an $R$-algebra map", "$S' \\to S$, an element $g' \\in S'$ such that", "$g' \\not \\in \\mathfrak q$ such that $S' \\to S$ induces", "an isomorphism $S'_{g'} \\cong S_{g'}$.", "(Note that $S'$ may not be unramified over $R$.)", "Thus we may assume that (a) $R$ is Noetherian, (b) $R \\to S$ is finite", "and (c) $R \\to S$ is unramified at $\\mathfrak q$", "(but no longer necessarily unramified at all primes).", "\\medskip\\noindent", "Step 4. Let $\\mathfrak p \\subset R$ be the prime corresponding", "to $\\mathfrak q$. Consider the fibre ring", "$S \\otimes_R \\kappa(\\mathfrak p)$. This is a finite algebra over", "$\\kappa(\\mathfrak p)$. Hence it is Artinian", "(see Lemma \\ref{lemma-finite-dimensional-algebra}) and", "so a finite product of local rings", "$$", "S \\otimes_R \\kappa(\\mathfrak p) = \\prod\\nolimits_{i = 1}^n A_i", "$$", "see Proposition \\ref{proposition-dimension-zero-ring}. One of the factors,", "say $A_1$, is the local ring $S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}$", "which is isomorphic to $\\kappa(\\mathfrak q)$,", "see Lemma \\ref{lemma-unramified-at-prime}. The other factors correspond to", "the other primes, say $\\mathfrak q_2, \\ldots, \\mathfrak q_n$ of", "$S$ lying over $\\mathfrak p$.", "\\medskip\\noindent", "Step 5. We may choose a nonzero element $\\alpha \\in \\kappa(\\mathfrak q)$ which", "generates the finite separable field extension", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$ (so even if the", "field extension is trivial we do not allow $\\alpha = 0$).", "Note that for any $\\lambda \\in \\kappa(\\mathfrak p)^*$ the", "element $\\lambda \\alpha$ also generates $\\kappa(\\mathfrak q)$", "over $\\kappa(\\mathfrak p)$. Consider the element", "$$", "\\overline{t} =", "(\\alpha, 0, \\ldots, 0) \\in", "\\prod\\nolimits_{i = 1}^n A_i =", "S \\otimes_R \\kappa(\\mathfrak p).", "$$", "After possibly replacing $\\alpha$ by $\\lambda \\alpha$ as above", "we may assume that $\\overline{t}$ is the image of $t \\in S$.", "Let $I \\subset R[x]$ be the kernel of the $R$-algebra", "map $R[x] \\to S$ which maps $x$ to $t$. Set $S' = R[x]/I$,", "so $S' \\subset S$. Here is a diagram", "$$", "\\xymatrix{", "R[x] \\ar[r] & S' \\ar[r] & S \\\\", "R \\ar[u] \\ar[ru] \\ar[rru] & &", "}", "$$", "By construction the primes $\\mathfrak q_j$, $j \\geq 2$ of $S$ all", "lie over the prime $(\\mathfrak p, x)$ of $R[x]$, whereas", "the prime $\\mathfrak q$ lies over a different prime of $R[x]$", "because $\\alpha \\not = 0$.", "\\medskip\\noindent", "Step 6. Denote $\\mathfrak q' \\subset S'$ the prime of $S'$", "corresponding to $\\mathfrak q$. By the above $\\mathfrak q$ is", "the only prime of $S$ lying over $\\mathfrak q'$. Thus we see that", "$S_{\\mathfrak q} = S_{\\mathfrak q'}$, see", "Lemma \\ref{lemma-unique-prime-over-localize-below} (we have", "going up for $S' \\to S$ by Lemma \\ref{lemma-integral-going-up}", "since $S' \\to S$ is finite as $R \\to S$ is finite).", "It follows that $S'_{\\mathfrak q'} \\to S_{\\mathfrak q}$ is finite", "and injective as the localization of the finite injective ring map", "$S' \\to S$. Consider the maps of local rings", "$$", "R_{\\mathfrak p} \\to S'_{\\mathfrak q'} \\to S_{\\mathfrak q}", "$$", "The second map is finite and injective. We have", "$S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q} = \\kappa(\\mathfrak q)$,", "see Lemma \\ref{lemma-unramified-at-prime}.", "Hence a fortiori", "$S_{\\mathfrak q}/\\mathfrak q'S_{\\mathfrak q} = \\kappa(\\mathfrak q)$.", "Since", "$$", "\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q') \\subset \\kappa(\\mathfrak q)", "$$", "and since $\\alpha$ is in the image of $\\kappa(\\mathfrak q')$ in", "$\\kappa(\\mathfrak q)$", "we conclude that $\\kappa(\\mathfrak q') = \\kappa(\\mathfrak q)$.", "Hence by Nakayama's Lemma \\ref{lemma-NAK} applied to the", "$S'_{\\mathfrak q'}$-module map $S'_{\\mathfrak q'} \\to S_{\\mathfrak q}$,", "the map $S'_{\\mathfrak q'} \\to S_{\\mathfrak q}$ is surjective.", "In other words,", "$S'_{\\mathfrak q'} \\cong S_{\\mathfrak q}$.", "\\medskip\\noindent", "Step 7. By Lemma \\ref{lemma-isomorphic-local-rings} there exist", "$g \\in S$, $g \\not \\in \\mathfrak q$ and", "$g' \\in S'$, $g' \\not \\in \\mathfrak q'$ such that $S'_{g'} \\cong S_g$.", "As $R$ is Noetherian the ring $S'$ is finite over $R$", "because it is an $R$-submodule", "of the finite $R$-module $S$. Hence after replacing $S$ by $S'$ we may", "assume that (a) $R$ is Noetherian, (b) $S$ finite over $R$, (c)", "$S$ is unramified over $R$ at $\\mathfrak q$, and (d) $S = R[x]/I$.", "\\medskip\\noindent", "Step 8. Consider the ring", "$S \\otimes_R \\kappa(\\mathfrak p) = \\kappa(\\mathfrak p)[x]/\\overline{I}$", "where $\\overline{I} = I \\cdot \\kappa(\\mathfrak p)[x]$ is the ideal generated", "by $I$ in $\\kappa(\\mathfrak p)[x]$. As $\\kappa(\\mathfrak p)[x]$ is a PID", "we know that $\\overline{I} = (\\overline{h})$ for some monic", "$\\overline{h} \\in \\kappa(\\mathfrak p)$. After replacing $\\overline{h}$", "by $\\lambda \\cdot \\overline{h}$ for some $\\lambda \\in \\kappa(\\mathfrak p)$", "we may assume that $\\overline{h}$ is the image of some $h \\in R[x]$.", "(The problem is that we do not know if we may choose $h$ monic.)", "Also, as in Step 4 we know that", "$S \\otimes_R \\kappa(\\mathfrak p) = A_1 \\times \\ldots \\times A_n$ with", "$A_1 = \\kappa(\\mathfrak q)$ a finite separable extension of", "$\\kappa(\\mathfrak p)$ and $A_2, \\ldots, A_n$ local. This implies", "that", "$$", "\\overline{h} = \\overline{h}_1 \\overline{h}_2^{e_2} \\ldots \\overline{h}_n^{e_n}", "$$", "for certain pairwise coprime irreducible monic polynomials", "$\\overline{h}_i \\in \\kappa(\\mathfrak p)[x]$ and certain", "$e_2, \\ldots, e_n \\geq 1$. Here the numbering is chosen so that", "$A_i = \\kappa(\\mathfrak p)[x]/(\\overline{h}_i^{e_i})$ as", "$\\kappa(\\mathfrak p)[x]$-algebras. Note that $\\overline{h}_1$ is", "the minimal polynomial of $\\alpha \\in \\kappa(\\mathfrak q)$ and hence", "is a separable polynomial (its derivative is prime to itself).", "\\medskip\\noindent", "Step 9. Let $m \\in I$ be a monic element; such an element exists", "because the ring extension $R \\to R[x]/I$ is finite hence integral.", "Denote $\\overline{m}$ the image in $\\kappa(\\mathfrak p)[x]$.", "We may factor", "$$", "\\overline{m} = \\overline{k}", "\\overline{h}_1^{d_1} \\overline{h}_2^{d_2} \\ldots \\overline{h}_n^{d_n}", "$$", "for some $d_1 \\geq 1$, $d_j \\geq e_j$, $j = 2, \\ldots, n$ and", "$\\overline{k} \\in \\kappa(\\mathfrak p)[x]$ prime to all the $\\overline{h}_i$.", "Set $f = m^l + h$ where $l \\deg(m) > \\deg(h)$, and $l \\geq 2$.", "Then $f$ is monic as a polynomial over $R$. Also, the image $\\overline{f}$", "of $f$ in $\\kappa(\\mathfrak p)[x]$ factors as", "$$", "\\overline{f} =", "\\overline{h}_1 \\overline{h}_2^{e_2} \\ldots \\overline{h}_n^{e_n}", "+", "\\overline{k}^l \\overline{h}_1^{ld_1} \\overline{h}_2^{ld_2}", "\\ldots \\overline{h}_n^{ld_n}", "=", "\\overline{h}_1(\\overline{h}_2^{e_2} \\ldots \\overline{h}_n^{e_n}", "+", "\\overline{k}^l", "\\overline{h}_1^{ld_1 - 1} \\overline{h}_2^{ld_2} \\ldots \\overline{h}_n^{ld_n})", "= \\overline{h}_1 \\overline{w}", "$$", "with $\\overline{w}$ a polynomial relatively prime to $\\overline{h}_1$.", "Set $g = f'$ (the derivative with respect to $x$).", "\\medskip\\noindent", "Step 10. The ring map $R[x] \\to S = R[x]/I$ has the properties:", "(1) it maps $f$ to zero, and", "(2) it maps $g$ to an element of $S \\setminus \\mathfrak q$.", "The first assertion is clear since $f$ is an element of $I$.", "For the second assertion we just have to show that $g$ does", "not map to zero in", "$\\kappa(\\mathfrak q) = \\kappa(\\mathfrak p)[x]/(\\overline{h}_1)$.", "The image of $g$ in $\\kappa(\\mathfrak p)[x]$ is the derivative", "of $\\overline{f}$. Thus (2) is clear because", "$$", "\\overline{g} =", "\\frac{\\text{d}\\overline{f}}{\\text{d}x} =", "\\overline{w}\\frac{\\text{d}\\overline{h}_1}{\\text{d}x} +", "\\overline{h}_1\\frac{\\text{d}\\overline{w}}{\\text{d}x},", "$$", "$\\overline{w}$ is prime to $\\overline{h}_1$ and", "$\\overline{h}_1$ is separable.", "\\medskip\\noindent", "Step 11.", "We conclude that $\\varphi : R[x]/(f) \\to S$ is a surjective ring map,", "$R[x]_g/(f)$ is \\'etale over $R$ (because it is standard \\'etale,", "see Lemma \\ref{lemma-standard-etale}) and $\\varphi(g) \\not \\in \\mathfrak q$.", "Thus the map $(R[x]/(f))_g \\to S_{\\varphi(g)}$ is the desired", "surjection." ], "refs": [ "algebra-proposition-etale-locally-standard", "algebra-definition-unramified", "algebra-lemma-unramified", "algebra-lemma-unramified-quasi-finite", "algebra-lemma-quasi-finite-open-integral-closure", "algebra-lemma-finite-dimensional-algebra", "algebra-proposition-dimension-zero-ring", "algebra-lemma-unramified-at-prime", "algebra-lemma-unique-prime-over-localize-below", "algebra-lemma-integral-going-up", "algebra-lemma-unramified-at-prime", "algebra-lemma-NAK", "algebra-lemma-isomorphic-local-rings", "algebra-lemma-standard-etale" ], "ref_ids": [ 1427, 1544, 1266, 1269, 1067, 642, 1410, 1268, 556, 500, 1268, 401, 1085, 1241 ] } ], "ref_ids": [] }, { "id": 1429, "type": "theorem", "label": "algebra-proposition-characterize-separable-field-extensions", "categories": [ "algebra" ], "title": "algebra-proposition-characterize-separable-field-extensions", "contents": [ "Let $k \\subset K$ be a field extension.", "If the characteristic of $k$ is zero then", "\\begin{enumerate}", "\\item $K$ is separable over $k$,", "\\item $K$ is geometrically reduced over $k$,", "\\item $K$ is formally smooth over $k$,", "\\item $H_1(L_{K/k}) = 0$, and", "\\item the map $K \\otimes_k \\Omega_{k/\\mathbf{Z}} \\to \\Omega_{K/\\mathbf{Z}}$", "is injective.", "\\end{enumerate}", "If the characteristic of $k$ is $p > 0$, then the following are", "equivalent:", "\\begin{enumerate}", "\\item $K$ is separable over $k$,", "\\item the ring $K \\otimes_k k^{1/p}$ is reduced,", "\\item $K$ is geometrically reduced over $k$,", "\\item the map $K \\otimes_k \\Omega_{k/\\mathbf{F}_p} \\to \\Omega_{K/\\mathbf{F}_p}$", "is injective,", "\\item $H_1(L_{K/k}) = 0$, and", "\\item $K$ is formally smooth over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a combination of", "Lemmas \\ref{lemma-characterize-separable-field-extensions},", "\\ref{lemma-fields-are-formally-smooth}", "\\ref{lemma-formally-smooth-implies-separable}, and", "\\ref{lemma-separable-differentials}." ], "refs": [ "algebra-lemma-characterize-separable-field-extensions", "algebra-lemma-fields-are-formally-smooth", "algebra-lemma-formally-smooth-implies-separable", "algebra-lemma-separable-differentials" ], "ref_ids": [ 569, 1321, 1318, 1317 ] } ], "ref_ids": [] }, { "id": 1430, "type": "theorem", "label": "algebra-proposition-nagata-universally-japanese", "categories": [ "algebra" ], "title": "algebra-proposition-nagata-universally-japanese", "contents": [ "Let $R$ be a ring. The following are equivalent:", "\\begin{enumerate}", "\\item $R$ is a Nagata ring,", "\\item any finite type $R$-algebra is Nagata, and", "\\item $R$ is universally Japanese and Noetherian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that a Noetherian universally Japanese ring is universally", "Nagata (i.e., condition (2) holds). Let $R$ be a Nagata ring.", "We will show that any finitely generated $R$-algebra $S$ is Nagata.", "This will prove the proposition.", "\\medskip\\noindent", "Step 1. There exists a sequence of ring maps", "$R = R_0 \\to R_1 \\to R_2 \\to \\ldots \\to R_n = S$ such that", "each $R_i \\to R_{i + 1}$ is generated by a single element.", "Hence by induction it suffices to prove $S$ is Nagata if", "$S \\cong R[x]/I$.", "\\medskip\\noindent", "Step 2. Let $\\mathfrak q \\subset S$ be a prime of $S$, and let", "$\\mathfrak p \\subset R$ be the corresponding prime of $R$.", "We have to show that $S/\\mathfrak q$ is N-2. Hence we have", "reduced to the proving the following:", "(*) Given a Nagata domain $R$ and a monogenic extension $R \\subset S$", "of domains then $S$ is N-2.", "\\medskip\\noindent", "Step 3. Let $R$ be a Nagata domain and $R \\subset S$ a monogenic", "extension of domains. Let $R \\subset R'$ be the integral closure", "of $R$ in its fraction field. Let $S'$ be the subring of the fraction field of", "$S$ generated by $R'$ and $S$. As $R'$ is finite over $R$", "(by the Nagata property) also $S'$ is finite over $S$.", "Since $S$ is Noetherian it suffices to prove that $S'$", "is N-2 (Lemma \\ref{lemma-finite-extension-N-2}).", "Hence we have reduced to proving the following:", "(**) Given a normal Nagata domain $R$ and a", "monogenic extension $R \\subset S$ of domains then $S$ is N-2.", "\\medskip\\noindent", "Step 4: Let $R$ be a normal Nagata domain and", "let $R \\subset S$ be a monogenic extension of domains.", "Suppose the induced extension of fraction fields of $R$ and $S$", "is purely transcendental. In this case $S = R[x]$. By", "Lemma \\ref{lemma-polynomial-ring-N-2} we see that $S$ is N-2.", "Hence we have reduced to proving the following:", "(**) Given a normal Nagata domain $R$ and a", "monogenic extension $R \\subset S$ of domains", "inducing a finite extension of fraction fields", "then $S$ is N-2.", "\\medskip\\noindent", "Step 5. Let $R$ be a normal Nagata domain and", "let $R \\subset S$ be a monogenic extension of domains", "inducing a finite extension of fraction fields $L/K$.", "Choose an element $x \\in S$", "which generates $S$ as an $R$-algebra. Let $L \\subset M$", "be a finite extension of fields.", "Let $R'$ be the integral closure of $R$ in $M$.", "Then the integral closure $S'$ of $S$ in $M$ is equal to the integral", "closure of $R'[x]$ in $M$.", "Also the fraction field of $R'$ is $M$ and $R \\subset R'$", "is finite (by the Nagata property of $R$).", "This implies that $R'$ is a Nagata ring", "(Lemma \\ref{lemma-quasi-finite-over-nagata}).", "To show that $S'$ is finite over $S$ is the same as showing that", "$S'$ is finite over $R'[x]$. Replace $R$ by $R'$ and $S$ by $R'[x]$", "to reduce to the following statement:", "(***) Given a normal Nagata domain $R$ with fraction field $K$,", "and $x \\in K$, the ring $S \\subset K$ generated by $R$ and $x$", "is N-1.", "\\medskip\\noindent", "Step 6. Let $R$ be a normal Nagata domain with fraction field $K$.", "Let $x = b/a \\in K$. We have to show that the ring $S \\subset K$", "generated by $R$ and $x$ is N-1. Note that $S_a \\cong R_a$ is normal.", "Hence by Lemma \\ref{lemma-characterize-N-1} it suffices to show that", "$S_{\\mathfrak m}$ is N-1 for every maximal ideal $\\mathfrak m$ of $S$.", "\\medskip\\noindent", "With assumptions as in the preceding paragraph, pick such a maximal", "ideal and set $\\mathfrak n = R \\cap \\mathfrak m$. The residue field", "extension $\\kappa(\\mathfrak n) \\subset \\kappa(\\mathfrak m)$ is finite", "(Theorem \\ref{theorem-nullstellensatz}) and generated by the image of $x$.", "Hence there exists a monic polynomial", "$f(X) = X^d + \\sum_{i = 1, \\ldots, d} a_iX^{d -i}$ with", "$f(x) \\in \\mathfrak m$. Let $K \\subset K''$ be a finite extension", "of fields such that $f(X)$ splits completely in $K''[X]$.", "Let $R'$ be the integral closure of $R$ in $K''$.", "Let $S' \\subset K''$ be the subring generated by $R'$ and $x$.", "As $R$ is Nagata we see $R'$ is finite over $R$ and Nagata", "(Lemma \\ref{lemma-quasi-finite-over-nagata}).", "Moreover, $S'$ is finite over $S$. If for every maximal ideal", "$\\mathfrak m'$ of $S'$ the local ring $S'_{\\mathfrak m'}$ is", "N-1, then $S'_{\\mathfrak m}$ is N-1 by", "Lemma \\ref{lemma-characterize-N-1}, which in turn", "implies that $S_{\\mathfrak m}$ is N-1 by", "Lemma \\ref{lemma-finite-extension-N-2}.", "After replacing $R$ by $R'$ and $S$ by $S'$, and $\\mathfrak m$ by", "any of the maximal ideals $\\mathfrak m'$ lying over $\\mathfrak m$", "we reach the situation where the polynomial $f$ above split completely:", "$f(X) = \\prod_{i = 1, \\ldots, d} (X - a_i)$ with $a_i \\in R$.", "Since $f(x) \\in \\mathfrak m$ we see that $x - a_i \\in \\mathfrak m$", "for some $i$. Finally, after replacing $x$ by $x - a_i$ we may assume", "that $x \\in \\mathfrak m$.", "\\medskip\\noindent", "To recapitulate: $R$ is a normal Nagata domain with fraction field $K$,", "$x \\in K$ and $S$ is the subring of $K$ generated by $x$ and $R$,", "finally $\\mathfrak m \\subset S$ is a maximal ideal with $x \\in \\mathfrak m$.", "We have to show $S_{\\mathfrak m}$ is N-1.", "\\medskip\\noindent", "We will show that Lemma \\ref{lemma-criterion-analytically-unramified}", "applies to the local ring", "$S_{\\mathfrak m}$ and the element $x$. This will imply that", "$S_{\\mathfrak m}$ is analytically unramified, whereupon we", "see that it is N-1 by Lemma \\ref{lemma-analytically-unramified-easy}.", "\\medskip\\noindent", "We have to check properties (1), (2), (3)(a) and (3)(b).", "Property (1) is trivial.", "Let $I = \\Ker(R[X] \\to S)$ where $X \\mapsto x$.", "We claim that $I$ is generated by all linear forms $aX - b$ such that", "$ax = b$ in $K$. Clearly all these linear forms are in $I$.", "If $g = a_d X^d + \\ldots a_1 X + a_0 \\in I$, then we see that", "$a_dx$ is integral over $R$ (Lemma \\ref{lemma-make-integral-trivial})", "and hence $b := a_dx \\in R$", "as $R$ is normal. Then $g - (a_dX - b)X^{d - 1} \\in I$ and we win by", "induction on the degree. As a consequence we see that", "$$", "S/xS = R[X]/(X, I) = R/J", "$$", "where", "$$", "J = \\{b \\in R \\mid ax = b \\text{ for some }a \\in R\\} = xR \\cap R", "$$", "By Lemma \\ref{lemma-normal-domain-intersection-localizations-height-1}", "we see that $S/xS = R/J$ has no embedded primes as an $R$-module, hence as", "an $R/J$-module, hence as an $S/xS$-module, hence as an $S$-module.", "This proves property (2).", "Take such an associated prime $\\mathfrak q \\subset S$ with the", "property $\\mathfrak q \\subset \\mathfrak m$ (so that it is an", "associated prime of $S_{\\mathfrak m}/xS_{\\mathfrak m}$ -- it does not", "matter for the arguments).", "Then $\\mathfrak q$ is minimal over $xS$ and hence has height $1$.", "By the sequence of equalities above we see that", "$\\mathfrak p = R \\cap \\mathfrak q$ is an associated", "prime of $R/J$, and so has height $1$", "(see Lemma \\ref{lemma-normal-domain-intersection-localizations-height-1}).", "Thus $R_{\\mathfrak p}$ is a discrete valuation ring and therefore", "$R_{\\mathfrak p} \\subset S_{\\mathfrak q}$ is an equality. This shows", "that $S_{\\mathfrak q}$ is regular. This proves property (3)(a).", "Finally, $(S/\\mathfrak q)_{\\mathfrak m}$ is a localization", "of $S/\\mathfrak q$, which is a quotient of $S/xS = R/J$.", "Hence $(S/\\mathfrak q)_{\\mathfrak m}$ is a localization of", "a quotient of the Nagata ring $R$, hence", "Nagata (Lemmas \\ref{lemma-quasi-finite-over-nagata}", "and \\ref{lemma-nagata-localize})", "and hence analytically unramified", "(Lemma \\ref{lemma-local-nagata-domain-analytically-unramified}).", "This shows (3)(b) holds and we are done." ], "refs": [ "algebra-lemma-finite-extension-N-2", "algebra-lemma-polynomial-ring-N-2", "algebra-lemma-quasi-finite-over-nagata", "algebra-lemma-characterize-N-1", "algebra-theorem-nullstellensatz", "algebra-lemma-quasi-finite-over-nagata", "algebra-lemma-characterize-N-1", "algebra-lemma-finite-extension-N-2", "algebra-lemma-criterion-analytically-unramified", "algebra-lemma-analytically-unramified-easy", "algebra-lemma-make-integral-trivial", "algebra-lemma-normal-domain-intersection-localizations-height-1", "algebra-lemma-normal-domain-intersection-localizations-height-1", "algebra-lemma-quasi-finite-over-nagata", "algebra-lemma-nagata-localize", "algebra-lemma-local-nagata-domain-analytically-unramified" ], "ref_ids": [ 1337, 1342, 1350, 1344, 316, 1350, 1344, 1337, 1356, 1354, 1057, 1313, 1313, 1350, 1351, 1357 ] } ], "ref_ids": [] }, { "id": 1431, "type": "theorem", "label": "algebra-proposition-ubiquity-nagata", "categories": [ "algebra" ], "title": "algebra-proposition-ubiquity-nagata", "contents": [ "The following types of rings are Nagata and in particular universally Japanese:", "\\begin{enumerate}", "\\item fields,", "\\item Noetherian complete local rings,", "\\item $\\mathbf{Z}$,", "\\item Dedekind domains with fraction field of characteristic zero,", "\\item finite type ring extensions of any of the above.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The Noetherian complete local ring case is", "Lemma \\ref{lemma-Noetherian-complete-local-Nagata}.", "In the other cases you just check if $R/\\mathfrak p$ is N-2 for every", "prime ideal $\\mathfrak p$ of the ring. This is clear whenever", "$R/\\mathfrak p$ is a field, i.e., $\\mathfrak p$ is maximal.", "Hence for the Dedekind ring case we only need to check it when", "$\\mathfrak p = (0)$. But since we assume the fraction field has", "characteristic zero Lemma \\ref{lemma-domain-char-zero-N-1-2} kicks in." ], "refs": [ "algebra-lemma-Noetherian-complete-local-Nagata", "algebra-lemma-domain-char-zero-N-1-2" ], "ref_ids": [ 1353, 1340 ] } ], "ref_ids": [] }, { "id": 1584, "type": "theorem", "label": "moduli-curves-theorem-stable-reduction", "categories": [ "moduli-curves" ], "title": "moduli-curves-theorem-stable-reduction", "contents": [ "\\begin{reference}", "\\cite[Corollary 2.7]{DM}", "\\end{reference}", "Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a", "smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$", "and genus $g \\geq 2$. Then", "\\begin{enumerate}", "\\item there exists an extension of discrete valuation rings $R \\subset R'$", "inducing a finite separable extension of fraction fields $K'/K$ and", "a stable family of curves $Y \\to \\Spec(R')$ of genus $g$ with", "$Y_{K'} \\cong C_{K'}$ over $K'$, and", "\\item there exists a finite separable extension $L/K$ and a stable", "family of curves $Y \\to \\Spec(A)$ of genus $g$ where $A \\subset L$", "is the integral closure of $R$ in $L$ such that", "$Y_L \\cong C_L$ over $L$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is an immediate consequence of Lemma \\ref{lemma-stable-reduction} and", "Semistable Reduction, Theorem \\ref{models-theorem-semistable-reduction}.", "\\medskip\\noindent", "Proof of (2). Let $L/K$ be the finite separable extension found in part (3) of", "Semistable Reduction, Theorem \\ref{models-theorem-semistable-reduction}.", "Let $A \\subset L$ be the integral closure of $R$.", "Recall that $A$ is a Dedekind domain finite over $R$ with", "finitely many maximal ideals $\\mathfrak m_1, \\ldots, \\mathfrak m_n$, see", "More on Algebra, Remark \\ref{more-algebra-remark-finite-separable-extension}.", "Set $S = \\Spec(A)$, $S_i = \\Spec(A_{\\mathfrak m_i})$,", "$U = \\Spec(L)$, and $U_i = S_i \\setminus \\{\\mathfrak m_i\\}$.", "Observe that $U \\cong U_i$ for $i = 1, \\ldots, n$.", "Set $X = C_L$ viewed as a scheme over the open subscheme $U$ of $S$.", "By our choice of $L$ and $A$ and Lemma \\ref{lemma-stable-reduction}", "we have a stable families of curves $X_i \\to S_i$ and isomorphisms", "$X \\times_U U_i \\cong X_i \\times_{S_i} U_i$.", "By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-glueing-near-multiple-closed-points}", "we can find a finitely presented morphism $Y \\to S$", "whose base change to $S_i$ is isomorphic to $X_i$ for $i = 1, \\ldots, n$.", "Alternatively, you can use that $S = \\bigcup_{i = 1, \\ldots, n} S_i$", "is an open covering of $S$ and $S_i \\cap S_j = U$ for $i \\not = j$", "and use $n - 1$ applications of", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-relative-glueing}", "to get $Y \\to S$ whose", "base change to $S_i$ is isomorphic to $X_i$ for $i = 1, \\ldots, n$.", "Clearly $Y \\to S$ is the stable family of curves we were looking for." ], "refs": [ "moduli-curves-lemma-stable-reduction", "models-theorem-semistable-reduction", "models-theorem-semistable-reduction", "more-algebra-remark-finite-separable-extension", "moduli-curves-lemma-stable-reduction", "spaces-limits-lemma-glueing-near-multiple-closed-points", "spaces-limits-lemma-relative-glueing" ], "ref_ids": [ 1639, 9190, 9190, 10676, 1639, 4637, 4634 ] } ], "ref_ids": [] }, { "id": 1585, "type": "theorem", "label": "moduli-curves-theorem-stable-smooth-proper", "categories": [ "moduli-curves" ], "title": "moduli-curves-theorem-stable-smooth-proper", "contents": [ "Let $g \\geq 2$. The algebraic stack $\\overline{\\mathcal{M}}_g$ is a", "Deligne-Mumford stack, proper and smooth over $\\Spec(\\mathbf{Z})$.", "Moreover, the locus $\\mathcal{M}_g$ parametrizing smooth curves", "is a dense open substack." ], "refs": [], "proofs": [ { "contents": [ "Most of the properties mentioned in the statement have already been shown.", "Smoothness is Lemma \\ref{lemma-stable-curves-smooth}.", "Deligne-Mumford is Lemma \\ref{lemma-stable-curves-deligne-mumford}.", "Openness of $\\mathcal{M}_g$ is Lemma \\ref{lemma-smooth-dense-in-stable}.", "We know that $\\overline{\\mathcal{M}}_g \\to \\Spec(\\mathbf{Z})$", "is separated by Lemma \\ref{lemma-stable-separated} and we know that", "$\\overline{\\mathcal{M}}_g$ is quasi-compact by", "Lemma \\ref{lemma-stable-quasi-compact}.", "Thus, to show that $\\overline{\\mathcal{M}}_g \\to \\Spec(\\mathbf{Z})$", "is proper and finish the proof, we may apply", "More on Morphisms of Stacks, Lemma", "\\ref{stacks-more-morphisms-lemma-refined-valuative-criterion-proper}", "to the morphisms $\\mathcal{M}_g \\to \\overline{\\mathcal{M}}_g$ and", "$\\overline{\\mathcal{M}}_g \\to \\Spec(\\mathbf{Z})$.", "Thus it suffices to check the following: given any $2$-commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d]_j &", "\\mathcal{M}_g \\ar[r] &", "\\overline{\\mathcal{M}}_g \\ar[d] \\\\", "\\Spec(A) \\ar[rr] & & \\Spec(\\mathbf{Z})", "}", "$$", "where $A$ is a discrete valuation ring with field of fractions $K$, there", "exist an extension $K'/K$ of fields, a valuation ring $A' \\subset K'$", "dominating $A$ such that the category of dotted arrows for the", "induced diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d]_{j'} & \\overline{\\mathcal{M}}_g \\ar[d] \\\\", "\\Spec(A') \\ar[r] \\ar@{..>}[ru] & \\Spec(\\mathbf{Z})", "}", "$$", "is nonempty (Morphisms of Stacks, Definition", "\\ref{stacks-morphisms-definition-fill-in-diagram}).", "(Observe that we don't need to worry about", "$2$-arrows too much, see Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-cat-dotted-arrows-independent}).", "Unwinding what this means using that", "$\\mathcal{M}_g$, resp.\\ $\\overline{\\mathcal{M}}_g$ are the algebraic", "stacks parametrizing smooth, resp.\\ stable families of genus $g$ curves,", "we find that what we have to prove is exactly the result contained", "in the stable reduction theorem, i.e., Theorem", "\\ref{theorem-stable-reduction}." ], "refs": [ "moduli-curves-lemma-stable-curves-smooth", "moduli-curves-lemma-stable-curves-deligne-mumford", "moduli-curves-lemma-smooth-dense-in-stable", "moduli-curves-lemma-stable-separated", "moduli-curves-lemma-stable-quasi-compact", "stacks-more-morphisms-lemma-refined-valuative-criterion-proper", "stacks-morphisms-definition-fill-in-diagram", "stacks-morphisms-lemma-cat-dotted-arrows-independent", "moduli-curves-theorem-stable-reduction" ], "ref_ids": [ 1631, 1632, 1633, 1641, 1642, 6911, 7628, 7572, 1584 ] } ], "ref_ids": [] }, { "id": 1586, "type": "theorem", "label": "moduli-curves-lemma-extend-curves-to-spaces", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-extend-curves-to-spaces", "contents": [ "Let $T \\to B$ be a morphism of algebraic spaces. The category", "$$", "\\Mor_B(T, B\\text{-}\\Curvesstack) = \\Mor(T, \\Curvesstack)", "$$", "is the category of families of curves over $T$." ], "refs": [], "proofs": [ { "contents": [ "A family of curves over $T$ is a morphism $f : X \\to T$ of algebraic", "spaces, which is flat, proper, of finite presentation, and has", "relative dimension $\\leq 1$ (Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-relative-dimension}).", "This is exactly the same as the definition in", "Quot, Situation \\ref{quot-situation-curves}", "except that $T$ the base is allowed to be an algebraic space.", "Our default base category for algebraic stacks/spaces", "is the category of schemes, hence the lemma does not follow", "immediately from the definitions. Having said this, we encourage", "the reader to skip the proof.", "\\medskip\\noindent", "By the product description of $B\\text{-}\\Curvesstack$ given above,", "it suffices to prove the lemma in the absolute case. Choose a scheme", "$U$ and a surjective \\'etale morphism $p : U \\to T$.", "Let $R = U \\times_T U$ with projections $s, t : R \\to U$.", "\\medskip\\noindent", "Let $v : T \\to \\Curvesstack$ be a morphism. Then $v \\circ p$", "corresponds to a family of curves $X_U \\to U$. The canonical", "$2$-morphism $v \\circ p \\circ t \\to v \\circ p \\circ s$", "is an isomorphism $\\varphi : X_U \\times_{U, s} R \\to X_U \\times_{U, t} R$.", "This isomorphism satisfies the cocycle condition on", "$R \\times_{s, t} R$.", "By Bootstrap, Lemma \\ref{bootstrap-lemma-descend-algebraic-space}", "we obtain a morphism of algebraic spaces $X \\to T$", "whose pullback to $U$ is equal to $X_U$ compatible with $\\varphi$.", "Since $\\{U \\to T\\}$ is an \\'etale covering, we see that", "$X \\to T$ is flat, proper, of finite presentation by", "Descent on Spaces, Lemmas", "\\ref{spaces-descent-lemma-descending-property-flat},", "\\ref{spaces-descent-lemma-descending-property-proper}, and", "\\ref{spaces-descent-lemma-descending-property-finite-presentation}.", "Also $X \\to T$ has relative dimension $\\leq 1$ because this is", "an \\'etale local property. Hence $X \\to T$ is a family of curves over $T$.", "\\medskip\\noindent", "Conversely, let $X \\to T$ be a family of curves. Then the", "base change $X_U$ determines a morphism $w : U \\to \\Curvesstack$", "and the canonical isomorphism $X_U \\times_{U, s} R \\to X_U \\times_{U, t} R$", "determines a $2$-arrow $w \\circ s \\to w \\circ t$ satisfying the", "cocycle condition. Thus a morphism $v : T = [U/R] \\to \\Curvesstack$", "by the universal property of the quotient $[U/R]$, see", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-2-coequalizer}.", "(Actually, it is much easier in this case to go back to before", "we introduced our abuse of language and direct construct", "the functor $\\Sch/T \\to \\Curvesstack$ which ``is'' the", "morphsim $T \\to \\Curvesstack$.)", "\\medskip\\noindent", "We omit the verification that the constructions given above", "extend to morphisms between objects and are mutually quasi-inverse." ], "refs": [ "spaces-morphisms-definition-relative-dimension", "bootstrap-lemma-descend-algebraic-space", "spaces-descent-lemma-descending-property-flat", "spaces-descent-lemma-descending-property-proper", "spaces-descent-lemma-descending-property-finite-presentation", "spaces-groupoids-lemma-quotient-stack-2-coequalizer" ], "ref_ids": [ 5010, 2627, 9393, 9399, 9392, 9327 ] } ], "ref_ids": [] }, { "id": 1587, "type": "theorem", "label": "moduli-curves-lemma-polarized-curves-in-polarized", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-polarized-curves-in-polarized", "contents": [ "The morphism", "$\\textit{PolarizedCurves} \\to", "\\Polarizedstack$ is an open and closed immersion." ], "refs": [], "proofs": [ { "contents": [ "This is true because the $1$-morphism", "$\\Curvesstack \\to \\Spacesstack'_{fp, flat, proper}$", "is representable by open and closed immersions, see", "Quot, Lemma \\ref{quot-lemma-curves-open-and-closed-in-spaces}." ], "refs": [ "quot-lemma-curves-open-and-closed-in-spaces" ], "ref_ids": [ 3215 ] } ], "ref_ids": [] }, { "id": 1588, "type": "theorem", "label": "moduli-curves-lemma-polarized-curves-over-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-polarized-curves-over-curves", "contents": [ "The morphism", "$\\textit{PolarizedCurves} \\to \\Curvesstack$", "is smooth and surjective." ], "refs": [], "proofs": [ { "contents": [ "Surjective. Given a field $k$ and a proper algebraic space", "$X$ over $k$ of dimension $\\leq 1$, i.e., an object of $\\Curvesstack$ over $k$.", "By Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-codim-1-point-in-schematic-locus}", "the algebraic space $X$ is a scheme. Hence $X$", "is a proper scheme of dimension $\\leq 1$ over $k$.", "By Varieties, Lemma \\ref{varieties-lemma-dim-1-proper-projective}", "we see that $X$ is H-projective over $\\kappa$.", "In particular, there exists an ample invertible $\\mathcal{O}_X$-module", "$\\mathcal{L}$ on $X$. Then $(X, \\mathcal{L})$ is an object", "of $\\textit{PolarizedCurves}$ over", "$k$ which maps to $X$.", "\\medskip\\noindent", "Smooth. Let $X \\to S$ be an object of $\\Curvesstack$, i.e., a", "morphism $S \\to \\Curvesstack$. It is clear that", "$$", "\\textit{PolarizedCurves}", "\\times_{\\Curvesstack} S", "\\subset \\Picardstack_{X/S}", "$$", "is the substack of objects $(T/S, \\mathcal{L}/X_T)$ such that", "$\\mathcal{L}$ is ample on $X_T/T$. This is an open substack by", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-ample-in-neighbourhood}.", "Since $\\Picardstack_{X/S} \\to S$ is smooth by", "Moduli Stacks, Lemma \\ref{moduli-lemma-pic-curves-smooth}", "we win." ], "refs": [ "spaces-over-fields-lemma-codim-1-point-in-schematic-locus", "varieties-lemma-dim-1-proper-projective", "spaces-descent-lemma-ample-in-neighbourhood", "moduli-lemma-pic-curves-smooth" ], "ref_ids": [ 12845, 11099, 9414, 1739 ] } ], "ref_ids": [] }, { "id": 1589, "type": "theorem", "label": "moduli-curves-lemma-etale-locally-scheme", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-etale-locally-scheme", "contents": [ "Let $X \\to S$ be a family of curves.", "Then there exists an \\'etale covering $\\{S_i \\to S\\}$", "such that $X_i = X \\times_S S_i$ is a scheme. We may even", "assume $X_i$ is H-projective over $S_i$." ], "refs": [], "proofs": [ { "contents": [ "This is an immediate corollary of", "Lemma \\ref{lemma-polarized-curves-over-curves}.", "Namely, unwinding the definitions, this lemma gives there is a", "surjective smooth morphism $S' \\to S$ such that $X' = X \\times_S S'$", "comes endowed with an invertible $\\mathcal{O}_{X'}$-module", "$\\mathcal{L}'$ which is ample on $X'/S'$.", "Then we can refine the smooth covering $\\{S' \\to S\\}$", "by an \\'etale covering $\\{S_i \\to S\\}$, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-etale-dominates-smooth}.", "After replacing $S_i$ by a suitable open covering we may assume", "$X_i \\to S_i$ is H-projective, see", "Morphisms, Lemmas \\ref{morphisms-lemma-proper-ample-locally-projective} and", "\\ref{morphisms-lemma-characterize-locally-projective}", "(this is also discussed in detail in", "More on Morphisms, Section \\ref{more-morphisms-section-projective})." ], "refs": [ "moduli-curves-lemma-polarized-curves-over-curves", "more-morphisms-lemma-etale-dominates-smooth", "morphisms-lemma-proper-ample-locally-projective", "morphisms-lemma-characterize-locally-projective" ], "ref_ids": [ 1588, 13880, 5423, 5421 ] } ], "ref_ids": [] }, { "id": 1590, "type": "theorem", "label": "moduli-curves-lemma-curves-diagonal-separated-fp", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-curves-diagonal-separated-fp", "contents": [ "The diagonal of $\\Curvesstack$ is separated", "and of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\Curvesstack$ is a limit preserving algebraic stack, see", "Quot, Lemma \\ref{quot-lemma-curves-limits}.", "By Limits of Stacks, Lemma \\ref{stacks-limits-lemma-limit-preserving-diagonal}", "this implies that", "$\\Delta : \\Polarizedstack \\to \\Polarizedstack \\times \\Polarizedstack$", "is limit preserving. Hence $\\Delta$ is locally of finite presentation", "by Limits of Stacks, Proposition", "\\ref{stacks-limits-proposition-characterize-locally-finite-presentation}.", "\\medskip\\noindent", "Let us prove that $\\Delta$ is separated. To see this, it suffices to show", "that given a scheme $U$ and two objects $Y \\to U$ and $X \\to U$ of", "$\\Curvesstack$ over $U$, the algebraic space", "$$", "\\mathit{Isom}_U(Y, X)", "$$", "is separated. This we have seen in", "Moduli Stacks, Lemmas \\ref{moduli-lemma-Mor-s-lfp} and", "\\ref{moduli-lemma-Isom-in-Mor} that the target is", "a separated algebraic space.", "\\medskip\\noindent", "To finish the proof we show that $\\Delta$ is quasi-compact. Since", "$\\Delta$ is representable by algebraic spaces, it suffices to check", "the base change of $\\Delta$ by a surjective smooth morphism", "$U \\to \\Curvesstack \\times \\Curvesstack$ is quasi-compact", "(see for example Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}).", "We choose $U = \\coprod U_i$ to be a disjoint union of affine opens", "with a surjective smooth morphism", "$$", "U \\longrightarrow", "\\textit{PolarizedCurves} \\times \\textit{PolarizedCurves}", "$$", "Then $U \\to \\Curvesstack \\times \\Curvesstack$ will be surjective", "and smooth since $\\textit{PolarizedCurves} \\to \\Curvesstack$", "is surjective and smooth by Lemma \\ref{lemma-polarized-curves-over-curves}.", "Since $\\textit{PolarizedCurves}$ is limit preserving", "(by Artin's Axioms, Lemma \\ref{artin-lemma-fibre-product-limit-preserving}", "and Quot, Lemmas \\ref{quot-lemma-curves-limits},", "\\ref{quot-lemma-polarized-limits}, and", "\\ref{quot-lemma-spaces-limits}), we", "see that $\\textit{PolarizedCurves} \\to \\Spec(\\mathbf{Z})$ is locally of", "finite presentation, hence $U_i \\to \\Spec(\\mathbf{Z})$ is", "locally of finite presentation", "(Limits of Stacks, Proposition", "\\ref{stacks-limits-proposition-characterize-locally-finite-presentation}", "and Morphisms of Stacks, Lemmas", "\\ref{stacks-morphisms-lemma-composition-finite-presentation} and", "\\ref{stacks-morphisms-lemma-smooth-locally-finite-presentation}).", "In particular, $U_i$ is Noetherian affine. This reduces us to the", "case discussed in the next paragraph.", "\\medskip\\noindent", "In this paragraph, given a Noetherian affine scheme $U$ and two objects", "$(Y, \\mathcal{N})$ and $(X, \\mathcal{L})$", "of $\\textit{PolarizedCurves}$ over $U$, we show the algebraic space", "$$", "\\mathit{Isom}_U(Y, X)", "$$", "is quasi-compact. Since the connected components of $U$ are open and closed", "we may replace $U$ by these. Thus we may and do assume $U$ is connected.", "Let $u \\in U$ be a point. Let $Q$, $P$ be the Hilbert polynomials", "of these families, i.e.,", "$$", "Q(n) = \\chi(Y_u, \\mathcal{N}_u^{\\otimes n})", "\\quad\\text{and}\\quad", "P(n) = \\chi(X_u, \\mathcal{L}_u^{\\otimes n})", "$$", "see Varieties, Lemma \\ref{varieties-lemma-numerical-polynomial-from-euler}.", "Since $U$ is connected and since", "the functions", "$u \\mapsto \\chi(Y_u, \\mathcal{N}_u^{\\otimes n})$ and", "$u \\mapsto \\chi(X_u, \\mathcal{L}_u^{\\otimes n})$", "are locally constant (see ", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-chi-locally-constant-geometric})", "we see that we get the same Hilbert polynomial in every point of $U$.", "Set", "$$", "\\mathcal{M} = \\text{pr}_1^*\\mathcal{N}", "\\otimes_{\\mathcal{O}_{Y \\times_U X}} \\text{pr}_2^*\\mathcal{L}", "$$", "on $Y \\times_U X$. Given $(f, \\varphi) \\in \\mathit{Isom}_U(Y, X)(T)$", "for some scheme $T$ over $U$ then for every $t \\in T$ we have", "\\begin{align*}", "\\chi(Y_t, (\\text{id} \\times f)^*\\mathcal{M}^{\\otimes n})", "& =", "\\chi(Y_t,", "\\mathcal{N}_t^{\\otimes n} \\otimes_{\\mathcal{O}_{Y_t}}", "f_t^*\\mathcal{L}_t^{\\otimes n}) \\\\", "& =", "n\\deg(\\mathcal{N}_t) + n\\deg(f_t^*\\mathcal{L}_t) +", "\\chi(Y_t, \\mathcal{O}_{Y_t}) \\\\", "& =", "Q(n) + n\\deg(\\mathcal{L}_t) \\\\", "& =", "Q(n) + P(n) - P(0)", "\\end{align*}", "by Riemann-Roch for proper curves, more precisely by", "Varieties, Definition \\ref{varieties-definition-degree-invertible-sheaf} and", "Lemma \\ref{varieties-lemma-degree-tensor-product}", "and the fact that $f_t$ is an isomorphism.", "Setting $P'(t) = Q(t) + P(t) - P(0)$ we find", "$$", "\\mathit{Isom}_U(Y, X) =", "\\mathit{Isom}_U(Y, X) \\cap \\mathit{Mor}^{P', \\mathcal{M}}_U(Y, X)", "$$", "The intersection is an intersection of open subspaces of", "$\\mathit{Mor}_U(Y, X)$, see", "Moduli Stacks, Lemma \\ref{moduli-lemma-Isom-in-Mor} and", "Remark \\ref{moduli-remark-Mor-numerical}.", "Now $\\mathit{Mor}^{P', \\mathcal{M}}_U(Y, X)$", "is a Noetherian algebraic space as it is of finite", "presentation over $U$ by", "Moduli Stacks, Lemma \\ref{moduli-lemma-Mor-qc-over-base}.", "Thus the intersection is a Noetherian algebraic space too", "and the proof is finished." ], "refs": [ "quot-lemma-curves-limits", "stacks-limits-lemma-limit-preserving-diagonal", "stacks-limits-proposition-characterize-locally-finite-presentation", "moduli-lemma-Mor-s-lfp", "moduli-lemma-Isom-in-Mor", "stacks-properties-lemma-check-property-covering", "moduli-curves-lemma-polarized-curves-over-curves", "artin-lemma-fibre-product-limit-preserving", "quot-lemma-curves-limits", "quot-lemma-polarized-limits", "quot-lemma-spaces-limits", "stacks-limits-proposition-characterize-locally-finite-presentation", "stacks-morphisms-lemma-composition-finite-presentation", "stacks-morphisms-lemma-smooth-locally-finite-presentation", "varieties-lemma-numerical-polynomial-from-euler", "perfect-lemma-chi-locally-constant-geometric", "varieties-definition-degree-invertible-sheaf", "varieties-lemma-degree-tensor-product", "moduli-lemma-Isom-in-Mor", "moduli-remark-Mor-numerical", "moduli-lemma-Mor-qc-over-base" ], "ref_ids": [ 3210, 15018, 15024, 1746, 1747, 8859, 1588, 11365, 3210, 3201, 3192, 15024, 7500, 7542, 11121, 7063, 11161, 11109, 1747, 1756, 1748 ] } ], "ref_ids": [] }, { "id": 1591, "type": "theorem", "label": "moduli-curves-lemma-curves-qs-lfp", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-curves-qs-lfp", "contents": [ "The morphism $\\Curvesstack \\to \\Spec(\\mathbf{Z})$ is quasi-separated and", "locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "To check $\\Curvesstack \\to \\Spec(\\mathbf{Z})$ is quasi-separated we have to", "show that its diagonal is quasi-compact and quasi-separated.", "This is immediate from Lemma \\ref{lemma-curves-diagonal-separated-fp}.", "To prove that $\\Curvesstack \\to \\Spec(\\mathbf{Z})$ is locally of finite", "presentation, it suffices to show that $\\Curvesstack$", "is limit preserving, see Limits of Stacks, Proposition", "\\ref{stacks-limits-proposition-characterize-locally-finite-presentation}.", "This is Quot, Lemma \\ref{quot-lemma-curves-limits}." ], "refs": [ "moduli-curves-lemma-curves-diagonal-separated-fp", "stacks-limits-proposition-characterize-locally-finite-presentation", "quot-lemma-curves-limits" ], "ref_ids": [ 1590, 15024, 3210 ] } ], "ref_ids": [] }, { "id": 1592, "type": "theorem", "label": "moduli-curves-lemma-DM-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-DM-curves", "contents": [ "There exist an open substack $\\Curvesstack^{DM} \\subset \\Curvesstack$", "with the following properties", "\\begin{enumerate}", "\\item $\\Curvesstack^{DM} \\subset \\Curvesstack$ is the maximal", "open substack which is DM,", "\\item given a family of curves $X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors through", "$\\Curvesstack^{DM}$,", "\\item the group algebraic space $\\mathit{Aut}_S(X)$ is unramified over $S$,", "\\end{enumerate}", "\\item given $X$ a proper scheme over a field $k$ of dimension $\\leq 1$", "the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through $\\Curvesstack^{DM}$,", "\\item $\\mathit{Aut}(X)$ is geometrically reduced over $k$ and", "has dimension $0$,", "\\item $\\mathit{Aut}(X) \\to \\Spec(k)$ is unramified.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The existence of an open substack with property (1) is", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-open-DM-locus}.", "The points of this open substack are characterized by (3)(c) by", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-points-DM-locus}.", "The equivalence of (3)(b) and (3)(c) is the statement that an", "algebraic space $G$ which is locally of finite type, geometrically reduced,", "and of dimension $0$ over a field $k$, is unramified over $k$.", "First, $G$ is a scheme by Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-locally-finite-type-dim-zero}.", "Then we can take an affine open in $G$ and observe", "that it will be proper over $k$ and apply", "Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}.", "Minor details omitted.", "\\medskip\\noindent", "Part (2) is true because (3) holds. Namely, the morphism", "$\\mathit{Aut}_S(X) \\to S$ is locally of finite type. Thus we can check whether", "$\\mathit{Aut}_S(X) \\to S$ is unramified at all points of", "$\\mathit{Aut}_S(X)$ by checking on fibres at points of the scheme $S$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-where-unramified}.", "But after base change to a point of $S$ we fall back into", "the equivalence of (3)(a) and (3)(c)." ], "refs": [ "stacks-morphisms-lemma-open-DM-locus", "stacks-morphisms-lemma-points-DM-locus", "spaces-over-fields-lemma-locally-finite-type-dim-zero", "varieties-lemma-proper-geometrically-reduced-global-sections", "spaces-morphisms-lemma-where-unramified" ], "ref_ids": [ 7482, 7483, 12843, 10948, 4903 ] } ], "ref_ids": [] }, { "id": 1593, "type": "theorem", "label": "moduli-curves-lemma-in-DM-locus-vector-fields", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-in-DM-locus-vector-fields", "contents": [ "Let $X$ be a proper scheme over a field $k$ of dimension $\\leq 1$.", "Then properties (3)(a), (b), (c) are also equivalent to", "$\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = 0$." ], "refs": [], "proofs": [ { "contents": [ "In the discussion above we have seen that $G = \\mathit{Aut}(X)$", "is a group scheme over $\\Spec(k)$ which is finite type and separated;", "this uses Lemma \\ref{lemma-curves-diagonal-separated-fp} and", "More on Groupoids in Spaces, Lemma", "\\ref{spaces-more-groupoids-lemma-group-space-scheme-locally-finite-type-over-k}.", "Then $G$ is unramified over $k$ if and only if $\\Omega_{G/k} = 0$", "(Morphisms, Lemma \\ref{morphisms-lemma-unramified-omega-zero}).", "By Groupoids, Lemma \\ref{groupoids-lemma-group-scheme-module-differentials}", "the vanishing holds if $T_{G/k, e} = 0$, where $T_{G/k, e}$ is the tangent", "space to $G$ at the identity element $e \\in G(k)$, see", "Varieties, Definition \\ref{varieties-definition-tangent-space}", "and the formula in", "Varieties, Lemma \\ref{varieties-lemma-tangent-space-cotangent-space}.", "Since $\\kappa(e) = k$ the tangent space is defined in terms of", "morphisms $\\alpha : \\Spec(k[\\epsilon]) \\to G = \\mathit{Aut}(X)$", "whose restriction to $\\Spec(k)$ is $e$.", "It follows that it suffices to show any automorphism", "$$", "\\alpha :", "X \\times_{\\Spec(k)} \\Spec(k[\\epsilon])", "\\longrightarrow", "X \\times_{\\Spec(k)} \\Spec(k[\\epsilon])", "$$", "over $\\Spec(k[\\epsilon])$ whose restriction to $\\Spec(k)$ is", "$\\text{id}_X$. Such automorphisms are", "called infinitesimal automorphisms.", "\\medskip\\noindent", "The infinitesimal automorphisms of $X$ correspond $1$-to-$1$", "with derivations of $\\mathcal{O}_X$ over $k$. This follows from", "More on Morphisms, Lemmas \\ref{more-morphisms-lemma-difference-derivation} and", "\\ref{more-morphisms-lemma-action-by-derivations} (we only need the first one", "as we don't care about the reverse direction; also, please look at", "More on Morphisms, Remark \\ref{more-morphisms-remark-another-special-case}", "for an elucidation). For a different argument proving this equality", "we refer the reader to", "Deformation Problems, Lemma \\ref{examples-defos-lemma-schemes-TI}." ], "refs": [ "moduli-curves-lemma-curves-diagonal-separated-fp", "spaces-more-groupoids-lemma-group-space-scheme-locally-finite-type-over-k", "morphisms-lemma-unramified-omega-zero", "groupoids-lemma-group-scheme-module-differentials", "varieties-definition-tangent-space", "varieties-lemma-tangent-space-cotangent-space", "more-morphisms-lemma-difference-derivation", "more-morphisms-lemma-action-by-derivations", "more-morphisms-remark-another-special-case", "examples-defos-lemma-schemes-TI" ], "ref_ids": [ 1590, 13190, 5343, 9585, 11149, 10972, 13716, 13717, 14129, 8746 ] } ], "ref_ids": [] }, { "id": 1594, "type": "theorem", "label": "moduli-curves-lemma-CM-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-CM-curves", "contents": [ "There exist an open substack $\\Curvesstack^{CM} \\subset \\Curvesstack$", "such that", "\\begin{enumerate}", "\\item given a family of curves $X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{CM}$,", "\\item the morphism $X \\to S$ is Cohen-Macaulay,", "\\end{enumerate}", "\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$", "the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through $\\Curvesstack^{CM}$,", "\\item $X$ is Cohen-Macaulay.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be a family of curves. By", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-flat-finite-presentation-CM-open}", "the set", "$$", "W = \\{x \\in |X| : f \\text{ is Cohen-Macaulay at }x\\}", "$$", "is open in $|X|$ and formation of this open commutes with arbitrary", "base change. Since $f$ is proper the subset", "$$", "S' = S \\setminus f(|X| \\setminus W)", "$$", "of $S$ is open and $X \\times_S S' \\to S'$ is Cohen-Macaulay.", "Moreover, formation of $S'$ commutes with arbitrary base", "change because this is true for $W$", "Thus we get the open substack with the desired properties", "by the method discussed in Section \\ref{section-open}." ], "refs": [ "spaces-more-morphisms-lemma-flat-finite-presentation-CM-open" ], "ref_ids": [ 145 ] } ], "ref_ids": [] }, { "id": 1595, "type": "theorem", "label": "moduli-curves-lemma-CM-1-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-CM-1-curves", "contents": [ "There exist an open substack $\\Curvesstack^{CM, 1} \\subset \\Curvesstack$", "such that", "\\begin{enumerate}", "\\item given a family of curves $X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{CM, 1}$,", "\\item the morphism $X \\to S$ is Cohen-Macaulay and has", "relative dimension $1$ (Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-relative-dimension}),", "\\end{enumerate}", "\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$", "the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through $\\Curvesstack^{CM, 1}$,", "\\item $X$ is Cohen-Macaulay and $X$ is equidimensional of", "dimension $1$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-relative-dimension" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-CM-curves} it is clear that we have", "$\\Curvesstack^{CM, 1} \\subset \\Curvesstack^{CM}$", "if it exists. Let $f : X \\to S$ be a family of curves", "such that $f$ is a Cohen-Macaulay morphism. By", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-lfp-CM-relative-dimension}", "we have a decomposition", "$$", "X = X_0 \\amalg X_1", "$$", "by open and closed subspaces such that $X_0 \\to S$ has relative", "dimension $0$ and $X_1 \\to S$ has relative dimension $1$.", "Since $f$ is proper the subset", "$$", "S' = S \\setminus f(|X_0|)", "$$", "of $S$ is open and $X \\times_S S' \\to S'$ is Cohen-Macaulay", "and has relative dimension $1$.", "Moreover, formation of $S'$ commutes with arbitrary base", "change because this is true for the decomposition above", "(as relative dimension behaves well with respect to base", "change, see Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}).", "Thus we get the open substack with the desired properties", "by the method discussed in Section \\ref{section-open}." ], "refs": [ "moduli-curves-lemma-CM-curves", "spaces-more-morphisms-lemma-lfp-CM-relative-dimension", "spaces-morphisms-lemma-dimension-fibre-after-base-change" ], "ref_ids": [ 1594, 146, 4872 ] } ], "ref_ids": [ 5010 ] }, { "id": 1596, "type": "theorem", "label": "moduli-curves-lemma-pre-genus-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-pre-genus-curves", "contents": [ "There exist an open substack $\\Curvesstack^{h0, 1} \\subset \\Curvesstack$", "such that", "\\begin{enumerate}", "\\item given a family of curves $f : X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{h0, 1}$,", "\\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$, this holds", "after arbitrary base change, and the fibres of $f$ have dimension $1$,", "\\end{enumerate}", "\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$", "the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through $\\Curvesstack^{h0, 1}$,", "\\item $H^0(X, \\mathcal{O}_X) = k$ and $\\dim(X) = 1$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Given a family of curves $X \\to S$ the set of $s \\in S$ where", "$\\kappa(s) = H^0(X_s, \\mathcal{O}_{X_s})$", "is open in $S$ by Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-jump-loci-geometric}.", "Also, the set of points in $S$ where the fibre has", "dimension $1$ is open by More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-dimension-fibres-proper-flat}.", "Moreover, if $f : X \\to S$ is a family of curves all of whose fibres", "have dimension $1$ (and in particular $f$ is surjective), then", "condition (1)(b) is equivalent to", "$\\kappa(s) = H^0(X_s, \\mathcal{O}_{X_s})$ for every $s \\in S$, see", "Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-proper-flat-h0}.", "Thus we see that the lemma follows from the general discussion in", "Section \\ref{section-open}." ], "refs": [ "spaces-perfect-lemma-jump-loci-geometric", "spaces-more-morphisms-lemma-dimension-fibres-proper-flat", "spaces-perfect-lemma-proper-flat-h0" ], "ref_ids": [ 2744, 164, 2749 ] } ], "ref_ids": [] }, { "id": 1597, "type": "theorem", "label": "moduli-curves-lemma-pre-genus-in-CM-1", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-pre-genus-in-CM-1", "contents": [ "We have $\\Curvesstack^{h0, 1} \\subset \\Curvesstack^{CM, 1}$", "as open substacks of $\\Curvesstack$." ], "refs": [], "proofs": [ { "contents": [ "See Algebraic Curves, Lemma \\ref{curves-lemma-automatic} and", "Lemmas \\ref{lemma-pre-genus-curves} and \\ref{lemma-CM-1-curves}." ], "refs": [ "curves-lemma-automatic", "moduli-curves-lemma-pre-genus-curves", "moduli-curves-lemma-CM-1-curves" ], "ref_ids": [ 6257, 1596, 1595 ] } ], "ref_ids": [] }, { "id": 1598, "type": "theorem", "label": "moduli-curves-lemma-genus", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-genus", "contents": [ "Let $f : X \\to S$ be a family of curves such that", "$\\kappa(s) = H^0(X_s, \\mathcal{O}_{X_s})$ for all $s \\in S$, i.e.,", "the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{h0, 1}$ (Lemma \\ref{lemma-pre-genus-curves}). Then", "\\begin{enumerate}", "\\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this holds universally,", "\\item $R^1f_*\\mathcal{O}_X$ is a finite locally free $\\mathcal{O}_S$-module,", "\\item for any morphism $h : S' \\to S$ if $f' : X' \\to S'$ is the base change,", "then $h^*(R^1f_*\\mathcal{O}_X) = R^1f'_*\\mathcal{O}_{X'}$.", "\\end{enumerate}" ], "refs": [ "moduli-curves-lemma-pre-genus-curves" ], "proofs": [ { "contents": [ "We apply Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-proper-flat-h0}.", "This proves part (1). It also implies that locally on $S$", "we can write $Rf_*\\mathcal{O}_X = \\mathcal{O}_S \\oplus P$", "where $P$ is perfect of tor amplitude in $[1, \\infty)$.", "Recall that formation of $Rf_*\\mathcal{O}_X$ commutes", "with arbitrary base change", "(Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-flat-proper-perfect-direct-image-general}).", "Thus for $s \\in S$ we have", "$$", "H^i(P \\otimes_{\\mathcal{O}_S}^\\mathbf{L} \\kappa(s)) =", "H^i(X_s, \\mathcal{O}_{X_s})", "\\text{ for }i \\geq 1", "$$", "This is zero unless $i = 1$ since $X_s$ is a $1$-dimensional", "Noetherian scheme, see", "Cohomology, Proposition \\ref{cohomology-proposition-vanishing-Noetherian}.", "Then $P = H^1(P)[-1]$ and $H^1(P)$ is finite locally free", "for example by More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-perfect-from-residue-field}.", "Since everything is compatible with base change we", "also see that (3) holds." ], "refs": [ "spaces-perfect-lemma-proper-flat-h0", "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "cohomology-proposition-vanishing-Noetherian", "more-algebra-lemma-lift-perfect-from-residue-field" ], "ref_ids": [ 2749, 2738, 2246, 10232 ] } ], "ref_ids": [ 1596 ] }, { "id": 1599, "type": "theorem", "label": "moduli-curves-lemma-pre-genus-one-piece-per-genus", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-pre-genus-one-piece-per-genus", "contents": [ "There is a decomposition into open and closed substacks", "$$", "\\Curvesstack^{h0, 1} = \\coprod\\nolimits_{g \\geq 0} \\Curvesstack_g", "$$", "where each $\\Curvesstack_g$ is characterized as follows:", "\\begin{enumerate}", "\\item given a family of curves $f : X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack_g$,", "\\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$, this holds after", "arbitrary base change, the fibres of $f$ have dimension $1$, and", "$R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module of rank $g$,", "\\end{enumerate}", "\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$", "the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through $\\Curvesstack_g$,", "\\item $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$, and", "the genus of $X$ is $g$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We already have the existence of $\\Curvesstack^{h0, 1}$ as an open", "substack of $\\Curvesstack$ characterized by the conditions of the", "lemma not involving $R^1f_*$ or $H^1$, see Lemma \\ref{lemma-pre-genus-curves}.", "The existence of the decomposition into open and closed substacks", "follows immediately from the discussion in Section \\ref{section-open}", "and Lemma \\ref{lemma-genus}. This proves the characterization in (1).", "The characterization in (2) follows from the definition of the", "genus in Algebraic Curves, Definition \\ref{curves-definition-genus}." ], "refs": [ "moduli-curves-lemma-pre-genus-curves", "moduli-curves-lemma-genus", "curves-definition-genus" ], "ref_ids": [ 1596, 1598, 6355 ] } ], "ref_ids": [] }, { "id": 1600, "type": "theorem", "label": "moduli-curves-lemma-geometrically-reduced-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-geometrically-reduced-curves", "contents": [ "There exist an open substack $\\Curvesstack^{geomred} \\subset \\Curvesstack$", "such that", "\\begin{enumerate}", "\\item given a family of curves $X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{geomred}$,", "\\item the fibres of the morphism $X \\to S$ are geometrically reduced", "(More on Morphisms of Spaces, Definition", "\\ref{spaces-more-morphisms-definition-geometrically-reduced-fibre}),", "\\end{enumerate}", "\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$", "the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through $\\Curvesstack^{geomred}$,", "\\item $X$ is geometrically reduced over $k$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [ "spaces-more-morphisms-definition-geometrically-reduced-fibre" ], "proofs": [ { "contents": [ "Let $f : X \\to S$ be a family of curves. By", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-geometrically-reduced-open}", "the set", "$$", "E = \\{s \\in S : \\text{the fibre of }X \\to S\\text{ at }s", "\\text{ is geometrically reduced}\\}", "$$", "is open in $S$. Formation of this open commutes with arbitrary", "base change by", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-base-change-fibres-geometrically-reduced}.", "Thus we get the open substack with the desired properties", "by the method discussed in Section \\ref{section-open}." ], "refs": [ "spaces-more-morphisms-lemma-geometrically-reduced-open", "spaces-more-morphisms-lemma-base-change-fibres-geometrically-reduced" ], "ref_ids": [ 158, 155 ] } ], "ref_ids": [ 294 ] }, { "id": 1601, "type": "theorem", "label": "moduli-curves-lemma-geomred-in-CM", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-geomred-in-CM", "contents": [ "We have $\\Curvesstack^{geomred} \\subset \\Curvesstack^{CM}$", "as open substacks of $\\Curvesstack$." ], "refs": [], "proofs": [ { "contents": [ "This is true because a reduced Noetherian scheme of", "dimension $\\leq 1$ is Cohen-Macaulay. See", "Algebra, Lemma \\ref{algebra-lemma-criterion-reduced}." ], "refs": [ "algebra-lemma-criterion-reduced" ], "ref_ids": [ 1310 ] } ], "ref_ids": [] }, { "id": 1602, "type": "theorem", "label": "moduli-curves-lemma-geometrically-reduced-connected-1-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-geometrically-reduced-connected-1-curves", "contents": [ "There exist an open substack $\\Curvesstack^{grc, 1} \\subset \\Curvesstack$", "such that", "\\begin{enumerate}", "\\item given a family of curves $X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{grc, 1}$,", "\\item the geometric fibres of the morphism $X \\to S$ are", "reduced, connected, and have dimension $1$,", "\\end{enumerate}", "\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$", "the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through $\\Curvesstack^{grc, 1}$,", "\\item $X$ is geometrically reduced, geometrically connected,", "and has dimension $1$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-geometrically-reduced-curves},", "\\ref{lemma-geomred-in-CM}, \\ref{lemma-CM-curves}, and \\ref{lemma-CM-1-curves}", "it is clear that we have", "$$", "\\Curvesstack^{grc, 1}", "\\subset", "\\Curvesstack^{geomred} \\cap \\Curvesstack^{CM, 1}", "$$", "if it exists. Let $f : X \\to S$ be a family of curves such that $f$ is", "Cohen-Macaulay, has geometrically reduced fibres, and", "has relative dimension $1$. By", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-stein-factorization-etale}", "in the Stein factorization", "$$", "X \\to T \\to S", "$$", "the morphism $T \\to S$ is \\'etale. This implies that", "there is an open and closed subscheme $S' \\subset S$", "such that $X \\times_S S' \\to S'$ has geometrically", "connected fibres (in the decomposition of", "Morphisms, Lemma \\ref{morphisms-lemma-finite-locally-free}", "for the finite locally free morphism $T \\to S$", "this corresponds to $S_1$).", "Formation of this open commutes with arbitrary base change", "because the number of connected components of geometric", "fibres is invariant under base change (it is also true", "that the Stein factorization commutes with base change", "in our particular case but we don't need this to conclude).", "Thus we get the open substack with the desired properties", "by the method discussed in Section \\ref{section-open}." ], "refs": [ "moduli-curves-lemma-geometrically-reduced-curves", "moduli-curves-lemma-geomred-in-CM", "moduli-curves-lemma-CM-curves", "moduli-curves-lemma-CM-1-curves", "spaces-more-morphisms-lemma-stein-factorization-etale", "morphisms-lemma-finite-locally-free" ], "ref_ids": [ 1600, 1601, 1594, 1595, 182, 5474 ] } ], "ref_ids": [] }, { "id": 1603, "type": "theorem", "label": "moduli-curves-lemma-geomredcon-in-h0-1", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-geomredcon-in-h0-1", "contents": [ "We have $\\Curvesstack^{grc, 1} \\subset \\Curvesstack^{h0, 1}$", "as open substacks of $\\Curvesstack$. In particular, given", "a family of curves $f : X \\to S$", "whose geometric fibres are reduced, connected and of dimension $1$, then", "$R^1f_*\\mathcal{O}_X$ is a finite locally free $\\mathcal{O}_S$-module", "whose formation commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "This follows from Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}", "and Lemmas \\ref{lemma-pre-genus-curves} and", "\\ref{lemma-geometrically-reduced-connected-1-curves}.", "The final statement follows from Lemma \\ref{lemma-genus}." ], "refs": [ "varieties-lemma-proper-geometrically-reduced-global-sections", "moduli-curves-lemma-pre-genus-curves", "moduli-curves-lemma-geometrically-reduced-connected-1-curves", "moduli-curves-lemma-genus" ], "ref_ids": [ 10948, 1596, 1602, 1598 ] } ], "ref_ids": [] }, { "id": 1604, "type": "theorem", "label": "moduli-curves-lemma-one-piece-per-genus", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-one-piece-per-genus", "contents": [ "There is a decomposition into open and closed substacks", "$$", "\\Curvesstack^{grc, 1} = \\coprod\\nolimits_{g \\geq 0} \\Curvesstack^{grc, 1}_g", "$$", "where each $\\Curvesstack^{grc, 1}_g$ is characterized as follows:", "\\begin{enumerate}", "\\item given a family of curves $f : X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{grc, 1}_g$,", "\\item the geometric fibres of the morphism $f : X \\to S$ are", "reduced, connected, of dimension $1$ and", "$R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module", "of rank $g$,", "\\end{enumerate}", "\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$", "the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through $\\Curvesstack^{grc, 1}_g$,", "\\item $X$ is geometrically reduced, geometrically connected,", "has dimension $1$, and has genus $g$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First proof: set", "$\\Curvesstack^{grc, 1}_g = \\Curvesstack^{grc, 1} \\cap \\Curvesstack_g$", "and combine Lemmas \\ref{lemma-geomredcon-in-h0-1} and", "\\ref{lemma-pre-genus-one-piece-per-genus}.", "Second proof:", "The existence of the decomposition into open and closed substacks", "follows immediately from the discussion in Section \\ref{section-open}", "and Lemma \\ref{lemma-geomredcon-in-h0-1}.", "This proves the characterization in (1).", "The characterization in (2) follows as well since", "the genus of a geometrically reduced and connected", "proper $1$-dimensional scheme $X/k$ is defined", "(Algebraic Curves, Definition \\ref{curves-definition-genus} and", "Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections})", "and is equal to $\\dim_k H^1(X, \\mathcal{O}_X)$." ], "refs": [ "moduli-curves-lemma-geomredcon-in-h0-1", "moduli-curves-lemma-pre-genus-one-piece-per-genus", "moduli-curves-lemma-geomredcon-in-h0-1", "curves-definition-genus", "varieties-lemma-proper-geometrically-reduced-global-sections" ], "ref_ids": [ 1603, 1599, 1603, 6355, 10948 ] } ], "ref_ids": [] }, { "id": 1605, "type": "theorem", "label": "moduli-curves-lemma-gorenstein-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-gorenstein-curves", "contents": [ "There exist an open substack $\\Curvesstack^{Gorenstein} \\subset \\Curvesstack$", "such that", "\\begin{enumerate}", "\\item given a family of curves $X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{Gorenstein}$,", "\\item the morphism $X \\to S$ is Gorenstein,", "\\end{enumerate}", "\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$", "the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through $\\Curvesstack^{Gorenstein}$,", "\\item $X$ is Gorenstein.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be a family of curves. By", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-flat-finite-presentation-gorenstein-open}", "the set", "$$", "W = \\{x \\in |X| : f \\text{ is Gorenstein at }x\\}", "$$", "is open in $|X|$ and formation of this open commutes with arbitrary", "base change. Since $f$ is proper the subset", "$$", "S' = S \\setminus f(|X| \\setminus W)", "$$", "of $S$ is open and $X \\times_S S' \\to S'$ is Gorenstein.", "Moreover, formation of $S'$ commutes with arbitrary base", "change because this is true for $W$", "Thus we get the open substack with the desired properties", "by the method discussed in Section \\ref{section-open}." ], "refs": [ "spaces-more-morphisms-lemma-flat-finite-presentation-gorenstein-open" ], "ref_ids": [ 152 ] } ], "ref_ids": [] }, { "id": 1606, "type": "theorem", "label": "moduli-curves-lemma-gorenstein-1-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-gorenstein-1-curves", "contents": [ "There exist an open substack", "$\\Curvesstack^{Gorenstein, 1} \\subset \\Curvesstack$ such that", "\\begin{enumerate}", "\\item given a family of curves $X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{Gorenstein, 1}$,", "\\item the morphism $X \\to S$ is Gorenstein and has", "relative dimension $1$ (Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-relative-dimension}),", "\\end{enumerate}", "\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$", "the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through $\\Curvesstack^{Gorenstein, 1}$,", "\\item $X$ is Gorenstein and $X$ is equidimensional of", "dimension $1$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-relative-dimension" ], "proofs": [ { "contents": [ "Recall that a Gorenstein scheme is Cohen-Macaulay", "(Duality for Schemes, Lemma \\ref{duality-lemma-gorenstein-CM})", "and that", "a Gorenstein morphism is a Cohen-Macaulay morphism", "(Duality for Schemes, Lemma \\ref{duality-lemma-gorenstein-CM-morphism}.", "Thus we can set", "$\\Curvesstack^{Gorenstein, 1}$ equal to the intersection", "of $\\Curvesstack^{Gorenstein}$ and $\\Curvesstack^{CM, 1}$", "inside of $\\Curvesstack$ and use", "Lemmas \\ref{lemma-gorenstein-curves} and \\ref{lemma-CM-1-curves}." ], "refs": [ "duality-lemma-gorenstein-CM", "duality-lemma-gorenstein-CM-morphism", "moduli-curves-lemma-gorenstein-curves", "moduli-curves-lemma-CM-1-curves" ], "ref_ids": [ 13589, 13596, 1605, 1595 ] } ], "ref_ids": [ 5010 ] }, { "id": 1607, "type": "theorem", "label": "moduli-curves-lemma-lci-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-lci-curves", "contents": [ "There exist an open substack $\\Curvesstack^{lci} \\subset \\Curvesstack$", "such that", "\\begin{enumerate}", "\\item given a family of curves $X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors through", "$\\Curvesstack^{lci}$,", "\\item $X \\to S$ is a local complete intersection morphism, and", "\\item $X \\to S$ is a syntomic morphism.", "\\end{enumerate}", "\\item given $X$ a proper scheme over a field $k$ of dimension $\\leq 1$", "the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through $\\Curvesstack^{lci}$,", "\\item $X$ is a local complete intersection over $k$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that being a syntomic morphism is the same as being flat and", "a local complete intersection morphism, see", "More on Morphisms of Spaces, Lemma \\ref{spaces-more-morphisms-lemma-flat-lci}.", "Thus (1)(b) is equivalent to (1)(c).", "In Section \\ref{section-open} we have seen", "it suffices to show that given a family of curves", "$f : X \\to S$, there is an open subscheme $S' \\subset S$", "such that $S' \\times_S X \\to S'$ is a local complete intersection", "morphism and such that formation of $S'$ commutes with arbitrary base change.", "This follows from the more general", "More on Morphisms of Spaces, Lemma \\ref{spaces-more-morphisms-lemma-where-lci}." ], "refs": [ "spaces-more-morphisms-lemma-flat-lci", "spaces-more-morphisms-lemma-where-lci" ], "ref_ids": [ 239, 253 ] } ], "ref_ids": [] }, { "id": 1608, "type": "theorem", "label": "moduli-curves-lemma-isolated-sings-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-isolated-sings-curves", "contents": [ "There exist an open substack", "$\\Curvesstack^{+} \\subset \\Curvesstack$", "such that", "\\begin{enumerate}", "\\item given a family of curves $X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors through", "$\\Curvesstack^{+}$,", "\\item the singular locus of $X \\to S$ endowed", "with any/some closed subspace structure is finite over $S$.", "\\end{enumerate}", "\\item given $X$ a proper scheme over a field $k$ of dimension $\\leq 1$", "the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through $\\Curvesstack^{+}$,", "\\item $X \\to \\Spec(k)$ is smooth except at finitely many points.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove the lemma it suffices to show that given a family of curves", "$f : X \\to S$, there is an open subscheme $S' \\subset S$", "such that the fibre of $S' \\times_S X \\to S'$ have property (2).", "(Formation of the open will automatically commute with base change.)", "By definition the locus $T \\subset |X|$ of points where $X \\to S$", "is not smooth is closed. Let $Z \\subset X$ be the closed subspace", "given by the reduced induced algebraic space structure on $T$", "(Properties of Spaces, Definition", "\\ref{spaces-properties-definition-reduced-induced-space}).", "Now if $s \\in S$ is a point where $Z_s$ is finite, then there", "is an open neighbourhood $U_s \\subset S$ of $s$ such that", "$Z \\cap f^{-1}(U_s) \\to U_s$ is finite, see", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood}.", "This proves the lemma." ], "refs": [ "spaces-properties-definition-reduced-induced-space", "spaces-more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood" ], "ref_ids": [ 11932, 174 ] } ], "ref_ids": [] }, { "id": 1609, "type": "theorem", "label": "moduli-curves-lemma-in-smooth-locus", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-in-smooth-locus", "contents": [ "In the situation above the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through the open where $\\Curvesstack \\to \\Spec(\\mathbf{Z})$ is smooth,", "\\item the deformation category $\\Deformationcategory_X$ is unobstructed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $\\Curvesstack \\longrightarrow \\Spec(\\mathbf{Z})$ is locally", "of finite presentation (Lemma \\ref{lemma-curves-qs-lfp})", "formation of the open substack where", "$\\Curvesstack \\longrightarrow \\Spec(\\mathbf{Z})$ is smooth commutes with", "flat base change", "(Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-where-smooth}).", "Since the Cohen ring $\\Lambda$ is flat over $\\mathbf{Z}$,", "we may work over $\\Lambda$. In other words, we are trying to prove that", "$$", "\\Lambda\\text{-}\\Curvesstack \\longrightarrow \\Spec(\\Lambda)", "$$", "is smooth in an open neighbourhood of the point", "$x_0 : \\Spec(k) \\to \\Lambda\\text{-}\\Curvesstack$", "defined by $X/k$ if and only if $\\Deformationcategory_X$ is unobstructed.", "\\medskip\\noindent", "The lemma now follows from", "Geometry of Stacks, Lemma \\ref{stacks-geometry-lemma-characterize-smoothness}", "and the equality", "$$", "\\Deformationcategory_X =", "\\mathcal{F}_{\\Lambda\\text{-}\\Curvesstack, k, x_0}", "$$", "This equality is not completely trivial to esthablish. Namely, on the left", "hand side we have the deformation category classifying all flat deformations", "$Y \\to \\Spec(A)$ of $X$ as a scheme over $A \\in \\Ob(\\mathcal{C}_\\Lambda)$.", "On the right hand side we have the deformation category classifying all", "flat morphisms $Y \\to \\Spec(A)$ with special fibre $X$", "where $Y$ is an algebraic space and", "$Y \\to \\Spec(A)$ is proper, of finite presentation, and of", "relative dimension $\\leq 1$. Since $A$ is Artinian, we find", "that $Y$ is a scheme for example by Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-codim-1-point-in-schematic-locus}.", "Thus it remains to show: a flat deformation $Y \\to \\Spec(A)$ of", "$X$ as a scheme over an Artinian local ring $A$ with residue field $k$", "is proper, of finite presentation, and of relative dimension $\\leq 1$.", "Relative dimension is defined in terms of fibres and hence holds", "automatically for $Y/A$ since it holds for $X/k$.", "The morphism $Y \\to \\Spec(A)$ is proper and locally of finite presentation", "as this is true for $X \\to \\Spec(k)$, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-deform-property}." ], "refs": [ "moduli-curves-lemma-curves-qs-lfp", "stacks-morphisms-lemma-where-smooth", "stacks-geometry-lemma-characterize-smoothness", "spaces-over-fields-lemma-codim-1-point-in-schematic-locus", "more-morphisms-lemma-deform-property" ], "ref_ids": [ 1591, 7543, 4670, 12845, 13725 ] } ], "ref_ids": [] }, { "id": 1610, "type": "theorem", "label": "moduli-curves-lemma-big-smooth-part-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-big-smooth-part-curves", "contents": [ "The open substack", "$$", "\\Curvesstack^{lci+} =", "\\Curvesstack^{lci} \\cap \\Curvesstack^{+}", "\\subset \\Curvesstack", "$$", "has the following properties", "\\begin{enumerate}", "\\item $\\Curvesstack^{lci+} \\to \\Spec(\\mathbf{Z})$ is smooth,", "\\item given a family of curves $X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors through", "$\\Curvesstack^{lci+}$,", "\\item $X \\to S$ is a local complete intersection morphism and", "the singular locus of $X \\to S$ endowed with any/some closed subspace", "structure is finite over $S$,", "\\end{enumerate}", "\\item given $X$ a proper scheme over a field $k$ of dimension $\\leq 1$", "the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through $\\Curvesstack^{lci+}$,", "\\item $X$ is a local complete intersection over $k$ and", "$X \\to \\Spec(k)$ is smooth except at finitely many points.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If we can show that there is an open substack $\\Curvesstack^{lci+}$", "whose points are characterized by (2), then we see that", "(1) holds by combining Lemma \\ref{lemma-in-smooth-locus} with", "Deformation Problems, Lemma \\ref{examples-defos-lemma-curve-isolated-lci}.", "Since", "$$", "\\Curvesstack^{lci+} = \\Curvesstack^{lci} \\cap \\Curvesstack^{+}", "$$", "inside $\\Curvesstack$, we conclude by", "Lemmas \\ref{lemma-lci-curves} and \\ref{lemma-isolated-sings-curves}." ], "refs": [ "moduli-curves-lemma-in-smooth-locus", "examples-defos-lemma-curve-isolated-lci", "moduli-curves-lemma-lci-curves", "moduli-curves-lemma-isolated-sings-curves" ], "ref_ids": [ 1609, 8777, 1607, 1608 ] } ], "ref_ids": [] }, { "id": 1611, "type": "theorem", "label": "moduli-curves-lemma-smooth-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-smooth-curves", "contents": [ "There exist an open substacks", "$$", "\\Curvesstack^{smooth, 1} \\subset \\Curvesstack^{smooth} \\subset \\Curvesstack", "$$", "such that", "\\begin{enumerate}", "\\item given a family of curves $f : X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{smooth}$, resp.\\ $\\Curvesstack^{smooth, 1}$,", "\\item $f$ is smooth, resp.\\ smooth of relative dimension $1$,", "\\end{enumerate}", "\\item given $X$ a scheme proper over a field $k$ with", "$\\dim(X) \\leq 1$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$", "factors through $\\Curvesstack^{smooth}$, resp.\\ $\\Curvesstack^{smooth, 1}$,", "\\item $X$ is smooth over $k$, resp.\\ $X$ is smooth over $k$ and", "$X$ is equidimensional of dimension $1$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove the statements regarding $\\Curvesstack^{smooth}$", "it suffices to show that given a family of curves", "$f : X \\to S$, there is an open subscheme $S' \\subset S$", "such that $S' \\times_S X \\to S'$ is smooth and such that the", "formation of this open commutes with base change.", "We know that there is a maximal open $U \\subset X$ such", "that $U \\to S$ is smooth and that formation of $U$ commutes", "with arbitrary base change, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-where-smooth}.", "If $T = |X| \\setminus |U|$ then $f(T)$ is closed in $S$ as $f$ is proper.", "Setting $S' = S \\setminus f(T)$ we obtain the desired open.", "\\medskip\\noindent", "Let $f : X \\to S$ be a family of curves with $f$ smooth.", "Then the fibres $X_s$ are smooth over $\\kappa(s)$ and hence", "Cohen-Macaulay (for example you can see this using", "Algebra, Lemmas \\ref{algebra-lemma-smooth-over-field} and", "\\ref{algebra-lemma-lci-CM}). Thus we see that we may set", "$$", "\\Curvesstack^{smooth, 1} = \\Curvesstack^{smooth} \\cap", "\\Curvesstack^{CM, 1}", "$$", "and the desired equivalences follow from what we've already", "shown for $\\Curvesstack^{smooth}$ and Lemma \\ref{lemma-CM-1-curves}." ], "refs": [ "spaces-morphisms-lemma-where-smooth", "algebra-lemma-smooth-over-field", "algebra-lemma-lci-CM", "moduli-curves-lemma-CM-1-curves" ], "ref_ids": [ 4893, 1192, 1166, 1595 ] } ], "ref_ids": [] }, { "id": 1612, "type": "theorem", "label": "moduli-curves-lemma-smooth-curves-smooth", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-smooth-curves-smooth", "contents": [ "The morphism $\\Curvesstack^{smooth} \\to \\Spec(\\mathbf{Z})$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from the observation that", "$\\Curvesstack^{smooth} \\subset \\Curvesstack^{lci+}$", "and Lemma \\ref{lemma-big-smooth-part-curves}." ], "refs": [ "moduli-curves-lemma-big-smooth-part-curves" ], "ref_ids": [ 1610 ] } ], "ref_ids": [] }, { "id": 1613, "type": "theorem", "label": "moduli-curves-lemma-smooth-curves-h0", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-smooth-curves-h0", "contents": [ "There exist an open substack", "$\\Curvesstack^{smooth, h0} \\subset \\Curvesstack$", "such that", "\\begin{enumerate}", "\\item given a family of curves $f : X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{smooth}$,", "\\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$, this holds after any base change,", "and $f$ is smooth of relative dimension $1$,", "\\end{enumerate}", "\\item given $X$ a scheme proper over a field $k$ with", "$\\dim(X) \\leq 1$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$", "factors through $\\Curvesstack^{smooth, h0}$,", "\\item $X$ is smooth, $\\dim(X) = 1$, and $k = H^0(X, \\mathcal{O}_X)$,", "\\item $X$ is smooth, $\\dim(X) = 1$, and $X$ is geometrically connected,", "\\item $X$ is smooth, $\\dim(X) = 1$, and $X$ is geometrically integral, and", "\\item $X_{\\overline{k}}$ is a smooth curve.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If we set", "$$", "\\Curvesstack^{smooth, h0} = \\Curvesstack^{smooth} \\cap", "\\Curvesstack^{h0, 1}", "$$", "then we see that (1) holds by", "Lemmas \\ref{lemma-pre-genus-curves} and \\ref{lemma-smooth-curves}.", "In fact, this also gives the equivalence of (2)(a) and (2)(b).", "To finish the proof we have to show that", "(2)(b) is equivalent to each of (2)(c), (2)(d), and (2)(e).", "\\medskip\\noindent", "A smooth scheme over a field is geometrically normal", "(Varieties, Lemma \\ref{varieties-lemma-smooth-geometrically-normal}),", "smoothness is preserved under base change", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-smooth}), and", "being smooth is fpqc local on the target", "(Descent, Lemma \\ref{descent-lemma-descending-property-smooth}).", "Keeping this in mind, the equivalence of (2)(b), (2)(c), 2(d), and (2)(e)", "follows from Varieties, Lemma \\ref{varieties-lemma-geometrically-normal-stein}." ], "refs": [ "moduli-curves-lemma-pre-genus-curves", "moduli-curves-lemma-smooth-curves", "varieties-lemma-smooth-geometrically-normal", "morphisms-lemma-base-change-smooth", "descent-lemma-descending-property-smooth", "varieties-lemma-geometrically-normal-stein" ], "ref_ids": [ 1596, 1611, 11005, 5327, 14692, 10956 ] } ], "ref_ids": [] }, { "id": 1614, "type": "theorem", "label": "moduli-curves-lemma-smooth-one-piece-per-genus", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-smooth-one-piece-per-genus", "contents": [ "There is a decomposition into open and closed substacks", "$$", "\\mathcal{M} = \\coprod\\nolimits_{g \\geq 0} \\mathcal{M}_g", "$$", "where each $\\mathcal{M}_g$ is characterized as follows:", "\\begin{enumerate}", "\\item given a family of curves $f : X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\mathcal{M}_g$,", "\\item $X \\to S$ is smooth, $f_*\\mathcal{O}_X = \\mathcal{O}_S$,", "this holds after any base change, and $R^1f_*\\mathcal{O}_X$", "is a locally free $\\mathcal{O}_S$-module of rank $g$,", "\\end{enumerate}", "\\item given $X$ a scheme proper over a field $k$ with", "$\\dim(X) \\leq 1$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$", "factors through $\\mathcal{M}_g$,", "\\item $X$ is smooth, $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$,", "and $X$ has genus $g$,", "\\item $X$ is smooth, $\\dim(X) = 1$, $X$ is geometrically connected, and", "$X$ has genus $g$,", "\\item $X$ is smooth, $\\dim(X) = 1$, $X$ is geometrically integral, and", "$X$ has genus $g$, and", "\\item $X_{\\overline{k}}$ is a smooth curve of genus $g$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-smooth-curves-h0} and", "\\ref{lemma-pre-genus-one-piece-per-genus}.", "You can also use", "Lemma \\ref{lemma-one-piece-per-genus}", "instead." ], "refs": [ "moduli-curves-lemma-smooth-curves-h0", "moduli-curves-lemma-pre-genus-one-piece-per-genus", "moduli-curves-lemma-one-piece-per-genus" ], "ref_ids": [ 1613, 1599, 1604 ] } ], "ref_ids": [] }, { "id": 1615, "type": "theorem", "label": "moduli-curves-lemma-smooth-curves-h0-smooth", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-smooth-curves-h0-smooth", "contents": [ "The morphisms $\\mathcal{M} \\to \\Spec(\\mathbf{Z})$ and", "$\\mathcal{M}_g \\to \\Spec(\\mathbf{Z})$", "are smooth." ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathcal{M}$ is an open substack of", "$\\Curvesstack^{lci+}$ this follows from", "Lemma \\ref{lemma-big-smooth-part-curves}." ], "refs": [ "moduli-curves-lemma-big-smooth-part-curves" ], "ref_ids": [ 1610 ] } ], "ref_ids": [] }, { "id": 1616, "type": "theorem", "label": "moduli-curves-lemma-smooth-dense", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-smooth-dense", "contents": [ "The inclusion", "$$", "|\\Curvesstack^{smooth}| \\subset |\\Curvesstack^{lci+}|", "$$", "is that of an open dense subset." ], "refs": [], "proofs": [ { "contents": [ "By the very construction of the topology on", "$|\\Curvesstack^{lci+}|$ in", "Properties of Stacks, Section \\ref{stacks-properties-section-points}", "we find that $|\\Curvesstack^{smooth}|$", "is an open subset. Let $\\xi \\in |\\Curvesstack^{lci+}|$ be a point.", "Then there exists a field $k$ and a scheme $X$ over $k$", "with $X$ proper over $k$, with $\\dim(X) \\leq 1$,", "with $X$ a local complete intersection over $k$, and", "with $X$ is smooth over $k$ except at finitely many points, such", "that $\\xi$ is the equivalence class of the", "classifying morphism $\\Spec(k) \\to \\Curvesstack^{lci+}$ determined by $X$.", "See Lemma \\ref{lemma-big-smooth-part-curves}.", "By Deformation Problems, Lemma", "\\ref{examples-defos-lemma-smoothing-proper-curve-isolated-lci}", "there exists a flat projective morphism $Y \\to \\Spec(k[[t]])$", "whose generic fibre is smooth and whose special fibre is", "isomorphic to $X$. Consider the classifying morphism", "$$", "\\Spec(k[[t]]) \\longrightarrow \\Curvesstack^{lci+}", "$$", "determined by $Y$. The image of the closed point is $\\xi$", "and the image of the generic point is in $|\\Curvesstack^{smooth}|$.", "Since the generic point specializes to the closed point in", "$|\\Spec(k[[t]])|$ we conclude that $\\xi$ is in the closure", "of $|\\Curvesstack^{smooth}|$ as desired." ], "refs": [ "moduli-curves-lemma-big-smooth-part-curves", "examples-defos-lemma-smoothing-proper-curve-isolated-lci" ], "ref_ids": [ 1610, 8783 ] } ], "ref_ids": [] }, { "id": 1617, "type": "theorem", "label": "moduli-curves-lemma-nodal-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-nodal-curves", "contents": [ "There exist an open substack $\\Curvesstack^{nodal} \\subset \\Curvesstack$", "such that", "\\begin{enumerate}", "\\item given a family of curves $f : X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{nodal}$,", "\\item $f$ is at-worst-nodal of relative dimension $1$,", "\\end{enumerate}", "\\item given $X$ a scheme proper over a field $k$ with", "$\\dim(X) \\leq 1$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors", "through $\\Curvesstack^{nodal}$,", "\\item the singularities of $X$ are at-worst-nodal and $X$", "is equidimensional of dimension $1$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In fact, it suffices to show that given a family of curves", "$f : X \\to S$, there is an open subscheme $S' \\subset S$", "such that $S' \\times_S X \\to S'$ is at-worst-nodal of relative dimension $1$", "and such that formation of $S'$ commutes with arbitrary base change.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-locus-where-nodal}", "there is a maximal open subspace $X' \\subset X$ such", "that $f|_{X'} : X' \\to S$ is at-worst-nodal of relative dimension $1$.", "Moreover, formation of $X'$ commutes with base change.", "Hence we can take", "$$", "S' = S \\setminus |f|(|X| \\setminus |X'|)", "$$", "This is open because a proper morphism is universally closed by", "definition." ], "refs": [ "spaces-more-morphisms-lemma-locus-where-nodal" ], "ref_ids": [ 277 ] } ], "ref_ids": [] }, { "id": 1618, "type": "theorem", "label": "moduli-curves-lemma-nodal-curves-smooth", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-nodal-curves-smooth", "contents": [ "The morphism $\\Curvesstack^{nodal} \\to \\Spec(\\mathbf{Z})$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from the observation that", "$\\Curvesstack^{nodal} \\subset \\Curvesstack^{lci+}$", "and Lemma \\ref{lemma-big-smooth-part-curves}." ], "refs": [ "moduli-curves-lemma-big-smooth-part-curves" ], "ref_ids": [ 1610 ] } ], "ref_ids": [] }, { "id": 1619, "type": "theorem", "label": "moduli-curves-lemma-CM-dualizing", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-CM-dualizing", "contents": [ "Let $X \\to S$ be a family of curves with Cohen-Macaulay fibres", "equidimensional of dimension $1$ (Lemma \\ref{lemma-CM-1-curves}).", "Then $\\omega_{X/S}^\\bullet = \\omega_{X/S}[1]$ where $\\omega_{X/S}$", "is a pseudo-coherent $\\mathcal{O}_X$-module flat over $S$ whose", "formation commutes with arbitrary base change." ], "refs": [ "moduli-curves-lemma-CM-1-curves" ], "proofs": [ { "contents": [ "We urge the reader to deduce this directly from the discussion above", "of what happens after base change to a field. Our proof will", "use a somewhat cumbersome reduction to the Noetherian schemes case.", "\\medskip\\noindent", "Once we show $\\omega_{X/S}^\\bullet = \\omega_{X/S}[1]$ with", "$\\omega_{X/S}$ flat over $S$, the statement on base change", "will follow as we already know that formation of $\\omega_{X/S}^\\bullet$", "commutes with arbitrary base change. Moreover, the pseudo-coherence", "will be automatic as $\\omega_{X/S}^\\bullet$ is pseudo-coherent", "by definition. Vanishing of the other cohomology sheaves and flatness ", "may be checked \\'etale locally. Thus we may assume $f : X \\to S$", "is a morphism of schemes with $S$ affine (see discussion above).", "Write $S = \\lim S_i$ as a cofiltered limit of affine schemes $S_i$", "of finite type over $\\mathbf{Z}$.", "Since $\\Curvesstack^{CM, 1}$ is locally of finite presentation over", "$\\mathbf{Z}$ (as an open substack of $\\Curvesstack$, see", "Lemmas \\ref{lemma-CM-1-curves} and \\ref{lemma-curves-qs-lfp}),", "we can find an $i$ and a family", "of curves $X_i \\to S_i$ whose pullback is $X \\to S$", "(Limits of Stacks, Lemma", "\\ref{stacks-limits-lemma-representable-by-spaces-limit-preserving}).", "After increasing $i$ if necessary we may assume $X_i$ is a scheme,", "see Limits of Spaces, Lemma \\ref{spaces-limits-lemma-limit-is-scheme}.", "Since formation of $\\omega_{X/S}^\\bullet$ commutes with", "arbitrary base change, we may replace $S$ by $S_i$.", "Doing so we may and do assume $S_i$ is Noetherian.", "Then $f$ is clearly a Cohen-Macaulay morphism", "(More on Morphisms, Definition \\ref{more-morphisms-definition-CM})", "by our assumption on the fibres.", "Also then $\\omega_{X/S}^\\bullet = f^!\\mathcal{O}_S$", "by the very construction of $f^!$ in", "Duality for Schemes, Section \\ref{duality-section-upper-shriek}.", "Thus the lemma by Duality for Schemes, Lemma", "\\ref{duality-lemma-affine-flat-Noetherian-CM}." ], "refs": [ "moduli-curves-lemma-CM-1-curves", "moduli-curves-lemma-curves-qs-lfp", "stacks-limits-lemma-representable-by-spaces-limit-preserving", "spaces-limits-lemma-limit-is-scheme", "more-morphisms-definition-CM", "duality-lemma-affine-flat-Noetherian-CM" ], "ref_ids": [ 1595, 1591, 15017, 4579, 14115, 13588 ] } ], "ref_ids": [ 1595 ] }, { "id": 1620, "type": "theorem", "label": "moduli-curves-lemma-gorenstein-dualizing", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-gorenstein-dualizing", "contents": [ "Let $X \\to S$ be a family of curves with Gorenstein fibres", "equidimensional of dimension $1$ (Lemma \\ref{lemma-gorenstein-1-curves}).", "Then the relative dualizing sheaf $\\omega_{X/S}$ is an", "invertible $\\mathcal{O}_X$-module whose", "formation commutes with arbitrary base change." ], "refs": [ "moduli-curves-lemma-gorenstein-1-curves" ], "proofs": [ { "contents": [ "This is true because the pullback of the relative dualizing module", "to a fibre is invertible by the discussion above. Alternatively, you", "can argue exactly as in the proof of", "Lemma \\ref{lemma-CM-dualizing} and deduce the result from", "Duality for Schemes, Lemma", "\\ref{duality-lemma-affine-flat-Noetherian-gorenstein}." ], "refs": [ "moduli-curves-lemma-CM-dualizing", "duality-lemma-affine-flat-Noetherian-gorenstein" ], "ref_ids": [ 1619, 13602 ] } ], "ref_ids": [ 1606 ] }, { "id": 1621, "type": "theorem", "label": "moduli-curves-lemma-prestable-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-prestable-curves", "contents": [ "There exist an open substack $\\Curvesstack^{prestable} \\subset \\Curvesstack$", "such that", "\\begin{enumerate}", "\\item given a family of curves $f : X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{prestable}$,", "\\item $X \\to S$ is a prestable family of curves,", "\\end{enumerate}", "\\item given $X$ a scheme proper over a field $k$ with", "$\\dim(X) \\leq 1$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$", "factors through $\\Curvesstack^{prestable}$,", "\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,", "and $k = H^0(X, \\mathcal{O}_X)$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Given a family of curves $X \\to S$ we see that it is prestable if", "and only if the classifying morphism factors both through", "$\\Curvesstack^{nodal}$ and $\\Curvesstack^{h0, 1}$. An alternative", "is to use $\\Curvesstack^{grc, 1}$ (since a nodal curve is geometrically", "reduced hence has $H^0$ equal to the ground field if and only if", "it is connected). In a formula", "$$", "\\Curvesstack^{prestable} =", "\\Curvesstack^{nodal} \\cap \\Curvesstack^{h0, 1} =", "\\Curvesstack^{nodal} \\cap \\Curvesstack^{grc, 1}", "$$", "Thus the lemma follows from", "Lemmas \\ref{lemma-pre-genus-curves} and \\ref{lemma-nodal-curves}." ], "refs": [ "moduli-curves-lemma-pre-genus-curves", "moduli-curves-lemma-nodal-curves" ], "ref_ids": [ 1596, 1617 ] } ], "ref_ids": [] }, { "id": 1622, "type": "theorem", "label": "moduli-curves-lemma-prestable-one-piece-per-genus", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-prestable-one-piece-per-genus", "contents": [ "There is a decomposition into open and closed substacks", "$$", "\\Curvesstack^{prestable} = \\coprod\\nolimits_{g \\geq 0}", "\\Curvesstack^{prestable}_g", "$$", "where each $\\Curvesstack^{prestable}_g$ is characterized as follows:", "\\begin{enumerate}", "\\item given a family of curves $f : X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{prestable}_g$,", "\\item $X \\to S$ is a prestable family of curves and", "$R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module of rank $g$,", "\\end{enumerate}", "\\item given $X$ a scheme proper over a field $k$ with", "$\\dim(X) \\leq 1$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$", "factors through $\\Curvesstack^{prestable}_g$,", "\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,", "$k = H^0(X, \\mathcal{O}_X)$, and the genus of $X$ is $g$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since we have seen that $\\Curvesstack^{prestable}$ is contained", "in $\\Curvesstack^{h0, 1}$, this", "follows from Lemmas \\ref{lemma-prestable-curves} and", "\\ref{lemma-pre-genus-one-piece-per-genus}." ], "refs": [ "moduli-curves-lemma-prestable-curves", "moduli-curves-lemma-pre-genus-one-piece-per-genus" ], "ref_ids": [ 1621, 1599 ] } ], "ref_ids": [] }, { "id": 1623, "type": "theorem", "label": "moduli-curves-lemma-prestable-curves-smooth", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-prestable-curves-smooth", "contents": [ "The morphisms", "$\\Curvesstack^{prestable} \\to \\Spec(\\mathbf{Z})$ and", "$\\Curvesstack^{prestable}_g \\to \\Spec(\\mathbf{Z})$ are", "smooth." ], "refs": [], "proofs": [ { "contents": [ "Since $\\Curvesstack^{prestable}$ is an open substack of", "$\\Curvesstack^{nodal}$ this follows from", "Lemma \\ref{lemma-nodal-curves-smooth}." ], "refs": [ "moduli-curves-lemma-nodal-curves-smooth" ], "ref_ids": [ 1618 ] } ], "ref_ids": [] }, { "id": 1624, "type": "theorem", "label": "moduli-curves-lemma-semistable", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-semistable", "contents": [ "Let $f : X \\to S$ be a prestable family of curves of genus $g \\geq 1$.", "Let $s \\in S$ be a point of the base scheme. Let $m \\geq 2$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X_s$ does not have a rational tail", "(Algebraic Curves, Example \\ref{curves-example-rational-tail}), and", "\\item $f^*f_*\\omega_{X/S}^{\\otimes m} \\to \\omega_{X/S}^{\\otimes m}$,", "is surjective over $f^{-1}(U)$ for some $s \\in U \\subset S$ open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (2). Using the material in Section \\ref{section-relative-dualizing}", "we conclude that $\\omega_{X_s}^{\\otimes m}$ is", "globally generated. However, if $C \\subset X_s$", "is a rational tail, then $\\deg(\\omega_{X_s}|_C) < 0$ by", "Algebraic Curves, Lemma \\ref{curves-lemma-rational-tail-negative}", "hence $H^0(C, \\omega_{X_s}|_C) = 0$ by", "Varieties, Lemma \\ref{varieties-lemma-check-invertible-sheaf-trivial}", "which contradicts the fact that it is globally generated.", "This proves (1).", "\\medskip\\noindent", "Assume (1). First assume that $g \\geq 2$. Assumption (1) ", "implies $\\omega_{X_s}^{\\otimes m}$ is globally generated,", "see Algebraic Curves, Lemma \\ref{curves-lemma-contracting-rational-tails}.", "Moreover, we have", "$$", "\\Hom_{\\kappa(s)}(H^1(X_s, \\omega_{X_s}^{\\otimes m}), \\kappa(s)) =", "H^0(X_s, \\omega_{X_s}^{\\otimes 1 - m})", "$$", "by duality, see Algebraic Curves, Lemma \\ref{curves-lemma-duality-dim-1-CM}.", "Since $\\omega_{X_s}^{\\otimes m}$ is globally generated we find", "that the restriction to each irreducible component has nonegative degree.", "Hence the restriction of $\\omega_{X_s}^{\\otimes 1 - m}$ to each", "irreducible component has nonpositive degree. Since", "$\\deg(\\omega_{X_s}^{\\otimes 1 - m}) = (1 - m)(2g - 2) < 0$ by Riemann-Roch", "(Algebraic Curves, Lemma \\ref{curves-lemma-rr}) we conclude that the $H^0$", "is zero by Varieties, Lemma \\ref{varieties-lemma-no-sections-dual-nef}.", "By cohomology and base change we conclude that", "$$", "E = Rf_*\\omega_{X/S}^{\\otimes m}", "$$", "is a perfect complex whose formation commutes with arbitrary base change", "(Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-flat-proper-perfect-direct-image-general}).", "The vanishing proved above tells us that $E \\otimes^\\mathbf{L} \\kappa(s)$", "is equal to $H^0(X_s, \\omega_{X_s}^{\\otimes m})$ placed in degree $0$.", "After shrinking $S$ we find $E = f_*\\omega_{X/S}^{\\otimes m}$", "is a locally free $\\mathcal{O}_S$-module placed in degree $0$", "(and its formation commutes with arbitrary base change as", "we've already said), see Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric}.", "The map $f^*f_*\\omega_{X/S}^{\\otimes m} \\to \\omega_{X/S}^{\\otimes m}$", "is surjective after restricting to $X_s$. Thus it is surjective in", "an open neighbourhood of $X_s$. Since $f$ is proper, this open", "neighbourhood contains $f^{-1}(U)$ for some open neighbourhood", "$U$ of $s$ in $S$.", "\\medskip\\noindent", "Assume (1) and $g = 1$. By", "Algebraic Curves, Lemma \\ref{curves-lemma-contracting-rational-tails}", "the assumption (1) means that $\\omega_{X_s}$ is isomorphic to", "$\\mathcal{O}_{X_s}$. If we can show that after shrinking $S$", "the invertible sheaf $\\omega_{X/S}$ because trivial, then", "we are done. We may assume $S$ is affine. After shrinking $S$", "further, we can write", "$$", "Rf_*\\mathcal{O}_X = (\\mathcal{O}_S \\xrightarrow{0} \\mathcal{O}_S)", "$$", "sitting in degrees $0$ and $1$", "compatibly with further base change, see Lemma \\ref{lemma-genus}.", "By duality this means that", "$$", "Rf_*\\omega_{X/S} = (\\mathcal{O}_S \\xrightarrow{0} \\mathcal{O}_S)", "$$", "sitting in degrees $0$ and $1$\\footnote{Use that", "$Rf_*\\omega_{X/S}^\\bullet =", "Rf_*R\\SheafHom_{\\mathcal{O}_X}(\\mathcal{O}_X. \\omega_{X/S}^\\bullet) =", "R\\SheafHom_{\\mathcal{O}_S}(Rf_*\\mathcal{O}_X, \\mathcal{O}_S)$", "by Duality for Spaces, Lemma \\ref{spaces-duality-lemma-iso-on-RSheafHom} and", "Remark \\ref{spaces-duality-remark-iso-on-RSheafHom}", "and then that $\\omega_{X/S}^\\bullet = \\omega_{X/S}[1]$ by", "our definitions in Section \\ref{section-relative-dualizing}.}.", "In particular we obtain an isomorphism $\\mathcal{O}_S \\to f_*\\omega_{X/S}$", "which is compatible with base change since", "formation of $Rf_*\\omega_{X/S}$ is compatible with base change", "(see reference given above).", "By adjointness, we get a global section $\\sigma \\in \\Gamma(X, \\omega_{X/S})$.", "The restriction of this section to the fibre $X_s$", "is nonzero (a basis element in fact) and as", "$\\omega_{X_s}$ is trivial on the fibres,", "this section is nonwhere zero on $X_s$.", "Thus it nowhere zero in", "an open neighbourhood of $X_s$. Since $f$ is proper, this open", "neighbourhood contains $f^{-1}(U)$ for some open neighbourhood", "$U$ of $s$ in $S$." ], "refs": [ "curves-lemma-rational-tail-negative", "varieties-lemma-check-invertible-sheaf-trivial", "curves-lemma-contracting-rational-tails", "curves-lemma-duality-dim-1-CM", "curves-lemma-rr", "varieties-lemma-no-sections-dual-nef", "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "spaces-perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric", "curves-lemma-contracting-rational-tails", "moduli-curves-lemma-genus", "spaces-duality-lemma-iso-on-RSheafHom", "spaces-duality-remark-iso-on-RSheafHom" ], "ref_ids": [ 6334, 11114, 6338, 6251, 6256, 11115, 2738, 2747, 6338, 1598, 11790, 11811 ] } ], "ref_ids": [] }, { "id": 1625, "type": "theorem", "label": "moduli-curves-lemma-semistable-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-semistable-curves", "contents": [ "There exist an open substack $\\Curvesstack^{semistable} \\subset \\Curvesstack$", "such that", "\\begin{enumerate}", "\\item given a family of curves $f : X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{semistable}$,", "\\item $X \\to S$ is a semistable family of curves,", "\\end{enumerate}", "\\item given $X$ a scheme proper over a field $k$ with", "$\\dim(X) \\leq 1$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$", "factors through $\\Curvesstack^{semistable}$,", "\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,", "$k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $\\geq 1$, and", "$X$ has no rational tails,", "\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,", "$k = H^0(X, \\mathcal{O}_X)$, and $\\omega_{X_s}^{\\otimes m}$ is", "globally generated for $m \\geq 2$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (2)(b) and (2)(c) is", "Algebraic Curves, Lemma \\ref{curves-lemma-contracting-rational-tails}.", "In the rest of the proof we will work with (2)(b)", "in accordance with Definition \\ref{definition-semistable}.", "\\medskip\\noindent", "By the discussion in Section \\ref{section-open}", "it suffices to look at families $f : X \\to S$ of", "prestable curves. By Lemma \\ref{lemma-semistable}", "we obtain the desired openness of the locus in question.", "Formation of this open commutes with arbitrary base change,", "because the (non)existence of rational tails is insensitive", "to ground field extensions by", "Algebraic Curves, Lemma \\ref{curves-lemma-contracting-rational-tails}." ], "refs": [ "curves-lemma-contracting-rational-tails", "moduli-curves-definition-semistable", "moduli-curves-lemma-semistable", "curves-lemma-contracting-rational-tails" ], "ref_ids": [ 6338, 1646, 1624, 6338 ] } ], "ref_ids": [] }, { "id": 1626, "type": "theorem", "label": "moduli-curves-lemma-semistable-one-piece-per-genus", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-semistable-one-piece-per-genus", "contents": [ "There is a decomposition into open and closed substacks", "$$", "\\Curvesstack^{semistable} = \\coprod\\nolimits_{g \\geq 1}", "\\Curvesstack^{semistable}_g", "$$", "where each $\\Curvesstack^{semistable}_g$ is characterized as follows:", "\\begin{enumerate}", "\\item given a family of curves $f : X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{semistable}_g$,", "\\item $X \\to S$ is a semistable family of curves and", "$R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module of rank $g$,", "\\end{enumerate}", "\\item given $X$ a scheme proper over a field $k$ with", "$\\dim(X) \\leq 1$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$", "factors through $\\Curvesstack^{semistable}_g$,", "\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,", "$k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $g$, and $X$", "has no rational tail,", "\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,", "$k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $g$, and", "$\\omega_{X_s}^{\\otimes m}$ is globally generated for $m \\geq 2$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-semistable-curves} and", "\\ref{lemma-prestable-one-piece-per-genus}." ], "refs": [ "moduli-curves-lemma-semistable-curves", "moduli-curves-lemma-prestable-one-piece-per-genus" ], "ref_ids": [ 1625, 1622 ] } ], "ref_ids": [] }, { "id": 1627, "type": "theorem", "label": "moduli-curves-lemma-semistable-curves-smooth", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-semistable-curves-smooth", "contents": [ "The morphisms", "$\\Curvesstack^{semistable} \\to \\Spec(\\mathbf{Z})$ and", "$\\Curvesstack^{semistable}_g \\to \\Spec(\\mathbf{Z})$", "are smooth." ], "refs": [], "proofs": [ { "contents": [ "Since $\\Curvesstack^{semistable}$ is an open substack of", "$\\Curvesstack^{nodal}$ this follows from", "Lemma \\ref{lemma-nodal-curves-smooth}." ], "refs": [ "moduli-curves-lemma-nodal-curves-smooth" ], "ref_ids": [ 1618 ] } ], "ref_ids": [] }, { "id": 1628, "type": "theorem", "label": "moduli-curves-lemma-stable", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-stable", "contents": [ "Let $f : X \\to S$ be a prestable family of curves of genus $g \\geq 2$.", "Let $s \\in S$ be a point of the base scheme.", "The following are equivalent", "\\begin{enumerate}", "\\item $X_s$ does not have a rational tail and does not have a", "rational bridge", "(Algebraic Curves, Examples", "\\ref{curves-example-rational-tail} and", "\\ref{curves-example-rational-bridge}), and", "\\item $\\omega_{X/S}$ is ample on $f^{-1}(U)$ for some $s \\in U \\subset S$ open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (2). Then $\\omega_{X_s}$ is ample on $X_s$.", "By Algebraic Curves, Lemmas \\ref{curves-lemma-rational-tail-negative} and", "\\ref{curves-lemma-rational-bridge-zero}", "we conclude that (1) holds (we also", "use the characterization of ample invertible sheaves", "in Varieties, Lemma", "\\ref{varieties-lemma-ampleness-in-terms-of-degrees-components}).", "\\medskip\\noindent", "Assume (1). Then $\\omega_{X_s}$ is ample on $X_s$ by", "Algebraic Curves, Lemmas \\ref{curves-lemma-contracting-rational-bridges}.", "We conclude by Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-ample-in-neighbourhood}." ], "refs": [ "curves-lemma-rational-tail-negative", "curves-lemma-rational-bridge-zero", "varieties-lemma-ampleness-in-terms-of-degrees-components", "curves-lemma-contracting-rational-bridges", "spaces-descent-lemma-ample-in-neighbourhood" ], "ref_ids": [ 6334, 6339, 11117, 6343, 9414 ] } ], "ref_ids": [] }, { "id": 1629, "type": "theorem", "label": "moduli-curves-lemma-stable-curves", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-stable-curves", "contents": [ "There exist an open substack $\\Curvesstack^{stable} \\subset \\Curvesstack$", "such that", "\\begin{enumerate}", "\\item given a family of curves $f : X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\Curvesstack^{stable}$,", "\\item $X \\to S$ is a stable family of curves,", "\\end{enumerate}", "\\item given $X$ a scheme proper over a field $k$ with", "$\\dim(X) \\leq 1$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$", "factors through $\\Curvesstack^{stable}$,", "\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,", "$k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $\\geq 2$, and", "$X$ has no rational tails or bridges,", "\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,", "$k = H^0(X, \\mathcal{O}_X)$, and $\\omega_{X_s}$ is ample.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By the discussion in Section \\ref{section-open}", "it suffices to look at families $f : X \\to S$ of", "prestable curves. By Lemma \\ref{lemma-stable}", "we obtain the desired openness of the locus in question.", "Formation of this open commutes with arbitrary base change,", "either because the (non)existence of rational tails or bridges", "is insensitive to ground field extensions by", "Algebraic Curves, Lemmas", "\\ref{curves-lemma-contracting-rational-tails} and", "\\ref{curves-lemma-contracting-rational-bridges}", "or because ampleness is insenstive to base field extensions by", "Descent, Lemma \\ref{descent-lemma-descending-property-ample}." ], "refs": [ "moduli-curves-lemma-stable", "curves-lemma-contracting-rational-tails", "curves-lemma-contracting-rational-bridges", "descent-lemma-descending-property-ample" ], "ref_ids": [ 1628, 6338, 6343, 14704 ] } ], "ref_ids": [] }, { "id": 1630, "type": "theorem", "label": "moduli-curves-lemma-stable-one-piece-per-genus", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-stable-one-piece-per-genus", "contents": [ "There is a decomposition into open and closed substacks", "$$", "\\overline{\\mathcal{M}} = \\coprod\\nolimits_{g \\geq 2} \\overline{\\mathcal{M}}_g", "$$", "where each $\\overline{\\mathcal{M}}_g$ is characterized as follows:", "\\begin{enumerate}", "\\item given a family of curves $f : X \\to S$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $S \\to \\Curvesstack$ factors", "through $\\overline{\\mathcal{M}}_g$,", "\\item $X \\to S$ is a stable family of curves and", "$R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module of rank $g$,", "\\end{enumerate}", "\\item given $X$ a scheme proper over a field $k$ with", "$\\dim(X) \\leq 1$ the following are equivalent", "\\begin{enumerate}", "\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$", "factors through $\\overline{\\mathcal{M}}_g$,", "\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,", "$k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $g$, and $X$", "has no rational tails or bridges.", "\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,", "$k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $g$, and", "$\\omega_{X_s}$ is ample.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-stable-curves} and", "\\ref{lemma-prestable-one-piece-per-genus}." ], "refs": [ "moduli-curves-lemma-stable-curves", "moduli-curves-lemma-prestable-one-piece-per-genus" ], "ref_ids": [ 1629, 1622 ] } ], "ref_ids": [] }, { "id": 1631, "type": "theorem", "label": "moduli-curves-lemma-stable-curves-smooth", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-stable-curves-smooth", "contents": [ "The morphisms", "$\\overline{\\mathcal{M}} \\to \\Spec(\\mathbf{Z})$ and", "$\\overline{\\mathcal{M}}_g \\to \\Spec(\\mathbf{Z})$", "are smooth." ], "refs": [], "proofs": [ { "contents": [ "Since $\\overline{\\mathcal{M}}$ is an open substack of", "$\\Curvesstack^{nodal}$ this follows from", "Lemma \\ref{lemma-nodal-curves-smooth}." ], "refs": [ "moduli-curves-lemma-nodal-curves-smooth" ], "ref_ids": [ 1618 ] } ], "ref_ids": [] }, { "id": 1632, "type": "theorem", "label": "moduli-curves-lemma-stable-curves-deligne-mumford", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-stable-curves-deligne-mumford", "contents": [ "The stacks $\\overline{\\mathcal{M}}$ and", "$\\overline{\\mathcal{M}}_g$", "are open substacks of $\\Curvesstack^{DM}$.", "In particular, $\\overline{\\mathcal{M}}$ and", "$\\overline{\\mathcal{M}}_g$ are DM", "(Morphisms of Stacks, Definition", "\\ref{stacks-morphisms-definition-absolute-separated})", "as well as Deligne-Mumford stacks", "(Algebraic Stacks, Definition \\ref{algebraic-definition-deligne-mumford})." ], "refs": [ "stacks-morphisms-definition-absolute-separated", "algebraic-definition-deligne-mumford" ], "proofs": [ { "contents": [ "Proof of the first assertion.", "Let $X$ be a scheme proper over a field $k$ whose singularities", "are at-worst-nodal, $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$,", "the genus of $X$ is $\\geq 2$, and $X$ has no rational tails or bridges.", "We have to show that the classifying morphism", "$\\Spec(k) \\to \\overline{\\mathcal{M}} \\to \\Curvesstack$", "factors through $\\Curvesstack^{DM}$.", "We may first replace $k$ by the algebraic closure", "(since we already know the relevant stacks are open", "substacks of the algebraic stack $\\Curvesstack$).", "By Lemmas \\ref{lemma-stable-curves}, \\ref{lemma-DM-curves}, and", "\\ref{lemma-in-DM-locus-vector-fields} it suffices to show that", "$\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = 0$.", "This is proven in", "Algebraic Curves, Lemma \\ref{curves-lemma-stable-vector-fields}.", "\\medskip\\noindent", "Since $\\Curvesstack^{DM}$ is the maximal open substack of", "$\\Curvesstack$ which is DM, we see this is true also for the", "open substack $\\overline{\\mathcal{M}}$ of $\\Curvesstack^{DM}$.", "Finally, a DM algebraic stack is Deligne-Mumford by", "Morphisms of Stacks, Theorem \\ref{stacks-morphisms-theorem-DM}." ], "refs": [ "moduli-curves-lemma-stable-curves", "moduli-curves-lemma-DM-curves", "moduli-curves-lemma-in-DM-locus-vector-fields", "curves-lemma-stable-vector-fields", "stacks-morphisms-theorem-DM" ], "ref_ids": [ 1629, 1592, 1593, 6349, 7389 ] } ], "ref_ids": [ 7602, 8485 ] }, { "id": 1633, "type": "theorem", "label": "moduli-curves-lemma-smooth-dense-in-stable", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-smooth-dense-in-stable", "contents": [ "Let $g \\geq 2$. The inclusion", "$$", "|\\mathcal{M}_g| \\subset |\\overline{\\mathcal{M}}_g|", "$$", "is that of an open dense subset." ], "refs": [], "proofs": [ { "contents": [ "Since $\\overline{\\mathcal{M}}_g \\subset \\Curvesstack^{lci+}$", "is open and since", "$\\Curvesstack^{smooth} \\cap \\overline{\\mathcal{M}}_g = \\mathcal{M}_g$", "this follows immediately from", "Lemma \\ref{lemma-smooth-dense}." ], "refs": [ "moduli-curves-lemma-smooth-dense" ], "ref_ids": [ 1616 ] } ], "ref_ids": [] }, { "id": 1634, "type": "theorem", "label": "moduli-curves-lemma-contract", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-contract", "contents": [ "Let $S$ be a scheme and $s \\in S$ a point.", "Let $f : X \\to S$ and $g : Y \\to S$ be families of curves.", "Let $c : X \\to Y$ be a morphism over $S$. If", "$c_{s, *}\\mathcal{O}_{X_s} = \\mathcal{O}_{Y_s}$ and", "$R^1c_{s, *}\\mathcal{O}_{X_s} = 0$, then", "after replacing $S$ by an open neighbourhood of $s$", "we have $\\mathcal{O}_Y = c_*\\mathcal{O}_X$ and $R^1c_*\\mathcal{O}_X = 0$", "and this remains true after base change by any morphism $S' \\to S$." ], "refs": [], "proofs": [ { "contents": [ "Let $(U, u) \\to (S, s)$ be an \\'etale neighbourhood such that", "$\\mathcal{O}_{Y_U} = (X_U \\to Y_U)_*\\mathcal{O}_{X_U}$ and", "$R^1(X_U \\to Y_U)_*\\mathcal{O}_{X_U} = 0$ and the same is true", "after base change by $U' \\to U$. Then we replace $S$ by the", "open image of $U \\to S$. Given $S' \\to S$ we set $U' = U \\times_S S'$", "and we obtain \\'etale coverings $\\{U' \\to S'\\}$ and", "$\\{Y_{U'} \\to Y_{S'}\\}$. Thus the truth of the statement for", "the base change of $c$ by $S' \\to S$ follows from the truth", "of the statement for the base change of $X_U \\to Y_U$ by", "$U' \\to U$. In other words, the question is local in the", "\\'etale topology on $S$.", "Thus by Lemma \\ref{lemma-etale-locally-scheme} we may assume", "$X$ and $Y$ are schemes. By", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-h1-fibre-zero-isom}", "there exists an open subscheme $V \\subset Y$ containing $Y_s$", "such that $c_*\\mathcal{O}_X|_V = \\mathcal{O}_V$ and", "$R^1c_*\\mathcal{O}_X|_V = 0$ and such that this remains true after", "any base change by $S' \\to S$. Since $g : Y \\to S$ is proper, we can", "find an open neighbourhood $U \\subset S$ of $s$ such that", "$g^{-1}(U) \\subset V$. Then $U$ works." ], "refs": [ "moduli-curves-lemma-etale-locally-scheme", "more-morphisms-lemma-h1-fibre-zero-isom" ], "ref_ids": [ 1589, 14075 ] } ], "ref_ids": [] }, { "id": 1635, "type": "theorem", "label": "moduli-curves-lemma-contract-basic-uniqueness", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-contract-basic-uniqueness", "contents": [ "Let $S$ be a scheme and $s \\in S$ a point.", "Let $f : X \\to S$ and $g_i : Y_i \\to S$, $i = 1, 2$ be families of curves.", "Let $c_i : X \\to Y_i$ be morphisms over $S$.", "Assume there is an isomorphism $Y_{1, s} \\cong Y_{2, s}$", "of fibres compatible with $c_{1, s}$ and $c_{2, s}$.", "If $c_{1, s, *}\\mathcal{O}_{X_s} = \\mathcal{O}_{Y_{1, s}}$ and", "$R^1c_{1, s, *}\\mathcal{O}_{X_s} = 0$, then there exist an", "open neighbourhood $U$ of $s$ and an isomorphism", "$Y_{1, U} \\cong Y_{2, U}$ of families of curves over $U$", "compatible with the given isomorphism of fibres and with", "$c_1$ and $c_2$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\mathcal{O}_{S, s} = \\colim \\mathcal{O}_S(U)$ where", "the colimit is over the system of affine neighbourhoods $U$ of $s$.", "Thus the category of algebraic spaces of finite presentation over", "the local ring is the colimit of the categories of algebraic spaces", "of finite presentation over the affine neighbourhoods of $s$.", "See Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-finite-presentation}.", "In this way we reduce to the case where $S$ is the spectrum of", "a local ring and $s$ is the closed point.", "\\medskip\\noindent", "Assume $S = \\Spec(A)$ where $A$ is a local ring and $s$ is the closed point.", "Write $A = \\colim A_j$ with $A_j$ local Noetherian (say essentially", "of finite type over $\\mathbf{Z}$) and local transition homomorphisms.", "Set $S_j = \\Spec(A_j)$ with closed point $s_j$. We can find a", "$j$ and families of curves $X_j \\to S_j$, $Y_{j, i} \\to S_j$,", "see Lemma \\ref{lemma-curves-qs-lfp} and", "Limits of Stacks, Lemma", "\\ref{stacks-limits-lemma-representable-by-spaces-limit-preserving}.", "After possibly increasing $j$ we can find morphisms", "$c_{j, i} : X_j \\to Y_{j, i}$ whose base change to $s$ is $c_i$, see", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-finite-presentation}.", "Since $\\kappa(s) = \\colim \\kappa(s_j)$ we can similarly", "assume there is an isomorphism $Y_{j, 1, s_j} \\cong Y_{j, 2, s_j}$", "compatible with $c_{j, 1, s_j}$ and $c_{j, 2, s_j}$.", "Finally, the assumptions", "$c_{1, s, *}\\mathcal{O}_{X_s} = \\mathcal{O}_{Y_{1, s}}$ and", "$R^1c_{1, s, *}\\mathcal{O}_{X_s} = 0$", "are inherited by $c_{j, 1, s_j}$ because", "$\\{s_j \\to s\\}$ is an fpqc covering and", "$c_{1, s}$ is the base of $c_{j, 1, s_j}$ by this covering (details omitted).", "In this way we reduce the lemma to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $S$ is the spectrum of a Noetherian local ring $\\Lambda$ and", "$s$ is the closed point. Consider the scheme theoretic image $Z$ of", "$$", "(c_1, c_2) : X \\longrightarrow Y_1 \\times_S Y_2", "$$", "The statement of the lemma is equivalent to the assertion that $Z$ maps", "isomorphically to $Y_1$ and $Y_2$ via the projection morphisms.", "Since taking the scheme theoretic image of this morphism commutes", "with flat base change (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image},", "we may replace $\\Lambda$ by its completion", "(More on Algebra, Section \\ref{more-algebra-section-permanence-completion}).", "\\medskip\\noindent", "Assume $S$ is the spectrum of a complete Noetherian local ring $\\Lambda$.", "Observe that $X$, $Y_1$, $Y_2$ are schemes in this case", "(More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-projective-over-complete}).", "Denote $X_n$, $Y_{1, n}$, $Y_{2, n}$ the base changes of", "$X$, $Y_1$, $Y_2$ to $\\Spec(\\Lambda/\\mathfrak m^{n + 1})$.", "Recall that the arrow", "$$", "\\Deformationcategory_{X_s \\to Y_{2, s}} \\cong", "\\Deformationcategory_{X_s \\to Y_{1, s}} \\longrightarrow", "\\Deformationcategory_{X_s}", "$$", "is an equivalence, see Deformation Problems, Lemma", "\\ref{examples-defos-lemma-schemes-morphisms-smooth-to-base}.", "Thus there is an isomorphism of formal objects", "$(X_n \\to Y_{1, n}) \\cong (X_n \\to Y_{2, n})$", "of $\\Deformationcategory_{X_s \\to Y_{1, s}}$.", "Finally, by Grothendieck's algebraization theorem", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-algebraize-morphism})", "this produces an isomorphism $Y_1 \\to Y_2$ compatible with $c_1$ and $c_2$." ], "refs": [ "spaces-limits-lemma-descend-finite-presentation", "moduli-curves-lemma-curves-qs-lfp", "stacks-limits-lemma-representable-by-spaces-limit-preserving", "spaces-limits-lemma-descend-finite-presentation", "spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image", "spaces-more-morphisms-lemma-projective-over-complete", "examples-defos-lemma-schemes-morphisms-smooth-to-base", "coherent-lemma-algebraize-morphism" ], "ref_ids": [ 4598, 1591, 15017, 4598, 4861, 213, 8757, 3398 ] } ], "ref_ids": [] }, { "id": 1636, "type": "theorem", "label": "moduli-curves-lemma-contract-basic", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-contract-basic", "contents": [ "Let $f : X \\to S$ be a family of curves. Let $s \\in S$ be a point.", "Let $h_0 : X_s \\to Y_0$ be a morphism to a proper scheme $Y_0$ over $\\kappa(s)$", "such that $h_{0, *}\\mathcal{O}_{X_s} = \\mathcal{O}_{Y_0}$ and", "$R^1h_{0, *}\\mathcal{O}_{X_s} = 0$. Then there exist an elementary", "\\'etale neighbourhood $(U, u) \\to (S, s)$, a family of curves $Y \\to U$,", "and a morphism $h : X_U \\to Y$ over $U$ whose fibre in $u$", "is isomorphic to $h_0$." ], "refs": [], "proofs": [ { "contents": [ "We first do some reductions; we urge the reader to skip ahead.", "The question is local on $S$, hence we may assume $S$ is affine.", "Write $S = \\lim S_i$ as a cofiltered limit of affine schemes $S_i$", "of finite type over $\\mathbf{Z}$.", "For some $i$ we can find a family of curves $X_i \\to S_i$", "whose base change is $X \\to S$. This follows from", "Lemma \\ref{lemma-curves-qs-lfp} and", "Limits of Stacks, Lemma", "\\ref{stacks-limits-lemma-representable-by-spaces-limit-preserving}.", "Let $s_i \\in S_i$ be the image of $s$. Observe that", "$\\kappa(s) = \\colim \\kappa(s_i)$ and that $X_s$ is a scheme", "(Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-codim-1-point-in-schematic-locus}).", "After increasing $i$ we may assume there exists a morphism", "$h_{i, 0} : X_{i, s_i} \\to Y_i$", "of finite type schemes over $\\kappa(s_i)$ whose base change to", "$\\kappa(s)$ is $h_0$, see", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}.", "After increasing $i$ we may assume $Y_i$ is proper over $\\kappa(s_i)$, see", "Limits, Lemma \\ref{limits-lemma-eventually-proper}.", "Let $g_{i, 0} : Y_0 \\to Y_{i, 0}$ be the projection. Observe that", "this is a faithfully flat morphism as the base change of", "$\\Spec(\\kappa(s)) \\to \\Spec(\\kappa(s_i))$.", "By flat base change we have", "$$", "h_{0, *}\\mathcal{O}_{X_s} = g_{i, 0}^*h_{i, 0, *}\\mathcal{O}_{X_{i, s_i}}", "\\quad\\text{and}\\quad", "R^1h_{0, *}\\mathcal{O}_{X_s} = g_{i, 0}^*Rh_{i, 0, *}\\mathcal{O}_{X_{i, s_i}}", "$$", "see Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology}.", "By faithful flatness we see that $X_i \\to S_i$, $s_i \\in S_i$, and", "$X_{i, s_i} \\to Y_i$ satisfies all the assumptions of the lemma.", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $S$ is affine of finite type over $\\mathbf{Z}$.", "Let $\\mathcal{O}_{S, s}^h$ be the henselization of the local", "ring of $S$ at $s$. Observe that $\\mathcal{O}_{S, s}^h$ is a G-ring by", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-G-ring} and", "Proposition \\ref{more-algebra-proposition-ubiquity-G-ring}.", "Suppose we can construct a family of curves", "$Y' \\to \\Spec(\\mathcal{O}_{S, s}^h)$ and a morphism", "$$", "h' : X \\times_S \\Spec(\\mathcal{O}_{S, s}^h) \\longrightarrow Y'", "$$", "over $\\Spec(\\mathcal{O}_{S, s}^h)$ whose base change to the closed", "point is $h_0$. This will be enough. Namely, first we use that", "$$", "\\mathcal{O}_{S, s}^h = \\colim_{(U, u)} \\mathcal{O}_U(U)", "$$", "where the colimit is over the filtered category of ", "elementary \\'etale neighbourhoods (More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-describe-henselization}).", "Next, we use again that given $Y'$ we can descend it to", "$Y \\to U$ for some $U$ (see references given above).", "Then we use", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "to descend $h'$ to some $h$. This reduces us to the case", "discussed in the next paragraph.", "\\medskip\\noindent", "Assume $S = \\Spec(\\Lambda)$ where $(\\Lambda, \\mathfrak m, \\kappa)$", "is a henselian Noetherian local G-ring and $s$ is the closed point of $S$.", "Recall that the map", "$$", "\\Deformationcategory_{X_s \\to Y_0} \\to \\Deformationcategory_{X_s}", "$$", "is an equivalence, see Deformation Problems, Lemma", "\\ref{examples-defos-lemma-schemes-morphisms-smooth-to-base}.", "(This is the only important step in the proof; everything else", "is technique.) Denote $\\Lambda^\\wedge$ the $\\mathfrak m$-adic completion.", "The pullbacks $X_n$ of $X$ to $\\Lambda/\\mathfrak m^{n + 1}$ define", "a formal object $\\xi$ of $\\Deformationcategory_{X_s}$ over $\\Lambda^\\wedge$.", "From the equivalence we obtain a formal object", "$\\xi'$ of $\\Deformationcategory_{X_s \\to Y_0}$ over $\\Lambda^\\wedge$.", "Thus we obtain a huge commutative diagram", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "X_n \\ar[r] \\ar[d] &", "X_{n - 1} \\ar[r] \\ar[d] &", "\\ldots \\ar[r] &", "X_s \\ar[d] \\\\", "\\ldots \\ar[r] &", "Y_n \\ar[r] \\ar[d] &", "Y_{n - 1} \\ar[r] \\ar[d] &", "\\ldots \\ar[r] &", "Y_0 \\ar[d] \\\\", "\\ldots \\ar[r] &", "\\Spec(\\Lambda/\\mathfrak m^{n + 1}) \\ar[r] &", "\\Spec(\\Lambda/\\mathfrak m^n) \\ar[r] &", "\\ldots \\ar[r] &", "\\Spec(\\kappa)", "}", "$$", "The formal object $(Y_n)$ comes from a family of curves", "$Y' \\to \\Spec(\\Lambda^\\wedge)$ by", "Quot, Lemma \\ref{quot-lemma-curves-existence}.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-algebraize-morphism}", "we get a morphism $h' : X_{\\Lambda^\\wedge} \\to Y'$ inducing", "the given morphisms $X_n \\to Y_n$ for all $n$", "and in particular the given morphism $X_s \\to Y_0$.", "\\medskip\\noindent", "To finish we do a standard algebraization/approximation argument.", "First, we observe that we can find a finitely generated $\\Lambda$-subalgebra", "$\\Lambda \\subset A \\subset \\Lambda^\\wedge$, a family of curves", "$Y'' \\to \\Spec(A)$ and a morphism $h'' : X_A \\to Y''$ over $A$", "whose base change to $\\Lambda^\\wedge$ is $h'$.", "This is true because $\\Lambda^\\wedge$ is the filtered colimit", "of these rings $A$ and we can argue as before using", "that $\\Curvesstack$ is locally of finite presentation", "(which gives us $Y''$ over $A$ by", "Limits of Stacks, Lemma", "\\ref{stacks-limits-lemma-representable-by-spaces-limit-preserving})", "and using ", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-descend-finite-presentation}", "to descend $h'$ to some $h''$.", "Then we can apply the approximation property for G-rings", "(in the form of Smoothing Ring Maps, Theorem", "\\ref{smoothing-theorem-approximation-property})", "to find a map $A \\to \\Lambda$ which induces the same map", "$A \\to \\kappa$ as we obtain from $A \\to \\Lambda^\\wedge$.", "Base changing $h''$ to $\\Lambda$ the proof is complete." ], "refs": [ "moduli-curves-lemma-curves-qs-lfp", "stacks-limits-lemma-representable-by-spaces-limit-preserving", "spaces-over-fields-lemma-codim-1-point-in-schematic-locus", "limits-lemma-descend-finite-presentation", "limits-lemma-eventually-proper", "coherent-lemma-flat-base-change-cohomology", "more-algebra-lemma-henselization-G-ring", "more-algebra-proposition-ubiquity-G-ring", "more-morphisms-lemma-describe-henselization", "limits-lemma-descend-finite-presentation", "examples-defos-lemma-schemes-morphisms-smooth-to-base", "quot-lemma-curves-existence", "spaces-more-morphisms-lemma-algebraize-morphism", "stacks-limits-lemma-representable-by-spaces-limit-preserving", "spaces-limits-lemma-descend-finite-presentation", "smoothing-theorem-approximation-property" ], "ref_ids": [ 1591, 15017, 12845, 15077, 15089, 3298, 10089, 10582, 13869, 15077, 8757, 3213, 211, 15017, 4598, 5606 ] } ], "ref_ids": [] }, { "id": 1637, "type": "theorem", "label": "moduli-curves-lemma-contract-prestable-to-stable", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-contract-prestable-to-stable", "contents": [ "Let $f : X \\to S$ be a prestable family of curves of genus $g \\geq 1$.", "There is a factorization $X \\to Y \\to S$ of $f$ where $g : Y \\to S$", "is a stable family of curves and $c : X \\to Y$ has the following", "properties", "\\begin{enumerate}", "\\item $\\mathcal{O}_Y = c_*\\mathcal{O}_X$ and $R^1c_*\\mathcal{O}_X = 0$", "and this remains true after base change by any morphism $S' \\to S$, and", "\\item for any $s \\in S$ the morphism $c_s : X_s \\to Y_s$ is the", "contraction of rational tails and bridges discussed in", "Algebraic Curves, Section \\ref{curves-section-contracting-to-stable}.", "\\end{enumerate}", "Moreover $c : X \\to Y$ is unique up to unique isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Let $s \\in S$. Let $c_0 : X_s \\to Y_0$ be the contraction of", "Algebraic Curves, Section \\ref{curves-section-contracting-to-stable}", "(more precisely Algebraic Curves, Lemma", "\\ref{curves-lemma-characterize-contraction-to-stable}).", "By Lemma \\ref{lemma-contract-basic}", "there exists an elementary \\'etale neighbourhood", "$(U, u)$ and a morphism $c : X_U \\to Y$", "of families of curves over $U$ which recovers", "$c_0$ as the fibre at $u$.", "Since $\\omega_{Y_0}$ is ample, after possibly shrinking $U$,", "we see that $Y \\to U$ is a stable family of genus $g$", "by the openness inherent in", "Lemmas \\ref{lemma-stable-curves} and \\ref{lemma-stable-one-piece-per-genus}.", "After possibly shrinking $U$ once more, assertion (1) of the lemma for", "$c : X_U \\to Y$ follows from Lemma \\ref{lemma-contract}.", "Moreover, part (2) holds by the uniqueness in Algebraic Curves, Lemma", "\\ref{curves-lemma-characterize-contraction-to-stable}.", "We conclude that a morphism $c$ as in the lemma exists \\'etale locally", "on $S$. More precisely, there exists an \\'etale covering", "$\\{U_i \\to S\\}$ and morphisms $c_i : X_{U_i} \\to Y_i$ over $U_i$", "where $Y_i \\to U_i$ is a stable family of curves", "having properties (1) and (2) stated in the lemma.", "\\medskip\\noindent", "To finish the proof it suffices to prove uniqueness of $c : X \\to Y$", "(up to unique isomorphism). Namely, once this is done, then we", "obtain isomorphisms", "$$", "\\varphi_{ij} :", "Y_i \\times_{U_i} (U_i \\times_S U_j)", "\\longrightarrow", "Y_i \\times_{U_j} (U_i \\times_S U_j)", "$$", "satisfying the cocycle condition (by uniqueness) over", "$U_i \\times U_j \\times U_k$. Since $\\overline{\\mathcal{M}_g}$", "is an algebraic stack, we have effectiveness of descent data", "and we obtain $Y \\to S$. The morphisms $c_i$ descend to a morphism", "$c : X \\to Y$ over $S$. Finally, properties (1) and (2) for $c$", "are immediate from properties (1) and (2) for $c_i$.", "\\medskip\\noindent", "Finally, if $c_1 : X \\to Y_i$, $i = 1, 2$ are two morphisms towards", "stably families of curves over $S$ satisfying (1) and (2), then", "we obtain a morphism $Y_1 \\to Y_2$ compatible with $c_1$ and $c_2$", "at least locally on $S$ by Lemma \\ref{lemma-contract-basic}.", "We omit the verification that these morphisms are unique", "(hint: this follows from the fact that the scheme theoretic image", "of $c_1$ is $Y_1$). Hence these locally given morphisms glue", "and the proof is complete." ], "refs": [ "curves-lemma-characterize-contraction-to-stable", "moduli-curves-lemma-contract-basic", "moduli-curves-lemma-stable-curves", "moduli-curves-lemma-stable-one-piece-per-genus", "moduli-curves-lemma-contract", "curves-lemma-characterize-contraction-to-stable", "moduli-curves-lemma-contract-basic" ], "ref_ids": [ 6345, 1636, 1629, 1630, 1634, 6345, 1636 ] } ], "ref_ids": [] }, { "id": 1638, "type": "theorem", "label": "moduli-curves-lemma-stabilization-morphism", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-stabilization-morphism", "contents": [ "Let $g \\geq 2$. There is a morphism of algebraic stacks over $\\mathbf{Z}$", "$$", "stabilization :", "\\Curvesstack^{prestable}_g", "\\longrightarrow", "\\overline{\\mathcal{M}}_g", "$$", "which sends a prestable family of curves $X \\to S$ of genus $g$", "to the stable family $Y \\to S$ asssociated to it in", "Lemma \\ref{lemma-contract-prestable-to-stable}." ], "refs": [ "moduli-curves-lemma-contract-prestable-to-stable" ], "proofs": [ { "contents": [ "To see this is true, it suffices to check that the construction of", "Lemma \\ref{lemma-contract-prestable-to-stable}", "is compatible with base change (and isomorphisms but that's immediate),", "see the (abuse of) language for algebraic stacks introduced", "in Properties of Stacks, Section \\ref{stacks-properties-section-conventions}.", "To see this it suffices to check properties (1) and (2) of", "Lemma \\ref{lemma-contract-prestable-to-stable} are stable", "under base change. This is immediately clear for (1).", "For (2) this follows either from the fact that the contractions of", "Algebraic Curves, Lemmas \\ref{curves-lemma-contracting-rational-tails} and", "\\ref{curves-lemma-contracting-rational-bridges}", "are stable under ground field extensions, or", "because the conditions characterizing the morphisms on fibres in", "Algebraic Curves, Lemma \\ref{curves-lemma-characterize-contraction-to-stable}", "are preserved under ground field extensions." ], "refs": [ "moduli-curves-lemma-contract-prestable-to-stable", "moduli-curves-lemma-contract-prestable-to-stable", "curves-lemma-contracting-rational-tails", "curves-lemma-contracting-rational-bridges", "curves-lemma-characterize-contraction-to-stable" ], "ref_ids": [ 1637, 1637, 6338, 6343, 6345 ] } ], "ref_ids": [ 1637 ] }, { "id": 1639, "type": "theorem", "label": "moduli-curves-lemma-stable-reduction", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-stable-reduction", "contents": [ "Let $R$ be a discrete valuation ring with fraction field $K$.", "Let $C$ be a smooth projective curve over $K$ with", "$K = H^0(C, \\mathcal{O}_C)$ having genus $g \\geq 2$.", "The following are equivalent", "\\begin{enumerate}", "\\item $C$ has semistable reduction", "(Semistable Reduction, Definition \\ref{models-definition-semistable}), or", "\\item there is a stable family of curves over $R$ with generic fibre $C$.", "\\end{enumerate}" ], "refs": [ "models-definition-semistable" ], "proofs": [ { "contents": [ "Since a stable family of curves is also prestable, it is immediate that", "(2) implies (1). Conversely, given a prestable family of curves over $R$", "with generic fibre $C$, we can contract it to a stable family of curves", "by Lemma \\ref{lemma-contract-prestable-to-stable}. Since the generic", "fibre already is stable, it does not get changed by this procedure and", "the proof is complete." ], "refs": [ "moduli-curves-lemma-contract-prestable-to-stable" ], "ref_ids": [ 1637 ] } ], "ref_ids": [ 9279 ] }, { "id": 1640, "type": "theorem", "label": "moduli-curves-lemma-unique-stable-model", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-unique-stable-model", "contents": [ "Let $R$ be a discrete valuation ring with fraction field $K$.", "Let $C$ be a smooth proper curve over $K$", "with $K = H^0(C, \\mathcal{O}_C)$ and genus $g$.", "If $X$ and $X'$ are models of $C$", "(Semistable Reduction, Section \\ref{models-section-models})", "and $X$ and $X'$ are stable families of genus $g$ curves over $R$,", "then there exists an unique isomorphism $X \\to X'$ of models." ], "refs": [], "proofs": [ { "contents": [ "Let $Y$ be the minimal model for $C$. Recall that $Y$ exists, is", "unique, and is at-worst-nodal of relative dimension $1$ over $R$, see", "Semistable Reduction,", "Proposition \\ref{models-proposition-exists-minimal-model} and", "Lemmas \\ref{models-lemma-minimal-model-unique} and", "\\ref{models-lemma-semistable} (applies because we have $X$).", "There is a contraction morphism", "$$", "Y \\longrightarrow Z", "$$", "such that $Z$ is a stable family of curves of genus $g$ over $R$", "(Lemma \\ref{lemma-contract-prestable-to-stable}). We claim", "there is a unique isomorphism of models $X \\to Z$.", "By symmetry the same is true for $X'$ and this will finish the proof.", "\\medskip\\noindent", "By Semistable Reduction, Lemma \\ref{models-lemma-blowup-at-worst-nodal}", "there exists a sequence", "$$", "X_m \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "such that $X_{i + 1} \\to X_i$ is the blowing up of a closed point", "$x_i$ where $X_i$ is singular, $X_i \\to \\Spec(R)$ is at-worst-nodal", "of relative dimension $1$, and $X_m$ is regular.", "By Semistable Reduction, Lemma \\ref{models-lemma-pre-exists-minimal-model}", "there is a sequence", "$$", "X_m = Y_n \\to Y_{n - 1} \\to \\ldots \\to Y_1 \\to Y_0 = Y", "$$", "of proper regular models of $C$, such that each morphism is a", "contraction of an exceptional curve of the first kind\\footnote{In fact", "we have $X_m = Y$, i.e., $X_m$ does not contain any exceptional curves", "of the first kind. We encourage the reader to think this through", "as it simplifies the proof somewhat.}.", "By Semistable Reduction, Lemma \\ref{models-lemma-blowdown-at-worst-nodal}", "each $Y_i$ is at-worst-nodal of relative dimension $1$ over $R$.", "To prove the claim it suffices to show that there is an isomorphism", "$X \\to Z$ compatible with the morphisms $X_m \\to X$", "and $X_m = Y_n \\to Y \\to Z$. Let $s \\in \\Spec(R)$ be the closed point.", "By either", "Lemma \\ref{lemma-contract-basic-uniqueness} or", "Lemma \\ref{lemma-contract-prestable-to-stable}", "we reduce to proving that the morphisms", "$X_{m, s} \\to X_s$ and", "$X_{m, s} \\to Z_s$", "are both equal to the canonical morphism of", "Algebraic Curves, Lemma \\ref{curves-lemma-characterize-contraction-to-stable}.", "\\medskip\\noindent", "For a morphism $c : U \\to V$ of schemes over $\\kappa(s)$", "we say $c$ has property (*) if $\\dim(U_v) \\leq 1$ for $v \\in V$,", "$\\mathcal{O}_V = c_*\\mathcal{O}_U$, and $R^1c_*\\mathcal{O}_U = 0$.", "This property is stable under composition.", "Since both $X_s$ and $Z_s$ are stable genus $g$ curves over $\\kappa(s)$,", "it suffices to show that each of the morphisms $Y_s \\to Z_s$,", "$X_{i + 1, s} \\to X_{i, s}$, and $Y_{i + 1, s} \\to Y_{i, s}$,", "satisfy property (*), see", "Algebraic Curves, Lemma \\ref{curves-lemma-characterize-contraction-to-stable}.", "\\medskip\\noindent", "Property (*) holds for $Y_s \\to Z_s$ by construction.", "\\medskip\\noindent", "The morphisms $c : X_{i + 1, s} \\to X_{i, s}$ are constructed and studied", "in the proof of", "Semistable Reduction, Lemma \\ref{models-lemma-blowup-at-worst-nodal}.", "It suffices to check (*) \\'etale locally on $X_{i, s}$.", "Hence it suffices to check (*) for the base change of the morphism", "``$X_1 \\to X_0$'' in Semistable Reduction, Example \\ref{models-example-blowup}", "to $R/\\pi R$.", "We leave the explicit calculation to the reader.", "\\medskip\\noindent", "The morphism $c : Y_{i + 1, s} \\to Y_{i, s}$ is the restriction", "of the blow down of an exceptional curve $E \\subset Y_{i + 1}$", "of the first kind, i.e., $b : Y_{i + 1} \\to Y_i$ is a contraction of $E$,", "i.e., $b$ is a blowing up of a regular point on the surface $Y_i$", "(Resolution of Surfaces, Section \\ref{resolve-section-minus-one}).", "Then $\\mathcal{O}_{Y_i} = b_*\\mathcal{O}_{Y_{i + 1}}$ and", "$R^1b_*\\mathcal{O}_{Y_{i + 1}} = 0$, see for example", "Resolution of Surfaces, Lemma", "\\ref{resolve-lemma-cohomology-of-blowup}.", "We conclude that $\\mathcal{O}_{Y_{i, s}} = c_*\\mathcal{O}_{Y_{i + 1, s}}$", "and $R^1c_*\\mathcal{O}_{Y_{i + 1, s}} = 0$ by", "More on Morphisms, Lemmas \\ref{more-morphisms-lemma-check-h1-fibre-zero},", "\\ref{more-morphisms-lemma-h1-fibre-zero}, and", "\\ref{more-morphisms-lemma-h1-fibre-zero-check-h0-kappa}", "(only gives surjectivity of", "$\\mathcal{O}_{Y_{i, s}} \\to c_*\\mathcal{O}_{Y_{i + 1, s}}$ but", "injectivity follows easily from the fact that $Y_{i, s}$ is reduced", "and $c$ changes things only over one closed point).", "This finishes the proof." ], "refs": [ "models-proposition-exists-minimal-model", "models-lemma-minimal-model-unique", "models-lemma-semistable", "moduli-curves-lemma-contract-prestable-to-stable", "models-lemma-blowup-at-worst-nodal", "models-lemma-pre-exists-minimal-model", "models-lemma-blowdown-at-worst-nodal", "moduli-curves-lemma-contract-basic-uniqueness", "moduli-curves-lemma-contract-prestable-to-stable", "curves-lemma-characterize-contraction-to-stable", "curves-lemma-characterize-contraction-to-stable", "models-lemma-blowup-at-worst-nodal", "resolve-lemma-cohomology-of-blowup", "more-morphisms-lemma-check-h1-fibre-zero", "more-morphisms-lemma-h1-fibre-zero", "more-morphisms-lemma-h1-fibre-zero-check-h0-kappa" ], "ref_ids": [ 9268, 9245, 9264, 1637, 9262, 9234, 9263, 1635, 1637, 6345, 6345, 9262, 11642, 14069, 14070, 14072 ] } ], "ref_ids": [] }, { "id": 1641, "type": "theorem", "label": "moduli-curves-lemma-stable-separated", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-stable-separated", "contents": [ "Let $g \\geq 2$. The stack $\\overline{\\mathcal{M}}_g$ is separated." ], "refs": [], "proofs": [ { "contents": [ "The statement means that the morphism", "$\\overline{\\mathcal{M}}_g \\to \\Spec(\\mathbf{Z})$ is separated.", "We will prove this using the refined Noetherian valuative criterion", "as stated in ", "More on Morphisms of Stacks, Lemma", "\\ref{stacks-more-morphisms-lemma-refined-valuative-criterion-separated}", "\\medskip\\noindent", "Since $\\overline{\\mathcal{M}}_g$ is an open substack of", "$\\Curvesstack$, we see $\\overline{\\mathcal{M}}_g \\to \\Spec(\\mathbf{Z})$", "is quasi-separated and", "locally of finite presentation by Lemma \\ref{lemma-curves-qs-lfp}.", "In particular the stack $\\overline{\\mathcal{M}}_g$ is locally", "Noetherian (Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-locally-finite-type-locally-noetherian}).", "By Lemma \\ref{lemma-smooth-dense-in-stable} the open immersion", "$\\mathcal{M}_g \\to \\overline{\\mathcal{M}}_g$", "has dense image. Also, $\\mathcal{M}_g \\to \\overline{\\mathcal{M}}_g$", "is quasi-compact (Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-locally-closed-in-noetherian}),", "hence of finite type. Thus all the preliminary assumptions", "of More on Morphisms of Stacks, Lemma", "\\ref{stacks-more-morphisms-lemma-refined-valuative-criterion-separated}", "are satisfied for the morphisms", "$$", "\\mathcal{M}_g \\to \\overline{\\mathcal{M}}_g", "\\quad\\text{and}\\quad", "\\overline{\\mathcal{M}}_g \\to \\Spec(\\mathbf{Z})", "$$", "and it suffices to check the following: given any $2$-commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] &", "\\mathcal{M}_g \\ar[r] &", "\\overline{\\mathcal{M}}_g \\ar[d] \\\\", "\\Spec(R) \\ar[rr] \\ar@{..>}[rru] & & \\Spec(\\mathbf{Z})", "}", "$$", "where $R$ is a discrete valuation ring with field of fractions $K$", "the category of dotted arrows is either empty or a setoid with exactly", "one isomorphism class. (Observe that we don't need to worry about", "$2$-arrows too much, see Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-cat-dotted-arrows-independent}).", "Unwinding what this means using that", "$\\mathcal{M}_g$, resp.\\ $\\overline{\\mathcal{M}}_g$ are", "the algebraic stacks parametrizing smooth, resp.\\ stable families", "of genus $g$ curves, we find that what we have to prove", "is exactly the uniqueness result stated and proved in", "Lemma \\ref{lemma-unique-stable-model}." ], "refs": [ "stacks-more-morphisms-lemma-refined-valuative-criterion-separated", "moduli-curves-lemma-curves-qs-lfp", "stacks-morphisms-lemma-locally-finite-type-locally-noetherian", "moduli-curves-lemma-smooth-dense-in-stable", "stacks-morphisms-lemma-locally-closed-in-noetherian", "stacks-more-morphisms-lemma-refined-valuative-criterion-separated", "stacks-morphisms-lemma-cat-dotted-arrows-independent", "moduli-curves-lemma-unique-stable-model" ], "ref_ids": [ 6912, 1591, 7462, 1633, 7431, 6912, 7572, 1640 ] } ], "ref_ids": [] }, { "id": 1642, "type": "theorem", "label": "moduli-curves-lemma-stable-quasi-compact", "categories": [ "moduli-curves" ], "title": "moduli-curves-lemma-stable-quasi-compact", "contents": [ "Let $g \\geq 2$. The stack $\\overline{\\mathcal{M}}_g$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "We will use the notation from Section \\ref{section-polarized-curves}.", "Consider the subset", "$$", "T \\subset |\\textit{PolarizedCurves}|", "$$", "of points $\\xi$ such that there exists a field $k$ and a pair", "$(X, \\mathcal{L})$ over $k$ representing $\\xi$", "with the following two properties", "\\begin{enumerate}", "\\item $X$ is a stable genus $g$ curve, and", "\\item $\\mathcal{L} = \\omega_X^{\\otimes 3}$.", "\\end{enumerate}", "Clearly, under the continuous map", "$$", "|\\textit{PolarizedCurves}|", "\\longrightarrow", "|\\Curvesstack|", "$$", "the image of the set $T$ is exactly the open subset", "$$", "|\\overline{\\mathcal{M}}_g| \\subset |\\Curvesstack|", "$$", "Thus it suffices to show that $T$ is quasi-compact.", "By Lemma \\ref{lemma-polarized-curves-in-polarized} we see that", "$$", "|\\textit{PolarizedCurves}| \\subset |\\Polarizedstack|", "$$", "is an open and closed immersion. Thus it suffices to", "prove quasi-compactness of $T$ as a subset of", "$|\\Polarizedstack|$. For this we use the criterion of", "Moduli Stacks, Lemma \\ref{moduli-lemma-bounded-polarized}.", "First, we observe that for $(X, \\mathcal{L})$", "as above the Hilbert polynomial $P$ is the function", "$P(t) = (6g - 6)t + (1 - g)$ by Riemann-Roch, see", "Algebraic Curves, Lemma \\ref{curves-lemma-rr}.", "Next, we observe that $H^1(X, \\mathcal{L}) = 0$", "and $\\mathcal{L}$ is very ample by", "Algebraic Curves, Lemma \\ref{curves-lemma-tricanonical}.", "This means exactly that with $n = P(3) - 1$", "there is a closed immersion", "$$", "i : X \\longrightarrow \\mathbf{P}^n_k", "$$", "such that $\\mathcal{L} = i^*\\mathcal{O}_{\\mathbf{P}^1_k}(1)$", "as desired." ], "refs": [ "moduli-curves-lemma-polarized-curves-in-polarized", "moduli-lemma-bounded-polarized", "curves-lemma-rr", "curves-lemma-tricanonical" ], "ref_ids": [ 1587, 1751, 6256, 6346 ] } ], "ref_ids": [] }, { "id": 1650, "type": "theorem", "label": "dpa-lemma-silly", "categories": [ "dpa" ], "title": "dpa-lemma-silly", "contents": [ "Let $A$ be a ring. Let $I$ be an ideal of $A$.", "\\begin{enumerate}", "\\item If $\\gamma$ is a divided power structure\\footnote{Here", "and in the following, $\\gamma$ stands short for a sequence", "of maps $\\gamma_1, \\gamma_2, \\gamma_3, \\ldots$ from $I$ to $I$.}", "on $I$, then", "$n! \\gamma_n(x) = x^n$ for $n \\geq 1$, $x \\in I$.", "\\end{enumerate}", "Assume $A$ is torsion free as a $\\mathbf{Z}$-module.", "\\begin{enumerate}", "\\item[(2)] A divided power structure on $I$, if it exists, is unique.", "\\item[(3)] If $\\gamma_n : I \\to I$ are maps then", "$$", "\\gamma\\text{ is a divided power structure}", "\\Leftrightarrow", "n! \\gamma_n(x) = x^n\\ \\forall x \\in I, n \\geq 1.", "$$", "\\item[(4)] The ideal $I$ has a divided power structure", "if and only if there exists", "a set of generators $x_i$ of $I$ as an ideal such that", "for all $n \\geq 1$ we have $x_i^n \\in (n!)I$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). If $\\gamma$ is a divided power structure, then condition", "(2) (applied to $1$ and $n-1$ instead of $n$ and $m$)", "implies that $n \\gamma_n(x) = \\gamma_1(x)\\gamma_{n - 1}(x)$. Hence", "by induction and condition (1) we get $n! \\gamma_n(x) = x^n$.", "\\medskip\\noindent", "Assume $A$ is torsion free as a $\\mathbf{Z}$-module.", "Proof of (2). This is clear from (1).", "\\medskip\\noindent", "Proof of (3). Assume that $n! \\gamma_n(x) = x^n$ for all $x \\in I$ and", "$n \\geq 1$. Since $A \\subset A \\otimes_{\\mathbf{Z}} \\mathbf{Q}$ it suffices", "to prove the axioms (1) -- (5) of Definition", "\\ref{definition-divided-powers} in case $A$ is a $\\mathbf{Q}$-algebra.", "In this case $\\gamma_n(x) = x^n/n!$ and it is straightforward", "to verify (1) -- (5); for example, (4) corresponds to the binomial", "formula", "$$", "(x + y)^n = \\sum_{i = 0, \\ldots, n} \\frac{n!}{i!(n - i)!} x^iy^{n - i}", "$$", "We encourage the reader to do the verifications", "to make sure that we have the coefficients correct.", "\\medskip\\noindent", "Proof of (4). Assume we have generators $x_i$ of $I$ as an ideal", "such that $x_i^n \\in (n!)I$ for all $n \\geq 1$. We claim that", "for all $x \\in I$ we have $x^n \\in (n!)I$. If the claim holds then", "we can set $\\gamma_n(x) = x^n/n!$ which is a divided power structure by (3).", "To prove the claim we note that it holds for $x = ax_i$. Hence we see", "that the claim holds for a set of generators of $I$ as an abelian group.", "By induction on the length of an expression in terms of these, it suffices", "to prove the claim for $x + y$ if it holds for $x$ and $y$. This", "follows immediately from the binomial theorem." ], "refs": [ "dpa-definition-divided-powers" ], "ref_ids": [ 1696 ] } ], "ref_ids": [] }, { "id": 1651, "type": "theorem", "label": "dpa-lemma-check-on-generators", "categories": [ "dpa" ], "title": "dpa-lemma-check-on-generators", "contents": [ "Let $A$ be a ring. Let $I$ be an ideal of $A$. Let $\\gamma_n : I \\to I$,", "$n \\geq 1$ be a sequence of maps. Assume", "\\begin{enumerate}", "\\item[(a)] (1), (3), and (4) of Definition \\ref{definition-divided-powers}", "hold for all $x, y \\in I$, and", "\\item[(b)] properties (2) and (5) hold for $x$ in", "some set of generators of $I$ as an ideal.", "\\end{enumerate}", "Then $\\gamma$ is a divided power structure on $I$." ], "refs": [ "dpa-definition-divided-powers" ], "proofs": [ { "contents": [ "The numbers (1), (2), (3), (4), (5) in this proof refer to the", "conditions listed in Definition \\ref{definition-divided-powers}.", "Applying (3) we see that if (2) and (5) hold for $x$ then (2) and (5)", "hold for $ax$ for all $a \\in A$. Hence we see (b) implies", "(2) and (5) hold for a set of generators", "of $I$ as an abelian group. Hence, by induction of the length", "of an expression in terms of these it suffices to prove that, given", "$x, y \\in I$ such that (2) and (5) hold for $x$ and $y$, then (2) and (5) hold", "for $x + y$.", "\\medskip\\noindent", "Proof of (2) for $x + y$. By (4) we have", "$$", "\\gamma_n(x + y)\\gamma_m(x + y) =", "\\sum\\nolimits_{i + j = n,\\ k + l = m}", "\\gamma_i(x)\\gamma_k(x)\\gamma_j(y)\\gamma_l(y)", "$$", "Using (2) for $x$ and $y$ this equals", "$$", "\\sum \\frac{(i + k)!}{i!k!}\\frac{(j + l)!}{j!l!}", "\\gamma_{i + k}(x)\\gamma_{j + l}(y)", "$$", "Comparing this with the expansion", "$$", "\\gamma_{n + m}(x + y) = \\sum \\gamma_a(x)\\gamma_b(y)", "$$", "we see that we have to prove that given $a + b = n + m$ we have", "$$", "\\sum\\nolimits_{i + k = a,\\ j + l = b,\\ i + j = n,\\ k + l = m}", "\\frac{(i + k)!}{i!k!}\\frac{(j + l)!}{j!l!}", "=", "\\frac{(n + m)!}{n!m!}.", "$$", "Instead of arguing this directly, we note that the result is true", "for the ideal $I = (x, y)$ in the polynomial ring $\\mathbf{Q}[x, y]$", "because $\\gamma_n(f) = f^n/n!$, $f \\in I$ defines a divided power", "structure on $I$. Hence the equality of rational numbers above is true.", "\\medskip\\noindent", "Proof of (5) for $x + y$ given that (1) -- (4) hold and that (5)", "holds for $x$ and $y$. We will again reduce the proof to an equality", "of rational numbers. Namely, using (4) we can write", "$\\gamma_n(\\gamma_m(x + y)) = \\gamma_n(\\sum \\gamma_i(x)\\gamma_j(y))$.", "Using (4) we can write", "$\\gamma_n(\\gamma_m(x + y))$ as a sum of terms which are products of", "factors of the form $\\gamma_k(\\gamma_i(x)\\gamma_j(y))$.", "If $i > 0$ then", "\\begin{align*}", "\\gamma_k(\\gamma_i(x)\\gamma_j(y)) & =", "\\gamma_j(y)^k\\gamma_k(\\gamma_i(x)) \\\\", "& = \\frac{(ki)!}{k!(i!)^k} \\gamma_j(y)^k \\gamma_{ki}(x) \\\\", "& =", "\\frac{(ki)!}{k!(i!)^k} \\frac{(kj)!}{(j!)^k} \\gamma_{ki}(x) \\gamma_{kj}(y)", "\\end{align*}", "using (3) in the first equality, (5) for $x$ in the second, and", "(2) exactly $k$ times in the third. Using (5) for $y$ we see the", "same equality holds when $i = 0$. Continuing like this using all", "axioms but (5) we see that we can write", "$$", "\\gamma_n(\\gamma_m(x + y)) =", "\\sum\\nolimits_{i + j = nm} c_{ij}\\gamma_i(x)\\gamma_j(y)", "$$", "for certain universal constants $c_{ij} \\in \\mathbf{Z}$. Again the fact", "that the equality is valid in the polynomial ring $\\mathbf{Q}[x, y]$", "implies that the coefficients $c_{ij}$ are all equal to $(nm)!/n!(m!)^n$", "as desired." ], "refs": [ "dpa-definition-divided-powers" ], "ref_ids": [ 1696 ] } ], "ref_ids": [ 1696 ] }, { "id": 1652, "type": "theorem", "label": "dpa-lemma-two-ideals", "categories": [ "dpa" ], "title": "dpa-lemma-two-ideals", "contents": [ "Let $A$ be a ring with two ideals $I, J \\subset A$.", "Let $\\gamma$ be a divided power structure on $I$ and let", "$\\delta$ be a divided power structure on $J$.", "Then", "\\begin{enumerate}", "\\item $\\gamma$ and $\\delta$ agree on $IJ$,", "\\item if $\\gamma$ and $\\delta$ agree on $I \\cap J$ then they are", "the restriction of a unique divided power structure $\\epsilon$", "on $I + J$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in I$ and $y \\in J$. Then", "$$", "\\gamma_n(xy) = y^n\\gamma_n(x) = n! \\delta_n(y) \\gamma_n(x) =", "\\delta_n(y) x^n = \\delta_n(xy).", "$$", "Hence $\\gamma$ and $\\delta$ agree on a set of (additive) generators", "of $IJ$. By property (4) of Definition \\ref{definition-divided-powers}", "it follows that they agree on all of $IJ$.", "\\medskip\\noindent", "Assume $\\gamma$ and $\\delta$ agree on $I \\cap J$.", "Let $z \\in I + J$. Write $z = x + y$ with $x \\in I$ and $y \\in J$.", "Then we set", "$$", "\\epsilon_n(z) = \\sum \\gamma_i(x)\\delta_{n - i}(y)", "$$", "for all $n \\geq 1$.", "To see that this is well defined, suppose that $z = x' + y'$ is another", "representation with $x' \\in I$ and $y' \\in J$. Then", "$w = x - x' = y' - y \\in I \\cap J$. Hence", "\\begin{align*}", "\\sum\\nolimits_{i + j = n} \\gamma_i(x)\\delta_j(y)", "& =", "\\sum\\nolimits_{i + j = n} \\gamma_i(x' + w)\\delta_j(y) \\\\", "& =", "\\sum\\nolimits_{i' + l + j = n} \\gamma_{i'}(x')\\gamma_l(w)\\delta_j(y) \\\\", "& =", "\\sum\\nolimits_{i' + l + j = n} \\gamma_{i'}(x')\\delta_l(w)\\delta_j(y) \\\\", "& =", "\\sum\\nolimits_{i' + j' = n} \\gamma_{i'}(x')\\delta_{j'}(y + w) \\\\", "& =", "\\sum\\nolimits_{i' + j' = n} \\gamma_{i'}(x')\\delta_{j'}(y')", "\\end{align*}", "as desired. Hence, we have defined maps", "$\\epsilon_n : I + J \\to I + J$ for all $n \\geq 1$; it is easy", "to see that $\\epsilon_n \\mid_{I} = \\gamma_n$ and", "$\\epsilon_n \\mid_{J} = \\delta_n$.", "Next, we prove conditions (1) -- (5) of", "Definition \\ref{definition-divided-powers} for the collection", "of maps $\\epsilon_n$.", "Properties (1) and (3) are clear. To see (4), suppose", "that $z = x + y$ and $z' = x' + y'$ with $x, x' \\in I$ and $y, y' \\in J$", "and compute", "\\begin{align*}", "\\epsilon_n(z + z') & =", "\\sum\\nolimits_{a + b = n} \\gamma_a(x + x')\\delta_b(y + y') \\\\", "& =", "\\sum\\nolimits_{i + i' + j + j' = n}", "\\gamma_i(x) \\gamma_{i'}(x')\\delta_j(y)\\delta_{j'}(y') \\\\", "& =", "\\sum\\nolimits_{k = 0, \\ldots, n}", "\\sum\\nolimits_{i+j=k} \\gamma_i(x)\\delta_j(y)", "\\sum\\nolimits_{i'+j'=n-k} \\gamma_{i'}(x')\\delta_{j'}(y') \\\\", "& =", "\\sum\\nolimits_{k = 0, \\ldots, n}\\epsilon_k(z)\\epsilon_{n-k}(z')", "\\end{align*}", "as desired. Now we see that it suffices to prove (2) and (5) for", "elements of $I$ or $J$, see Lemma \\ref{lemma-check-on-generators}.", "This is clear because $\\gamma$ and $\\delta$ are divided power", "structures.", "\\medskip\\noindent", "The existence of a divided power structure $\\epsilon$ on $I+J$", "whose restrictions to $I$ and $J$ are $\\gamma$ and $\\delta$ is", "thus proven; its uniqueness is rather clear." ], "refs": [ "dpa-definition-divided-powers", "dpa-definition-divided-powers", "dpa-lemma-check-on-generators" ], "ref_ids": [ 1696, 1696, 1651 ] } ], "ref_ids": [] }, { "id": 1653, "type": "theorem", "label": "dpa-lemma-nil", "categories": [ "dpa" ], "title": "dpa-lemma-nil", "contents": [ "Let $p$ be a prime number. Let $A$ be a ring, let $I \\subset A$ be an ideal,", "and let $\\gamma$ be a divided power structure on $I$. Assume $p$ is nilpotent", "in $A/I$. Then $I$ is locally nilpotent if and only if $p$ is nilpotent in $A$." ], "refs": [], "proofs": [ { "contents": [ "If $p^N = 0$ in $A$, then for $x \\in I$ we have", "$x^{pN} = (pN)!\\gamma_{pN}(x) = 0$ because $(pN)!$ is", "divisible by $p^N$. Conversely, assume $I$ is locally nilpotent.", "We've also assumed that $p$ is nilpotent in $A/I$, hence", "$p^r \\in I$ for some $r$, hence $p^r$ nilpotent, hence $p$ nilpotent." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1654, "type": "theorem", "label": "dpa-lemma-limits", "categories": [ "dpa" ], "title": "dpa-lemma-limits", "contents": [ "The category of divided power rings has all limits and they agree with", "limits in the category of rings." ], "refs": [], "proofs": [ { "contents": [ "The empty limit is the zero ring (that's weird but we need it).", "The product of a collection of divided power rings $(A_t, I_t, \\gamma_t)$,", "$t \\in T$ is given by $(\\prod A_t, \\prod I_t, \\gamma)$ where", "$\\gamma_n((x_t)) = (\\gamma_{t, n}(x_t))$.", "The equalizer of $\\alpha, \\beta : (A, I, \\gamma) \\to (B, J, \\delta)$", "is just $C = \\{a \\in A \\mid \\alpha(a) = \\beta(a)\\}$ with ideal $C \\cap I$", "and induced divided powers. It follows that all limits exist, see", "Categories, Lemma \\ref{categories-lemma-limits-products-equalizers}." ], "refs": [ "categories-lemma-limits-products-equalizers" ], "ref_ids": [ 12213 ] } ], "ref_ids": [] }, { "id": 1655, "type": "theorem", "label": "dpa-lemma-a-version-of-brown", "categories": [ "dpa" ], "title": "dpa-lemma-a-version-of-brown", "contents": [ "Let $\\mathcal{C}$ be the category of divided power rings. Let", "$F : \\mathcal{C} \\to \\textit{Sets}$ be a functor.", "Assume that", "\\begin{enumerate}", "\\item there exists a cardinal $\\kappa$ such that for every", "$f \\in F(A, I, \\gamma)$ there exists a morphism", "$(A', I', \\gamma') \\to (A, I, \\gamma)$ of $\\mathcal{C}$ such that $f$", "is the image of $f' \\in F(A', I', \\gamma')$ and $|A'| \\leq \\kappa$, and", "\\item $F$ commutes with limits.", "\\end{enumerate}", "Then $F$ is representable, i.e., there exists an object $(B, J, \\delta)$", "of $\\mathcal{C}$ such that", "$$", "F(A, I, \\gamma) = \\Hom_\\mathcal{C}((B, J, \\delta), (A, I, \\gamma))", "$$", "functorially in $(A, I, \\gamma)$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Categories, Lemma \\ref{categories-lemma-a-version-of-brown}." ], "refs": [ "categories-lemma-a-version-of-brown" ], "ref_ids": [ 12253 ] } ], "ref_ids": [] }, { "id": 1656, "type": "theorem", "label": "dpa-lemma-colimits", "categories": [ "dpa" ], "title": "dpa-lemma-colimits", "contents": [ "The category of divided power rings has all colimits." ], "refs": [], "proofs": [ { "contents": [ "The empty colimit is $\\mathbf{Z}$ with divided power ideal $(0)$.", "Let's discuss general colimits. Let $\\mathcal{C}$ be a category and let", "$c \\mapsto (A_c, I_c, \\gamma_c)$ be a diagram. Consider the functor", "$$", "F(B, J, \\delta) = \\lim_{c \\in \\mathcal{C}}", "Hom((A_c, I_c, \\gamma_c), (B, J, \\delta))", "$$", "Note that any $f = (f_c)_{c \\in C} \\in F(B, J, \\delta)$ has the property", "that all the images $f_c(A_c)$ generate a subring $B'$ of $B$ of bounded", "cardinality $\\kappa$ and that all the images $f_c(I_c)$ generate a", "divided power sub ideal $J'$ of $B'$. And we get a factorization of", "$f$ as a $f'$ in $F(B')$ followed by the inclusion $B' \\to B$. Also,", "$F$ commutes with limits. Hence we may apply", "Lemma \\ref{lemma-a-version-of-brown}", "to see that $F$ is representable and we win." ], "refs": [ "dpa-lemma-a-version-of-brown" ], "ref_ids": [ 1655 ] } ], "ref_ids": [] }, { "id": 1657, "type": "theorem", "label": "dpa-lemma-gamma-extends", "categories": [ "dpa" ], "title": "dpa-lemma-gamma-extends", "contents": [ "Let $(A, I, \\gamma)$ be a divided power ring.", "Let $A \\to B$ be a ring map.", "If $\\gamma$ extends to $B$ then it extends uniquely.", "Assume (at least) one of the following conditions holds", "\\begin{enumerate}", "\\item $IB = 0$,", "\\item $I$ is principal, or", "\\item $A \\to B$ is flat.", "\\end{enumerate}", "Then $\\gamma$ extends to $B$." ], "refs": [], "proofs": [ { "contents": [ "Any element of $IB$ can be written as a finite sum", "$\\sum\\nolimits_{i=1}^t b_ix_i$ with", "$b_i \\in B$ and $x_i \\in I$. If $\\gamma$ extends to $\\bar\\gamma$ on $IB$", "then $\\bar\\gamma_n(x_i) = \\gamma_n(x_i)$.", "Thus, conditions (3) and (4) in", "Definition \\ref{definition-divided-powers} imply that", "$$", "\\bar\\gamma_n(\\sum\\nolimits_{i=1}^t b_ix_i) =", "\\sum\\nolimits_{n_1 + \\ldots + n_t = n}", "\\prod\\nolimits_{i = 1}^t b_i^{n_i}\\gamma_{n_i}(x_i)", "$$", "Thus we see that $\\bar\\gamma$ is unique if it exists.", "\\medskip\\noindent", "If $IB = 0$ then setting $\\bar\\gamma_n(0) = 0$ works. If $I = (x)$", "then we define $\\bar\\gamma_n(bx) = b^n\\gamma_n(x)$. This is well defined:", "if $b'x = bx$, i.e., $(b - b')x = 0$ then", "\\begin{align*}", "b^n\\gamma_n(x) - (b')^n\\gamma_n(x)", "& =", "(b^n - (b')^n)\\gamma_n(x) \\\\", "& =", "(b^{n - 1} + \\ldots + (b')^{n - 1})(b - b')\\gamma_n(x) = 0", "\\end{align*}", "because $\\gamma_n(x)$ is divisible by $x$ (since", "$\\gamma_n(I) \\subset I$) and hence annihilated by $b - b'$.", "Next, we prove conditions (1) -- (5) of", "Definition \\ref{definition-divided-powers}.", "Parts (1), (2), (3), (5) are obvious from the construction.", "For (4) suppose that $y, z \\in IB$, say $y = bx$ and $z = cx$. Then", "$y + z = (b + c)x$ hence", "\\begin{align*}", "\\bar\\gamma_n(y + z)", "& =", "(b + c)^n\\gamma_n(x) \\\\", "& =", "\\sum \\frac{n!}{i!(n - i)!}b^ic^{n -i}\\gamma_n(x) \\\\", "& =", "\\sum b^ic^{n - i}\\gamma_i(x)\\gamma_{n - i}(x) \\\\", "& =", "\\sum \\bar\\gamma_i(y)\\bar\\gamma_{n -i}(z)", "\\end{align*}", "as desired.", "\\medskip\\noindent", "Assume $A \\to B$ is flat. Suppose that $b_1, \\ldots, b_r \\in B$ and", "$x_1, \\ldots, x_r \\in I$. Then", "$$", "\\bar\\gamma_n(\\sum b_ix_i) =", "\\sum b_1^{e_1} \\ldots b_r^{e_r} \\gamma_{e_1}(x_1) \\ldots \\gamma_{e_r}(x_r)", "$$", "where the sum is over $e_1 + \\ldots + e_r = n$", "if $\\bar\\gamma_n$ exists. Next suppose that we have $c_1, \\ldots, c_s \\in B$", "and $a_{ij} \\in A$ such that $b_i = \\sum a_{ij}c_j$.", "Setting $y_j = \\sum a_{ij}x_i$ we claim that", "$$", "\\sum b_1^{e_1} \\ldots b_r^{e_r} \\gamma_{e_1}(x_1) \\ldots \\gamma_{e_r}(x_r) =", "\\sum c_1^{d_1} \\ldots c_s^{d_s} \\gamma_{d_1}(y_1) \\ldots \\gamma_{d_s}(y_s)", "$$", "in $B$ where on the right hand side we are summing over", "$d_1 + \\ldots + d_s = n$. Namely, using the axioms of a divided power", "structure we can expand both sides into a sum with coefficients", "in $\\mathbf{Z}[a_{ij}]$ of terms of the form", "$c_1^{d_1} \\ldots c_s^{d_s}\\gamma_{e_1}(x_1) \\ldots \\gamma_{e_r}(x_r)$.", "To see that the coefficients agree we note that the result is true", "in $\\mathbf{Q}[x_1, \\ldots, x_r, c_1, \\ldots, c_s, a_{ij}]$ with", "$\\gamma$ the unique divided power structure on $(x_1, \\ldots, x_r)$.", "By Lazard's theorem (Algebra, Theorem \\ref{algebra-theorem-lazard})", "we can write $B$ as a directed colimit of finite free $A$-modules.", "In particular, if $z \\in IB$ is written as $z = \\sum x_ib_i$ and", "$z = \\sum x'_{i'}b'_{i'}$, then we can find $c_1, \\ldots, c_s \\in B$", "and $a_{ij}, a'_{i'j} \\in A$ such that $b_i = \\sum a_{ij}c_j$", "and $b'_{i'} = \\sum a'_{i'j}c_j$ such that", "$y_j = \\sum x_ia_{ij} = \\sum x'_{i'}a'_{i'j}$ holds\\footnote{This", "can also be proven without recourse to", "Algebra, Theorem \\ref{algebra-theorem-lazard}. Indeed, if", "$z = \\sum x_ib_i$ and $z = \\sum x'_{i'}b'_{i'}$, then", "$\\sum x_ib_i - \\sum x'_{i'}b'_{i'} = 0$ is a relation in the", "$A$-module $B$. Thus, Algebra, Lemma \\ref{algebra-lemma-flat-eq}", "(applied to the $x_i$ and $x'_{i'}$ taking the place of the $f_i$,", "and the $b_i$ and $b'_{i'}$ taking the role of the $x_i$) yields", "the existence of the $c_1, \\ldots, c_s \\in B$", "and $a_{ij}, a'_{i'j} \\in A$ as required.}.", "Hence the procedure above gives a well defined map $\\bar\\gamma_n$", "on $IB$. By construction $\\bar\\gamma$ satisfies conditions (1), (3), and", "(4). Moreover, for $x \\in I$ we have $\\bar\\gamma_n(x) = \\gamma_n(x)$. Hence", "it follows from Lemma \\ref{lemma-check-on-generators} that $\\bar\\gamma$", "is a divided power structure on $IB$." ], "refs": [ "dpa-definition-divided-powers", "dpa-definition-divided-powers", "algebra-theorem-lazard", "algebra-theorem-lazard", "algebra-lemma-flat-eq", "dpa-lemma-check-on-generators" ], "ref_ids": [ 1696, 1696, 318, 318, 531, 1651 ] } ], "ref_ids": [] }, { "id": 1658, "type": "theorem", "label": "dpa-lemma-kernel", "categories": [ "dpa" ], "title": "dpa-lemma-kernel", "contents": [ "Let $(A, I, \\gamma)$ be a divided power ring.", "\\begin{enumerate}", "\\item If $\\varphi : (A, I, \\gamma) \\to (B, J, \\delta)$ is a", "homomorphism of divided power rings, then $\\Ker(\\varphi) \\cap I$", "is preserved by $\\gamma_n$ for all $n \\geq 1$.", "\\item Let $\\mathfrak a \\subset A$ be an ideal and set", "$I' = I \\cap \\mathfrak a$. The following are equivalent", "\\begin{enumerate}", "\\item $I'$ is preserved by $\\gamma_n$ for all $n > 0$,", "\\item $\\gamma$ extends to $A/\\mathfrak a$, and", "\\item there exist a set of generators $x_i$ of $I'$ as an ideal", "such that $\\gamma_n(x_i) \\in I'$ for all $n > 0$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). This is clear. Assume (2)(a). Define", "$\\bar\\gamma_n(x \\bmod I') = \\gamma_n(x) \\bmod I'$ for $x \\in I$.", "This is well defined since $\\gamma_n(x + y) = \\gamma_n(x) \\bmod I'$", "for $y \\in I'$ by Definition \\ref{definition-divided-powers} (4) and", "the fact that $\\gamma_j(y) \\in I'$ by assumption. It is clear that", "$\\bar\\gamma$ is a divided power structure as $\\gamma$ is one.", "Hence (2)(b) holds. Also, (2)(b) implies (2)(a) by part (1).", "It is clear that (2)(a) implies (2)(c). Assume (2)(c).", "Note that $\\gamma_n(x) = a^n\\gamma_n(x_i) \\in I'$ for $x = ax_i$.", "Hence we see that $\\gamma_n(x) \\in I'$ for a set of generators of $I'$", "as an abelian group. By induction on the length of an expression in", "terms of these, it suffices to prove $\\forall n : \\gamma_n(x + y) \\in I'$", "if $\\forall n : \\gamma_n(x), \\gamma_n(y) \\in I'$. This", "follows immediately from the fourth axiom of a divided power structure." ], "refs": [ "dpa-definition-divided-powers" ], "ref_ids": [ 1696 ] } ], "ref_ids": [] }, { "id": 1659, "type": "theorem", "label": "dpa-lemma-sub-dp-ideal", "categories": [ "dpa" ], "title": "dpa-lemma-sub-dp-ideal", "contents": [ "Let $(A, I, \\gamma)$ be a divided power ring.", "Let $E \\subset I$ be a subset.", "Then the smallest ideal $J \\subset I$ preserved by $\\gamma$", "and containing all $f \\in E$ is the ideal $J$ generated by", "$\\gamma_n(f)$, $n \\geq 1$, $f \\in E$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-kernel}." ], "refs": [ "dpa-lemma-kernel" ], "ref_ids": [ 1658 ] } ], "ref_ids": [] }, { "id": 1660, "type": "theorem", "label": "dpa-lemma-extend-to-completion", "categories": [ "dpa" ], "title": "dpa-lemma-extend-to-completion", "contents": [ "Let $(A, I, \\gamma)$ be a divided power ring. Let $p$ be a prime.", "If $p$ is nilpotent in $A/I$, then", "\\begin{enumerate}", "\\item the $p$-adic completion $A^\\wedge = \\lim_e A/p^eA$ surjects onto $A/I$,", "\\item the kernel of this map is the $p$-adic completion $I^\\wedge$ of $I$, and", "\\item each $\\gamma_n$ is continuous for the $p$-adic topology and extends", "to $\\gamma_n^\\wedge : I^\\wedge \\to I^\\wedge$ defining a divided power", "structure on $I^\\wedge$.", "\\end{enumerate}", "If moreover $A$ is a $\\mathbf{Z}_{(p)}$-algebra, then", "\\begin{enumerate}", "\\item[(4)] for $e$ large enough the ideal $p^eA \\subset I$ is preserved by the", "divided power structure $\\gamma$ and", "$$", "(A^\\wedge, I^\\wedge, \\gamma^\\wedge) = \\lim_e (A/p^eA, I/p^eA, \\bar\\gamma)", "$$", "in the category of divided power rings.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $t \\geq 1$ be an integer such that $p^tA/I = 0$, i.e., $p^tA \\subset I$.", "The map $A^\\wedge \\to A/I$ is the composition $A^\\wedge \\to A/p^tA \\to A/I$", "which is surjective (for example by", "Algebra, Lemma \\ref{algebra-lemma-completion-generalities}).", "As $p^eI \\subset p^eA \\cap I \\subset p^{e - t}I$ for $e \\geq t$ we see", "that the kernel of the composition $A^\\wedge \\to A/I$ is the $p$-adic", "completion of $I$. The map $\\gamma_n$ is continuous because", "$$", "\\gamma_n(x + p^ey) =", "\\sum\\nolimits_{i + j = n} p^{je}\\gamma_i(x)\\gamma_j(y) =", "\\gamma_n(x) \\bmod p^eI", "$$", "by the axioms of a divided power structure. It is clear that the axioms", "for divided power structures are inherited by the maps $\\gamma_n^\\wedge$", "from the maps $\\gamma_n$. Finally, to see the last statement say $e > t$.", "Then $p^eA \\subset I$ and $\\gamma_1(p^eA) \\subset p^eA$ and for $n > 1$", "we have", "$$", "\\gamma_n(p^ea) = p^n \\gamma_n(p^{e - 1}a) = \\frac{p^n}{n!} p^{n(e - 1)}a^n", "\\in p^e A", "$$", "as $p^n/n! \\in \\mathbf{Z}_{(p)}$ and as $n \\geq 2$ and $e \\geq 2$ so", "$n(e - 1) \\geq e$.", "This proves that $\\gamma$ extends to $A/p^eA$, see Lemma \\ref{lemma-kernel}.", "The statement on limits is clear from the construction of limits in", "the proof of Lemma \\ref{lemma-limits}." ], "refs": [ "algebra-lemma-completion-generalities", "dpa-lemma-kernel", "dpa-lemma-limits" ], "ref_ids": [ 858, 1658, 1654 ] } ], "ref_ids": [] }, { "id": 1661, "type": "theorem", "label": "dpa-lemma-divided-power-polynomial-algebra", "categories": [ "dpa" ], "title": "dpa-lemma-divided-power-polynomial-algebra", "contents": [ "Let $(A, I, \\gamma)$ be a divided power ring.", "There exists a unique divided power structure $\\delta$ on", "$$", "J = IA\\langle x_1, \\ldots, x_t \\rangle + A\\langle x_1, \\ldots, x_t \\rangle_{+}", "$$", "such that", "\\begin{enumerate}", "\\item $\\delta_n(x_i) = x_i^{[n]}$, and", "\\item $(A, I, \\gamma) \\to (A\\langle x_1, \\ldots, x_t \\rangle, J, \\delta)$", "is a homomorphism of divided power rings.", "\\end{enumerate}", "Moreover, $(A\\langle x_1, \\ldots, x_t \\rangle, J, \\delta)$ has the", "following universal property: A homomorphism of divided power rings", "$\\varphi : (A\\langle x_1, \\ldots, x_t \\rangle, J, \\delta) \\to", "(C, K, \\epsilon)$ is", "the same thing as a homomorphism of divided power rings", "$A \\to C$ and elements $k_1, \\ldots, k_t \\in K$." ], "refs": [], "proofs": [ { "contents": [ "We will prove the lemma in case of a divided power polynomial algebra", "in one variable. The result for the general case can be argued in exactly", "the same way, or by noting that $A\\langle x_1, \\ldots, x_t\\rangle$ is", "isomorphic to the ring obtained by adjoining the divided power variables", "$x_1, \\ldots, x_t$ one by one.", "\\medskip\\noindent", "Let $A\\langle x \\rangle_{+}$ be the ideal generated by", "$x, x^{[2]}, x^{[3]}, \\ldots$.", "Note that $J = IA\\langle x \\rangle + A\\langle x \\rangle_{+}$", "and that", "$$", "IA\\langle x \\rangle \\cap A\\langle x \\rangle_{+} =", "IA\\langle x \\rangle \\cdot A\\langle x \\rangle_{+}", "$$", "Hence by Lemma \\ref{lemma-two-ideals} it suffices to show that there", "exist divided power structures on the ideals $IA\\langle x \\rangle$ and", "$A\\langle x \\rangle_{+}$. The existence of the first follows from", "Lemma \\ref{lemma-gamma-extends} as $A \\to A\\langle x \\rangle$ is flat.", "For the second, note that if $A$ is torsion free, then we can apply", "Lemma \\ref{lemma-silly} (4) to see that $\\delta$ exists. Namely, choosing", "as generators the elements $x^{[m]}$ we see that", "$(x^{[m]})^n = \\frac{(nm)!}{(m!)^n} x^{[nm]}$", "and $n!$ divides the integer $\\frac{(nm)!}{(m!)^n}$.", "In general write $A = R/\\mathfrak a$ for some torsion free ring $R$", "(e.g., a polynomial ring over $\\mathbf{Z}$). The kernel of", "$R\\langle x \\rangle \\to A\\langle x \\rangle$ is", "$\\bigoplus \\mathfrak a x^{[m]}$. Applying criterion (2)(c) of", "Lemma \\ref{lemma-kernel} we see that the divided power structure", "on $R\\langle x \\rangle_{+}$ extends to $A\\langle x \\rangle$ as", "desired.", "\\medskip\\noindent", "Proof of the universal property. Given a homomorphism $\\varphi : A \\to C$", "of divided power rings and $k_1, \\ldots, k_t \\in K$ we consider", "$$", "A\\langle x_1, \\ldots, x_t \\rangle \\to C,\\quad", "x_1^{[n_1]} \\ldots x_t^{[n_t]} \\longmapsto", "\\epsilon_{n_1}(k_1) \\ldots \\epsilon_{n_t}(k_t)", "$$", "using $\\varphi$ on coefficients. The only thing to check is that", "this is an $A$-algebra homomorphism (details omitted). The inverse", "construction is clear." ], "refs": [ "dpa-lemma-two-ideals", "dpa-lemma-gamma-extends", "dpa-lemma-silly", "dpa-lemma-kernel" ], "ref_ids": [ 1652, 1657, 1650, 1658 ] } ], "ref_ids": [] }, { "id": 1662, "type": "theorem", "label": "dpa-lemma-need-only-gamma-p", "categories": [ "dpa" ], "title": "dpa-lemma-need-only-gamma-p", "contents": [ "Let $p$ be a prime number. Let $A$ be a ring such that every integer $n$", "not divisible by $p$ is invertible, i.e., $A$ is a $\\mathbf{Z}_{(p)}$-algebra.", "Let $I \\subset A$ be an ideal. Two divided power structures", "$\\gamma, \\gamma'$ on $I$ are equal if and only if $\\gamma_p = \\gamma'_p$.", "Moreover, given a map $\\delta : I \\to I$ such that", "\\begin{enumerate}", "\\item $p!\\delta(x) = x^p$ for all $x \\in I$,", "\\item $\\delta(ax) = a^p\\delta(x)$ for all $a \\in A$, $x \\in I$, and", "\\item", "$\\delta(x + y) =", "\\delta(x) +", "\\sum\\nolimits_{i + j = p, i,j \\geq 1} \\frac{1}{i!j!} x^i y^j +", "\\delta(y)$ for all $x, y \\in I$,", "\\end{enumerate}", "then there exists a unique divided power structure $\\gamma$ on $I$ such", "that $\\gamma_p = \\delta$." ], "refs": [], "proofs": [ { "contents": [ "If $n$ is not divisible by $p$, then $\\gamma_n(x) = c x \\gamma_{n - 1}(x)$", "where $c$ is a unit in $\\mathbf{Z}_{(p)}$. Moreover,", "$$", "\\gamma_{pm}(x) = c \\gamma_m(\\gamma_p(x))", "$$", "where $c$ is a unit in $\\mathbf{Z}_{(p)}$. Thus the first assertion is clear.", "For the second assertion, we can, working backwards, use these equalities", "to define all $\\gamma_n$. More precisely, if", "$n = a_0 + a_1p + \\ldots + a_e p^e$ with $a_i \\in \\{0, \\ldots, p - 1\\}$ then", "we set", "$$", "\\gamma_n(x) = c_n x^{a_0} \\delta(x)^{a_1} \\ldots \\delta^e(x)^{a_e}", "$$", "for $c_n \\in \\mathbf{Z}_{(p)}$ defined by", "$$", "c_n =", "{(p!)^{a_1 + a_2(1 + p) + \\ldots + a_e(1 + \\ldots + p^{e - 1})}}/{n!}.", "$$", "Now we have to show the axioms (1) -- (5) of a divided power structure, see", "Definition \\ref{definition-divided-powers}. We observe that (1) and (3) are", "immediate. Verification of (2) and (5) is by a direct calculation which", "we omit. Let $x, y \\in I$. We claim there is a ring map", "$$", "\\varphi : \\mathbf{Z}_{(p)}\\langle u, v \\rangle \\longrightarrow A", "$$", "which maps $u^{[n]}$ to $\\gamma_n(x)$ and $v^{[n]}$ to $\\gamma_n(y)$.", "By construction of $\\mathbf{Z}_{(p)}\\langle u, v \\rangle$ this means", "we have to check that", "$$", "\\gamma_n(x)\\gamma_m(x) = \\frac{(n + m)!}{n!m!} \\gamma_{n + m}(x)", "$$", "in $A$ and similarly for $y$. This is true because (2) holds for $\\gamma$.", "Let $\\epsilon$ denote the divided power structure on the", "ideal $\\mathbf{Z}_{(p)}\\langle u, v\\rangle_{+}$ of", "$\\mathbf{Z}_{(p)}\\langle u, v\\rangle$.", "Next, we claim that $\\varphi(\\epsilon_n(f)) = \\gamma_n(\\varphi(f))$", "for $f \\in \\mathbf{Z}_{(p)}\\langle u, v\\rangle_{+}$ and all $n$.", "This is clear for $n = 0, 1, \\ldots, p - 1$. For $n = p$ it suffices", "to prove it for a set of generators of the ideal", "$\\mathbf{Z}_{(p)}\\langle u, v\\rangle_{+}$ because both $\\epsilon_p$", "and $\\gamma_p = \\delta$ satisfy properties (1) and (3) of the lemma.", "Hence it suffices to prove that", "$\\gamma_p(\\gamma_n(x)) = \\frac{(pn)!}{p!(n!)^p}\\gamma_{pn}(x)$ and", "similarly for $y$, which follows as (5) holds for $\\gamma$.", "Now, if $n = a_0 + a_1p + \\ldots + a_e p^e$", "is an arbitrary integer written in $p$-adic expansion as above, then", "$$", "\\epsilon_n(f) =", "c_n f^{a_0} \\gamma_p(f)^{a_1} \\ldots \\gamma_p^e(f)^{a_e}", "$$", "because $\\epsilon$ is a divided power structure. Hence we see that", "$\\varphi(\\epsilon_n(f)) = \\gamma_n(\\varphi(f))$ holds for all $n$.", "Applying this for $f = u + v$ we see that axiom (4) for $\\gamma$", "follows from the fact that $\\epsilon$ is a divided power structure." ], "refs": [ "dpa-definition-divided-powers" ], "ref_ids": [ 1696 ] } ], "ref_ids": [] }, { "id": 1663, "type": "theorem", "label": "dpa-lemma-dpdga-good", "categories": [ "dpa" ], "title": "dpa-lemma-dpdga-good", "contents": [ "Let $(A, \\text{d}, \\gamma)$ and $(B, \\text{d}, \\gamma)$ be as in", "Definition \\ref{definition-divided-powers-dga}. Let $f : A \\to B$", "be a map of differential graded algebras compatible with divided", "power structures. Assume", "\\begin{enumerate}", "\\item $H_k(A) = 0$ for $k > 0$, and", "\\item $f$ is surjective.", "\\end{enumerate}", "Then $\\gamma$ induces a divided power structure on the graded", "$R$-algebra $H(B)$." ], "refs": [ "dpa-definition-divided-powers-dga" ], "proofs": [ { "contents": [ "Suppose that $x$ and $x'$ are homogeneous of the same degree $2d$", "and define the same cohomology class in $H(B)$. Say $x' - x = \\text{d}(w)$.", "Choose a lift $y \\in A_{2d}$ of $x$ and a lift $z \\in A_{2d + 1}$", "of $w$. Then $y' = y + \\text{d}(z)$ is a lift of $x'$.", "Hence", "$$", "\\gamma_n(y') = \\sum \\gamma_i(y) \\gamma_{n - i}(\\text{d}(z))", "= \\gamma_n(y) +", "\\sum\\nolimits_{i < n} \\gamma_i(y) \\gamma_{n - i}(\\text{d}(z))", "$$", "Since $A$ is acyclic in positive degrees and since", "$\\text{d}(\\gamma_j(\\text{d}(z))) = 0$ for all $j$ we can write", "this as", "$$", "\\gamma_n(y') = \\gamma_n(y) +", "\\sum\\nolimits_{i < n} \\gamma_i(y) \\text{d}(z_i)", "$$", "for some $z_i$ in $A$. Moreover, for $0 < i < n$ we have", "$$", "\\text{d}(\\gamma_i(y) z_i) =", "\\text{d}(\\gamma_i(y))z_i + \\gamma_i(y)\\text{d}(z_i) =", "\\text{d}(y) \\gamma_{i - 1}(y) z_i + \\gamma_i(y)\\text{d}(z_i)", "$$", "and the first term maps to zero in $B$ as $\\text{d}(y)$ maps to zero in $B$.", "Hence $\\gamma_n(x')$ and $\\gamma_n(x)$ map to the same element of $H(B)$.", "Thus we obtain a well defined map $\\gamma_n : H_{2d}(B) \\to H_{2nd}(B)$", "for all $d > 0$ and $n > 0$. We omit the verification that this", "defines a divided power structure on $H(B)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1700 ] }, { "id": 1664, "type": "theorem", "label": "dpa-lemma-base-change-div", "categories": [ "dpa" ], "title": "dpa-lemma-base-change-div", "contents": [ "Let $(A, \\text{d}, \\gamma)$ be as in", "Definition \\ref{definition-divided-powers-dga}.", "Let $R \\to R'$ be a ring map.", "Then $\\text{d}$ and $\\gamma$ induce similar structures on", "$A' = A \\otimes_R R'$ such that $(A', \\text{d}, \\gamma)$ is as in", "Definition \\ref{definition-divided-powers-dga}." ], "refs": [ "dpa-definition-divided-powers-dga", "dpa-definition-divided-powers-dga" ], "proofs": [ { "contents": [ "Observe that $A'_{even} = A_{even} \\otimes_R R'$ and", "$A'_{even, +} = A_{even, +} \\otimes_R R'$. Hence we are trying to", "show that the divided powers $\\gamma$ extend to $A'_{even}$", "(terminology as in Definition \\ref{definition-extends}).", "Once we have shown $\\gamma$ extends it follows easily that this", "extension has all the desired properties.", "\\medskip\\noindent", "Choose a polynomial $R$-algebra $P$ (on any set of generators)", "and a surjection of $R$-algebras", "$P \\to R'$. The ring map $A_{even} \\to A_{even} \\otimes_R P$ is flat,", "hence the divided powers $\\gamma$ extend to $A_{even} \\otimes_R P$", "uniquely by Lemma \\ref{lemma-gamma-extends}.", "Let $J = \\Ker(P \\to R')$. To show that $\\gamma$ extends", "to $A \\otimes_R R'$ it suffices to show that", "$I' = \\Ker(A_{even, +} \\otimes_R P \\to A_{even, +} \\otimes_R R')$", "is generated by elements $z$ such that $\\gamma_n(z) \\in I'$", "for all $n > 0$. This is clear as $I'$ is generated by elements", "of the form $x \\otimes f$ with", "$x \\in A_{even, +}$ and $f \\in \\Ker(P \\to R')$." ], "refs": [ "dpa-definition-extends", "dpa-lemma-gamma-extends" ], "ref_ids": [ 1698, 1657 ] } ], "ref_ids": [ 1700, 1700 ] }, { "id": 1665, "type": "theorem", "label": "dpa-lemma-extend-differential", "categories": [ "dpa" ], "title": "dpa-lemma-extend-differential", "contents": [ "Let $(A, \\text{d}, \\gamma)$ be as in", "Definition \\ref{definition-divided-powers-dga}.", "Let $d \\geq 1$ be an integer.", "Let $A\\langle T \\rangle$ be the graded divided power polynomial algebra", "on $T$ with $\\deg(T) = d$", "constructed in Example \\ref{example-adjoining-odd} or", "\\ref{example-adjoining-even}.", "Let $f \\in A_{d - 1}$ be an element with $\\text{d}(f) = 0$.", "There exists a unique differential $\\text{d}$", "on $A\\langle T\\rangle$ such that $\\text{d}(T) = f$ and", "such that $\\text{d}$ is compatible with the divided power", "structure on $A\\langle T \\rangle$." ], "refs": [ "dpa-definition-divided-powers-dga" ], "proofs": [ { "contents": [ "This is proved by a direct computation which is omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1700 ] }, { "id": 1666, "type": "theorem", "label": "dpa-lemma-tate-resolution", "categories": [ "dpa" ], "title": "dpa-lemma-tate-resolution", "contents": [ "Let $R \\to S$ be a homomorphism of commutative rings.", "There exists a factorization", "$$", "R \\to A \\to S", "$$", "with the following properties:", "\\begin{enumerate}", "\\item $(A, \\text{d}, \\gamma)$ is as in", "Definition \\ref{definition-divided-powers-dga},", "\\item $A \\to S$ is a quasi-isomorphism (if we endow $S$ with", "the zero differential),", "\\item $A_0 = R[x_j: j\\in J] \\to S$ is any surjection of a polynomial", "ring onto $S$, and", "\\item $A$ is a graded divided power polynomial algebra over $R$.", "\\end{enumerate}", "The last condition means that $A$ is constructed out of $A_0$ by", "successively adjoining a set of variables $T$ in each degree $> 0$ as in", "Example \\ref{example-adjoining-odd} or \\ref{example-adjoining-even}.", "Moreover, if $R$ is Noetherian and $R\\to S$ is of finite type,", "then $A$ can be taken to have only finitely many generators in", "each degree." ], "refs": [ "dpa-definition-divided-powers-dga" ], "proofs": [ { "contents": [ "We write out the construction for the case that $R$ is Noetherian", "and $R\\to S$ is of finite type. Without those assumptions, the proof", "is the same, except that we have to use some set (possibly", "infinite) of generators in each degree.", "\\medskip\\noindent", "Start of the construction: Let $A(0) = R[x_1, \\ldots, x_n]$ be", "a (usual) polynomial ring and let $A(0) \\to S$ be a surjection.", "As grading we take $A(0)_0 = A(0)$ and $A(0)_d = 0$ for $d \\not = 0$.", "Thus $\\text{d} = 0$ and $\\gamma_n$, $n > 0$, is zero as well.", "\\medskip\\noindent", "Choose generators $f_1, \\ldots, f_m \\in R[x_1, \\ldots, x_n]$", "for the kernel of the given map $A(0) = R[x_1, \\ldots, x_n] \\to S$.", "We apply Example \\ref{example-adjoining-odd} $m$ times to get", "$$", "A(1) = A(0)\\langle T_1, \\ldots, T_m\\rangle", "$$", "with $\\deg(T_i) = 1$ as a graded divided power polynomial algebra.", "We set $\\text{d}(T_i) = f_i$. Since $A(1)$ is a divided power polynomial", "algebra over $A(0)$ and since $\\text{d}(f_i) = 0$", "this extends uniquely to a differential on $A(1)$ by", "Lemma \\ref{lemma-extend-differential}.", "\\medskip\\noindent", "Induction hypothesis: Assume we are given factorizations", "$$", "R \\to A(0) \\to A(1) \\to \\ldots \\to A(m) \\to S", "$$", "where $A(0)$ and $A(1)$ are as above and each $R \\to A(m') \\to S$", "for $2 \\leq m' \\leq m$ satisfies properties (1) and (4)", "of the statement of the lemma and (2) replaced by the condition that", "$H_i(A(m')) \\to H_i(S)$ is an isomorphism for", "$m' > i \\geq 0$. The base case is $m = 1$.", "\\medskip\\noindent", "Induction step: Assume we have $R \\to A(m) \\to S$", "as in the induction hypothesis. Consider the", "group $H_m(A(m))$. This is a module over $H_0(A(m)) = S$.", "In fact, it is a subquotient of $A(m)_m$ which is a finite", "type module over $A(m)_0 = R[x_1, \\ldots, x_n]$.", "Thus we can pick finitely many elements", "$$", "e_1, \\ldots, e_t \\in \\Ker(\\text{d} : A(m)_m \\to A(m)_{m - 1})", "$$", "which map to generators of this module. Applying", "Example \\ref{example-adjoining-odd} or", "\\ref{example-adjoining-even} $t$ times we get", "$$", "A(m + 1) = A(m)\\langle T_1, \\ldots, T_t\\rangle", "$$", "with $\\deg(T_i) = m + 1$ as a graded divided power algebra. We set", "$\\text{d}(T_i) = e_i$. Since $A(m+1)$ is a divided power polynomial", "algebra over $A(m)$ and since $\\text{d}(e_i) = 0$", "this extends uniquely to a differential on $A(m + 1)$", "compatible with the divided power structure.", "Since we've added only material in degree $m + 1$ and higher we see", "that $H_i(A(m + 1)) = H_i(A(m))$ for $i < m$. Moreover, it is", "clear that $H_m(A(m + 1)) = 0$ by construction.", "\\medskip\\noindent", "To finish the proof we observe that we have shown there exists", "a sequence of maps", "$$", "R \\to A(0) \\to A(1) \\to \\ldots \\to A(m) \\to A(m + 1) \\to \\ldots \\to S", "$$", "and to finish the proof we set $A = \\colim A(m)$." ], "refs": [ "dpa-lemma-extend-differential" ], "ref_ids": [ 1665 ] } ], "ref_ids": [ 1700 ] }, { "id": 1667, "type": "theorem", "label": "dpa-lemma-tate-resoluton-pseudo-coherent-ring-map", "categories": [ "dpa" ], "title": "dpa-lemma-tate-resoluton-pseudo-coherent-ring-map", "contents": [ "Let $R \\to S$ be a pseudo-coherent ring map (More on Algebra, Definition", "\\ref{more-algebra-definition-pseudo-coherent-perfect}). Then", "Lemma \\ref{lemma-tate-resolution} holds, with the resolution $A$ of $S$", "having finitely many generators in each degree." ], "refs": [ "more-algebra-definition-pseudo-coherent-perfect", "dpa-lemma-tate-resolution" ], "proofs": [ { "contents": [ "This is proved in exactly the same way as Lemma \\ref{lemma-tate-resolution}.", "The only additional twist is that, given $A(m) \\to S$ we have to", "show that $H_m = H_m(A(m))$ is a finite $R[x_1, \\ldots, x_m]$-module", "(so that in the next step we need only add finitely many variables).", "Consider the complex", "$$", "\\ldots \\to A(m)_{m - 1} \\to A(m)_m \\to A(m)_{m - 1} \\to", "\\ldots \\to A(m)_0 \\to S \\to 0", "$$", "Since $S$ is a pseudo-coherent $R[x_1, \\ldots, x_n]$-module", "and since $A(m)_i$ is a finite free $R[x_1, \\ldots, x_n]$-module", "we conclude that this is a pseudo-coherent complex, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-complex-pseudo-coherent-modules}.", "Since the complex is exact in (homological) degrees $> m$", "we conclude that $H_m$ is a finite $R$-module by", "More on Algebra, Lemma \\ref{more-algebra-lemma-finite-cohomology}." ], "refs": [ "dpa-lemma-tate-resolution", "more-algebra-lemma-complex-pseudo-coherent-modules", "more-algebra-lemma-finite-cohomology" ], "ref_ids": [ 1666, 10152, 10146 ] } ], "ref_ids": [ 10631, 1666 ] }, { "id": 1668, "type": "theorem", "label": "dpa-lemma-uniqueness-tate-resolution", "categories": [ "dpa" ], "title": "dpa-lemma-uniqueness-tate-resolution", "contents": [ "Let $R$ be a commutative ring. Suppose that $(A, \\text{d}, \\gamma)$ and", "$(B, \\text{d}, \\gamma)$ are as in", "Definition \\ref{definition-divided-powers-dga}.", "Let $\\overline{\\varphi} : H_0(A) \\to H_0(B)$ be an $R$-algebra map.", "Assume", "\\begin{enumerate}", "\\item $A$ is a graded divided power polynomial algebra over $R$.", "\\item $H_k(B) = 0$ for $k > 0$.", "\\end{enumerate}", "Then there exists a map $\\varphi : A \\to B$ of differential", "graded $R$-algebras compatible with divided powers", "that lifts $\\overline{\\varphi}$." ], "refs": [ "dpa-definition-divided-powers-dga" ], "proofs": [ { "contents": [ "The assumption means that $A$ is obtained from $R$ by successively adjoining", "some set of polynomial generators in degree zero, exterior generators", "in positive odd degrees, and divided power generators", "in positive even degrees. So we have a filtration", "$R \\subset A(0) \\subset A(1) \\subset \\ldots$", "of $A$ such that $A(m + 1)$ is obtained from $A(m)$ by adjoining", "generators of the appropriate type (which we simply call", "``divided power generators'') in degree $m + 1$.", "In particular, $A(0) \\to H_0(A)$ is a surjection from a (usual) polynomial", "algebra over $R$ onto $H_0(A)$. Thus we can lift $\\overline{\\varphi}$", "to an $R$-algebra map $\\varphi(0) : A(0) \\to B_0$.", "\\medskip\\noindent", "Write $A(1) = A(0)\\langle T_j:j\\in J\\rangle$ for some", "set $J$ of divided power variables $T_j$ of degree $1$. Let $f_j \\in B_0$", "be $f_j = \\varphi(0)(\\text{d}(T_j))$. Observe that $f_j$", "maps to zero in $H_0(B)$ as $\\text{d}T_j$ maps to zero in $H_0(A)$.", "Thus we can find $b_j \\in B_1$ with $\\text{d}(b_j) = f_j$.", "By the universal property of divided power polynomial algebras from", "Lemma \\ref{lemma-divided-power-polynomial-algebra},", "we find a lift $\\varphi(1) : A(1) \\to B$ of $\\varphi(0)$", "mapping $T_j$ to $f_j$.", "\\medskip\\noindent", "Having constructed $\\varphi(m)$ for some $m \\geq 1$ we can construct", "$\\varphi(m + 1) : A(m + 1) \\to B$ in exactly the same manner.", "We omit the details." ], "refs": [ "dpa-lemma-divided-power-polynomial-algebra" ], "ref_ids": [ 1661 ] } ], "ref_ids": [ 1700 ] }, { "id": 1669, "type": "theorem", "label": "dpa-lemma-divided-powers-on-tor", "categories": [ "dpa" ], "title": "dpa-lemma-divided-powers-on-tor", "contents": [ "Let $R$ be a commutative ring. Let $S$ and $T$ be commutative $R$-algebras.", "Then there is a canonical structure", "of a strictly graded commutative $R$-algebra with divided powers on", "$$", "\\operatorname{Tor}_*^R(S, T).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose a factorization $R \\to A \\to S$ as above. Since $A \\to S$", "is a quasi-isomorphism and since $A_d$ is a free $R$-module,", "we see that the differential graded algebra $B = A \\otimes_R T$ computes", "the Tor groups displayed in the lemma. Choose a surjection", "$R[y_j:j\\in J] \\to T$. Then we see that", "$B$ is a quotient of the differential graded algebra", "$A[y_j:j\\in J]$ whose homology sits in degree $0$ (it is equal", "to $S[y_j:j\\in J]$).", "By Lemma \\ref{lemma-base-change-div} the differential graded algebras $B$ and", "$A[y_j:j\\in J]$ have divided power structures compatible", "with the differentials. Hence we obtain our divided", "power structure on $H(B)$ by Lemma \\ref{lemma-dpdga-good}.", "\\medskip\\noindent", "The divided power algebra structure constructed in this way is independent", "of the choice of $A$. Namely, if $A'$ is a second choice, then", "Lemma \\ref{lemma-uniqueness-tate-resolution}", "implies there is a map $A \\to A'$ preserving all structure and the", "augmentations towards $S$. Then the induced map", "$B = A \\otimes_R T \\to A' \\otimes_R T' = B'$ also preserves", "all structure", "and is a quasi-isomorphism. The induced isomorphism of", "Tor algebras is therefore compatible with products", "and divided powers." ], "refs": [ "dpa-lemma-base-change-div", "dpa-lemma-dpdga-good", "dpa-lemma-uniqueness-tate-resolution" ], "ref_ids": [ 1664, 1663, 1668 ] } ], "ref_ids": [] }, { "id": 1670, "type": "theorem", "label": "dpa-lemma-get-derivation", "categories": [ "dpa" ], "title": "dpa-lemma-get-derivation", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d}, \\gamma)$ be as in", "Definition \\ref{definition-divided-powers-dga}.", "Let $R' \\to R$ be a surjection of rings whose kernel", "has square zero and is generated by one element $f$.", "If $A$ is a graded divided power polynomial algebra over $R$", "with finitely many variables in each degree,", "then we obtain a derivation", "$\\theta : A/IA \\to A/IA$ where $I$ is the annihilator", "of $f$ in $R$." ], "refs": [ "dpa-definition-divided-powers-dga" ], "proofs": [ { "contents": [ "Since $A$ is a divided power polynomial algebra, we can find a divided", "power polynomial algebra $A'$ over $R'$ such that $A = A' \\otimes_R R'$.", "Moreover, we can lift $\\text{d}$ to an $R$-linear", "operator $\\text{d}$ on $A'$ such that", "\\begin{enumerate}", "\\item $\\text{d}(xy) = \\text{d}(x)y + (-1)^{\\deg(x)}x \\text{d}(y)$", "for $x, y \\in A'$ homogeneous, and", "\\item $\\text{d}(\\gamma_n(x)) = \\text{d}(x) \\gamma_{n - 1}(x)$ for", "$x \\in A'_{even, +}$.", "\\end{enumerate}", "We omit the details (hint: proceed one variable at the time).", "However, it may not be the case that $\\text{d}^2$", "is zero on $A'$. It is clear that $\\text{d}^2$ maps $A'$ into", "$fA' \\cong A/IA$. Hence $\\text{d}^2$ annihilates $fA'$ and factors", "as a map $A \\to A/IA$. Since $\\text{d}^2$ is $R$-linear we obtain", "our map $\\theta : A/IA \\to A/IA$. The verification of the properties", "of a derivation is immediate." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1700 ] }, { "id": 1671, "type": "theorem", "label": "dpa-lemma-compute-theta", "categories": [ "dpa" ], "title": "dpa-lemma-compute-theta", "contents": [ "Assumption and notation as in Lemma \\ref{lemma-get-derivation}.", "Suppose $S = H_0(A)$ is isomorphic to", "$R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$", "for some $n$, $m$, and $f_j \\in R[x_1, \\ldots, x_n]$.", "Moreover, suppose given a relation", "$$", "\\sum r_j f_j = 0", "$$", "with $r_j \\in R[x_1, \\ldots, x_n]$.", "Choose $r'_j, f'_j \\in R'[x_1, \\ldots, x_n]$ lifting $r_j, f_j$.", "Write $\\sum r'_j f'_j = gf$ for some $g \\in R/I[x_1, \\ldots, x_n]$.", "If $H_1(A) = 0$ and all the coefficients of each $r_j$ are in $I$, then", "there exists an element $\\xi \\in H_2(A/IA)$ such that", "$\\theta(\\xi) = g$ in $S/IS$." ], "refs": [ "dpa-lemma-get-derivation" ], "proofs": [ { "contents": [ "Let $A(0) \\subset A(1) \\subset A(2) \\subset \\ldots$ be the filtration", "of $A$ such that $A(m)$ is gotten from $A(m - 1)$ by adjoining divided", "power variables of degree $m$. Then $A(0)$ is a polynomial algebra", "over $R$ equipped with an $R$-algebra surjection $A(0) \\to S$.", "Thus we can choose a map", "$$", "\\varphi : R[x_1, \\ldots, x_n] \\to A(0)", "$$", "lifting the augmentations to $S$.", "Next, $A(1) = A(0)\\langle T_1, \\ldots, T_t \\rangle$ for some divided", "power variables $T_i$ of degree $1$. Since $H_0(A) = S$ we", "can pick $\\xi_j \\in \\sum A(0)T_i$ with $\\text{d}(\\xi_j) = \\varphi(f_j)$.", "Then", "$$", "\\text{d}\\left(\\sum \\varphi(r_j) \\xi_j\\right) =", "\\sum \\varphi(r_j) \\varphi(f_j) = \\sum \\varphi(r_jf_j) = 0", "$$", "Since $H_1(A) = 0$ we can pick $\\xi \\in A_2$ with", "$\\text{d}(\\xi) = \\sum \\varphi(r_j) \\xi_j$.", "If the coefficients of $r_j$ are in $I$, then the same", "is true for $\\varphi(r_j)$. In this case", "$\\text{d}(\\xi)$ dies in $A_1/IA_1$ and", "hence $\\xi$ defines a class in $H_2(A/IA)$.", "\\medskip\\noindent", "The construction of $\\theta$ in the proof of Lemma \\ref{lemma-get-derivation}", "proceeds by successively lifting $A(i)$ to $A'(i)$ and lifting the", "differential $\\text{d}$. We lift $\\varphi$", "to $\\varphi' : R'[x_1, \\ldots, x_n] \\to A'(0)$.", "Next, we have $A'(1) = A'(0)\\langle T_1, \\ldots, T_t\\rangle$.", "Moreover, we can lift $\\xi_j$ to $\\xi'_j \\in \\sum A'(0)T_i$.", "Then $\\text{d}(\\xi'_j) = \\varphi'(f'_j) + f a_j$ for some", "$a_j \\in A'(0)$.", "Consider a lift $\\xi' \\in A'_2$ of $\\xi$.", "Then we know that", "$$", "\\text{d}(\\xi') = \\sum \\varphi'(r'_j)\\xi'_j + \\sum fb_iT_i", "$$", "for some $b_i \\in A(0)$. Applying $\\text{d}$ again we find", "$$", "\\theta(\\xi) = \\sum \\varphi'(r'_j)\\varphi'(f'_j) +", "\\sum f \\varphi'(r'_j) a_j + \\sum fb_i \\text{d}(T_i)", "$$", "The first term gives us what we want. The second term is zero", "because the coefficients of $r_j$ are in $I$ and hence are", "annihilated by $f$. The third term maps to zero in $H_0$", "because $\\text{d}(T_i)$ maps to zero." ], "refs": [ "dpa-lemma-get-derivation" ], "ref_ids": [ 1670 ] } ], "ref_ids": [ 1670 ] }, { "id": 1672, "type": "theorem", "label": "dpa-lemma-not-finite-pd", "categories": [ "dpa" ], "title": "dpa-lemma-not-finite-pd", "contents": [ "Let $R' \\to R$ be a surjection of Noetherian rings whose kernel has square", "zero and is generated by one element $f$. Let", "$S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$.", "Let $\\sum r_j f_j = 0$ be a relation in $R[x_1, \\ldots, x_n]$.", "Assume that", "\\begin{enumerate}", "\\item each $r_j$ has coefficients in the annihilator $I$ of $f$ in $R$,", "\\item for some lifts $r'_j, f'_j \\in R'[x_1, \\ldots, x_n]$ we have", "$\\sum r'_j f'_j = gf$ where $g$ is not nilpotent in $S$.", "\\end{enumerate}", "Then $S$ does not have finite tor dimension over $R$ (i.e., $S$ is not", "a perfect $R$-algebra)." ], "refs": [], "proofs": [ { "contents": [ "Choose a Tate resolution $R \\to A \\to S$ as in", "Lemma \\ref{lemma-tate-resolution}.", "Let $\\xi \\in H_2(A/IA)$ and $\\theta : A/IA \\to A/IA$ be the element", "and derivation found in Lemmas \\ref{lemma-get-derivation} and", "\\ref{lemma-compute-theta}.", "Observe that", "$$", "\\theta^n(\\gamma_n(\\xi)) = g^n", "$$", "Hence if $g$ is not nilpotent, then $\\xi^n$ is nonzero in", "$H_{2n}(A/IA)$ for all $n > 0$. Since", "$H_{2n}(A/IA) = \\text{Tor}^R_{2n}(S, R/I)$ we conclude." ], "refs": [ "dpa-lemma-tate-resolution", "dpa-lemma-get-derivation", "dpa-lemma-compute-theta" ], "ref_ids": [ 1666, 1670, 1671 ] } ], "ref_ids": [] }, { "id": 1673, "type": "theorem", "label": "dpa-lemma-injective", "categories": [ "dpa" ], "title": "dpa-lemma-injective", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let", "$I \\subset J \\subset A$ be proper ideals. If $A/J$ has finite", "tor dimension over $A/I$, then $I/\\mathfrak m I \\to J/\\mathfrak m J$", "is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $f \\in I$ be an element mapping to a nonzero element of $I/\\mathfrak m I$", "which is mapped to zero in $J/\\mathfrak mJ$. We can choose an ideal $I'$", "with $\\mathfrak mI \\subset I' \\subset I$ such that $I/I'$ is generated by", "the image of $f$. Set $R = A/I$ and $R' = A/I'$. Let $J = (a_1, \\ldots, a_m)$", "for some $a_j \\in A$. Then $f = \\sum b_j a_j$ for some $b_j \\in \\mathfrak m$.", "Let $r_j, f_j \\in R$ resp.\\ $r'_j, f'_j \\in R'$ be the image of $b_j, a_j$.", "Then we see we are", "in the situation of Lemma \\ref{lemma-not-finite-pd}", "(with the ideal $I$ of that lemma equal to $\\mathfrak m_R$)", "and the lemma is proved." ], "refs": [ "dpa-lemma-not-finite-pd" ], "ref_ids": [ 1672 ] } ], "ref_ids": [] }, { "id": 1674, "type": "theorem", "label": "dpa-lemma-regular-sequence", "categories": [ "dpa" ], "title": "dpa-lemma-regular-sequence", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let", "$I \\subset J \\subset A$ be proper ideals. Assume", "\\begin{enumerate}", "\\item $A/J$ has finite tor dimension over $A/I$, and", "\\item $J$ is generated by a regular sequence.", "\\end{enumerate}", "Then $I$ is generated by a regular sequence and $J/I$", "is generated by a regular sequence." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-injective} we see that", "$I/\\mathfrak m I \\to J/\\mathfrak m J$", "is injective. Thus we can find $s \\leq r$ and a minimal system of", "generators $f_1, \\ldots, f_r$ of $J$ such that $f_1, \\ldots, f_s$ are in $I$", "and form a minimal system of generators of $I$.", "The lemma follows as any minimal system of generators of $J$", "is a regular sequence by", "More on Algebra, Lemmas", "\\ref{more-algebra-lemma-independence-of-generators} and", "\\ref{more-algebra-lemma-noetherian-finite-all-equivalent}." ], "refs": [ "dpa-lemma-injective", "more-algebra-lemma-independence-of-generators", "more-algebra-lemma-noetherian-finite-all-equivalent" ], "ref_ids": [ 1673, 9986, 9978 ] } ], "ref_ids": [] }, { "id": 1675, "type": "theorem", "label": "dpa-lemma-perfect-map-ci", "categories": [ "dpa" ], "title": "dpa-lemma-perfect-map-ci", "contents": [ "Let $R \\to S$ be a local ring map of Noetherian local rings.", "Let $I \\subset R$ and $J \\subset S$ be ideals with", "$IS \\subset J$. If $R \\to S$ is flat and $S/\\mathfrak m_RS$ is", "regular, then the following are equivalent", "\\begin{enumerate}", "\\item $J$ is generated by a regular sequence and", "$S/J$ has finite tor dimension as a module over $R/I$,", "\\item $J$ is generated by a regular sequence and", "$\\text{Tor}^{R/I}_p(S/J, R/\\mathfrak m_R)$ is nonzero", "for only finitely many $p$,", "\\item $I$ is generated by a regular sequence", "and $J/IS$ is generated by a regular sequence in $S/IS$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (3) holds, then $J$ is generated by a regular sequence, see for example", "More on Algebra, Lemmas", "\\ref{more-algebra-lemma-join-koszul-regular-sequences} and", "\\ref{more-algebra-lemma-noetherian-finite-all-equivalent}.", "Moreover, if (3) holds, then $S/J = (S/I)/(J/I)$", "has finite projective dimension over $S/IS$ because the Koszul", "complex will be a finite free resolution of $S/J$ over $S/IS$.", "Since $R/I \\to S/IS$ is flat, it then follows that $S/J$ has finite", "tor dimension over $R/I$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-push-tor-amplitude}.", "Thus (3) implies (1).", "\\medskip\\noindent", "The implication (1) $\\Rightarrow$ (2) is trivial.", "Assume (2). By", "More on Algebra, Lemma \\ref{more-algebra-lemma-perfect-over-regular-local-ring}", "we find that $S/J$ has finite tor dimension over $S/IS$.", "Thus we can apply Lemma \\ref{lemma-regular-sequence}", "to conclude that $IS$ and $J/IS$ are generated by regular sequences.", "Let $f_1, \\ldots, f_r \\in I$ be a minimal system of generators of $I$.", "Since $R \\to S$ is flat, we see that $f_1, \\ldots, f_r$ form a minimal", "system of generators for $IS$ in $S$. Thus $f_1, \\ldots, f_r \\in R$", "is a sequence of elements whose images in $S$ form a regular sequence", "by More on Algebra, Lemmas", "\\ref{more-algebra-lemma-independence-of-generators} and", "\\ref{more-algebra-lemma-noetherian-finite-all-equivalent}.", "Thus $f_1, \\ldots, f_r$ is a regular sequence in $R$ by", "Algebra, Lemma \\ref{algebra-lemma-flat-increases-depth}." ], "refs": [ "more-algebra-lemma-join-koszul-regular-sequences", "more-algebra-lemma-noetherian-finite-all-equivalent", "more-algebra-lemma-flat-push-tor-amplitude", "more-algebra-lemma-perfect-over-regular-local-ring", "dpa-lemma-regular-sequence", "more-algebra-lemma-independence-of-generators", "more-algebra-lemma-noetherian-finite-all-equivalent", "algebra-lemma-flat-increases-depth" ], "ref_ids": [ 9984, 9978, 10178, 10246, 1674, 9986, 9978, 740 ] } ], "ref_ids": [] }, { "id": 1676, "type": "theorem", "label": "dpa-lemma-ci-well-defined", "categories": [ "dpa" ], "title": "dpa-lemma-ci-well-defined", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian complete local ring.", "The following are equivalent", "\\begin{enumerate}", "\\item for every surjection of local rings $R \\to A$ with $R$", "a regular local ring, the kernel of $R \\to A$ is generated", "by a regular sequence, and", "\\item for some surjection of local rings $R \\to A$ with $R$", "a regular local ring, the kernel of $R \\to A$ is generated", "by a regular sequence.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $k$ be the residue field of $A$. If the characteristic of", "$k$ is $p > 0$, then we denote $\\Lambda$ a Cohen ring", "(Algebra, Definition \\ref{algebra-definition-cohen-ring})", "with residue field $k$ (Algebra, Lemma \\ref{algebra-lemma-cohen-rings-exist}).", "If the characteristic of $k$ is $0$ we set $\\Lambda = k$.", "Recall that $\\Lambda[[x_1, \\ldots, x_n]]$ for any $n$", "is formally smooth over $\\mathbf{Z}$, resp.\\ $\\mathbf{Q}$", "in the $\\mathfrak m$-adic topology, see", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-power-series-ring-over-Cohen-fs}.", "Fix a surjection $\\Lambda[[x_1, \\ldots, x_n]] \\to A$ as in", "the Cohen structure theorem", "(Algebra, Theorem \\ref{algebra-theorem-cohen-structure-theorem}).", "\\medskip\\noindent", "Let $R \\to A$ be a surjection from a regular local ring $R$.", "Let $f_1, \\ldots, f_r$ be a minimal sequence of generators", "of $\\Ker(R \\to A)$. We will use without further mention", "that an ideal in a Noetherian local ring is generated by a regular", "sequence if and only if any minimal set of generators is a", "regular sequence. Observe that $f_1, \\ldots, f_r$", "is a regular sequence in $R$ if and only if $f_1, \\ldots, f_r$", "is a regular sequence in the completion $R^\\wedge$ by", "Algebra, Lemmas \\ref{algebra-lemma-flat-increases-depth} and", "\\ref{algebra-lemma-completion-flat}.", "Moreover, we have", "$$", "R^\\wedge/(f_1, \\ldots, f_r)R^\\wedge =", "(R/(f_1, \\ldots, f_n))^\\wedge = A^\\wedge = A", "$$", "because $A$ is $\\mathfrak m_A$-adically complete (first equality by", "Algebra, Lemma \\ref{algebra-lemma-completion-tensor}). Finally,", "the ring $R^\\wedge$ is regular since $R$ is regular", "(More on Algebra, Lemma \\ref{more-algebra-lemma-completion-regular}).", "Hence we may assume $R$ is complete.", "\\medskip\\noindent", "If $R$ is complete we can choose a map", "$\\Lambda[[x_1, \\ldots, x_n]] \\to R$ lifting the given map", "$\\Lambda[[x_1, \\ldots, x_n]] \\to A$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-lift-continuous}.", "By adding some more variables $y_1, \\ldots, y_m$ mapping", "to generators of the kernel of $R \\to A$ we may assume that", "$\\Lambda[[x_1, \\ldots, x_n, y_1, \\ldots, y_m]] \\to R$ is surjective", "(some details omitted). Then we can consider the commutative diagram", "$$", "\\xymatrix{", "\\Lambda[[x_1, \\ldots, x_n, y_1, \\ldots, y_m]] \\ar[r] \\ar[d] & R \\ar[d] \\\\", "\\Lambda[[x_1, \\ldots, x_n]] \\ar[r] & A", "}", "$$", "By Algebra, Lemma \\ref{algebra-lemma-ci-well-defined} we see that", "the condition for $R \\to A$ is equivalent to the condition for", "the fixed chosen map", "$\\Lambda[[x_1, \\ldots, x_n]] \\to A$. This finishes the proof of the lemma." ], "refs": [ "algebra-definition-cohen-ring", "algebra-lemma-cohen-rings-exist", "more-algebra-lemma-power-series-ring-over-Cohen-fs", "algebra-theorem-cohen-structure-theorem", "algebra-lemma-flat-increases-depth", "algebra-lemma-completion-flat", "algebra-lemma-completion-tensor", "more-algebra-lemma-completion-regular", "more-algebra-lemma-lift-continuous", "algebra-lemma-ci-well-defined" ], "ref_ids": [ 1550, 1329, 10027, 327, 740, 870, 869, 10045, 10016, 1168 ] } ], "ref_ids": [] }, { "id": 1677, "type": "theorem", "label": "dpa-lemma-quotient-regular-ring-by-regular-sequence", "categories": [ "dpa" ], "title": "dpa-lemma-quotient-regular-ring-by-regular-sequence", "contents": [ "Let $R$ be a regular ring. Let $\\mathfrak p \\subset R$ be a prime.", "Let $f_1, \\ldots, f_r \\in \\mathfrak p$ be a regular sequence.", "Then the completion of", "$$", "A = (R/(f_1, \\ldots, f_r))_\\mathfrak p =", "R_\\mathfrak p/(f_1, \\ldots, f_r)R_\\mathfrak p", "$$", "is a complete intersection in the sense defined above." ], "refs": [], "proofs": [ { "contents": [ "The completion of $A$ is equal to", "$A^\\wedge = R_\\mathfrak p^\\wedge/(f_1, \\ldots, f_r)R_\\mathfrak p^\\wedge$", "because completion for finite modules over the Noetherian ring", "$R_\\mathfrak p$ is exact", "(Algebra, Lemma \\ref{algebra-lemma-completion-tensor}).", "The image of the sequence $f_1, \\ldots, f_r$ in $R_\\mathfrak p$", "is a regular sequence by", "Algebra, Lemmas \\ref{algebra-lemma-completion-flat} and", "\\ref{algebra-lemma-flat-increases-depth}.", "Moreover, $R_\\mathfrak p^\\wedge$ is a regular local ring by", "More on Algebra, Lemma \\ref{more-algebra-lemma-completion-regular}.", "Hence the result holds by our definition of complete", "intersection for complete local rings." ], "refs": [ "algebra-lemma-completion-tensor", "algebra-lemma-completion-flat", "algebra-lemma-flat-increases-depth", "more-algebra-lemma-completion-regular" ], "ref_ids": [ 869, 870, 740, 10045 ] } ], "ref_ids": [] }, { "id": 1678, "type": "theorem", "label": "dpa-lemma-quotient-regular-ring", "categories": [ "dpa" ], "title": "dpa-lemma-quotient-regular-ring", "contents": [ "Let $R$ be a regular ring. Let $\\mathfrak p \\subset R$ be a prime.", "Let $I \\subset \\mathfrak p$ be an ideal.", "Set $A = (R/I)_\\mathfrak p = R_\\mathfrak p/I_\\mathfrak p$.", "The following are equivalent", "\\begin{enumerate}", "\\item the completion of $A$", "is a complete intersection in the sense above,", "\\item $I_\\mathfrak p \\subset R_\\mathfrak p$ is generated", "by a regular sequence,", "\\item the module $(I/I^2)_\\mathfrak p$ can be generated by", "$\\dim(R_\\mathfrak p) - \\dim(A)$ elements,", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may and do replace $R$ by its localization at $\\mathfrak p$.", "Then $\\mathfrak p = \\mathfrak m$ is the maximal ideal of $R$", "and $A = R/I$. Let $f_1, \\ldots, f_r \\in I$ be a minimal sequence", "of generators. The completion of $A$ is equal to", "$A^\\wedge = R^\\wedge/(f_1, \\ldots, f_r)R^\\wedge$", "because completion for finite modules over the Noetherian ring", "$R_\\mathfrak p$ is exact", "(Algebra, Lemma \\ref{algebra-lemma-completion-tensor}).", "\\medskip\\noindent", "If (1) holds, then the image of the sequence $f_1, \\ldots, f_r$ in $R^\\wedge$", "is a regular sequence by assumption. Hence it is a regular sequence", "in $R$ by Algebra, Lemmas \\ref{algebra-lemma-completion-flat} and", "\\ref{algebra-lemma-flat-increases-depth}. Thus (1) implies (2).", "\\medskip\\noindent", "Assume (3) holds. Set $c = \\dim(R) - \\dim(A)$ and let $f_1, \\ldots, f_c \\in I$", "map to generators of $I/I^2$. by Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK})", "we see that $I = (f_1, \\ldots, f_c)$. Since $R$ is regular and hence", "Cohen-Macaulay (Algebra, Proposition \\ref{algebra-proposition-CM-module})", "we see that $f_1, \\ldots, f_c$ is a regular sequence by", "Algebra, Proposition \\ref{algebra-proposition-CM-module}.", "Thus (3) implies (2).", "Finally, (2) implies (1) by", "Lemma \\ref{lemma-quotient-regular-ring-by-regular-sequence}." ], "refs": [ "algebra-lemma-completion-tensor", "algebra-lemma-completion-flat", "algebra-lemma-flat-increases-depth", "algebra-lemma-NAK", "algebra-proposition-CM-module", "algebra-proposition-CM-module", "dpa-lemma-quotient-regular-ring-by-regular-sequence" ], "ref_ids": [ 869, 870, 740, 401, 1420, 1420, 1677 ] } ], "ref_ids": [] }, { "id": 1679, "type": "theorem", "label": "dpa-lemma-ci-good", "categories": [ "dpa" ], "title": "dpa-lemma-ci-good", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let", "$\\mathfrak p \\subset A$ be a prime ideal. If $A$ is a complete", "intersection, then $A_\\mathfrak p$ is a complete intersection too." ], "refs": [], "proofs": [ { "contents": [ "Choose a prime $\\mathfrak q$ of $A^\\wedge$ lying over $\\mathfrak p$", "(this is possible as $A \\to A^\\wedge$ is faithfully flat by", "Algebra, Lemma \\ref{algebra-lemma-completion-faithfully-flat}).", "Then $A_\\mathfrak p \\to (A^\\wedge)_\\mathfrak q$ is a flat local", "ring homomorphism. Thus by Proposition \\ref{proposition-avramov}", "we see that $A_\\mathfrak p$ is a complete intersection if and only if", "$(A^\\wedge)_\\mathfrak q$ is a complete intersection. Thus it suffices", "to prove the lemma in case $A$ is complete (this is the key step", "of the proof).", "\\medskip\\noindent", "Assume $A$ is complete. By definition we may write", "$A = R/(f_1, \\ldots, f_r)$ for some regular sequence", "$f_1, \\ldots, f_r$ in a regular local ring $R$.", "Let $\\mathfrak q \\subset R$ be the prime corresponding to $\\mathfrak p$.", "Observe that $f_1, \\ldots, f_r \\in \\mathfrak q$ and that", "$A_\\mathfrak p = R_\\mathfrak q/(f_1, \\ldots, f_r)R_\\mathfrak q$.", "Hence $A_\\mathfrak p$ is a complete intersection by", "Lemma \\ref{lemma-quotient-regular-ring-by-regular-sequence}." ], "refs": [ "algebra-lemma-completion-faithfully-flat", "dpa-proposition-avramov", "dpa-lemma-quotient-regular-ring-by-regular-sequence" ], "ref_ids": [ 871, 1693, 1677 ] } ], "ref_ids": [] }, { "id": 1680, "type": "theorem", "label": "dpa-lemma-check-lci-at-maximal-ideals", "categories": [ "dpa" ], "title": "dpa-lemma-check-lci-at-maximal-ideals", "contents": [ "Let $A$ be a Noetherian ring. Then $A$ is a local complete intersection", "if and only if $A_\\mathfrak m$ is a complete intersection for every", "maximal ideal $\\mathfrak m$ of $A$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from Lemma \\ref{lemma-ci-good} and the definitions." ], "refs": [ "dpa-lemma-ci-good" ], "ref_ids": [ 1679 ] } ], "ref_ids": [] }, { "id": 1681, "type": "theorem", "label": "dpa-lemma-check-lci-agrees", "categories": [ "dpa" ], "title": "dpa-lemma-check-lci-agrees", "contents": [ "Let $S$ be a finite type algebra over a field $k$.", "\\begin{enumerate}", "\\item for a prime $\\mathfrak q \\subset S$ the local ring $S_\\mathfrak q$", "is a complete intersection in the sense of", "Algebra, Definition \\ref{algebra-definition-lci-local-ring}", "if and only if $S_\\mathfrak q$ is a complete", "intersection in the sense of Definition \\ref{definition-lci}, and", "\\item $S$ is a local complete intersection in the sense of", "Algebra, Definition \\ref{algebra-definition-lci-field}", "if and only if $S$ is a local complete", "intersection in the sense of Definition \\ref{definition-lci}.", "\\end{enumerate}" ], "refs": [ "algebra-definition-lci-local-ring", "dpa-definition-lci", "algebra-definition-lci-field", "dpa-definition-lci" ], "proofs": [ { "contents": [ "Proof of (1). Let $k[x_1, \\ldots, x_n] \\to S$ be a surjection.", "Let $\\mathfrak p \\subset k[x_1, \\ldots, x_n]$ be the prime ideal", "corresponding to $\\mathfrak q$.", "Let $I \\subset k[x_1, \\ldots, x_n]$ be the kernel of our surjection.", "Note that $k[x_1, \\ldots, x_n]_\\mathfrak p \\to S_\\mathfrak q$", "is surjective with kernel $I_\\mathfrak p$. Observe that", "$k[x_1, \\ldots, x_n]$ is a regular ring by", "Algebra, Proposition \\ref{algebra-proposition-finite-gl-dim-polynomial-ring}.", "Hence the equivalence of the two notions in (1) follows by", "combining", "Lemma \\ref{lemma-quotient-regular-ring}", "with Algebra, Lemma \\ref{algebra-lemma-lci-local}.", "\\medskip\\noindent", "Having proved (1) the equivalence in (2) follows from the", "definition and Algebra, Lemma \\ref{algebra-lemma-lci-global}." ], "refs": [ "algebra-proposition-finite-gl-dim-polynomial-ring", "dpa-lemma-quotient-regular-ring", "algebra-lemma-lci-local", "algebra-lemma-lci-global" ], "ref_ids": [ 1424, 1678, 1169, 1171 ] } ], "ref_ids": [ 1531, 1701, 1530, 1701 ] }, { "id": 1682, "type": "theorem", "label": "dpa-lemma-avramov", "categories": [ "dpa" ], "title": "dpa-lemma-avramov", "contents": [ "Let $A \\to B$ be a flat local homomorphism of Noetherian local rings.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $B$ is a complete intersection,", "\\item $A$ and $B/\\mathfrak m_A B$ are complete intersections.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Now that the definition makes sense this is a trivial reformulation", "of the (nontrivial) Proposition \\ref{proposition-avramov}." ], "refs": [ "dpa-proposition-avramov" ], "ref_ids": [ 1693 ] } ], "ref_ids": [] }, { "id": 1683, "type": "theorem", "label": "dpa-lemma-ci-map-well-defined", "categories": [ "dpa" ], "title": "dpa-lemma-ci-map-well-defined", "contents": [ "Let $A \\to B$ be a local homomorphism of Noetherian complete local rings.", "The following are equivalent", "\\begin{enumerate}", "\\item for some good factorization $A \\to S \\to B$ the kernel of", "$S \\to B$ is generated by a regular sequence, and", "\\item for every good factorization $A \\to S \\to B$ the kernel of", "$S \\to B$ is generated by a regular sequence.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $A \\to S \\to B$ be a good factorization.", "As $B$ is complete we obtain a factorization", "$A \\to S^\\wedge \\to B$ where $S^\\wedge$ is the completion of $S$.", "Note that this is also a good factorization:", "The ring map $S \\to S^\\wedge$ is flat", "(Algebra, Lemma \\ref{algebra-lemma-completion-flat}),", "hence $A \\to S^\\wedge$ is flat.", "The ring $S^\\wedge/\\mathfrak m_A S^\\wedge = (S/\\mathfrak m_A S)^\\wedge$", "is regular since $S/\\mathfrak m_A S$ is regular", "(More on Algebra, Lemma \\ref{more-algebra-lemma-completion-regular}).", "Let $f_1, \\ldots, f_r$ be a minimal sequence of generators", "of $\\Ker(S \\to B)$. We will use without further mention", "that an ideal in a Noetherian local ring is generated by a regular", "sequence if and only if any minimal set of generators is a", "regular sequence. Observe that $f_1, \\ldots, f_r$", "is a regular sequence in $S$ if and only if $f_1, \\ldots, f_r$", "is a regular sequence in the completion $S^\\wedge$ by", "Algebra, Lemma \\ref{algebra-lemma-flat-increases-depth}.", "Moreover, we have", "$$", "S^\\wedge/(f_1, \\ldots, f_r)R^\\wedge =", "(S/(f_1, \\ldots, f_n))^\\wedge = B^\\wedge = B", "$$", "because $B$ is $\\mathfrak m_B$-adically complete (first equality by", "Algebra, Lemma \\ref{algebra-lemma-completion-tensor}).", "Thus the kernel of $S \\to B$ is generated by a regular sequence", "if and only if the kernel of $S^\\wedge \\to B$ is generated by a", "regular sequence.", "Hence it suffices to consider good factorizations where $S$ is complete.", "\\medskip\\noindent", "Assume we have two factorizations $A \\to S \\to B$ and", "$A \\to S' \\to B$ with $S$ and $S'$ complete. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-dominate-two-surjections}", "the ring $S \\times_B S'$ is a Noetherian complete local ring.", "Hence, using More on Algebra, Lemma", "\\ref{more-algebra-lemma-embed-map-Noetherian-complete-local-rings}", "we can choose a good factorization $A \\to S'' \\to S \\times_B S'$", "with $S''$ complete. Thus it suffices to show:", "If $A \\to S' \\to S \\to B$ are comparable good factorizations,", "then $\\Ker(S \\to B)$ is generated by a regular sequence", "if and only if $\\Ker(S' \\to B)$ is generated by a regular sequence.", "\\medskip\\noindent", "Let $A \\to S' \\to S \\to B$ be comparable good factorizations.", "First, since $S'/\\mathfrak m_R S' \\to S/\\mathfrak m_R S$ is", "a surjection of regular local rings, the kernel is generated", "by a regular sequence", "$\\overline{x}_1, \\ldots, \\overline{x}_c \\in", "\\mathfrak m_{S'}/\\mathfrak m_R S'$", "which can be extended to a regular system of parameters for", "the regular local ring $S'/\\mathfrak m_R S'$, see", "(Algebra, Lemma \\ref{algebra-lemma-regular-quotient-regular}).", "Set $I = \\Ker(S' \\to S)$. By flatness of $S$ over $R$ we have", "$$", "I/\\mathfrak m_R I =", "\\Ker(S'/\\mathfrak m_R S' \\to S/\\mathfrak m_R S) =", "(\\overline{x}_1, \\ldots, \\overline{x}_c).", "$$", "Choose lifts $x_1, \\ldots, x_c \\in I$. These lifts form a regular sequence", "generating $I$ as $S'$ is flat over $R$, see", "Algebra, Lemma \\ref{algebra-lemma-grothendieck-regular-sequence}.", "\\medskip\\noindent", "We conclude that if also $\\Ker(S \\to B)$ is generated by a", "regular sequence, then so is $\\Ker(S' \\to B)$, see", "More on Algebra, Lemmas", "\\ref{more-algebra-lemma-join-koszul-regular-sequences} and", "\\ref{more-algebra-lemma-noetherian-finite-all-equivalent}.", "\\medskip\\noindent", "Conversely, assume that $J = \\Ker(S' \\to B)$ is generated", "by a regular sequence. Because the generators $x_1, \\ldots, x_c$", "of $I$ map to linearly independent elements of", "$\\mathfrak m_{S'}/\\mathfrak m_{S'}^2$ we see that", "$I/\\mathfrak m_{S'}I \\to J/\\mathfrak m_{S'}J$ is injective.", "Hence there exists a minimal system of generators", "$x_1, \\ldots, x_c, y_1, \\ldots, y_d$ for $J$.", "Then $x_1, \\ldots, x_c, y_1, \\ldots, y_d$ is a regular sequence", "and it follows that the images of $y_1, \\ldots, y_d$ in $S$", "form a regular sequence generating $\\Ker(S \\to B)$.", "This finishes the proof of the lemma." ], "refs": [ "algebra-lemma-completion-flat", "more-algebra-lemma-completion-regular", "algebra-lemma-flat-increases-depth", "algebra-lemma-completion-tensor", "more-algebra-lemma-dominate-two-surjections", "more-algebra-lemma-embed-map-Noetherian-complete-local-rings", "algebra-lemma-regular-quotient-regular", "algebra-lemma-grothendieck-regular-sequence", "more-algebra-lemma-join-koszul-regular-sequences", "more-algebra-lemma-noetherian-finite-all-equivalent" ], "ref_ids": [ 870, 10045, 740, 869, 10030, 10029, 942, 885, 9984, 9978 ] } ], "ref_ids": [] }, { "id": 1684, "type": "theorem", "label": "dpa-lemma-well-defined-if-you-can-find-good-factorization", "categories": [ "dpa" ], "title": "dpa-lemma-well-defined-if-you-can-find-good-factorization", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "S \\ar[r] & B \\\\", "& A \\ar[lu] \\ar[u]", "}", "$$", "of Noetherian local rings with $S \\to B$ surjective, $A \\to S$ flat, and", "$S/\\mathfrak m_A S$ a regular local ring. The following are equivalent", "\\begin{enumerate}", "\\item $\\Ker(S \\to B)$ is generated by a regular sequence, and", "\\item $A^\\wedge \\to B^\\wedge$ is a complete intersection homomorphism", "as defined above.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: the proof is identical to the argument given in", "the first paragraph of the proof of Lemma \\ref{lemma-ci-map-well-defined}." ], "refs": [ "dpa-lemma-ci-map-well-defined" ], "ref_ids": [ 1683 ] } ], "ref_ids": [] }, { "id": 1685, "type": "theorem", "label": "dpa-lemma-finite-type-lci-map", "categories": [ "dpa" ], "title": "dpa-lemma-finite-type-lci-map", "contents": [ "Let $A$ be a Noetherian ring.", "Let $A \\to B$ be a finite type ring map.", "The following are equivalent", "\\begin{enumerate}", "\\item $A \\to B$ is a local complete intersection in the sense of", "More on Algebra, Definition", "\\ref{more-algebra-definition-local-complete-intersection},", "\\item for every prime $\\mathfrak q \\subset B$ and with", "$\\mathfrak p = A \\cap \\mathfrak q$ the ring map", "$(A_\\mathfrak p)^\\wedge \\to (B_\\mathfrak q)^\\wedge$ is", "a complete intersection homomorphism in the sense defined above.", "\\end{enumerate}" ], "refs": [ "more-algebra-definition-local-complete-intersection" ], "proofs": [ { "contents": [ "Choose a surjection $R = A[x_1, \\ldots, x_n] \\to B$.", "Observe that $A \\to R$ is flat with regular fibres.", "Let $I$ be the kernel of $R \\to B$.", "Assume (2). Then we see that", "$I$ is locally generated by a regular sequence", "by", "Lemma \\ref{lemma-well-defined-if-you-can-find-good-factorization}", "and", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-neighbourhood}.", "In other words, (1) holds.", "Conversely, assume (1). Then after localizing on $R$ and $B$", "we can assume that $I$ is generated by a Koszul regular sequence.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-noetherian-finite-all-equivalent}", "we find that $I$ is locally generated by a regular sequence.", "Hence (2) hold by", "Lemma \\ref{lemma-well-defined-if-you-can-find-good-factorization}.", "Some details omitted." ], "refs": [ "dpa-lemma-well-defined-if-you-can-find-good-factorization", "algebra-lemma-regular-sequence-in-neighbourhood", "more-algebra-lemma-noetherian-finite-all-equivalent", "dpa-lemma-well-defined-if-you-can-find-good-factorization" ], "ref_ids": [ 1684, 741, 9978, 1684 ] } ], "ref_ids": [ 10609 ] }, { "id": 1686, "type": "theorem", "label": "dpa-lemma-avramov-map-finite-type", "categories": [ "dpa" ], "title": "dpa-lemma-avramov-map-finite-type", "contents": [ "Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map", "such that the image of $\\Spec(B) \\to \\Spec(A)$ contains all closed", "points of $\\Spec(A)$. Then the following are equivalent", "\\begin{enumerate}", "\\item $B$ is a complete intersection and $A \\to B$ has finite", "tor dimension,", "\\item $A$ is a complete intersection and $A \\to B$ is a local complete", "intersection in the sense of More on Algebra, Definition", "\\ref{more-algebra-definition-local-complete-intersection}.", "\\end{enumerate}" ], "refs": [ "more-algebra-definition-local-complete-intersection" ], "proofs": [ { "contents": [ "This is a reformulation of Proposition \\ref{proposition-avramov-map}", "via Lemma \\ref{lemma-finite-type-lci-map}.", "We omit the details." ], "refs": [ "dpa-proposition-avramov-map", "dpa-lemma-finite-type-lci-map" ], "ref_ids": [ 1694, 1685 ] } ], "ref_ids": [ 10609 ] }, { "id": 1687, "type": "theorem", "label": "dpa-lemma-local-perfect-diagonal", "categories": [ "dpa" ], "title": "dpa-lemma-local-perfect-diagonal", "contents": [ "Let $A \\to B$ be a local ring homomorphism of Noetherian local rings such that", "$B$ is flat and essentially of finite type over $A$. If", "$$", "B \\otimes_A B \\longrightarrow B", "$$", "is a perfect ring map, i.e., if $B$ has finite tor dimension over", "$B \\otimes_A B$, then $B$ is the localization of a smooth $A$-algebra." ], "refs": [], "proofs": [ { "contents": [ "As $B$ is essentially of finite type over $A$, so is $B \\otimes_A B$ and", "in particular $B \\otimes_A B$ is Noetherian. Hence the quotient $B$ of", "$B \\otimes_A B$ is pseudo-coherent over $B \\otimes_A B$", "(More on Algebra, Lemma \\ref{more-algebra-lemma-Noetherian-pseudo-coherent})", "which explains why perfectness of the ring map (More on Algebra, Definition", "\\ref{more-algebra-definition-pseudo-coherent-perfect}) agrees with the", "condition of finite tor dimension.", "\\medskip\\noindent", "We may write $B = R/K$ where $R$ is the localization of $A[x_1, \\ldots, x_n]$", "at a prime ideal and $K \\subset R$ is an ideal. Denote", "$\\mathfrak m \\subset R \\otimes_A R$ the maximal ideal which is the inverse", "image of the maximal ideal of $B$ via the surjection", "$R \\otimes_A R \\to B \\otimes_A B \\to B$. Then we have surjections", "$$", "(R \\otimes_A R)_\\mathfrak m \\to (B \\otimes_A B)_\\mathfrak m \\to B", "$$", "and hence ideals $I \\subset J \\subset (R \\otimes_A R)_\\mathfrak m$", "as in Lemma \\ref{lemma-injective}. We conclude that", "$I/\\mathfrak m I \\to J/\\mathfrak m J$ is injective.", "\\medskip\\noindent", "Let $K = (f_1, \\ldots, f_r)$ with $r$ minimal. We may and do assume that", "$f_i \\in R$ is the image of an element of $A[x_1, \\ldots, x_n]$ which we", "also denote $f_i$. Observe that $I$ is generated", "by $f_1 \\otimes 1, \\ldots, f_r \\otimes 1$ and", "$1 \\otimes f_1, \\ldots, 1 \\otimes f_r$. We claim that this is a minimal", "set of generators of $I$. Namely, if $\\kappa$ is the common residue field", "of $R$, $B$, $(R \\otimes_A R)_\\mathfrak m$, and $(B \\otimes_A B)_\\mathfrak m$", "then we have a map", "$R \\otimes_A R \\to R \\otimes_A \\kappa \\oplus \\kappa \\otimes_A R$", "which factors through $(R \\otimes_A R)_\\mathfrak m$. Since $B$ is", "flat over $A$ and since we have the short exact sequence", "$0 \\to K \\to R \\to B \\to 0$ we see that", "$K \\otimes_A \\kappa \\subset R \\otimes_A \\kappa$, see", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}.", "Thus restricting the map", "$(R \\otimes_A R)_\\mathfrak m \\to R \\otimes_A \\kappa \\oplus \\kappa \\otimes_A R$", "to $I$ we obtain a map", "$$", "I \\to K \\otimes_A \\kappa \\oplus \\kappa \\otimes_A K \\to", "K \\otimes_B \\kappa \\oplus \\kappa \\otimes_B K.", "$$", "The elements", "$f_1 \\otimes 1, \\ldots, f_r \\otimes 1, 1 \\otimes f_1, \\ldots, 1 \\otimes f_r$", "map to a basis of the target of this map, since by Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK})", "$f_1, \\ldots, f_r$ map to a basis of $K \\otimes_B \\kappa$.", "This proves our claim.", "\\medskip\\noindent", "The ideal $J$ is generated by $f_1 \\otimes 1, \\ldots, f_r \\otimes 1$", "and the elements $x_1 \\otimes 1 - 1 \\otimes x_1, \\ldots,", "x_n \\otimes 1 - 1 \\otimes x_n$ (for the proof it suffices to", "see that these elements are contained in the ideal $J$).", "Now we can write", "$$", "f_i \\otimes 1 - 1 \\otimes f_i =", "\\sum g_{ij} (x_j \\otimes 1 - 1 \\otimes x_j)", "$$", "for some $g_{ij}$ in $(R \\otimes_A R)_\\mathfrak m$. This is a general", "fact about elements of $A[x_1, \\ldots, x_n]$ whose proof we omit.", "Denote $a_{ij} \\in \\kappa$ the image of $g_{ij}$. Another computation", "shows that $a_{ij}$ is the image of $\\partial f_i / \\partial x_j$ in $\\kappa$.", "The injectivity of $I/\\mathfrak m I \\to J/\\mathfrak m J$ and the remarks", "made above forces the matrix $(a_{ij})$ to have maximal rank $r$.", "Set", "$$", "C = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_r)", "$$", "and consider the naive cotangent complex", "$\\NL_{C/A} \\cong (C^{\\oplus r} \\to C^{\\oplus n})$", "where the map is given by the matrix of partial derivatives.", "Thus $\\NL_{C/A} \\otimes_A B$", "is isomorphic to a free $B$-module of rank $n - r$ placed in degree $0$.", "Hence $C_g$ is smooth over $A$ for some $g \\in C$ mapping to a unit", "in $B$, see Algebra, Lemma \\ref{algebra-lemma-smooth-at-point}.", "This finishes the proof." ], "refs": [ "more-algebra-lemma-Noetherian-pseudo-coherent", "more-algebra-definition-pseudo-coherent-perfect", "dpa-lemma-injective", "algebra-lemma-flat-tor-zero", "algebra-lemma-NAK", "algebra-lemma-smooth-at-point" ], "ref_ids": [ 10160, 10631, 1673, 532, 401, 1196 ] } ], "ref_ids": [] }, { "id": 1688, "type": "theorem", "label": "dpa-lemma-perfect-diagonal", "categories": [ "dpa" ], "title": "dpa-lemma-perfect-diagonal", "contents": [ "Let $A \\to B$ be a flat finite type ring map of Noetherian rings. If", "$$", "B \\otimes_A B \\longrightarrow B", "$$", "is a perfect ring map, i.e., if $B$ has finite tor dimension over", "$B \\otimes_A B$, then $B$ is a smooth $A$-algebra." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-local-perfect-diagonal}", "and general facts about smooth ring maps, see", "Algebra, Lemmas \\ref{algebra-lemma-smooth-at-point} and", "\\ref{algebra-lemma-locally-smooth}.", "Alternatively, the reader can slightly modify the proof of", "Lemma \\ref{lemma-local-perfect-diagonal} to prove", "this lemma." ], "refs": [ "dpa-lemma-local-perfect-diagonal", "algebra-lemma-smooth-at-point", "algebra-lemma-locally-smooth", "dpa-lemma-local-perfect-diagonal" ], "ref_ids": [ 1687, 1196, 1197, 1687 ] } ], "ref_ids": [] }, { "id": 1689, "type": "theorem", "label": "dpa-lemma-free-summand-in-ideal-finite-proj-dim", "categories": [ "dpa" ], "title": "dpa-lemma-free-summand-in-ideal-finite-proj-dim", "contents": [ "\\begin{reference}", "\\cite{Vasconcelos}", "\\end{reference}", "Let $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal", "of finite projective dimension over $R$. If $F \\subset I/I^2$ is a", "direct summand isomorphic to $R/I$, then there exists a nonzerodivisor", "$x \\in I$ such that the image of $x$ in $I/I^2$ generates $F$." ], "refs": [], "proofs": [ { "contents": [ "By assumption we may choose a finite free resolution", "$$", "0 \\to R^{\\oplus n_e} \\to R^{\\oplus n_{e-1}} \\to \\ldots \\to", "R^{\\oplus n_1} \\to R \\to R/I \\to 0", "$$", "Then $\\varphi_1 : R^{\\oplus n_1} \\to R$ has rank $1$ and", "we see that $I$ contains a nonzerodivisor $y$ by", "Algebra, Proposition \\ref{algebra-proposition-what-exact}.", "Let $\\mathfrak p_1, \\ldots, \\mathfrak p_n$ be the associated", "primes of $R$, see Algebra, Lemma \\ref{algebra-lemma-finite-ass}.", "Let $I^2 \\subset J \\subset I$ be an ideal such that $J/I^2 = F$.", "Then $J \\not \\subset \\mathfrak p_i$ for all $i$", "as $y^2 \\in J$ and $y^2 \\not \\in \\mathfrak p_i$, see", "Algebra, Lemma \\ref{algebra-lemma-ass-zero-divisors}.", "By Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "we have $J \\not \\subset \\mathfrak m J + I^2$.", "By Algebra, Lemma \\ref{algebra-lemma-silly}", "we can pick $x \\in J$, $x \\not \\in \\mathfrak m J + I^2$ and", "$x \\not \\in \\mathfrak p_i$ for $i = 1, \\ldots, n$.", "Then $x$ is a nonzerodivisor and the image", "of $x$ in $I/I^2$ generates (by Nakayama's lemma)", "the summand $J/I^2 \\cong R/I$." ], "refs": [ "algebra-proposition-what-exact", "algebra-lemma-finite-ass", "algebra-lemma-ass-zero-divisors", "algebra-lemma-NAK", "algebra-lemma-silly" ], "ref_ids": [ 1419, 701, 704, 401, 378 ] } ], "ref_ids": [] }, { "id": 1690, "type": "theorem", "label": "dpa-lemma-vasconcelos", "categories": [ "dpa" ], "title": "dpa-lemma-vasconcelos", "contents": [ "\\begin{reference}", "Local version of \\cite[Theorem 1.1]{Vasconcelos}", "\\end{reference}", "Let $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal", "of finite projective dimension over $R$. If $F \\subset I/I^2$", "is a direct summand free of rank $r$, then there exists a regular sequence", "$x_1, \\ldots, x_r \\in I$ such that $x_1 \\bmod I^2, \\ldots, x_r \\bmod I^2$", "generate $F$." ], "refs": [], "proofs": [ { "contents": [ "If $r = 0$ there is nothing to prove. Assume $r > 0$. We may pick", "$x \\in I$ such that $x$ is a nonzerodivisor and $x \\bmod I^2$", "generates a summand of $F$ isomorphic to $R/I$, see", "Lemma \\ref{lemma-free-summand-in-ideal-finite-proj-dim}.", "Consider the ring $R' = R/(x)$ and the ideal $I' = I/(x)$.", "Of course $R'/I' = R/I$. The short exact sequence", "$$", "0 \\to R/I \\xrightarrow{x} I/xI \\to I' \\to 0", "$$", "splits because the map $I/xI \\to I/I^2$ sends $xR/xI$", "to a direct summand. Now $I/xI = I \\otimes_R^\\mathbf{L} R'$ has", "finite projective dimension over $R'$, see", "More on Algebra, Lemmas \\ref{more-algebra-lemma-perfect-module} and", "\\ref{more-algebra-lemma-pull-perfect}.", "Hence the summand $I'$ has finite projective dimension over $R'$.", "On the other hand, we have the short exact sequence", "$0 \\to xR/xI \\to I/I^2 \\to I'/(I')^2 \\to 0$ and we conclude", "$I'/(I')^2$ has the free direct summand $F' = F/(R/I \\cdot x)$", "of rank $r - 1$. By induction on $r$ we may", "we pick a regular sequence $x'_2, \\ldots, x'_r \\in I'$", "such that there congruence classes freely generate $F'$.", "If $x_1 = x$ and $x_2, \\ldots, x_r$ are any elements lifting", "$x'_1, \\ldots, x'_r$ in $R$, then we see that the lemma holds." ], "refs": [ "dpa-lemma-free-summand-in-ideal-finite-proj-dim", "more-algebra-lemma-perfect-module", "more-algebra-lemma-pull-perfect" ], "ref_ids": [ 1689, 10213, 10219 ] } ], "ref_ids": [] }, { "id": 1691, "type": "theorem", "label": "dpa-lemma-perfect-NL-lci", "categories": [ "dpa" ], "title": "dpa-lemma-perfect-NL-lci", "contents": [ "Let $A \\to B$ be a perfect (More on Algebra, Definition", "\\ref{more-algebra-definition-pseudo-coherent-perfect})", "ring homomorphism of Noetherian rings. Then the following are equivalent", "\\begin{enumerate}", "\\item $\\NL_{B/A}$ has tor-amplitude in $[-1, 0]$,", "\\item $\\NL_{B/A}$ is a perfect object of $D(B)$", "with tor-amplitude in $[-1, 0]$, and", "\\item $A \\to B$ is a local complete intersection", "(More on Algebra, Definition", "\\ref{more-algebra-definition-local-complete-intersection}).", "\\end{enumerate}" ], "refs": [ "more-algebra-definition-pseudo-coherent-perfect", "more-algebra-definition-local-complete-intersection" ], "proofs": [ { "contents": [ "Write $B = A[x_1, \\ldots, x_n]/I$. Then $\\NL_{B/A}$ is represented by", "the complex", "$$", "I/I^2 \\longrightarrow \\bigoplus B \\text{d}x_i", "$$", "of $B$-modules with $I/I^2$ placed in degree $-1$. Since the term in", "degree $0$ is finite free, this complex has tor-amplitude in $[-1, 0]$ if and", "only if $I/I^2$ is a flat $B$-module, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-last-one-flat}.", "Since $I/I^2$ is a finite $B$-module and $B$ is Noetherian, this is true", "if and only if $I/I^2$ is a finite locally free $B$-module", "(Algebra, Lemma \\ref{algebra-lemma-finite-projective}).", "Thus the equivalence of (1) and (2) is clear. Moreover, the equivalence", "of (1) and (3) also follows if we apply", "Proposition \\ref{proposition-regular-ideal}", "(and the observation that a regular ideal is a Koszul regular", "ideal as well as a quasi-regular ideal, see", "More on Algebra, Section \\ref{more-algebra-section-ideals})." ], "refs": [ "more-algebra-lemma-last-one-flat", "algebra-lemma-finite-projective", "dpa-proposition-regular-ideal" ], "ref_ids": [ 10169, 795, 1695 ] } ], "ref_ids": [ 10631, 10609 ] }, { "id": 1692, "type": "theorem", "label": "dpa-lemma-flat-fp-NL-lci", "categories": [ "dpa" ], "title": "dpa-lemma-flat-fp-NL-lci", "contents": [ "Let $A \\to B$ be a flat ring map of finite presentation.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $\\NL_{B/A}$ has tor-amplitude in $[-1, 0]$,", "\\item $\\NL_{B/A}$ is a perfect object of $D(B)$", "with tor-amplitude in $[-1, 0]$,", "\\item $A \\to B$ is syntomic", "(Algebra, Definition \\ref{algebra-definition-lci}), and", "\\item $A \\to B$ is a local complete intersection", "(More on Algebra, Definition", "\\ref{more-algebra-definition-local-complete-intersection}).", "\\end{enumerate}" ], "refs": [ "algebra-definition-lci", "more-algebra-definition-local-complete-intersection" ], "proofs": [ { "contents": [ "The equivalence of (3) and (4) is More on Algebra, Lemma", "\\ref{more-algebra-lemma-syntomic-lci}.", "\\medskip\\noindent", "If $A \\to B$ is syntomic, then we can find a cocartesian diagram", "$$", "\\xymatrix{", "B_0 \\ar[r] & B \\\\", "A_0 \\ar[r] \\ar[u] & A \\ar[u]", "}", "$$", "such that $A_0 \\to B_0$ is syntomic and $A_0$ is Noetherian, see", "Algebra, Lemmas \\ref{algebra-lemma-limit-module-finite-presentation} and", "\\ref{algebra-lemma-colimit-lci}. By Lemma \\ref{lemma-perfect-NL-lci}", "we see that $\\NL_{B_0/A_0}$ is perfect of tor-amplitude in $[-1, 0]$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-NL-flat}", "we conclude the same thing is true for", "$\\NL_{B/A} = \\NL_{B_0/A_0} \\otimes_{B_0}^\\mathbf{L} B$ (see", "also More on Algebra, Lemmas \\ref{more-algebra-lemma-pull-tor-amplitude} and", "\\ref{more-algebra-lemma-pull-perfect}).", "This proves that (3) implies (2).", "\\medskip\\noindent", "Assume (1). By More on Algebra, Lemma", "\\ref{more-algebra-lemma-base-change-NL-flat}", "for every ring map $A \\to k$ where", "$k$ is a field, we see that $\\NL_{B \\otimes_A k/k}$ has", "tor-amplitude in $[-1, 0]$ (see", "More on Algebra, Lemma \\ref{more-algebra-lemma-pull-tor-amplitude}).", "Hence by Lemma \\ref{lemma-perfect-NL-lci} we see that $k \\to B \\otimes_A k$ is", "a local complete intersection homomorphism. Thus $A \\to B$", "is syntomic by definition. This proves (1) implies (3)", "and finishes the proof." ], "refs": [ "more-algebra-lemma-syntomic-lci", "algebra-lemma-limit-module-finite-presentation", "algebra-lemma-colimit-lci", "dpa-lemma-perfect-NL-lci", "more-algebra-lemma-base-change-NL-flat", "more-algebra-lemma-pull-tor-amplitude", "more-algebra-lemma-pull-perfect", "more-algebra-lemma-base-change-NL-flat", "more-algebra-lemma-pull-tor-amplitude", "dpa-lemma-perfect-NL-lci" ], "ref_ids": [ 10002, 1106, 1397, 1691, 10310, 10180, 10219, 10310, 10180, 1691 ] } ], "ref_ids": [ 1532, 10609 ] }, { "id": 1693, "type": "theorem", "label": "dpa-proposition-avramov", "categories": [ "dpa" ], "title": "dpa-proposition-avramov", "contents": [ "Let $A \\to B$ be a flat local homomorphism of Noetherian local rings.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $B^\\wedge$ is a complete intersection,", "\\item $A^\\wedge$ and $(B/\\mathfrak m_A B)^\\wedge$ are complete intersections.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the diagram", "$$", "\\xymatrix{", "B \\ar[r] & B^\\wedge \\\\", "A \\ar[u] \\ar[r] & A^\\wedge \\ar[u]", "}", "$$", "Since the horizontal maps are faithfully flat", "(Algebra, Lemma \\ref{algebra-lemma-completion-faithfully-flat})", "we conclude that the right vertical arrow is flat", "(for example by Algebra, Lemma", "\\ref{algebra-lemma-criterion-flatness-fibre-Noetherian}).", "Moreover, we have", "$(B/\\mathfrak m_A B)^\\wedge = B^\\wedge/\\mathfrak m_{A^\\wedge} B^\\wedge$", "by Algebra, Lemma \\ref{algebra-lemma-completion-tensor}.", "Thus we may assume $A$ and $B$ are complete local Noetherian rings.", "\\medskip\\noindent", "Assume $A$ and $B$ are complete local Noetherian rings.", "Choose a diagram", "$$", "\\xymatrix{", "S \\ar[r] & B \\\\", "R \\ar[u] \\ar[r] & A \\ar[u]", "}", "$$", "as in More on Algebra, Lemma", "\\ref{more-algebra-lemma-embed-map-Noetherian-complete-local-rings}.", "Let $I = \\Ker(R \\to A)$ and $J = \\Ker(S \\to B)$.", "Note that since $R/I = A \\to B = S/J$ is flat the map", "$J/IS \\otimes_R R/\\mathfrak m_R \\to J/J \\cap \\mathfrak m_R S$", "is an isomorphism. Hence a minimal system of generators of $J/IS$", "maps to a minimal system of generators of", "$\\Ker(S/\\mathfrak m_R S \\to B/\\mathfrak m_A B)$.", "Finally, $S/\\mathfrak m_R S$ is a regular local ring.", "\\medskip\\noindent", "Assume (1) holds, i.e., $J$ is generated by a regular sequence.", "Since $A = R/I \\to B = S/J$ is flat we see", "Lemma \\ref{lemma-perfect-map-ci} applies and we deduce", "that $I$ and $J/IS$ are generated by regular sequences.", "We have $\\dim(B) = \\dim(A) + \\dim(B/\\mathfrak m_A B)$ and", "$\\dim(S/IS) = \\dim(A) + \\dim(S/\\mathfrak m_R S)$", "(Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}).", "Thus $J/IS$ is generated by", "$$", "\\dim(S/J) - \\dim(S/IS) = \\dim(S/\\mathfrak m_R S) - \\dim(B/\\mathfrak m_A B)", "$$", "elements (Algebra, Lemma \\ref{algebra-lemma-one-equation}).", "It follows that $\\Ker(S/\\mathfrak m_R S \\to B/\\mathfrak m_A B)$", "is generated by the same number of elements (see above).", "Hence $\\Ker(S/\\mathfrak m_R S \\to B/\\mathfrak m_A B)$", "is generated by a regular sequence, see for example", "Lemma \\ref{lemma-quotient-regular-ring}.", "In this way we see that (2) holds.", "\\medskip\\noindent", "If (2) holds, then $I$ and $J/J \\cap \\mathfrak m_RS$", "are generated by regular sequences. Lifting these generators", "(see above), using flatness of $R/I \\to S/IS$,", "and using Grothendieck's lemma", "(Algebra, Lemma \\ref{algebra-lemma-grothendieck-regular-sequence})", "we find that $J/IS$ is generated by a regular sequence in $S/IS$.", "Thus Lemma \\ref{lemma-perfect-map-ci} tells us that $J$", "is generated by a regular sequence, whence (1) holds." ], "refs": [ "algebra-lemma-completion-faithfully-flat", "algebra-lemma-criterion-flatness-fibre-Noetherian", "algebra-lemma-completion-tensor", "more-algebra-lemma-embed-map-Noetherian-complete-local-rings", "dpa-lemma-perfect-map-ci", "algebra-lemma-dimension-base-fibre-equals-total", "dpa-lemma-quotient-regular-ring", "algebra-lemma-grothendieck-regular-sequence", "dpa-lemma-perfect-map-ci" ], "ref_ids": [ 871, 897, 869, 10029, 1675, 987, 1678, 885, 1675 ] } ], "ref_ids": [] }, { "id": 1694, "type": "theorem", "label": "dpa-proposition-avramov-map", "categories": [ "dpa" ], "title": "dpa-proposition-avramov-map", "contents": [ "Let $A \\to B$ be a local homomorphism of Noetherian local rings.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $B$ is a complete intersection and", "$\\text{Tor}^A_p(B, A/\\mathfrak m_A)$ is nonzero for only finitely many $p$,", "\\item $A$ is a complete intersection and", "$A^\\wedge \\to B^\\wedge$ is a complete intersection homomorphism", "in the sense defined above.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $F_\\bullet \\to A/\\mathfrak m_A$ be a resolution by finite", "free $A$-modules. Observe that", "$\\text{Tor}^A_p(B, A/\\mathfrak m_A)$", "is the $p$th homology of the complex $F_\\bullet \\otimes_A B$.", "Let $F_\\bullet^\\wedge = F_\\bullet \\otimes_A A^\\wedge$ be the completion.", "Then $F_\\bullet^\\wedge$ is a resolution of $A^\\wedge/\\mathfrak m_{A^\\wedge}$", "by finite free $A^\\wedge$-modules (as $A \\to A^\\wedge$ is flat and completion", "on finite modules is exact, see", "Algebra, Lemmas \\ref{algebra-lemma-completion-tensor} and", "\\ref{algebra-lemma-completion-flat}).", "It follows that", "$$", "F_\\bullet^\\wedge \\otimes_{A^\\wedge} B^\\wedge =", "F_\\bullet \\otimes_A B \\otimes_B B^\\wedge", "$$", "By flatness of $B \\to B^\\wedge$ we conclude that", "$$", "\\text{Tor}^{A^\\wedge}_p(B^\\wedge, A^\\wedge/\\mathfrak m_{A^\\wedge}) =", "\\text{Tor}^A_p(B, A/\\mathfrak m_A) \\otimes_B B^\\wedge", "$$", "In this way we see that the condition in (1) on the local ring map $A \\to B$", "is equivalent to the same condition for the local ring map", "$A^\\wedge \\to B^\\wedge$.", "Thus we may assume $A$ and $B$ are complete local Noetherian rings", "(since the other conditions are formulated in terms of the completions", "in any case).", "\\medskip\\noindent", "Assume $A$ and $B$ are complete local Noetherian rings.", "Choose a diagram", "$$", "\\xymatrix{", "S \\ar[r] & B \\\\", "R \\ar[u] \\ar[r] & A \\ar[u]", "}", "$$", "as in More on Algebra, Lemma", "\\ref{more-algebra-lemma-embed-map-Noetherian-complete-local-rings}.", "Let $I = \\Ker(R \\to A)$ and $J = \\Ker(S \\to B)$.", "The proposition now follows from Lemma \\ref{lemma-perfect-map-ci}." ], "refs": [ "algebra-lemma-completion-tensor", "algebra-lemma-completion-flat", "more-algebra-lemma-embed-map-Noetherian-complete-local-rings", "dpa-lemma-perfect-map-ci" ], "ref_ids": [ 869, 870, 10029, 1675 ] } ], "ref_ids": [] }, { "id": 1695, "type": "theorem", "label": "dpa-proposition-regular-ideal", "categories": [ "dpa" ], "title": "dpa-proposition-regular-ideal", "contents": [ "\\begin{reference}", "Variant of \\cite[Corollary 1]{Vasconcelos}. See also", "\\cite{Iyengar} and \\cite{Ferrand-lci}.", "\\end{reference}", "Let $R$ be a Noetherian ring. Let $I \\subset R$ be an ideal", "which has finite projective dimension and such that $I/I^2$ is", "finite locally free over $R/I$. Then $I$ is a regular ideal", "(More on Algebra, Definition \\ref{more-algebra-definition-regular-ideal})." ], "refs": [ "more-algebra-definition-regular-ideal" ], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-neighbourhood}", "it suffices to show that $I_\\mathfrak p \\subset R_\\mathfrak p$ is generated", "by a regular sequence for every $\\mathfrak p \\supset I$. Thus we may", "assume $R$ is local. If $I/I^2$ has rank $r$, then by", "Lemma \\ref{lemma-vasconcelos} we find a regular sequence", "$x_1, \\ldots, x_r \\in I$ generating $I/I^2$. By", "Nakayama (Algebra, Lemma \\ref{algebra-lemma-NAK})", "we conclude that $I$ is generated by $x_1, \\ldots, x_r$." ], "refs": [ "algebra-lemma-regular-sequence-in-neighbourhood", "dpa-lemma-vasconcelos", "algebra-lemma-NAK" ], "ref_ids": [ 741, 1690, 401 ] } ], "ref_ids": [ 10608 ] }, { "id": 1706, "type": "theorem", "label": "moduli-lemma-coherent-diagonal-affine-fp", "categories": [ "moduli" ], "title": "moduli-lemma-coherent-diagonal-affine-fp", "contents": [ "The diagonal of $\\Cohstack_{X/B}$ over $B$ is affine", "and of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "The representability of the diagonal (by algebraic spaces)", "was shown in Quot, Lemma \\ref{quot-lemma-coherent-diagonal}.", "From the proof we find that we have to show", "$\\mathit{Isom}(\\mathcal{F}, \\mathcal{G}) \\to T$", "is affine and of finite presentation for a pair of", "finitely presented $\\mathcal{O}_{X_T}$-modules", "$\\mathcal{F}$, $\\mathcal{G}$ flat over $T$ with support", "proper over $T$. This was discussed in Section \\ref{section-hom-isom}." ], "refs": [ "quot-lemma-coherent-diagonal" ], "ref_ids": [ 3161 ] } ], "ref_ids": [] }, { "id": 1707, "type": "theorem", "label": "moduli-lemma-coherent-qs-lfp", "categories": [ "moduli" ], "title": "moduli-lemma-coherent-qs-lfp", "contents": [ "The morphism $\\Cohstack_{X/B} \\to B$ is quasi-separated and", "locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "To check $\\Cohstack_{X/B} \\to B$ is quasi-separated we have to", "show that its diagonal is quasi-compact and quasi-separated.", "This is immediate from Lemma \\ref{lemma-coherent-diagonal-affine-fp}.", "To prove that $\\Cohstack_{X/B} \\to B$ is locally of finite", "presentation, we have to show that $\\Cohstack_{X/B} \\to B$", "is limit preserving, see", "Limits of Stacks, Proposition", "\\ref{stacks-limits-proposition-characterize-locally-finite-presentation}.", "This follows from Quot, Lemma \\ref{quot-lemma-coherent-limits}", "(small detail omitted)." ], "refs": [ "moduli-lemma-coherent-diagonal-affine-fp", "stacks-limits-proposition-characterize-locally-finite-presentation", "quot-lemma-coherent-limits" ], "ref_ids": [ 1706, 15024, 3163 ] } ], "ref_ids": [] }, { "id": 1708, "type": "theorem", "label": "moduli-lemma-coherent-existence-part", "categories": [ "moduli" ], "title": "moduli-lemma-coherent-existence-part", "contents": [ "Assume $X \\to B$ is proper as well as of finite presentation.", "Then $\\Cohstack_{X/B} \\to B$ satisfies the existence part", "of the valuative criterion (Morphisms of Stacks, Definition", "\\ref{stacks-morphisms-definition-existence})." ], "refs": [ "stacks-morphisms-definition-existence" ], "proofs": [ { "contents": [ "Taking base change, this immediately reduces to the following", "problem: given a valuation ring $R$ with fraction field $K$ and", "an algebraic space $X$ proper over $R$ and a coherent", "$\\mathcal{O}_{X_K}$-module $\\mathcal{F}_K$, show there exists", "a finitely presented $\\mathcal{O}_X$-module $\\mathcal{F}$", "flat over $R$ whose generic fibre is $\\mathcal{F}_K$.", "Observe that by Flatness on Spaces, Theorem", "\\ref{spaces-flat-theorem-finite-type-flat}", "any finite type quasi-coherent $\\mathcal{O}_X$-module", "$\\mathcal{F}$ flat over $R$ is of finite presentation.", "Denote $j : X_K \\to X$ the embedding of the generic fibre.", "As a base change of the affine morphism $\\Spec(K) \\to \\Spec(R)$", "the morphism $j$ is affine. Thus $j_*\\mathcal{F}_K$ is", "quasi-coherent. Write", "$$", "j_*\\mathcal{F}_K = \\colim \\mathcal{F}_i", "$$", "as a filtered colimit of its finite type quasi-coherent", "$\\mathcal{O}_X$-submodules, see", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-directed-colimit-finite-type}.", "Since $j_*\\mathcal{F}_K$ is a sheaf of $K$-vector spaces over $X$,", "it is flat over $\\Spec(R)$. Thus each $\\mathcal{F}_i$ is flat", "over $R$ as flatness over a valuation ring is the same as being", "torsion free", "(More on Algebra, Lemma", "\\ref{more-algebra-lemma-valuation-ring-torsion-free-flat})", "and torsion freeness is inherited by submodules.", "Finally, we have to show that the map", "$j^*\\mathcal{F}_i \\to \\mathcal{F}_K$", "is an isomorphism for some $i$.", "Since $j^*j_*\\mathcal{F}_K = \\mathcal{F}_K$ (small detail omitted)", "and since $j^*$ is exact, we see that $j^*\\mathcal{F}_i \\to \\mathcal{F}_K$", "is injective for all $i$.", "Since $j^*$ commutes with colimits, we have", "$\\mathcal{F}_K = j^*j_*\\mathcal{F}_K = \\colim j^*\\mathcal{F}_i$.", "Since $\\mathcal{F}_K$ is coherent (i.e., finitely presented),", "there is an $i$ such that $j^*\\mathcal{F}_i$ contains all the", "(finitely many) generators over an affine \\'etale cover of $X$.", "Thus we get surjectivity of $j^*\\mathcal{F}_i \\to \\mathcal{F}_K$", "for $i$ large enough." ], "refs": [ "spaces-flat-theorem-finite-type-flat", "spaces-limits-lemma-directed-colimit-finite-type", "more-algebra-lemma-valuation-ring-torsion-free-flat" ], "ref_ids": [ 7144, 4602, 9920 ] } ], "ref_ids": [ 7630 ] }, { "id": 1709, "type": "theorem", "label": "moduli-lemma-coherent-functorial", "categories": [ "moduli" ], "title": "moduli-lemma-coherent-functorial", "contents": [ "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be a quasi-finite", "morphism of algebraic spaces which are separated and of finite presentation", "over $B$. Then $\\pi_*$ induces a morphism", "$\\Cohstack_{X/B} \\to \\Cohstack_{Y/B}$." ], "refs": [], "proofs": [ { "contents": [ "Let $(T \\to B, \\mathcal{F})$ be an object of $\\Cohstack_{X/B}$.", "We claim", "\\begin{enumerate}", "\\item[(a)] $(T \\to B, \\pi_{T, *}\\mathcal{F})$ is an object", "of $\\Cohstack_{Y/B}$ and", "\\item[(b)] for $T' \\to T$ we have", "$\\pi_{T', *}(X_{T'} \\to X_T)^*\\mathcal{F} =", "(Y_{T'} \\to Y_T)^*\\pi_{T, *}\\mathcal{F}$.", "\\end{enumerate}", "Part (b) guarantees that this construction defines a functor", "$\\Cohstack_{X/B} \\to \\Cohstack_{Y/B}$ as desired.", "\\medskip\\noindent", "Let $i : Z \\to X_T$ be the closed subspace cut out by the zeroth", "fitting ideal of $\\mathcal{F}$", "(Divisors on Spaces, Section", "\\ref{spaces-divisors-section-fitting-ideals}).", "Then $Z \\to B$ is proper by assumption (see", "Derived Categories of Spaces, Section", "\\ref{spaces-perfect-section-proper-over-base}).", "On the other hand $i$ is of finite presentation", "(Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-fitting-ideal-of-finitely-presented} and", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-closed-immersion-finite-presentation}).", "There exists a quasi-coherent $\\mathcal{O}_Z$-module", "$\\mathcal{G}$ of finite type with $i_*\\mathcal{G} = \\mathcal{F}$", "(Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-on-subscheme-cut-out-by-Fit-0}).", "In fact $\\mathcal{G}$ is of finite presentation as an $\\mathcal{O}_Z$-module", "by Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-finite-finitely-presented-module}.", "Observe that $\\mathcal{G}$ is flat over $B$, for example", "because the stalks of $\\mathcal{G}$ and $\\mathcal{F}$ agree", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-stalk-push-closed}).", "Observe that $\\pi_T \\circ i : Z \\to Y_T$ is quasi-finite as a composition", "of quasi-finite morphisms and that", "$\\pi_{T, *}\\mathcal{F} = (\\pi_T \\circ i)_*\\mathcal{G})$.", "Since $i$ is affine, formation of $i_*$ commutes with base change", "(Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-affine-base-change}).", "Therefore we may replace $B$ by $T$, $X$ by $Z$,", "$\\mathcal{F}$ by $\\mathcal{G}$, and $Y$ by $Y_T$", "to reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume that $X \\to B$ is proper. Then $\\pi$ is proper", "by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-permanence}", "and hence finite by", "More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-characterize-finite}.", "Since a finite morphism is affine we see that (b) holds by", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-affine-base-change}.", "On the other hand, $\\pi$ is of finite presentation by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-permanence}.", "Thus $\\pi_{T, *}\\mathcal{F}$ is of finite presentation by", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-finite-finitely-presented-module}.", "Finally, $\\pi_{T, *}\\mathcal{F} $ is flat over $B$ for example", "by looking at stalks using", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-stalk-push-finite}." ], "refs": [ "spaces-divisors-lemma-fitting-ideal-of-finitely-presented", "spaces-morphisms-lemma-closed-immersion-finite-presentation", "spaces-divisors-lemma-on-subscheme-cut-out-by-Fit-0", "spaces-descent-lemma-finite-finitely-presented-module", "spaces-morphisms-lemma-stalk-push-closed", "spaces-cohomology-lemma-affine-base-change", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-more-morphisms-lemma-characterize-finite", "spaces-cohomology-lemma-affine-base-change", "spaces-morphisms-lemma-finite-presentation-permanence", "spaces-descent-lemma-finite-finitely-presented-module", "spaces-cohomology-lemma-stalk-push-finite" ], "ref_ids": [ 12929, 4849, 12930, 9365, 4769, 11295, 4920, 173, 11295, 4846, 9365, 11274 ] } ], "ref_ids": [] }, { "id": 1710, "type": "theorem", "label": "moduli-lemma-coherent-open", "categories": [ "moduli" ], "title": "moduli-lemma-coherent-open", "contents": [ "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be an open immersion", "of algebraic spaces which are separated and of finite presentation over $B$.", "Then the morphism $\\Cohstack_{X/B} \\to \\Cohstack_{Y/B}$ of", "Lemma \\ref{lemma-coherent-functorial} is an open immersion." ], "refs": [ "moduli-lemma-coherent-functorial" ], "proofs": [ { "contents": [ "Omitted. Hint: If $\\mathcal{F}$ is an object of $\\Cohstack_{Y/B}$ over $T$", "and for $t \\in T$ we have $\\text{Supp}(\\mathcal{F}_t) \\subset |X_t|$,", "then the same is true for $t' \\in T$ in a neighbourhood of $t$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1709 ] }, { "id": 1711, "type": "theorem", "label": "moduli-lemma-coherent-closed", "categories": [ "moduli" ], "title": "moduli-lemma-coherent-closed", "contents": [ "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be a closed immersion", "of algebraic spaces which are separated and of finite presentation over $B$.", "Then the morphism $\\Cohstack_{X/B} \\to \\Cohstack_{Y/B}$ of", "Lemma \\ref{lemma-coherent-functorial} is a closed immersion." ], "refs": [ "moduli-lemma-coherent-functorial" ], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\subset \\mathcal{O}_Y$ be the sheaf of ideals cutting", "out $X$ as a closed subspace of $Y$. Recall that $\\pi_*$ induces", "an equivalence between the category of quasi-coherent $\\mathcal{O}_X$-modules", "and the category of quasi-coherent $\\mathcal{O}_Y$-modules annihilated", "by $\\mathcal{I}$, see Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-i-star-equivalence}.", "The same, mutatis mutandis, is true after base by $T \\to B$ with", "$\\mathcal{I}$ replaced by the ideal sheaf", "$\\mathcal{I}_T = \\Im((Y_T \\to Y)^*\\mathcal{I} \\to \\mathcal{O}_{Y_T})$.", "Analyzing the proof of Lemma \\ref{lemma-coherent-functorial}", "we find that the essential image of", "$\\Cohstack_{X/B} \\to \\Cohstack_{Y/B}$", "is exactly the objects $\\xi = (T \\to B, \\mathcal{F})$", "where $\\mathcal{F}$ is annihilated by $\\mathcal{I}_T$.", "In other words, $\\xi$ is in the essential image if and only if", "the multiplication map", "$$", "\\mathcal{F} \\otimes_{\\mathcal{O}_{Y_T}} (Y_T \\to Y)^*\\mathcal{I}", "\\longrightarrow", "\\mathcal{F}", "$$", "is zero and similarly after any further base change $T' \\to T$.", "Note that", "$$", "(Y_{T'} \\to Y_T)^*(", "\\mathcal{F} \\otimes_{\\mathcal{O}_{Y_T}} (Y_T \\to Y)^*\\mathcal{I}) =", "(Y_{T'} \\to Y_T)^*\\mathcal{F} \\otimes_{\\mathcal{O}_{Y_{T'}}}", "(Y_{T'} \\to Y)^*\\mathcal{I})", "$$", "Hence the vanishing of the multiplication map on $T'$", "is representable by a closed subspace of $T$ by", "Flatness on Spaces, Lemma \\ref{spaces-flat-lemma-F-zero-closed-proper}." ], "refs": [ "spaces-morphisms-lemma-i-star-equivalence", "moduli-lemma-coherent-functorial", "spaces-flat-lemma-F-zero-closed-proper" ], "ref_ids": [ 4771, 1709, 7182 ] } ], "ref_ids": [ 1709 ] }, { "id": 1712, "type": "theorem", "label": "moduli-lemma-open-P", "categories": [ "moduli" ], "title": "moduli-lemma-open-P", "contents": [ "In Situation \\ref{situation-numerical} the stack", "$\\Cohstack^P_{X/B}$ is algebraic and", "$$", "\\Cohstack^P_{X/B} \\longrightarrow \\Cohstack_{X/B}", "$$", "is a flat closed immersion. If $I$ is finite or $B$ is locally", "Noetherian, then $\\Cohstack^P_{X/B}$ is an open and closed substack of", "$\\Cohstack_{X/B}$." ], "refs": [], "proofs": [ { "contents": [ "This is immediately clear if $I$ is finite, because the functions", "$t \\mapsto \\chi_i(t)$ are locally constant. If $I$ is infinite, then", "we write", "$$", "I = \\bigcup\\nolimits_{I' \\subset I\\text{ finite}} I'", "$$", "and we denote $P' = P|_{I'}$. Then we have", "$$", "\\Cohstack^P_{X/B} = \\bigcap\\nolimits_{I' \\subset I\\text{ finite}}", "\\Cohstack^{P'}_{X/B}", "$$", "Therefore, $\\Cohstack^P_{X/B}$ is always an algebraic stack and the morphism", "$\\Cohstack^P_{X/B} \\subset \\Cohstack_{X/B}$ is always a flat closed immersion,", "but it may no longer be an open substack. (We leave it to the reader to", "make examples). However, if $B$ is locally Noetherian, then so", "is $\\Cohstack_{X/B}$ by Lemma \\ref{lemma-coherent-qs-lfp} and", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-locally-finite-type-locally-noetherian}.", "Hence if $U \\to \\Cohstack_{X/B}$ is a smooth surjective morphism", "where $U$ is a locally Noetherian scheme, then the inverse images of", "the open and closed substacks $\\Cohstack^{P'}_{X/B}$", "have an open intersection in $U$ (because connected components of", "locally Noetherian topological spaces are open).", "Thus the result in this case." ], "refs": [ "moduli-lemma-coherent-qs-lfp", "stacks-morphisms-lemma-locally-finite-type-locally-noetherian" ], "ref_ids": [ 1707, 7462 ] } ], "ref_ids": [] }, { "id": 1713, "type": "theorem", "label": "moduli-lemma-finite-list-perfect-objects", "categories": [ "moduli" ], "title": "moduli-lemma-finite-list-perfect-objects", "contents": [ "Let $f : X \\to B$ be as in the introduction to this section.", "Let $E_1, \\ldots, E_r \\in D(\\mathcal{O}_X)$ be perfect.", "Let $I = \\mathbf{Z}^{\\oplus r}$ and consider the map", "$$", "I \\longrightarrow D(\\mathcal{O}_X),\\quad", "(n_1, \\ldots, n_r) \\longmapsto", "E_1^{\\otimes n_1}", "\\otimes \\ldots \\otimes", "E_r^{\\otimes n_r}", "$$", "Let $P : I \\to \\mathbf{Z}$ be a map. Then", "$\\Cohstack^P_{X/B} \\subset \\Cohstack_{X/B}$", "as defined in Situation \\ref{situation-numerical}", "is an open and closed substack." ], "refs": [], "proofs": [ { "contents": [ "We may work \\'etale locally on $B$, hence we may assume that $B$ is affine.", "In this case we may perform absolute Noetherian reduction; we suggest", "the reader skip the proof. Namely, say $B = \\Spec(\\Lambda)$.", "Write $\\Lambda = \\colim \\Lambda_i$ as a filtered colimit with each $\\Lambda_i$", "of finite type over $\\mathbf{Z}$. For some $i$ we can find", "a morphism of algebraic spaces $X_i \\to \\Spec(\\Lambda_i)$", "which is separated and of finite presentation and whose base change", "to $\\Lambda$ is $X$. See Limits of Spaces, Lemmas", "\\ref{spaces-limits-lemma-descend-finite-presentation} and", "\\ref{spaces-limits-lemma-descend-separated-morphism}.", "Then after increasing $i$ we may assume there exist", "perfect objects $E_{1, i}, \\ldots, E_{r, i}$", "in $D(\\mathcal{O}_{X_i})$ whose derived pullback to $X$", "are isomorphic to $E_1, \\ldots, E_r$, see", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-perfect-on-limit}.", "Clearly we have a cartesian square", "$$", "\\xymatrix{", "\\Cohstack^P_{X/B} \\ar[r] \\ar[d] &", "\\Cohstack_{X/B} \\ar[d] \\\\", "\\Cohstack^P_{X_i/\\Spec(\\Lambda_i)} \\ar[r] &", "\\Cohstack_{X_i/\\Spec(\\Lambda_i)}", "}", "$$", "and hence we may appeal to Lemma \\ref{lemma-open-P}", "to finish the proof." ], "refs": [ "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-separated-morphism", "spaces-perfect-lemma-perfect-on-limit", "moduli-lemma-open-P" ], "ref_ids": [ 4598, 4592, 2735, 1712 ] } ], "ref_ids": [] }, { "id": 1714, "type": "theorem", "label": "moduli-lemma-quot-diagonal-closed", "categories": [ "moduli" ], "title": "moduli-lemma-quot-diagonal-closed", "contents": [ "The diagonal of $\\Quotfunctor_{\\mathcal{F}/X/B} \\to B$ is a closed immersion.", "If $\\mathcal{F}$ is of finite type, then the diagonal is a closed", "immersion of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Suppose we have a scheme $T/B$ and two quotients", "$\\mathcal{F}_T \\to \\mathcal{Q}_i$, $i = 1, 2$ corresponding", "to $T$-valued points of $\\Quotfunctor_{\\mathcal{F}/X/B}$ over $B$.", "Denote $\\mathcal{K}_1$ the kernel of the first one and set", "$u : \\mathcal{K}_1 \\to \\mathcal{Q}_2$ the composition.", "By Flatness on Spaces, Lemma \\ref{spaces-flat-lemma-F-zero-closed-proper}", "there is a closed subspace of $T$ such that $T' \\to T$", "factors through it if and only if the pullback $u_{T'}$ is zero.", "This proves the diagonal is a closed immersion.", "Moreover, if $\\mathcal{F}$ is of finite type, then", "$\\mathcal{K}_1$ is of finite type", "(Modules on Sites, Lemma", "\\ref{sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation})", "and we see that the diagonal is of finite presentation by", "the same lemma." ], "refs": [ "spaces-flat-lemma-F-zero-closed-proper", "sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation" ], "ref_ids": [ 7182, 14187 ] } ], "ref_ids": [] }, { "id": 1715, "type": "theorem", "label": "moduli-lemma-quot-s-lfp", "categories": [ "moduli" ], "title": "moduli-lemma-quot-s-lfp", "contents": [ "The morphism $\\Quotfunctor_{\\mathcal{F}/X/B} \\to B$ is separated.", "If $\\mathcal{F}$ is of finite presentation, then it is also", "locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "To check $\\Quotfunctor_{\\mathcal{F}/X/B} \\to B$ is separated we have to", "show that its diagonal is a closed immersion. This", "is true by Lemma \\ref{lemma-quot-diagonal-closed}.", "The second statement is part of", "Quot, Proposition \\ref{quot-proposition-quot}." ], "refs": [ "moduli-lemma-quot-diagonal-closed", "quot-proposition-quot" ], "ref_ids": [ 1714, 3227 ] } ], "ref_ids": [] }, { "id": 1716, "type": "theorem", "label": "moduli-lemma-quot-existence-part", "categories": [ "moduli" ], "title": "moduli-lemma-quot-existence-part", "contents": [ "Assume $X \\to B$ is proper as well as of finite presentation", "and $\\mathcal{F}$ quasi-coherent of finite type.", "Then $\\Quotfunctor_{\\mathcal{F}/X/B} \\to B$ satisfies the existence part", "of the valuative criterion (Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-valuative-criterion})." ], "refs": [ "spaces-morphisms-definition-valuative-criterion" ], "proofs": [ { "contents": [ "Taking base change, this immediately reduces to the following", "problem: given a valuation ring $R$ with fraction field $K$,", "an algebraic space $X$ proper over $R$, a finite type quasi-coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$, and a coherent", "quotient $\\mathcal{F}_K \\to \\mathcal{Q}_K$, show there exists", "a quotient $\\mathcal{F} \\to \\mathcal{Q}$ where $\\mathcal{Q}$ is a", "finitely presented $\\mathcal{O}_X$-module", "flat over $R$ whose generic fibre is $\\mathcal{Q}_K$.", "Observe that by Flatness on Spaces, Theorem", "\\ref{spaces-flat-theorem-finite-type-flat}", "any finite type quasi-coherent $\\mathcal{O}_X$-module", "$\\mathcal{F}$ flat over $R$ is of finite presentation.", "We first solve the existence of $\\mathcal{Q}$ affine locally.", "\\medskip\\noindent", "Affine locally we arrive at the following problem:", "let $R \\to A$ be a finitely presented ring map,", "let $M$ be a finite $A$-module, let $\\varphi : M_K \\to N_K$ be", "an $A_K$-quotient module. Then we may consider", "$$", "L = \\{x \\in M \\mid \\varphi(x \\otimes 1) = 0 \\}", "$$", "The $M \\to M/L$ is an $A$-module quotient which is", "torsion free as an $R$-module. Hence it is flat as an", "$R$-module (More on Algebra, Lemma", "\\ref{more-algebra-lemma-valuation-ring-torsion-free-flat}).", "Since $M$ is finite as an $A$-module so is $L$ and we", "conclude that $L$ is of finite presentation as an $A$-module", "(by the reference above). Clearly $M/L$ is the unqiue such", "quotient with $(M/L)_K = N_K$.", "\\medskip\\noindent", "The uniqueness in the construction of the previous paragraph", "guarantees these quotients glue and give the desired $\\mathcal{Q}$.", "Here is a bit more detail. Choose a surjective \\'etale morphism", "$U \\to X$ where $U$ is an affine scheme. Use the above construction", "to construct a quotient $\\mathcal{F}|_U \\to \\mathcal{Q}_U$", "which is quasi-coherent, is flat over $R$, and recovers $\\mathcal{Q}_K|U$", "on the generic fibre. Since $X$ is separated, we see that", "$U \\times_X U$ is an affine scheme \\'etale over $X$ as well.", "Then $\\mathcal{F}|_{U \\times_X U} \\to \\text{pr}_1^*\\mathcal{Q}_U$ and", "$\\mathcal{F}|_{U \\times_X U} \\to \\text{pr}_2^*\\mathcal{Q}_U$", "agree as quotients by the uniquess in the construction. Hence we may descend", "$\\mathcal{F}|_U \\to \\mathcal{Q}_U$ to a surjection", "$\\mathcal{F} \\to \\mathcal{Q}$ as desired (Properties of Spaces,", "Proposition \\ref{spaces-properties-proposition-quasi-coherent})." ], "refs": [ "spaces-flat-theorem-finite-type-flat", "more-algebra-lemma-valuation-ring-torsion-free-flat", "spaces-properties-proposition-quasi-coherent" ], "ref_ids": [ 7144, 9920, 11920 ] } ], "ref_ids": [ 5016 ] }, { "id": 1717, "type": "theorem", "label": "moduli-lemma-quot-functorial", "categories": [ "moduli" ], "title": "moduli-lemma-quot-functorial", "contents": [ "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be an affine quasi-finite", "morphism of algebraic spaces which are separated and of finite presentation", "over $B$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then $\\pi_*$ induces a morphism", "$\\Quotfunctor_{\\mathcal{F}/X/B} \\to \\Quotfunctor_{\\pi_*\\mathcal{F}/Y/B}$." ], "refs": [], "proofs": [ { "contents": [ "Set $\\mathcal{G} = \\pi_*\\mathcal{F}$. Since $\\pi$ is affine we see that for", "any scheme $T$ over $B$ we have $\\mathcal{G}_T = \\pi_{T, *}\\mathcal{F}_T$ by", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-affine-base-change}.", "Moreover $\\pi_T$ is affine, hence $\\pi_{T, *}$ is exact and transforms", "quotients into quotients. Observe that a quasi-coherent quotient", "$\\mathcal{F}_T \\to \\mathcal{Q}$ defines a point of $\\Quotfunctor_{X/B}$", "if and only if $\\mathcal{Q}$ defines an object of $\\Cohstack_{X/B}$", "over $T$ (similarly for $\\mathcal{G}$ and $Y$). Since we've seen in", "Lemma \\ref{lemma-coherent-functorial}", "that $\\pi_*$ induces a morphism $\\Cohstack_{X/B} \\to \\Cohstack_{Y/B}$", "we see that if $\\mathcal{F}_T \\to \\mathcal{Q}$ is in", "$\\Quotfunctor_{\\mathcal{F}/X/B}(T)$, then", "$\\mathcal{G}_T \\to \\pi_{T, *}\\mathcal{Q}$ is", "in $\\Quotfunctor_{\\mathcal{G}/Y/B}(T)$." ], "refs": [ "spaces-cohomology-lemma-affine-base-change", "moduli-lemma-coherent-functorial" ], "ref_ids": [ 11295, 1709 ] } ], "ref_ids": [] }, { "id": 1718, "type": "theorem", "label": "moduli-lemma-quot-open", "categories": [ "moduli" ], "title": "moduli-lemma-quot-open", "contents": [ "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be an affine open immersion", "of algebraic spaces which are separated and of finite presentation over $B$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then the morphism", "$\\Quotfunctor_{\\mathcal{F}/X/B} \\to \\Quotfunctor_{\\pi_*\\mathcal{F}/Y/B}$ of", "Lemma \\ref{lemma-quot-functorial} is an open immersion." ], "refs": [ "moduli-lemma-quot-functorial" ], "proofs": [ { "contents": [ "Omitted. Hint: If $(\\pi_*\\mathcal{F})_T \\to \\mathcal{Q}$ is an element of", "$\\Quotfunctor_{\\pi_*\\mathcal{F}/Y/B}(T)$", "and for $t \\in T$ we have $\\text{Supp}(\\mathcal{Q}_t) \\subset |X_t|$,", "then the same is true for $t' \\in T$ in a neighbourhood of $t$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1717 ] }, { "id": 1719, "type": "theorem", "label": "moduli-lemma-quot-better-open", "categories": [ "moduli" ], "title": "moduli-lemma-quot-better-open", "contents": [ "Let $B$ be an algebraic space. Let $j : X \\to Y$ be an open immersion", "of algebraic spaces which are separated and of finite presentation over $B$.", "Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module and set", "$\\mathcal{F} = j^*\\mathcal{G}$. Then there is an open immersion", "$$", "\\Quotfunctor_{\\mathcal{F}/X/B}", "\\longrightarrow", "\\Quotfunctor_{\\mathcal{G}/Y/B}", "$$", "of algebraic spaces over $B$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{F}_T \\to \\mathcal{Q}$ is an element of", "$\\Quotfunctor_{\\mathcal{F}/X/B}(T)$ then we can consider", "$\\mathcal{G}_T \\to j_{T, *}\\mathcal{F}_T \\to j_{T, *}\\mathcal{Q}$.", "Looking at stalks one finds that this is surjective.", "By Lemma \\ref{lemma-coherent-functorial}", "we see that $j_{T, *}\\mathcal{Q}$ is finitely presented, flat over $B$", "with support proper over $B$. Thus we obtain a $T$-valued", "point of $\\Quotfunctor_{\\mathcal{G}/Y/B}$.", "This defines the morphism of the lemma.", "We omit the proof that this is an open immersion. Hint:", "If $\\mathcal{G}_T \\to \\mathcal{Q}$ is an element of", "$\\Quotfunctor_{\\mathcal{G}/Y/B}(T)$", "and for $t \\in T$ we have $\\text{Supp}(\\mathcal{Q}_t) \\subset |X_t|$,", "then the same is true for $t' \\in T$ in a neighbourhood of $t$." ], "refs": [ "moduli-lemma-coherent-functorial" ], "ref_ids": [ 1709 ] } ], "ref_ids": [] }, { "id": 1720, "type": "theorem", "label": "moduli-lemma-quot-closed", "categories": [ "moduli" ], "title": "moduli-lemma-quot-closed", "contents": [ "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be a closed immersion", "of algebraic spaces which are separated and of finite presentation over $B$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then the morphism", "$\\Quotfunctor_{\\mathcal{F}/X/B} \\to \\Quotfunctor_{\\pi_*\\mathcal{F}/Y/B}$ of", "Lemma \\ref{lemma-quot-functorial} is an isomorphism." ], "refs": [ "moduli-lemma-quot-functorial" ], "proofs": [ { "contents": [ "For every scheme $T$ over $B$ the morphism $\\pi_T : X_T \\to Y_T$", "is a closed immersion. Then $\\pi_{T, *}$ is an equivalence of", "categories between $\\QCoh(\\mathcal{O}_{X_T})$ and the full subcategory", "of $\\QCoh(\\mathcal{O}_{Y_T})$ whose objects are those quasi-coherent", "modules annihilated by the ideal sheaf of $X_T$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-i-star-equivalence}.", "Since a qotient of", "$(\\pi_*\\mathcal{F})_T$ is annihilated by this ideal we obtain the", "bijectivity of the map", "$\\Quotfunctor_{\\mathcal{F}/X/B}(T) \\to \\Quotfunctor_{\\pi_*\\mathcal{F}/Y/B}(T)$", "for all $T$ as desired." ], "refs": [ "spaces-morphisms-lemma-i-star-equivalence" ], "ref_ids": [ 4771 ] } ], "ref_ids": [ 1717 ] }, { "id": 1721, "type": "theorem", "label": "moduli-lemma-quot-quotient", "categories": [ "moduli" ], "title": "moduli-lemma-quot-quotient", "contents": [ "Let $X \\to B$ be as in the introduction to this section. Let", "$\\mathcal{F} \\to \\mathcal{G}$ be a surjection of quasi-coherent", "$\\mathcal{O}_X$-modules. Then there is a canonical closed immersion", "$\\Quotfunctor_{\\mathcal{G}/X/B} \\to \\Quotfunctor_{\\mathcal{F}/X/B}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{K} = \\Ker(\\mathcal{F} \\to \\mathcal{G})$. By right", "exactness of pullbacks we find that", "$\\mathcal{K}_T \\to \\mathcal{F}_T \\to \\mathcal{G}_T \\to 0$", "is an exact sequecnce for all schemes $T$ over $B$.", "In particular, a quotient of $\\mathcal{G}_T$", "determines a quotient of $\\mathcal{F}_T$ and we obtain our transformation", "of functors", "$\\Quotfunctor_{\\mathcal{G}/X/B} \\to \\Quotfunctor_{\\mathcal{F}/X/B}$.", "This transformation is a closed immersion by", "Flatness on Spaces, Lemma \\ref{spaces-flat-lemma-F-zero-closed-proper}.", "Namely, given an element $\\mathcal{F}_T \\to \\mathcal{Q}$ of", "$\\Quotfunctor_{\\mathcal{F}/X/B}(T)$, then we see that the pull", "back to $T'/T$ is in the image of the transformation if and", "only if $\\mathcal{K}_{T'} \\to \\mathcal{Q}_{T'}$ is zero." ], "refs": [ "spaces-flat-lemma-F-zero-closed-proper" ], "ref_ids": [ 7182 ] } ], "ref_ids": [] }, { "id": 1722, "type": "theorem", "label": "moduli-lemma-quot-tensor-invertible", "categories": [ "moduli" ], "title": "moduli-lemma-quot-tensor-invertible", "contents": [ "Let $f : X \\to B$ and $\\mathcal{F}$ be as in the introduction to this section.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Then tensoring with $\\mathcal{L}$ defines an isomophism", "$$", "\\Quotfunctor_{\\mathcal{F}/X/B}", "\\longrightarrow", "\\Quotfunctor_{\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}/X/B}", "$$", "Given a numerical polynomial $P(t)$, then setting $P'(t) = P(t + 1)$", "this map induces an isomorphism", "$\\Quotfunctor^P_{\\mathcal{F}/X/B}", "\\longrightarrow", "\\Quotfunctor^{P'}_{\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}/X/B}$", "of open and closed substacks." ], "refs": [], "proofs": [ { "contents": [ "Set $\\mathcal{G} = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}$.", "Observe that", "$\\mathcal{G}_T = \\mathcal{F}_T \\otimes_{\\mathcal{O}_{X_T}} \\mathcal{L}_T$.", "If $\\mathcal{F}_T \\to \\mathcal{Q}$ is an element of", "$\\Quotfunctor_{\\mathcal{F}/X/B}(T)$, then we send it", "to the element", "$\\mathcal{G}_T \\to \\mathcal{Q} \\otimes_{\\mathcal{O}_{X_T}} \\mathcal{L}_T$", "of", "$\\Quotfunctor_{\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}/X/B}(T)$.", "This is compatible with pullbacks and hence", "defines a transformation of functors as desired.", "Since there is an obvious inverse transformation,", "it is an isomorphism. We omit the proof of the final statement." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1723, "type": "theorem", "label": "moduli-lemma-quot-power-invertible", "categories": [ "moduli" ], "title": "moduli-lemma-quot-power-invertible", "contents": [ "Let $f : X \\to B$ and $\\mathcal{F}$ be as in the introduction to this section.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Then", "$$", "\\Quotfunctor^{P, \\mathcal{L}}_{\\mathcal{F}/X/B} =", "\\Quotfunctor^{P', \\mathcal{L}^{\\otimes n}}_{\\mathcal{F}/X/B}", "$$", "where $P'(t) = P(nt)$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately after unwinding all the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1724, "type": "theorem", "label": "moduli-lemma-quot-Pn", "categories": [ "moduli" ], "title": "moduli-lemma-quot-Pn", "contents": [ "Let $n \\geq 0$, $r \\geq 1$, $P \\in \\mathbf{Q}[t]$.", "The algebraic space", "$$", "X = \\Quotfunctor^P_{\\mathcal{O}^{\\oplus r}_{\\mathbf{P}^n_\\mathbf{Z}}/", "\\mathbf{P}^n_\\mathbf{Z}/\\mathbf{Z}}", "$$", "parametrizing quotients of $\\mathcal{O}_{\\mathbf{}P^n_\\mathbf{Z}}^{\\oplus r}$", "with Hilbert polynomial $P$ is proper over $\\Spec(\\mathbf{Z})$." ], "refs": [], "proofs": [ { "contents": [ "We already know that $X \\to \\Spec(\\mathbf{Z})$ is separated and", "locally of finite presentation (Lemma \\ref{lemma-quot-s-lfp}).", "We also know that $X \\to \\Spec(\\mathbf{Z})$ satisfies the", "existence part of the valuative criterion, see", "Lemma \\ref{lemma-quot-existence-part}.", "By the valuative criterion for properness, it suffices to", "prove our Quot space is quasi-compact, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-characterize-proper}.", "Thus it suffices to find a quasi-compact scheme $T$ and a surjective", "morphism $T \\to X$. Let $m$ be the integer found in", "Varieties, Lemma \\ref{varieties-lemma-bound-quotients-free}.", "Let", "$$", "N = r{m + n \\choose n} - P(m)", "$$", "We will write $\\mathbf{P}^n$ for", "$\\mathbf{P}^n_\\mathbf{Z} = \\text{Proj}(\\mathbf{Z}[T_0, \\ldots, T_n])$", "and unadorned products will mean products over $\\Spec(\\mathbf{Z})$.", "The idea of the proof is to construct a ``universal'' map", "$$", "\\Psi :", "\\mathcal{O}_{T \\times \\mathbf{P}^n}(-m)^{\\oplus N}", "\\longrightarrow", "\\mathcal{O}_{T \\times \\mathbf{P}^n}^{\\oplus r}", "$$", "over an affine scheme $T$ and show that every point of $X$", "corresponds to a cokernel of this in some point of $T$.", "\\medskip\\noindent", "Definition of $T$ and $\\Psi$. We take $T = \\Spec(A)$ where", "$$", "A = \\mathbf{Z}[a_{i, j, E}]", "$$", "where $i \\in \\{1, \\ldots, r\\}$, $j \\in \\{1, \\ldots, N\\}$", "and $E = (e_0, \\ldots, e_n)$ runs through the multi-indices", "of total degree $|E| = \\sum_{k = 0, \\ldots n} e_k = m$.", "Then we define $\\Psi$ to be the map whose $(i, j)$ matrix", "entry is the map", "$$", "\\sum\\nolimits_{E = (e_0, \\ldots, e_n)}", "a_{i, j, E} T_0^{e_0} \\ldots T_n^{e_n} :", "\\mathcal{O}_{T \\times \\mathbf{P}^n}(-m)", "\\longrightarrow", "\\mathcal{O}_{T \\times \\mathbf{P}^n}", "$$", "where the sum is over $E$ as above (but $i$ and $j$ are fixed of course).", "\\medskip\\noindent", "Consider the quotient $\\mathcal{Q} = \\Coker(\\Psi)$ on $T \\times \\mathbf{P}^n$.", "By More on Flatness, Lemma \\ref{flat-lemma-generic-flatness-stratification}", "there exists a $t \\geq 0$ and closed subschemes", "$$", "T = T_0 \\supset T_1 \\supset \\ldots \\supset T_t = \\emptyset", "$$", "such that the pullback $\\mathcal{Q}_p$ of $\\mathcal{Q}$ to", "$(T_p \\setminus T_{p + 1}) \\times \\mathbf{P}^n$ is flat over", "$T_p \\setminus T_{p + 1}$. Observe that we", "have an exact sequence", "$$", "\\mathcal{O}_{(T_p \\setminus T_{p + 1}) \\times \\mathbf{P}^n}(-m)^{\\oplus N}", "\\to", "\\mathcal{O}_{(T_p \\setminus T_{p + 1}) \\times \\mathbf{P}^n}^{\\oplus r}", "\\to", "\\mathcal{Q}_p", "\\to", "0", "$$", "by pulling back the exact sequence defining $\\mathcal{Q} = \\Coker(\\Psi)$.", "Therefore we obtain a morphism", "$$", "\\coprod (T_p \\setminus T_{p + 1})", "\\longrightarrow", "\\Quotfunctor_{\\mathcal{O}^{\\oplus r}/\\mathbf{P}/\\mathbf{Z}}", "\\supset", "\\Quotfunctor^P_{\\mathcal{O}^{\\oplus r}/\\mathbf{P}/\\mathbf{Z}} = X", "$$", "Since the left hand side is a Noetherian scheme and the inclusion", "on the right hand side is open, it", "suffices to show that any point of $X$ is in the image of this morphism.", "\\medskip\\noindent", "Let $k$ be a field and let $x \\in X(k)$. Then $x$ corresponds to", "a surjection $\\mathcal{O}_{\\mathbf{P}^n_k}^{\\oplus r} \\to \\mathcal{F}$", "of coherent $\\mathcal{O}_{\\mathbf{P}^n_k}$-modules", "such that the Hilbert polynomial of $\\mathcal{F}$ is $P$.", "Consider the short exact sequence", "$$", "0 \\to \\mathcal{K} \\to", "\\mathcal{O}_{\\mathbf{P}^n_k}^{\\oplus r} \\to", "\\mathcal{F} \\to 0", "$$", "By Varieties, Lemma \\ref{varieties-lemma-bound-quotients-free}", "and our choice of $m$ we see that $\\mathcal{K}$ is $m$-regular.", "By Varieties, Lemma \\ref{varieties-lemma-m-regular-globally-generated}", "we see that $\\mathcal{K}(m)$ is globally generated.", "By Varieties, Lemma \\ref{varieties-lemma-m-regular-up}", "and the definition of $m$-regularity we see that", "$H^i(\\mathbf{P}^n_k, \\mathcal{K}(m)) = 0$ for $i > 0$.", "Hence we see that", "$$", "\\dim_k H^0(\\mathbf{P}^n_k, \\mathcal{K}(m)) =", "\\chi(\\mathcal{K}(m)) =", "\\chi(\\mathcal{O}_{\\mathbf{P}^n_k}(m)^{\\oplus r}) -", "\\chi(\\mathcal{F}(m)) = N", "$$", "by our choice of $N$. This gives a surjection", "$$", "\\mathcal{O}_{\\mathbf{P}^n_k}^{\\oplus N}", "\\longrightarrow", "\\mathcal{K}(m)", "$$", "Twisting back down and using the short exact sequence above", "we see that $\\mathcal{F}$ is the cokernel of a map", "$$", "\\Psi_x :", "\\mathcal{O}_{\\mathbf{P}^n_k}(-m)^{\\oplus N}", "\\to", "\\mathcal{O}_{\\mathbf{P}^n_k}^{\\oplus r}", "$$", "There is a unique ring map $\\tau : A \\to k$ such that the base change", "of $\\Psi$ by the corresponding morphism $t = \\Spec(\\tau) : \\Spec(k) \\to T$", "is $\\Psi_x$. This is true because the entries of the $N \\times r$", "matrix defining $\\Psi_x$ are homogeneous polynomials", "$\\sum \\lambda_{i, j, E} T_0^{e_0} \\ldots T_n^{e_n}$", "of degree $m$ in $T_0, \\ldots, T_n$ with coefficients", "$\\lambda_{i, j, E} \\in k$ and we can set", "$\\tau(a_{i, j, E}) = \\lambda_{i, j, E}$.", "Then $t \\in T_p \\setminus T_{p + 1}$ for some $p$ and", "the image of $t$ under the morphism above is $x$ as desired." ], "refs": [ "moduli-lemma-quot-s-lfp", "moduli-lemma-quot-existence-part", "spaces-morphisms-lemma-characterize-proper", "varieties-lemma-bound-quotients-free", "flat-lemma-generic-flatness-stratification", "varieties-lemma-bound-quotients-free", "varieties-lemma-m-regular-globally-generated", "varieties-lemma-m-regular-up" ], "ref_ids": [ 1715, 1716, 4938, 11046, 6084, 11046, 11043, 11041 ] } ], "ref_ids": [] }, { "id": 1725, "type": "theorem", "label": "moduli-lemma-quot-Pn-over-base", "categories": [ "moduli" ], "title": "moduli-lemma-quot-Pn-over-base", "contents": [ "Let $B$ be an algebraic space. Let $X = B \\times \\mathbf{P}^n_\\mathbf{Z}$.", "Let $\\mathcal{L}$ be the pullback of $\\mathcal{O}_{\\mathbf{P}^n}(1)$ to $X$.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite", "presentation. The algebraic space $\\Quotfunctor^P_{\\mathcal{F}/X/B}$", "parametrizing quotients of $\\mathcal{F}$", "having Hilbert polynomial $P$ with respect to $\\mathcal{L}$", "is proper over $B$." ], "refs": [], "proofs": [ { "contents": [ "The question is \\'etale local over $B$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-proper-local}.", "Thus we may assume $B$ is an affine scheme.", "In this case $\\mathcal{L}$ is an ample invertible module on $X$", "(by Constructions, Lemma \\ref{constructions-lemma-ample-on-proj}", "and the definition of ample invertible modules in", "Properties, Definition \\ref{properties-definition-ample}).", "Thus we can find $r' \\geq 0$ and $r \\geq 0$ and a surjection", "$$", "\\mathcal{O}_X^{\\oplus r} \\longrightarrow", "\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes r'}", "$$", "by Properties, Proposition \\ref{properties-proposition-characterize-ample}.", "By Lemma \\ref{lemma-quot-tensor-invertible}", "we may replace $\\mathcal{F}$ by", "$\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes r'}$", "and $P(t)$ by $P(t + r')$.", "By Lemma \\ref{lemma-quot-quotient}", "we obtain a closed immersion", "$$", "\\Quotfunctor^P_{\\mathcal{F}/X/B}", "\\longrightarrow", "\\Quotfunctor^P_{\\mathcal{O}_X^{\\oplus r}/X/B}", "$$", "Since we've shown that $\\Quotfunctor^P_{\\mathcal{O}_X^{\\oplus r}/X/B} \\to B$", "is proper in Lemma \\ref{lemma-quot-Pn} we conclude." ], "refs": [ "spaces-morphisms-lemma-proper-local", "constructions-lemma-ample-on-proj", "properties-definition-ample", "properties-proposition-characterize-ample", "moduli-lemma-quot-tensor-invertible", "moduli-lemma-quot-quotient", "moduli-lemma-quot-Pn" ], "ref_ids": [ 4916, 12606, 3088, 3067, 1722, 1721, 1724 ] } ], "ref_ids": [] }, { "id": 1726, "type": "theorem", "label": "moduli-lemma-quot-proper-over-base", "categories": [ "moduli" ], "title": "moduli-lemma-quot-proper-over-base", "contents": [ "Let $f : X \\to B$ be a proper morphism of finite presentation", "of algebraic spaces. Let $\\mathcal{F}$ be a finitely presented", "$\\mathcal{O}_X$-module. Let $\\mathcal{L}$ be an invertible", "$\\mathcal{O}_X$-module ample on $X/B$, see", "Divisors on Spaces, Definition", "\\ref{spaces-divisors-definition-relatively-ample}.", "The algebraic space $\\Quotfunctor^P_{\\mathcal{F}/X/B}$", "parametrizing quotients of $\\mathcal{F}$", "having Hilbert polynomial $P$ with respect to $\\mathcal{L}$", "is proper over $B$." ], "refs": [ "spaces-divisors-definition-relatively-ample" ], "proofs": [ { "contents": [ "The question is \\'etale local over $B$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-proper-local}.", "Thus we may assume $B$ is an affine scheme.", "Then we can find a closed immersion $i : X \\to \\mathbf{P}^n_B$", "such that $i^*\\mathcal{O}_{\\mathbf{P}^n_B}(1) \\cong \\mathcal{L}^{\\otimes d}$", "for some $d \\geq 1$. See", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-projective-finite-type-over-S}.", "Changing $\\mathcal{L}$ into $\\mathcal{L}^{\\otimes d}$ and", "the numerical polynomial $P(t)$ into $P(dt)$ leaves", "$\\Quotfunctor^P_{\\mathcal{F}/X/B}$ unaffected; some details omitted.", "Hence we may assume $\\mathcal{L} = i^*\\mathcal{O}_{\\mathbf{P}^n_B}(1)$.", "Then the isomorphism", "$\\Quotfunctor_{\\mathcal{F}/X/B} \\to", "\\Quotfunctor_{i_*\\mathcal{F}/\\mathbf{P}^n_B/B}$ of", "Lemma \\ref{lemma-quot-closed} induces an isomorphism", "$\\Quotfunctor^P_{\\mathcal{F}/X/B} \\cong", "\\Quotfunctor^P_{i_*\\mathcal{F}/\\mathbf{P}^n_B/B}$.", "Since $\\Quotfunctor^P_{i_*\\mathcal{F}/\\mathbf{P}^n_B/B}$", "is proper over $B$ by Lemma \\ref{lemma-quot-Pn-over-base}", "we conclude." ], "refs": [ "spaces-morphisms-lemma-proper-local", "morphisms-lemma-quasi-projective-finite-type-over-S", "moduli-lemma-quot-closed", "moduli-lemma-quot-Pn-over-base" ], "ref_ids": [ 4916, 5393, 1720, 1725 ] } ], "ref_ids": [ 13028 ] }, { "id": 1727, "type": "theorem", "label": "moduli-lemma-quot-qc-over-base", "categories": [ "moduli" ], "title": "moduli-lemma-quot-qc-over-base", "contents": [ "Let $f : X \\to B$ be a separated morphism of finite presentation", "of algebraic spaces. Let $\\mathcal{F}$ be a finitely presented", "$\\mathcal{O}_X$-module. Let $\\mathcal{L}$ be an invertible", "$\\mathcal{O}_X$-module ample on $X/B$, see", "Divisors on Spaces, Definition", "\\ref{spaces-divisors-definition-relatively-ample}.", "The algebraic space $\\Quotfunctor^P_{\\mathcal{F}/X/B}$", "parametrizing quotients of $\\mathcal{F}$", "having Hilbert polynomial $P$ with respect to $\\mathcal{L}$", "is separated of finite presentation over $B$." ], "refs": [ "spaces-divisors-definition-relatively-ample" ], "proofs": [ { "contents": [ "We have already seen that $\\Quotfunctor_{\\mathcal{F}/X/B} \\to B$", "is separated and locally of finite presentation, see", "Lemma \\ref{lemma-quot-s-lfp}. Thus it suffices to show that", "the open subspace $\\Quotfunctor^P_{\\mathcal{F}/X/B}$", "of Remark \\ref{remark-quot-numerical} is quasi-compact over $B$.", "\\medskip\\noindent", "The question is \\'etale local on $B$", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-quasi-compact-local}).", "Thus we may assume $B$ is affine.", "\\medskip\\noindent", "Assume $B = \\Spec(\\Lambda)$. Write $\\Lambda = \\colim \\Lambda_i$ as the", "colimit of its finite type $\\mathbf{Z}$-subalgebras. Then", "we can find an $i$ and a system $X_i, \\mathcal{F}_i, \\mathcal{L}_i$", "as in the lemma over $B_i = \\Spec(\\Lambda_i)$ whose base change to", "$B$ gives $X, \\mathcal{F}, \\mathcal{L}$.", "This follows from", "Limits of Spaces, Lemmas", "\\ref{spaces-limits-lemma-descend-finite-presentation} (to find $X_i$),", "\\ref{spaces-limits-lemma-descend-modules-finite-presentation} (to find", "$\\mathcal{F}_i$), \\ref{spaces-limits-lemma-descend-invertible-modules}", "(to find $\\mathcal{L}_i$), and \\ref{spaces-limits-lemma-descend-separated}", "(to make $X_i$ separated). Because", "$$", "\\Quotfunctor_{\\mathcal{F}/X/B} = B \\times_{B_i}", "\\Quotfunctor_{\\mathcal{F}_i/X_i/B_i}", "$$", "and similarly for $\\Quotfunctor^P_{\\mathcal{F}/X/B}$ we reduce", "to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $B$ is affine and Noetherian. We may replace $\\mathcal{L}$", "by a positive power, see Lemma \\ref{lemma-quot-power-invertible}.", "Thus we may assume there exists an immersion $i : X \\to \\mathbf{P}^n_B$", "such that $i^*\\mathcal{O}_{\\mathbf{P}^n}(1) = \\mathcal{L}$. By", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-immersion}", "there exists a closed subscheme $X' \\subset \\mathbf{P}^n_B$", "such that $i$ factors through an open immersion $j : X \\to X'$.", "By Properties, Lemma \\ref{properties-lemma-lift-finite-presentation}", "there exists a finitely presented $\\mathcal{O}_{X'}$-module", "$\\mathcal{G}$ such that $j^*\\mathcal{G} = \\mathcal{F}$.", "Thus we obtain an open immersion", "$$", "\\Quotfunctor_{\\mathcal{F}/X/B}", "\\longrightarrow", "\\Quotfunctor_{\\mathcal{G}/X'/B}", "$$", "by Lemma \\ref{lemma-quot-better-open}. Clearly this open immersion", "sends $\\Quotfunctor^P_{\\mathcal{F}/X/B}$ into", "$\\Quotfunctor^P_{\\mathcal{G}/X'/B}$. Now", "$\\Quotfunctor^P_{\\mathcal{G}/X'/B}$ is proper over $B$ by", "Lemma \\ref{lemma-quot-proper-over-base}.", "Therefore it is Noetherian and since any open of a Noetherian", "algebraic space is quasi-compact we win." ], "refs": [ "moduli-lemma-quot-s-lfp", "moduli-remark-quot-numerical", "spaces-morphisms-lemma-quasi-compact-local", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-modules-finite-presentation", "spaces-limits-lemma-descend-invertible-modules", "spaces-limits-lemma-descend-separated", "moduli-lemma-quot-power-invertible", "morphisms-lemma-quasi-compact-immersion", "properties-lemma-lift-finite-presentation", "moduli-lemma-quot-better-open", "moduli-lemma-quot-proper-over-base" ], "ref_ids": [ 1715, 1754, 4742, 4598, 4599, 4600, 4577, 1723, 5154, 3022, 1719, 1726 ] } ], "ref_ids": [ 13028 ] }, { "id": 1728, "type": "theorem", "label": "moduli-lemma-hilb-diagonal-closed", "categories": [ "moduli" ], "title": "moduli-lemma-hilb-diagonal-closed", "contents": [ "The diagonal of $\\Hilbfunctor_{X/B} \\to B$ is a closed immersion", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "In Quot, Lemma \\ref{quot-lemma-hilb-is-quot} we have seen that", "$\\Hilbfunctor_{X/B} = \\Quotfunctor_{\\mathcal{O}_X/X/B}$.", "Hence this follows from Lemma \\ref{lemma-quot-diagonal-closed}." ], "refs": [ "quot-lemma-hilb-is-quot", "moduli-lemma-quot-diagonal-closed" ], "ref_ids": [ 3176, 1714 ] } ], "ref_ids": [] }, { "id": 1729, "type": "theorem", "label": "moduli-lemma-hilb-s-lfp", "categories": [ "moduli" ], "title": "moduli-lemma-hilb-s-lfp", "contents": [ "The morphism $\\Hilbfunctor_{X/B} \\to B$ is separated", "and locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "To check $\\Hilbfunctor_{X/B} \\to B$ is separated we have to", "show that its diagonal is a closed immersion. This", "is true by Lemma \\ref{lemma-hilb-diagonal-closed}.", "The second statement is part of", "Quot, Proposition \\ref{quot-proposition-hilb}." ], "refs": [ "moduli-lemma-hilb-diagonal-closed", "quot-proposition-hilb" ], "ref_ids": [ 1728, 3228 ] } ], "ref_ids": [] }, { "id": 1730, "type": "theorem", "label": "moduli-lemma-hilb-existence-part", "categories": [ "moduli" ], "title": "moduli-lemma-hilb-existence-part", "contents": [ "Assume $X \\to B$ is proper as well as of finite presentation.", "Then $\\Hilbfunctor_{X/B} \\to B$ satisfies the existence part", "of the valuative criterion (Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-valuative-criterion})." ], "refs": [ "spaces-morphisms-definition-valuative-criterion" ], "proofs": [ { "contents": [ "In Quot, Lemma \\ref{quot-lemma-hilb-is-quot} we have seen that", "$\\Hilbfunctor_{X/B} = \\Quotfunctor_{\\mathcal{O}_X/X/B}$.", "Hence this follows from Lemma \\ref{lemma-quot-existence-part}." ], "refs": [ "quot-lemma-hilb-is-quot", "moduli-lemma-quot-existence-part" ], "ref_ids": [ 3176, 1716 ] } ], "ref_ids": [ 5016 ] }, { "id": 1731, "type": "theorem", "label": "moduli-lemma-hilb-open", "categories": [ "moduli" ], "title": "moduli-lemma-hilb-open", "contents": [ "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be an open immersion", "of algebraic spaces which are separated and of finite presentation over $B$.", "Then $\\pi$ induces an open immersion", "$\\Hilbfunctor_{X/B} \\to \\Hilbfunctor_{Y/B}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: If $Z \\subset X_T$ is a closed subscheme which is", "proper over $T$, then $Z$ is also closed in $Y_T$. Thus we obtain", "the transformation $\\Hilbfunctor_{X/B} \\to \\Hilbfunctor_{Y/B}$.", "If $Z \\subset Y_T$ is an element of $\\Hilbfunctor_{Y/B}(T)$", "and for $t \\in T$ we have $|Z_t| \\subset |X_t|$,", "then the same is true for $t' \\in T$ in a neighbourhood of $t$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1732, "type": "theorem", "label": "moduli-lemma-hilb-closed", "categories": [ "moduli" ], "title": "moduli-lemma-hilb-closed", "contents": [ "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be a closed immersion", "of algebraic spaces which are separated and of finite presentation", "over $B$. Then $\\pi$ induces a closed immersion", "$\\Hilbfunctor_{X/B} \\to \\Hilbfunctor_{Y/B}$." ], "refs": [], "proofs": [ { "contents": [ "Since $\\pi$ is a closed immersion, it is immediate that given a", "closed subscheme $Z \\subset X_T$,", "we can view $Z$ as a closed subscheme of $X_T$. Thus we obtain", "the transformation $\\Hilbfunctor_{X/B} \\to \\Hilbfunctor_{Y/B}$.", "This transformation is immediately seen to be a monomorphism.", "To prove that it is a closed immersion, you can use", "Lemma \\ref{lemma-quot-quotient} for the map", "$\\mathcal{O}_Y \\to \\mathcal{O}_X$ and the identifications", "$\\Hilbfunctor_{X/B} = \\Quotfunctor_{\\mathcal{O}_X/X/B}$,", "$\\Hilbfunctor_{Y/B} = \\Quotfunctor_{\\mathcal{O}_Y/Y/B}$", "of Quot, Lemma \\ref{quot-lemma-hilb-is-quot}." ], "refs": [ "moduli-lemma-quot-quotient", "quot-lemma-hilb-is-quot" ], "ref_ids": [ 1721, 3176 ] } ], "ref_ids": [] }, { "id": 1733, "type": "theorem", "label": "moduli-lemma-hilb-proper-over-base", "categories": [ "moduli" ], "title": "moduli-lemma-hilb-proper-over-base", "contents": [ "Let $f : X \\to B$ be a proper morphism of finite presentation", "of algebraic spaces. Let $\\mathcal{L}$ be an invertible", "$\\mathcal{O}_X$-module ample on $X/B$, see", "Divisors on Spaces, Definition", "\\ref{spaces-divisors-definition-relatively-ample}.", "The algebraic space $\\Hilbfunctor^P_{X/B}$", "parametrizing closed subschemes", "having Hilbert polynomial $P$ with respect to $\\mathcal{L}$", "is proper over $B$." ], "refs": [ "spaces-divisors-definition-relatively-ample" ], "proofs": [ { "contents": [ "Recall that $\\Hilbfunctor_{X/B} = \\Quotfunctor_{\\mathcal{O}_X/X/B}$, see", "Quot, Lemma \\ref{quot-lemma-hilb-is-quot}.", "Thus this lemma is an immediate consequence of", "Lemma \\ref{lemma-quot-proper-over-base}." ], "refs": [ "quot-lemma-hilb-is-quot", "moduli-lemma-quot-proper-over-base" ], "ref_ids": [ 3176, 1726 ] } ], "ref_ids": [ 13028 ] }, { "id": 1734, "type": "theorem", "label": "moduli-lemma-hilb-qc-over-base", "categories": [ "moduli" ], "title": "moduli-lemma-hilb-qc-over-base", "contents": [ "Let $f : X \\to B$ be a separated morphism of finite presentation", "of algebraic spaces. Let $\\mathcal{L}$ be an invertible", "$\\mathcal{O}_X$-module ample on $X/B$, see", "Divisors on Spaces, Definition", "\\ref{spaces-divisors-definition-relatively-ample}.", "The algebraic space $\\Hilbfunctor^P_{X/B}$", "parametrizing closed subschemes", "having Hilbert polynomial $P$ with respect to $\\mathcal{L}$", "is separated of finite presentation over $B$." ], "refs": [ "spaces-divisors-definition-relatively-ample" ], "proofs": [ { "contents": [ "Recall that $\\Hilbfunctor_{X/B} = \\Quotfunctor_{\\mathcal{O}_X/X/B}$, see", "Quot, Lemma \\ref{quot-lemma-hilb-is-quot}.", "Thus this lemma is an immediate consequence of", "Lemma \\ref{lemma-quot-qc-over-base}." ], "refs": [ "quot-lemma-hilb-is-quot", "moduli-lemma-quot-qc-over-base" ], "ref_ids": [ 3176, 1727 ] } ], "ref_ids": [ 13028 ] }, { "id": 1735, "type": "theorem", "label": "moduli-lemma-pic-diagonal-affine-fp", "categories": [ "moduli" ], "title": "moduli-lemma-pic-diagonal-affine-fp", "contents": [ "The diagonal of $\\Picardstack_{X/B}$ over $B$ is affine", "and of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "In Quot, Lemma \\ref{quot-lemma-picard-stack-open-in-coh} we have seen that", "$\\Picardstack_{X/B}$ is an open substack of", "$\\Cohstack_{X/B}$. Hence this follows from", "Lemma \\ref{lemma-coherent-diagonal-affine-fp}." ], "refs": [ "quot-lemma-picard-stack-open-in-coh", "moduli-lemma-coherent-diagonal-affine-fp" ], "ref_ids": [ 3178, 1706 ] } ], "ref_ids": [] }, { "id": 1736, "type": "theorem", "label": "moduli-lemma-pic-qs-lfp", "categories": [ "moduli" ], "title": "moduli-lemma-pic-qs-lfp", "contents": [ "The morphism $\\Picardstack_{X/B} \\to B$ is quasi-separated and", "locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "In Quot, Lemma \\ref{quot-lemma-picard-stack-open-in-coh} we have seen that", "$\\Picardstack_{X/B}$ is an open substack of", "$\\Cohstack_{X/B}$. Hence this follows from", "Lemma \\ref{lemma-coherent-qs-lfp}." ], "refs": [ "quot-lemma-picard-stack-open-in-coh", "moduli-lemma-coherent-qs-lfp" ], "ref_ids": [ 3178, 1707 ] } ], "ref_ids": [] }, { "id": 1737, "type": "theorem", "label": "moduli-lemma-pic-existence-part", "categories": [ "moduli" ], "title": "moduli-lemma-pic-existence-part", "contents": [ "Assume $X \\to B$ is smooth in addition to being proper.", "Then $\\Picardstack_{X/B} \\to B$ satisfies the existence part", "of the valuative criterion (Morphisms of Stacks, Definition", "\\ref{stacks-morphisms-definition-existence})." ], "refs": [ "stacks-morphisms-definition-existence" ], "proofs": [ { "contents": [ "Taking base change, this immediately reduces to the following", "problem: given a valuation ring $R$ with fraction field $K$ and", "an algebraic space $X$ proper and smooth over $R$ and an invertible", "$\\mathcal{O}_{X_K}$-module $\\mathcal{L}_K$, show there exists", "an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$", "whose generic fibre is $\\mathcal{L}_K$.", "Observe that $X_K$ is Noetherian, separated, and regular", "(use Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-noetherian}", "and", "Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-smooth-regular}).", "Thus we can write", "$\\mathcal{L}_K$ as the difference in the Picard group of", "$\\mathcal{O}_{X_K}(D_K)$ and $\\mathcal{O}_{X_K}(D'_K)$", "for two effective Cartier divisors $D_K, D'_K$ in $X_K$, see", "Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-Noetherian-regular-separated-pic-effective-Cartier}.", "Finally, we know that $D_K$ and $D'_K$ are restrictions of", "effective Cartier divisors $D, D' \\subset X$, see", "Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-smooth-over-valuation-ring-effective-Cartier}." ], "refs": [ "spaces-morphisms-lemma-finite-presentation-noetherian", "spaces-over-fields-lemma-smooth-regular", "spaces-divisors-lemma-Noetherian-regular-separated-pic-effective-Cartier", "spaces-divisors-lemma-smooth-over-valuation-ring-effective-Cartier" ], "ref_ids": [ 4843, 12872, 12952, 12953 ] } ], "ref_ids": [ 7630 ] }, { "id": 1738, "type": "theorem", "label": "moduli-lemma-pic-inertia", "categories": [ "moduli" ], "title": "moduli-lemma-pic-inertia", "contents": [ "Assume $f_{T, *}\\mathcal{O}_{X_T} \\cong \\mathcal{O}_T$ for all", "schemes $T$ over $B$. Then the inertia stack of $\\Picardstack_{X/B}$", "is equal to $\\mathbf{G}_m \\times \\Picardstack_{X/B}$." ], "refs": [], "proofs": [ { "contents": [ "This is explained in Examples of Stacks, Example", "\\ref{examples-stacks-example-inertia-stack-of-picard}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1739, "type": "theorem", "label": "moduli-lemma-pic-curves-smooth", "categories": [ "moduli" ], "title": "moduli-lemma-pic-curves-smooth", "contents": [ "Assume $f : X \\to B$ has relative dimension $\\leq 1$ in addition to", "the other assumptions in this section. Then $\\Picardstack_{X/B} \\to B$", "is smooth." ], "refs": [], "proofs": [ { "contents": [ "We already know that $\\Picardstack_{X/B} \\to B$ is", "locally of finite presentation, see Lemma \\ref{lemma-pic-qs-lfp}.", "Thus it suffices to show that $\\Picardstack_{X/B} \\to B$ is", "formally smooth, see More on Morphisms of Stacks, Lemma", "\\ref{stacks-more-morphisms-lemma-smooth-formally-smooth}.", "Taking base change, this immediately reduces to the following", "problem: given a first order thickening $T \\subset T'$", "of affine schemes, given $X' \\to T'$ proper, flat, of finite", "presentation and of relative dimension $\\leq 1$, and", "for $X = T \\times_{T'} X'$ given an invertible $\\mathcal{O}_X$-module", "$\\mathcal{L}$, prove that there exists an invertible", "$\\mathcal{O}_{X'}$-module $\\mathcal{L}'$ whose", "restriction to $X$ is $\\mathcal{L}$.", "Since $T \\subset T'$ is a first order thickening, the", "same is true for $X \\subset X'$, see", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-base-change-thickening}.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-picard-group-first-order-thickening}", "we see that it suffices to show $H^2(X, \\mathcal{I}) = 0$", "where $\\mathcal{I}$ is the quasi-coherent ideal cutting out $X$ in $X'$.", "Denote $f : X \\to T$ the structure morphism.", "By Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-higher-direct-images-zero-above-dimension-fibre}", "we see that $R^pf_*\\mathcal{I} = 0$ for $p > 1$.", "Hence we get the desired vanishing by", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-quasi-coherence-higher-direct-images-application}", "(here we finally use that $T$ is affine)." ], "refs": [ "moduli-lemma-pic-qs-lfp", "stacks-more-morphisms-lemma-smooth-formally-smooth", "spaces-more-morphisms-lemma-base-change-thickening", "spaces-more-morphisms-lemma-picard-group-first-order-thickening", "spaces-cohomology-lemma-higher-direct-images-zero-above-dimension-fibre", "spaces-cohomology-lemma-quasi-coherence-higher-direct-images-application" ], "ref_ids": [ 1736, 6906, 52, 58, 11342, 11272 ] } ], "ref_ids": [] }, { "id": 1740, "type": "theorem", "label": "moduli-lemma-pic-gerbe-over-pic-functor", "categories": [ "moduli" ], "title": "moduli-lemma-pic-gerbe-over-pic-functor", "contents": [ "The morphism $\\Picardstack_{X/B} \\to \\Picardfunctor_{X/B}$", "turns the Picard stack into a gerbe over the Picard functor." ], "refs": [], "proofs": [ { "contents": [ "The definition of $\\Picardstack_{X/B} \\to \\Picardfunctor_{X/B}$ being", "a gerbe is given in Morphisms of Stacks, Definition", "\\ref{stacks-morphisms-definition-gerbe}, which in turn refers to", "Stacks, Definition \\ref{stacks-definition-gerbe-over-stack-in-groupoids}.", "To prove it, we will check conditions (2)(a) and (2)(b) of", "Stacks, Lemma \\ref{stacks-lemma-when-gerbe}. This follows immediately from", "Quot, Lemma \\ref{quot-lemma-pic-over-pic}; here is a detailed explanation.", "\\medskip\\noindent", "Condition (2)(a).", "Suppose that $\\xi \\in \\Picardfunctor_{X/B}(U)$ for some scheme $U$ over $B$.", "Since $\\Picardfunctor_{X/B}$ is the fppf sheafification of the rule", "$T \\mapsto \\Pic(X_T)$ on schemes over $B$", "(Quot, Situation \\ref{quot-situation-pic}), we see that there exists an", "fppf covering $\\{U_i \\to U\\}$ such that $\\xi|_{U_i}$ corresponds", "to some invertible module $\\mathcal{L}_i$ on $X_{U_i}$.", "Then $(U_i \\to B, \\mathcal{L}_i)$ is an object of", "$\\Picardstack_{X/B}$ over $U_i$ mapping to $\\xi|_{U_i}$.", "\\medskip\\noindent", "Condition (2)(b). Suppose that $U$ is a scheme over $B$ and", "$\\mathcal{L}, \\mathcal{N}$ are invertible modules on $X_U$", "which map to the same element of $\\Picardfunctor_{X/B}(U)$.", "Then there exists an fppf covering $\\{U_i \\to U\\}$", "such that $\\mathcal{L}|_{X_{U_i}}$ is isomorphic to $\\mathcal{N}|_{X_{U_i}}$.", "Thus we find isomorphisms between", "$(U \\to B, \\mathcal{L})|_{U_i} \\to (U \\to B, \\mathcal{N})|_{U_i}$", "as desired." ], "refs": [ "stacks-morphisms-definition-gerbe", "stacks-definition-gerbe-over-stack-in-groupoids", "stacks-lemma-when-gerbe", "quot-lemma-pic-over-pic" ], "ref_ids": [ 7621, 9004, 8975, 3179 ] } ], "ref_ids": [] }, { "id": 1741, "type": "theorem", "label": "moduli-lemma-pic-functor-diagonal-qc-immersion", "categories": [ "moduli" ], "title": "moduli-lemma-pic-functor-diagonal-qc-immersion", "contents": [ "The diagonal of $\\Picardfunctor_{X/B}$ over $B$ is a quasi-compact immersion." ], "refs": [], "proofs": [ { "contents": [ "The diagonal is an immersion by Quot, Lemma \\ref{quot-lemma-diagonal-pic}.", "To finish we show that the diagonal is quasi-compact.", "The diagonal of $\\Picardstack_{X/B}$ is quasi-compact", "by Lemma \\ref{lemma-pic-diagonal-affine-fp} and", "$\\Picardstack_{X/B}$ is a gerbe over $\\Picardfunctor_{X/B}$ by", "Lemma \\ref{lemma-pic-gerbe-over-pic-functor}.", "We conclude by Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-gerbe-diagonal-quasi-compact}." ], "refs": [ "quot-lemma-diagonal-pic", "moduli-lemma-pic-diagonal-affine-fp", "moduli-lemma-pic-gerbe-over-pic-functor", "stacks-morphisms-lemma-gerbe-diagonal-quasi-compact" ], "ref_ids": [ 3185, 1735, 1740, 7526 ] } ], "ref_ids": [] }, { "id": 1742, "type": "theorem", "label": "moduli-lemma-pic-functor-qs-lfp", "categories": [ "moduli" ], "title": "moduli-lemma-pic-functor-qs-lfp", "contents": [ "The morphism $\\Picardfunctor_{X/B} \\to B$ is quasi-separated and", "locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "To check $\\Picardfunctor_{X/B} \\to B$ is quasi-separated we have to", "show that its diagonal is quasi-compact. This is immediate from", "Lemma \\ref{lemma-pic-functor-diagonal-qc-immersion}.", "Since the morphism $\\Picardstack_{X/B} \\to \\Picardfunctor_{X/B}$", "is surjective, flat, and locally of finite presentation", "(by Lemma \\ref{lemma-pic-gerbe-over-pic-functor} and", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-gerbe-fppf})", "it suffices to prove that $\\Picardstack_{X/B} \\to B$", "is locally of finite presentation, see", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-flat-finite-presentation-permanence}.", "This follows", "from Lemma \\ref{lemma-pic-qs-lfp}." ], "refs": [ "moduli-lemma-pic-functor-diagonal-qc-immersion", "moduli-lemma-pic-gerbe-over-pic-functor", "stacks-morphisms-lemma-gerbe-fppf", "stacks-morphisms-lemma-flat-finite-presentation-permanence", "moduli-lemma-pic-qs-lfp" ], "ref_ids": [ 1741, 1740, 7522, 7510, 1736 ] } ], "ref_ids": [] }, { "id": 1743, "type": "theorem", "label": "moduli-lemma-pic-functor-uniqueness-part", "categories": [ "moduli" ], "title": "moduli-lemma-pic-functor-uniqueness-part", "contents": [ "Assume the geometric fibres of $X \\to B$ are integral", "in addition to the other assumptions in this section.", "Then $\\Picardfunctor_{X/B} \\to B$ is separated." ], "refs": [], "proofs": [ { "contents": [ "Since $\\Picardfunctor_{X/B} \\to B$ is quasi-separated, it suffices", "to check the uniqueness part of the valuative criterion, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-valuative-criterion-separatedness}.", "This immediately reduces to the following problem: given", "\\begin{enumerate}", "\\item a valuation ring $R$ with fraction field $K$,", "\\item an algebraic space $X$ proper and flat over $R$", "with integral geometric fibre,", "\\item an element $a \\in \\Picardfunctor_{X/R}(R)$ with", "$a|_{\\Spec(K)} = 0$,", "\\end{enumerate}", "then we have to prove $a = 0$. Applying", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-lift-valuation-ring-through-flat-morphism}", "to the surjective flat morphism", "$\\Picardstack_{X/R} \\to \\Picardfunctor_{X/R}$", "(surjective and flat by Lemma \\ref{lemma-pic-gerbe-over-pic-functor} and", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-gerbe-fppf})", "after replacing $R$ by an extension we may assume", "$a$ is given by an invertible $\\mathcal{O}_X$-module", "$\\mathcal{L}$. Since $a|_{\\Spec(K)} = 0$ we find", "$\\mathcal{L}_K \\cong \\mathcal{O}_{X_K}$ by", "Quot, Lemma \\ref{quot-lemma-flat-geometrically-connected-fibres}.", "\\medskip\\noindent", "Denote $f : X \\to \\Spec(R)$ the structure morphism.", "Let $\\eta, 0 \\in \\Spec(R)$ be the generic and closed point.", "Consider the perfect complexes", "$K = Rf_*\\mathcal{L}$ and $M = Rf_*(\\mathcal{L}^{\\otimes -1})$", "on $\\Spec(R)$, see Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-flat-proper-perfect-direct-image-general}.", "Consider the functions", "$\\beta_{K, i}, \\beta_{M, i} : \\Spec(R) \\to \\mathbf{Z}$", "of Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-jump-loci} associated to $K$ and $M$.", "Since the formation of $K$ amd $M$ commutes with", "base change (see lemma cited above) we find", "$\\beta_{K, 0}(\\eta) = \\beta_{M, 0}(\\beta) = 1$ by", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-proper-geometrically-reduced-global-sections}", "and our assumption on the fibres of $f$.", "By upper semi-continuity we find", "$\\beta_{K, 0}(0) \\geq 1$ and $\\beta_{M, 0} \\geq 1$.", "By ", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-characterize-trivial-pic-integral}", "we conclude that the restriction of $\\mathcal{L}$", "to the special fibre $X_0$ is trivial. In turn this gives", "$\\beta_{K, 0}(0) = \\beta_{M, 0} = 1$ as above.", "Then by More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-pseudo-coherent-from-residue-field}", "we can represent $K$ by a complex of the form", "$$", "\\ldots \\to 0 \\to R \\to R^{\\oplus \\beta_{K, 1}(0)} \\to", "R^{\\oplus \\beta_{K, 2}(0)} \\to \\ldots", "$$", "Now $R \\to R^{\\oplus \\beta_{K, 1}(0)}$ is zero", "because $\\beta_{K, 0}(\\eta) = 1$. In other words", "$K = R \\oplus \\tau_{\\geq 1}(K)$ in $D(R)$ where $\\tau_{\\geq 1}(K)$", "has tor amplitude in $[1, b]$ for some $b \\in \\mathbf{Z}$.", "Hence there is a global section $s \\in H^0(X, \\mathcal{L})$", "whose restriction $s_0$", "to $X_0$ is nonvanishing (again because formation of $K$", "commutes with base change). Then $s : \\mathcal{O}_X \\to \\mathcal{L}$", "is a map of invertible sheaves whose restriction to $X_0$", "is an isomorphism and hence is an isomorphism as desired." ], "refs": [ "spaces-morphisms-lemma-valuative-criterion-separatedness", "stacks-morphisms-lemma-lift-valuation-ring-through-flat-morphism", "moduli-lemma-pic-gerbe-over-pic-functor", "stacks-morphisms-lemma-gerbe-fppf", "quot-lemma-flat-geometrically-connected-fibres", "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "spaces-perfect-lemma-jump-loci", "spaces-over-fields-lemma-proper-geometrically-reduced-global-sections", "spaces-over-fields-lemma-characterize-trivial-pic-integral", "more-algebra-lemma-lift-pseudo-coherent-from-residue-field" ], "ref_ids": [ 4936, 7498, 1740, 7522, 3180, 2738, 2743, 12868, 12869, 10231 ] } ], "ref_ids": [] }, { "id": 1744, "type": "theorem", "label": "moduli-lemma-pic-functor-curves-smooth", "categories": [ "moduli" ], "title": "moduli-lemma-pic-functor-curves-smooth", "contents": [ "Assume $f : X \\to B$ has relative dimension $\\leq 1$ in addition to", "the other assumptions in this section. Then $\\Picardfunctor_{X/B} \\to B$", "is smooth." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-pic-curves-smooth} we know that", "$\\Picardstack_{X/B} \\to B$ is smooth. The morphism", "$\\Picardstack_{X/B} \\to \\Picardfunctor_{X/B}$ is surjective", "and smooth by combining Lemma \\ref{lemma-pic-gerbe-over-pic-functor} with", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-gerbe-smooth}.", "Thus if $U$ is a scheme and $U \\to \\Picardstack_{X/B}$ is surjective", "and smooth, then $U \\to \\Picardfunctor_{X/B}$ is surjective and smooth", "and $U \\to B$ is surjective and smooth (because these properties", "are preserved by composition). Thus $\\Picardfunctor_{X/B} \\to B$", "is smooth for example by", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-syntomic-smooth-etale-permanence}." ], "refs": [ "moduli-lemma-pic-curves-smooth", "moduli-lemma-pic-gerbe-over-pic-functor", "stacks-morphisms-lemma-gerbe-smooth", "spaces-descent-lemma-syntomic-smooth-etale-permanence" ], "ref_ids": [ 1739, 1740, 7545, 9370 ] } ], "ref_ids": [] }, { "id": 1745, "type": "theorem", "label": "moduli-lemma-Mor-diagonal-closed", "categories": [ "moduli" ], "title": "moduli-lemma-Mor-diagonal-closed", "contents": [ "The diagonal of $\\mathit{Mor}_B(Y, X) \\to B$ is a closed immersion", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "There is an open immersion", "$\\mathit{Mor}_B(Y, X) \\to \\Hilbfunctor_{Y \\times_B X/B}$, see", "Quot, Lemma \\ref{quot-lemma-Mor-into-Hilb-open}.", "Thus the lemma follows from", "Lemma \\ref{lemma-hilb-diagonal-closed}." ], "refs": [ "quot-lemma-Mor-into-Hilb-open", "moduli-lemma-hilb-diagonal-closed" ], "ref_ids": [ 3187, 1728 ] } ], "ref_ids": [] }, { "id": 1746, "type": "theorem", "label": "moduli-lemma-Mor-s-lfp", "categories": [ "moduli" ], "title": "moduli-lemma-Mor-s-lfp", "contents": [ "The morphism $\\mathit{Mor}_B(Y, X) \\to B$ is separated", "and locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "To check $\\mathit{Mor}_B(Y, X) \\to B$ is separated we have to", "show that its diagonal is a closed immersion. This", "is true by Lemma \\ref{lemma-Mor-diagonal-closed}.", "The second statement is part of", "Quot, Proposition \\ref{quot-proposition-Mor}." ], "refs": [ "moduli-lemma-Mor-diagonal-closed", "quot-proposition-Mor" ], "ref_ids": [ 1745, 3231 ] } ], "ref_ids": [] }, { "id": 1747, "type": "theorem", "label": "moduli-lemma-Isom-in-Mor", "categories": [ "moduli" ], "title": "moduli-lemma-Isom-in-Mor", "contents": [ "With $B, X, Y$ as in the introduction of this section, in addition", "assume $X \\to B$ is proper. Then the", "subfunctor $\\mathit{Isom}_B(Y, X) \\subset \\mathit{Mor}_B(Y, X)$", "of isomorphisms is an open subspace." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-where-isomorphism}." ], "refs": [ "spaces-more-morphisms-lemma-where-isomorphism" ], "ref_ids": [ 252 ] } ], "ref_ids": [] }, { "id": 1748, "type": "theorem", "label": "moduli-lemma-Mor-qc-over-base", "categories": [ "moduli" ], "title": "moduli-lemma-Mor-qc-over-base", "contents": [ "With $B, X, Y$ as in the introduction of this section, let", "$\\mathcal{L}$ be ample on $X/B$ and let $\\mathcal{N}$ be ample on $Y/B$.", "See Divisors on Spaces, Definition", "\\ref{spaces-divisors-definition-relatively-ample}.", "Let $P$ be a numerical polynomial. Then", "$$", "\\mathit{Mor}^{P, \\mathcal{M}}_B(Y, X) \\longrightarrow B", "$$", "is separated and of finite presentation where", "$\\mathcal{M} = \\text{pr}_1^*\\mathcal{N}", "\\otimes_{\\mathcal{O}_{Y \\times_B X}} \\text{pr}_2^*\\mathcal{L}$." ], "refs": [ "spaces-divisors-definition-relatively-ample" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-Mor-s-lfp} the morphism $\\mathit{Mor}_B(Y, X) \\to B$", "is separated and locally of finite presentation. Thus it suffices to", "show that the open and closed subspace $\\mathit{Mor}^{P, \\mathcal{M}}_B(Y, X)$", "of Remark \\ref{remark-Mor-numerical} is quasi-compact over $B$.", "\\medskip\\noindent", "The question is \\'etale local on $B$", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-quasi-compact-local}).", "Thus we may assume $B$ is affine.", "\\medskip\\noindent", "Assume $B = \\Spec(\\Lambda)$. Note that $X$ and", "$Y$ are schemes and that $\\mathcal{L}$ and $\\mathcal{N}$ are ample", "invertible sheaves on $X$ and $Y$ (this follows immediately from the", "definitions). Write $\\Lambda = \\colim \\Lambda_i$ as the", "colimit of its finite type $\\mathbf{Z}$-subalgebras. Then", "we can find an $i$ and a system $X_i, Y_i, \\mathcal{L}_i, \\mathcal{N}_i$", "as in the lemma over $B_i = \\Spec(\\Lambda_i)$ whose base change to", "$B$ gives $X, Y, \\mathcal{L}, \\mathcal{N}$. This follows from", "Limits, Lemmas", "\\ref{limits-lemma-descend-finite-presentation} (to find $X_i$, $Y_i$),", "\\ref{limits-lemma-descend-invertible-modules} (to find $\\mathcal{L}_i$,", "$\\mathcal{N}_i$), \\ref{limits-lemma-descend-separated-finite-presentation}", "(to make $X_i \\to B_i$ separated), \\ref{limits-lemma-eventually-proper}", "(to make $Y_i \\to B_i$ proper), and \\ref{limits-lemma-limit-ample}", "(to make $\\mathcal{L}_i$, $\\mathcal{N}_i$ ample).", "Because", "$$", "\\mathit{Mor}_B(Y, X) = B \\times_{B_i} \\mathit{Mor}_{B_i}(Y_i, X_i)", "$$", "and similarly for $\\mathit{Mor}^P_B(Y, X)$ we reduce", "to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $B$ is a Noetherian affine scheme. By", "Properties, Lemma \\ref{properties-lemma-ample-on-product}", "we see that $\\mathcal{M}$ is ample. By Lemma \\ref{lemma-hilb-qc-over-base}", "we see that $\\Hilbfunctor^{P, \\mathcal{M}}_{Y \\times_B X/B}$ is of", "finite presentation over $B$ and hence Noetherian.", "By construction", "$$", "\\mathit{Mor}^{P, \\mathcal{M}}_B(Y, X) =", "\\mathit{Mor}_B(Y, X) \\cap", "\\Hilbfunctor^{P, \\mathcal{M}}_{Y \\times_B X/B}", "$$", "is an open subspace of $\\Hilbfunctor^{P, \\mathcal{M}}_{Y \\times_B X/B}$ and", "hence quasi-compact (as an open of a Noetherian algebraic space", "is quasi-compact)." ], "refs": [ "moduli-lemma-Mor-s-lfp", "moduli-remark-Mor-numerical", "spaces-morphisms-lemma-quasi-compact-local", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-invertible-modules", "limits-lemma-descend-separated-finite-presentation", "limits-lemma-eventually-proper", "limits-lemma-limit-ample", "properties-lemma-ample-on-product", "moduli-lemma-hilb-qc-over-base" ], "ref_ids": [ 1746, 1756, 4742, 15077, 15079, 15061, 15089, 15045, 3052, 1734 ] } ], "ref_ids": [ 13028 ] }, { "id": 1749, "type": "theorem", "label": "moduli-lemma-polarized-diagonal-separated-fp", "categories": [ "moduli" ], "title": "moduli-lemma-polarized-diagonal-separated-fp", "contents": [ "The diagonal of $\\Polarizedstack$ is separated", "and of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\Polarizedstack$ is a limit preserving algebraic stack, see", "Quot, Lemma \\ref{quot-lemma-polarized-limits}.", "By Limits of Stacks, Lemma \\ref{stacks-limits-lemma-limit-preserving-diagonal}", "this implies that", "$\\Delta : \\Polarizedstack \\to \\Polarizedstack \\times \\Polarizedstack$", "is limit preserving. Hence $\\Delta$ is locally of finite presentation", "by Limits of Stacks, Proposition", "\\ref{stacks-limits-proposition-characterize-locally-finite-presentation}.", "\\medskip\\noindent", "Let us prove that $\\Delta$ is separated. To see this, it suffices to show", "that given an affine scheme $U$ and two objects", "$\\upsilon = (Y, \\mathcal{N})$ and $\\chi = (X, \\mathcal{L})$", "of $\\Polarizedstack$ over $U$, the algebraic", "space", "$$", "\\mathit{Isom}_{\\Polarizedstack}(\\upsilon, \\chi)", "$$", "is separated. The rule which to an isomorphism $\\upsilon_T \\to \\chi_T$", "assigns the underlying isomorphism $Y_T \\to X_T$ defines a morphism", "$$", "\\mathit{Isom}_{\\Polarizedstack}(\\upsilon, \\chi)", "\\longrightarrow", "\\mathit{Isom}_U(Y, X)", "$$", "Since we have seen in Lemmas \\ref{lemma-Mor-s-lfp} and", "\\ref{lemma-Isom-in-Mor} that the target is", "a separated algebraic space, it suffices to prove that this morphism", "is separated. Given an isomorphism $f : Y_T \\to X_T$", "over some scheme $T/U$, then clearly", "$$", "\\mathit{Isom}_{\\Polarizedstack}(\\upsilon, \\chi)", "\\times_{\\mathit{Isom}_U(Y, X), [f]} T", "=", "\\mathit{Isom}(\\mathcal{N}_T, f^*\\mathcal{L}_T)", "$$", "Here $[f] : T \\to \\mathit{Isom}_U(Y, X)$ indicates the $T$-valued", "point corresponding to $f$ and", "$\\mathit{Isom}(\\mathcal{N}_T, f^*\\mathcal{L}_T)$", "is the algebraic space discussed in Section \\ref{section-hom-isom}.", "Since this algebraic space is affine over $U$, the claim implies", "$\\Delta$ is separated.", "\\medskip\\noindent", "To finish the proof we show that $\\Delta$ is quasi-compact. Since", "$\\Delta$ is representable by algebraic spaces, it suffice to check", "the base change of $\\Delta$ by a surjective smooth morphism", "$U \\to \\Polarizedstack \\times \\Polarizedstack$ is quasi-compact", "(see for example Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}).", "We can assume $U = \\coprod U_i$ is a disjoint union of affine opens.", "Since $\\Polarizedstack$ is limit preserving (see above), we", "see that $\\Polarizedstack \\to \\Spec(\\mathbf{Z})$ is locally of", "finite presentation, hence $U_i \\to \\Spec(\\mathbf{Z})$ is", "locally of finite presentation", "(Limits of Stacks, Proposition", "\\ref{stacks-limits-proposition-characterize-locally-finite-presentation}", "and Morphisms of Stacks, Lemmas", "\\ref{stacks-morphisms-lemma-composition-finite-presentation} and", "\\ref{stacks-morphisms-lemma-smooth-locally-finite-presentation}).", "In particular, $U_i$ is Noetherian affine. This reduces us to the", "case discussed in the next paragraph.", "\\medskip\\noindent", "In this paragraph, given a Noetherian affine scheme $U$ and two objects", "$\\upsilon = (Y, \\mathcal{N})$ and $\\chi = (X, \\mathcal{L})$", "of $\\Polarizedstack$ over $U$, we show the algebraic space", "$$", "\\mathit{Isom}_{\\Polarizedstack}(\\upsilon, \\chi)", "$$", "is quasi-compact. Since the connected components of $U$ are open and closed", "we may replace $U$ by these. Thus we may and do assume $U$ is connected.", "Let $u \\in U$ be a point. Let $P$ be the Hilbert polynomial", "$n \\mapsto \\chi(Y_u, \\mathcal{N}_u^{\\otimes n})$, see", "Varieties, Lemma \\ref{varieties-lemma-numerical-polynomial-from-euler}.", "Since $U$ is connected and since", "the functions $u \\mapsto \\chi(Y_u, \\mathcal{N}_u^{\\otimes n})$", "are locally constant (see ", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-chi-locally-constant-geometric})", "we see that we get the same Hilbert polynomial in every point of $U$.", "Set", "$\\mathcal{M} = \\text{pr}_1^*\\mathcal{N}", "\\otimes_{\\mathcal{O}_{Y \\times_U X}} \\text{pr}_2^*\\mathcal{L}$", "on $Y \\times_U X$. Given", "$(f, \\varphi) \\in \\mathit{Isom}_{\\Polarizedstack}(\\upsilon, \\chi)(T)$", "for some scheme $T$ over $U$ then for every $t \\in T$ we have", "$$", "\\chi(Y_t, (\\text{id} \\times f)^*\\mathcal{M}^{\\otimes n}) =", "\\chi(Y_t,", "\\mathcal{N}_t^{\\otimes n} \\otimes_{\\mathcal{O}_{Y_t}}", "f_t^*\\mathcal{L}_t^{\\otimes n}) =", "\\chi(Y_t, \\mathcal{N}_t^{\\otimes 2n}) = P(2n)", "$$", "where in the middle equality we use the isomorphism", "$\\varphi : f^*\\mathcal{L}_T \\to \\mathcal{N}_T$.", "Setting $P'(t) = P(2t)$ we find that the morphism", "$$", "\\mathit{Isom}_{\\Polarizedstack}(\\upsilon, \\chi)", "\\longrightarrow", "\\mathit{Isom}_U(Y, X)", "$$", "(see earlier) has image contained in the intersection", "$$", "\\mathit{Isom}_U(Y, X) \\cap \\mathit{Mor}^{P', \\mathcal{M}}_U(Y, X)", "$$", "The intersection is an intersection of open subspaces of", "$\\mathit{Mor}_U(Y, X)$ (see Lemma \\ref{lemma-Isom-in-Mor} and", "Remark \\ref{remark-Mor-numerical}).", "Now $\\mathit{Mor}^{P', \\mathcal{M}}_U(Y, X)$", "is a Noetherian algebraic space as it is of finite", "presentation over $U$ by Lemma \\ref{lemma-Mor-qc-over-base}.", "Thus the intersection", "is a Noetherian algebraic space too. Since the morphism", "$$", "\\mathit{Isom}_{\\Polarizedstack}(\\upsilon, \\chi)", "\\longrightarrow", "\\mathit{Isom}_U(Y, X) \\cap \\mathit{Mor}^{P', \\mathcal{M}}_U(Y, X)", "$$", "is affine (see above) we conclude." ], "refs": [ "quot-lemma-polarized-limits", "stacks-limits-lemma-limit-preserving-diagonal", "stacks-limits-proposition-characterize-locally-finite-presentation", "moduli-lemma-Mor-s-lfp", "moduli-lemma-Isom-in-Mor", "stacks-properties-lemma-check-property-covering", "stacks-limits-proposition-characterize-locally-finite-presentation", "stacks-morphisms-lemma-composition-finite-presentation", "stacks-morphisms-lemma-smooth-locally-finite-presentation", "varieties-lemma-numerical-polynomial-from-euler", "perfect-lemma-chi-locally-constant-geometric", "moduli-lemma-Isom-in-Mor", "moduli-remark-Mor-numerical", "moduli-lemma-Mor-qc-over-base" ], "ref_ids": [ 3201, 15018, 15024, 1746, 1747, 8859, 15024, 7500, 7542, 11121, 7063, 1747, 1756, 1748 ] } ], "ref_ids": [] }, { "id": 1750, "type": "theorem", "label": "moduli-lemma-polarized-qs-lfp", "categories": [ "moduli" ], "title": "moduli-lemma-polarized-qs-lfp", "contents": [ "The morphism $\\Polarizedstack \\to \\Spec(\\mathbf{Z})$ is quasi-separated and", "locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "To check $\\Polarizedstack \\to \\Spec(\\mathbf{Z})$ is quasi-separated we have to", "show that its diagonal is quasi-compact and quasi-separated.", "This is immediate from Lemma \\ref{lemma-polarized-diagonal-separated-fp}.", "To prove that $\\Polarizedstack \\to \\Spec(\\mathbf{Z})$ is locally of finite", "presentation, it suffices to show that $\\Polarizedstack$", "is limit preserving, see Limits of Stacks, Proposition", "\\ref{stacks-limits-proposition-characterize-locally-finite-presentation}.", "This is Quot, Lemma \\ref{quot-lemma-polarized-limits}." ], "refs": [ "moduli-lemma-polarized-diagonal-separated-fp", "stacks-limits-proposition-characterize-locally-finite-presentation", "quot-lemma-polarized-limits" ], "ref_ids": [ 1749, 15024, 3201 ] } ], "ref_ids": [] }, { "id": 1751, "type": "theorem", "label": "moduli-lemma-bounded-polarized", "categories": [ "moduli" ], "title": "moduli-lemma-bounded-polarized", "contents": [ "Let $n \\geq 1$ be an integer and let $P$ be a numerical polynomial.", "Let", "$$", "T \\subset |\\Polarizedstack|", "$$", "be a subset with the following property: for every $\\xi \\in T$", "there exists a field $k$ and an object $(X, \\mathcal{L})$", "of $\\Polarizedstack$ over $k$ representing $\\xi$ such that", "\\begin{enumerate}", "\\item the Hilbert polynomial of $\\mathcal{L}$ on $X$ is $P$, and", "\\item there exists a closed immersion $i : X \\to \\mathbf{P}^n_k$", "such that $i^*\\mathcal{O}_{\\mathbf{P}^n}(1) \\cong \\mathcal{L}$.", "\\end{enumerate}", "Then $T$ is a Noetherian topological space, in particular quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Observe that $|\\Polarizedstack|$ is a locally Noetherian topological", "space, see Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-Noetherian-topology}", "(this also uses that $\\Spec(\\mathbf{Z})$ is Noetherian and", "hence $\\Polarizedstack$ is a locally Noetherian algebraic stack", "by Lemma \\ref{lemma-polarized-qs-lfp} and", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-locally-finite-type-locally-noetherian}).", "Thus any quasi-compact subset of $|\\Polarizedstack|$ is", "a Noetherian topological space and any subset of such is", "also Noetherian, see", "Topology, Lemmas \\ref{topology-lemma-finite-union-Noetherian} and", "\\ref{topology-lemma-Noetherian}.", "Thus all we have to do is a find a quasi-compact subset", "containing $T$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-hilb-proper-over-base} the algebraic space", "$$", "H =", "\\Hilbfunctor^{P, \\mathcal{O}(1)}_{\\mathbf{P}^n_\\mathbf{Z}/\\Spec(\\mathbf{Z})}", "$$", "is proper over $\\Spec(\\mathbf{Z})$. By", "Quot, Lemma \\ref{quot-lemma-extend-hilb-to-spaces}\\footnote{We will see", "later (insert future reference here) that $H$ is a scheme and hence the", "use of this lemma and Quot, Lemma \\ref{quot-lemma-extend-polarized-to-spaces}", "isn't necessary.} the identity morphism of $H$ corresponds", "to a closed subspace", "$$", "Z \\subset \\mathbf{P}^n_H", "$$", "which is proper, flat, and of finite presentation over $H$ and", "such that the restriction $\\mathcal{N} = \\mathcal{O}(1)|_Z$", "is relatively ample on $Z/H$ and has Hilbert polynomial $P$", "on the fibres of $Z \\to H$. In particular, the pair $(Z \\to H, \\mathcal{N})$", "defines a morphism", "$$", "H \\longrightarrow \\Polarizedstack", "$$", "which sends a morphism of schemes $U \\to H$ to the classifying morphism", "of the family $(Z_U \\to U, \\mathcal{N}_U)$, see", "Quot, Lemma \\ref{quot-lemma-extend-polarized-to-spaces}.", "Since $H$ is a Noetherian algebraic space", "(as it is proper over $\\mathbf{Z})$)", "we see that $|H|$ is Noetherian and hence quasi-compact. The map", "$$", "|H| \\longrightarrow |\\Polarizedstack|", "$$", "is continuous, hence the image is quasi-compact.", "Thus it suffices to prove $T$ is contained", "in the image of $|H| \\to |\\Polarizedstack|$.", "However, assumptions (1) and (2) exactly express the fact", "that this is the case: any choice of a closed immersion", "$i : X \\to \\mathbf{P}^n_k$ with", "$i^*\\mathcal{O}_{\\mathbf{P}^n}(1) \\cong \\mathcal{L}$ we get a", "$k$-valued point of $H$ by the moduli interpretation of $H$.", "This finishes the proof of the lemma." ], "refs": [ "stacks-morphisms-lemma-Noetherian-topology", "moduli-lemma-polarized-qs-lfp", "stacks-morphisms-lemma-locally-finite-type-locally-noetherian", "topology-lemma-finite-union-Noetherian", "topology-lemma-Noetherian", "moduli-lemma-hilb-proper-over-base", "quot-lemma-extend-hilb-to-spaces", "quot-lemma-extend-polarized-to-spaces", "quot-lemma-extend-polarized-to-spaces" ], "ref_ids": [ 7432, 1750, 7462, 8222, 8220, 1733, 3177, 3198, 3198 ] } ], "ref_ids": [] }, { "id": 1752, "type": "theorem", "label": "moduli-lemma-complexes-diagonal-affine-fp", "categories": [ "moduli" ], "title": "moduli-lemma-complexes-diagonal-affine-fp", "contents": [ "The diagonal of $\\Complexesstack_{X/B}$ over $B$ is affine", "and of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "The representability of the diagonal (by algebraic spaces)", "was shown in Quot, Lemma \\ref{quot-lemma-complexes-diagonal}.", "From the proof we find that we have to show:", "given a scheme $T$ over $B$ and objects", "$E, E' \\in D(\\mathcal{O}_{X_T})$ such that", "$(T, E)$ and $(T, E')$ are objects of the fibre category", "of $\\Complexesstack_{X/B}$ over $T$, then", "$\\mathit{Isom}(E, E') \\to T$", "is affine and of finite presentation.", "Here $\\mathit{Isom}(E, E')$ is the functor", "$$", "(\\Sch/T)^{opp} \\to \\textit{Sets},\\quad", "T' \\mapsto \\{\\varphi : E_{T'} \\to E'_{T'}", "\\text{ isomorphism in }D(\\mathcal{O}_{X_{T'}})\\}", "$$", "where $E_{T'}$ and $E'_{T'}$ are the derived pullbacks of $E$ and $E'$", "to $X_{T'}$. Consider the functor $H = \\SheafHom(E, E')$ defined", "by the rule", "$$", "(\\Sch/T)^{opp} \\to \\textit{Sets},\\quad", "T' \\mapsto \\Hom_{\\mathcal{O}_{X_{T'}}}(E_T, E'_T)", "$$", "By Quot, Lemma \\ref{quot-lemma-complexes-open-neg-exts-vanishing}", "this is an algebraic space affine and of finite presentation over $T$.", "The same is true for $H' = \\SheafHom(E', E)$, $I = \\SheafHom(E, E)$, and", "$I' = \\SheafHom(E', E')$. Therefore we see that", "$$", "\\mathit{Isom}(E, E') = (H' \\times_T H) \\times_{c, I \\times_T I', \\sigma} T", "$$", "where $c(\\varphi', \\varphi) = (\\varphi \\circ \\varphi', \\varphi' \\circ \\varphi)$", "and $\\sigma = (\\text{id}, \\text{id})$ (compare with the proof of", "Quot, Proposition \\ref{quot-proposition-isom}). Thus", "$\\mathit{Isom}(E, E')$ is affine over $T$ as a fibre product of", "schemes affine over $T$. Similarly, $\\mathit{Isom}(E, E')$ is", "of finite presentation over $T$." ], "refs": [ "quot-lemma-complexes-diagonal", "quot-lemma-complexes-open-neg-exts-vanishing", "quot-proposition-isom" ], "ref_ids": [ 3219, 3216, 3226 ] } ], "ref_ids": [] }, { "id": 1753, "type": "theorem", "label": "moduli-lemma-complexes-qs-lfp", "categories": [ "moduli" ], "title": "moduli-lemma-complexes-qs-lfp", "contents": [ "The morphism $\\Complexesstack_{X/B} \\to B$ is quasi-separated and", "locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "To check $\\Complexesstack_{X/B} \\to B$ is quasi-separated we have to", "show that its diagonal is quasi-compact and quasi-separated.", "This is immediate from Lemma \\ref{lemma-complexes-diagonal-affine-fp}.", "To prove that $\\Complexesstack_{X/B} \\to B$ is locally of finite", "presentation, we have to show that $\\Complexesstack_{X/B} \\to B$", "is limit preserving, see", "Limits of Stacks, Proposition", "\\ref{stacks-limits-proposition-characterize-locally-finite-presentation}.", "This follows from Quot, Lemma \\ref{quot-lemma-complexes-limits}", "(small detail omitted)." ], "refs": [ "moduli-lemma-complexes-diagonal-affine-fp", "stacks-limits-proposition-characterize-locally-finite-presentation", "quot-lemma-complexes-limits" ], "ref_ids": [ 1752, 15024, 3221 ] } ], "ref_ids": [] }, { "id": 1757, "type": "theorem", "label": "derived-lemma-composition-zero", "categories": [ "derived" ], "title": "derived-lemma-composition-zero", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "Let $(X, Y, Z, f, g, h)$ be a distinguished triangle.", "Then $g \\circ f = 0$,", "$h \\circ g = 0$ and $f[1] \\circ h = 0$." ], "refs": [], "proofs": [ { "contents": [ "By TR1 we know $(X, X, 0, 1, 0, 0)$ is a distinguished triangle.", "Apply TR3 to", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d]^1 &", "X \\ar[r] \\ar[d]^f &", "0 \\ar[r] \\ar@{-->}[d] &", "X[1] \\ar[d]^{1[1]} \\\\", "X \\ar[r]^f &", "Y \\ar[r]^g &", "Z \\ar[r]^h &", "X[1]", "}", "$$", "Of course the dotted arrow is the zero map. Hence the commutativity of", "the diagram implies that $g \\circ f = 0$. For the other cases", "rotate the triangle, i.e., apply TR2." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1758, "type": "theorem", "label": "derived-lemma-representable-homological", "categories": [ "derived" ], "title": "derived-lemma-representable-homological", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "For any object $W$ of $\\mathcal{D}$ the functor", "$\\Hom_\\mathcal{D}(W, -)$ is homological, and the functor", "$\\Hom_\\mathcal{D}(-, W)$ is cohomological." ], "refs": [], "proofs": [ { "contents": [ "Consider a distinguished triangle $(X, Y, Z, f, g, h)$.", "We have already seen that $g \\circ f = 0$, see", "Lemma \\ref{lemma-composition-zero}.", "Suppose $a : W \\to Y$ is a morphism such that $g \\circ a = 0$.", "Then we get a commutative diagram", "$$", "\\xymatrix{", "W \\ar[r]_1 \\ar@{..>}[d]^b &", "W \\ar[r] \\ar[d]^a &", "0 \\ar[r] \\ar[d]^0 &", "W[1] \\ar@{..>}[d]^{b[1]} \\\\", "X \\ar[r] & Y \\ar[r] & Z \\ar[r] & X[1]", "}", "$$", "Both rows are distinguished triangles (use TR1 for the top row).", "Hence we can fill the dotted arrow $b$ (first rotate using TR2,", "then apply TR3, and then rotate back). This proves the lemma." ], "refs": [ "derived-lemma-composition-zero" ], "ref_ids": [ 1757 ] } ], "ref_ids": [] }, { "id": 1759, "type": "theorem", "label": "derived-lemma-third-isomorphism-triangle", "categories": [ "derived" ], "title": "derived-lemma-third-isomorphism-triangle", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "Let", "$$", "(a, b, c) : (X, Y, Z, f, g, h) \\to (X', Y', Z', f', g', h')", "$$", "be a morphism of distinguished triangles. If two among $a, b, c$", "are isomorphisms so is the third." ], "refs": [], "proofs": [ { "contents": [ "Assume that $a$ and $c$ are isomorphisms.", "For any object $W$ of $\\mathcal{D}$ write", "$H_W( - ) = \\Hom_\\mathcal{D}(W, -)$.", "Then we get a commutative diagram of abelian groups", "$$", "\\xymatrix{", "H_W(Z[-1]) \\ar[r] \\ar[d] &", "H_W(X) \\ar[r] \\ar[d] &", "H_W(Y) \\ar[r] \\ar[d] &", "H_W(Z) \\ar[r] \\ar[d] &", "H_W(X[1]) \\ar[d] \\\\", "H_W(Z'[-1]) \\ar[r] &", "H_W(X') \\ar[r] &", "H_W(Y') \\ar[r] &", "H_W(Z') \\ar[r] &", "H_W(X'[1])", "}", "$$", "By assumption the right two and left two vertical arrows are bijective.", "As $H_W$ is homological by", "Lemma \\ref{lemma-representable-homological}", "and the five lemma", "(Homology, Lemma \\ref{homology-lemma-five-lemma})", "it follows that the middle vertical arrow is an isomorphism.", "Hence by Yoneda's lemma, see", "Categories, Lemma \\ref{categories-lemma-yoneda}", "we see that $b$ is an isomorphism.", "This implies the other cases by rotating (using TR2)." ], "refs": [ "derived-lemma-representable-homological", "homology-lemma-five-lemma", "categories-lemma-yoneda" ], "ref_ids": [ 1758, 12030, 12203 ] } ], "ref_ids": [] }, { "id": 1760, "type": "theorem", "label": "derived-lemma-third-map-square-zero", "categories": [ "derived" ], "title": "derived-lemma-third-map-square-zero", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "Let", "$$", "(0, b, 0), (0, b', 0) : (X, Y, Z, f, g, h) \\to (X, Y, Z, f, g, h)", "$$", "be endomorphisms of a distinguished triangle. Then $bb' = 0$." ], "refs": [], "proofs": [ { "contents": [ "Picture", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d]^0 &", "Y \\ar[r] \\ar[d]^{b, b'} \\ar@{..>}[ld]^\\alpha &", "Z \\ar[r] \\ar[d]^0 \\ar@{..>}[ld]^\\beta &", "X[1] \\ar[d]^0 \\\\", "X \\ar[r] & Y \\ar[r] & Z \\ar[r] & X[1]", "}", "$$", "Applying", "Lemma \\ref{lemma-representable-homological}", "we find dotted arrows $\\alpha$ and $\\beta$ such that", "$b' = f \\circ \\alpha$ and $b = \\beta \\circ g$. Then", "$bb' = \\beta \\circ g \\circ f \\circ \\alpha = 0$", "as $g \\circ f = 0$ by", "Lemma \\ref{lemma-composition-zero}." ], "refs": [ "derived-lemma-representable-homological", "derived-lemma-composition-zero" ], "ref_ids": [ 1758, 1757 ] } ], "ref_ids": [] }, { "id": 1761, "type": "theorem", "label": "derived-lemma-third-map-idempotent", "categories": [ "derived" ], "title": "derived-lemma-third-map-idempotent", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "Let $(X, Y, Z, f, g, h)$ be a distinguished triangle.", "If", "$$", "\\xymatrix{", "Z \\ar[r]_h \\ar[d]_c & X[1] \\ar[d]^{a[1]} \\\\", "Z \\ar[r]^h & X[1]", "}", "$$", "is commutative and $a^2 = a$, $c^2 = c$, then there exists a", "morphism $b : Y \\to Y$ with $b^2 = b$ such that", "$(a, b, c)$ is an endomorphism of the triangle $(X, Y, Z, f, g, h)$." ], "refs": [], "proofs": [ { "contents": [ "By TR3 there exists a morphism $b'$ such that", "$(a, b', c)$ is an endomorphism of $(X, Y, Z, f, g, h)$.", "Then $(0, (b')^2 - b', 0)$ is also an endomorphism. By", "Lemma \\ref{lemma-third-map-square-zero}", "we see that $(b')^2 - b'$ has square zero.", "Set $b = b' - (2b' - 1)((b')^2 - b') = 3(b')^2 - 2(b')^3$.", "A computation shows that $(a, b, c)$ is an endomorphism and", "that $b^2 - b = (4(b')^2 - 4b' - 3)((b')^2 - b')^2 = 0$." ], "refs": [ "derived-lemma-third-map-square-zero" ], "ref_ids": [ 1760 ] } ], "ref_ids": [] }, { "id": 1762, "type": "theorem", "label": "derived-lemma-cone-triangle-unique-isomorphism", "categories": [ "derived" ], "title": "derived-lemma-cone-triangle-unique-isomorphism", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "Let $f : X \\to Y$ be a morphism of $\\mathcal{D}$.", "There exists a distinguished triangle $(X, Y, Z, f, g, h)$ which", "is unique up to (nonunique) isomorphism of triangles.", "More precisely, given a second such distinguished triangle", "$(X, Y, Z', f, g', h')$ there exists an isomorphism", "$$", "(1, 1, c) : (X, Y, Z, f, g, h) \\longrightarrow (X, Y, Z', f, g', h')", "$$" ], "refs": [], "proofs": [ { "contents": [ "Existence by TR1. Uniqueness up to isomorphism by TR3 and", "Lemma \\ref{lemma-third-isomorphism-triangle}." ], "refs": [ "derived-lemma-third-isomorphism-triangle" ], "ref_ids": [ 1759 ] } ], "ref_ids": [] }, { "id": 1763, "type": "theorem", "label": "derived-lemma-uniqueness-third-arrow", "categories": [ "derived" ], "title": "derived-lemma-uniqueness-third-arrow", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category. Let", "$$", "(a, b, c) : (X, Y, Z, f, g, h) \\to (X', Y', Z', f', g', h')", "$$", "be a morphism of distinguished triangles. If one of the following", "conditions holds", "\\begin{enumerate}", "\\item $\\Hom(Y, X') = 0$,", "\\item $\\Hom(Z, Y') = 0$,", "\\item $\\Hom(X, X') = \\Hom(Z, X') = 0$,", "\\item $\\Hom(Z, X') = \\Hom(Z, Z') = 0$, or", "\\item $\\Hom(X[1], Z') = \\Hom(Z, X') = 0$", "\\end{enumerate}", "then $b$ is the unique morphism from $Y \\to Y'$ such that", "$(a, b, c)$ is a morphism of triangles." ], "refs": [], "proofs": [ { "contents": [ "If we have a second morphism of triangles $(a, b', c)$", "then $(0, b - b', 0)$ is a morphism of triangles. Hence we", "have to show: the only morphism $b : Y \\to Y'$ such that the ", "$X \\to Y \\to Y'$ and $Y \\to Y' \\to Z'$ are zero is $0$.", "We will use Lemma \\ref{lemma-representable-homological}", "without further mention. In particular, condition (3) implies (1).", "Given condition (1) if the composition $b : Y \\to Y' \\to Z'$", "is zero, then $b$ lifts to a morphism $Y \\to X'$ which has to be zero.", "This proves (1).", "\\medskip\\noindent", "The proof of (2) and (4) are dual to this argument.", "\\medskip\\noindent", "Assume (5). Consider the diagram", "$$", "\\xymatrix{", "X \\ar[r]_f \\ar[d]^0 &", "Y \\ar[r]_g \\ar[d]^b &", "Z \\ar[r]_h \\ar[d]^0 \\ar@{..>}[ld]^\\epsilon &", "X[1] \\ar[d]^0 \\\\", "X' \\ar[r]^{f'} &", "Y' \\ar[r]^{g'} &", "Z' \\ar[r]^{h'} &", "X'[1]", "}", "$$", "We may choose $\\epsilon$ such that $b = g \\circ \\epsilon$.", "Then $g' \\circ \\epsilon \\circ g = 0$ which implies that", "$g' \\circ \\epsilon = \\delta \\circ h$ for some", "$\\delta \\in \\Hom(X[1], Z')$. Since $\\Hom(X[1], Z') = 0$", "we conclude that $g' \\circ \\epsilon = 0$. Hence", "$\\epsilon = f' \\circ \\gamma$ for some $\\gamma \\in \\Hom(Z, X')$.", "Since $\\Hom(Z, X') = 0$ we conclude that $\\epsilon = 0$", "and hence $b = 0$ as desired." ], "refs": [ "derived-lemma-representable-homological" ], "ref_ids": [ 1758 ] } ], "ref_ids": [] }, { "id": 1764, "type": "theorem", "label": "derived-lemma-third-object-zero", "categories": [ "derived" ], "title": "derived-lemma-third-object-zero", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "Let $f : X \\to Y$ be a morphism of $\\mathcal{D}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is an isomorphism,", "\\item $(X, Y, 0, f, 0, 0)$ is a distinguished triangle, and", "\\item for any distinguished triangle $(X, Y, Z, f, g, h)$ we have $Z = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By TR1 the triangle $(X, X, 0, 1, 0, 0)$ is distinguished.", "Let $(X, Y, Z, f, g, h)$ be a distinguished triangle.", "By TR3 there is a map of distinguished triangles", "$(1, f, 0) : (X, X, 0) \\to (X, Y, Z)$.", "If $f$ is an isomorphism, then $(1, f, 0)$ is an isomorphism", "of triangles by Lemma \\ref{lemma-third-isomorphism-triangle}", "and $Z = 0$. Conversely, if $Z = 0$, then $(1, f, 0)$ is an", "isomorphism of triangles as well, hence $f$ is an isomorphism." ], "refs": [ "derived-lemma-third-isomorphism-triangle" ], "ref_ids": [ 1759 ] } ], "ref_ids": [] }, { "id": 1765, "type": "theorem", "label": "derived-lemma-direct-sum-triangles", "categories": [ "derived" ], "title": "derived-lemma-direct-sum-triangles", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "Let $(X, Y, Z, f, g, h)$ and $(X', Y', Z', f', g', h')$ be triangles.", "The following are equivalent", "\\begin{enumerate}", "\\item $(X \\oplus X', Y \\oplus Y', Z \\oplus Z',", "f \\oplus f', g \\oplus g', h \\oplus h')$", "is a distinguished triangle,", "\\item both $(X, Y, Z, f, g, h)$ and $(X', Y', Z', f', g', h')$ are", "distinguished triangles.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (2). By TR1 we may choose a distinguished triangle", "$(X \\oplus X', Y \\oplus Y', Q, f \\oplus f', g'', h'')$.", "By TR3 we can find morphisms of distinguished triangles", "$(X, Y, Z, f, g, h) \\to", "(X \\oplus X', Y \\oplus Y', Q, f \\oplus f', g'', h'')$", "and", "$(X', Y', Z', f', g', h') \\to", "(X \\oplus X', Y \\oplus Y', Q, f \\oplus f', g'', h'')$.", "Taking the direct sum of these morphisms", "we obtain a morphism of triangles", "$$", "\\xymatrix{", "(X \\oplus X', Y \\oplus Y', Z \\oplus Z',", "f \\oplus f', g \\oplus g', h \\oplus h')", "\\ar[d]^{(1, 1, c)} \\\\", "(X \\oplus X', Y \\oplus Y', Q, f \\oplus f', g'', h'').", "}", "$$", "In the terminology of Remark \\ref{remark-special-triangles}", "this is a map of special triangles (because a direct sum of special", "triangles is special) and we conclude", "that $c$ is an isomorphism. Thus (1) holds.", "\\medskip\\noindent", "Assume (1). We will show that $(X, Y, Z, f, g, h)$ is a distinguished", "triangle. First observe that $(X, Y, Z, f, g, h)$ is a special triangle", "(terminology from Remark \\ref{remark-special-triangles})", "as a direct summand of the distinguished hence special", "triangle $(X \\oplus X', Y \\oplus Y', Z \\oplus Z',", "f \\oplus f', g \\oplus g', h \\oplus h')$. Using TR1 let", "$(X, Y, Q, f, g'', h'')$ be a distinguished triangle. By TR3 there exists", "a morphism of distinguished triangles", " $(X \\oplus X', Y \\oplus Y', Z \\oplus Z',", "f \\oplus f', g \\oplus g', h \\oplus h') \\to (X, Y, Q, f, g'', h'')$.", "Composing this with the inclusion map we get a morphism of triangles", "$$", "(1, 1, c) :", "(X, Y, Z, f, g, h)", "\\longrightarrow", "(X, Y, Q, f, g'', h'')", "$$", "By Remark \\ref{remark-special-triangles}", "we find that $c$ is an isomorphism and we conclude", "that (2) holds." ], "refs": [ "derived-remark-special-triangles", "derived-remark-special-triangles", "derived-remark-special-triangles" ], "ref_ids": [ 2009, 2009, 2009 ] } ], "ref_ids": [] }, { "id": 1766, "type": "theorem", "label": "derived-lemma-split", "categories": [ "derived" ], "title": "derived-lemma-split", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "Let $(X, Y, Z, f, g, h)$ be a distinguished triangle.", "\\begin{enumerate}", "\\item If $h = 0$, then there exists a right inverse $s : Z \\to Y$ to $g$.", "\\item For any right inverse $s : Z \\to Y$ of $g$ the map", "$f \\oplus s : X \\oplus Z \\to Y$ is an isomorphism.", "\\item For any objects $X', Z'$ of $\\mathcal{D}$ the triangle", "$(X', X' \\oplus Z', Z', (1, 0), (0, 1), 0)$ is distinguished.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To see (1) use that", "$\\Hom_\\mathcal{D}(Z, Y) \\to \\Hom_\\mathcal{D}(Z, Z) \\to", "\\Hom_\\mathcal{D}(Z, X[1])$", "is exact by", "Lemma \\ref{lemma-representable-homological}.", "By the same token, if $s$ is as in (2), then $h = 0$ and the sequence", "$$", "0 \\to \\Hom_\\mathcal{D}(W, X) \\to \\Hom_\\mathcal{D}(W, Y)", "\\to \\Hom_\\mathcal{D}(W, Z) \\to 0", "$$", "is split exact (split by $s : Z \\to Y$). Hence by Yoneda's lemma we", "see that $X \\oplus Z \\to Y$ is an isomorphism. The last assertion follows", "from TR1 and", "Lemma \\ref{lemma-direct-sum-triangles}." ], "refs": [ "derived-lemma-representable-homological", "derived-lemma-direct-sum-triangles" ], "ref_ids": [ 1758, 1765 ] } ], "ref_ids": [] }, { "id": 1767, "type": "theorem", "label": "derived-lemma-when-split", "categories": [ "derived" ], "title": "derived-lemma-when-split", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "Let $f : X \\to Y$ be a morphism of $\\mathcal{D}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ has a kernel,", "\\item $f$ has a cokernel,", "\\item $f$ is the isomorphic to a composition", "$K \\oplus Z \\to Z \\to Z \\oplus Q$ of a projection and coprojection", "for some objects $K, Z, Q$ of $\\mathcal{D}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Any morphism isomorphic to a map of the form", "$X' \\oplus Z \\to Z \\oplus Y'$ has both a kernel and a cokernel.", "Hence (3) $\\Rightarrow$ (1), (2).", "Next we prove (1) $\\Rightarrow$ (3).", "Suppose first that $f : X \\to Y$ is a monomorphism, i.e., its kernel is zero.", "By TR1 there exists a distinguished triangle $(X, Y, Z, f, g, h)$.", "By Lemma \\ref{lemma-composition-zero} the composition", "$f \\circ h[-1] = 0$. As $f$ is a monomorphism we see that $h[-1] = 0$", "and hence $h = 0$. Then", "Lemma \\ref{lemma-split}", "implies that $Y = X \\oplus Z$, i.e., we see that (3) holds.", "Next, assume $f$ has a kernel $K$. As $K \\to X$ is a monomorphism we", "conclude $X = K \\oplus X'$ and $f|_{X'} : X' \\to Y$ is a monomorphism.", "Hence $Y = X' \\oplus Y'$ and we win.", "The implication (2) $\\Rightarrow$ (3) is dual to this." ], "refs": [ "derived-lemma-composition-zero", "derived-lemma-split" ], "ref_ids": [ 1757, 1766 ] } ], "ref_ids": [] }, { "id": 1768, "type": "theorem", "label": "derived-lemma-products-sums-shifts-triangles", "categories": [ "derived" ], "title": "derived-lemma-products-sums-shifts-triangles", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "Let $I$ be a set.", "\\begin{enumerate}", "\\item Let $X_i$, $i \\in I$ be a family of objects of $\\mathcal{D}$.", "\\begin{enumerate}", "\\item If $\\prod X_i$ exists, then $(\\prod X_i)[1] = \\prod X_i[1]$.", "\\item If $\\bigoplus X_i$ exists, then $(\\bigoplus X_i)[1] = \\bigoplus X_i[1]$.", "\\end{enumerate}", "\\item Let $X_i \\to Y_i \\to Z_i \\to X_i[1]$ be a family of distinguished", "triangles of $\\mathcal{D}$.", "\\begin{enumerate}", "\\item If $\\prod X_i$, $\\prod Y_i$, $\\prod Z_i$ exist, then", "$\\prod X_i \\to \\prod Y_i \\to \\prod Z_i \\to \\prod X_i[1]$", "is a distinguished triangle.", "\\item If $\\bigoplus X_i$, $\\bigoplus Y_i$,", "$\\bigoplus Z_i$ exist, then", "$\\bigoplus X_i \\to \\bigoplus Y_i \\to \\bigoplus Z_i \\to \\bigoplus X_i[1]$", "is a distinguished triangle.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is true because $[1]$ is an autoequivalence of $\\mathcal{D}$", "and because direct sums and products are defined in terms of the", "category structure. Let us prove (2)(a). Choose a distinguished triangle", "$\\prod X_i \\to \\prod Y_i \\to Z \\to \\prod X_i[1]$. For each $j$ we can", "use TR3 to choose a morphism $p_j : Z \\to Z_j$", "fitting into a morphism of distinguished", "triangles with the projection maps $\\prod X_i \\to X_j$ and $\\prod Y_i \\to Y_j$.", "Using the definition of products we obtain a map", "$\\prod p_i : Z \\to \\prod Z_i$ fitting into a morphism", "of triangles from the distinguished triangle to the triangle", "made out of the products. Observe that the ``product'' triangle", "$\\prod X_i \\to \\prod Y_i \\to \\prod Z_i \\to \\prod X_i[1]$", "is special in the terminology of Remark \\ref{remark-special-triangles}", "because products of exact sequences of abelian groups are exact.", "Hence Remark \\ref{remark-special-triangles} shows that", "the morphism of triangles is an isomorphism and we conclude by TR1.", "The proof of (2)(b) is dual." ], "refs": [ "derived-remark-special-triangles", "derived-remark-special-triangles" ], "ref_ids": [ 2009, 2009 ] } ], "ref_ids": [] }, { "id": 1769, "type": "theorem", "label": "derived-lemma-projectors-have-images-triangulated", "categories": [ "derived" ], "title": "derived-lemma-projectors-have-images-triangulated", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "If $\\mathcal{D}$ has countable products, then $\\mathcal{D}$", "is Karoubian.", "If $\\mathcal{D}$ has countable coproducts, then $\\mathcal{D}$", "is Karoubian." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{D}$ has countable products. By", "Homology, Lemma \\ref{homology-lemma-projectors-have-images}", "it suffices to check that morphisms which have a right inverse have kernels.", "Any morphism which has a right inverse is an epimorphism, hence", "has a kernel by", "Lemma \\ref{lemma-when-split}.", "The second statement is dual to the first." ], "refs": [ "homology-lemma-projectors-have-images", "derived-lemma-when-split" ], "ref_ids": [ 12015, 1767 ] } ], "ref_ids": [] }, { "id": 1770, "type": "theorem", "label": "derived-lemma-easier-axiom-four", "categories": [ "derived" ], "title": "derived-lemma-easier-axiom-four", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "In order to prove TR4 it suffices to show that given", "any pair of composable morphisms", "$f : X \\to Y$ and $g : Y \\to Z$ there exist", "\\begin{enumerate}", "\\item isomorphisms $i : X' \\to X$, $j : Y' \\to Y$ and", "$k : Z' \\to Z$, and then setting $f' = j^{-1}fi : X' \\to Y'$ and", "$g' = k^{-1}gj : Y' \\to Z'$ there exist", "\\item distinguished triangles", "$(X', Y', Q_1, f', p_1, d_1)$,", "$(X', Z', Q_2, g' \\circ f', p_2, d_2)$", "and", "$(Y', Z', Q_3, g', p_3, d_3)$,", "such that the assertion of TR4 holds.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The replacement of $X, Y, Z$ by $X', Y', Z'$ is harmless by our", "definition of distinguished triangles and their isomorphisms.", "The lemma follows from the fact that the distinguished triangles", "$(X', Y', Q_1, f', p_1, d_1)$,", "$(X', Z', Q_2, g' \\circ f', p_2, d_2)$", "and", "$(Y', Z', Q_3, g', p_3, d_3)$", "are unique up to isomorphism by", "Lemma \\ref{lemma-cone-triangle-unique-isomorphism}." ], "refs": [ "derived-lemma-cone-triangle-unique-isomorphism" ], "ref_ids": [ 1762 ] } ], "ref_ids": [] }, { "id": 1771, "type": "theorem", "label": "derived-lemma-triangulated-subcategory", "categories": [ "derived" ], "title": "derived-lemma-triangulated-subcategory", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "Assume that $\\mathcal{D}'$ is an additive full subcategory of $\\mathcal{D}$.", "The following are equivalent", "\\begin{enumerate}", "\\item there exists a set of triangles $\\mathcal{T}'$ such that", "$(\\mathcal{D}', \\mathcal{T}')$ is a pre-triangulated subcategory", "of $\\mathcal{D}$,", "\\item $\\mathcal{D}'$ is preserved under $[1], [-1]$ and", "given any morphism $f : X \\to Y$ in $\\mathcal{D}'$ there exists", "a distinguished triangle $(X, Y, Z, f, g, h)$ in $\\mathcal{D}$", "such that $Z$ is isomorphic to an object of $\\mathcal{D}'$.", "\\end{enumerate}", "In this case $\\mathcal{T}'$ as in (1) is the set of distinguished triangles", "$(X, Y, Z, f, g, h)$ of $\\mathcal{D}$ such that", "$X, Y, Z \\in \\Ob(\\mathcal{D}')$. Finally, if $\\mathcal{D}$", "is a triangulated category, then (1) and (2) are also equivalent to", "\\begin{enumerate}", "\\item[(3)] $\\mathcal{D}'$ is a triangulated subcategory.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1772, "type": "theorem", "label": "derived-lemma-exact-functor-additive", "categories": [ "derived" ], "title": "derived-lemma-exact-functor-additive", "contents": [ "An exact functor of pre-triangulated categories is additive." ], "refs": [], "proofs": [ { "contents": [ "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of", "pre-triangulated categories. Since", "$(0, 0, 0, 1_0, 1_0, 0)$ is a distinguished triangle of $\\mathcal{D}$", "the triangle", "$$", "(F(0), F(0), F(0), 1_{F(0)}, 1_{F(0)}, F(0))", "$$", "is distinguished in $\\mathcal{D}'$.", "This implies that $1_{F(0)} \\circ 1_{F(0)}$ is zero, see", "Lemma \\ref{lemma-composition-zero}.", "Hence $F(0)$ is the zero object of $\\mathcal{D}'$. This also implies", "that $F$ applied to any zero morphism is zero (since a morphism in", "an additive category is zero if and only if it factors through the", "zero object). Next, using that", "$(X, X \\oplus Y, Y, (1, 0), (0, 1), 0)$ is a distinguished triangle,", "we see that $(F(X), F(X \\oplus Y), F(Y), F(1, 0), F(0, 1), 0)$ is", "one too. This implies that the map", "$F(1, 0) \\oplus F(0, 1) : F(X) \\oplus F(Y) \\to F(X \\oplus Y)$", "is an isomorphism, see", "Lemma \\ref{lemma-split}.", "We omit the rest of the argument." ], "refs": [ "derived-lemma-composition-zero", "derived-lemma-split" ], "ref_ids": [ 1757, 1766 ] } ], "ref_ids": [] }, { "id": 1773, "type": "theorem", "label": "derived-lemma-exact-equivalence", "categories": [ "derived" ], "title": "derived-lemma-exact-equivalence", "contents": [ "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be a fully faithful exact functor", "of pre-triangulated categories. Then a triangle $(X, Y, Z, f, g, h)$", "of $\\mathcal{D}$ is distinguished if and only if", "$(F(X), F(Y), F(Z), F(f), F(g), F(h))$ is distinguished in $\\mathcal{D}'$." ], "refs": [], "proofs": [ { "contents": [ "The ``only if'' part is clear. Assume $(F(X), F(Y), F(Z))$ is", "distinguished in $\\mathcal{D}'$. Pick a distinguished triangle", "$(X, Y, Z', f, g', h')$ in $\\mathcal{D}$. By", "Lemma \\ref{lemma-cone-triangle-unique-isomorphism}", "there exists an isomorphism of triangles", "$$", "(1, 1, c') : (F(X), F(Y), F(Z)) \\longrightarrow (F(X), F(Y), F(Z')).", "$$", "Since $F$ is fully faithful, there exists a morphism $c : Z \\to Z'$", "such that $F(c) = c'$. Then $(1, 1, c)$ is an isomorphism between", "$(X, Y, Z)$ and $(X, Y, Z')$. Hence $(X, Y, Z)$ is distinguished", "by TR1." ], "refs": [ "derived-lemma-cone-triangle-unique-isomorphism" ], "ref_ids": [ 1762 ] } ], "ref_ids": [] }, { "id": 1774, "type": "theorem", "label": "derived-lemma-composition-exact", "categories": [ "derived" ], "title": "derived-lemma-composition-exact", "contents": [ "Let $\\mathcal{D}, \\mathcal{D}', \\mathcal{D}''$ be pre-triangulated categories.", "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ and", "$F' : \\mathcal{D}' \\to \\mathcal{D}''$ be exact functors.", "Then $F' \\circ F$ is an exact functor." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1775, "type": "theorem", "label": "derived-lemma-exact-compose-homological-functor", "categories": [ "derived" ], "title": "derived-lemma-exact-compose-homological-functor", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "Let $\\mathcal{A}$ be an abelian category.", "Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor.", "\\begin{enumerate}", "\\item Let $\\mathcal{D}'$ be a pre-triangulated category.", "Let $F : \\mathcal{D}' \\to \\mathcal{D}$ be an exact functor.", "Then the composition $H \\circ F$ is a homological functor as well.", "\\item Let $\\mathcal{A}'$ be an abelian category. Let", "$G : \\mathcal{A} \\to \\mathcal{A}'$ be an exact functor.", "Then $G \\circ H$ is a homological functor as well.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1776, "type": "theorem", "label": "derived-lemma-exact-compose-delta-functor", "categories": [ "derived" ], "title": "derived-lemma-exact-compose-delta-functor", "contents": [ "Let $\\mathcal{D}$ be a triangulated category.", "Let $\\mathcal{A}$ be an abelian category.", "Let $G : \\mathcal{A} \\to \\mathcal{D}$ be a $\\delta$-functor.", "\\begin{enumerate}", "\\item Let $\\mathcal{D}'$ be a triangulated category.", "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor.", "Then the composition $F \\circ G$ is a $\\delta$-functor as well.", "\\item Let $\\mathcal{A}'$ be an abelian category. Let", "$H : \\mathcal{A}' \\to \\mathcal{A}$ be an exact functor.", "Then $G \\circ H$ is a $\\delta$-functor as well.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1777, "type": "theorem", "label": "derived-lemma-compose-delta-functor-homological", "categories": [ "derived" ], "title": "derived-lemma-compose-delta-functor-homological", "contents": [ "Let $\\mathcal{D}$ be a triangulated category.", "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.", "Let $G : \\mathcal{A} \\to \\mathcal{D}$ be a $\\delta$-functor.", "Let $H : \\mathcal{D} \\to \\mathcal{B}$ be a homological functor.", "Assume that $H^{-1}(G(A)) = 0$ for all $A$ in $\\mathcal{A}$.", "Then the collection", "$$", "\\{H^n \\circ G, H^n(\\delta_{A \\to B \\to C})\\}_{n \\geq 0}", "$$", "is a $\\delta$-functor from $\\mathcal{A} \\to \\mathcal{B}$, see", "Homology, Definition \\ref{homology-definition-cohomological-delta-functor}." ], "refs": [ "homology-definition-cohomological-delta-functor" ], "proofs": [ { "contents": [ "The notation signifies the following. If", "$0 \\to A \\xrightarrow{a} B \\xrightarrow{b} C \\to 0$ is", "a short exact sequence in $\\mathcal{A}$, then", "$$", "\\delta = \\delta_{A \\to B \\to C} : G(C) \\to G(A)[1]", "$$", "is a morphism in $\\mathcal{D}$ such that", "$(G(A), G(B), G(C), a, b, \\delta)$ is", "a distinguished triangle, see", "Definition \\ref{definition-delta-functor}.", "Then $H^n(\\delta) : H^n(G(C)) \\to H^n(G(A)[1]) = H^{n + 1}(G(A))$", "is clearly functorial in the short exact sequence.", "Finally, the long exact cohomology sequence", "(\\ref{equation-long-exact-cohomology-sequence})", "combined with the vanishing of $H^{-1}(G(C))$", "gives a long exact sequence", "$$", "0 \\to H^0(G(A)) \\to H^0(G(B)) \\to H^0(G(C))", "\\xrightarrow{H^0(\\delta)} H^1(G(A)) \\to \\ldots", "$$", "in $\\mathcal{B}$ as desired." ], "refs": [ "derived-definition-delta-functor" ], "ref_ids": [ 1972 ] } ], "ref_ids": [ 12149 ] }, { "id": 1778, "type": "theorem", "label": "derived-lemma-localization-conditions", "categories": [ "derived" ], "title": "derived-lemma-localization-conditions", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "Let $S$ be a set of morphisms of $\\mathcal{D}$ and assume that axioms", "MS1, MS5, MS6 hold (see", "Categories, Definition \\ref{categories-definition-multiplicative-system}", "and", "Definition \\ref{definition-localization}).", "Then MS2 holds." ], "refs": [ "categories-definition-multiplicative-system", "derived-definition-localization" ], "proofs": [ { "contents": [ "Suppose that $f : X \\to Y$ is a morphism of $\\mathcal{D}$ and", "$t : X \\to X'$ an element of $S$. Choose a distinguished triangle", "$(X, Y, Z, f, g, h)$. Next, choose a distinguished triangle", "$(X', Y', Z, f', g', t[1] \\circ h)$ (here we use TR1 and TR2).", "By MS5, MS6 (and TR2 to rotate) we can find the dotted arrow", "in the commutative diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d]^t &", "Y \\ar[r] \\ar@{..>}[d]^{s'} &", "Z \\ar[r] \\ar[d]^1 &", "X[1] \\ar[d]^{t[1]} \\\\", "X' \\ar[r] &", "Y' \\ar[r] &", "Z \\ar[r] &", "X'[1]", "}", "$$", "with moreover $s' \\in S$. This proves LMS2. The proof of RMS2 is dual." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12373, 1973 ] }, { "id": 1779, "type": "theorem", "label": "derived-lemma-triangle-functor-localize", "categories": [ "derived" ], "title": "derived-lemma-triangle-functor-localize", "contents": [ "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of", "pre-triangulated categories. Let", "$$", "S = \\{f \\in \\text{Arrows}(\\mathcal{D})", "\\mid F(f)\\text{ is an isomorphism}\\}", "$$", "Then $S$ is a saturated (see", "Categories,", "Definition \\ref{categories-definition-saturated-multiplicative-system})", "multiplicative system compatible with the", "triangulated structure on $\\mathcal{D}$." ], "refs": [ "categories-definition-saturated-multiplicative-system" ], "proofs": [ { "contents": [ "We have to prove axioms MS1 -- MS6, see", "Categories, Definitions \\ref{categories-definition-multiplicative-system} and", "\\ref{categories-definition-saturated-multiplicative-system}", "and", "Definition \\ref{definition-localization}.", "MS1, MS4, and MS5 are direct from the definitions. MS6 follows from TR3 and", "Lemma \\ref{lemma-third-isomorphism-triangle}.", "By", "Lemma \\ref{lemma-localization-conditions}", "we conclude that MS2 holds. To finish the proof we have to show that", "MS3 holds. To do this let $f, g : X \\to Y$ be morphisms of $\\mathcal{D}$,", "and let $t : Z \\to X$ be an element of $S$ such that $f \\circ t = g \\circ t$.", "As $\\mathcal{D}$ is additive this simply means that $a \\circ t = 0$ with", "$a = f - g$. Choose a distinguished triangle $(Z, X, Q, t, d, h)$ using TR1.", "Since $a \\circ t = 0$ we see by", "Lemma \\ref{lemma-representable-homological}", "there exists a morphism $i : Q \\to Y$ such that $i \\circ d = a$.", "Finally, using TR1 again we can choose a triangle", "$(Q, Y, W, i, j, k)$. Here is a picture", "$$", "\\xymatrix{", "Z \\ar[r]_t & X \\ar[r]_d \\ar[d]^1 & Q \\ar[r] \\ar[d]^i & Z[1] \\\\", "& X \\ar[r]_a & Y \\ar[d]^j \\\\", "& & W", "}", "$$", "OK, and now we apply the functor $F$ to this diagram.", "Since $t \\in S$ we see that $F(Q) = 0$, see", "Lemma \\ref{lemma-third-object-zero}.", "Hence $F(j)$ is an isomorphism by the same lemma, i.e., $j \\in S$.", "Finally, $j \\circ a = j \\circ i \\circ d = 0$ as $j \\circ i = 0$.", "Thus $j \\circ f = j \\circ g$ and we see that LMS3 holds.", "The proof of RMS3 is dual." ], "refs": [ "categories-definition-multiplicative-system", "categories-definition-saturated-multiplicative-system", "derived-definition-localization", "derived-lemma-third-isomorphism-triangle", "derived-lemma-localization-conditions", "derived-lemma-representable-homological", "derived-lemma-third-object-zero" ], "ref_ids": [ 12373, 12376, 1973, 1759, 1778, 1758, 1764 ] } ], "ref_ids": [ 12376 ] }, { "id": 1780, "type": "theorem", "label": "derived-lemma-homological-functor-localize", "categories": [ "derived" ], "title": "derived-lemma-homological-functor-localize", "contents": [ "Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor between a", "pre-triangulated category and an abelian category. Let", "$$", "S = \\{f \\in \\text{Arrows}(\\mathcal{D})", "\\mid H^i(f)\\text{ is an isomorphism for all }i \\in \\mathbf{Z}\\}", "$$", "Then $S$ is a saturated (see", "Categories,", "Definition \\ref{categories-definition-saturated-multiplicative-system})", "multiplicative system compatible with the", "triangulated structure on $\\mathcal{D}$." ], "refs": [ "categories-definition-saturated-multiplicative-system" ], "proofs": [ { "contents": [ "We have to prove axioms MS1 -- MS6, see", "Categories, Definitions \\ref{categories-definition-multiplicative-system} and", "\\ref{categories-definition-saturated-multiplicative-system}", "and", "Definition \\ref{definition-localization}.", "MS1, MS4, and MS5 are direct from the definitions.", "MS6 follows from TR3 and the long exact cohomology sequence", "(\\ref{equation-long-exact-cohomology-sequence}).", "By", "Lemma \\ref{lemma-localization-conditions}", "we conclude that MS2 holds. To finish the proof we have to show that", "MS3 holds. To do this let $f, g : X \\to Y$ be morphisms of $\\mathcal{D}$,", "and let $t : Z \\to X$ be an element of $S$ such that $f \\circ t = g \\circ t$.", "As $\\mathcal{D}$ is additive this simply means that $a \\circ t = 0$ with", "$a = f - g$. Choose a distinguished triangle $(Z, X, Q, t, g, h)$ using", "TR1 and TR2. Since $a \\circ t = 0$ we see by", "Lemma \\ref{lemma-representable-homological}", "there exists a morphism $i : Q \\to Y$ such that $i \\circ g = a$.", "Finally, using TR1 again we can choose a triangle", "$(Q, Y, W, i, j, k)$. Here is a picture", "$$", "\\xymatrix{", "Z \\ar[r]_t & X \\ar[r]_g \\ar[d]^1 & Q \\ar[r] \\ar[d]^i & Z[1] \\\\", "& X \\ar[r]_a & Y \\ar[d]^j \\\\", "& & W", "}", "$$", "OK, and now we apply the functors $H^i$ to this diagram.", "Since $t \\in S$ we see that $H^i(Q) = 0$ by the long exact cohomology", "sequence (\\ref{equation-long-exact-cohomology-sequence}).", "Hence $H^i(j)$ is an isomorphism for all $i$ by the same argument,", "i.e., $j \\in S$. Finally, $j \\circ a = j \\circ i \\circ g = 0$ as", "$j \\circ i = 0$. Thus $j \\circ f = j \\circ g$ and we see that LMS3 holds.", "The proof of RMS3 is dual." ], "refs": [ "categories-definition-multiplicative-system", "categories-definition-saturated-multiplicative-system", "derived-definition-localization", "derived-lemma-localization-conditions", "derived-lemma-representable-homological" ], "ref_ids": [ 12373, 12376, 1973, 1778, 1758 ] } ], "ref_ids": [ 12376 ] }, { "id": 1781, "type": "theorem", "label": "derived-lemma-universal-property-localization", "categories": [ "derived" ], "title": "derived-lemma-universal-property-localization", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category. Let $S$ be a multiplicative", "system compatible with the triangulated category. Let", "$Q : \\mathcal{D} \\to S^{-1}\\mathcal{D}$ be the localization functor, see", "Proposition \\ref{proposition-construct-localization}.", "\\begin{enumerate}", "\\item If $H : \\mathcal{D} \\to \\mathcal{A}$ is a homological functor into", "an abelian category $\\mathcal{A}$ such that $H(s)$ is an isomorphism for", "all $s \\in S$, then the unique factorization", "$H' : S^{-1}\\mathcal{D} \\to \\mathcal{A}$ such that $H = H' \\circ Q$ (see", "Categories, Lemma \\ref{categories-lemma-properties-left-localization})", "is a homological functor too.", "\\item If $F : \\mathcal{D} \\to \\mathcal{D}'$ is an exact functor into", "a pre-triangulated category $\\mathcal{D}'$ such that $F(s)$ is an isomorphism", "for all $s \\in S$, then the unique factorization", "$F' : S^{-1}\\mathcal{D} \\to \\mathcal{D}'$ such that $F = F' \\circ Q$ (see", "Categories, Lemma \\ref{categories-lemma-properties-left-localization})", "is an exact functor too.", "\\end{enumerate}" ], "refs": [ "derived-proposition-construct-localization", "categories-lemma-properties-left-localization", "categories-lemma-properties-left-localization" ], "proofs": [ { "contents": [ "This lemma proves itself. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1959, 12258, 12258 ] }, { "id": 1782, "type": "theorem", "label": "derived-lemma-kernel-localization", "categories": [ "derived" ], "title": "derived-lemma-kernel-localization", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category. Let $S$ be a multiplicative", "system compatible with the triangulated structure. Let $Z$ be an object", "of $\\mathcal{D}$. The following are equivalent", "\\begin{enumerate}", "\\item $Q(Z) = 0$ in $S^{-1}\\mathcal{D}$,", "\\item there exists $Z' \\in \\Ob(\\mathcal{D})$ such that", "$0 : Z \\to Z'$ is an element of $S$,", "\\item there exists $Z' \\in \\Ob(\\mathcal{D})$ such that", "$0 : Z' \\to Z$ is an element of $S$, and", "\\item there exists an object $Z'$ and a distinguished triangle", "$(X, Y, Z \\oplus Z', f, g, h)$ such that $f \\in S$.", "\\end{enumerate}", "If $S$ is saturated, then these are also equivalent to", "\\begin{enumerate}", "\\item[(5)] the morphism $0 \\to Z$ is an element of $S$,", "\\item[(6)] the morphism $Z \\to 0$ is an element of $S$,", "\\item[(7)] there exists a distinguished triangle $(X, Y, Z, f, g, h)$", "such that $f \\in S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1), (2), and (3) is", "Homology, Lemma \\ref{homology-lemma-kernel-localization}.", "If (2) holds, then $(Z'[-1], Z'[-1] \\oplus Z, Z, (1, 0), (0, 1), 0)$", "is a distinguished triangle (see", "Lemma \\ref{lemma-split})", "with ``$0 \\in S$''. By rotating we conclude that (4) holds.", "If $(X, Y, Z \\oplus Z', f, g, h)$ is a distinguished triangle with $f \\in S$", "then $Q(f)$ is an isomorphism hence $Q(Z \\oplus Z') = 0$ hence $Q(Z) = 0$.", "Thus (1) -- (4) are all equivalent.", "\\medskip\\noindent", "Next, assume that $S$ is saturated. Note that each of (5), (6), (7)", "implies one of the equivalent conditions (1) -- (4). Suppose that", "$Q(Z) = 0$. Then $0 \\to Z$ is a morphism of $\\mathcal{D}$ which becomes", "an isomorphism in $S^{-1}\\mathcal{D}$. According to", "Categories, Lemma \\ref{categories-lemma-what-gets-inverted}", "the fact that $S$ is saturated implies that $0 \\to Z$ is in $S$.", "Hence (1) $\\Rightarrow$ (5). Dually (1) $\\Rightarrow$ (6).", "Finally, if $0 \\to Z$ is in $S$, then the triangle", "$(0, Z, Z, 0, \\text{id}_Z, 0)$ is distinguished by TR1 and TR2 and", "is a triangle as in (4)." ], "refs": [ "homology-lemma-kernel-localization", "derived-lemma-split", "categories-lemma-what-gets-inverted" ], "ref_ids": [ 12039, 1766, 12268 ] } ], "ref_ids": [] }, { "id": 1783, "type": "theorem", "label": "derived-lemma-limit-triangles", "categories": [ "derived" ], "title": "derived-lemma-limit-triangles", "contents": [ "Let $\\mathcal{D}$ be a triangulated category.", "Let $S$ be a saturated multiplicative system in $\\mathcal{D}$", "that is compatible with the triangulated structure.", "Let $(X, Y, Z, f, g, h)$ be a distinguished triangle in $\\mathcal{D}$.", "Consider the category of morphisms of triangles", "$$", "\\mathcal{I} =", "\\{(s, s', s'') : (X, Y, Z, f, g, h) \\to (X', Y', Z', f', g', h')", "\\mid s, s', s'' \\in S\\}", "$$", "Then $\\mathcal{I}$ is a filtered category and the functors", "$\\mathcal{I} \\to X/S$, $\\mathcal{I} \\to Y/S$, and $\\mathcal{I} \\to Z/S$", "are cofinal." ], "refs": [], "proofs": [ { "contents": [ "We strongly suggest the reader skip the proof of this lemma and instead", "work it out on a napkin.", "\\medskip\\noindent", "The first remark is that using rotation of distinguished triangles (TR2)", "gives an equivalence of categories between $\\mathcal{I}$ and the", "corresponding category for the distinguished triangle", "$(Y, Z, X[1], g, h, -f[1])$. Using this we see for example that if", "we prove the functor $\\mathcal{I} \\to X/S$ is cofinal, then", "the same thing is true for the functors $\\mathcal{I} \\to Y/S$ and", "$\\mathcal{I} \\to Z/S$.", "\\medskip\\noindent", "Note that if $s : X \\to X'$ is a morphism of $S$, then using", "MS2 we can find $s' : Y \\to Y'$ and $f' : X' \\to Y'$ such that", "$f' \\circ s = s' \\circ f$, whereupon we can use MS6 to complete", "this into an object of $\\mathcal{I}$. Hence the functor", "$\\mathcal{I} \\to X/S$ is surjective on objects. Using rotation as above", "this implies the same thing is true for the functors", "$\\mathcal{I} \\to Y/S$ and $\\mathcal{I} \\to Z/S$.", "\\medskip\\noindent", "Suppose given objects $s_1 : X \\to X_1 $ and $s_2 : X \\to X_2$ in", "$X/S$ and a morphism $a : X_1 \\to X_2$ in $X/S$. Since $S$ is saturated,", "we see that $a \\in S$, see", "Categories, Lemma \\ref{categories-lemma-what-gets-inverted}.", "By the argument of the previous paragraph we can complete", "$s_1 : X \\to X_1$ to an object", "$(s_1, s'_1, s''_1) : (X, Y, Z, f, g, h) \\to (X_1, Y_1, Z_1, f_1, g_1, h_1)$", "in $\\mathcal{I}$. Then we can repeat and find", "$(a, b, c) : (X_1, Y_1, Z_1, f_1, g_1, h_1) \\to (X_2, Y_2, Z_2, f_2, g_2, h_2)$", "with $a, b, c \\in S$ completing the given $a : X_1 \\to X_2$.", "But then $(a, b, c)$ is a morphism in $\\mathcal{I}$.", "In this way we conclude that the functor $\\mathcal{I} \\to X/S$ is", "also surjective on arrows. Using rotation as above,", "this implies the same thing is true for the functors", "$\\mathcal{I} \\to Y/S$ and $\\mathcal{I} \\to Z/S$.", "\\medskip\\noindent", "The category $\\mathcal{I}$ is nonempty as the identity provides an object.", "This proves the condition (1) of the definition of a filtered category, see", "Categories, Definition \\ref{categories-definition-directed}.", "\\medskip\\noindent", "We check condition (2) of", "Categories, Definition \\ref{categories-definition-directed}", "for the category $\\mathcal{I}$. Suppose given objects", "$(s_1, s'_1, s''_1) : (X, Y, Z, f, g, h) \\to (X_1, Y_1, Z_1, f_1, g_1, h_1)$", "and", "$(s_2, s'_2, s''_2) : (X, Y, Z, f, g, h) \\to (X_2, Y_2, Z_2, f_2, g_2, h_2)$", "in $\\mathcal{I}$. We want to find an object of $\\mathcal{I}$", "which is the target of an arrow from both", "$(X_1, Y_1, Z_1, f_1, g_1, h_1)$ and $(X_2, Y_2, Z_2, f_2, g_2, h_2)$.", "By Categories, Remark", "\\ref{categories-remark-left-localization-morphisms-colimit}", "the categories $X/S$, $Y/S$, $Z/S$ are filtered.", "Thus we can find $X \\to X_3$ in $X/S$ and morphisms", "$s : X_2 \\to X_3$ and $a : X_1 \\to X_3$. By the above we can find a morphism", "$(s, s', s'') : (X_2, Y_2, Z_2, f_2, g_2, h_2) \\to", "(X_3, Y_3, Z_3, f_3, g_3, h_3)$ with $s', s'' \\in S$.", "After replacing $(X_2, Y_2, Z_2)$ by $(X_3, Y_3, Z_3)$ we may", "assume that there exists a morphism $a : X_1 \\to X_2$ in $X/S$.", "Repeating the argument for $Y$ and $Z$ (by rotating as above)", "we may assume there is a morphism", "$a : X_1 \\to X_2$ in $X/S$,", "$b : Y_1 \\to Y_2$ in $Y/S$, and", "$c : Z_1 \\to Z_2$ in $Z/S$.", "However, these morphisms do not necessarily give rise to a morphism of", "distinguished triangles. On the other hand, the necessary diagrams", "do commute in $S^{-1}\\mathcal{D}$. Hence we see (for example) that", "there exists a morphism $s'_2 : Y_2 \\to Y_3$ in $S$ such that", "$s'_2 \\circ f_2 \\circ a = s'_2 \\circ b \\circ f_1$. Another replacement", "of $(X_2, Y_2, Z_2)$ as above then gets us to the situation where", "$f_2 \\circ a = b \\circ f_1$. Rotating and applying the same argument", "two more times we see that we may assume $(a, b, c)$ is a morphism", "of triangles. This proves condition (2).", "\\medskip\\noindent", "Next we check condition (3) of", "Categories, Definition \\ref{categories-definition-directed}.", "Suppose $(s_1, s_1', s_1'') : (X, Y, Z) \\to (X_1, Y_1, Z_1)$ and", "$(s_2, s_2', s_2'') : (X, Y, Z) \\to (X_2, Y_2, Z_2)$", "are objects of $\\mathcal{I}$, and suppose $(a, b, c), (a', b', c')$", "are two morphisms between them. Since $a \\circ s_1 = a' \\circ s_1$", "there exists a morphism $s_3 : X_2 \\to X_3$ such that", "$s_3 \\circ a = s_3 \\circ a'$. Using the surjectivity statement", "we can complete this to a morphism of triangles", "$(s_3, s_3', s_3'') : (X_2, Y_2, Z_2) \\to (X_3, Y_3, Z_3)$", "with $s_3, s_3', s_3'' \\in S$. Thus", "$(s_3 \\circ s_2, s_3' \\circ s_2', s_3'' \\circ s_2'') :", "(X, Y, Z) \\to (X_3, Y_3, Z_3)$ is also an object of $\\mathcal{I}$", "and after composing the maps $(a, b, c), (a', b', c')$ with", "$(s_3, s_3', s_3'')$ we obtain $a = a'$. By rotating we may do the", "same to get $b = b'$ and $c = c'$.", "\\medskip\\noindent", "Finally, we check that $\\mathcal{I} \\to X/S$ is cofinal, see", "Categories, Definition \\ref{categories-definition-cofinal}.", "The first condition is true as the functor is surjective.", "Suppose that we have an object $s : X \\to X'$ in $X/S$ and", "two objects", "$(s_1, s'_1, s''_1) : (X, Y, Z, f, g, h) \\to (X_1, Y_1, Z_1, f_1, g_1, h_1)$", "and", "$(s_2, s'_2, s''_2) : (X, Y, Z, f, g, h) \\to (X_2, Y_2, Z_2, f_2, g_2, h_2)$", "in $\\mathcal{I}$ as well as morphisms $t_1 : X' \\to X_1$ and", "$t_2 : X' \\to X_2$ in $X/S$. By property (2) of $\\mathcal{I}$", "proved above we can find morphisms", "$(s_3, s'_3, s''_3) : (X_1, Y_1, Z_1, f_1, g_1, h_1) \\to", "(X_3, Y_3, Z_3, f_3, g_3, h_3)$", "and", "$(s_4, s'_4, s''_4) : (X_2, Y_2, Z_2, f_2, g_2, h_2) \\to", "(X_3, Y_3, Z_3, f_3, g_3, h_3)$ in $\\mathcal{I}$.", "We would be done if the compositions", "$X' \\to X_1 \\to X_3$ and $X' \\to X_1 \\to X_3$ where equal", "(see displayed equation in ", "Categories, Definition \\ref{categories-definition-cofinal}).", "If not, then, because $X/S$ is filtered, we can choose", "a morphism $X_3 \\to X_4$ in $S$ such that the compositions", "$X' \\to X_1 \\to X_3 \\to X_4$ and $X' \\to X_1 \\to X_3 \\to X_4$ are equal.", "Then we finally complete $X_3 \\to X_4$ to a morphism", "$(X_3, Y_3, Z_3) \\to (X_4, Y_4, Z_4)$ in $\\mathcal{I}$", "and compose with that morphism to see that the result is true." ], "refs": [ "categories-lemma-what-gets-inverted", "categories-definition-directed", "categories-definition-directed", "categories-remark-left-localization-morphisms-colimit", "categories-definition-directed", "categories-definition-cofinal", "categories-definition-cofinal" ], "ref_ids": [ 12268, 12363, 12363, 12424, 12363, 12361, 12361 ] } ], "ref_ids": [] }, { "id": 1784, "type": "theorem", "label": "derived-lemma-triangle-functor-kernel", "categories": [ "derived" ], "title": "derived-lemma-triangle-functor-kernel", "contents": [ "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of", "pre-triangulated categories. Let $\\mathcal{D}''$ be the full subcategory", "of $\\mathcal{D}$ with objects", "$$", "\\Ob(\\mathcal{D}'') =", "\\{X \\in \\Ob(\\mathcal{D}) \\mid F(X) = 0\\}", "$$", "Then $\\mathcal{D}''$ is a strictly full saturated pre-triangulated", "subcategory of $\\mathcal{D}$. If $\\mathcal{D}$ is a triangulated category,", "then $\\mathcal{D}''$ is a triangulated subcategory." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $\\mathcal{D}''$ is preserved under $[1]$ and $[-1]$.", "If $(X, Y, Z, f, g, h)$ is a distinguished triangle of $\\mathcal{D}$", "and $F(X) = F(Y) = 0$, then also $F(Z) = 0$ as", "$(F(X), F(Y), F(Z), F(f), F(g), F(h))$ is distinguished.", "Hence we may apply", "Lemma \\ref{lemma-triangulated-subcategory}", "to see that $\\mathcal{D}''$ is a pre-triangulated subcategory (respectively", "a triangulated subcategory if $\\mathcal{D}$ is a triangulated category).", "The final assertion of being saturated follows from", "$F(X) \\oplus F(Y) = 0 \\Rightarrow F(X) = F(Y) = 0$." ], "refs": [ "derived-lemma-triangulated-subcategory" ], "ref_ids": [ 1771 ] } ], "ref_ids": [] }, { "id": 1785, "type": "theorem", "label": "derived-lemma-homological-functor-kernel", "categories": [ "derived" ], "title": "derived-lemma-homological-functor-kernel", "contents": [ "Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor of", "a pre-triangulated category into an abelian category.", "Let $\\mathcal{D}'$ be the full subcategory of $\\mathcal{D}$ with objects", "$$", "\\Ob(\\mathcal{D}') =", "\\{X \\in \\Ob(\\mathcal{D}) \\mid", "H(X[n]) = 0\\text{ for all }n \\in \\mathbf{Z}\\}", "$$", "Then $\\mathcal{D}'$ is a strictly full saturated pre-triangulated subcategory", "of $\\mathcal{D}$. If $\\mathcal{D}$ is a triangulated category, then", "$\\mathcal{D}'$ is a triangulated subcategory." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $\\mathcal{D}'$ is preserved under $[1]$ and $[-1]$.", "If $(X, Y, Z, f, g, h)$ is a distinguished triangle of $\\mathcal{D}$", "and $H(X[n]) = H(Y[n]) = 0$ for all $n$, then also $H(Z[n]) = 0$ for all $n$", "by the long exact sequence (\\ref{equation-long-exact-cohomology-sequence}).", "Hence we may apply", "Lemma \\ref{lemma-triangulated-subcategory}", "to see that $\\mathcal{D}'$ is a pre-triangulated subcategory (respectively", "a triangulated subcategory if $\\mathcal{D}$ is a triangulated category).", "The assertion of being saturated follows from", "\\begin{align*}", "H((X \\oplus Y)[n]) = 0 & \\Rightarrow H(X[n] \\oplus Y[n]) = 0 \\\\", "& \\Rightarrow H(X[n]) \\oplus H(Y[n]) = 0 \\\\", "& \\Rightarrow H(X[n]) = H(Y[n]) = 0", "\\end{align*}", "for all $n \\in \\mathbf{Z}$." ], "refs": [ "derived-lemma-triangulated-subcategory" ], "ref_ids": [ 1771 ] } ], "ref_ids": [] }, { "id": 1786, "type": "theorem", "label": "derived-lemma-homological-functor-bounded", "categories": [ "derived" ], "title": "derived-lemma-homological-functor-bounded", "contents": [ "Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor of", "a pre-triangulated category into an abelian category.", "Let $\\mathcal{D}_H^{+}, \\mathcal{D}_H^{-}, \\mathcal{D}_H^b$", "be the full subcategory of $\\mathcal{D}$ with objects", "$$", "\\begin{matrix}", "\\Ob(\\mathcal{D}_H^{+}) =", "\\{X \\in \\Ob(\\mathcal{D}) \\mid", "H(X[n]) = 0\\text{ for all }n \\ll 0\\} \\\\", "\\Ob(\\mathcal{D}_H^{-}) =", "\\{X \\in \\Ob(\\mathcal{D}) \\mid", "H(X[n]) = 0\\text{ for all }n \\gg 0\\} \\\\", "\\Ob(\\mathcal{D}_H^b) =", "\\{X \\in \\Ob(\\mathcal{D}) \\mid", "H(X[n]) = 0\\text{ for all }|n| \\gg 0\\}", "\\end{matrix}", "$$", "Each of these is a strictly full saturated pre-triangulated subcategory", "of $\\mathcal{D}$. If $\\mathcal{D}$ is a triangulated category, then", "each is a triangulated subcategory." ], "refs": [], "proofs": [ { "contents": [ "Let us prove this for $\\mathcal{D}_H^{+}$.", "It is clear that it is preserved under $[1]$ and $[-1]$.", "If $(X, Y, Z, f, g, h)$ is a distinguished triangle of $\\mathcal{D}$", "and $H(X[n]) = H(Y[n]) = 0$ for all $n \\ll 0$, then also $H(Z[n]) = 0$", "for all $n \\ll 0$ by the long exact sequence", "(\\ref{equation-long-exact-cohomology-sequence}).", "Hence we may apply", "Lemma \\ref{lemma-triangulated-subcategory}", "to see that $\\mathcal{D}_H^{+}$ is a pre-triangulated subcategory", "(respectively a triangulated subcategory if $\\mathcal{D}$ is a", "triangulated category). The assertion of being saturated follows from", "\\begin{align*}", "H((X \\oplus Y)[n]) = 0 & \\Rightarrow H(X[n] \\oplus Y[n]) = 0 \\\\", "& \\Rightarrow H(X[n]) \\oplus H(Y[n]) = 0 \\\\", "& \\Rightarrow H(X[n]) = H(Y[n]) = 0", "\\end{align*}", "for all $n \\in \\mathbf{Z}$." ], "refs": [ "derived-lemma-triangulated-subcategory" ], "ref_ids": [ 1771 ] } ], "ref_ids": [] }, { "id": 1787, "type": "theorem", "label": "derived-lemma-construct-multiplicative-system", "categories": [ "derived" ], "title": "derived-lemma-construct-multiplicative-system", "contents": [ "Let $\\mathcal{D}$ be a triangulated category.", "Let $\\mathcal{D}' \\subset \\mathcal{D}$ be a full triangulated", "subcategory. Set", "\\begin{equation}", "\\label{equation-multiplicative-system}", "S =", "\\left\\{", "\\begin{matrix}", "f \\in \\text{Arrows}(\\mathcal{D})", "\\text{ such that there exists a distinguished triangle }\\\\", "(X, Y, Z, f, g, h) \\text{ of }\\mathcal{D}\\text{ with }", "Z\\text{ isomorphic to an object of }\\mathcal{D}'", "\\end{matrix}", "\\right\\}", "\\end{equation}", "Then $S$ is a multiplicative system compatible with the triangulated", "structure on $\\mathcal{D}$. In this situation the following are equivalent", "\\begin{enumerate}", "\\item $S$ is a saturated multiplicative system,", "\\item $\\mathcal{D}'$ is a saturated triangulated subcategory.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove the first assertion we have to prove that", "MS1, MS2, MS3 and MS5, MS6 hold.", "\\medskip\\noindent", "Proof of MS1. It is clear that identities are in $S$ because", "$(X, X, 0, 1, 0, 0)$ is distinguished for every object $X$ of $\\mathcal{D}$", "and because $0$ is an object of $\\mathcal{D}'$. Let $f : X \\to Y$", "and $g : Y \\to Z$ be composable morphisms contained in $S$.", "Choose distinguished triangles $(X, Y, Q_1, f, p_1, d_1)$,", "$(X, Z, Q_2, g \\circ f, p_2, d_2)$, and $(Y, Z, Q_3, g, p_3, d_3)$.", "By assumption we know that $Q_1$ and $Q_3$ are isomorphic to objects", "of $\\mathcal{D}'$. By TR4 we know there exists a distinguished", "triangle $(Q_1, Q_2, Q_3, a, b, c)$. Since $\\mathcal{D}'$ is a", "triangulated subcategory we conclude that $Q_2$ is isomorphic to", "an object of $\\mathcal{D}'$. Hence $g \\circ f \\in S$.", "\\medskip\\noindent", "Proof of MS3. Let $a : X \\to Y$ be a morphism and let $t : Z \\to X$ be", "an element of $S$ such that $a \\circ t = 0$. To prove LMS3 it suffices to", "find an $s \\in S$ such that $s \\circ a = 0$, compare with the proof of", "Lemma \\ref{lemma-triangle-functor-localize}. Choose a distinguished", "triangle $(Z, X, Q, t, g, h)$ using TR1 and TR2. Since $a \\circ t = 0$", "we see by", "Lemma \\ref{lemma-representable-homological}", "there exists a morphism $i : Q \\to Y$ such that $i \\circ g = a$.", "Finally, using TR1 again we can choose a triangle", "$(Q, Y, W, i, s, k)$. Here is a picture", "$$", "\\xymatrix{", "Z \\ar[r]_t & X \\ar[r]_g \\ar[d]^1 & Q \\ar[r] \\ar[d]^i & Z[1] \\\\", "& X \\ar[r]_a & Y \\ar[d]^s \\\\", "& & W", "}", "$$", "Since $t \\in S$ we see that $Q$ is isomorphic to an object of $\\mathcal{D}'$.", "Hence $s \\in S$. Finally, $s \\circ a = s \\circ i \\circ g = 0$ as", "$s \\circ i = 0$ by Lemma \\ref{lemma-composition-zero}.", "We conclude that LMS3 holds.", "The proof of RMS3 is dual.", "\\medskip\\noindent", "Proof of MS5. Follows as distinguished triangles and $\\mathcal{D}'$", "are stable under translations", "\\medskip\\noindent", "Proof of MS6. Suppose given a commutative diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d]^s &", "Y \\ar[d]^{s'} \\\\", "X' \\ar[r] &", "Y'", "}", "$$", "with $s, s' \\in S$. By", "Proposition \\ref{proposition-9}", "we can extend this to a nine square diagram. As $s, s'$ are elements of $S$", "we see that $X'', Y''$ are isomorphic to objects of $\\mathcal{D}'$.", "Since $\\mathcal{D}'$ is a full triangulated subcategory we see that", "$Z''$ is also isomorphic to an object of $\\mathcal{D}'$.", "Whence the morphism $Z \\to Z'$", "is an element of $S$. This proves MS6.", "\\medskip\\noindent", "MS2 is a formal consequence of MS1, MS5, and MS6, see", "Lemma \\ref{lemma-localization-conditions}.", "This finishes the proof of the first assertion of the lemma.", "\\medskip\\noindent", "Let's assume that $S$ is saturated. (In the following we will use", "rotation of distinguished triangles without further mention.)", "Let $X \\oplus Y$ be an object isomorphic to an object of $\\mathcal{D}'$.", "Consider the morphism $f : 0 \\to X$. The composition", "$0 \\to X \\to X \\oplus Y$ is an element", "of $S$ as $(0, X \\oplus Y, X \\oplus Y, 0, 1, 0)$ is a distinguished", "triangle. The composition $Y[-1] \\to 0 \\to X$ is an element of $S$", "as $(X, X \\oplus Y, Y, (1, 0), (0, 1), 0)$ is a distinguished triangle, see", "Lemma \\ref{lemma-split}.", "Hence $0 \\to X$ is an element of $S$ (as $S$ is saturated).", "Thus $X$ is isomorphic to an object of $\\mathcal{D}'$ as desired.", "\\medskip\\noindent", "Finally, assume $\\mathcal{D}'$ is a saturated triangulated subcategory.", "Let", "$$", "W \\xrightarrow{h}", "X \\xrightarrow{g}", "Y \\xrightarrow{f} Z", "$$", "be composable morphisms of $\\mathcal{D}$ such that $fg, gh \\in S$.", "We will build up a picture of objects as in the diagram below.", "$$", "\\xymatrix{", " & &", "Q_{12} \\ar[rd] & &", "Q_{23} \\ar[rd] \\\\", " &", "Q_1 \\ar[ld]_{\\! + \\! 1} \\ar[ru] & &", "Q_2 \\ar[ld]_{\\! + \\! 1} \\ar[ll]_{\\! + \\! 1} \\ar[ru] & &", "Q_3 \\ar[ld]_{\\! + \\! 1} \\ar[ll]_{\\! + \\! 1} \\\\", "W \\ar[rr] & &", "X \\ar[lu] \\ar[rr] & &", "Y \\ar[lu] \\ar[rr] & &", "Z \\ar[lu]", "}", "$$", "First choose distinguished triangles", "$(W, X, Q_1)$, $(X, Y, Q_2)$, $(Y, Z, Q_3)$ $(W, Y, Q_{12})$, and", "$(X, Z, Q_{23})$. Denote $s : Q_2 \\to Q_1[1]$ the composition", "$Q_2 \\to X[1] \\to Q_1[1]$. Denote $t : Q_3 \\to Q_2[1]$ the", "composition $Q_3 \\to Y[1] \\to Q_2[1]$.", "By TR4 applied to the composition $W \\to X \\to Y$", "and the composition $X \\to Y \\to Z$ there exist", "distinguished triangles $(Q_1, Q_{12}, Q_2)$ and $(Q_2, Q_{23}, Q_3)$", "which use the morphisms $s$ and $t$.", "The objects $Q_{12}$ and $Q_{23}$ are isomorphic to objects of", "$\\mathcal{D}'$ as $W \\to Y$ and $X \\to Z$ are assumed in $S$.", "Hence also $s[1]t$ is an element of $S$ as $S$ is closed under compositions", "and shifts.", "Note that $s[1]t = 0$ as $Y[1] \\to Q_2[1] \\to X[2]$ is zero, see", "Lemma \\ref{lemma-composition-zero}.", "Hence $Q_3[1] \\oplus Q_1[2]$ is isomorphic to an object of", "$\\mathcal{D}'$, see Lemma \\ref{lemma-split}.", "By assumption on $\\mathcal{D}'$ we conclude that $Q_3$ and $Q_1$ are isomorphic", "to objects of $\\mathcal{D}'$. Looking at the distinguished triangle", "$(Q_1, Q_{12}, Q_2)$ we conclude that $Q_2$ is also isomorphic to", "an object of $\\mathcal{D}'$. Looking at the distinguished triangle", "$(X, Y, Q_2)$ we finally conclude that $g \\in S$. (It is also", "follows that $h, f \\in S$, but we don't need this.)" ], "refs": [ "derived-lemma-triangle-functor-localize", "derived-lemma-representable-homological", "derived-lemma-composition-zero", "derived-proposition-9", "derived-lemma-localization-conditions", "derived-lemma-split", "derived-lemma-composition-zero", "derived-lemma-split" ], "ref_ids": [ 1779, 1758, 1757, 1958, 1778, 1766, 1757, 1766 ] } ], "ref_ids": [] }, { "id": 1788, "type": "theorem", "label": "derived-lemma-universal-property-quotient", "categories": [ "derived" ], "title": "derived-lemma-universal-property-quotient", "contents": [ "\\begin{slogan}", "The universal property of the Verdier quotient.", "\\end{slogan}", "Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{B}$", "be a full triangulated subcategory of $\\mathcal{D}$. Let", "$Q : \\mathcal{D} \\to \\mathcal{D}/\\mathcal{B}$ be the quotient functor.", "\\begin{enumerate}", "\\item If $H : \\mathcal{D} \\to \\mathcal{A}$ is a homological functor into", "an abelian category $\\mathcal{A}$ such that", "$\\mathcal{B} \\subset \\Ker(H)$ then there exists a unique factorization", "$H' : \\mathcal{D}/\\mathcal{B} \\to \\mathcal{A}$ such that $H = H' \\circ Q$", "and $H'$ is a homological functor too.", "\\item If $F : \\mathcal{D} \\to \\mathcal{D}'$ is an exact functor into", "a pre-triangulated category $\\mathcal{D}'$ such that", "$\\mathcal{B} \\subset \\Ker(F)$ then there exists a unique factorization", "$F' : \\mathcal{D}/\\mathcal{B} \\to \\mathcal{D}'$ such that $F = F' \\circ Q$", "and $F'$ is an exact functor too.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma follows from", "Lemma \\ref{lemma-universal-property-localization}.", "Namely, if $f : X \\to Y$ is a morphism of $\\mathcal{D}$", "such that for some distinguished triangle $(X, Y, Z, f, g, h)$", "the object $Z$ is isomorphic to an object of $\\mathcal{B}$, then", "$H(f)$, resp.\\ $F(f)$ is an isomorphism under the assumptions of", "(1), resp.\\ (2). Details omitted." ], "refs": [ "derived-lemma-universal-property-localization" ], "ref_ids": [ 1781 ] } ], "ref_ids": [] }, { "id": 1789, "type": "theorem", "label": "derived-lemma-kernel-quotient", "categories": [ "derived" ], "title": "derived-lemma-kernel-quotient", "contents": [ "Let $\\mathcal{D}$ be a triangulated category.", "Let $\\mathcal{B}$ be a full triangulated subcategory.", "The kernel of the quotient functor", "$Q : \\mathcal{D} \\to \\mathcal{D}/\\mathcal{B}$", "is the strictly full subcategory of $\\mathcal{D}$ whose objects are", "$$", "\\Ob(\\Ker(Q)) =", "\\left\\{", "\\begin{matrix}", "Z \\in \\Ob(\\mathcal{D})", "\\text{ such that there exists a }Z' \\in \\Ob(\\mathcal{D}) \\\\", "\\text{ such that }Z \\oplus Z'\\text{ is isomorphic to an object of }\\mathcal{B}", "\\end{matrix}", "\\right\\}", "$$", "In other words it is the smallest strictly full saturated triangulated", "subcategory of $\\mathcal{D}$ containing $\\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "First note that the kernel is automatically a strictly full", "triangulated subcategory containing summands of any of its objects, see", "Lemma \\ref{lemma-triangle-functor-kernel}.", "The description of its objects follows from the definitions and", "Lemma \\ref{lemma-kernel-localization} part (4)." ], "refs": [ "derived-lemma-triangle-functor-kernel", "derived-lemma-kernel-localization" ], "ref_ids": [ 1784, 1782 ] } ], "ref_ids": [] }, { "id": 1790, "type": "theorem", "label": "derived-lemma-operations", "categories": [ "derived" ], "title": "derived-lemma-operations", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. The operations described above", "have the following properties", "\\begin{enumerate}", "\\item $S(\\mathcal{B}(S))$ is the ``saturation'' of $S$, i.e., it is the", "smallest saturated multiplicative system in $\\mathcal{D}$ containing $S$, and", "\\item $\\mathcal{B}(S(\\mathcal{B}))$ is the ``saturation'' of $\\mathcal{B}$,", "i.e., it is the smallest strictly full saturated triangulated subcategory of", "$\\mathcal{D}$ containing $\\mathcal{B}$.", "\\end{enumerate}", "In particular, the constructions define mutually inverse maps between", "the (partially ordered) set of saturated multiplicative systems in", "$\\mathcal{D}$ compatible with the triangulated structure on $\\mathcal{D}$", "and", "the (partially ordered) set of strictly full saturated triangulated", "subcategories of $\\mathcal{D}$." ], "refs": [], "proofs": [ { "contents": [ "First, let's start with a full triangulated subcategory $\\mathcal{B}$. Then", "$\\mathcal{B}(S(\\mathcal{B})) =", "\\Ker(Q : \\mathcal{D} \\to \\mathcal{D}/\\mathcal{B})$", "and hence (2) is the content of", "Lemma \\ref{lemma-kernel-quotient}.", "\\medskip\\noindent", "Next, suppose that $S$ is multiplicative system in $\\mathcal{D}$ compatible", "with the triangulation on $\\mathcal{D}$. Then", "$\\mathcal{B}(S) = \\Ker(Q : \\mathcal{D} \\to S^{-1}\\mathcal{D})$.", "Hence (using", "Lemma \\ref{lemma-third-object-zero}", "in the localized category)", "\\begin{align*}", "S(\\mathcal{B}(S))", "& =", "\\left\\{", "\\begin{matrix}", "f \\in \\text{Arrows}(\\mathcal{D})", "\\text{ such that there exists a distinguished}\\\\", "\\text{triangle }(X, Y, Z, f, g, h) \\text{ of }\\mathcal{D}\\text{ with }Q(Z) = 0", "\\end{matrix}", "\\right\\}", "\\\\", "& =", "\\{f \\in \\text{Arrows}(\\mathcal{D}) \\mid Q(f)\\text{ is an isomorphism}\\} \\\\", "& = \\hat S = S'", "\\end{align*}", "in the notation of", "Categories, Lemma \\ref{categories-lemma-what-gets-inverted}.", "The final statement of that lemma finishes the proof." ], "refs": [ "derived-lemma-kernel-quotient", "derived-lemma-third-object-zero", "categories-lemma-what-gets-inverted" ], "ref_ids": [ 1789, 1764, 12268 ] } ], "ref_ids": [] }, { "id": 1791, "type": "theorem", "label": "derived-lemma-acyclic-general", "categories": [ "derived" ], "title": "derived-lemma-acyclic-general", "contents": [ "Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor from a", "triangulated category $\\mathcal{D}$ to an abelian category $\\mathcal{A}$, see", "Definition \\ref{definition-homological}.", "The subcategory $\\Ker(H)$ of $\\mathcal{D}$ is a strictly full", "saturated triangulated subcategory of $\\mathcal{D}$ whose corresponding", "saturated multiplicative system (see", "Lemma \\ref{lemma-operations})", "is the set", "$$", "S = \\{f \\in \\text{Arrows}(\\mathcal{D}) \\mid", "H^i(f)\\text{ is an isomorphism for all }i \\in \\mathbf{Z}\\}.", "$$", "The functor $H$ factors through the quotient functor", "$Q : \\mathcal{D} \\to \\mathcal{D}/\\Ker(H)$." ], "refs": [ "derived-definition-homological", "derived-lemma-operations" ], "proofs": [ { "contents": [ "The category $\\Ker(H)$ is a strictly full saturated triangulated", "subcategory of $\\mathcal{D}$ by", "Lemma \\ref{lemma-homological-functor-kernel}.", "The set $S$ is a saturated multiplicative system compatible with the", "triangulated structure by", "Lemma \\ref{lemma-homological-functor-localize}.", "Recall that the multiplicative system corresponding to", "$\\Ker(H)$ is the set", "$$", "\\left\\{", "\\begin{matrix}", "f \\in \\text{Arrows}(\\mathcal{D})", "\\text{ such that there exists a distinguished triangle }\\\\", "(X, Y, Z, f, g, h)\\text{ with } H^i(Z) = 0 \\text{ for all }i", "\\end{matrix}", "\\right\\}", "$$", "By the long exact cohomology sequence, see", "(\\ref{equation-long-exact-cohomology-sequence}),", "it is clear that $f$ is an element of this set if and only if $f$ is", "an element of $S$. Finally, the factorization of $H$ through $Q$ is a", "consequence of", "Lemma \\ref{lemma-universal-property-quotient}." ], "refs": [ "derived-lemma-homological-functor-kernel", "derived-lemma-homological-functor-localize", "derived-lemma-universal-property-quotient" ], "ref_ids": [ 1785, 1780, 1788 ] } ], "ref_ids": [ 1971, 1790 ] }, { "id": 1792, "type": "theorem", "label": "derived-lemma-adjoint-is-exact", "categories": [ "derived" ], "title": "derived-lemma-adjoint-is-exact", "contents": [ "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor between", "triangulated categories. If $F$ admits a right adjoint", "$G: \\mathcal{D'} \\to \\mathcal{D}$, then $G$ is also an exact functor." ], "refs": [], "proofs": [ { "contents": [ "Let $X$ be an object of $\\mathcal{D}$ and", "$A$ an object of $\\mathcal{D}'$. Since $F$ is an exact functor we see that", "\\begin{align*}", "\\Mor_\\mathcal{D}(X, G(A[1])", "& =", "\\Mor_{\\mathcal{D}'}(F(X), A[1]) \\\\", "& =", "\\Mor_{\\mathcal{D}'}(F(X)[-1], A) \\\\", "& =", "\\Mor_{\\mathcal{D}'}(F(X[-1]), A) \\\\", "& =", "\\Mor_\\mathcal{D}(X[-1], G(A)) \\\\", "& =", "\\Mor_\\mathcal{D}(X, G(A)[1])", "\\end{align*}", "By Yoneda's lemma (Categories, Lemma \\ref{categories-lemma-yoneda})", "we obtain a canonical isomorphism $G(A)[1] = G(A[1])$.", "Let $A \\to B \\to C \\to A[1]$ be a distinguished triangle in $\\mathcal{D}'$.", "Choose a distinguished triangle", "$$", "G(A) \\to G(B) \\to X \\to G(A)[1]", "$$", "in $\\mathcal{D}$. Then $F(G(A)) \\to F(G(B)) \\to F(X) \\to F(G(A))[1]$", "is a distinguished triangle in $\\mathcal{D}'$. By TR3 we can choose", "a morphism of distinguished triangles", "$$", "\\xymatrix{", "F(G(A)) \\ar[r] \\ar[d] & F(G(B)) \\ar[r] \\ar[d] & F(X) \\ar[r] \\ar[d] &", "F(G(A))[1] \\ar[d] \\\\", "A \\ar[r] & B \\ar[r] & C \\ar[r] & A[1]", "}", "$$", "Since $G$ is the adjoint the new morphism determines a morphism $X \\to G(C)$", "such that the diagram", "$$", "\\xymatrix{", "G(A) \\ar[r] \\ar[d] & G(B) \\ar[r] \\ar[d] & X \\ar[r] \\ar[d] & G(A)[1] \\ar[d] \\\\", "G(A) \\ar[r] & G(B) \\ar[r] & G(C) \\ar[r] & G(A)[1]", "}", "$$", "commutes. Applying the homological functor $\\Hom_{\\mathcal{D}'}(W, -)$", "for an object $W$ of $\\mathcal{D}'$ we deduce from the $5$ lemma that", "$$", "\\Hom_{\\mathcal{D}'}(W, X) \\to \\Hom_{\\mathcal{D}'}(W, G(C))", "$$", "is a bijection and using the Yoneda lemma once more we conclude that", "$X \\to G(C)$ is an isomorphism. Hence we conclude that", "$G(A) \\to G(B) \\to G(C) \\to G(A)[1]$ is a distinguished triangle", "which is what we wanted to show." ], "refs": [ "categories-lemma-yoneda" ], "ref_ids": [ 12203 ] } ], "ref_ids": [] }, { "id": 1793, "type": "theorem", "label": "derived-lemma-fully-faithful-adjoint-kernel-zero", "categories": [ "derived" ], "title": "derived-lemma-fully-faithful-adjoint-kernel-zero", "contents": [ "Let $\\mathcal{D}$, $\\mathcal{D}'$ be triangulated categories.", "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ and", "$G : \\mathcal{D}' \\to \\mathcal{D}$ be functors. Assume that", "\\begin{enumerate}", "\\item $F$ and $G$ are exact functors,", "\\item $F$ is fully faithful,", "\\item $G$ is a right adjoint to $F$, and", "\\item the kernel of $G$ is zero.", "\\end{enumerate}", "Then $F$ is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "Since $F$ is fully faithful the adjunction map $\\text{id} \\to G \\circ F$", "is an isomorphism (Categories, Lemma", "\\ref{categories-lemma-adjoint-fully-faithful}).", "Let $X$ be an object of $\\mathcal{D}'$.", "Choose a distinguished triangle", "$$", "F(G(X)) \\to X \\to Y \\to F(G(X))[1]", "$$", "in $\\mathcal{D}'$. Applying $G$ and using that $G(F(G(X))) = G(X)$", "we find a distinguished triangle", "$$", "G(X) \\to G(X) \\to G(Y) \\to G(X)[1]", "$$", "Hence $G(Y) = 0$. Thus $Y = 0$. Thus $F(G(X)) \\to X$ is an isomorphism." ], "refs": [ "categories-lemma-adjoint-fully-faithful" ], "ref_ids": [ 12248 ] } ], "ref_ids": [] }, { "id": 1794, "type": "theorem", "label": "derived-lemma-functorial-cone", "categories": [ "derived" ], "title": "derived-lemma-functorial-cone", "contents": [ "Suppose that", "$$", "\\xymatrix{", "K_1^\\bullet \\ar[r]_{f_1} \\ar[d]_a & L_1^\\bullet \\ar[d]^b \\\\", "K_2^\\bullet \\ar[r]^{f_2} & L_2^\\bullet", "}", "$$", "is a diagram of morphisms of complexes which is commutative", "up to homotopy. Then there exists a morphism", "$c : C(f_1)^\\bullet \\to C(f_2)^\\bullet$ which gives rise to", "a morphism of triangles", "$(a, b, c) : (K_1^\\bullet, L_1^\\bullet, C(f_1)^\\bullet, f_1, i_1, p_1)", "\\to", "(K_2^\\bullet, L_2^\\bullet, C(f_2)^\\bullet, f_2, i_2, p_2)$", "of $K(\\mathcal{A})$." ], "refs": [], "proofs": [ { "contents": [ "Let $h^n : K_1^n \\to L_2^{n - 1}$ be a family of morphisms such that", "$b \\circ f_1 - f_2 \\circ a= d \\circ h + h \\circ d$.", "Define $c^n$ by the matrix", "$$", "c^n =", "\\left(", "\\begin{matrix}", "b^n & h^{n + 1} \\\\", "0 & a^{n + 1}", "\\end{matrix}", "\\right) :", "L_1^n \\oplus K_1^{n + 1} \\to L_2^n \\oplus K_2^{n + 1}", "$$", "A matrix computation show that $c$ is a morphism of complexes.", "It is trivial that $c \\circ i_1 = i_2 \\circ b$, and it is", "trivial also to check that $p_2 \\circ c = a \\circ p_1$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1795, "type": "theorem", "label": "derived-lemma-map-from-cone", "categories": [ "derived" ], "title": "derived-lemma-map-from-cone", "contents": [ "Suppose that $f: K^\\bullet \\to L^\\bullet$ and $g : L^\\bullet \\to M^\\bullet$", "are morphisms of complexes such that $g \\circ f$ is homotopic to zero.", "Then", "\\begin{enumerate}", "\\item $g$ factors through a morphism $C(f)^\\bullet \\to M^\\bullet$, and", "\\item $f$ factors through a morphism $K^\\bullet \\to C(g)^\\bullet[-1]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The assumptions say that the diagram", "$$", "\\xymatrix{", "K^\\bullet \\ar[r]_f \\ar[d] & L^\\bullet \\ar[d]^g \\\\", "0 \\ar[r] & M^\\bullet", "}", "$$", "commutes up to homotopy.", "Since the cone on $0 \\to M^\\bullet$ is $M^\\bullet$ the", "map $C(f)^\\bullet \\to C(0 \\to M^\\bullet) = M^\\bullet$", "of Lemma \\ref{lemma-functorial-cone}", "is the map in (1). The cone on $K^\\bullet \\to 0$ is", "$K^\\bullet[1]$ and applying Lemma \\ref{lemma-functorial-cone}", "gives a map $K^\\bullet[1] \\to C(g)^\\bullet$. Applying", "$[-1]$ we obtain the map in (2)." ], "refs": [ "derived-lemma-functorial-cone", "derived-lemma-functorial-cone" ], "ref_ids": [ 1794, 1794 ] } ], "ref_ids": [] }, { "id": 1796, "type": "theorem", "label": "derived-lemma-make-commute-map", "categories": [ "derived" ], "title": "derived-lemma-make-commute-map", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let", "$$", "\\xymatrix{", "A^\\bullet \\ar[r]_f \\ar[d]_a & B^\\bullet \\ar[d]^b \\\\", "C^\\bullet \\ar[r]^g & D^\\bullet", "}", "$$", "be a diagram of morphisms of complexes commuting up to homotopy.", "If $f$ is a termwise split injection, then $b$ is homotopic to a", "morphism which makes the diagram commute.", "If $g$ is a split surjection, then $a$ is homotopic to a", "morphism which makes the diagram commute." ], "refs": [], "proofs": [ { "contents": [ "Let $h^n : A^n \\to D^{n - 1}$ be a collection of morphisms", "such that $bf - ga = dh + hd$. Suppose that $\\pi^n : B^n \\to A^n$", "are morphisms splitting the morphisms $f^n$.", "Take $b' = b - dh\\pi - h\\pi d$.", "Suppose $s^n : D^n \\to C^n$ are morphisms splitting the morphisms", "$g^n : C^n \\to D^n$. Take $a' = a + dsh + shd$.", "Computations omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1797, "type": "theorem", "label": "derived-lemma-make-injective", "categories": [ "derived" ], "title": "derived-lemma-make-injective", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let $\\alpha : K^\\bullet \\to L^\\bullet$ be a morphism", "of complexes of $\\mathcal{A}$.", "There exists a factorization", "$$", "\\xymatrix{", "K^\\bullet \\ar[r]^{\\tilde \\alpha} \\ar@/_1pc/[rr]_\\alpha &", "\\tilde L^\\bullet \\ar[r]^\\pi &", "L^\\bullet", "}", "$$", "such that", "\\begin{enumerate}", "\\item $\\tilde \\alpha$ is a termwise split injection (see", "Definition \\ref{definition-termwise-split-map}),", "\\item there is a map of complexes $s : L^\\bullet \\to \\tilde L^\\bullet$", "such that $\\pi \\circ s = \\text{id}_{L^\\bullet}$ and such that", "$s \\circ \\pi$ is homotopic to $\\text{id}_{\\tilde L^\\bullet}$.", "\\end{enumerate}", "Moreover, if both $K^\\bullet$ and $L^\\bullet$ are in", "$K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, or $K^b(\\mathcal{A})$,", "then so is $\\tilde L^\\bullet$." ], "refs": [ "derived-definition-termwise-split-map" ], "proofs": [ { "contents": [ "We set", "$$", "\\tilde L^n = L^n \\oplus K^n \\oplus K^{n + 1}", "$$", "and we define", "$$", "d^n_{\\tilde L} =", "\\left(", "\\begin{matrix}", "d^n_L & 0 & 0 \\\\", "0 & d^n_K & \\text{id}_{K^{n + 1}} \\\\", "0 & 0 & -d^{n + 1}_K", "\\end{matrix}", "\\right)", "$$", "In other words, $\\tilde L^\\bullet = L^\\bullet \\oplus C(1_{K^\\bullet})$.", "Moreover, we set", "$$", "\\tilde \\alpha =", "\\left(", "\\begin{matrix}", "\\alpha \\\\", "\\text{id}_{K^n} \\\\", "0", "\\end{matrix}", "\\right)", "$$", "which is clearly a split injection. It is also clear that it defines a morphism", "of complexes. We define", "$$", "\\pi =", "\\left(", "\\begin{matrix}", "\\text{id}_{L^n} &", "0 &", "0", "\\end{matrix}", "\\right)", "$$", "so that clearly $\\pi \\circ \\tilde \\alpha = \\alpha$. We set", "$$", "s =", "\\left(", "\\begin{matrix}", "\\text{id}_{L^n} \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "so that $\\pi \\circ s = \\text{id}_{L^\\bullet}$. Finally,", "let $h^n : \\tilde L^n \\to \\tilde L^{n - 1}$ be the map", "which maps the summand $K^n$ of $\\tilde L^n$ via the identity morphism", "to the summand $K^n$ of $\\tilde L^{n - 1}$. Then it is a trivial matter", "(see computations in remark below) to prove that", "$$", "\\text{id}_{\\tilde L^\\bullet} - s \\circ \\pi", "=", "d \\circ h + h \\circ d", "$$", "which finishes the proof of the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1979 ] }, { "id": 1798, "type": "theorem", "label": "derived-lemma-make-surjective", "categories": [ "derived" ], "title": "derived-lemma-make-surjective", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let $\\alpha : K^\\bullet \\to L^\\bullet$ be a morphism", "of complexes of $\\mathcal{A}$.", "There exists a factorization", "$$", "\\xymatrix{", "K^\\bullet \\ar[r]^i \\ar@/_1pc/[rr]_\\alpha &", "\\tilde K^\\bullet \\ar[r]^{\\tilde \\alpha} &", "L^\\bullet", "}", "$$", "such that", "\\begin{enumerate}", "\\item $\\tilde \\alpha$ is a termwise split surjection (see", "Definition \\ref{definition-termwise-split-map}),", "\\item there is a map of complexes $s : \\tilde K^\\bullet \\to K^\\bullet$", "such that $s \\circ i = \\text{id}_{K^\\bullet}$ and such that", "$i \\circ s$ is homotopic to $\\text{id}_{\\tilde K^\\bullet}$.", "\\end{enumerate}", "Moreover, if both $K^\\bullet$ and $L^\\bullet$ are in", "$K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, or $K^b(\\mathcal{A})$,", "then so is $\\tilde K^\\bullet$." ], "refs": [ "derived-definition-termwise-split-map" ], "proofs": [ { "contents": [ "Dual to Lemma \\ref{lemma-make-injective}.", "Take", "$$", "\\tilde K^n = K^n \\oplus L^{n - 1} \\oplus L^n", "$$", "and we define", "$$", "d^n_{\\tilde K} =", "\\left(", "\\begin{matrix}", "d^n_K & 0 & 0 \\\\", "0 & - d^{n - 1}_L & \\text{id}_{L^n} \\\\", "0 & 0 & d^n_L", "\\end{matrix}", "\\right)", "$$", "in other words $\\tilde K^\\bullet = K^\\bullet \\oplus C(1_{L^\\bullet[-1]})$.", "Moreover, we set", "$$", "\\tilde \\alpha =", "\\left(", "\\begin{matrix}", "\\alpha &", "0 &", "\\text{id}_{L^n}", "\\end{matrix}", "\\right)", "$$", "which is clearly a split surjection. It is also clear that it defines a", "morphism of complexes. We define", "$$", "i =", "\\left(", "\\begin{matrix}", "\\text{id}_{K^n} \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "so that clearly $\\tilde \\alpha \\circ i = \\alpha$. We set", "$$", "s =", "\\left(", "\\begin{matrix}", "\\text{id}_{K^n} &", "0 &", "0", "\\end{matrix}", "\\right)", "$$", "so that $s \\circ i = \\text{id}_{K^\\bullet}$. Finally,", "let $h^n : \\tilde K^n \\to \\tilde K^{n - 1}$ be the map", "which maps the summand $L^{n - 1}$ of $\\tilde K^n$ via the identity morphism", "to the summand $L^{n - 1}$ of $\\tilde K^{n - 1}$. Then it is a trivial matter", "to prove that", "$$", "\\text{id}_{\\tilde K^\\bullet} - i \\circ s", "=", "d \\circ h + h \\circ d", "$$", "which finishes the proof of the lemma." ], "refs": [ "derived-lemma-make-injective" ], "ref_ids": [ 1797 ] } ], "ref_ids": [ 1979 ] }, { "id": 1799, "type": "theorem", "label": "derived-lemma-triangle-independent-splittings", "categories": [ "derived" ], "title": "derived-lemma-triangle-independent-splittings", "contents": [ "Let $\\mathcal{A}$ be an additive category. Let", "$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$", "be termwise split exact sequences as in", "Definition \\ref{definition-split-ses}.", "Let $(\\pi')^n$, $(s')^n$ be a second collection of splittings.", "Denote $\\delta' : C^\\bullet \\longrightarrow A^\\bullet[1]$ the", "morphism associated to this second set of splittings.", "Then", "$$", "(1, 1, 1) :", "(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta)", "\\longrightarrow", "(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta')", "$$", "is an isomorphism of triangles in $K(\\mathcal{A})$." ], "refs": [ "derived-definition-split-ses" ], "proofs": [ { "contents": [ "The statement simply means that $\\delta$ and $\\delta'$ are", "homotopic maps of complexes. This is", "Homology, Lemma \\ref{homology-lemma-ses-termwise-split-homotopy-cochain}." ], "refs": [ "homology-lemma-ses-termwise-split-homotopy-cochain" ], "ref_ids": [ 12069 ] } ], "ref_ids": [ 1980 ] }, { "id": 1800, "type": "theorem", "label": "derived-lemma-nilpotent", "categories": [ "derived" ], "title": "derived-lemma-nilpotent", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let $0 \\to A_i^\\bullet \\to B_i^\\bullet \\to C_i^\\bullet \\to 0$, $i = 1, 2, 3$", "be termwise split exact sequences of complexes. Let", "$b : B_1^\\bullet \\to B_2^\\bullet$ and $b' : B_2^\\bullet \\to B_3^\\bullet$", "be morphisms of complexes such that", "$$", "\\vcenter{", "\\xymatrix{", "A_1^\\bullet \\ar[d]_0 \\ar[r] &", "B_1^\\bullet \\ar[r] \\ar[d]_b &", "C_1^\\bullet \\ar[d]_0 \\\\", "A_2^\\bullet \\ar[r] & B_2^\\bullet \\ar[r] & C_2^\\bullet", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "A_2^\\bullet \\ar[d]^0 \\ar[r] &", "B_2^\\bullet \\ar[r] \\ar[d]^{b'} &", "C_2^\\bullet \\ar[d]^0 \\\\", "A_3^\\bullet \\ar[r] & B_3^\\bullet \\ar[r] & C_3^\\bullet", "}", "}", "$$", "commute in $K(\\mathcal{A})$. Then $b' \\circ b = 0$ in $K(\\mathcal{A})$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-make-commute-map} we can replace $b$ and $b'$ by homotopic", "maps such that the right square of the left diagram commutes and the", "left square of the right diagram commutes. In other words, we have", "$\\Im(b^n) \\subset \\Im(A_2^n \\to B_2^n)$ and", "$\\Ker((b')^n) \\supset \\Im(A_2^n \\to B_2^n)$.", "Then $b \\circ b' = 0$ as a map of complexes." ], "refs": [ "derived-lemma-make-commute-map" ], "ref_ids": [ 1796 ] } ], "ref_ids": [] }, { "id": 1801, "type": "theorem", "label": "derived-lemma-third-isomorphism", "categories": [ "derived" ], "title": "derived-lemma-third-isomorphism", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let $f_1 : K_1^\\bullet \\to L_1^\\bullet$ and", "$f_2 : K_2^\\bullet \\to L_2^\\bullet$ be morphisms of complexes.", "Let", "$$", "(a, b, c) :", "(K_1^\\bullet, L_1^\\bullet, C(f_1)^\\bullet, f_1, i_1, p_1)", "\\longrightarrow", "(K_2^\\bullet, L_2^\\bullet, C(f_2)^\\bullet, f_2, i_2, p_2)", "$$", "be any morphism of triangles of $K(\\mathcal{A})$.", "If $a$ and $b$ are homotopy equivalences then so is $c$." ], "refs": [], "proofs": [ { "contents": [ "Let $a^{-1} : K_2^\\bullet \\to K_1^\\bullet$ be a morphism of complexes which", "is inverse to $a$ in $K(\\mathcal{A})$.", "Let $b^{-1} : L_2^\\bullet \\to L_1^\\bullet$ be a morphism of complexes which", "is inverse to $b$ in $K(\\mathcal{A})$.", "Let $c' : C(f_2)^\\bullet \\to C(f_1)^\\bullet$", "be the morphism from Lemma \\ref{lemma-functorial-cone} applied", "to $f_1 \\circ a^{-1} = b^{-1} \\circ f_2$. If we can show that", "$c \\circ c'$ and $c' \\circ c$ are isomorphisms in $K(\\mathcal{A})$", "then we win. Hence it suffices to prove the following: Given", "a morphism of triangles", "$(1, 1, c) : (K^\\bullet, L^\\bullet, C(f)^\\bullet, f, i, p)$", "in $K(\\mathcal{A})$ the morphism $c$ is an isomorphism in $K(\\mathcal{A})$.", "By assumption the two squares in the diagram", "$$", "\\xymatrix{", "L^\\bullet \\ar[r] \\ar[d]_1 &", "C(f)^\\bullet \\ar[r] \\ar[d]_c &", "K^\\bullet[1] \\ar[d]_1 \\\\", "L^\\bullet \\ar[r] &", "C(f)^\\bullet \\ar[r] &", "K^\\bullet[1]", "}", "$$", "commute up to homotopy. By construction of $C(f)^\\bullet$ the rows", "form termwise split sequences of complexes. Thus we see that", "$(c - 1)^2 = 0$ in $K(\\mathcal{A})$ by Lemma \\ref{lemma-nilpotent}.", "Hence $c$ is an isomorphism in $K(\\mathcal{A})$ with inverse $2 - c$." ], "refs": [ "derived-lemma-functorial-cone", "derived-lemma-nilpotent" ], "ref_ids": [ 1794, 1800 ] } ], "ref_ids": [] }, { "id": 1802, "type": "theorem", "label": "derived-lemma-the-same-up-to-isomorphisms", "categories": [ "derived" ], "title": "derived-lemma-the-same-up-to-isomorphisms", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "\\begin{enumerate}", "\\item Given a termwise split sequence of complexes", "$(\\alpha : A^\\bullet \\to B^\\bullet,", "\\beta : B^\\bullet \\to C^\\bullet, s^n, \\pi^n)$", "there exists a homotopy equivalence $C(\\alpha)^\\bullet \\to C^\\bullet$", "such that the diagram", "$$", "\\xymatrix{", "A^\\bullet \\ar[r] \\ar[d] & B^\\bullet \\ar[d] \\ar[r] &", "C(\\alpha)^\\bullet \\ar[r]_{-p} \\ar[d] & A^\\bullet[1] \\ar[d] \\\\", "A^\\bullet \\ar[r] & B^\\bullet \\ar[r] &", "C^\\bullet \\ar[r]^\\delta & A^\\bullet[1]", "}", "$$", "defines an isomorphism of triangles in $K(\\mathcal{A})$.", "\\item Given a morphism of complexes $f : K^\\bullet \\to L^\\bullet$", "there exists an isomorphism of triangles", "$$", "\\xymatrix{", "K^\\bullet \\ar[r] \\ar[d] & \\tilde L^\\bullet \\ar[d] \\ar[r] &", "M^\\bullet \\ar[r]_{\\delta} \\ar[d] & K^\\bullet[1] \\ar[d] \\\\", "K^\\bullet \\ar[r] & L^\\bullet \\ar[r] &", "C(f)^\\bullet \\ar[r]^{-p} & K^\\bullet[1]", "}", "$$", "where the upper triangle is the triangle associated to a", "termwise split exact sequence $K^\\bullet \\to \\tilde L^\\bullet \\to M^\\bullet$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). We have $C(\\alpha)^n = B^n \\oplus A^{n + 1}$", "and we simply define $C(\\alpha)^n \\to C^n$ via the projection", "onto $B^n$ followed by $\\beta^n$. This defines", "a morphism of complexes because the compositions", "$A^{n + 1} \\to B^{n + 1} \\to C^{n + 1}$ are zero.", "To get a homotopy inverse we take", "$C^\\bullet \\to C(\\alpha)^\\bullet$ given by", "$(s^n , -\\delta^n)$ in degree $n$. This is a morphism of complexes", "because the morphism $\\delta^n$ can be characterized as the", "unique morphism $C^n \\to A^{n + 1}$ such that", "$d \\circ s^n - s^{n + 1} \\circ d = \\alpha \\circ \\delta^n$,", "see proof of", "Homology, Lemma \\ref{homology-lemma-ses-termwise-split-cochain}.", "The composition", "$C^\\bullet \\to C(\\alpha)^\\bullet \\to C^\\bullet$ is the identity.", "The composition $C(\\alpha)^\\bullet \\to C^\\bullet \\to C(\\alpha)^\\bullet$", "is equal to the morphism", "$$", "\\left(", "\\begin{matrix}", "s^n \\circ \\beta^n & 0 \\\\", "-\\delta^n \\circ \\beta^n & 0", "\\end{matrix}", "\\right)", "$$", "To see that this is homotopic to the identity map", "use the homotopy $h^n : C(\\alpha)^n \\to C(\\alpha)^{n - 1}$", "given by the matrix", "$$", "\\left(", "\\begin{matrix}", "0 & 0 \\\\", "\\pi^n & 0", "\\end{matrix}", "\\right) : C(\\alpha)^n = B^n \\oplus A^{n + 1} \\to", "B^{n - 1} \\oplus A^n = C(\\alpha)^{n - 1}", "$$", "It is trivial to verify that", "$$", "\\left(", "\\begin{matrix}", "1 & 0 \\\\", "0 & 1", "\\end{matrix}", "\\right)", "-", "\\left(", "\\begin{matrix}", "s^n \\\\", "-\\delta^n", "\\end{matrix}", "\\right)", "\\left(", "\\begin{matrix}", "\\beta^n & 0", "\\end{matrix}", "\\right)", "=", "\\left(", "\\begin{matrix}", "d & \\alpha^n \\\\", "0 & -d", "\\end{matrix}", "\\right)", "\\left(", "\\begin{matrix}", "0 & 0 \\\\", "\\pi^n & 0", "\\end{matrix}", "\\right)", "+", "\\left(", "\\begin{matrix}", "0 & 0 \\\\", "\\pi^{n + 1} & 0", "\\end{matrix}", "\\right)", "\\left(", "\\begin{matrix}", "d & \\alpha^{n + 1} \\\\", "0 & -d", "\\end{matrix}", "\\right)", "$$", "To finish the proof of (1) we have to show that the morphisms", "$-p : C(\\alpha)^\\bullet \\to A^\\bullet[1]$ (see", "Definition \\ref{definition-cone})", "and $C(\\alpha)^\\bullet \\to C^\\bullet \\to A^\\bullet[1]$ agree up", "to homotopy. This is clear from the above. Namely, we can use the homotopy", "inverse $(s, -\\delta) : C^\\bullet \\to C(\\alpha)^\\bullet$", "and check instead that the two maps", "$C^\\bullet \\to A^\\bullet[1]$ agree. And note that", "$p \\circ (s, -\\delta) = -\\delta$ as desired.", "\\medskip\\noindent", "Proof of (2). We let $\\tilde f : K^\\bullet \\to \\tilde L^\\bullet$,", "$s : L^\\bullet \\to \\tilde L^\\bullet$", "and $\\pi : \\tilde L^\\bullet \\to L^\\bullet$ be as in", "Lemma \\ref{lemma-make-injective}. By", "Lemmas \\ref{lemma-functorial-cone} and \\ref{lemma-third-isomorphism}", "the triangles $(K^\\bullet, L^\\bullet, C(f), i, p)$ and", "$(K^\\bullet, \\tilde L^\\bullet, C(\\tilde f), \\tilde i, \\tilde p)$", "are isomorphic. Note that we can compose isomorphisms of", "triangles. Thus we may replace $L^\\bullet$ by", "$\\tilde L^\\bullet$ and $f$ by $\\tilde f$. In other words", "we may assume that $f$ is a termwise split injection.", "In this case the result follows from part (1)." ], "refs": [ "homology-lemma-ses-termwise-split-cochain", "derived-definition-cone", "derived-lemma-make-injective", "derived-lemma-functorial-cone", "derived-lemma-third-isomorphism" ], "ref_ids": [ 12067, 1978, 1797, 1794, 1801 ] } ], "ref_ids": [] }, { "id": 1803, "type": "theorem", "label": "derived-lemma-sequence-maps-split", "categories": [ "derived" ], "title": "derived-lemma-sequence-maps-split", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let $A_1^\\bullet \\to A_2^\\bullet \\to \\ldots \\to A_n^\\bullet$", "be a sequence of composable morphisms of complexes.", "There exists a commutative diagram", "$$", "\\xymatrix{", "A_1^\\bullet \\ar[r] &", "A_2^\\bullet \\ar[r] &", "\\ldots \\ar[r] &", "A_n^\\bullet \\\\", "B_1^\\bullet \\ar[r] \\ar[u] &", "B_2^\\bullet \\ar[r] \\ar[u] &", "\\ldots \\ar[r] &", "B_n^\\bullet \\ar[u]", "}", "$$", "such that each morphism $B_i^\\bullet \\to B_{i + 1}^\\bullet$", "is a split injection and each $B_i^\\bullet \\to A_i^\\bullet$", "is a homotopy equivalence. Moreover, if all $A_i^\\bullet$ are in", "$K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, or $K^b(\\mathcal{A})$,", "then so are the $B_i^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "The case $n = 1$ is without content.", "Lemma \\ref{lemma-make-injective} is the case $n = 2$.", "Suppose we have constructed the diagram", "except for $B_n^\\bullet$. Apply Lemma \\ref{lemma-make-injective} to", "the composition $B_{n - 1}^\\bullet \\to A_{n - 1}^\\bullet \\to A_n^\\bullet$.", "The result is a factorization", "$B_{n - 1}^\\bullet \\to B_n^\\bullet \\to A_n^\\bullet$", "as desired." ], "refs": [ "derived-lemma-make-injective", "derived-lemma-make-injective" ], "ref_ids": [ 1797, 1797 ] } ], "ref_ids": [] }, { "id": 1804, "type": "theorem", "label": "derived-lemma-rotate-triangle", "categories": [ "derived" ], "title": "derived-lemma-rotate-triangle", "contents": [ "Let $\\mathcal{A}$ be an additive category. Let", "$(\\alpha : A^\\bullet \\to B^\\bullet, \\beta : B^\\bullet \\to C^\\bullet, s^n,", "\\pi^n)$ be a termwise split sequence of complexes.", "Let $(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta)$", "be the associated triangle.", "Then the triangle", "$(C^\\bullet[-1], A^\\bullet, B^\\bullet, \\delta[-1], \\alpha, \\beta)$", "is isomorphic to the triangle", "$(C^\\bullet[-1], A^\\bullet, C(\\delta[-1])^\\bullet, \\delta[-1], i, p)$." ], "refs": [], "proofs": [ { "contents": [ "We write $B^n = A^n \\oplus C^n$ and we identify $\\alpha^n$ and $\\beta^n$", "with the natural inclusion and projection maps. By construction of $\\delta$ we", "have", "$$", "d_B^n =", "\\left(", "\\begin{matrix}", "d_A^n & \\delta^n \\\\", "0 & d_C^n", "\\end{matrix}", "\\right)", "$$", "On the other hand the cone of $\\delta[-1] : C^\\bullet[-1] \\to A^\\bullet$", "is given as $C(\\delta[-1])^n = A^n \\oplus C^n$ with differential identical", "with the matrix above! Whence the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1805, "type": "theorem", "label": "derived-lemma-rotate-cone", "categories": [ "derived" ], "title": "derived-lemma-rotate-cone", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let $f : K^\\bullet \\to L^\\bullet$ be a morphism of complexes.", "The triangle $(L^\\bullet, C(f)^\\bullet, K^\\bullet[1], i, p, f[1])$ is", "the triangle associated to the termwise split sequence", "$$", "0 \\to L^\\bullet \\to C(f)^\\bullet \\to K^\\bullet[1] \\to 0", "$$", "coming from the definition of the cone of $f$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1806, "type": "theorem", "label": "derived-lemma-two-split-injections", "categories": [ "derived" ], "title": "derived-lemma-two-split-injections", "contents": [ "Let $\\mathcal{A}$ be an additive category. Suppose that", "$\\alpha : A^\\bullet \\to B^\\bullet$ and $\\beta : B^\\bullet \\to C^\\bullet$", "are split injections of complexes. Then there exist distinguished triangles", "$(A^\\bullet, B^\\bullet, Q_1^\\bullet, \\alpha, p_1, d_1)$,", "$(A^\\bullet, C^\\bullet, Q_2^\\bullet, \\beta \\circ \\alpha, p_2, d_2)$", "and", "$(B^\\bullet, C^\\bullet, Q_3^\\bullet, \\beta, p_3, d_3)$", "for which TR4 holds." ], "refs": [], "proofs": [ { "contents": [ "Say $\\pi_1^n : B^n \\to A^n$, and $\\pi_3^n : C^n \\to B^n$ are the splittings.", "Then also $A^\\bullet \\to C^\\bullet$ is a split injection with splittings", "$\\pi_2^n = \\pi_1^n \\circ \\pi_3^n$. Let us write $Q_1^\\bullet$, $Q_2^\\bullet$", "and $Q_3^\\bullet$ for the ``quotient'' complexes. In other words,", "$Q_1^n = \\Ker(\\pi_1^n)$, $Q_3^n = \\Ker(\\pi_3^n)$ and", "$Q_2^n = \\Ker(\\pi_2^n)$. Note that the kernels exist. Then", "$B^n = A^n \\oplus Q_1^n$ and $C_n = B^n \\oplus Q_3^n$, where we think of $A^n$", "as a subobject of $B^n$ and so on. This implies", "$C^n = A^n \\oplus Q_1^n \\oplus Q_3^n$. Note that", "$\\pi_2^n = \\pi_1^n \\circ \\pi_3^n$ is zero on both $Q_1^n$ and $Q_3^n$. Hence", "$Q_2^n = Q_1^n \\oplus Q_3^n$. Consider the commutative diagram", "$$", "\\begin{matrix}", "0 & \\to & A^\\bullet & \\to & B^\\bullet & \\to & Q_1^\\bullet & \\to & 0 \\\\", " & & \\downarrow & & \\downarrow & & \\downarrow & \\\\", "0 & \\to & A^\\bullet & \\to & C^\\bullet & \\to & Q_2^\\bullet & \\to & 0 \\\\", " & & \\downarrow & & \\downarrow & & \\downarrow & \\\\", "0 & \\to & B^\\bullet & \\to & C^\\bullet & \\to & Q_3^\\bullet & \\to & 0", "\\end{matrix}", "$$", "The rows of this diagram are termwise split exact sequences, and", "hence determine distinguished triangles by", "definition. Moreover downward arrows in the diagram above", "are compatible with the chosen splittings and hence", "define morphisms of triangles", "$$", "(A^\\bullet \\to B^\\bullet \\to Q_1^\\bullet \\to A^\\bullet[1])", "\\longrightarrow", "(A^\\bullet \\to C^\\bullet \\to Q_2^\\bullet \\to A^\\bullet[1])", "$$", "and", "$$", "(A^\\bullet \\to C^\\bullet \\to Q_2^\\bullet \\to A^\\bullet[1])", "\\longrightarrow", "(B^\\bullet \\to C^\\bullet \\to Q_3^\\bullet \\to B^\\bullet[1]).", "$$", "Note that the splittings $Q_3^n \\to C^n$", "of the bottom split sequence in the diagram provides a splitting", "for the split sequence", "$0 \\to Q_1^\\bullet \\to Q_2^\\bullet \\to Q_3^\\bullet \\to 0$", "upon composing with $C^n \\to Q_2^n$. It follows easily from this", "that the morphism $\\delta : Q_3^\\bullet \\to Q_1^\\bullet[1]$", "in the corresponding distinguished triangle", "$$", "(Q_1^\\bullet \\to Q_2^\\bullet \\to Q_3^\\bullet \\to Q_1^\\bullet[1])", "$$", "is equal to the composition $Q_3^\\bullet \\to B^\\bullet[1] \\to Q_1^\\bullet[1]$.", "Hence we get a structure as in the conclusion of axiom TR4." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1807, "type": "theorem", "label": "derived-lemma-bounded-triangulated-subcategories", "categories": [ "derived" ], "title": "derived-lemma-bounded-triangulated-subcategories", "contents": [ "Let $\\mathcal{A}$ be an additive category. The categories", "$K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, and $K^b(\\mathcal{A})$", "are full triangulated subcategories of $K(\\mathcal{A})$." ], "refs": [], "proofs": [ { "contents": [ "Each of the categories mentioned is a full additive subcategory.", "We use the criterion of", "Lemma \\ref{lemma-triangulated-subcategory}", "to show that they are triangulated subcategories.", "It is clear that each of the categories", "$K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, and $K^b(\\mathcal{A})$", "is preserved under the shift functors $[1], [-1]$.", "Finally, suppose that $f : A^\\bullet \\to B^\\bullet$ is a morphism", "in $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, or $K^b(\\mathcal{A})$.", "Then $(A^\\bullet, B^\\bullet, C(f)^\\bullet, f, i, -p)$ is a distinguished", "triangle of $K(\\mathcal{A})$ with $C(f)^\\bullet \\in K^{+}(\\mathcal{A})$,", "$K^{-}(\\mathcal{A})$, or $K^b(\\mathcal{A})$ as is clear from the construction", "of the cone. Thus the lemma is proved. (Alternatively,", "$K^\\bullet \\to L^\\bullet$ is isomorphic to an termwise split injection", "of complexes in $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, or", "$K^b(\\mathcal{A})$, see", "Lemma \\ref{lemma-make-injective}", "and then one can directly take the associated", "distinguished triangle.)" ], "refs": [ "derived-lemma-triangulated-subcategory", "derived-lemma-make-injective" ], "ref_ids": [ 1771, 1797 ] } ], "ref_ids": [] }, { "id": 1808, "type": "theorem", "label": "derived-lemma-additive-exact-homotopy-category", "categories": [ "derived" ], "title": "derived-lemma-additive-exact-homotopy-category", "contents": [ "Let $\\mathcal{A}$, $\\mathcal{B}$ be additive categories.", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor.", "The induced functors", "$$", "\\begin{matrix}", "F : K(\\mathcal{A}) \\longrightarrow K(\\mathcal{B}) \\\\", "F : K^{+}(\\mathcal{A}) \\longrightarrow K^{+}(\\mathcal{B}) \\\\", "F : K^{-}(\\mathcal{A}) \\longrightarrow K^{-}(\\mathcal{B}) \\\\", "F : K^b(\\mathcal{A}) \\longrightarrow K^b(\\mathcal{B})", "\\end{matrix}", "$$", "are exact functors of triangulated categories." ], "refs": [], "proofs": [ { "contents": [ "Suppose $A^\\bullet \\to B^\\bullet \\to C^\\bullet$", "is a termwise split sequence of complexes of $\\mathcal{A}$ with splittings", "$(s^n, \\pi^n)$ and associated morphism $\\delta : C^\\bullet \\to A^\\bullet[1]$,", "see Definition \\ref{definition-split-ses}. Then", "$F(A^\\bullet) \\to F(B^\\bullet) \\to F(C^\\bullet)$", "is a termwise split sequence of complexes with splittings", "$(F(s^n), F(\\pi^n))$ and associated morphism", "$F(\\delta) : F(C^\\bullet) \\to F(A^\\bullet)[1]$.", "Thus $F$ transforms distinguished triangles into distinguished triangles." ], "refs": [ "derived-definition-split-ses" ], "ref_ids": [ 1980 ] } ], "ref_ids": [] }, { "id": 1809, "type": "theorem", "label": "derived-lemma-improve-distinguished-triangle-homotopy", "categories": [ "derived" ], "title": "derived-lemma-improve-distinguished-triangle-homotopy", "contents": [ "Let $\\mathcal{A}$ be an additive category. Let", "$(A^\\bullet, B^\\bullet, C^\\bullet, a, b, c)$ be a distinguished triangle in", "$K(\\mathcal{A})$. Then there exists an isomorphic distinguished triangle", "$(A^\\bullet, (B')^\\bullet, C^\\bullet, a', b', c)$ such that", "$0 \\to A^n \\to (B')^n \\to C^n \\to 0$ is a split short exact sequence", "for all $n$." ], "refs": [], "proofs": [ { "contents": [ "We will use that $K(\\mathcal{A})$ is a triangulated category by", "Proposition \\ref{proposition-homotopy-category-triangulated}.", "Let $W^\\bullet$ be the cone on $c : C^\\bullet \\to A^\\bullet[1]$ with its maps", "$i : A^\\bullet[1] \\to W^\\bullet$ and $p : W^\\bullet \\to C^\\bullet[1]$. Then", "$(C^\\bullet, A^\\bullet[1], W^\\bullet, c, i, -p)$ is a distinguished triangle", "by Lemma \\ref{lemma-the-same-up-to-isomorphisms}. Rotating backwards twice", "we see that $(A^\\bullet, W^\\bullet[-1], C^\\bullet, -i[-1], p[-1], c)$", "is a distinguished triangle. By TR3 there is a morphism of distinguished", "triangles", "$(\\text{id}, \\beta, \\text{id}) : (A^\\bullet, B^\\bullet, C^\\bullet, a, b, c) \\to", "(A^\\bullet, W^\\bullet[-1], C^\\bullet, -i[-1], p[-1], c)$", "which must be an isomorphism by Lemma \\ref{lemma-third-isomorphism-triangle}.", "This finishes the proof because", "$0 \\to A^\\bullet \\to W^\\bullet[-1] \\to C^\\bullet \\to 0$", "is a termwise split short exact sequence of complexes", "by the very construction of cones in Section \\ref{section-cones}." ], "refs": [ "derived-proposition-homotopy-category-triangulated", "derived-lemma-the-same-up-to-isomorphisms", "derived-lemma-third-isomorphism-triangle" ], "ref_ids": [ 1960, 1802, 1759 ] } ], "ref_ids": [] }, { "id": 1810, "type": "theorem", "label": "derived-lemma-cohomology-homological", "categories": [ "derived" ], "title": "derived-lemma-cohomology-homological", "contents": [ "Let $\\mathcal{A}$ be an abelian category. The functor", "$$", "H^0 : K(\\mathcal{A}) \\longrightarrow \\mathcal{A}", "$$", "is homological." ], "refs": [], "proofs": [ { "contents": [ "Because $H^0$ is a functor, and by our definition of distinguished triangles", "it suffices to prove that given a termwise split short exact sequence", "of complexes $0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$", "the sequence $H^0(A^\\bullet) \\to H^0(B^\\bullet) \\to H^0(C^\\bullet)$", "is exact. This follows from", "Homology, Lemma \\ref{homology-lemma-long-exact-sequence-cochain}." ], "refs": [ "homology-lemma-long-exact-sequence-cochain" ], "ref_ids": [ 12061 ] } ], "ref_ids": [] }, { "id": 1811, "type": "theorem", "label": "derived-lemma-acyclic", "categories": [ "derived" ], "title": "derived-lemma-acyclic", "contents": [ "Let $\\mathcal{A}$ be an abelian category. The full subcategory", "$\\text{Ac}(\\mathcal{A})$ of $K(\\mathcal{A})$ consisting of acyclic complexes", "is a strictly full saturated triangulated subcategory of $K(\\mathcal{A})$.", "The corresponding saturated multiplicative system (see", "Lemma \\ref{lemma-operations})", "of $K(\\mathcal{A})$ is the set $\\text{Qis}(\\mathcal{A})$", "of quasi-isomorphisms. In particular, the kernel of the localization", "functor $Q : K(\\mathcal{A}) \\to \\text{Qis}(\\mathcal{A})^{-1}K(\\mathcal{A})$", "is $\\text{Ac}(\\mathcal{A})$ and the functor $H^0$ factors through $Q$." ], "refs": [ "derived-lemma-operations" ], "proofs": [ { "contents": [ "We know that $H^0$ is a homological functor by", "Lemma \\ref{lemma-cohomology-homological}.", "Thus this lemma is a special case of", "Lemma \\ref{lemma-acyclic-general}." ], "refs": [ "derived-lemma-cohomology-homological", "derived-lemma-acyclic-general" ], "ref_ids": [ 1810, 1791 ] } ], "ref_ids": [ 1790 ] }, { "id": 1812, "type": "theorem", "label": "derived-lemma-complex-cohomology-bounded", "categories": [ "derived" ], "title": "derived-lemma-complex-cohomology-bounded", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $K^\\bullet$ be a complex.", "\\begin{enumerate}", "\\item If $H^n(K^\\bullet) = 0$ for all $n \\ll 0$, then there exists", "a quasi-isomorphism $K^\\bullet \\to L^\\bullet$ with $L^\\bullet$", "bounded below.", "\\item If $H^n(K^\\bullet) = 0$ for all $n \\gg 0$, then there exists", "a quasi-isomorphism $M^\\bullet \\to K^\\bullet$ with $M^\\bullet$", "bounded above.", "\\item If $H^n(K^\\bullet) = 0$ for all $|n| \\gg 0$, then there exists", "a commutative diagram of morphisms of complexes", "$$", "\\xymatrix{", "K^\\bullet \\ar[r] & L^\\bullet \\\\", "M^\\bullet \\ar[u] \\ar[r] & N^\\bullet \\ar[u]", "}", "$$", "where all the arrows are quasi-isomorphisms, $L^\\bullet$", "bounded below, $M^\\bullet$ bounded above, and $N^\\bullet$ a bounded", "complex.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Pick $a \\ll 0 \\ll b$ and set $M^\\bullet = \\tau_{\\leq b}K^\\bullet$,", "$L^\\bullet = \\tau_{\\geq a}K^\\bullet$, and", "$N^\\bullet = \\tau_{\\leq b}L^\\bullet = \\tau_{\\geq a}M^\\bullet$.", "See", "Homology, Section \\ref{homology-section-truncations}", "for the truncation functors." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1813, "type": "theorem", "label": "derived-lemma-bounded-derived", "categories": [ "derived" ], "title": "derived-lemma-bounded-derived", "contents": [ "Let $\\mathcal{A}$ be an abelian category. The subcategories", "$\\text{Ac}^{+}(\\mathcal{A})$, $\\text{Ac}^{-}(\\mathcal{A})$,", "resp.\\ $\\text{Ac}^b(\\mathcal{A})$", "are strictly full saturated triangulated subcategories", "of $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, resp.\\ $K^b(\\mathcal{A})$.", "The corresponding saturated multiplicative systems (see", "Lemma \\ref{lemma-operations})", "are the sets $\\text{Qis}^{+}(\\mathcal{A})$, $\\text{Qis}^{-}(\\mathcal{A})$,", "resp.\\ $\\text{Qis}^b(\\mathcal{A})$.", "\\begin{enumerate}", "\\item The kernel of the functor $K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{A})$", "is $\\text{Ac}^{+}(\\mathcal{A})$ and this induces an equivalence", "of triangulated categories", "$$", "K^{+}(\\mathcal{A})/\\text{Ac}^{+}(\\mathcal{A}) =", "\\text{Qis}^{+}(\\mathcal{A})^{-1}K^{+}(\\mathcal{A})", "\\longrightarrow", "D^{+}(\\mathcal{A})", "$$", "\\item The kernel of the functor $K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{A})$", "is $\\text{Ac}^{-}(\\mathcal{A})$ and this induces an equivalence", "of triangulated categories", "$$", "K^{-}(\\mathcal{A})/\\text{Ac}^{-}(\\mathcal{A}) =", "\\text{Qis}^{-}(\\mathcal{A})^{-1}K^{-}(\\mathcal{A})", "\\longrightarrow", "D^{-}(\\mathcal{A})", "$$", "\\item The kernel of the functor $K^b(\\mathcal{A}) \\to D^b(\\mathcal{A})$", "is $\\text{Ac}^b(\\mathcal{A})$ and this induces an equivalence", "of triangulated categories", "$$", "K^b(\\mathcal{A})/\\text{Ac}^b(\\mathcal{A}) =", "\\text{Qis}^b(\\mathcal{A})^{-1}K^b(\\mathcal{A})", "\\longrightarrow", "D^b(\\mathcal{A})", "$$", "\\end{enumerate}" ], "refs": [ "derived-lemma-operations" ], "proofs": [ { "contents": [ "The initial statements follow from", "Lemma \\ref{lemma-acyclic-general}", "by considering the restriction of the homological functor $H^0$.", "The statement on kernels in (1), (2), (3) is a consequence of the", "definitions in each case.", "Each of the functors is essentially surjective by", "Lemma \\ref{lemma-complex-cohomology-bounded}.", "To finish the proof we have to show the functors are fully faithful.", "We first do this for the bounded below version.", "\\medskip\\noindent", "Suppose that $K^\\bullet, L^\\bullet$ are bounded above complexes.", "A morphism between these in $D(\\mathcal{A})$ is of the form", "$s^{-1}f$ for a pair", "$f : K^\\bullet \\to (L')^\\bullet$, $s : L^\\bullet \\to (L')^\\bullet$", "where $s$ is a quasi-isomorphism. This implies that $(L')^\\bullet$", "has cohomology bounded below. Hence by", "Lemma \\ref{lemma-complex-cohomology-bounded}", "we can choose a quasi-isomorphism", "$s' : (L')^\\bullet \\to (L'')^\\bullet$", "with $(L'')^\\bullet$ bounded below. Then the pair $(s' \\circ f, s' \\circ s)$", "defines a morphism in $\\text{Qis}^{+}(\\mathcal{A})^{-1}K^{+}(\\mathcal{A})$.", "Hence the functor is ``full''. Finally, suppose that the pair", "$f : K^\\bullet \\to (L')^\\bullet$, $s : L^\\bullet \\to (L')^\\bullet$", "defines a morphism in $\\text{Qis}^{+}(\\mathcal{A})^{-1}K^{+}(\\mathcal{A})$", "which is zero in $D(\\mathcal{A})$. This means that there exists a", "quasi-isomorphism $s' : (L')^\\bullet \\to (L'')^\\bullet$", "such that $s' \\circ f = 0$. Using", "Lemma \\ref{lemma-complex-cohomology-bounded}", "once more we obtain a quasi-isomorphism", "$s'' : (L'')^\\bullet \\to (L''')^\\bullet$", "with $(L''')^\\bullet$ bounded below.", "Thus we see that $s'' \\circ s' \\circ f = 0$ which implies that", "$s^{-1}f$ is zero in $\\text{Qis}^{+}(\\mathcal{A})^{-1}K^{+}(\\mathcal{A})$.", "This finishes the proof that the functor in (1) is an equivalence.", "\\medskip\\noindent", "The proof of (2) is dual to the proof of (1).", "To prove (3) we may use the result of (2). Hence it suffices to", "prove that the functor", "$\\text{Qis}^b(\\mathcal{A})^{-1}K^b(\\mathcal{A})", "\\to \\text{Qis}^{-}(\\mathcal{A})^{-1}K^{-}(\\mathcal{A})$", "is fully faithful. The argument given in the previous paragraph", "applies directly to show this where we consistently work with complexes", "which are already bounded above." ], "refs": [ "derived-lemma-acyclic-general", "derived-lemma-complex-cohomology-bounded", "derived-lemma-complex-cohomology-bounded", "derived-lemma-complex-cohomology-bounded" ], "ref_ids": [ 1791, 1812, 1812, 1812 ] } ], "ref_ids": [ 1790 ] }, { "id": 1814, "type": "theorem", "label": "derived-lemma-derived-canonical-delta-functor", "categories": [ "derived" ], "title": "derived-lemma-derived-canonical-delta-functor", "contents": [ "Let $\\mathcal{A}$ be an abelian category. The functor", "$\\text{Comp}(\\mathcal{A}) \\to D(\\mathcal{A})$", "defined has the natural structure of a $\\delta$-functor,", "with", "$$", "\\delta_{A^\\bullet \\to B^\\bullet \\to C^\\bullet} = - p \\circ q^{-1}", "$$", "with $p$ and $q$ as explained above. The same construction turns the", "functors", "$\\text{Comp}^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{A})$,", "$\\text{Comp}^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{A})$, and", "$\\text{Comp}^b(\\mathcal{A}) \\to D^b(\\mathcal{A})$", "into $\\delta$-functors." ], "refs": [], "proofs": [ { "contents": [ "We have already seen that this choice leads to a distinguished", "triangle whenever given a short exact sequence of complexes.", "We have to show that given a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "A^\\bullet \\ar[r]_a \\ar[d]_f &", "B^\\bullet \\ar[r]_b \\ar[d]_g &", "C^\\bullet \\ar[r] \\ar[d]_h &", "0 \\\\", "0 \\ar[r] &", "(A')^\\bullet \\ar[r]^{a'} &", "(B')^\\bullet \\ar[r]^{b'} &", "(C')^\\bullet \\ar[r] &", "0", "}", "$$", "we get the desired commutative diagram of", "Definition \\ref{definition-delta-functor} (2).", "By Lemma \\ref{lemma-functorial-cone}", "the pair $(f, g)$ induces a canonical morphism", "$c : C(a)^\\bullet \\to C(a')^\\bullet$. It is a simple computation", "to show that $q' \\circ c = h \\circ q$ and", "$f[1] \\circ p = p' \\circ c$. From this the result follows directly." ], "refs": [ "derived-definition-delta-functor", "derived-lemma-functorial-cone" ], "ref_ids": [ 1972, 1794 ] } ], "ref_ids": [] }, { "id": 1815, "type": "theorem", "label": "derived-lemma-derived-compare-triangles-ses", "categories": [ "derived" ], "title": "derived-lemma-derived-compare-triangles-ses", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let", "$$", "\\xymatrix{", "0 \\ar[r] &", "A^\\bullet \\ar[r] \\ar[d] &", "B^\\bullet \\ar[r] \\ar[d] &", "C^\\bullet \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "D^\\bullet \\ar[r] &", "E^\\bullet \\ar[r] &", "F^\\bullet \\ar[r] &", "0", "}", "$$", "be a commutative diagram of morphisms of complexes", "such that the rows are short exact sequences of complexes, and", "the vertical arrows are quasi-isomorphisms.", "The $\\delta$-functor of", "Lemma \\ref{lemma-derived-canonical-delta-functor}", "above maps the short exact sequences", "$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$", "and", "$0 \\to D^\\bullet \\to E^\\bullet \\to F^\\bullet \\to 0$", "to isomorphic distinguished triangles." ], "refs": [ "derived-lemma-derived-canonical-delta-functor" ], "proofs": [ { "contents": [ "Trivial from the fact that $K(\\mathcal{A}) \\to D(\\mathcal{A})$", "transforms quasi-isomorphisms into isomorphisms and that the", "associated distinguished triangles are functorial." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1814 ] }, { "id": 1816, "type": "theorem", "label": "derived-lemma-derived-compare-triangles-split-case", "categories": [ "derived" ], "title": "derived-lemma-derived-compare-triangles-split-case", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let", "$$", "\\xymatrix{", "0 \\ar[r] &", "A^\\bullet \\ar[r] &", "B^\\bullet \\ar[r] &", "C^\\bullet \\ar[r] &", "0", "}", "$$", "be a short exact sequences of complexes.", "Assume this short exact sequence is termwise split. Let", "$(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta)$", "be the distinguished triangle of $K(\\mathcal{A})$", "associated to the sequence. The $\\delta$-functor of", "Lemma \\ref{lemma-derived-canonical-delta-functor}", "above maps the short exact sequences", "$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$", "to a triangle isomorphic to the distinguished triangle", "$$", "(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta).", "$$" ], "refs": [ "derived-lemma-derived-canonical-delta-functor" ], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-the-same-up-to-isomorphisms}." ], "refs": [ "derived-lemma-the-same-up-to-isomorphisms" ], "ref_ids": [ 1802 ] } ], "ref_ids": [ 1814 ] }, { "id": 1817, "type": "theorem", "label": "derived-lemma-trick-vanishing-composition", "categories": [ "derived" ], "title": "derived-lemma-trick-vanishing-composition", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let", "$$", "K_0^\\bullet \\to K_1^\\bullet \\to \\ldots \\to K_n^\\bullet", "$$", "be maps of complexes such that", "\\begin{enumerate}", "\\item $H^i(K_0^\\bullet) = 0$ for $i > 0$,", "\\item $H^{-j}(K_j^\\bullet) \\to H^{-j}(K_{j + 1}^\\bullet)$ is zero.", "\\end{enumerate}", "Then the composition $K_0^\\bullet \\to K_n^\\bullet$ factors through", "$\\tau_{\\leq -n}K_n^\\bullet \\to K_n^\\bullet$ in $D(\\mathcal{A})$.", "Dually, given maps of complexes", "$$", "K_n^\\bullet \\to K_{n - 1}^\\bullet \\to \\ldots \\to K_0^\\bullet", "$$", "such that", "\\begin{enumerate}", "\\item $H^i(K_0^\\bullet) = 0$ for $i < 0$,", "\\item $H^j(K_{j + 1}^\\bullet) \\to H^j(K_j^\\bullet)$ is zero,", "\\end{enumerate}", "then the composition $K_n^\\bullet \\to K_0^\\bullet$ factors through", "$K_n^\\bullet \\to \\tau_{\\geq n}K_n^\\bullet$ in $D(\\mathcal{A})$." ], "refs": [], "proofs": [ { "contents": [ "The case $n = 1$. Since $\\tau_{\\leq 0}K_0^\\bullet = K_0^\\bullet$", "in $D(\\mathcal{A})$ we can replace", "$K_0^\\bullet$ by $\\tau_{\\leq 0}K_0^\\bullet$ and", "$K_1^\\bullet$ by $\\tau_{\\leq 0}K_1^\\bullet$.", "Consider the distinguished triangle", "$$", "\\tau_{\\leq -1}K_1^\\bullet \\to K_1^\\bullet \\to", "H^0(K_1^\\bullet)[0] \\to (\\tau_{\\leq -1}K_1^\\bullet)[1]", "$$", "(Remark \\ref{remark-truncation-distinguished-triangle}).", "The composition $K_0^\\bullet \\to K_1^\\bullet \\to H^0(K_1^\\bullet)[0]$", "is zero as it is equal to $K_0^\\bullet \\to H^0(K_0^\\bullet)[0] \\to", "H^0(K_1^\\bullet)[0]$ which is zero by assumption.", "The fact that $\\Hom_{D(\\mathcal{A})}(K_0^\\bullet, -)$", "is a homological functor (Lemma \\ref{lemma-representable-homological}),", "allows us to find the desired factorization.", "For $n = 2$ we get a factorization", "$K_0^\\bullet \\to \\tau_{\\leq -1}K_1^\\bullet$ by the case $n = 1$", "and we can apply the case $n = 1$ to the map of complexes", "$\\tau_{\\leq -1}K_1^\\bullet \\to \\tau_{\\leq -1}K_2^\\bullet$", "to get a factorization", "$\\tau_{\\leq -1}K_1^\\bullet \\to \\tau_{\\leq -2}K_2^\\bullet$.", "The general case is proved in exactly the same manner." ], "refs": [ "derived-remark-truncation-distinguished-triangle", "derived-lemma-representable-homological" ], "ref_ids": [ 2016, 1758 ] } ], "ref_ids": [] }, { "id": 1818, "type": "theorem", "label": "derived-lemma-filtered-cohomology-homological", "categories": [ "derived" ], "title": "derived-lemma-filtered-cohomology-homological", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item The functor", "$K(\\text{Fil}^f(\\mathcal{A})) \\longrightarrow \\text{Gr}(\\mathcal{A})$,", "$K^\\bullet \\longmapsto H^0(\\text{gr}(K^\\bullet))$", "is homological.", "\\item The functor", "$K(\\text{Fil}^f(\\mathcal{A})) \\rightarrow \\mathcal{A}$,", "$K^\\bullet \\longmapsto H^0(\\text{gr}^p(K^\\bullet))$", "is homological.", "\\item The functor", "$K(\\text{Fil}^f(\\mathcal{A})) \\longrightarrow \\mathcal{A}$,", "$K^\\bullet \\longmapsto H^0((\\text{forget }F)K^\\bullet)$", "is homological.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that", "$H^0 : K(\\mathcal{A}) \\to \\mathcal{A}$ is homological, see", "Lemma \\ref{lemma-cohomology-homological}", "and the fact that the functors $\\text{gr}, \\text{gr}^p, (\\text{forget }F)$", "are exact functors of triangulated categories. See", "Lemma \\ref{lemma-exact-compose-homological-functor}." ], "refs": [ "derived-lemma-cohomology-homological", "derived-lemma-exact-compose-homological-functor" ], "ref_ids": [ 1810, 1775 ] } ], "ref_ids": [] }, { "id": 1819, "type": "theorem", "label": "derived-lemma-filtered-acyclic", "categories": [ "derived" ], "title": "derived-lemma-filtered-acyclic", "contents": [ "Let $\\mathcal{A}$ be an abelian category. The full subcategory", "$\\text{FAc}(\\mathcal{A})$ of $K(\\text{Fil}^f(\\mathcal{A}))$", "consisting of filtered acyclic complexes is a strictly full saturated", "triangulated subcategory of $K(\\text{Fil}^f(\\mathcal{A}))$.", "The corresponding saturated multiplicative system (see", "Lemma \\ref{lemma-operations})", "of $K(\\text{Fil}^f(\\mathcal{A}))$ is the set", "$\\text{FQis}(\\mathcal{A})$ of filtered quasi-isomorphisms.", "In particular, the kernel of the localization", "functor", "$$", "Q :", "K(\\text{Fil}^f(\\mathcal{A}))", "\\longrightarrow", "\\text{FQis}(\\mathcal{A})^{-1}K(\\text{Fil}^f(\\mathcal{A}))", "$$", "is $\\text{FAc}(\\mathcal{A})$ and the functor $H^0 \\circ \\text{gr}$", "factors through $Q$." ], "refs": [ "derived-lemma-operations" ], "proofs": [ { "contents": [ "We know that $H^0 \\circ \\text{gr}$ is a homological functor by", "Lemma \\ref{lemma-filtered-cohomology-homological}.", "Thus this lemma is a special case of", "Lemma \\ref{lemma-acyclic-general}." ], "refs": [ "derived-lemma-filtered-cohomology-homological", "derived-lemma-acyclic-general" ], "ref_ids": [ 1818, 1791 ] } ], "ref_ids": [ 1790 ] }, { "id": 1820, "type": "theorem", "label": "derived-lemma-filtered-derived-functors", "categories": [ "derived" ], "title": "derived-lemma-filtered-derived-functors", "contents": [ "The functors $\\text{gr}^p, \\text{gr}, (\\text{forget }F)$ induce", "canonical exact functors", "$$", "\\text{gr}^p, \\text{gr}, (\\text{forget }F):", "DF(\\mathcal{A})", "\\longrightarrow", "D(\\mathcal{A})", "$$", "which commute with the localization functors." ], "refs": [], "proofs": [ { "contents": [ "This follows from the universal property of localization, see", "Lemma \\ref{lemma-universal-property-localization},", "provided we can show that a filtered quasi-isomorphism is turned", "into a quasi-isomorphism by each of the functors", "$\\text{gr}^p, \\text{gr}, (\\text{forget }F)$. This is true by definition", "for the first two. For the last one the statement we have to do a little", "bit of work. Let $f : K^\\bullet \\to L^\\bullet$ be a filtered", "quasi-isomorphism in $K(\\text{Fil}^f(\\mathcal{A}))$.", "Choose a distinguished triangle $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$", "which contains $f$. Then $M^\\bullet$ is filtered acyclic, see", "Lemma \\ref{lemma-filtered-acyclic}.", "Hence by the corresponding lemma for $K(\\mathcal{A})$ it suffices", "to show that a filtered acyclic complex is an acyclic complex if", "we forget the filtration.", "This follows from", "Homology, Lemma \\ref{homology-lemma-filtered-acyclic}." ], "refs": [ "derived-lemma-universal-property-localization", "derived-lemma-filtered-acyclic", "homology-lemma-filtered-acyclic" ], "ref_ids": [ 1781, 1819, 12086 ] } ], "ref_ids": [] }, { "id": 1821, "type": "theorem", "label": "derived-lemma-filtered-complex-cohomology-bounded", "categories": [ "derived" ], "title": "derived-lemma-filtered-complex-cohomology-bounded", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $K^\\bullet \\in K(\\text{Fil}^f(\\mathcal{A}))$.", "\\begin{enumerate}", "\\item If $H^n(\\text{gr}(K^\\bullet)) = 0$ for all $n < a$, then there exists", "a filtered quasi-isomorphism $K^\\bullet \\to L^\\bullet$ with", "$L^n = 0$ for all $n < a$.", "\\item If $H^n(\\text{gr}(K^\\bullet)) = 0$ for all $n > b$, then there exists", "a filtered quasi-isomorphism $M^\\bullet \\to K^\\bullet$ with", "$M^n = 0$ for all $n > b$.", "\\item If $H^n(\\text{gr}(K^\\bullet)) = 0$ for all $|n| \\gg 0$, then there", "exists a commutative diagram of morphisms of complexes", "$$", "\\xymatrix{", "K^\\bullet \\ar[r] & L^\\bullet \\\\", "M^\\bullet \\ar[u] \\ar[r] & N^\\bullet \\ar[u]", "}", "$$", "where all the arrows are filtered quasi-isomorphisms, $L^\\bullet$", "bounded below, $M^\\bullet$ bounded above, and $N^\\bullet$ a bounded", "complex.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $H^n(\\text{gr}(K^\\bullet)) = 0$ for all $n < a$. By", "Homology, Lemma \\ref{homology-lemma-filtered-acyclic}", "the sequence", "$$", "K^{a - 1} \\xrightarrow{d^{a - 2}} K^{a - 1} \\xrightarrow{d^{a - 1}} K^a", "$$", "is an exact sequence of objects of $\\mathcal{A}$ and the morphisms", "$d^{a - 2}$ and $d^{a - 1}$ are strict. Hence", "$\\Coim(d^{a - 1}) = \\Im(d^{a - 1})$ in $\\text{Fil}^f(\\mathcal{A})$", "and the map $\\text{gr}(\\Im(d^{a - 1})) \\to \\text{gr}(K^a)$", "is injective with image equal to the image of", "$\\text{gr}(K^{a - 1}) \\to \\text{gr}(K^a)$, see", "Homology, Lemma \\ref{homology-lemma-characterize-strict}.", "This means that the map $K^\\bullet \\to \\tau_{\\geq a}K^\\bullet$", "into the truncation", "$$", "\\tau_{\\geq a}K^\\bullet =", "(\\ldots \\to 0 \\to K^a/\\Im(d^{a - 1}) \\to K^{a + 1} \\to \\ldots)", "$$", "is a filtered quasi-isomorphism. This proves (1). The proof of (2)", "is dual to the proof of (1). Part (3) follows formally from (1) and (2)." ], "refs": [ "homology-lemma-filtered-acyclic", "homology-lemma-characterize-strict" ], "ref_ids": [ 12086, 12084 ] } ], "ref_ids": [] }, { "id": 1822, "type": "theorem", "label": "derived-lemma-filtered-bounded-derived", "categories": [ "derived" ], "title": "derived-lemma-filtered-bounded-derived", "contents": [ "Let $\\mathcal{A}$ be an abelian category. The subcategories", "$\\text{FAc}^{+}(\\mathcal{A})$, $\\text{FAc}^{-}(\\mathcal{A})$,", "resp.\\ $\\text{FAc}^b(\\mathcal{A})$", "are strictly full saturated triangulated subcategories", "of $K^{+}(\\text{Fil}^f\\mathcal{A})$, $K^{-}(\\text{Fil}^f\\mathcal{A})$,", "resp.\\ $K^b(\\text{Fil}^f\\mathcal{A})$.", "The corresponding saturated multiplicative systems (see", "Lemma \\ref{lemma-operations})", "are the sets $\\text{FQis}^{+}(\\mathcal{A})$, $\\text{FQis}^{-}(\\mathcal{A})$,", "resp.\\ $\\text{FQis}^b(\\mathcal{A})$.", "\\begin{enumerate}", "\\item The kernel of the functor", "$K^{+}(\\text{Fil}^f\\mathcal{A}) \\to DF^{+}(\\mathcal{A})$", "is $\\text{FAc}^{+}(\\mathcal{A})$ and this induces an equivalence", "of triangulated categories", "$$", "K^{+}(\\text{Fil}^f\\mathcal{A})/\\text{FAc}^{+}(\\mathcal{A}) =", "\\text{FQis}^{+}(\\mathcal{A})^{-1}K^{+}(\\text{Fil}^f\\mathcal{A})", "\\longrightarrow", "DF^{+}(\\mathcal{A})", "$$", "\\item The kernel of the functor", "$K^{-}(\\text{Fil}^f\\mathcal{A}) \\to DF^{-}(\\mathcal{A})$", "is $\\text{FAc}^{-}(\\mathcal{A})$ and this induces an equivalence", "of triangulated categories", "$$", "K^{-}(\\text{Fil}^f\\mathcal{A})/\\text{FAc}^{-}(\\mathcal{A}) =", "\\text{FQis}^{-}(\\mathcal{A})^{-1}K^{-}(\\text{Fil}^f\\mathcal{A})", "\\longrightarrow", "DF^{-}(\\mathcal{A})", "$$", "\\item The kernel of the functor", "$K^b(\\text{Fil}^f\\mathcal{A}) \\to DF^b(\\mathcal{A})$", "is $\\text{FAc}^b(\\mathcal{A})$ and this induces an equivalence", "of triangulated categories", "$$", "K^b(\\text{Fil}^f\\mathcal{A})/\\text{FAc}^b(\\mathcal{A}) =", "\\text{FQis}^b(\\mathcal{A})^{-1}K^b(\\text{Fil}^f\\mathcal{A})", "\\longrightarrow", "DF^b(\\mathcal{A})", "$$", "\\end{enumerate}" ], "refs": [ "derived-lemma-operations" ], "proofs": [ { "contents": [ "This follows from the results above, in particular", "Lemma \\ref{lemma-filtered-complex-cohomology-bounded},", "by exactly the same arguments as used in the proof of", "Lemma \\ref{lemma-bounded-derived}." ], "refs": [ "derived-lemma-filtered-complex-cohomology-bounded", "derived-lemma-bounded-derived" ], "ref_ids": [ 1821, 1813 ] } ], "ref_ids": [ 1790 ] }, { "id": 1823, "type": "theorem", "label": "derived-lemma-derived-functor", "categories": [ "derived" ], "title": "derived-lemma-derived-functor", "contents": [ "Assumptions and notation as in", "Situation \\ref{situation-derived-functor}.", "Let $f : X \\to Y$ be a morphism of $\\mathcal{D}$.", "\\begin{enumerate}", "\\item If $RF$ is defined at $X$ and $Y$ then there exists a unique", "morphism $RF(f) : RF(X) \\to RF(Y)$ between the values such that", "for any commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f \\ar[r]_s & X' \\ar[d]^{f'} \\\\", "Y \\ar[r]^{s'} & Y'", "}", "$$", "with $s, s' \\in S$ the diagram", "$$", "\\xymatrix{", "F(X) \\ar[d] \\ar[r] & F(X') \\ar[d] \\ar[r] & RF(X) \\ar[d] \\\\", "F(Y) \\ar[r] & F(Y') \\ar[r] & RF(Y)", "}", "$$", "commutes.", "\\item If $LF$ is defined at $X$ and $Y$ then there exists a unique", "morphism $LF(f) : LF(X) \\to LF(Y)$ between the values such that", "for any commutative diagram", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_s & X \\ar[d]^f \\\\", "Y' \\ar[r]^{s'} & Y", "}", "$$", "with $s, s'$ in $S$ the diagram", "$$", "\\xymatrix{", "LF(X) \\ar[d] \\ar[r] & F(X') \\ar[d] \\ar[r] & F(X) \\ar[d] \\\\", "LF(Y) \\ar[r] & F(Y') \\ar[r] & F(Y)", "}", "$$", "commutes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) holds if we only assume that the colimits", "$$", "RF(X) = \\colim_{s : X \\to X'} F(X')", "\\quad\\text{and}\\quad", "RF(Y) = \\colim_{s' : Y \\to Y'} F(Y')", "$$", "exist. Namely, to give a morphism $RF(X) \\to RF(Y)$ between the colimits", "is the same thing as giving for each $s : X \\to X'$ in $\\Ob(X/S)$", "a morphism $F(X') \\to RF(Y)$ compatible with morphisms in the category", "$X/S$. To get the morphism we choose a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f \\ar[r]_s & X' \\ar[d]^{f'} \\\\", "Y \\ar[r]^{s'} & Y'", "}", "$$", "with $s, s'$ in $S$ as is possible by MS2 and we set", "$F(X') \\to RF(Y)$ equal to the composition $F(X') \\to F(Y') \\to RF(Y)$.", "To see that this is independent of the choice of the diagram above use", "MS3. Details omitted. The proof of (2) is dual." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1824, "type": "theorem", "label": "derived-lemma-derived-inverts", "categories": [ "derived" ], "title": "derived-lemma-derived-inverts", "contents": [ "Assumptions and notation as in", "Situation \\ref{situation-derived-functor}.", "Let $s : X \\to Y$ be an element of $S$.", "\\begin{enumerate}", "\\item $RF$ is defined at $X$ if and only if it is defined at $Y$.", "In this case the map $RF(s) : RF(X) \\to RF(Y)$ between values", "is an isomorphism.", "\\item $LF$ is defined at $X$ if and only if it is defined at $Y$.", "In this case the map $LF(s) : LF(X) \\to LF(Y)$ between values", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1825, "type": "theorem", "label": "derived-lemma-derived-shift", "categories": [ "derived" ], "title": "derived-lemma-derived-shift", "contents": [ "\\begin{slogan}", "Derived functors are compatible with shifts", "\\end{slogan}", "Assumptions and notation as in", "Situation \\ref{situation-derived-functor}.", "Let $X$ be an object of $\\mathcal{D}$ and $n \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item $RF$ is defined at $X$ if and only if it is defined at $X[n]$.", "In this case there is a canonical isomorphism", "$RF(X)[n]= RF(X[n])$ between values.", "\\item $LF$ is defined at $X$ if and only if it is defined at $X[n]$.", "In this case there is a canonical isomorphism", "$LF(X)[n] \\to LF(X[n])$ between values.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1826, "type": "theorem", "label": "derived-lemma-2-out-of-3-defined", "categories": [ "derived" ], "title": "derived-lemma-2-out-of-3-defined", "contents": [ "Assumptions and notation as in", "Situation \\ref{situation-derived-functor}.", "Let $(X, Y, Z, f, g, h)$ be a distinguished triangle of $\\mathcal{D}$.", "If $RF$ is defined at two out of three of $X, Y, Z$, then it is defined", "at the third. Moreover, in this case", "$$", "(RF(X), RF(Y), RF(Z), RF(f), RF(g), RF(h))", "$$", "is a distinguished triangle in $\\mathcal{D}'$. Similarly for $LF$." ], "refs": [], "proofs": [ { "contents": [ "Say $RF$ is defined at $X, Y$ with values $A, B$.", "Let $RF(f) : A \\to B$ be the induced morphism, see", "Lemma \\ref{lemma-derived-functor}.", "We may choose a distinguished triangle", "$(A, B, C, RF(f), b, c)$", "in $\\mathcal{D}'$. We claim that $C$ is a value of $RF$ at $Z$.", "\\medskip\\noindent", "To see this pick $s : X \\to X'$ in $S$ such that there exists a morphism", "$\\alpha : A \\to F(X')$ as in", "Categories,", "Definition \\ref{categories-definition-essentially-constant-diagram}.", "We may choose a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f \\ar[r]_s & X' \\ar[d]^{f'} \\\\", "Y \\ar[r]^{s'} & Y'", "}", "$$", "with $s' \\in S$ by MS2. Using that $Y/S$ is filtered we can (after replacing", "$s'$ by some $s'' : Y \\to Y''$ in $S$) assume that there exists", "a morphism $\\beta : B \\to F(Y')$ as in", "Categories,", "Definition \\ref{categories-definition-essentially-constant-diagram}.", "Picture", "$$", "\\xymatrix{", "A \\ar[d]_{RF(f)} \\ar[r]_-\\alpha &", "F(X') \\ar[r] \\ar[d]^{F(f')} &", "A \\ar[d]^{RF(f)} \\\\", "B \\ar[r]^-\\beta & F(Y') \\ar[r] & B", "}", "$$", "It may not be true that the left square commutes, but the outer and", "right squares commute.", "The assumption that the ind-object $\\{F(Y')\\}_{s' : Y' \\to Y}$", "is essentially constant means that there exists a $s'' : Y \\to Y''$", "in $S$ and a morphism $h : Y' \\to Y''$ such that $s'' = h \\circ s'$ and", "such that $F(h)$ equal to $F(Y') \\to B \\to F(Y') \\to F(Y'')$. Hence", "after replacing $Y'$ by $Y''$ and $\\beta$ by $F(h) \\circ \\beta$ the", "diagram will commute (by direct computation with arrows).", "\\medskip\\noindent", "Using MS6 choose a morphism of triangles", "$$", "(s, s', s'') : (X, Y, Z, f, g, h) \\longrightarrow (X', Y', Z', f', g', h')", "$$", "with $s'' \\in S$. By TR3 choose a morphism of triangles", "$$", "(\\alpha, \\beta, \\gamma) :", "(A, B, C, RF(f), b, c)", "\\longrightarrow", "(F(X'), F(Y'), F(Z'), F(f'), F(g'), F(h'))", "$$", "\\medskip\\noindent", "By", "Lemma \\ref{lemma-derived-inverts}", "it suffices to prove that $RF(Z')$ is defined and has value $C$.", "Consider the category $\\mathcal{I}$ of", "Lemma \\ref{lemma-limit-triangles}", "of triangles", "$$", "\\mathcal{I} =", "\\{(t, t', t'') : (X', Y', Z', f', g', h') \\to (X'', Y'', Z'', f'', g'', h'')", "\\mid (t, t', t'') \\in S\\}", "$$", "To show that the system $F(Z'')$ is essentially constant over the category", "$Z'/S$ is equivalent to showing that the system of $F(Z'')$ is essentially", "constant over $\\mathcal{I}$ because $\\mathcal{I} \\to Z'/S$ is cofinal, see", "Categories, Lemma \\ref{categories-lemma-cofinal-essentially-constant}", "(cofinality is proven in Lemma \\ref{lemma-limit-triangles}).", "For any object $W$ in $\\mathcal{D}'$ we", "consider the diagram", "$$", "\\xymatrix{", "\\colim_\\mathcal{I} \\Mor_{\\mathcal{D}'}(W, F(X'')) &", "\\Mor_{\\mathcal{D}'}(W, A) \\ar[l] \\\\", "\\colim_\\mathcal{I} \\Mor_{\\mathcal{D}'}(W, F(Y'')) \\ar[u] &", "\\Mor_{\\mathcal{D}'}(W, B) \\ar[u] \\ar[l] \\\\", "\\colim_\\mathcal{I} \\Mor_{\\mathcal{D}'}(W, F(Z'')) \\ar[u] &", "\\Mor_{\\mathcal{D}'}(W, C) \\ar[u] \\ar[l] \\\\", "\\colim_\\mathcal{I} \\Mor_{\\mathcal{D}'}(W, F(X''[1])) \\ar[u] &", "\\Mor_{\\mathcal{D}'}(W, A[1]) \\ar[u] \\ar[l] \\\\", "\\colim_\\mathcal{I} \\Mor_{\\mathcal{D}'}(W, F(Y''[1])) \\ar[u] &", "\\Mor_{\\mathcal{D}'}(W, B[1]) \\ar[u] \\ar[l]", "}", "$$", "where the horizontal arrows are given by composing with", "$(\\alpha, \\beta, \\gamma)$. Since filtered colimits are exact", "(Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}) the left column", "is an exact sequence. Thus the $5$ lemma", "(Homology, Lemma \\ref{homology-lemma-five-lemma}) tells us the", "$$", "\\colim_\\mathcal{I} \\Mor_{\\mathcal{D}'}(W, F(Z''))", "\\longrightarrow", "\\Mor_{\\mathcal{D}'}(W, C)", "$$", "is bijective. Choose an object", "$(t, t', t'') : (X', Y', Z') \\to (X'', Y'', Z'')$ of $\\mathcal{I}$.", "Applying what we just showed to $W = F(Z'')$ and the element", "$\\text{id}_{F(X'')}$ of the colimit we find a unique morphism", "$c_{(X'', Y'', Z'')} : F(Z'') \\to C$ such that for some", "$(X'', Y'', Z'') \\to (X''', Y''', Z'')$ in $\\mathcal{I}$", "$$", "F(Z'') \\xrightarrow{c_{(X'', Y'', Z'')}} C \\xrightarrow{\\gamma}", "F(Z') \\to F(Z'') \\to F(Z''')", "\\quad\\text{equals}\\quad", "F(Z'') \\to F(Z''')", "$$", "The family of morphisms $c_{(X'', Y'', Z'')}$ form an element $c$ of", "$\\lim_\\mathcal{I} \\Mor_{\\mathcal{D}'}(F(Z''), C)$ by uniqueness", "(computation omitted). Finally, we show that", "$\\colim_\\mathcal{I} F(Z'') = C$ via the morphisms $c_{(X'', Y'', Z'')}$", "which will finish the proof by ", "Categories, Lemma \\ref{categories-lemma-characterize-essentially-constant-ind}.", "Namely, let $W$ be an object of $\\mathcal{D}'$ and let", "$d_{(X'', Y'', Z'')} : F(Z'') \\to W$ be a family of maps corresponding", "to an element of $\\lim_\\mathcal{I} \\Mor_{\\mathcal{D}'}(F(Z''), W)$.", "If $d_{(X', Y', Z')} \\circ \\gamma = 0$, then for every object", "$(X'', Y'', Z'')$ of $\\mathcal{I}$ the morphism $d_{(X'', Y'', Z'')}$", "is zero by the existence of $c_{(X'', Y'', Z'')}$ and the", "morphism $(X'', Y'', Z'') \\to (X''', Y''', Z'')$ in $\\mathcal{I}$", "satisfying the displayed equality above. Hence the map", "$$", "\\lim_\\mathcal{I} \\Mor_{\\mathcal{D}'}(F(Z''), W)", "\\longrightarrow", "\\Mor_{\\mathcal{D}'}(C, W)", "$$", "(coming from precomposing by $\\gamma$) is injective. However, it is", "also surjective because the element $c$ gives a left inverse. We conclude", "that $C$ is the colimit by", "Categories, Remark \\ref{categories-remark-limit-colim}." ], "refs": [ "derived-lemma-derived-functor", "categories-definition-essentially-constant-diagram", "categories-definition-essentially-constant-diagram", "derived-lemma-derived-inverts", "derived-lemma-limit-triangles", "categories-lemma-cofinal-essentially-constant", "derived-lemma-limit-triangles", "algebra-lemma-directed-colimit-exact", "homology-lemma-five-lemma", "categories-lemma-characterize-essentially-constant-ind", "categories-remark-limit-colim" ], "ref_ids": [ 1823, 12368, 12368, 1824, 1783, 12242, 1783, 343, 12030, 12240, 12415 ] } ], "ref_ids": [] }, { "id": 1827, "type": "theorem", "label": "derived-lemma-direct-sum-defined", "categories": [ "derived" ], "title": "derived-lemma-direct-sum-defined", "contents": [ "Assumptions and notation as in Situation \\ref{situation-derived-functor}.", "Let $X, Y$ be objects of $\\mathcal{D}$.", "\\begin{enumerate}", "\\item If $RF$ is defined at $X$ and $Y$, then $RF$ is defined at $X \\oplus Y$.", "\\item If $\\mathcal{D}'$ is Karoubian and $RF$ is defined at $X \\oplus Y$,", "then $RF$ is defined at both $X$ and $Y$.", "\\end{enumerate}", "In either case we have $RF(X \\oplus Y) = RF(X) \\oplus RF(Y)$.", "Similarly for $LF$." ], "refs": [], "proofs": [ { "contents": [ "If $RF$ is defined at $X$ and $Y$, then the distinguished triangle", "$X \\to X \\oplus Y \\to Y \\to X[1]$ (Lemma \\ref{lemma-split}) and", "Lemma \\ref{lemma-2-out-of-3-defined}", "shows that $RF$ is defined at $X \\oplus Y$ and that we", "have a distinguished triangle", "$RF(X) \\to RF(X \\oplus Y) \\to RF(Y) \\to RF(X)[1]$.", "Applying Lemma \\ref{lemma-split} to this once more we find", "that $RF(X \\oplus Y) = RF(X) \\oplus RF(Y)$.", "This proves (1) and the final assertion.", "\\medskip\\noindent", "Conversely, assume that $RF$ is defined at $X \\oplus Y$ and that $\\mathcal{D}'$", "is Karoubian. Since $S$ is a saturated system $S$ is the set of arrows which", "become invertible under the additive localization functor", "$Q : \\mathcal{D} \\to S^{-1}\\mathcal{D}$, see", "Categories, Lemma \\ref{categories-lemma-what-gets-inverted}.", "Thus for any $s : X \\to X'$ and $s' : Y \\to Y'$ in $S$ the morphism", "$s \\oplus s' : X \\oplus Y \\to X' \\oplus Y'$ is an element of $S$.", "In this way we obtain a functor", "$$", "X/S \\times Y/S \\longrightarrow (X \\oplus Y)/S", "$$", "Recall that the categories $X/S, Y/S, (X \\oplus Y)/S$ are filtered", "(Categories, Remark", "\\ref{categories-remark-left-localization-morphisms-colimit}).", "By Categories, Lemma \\ref{categories-lemma-essentially-constant-over-product}", "$X/S \\times Y/S$ is filtered and", "$F|_{X/S} : X/S \\to \\mathcal{D}'$ (resp.\\ $G|_{Y/S} : Y/S \\to \\mathcal{D}'$)", "is essentially constant if and only if", "$F|_{X/S} \\circ \\text{pr}_1 : X/S \\times Y/S \\to \\mathcal{D}'$", "(resp.\\ $G|_{Y/S} \\circ \\text{pr}_2 : X/S \\times Y/S \\to \\mathcal{D}'$)", "is essentially constant. Below we will show that the displayed functor", "is cofinal, hence by", "Categories, Lemma \\ref{categories-lemma-cofinal-essentially-constant}.", "we see that $F|_{(X \\oplus Y)/S}$ is essentially constant implies that", "$F|_{X/S} \\circ \\text{pr}_1 \\oplus F|_{Y/S} \\circ \\text{pr}_2 :", "X/S \\times Y/S \\to \\mathcal{D}'$", "is essentially constant. By Homology, Lemma", "\\ref{homology-lemma-direct-sum-from-product-essentially-constant}", "(and this is where we use that $\\mathcal{D}'$ is Karoubian)", "we see that ", "$F|_{X/S} \\circ \\text{pr}_1 \\oplus F|_{Y/S} \\circ \\text{pr}_2$", "being essentially constant implies", "$F|_{X/S} \\circ \\text{pr}_1$ and", "$F|_{Y/S} \\circ \\text{pr}_2$ are essentially constant proving that $RF$ is", "defined at $X$ and $Y$.", "\\medskip\\noindent", "Proof that the displayed functor is cofinal.", "To do this pick any $t : X \\oplus Y \\to Z$ in $S$.", "Using MS2 we can find morphisms $Z \\to X'$, $Z \\to Y'$", "and $s : X \\to X'$, $s' : Y \\to Y'$ in $S$ such that", "$$", "\\xymatrix{", "X \\ar[d]^s & X \\oplus Y \\ar[d] \\ar[l] \\ar[r] & Y \\ar[d]_{s'} \\\\", "X' & Z \\ar[l] \\ar[r] & Y'", "}", "$$", "commutes. This proves there is a map $Z \\to X' \\oplus Y'$ in", "$(X \\oplus Y)/S$, i.e., we get part (1) of Categories, Definition", "\\ref{categories-definition-cofinal}. To prove part (2) it suffices", "to prove that given $t : X \\oplus Y \\to Z$ and morphisms", "$s_i \\oplus s'_i : Z \\to X'_i \\oplus Y'_i$, $i = 1, 2$ in $(X \\oplus Y)/S$", "we can find morphisms $a : X'_1 \\to X'$, $b : X'_2 \\to X'$,", "$c : Y'_1 \\to Y'$, $d : Y'_2 \\to Y'$ in $S$ such that", "$a \\circ s_1 = b \\circ s_2$ and $c \\circ s'_1 = d \\circ s'_2$.", "To do this we first choose any $X'$ and $Y'$ and maps $a, b, c, d$", "in $S$; this is possible as $X/S$ and $Y/S$ are filtered. Then the", "two maps $a \\circ s_1, b \\circ s_2 : Z \\to X'$ become equal in", "$S^{-1}\\mathcal{D}$. Hence we can find a morphism", "$X' \\to X''$ in $S$ equalizing them. Similarly we find $Y' \\to Y''$ in $S$", "equalizing $c \\circ s'_1$ and $d \\circ s'_2$. Replacing $X'$ by $X''$ and", "$Y'$ by $Y''$ we get $a \\circ s_1 = b \\circ s_2$ and", "$c \\circ s'_1 = d \\circ s'_2$.", "\\medskip\\noindent", "The proof of the corresponding statements for $LF$ are dual." ], "refs": [ "derived-lemma-split", "derived-lemma-2-out-of-3-defined", "derived-lemma-split", "categories-lemma-what-gets-inverted", "categories-remark-left-localization-morphisms-colimit", "categories-lemma-essentially-constant-over-product", "categories-lemma-cofinal-essentially-constant", "homology-lemma-direct-sum-from-product-essentially-constant", "categories-definition-cofinal" ], "ref_ids": [ 1766, 1826, 1766, 12268, 12424, 12243, 12242, 12122, 12361 ] } ], "ref_ids": [] }, { "id": 1828, "type": "theorem", "label": "derived-lemma-computes-shift", "categories": [ "derived" ], "title": "derived-lemma-computes-shift", "contents": [ "Assumptions and notation as in", "Situation \\ref{situation-derived-functor}.", "Let $X$ be an object of $\\mathcal{D}$ and $n \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item $X$ computes $RF$ if and only if $X[n]$ computes $RF$.", "\\item $X$ computes $LF$ if and only if $X[n]$ computes $LF$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1829, "type": "theorem", "label": "derived-lemma-2-out-of-3-computes", "categories": [ "derived" ], "title": "derived-lemma-2-out-of-3-computes", "contents": [ "Assumptions and notation as in", "Situation \\ref{situation-derived-functor}.", "Let $(X, Y, Z, f, g, h)$ be a distinguished triangle of $\\mathcal{D}$.", "If $X, Y$ compute $RF$ then so does $Z$. Similar for $LF$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-2-out-of-3-defined}", "we know that $RF$ is defined at $Z$ and that $RF$ applied to the", "triangle produces a distinguished triangle.", "Consider the morphism of distinguished triangles", "$$", "\\xymatrix{", "(F(X), F(Y), F(Z), F(f), F(g), F(h)) \\ar[d] \\\\", "(RF(X), RF(Y), RF(Z), RF(f), RF(g), RF(h))", "}", "$$", "Two out of three maps are isomorphisms, hence so is the third." ], "refs": [ "derived-lemma-2-out-of-3-defined" ], "ref_ids": [ 1826 ] } ], "ref_ids": [] }, { "id": 1830, "type": "theorem", "label": "derived-lemma-direct-sum-computes", "categories": [ "derived" ], "title": "derived-lemma-direct-sum-computes", "contents": [ "Assumptions and notation as in Situation \\ref{situation-derived-functor}.", "Let $X, Y$ be objects of $\\mathcal{D}$. If $X \\oplus Y$ computes $RF$, then", "$X$ and $Y$ compute $RF$. Similarly for $LF$." ], "refs": [], "proofs": [ { "contents": [ "If $X \\oplus Y$ computes $RF$, then $RF(X \\oplus Y) = F(X) \\oplus F(Y)$.", "In the proof of Lemma \\ref{lemma-direct-sum-defined} we have seen that", "the functor $X/S \\times Y/S \\to (X \\oplus Y)/S$, $(s, s') \\mapsto s \\oplus s'$", "is cofinal. We will use this without further mention. Let $s : X \\to X'$ be", "an element of $S$. Then $F(X) \\to F(X')$ has a section, namely,", "$$", "F(X') \\to F(X' \\oplus Y) \\to RF(X' \\oplus Y) =", "RF(X \\oplus Y) = F(X) \\oplus F(Y) \\to F(X).", "$$", "where we have used Lemma \\ref{lemma-derived-inverts}.", "Hence $F(X') = F(X) \\oplus E$ for some object $E$ of $\\mathcal{D}'$", "such that $E \\to F(X' \\oplus Y) \\to RF(X'\\oplus Y) = RF(X \\oplus Y)$", "is zero (Lemma \\ref{lemma-when-split}).", "Because $RF$ is defined at $X' \\oplus Y$ with value", "$F(X) \\oplus F(Y)$ we can find a morphism $t : X' \\oplus Y \\to Z$", "of $S$ such that $F(t)$ annihilates $E$. We may assume", "$Z = X'' \\oplus Y''$ and $t = t' \\oplus t''$ with $t', t'' \\in S$.", "Then $F(t')$ annihilates $E$. It follows that $F$ is essentially constant", "on $X/S$ with value $F(X)$ as desired." ], "refs": [ "derived-lemma-direct-sum-defined", "derived-lemma-derived-inverts", "derived-lemma-when-split" ], "ref_ids": [ 1827, 1824, 1767 ] } ], "ref_ids": [] }, { "id": 1831, "type": "theorem", "label": "derived-lemma-existence-computes", "categories": [ "derived" ], "title": "derived-lemma-existence-computes", "contents": [ "Assumptions and notation as in", "Situation \\ref{situation-derived-functor}.", "\\begin{enumerate}", "\\item If for every object $X \\in \\Ob(\\mathcal{D})$", "there exists an arrow $s : X \\to X'$ in $S$ such that $X'$ computes", "$RF$, then $RF$ is everywhere defined.", "\\item If for every object $X \\in \\Ob(\\mathcal{D})$", "there exists an arrow $s : X' \\to X$ in $S$ such that $X'$ computes", "$LF$, then $LF$ is everywhere defined.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1832, "type": "theorem", "label": "derived-lemma-find-existence-computes", "categories": [ "derived" ], "title": "derived-lemma-find-existence-computes", "contents": [ "Assumptions and notation as in", "Situation \\ref{situation-derived-functor}.", "If there exists a subset $\\mathcal{I} \\subset \\Ob(\\mathcal{D})$", "such that", "\\begin{enumerate}", "\\item for all $X \\in \\Ob(\\mathcal{D})$", "there exists $s : X \\to X'$ in $S$ with $X' \\in \\mathcal{I}$,", "and", "\\item for every arrow $s : X \\to X'$ in $S$ with $X, X' \\in \\mathcal{I}$", "the map $F(s) : F(X) \\to F(X')$ is an isomorphism,", "\\end{enumerate}", "then $RF$ is everywhere defined and every $X \\in \\mathcal{I}$", "computes $RF$. Dually, if there exists a subset", "$\\mathcal{P} \\subset \\Ob(\\mathcal{D})$", "such that", "\\begin{enumerate}", "\\item for all $X \\in \\Ob(\\mathcal{D})$", "there exists $s : X' \\to X$ in $S$ with $X' \\in \\mathcal{P}$,", "and", "\\item for every arrow $s : X \\to X'$ in $S$ with $X, X' \\in \\mathcal{P}$", "the map $F(s) : F(X) \\to F(X')$ is an isomorphism,", "\\end{enumerate}", "then $LF$ is everywhere defined and every $X \\in \\mathcal{P}$", "computes $LF$." ], "refs": [], "proofs": [ { "contents": [ "Let $X$ be an object of $\\mathcal{D}$.", "Assumption (1) implies that the arrows $s : X \\to X'$ in $S$ with", "$X' \\in \\mathcal{I}$ are cofinal in the category $X/S$. Assumption", "(2) implies that $F$ is constant on this cofinal subcategory.", "Clearly this implies that $F : (X/S) \\to \\mathcal{D}'$ is essentially", "constant with value $F(X')$ for any $s : X \\to X'$ in $S$", "with $X' \\in \\mathcal{I}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1833, "type": "theorem", "label": "derived-lemma-compose-derived-functors-general", "categories": [ "derived" ], "title": "derived-lemma-compose-derived-functors-general", "contents": [ "Let $\\mathcal{A}, \\mathcal{B}, \\mathcal{C}$ be triangulated categories.", "Let $S$, resp.\\ $S'$ be a saturated multiplicative system in", "$\\mathcal{A}$, resp.\\ $\\mathcal{B}$ compatible with the triangulated structure.", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{C}$", "be exact functors. Denote $F' : \\mathcal{A} \\to (S')^{-1}\\mathcal{B}$ the", "composition of $F$ with the localization functor.", "\\begin{enumerate}", "\\item If $RF'$, $RG$, $R(G \\circ F)$ are everywhere defined, then there", "is a canonical transformation of functors", "$t : R(G \\circ F) \\longrightarrow RG \\circ RF'$.", "\\item If $LF'$, $LG$, $L(G \\circ F)$ are everywhere defined, then there", "is a canonical transformation of functors", "$t : LG \\circ LF' \\to L(G \\circ F)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In this proof we try to be careful. Hence let us think of", "the derived functors as the functors", "$$", "RF' : S^{-1}\\mathcal{A} \\to (S')^{-1}\\mathcal{B}, \\quad", "R(G \\circ F) : S^{-1}\\mathcal{A} \\to \\mathcal{C}, \\quad", "RG : (S')^{-1}\\mathcal{B} \\to \\mathcal{C}.", "$$", "Let us denote", "$Q_A : \\mathcal{A} \\to S^{-1}\\mathcal{A}$ and", "$Q_B : \\mathcal{B} \\to (S')^{-1}\\mathcal{B}$", "the localization functors. Then $F' = Q_B \\circ F$. Note that for", "every object $Y$ of $\\mathcal{B}$ there is a canonical map", "$$", "G(Y) \\longrightarrow RG(Q_B(Y))", "$$", "in other words, there is a transformation of functors", "$t' : G \\to RG \\circ Q_B$. Let $X$ be an object of $\\mathcal{A}$.", "We have", "\\begin{align*}", "R(G \\circ F)(Q_A(X))", "& = \\colim_{s : X \\to X' \\in S} G(F(X')) \\\\", "& \\xrightarrow{t'} \\colim_{s : X \\to X' \\in S} RG(Q_B(F(X'))) \\\\", "& = \\colim_{s : X \\to X' \\in S} RG(F'(X')) \\\\", "& = RG(\\colim_{s : X \\to X' \\in S} F'(X')) \\\\", "& = RG(RF'(X)).", "\\end{align*}", "The system $F'(X')$ is essentially constant in the category", "$(S')^{-1}\\mathcal{B}$. Hence we may pull the colimit inside the", "functor $RG$ in the third equality of the diagram above, see", "Categories, Lemma \\ref{categories-lemma-image-essentially-constant}", "and its proof. We omit the proof this defines a transformation", "of functors. The case of left derived functors is similar." ], "refs": [ "categories-lemma-image-essentially-constant" ], "ref_ids": [ 12239 ] } ], "ref_ids": [] }, { "id": 1834, "type": "theorem", "label": "derived-lemma-irrelevant", "categories": [ "derived" ], "title": "derived-lemma-irrelevant", "contents": [ "In", "Situation \\ref{situation-classical}.", "\\begin{enumerate}", "\\item Let $X$ be an object of $K^{+}(\\mathcal{A})$.", "The right derived functor of $K(\\mathcal{A}) \\to D(\\mathcal{B})$", "is defined at $X$ if and only if the right derived functor of", "$K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is defined at $X$.", "Moreover, the values are canonically isomorphic.", "\\item Let $X$ be an object of $K^{+}(\\mathcal{A})$.", "Then $X$ computes the right derived functor of", "$K(\\mathcal{A}) \\to D(\\mathcal{B})$", "if and only if $X$ computes the right derived functor of", "$K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$.", "\\item Let $X$ be an object of $K^{-}(\\mathcal{A})$.", "The left derived functor of $K(\\mathcal{A}) \\to D(\\mathcal{B})$", "is defined at $X$ if and only if the left derived functor of", "$K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{B})$ is defined at $X$.", "Moreover, the values are canonically isomorphic.", "\\item Let $X$ be an object of $K^{-}(\\mathcal{A})$.", "Then $X$ computes the left derived functor of", "$K(\\mathcal{A}) \\to D(\\mathcal{B})$ if and only if $X$ computes", "the left derived functor of $K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{B})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X$ be an object of $K^{+}(\\mathcal{A})$.", "Consider a quasi-isomorphism $s : X \\to X'$ in $K(\\mathcal{A})$.", "By", "Lemma \\ref{lemma-complex-cohomology-bounded}", "there exists quasi-isomorphism $X' \\to X''$ with $X''$ bounded below.", "Hence we see that $X/\\text{Qis}^+(\\mathcal{A})$ is cofinal", "in $X/\\text{Qis}(\\mathcal{A})$. Thus it is clear that (1) holds.", "Part (2) follows directly from part (1).", "Parts (3) and (4) are dual to parts (1) and (2)." ], "refs": [ "derived-lemma-complex-cohomology-bounded" ], "ref_ids": [ 1812 ] } ], "ref_ids": [] }, { "id": 1835, "type": "theorem", "label": "derived-lemma-subcategory-left-resolution", "categories": [ "derived" ], "title": "derived-lemma-subcategory-left-resolution", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let", "$\\mathcal{P} \\subset \\Ob(\\mathcal{A})$ be a subset containing $0$", "such that every object of $\\mathcal{A}$ is a quotient of an element of", "$\\mathcal{P}$. Let $a \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item Given $K^\\bullet$ with $K^n = 0$ for $n > a$", "there exists a quasi-isomorphism $P^\\bullet \\to K^\\bullet$", "with $P^n \\in \\mathcal{P}$ and $P^n \\to K^n$ surjective", "for all $n$ and $P^n = 0$ for $n > a$.", "\\item Given $K^\\bullet$ with $H^n(K^\\bullet) = 0$ for $n > a$", "there exists a quasi-isomorphism $P^\\bullet \\to K^\\bullet$", "with $P^n \\in \\mathcal{P}$ for all $n$ and $P^n = 0$ for $n > a$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of part (1). Consider the following induction hypothesis $IH_n$:", "There are $P^j \\in \\mathcal{P}$, $j \\geq n$, with $P^j = 0$ for $j > a$,", "maps $d^j : P^j \\to P^{j + 1}$ for $j \\geq n$, and surjective maps", "$\\alpha^j : P^j \\to K^j$ for $j \\geq n$ such that the diagram", "$$", "\\xymatrix{", "& &", "P^n \\ar[d]^\\alpha \\ar[r] &", "P^{n + 1} \\ar[d]^\\alpha \\ar[r] &", "P^{n + 2} \\ar[d]^\\alpha \\ar[r] &", "\\ldots \\\\", "\\ldots \\ar[r] &", "K^{n - 1} \\ar[r] &", "K^n \\ar[r] &", "K^{n + 1} \\ar[r] &", "K^{n + 2} \\ar[r] &", "\\ldots", "}", "$$", "is commutative, such that $d^{j + 1} \\circ d^j = 0$ for $j \\geq n$,", "such that $\\alpha$ induces isomorphisms", "$H^j(K^\\bullet) \\to \\Ker(d^j)/\\Im(d^{j - 1})$", "for $j > n$, and such that $\\alpha : \\Ker(d^n) \\to \\Ker(d_K^n)$", "is surjective. Then we choose a surjection", "$$", "P^{n - 1}", "\\longrightarrow", "K^{n - 1} \\times_{K^n} \\Ker(d^n) =", "K^{n - 1} \\times_{\\Ker(d_K^n)} \\Ker(d^n)", "$$", "with $P^{n - 1}$ in $\\mathcal{P}$. This allows us to extend the diagram", "above to", "$$", "\\xymatrix{", "& P^{n - 1} \\ar[d]^\\alpha \\ar[r] &", "P^n \\ar[d]^\\alpha \\ar[r] &", "P^{n + 1} \\ar[d]^\\alpha \\ar[r] &", "P^{n + 2} \\ar[d]^\\alpha \\ar[r] &", "\\ldots \\\\", "\\ldots \\ar[r] &", "K^{n - 1} \\ar[r] &", "K^n \\ar[r] &", "K^{n + 1} \\ar[r] &", "K^{n + 2} \\ar[r] &", "\\ldots", "}", "$$", "The reader easily checks that $IH_{n - 1}$ holds with this choice.", "\\medskip\\noindent", "We finish the proof of (1) as follows.", "First we note that $IH_n$ is true for $n = a + 1$ since", "we can just take $P^j = 0$ for $j > a$. Hence we see that", "proceeding by descending induction we produce a complex $P^\\bullet$", "with $P^n = 0$ for $n > a$", "consisting of objects from $\\mathcal{P}$, and a termwise", "surjective quasi-isomorphism $\\alpha : P^\\bullet \\to K^\\bullet$ as desired.", "\\medskip\\noindent", "Proof of part (2). The assumption implies that the morphism", "$\\tau_{\\leq a}K^\\bullet \\to K^\\bullet$", "(Homology, Section \\ref{homology-section-truncations})", "is a quasi-isomorphism.", "Apply part (1) to find $P^\\bullet \\to \\tau_{\\leq a}K^\\bullet$.", "The composition $P^\\bullet \\to K^\\bullet$ is the desired quasi-isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1836, "type": "theorem", "label": "derived-lemma-subcategory-right-resolution", "categories": [ "derived" ], "title": "derived-lemma-subcategory-right-resolution", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let", "$\\mathcal{I} \\subset \\Ob(\\mathcal{A})$ be a subset containing $0$", "such that every object of $\\mathcal{A}$ is a subobject of an element of", "$\\mathcal{I}$. Let $a \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item Given $K^\\bullet$ with $K^n = 0$ for $n < a$", "there exists a quasi-isomorphism $K^\\bullet \\to I^\\bullet$", "with $K^n \\to I^n$ injective and $I^n \\in \\mathcal{I}$ for all $n$", "and $I^n = 0$ for $n < a$,", "\\item Given $K^\\bullet$ with $H^n(K^\\bullet) = 0$", "for $n < a$ there exists a quasi-isomorphism $K^\\bullet \\to I^\\bullet$", "with $I^n \\in \\mathcal{I}$ and $I^n = 0$ for $n < a$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma is dual to Lemma \\ref{lemma-subcategory-left-resolution}." ], "refs": [ "derived-lemma-subcategory-left-resolution" ], "ref_ids": [ 1835 ] } ], "ref_ids": [] }, { "id": 1837, "type": "theorem", "label": "derived-lemma-subcategory-right-acyclics", "categories": [ "derived" ], "title": "derived-lemma-subcategory-right-acyclics", "contents": [ "In", "Situation \\ref{situation-classical}.", "Let $\\mathcal{I} \\subset \\Ob(\\mathcal{A})$ be a subset with the", "following properties:", "\\begin{enumerate}", "\\item every object of $\\mathcal{A}$ is a subobject of an element of", "$\\mathcal{I}$,", "\\item for any short exact sequence $0 \\to P \\to Q \\to R \\to 0$ of", "$\\mathcal{A}$ with $P, Q \\in \\mathcal{I}$, then $R \\in \\mathcal{I}$,", "and $0 \\to F(P) \\to F(Q) \\to F(R) \\to 0$ is exact.", "\\end{enumerate}", "Then every object of $\\mathcal{I}$ is acyclic for $RF$." ], "refs": [], "proofs": [ { "contents": [ "We may add $0$ to $\\mathcal{I}$ if necessary. Pick $A \\in \\mathcal{I}$.", "Let $A[0] \\to K^\\bullet$ be a quasi-isomorphism with $K^\\bullet$", "bounded below. Then we can find a quasi-isomorphism", "$K^\\bullet \\to I^\\bullet$ with $I^\\bullet$ bounded below and", "each $I^n \\in \\mathcal{I}$, see", "Lemma \\ref{lemma-subcategory-right-resolution}.", "Hence we see that these resolutions are cofinal in the category", "$A[0]/\\text{Qis}^{+}(\\mathcal{A})$. To finish the proof it therefore", "suffices to show that for any quasi-isomorphism", "$A[0] \\to I^\\bullet$ with $I^\\bullet$ bounded above and $I^n \\in \\mathcal{I}$", "we have $F(A)[0] \\to F(I^\\bullet)$ is a quasi-isomorphism.", "To see this suppose that $I^n = 0$ for $n < n_0$. Of course we may assume", "that $n_0 < 0$. Starting with $n = n_0$ we prove inductively that", "$\\Im(d^{n - 1}) = \\Ker(d^n)$ and $\\Im(d^{-1})$", "are elements of $\\mathcal{I}$ using property (2) and the exact sequences", "$$", "0 \\to \\Ker(d^n) \\to I^n \\to \\Im(d^n) \\to 0.", "$$", "Moreover, property (2) also guarantees that the complex", "$$", "0 \\to F(I^{n_0}) \\to F(I^{n_0 + 1}) \\to \\ldots \\to F(I^{-1}) \\to", "F(\\Im(d^{-1})) \\to 0", "$$", "is exact. The exact sequence", "$0 \\to \\Im(d^{-1}) \\to I^0 \\to I^0/\\Im(d^{-1}) \\to 0$", "implies that $I^0/\\Im(d^{-1})$ is an element of $\\mathcal{I}$.", "The exact sequence $0 \\to A \\to I^0/\\Im(d^{-1}) \\to \\Im(d^0) \\to 0$", "then implies that $\\Im(d^0) = \\Ker(d^1)$ is an elements of", "$\\mathcal{I}$ and from then on one continues as before to show that", "$\\Im(d^{n - 1}) = \\Ker(d^n)$ is an element of $\\mathcal{I}$", "for all $n > 0$. Applying $F$ to each of the short exact sequences", "mentioned above and using (2) we observe that $F(A)[0] \\to F(I^\\bullet)$", "is an isomorphism as desired." ], "refs": [ "derived-lemma-subcategory-right-resolution" ], "ref_ids": [ 1836 ] } ], "ref_ids": [] }, { "id": 1838, "type": "theorem", "label": "derived-lemma-subcategory-left-acyclics", "categories": [ "derived" ], "title": "derived-lemma-subcategory-left-acyclics", "contents": [ "In", "Situation \\ref{situation-classical}.", "Let $\\mathcal{P} \\subset \\Ob(\\mathcal{A})$ be a subset with the", "following properties:", "\\begin{enumerate}", "\\item every object of $\\mathcal{A}$ is a quotient of an element of", "$\\mathcal{P}$,", "\\item for any short exact sequence $0 \\to P \\to Q \\to R \\to 0$ of", "$\\mathcal{A}$ with $Q, R \\in \\mathcal{P}$, then $P \\in \\mathcal{P}$,", "and $0 \\to F(P) \\to F(Q) \\to F(R) \\to 0$ is exact.", "\\end{enumerate}", "Then every object of $\\mathcal{P}$ is acyclic for $LF$." ], "refs": [], "proofs": [ { "contents": [ "Dual to the proof of", "Lemma \\ref{lemma-subcategory-right-acyclics}." ], "refs": [ "derived-lemma-subcategory-right-acyclics" ], "ref_ids": [ 1837 ] } ], "ref_ids": [] }, { "id": 1839, "type": "theorem", "label": "derived-lemma-negative-vanishing", "categories": [ "derived" ], "title": "derived-lemma-negative-vanishing", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor", "between abelian categories. Let $K^\\bullet$ be a complex of $\\mathcal{A}$", "and $a \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item If $H^i(K^\\bullet) = 0$ for all $i < a$ and $RF$ is defined at", "$K^\\bullet$, then $H^i(RF(K^\\bullet)) = 0$ for all $i < a$.", "\\item If $RF$ is defined at $K^\\bullet$ and $\\tau_{\\leq a}K^\\bullet$,", "then $H^i(RF(\\tau_{\\leq a}K^\\bullet)) = H^i(RF(K^\\bullet))$", "for all $i \\leq a$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $K^\\bullet$ satisfies the assumptions of (1).", "Let $K^\\bullet \\to L^\\bullet$ be any quasi-isomorphism.", "Then it is also true that $K^\\bullet \\to \\tau_{\\geq a}L^\\bullet$", "is a quasi-isomorphism by our assumption on $K^\\bullet$.", "Hence in the category $K^\\bullet/\\text{Qis}^{+}(\\mathcal{A})$ the", "quasi-isomorphisms $s : K^\\bullet \\to L^\\bullet$ with $L^n = 0$ for $n < a$", "are cofinal. Thus $RF$ is the value of the essentially constant", "ind-object $F(L^\\bullet)$ for these $s$ it follows that", "$H^i(RF(K^\\bullet)) = 0$ for $i < 0$.", "\\medskip\\noindent", "To prove (2) we use the distinguished triangle", "$$", "\\tau_{\\leq a}K^\\bullet \\to K^\\bullet \\to \\tau_{\\geq a + 1}K^\\bullet", "\\to (\\tau_{\\leq a}K^\\bullet)[1]", "$$", "of Remark \\ref{remark-truncation-distinguished-triangle} to conclude", "via Lemma \\ref{lemma-2-out-of-3-defined} that", "$RF$ is defined at $\\tau_{\\geq a + 1}K^\\bullet$ as well and that we have", "a distinguished triangle", "$$", "RF(\\tau_{\\leq a}K^\\bullet) \\to RF(K^\\bullet) \\to RF(\\tau_{\\geq a + 1}K^\\bullet)", "\\to RF(\\tau_{\\leq a}K^\\bullet)[1]", "$$", "in $D(\\mathcal{B})$. By part (1) we see that $RF(\\tau_{\\geq a + 1}K^\\bullet)$", "has vanishing cohomology in degrees $< a + 1$. The long exact cohomology", "sequence of this distinguished triangle then shows what we want." ], "refs": [ "derived-remark-truncation-distinguished-triangle", "derived-lemma-2-out-of-3-defined" ], "ref_ids": [ 2016, 1826 ] } ], "ref_ids": [] }, { "id": 1840, "type": "theorem", "label": "derived-lemma-left-exact-higher-derived", "categories": [ "derived" ], "title": "derived-lemma-left-exact-higher-derived", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor", "between abelian categories and assume", "$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere", "defined.", "\\begin{enumerate}", "\\item We have $R^iF = 0$ for $i < 0$,", "\\item $R^0F$ is left exact,", "\\item the map $F \\to R^0F$ is an isomorphism if and", "only if $F$ is left exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $A$ be an object of $\\mathcal{A}$. Let $A[0] \\to K^\\bullet$", "be any quasi-isomorphism. Then it is also true that", "$A[0] \\to \\tau_{\\geq 0}K^\\bullet$ is a quasi-isomorphism.", "Hence in the category $A[0]/\\text{Qis}^{+}(\\mathcal{A})$ the", "quasi-isomorphisms $s : A[0] \\to K^\\bullet$ with $K^n = 0$ for $n < 0$", "are cofinal. Thus it is clear that $H^i(RF(A[0])) = 0$ for $i < 0$.", "Moreover, for such an $s$ the sequence", "$$", "0 \\to A \\to K^0 \\to K^1", "$$", "is exact. Hence if $F$ is left exact, then $0 \\to F(A) \\to F(K^0) \\to F(K^1)$", "is exact as well, and we see that $F(A) \\to H^0(F(K^\\bullet))$ is an", "isomorphism for every $s : A[0] \\to K^\\bullet$ as above which implies", "that $H^0(RF(A[0])) = F(A)$.", "\\medskip\\noindent", "Let $0 \\to A \\to B \\to C \\to 0$ be a short exact sequence of $\\mathcal{A}$.", "By", "Lemma \\ref{lemma-derived-canonical-delta-functor}", "we obtain a distinguished triangle", "$(A[0], B[0], C[0], a, b, c)$ in $K^{+}(\\mathcal{A})$.", "From the long exact cohomology sequence (and the vanishing for $i < 0$", "proved above) we deduce that $0 \\to R^0F(A) \\to R^0F(B) \\to R^0F(C)$", "is exact. Hence $R^0F$ is left exact. Of course this also proves that if", "$F \\to R^0F$ is an isomorphism, then $F$ is left exact." ], "refs": [ "derived-lemma-derived-canonical-delta-functor" ], "ref_ids": [ 1814 ] } ], "ref_ids": [] }, { "id": 1841, "type": "theorem", "label": "derived-lemma-F-acyclic", "categories": [ "derived" ], "title": "derived-lemma-F-acyclic", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor", "between abelian categories and assume", "$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere", "defined. Let $A$ be an object of $\\mathcal{A}$.", "\\begin{enumerate}", "\\item $A$ is right acyclic for $F$ if and only if", "$F(A) \\to R^0F(A)$ is an isomorphism and $R^iF(A) = 0$ for all $i > 0$,", "\\item if $F$ is left exact, then $A$ is right acyclic for $F$", "if and only if $R^iF(A) = 0$ for all $i > 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $A$ is right acyclic for $F$, then $RF(A[0]) = F(A)[0]$ and in", "particular $F(A) \\to R^0F(A)$ is an isomorphism and", "$R^iF(A) = 0$ for $i \\not = 0$. Conversely, if $F(A) \\to R^0F(A)$", "is an isomorphism and $R^iF(A) = 0$ for all $i > 0$ then", "$F(A[0]) \\to RF(A[0])$ is a quasi-isomorphism by", "Lemma \\ref{lemma-left-exact-higher-derived} part (1)", "and hence $A$ is acyclic. If $F$ is left exact then $F = R^0F$, see", "Lemma \\ref{lemma-left-exact-higher-derived}." ], "refs": [ "derived-lemma-left-exact-higher-derived", "derived-lemma-left-exact-higher-derived" ], "ref_ids": [ 1840, 1840 ] } ], "ref_ids": [] }, { "id": 1842, "type": "theorem", "label": "derived-lemma-F-acyclic-ses", "categories": [ "derived" ], "title": "derived-lemma-F-acyclic-ses", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor", "between abelian categories and assume", "$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere", "defined. Let $0 \\to A \\to B \\to C \\to 0$ be a short exact sequence", "of $\\mathcal{A}$.", "\\begin{enumerate}", "\\item If $A$ and $C$ are right acyclic for $F$ then so is $B$.", "\\item If $A$ and $B$ are right acyclic for $F$ then so is $C$.", "\\item If $B$ and $C$ are right acyclic for $F$ and $F(B) \\to F(C)$ is", "surjective then $A$ is right acyclic for $F$.", "\\end{enumerate}", "In each of the three cases", "$$", "0 \\to F(A) \\to F(B) \\to F(C) \\to 0", "$$", "is a short exact sequence of $\\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-derived-canonical-delta-functor}", "we obtain a distinguished triangle", "$(A[0], B[0], C[0], a, b, c)$ in $K^{+}(\\mathcal{A})$.", "As $RF$ is an exact functor and since", "$R^iF = 0$ for $i < 0$ and $R^0F = F$", "(Lemma \\ref{lemma-left-exact-higher-derived})", "we obtain an exact cohomology sequence", "$$", "0 \\to F(A) \\to F(B) \\to F(C) \\to R^1F(A) \\to \\ldots", "$$", "in the abelian category $\\mathcal{B}$. Thus the lemma follows from", "the characterization of acyclic objects in", "Lemma \\ref{lemma-F-acyclic}." ], "refs": [ "derived-lemma-derived-canonical-delta-functor", "derived-lemma-left-exact-higher-derived", "derived-lemma-F-acyclic" ], "ref_ids": [ 1814, 1840, 1841 ] } ], "ref_ids": [] }, { "id": 1843, "type": "theorem", "label": "derived-lemma-right-derived-delta-functor", "categories": [ "derived" ], "title": "derived-lemma-right-derived-delta-functor", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor", "between abelian categories and assume", "$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined.", "\\begin{enumerate}", "\\item The functors $R^iF$, $i \\geq 0$ come equipped with a canonical", "structure of a $\\delta$-functor from $\\mathcal{A} \\to \\mathcal{B}$, see", "Homology, Definition \\ref{homology-definition-cohomological-delta-functor}.", "\\item If every object of $\\mathcal{A}$ is a subobject of a right", "acyclic object for $F$, then $\\{R^iF, \\delta\\}_{i \\geq 0}$ is a", "universal $\\delta$-functor, see", "Homology, Definition \\ref{homology-definition-universal-delta-functor}.", "\\end{enumerate}" ], "refs": [ "homology-definition-cohomological-delta-functor", "homology-definition-universal-delta-functor" ], "proofs": [ { "contents": [ "The functor $\\mathcal{A} \\to \\text{Comp}^{+}(\\mathcal{A})$,", "$A \\mapsto A[0]$ is exact. The functor", "$\\text{Comp}^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{A})$", "is a $\\delta$-functor, see", "Lemma \\ref{lemma-derived-canonical-delta-functor}.", "The functor $RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is exact.", "Finally, the functor $H^0 : D^{+}(\\mathcal{B}) \\to \\mathcal{B}$", "is a homological functor, see", "Definition \\ref{definition-unbounded-derived-category}.", "Hence we get the structure of a $\\delta$-functor from", "Lemma \\ref{lemma-compose-delta-functor-homological}", "and", "Lemma \\ref{lemma-exact-compose-delta-functor}.", "Part (2) follows from", "Homology, Lemma \\ref{homology-lemma-efface-implies-universal}", "and the description of acyclics in", "Lemma \\ref{lemma-F-acyclic}." ], "refs": [ "derived-lemma-derived-canonical-delta-functor", "derived-definition-unbounded-derived-category", "derived-lemma-compose-delta-functor-homological", "derived-lemma-exact-compose-delta-functor", "homology-lemma-efface-implies-universal", "derived-lemma-F-acyclic" ], "ref_ids": [ 1814, 1982, 1777, 1776, 12052, 1841 ] } ], "ref_ids": [ 12149, 12151 ] }, { "id": 1844, "type": "theorem", "label": "derived-lemma-leray-acyclicity", "categories": [ "derived" ], "title": "derived-lemma-leray-acyclicity", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor", "between abelian categories and assume", "$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined.", "Let $A^\\bullet$ be a bounded below complex of $F$-acyclic objects.", "The canonical map", "$$", "F(A^\\bullet) \\longrightarrow RF(A^\\bullet)", "$$", "is an isomorphism in $D^{+}(\\mathcal{B})$, i.e., $A^\\bullet$ computes", "$RF$." ], "refs": [], "proofs": [ { "contents": [ "First we claim the lemma holds for a bounded complex of acyclic objects.", "Namely, it holds for complexes with at most one nonzero object by definition.", "Suppose that $A^\\bullet$ is a complex with $A^n = 0$ for", "$n \\not \\in [a, b]$. Using the ``stupid'' truncations we obtain", "a termwise split short exact sequence of complexes", "$$", "0 \\to \\sigma_{\\geq a + 1} A^\\bullet \\to A^\\bullet \\to", "\\sigma_{\\leq a} A^\\bullet \\to 0", "$$", "see", "Homology, Section \\ref{homology-section-truncations}.", "Thus a distinguished triangle", "$(\\sigma_{\\geq a + 1} A^\\bullet, A^\\bullet, \\sigma_{\\leq a} A^\\bullet)$.", "By induction hypothesis the two outer complexes compute $RF$.", "Then the middle one does too by", "Lemma \\ref{lemma-2-out-of-3-computes}.", "\\medskip\\noindent", "Suppose that $A^\\bullet$ is a bounded below complex of acyclic objects.", "To show that $F(A) \\to RF(A)$ is an isomorphism in $D^{+}(\\mathcal{B})$", "it suffices to show that $H^i(F(A)) \\to H^i(RF(A))$ is an isomorphism for", "all $i$. Pick $i$. Consider the termwise split short exact sequence of", "complexes", "$$", "0 \\to \\sigma_{\\geq i + 2} A^\\bullet \\to A^\\bullet \\to", "\\sigma_{\\leq i + 1} A^\\bullet \\to 0.", "$$", "Note that this induces a termwise split short exact sequence", "$$", "0 \\to \\sigma_{\\geq i + 2} F(A^\\bullet) \\to F(A^\\bullet) \\to", "\\sigma_{\\leq i + 1} F(A^\\bullet) \\to 0.", "$$", "Hence we get distinguished triangles", "$$", "\\begin{matrix}", "(\\sigma_{\\geq i + 2} A^\\bullet, A^\\bullet,", "\\sigma_{\\leq i + 1} A^\\bullet) \\\\", "(\\sigma_{\\geq i + 2} F(A^\\bullet), F(A^\\bullet),", "\\sigma_{\\leq i + 1} F(A^\\bullet)) \\\\", "(RF(\\sigma_{\\geq i + 2} A^\\bullet), RF(A^\\bullet),", "RF(\\sigma_{\\leq i + 1} A^\\bullet))", "\\end{matrix}", "$$", "Using the last two we obtain a map of exact sequences", "$$", "\\xymatrix{", "H^i(\\sigma_{\\geq i + 2} F(A^\\bullet)) \\ar[r] \\ar[d] &", "H^i(F(A^\\bullet)) \\ar[r] \\ar[d]^\\alpha &", "H^i(\\sigma_{\\leq i + 1} F(A^\\bullet)) \\ar[r] \\ar[d]^\\beta &", "H^{i + 1}(\\sigma_{\\geq i + 2} F(A^\\bullet)) \\ar[d] \\\\", "H^i(RF(\\sigma_{\\geq i + 2} A^\\bullet)) \\ar[r] &", "H^i(RF(A^\\bullet)) \\ar[r] &", "H^i(RF(\\sigma_{\\leq i + 1} A^\\bullet)) \\ar[r] &", "H^{i + 1}(RF(\\sigma_{\\geq i + 2} A^\\bullet))", "}", "$$", "By the results of the first paragraph the map $\\beta$ is an isomorphism.", "By inspection the objects on the upper left and the upper right", "are zero. Hence to finish the proof it suffices to show that", "$H^i(RF(\\sigma_{\\geq i + 2} A^\\bullet)) = 0$ and", "$H^{i + 1}(RF(\\sigma_{\\geq i + 2} A^\\bullet)) = 0$.", "This follows immediately from", "Lemma \\ref{lemma-negative-vanishing}." ], "refs": [ "derived-lemma-2-out-of-3-computes", "derived-lemma-negative-vanishing" ], "ref_ids": [ 1829, 1839 ] } ], "ref_ids": [] }, { "id": 1845, "type": "theorem", "label": "derived-lemma-right-derived-exact-functor", "categories": [ "derived" ], "title": "derived-lemma-right-derived-exact-functor", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor of", "abelian categories. Then", "\\begin{enumerate}", "\\item every object of $\\mathcal{A}$ is right acyclic for $F$,", "\\item $RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined,", "\\item $RF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ is everywhere defined,", "\\item every complex computes $RF$, in other words, the canonical", "map $F(K^\\bullet) \\to RF(K^\\bullet)$ is an isomorphism for all complexes, and", "\\item $R^iF = 0$ for $i \\not = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is true because $F$ transforms acyclic complexes into acyclic complexes", "and quasi-isomorphisms into quasi-isomorphisms. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1846, "type": "theorem", "label": "derived-lemma-cohomology-in-serre-subcategory", "categories": [ "derived" ], "title": "derived-lemma-cohomology-in-serre-subcategory", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $\\mathcal{B} \\subset \\mathcal{A}$ be a weak Serre subcategory.", "The category $D_\\mathcal{B}(\\mathcal{A})$ is a strictly full", "saturated triangulated subcategory of $D(\\mathcal{A})$.", "Similarly for the bounded versions." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $D_\\mathcal{B}(\\mathcal{A})$ is an additive subcategory", "preserved under the translation functors.", "If $X \\oplus Y$ is in $D_\\mathcal{B}(\\mathcal{A})$, then", "both $H^n(X)$ and $H^n(Y)$ are kernels of maps between maps of objects", "of $\\mathcal{B}$ as $H^n(X \\oplus Y) = H^n(X) \\oplus H^n(Y)$.", "Hence both $X$ and $Y$ are in $D_\\mathcal{B}(\\mathcal{A})$. By", "Lemma \\ref{lemma-triangulated-subcategory}", "it therefore suffices to show that given a distinguished triangle", "$(X, Y, Z, f, g, h)$ such that $X$ and $Y$ are in $D_\\mathcal{B}(\\mathcal{A})$", "then $Z$ is an object of $D_\\mathcal{B}(\\mathcal{A})$. The long exact", "cohomology sequence (\\ref{equation-long-exact-cohomology-sequence-D})", "and the definition of a weak Serre subcategory (see", "Homology, Definition \\ref{homology-definition-serre-subcategory})", "show that $H^n(Z)$ is an object of $\\mathcal{B}$ for all $n$.", "Thus $Z$ is an object of $D_\\mathcal{B}(\\mathcal{A})$." ], "refs": [ "derived-lemma-triangulated-subcategory", "homology-definition-serre-subcategory" ], "ref_ids": [ 1771, 12146 ] } ], "ref_ids": [] }, { "id": 1847, "type": "theorem", "label": "derived-lemma-derived-of-quotient", "categories": [ "derived" ], "title": "derived-lemma-derived-of-quotient", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $\\mathcal{B} \\subset \\mathcal{A}$ be a Serre subcategory.", "Then $D(\\mathcal{A}) \\to D(\\mathcal{A}/\\mathcal{B})$", "is essentially surjective." ], "refs": [], "proofs": [ { "contents": [ "We will use the description of the category $\\mathcal{A}/\\mathcal{B}$", "in the proof of", "Homology, Lemma \\ref{homology-lemma-serre-subcategory-is-kernel}.", "Let $(X^\\bullet, d^\\bullet)$ be a complex of $\\mathcal{A}/\\mathcal{B}$.", "This means that $X^i$ is an object of $\\mathcal{A}$ and", "$d^i : X^i \\to X^{i + 1}$ is a morphism in $\\mathcal{A}/\\mathcal{B}$", "such that $d^i \\circ d^{i - 1} = 0$ in $\\mathcal{A}/\\mathcal{B}$.", "\\medskip\\noindent", "For $i \\geq 0$ we may write $d^i = (s^i, f^i)$ where $s^i : Y^i \\to X^i$", "is a morphism of $\\mathcal{A}$ whose kernel and cokernel are in $\\mathcal{B}$", "(equivalently $s^i$ becomes an isomorphism in the quotient category)", "and $f^i : Y^i \\to X^{i + 1}$ is a morphism of $\\mathcal{A}$.", "By induction we will construct a commutative diagram", "$$", "\\xymatrix{", "& (X')^1 \\ar@{..>}[r] & (X')^2 \\ar@{..>}[r] & \\ldots \\\\", "X^0 \\ar@{..>}[ru] &", "X^1 \\ar@{..>}[u] &", "X^2 \\ar@{..>}[u] &", "\\ldots \\\\", "Y^0 \\ar[u]_{s^0} \\ar[ru]_{f^0} &", "Y^1 \\ar[u]_{s^1} \\ar[ru]_{f^1} &", "Y^2 \\ar[u]_{s^2} \\ar[ru]_{f^2} &", "\\ldots", "}", "$$", "where the vertical arrows $X^i \\to (X')^i$ become isomorphisms", "in the quotient category. Namely, we first let", "$(X')^1 = \\Coker(Y^0 \\to X^0 \\oplus X^1)$ (or rather the", "pushout of the diagram with arrows $s^0$ and $f^0$) which gives the", "first commutative diagram. Next, we take", "$(X')^2 = \\Coker(Y^1 \\to (X')^1 \\oplus X^2)$. And so on.", "Setting additionally $(X')^n = X^n$ for $n \\leq 0$ we see that the map", "$(X^\\bullet, d^\\bullet) \\to ((X')^\\bullet, (d')^\\bullet)$", "is an isomorphism of complexes in $\\mathcal{A}/\\mathcal{B}$.", "Hence we may assume $d^n : X^n \\to X^{n + 1}$ is given", "by a map $X^n \\to X^{n + 1}$ in $\\mathcal{A}$ for $n \\geq 0$.", "\\medskip\\noindent", "Dually, for $i < 0$ we may write $d^i = (g^i, t^{i + 1})$ where", "$t^{i + 1} : X^{i + 1} \\to Z^{i + 1}$ is an isomorphism in the", "quotient category and $g^i : X^i \\to Z^{i + 1}$ is a morphism.", "By induction we will construct a commutative diagram", "$$", "\\xymatrix{", "\\ldots &", "Z^{-2} &", "Z^{-1} &", "Z^0 \\\\", "\\ldots &", "X^{-2} \\ar[u]_{t_{-2}} \\ar[ru]_{g_{-2}} &", "X^{-1} \\ar[u]_{t_{-1}} \\ar[ru]_{g_{-1}} &", "X^0 \\ar[u]_{t^0} \\\\", "\\ldots &", "(X')^{-2} \\ar@{..>}[u] \\ar@{..>}[r] &", "(X')^{-1} \\ar@{..>}[u] \\ar@{..>}[ru]", "}", "$$", "where the vertical arrows $(X')^i \\to X^i$ become isomorphisms", "in the quotient category. Namely, we take", "$(X')^{-1} = X^{-1} \\times_{Z^0} X^0$. Then we take", "$(X')^{-2} = X^{-2} \\times_{Z^{-1}} (X')^{-1}$. And so on.", "Setting additionally $(X')^n = X^n$ for $n \\geq 0$ we see that the map", "$((X')^\\bullet, (d')^\\bullet) \\to (X^\\bullet, d^\\bullet)$", "is an isomorphism of complexes in $\\mathcal{A}/\\mathcal{B}$.", "Hence we may assume $d^n : X^n \\to X^{n + 1}$ is given", "by a map $d^n : X^n \\to X^{n + 1}$ in $\\mathcal{A}$", "for all $n \\in \\mathbf{Z}$.", "\\medskip\\noindent", "In this case we know the compositions $d^n \\circ d^{n - 1}$", "are zero in $\\mathcal{A}/\\mathcal{B}$. If for $n > 0$ we replace", "$X^n$ by", "$$", "(X')^n = X^n/\\sum\\nolimits_{0 < k \\leq n} \\Im(\\Im(X^{k - 2} \\to X^k) \\to X^n)", "$$", "then the compositions $d^n \\circ d^{n - 1}$ are zero for $n \\geq 0$.", "(Similarly to the second paragraph above we obtain an isomorphism of", "complexes", "$(X^\\bullet, d^\\bullet) \\to ((X')^\\bullet, (d')^\\bullet)$.)", "Finally, for $n < 0$ we replace $X^n$ by", "$$", "(X')^n = \\bigcap\\nolimits_{n \\leq k < 0}", "(X^n \\to X^k)^{-1}\\Ker(X^k \\to X^{k + 2})", "$$", "and we argue in the same manner to get a complex in $\\mathcal{A}$", "whose image in $\\mathcal{A}/\\mathcal{B}$ is isomorphic to the given one." ], "refs": [ "homology-lemma-serre-subcategory-is-kernel" ], "ref_ids": [ 12048 ] } ], "ref_ids": [] }, { "id": 1848, "type": "theorem", "label": "derived-lemma-quotient-by-serre-easy", "categories": [ "derived" ], "title": "derived-lemma-quotient-by-serre-easy", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $\\mathcal{B} \\subset \\mathcal{A}$ be a Serre subcategory.", "Suppose that the functor $v : \\mathcal{A} \\to \\mathcal{A}/\\mathcal{B}$", "has a left adjoint $u : \\mathcal{A}/\\mathcal{B} \\to \\mathcal{A}$", "such that $vu \\cong \\text{id}$. Then", "$$", "D(\\mathcal{A})/D_\\mathcal{B}(\\mathcal{A}) = D(\\mathcal{A}/\\mathcal{B})", "$$", "and similarly for the bounded versions." ], "refs": [], "proofs": [ { "contents": [ "The functor $D(v) : D(\\mathcal{A}) \\to D(\\mathcal{A}/\\mathcal{B})$", "is essentially surjective by", "Lemma \\ref{lemma-derived-of-quotient}.", "For an object $X$ of $D(\\mathcal{A})$ the adjunction mapping", "$c_X : uvX \\to X$ maps to an isomorphism in $D(\\mathcal{A}/\\mathcal{B})$", "because $vuv \\cong v$ by the assumption that $vu \\cong \\text{id}$.", "Thus in a distinguished triangle $(uvX, X, Z, c_X, g, h)$ the object", "$Z$ is an object of $D_\\mathcal{B}(\\mathcal{A})$ as we see by looking", "at the long exact cohomology sequence.", "Hence $c_X$ is an element of the multiplicative system used to define", "the quotient category $D(\\mathcal{A})/D_\\mathcal{B}(\\mathcal{A})$.", "Thus $uvX \\cong X$ in $D(\\mathcal{A})/D_\\mathcal{B}(\\mathcal{A})$.", "For $X, Y \\in \\Ob(\\mathcal{A}))$ the map", "$$", "\\Hom_{D(\\mathcal{A})/D_\\mathcal{B}(\\mathcal{A})}(X, Y)", "\\longrightarrow", "\\Hom_{D(\\mathcal{A}/\\mathcal{B})}(vX, vY)", "$$", "is bijective because $u$ gives an inverse (by the remarks above)." ], "refs": [ "derived-lemma-derived-of-quotient" ], "ref_ids": [ 1847 ] } ], "ref_ids": [] }, { "id": 1849, "type": "theorem", "label": "derived-lemma-fully-faithful-embedding", "categories": [ "derived" ], "title": "derived-lemma-fully-faithful-embedding", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{B} \\subset \\mathcal{A}$", "be a Serre subcategory. Assume that for every surjection $X \\to Y$", "with $X \\in \\Ob(\\mathcal{A})$ and $Y \\in \\Ob(\\mathcal{B})$ there exists", "$X' \\subset X$, $X' \\in \\Ob(\\mathcal{B})$ which surjects onto $Y$.", "Then the functor $D^-(\\mathcal{B}) \\to D^-_\\mathcal{B}(\\mathcal{A})$ of", "(\\ref{equation-compare}) is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Let $X^\\bullet$ be a bounded above complex of $\\mathcal{A}$ such that", "$H^i(X^\\bullet) \\in \\Ob(\\mathcal{B})$ for all $i \\in \\mathbf{Z}$.", "Moreover, suppose we are given $B^i \\subset X^i$, $B^i \\in \\Ob(\\mathcal{B})$", "for all $i \\in \\mathbf{Z}$. Claim: there exists a subcomplex", "$Y^\\bullet \\subset X^\\bullet$ such that", "\\begin{enumerate}", "\\item $Y^\\bullet \\to X^\\bullet$ is a quasi-isomorphism,", "\\item $Y^i \\in \\Ob(\\mathcal{B})$ for all $i \\in \\mathbf{Z}$, and", "\\item $B^i \\subset Y^i$ for all $i \\in \\mathbf{Z}$.", "\\end{enumerate}", "To prove the claim, using the assumption of the lemma we first choose", "$C^i \\subset \\Ker(d^i : X^i \\to X^{i + 1})$, $C^i \\in \\Ob(\\mathcal{B})$", "surjecting onto $H^i(X^\\bullet)$. Setting", "$D^i = C^i + d^{i - 1}(B^{i - 1}) + B^i$ we find a subcomplex", "$D^\\bullet$ satisfying (2) and (3) such that", "$H^i(D^\\bullet) \\to H^i(X^\\bullet)$ is surjective for all $i \\in \\mathbf{Z}$.", "For any choice of $E^i \\subset X^i$ with $E^i \\in \\Ob(\\mathcal{B})$ and", "$d^i(E^i) \\subset D^{i + 1} + E^{i + 1}$ we see that setting", "$Y^i = D^i + E^i$ gives a subcomplex whose terms are in $\\mathcal{B}$ and", "whose cohomology surjects onto the cohomology of $X^\\bullet$. Clearly, if", "$d^i(E^i) = (D^{i + 1} + E^{i + 1}) \\cap \\text{Im}(d^i)$ then we see that", "the map on cohomology is also injective. For $n \\gg 0$ we can", "take $E^n$ equal to $0$. By descending induction", "we can choose $E^i$ for all $i$ with the desired property.", "Namely, given $E^{i + 1}, E^{i + 2}, \\ldots$ we choose $E^i \\subset X^i$", "such that $d^i(E^i) = (D^{i + 1} + E^{i + 1}) \\cap \\text{Im}(d^i)$.", "This is possible by our assumption in the lemma combined with", "the fact that $(D^{i + 1} + E^{i + 1}) \\cap \\text{Im}(d^i)$ is", "in $\\mathcal{B}$ as $\\mathcal{B}$ is a Serre subcategory of $\\mathcal{A}$.", "\\medskip\\noindent", "The claim above implies the lemma. Essential surjectivity is immediate", "from the claim. Let us prove faithfulness. Namely, suppose we have", "a morphism $f : U^\\bullet \\to V^\\bullet$ of bounded above complexes", "of $\\mathcal{B}$ whose image in $D(\\mathcal{A})$ is zero. Then", "there exists a quasi-isomorphism $s : V^\\bullet \\to X^\\bullet$", "into a bounded above complex of $\\mathcal{A}$ such that", "$s \\circ f$ is homotopic to zero. Choose a homotopy", "$h^i : U^i \\to X^{i - 1}$ between $0$ and $s \\circ f$.", "Apply the claim with $B^i = h^{i + 1}(U^{i + 1}) + s^i(V^i)$.", "The resulting map $s' : V^\\bullet \\to Y^\\bullet$", "is a quasi-isomorphism as well and $s' \\circ f$ is homotopic", "to zero as is clear from the fact that $h^i$ factors through $Y^{i - 1}$.", "This proves faithfulness. Fully faithfulness is proved in the", "exact same manner." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1850, "type": "theorem", "label": "derived-lemma-cohomology-bounded-below", "categories": [ "derived" ], "title": "derived-lemma-cohomology-bounded-below", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $K^\\bullet$ be a complex of $\\mathcal{A}$.", "\\begin{enumerate}", "\\item If $K^\\bullet$ has an injective resolution then", "$H^n(K^\\bullet) = 0$ for $n \\ll 0$.", "\\item If $H^n(K^\\bullet) = 0$ for all $n \\ll 0$ then there", "exists a quasi-isomorphism $K^\\bullet \\to L^\\bullet$", "with $L^\\bullet$ bounded below.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. For the second statement use", "$L^\\bullet = \\tau_{\\geq n}K^\\bullet$ for", "some $n \\ll 0$. See", "Homology, Section \\ref{homology-section-truncations}", "for the definition of the truncation $\\tau_{\\geq n}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1851, "type": "theorem", "label": "derived-lemma-injective-resolutions-exist", "categories": [ "derived" ], "title": "derived-lemma-injective-resolutions-exist", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Assume $\\mathcal{A}$ has enough injectives.", "\\begin{enumerate}", "\\item Any object of $\\mathcal{A}$ has an injective resolution.", "\\item If $H^n(K^\\bullet) = 0$ for all $n \\ll 0$ then", "$K^\\bullet$ has an injective resolution.", "\\item If $K^\\bullet$ is a complex with $K^n = 0$ for $n < a$, then", "there exists an injective resolution $\\alpha : K^\\bullet \\to I^\\bullet$", "with $I^n = 0$ for $n < a$ such that each $\\alpha^n : K^n \\to I^n$ is", "injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). First choose an injection $A \\to I^0$ of $A$ into an", "injective object of $\\mathcal{A}$. Next, choose an injection", "$I_0/A \\to I^1$ into an injective object of $\\mathcal{A}$.", "Denote $d^0$ the induced map $I^0 \\to I^1$.", "Next, choose an injection $I^1/\\Im(d^0) \\to I^2$ into", "an injective object of $\\mathcal{A}$. Denote $d^1$ the induced", "map $I^1 \\to I^2$. And so on.", "By Lemma \\ref{lemma-cohomology-bounded-below} part (2) follows from part (3).", "Part (3) is a special case of", "Lemma \\ref{lemma-subcategory-right-resolution}." ], "refs": [ "derived-lemma-cohomology-bounded-below", "derived-lemma-subcategory-right-resolution" ], "ref_ids": [ 1850, 1836 ] } ], "ref_ids": [] }, { "id": 1852, "type": "theorem", "label": "derived-lemma-acyclic-is-zero", "categories": [ "derived" ], "title": "derived-lemma-acyclic-is-zero", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $K^\\bullet$ be an acyclic complex.", "Let $I^\\bullet$ be bounded below and consisting of injective objects.", "Any morphism $K^\\bullet \\to I^\\bullet$ is homotopic to zero." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha : K^\\bullet \\to I^\\bullet$ be a morphism of", "complexes. Assume that $\\alpha^j = 0$ for $j < n$.", "We will show that there exists a morphism $h : K^{n + 1} \\to I^n$", "such that $\\alpha^n = h \\circ d$. Thus $\\alpha$ will be homotopic", "to the morphism of complexes $\\beta$ defined by", "$$", "\\beta^j =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & j \\leq n \\\\", "\\alpha^{n + 1} - d \\circ h & \\text{if} & j = n + 1 \\\\", "\\alpha^j & \\text{if} & j > n + 1", "\\end{matrix}", "\\right.", "$$", "This will clearly prove the lemma (by induction).", "To prove the existence of $h$ note that", "$\\alpha^n|_{d^{n - 1}(K^{n - 1})} = 0$ since", "$\\alpha^{n - 1} = 0$. Since $K^\\bullet$ is acyclic we", "have $d^{n - 1}(K^{n - 1}) = \\Ker(K^n \\to K^{n + 1})$.", "Hence we can think of $\\alpha^n$ as a map into $I^n$ defined", "on the subobject $\\Im(K^n \\to K^{n + 1})$ of $K^{n + 1}$.", "By injectivity of the object $I^n$ we can extend this to", "a map $h : K^{n + 1} \\to I^n$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1853, "type": "theorem", "label": "derived-lemma-morphisms-lift", "categories": [ "derived" ], "title": "derived-lemma-morphisms-lift", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Consider a solid diagram", "$$", "\\xymatrix{", "K^\\bullet \\ar[r]_\\alpha \\ar[d]_\\gamma & L^\\bullet \\ar@{-->}[dl]^\\beta \\\\", "I^\\bullet", "}", "$$", "where $I^\\bullet$ is bounded below and consists of injective", "objects, and $\\alpha$ is a quasi-isomorphism.", "\\begin{enumerate}", "\\item There exists a map of complexes $\\beta$ making the diagram", "commute up to homotopy.", "\\item If $\\alpha$ is injective in every degree", "then we can find a $\\beta$ which makes the diagram commute.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The ``correct'' proof of part (1) is explained in", "Remark \\ref{remark-easier-proofs}.", "We also give a direct proof here.", "\\medskip\\noindent", "We first show that (2) implies (1). Namely, let", "$\\tilde \\alpha : K \\to \\tilde L^\\bullet$, $\\pi$, $s$ be as in", "Lemma \\ref{lemma-make-injective}. Since $\\tilde \\alpha$ is injective", "by (2) there exists a morphism $\\tilde \\beta : \\tilde L^\\bullet \\to I^\\bullet$", "such that $\\gamma = \\tilde \\beta \\circ \\tilde \\alpha$. Set", "$\\beta = \\tilde \\beta \\circ s$. Then we have", "$$", "\\beta \\circ \\alpha", "=", "\\tilde \\beta \\circ s \\circ \\pi \\circ \\tilde \\alpha", "\\sim", "\\tilde \\beta \\circ \\tilde \\alpha", "=", "\\gamma", "$$", "as desired.", "\\medskip\\noindent", "Assume that $\\alpha : K^\\bullet \\to L^\\bullet$ is injective.", "Suppose we have already defined $\\beta$ in all degrees", "$\\leq n - 1$ compatible with differentials and such that", "$\\gamma^j = \\beta^j \\circ \\alpha^j$ for all $j \\leq n - 1$.", "Consider the commutative solid diagram", "$$", "\\xymatrix{", "K^{n - 1} \\ar[r] \\ar@/_2pc/[dd]_\\gamma \\ar[d]^\\alpha &", "K^n \\ar@/^2pc/[dd]^\\gamma \\ar[d]^\\alpha \\\\", "L^{n - 1} \\ar[r] \\ar[d]^\\beta &", "L^n \\ar@{-->}[d] \\\\", "I^{n - 1} \\ar[r] &", "I^n", "}", "$$", "Thus we see that the dotted arrow is prescribed on the subobjects", "$\\alpha(K^n)$ and $d^{n - 1}(L^{n - 1})$. Moreover, these two arrows", "agree on $\\alpha(d^{n - 1}(K^{n - 1}))$. Hence if", "\\begin{equation}", "\\label{equation-qis}", "\\alpha(d^{n - 1}(K^{n - 1}))", "=", "\\alpha(K^n) \\cap d^{n - 1}(L^{n - 1})", "\\end{equation}", "then these morphisms glue to a morphism", "$\\alpha(K^n) + d^{n - 1}(L^{n - 1}) \\to I^n$ and, using the injectivity", "of $I^n$, we can extend this to a morphism from all of $L^n$ into $I^n$.", "After this by induction we get the morphism $\\beta$ for all $n$ simultaneously", "(note that we can set $\\beta^n = 0$ for all $n \\ll 0$ since $I^\\bullet$", "is bounded below -- in this way starting the induction).", "\\medskip\\noindent", "It remains to prove the equality (\\ref{equation-qis}).", "The reader is encouraged to argue this for themselves with a suitable", "diagram chase. Nonetheless here is our argument.", "Note that the inclusion", "$\\alpha(d^{n - 1}(K^{n - 1})) \\subset \\alpha(K^n) \\cap d^{n - 1}(L^{n - 1})$", "is obvious. Take an object $T$ of $\\mathcal{A}$ and a morphism", "$x : T \\to L^n$ whose image is contained in the subobject", "$\\alpha(K^n) \\cap d^{n - 1}(L^{n - 1})$.", "Since $\\alpha$ is injective we see that $x = \\alpha \\circ x'$ for", "some $x' : T \\to K^n$. Moreover, since $x$ lies in $d^{n - 1}(L^{n - 1})$", "we see that $d^n \\circ x = 0$. Hence using injectivity of $\\alpha$ again", "we see that $d^n \\circ x' = 0$. Thus $x'$ gives a morphism", "$[x'] : T \\to H^n(K^\\bullet)$. On the other hand the corresponding", "map $[x] : T \\to H^n(L^\\bullet)$ induced by $x$ is zero by assumption.", "Since $\\alpha$ is a quasi-isomorphism we conclude that $[x'] = 0$.", "This of course means exactly that the image of $x'$ is", "contained in $d^{n - 1}(K^{n - 1})$ and we win." ], "refs": [ "derived-remark-easier-proofs", "derived-lemma-make-injective" ], "ref_ids": [ 2017, 1797 ] } ], "ref_ids": [] }, { "id": 1854, "type": "theorem", "label": "derived-lemma-morphisms-equal-up-to-homotopy", "categories": [ "derived" ], "title": "derived-lemma-morphisms-equal-up-to-homotopy", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Consider a solid diagram", "$$", "\\xymatrix{", "K^\\bullet \\ar[r]_\\alpha \\ar[d]_\\gamma & L^\\bullet \\ar@{-->}[dl]^{\\beta_i} \\\\", "I^\\bullet", "}", "$$", "where $I^\\bullet$ is bounded below and consists of injective", "objects, and $\\alpha$ is a quasi-isomorphism.", "Any two morphisms $\\beta_1, \\beta_2$ making the diagram commute", "up to homotopy are homotopic." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Remark \\ref{remark-easier-proofs}.", "We also give a direct argument here.", "\\medskip\\noindent", "Let $\\tilde \\alpha : K \\to \\tilde L^\\bullet$, $\\pi$, $s$ be as in", "Lemma \\ref{lemma-make-injective}. If we can show that $\\beta_1 \\circ\\pi$", "is homotopic to $\\beta_2 \\circ \\pi$, then we deduce that", "$\\beta_1 \\sim \\beta_2$ because $\\pi \\circ s$ is the identity.", "Hence we may assume $\\alpha^n : K^n \\to L^n$ is the", "inclusion of a direct summand for all $n$. Thus we get a", "short exact sequence of complexes", "$$", "0 \\to K^\\bullet \\to L^\\bullet \\to M^\\bullet \\to 0", "$$", "which is termwise split and such that $M^\\bullet$ is acyclic.", "We choose splittings $L^n = K^n \\oplus M^n$, so we have", "$\\beta_i^n : K^n \\oplus M^n \\to I^n$ and $\\gamma^n : K^n \\to I^n$.", "In this case the condition on $\\beta_i$ is that there are morphisms", "$h_i^n : K^n \\to I^{n - 1}$ such that", "$$", "\\gamma^n - \\beta_i^n|_{K^n} = d \\circ h_i^n + h_i^{n + 1} \\circ d", "$$", "Thus we see that", "$$", "\\beta_1^n|_{K^n} - \\beta_2^n|_{K^n}", "=", "d \\circ (h_1^n - h_2^n) + (h_1^{n + 1} - h_2^{n + 1}) \\circ d", "$$", "Consider the map $h^n : K^n \\oplus M^n \\to I^{n - 1}$ which", "equals $h_1^n - h_2^n$ on the first summand and zero on the second.", "Then we see that", "$$", "\\beta_1^n - \\beta_2^n", "-", "(d \\circ h^n + h^{n + 1}) \\circ d", "$$", "is a morphism of complexes $L^\\bullet \\to I^\\bullet$", "which is identically zero on the subcomplex $K^\\bullet$.", "Hence it factors as $L^\\bullet \\to M^\\bullet \\to I^\\bullet$.", "Thus the result of the lemma follows from Lemma \\ref{lemma-acyclic-is-zero}." ], "refs": [ "derived-remark-easier-proofs", "derived-lemma-make-injective", "derived-lemma-acyclic-is-zero" ], "ref_ids": [ 2017, 1797, 1852 ] } ], "ref_ids": [] }, { "id": 1855, "type": "theorem", "label": "derived-lemma-morphisms-into-injective-complex", "categories": [ "derived" ], "title": "derived-lemma-morphisms-into-injective-complex", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $I^\\bullet$ be bounded below complex consisting of injective", "objects. Let $L^\\bullet \\in K(\\mathcal{A})$. Then", "$$", "\\Mor_{K(\\mathcal{A})}(L^\\bullet, I^\\bullet)", "=", "\\Mor_{D(\\mathcal{A})}(L^\\bullet, I^\\bullet).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $a$ be an element of the right hand side.", "We may represent $a = \\gamma\\alpha^{-1}$ where", "$\\alpha : K^\\bullet \\to L^\\bullet$", "is a quasi-isomorphism and $\\gamma : K^\\bullet \\to I^\\bullet$ is a map", "of complexes. By", "Lemma \\ref{lemma-morphisms-lift}", "we can find a morphism $\\beta : L^\\bullet \\to I^\\bullet$ such that", "$\\beta \\circ \\alpha$ is homotopic to $\\gamma$. This proves that the", "map is surjective. Let $b$ be an element of the left hand side", "which maps to zero in the right hand side. Then $b$ is the homotopy class", "of a morphism $\\beta : L^\\bullet \\to I^\\bullet$ such that there exists a", "quasi-isomorphism $\\alpha : K^\\bullet \\to L^\\bullet$ with", "$\\beta \\circ \\alpha$ homotopic to zero. Then", "Lemma \\ref{lemma-morphisms-equal-up-to-homotopy}", "shows that $\\beta$ is homotopic to zero also, i.e., $b = 0$." ], "refs": [ "derived-lemma-morphisms-lift", "derived-lemma-morphisms-equal-up-to-homotopy" ], "ref_ids": [ 1853, 1854 ] } ], "ref_ids": [] }, { "id": 1856, "type": "theorem", "label": "derived-lemma-injective-resolution-ses", "categories": [ "derived" ], "title": "derived-lemma-injective-resolution-ses", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Assume $\\mathcal{A}$ has enough injectives.", "For any short exact sequence", "$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$", "of $\\text{Comp}^{+}(\\mathcal{A})$ there exists a", "commutative diagram in $\\text{Comp}^{+}(\\mathcal{A})$", "$$", "\\xymatrix{", "0 \\ar[r] &", "A^\\bullet \\ar[r] \\ar[d] &", "B^\\bullet \\ar[r] \\ar[d] &", "C^\\bullet \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "I_1^\\bullet \\ar[r] &", "I_2^\\bullet \\ar[r] &", "I_3^\\bullet \\ar[r] &", "0", "}", "$$", "where the vertical arrows are injective resolutions and", "the rows are short exact sequences of complexes.", "In fact, given any injective resolution $A^\\bullet \\to I^\\bullet$", "we may assume $I_1^\\bullet = I^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Step 1. Choose an injective resolution $A^\\bullet \\to I^\\bullet$ (see", "Lemma \\ref{lemma-injective-resolutions-exist}) or use the given one.", "Recall that $\\text{Comp}^{+}(\\mathcal{A})$ is an", "abelian category, see", "Homology, Lemma \\ref{homology-lemma-cat-cochain-abelian}.", "Hence we may form the pushout along", "the injective map $A^\\bullet \\to I^\\bullet$ to get", "$$", "\\xymatrix{", "0 \\ar[r] &", "A^\\bullet \\ar[r] \\ar[d] &", "B^\\bullet \\ar[r] \\ar[d] &", "C^\\bullet \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "I^\\bullet \\ar[r] &", "E^\\bullet \\ar[r] &", "C^\\bullet \\ar[r] &", "0", "}", "$$", "Note that the lower short exact sequence is termwise split, see", "Homology, Lemma \\ref{homology-lemma-characterize-injectives}.", "Hence it suffices to prove the lemma when", "$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$ is", "termwise split.", "\\medskip\\noindent", "Step 2. Choose splittings. In other words, write", "$B^n = A^n \\oplus C^n$. Denote $\\delta : C^\\bullet \\to A^\\bullet[1]$", "the morphism as in", "Homology, Lemma \\ref{homology-lemma-ses-termwise-split-cochain}.", "Choose injective resolutions $f_1 : A^\\bullet \\to I_1^\\bullet$", "and $f_3 : C^\\bullet \\to I_3^\\bullet$. (If $A^\\bullet$ is a complex of", "injectives, then use $I_1^\\bullet = A^\\bullet$.)", "We may assume $f_3$ is injective in", "every degree. By Lemma \\ref{lemma-morphisms-lift} we may find", "a morphism $\\delta' : I_3^\\bullet \\to I_1^\\bullet[1]$ such", "that $\\delta' \\circ f_3 = f_1[1] \\circ \\delta$ (equality of", "morphisms of complexes). Set $I_2^n = I_1^n \\oplus I_3^n$.", "Define", "$$", "d_{I_2}^n =", "\\left(", "\\begin{matrix}", "d_{I_1}^n & (\\delta')^n \\\\", "0 & d_{I_3}^n", "\\end{matrix}", "\\right)", "$$", "and define the maps $B^n \\to I_2^n$ to be given as the", "sum of the maps $A^n \\to I_1^n$ and $C^n \\to I_3^n$.", "Everything is clear." ], "refs": [ "derived-lemma-injective-resolutions-exist", "homology-lemma-cat-cochain-abelian", "homology-lemma-characterize-injectives", "homology-lemma-ses-termwise-split-cochain", "derived-lemma-morphisms-lift" ], "ref_ids": [ 1851, 12059, 12112, 12067, 1853 ] } ], "ref_ids": [] }, { "id": 1857, "type": "theorem", "label": "derived-lemma-cohomology-bounded-above", "categories": [ "derived" ], "title": "derived-lemma-cohomology-bounded-above", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $K^\\bullet$ be a complex of $\\mathcal{A}$.", "\\begin{enumerate}", "\\item If $K^\\bullet$ has a projective resolution then", "$H^n(K^\\bullet) = 0$ for $n \\gg 0$.", "\\item If $H^n(K^\\bullet) = 0$ for $n \\gg 0$ then there", "exists a quasi-isomorphism $L^\\bullet \\to K^\\bullet$", "with $L^\\bullet$ bounded above.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Dual to", "Lemma \\ref{lemma-cohomology-bounded-below}." ], "refs": [ "derived-lemma-cohomology-bounded-below" ], "ref_ids": [ 1850 ] } ], "ref_ids": [] }, { "id": 1858, "type": "theorem", "label": "derived-lemma-projective-resolutions-exist", "categories": [ "derived" ], "title": "derived-lemma-projective-resolutions-exist", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Assume $\\mathcal{A}$ has enough projectives.", "\\begin{enumerate}", "\\item Any object of $\\mathcal{A}$ has a projective resolution.", "\\item If $H^n(K^\\bullet) = 0$ for all $n \\gg 0$ then", "$K^\\bullet$ has a projective resolution.", "\\item If $K^\\bullet$ is a complex with $K^n = 0$ for $n > a$, then", "there exists a projective resolution $\\alpha : P^\\bullet \\to K^\\bullet$", "with $P^n = 0$ for $n > a$ such that each $\\alpha^n : P^n \\to K^n$ is", "surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Dual to", "Lemma \\ref{lemma-injective-resolutions-exist}." ], "refs": [ "derived-lemma-injective-resolutions-exist" ], "ref_ids": [ 1851 ] } ], "ref_ids": [] }, { "id": 1859, "type": "theorem", "label": "derived-lemma-projective-into-acyclic-is-zero", "categories": [ "derived" ], "title": "derived-lemma-projective-into-acyclic-is-zero", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $K^\\bullet$ be an acyclic complex.", "Let $P^\\bullet$ be bounded above and consisting of projective objects.", "Any morphism $P^\\bullet \\to K^\\bullet$ is homotopic to zero." ], "refs": [], "proofs": [ { "contents": [ "Dual to", "Lemma \\ref{lemma-acyclic-is-zero}." ], "refs": [ "derived-lemma-acyclic-is-zero" ], "ref_ids": [ 1852 ] } ], "ref_ids": [] }, { "id": 1860, "type": "theorem", "label": "derived-lemma-morphisms-lift-projective", "categories": [ "derived" ], "title": "derived-lemma-morphisms-lift-projective", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Consider a solid diagram", "$$", "\\xymatrix{", "K^\\bullet & L^\\bullet \\ar[l]^\\alpha \\\\", "P^\\bullet \\ar[u] \\ar@{-->}[ru]_\\beta", "}", "$$", "where $P^\\bullet$ is bounded above and consists of projective", "objects, and $\\alpha$ is a quasi-isomorphism.", "\\begin{enumerate}", "\\item There exists a map of complexes $\\beta$ making the diagram", "commute up to homotopy.", "\\item If $\\alpha$ is surjective in every degree", "then we can find a $\\beta$ which makes the diagram commute.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Dual to", "Lemma \\ref{lemma-morphisms-lift}." ], "refs": [ "derived-lemma-morphisms-lift" ], "ref_ids": [ 1853 ] } ], "ref_ids": [] }, { "id": 1861, "type": "theorem", "label": "derived-lemma-morphisms-equal-up-to-homotopy-projective", "categories": [ "derived" ], "title": "derived-lemma-morphisms-equal-up-to-homotopy-projective", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Consider a solid diagram", "$$", "\\xymatrix{", "K^\\bullet & L^\\bullet \\ar[l]^\\alpha \\\\", "P^\\bullet \\ar[u] \\ar@{-->}[ru]_{\\beta_i}", "}", "$$", "where $P^\\bullet$ is bounded above and consists of projective", "objects, and $\\alpha$ is a quasi-isomorphism.", "Any two morphisms $\\beta_1, \\beta_2$ making the diagram commute", "up to homotopy are homotopic." ], "refs": [], "proofs": [ { "contents": [ "Dual to", "Lemma \\ref{lemma-morphisms-equal-up-to-homotopy}." ], "refs": [ "derived-lemma-morphisms-equal-up-to-homotopy" ], "ref_ids": [ 1854 ] } ], "ref_ids": [] }, { "id": 1862, "type": "theorem", "label": "derived-lemma-morphisms-from-projective-complex", "categories": [ "derived" ], "title": "derived-lemma-morphisms-from-projective-complex", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $P^\\bullet$ be bounded above complex consisting of projective", "objects. Let $L^\\bullet \\in K(\\mathcal{A})$. Then", "$$", "\\Mor_{K(\\mathcal{A})}(P^\\bullet, L^\\bullet)", "=", "\\Mor_{D(\\mathcal{A})}(P^\\bullet, L^\\bullet).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Dual to", "Lemma \\ref{lemma-morphisms-into-injective-complex}." ], "refs": [ "derived-lemma-morphisms-into-injective-complex" ], "ref_ids": [ 1855 ] } ], "ref_ids": [] }, { "id": 1863, "type": "theorem", "label": "derived-lemma-projective-resolution-ses", "categories": [ "derived" ], "title": "derived-lemma-projective-resolution-ses", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Assume $\\mathcal{A}$ has enough projectives.", "For any short exact sequence", "$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$", "of $\\text{Comp}^{+}(\\mathcal{A})$ there exists a", "commutative diagram in $\\text{Comp}^{+}(\\mathcal{A})$", "$$", "\\xymatrix{", "0 \\ar[r] &", "P_1^\\bullet \\ar[r] \\ar[d] &", "P_2^\\bullet \\ar[r] \\ar[d] &", "P_3^\\bullet \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "A^\\bullet \\ar[r] &", "B^\\bullet \\ar[r] &", "C^\\bullet \\ar[r] &", "0", "}", "$$", "where the vertical arrows are projective resolutions and", "the rows are short exact sequences of complexes.", "In fact, given any projective resolution $P^\\bullet \\to C^\\bullet$", "we may assume $P_3^\\bullet = P^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Dual to", "Lemma \\ref{lemma-injective-resolution-ses}." ], "refs": [ "derived-lemma-injective-resolution-ses" ], "ref_ids": [ 1856 ] } ], "ref_ids": [] }, { "id": 1864, "type": "theorem", "label": "derived-lemma-precise-vanishing", "categories": [ "derived" ], "title": "derived-lemma-precise-vanishing", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $P^\\bullet$, $K^\\bullet$ be complexes.", "Let $n \\in \\mathbf{Z}$. Assume that", "\\begin{enumerate}", "\\item $P^\\bullet$ is a bounded complex consisting of projective", "objects,", "\\item $P^i = 0$ for $i < n$, and", "\\item $H^i(K^\\bullet) = 0$ for $i \\geq n$.", "\\end{enumerate}", "Then", "$\\Hom_{K(\\mathcal{A})}(P^\\bullet, K^\\bullet) =", "\\Hom_{D(\\mathcal{A})}(P^\\bullet, K^\\bullet) = 0$." ], "refs": [], "proofs": [ { "contents": [ "The first equality follows from", "Lemma \\ref{lemma-morphisms-from-projective-complex}.", "Note that there is a distinguished triangle", "$$", "(\\tau_{\\leq n - 1}K^\\bullet, K^\\bullet, \\tau_{\\geq n}K^\\bullet, f, g, h)", "$$", "by Remark \\ref{remark-truncation-distinguished-triangle}. Hence, by", "Lemma \\ref{lemma-representable-homological}", "it suffices to prove", "$\\Hom_{K(\\mathcal{A})}(P^\\bullet, \\tau_{\\leq n - 1}K^\\bullet) = 0$ and", "$\\Hom_{K(\\mathcal{A})}(P^\\bullet, \\tau_{\\geq n} K^\\bullet) = 0$.", "The first vanishing is trivial and the second is", "Lemma \\ref{lemma-projective-into-acyclic-is-zero}." ], "refs": [ "derived-lemma-morphisms-from-projective-complex", "derived-remark-truncation-distinguished-triangle", "derived-lemma-representable-homological", "derived-lemma-projective-into-acyclic-is-zero" ], "ref_ids": [ 1862, 2016, 1758, 1859 ] } ], "ref_ids": [] }, { "id": 1865, "type": "theorem", "label": "derived-lemma-lift-map", "categories": [ "derived" ], "title": "derived-lemma-lift-map", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $\\beta : P^\\bullet \\to L^\\bullet$ and", "$\\alpha : E^\\bullet \\to L^\\bullet$ be", "maps of complexes. Let $n \\in \\mathbf{Z}$. Assume", "\\begin{enumerate}", "\\item $P^\\bullet$ is a bounded complex of projectives and", "$P^i = 0$ for $i < n$,", "\\item $H^i(\\alpha)$ is an isomorphism for $i > n$ and surjective", "for $i = n$.", "\\end{enumerate}", "Then there exists a map of complexes $\\gamma : P^\\bullet \\to E^\\bullet$", "such that $\\alpha \\circ \\gamma$ and $\\beta$ are homotopic." ], "refs": [], "proofs": [ { "contents": [ "Consider the cone $C^\\bullet = C(\\alpha)^\\bullet$ with map", "$i : L^\\bullet \\to C^\\bullet$.", "Note that $i \\circ \\beta$ is zero by", "Lemma \\ref{lemma-precise-vanishing}.", "Hence we can lift $\\beta$ to $E^\\bullet$ by", "Lemma \\ref{lemma-representable-homological}." ], "refs": [ "derived-lemma-precise-vanishing", "derived-lemma-representable-homological" ], "ref_ids": [ 1864, 1758 ] } ], "ref_ids": [] }, { "id": 1866, "type": "theorem", "label": "derived-lemma-injective-acyclic", "categories": [ "derived" ], "title": "derived-lemma-injective-acyclic", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $I \\in \\Ob(\\mathcal{A})$ be an injective object.", "Let $I^\\bullet$ be a bounded below complex of injectives in $\\mathcal{A}$.", "\\begin{enumerate}", "\\item $I^\\bullet$ computes $RF$ relative to $\\text{Qis}^{+}(\\mathcal{A})$", "for any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$", "into any triangulated category $\\mathcal{D}$.", "\\item $I$ is right acyclic for any additive functor", "$F : \\mathcal{A} \\to \\mathcal{B}$ into any abelian category $\\mathcal{B}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) is a direct consequences of part (1) and", "Definition \\ref{definition-derived-functor}.", "To prove (1) let $\\alpha : I^\\bullet \\to K^\\bullet$ be a quasi-isomorphism", "into a complex. By", "Lemma \\ref{lemma-morphisms-lift}", "we see that $\\alpha$ has a left inverse. Hence the category", "$I^\\bullet/\\text{Qis}^{+}(\\mathcal{A})$ is essentially constant with value", "$\\text{id} : I^\\bullet \\to I^\\bullet$. Thus also the ind-object", "$$", "I^\\bullet/\\text{Qis}^{+}(\\mathcal{A}) \\longrightarrow \\mathcal{D}, \\quad", "(I^\\bullet \\to K^\\bullet) \\longmapsto F(K^\\bullet)", "$$", "is essentially constant with value $F(I^\\bullet)$. This proves (1), see", "Definitions \\ref{definition-right-derived-functor-defined} and", "\\ref{definition-computes}." ], "refs": [ "derived-definition-derived-functor", "derived-lemma-morphisms-lift", "derived-definition-right-derived-functor-defined", "derived-definition-computes" ], "ref_ids": [ 1990, 1853, 1987, 1989 ] } ], "ref_ids": [] }, { "id": 1867, "type": "theorem", "label": "derived-lemma-enough-injectives-right-derived", "categories": [ "derived" ], "title": "derived-lemma-enough-injectives-right-derived", "contents": [ "Let $\\mathcal{A}$ be an abelian category with enough injectives.", "\\begin{enumerate}", "\\item For any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$", "into a triangulated category $\\mathcal{D}$ the right derived", "functor", "$$", "RF : D^{+}(\\mathcal{A}) \\longrightarrow \\mathcal{D}", "$$", "is everywhere defined.", "\\item For any additive functor $F : \\mathcal{A} \\to \\mathcal{B}$ into an", "abelian category $\\mathcal{B}$ the right derived functor", "$$", "RF : D^{+}(\\mathcal{A}) \\longrightarrow D^{+}(\\mathcal{B})", "$$", "is everywhere defined.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemma \\ref{lemma-injective-acyclic}", "and", "Proposition \\ref{proposition-enough-acyclics}", "for the second assertion. To see the first assertion combine", "Lemma \\ref{lemma-injective-resolutions-exist},", "Lemma \\ref{lemma-injective-acyclic},", "Lemma \\ref{lemma-existence-computes},", "and Equation (\\ref{equation-everywhere})." ], "refs": [ "derived-lemma-injective-acyclic", "derived-proposition-enough-acyclics", "derived-lemma-injective-resolutions-exist", "derived-lemma-injective-acyclic", "derived-lemma-existence-computes" ], "ref_ids": [ 1866, 1962, 1851, 1866, 1831 ] } ], "ref_ids": [] }, { "id": 1868, "type": "theorem", "label": "derived-lemma-right-derived-properties", "categories": [ "derived" ], "title": "derived-lemma-right-derived-properties", "contents": [ "Let $\\mathcal{A}$ be an abelian category with enough injectives.", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor.", "\\begin{enumerate}", "\\item The functor $RF$ is an exact functor", "$D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$.", "\\item The functor $RF$ induces an exact functor", "$K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$.", "\\item The functor $RF$ induces a $\\delta$-functor", "$\\text{Comp}^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$.", "\\item The functor $RF$ induces a $\\delta$-functor", "$\\mathcal{A} \\to D^{+}(\\mathcal{B})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma simply reviews some of the results obtained so far.", "Note that by", "Lemma \\ref{lemma-enough-injectives-right-derived}", "$RF$ is everywhere defined. Here are some references:", "\\begin{enumerate}", "\\item The derived functor is exact: This boils down to", "Lemma \\ref{lemma-2-out-of-3-defined}.", "\\item This is true because $K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{A})$", "is exact and compositions of exact functors are exact.", "\\item This is true because", "$\\text{Comp}^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{A})$ is", "a $\\delta$-functor, see", "Lemma \\ref{lemma-derived-canonical-delta-functor}.", "\\item This is true because $\\mathcal{A} \\to \\text{Comp}^{+}(\\mathcal{A})$", "is exact and precomposing a $\\delta$-functor by an exact functor gives", "a $\\delta$-functor.", "\\end{enumerate}" ], "refs": [ "derived-lemma-enough-injectives-right-derived", "derived-lemma-2-out-of-3-defined", "derived-lemma-derived-canonical-delta-functor" ], "ref_ids": [ 1867, 1826, 1814 ] } ], "ref_ids": [] }, { "id": 1869, "type": "theorem", "label": "derived-lemma-higher-derived-functors", "categories": [ "derived" ], "title": "derived-lemma-higher-derived-functors", "contents": [ "Let $\\mathcal{A}$ be an abelian category with enough injectives.", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor.", "\\begin{enumerate}", "\\item For any short exact sequence", "$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$", "of complexes in $\\text{Comp}^{+}(\\mathcal{A})$ there", "is an associated long exact sequence", "$$", "\\ldots \\to", "H^i(RF(A^\\bullet)) \\to", "H^i(RF(B^\\bullet)) \\to", "H^i(RF(C^\\bullet)) \\to", "H^{i + 1}(RF(A^\\bullet)) \\to \\ldots", "$$", "\\item The functors $R^iF : \\mathcal{A} \\to \\mathcal{B}$", "are zero for $i < 0$. Also $R^0F = F : \\mathcal{A} \\to \\mathcal{B}$.", "\\item We have $R^iF(I) = 0$ for $i > 0$ and $I$ injective.", "\\item The sequence $(R^iF, \\delta)$ forms a universal $\\delta$-functor (see", "Homology, Definition \\ref{homology-definition-universal-delta-functor})", "from $\\mathcal{A}$ to $\\mathcal{B}$.", "\\end{enumerate}" ], "refs": [ "homology-definition-universal-delta-functor" ], "proofs": [ { "contents": [ "This lemma simply reviews some of the results obtained so far.", "Note that by", "Lemma \\ref{lemma-enough-injectives-right-derived}", "$RF$ is everywhere defined. Here are some references:", "\\begin{enumerate}", "\\item This follows from", "Lemma \\ref{lemma-right-derived-properties} part (3)", "combined with the long exact cohomology sequence", "(\\ref{equation-long-exact-cohomology-sequence-D}) for", "$D^{+}(\\mathcal{B})$.", "\\item This is", "Lemma \\ref{lemma-left-exact-higher-derived}.", "\\item This is the fact that injective objects are acyclic.", "\\item This is", "Lemma \\ref{lemma-right-derived-delta-functor}.", "\\end{enumerate}" ], "refs": [ "derived-lemma-enough-injectives-right-derived", "derived-lemma-right-derived-properties", "derived-lemma-left-exact-higher-derived", "derived-lemma-right-derived-delta-functor" ], "ref_ids": [ 1867, 1868, 1840, 1843 ] } ], "ref_ids": [ 12151 ] }, { "id": 1870, "type": "theorem", "label": "derived-lemma-cartan-eilenberg", "categories": [ "derived" ], "title": "derived-lemma-cartan-eilenberg", "contents": [ "Let $\\mathcal{A}$ be an abelian category with enough injectives.", "Let $K^\\bullet$ be a bounded below complex.", "There exists a Cartan-Eilenberg resolution of $K^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $K^p = 0$ for $p < n$. Decompose $K^\\bullet$ into", "short exact sequences as follows: Set $Z^p = \\Ker(d^p)$,", "$B^p = \\Im(d^{p - 1})$, $H^p = Z^p/B^p$, and consider", "$$", "\\begin{matrix}", "0 \\to Z^n \\to K^n \\to B^{n + 1} \\to 0 \\\\", "0 \\to B^{n + 1} \\to Z^{n + 1} \\to H^{n + 1} \\to 0 \\\\", "0 \\to Z^{n + 1} \\to K^{n + 1} \\to B^{n + 2} \\to 0 \\\\", "0 \\to B^{n + 2} \\to Z^{n + 2} \\to H^{n + 2} \\to 0 \\\\", "\\ldots", "\\end{matrix}", "$$", "Set $I^{p, q} = 0$ for $p < n$. Inductively we choose", "injective resolutions as follows:", "\\begin{enumerate}", "\\item Choose an injective resolution $Z^n \\to J_Z^{n, \\bullet}$.", "\\item Using Lemma \\ref{lemma-injective-resolution-ses} choose injective", "resolutions $K^n \\to I^{n, \\bullet}$, $B^{n + 1} \\to J_B^{n + 1, \\bullet}$,", "and an exact sequence of complexes", "$0 \\to J_Z^{n, \\bullet} \\to I^{n, \\bullet} \\to J_B^{n + 1, \\bullet} \\to 0$", "compatible with the short exact sequence", "$0 \\to Z^n \\to K^n \\to B^{n + 1} \\to 0$.", "\\item Using Lemma \\ref{lemma-injective-resolution-ses} choose injective", "resolutions $Z^{n + 1} \\to J_Z^{n + 1, \\bullet}$,", "$H^{n + 1} \\to J_H^{n + 1, \\bullet}$,", "and an exact sequence of complexes", "$0 \\to J_B^{n + 1, \\bullet} \\to J_Z^{n + 1, \\bullet}", "\\to J_H^{n + 1, \\bullet} \\to 0$", "compatible with the short exact sequence", "$0 \\to B^{n + 1} \\to Z^{n + 1} \\to H^{n + 1} \\to 0$.", "\\item Etc.", "\\end{enumerate}", "Taking as maps $d_1^\\bullet : I^{p, \\bullet} \\to I^{p + 1, \\bullet}$", "the compositions", "$I^{p, \\bullet} \\to J_B^{p + 1, \\bullet} \\to", "J_Z^{p + 1, \\bullet} \\to I^{p + 1, \\bullet}$ everything is clear." ], "refs": [ "derived-lemma-injective-resolution-ses", "derived-lemma-injective-resolution-ses" ], "ref_ids": [ 1856, 1856 ] } ], "ref_ids": [] }, { "id": 1871, "type": "theorem", "label": "derived-lemma-two-ss-complex-functor", "categories": [ "derived" ], "title": "derived-lemma-two-ss-complex-functor", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor of", "abelian categories.", "Let $K^\\bullet$ be a bounded below complex of $\\mathcal{A}$.", "Let $I^{\\bullet, \\bullet}$ be a Cartan-Eilenberg resolution", "for $K^\\bullet$. The spectral sequences", "$({}'E_r, {}'d_r)_{r \\geq 0}$ and $({}''E_r, {}''d_r)_{r \\geq 0}$", "associated to the double complex $F(I^{\\bullet, \\bullet})$", "satisfy the relations", "$$", "{}'E_1^{p, q} = R^qF(K^p)", "\\quad", "\\text{and}", "\\quad", "{}''E_2^{p, q} = R^pF(H^q(K^\\bullet))", "$$", "Moreover, these spectral sequences are bounded, converge to", "$H^*(RF(K^\\bullet))$, and the associated induced filtrations on", "$H^n(RF(K^\\bullet))$ are finite." ], "refs": [], "proofs": [ { "contents": [ "We will use the following remarks without further mention:", "\\begin{enumerate}", "\\item As $I^{p, \\bullet}$ is an injective resolution of", "$K^p$ we see that $RF$ is defined at $K^p[0]$", "with value $F(I^{p, \\bullet})$.", "\\item As $H^p_I(I^{\\bullet, \\bullet})$ is an injective resolution of", "$H^p(K^\\bullet)$ the derived functor $RF$ is defined at $H^p(K^\\bullet)[0]$", "with value $F(H^p_I(I^{\\bullet, \\bullet}))$.", "\\item By", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}", "the total complex $\\text{Tot}(I^{\\bullet, \\bullet})$", "is an injective resolution of", "$K^\\bullet$. Hence $RF$ is defined at $K^\\bullet$ with value", "$F(\\text{Tot}(I^{\\bullet, \\bullet}))$.", "\\end{enumerate}", "Consider the two spectral sequences associated to the double complex", "$L^{\\bullet, \\bullet} = F(I^{\\bullet, \\bullet})$, see", "Homology, Lemma \\ref{homology-lemma-ss-double-complex}.", "These are both bounded, converge to $H^*(\\text{Tot}(L^{\\bullet, \\bullet}))$,", "and induce finite filtrations on $H^n(\\text{Tot}(L^{\\bullet, \\bullet}))$, see", "Homology, Lemma \\ref{homology-lemma-first-quadrant-ss}.", "Since", "$\\text{Tot}(L^{\\bullet, \\bullet}) =", "\\text{Tot}(F(I^{\\bullet, \\bullet})) =", "F(\\text{Tot}(I^{\\bullet, \\bullet}))$ computes", "$H^n(RF(K^\\bullet))$ we find the final assertion of the lemma holds true.", "\\medskip\\noindent", "Computation of the first spectral sequence. We have", "${}'E_1^{p, q} = H^q(L^{p, \\bullet})$ in other words", "$$", "{}'E_1^{p, q} = H^q(F(I^{p, \\bullet})) = R^qF(K^p)", "$$", "as desired. Observe for later use that the maps", "${}'d_1^{p, q} : {}'E_1^{p, q} \\to {}'E_1^{p + 1, q}$ are the maps", "$R^qF(K^p) \\to R^qF(K^{p + 1})$ induced by $K^p \\to K^{p + 1}$", "and the fact that $R^qF$ is a functor.", "\\medskip\\noindent", "Computation of the second spectral sequence. We have", "${}''E_1^{p, q} = H^q(L^{\\bullet, p}) = H^q(F(I^{\\bullet, p}))$.", "Note that the complex $I^{\\bullet, p}$ is bounded below,", "consists of injectives, and moreover each kernel, image, and", "cohomology group of the differentials is an injective object", "of $\\mathcal{A}$. Hence we can split the differentials, i.e.,", "each differential is a split surjection onto a direct summand.", "It follows that the same is true after applying $F$. Hence", "${}''E_1^{p, q} = F(H^q(I^{\\bullet, p})) = F(H^q_I(I^{\\bullet, p}))$.", "The differentials on this are $(-1)^q$ times $F$ applied to", "the differential of the complex $H^p_I(I^{\\bullet, \\bullet})$", "which is an injective resolution of $H^p(K^\\bullet)$. Hence the", "description of the $E_2$ terms." ], "refs": [ "homology-lemma-double-complex-gives-resolution", "homology-lemma-ss-double-complex", "homology-lemma-first-quadrant-ss" ], "ref_ids": [ 12106, 12104, 12105 ] } ], "ref_ids": [] }, { "id": 1872, "type": "theorem", "label": "derived-lemma-compose-derived-functors", "categories": [ "derived" ], "title": "derived-lemma-compose-derived-functors", "contents": [ "Let $\\mathcal{A}, \\mathcal{B}, \\mathcal{C}$ be abelian categories.", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{C}$", "be left exact functors. Assume $\\mathcal{A}$, $\\mathcal{B}$ have", "enough injectives. The following are equivalent", "\\begin{enumerate}", "\\item $F(I)$ is right acyclic for $G$ for each injective object $I$", "of $\\mathcal{A}$, and", "\\item the canonical map", "$$", "t : R(G \\circ F) \\longrightarrow RG \\circ RF.", "$$", "is isomorphism of functors from $D^{+}(\\mathcal{A})$ to $D^{+}(\\mathcal{C})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (2) holds, then (1) follows by evaluating the isomorphism", "$t$ on $RF(I) = F(I)$. Conversely, assume (1) holds.", "Let $A^\\bullet$ be a bounded below complex of $\\mathcal{A}$.", "Choose an injective resolution $A^\\bullet \\to I^\\bullet$.", "The map $t$ is given (see proof of", "Lemma \\ref{lemma-compose-derived-functors-general})", "by the maps", "$$", "R(G \\circ F)(A^\\bullet) =", "(G \\circ F)(I^\\bullet) =", "G(F(I^\\bullet))) \\to", "RG(F(I^\\bullet)) =", "RG(RF(A^\\bullet))", "$$", "where the arrow is an isomorphism by", "Lemma \\ref{lemma-leray-acyclicity}." ], "refs": [ "derived-lemma-compose-derived-functors-general", "derived-lemma-leray-acyclicity" ], "ref_ids": [ 1833, 1844 ] } ], "ref_ids": [] }, { "id": 1873, "type": "theorem", "label": "derived-lemma-grothendieck-spectral-sequence", "categories": [ "derived" ], "title": "derived-lemma-grothendieck-spectral-sequence", "contents": [ "With assumptions as in Lemma \\ref{lemma-compose-derived-functors}", "and assuming the equivalent conditions (1) and (2) hold.", "Let $X$ be an object of $D^{+}(\\mathcal{A})$.", "There exists a spectral sequence $(E_r, d_r)_{r \\geq 0}$", "consisting of bigraded objects $E_r$ of $\\mathcal{C}$ with", "$d_r$ of bidegree $(r, - r + 1)$ and with", "$$", "E_2^{p, q} = R^pG(H^q(RF(X)))", "$$", "Moreover, this spectral sequence is bounded, converges to", "$H^*(R(G \\circ F)(X))$, and induces a finite filtration", "on each $H^n(R(G \\circ F)(X))$." ], "refs": [ "derived-lemma-compose-derived-functors" ], "proofs": [ { "contents": [ "We may represent $X$ by a bounded below complex $A^\\bullet$.", "Choose an injective resolution $A^\\bullet \\to I^\\bullet$.", "Choose a Cartan-Eilenberg resolution", "$F(I^\\bullet) \\to I^{\\bullet, \\bullet}$ using", "Lemma \\ref{lemma-cartan-eilenberg}.", "Apply the second spectral sequence of", "Lemma \\ref{lemma-two-ss-complex-functor}." ], "refs": [ "derived-lemma-cartan-eilenberg", "derived-lemma-two-ss-complex-functor" ], "ref_ids": [ 1870, 1871 ] } ], "ref_ids": [ 1872 ] }, { "id": 1874, "type": "theorem", "label": "derived-lemma-resolution-functor", "categories": [ "derived" ], "title": "derived-lemma-resolution-functor", "contents": [ "Let $\\mathcal{A}$ be an abelian category with enough injectives.", "Given a resolution functor $(j, i)$ there is a unique way to", "turn $j$ into a functor and $i$ into a $2$-isomorphism", "producing a $2$-commutative diagram", "$$", "\\xymatrix{", "K^{+}(\\mathcal{A}) \\ar[rd] \\ar[rr]_j & & K^{+}(\\mathcal{I}) \\ar[ld] \\\\", "& D^{+}(\\mathcal{A})", "}", "$$", "where $\\mathcal{I}$ is the full additive subcategory of $\\mathcal{A}$", "consisting of injective objects." ], "refs": [], "proofs": [ { "contents": [ "For every morphism $\\alpha : K^\\bullet \\to L^\\bullet$ of $K^{+}(\\mathcal{A})$", "there is a unique morphism", "$j(\\alpha) : j(K^\\bullet) \\to j(L^\\bullet)$ in $K^{+}(\\mathcal{I})$", "such that", "$$", "\\xymatrix{", "K^\\bullet \\ar[r]_\\alpha \\ar[d]_{i_{K^\\bullet}} &", "L^\\bullet \\ar[d]^{i_{L^\\bullet}} \\\\", "j(K^\\bullet) \\ar[r]^{j(\\alpha)} & j(L^\\bullet)", "}", "$$", "is commutative in $K^{+}(\\mathcal{A})$. To see this either use", "Lemmas \\ref{lemma-morphisms-lift} and", "\\ref{lemma-morphisms-equal-up-to-homotopy}", "or the equivalent", "Lemma \\ref{lemma-morphisms-into-injective-complex}.", "The uniqueness implies that $j$ is a functor, and the commutativity of", "the diagram implies that $i$ gives a $2$-morphism which witnesses the", "$2$-commutativity of the diagram of categories in the statement of", "the lemma." ], "refs": [ "derived-lemma-morphisms-lift", "derived-lemma-morphisms-equal-up-to-homotopy", "derived-lemma-morphisms-into-injective-complex" ], "ref_ids": [ 1853, 1854, 1855 ] } ], "ref_ids": [] }, { "id": 1875, "type": "theorem", "label": "derived-lemma-into-derived-category", "categories": [ "derived" ], "title": "derived-lemma-into-derived-category", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Assume $\\mathcal{A}$ has enough injectives.", "Then a resolution functor $j$ exists and is", "unique up to unique isomorphism of functors." ], "refs": [], "proofs": [ { "contents": [ "Consider the set of all objects $K^\\bullet$ of $K^{+}(\\mathcal{A})$.", "(Recall that by our conventions any category has a set of", "objects unless mentioned otherwise.)", "By Lemma \\ref{lemma-injective-resolutions-exist} every object", "has an injective resolution.", "By the axiom of choice we can choose for each $K^\\bullet$", "an injective resolution $i_{K^\\bullet} : K^\\bullet \\to j(K^\\bullet)$." ], "refs": [ "derived-lemma-injective-resolutions-exist" ], "ref_ids": [ 1851 ] } ], "ref_ids": [] }, { "id": 1876, "type": "theorem", "label": "derived-lemma-j-is-exact", "categories": [ "derived" ], "title": "derived-lemma-j-is-exact", "contents": [ "Let $\\mathcal{A}$ be an abelian category with enough injectives.", "Any resolution functor", "$j : K^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{I})$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "Denote $i_{K^\\bullet} : K^\\bullet \\to j(K^\\bullet)$ the", "canonical maps of Definition \\ref{definition-localization-functor}.", "First we discuss the existence of the functorial isomorphism", "$j(K^\\bullet[1]) \\to j(K^\\bullet)[1]$.", "Consider the diagram", "$$", "\\xymatrix{", "K^\\bullet[1] \\ar[d]^{i_{K^\\bullet[1]}} \\ar@{=}[rr] & &", "K^\\bullet[1] \\ar[d]^{i_{K^\\bullet}[1]} \\\\", "j(K^\\bullet[1]) \\ar@{..>}[rr]^{\\xi_{K^\\bullet}} & & j(K^\\bullet)[1]", "}", "$$", "By Lemmas \\ref{lemma-morphisms-lift}", "and \\ref{lemma-morphisms-equal-up-to-homotopy}", "there exists a unique dotted arrow $\\xi_{K^\\bullet}$ in $K^{+}(\\mathcal{I})$", "making the diagram commute in $K^{+}(\\mathcal{A})$.", "We omit the verification that this gives a functorial isomorphism.", "(Hint: use Lemma \\ref{lemma-morphisms-equal-up-to-homotopy} again.)", "\\medskip\\noindent", "Let $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$", "be a distinguished triangle of $K^{+}(\\mathcal{A})$.", "We have to show that", "$(j(K^\\bullet), j(L^\\bullet), j(M^\\bullet), j(f), j(g),", "\\xi_{K^\\bullet} \\circ j(h))$ is", "a distinguished triangle of $K^{+}(\\mathcal{I})$.", "Note that we have a commutative diagram", "$$", "\\xymatrix{", "K^\\bullet \\ar[r]_f \\ar[d] &", "L^\\bullet \\ar[r]_g \\ar[d] &", "M^\\bullet \\ar[rr]_h \\ar[d] & &", "K^\\bullet[1] \\ar[d] \\\\", "j(K^\\bullet) \\ar[r]^{j(f)} &", "j(L^\\bullet) \\ar[r]^{j(g)} &", "j(M^\\bullet) \\ar[rr]^{\\xi_{K^\\bullet} \\circ j(h)} & &", "j(K^\\bullet)[1]", "}", "$$", "in $K^{+}(\\mathcal{A})$ whose vertical arrows are the quasi-isomorphisms", "$i_K, i_L, i_M$. Hence we see that the image of", "$(j(K^\\bullet), j(L^\\bullet), j(M^\\bullet), j(f), j(g),", "\\xi_{K^\\bullet} \\circ j(h))$", "in $D^{+}(\\mathcal{A})$ is isomorphic to a distinguished triangle", "and hence a distinguished triangle by TR1. Thus we see from", "Lemma \\ref{lemma-exact-equivalence}", "that $(j(K^\\bullet), j(L^\\bullet), j(M^\\bullet), j(f), j(g),", "\\xi_{K^\\bullet} \\circ j(h))$ is a distinguished triangle in", "$K^{+}(\\mathcal{I})$." ], "refs": [ "derived-definition-localization-functor", "derived-lemma-morphisms-lift", "derived-lemma-morphisms-equal-up-to-homotopy", "derived-lemma-morphisms-equal-up-to-homotopy", "derived-lemma-exact-equivalence" ], "ref_ids": [ 1995, 1853, 1854, 1854, 1773 ] } ], "ref_ids": [] }, { "id": 1877, "type": "theorem", "label": "derived-lemma-resolution-functor-quasi-inverse", "categories": [ "derived" ], "title": "derived-lemma-resolution-functor-quasi-inverse", "contents": [ "Let $\\mathcal{A}$ be an abelian category which has enough injectives.", "Let $j$ be a resolution functor. Write", "$Q : K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{A})$ for the natural functor.", "Then $j = j' \\circ Q$ for a unique", "functor $j' : D^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{I})$ which", "is quasi-inverse to the canonical functor", "$K^{+}(\\mathcal{I}) \\to D^{+}(\\mathcal{A})$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-bounded-derived}", "$Q$ is a localization functor.", "To prove the existence of $j'$ it suffices to show that any element of", "$\\text{Qis}^{+}(\\mathcal{A})$ is mapped to an isomorphism under", "the functor $j$, see", "Lemma \\ref{lemma-universal-property-localization}.", "This is true by the remarks following", "Definition \\ref{definition-localization-functor}." ], "refs": [ "derived-lemma-bounded-derived", "derived-lemma-universal-property-localization", "derived-definition-localization-functor" ], "ref_ids": [ 1813, 1781, 1995 ] } ], "ref_ids": [] }, { "id": 1878, "type": "theorem", "label": "derived-lemma-functorial-injective-resolutions", "categories": [ "derived" ], "title": "derived-lemma-functorial-injective-resolutions", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Assume $\\mathcal{A}$ has functorial injective embeddings, see", "Homology, Definition \\ref{homology-definition-functorial-injective-embedding}.", "\\begin{enumerate}", "\\item There exists a functor", "$inj : \\text{Comp}^{+}(\\mathcal{A}) \\to \\text{InjRes}(\\mathcal{A})$", "such that $s \\circ inj = \\text{id}$.", "\\item For any functor", "$inj : \\text{Comp}^{+}(\\mathcal{A}) \\to \\text{InjRes}(\\mathcal{A})$", "such that $s \\circ inj = \\text{id}$ we obtain a resolution functor, see", "Definition \\ref{definition-localization-functor}.", "\\end{enumerate}" ], "refs": [ "homology-definition-functorial-injective-embedding", "derived-definition-localization-functor" ], "proofs": [ { "contents": [ "Let $A \\mapsto (A \\to J(A))$ be a functorial injective embedding, see", "Homology, Definition \\ref{homology-definition-functorial-injective-embedding}.", "We first note that we may assume $J(0) = 0$. Namely, if not then", "for any object $A$ we have $0 \\to A \\to 0$ which gives", "a direct sum decomposition $J(A) = J(0) \\oplus \\Ker(J(A) \\to J(0))$.", "Note that the functorial morphism $A \\to J(A)$ has to map", "into the second summand. Hence we can replace our functor", "by $J'(A) = \\Ker(J(A) \\to J(0))$ if needed.", "\\medskip\\noindent", "Let $K^\\bullet$ be a bounded below complex of $\\mathcal{A}$.", "Say $K^p = 0$ if $p < B$.", "We are going to construct a double complex $I^{\\bullet, \\bullet}$", "of injectives, together with a map $\\alpha : K^\\bullet \\to I^{\\bullet, 0}$", "such that $\\alpha$ induces a quasi-isomorphism of $K^\\bullet$", "with the associated total complex of $I^{\\bullet, \\bullet}$.", "First we set $I^{p, q} = 0$ whenever $q < 0$.", "Next, we set $I^{p, 0} = J(K^p)$ and $\\alpha^p : K^p \\to I^{p, 0}$", "the functorial embedding. Since $J$ is a functor we see that", "$I^{\\bullet, 0}$ is a complex and that $\\alpha$ is a", "morphism of complexes. Each $\\alpha^p$ is injective. And", "$I^{p, 0} = 0$ for $p < B$ because $J(0) = 0$. Next, we set", "$I^{p, 1} = J(\\Coker(K^p \\to I^{p, 0}))$. Again by functoriality", "we see that $I^{\\bullet, 1}$ is a complex. And again we get", "that $I^{p, 1} = 0$ for $p < B$. It is also clear that", "$K^p$ maps isomorphically onto $\\Ker(I^{p, 0} \\to I^{p, 1})$.", "As our third step we take $I^{p, 2} = J(\\Coker(I^{p, 0} \\to I^{p, 1}))$.", "And so on and so forth.", "\\medskip\\noindent", "At this point we can apply", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}", "to get that the map", "$$", "\\alpha : K^\\bullet \\longrightarrow \\text{Tot}(I^{\\bullet, \\bullet})", "$$", "is a quasi-isomorphism. To prove we get a functor $inj$ it", "rests to show that the construction above", "is functorial. This verification is omitted.", "\\medskip\\noindent", "Suppose we have a functor $inj$ such that $s \\circ inj = \\text{id}$.", "For every object $K^\\bullet$ of $\\text{Comp}^{+}(\\mathcal{A})$", "we can write", "$$", "inj(K^\\bullet) = (i_{K^\\bullet} : K^\\bullet \\to j(K^\\bullet))", "$$", "This provides us with a resolution functor as in", "Definition \\ref{definition-localization-functor}." ], "refs": [ "homology-definition-functorial-injective-embedding", "homology-lemma-double-complex-gives-resolution", "derived-definition-localization-functor" ], "ref_ids": [ 12184, 12106, 1995 ] } ], "ref_ids": [ 12184, 1995 ] }, { "id": 1879, "type": "theorem", "label": "derived-lemma-right-derived-functor", "categories": [ "derived" ], "title": "derived-lemma-right-derived-functor", "contents": [ "Let $\\mathcal{A}$ be an abelian category with enough injectives", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor into", "an abelian category. Let $(i, j)$ be a resolution functor, see", "Definition \\ref{definition-localization-functor}.", "The right derived functor $RF$ of $F$ fits into the following", "$2$-commutative diagram", "$$", "\\xymatrix{", "D^{+}(\\mathcal{A}) \\ar[rd]_{RF} \\ar[rr]^{j'} & &", "K^{+}(\\mathcal{I}) \\ar[ld]^F \\\\", "& D^{+}(\\mathcal{B})", "}", "$$", "where $j'$ is the functor from", "Lemma \\ref{lemma-resolution-functor-quasi-inverse}." ], "refs": [ "derived-definition-localization-functor", "derived-lemma-resolution-functor-quasi-inverse" ], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-injective-acyclic}", "we have $RF(K^\\bullet) = F(j(K^\\bullet))$." ], "refs": [ "derived-lemma-injective-acyclic" ], "ref_ids": [ 1866 ] } ], "ref_ids": [ 1995, 1877 ] }, { "id": 1880, "type": "theorem", "label": "derived-lemma-filtered-injective", "categories": [ "derived" ], "title": "derived-lemma-filtered-injective", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "An object $I$ of $\\text{Fil}^f(\\mathcal{A})$ is filtered injective", "if and only if", "there exist $a \\leq b$, injective objects $I_n$, $a \\leq n \\leq b$", "of $\\mathcal{A}$ and an isomorphism $I \\cong \\bigoplus_{a \\leq n \\leq b} I_n$", "such that $F^pI = \\bigoplus_{n \\geq p} I_n$." ], "refs": [], "proofs": [ { "contents": [ "Follows from the fact that any injection $J \\to M$ of $\\mathcal{A}$", "is split if $J$ is an injective object. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1881, "type": "theorem", "label": "derived-lemma-split-strict-monomorphism", "categories": [ "derived" ], "title": "derived-lemma-split-strict-monomorphism", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Any strict monomorphism $u : I \\to A$ of $\\text{Fil}^f(\\mathcal{A})$", "where $I$ is a filtered injective object is a split injection." ], "refs": [], "proofs": [ { "contents": [ "Let $p$ be the largest integer such that $F^pI \\not = 0$.", "In particular $\\text{gr}^p(I) = F^pI$.", "Let $I'$ be the object of $\\text{Fil}^f(\\mathcal{A})$ whose", "underlying object of $\\mathcal{A}$ is $F^pI$ and with filtration", "given by $F^nI' = 0$ for $n > p$ and $F^nI' = I' = F^pI$ for", "$n \\leq p$. Note that $I' \\to I$ is a strict monomorphism too.", "The fact that $u$ is a strict monomorphism implies that", "$F^pI \\to A/F^{p + 1}(A)$ is injective, see", "Homology, Lemma \\ref{homology-lemma-characterize-strict}.", "Choose a splitting $s : A/F^{p + 1}A \\to F^pI$ in $\\mathcal{A}$.", "The induced morphism $s' : A \\to I'$ is a strict morphism of", "filtered objects splitting the composition $I' \\to I \\to A$.", "Hence we can write $A = I' \\oplus \\Ker(s')$ and", "$I = I' \\oplus \\Ker(s'|_I)$. Note that", "$\\Ker(s'|_I) \\to \\Ker(s')$ is a strict monomorphism", "and that $\\Ker(s'|_I)$ is a filtered injective object.", "By induction on the length of the filtration on $I$ the map", "$\\Ker(s'|_I) \\to \\Ker(s')$ is a split injection.", "Thus we win." ], "refs": [ "homology-lemma-characterize-strict" ], "ref_ids": [ 12084 ] } ], "ref_ids": [] }, { "id": 1882, "type": "theorem", "label": "derived-lemma-injective-property-filtered-injective", "categories": [ "derived" ], "title": "derived-lemma-injective-property-filtered-injective", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $u : A \\to B$ be a strict monomorphism", "of $\\text{Fil}^f(\\mathcal{A})$", "and $f : A \\to I$ a morphism from $A$ into a filtered injective object", "in $\\text{Fil}^f(\\mathcal{A})$.", "Then there exists a morphism $g : B \\to I$ such that $f = g \\circ u$." ], "refs": [], "proofs": [ { "contents": [ "The pushout $f' : I \\to I \\amalg_A B$ of $f$ by $u$ is a strict", "monomorphism, see", "Homology, Lemma \\ref{homology-lemma-pushout-filtered}.", "Hence the result follows formally from", "Lemma \\ref{lemma-split-strict-monomorphism}." ], "refs": [ "homology-lemma-pushout-filtered", "derived-lemma-split-strict-monomorphism" ], "ref_ids": [ 12081, 1881 ] } ], "ref_ids": [] }, { "id": 1883, "type": "theorem", "label": "derived-lemma-strict-monomorphism-into-filtered-injective", "categories": [ "derived" ], "title": "derived-lemma-strict-monomorphism-into-filtered-injective", "contents": [ "Let $\\mathcal{A}$ be an abelian category with enough injectives.", "For any object $A$ of $\\text{Fil}^f(\\mathcal{A})$ there exists", "a strict monomorphism $A \\to I$", "where $I$ is a filtered injective object." ], "refs": [], "proofs": [ { "contents": [ "Pick $a \\leq b$ such that $\\text{gr}^p(A) = 0$ unless", "$p \\in \\{a, a + 1, \\ldots, b\\}$. For each", "$n \\in \\{a, a + 1, \\ldots, b\\}$ choose an injection", "$u_n : A/F^{n + 1}A \\to I_n$ with $I_n$ an injective object.", "Set $I = \\bigoplus_{a \\leq n \\leq b} I_n$ with filtration", "$F^pI = \\bigoplus_{n \\geq p} I_n$ and set $u : A \\to I$ equal to", "the direct sum of the maps $u_n$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1884, "type": "theorem", "label": "derived-lemma-filtered-injective-right-resolution-single-object", "categories": [ "derived" ], "title": "derived-lemma-filtered-injective-right-resolution-single-object", "contents": [ "Let $\\mathcal{A}$ be an abelian category with enough injectives.", "For any object $A$ of $\\text{Fil}^f(\\mathcal{A})$ there exists", "a filtered quasi-isomorphism $A[0] \\to I^\\bullet$", "where $I^\\bullet$ is a complex of filtered injective objects", "with $I^n = 0$ for $n < 0$." ], "refs": [], "proofs": [ { "contents": [ "First choose a strict monomorphism $u_0 : A \\to I^0$ of $A$ into a filtered", "injective object, see", "Lemma \\ref{lemma-strict-monomorphism-into-filtered-injective}.", "Next, choose a strict monomorphism", "$u_1 : \\Coker(u_0) \\to I^1$ into a filtered injective object of", "$\\mathcal{A}$. Denote $d^0$ the induced map $I^0 \\to I^1$.", "Next, choose a strict monomorphism $u_2 : \\Coker(u_1) \\to I^2$ into", "a filtered injective object of $\\mathcal{A}$. Denote $d^1$ the induced", "map $I^1 \\to I^2$. And so on. This works because each", "of the sequences", "$$", "0 \\to \\Coker(u_n) \\to I^{n + 1} \\to \\Coker(u_{n + 1}) \\to 0", "$$", "is short exact, i.e., induces a short exact sequence on applying", "$\\text{gr}$. To see this use", "Homology, Lemma \\ref{homology-lemma-characterize-strict}." ], "refs": [ "derived-lemma-strict-monomorphism-into-filtered-injective", "homology-lemma-characterize-strict" ], "ref_ids": [ 1883, 12084 ] } ], "ref_ids": [] }, { "id": 1885, "type": "theorem", "label": "derived-lemma-filtered-injective-right-resolution-map", "categories": [ "derived" ], "title": "derived-lemma-filtered-injective-right-resolution-map", "contents": [ "Let $\\mathcal{A}$ be an abelian category with enough injectives.", "Let $f : A \\to B$ be a morphism of $\\text{Fil}^f(\\mathcal{A})$.", "Given filtered quasi-isomorphisms $A[0] \\to I^\\bullet$ and", "$B[0] \\to J^\\bullet$ where $I^\\bullet, J^\\bullet$ are complexes of", "filtered injective objects with $I^n = J^n = 0$ for $n < 0$, then", "there exists a commutative diagram", "$$", "\\xymatrix{", "A[0] \\ar[r] \\ar[d] &", "B[0] \\ar[d] \\\\", "I^\\bullet \\ar[r] &", "J^\\bullet", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "As $A[0] \\to I^\\bullet$ and $C[0] \\to J^\\bullet$ are filtered", "quasi-isomorphisms we conclude that $a : A \\to I^0$, $b : B \\to J^0$", "and all the morphisms $d_I^n$, $d_J^n$ are strict, see", "Homology, Lemma \\ref{homology-lemma-filtered-acyclic}.", "We will inductively construct the maps $f^n$ in the following", "commutative diagram", "$$", "\\xymatrix{", "A \\ar[r]_a \\ar[d]_f &", "I^0 \\ar[r] \\ar[d]^{f^0} &", "I^1 \\ar[r] \\ar[d]^{f^1} &", "I^2 \\ar[r] \\ar[d]^{f^2} &", "\\ldots \\\\", "B \\ar[r]^b &", "J^0 \\ar[r] &", "J^1 \\ar[r] &", "J^2 \\ar[r] &", "\\ldots", "}", "$$", "Because $A \\to I^0$ is a strict monomorphism and because", "$J^0$ is filtered injective, we can find a morphism $f^0 : I^0 \\to J^0$", "such that $f^0 \\circ a = b \\circ f$, see", "Lemma \\ref{lemma-injective-property-filtered-injective}.", "The composition $d_J^0 \\circ b \\circ f$ is zero, hence", "$d_J^0 \\circ f^0 \\circ a = 0$, hence $d_J^0 \\circ f^0$ factors", "through a unique morphism", "$$", "\\Coker(a) = \\Coim(d_I^0) = \\Im(d_I^0) \\longrightarrow J^1.", "$$", "As $\\Im(d_I^0) \\to I^1$ is a strict monomorphism we can extend the", "displayed arrow to a morphism $f^1 : I^1 \\to J^1$ by", "Lemma \\ref{lemma-injective-property-filtered-injective}", "again. And so on." ], "refs": [ "homology-lemma-filtered-acyclic", "derived-lemma-injective-property-filtered-injective", "derived-lemma-injective-property-filtered-injective" ], "ref_ids": [ 12086, 1882, 1882 ] } ], "ref_ids": [] }, { "id": 1886, "type": "theorem", "label": "derived-lemma-filtered-injective-right-resolution-ses", "categories": [ "derived" ], "title": "derived-lemma-filtered-injective-right-resolution-ses", "contents": [ "Let $\\mathcal{A}$ be an abelian category with enough injectives.", "Let $0 \\to A \\to B \\to C \\to 0$ be a short exact sequence in", "$\\text{Fil}^f(\\mathcal{A})$.", "Given filtered quasi-isomorphisms $A[0] \\to I^\\bullet$ and", "$C[0] \\to J^\\bullet$ where $I^\\bullet, J^\\bullet$ are complexes of", "filtered injective objects with $I^n = J^n = 0$ for $n < 0$, then", "there exists a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "A[0] \\ar[r] \\ar[d] &", "B[0] \\ar[r] \\ar[d] &", "C[0] \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "I^\\bullet \\ar[r] &", "M^\\bullet \\ar[r] &", "J^\\bullet \\ar[r] &", "0", "}", "$$", "where the lower row is a termwise split sequence of complexes." ], "refs": [], "proofs": [ { "contents": [ "As $A[0] \\to I^\\bullet$ and $C[0] \\to J^\\bullet$ are filtered", "quasi-isomorphisms we conclude that $a : A \\to I^0$, $c : C \\to J^0$", "and all the morphisms $d_I^n$, $d_J^n$ are strict, see", "Homology, Lemma \\ref{lemma-filtered-acyclic}.", "We are going to step by step construct the south-east and the south", "arrows in the following commutative diagram", "$$", "\\xymatrix{", "B \\ar[r]_\\beta \\ar[rd]^b &", "C \\ar[r]_c \\ar[rd]^{\\overline{b}} &", "J^0 \\ar[d]^{\\delta^0} \\ar[r] &", "J^1 \\ar[d]^{\\delta^1} \\ar[r] & \\ldots \\\\", "A \\ar[u]^\\alpha \\ar[r]^a &", "I^0 \\ar[r] &", "I^1 \\ar[r] &", "I^2 \\ar[r] & \\ldots", "}", "$$", "As $A \\to B$ is a strict monomorphism, we can find a morphism", "$b : B \\to I^0$ such that $b \\circ \\alpha = a$, see", "Lemma \\ref{lemma-injective-property-filtered-injective}.", "As $A$ is the kernel of the strict morphism $I^0 \\to I^1$", "and $\\beta = \\Coker(\\alpha)$ we obtain a unique morphism", "$\\overline{b} : C \\to I^1$ fitting into the diagram.", "As $c$ is a strict monomorphism and $I^1$ is filtered injective", "we can find $\\delta^0 : J^0 \\to I^1$, see", "Lemma \\ref{lemma-injective-property-filtered-injective}.", "Because $B \\to C$ is a strict epimorphism and because", "$B \\to I^0 \\to I^1 \\to I^2$ is zero, we see that", "$C \\to I^1 \\to I^2$ is zero. Hence $d_I^1 \\circ \\delta^0$", "is zero on $C \\cong \\Im(c)$.", "Hence $d_I^1 \\circ \\delta^0$ factors through a unique morphism", "$$", "\\Coker(c) = \\Coim(d_J^0) = \\Im(d_J^0) \\longrightarrow I^2.", "$$", "As $I^2$ is filtered injective and $\\Im(d_J^0) \\to J^1$ is a", "strict monomorphism we can extend the displayed morphism to a morphism", "$\\delta^1 : J^1 \\to I^2$, see", "Lemma \\ref{lemma-injective-property-filtered-injective}.", "And so on. We set $M^\\bullet = I^\\bullet \\oplus J^\\bullet$", "with differential", "$$", "d_M^n =", "\\left(", "\\begin{matrix}", "d_I^n & (-1)^{n + 1}\\delta^n \\\\", "0 & d_J^n", "\\end{matrix}", "\\right)", "$$", "Finally, the map $B[0] \\to M^\\bullet$ is given by", "$b \\oplus c \\circ \\beta : M \\to I^0 \\oplus J^0$." ], "refs": [ "derived-lemma-filtered-acyclic", "derived-lemma-injective-property-filtered-injective", "derived-lemma-injective-property-filtered-injective", "derived-lemma-injective-property-filtered-injective" ], "ref_ids": [ 1819, 1882, 1882, 1882 ] } ], "ref_ids": [] }, { "id": 1887, "type": "theorem", "label": "derived-lemma-right-resolution-by-filtered-injectives", "categories": [ "derived" ], "title": "derived-lemma-right-resolution-by-filtered-injectives", "contents": [ "Let $\\mathcal{A}$ be an abelian category with enough injectives.", "For every $K^\\bullet \\in K^{+}(\\text{Fil}^f(\\mathcal{A}))$", "there exists a filtered quasi-isomorphism $K^\\bullet \\to I^\\bullet$", "with $I^\\bullet$ bounded below,", "each $I^n$ a filtered injective object, and", "each $K^n \\to I^n$ a strict monomorphism." ], "refs": [], "proofs": [ { "contents": [ "After replacing $K^\\bullet$ by a shift (which is harmless for the proof)", "we may assume that $K^n = 0$ for $n < 0$. Consider the", "short exact sequences", "$$", "\\begin{matrix}", "0 \\to \\Ker(d_K^0) \\to K^0 \\to \\Coim(d_K^0) \\to 0 \\\\", "0 \\to \\Ker(d_K^1) \\to K^1 \\to \\Coim(d_K^1) \\to 0 \\\\", "0 \\to \\Ker(d_K^2) \\to K^2 \\to \\Coim(d_K^2) \\to 0 \\\\", "\\ldots", "\\end{matrix}", "$$", "of the exact category $\\text{Fil}^f(\\mathcal{A})$", "and the maps $u_i : \\Coim(d_K^i) \\to \\Ker(d_K^{i + 1})$.", "For each $i \\geq 0$ we may choose filtered quasi-isomorphisms", "$$", "\\begin{matrix}", "\\Ker(d_K^i)[0] \\to I_{ker, i}^\\bullet \\\\", "\\Coim(d_K^i)[0] \\to I_{coim, i}^\\bullet", "\\end{matrix}", "$$", "with $I_{ker, i}^n, I_{coim, i}^n$ filtered injective and zero for $n < 0$, see", "Lemma \\ref{lemma-filtered-injective-right-resolution-single-object}.", "By", "Lemma \\ref{lemma-filtered-injective-right-resolution-map}", "we may lift $u_i$ to a morphism of complexes", "$u_i^\\bullet : I_{coim, i}^\\bullet \\to I_{ker, i + 1}^\\bullet$.", "Finally, for each $i \\geq 0$ we may complete the diagrams", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Ker(d_K^i)[0] \\ar[r] \\ar[d] &", "K^i[0] \\ar[r] \\ar[d] &", "\\Coim(d_K^i)[0] \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "I_{ker, i}^\\bullet \\ar[r]^{\\alpha_i} &", "I_i^\\bullet \\ar[r]^{\\beta_i} &", "I_{coim, i}^\\bullet \\ar[r] &", "0", "}", "$$", "with the lower sequence a termwise split exact sequence, see", "Lemma \\ref{lemma-filtered-injective-right-resolution-ses}.", "For $i \\geq 0$ set $d_i : I_i^\\bullet \\to I_{i + 1}^\\bullet$", "equal to $d_i = \\alpha_{i + 1} \\circ u_i^\\bullet \\circ \\beta_i$.", "Note that $d_i \\circ d_{i - 1} = 0$ because", "$\\beta_i \\circ \\alpha_i = 0$. Hence we have constructed", "a commutative diagram", "$$", "\\xymatrix{", "I_0^\\bullet \\ar[r] &", "I_1^\\bullet \\ar[r] &", "I_2^\\bullet \\ar[r] & \\ldots \\\\", "K^0[0] \\ar[r] \\ar[u] &", "K^1[0] \\ar[r] \\ar[u] &", "K^2[0] \\ar[r] \\ar[u] &", "\\ldots", "}", "$$", "Here the vertical arrows are filtered quasi-isomorphisms.", "The upper row is a complex of complexes and each complex consists of", "filtered injective objects with no nonzero objects in degree $< 0$.", "Thus we obtain a double complex by setting $I^{a, b} = I_a^b$ and using", "$$", "d_1^{a, b} : I^{a, b} = I_a^b \\to I_{a + 1}^b = I^{a + 1, b}", "$$", "the map $d_a^b$ and using for", "$$", "d_2^{a, b} : I^{a, b} = I_a^b \\to I_a^{b + 1} = I^{a, b + 1}", "$$", "the map $d_{I_a}^b$. Denote $\\text{Tot}(I^{\\bullet, \\bullet})$", "the total complex associated to this double complex, see", "Homology, Definition \\ref{homology-definition-associated-simple-complex}.", "Observe that the maps $K^n[0] \\to I_n^\\bullet$ come from maps", "$K^n \\to I^{n, 0}$ which give rise to a map of complexes", "$$", "K^\\bullet \\longrightarrow \\text{Tot}(I^{\\bullet, \\bullet})", "$$", "We claim this is a filtered quasi-isomorphism.", "As $\\text{gr}(-)$ is an additive functor, we see that", "$\\text{gr}(\\text{Tot}(I^{\\bullet, \\bullet})) =", "\\text{Tot}(\\text{gr}(I^{\\bullet, \\bullet}))$.", "Thus we can use", "Homology,", "Lemma \\ref{homology-lemma-double-complex-gives-resolution}", "to conclude that", "$\\text{gr}(K^\\bullet) \\to \\text{gr}(\\text{Tot}(I^{\\bullet, \\bullet}))$", "is a quasi-isomorphism as desired." ], "refs": [ "derived-lemma-filtered-injective-right-resolution-single-object", "derived-lemma-filtered-injective-right-resolution-map", "derived-lemma-filtered-injective-right-resolution-ses", "homology-definition-associated-simple-complex", "homology-lemma-double-complex-gives-resolution" ], "ref_ids": [ 1884, 1885, 1886, 12164, 12106 ] } ], "ref_ids": [] }, { "id": 1888, "type": "theorem", "label": "derived-lemma-filtered-acyclic-is-zero", "categories": [ "derived" ], "title": "derived-lemma-filtered-acyclic-is-zero", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $K^\\bullet, I^\\bullet \\in K(\\text{Fil}^f(\\mathcal{A}))$.", "Assume $K^\\bullet$ is filtered acyclic and", "$I^\\bullet$ bounded below and consisting of filtered injective objects.", "Any morphism $K^\\bullet \\to I^\\bullet$ is homotopic to zero:", "$\\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(K^\\bullet, I^\\bullet) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha : K^\\bullet \\to I^\\bullet$ be a morphism of", "complexes. Assume that $\\alpha^j = 0$ for $j < n$.", "We will show that there exists a morphism $h : K^{n + 1} \\to I^n$", "such that $\\alpha^n = h \\circ d$. Thus $\\alpha$ will be homotopic", "to the morphism of complexes $\\beta$ defined by", "$$", "\\beta^j =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & j \\leq n \\\\", "\\alpha^{n + 1} - d \\circ h & \\text{if} & j = n + 1 \\\\", "\\alpha^j & \\text{if} & j > n + 1", "\\end{matrix}", "\\right.", "$$", "This will clearly prove the lemma (by induction).", "To prove the existence of $h$ note that", "$\\alpha^n \\circ d_K^{n - 1} = 0$ since", "$\\alpha^{n - 1} = 0$. Since $K^\\bullet$ is filtered acyclic", "we see that $d_K^{n - 1}$ and $d_K^n$ are strict and that", "$$", "0 \\to \\Im(d_K^{n - 1}) \\to K^n \\to \\Im(d_K^n) \\to 0", "$$", "is an exact sequence of the exact category $\\text{Fil}^f(\\mathcal{A})$, see", "Homology, Lemma \\ref{homology-lemma-filtered-acyclic}.", "Hence we can think of $\\alpha^n$ as a map into $I^n$ defined", "on $\\Im(d_K^n)$.", "Using that $\\Im(d_K^n) \\to K^{n + 1}$ is a strict monomorphism", "and that $I^n$ is filtered injective we may lift this map to a map", "$h : K^{n + 1} \\to I^n$ as desired, see", "Lemma \\ref{lemma-injective-property-filtered-injective}." ], "refs": [ "homology-lemma-filtered-acyclic", "derived-lemma-injective-property-filtered-injective" ], "ref_ids": [ 12086, 1882 ] } ], "ref_ids": [] }, { "id": 1889, "type": "theorem", "label": "derived-lemma-morphisms-into-filtered-injective-complex", "categories": [ "derived" ], "title": "derived-lemma-morphisms-into-filtered-injective-complex", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $I^\\bullet \\in K(\\text{Fil}^f(\\mathcal{A}))$", "be a bounded below complex consisting of", "filtered injective objects.", "\\begin{enumerate}", "\\item Let $\\alpha : K^\\bullet \\to L^\\bullet$ in $K(\\text{Fil}^f(\\mathcal{A}))$", "be a filtered quasi-isomorphism.", "Then the map", "$$", "\\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(L^\\bullet, I^\\bullet)", "\\to", "\\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(K^\\bullet, I^\\bullet)", "$$", "is bijective.", "\\item Let $L^\\bullet \\in K(\\text{Fil}^f(\\mathcal{A}))$. Then", "$$", "\\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(L^\\bullet, I^\\bullet)", "=", "\\Hom_{DF(\\mathcal{A})}(L^\\bullet, I^\\bullet).", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Note that", "$$", "(K^\\bullet, L^\\bullet, C(\\alpha)^\\bullet, \\alpha, i, -p)", "$$", "is a distinguished triangle in $K(\\text{Fil}^f(\\mathcal{A}))$", "(Lemma \\ref{lemma-the-same-up-to-isomorphisms})", "and $C(\\alpha)^\\bullet$ is a filtered acyclic complex", "(Lemma \\ref{lemma-filtered-acyclic}).", "Then", "$$", "\\xymatrix{", "\\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(C(\\alpha)^\\bullet, I^\\bullet) \\ar[r] &", "\\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(L^\\bullet, I^\\bullet) \\ar[r] &", "\\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(K^\\bullet, I^\\bullet) \\ar[lld] \\\\", "\\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(C(\\alpha)^\\bullet[-1], I^\\bullet)", "}", "$$", "is an exact sequence of abelian groups, see", "Lemma \\ref{lemma-representable-homological}.", "At this point", "Lemma \\ref{lemma-filtered-acyclic-is-zero}", "guarantees that the outer two groups are zero and hence", "$\\Hom_{K(\\mathcal{A})}(L^\\bullet, I^\\bullet) =", "\\Hom_{K(\\mathcal{A})}(K^\\bullet, I^\\bullet)$.", "\\medskip\\noindent", "Proof of (2).", "Let $a$ be an element of the right hand side.", "We may represent $a = \\gamma\\alpha^{-1}$ where", "$\\alpha : K^\\bullet \\to L^\\bullet$", "is a filtered quasi-isomorphism and $\\gamma : K^\\bullet \\to I^\\bullet$", "is a map of complexes. By part (1)", "we can find a morphism $\\beta : L^\\bullet \\to I^\\bullet$ such that", "$\\beta \\circ \\alpha$ is homotopic to $\\gamma$. This proves that the", "map is surjective. Let $b$ be an element of the left hand side", "which maps to zero in the right hand side. Then $b$ is the homotopy class", "of a morphism $\\beta : L^\\bullet \\to I^\\bullet$ such that there exists a", "filtered quasi-isomorphism $\\alpha : K^\\bullet \\to L^\\bullet$ with", "$\\beta \\circ \\alpha$ homotopic to zero. Then part (1)", "shows that $\\beta$ is homotopic to zero also, i.e., $b = 0$." ], "refs": [ "derived-lemma-the-same-up-to-isomorphisms", "derived-lemma-filtered-acyclic", "derived-lemma-representable-homological", "derived-lemma-filtered-acyclic-is-zero" ], "ref_ids": [ 1802, 1819, 1758, 1888 ] } ], "ref_ids": [] }, { "id": 1890, "type": "theorem", "label": "derived-lemma-filtered-localization-functor", "categories": [ "derived" ], "title": "derived-lemma-filtered-localization-functor", "contents": [ "Let $\\mathcal{A}$ be an abelian category with enough injectives.", "Let $\\mathcal{I}^f \\subset \\text{Fil}^f(\\mathcal{A})$", "denote the strictly full additive subcategory whose objects are", "the filtered injective objects. The canonical functor", "$$", "K^{+}(\\mathcal{I}^f)", "\\longrightarrow", "DF^{+}(\\mathcal{A})", "$$", "is exact, fully faithful and essentially surjective, i.e., an", "equivalence of triangulated categories. Furthermore the diagrams", "$$", "\\xymatrix{", "K^{+}(\\mathcal{I}^f) \\ar[d]_{\\text{gr}^p} \\ar[r] &", "DF^{+}(\\mathcal{A}) \\ar[d]_{\\text{gr}^p} \\\\", "K^{+}(\\mathcal{I}) \\ar[r] &", "D^{+}(\\mathcal{A})", "}", "\\quad", "\\xymatrix{", "K^{+}(\\mathcal{I}^f) \\ar[d]^{\\text{forget }F} \\ar[r] &", "DF^{+}(\\mathcal{A}) \\ar[d]^{\\text{forget }F} \\\\", "K^{+}(\\mathcal{I}) \\ar[r] &", "D^{+}(\\mathcal{A})", "}", "$$", "are commutative, where $\\mathcal{I} \\subset \\mathcal{A}$ is the", "strictly full additive subcategory whose objects are", "the injective objects." ], "refs": [], "proofs": [ { "contents": [ "The functor $K^{+}(\\mathcal{I}^f) \\to DF^{+}(\\mathcal{A})$", "is essentially surjective by", "Lemma \\ref{lemma-right-resolution-by-filtered-injectives}.", "It is fully faithful by", "Lemma \\ref{lemma-morphisms-into-filtered-injective-complex}.", "It is an exact functor by our definitions regarding distinguished", "triangles.", "The commutativity of the squares is immediate." ], "refs": [ "derived-lemma-right-resolution-by-filtered-injectives", "derived-lemma-morphisms-into-filtered-injective-complex" ], "ref_ids": [ 1887, 1889 ] } ], "ref_ids": [] }, { "id": 1891, "type": "theorem", "label": "derived-lemma-ss-filtered-derived", "categories": [ "derived" ], "title": "derived-lemma-ss-filtered-derived", "contents": [ "Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories. Let", "$T : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor.", "Assume $\\mathcal{A}$ has enough injectives.", "Let $(K^\\bullet, F)$ be an object of", "$\\text{Comp}^{+}(\\text{Fil}^f(\\mathcal{A}))$.", "There exists a spectral sequence $(E_r, d_r)_{r\\geq 0}$", "consisting of bigraded objects $E_r$ of $\\mathcal{B}$", "and $d_r$ of bidegree $(r, - r + 1)$ and with", "$$", "E_1^{p, q} = R^{p + q}T(\\text{gr}^p(K^\\bullet))", "$$", "Moreover, this spectral sequence is bounded, converges", "to $R^*T(K^\\bullet)$, and induces a finite", "filtration on each $R^nT(K^\\bullet)$. The construction", "of this spectral sequence is functorial in the object", "$K^\\bullet$ of $\\text{Comp}^{+}(\\text{Fil}^f(\\mathcal{A}))$", "and the terms $(E_r, d_r)$ for $r \\geq 1$ do not depend", "on any choices." ], "refs": [], "proofs": [ { "contents": [ "Choose a filtered quasi-isomorphism $K^\\bullet \\to I^\\bullet$", "with $I^\\bullet$ a bounded below complex of filtered injective objects, see", "Lemma \\ref{lemma-right-resolution-by-filtered-injectives}.", "Consider the complex $RT(K^\\bullet) = T_{ext}(I^\\bullet)$, see", "(\\ref{equation-definition-filtered-derived-functor}).", "Thus we can consider the spectral sequence", "$(E_r, d_r)_{r \\geq 0}$ associated to", "this as a filtered complex in $\\mathcal{B}$, see", "Homology, Section \\ref{homology-section-filtered-complex}.", "By", "Homology, Lemma \\ref{homology-lemma-spectral-sequence-filtered-complex}", "we have $E_1^{p, q} = H^{p + q}(\\text{gr}^p(T(I^\\bullet)))$.", "By Equation (\\ref{equation-decompose}) we have", "$E_1^{p, q} = H^{p + q}(T(\\text{gr}^p(I^\\bullet)))$, and", "by definition of a filtered injective resolution the", "map $\\text{gr}^p(K^\\bullet) \\to \\text{gr}^p(I^\\bullet)$", "is an injective resolution. Hence", "$E_1^{p, q} = R^{p + q}T(\\text{gr}^p(K^\\bullet))$.", "\\medskip\\noindent", "On the other hand, each $I^n$ has a finite filtration and hence", "each $T(I^n)$ has a finite filtration. Thus we may apply", "Homology, Lemma \\ref{homology-lemma-biregular-ss-converges}", "to conclude that the spectral sequence is bounded, converges to", "$H^n(T(I^\\bullet)) = R^nT(K^\\bullet)$", "moreover inducing finite filtrations on each of the terms.", "\\medskip\\noindent", "Suppose that $K^\\bullet \\to L^\\bullet$ is a morphism of", "$\\text{Comp}^{+}(\\text{Fil}^f(\\mathcal{A}))$.", "Choose a filtered quasi-isomorphism $L^\\bullet \\to J^\\bullet$", "with $J^\\bullet$ a bounded below complex of filtered injective", "objects, see", "Lemma \\ref{lemma-right-resolution-by-filtered-injectives}.", "By our results above,", "for example", "Lemma \\ref{lemma-morphisms-into-filtered-injective-complex},", "there exists a diagram", "$$", "\\xymatrix{", "K^\\bullet \\ar[r] \\ar[d] & L^\\bullet \\ar[d] \\\\", "I^\\bullet \\ar[r] & J^\\bullet", "}", "$$", "which commutes up to homotopy. Hence we get a morphism of filtered", "complexes $T(I^\\bullet) \\to T(J^\\bullet)$ which gives rise to the", "morphism of spectral sequences, see", "Homology,", "Lemma \\ref{homology-lemma-spectral-sequence-filtered-complex-functorial}.", "The last statement follows from this." ], "refs": [ "derived-lemma-right-resolution-by-filtered-injectives", "homology-lemma-spectral-sequence-filtered-complex", "homology-lemma-biregular-ss-converges", "derived-lemma-right-resolution-by-filtered-injectives", "derived-lemma-morphisms-into-filtered-injective-complex", "homology-lemma-spectral-sequence-filtered-complex-functorial" ], "ref_ids": [ 1887, 12095, 12101, 1887, 1889, 12097 ] } ], "ref_ids": [] }, { "id": 1892, "type": "theorem", "label": "derived-lemma-compute-ext-resolutions", "categories": [ "derived" ], "title": "derived-lemma-compute-ext-resolutions", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $X^\\bullet, Y^\\bullet \\in \\Ob(K(\\mathcal{A}))$.", "\\begin{enumerate}", "\\item Let $Y^\\bullet \\to I^\\bullet$ be an injective resolution", "(Definition \\ref{definition-injective-resolution}). Then", "$$", "\\Ext^i_\\mathcal{A}(X^\\bullet, Y^\\bullet) =", "\\Hom_{K(\\mathcal{A})}(X^\\bullet, I^\\bullet[i]).", "$$", "\\item Let $P^\\bullet \\to X^\\bullet$ be a projective resolution", "(Definition \\ref{definition-projective-resolution}). Then", "$$", "\\Ext^i_\\mathcal{A}(X^\\bullet, Y^\\bullet) =", "\\Hom_{K(\\mathcal{A})}(P^\\bullet[-i], Y^\\bullet).", "$$", "\\end{enumerate}" ], "refs": [ "derived-definition-injective-resolution", "derived-definition-projective-resolution" ], "proofs": [ { "contents": [ "Follows immediately from", "Lemma \\ref{lemma-morphisms-into-injective-complex}", "and", "Lemma \\ref{lemma-morphisms-from-projective-complex}." ], "refs": [ "derived-lemma-morphisms-into-injective-complex", "derived-lemma-morphisms-from-projective-complex" ], "ref_ids": [ 1855, 1862 ] } ], "ref_ids": [ 1992, 1993 ] }, { "id": 1893, "type": "theorem", "label": "derived-lemma-negative-exts", "categories": [ "derived" ], "title": "derived-lemma-negative-exts", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item Let $X$, $Y$ be objects of $D(\\mathcal{A})$. Given $a, b \\in \\mathbf{Z}$", "such that $H^i(X) = 0$ for $i > a$ and $H^j(Y) = 0$", "for $j < b$, we have $\\Ext^n_\\mathcal{A}(X, Y) = 0$ for", "$n < b - a$ and", "$$", "\\Ext^{b - a}_\\mathcal{A}(X, Y) = \\Hom_\\mathcal{A}(H^a(X), H^b(Y))", "$$", "\\item Let $A, B \\in \\Ob(\\mathcal{A})$.", "For $i < 0$ we have $\\Ext^i_\\mathcal{A}(B, A) = 0$.", "We have $\\Ext^0_\\mathcal{A}(B, A) = \\Hom_\\mathcal{A}(B, A)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose complexes $X^\\bullet$ and $Y^\\bullet$ representing $X$ and $Y$.", "Since $Y^\\bullet \\to \\tau_{\\geq b}Y^\\bullet$ is a quasi-isomorphism,", "we may assume that $Y^j = 0$ for $j < b$.", "Let $L^\\bullet \\to X^\\bullet$ be any quasi-isomorphism.", "Then $\\tau_{\\leq a}L^\\bullet \\to X^\\bullet$", "is a quasi-isomorphism. Hence a morphism $X \\to Y[n]$", "in $D(\\mathcal{A})$ can be represented as $fs^{-1}$ where", "$s : L^\\bullet \\to X^\\bullet$ is a quasi-isomorphism,", "$f : L^\\bullet \\to Y^\\bullet[n]$ a morphism, and", "$L^i = 0$ for $i < a$. Note that $f$ maps $L^i$ to $Y^{i + n}$.", "Thus $f = 0$ if $n < b - a$ because always either $L^i$ or $Y^{i + n}$ is", "zero. If $n = b - a$, then $f$ corresponds exactly to a morphism", "$H^a(X) \\to H^b(Y)$. Part (2) is a special case of (1)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1894, "type": "theorem", "label": "derived-lemma-yoneda-extension", "categories": [ "derived" ], "title": "derived-lemma-yoneda-extension", "contents": [ "Let $\\mathcal{A}$ be an abelian category with objects $A$, $B$.", "Any element in $\\Ext^i_\\mathcal{A}(B, A)$ is $\\delta(E)$", "for some degree $i$ Yoneda extension of $B$ by $A$.", "Given two Yoneda extensions $E$, $E'$ of the same degree", "then $E$ is equivalent to $E'$ if and only if $\\delta(E) = \\delta(E')$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\xi : B[0] \\to A[i]$ be an element of $\\Ext^i_\\mathcal{A}(B, A)$.", "We may write $\\xi = f s^{-1}$ for some quasi-isomorphism", "$s : L^\\bullet \\to B[0]$ and map $f : L^\\bullet \\to A[i]$.", "After replacing $L^\\bullet$ by $\\tau_{\\leq 0}L^\\bullet$ we may assume", "that $L^j = 0$ for $j > 0$. Picture", "$$", "\\xymatrix{", "L^{- i - 1} \\ar[r] & L^{-i} \\ar[r] \\ar[d] & \\ldots \\ar[r] &", "L^0 \\ar[r] & B \\ar[r] & 0 \\\\", "& A", "}", "$$", "Then setting $Z_{i - 1} = (L^{- i + 1} \\oplus A)/L^{-i}$ and", "$Z_j = L^{-j}$ for $j = i - 2, \\ldots, 0$ we see that we obtain a", "degree $i$ extension $E$ of $B$ by $A$ whose class $\\delta(E)$ equals", "$\\xi$.", "\\medskip\\noindent", "It is immediate from the definitions that equivalent Yoneda extensions", "have the same class. Suppose that", "$E : 0 \\to A \\to Z_{i - 1} \\to Z_{i - 2} \\to \\ldots \\to Z_0 \\to B \\to 0$ and", "$E' : 0 \\to A \\to Z'_{i - 1} \\to Z'_{i - 2} \\to \\ldots \\to Z'_0 \\to B \\to 0$", "are Yoneda extensions with the same class.", "By construction of $D(\\mathcal{A})$ as the localization", "of $K(\\mathcal{A})$ at the set of quasi-isomorphisms, this means there", "exists a complex $L^\\bullet$ and quasi-isomorphisms", "$$", "t : L^\\bullet \\to", "(\\ldots \\to 0 \\to A \\to Z_{i - 1} \\to \\ldots \\to Z_0 \\to 0 \\to \\ldots)", "$$", "and", "$$", "t' : L^\\bullet \\to", "(\\ldots \\to 0 \\to A \\to Z'_{i - 1} \\to \\ldots \\to Z'_0 \\to 0 \\to \\ldots)", "$$", "such that $s \\circ t = s' \\circ t'$ and $f \\circ t = f' \\circ t'$, see", "Categories, Section \\ref{categories-section-localization}.", "Let $E''$ be the degree $i$ extension of $B$ by $A$ constructed from", "the pair $L^\\bullet \\to B[0]$ and $L^\\bullet \\to A[i]$ in the first", "paragraph of the proof. Then the reader sees readily that there exists", "``morphisms'' of degree $i$ Yoneda extensions $E'' \\to E$ and $E'' \\to E'$", "as in the definition of equivalent Yoneda extensions (details omitted).", "This finishes the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1895, "type": "theorem", "label": "derived-lemma-ext-1", "categories": [ "derived" ], "title": "derived-lemma-ext-1", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $A$, $B$ be objects", "of $\\mathcal{A}$. Then $\\Ext^1_\\mathcal{A}(B, A)$ is", "the group $\\Ext_\\mathcal{A}(B, A)$ constructed in", "Homology, Definition \\ref{homology-definition-ext-group}." ], "refs": [ "homology-definition-ext-group" ], "proofs": [ { "contents": [ "This is the case $i = 1$ of", "Lemma \\ref{lemma-yoneda-extension}." ], "refs": [ "derived-lemma-yoneda-extension" ], "ref_ids": [ 1894 ] } ], "ref_ids": [ 12142 ] }, { "id": 1896, "type": "theorem", "label": "derived-lemma-higher-ext-zero", "categories": [ "derived" ], "title": "derived-lemma-higher-ext-zero", "contents": [ "Let $\\mathcal{A}$ be an abelian category and let $p \\geq 0$.", "If $\\Ext^p_\\mathcal{A}(B, A) = 0$ for any pair of objects $A$, $B$", "of $\\mathcal{A}$, then $\\Ext^i_\\mathcal{A}(B, A) = 0$ for", "$i \\geq p$ and any pair of objects $A$, $B$ of $\\mathcal{A}$." ], "refs": [], "proofs": [ { "contents": [ "For $i > p$ write any class $\\xi$ as $\\delta(E)$", "where $E$ is a Yoneda extension", "$$", "E : 0 \\to A \\to Z_{i - 1} \\to Z_{i - 2} \\to \\ldots \\to Z_0 \\to B \\to 0", "$$", "This is possible by Lemma \\ref{lemma-yoneda-extension}.", "Set $C = \\Ker(Z_{p - 1} \\to Z_p) = \\Im(Z_p \\to Z_{p - 1})$.", "Then $\\delta(E)$ is the composition of $\\delta(E')$ and $\\delta(E'')$", "where", "$$", "E' : 0 \\to C \\to Z_{p - 1} \\to \\ldots \\to Z_0 \\to B \\to 0", "$$", "and", "$$", "E'' : 0 \\to A \\to Z_{i - 1} \\to Z_{i - 2} \\to \\ldots \\to Z_p \\to C \\to 0", "$$", "Since $\\delta(E') \\in \\Ext^p_\\mathcal{A}(B, C) = 0$", "we conclude." ], "refs": [ "derived-lemma-yoneda-extension" ], "ref_ids": [ 1894 ] } ], "ref_ids": [] }, { "id": 1897, "type": "theorem", "label": "derived-lemma-ext-2-zero", "categories": [ "derived" ], "title": "derived-lemma-ext-2-zero", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Assume $\\Ext^2_\\mathcal{A}(B, A) = 0$", "for any pair of objects $A$, $B$ of $\\mathcal{A}$.", "Then any object $K$ of $D^b(\\mathcal{A})$ is isomorphic to the direct", "sum of its cohomologies: $K \\cong \\bigoplus H^i(K)[-i]$." ], "refs": [], "proofs": [ { "contents": [ "Choose $a, b$ such that $H^i(K) = 0$ for $i \\not \\in [a, b]$.", "We will prove the lemma by induction on $b - a$. If $b - a \\leq 0$,", "then the result is clear. If $b - a > 0$, then we look at the", "distinguished triangle of truncations", "$$", "\\tau_{\\leq b - 1}K \\to K \\to H^b(K)[-b] \\to (\\tau_{\\leq b - 1}K)[1]", "$$", "see Remark \\ref{remark-truncation-distinguished-triangle}.", "By Lemma \\ref{lemma-split} if the last arrow is zero, then", "$K \\cong \\tau_{\\leq b - 1}K \\oplus H^b(K)[-b]$ and we win", "by induction. Again using induction we see that", "$$", "\\Hom_{D(\\mathcal{A})}(H^b(K)[-b], (\\tau_{\\leq b - 1}K)[1]) =", "\\bigoplus\\nolimits_{i < b} \\Ext_\\mathcal{A}^{b - i + 1}(H^b(K), H^i(K))", "$$", "Since $\\Ext^i_\\mathcal{A}(B, A) = 0$ for $i \\geq 2$", "and any pair of objects $A, B$ of $\\mathcal{A}$ by", "our assumption and Lemma \\ref{lemma-higher-ext-zero} we are done." ], "refs": [ "derived-remark-truncation-distinguished-triangle", "derived-lemma-split", "derived-lemma-higher-ext-zero" ], "ref_ids": [ 2016, 1766, 1896 ] } ], "ref_ids": [] }, { "id": 1898, "type": "theorem", "label": "derived-lemma-K-bounded-derived", "categories": [ "derived" ], "title": "derived-lemma-K-bounded-derived", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Then there is a canonical", "identification $K_0(D^b(\\mathcal{A})) = K_0(\\mathcal{A})$", "of zeroth $K$-groups." ], "refs": [], "proofs": [ { "contents": [ "Given an object $A$ of $\\mathcal{A}$ denote $A[0]$ the object $A$", "viewed as a complex sitting in degree $0$.", "If $0 \\to A \\to A' \\to A'' \\to 0$ is a short", "exact sequence, then we get a distinguished triangle", "$A[0] \\to A'[0] \\to A''[0] \\to A[1]$, see", "Section \\ref{section-canonical-delta-functor}.", "This shows that we obtain a map $K_0(\\mathcal{A}) \\to K_0(D^b(\\mathcal{A}))$", "by sending $[A]$ to $[A[0]]$ with apologies for the horrendous notation.", "\\medskip\\noindent", "On the other hand, given an object $X$ of $D^b(\\mathcal{A})$ we can", "consider the element", "$$", "c(X) = \\sum (-1)^i[H^i(X)] \\in K_0(\\mathcal{A})", "$$", "Given a distinguished triangle $X \\to Y \\to Z$ the long exact sequence", "of cohomology (\\ref{equation-long-exact-cohomology-sequence-D})", "and the relations in $K_0(\\mathcal{A})$ show that", "$c(Y) = c(X) + c(Z)$. Thus $c$ factors through a map", "$c : K_0(D^b(\\mathcal{A})) \\to K_0(\\mathcal{A})$.", "\\medskip\\noindent", "We want to show that the two maps above are mutually inverse.", "It is clear that the composition $K_0(\\mathcal{A}) \\to", "K_0(D^b(\\mathcal{A})) \\to K_0(\\mathcal{A})$ is the identity.", "Suppose that $X^\\bullet$ is a bounded complex of $\\mathcal{A}$.", "The existence of the distinguished triangles of ``stupid truncations'' (see", "Homology, Section \\ref{homology-section-truncations})", "$$", "\\sigma_{\\geq n}X^\\bullet \\to \\sigma_{\\geq n - 1}X^\\bullet \\to", "X^{n - 1}[-n + 1] \\to (\\sigma_{\\geq n}X^\\bullet)[1]", "$$", "and induction show that", "$$", "[X^\\bullet] = \\sum (-1)^i[X^i[0]]", "$$", "in $K_0(D^b(\\mathcal{A}))$ (with again apologies for the notation).", "It follows that the composition", "$K_0(\\mathcal{A}) \\to K_0(D^b(\\mathcal{A}))$ is surjective", "which finishes the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1899, "type": "theorem", "label": "derived-lemma-map-K", "categories": [ "derived" ], "title": "derived-lemma-map-K", "contents": [ "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of triangulated", "categories. Then $F$ induces a group homomorphism", "$K_0(\\mathcal{D}) \\to K_0(\\mathcal{D}')$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1900, "type": "theorem", "label": "derived-lemma-homological-map-K", "categories": [ "derived" ], "title": "derived-lemma-homological-map-K", "contents": [ "Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor", "from a triangulated category to an abelian category. Assume that", "for any $X$ in $\\mathcal{D}$ only a finite number of the objects", "$H(X[i])$ are nonzero in $\\mathcal{A}$. Then $H$ induces a group homomorphism", "$K_0(\\mathcal{D}) \\to K_0(\\mathcal{A})$ sending $[X]$ to", "$\\sum (-1)^i[H(X[i])]$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1901, "type": "theorem", "label": "derived-lemma-DBA-map-K", "categories": [ "derived" ], "title": "derived-lemma-DBA-map-K", "contents": [ "Let $\\mathcal{B}$ be a weak Serre subcategory of the abelian category", "$\\mathcal{A}$. Then there are canonical maps", "$$", "K_0(\\mathcal{B}) \\longrightarrow", "K_0(D^b_\\mathcal{B}(\\mathcal{A})) \\longrightarrow", "K_0(\\mathcal{B})", "$$", "whose composition is zero. The second arrow", "sends the class $[X]$ of the object $X$ to the element", "$\\sum (-1)^i[H^i(X)]$ of $K_0(\\mathcal{B})$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1902, "type": "theorem", "label": "derived-lemma-bilinear-map-K", "categories": [ "derived" ], "title": "derived-lemma-bilinear-map-K", "contents": [ "Let $\\mathcal{D}$, $\\mathcal{D}'$, $\\mathcal{D}''$ be triangulated categories.", "Let", "$$", "\\otimes : \\mathcal{D} \\times \\mathcal{D}' \\longrightarrow \\mathcal{D}''", "$$", "be a functor such that for fixed $X$ in $\\mathcal{D}$ the functor", "$X \\otimes - : \\mathcal{D}' \\to \\mathcal{D}''$ is an exact functor and", "for fixed $X'$ in $\\mathcal{D}'$ the functor", "$- \\otimes X' : \\mathcal{D} \\to \\mathcal{D}''$ is an exact functor. Then", "$\\otimes$ induces a bilinear map", "$K_0(\\mathcal{D}) \\times K_0(\\mathcal{D}') \\to K_0(\\mathcal{D}'')$", "which sends $([X], [X'])$ to $[X \\otimes X']$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1903, "type": "theorem", "label": "derived-lemma-special-direct-system", "categories": [ "derived" ], "title": "derived-lemma-special-direct-system", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let", "$\\mathcal{P} \\subset \\Ob(\\mathcal{A})$ be a subset.", "Assume $\\mathcal{P}$ contains $0$, is closed under (finite) direct sums,", "and every object of $\\mathcal{A}$ is a quotient of an", "element of $\\mathcal{P}$. Let $K^\\bullet$ be a complex.", "There exists a commutative diagram", "$$", "\\xymatrix{", "P_1^\\bullet \\ar[d] \\ar[r] & P_2^\\bullet \\ar[d] \\ar[r] & \\ldots \\\\", "\\tau_{\\leq 1}K^\\bullet \\ar[r] & \\tau_{\\leq 2}K^\\bullet \\ar[r] & \\ldots", "}", "$$", "in the category of complexes such that", "\\begin{enumerate}", "\\item the vertical arrows are quasi-isomorphisms and termwise surjective,", "\\item $P_n^\\bullet$ is a bounded above complex with terms in", "$\\mathcal{P}$,", "\\item the arrows $P_n^\\bullet \\to P_{n + 1}^\\bullet$", "are termwise split injections and each cokernel", "$P^i_{n + 1}/P^i_n$ is an element of $\\mathcal{P}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We are going to use that the homotopy category $K(\\mathcal{A})$ is a", "triangulated category, see Proposition", "\\ref{proposition-homotopy-category-triangulated}.", "By Lemma \\ref{lemma-subcategory-left-resolution} we can find a", "termwise surjective map of complexes $P_1^\\bullet \\to \\tau_{\\leq 1}K^\\bullet$", "which is a quasi-isomorphism", "such that the terms of $P_1^\\bullet$ are in $\\mathcal{P}$.", "By induction it suffices, given", "$P_1^\\bullet, \\ldots, P_n^\\bullet$ to construct", "$P_{n + 1}^\\bullet$ and the maps", "$P_n^\\bullet \\to P_{n + 1}^\\bullet$ and", "$P_{n + 1}^\\bullet \\to \\tau_{\\leq n + 1}K^\\bullet$.", "\\medskip\\noindent", "Choose a distinguished triangle", "$P_n^\\bullet \\to \\tau_{\\leq n + 1}K^\\bullet \\to C^\\bullet \\to P_n^\\bullet[1]$", "in $K(\\mathcal{A})$. Applying", "Lemma \\ref{lemma-subcategory-left-resolution} we choose a", "map of complexes $Q^\\bullet \\to C^\\bullet$", "which is a quasi-isomorphism such that the terms of $Q^\\bullet$", "are in $\\mathcal{P}$. By the axioms of triangulated categories", "we may fit the composition $Q^\\bullet \\to C^\\bullet \\to P_n^\\bullet[1]$ into", "a distinguished triangle", "$P_n^\\bullet \\to P_{n + 1}^\\bullet \\to Q^\\bullet \\to P_n^\\bullet[1]$", "in $K(\\mathcal{A})$.", "By Lemma \\ref{lemma-improve-distinguished-triangle-homotopy}", "we may and do assume", "$0 \\to P_n^\\bullet \\to P_{n + 1}^\\bullet \\to Q^\\bullet \\to 0$", "is a termwise split short exact sequence. This implies that", "the terms of $P_{n + 1}^\\bullet$ are in $\\mathcal{P}$ and that", "$P_n^\\bullet \\to P_{n + 1}^\\bullet$ is a termwise split injection", "whose cokernels are in $\\mathcal{P}$.", "By the axioms of triangulated categories we obtain a map", "of distinguished triangles", "$$", "\\xymatrix{", "P_n^\\bullet \\ar[r] \\ar[d] &", "P_{n + 1}^\\bullet \\ar[r] \\ar[d] &", "Q^\\bullet \\ar[r] \\ar[d] &", "P_n^\\bullet[1] \\ar[d] \\\\", "P_n^\\bullet \\ar[r] &", "\\tau_{\\leq n + 1}K^\\bullet \\ar[r] &", "C^\\bullet \\ar[r] &", "P_n^\\bullet[1]", "}", "$$", "in the triangulated category $K(\\mathcal{A})$. Choose an actual morphism of", "complexes $f : P_{n + 1}^\\bullet \\to \\tau_{\\leq n + 1}K^\\bullet$.", "The left square of the diagram above commutes up to homotopy, but as", "$P_n^\\bullet \\to P_{n + 1}^\\bullet$ is a termwise split injection", "we can lift the homotopy and modify our choice of $f$ to make it commute.", "Finally, $f$ is a quasi-isomorphism, because both $P_n^\\bullet \\to P_n^\\bullet$", "and $Q^\\bullet \\to C^\\bullet$ are.", "\\medskip\\noindent", "At this point we have all the properties we want, except we don't know", "that the map $f : P_{n + 1}^\\bullet \\to \\tau_{\\leq n + 1}K^\\bullet$", "is termwise surjective. Since we have the commutative diagram", "$$", "\\xymatrix{", "P_n^\\bullet \\ar[d] \\ar[r] & P_{n + 1}^\\bullet \\ar[d] \\\\", "\\tau_{\\leq n}K^\\bullet \\ar[r] & \\tau_{\\leq n + 1}K^\\bullet", "}", "$$", "of complexes, by induction hypothesis we see that $f$ is surjective", "on terms in all degrees except possibly $n$ and $n + 1$. Choose", "an object $P \\in \\mathcal{P}$ and a surjection $q : P \\to K^n$.", "Consider the map", "$$", "g :", "P^\\bullet = (\\ldots \\to 0 \\to P \\xrightarrow{1} P \\to 0 \\to \\ldots)", "\\longrightarrow", "\\tau_{\\leq n + 1}K^\\bullet", "$$", "with first copy of $P$ in degree $n$ and maps given by", "$q$ in degree $n$ and $d_K \\circ q$ in degree $n + 1$.", "This is a surjection in degree $n$ and the cokernel in", "degree $n + 1$ is $H^{n + 1}(\\tau_{\\leq n + 1}K^\\bullet)$;", "to see this recall that $\\tau_{\\leq n + 1}K^\\bullet$ has", "$\\Ker(d_K^{n + 1})$ in degree $n + 1$.", "However, since $f$ is a quasi-isomorphism we know that", "$H^{n + 1}(f)$ is surjective. Hence after replacing ", "$f : P_{n + 1}^\\bullet \\to \\tau_{\\leq n + 1}K^\\bullet$", "by", "$f \\oplus g : P_{n + 1}^\\bullet \\oplus P^\\bullet \\to \\tau_{\\leq n + 1}K^\\bullet$", "we win." ], "refs": [ "derived-proposition-homotopy-category-triangulated", "derived-lemma-subcategory-left-resolution", "derived-lemma-subcategory-left-resolution", "derived-lemma-improve-distinguished-triangle-homotopy" ], "ref_ids": [ 1960, 1835, 1835, 1809 ] } ], "ref_ids": [] }, { "id": 1904, "type": "theorem", "label": "derived-lemma-special-inverse-system", "categories": [ "derived" ], "title": "derived-lemma-special-inverse-system", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let", "$\\mathcal{I} \\subset \\Ob(\\mathcal{A})$ be a subset.", "Assume $\\mathcal{I}$ contains $0$, is closed under (finite) products,", "and every object of $\\mathcal{A}$ is a subobject of an", "element of $\\mathcal{I}$. Let $K^\\bullet$ be a complex.", "There exists a commutative diagram", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "\\tau_{\\geq -2}K^\\bullet \\ar[r] \\ar[d] &", "\\tau_{\\geq -1}K^\\bullet \\ar[d] \\\\", "\\ldots \\ar[r] & I_2^\\bullet \\ar[r] & I_1^\\bullet", "}", "$$", "in the category of complexes such that", "\\begin{enumerate}", "\\item the vertical arrows are quasi-isomorphisms and termwise injective,", "\\item $I_n^\\bullet$ is a bounded below complex with terms in $\\mathcal{I}$,", "\\item the arrows $I_{n + 1}^\\bullet \\to I_n^\\bullet$ are termwise split", "surjections and $\\Ker(I^i_{n + 1} \\to I^i_n)$ is an element of $\\mathcal{I}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma is dual to", "Lemma \\ref{lemma-special-direct-system}." ], "refs": [ "derived-lemma-special-direct-system" ], "ref_ids": [ 1903 ] } ], "ref_ids": [] }, { "id": 1905, "type": "theorem", "label": "derived-lemma-pre-derived-adjoint-functors-general", "categories": [ "derived" ], "title": "derived-lemma-pre-derived-adjoint-functors-general", "contents": [ "In the situation above assume $F$ is right adjoint", "to $G$. Let $K \\in \\Ob(\\mathcal{D})$ and", "$M \\in \\Ob(\\mathcal{D}')$. If $RF$ is defined at $K$", "and $LG$ is defined at $M$, then there is a canonical isomorphism", "$$", "\\Hom_{(S')^{-1}\\mathcal{D}'}(M, RF(K)) =", "\\Hom_{S^{-1}\\mathcal{D}}(LG(M), K)", "$$", "This isomorphism is functorial in both variables on the triangulated", "subcategories of $S^{-1}\\mathcal{D}$ and $(S')^{-1}\\mathcal{D}'$", "where $RF$ and $LG$ are defined." ], "refs": [], "proofs": [ { "contents": [ "Since $RF$ is defined at $K$, we see that the rule which assigns to a", "$s : K \\to I$ in $S$ the object $F(I)$ is essentially", "constant as an ind-object of $(S')^{-1}\\mathcal{D}'$ with value $RF(K)$.", "Similarly, the rule which assigns to a $t : P \\to M$ in $S'$", "the object $G(P)$ is essentially constant as a pro-object of", "$S^{-1}\\mathcal{D}$ with value $LG(M)$. Thus we have", "\\begin{align*}", "\\Hom_{(S')^{-1}\\mathcal{D}'}(M, RF(K))", "& =", "\\colim_{s : K \\to I} \\Hom_{(S')^{-1}\\mathcal{D}'}(M, F(I)) \\\\", "& =", "\\colim_{s : K \\to I} \\colim_{t : P \\to M} \\Hom_{\\mathcal{D}'}(P, F(I)) \\\\", "& =", "\\colim_{t : P \\to M} \\colim_{s : K \\to I} \\Hom_{\\mathcal{D}'}(P, F(I)) \\\\", "& =", "\\colim_{t : P \\to M} \\colim_{s : K \\to I} \\Hom_{\\mathcal{D}}(G(P), I) \\\\", "& =", "\\colim_{t : P \\to M} \\Hom_{S^{-1}\\mathcal{D}}(G(P), K) \\\\", "& =", "\\Hom_{S^{-1}\\mathcal{D}}(LG(M), K)", "\\end{align*}", "The first equality holds by", "Categories, Lemma \\ref{categories-lemma-characterize-essentially-constant-ind}.", "The second equality holds by the definition of morphisms in", "$D(\\mathcal{B})$, see Categories, Remark", "\\ref{categories-remark-right-localization-morphisms-colimit}.", "The third equality holds by", "Categories, Lemma \\ref{categories-lemma-colimits-commute}.", "The fourth equality holds because $F$ and $G$ are adjoint.", "The fifth equality holds by definition of morphism", "in $D(\\mathcal{A})$, see Categories, Remark", "\\ref{categories-remark-left-localization-morphisms-colimit}.", "The sixth equality holds by", "Categories, Lemma \\ref{categories-lemma-characterize-essentially-constant-pro}.", "We omit the proof of functoriality." ], "refs": [ "categories-lemma-characterize-essentially-constant-ind", "categories-remark-right-localization-morphisms-colimit", "categories-lemma-colimits-commute", "categories-remark-left-localization-morphisms-colimit", "categories-lemma-characterize-essentially-constant-pro" ], "ref_ids": [ 12240, 12425, 12212, 12424, 12241 ] } ], "ref_ids": [] }, { "id": 1906, "type": "theorem", "label": "derived-lemma-pre-derived-adjoint-functors", "categories": [ "derived" ], "title": "derived-lemma-pre-derived-adjoint-functors", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{A}$", "be functors of abelian categories such that $F$ is a right adjoint to $G$.", "Let $K^\\bullet$ be a complex of $\\mathcal{A}$ and let $M^\\bullet$ be", "a complex of $\\mathcal{B}$. If $RF$ is defined at $K^\\bullet$", "and $LG$ is defined at $M^\\bullet$, then there is a canonical isomorphism", "$$", "\\Hom_{D(\\mathcal{B})}(M^\\bullet, RF(K^\\bullet)) =", "\\Hom_{D(\\mathcal{A})}(LG(M^\\bullet), K^\\bullet)", "$$", "This isomorphism is functorial in both variables on the triangulated", "subcategories of $D(\\mathcal{A})$ and $D(\\mathcal{B})$", "where $RF$ and $LG$ are defined." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of the very general", "Lemma \\ref{lemma-pre-derived-adjoint-functors-general}." ], "refs": [ "derived-lemma-pre-derived-adjoint-functors-general" ], "ref_ids": [ 1905 ] } ], "ref_ids": [] }, { "id": 1907, "type": "theorem", "label": "derived-lemma-derived-adjoint-functors", "categories": [ "derived" ], "title": "derived-lemma-derived-adjoint-functors", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{A}$", "be functors of abelian categories such that $F$ is a right adjoint to $G$.", "If the derived functors $RF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ and", "$LG : D(\\mathcal{B}) \\to D(\\mathcal{A})$ exist, then", "$RF$ is a right adjoint to $LG$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-pre-derived-adjoint-functors}." ], "refs": [ "derived-lemma-pre-derived-adjoint-functors" ], "ref_ids": [ 1906 ] } ], "ref_ids": [] }, { "id": 1908, "type": "theorem", "label": "derived-lemma-K-injective", "categories": [ "derived" ], "title": "derived-lemma-K-injective", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $I^\\bullet$ be a complex. The following are equivalent", "\\begin{enumerate}", "\\item $I^\\bullet$ is K-injective,", "\\item for every quasi-isomorphism $M^\\bullet \\to N^\\bullet$ the map", "$$", "\\Hom_{K(\\mathcal{A})}(N^\\bullet, I^\\bullet)", "\\to \\Hom_{K(\\mathcal{A})}(M^\\bullet, I^\\bullet)", "$$", "is bijective, and", "\\item for every complex $N^\\bullet$ the map", "$$", "\\Hom_{K(\\mathcal{A})}(N^\\bullet, I^\\bullet)", "\\to \\Hom_{D(\\mathcal{A})}(N^\\bullet, I^\\bullet)", "$$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Then (2) holds because the functor", "$\\Hom_{K(\\mathcal{A})}( - , I^\\bullet)$ is cohomological", "and the cone on a quasi-isomorphism is acyclic.", "\\medskip\\noindent", "Assume (2). A morphism $N^\\bullet \\to I^\\bullet$ in $D(\\mathcal{A})$", "is of the form $fs^{-1} : N^\\bullet \\to I^\\bullet$ where", "$s : M^\\bullet \\to N^\\bullet$ is a quasi-isomorphism and", "$f : M^\\bullet \\to I^\\bullet$ is a map. By (2) this corresponds to", "a unique morphism $N^\\bullet \\to I^\\bullet$ in $K(\\mathcal{A})$, i.e.,", "(3) holds.", "\\medskip\\noindent", "Assume (3). If $M^\\bullet$ is acyclic then $M^\\bullet$ is isomorphic", "to the zero complex in $D(\\mathcal{A})$ hence", "$\\Hom_{D(\\mathcal{A})}(M^\\bullet, I^\\bullet) = 0$, whence", "$\\Hom_{K(\\mathcal{A})}(M^\\bullet, I^\\bullet) = 0$ by (3),", "i.e., (1) holds." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1909, "type": "theorem", "label": "derived-lemma-triangle-K-injective", "categories": [ "derived" ], "title": "derived-lemma-triangle-K-injective", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $(K, L, M, f, g, h)$", "be a distinguished triangle of $K(\\mathcal{A})$. If two out of", "$K$, $L$, $M$ are K-injective complexes, then the third is too." ], "refs": [], "proofs": [ { "contents": [ "Follows from the definition,", "Lemma \\ref{lemma-representable-homological}, and", "the fact that $K(\\mathcal{A})$ is a triangulated category", "(Proposition \\ref{proposition-homotopy-category-triangulated})." ], "refs": [ "derived-lemma-representable-homological", "derived-proposition-homotopy-category-triangulated" ], "ref_ids": [ 1758, 1960 ] } ], "ref_ids": [] }, { "id": 1910, "type": "theorem", "label": "derived-lemma-bounded-below-injectives-K-injective", "categories": [ "derived" ], "title": "derived-lemma-bounded-below-injectives-K-injective", "contents": [ "Let $\\mathcal{A}$ be an abelian category. A bounded below complex of", "injectives is K-injective." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemmas \\ref{lemma-K-injective} and", "\\ref{lemma-morphisms-into-injective-complex}." ], "refs": [ "derived-lemma-K-injective", "derived-lemma-morphisms-into-injective-complex" ], "ref_ids": [ 1908, 1855 ] } ], "ref_ids": [] }, { "id": 1911, "type": "theorem", "label": "derived-lemma-product-K-injective", "categories": [ "derived" ], "title": "derived-lemma-product-K-injective", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $T$ be a set and for", "each $t \\in T$ let $I_t^\\bullet$ be a K-injective complex. If", "$I^n = \\prod_t I_t^n$ exists for all $n$, then $I^\\bullet$ is", "a K-injective complex. Moreover, $I^\\bullet$ represents the", "product of the objects $I_t^\\bullet$ in $D(\\mathcal{A})$." ], "refs": [], "proofs": [ { "contents": [ "Let $K^\\bullet$ be an complex. Observe that the complex", "$$", "C :", "\\prod\\nolimits_b \\Hom(K^{-b}, I^{b - 1}) \\to", "\\prod\\nolimits_b \\Hom(K^{-b}, I^b) \\to", "\\prod\\nolimits_b \\Hom(K^{-b}, I^{b + 1})", "$$", "has cohomology $\\Hom_{K(\\mathcal{A})}(K^\\bullet, I^\\bullet)$", "in the middle. Similarly, the complex", "$$", "C_t :", "\\prod\\nolimits_b \\Hom(K^{-b}, I_t^{b - 1}) \\to", "\\prod\\nolimits_b \\Hom(K^{-b}, I_t^b) \\to", "\\prod\\nolimits_b \\Hom(K^{-b}, I_t^{b + 1})", "$$", "computes $\\Hom_{K(\\mathcal{A})}(K^\\bullet, I_t^\\bullet)$.", "Next, observe that we have", "$$", "C = \\prod\\nolimits_{t \\in T} C_t", "$$", "as complexes of abelian groups by our choice of $I$.", "Taking products is an exact functor on the", "category of abelian groups. Hence if $K^\\bullet$ is acyclic, then", "$\\Hom_{K(\\mathcal{A})}(K^\\bullet, I_t^\\bullet) = 0$, hence", "$C_t$ is acyclic, hence $C$ is acyclic, hence we get", "$\\Hom_{K(\\mathcal{A})}(K^\\bullet, I^\\bullet) = 0$.", "Thus we find that $I^\\bullet$ is K-injective.", "Having said this, we can use Lemma \\ref{lemma-K-injective}", "to conclude that", "$$", "\\Hom_{D(\\mathcal{A})}(K^\\bullet, I^\\bullet)", "=", "\\prod\\nolimits_{t \\in T} \\Hom_{D(\\mathcal{A})}(K^\\bullet, I_t^\\bullet)", "$$", "and indeed $I^\\bullet$ represents the product in the derived category." ], "refs": [ "derived-lemma-K-injective" ], "ref_ids": [ 1908 ] } ], "ref_ids": [] }, { "id": 1912, "type": "theorem", "label": "derived-lemma-K-injective-defined", "categories": [ "derived" ], "title": "derived-lemma-K-injective-defined", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $F : K(\\mathcal{A}) \\to \\mathcal{D}'$ be an exact functor", "of triangulated categories. Then $RF$ is defined at every complex", "in $K(\\mathcal{A})$ which is quasi-isomorphic to a", "K-injective complex. In fact, every K-injective complex computes $RF$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-derived-inverts}", "it suffices to show that $RF$ is defined at a K-injective complex,", "i.e., it suffices to show a K-injective complex $I^\\bullet$ computes $RF$.", "Any quasi-isomorphism $I^\\bullet \\to N^\\bullet$ is a homotopy equivalence", "as it has an inverse by", "Lemma \\ref{lemma-K-injective}.", "Thus $I^\\bullet \\to I^\\bullet$ is a final object of", "$I^\\bullet/\\text{Qis}(\\mathcal{A})$ and we win." ], "refs": [ "derived-lemma-derived-inverts", "derived-lemma-K-injective" ], "ref_ids": [ 1824, 1908 ] } ], "ref_ids": [] }, { "id": 1913, "type": "theorem", "label": "derived-lemma-enough-K-injectives-implies", "categories": [ "derived" ], "title": "derived-lemma-enough-K-injectives-implies", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Assume every complex has a quasi-isomorphism towards a K-injective complex.", "Then any exact functor $F : K(\\mathcal{A}) \\to \\mathcal{D}'$ of triangulated", "categories has a right derived functor", "$$", "RF : D(\\mathcal{A}) \\longrightarrow \\mathcal{D}'", "$$", "and $RF(I^\\bullet) = F(I^\\bullet)$ for K-injective complexes $I^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "To see this we apply", "Lemma \\ref{lemma-find-existence-computes}", "with $\\mathcal{I}$ the collection of K-injective complexes. Since (1)", "holds by assumption, it suffices to prove that if $I^\\bullet \\to J^\\bullet$", "is a quasi-isomorphism of K-injective complexes, then", "$F(I^\\bullet) \\to F(J^\\bullet)$ is an isomorphism. This is clear because", "$I^\\bullet \\to J^\\bullet$ is a homotopy equivalence, i.e., an", "isomorphism in $K(\\mathcal{A})$, by", "Lemma \\ref{lemma-K-injective}." ], "refs": [ "derived-lemma-find-existence-computes", "derived-lemma-K-injective" ], "ref_ids": [ 1832, 1908 ] } ], "ref_ids": [] }, { "id": 1914, "type": "theorem", "label": "derived-lemma-limit-K-injectives", "categories": [ "derived" ], "title": "derived-lemma-limit-K-injectives", "contents": [ "\\begin{slogan}", "The limit of a ``split'' tower of K-injective complexes is K-injective.", "\\end{slogan}", "Let $\\mathcal{A}$ be an abelian category. Let", "$$", "\\ldots \\to I_3^\\bullet \\to I_2^\\bullet \\to I_1^\\bullet", "$$", "be an inverse system of complexes. Assume", "\\begin{enumerate}", "\\item each $I_n^\\bullet$ is $K$-injective,", "\\item each map $I_{n + 1}^m \\to I_n^m$ is a split surjection,", "\\item the limits $I^m = \\lim I_n^m$ exist.", "\\end{enumerate}", "Then the complex $I^\\bullet$ is K-injective." ], "refs": [], "proofs": [ { "contents": [ "We urge the reader to skip the proof of this lemma.", "Let $M^\\bullet$ be an acyclic complex. Let us abbreviate", "$H_n(a, b) = \\Hom_\\mathcal{A}(M^a, I_n^b)$. With this notation", "$\\Hom_{K(\\mathcal{A})}(M^\\bullet, I^\\bullet)$ is the cohomology", "of the complex", "$$", "\\prod_m \\lim\\limits_n H_n(m, m - 2)", "\\to", "\\prod_m \\lim\\limits_n H_n(m, m - 1)", "\\to", "\\prod_m \\lim\\limits_n H_n(m, m)", "\\to", "\\prod_m \\lim\\limits_n H_n(m, m + 1)", "$$", "in the third spot from the left.", "We may exchange the order of $\\prod$ and $\\lim$ and each of the complexes", "$$", "\\prod_m H_n(m, m - 2)", "\\to", "\\prod_m H_n(m, m - 1)", "\\to", "\\prod_m H_n(m, m)", "\\to", "\\prod_m H_n(m, m + 1)", "$$", "is exact by assumption (1). By assumption (2) the maps in the systems", "$$", "\\ldots \\to", "\\prod_m H_3(m, m - 2) \\to", "\\prod_m H_2(m, m - 2) \\to", "\\prod_m H_1(m, m - 2)", "$$", "are surjective. Thus the lemma follows from", "Homology, Lemma \\ref{homology-lemma-apply-Mittag-Leffler}." ], "refs": [ "homology-lemma-apply-Mittag-Leffler" ], "ref_ids": [ 12125 ] } ], "ref_ids": [] }, { "id": 1915, "type": "theorem", "label": "derived-lemma-adjoint-preserve-K-injectives", "categories": [ "derived" ], "title": "derived-lemma-adjoint-preserve-K-injectives", "contents": [ "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.", "Let $u : \\mathcal{A} \\to \\mathcal{B}$ and", "$v : \\mathcal{B} \\to \\mathcal{A}$ be additive functors. Assume", "\\begin{enumerate}", "\\item $u$ is right adjoint to $v$, and", "\\item $v$ is exact.", "\\end{enumerate}", "Then $u$ transforms K-injective complexes into K-injective complexes." ], "refs": [], "proofs": [ { "contents": [ "Let $I^\\bullet$ be a K-injective complex of $\\mathcal{A}$.", "Let $M^\\bullet$ be a acyclic complex of $\\mathcal{B}$.", "As $v$ is exact we see that $v(M^\\bullet)$ is an acyclic complex.", "By adjointness we get", "$$", "0 = \\Hom_{K(\\mathcal{A})}(v(M^\\bullet), I^\\bullet) =", "\\Hom_{K(\\mathcal{B})}(M^\\bullet, u(I^\\bullet))", "$$", "hence the lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1916, "type": "theorem", "label": "derived-lemma-replace-resolution", "categories": [ "derived" ], "title": "derived-lemma-replace-resolution", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let", "$d : \\Ob(\\mathcal{A}) \\to \\{0, 1, 2, \\ldots, \\infty\\}$ be a function.", "Assume that", "\\begin{enumerate}", "\\item every object of $\\mathcal{A}$ is a subobject of an", "object $A$ with $d(A) = 0$,", "\\item $d(A \\oplus B) \\leq \\max \\{d(A), d(B)\\}$ for $A, B \\in \\mathcal{A}$, and", "\\item if $0 \\to A \\to B \\to C \\to 0$ is short exact, then", "$d(C) \\leq \\max\\{d(A) - 1, d(B)\\}$.", "\\end{enumerate}", "Let $K^\\bullet$ be a complex such that $n + d(K^n)$ tends to $-\\infty$", "as $n \\to -\\infty$. Then there exists a quasi-isomorphism", "$K^\\bullet \\to L^\\bullet$ with $d(L^n) = 0$ for all $n \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-subcategory-right-resolution} we can find a", "quasi-isomorphism $\\sigma_{\\geq 0}K^\\bullet \\to M^\\bullet$ with", "$M^n = 0$ for $n < 0$ and $d(M^n) = 0$ for $n \\geq 0$. Then $K^\\bullet$", "is quasi-isomorphic to the complex", "$$", "\\ldots \\to K^{-2} \\to K^{-1} \\to M^0 \\to M^1 \\to \\ldots", "$$", "Hence we may assume that $d(K^n) = 0$ for $n \\gg 0$. Note that", "the condition $n + d(K^n) \\to -\\infty$ as $n \\to -\\infty$ is not", "violated by this replacement.", "\\medskip\\noindent", "We are going to improve $K^\\bullet$ by an (infinite) sequence of", "elementary replacements. An {\\it elementary replacement} is the following.", "Choose an index $n$ such that $d(K^n) > 0$. Choose an injection", "$K^n \\to M$ where $d(M) = 0$. Set", "$M' = \\Coker(K^n \\to M \\oplus K^{n + 1})$. Consider the map of complexes", "$$", "\\xymatrix{", "K^\\bullet : \\ar[d] &", "K^{n - 1} \\ar[d] \\ar[r] &", "K^n \\ar[d] \\ar[r] &", "K^{n + 1} \\ar[d] \\ar[r] &", "K^{n + 2} \\ar[d] \\\\", "(K')^\\bullet : &", "K^{n - 1} \\ar[r] &", "M \\ar[r] &", "M' \\ar[r] &", "K^{n + 2}", "}", "$$", "It is clear that $K^\\bullet \\to (K')^\\bullet$ is a quasi-isomorphism.", "Moreover, it is clear that $d((K')^n) = 0$ and", "$$", "d((K')^{n + 1}) \\leq \\max\\{d(K^n) - 1, d(M \\oplus K^{n + 1})\\} \\leq", "\\max\\{d(K^n) - 1, d(K^{n + 1})\\}", "$$", "and the other values are unchanged.", "\\medskip\\noindent", "To finish the proof we carefuly choose the order in which to do", "the elementary replacements so that for every integer $m$ the complex", "$\\sigma_{\\geq m}K^\\bullet$ is changed only a finite number of times.", "To do this set", "$$", "\\xi(K^\\bullet) = \\max \\{n + d(K^n) \\mid d(K^n) > 0\\}", "$$", "and", "$$", "I = \\{n \\in \\mathbf{Z} \\mid \\xi(K^\\bullet) = n + d(K^n)", "\\text{ and }", " d(K^n) > 0\\}", "$$", "Our assumption that $n + d(K^n)$ tends to $-\\infty$ as $n \\to -\\infty$", "and the fact that $d(K^n) = 0$ for $n >> 0$", "implies $\\xi(K^\\bullet) < +\\infty$ and that $I$ is a finite set.", "It is clear that $\\xi((K')^\\bullet) \\leq \\xi(K^\\bullet)$ for an", "elementary transformation as above. An elementary transformation", "changes the complex in degrees $\\leq \\xi(K^\\bullet) + 1$. Hence if we can", "find finite sequence of elementary transformations which", "decrease $\\xi(K^\\bullet)$, then we win. However, note that if we", "do an elementary transformation starting with the smallest element", "$n \\in I$, then we either decrease the size of $I$, or we increase", "$\\min I$. Since every element of $I$ is $\\leq \\xi(K^\\bullet)$ we see", "that we win after a finite number of steps." ], "refs": [ "derived-lemma-subcategory-right-resolution" ], "ref_ids": [ 1836 ] } ], "ref_ids": [] }, { "id": 1917, "type": "theorem", "label": "derived-lemma-unbounded-right-derived", "categories": [ "derived" ], "title": "derived-lemma-unbounded-right-derived", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor of", "abelian categories. Assume", "\\begin{enumerate}", "\\item every object of $\\mathcal{A}$ is a subobject of an object", "which is right acyclic for $F$,", "\\item there exists an integer $n \\geq 0$ such that $R^nF = 0$,", "\\end{enumerate}", "Then", "\\begin{enumerate}", "\\item $RF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ exists,", "\\item any complex consisting of right acyclic objects for $F$ computes $RF$,", "\\item any complex is the source of a quasi-isomorphism into a complex", "consisting of right acyclic objects for $F$,", "\\item for $E \\in D(\\mathcal{A})$", "\\begin{enumerate}", "\\item $H^i(RF(\\tau_{\\leq a}E) \\to H^i(RF(E))$ is an isomorphism", "for $i \\leq a$,", "\\item $H^i(RF(E)) \\to H^i(RF(\\tau_{\\geq b - n + 1}E))$ is an isomorphism", "for $i \\geq b$,", "\\item if $H^i(E) = 0$ for $i \\not \\in [a, b]$ for some", "$-\\infty \\leq a \\leq b \\leq \\infty$, then $H^i(RF(E)) = 0$", "for $i \\not \\in [a, b + n - 1]$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that the first assumption implies that", "$RF : D^+(\\mathcal{A}) \\to D^+(\\mathcal{B})$ exists, see", "Proposition \\ref{proposition-enough-acyclics}.", "Let $A$ be an object of $\\mathcal{A}$. Choose an injection $A \\to A'$", "with $A'$ acyclic. Then we see that $R^{n + 1}F(A) = R^nF(A'/A) = 0$ by", "the long exact cohomology sequence. Hence we conclude that $R^{n + 1}F = 0$.", "Continuing like this using induction we find that $R^mF = 0$ for all", "$m \\geq n$.", "\\medskip\\noindent", "We are going to use Lemma \\ref{lemma-replace-resolution} with the function", "$d : \\Ob(\\mathcal{A}) \\to \\{0, 1, 2, \\ldots \\}$ given by", "$d(A) = \\max \\{0\\} \\cup \\{i \\mid R^iF(A) \\not = 0\\}$.", "The first assumption of Lemma \\ref{lemma-replace-resolution}", "is our assumption (1). The second assumption of", "Lemma \\ref{lemma-replace-resolution} follows from the fact", "that $RF(A \\oplus B) = RF(A) \\oplus RF(B)$. The third assumption of", "Lemma \\ref{lemma-replace-resolution} follows from the long exact", "cohomology sequence. Hence for every complex $K^\\bullet$ there exists a", "quasi-isomorphism $K^\\bullet \\to L^\\bullet$ into a complex of", "objects right acyclic for $F$. This proves statement (3).", "\\medskip\\noindent", "We claim that if $L^\\bullet \\to M^\\bullet$ is a quasi-isomorphism of", "complexes of right acyclic objects for $F$, then", "$F(L^\\bullet) \\to F(M^\\bullet)$", "is a quasi-isomorphism. If we prove this claim then we get statements", "(1) and (2) of the lemma by", "Lemma \\ref{lemma-find-existence-computes}.", "To prove the claim pick an integer $i \\in \\mathbf{Z}$.", "Consider the distinguished triangle", "$$", "\\sigma_{\\geq i - n - 1}L^\\bullet \\to", "\\sigma_{\\geq i - n - 1}M^\\bullet \\to Q^\\bullet,", "$$", "i.e., let $Q^\\bullet$ be the cone of the first map.", "Note that $Q^\\bullet$ is bounded below and that", "$H^j(Q^\\bullet)$ is zero except possibly for $j = i - n - 1$", "or $j = i - n - 2$. We may apply $RF$ to $Q^\\bullet$.", "Using the second spectral sequence of", "Lemma \\ref{lemma-two-ss-complex-functor}", "and the assumed vanishing of cohomology (2) we conclude", "that $H^j(RF(Q^\\bullet))$ is zero except possibly for", "$j \\in \\{i - n - 2, \\ldots, i - 1\\}$. Hence we see that", "$RF(\\sigma_{\\geq i - n - 1}L^\\bullet) \\to RF(\\sigma_{\\geq i - n - 1}M^\\bullet)$", "induces an isomorphism of cohomology objects in degrees $\\geq i$.", "By Proposition \\ref{proposition-enough-acyclics} we know that", "$RF(\\sigma_{\\geq i - n - 1}L^\\bullet) = \\sigma_{\\geq i - n - 1}F(L^\\bullet)$", "and", "$RF(\\sigma_{\\geq i - n - 1}M^\\bullet) = \\sigma_{\\geq i - n - 1}F(M^\\bullet)$.", "We conclude that $F(L^\\bullet) \\to F(M^\\bullet)$", "is an isomorphism in degree $i$ as desired.", "\\medskip\\noindent", "Part (4)(a) follows from Lemma \\ref{lemma-negative-vanishing}.", "\\medskip\\noindent", "For part (4)(b) let $E$ be represented by the complex $L^\\bullet$", "of objects right acyclic for $F$. By part (2) $RF(E)$ is represented", "by the complex $F(L^\\bullet)$ and $RF(\\sigma_{\\geq c}L^\\bullet)$", "is represented by $\\sigma_{\\geq c}F(L^\\bullet)$. Consider the", "distinguished triangle", "$$", "H^{b - n}(L^\\bullet)[n - b] \\to", "\\tau_{\\geq b - n}L^\\bullet \\to", "\\tau_{\\geq b - n + 1}L^\\bullet", "$$", "of Remark \\ref{remark-truncation-distinguished-triangle}.", "The vanishing established above gives that", "$H^i(RF(\\tau_{\\geq b - n}L^\\bullet))$ agrees with", "$H^i(RF(\\tau_{\\geq b - n + 1}L^\\bullet))$ for $i \\geq b$.", "Consider the short exact sequence of complexes", "$$", "0 \\to", "\\Im(L^{b - n - 1} \\to L^{b - n})[n - b] \\to", "\\sigma_{\\geq b - n}L^\\bullet \\to", "\\tau_{\\geq b - n}L^\\bullet \\to 0", "$$", "Using the distinguished triangle associated to this", "(see Section \\ref{section-canonical-delta-functor})", "and the vanishing as before we conclude that", "$H^i(RF(\\tau_{\\geq b - n}L^\\bullet))$ agrees with", "$H^i(RF(\\sigma_{\\geq b - n}L^\\bullet))$ for $i \\geq b$.", "Since the map $RF(\\sigma_{\\geq b - n}L^\\bullet) \\to RF(L^\\bullet)$", "is represented by $\\sigma_{\\geq b - n}F(L^\\bullet) \\to F(L^\\bullet)$", "we conclude that this in turn agrees with $H^i(RF(L^\\bullet))$", "for $i \\geq b$ as desired.", "\\medskip\\noindent", "Proof of (4)(c). Under the assumption on $E$ we have", "$\\tau_{\\leq a - 1}E = 0$ and we get the vanishing", "of $H^i(RF(E))$ for $i \\leq a - 1$ from part (4)(a).", "Similarly, we have $\\tau_{\\geq b + 1}E = 0$ and hence", "we get the vanishing of $H^i(RF(E))$ for $i \\geq b + n$ from", "part (4)(b)." ], "refs": [ "derived-proposition-enough-acyclics", "derived-lemma-replace-resolution", "derived-lemma-replace-resolution", "derived-lemma-replace-resolution", "derived-lemma-replace-resolution", "derived-lemma-find-existence-computes", "derived-lemma-two-ss-complex-functor", "derived-proposition-enough-acyclics", "derived-lemma-negative-vanishing", "derived-remark-truncation-distinguished-triangle" ], "ref_ids": [ 1962, 1916, 1916, 1916, 1916, 1832, 1871, 1962, 1839, 2016 ] } ], "ref_ids": [] }, { "id": 1918, "type": "theorem", "label": "derived-lemma-unbounded-left-derived", "categories": [ "derived" ], "title": "derived-lemma-unbounded-left-derived", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a right exact functor of", "abelian categories. If", "\\begin{enumerate}", "\\item every object of $\\mathcal{A}$ is a quotient of an object", "which is left acyclic for $F$,", "\\item there exists an integer $n \\geq 0$ such that $L^nF = 0$,", "\\end{enumerate}", "Then", "\\begin{enumerate}", "\\item $LF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ exists,", "\\item any complex consisting of left acyclic objects for $F$ computes $LF$,", "\\item any complex is the target of a quasi-isomorphism from a complex", "consisting of left acyclic objects for $F$,", "\\item for $E \\in D(\\mathcal{A})$", "\\begin{enumerate}", "\\item $H^i(LF(\\tau_{\\leq a + n - 1}E) \\to H^i(LF(E))$ is an isomorphism", "for $i \\leq a$,", "\\item $H^i(LF(E)) \\to H^i(LF(\\tau_{\\geq b}E))$ is an isomorphism", "for $i \\geq b$,", "\\item if $H^i(E) = 0$ for $i \\not \\in [a, b]$ for some", "$-\\infty \\leq a \\leq b \\leq \\infty$, then $H^i(LF(E)) = 0$", "for $i \\not \\in [a - n + 1, b]$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is dual to Lemma \\ref{lemma-unbounded-right-derived}." ], "refs": [ "derived-lemma-unbounded-right-derived" ], "ref_ids": [ 1917 ] } ], "ref_ids": [] }, { "id": 1919, "type": "theorem", "label": "derived-lemma-hocolim-subsequence", "categories": [ "derived" ], "title": "derived-lemma-hocolim-subsequence", "contents": [ "Let $\\mathcal{D}$ be a triangulated category.", "Let $(K_n, f_n)$ be a system of objects of $\\mathcal{D}$.", "Let $n_1 < n_2 < n_3 < \\ldots$ be a sequence of integers.", "Assume $\\bigoplus K_n$ and $\\bigoplus K_{n_i}$ exist.", "Then there exists an isomorphism", "$\\text{hocolim} K_{n_i} \\to \\text{hocolim} K_n$", "such that", "$$", "\\xymatrix{", "K_{n_i} \\ar[r] \\ar[d]_{\\text{id}} & \\text{hocolim} K_{n_i} \\ar[d] \\\\", "K_{n_i} \\ar[r] & \\text{hocolim} K_n", "}", "$$", "commutes for all $i$." ], "refs": [], "proofs": [ { "contents": [ "Let $g_i : K_{n_i} \\to K_{n_{i + 1}}$ be the composition", "$f_{n_{i + 1} - 1} \\circ \\ldots \\circ f_{n_i}$.", "We construct commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "\\bigoplus\\nolimits_i K_{n_i} \\ar[r]_{1 - g_i} \\ar[d]_b &", "\\bigoplus\\nolimits_i K_{n_i} \\ar[d]^a \\\\", "\\bigoplus\\nolimits_n K_n \\ar[r]^{1 - f_n} &", "\\bigoplus\\nolimits_n K_n", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "\\bigoplus\\nolimits_n K_n \\ar[r]_{1 - f_n} \\ar[d]_d &", "\\bigoplus\\nolimits_n K_n \\ar[d]^c \\\\", "\\bigoplus\\nolimits_i K_{n_i} \\ar[r]^{1 - g_i} &", "\\bigoplus\\nolimits_i K_{n_i}", "}", "}", "$$", "as follows. Let $a_i = a|_{K_{n_i}}$ be the inclusion of $K_{n_i}$", "into the direct sum. In other words, $a$ is the natural inclusion.", "Let $b_i = b|_{K_{n_i}}$ be the map", "$$", "K_{n_i}", "\\xrightarrow{1,\\ f_{n_i},\\ f_{n_i + 1} \\circ f_{n_i},", "\\ \\ldots,\\ f_{n_{i + 1} - 2} \\circ \\ldots \\circ f_{n_i}}", "K_{n_i} \\oplus K_{n_i + 1} \\oplus \\ldots \\oplus K_{n_{i + 1} - 1}", "$$", "If $n_{i - 1} < j \\leq n_i$, then we let $c_j = c|_{K_j}$", "be the map", "$$", "K_j \\xrightarrow{f_{n_i - 1} \\circ \\ldots \\circ f_j} K_{n_i}", "$$", "We let $d_j = d|_{K_j}$ be zero if $j \\not = n_i$ for any $i$", "and we let $d_{n_i}$ be the natural inclusion of $K_{n_i}$", "into the direct sum. In other words, $d$ is the natural projection.", "By TR3 these diagrams define morphisms", "$$", "\\varphi : \\text{hocolim} K_{n_i} \\to \\text{hocolim} K_n", "\\quad\\text{and}\\quad", "\\psi : \\text{hocolim} K_n \\to \\text{hocolim} K_{n_i}", "$$", "Since $c \\circ a$ and $d \\circ b$ are the identity maps we see that", "$\\varphi \\circ \\psi$ is an isomorphism by", "Lemma \\ref{lemma-third-isomorphism-triangle}.", "The other way around we get the morphisms $a \\circ c$ and $b \\circ d$.", "Consider the morphism", "$h = (h_j) : \\bigoplus K_n \\to \\bigoplus K_n$ given by", "the rule: for $n_{i - 1} < j < n_i$ we set", "$$", "h_j : K_j", "\\xrightarrow{1,\\ f_j,\\ f_{j + 1} \\circ f_j,", "\\ \\ldots,\\ f_{n_i - 1} \\circ \\ldots \\circ f_j}", "K_j \\oplus \\ldots \\oplus K_{n_i}", "$$", "Then the reader verifies that $(1 - f) \\circ h = \\text{id} - a \\circ c$", "and $h \\circ (1 - f) = \\text{id} - b \\circ d$. This means that", "$\\text{id} - \\psi \\circ \\varphi$ has square zero by", "Lemma \\ref{lemma-third-map-square-zero} (small argument omitted).", "In other words, $\\psi \\circ \\varphi$ differs from the identity", "by a nilpotent endomorphism, hence is an isomorphism. Thus", "$\\varphi$ and $\\psi$ are isomorphisms as desired." ], "refs": [ "derived-lemma-third-isomorphism-triangle", "derived-lemma-third-map-square-zero" ], "ref_ids": [ 1759, 1760 ] } ], "ref_ids": [] }, { "id": 1920, "type": "theorem", "label": "derived-lemma-direct-sums", "categories": [ "derived" ], "title": "derived-lemma-direct-sums", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "If $\\mathcal{A}$ has exact countable direct sums, then", "$D(\\mathcal{A})$ has countable direct sums. In fact given", "a collection of complexes $K_i^\\bullet$ indexed by a countable", "index set $I$ the termwise direct sum $\\bigoplus K_i^\\bullet$", "is the direct sum of $K_i^\\bullet$ in $D(\\mathcal{A})$." ], "refs": [], "proofs": [ { "contents": [ "Let $L^\\bullet$ be a complex. Suppose given maps", "$\\alpha_i : K_i^\\bullet \\to L^\\bullet$ in $D(\\mathcal{A})$.", "This means there exist quasi-isomorphisms", "$s_i : M_i^\\bullet \\to K_i^\\bullet$", "of complexes and maps of complexes $f_i : M_i^\\bullet \\to L^\\bullet$", "such that $\\alpha_i = f_is_i^{-1}$. By assumption the map of complexes", "$$", "s : \\bigoplus M_i^\\bullet \\longrightarrow \\bigoplus K_i^\\bullet", "$$", "is a quasi-isomorphism. Hence setting $f = \\bigoplus f_i$ we see that", "$\\alpha = fs^{-1}$ is a map in $D(\\mathcal{A})$ whose composition", "with the coprojection $K_i^\\bullet \\to \\bigoplus K_i^\\bullet$ is $\\alpha_i$.", "We omit the verification that $\\alpha$ is unique." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1921, "type": "theorem", "label": "derived-lemma-compute-colimit", "categories": [ "derived" ], "title": "derived-lemma-compute-colimit", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Assume colimits over $\\mathbf{N}$", "exist and are exact. Then countable direct sums exists and are exact.", "Moreover, if $(A_n, f_n)$ is a system over $\\mathbf{N}$, then there is", "a short exact sequence", "$$", "0 \\to \\bigoplus A_n \\to \\bigoplus A_n \\to \\colim A_n \\to 0", "$$", "where the first map in degree $n$ is given by $1 - f_n$." ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from", "$\\bigoplus A_n = \\colim (A_1 \\oplus \\ldots \\oplus A_n)$.", "For the second, note that for each $n$ we have the short exact sequence", "$$", "0 \\to", "A_1 \\oplus \\ldots \\oplus A_{n - 1} \\to", "A_1 \\oplus \\ldots \\oplus A_n \\to A_n \\to 0", "$$", "where the first map is given by the maps $1 - f_i$ and the", "second map is the sum of the transition maps.", "Take the colimit to get the sequence of the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1922, "type": "theorem", "label": "derived-lemma-colim-hocolim", "categories": [ "derived" ], "title": "derived-lemma-colim-hocolim", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $L_n^\\bullet$", "be a system of complexes of $\\mathcal{A}$. Assume", "colimits over $\\mathbf{N}$ exist and are exact in $\\mathcal{A}$.", "Then the termwise", "colimit $L^\\bullet = \\colim L_n^\\bullet$ is a homotopy colimit of the", "system in $D(\\mathcal{A})$." ], "refs": [], "proofs": [ { "contents": [ "We have an exact sequence of complexes", "$$", "0 \\to \\bigoplus L_n^\\bullet \\to \\bigoplus L_n^\\bullet \\to L^\\bullet \\to 0", "$$", "by Lemma \\ref{lemma-compute-colimit}.", "The direct sums are direct sums in $D(\\mathcal{A})$ by", "Lemma \\ref{lemma-direct-sums}.", "Thus the result follows from the definition", "of derived colimits in", "Definition \\ref{definition-derived-colimit}", "and the fact that a short exact sequence of complexes", "gives a distinguished triangle", "(Lemma \\ref{lemma-derived-canonical-delta-functor})." ], "refs": [ "derived-lemma-compute-colimit", "derived-lemma-direct-sums", "derived-definition-derived-colimit", "derived-lemma-derived-canonical-delta-functor" ], "ref_ids": [ 1921, 1920, 2001, 1814 ] } ], "ref_ids": [] }, { "id": 1923, "type": "theorem", "label": "derived-lemma-cohomology-of-hocolim", "categories": [ "derived" ], "title": "derived-lemma-cohomology-of-hocolim", "contents": [ "Let $\\mathcal{D}$ be a triangulated category having countable", "direct sums. Let $\\mathcal{A}$ be an abelian category with exact", "colimits over $\\mathbf{N}$.", "Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor", "commuting with countable direct sums.", "Then $H(\\text{hocolim} K_n) = \\colim H(K_n)$", "for any system of objects of $\\mathcal{D}$." ], "refs": [], "proofs": [ { "contents": [ "Write $K = \\text{hocolim} K_n$. Apply $H$ to the defining", "distinguished triangle to get", "$$", "\\bigoplus H(K_n) \\to \\bigoplus H(K_n)", "\\to H(K) \\to", "\\bigoplus H(K_n[1]) \\to \\bigoplus H(K_n[1])", "$$", "where the first map is given by $1 - H(f_n)$ and the last map", "is given by $1 - H(f_n[1])$.", "Apply Lemma \\ref{lemma-compute-colimit} to see that this proves the lemma." ], "refs": [ "derived-lemma-compute-colimit" ], "ref_ids": [ 1921 ] } ], "ref_ids": [] }, { "id": 1924, "type": "theorem", "label": "derived-lemma-commutes-with-countable-sums", "categories": [ "derived" ], "title": "derived-lemma-commutes-with-countable-sums", "contents": [ "Let $\\mathcal{D}$ be a triangulated category with countable direct sums.", "Let $K \\in \\mathcal{D}$ be an object such that for every", "countable set of objects $E_n \\in \\mathcal{D}$ the canonical map", "$$", "\\bigoplus \\Hom_\\mathcal{D}(K, E_n)", "\\longrightarrow", "\\Hom_\\mathcal{D}(K, \\bigoplus E_n)", "$$", "is a bijection. Then, given any system $L_n$ of $\\mathcal{D}$ over", "$\\mathbf{N}$ whose derived colimit $L = \\text{hocolim} L_n$", "exists we have that", "$$", "\\colim \\Hom_\\mathcal{D}(K, L_n) \\longrightarrow \\Hom_\\mathcal{D}(K, L)", "$$", "is a bijection." ], "refs": [], "proofs": [ { "contents": [ "Consider the defining distinguished triangle", "$$", "\\bigoplus L_n \\to \\bigoplus L_n \\to L \\to \\bigoplus L_n[1]", "$$", "Apply the cohomological functor $\\Hom_\\mathcal{D}(K, -)$", "(see Lemma \\ref{lemma-representable-homological}).", "By elementary considerations concerning colimits of abelian groups", "we get the result." ], "refs": [ "derived-lemma-representable-homological" ], "ref_ids": [ 1758 ] } ], "ref_ids": [] }, { "id": 1925, "type": "theorem", "label": "derived-lemma-products", "categories": [ "derived" ], "title": "derived-lemma-products", "contents": [ "Let $\\mathcal{A}$ be an abelian category with exact", "countable products. Then", "\\begin{enumerate}", "\\item $D(\\mathcal{A})$ has countable products,", "\\item countable products $\\prod K_i$ in $D(\\mathcal{A})$ are obtained by", "taking termwise products of any complexes representing the $K_i$, and", "\\item $H^p(\\prod K_i) = \\prod H^p(K_i)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $K_i^\\bullet$ be a complex representing $K_i$ in $D(\\mathcal{A})$.", "Let $L^\\bullet$ be a complex. Suppose given maps", "$\\alpha_i : L^\\bullet \\to K_i^\\bullet$ in $D(\\mathcal{A})$.", "This means there exist quasi-isomorphisms $s_i : K_i^\\bullet \\to M_i^\\bullet$", "of complexes and maps of complexes $f_i : L^\\bullet \\to M_i^\\bullet$", "such that $\\alpha_i = s_i^{-1}f_i$. By assumption the map of complexes", "$$", "s : \\prod K_i^\\bullet \\longrightarrow \\prod M_i^\\bullet", "$$", "is a quasi-isomorphism. Hence setting $f = \\prod f_i$ we see that", "$\\alpha = s^{-1}f$ is a map in $D(\\mathcal{A})$ whose composition", "with the projection $\\prod K_i^\\bullet \\to K_i^\\bullet$ is $\\alpha_i$.", "We omit the verification that $\\alpha$ is unique." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1926, "type": "theorem", "label": "derived-lemma-inverse-limit-bounded-below", "categories": [ "derived" ], "title": "derived-lemma-inverse-limit-bounded-below", "contents": [ "Let $\\mathcal{A}$ be an abelian category with countable products and", "enough injectives. Let $(K_n)$ be an inverse system of $D^+(\\mathcal{A})$.", "Then $R\\lim K_n$ exists." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that $\\prod K_n$ exists in $D(\\mathcal{A})$.", "For every $n$ we can represent $K_n$ by a bounded below complex", "$I_n^\\bullet$ of injectives (Lemma \\ref{lemma-injective-resolutions-exist}).", "Then $\\prod K_n$ is represented by $\\prod I_n^\\bullet$, see", "Lemma \\ref{lemma-product-K-injective}." ], "refs": [ "derived-lemma-injective-resolutions-exist", "derived-lemma-product-K-injective" ], "ref_ids": [ 1851, 1911 ] } ], "ref_ids": [] }, { "id": 1927, "type": "theorem", "label": "derived-lemma-difficulty-K-injectives", "categories": [ "derived" ], "title": "derived-lemma-difficulty-K-injectives", "contents": [ "Let $\\mathcal{A}$ be an abelian category with countable products and", "enough injectives. Let $K^\\bullet$ be a complex. Let $I_n^\\bullet$ be", "the inverse system of bounded below complexes of injectives produced by", "Lemma \\ref{lemma-special-inverse-system}. Then", "$I^\\bullet = \\lim I_n^\\bullet$ exists, is K-injective, and", "the following are equivalent", "\\begin{enumerate}", "\\item the map $K^\\bullet \\to I^\\bullet$ is a quasi-isomorphism,", "\\item the canonical map $K^\\bullet \\to R\\lim \\tau_{\\geq -n}K^\\bullet$", "is an isomorphism in $D(\\mathcal{A})$.", "\\end{enumerate}" ], "refs": [ "derived-lemma-special-inverse-system" ], "proofs": [ { "contents": [ "The statement of the lemma makes sense as $R\\lim \\tau_{\\geq -n}K^\\bullet$", "exists by Lemma \\ref{lemma-inverse-limit-bounded-below}.", "Each complex $I_n^\\bullet$ is K-injective by", "Lemma \\ref{lemma-bounded-below-injectives-K-injective}.", "Choose direct sum decompositions $I_{n + 1}^p = C_{n + 1}^p \\oplus I_n^p$", "for all $n \\geq 1$. Set $C_1^p = I_1^p$. The complex", "$I^\\bullet = \\lim I_n^\\bullet$ exists because we can take", "$I^p = \\prod_{n \\geq 1} C_n^p$. Fix $p \\in \\mathbf{Z}$.", "We claim there is a split short exact sequence", "$$", "0 \\to I^p \\to \\prod I_n^p \\to \\prod I_n^p \\to 0", "$$", "of objects of $\\mathcal{A}$. Here the first map is given by", "the projection maps $I^p \\to I_n^p$ and the second map", "by $(x_n) \\mapsto (x_n - f^p_{n + 1}(x_{n + 1}))$ where", "$f^p_n : I_n^p \\to I_{n - 1}^p$ are the transition maps.", "The splitting comes from the map $\\prod I_n^p \\to \\prod C_n^p = I^p$.", "We obtain a termwise split short exact sequence of complexes", "$$", "0 \\to I^\\bullet \\to \\prod I_n^\\bullet \\to \\prod I_n^\\bullet \\to 0", "$$", "Hence a corresponding distinguished triangle in $K(\\mathcal{A})$", "and $D(\\mathcal{A})$. By Lemma \\ref{lemma-product-K-injective}", "the products are K-injective and represent the corresponding", "products in $D(\\mathcal{A})$.", "It follows that $I^\\bullet$ represents $R\\lim I_n^\\bullet$", "(Definition \\ref{definition-derived-limit}).", "Moreover, it follows that $I^\\bullet$ is K-injective by", "Lemma \\ref{lemma-triangle-K-injective}.", "By the commutative diagram of Lemma \\ref{lemma-special-inverse-system}", "we obtain a corresponding commutative diagram", "$$", "\\xymatrix{", "K^\\bullet \\ar[r] \\ar[d] & R\\lim \\tau_{\\geq -n} K^\\bullet \\ar[d] \\\\", "I^\\bullet \\ar[r] & R\\lim I_n^\\bullet", "}", "$$", "in $D(\\mathcal{A})$. Since the right vertical arrow is an isomorphism", "(as derived limits are defined on the level of the derived category", "and since $\\tau_{\\geq -n}K^\\bullet \\to I_n^\\bullet$ is a quasi-isomorphism),", "the lemma follows." ], "refs": [ "derived-lemma-inverse-limit-bounded-below", "derived-lemma-bounded-below-injectives-K-injective", "derived-lemma-product-K-injective", "derived-definition-derived-limit", "derived-lemma-triangle-K-injective", "derived-lemma-special-inverse-system" ], "ref_ids": [ 1926, 1910, 1911, 2002, 1909, 1904 ] } ], "ref_ids": [ 1904 ] }, { "id": 1928, "type": "theorem", "label": "derived-lemma-enough-K-injectives-Ab4-star", "categories": [ "derived" ], "title": "derived-lemma-enough-K-injectives-Ab4-star", "contents": [ "Let $\\mathcal{A}$ be an abelian category having enough injectives", "and exact countable products. Then for every complex", "there is a quasi-isomorphism to a K-injective complex." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-difficulty-K-injectives} it suffices to show that", "$K \\to R\\lim\\tau_{\\geq -n}K$ is an isomorphism for all $K$ in $D(\\mathcal{A})$.", "Consider the defining distinguished triangle", "$$", "R\\lim\\tau_{\\geq -n}K \\to", "\\prod \\tau_{\\geq -n}K \\to", "\\prod \\tau_{\\geq -n}K \\to", "(R\\lim\\tau_{\\geq -n}K)[1]", "$$", "By Lemma \\ref{lemma-products} we have", "$$", "H^p(\\prod \\tau_{\\geq -n}K) = \\prod\\nolimits_{p \\geq -n} H^p(K)", "$$", "It follows in a straightforward manner from the long exact cohomology", "sequence of the displayed distinguished triangle", "that $H^p(R\\lim \\tau_{\\geq -n}K) = H^p(K)$." ], "refs": [ "derived-lemma-difficulty-K-injectives", "derived-lemma-products" ], "ref_ids": [ 1927, 1925 ] } ], "ref_ids": [] }, { "id": 1929, "type": "theorem", "label": "derived-lemma-associativity-star", "categories": [ "derived" ], "title": "derived-lemma-associativity-star", "contents": [ "Let $\\mathcal{T}$ be a triangulated category.", "Given full subcategories $\\mathcal{A}$, $\\mathcal{B}$, $\\mathcal{C}$", "we have $(\\mathcal{A} \\star \\mathcal{B}) \\star \\mathcal{C} =", "\\mathcal{A} \\star (\\mathcal{B} \\star \\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "If we have distinguished triangles $A \\to X \\to B$ and $X \\to Y \\to C$", "then by Axiom TR4 we have distinguished triangles", "$A \\to Y \\to Z$ and $B \\to Z \\to C$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1930, "type": "theorem", "label": "derived-lemma-smd-star", "categories": [ "derived" ], "title": "derived-lemma-smd-star", "contents": [ "Let $\\mathcal{T}$ be a triangulated category.", "Given full subcategories $\\mathcal{A}$, $\\mathcal{B}$", "we have", "$smd(\\mathcal{A}) \\star smd(\\mathcal{B}) \\subset", "smd(\\mathcal{A} \\star \\mathcal{B})$ and", "$smd(smd(\\mathcal{A}) \\star smd(\\mathcal{B})) =", "smd(\\mathcal{A} \\star \\mathcal{B})$." ], "refs": [], "proofs": [ { "contents": [ "Suppose we have a distinguished triangle $A_1 \\to X \\to B_1$ where", "$A_1 \\oplus A_2 \\in \\Ob(\\mathcal{A})$ and $B_1 \\oplus B_2 \\in \\Ob(\\mathcal{B})$.", "Then we obtain a distinguished triangle", "$A_1 \\oplus A_2 \\to A_2 \\oplus X \\oplus B_2 \\to B_1 \\oplus B_2$", "which proves that $X$ is in $smd(\\mathcal{A} \\star \\mathcal{B})$.", "This proves the inclusion. The equality follows trivially from this." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1931, "type": "theorem", "label": "derived-lemma-add-star", "categories": [ "derived" ], "title": "derived-lemma-add-star", "contents": [ "Let $\\mathcal{T}$ be a triangulated category. Given full subcategories", "$\\mathcal{A}$, $\\mathcal{B}$ the full subcategories", "$add(\\mathcal{A}) \\star add(\\mathcal{B})$ and", "$smd(add(\\mathcal{A}))$ are closed under direct sums." ], "refs": [], "proofs": [ { "contents": [ "Namely, if $A \\to X \\to B$ and $A' \\to X' \\to B'$ are distinguished triangles", "and $A, A' \\in add(\\mathcal{A})$ and $B, B' \\in add(\\mathcal{B})$ then", "$A \\oplus A' \\to X \\oplus X' \\to B \\oplus B'$ is a distinguished triangle", "with $A \\oplus A' \\in add(\\mathcal{A})$ and $B \\oplus B' \\in add(\\mathcal{B})$.", "The result for $smd(add(\\mathcal{A}))$ is trivial." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1932, "type": "theorem", "label": "derived-lemma-cone-n", "categories": [ "derived" ], "title": "derived-lemma-cone-n", "contents": [ "Let $\\mathcal{T}$ be a triangulated category. Given a full subcategory", "$\\mathcal{A}$ for $n \\geq 1$ the subcategory", "$$", "\\mathcal{C}_n = smd(add(\\mathcal{A})^{\\star n}) =", "smd(add(\\mathcal{A}) \\star \\ldots \\star add(\\mathcal{A}))", "$$", "defined above is a strictly full subcategory of $\\mathcal{T}$", "closed under direct sums and direct summands and", "$\\mathcal{C}_{n + m} = smd(\\mathcal{C}_n \\star \\mathcal{C}_m)$", "for all $n, m \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemmas \\ref{lemma-associativity-star}, \\ref{lemma-smd-star}, and", "\\ref{lemma-add-star}." ], "refs": [ "derived-lemma-associativity-star", "derived-lemma-smd-star", "derived-lemma-add-star" ], "ref_ids": [ 1929, 1930, 1931 ] } ], "ref_ids": [] }, { "id": 1933, "type": "theorem", "label": "derived-lemma-in-cone-n", "categories": [ "derived" ], "title": "derived-lemma-in-cone-n", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{D} = D(\\mathcal{A})$.", "Let $\\mathcal{E} \\subset \\Ob(\\mathcal{A})$ be a subset which we view as", "a subset of $\\Ob(\\mathcal{D})$ also. Let $K$ be an object of $\\mathcal{D}$.", "\\begin{enumerate}", "\\item Let $b \\geq a$ and assume $H^i(K)$ is zero for $i \\not \\in [a, b]$", "and $H^i(K) \\in \\mathcal{E}$ if $i \\in [a, b]$. Then $K$ is in", "$smd(add(\\mathcal{E}[a, b])^{\\star (b - a + 1)})$.", "\\item Let $b \\geq a$ and assume $H^i(K)$ is zero for $i \\not \\in [a, b]$", "and $H^i(K) \\in smd(add(\\mathcal{E}))$ if $i \\in [a, b]$. Then $K$ is in", "$smd(add(\\mathcal{E}[a, b])^{\\star (b - a + 1)})$.", "\\item Let $b \\geq a$ and assume $K$ can be represented by a complex $K^\\bullet$", "with $K^i = 0$ for $i \\not \\in [a, b]$ and $K^i \\in \\mathcal{E}$ for", "$i \\in [a, b]$. Then $K$ is in", "$smd(add(\\mathcal{E}[a, b])^{\\star (b - a + 1)})$.", "\\item Let $b \\geq a$ and assume $K$ can be represented by a complex $K^\\bullet$", "with $K^i = 0$ for $i \\not \\in [a, b]$ and $K^i \\in smd(add(\\mathcal{E}))$ for", "$i \\in [a, b]$. Then $K$ is in", "$smd(add(\\mathcal{E}[a, b])^{\\star (b - a + 1)})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-cone-n} without further mention.", "We will prove (2) which trivially implies (1). We use induction", "on $b - a$. If $b - a = 0$, then $K$ is isomorphic to $H^i(K)[-a]$", "in $\\mathcal{D}$ and the result is immediate. If $b - a > 0$, then", "we consider the distinguished triangle", "$$", "\\tau_{\\leq b - 1}K^\\bullet \\to K^\\bullet \\to K^b[-b]", "$$", "and we conclude by induction on $b - a$. We omit the proof of (3) and (4)." ], "refs": [ "derived-lemma-cone-n" ], "ref_ids": [ 1932 ] } ], "ref_ids": [] }, { "id": 1934, "type": "theorem", "label": "derived-lemma-forward-cone-n", "categories": [ "derived" ], "title": "derived-lemma-forward-cone-n", "contents": [ "Let $\\mathcal{T}$ be a triangulated category. Let", "$H : \\mathcal{T} \\to \\mathcal{A}$ be a homological functor", "to an abelian category $\\mathcal{A}$.", "Let $a \\leq b$ and $\\mathcal{E} \\subset \\Ob(\\mathcal{T})$", "be a subset such that $H^i(E) = 0$ for $E \\in \\mathcal{E}$", "and $i \\not \\in [a, b]$.", "Then for $X \\in smd(add(\\mathcal{E}[-m, m])^{\\star n})$", "we have $H^i(X) = 0$ for $i \\not \\in [-m + na, m + nb]$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Pleasant exercise in the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1935, "type": "theorem", "label": "derived-lemma-generated-by-E-explicit", "categories": [ "derived" ], "title": "derived-lemma-generated-by-E-explicit", "contents": [ "Let $\\mathcal{T}$ be a triangulated category. Let $E$ be an object", "of $\\mathcal{T}$. For $n \\geq 1$ we have", "$$", "\\langle E \\rangle_n =", "smd(\\langle E \\rangle_1 \\star \\ldots \\star \\langle E \\rangle_1) =", "smd({\\langle E \\rangle_1}^{\\star n}) =", "\\bigcup\\nolimits_{m \\geq 1} smd(add(E[-m, m])^{\\star n})", "$$", "For $n, n' \\geq 1$ we have $\\langle E \\rangle_{n + n'} =", "smd(\\langle E \\rangle_n \\star \\langle E \\rangle_{n'})$." ], "refs": [], "proofs": [ { "contents": [ "The left equality in the displayed formula follows from", "Lemmas \\ref{lemma-associativity-star} and \\ref{lemma-smd-star}", "and induction. The middle equality is a matter of notation.", "Since $\\langle E \\rangle_1 = smd(add(E[-\\infty, \\infty])])$", "and since $E[-\\infty, \\infty] = \\bigcup_{m \\geq 1} E[-m, m]$", "we see from Remark \\ref{remark-operations-unions} and", "Lemma \\ref{lemma-smd-star} that we get the equality on the right.", "Then the final statement follows from the remark and the", "corresponding statement of Lemma \\ref{lemma-cone-n}." ], "refs": [ "derived-lemma-associativity-star", "derived-lemma-smd-star", "derived-remark-operations-unions", "derived-lemma-smd-star", "derived-lemma-cone-n" ], "ref_ids": [ 1929, 1930, 2029, 1930, 1932 ] } ], "ref_ids": [] }, { "id": 1936, "type": "theorem", "label": "derived-lemma-find-smallest-containing-E", "categories": [ "derived" ], "title": "derived-lemma-find-smallest-containing-E", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let $E$ be an object", "of $\\mathcal{D}$. The subcategory", "$$", "\\langle E \\rangle = \\bigcup\\nolimits_n \\langle E \\rangle_n", "= \\bigcup\\nolimits_{n, m \\geq 1} smd(add(E[-m, m])^{\\star n})", "$$", "is a strictly full, saturated, triangulated subcategory of $\\mathcal{D}$", "and it is the smallest such subcategory of $\\mathcal{D}$ containing", "the object $E$." ], "refs": [], "proofs": [ { "contents": [ "The equality on the right follows from", "Lemma \\ref{lemma-generated-by-E-explicit}.", "It is clear that $\\langle E \\rangle = \\bigcup \\langle E \\rangle_n$", "contains $E$, is preserved under shifts, direct sums, direct summands.", "If $A \\in \\langle E \\rangle_a$ and $B \\in \\langle E \\rangle_b$", "and if $A \\to X \\to B \\to A[1]$ is a distinguished triangle, then", "$X \\in \\langle E \\rangle_{a + b}$ by Lemma \\ref{lemma-generated-by-E-explicit}.", "Hence $\\bigcup \\langle E \\rangle_n$ is also preserved under extensions", "and it follows that it is a triangulated subcategory. ", "\\medskip\\noindent", "Finally, let $\\mathcal{D}' \\subset \\mathcal{D}$ be a ", "strictly full, saturated, triangulated subcategory of $\\mathcal{D}$", "containing $E$. Then", "$\\mathcal{D}'[-\\infty, \\infty] \\subset \\mathcal{D}'$,", "$add(\\mathcal{D}) \\subset \\mathcal{D}'$,", "$smd(\\mathcal{D}') \\subset \\mathcal{D}'$, and", "$\\mathcal{D}' \\star \\mathcal{D}' \\subset \\mathcal{D}'$.", "In other words, all the operations we used to construct", "$\\langle E \\rangle$ out of $E$ preserve $\\mathcal{D}'$.", "Hence $\\langle E \\rangle \\subset \\mathcal{D}'$ and this", "finishes the proof." ], "refs": [ "derived-lemma-generated-by-E-explicit", "derived-lemma-generated-by-E-explicit" ], "ref_ids": [ 1935, 1935 ] } ], "ref_ids": [] }, { "id": 1937, "type": "theorem", "label": "derived-lemma-right-orthogonal", "categories": [ "derived" ], "title": "derived-lemma-right-orthogonal", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let $E, K$ be objects", "of $\\mathcal{D}$. The following are equivalent", "\\begin{enumerate}", "\\item $\\Hom(E, K[i]) = 0$ for all $i \\in \\mathbf{Z}$,", "\\item $\\Hom(E', K) = 0$ for all $E' \\in \\langle E \\rangle$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is immediate. Conversely,", "assume (1). Then $\\Hom(X, K) = 0$ for all $X$ in $\\langle E \\rangle_1$.", "Arguing by induction on $n$ and using", "Lemma \\ref{lemma-representable-homological}", "we see that $\\Hom(X, K) = 0$ for all $X$", "in $\\langle E \\rangle_n$." ], "refs": [ "derived-lemma-representable-homological" ], "ref_ids": [ 1758 ] } ], "ref_ids": [] }, { "id": 1938, "type": "theorem", "label": "derived-lemma-classical-generator-generator", "categories": [ "derived" ], "title": "derived-lemma-classical-generator-generator", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let $E$ be an object", "of $\\mathcal{D}$. If $E$ is a classical generator of $\\mathcal{D}$,", "then $E$ is a generator." ], "refs": [], "proofs": [ { "contents": [ "Assume $E$ is a classical generator. Let $K$ be an object of $\\mathcal{D}$", "such that $\\Hom(E, K[i]) = 0$ for all $i \\in \\mathbf{Z}$. By", "Lemma \\ref{lemma-right-orthogonal}", "$\\Hom(E', K) = 0$ for all $E'$ in $\\langle E \\rangle$. However, since", "$\\mathcal{D} = \\langle E \\rangle$ we conclude that $\\text{id}_K = 0$,", "i.e., $K = 0$." ], "refs": [ "derived-lemma-right-orthogonal" ], "ref_ids": [ 1937 ] } ], "ref_ids": [] }, { "id": 1939, "type": "theorem", "label": "derived-lemma-classical-generator-strong-generator", "categories": [ "derived" ], "title": "derived-lemma-classical-generator-strong-generator", "contents": [ "Let $\\mathcal{D}$ be a triangulated category which has a strong generator.", "Let $E$ be an object of $\\mathcal{D}$. If $E$ is a classical generator of", "$\\mathcal{D}$, then $E$ is a strong generator." ], "refs": [], "proofs": [ { "contents": [ "Let $E'$ be an object of $\\mathcal{D}$ such that", "$\\mathcal{D} = \\langle E' \\rangle_n$. Since", "$\\mathcal{D} = \\langle E \\rangle$ we see that $E' \\in \\langle E \\rangle_m$", "for some $m \\geq 1$ by Lemma \\ref{lemma-find-smallest-containing-E}.", "Then $\\langle E' \\rangle_1 \\subset \\langle E \\rangle_m$ hence", "$$", "\\mathcal{D} =", "\\langle E' \\rangle_n = smd(", "\\langle E' \\rangle_1 \\star \\ldots \\star \\langle E' \\rangle_1)", "\\subset", "smd(", "\\langle E \\rangle_m \\star \\ldots \\star \\langle E \\rangle_m)", "=", "\\langle E \\rangle_{nm}", "$$", "as desired. Here we used Lemma \\ref{lemma-generated-by-E-explicit}." ], "refs": [ "derived-lemma-find-smallest-containing-E", "derived-lemma-generated-by-E-explicit" ], "ref_ids": [ 1936, 1935 ] } ], "ref_ids": [] }, { "id": 1940, "type": "theorem", "label": "derived-lemma-compact-objects-subcategory", "categories": [ "derived" ], "title": "derived-lemma-compact-objects-subcategory", "contents": [ "Let $\\mathcal{D}$ be a (pre-)triangulated category with direct sums.", "Then the compact objects of $\\mathcal{D}$ form the objects of a", "Karoubian, saturated, strictly full, (pre-)triangulated subcategory", "$\\mathcal{D}_c$ of $\\mathcal{D}$." ], "refs": [], "proofs": [ { "contents": [ "Let $(X, Y, Z, f, g, h)$ be a distinguished triangle of $\\mathcal{D}$", "with $X$ and $Y$ compact. Then it follows from", "Lemma \\ref{lemma-representable-homological}", "and the five lemma", "(Homology, Lemma \\ref{homology-lemma-five-lemma})", "that $Z$ is a compact object too. It is clear that if $X \\oplus Y$", "is compact, then $X$, $Y$ are compact objects too. Hence", "$\\mathcal{D}_c$ is a saturated triangulated subcategory.", "Since $\\mathcal{D}$ is Karoubian by", "Lemma \\ref{lemma-projectors-have-images-triangulated}", "we conclude that the same is true for $\\mathcal{D}_c$." ], "refs": [ "derived-lemma-representable-homological", "homology-lemma-five-lemma", "derived-lemma-projectors-have-images-triangulated" ], "ref_ids": [ 1758, 12030, 1769 ] } ], "ref_ids": [] }, { "id": 1941, "type": "theorem", "label": "derived-lemma-write-as-colimit", "categories": [ "derived" ], "title": "derived-lemma-write-as-colimit", "contents": [ "Let $\\mathcal{D}$ be a triangulated category with direct sums.", "Let $E_i$, $i \\in I$ be a family of compact objects of $\\mathcal{D}$", "such that $\\bigoplus E_i$ generates $\\mathcal{D}$.", "Then every object $X$ of $\\mathcal{D}$ can be written as", "$$", "X = \\text{hocolim} X_n", "$$", "where $X_1$ is a direct sum of shifts of the $E_i$ and each transition", "morphism fits into a distinguished triangle", "$Y_n \\to X_n \\to X_{n + 1} \\to Y_n[1]$", "where $Y_n$ is a direct sum of shifts of the $E_i$." ], "refs": [], "proofs": [ { "contents": [ "Set $X_1 = \\bigoplus_{(i, m, \\varphi)} E_i[m]$ where the direct sum is over", "all triples $(i, m, \\varphi)$ such that $i \\in I$, $m \\in \\mathbf{Z}$", "and $\\varphi : E_i[m] \\to X$. Then $X_1$ comes equipped with a canonical", "morphism $X_1 \\to X$. Given $X_n \\to X$ we set", "$Y_n = \\bigoplus_{(i, m, \\varphi)} E_i[m]$ where the direct sum is over", "all triples $(i, m, \\varphi)$ such that $i \\in I$, $m \\in \\mathbf{Z}$, and", "$\\varphi : E_i[m] \\to X_n$ is a morphism such that $E_i[m] \\to X_n \\to X$", "is zero. Choose a distinguished triangle", "$Y_n \\to X_n \\to X_{n + 1} \\to Y_n[1]$", "and let $X_{n + 1} \\to X$ be any morphism such that $X_n \\to X_{n + 1} \\to X$", "is the given one; such a morphism exists by our choice of $Y_n$.", "We obtain a morphism $\\text{hocolim} X_n \\to X$ by the construction", "of our maps $X_n \\to X$. Choose a distinguished triangle", "$$", "C \\to \\text{hocolim} X_n \\to X \\to C[1]", "$$", "Let $E_i[m] \\to C$ be a morphism. Since $E_i$ is compact, the", "composition $E_i[m] \\to C \\to \\text{hocolim} X_n$ factors through", "$X_n$ for some $n$, say by $E_i[m] \\to X_n$. Then the", "construction of $Y_n$ shows that the composition", "$E_i[m] \\to X_n \\to X_{n + 1}$ is zero. In other words, the composition", "$E_i[m] \\to C \\to \\text{hocolim} X_n$ is zero. This means that our", "morphism $E_i[m] \\to C$ comes from a morphism $E_i[m] \\to X[-1]$.", "The construction of $X_1$ then shows that such morphism lifts to", "$\\text{hocolim} X_n$ and we conclude that our morphism $E_i[m] \\to C$", "is zero. The assumption that $\\bigoplus E_i$ generates $\\mathcal{D}$", "implies that $C$ is zero and the proof is done." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1942, "type": "theorem", "label": "derived-lemma-factor-through", "categories": [ "derived" ], "title": "derived-lemma-factor-through", "contents": [ "With assumptions and notation as in Lemma \\ref{lemma-write-as-colimit}.", "If $C$ is a compact object and $C \\to X_n$ is a morphism, then", "there is a factorization $C \\to E \\to X_n$ where", "$E$ is an object of $\\langle E_{i_1} \\oplus \\ldots \\oplus E_{i_t} \\rangle$", "for some $i_1, \\ldots, i_t \\in I$." ], "refs": [ "derived-lemma-write-as-colimit" ], "proofs": [ { "contents": [ "We prove this by induction on $n$. The base case $n = 1$ is clear.", "If $n > 1$ consider the composition $C \\to X_n \\to Y_{n - 1}[1]$.", "This can be factored through some $E'[1] \\to Y_{n - 1}[1]$ where", "$E'$ is a finite direct sum of shifts of the $E_i$. Let $I' \\subset I$", "be the finite set of indices that occur in this direct sum. Thus we obtain", "$$", "\\xymatrix{", "E' \\ar[r] \\ar[d] &", "C' \\ar[r] \\ar[d] &", "C \\ar[r] \\ar[d] &", "E'[1] \\ar[d] \\\\", "Y_{n - 1} \\ar[r] &", "X_{n - 1} \\ar[r] &", "X_n \\ar[r] &", "Y_{n - 1}[1]", "}", "$$", "By induction the morphism $C' \\to X_{n - 1}$ factors through", "$E'' \\to X_{n - 1}$ with $E''$ an object of", "$\\langle \\bigoplus_{i \\in I''} E_i \\rangle$", "for some finite subset $I'' \\subset I$. Choose a distinguished", "triangle", "$$", "E' \\to E'' \\to E \\to E'[1]", "$$", "then $E$ is an object of $\\langle \\bigoplus_{i \\in I' \\cup I''} E_i \\rangle$.", "By construction and the axioms of a triangulated category we can choose", "morphisms $C \\to E$ and a morphism $E \\to X_n$ fitting into morphisms", "of triangles $(E', C', C) \\to (E', E'', E)$ and", "$(E', E'', E) \\to (Y_{n - 1}, X_{n - 1}, X_n)$. The composition", "$C \\to E \\to X_n$ may not equal the given morphism $C \\to X_n$, but", "the compositions into $Y_{n - 1}$ are equal. Let $C \\to X_{n - 1}$", "be a morphism that lifts the difference. By induction assumption we", "can factor this through a morphism $E''' \\to X_{n - 1}$ with", "$E''$ an object of $\\langle \\bigoplus_{i \\in I'''} E_i \\rangle$", "for some finite subset $I' \\subset I$. Thus we see that we get", "a solution on considering $E \\oplus E''' \\to X_n$ because", "$E \\oplus E'''$ is an object of", "$\\langle \\bigoplus_{i \\in I' \\cup I'' \\cup I'''} E_i \\rangle$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1941 ] }, { "id": 1943, "type": "theorem", "label": "derived-lemma-brown", "categories": [ "derived" ], "title": "derived-lemma-brown", "contents": [ "\\begin{reference}", "\\cite[Theorem 3.1]{Neeman-Grothendieck}.", "\\end{reference}", "Let $\\mathcal{D}$ be a triangulated category with direct sums which is", "compactly generated. Let $H : \\mathcal{D} \\to \\textit{Ab}$ be a contravariant", "cohomological functor which transforms direct sums into products.", "Then $H$ is representable." ], "refs": [], "proofs": [ { "contents": [ "Let $E_i$, $i \\in I$ be a set of compact objects such that", "$\\bigoplus_{i \\in I} E_i$ generates $\\mathcal{D}$. We may and do assume", "that the set of objects $\\{E_i\\}$ is preserved under shifts. Consider pairs", "$(i, a)$ where $i \\in I$ and $a \\in H(E_i)$ and set", "$$", "X_1 = \\bigoplus\\nolimits_{(i, a)} E_i", "$$", "Since $H(X_1) = \\prod_{(i, a)} H(E_i)$ we see that $(a)_{(i, a)}$", "defines an element $a_1 \\in H(X_1)$. Set $H_1 = \\Hom_\\mathcal{D}(- , X_1)$.", "By Yoneda's lemma (Categories, Lemma \\ref{categories-lemma-yoneda})", "the element $a_1$ defines a natural transformation $H_1 \\to H$.", "\\medskip\\noindent", "We are going to inductively construct $X_n$ and transformations", "$a_n : H_n \\to H$ where $H_n = \\Hom_\\mathcal{D}(-, X_n)$.", "Namely, we apply the procedure", "above to the functor $\\Ker(H_n \\to H)$ to get an object", "$$", "K_{n + 1} = \\bigoplus\\nolimits_{(i, k),\\ k \\in \\Ker(H_n(E_i) \\to H(E_i))} E_i", "$$", "and a transformation $\\Hom_\\mathcal{D}(-, K_{n + 1}) \\to \\Ker(H_n \\to H)$.", "By Yoneda's lemma the composition $\\Hom_\\mathcal{D}(-, K_{n + 1}) \\to H_n$", "gives a morphism $K_{n + 1} \\to X_n$. We choose", "a distinguished triangle", "$$", "K_{n + 1} \\to X_n \\to X_{n + 1} \\to K_{n + 1}[1]", "$$", "in $\\mathcal{D}$. The element $a_n \\in H(X_n)$ maps to zero", "in $H(K_{n + 1})$ by construction. Since $H$ is cohomological", "we can lift it to an element $a_{n + 1} \\in H(X_{n + 1})$.", "\\medskip\\noindent", "We claim that $X = \\text{hocolim} X_n$ represents $H$. Applying $H$", "to the defining distinguished triangle", "$$", "\\bigoplus X_n \\to", "\\bigoplus X_n \\to X \\to", "\\bigoplus X_n[1]", "$$", "we obtain an exact sequence", "$$", "\\prod H(X_n) \\leftarrow ", "\\prod H(X_n) \\leftarrow ", "H(X)", "$$", "Thus there exists an element $a \\in H(X)$ mapping to $(a_n)$", "in $\\prod H(X_n)$. Hence a natural transformation", "$\\Hom_\\mathcal{D}(- , X) \\to H$ such that", "$$", "\\Hom_\\mathcal{D}(-, X_1) \\to", "\\Hom_\\mathcal{D}(-, X_2) \\to", "\\Hom_\\mathcal{D}(-, X_3) \\to \\ldots \\to", "\\Hom_\\mathcal{D}(-, X) \\to H", "$$", "commutes. For each $i$ the map $\\Hom_\\mathcal{D}(E_i, X) \\to H(E_i)$", "is surjective, by construction of $X_1$. On the other hand, by construction", "of $X_n \\to X_{n + 1}$ the kernel of $\\Hom_\\mathcal{D}(E_i, X_n) \\to H(E_i)$", "is killed by the map", "$\\Hom_\\mathcal{D}(E_i, X_n) \\to \\Hom_\\mathcal{D}(E_i, X_{n + 1})$.", "Since", "$$", "\\Hom_\\mathcal{D}(E_i, X) = \\colim \\Hom_\\mathcal{D}(E_i, X_n)", "$$", "by Lemma \\ref{lemma-commutes-with-countable-sums}", "we see that $\\Hom_\\mathcal{D}(E_i, X) \\to H(E_i)$ is injective.", "\\medskip\\noindent", "To finish the proof, consider the subcategory", "$$", "\\mathcal{D}' =", "\\{Y \\in \\Ob(\\mathcal{D}) \\mid \\Hom_\\mathcal{D}(Y[n], X) \\to H(Y[n])", "\\text{ is an isomorphism for all }n\\}", "$$", "As $\\Hom_\\mathcal{D}(-, X) \\to H$ is a transformation between", "cohomological functors,", "the subcategory $\\mathcal{D}'$ is a strictly full, saturated, triangulated", "subcategory of $\\mathcal{D}$ (details omitted; see proof of", "Lemma \\ref{lemma-homological-functor-kernel}). Moreover, as both", "$H$ and $\\Hom_\\mathcal{D}(-, X)$ transform direct sums into products,", "we see that direct sums of objects of $\\mathcal{D}'$ are in $\\mathcal{D}'$.", "Thus derived colimits of objects of $\\mathcal{D}'$ are in $\\mathcal{D}'$.", "Since $\\{E_i\\}$ is preserved under shifts, we see that $E_i$", "is an object of $\\mathcal{D}'$ for all $i$. It follows from", "Lemma \\ref{lemma-write-as-colimit} that $\\mathcal{D}' = \\mathcal{D}$", "and the proof is complete." ], "refs": [ "categories-lemma-yoneda", "derived-lemma-commutes-with-countable-sums", "derived-lemma-homological-functor-kernel", "derived-lemma-write-as-colimit" ], "ref_ids": [ 12203, 1924, 1785, 1941 ] } ], "ref_ids": [] }, { "id": 1944, "type": "theorem", "label": "derived-lemma-pre-prepare-adjoint", "categories": [ "derived" ], "title": "derived-lemma-pre-prepare-adjoint", "contents": [ "Let $\\mathcal{D}$ be a triangulated category.", "Let $\\mathcal{A} \\subset \\mathcal{D}$", "be a full subcategory invariant under all shifts.", "Consider a distinguished triangle", "$$", "X \\to Y \\to Z \\to X[1]", "$$", "of $\\mathcal{D}$. The following are equivalent", "\\begin{enumerate}", "\\item $Z$ is in $\\mathcal{A}^\\perp$, and", "\\item $\\Hom(A, X) = \\Hom(A, Y)$ for all $A \\in \\Ob(\\mathcal{A})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-representable-homological} the functor", "$\\Hom(A, -)$ is homological and hence we get a long exact sequence", "as in (\\ref{equation-long-exact-cohomology-sequence}).", "Assume (1) and let $A \\in \\Ob(\\mathcal{A})$.", "Then we consider the exact sequence", "$$", "\\Hom(A[1], Z) \\to \\Hom(A, X) \\to \\Hom(A, Y) \\to \\Hom(A, Z)", "$$", "Since $A[1] \\in \\Ob(\\mathcal{A})$", "we see that the first and last groups are zero.", "Thus we get (2). Assume (2) and let $A \\in \\Ob(\\mathcal{A})$.", "Then we consider the exact sequence", "$$", "\\Hom(A, X) \\to \\Hom(A, Y) \\to \\Hom(A, Z) \\to \\Hom(A[-1], X) \\to \\Hom(A[-1], Y)", "$$", "and we conclude that $\\Hom(A, Z) = 0$ as desired." ], "refs": [ "derived-lemma-representable-homological" ], "ref_ids": [ 1758 ] } ], "ref_ids": [] }, { "id": 1945, "type": "theorem", "label": "derived-lemma-orthogonal-triangulated", "categories": [ "derived" ], "title": "derived-lemma-orthogonal-triangulated", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let", "$\\mathcal{A} \\subset \\mathcal{D}$ be a full subcategory invariant", "under all shifts. Then both the right orthogonal $\\mathcal{A}^\\perp$ and", "the left orthogonal ${}^\\perp\\mathcal{A}$ of $\\mathcal{A}$", "are strictly full, saturated\\footnote{Definition \\ref{definition-saturated}.},", "triangulated subcagories of $\\mathcal{D}$." ], "refs": [ "derived-definition-saturated" ], "proofs": [ { "contents": [ "It is immediate from the definitions that the orthogonals are preserved", "under taking shifts, direct sums, and direct summands.", "Consider a distinguished triangle", "$$", "X \\to Y \\to Z \\to X[1]", "$$", "of $\\mathcal{D}$. By Lemma \\ref{lemma-triangulated-subcategory} it", "suffices to show that if $X$ and $Y$ are in $\\mathcal{A}^\\perp$, then", "$Z$ is in $\\mathcal{A}^\\perp$. This is immediate from", "Lemma \\ref{lemma-pre-prepare-adjoint}." ], "refs": [ "derived-lemma-triangulated-subcategory", "derived-lemma-pre-prepare-adjoint" ], "ref_ids": [ 1771, 1944 ] } ], "ref_ids": [ 1974 ] }, { "id": 1946, "type": "theorem", "label": "derived-lemma-prepare-adjoint", "categories": [ "derived" ], "title": "derived-lemma-prepare-adjoint", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{A}$", "be a full triangulated subcategory of $\\mathcal{D}$. For an object $X$", "of $\\mathcal{D}$ consider the property $P(X)$: there exists a", "distinguished triangle $A \\to X \\to B \\to A[1]$", "in $\\mathcal{D}$ with $A$ in $\\mathcal{A}$ and $B$ in $\\mathcal{A}^\\perp$.", "\\begin{enumerate}", "\\item If $X_1 \\to X_2 \\to X_3 \\to X_1[1]$ is a distinguished triangle", "and $P$ holds for two out of three, then it holds for the third.", "\\item If $P$ holds for $X_1$ and $X_2$, then it holds for $X_1 \\oplus X_2$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X_1 \\to X_2 \\to X_3 \\to X_1[1]$ be a distinguished triangle", "and assume $P$ holds for $X_1$ and $X_2$. Choose distinguished triangles", "$$", "A_1 \\to X_1 \\to B_1 \\to A_1[1]", "\\quad\\text{and}\\quad", "A_2 \\to X_2 \\to B_2 \\to A_2[1]", "$$", "as in condition $P$. Since", "$\\Hom(A_1, A_2) = \\Hom(A_1, X_2)$ by Lemma \\ref{lemma-pre-prepare-adjoint}", "there is a unique morphism $A_1 \\to A_2$ such that the diagram", "$$", "\\xymatrix{", "A_1 \\ar[d] \\ar[r] & X_1 \\ar[d] \\\\", "A_2 \\ar[r] & X_2", "}", "$$", "commutes. Choose an extension of this to a diagram", "$$", "\\xymatrix{", "A_1 \\ar[r] \\ar[d] & X_1 \\ar[r] \\ar[d] & Q_1 \\ar[r] \\ar[d] & A_1[1] \\ar[d] \\\\", "A_2 \\ar[r] \\ar[d] & X_2 \\ar[r] \\ar[d] & Q_2 \\ar[r] \\ar[d] & A_2[1] \\ar[d] \\\\", "A_3 \\ar[r] \\ar[d] & X_3 \\ar[r] \\ar[d] & Q_3 \\ar[r] \\ar[d] & A_3[1] \\ar[d] \\\\", "A_1[1] \\ar[r] & X_1[1] \\ar[r] & Q_1[1] \\ar[r] & A_1[2]", "}", "$$", "as in Proposition \\ref{proposition-9}. By TR3 we see that", "$Q_1 \\cong B_1$ and $Q_2 \\cong B_2$ and hence", "$Q_1, Q_2 \\in \\Ob(\\mathcal{A}^\\perp)$.", "As $Q_1 \\to Q_2 \\to Q_3 \\to Q_1[1]$", "is a distinguished triangle we see that $Q_3 \\in \\Ob(\\mathcal{A}^\\perp)$", "by Lemma \\ref{lemma-orthogonal-triangulated}.", "Since $\\mathcal{A}$ is a full triangulated subcategory, we see that", "$A_3$ is isomorphic to an object of $\\mathcal{A}$.", "Thus $X_3$ satisfies $P$. The other cases of (1) follow from this", "case by translation. Part (2) is a special case of (1)", "via Lemma \\ref{lemma-split}." ], "refs": [ "derived-lemma-pre-prepare-adjoint", "derived-proposition-9", "derived-lemma-orthogonal-triangulated", "derived-lemma-split" ], "ref_ids": [ 1944, 1958, 1945, 1766 ] } ], "ref_ids": [] }, { "id": 1947, "type": "theorem", "label": "derived-lemma-right-adjoint", "categories": [ "derived" ], "title": "derived-lemma-right-adjoint", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let", "$\\mathcal{A} \\subset \\mathcal{D}$ be a full triangulated subcategory.", "The following are equivalent", "\\begin{enumerate}", "\\item the inclusion functor $\\mathcal{A} \\to \\mathcal{D}$", "has a right adjoint, and", "\\item for every $X$ in $\\mathcal{D}$ there exists a distinguished", "triangle", "$$", "A \\to X \\to B \\to X'[1]", "$$", "in $\\mathcal{D}$ with $A \\in \\Ob(\\mathcal{A})$ and", "$B \\in \\Ob(\\mathcal{A}^\\perp)$.", "\\end{enumerate}", "If this holds, then $\\mathcal{A}$ is saturated", "(Definition \\ref{definition-saturated}) and if $\\mathcal{A}$", "is strictly full in $\\mathcal{D}$, then", "$\\mathcal{A} = {}^\\perp(\\mathcal{A}^\\perp)$." ], "refs": [ "derived-definition-saturated" ], "proofs": [ { "contents": [ "Assume (1) and denote $v : \\mathcal{D} \\to \\mathcal{A}$ the right adjoint.", "Let $X \\in \\Ob(\\mathcal{D})$. Set $A = v(X)$. We may extend the", "adjunction mapping $A \\to X$ to a distinguished triangle", "$A \\to X \\to B \\to A[1]$. Since", "$$", "\\Hom_\\mathcal{A}(A', A) =", "\\Hom_\\mathcal{A}(A', v(X)) =", "\\Hom_\\mathcal{D}(A', X)", "$$", "for $A' \\in \\Ob(\\mathcal{A})$, we conclude that $B \\in \\Ob(\\mathcal{A}^\\perp)$", "by Lemma \\ref{lemma-pre-prepare-adjoint}.", "\\medskip\\noindent", "Assume (2). We will contruct the adjoint $v$ explictly.", "Let $X \\in \\Ob(\\mathcal{D})$. Choose $A \\to X \\to B \\to A[1]$ as in (2).", "Set $v(X) = A$. Let $f : X \\to Y$ be a morphism in $\\mathcal{D}$.", "Choose $A' \\to Y \\to B' \\to A'[1]$ as in (2). Since", "$\\Hom(A, A') = \\Hom(A, Y)$ by Lemma \\ref{lemma-pre-prepare-adjoint}", "there is a unique morphism $f' : A \\to A'$ such that the diagram", "$$", "\\xymatrix{", "A \\ar[d]_{f'} \\ar[r] & X \\ar[d]^f \\\\", "A' \\ar[r] & Y", "}", "$$", "commutes. Hence we can set $v(f) = f'$ to get a functor.", "To see that $v$ is adjoint to the inclusion morphism use", "Lemma \\ref{lemma-pre-prepare-adjoint} again.", "\\medskip\\noindent", "Proof of the final statement. In other to prove that $\\mathcal{A}$", "is saturated we may replace $\\mathcal{A}$ by the strictly full", "subcategory having the same isomorphism classes as $\\mathcal{A}$;", "details omitted. Assume $\\mathcal{A}$ is strictly full. If we show that", "$\\mathcal{A} = {}^\\perp(\\mathcal{A}^\\perp)$, then", "$\\mathcal{A}$ will be saturated by Lemma \\ref{lemma-orthogonal-triangulated}.", "Since the incusion $\\mathcal{A} \\subset {}^\\perp(\\mathcal{A}^\\perp)$", "is clear it suffices to prove the other inclusion.", "Let $X$ be an object of ${}^\\perp(\\mathcal{A}^\\perp)$.", "Choose a distinguished triangle $A \\to X \\to B \\to A[1]$", "as in (2). As $\\Hom(X, B) = 0$ by assumption we see that", "$A \\cong X \\oplus B[-1]$ by Lemma \\ref{lemma-split}.", "Since $\\Hom(A, B[-1]) = 0$ as $B \\in \\mathcal{A}^\\perp$", "this implies $B[-1] = 0$ and $A \\cong X$ as desired." ], "refs": [ "derived-lemma-pre-prepare-adjoint", "derived-lemma-pre-prepare-adjoint", "derived-lemma-pre-prepare-adjoint", "derived-lemma-orthogonal-triangulated", "derived-lemma-split" ], "ref_ids": [ 1944, 1944, 1944, 1945, 1766 ] } ], "ref_ids": [ 1974 ] }, { "id": 1948, "type": "theorem", "label": "derived-lemma-left-adjoint", "categories": [ "derived" ], "title": "derived-lemma-left-adjoint", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let", "$\\mathcal{A} \\subset \\mathcal{D}$ be a full triangulated subcategory.", "The following are equivalent", "\\begin{enumerate}", "\\item the inclusion functor $\\mathcal{A} \\to \\mathcal{D}$", "has a left adjoint, and", "\\item for every $X$ in $\\mathcal{D}$ there exists a distinguished", "triangle", "$$", "B \\to X \\to A \\to K[1]", "$$", "in $\\mathcal{D}$ with $A \\in \\Ob(\\mathcal{A})$ and", "$B \\in \\Ob({}^\\perp\\mathcal{A})$.", "\\end{enumerate}", "If this holds, then $\\mathcal{A}$ is saturated", "(Definition \\ref{definition-saturated}) and if $\\mathcal{A}$", "is strictly full in $\\mathcal{D}$, then", "$\\mathcal{A} = ({}^\\perp\\mathcal{A})^\\perp$." ], "refs": [ "derived-definition-saturated" ], "proofs": [ { "contents": [ "Omitted. Dual to Lemma \\ref{lemma-right-adjoint}." ], "refs": [ "derived-lemma-right-adjoint" ], "ref_ids": [ 1947 ] } ], "ref_ids": [ 1974 ] }, { "id": 1949, "type": "theorem", "label": "derived-lemma-postnikov-system-small-cases", "categories": [ "derived" ], "title": "derived-lemma-postnikov-system-small-cases", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Consider", "Postnikov systems for complexes of length $n$.", "\\begin{enumerate}", "\\item For $n = 0$ Postnikov systems always exist and", "any morphism (\\ref{equation-map-complexes}) of complexes", "extends to a unique morphism of Postnikov systems.", "\\item For $n = 1$ Postnikov systems always exist and", "any morphism (\\ref{equation-map-complexes}) of complexes", "extends to a (nonunique) morphism of Postnikov systems.", "\\item For $n = 2$ Postnikov systems always exist but", "morphisms (\\ref{equation-map-complexes}) of complexes", "in general do not extend to morphisms of Postnikov systems.", "\\item For $n > 2$ Postnikov systems do not always exist.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The case $n = 0$ is immediate as isomorphisms are invertible.", "The case $n = 1$ follows immediately from TR1 (existence of triangles)", "and TR3 (extending morphisms to triangles).", "For the case $n = 2$ we argue as follows.", "Set $Y_0 = X_0$. By the case $n = 1$ we can choose", "a Postnikov system", "$$", "Y_1 \\to X_1 \\to Y_0 \\to Y_1[1]", "$$", "Since the composition $X_2 \\to X_1 \\to X_0$ is zero, we can factor", "$X_2 \\to X_1$ (nonuniquely) as $X_2 \\to Y_1 \\to X_1$ by", "Lemma \\ref{lemma-representable-homological}.", "Then we simply fit the morphism $X_2 \\to Y_1$ into a distinguished", "triangle", "$$", "Y_2 \\to X_2 \\to Y_1 \\to Y_2[1]", "$$", "to get the Postnikov system for $n = 2$.", "For $n > 2$ we cannot argue similarly, as we do not", "know whether the composition $X_n \\to X_{n - 1} \\to Y_{n - 1}$", "is zero in $\\mathcal{D}$." ], "refs": [ "derived-lemma-representable-homological" ], "ref_ids": [ 1758 ] } ], "ref_ids": [] }, { "id": 1950, "type": "theorem", "label": "derived-lemma-maps-postnikov-systems-vanishing", "categories": [ "derived" ], "title": "derived-lemma-maps-postnikov-systems-vanishing", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Given a map", "(\\ref{equation-map-complexes}) consider the condition", "\\begin{equation}", "\\label{equation-P}", "\\Hom(X_i[i - j - 1], X'_j) = 0 \\text{ for }i > j + 1", "\\end{equation}", "Then", "\\begin{enumerate}", "\\item If we have a Postnikov system for", "$X'_n \\to X'_{n - 1} \\to \\ldots \\to X'_0$ then", "property (\\ref{equation-P}) implies that", "$$", "\\Hom(X_i[i - j - 1], Y'_j) = 0 \\text{ for }i > j + 1", "$$", "\\item If we are given Postnikov systems for both complexes and", "we have (\\ref{equation-P}), then the map extends to a (nonunique) map", "of Postnikov systems.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We first prove (1) by induction on $j$. For the base case $j = 0$", "there is nothing to prove as $Y'_0 \\to X'_0$ is an isomorphism.", "Say the result holds for $j - 1$. We consider the distinguished triangle", "$$", "Y'_j \\to X'_j \\to Y'_{j - 1} \\to Y'_j[1]", "$$", "The long exact sequence of Lemma \\ref{lemma-representable-homological}", "gives an exact sequence", "$$", "\\Hom(X_i[i - j - 1], Y'_{j - 1}[-1]) \\to", "\\Hom(X_i[i - j - 1], Y'_j) \\to", "\\Hom(X_i[i - j - 1], X'_j)", "$$", "From the induction hypothesis and (\\ref{equation-P}) we conclude the outer", "groups are zero and we win.", "\\medskip\\noindent", "Proof of (2). For $n = 1$ the existence of morphisms has been", "established in Lemma \\ref{lemma-postnikov-system-small-cases}.", "For $n > 1$ by induction, we may assume given the map of", "Postnikov systems of length $n - 1$. The problem is that we do", "not know whether the diagram", "$$", "\\xymatrix{", "X_n \\ar[r] \\ar[d] & Y_{n - 1} \\ar[d] \\\\", "X'_n \\ar[r] & Y'_{n - 1}", "}", "$$", "is commutative. Denote $\\alpha : X_n \\to Y'_{n - 1}$ the difference.", "Then we do know that the composition of $\\alpha$ with", "$Y'_{n - 1} \\to X'_{n - 1}$ is zero (because of what it means", "to be a map of Postnikov systems of length $n - 1$).", "By the distinguished triangle", "$Y'_{n - 1} \\to X'_{n - 1} \\to Y'_{n - 2} \\to Y'_{n - 1}[1]$,", "this means that $\\alpha$ is the composition of", "$Y'_{n - 2}[-1] \\to Y'_{n - 1}$ with", "a map $\\alpha' : X_n \\to Y'_{n - 2}[-1]$. Then (\\ref{equation-P}) guarantees", "$\\alpha'$ is zero by part (1) of the lemma. Thus $\\alpha$ is zero.", "To finish the proof of existence, the commutativity guarantees", "we can choose the dotted arrow fitting into the diagram", "$$", "\\xymatrix{", "Y_{n - 1}[-1] \\ar[d] \\ar[r] &", "Y_n \\ar[r] \\ar@{..>}[d] &", "X_n \\ar[r] \\ar[d] &", "Y_{n - 1} \\ar[d] \\\\", "Y'_{n - 1}[-1] \\ar[r] &", "Y'_n \\ar[r] &", "X'_n \\ar[r] &", "Y'_{n - 1}", "}", "$$", "by TR3." ], "refs": [ "derived-lemma-representable-homological", "derived-lemma-postnikov-system-small-cases" ], "ref_ids": [ 1758, 1949 ] } ], "ref_ids": [] }, { "id": 1951, "type": "theorem", "label": "derived-lemma-uniqueness-maps-postnikov-systems", "categories": [ "derived" ], "title": "derived-lemma-uniqueness-maps-postnikov-systems", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Given a map", "(\\ref{equation-map-complexes}) assume we are given", "Postnikov systems for both complexes. If", "\\begin{enumerate}", "\\item $\\Hom(X_i[i], Y'_n[n]) = 0$ for $i = 1, \\ldots, n$, or", "\\item $\\Hom(Y_n[n], X'_{n - i}[n - i]) = 0$ for $i = 1, \\ldots, n$, or", "\\item $\\Hom(X_{j - i}[-i + 1], X'_j) = 0$ and", "$\\Hom(X_j, X'_{j - i}[-i]) = 0$ for $j \\geq i > 0$,", "\\end{enumerate}", "then there exists at most one morphism between these Postnikov systems." ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Look at the following diagram", "$$", "\\xymatrix{", "Y_0 \\ar[r] \\ar[d] &", "Y_1[1] \\ar[r] \\ar[ld] &", "Y_2[2] \\ar[r] \\ar[lld] &", "\\ldots \\ar[r] &", "Y_n[n] \\ar[lllld] \\\\", "Y'_n[n]", "}", "$$", "The arrows are the composition of the morphism $Y_n[n] \\to Y'_n[n]$", "and the morphism $Y_i[i] \\to Y_n[n]$. The arrow $Y_0 \\to Y'_n[n]$", "is determined as it is the composition $Y_0 = X_0 \\to X'_0 = Y'_0 \\to Y'_n[n]$.", "Since we have the distinguished triangle $Y_0 \\to Y_1[1] \\to X_1[1]$", "we see that $\\Hom(X_1[1], Y'_n[n]) = 0$ guarantees that the second vertical", "arrow is unique. Since we have the distinguished triangle", "$Y_1[1] \\to Y_2[2] \\to X_2[2]$ we see that $\\Hom(X_2[2], Y'_n[n]) = 0$", "guarantees that the third vertical arrow is unique. And so on.", "\\medskip\\noindent", "Proof of (2). The composition $Y_n[n] \\to Y'_n[n] \\to X_n[n]$ is", "the same as the composition $Y_n[n] \\to X_n[n] \\to X'_n[n]$ and hence", "is unique. Then using the distinguished triangle", "$Y'_{n - 1}[n - 1] \\to Y'_n[n] \\to X'_n[n]$ we see that it suffices", "to show $\\Hom(Y_n[n], Y'_{n - 1}[n - 1]) = 0$. Using the distinguished", "triangles", "$$", "Y'_{n - i - 1}[n - i - 1] \\to Y'_{n - i}[n - i] \\to X'_{n - i}[n - i]", "$$", "we get this vanishing from our assumption. Small details omitted.", "\\medskip\\noindent", "Proof of (3). Looking at the proof of", "Lemma \\ref{lemma-maps-postnikov-systems-vanishing}", "and arguing by induction on $n$ it suffices to show that the dotted arrow", "in the morphism of triangles", "$$", "\\xymatrix{", "Y_{n - 1}[-1] \\ar[d] \\ar[r] &", "Y_n \\ar[r] \\ar@{..>}[d] &", "X_n \\ar[r] \\ar[d] &", "Y_{n - 1} \\ar[d] \\\\", "Y'_{n - 1}[-1] \\ar[r] &", "Y'_n \\ar[r] &", "X'_n \\ar[r] &", "Y'_{n - 1}", "}", "$$", "is unique. By Lemma \\ref{lemma-uniqueness-third-arrow} part (5)", "it suffices to show that $\\Hom(Y_{n - 1}, X'_n) = 0$ and", "$\\Hom(X_n, Y'_{n - 1}[-1]) = 0$.", "To prove the first vanishing we use the distinguished triangles", "$Y_{n - i - 1}[-i] \\to Y_{n - i}[-(i - 1)] \\to X_{n - i}[-(i - 1)]$", "for $i > 0$ and induction on $i$ to see that the assumed", "vanishing of $\\Hom(X_{n - i}[-i + 1], X'_n)$ is enough.", "For the second we similarly use the distinguished triangles", "$Y'_{n - i - 1}[-i - 1] \\to Y'_{n - i}[-i] \\to X'_{n - i}[-i]$", "to see that the assumed vanishing of", "$\\Hom(X_n, X'_{n - i}[-i])$ is enough as well." ], "refs": [ "derived-lemma-maps-postnikov-systems-vanishing", "derived-lemma-uniqueness-third-arrow" ], "ref_ids": [ 1950, 1763 ] } ], "ref_ids": [] }, { "id": 1952, "type": "theorem", "label": "derived-lemma-existence-postnikov-system", "categories": [ "derived" ], "title": "derived-lemma-existence-postnikov-system", "contents": [ "Let $\\mathcal{D}$ be a triangulated category.", "Let $X_n \\to X_{n - 1} \\to \\ldots \\to X_0$ be", "a complex in $\\mathcal{D}$. If", "$$", "\\Hom(X_i[i - j - 2], X_j) = 0 \\text{ for }i > j + 2", "$$", "then there exists a Postnikov system. If we have", "$$", "\\Hom(X_i[i - j - 1], X_j) = 0 \\text{ for }i > j + 1", "$$", "then any two Postnikov systems are isomorphic." ], "refs": [], "proofs": [ { "contents": [ "We argue by induction on $n$. The cases $n = 0, 1, 2$", "follow from Lemma \\ref{lemma-postnikov-system-small-cases}.", "Assume $n > 2$.", "Suppose given a Postnikov system for the complex", "$X_{n - 1} \\to X_{n - 2} \\to \\ldots \\to X_0$.", "The only obstruction to extending this to a Postnikov system", "of length $n$ is that we have to find a morphism", "$X_n \\to Y_{n - 1}$ such that the composition", "$X_n \\to Y_{n - 1} \\to X_{n - 1}$ is equal to the given map", "$X_n \\to X_{n - 1}$. Considering the distinguished triangle", "$$", "Y_{n - 1} \\to X_{n - 1} \\to Y_{n - 2} \\to Y_{n - 1}[1]", "$$", "and the associated long exact sequence coming from this", "and the functor $\\Hom(X_n, -)$", "(see Lemma \\ref{lemma-representable-homological})", "we find that it suffices to show that the composition", "$X_n \\to X_{n - 1} \\to Y_{n - 2}$ is zero.", "Since we know that $X_n \\to X_{n - 1} \\to X_{n - 2}$ is zero", "we can apply the distinguished triangle", "$$", "Y_{n - 2} \\to X_{n - 2} \\to Y_{n - 3} \\to Y_{n - 2}[1]", "$$", "to see that it suffices if $\\Hom(X_n, Y_{n - 3}[-1]) = 0$.", "Arguing exactly as in the proof of", "Lemma \\ref{lemma-maps-postnikov-systems-vanishing} part (1)", "the reader easily sees this follows from the condition", "stated in the lemma.", "\\medskip\\noindent", "The statement on isomorphisms follows from the existence of a map", "between the Postnikov systems extending the identity on the complex", "proven in Lemma \\ref{lemma-maps-postnikov-systems-vanishing} part (2)", "and Lemma \\ref{lemma-third-isomorphism-triangle} to show all the maps are", "isomorphisms." ], "refs": [ "derived-lemma-postnikov-system-small-cases", "derived-lemma-representable-homological", "derived-lemma-maps-postnikov-systems-vanishing", "derived-lemma-maps-postnikov-systems-vanishing", "derived-lemma-third-isomorphism-triangle" ], "ref_ids": [ 1949, 1758, 1950, 1950, 1759 ] } ], "ref_ids": [] }, { "id": 1953, "type": "theorem", "label": "derived-lemma-essentially-constant", "categories": [ "derived" ], "title": "derived-lemma-essentially-constant", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let $(A_i)$ be an inverse system", "in $\\mathcal{D}$. Then $(A_i)$ is essentially constant (see", "Categories, Definition", "\\ref{categories-definition-essentially-constant-diagram})", "if and only if there exists an $i$ and for all $j \\geq i$ a direct sum", "decomposition $A_j = A \\oplus Z_j$ such that", "(a) the maps $A_{j'} \\to A_j$ are compatible with the direct sum", "decompositions and identity on $A$, (b) for all $j \\geq i$ there exists some", "$j' \\geq j$ such that $Z_{j'} \\to Z_j$ is zero." ], "refs": [ "categories-definition-essentially-constant-diagram" ], "proofs": [ { "contents": [ "Assume $(A_i)$ is essentially constant with value $A$. Then $A = \\lim A_i$", "and there exists an $i$ and a morphism $A_i \\to A$ such that (1)", "the composition $A \\to A_i \\to A$ is the identity on $A$ and (2) for all", "$j \\geq i$ there exists a $j' \\geq j$ such that $A_{j'} \\to A_j$ factors as", "$A_{j'} \\to A_i \\to A \\to A_j$. From (1) we conclude that for $j \\geq i$", "the maps $A \\to A_j$ and $A_j \\to A_i \\to A$ compose to the identity on $A$.", "It follows that $A_j \\to A$ has a kernel $Z_j$ and that", "the map $A \\oplus Z_j \\to A_j$ is an isomorphism, see", "Lemmas \\ref{lemma-when-split} and \\ref{lemma-split}.", "These direct sum decompositions clearly satisfy (a).", "From (2) we conclude that for all $j$ there is a $j' \\geq j$ such that", "$Z_{j'} \\to Z_j$ is zero, so (b) holds. Proof of the converse is omitted." ], "refs": [ "derived-lemma-when-split", "derived-lemma-split" ], "ref_ids": [ 1767, 1766 ] } ], "ref_ids": [ 12368 ] }, { "id": 1954, "type": "theorem", "label": "derived-lemma-essentially-constant-2-out-of-3", "categories": [ "derived" ], "title": "derived-lemma-essentially-constant-2-out-of-3", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let", "$$", "A_n \\to B_n \\to C_n \\to A_n[1]", "$$", "be an inverse system of distinguished triangles in $\\mathcal{D}$.", "If $(A_n)$ and $(C_n)$ are essentially constant, then", "$(B_n)$ is essentially constant and their values fit into", "a distinguished triangle $A \\to B \\to C \\to A[1]$ such that for", "some $n \\geq 1$ there is a map", "$$", "\\xymatrix{", "A_n \\ar[d] \\ar[r] &", "B_n \\ar[d] \\ar[r] &", "C_n \\ar[d] \\ar[r] &", "A_n[1] \\ar[d] \\\\", "A \\ar[r] &", "B \\ar[r] &", "C \\ar[r] &", "A[1]", "}", "$$", "of distinguished triangles which induces an isomorphism", "$\\lim_{n' \\geq n} A_{n'} \\to A$ and similarly for $B$ and $C$." ], "refs": [], "proofs": [ { "contents": [ "After renumbering we may assume that $A_n = A \\oplus A'_n$ and", "$C_n = C \\oplus C'_n$ for inverse systems $(A'_n)$ and $(C'_n)$", "which are essentially zero, see Lemma \\ref{lemma-essentially-constant}.", "In particular, the morphism", "$$", "C \\oplus C'_n \\to (A \\oplus A'_n)[1]", "$$", "maps the summand $C$ into the summand $A[1]$ for all $n$ by a map", "$\\delta : C \\to A[1]$ which is independent of $n$. Choose a distinguished", "triangle", "$$", "A \\to B \\to C \\xrightarrow{\\delta} A[1]", "$$", "Next, choose a morphism of distingished triangles", "$$", "(A_1 \\to B_1 \\to C_1 \\to A_1[1]) \\to", "(A \\to B \\to C \\to A[1])", "$$", "which is possible by TR3. For any object $D$ of $\\mathcal{D}$ this induces", "a commutative diagram", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "\\Hom_\\mathcal{D}(C, D) \\ar[r] \\ar[d] &", "\\Hom_\\mathcal{D}(B, D) \\ar[r] \\ar[d] &", "\\Hom_\\mathcal{D}(A, D) \\ar[r] \\ar[d] &", "\\ldots \\\\", "\\ldots \\ar[r] &", "\\colim \\Hom_\\mathcal{D}(C_n, D) \\ar[r] &", "\\colim \\Hom_\\mathcal{D}(B_n, D) \\ar[r] &", "\\colim \\Hom_\\mathcal{D}(A_n, D) \\ar[r] &", "\\ldots", "}", "$$", "The left and right vertical arrows are isomorphisms and so are the ones", "to the left and right of those. Thus by the 5-lemma we conclude that", "the middle arrow is an isomorphism. It follows that", "$(B_n)$ is isomorphic to the constant inverse system with value $B$", "by the discussion in", "Categories, Remark \\ref{categories-remark-pro-category-copresheaves}.", "Since this is equivalent to $(B_n)$ being essentially constant", "with value $B$ by", "Categories, Remark \\ref{categories-remark-pro-category}", "the proof is complete." ], "refs": [ "derived-lemma-essentially-constant", "categories-remark-pro-category-copresheaves", "categories-remark-pro-category" ], "ref_ids": [ 1953, 12421, 12420 ] } ], "ref_ids": [] }, { "id": 1955, "type": "theorem", "label": "derived-lemma-essentially-constant-cohomology", "categories": [ "derived" ], "title": "derived-lemma-essentially-constant-cohomology", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $A_n$ be an inverse", "system of objects of $D(\\mathcal{A})$. Assume", "\\begin{enumerate}", "\\item there exist integers $a \\leq b$ such that $H^i(A_n) = 0$", "for $i \\not \\in [a, b]$, and", "\\item the inverse systems $H^i(A_n)$ of $\\mathcal{A}$ are essentially constant", "for all $i \\in \\mathbf{Z}$.", "\\end{enumerate}", "Then $A_n$ is an essentially constant system of $D(\\mathcal{A})$ whose", "value $A$ satisfies that $H^i(A)$ is the value of the constant system", "$H^i(A_n)$ for each $i \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "By Remark \\ref{remark-truncation-distinguished-triangle} we obtain", "an inverse system of distinguished triangles", "$$", "\\tau_{\\leq a}A_n \\to A_n \\to \\tau_{\\geq a + 1}A_n \\to (\\tau_{\\leq a}A_n)[1]", "$$", "Of course we have $\\tau_{\\leq a}A_n = H^a(A_n)[-a]$ in $D(\\mathcal{A})$.", "Thus by assumption these form an essentially constant system.", "By induction on $b - a$ we find that the inverse system", "$\\tau_{\\geq a + 1}A_n$ is essentially constant, say with value $A'$.", "By Lemma \\ref{lemma-essentially-constant-2-out-of-3} we find that", "$A_n$ is an essentially constant system. We omit the proof of", "the statement on cohomologies (hint: use the final part of", "Lemma \\ref{lemma-essentially-constant-2-out-of-3})." ], "refs": [ "derived-remark-truncation-distinguished-triangle", "derived-lemma-essentially-constant-2-out-of-3", "derived-lemma-essentially-constant-2-out-of-3" ], "ref_ids": [ 2016, 1954, 1954 ] } ], "ref_ids": [] }, { "id": 1956, "type": "theorem", "label": "derived-lemma-pro-isomorphism", "categories": [ "derived" ], "title": "derived-lemma-pro-isomorphism", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let", "$$", "A_n \\to B_n \\to C_n \\to A_n[1]", "$$", "be an inverse system of distinguished triangles. If the system $C_n$", "is pro-zero (essentially constant with value $0$), then the maps", "$A_n \\to B_n$ determine a pro-isomorphism between the pro-object $(A_n)$", "and the pro-object $(B_n)$." ], "refs": [], "proofs": [ { "contents": [ "For any object $X$ of $\\mathcal{D}$ consider the exact sequence", "$$", "\\colim \\Hom(C_n, X) \\to", "\\colim \\Hom(B_n, X) \\to", "\\colim \\Hom(A_n, X) \\to", "\\colim \\Hom(C_n[-1], X) \\to", "$$", "Exactness follows from Lemma \\ref{lemma-representable-homological}", "combined with", "Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}.", "By assumption the first and last term are zero. Hence the map", "$\\colim \\Hom(B_n, X) \\to \\colim \\Hom(A_n, X)$ is an isomorphism", "for all $X$. The lemma follows from this and", "Categories, Remark \\ref{categories-remark-pro-category-copresheaves}." ], "refs": [ "derived-lemma-representable-homological", "algebra-lemma-directed-colimit-exact", "categories-remark-pro-category-copresheaves" ], "ref_ids": [ 1758, 343, 12421 ] } ], "ref_ids": [] }, { "id": 1957, "type": "theorem", "label": "derived-lemma-pro-isomorphism-bis", "categories": [ "derived" ], "title": "derived-lemma-pro-isomorphism-bis", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "$$", "A_n \\to B_n", "$$", "be an inverse system of maps of $D(\\mathcal{A})$. Assume", "\\begin{enumerate}", "\\item there exist integers $a \\leq b$ such that $H^i(A_n) = 0$", "and $H^i(B_n) = 0$ for $i \\not \\in [a, b]$, and", "\\item the inverse system of maps $H^i(A_n) \\to H^i(B_n)$ of $\\mathcal{A}$", "define an isomorphism of pro-objects of $\\mathcal{A}$", "for all $i \\in \\mathbf{Z}$.", "\\end{enumerate}", "Then the maps $A_n \\to B_n$", "determine a pro-isomorphism between the pro-object $(A_n)$", "and the pro-object $(B_n)$." ], "refs": [], "proofs": [ { "contents": [ "We can inductively extend the maps $A_n \\to B_n$ to an inverse system of", "distinguished triangles $A_n \\to B_n \\to C_n \\to A_n[1]$ by", "axiom TR3. By Lemma \\ref{lemma-pro-isomorphism} it suffices to prove", "that $C_n$ is pro-zero. By Lemma \\ref{lemma-essentially-constant-cohomology}", "it suffices to show that $H^p(C_n)$ is pro-zero for each $p$.", "This follows from assumption (2) and the long exact sequences", "$$", "H^p(A_n) \\xrightarrow{\\alpha_n} H^p(B_n)", "\\xrightarrow{\\beta_n}", "H^p(C_n) \\xrightarrow{\\delta_n} H^{p + 1}(A_n)", "\\xrightarrow{\\epsilon_n}", "H^{p + 1}(B_n)", "$$", "Namely, for every $n$ we can find an $m > n$ such that", "$\\Im(\\beta_m)$ maps to zero in $H^p(C_n)$ because we may choose", "$m$ such that $H^p(B_m) \\to H^p(B_n)$ factors through", "$\\alpha_n : H^p(A_n) \\to H^p(B_n)$. For a similar reason we may", "then choose $k > m$ such that $\\Im(\\delta_k)$ maps to zero", "in $H^{p + 1}(A_m)$. Then $H^p(C_k) \\to H^p(C_n)$ is zero because", "$H^p(C_k) \\to H^p(C_m)$ maps into $\\Ker(\\delta_m)$ and $H^p(C_m) \\to H^p(C_n)$", "annihilates $\\Ker(\\delta_m) = \\Im(\\beta_m)$." ], "refs": [ "derived-lemma-pro-isomorphism", "derived-lemma-essentially-constant-cohomology" ], "ref_ids": [ 1956, 1955 ] } ], "ref_ids": [] }, { "id": 1958, "type": "theorem", "label": "derived-proposition-9", "categories": [ "derived" ], "title": "derived-proposition-9", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Any commutative diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d] & Y \\ar[d] \\\\", "X' \\ar[r] & Y'", "}", "$$", "can be extended to a diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d] & Y \\ar[r] \\ar[d] & Z \\ar[r] \\ar[d] & X[1] \\ar[d] \\\\", "X' \\ar[r] \\ar[d] & Y' \\ar[r] \\ar[d] & Z' \\ar[r] \\ar[d] & X'[1] \\ar[d] \\\\", "X'' \\ar[r] \\ar[d] & Y'' \\ar[r] \\ar[d] & Z'' \\ar[r] \\ar[d] & X''[1] \\ar[d] \\\\", "X[1] \\ar[r] & Y[1] \\ar[r] & Z[1] \\ar[r] & X[2]", "}", "$$", "where all the squares are commutative, except for the lower right square", "which is anticommutative. Moreover, each of the rows and columns are", "distinguished triangles. Finally, the morphisms on the bottom row", "(resp.\\ right column) are obtained from the morphisms of the top row", "(resp.\\ left column) by applying $[1]$." ], "refs": [], "proofs": [ { "contents": [ "During this proof we avoid writing the arrows in order to make the proof", "legible. Choose distinguished triangles", "$(X, Y, Z)$, $(X', Y', Z')$, $(X, X', X'')$, $(Y, Y', Y'')$, and", "$(X, Y', A)$. Note that the morphism $X \\to Y'$ is both equal", "to the composition $X \\to Y \\to Y'$ and equal to the composition", "$X \\to X' \\to Y'$. Hence, we can find morphisms", "\\begin{enumerate}", "\\item $a : Z \\to A$ and $b : A \\to Y''$, and", "\\item $a' : X'' \\to A$ and $b' : A \\to Z'$", "\\end{enumerate}", "as in TR4. Denote $c : Y'' \\to Z[1]$ the composition", "$Y'' \\to Y[1] \\to Z[1]$ and denote $c' : Z' \\to X''[1]$ the composition", "$Z' \\to X'[1] \\to X''[1]$. The conclusion of our application TR4", "are that", "\\begin{enumerate}", "\\item $(Z, A, Y'', a, b, c)$, $(X'', A, Z', a', b', c')$", "are distinguished triangles,", "\\item $(X, Y, Z) \\to (X, Y', A)$,", "$(X, Y', A) \\to (Y, Y', Y'')$,", "$(X, X', X'') \\to (X, Y', A)$,", "$(X, Y', A) \\to (X', Y', Z')$", "are morphisms of triangles.", "\\end{enumerate}", "First using that", "$(X, X', X'') \\to (X, Y', A)$ and $(X, Y', A) \\to (Y, Y', Y'')$.", "are morphisms of triangles we see the first of the diagrams", "$$", "\\vcenter{", "\\xymatrix{", "X' \\ar[r] \\ar[d] & Y' \\ar[d] \\\\", "X'' \\ar[r]^{b \\circ a'} \\ar[d] & Y'' \\ar[d] \\\\", "X[1] \\ar[r] & Y[1]", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "Y \\ar[r] \\ar[d] & Z \\ar[d]^{b' \\circ a} \\ar[r] & X[1] \\ar[d] \\\\", "Y' \\ar[r] & Z' \\ar[r] & X'[1]", "}", "}", "$$", "is commutative. The second is commutative too using that", "$(X, Y, Z) \\to (X, Y', A)$ and $(X, Y', A) \\to (X', Y', Z')$ are morphisms", "of triangles. At this point we choose a distinguished triangle", "$(X'', Y'' , Z'')$ starting with the map $b \\circ a' : X'' \\to Y''$.", "\\medskip\\noindent", "Next we apply TR4 one more time to the morphisms", "$X'' \\to A \\to Y''$ and the triangles", "$(X'', A, Z', a', b', c')$,", "$(X'', Y'', Z'')$, and", "$(A, Y'', Z[1], b, c , -a[1])$ to get morphisms", "$a'' : Z' \\to Z''$ and $b'' : Z'' \\to Z[1]$.", "Then $(Z', Z'', Z[1], a'', b'', - b'[1] \\circ a[1])$ is a distinguished", "triangle, hence also $(Z, Z', Z'', -b' \\circ a, a'', -b'')$", "and hence also $(Z, Z', Z'', b' \\circ a, a'', b'')$.", "Moreover, $(X'', A, Z') \\to (X'', Y'', Z'')$ and", "$(X'', Y'', Z'') \\to (A, Y'', Z[1], b, c , -a[1])$", "are morphisms of triangles.", "At this point we have defined all the distinguished triangles", "and all the morphisms, and all that's left is to verify some", "commutativity relations.", "\\medskip\\noindent", "To see that the middle square in the diagram commutes, note", "that the arrow $Y' \\to Z'$ factors as $Y' \\to A \\to Z'$", "because $(X, Y', A) \\to (X', Y', Z')$ is a morphism of triangles.", "Similarly, the morphism $Y' \\to Y''$ factors as", "$Y' \\to A \\to Y''$ because $(X, Y', A) \\to (Y, Y', Y'')$ is a", "morphism of triangles. Hence the middle square commutes because", "the square with sides $(A, Z', Z'', Y'')$ commutes as", "$(X'', A, Z') \\to (X'', Y'', Z'')$ is a morphism of triangles (by TR4).", "The square with sides $(Y'', Z'', Y[1], Z[1])$ commutes", "because $(X'', Y'', Z'') \\to (A, Y'', Z[1], b, c , -a[1])$", "is a morphism of triangles and $c : Y'' \\to Z[1]$ is the composition", "$Y'' \\to Y[1] \\to Z[1]$.", "The square with sides $(Z', X'[1], X''[1], Z'')$ is commutative", "because $(X'', A, Z') \\to (X'', Y'', Z'')$ is a morphism of triangles", "and $c' : Z' \\to X''[1]$ is the composition $Z' \\to X'[1] \\to X''[1]$.", "Finally, we have to show that the square with sides", "$(Z'', X''[1], Z[1], X[2])$ anticommutes. This holds because", "$(X'', Y'', Z'') \\to (A, Y'', Z[1], b, c , -a[1])$", "is a morphism of triangles and we're done." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1959, "type": "theorem", "label": "derived-proposition-construct-localization", "categories": [ "derived" ], "title": "derived-proposition-construct-localization", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category. Let $S$ be a multiplicative", "system compatible with the triangulated structure.", "Then there exists a unique structure of a pre-triangulated category on", "$S^{-1}\\mathcal{D}$ such that the localization functor", "$Q : \\mathcal{D} \\to S^{-1}\\mathcal{D}$ is exact.", "Moreover, if $\\mathcal{D}$ is a triangulated category, so is", "$S^{-1}\\mathcal{D}$." ], "refs": [], "proofs": [ { "contents": [ "We have seen that $S^{-1}\\mathcal{D}$ is an additive category", "and that the localization functor $Q$ is additive in", "Homology, Lemma \\ref{homology-lemma-localization-additive}.", "It is clear that we may define $Q(X)[n] = Q(X[n])$ since", "$\\mathcal{S}$ is preserved under the shift functors $[n]$ by", "MS5. Finally, we say a triangle of $S^{-1}\\mathcal{D}$ is distinguished", "if it is isomorphic to the image of a distinguished triangle under", "the localization functor $Q$.", "\\medskip\\noindent", "Proof of TR1. The only thing to prove here is that if", "$a : Q(X) \\to Q(Y)$ is a morphism of $S^{-1}\\mathcal{D}$, then", "$a$ fits into a distinguished triangle. Write $a = Q(s)^{-1} \\circ Q(f)$ for", "some $s : Y \\to Y'$ in $S$ and $f : X \\to Y'$. Choose a distinguished", "triangle $(X, Y', Z, f, g, h)$ in $\\mathcal{D}$. Then we see that", "$(Q(X), Q(Y), Q(Z), a, Q(g) \\circ Q(s), Q(h))$ is a distinguished triangle", "of $S^{-1}\\mathcal{D}$.", "\\medskip\\noindent", "Proof of TR2. This is immediate from the definitions.", "\\medskip\\noindent", "Proof of TR3. Note that the existence of the dotted arrow which is", "required to exist may be proven after replacing the two triangles", "by isomorphic triangles. Hence we may assume given distinguished", "triangles $(X, Y, Z, f, g, h)$ and $(X', Y', Z', f', g', h')$ of", "$\\mathcal{D}$ and a commutative diagram", "$$", "\\xymatrix{", "Q(X) \\ar[r]_{Q(f)} \\ar[d]_a & Q(Y) \\ar[d]^b \\\\", "Q(X') \\ar[r]^{Q(f')} & Q(Y')", "}", "$$", "in $S^{-1}\\mathcal{D}$. Now we apply", "Categories, Lemma \\ref{categories-lemma-left-localization-diagram}", "to find a morphism $f'' : X'' \\to Y''$ in $\\mathcal{D}$ and a commutative", "diagram", "$$", "\\xymatrix{", "X \\ar[d]_f \\ar[r]_k & X'' \\ar[d]^{f''} & X' \\ar[d]^{f'} \\ar[l]^s \\\\", "Y \\ar[r]^l & Y'' & Y' \\ar[l]_t", "}", "$$", "in $\\mathcal{D}$ with $s, t \\in S$ and $a = s^{-1}k$, $b = t^{-1}l$.", "At this point we can use TR3 for $\\mathcal{D}$ and MS6 to find", "a commutative diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d]^k &", "Y \\ar[r] \\ar[d]^l &", "Z \\ar[r] \\ar[d]^m &", "X[1] \\ar[d]^{g[1]} \\\\", "X'' \\ar[r] &", "Y'' \\ar[r] &", "Z'' \\ar[r] &", "X''[1] \\\\", "X' \\ar[r] \\ar[u]_s &", "Y' \\ar[r] \\ar[u]_t &", "Z' \\ar[r] \\ar[u]_r &", "X'[1] \\ar[u]_{s[1]}", "}", "$$", "with $r \\in S$. It follows that setting $c = Q(r)^{-1}Q(m)$ we obtain", "the desired morphism of triangles", "$$", "\\xymatrix{", "(Q(X), Q(Y), Q(Z), Q(f), Q(g), Q(h))", "\\ar[d]^{(a, b, c)} \\\\", "(Q(X'), Q(Y'), Q(Z'), Q(f'), Q(g'), Q(h'))", "}", "$$", "\\medskip\\noindent", "This proves the first statement of the lemma. If $\\mathcal{D}$ is also", "a triangulated category, then we still have to prove TR4 in order to show", "that $S^{-1}\\mathcal{D}$ is triangulated as well. To do this we reduce by", "Lemma \\ref{lemma-easier-axiom-four}", "to the following statement: Given composable morphisms", "$a : Q(X) \\to Q(Y)$ and $b : Q(Y) \\to Q(Z)$ we have to produce", "an octahedron after possibly replacing $Q(X), Q(Y), Q(Z)$ by isomorphic", "objects. To do this we may first replace $Y$ by an object such that", "$a = Q(f)$ for some morphism $f : X \\to Y$ in $\\mathcal{D}$. (More precisely,", "write $a = s^{-1}f$ with $s : Y \\to Y'$ in $S$ and $f : X \\to Y'$. Then", "replace $Y$ by $Y'$.) After this we similarly replace $Z$ by an object such", "that $b = Q(g)$ for some morphism $g : Y \\to Z$. Now we can find", "distinguished triangles $(X, Y, Q_1, f, p_1, d_1)$,", "$(X, Z, Q_2, g \\circ f, p_2, d_2)$, and", "$(Y, Z, Q_3, g, p_3, d_3)$ in $\\mathcal{D}$ (by TR1), and", "morphisms $a : Q_1 \\to Q_2$ and $b : Q_2 \\to Q_3$ as in TR4.", "Then it is immediately verified that applying the functor $Q$ to", "all these data gives a corresponding structure in $S^{-1}\\mathcal{D}$" ], "refs": [ "homology-lemma-localization-additive", "categories-lemma-left-localization-diagram", "derived-lemma-easier-axiom-four" ], "ref_ids": [ 12038, 12260, 1770 ] } ], "ref_ids": [] }, { "id": 1960, "type": "theorem", "label": "derived-proposition-homotopy-category-triangulated", "categories": [ "derived" ], "title": "derived-proposition-homotopy-category-triangulated", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "The category $K(\\mathcal{A})$ of complexes up to", "homotopy with its natural translation functors", "and distinguished triangles as defined above", "is a triangulated category." ], "refs": [], "proofs": [ { "contents": [ "Proof of TR1. By definition every triangle isomorphic to a distinguished", "one is distinguished. Also, any triangle $(A^\\bullet, A^\\bullet, 0, 1, 0, 0)$", "is distinguished since $0 \\to A^\\bullet \\to A^\\bullet \\to 0 \\to 0$ is", "a termwise split sequence of complexes. Finally, given any morphism of", "complexes $f : K^\\bullet \\to L^\\bullet$ the triangle", "$(K, L, C(f), f, i, -p)$ is distinguished by", "Lemma \\ref{lemma-the-same-up-to-isomorphisms}.", "\\medskip\\noindent", "Proof of TR2. Let $(X, Y, Z, f, g, h)$ be a triangle.", "Assume $(Y, Z, X[1], g, h, -f[1])$ is distinguished.", "Then there exists a termwise split sequence of complexes", "$A^\\bullet \\to B^\\bullet \\to C^\\bullet$ such that the associated", "triangle $(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta)$", "is isomorphic to $(Y, Z, X[1], g, h, -f[1])$. Rotating back we see", "that $(X, Y, Z, f, g, h)$ is isomorphic to", "$(C^\\bullet[-1], A^\\bullet, B^\\bullet, -\\delta[-1], \\alpha, \\beta)$.", "It follows from Lemma \\ref{lemma-rotate-triangle} that the triangle", "$(C^\\bullet[-1], A^\\bullet, B^\\bullet, \\delta[-1], \\alpha, \\beta)$", "is isomorphic to", "$(C^\\bullet[-1], A^\\bullet, C(\\delta[-1])^\\bullet, \\delta[-1], i, p)$.", "Precomposing the previous isomorphism of triangles with $-1$ on $Y$", "it follows that $(X, Y, Z, f, g, h)$ is isomorphic to", "$(C^\\bullet[-1], A^\\bullet, C(\\delta[-1])^\\bullet, \\delta[-1], i, -p)$.", "Hence it is distinguished by", "Lemma \\ref{lemma-the-same-up-to-isomorphisms}.", "On the other hand, suppose that $(X, Y, Z, f, g, h)$ is distinguished.", "By Lemma \\ref{lemma-the-same-up-to-isomorphisms} this means that it is", "isomorphic to a triangle of the form", "$(K^\\bullet, L^\\bullet, C(f), f, i, -p)$ for some morphism of", "complexes $f$. Then the rotated triangle $(Y, Z, X[1], g, h, -f[1])$ is", "isomorphic to $(L^\\bullet, C(f), K^\\bullet[1], i, -p, -f[1])$ which is", "isomorphic to the triangle $(L^\\bullet, C(f), K^\\bullet[1], i, p, f[1])$.", "By Lemma \\ref{lemma-rotate-cone} this triangle is distinguished.", "Hence $(Y, Z, X[1], g, h, -f[1])$ is distinguished as desired.", "\\medskip\\noindent", "Proof of TR3. Let", "$(X, Y, Z, f, g, h)$ and $(X', Y', Z', f', g', h')$", "be distinguished triangles of $K(\\mathcal{A})$", "and let $a : X \\to X'$ and $b : Y \\to Y'$ be morphisms", "such that $f' \\circ a = b \\circ f$. By", "Lemma \\ref{lemma-the-same-up-to-isomorphisms} we may assume that", "$(X, Y, Z, f, g, h) = (X, Y, C(f), f, i, -p)$ and", "$(X', Y', Z', f', g', h') = (X', Y', C(f'), f', i', -p')$.", "At this point we simply apply Lemma \\ref{lemma-functorial-cone}", "to the commutative diagram given by $f, f', a, b$.", "\\medskip\\noindent", "Proof of TR4. At this point we know that $K(\\mathcal{A})$", "is a pre-triangulated category. Hence we can use", "Lemma \\ref{lemma-easier-axiom-four}. Let $A^\\bullet \\to B^\\bullet$", "and $B^\\bullet \\to C^\\bullet$ be composable morphisms of", "$K(\\mathcal{A})$. By Lemma \\ref{lemma-sequence-maps-split} we may assume that", "$A^\\bullet \\to B^\\bullet$ and $B^\\bullet \\to C^\\bullet$", "are split injective morphisms. In this case the result follows", "from Lemma \\ref{lemma-two-split-injections}." ], "refs": [ "derived-lemma-the-same-up-to-isomorphisms", "derived-lemma-rotate-triangle", "derived-lemma-the-same-up-to-isomorphisms", "derived-lemma-the-same-up-to-isomorphisms", "derived-lemma-rotate-cone", "derived-lemma-the-same-up-to-isomorphisms", "derived-lemma-functorial-cone", "derived-lemma-easier-axiom-four", "derived-lemma-sequence-maps-split", "derived-lemma-two-split-injections" ], "ref_ids": [ 1802, 1804, 1802, 1802, 1805, 1802, 1794, 1770, 1803, 1806 ] } ], "ref_ids": [] }, { "id": 1961, "type": "theorem", "label": "derived-proposition-derived-functor", "categories": [ "derived" ], "title": "derived-proposition-derived-functor", "contents": [ "Assumptions and notation as in Situation \\ref{situation-derived-functor}.", "\\begin{enumerate}", "\\item The full subcategory $\\mathcal{E}$ of $\\mathcal{D}$ consisting of", "objects at which $RF$ is defined is a strictly full triangulated", "subcategory of $\\mathcal{D}$.", "\\item We obtain an exact functor", "$RF : \\mathcal{E} \\longrightarrow \\mathcal{D}'$", "of triangulated categories.", "\\item Elements of $S$ with either source or target", "in $\\mathcal{E}$ are morphisms of $\\mathcal{E}$.", "\\item The functor $S_\\mathcal{E}^{-1}\\mathcal{E} \\to S^{-1}\\mathcal{D}$", "is a fully faithful exact functor of triangulated categories.", "\\item Any element of $S_\\mathcal{E} = \\text{Arrows}(\\mathcal{E}) \\cap S$", "is mapped to an isomorphism by $RF$.", "\\item We obtain an exact functor", "$$", "RF : S_\\mathcal{E}^{-1}\\mathcal{E} \\longrightarrow \\mathcal{D}'.", "$$", "\\item If $\\mathcal{D}'$ is Karoubian, then $\\mathcal{E}$ is a saturated", "triangulated subcategory of $\\mathcal{D}$.", "\\end{enumerate}", "A similar result holds for $LF$." ], "refs": [], "proofs": [ { "contents": [ "Since $S$ is saturated it contains all isomorphisms (see remark", "following Categories, Definition", "\\ref{categories-definition-saturated-multiplicative-system}). Hence", "(1) follows from Lemmas \\ref{lemma-derived-inverts},", "\\ref{lemma-2-out-of-3-defined}, and", "\\ref{lemma-derived-shift}. We get (2) from", "Lemmas \\ref{lemma-derived-functor}, \\ref{lemma-derived-shift}, and", "\\ref{lemma-2-out-of-3-defined}. We get (3) from", "Lemma \\ref{lemma-derived-inverts}. The fully faithfulness in (4) follows", "from (3) and the definitions. The fact that", "$S_\\mathcal{E}^{-1}\\mathcal{E} \\to S^{-1}\\mathcal{D}$ is exact", "follows from the fact that a triangle in $S_\\mathcal{E}^{-1}\\mathcal{E}$", "is distinguished if and only if it is isomorphic to the image of a", "distinguished triangle in $\\mathcal{E}$, see proof of", "Proposition \\ref{proposition-construct-localization}.", "Part (5) follows from Lemma \\ref{lemma-derived-inverts}.", "The factorization of $RF : \\mathcal{E} \\to \\mathcal{D}'$", "through an exact functor $S_\\mathcal{E}^{-1}\\mathcal{E} \\to \\mathcal{D}'$", "follows from Lemma \\ref{lemma-universal-property-localization}.", "Part (7) follows from Lemma \\ref{lemma-direct-sum-defined}." ], "refs": [ "categories-definition-saturated-multiplicative-system", "derived-lemma-derived-inverts", "derived-lemma-2-out-of-3-defined", "derived-lemma-derived-shift", "derived-lemma-derived-functor", "derived-lemma-derived-shift", "derived-lemma-2-out-of-3-defined", "derived-lemma-derived-inverts", "derived-proposition-construct-localization", "derived-lemma-derived-inverts", "derived-lemma-universal-property-localization", "derived-lemma-direct-sum-defined" ], "ref_ids": [ 12376, 1824, 1826, 1825, 1823, 1825, 1826, 1824, 1959, 1824, 1781, 1827 ] } ], "ref_ids": [] }, { "id": 1962, "type": "theorem", "label": "derived-proposition-enough-acyclics", "categories": [ "derived" ], "title": "derived-proposition-enough-acyclics", "contents": [ "\\begin{slogan}", "A functor on an Abelian categories is extended to the (bounded below or above)", "derived category by resolving with a complex that is acyclic for that functor.", "\\end{slogan}", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor of", "abelian categories.", "\\begin{enumerate}", "\\item If every object of $\\mathcal{A}$ injects into an object acyclic", "for $RF$, then $RF$ is defined on all of $K^{+}(\\mathcal{A})$", "and we obtain an exact functor", "$$", "RF : D^{+}(\\mathcal{A}) \\longrightarrow D^{+}(\\mathcal{B})", "$$", "see (\\ref{equation-everywhere}). Moreover, any bounded below complex", "$A^\\bullet$ whose terms are acyclic for $RF$ computes $RF$.", "\\item If every object of $\\mathcal{A}$ is quotient of", "an object acyclic for $LF$, then $LF$ is defined on all of", "$K^{-}(\\mathcal{A})$ and we obtain an exact functor", "$$", "LF : D^{-}(\\mathcal{A}) \\longrightarrow D^{-}(\\mathcal{B})", "$$", "see (\\ref{equation-everywhere}). Moreover, any bounded above complex", "$A^\\bullet$ whose terms are acyclic for $LF$ computes $LF$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume every object of $\\mathcal{A}$ injects into an object acyclic", "for $RF$. Let $\\mathcal{I}$ be the set of objects acyclic for $RF$.", "Let $K^\\bullet$ be a bounded below complex in $\\mathcal{A}$. By", "Lemma \\ref{lemma-subcategory-right-resolution}", "there exists a quasi-isomorphism $\\alpha : K^\\bullet \\to I^\\bullet$ with", "$I^\\bullet$ bounded below and $I^n \\in \\mathcal{I}$. Hence in order to", "prove (1) it suffices to show that", "$F(I^\\bullet) \\to F((I')^\\bullet)$ is a quasi-isomorphism when", "$s : I^\\bullet \\to (I')^\\bullet$ is a quasi-isomorphism of bounded", "below complexes of objects from $\\mathcal{I}$, see", "Lemma \\ref{lemma-find-existence-computes}.", "Note that the cone $C(s)^\\bullet$ is an acyclic bounded below complex", "all of whose terms are in $\\mathcal{I}$.", "Hence it suffices to show: given an acyclic bounded below complex", "$I^\\bullet$ all of whose terms are in $\\mathcal{I}$ the complex", "$F(I^\\bullet)$ is acyclic.", "\\medskip\\noindent", "Say $I^n = 0$ for $n < n_0$. Setting $J^n = \\Im(d^n)$ we break", "$I^\\bullet$ into short exact sequences", "$0 \\to J^n \\to I^{n + 1} \\to J^{n + 1} \\to 0$", "for $n \\geq n_0$. These sequences induce distinguished triangles", "$(J^n, I^{n + 1}, J^{n + 1})$ in $D^+(\\mathcal{A})$ by", "Lemma \\ref{lemma-derived-canonical-delta-functor}.", "For each $k \\in \\mathbf{Z}$ denote $H_k$ the assertion:", "For all $n \\leq k$ the right derived functor", "$RF$ is defined at $J^n$ and $R^iF(J^n) = 0$ for $i \\not = 0$.", "Then $H_k$ holds trivially for $k \\leq n_0$. If $H_n$ holds,", "then, using Proposition \\ref{proposition-derived-functor},", "we see that $RF$ is defined at $J^{n + 1}$ and", "$(RF(J^n), RF(I^{n + 1}), RF(J^{n + 1}))$ is a distinguished", "triangle of $D^+(\\mathcal{B})$. Thus the long exact cohomology sequence", "(\\ref{equation-long-exact-cohomology-sequence-D})", "associated to this triangle gives an exact sequence", "$$", "0 \\to R^{-1}F(J^{n + 1}) \\to R^0F(J^n) \\to", "F(I^{n + 1}) \\to R^0F(J^{n + 1}) \\to 0", "$$", "and gives that $R^iF(J^{n + 1}) = 0$ for $i \\not \\in \\{-1, 0\\}$.", "By Lemma \\ref{lemma-negative-vanishing} we see that $R^{-1}F(J^{n + 1}) = 0$.", "This proves that $H_{n + 1}$ is true hence $H_k$ holds for all $k$.", "We also conclude that", "$$", "0 \\to R^0F(J^n) \\to F(I^{n + 1}) \\to R^0F(J^{n + 1}) \\to 0", "$$", "is short exact for all $n$. This in turn proves that $F(I^\\bullet)$ is exact.", "\\medskip\\noindent", "The proof in the case of $LF$ is dual." ], "refs": [ "derived-lemma-subcategory-right-resolution", "derived-lemma-find-existence-computes", "derived-lemma-derived-canonical-delta-functor", "derived-proposition-derived-functor", "derived-lemma-negative-vanishing" ], "ref_ids": [ 1836, 1832, 1814, 1961, 1839 ] } ], "ref_ids": [] }, { "id": 1963, "type": "theorem", "label": "derived-proposition-derived-category", "categories": [ "derived" ], "title": "derived-proposition-derived-category", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Assume $\\mathcal{A}$ has enough injectives.", "Denote $\\mathcal{I} \\subset \\mathcal{A}$ the strictly full", "additive subcategory whose objects are the injective objects of", "$\\mathcal{A}$.", "The functor", "$$", "K^{+}(\\mathcal{I}) \\longrightarrow D^{+}(\\mathcal{A})", "$$", "is exact, fully faithful and essentially surjective, i.e.,", "an equivalence of triangulated categories." ], "refs": [], "proofs": [ { "contents": [ "It is clear that the functor is exact.", "It is essentially surjective by", "Lemma \\ref{lemma-injective-resolutions-exist}.", "Fully faithfulness is a consequence of", "Lemma \\ref{lemma-morphisms-into-injective-complex}." ], "refs": [ "derived-lemma-injective-resolutions-exist", "derived-lemma-morphisms-into-injective-complex" ], "ref_ids": [ 1851, 1855 ] } ], "ref_ids": [] }, { "id": 1964, "type": "theorem", "label": "derived-proposition-left-derived-exists", "categories": [ "derived" ], "title": "derived-proposition-left-derived-exists", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a right exact functor", "of abelian categories. Let $\\mathcal{P} \\subset \\Ob(\\mathcal{A})$ be a", "subset. Assume", "\\begin{enumerate}", "\\item $\\mathcal{P}$ contains $0$, is closed under (finite) direct sums,", "and every object of $\\mathcal{A}$ is a quotient of an", "element of $\\mathcal{P}$,", "\\item for any bounded above acyclic complex $P^\\bullet$ of", "$\\mathcal{A}$ with $P^n \\in \\mathcal{P}$ for all $n$ the", "complex $F(P^\\bullet)$ is exact,", "\\item $\\mathcal{A}$ and $\\mathcal{B}$ have colimits", "of systems over $\\mathbf{N}$,", "\\item colimits over $\\mathbf{N}$ are exact in both", "$\\mathcal{A}$ and $\\mathcal{B}$, and", "\\item $F$ commutes with colimits over $\\mathbf{N}$.", "\\end{enumerate}", "Then $LF$ is defined on all of $D(\\mathcal{A})$." ], "refs": [], "proofs": [ { "contents": [ "By (1) and Lemma \\ref{lemma-subcategory-left-resolution} for any bounded", "above complex $K^\\bullet$ there exists a quasi-isomorphism", "$P^\\bullet \\to K^\\bullet$ with $P^\\bullet$ bounded above and", "$P^n \\in \\mathcal{P}$ for all $n$. Suppose that", "$s : P^\\bullet \\to (P')^\\bullet$ is a quasi-isomorphism of bounded", "above complexes consisting of objects of $\\mathcal{P}$. Then", "$F(P^\\bullet) \\to F((P')^\\bullet)$ is a quasi-isomorphism because", "$F(C(s)^\\bullet)$ is acyclic by assumption (2). This already shows that", "$LF$ is defined on $D^{-}(\\mathcal{A})$ and that a bounded above", "complex consisting of objects of $\\mathcal{P}$ computes $LF$, see", "Lemma \\ref{lemma-find-existence-computes}.", "\\medskip\\noindent", "Next, let $K^\\bullet$ be an arbitrary complex of $\\mathcal{A}$.", "Choose a diagram", "$$", "\\xymatrix{", "P_1^\\bullet \\ar[d] \\ar[r] & P_2^\\bullet \\ar[d] \\ar[r] & \\ldots \\\\", "\\tau_{\\leq 1}K^\\bullet \\ar[r] & \\tau_{\\leq 2}K^\\bullet \\ar[r] & \\ldots", "}", "$$", "as in Lemma \\ref{lemma-special-direct-system}. Note that", "the map $\\colim P_n^\\bullet \\to K^\\bullet$ is a quasi-isomorphism", "because colimits over $\\mathbf{N}$ in $\\mathcal{A}$ are exact", "and $H^i(P_n^\\bullet) = H^i(K^\\bullet)$ for $n > i$. We claim that", "$$", "F(\\colim P_n^\\bullet) = \\colim F(P_n^\\bullet)", "$$", "(termwise colimits) is $LF(K^\\bullet)$, i.e., that $\\colim P_n^\\bullet$", "computes $LF$. To see this, by Lemma \\ref{lemma-find-existence-computes},", "it suffices to prove the following claim. Suppose that", "$$", "\\colim Q_n^\\bullet = Q^\\bullet", "\\xrightarrow{\\ \\alpha\\ }", "P^\\bullet = \\colim P_n^\\bullet", "$$", "is a quasi-isomorphism of complexes, such that each", "$P_n^\\bullet$, $Q_n^\\bullet$ is a bounded above complex whose terms are", "in $\\mathcal{P}$ and the maps $P_n^\\bullet \\to \\tau_{\\leq n}P^\\bullet$ and", "$Q_n^\\bullet \\to \\tau_{\\leq n}Q^\\bullet$ are quasi-isomorphisms.", "Claim: $F(\\alpha)$ is a quasi-isomorphism.", "\\medskip\\noindent", "The problem is that we do not assume that $\\alpha$ is given as a colimit", "of maps between the complexes $P_n^\\bullet$ and $Q_n^\\bullet$. However,", "for each $n$ we know that the solid arrows in the diagram", "$$", "\\xymatrix{", "& R^\\bullet \\ar@{..>}[d] \\\\", "P_n^\\bullet \\ar[d] &", "L^\\bullet \\ar@{..>}[l] \\ar@{..>}[r] &", "Q_n^\\bullet \\ar[d] \\\\", "\\tau_{\\leq n}P^\\bullet \\ar[rr]^{\\tau_{\\leq n}\\alpha} & &", "\\tau_{\\leq n}Q^\\bullet", "}", "$$", "are quasi-isomorphisms. Because quasi-isomorphisms form a multiplicative", "system in $K(\\mathcal{A})$ (see Lemma \\ref{lemma-acyclic})", "we can find a quasi-isomorphism", "$L^\\bullet \\to P_n^\\bullet$ and map of complexes $L^\\bullet \\to Q_n^\\bullet$", "such that the diagram above commutes up to homotopy. Then", "$\\tau_{\\leq n}L^\\bullet \\to L^\\bullet$ is a quasi-isomorphism.", "Hence (by the first part of the proof) we can find a bounded above", "complex $R^\\bullet$ whose terms are in $\\mathcal{P}$ and a quasi-isomorphism", "$R^\\bullet \\to L^\\bullet$ (as indicated in the diagram). Using the result", "of the first paragraph of the proof we see that", "$F(R^\\bullet) \\to F(P_n^\\bullet)$ and $F(R^\\bullet) \\to F(Q_n^\\bullet)$", "are quasi-isomorphisms. Thus we obtain a isomorphisms", "$H^i(F(P_n^\\bullet)) \\to H^i(F(Q_n^\\bullet))$ fitting into the commutative", "diagram", "$$", "\\xymatrix{", "H^i(F(P_n^\\bullet)) \\ar[r] \\ar[d] &", "H^i(F(Q_n^\\bullet)) \\ar[d] \\\\", "H^i(F(P^\\bullet)) \\ar[r] &", "H^i(F(Q^\\bullet))", "}", "$$", "The exact same argument shows that these maps are also compatible", "as $n$ varies. Since by (4) and (5) we have", "$$", "H^i(F(P^\\bullet)) =", "H^i(F(\\colim P_n^\\bullet)) =", "H^i(\\colim F(P_n^\\bullet)) = \\colim H^i(F(P_n^\\bullet))", "$$", "and similarly for $Q^\\bullet$ we conclude that", "$H^i(\\alpha) : H^i(F(P^\\bullet) \\to H^i(F(Q^\\bullet)$ is an isomorphism", "and the claim follows." ], "refs": [ "derived-lemma-subcategory-left-resolution", "derived-lemma-find-existence-computes", "derived-lemma-special-direct-system", "derived-lemma-find-existence-computes", "derived-lemma-acyclic" ], "ref_ids": [ 1835, 1832, 1903, 1832, 1811 ] } ], "ref_ids": [] }, { "id": 1965, "type": "theorem", "label": "derived-proposition-generator-versus-classical-generator", "categories": [ "derived" ], "title": "derived-proposition-generator-versus-classical-generator", "contents": [ "Let $\\mathcal{D}$ be a triangulated category with direct sums.", "Let $E$ be a compact object of $\\mathcal{D}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $E$ is a classical generator for $\\mathcal{D}_c$ and", "$\\mathcal{D}$ is compactly generated, and", "\\item $E$ is a generator for $\\mathcal{D}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $E$ is a classical generator for $\\mathcal{D}_c$, then", "$\\mathcal{D}_c = \\langle E \\rangle$. It follows formally", "from the assumption that $\\mathcal{D}$ is compactly generated", "and Lemma \\ref{lemma-right-orthogonal} that $E$ is a generator", "for $\\mathcal{D}$.", "\\medskip\\noindent", "The converse is more interesting. Assume that $E$ is a generator", "for $\\mathcal{D}$. Let $X$ be a compact object of $\\mathcal{D}$.", "Apply Lemma \\ref{lemma-write-as-colimit} with $I = \\{1\\}$ and", "$E_1 = E$ to write", "$$", "X = \\text{hocolim} X_n", "$$", "as in the lemma. Since $X$ is compact we", "find that $X \\to \\text{hocolim} X_n$ factors through $X_n$ for", "some $n$ (Lemma \\ref{lemma-commutes-with-countable-sums}).", "Thus $X$ is a direct summand of $X_n$.", "By Lemma \\ref{lemma-factor-through} we see that $X$ is an", "object of $\\langle E \\rangle$ and the lemma is proven." ], "refs": [ "derived-lemma-right-orthogonal", "derived-lemma-write-as-colimit", "derived-lemma-commutes-with-countable-sums", "derived-lemma-factor-through" ], "ref_ids": [ 1937, 1941, 1924, 1942 ] } ], "ref_ids": [] }, { "id": 1966, "type": "theorem", "label": "derived-proposition-brown", "categories": [ "derived" ], "title": "derived-proposition-brown", "contents": [ "\\begin{reference}", "\\cite[Theorem 4.1]{Neeman-Grothendieck}.", "\\end{reference}", "Let $\\mathcal{D}$ be a triangulated category with direct sums which is", "compactly generated. Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an", "exact functor of triangulated categories which transforms direct sums", "into direct sums. Then $F$ has an exact right adjoint." ], "refs": [], "proofs": [ { "contents": [ "For an object $Y$ of $\\mathcal{D}'$ consider the contravariant functor", "$$", "\\mathcal{D} \\to \\textit{Ab},\\quad W \\mapsto \\Hom_{\\mathcal{D}'}(F(W), Y)", "$$", "This is a cohomological functor as $F$ is exact and transforms direct sums", "into products as $F$ transforms direct sums into direct sums. Thus by", "Lemma \\ref{lemma-brown} we find an object $X$ of $\\mathcal{D}$ such that", "$\\Hom_\\mathcal{D}(W, X) = \\Hom_{\\mathcal{D}'}(F(W), Y)$.", "The existence of the adjoint follows from", "Categories, Lemma \\ref{categories-lemma-adjoint-exists}.", "Exactness follows from Lemma \\ref{lemma-adjoint-is-exact}." ], "refs": [ "derived-lemma-brown", "categories-lemma-adjoint-exists", "derived-lemma-adjoint-is-exact" ], "ref_ids": [ 1943, 12246, 1792 ] } ], "ref_ids": [] }, { "id": 2031, "type": "theorem", "label": "cohomology-theorem-proper-base-change", "categories": [ "cohomology" ], "title": "cohomology-theorem-proper-base-change", "contents": [ "\\begin{reference}", "\\cite[Expose V bis, 4.1.1]{SGA4}", "\\end{reference}", "Consider a cartesian square of topological spaces", "$$", "\\xymatrix{", "X' = Y' \\times_Y X \\ar[d]_{f'} \\ar[r]_-{g'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "Assume that $f$ is proper and separated.", "Let $E$ be an object of $D^+(X)$. Then the base change map", "$$", "g^{-1}Rf_*E \\longrightarrow Rf'_*(g')^{-1}E", "$$", "of Lemma \\ref{lemma-base-change-map-flat-case} is an isomorphism", "in $D^+(Y')$." ], "refs": [ "cohomology-lemma-base-change-map-flat-case" ], "proofs": [ { "contents": [ "Let $y' \\in Y'$ be a point with image $y \\in Y$. It suffices to show that", "the base change map induces an isomorphism on stalks at $y'$.", "As $f$ is proper it follows that $f'$ is proper, the", "fibres of $f$ and $f'$ are quasi-compact and $f$ and $f'$ are closed, see", "Topology, Theorem \\ref{topology-theorem-characterize-proper}.", "Moreover $f'$ is separated by", "Topology, Lemma \\ref{topology-lemma-base-change-separated}.", "Thus we can apply Lemma \\ref{lemma-proper-base-change} twice to see that", "$$", "(Rf'_*(g')^{-1}E)_{y'} = R\\Gamma((f')^{-1}(y'), (g')^{-1}E|_{(f')^{-1}(y')})", "$$", "and", "$$", "(Rf_*E)_y = R\\Gamma(f^{-1}(y), E|_{f^{-1}(y)})", "$$", "The induced map of fibres $(f')^{-1}(y') \\to f^{-1}(y)$ is", "a homeomorphism of topological spaces and the pull back of", "$E|_{f^{-1}(y)}$ is $(g')^{-1}E|_{(f')^{-1}(y')}$. The", "desired result follows." ], "refs": [ "topology-theorem-characterize-proper", "topology-lemma-base-change-separated", "cohomology-lemma-proper-base-change" ], "ref_ids": [ 8189, 8196, 2080 ] } ], "ref_ids": [ 2079 ] }, { "id": 2032, "type": "theorem", "label": "cohomology-theorem-glueing-bbd-general", "categories": [ "cohomology" ], "title": "cohomology-theorem-glueing-bbd-general", "contents": [ "\\begin{reference}", "Special case of \\cite[Theorem 3.2.4]{BBD}", "without boundedness assumption.", "\\end{reference}", "In Situation \\ref{situation-locally-given} assume", "\\begin{enumerate}", "\\item $X = \\bigcup_{U \\in \\mathcal{B}} U$,", "\\item for $U, V \\in \\mathcal{B}$ we have", "$U \\cap V = \\bigcup_{W \\in \\mathcal{B}, W \\subset U \\cap V} W$,", "\\item for any $U \\in \\mathcal{B}$ we have $\\Ext^i(K_U, K_U) = 0$", "for $i < 0$.", "\\end{enumerate}", "Then there exists an object $K$ of $D(\\mathcal{O}_X)$", "and isomorphisms $\\rho_U : K|_U \\to K_U$ in $D(\\mathcal{O}_U)$ for", "$U \\in \\mathcal{B}$ such that $\\rho^U_V \\circ \\rho_U|_V = \\rho_V$", "for all $V \\subset U$ with $U, V \\in \\mathcal{B}$.", "The pair $(K, \\rho_U)$ is unique up to unique isomorphism." ], "refs": [], "proofs": [ { "contents": [ "A pair $(K, \\rho_U)$ is called a solution in the text above.", "The uniqueness follows from Lemma \\ref{lemma-uniqueness}.", "If $X$ has a finite covering by elements of $\\mathcal{B}$", "(for example if $X$ is quasi-compact), then the theorem", "is a consequence of Lemma \\ref{lemma-solution-in-finite-case}.", "In the general case we argue in exactly the same manner,", "using transfinite induction and", "Lemma \\ref{lemma-glueing-increasing-union}.", "\\medskip\\noindent", "First we use transfinite induction to choose opens $W_\\alpha \\subset X$", "for any ordinal $\\alpha$. Namely, we set $W_0 = \\emptyset$.", "If $\\alpha = \\beta + 1$ is a successor, then either $W_\\beta = X$", "and we set $W_\\alpha = X$ or $W_\\beta \\not = X$ and we set", "$W_\\alpha = W_\\beta \\cup U_\\alpha$ where", "$U_\\alpha \\in \\mathcal{B}$ is not contained in $W_\\beta$.", "If $\\alpha$ is a limit ordinal we set", "$W_\\alpha = \\bigcup_{\\beta < \\alpha} W_\\beta$.", "Then for large enough $\\alpha$ we have $W_\\alpha = X$.", "Observe that for every $\\alpha$ the open $W_\\alpha$ is", "a union of elements of $\\mathcal{B}$. Hence if", "$\\mathcal{B}_\\alpha = \\{U \\in \\mathcal{B}, U \\subset W_\\alpha\\}$, then", "$$", "S_\\alpha = (\\{K_U\\}_{U \\in \\mathcal{B}_\\alpha},", "\\{\\rho_V^U\\}_{V \\subset U\\text{ with }U, V \\in \\mathcal{B}_\\alpha})", "$$", "is a system as in Lemma \\ref{lemma-uniqueness} on the ringed space $W_\\alpha$.", "\\medskip\\noindent", "We will show by transfinite induction that for every $\\alpha$", "the system $S_\\alpha$ has a solution. This will prove the theorem", "as this system is the system given in the theorem for large $\\alpha$.", "\\medskip\\noindent", "The case where $\\alpha = \\beta + 1$ is a successor ordinal.", "(This case was already treated in the proof of the lemma above", "but for clarity we repeat the argument.)", "Recall that $W_\\alpha = W_\\beta \\cup U_\\alpha$ for some", "$U_\\alpha \\in \\mathcal{B}$ in this case.", "By induction hypothesis we have a solution", "$(K_{W_\\beta}, \\{\\rho^{W_\\beta}_U\\}_{U \\in \\mathcal{B}_\\beta})$", "for the system $S_\\beta$.", "Then we can consider the collection", "$\\mathcal{B}_\\alpha^* = \\mathcal{B}_\\alpha \\cup \\{W_\\beta\\}$", "of opens of $W_\\alpha$ and we see that we obtain a system", "$(\\{K_U\\}_{U \\in \\mathcal{B}_\\alpha^*},", "\\{\\rho_V^U\\}_{V \\subset U\\text{ with }U, V \\in \\mathcal{B}_\\alpha^*})$.", "Note that this new system also satisfies condition (3)", "by Lemma \\ref{lemma-uniqueness} applied to the solution $K_{W_\\beta}$.", "For this system we have $W_\\alpha = W_\\beta \\cup U_\\alpha$.", "This reduces us to the case handled in", "Lemma \\ref{lemma-solution-in-finite-case}.", "\\medskip\\noindent", "The case where $\\alpha$ is a limit ordinal. Recall that", "$W_\\alpha = \\bigcup_{\\beta < \\alpha} W_\\beta$ in this case.", "For $\\beta < \\alpha$ let", "$(K_{W_\\beta}, \\{\\rho^{W_\\beta}_U\\}_{U \\in \\mathcal{B}_\\beta})$", "be the solution for $S_\\beta$.", "For $\\gamma < \\beta < \\alpha$ the restriction", "$K_{W_\\beta}|_{W_\\gamma}$ endowed with the maps", "$\\rho^{W_\\beta}_U$, $U \\in \\mathcal{B}_\\gamma$", "is a solution for $S_\\gamma$. By uniqueness we get unique isomorphisms", "$\\rho_{W_\\gamma}^{W_\\beta} : K_{W_\\beta}|_{W_\\gamma} \\to K_{W_\\gamma}$", "compatible with the maps $\\rho^{W_\\beta}_U$ and $\\rho^{W_\\gamma}_U$", "for $U \\in \\mathcal{B}_\\gamma$. These maps compose in the correct manner,", "i.e., $\\rho_{W_\\delta}^{W_\\gamma} \\circ \\rho_{W_\\gamma}^{W_\\beta}|_{W_\\delta}", "= \\rho^{W_\\delta}_{W_\\beta}$ for $\\delta < \\gamma < \\beta < \\alpha$.", "Thus we may apply Lemma \\ref{lemma-glueing-increasing-union}", "(note that the vanishing of negative exts is true for", "$K_{W_\\beta}$ by Lemma \\ref{lemma-uniqueness} applied", "to the solution $K_{W_\\beta}$)", "to obtain $K_{W_\\alpha}$ and isomorphisms", "$$", "\\rho_{W_\\beta}^{W_\\alpha} :", "K_{W_\\alpha}|_{W_\\beta}", "\\longrightarrow", "K_{W_\\beta}", "$$", "compatible with the maps $\\rho_{W_\\gamma}^{W_\\beta}$ for", "$\\gamma < \\beta < \\alpha$.", "\\medskip\\noindent", "To show that $K_{W_\\alpha}$ is a solution we still need to construct the", "isomorphisms $\\rho_U^{W_\\alpha} : K_{W_\\alpha}|_U \\to K_U$ for", "$U \\in \\mathcal{B}_\\alpha$ satisfying certain compatibilities.", "We choose $\\rho_U^{W_\\alpha}$ to be the unique map such that", "for any $\\beta < \\alpha$ and any $V \\in \\mathcal{B}_\\beta$", "with $V \\subset U$ the diagram", "$$", "\\xymatrix{", "K_{W_\\alpha}|_V \\ar[r]_{\\rho_U^{W_\\alpha}|_V}", "\\ar[d]_{\\rho_{W_\\beta}^{W_\\alpha}|_V}", "& K_U|_V \\ar[d]^{\\rho_U^V} \\\\", "K_{W_\\beta} \\ar[r]^{\\rho_V^{W_\\beta}}", "& K_V", "}", "$$", "commutes. This makes sense because", "$$", "(\\{K_V\\}_{V \\subset U, V \\in \\mathcal{B}_\\beta\\text{ for some }\\beta < \\alpha},", "\\{\\rho_V^{V'}\\}_{V \\subset V'\\text{ with }V, V' \\subset U", "\\text{ and }V, V' \\in \\mathcal{B}_\\beta\\text{ for some }\\beta < \\alpha})", "$$", "is a system as in Lemma \\ref{lemma-uniqueness} on the ringed space $U$", "and because $(K_U, \\rho^U_V)$ and", "$(K_{W_\\alpha}|_U, \\rho_V^{W_\\beta}\\circ \\rho_{W_\\beta}^{W_\\alpha}|_V)$", "are both solutions for this system. This gives existence and uniqueness.", "We omit the proof that these", "maps satisfy the desired compatibilities (it is just bookkeeping)." ], "refs": [ "cohomology-lemma-uniqueness", "cohomology-lemma-solution-in-finite-case", "cohomology-lemma-glueing-increasing-union", "cohomology-lemma-uniqueness", "cohomology-lemma-uniqueness", "cohomology-lemma-solution-in-finite-case", "cohomology-lemma-glueing-increasing-union", "cohomology-lemma-uniqueness", "cohomology-lemma-uniqueness" ], "ref_ids": [ 2193, 2194, 2195, 2193, 2193, 2194, 2195, 2193, 2193 ] } ], "ref_ids": [] }, { "id": 2033, "type": "theorem", "label": "cohomology-lemma-trivial-torsor", "categories": [ "cohomology" ], "title": "cohomology-lemma-trivial-torsor", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $X$.", "A $\\mathcal{G}$-torsor $\\mathcal{F}$ is trivial if and only if", "$\\mathcal{F}(X) \\not = \\emptyset$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2034, "type": "theorem", "label": "cohomology-lemma-torsors-h1", "categories": [ "cohomology" ], "title": "cohomology-lemma-torsors-h1", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{H}$ be an abelian sheaf on $X$.", "There is a canonical bijection between the set of isomorphism", "classes of $\\mathcal{H}$-torsors and $H^1(X, \\mathcal{H})$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a $\\mathcal{H}$-torsor.", "Consider the free abelian sheaf $\\mathbf{Z}[\\mathcal{F}]$", "on $\\mathcal{F}$. It is the sheafification of the rule", "which associates to $U \\subset X$ open the collection of finite", "formal sums $\\sum n_i[s_i]$ with $n_i \\in \\mathbf{Z}$", "and $s_i \\in \\mathcal{F}(U)$. There is a natural map", "$$", "\\sigma : \\mathbf{Z}[\\mathcal{F}] \\longrightarrow \\underline{\\mathbf{Z}}", "$$", "which to a local section $\\sum n_i[s_i]$ associates $\\sum n_i$.", "The kernel of $\\sigma$ is generated by the local section of the form", "$[s] - [s']$. There is a canonical map", "$a : \\Ker(\\sigma) \\to \\mathcal{H}$", "which maps $[s] - [s'] \\mapsto h$ where $h$ is the local section of", "$\\mathcal{H}$ such that $h \\cdot s = s'$. Consider the pushout diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Ker(\\sigma) \\ar[r] \\ar[d]^a &", "\\mathbf{Z}[\\mathcal{F}] \\ar[r] \\ar[d] &", "\\underline{\\mathbf{Z}} \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{H} \\ar[r] &", "\\mathcal{E} \\ar[r] &", "\\underline{\\mathbf{Z}} \\ar[r] &", "0", "}", "$$", "Here $\\mathcal{E}$ is the extension obtained by pushout.", "From the long exact cohomology sequence associated to the lower", "short exact sequence we obtain an element", "$\\xi = \\xi_\\mathcal{F} \\in H^1(X, \\mathcal{H})$", "by applying the boundary operator to $1 \\in H^0(X, \\underline{\\mathbf{Z}})$.", "\\medskip\\noindent", "Conversely, given $\\xi \\in H^1(X, \\mathcal{H})$ we can associate to", "$\\xi$ a torsor as follows. Choose an embedding $\\mathcal{H} \\to \\mathcal{I}$", "of $\\mathcal{H}$ into an injective abelian sheaf $\\mathcal{I}$. We set", "$\\mathcal{Q} = \\mathcal{I}/\\mathcal{H}$ so that we have a short exact", "sequence", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{H} \\ar[r] &", "\\mathcal{I} \\ar[r] &", "\\mathcal{Q} \\ar[r] &", "0", "}", "$$", "The element $\\xi$ is the image of a global section $q \\in H^0(X, \\mathcal{Q})$", "because $H^1(X, \\mathcal{I}) = 0$ (see", "Derived Categories, Lemma \\ref{derived-lemma-higher-derived-functors}).", "Let $\\mathcal{F} \\subset \\mathcal{I}$ be the subsheaf (of sets) of sections", "that map to $q$ in the sheaf $\\mathcal{Q}$. It is easy to verify that", "$\\mathcal{F}$ is a torsor.", "\\medskip\\noindent", "We omit the verification that the two constructions given", "above are mutually inverse." ], "refs": [ "derived-lemma-higher-derived-functors" ], "ref_ids": [ 1869 ] } ], "ref_ids": [] }, { "id": 2035, "type": "theorem", "label": "cohomology-lemma-h1-extensions", "categories": [ "cohomology" ], "title": "cohomology-lemma-h1-extensions", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of", "$\\mathcal{O}_X$-modules. There is a canonical bijection", "$$", "\\Ext^1_{\\textit{Mod}(\\mathcal{O}_X)}(\\mathcal{O}_X, \\mathcal{F})", "\\longrightarrow", "H^1(X, \\mathcal{F})", "$$", "which associates to the extension", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{E} \\to \\mathcal{O}_X \\to 0", "$$", "the image of $1 \\in \\Gamma(X, \\mathcal{O}_X)$ in $H^1(X, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Let us construct the inverse of the map given in the lemma. Let", "$\\xi \\in H^1(X, \\mathcal{F})$. Choose an injection", "$\\mathcal{F} \\subset \\mathcal{I}$ with $\\mathcal{I}$ injective in", "$\\textit{Mod}(\\mathcal{O}_X)$.", "Set $\\mathcal{Q} = \\mathcal{I}/\\mathcal{F}$.", "By the long exact sequence of cohomology, we see that", "$\\xi$ is the image of a section", "$\\tilde \\xi \\in \\Gamma(X, \\mathcal{Q}) =", "\\Hom_{\\mathcal{O}_X}(\\mathcal{O}_X, \\mathcal{Q})$.", "Now, we just form the pullback", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{F} \\ar[r] \\ar@{=}[d] &", "\\mathcal{E} \\ar[r] \\ar[d] &", "\\mathcal{O}_X \\ar[r] \\ar[d]^{\\tilde \\xi} &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{F} \\ar[r] &", "\\mathcal{I} \\ar[r] &", "\\mathcal{Q} \\ar[r] &", "0", "}", "$$", "see Homology, Section \\ref{homology-section-extensions}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2036, "type": "theorem", "label": "cohomology-lemma-h1-invertible", "categories": [ "cohomology" ], "title": "cohomology-lemma-h1-invertible", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a locally ringed space.", "There is a canonical isomorphism", "$$", "H^1(X, \\mathcal{O}_X^*) = \\Pic(X).", "$$", "of abelian groups." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Consider the presheaf $\\mathcal{L}^*$ defined by the rule", "$$", "U \\longmapsto \\{s \\in \\mathcal{L}(U)", "\\text{ such that } \\mathcal{O}_U \\xrightarrow{s \\cdot -} \\mathcal{L}_U", "\\text{ is an isomorphism}\\}", "$$", "This presheaf satisfies the sheaf condition. Moreover, if", "$f \\in \\mathcal{O}_X^*(U)$ and $s \\in \\mathcal{L}^*(U)$, then clearly", "$fs \\in \\mathcal{L}^*(U)$. By the same token, if $s, s' \\in \\mathcal{L}^*(U)$", "then there exists a unique $f \\in \\mathcal{O}_X^*(U)$ such that", "$fs = s'$. Moreover, the sheaf $\\mathcal{L}^*$ has sections locally", "by Modules, Lemma \\ref{modules-lemma-invertible-is-locally-free-rank-1}.", "In other words we", "see that $\\mathcal{L}^*$ is a $\\mathcal{O}_X^*$-torsor. Thus we get", "a map", "$$", "\\begin{matrix}", "\\text{invertible sheaves on }(X, \\mathcal{O}_X) \\\\", "\\text{ up to isomorphism}", "\\end{matrix}", "\\longrightarrow", "\\begin{matrix}", "\\mathcal{O}_X^*\\text{-torsors} \\\\", "\\text{ up to isomorphism}", "\\end{matrix}", "$$", "We omit the verification that this is a homomorphism of abelian groups.", "By", "Lemma \\ref{lemma-torsors-h1}", "the right hand side is canonically", "bijective to $H^1(X, \\mathcal{O}_X^*)$.", "Thus we have to show this map is injective and surjective.", "\\medskip\\noindent", "Injective. If the torsor $\\mathcal{L}^*$ is trivial, this means by", "Lemma \\ref{lemma-trivial-torsor}", "that $\\mathcal{L}^*$ has a global section.", "Hence this means exactly that $\\mathcal{L} \\cong \\mathcal{O}_X$ is", "the neutral element in $\\Pic(X)$.", "\\medskip\\noindent", "Surjective. Let $\\mathcal{F}$ be an $\\mathcal{O}_X^*$-torsor.", "Consider the presheaf of sets", "$$", "\\mathcal{L}_1 : U \\longmapsto", "(\\mathcal{F}(U) \\times \\mathcal{O}_X(U))/\\mathcal{O}_X^*(U)", "$$", "where the action of $f \\in \\mathcal{O}_X^*(U)$ on", "$(s, g)$ is $(fs, f^{-1}g)$. Then $\\mathcal{L}_1$ is a presheaf", "of $\\mathcal{O}_X$-modules by setting", "$(s, g) + (s', g') = (s, g + (s'/s)g')$ where $s'/s$ is the local", "section $f$ of $\\mathcal{O}_X^*$ such that $fs = s'$, and", "$h(s, g) = (s, hg)$ for $h$ a local section of $\\mathcal{O}_X$.", "We omit the verification that the sheafification", "$\\mathcal{L} = \\mathcal{L}_1^\\#$ is an invertible $\\mathcal{O}_X$-module", "whose associated $\\mathcal{O}_X^*$-torsor $\\mathcal{L}^*$ is isomorphic", "to $\\mathcal{F}$." ], "refs": [ "modules-lemma-invertible-is-locally-free-rank-1", "cohomology-lemma-torsors-h1", "cohomology-lemma-trivial-torsor" ], "ref_ids": [ 13302, 2034, 2033 ] } ], "ref_ids": [] }, { "id": 2037, "type": "theorem", "label": "cohomology-lemma-cohomology-of-open", "categories": [ "cohomology" ], "title": "cohomology-lemma-cohomology-of-open", "contents": [ "Let $X$ be a ringed space.", "Let $U \\subset X$ be an open subspace.", "\\begin{enumerate}", "\\item If $\\mathcal{I}$ is an injective $\\mathcal{O}_X$-module", "then $\\mathcal{I}|_U$ is an injective $\\mathcal{O}_U$-module.", "\\item For any sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ we have", "$H^p(U, \\mathcal{F}) = H^p(U, \\mathcal{F}|_U)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $j : U \\to X$ the open immersion.", "Recall that the functor $j^{-1}$ of restriction to $U$ is a right adjoint", "to the functor $j_!$ of extension by $0$, see", "Sheaves, Lemma \\ref{sheaves-lemma-j-shriek-modules}.", "Moreover, $j_!$ is exact. Hence (1) follows from", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}.", "\\medskip\\noindent", "By definition $H^p(U, \\mathcal{F}) = H^p(\\Gamma(U, \\mathcal{I}^\\bullet))$", "where $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ is an injective resolution", "in $\\textit{Mod}(\\mathcal{O}_X)$.", "By the above we see that $\\mathcal{F}|_U \\to \\mathcal{I}^\\bullet|_U$", "is an injective resolution in $\\textit{Mod}(\\mathcal{O}_U)$.", "Hence $H^p(U, \\mathcal{F}|_U)$ is equal to", "$H^p(\\Gamma(U, \\mathcal{I}^\\bullet|_U))$.", "Of course $\\Gamma(U, \\mathcal{F}) = \\Gamma(U, \\mathcal{F}|_U)$ for", "any sheaf $\\mathcal{F}$ on $X$.", "Hence the equality", "in (2)." ], "refs": [ "sheaves-lemma-j-shriek-modules", "homology-lemma-adjoint-preserve-injectives" ], "ref_ids": [ 14546, 12116 ] } ], "ref_ids": [] }, { "id": 2038, "type": "theorem", "label": "cohomology-lemma-kill-cohomology-class-on-covering", "categories": [ "cohomology" ], "title": "cohomology-lemma-kill-cohomology-class-on-covering", "contents": [ "Let $X$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "Let $U \\subset X$ be an open subspace.", "Let $n > 0$ and let $\\xi \\in H^n(U, \\mathcal{F})$.", "Then there exists an open covering", "$U = \\bigcup_{i\\in I} U_i$ such that $\\xi|_{U_i} = 0$ for", "all $i \\in I$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ be an injective resolution.", "Then", "$$", "H^n(U, \\mathcal{F}) =", "\\frac{\\Ker(\\mathcal{I}^n(U) \\to \\mathcal{I}^{n + 1}(U))}", "{\\Im(\\mathcal{I}^{n - 1}(U) \\to \\mathcal{I}^n(U))}.", "$$", "Pick an element $\\tilde \\xi \\in \\mathcal{I}^n(U)$ representing the", "cohomology class in the presentation above. Since $\\mathcal{I}^\\bullet$", "is an injective resolution of $\\mathcal{F}$ and $n > 0$ we see that", "the complex $\\mathcal{I}^\\bullet$ is exact in degree $n$. Hence", "$\\Im(\\mathcal{I}^{n - 1} \\to \\mathcal{I}^n) =", "\\Ker(\\mathcal{I}^n \\to \\mathcal{I}^{n + 1})$ as sheaves.", "Since $\\tilde \\xi$ is a section of the kernel sheaf over $U$", "we conclude there exists an open covering $U = \\bigcup_{i \\in I} U_i$", "such that $\\tilde \\xi|_{U_i}$ is the image under $d$ of a section", "$\\xi_i \\in \\mathcal{I}^{n - 1}(U_i)$. By our definition of the", "restriction $\\xi|_{U_i}$ as corresponding to the class of", "$\\tilde \\xi|_{U_i}$ we conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2039, "type": "theorem", "label": "cohomology-lemma-describe-higher-direct-images", "categories": [ "cohomology" ], "title": "cohomology-lemma-describe-higher-direct-images", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Let $\\mathcal{F}$ be a $\\mathcal{O}_X$-module.", "The sheaves $R^if_*\\mathcal{F}$ are the sheaves", "associated to the presheaves", "$$", "V \\longmapsto H^i(f^{-1}(V), \\mathcal{F})", "$$", "with restriction mappings as in Equation (\\ref{equation-restriction-mapping}).", "There is a similar statement for $R^if_*$ applied to a", "bounded below complex $\\mathcal{F}^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ be an injective resolution.", "Then $R^if_*\\mathcal{F}$ is by definition the $i$th cohomology sheaf", "of the complex", "$$", "f_*\\mathcal{I}^0 \\to f_*\\mathcal{I}^1 \\to f_*\\mathcal{I}^2 \\to \\ldots", "$$", "By definition of the abelian category structure on $\\mathcal{O}_Y$-modules", "this cohomology sheaf is the sheaf associated to the presheaf", "$$", "V", "\\longmapsto", "\\frac{\\Ker(f_*\\mathcal{I}^i(V) \\to f_*\\mathcal{I}^{i + 1}(V))}", "{\\Im(f_*\\mathcal{I}^{i - 1}(V) \\to f_*\\mathcal{I}^i(V))}", "$$", "and this is obviously equal to", "$$", "\\frac{\\Ker(\\mathcal{I}^i(f^{-1}(V)) \\to \\mathcal{I}^{i + 1}(f^{-1}(V)))}", "{\\Im(\\mathcal{I}^{i - 1}(f^{-1}(V)) \\to \\mathcal{I}^i(f^{-1}(V)))}", "$$", "which is equal to $H^i(f^{-1}(V), \\mathcal{F})$", "and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2040, "type": "theorem", "label": "cohomology-lemma-localize-higher-direct-images", "categories": [ "cohomology" ], "title": "cohomology-lemma-localize-higher-direct-images", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "Let $V \\subset Y$ be an open subspace.", "Denote $g : f^{-1}(V) \\to V$ the restriction of $f$.", "Then we have", "$$", "R^pg_*(\\mathcal{F}|_{f^{-1}(V)}) = (R^pf_*\\mathcal{F})|_V", "$$", "There is a similar statement for the", "derived image $Rf_*\\mathcal{F}^\\bullet$ where $\\mathcal{F}^\\bullet$", "is a bounded below complex of $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "First proof. Apply Lemmas \\ref{lemma-describe-higher-direct-images}", "and \\ref{lemma-cohomology-of-open} to see the displayed equality.", "Second proof. Choose an injective resolution", "$\\mathcal{F} \\to \\mathcal{I}^\\bullet$", "and use that $\\mathcal{F}|_{f^{-1}(V)} \\to \\mathcal{I}^\\bullet|_{f^{-1}(V)}$", "is an injective resolution also." ], "refs": [ "cohomology-lemma-describe-higher-direct-images", "cohomology-lemma-cohomology-of-open" ], "ref_ids": [ 2039, 2037 ] } ], "ref_ids": [] }, { "id": 2041, "type": "theorem", "label": "cohomology-lemma-injective-restriction-surjective", "categories": [ "cohomology" ], "title": "cohomology-lemma-injective-restriction-surjective", "contents": [ "\\begin{slogan}", "Local sections in injective sheaves can be extended globally.", "\\end{slogan}", "Let $X$ be a ringed space.", "Let $U' \\subset U \\subset X$ be open subspaces.", "For any injective $\\mathcal{O}_X$-module $\\mathcal{I}$ the", "restriction mapping", "$\\mathcal{I}(U) \\to \\mathcal{I}(U')$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $j : U \\to X$ and $j' : U' \\to X$ be the open immersions.", "Recall that $j_!\\mathcal{O}_U$ is the extension by zero of", "$\\mathcal{O}_U = \\mathcal{O}_X|_U$, see", "Sheaves, Section \\ref{sheaves-section-open-immersions}.", "Since $j_!$ is a left adjoint to restriction we see that", "for any sheaf $\\mathcal{F}$ of $\\mathcal{O}_X$-modules", "$$", "\\Hom_{\\mathcal{O}_X}(j_!\\mathcal{O}_U, \\mathcal{F})", "=", "\\Hom_{\\mathcal{O}_U}(\\mathcal{O}_U, \\mathcal{F}|_U)", "=", "\\mathcal{F}(U)", "$$", "see Sheaves, Lemma \\ref{sheaves-lemma-j-shriek-modules}.", "Similarly, the sheaf $j'_!\\mathcal{O}_{U'}$ represents the", "functor $\\mathcal{F} \\mapsto \\mathcal{F}(U')$.", "Moreover there", "is an obvious canonical map of $\\mathcal{O}_X$-modules", "$$", "j'_!\\mathcal{O}_{U'} \\longrightarrow j_!\\mathcal{O}_U", "$$", "which corresponds to the restriction mapping", "$\\mathcal{F}(U) \\to \\mathcal{F}(U')$ via Yoneda's lemma", "(Categories, Lemma \\ref{categories-lemma-yoneda}). By the description", "of the stalks of the sheaves", "$j'_!\\mathcal{O}_{U'}$, $j_!\\mathcal{O}_U$", "we see that the displayed map above is injective (see lemma cited above).", "Hence if $\\mathcal{I}$ is an injective $\\mathcal{O}_X$-module,", "then the map", "$$", "\\Hom_{\\mathcal{O}_X}(j_!\\mathcal{O}_U, \\mathcal{I})", "\\longrightarrow", "\\Hom_{\\mathcal{O}_X}(j'_!\\mathcal{O}_{U'}, \\mathcal{I})", "$$", "is surjective, see", "Homology, Lemma \\ref{homology-lemma-characterize-injectives}.", "Putting everything together we obtain the lemma." ], "refs": [ "sheaves-lemma-j-shriek-modules", "categories-lemma-yoneda", "homology-lemma-characterize-injectives" ], "ref_ids": [ 14546, 12203, 12112 ] } ], "ref_ids": [] }, { "id": 2042, "type": "theorem", "label": "cohomology-lemma-mayer-vietoris", "categories": [ "cohomology" ], "title": "cohomology-lemma-mayer-vietoris", "contents": [ "Let $X$ be a ringed space. Suppose that $X = U \\cup V$ is a", "union of two open subsets. For every $\\mathcal{O}_X$-module $\\mathcal{F}$", "there exists a long exact cohomology sequence", "$$", "0 \\to", "H^0(X, \\mathcal{F}) \\to", "H^0(U, \\mathcal{F}) \\oplus H^0(V, \\mathcal{F}) \\to", "H^0(U \\cap V, \\mathcal{F}) \\to", "H^1(X, \\mathcal{F}) \\to \\ldots", "$$", "This long exact sequence is functorial in $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "The sheaf condition says that the kernel of", "$(1, -1) : \\mathcal{F}(U) \\oplus \\mathcal{F}(V) \\to \\mathcal{F}(U \\cap V)$", "is equal to the image of $\\mathcal{F}(X)$ by the first map", "for any abelian sheaf $\\mathcal{F}$.", "Lemma \\ref{lemma-injective-restriction-surjective} above implies that the map", "$(1, -1) : \\mathcal{I}(U) \\oplus \\mathcal{I}(V) \\to \\mathcal{I}(U \\cap V)$", "is surjective whenever $\\mathcal{I}$ is an injective $\\mathcal{O}_X$-module.", "Hence if $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ is an injective resolution", "of $\\mathcal{F}$, then we get a short exact sequence of complexes", "$$", "0 \\to", "\\mathcal{I}^\\bullet(X) \\to", "\\mathcal{I}^\\bullet(U) \\oplus \\mathcal{I}^\\bullet(V) \\to", "\\mathcal{I}^\\bullet(U \\cap V) \\to", "0.", "$$", "Taking cohomology gives the result (use", "Homology, Lemma \\ref{homology-lemma-long-exact-sequence-cochain}).", "We omit the proof of the functoriality of the sequence." ], "refs": [ "cohomology-lemma-injective-restriction-surjective", "homology-lemma-long-exact-sequence-cochain" ], "ref_ids": [ 2041, 12061 ] } ], "ref_ids": [] }, { "id": 2043, "type": "theorem", "label": "cohomology-lemma-relative-mayer-vietoris", "categories": [ "cohomology" ], "title": "cohomology-lemma-relative-mayer-vietoris", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Suppose that $X = U \\cup V$ is a union of two open subsets.", "Denote $a = f|_U : U \\to Y$, $b = f|_V : V \\to Y$, and", "$c = f|_{U \\cap V} : U \\cap V \\to Y$.", "For every $\\mathcal{O}_X$-module $\\mathcal{F}$", "there exists a long exact sequence", "$$", "0 \\to", "f_*\\mathcal{F} \\to", "a_*(\\mathcal{F}|_U) \\oplus b_*(\\mathcal{F}|_V) \\to", "c_*(\\mathcal{F}|_{U \\cap V}) \\to", "R^1f_*\\mathcal{F} \\to \\ldots", "$$", "This long exact sequence is functorial in $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ be an injective resolution", "of $\\mathcal{F}$. We claim that we", "get a short exact sequence of complexes", "$$", "0 \\to", "f_*\\mathcal{I}^\\bullet \\to", "a_*\\mathcal{I}^\\bullet|_U \\oplus b_*\\mathcal{I}^\\bullet|_V \\to", "c_*\\mathcal{I}^\\bullet|_{U \\cap V} \\to", "0.", "$$", "Namely, for any open $W \\subset Y$, and for any $n \\geq 0$ the", "corresponding sequence of groups of sections over $W$", "$$", "0 \\to", "\\mathcal{I}^n(f^{-1}(W)) \\to", "\\mathcal{I}^n(U \\cap f^{-1}(W))", "\\oplus \\mathcal{I}^n(V \\cap f^{-1}(W)) \\to", "\\mathcal{I}^n(U \\cap V \\cap f^{-1}(W)) \\to", "0", "$$", "was shown to be short exact in the proof of Lemma \\ref{lemma-mayer-vietoris}.", "The lemma follows by taking cohomology sheaves and using the fact that", "$\\mathcal{I}^\\bullet|_U$ is an injective resolution of $\\mathcal{F}|_U$", "and similarly for $\\mathcal{I}^\\bullet|_V$, $\\mathcal{I}^\\bullet|_{U \\cap V}$", "see Lemma \\ref{lemma-cohomology-of-open}." ], "refs": [ "cohomology-lemma-mayer-vietoris", "cohomology-lemma-cohomology-of-open" ], "ref_ids": [ 2042, 2037 ] } ], "ref_ids": [] }, { "id": 2044, "type": "theorem", "label": "cohomology-lemma-cech-h0", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-h0", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{F}$ be an abelian presheaf on $X$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is an abelian sheaf and", "\\item for every open covering $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$", "the natural map", "$$", "\\mathcal{F}(U) \\to \\check{H}^0(\\mathcal{U}, \\mathcal{F})", "$$", "is bijective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is true since the sheaf condition is exactly that", "$\\mathcal{F}(U) \\to \\check{H}^0(\\mathcal{U}, \\mathcal{F})$", "is bijective for every open covering." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2045, "type": "theorem", "label": "cohomology-lemma-cech-trivial", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-trivial", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{F}$ be an abelian presheaf on $X$.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering. If", "$U_i = U$ for some $i \\in I$, then the extended {\\v C}ech complex", "$$", "\\mathcal{F}(U) \\to \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "$$", "obtained by putting $\\mathcal{F}(U)$ in degree $-1$ with differential given by", "the canonical map of $\\mathcal{F}(U)$ into", "$\\check{\\mathcal{C}}^0(\\mathcal{U}, \\mathcal{F})$", "is homotopy equivalent to $0$." ], "refs": [], "proofs": [ { "contents": [ "Fix an element $i \\in I$ with $U = U_i$. Observe that", "$U_{i_0 \\ldots i_p} = U_{i_0 \\ldots \\hat i_j \\ldots i_p}$ if $i_j = i$.", "Let us define a homotopy", "$$", "h :", "\\prod\\nolimits_{i_0 \\ldots i_{p + 1}} \\mathcal{F}(U_{i_0 \\ldots i_{p + 1}})", "\\longrightarrow", "\\prod\\nolimits_{i_0 \\ldots i_p} \\mathcal{F}(U_{i_0 \\ldots i_p})", "$$", "by the rule", "$$", "h(s)_{i_0 \\ldots i_p} = s_{i i_0 \\ldots i_p}", "$$", "In other words, $h : \\prod_{i_0} \\mathcal{F}(U_{i_0}) \\to \\mathcal{F}(U)$", "is projection onto the factor $\\mathcal{F}(U_i) = \\mathcal{F}(U)$", "and in general the map $h$ equals the projection onto the factors", "$\\mathcal{F}(U_{i i_1 \\ldots i_{p + 1}}) =", "\\mathcal{F}(U_{i_1 \\ldots i_{p + 1}})$.", "We compute", "\\begin{align*}", "(dh + hd)(s)_{i_0 \\ldots i_p}", "& =", "\\sum\\nolimits_{j = 0}^p", "(-1)^j", "h(s)_{i_0 \\ldots \\hat i_j \\ldots i_p}", "+", "d(s)_{i i_0 \\ldots i_p}\\\\", "& =", "\\sum\\nolimits_{j = 0}^p", "(-1)^j", "s_{i i_0 \\ldots \\hat i_j \\ldots i_p}", "+", "s_{i_0 \\ldots i_p}", "+", "\\sum\\nolimits_{j = 0}^p", "(-1)^{j + 1}", "s_{i i_0 \\ldots \\hat i_j \\ldots i_p} \\\\", "& =", "s_{i_0 \\ldots i_p}", "\\end{align*}", "This proves the identity map is homotopic to zero as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2046, "type": "theorem", "label": "cohomology-lemma-cech-exact-presheaves", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-exact-presheaves", "contents": [ "The functor given by Equation (\\ref{equation-cech-functor})", "is an exact functor (see Homology, Lemma \\ref{homology-lemma-exact-functor})." ], "refs": [ "homology-lemma-exact-functor" ], "proofs": [ { "contents": [ "For any open $W \\subset U$ the functor", "$\\mathcal{F} \\mapsto \\mathcal{F}(W)$ is an additive exact functor", "from $\\textit{PMod}(\\mathcal{O}_X)$ to $\\text{Mod}_{\\mathcal{O}_X(U)}$.", "The terms", "$\\check{\\mathcal{C}}^p(\\mathcal{U}, \\mathcal{F})$", "of the complex are products of these exact functors and hence exact.", "Moreover a sequence of complexes is exact if and only if the sequence", "of terms in a given degree is exact. Hence the lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12034 ] }, { "id": 2047, "type": "theorem", "label": "cohomology-lemma-cech-cohomology-delta-functor-presheaves", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-cohomology-delta-functor-presheaves", "contents": [ "Let $X$ be a ringed space.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.", "The functors $\\mathcal{F} \\mapsto \\check{H}^n(\\mathcal{U}, \\mathcal{F})$", "form a $\\delta$-functor from the abelian category of", "presheaves of $\\mathcal{O}_X$-modules to the category", "of $\\mathcal{O}_X(U)$-modules (see", "Homology, Definition \\ref{homology-definition-cohomological-delta-functor})." ], "refs": [ "homology-definition-cohomological-delta-functor" ], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-cech-exact-presheaves}", "a short exact sequence of presheaves of", "$\\mathcal{O}_X$-modules", "$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "is turned into a short exact sequence of complexes of", "$\\mathcal{O}_X(U)$-modules. Hence we can use", "Homology, Lemma \\ref{homology-lemma-long-exact-sequence-cochain}", "to get the boundary maps", "$\\delta_{\\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3} :", "\\check{H}^n(\\mathcal{U}, \\mathcal{F}_3) \\to", "\\check{H}^{n + 1}(\\mathcal{U}, \\mathcal{F}_1)$", "and a corresponding long exact sequence. We omit the verification", "that these maps are compatible with maps between short exact", "sequences of presheaves." ], "refs": [ "cohomology-lemma-cech-exact-presheaves", "homology-lemma-long-exact-sequence-cochain" ], "ref_ids": [ 2046, 12061 ] } ], "ref_ids": [ 12149 ] }, { "id": 2048, "type": "theorem", "label": "cohomology-lemma-cech-map-into", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-map-into", "contents": [ "Let $X$ be a ringed space.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be a covering.", "Denote $j_{i_0\\ldots i_p} : U_{i_0 \\ldots i_p} \\to X$ the open immersion.", "Consider the chain complex $K(\\mathcal{U})_\\bullet$", "of presheaves of $\\mathcal{O}_X$-modules", "$$", "\\ldots", "\\to", "\\bigoplus_{i_0i_1i_2} (j_{i_0i_1i_2})_{p!}\\mathcal{O}_{U_{i_0i_1i_2}}", "\\to", "\\bigoplus_{i_0i_1} (j_{i_0i_1})_{p!}\\mathcal{O}_{U_{i_0i_1}}", "\\to", "\\bigoplus_{i_0} (j_{i_0})_{p!}\\mathcal{O}_{U_{i_0}}", "\\to 0 \\to \\ldots", "$$", "where the last nonzero term is placed in degree $0$", "and where the map", "$$", "(j_{i_0\\ldots i_{p + 1}})_{p!}\\mathcal{O}_{U_{i_0\\ldots i_{p + 1}}}", "\\longrightarrow", "(j_{i_0\\ldots \\hat i_j \\ldots i_{p + 1}})_{p!}", "\\mathcal{O}_{U_{i_0\\ldots \\hat i_j \\ldots i_{p + 1}}}", "$$", "is given by $(-1)^j$ times the canonical map.", "Then there is an isomorphism", "$$", "\\Hom_{\\mathcal{O}_X}(K(\\mathcal{U})_\\bullet, \\mathcal{F})", "=", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "$$", "functorial in $\\mathcal{F} \\in \\Ob(\\textit{PMod}(\\mathcal{O}_X))$." ], "refs": [], "proofs": [ { "contents": [ "We saw in the discussion just above the lemma that", "$$", "\\Hom_{\\mathcal{O}_X}(", "(j_{i_0\\ldots i_p})_{p!}\\mathcal{O}_{U_{i_0\\ldots i_p}},", "\\mathcal{F})", "=", "\\mathcal{F}(U_{i_0\\ldots i_p}).", "$$", "Hence we see that it is indeed the case that the direct sum", "$$", "\\bigoplus\\nolimits_{i_0 \\ldots i_p}", "(j_{i_0 \\ldots i_p})_{p!}\\mathcal{O}_{U_{i_0 \\ldots i_p}}", "$$", "represents the functor", "$$", "\\mathcal{F}", "\\longmapsto", "\\prod\\nolimits_{i_0\\ldots i_p} \\mathcal{F}(U_{i_0\\ldots i_p}).", "$$", "Hence by Categories, Yoneda Lemma \\ref{categories-lemma-yoneda}", "we see that there is a complex $K(\\mathcal{U})_\\bullet$ with terms", "as given. It is a simple matter to see that the maps are as given", "in the lemma." ], "refs": [ "categories-lemma-yoneda" ], "ref_ids": [ 12203 ] } ], "ref_ids": [] }, { "id": 2049, "type": "theorem", "label": "cohomology-lemma-homology-complex", "categories": [ "cohomology" ], "title": "cohomology-lemma-homology-complex", "contents": [ "Let $X$ be a ringed space.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be a covering.", "Let $\\mathcal{O}_\\mathcal{U} \\subset \\mathcal{O}_X$", "be the image presheaf of the map", "$\\bigoplus j_{p!}\\mathcal{O}_{U_i} \\to \\mathcal{O}_X$.", "The chain complex $K(\\mathcal{U})_\\bullet$ of presheaves", "of Lemma \\ref{lemma-cech-map-into} above has homology presheaves", "$$", "H_i(K(\\mathcal{U})_\\bullet) =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & i \\not = 0 \\\\", "\\mathcal{O}_\\mathcal{U} & \\text{if} & i = 0", "\\end{matrix}", "\\right.", "$$" ], "refs": [ "cohomology-lemma-cech-map-into" ], "proofs": [ { "contents": [ "Consider the extended complex $K^{ext}_\\bullet$ one gets by putting", "$\\mathcal{O}_\\mathcal{U}$ in degree $-1$ with the obvious map", "$K(\\mathcal{U})_0 =", "\\bigoplus_{i_0} (j_{i_0})_{p!}\\mathcal{O}_{U_{i_0}} \\to", "\\mathcal{O}_\\mathcal{U}$.", "It suffices to show that taking sections of this extended complex over", "any open $W \\subset X$ leads to an acyclic complex.", "In fact, we claim that for every $W \\subset X$ the complex", "$K^{ext}_\\bullet(W)$ is homotopy equivalent to the zero complex.", "Write $I = I_1 \\amalg I_2$ where $W \\subset U_i$ if and only", "if $i \\in I_1$.", "\\medskip\\noindent", "If $I_1 = \\emptyset$, then the complex $K^{ext}_\\bullet(W) = 0$ so there is", "nothing to prove.", "\\medskip\\noindent", "If $I_1 \\not = \\emptyset$, then", "$\\mathcal{O}_\\mathcal{U}(W) = \\mathcal{O}_X(W)$", "and", "$$", "K^{ext}_p(W) =", "\\bigoplus\\nolimits_{i_0 \\ldots i_p \\in I_1} \\mathcal{O}_X(W).", "$$", "This is true because of the simple description of the presheaves", "$(j_{i_0 \\ldots i_p})_{p!}\\mathcal{O}_{U_{i_0 \\ldots i_p}}$.", "Moreover, the differential of the complex $K^{ext}_\\bullet(W)$", "is given by", "$$", "d(s)_{i_0 \\ldots i_p} =", "\\sum\\nolimits_{j = 0, \\ldots, p + 1} \\sum\\nolimits_{i \\in I_1}", "(-1)^j s_{i_0 \\ldots i_{j - 1} i i_j \\ldots i_p}.", "$$", "The sum is finite as the element $s$ has finite support.", "Fix an element $i_{\\text{fix}} \\in I_1$. Define a map", "$$", "h : K^{ext}_p(W) \\longrightarrow K^{ext}_{p + 1}(W)", "$$", "by the rule", "$$", "h(s)_{i_0 \\ldots i_{p + 1}} =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & i_0 \\not = i \\\\", "s_{i_1 \\ldots i_{p + 1}} & \\text{if} & i_0 = i_{\\text{fix}}", "\\end{matrix}", "\\right.", "$$", "We will use the shorthand", "$h(s)_{i_0 \\ldots i_{p + 1}} = (i_0 = i_{\\text{fix}}) s_{i_1 \\ldots i_p}$", "for this. Then we compute", "\\begin{eqnarray*}", "& & (dh + hd)(s)_{i_0 \\ldots i_p} \\\\", "& = &", "\\sum_j \\sum_{i \\in I_1} (-1)^j h(s)_{i_0 \\ldots i_{j - 1} i i_j \\ldots i_p}", "+", "(i = i_0) d(s)_{i_1 \\ldots i_p} \\\\", "& = &", "s_{i_0 \\ldots i_p} +", "\\sum_{j \\geq 1}\\sum_{i \\in I_1}", "(-1)^j (i_0 = i_{\\text{fix}}) s_{i_1 \\ldots i_{j - 1} i i_j \\ldots i_p}", "+", "(i_0 = i_{\\text{fix}}) d(s)_{i_1 \\ldots i_p}", "\\end{eqnarray*}", "which is equal to $s_{i_0 \\ldots i_p}$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 2048 ] }, { "id": 2050, "type": "theorem", "label": "cohomology-lemma-cech-cohomology-derived-presheaves", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-cohomology-derived-presheaves", "contents": [ "Let $X$ be a ringed space.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$", "be an open covering of $U \\subset X$.", "The {\\v C}ech cohomology functors $\\check{H}^p(\\mathcal{U}, -)$", "are canonically isomorphic as a $\\delta$-functor to", "the right derived functors of the functor", "$$", "\\check{H}^0(\\mathcal{U}, -) :", "\\textit{PMod}(\\mathcal{O}_X)", "\\longrightarrow", "\\text{Mod}_{\\mathcal{O}_X(U)}.", "$$", "Moreover, there is a functorial quasi-isomorphism", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\longrightarrow", "R\\check{H}^0(\\mathcal{U}, \\mathcal{F})", "$$", "where the right hand side indicates the right derived functor", "$$", "R\\check{H}^0(\\mathcal{U}, -) :", "D^{+}(\\textit{PMod}(\\mathcal{O}_X))", "\\longrightarrow", "D^{+}(\\mathcal{O}_X(U))", "$$", "of the left exact functor $\\check{H}^0(\\mathcal{U}, -)$." ], "refs": [], "proofs": [ { "contents": [ "Note that the category of presheaves of $\\mathcal{O}_X$-modules", "has enough injectives, see", "Injectives, Proposition \\ref{injectives-proposition-presheaves-modules}.", "Note that $\\check{H}^0(\\mathcal{U}, -)$ is a left exact functor", "from the category of presheaves of $\\mathcal{O}_X$-modules", "to the category of $\\mathcal{O}_X(U)$-modules.", "Hence the derived functor and the right derived functor exist, see", "Derived Categories, Section \\ref{derived-section-right-derived-functor}.", "\\medskip\\noindent", "Let $\\mathcal{I}$ be a injective presheaf of $\\mathcal{O}_X$-modules.", "In this case the functor $\\Hom_{\\mathcal{O}_X}(-, \\mathcal{I})$", "is exact on $\\textit{PMod}(\\mathcal{O}_X)$. By", "Lemma \\ref{lemma-cech-map-into} we have", "$$", "\\Hom_{\\mathcal{O}_X}(K(\\mathcal{U})_\\bullet, \\mathcal{I})", "=", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}).", "$$", "By Lemma \\ref{lemma-homology-complex} we have that $K(\\mathcal{U})_\\bullet$ is", "quasi-isomorphic to $\\mathcal{O}_\\mathcal{U}[0]$. Hence by", "the exactness of Hom into $\\mathcal{I}$ mentioned above we see", "that $\\check{H}^i(\\mathcal{U}, \\mathcal{I}) = 0$ for all", "$i > 0$. Thus the $\\delta$-functor $(\\check{H}^n, \\delta)$", "(see Lemma \\ref{lemma-cech-cohomology-delta-functor-presheaves})", "satisfies the assumptions of", "Homology, Lemma \\ref{homology-lemma-efface-implies-universal},", "and hence is a universal $\\delta$-functor.", "\\medskip\\noindent", "By", "Derived Categories, Lemma \\ref{derived-lemma-higher-derived-functors}", "also the sequence $R^i\\check{H}^0(\\mathcal{U}, -)$", "forms a universal $\\delta$-functor. By the uniqueness of universal", "$\\delta$-functors, see", "Homology, Lemma \\ref{homology-lemma-uniqueness-universal-delta-functor}", "we conclude that", "$R^i\\check{H}^0(\\mathcal{U}, -) = \\check{H}^i(\\mathcal{U}, -)$.", "This is enough for most applications", "and the reader is suggested to skip the rest of the proof.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be any presheaf of $\\mathcal{O}_X$-modules.", "Choose an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$", "in the category $\\textit{PMod}(\\mathcal{O}_X)$.", "Consider the double complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet)$ with terms", "$\\check{\\mathcal{C}}^p(\\mathcal{U}, \\mathcal{I}^q)$.", "Consider the associated total complex", "$\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet))$,", "see Homology, Definition \\ref{homology-definition-associated-simple-complex}.", "There is a map of complexes", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\longrightarrow", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet))", "$$", "coming from the maps", "$\\check{\\mathcal{C}}^p(\\mathcal{U}, \\mathcal{F})", "\\to \\check{\\mathcal{C}}^p(\\mathcal{U}, \\mathcal{I}^0)$", "and there is a map of complexes", "$$", "\\check{H}^0(\\mathcal{U}, \\mathcal{I}^\\bullet)", "\\longrightarrow", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet))", "$$", "coming from the maps", "$\\check{H}^0(\\mathcal{U}, \\mathcal{I}^q) \\to", "\\check{\\mathcal{C}}^0(\\mathcal{U}, \\mathcal{I}^q)$.", "Both of these maps are quasi-isomorphisms by an application of", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}.", "Namely, the columns of the double complex are exact in positive degrees", "because the {\\v C}ech complex as a functor is exact", "(Lemma \\ref{lemma-cech-exact-presheaves})", "and the rows of the double complex are exact in positive degrees", "since as we just saw the higher {\\v C}ech cohomology groups of the injective", "presheaves $\\mathcal{I}^q$ are zero.", "Since quasi-isomorphisms become invertible", "in $D^{+}(\\mathcal{O}_X(U))$ this gives the last displayed morphism", "of the lemma. We omit the verification that this morphism is", "functorial." ], "refs": [ "injectives-proposition-presheaves-modules", "cohomology-lemma-cech-map-into", "cohomology-lemma-homology-complex", "cohomology-lemma-cech-cohomology-delta-functor-presheaves", "homology-lemma-efface-implies-universal", "derived-lemma-higher-derived-functors", "homology-lemma-uniqueness-universal-delta-functor", "homology-definition-associated-simple-complex", "homology-lemma-double-complex-gives-resolution", "cohomology-lemma-cech-exact-presheaves" ], "ref_ids": [ 7806, 2048, 2049, 2047, 12052, 1869, 12053, 12164, 12106, 2046 ] } ], "ref_ids": [] }, { "id": 2051, "type": "theorem", "label": "cohomology-lemma-injective-trivial-cech", "categories": [ "cohomology" ], "title": "cohomology-lemma-injective-trivial-cech", "contents": [ "Let $X$ be a ringed space.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be a covering.", "Let $\\mathcal{I}$ be an injective $\\mathcal{O}_X$-module.", "Then", "$$", "\\check{H}^p(\\mathcal{U}, \\mathcal{I}) =", "\\left\\{", "\\begin{matrix}", "\\mathcal{I}(U) & \\text{if} & p = 0 \\\\", "0 & \\text{if} & p > 0", "\\end{matrix}", "\\right.", "$$" ], "refs": [], "proofs": [ { "contents": [ "An injective $\\mathcal{O}_X$-module is also injective as an object in", "the category $\\textit{PMod}(\\mathcal{O}_X)$ (for example since", "sheafification is an exact left adjoint to the inclusion functor,", "using Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}).", "Hence we can apply Lemma \\ref{lemma-cech-cohomology-derived-presheaves}", "(or its proof) to see the result." ], "refs": [ "homology-lemma-adjoint-preserve-injectives", "cohomology-lemma-cech-cohomology-derived-presheaves" ], "ref_ids": [ 12116, 2050 ] } ], "ref_ids": [] }, { "id": 2052, "type": "theorem", "label": "cohomology-lemma-cech-cohomology", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-cohomology", "contents": [ "Let $X$ be a ringed space.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be a covering.", "There is a transformation", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, -)", "\\longrightarrow", "R\\Gamma(U, -)", "$$", "of functors", "$\\textit{Mod}(\\mathcal{O}_X) \\to D^{+}(\\mathcal{O}_X(U))$.", "In particular this provides canonical maps", "$\\check{H}^p(\\mathcal{U}, \\mathcal{F}) \\to H^p(U, \\mathcal{F})$ for", "$\\mathcal{F}$ ranging over $\\textit{Mod}(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Choose an injective resolution", "$\\mathcal{F} \\to \\mathcal{I}^\\bullet$. Consider the double complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet)$ with terms", "$\\check{\\mathcal{C}}^p(\\mathcal{U}, \\mathcal{I}^q)$.", "There is a map of complexes", "$$", "\\alpha :", "\\Gamma(U, \\mathcal{I}^\\bullet)", "\\longrightarrow", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet))", "$$", "coming from the maps", "$\\mathcal{I}^q(U) \\to \\check{H}^0(\\mathcal{U}, \\mathcal{I}^q)$", "and a map of complexes", "$$", "\\beta :", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\longrightarrow", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet))", "$$", "coming from the map $\\mathcal{F} \\to \\mathcal{I}^0$.", "We can apply", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}", "to see that $\\alpha$ is a quasi-isomorphism.", "Namely, Lemma \\ref{lemma-injective-trivial-cech} implies that", "the $q$th row of the double complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet)$ is a", "resolution of $\\Gamma(U, \\mathcal{I}^q)$.", "Hence $\\alpha$ becomes invertible in $D^{+}(\\mathcal{O}_X(U))$ and", "the transformation of the lemma is the composition of $\\beta$", "followed by the inverse of $\\alpha$. We omit the verification", "that this is functorial." ], "refs": [ "homology-lemma-double-complex-gives-resolution", "cohomology-lemma-injective-trivial-cech" ], "ref_ids": [ 12106, 2051 ] } ], "ref_ids": [] }, { "id": 2053, "type": "theorem", "label": "cohomology-lemma-cech-h1", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-h1", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{H}$ be an abelian sheaf", "on $X$. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be an open covering.", "The map", "$$", "\\check{H}^1(\\mathcal{U}, \\mathcal{H}) \\longrightarrow H^1(X, \\mathcal{H})", "$$", "is injective and identifies $\\check{H}^1(\\mathcal{U}, \\mathcal{H})$ via", "the bijection of Lemma \\ref{lemma-torsors-h1}", "with the set of isomorphism classes of $\\mathcal{H}$-torsors", "which restrict to trivial torsors over each $U_i$." ], "refs": [ "cohomology-lemma-torsors-h1" ], "proofs": [ { "contents": [ "To see this we construct an inverse map. Namely, let $\\mathcal{F}$ be a", "$\\mathcal{H}$-torsor whose restriction to $U_i$ is trivial. By", "Lemma \\ref{lemma-trivial-torsor} this means there", "exists a section $s_i \\in \\mathcal{F}(U_i)$. On $U_{i_0} \\cap U_{i_1}$", "there is a unique section $s_{i_0i_1}$ of $\\mathcal{H}$ such that", "$s_{i_0i_1} \\cdot s_{i_0}|_{U_{i_0} \\cap U_{i_1}} =", "s_{i_1}|_{U_{i_0} \\cap U_{i_1}}$. A computation shows", "that $s_{i_0i_1}$ is a {\\v C}ech cocycle and that its class is well", "defined (i.e., does not depend on the choice of the sections $s_i$).", "The inverse maps the isomorphism class of $\\mathcal{F}$ to the cohomology", "class of the cocycle $(s_{i_0i_1})$.", "We omit the verification that this map is indeed an inverse." ], "refs": [ "cohomology-lemma-trivial-torsor" ], "ref_ids": [ 2033 ] } ], "ref_ids": [ 2034 ] }, { "id": 2054, "type": "theorem", "label": "cohomology-lemma-include", "categories": [ "cohomology" ], "title": "cohomology-lemma-include", "contents": [ "Let $X$ be a ringed space.", "Consider the functor", "$i : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{PMod}(\\mathcal{O}_X)$.", "It is a left exact functor with right derived functors given by", "$$", "R^pi(\\mathcal{F}) = \\underline{H}^p(\\mathcal{F}) :", "U \\longmapsto H^p(U, \\mathcal{F})", "$$", "see discussion in Section \\ref{section-locality}." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $i$ is left exact.", "Choose an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$.", "By definition $R^pi$ is the $p$th cohomology {\\it presheaf}", "of the complex $\\mathcal{I}^\\bullet$. In other words, the", "sections of $R^pi(\\mathcal{F})$ over an open $U$ are given by", "$$", "\\frac{\\Ker(\\mathcal{I}^n(U) \\to \\mathcal{I}^{n + 1}(U))}", "{\\Im(\\mathcal{I}^{n - 1}(U) \\to \\mathcal{I}^n(U))}.", "$$", "which is the definition of $H^p(U, \\mathcal{F})$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2055, "type": "theorem", "label": "cohomology-lemma-cech-spectral-sequence", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-spectral-sequence", "contents": [ "Let $X$ be a ringed space.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be a covering.", "For any sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ there", "is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with", "$$", "E_2^{p, q} = \\check{H}^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F}))", "$$", "converging to $H^{p + q}(U, \\mathcal{F})$.", "This spectral sequence is functorial in $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "This is a Grothendieck spectral sequence", "(see", "Derived Categories, Lemma \\ref{derived-lemma-grothendieck-spectral-sequence})", "for the functors", "$$", "i : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{PMod}(\\mathcal{O}_X)", "\\quad\\text{and}\\quad", "\\check{H}^0(\\mathcal{U}, - ) : \\textit{PMod}(\\mathcal{O}_X)", "\\to \\text{Mod}_{\\mathcal{O}_X(U)}.", "$$", "Namely, we have $\\check{H}^0(\\mathcal{U}, i(\\mathcal{F})) = \\mathcal{F}(U)$", "by Lemma \\ref{lemma-cech-h0}. We have that $i(\\mathcal{I})$ is", "{\\v C}ech acyclic by Lemma \\ref{lemma-injective-trivial-cech}. And we", "have that $\\check{H}^p(\\mathcal{U}, -) = R^p\\check{H}^0(\\mathcal{U}, -)$", "as functors on $\\textit{PMod}(\\mathcal{O}_X)$", "by Lemma \\ref{lemma-cech-cohomology-derived-presheaves}.", "Putting everything together gives the lemma." ], "refs": [ "derived-lemma-grothendieck-spectral-sequence", "cohomology-lemma-cech-h0", "cohomology-lemma-injective-trivial-cech", "cohomology-lemma-cech-cohomology-derived-presheaves" ], "ref_ids": [ 1873, 2044, 2051, 2050 ] } ], "ref_ids": [] }, { "id": 2056, "type": "theorem", "label": "cohomology-lemma-cech-spectral-sequence-application", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-spectral-sequence-application", "contents": [ "Let $X$ be a ringed space.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be a covering.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "Assume that $H^i(U_{i_0 \\ldots i_p}, \\mathcal{F}) = 0$", "for all $i > 0$, all $p \\geq 0$ and all $i_0, \\ldots, i_p \\in I$.", "Then $\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = H^p(U, \\mathcal{F})$", "as $\\mathcal{O}_X(U)$-modules." ], "refs": [], "proofs": [ { "contents": [ "We will use the spectral sequence of", "Lemma \\ref{lemma-cech-spectral-sequence}.", "The assumptions mean that $E_2^{p, q} = 0$ for all $(p, q)$ with", "$q \\not = 0$. Hence the spectral sequence degenerates at $E_2$", "and the result follows." ], "refs": [ "cohomology-lemma-cech-spectral-sequence" ], "ref_ids": [ 2055 ] } ], "ref_ids": [] }, { "id": 2057, "type": "theorem", "label": "cohomology-lemma-ses-cech-h1", "categories": [ "cohomology" ], "title": "cohomology-lemma-ses-cech-h1", "contents": [ "Let $X$ be a ringed space.", "Let", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0", "$$", "be a short exact sequence of $\\mathcal{O}_X$-modules.", "Let $U \\subset X$ be an open subset.", "If there exists a cofinal system of open coverings $\\mathcal{U}$", "of $U$ such that $\\check{H}^1(\\mathcal{U}, \\mathcal{F}) = 0$,", "then the map $\\mathcal{G}(U) \\to \\mathcal{H}(U)$ is", "surjective." ], "refs": [], "proofs": [ { "contents": [ "Take an element $s \\in \\mathcal{H}(U)$. Choose an open covering", "$\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ such that", "(a) $\\check{H}^1(\\mathcal{U}, \\mathcal{F}) = 0$ and (b)", "$s|_{U_i}$ is the image of a section $s_i \\in \\mathcal{G}(U_i)$.", "Since we can certainly find a covering such that (b) holds", "it follows from the assumptions of the lemma that we can find", "a covering such that (a) and (b) both hold.", "Consider the sections", "$$", "s_{i_0i_1} = s_{i_1}|_{U_{i_0i_1}} - s_{i_0}|_{U_{i_0i_1}}.", "$$", "Since $s_i$ lifts $s$ we see that $s_{i_0i_1} \\in \\mathcal{F}(U_{i_0i_1})$.", "By the vanishing of $\\check{H}^1(\\mathcal{U}, \\mathcal{F})$ we can", "find sections $t_i \\in \\mathcal{F}(U_i)$ such that", "$$", "s_{i_0i_1} = t_{i_1}|_{U_{i_0i_1}} - t_{i_0}|_{U_{i_0i_1}}.", "$$", "Then clearly the sections $s_i - t_i$ satisfy the sheaf condition", "and glue to a section of $\\mathcal{G}$ over $U$ which maps to $s$.", "Hence we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2058, "type": "theorem", "label": "cohomology-lemma-cech-vanish", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-vanish", "contents": [ "\\begin{slogan}", "If higher {\\v C}ech cohomology of an abelian sheaf vanishes for all open covers,", "then higher cohomology vanishes.", "\\end{slogan}", "Let $X$ be a ringed space.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module such that", "$$", "\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0", "$$", "for all $p > 0$ and any open covering $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$", "of an open of $X$. Then $H^p(U, \\mathcal{F}) = 0$ for all $p > 0$", "and any open $U \\subset X$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a sheaf satisfying the assumption of the lemma.", "We will indicate this by saying ``$\\mathcal{F}$ has vanishing higher", "{\\v C}ech cohomology for any open covering''.", "Choose an embedding $\\mathcal{F} \\to \\mathcal{I}$ into an", "injective $\\mathcal{O}_X$-module.", "By Lemma \\ref{lemma-injective-trivial-cech} $\\mathcal{I}$ has vanishing higher", "{\\v C}ech cohomology for any open covering.", "Let $\\mathcal{Q} = \\mathcal{I}/\\mathcal{F}$", "so that we have a short exact sequence", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{I} \\to \\mathcal{Q} \\to 0.", "$$", "By Lemma \\ref{lemma-ses-cech-h1} and our assumptions", "this sequence is actually exact as a sequence of presheaves!", "In particular we have a long exact sequence of {\\v C}ech cohomology", "groups for any open covering $\\mathcal{U}$, see", "Lemma \\ref{lemma-cech-cohomology-delta-functor-presheaves}", "for example. This implies that $\\mathcal{Q}$ is also an $\\mathcal{O}_X$-module", "with vanishing higher {\\v C}ech cohomology for all open coverings.", "\\medskip\\noindent", "Next, we look at the long exact cohomology sequence", "$$", "\\xymatrix{", "0 \\ar[r] &", "H^0(U, \\mathcal{F}) \\ar[r] &", "H^0(U, \\mathcal{I}) \\ar[r] &", "H^0(U, \\mathcal{Q}) \\ar[lld] \\\\", "&", "H^1(U, \\mathcal{F}) \\ar[r] &", "H^1(U, \\mathcal{I}) \\ar[r] &", "H^1(U, \\mathcal{Q}) \\ar[lld] \\\\", "&", "\\ldots & \\ldots & \\ldots \\\\", "}", "$$", "for any open $U \\subset X$. Since $\\mathcal{I}$ is injective we", "have $H^n(U, \\mathcal{I}) = 0$ for $n > 0$ (see", "Derived Categories, Lemma \\ref{derived-lemma-higher-derived-functors}).", "By the above we see that $H^0(U, \\mathcal{I}) \\to H^0(U, \\mathcal{Q})$", "is surjective and hence $H^1(U, \\mathcal{F}) = 0$.", "Since $\\mathcal{F}$ was an arbitrary $\\mathcal{O}_X$-module with", "vanishing higher {\\v C}ech cohomology we conclude that also", "$H^1(U, \\mathcal{Q}) = 0$ since $\\mathcal{Q}$ is another of these", "sheaves (see above). By the long exact sequence this in turn implies", "that $H^2(U, \\mathcal{F}) = 0$. And so on and so forth." ], "refs": [ "cohomology-lemma-injective-trivial-cech", "cohomology-lemma-ses-cech-h1", "cohomology-lemma-cech-cohomology-delta-functor-presheaves", "derived-lemma-higher-derived-functors" ], "ref_ids": [ 2051, 2057, 2047, 1869 ] } ], "ref_ids": [] }, { "id": 2059, "type": "theorem", "label": "cohomology-lemma-cech-vanish-basis", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-vanish-basis", "contents": [ "(Variant of Lemma \\ref{lemma-cech-vanish}.)", "Let $X$ be a ringed space.", "Let $\\mathcal{B}$ be a basis for the topology on $X$.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "Assume there exists a set of open coverings $\\text{Cov}$", "with the following properties:", "\\begin{enumerate}", "\\item For every $\\mathcal{U} \\in \\text{Cov}$", "with $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ we have", "$U, U_i \\in \\mathcal{B}$ and every $U_{i_0 \\ldots i_p} \\in \\mathcal{B}$.", "\\item For every $U \\in \\mathcal{B}$ the open coverings of $U$", "occurring in $\\text{Cov}$ is a cofinal system of open coverings", "of $U$.", "\\item For every $\\mathcal{U} \\in \\text{Cov}$ we have", "$\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0$ for all $p > 0$.", "\\end{enumerate}", "Then $H^p(U, \\mathcal{F}) = 0$ for all $p > 0$ and any $U \\in \\mathcal{B}$." ], "refs": [ "cohomology-lemma-cech-vanish" ], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ and $\\text{Cov}$ be as in the lemma.", "We will indicate this by saying ``$\\mathcal{F}$ has vanishing higher", "{\\v C}ech cohomology for any $\\mathcal{U} \\in \\text{Cov}$''.", "Choose an embedding $\\mathcal{F} \\to \\mathcal{I}$ into an", "injective $\\mathcal{O}_X$-module.", "By Lemma \\ref{lemma-injective-trivial-cech} $\\mathcal{I}$", "has vanishing higher {\\v C}ech cohomology for any $\\mathcal{U} \\in \\text{Cov}$.", "Let $\\mathcal{Q} = \\mathcal{I}/\\mathcal{F}$", "so that we have a short exact sequence", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{I} \\to \\mathcal{Q} \\to 0.", "$$", "By Lemma \\ref{lemma-ses-cech-h1} and our assumption (2)", "this sequence gives rise to an exact sequence", "$$", "0 \\to \\mathcal{F}(U) \\to \\mathcal{I}(U) \\to \\mathcal{Q}(U) \\to 0.", "$$", "for every $U \\in \\mathcal{B}$. Hence for any $\\mathcal{U} \\in \\text{Cov}$", "we get a short exact sequence of {\\v C}ech complexes", "$$", "0 \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}) \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{Q}) \\to 0", "$$", "since each term in the {\\v C}ech complex is made up out of a product of", "values over elements of $\\mathcal{B}$ by assumption (1).", "In particular we have a long exact sequence of {\\v C}ech cohomology", "groups for any open covering $\\mathcal{U} \\in \\text{Cov}$.", "This implies that $\\mathcal{Q}$ is also an $\\mathcal{O}_X$-module", "with vanishing higher {\\v C}ech cohomology for all", "$\\mathcal{U} \\in \\text{Cov}$.", "\\medskip\\noindent", "Next, we look at the long exact cohomology sequence", "$$", "\\xymatrix{", "0 \\ar[r] &", "H^0(U, \\mathcal{F}) \\ar[r] &", "H^0(U, \\mathcal{I}) \\ar[r] &", "H^0(U, \\mathcal{Q}) \\ar[lld] \\\\", "&", "H^1(U, \\mathcal{F}) \\ar[r] &", "H^1(U, \\mathcal{I}) \\ar[r] &", "H^1(U, \\mathcal{Q}) \\ar[lld] \\\\", "&", "\\ldots & \\ldots & \\ldots \\\\", "}", "$$", "for any $U \\in \\mathcal{B}$. Since $\\mathcal{I}$ is injective we", "have $H^n(U, \\mathcal{I}) = 0$ for $n > 0$ (see", "Derived Categories, Lemma \\ref{derived-lemma-higher-derived-functors}).", "By the above we see that $H^0(U, \\mathcal{I}) \\to H^0(U, \\mathcal{Q})$", "is surjective and hence $H^1(U, \\mathcal{F}) = 0$.", "Since $\\mathcal{F}$ was an arbitrary $\\mathcal{O}_X$-module with", "vanishing higher {\\v C}ech cohomology for all $\\mathcal{U} \\in \\text{Cov}$", "we conclude that also $H^1(U, \\mathcal{Q}) = 0$ since $\\mathcal{Q}$ is", "another of these sheaves (see above). By the long exact sequence this in", "turn implies that $H^2(U, \\mathcal{F}) = 0$. And so on and so forth." ], "refs": [ "cohomology-lemma-injective-trivial-cech", "cohomology-lemma-ses-cech-h1", "derived-lemma-higher-derived-functors" ], "ref_ids": [ 2051, 2057, 1869 ] } ], "ref_ids": [ 2058 ] }, { "id": 2060, "type": "theorem", "label": "cohomology-lemma-pushforward-injective", "categories": [ "cohomology" ], "title": "cohomology-lemma-pushforward-injective", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Let $\\mathcal{I}$ be an injective $\\mathcal{O}_X$-module.", "Then", "\\begin{enumerate}", "\\item $\\check{H}^p(\\mathcal{V}, f_*\\mathcal{I}) = 0$", "for all $p > 0$ and any open covering", "$\\mathcal{V} : V = \\bigcup_{j \\in J} V_j$ of $Y$.", "\\item $H^p(V, f_*\\mathcal{I}) = 0$ for all $p > 0$ and", "every open $V \\subset Y$.", "\\end{enumerate}", "In other words, $f_*\\mathcal{I}$ is right acyclic for $\\Gamma(V, -)$", "(see", "Derived Categories, Definition \\ref{derived-definition-derived-functor})", "for any $V \\subset Y$ open." ], "refs": [ "derived-definition-derived-functor" ], "proofs": [ { "contents": [ "Set $\\mathcal{U} : f^{-1}(V) = \\bigcup_{j \\in J} f^{-1}(V_j)$.", "It is an open covering of $X$ and", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, f_*\\mathcal{I}) =", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}).", "$$", "This is true because", "$$", "f_*\\mathcal{I}(V_{j_0 \\ldots j_p})", "= \\mathcal{I}(f^{-1}(V_{j_0 \\ldots j_p})) =", "\\mathcal{I}(f^{-1}(V_{j_0}) \\cap \\ldots \\cap f^{-1}(V_{j_p}))", "= \\mathcal{I}(U_{j_0 \\ldots j_p}).", "$$", "Thus the first statement of the lemma follows from", "Lemma \\ref{lemma-injective-trivial-cech}. The second statement", "follows from the first and Lemma \\ref{lemma-cech-vanish}." ], "refs": [ "cohomology-lemma-injective-trivial-cech", "cohomology-lemma-cech-vanish" ], "ref_ids": [ 2051, 2058 ] } ], "ref_ids": [ 1990 ] }, { "id": 2061, "type": "theorem", "label": "cohomology-lemma-pushforward-injective-flat", "categories": [ "cohomology" ], "title": "cohomology-lemma-pushforward-injective-flat", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Assume $f$ is flat.", "Then $f_*\\mathcal{I}$ is an injective $\\mathcal{O}_Y$-module", "for any injective $\\mathcal{O}_X$-module $\\mathcal{I}$." ], "refs": [], "proofs": [ { "contents": [ "In this case the functor $f^*$ transforms injections into injections", "(Modules, Lemma \\ref{modules-lemma-pullback-flat}).", "Hence the result follows from", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}." ], "refs": [ "modules-lemma-pullback-flat", "homology-lemma-adjoint-preserve-injectives" ], "ref_ids": [ 13287, 12116 ] } ], "ref_ids": [] }, { "id": 2062, "type": "theorem", "label": "cohomology-lemma-cohomology-products", "categories": [ "cohomology" ], "title": "cohomology-lemma-cohomology-products", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $I$ be a set.", "For $i \\in I$ let $\\mathcal{F}_i$ be an $\\mathcal{O}_X$-module.", "Let $U \\subset X$ be open. The canonical map", "$$", "H^p(U, \\prod\\nolimits_{i \\in I} \\mathcal{F}_i)", "\\longrightarrow", "\\prod\\nolimits_{i \\in I} H^p(U, \\mathcal{F}_i)", "$$", "is an isomorphism for $p = 0$ and injective for $p = 1$." ], "refs": [], "proofs": [ { "contents": [ "The statement for $p = 0$ is true because the product of sheaves", "is equal to the product of the underlying presheaves, see", "Sheaves, Section \\ref{sheaves-section-limits-sheaves}.", "Proof for $p = 1$. Set $\\mathcal{F} = \\prod \\mathcal{F}_i$.", "Let $\\xi \\in H^1(U, \\mathcal{F})$ map to zero in", "$\\prod H^1(U, \\mathcal{F}_i)$. By locality of cohomology, see", "Lemma \\ref{lemma-kill-cohomology-class-on-covering},", "there exists an open covering $\\mathcal{U} : U = \\bigcup U_j$ such that", "$\\xi|_{U_j} = 0$ for all $j$. By Lemma \\ref{lemma-cech-h1} this means", "$\\xi$ comes from an element", "$\\check \\xi \\in \\check H^1(\\mathcal{U}, \\mathcal{F})$.", "Since the maps", "$\\check H^1(\\mathcal{U}, \\mathcal{F}_i) \\to H^1(U, \\mathcal{F}_i)$", "are injective for all $i$ (by Lemma \\ref{lemma-cech-h1}), and since", "the image of $\\xi$ is zero in $\\prod H^1(U, \\mathcal{F}_i)$ we see", "that the image", "$\\check \\xi_i = 0$ in $\\check H^1(\\mathcal{U}, \\mathcal{F}_i)$.", "However, since $\\mathcal{F} = \\prod \\mathcal{F}_i$ we see that", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$ is the", "product of the complexes", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}_i)$,", "hence by", "Homology, Lemma \\ref{homology-lemma-product-abelian-groups-exact}", "we conclude that $\\check \\xi = 0$ as desired." ], "refs": [ "cohomology-lemma-kill-cohomology-class-on-covering", "cohomology-lemma-cech-h1", "cohomology-lemma-cech-h1", "homology-lemma-product-abelian-groups-exact" ], "ref_ids": [ 2038, 2053, 2053, 12130 ] } ], "ref_ids": [] }, { "id": 2063, "type": "theorem", "label": "cohomology-lemma-injective-flasque", "categories": [ "cohomology" ], "title": "cohomology-lemma-injective-flasque", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Then any injective $\\mathcal{O}_X$-module is flasque." ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of Lemma \\ref{lemma-injective-restriction-surjective}." ], "refs": [ "cohomology-lemma-injective-restriction-surjective" ], "ref_ids": [ 2041 ] } ], "ref_ids": [] }, { "id": 2064, "type": "theorem", "label": "cohomology-lemma-flasque-acyclic", "categories": [ "cohomology" ], "title": "cohomology-lemma-flasque-acyclic", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Any flasque $\\mathcal{O}_X$-module", "is acyclic for $R\\Gamma(X, -)$ as well as $R\\Gamma(U, -)$ for any", "open $U$ of $X$." ], "refs": [], "proofs": [ { "contents": [ "We will prove this using", "Derived Categories, Lemma \\ref{derived-lemma-subcategory-right-acyclics}.", "Since every injective module is flasque we see that we can embed", "every $\\mathcal{O}_X$-module into a flasque module, see", "Injectives, Lemma \\ref{injectives-lemma-abelian-sheaves-space}.", "Thus it suffices to show that given a short exact sequence", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0", "$$", "with $\\mathcal{F}$, $\\mathcal{G}$ flasque, then $\\mathcal{H}$", "is flasque and the sequence remains short exact after taking sections", "on any open of $X$. In fact, the second statement implies the first.", "Thus, let $U \\subset X$ be an open subspace. Let $s \\in \\mathcal{H}(U)$.", "We will show that we can lift $s$ to a section of $\\mathcal{G}$", "over $U$. To do this consider the set $T$ of pairs $(V, t)$", "where $V \\subset U$ is open and $t \\in \\mathcal{G}(V)$ is a section", "mapping to $s|_V$ in $\\mathcal{H}$.", "We put a partial ordering on $T$ by setting", "$(V, t) \\leq (V', t')$ if and only if $V \\subset V'$ and $t'|_V = t$.", "If $(V_\\alpha, t_\\alpha)$, $\\alpha \\in A$", "is a totally ordered subset of $T$, then $V = \\bigcup V_\\alpha$", "is open and there is a unique section $t \\in \\mathcal{G}(V)$", "restricting to $t_\\alpha$ over $V_\\alpha$ by the sheaf condition on", "$\\mathcal{G}$. Thus by Zorn's lemma there exists a maximal element", "$(V, t)$ in $T$. We will show that $V = U$ thereby finishing the proof.", "Namely, pick any $x \\in U$. We can find a small open neighbourhood", "$W \\subset U$ of $x$ and $t' \\in \\mathcal{G}(W)$ mapping to $s|_W$", "in $\\mathcal{H}$. Then $t'|_{W \\cap V} - t|_{W \\cap V}$ maps to", "zero in $\\mathcal{H}$, hence comes from some section", "$r' \\in \\mathcal{F}(W \\cap V)$. Using that $\\mathcal{F}$ is flasque", "we find a section $r \\in \\mathcal{F}(W)$ restricting to $r'$", "over $W \\cap V$. Modifying $t'$ by the image of $r$ we may", "assume that $t$ and $t'$ restrict to the same section over", "$W \\cap V$. By the sheaf condition of $\\mathcal{G}$", "we can find a section $\\tilde t$ of $\\mathcal{G}$ over", "$W \\cup V$ restricting to $t$ and $t'$.", "By maximality of $(V, t)$ we see that $V \\cup W = V$.", "Thus $x \\in V$ and we are done." ], "refs": [ "derived-lemma-subcategory-right-acyclics", "injectives-lemma-abelian-sheaves-space" ], "ref_ids": [ 1837, 7774 ] } ], "ref_ids": [] }, { "id": 2065, "type": "theorem", "label": "cohomology-lemma-flasque-acyclic-cech", "categories": [ "cohomology" ], "title": "cohomology-lemma-flasque-acyclic-cech", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "Let $\\mathcal{U} : U = \\bigcup U_i$ be an open covering.", "If $\\mathcal{F}$ is flasque, then", "$\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0$ for $p > 0$." ], "refs": [], "proofs": [ { "contents": [ "The presheaves $\\underline{H}^q(\\mathcal{F})$ used in the statement", "of Lemma \\ref{lemma-cech-spectral-sequence} are zero by", "Lemma \\ref{lemma-flasque-acyclic}.", "Hence $\\check{H}^p(U, \\mathcal{F}) = H^p(U, \\mathcal{F}) = 0$", "by Lemma \\ref{lemma-flasque-acyclic} again." ], "refs": [ "cohomology-lemma-cech-spectral-sequence", "cohomology-lemma-flasque-acyclic", "cohomology-lemma-flasque-acyclic" ], "ref_ids": [ 2055, 2064, 2064 ] } ], "ref_ids": [] }, { "id": 2066, "type": "theorem", "label": "cohomology-lemma-flasque-acyclic-pushforward", "categories": [ "cohomology" ], "title": "cohomology-lemma-flasque-acyclic-pushforward", "contents": [ "Let $(X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism", "of ringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "If $\\mathcal{F}$ is flasque, then", "$R^pf_*\\mathcal{F} = 0$ for $p > 0$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from ", "Lemma \\ref{lemma-describe-higher-direct-images} and", "Lemma \\ref{lemma-flasque-acyclic}." ], "refs": [ "cohomology-lemma-describe-higher-direct-images", "cohomology-lemma-flasque-acyclic" ], "ref_ids": [ 2039, 2064 ] } ], "ref_ids": [] }, { "id": 2067, "type": "theorem", "label": "cohomology-lemma-vanishing-ravi", "categories": [ "cohomology" ], "title": "cohomology-lemma-vanishing-ravi", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{F}$ be an abelian sheaf", "on $X$. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an", "open covering. Assume the restriction mappings", "$\\mathcal{F}(U) \\to \\mathcal{F}(U')$ are surjective", "for $U'$ an arbitrary union of opens of the form $U_{i_0 \\ldots i_p}$.", "Then $\\check{H}^p(\\mathcal{U}, \\mathcal{F})$", "vanishes for $p > 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $Y$ be the set of nonempty subsets of $I$. We will use the letters", "$A, B, C, \\ldots$ to denote elements of $Y$, i.e., nonempty subsets of $I$.", "For a finite nonempty subset $J \\subset I$ let", "$$", "V_J = \\{A \\in Y \\mid J \\subset A\\}", "$$", "This means that $V_{\\{i\\}} = \\{A \\in Y \\mid i \\in A\\}$ and", "$V_J = \\bigcap_{j \\in J} V_{\\{j\\}}$.", "Then $V_J \\subset V_K$ if and only if $J \\supset K$.", "There is a unique topology on $Y$ such that the collection of", "subsets $V_J$ is a basis for the topology on $Y$. Any open is of the form", "$$", "V = \\bigcup\\nolimits_{t \\in T} V_{J_t}", "$$", "for some family of finite subsets $J_t$. If $J_t \\subset J_{t'}$", "then we may remove $J_{t'}$ from the family without changing $V$.", "Thus we may assume there are no inclusions among the $J_t$.", "In this case the minimal elements of $V$ are the sets $A = J_t$.", "Hence we can read off the family $(J_t)_{t \\in T}$ from the open $V$.", "\\medskip\\noindent", "We can completely understand open coverings in $Y$. First, because", "the elements $A \\in Y$ are nonempty subsets of $I$ we have", "$$", "Y = \\bigcup\\nolimits_{i \\in I} V_{\\{i\\}}", "$$", "To understand other coverings, let $V$ be as above and let $V_s \\subset Y$", "be an open corresponding to the family $(J_{s, t})_{t \\in T_s}$. Then", "$$", "V = \\bigcup\\nolimits_{s \\in S} V_s", "$$", "if and only if for each $t \\in T$ there exists an $s \\in S$ and", "$t_s \\in T_s$ such that $J_t = J_{s, t_s}$. Namely, as the family", "$(J_t)_{t \\in T}$ is minimal, the minimal element $A = J_t$", "has to be in $V_s$ for some $s$, hence $A \\in V_{J_{t_s}}$ for some", "$t_s \\in T_s$. But since $A$ is also minimal in $V_s$ we conclude", "that $J_{t_s} = J_t$.", "\\medskip\\noindent", "Next we map the set of opens of $Y$ to opens of $X$. Namely, we send", "$Y$ to $U$, we use the rule", "$$", "V_J \\mapsto U_J = \\bigcap\\nolimits_{i \\in J} U_i", "$$", "on the opens $V_J$, and we extend it to arbitrary opens $V$ by the rule", "$$", "V = \\bigcup\\nolimits_{t \\in T} V_{J_t}", "\\mapsto", "\\bigcup\\nolimits_{t \\in T} U_{J_t}", "$$", "The classification of open coverings of $Y$ given above shows that", "this rule transforms open coverings into open coverings. Thus we obtain", "an abelian sheaf $\\mathcal{G}$ on $Y$ by setting", "$\\mathcal{G}(Y) = \\mathcal{F}(U)$ and for", "$V = \\bigcup\\nolimits_{t \\in T} V_{J_t}$ setting", "$$", "\\mathcal{G}(V) = \\mathcal{F}\\left(\\bigcup\\nolimits_{t \\in T} U_{J_t}\\right)", "$$", "and using the restriction maps of $\\mathcal{F}$.", "\\medskip\\noindent", "With these preliminaries out of the way we can prove our lemma as follows.", "We have an open covering", "$\\mathcal{V} : Y = \\bigcup_{i \\in I} V_{\\{i\\}}$ of $Y$.", "By construction we have an equality", "$$", "\\check{C}^\\bullet(\\mathcal{V}, \\mathcal{G}) =", "\\check{C}^\\bullet(\\mathcal{U}, \\mathcal{F})", "$$", "of {\\v C}ech complexes. Since the sheaf $\\mathcal{G}$ is flasque on $Y$", "(by our assumption on $\\mathcal{F}$ in the statement of the lemma)", "the vanishing follows from", "Lemma \\ref{lemma-flasque-acyclic-cech}." ], "refs": [ "cohomology-lemma-flasque-acyclic-cech" ], "ref_ids": [ 2065 ] } ], "ref_ids": [] }, { "id": 2068, "type": "theorem", "label": "cohomology-lemma-before-Leray", "categories": [ "cohomology" ], "title": "cohomology-lemma-before-Leray", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "There is a commutative diagram", "$$", "\\xymatrix{", "D^{+}(X) \\ar[rr]_-{R\\Gamma(X, -)} \\ar[d]_{Rf_*} & &", "D^{+}(\\mathcal{O}_X(X)) \\ar[d]^{\\text{restriction}} \\\\", "D^{+}(Y) \\ar[rr]^-{R\\Gamma(Y, -)} & &", "D^{+}(\\mathcal{O}_Y(Y))", "}", "$$", "More generally for any $V \\subset Y$ open and $U = f^{-1}(V)$ there", "is a commutative diagram", "$$", "\\xymatrix{", "D^{+}(X) \\ar[rr]_-{R\\Gamma(U, -)} \\ar[d]_{Rf_*} & &", "D^{+}(\\mathcal{O}_X(U)) \\ar[d]^{\\text{restriction}} \\\\", "D^{+}(Y) \\ar[rr]^-{R\\Gamma(V, -)} & &", "D^{+}(\\mathcal{O}_Y(V))", "}", "$$", "See also Remark \\ref{remark-elucidate-lemma} for more explanation." ], "refs": [ "cohomology-remark-elucidate-lemma" ], "proofs": [ { "contents": [ "Let", "$\\Gamma_{res} : \\textit{Mod}(\\mathcal{O}_X) \\to \\text{Mod}_{\\mathcal{O}_Y(Y)}$", "be the functor which associates to an $\\mathcal{O}_X$-module $\\mathcal{F}$", "the global sections of $\\mathcal{F}$ viewed as a $\\mathcal{O}_Y(Y)$-module", "via the map $f^\\sharp : \\mathcal{O}_Y(Y) \\to \\mathcal{O}_X(X)$. Let", "$restriction : \\text{Mod}_{\\mathcal{O}_X(X)} \\to \\text{Mod}_{\\mathcal{O}_Y(Y)}$", "be the restriction functor induced by", "$f^\\sharp : \\mathcal{O}_Y(Y) \\to \\mathcal{O}_X(X)$. Note that $restriction$", "is exact so that", "its right derived functor is computed by simply applying the restriction", "functor, see", "Derived Categories, Lemma \\ref{derived-lemma-right-derived-exact-functor}.", "It is clear that", "$$", "\\Gamma_{res}", "=", "restriction \\circ \\Gamma(X, -)", "=", "\\Gamma(Y, -) \\circ f_*", "$$", "We claim that", "Derived Categories, Lemma \\ref{derived-lemma-compose-derived-functors}", "applies to both compositions. For the first this is clear by our remarks", "above. For the second, it follows from", "Lemma \\ref{lemma-pushforward-injective} which implies that", "injective $\\mathcal{O}_X$-modules are mapped to $\\Gamma(Y, -)$-acyclic", "sheaves on $Y$." ], "refs": [ "derived-lemma-right-derived-exact-functor", "derived-lemma-compose-derived-functors", "cohomology-lemma-pushforward-injective" ], "ref_ids": [ 1845, 1872, 2060 ] } ], "ref_ids": [ 2262 ] }, { "id": 2069, "type": "theorem", "label": "cohomology-lemma-modules-abelian", "categories": [ "cohomology" ], "title": "cohomology-lemma-modules-abelian", "contents": [ "Let $X$ be a ringed space.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item The cohomology groups $H^i(U, \\mathcal{F})$ for $U \\subset X$ open", "of $\\mathcal{F}$ computed as an $\\mathcal{O}_X$-module, or computed as an", "abelian sheaf are identical.", "\\item Let $f : X \\to Y$ be a morphism of ringed spaces.", "The higher direct images $R^if_*\\mathcal{F}$ of $\\mathcal{F}$", "computed as an $\\mathcal{O}_X$-module, or computed as an abelian sheaf", "are identical.", "\\end{enumerate}", "There are similar statements in the case of bounded below", "complexes of $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "Consider the morphism of ringed spaces", "$(X, \\mathcal{O}_X) \\to (X, \\underline{\\mathbf{Z}}_X)$ given", "by the identity on the underlying topological space and by", "the unique map of sheaves of rings", "$\\underline{\\mathbf{Z}}_X \\to \\mathcal{O}_X$.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "Denote $\\mathcal{F}_{ab}$ the same sheaf seen as an", "$\\underline{\\mathbf{Z}}_X$-module, i.e., seen as a sheaf of", "abelian groups. Let", "$\\mathcal{F} \\to \\mathcal{I}^\\bullet$ be an injective resolution.", "By Remark \\ref{remark-elucidate-lemma} we see that", "$\\Gamma(X, \\mathcal{I}^\\bullet)$ computes both", "$R\\Gamma(X, \\mathcal{F})$ and $R\\Gamma(X, \\mathcal{F}_{ab})$.", "This proves (1).", "\\medskip\\noindent", "To prove (2) we use (1) and Lemma \\ref{lemma-describe-higher-direct-images}.", "The result follows immediately." ], "refs": [ "cohomology-remark-elucidate-lemma", "cohomology-lemma-describe-higher-direct-images" ], "ref_ids": [ 2262, 2039 ] } ], "ref_ids": [] }, { "id": 2070, "type": "theorem", "label": "cohomology-lemma-Leray", "categories": [ "cohomology" ], "title": "cohomology-lemma-Leray", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Let $\\mathcal{F}^\\bullet$ be", "a bounded below complex of $\\mathcal{O}_X$-modules.", "There is a spectral sequence", "$$", "E_2^{p, q} = H^p(Y, R^qf_*(\\mathcal{F}^\\bullet))", "$$", "converging to $H^{p + q}(X, \\mathcal{F}^\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "This is just the Grothendieck spectral sequence", "Derived Categories, Lemma \\ref{derived-lemma-grothendieck-spectral-sequence}", "coming from the composition of functors", "$\\Gamma_{res} = \\Gamma(Y, -) \\circ f_*$ where $\\Gamma_{res}$ is as", "in the proof of Lemma \\ref{lemma-before-Leray}.", "To see that the assumptions of", "Derived Categories, Lemma \\ref{derived-lemma-grothendieck-spectral-sequence}", "are satisfied, see the proof of Lemma \\ref{lemma-before-Leray} or", "Remark \\ref{remark-elucidate-lemma}." ], "refs": [ "derived-lemma-grothendieck-spectral-sequence", "cohomology-lemma-before-Leray", "derived-lemma-grothendieck-spectral-sequence", "cohomology-lemma-before-Leray", "cohomology-remark-elucidate-lemma" ], "ref_ids": [ 1873, 2068, 1873, 2068, 2262 ] } ], "ref_ids": [] }, { "id": 2071, "type": "theorem", "label": "cohomology-lemma-apply-Leray", "categories": [ "cohomology" ], "title": "cohomology-lemma-apply-Leray", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item If $R^qf_*\\mathcal{F} = 0$ for $q > 0$, then", "$H^p(X, \\mathcal{F}) = H^p(Y, f_*\\mathcal{F})$ for all $p$.", "\\item If $H^p(Y, R^qf_*\\mathcal{F}) = 0$ for all $q$ and $p > 0$, then", "$H^q(X, \\mathcal{F}) = H^0(Y, R^qf_*\\mathcal{F})$ for all $q$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "These are two simple conditions that force the Leray spectral sequence to", "degenerate at $E_2$. You can also prove these facts directly (without using", "the spectral sequence) which is a good exercise in cohomology of sheaves." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2072, "type": "theorem", "label": "cohomology-lemma-higher-direct-images-compose", "categories": [ "cohomology" ], "title": "cohomology-lemma-higher-direct-images-compose", "contents": [ "\\begin{slogan}", "The total derived functor of a composition is the", "composition of the total derived functors.", "\\end{slogan}", "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces.", "In this case $Rg_* \\circ Rf_* = R(g \\circ f)_*$ as functors", "from $D^{+}(X) \\to D^{+}(Z)$." ], "refs": [], "proofs": [ { "contents": [ "We are going to apply", "Derived Categories, Lemma \\ref{derived-lemma-compose-derived-functors}.", "It is clear that $g_* \\circ f_* = (g \\circ f)_*$, see", "Sheaves, Lemma \\ref{sheaves-lemma-pushforward-composition}.", "It remains to show that $f_*\\mathcal{I}$ is $g_*$-acyclic.", "This follows from Lemma \\ref{lemma-pushforward-injective}", "and the description of the", "higher direct images $R^ig_*$ in", "Lemma \\ref{lemma-describe-higher-direct-images}." ], "refs": [ "derived-lemma-compose-derived-functors", "sheaves-lemma-pushforward-composition", "cohomology-lemma-pushforward-injective", "cohomology-lemma-describe-higher-direct-images" ], "ref_ids": [ 1872, 14504, 2060, 2039 ] } ], "ref_ids": [] }, { "id": 2073, "type": "theorem", "label": "cohomology-lemma-relative-Leray", "categories": [ "cohomology" ], "title": "cohomology-lemma-relative-Leray", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "There is a spectral sequence with", "$$", "E_2^{p, q} = R^pg_*(R^qf_*\\mathcal{F})", "$$", "converging to $R^{p + q}(g \\circ f)_*\\mathcal{F}$.", "This spectral sequence is functorial in $\\mathcal{F}$, and there", "is a version for bounded below complexes of $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "This is a Grothendieck spectral sequence for composition of functors", "and follows from Lemma \\ref{lemma-higher-direct-images-compose} and", "Derived Categories, Lemma \\ref{derived-lemma-grothendieck-spectral-sequence}." ], "refs": [ "cohomology-lemma-higher-direct-images-compose", "derived-lemma-grothendieck-spectral-sequence" ], "ref_ids": [ 2072, 1873 ] } ], "ref_ids": [] }, { "id": 2074, "type": "theorem", "label": "cohomology-lemma-functoriality", "categories": [ "cohomology" ], "title": "cohomology-lemma-functoriality", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Let $\\mathcal{G}^\\bullet$, resp.\\ $\\mathcal{F}^\\bullet$ be", "a bounded below complex of $\\mathcal{O}_Y$-modules,", "resp.\\ $\\mathcal{O}_X$-modules. Let", "$\\varphi : \\mathcal{G}^\\bullet \\to f_*\\mathcal{F}^\\bullet$", "be a morphism of complexes. There is a canonical morphism", "$$", "\\mathcal{G}^\\bullet", "\\longrightarrow", "Rf_*(\\mathcal{F}^\\bullet)", "$$", "in $D^{+}(Y)$. Moreover this construction is functorial in the triple", "$(\\mathcal{G}^\\bullet, \\mathcal{F}^\\bullet, \\varphi)$." ], "refs": [], "proofs": [ { "contents": [ "Choose an injective resolution $\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$.", "By definition $Rf_*(\\mathcal{F}^\\bullet)$ is represented by", "$f_*\\mathcal{I}^\\bullet$ in $K^{+}(\\mathcal{O}_Y)$.", "The composition", "$$", "\\mathcal{G}^\\bullet \\to f_*\\mathcal{F}^\\bullet \\to f_*\\mathcal{I}^\\bullet", "$$", "is a morphism in $K^{+}(Y)$ which turns", "into the morphism of the lemma upon applying the", "localization functor $j_Y : K^{+}(Y) \\to D^{+}(Y)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2075, "type": "theorem", "label": "cohomology-lemma-functoriality-cech", "categories": [ "cohomology" ], "title": "cohomology-lemma-functoriality-cech", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Let $\\varphi : f^*\\mathcal{G} \\to \\mathcal{F}$ be an $f$-map", "from an $\\mathcal{O}_Y$-module $\\mathcal{G}$ to an", "$\\mathcal{O}_X$-module $\\mathcal{F}$.", "Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ and", "$\\mathcal{V} : Y = \\bigcup_{j \\in J} V_j$ be open coverings.", "Assume that $\\mathcal{U}$ is a refinement of", "$f^{-1}\\mathcal{V} : X = \\bigcup_{j \\in J} f^{-1}(V_j)$.", "In this case there exists a commutative diagram", "$$", "\\xymatrix{", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\ar[r] &", "R\\Gamma(X, \\mathcal{F}) \\\\", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G}) \\ar[r]", "\\ar[u]^\\gamma &", "R\\Gamma(Y, \\mathcal{G}) \\ar[u]", "}", "$$", "in $D^{+}(\\mathcal{O}_X(X))$ with horizontal arrows given by", "Lemma \\ref{lemma-cech-cohomology} and right vertical arrow by", "(\\ref{equation-functorial-derived}).", "In particular we get commutative diagrams of cohomology groups", "$$", "\\xymatrix{", "\\check{H}^p(\\mathcal{U}, \\mathcal{F}) \\ar[r] &", "H^p(X, \\mathcal{F}) \\\\", "\\check{H}^p(\\mathcal{V}, \\mathcal{G}) \\ar[r]", "\\ar[u]^\\gamma &", "H^p(Y, \\mathcal{G}) \\ar[u]", "}", "$$", "where the right vertical arrow is (\\ref{equation-functorial})" ], "refs": [ "cohomology-lemma-cech-cohomology" ], "proofs": [ { "contents": [ "We first define the left vertical arrow. Namely, choose a map", "$c : I \\to J$ such that $U_i \\subset f^{-1}(V_{c(i)})$ for all", "$i \\in I$. In degree $p$ we define the map by the rule", "$$", "\\gamma(s)_{i_0 \\ldots i_p} = \\varphi(s)_{c(i_0) \\ldots c(i_p)}", "$$", "This makes sense because $\\varphi$ does indeed induce maps", "$\\mathcal{G}(V_{c(i_0) \\ldots c(i_p)}) \\to \\mathcal{F}(U_{i_0 \\ldots i_p})$", "by assumption. It is also clear that this defines a morphism of complexes.", "Choose injective resolutions", "$\\mathcal{F} \\to \\mathcal{I}^\\bullet$ on $X$ and", "$\\mathcal{G} \\to J^\\bullet$ on $Y$. According to", "the proof of Lemma \\ref{lemma-cech-cohomology} we introduce the double", "complexes $A^{\\bullet, \\bullet}$ and $B^{\\bullet, \\bullet}$", "with terms", "$$", "B^{p, q} = \\check{\\mathcal{C}}^p(\\mathcal{V}, \\mathcal{J}^q)", "\\quad", "\\text{and}", "\\quad", "A^{p, q} = \\check{\\mathcal{C}}^p(\\mathcal{U}, \\mathcal{I}^q).", "$$", "As in Remark \\ref{remark-explain-arrow} above we also choose an", "injective resolution", "$f_*\\mathcal{I} \\to (\\mathcal{J}')^\\bullet$ on $Y$ and a morphism", "of complexes $\\beta : \\mathcal{J} \\to (\\mathcal{J}')^\\bullet$", "making (\\ref{equation-choice}) commutes. We introduce some more", "double complexes, namely $(B')^{\\bullet, \\bullet}$ and", "$(B''){\\bullet, \\bullet}$ with", "$$", "(B')^{p, q} = \\check{\\mathcal{C}}^p(\\mathcal{V}, (\\mathcal{J}')^q)", "\\quad", "\\text{and}", "\\quad", "(B'')^{p, q} = \\check{\\mathcal{C}}^p(\\mathcal{V}, f_*\\mathcal{I}^q).", "$$", "Note that there is an $f$-map of complexes from", "$f_*\\mathcal{I}^\\bullet$ to $\\mathcal{I}^\\bullet$. Hence", "it is clear that the same rule as above defines a morphism", "of double complexes", "$$", "\\gamma : (B'')^{\\bullet, \\bullet} \\longrightarrow A^{\\bullet, \\bullet}.", "$$", "Consider the diagram of complexes", "$$", "\\xymatrix{", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\ar[r] &", "\\text{Tot}(A^{\\bullet, \\bullet}) & & &", "\\Gamma(X, \\mathcal{I}^\\bullet) \\ar[lll]^{qis}", "\\ar@{=}[ddl]\\\\", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G})", "\\ar[r] \\ar[u]^\\gamma &", "\\text{Tot}(B^{\\bullet, \\bullet}) \\ar[r]^\\beta &", "\\text{Tot}((B')^{\\bullet, \\bullet}) &", "\\text{Tot}((B'')^{\\bullet, \\bullet}) \\ar[l] \\ar[llu]_{s\\gamma} \\\\", "& \\Gamma(Y, \\mathcal{J}^\\bullet) \\ar[u]^{qis} \\ar[r]^\\beta &", "\\Gamma(Y, (\\mathcal{J}')^\\bullet) \\ar[u] &", "\\Gamma(Y, f_*\\mathcal{I}^\\bullet) \\ar[u] \\ar[l]_{qis}", "}", "$$", "The two horizontal arrows with targets $\\text{Tot}(A^{\\bullet, \\bullet})$ and", "$\\text{Tot}(B^{\\bullet, \\bullet})$", "are the ones explained in Lemma \\ref{lemma-cech-cohomology}.", "The left upper shape (a pentagon) is commutative simply", "because (\\ref{equation-choice}) is commutative.", "The two lower squares are trivially commutative.", "It is also immediate from the definitions that the", "right upper shape (a square) is commutative.", "The result of the lemma now follows from the definitions", "and the fact that going around the diagram on the outer sides", "from $\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G})$", "to $\\Gamma(X, \\mathcal{I}^\\bullet)$ either on top or on bottom", "is the same (where you have to invert any quasi-isomorphisms along the way)." ], "refs": [ "cohomology-lemma-cech-cohomology", "cohomology-remark-explain-arrow", "cohomology-lemma-cech-cohomology" ], "ref_ids": [ 2052, 2264, 2052 ] } ], "ref_ids": [ 2052 ] }, { "id": 2076, "type": "theorem", "label": "cohomology-lemma-cech-always", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-always", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{F}$ be an abelian sheaf. Then", "the map $\\check{H}^1(X, \\mathcal{F}) \\to H^1(X, \\mathcal{F})$ defined", "in (\\ref{equation-cech-to-cohomology}) is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{U}$ be an open covering of $X$.", "By Lemma \\ref{lemma-cech-spectral-sequence}", "there is an exact sequence", "$$", "0 \\to \\check{H}^1(\\mathcal{U}, \\mathcal{F}) \\to H^1(X, \\mathcal{F})", "\\to \\check{H}^0(\\mathcal{U}, \\underline{H}^1(\\mathcal{F}))", "$$", "Thus the map is injective. To show surjectivity it suffices to show that", "any element of $\\check{H}^0(\\mathcal{U}, \\underline{H}^1(\\mathcal{F}))$", "maps to zero after replacing $\\mathcal{U}$ by a refinement.", "This is immediate from the definitions and the fact that", "$\\underline{H}^1(\\mathcal{F})$ is a presheaf of abelian groups", "whose sheafification is zero by locality of cohomology, see", "Lemma \\ref{lemma-kill-cohomology-class-on-covering}." ], "refs": [ "cohomology-lemma-cech-spectral-sequence", "cohomology-lemma-kill-cohomology-class-on-covering" ], "ref_ids": [ 2055, 2038 ] } ], "ref_ids": [] }, { "id": 2077, "type": "theorem", "label": "cohomology-lemma-cech-Hausdorff-quasi-compact", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-Hausdorff-quasi-compact", "contents": [ "Let $X$ be a Hausdorff and quasi-compact topological space. Let", "$\\mathcal{F}$ be an abelian sheaf on $X$. Then", "the map $\\check{H}^n(X, \\mathcal{F}) \\to H^n(X, \\mathcal{F})$ defined", "in (\\ref{equation-cech-to-cohomology}) is an isomorphism for", "all $n$." ], "refs": [], "proofs": [ { "contents": [ "We already know that $\\check{H}^n(X, -) \\to H^p(X, -)$", "is an isomorphism of functors for $n = 0, 1$, see", "Lemma \\ref{lemma-cech-always}.", "The functors $H^n(X, -)$ form a universal $\\delta$-functor, see", "Derived Categories, Lemma \\ref{derived-lemma-higher-derived-functors}.", "If we show that $\\check{H}^n(X, -)$ forms a universal $\\delta$-functor", "and that $\\check{H}^n(X, -) \\to H^n(X, -)$ is compatible with boundary", "maps, then the map will automatically be an isomorphism by uniqueness", "of universal $\\delta$-functors, see", "Homology, Lemma \\ref{homology-lemma-uniqueness-universal-delta-functor}.", "\\medskip\\noindent", "Let $0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0$", "be a short exact sequence of abelian sheaves on $X$.", "Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be an open covering.", "This gives a complex of complexes", "$$", "0 \\to \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{G}) \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{H}) \\to 0", "$$", "which is in general not exact on the right. The sequence defines", "the maps", "$$", "\\check{H}^n(\\mathcal{U}, \\mathcal{F}) \\to", "\\check{H}^n(\\mathcal{U}, \\mathcal{G}) \\to", "\\check{H}^n(\\mathcal{U}, \\mathcal{H})", "$$", "but isn't good enough to define a boundary operator", "$\\delta : \\check{H}^n(\\mathcal{U}, \\mathcal{H}) \\to", "\\check{H}^{n + 1}(\\mathcal{U}, \\mathcal{F})$. Indeed", "such a thing will not exist in general. However, given an", "element $\\overline{h} \\in \\check{H}^n(\\mathcal{U}, \\mathcal{H})$", "which is the cohomology class of a cocycle", "$h = (h_{i_0 \\ldots i_n})$", "we can choose open coverings", "$$", "U_{i_0 \\ldots i_n} = \\bigcup W_{i_0 \\ldots i_n, k}", "$$", "such that $h_{i_0 \\ldots i_n}|_{W_{i_0 \\ldots i_n, k}}$", "lifts to a section of $\\mathcal{G}$ over $W_{i_0 \\ldots i_n, k}$.", "By Topology, Lemma \\ref{topology-lemma-refine-covering}", "(this is where we use the assumption that $X$ is hausdorff and quasi-compact)", "we can choose an open covering $\\mathcal{V} : X = \\bigcup_{j \\in J} V_j$", "and $\\alpha : J \\to I$ such that $V_j \\subset U_{\\alpha(j)}$", "(it is a refinement) and such that for all $j_0, \\ldots, j_n \\in J$", "there is a $k$ such that", "$V_{j_0 \\ldots j_n} \\subset W_{\\alpha(j_0) \\ldots \\alpha(j_n), k}$.", "We obtain maps of complexes", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\ar[d] \\ar[r] &", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{G}) \\ar[d] \\ar[r] &", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{H}) \\ar[d] \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{F}) \\ar[r] &", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G}) \\ar[r] &", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{H}) \\ar[r] &", "0", "}", "$$", "In fact, the vertical arrows are the maps of complexes used", "to define the transition maps between the {\\v C}ech cohomology groups.", "Our choice of refinement shows that we may choose", "$$", "g_{j_0 \\ldots j_n} \\in", "\\mathcal{G}(V_{j_0 \\ldots j_n}),\\quad", "g_{j_0 \\ldots j_n} \\longmapsto", "h_{\\alpha(j_0) \\ldots \\alpha(j_n)}|_{V_{j_0 \\ldots j_n}}", "$$", "The cochain $g = (g_{j_0 \\ldots j_n})$ is not a cocycle", "in general but we know that its {\\v C}ech boundary $\\text{d}(g)$", "maps to zero in $\\check{\\mathcal{C}}^{n + 1}(\\mathcal{V}, \\mathcal{H})$", "(by the commutative diagram above and the fact that $h$ is a cocycle).", "Hence $\\text{d}(g)$ is a cocycle in", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{F})$.", "This allows us to define", "$$", "\\delta(\\overline{h}) = \\text{class of }\\text{d}(g)\\text{ in }", "\\check{H}^{n + 1}(\\mathcal{V}, \\mathcal{F})", "$$", "Now, given an element $\\xi \\in \\check{H}^n(X, \\mathcal{G})$", "we choose an open covering $\\mathcal{U}$ and an element", "$\\overline{h} \\in \\check{H}^n(\\mathcal{U}, \\mathcal{G})$", "mapping to $\\xi$ in the colimit defining {\\v C}ech cohomology.", "Then we choose $\\mathcal{V}$ and $g$ as above and set", "$\\delta(\\xi)$ equal to the image of $\\delta(\\overline{h})$", "in $\\check{H}^n(X, \\mathcal{F})$.", "At this point a lot of properties have to be checked, all of which", "are straightforward. For example, we need to check that our construction", "is independent of the choice of", "$\\mathcal{U}, \\overline{h}, \\mathcal{V}, \\alpha : J \\to I, g$.", "The class of $\\text{d}(g)$ is independent of the choice of the lifts", "$g_{i_0 \\ldots i_n}$ because the difference will be a coboundary.", "Independence of $\\alpha$ holds\\footnote{This is an important", "check because the nonuniqueness of $\\alpha$ is the only thing preventing", "us from taking the colimit of {\\v C}ech complexes over all open", "coverings of $X$ to get a short exact sequence of complexes computing", "{\\v C}ech cohomology.}", "because a different choice", "of $\\alpha$ determines homotopic vertical maps of complexes", "in the diagram above, see Section \\ref{section-refinements-cech}.", "For the other choices we use that given a finite collection", "of coverings of $X$ we can always find a covering refining all", "of them. We also need to check additivity which is shown in the same manner.", "Finally, we need to check that the maps", "$\\check{H}^n(X, -) \\to H^n(X, -)$ are compatible", "with boundary maps. To do this we choose injective", "resolutions", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{F} \\ar[r] \\ar[d] &", "\\mathcal{G} \\ar[r] \\ar[d] &", "\\mathcal{H} \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{I}_1^\\bullet \\ar[r] &", "\\mathcal{I}_2^\\bullet \\ar[r] &", "\\mathcal{I}_3^\\bullet \\ar[r] &", "0", "}", "$$", "as in Derived Categories, Lemma \\ref{derived-lemma-injective-resolution-ses}.", "This will give a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\ar[r] \\ar[d] &", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\ar[r] \\ar[d] &", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}_1^\\bullet))", "\\ar[r] &", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}_2^\\bullet))", "\\ar[r] &", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}_3^\\bullet))", "\\ar[r] &", "0", "}", "$$", "Here $\\mathcal{U}$ is an open covering as above and", "the vertical maps are those used to define the maps", "$\\check{H}^n(\\mathcal{U}, -) \\to H^n(X, -)$, see", "Lemma \\ref{lemma-cech-cohomology}.", "The bottom complex is exact as the sequence of", "complexes of injectives is termwise split exact.", "Hence the boundary map in cohomology is computed", "by the usual procedure for this lower exact sequence, see", "Homology, Lemma \\ref{homology-lemma-long-exact-sequence-cochain}.", "The same will be true after passing to the refinement", "$\\mathcal{V}$ where the boundary map for {\\v C}ech cohomology", "was defined. Hence the boundary maps agree because they", "use the same construction (whenever the first one is defined", "on an element in {\\v C}ech cohomology on a given covering).", "This finishes our discussion of the construction of", "the structure of a $\\delta$-functor on {\\v C}ech cohomology", "and why this structure is compatible with the given", "$\\delta$-functor structure on usual cohomology.", "\\medskip\\noindent", "Finally, we may apply Lemma \\ref{lemma-injective-trivial-cech}", "to see that higher {\\v C}ech cohomology is trivial on injective", "sheaves. Hence we see that {\\v C}ech cohomology is a universal", "$\\delta$-functor by", "Homology, Lemma \\ref{homology-lemma-efface-implies-universal}." ], "refs": [ "cohomology-lemma-cech-always", "derived-lemma-higher-derived-functors", "homology-lemma-uniqueness-universal-delta-functor", "topology-lemma-refine-covering", "derived-lemma-injective-resolution-ses", "cohomology-lemma-cech-cohomology", "homology-lemma-long-exact-sequence-cochain", "cohomology-lemma-injective-trivial-cech", "homology-lemma-efface-implies-universal" ], "ref_ids": [ 2076, 1869, 12053, 8245, 1856, 2052, 12061, 2051, 12052 ] } ], "ref_ids": [] }, { "id": 2078, "type": "theorem", "label": "cohomology-lemma-cohomology-of-closed", "categories": [ "cohomology" ], "title": "cohomology-lemma-cohomology-of-closed", "contents": [ "\\begin{reference}", "\\cite[Expose V bis, 4.1.3]{SGA4}", "\\end{reference}", "Let $X$ be a topological space. Let $Z \\subset X$ be a quasi-compact subset", "such that any two points of $Z$ have disjoint open neighbourhoods in $X$.", "For every abelian sheaf $\\mathcal{F}$ on $X$ the canonical", "map", "$$", "\\colim H^p(U, \\mathcal{F})", "\\longrightarrow", "H^p(Z, \\mathcal{F}|_Z)", "$$", "where the colimit is over open neighbourhoods $U$ of $Z$ in $X$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "We first prove this for $p = 0$. Injectivity follows from", "the definition of $\\mathcal{F}|_Z$ and holds in general", "(for any subset of any topological space $X$). Next, suppose that", "$s \\in H^0(Z, \\mathcal{F}|_Z)$. Then we can find opens $U_i \\subset X$", "such that $Z \\subset \\bigcup U_i$ and such that $s|_{Z \\cap U_i}$", "comes from $s_i \\in \\mathcal{F}(U_i)$. It follows that", "there exist opens $W_{ij} \\subset U_i \\cap U_j$ with", "$W_{ij} \\cap Z = U_i \\cap U_j \\cap Z$ such that", "$s_i|_{W_{ij}} = s_j|_{W_{ij}}$. Applying", "Topology, Lemma", "\\ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset}", "we find opens $V_i$ of $X$ such that $V_i \\subset U_i$ and", "such that $V_i \\cap V_j \\subset W_{ij}$. Hence we see that", "$s_i|_{V_i}$ glue to a section of $\\mathcal{F}$ over the", "open neighbourhood $\\bigcup V_i$ of $Z$.", "\\medskip\\noindent", "To finish the proof, it suffices to show that if $\\mathcal{I}$ is an", "injective abelian sheaf on $X$, then $H^p(Z, \\mathcal{I}|_Z) = 0$", "for $p > 0$. This follows using short exact sequences and dimension", "shifting; details omitted. Thus, suppose $\\overline{\\xi}$ is an element", "of $H^p(Z, \\mathcal{I}|_Z)$ for some $p > 0$.", "By Lemma \\ref{lemma-cech-Hausdorff-quasi-compact}", "the element $\\overline{\\xi}$ comes from", "$\\check{H}^p(\\mathcal{V}, \\mathcal{I}|_Z)$", "for some open covering $\\mathcal{V} : Z = \\bigcup V_i$ of $Z$.", "Say $\\overline{\\xi}$ is the image of the class of a cocycle", "$\\xi = (\\xi_{i_0 \\ldots i_p})$ in", "$\\check{\\mathcal{C}}^p(\\mathcal{V}, \\mathcal{I}|_Z)$.", "\\medskip\\noindent", "Let $\\mathcal{I}' \\subset \\mathcal{I}|_Z$ be the subpresheaf", "defined by the rule", "$$", "\\mathcal{I}'(V) =", "\\{s \\in \\mathcal{I}|_Z(V) \\mid", "\\exists (U, t),\\ U \\subset X\\text{ open},", "\\ t \\in \\mathcal{I}(U),\\ V = Z \\cap U,\\ s = t|_{Z \\cap U} \\}", "$$", "Then $\\mathcal{I}|_Z$ is the sheafification of $\\mathcal{I}'$.", "Thus for every $(p + 1)$-tuple $i_0 \\ldots i_p$ we can find an", "open covering $V_{i_0 \\ldots i_p} = \\bigcup W_{i_0 \\ldots i_p, k}$", "such that $\\xi_{i_0 \\ldots i_p}|_{W_{i_0 \\ldots i_p, k}}$ is", "a section of $\\mathcal{I}'$. Applying", "Topology, Lemma \\ref{topology-lemma-refine-covering}", "we may after refining $\\mathcal{V}$ assume that each", "$\\xi_{i_0 \\ldots i_p}$ is a section of the presheaf $\\mathcal{I}'$.", "\\medskip\\noindent", "Write $V_i = Z \\cap U_i$ for some opens $U_i \\subset X$.", "Since $\\mathcal{I}$ is flasque (Lemma \\ref{lemma-injective-flasque})", "and since $\\xi_{i_0 \\ldots i_p}$ is a section of $\\mathcal{I}'$", "for every $(p + 1)$-tuple $i_0 \\ldots i_p$ we can choose", "a section $s_{i_0 \\ldots i_p} \\in \\mathcal{I}(U_{i_0 \\ldots i_p})$", "which restricts to $\\xi_{i_0 \\ldots i_p}$ on", "$V_{i_0 \\ldots i_p} = Z \\cap U_{i_0 \\ldots i_p}$.", "(This appeal to injectives being flasque can be avoided by an", "additional application of", "Topology, Lemma", "\\ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset}.)", "Let $s = (s_{i_0 \\ldots i_p})$ be the corresponding cochain", "for the open covering $U = \\bigcup U_i$.", "Since $\\text{d}(\\xi) = 0$ we see that the sections", "$\\text{d}(s)_{i_0 \\ldots i_{p + 1}}$ restrict to zero", "on $Z \\cap U_{i_0 \\ldots i_{p + 1}}$. Hence, by the initial", "remarks of the proof, there exists open subsets", "$W_{i_0 \\ldots i_{p + 1}} \\subset U_{i_0 \\ldots i_{p + 1}}$", "with $Z \\cap W_{i_0 \\ldots i_{p + 1}} = Z \\cap U_{i_0 \\ldots i_{p + 1}}$", "such that $\\text{d}(s)_{i_0 \\ldots i_{p + 1}}|_{W_{i_0 \\ldots i_{p + 1}}} = 0$.", "By Topology, Lemma", "\\ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset}", "we can find $U'_i \\subset U_i$ such that $Z \\subset \\bigcup U'_i$", "and such that $U'_{i_0 \\ldots i_{p + 1}} \\subset W_{i_0 \\ldots i_{p + 1}}$.", "Then $s' = (s'_{i_0 \\ldots i_p})$ with", "$s'_{i_0 \\ldots i_p} = s_{i_0 \\ldots i_p}|_{U'_{i_0 \\ldots i_p}}$", "is a cocycle for $\\mathcal{I}$ for the open covering", "$U' = \\bigcup U'_i$ of an open neighbourhood of $Z$.", "Since $\\mathcal{I}$ has trivial higher {\\v C}ech cohomology groups", "(Lemma \\ref{lemma-injective-trivial-cech})", "we conclude that $s'$ is a coboundary. It follows that the image of", "$\\xi$ in the {\\v C}ech complex for the open covering", "$Z = \\bigcup Z \\cap U'_i$ is a coboundary and we are done." ], "refs": [ "topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset", "cohomology-lemma-cech-Hausdorff-quasi-compact", "topology-lemma-refine-covering", "cohomology-lemma-injective-flasque", "topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset", "topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset", "cohomology-lemma-injective-trivial-cech" ], "ref_ids": [ 8247, 2077, 8245, 2063, 8247, 8247, 2051 ] } ], "ref_ids": [] }, { "id": 2079, "type": "theorem", "label": "cohomology-lemma-base-change-map-flat-case", "categories": [ "cohomology" ], "title": "cohomology-lemma-base-change-map-flat-case", "contents": [ "Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "S' \\ar[r]^g &", "S", "}", "$$", "be a commutative diagram of ringed spaces.", "Let $\\mathcal{F}^\\bullet$ be a bounded below complex of", "$\\mathcal{O}_X$-modules.", "Assume both $g$ and $g'$ are flat.", "Then there exists a canonical base change map", "$$", "g^*Rf_*\\mathcal{F}^\\bullet", "\\longrightarrow", "R(f')_*(g')^*\\mathcal{F}^\\bullet", "$$", "in $D^{+}(S')$." ], "refs": [], "proofs": [ { "contents": [ "Choose injective resolutions $\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$", "and $(g')^*\\mathcal{F}^\\bullet \\to \\mathcal{J}^\\bullet$.", "By Lemma \\ref{lemma-pushforward-injective-flat} we see that", "$(g')_*\\mathcal{J}^\\bullet$ is a complex of injectives representing", "$R(g')_*(g')^*\\mathcal{F}^\\bullet$. Hence by", "Derived Categories, Lemmas \\ref{derived-lemma-morphisms-lift}", "and \\ref{derived-lemma-morphisms-equal-up-to-homotopy}", "the arrow $\\beta$ in the diagram", "$$", "\\xymatrix{", "(g')_*(g')^*\\mathcal{F}^\\bullet \\ar[r] &", "(g')_*\\mathcal{J}^\\bullet \\\\", "\\mathcal{F}^\\bullet \\ar[u]^{adjunction} \\ar[r] &", "\\mathcal{I}^\\bullet \\ar[u]_\\beta", "}", "$$", "exists and is unique up to homotopy.", "Pushing down to $S$ we get", "$$", "f_*\\beta :", "f_*\\mathcal{I}^\\bullet", "\\longrightarrow", "f_*(g')_*\\mathcal{J}^\\bullet", "=", "g_*(f')_*\\mathcal{J}^\\bullet", "$$", "By adjunction of $g^*$ and $g_*$ we get a map of complexes", "$g^*f_*\\mathcal{I}^\\bullet \\to (f')_*\\mathcal{J}^\\bullet$.", "Note that this map is unique up to homotopy since the only", "choice in the whole process was the choice of the map $\\beta$", "and everything was done on the level of complexes." ], "refs": [ "cohomology-lemma-pushforward-injective-flat", "derived-lemma-morphisms-lift", "derived-lemma-morphisms-equal-up-to-homotopy" ], "ref_ids": [ 2061, 1853, 1854 ] } ], "ref_ids": [] }, { "id": 2080, "type": "theorem", "label": "cohomology-lemma-proper-base-change", "categories": [ "cohomology" ], "title": "cohomology-lemma-proper-base-change", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of", "ringed spaces. Let $y \\in Y$. Assume that", "\\begin{enumerate}", "\\item $f$ is closed,", "\\item $f$ is separated, and", "\\item $f^{-1}(y)$ is quasi-compact.", "\\end{enumerate}", "Then for $E$ in $D^+(\\mathcal{O}_X)$", "we have $(Rf_*E)_y = R\\Gamma(f^{-1}(y), E|_{f^{-1}(y)})$ in", "$D^+(\\mathcal{O}_{Y, y})$." ], "refs": [], "proofs": [ { "contents": [ "The base change map of Lemma \\ref{lemma-base-change-map-flat-case}", "gives a canonical map $(Rf_*E)_y \\to R\\Gamma(f^{-1}(y), E|_{f^{-1}(y)})$.", "To prove this map is an isomorphism, we represent $E$ by a bounded", "below complex of injectives $\\mathcal{I}^\\bullet$.", "Set $Z = f^{-1}(\\{y\\})$. The assumptions of", "Lemma \\ref{lemma-cohomology-of-closed}", "are satisfied, see Topology, Lemma \\ref{topology-lemma-separated}.", "Hence the restrictions", "$\\mathcal{I}^n|_Z$ are acyclic for $\\Gamma(Z, -)$.", "Thus $R\\Gamma(Z, E|_Z)$ is represented by the", "complex $\\Gamma(Z, \\mathcal{I}^\\bullet|_Z)$, see", "Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity}.", "In other words, we have to show the map", "$$", "\\colim_V \\mathcal{I}^\\bullet(f^{-1}(V))", "\\longrightarrow", "\\Gamma(Z, \\mathcal{I}^\\bullet|_Z)", "$$", "is an isomorphism. Using Lemma \\ref{lemma-cohomology-of-closed}", "we see that it suffices to show that the collection of open neighbourhoods", "$f^{-1}(V)$ of $Z = f^{-1}(\\{y\\})$", "is cofinal in the system of all open neighbourhoods.", "If $f^{-1}(\\{y\\}) \\subset U$ is an open neighbourhood, then as $f$ is closed", "the set $V = Y \\setminus f(X \\setminus U)$ is an open neighbourhood", "of $y$ with $f^{-1}(V) \\subset U$. This proves the lemma." ], "refs": [ "cohomology-lemma-base-change-map-flat-case", "cohomology-lemma-cohomology-of-closed", "topology-lemma-separated", "derived-lemma-leray-acyclicity", "cohomology-lemma-cohomology-of-closed" ], "ref_ids": [ 2079, 2078, 8194, 1844, 2078 ] } ], "ref_ids": [] }, { "id": 2081, "type": "theorem", "label": "cohomology-lemma-proper-base-change-sheaves-of-sets", "categories": [ "cohomology" ], "title": "cohomology-lemma-proper-base-change-sheaves-of-sets", "contents": [ "Consider a cartesian square of topological spaces", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_-{g'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "Assume that $f$ is proper and separated. Then", "$g^{-1}f_*\\mathcal{F} = f'_*(g')^{-1}\\mathcal{F}$", "for any sheaf of sets $\\mathcal{F}$ on $X$." ], "refs": [], "proofs": [ { "contents": [ "We argue exactly as in the proof of Theorem \\ref{theorem-proper-base-change}", "and we find it suffices to show", "$(f_*\\mathcal{F})_y = \\Gamma(X_y, \\mathcal{F}|_{X_y})$.", "Then we argue as in Lemma \\ref{lemma-proper-base-change}", "to reduce this to the $p = 0$ case of Lemma \\ref{lemma-cohomology-of-closed}", "for sheaves of sets. The first part of the proof of", "Lemma \\ref{lemma-cohomology-of-closed}", "works for sheaves of sets and this finishes the proof.", "Some details omitted." ], "refs": [ "cohomology-theorem-proper-base-change", "cohomology-lemma-proper-base-change", "cohomology-lemma-cohomology-of-closed", "cohomology-lemma-cohomology-of-closed" ], "ref_ids": [ 2031, 2080, 2078, 2078 ] } ], "ref_ids": [] }, { "id": 2082, "type": "theorem", "label": "cohomology-lemma-quasi-separated-cohomology-colimit", "categories": [ "cohomology" ], "title": "cohomology-lemma-quasi-separated-cohomology-colimit", "contents": [ "Let $X$ be a ringed space. Assume that the underlying topological space", "of $X$ has the following properties:", "\\begin{enumerate}", "\\item there exists a basis of quasi-compact open subsets, and", "\\item the intersection of any two quasi-compact opens is quasi-compact.", "\\end{enumerate}", "Then for any directed system $(\\mathcal{F}_i, \\varphi_{ii'})$", "of sheaves of $\\mathcal{O}_X$-modules and for any quasi-compact open", "$U \\subset X$ the canonical map", "$$", "\\colim_i H^q(U, \\mathcal{F}_i)", "\\longrightarrow", "H^q(U, \\colim_i \\mathcal{F}_i)", "$$", "is an isomorphism for every $q \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "It is important in this proof to argue for all quasi-compact opens", "$U \\subset X$ at the same time.", "The result is true for $i = 0$ and any quasi-compact open $U \\subset X$ by", "Sheaves, Lemma \\ref{sheaves-lemma-directed-colimits-sections}", "(combined with", "Topology, Lemma \\ref{topology-lemma-topology-quasi-separated-scheme}).", "Assume that we have proved the result for all $q \\leq q_0$ and let", "us prove the result for $q = q_0 + 1$.", "\\medskip\\noindent", "By our conventions on directed systems the index set $I$ is directed,", "and any system of $\\mathcal{O}_X$-modules $(\\mathcal{F}_i, \\varphi_{ii'})$", "over $I$ is directed.", "By Injectives, Lemma \\ref{injectives-lemma-sheaves-modules-space} the category", "of $\\mathcal{O}_X$-modules has functorial injective embeddings.", "Thus for any system $(\\mathcal{F}_i, \\varphi_{ii'})$ there exists a", "system $(\\mathcal{I}_i, \\varphi_{ii'})$ with each $\\mathcal{I}_i$ an", "injective $\\mathcal{O}_X$-module and a morphism of systems given", "by injective $\\mathcal{O}_X$-module maps", "$\\mathcal{F}_i \\to \\mathcal{I}_i$. Denote $\\mathcal{Q}_i$ the", "cokernel so that we have short exact sequences", "$$", "0 \\to", "\\mathcal{F}_i \\to", "\\mathcal{I}_i \\to", "\\mathcal{Q}_i \\to 0.", "$$", "We claim that the sequence", "$$", "0 \\to", "\\colim_i \\mathcal{F}_i \\to", "\\colim_i \\mathcal{I}_i \\to", "\\colim_i \\mathcal{Q}_i \\to 0.", "$$", "is also a short exact sequence of $\\mathcal{O}_X$-modules.", "We may check this on stalks. By", "Sheaves, Sections \\ref{sheaves-section-limits-presheaves}", "and \\ref{sheaves-section-limits-sheaves}", "taking stalks commutes with colimits. Since a directed colimit", "of short exact sequences of abelian groups is short exact", "(see Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact})", "we deduce the result. We claim that", "$H^q(U, \\colim_i \\mathcal{I}_i) = 0$ for all quasi-compact", "open $U \\subset X$ and all $q \\geq 1$. Accepting this claim", "for the moment consider the diagram", "$$", "\\xymatrix{", "\\colim_i H^{q_0}(U, \\mathcal{I}_i) \\ar[d] \\ar[r] &", "\\colim_i H^{q_0}(U, \\mathcal{Q}_i) \\ar[d] \\ar[r] &", "\\colim_i H^{q_0 + 1}(U, \\mathcal{F}_i) \\ar[d] \\ar[r] &", "0 \\ar[d] \\\\", "H^{q_0}(U, \\colim_i \\mathcal{I}_i) \\ar[r] &", "H^{q_0}(U, \\colim_i \\mathcal{Q}_i) \\ar[r] &", "H^{q_0 + 1}(U, \\colim_i \\mathcal{F}_i) \\ar[r] &", "0", "}", "$$", "The zero at the lower right corner comes from the claim and the", "zero at the upper right corner comes from the fact that the sheaves", "$\\mathcal{I}_i$ are injective.", "The top row is exact by an application of", "Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}.", "Hence by the snake lemma we deduce the", "result for $q = q_0 + 1$.", "\\medskip\\noindent", "It remains to show that the claim is true. We will use", "Lemma \\ref{lemma-cech-vanish-basis}.", "Let $\\mathcal{B}$ be the collection of all quasi-compact open", "subsets of $X$. This is a basis for the topology on $X$ by assumption.", "Let $\\text{Cov}$ be the collection of finite open coverings", "$\\mathcal{U} : U = \\bigcup_{j = 1, \\ldots, m} U_j$ with each", "of $U$, $U_j$ quasi-compact open in $X$. By the result for $q = 0$", "we see that for $\\mathcal{U} \\in \\text{Cov}$ we have", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\colim_i \\mathcal{I}_i)", "=", "\\colim_i \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}_i)", "$$", "because all the multiple intersections $U_{j_0 \\ldots j_p}$", "are quasi-compact. By Lemma \\ref{lemma-injective-trivial-cech}", "each of the complexes in the colimit of {\\v C}ech complexes is", "acyclic in degree $\\geq 1$. Hence by", "Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}", "we see that also the {\\v C}ech complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\colim_i \\mathcal{I}_i)$", "is acyclic in degrees $\\geq 1$. In other words we see that", "$\\check{H}^p(\\mathcal{U}, \\colim_i \\mathcal{I}_i) = 0$", "for all $p \\geq 1$. Thus the assumptions of", "Lemma \\ref{lemma-cech-vanish-basis} are satisfied and the claim follows." ], "refs": [ "sheaves-lemma-directed-colimits-sections", "topology-lemma-topology-quasi-separated-scheme", "injectives-lemma-sheaves-modules-space", "algebra-lemma-directed-colimit-exact", "algebra-lemma-directed-colimit-exact", "cohomology-lemma-cech-vanish-basis", "cohomology-lemma-injective-trivial-cech", "algebra-lemma-directed-colimit-exact", "cohomology-lemma-cech-vanish-basis" ], "ref_ids": [ 14526, 8333, 7775, 343, 343, 2059, 2051, 343, 2059 ] } ], "ref_ids": [] }, { "id": 2083, "type": "theorem", "label": "cohomology-lemma-colimit", "categories": [ "cohomology" ], "title": "cohomology-lemma-colimit", "contents": [ "In the situation discussed above.", "Let $i \\in \\Ob(\\mathcal{I})$ and let $U_i \\subset X_i$ be quasi-compact open.", "Then", "$$", "\\colim_{a : j \\to i} H^p(f_a^{-1}(U_i), \\mathcal{F}_j) =", "H^p(p_i^{-1}(U_i), \\mathcal{F})", "$$", "for all $p \\geq 0$. In particular we have", "$H^p(X, \\mathcal{F}) = \\colim H^p(X_i, \\mathcal{F}_i)$." ], "refs": [], "proofs": [ { "contents": [ "The case $p = 0$ is Sheaves, Lemma \\ref{sheaves-lemma-descend-opens}.", "\\medskip\\noindent", "In this paragraph we show that we can find a map of systems", "$(\\gamma_i) : (\\mathcal{F}_i, \\varphi_a) \\to (\\mathcal{G}_i, \\psi_a)$", "with $\\mathcal{G}_i$ an injective abelian sheaf and $\\gamma_i$ injective.", "For each $i$ we pick an injection $\\mathcal{F}_i \\to \\mathcal{I}_i$", "where $\\mathcal{I}_i$ is an injective abelian sheaf on $X_i$.", "Then we can consider the family of maps", "$$", "\\gamma_i :", "\\mathcal{F}_i", "\\longrightarrow", "\\prod\\nolimits_{b : k \\to i} f_{b, *}\\mathcal{I}_k = \\mathcal{G}_i", "$$", "where the component maps are the maps adjoint to the maps", "$f_b^{-1}\\mathcal{F}_i \\to \\mathcal{F}_k \\to \\mathcal{I}_k$.", "For $a : j \\to i$ in $\\mathcal{I}$ there is a canonical map", "$$", "\\psi_a : f_a^{-1}\\mathcal{G}_i \\to \\mathcal{G}_j", "$$", "whose components are the canonical maps", "$f_b^{-1}f_{a \\circ b, *}\\mathcal{I}_k \\to f_{b, *}\\mathcal{I}_k$", "for $b : k \\to j$. Thus we find an injection", "$\\{\\gamma_i\\} : \\{\\mathcal{F}_i, \\varphi_a) \\to (\\mathcal{G}_i, \\psi_a)$", "of systems of abelian sheaves. Note that $\\mathcal{G}_i$ is an injective", "sheaf of abelian groups on $X_i$, see", "Lemma \\ref{lemma-pushforward-injective-flat} and", "Homology, Lemma \\ref{homology-lemma-product-injectives}.", "This finishes the construction.", "\\medskip\\noindent", "Arguing exactly as in the proof of", "Lemma \\ref{lemma-quasi-separated-cohomology-colimit}", "we see that it suffices to prove that", "$H^p(X, \\colim f_i^{-1}\\mathcal{G}_i) = 0$ for $p > 0$.", "\\medskip\\noindent", "Set $\\mathcal{G} = \\colim f_i^{-1}\\mathcal{G}_i$.", "To show vanishing of cohomology of $\\mathcal{G}$ on every quasi-compact", "open of $X$, it suffices to show that the {\\v C}ech cohomology of", "$\\mathcal{G}$ for any covering $\\mathcal{U}$ of a quasi-compact open of", "$X$ by finitely many quasi-compact opens is zero, see", "Lemma \\ref{lemma-cech-vanish-basis}.", "Such a covering is the inverse by $p_i$ of such a covering $\\mathcal{U}_i$", "on the space $X_i$ for some $i$ by", "Topology, Lemma \\ref{topology-lemma-descend-opens}. We have", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{G}) =", "\\colim_{a : j \\to i}", "\\check{\\mathcal{C}}^\\bullet(f_a^{-1}(\\mathcal{U}_i), \\mathcal{G}_j)", "$$", "by the case $p = 0$. The right hand side is a filtered colimit of", "complexes each of which is acyclic in positive degrees by", "Lemma \\ref{lemma-injective-trivial-cech}. Thus we conclude by", "Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}." ], "refs": [ "sheaves-lemma-descend-opens", "cohomology-lemma-pushforward-injective-flat", "homology-lemma-product-injectives", "cohomology-lemma-quasi-separated-cohomology-colimit", "cohomology-lemma-cech-vanish-basis", "topology-lemma-descend-opens", "cohomology-lemma-injective-trivial-cech", "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 14528, 2061, 12113, 2082, 2059, 8322, 2051, 343 ] } ], "ref_ids": [] }, { "id": 2084, "type": "theorem", "label": "cohomology-lemma-cohomology-and-closed-immersions", "categories": [ "cohomology" ], "title": "cohomology-lemma-cohomology-and-closed-immersions", "contents": [ "Let $i : Z \\to X$ be a closed immersion of topological spaces.", "For any abelian sheaf $\\mathcal{F}$ on $Z$ we have", "$H^p(Z, \\mathcal{F}) = H^p(X, i_*\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "This is true because $i_*$ is exact (see", "Modules, Lemma \\ref{modules-lemma-i-star-exact}),", "and hence $R^pi_* = 0$ as a functor", "(Derived Categories, Lemma \\ref{derived-lemma-right-derived-exact-functor}).", "Thus we may apply Lemma \\ref{lemma-apply-Leray}." ], "refs": [ "modules-lemma-i-star-exact", "derived-lemma-right-derived-exact-functor", "cohomology-lemma-apply-Leray" ], "ref_ids": [ 13232, 1845, 2071 ] } ], "ref_ids": [] }, { "id": 2085, "type": "theorem", "label": "cohomology-lemma-irreducible-constant-cohomology-zero", "categories": [ "cohomology" ], "title": "cohomology-lemma-irreducible-constant-cohomology-zero", "contents": [ "Let $X$ be an irreducible topological space.", "Then $H^p(X, \\underline{A}) = 0$ for all $p > 0$", "and any abelian group $A$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\underline{A}$ is the constant sheaf as defined", "in Sheaves, Definition \\ref{sheaves-definition-constant-sheaf}.", "It is clear that for any nonempty", "open $U \\subset X$ we have $\\underline{A}(U) = A$ as $X$ is", "irreducible (and hence $U$ is connected).", "We will show that the higher {\\v C}ech cohomology groups", "$\\check{H}^p(\\mathcal{U}, \\underline{A})$ are zero for", "any open covering $\\mathcal{U} : U = \\bigcup_{i\\in I} U_i$", "of an open $U \\subset X$. Then the lemma will follow", "from Lemma \\ref{lemma-cech-vanish}.", "\\medskip\\noindent", "Recall that the value of an abelian", "sheaf on the empty open set is $0$. Hence we may clearly assume", "$U_i \\not = \\emptyset$ for all $i \\in I$. In this case we see", "that $U_i \\cap U_{i'} \\not = \\emptyset$ for all $i, i' \\in I$.", "Hence we see that the {\\v C}ech complex is simply the complex", "$$", "\\prod_{i_0 \\in I} A \\to", "\\prod_{(i_0, i_1) \\in I^2} A \\to", "\\prod_{(i_0, i_1, i_2) \\in I^3} A \\to", "\\ldots", "$$", "We have to see this has trivial higher cohomology groups.", "We can see this for example because this is the {\\v C}ech complex for the", "covering of a $1$-point space and {\\v C}ech cohomology agrees with cohomology", "on such a space. (You can also directly verify it", "by writing an explicit homotopy.)" ], "refs": [ "sheaves-definition-constant-sheaf", "cohomology-lemma-cech-vanish" ], "ref_ids": [ 14566, 2058 ] } ], "ref_ids": [] }, { "id": 2086, "type": "theorem", "label": "cohomology-lemma-subsheaf-of-constant-sheaf", "categories": [ "cohomology" ], "title": "cohomology-lemma-subsheaf-of-constant-sheaf", "contents": [ "\\begin{reference}", "\\cite[Page 168]{Tohoku}.", "\\end{reference}", "Let $X$ be a topological space such that the intersection of any", "two quasi-compact opens is quasi-compact. Let", "$\\mathcal{F} \\subset \\underline{\\mathbf{Z}}$", "be a subsheaf generated by finitely many sections over quasi-compact opens.", "Then there exists a finite filtration", "$$", "(0) = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset \\ldots \\subset", "\\mathcal{F}_n = \\mathcal{F}", "$$", "by abelian subsheaves such that for each $0 < i \\leq n$", "there exists a short exact sequence", "$$", "0 \\to j'_!\\underline{\\mathbf{Z}}_V \\to j_!\\underline{\\mathbf{Z}}_U \\to", "\\mathcal{F}_i/\\mathcal{F}_{i - 1} \\to 0", "$$", "with $j : U \\to X$ and $j' : V \\to X$ the inclusion of quasi-compact opens", "into $X$." ], "refs": [], "proofs": [ { "contents": [ "Say $\\mathcal{F}$ is generated by the sections $s_1, \\ldots, s_t$ over the", "quasi-compact opens $U_1, \\ldots, U_t$. Since $U_i$ is quasi-compact and", "$s_i$ a locally constant function to $\\mathbf{Z}$ we may assume, after", "possibly replacing $U_i$ by the parts of a finite decomposition into open", "and closed subsets, that $s_i$ is a constant section.", "Say $s_i = n_i$ with $n_i \\in \\mathbf{Z}$. Of course we can remove", "$(U_i, n_i)$ from the list if $n_i = 0$. Flipping signs if necessary", "we may also assume $n_i > 0$. Next, for any subset $I \\subset \\{1, \\ldots, t\\}$", "we may add $\\bigcup_{i \\in I} U_i$ and $\\gcd(n_i, i \\in I)$ to the list.", "After doing this we see that our list $(U_1, n_1), \\ldots, (U_t, n_t)$", "satisfies the following property:", "For $x \\in X$ set $I_x = \\{i \\in \\{1, \\ldots, t\\} \\mid x \\in U_i\\}$.", "Then $\\gcd(n_i, i \\in I_x)$ is attained by $n_i$ for some $i \\in I_x$.", "\\medskip\\noindent", "As our filtration we take $\\mathcal{F}_0 = (0)$ and", "$\\mathcal{F}_n$ generated by the sections $n_i$ over $U_i$ for those", "$i$ such that $n_i \\leq n$. It is clear that", "$\\mathcal{F}_n = \\mathcal{F}$ for $n \\gg 0$. Moreover, the quotient", "$\\mathcal{F}_n/\\mathcal{F}_{n - 1}$ is generated by the section", "$n$ over $U = \\bigcup_{n_i \\leq n} U_i$ and the kernel of the map", "$j_!\\underline{\\mathbf{Z}}_U \\to \\mathcal{F}_n/\\mathcal{F}_{n - 1}$", "is generated by the section $n$ over $V = \\bigcup_{n_i \\leq n - 1} U_i$.", "Thus a short exact sequence as in the statement of the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2087, "type": "theorem", "label": "cohomology-lemma-vanishing-generated-one-section", "categories": [ "cohomology" ], "title": "cohomology-lemma-vanishing-generated-one-section", "contents": [ "\\begin{reference}", "This is a special case of \\cite[Proposition 3.6.1]{Tohoku}.", "\\end{reference}", "Let $X$ be a topological space. Let $d \\geq 0$ be an integer. Assume", "\\begin{enumerate}", "\\item $X$ is quasi-compact,", "\\item the quasi-compact opens form a basis for $X$, and", "\\item the intersection of two quasi-compact opens is quasi-compact.", "\\item $H^p(X, j_!\\underline{\\mathbf{Z}}_U) = 0$ for all $p > d$", "and any quasi-compact open $j : U \\to X$.", "\\end{enumerate}", "Then $H^p(X, \\mathcal{F}) = 0$ for all $p > d$", "and any abelian sheaf $\\mathcal{F}$ on $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $S = \\coprod_{U \\subset X} \\mathcal{F}(U)$ where $U$ runs over the", "quasi-compact opens of $X$.", "For any finite subset $A = \\{s_1, \\ldots, s_n\\} \\subset S$,", "let $\\mathcal{F}_A$ be the subsheaf of $\\mathcal{F}$ generated", "by all $s_i$ (see", "Modules, Definition \\ref{modules-definition-generated-by-local-sections}).", "Note that if $A \\subset A'$, then $\\mathcal{F}_A \\subset \\mathcal{F}_{A'}$.", "Hence $\\{\\mathcal{F}_A\\}$ forms a system over the", "directed partially ordered set of finite subsets of $S$.", "By Modules, Lemma \\ref{modules-lemma-generated-by-local-sections-stalk}", "it is clear that", "$$", "\\colim_A \\mathcal{F}_A = \\mathcal{F}", "$$", "by looking at stalks. By", "Lemma \\ref{lemma-quasi-separated-cohomology-colimit} we have", "$$", "H^p(X, \\mathcal{F}) =", "\\colim_A H^p(X, \\mathcal{F}_A)", "$$", "Hence it suffices to prove the vanishing for the abelian sheaves", "$\\mathcal{F}_A$. In other words, it suffices to prove the", "result when $\\mathcal{F}$ is generated by finitely many local sections", "over quasi-compact opens of $X$.", "\\medskip\\noindent", "Suppose that $\\mathcal{F}$ is generated by the local sections", "$s_1, \\ldots, s_n$. Let $\\mathcal{F}' \\subset \\mathcal{F}$", "be the subsheaf generated by $s_1, \\ldots, s_{n - 1}$.", "Then we have a short exact sequence", "$$", "0 \\to \\mathcal{F}' \\to \\mathcal{F} \\to \\mathcal{F}/\\mathcal{F}' \\to 0", "$$", "From the long exact sequence of cohomology we see that it suffices", "to prove the vanishing for the abelian sheaves $\\mathcal{F}'$", "and $\\mathcal{F}/\\mathcal{F}'$ which are generated by fewer than", "$n$ local sections. Hence it suffices to prove the vanishing", "for sheaves generated by at most one local section. These sheaves", "are exactly the quotients of the sheaves $j_!\\underline{\\mathbf{Z}}_U$", "where $U$ is a quasi-compact open of $X$.", "\\medskip\\noindent", "Assume now that we have a short exact sequence", "$$", "0 \\to \\mathcal{K} \\to j_!\\underline{\\mathbf{Z}}_U \\to \\mathcal{F} \\to 0", "$$", "with $U$ quasi-compact open in $X$.", "It suffices to show that $H^q(X, \\mathcal{K})$ is zero for $q \\geq d + 1$.", "As above we can write $\\mathcal{K}$ as the filtered colimit of", "subsheaves $\\mathcal{K}'$ generated by finitely many sections over", "quasi-compact opens. Then $\\mathcal{F}$ is the filtered colimit of the", "sheaves $j_!\\underline{\\mathbf{Z}}_U/\\mathcal{K}'$. In this way we", "reduce to the case that $\\mathcal{K}$ is generated by finitely many", "sections over quasi-compact opens. Note that $\\mathcal{K}$", "is a subsheaf of $\\underline{\\mathbf{Z}}_X$. Thus by", "Lemma \\ref{lemma-subsheaf-of-constant-sheaf} there exists a finite", "filtration of $\\mathcal{K}$ whose successive quotients $\\mathcal{Q}$ fit", "into a short exact sequence", "$$", "0 \\to j''_!\\underline{\\mathbf{Z}}_W \\to", "j'_!\\underline{\\mathbf{Z}}_V \\to \\mathcal{Q} \\to 0", "$$", "with $j'' : W \\to X$ and $j' : V \\to X$ the inclusions of quasi-compact opens.", "Hence the vanishing of $H^p(X, \\mathcal{Q})$ for $p > d$ follows", "from our assumption (in the lemma) on the vanishing of the cohomology groups", "of $j''_!\\underline{\\mathbf{Z}}_W$ and $j'_!\\underline{\\mathbf{Z}}_V$.", "Returning to $\\mathcal{K}$ this, via an induction argument using the", "long exact cohomology sequence, implies the desired vanishing for it as well." ], "refs": [ "modules-definition-generated-by-local-sections", "modules-lemma-generated-by-local-sections-stalk", "cohomology-lemma-quasi-separated-cohomology-colimit", "cohomology-lemma-subsheaf-of-constant-sheaf" ], "ref_ids": [ 13333, 13229, 2082, 2086 ] } ], "ref_ids": [] }, { "id": 2088, "type": "theorem", "label": "cohomology-lemma-subsheaf-irreducible", "categories": [ "cohomology" ], "title": "cohomology-lemma-subsheaf-irreducible", "contents": [ "Let $X$ be an irreducible topological space.", "Let $\\mathcal{H} \\subset \\underline{\\mathbf{Z}}$ be", "an abelian subsheaf of the constant sheaf.", "Then there exists a nonempty open $U \\subset X$ such", "that $\\mathcal{H}|_U = \\underline{d\\mathbf{Z}}_U$", "for some $d \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\underline{\\mathbf{Z}}(V) = \\mathbf{Z}$", "for any nonempty open $V$ of $X$ (see proof of", "Lemma \\ref{lemma-irreducible-constant-cohomology-zero}).", "If $\\mathcal{H} = 0$, then the lemma holds with $d = 0$.", "If $\\mathcal{H} \\not = 0$, then there exists a nonempty open", "$U \\subset X$ such that $\\mathcal{H}(U) \\not = 0$.", "Say $\\mathcal{H}(U) = n\\mathbf{Z}$ for some $n \\geq 1$.", "Hence we see that", "$\\underline{n\\mathbf{Z}}_U", "\\subset \\mathcal{H}|_U \\subset", "\\underline{\\mathbf{Z}}_U$. If the first inclusion is strict we", "can find a nonempty $U' \\subset U$ and an integer $1 \\leq n' < n$", "such that", "$\\underline{n'\\mathbf{Z}}_{U'}", "\\subset \\mathcal{H}|_{U'} \\subset", "\\underline{\\mathbf{Z}}_{U'}$.", "This process has to stop after a finite number of steps, and", "hence we get the lemma." ], "refs": [ "cohomology-lemma-irreducible-constant-cohomology-zero" ], "ref_ids": [ 2085 ] } ], "ref_ids": [] }, { "id": 2089, "type": "theorem", "label": "cohomology-lemma-sections-with-support-acyclic", "categories": [ "cohomology" ], "title": "cohomology-lemma-sections-with-support-acyclic", "contents": [ "Let $i : Z \\to X$ be the inclusion of a closed subset.", "Let $\\mathcal{I}$ be an injective abelian sheaf on $X$.", "Then $\\mathcal{H}_Z(\\mathcal{I})$ is an injective abelian sheaf on $Z$." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}", "as $\\mathcal{H}_Z(-)$ is right adjoint to the exact functor $i_*$.", "See Modules, Lemmas \\ref{modules-lemma-i-star-exact} and", "\\ref{modules-lemma-i-star-right-adjoint}." ], "refs": [ "homology-lemma-adjoint-preserve-injectives", "modules-lemma-i-star-exact", "modules-lemma-i-star-right-adjoint" ], "ref_ids": [ 12116, 13232, 13233 ] } ], "ref_ids": [] }, { "id": 2090, "type": "theorem", "label": "cohomology-lemma-cohomology-of-neighbourhoods-of-closed", "categories": [ "cohomology" ], "title": "cohomology-lemma-cohomology-of-neighbourhoods-of-closed", "contents": [ "Let $X$ be a spectral space. Let $\\mathcal{F}$ be an abelian sheaf on $X$.", "Let $E \\subset X$ be a quasi-compact subset. Let $W \\subset X$ be the set of", "points of $X$ which specialize to a point of $E$.", "\\begin{enumerate}", "\\item $H^p(W, \\mathcal{F}|_W) = \\colim H^p(U, \\mathcal{F})$", "where the colimit is over quasi-compact open neighbourhoods of $E$,", "\\item $H^p(W \\setminus E, \\mathcal{F}|_{W \\setminus E}) =", "\\colim H^p(U \\setminus E, \\mathcal{F}|_{U \\setminus E})$", "if $E$ is a constructible subset.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "From Topology, Lemma \\ref{topology-lemma-make-spectral-space}", "we see that $W = \\lim U$ where the limit is over the quasi-compact", "opens containing $E$. Each $U$ is a spectral space by", "Topology, Lemma \\ref{topology-lemma-spectral-sub}.", "Thus we may apply Lemma \\ref{lemma-colimit} to conclude that (1) holds.", "The same proof works for part (2) except we use", "Topology, Lemma \\ref{topology-lemma-make-spectral-space-minus}." ], "refs": [ "topology-lemma-make-spectral-space", "topology-lemma-spectral-sub", "cohomology-lemma-colimit", "topology-lemma-make-spectral-space-minus" ], "ref_ids": [ 8323, 8306, 2083, 8324 ] } ], "ref_ids": [] }, { "id": 2091, "type": "theorem", "label": "cohomology-lemma-proper-base-change-spectral", "categories": [ "cohomology" ], "title": "cohomology-lemma-proper-base-change-spectral", "contents": [ "Let $f : X \\to Y$ be a spectral map of spectral spaces. Let $y \\in Y$.", "Let $E \\subset Y$ be the set of points specializing to $y$.", "Let $\\mathcal{F}$ be an abelian sheaf on $X$.", "Then $(R^pf_*\\mathcal{F})_y = H^p(f^{-1}(E), \\mathcal{F}|_{f^{-1}(E)})$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $E = \\bigcap V$ where $V$ runs over the quasi-compact", "open neighbourhoods of $y$ in $Y$. Hence $f^{-1}(E) = \\bigcap f^{-1}(V)$.", "This implies that $f^{-1}(E) = \\lim f^{-1}(V)$ as topological spaces.", "Since $f$ is spectral, each $f^{-1}(V)$ is a spectral space too", "(Topology, Lemma \\ref{topology-lemma-spectral-sub}).", "We conclude that $f^{-1}(E)$ is a spectral space and that", "$$", "H^p(f^{-1}(E), \\mathcal{F}|_{f^{-1}(E)}) =", "\\colim H^p(f^{-1}(V), \\mathcal{F})", "$$", "by Lemma \\ref{lemma-colimit}. On the other hand, the stalk of", "$R^pf_*\\mathcal{F}$ at $y$ is given by the colimit on the right." ], "refs": [ "topology-lemma-spectral-sub", "cohomology-lemma-colimit" ], "ref_ids": [ 8306, 2083 ] } ], "ref_ids": [] }, { "id": 2092, "type": "theorem", "label": "cohomology-lemma-vanishing-for-profinite", "categories": [ "cohomology" ], "title": "cohomology-lemma-vanishing-for-profinite", "contents": [ "Let $X$ be a profinite topological space. Then $H^q(X, \\mathcal{F}) = 0$", "for all $q > 0$ and all abelian sheaves $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Any open covering of $X$ can be refined by a finite disjoint union", "decomposition with open parts, see", "Topology, Lemma \\ref{topology-lemma-profinite-refine-open-covering}.", "Hence if $\\mathcal{F} \\to \\mathcal{G}$ is a surjection of abelian", "sheaves on $X$, then $\\mathcal{F}(X) \\to \\mathcal{G}(X)$ is surjective.", "In other words, the global sections functor is an exact functor.", "Therefore its higher derived functors are zero, see", "Derived Categories, Lemma \\ref{derived-lemma-right-derived-exact-functor}." ], "refs": [ "topology-lemma-profinite-refine-open-covering", "derived-lemma-right-derived-exact-functor" ], "ref_ids": [ 8301, 1845 ] } ], "ref_ids": [] }, { "id": 2093, "type": "theorem", "label": "cohomology-lemma-ordered-alternating", "categories": [ "cohomology" ], "title": "cohomology-lemma-ordered-alternating", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.", "Assume $I$ comes equipped with a total ordering.", "The map $c$ is a morphism of complexes. In fact it induces", "an isomorphism", "$$", "c : \\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\to \\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})", "$$", "of complexes." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2094, "type": "theorem", "label": "cohomology-lemma-project-to-ordered", "categories": [ "cohomology" ], "title": "cohomology-lemma-project-to-ordered", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.", "Assume $I$ comes equipped with a total ordering.", "The map $\\pi : \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\to \\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "is a morphism of complexes. It induces an isomorphism", "$$", "\\pi : \\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\to \\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})", "$$", "of complexes which is a left inverse to the morphism $c$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2095, "type": "theorem", "label": "cohomology-lemma-alternating-usual", "categories": [ "cohomology" ], "title": "cohomology-lemma-alternating-usual", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.", "Assume $I$ comes equipped with a total ordering.", "The map $c \\circ \\pi$ is homotopic to the identity on", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$.", "In particular the inclusion map", "$\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "is a homotopy equivalence." ], "refs": [], "proofs": [ { "contents": [ "For any multi-index $(i_0, \\ldots, i_p) \\in I^{p + 1}$ there exists", "a unique permutation $\\sigma : \\{0, \\ldots, p\\} \\to \\{0, \\ldots, p\\}$", "such that", "$$", "i_{\\sigma(0)} \\leq i_{\\sigma(1)} \\leq \\ldots \\leq i_{\\sigma(p)}", "\\quad", "\\text{and}", "\\quad", "\\sigma(j) < \\sigma(j + 1)", "\\quad", "\\text{if}", "\\quad", "i_{\\sigma(j)} = i_{\\sigma(j + 1)}.", "$$", "We denote this permutation $\\sigma = \\sigma^{i_0 \\ldots i_p}$.", "\\medskip\\noindent", "For any permutation $\\sigma : \\{0, \\ldots, p\\} \\to \\{0, \\ldots, p\\}$", "and any $a$, $0 \\leq a \\leq p$ we denote $\\sigma_a$", "the permutation of $\\{0, \\ldots, p\\}$ such that", "$$", "\\sigma_a(j) =", "\\left\\{", "\\begin{matrix}", "\\sigma(j) & \\text{if} & 0 \\leq j < a, \\\\", "\\min\\{j' \\mid j' > \\sigma_a(j - 1), j' \\not = \\sigma(k), \\forall k < a\\}", "& \\text{if} & a \\leq j", "\\end{matrix}", "\\right.", "$$", "So if $p = 3$ and $\\sigma$, $\\tau$ are given by", "$$", "\\begin{matrix}", "\\text{id} & 0 & 1 & 2 & 3 \\\\", "\\sigma & 3 & 2 & 1 & 0", "\\end{matrix}", "\\quad \\text{and} \\quad", "\\begin{matrix}", "\\text{id} & 0 & 1 & 2 & 3 \\\\", "\\tau & 3 & 0 & 2 & 1", "\\end{matrix}", "$$", "then we have", "$$", "\\begin{matrix}", "\\text{id} & 0 & 1 & 2 & 3 \\\\", "\\sigma_0 & 0 & 1 & 2 & 3 \\\\", "\\sigma_1 & 3 & 0 & 1 & 2 \\\\", "\\sigma_2 & 3 & 2 & 0 & 1 \\\\", "\\sigma_3 & 3 & 2 & 1 & 0 \\\\", "\\end{matrix}", "\\quad \\text{and} \\quad", "\\begin{matrix}", "\\text{id} & 0 & 1 & 2 & 3 \\\\", "\\tau_0 & 0 & 1 & 2 & 3 \\\\", "\\tau_1 & 3 & 0 & 1 & 2 \\\\", "\\tau_2 & 3 & 0 & 1 & 2 \\\\", "\\tau_3 & 3 & 0 & 2 & 1 \\\\", "\\end{matrix}", "$$", "It is clear that always $\\sigma_0 = \\text{id}$ and $\\sigma_p = \\sigma$.", "\\medskip\\noindent", "Having introduced this notation we define for", "$s \\in \\check{\\mathcal{C}}^{p + 1}(\\mathcal{U}, \\mathcal{F})$", "the element $h(s) \\in \\check{\\mathcal{C}}^p(\\mathcal{U}, \\mathcal{F})$", "to be the element with components", "\\begin{equation}", "\\label{equation-first-homotopy}", "h(s)_{i_0\\ldots i_p} =", "\\sum\\nolimits_{0 \\leq a \\leq p}", "(-1)^a \\text{sign}(\\sigma_a)", "s_{i_{\\sigma(0)} \\ldots i_{\\sigma(a)} i_{\\sigma_a(a)} \\ldots i_{\\sigma_a(p)}}", "\\end{equation}", "where $\\sigma = \\sigma^{i_0 \\ldots i_p}$. The index", "$i_{\\sigma(a)}$ occurs twice in", "$i_{\\sigma(0)} \\ldots i_{\\sigma(a)} i_{\\sigma_a(a)} \\ldots i_{\\sigma_a(p)}$", "once in the first group of $a + 1$ indices and once in the second group", "of $p - a + 1$ indices since $\\sigma_a(j) = \\sigma(a)$ for some", "$j \\geq a$ by definition of $\\sigma_a$. Hence the sum makes sense since each", "of the elements", "$s_{i_{\\sigma(0)} \\ldots i_{\\sigma(a)} i_{\\sigma_a(a)} \\ldots i_{\\sigma_a(p)}}$", "is defined over the open $U_{i_0 \\ldots i_p}$.", "Note also that for $a = 0$ we get $s_{i_0 \\ldots i_p}$ and", "for $a = p$ we get", "$(-1)^p \\text{sign}(\\sigma) s_{i_{\\sigma(0)} \\ldots i_{\\sigma(p)}}$.", "\\medskip\\noindent", "We claim that", "$$", "(dh + hd)(s)_{i_0 \\ldots i_p} =", "s_{i_0 \\ldots i_p} -", "\\text{sign}(\\sigma) s_{i_{\\sigma(0)} \\ldots i_{\\sigma(p)}}", "$$", "where $\\sigma = \\sigma^{i_0 \\ldots i_p}$. We omit the verification", "of this claim. (There is a PARI/gp script called first-homotopy.gp", "in the stacks-project subdirectory scripts which can be used to check", "finitely many instances of this claim.", "We wrote this script to make sure the signs are correct.)", "Write", "$$", "\\kappa :", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\longrightarrow", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "$$", "for the operator given by the rule", "$$", "\\kappa(s)_{i_0 \\ldots i_p} =", "\\text{sign}(\\sigma^{i_0 \\ldots i_p}) s_{i_{\\sigma(0)} \\ldots i_{\\sigma(p)}}.", "$$", "The claim above implies that $\\kappa$ is a morphism of complexes and that", "$\\kappa$ is homotopic to the identity map of the {\\v C}ech complex.", "This does not immediately imply the lemma since", "the image of the operator $\\kappa$ is not the alternating subcomplex.", "Namely, the image of $\\kappa$ is the ``semi-alternating'' complex", "$\\check{\\mathcal{C}}_{semi\\text{-}alt}^p(\\mathcal{U}, \\mathcal{F})$", "where $s$ is a $p$-cochain of this complex if and only if", "$$", "s_{i_0 \\ldots i_p} = \\text{sign}(\\sigma) s_{i_{\\sigma(0)} \\ldots i_{\\sigma(p)}}", "$$", "for any $(i_0, \\ldots, i_p) \\in I^{p + 1}$ with", "$\\sigma = \\sigma^{i_0 \\ldots i_p}$.", "We introduce yet another variant {\\v C}ech complex, namely the semi-ordered", "{\\v C}ech complex defined by", "$$", "\\check{\\mathcal{C}}_{semi\\text{-}ord}^p(\\mathcal{U}, \\mathcal{F})", "=", "\\prod\\nolimits_{i_0 \\leq i_1 \\leq \\ldots \\leq i_p}", "\\mathcal{F}(U_{i_0 \\ldots i_p})", "$$", "It is easy to see that Equation (\\ref{equation-d-cech}) also defines", "a differential and hence that we get a complex. It is also clear", "(analogous to Lemma \\ref{lemma-project-to-ordered}) that the projection map", "$$", "\\check{\\mathcal{C}}_{semi\\text{-}alt}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\longrightarrow", "\\check{\\mathcal{C}}_{semi\\text{-}ord}^\\bullet(\\mathcal{U}, \\mathcal{F})", "$$", "is an isomorphism of complexes.", "\\medskip\\noindent", "Hence the Lemma follows if we can show that the obvious inclusion map", "$$", "\\check{\\mathcal{C}}_{ord}^p(\\mathcal{U}, \\mathcal{F})", "\\longrightarrow", "\\check{\\mathcal{C}}_{semi\\text{-}ord}^p(\\mathcal{U}, \\mathcal{F})", "$$", "is a homotopy equivalence. To see this we use the homotopy", "\\begin{equation}", "\\label{equation-second-homotopy}", "h(s)_{i_0 \\ldots i_p} =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & i_0 < i_1 < \\ldots < i_p \\\\", "(-1)^a s_{i_0 \\ldots i_{a - 1} i_a i_a i_{a + 1} \\ldots i_p}", "& \\text{if} & i_0 < i_1 < \\ldots < i_{a - 1} < i_a = i_{a + 1}", "\\end{matrix}", "\\right.", "\\end{equation}", "We claim that", "$$", "(dh + hd)(s)_{i_0 \\ldots i_p} =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & i_0 < i_1 < \\ldots < i_p \\\\", "s_{i_0 \\ldots i_p}", "& \\text{else} &", "\\end{matrix}", "\\right.", "$$", "We omit the verification. (There is a PARI/gp script called second-homotopy.gp", "in the stacks-project subdirectory scripts which can be used to check", "finitely many instances of this claim.", "We wrote this script to make sure the signs are correct.)", "The claim clearly shows that the composition", "$$", "\\check{\\mathcal{C}}_{semi\\text{-}ord}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\longrightarrow", "\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\longrightarrow", "\\check{\\mathcal{C}}_{semi\\text{-}ord}^\\bullet(\\mathcal{U}, \\mathcal{F})", "$$", "of the projection with the natural inclusion", "is homotopic to the identity map as desired." ], "refs": [ "cohomology-lemma-project-to-ordered" ], "ref_ids": [ 2094 ] } ], "ref_ids": [] }, { "id": 2096, "type": "theorem", "label": "cohomology-lemma-alternating-cech-trivial", "categories": [ "cohomology" ], "title": "cohomology-lemma-alternating-cech-trivial", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{F}$ be an abelian presheaf on $X$.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering. If", "$U_i = U$ for some $i \\in I$, then the extended alternating {\\v C}ech complex", "$$", "\\mathcal{F}(U) \\to \\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})", "$$", "obtained by putting $\\mathcal{F}(U)$ in degree $-1$ with differential given by", "the canonical map of $\\mathcal{F}(U)$ into", "$\\check{\\mathcal{C}}^0(\\mathcal{U}, \\mathcal{F})$", "is homotopy equivalent to $0$. Similarly, for any total ordering on $I$", "the extended ordered {\\v C}ech complex", "$$", "\\mathcal{F}(U) \\to", "\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})", "$$", "is homotopy equivalent to $0$." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "Combine Lemmas \\ref{lemma-cech-trivial} and \\ref{lemma-alternating-usual}." ], "refs": [ "cohomology-lemma-cech-trivial", "cohomology-lemma-alternating-usual" ], "ref_ids": [ 2045, 2095 ] } ], "ref_ids": [] }, { "id": 2097, "type": "theorem", "label": "cohomology-lemma-covering-resolution", "categories": [ "cohomology" ], "title": "cohomology-lemma-covering-resolution", "contents": [ "Let $X$ be a ringed space. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$", "be an open covering of $X$. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "Denote $\\mathcal{F}_{i_0 \\ldots i_p}$ the restriction of", "$\\mathcal{F}$ to $U_{i_0 \\ldots i_p}$. There exists a complex", "${\\mathfrak C}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "of $\\mathcal{O}_X$-modules with", "$$", "{\\mathfrak C}^p(\\mathcal{U}, \\mathcal{F}) =", "\\prod\\nolimits_{i_0 \\ldots i_p}", "(j_{i_0 \\ldots i_p})_* \\mathcal{F}_{i_0 \\ldots i_p}", "$$", "and differential", "$d : {\\mathfrak C}^p(\\mathcal{U}, \\mathcal{F})", "\\to {\\mathfrak C}^{p + 1}(\\mathcal{U}, \\mathcal{F})$", "as in Equation (\\ref{equation-d-cech}). Moreover, there exists a canonical", "map", "$$", "\\mathcal{F} \\to {\\mathfrak C}^\\bullet(\\mathcal{U}, \\mathcal{F})", "$$", "which is a quasi-isomorphism, i.e.,", "${\\mathfrak C}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "is a resolution of $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "We check", "$$", "0 \\to \\mathcal{F} \\to \\mathfrak{C}^0(\\mathcal{U}, \\mathcal{F}) \\to", "\\mathfrak{C}^1(\\mathcal{U}, \\mathcal{F}) \\to \\ldots", "$$", "is exact on stalks. Let $x \\in X$ and choose $i_{\\text{fix}} \\in I$", "such that $x \\in U_{i_{\\text{fix}}}$. Then define ", "$$", "h : \\mathfrak{C}^p(\\mathcal{U}, \\mathcal{F})_x", "\\to \\mathfrak{C}^{p - 1}(\\mathcal{U}, \\mathcal{F})_x", "$$", "as follows: If $s \\in \\mathfrak{C}^p(\\mathcal{U}, \\mathcal{F})_x$, take", "a representative", "$$", "\\widetilde{s} \\in", "\\mathfrak{C}^p(\\mathcal{U}, \\mathcal{F})(V) =", "\\prod\\nolimits_{i_0 \\ldots i_p}", "\\mathcal{F}(V \\cap U_{i_0} \\cap \\ldots \\cap U_{i_p})", "$$", "defined on some neighborhood $V$ of $x$, and set", "$$", "h(s)_{i_0 \\ldots i_{p - 1}} =", "\\widetilde{s}_{i_{\\text{fix}} i_0 \\ldots i_{p - 1}, x}.", "$$", "By the same formula (for $p = 0$) we get a map", "$\\mathfrak{C}^{0}(\\mathcal{U},\\mathcal{F})_x \\to \\mathcal{F}_x$.", "We compute formally as follows:", "\\begin{align*}", "(dh + hd)(s)_{i_0 \\ldots i_p}", "& =", "\\sum\\nolimits_{j = 0}^p", "(-1)^j", "h(s)_{i_0 \\ldots \\hat i_j \\ldots i_p}", "+", "d(s)_{i_{\\text{fix}} i_0 \\ldots i_p}\\\\", "& =", "\\sum\\nolimits_{j = 0}^p", "(-1)^j", "s_{i_{\\text{fix}} i_0 \\ldots \\hat i_j \\ldots i_p}", "+", "s_{i_0 \\ldots i_p}", "+", "\\sum\\nolimits_{j = 0}^p", "(-1)^{j + 1}", "s_{i_{\\text{fix}} i_0 \\ldots \\hat i_j \\ldots i_p} \\\\", "& =", "s_{i_0 \\ldots i_p}", "\\end{align*}", "This shows $h$ is a homotopy from the identity map of", "the extended complex", "$$", "0 \\to \\mathcal{F}_x \\to \\mathfrak{C}^0(\\mathcal{U}, \\mathcal{F})_x", "\\to \\mathfrak{C}^1(\\mathcal{U}, \\mathcal{F})_x \\to \\ldots", "$$", "to zero and we conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2098, "type": "theorem", "label": "cohomology-lemma-cech-complex-complex", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-complex-complex", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be", "an open covering. For a bounded below complex $\\mathcal{F}^\\bullet$", "of $\\mathcal{O}_X$-modules there is a canonical map", "$$", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))", "\\longrightarrow", "R\\Gamma(X, \\mathcal{F}^\\bullet)", "$$", "functorial in $\\mathcal{F}^\\bullet$ and compatible with", "(\\ref{equation-global-sections-to-cech}) and (\\ref{equation-transformation}).", "There is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with", "$$", "E_2^{p, q} =", "H^p(\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U},", "\\underline{H}^q(\\mathcal{F}^\\bullet)))", "$$", "converging to $H^{p + q}(X, \\mathcal{F}^\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "Let ${\\mathcal I}^\\bullet$ be a bounded below complex of injectives.", "The map (\\ref{equation-global-sections-to-cech}) for", "$\\mathcal{I}^\\bullet$ is a map", "$\\Gamma(X, {\\mathcal I}^\\bullet) \\to", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet({\\mathcal U}, {\\mathcal I}^\\bullet))$.", "This is a quasi-isomorphism of complexes of abelian groups", "as follows from", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}", "applied to the double complex", "$\\check{\\mathcal{C}}^\\bullet({\\mathcal U}, {\\mathcal I}^\\bullet)$ using", "Lemma \\ref{lemma-injective-trivial-cech}.", "Suppose ${\\mathcal F}^\\bullet \\to {\\mathcal I}^\\bullet$ is a quasi-isomorphism", "of ${\\mathcal F}^\\bullet$ into a bounded below complex of injectives.", "Since $R\\Gamma(X, {\\mathcal F}^\\bullet)$ is represented by the complex", "$\\Gamma(X, {\\mathcal I}^\\bullet)$ we obtain the map of the lemma", "using", "$$", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet({\\mathcal U}, {\\mathcal F}^\\bullet))", "\\longrightarrow", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet({\\mathcal U}, {\\mathcal I}^\\bullet)).", "$$", "We omit the verification of functoriality and compatibilities.", "To construct the spectral sequence of the lemma, choose a Cartan-Eilenberg", "resolution $\\mathcal{F}^\\bullet \\to \\mathcal{I}^{\\bullet, \\bullet}$, see", "Derived Categories, Lemma \\ref{derived-lemma-cartan-eilenberg}. In this", "case $\\mathcal{F}^\\bullet \\to \\text{Tot}(\\mathcal{I}^{\\bullet, \\bullet})$", "is an injective resolution and hence", "$$", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet({\\mathcal U},", "\\text{Tot}({\\mathcal I}^{\\bullet, \\bullet})))", "$$", "computes $R\\Gamma(X, \\mathcal{F}^\\bullet)$ as we've seen above.", "By Homology, Remark \\ref{homology-remark-triple-complex}", "we can view this as the total complex associated to the", "triple complex", "$\\check{\\mathcal{C}}^\\bullet({\\mathcal U}, {\\mathcal I}^{\\bullet, \\bullet})$", "hence, using the same remark we can view it as the total complex", "associate to the double complex $A^{\\bullet, \\bullet}$ with terms", "$$", "A^{n, m} =", "\\bigoplus\\nolimits_{p + q = n}", "\\check{\\mathcal{C}}^p({\\mathcal U}, \\mathcal{I}^{q, m})", "$$", "Since $\\mathcal{I}^{q, \\bullet}$ is an injective resolution of", "$\\mathcal{F}^q$ we can apply the first spectral sequence associated to", "$A^{\\bullet, \\bullet}$", "(Homology, Lemma \\ref{homology-lemma-ss-double-complex})", "to get a spectral sequence with", "$$", "E_1^{n, m} =", "\\bigoplus\\nolimits_{p + q = n}", "\\check{\\mathcal{C}}^p(\\mathcal{U}, \\underline{H}^m(\\mathcal{F}^q))", "$$", "which is the $n$th term of the complex", "$\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U},", "\\underline{H}^m(\\mathcal{F}^\\bullet))$. Hence we obtain", "$E_2$ terms as described in the lemma. Convergence by", "Homology, Lemma \\ref{homology-lemma-first-quadrant-ss}." ], "refs": [ "homology-lemma-double-complex-gives-resolution", "cohomology-lemma-injective-trivial-cech", "derived-lemma-cartan-eilenberg", "homology-remark-triple-complex", "homology-lemma-ss-double-complex", "homology-lemma-first-quadrant-ss" ], "ref_ids": [ 12106, 2051, 1870, 12191, 12104, 12105 ] } ], "ref_ids": [] }, { "id": 2099, "type": "theorem", "label": "cohomology-lemma-cech-complex-complex-computes", "categories": [ "cohomology" ], "title": "cohomology-lemma-cech-complex-complex-computes", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be", "an open covering. Let $\\mathcal{F}^\\bullet$ be a bounded below complex", "of $\\mathcal{O}_X$-modules. If $H^i(U_{i_0 \\ldots i_p}, \\mathcal{F}^q) = 0$", "for all $i > 0$ and all $p, i_0, \\ldots, i_p, q$, then the map", "$", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))", "\\to", "R\\Gamma(X, \\mathcal{F}^\\bullet)", "$", "of Lemma \\ref{lemma-cech-complex-complex} is an isomorphism." ], "refs": [ "cohomology-lemma-cech-complex-complex" ], "proofs": [ { "contents": [ "Immediate from the spectral sequence of Lemma \\ref{lemma-cech-complex-complex}." ], "refs": [ "cohomology-lemma-cech-complex-complex" ], "ref_ids": [ 2098 ] } ], "ref_ids": [ 2098 ] }, { "id": 2100, "type": "theorem", "label": "cohomology-lemma-compute-sign-cup-product-boundaries", "categories": [ "cohomology" ], "title": "cohomology-lemma-compute-sign-cup-product-boundaries", "contents": [ "In the situation above, assume {\\v C}ech cohomology agrees with cohomology", "for the sheaves $\\mathcal{F}_i^p$ and $\\mathcal{G}_j^q$.", "Let $a_3 \\in H^n(X, \\mathcal{F}_3^\\bullet)$ and", "$b_1 \\in H^m(X, \\mathcal{G}_1^\\bullet)$. Then we have", "$$", "\\gamma_1( \\partial a_3 \\cup b_1) =", "(-1)^{n + 1} \\gamma_3( a_3 \\cup \\partial b_1)", "$$", "in $H^{n + m}(X, \\mathcal{H}^\\bullet)$ where $\\partial$ indicates the", "boundary map on cohomology associated to the short exact sequences of", "complexes above." ], "refs": [], "proofs": [ { "contents": [ "We will use the following conventions and notation. We think of", "${\\mathcal F}_1^p$ as a subsheaf of ${\\mathcal F}_2^p$ and we think of", "${\\mathcal G}_3^q$ as a subsheaf of ${\\mathcal G}_2^q$. Hence if $s$ is", "a local section of ${\\mathcal F}_1^p$ we use $s$ to denote", "the corresponding section of ${\\mathcal F}_2^p$ as well. Similarly", "for local sections of ${\\mathcal G}_3^q$. Furthermore,", "if $s$ is a local section of ${\\mathcal F}_2^p$ then we denote", "$\\bar s$ its image in ${\\mathcal F}_3^p$. Similarly for the", "map ${\\mathcal G}_2^q \\to {\\mathcal G}^q_1$. In particular if", "$s$ is a local section of ${\\mathcal F}_2^p$ and $\\bar s = 0$", "then $s$ is a local section of ${\\mathcal F}_1^p$. The commutativity", "of the diagrams above implies, for local sections $s$ of", "${\\mathcal F}_2^p$ and $t$ of ${\\mathcal G}_3^q$ that", "$\\gamma_2(s \\otimes t) = \\gamma_3(\\bar s \\otimes t)$ as sections of", "${\\mathcal H}^{p + q}$.", "\\medskip\\noindent", "Let ${\\mathcal U} : X = \\bigcup_{i \\in I} U_i$", "be an open covering of $X$. Suppose that $\\alpha_3$,", "resp.\\ $\\beta_1$ is a degree $n$, resp.\\ $m$ cocycle of", "$\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet({\\mathcal U}, {\\mathcal F}_3^\\bullet))$,", "resp.\\ $\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet({\\mathcal U}, {\\mathcal G}_1^\\bullet))$", "representing $a_3$, resp.\\ $b_1$. After refining $\\mathcal{U}$ if necessary,", "we can find cochains $\\alpha_2$, resp.\\ $\\beta_2$ of", "degree $n$, resp.\\ $m$ in", "$\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet({\\mathcal U}, {\\mathcal F}_2^\\bullet))$,", "resp.\\ $\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet({\\mathcal U}, {\\mathcal G}_2^\\bullet))$", "mapping to $\\alpha_3$, resp.\\ $\\beta_1$.", "Then we see that", "$$", "\\overline{d(\\alpha_2)} = d(\\bar \\alpha_2) = 0", "\\quad\\text{and}\\quad", "\\overline{d(\\beta_2)} = d(\\bar \\beta_2) = 0.", "$$", "This means that $\\alpha_1 = d(\\alpha_2)$ is a degree $n + 1$ cocycle in", "$\\text{Tot}(\\check{\\mathcal{C}}^\\bullet({\\mathcal U}, {\\mathcal F}_1^\\bullet))$", "representing $\\partial a_3$. Similarly, $\\beta_3 = d(\\beta_2)$ is", "a degree $m + 1$ cocycle in", "$\\text{Tot}(\\check{\\mathcal{C}}^\\bullet({\\mathcal U}, {\\mathcal G}_3^\\bullet))$", "representing $\\partial b_1$.", "Thus we may compute", "\\begin{align*}", "d(\\gamma_2(\\alpha_2 \\cup \\beta_2))", "& =", "\\gamma_2(d(\\alpha_2 \\cup \\beta_2))", "\\\\", "& =", "\\gamma_2(d(\\alpha_2) \\cup \\beta_2 + (-1)^n \\alpha_2 \\cup d(\\beta_2) )", "\\\\", "& =", "\\gamma_2( \\alpha_1 \\cup \\beta_2) + (-1)^n \\gamma_2( \\alpha_2 \\cup \\beta_3)", "\\\\", "& =", "\\gamma_1(\\alpha_1 \\cup \\beta_1) + (-1)^n \\gamma_3(\\alpha_3 \\cup \\beta_3)", "\\end{align*}", "So this even tells us that the sign is $(-1)^{n + 1}$ as indicated", "in the lemma\\footnote{The sign depends on the convention for the", "signs in the long exact sequence in cohomology associated to a triangle", "in $D(X)$. The conventions in the Stacks project are (a) distinguished", "triangles correspond to termwise split exact sequences and (b) the boundary", "maps in the long exact sequence are given by the maps in the snake lemma", "without the intervention of signs. See", "Derived Categories, Section \\ref{derived-section-homotopy-triangulated}.}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2101, "type": "theorem", "label": "cohomology-lemma-boundary-derivation-over-cup-product", "categories": [ "cohomology" ], "title": "cohomology-lemma-boundary-derivation-over-cup-product", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{O}' \\to \\mathcal{O}$ be a", "surjection of sheaves of rings whose kernel $\\mathcal{I} \\subset \\mathcal{O}'$", "has square zero. Then $M = H^1(X, \\mathcal{I})$ is a", "$R = H^0(X, \\mathcal{O})$-module and the boundary map", "$\\partial : R \\to M$ associated to the short exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}' \\to \\mathcal{O} \\to 0", "$$", "is a derivation (Algebra, Definition \\ref{algebra-definition-derivation})." ], "refs": [ "algebra-definition-derivation" ], "proofs": [ { "contents": [ "The map $\\mathcal{O}' \\to \\SheafHom(\\mathcal{I}, \\mathcal{I})$", "factors through $\\mathcal{O}$ as $\\mathcal{I} \\cdot \\mathcal{I} = 0$", "by assumption. Hence $\\mathcal{I}$ is a sheaf of $\\mathcal{O}$-modules", "and this defines the $R$-module structure on $M$.", "The boundary map is additive hence it suffices to prove", "the Leibniz rule. Let $f \\in R$. Choose an open covering", "$\\mathcal{U} : X = \\bigcup U_i$ such that there exist", "$f_i \\in \\mathcal{O}'(U_i)$ lifting $f|_{U_i} \\in \\mathcal{O}(U_i)$.", "Observe that $f_i - f_j$ is an element of $\\mathcal{I}(U_i \\cap U_j)$.", "Then $\\partial(f)$ corresponds to the {\\v C}ech cohomology class of", "the $1$-cocycle $\\alpha$ with $\\alpha_{i_0i_1} = f_{i_0} - f_{i_1}$.", "(Observe that by Lemma \\ref{lemma-cech-h1} the first {\\v C}ech cohomology", "group with respect to $\\mathcal{U}$ is a submodule of $M$.)", "Next, let $g \\in R$ be a second element and assume (after possibly", "refining the open covering) that $g_i \\in \\mathcal{O}'(U_i)$ lifts", "$g|_{U_i} \\in \\mathcal{O}(U_i)$. Then we see that", "$\\partial(g)$ is given by the cocycle $\\beta$ with", "$\\beta_{i_0i_1} = g_{i_0} - g_{i_1}$. Since $f_ig_i \\in \\mathcal{O}'(U_i)$", "lifts $fg|_{U_i}$ we see that", "$\\partial(fg)$ is given by the cocycle $\\gamma$ with", "$$", "\\gamma_{i_0i_1} = f_{i_0}g_{i_0} - f_{i_1}g_{i_1} =", "(f_{i_0} - f_{i_1})g_{i_0} + f_{i_1}(g_{i_0} - g_{i_1}) =", "\\alpha_{i_0i_1}g + f\\beta_{i_0i_1}", "$$", "by our definition of the $\\mathcal{O}$-module structure on $\\mathcal{I}$.", "This proves the Leibniz rule and the proof is complete." ], "refs": [ "cohomology-lemma-cech-h1" ], "ref_ids": [ 2053 ] } ], "ref_ids": [ 1525 ] }, { "id": 2102, "type": "theorem", "label": "cohomology-lemma-derived-tor-exact", "categories": [ "cohomology" ], "title": "cohomology-lemma-derived-tor-exact", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{G}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules.", "The functors", "$$", "K(\\textit{Mod}(\\mathcal{O}_X))", "\\longrightarrow", "K(\\textit{Mod}(\\mathcal{O}_X)),", "\\quad", "\\mathcal{F}^\\bullet \\longmapsto", "\\text{Tot}(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{F}^\\bullet)", "$$", "and", "$$", "K(\\textit{Mod}(\\mathcal{O}_X))", "\\longrightarrow", "K(\\textit{Mod}(\\mathcal{O}_X)),", "\\quad", "\\mathcal{F}^\\bullet \\longmapsto", "\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)", "$$", "are exact functors of triangulated categories." ], "refs": [], "proofs": [ { "contents": [ "This follows from Derived Categories, Remark", "\\ref{derived-remark-double-complex-as-tensor-product-of}." ], "refs": [ "derived-remark-double-complex-as-tensor-product-of" ], "ref_ids": [ 2014 ] } ], "ref_ids": [] }, { "id": 2103, "type": "theorem", "label": "cohomology-lemma-K-flat-quasi-isomorphism", "categories": [ "cohomology" ], "title": "cohomology-lemma-K-flat-quasi-isomorphism", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{K}^\\bullet$ be a K-flat complex.", "Then the functor", "$$", "K(\\textit{Mod}(\\mathcal{O}_X))", "\\longrightarrow", "K(\\textit{Mod}(\\mathcal{O}_X)), \\quad", "\\mathcal{F}^\\bullet", "\\longmapsto", "\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{K}^\\bullet)", "$$", "transforms quasi-isomorphisms into quasi-isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-derived-tor-exact}", "and the fact that quasi-isomorphisms are characterized by having", "acyclic cones." ], "refs": [ "cohomology-lemma-derived-tor-exact" ], "ref_ids": [ 2102 ] } ], "ref_ids": [] }, { "id": 2104, "type": "theorem", "label": "cohomology-lemma-check-K-flat-stalks", "categories": [ "cohomology" ], "title": "cohomology-lemma-check-K-flat-stalks", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{K}^\\bullet$", "be a complex of $\\mathcal{O}_X$-modules. Then $\\mathcal{K}^\\bullet$", "is K-flat if and only if for all $x \\in X$ the complex", "$\\mathcal{K}_x^\\bullet$ of $\\mathcal{O}_{X, x}$-modules is K-flat", "(More on Algebra, Definition \\ref{more-algebra-definition-K-flat})." ], "refs": [ "more-algebra-definition-K-flat" ], "proofs": [ { "contents": [ "If $\\mathcal{K}_x^\\bullet$ is K-flat for all $x \\in X$ then we see", "that $\\mathcal{K}^\\bullet$ is K-flat because $\\otimes$ and", "direct sums commute with taking stalks and because we can check exactness", "at stalks, see", "Modules, Lemma \\ref{modules-lemma-abelian}.", "Conversely, assume $\\mathcal{K}^\\bullet$ is K-flat. Pick $x \\in X$", "$M^\\bullet$ be an acyclic complex of $\\mathcal{O}_{X, x}$-modules.", "Then $i_{x, *}M^\\bullet$ is an acyclic complex of $\\mathcal{O}_X$-modules.", "Thus $\\text{Tot}(i_{x, *}M^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{K}^\\bullet)$", "is acyclic. Taking stalks at $x$ shows that", "$\\text{Tot}(M^\\bullet \\otimes_{\\mathcal{O}_{X, x}} \\mathcal{K}_x^\\bullet)$", "is acyclic." ], "refs": [ "modules-lemma-abelian" ], "ref_ids": [ 13221 ] } ], "ref_ids": [ 10620 ] }, { "id": 2105, "type": "theorem", "label": "cohomology-lemma-tensor-product-K-flat", "categories": [ "cohomology" ], "title": "cohomology-lemma-tensor-product-K-flat", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "If $\\mathcal{K}^\\bullet$, $\\mathcal{L}^\\bullet$ are K-flat complexes", "of $\\mathcal{O}_X$-modules, then", "$\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet)$", "is a K-flat complex of $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "Follows from the isomorphism", "$$", "\\text{Tot}(\\mathcal{M}^\\bullet \\otimes_{\\mathcal{O}_X}", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet))", "=", "\\text{Tot}(\\text{Tot}(\\mathcal{M}^\\bullet \\otimes_{\\mathcal{O}_X}", "\\mathcal{K}^\\bullet) \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet)", "$$", "and the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2106, "type": "theorem", "label": "cohomology-lemma-K-flat-two-out-of-three", "categories": [ "cohomology" ], "title": "cohomology-lemma-K-flat-two-out-of-three", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $(\\mathcal{K}_1^\\bullet, \\mathcal{K}_2^\\bullet, \\mathcal{K}_3^\\bullet)$", "be a distinguished triangle in $K(\\textit{Mod}(\\mathcal{O}_X))$.", "If two out of three of $\\mathcal{K}_i^\\bullet$ are K-flat, so is the third." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-derived-tor-exact}", "and the fact that in a distinguished triangle in", "$K(\\textit{Mod}(\\mathcal{O}_X))$", "if two out of three are acyclic, so is the third." ], "refs": [ "cohomology-lemma-derived-tor-exact" ], "ref_ids": [ 2102 ] } ], "ref_ids": [] }, { "id": 2107, "type": "theorem", "label": "cohomology-lemma-K-flat-two-out-of-three-ses", "categories": [ "cohomology" ], "title": "cohomology-lemma-K-flat-two-out-of-three-ses", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let", "$0 \\to \\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet \\to", "\\mathcal{K}_3^\\bullet \\to 0$ be a short exact sequence of complexes", "such that the terms of $\\mathcal{K}_3^\\bullet$ are flat $\\mathcal{O}_X$-modules.", "If two out of three of $\\mathcal{K}_i^\\bullet$ are K-flat, so is the third." ], "refs": [], "proofs": [ { "contents": [ "By Modules, Lemma \\ref{modules-lemma-flat-tor-zero}", "for every complex $\\mathcal{L}^\\bullet$", "we obtain a short exact sequence", "$$", "0 \\to", "\\text{Tot}(\\mathcal{L}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{K}_1^\\bullet)", "\\to", "\\text{Tot}(\\mathcal{L}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{K}_1^\\bullet)", "\\to", "\\text{Tot}(\\mathcal{L}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{K}_1^\\bullet)", "\\to 0", "$$", "of complexes. Hence the lemma follows from the long exact sequence of", "cohomology sheaves and the definition of K-flat complexes." ], "refs": [ "modules-lemma-flat-tor-zero" ], "ref_ids": [ 13277 ] } ], "ref_ids": [] }, { "id": 2108, "type": "theorem", "label": "cohomology-lemma-pullback-K-flat", "categories": [ "cohomology" ], "title": "cohomology-lemma-pullback-K-flat", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of", "ringed spaces. The pullback of a K-flat complex of $\\mathcal{O}_Y$-modules", "is a K-flat complex of $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "We can check this on stalks, see", "Lemma \\ref{lemma-check-K-flat-stalks}.", "Hence this follows from", "Sheaves, Lemma \\ref{sheaves-lemma-stalk-pullback-modules}", "and", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-K-flat}." ], "refs": [ "cohomology-lemma-check-K-flat-stalks", "sheaves-lemma-stalk-pullback-modules", "more-algebra-lemma-base-change-K-flat" ], "ref_ids": [ 2104, 14523, 10124 ] } ], "ref_ids": [] }, { "id": 2109, "type": "theorem", "label": "cohomology-lemma-bounded-flat-K-flat", "categories": [ "cohomology" ], "title": "cohomology-lemma-bounded-flat-K-flat", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. A bounded above complex", "of flat $\\mathcal{O}_X$-modules is K-flat." ], "refs": [], "proofs": [ { "contents": [ "We can check this on stalks, see", "Lemma \\ref{lemma-check-K-flat-stalks}.", "Thus this lemma follows from", "Modules, Lemma \\ref{modules-lemma-flat-stalks-flat}", "and", "More on Algebra, Lemma \\ref{more-algebra-lemma-derived-tor-quasi-isomorphism}." ], "refs": [ "cohomology-lemma-check-K-flat-stalks", "modules-lemma-flat-stalks-flat", "more-algebra-lemma-derived-tor-quasi-isomorphism" ], "ref_ids": [ 2104, 13273, 10128 ] } ], "ref_ids": [] }, { "id": 2110, "type": "theorem", "label": "cohomology-lemma-colimit-K-flat", "categories": [ "cohomology" ], "title": "cohomology-lemma-colimit-K-flat", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet \\to \\ldots$", "be a system of K-flat complexes.", "Then $\\colim_i \\mathcal{K}_i^\\bullet$ is K-flat." ], "refs": [], "proofs": [ { "contents": [ "Because we are taking termwise colimits it is clear that", "$$", "\\colim_i \\text{Tot}(", "\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{K}_i^\\bullet)", "=", "\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X}", "\\colim_i \\mathcal{K}_i^\\bullet)", "$$", "Hence the lemma follows from the fact that filtered colimits are", "exact." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2111, "type": "theorem", "label": "cohomology-lemma-resolution-by-direct-sums-extensions-by-zero", "categories": [ "cohomology" ], "title": "cohomology-lemma-resolution-by-direct-sums-extensions-by-zero", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "For any complex $\\mathcal{G}^\\bullet$ of $\\mathcal{O}_X$-modules", "there exists a commutative diagram of complexes of $\\mathcal{O}_X$-modules", "$$", "\\xymatrix{", "\\mathcal{K}_1^\\bullet \\ar[d] \\ar[r] &", "\\mathcal{K}_2^\\bullet \\ar[d] \\ar[r] & \\ldots \\\\", "\\tau_{\\leq 1}\\mathcal{G}^\\bullet \\ar[r] &", "\\tau_{\\leq 2}\\mathcal{G}^\\bullet \\ar[r] & \\ldots", "}", "$$", "with the following properties: (1) the vertical arrows are quasi-isomorphisms", "and termwise surjective,", "(2) each $\\mathcal{K}_n^\\bullet$ is a bounded above complex whose terms", "are direct sums of $\\mathcal{O}_X$-modules of the form", "$j_{U!}\\mathcal{O}_U$, and", "(3) the maps $\\mathcal{K}_n^\\bullet \\to \\mathcal{K}_{n + 1}^\\bullet$ are", "termwise split injections whose cokernels are direct sums of", "$\\mathcal{O}_X$-modules of the form $j_{U!}\\mathcal{O}_U$. Moreover, the map", "$\\colim \\mathcal{K}_n^\\bullet \\to \\mathcal{G}^\\bullet$ is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The existence of the diagram and properties (1), (2), (3) follows immediately", "from", "Modules, Lemma \\ref{modules-lemma-module-quotient-flat}", "and", "Derived Categories, Lemma \\ref{derived-lemma-special-direct-system}.", "The induced map", "$\\colim \\mathcal{K}_n^\\bullet \\to \\mathcal{G}^\\bullet$", "is a quasi-isomorphism because filtered colimits are exact." ], "refs": [ "modules-lemma-module-quotient-flat", "derived-lemma-special-direct-system" ], "ref_ids": [ 13276, 1903 ] } ], "ref_ids": [] }, { "id": 2112, "type": "theorem", "label": "cohomology-lemma-K-flat-resolution", "categories": [ "cohomology" ], "title": "cohomology-lemma-K-flat-resolution", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "For any complex $\\mathcal{G}^\\bullet$ there exists a $K$-flat complex", "$\\mathcal{K}^\\bullet$ whose terms are flat $\\mathcal{O}_X$-modules", "and a quasi-isomorphism $\\mathcal{K}^\\bullet \\to \\mathcal{G}^\\bullet$", "which is termwise surjective." ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram as in", "Lemma \\ref{lemma-resolution-by-direct-sums-extensions-by-zero}.", "Each complex $\\mathcal{K}_n^\\bullet$ is a bounded", "above complex of flat modules, see", "Modules, Lemma \\ref{modules-lemma-j-shriek-flat}.", "Hence $\\mathcal{K}_n^\\bullet$ is K-flat by", "Lemma \\ref{lemma-bounded-flat-K-flat}.", "Thus $\\colim \\mathcal{K}_n^\\bullet$ is K-flat by", "Lemma \\ref{lemma-colimit-K-flat}.", "The induced map", "$\\colim \\mathcal{K}_n^\\bullet \\to \\mathcal{G}^\\bullet$", "is a quasi-isomorphism and termwise surjective by construction.", "Property (3) of Lemma \\ref{lemma-resolution-by-direct-sums-extensions-by-zero}", "shows that $\\colim \\mathcal{K}_n^m$ is a direct sum of", "flat modules and hence flat which proves the final assertion." ], "refs": [ "cohomology-lemma-resolution-by-direct-sums-extensions-by-zero", "modules-lemma-j-shriek-flat", "cohomology-lemma-bounded-flat-K-flat", "cohomology-lemma-colimit-K-flat", "cohomology-lemma-resolution-by-direct-sums-extensions-by-zero" ], "ref_ids": [ 2111, 13275, 2109, 2110, 2111 ] } ], "ref_ids": [] }, { "id": 2113, "type": "theorem", "label": "cohomology-lemma-derived-tor-quasi-isomorphism-other-side", "categories": [ "cohomology" ], "title": "cohomology-lemma-derived-tor-quasi-isomorphism-other-side", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let", "$\\alpha : \\mathcal{P}^\\bullet \\to \\mathcal{Q}^\\bullet$ be a", "quasi-isomorphism of K-flat complexes of $\\mathcal{O}_X$-modules.", "For every complex $\\mathcal{F}^\\bullet$ of $\\mathcal{O}_X$-modules", "the induced map", "$$", "\\text{Tot}(\\text{id}_{\\mathcal{F}^\\bullet} \\otimes \\alpha) :", "\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{P}^\\bullet)", "\\longrightarrow", "\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{Q}^\\bullet)", "$$", "is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Choose a quasi-isomorphism $\\mathcal{K}^\\bullet \\to \\mathcal{F}^\\bullet$", "with $\\mathcal{K}^\\bullet$ a K-flat complex, see", "Lemma \\ref{lemma-K-flat-resolution}.", "Consider the commutative diagram", "$$", "\\xymatrix{", "\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X} \\mathcal{P}^\\bullet) \\ar[r] \\ar[d] &", "\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X} \\mathcal{Q}^\\bullet) \\ar[d] \\\\", "\\text{Tot}(\\mathcal{F}^\\bullet", "\\otimes_{\\mathcal{O}_X} \\mathcal{P}^\\bullet) \\ar[r] &", "\\text{Tot}(\\mathcal{F}^\\bullet", "\\otimes_{\\mathcal{O}_X} \\mathcal{Q}^\\bullet)", "}", "$$", "The result follows as by", "Lemma \\ref{lemma-K-flat-quasi-isomorphism}", "the vertical arrows and the top horizontal arrow are quasi-isomorphisms." ], "refs": [ "cohomology-lemma-K-flat-resolution", "cohomology-lemma-K-flat-quasi-isomorphism" ], "ref_ids": [ 2112, 2103 ] } ], "ref_ids": [] }, { "id": 2114, "type": "theorem", "label": "cohomology-lemma-flat-tor-zero", "categories": [ "cohomology" ], "title": "cohomology-lemma-flat-tor-zero", "contents": [ "\\begin{slogan}", "Tor measures the deviation of flatness.", "\\end{slogan}", "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a flat $\\mathcal{O}_X$-module, and", "\\item $\\text{Tor}_1^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) = 0$", "for every $\\mathcal{O}_X$-module $\\mathcal{G}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{F}$ is flat, then $\\mathcal{F} \\otimes_{\\mathcal{O}_X} -$", "is an exact functor and the satellites vanish. Conversely assume (2)", "holds. Then if $\\mathcal{G} \\to \\mathcal{H}$ is injective with cokernel", "$\\mathcal{Q}$, the long exact sequence of $\\text{Tor}$ shows that", "the kernel of", "$\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to", "\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{H}$", "is a quotient of", "$\\text{Tor}_1^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{Q})$", "which is zero by assumption. Hence $\\mathcal{F}$ is flat." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2115, "type": "theorem", "label": "cohomology-lemma-factor-through-K-flat", "categories": [ "cohomology" ], "title": "cohomology-lemma-factor-through-K-flat", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $a : \\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet$ be a map of complexes", "of $\\mathcal{O}_X$-modules. If $\\mathcal{K}^\\bullet$ is K-flat, then", "there exist a complex $\\mathcal{N}^\\bullet$ and maps of complexes", "$b : \\mathcal{K}^\\bullet \\to \\mathcal{N}^\\bullet$", "and $c : \\mathcal{N}^\\bullet \\to \\mathcal{L}^\\bullet$ such that", "\\begin{enumerate}", "\\item $\\mathcal{N}^\\bullet$ is K-flat,", "\\item $c$ is a quasi-isomorphism,", "\\item $a$ is homotopic to $c \\circ b$.", "\\end{enumerate}", "If the terms of $\\mathcal{K}^\\bullet$ are flat, then we may choose", "$\\mathcal{N}^\\bullet$, $b$, and $c$", "such that the same is true for $\\mathcal{N}^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "We will use that the homotopy category $K(\\textit{Mod}(\\mathcal{O}_X))$", "is a triangulated category, see Derived Categories, Proposition", "\\ref{derived-proposition-homotopy-category-triangulated}.", "Choose a distinguished triangle", "$\\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet \\to", "\\mathcal{C}^\\bullet \\to \\mathcal{K}^\\bullet[1]$.", "Choose a quasi-isomorphism $\\mathcal{M}^\\bullet \\to \\mathcal{C}^\\bullet$ with", "$\\mathcal{M}^\\bullet$ K-flat with flat terms, see", "Lemma \\ref{lemma-K-flat-resolution}.", "By the axioms of triangulated categories,", "we may fit the composition", "$\\mathcal{M}^\\bullet \\to \\mathcal{C}^\\bullet \\to \\mathcal{K}^\\bullet[1]$", "into a distinguished triangle", "$\\mathcal{K}^\\bullet \\to \\mathcal{N}^\\bullet \\to", "\\mathcal{M}^\\bullet \\to \\mathcal{K}^\\bullet[1]$.", "By Lemma \\ref{lemma-K-flat-two-out-of-three} we see that", "$\\mathcal{N}^\\bullet$ is K-flat.", "Again using the axioms of triangulated categories,", "we can choose a map $\\mathcal{N}^\\bullet \\to \\mathcal{L}^\\bullet$ fitting into", "the following morphism of distinghuised triangles", "$$", "\\xymatrix{", "\\mathcal{K}^\\bullet \\ar[r] \\ar[d] &", "\\mathcal{N}^\\bullet \\ar[r] \\ar[d] &", "\\mathcal{M}^\\bullet \\ar[r] \\ar[d] &", "\\mathcal{K}^\\bullet[1] \\ar[d] \\\\", "\\mathcal{K}^\\bullet \\ar[r] &", "\\mathcal{L}^\\bullet \\ar[r] &", "\\mathcal{C}^\\bullet \\ar[r] &", "\\mathcal{K}^\\bullet[1]", "}", "$$", "Since two out of three of the arrows are quasi-isomorphisms, so is", "the third arrow $\\mathcal{N}^\\bullet \\to \\mathcal{L}^\\bullet$", "by the long exact sequences", "of cohomology associated to these distinguished triangles", "(or you can look at the image of this diagram in $D(\\mathcal{O}_X)$ and use", "Derived Categories, Lemma \\ref{derived-lemma-third-isomorphism-triangle}", "if you like). This finishes the proof of (1), (2), and (3).", "To prove the final assertion, we may choose $\\mathcal{N}^\\bullet$", "such that $\\mathcal{N}^n \\cong \\mathcal{M}^n \\oplus \\mathcal{K}^n$, see", "Derived Categories, Lemma", "\\ref{derived-lemma-improve-distinguished-triangle-homotopy}.", "Hence we get the desired flatness", "if the terms of $\\mathcal{K}^\\bullet$ are flat." ], "refs": [ "derived-proposition-homotopy-category-triangulated", "cohomology-lemma-K-flat-resolution", "cohomology-lemma-K-flat-two-out-of-three", "derived-lemma-third-isomorphism-triangle", "derived-lemma-improve-distinguished-triangle-homotopy" ], "ref_ids": [ 1960, 2112, 2106, 1759, 1809 ] } ], "ref_ids": [] }, { "id": 2116, "type": "theorem", "label": "cohomology-lemma-derived-base-change", "categories": [ "cohomology" ], "title": "cohomology-lemma-derived-base-change", "contents": [ "The construction above is independent of choices and defines an exact", "functor of triangulated categories", "$Lf^* : D(\\mathcal{O}_Y) \\to D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "To see this we use the general theory developed in", "Derived Categories, Section \\ref{derived-section-derived-functors}.", "Set $\\mathcal{D} = K(\\mathcal{O}_Y)$ and $\\mathcal{D}' = D(\\mathcal{O}_X)$.", "Let us write $F : \\mathcal{D} \\to \\mathcal{D}'$ the exact functor", "of triangulated categories defined by the rule", "$F(\\mathcal{G}^\\bullet) = f^*\\mathcal{G}^\\bullet$.", "We let $S$ be the set of quasi-isomorphisms in", "$\\mathcal{D} = K(\\mathcal{O}_Y)$.", "This gives a situation as in", "Derived Categories, Situation \\ref{derived-situation-derived-functor}", "so that", "Derived Categories, Definition", "\\ref{derived-definition-right-derived-functor-defined}", "applies. We claim that $LF$ is everywhere defined.", "This follows from", "Derived Categories, Lemma \\ref{derived-lemma-find-existence-computes}", "with $\\mathcal{P} \\subset \\Ob(\\mathcal{D})$ the collection", "of $K$-flat complexes: (1) follows from", "Lemma \\ref{lemma-K-flat-resolution}", "and to see (2) we have to show that for a quasi-isomorphism", "$\\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet$ between", "K-flat complexes of $\\mathcal{O}_Y$-modules the map", "$f^*\\mathcal{K}_1^\\bullet \\to f^*\\mathcal{K}_2^\\bullet$ is a", "quasi-isomorphism. To see this write this as", "$$", "f^{-1}\\mathcal{K}_1^\\bullet \\otimes_{f^{-1}\\mathcal{O}_Y} \\mathcal{O}_X", "\\longrightarrow", "f^{-1}\\mathcal{K}_2^\\bullet \\otimes_{f^{-1}\\mathcal{O}_Y} \\mathcal{O}_X", "$$", "The functor $f^{-1}$ is exact, hence the map", "$f^{-1}\\mathcal{K}_1^\\bullet \\to f^{-1}\\mathcal{K}_2^\\bullet$ is a", "quasi-isomorphism. By", "Lemma \\ref{lemma-pullback-K-flat}", "applied to the morphism $(X, f^{-1}\\mathcal{O}_Y) \\to (Y, \\mathcal{O}_Y)$", "the complexes $f^{-1}\\mathcal{K}_1^\\bullet$ and $f^{-1}\\mathcal{K}_2^\\bullet$", "are K-flat complexes of $f^{-1}\\mathcal{O}_Y$-modules. Hence", "Lemma \\ref{lemma-derived-tor-quasi-isomorphism-other-side}", "guarantees that the displayed map is a quasi-isomorphism.", "Thus we obtain a derived functor", "$$", "LF :", "D(\\mathcal{O}_Y) = S^{-1}\\mathcal{D}", "\\longrightarrow", "\\mathcal{D}' = D(\\mathcal{O}_X)", "$$", "see", "Derived Categories, Equation (\\ref{derived-equation-everywhere}).", "Finally,", "Derived Categories, Lemma \\ref{derived-lemma-find-existence-computes}", "also guarantees that", "$LF(\\mathcal{K}^\\bullet) = F(\\mathcal{K}^\\bullet) = f^*\\mathcal{K}^\\bullet$", "when $\\mathcal{K}^\\bullet$ is K-flat, i.e., $Lf^* = LF$ is", "indeed computed in the way described above." ], "refs": [ "derived-definition-right-derived-functor-defined", "derived-lemma-find-existence-computes", "cohomology-lemma-K-flat-resolution", "cohomology-lemma-pullback-K-flat", "cohomology-lemma-derived-tor-quasi-isomorphism-other-side", "derived-lemma-find-existence-computes" ], "ref_ids": [ 1987, 1832, 2112, 2108, 2113, 1832 ] } ], "ref_ids": [] }, { "id": 2117, "type": "theorem", "label": "cohomology-lemma-derived-pullback-composition", "categories": [ "cohomology" ], "title": "cohomology-lemma-derived-pullback-composition", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces.", "Then $Lf^* \\circ Lg^* = L(g \\circ f)^*$ as functors", "$D(\\mathcal{O}_Z) \\to D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Let $E$ be an object of $D(\\mathcal{O}_Z)$.", "By construction $Lg^*E$ is computed by choosing a K-flat complex", "$\\mathcal{K}^\\bullet$ representing $E$ on $Z$ and", "setting $Lg^*E = g^*\\mathcal{K}^\\bullet$.", "By Lemma \\ref{lemma-pullback-K-flat} we see that $g^*\\mathcal{K}^\\bullet$", "is K-flat on $Y$. Then $Lf^*Lg^*E$ is given by", "$f^*g^*\\mathcal{K}^\\bullet = (g \\circ f)^*\\mathcal{K}^\\bullet$", "which also represents $L(g \\circ f)^*E$." ], "refs": [ "cohomology-lemma-pullback-K-flat" ], "ref_ids": [ 2108 ] } ], "ref_ids": [] }, { "id": 2118, "type": "theorem", "label": "cohomology-lemma-pullback-tensor-product", "categories": [ "cohomology" ], "title": "cohomology-lemma-pullback-tensor-product", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces. There is a canonical bifunctorial", "isomorphism", "$$", "Lf^*(", "\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_Y}^{\\mathbf{L}} \\mathcal{G}^\\bullet", ") =", "Lf^*\\mathcal{F}^\\bullet ", "\\otimes_{\\mathcal{O}_X}^{\\mathbf{L}}", "Lf^*\\mathcal{G}^\\bullet ", "$$", "for $\\mathcal{F}^\\bullet, \\mathcal{G}^\\bullet \\in \\Ob(D(X))$." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $\\mathcal{F}^\\bullet$ and $\\mathcal{G}^\\bullet$", "are K-flat complexes. In this case", "$\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_Y}^{\\mathbf{L}} \\mathcal{G}^\\bullet$", "is just the total complex associated to the double complex", "$\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_Y} \\mathcal{G}^\\bullet$.", "By", "Lemma \\ref{lemma-tensor-product-K-flat}", "$\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_Y} \\mathcal{G}^\\bullet)$", "is K-flat also. Hence the isomorphism of the lemma comes from the", "isomorphism", "$$", "\\text{Tot}(f^*\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X}", "f^*\\mathcal{G}^\\bullet)", "\\longrightarrow", "f^*\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_Y} \\mathcal{G}^\\bullet)", "$$", "whose constituents are the isomorphisms", "$f^*\\mathcal{F}^p \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}^q \\to", "f^*(\\mathcal{F}^p \\otimes_{\\mathcal{O}_Y} \\mathcal{G}^q)$ of", "Modules, Lemma \\ref{modules-lemma-tensor-product-pullback}." ], "refs": [ "cohomology-lemma-tensor-product-K-flat", "modules-lemma-tensor-product-pullback" ], "ref_ids": [ 2105, 13270 ] } ], "ref_ids": [] }, { "id": 2119, "type": "theorem", "label": "cohomology-lemma-variant-derived-pullback", "categories": [ "cohomology" ], "title": "cohomology-lemma-variant-derived-pullback", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces. There is a canonical bifunctorial", "isomorphism", "$$", "\\mathcal{F}^\\bullet", "\\otimes_{\\mathcal{O}_X}^{\\mathbf{L}}", "Lf^*\\mathcal{G}^\\bullet", "=", "\\mathcal{F}^\\bullet ", "\\otimes_{f^{-1}\\mathcal{O}_Y}^{\\mathbf{L}}", "f^{-1}\\mathcal{G}^\\bullet ", "$$", "for $\\mathcal{F}^\\bullet$ in $D(X)$ and $\\mathcal{G}^\\bullet$ in $D(Y)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module and let $\\mathcal{G}$", "be an $\\mathcal{O}_Y$-module. Then", "$\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G} =", "\\mathcal{F} \\otimes_{f^{-1}\\mathcal{O}_Y} f^{-1}\\mathcal{G}$", "because", "$f^*\\mathcal{G} =", "\\mathcal{O}_X \\otimes_{f^{-1}\\mathcal{O}_Y} f^{-1}\\mathcal{G}$.", "The lemma follows from this and the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2120, "type": "theorem", "label": "cohomology-lemma-tensor-pull-compatibility", "categories": [ "cohomology" ], "title": "cohomology-lemma-tensor-pull-compatibility", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism", "of ringed spaces. Let $\\mathcal{K}^\\bullet$ and $\\mathcal{M}^\\bullet$", "be complexes of $\\mathcal{O}_Y$-modules. The diagram", "$$", "\\xymatrix{", "Lf^*(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "\\mathcal{M}^\\bullet) \\ar[r] \\ar[d] &", "Lf^*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}", "\\mathcal{M}^\\bullet) \\ar[d] \\\\", "Lf^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "Lf^*\\mathcal{M}^\\bullet \\ar[d] &", "f^*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}", "\\mathcal{M}^\\bullet) \\ar[d] \\\\", "f^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "f^*\\mathcal{M}^\\bullet \\ar[r] &", "\\text{Tot}(f^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}", "f^*\\mathcal{M}^\\bullet)", "}", "$$", "commutes." ], "refs": [], "proofs": [ { "contents": [ "We will use the existence of K-flat resolutions as in", "Lemma \\ref{lemma-pullback-K-flat}. If we choose such", "resolutions $\\mathcal{P}^\\bullet \\to \\mathcal{K}^\\bullet$", "and $\\mathcal{Q}^\\bullet \\to \\mathcal{M}^\\bullet$, then", "we see that", "$$", "\\xymatrix{", "Lf^*\\text{Tot}(\\mathcal{P}^\\bullet", "\\otimes_{\\mathcal{O}_Y}", "\\mathcal{Q}^\\bullet) \\ar[r] \\ar[d] &", "Lf^*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}", "\\mathcal{M}^\\bullet) \\ar[d] \\\\", "f^*\\text{Tot}(\\mathcal{P}^\\bullet", "\\otimes_{\\mathcal{O}_Y}", "\\mathcal{Q}^\\bullet) \\ar[d] \\ar[r] &", "f^*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}", "\\mathcal{M}^\\bullet) \\ar[d] \\\\", "\\text{Tot}(f^*\\mathcal{P}^\\bullet \\otimes_{\\mathcal{O}_X}", "f^*\\mathcal{Q}^\\bullet) \\ar[r] &", "\\text{Tot}(f^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}", "f^*\\mathcal{M}^\\bullet)", "}", "$$", "commutes. However, now the left hand side of the diagram", "is the left hand side of the diagram by our choice of", "$\\mathcal{P}^\\bullet$ and $\\mathcal{Q}^\\bullet$ and", "Lemma \\ref{lemma-tensor-product-K-flat}." ], "refs": [ "cohomology-lemma-pullback-K-flat", "cohomology-lemma-tensor-product-K-flat" ], "ref_ids": [ 2108, 2105 ] } ], "ref_ids": [] }, { "id": 2121, "type": "theorem", "label": "cohomology-lemma-adjoint", "categories": [ "cohomology" ], "title": "cohomology-lemma-adjoint", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of", "ringed spaces. The functor $Rf_*$ defined above and the functor $Lf^*$", "defined in Lemma \\ref{lemma-derived-base-change} are adjoint:", "$$", "\\Hom_{D(X)}(Lf^*\\mathcal{G}^\\bullet, \\mathcal{F}^\\bullet)", "=", "\\Hom_{D(Y)}(\\mathcal{G}^\\bullet, Rf_*\\mathcal{F}^\\bullet)", "$$", "bifunctorially in $\\mathcal{F}^\\bullet \\in \\Ob(D(X))$ and", "$\\mathcal{G}^\\bullet \\in \\Ob(D(Y))$." ], "refs": [ "cohomology-lemma-derived-base-change" ], "proofs": [ { "contents": [ "This follows formally from the fact that $Rf_*$ and $Lf^*$ exist, see", "Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors}." ], "refs": [ "derived-lemma-derived-adjoint-functors" ], "ref_ids": [ 1907 ] } ], "ref_ids": [ 2116 ] }, { "id": 2122, "type": "theorem", "label": "cohomology-lemma-derived-pushforward-composition", "categories": [ "cohomology" ], "title": "cohomology-lemma-derived-pushforward-composition", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces.", "Then $Rg_* \\circ Rf_* = R(g \\circ f)_*$ as functors", "$D(\\mathcal{O}_X) \\to D(\\mathcal{O}_Z)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-adjoint} we see that $Rg_* \\circ Rf_*$", "is adjoint to $Lf^* \\circ Lg^*$. We have", "$Lf^* \\circ Lg^* = L(g \\circ f)^*$ by", "Lemma \\ref{lemma-derived-pullback-composition}", "and hence by", "uniqueness of adjoint functors we have $Rg_* \\circ Rf_* = R(g \\circ f)_*$." ], "refs": [ "cohomology-lemma-adjoint", "cohomology-lemma-derived-pullback-composition" ], "ref_ids": [ 2121, 2117 ] } ], "ref_ids": [] }, { "id": 2123, "type": "theorem", "label": "cohomology-lemma-adjoints-push-pull-compatibility", "categories": [ "cohomology" ], "title": "cohomology-lemma-adjoints-push-pull-compatibility", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces. Let $\\mathcal{K}^\\bullet$", "be a complex of $\\mathcal{O}_X$-modules.", "The diagram", "$$", "\\xymatrix{", "Lf^*f_*\\mathcal{K}^\\bullet \\ar[r] \\ar[d] &", "f^*f_*\\mathcal{K}^\\bullet \\ar[d] \\\\", "Lf^*Rf_*\\mathcal{K}^\\bullet \\ar[r] &", "\\mathcal{K}^\\bullet", "}", "$$", "coming from $Lf^* \\to f^*$ on complexes, $f_* \\to Rf_*$ on complexes,", "and adjunction $Lf^* \\circ Rf_* \\to \\text{id}$", "commutes in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "We will use the existence of K-flat resolutions and", "K-injective resolutions, see Lemma \\ref{lemma-pullback-K-flat}", "and the discussion above. Choose a quasi-isomorphism", "$\\mathcal{K}^\\bullet \\to \\mathcal{I}^\\bullet$ where $\\mathcal{I}^\\bullet$", "is K-injective as a complex of $\\mathcal{O}_X$-modules.", "Choose a quasi-isomorphism $\\mathcal{Q}^\\bullet \\to f_*\\mathcal{I}^\\bullet$", "where $\\mathcal{Q}^\\bullet$ is K-flat as a complex of", "$\\mathcal{O}_Y$-modules. We can choose a K-flat complex of", "$\\mathcal{O}_Y$-modules $\\mathcal{P}^\\bullet$", "and a diagram of morphisms of complexes", "$$", "\\xymatrix{", "\\mathcal{P}^\\bullet \\ar[r] \\ar[d] &", "f_*\\mathcal{K}^\\bullet \\ar[d] \\\\", "\\mathcal{Q}^\\bullet \\ar[r] & f_*\\mathcal{I}^\\bullet", "}", "$$", "commutative up to homotopy where the top horizontal arrow", "is a quasi-isomorphism. Namely, we can first choose such a", "diagram for some complex $\\mathcal{P}^\\bullet$ because", "the quasi-isomorphisms form a multiplicative system in", "the homotopy category of complexes and then we can replace", "$\\mathcal{P}^\\bullet$ by a K-flat complex.", "Taking pullbacks we obtain a diagram of morphisms of complexes", "$$", "\\xymatrix{", "f^*\\mathcal{P}^\\bullet \\ar[r] \\ar[d] &", "f^*f_*\\mathcal{K}^\\bullet \\ar[d] \\ar[r] &", "\\mathcal{K}^\\bullet \\ar[d] \\\\", "f^*\\mathcal{Q}^\\bullet \\ar[r] &", "f^*f_*\\mathcal{I}^\\bullet \\ar[r] &", "\\mathcal{I}^\\bullet", "}", "$$", "commutative up to homotopy. The outer rectangle witnesses the", "truth of the statement in the lemma." ], "refs": [ "cohomology-lemma-pullback-K-flat" ], "ref_ids": [ 2108 ] } ], "ref_ids": [] }, { "id": 2124, "type": "theorem", "label": "cohomology-lemma-spectral-sequence-filtered-object", "categories": [ "cohomology" ], "title": "cohomology-lemma-spectral-sequence-filtered-object", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}^\\bullet$ be a", "filtered complex of $\\mathcal{O}_X$-modules. There exists a canonical", "spectral sequence $(E_r, \\text{d}_r)_{r \\geq 1}$ of bigraded", "$\\Gamma(X, \\mathcal{O}_X)$-modules with $d_r$ of bidegree $(r, -r + 1)$ and", "$$", "E_1^{p, q} = H^{p + q}(X, \\text{gr}^p\\mathcal{F}^\\bullet)", "$$", "If for every $n$ we have", "$$", "H^n(X, F^p\\mathcal{F}^\\bullet) = 0\\text{ for }p \\gg 0", "\\quad\\text{and}\\quad", "H^n(X, F^p\\mathcal{F}^\\bullet) = H^n(X, \\mathcal{F}^\\bullet)\\text{ for }p \\ll 0", "$$", "then the spectral sequence is bounded and converges to", "$H^*(X, \\mathcal{F}^\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "(For a proof in case the complex is a bounded below complex", "of modules with finite filtrations, see the remark below.)", "Choose an map of filtered complexes", "$j : \\mathcal{F}^\\bullet \\to \\mathcal{J}^\\bullet$ as in", "Injectives, Lemma", "\\ref{injectives-lemma-K-injective-embedding-filtration}.", "The spectral sequence is the spectral sequence of", "Homology, Section \\ref{homology-section-filtered-complex}", "associated to the filtered complex", "$$", "\\Gamma(X, \\mathcal{J}^\\bullet)", "\\quad\\text{with}\\quad", "F^p\\Gamma(X, \\mathcal{J}^\\bullet) = \\Gamma(X, F^p\\mathcal{J}^\\bullet)", "$$", "Since cohomology is computed by evaluating on K-injective representatives", "we see that the $E_1$ page is as stated in the lemma.", "The convergence and boundedness under the stated conditions", "follows from Homology, Lemma \\ref{homology-lemma-ss-converges-trivial}." ], "refs": [ "injectives-lemma-K-injective-embedding-filtration", "homology-lemma-ss-converges-trivial" ], "ref_ids": [ 7797, 12103 ] } ], "ref_ids": [] }, { "id": 2125, "type": "theorem", "label": "cohomology-lemma-relative-spectral-sequence-filtered-object", "categories": [ "cohomology" ], "title": "cohomology-lemma-relative-spectral-sequence-filtered-object", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of", "ringed spaces. Let $\\mathcal{F}^\\bullet$ be a filtered complex of", "$\\mathcal{O}_X$-modules. There exists a canonical spectral sequence", "$(E_r, \\text{d}_r)_{r \\geq 1}$ of bigraded", "$\\mathcal{O}_Y$-modules with $d_r$ of bidegree $(r, -r + 1)$ and", "$$", "E_1^{p, q} = R^{p + q}f_*\\text{gr}^p\\mathcal{F}^\\bullet", "$$", "If for every $n$ we have", "$$", "R^nf_*F^p\\mathcal{F}^\\bullet = 0 \\text{ for }p \\gg 0", "\\quad\\text{and}\\quad", "R^nf_*F^p\\mathcal{F}^\\bullet = R^nf_*\\mathcal{F}^\\bullet \\text{ for }p \\ll 0", "$$", "then the spectral sequence is bounded and converges to", "$Rf_*\\mathcal{F}^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Lemma \\ref{lemma-spectral-sequence-filtered-object}." ], "refs": [ "cohomology-lemma-spectral-sequence-filtered-object" ], "ref_ids": [ 2124 ] } ], "ref_ids": [] }, { "id": 2126, "type": "theorem", "label": "cohomology-lemma-godement-resolution", "categories": [ "cohomology" ], "title": "cohomology-lemma-godement-resolution", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. For every sheaf of", "$\\mathcal{O}_X$-modules $\\mathcal{F}$ there is a resolution", "$$", "0 \\to", "\\mathcal{F} \\to", "f_*f^*\\mathcal{F} \\to", "f_*f^*f_*f^*\\mathcal{F} \\to", "f_*f^*f_*f^*f_*f^*\\mathcal{F} \\to \\ldots", "$$", "functorial in $\\mathcal{F}$ such that each term", "$f_*f^* \\ldots f_*f^*\\mathcal{F}$ is a flasque", "$\\mathcal{O}_X$-module and such that for all $x \\in X$ the", "map", "$$", "\\mathcal{F}_x[0] \\to \\Big(", "(f_*f^*\\mathcal{F})_x \\to", "(f_*f^*f_*f^*\\mathcal{F})_x \\to", "(f_*f^*f_*f^*f_*f^*\\mathcal{F})_x \\to \\ldots", "\\Big)", "$$", "is a homotopy equivalence in the category of complexes", "of $\\mathcal{O}_{X, x}$-modules." ], "refs": [], "proofs": [ { "contents": [ "The complex $f_*f^*\\mathcal{F} \\to f_*f^*f_*f^*\\mathcal{F} \\to", "f_*f^*f_*f^*f_*f^*\\mathcal{F} \\to \\ldots$ is the complex associated", "to the cosimplicial object with terms", "$f_*f^*\\mathcal{F}, f_*f^*f_*f^*\\mathcal{F},", "f_*f^*f_*f^*f_*f^*\\mathcal{F}, \\ldots$ described above, see", "Simplicial, Section \\ref{simplicial-section-dold-kan-cosimplicial}.", "The augmentation gives rise to the map $\\mathcal{F} \\to f_*f^*\\mathcal{F}$", "as indicated. For any abelian sheaf $\\mathcal{H}$ on $X_{disc}$ the", "pushforward $f_*\\mathcal{H}$ is flasque because $X_{disc}$", "is a discrete space and the pushforward of a flasque sheaf is flasque.", "Hence the terms of the complex are flasque $\\mathcal{O}_X$-modules.", "\\medskip\\noindent", "If $x \\in X_{disc} = X$ is a point, then $(f^*\\mathcal{G})_x = \\mathcal{G}_x$", "for any $\\mathcal{O}_X$-module $\\mathcal{G}$. Hence $f^*$ is an exact functor", "and a complex of $\\mathcal{O}_X$-modules", "$\\mathcal{G}_1 \\to \\mathcal{G}_2 \\to \\mathcal{G}_3$", "is exact if and only if", "$f^*\\mathcal{G}_1 \\to f^*\\mathcal{G}_2 \\to f^*\\mathcal{G}_3$", "is exact (see Modules, Lemma \\ref{modules-lemma-abelian}).", "The result mentioned in the introduction to this section", "proves the pullback by $f^*$ gives a homotopy equivalence from", "the constant cosimplicial object $f^*\\mathcal{F}$ to the", "cosimplicial object with terms", "$f_*f^*\\mathcal{F}, f_*f^*f_*f^*\\mathcal{F},", "f_*f^*f_*f^*f_*f^*\\mathcal{F}, \\ldots$.", "By Simplicial, Lemma \\ref{simplicial-lemma-homotopy-equivalence-s-Q}", "we obtain that", "$$", "f^*\\mathcal{F}[0] \\to \\Big(", "f^*f_*f^*\\mathcal{F} \\to", "f^*f_*f^*f_*f^*\\mathcal{F} \\to", "f^*f_*f^*f_*f^*f_*f^*\\mathcal{F} \\to \\ldots", "\\Big)", "$$", "is a homotopy equivalence. This immediately implies the two remaining", "statements of the lemma." ], "refs": [ "modules-lemma-abelian", "simplicial-lemma-homotopy-equivalence-s-Q" ], "ref_ids": [ 13221, 14882 ] } ], "ref_ids": [] }, { "id": 2127, "type": "theorem", "label": "cohomology-lemma-godement-resolution-bounded-below", "categories": [ "cohomology" ], "title": "cohomology-lemma-godement-resolution-bounded-below", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let", "$\\mathcal{F}^\\bullet$ be a bounded below complex of", "$\\mathcal{O}_X$-modules. There exists a quasi-isomorphism", "$\\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$", "where $\\mathcal{F}^\\bullet$ be a bounded below complex of flasque", "$\\mathcal{O}_X$-modules and for all $x \\in X$ the", "map $\\mathcal{F}^\\bullet_x \\to \\mathcal{G}^\\bullet_x$", "is a homotopy equivalence in the category of complexes", "of $\\mathcal{O}_{X, x}$-modules." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{A}$ be the category of complexes of $\\mathcal{O}_X$-modules", "and let $\\mathcal{B}$ be the category of complexes of $\\mathcal{O}_X$-modules.", "Then we can apply the discussion above to the adjoint functors", "$f^*$ and $f_*$ between $\\mathcal{A}$ and $\\mathcal{B}$.", "Arguing exactly as in the proof of", "Lemma \\ref{lemma-godement-resolution}", "we get a resolution", "$$", "0 \\to", "\\mathcal{F}^\\bullet \\to", "f_*f^*\\mathcal{F}^\\bullet \\to", "f_*f^*f_*f^*\\mathcal{F}^\\bullet \\to", "f_*f^*f_*f^*f_*f^*\\mathcal{F}^\\bullet \\to \\ldots", "$$", "in the abelian category $\\mathcal{A}$ such that each term of each", "$f_*f^*\\ldots f_*f^*\\mathcal{F}^\\bullet$ is a flasque", "$\\mathcal{O}_X$-module and such that for all $x \\in X$ the", "map", "$$", "\\mathcal{F}^\\bullet_x[0] \\to \\Big(", "(f_*f^*\\mathcal{F}^\\bullet)_x \\to", "(f_*f^*f_*f^*\\mathcal{F}^\\bullet)_x \\to", "(f_*f^*f_*f^*f_*f^*\\mathcal{F}^\\bullet)_x \\to \\ldots", "\\Big)", "$$", "is a homotopy equivalence in the category of complexes of complexes", "of $\\mathcal{O}_{X, x}$-modules. Since a complex of complexes is the", "same thing as a double complex, we can consider the induced map", "$$", "\\mathcal{F}^\\bullet \\to", "\\mathcal{G}^\\bullet =", "\\text{Tot}(", "f_*f^*\\mathcal{F}^\\bullet \\to", "f_*f^*f_*f^*\\mathcal{F}^\\bullet \\to", "f_*f^*f_*f^*f_*f^*\\mathcal{F}^\\bullet \\to \\ldots", ")", "$$", "Since the complex $\\mathcal{F}^\\bullet$ is bounded below, the", "same is true for $\\mathcal{G}^\\bullet$ and in fact each term", "of $\\mathcal{G}^\\bullet$ is a finite direct sum of", "terms of the complexes $f_*f^*\\ldots f_*f^*\\mathcal{F}^\\bullet$", "and hence is flasque. The final assertion of the lemma", "now follows from", "Homology, Lemma \\ref{homology-lemma-homotopy-complex-complexes}.", "Since this in particular shows that", "$\\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$", "is a quasi-isomorphism, the proof is complete." ], "refs": [ "cohomology-lemma-godement-resolution", "homology-lemma-homotopy-complex-complexes" ], "ref_ids": [ 2126, 12107 ] } ], "ref_ids": [] }, { "id": 2128, "type": "theorem", "label": "cohomology-lemma-second-cup-equals-first", "categories": [ "cohomology" ], "title": "cohomology-lemma-second-cup-equals-first", "contents": [ "This construction gives the cup product." ], "refs": [], "proofs": [ { "contents": [ "With $f : (X, \\mathcal{O}_X) \\to (pt, A)$ as above we have", "$Rf_*(-) = R\\Gamma(X, -)$ and our map $\\mu$ is adjoint to the map", "$$", "Lf^*(Rf_*K \\otimes_A^\\mathbf{L} Rf_*M) =", "Lf^*Rf_*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*Rf_*M", "\\xrightarrow{\\epsilon_K \\otimes \\epsilon_M}", "K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M", "$$", "where $\\epsilon$ is the counit of the adjunction between", "$Lf^*$ and $Rf_*$.", "If we think of $\\xi$ and $\\eta$ as maps $\\xi : A[-i] \\to R\\Gamma(X, K)$", "and $\\eta : A[-j] \\to R\\Gamma(X, M)$, then", "the tensor $\\xi \\otimes \\eta$ corresponds to the map\\footnote{There", "is a sign hidden here, namely, the equality is defined by", "the composition", "$$", "A[-i - j] \\to (A \\otimes_A^\\mathbf{L} A)[-i - j] \\to", "A[-i] \\otimes_A^\\mathbf{L} A[-j]", "$$", "where in the second step we use the identification of", "More on Algebra, Item (\\ref{more-algebra-item-shift-tensor})", "which uses a sign in principle.", "Except, in this case the sign is $+1$ by our convention and even if it wasn't", "$+1$ it wouldn't matter since we used the same sign", "in the identification", "$\\mathcal{O}_X[-i - j] =", "\\mathcal{O}_X[-i] \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_X[-j]$.}", "$$", "A[-i - j] = A[-i] \\otimes_A^\\mathbf{L} A[-j]", "\\xrightarrow{\\xi \\otimes \\eta}", "R\\Gamma(X, K) \\otimes_A^\\mathbf{L} R\\Gamma(X, M)", "$$", "By definition the cup product $\\xi \\cup \\eta$ is the map", "$A[-i - j] \\to R\\Gamma(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)$", "which is adjoint to", "$$", "(\\epsilon_K \\otimes \\epsilon_M) \\circ Lf^*(\\xi \\otimes \\eta) =", "(\\epsilon_K \\circ Lf^*\\xi) \\otimes (\\epsilon_M \\circ Lf^*\\eta)", "$$", "However, it is easy to see that", "$\\epsilon_K \\circ Lf^*\\xi = \\tilde \\xi$ and", "$\\epsilon_M \\circ Lf^*\\eta = \\tilde \\eta$.", "We conclude that $\\widetilde{\\xi \\cup \\eta} = \\tilde \\xi \\otimes \\tilde \\eta$", "which means we have the desired agreement." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2129, "type": "theorem", "label": "cohomology-lemma-cup-compatible-with-naive", "categories": [ "cohomology" ], "title": "cohomology-lemma-cup-compatible-with-naive", "contents": [ "In the situation above the following diagram commutes", "$$", "\\xymatrix{", "f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "f_*\\mathcal{M}^\\bullet \\ar[r] \\ar[d]", "&", "Rf_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "Rf_*\\mathcal{M}^\\bullet \\ar[d]^{\\text{Remark \\ref{remark-cup-product}}} \\\\", "\\text{Tot}(", "f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}", "f_*\\mathcal{M}^\\bullet) \\ar[d]_{\\text{naive cup product}} &", "Rf_*(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "\\mathcal{M}^\\bullet) \\ar[d] \\\\", "f_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet) \\ar[r] &", "Rf_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet)", "}", "$$" ], "refs": [ "cohomology-remark-cup-product" ], "proofs": [ { "contents": [ "By the construction in Remark \\ref{remark-cup-product} we see that", "going around the diagram clockwise the map", "$$", "f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "f_*\\mathcal{M}^\\bullet ", "\\longrightarrow", "Rf_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet)", "$$", "is adjoint to the map", "\\begin{align*}", "Lf^*(f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "f_*\\mathcal{M}^\\bullet)", "& =", "Lf^*f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "Lf^*f_*\\mathcal{M}^\\bullet \\\\", "& \\to", "Lf^*Rf_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "Lf^*Rf_*\\mathcal{M}^\\bullet \\\\", "& \\to", "\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "\\mathcal{M}^\\bullet \\\\", "& \\to", "\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet)", "\\end{align*}", "By Lemma \\ref{lemma-adjoints-push-pull-compatibility} this is also equal to", "\\begin{align*}", "Lf^*(f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "f_*\\mathcal{M}^\\bullet)", "& =", "Lf^*f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "Lf^*f_*\\mathcal{M}^\\bullet \\\\", "& \\to", "f^*f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "f^*f_*\\mathcal{M}^\\bullet \\\\", "& \\to", "\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "\\mathcal{M}^\\bullet \\\\", "& \\to", "\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet)", "\\end{align*}", "Going around anti-clockwise we obtain the map adjoint to the map", "\\begin{align*}", "Lf^*(f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "f_*\\mathcal{M}^\\bullet)", "& \\to", "Lf^*\\text{Tot}(", "f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}", "f_*\\mathcal{M}^\\bullet) \\\\", "& \\to", "Lf^*f_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet) \\\\", "& \\to", "Lf^*Rf_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet) \\\\", "& \\to", "\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet)", "\\end{align*}", "By Lemma \\ref{lemma-adjoints-push-pull-compatibility} this is also equal to", "\\begin{align*}", "Lf^*(f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "f_*\\mathcal{M}^\\bullet)", "& \\to", "Lf^*\\text{Tot}(", "f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}", "f_*\\mathcal{M}^\\bullet) \\\\", "& \\to", "Lf^*f_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet) \\\\", "& \\to", "f^*f_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet) \\\\", "& \\to", "\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet)", "\\end{align*}", "Now the proof is finished by a contemplation of the diagram", "$$", "\\xymatrix{", "Lf^*(f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "f_*\\mathcal{M}^\\bullet) \\ar[d] \\ar[rr] & &", "Lf^*f_*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "Lf^*f_*\\mathcal{M}^\\bullet \\ar[d] \\\\", "Lf^*\\text{Tot}(", "f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}", "f_*\\mathcal{M}^\\bullet) \\ar[d]_{naive} \\ar[r] &", "f^*\\text{Tot}(", "f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_Y}", "f_*\\mathcal{M}^\\bullet) \\ar[ldd]^{naive} \\ar[dd] &", "f^*f_*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "f^*f_*\\mathcal{M}^\\bullet \\ar[dd] \\ar[ldd] \\\\", "Lf^*f_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet) \\ar[d] \\\\", "f^*f_*\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet) \\ar[rd] &", "\\text{Tot}(f^*f_*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}", "f^*f_*\\mathcal{M}^\\bullet) \\ar[d] &", "\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "\\mathcal{M}^\\bullet \\ar[ld] \\\\", "& \\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet)", "}", "$$", "All of the polygons in this diagram commute. The top one commutes", "by Lemma \\ref{lemma-tensor-pull-compatibility}.", "The square with the two naive cup products commutes because", "$Lf^* \\to f^*$ is functorial in the complex of modules.", "Similarly with the square involving the two maps", "$\\mathcal{A}^\\bullet \\otimes^\\mathbf{L} \\mathcal{B}^\\bullet \\to", "\\text{Tot}(\\mathcal{A}^\\bullet \\otimes \\mathcal{B}^\\bullet)$.", "Finally, the commutativity of the remaining square", "is true on the level of complexes and may be viewed as the", "definiton of the naive cup product (by the adjointness", "of $f^*$ and $f_*$). The proof is finished because", "going around the diagram on the outside are the two maps", "given above." ], "refs": [ "cohomology-remark-cup-product", "cohomology-lemma-adjoints-push-pull-compatibility", "cohomology-lemma-adjoints-push-pull-compatibility", "cohomology-lemma-tensor-pull-compatibility" ], "ref_ids": [ 2272, 2123, 2123, 2120 ] } ], "ref_ids": [ 2272 ] }, { "id": 2130, "type": "theorem", "label": "cohomology-lemma-diagrams-commute", "categories": [ "cohomology" ], "title": "cohomology-lemma-diagrams-commute", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let", "$\\mathcal{K}^\\bullet$ and $\\mathcal{M}^\\bullet$", "be bounded below complexes of $\\mathcal{O}_X$-modules.", "Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be an open covering", "Then", "$$", "\\xymatrix{", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{K}^\\bullet))", "\\otimes_A^\\mathbf{L}", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{M}^\\bullet))", "\\ar[d] \\ar[r] &", "R\\Gamma(X, \\mathcal{K}^\\bullet)", "\\otimes_A^\\mathbf{L}", "R\\Gamma(X, \\mathcal{M}^\\bullet) \\ar[d]^\\mu \\\\", "\\text{Tot}(", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{K}^\\bullet))", "\\otimes_A", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{M}^\\bullet)))", "\\ar[d]^{(\\ref{equation-needs-signs})} &", "R\\Gamma(X,", "\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{M}^\\bullet)", "\\ar[d] \\\\", "\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet({\\mathcal U},", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{M}^\\bullet)", ")) \\ar[r] &", "R\\Gamma(X,", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{M}^\\bullet))", "}", "$$", "where the horizontal arrows are the ones in", "Lemma \\ref{lemma-cech-complex-complex}", "commutes in $D(A)$." ], "refs": [ "cohomology-lemma-cech-complex-complex" ], "proofs": [ { "contents": [ "Choose quasi-isomorphisms of complexes", "$a : \\mathcal{K}^\\bullet \\to \\mathcal{K}_1^\\bullet$ and", "$b : \\mathcal{M}^\\bullet \\to \\mathcal{M}_1^\\bullet$", "as in Lemma \\ref{lemma-godement-resolution-bounded-below}.", "Since the maps $a$ and $b$ on stalks are homotopy equivalences", "we see that the induced map", "$$", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{M}^\\bullet)", "\\to", "\\text{Tot}(\\mathcal{K}_1^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{M}_1^\\bullet)", "$$", "is a homotopy equivalence on stalks too (More on Algebra, Lemma", "\\ref{more-algebra-lemma-derived-tor-homotopy}) and hence a quasi-isomorphism.", "Thus the targets", "$$", "R\\Gamma(X,", "\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X} \\mathcal{M}^\\bullet)) =", "R\\Gamma(X,", "\\text{Tot}(\\mathcal{K}_1^\\bullet", "\\otimes_{\\mathcal{O}_X} \\mathcal{M}_1^\\bullet))", "$$", "of the two diagrams are the same in $D(A)$. It follows that it suffices", "to prove the diagram commutes for $\\mathcal{K}$ and $\\mathcal{M}$", "replaced by $\\mathcal{K}_1$ and $\\mathcal{M}_1$. This reduces us to", "the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $\\mathcal{K}^\\bullet$ and $\\mathcal{M}^\\bullet$ are bounded", "below complexes of flasque $\\mathcal{O}_X$-modules and", "consider the diagram relating the cup product with the cup product", "(\\ref{equation-needs-signs}) on {\\v C}ech complexes.", "Then we can consider the commutative diagram", "$$", "\\xymatrix{", "\\Gamma(X, \\mathcal{K}^\\bullet)", "\\otimes_A^\\mathbf{L}", "\\Gamma(X, \\mathcal{M}^\\bullet) \\ar[d] \\ar[r] &", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{K}^\\bullet))", "\\otimes_A^\\mathbf{L}", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{M}^\\bullet))", "\\ar[d] \\\\", "\\text{Tot}(\\Gamma(X, \\mathcal{K}^\\bullet)", "\\otimes_A", "\\Gamma(X, \\mathcal{M}^\\bullet)) \\ar[d] \\ar[r] &", "\\text{Tot}(", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{K}^\\bullet))", "\\otimes_A", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{M}^\\bullet)))", "\\ar[d]^{(\\ref{equation-needs-signs})} \\\\", "\\Gamma(X, \\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{M}^\\bullet)) \\ar[r] &", "\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet({\\mathcal U},", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{M}^\\bullet)", "))", "}", "$$", "In this diagram the horizontal arrows are isomorphisms in $D(A)$ because", "for a bounded below complex of flasque modules such as $\\mathcal{K}^\\bullet$", "we have", "$$", "\\Gamma(X, \\mathcal{K}^\\bullet) =", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{K}^\\bullet)) =", "R\\Gamma(X, \\mathcal{K}^\\bullet)", "$$", "in $D(A)$. This follows from", "Lemma \\ref{lemma-flasque-acyclic},", "Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity}, and", "Lemma \\ref{lemma-cech-complex-complex-computes}.", "Hence the commutativity of the diagram of the lemma involving", "(\\ref{equation-needs-signs}) follows from the already proven", "commutativity of Lemma \\ref{lemma-cup-compatible-with-naive}", "where $f$ is the morphism to a point (see discussion", "following Lemma \\ref{lemma-cup-compatible-with-naive})." ], "refs": [ "cohomology-lemma-godement-resolution-bounded-below", "more-algebra-lemma-derived-tor-homotopy", "cohomology-lemma-flasque-acyclic", "derived-lemma-leray-acyclicity", "cohomology-lemma-cech-complex-complex-computes", "cohomology-lemma-cup-compatible-with-naive", "cohomology-lemma-cup-compatible-with-naive" ], "ref_ids": [ 2127, 10121, 2064, 1844, 2099, 2129, 2129 ] } ], "ref_ids": [ 2098 ] }, { "id": 2131, "type": "theorem", "label": "cohomology-lemma-cup-product-associative", "categories": [ "cohomology" ], "title": "cohomology-lemma-cup-product-associative", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces. The relative cup product of", "Remark \\ref{remark-cup-product} is associative in the sense that", "the diagram", "$$", "\\xymatrix{", "Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "Rf_*L \\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "Rf_*M \\ar[r] \\ar[d] &", "Rf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*M \\ar[d] \\\\", "Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "Rf_*(L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\ar[r] &", "Rf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} ", "L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)", "}", "$$", "is commutative in $D(\\mathcal{O}_Y)$ for all $K, L, M$ in $D(\\mathcal{O}_X)$." ], "refs": [ "cohomology-remark-cup-product" ], "proofs": [ { "contents": [ "Going around either side we obtain the map adjoint to the obvious map", "\\begin{align*}", "Lf^*(Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "Rf_*L \\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "Rf_*M) & =", "Lf^*(Rf_*K) \\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "Lf^*(Rf_*L) \\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "Lf^*(Rf_*M) \\\\", "& \\to", "K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} ", "L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M", "\\end{align*}", "in $D(\\mathcal{O}_X)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 2272 ] }, { "id": 2132, "type": "theorem", "label": "cohomology-lemma-cup-product-commutative", "categories": [ "cohomology" ], "title": "cohomology-lemma-cup-product-commutative", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces. The relative cup product of", "Remark \\ref{remark-cup-product} is commutative in the sense that", "the diagram", "$$", "\\xymatrix{", "Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L \\ar[r] \\ar[d]_\\psi &", "Rf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) \\ar[d]^{Rf_*\\psi} \\\\", "Rf_*L \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*K \\ar[r] &", "Rf_*(L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K)", "}", "$$", "is commutative in $D(\\mathcal{O}_Y)$ for all $K, L$ in $D(\\mathcal{O}_X)$.", "Here $\\psi$ is the commutativity constraint on the derived category", "(Lemma \\ref{lemma-symmetric-monoidal-derived})." ], "refs": [ "cohomology-remark-cup-product", "cohomology-lemma-symmetric-monoidal-derived" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 2272, 2234 ] }, { "id": 2133, "type": "theorem", "label": "cohomology-lemma-compose-cup-product", "categories": [ "cohomology" ], "title": "cohomology-lemma-compose-cup-product", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ and", "$g : (Y, \\mathcal{O}_Y) \\to (Z, \\mathcal{O}_Z)$", "be morphisms of ringed spaces. The relative cup product of", "Remark \\ref{remark-cup-product} is compatible with compositions", "in the sense that the diagram", "$$", "\\xymatrix{", "R(g \\circ f)_*K \\otimes_{\\mathcal{O}_Z}^\\mathbf{L} R(g \\circ f)_*L", "\\ar@{=}[rr] \\ar[d] & &", "Rg_*Rf_*K \\otimes_{\\mathcal{O}_Z}^\\mathbf{L} Rg_*Rf_*L \\ar[d] \\\\", "R(g \\circ f)_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) \\ar@{=}[r] &", "Rg_*Rf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) &", "Rg_*(Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L) \\ar[l]", "}", "$$", "is commutative in $D(\\mathcal{O}_Z)$ for all $K, L$ in $D(\\mathcal{O}_X)$." ], "refs": [ "cohomology-remark-cup-product" ], "proofs": [ { "contents": [ "This is true because going around the diagram either way we obtain the map", "adjoint to the map", "\\begin{align*}", "& L(g \\circ f)^*\\left(R(g \\circ f)_*K", "\\otimes_{\\mathcal{O}_Z}^\\mathbf{L}", "R(g \\circ f)_*L\\right) \\\\", "& =", "L(g \\circ f)^*R(g \\circ f)_*K", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "L(g \\circ f)^*R(g \\circ f)_*L) \\\\", "& \\to", "K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L", "\\end{align*}", "in $D(\\mathcal{O}_X)$. To see this one uses that the composition", "of the counits like so", "$$", "L(g \\circ f)^*R(g \\circ f)_* =", "Lf^* Lg^* Rg_* Rf_* \\to", "Lf^* Rf_* \\to \\text{id}", "$$", "is the counit for $L(g \\circ f)^*$ and $R(g \\circ f)_*$. See", "Categories, Lemma \\ref{categories-lemma-compose-counits}." ], "refs": [ "categories-lemma-compose-counits" ], "ref_ids": [ 12252 ] } ], "ref_ids": [ 2272 ] }, { "id": 2134, "type": "theorem", "label": "cohomology-lemma-restrict-K-injective-to-open", "categories": [ "cohomology" ], "title": "cohomology-lemma-restrict-K-injective-to-open", "contents": [ "Let $X$ be a ringed space. Let $U \\subset X$ be an open subspace.", "The restriction of a K-injective complex of $\\mathcal{O}_X$-modules", "to $U$ is a K-injective complex of $\\mathcal{O}_U$-modules." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}", "and the fact that the restriction functor has the", "exact left adjoint $j_!$.", "For the construction of $j_!$ see", "Sheaves, Section \\ref{sheaves-section-open-immersions}", "and for exactness see Modules, Lemma \\ref{modules-lemma-j-shriek-exact}." ], "refs": [ "derived-lemma-adjoint-preserve-K-injectives", "modules-lemma-j-shriek-exact" ], "ref_ids": [ 1915, 13224 ] } ], "ref_ids": [] }, { "id": 2135, "type": "theorem", "label": "cohomology-lemma-unbounded-cohomology-of-open", "categories": [ "cohomology" ], "title": "cohomology-lemma-unbounded-cohomology-of-open", "contents": [ "Let $X$ be a ringed space. Let $U \\subset X$ be an open subspace.", "For $K$ in $D(\\mathcal{O}_X)$ we have", "$H^p(U, K) = H^p(U, K|_U)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I}^\\bullet$ be a K-injective complex of $\\mathcal{O}_X$-modules", "representing $K$. Then", "$$", "H^q(U, K) = H^q(\\Gamma(U, \\mathcal{I}^\\bullet)) =", "H^q(\\Gamma(U, \\mathcal{I}^\\bullet|_U))", "$$", "by construction of cohomology. By Lemma \\ref{lemma-restrict-K-injective-to-open}", "the complex $\\mathcal{I}^\\bullet|_U$ is a K-injective complex", "representing $K|_U$ and the lemma follows." ], "refs": [ "cohomology-lemma-restrict-K-injective-to-open" ], "ref_ids": [ 2134 ] } ], "ref_ids": [] }, { "id": 2136, "type": "theorem", "label": "cohomology-lemma-sheafification-cohomology", "categories": [ "cohomology" ], "title": "cohomology-lemma-sheafification-cohomology", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K$ be an object of", "$D(\\mathcal{O}_X)$. The sheafification of", "$$", "U \\mapsto H^q(U, K) = H^q(U, K|_U)", "$$", "is the $q$th cohomology sheaf $H^q(K)$ of $K$." ], "refs": [], "proofs": [ { "contents": [ "The equality $H^q(U, K) = H^q(U, K|_U)$ holds by", "Lemma \\ref{lemma-unbounded-cohomology-of-open}.", "Choose a K-injective complex $\\mathcal{I}^\\bullet$ representing $K$.", "Then", "$$", "H^q(U, K) =", "\\frac{\\Ker(\\mathcal{I}^q(U) \\to \\mathcal{I}^{q + 1}(U))}", "{\\Im(\\mathcal{I}^{q - 1}(U) \\to \\mathcal{I}^q(U))}.", "$$", "by our construction of cohomology. Since", "$H^q(K) = \\Ker(\\mathcal{I}^q \\to \\mathcal{I}^{q + 1})/", "\\Im(\\mathcal{I}^{q - 1} \\to \\mathcal{I}^q)$ the result is clear." ], "refs": [ "cohomology-lemma-unbounded-cohomology-of-open" ], "ref_ids": [ 2135 ] } ], "ref_ids": [] }, { "id": 2137, "type": "theorem", "label": "cohomology-lemma-restrict-direct-image-open", "categories": [ "cohomology" ], "title": "cohomology-lemma-restrict-direct-image-open", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed", "spaces. Given an open subspace $V \\subset Y$, set $U = f^{-1}(V)$ and denote", "$g : U \\to V$ the induced morphism. Then", "$(Rf_*E)|_V = Rg_*(E|_U)$ for $E$ in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Represent $E$ by a K-injective complex $\\mathcal{I}^\\bullet$ of", "$\\mathcal{O}_X$-modules. Then $Rf_*(E) = f_*\\mathcal{I}^\\bullet$", "and $Rg_*(E|_U) = g_*(\\mathcal{I}^\\bullet|_U)$ by", "Lemma \\ref{lemma-restrict-K-injective-to-open}.", "Since it is clear that $(f_*\\mathcal{F})|_V = g_*(\\mathcal{F}|_U)$", "for any sheaf $\\mathcal{F}$ on $X$ the result follows." ], "refs": [ "cohomology-lemma-restrict-K-injective-to-open" ], "ref_ids": [ 2134 ] } ], "ref_ids": [] }, { "id": 2138, "type": "theorem", "label": "cohomology-lemma-Leray-unbounded", "categories": [ "cohomology" ], "title": "cohomology-lemma-Leray-unbounded", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Then $R\\Gamma(Y, -) \\circ Rf_* = R\\Gamma(X, -)$ as functors", "$D(\\mathcal{O}_X) \\to D(\\Gamma(Y, \\mathcal{O}_Y))$.", "More generally for $V \\subset Y$ open and $U = f^{-1}(V)$", "we have $R\\Gamma(U, -) = R\\Gamma(V, -) \\circ Rf_*$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z$ be the ringed space consisting of a singleton", "space with $\\Gamma(Z, \\mathcal{O}_Z) = \\Gamma(Y, \\mathcal{O}_Y)$.", "There is a canonical morphism $Y \\to Z$ of ringed spaces", "inducing the identification on global sections of structure sheaves.", "Then $D(\\mathcal{O}_Z) = D(\\Gamma(Y, \\mathcal{O}_Y))$.", "Hence the assertion $R\\Gamma(Y, -) \\circ Rf_* = R\\Gamma(X, -)$", "follows from Lemma \\ref{lemma-derived-pushforward-composition}", "applied to $X \\to Y \\to Z$.", "\\medskip\\noindent", "The second (more general) statement follows from the first statement", "after applying Lemma \\ref{lemma-restrict-direct-image-open}." ], "refs": [ "cohomology-lemma-derived-pushforward-composition", "cohomology-lemma-restrict-direct-image-open" ], "ref_ids": [ 2122, 2137 ] } ], "ref_ids": [] }, { "id": 2139, "type": "theorem", "label": "cohomology-lemma-unbounded-describe-higher-direct-images", "categories": [ "cohomology" ], "title": "cohomology-lemma-unbounded-describe-higher-direct-images", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed", "spaces. Let $K$ be in $D(\\mathcal{O}_X)$. Then $H^i(Rf_*K)$ is the sheaf", "associated to the presheaf", "$$", "V \\mapsto H^i(f^{-1}(V), K) = H^i(V, Rf_*K)", "$$" ], "refs": [], "proofs": [ { "contents": [ "The equality $H^i(f^{-1}(V), K) = H^i(V, Rf_*K)$ follows upon taking", "cohomology from the second statement in", "Lemma \\ref{lemma-Leray-unbounded}. Then the statement on sheafification", "follows from Lemma \\ref{lemma-sheafification-cohomology}." ], "refs": [ "cohomology-lemma-Leray-unbounded", "cohomology-lemma-sheafification-cohomology" ], "ref_ids": [ 2138, 2136 ] } ], "ref_ids": [] }, { "id": 2140, "type": "theorem", "label": "cohomology-lemma-modules-abelian-unbounded", "categories": [ "cohomology" ], "title": "cohomology-lemma-modules-abelian-unbounded", "contents": [ "Let $X$ be a ringed space. Let $K$ be an object of $D(\\mathcal{O}_X)$", "and denote $K_{ab}$ its image in $D(\\underline{\\mathbf{Z}}_X)$.", "\\begin{enumerate}", "\\item For any open $U \\subset X$ there is a canonical map", "$R\\Gamma(U, K) \\to R\\Gamma(U, K_{ab})$", "which is an isomorphism in $D(\\textit{Ab})$.", "\\item Let $f : X \\to Y$ be a morphism of ringed spaces.", "There is a canonical map $Rf_*K \\to Rf_*(K_{ab})$ which", "is an isomorphism in $D(\\underline{\\mathbf{Z}}_Y)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The map is constructed as follows. Choose a K-injective complex", "$\\mathcal{I}^\\bullet$ representing $K$. Choose a quasi-isomorpism", "$\\mathcal{I}^\\bullet \\to \\mathcal{J}^\\bullet$ where $\\mathcal{J}^\\bullet$", "is a K-injective complex of abelian groups. Then the map in (1)", "is given by $\\Gamma(U, \\mathcal{I}^\\bullet) \\to \\Gamma(U, \\mathcal{J}^\\bullet)$", "and the map in (2) is given by", "$f_*\\mathcal{I}^\\bullet \\to f_*\\mathcal{J}^\\bullet$.", "To show that these maps are isomorphisms, it suffices to prove", "they induce isomorphisms on cohomology groups and cohomology sheaves.", "By Lemmas \\ref{lemma-unbounded-cohomology-of-open} and", "\\ref{lemma-unbounded-describe-higher-direct-images}", "it suffices to show that the map", "$$", "H^0(X, K) \\longrightarrow H^0(X, K_{ab})", "$$", "is an isomorphism. Observe that", "$$", "H^0(X, K) = \\Hom_{D(\\mathcal{O}_X)}(\\mathcal{O}_X, K)", "$$", "and similarly for the other group. Choose any complex $\\mathcal{K}^\\bullet$", "of $\\mathcal{O}_X$-modules representing $K$. By construction of the", "derived category as a localization we have", "$$", "\\Hom_{D(\\mathcal{O}_X)}(\\mathcal{O}_X, K) =", "\\colim_{s : \\mathcal{F}^\\bullet \\to \\mathcal{O}_X}", "\\Hom_{K(\\mathcal{O}_X)}(\\mathcal{F}^\\bullet, \\mathcal{K}^\\bullet)", "$$", "where the colimit is over quasi-isomorphisms $s$ of complexes of", "$\\mathcal{O}_X$-modules. Similarly, we have", "$$", "\\Hom_{D(\\underline{\\mathbf{Z}}_X)}(\\underline{\\mathbf{Z}}_X, K) =", "\\colim_{s : \\mathcal{G}^\\bullet \\to \\underline{\\mathbf{Z}}_X}", "\\Hom_{K(\\underline{\\mathbf{Z}}_X)}(\\mathcal{G}^\\bullet, \\mathcal{K}^\\bullet)", "$$", "Next, we observe that the quasi-isomorphisms", "$s : \\mathcal{G}^\\bullet \\to \\underline{\\mathbf{Z}}_X$", "with $\\mathcal{G}^\\bullet$ bounded above complex of flat", "$\\underline{\\mathbf{Z}}_X$-modules is cofinal in the system.", "(This follows from Modules, Lemma \\ref{modules-lemma-module-quotient-flat} and", "Derived Categories, Lemma \\ref{derived-lemma-subcategory-left-resolution};", "see discussion in Section \\ref{section-flat}.)", "Hence we can construct an inverse to the map", "$H^0(X, K) \\longrightarrow H^0(X, K_{ab})$", "by representing an element $\\xi \\in H^0(X, K_{ab})$ by a pair", "$$", "(s : \\mathcal{G}^\\bullet \\to \\underline{\\mathbf{Z}}_X,", "a : \\mathcal{G}^\\bullet \\to \\mathcal{K}^\\bullet)", "$$", "with $\\mathcal{G}^\\bullet$ a bounded above complex of flat", "$\\underline{\\mathbf{Z}}_X$-modules and sending this to", "$$", "(\\mathcal{G}^\\bullet \\otimes_{\\underline{\\mathbf{Z}}_X} \\mathcal{O}_X", "\\to \\mathcal{O}_X,", "\\mathcal{G}^\\bullet \\otimes_{\\underline{\\mathbf{Z}}_X} \\mathcal{O}_X", "\\to \\mathcal{K}^\\bullet)", "$$", "The only thing to note here is that the first arrow", "is a quasi-isomorphism by", "Lemmas \\ref{lemma-derived-tor-quasi-isomorphism-other-side} and", "\\ref{lemma-bounded-flat-K-flat}.", "We omit the detailed verification that this construction", "is indeed an inverse." ], "refs": [ "cohomology-lemma-unbounded-cohomology-of-open", "cohomology-lemma-unbounded-describe-higher-direct-images", "modules-lemma-module-quotient-flat", "derived-lemma-subcategory-left-resolution", "cohomology-lemma-derived-tor-quasi-isomorphism-other-side", "cohomology-lemma-bounded-flat-K-flat" ], "ref_ids": [ 2135, 2139, 13276, 1835, 2113, 2109 ] } ], "ref_ids": [] }, { "id": 2141, "type": "theorem", "label": "cohomology-lemma-adjoint-lower-shriek-restrict", "categories": [ "cohomology" ], "title": "cohomology-lemma-adjoint-lower-shriek-restrict", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $U \\subset X$ be an", "open subset. Denote $j : (U, \\mathcal{O}_U) \\to (X, \\mathcal{O}_X)$", "the corresponding open immersion. The restriction functor", "$D(\\mathcal{O}_X) \\to D(\\mathcal{O}_U)$ is a right adjoint to", "extension by zero $j_! : D(\\mathcal{O}_U) \\to D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "This follows formally from the fact that $j_!$ and $j^*$ are adjoint and", "exact (and hence $Lj_! = j_!$ and $Rj^* = j^*$ exist), see", "Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors}." ], "refs": [ "derived-lemma-derived-adjoint-functors" ], "ref_ids": [ 1907 ] } ], "ref_ids": [] }, { "id": 2142, "type": "theorem", "label": "cohomology-lemma-K-injective-flat", "categories": [ "cohomology" ], "title": "cohomology-lemma-K-injective-flat", "contents": [ "Let $f : X \\to Y$ be a flat morphism of ringed spaces.", "If $\\mathcal{I}^\\bullet$ is a K-injective complex of $\\mathcal{O}_X$-modules,", "then $f_*\\mathcal{I}^\\bullet$ is K-injective as a complex of", "$\\mathcal{O}_Y$-modules." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$$", "\\Hom_{K(\\mathcal{O}_Y)}(\\mathcal{F}^\\bullet, f_*\\mathcal{I}^\\bullet)", "=", "\\Hom_{K(\\mathcal{O}_X)}(f^*\\mathcal{F}^\\bullet, \\mathcal{I}^\\bullet)", "$$", "by", "Sheaves, Lemma", "\\ref{sheaves-lemma-adjoint-pullback-pushforward-modules}", "and the fact that $f^*$ is exact as $f$ is assumed to be flat." ], "refs": [ "sheaves-lemma-adjoint-pullback-pushforward-modules" ], "ref_ids": [ 14521 ] } ], "ref_ids": [] }, { "id": 2143, "type": "theorem", "label": "cohomology-lemma-exact-sequence-lower-shriek", "categories": [ "cohomology" ], "title": "cohomology-lemma-exact-sequence-lower-shriek", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $X = U \\cup V$ be the union of two open subspaces.", "For any object $E$ of $D(\\mathcal{O}_X)$ we have a distinguished", "triangle", "$$", "j_{U \\cap V!}E|_{U \\cap V} \\to", "j_{U!}E|_U \\oplus j_{V!}E|_V \\to E \\to ", "j_{U \\cap V!}E|_{U \\cap V}[1]", "$$", "in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "We have seen in Section \\ref{section-properties-K-injective}", "that the restriction functors and the extension", "by zero functors are computed by just applying the functors to", "any complex. Let $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules", "representing $E$. The distinguished triangle of the lemma is the", "distinguished triangle associated (by", "Derived Categories, Section", "\\ref{derived-section-canonical-delta-functor} and especially", "Lemma \\ref{derived-lemma-derived-canonical-delta-functor})", "to the short exact sequence of complexes of $\\mathcal{O}_X$-modules", "$$", "0 \\to j_{U \\cap V!}\\mathcal{E}^\\bullet|_{U \\cap V} \\to", "j_{U!}\\mathcal{E}^\\bullet|_U \\oplus j_{V!}\\mathcal{E}^\\bullet|_V", "\\to \\mathcal{E}^\\bullet \\to 0", "$$", "To see this sequence is exact one checks on stalks using", "Sheaves, Lemma \\ref{sheaves-lemma-j-shriek-modules}", "(computation omitted)." ], "refs": [ "derived-lemma-derived-canonical-delta-functor", "sheaves-lemma-j-shriek-modules" ], "ref_ids": [ 1814, 14546 ] } ], "ref_ids": [] }, { "id": 2144, "type": "theorem", "label": "cohomology-lemma-exact-sequence-j-star", "categories": [ "cohomology" ], "title": "cohomology-lemma-exact-sequence-j-star", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $X = U \\cup V$ be the union of two open subspaces.", "For any object $E$ of $D(\\mathcal{O}_X)$ we have a distinguished", "triangle", "$$", "E \\to ", "Rj_{U, *}E|_U \\oplus Rj_{V, *}E|_V \\to", "Rj_{U \\cap V, *}E|_{U \\cap V} \\to", "E[1]", "$$", "in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $\\mathcal{I}^\\bullet$ representing $E$", "whose terms $\\mathcal{I}^n$ are injective objects of", "$\\textit{Mod}(\\mathcal{O}_X)$, see Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "We have seen that $\\mathcal{I}^\\bullet|U$ is a K-injective complex", "as well (Lemma \\ref{lemma-restrict-K-injective-to-open}). Hence", "$Rj_{U, *}E|_U$ is represented by $j_{U, *}\\mathcal{I}^\\bullet|_U$.", "Similarly for $V$ and $U \\cap V$. Hence the distinguished triangle", "of the lemma is the distinguished triangle associated (by", "Derived Categories, Section", "\\ref{derived-section-canonical-delta-functor} and especially", "Lemma \\ref{derived-lemma-derived-canonical-delta-functor})", "to the short exact sequence of complexes", "$$", "0 \\to", "\\mathcal{I}^\\bullet \\to", "j_{U, *}\\mathcal{I}^\\bullet|_U \\oplus j_{V, *}\\mathcal{I}^\\bullet|_V \\to", "j_{U \\cap V, *}\\mathcal{I}^\\bullet|_{U \\cap V} \\to", "0.", "$$", "This sequence is exact because for any $W \\subset X$ open", "and any $n$ the sequence", "$$", "0 \\to", "\\mathcal{I}^n(W) \\to", "\\mathcal{I}^n(W \\cap U) \\oplus \\mathcal{I}^n(W \\cap V) \\to", "\\mathcal{I}^n(W \\cap U \\cap V) \\to", "0", "$$", "is exact (see proof of Lemma \\ref{lemma-mayer-vietoris})." ], "refs": [ "injectives-theorem-K-injective-embedding-grothendieck", "cohomology-lemma-restrict-K-injective-to-open", "derived-lemma-derived-canonical-delta-functor", "cohomology-lemma-mayer-vietoris" ], "ref_ids": [ 7768, 2134, 1814, 2042 ] } ], "ref_ids": [] }, { "id": 2145, "type": "theorem", "label": "cohomology-lemma-mayer-vietoris-hom", "categories": [ "cohomology" ], "title": "cohomology-lemma-mayer-vietoris-hom", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $X = U \\cup V$ be", "the union of two open subspaces of $X$.", "For objects $E$, $F$ of $D(\\mathcal{O}_X)$ we have a", "Mayer-Vietoris sequence", "$$", "\\xymatrix{", "& \\ldots \\ar[r] & \\Ext^{-1}(E_{U \\cap V}, F_{U \\cap V}) \\ar[lld] \\\\", "\\Hom(E, F) \\ar[r] &", "\\Hom(E_U, F_U) \\oplus", "\\Hom(E_V, F_V) \\ar[r] &", "\\Hom(E_{U \\cap V}, F_{U \\cap V})", "}", "$$", "where the subscripts denote restrictions to the relevant opens", "and the $\\Hom$'s and $\\Ext$'s are taken in the relevant", "derived categories." ], "refs": [], "proofs": [ { "contents": [ "Use the distinguished triangle of", "Lemma \\ref{lemma-exact-sequence-lower-shriek}", "to obtain a long exact sequence of $\\Hom$'s", "(from Derived Categories, Lemma \\ref{derived-lemma-representable-homological})", "and use that", "$$", "\\Hom_{D(\\mathcal{O}_X)}(j_{U!}E|_U, F) =", "\\Hom_{D(\\mathcal{O}_U)}(E|_U, F|_U)", "$$", "by Lemma \\ref{lemma-adjoint-lower-shriek-restrict}." ], "refs": [ "cohomology-lemma-exact-sequence-lower-shriek", "derived-lemma-representable-homological", "cohomology-lemma-adjoint-lower-shriek-restrict" ], "ref_ids": [ 2143, 1758, 2141 ] } ], "ref_ids": [] }, { "id": 2146, "type": "theorem", "label": "cohomology-lemma-unbounded-mayer-vietoris", "categories": [ "cohomology" ], "title": "cohomology-lemma-unbounded-mayer-vietoris", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Suppose that", "$X = U \\cup V$ is a union of two open subsets. For an object $E$", "of $D(\\mathcal{O}_X)$ we have a distinguished triangle", "$$", "R\\Gamma(X, E) \\to R\\Gamma(U, E) \\oplus R\\Gamma(V, E) \\to", "R\\Gamma(U \\cap V, E) \\to R\\Gamma(X, E)[1]", "$$", "and in particular a long exact cohomology sequence", "$$", "\\ldots \\to", "H^n(X, E) \\to", "H^n(U, E) \\oplus H^0(V, E) \\to", "H^n(U \\cap V, E) \\to", "H^{n + 1}(X, E) \\to \\ldots", "$$", "The construction of the distinguished triangle and the", "long exact sequence is functorial in $E$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $\\mathcal{I}^\\bullet$", "representing $E$. We may assume $\\mathcal{I}^n$ is an injective", "object of $\\textit{Mod}(\\mathcal{O}_X)$ for all $n$, see", "Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "Then $R\\Gamma(X, E)$ is computed by $\\Gamma(X, \\mathcal{I}^\\bullet)$.", "Similarly for $U$, $V$, and $U \\cap V$ by", "Lemma \\ref{lemma-restrict-K-injective-to-open}.", "Hence the distinguished triangle of the lemma is the distinguished", "triangle associated (by", "Derived Categories, Section", "\\ref{derived-section-canonical-delta-functor} and especially", "Lemma \\ref{derived-lemma-derived-canonical-delta-functor})", "to the short exact sequence of complexes", "$$", "0 \\to", "\\mathcal{I}^\\bullet(X) \\to", "\\mathcal{I}^\\bullet(U) \\oplus \\mathcal{I}^\\bullet(V) \\to", "\\mathcal{I}^\\bullet(U \\cap V) \\to", "0.", "$$", "We have seen this is a short exact sequence in the proof of", "Lemma \\ref{lemma-mayer-vietoris}.", "The final statement follows from the functoriality of the construction", "in Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}." ], "refs": [ "injectives-theorem-K-injective-embedding-grothendieck", "cohomology-lemma-restrict-K-injective-to-open", "derived-lemma-derived-canonical-delta-functor", "cohomology-lemma-mayer-vietoris", "injectives-theorem-K-injective-embedding-grothendieck" ], "ref_ids": [ 7768, 2134, 1814, 2042, 7768 ] } ], "ref_ids": [] }, { "id": 2147, "type": "theorem", "label": "cohomology-lemma-unbounded-relative-mayer-vietoris", "categories": [ "cohomology" ], "title": "cohomology-lemma-unbounded-relative-mayer-vietoris", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Suppose that $X = U \\cup V$ is a union of two open subsets.", "Denote $a = f|_U : U \\to Y$, $b = f|_V : V \\to Y$, and", "$c = f|_{U \\cap V} : U \\cap V \\to Y$.", "For every object $E$ of $D(\\mathcal{O}_X)$ there exists a", "distinguished triangle", "$$", "Rf_*E \\to", "Ra_*(E|_U) \\oplus Rb_*(E|_V) \\to", "Rc_*(E|_{U \\cap V}) \\to", "Rf_*E[1]", "$$", "This triangle is functorial in $E$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $\\mathcal{I}^\\bullet$", "representing $E$. We may assume $\\mathcal{I}^n$ is an injective", "object of $\\textit{Mod}(\\mathcal{O}_X)$ for all $n$, see", "Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "Then $Rf_*E$ is computed by $f_*\\mathcal{I}^\\bullet$.", "Similarly for $U$, $V$, and $U \\cap V$ by", "Lemma \\ref{lemma-restrict-K-injective-to-open}.", "Hence the distinguished triangle of the lemma is the distinguished", "triangle associated (by", "Derived Categories, Section", "\\ref{derived-section-canonical-delta-functor} and especially", "Lemma \\ref{derived-lemma-derived-canonical-delta-functor})", "to the short exact sequence of complexes", "$$", "0 \\to", "f_*\\mathcal{I}^\\bullet \\to", "a_*\\mathcal{I}^\\bullet|_U \\oplus b_*\\mathcal{I}^\\bullet|_V \\to", "c_*\\mathcal{I}^\\bullet|_{U \\cap V} \\to", "0.", "$$", "This is a short exact sequence of complexes by", "Lemma \\ref{lemma-relative-mayer-vietoris}", "and the fact that $R^1f_*\\mathcal{I} = 0$", "for an injective object $\\mathcal{I}$ of $\\textit{Mod}(\\mathcal{O}_X)$.", "The final statement follows from the functoriality of the construction", "in Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}." ], "refs": [ "injectives-theorem-K-injective-embedding-grothendieck", "cohomology-lemma-restrict-K-injective-to-open", "derived-lemma-derived-canonical-delta-functor", "cohomology-lemma-relative-mayer-vietoris", "injectives-theorem-K-injective-embedding-grothendieck" ], "ref_ids": [ 7768, 2134, 1814, 2043, 7768 ] } ], "ref_ids": [] }, { "id": 2148, "type": "theorem", "label": "cohomology-lemma-pushforward-restriction", "categories": [ "cohomology" ], "title": "cohomology-lemma-pushforward-restriction", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $j : U \\to X$ be an", "open subspace. Let $T \\subset X$ be a closed subset contained in $U$.", "\\begin{enumerate}", "\\item If $E$ is an object of $D(\\mathcal{O}_X)$ whose cohomology sheaves", "are supported on $T$, then $E \\to Rj_*(E|_U)$ is an isomorphism.", "\\item If $F$ is an object of $D(\\mathcal{O}_U)$ whose cohomology sheaves", "are supported on $T$, then $j_!F \\to Rj_*F$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $V = X \\setminus T$ and $W = U \\cap V$. Note that $X = U \\cup V$ is an", "open covering of $X$. Denote $j_W : W \\to V$ the open immersion.", "Let $E$ be an object of $D(\\mathcal{O}_X)$ whose cohomology sheaves are", "supported on $T$. By", "Lemma \\ref{lemma-restrict-direct-image-open} we have", "$(Rj_*E|_U)|_V = Rj_{W, *}(E|_W) = 0$ because $E|_W = 0$ by our assumption.", "On the other hand, $Rj_*(E|_U)|_U = E|_U$. Thus (1) is clear.", "Let $F$ be an object of $D(\\mathcal{O}_U)$ whose cohomology sheaves", "are supported on $T$. By", "Lemma \\ref{lemma-restrict-direct-image-open} we have", "$(Rj_*F)|_V = Rj_{W, *}(F|_W) = 0$ because $F|_W = 0$ by our assumption.", "We also have $(j_!F)|_V = j_{W!}(F|_W) = 0$ (the first equality is immediate", "from the definition of extension by zero). Since both", "$(Rj_*F)|_U = F$ and $(j_!F)|_U = F$ we see that (2) holds." ], "refs": [ "cohomology-lemma-restrict-direct-image-open", "cohomology-lemma-restrict-direct-image-open" ], "ref_ids": [ 2137, 2137 ] } ], "ref_ids": [] }, { "id": 2149, "type": "theorem", "label": "cohomology-lemma-mayer-vietoris-cup", "categories": [ "cohomology" ], "title": "cohomology-lemma-mayer-vietoris-cup", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Set $A = \\Gamma(X, \\mathcal{O}_X)$.", "Suppose that $X = U \\cup V$ is a union of two open subsets. For objects", "$K$ and $M$ of $D(\\mathcal{O}_X)$ we have a map of distinguished triangles", "$$", "\\xymatrix{", "R\\Gamma(X, K) \\otimes_A^\\mathbf{L} R\\Gamma(X, M) \\ar[r] \\ar[d] &", "R\\Gamma(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\ar[d] \\\\", "R\\Gamma(X, K) \\otimes_A^\\mathbf{L}", "(R\\Gamma(U, M) \\oplus R\\Gamma(V, M)) \\ar[r] \\ar[d] &", "R\\Gamma(U, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)", "\\oplus R\\Gamma(V, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)) \\ar[d] \\\\", "R\\Gamma(X, K) \\otimes_A^\\mathbf{L} R\\Gamma(U \\cap V, M) \\ar[r] \\ar[d] &", "R\\Gamma(U \\cap V, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\ar[d] \\\\", "R\\Gamma(X, K) \\otimes_A^\\mathbf{L} R\\Gamma(X, M)[1] \\ar[r] &", "R\\Gamma(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)[1]", "}", "$$", "where", "\\begin{enumerate}", "\\item the horizontal arrows are given by cup product,", "\\item on the right hand side we have the distinguished triangle", "of Lemma \\ref{lemma-unbounded-mayer-vietoris} for", "$K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M$, and", "\\item on the left hand side we have the exact functor", "$R\\Gamma(X, K) \\otimes_A^\\mathbf{L} - $ applied to the", "distinguished triangle of Lemma \\ref{lemma-unbounded-mayer-vietoris} for $M$.", "\\end{enumerate}" ], "refs": [ "cohomology-lemma-unbounded-mayer-vietoris", "cohomology-lemma-unbounded-mayer-vietoris" ], "proofs": [ { "contents": [ "Choose a K-flat complex $T^\\bullet$ of flat $A$-modules representing", "$R\\Gamma(X, K)$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-K-flat-resolution}.", "Denote $T^\\bullet \\otimes_A \\mathcal{O}_X$ the pullback of $T^\\bullet$", "by the morphism of ringed spaces $(X, \\mathcal{O}_X) \\to (pt, A)$.", "There is a natural adjunction map", "$\\epsilon : T^\\bullet \\otimes_A \\mathcal{O}_X \\to K$ in $D(\\mathcal{O}_X)$.", "Observe that $T^\\bullet \\otimes_A \\mathcal{O}_X$ is a K-flat", "complex of $\\mathcal{O}_X$-modules with flat terms, see", "Lemma \\ref{lemma-pullback-K-flat} and", "Modules, Lemma \\ref{modules-lemma-pullback-flat}.", "By Lemma \\ref{lemma-factor-through-K-flat} we can find a morphism of complexes", "$$", "T^\\bullet \\otimes_A \\mathcal{O}_X \\longrightarrow \\mathcal{K}^\\bullet", "$$", "of $\\mathcal{O}_X$-modules representing $\\epsilon$", "such that $\\mathcal{K}^\\bullet$ is a", "K-flat complex with flat terms. Namely, by the construction of", "$D(\\mathcal{O}_X)$ we can first represent $\\epsilon$ by some map of complexes", "$e : T^\\bullet \\otimes_A \\mathcal{O}_X \\to \\mathcal{L}^\\bullet$", "of $\\mathcal{O}_X$-modules representing $\\epsilon$", "and then we can apply the lemma to $e$. Choose a K-injective", "complex $\\mathcal{I}^\\bullet$ whose terms are injective $\\mathcal{O}_X$-modules", "representing $M$. Finally, choose a quasi-isomorphism", "$$", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{I}^\\bullet)", "\\longrightarrow", "\\mathcal{J}^\\bullet", "$$", "into a K-injective complex whose terms are injective $\\mathcal{O}_X$-modules.", "Observe that source and target of this arrow represent", "$K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M$ in $D(\\mathcal{O}_X)$.", "At this point, for any open $W \\subset X$ we obtain a map of complexes", "$$", "\\text{Tot}(T^\\bullet \\otimes_A \\mathcal{I}^\\bullet(W))", "\\to", "\\text{Tot}(\\mathcal{K}^\\bullet(W) \\otimes_A \\mathcal{I}^\\bullet(W))", "\\to", "\\mathcal{J}^\\bullet(W)", "$$", "of $A$-modules whose composition represents the map", "$$", "R\\Gamma(X, K) \\otimes_A^\\mathbf{L} R\\Gamma(W, M)", "\\longrightarrow", "R\\Gamma(W, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)", "$$", "in $D(A)$. Clearly, these maps are compatible with restriction mappings.", "OK, so now we can consider the following commutative(!) diagram", "of complexes of $A$-modules", "$$", "\\xymatrix{", "0 \\ar[d] & 0 \\ar[d] \\\\", "\\text{Tot}(T^\\bullet \\otimes_A \\mathcal{I}^\\bullet(X)) \\ar[d] \\ar[r] &", "\\mathcal{J}^\\bullet(X) \\ar[d] \\\\", "\\text{Tot}(T^\\bullet \\otimes_A", "(\\mathcal{I}^\\bullet(U) \\oplus \\mathcal{I}^\\bullet(V)) \\ar[d] \\ar[r] &", "\\mathcal{J}^\\bullet(U) \\oplus \\mathcal{J}^\\bullet(V) \\ar[d] \\\\", "\\text{Tot}(T^\\bullet \\otimes_A \\mathcal{I}^\\bullet(U \\cap V)) \\ar[r] \\ar[d] &", "\\mathcal{J}^\\bullet(U \\cap V) \\ar[d] \\\\", "0 & 0", "}", "$$", "By the proof of Lemma \\ref{lemma-mayer-vietoris} the columns are", "exact sequences of complexes of $A$-modules (this also uses that", "$\\text{Tot}(T^\\bullet \\otimes_A -)$ transforms short exact sequences", "of complexes of $A$-modules into short exact sequences as the terms", "of $T^\\bullet$ are flat $A$-modules). Since the distinguished triangles", "of Lemma \\ref{lemma-unbounded-mayer-vietoris}", "are the distinguished triangles associated to these", "short exact sequences of complexes, the desired result follows from", "the functoriality of ``taking the associated distinguished triangle''", "discussed in", "Derived Categories, Section \\ref{derived-section-canonical-delta-functor}." ], "refs": [ "more-algebra-lemma-K-flat-resolution", "cohomology-lemma-pullback-K-flat", "modules-lemma-pullback-flat", "cohomology-lemma-factor-through-K-flat", "cohomology-lemma-mayer-vietoris", "cohomology-lemma-unbounded-mayer-vietoris" ], "ref_ids": [ 10131, 2108, 13287, 2115, 2042, 2146 ] } ], "ref_ids": [ 2146, 2146 ] }, { "id": 2150, "type": "theorem", "label": "cohomology-lemma-cohomology-with-support-sheaf-on-support", "categories": [ "cohomology" ], "title": "cohomology-lemma-cohomology-with-support-sheaf-on-support", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the", "inclusion of a closed subset.", "\\begin{enumerate}", "\\item $R\\mathcal{H}_Z : D(\\mathcal{O}_X) \\to D(\\mathcal{O}_X|_Z)$", "is right adjoint to $i_* : D(\\mathcal{O}_X|_Z) \\to D(\\mathcal{O}_X)$.", "\\item For $K$ in $D(\\mathcal{O}_X|_Z)$ we have $R\\mathcal{H}_Z(i_*K) = K$.", "\\item Let $\\mathcal{G}$ be a sheaf of", "$\\mathcal{O}_X|_Z$-modules on $Z$. Then", "$\\mathcal{H}^p_Z(i_*\\mathcal{G}) = 0$ for $p > 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The functor $i_*$ is exact, so $i_* = Ri_* = Li_*$. Hence part (1)", "of the lemma follows from", "Modules, Lemma \\ref{modules-lemma-adjoint-section-with-support}", "and", "Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors}.", "Let $K$ be as in (2). We can represent $K$ by a K-injective complex", "$\\mathcal{I}^\\bullet$ of $\\mathcal{O}_X|_Z$-modules. By", "Lemma \\ref{lemma-K-injective-flat}", "the complex $i_*\\mathcal{I}^\\bullet$, which represents $i_*K$,", "is a K-injective complex of $\\mathcal{O}_X$-modules. Thus", "$R\\mathcal{H}_Z(i_*K)$ is computed by", "$\\mathcal{H}_Z(i_*\\mathcal{I}^\\bullet) = \\mathcal{I}^\\bullet$", "which proves (2). Part (3) is a special case of (2)." ], "refs": [ "derived-lemma-derived-adjoint-functors", "cohomology-lemma-K-injective-flat" ], "ref_ids": [ 1907, 2142 ] } ], "ref_ids": [] }, { "id": 2151, "type": "theorem", "label": "cohomology-lemma-complexes-with-support-on-closed", "categories": [ "cohomology" ], "title": "cohomology-lemma-complexes-with-support-on-closed", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the", "inclusion of a closed subset.", "\\begin{enumerate}", "\\item For $K$ in $D(\\mathcal{O}_X|_Z)$ we have $i_*K$ in $D_Z(\\mathcal{O}_X)$.", "\\item The functor $i_* : D(\\mathcal{O}_X|_Z) \\to D_Z(\\mathcal{O}_X)$", "is an equivalence with quasi-inverse", "$i^{-1}|_{D_Z(\\mathcal{O}_X)} = R\\mathcal{H}_Z|_{D_Z(\\mathcal{O}_X)}$.", "\\item The functor", "$i_* \\circ R\\mathcal{H}_Z : D(\\mathcal{O}_X) \\to D_Z(\\mathcal{O}_X)$", "is right adjoint to the inclusion functor", "$D_Z(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is immediate from the definitions. Part (3) is a formal", "consequence of part (2) and", "Lemma \\ref{lemma-cohomology-with-support-sheaf-on-support}.", "In the rest of the proof we prove part (2).", "\\medskip\\noindent", "Let us think of $i$ as the morphism of ringed spaces", "$i : (Z, \\mathcal{O}_X|_Z) \\to (X, \\mathcal{O}_X)$.", "Recall that $i^*$ and $i_*$ is an adjoint pair of functors.", "Since $i$ is a closed immersion, $i_*$ is exact.", "Since $i^{-1}\\mathcal{O}_X = \\mathcal{O}_X|_Z$ is the structure", "sheaf of $(Z, \\mathcal{O}_X|_Z)$ we see that $i^* = i^{-1}$", "is exact and we see that that $i^*i_* = i^{-1}i_*$", "is isomorphic to the identify functor. See", "Modules, Lemmas \\ref{modules-lemma-exactness-pushforward-pullback} and", "\\ref{modules-lemma-i-star-exact}. Thus", "$i_* : D(\\mathcal{O}_X|_Z) \\to D_Z(\\mathcal{O}_X)$", "is fully faithful and $i^{-1}$ determines", "a left inverse. On the other hand, suppose that $K$ is an object of", "$D_Z(\\mathcal{O}_X)$ and consider the adjunction map", "$K \\to i_*i^{-1}K$.", "Using exactness of $i_*$ and $i^{-1}$ this induces the adjunction maps", "$H^n(K) \\to i_*i^{-1}H^n(K)$ on cohomology sheaves. Since these cohomology", "sheaves are supported on $Z$ we see these adjunction maps are isomorphisms", "and we conclude that $i_* : D(\\mathcal{O}_X|_Z) \\to D_Z(\\mathcal{O}_X)$", "is an equivalence.", "\\medskip\\noindent", "To finish the proof it suffices to show that $R\\mathcal{H}_Z(K) = i^{-1}K$ if", "$K$ is an object of $D_Z(\\mathcal{O}_X)$. To do this we can use that", "$K = i_*i^{-1}K$ as we've just proved this is the case. Then", "Lemma \\ref{lemma-cohomology-with-support-sheaf-on-support}", "tells us what we want." ], "refs": [ "cohomology-lemma-cohomology-with-support-sheaf-on-support", "modules-lemma-exactness-pushforward-pullback", "modules-lemma-i-star-exact", "cohomology-lemma-cohomology-with-support-sheaf-on-support" ], "ref_ids": [ 2150, 13223, 13232, 2150 ] } ], "ref_ids": [] }, { "id": 2152, "type": "theorem", "label": "cohomology-lemma-sections-with-support-K-injective", "categories": [ "cohomology" ], "title": "cohomology-lemma-sections-with-support-K-injective", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$", "be the inclusion of a closed subset. If $\\mathcal{I}^\\bullet$ is a K-injective", "complex of $\\mathcal{O}_X$-modules, then", "$\\mathcal{H}_Z(\\mathcal{I}^\\bullet)$ is K-injective complex of", "$\\mathcal{O}_X|_Z$-modules." ], "refs": [], "proofs": [ { "contents": [ "Since $i_* : \\textit{Mod}(\\mathcal{O}_X|_Z) \\to \\textit{Mod}(\\mathcal{O}_X)$", "is exact and left adjoint to $\\mathcal{H}_Z$", "(Modules, Lemma \\ref{modules-lemma-adjoint-section-with-support})", "this follows from", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}." ], "refs": [ "derived-lemma-adjoint-preserve-K-injectives" ], "ref_ids": [ 1915 ] } ], "ref_ids": [] }, { "id": 2153, "type": "theorem", "label": "cohomology-lemma-local-to-global-sections-with-support", "categories": [ "cohomology" ], "title": "cohomology-lemma-local-to-global-sections-with-support", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the", "inclusion of a closed subset. Then", "$R\\Gamma(Z, - ) \\circ R\\mathcal{H}_Z = R\\Gamma_Z(X, - )$", "as functors $D(\\mathcal{O}_X) \\to D(\\mathcal{O}_X(X))$." ], "refs": [], "proofs": [ { "contents": [ "Follows from the construction of right derived functors using", "K-injective resolutions, Lemma \\ref{lemma-sections-with-support-K-injective},", "and the fact that $\\Gamma_Z(X, -) = \\Gamma(Z, -) \\circ \\mathcal{H}_Z$." ], "refs": [ "cohomology-lemma-sections-with-support-K-injective" ], "ref_ids": [ 2152 ] } ], "ref_ids": [] }, { "id": 2154, "type": "theorem", "label": "cohomology-lemma-triangle-sections-with-support", "categories": [ "cohomology" ], "title": "cohomology-lemma-triangle-sections-with-support", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the", "inclusion of a closed subset. Let $U = X \\setminus Z$.", "There is a distinguished triangle", "$$", "R\\Gamma_Z(X, K) \\to R\\Gamma(X, K) \\to R\\Gamma(U, K) \\to", "R\\Gamma_Z(X, K)[1]", "$$", "in $D(\\mathcal{O}_X(X))$ functorial for $K$ in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $\\mathcal{I}^\\bullet$ all of whose terms", "are injective $\\mathcal{O}_X$-modules representing $K$. See", "Section \\ref{section-unbounded}. Recall that $\\mathcal{I}^\\bullet|_U$", "is a K-injective complex of $\\mathcal{O}_U$-modules, see", "Lemma \\ref{lemma-restrict-K-injective-to-open}. Hence each", "of the derived functors in the distinguished triangle is gotten", "by applying the underlying functor to $\\mathcal{I}^\\bullet$.", "Hence we find that it suffices to prove that", "for an injective $\\mathcal{O}_X$-module $\\mathcal{I}$ we have", "a short exact sequence", "$$", "0 \\to \\Gamma_Z(X, \\mathcal{I}) \\to \\Gamma(X, \\mathcal{I})", "\\to \\Gamma(U, \\mathcal{I}) \\to 0", "$$", "This follows from Lemma \\ref{lemma-injective-restriction-surjective}", "and the definitions." ], "refs": [ "cohomology-lemma-restrict-K-injective-to-open", "cohomology-lemma-injective-restriction-surjective" ], "ref_ids": [ 2134, 2041 ] } ], "ref_ids": [] }, { "id": 2155, "type": "theorem", "label": "cohomology-lemma-triangle-sections-with-support-sheaves", "categories": [ "cohomology" ], "title": "cohomology-lemma-triangle-sections-with-support-sheaves", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the", "inclusion of a closed subset. Denote $j : U = X \\setminus Z \\to X$", "the inclusion of the complement. There is a distinguished triangle", "$$", "i_*R\\mathcal{H}_Z(K) \\to K \\to Rj_*(K|_U) \\to", "i_*R\\mathcal{H}_Z(K)[1]", "$$", "in $D(\\mathcal{O}_X)$ functorial for $K$ in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $\\mathcal{I}^\\bullet$ all of whose terms", "are injective $\\mathcal{O}_X$-modules representing $K$. See", "Section \\ref{section-unbounded}. Recall that $\\mathcal{I}^\\bullet|_U$", "is a K-injective complex of $\\mathcal{O}_U$-modules, see", "Lemma \\ref{lemma-restrict-K-injective-to-open}. Hence each", "of the derived functors in the distinguished triangle is gotten", "by applying the underlying functor to $\\mathcal{I}^\\bullet$.", "Hence it suffices to prove that", "for an injective $\\mathcal{O}_X$-module $\\mathcal{I}$ we have", "a short exact sequence", "$$", "0 \\to i_*\\mathcal{H}_Z(\\mathcal{I}) \\to \\mathcal{I}", "\\to j_*(\\mathcal{I}|_U) \\to 0", "$$", "This follows from Lemma \\ref{lemma-injective-restriction-surjective}", "and the definitions." ], "refs": [ "cohomology-lemma-restrict-K-injective-to-open", "cohomology-lemma-injective-restriction-surjective" ], "ref_ids": [ 2134, 2041 ] } ], "ref_ids": [] }, { "id": 2156, "type": "theorem", "label": "cohomology-lemma-sections-support-in-closed-disjoint-open", "categories": [ "cohomology" ], "title": "cohomology-lemma-sections-support-in-closed-disjoint-open", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $Z \\subset X$", "be a closed subset. Let $j : U \\to X$ be the inclusion of", "an open subset with $U \\cap Z = \\emptyset$. Then", "$R\\mathcal{H}_Z(Rj_*K) = 0$ for all $K$ in $D(\\mathcal{O}_U)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $\\mathcal{I}^\\bullet$ of $\\mathcal{O}_U$-modules", "representing $K$. Then $j_*\\mathcal{I}^\\bullet$ represents $Rj_*K$. By", "Lemma \\ref{lemma-K-injective-flat} the complex $j_*\\mathcal{I}^\\bullet$ is a", "K-injective complex of $\\mathcal{O}_X$-modules. Hence", "$\\mathcal{H}_Z(j_*\\mathcal{I}^\\bullet)$ represents $R\\mathcal{H}_Z(Rj_*K)$.", "Thus it suffices to show that $\\mathcal{H}_Z(j_*\\mathcal{G}) = 0$", "for any abelian sheaf $\\mathcal{G}$ on $U$. Thus we have to show that", "a section $s$ of $j_*\\mathcal{G}$ over some open $W$ which is supported", "on $W \\cap Z$ is zero. The support condition means that", "$s|_{W \\setminus W \\cap Z} = 0$. Since $j_*\\mathcal{G}(W) =", "\\mathcal{G}(U \\cap W) = j_*\\mathcal{G}(W \\setminus W \\cap Z)$", "this implies that $s$ is zero as desired." ], "refs": [ "cohomology-lemma-K-injective-flat" ], "ref_ids": [ 2142 ] } ], "ref_ids": [] }, { "id": 2157, "type": "theorem", "label": "cohomology-lemma-sections-support-abelian-unbounded", "categories": [ "cohomology" ], "title": "cohomology-lemma-sections-support-abelian-unbounded", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $Z \\subset X$", "be a closed subset. Let $K$ be an object of $D(\\mathcal{O}_X)$", "and denote $K_{ab}$ its image in $D(\\underline{\\mathbf{Z}}_X)$.", "\\begin{enumerate}", "\\item There is a canonical map $R\\Gamma_Z(X, K) \\to R\\Gamma_Z(X, K_{ab})$", "which is an isomorphism in $D(\\textit{Ab})$.", "\\item There is a canonical map", "$R\\mathcal{H}_Z(K) \\to R\\mathcal{H}_Z(K_{ab})$", "which is an isomorphism in $D(\\underline{\\mathbf{Z}}_Z)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). The map is constructed as follows. Choose a K-injective complex", "of $\\mathcal{O}_X$-modules $\\mathcal{I}^\\bullet$ representing $K$.", "Choose a quasi-isomorpism", "$\\mathcal{I}^\\bullet \\to \\mathcal{J}^\\bullet$ where $\\mathcal{J}^\\bullet$", "is a K-injective complex of abelian groups. Then the map in (1)", "is given by", "$$", "\\Gamma_Z(X, \\mathcal{I}^\\bullet) \\to \\Gamma_Z(X, \\mathcal{J}^\\bullet)", "$$", "determined by the fact that $\\Gamma_Z$ is a functor on abelian sheaves.", "An easy check shows that the resulting map combined with the canonical", "maps of Lemma \\ref{lemma-modules-abelian-unbounded}", "fit into a morphism of distinguished triangles", "$$", "\\xymatrix{", "R\\Gamma_Z(X, K) \\ar[r] \\ar[d] &", "R\\Gamma(X, K) \\ar[r] \\ar[d] &", "R\\Gamma(U, K) \\ar[d] \\\\", "R\\Gamma_Z(X, K_{ab}) \\ar[r] &", "R\\Gamma(X, K_{ab}) \\ar[r] &", "R\\Gamma(U, K_{ab})", "}", "$$", "of Lemma \\ref{lemma-triangle-sections-with-support}.", "Since two of the three arrows are isomorphisms by the lemma cited,", "we conclude by Derived Categories, Lemma", "\\ref{derived-lemma-third-isomorphism-triangle}.", "\\medskip\\noindent", "The proof of (2) is omitted. Hint: use the same argument with", "Lemma \\ref{lemma-triangle-sections-with-support-sheaves}", "for the distinguished triangle." ], "refs": [ "cohomology-lemma-modules-abelian-unbounded", "cohomology-lemma-triangle-sections-with-support", "derived-lemma-third-isomorphism-triangle", "cohomology-lemma-triangle-sections-with-support-sheaves" ], "ref_ids": [ 2140, 2154, 1759, 2155 ] } ], "ref_ids": [] }, { "id": 2158, "type": "theorem", "label": "cohomology-lemma-support-cup-product", "categories": [ "cohomology" ], "title": "cohomology-lemma-support-cup-product", "contents": [ "With notation as in Remark \\ref{remark-support-cup-product} the diagram", "$$", "\\xymatrix{", "H^i(X, K) \\times H^j_Z(X, M) \\ar[r] \\ar[d] &", "H^{i + j}_Z(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\ar[d] \\\\", "H^i(X, K) \\times H^j(X, M) \\ar[r] &", "H^{i + j}(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)", "}", "$$", "commutes where the top horizontal arrow is the cup product of", "Remark \\ref{remark-support-cup-product-global}." ], "refs": [ "cohomology-remark-support-cup-product", "cohomology-remark-support-cup-product-global" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 2275, 2276 ] }, { "id": 2159, "type": "theorem", "label": "cohomology-lemma-support-functorial", "categories": [ "cohomology" ], "title": "cohomology-lemma-support-functorial", "contents": [ "With notation and assumptions as in Remark \\ref{remark-support-functorial}", "the diagram", "$$", "\\xymatrix{", "H^p_Z(X, K) \\ar[r] \\ar[d] & H^p_{Z'}(X, Lf^*K) \\ar[d] \\\\", "H^p(X, K) \\ar[r] & H^p(X', Lf^*K)", "}", "$$", "commutes. Here the top horizontal arrow comes from the identifications", "$H^p_Z(X, K) = H^p(Z, R\\mathcal{H}_Z(K))$ and", "$H^p_{Z'}(X', Lf^*K) = H^p(Z', R\\mathcal{H}_{Z'}(K'))$,", "the pullback map", "$H^p(Z, R\\mathcal{H}_Z(K)) \\to H^p(Z', L(f|_{Z'})^*R\\mathcal{H}_Z(K))$,", "and the map constructed in Remark \\ref{remark-support-functorial}." ], "refs": [ "cohomology-remark-support-functorial", "cohomology-remark-support-functorial" ], "proofs": [ { "contents": [ "Omitted. Hints:", "Using that $H^p(Z, R\\mathcal{H}_Z(K)) = H^p(X, i_*R\\mathcal{H}_Z(K))$", "and similarly for $R\\mathcal{H}_{Z'}(Lf^*K)$ this follows from", "the functoriality of the pullback maps and the commutative diagram", "used to define the map of Remark \\ref{remark-support-functorial}." ], "refs": [ "cohomology-remark-support-functorial" ], "ref_ids": [ 2277 ] } ], "ref_ids": [ 2277, 2277 ] }, { "id": 2160, "type": "theorem", "label": "cohomology-lemma-RGamma-commutes-with-Rlim", "categories": [ "cohomology" ], "title": "cohomology-lemma-RGamma-commutes-with-Rlim", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. For $U \\subset X$ open the", "functor $R\\Gamma(U, -)$ commutes with $R\\lim$. Moreover, there are", "short exact sequences", "$$", "0 \\to", "R^1\\lim H^{m - 1}(U, K_n) \\to H^m(U, R\\lim K_n) \\to", "\\lim H^m(U, K_n) \\to 0", "$$", "for any inverse system $(K_n)$ in $D(\\mathcal{O}_X)$ and any $m \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from", "Injectives, Lemma \\ref{injectives-lemma-RF-commutes-with-Rlim}.", "Then we may apply ", "More on Algebra, Remark \\ref{more-algebra-remark-compare-derived-limit}", "to $R\\lim R\\Gamma(U, K_n) = R\\Gamma(U, R\\lim K_n)$ to get the short", "exact sequences." ], "refs": [ "injectives-lemma-RF-commutes-with-Rlim", "more-algebra-remark-compare-derived-limit" ], "ref_ids": [ 7796, 10658 ] } ], "ref_ids": [] }, { "id": 2161, "type": "theorem", "label": "cohomology-lemma-Rf-commutes-with-Rlim", "categories": [ "cohomology" ], "title": "cohomology-lemma-Rf-commutes-with-Rlim", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed", "spaces. Then $Rf_*$ commutes with $R\\lim$, i.e., $Rf_*$ commutes with", "derived limits." ], "refs": [], "proofs": [ { "contents": [ "Let $(K_n)$ be an inverse system in $D(\\mathcal{O}_X)$. Consider the defining", "distinguished triangle", "$$", "R\\lim K_n \\to \\prod K_n \\to \\prod K_n", "$$", "in $D(\\mathcal{O}_X)$. Applying the exact functor $Rf_*$ we obtain", "the distinguished triangle", "$$", "Rf_*(R\\lim K_n) \\to Rf_*\\left(\\prod K_n\\right) \\to Rf_*\\left(\\prod K_n\\right)", "$$", "in $D(\\mathcal{O}_Y)$. Thus we see that it suffices to prove that", "$Rf_*$ commutes with products in the derived category (which are not just", "given by products of complexes, see", "Injectives, Lemma \\ref{injectives-lemma-derived-products}).", "However, since $Rf_*$ is a right adjoint by Lemma \\ref{lemma-adjoint}", "this follows formally (see", "Categories, Lemma \\ref{categories-lemma-adjoint-exact}).", "Caution: Note that we cannot apply", "Categories, Lemma \\ref{categories-lemma-adjoint-exact}", "directly as $R\\lim K_n$ is not a limit in $D(\\mathcal{O}_X)$." ], "refs": [ "injectives-lemma-derived-products", "cohomology-lemma-adjoint", "categories-lemma-adjoint-exact", "categories-lemma-adjoint-exact" ], "ref_ids": [ 7795, 2121, 12249, 12249 ] } ], "ref_ids": [] }, { "id": 2162, "type": "theorem", "label": "cohomology-lemma-inverse-limit-is-derived-limit", "categories": [ "cohomology" ], "title": "cohomology-lemma-inverse-limit-is-derived-limit", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(\\mathcal{F}_n)$ be an", "inverse system of $\\mathcal{O}_X$-modules. Let $\\mathcal{B}$ be a set", "of opens of $X$. Assume", "\\begin{enumerate}", "\\item every open of $X$ has a covering whose members are elements of", "$\\mathcal{B}$,", "\\item $H^p(U, \\mathcal{F}_n) = 0$ for $p > 0$ and $U \\in \\mathcal{B}$,", "\\item the inverse system $\\mathcal{F}_n(U)$ has vanishing $R^1\\lim$", "for $U \\in \\mathcal{B}$.", "\\end{enumerate}", "Then $R\\lim \\mathcal{F}_n = \\lim \\mathcal{F}_n$ and we have", "$H^p(U, \\lim \\mathcal{F}_n) = 0$ for $p > 0$ and $U \\in \\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "Set $K_n = \\mathcal{F}_n$ and $K = R\\lim \\mathcal{F}_n$. Using the notation", "of Remark \\ref{remark-discuss-derived-limit} and assumption (2) we see that for", "$U \\in \\mathcal{B}$ we have $\\underline{\\mathcal{H}}_n^m(U) = 0$", "when $m \\not = 0$ and $\\underline{\\mathcal{H}}_n^0(U) = \\mathcal{F}_n(U)$.", "From Equation (\\ref{equation-ses-Rlim-over-U}) and assumption (3)", "we see that $\\underline{\\mathcal{H}}^m(U) = 0$", "when $m \\not = 0$ and equal to $\\lim \\mathcal{F}_n(U)$", "when $m = 0$. Sheafifying using (1) we find that", "$\\mathcal{H}^m = 0$ when $m \\not = 0$ and equal to", "$\\lim \\mathcal{F}_n$ when $m = 0$. Hence $K = \\lim \\mathcal{F}_n$.", "Since $H^m(U, K) = \\underline{\\mathcal{H}}^m(U) = 0$ for $m > 0$", "(see above) we see that the second assertion holds." ], "refs": [ "cohomology-remark-discuss-derived-limit" ], "ref_ids": [ 2278 ] } ], "ref_ids": [] }, { "id": 2163, "type": "theorem", "label": "cohomology-lemma-cohomology-derived-limit-injective", "categories": [ "cohomology" ], "title": "cohomology-lemma-cohomology-derived-limit-injective", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(K_n)$ be an", "inverse system in $D(\\mathcal{O}_X)$. Let $x \\in X$ and $m \\in \\mathbf{Z}$.", "Assume there exist an integer $n(x)$ and a fundamental system $\\mathfrak{U}_x$", "of open neighbourhoods of $x$ such that for $U \\in \\mathfrak{U}_x$", "\\begin{enumerate}", "\\item $R^1\\lim H^{m - 1}(U, K_n) = 0$, and", "\\item $H^m(U, K_n) \\to H^m(U, K_{n(x)})$ is injective", "for $n \\geq n(x)$.", "\\end{enumerate}", "Then the map on stalks $H^m(R\\lim K_n)_x \\to H^m(K_{n(x)})_x$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\gamma$ be an element of $H^m(R\\lim K_n)_x$ which maps to zero", "in $H^m(K_{n(x)})_x$. Since $H^m(R\\lim K_n)$ is the sheafification", "of $U \\mapsto H^m(U, R\\lim K_n)$", "(by Lemma \\ref{lemma-sheafification-cohomology})", "we can choose $U \\in \\mathfrak{U}_x$", "and an element $\\tilde \\gamma \\in H^m(U, R\\lim K_n)$ mapping to $\\gamma$.", "Then $\\tilde\\gamma$ maps to $\\tilde\\gamma_{n(x)} \\in H^m(U, K_{n(x)})$.", "Using that $H^m(K_{n(x)})$ is the sheafification of", "$U \\mapsto H^m(U, K_{n(x)})$", "(by Lemma \\ref{lemma-sheafification-cohomology} again)", "we see that after shrinking $U$ we may assume that $\\tilde\\gamma_{n(x)} = 0$.", "For this $U$ we consider the short exact sequence", "$$", "0 \\to", "R^1\\lim H^{m - 1}(U, K_n) \\to H^m(U, R\\lim K_n) \\to", "\\lim H^m(U, K_n) \\to 0", "$$", "of Lemma \\ref{lemma-RGamma-commutes-with-Rlim}.", "By assumption (1) the group on the left is zero and by", "assumption (2) the group on the right maps injectively", "into $H^m(U, K_{n(x)})$. We conclude $\\tilde\\gamma = 0$", "and hence $\\gamma = 0$ as desired." ], "refs": [ "cohomology-lemma-sheafification-cohomology", "cohomology-lemma-sheafification-cohomology", "cohomology-lemma-RGamma-commutes-with-Rlim" ], "ref_ids": [ 2136, 2136, 2160 ] } ], "ref_ids": [] }, { "id": 2164, "type": "theorem", "label": "cohomology-lemma-is-limit-per-point", "categories": [ "cohomology" ], "title": "cohomology-lemma-is-limit-per-point", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E \\in D(\\mathcal{O}_X)$.", "Assume that for every $x \\in X$ there exist", "a function $p(x, -) : \\mathbf{Z} \\to \\mathbf{Z}$ and", "a fundamental system $\\mathfrak{U}_x$ of open neighbourhoods of $x$", "such that", "$$", "H^p(U, H^{m - p}(E)) = 0 \\text{ for }", "U \\in \\mathfrak{U}_x \\text{ and } p > p(x, m)", "$$", "Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$", "is an isomorphism in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Set $K_n = \\tau_{\\geq -n}E$ and $K = R\\lim K_n$.", "The canonical map $E \\to K$", "comes from the canonical maps $E \\to K_n = \\tau_{\\geq -n}E$.", "We have to show that $E \\to K$ induces an isomorphism", "$H^m(E) \\to H^m(K)$ of cohomology sheaves. In the rest of the", "proof we fix $m$. If $n \\geq -m$, then", "the map $E \\to \\tau_{\\geq -n}E = K_n$ induces an isomorphism", "$H^m(E) \\to H^m(K_n)$.", "To finish the proof it suffices to show that for every $x \\in X$", "there exists an integer $n(x) \\geq -m$ such that the map", "$H^m(K)_x \\to H^m(K_{n(x)})_x$ is injective. Namely, then", "the composition", "$$", "H^m(E)_x \\to H^m(K)_x \\to H^m(K_{n(x)})_x", "$$", "is a bijection and the second arrow is injective, hence the", "first arrow is bijective. Set", "$$", "n(x) = 1 + \\max\\{-m, p(x, m - 1) - m, -1 + p(x, m) - m, -2 + p(x, m + 1) - m\\}.", "$$", "so that in any case $n(x) \\geq -m$. Claim: the maps", "$$", "H^{m - 1}(U, K_{n + 1}) \\to H^{m - 1}(U, K_n)", "\\quad\\text{and}\\quad", "H^m(U, K_{n + 1}) \\to H^m(U, K_n)", "$$", "are isomorphisms for $n \\geq n(x)$ and $U \\in \\mathfrak{U}_x$.", "The claim implies conditions", "(1) and (2) of Lemma \\ref{lemma-cohomology-derived-limit-injective}", "are satisfied and hence implies the desired injectivity.", "Recall (Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle})", "that we have distinguished triangles", "$$", "H^{-n - 1}(E)[n + 1] \\to", "K_{n + 1} \\to K_n \\to H^{-n - 1}(E)[n + 2]", "$$", "Looking at the asssociated long exact cohomology sequence the claim follows if", "$$", "H^{m + n}(U, H^{-n - 1}(E)),\\quad", "H^{m + n + 1}(U, H^{-n - 1}(E)),\\quad", "H^{m + n + 2}(U, H^{-n - 1}(E))", "$$", "are zero for $n \\geq n(x)$ and $U \\in \\mathfrak{U}_x$.", "This follows from our choice of $n(x)$", "and the assumption in the lemma." ], "refs": [ "cohomology-lemma-cohomology-derived-limit-injective", "derived-remark-truncation-distinguished-triangle" ], "ref_ids": [ 2163, 2016 ] } ], "ref_ids": [] }, { "id": 2165, "type": "theorem", "label": "cohomology-lemma-is-limit-spaltenstein", "categories": [ "cohomology" ], "title": "cohomology-lemma-is-limit-spaltenstein", "contents": [ "\\begin{reference}", "\\cite[Proposition 3.13]{Spaltenstein}", "\\end{reference}", "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E \\in D(\\mathcal{O}_X)$.", "Assume that for every $x \\in X$ there exist an integer $d_x \\geq 0$ and", "a fundamental system $\\mathfrak{U}_x$ of open neighbourhoods of $x$", "such that", "$$", "H^p(U, H^q(E)) = 0 \\text{ for }", "U \\in \\mathfrak{U}_x,\\ p > d_x, \\text{ and }q < 0", "$$", "Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$", "is an isomorphism in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-is-limit-per-point}", "with $p(x, m) = d_x + \\max(0, m)$." ], "refs": [ "cohomology-lemma-is-limit-per-point" ], "ref_ids": [ 2164 ] } ], "ref_ids": [] }, { "id": 2166, "type": "theorem", "label": "cohomology-lemma-is-limit", "categories": [ "cohomology" ], "title": "cohomology-lemma-is-limit", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E \\in D(\\mathcal{O}_X)$.", "Assume there exist a function $p(-) : \\mathbf{Z} \\to \\mathbf{Z}$", "and a set $\\mathcal{B}$ of opens of $X$ such that", "\\begin{enumerate}", "\\item every open in $X$ has a covering whose members are", "elements of $\\mathcal{B}$, and", "\\item $H^p(U, H^{m - p}(E)) = 0$ for $p > p(m)$ and $U \\in \\mathcal{B}$.", "\\end{enumerate}", "Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$", "is an isomorphism in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-is-limit-per-point}", "with $p(x, m) = p(m)$ and", "$\\mathfrak{U}_x = \\{U \\in \\mathcal{B} \\mid x \\in U\\}$." ], "refs": [ "cohomology-lemma-is-limit-per-point" ], "ref_ids": [ 2164 ] } ], "ref_ids": [] }, { "id": 2167, "type": "theorem", "label": "cohomology-lemma-is-limit-dimension", "categories": [ "cohomology" ], "title": "cohomology-lemma-is-limit-dimension", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E \\in D(\\mathcal{O}_X)$.", "Assume there exist an integer $d \\geq 0$ and a basis $\\mathcal{B}$ for the", "topology of $X$ such that", "$$", "H^p(U, H^q(E)) = 0 \\text{ for }", "U \\in \\mathcal{B},\\ p > d, \\text{ and }q < 0", "$$", "Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$", "is an isomorphism in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-is-limit-spaltenstein} with $d_x = d$", "and $\\mathfrak{U}_x = \\{U \\in \\mathcal{B} \\mid x \\in U\\}$." ], "refs": [ "cohomology-lemma-is-limit-spaltenstein" ], "ref_ids": [ 2165 ] } ], "ref_ids": [] }, { "id": 2168, "type": "theorem", "label": "cohomology-lemma-cohomology-over-U-trivial", "categories": [ "cohomology" ], "title": "cohomology-lemma-cohomology-over-U-trivial", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K$", "be an object of $D(\\mathcal{O}_X)$.", "Let $\\mathcal{B}$ be a set of opens of $X$. Assume", "\\begin{enumerate}", "\\item every open of $X$ has a covering whose members are", "elements of $\\mathcal{B}$,", "\\item $H^p(U, H^q(K)) = 0$ for all $p > 0$, $q \\in \\mathbf{Z}$, and", "$U \\in \\mathcal{B}$.", "\\end{enumerate}", "Then $H^q(U, K) = H^0(U, H^q(K))$ for $q \\in \\mathbf{Z}$", "and $U \\in \\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $K = R\\lim \\tau_{\\geq -n} K$ by", "Lemma \\ref{lemma-is-limit-dimension} with $d = 0$.", "Let $U \\in \\mathcal{B}$. By Equation (\\ref{equation-ses-Rlim-over-U})", "we get a short exact sequence", "$$", "0 \\to R^1\\lim H^{q - 1}(U, \\tau_{\\geq -n}K) \\to", "H^q(U, K) \\to \\lim H^q(U, \\tau_{\\geq -n}K) \\to 0", "$$", "Condition (2) implies", "$H^q(U, \\tau_{\\geq -n} K) = H^0(U, H^q(\\tau_{\\geq -n} K))$", "for all $q$ by using the spectral sequence of", "Example \\ref{example-spectral-sequence}.", "The spectral sequence converges because $\\tau_{\\geq -n}K$ is bounded", "below. If $n > -q$ then we have $H^q(\\tau_{\\geq -n}K) = H^q(K)$.", "Thus the systems on the left and the right of the displayed", "short exact sequence are eventually constant with values", "$H^0(U, H^{q - 1}(K))$ and $H^0(U, H^q(K))$. The lemma follows." ], "refs": [ "cohomology-lemma-is-limit-dimension" ], "ref_ids": [ 2167 ] } ], "ref_ids": [] }, { "id": 2169, "type": "theorem", "label": "cohomology-lemma-derived-limit-suitable-system", "categories": [ "cohomology" ], "title": "cohomology-lemma-derived-limit-suitable-system", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(K_n)$", "be an inverse system of objects of $D(\\mathcal{O}_X)$.", "Let $\\mathcal{B}$ be a set of opens of $X$. Assume", "\\begin{enumerate}", "\\item every open of $X$ has a covering whose members are", "elements of $\\mathcal{B}$,", "\\item for all $U \\in \\mathcal{B}$ and all $q \\in \\mathbf{Z}$ we have", "\\begin{enumerate}", "\\item $H^p(U, H^q(K_n)) = 0$ for $p > 0$,", "\\item the inverse system $H^0(U, H^q(K_n))$ has vanishing $R^1\\lim$.", "\\end{enumerate}", "\\end{enumerate}", "Then $H^q(R\\lim K_n) = \\lim H^q(K_n)$ for $q \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Set $K = R\\lim K_n$. We will use notation as in", "Remark \\ref{remark-discuss-derived-limit}. Let $U \\in \\mathcal{B}$.", "By Lemma \\ref{lemma-cohomology-over-U-trivial} and (2)(a)", "we have $H^q(U, K_n) = H^0(U, H^q(K_n))$.", "Using that the functor $R\\Gamma(U, -)$ commutes with", "derived limits we have", "$$", "H^q(U, K) = H^q(R\\lim R\\Gamma(U, K_n)) = \\lim H^0(U, H^q(K_n))", "$$", "where the final equality follows from", "More on Algebra, Remark \\ref{more-algebra-remark-compare-derived-limit}", "and assumption (2)(b). Thus $H^q(U, K)$ is the inverse limit", "the sections of the sheaves $H^q(K_n)$ over $U$. Since", "$\\lim H^q(K_n)$ is a sheaf we find using assumption (1) that $H^q(K)$,", "which is the sheafification of the presheaf $U \\mapsto H^q(U, K)$,", "is equal to $\\lim H^q(K_n)$. This proves the lemma." ], "refs": [ "cohomology-remark-discuss-derived-limit", "cohomology-lemma-cohomology-over-U-trivial", "more-algebra-remark-compare-derived-limit" ], "ref_ids": [ 2278, 2168, 10658 ] } ], "ref_ids": [] }, { "id": 2170, "type": "theorem", "label": "cohomology-lemma-K-injective", "categories": [ "cohomology" ], "title": "cohomology-lemma-K-injective", "contents": [ "In the situation described above.", "Denote $\\mathcal{H}^m = H^m(\\mathcal{F}^\\bullet)$ the $m$th cohomology sheaf.", "Let $\\mathcal{B}$ be a set of open subsets of $X$.", "Let $d \\in \\mathbf{N}$.", "Assume", "\\begin{enumerate}", "\\item every open in $X$ has a covering whose members are", "elements of $\\mathcal{B}$,", "\\item for every $U \\in \\mathcal{B}$ we have $H^p(U, \\mathcal{H}^q) = 0$", "for $p > d$ and $q < 0$\\footnote{It suffices if", "$\\forall m$, $\\exists p(m)$, $H^p(U. \\mathcal{H}^{m - p}) = 0$ for", "$p > p(m)$, see Lemma \\ref{lemma-is-limit}.}.", "\\end{enumerate}", "Then (\\ref{equation-into-candidate-K-injective}) is a quasi-isomorphism." ], "refs": [ "cohomology-lemma-is-limit" ], "proofs": [ { "contents": [ "By Derived Categories, Lemma \\ref{derived-lemma-difficulty-K-injectives}", "it suffices to show that the canonical map", "$\\mathcal{F}^\\bullet \\to R\\lim \\tau_{\\geq -n} \\mathcal{F}^\\bullet$", "is an isomorphism. This is Lemma \\ref{lemma-is-limit-dimension}." ], "refs": [ "derived-lemma-difficulty-K-injectives", "cohomology-lemma-is-limit-dimension" ], "ref_ids": [ 1927, 2167 ] } ], "ref_ids": [ 2166 ] }, { "id": 2171, "type": "theorem", "label": "cohomology-lemma-inverse-limit-complexes", "categories": [ "cohomology" ], "title": "cohomology-lemma-inverse-limit-complexes", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(\\mathcal{F}_n^\\bullet)$", "be an inverse system of complexes of $\\mathcal{O}_X$-modules.", "Let $m \\in \\mathbf{Z}$. Assume there exist a set $\\mathcal{B}$", "of open subsets of $X$ and an integer $n_0$ such that", "\\begin{enumerate}", "\\item every open in $X$ has a covering whose members are", "elements of $\\mathcal{B}$,", "\\item for every $U \\in \\mathcal{B}$", "\\begin{enumerate}", "\\item the systems of abelian groups", "$\\mathcal{F}_n^{m - 2}(U)$ and $\\mathcal{F}_n^{m - 1}(U)$", "have vanishing $R^1\\lim$ (for example these have the Mittag-Leffler", "condition),", "\\item the system of abelian groups $H^{m - 1}(\\mathcal{F}_n^\\bullet(U))$", "has vanishing $R^1\\lim$ (for example it has the Mittag-Leffler condition), and", "\\item we have", "$H^m(\\mathcal{F}_n^\\bullet(U)) = H^m(\\mathcal{F}_{n_0}^\\bullet(U))$", "for all $n \\geq n_0$.", "\\end{enumerate}", "\\end{enumerate}", "Then the maps", "$H^m(\\mathcal{F}^\\bullet) \\to \\lim H^m(\\mathcal{F}_n^\\bullet) \\to", "H^m(\\mathcal{F}_{n_0}^\\bullet)$", "are isomorphisms of sheaves where", "$\\mathcal{F}^\\bullet = \\lim \\mathcal{F}_n^\\bullet$ is the termwise", "inverse limit." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\in \\mathcal{B}$. Note that $H^m(\\mathcal{F}^\\bullet(U))$ is the", "cohomology of", "$$", "\\lim_n \\mathcal{F}_n^{m - 2}(U) \\to", "\\lim_n \\mathcal{F}_n^{m - 1}(U) \\to", "\\lim_n \\mathcal{F}_n^m(U) \\to", "\\lim_n \\mathcal{F}_n^{m + 1}(U)", "$$", "in the third spot from the left. By assumptions (2)(a) and (2)(b)", "we may apply", "More on Algebra, Lemma \\ref{more-algebra-lemma-apply-Mittag-Leffler-again}", "to conclude that", "$$", "H^m(\\mathcal{F}^\\bullet(U)) = \\lim H^m(\\mathcal{F}_n^\\bullet(U))", "$$", "By assumption (2)(c) we conclude", "$$", "H^m(\\mathcal{F}^\\bullet(U)) = H^m(\\mathcal{F}_n^\\bullet(U))", "$$", "for all $n \\geq n_0$. By assumption (1) we conclude that the sheafification of", "$U \\mapsto H^m(\\mathcal{F}^\\bullet(U))$ is equal to the sheafification", "of $U \\mapsto H^m(\\mathcal{F}_n^\\bullet(U))$ for all $n \\geq n_0$.", "Thus the inverse system of sheaves $H^m(\\mathcal{F}_n^\\bullet)$ is", "constant for $n \\geq n_0$ with value $H^m(\\mathcal{F}^\\bullet)$ which", "proves the lemma." ], "refs": [ "more-algebra-lemma-apply-Mittag-Leffler-again" ], "ref_ids": [ 10314 ] } ], "ref_ids": [] }, { "id": 2172, "type": "theorem", "label": "cohomology-lemma-alternating-cech-complex-complex", "categories": [ "cohomology" ], "title": "cohomology-lemma-alternating-cech-complex-complex", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be", "a finite open covering. For a complex $\\mathcal{F}^\\bullet$", "of $\\mathcal{O}_X$-modules there is a canonical map", "$$", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet_{alt}(\\mathcal{U}, \\mathcal{F}^\\bullet))", "\\longrightarrow", "R\\Gamma(X, \\mathcal{F}^\\bullet)", "$$", "functorial in $\\mathcal{F}^\\bullet$ and compatible with", "(\\ref{equation-global-sections-to-alternating-cech})." ], "refs": [], "proofs": [ { "contents": [ "Let ${\\mathcal I}^\\bullet$ be a K-injective complex whose terms", "are injective $\\mathcal{O}_X$-modules.", "The map (\\ref{equation-global-sections-to-alternating-cech}) for", "$\\mathcal{I}^\\bullet$ is a map", "$\\Gamma(X, {\\mathcal I}^\\bullet) \\to", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet_{alt}({\\mathcal U},", "{\\mathcal I}^\\bullet))$.", "This is a quasi-isomorphism of complexes of abelian groups", "as follows from", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}", "applied to the double complex", "$\\check{\\mathcal{C}}^\\bullet_{alt}({\\mathcal U},", "{\\mathcal I}^\\bullet)$ using", "Lemmas \\ref{lemma-injective-trivial-cech} and \\ref{lemma-alternating-usual}.", "Suppose ${\\mathcal F}^\\bullet \\to {\\mathcal I}^\\bullet$ is a quasi-isomorphism", "of ${\\mathcal F}^\\bullet$ into a K-injective complex whose terms", "are injectives (Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}).", "Since $R\\Gamma(X, {\\mathcal F}^\\bullet)$ is represented by the complex", "$\\Gamma(X, {\\mathcal I}^\\bullet)$ we obtain the map of the lemma", "using", "$$", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet_{alt}({\\mathcal U},", "{\\mathcal F}^\\bullet))", "\\longrightarrow", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet_{alt}({\\mathcal U},", "{\\mathcal I}^\\bullet)).", "$$", "We omit the verification of functoriality and compatibilities." ], "refs": [ "homology-lemma-double-complex-gives-resolution", "cohomology-lemma-injective-trivial-cech", "cohomology-lemma-alternating-usual", "injectives-theorem-K-injective-embedding-grothendieck" ], "ref_ids": [ 12106, 2051, 2095, 7768 ] } ], "ref_ids": [] }, { "id": 2173, "type": "theorem", "label": "cohomology-lemma-alternating-cech-complex-complex-ss", "categories": [ "cohomology" ], "title": "cohomology-lemma-alternating-cech-complex-complex-ss", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let", "$\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be a finite open covering. Let", "$\\mathcal{F}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules.", "Let $\\mathcal{B}$ be a set of open subsets of $X$. Assume", "\\begin{enumerate}", "\\item every open in $X$ has a covering whose members are", "elements of $\\mathcal{B}$,", "\\item we have $U_{i_0\\ldots i_p} \\in \\mathcal{B}$ for all", "$i_0, \\ldots, i_p \\in I$,", "\\item for every $U \\in \\mathcal{B}$ and $p > 0$ we have", "\\begin{enumerate}", "\\item $H^p(U, \\mathcal{F}^q) = 0$,", "\\item $H^p(U, \\Coker(\\mathcal{F}^{q - 1} \\to \\mathcal{F}^q)) = 0$, and", "\\item $H^p(U, H^q(\\mathcal{F})) = 0$.", "\\end{enumerate}", "\\end{enumerate}", "Then the map", "$$", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet_{alt}(\\mathcal{U}, \\mathcal{F}^\\bullet))", "\\longrightarrow", "R\\Gamma(X, \\mathcal{F}^\\bullet)", "$$", "of Lemma \\ref{lemma-alternating-cech-complex-complex}", "is an isomorphism in $D(\\textit{Ab})$." ], "refs": [ "cohomology-lemma-alternating-cech-complex-complex" ], "proofs": [ { "contents": [ "First assume $\\mathcal{F}^\\bullet$ is bounded below. In this case the map", "$$", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet_{alt}(\\mathcal{U}, \\mathcal{F}^\\bullet))", "\\longrightarrow", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))", "$$", "is a quasi-isomorphism by Lemma \\ref{lemma-alternating-usual}.", "Namely, the map of double complexes", "$\\check{\\mathcal{C}}^\\bullet_{alt}(\\mathcal{U}, \\mathcal{F}^\\bullet) \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet)$", "induces an isomorphism between the first pages of the second spectral sequences", "associated to these complexes", "(by Homology, Lemma \\ref{homology-lemma-ss-double-complex})", "and these spectral sequences converge", "(Homology, Lemma \\ref{homology-lemma-first-quadrant-ss}).", "Thus the conclusion in this case by", "Lemma \\ref{lemma-cech-complex-complex-computes} and assumption (3)(a).", "\\medskip\\noindent", "In general, by assumption (3)(c) we may choose a resolution", "$\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet = \\lim \\mathcal{I}_n^\\bullet$", "as in Lemma \\ref{lemma-K-injective}.", "Then the map of the lemma becomes", "$$", "\\lim_n ", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet_{alt}(\\mathcal{U},", "\\tau_{\\geq -n}\\mathcal{F}^\\bullet))", "\\longrightarrow", "\\Gamma(X, \\mathcal{I}^\\bullet) =", "\\lim_n \\Gamma(X, \\mathcal{I}_n^\\bullet)", "$$", "Here the arrow is in the derived category, but the equality on the", "right holds on the level of complexes.", "Note that (3)(b) shows that $\\tau_{\\geq -n}\\mathcal{F}^\\bullet$", "is a bounded below complex satisfying the hypothesis of the lemma.", "Thus the case of bounded below complexes shows each of the maps", "$$", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet_{alt}(\\mathcal{U},", "\\tau_{\\geq -n}\\mathcal{F}^\\bullet))", "\\longrightarrow", "\\Gamma(X, \\mathcal{I}_n^\\bullet)", "$$", "is a quasi-isomorphism. The cohomologies of the complexes on the left", "hand side in given degree are eventually", "constant (as the alternating {\\v C}ech complex is finite).", "Hence the same is true on the right hand side.", "Thus the cohomology of the limit on the right hand side is", "this constant value by", "Homology, Lemma \\ref{homology-lemma-apply-Mittag-Leffler-again}", "(or the stronger More on Algebra, Lemma", "\\ref{more-algebra-lemma-apply-Mittag-Leffler-again})", "and we win." ], "refs": [ "cohomology-lemma-alternating-usual", "homology-lemma-ss-double-complex", "homology-lemma-first-quadrant-ss", "cohomology-lemma-cech-complex-complex-computes", "cohomology-lemma-K-injective", "homology-lemma-apply-Mittag-Leffler-again", "more-algebra-lemma-apply-Mittag-Leffler-again" ], "ref_ids": [ 2095, 12104, 12105, 2099, 2170, 12128, 10314 ] } ], "ref_ids": [ 2172 ] }, { "id": 2174, "type": "theorem", "label": "cohomology-lemma-compose", "categories": [ "cohomology" ], "title": "cohomology-lemma-compose", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Given complexes $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$", "of $\\mathcal{O}_X$-modules there is an isomorphism", "$$", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet,", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet))", "=", "\\SheafHom^\\bullet(\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}", "\\mathcal{L}^\\bullet), \\mathcal{M}^\\bullet)", "$$", "of complexes of $\\mathcal{O}_X$-modules functorial in", "$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is proved in exactly the same way as", "More on Algebra, Lemma \\ref{more-algebra-lemma-compose}." ], "refs": [ "more-algebra-lemma-compose" ], "ref_ids": [ 10198 ] } ], "ref_ids": [] }, { "id": 2175, "type": "theorem", "label": "cohomology-lemma-composition", "categories": [ "cohomology" ], "title": "cohomology-lemma-composition", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given complexes", "$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$", "of $\\mathcal{O}_X$-modules there is a canonical morphism", "$$", "\\text{Tot}\\left(", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet)", "\\otimes_{\\mathcal{O}_X}", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet)", "\\right)", "\\longrightarrow", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{M}^\\bullet)", "$$", "of complexes of $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is proved in exactly the same way as", "More on Algebra, Lemma \\ref{more-algebra-lemma-composition}." ], "refs": [ "more-algebra-lemma-composition" ], "ref_ids": [ 10199 ] } ], "ref_ids": [] }, { "id": 2176, "type": "theorem", "label": "cohomology-lemma-diagonal-better", "categories": [ "cohomology" ], "title": "cohomology-lemma-diagonal-better", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given complexes", "$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$", "of $\\mathcal{O}_X$-modules there is a canonical morphism", "$$", "\\text{Tot}\\left(", "\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}", "\\SheafHom^\\bullet(\\mathcal{M}^\\bullet, \\mathcal{L}^\\bullet)", "\\right)", "\\longrightarrow", "\\SheafHom^\\bullet(\\mathcal{M}^\\bullet,", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet))", "$$", "of complexes of $\\mathcal{O}_X$-modules functorial in all three complexes." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is proved in exactly the same way as", "More on Algebra, Lemma \\ref{more-algebra-lemma-diagonal-better}." ], "refs": [ "more-algebra-lemma-diagonal-better" ], "ref_ids": [ 10200 ] } ], "ref_ids": [] }, { "id": 2177, "type": "theorem", "label": "cohomology-lemma-diagonal", "categories": [ "cohomology" ], "title": "cohomology-lemma-diagonal", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given complexes", "$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet$", "of $\\mathcal{O}_X$-modules there is a canonical morphism", "$$", "\\mathcal{K}^\\bullet", "\\longrightarrow", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet,", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet))", "$$", "of complexes of $\\mathcal{O}_X$-modules functorial in both complexes." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is proved in exactly the same way as", "More on Algebra, Lemma \\ref{more-algebra-lemma-diagonal}." ], "refs": [ "more-algebra-lemma-diagonal" ], "ref_ids": [ 10201 ] } ], "ref_ids": [] }, { "id": 2178, "type": "theorem", "label": "cohomology-lemma-evaluate-and-more", "categories": [ "cohomology" ], "title": "cohomology-lemma-evaluate-and-more", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given complexes", "$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$", "of $\\mathcal{O}_X$-modules there is a canonical morphism", "$$", "\\text{Tot}(\\SheafHom^\\bullet(\\mathcal{L}^\\bullet,", "\\mathcal{M}^\\bullet) \\otimes_{\\mathcal{O}_X} \\mathcal{K}^\\bullet)", "\\longrightarrow", "\\SheafHom^\\bullet(\\SheafHom^\\bullet(\\mathcal{K}^\\bullet,", "\\mathcal{L}^\\bullet), \\mathcal{M}^\\bullet)", "$$", "of complexes of $\\mathcal{O}_X$-modules functorial in all three complexes." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is proved in exactly the same way as", "More on Algebra, Lemma \\ref{more-algebra-lemma-evaluate-and-more}." ], "refs": [ "more-algebra-lemma-evaluate-and-more" ], "ref_ids": [ 10202 ] } ], "ref_ids": [] }, { "id": 2179, "type": "theorem", "label": "cohomology-lemma-RHom-into-K-injective", "categories": [ "cohomology" ], "title": "cohomology-lemma-RHom-into-K-injective", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{I}^\\bullet$", "be a K-injective complex of $\\mathcal{O}_X$-modules. Let", "$\\mathcal{L}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules.", "Then", "$$", "H^0(\\Gamma(U, \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet))) =", "\\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U)", "$$", "for all $U \\subset X$ open." ], "refs": [], "proofs": [ { "contents": [ "We have", "\\begin{align*}", "H^0(\\Gamma(U, \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet)))", "& =", "\\Hom_{K(\\mathcal{O}_U)}(L|_U, M|_U) \\\\", "& =", "\\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U)", "\\end{align*}", "The first equality is (\\ref{equation-cohomology-hom-complex}).", "The second equality is true because $\\mathcal{I}^\\bullet|_U$", "is K-injective by Lemma \\ref{lemma-restrict-K-injective-to-open}." ], "refs": [ "cohomology-lemma-restrict-K-injective-to-open" ], "ref_ids": [ 2134 ] } ], "ref_ids": [] }, { "id": 2180, "type": "theorem", "label": "cohomology-lemma-RHom-well-defined", "categories": [ "cohomology" ], "title": "cohomology-lemma-RHom-well-defined", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let", "$(\\mathcal{I}')^\\bullet \\to \\mathcal{I}^\\bullet$", "be a quasi-isomorphism of K-injective complexes of $\\mathcal{O}_X$-modules.", "Let $(\\mathcal{L}')^\\bullet \\to \\mathcal{L}^\\bullet$", "be a quasi-isomorphism of complexes of $\\mathcal{O}_X$-modules.", "Then", "$$", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, (\\mathcal{I}')^\\bullet)", "\\longrightarrow", "\\SheafHom^\\bullet((\\mathcal{L}')^\\bullet, \\mathcal{I}^\\bullet)", "$$", "is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be the object of $D(\\mathcal{O}_X)$ represented by", "$\\mathcal{I}^\\bullet$ and $(\\mathcal{I}')^\\bullet$.", "Let $L$ be the object of $D(\\mathcal{O}_X)$ represented by", "$\\mathcal{L}^\\bullet$ and $(\\mathcal{L}')^\\bullet$.", "By Lemma \\ref{lemma-RHom-into-K-injective}", "we see that the sheaves", "$$", "H^0(\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, (\\mathcal{I}')^\\bullet))", "\\quad\\text{and}\\quad", "H^0(\\SheafHom^\\bullet((\\mathcal{L}')^\\bullet, \\mathcal{I}^\\bullet))", "$$", "are both equal to the sheaf associated to the presheaf", "$$", "U \\longmapsto \\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U)", "$$", "Thus the map is a quasi-isomorphism." ], "refs": [ "cohomology-lemma-RHom-into-K-injective" ], "ref_ids": [ 2179 ] } ], "ref_ids": [] }, { "id": 2181, "type": "theorem", "label": "cohomology-lemma-RHom-from-K-flat-into-K-injective", "categories": [ "cohomology" ], "title": "cohomology-lemma-RHom-from-K-flat-into-K-injective", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{I}^\\bullet$", "be a K-injective complex of $\\mathcal{O}_X$-modules. Let", "$\\mathcal{L}^\\bullet$ be a K-flat complex of $\\mathcal{O}_X$-modules.", "Then $\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet)$", "is a K-injective complex of $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "Namely, if $\\mathcal{K}^\\bullet$ is an acyclic complex of", "$\\mathcal{O}_X$-modules, then", "\\begin{align*}", "\\Hom_{K(\\mathcal{O}_X)}(\\mathcal{K}^\\bullet,", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet))", "& =", "H^0(\\Gamma(X,", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet,", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet)))) \\\\", "& =", "H^0(\\Gamma(X, \\SheafHom^\\bullet(\\text{Tot}(", "\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet),", "\\mathcal{I}^\\bullet))) \\\\", "& =", "\\Hom_{K(\\mathcal{O}_X)}(", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet),", "\\mathcal{I}^\\bullet) \\\\", "& =", "0", "\\end{align*}", "The first equality by (\\ref{equation-cohomology-hom-complex}).", "The second equality by Lemma \\ref{lemma-compose}.", "The third equality by (\\ref{equation-cohomology-hom-complex}).", "The final equality because", "$\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet)$", "is acyclic because $\\mathcal{L}^\\bullet$ is K-flat", "(Definition \\ref{definition-K-flat}) and because $\\mathcal{I}^\\bullet$", "is K-injective." ], "refs": [ "cohomology-lemma-compose", "cohomology-definition-K-flat" ], "ref_ids": [ 2174, 2254 ] } ], "ref_ids": [] }, { "id": 2182, "type": "theorem", "label": "cohomology-lemma-section-RHom-over-U", "categories": [ "cohomology" ], "title": "cohomology-lemma-section-RHom-over-U", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $L, M$ be objects", "of $D(\\mathcal{O}_X)$. For every open $U$ we have", "$$", "H^0(U, R\\SheafHom(L, M)) =", "\\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U)", "$$", "and in particular $H^0(X, R\\SheafHom(L, M)) = \\Hom_{D(\\mathcal{O}_X)}(L, M)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $\\mathcal{I}^\\bullet$ of", "$\\mathcal{O}_X$-modules representing $M$ and a K-flat complex", "$\\mathcal{L}^\\bullet$ representing $L$. Then", "$\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet)$", "is K-injective by Lemma \\ref{lemma-RHom-from-K-flat-into-K-injective}.", "Hence we can compute cohomology over $U$ by simply taking sections over $U$", "and the result follows from Lemma \\ref{lemma-RHom-into-K-injective}." ], "refs": [ "cohomology-lemma-RHom-from-K-flat-into-K-injective", "cohomology-lemma-RHom-into-K-injective" ], "ref_ids": [ 2181, 2179 ] } ], "ref_ids": [] }, { "id": 2183, "type": "theorem", "label": "cohomology-lemma-internal-hom", "categories": [ "cohomology" ], "title": "cohomology-lemma-internal-hom", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L, M$ be objects", "of $D(\\mathcal{O}_X)$. With the construction as described above", "there is a canonical isomorphism", "$$", "R\\SheafHom(K, R\\SheafHom(L, M)) =", "R\\SheafHom(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L, M)", "$$", "in $D(\\mathcal{O}_X)$ functorial in $K, L, M$", "which recovers (\\ref{equation-internal-hom}) by taking $H^0(X, -)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $\\mathcal{I}^\\bullet$ representing", "$M$ and a K-flat complex of $\\mathcal{O}_X$-modules $\\mathcal{L}^\\bullet$", "representing $L$. Let $\\mathcal{H}^\\bullet$ be the complex described above.", "For any complex of $\\mathcal{O}_X$-modules $\\mathcal{K}^\\bullet$", "we have", "$$", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet,", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet))", "=", "\\SheafHom^\\bullet(", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet),", "\\mathcal{I}^\\bullet)", "$$", "by Lemma \\ref{lemma-compose}.", "Note that the left hand side represents", "$R\\SheafHom(K, R\\SheafHom(L, M))$ (use", "Lemma \\ref{lemma-RHom-from-K-flat-into-K-injective})", "and that the right hand side represents", "$R\\SheafHom(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L, M)$.", "This proves the displayed formula of the lemma.", "Taking global sections and using Lemma \\ref{lemma-section-RHom-over-U}", "we obtain (\\ref{equation-internal-hom})." ], "refs": [ "cohomology-lemma-compose", "cohomology-lemma-RHom-from-K-flat-into-K-injective" ], "ref_ids": [ 2174, 2181 ] } ], "ref_ids": [] }, { "id": 2184, "type": "theorem", "label": "cohomology-lemma-restriction-RHom-to-U", "categories": [ "cohomology" ], "title": "cohomology-lemma-restriction-RHom-to-U", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L$ be objects", "of $D(\\mathcal{O}_X)$. The construction of $R\\SheafHom(K, L)$", "commutes with restrictions to opens, i.e.,", "for every open $U$ we have", "$R\\SheafHom(K|_U, L|_U) = R\\SheafHom(K, L)|_U$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the construction and", "Lemma \\ref{lemma-restrict-K-injective-to-open}." ], "refs": [ "cohomology-lemma-restrict-K-injective-to-open" ], "ref_ids": [ 2134 ] } ], "ref_ids": [] }, { "id": 2185, "type": "theorem", "label": "cohomology-lemma-RHom-triangulated", "categories": [ "cohomology" ], "title": "cohomology-lemma-RHom-triangulated", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. The bifunctor $R\\SheafHom(- , -)$", "transforms distinguished triangles into distinguished triangles in both", "variables." ], "refs": [], "proofs": [ { "contents": [ "This follows from the observation that the assignment", "$$", "(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet) \\longmapsto", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet)", "$$", "transforms a termwise split short exact sequences of complexes in either", "variable into a termwise split short exact sequence. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2186, "type": "theorem", "label": "cohomology-lemma-internal-hom-composition", "categories": [ "cohomology" ], "title": "cohomology-lemma-internal-hom-composition", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given $K, L, M$ in", "$D(\\mathcal{O}_X)$ there is a canonical morphism", "$$", "R\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(K, L)", "\\longrightarrow R\\SheafHom(K, M)", "$$", "in $D(\\mathcal{O}_X)$ functorial in $K, L, M$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $\\mathcal{I}^\\bullet$ representing $M$,", "a K-injective complex $\\mathcal{J}^\\bullet$ representing $L$, and", "any complex of $\\mathcal{O}_X$-modules $\\mathcal{K}^\\bullet$ representing $K$.", "By Lemma \\ref{lemma-composition} there is a map of complexes", "$$", "\\text{Tot}\\left(", "\\SheafHom^\\bullet(\\mathcal{J}^\\bullet, \\mathcal{I}^\\bullet)", "\\otimes_{\\mathcal{O}_X}", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{J}^\\bullet)", "\\right)", "\\longrightarrow", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{I}^\\bullet)", "$$", "The complexes of $\\mathcal{O}_X$-modules", "$\\SheafHom^\\bullet(\\mathcal{J}^\\bullet, \\mathcal{I}^\\bullet)$,", "$\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{J}^\\bullet)$, and", "$\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{I}^\\bullet)$", "represent $R\\SheafHom(L, M)$, $R\\SheafHom(K, L)$, and $R\\SheafHom(K, M)$.", "If we choose a K-flat complex $\\mathcal{H}^\\bullet$ and a quasi-isomorphism", "$\\mathcal{H}^\\bullet \\to", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{J}^\\bullet)$,", "then there is a map", "$$", "\\text{Tot}\\left(", "\\SheafHom^\\bullet(\\mathcal{J}^\\bullet, \\mathcal{I}^\\bullet)", "\\otimes_{\\mathcal{O}_X} \\mathcal{H}^\\bullet", "\\right)", "\\longrightarrow", "\\text{Tot}\\left(", "\\SheafHom^\\bullet(\\mathcal{J}^\\bullet, \\mathcal{I}^\\bullet)", "\\otimes_{\\mathcal{O}_X}", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{J}^\\bullet)", "\\right)", "$$", "whose source represents", "$R\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(K, L)$.", "Composing the two displayed arrows gives the desired map. We omit the", "proof that the construction is functorial." ], "refs": [ "cohomology-lemma-composition" ], "ref_ids": [ 2175 ] } ], "ref_ids": [] }, { "id": 2187, "type": "theorem", "label": "cohomology-lemma-internal-hom-diagonal-better", "categories": [ "cohomology" ], "title": "cohomology-lemma-internal-hom-diagonal-better", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given $K, L, M$", "in $D(\\mathcal{O}_X)$ there is a canonical morphism", "$$", "K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(M, L)", "\\longrightarrow", "R\\SheafHom(M, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)", "$$", "in $D(\\mathcal{O}_X)$ functorial in $K, L, M$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-flat complex $\\mathcal{K}^\\bullet$ representing $K$,", "and a K-injective complex $\\mathcal{I}^\\bullet$ representing $L$, and", "choose any complex of $\\mathcal{O}_X$-modules $\\mathcal{M}^\\bullet$", "representing $M$. Choose a quasi-isomorphism", "$\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{I}^\\bullet)", "\\to \\mathcal{J}^\\bullet$", "where $\\mathcal{J}^\\bullet$ is K-injective. Then we use the map", "$$", "\\text{Tot}\\left(", "\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}", "\\SheafHom^\\bullet(\\mathcal{M}^\\bullet, \\mathcal{I}^\\bullet)", "\\right)", "\\to", "\\SheafHom^\\bullet(\\mathcal{M}^\\bullet,", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{I}^\\bullet))", "\\to", "\\SheafHom^\\bullet(\\mathcal{M}^\\bullet, \\mathcal{J}^\\bullet)", "$$", "where the first map is the map from Lemma \\ref{lemma-diagonal-better}." ], "refs": [ "cohomology-lemma-diagonal-better" ], "ref_ids": [ 2176 ] } ], "ref_ids": [] }, { "id": 2188, "type": "theorem", "label": "cohomology-lemma-internal-hom-diagonal", "categories": [ "cohomology" ], "title": "cohomology-lemma-internal-hom-diagonal", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given $K, L$ in $D(\\mathcal{O}_X)$", "there is a canonical morphism", "$$", "K \\longrightarrow R\\SheafHom(L, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)", "$$", "in $D(\\mathcal{O}_X)$ functorial in both $K$ and $L$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-flat complex $\\mathcal{K}^\\bullet$ representing $K$", "and any complex of $\\mathcal{O}_X$-modules $\\mathcal{L}^\\bullet$", "representing $L$. Choose a K-injective complex $\\mathcal{J}^\\bullet$", "and a quasi-isomorphism", "$\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet)", "\\to \\mathcal{J}^\\bullet$. Then we use", "$$", "\\mathcal{K}^\\bullet \\to", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet,", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet))", "\\to", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{J}^\\bullet)", "$$", "where the first map comes from Lemma \\ref{lemma-diagonal}." ], "refs": [ "cohomology-lemma-diagonal" ], "ref_ids": [ 2177 ] } ], "ref_ids": [] }, { "id": 2189, "type": "theorem", "label": "cohomology-lemma-dual", "categories": [ "cohomology" ], "title": "cohomology-lemma-dual", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $L$ be an", "object of $D(\\mathcal{O}_X)$. Set $L^\\vee = R\\SheafHom(L, \\mathcal{O}_X)$.", "For $M$ in $D(\\mathcal{O}_X)$ there is a canonical map", "\\begin{equation}", "\\label{equation-eval}", "M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} L^\\vee", "\\longrightarrow", "R\\SheafHom(L, M)", "\\end{equation}", "which induces a canonical map", "$$", "H^0(X, M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} L^\\vee)", "\\longrightarrow", "\\Hom_{D(\\mathcal{O}_X)}(L, M)", "$$", "functorial in $M$ in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "The map (\\ref{equation-eval}) is a special case of", "Lemma \\ref{lemma-internal-hom-composition}", "using the identification $M = R\\SheafHom(\\mathcal{O}_X, M)$." ], "refs": [ "cohomology-lemma-internal-hom-composition" ], "ref_ids": [ 2186 ] } ], "ref_ids": [] }, { "id": 2190, "type": "theorem", "label": "cohomology-lemma-internal-hom-evaluate", "categories": [ "cohomology" ], "title": "cohomology-lemma-internal-hom-evaluate", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L, M$ be objects of", "$D(\\mathcal{O}_X)$. There is a canonical morphism", "$$", "R\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K", "\\longrightarrow", "R\\SheafHom(R\\SheafHom(K, L), M)", "$$", "in $D(\\mathcal{O}_X)$ functorial in $K, L, M$." ], "refs": [], "proofs": [ { "contents": [ "Choose", "a K-injective complex $\\mathcal{I}^\\bullet$ representing $M$,", "a K-injective complex $\\mathcal{J}^\\bullet$ representing $L$, and", "a K-flat complex $\\mathcal{K}^\\bullet$ representing $K$.", "The map is defined using the map", "$$", "\\text{Tot}(\\SheafHom^\\bullet(\\mathcal{J}^\\bullet,", "\\mathcal{I}^\\bullet) \\otimes_{\\mathcal{O}_X} \\mathcal{K}^\\bullet)", "\\longrightarrow", "\\SheafHom^\\bullet(\\SheafHom^\\bullet(\\mathcal{K}^\\bullet,", "\\mathcal{J}^\\bullet), \\mathcal{I}^\\bullet)", "$$", "of Lemma \\ref{lemma-evaluate-and-more}. By our particular", "choice of complexes the left hand side represents", "$R\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K$", "and the right hand side represents", "$R\\SheafHom(R\\SheafHom(K, L), M)$. We omit the proof that", "this is functorial in all three objects of $D(\\mathcal{O}_X)$." ], "refs": [ "cohomology-lemma-evaluate-and-more" ], "ref_ids": [ 2178 ] } ], "ref_ids": [] }, { "id": 2191, "type": "theorem", "label": "cohomology-lemma-glue", "categories": [ "cohomology" ], "title": "cohomology-lemma-glue", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $X = U \\cup V$ be", "the union of two open subspaces of $X$. Suppose given", "\\begin{enumerate}", "\\item an object $A$ of $D(\\mathcal{O}_U)$,", "\\item an object $B$ of $D(\\mathcal{O}_V)$, and", "\\item an isomorphism $c : A|_{U \\cap V} \\to B|_{U \\cap V}$.", "\\end{enumerate}", "Then there exists an object $F$ of $D(\\mathcal{O}_X)$", "and isomorphisms $f : F|_U \\to A$, $g : F|_V \\to B$ such", "that $c = g|_{U \\cap V} \\circ f^{-1}|_{U \\cap V}$.", "Moreover, given", "\\begin{enumerate}", "\\item an object $E$ of $D(\\mathcal{O}_X)$,", "\\item a morphism $a : A \\to E|_U$ of $D(\\mathcal{O}_U)$,", "\\item a morphism $b : B \\to E|_V$ of $D(\\mathcal{O}_V)$, ", "\\end{enumerate}", "such that", "$$", "a|_{U \\cap V} = b|_{U \\cap V} \\circ c.", "$$", "Then there exists a morphism $F \\to E$ in $D(\\mathcal{O}_X)$", "whose restriction to $U$ is $a \\circ f$", "and whose restriction to $V$ is $b \\circ g$." ], "refs": [], "proofs": [ { "contents": [ "Denote $j_U$, $j_V$, $j_{U \\cap V}$ the corresponding open immersions.", "Choose a distinguished triangle", "$$", "F \\to Rj_{U, *}A \\oplus Rj_{V, *}B \\to Rj_{U \\cap V, *}(B|_{U \\cap V})", "\\to F[1]", "$$", "where the map $Rj_{V, *}B \\to Rj_{U \\cap V, *}(B|_{U \\cap V})$ is the", "obvious one and where", "$Rj_{U, *}A \\to Rj_{U \\cap V, *}(B|_{U \\cap V})$", "is the composition of", "$Rj_{U, *}A \\to Rj_{U \\cap V, *}(A|_{U \\cap V})$", "with $Rj_{U \\cap V, *}c$. Restricting to $U$ we obtain", "$$", "F|_U \\to A \\oplus (Rj_{V, *}B)|_U \\to (Rj_{U \\cap V, *}(B|_{U \\cap V}))|_U", "\\to F|_U[1]", "$$", "Denote $j : U \\cap V \\to U$. Compatibility of restriction to opens and", "cohomology shows that both", "$(Rj_{V, *}B)|_U$ and $(Rj_{U \\cap V, *}(B|_{U \\cap V}))|_U$", "are canonically isomorphic to $Rj_*(B|_{U \\cap V})$.", "Hence the second arrow of the last displayed diagram has", "a section, and we conclude that the morphism $F|_U \\to A$ is", "an isomorphism. Similarly, the morphism $F|_V \\to B$ is an", "isomorphism. The existence of the morphism $F \\to E$ follows", "from the Mayer-Vietoris sequence for $\\Hom$, see", "Lemma \\ref{lemma-mayer-vietoris-hom}." ], "refs": [ "cohomology-lemma-mayer-vietoris-hom" ], "ref_ids": [ 2145 ] } ], "ref_ids": [] }, { "id": 2192, "type": "theorem", "label": "cohomology-lemma-vanishing-and-glueing", "categories": [ "cohomology" ], "title": "cohomology-lemma-vanishing-and-glueing", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism", "of ringed spaces. Let $\\mathcal{B}$ be a basis for the topology on $Y$.", "\\begin{enumerate}", "\\item Assume $K$ is in $D(\\mathcal{O}_X)$ such that", "for $V \\in \\mathcal{B}$ we have $H^i(f^{-1}(V), K) = 0$ for $i < 0$.", "Then $Rf_*K$ has vanishing cohomology sheaves in negative degrees,", "$H^i(f^{-1}(V), K) = 0$ for $i < 0$ for all opens $V \\subset Y$, and", "the rule $V \\mapsto H^0(f^{-1}V, K)$ is a sheaf on $Y$.", "\\item Assume $K, L$ are in $D(\\mathcal{O}_X)$ such that", "for $V \\in \\mathcal{B}$ we have", "$\\Ext^i(K|_{f^{-1}V}, L|_{f^{-1}V}) = 0$ for $i < 0$.", "Then $\\Ext^i(K|_{f^{-1}V}, L|_{f^{-1}V}) = 0$ for $i < 0$", "for all opens $V \\subset Y$ and", "the rule $V \\mapsto \\Hom(K|_{f^{-1}V}, L|_{f^{-1}V})$ is a sheaf on $Y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Lemma \\ref{lemma-unbounded-describe-higher-direct-images} tells us", "$H^i(Rf_*K)$ is the sheaf associated to the presheaf", "$V \\mapsto H^i(f^{-1}(V), K) = H^i(V, Rf_*K)$.", "The assumptions in (1) imply that $Rf_*K$ has vanishing cohomology", "sheaves in degrees $< 0$. We conclude that for any open $V \\subset Y$", "the cohomology group $H^i(V, Rf_*K)$ is zero for $i < 0$ and is equal to", "$H^0(V, H^0(Rf_*K))$ for $i = 0$. This proves (1).", "\\medskip\\noindent", "To prove (2) apply (1) to the complex $R\\SheafHom(K, L)$ using", "Lemma \\ref{lemma-section-RHom-over-U} to do the translation." ], "refs": [ "cohomology-lemma-unbounded-describe-higher-direct-images" ], "ref_ids": [ 2139 ] } ], "ref_ids": [] }, { "id": 2193, "type": "theorem", "label": "cohomology-lemma-uniqueness", "categories": [ "cohomology" ], "title": "cohomology-lemma-uniqueness", "contents": [ "In Situation \\ref{situation-locally-given} assume", "\\begin{enumerate}", "\\item $X = \\bigcup_{U \\in \\mathcal{B}} U$ and", "for $U, V \\in \\mathcal{B}$ we have", "$U \\cap V = \\bigcup_{W \\in \\mathcal{B}, W \\subset U \\cap V} W$,", "\\item for any $U \\in \\mathcal{B}$ we have $\\Ext^i(K_U, K_U) = 0$", "for $i < 0$.", "\\end{enumerate}", "If a solution $(K, \\rho_U)$ exists, then it is unique up to unique isomorphism", "and moreover $\\Ext^i(K, K) = 0$ for $i < 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $(K, \\rho_U)$ and $(K', \\rho'_U)$ be a pair of solutions.", "Let $f : X \\to Y$ be the continuous map constructed", "in Topology, Lemma \\ref{topology-lemma-create-map-from-subcollection}.", "Set $\\mathcal{O}_Y = f_*\\mathcal{O}_X$.", "Then $K, K'$ and $\\mathcal{B}$ are as in", "Lemma \\ref{lemma-vanishing-and-glueing} part (2).", "Hence we obtain the vanishing of negative exts for $K$ and we see that", "the rule", "$$", "V \\longmapsto \\Hom(K|_{f^{-1}V}, K'|_{f^{-1}V})", "$$", "is a sheaf on $Y$. As both $(K, \\rho_U)$ and $(K', \\rho'_U)$ are solutions", "the maps", "$$", "(\\rho'_U)^{-1} \\circ \\rho_U : K|_U \\longrightarrow K'|_U", "$$", "over $U = f^{-1}(f(U))$ agree on overlaps. Hence we get a unique global", "section of the sheaf above which defines the desired isomorphism", "$K \\to K'$ compatible with all structure available." ], "refs": [ "topology-lemma-create-map-from-subcollection", "cohomology-lemma-vanishing-and-glueing" ], "ref_ids": [ 8200, 2192 ] } ], "ref_ids": [] }, { "id": 2194, "type": "theorem", "label": "cohomology-lemma-solution-in-finite-case", "categories": [ "cohomology" ], "title": "cohomology-lemma-solution-in-finite-case", "contents": [ "In Situation \\ref{situation-locally-given} assume", "\\begin{enumerate}", "\\item $X = U_1 \\cup \\ldots \\cup U_n$ with $U_i \\in \\mathcal{B}$,", "\\item for $U, V \\in \\mathcal{B}$ we have", "$U \\cap V = \\bigcup_{W \\in \\mathcal{B}, W \\subset U \\cap V} W$,", "\\item for any $U \\in \\mathcal{B}$ we have $\\Ext^i(K_U, K_U) = 0$", "for $i < 0$.", "\\end{enumerate}", "Then a solution exists and is unique up to unique isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Uniqueness was seen in Lemma \\ref{lemma-uniqueness}. We may prove the lemma", "by induction on $n$. The case $n = 1$ is immediate.", "\\medskip\\noindent", "The case $n = 2$.", "Consider the isomorphism", "$\\rho_{U_1, U_2} : K_{U_1}|_{U_1 \\cap U_2} \\to K_{U_2}|_{U_1 \\cap U_2}$", "constructed in Remark \\ref{remark-uniqueness}.", "By Lemma \\ref{lemma-glue} we obtain an object $K$ in $D(\\mathcal{O}_X)$", "and isomorphisms $\\rho_{U_1} : K|_{U_1} \\to K_{U_1}$ and", "$\\rho_{U_2} : K|_{U_2} \\to K_{U_2}$ compatible with $\\rho_{U_1, U_2}$.", "Take $U \\in \\mathcal{B}$. We will construct an isomorphism", "$\\rho_U : K|_U \\to K_U$ and we will leave it to the reader to verify", "that $(K, \\rho_U)$ is a solution. Consider the set $\\mathcal{B}'$", "of elements of $\\mathcal{B}$ contained in either $U \\cap U_1$ or contained in", "$U \\cap U_2$. Then $(K_U, \\rho^U_{U'})$ is a solution for the system", "$(\\{K_{U'}\\}_{U' \\in \\mathcal{B}'},", "\\{\\rho_{V'}^{U'}\\}_{V' \\subset U'\\text{ with }U', V' \\in \\mathcal{B}'})$", "on the ringed space $U$.", "We claim that $(K|_U, \\tau_{U'})$ is another solution where", "$\\tau_{U'}$ for $U' \\in \\mathcal{B}'$ is chosen as follows:", "if $U' \\subset U_1$ then we take the composition", "$$", "K|_{U'} \\xrightarrow{\\rho_{U_1}|_{U'}}", "K_{U_1}|_{U'} \\xrightarrow{\\rho^{U_1}_{U'}}", "K_{U'}", "$$", "and if $U' \\subset U_2$ then we take the composition", "$$", "K|_{U'} \\xrightarrow{\\rho_{U_2}|_{U'}}", "K_{U_2}|_{U'} \\xrightarrow{\\rho^{U_2}_{U'}}", "K_{U'}.", "$$", "To verify this is a solution use the property of the map $\\rho_{U_1, U_2}$", "described in Remark \\ref{remark-uniqueness} and the compatibility of", "$\\rho_{U_1}$ and $\\rho_{U_2}$ with $\\rho_{U_1, U_2}$. Having said this", "we apply Lemma \\ref{lemma-uniqueness} to see that we obtain a unique", "isomorphism $K|_{U'} \\to K_{U'}$ compatible with the maps $\\tau_{U'}$ and", "$\\rho^U_{U'}$ for $U' \\in \\mathcal{B}'$.", "\\medskip\\noindent", "The case $n > 2$. Consider the open subspace", "$X' = U_1 \\cup \\ldots \\cup U_{n - 1}$ and let $\\mathcal{B}'$ be the set of", "elements of $\\mathcal{B}$ contained in $X'$. Then we find a system", "$(\\{K_U\\}_{U \\in \\mathcal{B}'}, \\{\\rho_V^U\\}_{U, V \\in \\mathcal{B}'})$", "on the ringed space $X'$ to which we may apply our induction hypothesis.", "We find a solution $(K_{X'}, \\rho^{X'}_U)$.", "Then we can consider the collection", "$\\mathcal{B}^* = \\mathcal{B} \\cup \\{X'\\}$ of opens of $X$ and we see that", "we obtain a system", "$(\\{K_U\\}_{U \\in \\mathcal{B}^*},", "\\{\\rho_V^U\\}_{V \\subset U\\text{ with }U, V \\in \\mathcal{B}^*})$.", "Note that this new system also satisfies condition (3)", "by Lemma \\ref{lemma-uniqueness} applied to the solution $K_{X'}$.", "For this system we have $X = X' \\cup U_n$.", "This reduces us to the case $n = 2$ we worked out above." ], "refs": [ "cohomology-lemma-uniqueness", "cohomology-remark-uniqueness", "cohomology-lemma-glue", "cohomology-remark-uniqueness", "cohomology-lemma-uniqueness", "cohomology-lemma-uniqueness" ], "ref_ids": [ 2193, 2284, 2191, 2284, 2193, 2193 ] } ], "ref_ids": [] }, { "id": 2195, "type": "theorem", "label": "cohomology-lemma-glueing-increasing-union", "categories": [ "cohomology" ], "title": "cohomology-lemma-glueing-increasing-union", "contents": [ "Let $X$ be a ringed space. Let $E$ be a well ordered set and let", "$$", "X = \\bigcup\\nolimits_{\\alpha \\in E} W_\\alpha", "$$", "be an open covering with $W_\\alpha \\subset W_{\\alpha + 1}$", "and $W_\\alpha = \\bigcup_{\\beta < \\alpha} W_\\beta$ if $\\alpha$ is not", "a successor. Let $K_\\alpha$ be an object of $D(\\mathcal{O}_{W_\\alpha})$", "with $\\Ext^i(K_\\alpha, K_\\alpha) = 0$ for $i < 0$.", "Assume given isomorphisms", "$\\rho_\\beta^\\alpha : K_\\alpha|_{W_\\beta} \\to K_\\beta$ in", "$D(\\mathcal{O}_{W_\\beta})$ for all $\\beta < \\alpha$ with", "$\\rho_\\gamma^\\alpha = \\rho_\\gamma^\\beta \\circ \\rho^\\alpha_\\beta|_{W_\\gamma}$", "for $\\gamma < \\beta < \\alpha$.", "Then there exists an object", "$K$ in $D(\\mathcal{O}_X)$ and isomorphisms", "$K|_{W_\\alpha} \\to K_\\alpha$ for $\\alpha \\in E$", "compatible with the isomorphisms $\\rho_\\beta^\\alpha$." ], "refs": [], "proofs": [ { "contents": [ "In this proof $\\alpha, \\beta, \\gamma, \\ldots$ represent elements of $E$.", "Choose a K-injective complex", "$I_\\alpha^\\bullet$ on $W_\\alpha$ representing $K_\\alpha$.", "For $\\beta < \\alpha$ denote $j_{\\beta, \\alpha} : W_\\beta \\to W_\\alpha$", "the inclusion morphism. By transfinite induction, we will construct for all", "$\\beta < \\alpha$ a map of complexes", "$$", "\\tau_{\\beta, \\alpha} :", "(j_{\\beta, \\alpha})_!I_\\beta^\\bullet", "\\longrightarrow", "I_\\alpha^\\bullet", "$$", "representing the adjoint to the inverse of the isomorphism", "$\\rho^\\alpha_\\beta : K_\\alpha|_{W_\\beta} \\to K_\\beta$.", "Moreover, we will do this in such that for", "$\\gamma < \\beta < \\alpha$ we have", "$$", "\\tau_{\\gamma, \\alpha} = \\tau_{\\beta, \\alpha} \\circ", "(j_{\\beta, \\alpha})_!\\tau_{\\gamma, \\beta}", "$$", "as maps of complexes. Namely, suppose already given $\\tau_{\\gamma, \\beta}$", "composing correctly for all $\\gamma < \\beta < \\alpha$.", "If $\\alpha = \\alpha' + 1$ is a successor, then we choose any map of complexes", "$$", "(j_{\\alpha', \\alpha})_!I_{\\alpha'}^\\bullet \\to I_\\alpha^\\bullet", "$$", "which is adjoint to the inverse of the isomorphism", "$\\rho^\\alpha_{\\alpha'} : K_\\alpha|_{W_{\\alpha'}} \\to K_{\\alpha'}$", "(possible because $I_\\alpha^\\bullet$ is K-injective)", "and for any $\\beta < \\alpha'$ we set", "$$", "\\tau_{\\beta, \\alpha} = \\tau_{\\alpha', \\alpha} \\circ", "(j_{\\alpha', \\alpha})_!\\tau_{\\beta, \\alpha'}", "$$", "If $\\alpha$ is not a successor, then", "we can consider the complex on $W_\\alpha$ given by", "$$", "C^\\bullet = \\colim_{\\beta < \\alpha} (j_{\\beta, \\alpha})_!I_\\beta^\\bullet", "$$", "(termwise colimit) where the transition maps of the sequence", "are given by the maps $\\tau_{\\beta', \\beta}$ for", "$\\beta' < \\beta < \\alpha$. We claim that $C^\\bullet$", "represents $K_\\alpha$. Namely, for $\\beta < \\alpha$ the restriction", "of the coprojection $(j_{\\beta, \\alpha})_!I_\\beta^\\bullet \\to C^\\bullet$", "gives a map", "$$", "\\sigma_\\beta : I_\\beta^\\bullet \\longrightarrow C^\\bullet|_{W_\\beta}", "$$", "which is a quasi-isomorphism: if $x \\in W_\\beta$ then looking", "at stalks we get", "$$", "(C^\\bullet)_x =", "\\colim_{\\beta' < \\alpha}", "\\left((j_{\\beta', \\alpha})_!I_{\\beta'}^\\bullet\\right)_x =", "\\colim_{\\beta \\leq \\beta' < \\alpha} (I_{\\beta'}^\\bullet)_x", "\\longleftarrow", "(I_\\beta^\\bullet)_x", "$$", "which is a quasi-isomorphism. Here we used that taking stalks", "commutes with colimits, that filtered colimits are exact, and", "that the maps $(I_\\beta^\\bullet)_x \\to (I_{\\beta'}^\\bullet)_x$", "are quasi-isomorphisms for $\\beta \\leq \\beta' < \\alpha$.", "Hence $(C^\\bullet, \\sigma_\\beta^{-1})$ is a solution to the", "system $(\\{K_\\beta\\}_{\\beta < \\alpha},", "\\{\\rho^\\beta_{\\beta'}\\}_{\\beta' < \\beta < \\alpha})$.", "Since $(K_\\alpha, \\rho^\\alpha_\\beta)$ is another solution", "we obtain a unique isomorphism $\\sigma : K_\\alpha \\to C^\\bullet$", "in $D(\\mathcal{O}_{W_\\alpha})$ compatible with all our maps, see", "Lemma \\ref{lemma-solution-in-finite-case}", "(this is where we use the vanishing of negative ext groups).", "Choose a morphism $\\tau : C^\\bullet \\to I_\\alpha^\\bullet$", "of complexes representing $\\sigma$. Then we set", "$$", "\\tau_{\\beta, \\alpha} = \\tau|_{W_\\beta} \\circ \\sigma_\\beta", "$$", "to get the desired maps. Finally, we take $K$ to be the object of the derived", "category represented by the complex", "$$", "K^\\bullet = \\colim_{\\alpha \\in E} (W_\\alpha \\to X)_!I_\\alpha^\\bullet", "$$", "where the transition maps are given by our carefully constructed", "maps $\\tau_{\\beta, \\alpha}$ for $\\beta < \\alpha$.", "Arguing exactly as above we see that for all $\\alpha$", "the restriction of the coprojection determines an isomorphism", "$$", "K|_{W_\\alpha} \\longrightarrow K_\\alpha", "$$", "compatible with the given maps $\\rho^\\alpha_\\beta$." ], "refs": [ "cohomology-lemma-solution-in-finite-case" ], "ref_ids": [ 2194 ] } ], "ref_ids": [] }, { "id": 2196, "type": "theorem", "label": "cohomology-lemma-cone", "categories": [ "cohomology" ], "title": "cohomology-lemma-cone", "contents": [ "The cone on a morphism of strictly perfect complexes is", "strictly perfect." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2197, "type": "theorem", "label": "cohomology-lemma-tensor", "categories": [ "cohomology" ], "title": "cohomology-lemma-tensor", "contents": [ "The total complex associated to the tensor product of two", "strictly perfect complexes is strictly perfect." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2198, "type": "theorem", "label": "cohomology-lemma-strictly-perfect-pullback", "categories": [ "cohomology" ], "title": "cohomology-lemma-strictly-perfect-pullback", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces. If $\\mathcal{F}^\\bullet$ is a strictly", "perfect complex of $\\mathcal{O}_Y$-modules, then", "$f^*\\mathcal{F}^\\bullet$ is a strictly perfect complex of", "$\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "The pullback of a finite free module is finite free. The functor", "$f^*$ is additive functor hence preserves direct summands. The lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2199, "type": "theorem", "label": "cohomology-lemma-local-lift-map", "categories": [ "cohomology" ], "title": "cohomology-lemma-local-lift-map", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Given a solid diagram of $\\mathcal{O}_X$-modules", "$$", "\\xymatrix{", "\\mathcal{E} \\ar@{..>}[dr] \\ar[r] & \\mathcal{F} \\\\", "& \\mathcal{G} \\ar[u]_p", "}", "$$", "with $\\mathcal{E}$ a direct summand of a finite free", "$\\mathcal{O}_X$-module and $p$ surjective, then a dotted arrow", "making the diagram commute exists locally on $X$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $\\mathcal{E} = \\mathcal{O}_X^{\\oplus n}$ for some $n$.", "In this case finding the dotted arrow is equivalent to lifting the", "images of the basis elements in $\\Gamma(X, \\mathcal{F})$. This is", "locally possible by the characterization of surjective maps of", "sheaves (Sheaves, Section \\ref{sheaves-section-exactness-points})." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2200, "type": "theorem", "label": "cohomology-lemma-local-homotopy", "categories": [ "cohomology" ], "title": "cohomology-lemma-local-homotopy", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "\\begin{enumerate}", "\\item Let $\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$", "be a morphism of complexes of $\\mathcal{O}_X$-modules", "with $\\mathcal{E}^\\bullet$ strictly perfect and $\\mathcal{F}^\\bullet$", "acyclic. Then $\\alpha$ is locally on $X$ homotopic to zero.", "\\item Let $\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$", "be a morphism of complexes of $\\mathcal{O}_X$-modules", "with $\\mathcal{E}^\\bullet$ strictly perfect, $\\mathcal{E}^i = 0$", "for $i < a$, and $H^i(\\mathcal{F}^\\bullet) = 0$ for $i \\geq a$.", "Then $\\alpha$ is locally on $X$ homotopic to zero.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from the second, hence we only prove (2).", "We will prove this by induction on the length of the complex", "$\\mathcal{E}^\\bullet$. If $\\mathcal{E}^\\bullet \\cong \\mathcal{E}[-n]$", "for some direct summand $\\mathcal{E}$ of a finite free", "$\\mathcal{O}_X$-module and integer $n \\geq a$, then the result follows from", "Lemma \\ref{lemma-local-lift-map} and the fact that", "$\\mathcal{F}^{n - 1} \\to \\Ker(\\mathcal{F}^n \\to \\mathcal{F}^{n + 1})$", "is surjective by the assumed vanishing of $H^n(\\mathcal{F}^\\bullet)$.", "If $\\mathcal{E}^i$ is zero except for $i \\in [a, b]$, then we have a", "split exact sequence of complexes", "$$", "0 \\to \\mathcal{E}^b[-b] \\to \\mathcal{E}^\\bullet \\to", "\\sigma_{\\leq b - 1}\\mathcal{E}^\\bullet \\to 0", "$$", "which determines a distinguished triangle in", "$K(\\mathcal{O}_X)$. Hence an exact sequence", "$$", "\\Hom_{K(\\mathcal{O}_X)}(", "\\sigma_{\\leq b - 1}\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)", "\\to", "\\Hom_{K(\\mathcal{O}_X)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)", "\\to", "\\Hom_{K(\\mathcal{O}_X)}(\\mathcal{E}^b[-b], \\mathcal{F}^\\bullet)", "$$", "by the axioms of triangulated categories. The composition", "$\\mathcal{E}^b[-b] \\to \\mathcal{F}^\\bullet$ is locally homotopic to", "zero, whence we may assume our map comes from an element in the", "left hand side of the displayed exact sequence above. This element", "is locally zero by induction hypothesis." ], "refs": [ "cohomology-lemma-local-lift-map" ], "ref_ids": [ 2199 ] } ], "ref_ids": [] }, { "id": 2201, "type": "theorem", "label": "cohomology-lemma-lift-through-quasi-isomorphism", "categories": [ "cohomology" ], "title": "cohomology-lemma-lift-through-quasi-isomorphism", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Given a solid diagram of complexes of $\\mathcal{O}_X$-modules", "$$", "\\xymatrix{", "\\mathcal{E}^\\bullet \\ar@{..>}[dr] \\ar[r]_\\alpha & \\mathcal{F}^\\bullet \\\\", "& \\mathcal{G}^\\bullet \\ar[u]_f", "}", "$$", "with $\\mathcal{E}^\\bullet$ strictly perfect, $\\mathcal{E}^j = 0$ for", "$j < a$ and $H^j(f)$ an isomorphism for $j > a$ and surjective for $j = a$,", "then a dotted arrow making the diagram commute up to homotopy", "exists locally on $X$." ], "refs": [], "proofs": [ { "contents": [ "Our assumptions on $f$ imply the cone $C(f)^\\bullet$ has vanishing", "cohomology sheaves in degrees $\\geq a$.", "Hence Lemma \\ref{lemma-local-homotopy} guarantees there is an open", "covering $X = \\bigcup U_i$ such that the composition", "$\\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet \\to C(f)^\\bullet$", "is homotopic to zero over $U_i$. Since", "$$", "\\mathcal{G}^\\bullet \\to \\mathcal{F}^\\bullet \\to C(f)^\\bullet \\to", "\\mathcal{G}^\\bullet[1]", "$$", "restricts to a distinguished triangle in $K(\\mathcal{O}_{U_i})$", "we see that we can lift $\\alpha|_{U_i}$ up to homotopy to a map", "$\\alpha_i : \\mathcal{E}^\\bullet|_{U_i} \\to \\mathcal{G}^\\bullet|_{U_i}$", "as desired." ], "refs": [ "cohomology-lemma-local-homotopy" ], "ref_ids": [ 2200 ] } ], "ref_ids": [] }, { "id": 2202, "type": "theorem", "label": "cohomology-lemma-local-actual", "categories": [ "cohomology" ], "title": "cohomology-lemma-local-actual", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes", "of $\\mathcal{O}_X$-modules with $\\mathcal{E}^\\bullet$ strictly perfect.", "\\begin{enumerate}", "\\item For any element", "$\\alpha \\in \\Hom_{D(\\mathcal{O}_X)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$", "there exists an open covering $X = \\bigcup U_i$ such that", "$\\alpha|_{U_i}$ is given by a morphism of complexes", "$\\alpha_i : \\mathcal{E}^\\bullet|_{U_i} \\to \\mathcal{F}^\\bullet|_{U_i}$.", "\\item Given a morphism of complexes", "$\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$", "whose image in the group", "$\\Hom_{D(\\mathcal{O}_X)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$", "is zero, there exists an open covering $X = \\bigcup U_i$ such that", "$\\alpha|_{U_i}$ is homotopic to zero.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1).", "By the construction of the derived category we can find a quasi-isomorphism", "$f : \\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$ and a map of complexes", "$\\beta : \\mathcal{E}^\\bullet \\to \\mathcal{G}^\\bullet$ such that", "$\\alpha = f^{-1}\\beta$. Thus the result follows from", "Lemma \\ref{lemma-lift-through-quasi-isomorphism}.", "We omit the proof of (2)." ], "refs": [ "cohomology-lemma-lift-through-quasi-isomorphism" ], "ref_ids": [ 2201 ] } ], "ref_ids": [] }, { "id": 2203, "type": "theorem", "label": "cohomology-lemma-Rhom-strictly-perfect", "categories": [ "cohomology" ], "title": "cohomology-lemma-Rhom-strictly-perfect", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes", "of $\\mathcal{O}_X$-modules with $\\mathcal{E}^\\bullet$ strictly perfect.", "Then the internal hom $R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$", "is represented by the complex $\\mathcal{H}^\\bullet$ with terms", "$$", "\\mathcal{H}^n =", "\\bigoplus\\nolimits_{n = p + q}", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}^{-q}, \\mathcal{F}^p)", "$$", "and differential as described in Section \\ref{section-internal-hom}." ], "refs": [], "proofs": [ { "contents": [ "Choose a quasi-isomorphism $\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$", "into a K-injective complex. Let $(\\mathcal{H}')^\\bullet$ be the", "complex with terms", "$$", "(\\mathcal{H}')^n =", "\\prod\\nolimits_{n = p + q}", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{L}^{-q}, \\mathcal{I}^p)", "$$", "which represents $R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$", "by the construction in Section \\ref{section-internal-hom}. It suffices", "to show that the map", "$$", "\\mathcal{H}^\\bullet \\longrightarrow (\\mathcal{H}')^\\bullet", "$$", "is a quasi-isomorphism. Given an open $U \\subset X$ we have", "by inspection", "$$", "H^0(\\mathcal{H}^\\bullet(U)) =", "\\Hom_{K(\\mathcal{O}_U)}(\\mathcal{E}^\\bullet|_U, \\mathcal{K}^\\bullet|_U)", "\\to", "H^0((\\mathcal{H}')^\\bullet(U)) =", "\\Hom_{D(\\mathcal{O}_U)}(\\mathcal{E}^\\bullet|_U, \\mathcal{K}^\\bullet|_U)", "$$", "By Lemma \\ref{lemma-local-actual} the sheafification of", "$U \\mapsto H^0(\\mathcal{H}^\\bullet(U))$", "is equal to the sheafification of", "$U \\mapsto H^0((\\mathcal{H}')^\\bullet(U))$. A similar argument can be", "given for the other cohomology sheaves. Thus $\\mathcal{H}^\\bullet$", "is quasi-isomorphic to $(\\mathcal{H}')^\\bullet$ which proves the lemma." ], "refs": [ "cohomology-lemma-local-actual" ], "ref_ids": [ 2202 ] } ], "ref_ids": [] }, { "id": 2204, "type": "theorem", "label": "cohomology-lemma-Rhom-complex-of-direct-summands-finite-free", "categories": [ "cohomology" ], "title": "cohomology-lemma-Rhom-complex-of-direct-summands-finite-free", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes", "of $\\mathcal{O}_X$-modules with", "\\begin{enumerate}", "\\item $\\mathcal{F}^n = 0$ for $n \\ll 0$,", "\\item $\\mathcal{E}^n = 0$ for $n \\gg 0$, and", "\\item $\\mathcal{E}^n$ isomorphic to a direct summand of a finite", "free $\\mathcal{O}_X$-module.", "\\end{enumerate}", "Then the internal hom $R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$", "is represented by the complex $\\mathcal{H}^\\bullet$ with terms", "$$", "\\mathcal{H}^n =", "\\bigoplus\\nolimits_{n = p + q}", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}^{-q}, \\mathcal{F}^p)", "$$", "and differential as described in Section \\ref{section-internal-hom}." ], "refs": [], "proofs": [ { "contents": [ "Choose a quasi-isomorphism $\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$", "where $\\mathcal{I}^\\bullet$ is a bounded below complex of injectives.", "Note that $\\mathcal{I}^\\bullet$ is K-injective", "(Derived Categories, Lemma", "\\ref{derived-lemma-bounded-below-injectives-K-injective}).", "Hence the construction in Section \\ref{section-internal-hom}", "shows that", "$R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$ is ", "represented by the complex $(\\mathcal{H}')^\\bullet$ with terms", "$$", "(\\mathcal{H}')^n =", "\\prod\\nolimits_{n = p + q}", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}^{-q}, \\mathcal{I}^p) =", "\\bigoplus\\nolimits_{n = p + q}", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}^{-q}, \\mathcal{I}^p)", "$$", "(equality because there are only finitely many nonzero terms).", "Note that $\\mathcal{H}^\\bullet$ is the total complex associated to", "the double complex with terms", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}^{-q}, \\mathcal{F}^p)$", "and similarly for $(\\mathcal{H}')^\\bullet$.", "The natural map $(\\mathcal{H}')^\\bullet \\to \\mathcal{H}^\\bullet$", "comes from a map of double complexes.", "Thus to show this map is a quasi-isomorphism, we may use the spectral", "sequence of a double complex", "(Homology, Lemma \\ref{homology-lemma-first-quadrant-ss})", "$$", "{}'E_1^{p, q} =", "H^p(\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}^{-q}, \\mathcal{F}^\\bullet))", "$$", "converging to $H^{p + q}(\\mathcal{H}^\\bullet)$ and similarly for", "$(\\mathcal{H}')^\\bullet$. To finish the proof of the lemma it", "suffices to show that $\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$", "induces an isomorphism", "$$", "H^p(\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}, \\mathcal{F}^\\bullet))", "\\longrightarrow", "H^p(\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}, \\mathcal{I}^\\bullet))", "$$", "on cohomology sheaves whenever $\\mathcal{E}$ is a direct summand of a", "finite free $\\mathcal{O}_X$-module. Since this is clear when $\\mathcal{E}$", "is finite free the result follows." ], "refs": [ "derived-lemma-bounded-below-injectives-K-injective", "homology-lemma-first-quadrant-ss" ], "ref_ids": [ 1910, 12105 ] } ], "ref_ids": [] }, { "id": 2205, "type": "theorem", "label": "cohomology-lemma-pseudo-coherent-independent-representative", "categories": [ "cohomology" ], "title": "cohomology-lemma-pseudo-coherent-independent-representative", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E$ be an object", "of $D(\\mathcal{O}_X)$.", "\\begin{enumerate}", "\\item If there exists an open covering $X = \\bigcup U_i$,", "strictly perfect complexes $\\mathcal{E}_i^\\bullet$ on $U_i$, and", "maps $\\alpha_i : \\mathcal{E}_i^\\bullet \\to E|_{U_i}$ in", "$D(\\mathcal{O}_{U_i})$ with $H^j(\\alpha_i)$ an isomorphism for $j > m$", "and $H^m(\\alpha_i)$ surjective, then $E$ is $m$-pseudo-coherent.", "\\item If $E$ is $m$-pseudo-coherent, then any complex representing", "$E$ is $m$-pseudo-coherent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}^\\bullet$ be any complex representing $E$", "and let $X = \\bigcup U_i$ and", "$\\alpha_i : \\mathcal{E}_i^\\bullet \\to E|_{U_i}$ be as in (1).", "We will show that $\\mathcal{F}^\\bullet$ is $m$-pseudo-coherent", "as a complex, which will prove (1) and (2) simultaneously.", "By Lemma \\ref{lemma-local-actual}", "we can after refining the open covering $X = \\bigcup U_i$", "represent the maps $\\alpha_i$ by maps of complexes", "$\\alpha_i : \\mathcal{E}_i^\\bullet \\to \\mathcal{F}^\\bullet|_{U_i}$.", "By assumption", "$H^j(\\alpha_i)$ are isomorphisms for $j > m$, and $H^m(\\alpha_i)$", "is surjective whence $\\mathcal{F}^\\bullet$ is", "$m$-pseudo-coherent." ], "refs": [ "cohomology-lemma-local-actual" ], "ref_ids": [ 2202 ] } ], "ref_ids": [] }, { "id": 2206, "type": "theorem", "label": "cohomology-lemma-pseudo-coherent-pullback", "categories": [ "cohomology" ], "title": "cohomology-lemma-pseudo-coherent-pullback", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces. Let $E$ be an object of", "$D(\\mathcal{O}_Y)$. If $E$ is $m$-pseudo-coherent,", "then $Lf^*E$ is $m$-pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "Represent $E$ by a complex $\\mathcal{E}^\\bullet$ of $\\mathcal{O}_Y$-modules", "and choose an open covering $Y = \\bigcup V_i$ and", "$\\alpha_i : \\mathcal{E}_i^\\bullet \\to \\mathcal{E}^\\bullet|_{V_i}$", "as in Definition \\ref{definition-pseudo-coherent}.", "Set $U_i = f^{-1}(V_i)$.", "By Lemma \\ref{lemma-pseudo-coherent-independent-representative}", "it suffices to show that $Lf^*\\mathcal{E}^\\bullet|_{U_i}$ is", "$m$-pseudo-coherent.", "Choose a distinguished triangle", "$$", "\\mathcal{E}_i^\\bullet \\to", "\\mathcal{E}^\\bullet|_{V_i} \\to", "C \\to", "\\mathcal{E}_i^\\bullet[1]", "$$", "The assumption on $\\alpha_i$ means exactly that the cohomology sheaves", "$H^j(C)$ are zero for all $j \\geq m$. Denote $f_i : U_i \\to V_i$", "the restriction of $f$. Note that $Lf^*\\mathcal{E}^\\bullet|_{U_i} = ", "Lf_i^*(\\mathcal{E}|_{V_i})$. Applying $Lf_i^*$ we obtain", "the distinguished triangle", "$$", "Lf_i^*\\mathcal{E}_i^\\bullet \\to", "Lf_i^*\\mathcal{E}|_{V_i} \\to", "Lf_i^*C \\to", "Lf_i^*\\mathcal{E}_i^\\bullet[1]", "$$", "By the construction of $Lf_i^*$ as a left derived functor we see that", "$H^j(Lf_i^*C) = 0$ for $j \\geq m$ (by the dual of Derived Categories, Lemma", "\\ref{derived-lemma-negative-vanishing}). Hence $H^j(Lf_i^*\\alpha_i)$ is an", "isomorphism for $j > m$ and $H^m(Lf^*\\alpha_i)$ is surjective.", "On the other hand,", "$Lf_i^*\\mathcal{E}_i^\\bullet = f_i^*\\mathcal{E}_i^\\bullet$.", "is strictly perfect by Lemma \\ref{lemma-strictly-perfect-pullback}.", "Thus we conclude." ], "refs": [ "cohomology-definition-pseudo-coherent", "cohomology-lemma-pseudo-coherent-independent-representative", "derived-lemma-negative-vanishing", "cohomology-lemma-strictly-perfect-pullback" ], "ref_ids": [ 2258, 2205, 1839, 2198 ] } ], "ref_ids": [] }, { "id": 2207, "type": "theorem", "label": "cohomology-lemma-cone-pseudo-coherent", "categories": [ "cohomology" ], "title": "cohomology-lemma-cone-pseudo-coherent", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space and $m \\in \\mathbf{Z}$.", "Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\\mathcal{O}_X)$.", "\\begin{enumerate}", "\\item If $K$ is $(m + 1)$-pseudo-coherent and $L$ is $m$-pseudo-coherent", "then $M$ is $m$-pseudo-coherent.", "\\item If $K$ and $M$ are $m$-pseudo-coherent, then $L$ is $m$-pseudo-coherent.", "\\item If $L$ is $(m + 1)$-pseudo-coherent and $M$", "is $m$-pseudo-coherent, then $K$ is $(m + 1)$-pseudo-coherent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Choose an open covering $X = \\bigcup U_i$ and", "maps $\\alpha_i : \\mathcal{K}_i^\\bullet \\to K|_{U_i}$ in $D(\\mathcal{O}_{U_i})$", "with $\\mathcal{K}_i^\\bullet$ strictly perfect and $H^j(\\alpha_i)$", "isomorphisms for $j > m + 1$ and surjective for $j = m + 1$.", "We may replace $\\mathcal{K}_i^\\bullet$ by", "$\\sigma_{\\geq m + 1}\\mathcal{K}_i^\\bullet$", "and hence we may assume that $\\mathcal{K}_i^j = 0$", "for $j < m + 1$. After refining the open covering we may choose", "maps $\\beta_i : \\mathcal{L}_i^\\bullet \\to L|_{U_i}$ in $D(\\mathcal{O}_{U_i})$", "with $\\mathcal{L}_i^\\bullet$ strictly perfect such that", "$H^j(\\beta)$ is an isomorphism for $j > m$ and", "surjective for $j = m$. By", "Lemma \\ref{lemma-lift-through-quasi-isomorphism}", "we can, after refining the covering, find maps of complexes", "$\\gamma_i : \\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet$", "such that the diagrams", "$$", "\\xymatrix{", "K|_{U_i} \\ar[r] & L|_{U_i} \\\\", "\\mathcal{K}_i^\\bullet \\ar[u]^{\\alpha_i} \\ar[r]^{\\gamma_i} &", "\\mathcal{L}_i^\\bullet \\ar[u]_{\\beta_i}", "}", "$$", "are commutative in $D(\\mathcal{O}_{U_i})$ (this requires representing the", "maps $\\alpha_i$, $\\beta_i$ and $K|_{U_i} \\to L|_{U_i}$", "by actual maps of complexes; some details omitted).", "The cone $C(\\gamma_i)^\\bullet$ is strictly perfect (Lemma \\ref{lemma-cone}).", "The commutativity of the diagram implies that there exists a morphism", "of distinguished triangles", "$$", "(\\mathcal{K}_i^\\bullet, \\mathcal{L}_i^\\bullet, C(\\gamma_i)^\\bullet)", "\\longrightarrow", "(K|_{U_i}, L|_{U_i}, M|_{U_i}).", "$$", "It follows from the induced map on long exact cohomology sequences and", "Homology, Lemmas \\ref{homology-lemma-four-lemma} and", "\\ref{homology-lemma-five-lemma}", "that $C(\\gamma_i)^\\bullet \\to M|_{U_i}$ induces an isomorphism", "on cohomology in degrees $> m$ and a surjection in degree $m$.", "Hence $M$ is $m$-pseudo-coherent by", "Lemma \\ref{lemma-pseudo-coherent-independent-representative}.", "\\medskip\\noindent", "Assertions (2) and (3) follow from (1) by rotating the distinguished", "triangle." ], "refs": [ "cohomology-lemma-lift-through-quasi-isomorphism", "cohomology-lemma-cone", "homology-lemma-four-lemma", "homology-lemma-five-lemma", "cohomology-lemma-pseudo-coherent-independent-representative" ], "ref_ids": [ 2201, 2196, 12029, 12030, 2205 ] } ], "ref_ids": [] }, { "id": 2208, "type": "theorem", "label": "cohomology-lemma-tensor-pseudo-coherent", "categories": [ "cohomology" ], "title": "cohomology-lemma-tensor-pseudo-coherent", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L$ be objects", "of $D(\\mathcal{O}_X)$.", "\\begin{enumerate}", "\\item If $K$ is $n$-pseudo-coherent and $H^i(K) = 0$ for $i > a$", "and $L$ is $m$-pseudo-coherent and $H^j(L) = 0$ for $j > b$, then", "$K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ is $t$-pseudo-coherent", "with $t = \\max(m + a, n + b)$.", "\\item If $K$ and $L$ are pseudo-coherent, then", "$K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ is pseudo-coherent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). By replacing $X$ by the members of an open covering", "we may assume there exist strictly perfect complexes $\\mathcal{K}^\\bullet$", "and $\\mathcal{L}^\\bullet$ and maps", "$\\alpha : \\mathcal{K}^\\bullet \\to K$ and", "$\\beta : \\mathcal{L}^\\bullet \\to L$ with $H^i(\\alpha)$ and isomorphism", "for $i > n$ and surjective for $i = n$ and with", "$H^i(\\beta)$ and isomorphism for $i > m$ and surjective for $i = m$.", "Then the map", "$$", "\\alpha \\otimes^\\mathbf{L} \\beta :", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet)", "\\to K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L", "$$", "induces isomorphisms on cohomology sheaves in degree $i$ for", "$i > t$ and a surjection for $i = t$. This follows from the", "spectral sequence of tors (details omitted).", "\\medskip\\noindent", "Proof of (2). We may first replace $X$ by the members of an open", "covering to reduce to the case that $K$ and $L$ are bounded above.", "Then the statement follows immediately from case (1)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2209, "type": "theorem", "label": "cohomology-lemma-summands-pseudo-coherent", "categories": [ "cohomology" ], "title": "cohomology-lemma-summands-pseudo-coherent", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $m \\in \\mathbf{Z}$.", "If $K \\oplus L$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)", "in $D(\\mathcal{O}_X)$ so are $K$ and $L$." ], "refs": [], "proofs": [ { "contents": [ "Assume that $K \\oplus L$ is $m$-pseudo-coherent.", "After replacing $X$ by the members of an open covering we may", "assume $K \\oplus L \\in D^-(\\mathcal{O}_X)$, hence", "$L \\in D^-(\\mathcal{O}_X)$.", "Note that there is a distinguished triangle", "$$", "(K \\oplus L, K \\oplus L, L \\oplus L[1]) =", "(K, K, 0) \\oplus (L, L, L \\oplus L[1])", "$$", "see", "Derived Categories, Lemma \\ref{derived-lemma-direct-sum-triangles}.", "By", "Lemma \\ref{lemma-cone-pseudo-coherent}", "we see that $L \\oplus L[1]$ is $m$-pseudo-coherent.", "Hence also $L[1] \\oplus L[2]$ is $m$-pseudo-coherent.", "By induction $L[n] \\oplus L[n + 1]$ is $m$-pseudo-coherent.", "Since $L$ is bounded above we see that $L[n]$ is $m$-pseudo-coherent", "for large $n$. Hence working backwards, using the distinguished triangles", "$$", "(L[n], L[n] \\oplus L[n - 1], L[n - 1])", "$$", "we conclude that $L[n - 1], L[n - 2], \\ldots, L$ are $m$-pseudo-coherent", "as desired." ], "refs": [ "derived-lemma-direct-sum-triangles", "cohomology-lemma-cone-pseudo-coherent" ], "ref_ids": [ 1765, 2207 ] } ], "ref_ids": [] }, { "id": 2210, "type": "theorem", "label": "cohomology-lemma-complex-pseudo-coherent-modules", "categories": [ "cohomology" ], "title": "cohomology-lemma-complex-pseudo-coherent-modules", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $m \\in \\mathbf{Z}$. Let $\\mathcal{F}^\\bullet$ be a (locally) bounded", "above complex of $\\mathcal{O}_X$-modules such that", "$\\mathcal{F}^i$ is $(m - i)$-pseudo-coherent for all $i$.", "Then $\\mathcal{F}^\\bullet$ is $m$-pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: use Lemma \\ref{lemma-cone-pseudo-coherent} and truncations", "as in the proof of", "More on Algebra, Lemma \\ref{more-algebra-lemma-complex-pseudo-coherent-modules}." ], "refs": [ "cohomology-lemma-cone-pseudo-coherent", "more-algebra-lemma-complex-pseudo-coherent-modules" ], "ref_ids": [ 2207, 10152 ] } ], "ref_ids": [] }, { "id": 2211, "type": "theorem", "label": "cohomology-lemma-cohomology-pseudo-coherent", "categories": [ "cohomology" ], "title": "cohomology-lemma-cohomology-pseudo-coherent", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $m \\in \\mathbf{Z}$. Let", "$E$ be an object of $D(\\mathcal{O}_X)$. If $E$ is (locally) bounded above", "and $H^i(E)$ is $(m - i)$-pseudo-coherent for all $i$, then", "$E$ is $m$-pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: use Lemma \\ref{lemma-cone-pseudo-coherent} and truncations", "as in the proof of", "More on Algebra, Lemma \\ref{more-algebra-lemma-cohomology-pseudo-coherent}." ], "refs": [ "cohomology-lemma-cone-pseudo-coherent", "more-algebra-lemma-cohomology-pseudo-coherent" ], "ref_ids": [ 2207, 10153 ] } ], "ref_ids": [] }, { "id": 2212, "type": "theorem", "label": "cohomology-lemma-finite-cohomology", "categories": [ "cohomology" ], "title": "cohomology-lemma-finite-cohomology", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $K$ be an object of $D(\\mathcal{O}_X)$.", "Let $m \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$", "for $i > m$, then $H^m(K)$ is a finite type $\\mathcal{O}_X$-module.", "\\item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$", "for $i > m + 1$, then $H^{m + 1}(K)$ is a finitely presented", "$\\mathcal{O}_X$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). We may work locally on $X$. Hence we may assume there exists", "a strictly perfect complex $\\mathcal{E}^\\bullet$ and a map", "$\\alpha : \\mathcal{E}^\\bullet \\to K$ which induces", "an isomorphism on cohomology in degrees $> m$ and a surjection in degree $m$.", "It suffices to prove the result for $\\mathcal{E}^\\bullet$.", "Let $n$ be the largest integer such that $\\mathcal{E}^n \\not = 0$.", "If $n = m$, then $H^m(\\mathcal{E}^\\bullet)$ is a quotient of", "$\\mathcal{E}^n$ and the result is clear.", "If $n > m$, then $\\mathcal{E}^{n - 1} \\to \\mathcal{E}^n$ is surjective as", "$H^n(E^\\bullet) = 0$. By Lemma \\ref{lemma-local-lift-map}", "we can locally find a section of this surjection and write", "$\\mathcal{E}^{n - 1} = \\mathcal{E}' \\oplus \\mathcal{E}^n$.", "Hence it suffices to prove the result for the complex", "$(\\mathcal{E}')^\\bullet$ which is the same as $\\mathcal{E}^\\bullet$", "except has $\\mathcal{E}'$ in degree $n - 1$ and $0$ in degree $n$.", "We win by induction on $n$.", "\\medskip\\noindent", "Proof of (2). We may work locally on $X$. Hence we may assume there exists", "a strictly perfect complex $\\mathcal{E}^\\bullet$ and a map", "$\\alpha : \\mathcal{E}^\\bullet \\to K$ which induces", "an isomorphism on cohomology in degrees $> m$ and a surjection in degree $m$.", "As in the proof of (1) we can reduce to the case that $\\mathcal{E}^i = 0$", "for $i > m + 1$. Then we see that", "$H^{m + 1}(K) \\cong H^{m + 1}(\\mathcal{E}^\\bullet) =", "\\Coker(\\mathcal{E}^m \\to \\mathcal{E}^{m + 1})$", "which is of finite presentation." ], "refs": [ "cohomology-lemma-local-lift-map" ], "ref_ids": [ 2199 ] } ], "ref_ids": [] }, { "id": 2213, "type": "theorem", "label": "cohomology-lemma-n-pseudo-module", "categories": [ "cohomology" ], "title": "cohomology-lemma-n-pseudo-module", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf", "of $\\mathcal{O}_X$-modules.", "\\begin{enumerate}", "\\item $\\mathcal{F}$ viewed as an object of $D(\\mathcal{O}_X)$ is", "$0$-pseudo-coherent if and only if $\\mathcal{F}$ is a finite type", "$\\mathcal{O}_X$-module, and", "\\item $\\mathcal{F}$ viewed as an object of $D(\\mathcal{O}_X)$ is", "$(-1)$-pseudo-coherent if and only if $\\mathcal{F}$ is an", "$\\mathcal{O}_X$-module of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Use Lemma \\ref{lemma-finite-cohomology}", "to prove the implications in one direction and", "Lemma \\ref{lemma-cohomology-pseudo-coherent} for the other." ], "refs": [ "cohomology-lemma-finite-cohomology", "cohomology-lemma-cohomology-pseudo-coherent" ], "ref_ids": [ 2212, 2211 ] } ], "ref_ids": [] }, { "id": 2214, "type": "theorem", "label": "cohomology-lemma-last-one-flat", "categories": [ "cohomology" ], "title": "cohomology-lemma-last-one-flat", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{E}^\\bullet$ be a bounded above complex of flat", "$\\mathcal{O}_X$-modules with tor-amplitude in $[a, b]$.", "Then $\\Coker(d_{\\mathcal{E}^\\bullet}^{a - 1})$ is a flat", "$\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "As $\\mathcal{E}^\\bullet$ is a bounded above complex of flat modules we see that", "$\\mathcal{E}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{F} =", "\\mathcal{E}^\\bullet \\otimes_{\\mathcal{O}_X}^{\\mathbf{L}} \\mathcal{F}$", "for any $\\mathcal{O}_X$-module $\\mathcal{F}$.", "Hence for every $\\mathcal{O}_X$-module $\\mathcal{F}$ the sequence", "$$", "\\mathcal{E}^{a - 2} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "\\mathcal{E}^{a - 1} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "\\mathcal{E}^a \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "$$", "is exact in the middle. Since", "$\\mathcal{E}^{a - 2} \\to \\mathcal{E}^{a - 1} \\to \\mathcal{E}^a \\to", "\\Coker(d^{a - 1}) \\to 0$", "is a flat resolution this implies that", "$\\text{Tor}_1^{\\mathcal{O}_X}(\\Coker(d^{a - 1}), \\mathcal{F}) = 0$", "for all $\\mathcal{O}_X$-modules $\\mathcal{F}$. This means that", "$\\Coker(d^{a - 1})$ is flat, see Lemma \\ref{lemma-flat-tor-zero}." ], "refs": [ "cohomology-lemma-flat-tor-zero" ], "ref_ids": [ 2114 ] } ], "ref_ids": [] }, { "id": 2215, "type": "theorem", "label": "cohomology-lemma-tor-amplitude", "categories": [ "cohomology" ], "title": "cohomology-lemma-tor-amplitude", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $E$ be an object of $D(\\mathcal{O}_X)$.", "Let $a, b \\in \\mathbf{Z}$ with $a \\leq b$. The following are equivalent", "\\begin{enumerate}", "\\item $E$ has tor-amplitude in $[a, b]$.", "\\item $E$ is represented by a complex", "$\\mathcal{E}^\\bullet$ of flat $\\mathcal{O}_X$-modules with", "$\\mathcal{E}^i = 0$ for $i \\not \\in [a, b]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (2) holds, then we may compute", "$E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F} =", "\\mathcal{E}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{F}$", "and it is clear that (1) holds.", "\\medskip\\noindent", "Assume that (1) holds. We may represent $E$ by a bounded above complex", "of flat $\\mathcal{O}_X$-modules $\\mathcal{K}^\\bullet$, see", "Section \\ref{section-flat}.", "Let $n$ be the largest integer such that $\\mathcal{K}^n \\not = 0$.", "If $n > b$, then $\\mathcal{K}^{n - 1} \\to \\mathcal{K}^n$ is surjective as", "$H^n(\\mathcal{K}^\\bullet) = 0$. As $\\mathcal{K}^n$ is flat we see that", "$\\Ker(\\mathcal{K}^{n - 1} \\to \\mathcal{K}^n)$ is flat", "(Modules, Lemma \\ref{modules-lemma-flat-ses}).", "Hence we may replace $\\mathcal{K}^\\bullet$ by", "$\\tau_{\\leq n - 1}\\mathcal{K}^\\bullet$. Thus, by induction on $n$, we", "reduce to the case that $K^\\bullet$ is a complex of flat", "$\\mathcal{O}_X$-modules with $\\mathcal{K}^i = 0$ for $i > b$.", "\\medskip\\noindent", "Set $\\mathcal{E}^\\bullet = \\tau_{\\geq a}\\mathcal{K}^\\bullet$.", "Everything is clear except that $\\mathcal{E}^a$ is flat", "which follows immediately from Lemma \\ref{lemma-last-one-flat}", "and the definitions." ], "refs": [ "modules-lemma-flat-ses", "cohomology-lemma-last-one-flat" ], "ref_ids": [ 13278, 2214 ] } ], "ref_ids": [] }, { "id": 2216, "type": "theorem", "label": "cohomology-lemma-tor-amplitude-pullback", "categories": [ "cohomology" ], "title": "cohomology-lemma-tor-amplitude-pullback", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed", "spaces. Let $E$ be an object of $D(\\mathcal{O}_Y)$.", "If $E$ has tor amplitude in $[a, b]$, then $Lf^*E$ has tor amplitude in", "$[a, b]$." ], "refs": [], "proofs": [ { "contents": [ "Assume $E$ has tor amplitude in $[a, b]$. By", "Lemma \\ref{lemma-tor-amplitude}", "we can represent $E$ by a complex of", "$\\mathcal{E}^\\bullet$ of flat $\\mathcal{O}$-modules with", "$\\mathcal{E}^i = 0$ for $i \\not \\in [a, b]$. Then", "$Lf^*E$ is represented by $f^*\\mathcal{E}^\\bullet$.", "By Modules, Lemma \\ref{modules-lemma-pullback-flat}", "the modules $f^*\\mathcal{E}^i$ are flat.", "Thus by Lemma \\ref{lemma-tor-amplitude}", "we conclude that $Lf^*E$ has tor amplitude in $[a, b]$." ], "refs": [ "cohomology-lemma-tor-amplitude", "modules-lemma-pullback-flat", "cohomology-lemma-tor-amplitude" ], "ref_ids": [ 2215, 13287, 2215 ] } ], "ref_ids": [] }, { "id": 2217, "type": "theorem", "label": "cohomology-lemma-tor-amplitude-stalk", "categories": [ "cohomology" ], "title": "cohomology-lemma-tor-amplitude-stalk", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $E$ be an object of $D(\\mathcal{O}_X)$.", "Let $a, b \\in \\mathbf{Z}$ with $a \\leq b$. The following are equivalent", "\\begin{enumerate}", "\\item $E$ has tor-amplitude in $[a, b]$.", "\\item for every $x \\in X$ the object $E_x$ of $D(\\mathcal{O}_{X, x})$", "has tor-amplitude in $[a, b]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Taking stalks at $x$ is the same thing as pulling back by the", "morphism of ringed spaces $(x, \\mathcal{O}_{X, x}) \\to (X, \\mathcal{O}_X)$.", "Hence the implication (1) $\\Rightarrow$ (2) follows from", "Lemma \\ref{lemma-tor-amplitude-pullback}.", "For the converse, note that taking stalks commutes with tensor", "products (Modules, Lemma \\ref{modules-lemma-stalk-tensor-product}).", "Hence", "$$", "(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F})_x =", "E_x \\otimes_{\\mathcal{O}_{X, x}}^\\mathbf{L} \\mathcal{F}_x", "$$", "On the other hand, taking stalks is exact, so", "$$", "H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F})_x =", "H^i((E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F})_x) =", "H^i(E_x \\otimes_{\\mathcal{O}_{X, x}}^\\mathbf{L} \\mathcal{F}_x)", "$$", "and we can check whether", "$H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F})$ is zero", "by checking whether all of its stalks are zero", "(Modules, Lemma \\ref{modules-lemma-abelian}). Thus (2) implies (1)." ], "refs": [ "cohomology-lemma-tor-amplitude-pullback", "modules-lemma-stalk-tensor-product", "modules-lemma-abelian" ], "ref_ids": [ 2216, 13267, 13221 ] } ], "ref_ids": [] }, { "id": 2218, "type": "theorem", "label": "cohomology-lemma-cone-tor-amplitude", "categories": [ "cohomology" ], "title": "cohomology-lemma-cone-tor-amplitude", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $(K, L, M, f, g, h)$ be a distinguished", "triangle in $D(\\mathcal{O}_X)$. Let $a, b \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item If $K$ has tor-amplitude in $[a + 1, b + 1]$ and", "$L$ has tor-amplitude in $[a, b]$ then $M$ has", "tor-amplitude in $[a, b]$.", "\\item If $K$ and $M$ have tor-amplitude in $[a, b]$, then", "$L$ has tor-amplitude in $[a, b]$.", "\\item If $L$ has tor-amplitude in $[a + 1, b + 1]$", "and $M$ has tor-amplitude in $[a, b]$, then", "$K$ has tor-amplitude in $[a + 1, b + 1]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This just follows from the long exact cohomology sequence", "associated to a distinguished triangle and the fact that", "$- \\otimes_{\\mathcal{O}_X}^{\\mathbf{L}} \\mathcal{F}$", "preserves distinguished triangles.", "The easiest one to prove is (2) and the others follow from it by", "translation." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2219, "type": "theorem", "label": "cohomology-lemma-tensor-tor-amplitude", "categories": [ "cohomology" ], "title": "cohomology-lemma-tensor-tor-amplitude", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L$ be objects of", "$D(\\mathcal{O}_X)$. If $K$ has tor-amplitude in $[a, b]$ and", "$L$ has tor-amplitude in $[c, d]$ then $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$", "has tor amplitude in $[a + c, b + d]$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: use the spectral sequence for tors." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2220, "type": "theorem", "label": "cohomology-lemma-summands-tor-amplitude", "categories": [ "cohomology" ], "title": "cohomology-lemma-summands-tor-amplitude", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $a, b \\in \\mathbf{Z}$.", "For $K$, $L$ objects of $D(\\mathcal{O}_X)$ if $K \\oplus L$ has tor", "amplitude in $[a, b]$ so do $K$ and $L$." ], "refs": [], "proofs": [ { "contents": [ "Clear from the fact that the Tor functors are additive." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2221, "type": "theorem", "label": "cohomology-lemma-perfect-independent-representative", "categories": [ "cohomology" ], "title": "cohomology-lemma-perfect-independent-representative", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $E$ be an object of $D(\\mathcal{O}_X)$.", "\\begin{enumerate}", "\\item If there exists an open covering $X = \\bigcup U_i$ and", "strictly perfect complexes $\\mathcal{E}_i^\\bullet$ on $U_i$", "such that $\\mathcal{E}_i^\\bullet$ represents $E|_{U_i}$ in", "$D(\\mathcal{O}_{U_i})$, then $E$ is perfect.", "\\item If $E$ is perfect, then any complex representing $E$ is perfect.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Identical to the proof of", "Lemma \\ref{lemma-pseudo-coherent-independent-representative}." ], "refs": [ "cohomology-lemma-pseudo-coherent-independent-representative" ], "ref_ids": [ 2205 ] } ], "ref_ids": [] }, { "id": 2222, "type": "theorem", "label": "cohomology-lemma-perfect-on-locally-ringed", "categories": [ "cohomology" ], "title": "cohomology-lemma-perfect-on-locally-ringed", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E$ be an object of", "$D(\\mathcal{O}_X)$. Assume that all stalks $\\mathcal{O}_{X, x}$", "are local rings. Then the following are equivalent", "\\begin{enumerate}", "\\item $E$ is perfect,", "\\item there exists an open covering $X = \\bigcup U_i$ such that", "$E|_{U_i}$ can be represented by a finite complex of finite locally", "free $\\mathcal{O}_{U_i}$-modules, and", "\\item there exists an open covering $X = \\bigcup U_i$ such that", "$E|_{U_i}$ can be represented by a finite complex of finite", "free $\\mathcal{O}_{U_i}$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-perfect-independent-representative}", "and the fact that on $X$ every direct summand of a finite free module", "is finite locally free. See Modules, Lemma", "\\ref{modules-lemma-direct-summand-of-locally-free-is-locally-free}." ], "refs": [ "cohomology-lemma-perfect-independent-representative", "modules-lemma-direct-summand-of-locally-free-is-locally-free" ], "ref_ids": [ 2221, 13266 ] } ], "ref_ids": [] }, { "id": 2223, "type": "theorem", "label": "cohomology-lemma-perfect-precise", "categories": [ "cohomology" ], "title": "cohomology-lemma-perfect-precise", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $E$ be an object of $D(\\mathcal{O}_X)$.", "Let $a \\leq b$ be integers. If $E$ has tor amplitude in $[a, b]$", "and is $(a - 1)$-pseudo-coherent, then $E$ is perfect." ], "refs": [], "proofs": [ { "contents": [ "After replacing $X$ by the members of an open covering we may assume there", "exists a strictly perfect complex $\\mathcal{E}^\\bullet$ and a map", "$\\alpha : \\mathcal{E}^\\bullet \\to E$ such that $H^i(\\alpha)$ is an isomorphism", "for $i \\geq a$. We may and do replace $\\mathcal{E}^\\bullet$ by", "$\\sigma_{\\geq a - 1}\\mathcal{E}^\\bullet$. Choose a distinguished triangle", "$$", "\\mathcal{E}^\\bullet \\to E \\to C \\to \\mathcal{E}^\\bullet[1]", "$$", "From the vanishing of cohomology sheaves of $E$ and $\\mathcal{E}^\\bullet$", "and the assumption on $\\alpha$ we obtain $C \\cong \\mathcal{K}[a - 2]$ with", "$\\mathcal{K} = \\Ker(\\mathcal{E}^{a - 1} \\to \\mathcal{E}^a)$.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "Applying $- \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}$", "the assumption that $E$ has tor amplitude in $[a, b]$", "implies $\\mathcal{K} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "\\mathcal{E}^{a - 1} \\otimes_{\\mathcal{O}_X} \\mathcal{F}$ has image", "$\\Ker(\\mathcal{E}^{a - 1} \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "\\to \\mathcal{E}^a \\otimes_{\\mathcal{O}_X} \\mathcal{F})$.", "It follows that $\\text{Tor}_1^{\\mathcal{O}_X}(\\mathcal{E}', \\mathcal{F}) = 0$", "where $\\mathcal{E}' = \\Coker(\\mathcal{E}^{a - 1} \\to \\mathcal{E}^a)$.", "Hence $\\mathcal{E}'$ is flat (Lemma \\ref{lemma-flat-tor-zero}).", "Thus $\\mathcal{E}'$ is locally a direct summand of a finite free module by", "Modules, Lemma \\ref{modules-lemma-flat-locally-finite-presentation}.", "Thus locally the complex", "$$", "\\mathcal{E}' \\to \\mathcal{E}^{a - 1} \\to \\ldots \\to \\mathcal{E}^b", "$$", "is quasi-isomorphic to $E$ and $E$ is perfect." ], "refs": [ "cohomology-lemma-flat-tor-zero", "modules-lemma-flat-locally-finite-presentation" ], "ref_ids": [ 2114, 13282 ] } ], "ref_ids": [] }, { "id": 2224, "type": "theorem", "label": "cohomology-lemma-perfect", "categories": [ "cohomology" ], "title": "cohomology-lemma-perfect", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $E$ be an object of $D(\\mathcal{O}_X)$.", "The following are equivalent", "\\begin{enumerate}", "\\item $E$ is perfect, and", "\\item $E$ is pseudo-coherent and locally has finite tor dimension.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). By definition this means there exists an open covering", "$X = \\bigcup U_i$ such that $E|_{U_i}$ is represented by a", "strictly perfect complex. Thus $E$ is pseudo-coherent (i.e.,", "$m$-pseudo-coherent for all $m$) by", "Lemma \\ref{lemma-pseudo-coherent-independent-representative}.", "Moreover, a direct summand of a finite free module is flat, hence", "$E|_{U_i}$ has finite Tor dimension by", "Lemma \\ref{lemma-tor-amplitude}. Thus (2) holds.", "\\medskip\\noindent", "Assume (2). After replacing $X$ by the members of an open covering", "we may assume there exist integers $a \\leq b$ such that $E$", "has tor amplitude in $[a, b]$. Since $E$ is $m$-pseudo-coherent", "for all $m$ we conclude using Lemma \\ref{lemma-perfect-precise}." ], "refs": [ "cohomology-lemma-pseudo-coherent-independent-representative", "cohomology-lemma-tor-amplitude", "cohomology-lemma-perfect-precise" ], "ref_ids": [ 2205, 2215, 2223 ] } ], "ref_ids": [] }, { "id": 2225, "type": "theorem", "label": "cohomology-lemma-perfect-pullback", "categories": [ "cohomology" ], "title": "cohomology-lemma-perfect-pullback", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed", "spaces. Let $E$ be an object of $D(\\mathcal{O}_Y)$. If $E$ is perfect in", "$D(\\mathcal{O}_Y)$, then $Lf^*E$ is perfect in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-perfect},", "\\ref{lemma-tor-amplitude-pullback}, and", "\\ref{lemma-pseudo-coherent-pullback}.", "(An alternative proof is to copy the proof of", "Lemma \\ref{lemma-pseudo-coherent-pullback}.)" ], "refs": [ "cohomology-lemma-perfect", "cohomology-lemma-tor-amplitude-pullback", "cohomology-lemma-pseudo-coherent-pullback", "cohomology-lemma-pseudo-coherent-pullback" ], "ref_ids": [ 2224, 2216, 2206, 2206 ] } ], "ref_ids": [] }, { "id": 2226, "type": "theorem", "label": "cohomology-lemma-two-out-of-three-perfect", "categories": [ "cohomology" ], "title": "cohomology-lemma-two-out-of-three-perfect", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(K, L, M, f, g, h)$", "be a distinguished triangle in $D(\\mathcal{O}_X)$. If two out of three of", "$K, L, M$ are perfect then the third is also perfect." ], "refs": [], "proofs": [ { "contents": [ "First proof: Combine", "Lemmas \\ref{lemma-perfect}, \\ref{lemma-cone-pseudo-coherent}, and", "\\ref{lemma-cone-tor-amplitude}.", "Second proof (sketch): Say $K$ and $L$ are perfect. After replacing", "$X$ by the members of an open covering we may assume that $K$ and $L$", "are represented by strictly perfect complexes $\\mathcal{K}^\\bullet$", "and $\\mathcal{L}^\\bullet$. After replacing $X$ by the members", "of an open covering we may assume the map $K \\to L$ is given by", "a map of complexes $\\alpha : \\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet$,", "see Lemma \\ref{lemma-local-actual}.", "Then $M$ is isomorphic to the cone of $\\alpha$ which is strictly", "perfect by Lemma \\ref{lemma-cone}." ], "refs": [ "cohomology-lemma-perfect", "cohomology-lemma-cone-pseudo-coherent", "cohomology-lemma-cone-tor-amplitude", "cohomology-lemma-local-actual", "cohomology-lemma-cone" ], "ref_ids": [ 2224, 2207, 2218, 2202, 2196 ] } ], "ref_ids": [] }, { "id": 2227, "type": "theorem", "label": "cohomology-lemma-tensor-perfect", "categories": [ "cohomology" ], "title": "cohomology-lemma-tensor-perfect", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "If $K, L$ are perfect objects of $D(\\mathcal{O}_X)$, then", "so is $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemmas \\ref{lemma-perfect}, \\ref{lemma-tensor-pseudo-coherent}, and", "\\ref{lemma-tensor-tor-amplitude}." ], "refs": [ "cohomology-lemma-perfect", "cohomology-lemma-tensor-pseudo-coherent", "cohomology-lemma-tensor-tor-amplitude" ], "ref_ids": [ 2224, 2208, 2219 ] } ], "ref_ids": [] }, { "id": 2228, "type": "theorem", "label": "cohomology-lemma-summands-perfect", "categories": [ "cohomology" ], "title": "cohomology-lemma-summands-perfect", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "If $K \\oplus L$ is a perfect object of $D(\\mathcal{O}_X)$, then", "so are $K$ and $L$." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemmas \\ref{lemma-perfect}, \\ref{lemma-summands-pseudo-coherent}, and", "\\ref{lemma-summands-tor-amplitude}." ], "refs": [ "cohomology-lemma-perfect", "cohomology-lemma-summands-pseudo-coherent", "cohomology-lemma-summands-tor-amplitude" ], "ref_ids": [ 2224, 2209, 2220 ] } ], "ref_ids": [] }, { "id": 2229, "type": "theorem", "label": "cohomology-lemma-pushforward-perfect", "categories": [ "cohomology" ], "title": "cohomology-lemma-pushforward-perfect", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $j : U \\to X$ be an", "open subspace. Let $E$ be a perfect object of $D(\\mathcal{O}_U)$", "whose cohomology", "sheaves are supported on a closed subset $T \\subset U$ with $j(T)$", "closed in $X$. Then $Rj_*E$ is a perfect object of $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Being a perfect complex is local on $X$. Thus it suffices to check that", "$Rj_*E$ is perfect when restricted to $U$ and $V = X \\setminus j(T)$.", "We have $Rj_*E|_U = E$ which is perfect. We have", " $Rj_*E|_V = 0$ because $E|_{U \\setminus T} = 0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2230, "type": "theorem", "label": "cohomology-lemma-symmetric-monoidal-cat-complexes", "categories": [ "cohomology" ], "title": "cohomology-lemma-symmetric-monoidal-cat-complexes", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. The category of complexes", "of $\\mathcal{O}_X$-modules with tensor product defined by", "$\\mathcal{F}^\\bullet \\otimes \\mathcal{G}^\\bullet =", "\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)$", "is a symmetric monoidal category (for sign rules, see", "More on Algebra, Section \\ref{more-algebra-section-sign-rules})." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints: as unit $\\mathbf{1}$ we take the complex having", "$\\mathcal{O}_X$ in degree $0$ and zero in other degrees with", "obvious isomorphisms", "$\\text{Tot}(\\mathbf{1} \\otimes_{\\mathcal{O}_X} \\mathcal{G}^\\bullet) =", "\\mathcal{G}^\\bullet$ and", "$\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathbf{1}) =", "\\mathcal{F}^\\bullet$.", "to prove the lemma you have to check the commutativity", "of various diagrams, see Categories, Definitions", "\\ref{categories-definition-monoidal-category} and", "\\ref{categories-definition-symmetric-monoidal-category}.", "The verifications are straightforward in each case." ], "refs": [ "categories-definition-monoidal-category", "categories-definition-symmetric-monoidal-category" ], "ref_ids": [ 12404, 12408 ] } ], "ref_ids": [] }, { "id": 2231, "type": "theorem", "label": "cohomology-lemma-left-dual-complex", "categories": [ "cohomology" ], "title": "cohomology-lemma-left-dual-complex", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}^\\bullet$", "be a complex of $\\mathcal{O}_X$-modules. If $\\mathcal{F}^\\bullet$", "has a left dual in the monoidal category of complexes of", "$\\mathcal{O}_X$-modules", "(Categories, Definition \\ref{categories-definition-dual})", "then $\\mathcal{F}^\\bullet$ is a locally bounded complex whose terms are", "locally direct summands of finite free $\\mathcal{O}_X$-modules", "and the left dual is as constructed in Example \\ref{example-dual}." ], "refs": [ "categories-definition-dual" ], "proofs": [ { "contents": [ "By uniqueness of left duals", "(Categories, Remark \\ref{categories-remark-left-dual-adjoint})", "we get the final statement provided we show that $\\mathcal{F}^\\bullet$", "is as stated. Let $\\mathcal{G}^\\bullet, \\eta, \\epsilon$ be a left dual.", "Write $\\eta = \\sum \\eta_n$ and $\\epsilon = \\sum \\epsilon_n$", "where $\\eta_n : \\mathcal{O}_X \\to", "\\mathcal{F}^n \\otimes_{\\mathcal{O}_X} \\mathcal{G}^{-n}$", "and", "$\\epsilon_n : \\mathcal{G}^{-n} \\otimes_{\\mathcal{O}_X} \\mathcal{F}^n", "\\to \\mathcal{O}_X$. Since", "$(1 \\otimes \\epsilon) \\circ (\\eta \\otimes 1) = \\text{id}_{\\mathcal{F}^\\bullet}$", "and", "$(\\epsilon \\otimes 1) \\circ (1 \\otimes \\eta) = \\text{id}_{\\mathcal{G}^\\bullet}$", "by Categories, Definition \\ref{categories-definition-dual} we see immediately", "that we have", "$(1 \\otimes \\epsilon_n) \\circ (\\eta_n \\otimes 1) = \\text{id}_{\\mathcal{F}^n}$", "and", "$(\\epsilon_n \\otimes 1) \\circ (1 \\otimes \\eta_n) =", "\\text{id}_{\\mathcal{G}^{-n}}$.", "Hence we see that $\\mathcal{F}^n$ is locally a direct summand of a finite", "free $\\mathcal{O}_X$-module by", "Modules, Lemma \\ref{modules-lemma-left-dual-module}.", "Since the sum $\\eta = \\sum \\eta_n$ is locally finite, we conclude that", "$\\mathcal{F}^\\bullet$ is locally bounded." ], "refs": [ "categories-remark-left-dual-adjoint", "categories-definition-dual", "modules-lemma-left-dual-module" ], "ref_ids": [ 12430, 12407, 13281 ] } ], "ref_ids": [ 12407 ] }, { "id": 2232, "type": "theorem", "label": "cohomology-lemma-internal-hom-evaluate-isom", "categories": [ "cohomology" ], "title": "cohomology-lemma-internal-hom-evaluate-isom", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L, M \\in D(\\mathcal{O}_X)$.", "If $K$ is perfect, then the map", "$$", "R\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K", "\\longrightarrow", "R\\SheafHom(R\\SheafHom(K, L), M)", "$$", "of Lemma \\ref{lemma-internal-hom-evaluate} is an isomorphism." ], "refs": [ "cohomology-lemma-internal-hom-evaluate" ], "proofs": [ { "contents": [ "Since the map is globally defined and since formation of the right and", "left hand side commute with localization", "(see Lemma \\ref{lemma-restriction-RHom-to-U}), to prove this we may work", "locally on $X$. Thus we may assume $K$ is represented by a strictly", "perfect complex $\\mathcal{E}^\\bullet$.", "\\medskip\\noindent", "If $K_1 \\to K_2 \\to K_3$ is a distinguished triangle in $D(\\mathcal{O}_X)$,", "then we get distinguished triangles", "$$", "R\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K_1 \\to", "R\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K_2 \\to", "R\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K_3", "$$", "and", "$$", "R\\SheafHom(R\\SheafHom(K_1, L), M) \\to", "R\\SheafHom(R\\SheafHom(K_2, L), M)", "R\\SheafHom(R\\SheafHom(K_3, L), M)", "$$", "See Section \\ref{section-flat} and", "Lemma \\ref{lemma-RHom-triangulated}.", "The arrow of Lemma \\ref{lemma-internal-hom-evaluate} is functorial in $K$", "hence we get a morphism between these distinguished triangles.", "Thus, if the result holds for $K_1$ and $K_3$, then the result holds for", "$K_2$ by Derived Categories, Lemma", "\\ref{derived-lemma-third-isomorphism-triangle}.", "\\medskip\\noindent", "Combining the remarks above with the distinguished triangles", "$$", "\\sigma_{\\geq n}\\mathcal{E}^\\bullet \\to \\mathcal{E}^\\bullet \\to", "\\sigma_{\\leq n - 1}\\mathcal{E}^\\bullet", "$$", "of stupid trunctions, we reduce to the case where $K$ consists", "of a direct summand of a finite free $\\mathcal{O}_X$-module placed", "in some degree. By an obvious compatibility of the problem with direct sums", "(similar to what was said above) and shifts this reduces us to the case", "where $K = \\mathcal{O}_X^{\\oplus n}$ for some integer $n$.", "This case is clear." ], "refs": [ "cohomology-lemma-restriction-RHom-to-U", "cohomology-lemma-RHom-triangulated", "cohomology-lemma-internal-hom-evaluate", "derived-lemma-third-isomorphism-triangle" ], "ref_ids": [ 2184, 2185, 2190, 1759 ] } ], "ref_ids": [ 2190 ] }, { "id": 2233, "type": "theorem", "label": "cohomology-lemma-dual-perfect-complex", "categories": [ "cohomology" ], "title": "cohomology-lemma-dual-perfect-complex", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K$ be a perfect object of", "$D(\\mathcal{O}_X)$. Then $K^\\vee = R\\SheafHom(K, \\mathcal{O}_X)$ is a", "perfect object too and $(K^\\vee)^\\vee \\cong K$. There are", "functorial isomorphisms", "$$", "M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} K^\\vee = R\\SheafHom(K, M)", "$$", "and", "$$", "H^0(X, M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} K^\\vee) =", "\\Hom_{D(\\mathcal{O}_X)}(K, M)", "$$", "for $M$ in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-internal-hom-evaluate} there is a canonical map", "$$", "K = R\\SheafHom(\\mathcal{O}_X, \\mathcal{O}_X)", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L} K \\longrightarrow", "R\\SheafHom(R\\SheafHom(K, \\mathcal{O}_X), \\mathcal{O}_X) =", "(K^\\vee)^\\vee", "$$", "which is an isomorphism by Lemma \\ref{lemma-internal-hom-evaluate-isom}.", "To check the other statements we will use without further mention that", "formation of internal hom commutes with restriction to opens", "(Lemma \\ref{lemma-restriction-RHom-to-U}).", "We may check $K^\\vee$ is perfect locally on $X$.", "By Lemma \\ref{lemma-dual}", "to see the final statement it suffices to check that the map", "(\\ref{equation-eval})", "$$", "M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} K^\\vee", "\\longrightarrow", "R\\SheafHom(K, M)", "$$", "is an isomorphism. This is local on $X$ as well.", "Hence it suffices to prove these two statements $K$ is represented", "by a strictly perfect complex.", "\\medskip\\noindent", "Assume $K$ is represented by the strictly perfect complex", "$\\mathcal{E}^\\bullet$. Then it follows from", "Lemma \\ref{lemma-Rhom-strictly-perfect}", "that $K^\\vee$ is represented by the complex whose terms are", "$(\\mathcal{E}^{-n})^\\vee =", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}^{-n}, \\mathcal{O}_X)$", "in degree $n$. Since $\\mathcal{E}^{-n}$ is a direct summand of a finite", "free $\\mathcal{O}_X$-module, so is $(\\mathcal{E}^{-n})^\\vee$.", "Hence $K^\\vee$ is represented by a strictly perfect complex too", "and we see that $K^\\vee$ is perfect.", "To see that (\\ref{equation-eval}) is an isomorphism, represent", "$M$ by a complex $\\mathcal{F}^\\bullet$.", "By Lemma \\ref{lemma-Rhom-strictly-perfect} the complex", "$R\\SheafHom(K, M)$ is represented by the complex with terms", "$$", "\\bigoplus\\nolimits_{n = p + q}", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}^{-q}, \\mathcal{F}^p)", "$$", "On the other hand, the object $M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} K^\\vee$", "is represented by the complex with terms", "$$", "\\bigoplus\\nolimits_{n = p + q}", "\\mathcal{F}^p \\otimes_{\\mathcal{O}_X} (\\mathcal{E}^{-q})^\\vee", "$$", "Thus the assertion that (\\ref{equation-eval}) is an isomorphism", "reduces to the assertion that the canonical map", "$$", "\\mathcal{F}", "\\otimes_{\\mathcal{O}_X}", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}, \\mathcal{O}_X)", "\\longrightarrow", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}, \\mathcal{F})", "$$", "is an isomorphism when $\\mathcal{E}$ is a direct summand of a finite", "free $\\mathcal{O}_X$-module and $\\mathcal{F}$ is any $\\mathcal{O}_X$-module.", "This follows immediately from the corresponding statement when", "$\\mathcal{E}$ is finite free." ], "refs": [ "cohomology-lemma-internal-hom-evaluate", "cohomology-lemma-internal-hom-evaluate-isom", "cohomology-lemma-restriction-RHom-to-U", "cohomology-lemma-dual", "cohomology-lemma-Rhom-strictly-perfect", "cohomology-lemma-Rhom-strictly-perfect" ], "ref_ids": [ 2190, 2232, 2184, 2189, 2203, 2203 ] } ], "ref_ids": [] }, { "id": 2234, "type": "theorem", "label": "cohomology-lemma-symmetric-monoidal-derived", "categories": [ "cohomology" ], "title": "cohomology-lemma-symmetric-monoidal-derived", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. The derived category", "$D(\\mathcal{O}_X)$ is a symmetric monoidal category with tensor product", "given by derived tensor product with usual associativity and", "commutativity constraints (for sign rules, see", "More on Algebra, Section \\ref{more-algebra-section-sign-rules})." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Compare with Lemma \\ref{lemma-symmetric-monoidal-cat-complexes}." ], "refs": [ "cohomology-lemma-symmetric-monoidal-cat-complexes" ], "ref_ids": [ 2230 ] } ], "ref_ids": [] }, { "id": 2235, "type": "theorem", "label": "cohomology-lemma-left-dual-derived", "categories": [ "cohomology" ], "title": "cohomology-lemma-left-dual-derived", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $M$ be an object", "of $D(\\mathcal{O}_X)$. If $M$ has a left dual in the monoidal category", "$D(\\mathcal{O}_X)$ (Categories, Definition \\ref{categories-definition-dual})", "then $M$ is perfect and the left dual is as constructed in", "Example \\ref{example-dual-derived}." ], "refs": [ "categories-definition-dual" ], "proofs": [ { "contents": [ "Let $x \\in X$. It suffices to find an open neighbourhood $U$ of $x$", "such that $M$ restricts to a perfect complex over $U$. Hence during the", "proof we can (finitely often) replace $X$ by an open neighbourhood of $x$.", "Let $N, \\eta, \\epsilon$ be a left dual.", "\\medskip\\noindent", "We are going to use the following argument several times. Choose any", "complex $\\mathcal{M}^\\bullet$", "of $\\mathcal{O}_X$-modules representing $M$. Choose a K-flat complex", "$\\mathcal{N}^\\bullet$ representing $N$ whose terms are flat", "$\\mathcal{O}_X$-modules, see Lemma \\ref{lemma-K-flat-resolution}.", "Consider the map", "$$", "\\eta : \\mathcal{O}_X \\to", "\\text{Tot}(\\mathcal{M}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{N}^\\bullet)", "$$", "After shrinking $X$ we can find an integer $N$ and for", "$i = 1, \\ldots, N$ integers $n_i \\in \\mathbf{Z}$ and sections", "$f_i$ and $g_i$ of $\\mathcal{M}^{n_i}$ and $\\mathcal{N}^{-n_i}$", "such that", "$$", "\\eta(1) = \\sum\\nolimits_i f_i \\otimes g_i", "$$", "Let $\\mathcal{K}^\\bullet \\subset \\mathcal{M}^\\bullet$ be any subcomplex", "of $\\mathcal{O}_X$-modules containing the sections $f_i$", "for $i = 1, \\ldots, N$.", "Since", "$\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{N}^\\bullet)", "\\subset", "\\text{Tot}(\\mathcal{M}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{N}^\\bullet)$", "by flatness of the modules $\\mathcal{N}^n$, we see that $\\eta$ factors through", "$$", "\\tilde \\eta :", "\\mathcal{O}_X \\to", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{N}^\\bullet)", "$$", "Denoting $K$ the object of $D(\\mathcal{O}_X)$ represented by", "$\\mathcal{K}^\\bullet$ we find a commutative diagram", "$$", "\\xymatrix{", "M \\ar[rr]_-{\\eta \\otimes 1} \\ar[rrd]_{\\tilde \\eta \\otimes 1} & &", "M \\otimes^\\mathbf{L} N \\otimes^\\mathbf{L} M", "\\ar[r]_-{1 \\otimes \\epsilon} &", "M \\\\", "& &", "K \\otimes^\\mathbf{L} N \\otimes^\\mathbf{L} M", "\\ar[u] \\ar[r]^-{1 \\otimes \\epsilon} &", "K \\ar[u]", "}", "$$", "Since the composition of the upper row is the identity on $M$", "we conclude that $M$ is a direct summand of $K$ in $D(\\mathcal{O}_X)$.", "\\medskip\\noindent", "As a first use of the argument above, we can choose the subcomplex", "$\\mathcal{K}^\\bullet = \\sigma_{\\geq a} \\tau_{\\leq b}\\mathcal{M}^\\bullet$", "with $a < n_i < b$ for $i = 1, \\ldots, N$. Thus $M$ is a direct", "summand in $D(\\mathcal{O}_X)$ of a bounded complex and we conclude", "we may assume $M$ is in $D^b(\\mathcal{O}_X)$. (Recall that the process", "above involves shrinking $X$.)", "\\medskip\\noindent", "Since $M$ is in $D^b(\\mathcal{O}_X)$ we may choose", "$\\mathcal{M}^\\bullet$ to be a bounded above complex of", "flat modules (by Modules, Lemma \\ref{modules-lemma-module-quotient-flat} and", "Derived Categories, Lemma \\ref{derived-lemma-subcategory-left-resolution}).", "Then we can choose $\\mathcal{K}^\\bullet = \\sigma_{\\geq a}\\mathcal{M}^\\bullet$", "with $a < n_i$ for $i = 1, \\ldots, N$ in the argument above.", "Thus we find that we may assume $M$ is a direct summand in", "$D(\\mathcal{O}_X)$ of a bounded complex of flat modules.", "In particular, $M$ has finite tor amplitude.", "\\medskip\\noindent", "Say $M$ has tor amplitude in $[a, b]$. Assuming $M$ is $m$-pseudo-coherent", "we are going to show that (after shrinking $X$) we may assume $M$", "is $(m - 1)$-pseudo-coherent. This will finish the proof by", "Lemma \\ref{lemma-perfect-precise} and the fact that", "$M$ is $(b + 1)$-pseudo-coherent in any case.", "After shrinking $X$ we may assume there exists a strictly perfect", "complex $\\mathcal{E}^\\bullet$ and a map $\\alpha : \\mathcal{E}^\\bullet \\to M$", "in $D(\\mathcal{O}_X)$ such that $H^i(\\alpha)$ is an isomorphism for", "$i > m$ and surjective for $i = m$. We may and do assume", "that $\\mathcal{E}^i = 0$ for $i < m$. Choose a distinguished triangle", "$$", "\\mathcal{E}^\\bullet \\to M \\to L \\to \\mathcal{E}^\\bullet[1]", "$$", "Observe that $H^i(L) = 0$ for $i \\geq m$. Thus we may represent", "$L$ by a complex $\\mathcal{L}^\\bullet$ with $\\mathcal{L}^i = 0$", "for $i \\geq m$. The map $L \\to \\mathcal{E}^\\bullet[1]$", "is given by a map of complexes", "$\\mathcal{L}^\\bullet \\to \\mathcal{E}^\\bullet[1]$", "which is zero in all degrees except in degree $m - 1$", "where we obtain a map $\\mathcal{L}^{m - 1} \\to \\mathcal{E}^m$, see", "Derived Categories, Lemma \\ref{derived-lemma-negative-exts}.", "Then $M$ is represented by the complex", "$$", "\\mathcal{M}^\\bullet :", "\\ldots \\to", "\\mathcal{L}^{m - 2} \\to", "\\mathcal{L}^{m - 1} \\to", "\\mathcal{E}^m \\to", "\\mathcal{E}^{m + 1} \\to \\ldots", "$$", "Apply the discussion in the second paragraph to this complex to get", "sections $f_i$ of $\\mathcal{M}^{n_i}$ for $i = 1, \\ldots, N$.", "For $n < m$ let $\\mathcal{K}^n \\subset \\mathcal{L}^n$", "be the $\\mathcal{O}_X$-submodule generated by the sections", "$f_i$ for $n_i = n$ and $d(f_i)$ for $n_i = n - 1$.", "For $n \\geq m$ set $\\mathcal{K}^n = \\mathcal{E}^n$.", "Clearly, we have a morphism of", "distinguished triangles", "$$", "\\xymatrix{", "\\mathcal{E}^\\bullet \\ar[r] &", "\\mathcal{M}^\\bullet \\ar[r] &", "\\mathcal{L}^\\bullet \\ar[r] &", "\\mathcal{E}^\\bullet[1] \\\\", "\\mathcal{E}^\\bullet \\ar[r] \\ar[u] &", "\\mathcal{K}^\\bullet \\ar[r] \\ar[u] &", "\\sigma_{\\leq m - 1}\\mathcal{K}^\\bullet \\ar[r] \\ar[u] &", "\\mathcal{E}^\\bullet[1] \\ar[u]", "}", "$$", "where all the morphisms are as indicated above.", "Denote $K$ the object of $D(\\mathcal{O}_X)$ corresponding to the complex", "$\\mathcal{K}^\\bullet$.", "By the arguments in the second paragraph of the proof we obtain", "a morphism $s : M \\to K$ in $D(\\mathcal{O}_X)$ such that the composition", "$M \\to K \\to M$ is the identity on $M$. We don't know that the", "diagram", "$$", "\\xymatrix{", "\\mathcal{E}^\\bullet \\ar[r] &", "\\mathcal{K}^\\bullet \\ar@{=}[r] &", "K \\\\", "\\mathcal{E}^\\bullet \\ar[u]^{\\text{id}} \\ar[r]^i &", "\\mathcal{M}^\\bullet \\ar@{=}[r] &", "M \\ar[u]_s", "}", "$$", "commutes, but we do know it commutes after composing with the", "map $K \\to M$. By Lemma \\ref{lemma-local-actual} after shrinking $X$ we may", "assume that $s \\circ i$ is given by a map of complexes", "$\\sigma : \\mathcal{E}^\\bullet \\to \\mathcal{K}^\\bullet$.", "By the same lemma we may assume the composition of $\\sigma$", "with the inclusion $\\mathcal{K}^\\bullet \\subset \\mathcal{M}^\\bullet$", "is homotopic to zero by some homotopy", "$\\{h^i : \\mathcal{E}^i \\to \\mathcal{M}^{i - 1}\\}$.", "Thus, after replacing $\\mathcal{K}^{m - 1}$ by", "$\\mathcal{K}^{m - 1} + \\Im(h^m)$ (note that after doing this", "it is still the case that $\\mathcal{K}^{m - 1}$ is generated", "by finitely many global sections), we see that", "$\\sigma$ itself is homotopic to zero!", "This means that we have a commutative solid diagram", "$$", "\\xymatrix{", "\\mathcal{E}^\\bullet \\ar[r] &", "M \\ar[r] &", "\\mathcal{L}^\\bullet \\ar[r] &", "\\mathcal{E}^\\bullet[1] \\\\", "\\mathcal{E}^\\bullet \\ar[r] \\ar[u] &", "K \\ar[r] \\ar[u] &", "\\sigma_{\\leq m - 1}\\mathcal{K}^\\bullet \\ar[r] \\ar[u] &", "\\mathcal{E}^\\bullet[1] \\ar[u] \\\\", "\\mathcal{E}^\\bullet \\ar[r] \\ar[u] &", "M \\ar[r] \\ar[u]^s &", "\\mathcal{L}^\\bullet \\ar[r] \\ar@{..>}[u] &", "\\mathcal{E}^\\bullet[1] \\ar[u]", "}", "$$", "By the axioms of triangulated categories we obtain a dotted", "arrow fitting into the diagram.", "Looking at cohomology sheaves in degree $m - 1$ we see that we obtain", "$$", "\\xymatrix{", "H^{m - 1}(M) \\ar[r] &", "H^{m - 1}(\\mathcal{L}^\\bullet) \\ar[r] &", "H^m(\\mathcal{E}^\\bullet) \\\\", "H^{m - 1}(K) \\ar[r] \\ar[u] &", "H^{m - 1}(\\sigma_{\\leq m - 1}\\mathcal{K}^\\bullet) \\ar[r] \\ar[u] &", "H^m(\\mathcal{E}^\\bullet) \\ar[u] \\\\", "H^{m - 1}(M) \\ar[r] \\ar[u] &", "H^{m - 1}(\\mathcal{L}^\\bullet) \\ar[r] \\ar[u] &", "H^m(\\mathcal{E}^\\bullet) \\ar[u]", "}", "$$", "Since the vertical compositions are the identity in both the", "left and right column, we conclude the vertical composition", "$H^{m - 1}(\\mathcal{L}^\\bullet) \\to", "H^{m - 1}(\\sigma_{\\leq m - 1}\\mathcal{K}^\\bullet) \\to", "H^{m - 1}(\\mathcal{L}^\\bullet)$ in the middle is surjective!", "In particular $H^{m - 1}(\\sigma_{\\leq m - 1}\\mathcal{K}^\\bullet) \\to", "H^{m - 1}(\\mathcal{L}^\\bullet)$ is surjective.", "Using the induced map of long exact sequences of cohomology", "sheaves from the morphism of triangles above, a diagram chase", "shows this implies $H^i(K) \\to H^i(M)$ is an isomorphism", "for $i \\geq m$ and surjective for $i = m - 1$.", "By construction we can choose an $r \\geq 0$ and a surjection", "$\\mathcal{O}_X^{\\oplus r} \\to \\mathcal{K}^{m - 1}$. Then the", "composition", "$$", "(\\mathcal{O}_X^{\\oplus r} \\to \\mathcal{E}^m \\to", "\\mathcal{E}^{m + 1} \\to \\ldots ) \\longrightarrow", "K \\longrightarrow M", "$$", "induces an isomorphism on cohomology sheaves in degrees $\\geq m$ and", "a surjection in degree $m - 1$ and the proof is complete." ], "refs": [ "cohomology-lemma-K-flat-resolution", "modules-lemma-module-quotient-flat", "derived-lemma-subcategory-left-resolution", "cohomology-lemma-perfect-precise", "derived-lemma-negative-exts", "cohomology-lemma-local-actual" ], "ref_ids": [ 2112, 13276, 1835, 2223, 1893, 2202 ] } ], "ref_ids": [ 12407 ] }, { "id": 2236, "type": "theorem", "label": "cohomology-lemma-colim-and-lim-of-duals", "categories": [ "cohomology" ], "title": "cohomology-lemma-colim-and-lim-of-duals", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let", "$(K_n)_{n \\in \\mathbf{N}}$ be a system of perfect objects of $D(\\mathcal{O}_X)$.", "Let $K = \\text{hocolim} K_n$ be the derived colimit", "(Derived Categories, Definition \\ref{derived-definition-derived-colimit}).", "Then for any object $E$ of $D(\\mathcal{O}_X)$ we have", "$$", "R\\SheafHom(K, E) = R\\lim E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} K_n^\\vee", "$$", "where $(K_n^\\vee)$ is the inverse system of dual perfect complexes." ], "refs": [ "derived-definition-derived-colimit" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-dual-perfect-complex} we have", "$R\\lim E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} K_n^\\vee =", "R\\lim R\\SheafHom(K_n, E)$", "which fits into the distinguished triangle", "$$", "R\\lim R\\SheafHom(K_n, E) \\to", "\\prod R\\SheafHom(K_n, E) \\to", "\\prod R\\SheafHom(K_n, E)", "$$", "Because $K$ similarly fits into the distinguished triangle", "$\\bigoplus K_n \\to \\bigoplus K_n \\to K$ it suffices to show that", "$\\prod R\\SheafHom(K_n, E) = R\\SheafHom(\\bigoplus K_n, E)$.", "This is a formal consequence of (\\ref{equation-internal-hom})", "and the fact that derived tensor product commutes with direct sums." ], "refs": [ "cohomology-lemma-dual-perfect-complex" ], "ref_ids": [ 2233 ] } ], "ref_ids": [ 2001 ] }, { "id": 2237, "type": "theorem", "label": "cohomology-lemma-ext-composition-is-cup", "categories": [ "cohomology" ], "title": "cohomology-lemma-ext-composition-is-cup", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K$ and $E$ be objects", "of $D(\\mathcal{O}_X)$ with $E$ perfect. The diagram", "$$", "\\xymatrix{", "H^0(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E^\\vee) \\times H^0(X, E)", "\\ar[r] \\ar[d] &", "H^0(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E^\\vee", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) \\ar[d] \\\\", "\\Hom_X(E, K) \\times H^0(X, E) \\ar[r] &", "H^0(X, K)", "}", "$$", "commutes where the top horizontal arrow is the cup product, the", "right vertical arrow uses", "$\\epsilon : E^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E \\to \\mathcal{O}_X$", "(Example \\ref{example-dual-derived}), the left vertical arrow uses", "Lemma \\ref{lemma-dual-perfect-complex}, and the bottom horizontal", "arrow is the obvious one." ], "refs": [ "cohomology-lemma-dual-perfect-complex" ], "proofs": [ { "contents": [ "We will abbreviate $\\otimes = \\otimes_{\\mathcal{O}_X}^\\mathbf{L}$", "and $\\mathcal{O} = \\mathcal{O}_X$. We will identify $E$ and $K$", "with $R\\SheafHom(\\mathcal{O}, E)$ and $R\\SheafHom(\\mathcal{O}, K)$", "and we will identify $E^\\vee$ with $R\\SheafHom(E, \\mathcal{O})$.", "\\medskip\\noindent", "Let $\\xi \\in H^0(X, K \\otimes E^\\vee)$ and $\\eta \\in H^0(X, E)$.", "Denote $\\tilde \\xi : \\mathcal{O} \\to K \\otimes E^\\vee$ and", "$\\tilde \\eta : \\mathcal{O} \\to E$ the corresponding maps in", "$D(\\mathcal{O})$. By Lemma \\ref{lemma-second-cup-equals-first}", "the cup product $\\xi \\cup \\eta$ corresponds to", "$\\tilde \\xi \\otimes \\tilde \\eta : \\mathcal{O} \\to", "K \\otimes E^\\vee \\otimes E$.", "\\medskip\\noindent", "We claim the map $\\xi' : E \\to K$ corresponding to $\\xi$ by", "Lemma \\ref{lemma-dual-perfect-complex} is the composition", "$$", "E = \\mathcal{O} \\otimes E", "\\xrightarrow{\\tilde \\xi \\otimes 1_E}", "K \\otimes E^\\vee \\otimes E", "\\xrightarrow{1_K \\otimes \\epsilon}", "K", "$$", "The construction in Lemma \\ref{lemma-dual-perfect-complex}", "uses the evaluation map (\\ref{equation-eval}) which in turn", "is constructed using the identification of $E$ with", "$R\\SheafHom(\\mathcal{O}, E)$ and the composition", "$\\underline{\\circ}$ constructed", "in Lemma \\ref{lemma-internal-hom-composition}.", "Hence $\\xi'$ is the composition", "\\begin{align*}", "E = \\mathcal{O} \\otimes", "R\\SheafHom(\\mathcal{O}, E)", "& \\xrightarrow{\\tilde \\xi \\otimes 1}", "R\\SheafHom(\\mathcal{O}, K) \\otimes", "R\\SheafHom(E, \\mathcal{O}) \\otimes", "R\\SheafHom(\\mathcal{O}, E) \\\\", "& \\xrightarrow{\\underline{\\circ} \\otimes 1}", "R\\SheafHom(E, K) \\otimes R\\SheafHom(\\mathcal{O}, E) \\\\", "& \\xrightarrow{\\underline{\\circ}}", "R\\SheafHom(\\mathcal{O}, K) = K", "\\end{align*}", "The claim follows immediately from this and the fact that", "the composition $\\underline{\\circ}$ constructed in", "Lemma \\ref{lemma-internal-hom-composition} is associative", "(insert future reference here) and the fact that $\\epsilon$", "is defined as the composition", "$\\underline{\\circ} : E^\\vee \\otimes E \\to \\mathcal{O}$ in", "Example \\ref{example-dual-derived}.", "\\medskip\\noindent", "Using the results from the previous two paragraphs, we find", "the statement of the lemma is that", "$(1_K \\otimes \\epsilon) \\circ (\\tilde \\xi \\otimes \\tilde \\eta)$", "is equal to", "$(1_K \\otimes \\epsilon) \\circ (\\tilde \\xi \\otimes 1_E)", "\\circ (1_\\mathcal{O} \\otimes \\tilde \\eta)$", "which is immediate." ], "refs": [ "cohomology-lemma-second-cup-equals-first", "cohomology-lemma-dual-perfect-complex", "cohomology-lemma-dual-perfect-complex", "cohomology-lemma-internal-hom-composition", "cohomology-lemma-internal-hom-composition" ], "ref_ids": [ 2128, 2233, 2233, 2186, 2186 ] } ], "ref_ids": [ 2233 ] }, { "id": 2238, "type": "theorem", "label": "cohomology-lemma-category-summands-finite-free", "categories": [ "cohomology" ], "title": "cohomology-lemma-category-summands-finite-free", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Set $R = \\Gamma(X, \\mathcal{O}_X)$. The category of", "$\\mathcal{O}_X$-modules which are summands of finite free", "$\\mathcal{O}_X$-modules is equivalent to the category of", "finite projective $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "Observe that a finite projective $R$-module is the same thing", "as a summand of a finite free $R$-module.", "The equivalence is given by the functor $\\mathcal{E} \\mapsto", "\\Gamma(X, \\mathcal{E})$. The inverse functor is given by the construction of", "Modules, Lemma \\ref{modules-lemma-construct-quasi-coherent-sheaves}." ], "refs": [ "modules-lemma-construct-quasi-coherent-sheaves" ], "ref_ids": [ 13245 ] } ], "ref_ids": [] }, { "id": 2239, "type": "theorem", "label": "cohomology-lemma-invertible-derived", "categories": [ "cohomology" ], "title": "cohomology-lemma-invertible-derived", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $M$ be an object", "of $D(\\mathcal{O}_X)$. The following are equivalent", "\\begin{enumerate}", "\\item $M$ is invertible in $D(\\mathcal{O}_X)$, see", "Categories, Definition \\ref{categories-definition-invertible}, and", "\\item there is a locally finite direct product decomposition", "$$", "\\mathcal{O}_X = \\prod\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_n", "$$", "and for each $n$ there is an invertible $\\mathcal{O}_n$-module", "$\\mathcal{H}^n$ (Modules, Definition \\ref{modules-definition-invertible})", "and $M = \\bigoplus \\mathcal{H}^n[-n]$ in $D(\\mathcal{O}_X)$.", "\\end{enumerate}", "If (1) and (2) hold, then $M$ is a perfect object of $D(\\mathcal{O}_X)$. If", "$\\mathcal{O}_{X, x}$ is a local ring for all $x \\in X$ these condition", "are also equivalent to", "\\begin{enumerate}", "\\item[(3)] there exists an open covering $X = \\bigcup U_i$", "and for each $i$ an integer $n_i$ such that $M|_{U_i}$", "is represented by an invertible $\\mathcal{O}_{U_i}$-module", "placed in degree $n_i$.", "\\end{enumerate}" ], "refs": [ "categories-definition-invertible", "modules-definition-invertible" ], "proofs": [ { "contents": [ "Assume (2). Consider the object $R\\SheafHom(M, \\mathcal{O}_X)$", "and the composition map", "$$", "R\\SheafHom(M, \\mathcal{O}_X) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M \\to", "\\mathcal{O}_X", "$$", "To prove this is an isomorphism, we may work locally. Thus we may", "assume $\\mathcal{O}_X = \\prod_{a \\leq n \\leq b} \\mathcal{O}_n$", "and $M = \\bigoplus_{a \\leq n \\leq b} \\mathcal{H}^n[-n]$.", "Then it suffices to show that", "$$", "R\\SheafHom(\\mathcal{H}^m, \\mathcal{O}_X)", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{H}^n", "$$", "is zero if $n \\not = m$ and equal to $\\mathcal{O}_n$ if $n = m$.", "The case $n \\not = m$ follows from the fact that $\\mathcal{O}_n$ and", "$\\mathcal{O}_m$ are flat $\\mathcal{O}_X$-algebras with", "$\\mathcal{O}_n \\otimes_{\\mathcal{O}_X} \\mathcal{O}_m = 0$.", "Using the local structure of invertible $\\mathcal{O}_X$-modules", "(Modules, Lemma \\ref{modules-lemma-invertible}) and working locally", "the isomorphism in case $n = m$ follows in a straightforward manner;", "we omit the details. Because $D(\\mathcal{O}_X)$ is symmetric monoidal,", "we conclude that $M$ is invertible.", "\\medskip\\noindent", "Assume (1). The description in (2) shows that we have a candidate", "for $\\mathcal{O}_n$, namely,", "$\\SheafHom_{\\mathcal{O}_X}(H^n(M), H^n(M))$.", "If this is a locally finite family of sheaves of rings", "and if $\\mathcal{O}_X = \\prod \\mathcal{O}_n$, then we immediately", "obtain the direct sum decomposition $M = \\bigoplus H^n(M)[-n]$", "using the idempotents in $\\mathcal{O}_X$ coming from the product", "decomposition.", "This shows that in order to prove (2) we may work locally on $X$.", "\\medskip\\noindent", "Choose an object $N$ of $D(\\mathcal{O}_X)$", "and an isomorphism", "$M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} N \\cong \\mathcal{O}_X$.", "Let $x \\in X$.", "Then $N$ is a left dual for $M$ in the monoidal category", "$D(\\mathcal{O}_X)$ and we conclude that $M$ is perfect by", "Lemma \\ref{lemma-left-dual-derived}. By symmetry we see that", "$N$ is perfect. After replacing $X$ by an open neighbourhood of $x$,", "we may assume $M$ and $N$ are represented by a strictly perfect", "complexes $\\mathcal{E}^\\bullet$ and $\\mathcal{F}^\\bullet$.", "Then $M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} N$ is represented by", "$\\text{Tot}(\\mathcal{E}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{F}^\\bullet)$.", "After another shinking of $X$ we may assume the mutually inverse", "isomorphisms", "$\\mathcal{O}_X \\to M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} N$ and", "$M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} N \\to \\mathcal{O}_X$", "are given by maps of complexes", "$$", "\\alpha : \\mathcal{O}_X \\to", "\\text{Tot}(\\mathcal{E}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{F}^\\bullet)", "\\quad\\text{and}\\quad", "\\beta :", "\\text{Tot}(\\mathcal{E}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{F}^\\bullet)", "\\to \\mathcal{O}_X", "$$", "See Lemma \\ref{lemma-local-actual}. Then $\\beta \\circ \\alpha = 1$", "as maps of complexes and $\\alpha \\circ \\beta = 1$ as a morphism", "in $D(\\mathcal{O}_X)$. After shrinking $X$", "we may assume the composition $\\alpha \\circ \\beta$ is homotopic to $1$", "by some homotopy $\\theta$ with components", "$$", "\\theta^n :", "\\text{Tot}^n(\\mathcal{E}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{F}^\\bullet)", "\\to", "\\text{Tot}^{n - 1}(", "\\mathcal{E}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{F}^\\bullet)", "$$", "by the same lemma as before. Set $R = \\Gamma(X, \\mathcal{O}_X)$. By", "Lemma \\ref{lemma-category-summands-finite-free}", "we find that we obtain", "\\begin{enumerate}", "\\item $M^\\bullet = \\Gamma(X, \\mathcal{E}^\\bullet)$ is a bounded complex", "of finite projective $R$-modules,", "\\item $N^\\bullet = \\Gamma(X, \\mathcal{F}^\\bullet)$ is a bounded complex", "of finite projective $R$-modules,", "\\item $\\alpha$ and $\\beta$ correspond to maps of complexes", "$a : R \\to \\text{Tot}(M^\\bullet \\otimes_R N^\\bullet)$ and", "$b : \\text{Tot}(M^\\bullet \\otimes_R N^\\bullet) \\to R$,", "\\item $\\theta^n$ corresponds to a map", "$h^n : \\text{Tot}^n(M^\\bullet \\otimes_R N^\\bullet) \\to", "\\text{Tot}^{n - 1}(M^\\bullet \\otimes_R N^\\bullet)$, and", "\\item $b \\circ a = 1$ and $b \\circ a - 1 = dh + hd$,", "\\end{enumerate}", "It follows that $M^\\bullet$ and $N^\\bullet$ define", "mutually inverse objects of $D(R)$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-invertible-derived}", "we find a product decomposition $R = \\prod_{a \\leq n \\leq b} R_n$", "and invertible $R_n$-modules $H^n$ such", "that $M^\\bullet \\cong \\bigoplus_{a \\leq n \\leq b} H^n[-n]$.", "This isomorphism in $D(R)$ can be lifted to an morphism", "$$", "\\bigoplus H^n[-n] \\longrightarrow M^\\bullet", "$$", "of complexes because each $H^n$ is projective as an $R$-module.", "Correspondingly, using Lemma \\ref{lemma-category-summands-finite-free} again,", "we obtain an morphism", "$$", "\\bigoplus H^n \\otimes_R \\mathcal{O}_X[-n] \\to \\mathcal{E}^\\bullet", "$$", "which is an isomorphism in $D(\\mathcal{O}_X)$. Setting", "$\\mathcal{O}_n = R_n \\otimes_R \\mathcal{O}_X$ we conclude (2) is true.", "\\medskip\\noindent", "If all stalks of $\\mathcal{O}_X$ are local, then it is straightforward", "to prove the equivalence of (2) and (3). We omit the details." ], "refs": [ "modules-lemma-invertible", "cohomology-lemma-left-dual-derived", "cohomology-lemma-local-actual", "cohomology-lemma-category-summands-finite-free", "more-algebra-lemma-invertible-derived", "cohomology-lemma-category-summands-finite-free" ], "ref_ids": [ 13300, 2235, 2202, 2238, 10575, 2238 ] } ], "ref_ids": [ 12406, 13349 ] }, { "id": 2240, "type": "theorem", "label": "cohomology-lemma-when-jshriek-compact", "categories": [ "cohomology" ], "title": "cohomology-lemma-when-jshriek-compact", "contents": [ "Let $X$ be a ringed space. Let $j : U \\to X$ be the", "inclusion of an open. The $\\mathcal{O}_X$-module $j_!\\mathcal{O}_U$ is a", "compact object of $D(\\mathcal{O}_X)$ if there exists an integer $d$ such that", "\\begin{enumerate}", "\\item $H^p(U, \\mathcal{F}) = 0$ for all $p > d$, and", "\\item the functors $\\mathcal{F} \\mapsto H^p(U, \\mathcal{F})$", "commute with direct sums.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and (2). Since", "$\\Hom(j_!\\mathcal{O}_U, \\mathcal{F}) = \\mathcal{F}(U)$", "by Sheaves, Lemma \\ref{sheaves-lemma-j-shriek-modules}", "we have $\\Hom(j_!\\mathcal{O}_U, K) = R\\Gamma(U, K)$ for", "$K$ in $D(\\mathcal{O}_X)$. Thus we have to show that $R\\Gamma(U, -)$", "commutes with direct sums. The first assumption means that the functor", "$F = H^0(U, -)$ has finite cohomological dimension. Moreover, the second", "assumption implies any direct sum of injective modules is acyclic for $F$.", "Let $K_i$ be a family of objects of $D(\\mathcal{O}_X)$.", "Choose K-injective representatives $I_i^\\bullet$ with injective terms", "representing $K_i$, see Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "Since we may compute $RF$ by applying $F$ to any complex of acyclics", "(Derived Categories, Lemma \\ref{derived-lemma-unbounded-right-derived})", "and since $\\bigoplus K_i$ is represented by $\\bigoplus I_i^\\bullet$", "(Injectives, Lemma \\ref{injectives-lemma-derived-products})", "we conclude that $R\\Gamma(U, \\bigoplus K_i)$ is represented by", "$\\bigoplus H^0(U, I_i^\\bullet)$. Hence $R\\Gamma(U, -)$ commutes", "with direct sums as desired." ], "refs": [ "sheaves-lemma-j-shriek-modules", "injectives-theorem-K-injective-embedding-grothendieck", "derived-lemma-unbounded-right-derived", "injectives-lemma-derived-products" ], "ref_ids": [ 14546, 7768, 1917, 7795 ] } ], "ref_ids": [] }, { "id": 2241, "type": "theorem", "label": "cohomology-lemma-perfect-is-compact", "categories": [ "cohomology" ], "title": "cohomology-lemma-perfect-is-compact", "contents": [ "Let $X$ be a ringed space. Assume that the underlying topological space", "of $X$ has the following properties:", "\\begin{enumerate}", "\\item $X$ is quasi-compact,", "\\item there exists a basis of quasi-compact open subsets, and", "\\item the intersection of any two quasi-compact opens is quasi-compact.", "\\end{enumerate}", "Let $K$ be a perfect object of $D(\\mathcal{O}_X)$. Then", "\\begin{enumerate}", "\\item[(a)] $K$ is a compact object of $D^+(\\mathcal{O}_X)$", "in the following sense: if $M = \\bigoplus_{i \\in I} M_i$ is", "bounded below, then $\\Hom(K, M) = \\bigoplus_{i \\in I} \\Hom(K, M_i)$.", "\\item[(b)] If $X$ has finite cohomological dimension, i.e., if there exists", "a $d$ such that $H^i(X, \\mathcal{F}) = 0$ for $i > d$, then", "$K$ is a compact object of $D(\\mathcal{O}_X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $K^\\vee$ be the dual of $K$, see", "Lemma \\ref{lemma-dual-perfect-complex}. Then we have", "$$", "\\Hom_{D(\\mathcal{O}_X)}(K, M) =", "H^0(X, K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)", "$$", "functorially in $M$ in $D(\\mathcal{O}_X)$.", "Since $K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} -$ commutes with", "direct sums it suffices", "to show that $R\\Gamma(X, -)$ commutes with the relevant direct sums.", "\\medskip\\noindent", "Proof of (b). Since $R\\Gamma(X, K) = R\\Hom(\\mathcal{O}_X, K)$", "and since $H^p(X, -)$ commutes with direct sums by", "Lemma \\ref{lemma-quasi-separated-cohomology-colimit}", "this is a special case of", "Lemma \\ref{lemma-when-jshriek-compact}", "\\medskip\\noindent", "Proof of (a). Let $\\mathcal{I}_i$, $i \\in I$ be a collection of injective", "$\\mathcal{O}_X$-modules. By Lemma \\ref{lemma-quasi-separated-cohomology-colimit}", "we see that", "$$", "H^p(X, \\bigoplus\\nolimits_{i \\in I} \\mathcal{I}_i) =", "\\bigoplus\\nolimits_{i \\in I} H^p(X, \\mathcal{I}_i) = 0", "$$", "for all $p$. Now if $M = \\bigoplus M_i$ is as in (a), then we", "see that there exists an $a \\in \\mathbf{Z}$ such that $H^n(M_i) = 0$", "for $n < a$. Thus we can choose complexes of injective $\\mathcal{O}_X$-modules", "$\\mathcal{I}_i^\\bullet$ representing $M_i$", "with $\\mathcal{I}_i^n = 0$ for $n < a$, see", "Derived Categories, Lemma \\ref{derived-lemma-injective-resolutions-exist}.", "By Injectives, Lemma \\ref{injectives-lemma-derived-products}", "we see that the direct sum complex $\\bigoplus \\mathcal{I}_i^\\bullet$", "represents $M$. By Leray acyclicity", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "we see that", "$$", "R\\Gamma(X, M) = \\Gamma(X, \\bigoplus \\mathcal{I}_i^\\bullet) =", "\\bigoplus \\Gamma(X, \\bigoplus \\mathcal{I}_i^\\bullet) =", "\\bigoplus R\\Gamma(X, M_i)", "$$", "as desired." ], "refs": [ "cohomology-lemma-dual-perfect-complex", "cohomology-lemma-quasi-separated-cohomology-colimit", "cohomology-lemma-when-jshriek-compact", "cohomology-lemma-quasi-separated-cohomology-colimit", "derived-lemma-injective-resolutions-exist", "injectives-lemma-derived-products", "derived-lemma-leray-acyclicity" ], "ref_ids": [ 2233, 2082, 2240, 2082, 1851, 7795, 1844 ] } ], "ref_ids": [] }, { "id": 2242, "type": "theorem", "label": "cohomology-lemma-injective-tensor-finite-locally-free", "categories": [ "cohomology" ], "title": "cohomology-lemma-injective-tensor-finite-locally-free", "contents": [ "Let $X$ be a ringed space.", "Let $\\mathcal{I}$ be an injective $\\mathcal{O}_X$-module.", "Let $\\mathcal{E}$ be an $\\mathcal{O}_X$-module.", "Assume $\\mathcal{E}$ is finite locally free on $X$, see", "Modules, Definition \\ref{modules-definition-locally-free}.", "Then $\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{I}$ is", "an injective $\\mathcal{O}_X$-module." ], "refs": [ "modules-definition-locally-free" ], "proofs": [ { "contents": [ "This is true because under the assumptions of the lemma we have", "$$", "\\Hom_{\\mathcal{O}_X}(\\mathcal{F},", "\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{I})", "=", "\\Hom_{\\mathcal{O}_X}(", "\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{E}^\\vee, \\mathcal{I})", "$$", "where", "$\\mathcal{E}^\\vee = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}, \\mathcal{O}_X)$", "is the dual of $\\mathcal{E}$ which is finite locally free also. Since tensoring", "with a finite locally free sheaf is an exact functor we win by", "Homology, Lemma \\ref{homology-lemma-characterize-injectives}." ], "refs": [ "homology-lemma-characterize-injectives" ], "ref_ids": [ 12112 ] } ], "ref_ids": [ 13342 ] }, { "id": 2243, "type": "theorem", "label": "cohomology-lemma-projection-formula", "categories": [ "cohomology" ], "title": "cohomology-lemma-projection-formula", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "Let $\\mathcal{E}$ be an $\\mathcal{O}_Y$-module.", "Assume $\\mathcal{E}$ is finite locally free on $Y$, see", "Modules, Definition \\ref{modules-definition-locally-free}.", "Then there exist isomorphisms", "$$", "\\mathcal{E} \\otimes_{\\mathcal{O}_Y} R^qf_*\\mathcal{F}", "\\longrightarrow", "R^qf_*(f^*\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{F})", "$$", "for all $q \\geq 0$. In fact there exists an isomorphism", "$$", "\\mathcal{E} \\otimes_{\\mathcal{O}_Y} Rf_*\\mathcal{F}", "\\longrightarrow", "Rf_*(f^*\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{F})", "$$", "in $D^{+}(Y)$ functorial in $\\mathcal{F}$." ], "refs": [ "modules-definition-locally-free" ], "proofs": [ { "contents": [ "Choose an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$", "on $X$. Note that $f^*\\mathcal{E}$ is finite locally free also, hence", "we get a resolution", "$$", "f^*\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "\\longrightarrow", "f^*\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{I}^\\bullet", "$$", "which is an injective resolution by", "Lemma \\ref{lemma-injective-tensor-finite-locally-free}.", "Apply $f_*$ to see that", "$$", "Rf_*(f^*\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{F})", "=", "f_*(f^*\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{I}^\\bullet).", "$$", "Hence the lemma follows if we can show that", "$f_*(f^*\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{F}) =", "\\mathcal{E} \\otimes_{\\mathcal{O}_Y} f_*(\\mathcal{F})$ functorially", "in the $\\mathcal{O}_X$-module $\\mathcal{F}$. This is clear when", "$\\mathcal{E} = \\mathcal{O}_Y^{\\oplus n}$, and follows in general", "by working locally on $Y$. Details omitted." ], "refs": [ "cohomology-lemma-injective-tensor-finite-locally-free" ], "ref_ids": [ 2242 ] } ], "ref_ids": [ 13342 ] }, { "id": 2244, "type": "theorem", "label": "cohomology-lemma-projection-formula-perfect", "categories": [ "cohomology" ], "title": "cohomology-lemma-projection-formula-perfect", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Let $E \\in D(\\mathcal{O}_X)$ and $K \\in D(\\mathcal{O}_Y)$.", "If $K$ is perfect, then", "$$", "Rf_*E \\otimes^\\mathbf{L}_{\\mathcal{O}_Y} K =", "Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} Lf^*K)", "$$", "in $D(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "To check (\\ref{equation-projection-formula-map}) is an isomorphism", "we may work locally on $Y$, i.e., we have to find a covering $\\{V_j \\to Y\\}$", "such that the map restricts to an isomorphism on $V_j$. By definition", "of perfect objects, this means we may assume $K$ is represented by", "a strictly perfect complex of $\\mathcal{O}_Y$-modules.", "Note that, completely generally, the statement is true for", "$K = K_1 \\oplus K_2$, if and only if the statement is true for", "$K_1$ and $K_2$. Hence we may assume $K$ is a finite", "complex of finite free $\\mathcal{O}_Y$-modules.", "In this case a simple argument involving stupid truncations reduces", "the statement to the case where $K$ is represented by a finite", "free $\\mathcal{O}_Y$-module. Since the statement is invariant", "under finite direct summands in the $K$ variable, we conclude", "it suffices to prove it for $K = \\mathcal{O}_Y[n]$", "in which case it is trivial." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2245, "type": "theorem", "label": "cohomology-lemma-projection-formula-closed-immersion", "categories": [ "cohomology" ], "title": "cohomology-lemma-projection-formula-closed-immersion", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces such that $f$ is a", "homeomorphism onto a closed subset. Then", "(\\ref{equation-projection-formula-map}) is an isomorphism always." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is a homeomorphism onto a closed subset, the functor $f_*$", "is exact (Modules, Lemma \\ref{modules-lemma-i-star-exact}). Hence", "$Rf_*$ is computed by applying $f_*$ to any representative complex.", "Choose a K-flat complex $\\mathcal{K}^\\bullet$ of $\\mathcal{O}_Y$-modules", "representing $K$ and choose any complex $\\mathcal{E}^\\bullet$", "of $\\mathcal{O}_X$-modules representing $E$. Then", "$Lf^*K$ is represented by $f^*\\mathcal{K}^\\bullet$ which is", "a K-flat complex of $\\mathcal{O}_X$-modules", "(Lemma \\ref{lemma-pullback-K-flat}). Thus the right hand side of", "(\\ref{equation-projection-formula-map}) is represented by", "$$", "f_*\\text{Tot}(\\mathcal{E}^\\bullet", "\\otimes_{\\mathcal{O}_X} f^*\\mathcal{K}^\\bullet)", "$$", "By the same reasoning we see that the left hand side is represented by", "$$", "\\text{Tot}(f_*\\mathcal{E}^\\bullet \\otimes_{\\mathcal{O}_Y} \\mathcal{K}^\\bullet)", "$$", "Since $f_*$ commutes with direct sums", "(Modules, Lemma \\ref{modules-lemma-i-star-right-adjoint})", "it suffices to show that", "$$", "f_*(\\mathcal{E} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{K}) =", "f_*\\mathcal{E} \\otimes_{\\mathcal{O}_Y} \\mathcal{K}", "$$", "for any $\\mathcal{O}_X$-module $\\mathcal{E}$ and $\\mathcal{O}_Y$-module", "$\\mathcal{K}$. We will check this by checking on stalks.", "Let $y \\in Y$. If $y \\not \\in f(X)$, then the stalks", "of both sides are zero. If $y = f(x)$, then we see that we have to show", "$$", "\\mathcal{E}_x \\otimes_{\\mathcal{O}_{X, x}}", "(\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{Y, y}} \\mathcal{F}_y) =", "\\mathcal{E}_x \\otimes_{\\mathcal{O}_{Y, y}} \\mathcal{F}_y", "$$", "(using Sheaves, Lemma \\ref{sheaves-lemma-stalks-closed-pushforward}", "and Lemma \\ref{sheaves-lemma-stalk-pullback-modules}).", "This equality holds and therefore the lemma has been proved." ], "refs": [ "modules-lemma-i-star-exact", "cohomology-lemma-pullback-K-flat", "modules-lemma-i-star-right-adjoint", "sheaves-lemma-stalks-closed-pushforward", "sheaves-lemma-stalk-pullback-modules" ], "ref_ids": [ 13232, 2108, 13233, 14551, 14523 ] } ], "ref_ids": [] }, { "id": 2246, "type": "theorem", "label": "cohomology-proposition-vanishing-Noetherian", "categories": [ "cohomology" ], "title": "cohomology-proposition-vanishing-Noetherian", "contents": [ "\\begin{reference}", "\\cite[Theorem 3.6.5]{Tohoku}.", "\\end{reference}", "Let $X$ be a Noetherian topological space.", "If $\\dim(X) \\leq d$, then $H^p(X, \\mathcal{F}) = 0$", "for all $p > d$ and any abelian sheaf $\\mathcal{F}$", "on $X$." ], "refs": [], "proofs": [ { "contents": [ "We prove this lemma by induction on $d$.", "So fix $d$ and assume the lemma holds for all", "Noetherian topological spaces of dimension $< d$.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be an abelian sheaf on $X$.", "Suppose $U \\subset X$ is an open. Let $Z \\subset X$", "denote the closed complement.", "Denote $j : U \\to X$ and $i : Z \\to X$ the inclusion maps.", "Then there is a short exact sequence", "$$", "0 \\to j_{!}j^*\\mathcal{F} \\to \\mathcal{F} \\to i_*i^*\\mathcal{F} \\to 0", "$$", "see Modules, Lemma \\ref{modules-lemma-canonical-exact-sequence}.", "Note that $j_!j^*\\mathcal{F}$ is supported on", "the topological closure $Z'$ of $U$, i.e., it is of", "the form $i'_*\\mathcal{F}'$ for some abelian sheaf $\\mathcal{F}'$", "on $Z'$, where $i' : Z' \\to X$ is the inclusion.", "\\medskip\\noindent", "We can use this to reduce to the case where $X$ is irreducible.", "Namely, according to", "Topology, Lemma \\ref{topology-lemma-Noetherian}", "$X$ has finitely", "many irreducible components. If $X$ has more than one irreducible", "component, then let $Z \\subset X$ be an irreducible component of $X$", "and set $U = X \\setminus Z$. By the above, and the long exact sequence", "of cohomology, it suffices to prove the vanishing of", "$H^p(X, i_*i^*\\mathcal{F})$ and $H^p(X, i'_*\\mathcal{F}')$ for $p > d$.", "By Lemma \\ref{lemma-cohomology-and-closed-immersions} it suffices to prove", "$H^p(Z, i^*\\mathcal{F})$ and $H^p(Z', \\mathcal{F}')$ vanish for $p > d$.", "Since $Z'$ and $Z$ have fewer irreducible components we indeed", "reduce to the case of an irreducible $X$.", "\\medskip\\noindent", "If $d = 0$ and $X = \\{*\\}$, then every sheaf is constant and", "higher cohomology", "groups vanish (for example by", "Lemma \\ref{lemma-irreducible-constant-cohomology-zero}).", "\\medskip\\noindent", "Suppose $X$ is irreducible of dimension $d$.", "By Lemma \\ref{lemma-vanishing-generated-one-section}", "we reduce to the case where", "$\\mathcal{F} = j_!\\underline{\\mathbf{Z}}_U$ for some open $U \\subset X$.", "In this case we look at the short exact sequence", "$$", "0 \\to j_!(\\underline{\\mathbf{Z}}_U) \\to", "\\underline{\\mathbf{Z}}_X \\to i_*\\underline{\\mathbf{Z}}_Z \\to 0", "$$", "where $Z = X \\setminus U$.", "By Lemma \\ref{lemma-irreducible-constant-cohomology-zero}", "we have the vanishing of $H^p(X, \\underline{\\mathbf{Z}}_X)$", "for all $p \\geq 1$. By induction we have", "$H^p(X, i_*\\underline{\\mathbf{Z}}_Z) = H^p(Z, \\underline{\\mathbf{Z}}_Z) = 0$", "for $p \\geq d$. Hence we win by the long exact cohomology sequence." ], "refs": [ "modules-lemma-canonical-exact-sequence", "topology-lemma-Noetherian", "cohomology-lemma-cohomology-and-closed-immersions", "cohomology-lemma-irreducible-constant-cohomology-zero", "cohomology-lemma-irreducible-constant-cohomology-zero" ], "ref_ids": [ 13234, 8220, 2084, 2085, 2085 ] } ], "ref_ids": [] }, { "id": 2247, "type": "theorem", "label": "cohomology-proposition-cohomological-dimension-spectral", "categories": [ "cohomology" ], "title": "cohomology-proposition-cohomological-dimension-spectral", "contents": [ "\\begin{reference}", "Part (1) is the main theorem of \\cite{Scheiderer}.", "\\end{reference}", "Let $X$ be a spectral space of Krull dimension $d$.", "Let $\\mathcal{F}$ be an abelian sheaf on $X$.", "\\begin{enumerate}", "\\item $H^q(X, \\mathcal{F}) = 0$ for $q > d$,", "\\item $H^d(X, \\mathcal{F}) \\to H^d(U, \\mathcal{F})$ is surjective", "for every quasi-compact open $U \\subset X$,", "\\item $H^q_Z(X, \\mathcal{F}) = 0$ for $q > d$ and any constructible", "closed subset $Z \\subset X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We prove this result by induction on $d$.", "\\medskip\\noindent", "If $d = 0$, then $X$ is a profinite space, see", "Topology, Lemma \\ref{topology-lemma-characterize-profinite-spectral}.", "Thus (1) holds by Lemma \\ref{lemma-vanishing-for-profinite}.", "If $U \\subset X$ is quasi-compact open, then $U$ is", "also closed as a quasi-compact subset of a Hausdorff space.", "Hence $X = U \\amalg (X \\setminus U)$ as a topological space", "and we see that (2) holds. Given $Z$ as in (3) we consider the", "long exact sequence", "$$", "H^{q - 1}(X, \\mathcal{F}) \\to", "H^{q - 1}(X \\setminus Z, \\mathcal{F}) \\to", "H^q_Z(X, \\mathcal{F}) \\to H^q(X, \\mathcal{F})", "$$", "Since $X$ and $U = X \\setminus Z$ are profinite (namely $U$ is quasi-compact", "because $Z$ is constructible) and since", "we have (2) and (1) we obtain the desired vanishing of the", "cohomology groups with support in $Z$.", "\\medskip\\noindent", "Induction step. Assume $d \\geq 1$ and assume", "the proposition is valid for all spectral", "spaces of dimension $< d$. We first prove part (2) for $X$.", "Let $U$ be a quasi-compact open. Let $\\xi \\in H^d(U, \\mathcal{F})$.", "Set $Z = X \\setminus U$. Let $W \\subset X$ be the set of points", "specializing to $Z$. By", "Lemma \\ref{lemma-cohomology-of-neighbourhoods-of-closed} we have", "$$", "H^d(W \\setminus Z, \\mathcal{F}|_{W \\setminus Z}) =", "\\colim_{Z \\subset V} H^d(V \\setminus Z, \\mathcal{F})", "$$", "where the colimit is over the quasi-compact open neighbourhoods $V$", "of $Z$ in $X$.", "By Topology, Lemma \\ref{topology-lemma-make-spectral-space} we see that", "$W \\setminus Z$ is a spectral space.", "Since every point of $W$ specializes to a point of $Z$, we see that", "$W \\setminus Z$ is a spectral space of Krull dimension $< d$.", "By induction hypothesis we see that the image of $\\xi$ in", "$H^d(W \\setminus Z, \\mathcal{F}|_{W \\setminus Z})$ is zero.", "By the displayed formula, there exists a $Z \\subset V \\subset X$", "quasi-compact open such that $\\xi|_{V \\setminus Z} = 0$.", "Since $V \\setminus Z = V \\cap U$ we conclude by the Mayer-Vietoris", "(Lemma \\ref{lemma-mayer-vietoris}) for the covering $X = U \\cup V$", "that there exists a $\\tilde \\xi \\in H^d(X, \\mathcal{F})$ which restricts", "to $\\xi$ on $U$ and to zero on $V$. In other words, part (2) is true.", "\\medskip\\noindent", "Proof of part (1) assuming (2). Choose an injective resolution", "$\\mathcal{F} \\to \\mathcal{I}^\\bullet$. Set", "$$", "\\mathcal{G} = \\Im(\\mathcal{I}^{d - 1} \\to \\mathcal{I}^d) =", "\\Ker(\\mathcal{I}^d \\to \\mathcal{I}^{d + 1})", "$$", "For $U \\subset X$ quasi-compact open we have a map of exact sequences", "as follows", "$$", "\\xymatrix{", "\\mathcal{I}^{d - 1}(X) \\ar[r] \\ar[d] &", "\\mathcal{G}(X) \\ar[r] \\ar[d] &", "H^d(X, \\mathcal{F}) \\ar[d] \\ar[r] & 0 \\\\", "\\mathcal{I}^{d - 1}(U) \\ar[r] &", "\\mathcal{G}(U) \\ar[r] &", "H^d(U, \\mathcal{F}) \\ar[r] & 0", "}", "$$", "The sheaf $\\mathcal{I}^{d - 1}$ is flasque by", "Lemma \\ref{lemma-injective-flasque} and the fact that $d \\geq 1$.", "By part (2) we see that the right vertical arrow is surjective.", "We conclude by a diagram chase that the map", "$\\mathcal{G}(X) \\to \\mathcal{G}(U)$ is surjective.", "By Lemma \\ref{lemma-vanishing-ravi} we conclude that", "$\\check{H}^q(\\mathcal{U}, \\mathcal{G}) = 0$ for $q > 0$ and", "any finite covering $\\mathcal{U} : U = U_1 \\cup \\ldots \\cup U_n$", "of a quasi-compact open by quasi-compact opens. Applying", "Lemma \\ref{lemma-cech-vanish-basis} we find that $H^q(U, \\mathcal{G}) = 0$", "for all $q > 0$ and all quasi-compact opens $U$ of $X$.", "By Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "we conclude that", "$$", "H^q(X, \\mathcal{F}) =", "H^q\\left(", "\\Gamma(X, \\mathcal{I}^0) \\to \\ldots \\to", "\\Gamma(X, \\mathcal{I}^{d - 1}) \\to \\Gamma(X, \\mathcal{G})", "\\right)", "$$", "In particular the cohomology group vanishes if $q > d$.", "\\medskip\\noindent", "Proof of (3). Given $Z$ as in (3) we consider the long exact sequence", "$$", "H^{q - 1}(X, \\mathcal{F}) \\to", "H^{q - 1}(X \\setminus Z, \\mathcal{F}) \\to", "H^q_Z(X, \\mathcal{F}) \\to H^q(X, \\mathcal{F})", "$$", "Since $X$ and $U = X \\setminus Z$ are spectral spaces", "(Topology, Lemma \\ref{topology-lemma-spectral-sub})", "of dimension $\\leq d$", "and since we have (2) and (1) we obtain the desired vanishing." ], "refs": [ "topology-lemma-characterize-profinite-spectral", "cohomology-lemma-vanishing-for-profinite", "cohomology-lemma-cohomology-of-neighbourhoods-of-closed", "topology-lemma-make-spectral-space", "cohomology-lemma-mayer-vietoris", "cohomology-lemma-injective-flasque", "cohomology-lemma-vanishing-ravi", "cohomology-lemma-cech-vanish-basis", "derived-lemma-leray-acyclicity", "topology-lemma-spectral-sub" ], "ref_ids": [ 8309, 2092, 2090, 8323, 2042, 2063, 2067, 2059, 1844, 8306 ] } ], "ref_ids": [] }, { "id": 2286, "type": "theorem", "label": "stacks-introduction-lemma-key-fact", "categories": [ "stacks-introduction" ], "title": "stacks-introduction-lemma-key-fact", "contents": [ "The functor $\\Sch^{opp} \\to \\textit{Sets}$,", "$T \\mapsto \\{(a, a', \\alpha)\\text{ as above}\\}$", "is representable by a scheme $S \\times_{\\mathcal{M}_{1, 1}} S'$." ], "refs": [], "proofs": [ { "contents": [ "Idea of proof. Relate this functor to", "$$", "\\mathit{Isom}_{S \\times S'}(E \\times S', S \\times E')", "$$", "and use Grothendieck's theory of Hilbert schemes." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2287, "type": "theorem", "label": "stacks-introduction-lemma-Weierstrass-smooth-cover", "categories": [ "stacks-introduction" ], "title": "stacks-introduction-lemma-Weierstrass-smooth-cover", "contents": [ "The morphism $W \\xrightarrow{(E_W, f_W, 0_W)} \\mathcal{M}_{1, 1}$ is smooth", "and surjective." ], "refs": [], "proofs": [ { "contents": [ "Surjectivity follows from the fact that every elliptic curve over a", "field has a Weierstrass equation. We give a rough sketch of one", "way to prove smoothness. Consider the sub group scheme", "$$", "H =", "\\left\\{", "\\left(", "\\begin{matrix}", "u^2 & s & 0 \\\\", "0 & u^3 & 0 \\\\", "r & t & 1", "\\end{matrix}", "\\right)", "\\middle|", "\\begin{matrix}", "u\\text{ unit} \\\\", "s, r, t\\text{ arbitrary}", "\\end{matrix}", "\\right\\}", "\\subset", "\\text{GL}_{3, \\mathbf{Z}}", "$$", "There is an action $H \\times W \\to W$ of $H$ on the Weierstrass scheme $W$.", "To find the equations for this action write out what a coordinate change", "given by a matrix in $H$ does to the general Weierstrass equation.", "Then it turns out the following statements hold", "\\begin{enumerate}", "\\item any elliptic curve $(E, f, 0)/S$ has Zariski locally on $S$", "a Weierstrass equation,", "\\item any two Weierstrass equations for $(E, f, 0)$ differ (Zariski locally)", "by an element of $H$.", "\\end{enumerate}", "Considering the fibre product", "$S \\times_{\\mathcal{M}_{1, 1}} W =", "\\mathit{Isom}_{S \\times W}(E \\times W, S \\times E_W)$", "we conclude that this means that the morphism", "$W \\to \\mathcal{M}_{1, 1}$ is an $H$-torsor.", "Since $H \\to \\Spec(\\mathbf{Z})$ is smooth, and since torsors", "over smooth group schemes are smooth we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2292, "type": "theorem", "label": "restricted-theorem-dilatations-general", "categories": [ "restricted" ], "title": "restricted-theorem-dilatations-general", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.", "Let $T \\subset |X|$ be a closed subset. Let $U \\subset X$ be the open subspace", "with $|U| = |X| \\setminus T$. The completion functor", "(\\ref{equation-completion-functor})", "$$", "\\left\\{", "\\begin{matrix}", "\\text{morphisms of algebraic spaces}\\\\", "f : X' \\to X\\text{ which are locally}\\\\", "\\text{of finite type and such that}\\\\", "f^{-1}U \\to U\\text{ is an isomorphism}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{morphisms }g : W \\to X_{/T}\\\\", "\\text{of formal algebraic spaces}\\\\", "\\text{with }W\\text{ locally Noetherian}\\\\", "\\text{and }g\\text{ rig-\\'etale}", "\\end{matrix}", "\\right\\}", "$$", "sending $f : X' \\to X$ to $f_{/T} : X'_{/T'} \\to X_{/T}$ is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "The functor is fully faithful by", "Lemma \\ref{lemma-completion-functor-fully-faithful}.", "Let $g : W \\to X_{/T}$ be a morphism of formal algebraic spaces", "with $W$ locally Noetherian and $g$ rig-\\'etale.", "We will prove $W$ is in the essential image to finish the proof.", "\\medskip\\noindent", "Choose an \\'etale covering $\\{X_i \\to X\\}$ with $X_i$ affine for all $i$.", "Denote $U_i \\subset X_i$ the inverse image of $U$ and denote", "$T_i \\subset X_i$ the inverse image of $T$.", "Recall that $(X_i)_{/T_i} = (X_i)_{/T} = (X_i \\times_X X)_{/T}$ and", "$W_i = X_i \\times_X W = (X_i)_{/T} \\times_{X_{/T}} W$, see", "Lemma \\ref{lemma-functoriality-completion-functor}.", "Observe that we obtain isomorphisms", "$$", "\\alpha_{ij} :", "W_i \\times_{X_{/T}} (X_j)_{/T}", "\\longrightarrow ", "(X_i)_{/T} \\times_{X_{/T}} W_j", "$$", "satisfying a suitable cocycle condition.", "By Lemma \\ref{lemma-dilatations-affine} applied to", "$X_i, T_i, U_i, W_i \\to (X_i)_{/T}$", "there exists a morphism $X'_i \\to X_i$ of algebraic spaces", "which is locally of finite type and an isomorphism over $U_i$", "and an isomorphism $\\beta_i : (X'_i)_{/T} \\cong W_i$ over $(X_i)_{/T}$.", "By fully faithfullness we find an isomorphism", "$$", "a_{ij} : X'_i \\times_X X_j \\longrightarrow X_i \\times_X X'_j", "$$", "over $X_i \\times_X X_j$ such that", "$\\alpha_{ij} = \\beta_j|_{X_i \\times_X X_j}", "\\circ (a_{ij})_{/T} \\circ \\beta_i^{-1}|_{X_i \\times_X X_j}$.", "By fully faithfulness again (this time over", "$X_i \\times_X X_j \\times_X X_k$)", "we see that these morphisms $a_{ij}$ satisfy the same", "cocycle condition as satisfied by the $\\alpha_{ij}$.", "In other words, we obtain a descent datum", "(as in Descent on Spaces, Definition", "\\ref{spaces-descent-definition-descent-datum-for-family-of-morphisms})", "$(X'_i, a_{ij})$ relative to the family $\\{X_i \\to X\\}$.", "By Bootstrap, Lemma \\ref{bootstrap-lemma-descend-algebraic-space},", "this descent datum is effective. Thus we find a morphism", "$f : X' \\to X$ of algebraic spaces and isomorphisms", "$h_i : X' \\times_X X_i \\to X'_i$ over $X_i$ such that", "$a_{ij} = h_j|_{X_i \\times_X X_j} \\circ h_i^{-1}|_{X_i \\times_X X_j}$.", "The reader can check that the ensuing isomorphisms", "$$", "(X' \\times_X X_i)_{/T}", "\\xrightarrow{\\beta_i \\circ (h_i)_{/T}}", "W_i", "$$", "over $X_i$ glue to an isomorphism $X'_{/T} \\to W$", "over $X_{/T}$; some details omitted." ], "refs": [ "restricted-lemma-completion-functor-fully-faithful", "restricted-lemma-functoriality-completion-functor", "restricted-lemma-dilatations-affine", "spaces-descent-definition-descent-datum-for-family-of-morphisms", "bootstrap-lemma-descend-algebraic-space" ], "ref_ids": [ 2420, 2419, 2421, 9445, 2627 ] } ], "ref_ids": [] }, { "id": 2293, "type": "theorem", "label": "restricted-theorem-dilatations", "categories": [ "restricted" ], "title": "restricted-theorem-dilatations", "contents": [ "\\begin{reference}", "\\cite[Theorem 3.2]{ArtinII}", "\\end{reference}", "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.", "Let $T \\subset |X|$ be a closed subset. Let", "$\\mathfrak X = X_{/T}$", "be the formal completion of $X$ along $T$. Let", "$$", "\\mathfrak f : \\mathfrak X' \\to \\mathfrak X", "$$", "be a formal modification (Definition \\ref{definition-formal-modification}).", "Then there exists a unique proper morphism $f : X' \\to X$ which is an", "isomorphism over the complement of $T$ in $X$ whose completion $f_{/T}$", "recovers $\\mathfrak f$." ], "refs": [ "restricted-definition-formal-modification" ], "proofs": [ { "contents": [ "This follows from Theorem \\ref{theorem-dilatations-general}", "and Lemma \\ref{lemma-output-proper}." ], "refs": [ "restricted-theorem-dilatations-general", "restricted-lemma-output-proper" ], "ref_ids": [ 2292, 2425 ] } ], "ref_ids": [ 2447 ] }, { "id": 2294, "type": "theorem", "label": "restricted-lemma-topologically-finite-type", "categories": [ "restricted" ], "title": "restricted-lemma-topologically-finite-type", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "The functor", "$$", "\\mathcal{C} \\longrightarrow \\mathcal{C}',\\quad", "(B_n) \\longmapsto B = \\lim B_n", "$$", "is a quasi-inverse to (\\ref{equation-from-complete-to-systems}).", "The completions $A[x_1, \\ldots, x_r]^\\wedge$ are in $\\mathcal{C}'$ and", "any object of $\\mathcal{C}'$ is of the form", "$$", "B = A[x_1, \\ldots, x_r]^\\wedge / J", "$$", "for some ideal $J \\subset A[x_1, \\ldots, x_r]^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "Let $(B_n)$ be an object of $\\mathcal{C}$. By", "Algebra, Lemma \\ref{algebra-lemma-limit-complete}", "we see that $B = \\lim B_n$ is $I$-adically complete", "and $B/I^nB = B_n$. Hence we see that $B$ is an object of", "$\\mathcal{C}'$ and that we can recover the object $(B_n)$ ", "by taking the quotients.", "Conversely, if $B$ is an object of $\\mathcal{C}'$, then", "$B = \\lim B/I^nB$ by assumption. Thus $B \\mapsto (B/I^nB)$ is a quasi-inverse", "to the functor of the lemma.", "\\medskip\\noindent", "Since $A[x_1, \\ldots, x_r]^\\wedge = \\lim A_n[x_1, \\ldots, x_r]$", "it is an object of $\\mathcal{C}'$ by the first statement of the lemma.", "Finally, let $B$ be an object of $\\mathcal{C}'$. Choose", "$b_1, \\ldots, b_r \\in B$ whose images in $B/IB$ generate", "$B/IB$ as an algebra over $A/I$. Since $B$ is $I$-adically", "complete, the $A$-algebra map $A[x_1, \\ldots, x_r] \\to B$, $x_i \\mapsto b_i$", "extends to an $A$-algebra map $A[x_1, \\ldots, x_r]^\\wedge \\to B$.", "To finish the proof we have to show this map is surjective", "which follows from Algebra, Lemma \\ref{algebra-lemma-completion-generalities}", "as our map $A[x_1, \\ldots, x_r] \\to B$ is surjective modulo $I$", "and as $B = B^\\wedge$." ], "refs": [ "algebra-lemma-limit-complete", "algebra-lemma-completion-generalities" ], "ref_ids": [ 880, 858 ] } ], "ref_ids": [] }, { "id": 2295, "type": "theorem", "label": "restricted-lemma-topologically-finite-type-Noetherian", "categories": [ "restricted" ], "title": "restricted-lemma-topologically-finite-type-Noetherian", "contents": [ "\\begin{reference}", "\\cite[Proposition 7.5.5]{EGA1}", "\\end{reference}", "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. Then", "\\begin{enumerate}", "\\item every object of the category $\\mathcal{C}'$", "(\\ref{equation-C-prime}) is Noetherian,", "\\item if $B \\in \\Ob(\\mathcal{C}')$ and $J \\subset B$ is an ideal,", "then $B/J$ is an object of $\\mathcal{C}'$,", "\\item for a finite type $A$-algebra $C$ the $I$-adic completion", "$C^\\wedge$ is in $\\mathcal{C}'$,", "\\item in particular the completion $A[x_1, \\ldots, x_r]^\\wedge$", "is in $\\mathcal{C}'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (4) follows from", "Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian-Noetherian}", "as $A[x_1, \\ldots, x_r]$ is Noetherian", "(Algebra, Lemma \\ref{algebra-lemma-Noetherian-permanence}).", "To see (1) by Lemma \\ref{lemma-topologically-finite-type}", "we reduce to the case of the completion of the polynomial ring", "which we just proved.", "Part (2) follows from Algebra, Lemma \\ref{algebra-lemma-completion-tensor}", "which tells us that ever finite $B$-module is $IB$-adically complete.", "Part (3) follows in the same manner as part (4)." ], "refs": [ "algebra-lemma-completion-Noetherian-Noetherian", "algebra-lemma-Noetherian-permanence", "restricted-lemma-topologically-finite-type", "algebra-lemma-completion-tensor" ], "ref_ids": [ 874, 448, 2294, 869 ] } ], "ref_ids": [] }, { "id": 2296, "type": "theorem", "label": "restricted-lemma-NL-up-to-homotopy", "categories": [ "restricted" ], "title": "restricted-lemma-NL-up-to-homotopy", "contents": [ "Let $A$ be a Noetherian ring and let $I \\subset A$ be a ideal.", "Let $B$ be an object of (\\ref{equation-C-prime}). The naive", "cotangent complex $\\NL_{B/A}^\\wedge$ is well defined in $K(B)$." ], "refs": [], "proofs": [ { "contents": [ "The lemma signifies that given a second presentation", "$B = A[y_1, \\ldots, y_s]^\\wedge / K$ the complexes of $B$-modules", "$$", "(J/J^2 \\to B\\text{d}x_i)", "\\quad\\text{and}\\quad", "(K/K^2 \\to \\bigoplus B\\text{d}y_j)", "$$", "are homotopy equivalent. To see this, we can argue exactly as in", "the proof of Algebra, Lemma \\ref{algebra-lemma-NL-homotopy}.", "\\medskip\\noindent", "Step 1. If we choose $g_i(y_1, \\ldots, y_s) \\in A[y_1, \\ldots, y_s]^\\wedge$", "mapping to the image of $x_i$ in $B$, then we obtain a (unique) continuous", "$A$-algebra homomorphism", "$$", "A[x_1, \\ldots, x_r]^\\wedge \\to A[y_1, \\ldots, y_s]^\\wedge,\\quad", "x_i \\mapsto g_i(y_1, \\ldots, y_s)", "$$", "compatible with the given surjections to $B$. Such a map is called", "a morphism of presentations. It induces a map from $J$ into $K$", "and hence induces a $B$-module map $J/J^2 \\to K/K^2$. Sending", "$\\text{d}x_i$ to $\\sum (\\partial g_i/\\partial y_j)\\text{d}y_j$", "we obtain a map of complexes", "$$", "(J/J^2 \\to \\bigoplus B\\text{d}x_i)", "\\longrightarrow", "(K/K^2 \\to \\bigoplus B\\text{d}y_j)", "$$", "Of course we can do the same thing with the roles of the two presentations", "exchanged to get a map of complexes in the other direction.", "\\medskip\\noindent", "Step 2. The construction above is compatible with compositions of morphsms", "of presentations. Hence to finish the proof it suffices to show: given", "$g_i(x_1, \\ldots, x_r) \\in A[x_1, \\ldots, x_n]^\\wedge$", "mapping to the image of $x_i$ in $B$, the induced map of complexes", "$$", "(J/J^2 \\to \\bigoplus B\\text{d}x_i)", "\\longrightarrow", "(J/J^2 \\to \\bigoplus B\\text{d}x_i)", "$$", "is homotopic to the identity map. To see this consider the map", "$h : \\bigoplus B \\text{d}x_i \\to J/J^2$ given by the rule", "$\\text{d}x_i \\mapsto g_i(x_1, \\ldots, x_n) - x_i$ and compute." ], "refs": [ "algebra-lemma-NL-homotopy" ], "ref_ids": [ 1151 ] } ], "ref_ids": [] }, { "id": 2297, "type": "theorem", "label": "restricted-lemma-NL-is-completion", "categories": [ "restricted" ], "title": "restricted-lemma-NL-is-completion", "contents": [ "Let $A$ be a Noetherian ring and let $I \\subset A$ be a ideal.", "Let $A \\to B$ be a finite type ring map. Choose a presentation", "$\\alpha : A[x_1, \\ldots, x_n] \\to B$. Then", "$\\NL_{B^\\wedge/A}^\\wedge = \\lim \\NL(\\alpha) \\otimes_B B^\\wedge$", "as complexes and", "$\\NL_{B^\\wedge/A}^\\wedge = \\NL_{B/A} \\otimes_B^\\mathbf{L} B^\\wedge$", "in $D(B^\\wedge)$." ], "refs": [], "proofs": [ { "contents": [ "The statement makes sense as $B^\\wedge$ is an object of", "(\\ref{equation-C-prime}) by", "Lemma \\ref{lemma-topologically-finite-type-Noetherian}.", "Let $J = \\Ker(\\alpha)$. The functor of taking", "$I$-adic completion is exact on finite modules over", "$A[x_1, \\ldots, x_n]$ and agrees with the functor", "$M \\mapsto M \\otimes_{A[x_1, \\ldots, x_n]} A[x_1, \\ldots, x_n]^\\wedge$, see", "Algebra, Lemmas \\ref{algebra-lemma-completion-tensor} and", "\\ref{algebra-lemma-completion-flat}. Moreover, the ring maps", "$A[x_1, \\ldots, x_n] \\to A[x_1, \\ldots, x_n]^\\wedge$", "and $B \\to B^\\wedge$ are flat. Hence", "$B^\\wedge = A[x_1, \\ldots, x_n]^\\wedge / J^\\wedge$ and", "$$", "(J/J^2) \\otimes_B B^\\wedge = (J/J^2)^\\wedge = J^\\wedge/(J^\\wedge)^2", "$$", "Since $\\NL(\\alpha) = (J/J^2 \\to \\bigoplus B\\text{d}x_i)$,", "see Algebra, Section \\ref{algebra-section-netherlander},", "we conclude the complex $\\NL_{B^\\wedge/A}^\\wedge$ is equal", "to $\\NL(\\alpha) \\otimes_B B^\\wedge$. The final statement", "follows as $\\NL_{B/A}$ is homotopy equivalent to $\\NL(\\alpha)$", "and because the ring map $B \\to B^\\wedge$ is flat (so derived", "base change along $B \\to B^\\wedge$ is just base change)." ], "refs": [ "restricted-lemma-topologically-finite-type-Noetherian", "algebra-lemma-completion-tensor", "algebra-lemma-completion-flat" ], "ref_ids": [ 2295, 869, 870 ] } ], "ref_ids": [] }, { "id": 2298, "type": "theorem", "label": "restricted-lemma-NL-is-limit", "categories": [ "restricted" ], "title": "restricted-lemma-NL-is-limit", "contents": [ "Let $A$ be a Noetherian ring and let $I \\subset A$ be a ideal.", "Let $B$ be an object of (\\ref{equation-C-prime}). Then", "\\begin{enumerate}", "\\item the pro-objects", "$\\{\\NL_{B/A}^\\wedge \\otimes_B B/I^nB\\}$ and $\\{\\NL_{B_n/A_n}\\}$", "of $D(B)$ are strictly isomorphic (see proof for elucidation),", "\\item $\\NL_{B/A}^\\wedge = R\\lim \\NL_{B_n/A_n}$ in $D(B)$.", "\\end{enumerate}", "Here $B_n$ and $A_n$ are as in Section \\ref{section-two-categories}." ], "refs": [], "proofs": [ { "contents": [ "The statement means the following: for every $n$ we have a well", "defined complex $\\NL_{B_n/A_n}$ of $B_n$-modules and we have", "transition maps $\\NL_{B_{n + 1}/A_{n + 1}} \\to \\NL_{B_n/A_n}$.", "See Algebra, Section \\ref{algebra-section-netherlander}.", "Thus we can consider", "$$", "\\ldots \\to \\NL_{B_3/A_3} \\to \\NL_{B_2/A_2} \\to \\NL_{B_1/A_1}", "$$", "as an inverse system of complexes of $B$-modules and a fortiori as", "an inverse system in $D(B)$. Furthermore $R\\lim \\NL_{B_n/A_n}$", "is a homotopy limit of this inverse system, see", "Derived Categories, Section \\ref{derived-section-derived-limit}.", "\\medskip\\noindent", "Choose a presentation $B = A[x_1, \\ldots, x_r]^\\wedge / J$. This", "defines presentations", "$$", "B_n = B/I^nB = A_n[x_1, \\ldots, x_r]/J_n", "$$", "where", "$$", "J_n = JA_n[x_1, \\ldots, x_r] =", "J/(J \\cap I^nA[x_1, \\ldots, x_r]^\\wedge)", "$$", "The two term complex $J_n/J_n^2 \\longrightarrow \\bigoplus B_n \\text{d}x_i$", "represents $\\NL_{B_n/A_n}$, see", "Algebra, Section \\ref{algebra-section-netherlander}.", "By Artin-Rees (Algebra, Lemma \\ref{algebra-lemma-Artin-Rees})", "in the Noetherian ring $A[x_1, \\ldots, x_r]^\\wedge$", "(Lemma \\ref{lemma-topologically-finite-type-Noetherian})", "we find a $c \\geq 0$ such that we have canonical surjections", "$$", "J/I^nJ \\to J_n \\to J/I^{n - c}J \\to J_{n - c},\\quad n \\geq c", "$$", "for all $n \\geq c$. A moment's thought shows that these maps are", "compatible with differentials and we obtain maps of complexes", "$$", "\\NL_{B/A}^\\wedge \\otimes_B B/I^nB \\to", "\\NL_{B_n/A_n} \\to", "\\NL_{B/A}^\\wedge \\otimes_B B/I^{n - c}B \\to", "\\NL_{B_{n - c}/A_{n - c}}", "$$", "compatible with the transition maps of the inverse systems", "$\\{\\NL_{B/A}^\\wedge \\otimes_B B/I^nB\\}$ and $\\{\\NL_{B_n/A_n}\\}$.", "This proves part (1) of the lemma.", "\\medskip\\noindent", "By part (1) and since pro-isomorphic", "systems have the same $R\\lim$ in order to prove (2)", "it suffices to show that $\\NL_{B/A}^\\wedge$ is", "equal to $R\\lim \\NL_{B/A}^\\wedge \\otimes_B B/I^nB$.", "However, $\\NL_{B/A}^\\wedge$ is a two term complex $M^\\bullet$", "of finite $B$-modules which are $I$-adically complete for example by", "Algebra, Lemma \\ref{algebra-lemma-completion-tensor}. Hence", "$M^\\bullet = \\lim M^\\bullet/I^nM^\\bullet = R\\lim M^\\bullet/I^n M^\\bullet$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-compute-Rlim-modules} and", "Remark \\ref{more-algebra-remark-how-unique}." ], "refs": [ "algebra-lemma-Artin-Rees", "restricted-lemma-topologically-finite-type-Noetherian", "algebra-lemma-completion-tensor", "more-algebra-lemma-compute-Rlim-modules", "more-algebra-remark-how-unique" ], "ref_ids": [ 625, 2295, 869, 10324, 10660 ] } ], "ref_ids": [] }, { "id": 2299, "type": "theorem", "label": "restricted-lemma-NL-base-change", "categories": [ "restricted" ], "title": "restricted-lemma-NL-base-change", "contents": [ "Let $(A_1, I_1) \\to (A_2, I_2)$ be as in", "Remark \\ref{remark-base-change} with $A_1$ and $A_2$ Noetherian.", "Let $B_1$ be in (\\ref{equation-C-prime}) for $(A_1, I_1)$.", "Let $B_2$ be the base change of $B_1$. Then there is a canonical map", "$$", "\\NL_{B_1/A_1} \\otimes_{B_2} B_1 \\to \\NL_{B_2/A_2}", "$$", "which induces and isomorphism on $H^0$ and a surjection on $H^{-1}$." ], "refs": [ "restricted-remark-base-change" ], "proofs": [ { "contents": [ "Choose a presentation $B_1 = A_1[x_1, \\ldots, x_r]^\\wedge/J_1$.", "Since", "$A_2/I_2^n[x_1, \\ldots, x_r] =", "A_1/I_1^{cn}[x_1, \\ldots, x_r] \\otimes_{A_1/I_1^{cn}} A_2/I_2^n$", "we have", "$$", "A_2[x_1, \\ldots, x_r]^\\wedge =", "(A_1[x_1, \\ldots, x_r]^\\wedge \\otimes_{A_1} A_2)^\\wedge", "$$", "where we use $I_2$-adic completion on both sides (but of course", "$I_1$-adic completion for $A_1[x_1, \\ldots, x_r]^\\wedge$).", "Set $J_2 = J_1 A_2[x_1, \\ldots, x_r]^\\wedge$. Arguing similarly", "we get the presentation", "\\begin{align*}", "B_2", "& =", "(B_1 \\otimes_{A_1} A_2)^\\wedge \\\\", "& =", "\\lim \\frac{A_1/I_1^{cn}[x_1, \\ldots, x_r]}{J_1(A_1/I_1^{cn}[x_1, \\ldots, x_r])}", "\\otimes_{A_1/I_1^{cn}} A_2/I_2^n \\\\", "& =", "\\lim \\frac{A_2/I_2^n[x_1, \\ldots, x_r]}{J_2(A_2/I_2^n[x_1, \\ldots, x_r])} \\\\", "& =", "A_2[x_1, \\ldots, x_r]^\\wedge/J_2", "\\end{align*}", "for $B_2$ over $A_2$. As a consequence obtain a commutative diagram", "$$", "\\xymatrix{", "\\NL^\\wedge_{B_1/A_1} : \\ar[d] &", "J_1/J_1^2 \\ar[r]_-{\\text{d}} \\ar[d] & \\bigoplus B_1\\text{d}x_i \\ar[d] \\\\", "\\NL^\\wedge_{B_2/A_2} : &", "J_2/J_2^2 \\ar[r]^-{\\text{d}} & \\bigoplus B_2\\text{d}x_i", "}", "$$", "The induced arrow $J_1/J_1^2 \\otimes_{B_1} B_2 \\to J_2/J_2^2$", "is surjective because $J_2$ is generated by the image of $J_1$.", "This determines the arrow displayed in the lemma. We omit the proof", "that this arrow is well defined up to homotopy (i.e., indepedent", "of the choice of the presentations up to homotopy). The statement", "about the induced map on cohomology modules follows easily", "from the discussion (details omitted)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 2448 ] }, { "id": 2300, "type": "theorem", "label": "restricted-lemma-exact-sequence-NL", "categories": [ "restricted" ], "title": "restricted-lemma-exact-sequence-NL", "contents": [ "Let $A$ be a Noetherian ring and let $I \\subset A$ be a ideal.", "Let $B \\to C$ be morphism of (\\ref{equation-C-prime}). Then", "there is an exact sequence", "$$", "\\xymatrix{", "C \\otimes_B H^0(\\NL_{B/A}^\\wedge) \\ar[r] &", "H^0(\\NL_{C/A}^\\wedge) \\ar[r] &", "H^0(\\NL_{C/B}^\\wedge) \\ar[r] & 0 \\\\", "H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C) \\ar[r] &", "H^{-1}(\\NL_{C/A}^\\wedge) \\ar[r] &", "H^{-1}(\\NL_{C/B}^\\wedge) \\ar[llu]", "}", "$$", "See proof for elucidation." ], "refs": [], "proofs": [ { "contents": [ "Observe that taking the tensor product $\\NL_{B/A}^\\wedge \\otimes_B C$", "makes sense as $\\NL_{B/A}^\\wedge$ is well defined up to homotopy by", "Lemma \\ref{lemma-NL-up-to-homotopy}.", "Also, $(B, IB)$ is pair where $B$ is a Noetherian ring", "(Lemma \\ref{lemma-topologically-finite-type-Noetherian})", "and $C$ is in the corresponding category (\\ref{equation-C-prime}).", "Thus all the terms in the $6$-term sequence are (well) defined.", "\\medskip\\noindent", "Choose a presentation $B = A[x_1, \\ldots, x_r]^\\wedge/J$.", "Choose a presentation $C = B[y_1, \\ldots, y_s]^\\wedge/J'$.", "Combinging these presentations gives a presentation", "$$", "C = A[x_1, \\ldots, x_r, y_1, \\ldots, y_s]^\\wedge/K", "$$", "Then the reader verifies that we obtain a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\bigoplus C \\text{d}x_i \\ar[r] &", "\\bigoplus C \\text{d}x_i \\oplus \\bigoplus C \\text{d}y_j \\ar[r] &", "\\bigoplus C \\text{d}y_j \\ar[r] &", "0 \\\\", "&", "J/J^2 \\otimes_B C \\ar[r] \\ar[u] &", "K/K^2 \\ar[r] \\ar[u] &", "J'/(J')^2 \\ar[r] \\ar[u] &", "0", "}", "$$", "with exact rows. Note that the vertical arrow on the left hand side", "is the tensor product of the arrow defining $\\NL_{B/A}^\\wedge$ with", "$\\text{id}_C$. The lemma follows by applying the snake lemma", "(Algebra, Lemma \\ref{algebra-lemma-snake})." ], "refs": [ "restricted-lemma-NL-up-to-homotopy", "restricted-lemma-topologically-finite-type-Noetherian", "algebra-lemma-snake" ], "ref_ids": [ 2296, 2295, 328 ] } ], "ref_ids": [] }, { "id": 2301, "type": "theorem", "label": "restricted-lemma-transitive-lci-at-end", "categories": [ "restricted" ], "title": "restricted-lemma-transitive-lci-at-end", "contents": [ "With assumptions as in Lemma \\ref{lemma-exact-sequence-NL}", "assume that $B/I^nB \\to C/I^nC$ is a local complete intersection", "homomorphism for all $n$. Then", "$H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C) \\to H^{-1}(\\NL_{C/A}^\\wedge)$", "is injective." ], "refs": [ "restricted-lemma-exact-sequence-NL" ], "proofs": [ { "contents": [ "For each $n \\geq 1$ we set $A_n = A/I^n$, $B_n = B/I^nB$, and", "$C_n = C/I^nC$. We have", "\\begin{align*}", "H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C)", "& =", "\\lim H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C_n) \\\\", "& =", "\\lim H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B B_n \\otimes_{B_n} C_n) \\\\", "& =", "\\lim H^{-1}(\\NL_{B_n/A_n} \\otimes_{B_n} C_n)", "\\end{align*}", "The first equality follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-consequence-Artin-Rees}", "and the fact that $H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C)$ is a finite", "$C$-module and hence $I$-adically complete for example by", "Algebra, Lemma \\ref{algebra-lemma-completion-tensor}.", "The second equality is trivial.", "The third holds by Lemma \\ref{lemma-NL-is-limit}.", "The maps $H^{-1}(\\NL_{B_n/A_n} \\otimes_{B_n} C_n) \\to", "H^{-1}(\\NL_{C_n/A_n})$ are injective by", "More on Algebra, Lemma \\ref{more-algebra-lemma-transitive-lci-at-end}.", "The proof is finished because we also have", "$H^{-1}(\\NL_{C/A}^\\wedge) = \\lim H^{-1}(\\NL_{C_n/A_n})$", "similarly to the above." ], "refs": [ "more-algebra-lemma-consequence-Artin-Rees", "algebra-lemma-completion-tensor", "restricted-lemma-NL-is-limit", "more-algebra-lemma-transitive-lci-at-end" ], "ref_ids": [ 10419, 869, 2298, 10003 ] } ], "ref_ids": [ 2300 ] }, { "id": 2302, "type": "theorem", "label": "restricted-lemma-equivalent-with-artin-smooth", "categories": [ "restricted" ], "title": "restricted-lemma-equivalent-with-artin-smooth", "contents": [ "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.", "Let $B$ be an object of (\\ref{equation-C-prime}). Write", "$B = A[x_1, \\ldots, x_r]^\\wedge/J$", "(Lemma \\ref{lemma-topologically-finite-type-Noetherian})", "and let $\\NL_{B/A}^\\wedge = (J/J^2 \\to \\bigoplus B\\text{d}x_i)$", "be its naive cotangent complex (\\ref{equation-NL}).", "The following are equivalent", "\\begin{enumerate}", "\\item $B$ is rig-smooth over $(A, I)$,", "\\item the object $\\NL_{B/A}^\\wedge$ of $D(B)$ satisfies the equivalent", "conditions (1) -- (4) of More on Algebra, Lemma", "\\ref{more-algebra-lemma-ext-1-annihilated} with respect to the ideal $IB$,", "\\item there exists a $c \\geq 0$ such that for all $a \\in I^c$", "there is a map $h : \\bigoplus B\\text{d}x_i \\to J/J^2$ such that", "$a : J/J^2 \\to J/J^2$ is equal to $h \\circ \\text{d}$,", "\\item there exist $b_1, \\ldots, b_s \\in B$ such that", "$V(b_1, \\ldots, b_s) \\subset V(IB)$ and such that for every", "$l = 1, \\ldots, s$ there exist $m \\geq 0$, $f_1, \\ldots, f_m \\in J$,", "and subset $T \\subset \\{1, \\ldots, n\\}$ with $|T| = m$ such that", "\\begin{enumerate}", "\\item $\\det_{i \\in T, j \\leq m}(\\partial f_j/ \\partial x_i)$", "divides $b_l$ in $B$, and", "\\item $b_l J \\subset (f_1, \\ldots, f_m) + J^2$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [ "restricted-lemma-topologically-finite-type-Noetherian", "more-algebra-lemma-ext-1-annihilated" ], "proofs": [ { "contents": [ "The equivalence of (1), (2), and (3) is immediate from", "More on Algebra, Lemma \\ref{more-algebra-lemma-ext-1-annihilated}.", "\\medskip\\noindent", "Assume $b_1, \\ldots, b_s$ are as in (4). Since $B$ is Noetherian the inclusion", "$V(b_1, \\ldots, b_s) \\subset V(IB)$ implies $I^cB \\subset (b_1, \\ldots, b_s)$", "for some $c \\geq 0$ (for example by", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-power-ideal-kills-module}).", "Pick $1 \\leq l \\leq s$ and $m \\geq 0$ and $f_1, \\ldots, f_m \\in J$", "and $T \\subset \\{1, \\ldots, n\\}$ with $|T| = m$ satisfying (4)(a) and (b).", "Then if we invert $b_l$ we see that", "$$", "\\NL_{B/A}^\\wedge \\otimes_B B_{b_l} =", "\\left(", "\\bigoplus\\nolimits_{j \\leq m} B_{b_l} f_j", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} B_{b_l} \\text{d}x_i", "\\right)", "$$", "and moreover the arrow is isomorphic to the inclusion of the direct", "summand $\\bigoplus_{i \\in T} B_{b_l} \\text{d}x_i$. We conclude that", "$H^{-1}(\\NL_{B/A}^\\wedge)$ is $b_l$-power torsion and that", "$H^0(\\NL_{B/A}^\\wedge)$ becomes finite free after inverting $b_l$.", "Combined with the inclusion $I^cB \\subset (b_1, \\ldots, b_s)$", "we see that $H^{-1}(\\NL_{B/A}^\\wedge)$ is $IB$-power torsion.", "Hence we see that condition (4) of", "More on Algebra, Lemma \\ref{more-algebra-lemma-ext-1-annihilated}", "holds. In this way we see that (4) implies (2).", "\\medskip\\noindent", "Assume the equivalent conditions (1), (2), and (3) hold. We will prove", "that (4) holds, but we strongly urge the reader to convince themselves", "of this. The complex $\\NL_{B/A}^\\wedge$ determines an object of", "$D^b_{\\textit{Coh}}(\\Spec(B))$ whose restriction to the Zariski open", "$U = \\Spec(B) \\setminus V(IB)$ is a finite locally free module", "$\\mathcal{E}$ placed in degree $0$ (this follows for example from the", "the fourth equivalent condition in", "More on Algebra, Lemma \\ref{more-algebra-lemma-ext-1-annihilated}).", "Choose generators $f_1, \\ldots, f_M$ for $J$.", "This determines an exact sequence", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, M} \\mathcal{O}_U \\cdot f_j \\to", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} \\mathcal{O}_U \\cdot \\text{d}x_i \\to", "\\mathcal{E} \\to 0", "$$", "Let $U = \\bigcup_{l = 1, \\ldots, s} U_l$", "be a finite affine open covering such that", "$\\mathcal{E}|_{U_l}$ is free of rank $r_l = n - m_l$ for some integer", "$n \\geq m_l \\geq 0$. After replacing", "each $U_l$ by an affine open covering we may assume there exists", "a subset $T_l \\subset \\{1, \\ldots, n\\}$ such that the elements", "$\\text{d}x_i$, $i \\in \\{1, \\ldots, n\\} \\setminus T_l$ map to a", "basis for $\\mathcal{E}|_{U_l}$. Repeating the argument, we may", "assume there exists a subset $T'_l \\subset \\{1, \\ldots, M\\}$", "of cardinality $m_l$ such that $f_j$, $j \\in T'_l$ map to a basis", "of the kernel of $\\mathcal{O}_{U_l} \\cdot \\text{d}x_i \\to", "\\mathcal{E}|_{U_l}$. Finally, since the open covering", "$U = \\bigcup U_l$ may be refined by a open covering by standard opens", "(Algebra, Lemma \\ref{algebra-lemma-Zariski-topology})", "we may assume $U_l = D(g_l)$ for some $g_l \\in B$.", "In particular we have $V(g_1, \\ldots, g_s) = V(IB)$.", "A linear algebra argument using our choices above shows that", "$\\det_{i \\in T_l, j \\in T'_l}(\\partial f_j/ \\partial x_i)$", "maps to an invertible element of $B_{b_l}$. Similarly, the vanishing", "of cohomology of $\\NL_{B/A}^\\wedge$ in degree $-1$ over $U_l$ shows that", "$J/J^2 + (f_j; j \\in T')$ is annihilated by a power of $b_l$.", "After replacing each $g_l$ by a suitable power we obtain", "conditions (4)(a) and (4)(b) of the lemma. Some details omitted." ], "refs": [ "more-algebra-lemma-ext-1-annihilated", "algebra-lemma-Noetherian-power-ideal-kills-module", "more-algebra-lemma-ext-1-annihilated", "more-algebra-lemma-ext-1-annihilated", "algebra-lemma-Zariski-topology" ], "ref_ids": [ 10305, 694, 10305, 10305, 389 ] } ], "ref_ids": [ 2295, 10305 ] }, { "id": 2303, "type": "theorem", "label": "restricted-lemma-rig-smooth", "categories": [ "restricted" ], "title": "restricted-lemma-rig-smooth", "contents": [ "Let $A$ be a Noetherian ring and let $I$ be an ideal.", "Let $B$ be a finite type $A$-algebra.", "\\begin{enumerate}", "\\item If $\\Spec(B) \\to \\Spec(A)$ is smooth over $\\Spec(A) \\setminus V(I)$,", "then $B^\\wedge$ is rig-smooth over $(A, I)$.", "\\item If $B^\\wedge$ is rig-smooth over $(A, I)$,", "then there exists $g \\in 1 + IB$ such that $\\Spec(B_g)$ is smooth", "over $\\Spec(A) \\setminus V(I)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-equivalent-with-artin-smooth}", "without further mention.", "\\medskip\\noindent", "Assume (1). Recall that formation of $\\NL_{B/A}$ commutes with", "localization, see Algebra, Lemma \\ref{algebra-lemma-localize-NL}.", "Hence by the very definition of smooth ring maps (in terms of", "the naive cotangent complex being quasi-isomorphic to a finite", "projective module placed in degree $0$), we see that", "$\\NL_{B/A}$ satisfies the fourth equivalent condition", "of More on Algebra, Lemma \\ref{more-algebra-lemma-ext-1-annihilated}", "with respect to the ideal $IB$ (small detail omitted).", "Since $\\NL_{B^\\wedge/A}^\\wedge = \\NL_{B/A} \\otimes_B B^\\wedge$", "by Lemma \\ref{lemma-NL-is-completion} we conclude (2) holds by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-base-change-property-ext-1-annihilated}.", "\\medskip\\noindent", "Assume (2). Choose a presentation", "$B = A[x_1, \\ldots, x_n]/J$, set $N = J/J^2$, and", "consider the element $\\xi \\in \\Ext^1_B(\\NL_{B/A}, J/J^2)$", "determined by the identity map on $J/J^2$.", "Using again that $\\NL_{B^\\wedge/A}^\\wedge = \\NL_{B/A} \\otimes_B B^\\wedge$", "we find that our assumption implies the image", "$$", "\\xi \\otimes 1 \\in", "\\Ext^1_{B^\\wedge}(\\NL_{B/A} \\otimes_B B^\\wedge, N \\otimes_B B^\\wedge) =", "\\Ext^1_{B^\\wedge}(\\NL_{B/A}, N) \\otimes_B B^\\wedge", "$$", "is annihilated by $I^c$ for some integer $c \\geq 0$.", "The equality holds for example by", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom}", "(but can also easily be deduced from the much simpler", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-pseudo-coherence-and-base-change-ext}).", "Thus $M = I^cB\\xi \\subset \\Ext^1_B(\\NL_{B/A}, N)$ is a finite submodule", "which maps to zero in $\\Ext^1_B(\\NL_{B/A}, N) \\otimes_B B^\\wedge$.", "Since $B \\to B^\\wedge$ is flat this means that", "$M \\otimes_B B^\\wedge$ is zero. By ", "Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "this means that $M = I^cB\\xi$ is annihilated by an element", "of the form $g = 1 + x$ with $x \\in IB$.", "This implies that for every $b \\in I^cB$ there is a", "$B$-linear dotted arrow making the diagram commute", "$$", "\\xymatrix{", "J/J^2 \\ar[r] \\ar[d]^b & \\bigoplus B\\text{d}x_i \\ar@{..>}[d]^h \\\\", "J/J^2 \\ar[r] & (J/J^2)_g", "}", "$$", "Thus $(\\NL_{B/A})_{gb}$ is quasi-isomorphic", "to a finite projective module; small detail omitted.", "Since $(\\NL_{B/A})_{gb} = \\NL_{B_{gb}/A}$ in $D(B_{gb})$", "this shows that $B_{gb}$ is smooth over $\\Spec(A)$.", "As this holds for all $b \\in I^cB$ we conclude", "that $\\Spec(B_g) \\to \\Spec(A)$ is smooth over $\\Spec(A) \\setminus V(I)$", "as desired." ], "refs": [ "restricted-lemma-equivalent-with-artin-smooth", "algebra-lemma-localize-NL", "more-algebra-lemma-ext-1-annihilated", "restricted-lemma-NL-is-completion", "more-algebra-lemma-base-change-property-ext-1-annihilated", "more-algebra-lemma-base-change-RHom", "more-algebra-lemma-pseudo-coherence-and-base-change-ext", "algebra-lemma-NAK" ], "ref_ids": [ 2302, 1161, 10305, 2297, 10302, 10418, 10165, 401 ] } ], "ref_ids": [] }, { "id": 2304, "type": "theorem", "label": "restricted-lemma-zero-ext-1-after-modding-out", "categories": [ "restricted" ], "title": "restricted-lemma-zero-ext-1-after-modding-out", "contents": [ "Let $(A_1, I_1) \\to (A_2, I_2)$ be as in", "Remark \\ref{remark-base-change} with $A_1$ and $A_2$ Noetherian.", "Let $B_1$ be in (\\ref{equation-C-prime}) for $(A_1, I_1)$.", "Let $B_2$ be the base change of $B_1$. Let $f_1 \\in B_1$", "with image $f_2 \\in B_2$.", "If $\\Ext^1_{B_1}(\\NL_{B_1/A_1}^\\wedge, N_1)$ is annihilated", "by $f_1$ for every $B_1$-module $N_1$, then", "$\\Ext^1_{B_2}(\\NL_{B_2/A_2}^\\wedge, N_2)$ is annihilated", "by $f_2$ for every $B_2$-module $N_2$." ], "refs": [ "restricted-remark-base-change" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-NL-base-change} there is a map", "$$", "\\NL_{B_1/A_1} \\otimes_{B_2} B_1 \\to \\NL_{B_2/A_2}", "$$", "which induces and isomorphism on $H^0$ and a surjection on $H^{-1}$.", "Thus the result by More on Algebra, Lemmas", "\\ref{more-algebra-lemma-two-term-base-change},", "\\ref{more-algebra-lemma-base-change-property-ext-1-annihilated}, and", "\\ref{more-algebra-lemma-surjection-property-ext-1-annihilated}", "the last two applied with the principal ideals $(f_1) \\subset B_1$ and", "$(f_2) \\subset B_2$." ], "refs": [ "restricted-lemma-NL-base-change", "more-algebra-lemma-two-term-base-change", "more-algebra-lemma-base-change-property-ext-1-annihilated", "more-algebra-lemma-surjection-property-ext-1-annihilated" ], "ref_ids": [ 2299, 10301, 10302, 10304 ] } ], "ref_ids": [ 2448 ] }, { "id": 2305, "type": "theorem", "label": "restricted-lemma-base-change-rig-smooth-homomorphism", "categories": [ "restricted" ], "title": "restricted-lemma-base-change-rig-smooth-homomorphism", "contents": [ "Let $A_1 \\to A_2$ be a map of Noetherian rings. Let $I_i \\subset A_i$", "be an ideal such that $V(I_1A_2) = V(I_2)$. Let $B_1$ be in", "(\\ref{equation-C-prime}) for $(A_1, I_1)$.", "Let $B_2$ be the base change of $B_1$ as in", "Remark \\ref{remark-base-change}.", "If $B_1$ is rig-smooth over $(A_1, I_1)$,", "then $B_2$ is rig-smooth over $(A_2, I_2)$." ], "refs": [ "restricted-remark-base-change" ], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-zero-ext-1-after-modding-out} and", "Definition \\ref{definition-rig-smooth-homomorphism}", "and the fact that $I_2^c$ is contained in $I_1A_2$ for some $c \\geq 0$", "as $A_2$ is Noetherian." ], "refs": [ "restricted-lemma-zero-ext-1-after-modding-out", "restricted-definition-rig-smooth-homomorphism" ], "ref_ids": [ 2304, 2434 ] } ], "ref_ids": [ 2448 ] }, { "id": 2306, "type": "theorem", "label": "restricted-lemma-get-morphism-general-better", "categories": [ "restricted" ], "title": "restricted-lemma-get-morphism-general-better", "contents": [ "Assume given the following data", "\\begin{enumerate}", "\\item an integer $c \\geq 0$,", "\\item an ideal $I$ of a Noetherian ring $A$,", "\\item $B$ in (\\ref{equation-C-prime}) for $(A, I)$ such that", "$I^c$ annihilates $\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$", "for any $B$-module $N$,", "\\item a Noetherian $I$-adically complete $A$-algebra $C$; denote", "$d = d(\\text{Gr}_I(C))$ and $q_0 = q(\\text{Gr}_I(C))$ the integers found in", "Local Cohomology, Section \\ref{local-cohomology-section-uniform},", "\\item an integer $n \\geq \\max(q_0 + (d + 1)c, 2(d + 1)c + 1)$, and", "\\item an $A$-algebra homomorphism $\\psi_n : B \\to C/I^nC$.", "\\end{enumerate}", "Then there exists a map $\\varphi : B \\to C$ of $A$-algebras such", "that $\\psi_n \\bmod I^{n - (d + 1)c} = \\varphi \\bmod I^{n - (d + 1)c}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the obstruction class", "$$", "o(\\psi_n) \\in \\Ext^1_B(\\NL_{B/A}^\\wedge, I^nC/I^{2n}C)", "$$", "of Remark \\ref{remark-improve-homomorphism}. For any $C/I^nC$-module", "$N$ we have", "\\begin{align*}", "\\Ext^1_B(\\NL_{B/A}^\\wedge, N)", "& =", "\\Ext^1_{C/I^nC}(\\NL_{B/A}^\\wedge \\otimes_B^\\mathbf{L} C/I^nC, N) \\\\", "& =", "\\Ext^1_{C/I^nC}(\\NL_{B/A}^\\wedge \\otimes_B C/I^nC, N)", "\\end{align*}", "The first equality by", "More on Algebra, Lemma \\ref{more-algebra-lemma-upgrade-adjoint-tensor-RHom}", "and the second one by", "More on Algebra, Lemma \\ref{more-algebra-lemma-two-term-base-change}.", "In particular, we see that", "$\\Ext^1_{C/I^nC}(\\NL_{B/A}^\\wedge \\otimes_B C/I^nC, N)$ is annihilated by", "$I^cC$ for all $C/I^nC$-modules $N$.", "It follows that we may apply", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-bound-two-term-complex}", "to see that $o(\\psi_n)$ maps to zero in", "$$", "\\Ext^1_{C/I^nC}(\\NL_{B/A}^\\wedge \\otimes_B C/I^nC, I^{n'}C/I^{2n'}C) =", "\\Ext^1_B(\\NL_{B/A}^\\wedge, I^{n'}C/I^{2n'}C) =", "$$", "where $n' = n - (d + 1)c$. By the discussion in", "Remark \\ref{remark-improve-homomorphism} we obtain a map", "$$", "\\psi'_{2n'} : B \\to C/I^{2n'}C", "$$", "which agrees with $\\psi_n$ modulo $I^{n'}$.", "Observe that $2n' > n$ because $n \\geq 2(d + 1)c + 1$.", "\\medskip\\noindent", "We may repeat this procedure. Starting with $n_0 = n$ and", "$\\psi^0 = \\psi_n$ we end up getting a strictly increasing", "sequence of integers", "$$", "n_0 < n_1 < n_2 < \\ldots", "$$", "and $A$-algebra homorphisms $\\psi^i : B \\to C/I^{n_i}C$", "such that $\\psi^{i + 1}$ and $\\psi^i$ agree modulo $I^{n_i - tc}$.", "Since $C$ is $I$-adically complete we can take $\\varphi$", "to be the limit of the maps", "$\\psi^i \\bmod I^{n_i - (d + 1)c} : B \\to C/I^{n_i - (d + 1)c}C$", "and the lemma follows." ], "refs": [ "restricted-remark-improve-homomorphism", "more-algebra-lemma-upgrade-adjoint-tensor-RHom", "more-algebra-lemma-two-term-base-change", "local-cohomology-lemma-bound-two-term-complex", "restricted-remark-improve-homomorphism" ], "ref_ids": [ 2451, 10417, 10301, 9784, 2451 ] } ], "ref_ids": [] }, { "id": 2307, "type": "theorem", "label": "restricted-lemma-get-morphism-nonzerodivisor", "categories": [ "restricted" ], "title": "restricted-lemma-get-morphism-nonzerodivisor", "contents": [ "Let $I = (a)$ be a principal ideal of a Noetherian ring $A$.", "Let $B$ be an object of (\\ref{equation-C-prime}).", "Assume given an integer $c \\geq 0$ such that $\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$", "is annihilated by $a^c$ for all $B$-modules $N$.", "Let $C$ be an $I$-adically complete $A$-algebra such that", "$a$ is a nonzerodivisor on $C$. Let $n > 2c$. For any $A$-algebra", "map $\\psi_n : B \\to C/a^nC$ there exists an $A$-algebra", "map $\\varphi : B \\to C$ such that", "$\\psi_n \\bmod a^{n - c}C = \\varphi \\bmod a^{n - c}C$." ], "refs": [], "proofs": [ { "contents": [ "Consider the obstruction class", "$$", "o(\\psi_n) \\in \\Ext^1_B(\\NL_{B/A}^\\wedge, a^nC/a^{2n}C)", "$$", "of Remark \\ref{remark-improve-homomorphism}. Since $a$ is a nonzerodivisor", "on $C$ the map $a^c : a^nC/a^{2n}C \\to a^nC/a^{2n}C$ is isomorphic to the", "map $a^nC/a^{2n}C \\to a^{n - c}C/a^{2n - c}C$ in the category of $C$-modules.", "Hence by our assumption on $\\NL_{B/A}^\\wedge$", "we conclude that the class $o(\\psi_n)$ maps to zero in", "$$", "\\Ext^1_B(\\NL_{B/A}^\\wedge, a^{n - c}C/a^{2n - c}C)", "$$", "and a fortiori in", "$$", "\\Ext^1_B(\\NL_{B/A}^\\wedge, a^{n - c}C/a^{2n - 2c}C)", "$$", "By the discussion in Remark \\ref{remark-improve-homomorphism} we obtain a map", "$$", "\\psi_{2n - 2c} : B \\to C/a^{2n - 2c}C", "$$", "which agrees with $\\psi_n$ modulo $a^{n - c}C$.", "Observe that $2n - 2c > n$ because $n > 2c$.", "\\medskip\\noindent", "We may repeat this procedure. Starting with $n_0 = n$ and", "$\\psi^0 = \\psi_n$ we end up getting a strictly increasing", "sequence of integers", "$$", "n_0 < n_1 < n_2 < \\ldots", "$$", "and $A$-algebra homorphisms $\\psi^i : B \\to C/a^{n_i}C$", "such that $\\psi^{i + 1}$ and $\\psi^i$ agree modulo $a^{n_i - c}C$.", "Since $C$ is $I$-adically complete we can take $\\varphi$", "to be the limit of the maps", "$\\psi^i \\bmod a^{n_i - c}C : B \\to C/a^{n_i - c}C$", "and the lemma follows." ], "refs": [ "restricted-remark-improve-homomorphism", "restricted-remark-improve-homomorphism" ], "ref_ids": [ 2451, 2451 ] } ], "ref_ids": [] }, { "id": 2308, "type": "theorem", "label": "restricted-lemma-get-morphism-principal", "categories": [ "restricted" ], "title": "restricted-lemma-get-morphism-principal", "contents": [ "Let $I = (a)$ be a principal ideal of a Noetherian ring $A$.", "Let $B$ be an object of (\\ref{equation-C-prime}).", "Assume given an integer $c \\geq 0$ such that $\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$", "is annihilated by $a^c$ for all $B$-modules $N$.", "Let $C$ be an $I$-adically complete $A$-algebra.", "Assume given an integer $d \\geq 0$ such that $C[a^\\infty] \\cap a^dC = 0$.", "Let $n > \\max(2c, c + d)$. For any $A$-algebra map", "$\\psi_n : B \\to C/a^nC$ there exists an $A$-algebra map", "$\\varphi : B \\to C$ such", "that $\\psi_n \\bmod a^{n - c} = \\varphi \\bmod a^{n - c}$." ], "refs": [], "proofs": [ { "contents": [ "Let $C \\to C'$ be the quotient of $C$ by $C[a^\\infty]$. The $A$-algebra", "$C'$ is $I$-adically complete by", "Algebra, Lemma \\ref{algebra-lemma-quotient-complete}", "and the fact that $\\bigcap (C[a^\\infty] + a^nC) = C[a^\\infty]$", "because for $n \\geq d$ the sum $C[a^\\infty] + a^nC$ is direct.", "For $m \\geq d$ the diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "C[a^\\infty] \\ar[r] \\ar[d] &", "C \\ar[r] \\ar[d] & C' \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "C[a^\\infty] \\ar[r] &", "C/a^m C \\ar[r] & C'/a^m C' \\ar[r] & 0", "}", "$$", "has exact rows. Thus $C$ is the fibre product of $C'$ and", "$C/a^mC$ over $C'/a^mC'$ for all $m \\geq d$. By", "Lemma \\ref{lemma-get-morphism-nonzerodivisor}", "we can choose a homomorphism $\\varphi' : B \\to C'$", "such that $\\varphi'$ and $\\psi_n$ agree as maps into $C'/a^{n - c}C'$.", "We obtain a homomorphism $(\\varphi', \\psi_n \\bmod a^{n - c}C) : B \\to", "C' \\times_{C'/a^{n - c}C'} C/a^{n - c}C$.", "Since $n - c \\geq d$ this is the same thing as a homomorphism", "$\\varphi : B \\to C$. This finishes the proof." ], "refs": [ "algebra-lemma-quotient-complete", "restricted-lemma-get-morphism-nonzerodivisor" ], "ref_ids": [ 866, 2307 ] } ], "ref_ids": [] }, { "id": 2309, "type": "theorem", "label": "restricted-lemma-close-enough", "categories": [ "restricted" ], "title": "restricted-lemma-close-enough", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$. Let $r \\geq 0$", "and write $P = A[x_1, \\ldots, x_r]$ the $I$-adic completion.", "Consider a resolution", "$$", "P^{\\oplus t} \\xrightarrow{K} P^{\\oplus m}", "\\xrightarrow{g_1, \\ldots, g_m} P \\to B \\to 0", "$$", "of a quotient of $P$. Assume $B$ is rig-smooth over $(A, I)$.", "Then there exists an integer $n$ such that for any complex", "$$", "P^{\\oplus t} \\xrightarrow{K'} P^{\\oplus m}", "\\xrightarrow{g'_1, \\ldots, g'_m} P", "$$", "with $g_i - g'_i \\in I^nP$ and $K - K' \\in I^n\\text{Mat}(m \\times t, P)$", "there exists an isomorphism $B \\to B'$ of $A$-algebras where", "$B' = P/(g'_1, \\ldots, g'_m)$." ], "refs": [], "proofs": [ { "contents": [ "(A) By Definition \\ref{definition-rig-smooth-homomorphism}", "we can choose a $c \\geq 0$ such that $I^c$ annihilates", "$\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$ for all $B$-modules $N$.", "\\medskip\\noindent", "(B) By More on Algebra, Lemmas \\ref{more-algebra-lemma-approximate-complex} and", "\\ref{more-algebra-lemma-approximate-complex-graded}", "there exists a constant $c_1 = c(g_1, \\ldots, g_m, K)$", "such that for $n \\geq c_1 + 1$ the complex", "$$", "P^{\\oplus t} \\xrightarrow{K'} P^{\\oplus m}", "\\xrightarrow{g'_1, \\ldots, g'_m} P \\to B' \\to 0", "$$", "is exact and $\\text{Gr}_I(B) \\cong \\text{Gr}_I(B')$.", "\\medskip\\noindent", "(C) Let $d_0 = d(\\text{Gr}_I(B))$ and $q_0 = q(\\text{Gr}_I(B))$", "be the integers found in", "Local Cohomology, Section \\ref{local-cohomology-section-uniform}.", "\\medskip\\noindent", "We claim that $n = \\max(c_1 + 1, q_0 + (d_0 + 1)c, 2(d_0 + 1)c + 1)$", "works where $c$ is as in (A), $c_1$ is as in (B), and $q_0, d_0$ are as", "in (C).", "\\medskip\\noindent", "Let $g'_1, \\ldots, g'_m$ and $K'$ be as in the lemma.", "Since $g_i = g'_i \\in I^nP$ we obtain a canonical", "$A$-algebra homomorphism", "$$", "\\psi_n : B \\longrightarrow B'/I^nB'", "$$", "which induces an isomorphism $B/I^nB \\to B'/I^nB'$. Since", "$\\text{Gr}_I(B) \\cong \\text{Gr}_I(B')$ we have", "$d_0 = d(\\text{Gr}_I(B'))$ and $q_0 = q(\\text{Gr}_I(B'))$", "and since $n \\geq \\max(q_0 + (1 + d_0)c, 2(d_0 + 1)c + 1)$", "we may apply Lemma \\ref{lemma-get-morphism-general-better}", "to find an $A$-algebra homomorphism", "$$", "\\varphi : B \\longrightarrow B'", "$$", "such that", "$\\varphi \\bmod I^{n - (d_0 + 1)c}B' = \\psi_n \\bmod I^{n - (d_0 + 1)c}B'$.", "Since $n - (d_0 + 1)c > 0$ we see that $\\varphi$ is an $A$-algebra", "homomorphism which modulo $I$ induces the isomorphism $B/IB \\to B'/IB'$", "we found above.", "The rest of the proof shows that these facts force $\\varphi$", "to be an isomorphism; we suggest the reader find their own proof of this.", "\\medskip\\noindent", "Namely, it follows that $\\varphi$ is surjective for example by applying", "Algebra, Lemma \\ref{algebra-lemma-completion-generalities} part (1)", "using the fact that $B$ and $B'$ are complete.", "Thus $\\varphi$ induces a surjection $\\text{Gr}_I(B) \\to \\text{Gr}_I(B')$", "which has to be an isomorphism because the source and target are", "isomorphic Noetherian rings, see", "Algebra, Lemma \\ref{algebra-lemma-surjective-endo-noetherian-ring-is-iso}", "(of course you can show $\\varphi$ induces the isomorphism we found above", "but that would need a tiny argument).", "Thus $\\varphi$ induces injective maps", "$I^eB/I^{e + 1}B \\to I^eB'/I^{e + 1}B'$ for all $e \\geq 0$.", "This implies $\\varphi$ is injective since for any $b \\in B$ there", "exists an $e \\geq 0$ such that $b \\in I^eB$, $b \\not \\in I^{e + 1}B$", "by Krull's intersection theorem (Algebra, Lemma", "\\ref{algebra-lemma-intersect-powers-ideal-module-zero}).", "This finishes the proof." ], "refs": [ "restricted-definition-rig-smooth-homomorphism", "more-algebra-lemma-approximate-complex", "more-algebra-lemma-approximate-complex-graded", "restricted-lemma-get-morphism-general-better", "algebra-lemma-completion-generalities", "algebra-lemma-surjective-endo-noetherian-ring-is-iso", "algebra-lemma-intersect-powers-ideal-module-zero" ], "ref_ids": [ 2434, 9811, 9812, 2306, 858, 457, 627 ] } ], "ref_ids": [] }, { "id": 2310, "type": "theorem", "label": "restricted-lemma-algebraize-easy", "categories": [ "restricted" ], "title": "restricted-lemma-algebraize-easy", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$. Let $C^h$ be the henselization", "of a finite type $A$-algebra $C$ with respect to the ideal $IC$. Let", "$J \\subset C^h$ be an ideal. Then there exists a finite type $A$-algebra", "$B$ such that $B^\\wedge \\cong (C^h/J)^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-henselization-Noetherian-pair}", "the ring $C^h$ is Noetherian. Say $J = (g_1, \\ldots, g_m)$.", "The ring $C^h$ is a filtered colimit of \\'etale $C$ algebras $C'$", "such that $C/IC \\to C'/IC'$ is an isomorphism", "(see proof of More on Algebra, Lemma \\ref{more-algebra-lemma-henselization}).", "Pick an $C'$ such that $g_1, \\ldots, g_m$ are the", "images of $g'_1, \\ldots, g'_m \\in C'$.", "Setting $B = C'/(g'_1, \\ldots, g'_m)$ we get a finite", "type $A$-algebra. Of course $(C, IC)$ and $C', IC')$ have", "the same henselizations and the same completions.", "It follows easily from this that $B^\\wedge = (C^h/J)^\\wedge$." ], "refs": [ "more-algebra-lemma-henselization-Noetherian-pair", "more-algebra-lemma-henselization" ], "ref_ids": [ 9874, 9871 ] } ], "ref_ids": [] }, { "id": 2311, "type": "theorem", "label": "restricted-lemma-fully-faithfulness", "categories": [ "restricted" ], "title": "restricted-lemma-fully-faithfulness", "contents": [ "Let $A$ be a Noetherian G-ring. Let $I \\subset A$ be an ideal.", "Let $B, C$ be finite type $A$-algebras. For any $A$-algebra map", "$\\varphi : B^\\wedge \\to C^\\wedge$ of $I$-adic completions and any", "$N \\geq 1$ there exist", "\\begin{enumerate}", "\\item an \\'etale ring map $C \\to C'$ which induces", "an isomorphism $C/IC \\to C'/IC'$,", "\\item an $A$-algebra map $\\varphi : B \\to C'$", "\\end{enumerate}", "such that $\\varphi$ and $\\psi$ agree modulo $I^N$", "into $C^\\wedge = (C')^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "The statement of the lemma makes sense as $C \\to C'$ is flat", "(Algebra, Lemma \\ref{algebra-lemma-etale}) hence induces an isomorphism", "$C/I^nC \\to C'/I^nC'$ for all $n$", "(More on Algebra, Lemma \\ref{more-algebra-lemma-neighbourhood-isomorphism})", "and hence an isomorphism on completions.", "Let $C^h$ be the henselization of the pair $(C, IC)$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization}.", "Then $C^h$ is the filtered colimit of the algebras $C'$", "and the maps", "$C \\to C' \\to C^h$ induce isomorphism on completions (More on Algebra,", "Lemma \\ref{more-algebra-lemma-henselization-Noetherian-pair}).", "Thus it suffices to prove there exists an $A$-algebra map", "$B \\to C^h$ which is congruent to $\\psi$ modulo $I^N$.", "Write $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$.", "The ring map $\\psi$ corresponds to elements", "$\\hat c_1, \\ldots, \\hat c_n \\in C^\\wedge$ with", "$f_j(\\hat c_1, \\ldots, \\hat c_n) = 0$ for $j = 1, \\ldots, m$.", "Namely, as $A$ is a Noetherian G-ring, so is $C$, see", "More on Algebra, Proposition", "\\ref{more-algebra-proposition-finite-type-over-G-ring}.", "Thus Smoothing Ring Maps,", "Lemma \\ref{smoothing-lemma-henselian-pair}", "applies to give elements $c_1, \\ldots, c_n \\in C^h$ such", "that $f_j(c_1, \\ldots, c_n) = 0$ for $j = 1, \\ldots, m$", "and such that $\\hat c_i - c_i \\in I^NC^h$.", "This determines the map $B \\to C^h$ as desired." ], "refs": [ "algebra-lemma-etale", "more-algebra-lemma-neighbourhood-isomorphism", "more-algebra-lemma-henselization", "more-algebra-lemma-henselization-Noetherian-pair", "more-algebra-proposition-finite-type-over-G-ring", "smoothing-lemma-henselian-pair" ], "ref_ids": [ 1231, 10340, 9871, 9874, 10581, 5644 ] } ], "ref_ids": [] }, { "id": 2312, "type": "theorem", "label": "restricted-lemma-presentation-rig-smooth", "categories": [ "restricted" ], "title": "restricted-lemma-presentation-rig-smooth", "contents": [ "Let $A$ be a ring. Let $f_1, \\ldots, f_m \\in A[x_1, \\ldots, x_n]$", "and set $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$. Assume $m \\leq n$", "and set $g = \\det_{1 \\leq i, j \\leq m}(\\partial f_j/\\partial x_i)$.", "Then", "\\begin{enumerate}", "\\item $g$ annihilates $\\Ext^1_B(\\NL_{B/A}, N)$ for every $B$-module $N$,", "\\item if $n = m$, then multiplication by $g$ on $\\NL_{B/A}$ is $0$ in $D(B)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be the $m \\times m$ matrix with entries $\\partial f_j/\\partial x_i$", "for $1 \\leq i, j \\leq n$. Let $K \\in D(B)$ be represented by the complex", "$T : B^{\\oplus m} \\to B^{\\oplus m}$ with terms sitting in degrees $-1$ and $0$.", "By More on Algebra, Lemmas \\ref{more-algebra-lemma-silly}", "we have $g : K \\to K$ is zero in $D(B)$. Set $J = (f_1, \\ldots, f_m)$.", "Recall that $\\NL_{B/A}$ is homotopy equivalent to", "$J/J^2 \\to \\bigoplus_{i = 1, \\ldots, n} B\\text{d}x_i$, see", "Algebra, Section \\ref{algebra-section-netherlander}.", "Denote $L$ the complex $J/J^2 \\to \\bigoplus_{i = 1, \\ldots, m} B\\text{d}x_i$", "to that we have the quotient map $\\NL_{B/A} \\to L$.", "We also have a surjective map of complexes $K \\to L$", "by sending the $j$th basis element in the term $B^{\\oplus m}$", "in degree $-1$ to the class of $f_j$ in $J/J^2$. Picture", "$$", "\\NL_{B/A} \\to L \\leftarrow K", "$$", "From", "More on Algebra, Lemma \\ref{more-algebra-lemma-two-term-surjection-map-zero}", "we conclude that multiplication by $g$ on $L$ is $0$ in $D(B)$.", "On the other hand, the distinguished triangle", "$B^{\\oplus n - m}[0] \\to \\NL_{B/A} \\to L$", "shows that $\\Ext^1_B(L, N) \\to \\Ext^1_B(\\NL_{B/A}, N)$ is surjective", "for every $B$-module $N$ and hence annihilated by $g$.", "This proves part (1).", "If $n = m$ then $\\NL_{B/A} = L$ and we see that (2) holds." ], "refs": [ "more-algebra-lemma-silly", "more-algebra-lemma-two-term-surjection-map-zero" ], "ref_ids": [ 10307, 10303 ] } ], "ref_ids": [] }, { "id": 2313, "type": "theorem", "label": "restricted-lemma-approximate-presentation-rig-smooth", "categories": [ "restricted" ], "title": "restricted-lemma-approximate-presentation-rig-smooth", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$. Let $B$ be an object", "of (\\ref{equation-C-prime}). Let $B = A[x_1, \\ldots, x_r]^\\wedge/J$", "be a presentation. Assume there exists an element", "$b \\in B$, $0 \\leq m \\leq r$, and $f_1, \\ldots, f_m \\in J$", "such that", "\\begin{enumerate}", "\\item $V(b) \\subset V(IB)$ in $\\Spec(B)$,", "\\item the image of", "$\\Delta = \\det_{1 \\leq i, j \\leq m}(\\partial f_j/\\partial x_i)$", "in $B$ divides $b$, and", "\\item $b J \\subset (f_1, \\ldots, f_m) + J^2$.", "\\end{enumerate}", "Then there exists a finite type $A$-algebra $C$ and an $A$-algebra", "isomorphism $B \\cong C^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "The conditions imply that $B$ is rig-smooth over $(A, I)$, see", "Lemma \\ref{lemma-equivalent-with-artin-smooth}.", "Write $b' \\Delta = b$ in $B$ for some $b' \\in B$.", "Say $I = (a_1, \\ldots, a_t)$. Since $V(b) \\subset V(IB)$ there", "exists an integer $c \\geq 0$ such that $I^cB \\subset bB$.", "Write $bb_i = a_i^c$ in $B$ for some $b_i \\in B$.", "\\medskip\\noindent", "Choose an integer $n \\gg 0$ (we will see later how large).", "Choose polynomials $f'_1, \\ldots, f'_m \\in A[x_1, \\ldots, x_r]$", "such that $f_i - f'_i \\in I^nA[x_1, \\ldots, x_r]^\\wedge$.", "We set $\\Delta' = \\det_{1 \\leq i, j \\leq m}(\\partial f'_j/\\partial x_i)$", "and we consider the finite type $A$-algebra", "$$", "C = A[x_1, \\ldots, x_r, z_1, \\ldots, z_t]/", "(f'_1, \\ldots, f'_m,", "z_1\\Delta' - a_1^c, \\ldots, z_t\\Delta' - a_t^c)", "$$", "We will apply Lemma \\ref{lemma-presentation-rig-smooth} to $C$.", "We compute", "$$", "\\det\\left(", "\\begin{matrix}", "\\text{matrix of partials of} \\\\", "f'_1, \\ldots, f'_m, z_1\\Delta' - a_1^c, \\ldots, z_t\\Delta' - a_t^c \\\\", "\\text{with respect to the variables} \\\\", "x_1, \\ldots, x_m, z_1, \\ldots, z_t", "\\end{matrix}", "\\right) =", "(\\Delta')^{t + 1}", "$$", "Hence we see that $\\Ext^1_C(\\NL_{C/A}, N)$ is annihilated by", "$(\\Delta')^{t + 1}$ for all $C$-modules $N$. Since $a_i^c$ is", "divisible by $\\Delta'$ in $C$ we see that $a_i^{(t + 1)c}$ annihilates", "these $\\Ext^1$'s also. Thus $I^{c_1}$ annihilates", "$\\Ext^1_C(\\NL_{C/A}, N)$ for all $C$-modules $N$", "where $c_1 = 1 + t((t + 1)c - 1)$. The exact value of $c_1$ doesn't matter for", "the rest of the argument; what matters is that it is independent of $n$.", "\\medskip\\noindent", "Since $\\NL_{C^\\wedge/A}^\\wedge = \\NL_{C/A} \\otimes_C C^\\wedge$ by", "Lemma \\ref{lemma-NL-is-completion} we conclude that multiplication", "by $I^{c_1}$ is zero on $\\Ext^1_{C^\\wedge}(\\NL_{C^\\wedge/A}^\\wedge, N)$", "for any $C^\\wedge$-module $N$ as well, see", "More on Algebra, Lemmas", "\\ref{more-algebra-lemma-base-change-property-ext-1-annihilated} and", "\\ref{more-algebra-lemma-two-term-base-change}.", "In particular $C^\\wedge$ is rig-smooth over $(A, I)$.", "\\medskip\\noindent", "Observe that we have a surjective $A$-algebra homomorphism", "$$", "\\psi_n : C \\longrightarrow B/I^nB", "$$", "sending the class of $x_i$ to the class of $x_i$ and sending the class of", "$z_i$ to the class of $b_ib'$. This works because of our choices of $b'$", "and $b_i$ in the first paragraph of the proof.", "\\medskip\\noindent", "Let $d = d(\\text{Gr}_I(B))$ and $q_0 = q(\\text{Gr}_I(B))$ be the integers", "found in ", "Local Cohomology, Section \\ref{local-cohomology-section-uniform}.", "By Lemma \\ref{lemma-get-morphism-general-better} if we take", "$n \\geq \\max(q_0 + (d + 1)c_1, 2(d + 1)c_1 + 1)$ we can find a homomorphism", "$\\varphi : C^\\wedge \\to B$ of $A$-algebras which is congruent to", "$\\psi_n$ modulo $I^{n - (d + 1)c_1}B$.", "\\medskip\\noindent", "Since $\\varphi : C^\\wedge \\to B$ is surjective modulo $I$", "we see that it is surjective (for example use", "Algebra, Lemma \\ref{algebra-lemma-completion-generalities}).", "To finish the proof it suffices to show that", "$\\Ker(\\varphi)/\\Ker(\\varphi)^2$ is annihilated by a power of $I$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-quotient-by-idempotent}.", "\\medskip\\noindent", "Since $\\varphi$ is surjective we see that", "$\\NL_{B/C^{\\wedge}}^\\wedge$ has cohomology modules", "$H^0(\\NL_{B/C^{\\wedge}}^\\wedge) = 0$ and", "$H^{-1}(\\NL_{B/C^{\\wedge}}^\\wedge) = \\Ker(\\varphi)/\\Ker(\\varphi)^2$.", "We have an exact sequence", "$$", "H^{-1}(\\NL_{C^\\wedge/A}^\\wedge \\otimes_{C^\\wedge} B) \\to", "H^{-1}(\\NL_{B/A}^\\wedge) \\to", "H^{-1}(\\NL_{B/C^{\\wedge}}^\\wedge) \\to", "H^0(\\NL_{C^\\wedge/A}^\\wedge \\otimes_{C^\\wedge} B) \\to", "H^0(\\NL_{B/A}^\\wedge) \\to 0", "$$", "by Lemma \\ref{lemma-exact-sequence-NL}. The first two modules are", "annihilated by a power of $I$ as $B$ and $C^\\wedge$ are rig-smooth over", "$(A, I)$. Hence it suffices", "to show that the kernel of the surjective map", "$H^0(\\NL_{C^\\wedge/A}^\\wedge \\otimes_{C^\\wedge} B) \\to", "H^0(\\NL_{B/A}^\\wedge)$ is annihilated by a power of $I$.", "For this it suffices to show that it is annihilated by", "a power of $b$. In other words, it suffices to show that", "$$", "H^0(\\NL_{C^\\wedge/A}^\\wedge) \\otimes_{C^\\wedge} B[1/b]", "\\longrightarrow", "H^0(\\NL_{B/A}^\\wedge) \\otimes_B B[1/b]", "$$", "is an isomorphism. However, both are free $B[1/b]$ modules", "of rank $r - m$ with basis $\\text{d}x_{m + 1}, \\ldots, \\text{d}x_r$", "and we conclude the proof." ], "refs": [ "restricted-lemma-equivalent-with-artin-smooth", "restricted-lemma-presentation-rig-smooth", "restricted-lemma-NL-is-completion", "more-algebra-lemma-base-change-property-ext-1-annihilated", "more-algebra-lemma-two-term-base-change", "restricted-lemma-get-morphism-general-better", "algebra-lemma-completion-generalities", "more-algebra-lemma-quotient-by-idempotent", "restricted-lemma-exact-sequence-NL" ], "ref_ids": [ 2302, 2312, 2297, 10302, 10301, 2306, 858, 10475, 2300 ] } ], "ref_ids": [] }, { "id": 2314, "type": "theorem", "label": "restricted-lemma-equivalent-with-artin", "categories": [ "restricted" ], "title": "restricted-lemma-equivalent-with-artin", "contents": [ "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.", "Let $B$ be an object of (\\ref{equation-C-prime}). Write", "$B = A[x_1, \\ldots, x_r]^\\wedge/J$", "(Lemma \\ref{lemma-topologically-finite-type-Noetherian})", "and let $\\NL_{B/A}^\\wedge = (J/J^2 \\to \\bigoplus B\\text{d}x_i)$", "be its naive cotangent complex (\\ref{equation-NL}).", "The following are equivalent", "\\begin{enumerate}", "\\item $B$ is rig-\\'etale over $(A, I)$,", "\\item", "\\label{item-zero-on-NL}", "there exists a $c \\geq 0$ such that for all $a \\in I^c$ multiplication by $a$", "on $\\NL_{B/A}^\\wedge$ is zero in $D(B)$,", "\\item", "\\label{item-zero-on-cohomology-NL}", "there exits a $c \\geq 0$ such that $H^i(\\NL_{B/A}^\\wedge)$, $i = -1, 0$ is", "annihilated by $I^c$,", "\\item", "\\label{item-zero-on-cohomology-NL-truncations}", "there exists a $c \\geq 0$ such that $H^i(\\NL_{B_n/A_n})$, $i = -1, 0$ is", "annihilated by $I^c$ for all $n \\geq 1$ where $A_n = A/I^n$ and $B_n = B/I^nB$,", "\\item", "\\label{item-condition-artin-pre-pre}", "for every $a \\in I$ there exists a $c \\geq 0$ such that", "\\begin{enumerate}", "\\item $a^c$ annihilates $H^0(\\NL_{B/A}^\\wedge)$, and", "\\item there exist $f_1, \\ldots, f_r \\in J$ such that", "$a^c J \\subset (f_1, \\ldots, f_r) + J^2$.", "\\end{enumerate}", "\\item", "\\label{item-condition-artin-pre}", "for every $a \\in I$ there exist $f_1, \\ldots, f_r \\in J$ and $c \\geq 0$", "such that", "\\begin{enumerate}", "\\item $\\det_{1 \\leq i, j \\leq r}(\\partial f_j/\\partial x_i)$ divides", "$a^c$ in $B$, and", "\\item $a^c J \\subset (f_1, \\ldots, f_r) + J^2$.", "\\end{enumerate}", "\\item", "\\label{item-condition-artin}", "choosing generaters $f_1, \\ldots, f_t$ for $J$ we have", "\\begin{enumerate}", "\\item the Jacobian ideal of $B$ over $A$, namely the ideal in $B$", "generated by the $r \\times r$ minors of the matrx", "$(\\partial f_j/\\partial x_i)_{1 \\leq i \\leq r, 1 \\leq j \\leq t}$,", "contains the ideal $I^cB$ for some $c$, and", "\\item the Cramer ideal of $B$ over $A$, namely the ideal in $B$", "generated by the image in $B$ of the $r$th Fitting ideal of $J$", "as an $A[x_1, \\ldots, x_r]^\\wedge$-module, contains $I^cB$ for some $c$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [ "restricted-lemma-topologically-finite-type-Noetherian" ], "proofs": [ { "contents": [ "The equivalence of (1) and (\\ref{item-zero-on-NL}) is a restatement of", "Definition \\ref{definition-rig-etale-homomorphism}.", "\\medskip\\noindent", "The equivalence of (\\ref{item-zero-on-NL}) and", "(\\ref{item-zero-on-cohomology-NL}) follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-zero-in-derived}.", "\\medskip\\noindent", "The equivalence of (\\ref{item-zero-on-cohomology-NL})", "and (\\ref{item-zero-on-cohomology-NL-truncations}) follows from the fact that", "the systems $\\{\\NL_{B_n/A_n}\\}$ and $\\NL_{B/A}^\\wedge \\otimes_B B_n$", "are strictly isomorphic, see Lemma \\ref{lemma-NL-is-limit}.", "Some details omitted.", "\\medskip\\noindent", "Assume (\\ref{item-zero-on-NL}). Let $a \\in I$.", "Let $c$ be such that multiplication by $a^c$ is zero on $\\NL_{B/A}^\\wedge$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-map-out-of-almost-free}", "part (1) there exists a map $\\alpha : \\bigoplus B\\text{d}x_i \\to J/J^2$", "such that $\\text{d} \\circ \\alpha$ and $\\alpha \\circ \\text{d}$ are both", "multiplication by $a^c$. Let $f_i \\in J$ be an element whose", "class modulo $J^2$ is equal to $\\alpha(\\text{d}x_i)$.", "A simple calculation gives that (\\ref{item-condition-artin-pre})(a), (b) hold.", "\\medskip\\noindent", "We omit the verification that (\\ref{item-condition-artin-pre})", "implies (\\ref{item-condition-artin-pre-pre}); it is just a statement", "on two term complexes over $B$ of the form $M \\to B^{\\oplus r}$.", "\\medskip\\noindent", "Assume (\\ref{item-condition-artin-pre-pre}) holds.", "Say $I = (a_1, \\ldots, a_t)$. Let $c_i \\geq 0$ be the integer such that", "(\\ref{item-condition-artin-pre-pre})(a), (b)", "hold for $a_i^{c_i}$. Then we see that $I^{\\sum c_i}$ annihilates", "$H^0(\\NL_{B/A}^\\wedge)$. Let $f_{i, 1}, \\ldots, f_{i, r} \\in J$", "be as in (\\ref{item-condition-artin-pre-pre})(b) for $a_i$.", "Consider the composition", "$$", "B^{\\oplus r} \\to J/J^2 \\to \\bigoplus B\\text{d}x_i", "$$", "where the $j$th basis vector is mapped to the class of $f_{i, j}$ in $J/J^2$.", "By (\\ref{item-condition-artin-pre-pre})(a) and (b) the cokernel of the", "composition is annihilated by $a_i^{2c_i}$. Thus this map is surjective", "after inverting $a_i^{c_i}$, and hence an isomorphism", "(Algebra, Lemma \\ref{algebra-lemma-fun}).", "Thus the kernel of $B^{\\oplus r} \\to \\bigoplus B\\text{d}x_i$ is", "$a_i$-power torsion, and hence", "$H^{-1}(\\NL_{B/A}^\\wedge) = \\Ker(J/J^2 \\to \\bigoplus B\\text{d}x_i)$", "is $a_i$-power torsion. Since $B$ is Noetherian", "(Lemma \\ref{lemma-topologically-finite-type-Noetherian}),", "all modules including $H^{-1}(\\NL_{B/A}^\\wedge)$ are finite.", "Thus $a_i^{d_i}$ annihilates $H^{-1}(\\NL_{B/A}^\\wedge)$ for some $d_i \\geq 0$.", "It follows that $I^{\\sum d_i}$ annihilates $H^{-1}(\\NL_{B/A}^\\wedge)$", "and we see that (\\ref{item-zero-on-cohomology-NL}) holds.", "\\medskip\\noindent", "Thus conditions", "(\\ref{item-zero-on-NL}),", "(\\ref{item-zero-on-cohomology-NL}),", "(\\ref{item-zero-on-cohomology-NL-truncations}),", "(\\ref{item-condition-artin-pre-pre}), and", "(\\ref{item-condition-artin-pre}) are equivalent.", "Thus it remains to show that these conditions are", "equivalent with (\\ref{item-condition-artin}).", "Observe that the Cramer ideal $\\text{Fit}_r(J) B$ is", "equal to $\\text{Fit}_r(J/J^2)$ as", "$J/J^2 = J \\otimes_{A[x_1, \\ldots, x_r]^\\wedge} B$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-fitting-ideal-basics} part (3).", "Also, observe that the Jacobian ideal is just", "$\\text{Fit}_0(H^0(\\NL_{B/A}^\\wedge))$.", "Thus we see that the equivalence of", "(\\ref{item-zero-on-cohomology-NL}) and (\\ref{item-condition-artin})", "is a purely algebraic question which we discuss in the next paragraph.", "\\medskip\\noindent", "Let $R$ be a Noetherian ring and let $I \\subset R$ be an ideal.", "Let $M \\xrightarrow{d} R^{\\oplus r}$ be a two term complex.", "We have to show that the following are equivalent", "\\begin{enumerate}", "\\item[(A)] the cohomology of $M \\to R^{\\oplus r}$ is annihilated by a power", "of $I$, and", "\\item[(B)] the ideals $\\text{Fit}_r(M)$ and $\\text{Fit}_0(\\text{Coker}(d))$", "contain a power of $I$.", "\\end{enumerate}", "Since $R$ is Noetherian, we can reformulate part (2) as an inclusion", "of the corresponding closed subschemes, see Algebra, Lemmas", "\\ref{algebra-lemma-Zariski-topology} and \\ref{algebra-lemma-Noetherian-power}.", "On the other hand, over the complement of $V(\\text{Fit}_0(\\Coker(d)))$", "the cokernel of $d$ vanishes and over the complement of", "$V(\\text{Fit}_r(M))$ the module $M$ is locally generated by $r$", "elements, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-fitting-ideal-generate-locally}.", "Thus (B) is equivalent to", "\\begin{enumerate}", "\\item[(C)] away from $V(I)$ the cokernel of $d$ vanishes and", "the module $M$ is locally generated by $\\leq r$ elements.", "\\end{enumerate}", "Of course this is equivalent to the condition that $M \\to R^{\\oplus r}$", "has vanishing cohomology over $\\Spec(R) \\setminus V(I)$ which in turn", "is equivalent to (A). This finishes the proof." ], "refs": [ "restricted-definition-rig-etale-homomorphism", "more-algebra-lemma-zero-in-derived", "restricted-lemma-NL-is-limit", "more-algebra-lemma-map-out-of-almost-free", "algebra-lemma-fun", "restricted-lemma-topologically-finite-type-Noetherian", "more-algebra-lemma-fitting-ideal-basics", "algebra-lemma-Zariski-topology", "algebra-lemma-Noetherian-power", "more-algebra-lemma-fitting-ideal-generate-locally" ], "ref_ids": [ 2435, 10306, 2298, 10299, 388, 2295, 9834, 389, 460, 9835 ] } ], "ref_ids": [ 2295 ] }, { "id": 2315, "type": "theorem", "label": "restricted-lemma-rig-etale-rig-smooth", "categories": [ "restricted" ], "title": "restricted-lemma-rig-etale-rig-smooth", "contents": [ "Let $A$ be a Noetherian ring and let $I$ be an ideal.", "Let $B$ be an object of (\\ref{equation-C-prime}).", "If $B$ is rig-\\'etale over $(A, I)$, then $B$ is rig-smooth over $(A, I)$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Definitions \\ref{definition-rig-smooth-homomorphism} and", "\\ref{definition-rig-etale-homomorphism}." ], "refs": [ "restricted-definition-rig-smooth-homomorphism", "restricted-definition-rig-etale-homomorphism" ], "ref_ids": [ 2434, 2435 ] } ], "ref_ids": [] }, { "id": 2316, "type": "theorem", "label": "restricted-lemma-rig-etale", "categories": [ "restricted" ], "title": "restricted-lemma-rig-etale", "contents": [ "Let $A$ be a Noetherian ring and let $I$ be an ideal.", "Let $B$ be a finite type $A$-algebra.", "\\begin{enumerate}", "\\item If $\\Spec(B) \\to \\Spec(A)$ is \\'etale over $\\Spec(A) \\setminus V(I)$,", "then $B^\\wedge$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-equivalent-with-artin}.", "\\item If $B^\\wedge$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-equivalent-with-artin},", "then there exists $g \\in 1 + IB$ such that $\\Spec(B_g)$ is \\'etale", "over $\\Spec(A) \\setminus V(I)$.", "\\end{enumerate}" ], "refs": [ "restricted-lemma-equivalent-with-artin", "restricted-lemma-equivalent-with-artin" ], "proofs": [ { "contents": [ "Assume $B^\\wedge$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-equivalent-with-artin}.", "The naive cotangent complex $\\NL_{B/A}$ is a complex of finite type", "$B$-modules and hence $H^{-1}$ and $H^0$ are finite $B$-modules.", "Completion is an exact functor on finite $B$-modules (Algebra,", "Lemma \\ref{algebra-lemma-completion-flat}) and $\\NL_{B^\\wedge/A}^\\wedge$", "is the completion of the complex $\\NL_{B/A}$ (this is easy to see", "by choosing presentations).", "Hence the assumption implies there exists a $c \\geq 0$ such that", "$H^{-1}/I^nH^{-1}$ and $H^0/I^nH^0$ are annihilated by $I^c$", "for all $n$. By Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "this means that $I^cH^{-1}$ and $I^cH^0$ are annihilated by an element", "of the form $g = 1 + x$ with $x \\in IB$. After inverting $g$", "(which does not change the quotients $B/I^nB$)", "we see that $\\NL_{B/A}$ has cohomology annihilated by $I^c$. Thus", "$A \\to B$ is \\'etale at any prime of $B$ not lying over $V(I)$", "by the definition of \\'etale ring maps, see", "Algebra, Definition \\ref{algebra-definition-etale}.", "\\medskip\\noindent", "Conversely, assume that $\\Spec(B) \\to \\Spec(A)$ is \\'etale over", "$\\Spec(A) \\setminus V(I)$. Then for every $a \\in I$ there exists", "a $c \\geq 0$ such that multiplication by $a^c$ is zero $\\NL_{B/A}$.", "Since $\\NL_{B^\\wedge/A}^\\wedge$ is the derived completion of", "$\\NL_{B/A}$ (see Lemma \\ref{lemma-NL-is-limit}) it follows that", "$B^\\wedge$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-equivalent-with-artin}." ], "refs": [ "restricted-lemma-equivalent-with-artin", "algebra-lemma-completion-flat", "algebra-lemma-NAK", "algebra-definition-etale", "restricted-lemma-NL-is-limit", "restricted-lemma-equivalent-with-artin" ], "ref_ids": [ 2314, 870, 401, 1539, 2298, 2314 ] } ], "ref_ids": [ 2314, 2314 ] }, { "id": 2317, "type": "theorem", "label": "restricted-lemma-zero-after-modding-out", "categories": [ "restricted" ], "title": "restricted-lemma-zero-after-modding-out", "contents": [ "Let $(A_1, I_1) \\to (A_2, I_2)$ be as in", "Remark \\ref{remark-base-change} with $A_1$ and $A_2$ Noetherian.", "Let $B_1$ be in (\\ref{equation-C-prime}) for $(A_1, I_1)$.", "Let $B_2$ be the base change of $B_1$.", "If multiplication by $f_1 \\in B_1$ on $\\NL^\\wedge_{B_1/A_1}$", "is zero in $D(B_1)$, then multiplication by", "the image $f_2 \\in B_2$ on $\\NL^\\wedge_{B_2/A_2}$ is zero", "in $D(B_2)$." ], "refs": [ "restricted-remark-base-change" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-NL-base-change} there is a map", "$$", "\\NL_{B_1/A_1} \\otimes_{B_2} B_1 \\to \\NL_{B_2/A_2}", "$$", "which induces and isomorphism on $H^0$ and a surjection on $H^{-1}$.", "Thus the result by", "More on Algebra, Lemma \\ref{more-algebra-lemma-two-term-surjection-map-zero}." ], "refs": [ "restricted-lemma-NL-base-change", "more-algebra-lemma-two-term-surjection-map-zero" ], "ref_ids": [ 2299, 10303 ] } ], "ref_ids": [ 2448 ] }, { "id": 2318, "type": "theorem", "label": "restricted-lemma-base-change-rig-etale-homomorphism", "categories": [ "restricted" ], "title": "restricted-lemma-base-change-rig-etale-homomorphism", "contents": [ "Let $A_1 \\to A_2$ be a map of Noetherian rings. Let $I_i \\subset A_i$", "be an ideal such that $V(I_1A_2) = V(I_2)$. Let $B_1$ be in", "(\\ref{equation-C-prime}) for $(A_1, I_1)$.", "Let $B_2$ be the base change of $B_1$ as in", "Remark \\ref{remark-base-change}.", "If $B_1$ is rig-\\'etale over $(A_1, I_1)$,", "then $B_2$ is rig-\\'etale over $(A_2, I_2)$." ], "refs": [ "restricted-remark-base-change" ], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-zero-after-modding-out} and", "Definition \\ref{definition-rig-etale-homomorphism} and the fact", "that $I_2^c \\subset I_1A_2$ for some $c \\geq 0$ as $A_2$ is Noetherian." ], "refs": [ "restricted-lemma-zero-after-modding-out", "restricted-definition-rig-etale-homomorphism" ], "ref_ids": [ 2317, 2435 ] } ], "ref_ids": [ 2448 ] }, { "id": 2319, "type": "theorem", "label": "restricted-lemma-fully-faithful-etale-over-complement", "categories": [ "restricted" ], "title": "restricted-lemma-fully-faithful-etale-over-complement", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.", "Let $B$ be a finite type $A$-algebra such that", "$\\Spec(B) \\to \\Spec(A)$ is \\'etale over $\\Spec(A) \\setminus V(I)$.", "Let $C$ be a Noetherian $A$-algebra. Then any $A$-algebra", "map $B^\\wedge \\to C^\\wedge$ of $I$-adic completions", "comes from a unique $A$-algebra map", "$$", "B \\longrightarrow C^h", "$$", "where $C^h$ is the henselization of the pair $(C, IC)$ as", "in More on Algebra, Lemma \\ref{more-algebra-lemma-henselization}.", "Moreover, any $A$-algebra homomorphism $B \\to C^h$ factors through", "some \\'etale $C$-algebra $C'$ such that $C/IC \\to C'/IC'$ is an isomorphism." ], "refs": [ "more-algebra-lemma-henselization" ], "proofs": [ { "contents": [ "Uniqueness follows from the fact that $C^h$ is a subring of", "$C^\\wedge$, see for example", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-Noetherian-pair}.", "The final assertion follows from the fact that $C^h$ is the filtered colimit", "of these $C$-algebras $C'$, see proof of", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization}.", "Having said this we now turn to the proof of existence.", "\\medskip\\noindent", "Let $\\varphi : B^\\wedge \\to C^\\wedge$ be the given map.", "This defines a section", "$$", "\\sigma : (B \\otimes_A C)^\\wedge \\longrightarrow C^\\wedge", "$$", "of the completion of the map $C \\to B \\otimes_A C$. We may", "replace $(A, I, B, C, \\varphi)$ by $(C, IC, B \\otimes_A C, C, \\sigma)$.", "In this way we see that we may assume that $A = C$.", "\\medskip\\noindent", "Proof of existence in the case $A = C$. In this case the map", "$\\varphi : B^\\wedge \\to A^\\wedge$ is necessarily surjective.", "By Lemmas \\ref{lemma-rig-etale} and \\ref{lemma-exact-sequence-NL}", "we see that the cohomology groups of", "$\\NL_{A^\\wedge/\\!_\\varphi B^\\wedge}^\\wedge$", "are annihilated by a power of $I$. Since $\\varphi$ is surjective,", "this implies that $\\Ker(\\varphi)/\\Ker(\\varphi)^2$ is annihilated", "by a power of $I$. Hence $\\varphi : B^\\wedge \\to A^\\wedge$", "is the completion of a finite type $B$-algebra $B \\to D$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-quotient-by-idempotent}.", "Hence $A \\to D$ is a finite type algebra map which induces an isomorphism", "$A^\\wedge \\to D^\\wedge$. By", "Lemma \\ref{lemma-rig-etale} we may replace $D$ by a localization", "and assume that $A \\to D$ is \\'etale away from $V(I)$.", "Since $A^\\wedge \\to D^\\wedge$ is an isomorphism, we see that", "$\\Spec(D) \\to \\Spec(A)$ is also \\'etale in a neighbourhood of $V(ID)$", "(for example by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds}).", "Thus $\\Spec(D) \\to \\Spec(A)$ is \\'etale. Therefore $D$ maps to", "$A^h$ and the lemma is proved." ], "refs": [ "more-algebra-lemma-henselization-Noetherian-pair", "more-algebra-lemma-henselization", "restricted-lemma-rig-etale", "restricted-lemma-exact-sequence-NL", "more-algebra-lemma-quotient-by-idempotent", "restricted-lemma-rig-etale", "more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds" ], "ref_ids": [ 9874, 9871, 2316, 2300, 10475, 2316, 13743 ] } ], "ref_ids": [ 9871 ] }, { "id": 2320, "type": "theorem", "label": "restricted-lemma-lift-approximation", "categories": [ "restricted" ], "title": "restricted-lemma-lift-approximation", "contents": [ "Let $A$ be a Noetherian ring and $I \\subset A$ an ideal.", "Let $J \\subset A$ be a nilpotent ideal. Consider a commutative diagram", "$$", "\\xymatrix{", "C \\ar[r] & C_0 \\ar@{=}[r] & C/JC \\\\", "& B_0 \\ar[u] \\\\", "A \\ar[r] \\ar[uu] & A_0 \\ar[u] \\ar@{=}[r] & A/J", "}", "$$", "whose vertical arrows are of finite type such that", "\\begin{enumerate}", "\\item $\\Spec(C) \\to \\Spec(A)$ is \\'etale over $\\Spec(A) \\setminus V(I)$,", "\\item $\\Spec(B_0) \\to \\Spec(A_0)$ is \\'etale over", "$\\Spec(A_0) \\setminus V(IA_0)$, and", "\\item $B_0 \\to C_0$ is \\'etale and induces an isomorphism", "$B_0/IB_0 = C_0/IC_0$.", "\\end{enumerate}", "Then we can fill in the diagram above to a commutative diagram", "$$", "\\xymatrix{", "C \\ar[r] & C/JC \\\\", "B \\ar[u] \\ar[r] & B_0 \\ar[u] \\\\", "A \\ar[r] \\ar[u] & A/J \\ar[u]", "}", "$$", "with $A \\to B$ of finite type, $B/JB = B_0$, $B \\to C$ \\'etale, and", "$\\Spec(B) \\to \\Spec(A)$ \\'etale over $\\Spec(A) \\setminus V(I)$." ], "refs": [], "proofs": [ { "contents": [ "Set $X = \\Spec(A)$, $X_0 = \\Spec(A_0)$, $Y_0 = \\Spec(B_0)$,", "$Z = \\Spec(C)$, $Z_0 = \\Spec(C_0)$. Furthermore, denote", "$U \\subset X$, $U_0 \\subset X_0$, $V_0 \\subset Y_0$,", "$W \\subset Z$, $W_0 \\subset Z_0$ the complement of the", "vanishing set of $I$. Here is a picture to help visualize the", "situation:", "$$", "\\xymatrix{", "Z \\ar[dd] & Z_0 \\ar[l] \\ar[d] \\\\", "& Y_0 \\ar[d] \\\\", "X & X_0 \\ar[l]", "}", "\\quad\\quad\\quad", "\\xymatrix{", "W \\ar[dd] & W_0 \\ar[l] \\ar[d] \\\\", "& V_0 \\ar[d] \\\\", "U & U_0 \\ar[l]", "}", "$$", "The conditions in the lemma guarantee that", "$$", "\\xymatrix{", "W_0 \\ar[r] \\ar[d] & Z_0 \\ar[d] \\\\", "V_0 \\ar[r] & Y_0", "}", "$$", "is an elementary distinguished square, see", "Derived Categories of Spaces, Definition", "\\ref{spaces-perfect-definition-elementary-distinguished-square}.", "In addition we know that", "$W_0 \\to U_0$ and $V_0 \\to U_0$ are \\'etale. The morphism", "$X_0 \\subset X$ is a finite order thickening as $J$ is assumed nilpotent.", "By the topological invariance of the \\'etale site", "we can find a unique \\'etale morphism $V \\to X$ of schemes", "with $V_0 = V \\times_X X_0$ and we can lift the given morphism", "$W_0 \\to V_0$ to a unique morphism $W \\to V$ over $X$.", "See \\'Etale Morphisms, Theorem", "\\ref{etale-theorem-remarkable-equivalence}.", "Since $W_0 \\to V_0$ is separated, the morphism $W \\to V$ is separated too,", "see for example", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-deform-property}.", "By Pushouts of Spaces, Lemma", "\\ref{spaces-pushouts-lemma-construct-elementary-distinguished-square}", "we can construct an elementary distinguished square", "$$", "\\xymatrix{", "W \\ar[r] \\ar[d] & Z \\ar[d] \\\\", "V \\ar[r] & Y", "}", "$$", "in the category of algebraic spaces over $X$. Since the base change", "of an elementary distinguished square is an elementary distinguished", "square (Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-make-more-elementary-distinguished-squares})", "we see that", "$$", "\\xymatrix{", "W_0 \\ar[r] \\ar[d] & Z_0 \\ar[d] \\\\", "V_0 \\ar[r] & Y \\times_X X_0", "}", "$$", "is an elementary distinguished square. It follows that there is a", "unique isomorphism $Y \\times_X X_0 = Y_0$ compatible with the two", "squares involving these spaces because", "elementary distinguished squares are pushouts (Pushouts of Spaces, Lemma", "\\ref{spaces-pushouts-lemma-elementary-distinguished-square-pushout}).", "It follows that $Y$ is affine by", "Limits of Spaces, Proposition \\ref{spaces-limits-proposition-affine}.", "Write $Y = \\Spec(B)$. It is clear that $B$ fits into the desired diagram", "and satisfies all the properties required of it." ], "refs": [ "spaces-perfect-definition-elementary-distinguished-square", "etale-theorem-remarkable-equivalence", "more-morphisms-lemma-deform-property", "spaces-pushouts-lemma-construct-elementary-distinguished-square", "spaces-perfect-lemma-make-more-elementary-distinguished-squares", "spaces-pushouts-lemma-elementary-distinguished-square-pushout", "spaces-limits-proposition-affine" ], "ref_ids": [ 2764, 10694, 13725, 10868, 2669, 10867, 4658 ] } ], "ref_ids": [] }, { "id": 2321, "type": "theorem", "label": "restricted-lemma-approximate-principal", "categories": [ "restricted" ], "title": "restricted-lemma-approximate-principal", "contents": [ "\\begin{reference}", "The rig-\\'etale case of \\cite[III Theorem 7]{Elkik}", "\\end{reference}", "Let $A$ be a Noetherian ring and $I = (a)$ a principal ideal.", "Let $B$ be an object of (\\ref{equation-C-prime}) which is", "rig-\\'etale over $(A, I)$.", "Then there exists a finite type $A$-algebra $C$ and an", "isomorphism $B \\cong C^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $B = A[x_1, \\ldots, x_r]^\\wedge/J$.", "By Lemma \\ref{lemma-equivalent-with-artin} part (6) we can find", "$c \\geq 0$ and $f_1, \\ldots, f_r \\in J$ such that", "$\\det_{1 \\leq i, j \\leq r}(\\partial f_j/\\partial x_i)$ divides", "$a^c$ in $B$ and", "$a^c J \\subset (f_1, \\ldots, f_r) + J^2$.", "Hence Lemma \\ref{lemma-approximate-presentation-rig-smooth} applies.", "This finishes the proof, but we'd like to point out that in", "this case the use of", "Lemma \\ref{lemma-get-morphism-general-better}", "can be replaced by the much easier", "Lemma \\ref{lemma-get-morphism-principal}." ], "refs": [ "restricted-lemma-equivalent-with-artin", "restricted-lemma-approximate-presentation-rig-smooth", "restricted-lemma-get-morphism-general-better", "restricted-lemma-get-morphism-principal" ], "ref_ids": [ 2314, 2313, 2306, 2308 ] } ], "ref_ids": [] }, { "id": 2322, "type": "theorem", "label": "restricted-lemma-approximate", "categories": [ "restricted" ], "title": "restricted-lemma-approximate", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.", "Let $B$ be an object of (\\ref{equation-C-prime}) which is", "rig-\\'etale over $(A, I)$.", "Then there exists a finite type $A$-algebra $C$ and an", "isomorphism $B \\cong C^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "We prove this lemma by induction on the number of generators of $I$.", "Say $I = (a_1, \\ldots, a_t)$. If $t = 0$, then $I = 0$ and there", "is nothing to prove. If $t = 1$, then the lemma follows from", "Lemma \\ref{lemma-approximate-principal}. Assume $t > 1$.", "\\medskip\\noindent", "For any $m \\geq 1$ set $\\bar A_m = A/(a_t^m)$. Consider the ideal", "$\\bar I_m = (\\bar a_1, \\ldots, \\bar a_{t - 1})$ in $\\bar A_m$.", "Observe that $V(I \\bar A_m) = V(\\bar I_m)$.", "Let $B_m = B/(a_t^m)$ be the base change of $B$ for the", "map $(A, I) \\to (\\bar A_m, \\bar I_m)$, see Remark \\ref{remark-take-bar}.", "By Lemma \\ref{lemma-base-change-rig-etale-homomorphism}", "we find that $B_m$ is rig-\\'etale over $(\\bar A_m, \\bar I_m)$.", "\\medskip\\noindent", "By induction hypothesis (on $t$) we can find a finite type", "$\\bar A_m$-algebra $C_m$ and a map $C_m \\to B_m$ which induces an", "isomorphism $C_m^\\wedge \\cong B_m$", "where the completion is with respect to $\\bar I_m$.", "By Lemma \\ref{lemma-rig-etale} we may assume that", "$\\Spec(C_m) \\to \\Spec(\\bar A_m)$ is \\'etale", "over $\\Spec(\\bar A_m) \\setminus V(\\bar I_m)$.", "\\medskip\\noindent", "We claim that we may choose $A_m \\to C_m \\to B_m$ as in the previous", "paragraph such that moreover there are isomorphisms", "$C_m/(a_t^{m - 1}) \\to C_{m - 1}$ compatible with the given", "$A$-algebra structure and the maps to $B_{m - 1} = B_m/(a_t^{m - 1})$.", "Namely, first fix a choice of $A_1 \\to C_1 \\to B_1$.", "Suppose we have found $C_{m - 1} \\to C_{m - 2} \\to \\ldots \\to C_1$", "with the desired properties.", "Note that $C_m/(a_t^{m - 1})$ is \\'etale over", "$\\Spec(\\bar A_{m - 1}) \\setminus V(\\bar I_{m - 1})$.", "Hence by Lemma \\ref{lemma-fully-faithful-etale-over-complement}", "there exists an \\'etale extension $C_{m - 1} \\to C'_{m - 1}$", "which induces an isomorphism modulo $\\bar I_{m - 1}$ and an", "$\\bar A_{m - 1}$-algebra map $C_m/(a_t^{m - 1}) \\to C'_{m - 1}$", "inducing the isomorphism $B_m/(a_t^{m - 1}) \\to B_{m - 1}$ on completions.", "Note that $C_m/(a_t^{m - 1}) \\to C'_{m - 1}$ is \\'etale over the complement", "of $V(\\bar I_{m - 1})$ by", "Morphisms, Lemma \\ref{morphisms-lemma-etale-permanence}", "and over $V(\\bar I_{m - 1})$ induces an isomorphism on completions", "hence is \\'etale there too (for example by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds}).", "Thus $C_m/(a_t^{m - 1}) \\to C'_{m - 1}$ is \\'etale. By the", "topological invariance of \\'etale morphisms", "(\\'Etale Morphisms, Theorem \\ref{etale-theorem-remarkable-equivalence})", "there exists an \\'etale ring map $C_m \\to C'_m$ such that", "$C_m/(a_t^{m - 1}) \\to C'_{m - 1}$ is isomorphic to", "$C_m/(a_t^{m - 1}) \\to C'_m/(a_t^{m - 1})$. Observe that the", "$\\bar I_m$-adic completion of $C'_m$ is equal to the $\\bar I_m$-adic", "completion of $C_m$, i.e., to $B_m$ (details omitted).", "We apply Lemma \\ref{lemma-lift-approximation} to the diagram", "$$", "\\xymatrix{", " & C'_m \\ar[r] & C'_m/(a_t^{m - 1}) \\\\", "C''_m \\ar@{..>}[ru] \\ar@{..>}[rr] & & C_{m - 1} \\ar[u] \\\\", " & \\bar A_m \\ar[r] \\ar[uu] \\ar@{..>}[lu] & \\bar A_{m - 1} \\ar[u]", "}", "$$", "to see that there exists a ``lift'' of $C''_m$ of $C_{m - 1}$", "to an algebra over $\\bar A_m$ with all the desired properties.", "\\medskip\\noindent", "By construction $(C_m)$ is an object of the category", "(\\ref{equation-C}) for the principal ideal $(a_t)$.", "Thus the inverse limit $B' = \\lim C_m$ is an $(a_t)$-adically", "complete $A$-algebra such that $B'/a_t B'$ is of finite type", "over $A/(a_t)$, see Lemma \\ref{lemma-topologically-finite-type}.", "By construction the $I$-adic completion of $B'$ is isomorphic to $B$", "(details omitted). Consider the complex $\\NL_{B'/A}^\\wedge$ constructed", "using the $(a_t)$-adic topology. Choosing a presentation for $B'$", "(which induces a similar presentation for $B$) the reader immediately", "sees that $\\NL_{B'/A}^\\wedge \\otimes_{B'} B = \\NL_{B/A}^\\wedge$.", "Since $a_t \\in I$ and since the cohomology modules of", "$\\NL_{B'/A}^\\wedge$ are finite $B'$-modules (hence complete for the", "$a_t$-adic topology), we conclude that $a_t^c$ acts as zero on", "these cohomologies as the same thing is true by assumption for", "$\\NL_{B/A}^\\wedge$. Thus $B'$ is rig-\\'etale over $(A, (a_t))$", "by Lemma \\ref{lemma-equivalent-with-artin}.", "Hence finally, we may apply Lemma \\ref{lemma-approximate-principal}", "to $B'$ over $(A, (a_t))$ to finish the proof." ], "refs": [ "restricted-lemma-approximate-principal", "restricted-remark-take-bar", "restricted-lemma-base-change-rig-etale-homomorphism", "restricted-lemma-rig-etale", "restricted-lemma-fully-faithful-etale-over-complement", "morphisms-lemma-etale-permanence", "more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds", "etale-theorem-remarkable-equivalence", "restricted-lemma-lift-approximation", "restricted-lemma-topologically-finite-type", "restricted-lemma-equivalent-with-artin", "restricted-lemma-approximate-principal" ], "ref_ids": [ 2321, 2449, 2318, 2316, 2319, 5375, 13743, 10694, 2320, 2294, 2314, 2321 ] } ], "ref_ids": [] }, { "id": 2323, "type": "theorem", "label": "restricted-lemma-approximate-by-etale-over-complement", "categories": [ "restricted" ], "title": "restricted-lemma-approximate-by-etale-over-complement", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.", "Let $B$ be an $I$-adically complete $A$-algebra with $A/I \\to B/IB$", "of finite type. The equivalent conditions of", "Lemma \\ref{lemma-equivalent-with-artin} are also equivalent to", "\\begin{enumerate}", "\\item[(8)]", "\\label{item-algebraize}", "there exists a finite type $A$-algebra $C$ such that", "$\\Spec(C) \\to \\Spec(A)$ is \\'etale over $\\Spec(A) \\setminus V(I)$", "and such that $B \\cong C^\\wedge$.", "\\end{enumerate}" ], "refs": [ "restricted-lemma-equivalent-with-artin" ], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-equivalent-with-artin}, \\ref{lemma-approximate}, and", "\\ref{lemma-rig-etale}. Small detail omitted." ], "refs": [ "restricted-lemma-equivalent-with-artin", "restricted-lemma-approximate", "restricted-lemma-rig-etale" ], "ref_ids": [ 2314, 2322, 2316 ] } ], "ref_ids": [ 2314 ] }, { "id": 2324, "type": "theorem", "label": "restricted-lemma-finite-type", "categories": [ "restricted" ], "title": "restricted-lemma-finite-type", "contents": [ "Let $A$ and $B$ be adic topological rings which have a finitely generated", "ideal of definition. Let $\\varphi : A \\to B$ be a continuous ring homomorphism.", "The following are equivalent:", "\\begin{enumerate}", "\\item $\\varphi$ is adic and $B$ is topologically of finite type over $A$,", "\\item $\\varphi$ is taut and $B$ is topologically of finite type over $A$,", "\\item there exists an ideal of definition $I \\subset A$ such that", "the topology on $B$ is the $I$-adic topology and there exist an ideal", "of definition $I' \\subset A$ such that $A/I' \\to B/I'B$ is of finite type,", "\\item for all ideals of definition $I \\subset A$ the topology on $B$", "is the $I$-adic topology and $A/I \\to B/IB$ is of finite type,", "\\item there exists an ideal of definition $I \\subset A$ such that", "the topology on $B$ is the $I$-adic topology and $B$ is in the category", "(\\ref{equation-C-prime}),", "\\item for all ideals of definition $I \\subset A$ the topology on $B$", "is the $I$-adic topology and $B$ is in the category (\\ref{equation-C-prime}),", "\\item $B$ as a topological $A$-algebra is the quotient of", "$A\\{x_1, \\ldots, x_r\\}$ by a closed ideal,", "\\item $B$ as a topological $A$-algebra is the quotient of", "$A[x_1, \\ldots, x_r]^\\wedge$ by a closed ideal where", "$A[x_1, \\ldots, x_r]^\\wedge$ is the completion of $A[x_1, \\ldots, x_r]$", "with respect to some ideal of definition of $A$, and", "\\item add more here.", "\\end{enumerate}", "Moreover, these equivalent conditions define", "a local property of morphisms of $\\text{WAdm}^{adic*}$ as defined in", "Formal Spaces, Remark \\ref{formal-spaces-remark-variant-adic-star}." ], "refs": [ "formal-spaces-remark-variant-adic-star" ], "proofs": [ { "contents": [ "Taut ring homomorphisms are defined in", "Formal Spaces, Definition \\ref{formal-spaces-definition-taut}.", "Adic ring homomorphisms are defined in", "Formal Spaces, Definition", "\\ref{formal-spaces-definition-adic-homomorphism}.", "The lemma follows from a combination of", "Formal Spaces, Lemmas", "\\ref{formal-spaces-lemma-quotient-restricted-power-series},", "\\ref{formal-spaces-lemma-quotient-restricted-power-series-admissible}, and", "\\ref{formal-spaces-lemma-adic-homomorphism}. We omit the details.", "To be sure, there is no difference between the topological rings", "$A[x_1, \\ldots, x_n]^\\wedge$ and $A\\{x_1, \\ldots, x_r\\}$, see", "Formal Spaces, Remark", "\\ref{formal-spaces-remark-I-adic-completion-and-restricted-power-series}." ], "refs": [ "formal-spaces-definition-taut", "formal-spaces-definition-adic-homomorphism", "formal-spaces-lemma-quotient-restricted-power-series", "formal-spaces-lemma-quotient-restricted-power-series-admissible", "formal-spaces-lemma-adic-homomorphism", "formal-spaces-remark-I-adic-completion-and-restricted-power-series" ], "ref_ids": [ 3976, 3987, 3951, 3952, 3929, 4017 ] } ], "ref_ids": [ 4007 ] }, { "id": 2325, "type": "theorem", "label": "restricted-lemma-base-change-finite-type", "categories": [ "restricted" ], "title": "restricted-lemma-base-change-finite-type", "contents": [ "Consider the property $P$ on arrows of $\\textit{WAdm}^{adic*}$ defined in", "Lemma \\ref{lemma-finite-type}. Then $P$ is stable under base change as", "defined in Formal Spaces, Remark", "\\ref{formal-spaces-remark-base-change-variant-adic-star}." ], "refs": [ "restricted-lemma-finite-type", "formal-spaces-remark-base-change-variant-adic-star" ], "proofs": [ { "contents": [ "The statement makes sense by Lemma \\ref{lemma-finite-type}.", "To see that it is true assume we have morphisms", "$B \\to A$ and $B \\to C$ in $\\textit{WAdm}^{adic*}$", "and that as a topological $B$-algebra we have", "$A = B\\{x_1, \\ldots, x_r\\}/J$ for some closed ideal $J$.", "Then $A \\widehat{\\otimes}_B C$ is isomorphic to the", "quotient of $C\\{x_1, \\ldots, x_r\\}/J'$ where", "$J'$ is the closure of $JC\\{x_1, \\ldots, x_r\\}$.", "Some details omitted." ], "refs": [ "restricted-lemma-finite-type" ], "ref_ids": [ 2324 ] } ], "ref_ids": [ 2324, 4009 ] }, { "id": 2326, "type": "theorem", "label": "restricted-lemma-composition-finite-type", "categories": [ "restricted" ], "title": "restricted-lemma-composition-finite-type", "contents": [ "Consider the property $P$ on arrows of $\\textit{WAdm}^{adic*}$ defined in", "Lemma \\ref{lemma-finite-type}. Then $P$ is stable under composition as", "defined in Formal Spaces, Remark", "\\ref{formal-spaces-remark-composition-variant-adic-star}." ], "refs": [ "restricted-lemma-finite-type", "formal-spaces-remark-composition-variant-adic-star" ], "proofs": [ { "contents": [ "The statement makes sense by Lemma \\ref{lemma-finite-type}.", "The easiest way to prove it is true is to show that", "(a) compositions of adic ring maps between adic topological rings", "are adic and (b) that compositions of continuous ring maps", "preserves the property of being topologically of finite type.", "We omit the details." ], "refs": [ "restricted-lemma-finite-type" ], "ref_ids": [ 2324 ] } ], "ref_ids": [ 2324, 4012 ] }, { "id": 2327, "type": "theorem", "label": "restricted-lemma-finite-type-morphisms", "categories": [ "restricted" ], "title": "restricted-lemma-finite-type-morphisms", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "locally adic* formal algebraic spaces over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$", "representable by algebraic spaces and \\'etale, the morphism $U \\to V$", "corresponds to an arrow of $\\textit{WAdm}^{adic*}$ which is", "adic and topologically of finite type,", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "and for each $j$", "a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "such that each $X_{ji} \\to Y_j$ corresponds", "to an arrow of $\\textit{WAdm}^{adic*}$ which is adic and", "topologically of finite type,", "\\item there exist a covering $\\{X_i \\to X\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$", "is an affine formal algebraic space, $Y_i \\to Y$ is representable", "by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds", "to an arrow of $\\textit{WAdm}^{adic*}$ which is adic and topologically", "of finite type, and", "\\item $f$ is locally of finite type.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "Immediate consequence of the equivalence of (1) and (2) in", "Lemma \\ref{lemma-finite-type} and", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-finite-type-local-property}." ], "refs": [ "restricted-lemma-finite-type", "formal-spaces-lemma-finite-type-local-property" ], "ref_ids": [ 2324, 3954 ] } ], "ref_ids": [ 3981, 3981, 3981 ] }, { "id": 2328, "type": "theorem", "label": "restricted-lemma-finite-type-red", "categories": [ "restricted" ], "title": "restricted-lemma-finite-type-red", "contents": [ "For an arrow $\\varphi : A \\to B$ in $\\text{WAdm}^{count}$ consider", "the property $P(\\varphi)=$``the induced ring homomorphism", "$A/\\mathfrak a \\to B/\\mathfrak b$ is of finite type''", "where $\\mathfrak a \\subset A$ and $\\mathfrak b \\subset B$ are the ideals", "of topologically nilpotent elements. Then $P$ is a local property", "as defined in", "Formal Spaces, Situation", "\\ref{formal-spaces-situation-local-property}." ], "refs": [], "proofs": [ { "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "B \\ar[r] & (B')^\\wedge \\\\", "A \\ar[r] \\ar[u]^\\varphi & (A')^\\wedge \\ar[u]_{\\varphi'}", "}", "$$", "as in Formal Spaces, Situation", "\\ref{formal-spaces-situation-local-property}.", "Taking $\\text{Spf}$ of this diagram we obtain", "$$", "\\xymatrix{", "\\text{Spf}(B) \\ar[d] &", "\\text{Spf}((B')^\\wedge) \\ar[l] \\ar[d] \\\\", "\\text{Spf}(A) &", "\\text{Spf}((A')^\\wedge) \\ar[l]", "}", "$$", "of affine formal algebraic spaces whose horizontal arrows are", "representable by algebraic spaces and \\'etale by", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-etale}.", "Hence we obtain a commutative diagram of affine schemes", "$$", "\\xymatrix{", "\\text{Spf}(B)_{red} \\ar[d]^f &", "\\text{Spf}((B')^\\wedge)_{red} \\ar[l]^g \\ar[d]^{f'} \\\\", "\\text{Spf}(A)_{red} &", "\\text{Spf}((A')^\\wedge)_{red} \\ar[l]", "}", "$$", "whose horizontal arrows are \\'etale by", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-reduction-smooth}.", "By Formal Spaces, Example", "\\ref{formal-spaces-example-reduction-affine-formal-spectrum} and", "Lemma \\ref{formal-spaces-lemma-etale-surjective}", "conditions (1), (2), and (3) of", "Formal Spaces, Situation", "\\ref{formal-spaces-situation-local-property}", "translate into the following statements", "\\begin{enumerate}", "\\item if $f$ is locally of finite type, then $f'$ is locally of finite type,", "\\item if $f'$ is locally of finite type and $g$ is surjective, then", "$f$ is locally of finite type, and", "\\item if $T_i \\to S$, $i = 1, \\ldots, n$ are locally of finite type, then", "$\\coprod_{i = 1, \\ldots, n} T_i \\to S$ is locally of finite type.", "\\end{enumerate}", "Properties (1) and (2) follow from the fact that being locally", "of finite type is local on the source and target in the", "\\'etale topology, see discussion in", "Morphisms of Spaces, Section \\ref{spaces-morphisms-section-finite-type}.", "Property (3) is a straightforward consequence of the definition." ], "refs": [ "formal-spaces-lemma-etale", "formal-spaces-lemma-reduction-smooth", "formal-spaces-lemma-etale-surjective" ], "ref_ids": [ 3913, 3880, 3914 ] } ], "ref_ids": [] }, { "id": 2329, "type": "theorem", "label": "restricted-lemma-base-change-finite-type-red", "categories": [ "restricted" ], "title": "restricted-lemma-base-change-finite-type-red", "contents": [ "Consider the property $P$ on arrows of $\\textit{WAdm}^{count}$ defined in", "Lemma \\ref{lemma-finite-type-red}. Then $P$ is stable under base change", "(Formal Spaces, Situation", "\\ref{formal-spaces-situation-base-change-local-property})." ], "refs": [ "restricted-lemma-finite-type-red" ], "proofs": [ { "contents": [ "The statement makes sense by Lemma \\ref{lemma-finite-type-red}.", "To see that it is true assume we have morphisms $B \\to A$ and $B \\to C$", "in $\\textit{WAdm}^{count}$ such that $B/\\mathfrak b \\to A/\\mathfrak a$", "is of finite type where $\\mathfrak b \\subset B$ and $\\mathfrak a \\subset A$", "are the ideals of topologically nilpotent elements.", "Since $A$ and $B$ are weakly admissible, the ideals", "$\\mathfrak a$ and $\\mathfrak b$ are open.", "Let $\\mathfrak c \\subset C$ be the (open) ideal", "of topologically nilpotent elements. Then we find a surjection", "$A \\widehat{\\otimes}_B C \\to", "A/\\mathfrak a \\otimes_{B/\\mathfrak b} C/\\mathfrak c$", "whose kernel is a weak ideal of definition and hence consists", "of topologically nilpotent elements", "(please compare with the proof of Formal Spaces,", "Lemma \\ref{formal-spaces-lemma-completed-tensor-product}). Since already", "$C/\\mathfrak c \\to A/\\mathfrak a \\otimes_{B/\\mathfrak b} C/\\mathfrak c$", "is of finite type as a base change of $B/\\mathfrak b \\to A/\\mathfrak a$", "we conclude." ], "refs": [ "restricted-lemma-finite-type-red", "formal-spaces-lemma-completed-tensor-product" ], "ref_ids": [ 2328, 3865 ] } ], "ref_ids": [ 2328 ] }, { "id": 2330, "type": "theorem", "label": "restricted-lemma-composition-finite-type-red", "categories": [ "restricted" ], "title": "restricted-lemma-composition-finite-type-red", "contents": [ "Consider the property $P$ on arrows of $\\textit{WAdm}^{count}$ defined in", "Lemma \\ref{lemma-finite-type-red}. Then $P$ is stable under composition", "(Formal Spaces, Situation", "\\ref{formal-spaces-situation-composition-local-property})." ], "refs": [ "restricted-lemma-finite-type-red" ], "proofs": [ { "contents": [ "Omitted. Hint: compositions of finite type ring maps are of finite type." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 2328 ] }, { "id": 2331, "type": "theorem", "label": "restricted-lemma-finite-type-finite-type-red", "categories": [ "restricted" ], "title": "restricted-lemma-finite-type-finite-type-red", "contents": [ "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{count}$.", "If $\\varphi$ is taut and topologically of finite type, then $\\varphi$", "satisfies the condition defined in Lemma \\ref{lemma-finite-type-red}." ], "refs": [ "restricted-lemma-finite-type-red" ], "proofs": [ { "contents": [ "This is an easy consequence of the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 2328 ] }, { "id": 2332, "type": "theorem", "label": "restricted-lemma-Noetherian-finite-type-red", "categories": [ "restricted" ], "title": "restricted-lemma-Noetherian-finite-type-red", "contents": [ "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$", "satisfying the condition defined in Lemma \\ref{lemma-finite-type-red}.", "Then $A \\to B$ is topologically of finite type." ], "refs": [ "restricted-lemma-finite-type-red" ], "proofs": [ { "contents": [ "Let $\\mathfrak b \\subset B$ be the", "ideal of topologically nilpotent elements. Choose $b_1, \\ldots, b_r \\in B$", "which map to generators of $B/\\mathfrak b$ over $A$.", "Choose generators $b_{r + 1}, \\ldots, b_s$ of the ideal", "$\\mathfrak b$. We claim that the image of", "$$", "\\varphi : A[x_1, \\ldots, x_s] \\longrightarrow B, \\quad", "x_i \\longmapsto b_i", "$$", "has dense image. Namely, if $b \\in \\mathfrak b^n$ for some $n \\geq 0$,", "then we can write", "$b = \\sum b_E b_{r + 1}^{e_{r + 1}} \\ldots b_s^{e_s}$ for multiindices", "$E = (e_{r + 1}, \\ldots, e_s)$ with $|E| = \\sum e_i = n$ and $b_E \\in B$.", "Next, we can write $b_E = f_E(b_1, \\ldots, b_r) + b'_E$", "with $b'_E \\in \\mathfrak b$ and $f_E \\in A[x_1, \\ldots, x_r]$.", "Combined we obtain $b \\in \\Im(\\varphi) + \\mathfrak b^{n + 1}$.", "By induction we see that $B = \\Im(\\varphi) + \\mathfrak b^n$ for all", "$n \\geq 0$ which mplies what we want as $\\mathfrak b$ is an ideal", "of definition of $B$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 2328 ] }, { "id": 2333, "type": "theorem", "label": "restricted-lemma-Noetherian-adic-finite-type-red", "categories": [ "restricted" ], "title": "restricted-lemma-Noetherian-adic-finite-type-red", "contents": [ "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$.", "If $\\varphi$ is adic the following are equivalent", "\\begin{enumerate}", "\\item $\\varphi$ satisfies the condition defined in", "Lemma \\ref{lemma-finite-type-red} and", "\\item $\\varphi$ satisfies the condition defined in", "Lemma \\ref{lemma-finite-type}.", "\\end{enumerate}" ], "refs": [ "restricted-lemma-finite-type-red", "restricted-lemma-finite-type" ], "proofs": [ { "contents": [ "Omitted. Hint: For the proof of (1) $\\Rightarrow$ (2) use", "Lemma \\ref{lemma-Noetherian-finite-type-red}." ], "refs": [ "restricted-lemma-Noetherian-finite-type-red" ], "ref_ids": [ 2332 ] } ], "ref_ids": [ 2328, 2324 ] }, { "id": 2334, "type": "theorem", "label": "restricted-lemma-finite-type-red-morphisms", "categories": [ "restricted" ], "title": "restricted-lemma-finite-type-red-morphisms", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "locally countably indexed formal algebraic spaces over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$", "representable by algebraic spaces and \\'etale, the morphism $U \\to V$", "corresponds to an arrow of $\\textit{WAdm}^{count}$ satisfying the", "property defined in Lemma \\ref{lemma-finite-type-red},", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "and for each $j$", "a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "such that each $X_{ji} \\to Y_j$ corresponds", "to an arrow of $\\textit{WAdm}^{count}$ satisfying the", "property defined in Lemma \\ref{lemma-finite-type-red},", "\\item there exist a covering $\\{X_i \\to X\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$", "is an affine formal algebraic space, $Y_i \\to Y$ is representable", "by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds", "to an arrow of $\\textit{WAdm}^{count}$ satisfying the", "property defined in Lemma \\ref{lemma-finite-type-red}, and", "\\item the morphism $f_{red} : X_{red} \\to Y_{red}$ is locally of finite type.", "\\end{enumerate}" ], "refs": [ "restricted-lemma-finite-type-red", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "restricted-lemma-finite-type-red", "formal-spaces-definition-formal-algebraic-space", "restricted-lemma-finite-type-red" ], "proofs": [ { "contents": [ "The equivalence of (1), (2), and (3) follows from", "Lemma \\ref{lemma-finite-type-red} and an application of", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-property-defines-property-morphisms}.", "Let $Y_j$ and $X_{ji}$ be as in (2). Then", "\\begin{itemize}", "\\item The families $\\{Y_{j, red} \\to Y_{red}\\}$ and", "$\\{X_{ji, red} \\to X_{red}\\}$ are \\'etale coverings by affine schemes.", "This follows from the discussion in the proof of", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-reduction-formal-algebraic-space}", "or directly from", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-reduction-smooth}.", "\\item If $X_{ji} \\to Y_j$ corresponds to the morphism", "$B_j \\to A_{ji}$ of $\\textit{WAdm}^{count}$, then", "$X_{ji, red} \\to Y_{j, red}$ corresponds to the ring map", "$B_j/\\mathfrak b_j \\to A_{ji}/\\mathfrak a_{ji}$ where", "$\\mathfrak b_j$ and $\\mathfrak a_{ji}$ are the ideals of", "topologically nilpotent elements. This follows from", "Formal Spaces, Example", "\\ref{formal-spaces-example-reduction-affine-formal-spectrum}.", "Hence $X_{ji, red} \\to Y_{j, red}$ is locally of finite type", "if and only if $B_j \\to A_{ji}$ satisfies the", "property defined in Lemma \\ref{lemma-finite-type-red}.", "\\end{itemize}", "The equivalence of (2) and (4) follows from these remarks because", "being locally of finite type is a property of morphisms of algebraic", "spaces which is \\'etale local on source and target, see discussion in", "Morphisms of Spaces, Section \\ref{spaces-morphisms-section-finite-type}." ], "refs": [ "restricted-lemma-finite-type-red", "formal-spaces-lemma-property-defines-property-morphisms", "formal-spaces-lemma-reduction-formal-algebraic-space", "formal-spaces-lemma-reduction-smooth", "restricted-lemma-finite-type-red" ], "ref_ids": [ 2328, 3923, 3879, 3880, 2328 ] } ], "ref_ids": [ 2328, 3981, 3981, 2328, 3981, 2328 ] }, { "id": 2335, "type": "theorem", "label": "restricted-lemma-flat-axioms", "categories": [ "restricted" ], "title": "restricted-lemma-flat-axioms", "contents": [ "The property $P(\\varphi)=$``$\\varphi$ is flat'' on arrows", "of $\\textit{WAdm}^{Noeth}$ is a local property as defined in", "Formal Spaces, Remark \\ref{formal-spaces-remark-variant-Noetherian}." ], "refs": [ "formal-spaces-remark-variant-Noetherian" ], "proofs": [ { "contents": [ "Let us recall what the statement signifies. First, ", "$\\textit{WAdm}^{Noeth}$ is the category whose objects are", "adic Noetherian topological rings and whose morphisms are", "continuous ring homomorphisms. Consider a commutative diagram", "$$", "\\xymatrix{", "B \\ar[r] & (B')^\\wedge \\\\", "A \\ar[r] \\ar[u]^\\varphi & (A')^\\wedge \\ar[u]_{\\varphi'}", "}", "$$", "satisfying the following conditions:", "$A$ and $B$ are adic Noetherian topological rings,", "$A \\to A'$ and $B \\to B'$ are \\'etale ring maps,", "$(A')^\\wedge = \\lim A'/I^nA'$ for some ideal of definition $I \\subset A$,", "$(B')^\\wedge = \\lim B'/J^nB'$ for some ideal of definition $J \\subset B$, and", "$\\varphi : A \\to B$ and $\\varphi' : (A')^\\wedge \\to (B')^\\wedge$", "are continuous. Note that $(A')^\\wedge$ and $(B')^\\wedge$ are", "adic Noetherian topological rings by", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-completion-in-sub}.", "We have to show", "\\begin{enumerate}", "\\item $\\varphi$ is flat $\\Rightarrow \\varphi'$ is flat,", "\\item if $B \\to B'$ faithfully flat, then $\\varphi'$ is flat", "$\\Rightarrow \\varphi$ is flat, and", "\\item if $A \\to B_i$ is flat for $i = 1, \\ldots, n$, then", "$A \\to \\prod_{i = 1, \\ldots, n} B_i$ is flat.", "\\end{enumerate}", "We will use without further mention that completions of Noetherian rings", "are flat (Algebra, Lemma \\ref{algebra-lemma-completion-flat}). Since", "of course $A \\to A'$ and $B \\to B'$ are flat, we see in", "particular that the horizontal arrows in the diagram are flat.", "\\medskip\\noindent", "Proof of (1). If $\\varphi$ is flat, then the composition", "$A \\to (A')^\\wedge \\to (B')^\\wedge$ is flat. Hence $A' \\to (B')^\\wedge$", "is flat by More on Flatness, Lemma \\ref{flat-lemma-etale-flat-up-down}.", "Hence we see that $(A')^\\wedge \\to (B')^\\wedge$ is flat by applying", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-after-completion}", "with $R = A'$, with ideal $I(A')$, and with $M = (B')^\\wedge = M^\\wedge$.", "\\medskip\\noindent", "Proof of (2). Assume $\\varphi'$ is flat and $B \\to B'$ is faithfully flat.", "Then the composition $A \\to (A')^\\wedge \\to (B')^\\wedge$ is flat.", "Also we see that $B \\to (B')^\\wedge$ is faithfully flat by", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-etale-surjective}.", "Hence by Algebra, Lemma \\ref{algebra-lemma-flatness-descends-more-general}", "we find that $\\varphi : A \\to B$ is flat.", "\\medskip\\noindent", "Proof of (3). Omitted." ], "refs": [ "formal-spaces-lemma-completion-in-sub", "algebra-lemma-completion-flat", "flat-lemma-etale-flat-up-down", "more-algebra-lemma-flat-after-completion", "formal-spaces-lemma-etale-surjective", "algebra-lemma-flatness-descends-more-general" ], "ref_ids": [ 3922, 870, 5980, 9956, 3914, 529 ] } ], "ref_ids": [ 4008 ] }, { "id": 2336, "type": "theorem", "label": "restricted-lemma-base-change-flat-continuous", "categories": [ "restricted" ], "title": "restricted-lemma-base-change-flat-continuous", "contents": [ "Denote $P$ the property of arrows of $\\textit{WAdm}^{Noeth}$", "defined in Lemma \\ref{lemma-flat-axioms}.", "Denote $Q$ the property defined in Lemma \\ref{lemma-finite-type-red}", "viewed as a property of arrows of $\\textit{WAdm}^{Noeth}$.", "Denote $R$ the property defined in Lemma \\ref{lemma-finite-type}", "viewed as a property of arrows of $\\textit{WAdm}^{Noeth}$. Then", "\\begin{enumerate}", "\\item $P$ is stable under base change by $Q$", "(Formal Spaces, Remark", "\\ref{formal-spaces-remark-base-change-variant-variant-Noetherian}), and", "\\item $P + R$ is stable under base change", "(Formal Spaces, Remark", "\\ref{formal-spaces-remark-base-change-variant-Noetherian}).", "\\end{enumerate}" ], "refs": [ "restricted-lemma-flat-axioms", "restricted-lemma-finite-type-red", "restricted-lemma-finite-type", "formal-spaces-remark-base-change-variant-variant-Noetherian", "formal-spaces-remark-base-change-variant-Noetherian" ], "proofs": [ { "contents": [ "The statement makes sense as each of the properties $P$, $Q$, and $R$", "is a local property of morphisms of $\\textit{WAdm}^{Noeth}$.", "Let $\\varphi : B \\to A$ and $\\psi : B \\to C$ be morphisms of", "$\\textit{WAdm}^{Noeth}$. If either $Q(\\varphi)$ or $Q(\\psi)$", "then we see that $A \\widehat{\\otimes}_B C$ is Noetherian by", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-completed-tensor-product}.", "Since $R$ implies $Q$ (Lemma \\ref{lemma-finite-type-finite-type-red}),", "we find that this holds in both cases (1) and (2).", "This is the first thing we have to check. It remains to show", "that $C \\to A \\widehat{\\otimes}_B C$ is flat.", "\\medskip\\noindent", "Proof of (1). Fix ideals of definition $I \\subset A$ and $J \\subset B$.", "By Lemma \\ref{lemma-Noetherian-finite-type-red} the ring map", "$B \\to C$ is topologically of finite type. Hence", "$B \\to C/J^n$ is of finite type for all $n \\geq 1$.", "Hence $A \\otimes_B C/J^n$ is Noetherian as a ring", "(because it is of finite type over $A$ and $A$ is Noetherian).", "Thus the $I$-adic completion $A \\widehat{\\otimes}_B C/J^n$", "of $A \\otimes_B C/J^n$ is flat over $C/J^n$ because", "$C/J^n \\to A \\otimes_B C/J^n$ is flat as a base change of", "$B \\to A$ and because", "$A \\otimes_B C/J^n \\to A \\widehat{\\otimes}_B C/J^n$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}", "Observe that $A \\widehat{\\otimes}_B C/J^n =", "(A \\widehat{\\otimes}_B C)/J^n(A \\widehat{\\otimes}_B C)$; details omitted.", "We conclude that $M = A \\widehat{\\otimes}_B C$ is a $C$-module", "which is complete with respect to the $J$-adic topology", "such that $M/J^nM$ is flat over $C/J^n$ for all $n \\geq 1$.", "This implies that $M$ is flat over $C$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-limit-flat}.", "\\medskip\\noindent", "Proof of (2). In this case $B \\to A$ is adic and hence we have just", "$A \\widehat{\\otimes}_B C = \\lim A \\otimes_B C/J^n$.", "The rings $A \\otimes_B C/J^n$ are Noetherian by an application of ", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-completed-tensor-product}", "with $C$ replaced by $C/J^n$.", "We conclude in the same manner as before." ], "refs": [ "formal-spaces-lemma-completed-tensor-product", "restricted-lemma-finite-type-finite-type-red", "restricted-lemma-Noetherian-finite-type-red", "algebra-lemma-completion-flat", "more-algebra-lemma-limit-flat", "formal-spaces-lemma-completed-tensor-product" ], "ref_ids": [ 3865, 2331, 2332, 870, 9955, 3865 ] } ], "ref_ids": [ 2335, 2328, 2324, 4011, 4010 ] }, { "id": 2337, "type": "theorem", "label": "restricted-lemma-composition-flat-continuous", "categories": [ "restricted" ], "title": "restricted-lemma-composition-flat-continuous", "contents": [ "Denote $P$ the property of arrows of $\\textit{WAdm}^{Noeth}$", "defined in Lemma \\ref{lemma-flat-axioms}.", "Then $P$ is stable under composition (Formal Spaces, Remark", "\\ref{formal-spaces-remark-composition-variant-Noetherian})." ], "refs": [ "restricted-lemma-flat-axioms", "formal-spaces-remark-composition-variant-Noetherian" ], "proofs": [ { "contents": [ "This is true because compositions of flat ring maps are flat." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 2335, 4013 ] }, { "id": 2338, "type": "theorem", "label": "restricted-lemma-flat-morphisms", "categories": [ "restricted" ], "title": "restricted-lemma-flat-morphisms", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "locally Noetherian formal algebraic spaces over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is flat,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$", "representable by algebraic spaces and \\'etale, the morphism $U \\to V$", "corresponds to a flat map in $\\textit{WAdm}^{Noeth}$,", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "and for each $j$", "a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "such that each $X_{ji} \\to Y_j$ corresponds", "to a flat map in $\\textit{WAdm}^{Noeth}$, and", "\\item there exist a covering $\\{X_i \\to X\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$", "is an affine formal algebraic space, $Y_i \\to Y$ is representable", "by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds", "to a flat map in $\\textit{WAdm}^{Noeth}$.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is Definition \\ref{definition-flat}.", "The equivalence of (2), (3), and (4) follows from the fact that", "being flat is a local property of arrows of", "$\\text{WAdm}^{Noeth}$ by Lemma \\ref{lemma-flat-axioms}", "and an application of the variant of", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-property-defines-property-morphisms}", "for morphisms between locally Noetherian algebraic spaces", "mentioned in", "Formal Spaces, Remark", "\\ref{formal-spaces-remark-variant-Noetherian}." ], "refs": [ "restricted-definition-flat", "restricted-lemma-flat-axioms", "formal-spaces-lemma-property-defines-property-morphisms", "formal-spaces-remark-variant-Noetherian" ], "ref_ids": [ 2436, 2335, 3923, 4008 ] } ], "ref_ids": [ 3981, 3981, 3981 ] }, { "id": 2339, "type": "theorem", "label": "restricted-lemma-base-change-flat", "categories": [ "restricted" ], "title": "restricted-lemma-base-change-flat", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$", "be morphisms of locally Noetherian formal algebraic spaces over $S$.", "\\begin{enumerate}", "\\item If $f$ is flat and $g_{red} : Z_{red} \\to Y_{red}$ is", "locally of finite type, then the base change", "$X \\times_Y Z \\to Z$ is flat.", "\\item If $f$ is flat and locally of finite type, then", "the base change $X \\times_Y Z \\to Z$ is flat and locally of finite type.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from a combination of", "Formal Spaces, Remark", "\\ref{formal-spaces-remark-base-change-variant-variant-Noetherian},", "Lemma \\ref{lemma-base-change-flat-continuous} part (1),", "Lemma \\ref{lemma-flat-morphisms}, and", "Lemma \\ref{lemma-finite-type-red-morphisms}.", "\\medskip\\noindent", "Part (2) follows from a combination of", "Formal Spaces, Remark", "\\ref{formal-spaces-remark-base-change-variant-Noetherian},", "Lemma \\ref{lemma-base-change-flat-continuous} part (2),", "Lemma \\ref{lemma-flat-morphisms}, and", "Lemma \\ref{lemma-finite-type-morphisms}." ], "refs": [ "formal-spaces-remark-base-change-variant-variant-Noetherian", "restricted-lemma-base-change-flat-continuous", "restricted-lemma-flat-morphisms", "restricted-lemma-finite-type-red-morphisms", "formal-spaces-remark-base-change-variant-Noetherian", "restricted-lemma-base-change-flat-continuous", "restricted-lemma-flat-morphisms", "restricted-lemma-finite-type-morphisms" ], "ref_ids": [ 4011, 2336, 2338, 2334, 4010, 2336, 2338, 2327 ] } ], "ref_ids": [] }, { "id": 2340, "type": "theorem", "label": "restricted-lemma-composition-flat", "categories": [ "restricted" ], "title": "restricted-lemma-composition-flat", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$", "be morphisms of locally Noetherian formal algebraic spaces over $S$.", "If $f$ and $g$ are flat, then so is $g \\circ f$." ], "refs": [], "proofs": [ { "contents": [ "Combine Formal Spaces, Remark", "\\ref{formal-spaces-remark-composition-variant-Noetherian}", "and Lemma \\ref{lemma-composition-flat-continuous}." ], "refs": [ "formal-spaces-remark-composition-variant-Noetherian", "restricted-lemma-composition-flat-continuous" ], "ref_ids": [ 4013, 2337 ] } ], "ref_ids": [] }, { "id": 2341, "type": "theorem", "label": "restricted-lemma-representable-flat", "categories": [ "restricted" ], "title": "restricted-lemma-representable-flat", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphisms of", "locally Noetherian formal algebraic spaces over $S$.", "If $f$ is representable by algebraic spaces and", "flat in the sense of Bootstrap, Definition", "\\ref{bootstrap-definition-property-transformation},", "then $f$ is flat in the sense of Definition \\ref{definition-flat}." ], "refs": [ "bootstrap-definition-property-transformation", "restricted-definition-flat" ], "proofs": [ { "contents": [ "This is a sanity check whose proof should be trivial but isn't quite.", "We urge the reader to skip the proof.", "Assume $f$ is representable by algebraic spaces and", "flat in the sense of Bootstrap, Definition", "\\ref{bootstrap-definition-property-transformation}.", "Consider a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$", "representable by algebraic spaces and \\'etale.", "Then the morphism $U \\to V$", "corresponds to a taut map $B \\to A$ of $\\textit{WAdm}^{Noeth}$ by", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-representable-local-property}.", "Observe that this means $B \\to A$ is adic (Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-adic-homomorphism})", "and in particular for any ideal of definition $J \\subset B$", "the topology on $A$ is the $J$-adic topology and the diagrams", "$$", "\\xymatrix{", "\\Spec(A/J^nA) \\ar[r] \\ar[d] & \\Spec(B/J^n) \\ar[d] \\\\", "U \\ar[r] & V", "}", "$$", "are cartesian.", "\\medskip\\noindent", "Let $T \\to V$ is a morphism where $T$ is a scheme. Then", "\\begin{align*}", "X \\times_Y T \\to T\\text{ is flat}", "& \\Rightarrow", "U \\times_Y T \\to T\\text{ is flat} \\\\", "& \\Rightarrow", "U \\times_V V \\times_Y T \\to T\\text{ is flat} \\\\", "& \\Rightarrow", "U \\times_V V \\times_Y T \\to V \\times_Y T\\text{ is flat} \\\\", "& \\Rightarrow", "U \\times_V T \\to T\\text{ is flat}", "\\end{align*}", "The first statement is the assumption on $f$.", "The first implication because $U \\to X$ is \\'etale and hence flat", "and compositions of flat morphisms of algebraic spaces are flat.", "The second impliciation because $U \\times_Y T = U \\times_V V \\times_Y T$.", "The third implication by", "More on Flatness, Lemma \\ref{flat-lemma-etale-flat-up-down}.", "The fourth implication because we can pullback by the morphism", "$T \\to V \\times_Y T$.", "We conclude that $U \\to V$ is flat in the sense of", "Bootstrap, Definition", "\\ref{bootstrap-definition-property-transformation}.", "In terms of the continuous ring map $B \\to A$", "this means the ring maps $B/J^n \\to A/J^nA$ are flat (see diagram above).", "\\medskip\\noindent", "Finally, we can conclude that $B \\to A$ is flat for example", "by More on Algebra, Lemma \\ref{more-algebra-lemma-limit-flat}." ], "refs": [ "bootstrap-definition-property-transformation", "formal-spaces-lemma-representable-local-property", "formal-spaces-lemma-adic-homomorphism", "flat-lemma-etale-flat-up-down", "bootstrap-definition-property-transformation", "more-algebra-lemma-limit-flat" ], "ref_ids": [ 2638, 3928, 3929, 5980, 2638, 9955 ] } ], "ref_ids": [ 2638, 2436 ] }, { "id": 2342, "type": "theorem", "label": "restricted-lemma-rig-point", "categories": [ "restricted" ], "title": "restricted-lemma-rig-point", "contents": [ "Let $A$ be a Noetherian adic topological ring. Let", "$\\mathfrak q \\subset A$ be a prime ideal. The following are", "equivalent", "\\begin{enumerate}", "\\item for some ideal of definition $I \\subset A$ we have", "$I \\not \\subset \\mathfrak q$ and $\\mathfrak q$ is maximal", "with respect to this property,", "\\item for some ideal of definition $I \\subset A$ the prime", "$\\mathfrak q$ defines a closed point of $\\Spec(A) \\setminus V(I)$,", "\\item for any ideal of definition $I \\subset A$ we have", "$I \\not \\subset \\mathfrak q$ and $\\mathfrak q$ is maximal", "with respect to this property,", "\\item for any ideal of definition $I \\subset A$ the prime", "$\\mathfrak q$ defines a closed point of $\\Spec(A) \\setminus V(I)$,", "\\item $\\dim(A/\\mathfrak q) = 1$ and for some ideal of definition", "$I \\subset A$ we have $I \\not \\subset \\mathfrak q$,", "\\item $\\dim(A/\\mathfrak q) = 1$ and for any ideal of definition", "$I \\subset A$ we have $I \\not \\subset \\mathfrak q$,", "\\item $\\dim(A/\\mathfrak q) = 1$ and the induced topology", "on $A/\\mathfrak q$ is nontrivial,", "\\item $A/\\mathfrak q$ is a $1$-dimensional Noetherian complete local domain", "whose maximal ideal is the radical of the image of any ideal of", "definition of $A$, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) and (2) are equivalent and for the same reason", "that (3) and (4) are equivalent.", "Since $V(I)$ is independent of the choice of the ideal of definition", "$I$ of $A$, we see that (2) and (4) are equivalent.", "\\medskip\\noindent", "Assume the equivalent conditions (1) -- (4) hold.", "If $\\dim(A/\\mathfrak q) > 1$ we can choose a maximal", "ideal $\\mathfrak q \\subset \\mathfrak m \\subset A$", "such that $\\dim((A/\\mathfrak q)_\\mathfrak m) > 1$.", "Then $\\Spec((A/\\mathfrak q)_\\mathfrak m) - V(I(A/\\mathfrak q)_\\mathfrak m)$", "would be infinite by Algebra, Lemma", "\\ref{algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens}.", "This contradicts the fact that $\\mathfrak q$ is closed in", "$\\Spec(A) \\setminus V(I)$.", "Hence we see that (6) holds. Trivially (6) implies (5).", "\\medskip\\noindent", "Conversely, assume (5) holds. Let $I \\subset A$ be an ideal of definition.", "Since $A/\\mathfrak q$ is complete", "with respect to $I(A/\\mathfrak q)$ (for example by", "Algebra, Lemma \\ref{algebra-lemma-completion-tensor})", "we see that all closed points of $\\Spec(A/\\mathfrak q)$ are", "contained in $V(IA/\\mathfrak q)$ by", "Algebra, Lemma \\ref{algebra-lemma-radical-completion}.", "Since $\\dim(A/\\mathfrak q) = 1$ and since $I \\not \\subset \\mathfrak q$", "we conclude two things: (a) $V(IA/\\mathfrak q)$ must contain", "all points distinct from the generic point of $\\Spec(A/\\mathfrak q)$, and", "(b) $V(IA/\\mathfrak q)$ must be a (finite) discrete set.", "From (a) we see that $\\mathfrak q$ is a closed point of", "$\\Spec(A) \\setminus V(I)$ and we conclude that (2) holds.", "\\medskip\\noindent", "Continuing to assume (5) we see that the finite discrete space", "$V(IA/\\mathfrak q)$ must be a singleton by More on Algebra, Lemma", "\\ref{more-algebra-lemma-irreducible-henselian-pair-connected}", "for example (and the fact that complete pairs are henselian pairs, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-complete-henselian}).", "Hence we see that (8) is true.", "Conversely, it is clear that (8) implies (5).", "\\medskip\\noindent", "At this point we know that (1) -- (6) and (8) are equivalent.", "We omit the verification that these are also equivalent to (7)." ], "refs": [ "algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens", "algebra-lemma-completion-tensor", "algebra-lemma-radical-completion", "more-algebra-lemma-irreducible-henselian-pair-connected", "more-algebra-lemma-complete-henselian" ], "ref_ids": [ 687, 869, 862, 9870, 9859 ] } ], "ref_ids": [] }, { "id": 2343, "type": "theorem", "label": "restricted-lemma-rig-closed-point-relative-residue-field", "categories": [ "restricted" ], "title": "restricted-lemma-rig-closed-point-relative-residue-field", "contents": [ "Let $\\varphi : A \\to B$ in $\\textit{WAdm}^{Noeth}$.", "Denote $\\mathfrak a \\subset A$ and $\\mathfrak b \\subset B$", "the ideals of topologically nilpotent elements. Assume", "$A/\\mathfrak a \\to B/\\mathfrak b$ is of finite type.", "Let $\\mathfrak q \\subset B$ be rig-closed.", "The residue field $\\kappa$ of the local ring $B/\\mathfrak q$", "is a finite type $A/\\mathfrak a$-algebra." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q \\subset \\mathfrak m \\subset B$ be the unique", "maximal ideal containing $\\mathfrak q$.", "Then $\\mathfrak b \\subset \\mathfrak m$. Hence", "$A/\\mathfrak a \\to B/\\mathfrak b \\to B/\\mathfrak m = \\kappa$ is", "of finite type." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2344, "type": "theorem", "label": "restricted-lemma-rig-closed-point-relative", "categories": [ "restricted" ], "title": "restricted-lemma-rig-closed-point-relative", "contents": [ "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$", "which is adic and topologically of finite type.", "Let $\\mathfrak q \\subset B$ be rig-closed.", "Let $\\mathfrak p = \\varphi^{-1}(\\mathfrak q) \\subset A$.", "Let $\\mathfrak a \\subset A$ be the ideal of topologically nilpotent", "elements.", "The following are equivalent", "\\begin{enumerate}", "\\item the residue field $\\kappa$ of $B/\\mathfrak q$ is finite", "over $A/\\mathfrak a$,", "\\item $\\mathfrak p \\subset A$ is rig-closed,", "\\item $A/\\mathfrak p \\subset B/\\mathfrak q$ is a finite extension", "of rings.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Recall that $B/\\mathfrak q$ is a Noetherian local ring", "of dimension $1$ whose topology is the adic topology coming", "from the maximal ideal. Since $\\varphi$ is adic, we see that", "$A \\to B/\\mathfrak q$ is adic. Hence $\\varphi(\\mathfrak a)$", "is a nonzero ideal in $B/\\mathfrak q$. Hence", "$B/\\mathfrak q + \\varphi(\\mathfrak a)$", "has finite length. Hence $B/\\mathfrak q + \\varphi(\\mathfrak a)$", "is finite as an $A/\\mathfrak a$-module by our assumption.", "Thus $B/\\mathfrak q$ is finite over $A$ by", "Algebra, Lemma \\ref{algebra-lemma-finite-over-complete-ring}.", "Thus (3) holds.", "\\medskip\\noindent", "Assume (3). Then $\\Spec(B/\\mathfrak q) \\to \\Spec(A/\\mathfrak p)$", "is surjective by", "Algebra, Lemma \\ref{algebra-lemma-integral-overring-surjective}.", "This implies (2).", "\\medskip\\noindent", "Assume (2). Denote $\\kappa'$ the residue field of $A/\\mathfrak p$.", "By Lemma \\ref{lemma-rig-closed-point-relative-residue-field}", "(and Lemma \\ref{lemma-finite-type-finite-type-red})", "the extension $\\kappa/\\kappa'$ is finitely generated as an algebra.", "By the Hilbert Nullstellensatz (Algebra, Lemma", "\\ref{algebra-lemma-field-finite-type-over-domain})", "we see that $\\kappa/\\kappa'$ is a finite extension.", "Hence we see that (1) holds." ], "refs": [ "algebra-lemma-finite-over-complete-ring", "algebra-lemma-integral-overring-surjective", "restricted-lemma-rig-closed-point-relative-residue-field", "restricted-lemma-finite-type-finite-type-red", "algebra-lemma-field-finite-type-over-domain" ], "ref_ids": [ 868, 495, 2343, 2331, 466 ] } ], "ref_ids": [] }, { "id": 2345, "type": "theorem", "label": "restricted-lemma-rig-closed-jacobson", "categories": [ "restricted" ], "title": "restricted-lemma-rig-closed-jacobson", "contents": [ "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$", "which is adic and topologically of finite type.", "Let $\\mathfrak q \\subset B$ be rig-closed.", "If $A/I$ is Jacobson for some ideal of definition $I \\subset A$, then", "$\\mathfrak p = \\varphi^{-1}(\\mathfrak q) \\subset A$", "is rig-closed." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-rig-closed-point-relative-residue-field}", "(combined with Lemma \\ref{lemma-finite-type-finite-type-red})", "the residue field $\\kappa$ of $B/\\mathfrak q$ is of finite type over", "$A/\\mathfrak a$. Since $A/\\mathfrak a$ is Jacobson, we", "see that $\\kappa$ is finite over $A/\\mathfrak a$ by", "Algebra, Lemma \\ref{algebra-lemma-silly-jacobson}.", "We conclude by Lemma \\ref{lemma-rig-closed-point-relative}." ], "refs": [ "restricted-lemma-rig-closed-point-relative-residue-field", "restricted-lemma-finite-type-finite-type-red", "algebra-lemma-silly-jacobson", "restricted-lemma-rig-closed-point-relative" ], "ref_ids": [ 2343, 2331, 477, 2344 ] } ], "ref_ids": [] }, { "id": 2346, "type": "theorem", "label": "restricted-lemma-rig-closed-point-in-image", "categories": [ "restricted" ], "title": "restricted-lemma-rig-closed-point-in-image", "contents": [ "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$", "which is adic and topologically of finite type.", "Let $\\mathfrak p \\subset A$ be rig-closed.", "Let $\\mathfrak a \\subset A$ and $\\mathfrak b \\subset B$", "be the ideals of topologically nilpotent elements. If $\\varphi$", "is flat, then the following are equivalent", "\\begin{enumerate}", "\\item the maximal ideal of $A/\\mathfrak p$ is in the image of", "$\\Spec(B/\\mathfrak b) \\to \\Spec(A/\\mathfrak a)$,", "\\item there exists a rig-closed prime ideal $\\mathfrak q \\subset B$", "such that $\\mathfrak p = \\varphi^{-1}(\\mathfrak q)$.", "\\end{enumerate}", "and if so then $\\varphi$, $\\mathfrak p$, and $\\mathfrak q$", "satisfy the conclusions of Lemma \\ref{lemma-rig-closed-point-relative}." ], "refs": [ "restricted-lemma-rig-closed-point-relative" ], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is immediate. Assume (1).", "To prove the existence of $\\mathfrak q$", "we may replace $A$ by $A/\\mathfrak p$ and $B$ by $B/\\mathfrak p B$", "(some details omitted). Thus we may assume $(A, \\mathfrak m, \\kappa)$", "is a local complete $1$-dimensional Noetherian ring,", "$\\mathfrak m = \\mathfrak a$, and $\\mathfrak p = (0)$. Condition (1)", "just says that $B_0 = B \\otimes_A \\kappa = B/\\mathfrak m B = B/\\mathfrak a B$", "is nonzero. Note that $B_0$ is of finite type over $\\kappa$.", "Hence we can use induction on $\\dim(B_0)$.", "If $\\dim(B_0) = 0$, then any minimal prime $\\mathfrak q \\subset B$", "will do (flatness of $A \\to B$ insures that $\\mathfrak q$ will", "lie over $\\mathfrak p = (0)$).", "If $\\dim(B_0) > 0$ then we can find an element $b \\in B$", "which maps to an element $b_0 \\in B_0$ which is a nonzerodivisor", "and a nonunit, see Algebra, Lemma", "\\ref{algebra-lemma-dim-not-zero-exists-nonzerodivisor-nonunit}.", "By Algebra, Lemma \\ref{algebra-lemma-grothendieck}", "the ring $B' = B/bB$ is flat over $A$. Since", "$B'_0 = B' \\otimes_A \\kappa = B_0/(b_0)$ is not zero,", "we may apply the induction hypothesis to $B'$ and conclude.", "The final statement of the lemma is clear from", "Lemma \\ref{lemma-rig-closed-point-relative}." ], "refs": [ "algebra-lemma-dim-not-zero-exists-nonzerodivisor-nonunit", "algebra-lemma-grothendieck", "restricted-lemma-rig-closed-point-relative" ], "ref_ids": [ 1123, 884, 2344 ] } ], "ref_ids": [ 2344 ] }, { "id": 2347, "type": "theorem", "label": "restricted-lemma-rig-closed-point-in-localization", "categories": [ "restricted" ], "title": "restricted-lemma-rig-closed-point-in-localization", "contents": [ "Let $A$ be an adic Noetherian topological ring.", "Let $\\mathfrak p \\subset A$ be a prime ideal.", "Let $f \\in A$ be an element mapping to a unit in $A/\\mathfrak p$.", "Then", "$$", "\\mathfrak p A_{\\{f\\}} =", "\\mathfrak p(A_f)^\\wedge =", "\\mathfrak p \\otimes_A (A_f)^\\wedge =", "(\\mathfrak p_f)^\\wedge", "$$", "is a prime ideal with quotient", "$$", "A/\\mathfrak p = (A/\\mathfrak p) \\otimes_A (A_f)^\\wedge =", "(A_f)^\\wedge / \\mathfrak p (A_f)^\\wedge = A_{\\{f\\}}/\\mathfrak p A_{\\{f\\}}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since $A_f$ is Noetherian the ring map $A \\to A_f \\to (A_f)^\\wedge$", "is flat. For any finite $A$-module $M$ we see that", "$M \\otimes_A (A_f)^\\wedge$ is the completion of $M_f$.", "If $f$ is a unit on $M$, then $M_f = M$ is already complete.", "See discussion in Algebra, Section \\ref{algebra-section-completion-noetherian}.", "From these observations the results follow easily." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2348, "type": "theorem", "label": "restricted-lemma-rig-closed-point-after-localization", "categories": [ "restricted" ], "title": "restricted-lemma-rig-closed-point-after-localization", "contents": [ "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$", "which is adic and topologically of finite type.", "Let $\\mathfrak q \\subset B$ be rig-closed.", "There exists an $f \\in A$ which maps to a unit in", "$B/\\mathfrak q$ such that we obtain a diagram", "$$", "\\vcenter{", "\\xymatrix{", "B \\ar[r] &", "B_{\\{f\\}} \\\\", "A \\ar[r] \\ar[u]_\\varphi &", "A_{\\{f\\}} \\ar[u]_{\\varphi_{\\{f\\}}}", "}", "}", "\\quad\\text{with primes}\\quad", "\\vcenter{", "\\xymatrix{", "\\mathfrak q \\ar@{-}[r] \\ar@{-}[d] &", "\\mathfrak q' \\ar@{-}[d] \\ar@{=}[r] &", "\\mathfrak q B_{\\{f\\}} \\\\", "\\mathfrak p \\ar@{-}[r] &", "\\mathfrak p'", "}", "}", "$$", "such that $\\mathfrak p'$ is rig-closed, i.e.,", "the map $A_{\\{f\\}} \\to B_{\\{f\\}}$ and the prime ideals", "$\\mathfrak q'$ and $\\mathfrak p'$ satisfy", "the equivalent conditions of Lemma \\ref{lemma-rig-closed-point-relative}." ], "refs": [ "restricted-lemma-rig-closed-point-relative" ], "proofs": [ { "contents": [ "Please see Lemma \\ref{lemma-rig-closed-point-in-localization}", "for the description of $\\mathfrak q'$. The only assertion the lemma makes", "is that for a suitable choice of $f$ the prime ideal $\\mathfrak p'$", "has the property $\\dim((A_f)^\\wedge/\\mathfrak p') = 1$.", "By Lemma \\ref{lemma-rig-closed-point-relative} this in turn", "just means that the residue field $\\kappa$ of", "$B/\\mathfrak q = (B_f)^\\wedge/\\mathfrak q'$ is finite over", "$(A_f)^\\wedge/\\mathfrak a' = (A/\\mathfrak a)_f$.", "By Lemma \\ref{lemma-rig-closed-point-relative-residue-field}", "we know that $A/\\mathfrak a \\to \\kappa$ is a finite type", "algebra homomorphism. By the Hilbert Nullstellensatz in", "the form of Algebra, Lemma \\ref{algebra-lemma-field-finite-type-over-domain}", "we can find an $f \\in A$ which maps to a unit in $\\kappa$", "such that $\\kappa$ is finite over $A_f$. This finishes the proof." ], "refs": [ "restricted-lemma-rig-closed-point-in-localization", "restricted-lemma-rig-closed-point-relative", "restricted-lemma-rig-closed-point-relative-residue-field", "algebra-lemma-field-finite-type-over-domain" ], "ref_ids": [ 2347, 2344, 2343, 466 ] } ], "ref_ids": [ 2344 ] }, { "id": 2349, "type": "theorem", "label": "restricted-lemma-rig-closed-point-variables", "categories": [ "restricted" ], "title": "restricted-lemma-rig-closed-point-variables", "contents": [ "Let $A$ be a Noetherian adic topological ring. Denote $A\\{x_1, \\ldots, x_n\\}$", "the restricted power series over $A$. Let", "$\\mathfrak q \\subset A\\{x_1, \\ldots, x_n\\}$ be a prime ideal.", "Set $\\mathfrak q' = A[x_1, \\ldots, x_n] \\cap \\mathfrak q$ and", "$\\mathfrak p = A \\cap \\mathfrak q$. If $\\mathfrak q$ and $\\mathfrak p$", "are rig-closed, then the map", "$$", "A[x_1, \\ldots, x_n]_{\\mathfrak q'}", "\\to", "A\\{x_1, \\ldots, x_n\\}_\\mathfrak q", "$$", "defines an isomorphism on completions with respect to their maximal ideals." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-rig-closed-point-relative} the ring map", "$A/\\mathfrak p \\to A\\{x_1, \\ldots, x_n\\}/\\mathfrak q$ is finite.", "For every $m \\geq 1$ the module $\\mathfrak q^m/\\mathfrak q^{m + 1}$", "is finite over $A$ as it is a finite", "$A\\{x_1, \\ldots, x_n\\}/\\mathfrak q$-module.", "Hence $A\\{x_1, \\ldots x_n\\}/\\mathfrak q^m$ is a finite $A$-module.", "Hence $A[x_1, \\ldots, x_n] \\to A\\{x_1, \\ldots, x_n\\}/\\mathfrak q^m$", "is surjective (as the image is dense and an $A$-submodule).", "It follows in a straightforward manner that", "$A[x_1, \\ldots, x_n]/(\\mathfrak q')^m \\to A\\{x_1, \\ldots, x_n\\}/\\mathfrak q^m$", "is an isomorphism for all $m$. From this the lemma easily follows.", "Hint: Pick a topologically nilpotent $g \\in A$ which is not contained", "in $\\mathfrak p$. Then the map of completions is the map", "$$", "\\lim_m \\left(A[x_1, \\ldots, x_n]/(\\mathfrak q')^m\\right)_g", "\\longrightarrow", "\\left(A\\{x_1, \\ldots, x_n\\}/\\mathfrak q^m\\right)_g", "$$", "Some details omitted." ], "refs": [ "restricted-lemma-rig-closed-point-relative" ], "ref_ids": [ 2344 ] } ], "ref_ids": [] }, { "id": 2350, "type": "theorem", "label": "restricted-lemma-rig-closed-point-etale", "categories": [ "restricted" ], "title": "restricted-lemma-rig-closed-point-etale", "contents": [ "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$.", "Assume $\\varphi$ is adic, topologically of finite type, flat,", "and $A/I \\to B/IB$ is \\'etale for some (resp.\\ any)", "ideal of definition $I \\subset A$. Let $\\mathfrak q \\subset B$", "be rig-closed such that $\\mathfrak p = A \\cap \\mathfrak q$", "is rig-closed as well. Then", "$\\mathfrak p B_\\mathfrak q = \\mathfrak q B_\\mathfrak q$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\kappa$ be the residue field of the $1$-dimensional complete", "local ring $A/\\mathfrak p$. Since $A/I \\to B/IB$ is \\'etale, we see that", "$B \\otimes_A \\kappa$ is a finite product of finite separable", "extensions of $\\kappa$, see", "Algebra, Lemma \\ref{algebra-lemma-etale-over-field}.", "One of these is the residue field of $B/\\mathfrak q$.", "By Algebra, Lemma \\ref{algebra-lemma-finite-over-complete-ring} we see that", "$B/\\mathfrak p B$ is a finite $A/\\mathfrak p$-algebra.", "It is also flat. Combining the above", "we see that $A/\\mathfrak p \\to B /\\mathfrak p B$", "is finite \\'etale, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-etale}.", "Hence $B/\\mathfrak p B$ is reduced, which implies the statement of", "the lemma (details omitted)." ], "refs": [ "algebra-lemma-etale-over-field", "algebra-lemma-finite-over-complete-ring", "algebra-lemma-characterize-etale" ], "ref_ids": [ 1232, 868, 1235 ] } ], "ref_ids": [] }, { "id": 2351, "type": "theorem", "label": "restricted-lemma-fibre-regular", "categories": [ "restricted" ], "title": "restricted-lemma-fibre-regular", "contents": [ "Let $A$ be an adic Noetherian topological ring.", "Let $\\mathfrak p \\subset A$ be a rig-closed prime.", "For any $n \\geq 1$ the ring map", "$$", "A/\\mathfrak p", "\\longrightarrow", "A\\{x_1, \\ldots, x_n\\} \\otimes_A A/\\mathfrak p =", "A/\\mathfrak p\\{x_1, \\ldots, x_n\\}", "$$", "is regular. In particular, the algebra", "$A\\{x_1, \\ldots, x_n\\} \\otimes_A \\kappa(\\mathfrak p)$", "is geometrically regular over $\\kappa(\\mathfrak p)$." ], "refs": [], "proofs": [ { "contents": [ "We will use some fact on regular ring maps the reader can find", "in More on Algebra, Section \\ref{more-algebra-section-regular}.", "Since $A/\\mathfrak p$ is a complete local Noetherian ring it", "is excellent (More on Algebra, Proposition", "\\ref{more-algebra-proposition-ubiquity-excellent}).", "Hence $A/\\mathfrak p[x_1, \\ldots, x_n]$ is excellent", "(by the same reference). Hence", "$A/\\mathfrak p[x_1, \\ldots, x_n] \\to A/\\mathfrak p\\{x_1, \\ldots, x_n\\}$", "is a regular ring homomorphism by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-map-G-ring-to-completion-regular}.", "Of course $A/\\mathfrak p \\to A/\\mathfrak p[x_1, \\ldots, x_n]$", "is smooth and hence regular. Since the composition of regular", "ring maps is regular the proof is complete." ], "refs": [ "more-algebra-proposition-ubiquity-excellent", "more-algebra-lemma-map-G-ring-to-completion-regular" ], "ref_ids": [ 10584, 10092 ] } ], "ref_ids": [] }, { "id": 2352, "type": "theorem", "label": "restricted-lemma-naively-rig-flat-continuous", "categories": [ "restricted" ], "title": "restricted-lemma-naively-rig-flat-continuous", "contents": [ "Let $\\varphi : A \\to B$ be a morphism in $\\textit{WAdm}^{adic*}$", "(Formal Spaces, Section \\ref{formal-spaces-section-morphisms-rings}).", "Assume $\\varphi$ is adic. The following are equivalent:", "\\begin{enumerate}", "\\item $B_f$ is flat over $A$ for all", "topologically nilpotent $f \\in A$,", "\\item $B_g$ is flat over $A$ for all", "topologically nilpotent $g \\in B$,", "\\item $B_\\mathfrak q$ is flat over $A$", "for all primes $\\mathfrak q \\subset B$ which do not contain", "an ideal of definition,", "\\item $B_\\mathfrak q$ is flat over $A$ for every rig-closed", "prime $\\mathfrak q \\subset B$, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows from the definitions and", "Algebra, Lemma \\ref{algebra-lemma-flat-localization}." ], "refs": [ "algebra-lemma-flat-localization" ], "ref_ids": [ 538 ] } ], "ref_ids": [] }, { "id": 2353, "type": "theorem", "label": "restricted-lemma-rig-flat-naive", "categories": [ "restricted" ], "title": "restricted-lemma-rig-flat-naive", "contents": [ "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$.", "If $A/I$ is Jacobson for some (equivalently any) ideal of definition", "$I \\subset A$ and $\\varphi$ is naively rig-flat, then $\\varphi$ is", "rig-flat." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\varphi$ is naively rig-flat. We first state some obvious", "consequences of the assumptions. Namely, let $f \\in A$.", "Then $A, B, A_{\\{f\\}}, B_{\\{f\\}}$", "are Noetherian adic topological rings. The maps", "$A \\to A_{\\{f\\}} \\to B_{\\{f\\}}$ and $A \\to B \\to B_{\\{f\\}}$", "are adic and topologically of finite type.", "The ring maps $A \\to A_{\\{f\\}}$ and $B \\to B_{\\{f\\}}$", "are flat as compositions of $A \\to A_f$ and $B \\to B_f$", "and the completion maps which are flat by", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}.", "The quotients of each of the rings", "$A, B, A_{\\{f\\}}, B_{\\{f\\}}$ by $I$ is of finite type", "over $A/I$ and hence Jacobson too", "(Algebra, Proposition \\ref{algebra-proposition-Jacobson-permanence}).", "\\medskip\\noindent", "Let $\\mathfrak q' \\subset B_{\\{f\\}}$ be rig-closed.", "It suffices to prove that $(B_{\\{f\\}})_{\\mathfrak q'}$", "is flat over $A_{\\{f\\}}$, see Lemma \\ref{lemma-naively-rig-flat-continuous}.", "By Lemma \\ref{lemma-rig-closed-jacobson} the primes", "$\\mathfrak q \\subset B$ and $\\mathfrak p' \\subset A_{\\{f\\}}$", "and $\\mathfrak p \\subset A$ lying under $\\mathfrak q'$ are rig-closed.", "We are going to apply", "Algebra, Lemma \\ref{algebra-lemma-yet-another-variant-local-criterion-flatness}", "to the diagram", "$$", "\\xymatrix{", "B_\\mathfrak q \\ar[r] &", "(B_{\\{f\\}})_{\\mathfrak q'} \\\\", "A_\\mathfrak p \\ar[u] \\ar[r] &", "(A_{\\{f\\}})_{\\mathfrak p'} \\ar[u]", "}", "$$", "with $M = B_\\mathfrak q$.", "The only assumption that hasn't been checked yet is the fact", "that $\\mathfrak p$ generates the maximal ideal of", "$(A_{\\{f\\}})_{\\mathfrak p'}$. This follows from", "Lemma \\ref{lemma-rig-closed-point-in-localization};", "here we use that $\\mathfrak p$ and $\\mathfrak p'$ are rig-closed", "to see that $f$ maps to a unit of $A/\\mathfrak p$", "(this is the only step in the proof that fails without", "the Jacobson assumption). Namely,", "this tells us that $A/\\mathfrak p \\to A_{\\{f\\}}/\\mathfrak p'$", "is a finite inclusion of local rings", "(Lemma \\ref{lemma-rig-closed-point-relative})", "and $f$ maps to a unit in the second one." ], "refs": [ "algebra-lemma-completion-flat", "algebra-proposition-Jacobson-permanence", "restricted-lemma-naively-rig-flat-continuous", "restricted-lemma-rig-closed-jacobson", "algebra-lemma-yet-another-variant-local-criterion-flatness", "restricted-lemma-rig-closed-point-in-localization", "restricted-lemma-rig-closed-point-relative" ], "ref_ids": [ 870, 1405, 2352, 2345, 899, 2347, 2344 ] } ], "ref_ids": [] }, { "id": 2354, "type": "theorem", "label": "restricted-lemma-rig-flat-base-change", "categories": [ "restricted" ], "title": "restricted-lemma-rig-flat-base-change", "contents": [ "Let $\\varphi : A \\to B$ and $A \\to C$ be arrows of $\\textit{WAdm}^{Noeth}$.", "Assume $\\varphi$ is rig-flat and $A \\to C$ adic and topologically", "of finite type. Then $C \\to B \\widehat{\\otimes}_A C$ is rig-flat." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\varphi$ is rig-flat. Let $f \\in C$ be an element.", "We have to show that", "$C_{\\{f\\}} \\to B \\widehat{\\otimes}_A C_{\\{f\\}}$", "is naively rig-flat. Since we can replace $C$ by $C_{\\{f\\}}$", "we it suffices to show that $C \\to B \\widehat{\\otimes}_A C$", "is naively rig-flat.", "\\medskip\\noindent", "If $A \\to C$ is surjective or more generally if $C$ is finite as", "an $A$-module, then $B \\otimes_A C = B \\widehat{\\otimes}_A C$", "as a finite module over a complete Noetherian ring is complete, see", "Algebra, Lemma \\ref{algebra-lemma-completion-tensor}.", "By the usual base change for flatness", "(Algebra, Lemma \\ref{algebra-lemma-flat-base-change})", "we see that naive rig-flatness of $\\varphi$ implies naive rig-flatness", "for $C \\to B \\times_A C$ in this case.", "\\medskip\\noindent", "In the general case, we can factor $A \\to C$ as", "$A \\to A\\{x_1, \\ldots, x_n\\} \\to C$", "where $A\\{x_1, \\ldots, x_n\\}$ is the restricted power series ring", "and $A\\{x_1, \\ldots, x_n\\} \\to C$ is surjective. Thus it", "suffices to show $C \\to B \\widehat{\\otimes}_A B$ is naively", "rig-flat in case $C = A\\{x_1, \\ldots, x_n\\}$.", "Since $A\\{x_1, \\ldots, x_n\\} = A\\{x_1, \\ldots, x_{n - 1}\\}\\{x_n\\}$", "by induction on $n$ we reduce to the case discussed in the next", "paragraph.", "\\medskip\\noindent", "Here $C = A\\{x\\}$. Note that $B \\widehat{\\otimes}_A C = B\\{x\\}$.", "We have to show that $A\\{x\\} \\to B\\{x\\}$ is naively rig-flat.", "Let $\\mathfrak q \\subset B\\{x\\}$ be a rig-closed prime ideal.", "We have to show that $B\\{x\\}_{\\mathfrak q}$ is flat over $A\\{x\\}$.", "Set $\\mathfrak p = A \\cap \\mathfrak q$.", "By Lemma \\ref{lemma-rig-closed-point-after-localization}", "we can find an $f \\in A$ such that", "$f$ maps to a unit in $B\\{x\\}/\\mathfrak q$ and such", "that the prime ideal $\\mathfrak p'$ in $A_{\\{f\\}}$ induced is rig-closed.", "Below we will use that $A_{\\{f\\}}\\{x\\} = A\\{x\\}_{\\{f\\}}$ and similarly", "for $B$; details omitted. Consider the diagram", "$$", "\\xymatrix{", "(B\\{x\\})_{\\mathfrak q} \\ar[r] &", "(B_{\\{f\\}}\\{x\\})_{\\mathfrak q'} \\\\", "A\\{x\\} \\ar[r] \\ar[u] &", "A_{\\{f\\}}\\{x\\} \\ar[u]", "}", "$$", "We want to show that the left vertical arrow is flat.", "The top horizontal arrow is faithfully flat as it is a local", "homomorphism of local rings and flat as $B_{\\{f\\}}\\{x\\}$", "is the completion of a localization of the Noetherian ring", "$B\\{x\\}$. Similarly the bottom horizontal arrow is flat.", "Hence it suffices to prove that the right vertical arrow is flat.", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Here $C = A\\{x\\}$, we have a rig-closed prime ideal", "$\\mathfrak q \\subset B\\{x\\}$ such that", "$\\mathfrak p = A \\cap \\mathfrak q$ is rig-closed as well.", "This implies, via Lemma \\ref{lemma-rig-closed-point-relative},", "that the intermediate primes $B \\cap \\mathfrak q$ and", "$A\\{x\\} \\cap \\mathfrak q$ are rig-closed as well.", "Consider the diagram", "$$", "\\xymatrix{", "(B[x])_{B[x] \\cap \\mathfrak q} \\ar[r] &", "(B\\{x\\})_{\\mathfrak q} \\\\", "(A[x])_{A[x] \\cap \\mathfrak q} \\ar[r] \\ar[u] &", "(A\\{x\\})_{A\\{x\\} \\cap \\mathfrak q} \\ar[u]", "}", "$$", "of local homomorphisms of Noetherian local rings.", "By Lemma \\ref{lemma-rig-closed-point-variables}", "the horizontal arrows define isomorphisms", "on completions. We already know that the left", "vertical arrow is flat (as $A \\to B$ is naively rig-flat", "and hence $A[x] \\to B[x]$ is flat away from the", "closed locus defined by an ideal of definition).", "Hence we finally conclude by", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-completion}." ], "refs": [ "algebra-lemma-completion-tensor", "algebra-lemma-flat-base-change", "restricted-lemma-rig-closed-point-after-localization", "restricted-lemma-rig-closed-point-relative", "restricted-lemma-rig-closed-point-variables", "more-algebra-lemma-flat-completion" ], "ref_ids": [ 869, 527, 2348, 2344, 2349, 10049 ] } ], "ref_ids": [] }, { "id": 2355, "type": "theorem", "label": "restricted-lemma-rig-flat-local-etale", "categories": [ "restricted" ], "title": "restricted-lemma-rig-flat-local-etale", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "B \\ar[r] & B' \\\\", "A \\ar[r] \\ar[u]^\\varphi & A' \\ar[u]_{\\varphi'}", "}", "$$", "in $\\textit{WAdm}^{Noeth}$ with all arrows adic and topologically", "of finite type. Assume $A \\to A'$ and $B \\to B'$ are flat.", "Let $I \\subset A$ be an ideal of definition.", "If $\\varphi$ is rig-flat and $A/I \\to A'/IA'$", "is \\'etale, then $\\varphi'$ is rig-flat." ], "refs": [], "proofs": [ { "contents": [ "Given $f \\in A'$ the assumptions of the lemma remain true for the digram", "$$", "\\xymatrix{", "B \\ar[r] & (B')_{\\{f\\}} \\\\", "A \\ar[r] \\ar[u]^\\varphi & (A')_{\\{f\\}} \\ar[u]", "}", "$$", "Hence it suffices to prove that $\\varphi'$ is naively rig-flat.", "\\medskip\\noindent", "Take a rig-closed prime ideal $\\mathfrak q' \\subset B'$.", "We have to show that $(B')_{\\mathfrak q'}$ is flat over $A'$.", "We can choose an $f \\in A$ which maps to a unit of $B'/\\mathfrak q'$", "such that the induced prime ideal $\\mathfrak p''$ of $A_{\\{f\\}}$ ", "is rig-closed, see Lemma \\ref{lemma-rig-closed-point-after-localization}.", "To be precise, here $\\mathfrak q'' = \\mathfrak q' B'_{\\{f\\}}$ and", "$\\mathfrak p'' = A_{\\{f\\}} \\cap \\mathfrak q''$.", "Consider the diagram", "$$", "\\xymatrix{", "B'_{\\mathfrak q'} \\ar[r] &", "(B'_{\\{f\\}})_{\\mathfrak q''} \\\\", "A \\ar[r] \\ar[u] &", "A_{\\{f\\}} \\ar[u]", "}", "$$", "We want to show that the left vertical arrow is flat.", "The top horizontal arrow is faithfully flat as it is a local", "homomorphism of local rings and flat as $B'_{\\{f\\}}$", "is the completion of a localization of the Noetherian ring", "$B'_f$. Similarly the bottom horizontal arrow is flat.", "Hence it suffices to prove that the right vertical arrow is flat.", "Finally, all the assumptions of the lemma remain true for the diagram", "$$", "\\xymatrix{", "B_{\\{f\\}} \\ar[r] &", "B'_{\\{f\\}} \\\\", "A_{\\{f\\}} \\ar[r] \\ar[u] &", "A'_{\\{f\\}} \\ar[u]", "}", "$$", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Take a rig-closed prime ideal $\\mathfrak q' \\subset B'$", "and assume $\\mathfrak p = A \\cap \\mathfrak q'$ is rig-closed as well.", "This implies also the primes $\\mathfrak q = B \\cap \\mathfrak q'$", "and $\\mathfrak p' = A' \\cap \\mathfrak q'$ are rig-closed, see", "Lemma \\ref{lemma-rig-closed-point-relative}.", "We are going to apply", "Algebra, Lemma \\ref{algebra-lemma-yet-another-variant-local-criterion-flatness}", "to the diagram", "$$", "\\xymatrix{", "B_\\mathfrak q \\ar[r] &", "B'_{\\mathfrak q'} \\\\", "A_\\mathfrak p \\ar[u] \\ar[r] &", "A'_{\\mathfrak p'} \\ar[u]", "}", "$$", "with $M = B_\\mathfrak q$. The only assumption that hasn't been checked yet", "is the fact that $\\mathfrak p$ generates the maximal ideal of", "$A'_{\\mathfrak p'}$. This follows from", "Lemma \\ref{lemma-rig-closed-point-etale}." ], "refs": [ "restricted-lemma-rig-closed-point-after-localization", "restricted-lemma-rig-closed-point-relative", "algebra-lemma-yet-another-variant-local-criterion-flatness", "restricted-lemma-rig-closed-point-etale" ], "ref_ids": [ 2348, 2344, 899, 2350 ] } ], "ref_ids": [] }, { "id": 2356, "type": "theorem", "label": "restricted-lemma-rig-flat-local-down", "categories": [ "restricted" ], "title": "restricted-lemma-rig-flat-local-down", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "B \\ar[r] & B' \\\\", "A \\ar[r] \\ar[u]^\\varphi & A' \\ar[u]_{\\varphi'}", "}", "$$", "in $\\textit{WAdm}^{Noeth}$ with all arrows adic and topologically", "of finite type. Assume $A \\to A'$ flat and $B \\to B'$ faithfully flat.", "If $\\varphi'$ is rig-flat, then $\\varphi$ is rig-flat." ], "refs": [], "proofs": [ { "contents": [ "Given $f \\in A$ the assumptions of the lemma remain true for the digram", "$$", "\\xymatrix{", "B_{\\{f\\}} \\ar[r] & (B')_{\\{f\\}} \\\\", "A_{\\{f\\}} \\ar[r] \\ar[u]^\\varphi & (A')_{\\{f\\}} \\ar[u]", "}", "$$", "(To check the condition on faithful flatness: faithful flatness", "of $B \\to B'$ is equivalent to $B \\to B'$ being flat and", "$\\Spec(B'/IB') \\to \\Spec(B/IB)$ being", "surjective for some ideal of definition $I \\subset A$.)", "Hence it suffices to prove that $\\varphi$ is naively rig-flat.", "However, we know that $\\varphi'$ is naively rig-flat and", "that $\\Spec(B') \\to \\Spec(B)$ is surjective. From this the", "result follows immediately." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2357, "type": "theorem", "label": "restricted-lemma-rig-flat-axioms", "categories": [ "restricted" ], "title": "restricted-lemma-rig-flat-axioms", "contents": [ "The property $P(\\varphi)=$``$\\varphi$ is rig-flat'' on arrows", "of $\\textit{WAdm}^{adic*}$ is a local property as defined in", "Formal Spaces, Remark \\ref{formal-spaces-remark-variant-adic-star}." ], "refs": [ "formal-spaces-remark-variant-adic-star" ], "proofs": [ { "contents": [ "Let us recall what the statement signifies. First, ", "$\\textit{WAdm}^{Noeth}$ is the category whose objects are", "adic Noetherian topological rings and whose morphisms are", "continuous ring homomorphisms. Consider a commutative diagram", "$$", "\\xymatrix{", "B \\ar[r] & (B')^\\wedge \\\\", "A \\ar[r] \\ar[u]^\\varphi & (A')^\\wedge \\ar[u]_{\\varphi'}", "}", "$$", "satisfying the following conditions:", "$A$ and $B$ are adic Noetherian topological rings,", "$A \\to A'$ and $B \\to B'$ are \\'etale ring maps,", "$(A')^\\wedge = \\lim A'/I^nA'$ for some ideal of definition $I \\subset A$,", "$(B')^\\wedge = \\lim B'/J^nB'$ for some ideal of definition $J \\subset B$, and", "$\\varphi : A \\to B$ and $\\varphi' : (A')^\\wedge \\to (B')^\\wedge$", "are continuous. Note that $(A')^\\wedge$ and $(B')^\\wedge$ are", "adic Noetherian topological rings by", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-completion-in-sub}.", "We have to show", "\\begin{enumerate}", "\\item $\\varphi$ is rig-flat $\\Rightarrow \\varphi'$ is rig-flat,", "\\item if $B \\to B'$ faithfully flat, then $\\varphi'$ is rig-flat", "$\\Rightarrow \\varphi$ is rig-flat, and", "\\item if $A \\to B_i$ is rig-flat for $i = 1, \\ldots, n$, then", "$A \\to \\prod_{i = 1, \\ldots, n} B_i$ is rig-flat.", "\\end{enumerate}", "Being adic and topologically of finite type satisfies", "conditions (1), (2), and (3), see Lemma \\ref{lemma-finite-type}.", "Thus in verifying (1), (2), and (3) for the property", "``rig-flat'' we may already assume our ring maps are all adic", "and topologically of finite type. Then (1) and (2) follow", "from Lemmas \\ref{lemma-rig-flat-local-etale} and", "\\ref{lemma-rig-flat-local-down}.", "We omit the trivial proof of (3)." ], "refs": [ "formal-spaces-lemma-completion-in-sub", "restricted-lemma-finite-type", "restricted-lemma-rig-flat-local-etale", "restricted-lemma-rig-flat-local-down" ], "ref_ids": [ 3922, 2324, 2355, 2356 ] } ], "ref_ids": [ 4007 ] }, { "id": 2358, "type": "theorem", "label": "restricted-lemma-composition-rig-flat-continuous", "categories": [ "restricted" ], "title": "restricted-lemma-composition-rig-flat-continuous", "contents": [ "The property $P(\\varphi)=$``$\\varphi$ is rig-flat''", "on arrows of $\\textit{WAdm}^{Noeth}$ is stable under composition", "as defined in Formal Spaces, Remark", "\\ref{formal-spaces-remark-composition-variant-Noetherian}." ], "refs": [ "formal-spaces-remark-composition-variant-Noetherian" ], "proofs": [ { "contents": [ "The statement makes sense by Lemma \\ref{lemma-rig-flat-axioms}.", "To see that it is true assume we have rig-flat morphisms", "$A \\to B$ and $B \\to C$ in $\\textit{WAdm}^{Noeth}$.", "Then $A \\to C$ is adic and topologically of finite type", "by Lemma \\ref{lemma-composition-finite-type}.", "To finish the proof we have to show that for all $f \\in A$ the map", "$A_{\\{f\\}} \\to C_{\\{f\\}}$ is naively rig-flat.", "Since $A_{\\{f\\}} \\to B_{\\{f\\}}$ and $B_{\\{f\\}} \\to C_{\\{f\\}}$", "are naively rig-flat, it suffices to show that", "compositions of naively rig-flat maps are naively rig-flat.", "This is a consequence of Algebra, Lemma \\ref{algebra-lemma-composition-flat}." ], "refs": [ "restricted-lemma-rig-flat-axioms", "restricted-lemma-composition-finite-type", "algebra-lemma-composition-flat" ], "ref_ids": [ 2357, 2326, 524 ] } ], "ref_ids": [ 4013 ] }, { "id": 2359, "type": "theorem", "label": "restricted-lemma-rig-flat-morphisms", "categories": [ "restricted" ], "title": "restricted-lemma-rig-flat-morphisms", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "locally Noetherian formal algebraic spaces over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is rig-flat,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$", "representable by algebraic spaces and \\'etale, the morphism $U \\to V$", "corresponds to a rig-flat map in $\\textit{WAdm}^{Noeth}$,", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "and for each $j$", "a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "such that each $X_{ji} \\to Y_j$ corresponds", "to a rig-flat map in $\\textit{WAdm}^{Noeth}$, and", "\\item there exist a covering $\\{X_i \\to X\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$", "is an affine formal algebraic space, $Y_i \\to Y$ is representable", "by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds", "to a rig-flat map in $\\textit{WAdm}^{Noeth}$.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is Definition \\ref{definition-rig-flat}.", "The equivalence of (2), (3), and (4) follows from the fact that", "being rig-flat is a local property of arrows of", "$\\text{WAdm}^{Noeth}$ by Lemma \\ref{lemma-rig-flat-axioms}", "and an application of the variant of", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-property-defines-property-morphisms}", "for morphisms between locally Noetherian algebraic spaces", "mentioned in", "Formal Spaces, Remark", "\\ref{formal-spaces-remark-variant-Noetherian}." ], "refs": [ "restricted-definition-rig-flat", "restricted-lemma-rig-flat-axioms", "formal-spaces-lemma-property-defines-property-morphisms", "formal-spaces-remark-variant-Noetherian" ], "ref_ids": [ 2441, 2357, 3923, 4008 ] } ], "ref_ids": [ 3981, 3981, 3981 ] }, { "id": 2360, "type": "theorem", "label": "restricted-lemma-base-change-rig-flat", "categories": [ "restricted" ], "title": "restricted-lemma-base-change-rig-flat", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$", "be morphisms of locally Noetherian formal algebraic spaces over $S$.", "If $f$ is rig-flat and $g$ is locally of finite type, then the base change", "$X \\times_Y Z \\to Z$ is rig-flat." ], "refs": [], "proofs": [ { "contents": [ "By Formal Spaces, Remark", "\\ref{formal-spaces-remark-base-change-variant-variant-Noetherian}", "and the discussion in Formal Spaces, Section", "\\ref{formal-spaces-section-adic},", "this follows from", "Lemma \\ref{lemma-rig-flat-base-change}." ], "refs": [ "formal-spaces-remark-base-change-variant-variant-Noetherian", "restricted-lemma-rig-flat-base-change" ], "ref_ids": [ 4011, 2354 ] } ], "ref_ids": [] }, { "id": 2361, "type": "theorem", "label": "restricted-lemma-composition-rig-flat", "categories": [ "restricted" ], "title": "restricted-lemma-composition-rig-flat", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$", "be morphisms of locally Noetherian formal algebraic spaces over $S$.", "If $f$ and $g$ are rig-flat, then so is $g \\circ f$." ], "refs": [], "proofs": [ { "contents": [ "By Formal Spaces, Remark", "\\ref{formal-spaces-remark-composition-variant-Noetherian}", "this follows from Lemma \\ref{lemma-composition-rig-flat-continuous}." ], "refs": [ "formal-spaces-remark-composition-variant-Noetherian", "restricted-lemma-composition-rig-flat-continuous" ], "ref_ids": [ 4013, 2358 ] } ], "ref_ids": [] }, { "id": 2362, "type": "theorem", "label": "restricted-lemma-rig-smooth-continuous", "categories": [ "restricted" ], "title": "restricted-lemma-rig-smooth-continuous", "contents": [ "Let $A \\to B$ be a morphism in $\\textit{WAdm}^{Noeth}$", "(Formal Spaces, Section \\ref{formal-spaces-section-morphisms-rings}).", "The following are equivalent:", "\\begin{enumerate}", "\\item[(a)] $A \\to B$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-finite-type} and there exists an ideal of definition", "$I \\subset B$ such that $B$ is rig-smooth over $(A, I)$, and", "\\item[(b)] $A \\to B$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-finite-type} and for all ideals of definition", "$I \\subset A$ the algebra $B$ is rig-smooth over $(A, I)$.", "\\end{enumerate}" ], "refs": [ "restricted-lemma-finite-type", "restricted-lemma-finite-type" ], "proofs": [ { "contents": [ "Let $I$ and $I'$ be ideals of definitions of $A$. Then there exists an", "integer $c \\geq 0$ such that $I^c \\subset I'$ and $(I')^c \\subset I$. Hence", "$B$ is rig-smooth over $(A, I)$ if and only if", "$B$ is rig-smooth over $(A, I')$. This follows from", "Definition \\ref{definition-rig-smooth-homomorphism},", "the inclusions $I^c \\subset I'$ and $(I')^c \\subset I$, and", "the fact that the naive cotangent complex $\\NL_{B/A}^\\wedge$", "is independent of the choice of ideal of definition of $A$ by", "Remark \\ref{remark-NL-well-defined-topological}." ], "refs": [ "restricted-definition-rig-smooth-homomorphism", "restricted-remark-NL-well-defined-topological" ], "ref_ids": [ 2434, 2453 ] } ], "ref_ids": [ 2324, 2324 ] }, { "id": 2363, "type": "theorem", "label": "restricted-lemma-rig-smooth-axioms", "categories": [ "restricted" ], "title": "restricted-lemma-rig-smooth-axioms", "contents": [ "The property $P(\\varphi)=$``$\\varphi$ is rig-smooth'' on arrows", "of $\\textit{WAdm}^{Noeth}$ is a local property as defined in", "Formal Spaces, Remark \\ref{formal-spaces-remark-variant-Noetherian}." ], "refs": [ "formal-spaces-remark-variant-Noetherian" ], "proofs": [ { "contents": [ "Let us recall what the statement signifies. First, ", "$\\textit{WAdm}^{Noeth}$ is the category whose objects are", "adic Noetherian topological rings and whose morphisms are", "continuous ring homomorphisms. Consider a commutative diagram", "$$", "\\xymatrix{", "B \\ar[r] & (B')^\\wedge \\\\", "A \\ar[r] \\ar[u]^\\varphi & (A')^\\wedge \\ar[u]_{\\varphi'}", "}", "$$", "satisfying the following conditions:", "$A$ and $B$ are adic Noetherian topological rings,", "$A \\to A'$ and $B \\to B'$ are \\'etale ring maps,", "$(A')^\\wedge = \\lim A'/I^nA'$ for some ideal of definition $I \\subset A$,", "$(B')^\\wedge = \\lim B'/J^nB'$ for some ideal of definition $J \\subset B$, and", "$\\varphi : A \\to B$ and $\\varphi' : (A')^\\wedge \\to (B')^\\wedge$", "are continuous. Note that $(A')^\\wedge$ and $(B')^\\wedge$ are", "adic Noetherian topological rings by", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-completion-in-sub}.", "We have to show", "\\begin{enumerate}", "\\item $\\varphi$ is rig-smooth $\\Rightarrow \\varphi'$ is rig-smooth,", "\\item if $B \\to B'$ faithfully flat, then $\\varphi'$ is rig-smooth", "$\\Rightarrow \\varphi$ is rig-smooth, and", "\\item if $A \\to B_i$ is rig-smooth for $i = 1, \\ldots, n$, then", "$A \\to \\prod_{i = 1, \\ldots, n} B_i$ is rig-smooth.", "\\end{enumerate}", "The equivalent conditions of Lemma \\ref{lemma-finite-type} satisfy", "conditions (1), (2), and (3).", "Thus in verifying (1), (2), and (3) for the property", "``rig-smooth'' we may already assume our ring maps satisfy", "the equivalent conditions of Lemma \\ref{lemma-finite-type}", "in each case.", "\\medskip\\noindent", "Pick an ideal of definition $I \\subset A$. By the remarks above", "the topology on each ring in the diagram is the $I$-adic topology", "and $B$, $(A')^\\wedge$, and $(B')^\\wedge$ are in the category", "(\\ref{equation-C-prime}) for $(A, I)$.", "Since $A \\to A'$ and $B \\to B'$ are \\'etale the complexes", "$\\NL_{A'/A}$ and $\\NL_{B'/B}$ are zero and hence", "$\\NL_{(A')^\\wedge/A}^\\wedge$ and $\\NL_{(B')^\\wedge/B}^\\wedge$", "are zero by Lemma \\ref{lemma-NL-is-completion}.", "Applying Lemma \\ref{lemma-exact-sequence-NL} to", "$A \\to (A')^\\wedge \\to (B')^\\wedge$ we get isomorphisms", "$$", "H^i(\\NL_{(B')^\\wedge/(A')^\\wedge}^\\wedge) \\to H^i(\\NL_{(B')^\\wedge/A}^\\wedge)", "$$", "Thus $\\NL_{(B')^\\wedge/A}^\\wedge \\to \\NL_{(B')^\\wedge/(A')^\\wedge}$", "is a quasi-isomorphism. The ring maps $B/I^nB \\to B'/I^nB'$ are \\'etale", "and hence are local complete intersections", "(Algebra, Lemma \\ref{algebra-lemma-etale-standard-smooth}).", "Hence we may apply", "Lemmas \\ref{lemma-exact-sequence-NL} and", "\\ref{lemma-transitive-lci-at-end} to", "$A \\to B \\to (B')^\\wedge$ and we get isomorphisms", "$$", "H^i(\\NL_{B/A}^\\wedge \\otimes_B (B')^\\wedge) \\to", "H^i(\\NL_{(B')^\\wedge/A}^\\wedge)", "$$", "We conclude that", "$\\NL_{B/A}^\\wedge \\otimes_B (B')^\\wedge \\to \\NL_{(B')^\\wedge/A}^\\wedge$", "is a quasi-isomorphism. Combining these two observations we obtain that", "$$", "\\NL_{(B')^\\wedge/(A')^\\wedge}^\\wedge \\cong", "\\NL_{B/A}^\\wedge \\otimes_B (B')^\\wedge", "$$", "in $D((B')^\\wedge)$.", "With these preparations out of the way we can start the actual proof.", "\\medskip\\noindent", "Proof of (1). Assume $\\varphi$ is rig-smooth. Then there exists a $c \\geq 0$", "such that $\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$ is annihilated by $I^c$", "for every $B$-module $N$. By", "More on Algebra, Lemmas \\ref{more-algebra-lemma-two-term-base-change} and", "\\ref{more-algebra-lemma-base-change-property-ext-1-annihilated}", "this property is preserved under base change by $B \\to (B')^\\wedge$.", "Hence $\\Ext^1_{(B')^\\wedge}(\\NL_{(B')^\\wedge/(A')^\\wedge}^\\wedge, N)$", "is annihilated by $I^c(A')^\\wedge$ for all $(B')^\\wedge$-modules $N$", "which tells us that $\\varphi'$ is rig-smooth.", "This proves (1).", "\\medskip\\noindent", "To prove (2) assume $B \\to B'$ is faithfully flat and that $\\varphi'$", "is rig-smooth. Then there exists a $c \\geq 0$ such that", "$\\Ext^1_{(B')^\\wedge}(\\NL_{(B')^\\wedge/(A')^\\wedge}^\\wedge, N')$", "is annihilated by $I^c(B')^\\wedge$ for every $(B')^\\wedge$-module $N'$.", "The composition $B \\to B' \\to (B')^\\wedge$ is flat", "(Algebra, Lemma \\ref{algebra-lemma-completion-flat})", "hence for any $B$-module $N$ we have", "$$", "\\Ext^1_B(\\NL_{B/A}^\\wedge, N) \\otimes_B (B')^\\wedge =", "\\Ext^1_{(B')^\\wedge}(\\NL_{B/A}^\\wedge \\otimes_B (B')^\\wedge,", "N \\otimes_B (B')^\\wedge)", "$$", "by More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom} part (3)", "(minor details omitted). Thus we see that this module is annihilated", "by $I^c$. However, $B \\to (B')^\\wedge$ is actually faithfully flat", "by our assumption that $B \\to B'$ is faithfully flat (Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-etale-surjective}). Thus we conclude that", "$\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$ is annihilated by $I^c$.", "Hence $\\varphi$ is rig-smooth. This proves (2).", "\\medskip\\noindent", "To prove (3), setting $B = \\prod_{i = 1, \\ldots, n} B_i$", "we just observe that $\\NL_{B/A}^\\wedge$ is the direct", "sum of the complexes $\\NL_{B_i/A}^\\wedge$ viewed as complexes", "of $B$-modules." ], "refs": [ "formal-spaces-lemma-completion-in-sub", "restricted-lemma-finite-type", "restricted-lemma-finite-type", "restricted-lemma-NL-is-completion", "restricted-lemma-exact-sequence-NL", "algebra-lemma-etale-standard-smooth", "restricted-lemma-exact-sequence-NL", "restricted-lemma-transitive-lci-at-end", "more-algebra-lemma-two-term-base-change", "more-algebra-lemma-base-change-property-ext-1-annihilated", "algebra-lemma-completion-flat", "more-algebra-lemma-base-change-RHom", "formal-spaces-lemma-etale-surjective" ], "ref_ids": [ 3922, 2324, 2324, 2297, 2300, 1230, 2300, 2301, 10301, 10302, 870, 10418, 3914 ] } ], "ref_ids": [ 4008 ] }, { "id": 2364, "type": "theorem", "label": "restricted-lemma-base-change-rig-smooth-continuous", "categories": [ "restricted" ], "title": "restricted-lemma-base-change-rig-smooth-continuous", "contents": [ "Consider the properties $P(\\varphi)=$``$\\varphi$ is rig-smooth''", "and $Q(\\varphi)$=``$\\varphi$ is adic'' on arrows of $\\textit{WAdm}^{Noeth}$.", "Then $P$ is stable under base change by $Q$ as defined in", "Formal Spaces, Remark", "\\ref{formal-spaces-remark-base-change-variant-variant-Noetherian}." ], "refs": [ "formal-spaces-remark-base-change-variant-variant-Noetherian" ], "proofs": [ { "contents": [ "The statement makes sense by Lemma \\ref{lemma-rig-smooth-continuous}.", "To see that it is true assume we have morphisms", "$B \\to A$ and $B \\to C$ in $\\textit{WAdm}^{Noeth}$", "and that $B \\to A$ is rig-smooth and $B \\to C$ is adic", "(Formal Spaces, Definition", "\\ref{formal-spaces-definition-adic-homomorphism}).", "Then we can choose an ideal of definition $I \\subset B$", "such that the topology on $A$ and $C$ is the $I$-adic topology.", "In this situation it follows immediately that", "$A \\widehat{\\otimes}_B C$ is rig-smooth over $(C, IC)$ by", "Lemma \\ref{lemma-base-change-rig-smooth-homomorphism}." ], "refs": [ "restricted-lemma-rig-smooth-continuous", "formal-spaces-definition-adic-homomorphism", "restricted-lemma-base-change-rig-smooth-homomorphism" ], "ref_ids": [ 2362, 3987, 2305 ] } ], "ref_ids": [ 4011 ] }, { "id": 2365, "type": "theorem", "label": "restricted-lemma-composition-rig-smooth-continuous", "categories": [ "restricted" ], "title": "restricted-lemma-composition-rig-smooth-continuous", "contents": [ "The property $P(\\varphi)=$``$\\varphi$ is rig-smooth''", "on arrows of $\\textit{WAdm}^{Noeth}$ is stable under composition", "as defined in Formal Spaces, Remark", "\\ref{formal-spaces-remark-composition-variant-Noetherian}." ], "refs": [ "formal-spaces-remark-composition-variant-Noetherian" ], "proofs": [ { "contents": [ "We strongly urge the reader to find their own proof and not read the proof", "that follows. The statement makes sense by", "Lemma \\ref{lemma-rig-smooth-continuous}.", "To see that it is true assume we have rig-smooth morphisms", "$A \\to B$ and $B \\to C$ in $\\textit{WAdm}^{Noeth}$.", "Then we can choose an ideal of definition $I \\subset A$", "such that the topology on $C$ and $B$ is the $I$-adic topology.", "By Lemma \\ref{lemma-exact-sequence-NL} we obtain an exact sequence", "$$", "\\xymatrix{", "C \\otimes_B H^0(\\NL_{B/A}^\\wedge) \\ar[r] &", "H^0(\\NL_{C/A}^\\wedge) \\ar[r] &", "H^0(\\NL_{C/B}^\\wedge) \\ar[r] & 0 \\\\", "H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C) \\ar[r] &", "H^{-1}(\\NL_{C/A}^\\wedge) \\ar[r] &", "H^{-1}(\\NL_{C/B}^\\wedge) \\ar[llu]", "}", "$$", "Observe that $H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C)$", "and $H^{-1}(\\NL_{C/B}^\\wedge)$ are annihilated by", "a power of $I$; this follows from", "Lemma \\ref{lemma-equivalent-with-artin-smooth} part (2)", "combined with", "More on Algebra, Lemmas \\ref{more-algebra-lemma-two-term-base-change} and", "\\ref{more-algebra-lemma-base-change-property-ext-1-annihilated}", "(to deal with the base change by $B \\to C$).", "Hence $H^{-1}(\\NL_{C/A}^\\wedge)$ is annihilated by a power of $I$.", "Next, by the characterization of rig-smooth algebras in", "Lemma \\ref{lemma-equivalent-with-artin-smooth} part (2)", "which in turn refers to", "More on Algebra, Lemma \\ref{more-algebra-lemma-ext-1-annihilated} part (5)", "we can choose $f_1, \\ldots, f_s \\in IB$ and $g_1, \\ldots, g_t \\in IC$", "such that $V(f_1, \\ldots, f_s) = V(IB)$ and", "$V(g_1, \\ldots, g_t) = V(IC)$ and such that", "$H^0(\\NL_{B/A}^\\wedge)_{f_i}$ is a finite projective $B_{f_i}$-module and", "$H^0(\\NL_{C/B}^\\wedge)_{g_j}$ is a finite projective $C_{g_j}$-module.", "Since the cohomologies in degree $-1$ vanish upon localization at", "$f_ig_j$ we get a short exact sequence", "$$", "0 \\to", "(C \\otimes_B H^0(\\NL_{B/A}^\\wedge))_{f_ig_j} \\to", "H^0(\\NL_{C/A}^\\wedge)_{f_ig_j} \\to", "H^0(\\NL_{C/B}^\\wedge)_{f_ig_j} \\to 0", "$$", "and we conclude that $H^0(\\NL_{C/A}^\\wedge)_{f_ig_j}$ is a finite", "projective $C_{f_ig_j}$-module as an extension of same.", "Thus by the criterion in", "Lemma \\ref{lemma-equivalent-with-artin-smooth} part (2)", "and via that the criterion in", "More on Algebra, Lemma \\ref{more-algebra-lemma-ext-1-annihilated} part (4)", "we conclude that $C$ is rig-smooth over $(A, I)$." ], "refs": [ "restricted-lemma-rig-smooth-continuous", "restricted-lemma-exact-sequence-NL", "restricted-lemma-equivalent-with-artin-smooth", "more-algebra-lemma-two-term-base-change", "more-algebra-lemma-base-change-property-ext-1-annihilated", "restricted-lemma-equivalent-with-artin-smooth", "more-algebra-lemma-ext-1-annihilated", "restricted-lemma-equivalent-with-artin-smooth", "more-algebra-lemma-ext-1-annihilated" ], "ref_ids": [ 2362, 2300, 2302, 10301, 10302, 2302, 10305, 2302, 10305 ] } ], "ref_ids": [ 4013 ] }, { "id": 2366, "type": "theorem", "label": "restricted-lemma-rig-smooth-rig-flat", "categories": [ "restricted" ], "title": "restricted-lemma-rig-smooth-rig-flat", "contents": [ "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$.", "If $\\varphi$ is rig-smooth, then $\\varphi$ is rig-flat, and", "for any presentation $B = A\\{x_1, \\ldots, x_n\\}/J$", "and prime $J \\subset \\mathfrak q \\subset A\\{x_1, \\ldots, x_n\\}$", "not containing an ideal of definition the ideal", "$J_\\mathfrak q \\subset A\\{x_1, \\ldots, x_n\\}_\\mathfrak q$", "is generated by a regular sequence." ], "refs": [], "proofs": [ { "contents": [ "Let $f \\in A$. To prove that $\\varphi$ is rig-flat we have to show", "that $\\varphi_{\\{f\\}} : A_{\\{f\\}} \\to B_{\\{f\\}}$ is naively rig-flat.", "Now either by viewing $\\varphi_{\\{f\\}}$ as a base change of $\\varphi$", "and using Lemma \\ref{lemma-base-change-rig-smooth-continuous}", "or by using the fact that being rig-smooth", "is a local property (Lemma \\ref{lemma-rig-smooth-axioms}) we see that", "$\\varphi_{\\{f\\}}$ is rig-smooth. Hence it suffices to show", "that $\\varphi$ is naively rig-flat.", "\\medskip\\noindent", "Choose a presentation $B = A\\{x_1, \\ldots, x_n\\}/J$.", "In order to check the second part of the lemma it suffices", "to check $J_\\mathfrak q \\subset A\\{x_1, \\ldots, x_n\\}_\\mathfrak q$", "is generated by a regular sequence for $J \\subset \\mathfrak q$", "for $\\mathfrak q$ maximal with respect to not containing", "an ideal of definition, see", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-neighbourhood}", "(which shows that the set of primes in $V(J)$ where there is", "a regular sequence generating $J$ is open).", "In other words, we may assume $\\mathfrak q$ is rig-closed", "in $A\\{x_1, \\ldots, x_n\\}$. And to check that", "$B$ is naively rig-flat, it also suffices to", "check that the corresponding localizations $B_\\mathfrak q$", "are flat over $A$.", "\\medskip\\noindent", "Let $\\mathfrak q \\subset A\\{x_1, \\ldots, x_n\\}$ be rig-closed with", "$J \\subset \\mathfrak q$. By", "Lemma \\ref{lemma-rig-closed-point-after-localization}", "we may choose an $f \\in A$ mapping to a unit in", "$A\\{x_1, \\ldots, x_n\\}/\\mathfrak q$ and such", "that the prime ideal $\\mathfrak p'$ in $A_{\\{f\\}}$ induced is rig-closed.", "Below we will use that", "$A_{\\{f\\}}\\{x_1, \\ldots, x_n\\} = A\\{x_1, \\ldots, x_n\\}_{\\{f\\}}$;", "details omitted. Consider the diagram", "$$", "\\xymatrix{", "A\\{x_1, \\ldots, x_n\\}_{\\mathfrak q} / J_\\mathfrak q \\ar[r] &", "A_{\\{f\\}}\\{x_1, \\ldots, x_n\\}_{\\mathfrak q'}/", "J A_{\\{f\\}}\\{x_1, \\ldots, x_n\\}_{\\mathfrak q'} \\\\", "A\\{x_1, \\ldots, x_n\\}_{\\mathfrak q} \\ar[r] \\ar[u] &", "A_{\\{f\\}}\\{x_1, \\ldots, x_n\\}_{\\mathfrak q'} \\ar[u] \\\\", "A \\ar[r] \\ar[u] &", "A_{\\{f\\}} \\ar[u]", "}", "$$", "The middle horizontal arrow is faithfully flat as it is a local", "homomorphism of local rings and flat as $A_{\\{f\\}}\\{x_1, \\ldots, x_n\\}$", "is the completion of a localization of the Noetherian ring", "$A\\{x_1, \\ldots, x_n\\}$. Similarly the bottom horizontal arrow is flat.", "Hence to show that $J_\\mathfrak q$ is generated by a regular sequence", "and that $A \\to A\\{x_1, \\ldots, x_n\\}_{\\mathfrak q} / J_\\mathfrak q$", "is flat, it suffices to prove the same things for", "$J A_{\\{f\\}}\\{x_1, \\ldots, x_n\\}_{\\mathfrak q'}$ and", "$A_{\\{f\\}} \\to A_{\\{f\\}}\\{x_1, \\ldots, x_n\\}_{\\mathfrak q'}/", "J A_{\\{f\\}}\\{x_1, \\ldots, x_n\\}_{\\mathfrak q'}$.", "See Algebra, Lemma \\ref{algebra-lemma-flat-increases-depth} or", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-descent-regular-ideal}", "for the statement on regular sequences. Finally, we have already seen that", "$A_{\\{f\\}} \\to B_{\\{f\\}}$ is rig-smooth.", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Let $\\mathfrak q \\subset A\\{x_1, \\ldots, x_n\\}$ be rig-closed with", "$J \\subset \\mathfrak q$ such that moreover $\\mathfrak p = A \\cap \\mathfrak q$", "is rig-closed as well. By the characterization of rig-smooth algebras", "given in Lemma \\ref{lemma-equivalent-with-artin-smooth}", "after reordering the variables $x_1, \\ldots, x_n$", "we can find $m \\geq 0$ and $f_1, \\ldots, f_m \\in J$ such that", "\\begin{enumerate}", "\\item $J_\\mathfrak q$ is generated by $f_1, \\ldots, f_m$, and", "\\item $\\det_{1 \\leq i, j \\leq m}(\\partial f_j/ \\partial x_i)$", "maps to a unit in $A\\{x_1, \\ldots, x_n\\}_\\mathfrak q$.", "\\end{enumerate}", "By Lemma \\ref{lemma-fibre-regular} the fibre ring", "$$", "F = A\\{x_1, \\ldots, x_n\\} \\otimes_A \\kappa(\\mathfrak p)", "$$", "is regular. Observe that the $A$-derivations $\\partial / \\partial x_i$", "extend (uniquely) to derivations $D_i : F \\to F$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-quotient-sequence-regular}", "we see that $f_1, \\ldots, f_m$ map to a regular sequence in", "$F_\\mathfrak q$. By flatness of $A \\to A\\{x_1, \\ldots, x_n\\}$", "and Algebra, Lemma \\ref{algebra-lemma-grothendieck-regular-sequence}", "this shows that $f_1, \\ldots, f_m$ map to a regular sequence in", "$A\\{x_1, \\ldots, x_m\\}_\\mathfrak q$ and the quotient by", "these elements is flat over $A$. This finishes the proof." ], "refs": [ "restricted-lemma-base-change-rig-smooth-continuous", "restricted-lemma-rig-smooth-axioms", "algebra-lemma-regular-sequence-in-neighbourhood", "restricted-lemma-rig-closed-point-after-localization", "algebra-lemma-flat-increases-depth", "more-algebra-lemma-flat-descent-regular-ideal", "restricted-lemma-equivalent-with-artin-smooth", "restricted-lemma-fibre-regular", "more-algebra-lemma-quotient-sequence-regular", "algebra-lemma-grothendieck-regular-sequence" ], "ref_ids": [ 2364, 2363, 741, 2348, 740, 9997, 2302, 2351, 10079, 885 ] } ], "ref_ids": [] }, { "id": 2367, "type": "theorem", "label": "restricted-lemma-exact-sequence-NL-rig-smooth", "categories": [ "restricted" ], "title": "restricted-lemma-exact-sequence-NL-rig-smooth", "contents": [ "Let $A \\to B \\to C$ be arrows in $\\textit{WAdm}^{Noeth}$", "which are adic and topologically of finite type. If $B \\to C$", "is rig-smooth, then the kernel of the map", "$$", "H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C) \\to H^{-1}(\\NL_{C/A}^\\wedge)", "$$", "(see Lemma \\ref{lemma-exact-sequence-NL})", "is annihilated by an ideal of definition." ], "refs": [ "restricted-lemma-exact-sequence-NL" ], "proofs": [ { "contents": [ "Let $\\overline{\\mathfrak q} \\subset C$ be a prime ideal which does not contain", "an ideal of definition. Since the modules in question are finite", "it suffices to show that", "$$", "H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C)_{\\overline{\\mathfrak q}} \\to", "H^{-1}(\\NL_{C/A}^\\wedge)_{\\overline{\\mathfrak q}}", "$$", "is injective. As in the proof of Lemma \\ref{lemma-exact-sequence-NL}", "choose presentations $B = A\\{x_1, \\ldots, x_r\\}/J$,", "$C = B\\{y_1, \\ldots, y_s\\}/J'$, and", "$C = A\\{x_1, \\ldots, x_r, y_1, \\ldots, y_s\\}/K$.", "Looking at the diagram in the proof of Lemma \\ref{lemma-exact-sequence-NL}", "we see that it suffices to show that $J/J^2 \\otimes_B C \\to K/K^2$", "is injective after localization at the prime ideal", "$\\mathfrak q \\subset A\\{x_1, \\ldots, x_r, y_1, \\ldots, y_s\\}$", "corresponding to $\\overline{\\mathfrak q}$. Please compare with", "More on Algebra, Lemma \\ref{more-algebra-lemma-transitive-lci-at-end}", "and its proof. This is the same as ", "asking $J/KJ \\to K/K^2$ to be injective after localization", "at $\\mathfrak q$. Equivalently, we have to show that", "$J_\\mathfrak q \\cap K^2_\\mathfrak q = (KJ)_\\mathfrak q$.", "By Lemma \\ref{lemma-rig-smooth-rig-flat}", "we know that $(K/J)_\\mathfrak q = J'_\\mathfrak q$", "is generated by a regular sequence.", "Hence the desired intersection property follows from", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-conormal-sequence-H1-regular-ideal}", "(and the fact that an ideal generated by a regular sequence", "is $H_1$-regular, see", "More on Algebra, Section \\ref{more-algebra-section-ideals})." ], "refs": [ "restricted-lemma-exact-sequence-NL", "restricted-lemma-exact-sequence-NL", "more-algebra-lemma-transitive-lci-at-end", "restricted-lemma-rig-smooth-rig-flat", "more-algebra-lemma-conormal-sequence-H1-regular-ideal" ], "ref_ids": [ 2300, 2300, 10003, 2366, 9998 ] } ], "ref_ids": [ 2300 ] }, { "id": 2368, "type": "theorem", "label": "restricted-lemma-rig-smooth-morphisms", "categories": [ "restricted" ], "title": "restricted-lemma-rig-smooth-morphisms", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "locally Noetherian formal algebraic spaces over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is rig-smooth,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$", "representable by algebraic spaces and \\'etale, the morphism $U \\to V$", "corresponds to a rig-smooth map in $\\textit{WAdm}^{Noeth}$,", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "and for each $j$", "a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "such that each $X_{ji} \\to Y_j$ corresponds", "to a rig-smooth map in $\\textit{WAdm}^{Noeth}$, and", "\\item there exist a covering $\\{X_i \\to X\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$", "is an affine formal algebraic space, $Y_i \\to Y$ is representable", "by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds", "to a rig-smooth map in $\\textit{WAdm}^{Noeth}$.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is Definition \\ref{definition-rig-smooth}.", "The equivalence of (2), (3), and (4) follows from the fact that", "being rig-smooth is a local property of arrows of", "$\\text{WAdm}^{Noeth}$ by Lemma \\ref{lemma-rig-smooth-axioms}", "and an application of the variant of", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-property-defines-property-morphisms}", "for morphisms between locally Noetherian algebraic spaces", "mentioned in", "Formal Spaces, Remark", "\\ref{formal-spaces-remark-variant-Noetherian}." ], "refs": [ "restricted-definition-rig-smooth", "restricted-lemma-rig-smooth-axioms", "formal-spaces-lemma-property-defines-property-morphisms", "formal-spaces-remark-variant-Noetherian" ], "ref_ids": [ 2443, 2363, 3923, 4008 ] } ], "ref_ids": [ 3981, 3981, 3981 ] }, { "id": 2369, "type": "theorem", "label": "restricted-lemma-base-change-rig-smooth", "categories": [ "restricted" ], "title": "restricted-lemma-base-change-rig-smooth", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$", "be morphisms of locally Noetherian formal algebraic spaces over $S$.", "If $f$ is rig-smooth and $g$ is adic, then the base change", "$X \\times_Y Z \\to Z$ is rig-smooth." ], "refs": [], "proofs": [ { "contents": [ "By Formal Spaces, Remark", "\\ref{formal-spaces-remark-base-change-variant-variant-Noetherian}", "and the discussion in Formal Spaces, Section", "\\ref{formal-spaces-section-adic},", "this follows from Lemma \\ref{lemma-base-change-rig-smooth-continuous}." ], "refs": [ "formal-spaces-remark-base-change-variant-variant-Noetherian", "restricted-lemma-base-change-rig-smooth-continuous" ], "ref_ids": [ 4011, 2364 ] } ], "ref_ids": [] }, { "id": 2370, "type": "theorem", "label": "restricted-lemma-composition-rig-smooth", "categories": [ "restricted" ], "title": "restricted-lemma-composition-rig-smooth", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$", "be morphisms of locally Noetherian formal algebraic spaces over $S$.", "If $f$ and $g$ are rig-smooth, then so is $g \\circ f$." ], "refs": [], "proofs": [ { "contents": [ "By Formal Spaces, Remark", "\\ref{formal-spaces-remark-composition-variant-Noetherian}", "this follows from Lemma \\ref{lemma-composition-rig-smooth-continuous}." ], "refs": [ "formal-spaces-remark-composition-variant-Noetherian", "restricted-lemma-composition-rig-smooth-continuous" ], "ref_ids": [ 4013, 2365 ] } ], "ref_ids": [] }, { "id": 2371, "type": "theorem", "label": "restricted-lemma-rig-smooth-rig-flat-morphism", "categories": [ "restricted" ], "title": "restricted-lemma-rig-smooth-rig-flat-morphism", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "locally Noetherian formal algebraic spaces over $S$.", "If $f$ is rig-smooth, then $f$ is rig-flat." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-rig-smooth-rig-flat}", "and the definitions." ], "refs": [ "restricted-lemma-rig-smooth-rig-flat" ], "ref_ids": [ 2366 ] } ], "ref_ids": [] }, { "id": 2372, "type": "theorem", "label": "restricted-lemma-rig-etale-continuous", "categories": [ "restricted" ], "title": "restricted-lemma-rig-etale-continuous", "contents": [ "Let $A \\to B$ be a morphism in $\\textit{WAdm}^{Noeth}$", "(Formal Spaces, Section \\ref{formal-spaces-section-morphisms-rings}).", "The following are equivalent:", "\\begin{enumerate}", "\\item[(a)] $A \\to B$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-finite-type} and there exists an ideal of definition", "$I \\subset B$ such that $B$ is rig-\\'etale over $(A, I)$, and", "\\item[(b)] $A \\to B$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-finite-type} and for all ideals of definition", "$I \\subset A$ the algebra $B$ is rig-\\'etale over $(A, I)$.", "\\end{enumerate}" ], "refs": [ "restricted-lemma-finite-type", "restricted-lemma-finite-type" ], "proofs": [ { "contents": [ "Let $I$ and $I'$ be ideals of definitions of $A$. Then there exists an", "integer $c \\geq 0$ such that $I^c \\subset I'$ and $(I')^c \\subset I$. Hence", "$B$ is rig-\\'etale over $(A, I)$ if and only if", "$B$ is rig-\\'etale over $(A, I')$. This follows from", "Definition \\ref{definition-rig-etale-homomorphism},", "the inclusions $I^c \\subset I'$ and $(I')^c \\subset I$, and", "the fact that the naive cotangent complex $\\NL_{B/A}^\\wedge$", "is independent of the choice of ideal of definition of $A$ by", "Remark \\ref{remark-NL-well-defined-topological}." ], "refs": [ "restricted-definition-rig-etale-homomorphism", "restricted-remark-NL-well-defined-topological" ], "ref_ids": [ 2435, 2453 ] } ], "ref_ids": [ 2324, 2324 ] }, { "id": 2373, "type": "theorem", "label": "restricted-lemma-rig-etale-axioms", "categories": [ "restricted" ], "title": "restricted-lemma-rig-etale-axioms", "contents": [ "The property $P(\\varphi)=$``$\\varphi$ is rig-\\'etale'' on arrows", "of $\\textit{WAdm}^{Noeth}$ is a local property as defined in", "Formal Spaces, Remark \\ref{formal-spaces-remark-variant-Noetherian}." ], "refs": [ "formal-spaces-remark-variant-Noetherian" ], "proofs": [ { "contents": [ "This proof is exactly the same as the proof of", "Lemma \\ref{lemma-rig-smooth-axioms}.", "Let us recall what the statement signifies. First, ", "$\\textit{WAdm}^{Noeth}$ is the category whose objects are", "adic Noetherian topological rings and whose morphisms are", "continuous ring homomorphisms. Consider a commutative diagram", "$$", "\\xymatrix{", "B \\ar[r] & (B')^\\wedge \\\\", "A \\ar[r] \\ar[u]^\\varphi & (A')^\\wedge \\ar[u]_{\\varphi'}", "}", "$$", "satisfying the following conditions:", "$A$ and $B$ are adic Noetherian topological rings,", "$A \\to A'$ and $B \\to B'$ are \\'etale ring maps,", "$(A')^\\wedge = \\lim A'/I^nA'$ for some ideal of definition $I \\subset A$,", "$(B')^\\wedge = \\lim B'/J^nB'$ for some ideal of definition $J \\subset B$, and", "$\\varphi : A \\to B$ and $\\varphi' : (A')^\\wedge \\to (B')^\\wedge$", "are continuous. Note that $(A')^\\wedge$ and $(B')^\\wedge$ are", "adic Noetherian topological rings by", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-completion-in-sub}.", "We have to show", "\\begin{enumerate}", "\\item $\\varphi$ is rig-\\'etale $\\Rightarrow \\varphi'$ is rig-\\'etale,", "\\item if $B \\to B'$ faithfully flat, then $\\varphi'$ is rig-\\'etale", "$\\Rightarrow \\varphi$ is rig-\\'etale, and", "\\item if $A \\to B_i$ is rig-\\'etale for $i = 1, \\ldots, n$, then", "$A \\to \\prod_{i = 1, \\ldots, n} B_i$ is rig-\\'etale.", "\\end{enumerate}", "The equivalent conditions of Lemma \\ref{lemma-finite-type} satisfy", "conditions (1), (2), and (3).", "Thus in verifying (1), (2), and (3) for the property", "``rig-\\'etale'' we may already assume our ring maps satisfy", "the equivalent conditions of Lemma \\ref{lemma-finite-type}", "in each case.", "\\medskip\\noindent", "Pick an ideal of definition $I \\subset A$. By the remarks above", "the topology on each ring in the diagram is the $I$-adic topology", "and $B$, $(A')^\\wedge$, and $(B')^\\wedge$ are in the category", "(\\ref{equation-C-prime}) for $(A, I)$.", "Since $A \\to A'$ and $B \\to B'$ are \\'etale the complexes", "$\\NL_{A'/A}$ and $\\NL_{B'/B}$ are zero and hence", "$\\NL_{(A')^\\wedge/A}^\\wedge$ and $\\NL_{(B')^\\wedge/B}^\\wedge$", "are zero by Lemma \\ref{lemma-NL-is-completion}.", "Applying Lemma \\ref{lemma-exact-sequence-NL} to", "$A \\to (A')^\\wedge \\to (B')^\\wedge$ we get isomorphisms", "$$", "H^i(\\NL_{(B')^\\wedge/(A')^\\wedge}^\\wedge) \\to H^i(\\NL_{(B')^\\wedge/A}^\\wedge)", "$$", "Thus $\\NL_{(B')^\\wedge/A}^\\wedge \\to \\NL_{(B')^\\wedge/(A')^\\wedge}$", "is a quasi-isomorphism. The ring maps $B/I^nB \\to B'/I^nB'$ are \\'etale", "and hence are local complete intersections", "(Algebra, Lemma \\ref{algebra-lemma-etale-standard-smooth}).", "Hence we may apply", "Lemmas \\ref{lemma-exact-sequence-NL} and", "\\ref{lemma-transitive-lci-at-end} to", "$A \\to B \\to (B')^\\wedge$ and we get isomorphisms", "$$", "H^i(\\NL_{B/A}^\\wedge \\otimes_B (B')^\\wedge) \\to", "H^i(\\NL_{(B')^\\wedge/A}^\\wedge)", "$$", "We conclude that", "$\\NL_{B/A}^\\wedge \\otimes_B (B')^\\wedge \\to \\NL_{(B')^\\wedge/A}^\\wedge$", "is a quasi-isomorphism. Combining these two observations we obtain that", "$$", "\\NL_{(B')^\\wedge/(A')^\\wedge}^\\wedge \\cong", "\\NL_{B/A}^\\wedge \\otimes_B (B')^\\wedge", "$$", "in $D((B')^\\wedge)$.", "With these preparations out of the way we can start the actual proof.", "\\medskip\\noindent", "Proof of (1). Assume $\\varphi$ is rig-\\'etale. Then there exists a $c \\geq 0$", "such that multiplication by $a \\in I^c$ is zero on $\\NL_{B/A}^\\wedge$", "in $D(B)$. This property is preserved under base change", "by $B \\to (B')^\\wedge$, see", "More on Algebra, Lemmas \\ref{more-algebra-lemma-two-term-base-change}.", "By the isomorphism above we find that $\\varphi'$ is rig-\\'etale.", "This proves (1).", "\\medskip\\noindent", "To prove (2) assume $B \\to B'$ is faithfully flat and that $\\varphi'$", "is rig-\\'etale. Then there exists a $c \\geq 0$ such that", "multiplication by $a \\in I^c$ is zero on", "$\\NL_{(B')^\\wedge/(A')^\\wedge}^\\wedge$ in $D((B')^\\wedge)$.", "By the isomorphism above we see that $a^c$ annihilates the", "cohomology modules of", "$\\NL_{B/A}^\\wedge \\otimes_B (B')^\\wedge$.", "The composition $B \\to (B')^\\wedge$ is faithfully flat", "by our assumption that $B \\to B'$ is faithfully flat, see", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-etale-surjective}.", "Hence the cohomology modules of $\\NL_{B/A}^\\wedge$ are annihilated", "by $I^c$. It follows from Lemma \\ref{lemma-equivalent-with-artin}", "that $\\varphi$ is rig-\\'etale.", "This proves (2).", "\\medskip\\noindent", "To prove (3), setting $B = \\prod_{i = 1, \\ldots, n} B_i$", "we just observe that $\\NL_{B/A}^\\wedge$ is the direct", "sum of the complexes $\\NL_{B_i/A}^\\wedge$ viewed as complexes", "of $B$-modules." ], "refs": [ "restricted-lemma-rig-smooth-axioms", "formal-spaces-lemma-completion-in-sub", "restricted-lemma-finite-type", "restricted-lemma-finite-type", "restricted-lemma-NL-is-completion", "restricted-lemma-exact-sequence-NL", "algebra-lemma-etale-standard-smooth", "restricted-lemma-exact-sequence-NL", "restricted-lemma-transitive-lci-at-end", "more-algebra-lemma-two-term-base-change", "formal-spaces-lemma-etale-surjective", "restricted-lemma-equivalent-with-artin" ], "ref_ids": [ 2363, 3922, 2324, 2324, 2297, 2300, 1230, 2300, 2301, 10301, 3914, 2314 ] } ], "ref_ids": [ 4008 ] }, { "id": 2374, "type": "theorem", "label": "restricted-lemma-base-change-rig-etale-continuous", "categories": [ "restricted" ], "title": "restricted-lemma-base-change-rig-etale-continuous", "contents": [ "Consider the properties $P(\\varphi)=$``$\\varphi$ is rig-\\'etale''", "and $Q(\\varphi)$=``$\\varphi$ is adic'' on arrows of $\\textit{WAdm}^{Noeth}$.", "Then $P$ is stable under base change by $Q$ as defined in", "Formal Spaces, Remark", "\\ref{formal-spaces-remark-base-change-variant-variant-Noetherian}." ], "refs": [ "formal-spaces-remark-base-change-variant-variant-Noetherian" ], "proofs": [ { "contents": [ "The statement makes sense by Lemma \\ref{lemma-rig-etale-continuous}.", "To see that it is true assume we have morphisms", "$B \\to A$ and $B \\to C$ in $\\textit{WAdm}^{Noeth}$", "and that $B \\to A$ is rig-\\'etale and $B \\to C$ is adic", "(Formal Spaces, Definition", "\\ref{formal-spaces-definition-adic-homomorphism}).", "Then we can choose an ideal of definition $I \\subset B$", "such that the topology on $A$ and $C$ is the $I$-adic topology.", "In this situation it follows immediately that", "$A \\widehat{\\otimes}_B C$ is rig-\\'etale over $(C, IC)$ by", "Lemma \\ref{lemma-base-change-rig-etale-homomorphism}." ], "refs": [ "restricted-lemma-rig-etale-continuous", "formal-spaces-definition-adic-homomorphism", "restricted-lemma-base-change-rig-etale-homomorphism" ], "ref_ids": [ 2372, 3987, 2318 ] } ], "ref_ids": [ 4011 ] }, { "id": 2375, "type": "theorem", "label": "restricted-lemma-composition-rig-etale-continuous", "categories": [ "restricted" ], "title": "restricted-lemma-composition-rig-etale-continuous", "contents": [ "The property $P(\\varphi)=$``$\\varphi$ is rig-\\'etale''", "on arrows of $\\textit{WAdm}^{Noeth}$ is stable under composition", "as defined in Formal Spaces, Remark", "\\ref{formal-spaces-remark-composition-variant-Noetherian}." ], "refs": [ "formal-spaces-remark-composition-variant-Noetherian" ], "proofs": [ { "contents": [ "The statement makes sense by", "Lemma \\ref{lemma-rig-etale-continuous}.", "To see that it is true assume we have rig-\\'etale morphisms", "$A \\to B$ and $B \\to C$ in $\\textit{WAdm}^{Noeth}$.", "Then we can choose an ideal of definition $I \\subset A$", "such that the topology on $C$ and $B$ is the $I$-adic topology.", "By Lemma \\ref{lemma-exact-sequence-NL} we obtain an exact sequence", "$$", "\\xymatrix{", "C \\otimes_B H^0(\\NL_{B/A}^\\wedge) \\ar[r] &", "H^0(\\NL_{C/A}^\\wedge) \\ar[r] &", "H^0(\\NL_{C/B}^\\wedge) \\ar[r] & 0 \\\\", "H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C) \\ar[r] &", "H^{-1}(\\NL_{C/A}^\\wedge) \\ar[r] &", "H^{-1}(\\NL_{C/B}^\\wedge) \\ar[llu]", "}", "$$", "There exists a $c \\geq 0$ such that for all $a \\in I$ ", "multiplication by $a^c$ is zero on $\\NL_{B/A}^\\wedge$ in $D(B)$ and", "$\\NL_{C/B}^\\wedge$ in $D(C)$. Then of course", "multiplication by $a^c$ is zero on $\\NL_{B/A}^\\wedge \\otimes_B C$", "in $D(C)$ too. Hence", "$H^0(\\NL_{B/A}^\\wedge) \\otimes_A C$, ", "$H^0(\\NL_{C/B}^\\wedge)$, ", "$H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C)$, and", "$H^{-1}(\\NL_{C/B}^\\wedge)$", "are annihilated by $a^c$. From the exact sequence", "we obtain that multiplication by $a^{2c}$ is zero on", "$H^0(\\NL_{C/A}^\\wedge)$ and $H^{-1}(\\NL_{C/A}^\\wedge)$.", "It follows from Lemma \\ref{lemma-equivalent-with-artin}", "that $C$ is rig-\\'etale over $(A, I)$ as desired." ], "refs": [ "restricted-lemma-rig-etale-continuous", "restricted-lemma-exact-sequence-NL", "restricted-lemma-equivalent-with-artin" ], "ref_ids": [ 2372, 2300, 2314 ] } ], "ref_ids": [ 4013 ] }, { "id": 2376, "type": "theorem", "label": "restricted-lemma-permanence-rig-etale-continuous", "categories": [ "restricted" ], "title": "restricted-lemma-permanence-rig-etale-continuous", "contents": [ "The property $P(\\varphi)=$``$\\varphi$ is rig-\\'etale''", "on arrows of $\\textit{WAdm}^{Noeth}$ has the cancellation property", "as defined in Formal Spaces, Remark", "\\ref{formal-spaces-remark-permanence-variant-Noetherian}." ], "refs": [ "formal-spaces-remark-permanence-variant-Noetherian" ], "proofs": [ { "contents": [ "The statement makes sense by", "Lemma \\ref{lemma-rig-etale-continuous}.", "To see that it is true assume we have maps", "$A \\to B$ and $B \\to C$ in $\\textit{WAdm}^{Noeth}$", "with $A \\to C$ and $A \\to B$ rig-\\'etale.", "We have to show that $B \\to C$ is rig-\\'etale.", "Then we can choose an ideal of definition $I \\subset A$", "such that the topology on $C$ and $B$ is the $I$-adic topology.", "By Lemma \\ref{lemma-exact-sequence-NL} we obtain an exact sequence", "$$", "\\xymatrix{", "C \\otimes_B H^0(\\NL_{B/A}^\\wedge) \\ar[r] &", "H^0(\\NL_{C/A}^\\wedge) \\ar[r] &", "H^0(\\NL_{C/B}^\\wedge) \\ar[r] & 0 \\\\", "H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C) \\ar[r] &", "H^{-1}(\\NL_{C/A}^\\wedge) \\ar[r] &", "H^{-1}(\\NL_{C/B}^\\wedge) \\ar[llu]", "}", "$$", "There exists a $c \\geq 0$ such that for all $a \\in I$ ", "multiplication by $a^c$ is zero on $\\NL_{B/A}^\\wedge$ in $D(B)$ and", "$\\NL_{C/A}^\\wedge$ in $D(C)$. Hence", "$H^0(\\NL_{B/A}^\\wedge) \\otimes_A C$,", "$H^0(\\NL_{C/A}^\\wedge)$, and", "$H^{-1}(\\NL_{C/A}^\\wedge)$", "are annihilated by $a^c$. From the exact sequence", "we obtain that multiplication by $a^{2c}$ is zero on", "$H^0(\\NL_{C/B}^\\wedge)$ and $H^{-1}(\\NL_{C/B}^\\wedge)$.", "It follows from Lemma \\ref{lemma-equivalent-with-artin}", "that $C$ is rig-\\'etale over $(B, IB)$ as desired." ], "refs": [ "restricted-lemma-rig-etale-continuous", "restricted-lemma-exact-sequence-NL", "restricted-lemma-equivalent-with-artin" ], "ref_ids": [ 2372, 2300, 2314 ] } ], "ref_ids": [ 4015 ] }, { "id": 2377, "type": "theorem", "label": "restricted-lemma-rig-etale-morphisms", "categories": [ "restricted" ], "title": "restricted-lemma-rig-etale-morphisms", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "locally Noetherian formal algebraic spaces over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is rig-\\'etale,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$", "representable by algebraic spaces and \\'etale, the morphism $U \\to V$", "corresponds to a rig-\\'etale map in $\\textit{WAdm}^{Noeth}$,", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "and for each $j$", "a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "such that each $X_{ji} \\to Y_j$ corresponds", "to a rig-\\'etale map in $\\textit{WAdm}^{Noeth}$, and", "\\item there exist a covering $\\{X_i \\to X\\}$ as in", "Formal Spaces,", "Definition \\ref{formal-spaces-definition-formal-algebraic-space}", "and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$", "is an affine formal algebraic space, $Y_i \\to Y$ is representable", "by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds", "to a rig-\\'etale map in $\\textit{WAdm}^{Noeth}$.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is Definition \\ref{definition-rig-etale}.", "The equivalence of (2), (3), and (4) follows from the fact that", "being rig-\\'etale is a local property of arrows of", "$\\text{WAdm}^{Noeth}$ by Lemma \\ref{lemma-rig-etale-axioms}", "and an application of the variant of", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-property-defines-property-morphisms}", "for morphisms between locally Noetherian algebraic spaces", "mentioned in Formal Spaces, Remark", "\\ref{formal-spaces-remark-variant-Noetherian}." ], "refs": [ "restricted-definition-rig-etale", "restricted-lemma-rig-etale-axioms", "formal-spaces-lemma-property-defines-property-morphisms", "formal-spaces-remark-variant-Noetherian" ], "ref_ids": [ 2445, 2373, 3923, 4008 ] } ], "ref_ids": [ 3981, 3981, 3981 ] }, { "id": 2378, "type": "theorem", "label": "restricted-lemma-rig-etale-finite-type", "categories": [ "restricted" ], "title": "restricted-lemma-rig-etale-finite-type", "contents": [ "A rig-\\'etale morphism of locally Noetherian formal algebraic spaces", "is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "The property $P$ in Lemma \\ref{lemma-rig-etale-axioms}", "implies the equivalent conditions (a), (b), (c), and (d) in", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-quotient-restricted-power-series}.", "Hence this follows from", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-finite-type-local-property}." ], "refs": [ "restricted-lemma-rig-etale-axioms", "formal-spaces-lemma-quotient-restricted-power-series", "formal-spaces-lemma-finite-type-local-property" ], "ref_ids": [ 2373, 3951, 3954 ] } ], "ref_ids": [] }, { "id": 2379, "type": "theorem", "label": "restricted-lemma-rig-etale-rig-smooth-morphism", "categories": [ "restricted" ], "title": "restricted-lemma-rig-etale-rig-smooth-morphism", "contents": [ "A rig-\\'etale morphism of locally Noetherian formal algebraic spaces", "is rig-smooth." ], "refs": [], "proofs": [ { "contents": [ "Follows from the definitions and", "Lemma \\ref{lemma-rig-etale-rig-smooth}." ], "refs": [ "restricted-lemma-rig-etale-rig-smooth" ], "ref_ids": [ 2315 ] } ], "ref_ids": [] }, { "id": 2380, "type": "theorem", "label": "restricted-lemma-base-change-rig-etale", "categories": [ "restricted" ], "title": "restricted-lemma-base-change-rig-etale", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$", "be morphisms of locally Noetherian formal algebraic spaces over $S$.", "If $f$ is rig-\\'etale and $g$ is adic, then the base change", "$X \\times_Y Z \\to Z$ is rig-\\'etale." ], "refs": [], "proofs": [ { "contents": [ "By Formal Spaces, Remark", "\\ref{formal-spaces-remark-base-change-variant-variant-Noetherian}", "and the discussion in Formal Spaces, Section", "\\ref{formal-spaces-section-adic},", "this follows from Lemma \\ref{lemma-base-change-rig-etale-continuous}." ], "refs": [ "formal-spaces-remark-base-change-variant-variant-Noetherian", "restricted-lemma-base-change-rig-etale-continuous" ], "ref_ids": [ 4011, 2374 ] } ], "ref_ids": [] }, { "id": 2381, "type": "theorem", "label": "restricted-lemma-composition-rig-etale", "categories": [ "restricted" ], "title": "restricted-lemma-composition-rig-etale", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$", "be morphisms of locally Noetherian formal algebraic spaces over $S$.", "If $f$ and $g$ are rig-\\'etale, then so is $g \\circ f$." ], "refs": [], "proofs": [ { "contents": [ "By Formal Spaces, Remark", "\\ref{formal-spaces-remark-composition-variant-Noetherian}", "this follows from Lemma \\ref{lemma-composition-rig-etale-continuous}." ], "refs": [ "formal-spaces-remark-composition-variant-Noetherian", "restricted-lemma-composition-rig-etale-continuous" ], "ref_ids": [ 4013, 2375 ] } ], "ref_ids": [] }, { "id": 2382, "type": "theorem", "label": "restricted-lemma-rig-etale-permanence", "categories": [ "restricted" ], "title": "restricted-lemma-rig-etale-permanence", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$", "be a morphism of locally Noetherian formal algebraic spaces over $S$.", "If $g \\circ f$ and $g$ are rig-\\'etale, then so is $f$." ], "refs": [], "proofs": [ { "contents": [ "By Formal Spaces, Remark", "\\ref{formal-spaces-remark-permanence-variant-Noetherian}", "this follows from Lemma \\ref{lemma-permanence-rig-etale-continuous}." ], "refs": [ "formal-spaces-remark-permanence-variant-Noetherian", "restricted-lemma-permanence-rig-etale-continuous" ], "ref_ids": [ 4015, 2376 ] } ], "ref_ids": [] }, { "id": 2383, "type": "theorem", "label": "restricted-lemma-rig-etale-alternative-permanence", "categories": [ "restricted" ], "title": "restricted-lemma-rig-etale-alternative-permanence", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$", "be morphisms of locally Noetherian formal algebraic spaces over $S$.", "If $g \\circ f$ is rig-\\'etale and $g$ is an adic monomorphism, then", "$f$ is rig-\\'etale." ], "refs": [], "proofs": [ { "contents": [ "Use Lemma \\ref{lemma-base-change-rig-etale} and that", "$f$ is the base change of $g \\circ f$ by $g$." ], "refs": [ "restricted-lemma-base-change-rig-etale" ], "ref_ids": [ 2380 ] } ], "ref_ids": [] }, { "id": 2384, "type": "theorem", "label": "restricted-lemma-closed-immersion-rig-smooth", "categories": [ "restricted" ], "title": "restricted-lemma-closed-immersion-rig-smooth", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces. Assume that $X$ and $Y$ are locally Noetherian and $f$ is a", "closed immersion. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is rig-smooth,", "\\item $f$ is rig-\\'etale,", "\\item for every affine formal algebraic space $V$ and every morphism", "$V \\to Y$ which is representable by algebraic spaces and \\'etale", "the morphism $X \\times_Y V \\to V$ corresponds to a surjective morphism", "$B \\to A$ in $\\textit{WAdm}^{Noeth}$ whose kernel $J$ has the following", "property: $I(J/J^2) = 0$ for some ideal of definition $I$ of $B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us observe that given $V$ and $V \\to Y$ as in (2) without any", "further assumption on $f$ we see that the morphism $X \\times_Y V \\to V$", "corresponds to a surjective morphism $B \\to A$ in $\\textit{WAdm}^{Noeth}$", "by Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-closed-immersion-into-countably-indexed}.", "\\medskip\\noindent", "We have (2) $\\Rightarrow$ (1) by", "Lemma \\ref{lemma-rig-etale-rig-smooth-morphism}.", "\\medskip\\noindent", "Proof of (3) $\\Rightarrow$ (2). Assume (3). By", "Lemma \\ref{lemma-rig-etale-morphisms}", "it suffices to show that the ring maps", "$B \\to A$ occuring in (3) are rig-\\'etale in the", "sense of Definition \\ref{definition-rig-etale-continuous-homomorphism}.", "Let $I$ be as in (3). The naive cotangent complex", "$\\NL_{A/B}^\\wedge$ of $A$ over $(B, I)$ is the complex of $A$-modules", "given by putting $J/J^2$ in degree $-1$. Hence $A$ is", "rig-\\'etale over $(B, I)$ by", "Definition \\ref{definition-rig-etale-homomorphism}.", "\\medskip\\noindent", "Assume (1) and let $V$ and $B \\to A$ be as in (3).", "By Definition \\ref{definition-rig-smooth} we see that", "$B \\to A$ is rig-smooth. Choose any ideal of definition $I \\subset B$.", "Then $A$ is rig-smooth over $(B, I)$.", "As above the complex $\\NL_{A/B}^\\wedge$ is ", "given by putting $J/J^2$ in degree $-1$.", "Hence by Lemma \\ref{lemma-equivalent-with-artin-smooth}", "we see that $J/J^2$ is annihilated by", "a power $I^n$ for some $n \\geq 1$. Since $B$ is adic, we see", "that $I^n$ is an ideal of definition of $B$ and the", "proof is complete." ], "refs": [ "formal-spaces-lemma-closed-immersion-into-countably-indexed", "restricted-lemma-rig-etale-rig-smooth-morphism", "restricted-lemma-rig-etale-morphisms", "restricted-definition-rig-etale-continuous-homomorphism", "restricted-definition-rig-etale-homomorphism", "restricted-definition-rig-smooth", "restricted-lemma-equivalent-with-artin-smooth" ], "ref_ids": [ 3950, 2379, 2377, 2444, 2435, 2443, 2302 ] } ], "ref_ids": [] }, { "id": 2385, "type": "theorem", "label": "restricted-lemma-composition-rig-surjective", "categories": [ "restricted" ], "title": "restricted-lemma-composition-rig-surjective", "contents": [ "\\begin{slogan}", "Rig-surjectivity of locally finite type morphisms is preserved under", "composition", "\\end{slogan}", "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of formal", "algebraic spaces over $S$. Assume $X$, $Y$, $Z$ are locally Noetherian and", "$f$ and $g$ locally of finite type. Then if $f$ and $g$ are rig-surjective,", "so is $g \\circ f$." ], "refs": [], "proofs": [ { "contents": [ "Follows in a straightforward manner from the definitions", "(and Formal Spaces, Lemma \\ref{formal-spaces-lemma-composition-finite-type})." ], "refs": [ "formal-spaces-lemma-composition-finite-type" ], "ref_ids": [ 3931 ] } ], "ref_ids": [] }, { "id": 2386, "type": "theorem", "label": "restricted-lemma-base-change-rig-surjective", "categories": [ "restricted" ], "title": "restricted-lemma-base-change-rig-surjective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $Z \\to Y$ be morphisms", "of formal algebraic spaces over $S$. Assume $X$, $Y$, $Z$ are locally", "Noetherian and $f$ and $g$ locally of finite type. If $f$ is", "rig-surjective, then the base change $Z \\times_Y X \\to Z$ is too." ], "refs": [], "proofs": [ { "contents": [ "Follows in a straightforward manner from the definitions (and", "Formal Spaces, Lemmas \\ref{formal-spaces-lemma-fibre-product-Noetherian} and", "\\ref{formal-spaces-lemma-base-change-finite-type})." ], "refs": [ "formal-spaces-lemma-fibre-product-Noetherian", "formal-spaces-lemma-base-change-finite-type" ], "ref_ids": [ 3936, 3932 ] } ], "ref_ids": [] }, { "id": 2387, "type": "theorem", "label": "restricted-lemma-rig-surjective-alternative-permanence", "categories": [ "restricted" ], "title": "restricted-lemma-rig-surjective-alternative-permanence", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$", "be morphisms locally of finite type of locally Noetherian", "formal algebraic spaces over $S$. If $g \\circ f$ is rig-surjective", "and $g$ is a monomorphism, then $f$ is rig-surjective." ], "refs": [], "proofs": [ { "contents": [ "Use Lemma \\ref{lemma-base-change-rig-surjective} and that", "$f$ is the base change of $g \\circ f$ by $g$." ], "refs": [ "restricted-lemma-base-change-rig-surjective" ], "ref_ids": [ 2386 ] } ], "ref_ids": [] }, { "id": 2388, "type": "theorem", "label": "restricted-lemma-permanence-rig-surjective", "categories": [ "restricted" ], "title": "restricted-lemma-permanence-rig-surjective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of", "formal algebraic spaces over $S$. Assume $X$, $Y$, $Z$ locally Noetherian", "and $f$ and $g$ locally of finite type. If $g \\circ f : X \\to Z$", "is rig-surjective, so is $g : Y \\to Z$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2389, "type": "theorem", "label": "restricted-lemma-etale-covering-rig-surjective", "categories": [ "restricted" ], "title": "restricted-lemma-etale-covering-rig-surjective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian", "formal algebraic spaces which is representable by algebraic spaces, \\'etale,", "and surjective. Then $f$ is rig-surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $p : \\text{Spf}(R) \\to Y$ be an adic morphism where $R$ is a complete", "discrete valuation ring. Let $Z = \\text{Spf}(R) \\times_Y X$. Then", "$Z \\to \\text{Spf}(R)$ is representable by algebraic spaces, \\'etale, and", "surjective. Hence $Z$ is nonempty. Pick a nonempty affine formal algebraic", "space $V$ and an \\'etale morphism $V \\to Z$ (possible by our definitions).", "Then $V \\to \\text{Spf}(R)$ corresponds to $R \\to A^\\wedge$ where", "$R \\to A$ is an \\'etale ring map, see Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-etale}. Since $A^\\wedge \\not = 0$", "(as $V \\not = \\emptyset$) we can find a maximal ideal $\\mathfrak m$", "of $A$ lying over $\\mathfrak m_R$. Then $A_\\mathfrak m$ is a discrete", "valuation ring (More on Algebra, Lemma", "\\ref{more-algebra-lemma-Dedekind-etale-extension}).", "Then $R' = A_\\mathfrak m^\\wedge$ is a complete discrete valuation ring", "(More on Algebra, Lemma \\ref{more-algebra-lemma-completion-dvr}).", "Applying Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-morphism-between-formal-spectra}.", "we find the desired morphism $\\text{Spf}(R') \\to V \\to Z \\to X$." ], "refs": [ "formal-spaces-lemma-etale", "more-algebra-lemma-Dedekind-etale-extension", "more-algebra-lemma-completion-dvr", "formal-spaces-lemma-morphism-between-formal-spectra" ], "ref_ids": [ 3913, 10054, 10046, 3871 ] } ], "ref_ids": [] }, { "id": 2390, "type": "theorem", "label": "restricted-lemma-upshot", "categories": [ "restricted" ], "title": "restricted-lemma-upshot", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally", "Noetherian formal algebraic spaces which is locally of finite type.", "Let $\\{g_i : Y_i \\to Y\\}$ be a family of morphisms of formal", "algebraic spaces which are representable by algebraic spaces and", "\\'etale such that $\\coprod g_i$ is surjective.", "Then $f$ is rig-surjective if and only if each", "$f_i : X \\times_Y Y_i \\to Y_i$ is rig-surjective." ], "refs": [], "proofs": [ { "contents": [ "Namely, if $f$ is rig-surjective, so is any base change", "(Lemma \\ref{lemma-base-change-rig-surjective}).", "Conversely, if all $f_i$ are rig-surjective, so is", "$\\coprod f_i : \\coprod X \\times_Y Y_i \\to \\coprod Y_i$.", "By Lemma \\ref{lemma-etale-covering-rig-surjective}", "the morphism $\\coprod g_i : \\coprod Y_i \\to Y$ is rig-surjective.", "Hence $\\coprod X \\times_Y Y_i \\to Y$ is rig-surjective", "(Lemma \\ref{lemma-composition-rig-surjective}).", "Since this morphism factors through $X \\to Y$ we see that $X \\to Y$", "is rig-surjective by Lemma \\ref{lemma-permanence-rig-surjective}." ], "refs": [ "restricted-lemma-base-change-rig-surjective", "restricted-lemma-etale-covering-rig-surjective", "restricted-lemma-composition-rig-surjective", "restricted-lemma-permanence-rig-surjective" ], "ref_ids": [ 2386, 2389, 2385, 2388 ] } ], "ref_ids": [] }, { "id": 2391, "type": "theorem", "label": "restricted-lemma-faithfully-flat-rig-surjective", "categories": [ "restricted" ], "title": "restricted-lemma-faithfully-flat-rig-surjective", "contents": [ "Let $A$ be a Noetherian ring complete with respect to an ideal $I$.", "Let $B$ be an $I$-adically complete $A$-algebra.", "If $A/I^n \\to B/I^nB$ is of finite type and flat for all $n$ and", "faithfully flat for $n = 1$, then $\\text{Spf}(B) \\to \\text{Spf}(A)$", "is rig-surjective." ], "refs": [], "proofs": [ { "contents": [ "We will use without further mention that morphisms between formal spectra", "are given by continuous maps between the corresponding topological rings, see", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-morphism-between-formal-spectra}.", "Let $\\varphi : A \\to R$ be a continuous map into a complete discrete", "valuation ring $A$. This implies that $\\varphi(I) \\subset \\mathfrak m_R$.", "On the other hand, since we only need to produce the lift", "$\\varphi' : B' \\to R'$ in the case that $\\varphi$ corresponds to an adic", "morphism, we may assume that $\\varphi(I) \\not = 0$. Thus we may consider", "the base change $C = B \\widehat{\\otimes}_A R$, see", "Remark \\ref{remark-base-change} for example.", "Then $C$ is an $\\mathfrak m_R$-adically complete $R$-algebra", "such that $C/\\mathfrak m_R^n C$ is of finite type and flat over", "$R/\\mathfrak m_R^n$ and such that $C/\\mathfrak m_R C$ is nonzero.", "Pick any maximal ideal $\\mathfrak m \\subset C$ lying over", "$\\mathfrak m_R$. By flatness (which implies going down) we see that", "$\\Spec(C_\\mathfrak m) \\setminus V(\\mathfrak m_R C_\\mathfrak m)$", "is a nonempty open. Hence", "We can pick a prime $\\mathfrak q \\subset \\mathfrak m$", "such that $\\mathfrak q$ defines a closed point of", "$\\Spec(C_\\mathfrak m) \\setminus \\{\\mathfrak m\\}$ and such that", "$\\mathfrak q \\not \\in V(IC_\\mathfrak m)$, see", "Properties, Lemma \\ref{properties-lemma-complement-closed-point-Jacobson}.", "Then $C/\\mathfrak q$ is a dimension $1$-local domain and we can find", "$C/\\mathfrak q \\subset R'$ with $R'$ a discrete valuation ring", "(Algebra, Lemma \\ref{algebra-lemma-exists-dvr}).", "By construction $\\mathfrak m_R R' \\subset \\mathfrak m_{R'}$", "and we see that $C \\to R'$ extends to a continuous map", "$C \\to (R')^\\wedge$ (in fact we can pick $R'$ such that", "$R' = (R')^\\wedge$ in our current situation but we do not need this).", "Since the completion of a discrete valuation ring is a discrete", "valuation ring, we see that the assumption gives a commutative", "diagram of rings", "$$", "\\xymatrix{", "(R')^\\wedge & C \\ar[l] & B \\ar[l] \\\\", "R \\ar[u] & R \\ar[l] \\ar[u] & A \\ar[l] \\ar[u]", "}", "$$", "which gives the desired lift." ], "refs": [ "formal-spaces-lemma-morphism-between-formal-spectra", "restricted-remark-base-change", "properties-lemma-complement-closed-point-Jacobson", "algebra-lemma-exists-dvr" ], "ref_ids": [ 3871, 2448, 2965, 1028 ] } ], "ref_ids": [] }, { "id": 2392, "type": "theorem", "label": "restricted-lemma-flat-rig-surjective", "categories": [ "restricted" ], "title": "restricted-lemma-flat-rig-surjective", "contents": [ "Let $A$ be a Noetherian ring complete with respect to an ideal $I$.", "Let $B$ be an $I$-adically complete $A$-algebra. Assume that", "\\begin{enumerate}", "\\item the $I$-torsion in $A$ is $0$,", "\\item $A/I^n \\to B/I^nB$ is flat and of finite type for all $n$.", "\\end{enumerate}", "Then $\\text{Spf}(B) \\to \\text{Spf}(A)$ is rig-surjective if and only", "if $A/I \\to B/IB$ is faithfully flat." ], "refs": [], "proofs": [ { "contents": [ "Faithful flatness implies rig-surjectivity by", "Lemma \\ref{lemma-faithfully-flat-rig-surjective}.", "To prove the converse we will use without further mention that the", "vanishing of $I$-torsion is equivalent to the vanishing of $I$-power torsion", "(More on Algebra, Lemma \\ref{more-algebra-lemma-torsion-free}).", "We will also use without further mention that morphisms between", "formal spectra are given by continuous maps between the corresponding", "topological rings, see", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-morphism-between-formal-spectra}.", "\\medskip\\noindent", "Assume $\\text{Spf}(B) \\to \\text{Spf}(A)$ is rig-surjective.", "Choose a maximal ideal $I \\subset \\mathfrak m \\subset A$.", "The open $U = \\Spec(A_\\mathfrak m) \\setminus V(I_\\mathfrak m)$", "of $\\Spec(A_\\mathfrak m)$ is nonempty as the $I_\\mathfrak m$-torsion of", "$A_\\mathfrak m$ is zero", "(use Algebra, Lemma \\ref{algebra-lemma-Noetherian-power-ideal-kills-module}).", "Thus we can find a prime $\\mathfrak q \\subset A_\\mathfrak m$ which defines", "a point of $U$ (i.e., $IA_\\mathfrak m \\not \\subset \\mathfrak q$)", "and which corresponds to a closed point", "of $\\Spec(A_\\mathfrak m) \\setminus \\{\\mathfrak m\\}$, see", "Properties, Lemma \\ref{properties-lemma-complement-closed-point-Jacobson}.", "Then $A_\\mathfrak m/\\mathfrak q$ is a dimension $1$ local domain.", "Thus we can find an injective local homomorphism of local rings", "$A_\\mathfrak m/\\mathfrak q \\subset R$ where $R$ is a discrete valuation ring", "(Algebra, Lemma \\ref{algebra-lemma-exists-dvr}).", "By construction $IR \\subset \\mathfrak m_R$ and we see that", "$A \\to R$ extends to a continuous map $A \\to R^\\wedge$.", "Since the completion of a discrete valuation ring is a discrete", "valuation ring, we see that the assumption gives a commutative", "diagram of rings", "$$", "\\xymatrix{", "R' & B \\ar[l] \\\\", "R^\\wedge \\ar[u] & A \\ar[l] \\ar[u]", "}", "$$", "Thus we find a prime ideal of $B$ lying over $\\mathfrak m$. It follows", "that $\\Spec(B/IB) \\to \\Spec(A/I)$ is surjective, whence $A/I \\to B/IB$", "is faithfully flat", "(Algebra, Lemma \\ref{algebra-lemma-ff-rings})." ], "refs": [ "restricted-lemma-faithfully-flat-rig-surjective", "more-algebra-lemma-torsion-free", "formal-spaces-lemma-morphism-between-formal-spectra", "algebra-lemma-Noetherian-power-ideal-kills-module", "properties-lemma-complement-closed-point-Jacobson", "algebra-lemma-exists-dvr", "algebra-lemma-ff-rings" ], "ref_ids": [ 2391, 10334, 3871, 694, 2965, 1028, 536 ] } ], "ref_ids": [] }, { "id": 2393, "type": "theorem", "label": "restricted-lemma-monomorphism-rig-surjective", "categories": [ "restricted" ], "title": "restricted-lemma-monomorphism-rig-surjective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces. Assume $X$ and $Y$ are locally Noetherian, $f$ locally of finite", "type, and $f$ a monomorphism. Then $f$ is rig surjective if and only if", "every adic morphism $\\text{Spf}(R) \\to Y$ where $R$ is a complete discrete", "valuation ring factors through $X$." ], "refs": [], "proofs": [ { "contents": [ "One direction is trivial. For the other, suppose that $\\text{Spf}(R) \\to Y$", "is an adic morphism such that there exists an extension of complete", "discrete valuation rings $R \\subset R'$ with", "$\\text{Spf}(R') \\to \\text{Spf}(R) \\to X$ factoring through $Y$. Then", "$\\Spec(R'/\\mathfrak m_R^n R') \\to \\Spec(R/\\mathfrak m_R^n)$ is surjective", "and flat, hence the morphisms $\\Spec(R/\\mathfrak m_R^n) \\to X$ factor", "through $X$ as $X$ satisfies the sheaf condition for fpqc coverings, see", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-sheaf-fpqc}.", "In other words, $\\text{Spf}(R) \\to Y$ factors through $X$." ], "refs": [ "formal-spaces-lemma-sheaf-fpqc" ], "ref_ids": [ 3961 ] } ], "ref_ids": [] }, { "id": 2394, "type": "theorem", "label": "restricted-lemma-closed-immersion-rig-surjective", "categories": [ "restricted" ], "title": "restricted-lemma-closed-immersion-rig-surjective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces. Assume that $X$ and $Y$ are locally Noetherian and $f$ is a", "closed immersion. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is rig-surjective, and", "\\item for every affine formal algebraic space $V$ and every morphism", "$V \\to Y$ which is representable by algebraic spaces and \\'etale", "the morphism $X \\times_Y V \\to V$ corresponds to a surjective morphism", "$B \\to A$ in $\\textit{WAdm}^{Noeth}$ whose kernel $J$ has the following", "property: $IJ^n = 0$ for some ideal of definition $I$ of $B$", "and some $n \\geq 1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us observe that given $V$ and $V \\to Y$ as in (2) without any", "further assumption on $f$ we see that the morphism $X \\times_Y V \\to V$", "corresponds to a surjective morphism $B \\to A$ in $\\textit{WAdm}^{Noeth}$", "by Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-closed-immersion-into-countably-indexed}.", "\\medskip\\noindent", "Assume (1). By Lemma \\ref{lemma-base-change-rig-surjective} we see that", "$\\text{Spf}(A) \\to \\text{Spf}(B)$ is rig-surjective.", "Let $I \\subset B$ be an ideal of definition. Since $B$ is adic,", "$I^m \\subset B$ is an ideal of definition for all $m \\geq 1$.", "If $I^m J^n \\not = 0$ for all $n, m \\geq 1$, then", "$IJ$ is not nilpotent, hence $V(IJ) \\not = \\Spec(B)$.", "Thus we can find a prime ideal $\\mathfrak p \\subset B$", "with $\\mathfrak p \\not \\in V(I) \\cup V(J)$.", "Observe that $I(B/\\mathfrak p) \\not = B/\\mathfrak p$", "hence we can find a maximal ideal", "$\\mathfrak p + I \\subset \\mathfrak m \\subset B$.", "By Algebra, Lemma \\ref{algebra-lemma-exists-dvr}", "we can find a discrete valuation ring $R$", "and an injective local ring homomorphism $(B/\\mathfrak p)_\\mathfrak m \\to R$.", "Clearly, the ring map $B \\to R$ cannot factor through $A = B/J$.", "According to Lemma \\ref{lemma-monomorphism-rig-surjective}", "this contradicts the fact that $\\text{Spf}(A) \\to \\text{Spf}(B)$", "is rig-surjective. Hence for some $n, m$ we do have", "$I^n J^m = 0$ which shows that (2) holds.", "\\medskip\\noindent", "Assume (2). By Lemma \\ref{lemma-upshot} it suffices to show", "that $\\text{Spf}(A) \\to \\text{Spf}(B)$ is rig-surjective.", "Pick an ideal of definition $I \\subset B$ and an integer $n$", "such that $I J^n = 0$.", "Consider a ring map $B \\to R$ where $R$ is a discrete valuation", "ring and the image of $I$ is nonzero. Since $R$ is a domain, we", "conclude the image of $J$ in $R$ is zero. Hence $B \\to R$", "factors through the surjection $B \\to A$ and we are done by", "definition of rig-surjective morphisms." ], "refs": [ "formal-spaces-lemma-closed-immersion-into-countably-indexed", "restricted-lemma-base-change-rig-surjective", "algebra-lemma-exists-dvr", "restricted-lemma-monomorphism-rig-surjective", "restricted-lemma-upshot" ], "ref_ids": [ 3950, 2386, 1028, 2393, 2390 ] } ], "ref_ids": [] }, { "id": 2395, "type": "theorem", "label": "restricted-lemma-closed-immersion-rig-smooth-rig-surjective", "categories": [ "restricted" ], "title": "restricted-lemma-closed-immersion-rig-smooth-rig-surjective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces. Assume that $X$ and $Y$ are locally Noetherian and $f$ is a", "closed immersion. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is rig-smooth and rig-surjective,", "\\item $f$ is rig-\\'etale and rig-surjective, and", "\\item for every affine formal algebraic space $V$ and every morphism", "$V \\to Y$ which is representable by algebraic spaces and \\'etale", "the morphism $X \\times_Y V \\to V$ corresponds to a surjective morphism", "$B \\to A$ in $\\textit{WAdm}^{Noeth}$ whose kernel $J$ has the following", "property: $IJ = 0$ for some ideal of definition $I$ of $B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $I$ and $J$ be ideals of a ring $B$ such that $IJ^n = 0$ and", "$I(J/J^2) = 0$. Then $I^nJ = 0$ (proof omitted).", "Hence this lemma follows from a trivial combination of", "Lemmas \\ref{lemma-closed-immersion-rig-smooth} and", "\\ref{lemma-closed-immersion-rig-surjective}." ], "refs": [ "restricted-lemma-closed-immersion-rig-smooth", "restricted-lemma-closed-immersion-rig-surjective" ], "ref_ids": [ 2384, 2394 ] } ], "ref_ids": [] }, { "id": 2396, "type": "theorem", "label": "restricted-lemma-rig-etale-descent", "categories": [ "restricted" ], "title": "restricted-lemma-rig-etale-descent", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$", "be morphisms of locally Noetherian formal algebraic spaces over $S$.", "Assume", "\\begin{enumerate}", "\\item $g$ is locally of finite type,", "\\item $f$ is rig-smooth (resp.\\ rig-\\'etale) and rig-surjective,", "\\item $g \\circ f$ is rig-smooth (resp.\\ rig-\\'etale)", "\\end{enumerate}", "then $g$ is rig-smooth (resp.\\ rig-\\'etale)." ], "refs": [], "proofs": [ { "contents": [ "We will prove this in the rig-smooth case and indicate the necessary", "changes to prove the rig-\\'etale case at the end of the proof.", "Consider a commutative diagram", "$$", "\\xymatrix{", "X \\times_Y V \\ar[r] \\ar[d] &", "V \\ar[d] \\ar[r] &", "W \\ar[d] \\\\", "X \\ar[r] &", "Y \\ar[r] &", "Z", "}", "$$", "with $V$ and $W$ affine formal algebraic spaces, $V \\to Y$ and $W \\to Z$", "representable by algebraic spaces and \\'etale. We have to show that", "$V \\to W$ corresponds to a rig-smooth map of adic Noetherian topological", "rings, see Definition \\ref{definition-rig-smooth}.", "We may write $V = \\text{Spf}(B)$ and $W = \\text{Spf}(C)$ and that", "$V \\to W$ corresponds to an adic ring map $C \\to B$ which is topologically", "of finite type, see Lemma \\ref{lemma-finite-type-morphisms}.", "\\medskip\\noindent", "We will use below without further mention that $X \\times_Y V \\to V$", "is rig-smooth and rig-surjective, see", "Lemmas \\ref{lemma-base-change-rig-smooth} and", "\\ref{lemma-base-change-rig-surjective}.", "Also, the composition $X \\times_Y V \\to V \\to W$ is rig-smooth", "since $g \\circ f$ is rig-smooth.", "\\medskip\\noindent", "Let $I \\subset C$ be an ideal of definition. The module", "Assume $C \\to B$ is not rig-smooth to get a contradiction.", "This means that there exists a prime ideal $\\mathfrak q \\subset B$", "not containing $IB$ such that either $H^{-1}(\\NL_{B/C}^\\wedge)_\\mathfrak p$", "is nonzero or $H^0(\\NL_{B/C}^\\wedge)_\\mathfrak p$ is not a finite free", "$B_\\mathfrak q$-module. See", "Lemma \\ref{lemma-equivalent-with-artin-smooth}; some details", "omitted. We may choose a maximal ideal", "$IB + \\mathfrak q \\subset \\mathfrak m$. By", "Algebra, Lemma \\ref{algebra-lemma-exists-dvr}", "we can find a complete discrete valuation ring $R$ and an injective", "local ring homomorphism $(B/\\mathfrak q)_\\mathfrak m \\to R$.", "\\medskip\\noindent", "After replacing $R$ by an extension, we may assume given a lift", "$\\text{Spf}(R) \\to X \\times_Y V$ of the adic morphism", "$\\text{Spf}(R) \\to V = \\text{Spf}(B)$. Choose an \\'etale", "covering $\\{\\text{Spf}(A_i) \\to X \\times_Y V\\}$ as in", "Formal Spaces, Definition", "\\ref{formal-spaces-definition-formal-algebraic-space}.", "By Lemma \\ref{lemma-etale-covering-rig-surjective}", "we may assume $\\text{Spf}(R) \\to X \\times_Y V$ lifts to a", "morphism $\\text{Spf}(R) \\to \\text{Spf}(A_i)$ for some $i$", "(this might require replacing $R$ by another extension).", "Set $A = A_i$. Consider the ring maps", "$$", "C \\to B \\to A \\to R", "$$", "Let $\\mathfrak p \\subset A$ be the kernel of the map", "$A \\to R$ and note that $\\mathfrak p$ lies over $\\mathfrak q$.", "We know that $C \\to A$ and $B \\to A$ are rig-smooth.", "In particular the ring map $B_\\mathfrak q \\to A_\\mathfrak p$", "is flat by Lemma \\ref{lemma-rig-smooth-rig-flat}.", "Consider the associated exact sequence", "$$", "\\xymatrix{", "&", "H^0(\\NL_{B/C}^\\wedge) \\otimes_B A_\\mathfrak p \\ar[r] &", "H^0(\\NL_{A/C}^\\wedge)_\\mathfrak p \\ar[r] &", "H^0(\\NL_{A/B}^\\wedge)_\\mathfrak p \\ar[r] & 0 \\\\", "0 \\ar[r] &", "H^{-1}(\\NL_{B/C}^\\wedge \\otimes_B A)_\\mathfrak p \\ar[r] &", "H^{-1}(\\NL_{A/C}^\\wedge)_\\mathfrak p \\ar[r] &", "H^{-1}(\\NL_{A/B}^\\wedge)_\\mathfrak p \\ar[llu]", "}", "$$", "of Lemmas \\ref{lemma-exact-sequence-NL} and", "\\ref{lemma-exact-sequence-NL-rig-smooth}.", "Given the rig-smoothness of $C \\to A$ and $B \\to A$", "we conclude that $H^{-1}(\\NL_{B/C}^\\wedge \\otimes_B A)_\\mathfrak p = 0$", "and that $H^0(\\NL_{B/C}^\\wedge) \\otimes_B A_\\mathfrak p$", "is finite free as a kernel of a surjection of finite free", "$A_\\mathfrak p$-modules. Since $B_\\mathfrak q \\to A_\\mathfrak p$", "is flat and hence faithfully flat, this implies that", "$H^{-1}(\\NL_{B/C}^\\wedge)_\\mathfrak q = 0$", "and that $H^0(\\NL_{B/C}^\\wedge)_\\mathfrak q$", "is finite free which is the contradiction we were looking for.", "\\medskip\\noindent", "In the rig-\\'etale case one argues in exactly the same manner", "but the conclusion obtained is that both", "$H^{-1}(\\NL_{B/C}^\\wedge)_\\mathfrak q$", "and $H^0(\\NL_{B/C}^\\wedge)_\\mathfrak q$ are zero." ], "refs": [ "restricted-definition-rig-smooth", "restricted-lemma-finite-type-morphisms", "restricted-lemma-base-change-rig-smooth", "restricted-lemma-base-change-rig-surjective", "restricted-lemma-equivalent-with-artin-smooth", "algebra-lemma-exists-dvr", "formal-spaces-definition-formal-algebraic-space", "restricted-lemma-etale-covering-rig-surjective", "restricted-lemma-rig-smooth-rig-flat", "restricted-lemma-exact-sequence-NL", "restricted-lemma-exact-sequence-NL-rig-smooth" ], "ref_ids": [ 2443, 2327, 2369, 2386, 2302, 1028, 3981, 2389, 2366, 2300, 2367 ] } ], "ref_ids": [] }, { "id": 2397, "type": "theorem", "label": "restricted-lemma-flat-locus", "categories": [ "restricted" ], "title": "restricted-lemma-flat-locus", "contents": [ "Let $X$ be a locally Noetherian formal algebraic space over", "a complete discrete valuation ring $A$.", "Then there exists a closed immersion $X' \\to X$", "of formal algebraic spaces such that $X'$ is flat over $A$", "and such that any morphism $Y \\to X$ of locally Noetherian formal algebraic", "spaces with $Y$ flat over $A$ factors through $X'$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\pi \\in A$ be the uniformizer. Recall that an $A$-module", "is flat if and only if the $\\pi$-power torsion is $0$.", "\\medskip\\noindent", "First assume that $X$ is an affine formal algebraic space.", "Then $X = \\text{Spf}(B)$ with $B$ an adic Noetherian $A$-algebra.", "In this case we set $X' = \\text{Spf}(B')$ where", "$B' = B/\\pi\\text{-power torsion}$. It is clear that $X'$ is flat", "over $A$ and that $X' \\to X$ is a closed immersion.", "Let $g : Y \\to X$ be a morphism of locally Noetherian formal algebraic spaces", "with $Y$ flat over $A$. Choose a covering $\\{Y_j \\to Y\\}$ as in", "Formal Spaces, Definition", "\\ref{formal-spaces-definition-formal-algebraic-space}.", "Then $Y_j = \\text{Spf}(C_j)$ with $C_j$ flat over $A$.", "Hence the morphism $Y_j \\to X$, which correspond to a continuous", "$R$-algebra map $B \\to C_j$, factors through $X'$ as clearly", "$B \\to C_j$ kills the $\\pi$-power torsion.", "Since $\\{Y_j \\to Y\\}$ is a covering and since $X' \\to X$", "is a monomorphism, we conclude that $g$ factors through $X'$.", "\\medskip\\noindent", "Let $X$ and $\\{X_i \\to X\\}_{i \\in I}$ be as in", "Formal Spaces, Definition", "\\ref{formal-spaces-definition-formal-algebraic-space}.", "For each $i$ let $X'_i \\to X_i$ be the flat part as", "constructed above. For $i, j \\in I$ the projection", "$X'_i \\times_X X_j \\to X'_i$ is an \\'etale (by assumption)", "morphism of schemes (by Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-presentation-representable}).", "Hence $X'_i \\times_X X_j$ is flat over $A$ as morphisms", "representable by algebraic spaces and \\'etale", "are flat (Lemma \\ref{lemma-representable-flat}).", "Thus the projection $X'_i \\times_X X_j \\to X_j$ factors", "through $X'_j$ by the universal property. We conclude that", "$$", "R_{ij} = X'_i \\times_X X_j = X'_i \\times_X X'_j = X_i \\times_X X'_j", "$$", "because the morphisms $X'_i \\to X_i$ are injections of sheaves.", "Set $U = \\coprod X'_i$, set", "$R = \\coprod R_{ij}$, and denote $s, t : R \\to U$ the two", "projections. As a sheaf $R = U \\times_X U$ and $s$ and $t$", "are \\'etale. Then $(t, s) : R \\to U$ defines an \\'etale equivalence", "relation by our observations above. Thus $X' = U/R$ is an", "algebraic space by Spaces, Theorem \\ref{spaces-theorem-presentation}.", "By construction the diagram", "$$", "\\xymatrix{", "\\coprod X'_i \\ar[r] \\ar[d] & \\coprod X_i \\ar[d] \\\\", "X' \\ar[r] & X", "}", "$$", "is cartesian. Since the right vertical arrow is \\'etale surjective", "and the top horizontal arrow is representable and a closed immersion", "we conclude that $X' \\to X$ is representable by", "Bootstrap, Lemma \\ref{bootstrap-lemma-after-fppf-sep-lqf}.", "Then we can use Spaces, Lemma", "\\ref{spaces-lemma-descent-representable-transformations-property}", "to conclude that $X' \\to X$ is a closed immersion.", "\\medskip\\noindent", "Finally, suppose that $Y \\to X$ is a morphism with", "$Y$ a locally Noetherian formal algebraic space flat over $A$.", "Then each $X_i \\times_X Y$ is \\'etale over $Y$ and", "therefore flat over $A$ (see above).", "Then $X_i \\times_X Y \\to X_i$ factors through $X'_i$.", "Hence $Y \\to X$ factors through $X'$ because", "$\\{X_i \\times_X Y \\to Y\\}$ is an \\'etale covering." ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-lemma-presentation-representable", "restricted-lemma-representable-flat", "spaces-theorem-presentation", "bootstrap-lemma-after-fppf-sep-lqf", "spaces-lemma-descent-representable-transformations-property" ], "ref_ids": [ 3981, 3981, 3872, 2341, 8124, 2619, 8134 ] } ], "ref_ids": [] }, { "id": 2398, "type": "theorem", "label": "restricted-lemma-flat-and-diagonal-rig-surjective", "categories": [ "restricted" ], "title": "restricted-lemma-flat-and-diagonal-rig-surjective", "contents": [ "Let $X$ be a locally Noetherian formal algebraic space which is", "locally of finite type over a complete discrete valuation ring $A$.", "Let $X' \\subset X$ be as in Lemma \\ref{lemma-flat-locus}.", "If $X \\to X \\times_{\\text{Spf}(A)} X$ is rig-\\'etale and rig-surjective,", "then $X' = \\text{Spf}(A)$ or $X' = \\emptyset$." ], "refs": [ "restricted-lemma-flat-locus" ], "proofs": [ { "contents": [ "(Aside: the diagonal is always locally of finite type by", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-diagonal-morphism-formal-algebraic-spaces}", "and $X \\times_{\\text{Spf}(A)} X$ is locally Noetherian by", "Formal Spaces, Lemmas", "\\ref{formal-spaces-lemma-base-change-finite-type} and", "\\ref{formal-spaces-lemma-locally-finite-type-locally-noetherian}.", "Thus imposing the conditions on the diagonal morphism makes sense.)", "The diagram", "$$", "\\xymatrix{", "X' \\ar[r] \\ar[d] & X' \\times_{\\text{Spf}(A)} X' \\ar[d] \\\\", "X \\ar[r] & X \\times_{\\text{Spf}(A)} X", "}", "$$", "is cartesian. Hence $X' \\to X' \\times_{\\text{Spf}(A)} X'$", "is rig-\\'etale and rig-surjective by", "Lemma \\ref{lemma-base-change-rig-surjective}.", "Choose an affine formal algebraic space $U$ and a morphism", "$U \\to X'$ which is representable by algebraic spaces and \\'etale.", "Then $U = \\text{Spf}(B)$ where $B$ is an adic Noetherian topological ring", "which is a flat $A$-algebra, whose topology is the $\\pi$-adic", "topology where $\\pi \\in A$ is a uniformizer, and such that", "$A/\\pi^n A \\to B/\\pi^n B$ is of finite type for each $n$.", "For later use, we remark that this in particular implies: if $B \\not = 0$,", "then the map $\\text{Spf}(B) \\to \\text{Spf}(A)$ is a surjection", "of sheaves (please recall that we are using the fppf topology", "as always). Repeating the argument above, we see that", "$$", "W = U \\times_{X'} U =", "X' \\times_{X' \\times_{\\text{Spf}(A)} X'} (U \\times_{\\text{Spf}(A)} U)", "\\longrightarrow", "U \\times_{\\text{Spf}(A)} U", "$$", "is a closed immersion and rig-\\'etale and rig-surjective. We have", "$U \\times_{\\text{Spf}(A)} U = \\text{Spf}(B \\widehat{\\otimes}_A B)$", "by Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-fibre-product-affines-over-separated}.", "Then $B \\widehat{\\otimes}_A B$ is a flat $A$-algebra", "as the $\\pi$-adic completion of the flat $A$-algebra $B \\otimes_A B$.", "Hence $W = U \\times_{\\text{Spf}(A)} U$ by", "Lemma \\ref{lemma-closed-immersion-rig-smooth-rig-surjective}.", "In other words, we have $U \\times_{X'} U = U \\times_{\\text{Spf}(A)} U$", "which in turn means that the image of $U \\to X'$ (as a map of sheaves)", "maps injectively to $\\text{Spf}(A)$.", "Choose a covering $\\{U_i \\to X'\\}$ as in Formal Spaces, Definition", "\\ref{formal-spaces-definition-formal-algebraic-space}.", "In particular $\\coprod U_i \\to X'$ is a surjection of sheaves.", "By applying the above to $U_i \\coprod U_j \\to X'$ (using the", "fact that $U_i \\amalg U_j$ is an affine formal algebraic space", "as well) we see that $X' \\to \\text{Spf}(A)$ is an injective map", "of fppf sheaves. Since $X'$ is flat over $A$, either $X'$ is empty", "(if $U_i$ is empty for all $i$) or the map is an isomorphism", "(if $U_i$ is nonempty for some $i$ when we have seen that", "$U_i \\to \\text{Spf}(A)$ is a surjective map of sheaves)", "and the proof is complete." ], "refs": [ "formal-spaces-lemma-diagonal-morphism-formal-algebraic-spaces", "formal-spaces-lemma-base-change-finite-type", "formal-spaces-lemma-locally-finite-type-locally-noetherian", "restricted-lemma-base-change-rig-surjective", "formal-spaces-lemma-fibre-product-affines-over-separated", "restricted-lemma-closed-immersion-rig-smooth-rig-surjective", "formal-spaces-definition-formal-algebraic-space" ], "ref_ids": [ 3893, 3932, 3935, 2386, 3896, 2395, 3981 ] } ], "ref_ids": [ 2397 ] }, { "id": 2399, "type": "theorem", "label": "restricted-lemma-rig-monomorphism-rig-surjective", "categories": [ "restricted" ], "title": "restricted-lemma-rig-monomorphism-rig-surjective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces. Assume", "\\begin{enumerate}", "\\item $X$ and $Y$ are locally Noetherian,", "\\item $f$ locally of finite type,", "\\item $\\Delta_f : X \\to X \\times_Y X$ is rig-\\'etale and rig-surjective.", "\\end{enumerate}", "Then $f$ is rig surjective if and only if every adic morphism", "$\\text{Spf}(R) \\to Y$ where $R$ is a complete discrete", "valuation ring lifts to a morphism $\\text{Spf}(R) \\to X$." ], "refs": [], "proofs": [ { "contents": [ "One direction is trivial. For the other, suppose that $\\text{Spf}(R) \\to Y$", "is an adic morphism such that there exists an extension of complete", "discrete valuation rings $R \\subset R'$ with", "$\\text{Spf}(R') \\to \\text{Spf}(R) \\to X$ factoring through $Y$.", "Consider the fibre product diagram", "$$", "\\xymatrix{", "\\text{Spf}(R') \\ar[r] \\ar[rd] &", "\\text{Spf}(R) \\times_Y X \\ar[r] \\ar[d]^p &", "X \\ar[d]^f \\\\", "&", "\\text{Spf}(R) \\ar[r] &", "Y", "}", "$$", "The morphism $p$ is locally of finite type as a base change of $f$, see", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-base-change-finite-type}.", "The diagonal morphism $\\Delta_p$ is the base change of", "$\\Delta_f$ and hence is rig-\\'etale and rig-surjective.", "By Lemma \\ref{lemma-flat-and-diagonal-rig-surjective}", "the flat locus of $\\text{Spf}(R) \\times_Y X$ over $R$", "is either $\\emptyset$ or equal to $\\text{Spf}(R)$.", "However, since $\\text{Spf}(R')$ factors through it", "we conclude it is not empty and hence", "we get a morphism $\\text{Spf}(R) \\to \\text{Spf}(R) \\times_Y X \\to X$", "as desired." ], "refs": [ "formal-spaces-lemma-base-change-finite-type", "restricted-lemma-flat-and-diagonal-rig-surjective" ], "ref_ids": [ 3932, 2398 ] } ], "ref_ids": [] }, { "id": 2400, "type": "theorem", "label": "restricted-lemma-map-completions-finite-type", "categories": [ "restricted" ], "title": "restricted-lemma-map-completions-finite-type", "contents": [ "In the situation above. If $f$ is locally of finite type, then", "$f_{/T}$ is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "(Finite type morphisms of formal algebraic spaces are discussed in", "Formal Spaces, Section \\ref{formal-spaces-section-finite-type}.)", "Namely, suppose that $Z \\to X$ is a morphism from a scheme into $X$ such", "that $|Z|$ maps into $T$. From the cartesian square above we see that", "$Z \\times_X X'$ is an algebraic space representing", "$Z \\times_{X_{/T}} X'_{/T'}$. Since $Z \\times_X X' \\to Z$", "is locally of finite type by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-finite-type} we conclude." ], "refs": [ "spaces-morphisms-lemma-base-change-finite-type" ], "ref_ids": [ 4815 ] } ], "ref_ids": [] }, { "id": 2401, "type": "theorem", "label": "restricted-lemma-map-completions-etale", "categories": [ "restricted" ], "title": "restricted-lemma-map-completions-etale", "contents": [ "In the situation above. If $f$ is \\'etale, then $f_{/T}$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "By the same argument as in the proof of", "Lemma \\ref{lemma-map-completions-finite-type} this follows from", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-etale}." ], "refs": [ "restricted-lemma-map-completions-finite-type", "spaces-morphisms-lemma-base-change-etale" ], "ref_ids": [ 2400, 4907 ] } ], "ref_ids": [] }, { "id": 2402, "type": "theorem", "label": "restricted-lemma-closed-immersion-gives-closed-immersion", "categories": [ "restricted" ], "title": "restricted-lemma-closed-immersion-gives-closed-immersion", "contents": [ "In the situation above. If $f$ is a closed immersion, then", "$f_{/T}$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "(Closed immersions of formal algebraic spaces are discussed in", "Formal Spaces, Section", "\\ref{formal-spaces-section-closed-immersions}.)", "By the same argument as in the proof of", "Lemma \\ref{lemma-map-completions-finite-type} this follows from", "Spaces, Lemma", "\\ref{spaces-lemma-base-change-immersions}." ], "refs": [ "restricted-lemma-map-completions-finite-type", "spaces-lemma-base-change-immersions" ], "ref_ids": [ 2400, 8161 ] } ], "ref_ids": [] }, { "id": 2403, "type": "theorem", "label": "restricted-lemma-proper-gives-proper", "categories": [ "restricted" ], "title": "restricted-lemma-proper-gives-proper", "contents": [ "In the situation above. If $f$ is proper, then $f_{/T}$ is proper." ], "refs": [], "proofs": [ { "contents": [ "(Proper morphisms of formal algebraic spaces are discussed in", "Formal Spaces, Section \\ref{formal-spaces-section-proper}.)", "By the same argument as in the proof of", "Lemma \\ref{lemma-map-completions-finite-type} this follows from", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-proper}." ], "refs": [ "restricted-lemma-map-completions-finite-type", "spaces-morphisms-lemma-base-change-proper" ], "ref_ids": [ 2400, 4917 ] } ], "ref_ids": [] }, { "id": 2404, "type": "theorem", "label": "restricted-lemma-quasi-compact-gives-quasi-compact", "categories": [ "restricted" ], "title": "restricted-lemma-quasi-compact-gives-quasi-compact", "contents": [ "In the situation above. If $f$ is quasi-compact, then", "$f_{/T}$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "(Quasi-compact morphisms of formal algebraic spaces are discussed in", "Formal Spaces, Section \\ref{formal-spaces-section-quasi-compact}.)", "We have to show that $(X'_{/T'})_{red} \\to (X_{/T})_{red}$ is a quasi-compact", "morphism of algebraic spaces. By", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-reduction-completion}", "this is the morphism $Z' \\to Z$ where $Z' \\subset X'$, resp.\\ $Z \\subset X$", "is the reduced induced algebraic space structure on $T'$, resp.\\ $T$.", "It follows that $Z' \\to f^{-1}Z = Z \\times_X X'$ is a thickening (a closed", "immersion defining an isomorphism on underlying topological spaces).", "Since $Z \\times_X X' \\to Z$ is quasi-compact as a base change of $f$", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-quasi-compact})", "we conclude that $Z' \\to Z$ is too by", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-thicken-property-morphisms}." ], "refs": [ "formal-spaces-lemma-reduction-completion", "spaces-morphisms-lemma-base-change-quasi-compact", "spaces-more-morphisms-lemma-thicken-property-morphisms" ], "ref_ids": [ 3886, 4738, 55 ] } ], "ref_ids": [] }, { "id": 2405, "type": "theorem", "label": "restricted-lemma-quasi-separated-gives-quasi-separated", "categories": [ "restricted" ], "title": "restricted-lemma-quasi-separated-gives-quasi-separated", "contents": [ "In the situation above. If $f$ is (quasi-)separated, then", "$f_{/T}$ is too." ], "refs": [], "proofs": [ { "contents": [ "(Separation conditions on morphisms of formal algebraic spaces are discussed in", "Formal Spaces, Section", "\\ref{formal-spaces-section-separation-axioms}.)", "We have to show that if $\\Delta_f$ is quasi-compact, resp.\\ a closed immersion,", "then the same is true for $\\Delta_{f_{/T}}$. This follows from the discussion", "in Remark \\ref{remark-diagonal-gives-diagonal} and", "Lemmas \\ref{lemma-quasi-compact-gives-quasi-compact} and", "\\ref{lemma-closed-immersion-gives-closed-immersion}." ], "refs": [ "restricted-remark-diagonal-gives-diagonal", "restricted-lemma-quasi-compact-gives-quasi-compact", "restricted-lemma-closed-immersion-gives-closed-immersion" ], "ref_ids": [ 2455, 2404, 2402 ] } ], "ref_ids": [] }, { "id": 2406, "type": "theorem", "label": "restricted-lemma-smooth-gives-rig-smooth", "categories": [ "restricted" ], "title": "restricted-lemma-smooth-gives-rig-smooth", "contents": [ "In the situation above. If $X$ is locally Noetherian,", "$f$ is locally of finite type, and $U' \\to U$ is smooth, then", "$f_{/T}$ is rig-smooth." ], "refs": [], "proofs": [ { "contents": [ "The strategy of the proof is this: reduce to the case where $X$ and $X'$", "are affine, translate the affine case into algebra, and finally apply", "Lemma \\ref{lemma-rig-smooth}. We urge the reader to skip the details.", "\\medskip\\noindent", "Choose a surjective \\'etale morphism $W \\to X$ with $W = \\coprod W_i$", "a disjoint union of affine schemes, see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-cover-by-union-affines}.", "For each $i$ choose a surjective \\'etale morphism $W'_i \\to W_i \\times_X X'$", "where $W'_i = \\coprod W'_{ij}$ is a disjoint union of affines.", "In particular $\\coprod W'_{ij} \\to X'$ is surjective and \\'etale.", "Denote $f_{ij} : W_{ij} \\to W_i$ the given morphism.", "Denote $T_i \\subset W_i$ and $T'_{ij} \\subset W_{ij}$ the inverse", "images of $T$. Since taking the completion along the inverse", "image of $T$ produces cartesian diagrams (see above) we", "have $(W_i)_{/T_i} = W_i \\times_X X_{/T}$ and similarly", "$(W'_{ij})_{/T'_{ij}} = W'_{ij} \\times_{X'} X'_{/T'}$.", "Moreover, recall that $(W_i)_{/T_i}$ and $(W'_{ij})_{/T'_{ij}}$", "are affine formal algebraic spaces.", "Hence $\\{W'_{ij})_{/T'_{ij}} \\to X'_{/T'}\\}$ is a covering", "as in Formal Spaces, Definition", "\\ref{formal-spaces-definition-formal-algebraic-space}.", "By Lemma \\ref{lemma-rig-smooth-morphisms}", "we see that it suffices to prove that", "$$", "(W'_{ij})_{/T'_{ij}} \\longrightarrow (W_i)_{/T_i}", "$$", "is rig-smooth. Observe that $W'_{ij} \\to W_i$", "is locally of finite type and induces a smooth morphism", "$W'_{ij} \\setminus T'_{ij} \\to W_i \\setminus T_i$", "(as this is true for $f$ and these properties of morphisms are", "\\'etale local on the source and target).", "Observe that $W_i$ is locally Noetherian (as $X$ is locally Noetherian", "and this property is \\'etale local on the algebraic space).", "Hence it suffices to prove the lemma when", "$X$ and $X'$ are affine schemes.", "\\medskip\\noindent", "Assume $X = \\Spec(A)$ and $X' = \\Spec(A')$ are affine schemes.", "Since $X$ is Noetherian, we see that $A$ is Noetherian.", "The morphism $f$ is given by a ring map $A \\to A'$ of finite type.", "Let $I \\subset A$ be an ideal cutting out $T$. Then $IA'$ cuts out $T'$.", "Also $\\Spec(A') \\to \\Spec(A)$ is smooth over $\\Spec(A) \\setminus T$.", "Let $A^\\wedge$ and $(A')^\\wedge$ be the $I$-adic", "completions. We have $X_{/T} = \\text{Spf}(A^\\wedge)$ and", "$X'_{/T'} = \\text{Spf}((A')^\\wedge)$, see proof of", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-formal-completion-types}.", "By Lemma \\ref{lemma-rig-smooth} we see that", "$(A')^\\wedge$ is rig-smooth over $(A. I)$", "which in turn means that $A^\\wedge \\to (A')^\\wedge$ is rig-smooth which", "finally implies that $X'_{/T'} \\to X_{/T}$ is rig smooth by", "Lemma \\ref{lemma-rig-smooth-morphisms}." ], "refs": [ "restricted-lemma-rig-smooth", "spaces-properties-lemma-cover-by-union-affines", "formal-spaces-definition-formal-algebraic-space", "restricted-lemma-rig-smooth-morphisms", "formal-spaces-lemma-formal-completion-types", "restricted-lemma-rig-smooth", "restricted-lemma-rig-smooth-morphisms" ], "ref_ids": [ 2303, 11830, 3981, 2368, 3919, 2303, 2368 ] } ], "ref_ids": [] }, { "id": 2407, "type": "theorem", "label": "restricted-lemma-etale-gives-rig-etale", "categories": [ "restricted" ], "title": "restricted-lemma-etale-gives-rig-etale", "contents": [ "In the situation above. If $X$ is locally Noetherian,", "$f$ is locally of finite type, and $U' \\to U$ is \\'etale, then", "$f_{/T}$ is rig-\\'etale." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Lemma \\ref{lemma-smooth-gives-rig-smooth} except with", "Lemmas \\ref{lemma-rig-smooth} and \\ref{lemma-rig-smooth-morphisms}", "replaced by", "Lemmas \\ref{lemma-rig-etale} and \\ref{lemma-rig-etale-morphisms}" ], "refs": [ "restricted-lemma-smooth-gives-rig-smooth", "restricted-lemma-rig-smooth", "restricted-lemma-rig-smooth-morphisms", "restricted-lemma-rig-etale", "restricted-lemma-rig-etale-morphisms" ], "ref_ids": [ 2406, 2303, 2368, 2316, 2377 ] } ], "ref_ids": [] }, { "id": 2408, "type": "theorem", "label": "restricted-lemma-completion-proper-surjective-rig-surjective", "categories": [ "restricted" ], "title": "restricted-lemma-completion-proper-surjective-rig-surjective", "contents": [ "In the situation above. If $X$ is locally Noetherian,", "$f$ is proper, and $U' \\to U$ is surjective, then $f_{/T}$ is rig-surjective." ], "refs": [], "proofs": [ { "contents": [ "(The statement makes sense by", "Lemma \\ref{lemma-map-completions-finite-type} and", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-formal-completion-types}.)", "Let $R$ be a complete discrete valuation ring with fraction field $K$.", "Let $p : \\text{Spf}(R) \\to X_{/T}$ be an adic morphism of", "formal algebraic spaces. By Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-adic-into-completion}", "the composition $\\text{Spf}(R) \\to X_{/T} \\to X$", "corresponds to a morphism $q : \\Spec(R) \\to X$", "which maps $\\Spec(K)$ into $U$. Since $U' \\to U$ is proper and surjective", "we see that $\\Spec(K) \\times_U U'$ is nonempty and proper over $K$.", "Hence we can choose a field extension $K'/K$ and a commutative", "diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & U' \\ar[r] \\ar[d] & X' \\ar[d] \\\\", "\\Spec(K) \\ar[r] & U \\ar[r] & X", "}", "$$", "Let $R' \\subset K'$ be a discrete valuation ring dominating $R$", "with fraction field $K'$, see Algebra, Lemma \\ref{algebra-lemma-exists-dvr}.", "Since $\\Spec(K) \\to X$ extends to $\\Spec(R) \\to X$ we see by the valuative", "criterion of properness", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-characterize-proper})", "that we can extend our $K'$-point of $U'$ to a morphism", "$\\Spec(R') \\to X'$ over $\\Spec(R) \\to X$.", "It follows that the inverse image of $T'$ in $\\Spec(R')$ is the", "closed point and we find an adic morphism", "$\\text{Spf}((R')^\\wedge) \\to X'_{/T'}$ lifting $p$", "as desired (note that $(R')^\\wedge$ is a complete discrete valuation ring", "by More on Algebra, Lemma \\ref{more-algebra-lemma-completion-dvr})." ], "refs": [ "restricted-lemma-map-completions-finite-type", "formal-spaces-lemma-formal-completion-types", "formal-spaces-lemma-adic-into-completion", "algebra-lemma-exists-dvr", "spaces-morphisms-lemma-characterize-proper", "more-algebra-lemma-completion-dvr" ], "ref_ids": [ 2400, 3919, 3965, 1028, 4938, 10046 ] } ], "ref_ids": [] }, { "id": 2409, "type": "theorem", "label": "restricted-lemma-separated-mono-open-diagonal-rig-surjective", "categories": [ "restricted" ], "title": "restricted-lemma-separated-mono-open-diagonal-rig-surjective", "contents": [ "In the situation above. If $X$ is locally Noetherian,", "$f$ is separated and locally of finite type, and $U' \\to U$ is", "a monomorphism, then $\\Delta_{f_{/T}}$ is rig-surjective." ], "refs": [], "proofs": [ { "contents": [ "The diagonal $\\Delta_f : X' \\to X' \\times_X X'$ is a closed", "immersion and the restriction $U' \\to U' \\times_U U'$ of $\\Delta_f$", "is surjective. Hence the lemma follows from the discussion in", "Remark \\ref{remark-diagonal-gives-diagonal} and", "Lemma \\ref{lemma-completion-proper-surjective-rig-surjective}." ], "refs": [ "restricted-remark-diagonal-gives-diagonal", "restricted-lemma-completion-proper-surjective-rig-surjective" ], "ref_ids": [ 2455, 2408 ] } ], "ref_ids": [] }, { "id": 2410, "type": "theorem", "label": "restricted-lemma-modification-gives-formal-modification", "categories": [ "restricted" ], "title": "restricted-lemma-modification-gives-formal-modification", "contents": [ "Let $S$, $f : X' \\to X$, $T \\subset |X|$, $U \\subset X$,", "$T' \\subset |X'|$, and $U' \\subset X'$ be as in", "Section \\ref{section-completion-functor}.", "If $X$ is locally Noetherian, $f$ is proper, and $U' \\to U$ is an isomorphism,", "then $f_{/T} : X'_{/T'} \\to X_{/T}$ is a formal modification." ], "refs": [], "proofs": [ { "contents": [ "By Formal Spaces, Lemmas \\ref{formal-spaces-lemma-formal-completion-types}", "the source and target of the arrow are locally Noetherian", "formal algebraic spaces.", "The other conditions follow from", "Lemmas \\ref{lemma-proper-gives-proper},", "\\ref{lemma-etale-gives-rig-etale},", "\\ref{lemma-completion-proper-surjective-rig-surjective}, and", "\\ref{lemma-separated-mono-open-diagonal-rig-surjective}." ], "refs": [ "formal-spaces-lemma-formal-completion-types", "restricted-lemma-proper-gives-proper", "restricted-lemma-etale-gives-rig-etale", "restricted-lemma-completion-proper-surjective-rig-surjective", "restricted-lemma-separated-mono-open-diagonal-rig-surjective" ], "ref_ids": [ 3919, 2403, 2407, 2408, 2409 ] } ], "ref_ids": [] }, { "id": 2411, "type": "theorem", "label": "restricted-lemma-base-change-formal-modification", "categories": [ "restricted" ], "title": "restricted-lemma-base-change-formal-modification", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian", "formal algebraic spaces over $S$ which is a formal modification.", "Then for any adic morphism $Y' \\to Y$ of locally Noetherian formal", "algebraic spaces, the base change $f' : X \\times_Y Y' \\to Y'$ is", "a formal modification." ], "refs": [], "proofs": [ { "contents": [ "The morphism $f'$ is proper by", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-base-change-proper}.", "The morphism $f'$ is rig-etale by", "Lemma \\ref{lemma-base-change-rig-etale}.", "Then morphism $f'$ is rig-surjective by", "Lemma \\ref{lemma-base-change-rig-surjective}. Set $X' = X \\times_ Y'$.", "The morphism $\\Delta_{f'}$ is the base change of", "$\\Delta_f$ by the adic morphism $X' \\times_{Y'} X' \\to X \\times_Y X$.", "Hence $\\Delta_{f'}$ is rig-surjective by", "Lemma \\ref{lemma-base-change-rig-surjective}." ], "refs": [ "formal-spaces-lemma-base-change-proper", "restricted-lemma-base-change-rig-etale", "restricted-lemma-base-change-rig-surjective", "restricted-lemma-base-change-rig-surjective" ], "ref_ids": [ 3960, 2380, 2386, 2386 ] } ], "ref_ids": [] }, { "id": 2412, "type": "theorem", "label": "restricted-lemma-algebraize-rig-etale-affine", "categories": [ "restricted" ], "title": "restricted-lemma-algebraize-rig-etale-affine", "contents": [ "Let $T \\subset X$ be a closed subset of a Noetherian affine scheme $X$.", "Let $W$ be a Noetherian affine formal algebraic space.", "Let $g : W \\to X_{/T}$ be a rig-\\'etale morphism. Then there exists", "an affine scheme $X'$ and a finite type morphism $f : X' \\to X$", "\\'etale over $X \\setminus T$ such that there is an isomorphism", "$X'_{/f^{-1}T} \\cong W$ compatible with $f_{/T}$ and $g$.", "Moreover, if $W \\to X_{/T}$ is \\'etale, then $X' \\to X$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "The existence of $X'$ is a restatement of", "Lemma \\ref{lemma-approximate-by-etale-over-complement}.", "The final statement follows from", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds}." ], "refs": [ "restricted-lemma-approximate-by-etale-over-complement", "more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds" ], "ref_ids": [ 2323, 13743 ] } ], "ref_ids": [] }, { "id": 2413, "type": "theorem", "label": "restricted-lemma-algebraize-morphism-rig-etale", "categories": [ "restricted" ], "title": "restricted-lemma-algebraize-morphism-rig-etale", "contents": [ "Assume we have", "\\begin{enumerate}", "\\item Noetherian affine schemes $X$, $X'$, and $Y$,", "\\item a closed subset $T \\subset |X|$,", "\\item a morphism $f : X' \\to X$ locally of finite type", "and \\'etale over $X \\setminus T$,", "\\item a morphism $h : Y \\to X$,", "\\item a morphism $\\alpha : Y_{/T} \\to X'_{/T}$ over $X_{/T}$", "(see proof for notation).", "\\end{enumerate}", "Then there exists an \\'etale morphism $b : Y' \\to Y$ of affine schemes", "which induces an isomorphism $b_{/T} : Y'_{/T} \\to Y_{/T}$", "and a morphism $a : Y' \\to X'$ over $X$", "such that $\\alpha = a_{/T} \\circ b_{/T}^{-1}$." ], "refs": [], "proofs": [ { "contents": [ "The notation using the subscript ${}_{/T}$ in the statement", "refers to the construction which to a morphism of schemes $g : V \\to X$", "associates the morphism $g_{/T} : V_{/g^{-1}T} \\to X_{/T}$ of formal", "algebraic spaces; it is a functor from the category of schemes over $X$", "to the category of formal algebraic spaces over $X_{/T}$, see", "Section \\ref{section-completion-functor}.", "Having said this, the lemma is just a reformulation of", "Lemma \\ref{lemma-fully-faithful-etale-over-complement}." ], "refs": [ "restricted-lemma-fully-faithful-etale-over-complement" ], "ref_ids": [ 2319 ] } ], "ref_ids": [] }, { "id": 2414, "type": "theorem", "label": "restricted-lemma-factor", "categories": [ "restricted" ], "title": "restricted-lemma-factor", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$ be morphisms", "of algebraic spaces. Let $T \\subset |X|$ be closed.", "Assume that", "\\begin{enumerate}", "\\item $X$ is locally Noetherian,", "\\item $g$ is a monomorphism and locally of finite type,", "\\item $f|_{X \\setminus T} : X \\setminus T \\to Y$ factors through $g$, and", "\\item $f_{/T} : X_{/T} \\to Y$ factors through $g$,", "\\end{enumerate}", "then $f$ factors through $g$." ], "refs": [], "proofs": [ { "contents": [ "Consider the fibre product $E = X \\times_Y Z \\to X$.", "By assumption the open immersion $X \\setminus T \\to X$", "factors through $E$ and any morphism $\\varphi : X' \\to X$ with", "$|\\varphi|(|X'|) \\subset T$ factors through $E$ as well, see", "Formal Spaces, Section \\ref{formal-spaces-section-completion}.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds}", "this implies that $E \\to X$ is \\'etale at every point of $E$", "mapping to a point of $T$. Hence $E \\to X$ is an \\'etale", "monomorphism, hence an open immersion", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-etale-universally-injective-open}).", "Then it follows that $E = X$ since our assumptions imply that $|X| = |E|$." ], "refs": [ "spaces-more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds", "spaces-morphisms-lemma-etale-universally-injective-open" ], "ref_ids": [ 121, 4973 ] } ], "ref_ids": [] }, { "id": 2415, "type": "theorem", "label": "restricted-lemma-faithful-general", "categories": [ "restricted" ], "title": "restricted-lemma-faithful-general", "contents": [ "Let $S$ be a scheme. Let $X$, $W$ be algebraic spaces over $S$ with", "$X$ locally Noetherian. Let $T \\subset |X|$ be a closed subset.", "Let $a, b : X \\to W$ be morphisms of algebraic spaces over $S$ such", "that $a|_{X \\setminus T} = b|_{X \\setminus T}$ and such that", "$a_{/T} = b_{/T}$ as morphisms $X_{/T} \\to W$. Then $a = b$." ], "refs": [], "proofs": [ { "contents": [ "Let $E$ be the equalizer of $a$ and $b$. Then $E$ is an algebraic space", "and $E \\to X$ is locally of finite type and a monomorphism, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-properties-diagonal}.", "Our assumptions imply we can apply Lemma \\ref{lemma-factor} to the two", "morphisms $f = \\text{id} : X \\to X$ and $g : E \\to X$ and the closed", "subset $T$ of $|X|$." ], "refs": [ "spaces-morphisms-lemma-properties-diagonal", "restricted-lemma-factor" ], "ref_ids": [ 4712, 2414 ] } ], "ref_ids": [] }, { "id": 2416, "type": "theorem", "label": "restricted-lemma-faithful", "categories": [ "restricted" ], "title": "restricted-lemma-faithful", "contents": [ "Let $S$ be a scheme. Let $X$, $Y$ be locally Noetherian algebraic spaces", "over $S$. Let $T \\subset |X|$ and $T' \\subset |Y|$ be closed subsets.", "Let $a, b : X \\to Y$ be morphisms of algebraic spaces over $S$ such", "that $a|_{X \\setminus T} = b|_{X \\setminus T}$, such that", "$|a|(T) \\subset T'$ and $|b|(T) \\subset T'$, and such that", "$a_{/T} = b_{/T}$ as morphisms $X_{/T} \\to Y_{/T'}$.", "Then $a = b$." ], "refs": [], "proofs": [ { "contents": [ "Consequence of the more general Lemma \\ref{lemma-faithful-general}." ], "refs": [ "restricted-lemma-faithful-general" ], "ref_ids": [ 2415 ] } ], "ref_ids": [] }, { "id": 2417, "type": "theorem", "label": "restricted-lemma-equivalence-relation", "categories": [ "restricted" ], "title": "restricted-lemma-equivalence-relation", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space", "over $S$. Let $T \\subset |X|$ be a closed subset.", "Let $s, t : R \\to U$ be two morphisms of algebraic spaces over $X$.", "Assume", "\\begin{enumerate}", "\\item $R$, $U$ are locally of finite type over $X$,", "\\item the base change of $s$ and $t$ to $X \\setminus T$", "is an \\'etale equivalence relation, and", "\\item the formal completion", "$(t_{/T}, s_{/T}) : R_{/T} \\to U_{/T} \\times_{X_{/T}} U_{/T}$", "is an equivalence relation too (see proof for notation).", "\\end{enumerate}", "Then $(t, s) : R \\to U \\times_X U$ is an \\'etale equivalence relation." ], "refs": [], "proofs": [ { "contents": [ "The notation using the subscript ${}_{/T}$ in the statement refers to the", "construction which to a morphism $f : X' \\to X$ of algebraic spaces", "associates the morphism $f_{/T} : X'_{/f^{-1}T} \\to X_{/T}$ of formal", "algebraic spaces, see Section \\ref{section-completion-functor}.", "The morphisms $s, t : R \\to U$ are \\'etale over $X \\setminus T$", "by assumption. Since the formal completions of the maps", "$s, t : R \\to U$ are \\'etale, we see that $s$ and $t$ are \\'etale", "for example by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds}.", "Applying Lemma \\ref{lemma-factor} to the morphisms", "$\\text{id} : R \\times_{U \\times_X U} R \\to R \\times_{U \\times_X U} R$", "and $\\Delta : R \\to R \\times_{U \\times_X U} R$ we conclude that", "$(t, s)$ is a monomorphism. Applying it again to", "$(t \\circ \\text{pr}_0, s \\circ \\text{pr}_1) :", "R \\times_{s, U, t} R \\to U \\times_X U$ and $(t, s) : R \\to U \\times_X U$", "we find that ``transitivity'' holds. We omit the proof of", "the other two axioms of an equivalence relation." ], "refs": [ "more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds", "restricted-lemma-factor" ], "ref_ids": [ 13743, 2414 ] } ], "ref_ids": [] }, { "id": 2418, "type": "theorem", "label": "restricted-lemma-smash-away-from-T", "categories": [ "restricted" ], "title": "restricted-lemma-smash-away-from-T", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$", "and let $T \\subset |X|$ be a closed subset. Let $f : X' \\to X$ be a morphism", "of algebraic spaces which is locally of finite type and \\'etale outside of $T$.", "There exists a factorization", "$$", "X' \\longrightarrow X'' \\longrightarrow X", "$$", "of $f$ with the following properties:", "$X'' \\to X$ is locally of finite type,", "$X'' \\to X$ is an isomorphism over $X \\setminus T$, and", "$X'_{/T} \\to X''_{/T}$ is an isomorphism (see proof for notation)." ], "refs": [], "proofs": [ { "contents": [ "The notation using the subscript ${}_{/T}$ in the statement refers to the", "construction which to a morphism $f : X' \\to X$ of algebraic spaces", "associates the morphism $f_{/T} : X'_{/f^{-1}T} \\to X_{/T}$ of formal", "algebraic spaces, see Section \\ref{section-completion-functor}.", "We will also use the notion $U \\subset X$ and $U' \\subset X'$ to denote", "the open subspaces with $|U| = |X| \\setminus T$ and", "$U' = |X'| \\setminus f^{-1}T$ introduced in", "Section \\ref{section-completion-functor}.", "\\medskip\\noindent", "After replacing $X'$ by $X' \\amalg U$ we may and do assume", "the image of $X' \\to X$ contains $U$.", "Let", "$$", "R = X' \\amalg_{U'} (U' \\times_U U')", "$$", "be the pushout of $U' \\to X'$ and the diagonal morphism", "$U' \\to U' \\times_U U' = U' \\times_X U'$. Since $U' \\to X$ is \\'etale,", "this diagonal is an open immersion and we see that $R$ is an algebraic space", "(this follows for example from", "Spaces, Lemma \\ref{spaces-lemma-glueing-algebraic-spaces}).", "The two projections $U' \\times_U U' \\to U'$ extend to $R$", "and we obtain two \\'etale morphisms $s, t : R \\to X'$.", "Checking on each piece separatedly we find that $R$", "is an \\'etale equivalence relation on $X'$. Set $X'' = X'/R$", "which is an algebraic space by", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}.", "By construction have the factorization as in the lemma and", "the morphism $X'' \\to X$ is locally of finite type (as this", "can be checked \\'etale locally, i.e., on $X'$).", "Since $U' \\to U$ is a surjective \\'etale morphism", "and since $s^{-1}(U') = t^{-1}(U') = U' \\times_U U'$", "we see that $U'' = U \\times_X X'' \\to U$ is an isomorphism.", "Finally, we have to show the morphism $X' \\to X''$ induces an isomorphism", "$X'_{/T} \\to X''_{/T}$. To see this, note that the formal completion of $R$", "along the inverse image of $T$ is equal to the formal completion of", "$X'$ along the inverse image of $T$ by our choice of $R$! By", "our construction of the formal completion in", "Formal Spaces, Section \\ref{formal-spaces-section-completion}", "we have $X''_{/T} = (X'_{/T}) / (R_{/T})$ as sheaves. Since", "$X'_{/T} = R_{/T}$ we conclude that $X'_{/T} = X''_{/T}$", "and this finishes the proof." ], "refs": [ "spaces-lemma-glueing-algebraic-spaces", "bootstrap-theorem-final-bootstrap" ], "ref_ids": [ 8148, 2602 ] } ], "ref_ids": [] }, { "id": 2419, "type": "theorem", "label": "restricted-lemma-functoriality-completion-functor", "categories": [ "restricted" ], "title": "restricted-lemma-functoriality-completion-functor", "contents": [ "In the situation above, let $X_1 \\to X$ be a morphism of algebraic", "spaces with $X_1$ locally Noetherian. Denote $T_1 \\subset |X_1|$", "the inverse image of $T$ and $U_1 \\subset X_1$ the inverse image of $U$.", "We denote", "\\begin{enumerate}", "\\item $\\mathcal{C}_{X, T}$ the category whose objects are", "morphisms of algebraic spaces $f : X' \\to X$ which are locally", "of finite type and such that $U' = f^{-1}U \\to U$ is an isomorphism,", "\\item $\\mathcal{C}_{X_1, T_1}$ the category whose objects are", "morphisms of algebraic spaces $f_1 : X_1' \\to X_1$ which are locally", "of finite type and such that $f_1^{-1}U_1 \\to U_1$ is an isomorphism,", "\\item $\\mathcal{C}_{X_{/T}}$ the category whose objects are", "morphisms $g : W \\to X_{/T}$ of formal algebraic spaces", "with $W$ locally Noetherian and $g$ rig-\\'etale,", "\\item $\\mathcal{C}_{X_{1, /T_1}}$ the category whose objects are", "morphisms $g_1 : W_1 \\to X_{1, /T_1}$ of formal algebraic spaces", "with $W_1$ locally Noetherian and $g_1$ rig-\\'etale.", "\\end{enumerate}", "Then the diagram", "$$", "\\xymatrix{", "\\mathcal{C}_{X, T} \\ar[d] \\ar[r] &", "\\mathcal{C}_{X_{/T}} \\ar[d] \\\\", "\\mathcal{C}_{X_1, T_1} \\ar[r] &", "\\mathcal{C}_{X_{1, /T_1}}", "}", "$$", "is commutative where the horizonal arrows are given by", "(\\ref{equation-completion-functor})", "and the vertical arrows by base change along", "$X_1 \\to X$ and along $X_{1, /T_1} \\to X_{/T}$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the fact that the completion functor", "$(h : Y \\to X) \\mapsto Y_{/T} = Y_{/|h|^{-1}T}$", "on the category of algebraic spaces over $X$", "commutes with fibre products." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2420, "type": "theorem", "label": "restricted-lemma-completion-functor-fully-faithful", "categories": [ "restricted" ], "title": "restricted-lemma-completion-functor-fully-faithful", "contents": [ "In the situation above. Let $f : X' \\to X$ be a morphism of algebraic spaces", "which is locally of finite type and an isomorphism over $U$. Let", "$g : Y \\to X$ be a morphism with $Y$ locally Noetherian. Then completion", "defines a bijection", "$$", "\\Mor_X(Y, X') \\longrightarrow \\Mor_{X_{/T}}(Y_{/T}, X'_{/T})", "$$", "In particular, the functor (\\ref{equation-completion-functor}) is", "fully faithful." ], "refs": [], "proofs": [ { "contents": [ "Let $a, b : Y \\to X'$ be morphisms over $X$ such that", "$a_{/T} = b_{/T}$. Then we see that $a$ and $b$ agree over the open", "subspace $g^{-1}U$ and after completion along $g^{-1}T$.", "Hence $a = b$ by Lemma \\ref{lemma-faithful}.", "In other words, the completion map is always injective.", "\\medskip\\noindent", "Let $\\alpha : Y_{/T} \\to X'_{/T}$ be a morphism of formal algebraic spaces", "over $X_{/T}$. We have to prove there exists a morphism $a : Y \\to X'$", "over $X$ such that $\\alpha = a_{/T}$. The proof proceeds by a standard", "but cumbersome reduction to the affine case and then applying", "Lemma \\ref{lemma-algebraize-morphism-rig-etale}.", "\\medskip\\noindent", "Let $\\{h_i : Y_i \\to Y\\}$ be an \\'etale covering of algebraic spaces.", "If we can find for each $i$ a morphism $a_i : Y_i \\to X'$ over $X$", "whose completion $(a_i)_{/T} : (Y_i)_{/T} \\to X'_{/T}$ is equal to", "$\\alpha \\circ (h_i)_{/T}$, then we get a morphism $a : Y \\to X'$", "with $\\alpha = a_{/T}$. Namely, we first observe that", "$(a_i)_{/T} \\circ \\text{pr}_1 = (a_j)_{/T} \\circ \\text{pr}_2$", "as morphisms $(Y_i \\times_Y Y_j)_{/T} \\to X'_{/T}$ by the", "agreement with $\\alpha$ (this uses that completion ${}_{/T}$ commutes", "with fibre products). By the injectivity already proven this shows that", "$a_i \\circ \\text{pr}_1 = a_j \\circ \\text{pr}_2$ as morphisms", "$Y_i \\times_Y Y_j \\to X'$. Since $X'$ is an fppf sheaf this means that", "the collection of morphisms $a_i$ descends to a morphism $a : Y \\to X'$.", "We have $\\alpha = a_{/T}$ because $\\{(a_i)_{/T} : (Y_i)_{/T} \\to X'_{/T}\\}$", "is an \\'etale covering.", "\\medskip\\noindent", "By the result of the previous paragraph, to prove existence,", "we may assume that $Y$ is affine and that $g : Y \\to X$ factors", "as $g_1 : Y \\to X_1$ and an \\'etale morphism $X_1 \\to X$ with $X_1$ affine.", "Then we can consider $T_1 \\subset |X_1|$ the inverse image of $T$", "and we can set $X'_1 = X' \\times_X X_1$ with projection $f_1 : X'_1 \\to X_1$", "and", "$$", "\\alpha_1 = (\\alpha, (g_1)_{/T_1}) :", "Y_{/T_1} = Y_{/T}", "\\longrightarrow", "X'_{/T} \\times_{X_{/T}} (X_1)_{/T_1} = (X'_1)_{/T_1}", "$$", "We conclude that it suffices to prove the existence for $\\alpha_1$", "over $X_1$, in other words, we may replace $X, T, X', Y, f, g, \\alpha$", "by $X_1, T_1, X'_1, Y, g_1, \\alpha_1$.", "This reduces us to the case described in the next paragraph.", "\\medskip\\noindent", "Assume $Y$ and $X$ are affine. Recall that $(Y_{/T})_{red}$", "is an affine scheme (isomorphic to the reduced induced scheme structure", "on $g^{-1}T \\subset Y$, see Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-reduction-completion}).", "Hence $\\alpha_{red} : (Y_{/T})_{red} \\to (X'_{/T})_{red}$", "has quasi-compact image $E$ in $f^{-1}T$ (this is the underlying", "topological space of $(X'_{/T})_{red}$ by the same lemma as above).", "Thus we can find an affine scheme $V$ and an \\'etale morpism", "$h : V \\to X'$ such that the image of $h$ contains $E$.", "Choose a solid cartesian diagram", "$$", "\\xymatrix{", "Y'_{/T} \\ar@{..>}[rd] \\ar@{..>}[r] &", "W \\ar[d] \\ar[r] & V_{/T} \\ar[d]^{h_{/T}} \\\\", "& Y_{/T} \\ar[r]^\\alpha & X'_{/T}", "}", "$$", "By construction, the morphism $W \\to Y_{/T}$ is representable by", "algebraic spaces, \\'etale, and surjective", "(surjectivity can be seen by looking at the reductions, see", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-reduction-surjective}).", "By Lemma \\ref{lemma-algebraize-rig-etale-affine}", "we can write $W = Y'_{/T}$ for $Y' \\to Y$ \\'etale and $Y'$ affine.", "This gives the dotted arrows in the diagram.", "Since $W \\to Y_{/T}$ is surjective, we see that", "the image of $Y' \\to Y$ contains $g^{-1}T$.", "Hence $\\{Y' \\to Y, Y \\setminus g^{-1}T \\to Y\\}$", "is an \\'etale covering. As $f$ is an isomorphism over $U$ we have a", "(unique) morphism $Y \\setminus g^{-1}T \\to X'$ over $X$", "agreeing with $\\alpha$ on completions", "(as the completion of $Y \\setminus g^{-1}T$ is empty).", "Thus it suffices to prove the existence for $Y'$", "which reduces us to the case studied in the next paragraph.", "\\medskip\\noindent", "By the result of the previous paragraph, we may assume that $Y$ is affine", "and that $\\alpha$ factors as $Y_{/T} \\to V_{/T} \\to X'_{/T}$", "where $V$ is an affine scheme \\'etale over $X'$. We may still replace $Y$ by", "the members of an affine \\'etale covering.", "By Lemma \\ref{lemma-algebraize-morphism-rig-etale}", "we may find an \\'etale morphism $b : Y' \\to Y$ of affine schemes", "which induces an isomorphism $b_{/T} : Y'_{/T} \\to Y_{/T}$", "and a morphism $c : Y' \\to V$ such that $c_{/T} \\circ b_{/T}^{-1}$", "is the given morphism $Y_{/T} \\to V_{/T}$. Setting", "$a' : Y' \\to X'$ equal to the composition of $c$ and $V \\to X'$", "we find that $a'_{/T} = \\alpha \\circ b_{/T}$, in other words, we have", "existence for $Y'$ and $\\alpha \\circ b_{/T}$.", "Then we are done by replacing considering once more", "the \\'etale covering $\\{Y' \\to Y, Y \\setminus g^{-1}T \\to Y\\}$." ], "refs": [ "restricted-lemma-faithful", "restricted-lemma-algebraize-morphism-rig-etale", "formal-spaces-lemma-reduction-completion", "formal-spaces-lemma-reduction-surjective", "restricted-lemma-algebraize-rig-etale-affine", "restricted-lemma-algebraize-morphism-rig-etale" ], "ref_ids": [ 2416, 2413, 3886, 3881, 2412, 2413 ] } ], "ref_ids": [] }, { "id": 2421, "type": "theorem", "label": "restricted-lemma-dilatations-affine", "categories": [ "restricted" ], "title": "restricted-lemma-dilatations-affine", "contents": [ "In the situation above. Assume $X$ is affine. Then the functor", "(\\ref{equation-completion-functor}) is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "We already know the functor is fully faithful, see", "Lemma \\ref{lemma-completion-functor-fully-faithful}.", "Essential surjectivity. Let $g : W \\to X_{/T}$ be a morphism", "of formal algebraic spaces with $W$ locally Noetherian", "and $g$ rig-\\'etale. We will prove $W$ is in the essential", "image in a number of steps.", "\\medskip\\noindent", "Step 1: $W$ is an affine formal algebraic space. Then we can find", "$U \\to X$ of finite type and \\'etale over $X \\setminus T$ such that", "$U_{/T}$ is isomorphic to $W$, see", "Lemma \\ref{lemma-algebraize-rig-etale-affine}.", "Thus we see that $W$ is in the essential image by", "Lemma \\ref{lemma-smash-away-from-T}.", "\\medskip\\noindent", "Step 2: $W$ is separated. Choose $\\{W_i \\to W\\}$ as in", "Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space}.", "By Step 1 the formal algebraic spaces $W_i$ and $W_i \\times_W W_j$", "are in the essential image.", "Say $W_i = (X'_i)_{/T}$ and $W_i \\times_W W_j = (X'_{ij})_{/T}$.", "By fully faithfulness we obtain morphisms $t_{ij} : X'_{ij} \\to X'_i$", "and $s_{ij} : X'_{ij} \\to X'_j$ matching the projections", "$W_i \\times_W W_j \\to W_i$ and $W_i \\times_W W_j \\to W_j$.", "Consider the structure", "$$", "R = \\coprod X'_{ij},\\quad", "V = \\coprod X'_i,\\quad", "s = \\coprod s_{ij},\\quad", "t = \\coprod t_{ij}", "$$", "(We can't use the letter $U$ as it has already been used.)", "Applying Lemma \\ref{lemma-equivalence-relation} we find that", "$(t, s) : R \\to V \\times_X V$ defines an \\'etale equivalence relation", "on $V$ over $X$. Thus we can take the quotient $X' = V/R$ and", "it is an algebraic space, see", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}.", "Since completion commutes with fibre products and taking", "quotient sheaves, we find that", "$X'_{/T} \\cong W$ as formal algebraic spaces over $X_{/T}$.", "\\medskip\\noindent", "Step 3: $W$ is general. Choose $\\{W_i \\to W\\}$ as in", "Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space}.", "The formal algebraic spaces $W_i$ and $W_i \\times_W W_j$ are separated.", "Hence by Step 2 the formal algebraic spaces $W_i$ and $W_i \\times_W W_j$", "are in the essential image. Then we argue exactly as in the previous", "paragraph to see that $W$ is in the essential image as well.", "This concludes the proof." ], "refs": [ "restricted-lemma-completion-functor-fully-faithful", "restricted-lemma-algebraize-rig-etale-affine", "restricted-lemma-smash-away-from-T", "formal-spaces-definition-formal-algebraic-space", "restricted-lemma-equivalence-relation", "bootstrap-theorem-final-bootstrap", "formal-spaces-definition-formal-algebraic-space" ], "ref_ids": [ 2420, 2412, 2418, 3981, 2417, 2602, 3981 ] } ], "ref_ids": [] }, { "id": 2422, "type": "theorem", "label": "restricted-lemma-output-quasi-compact", "categories": [ "restricted" ], "title": "restricted-lemma-output-quasi-compact", "contents": [ "With assumptions and notation as in Theorem \\ref{theorem-dilatations-general}", "let $f : X' \\to X$ correspond to $g : W \\to X_{/T}$.", "Then $f$ is quasi-compact if and only if $g$ is quasi-compact." ], "refs": [ "restricted-theorem-dilatations-general" ], "proofs": [ { "contents": [ "If $f$ is quasi-compact, then $g$ is quasi-compact by", "Lemma \\ref{lemma-quasi-compact-gives-quasi-compact}.", "Conversely, assume $g$ is quasi-compact.", "Choose an \\'etale covering $\\{X_i \\to X\\}$ with $X_i$ affine.", "It suffices to prove that the base change $X' \\times_X X_i \\to X_i$", "is quasi-compact, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-quasi-compact-local}.", "By Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-characterize-quasi-compact-morphism}", "the base changes $W_i \\times_{X_{/T}} (X_i)_{/T} \\to (X_i)_{/T}$", "are quasi-compact.", "By Lemma \\ref{lemma-functoriality-completion-functor}", "we reduce to the case described in the next paragraph.", "\\medskip\\noindent", "Assume $X$ is affine and $g : W \\to X_{/T}$ quasi-compact.", "We have to show that $X'$ is quasi-compact.", "Let $V \\to X'$ be a surjective \\'etale morphism", "where $V = \\coprod_{j \\in J} V_j$ is a disjoint union of affines. Then", "$V_{/T} \\to X'_{/T} = W$ is a surjective \\'etale morphism.", "Since $W$ is quasi-compact, then we can find a finite subset", "$J' \\subset J$ such that $\\coprod_{j \\in J'} (V_j)_{/T} \\to W$ is surjective.", "Then it follows that", "$$", "U \\amalg \\coprod\\nolimits_{j \\in J'} V_j \\longrightarrow X'", "$$", "is surjective (and hence $X'$ is quasi-compact).", "Namely, we have $|X'| = |U| \\amalg |W_{red}|$ as $X'_{/T} = W$." ], "refs": [ "restricted-lemma-quasi-compact-gives-quasi-compact", "spaces-morphisms-lemma-quasi-compact-local", "formal-spaces-lemma-characterize-quasi-compact-morphism", "restricted-lemma-functoriality-completion-functor" ], "ref_ids": [ 2404, 4742, 3899, 2419 ] } ], "ref_ids": [ 2292 ] }, { "id": 2423, "type": "theorem", "label": "restricted-lemma-output-quasi-separated", "categories": [ "restricted" ], "title": "restricted-lemma-output-quasi-separated", "contents": [ "With assumptions and notation as in Theorem \\ref{theorem-dilatations-general}", "let $f : X' \\to X$ correspond to $g : W \\to X_{/T}$.", "Then $f$ is quasi-separated if and only if $g$ is so." ], "refs": [ "restricted-theorem-dilatations-general" ], "proofs": [ { "contents": [ "If $f$ is quasi-separated, then $g$ is quasi-separated by", "Lemma \\ref{lemma-quasi-separated-gives-quasi-separated}.", "Conversely, assume $g$ is quasi-separated. We have to show", "that $f$ is quasi-separated. Exactly as in the proof", "of Lemma \\ref{lemma-output-quasi-compact} we may check", "this over the members of a \\'etale covering of $X$ by affine schemes", "using Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-separated-local}", "and Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-separated-local}.", "Thus we may and do assume $X$ is affine.", "\\medskip\\noindent", "Let $V \\to X'$ be a surjective \\'etale morphism", "where $V = \\coprod_{j \\in J} V_j$ is a disjoint union of affines.", "To show that $X'$ is quasi-separated, it suffices to show that", "$V_j \\times_{X'} V_{j'}$ is quasi-compact for all $j, j' \\in J$.", "Since $W$ is quasi-separated the fibre products", "$(V_j \\times_Y V_{j'})_{/T} = (V_j)_{/T} \\times_{X'_{/T}} (V_{j'})_{/T}$", "are quasi-compact for all $j, j' \\in J$. Since $X$ is Noetherian affine", "and $U' \\to U$ is an isomorphism, we see that", "$$", "(V_j \\times_{X'} V_{j'}) \\times_X U =", "(V_j \\times_X V_{j'}) \\times_X U", "$$", "is quasi-compact. Hence we conclude by the equality", "$$", "|V_j \\times_{X'} V_{j'}| =", "|(V_j \\times_{X'} V_{j'}) \\times_X U| \\amalg", "|(V_j \\times_{X'} V_{j'})_{/T, red}|", "$$", "and the fact that a formal algebraic space is quasi-compact if and", "only if its associated reduced algebraic space is so." ], "refs": [ "restricted-lemma-quasi-separated-gives-quasi-separated", "restricted-lemma-output-quasi-compact", "spaces-morphisms-lemma-separated-local", "formal-spaces-lemma-separated-local" ], "ref_ids": [ 2405, 2422, 4722, 3958 ] } ], "ref_ids": [ 2292 ] }, { "id": 2424, "type": "theorem", "label": "restricted-lemma-output-separated", "categories": [ "restricted" ], "title": "restricted-lemma-output-separated", "contents": [ "With assumptions and notation as in Theorem \\ref{theorem-dilatations-general}", "let $f : X' \\to X$ correspond to $g : W \\to X_{/T}$.", "Then $f$ is separated $\\Leftrightarrow$ $g$ is", "separated and $\\Delta_g : W \\to W \\times_{X_{/T}} W$ is rig-surjective." ], "refs": [ "restricted-theorem-dilatations-general" ], "proofs": [ { "contents": [ "If $f$ is separated, then $g$ is separated and $\\Delta_g$", "is rig-surjective by", "Lemmas \\ref{lemma-quasi-separated-gives-quasi-separated} and", "\\ref{lemma-separated-mono-open-diagonal-rig-surjective}.", "Assume $g$ is separated and $\\Delta_g$ is rig-surjective.", "Exactly as in the proof of", "Lemma \\ref{lemma-output-quasi-compact}", "we may check this over the members of a \\'etale covering of $X$", "by affine schemes using", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-separated}", "(locality on the base of being separated for morphisms of algebraic spaces),", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-base-change-separated}", "(being separated for morphisms of formal algebraic spaces is preserved", "by base change), and", "Lemma \\ref{lemma-base-change-rig-surjective} (being rig-surjective", "is preserved by base change).", "Thus we may and do assume $X$ is affine. Furthermore,", "we already know that $f : X' \\to X$ is quasi-separated by", "Lemma \\ref{lemma-output-quasi-separated}.", "\\medskip\\noindent", "By Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-check-separated-dvr} and", "Remark \\ref{spaces-cohomology-remark-variant}", "it suffices to show that given any commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X' \\ar[d] \\\\", "\\Spec(R) \\ar[r]^p \\ar@{-->}[ru] & X' \\times_X X'", "}", "$$", "where $R$ is a complete discrete valuation ring with fraction field $K$,", "there is a dotted arrow making the diagram commute (as this will", "give the uniqueness part of the valuative criterion). Let", "$h : \\Spec(R) \\to X$ be the composition of $p$ with the morphism", "$Y \\times_X Y \\to X$. There are three cases:", "Case I: $h(\\Spec(R)) \\subset U$. This case is trivial", "because $U' = X' \\times_X U \\to U$ is an isomorphism.", "Case II: $h$ maps $\\Spec(R)$ into $T$. This case follows", "from our assumption that $g : W \\to X_{/T}$ is separated. Namely,", "if $Z$ denotes the reduced induced closed subspace structure", "on $T$, then $h$ factors through $Z$ and", "$$", "W \\times_{X_{/T}} Z = X' \\times_X Z \\longrightarrow Z", "$$", "is separated by assumption (and for example", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-separated-local})", "which implies we get the lifting property by", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-check-separated-dvr}", "applied to the displayed arrow. Case III: $h(\\Spec(K))$ is not in $T$", "but $h$ maps the closed point of $\\Spec(R)$ into $T$. In this case", "the corresponding morphism", "$$", "p_{/T} :", "\\text{Spf}(R)", "\\longrightarrow", "(X' \\times_X X')_{/T} =", "W \\times_{X_{/T}} W", "$$", "is an adic morphism (by", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-map-completions-representable} and", "Definition \\ref{formal-spaces-definition-adic-morphism}).", "Hence our assumption that", "$\\Delta_g : W \\to W \\times_{X_{/T}} W$ is rig-surjective implies we can lift", "$p_{/T}$ to a morphism $\\text{Spf}(R) \\to W = X'_{/T}$, see", "Lemma \\ref{lemma-monomorphism-rig-surjective}.", "Algebraizing the composition $\\text{Spf}(R) \\to X'$ using", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-map-into-algebraic-space}", "we find a morphism $\\Spec(R) \\to X'$ lifting $p$ as desired." ], "refs": [ "restricted-lemma-quasi-separated-gives-quasi-separated", "restricted-lemma-separated-mono-open-diagonal-rig-surjective", "restricted-lemma-output-quasi-compact", "spaces-morphisms-lemma-base-change-separated", "formal-spaces-lemma-base-change-separated", "restricted-lemma-base-change-rig-surjective", "restricted-lemma-output-quasi-separated", "spaces-cohomology-lemma-check-separated-dvr", "spaces-cohomology-remark-variant", "formal-spaces-lemma-separated-local", "spaces-cohomology-lemma-check-separated-dvr", "formal-spaces-lemma-map-completions-representable", "formal-spaces-definition-adic-morphism", "restricted-lemma-monomorphism-rig-surjective", "formal-spaces-lemma-map-into-algebraic-space" ], "ref_ids": [ 2405, 2409, 2422, 4714, 3955, 2386, 2423, 11328, 11349, 3958, 11328, 3885, 3988, 2393, 3964 ] } ], "ref_ids": [ 2292 ] }, { "id": 2425, "type": "theorem", "label": "restricted-lemma-output-proper", "categories": [ "restricted" ], "title": "restricted-lemma-output-proper", "contents": [ "With assumptions and notation as in Theorem \\ref{theorem-dilatations-general}", "let $f : X' \\to X$ correspond to $g : W \\to X_{/T}$.", "Then $f$ is proper if and only if $g$ is a formal modification", "(Definition \\ref{definition-formal-modification})." ], "refs": [ "restricted-theorem-dilatations-general", "restricted-definition-formal-modification" ], "proofs": [ { "contents": [ "If $f$ is proper, then $g$ is a formal modification by", "Lemma \\ref{lemma-modification-gives-formal-modification}.", "Assume $g$ is a formal modification. By", "Lemmas \\ref{lemma-output-quasi-compact} and \\ref{lemma-output-separated}", "we see that $f$ is quasi-compact and separated.", "\\medskip\\noindent", "By Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-check-proper-dvr}", "and Remark \\ref{spaces-cohomology-remark-variant}", "it suffices to show that given any commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X' \\ar[d]^f \\\\", "\\Spec(R) \\ar[r]^p \\ar@{-->}[ru] & X", "}", "$$", "where $R$ is a complete discrete valuation ring with fraction field $K$,", "there is a dotted arrow making the diagram commute.", "There are three cases:", "Case I: $p(\\Spec(R)) \\subset U$. This case is trivial", "because $U' \\to U$ is an isomorphism.", "Case II: $p$ maps $\\Spec(R)$ into $T$. This case follows", "from our assumption that $g : W \\to X_{/T}$ is proper. Namely,", "if $Z$ denotes the reduced induced closed subspace structure", "on $T$, then $p$ factors through $Z$ and", "$$", "W \\times_{X_{/T}} Z = X' \\times_X Z \\longrightarrow Z", "$$", "is proper by assumption which implies we get the lifting property by", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-check-proper-dvr}", "applied to the displayed arrow. Case III: $p(\\Spec(K))$ is not in $T$", "but $p$ maps the closed point of $\\Spec(R)$ into $T$. In this case", "the corresponding morphism", "$$", "p_{/T} : \\text{Spf}(R) \\longrightarrow X'_{/T} = W", "$$", "is an adic morphism (by", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-map-completions-representable} and", "Definition \\ref{formal-spaces-definition-adic-morphism}).", "Hence our assumption that $g : W \\to X_{/T}$ be rig-surjective", "implies we can lift $g_{/T}$ to a morphism", "$\\text{Spf}(R') \\to W = X'_{/T}$", "for some extension of complete discrete valuation rings $R \\subset R'$.", "Algebraizing the composition $\\text{Spf}(R') \\to X'$ using", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-map-into-algebraic-space}", "we find a morphism $\\Spec(R') \\to X'$ lifting $p$ as desired." ], "refs": [ "restricted-lemma-modification-gives-formal-modification", "restricted-lemma-output-quasi-compact", "restricted-lemma-output-separated", "spaces-cohomology-lemma-check-proper-dvr", "spaces-cohomology-remark-variant", "spaces-cohomology-lemma-check-proper-dvr", "formal-spaces-lemma-map-completions-representable", "formal-spaces-definition-adic-morphism", "formal-spaces-lemma-map-into-algebraic-space" ], "ref_ids": [ 2410, 2422, 2424, 11329, 11349, 11329, 3885, 3988, 3964 ] } ], "ref_ids": [ 2292, 2447 ] }, { "id": 2426, "type": "theorem", "label": "restricted-lemma-output-etale", "categories": [ "restricted" ], "title": "restricted-lemma-output-etale", "contents": [ "With assumptions and notation as in Theorem \\ref{theorem-dilatations-general}", "let $f : X' \\to X$ correspond to $g : W \\to X_{/T}$.", "Then $f$ is \\'etale if and only if $g$ is \\'etale." ], "refs": [ "restricted-theorem-dilatations-general" ], "proofs": [ { "contents": [ "If $f$ is \\'etale, then $g$ is \\'etale by", "Lemma \\ref{lemma-map-completions-etale}.", "Conversely, assume $g$ is \\'etale.", "Since $f$ is an isomorphism over $U$ we see that $f$ is \\'etale", "over $U$. Thus it suffices to prove that $f$ is \\'etale", "at any point of $X'$ lying over $T$. Denote $Z \\subset X$", "the reduced closed subspace whose underlying topological", "space is $|Z| = T \\subset |X|$, see", "Properties of Spaces, Definition", "\\ref{spaces-properties-definition-reduced-induced-space}.", "Letting $Z_n \\subset X$ be the $n$th infinitesimal neighbourhood", "we have $X_{/T} = \\colim Z_n$. Since $X'_{/T} = W \\to X_{/T}$", "we conclude that $f^{-1}(Z_n) = X' \\times_X Z_n \\to Z_n$", "is \\'etale by the assumed \\'etaleness of $g$.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds}", "we conclude that $f$ is \\'etale at points lying over $T$." ], "refs": [ "restricted-lemma-map-completions-etale", "spaces-properties-definition-reduced-induced-space", "spaces-more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds" ], "ref_ids": [ 2401, 11932, 121 ] } ], "ref_ids": [ 2292 ] }, { "id": 2427, "type": "theorem", "label": "restricted-lemma-formal-modifications-locally-algebraic", "categories": [ "restricted" ], "title": "restricted-lemma-formal-modifications-locally-algebraic", "contents": [ "Let $S$ be a scheme. Let $\\mathfrak X' \\to \\mathfrak X$", "be a formal modification (Definition \\ref{definition-formal-modification})", "of locally Noetherian formal algebraic spaces over $S$. Given", "\\begin{enumerate}", "\\item any adic Noetherian topological ring $A$,", "\\item any adic morphism $\\text{Spf}(A) \\longrightarrow \\mathfrak X$", "\\end{enumerate}", "there exists a proper morphism $X \\to \\Spec(A)$ of algebraic spaces", "and an isomorphism", "$$", "\\text{Spf}(A) \\times_{\\mathfrak X} \\mathfrak X'", "\\longrightarrow", "X_{/Z}", "$$", "over $\\text{Spf}(A)$ of the base change of $\\mathfrak X$", "with the formal completion of $X$ along the ``closed fibre''", "$Z = X \\times_{\\Spec(A)} \\text{Spf}(A)_{red}$ of $X$ over $A$." ], "refs": [ "restricted-definition-formal-modification" ], "proofs": [ { "contents": [ "The morphism $\\text{Spf}(A) \\times_{\\mathfrak X} \\mathfrak X'", "\\to \\text{Spf}(A)$ is a formal modification by", "Lemma \\ref{lemma-base-change-formal-modification}.", "Hence this follows from Theorem \\ref{theorem-dilatations}." ], "refs": [ "restricted-lemma-base-change-formal-modification", "restricted-theorem-dilatations" ], "ref_ids": [ 2411, 2293 ] } ], "ref_ids": [ 2447 ] }, { "id": 2428, "type": "theorem", "label": "restricted-lemma-Noetherian-local-ring", "categories": [ "restricted" ], "title": "restricted-lemma-Noetherian-local-ring", "contents": [ "Let $A \\to B$ be a ring homomorphism of Noetherian rings inducing an", "isomorphism on $I$-adic completions for some ideal $I \\subset A$", "(for example if $B$ is the $I$-adic completion of $A$).", "Then base change defines an equivalence of categories between the", "category (\\ref{equation-modification}) for $(A, I)$", "with the category (\\ref{equation-modification}) for $(B, IB)$." ], "refs": [], "proofs": [ { "contents": [ "Set $X = \\Spec(A)$ and $T = V(I)$.", "Set $X_1 = \\Spec(B)$ and $T_1 = V(IB)$.", "By Theorem \\ref{theorem-dilatations-general} (in fact we only need", "the affine case treated in Lemma \\ref{lemma-dilatations-affine})", "the category (\\ref{equation-modification}) for $X$ and $T$", "is equivalent to the the category of rig-\\'etale morphisms", "$W \\to X_{/T}$ of locally Noetherian formal algebraic spaces.", "Similarly, the the category (\\ref{equation-modification})", "for $X_1$ and $T_1$ is equivalent to the category of rig-\\'etale", "morphisms $W_1 \\to X_{1, /T_1}$ of locally Noetherian formal", "algebraic spaces. Since $X_{/T} = \\text{Spf}(A^\\wedge)$", "and $X_{1, /T_1} = \\text{Spf}(B^\\wedge)$ (Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-affine-formal-completion-types}) we see that", "these categories are equivalent by our assumption that", "$A^\\wedge \\to B^\\wedge$ is an isomorphism. We omit the verification", "that this equivalence is given by base change." ], "refs": [ "restricted-theorem-dilatations-general", "restricted-lemma-dilatations-affine", "formal-spaces-lemma-affine-formal-completion-types" ], "ref_ids": [ 2292, 2421, 3887 ] } ], "ref_ids": [] }, { "id": 2429, "type": "theorem", "label": "restricted-lemma-Noetherian-local-ring-properties", "categories": [ "restricted" ], "title": "restricted-lemma-Noetherian-local-ring-properties", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-Noetherian-local-ring}.", "Let $f : X' \\to \\Spec(A)$ correspond to $g : Y' \\to \\Spec(B)$", "via the equivalence. Then $f$ is quasi-compact, quasi-separated, separated,", "proper, finite, and add more here if and only if $g$ is so." ], "refs": [ "restricted-lemma-Noetherian-local-ring" ], "proofs": [ { "contents": [ "You can deduce this for the statements", "quasi-compact, quasi-separated, separated, and proper", "by using Lemmas \\ref{lemma-output-quasi-compact}", "\\ref{lemma-output-quasi-separated},", "\\ref{lemma-output-separated},", "\\ref{lemma-output-quasi-separated}, and", "\\ref{lemma-output-proper}", "to translate the corresponding property into a property", "of the formal completion and using the argument of the proof", "of Lemma \\ref{lemma-Noetherian-local-ring}.", "However, there is a direct argument using fpqc descent as follows.", "First, you can reduce to proving the lemma for $A \\to A^\\wedge$", "and $B \\to B^\\wedge$ since $A^\\wedge \\to B^\\wedge$ is an isomorphism.", "Then note that $\\{U \\to \\Spec(A), \\Spec(A^\\wedge) \\to \\Spec(A)\\}$ is an", "fpqc covering with $U = \\Spec(A) \\setminus V(I)$ as before.", "The base change of $f$ by $U \\to \\Spec(A)$ is $\\text{id}_U$", "by definition of our category (\\ref{equation-modification}).", "Let $P$ be a property of morphisms of algebraic spaces which", "is fpqc local on the base (Descent on Spaces, Definition", "\\ref{spaces-descent-definition-property-morphisms-local})", "such that $P$ holds for identity morphisms.", "Then we see that $P$ holds for $f$ if and only if $P$ holds for $g$.", "This applies to $P$ equal to", "quasi-compact, quasi-separated, separated, proper, and finite", "by", "Descent on Spaces, Lemmas", "\\ref{spaces-descent-lemma-descending-property-quasi-compact},", "\\ref{spaces-descent-lemma-descending-property-quasi-separated},", "\\ref{spaces-descent-lemma-descending-property-separated},", "\\ref{spaces-descent-lemma-descending-property-proper}, and", "\\ref{spaces-descent-lemma-descending-property-finite}." ], "refs": [ "restricted-lemma-output-quasi-compact", "restricted-lemma-output-quasi-separated", "restricted-lemma-output-separated", "restricted-lemma-output-quasi-separated", "restricted-lemma-output-proper", "restricted-lemma-Noetherian-local-ring", "spaces-descent-definition-property-morphisms-local", "spaces-descent-lemma-descending-property-quasi-compact", "spaces-descent-lemma-descending-property-quasi-separated", "spaces-descent-lemma-descending-property-separated", "spaces-descent-lemma-descending-property-proper", "spaces-descent-lemma-descending-property-finite" ], "ref_ids": [ 2422, 2423, 2424, 2423, 2425, 2428, 9440, 9381, 9382, 9398, 9399, 9403 ] } ], "ref_ids": [ 2428 ] }, { "id": 2430, "type": "theorem", "label": "restricted-lemma-equivalence-to-completion", "categories": [ "restricted" ], "title": "restricted-lemma-equivalence-to-completion", "contents": [ "Let $A \\to B$ be a local map of local Noetherian rings such that", "\\begin{enumerate}", "\\item $A \\to B$ is flat,", "\\item $\\mathfrak m_B = \\mathfrak m_A B$, and", "\\item $\\kappa(\\mathfrak m_A) = \\kappa(\\mathfrak m_B)$", "\\end{enumerate}", "Then the base change functor from the category", "(\\ref{equation-modification}) for $(A, \\mathfrak m_A)$ to the category", "(\\ref{equation-modification}) for $(B, \\mathfrak m_B)$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "The conditions signify that $A \\to B$ induces an isomorphism on", "completions, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-unramified}.", "Hence this lemma is a special case of", "Lemma \\ref{lemma-Noetherian-local-ring}." ], "refs": [ "more-algebra-lemma-flat-unramified", "restricted-lemma-Noetherian-local-ring" ], "ref_ids": [ 10050, 2428 ] } ], "ref_ids": [] }, { "id": 2431, "type": "theorem", "label": "restricted-lemma-dominate-by-admissible-blowup", "categories": [ "restricted" ], "title": "restricted-lemma-dominate-by-admissible-blowup", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Let $f : X \\to S$ be an object of (\\ref{equation-modification}).", "Then there exists a $U$-admissible blowup $S' \\to S$", "which dominates $X$." ], "refs": [], "proofs": [ { "contents": [ "Special case of More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-dominate-modification-by-blowup}." ], "refs": [ "spaces-more-morphisms-lemma-dominate-modification-by-blowup" ], "ref_ids": [ 193 ] } ], "ref_ids": [] }, { "id": 2432, "type": "theorem", "label": "restricted-proposition-approximate", "categories": [ "restricted" ], "title": "restricted-proposition-approximate", "contents": [ "Let $I$ be an ideal of a Noetherian G-ring $A$. Let $B$ be an", "object of (\\ref{equation-C-prime}). If $B$ is rig-smooth", "over $(A, I)$, then there exists a finite type $A$-algebra", "$C$ and an isomorphism $B \\cong C^\\wedge$ of $A$-algebras." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $B = A[x_1, \\ldots, x_r]^\\wedge/J$. Write", "$P = A[x_1, \\ldots, x_r]^\\wedge$. Choose generators $g_1, \\ldots, g_m \\in J$.", "Choose generators $k_1, \\ldots, k_t$ of the module", "of relations between $g_1, \\ldots, g_m$, i.e., such that", "$$", "P^{\\oplus t} \\xrightarrow{k_1, \\ldots, k_t}", "P^{\\oplus m} \\xrightarrow{g_1, \\ldots, g_m}", "P \\to B \\to 0", "$$", "is a resolution. Write $k_i = (k_{i1}, \\ldots, k_{im})$ so that we have", "\\begin{equation}", "\\label{equation-relations-straight-up}", "\\sum\\nolimits_j k_{ij}g_j = 0", "\\end{equation}", "for $i = 1, \\ldots, t$. Denote $K = (k_{ij})$ the $m \\times t$-matrix", "with entries $k_{ij}$.", "\\medskip\\noindent", "Let $A[x_1, \\ldots, x_r]^h$ be the henselization of the", "pair $(A[x_1, \\ldots, x_r], IA[x_1, \\ldots, x_r])$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization}.", "We may and do think of $A[x_1, \\ldots, x_r]^h$ as a subring", "of $P = A[x_1, \\ldots, x_r]^\\wedge$, see", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-henselization-Noetherian-pair}.", "Since $A$ is a Noetherian G-ring, so is $A[x_1, \\ldots, x_r]$, see", "More on Algebra, Proposition", "\\ref{more-algebra-proposition-finite-type-over-G-ring}.", "Hence we have approximation for the map", "$A[x_1, \\ldots, x_r]^h \\to A[x_1, \\ldots, x_r]^\\wedge = P$", "with respect to the ideal generated by $I$, see", "Smoothing Ring Maps, Lemma \\ref{smoothing-lemma-henselian-pair}.", "Choose a large enough integer $n$ as in Lemma \\ref{lemma-close-enough}.", "By the approximation property we may choose", "$g'_1, \\ldots, g'_m \\in A[x_1, \\ldots, x_r]^h$", "and a matrix", "$K' = (k'_{ij}) \\in \\text{Mat}(m \\times t, A[x_1, \\ldots, x_r]^h)$", "such that $\\sum\\nolimits_j k'_{ij}g'_j = 0$ in", "$A[x_1, \\ldots, x_r]^h$ and such that", "$g_i - g'_i \\in I^nP$ and $K - K' \\in I^n\\text{Mat}(m \\times t, P)$.", "By our choice of $n$ we conclude that there is an isomorphism", "$$", "B \\to P/(g'_1, \\ldots, g'_m) =", "\\left(A[x_1, \\ldots, x_r]^h/(g'_1, \\ldots, g'_m)\\right)^\\wedge", "$$", "This finishes the proof by Lemma \\ref{lemma-algebraize-easy}." ], "refs": [ "more-algebra-lemma-henselization", "more-algebra-lemma-henselization-Noetherian-pair", "more-algebra-proposition-finite-type-over-G-ring", "smoothing-lemma-henselian-pair", "restricted-lemma-close-enough", "restricted-lemma-algebraize-easy" ], "ref_ids": [ 9871, 9874, 10581, 5644, 2309, 2310 ] } ], "ref_ids": [] }, { "id": 2433, "type": "theorem", "label": "restricted-proposition-glue-modification", "categories": [ "restricted" ], "title": "restricted-proposition-glue-modification", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.", "Let $T \\subset |X|$ be a closed subset with complementary open subspace", "$U \\subset X$. Let $f : X' \\to X$ be a proper morphism", "of algebraic spaces such that $f^{-1}(U) \\to U$ is an isomorphism.", "For any algebraic space $W$ over $S$ the map", "$$", "\\Mor_S(X, W) \\longrightarrow", "\\Mor_S(X', W) \\times_{\\Mor_S(X'_{/T}, W)} \\Mor_S(X_{/T}, W)", "$$", "is bijective." ], "refs": [], "proofs": [ { "contents": [ "Let $w' : X' \\to W$ and $\\hat w : X_{/T} \\to W$ be morphisms", "which determine the same morphism $X'_{/T} \\to W$ by composition", "with $X'_{/T} \\to X$ and $X'_{/T} \\to X_{/T}$. We have to prove", "there exists a unique morphism $w : X \\to W$ whose composition", "with $X' \\to X$ and $X_{/T} \\to X$ recovers $w'$ and $\\hat w$.", "The uniqueness is immediate from Lemma \\ref{lemma-faithful-general}.", "\\medskip\\noindent", "The assumptions on $T$ and $f$ are preserved by base change", "by any \\'etale morphism $X_1 \\to X$ of algebraic spaces.", "Since formal algebraic spaces are sheaves for the", "\\'etale topology and since we aready have the uniqueness,", "it suffices to prove existence after replacing $X$ by", "the members of an \\'etale covering. Thus we may assume $X$ is", "an affine Noetherian scheme.", "\\medskip\\noindent", "Assume $X$ is an affine Noetherian scheme. We will construct", "the morphism $w : X \\to W$ using the material in", "Pushouts of Spaces, Section \\ref{spaces-pushouts-section-coequalizer-glue}.", "It makes sense to read a little bit of the material", "in that section before continuing the read the proof.", "\\medskip\\noindent", "Set $X'' = X' \\times_X X'$ and consider the two morphisms", "$a = w' \\circ \\text{pr}_1 : X'' \\to W$ and ", "$b = w' \\circ \\text{pr}_2 : X'' \\to W$.", "Then we see that $a$ and $b$ agree over the open $U$", "and that $a_{/T}$ and $b_{a/T}$ agree (as these are", "both equal to the composition $X''_{/T} \\to X_{/T} \\to W$", "where the second arrow is $\\hat w$).", "Thus by Lemma \\ref{lemma-faithful-general}", "we see $a = b$.", "\\medskip\\noindent", "Denote $Z \\subset X$ the reduced induced closed subscheme structure", "on $T$. For $n \\geq 1$ denote $Z_n \\subset X$ the $n$th infinitesimal", "neighbourhood of $Z$. Denote $w_n = \\hat w|_{Z_n} : Z_n \\to W$", "so that we have $\\hat w = \\colim w_n$ on $X_{/T} = \\colim Z_n$.", "Set $Y_n = X' \\amalg Z_n$. Consider the two projections", "$$", "s_n, t_n : R_n = Y_n \\times_X Y_n \\longrightarrow Y_n", "$$", "Let $Y_n \\to X_n \\to X$ be the coequalizer of $s_n$ and $t_n$ as in", "Pushouts of Spaces, Section \\ref{spaces-pushouts-section-coequalizer-glue}", "(in particular this coequalizer exists, has good properties, etc, see", "Pushouts of Spaces, Lemma \\ref{spaces-pushouts-lemma-coequalizer}).", "By the result $a = b$ of the previous parapgraph and the agreement", "of $w'$ and $\\hat w$ over $X'_{/T}$ we see that the morphism", "$$", "w' \\amalg w_n : Y_n \\longrightarrow W", "$$", "equalizes the morphisms $s_n$ and $t_n$. Hence we see that for all $n \\geq 1$", "there is a morphism $w^n : X_n \\to W$ compatible with $w'$ and $w_n$.", "Moreover, for $m \\geq 1$ the composition", "$$", "X_n \\to X_{n + m} \\xrightarrow{w^{n + m}} W", "$$", "is equal to $w^n$ by construction (as the corresponding statement holds", "for $w' \\amalg w_{n + m}$ and $w' \\amalg w_n$). By", "Pushouts of Spaces, Lemma", "\\ref{spaces-pushouts-lemma-essentially-constant} and", "Remark \\ref{spaces-pushouts-remark-essentially-constant}", "the system of algebraic spaces $X_n$ is essentially constant", "with value $X$ and we conclude." ], "refs": [ "restricted-lemma-faithful-general", "restricted-lemma-faithful-general", "spaces-pushouts-lemma-coequalizer", "spaces-pushouts-lemma-essentially-constant", "spaces-pushouts-remark-essentially-constant" ], "ref_ids": [ 2415, 2415, 10887, 10889, 10899 ] } ], "ref_ids": [] }, { "id": 2457, "type": "theorem", "label": "more-groupoids-lemma-sheaf-differentials", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-sheaf-differentials", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "The sheaf of differentials of $R$ seen as a scheme over", "$U$ via $t$ is a quotient of the pullback via $t$ of the conormal sheaf of", "the immersion $e : U \\to R$. In a formula: there is a canonical surjection", "$t^*\\mathcal{C}_{U/R} \\to \\Omega_{R/U}$. If $s$ is flat, then", "this map is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Note that $e : U \\to R$ is an immersion as it is a section", "of the morphism $s$, see", "Schemes, Lemma \\ref{schemes-lemma-section-immersion}.", "Consider the following diagram", "$$", "\\xymatrix{", "R \\ar[r]_-{(1, i)} \\ar[d]_t &", "R \\times_{s, U, t} R \\ar[d]^c \\ar[rr]_{(\\text{pr}_0, i \\circ \\text{pr}_1)} & &", "R \\times_{t, U, t} R \\\\", "U \\ar[r]^e &", "R", "}", "$$", "The square on the left is cartesian, because if $a \\circ b = e$, then", "$b = i(a)$. The composition of the horizontal maps is the diagonal", "morphism of $t : R \\to U$. The right top horizontal arrow is an", "isomorphism. Hence since $\\Omega_{R/U}$ is the conormal sheaf of the", "composition it is isomorphic to the conormal sheaf of", "$(1, i)$. By", "Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial-flat}", "we get the surjection $t^*\\mathcal{C}_{U/R} \\to \\Omega_{R/U}$", "and if $c$ is flat, then this is an isomorphism. Since $c$ is a base change", "of $s$ by the properties of Diagram (\\ref{equation-pull})", "we conclude that if $s$ is flat, then $c$ is flat, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat}." ], "refs": [ "morphisms-lemma-conormal-functorial-flat", "morphisms-lemma-base-change-flat" ], "ref_ids": [ 5305, 5265 ] } ], "ref_ids": [] }, { "id": 2458, "type": "theorem", "label": "more-groupoids-lemma-first-order-structure-c", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-first-order-structure-c", "contents": [ "The map $I/I^2 \\to J/J^2$ induced by $c$ is the composition", "$$", "I/I^2 \\xrightarrow{(1, 1)} I/I^2 \\oplus I/I^2 \\to J/J^2", "$$", "where the second arrow comes from the equality", "$J = (I \\otimes B + B \\otimes I)C$.", "The map $i : B \\to B$ induces the map $-1 : I/I^2 \\to I/I^2$." ], "refs": [], "proofs": [ { "contents": [ "To describe a local homomorphism from $C$ to another local ring", "it is enough to say what happens to elements of the form", "$b_1 \\otimes b_2$. Keeping this in mind we have the two canonical maps", "$$", "e_2 : C \\to B,\\ b_1 \\otimes b_2 \\mapsto b_1s(e(b_2)),\\quad", "e_1 : C \\to B,\\ b_1 \\otimes b_2 \\mapsto t(e(b_1))b_2", "$$", "corresponding to the embeddings", "$R \\to R \\times_{s, U, t} R$ given by", "$r \\mapsto (r, e(s(r)))$ and $r \\mapsto (e(t(r)), r)$.", "These maps define maps $J/J^2 \\to I/I^2$ which jointly", "give an inverse to the map $I/I^2 \\oplus I/I^2 \\to J/J^2$", "of the lemma. Thus to prove statement we only have to show", "that $e_1 \\circ c : B \\to B$ and $e_2 \\circ c : B \\to B$", "are the identity maps. This follows from the fact that both", "compositions $R \\to R \\times_{s, U, t} R \\to R$ are identities.", "\\medskip\\noindent", "The statement on $i$ follows from the statement on $c$ and the", "fact that $c \\circ (1, i) = e \\circ t$. Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2459, "type": "theorem", "label": "more-groupoids-lemma-local-source", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-local-source", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c, e, i)$ be a groupoid over $S$.", "Let $g : U' \\to U$ be a morphism of schemes.", "Denote $h$ the composition", "$$", "\\xymatrix{", "h : U' \\times_{g, U, t} R \\ar[r]_-{\\text{pr}_1} & R \\ar[r]_s & U.", "}", "$$", "Let $\\mathcal{P}, \\mathcal{Q}, \\mathcal{R}$ be properties of morphisms", "of schemes. Assume", "\\begin{enumerate}", "\\item $\\mathcal{R} \\Rightarrow \\mathcal{Q}$,", "\\item $\\mathcal{Q}$ is preserved under base change and composition,", "\\item for any morphism $f : X \\to Y$ which has $\\mathcal{Q}$ there exists a", "largest open $W(\\mathcal{P}, f) \\subset X$ such that $f|_{W(\\mathcal{P}, f)}$", "has $\\mathcal{P}$, and", "\\item for any morphism $f : X \\to Y$ which has $\\mathcal{Q}$,", "and any morphism $Y' \\to Y$ which has $\\mathcal{R}$ we have", "$Y' \\times_Y W(\\mathcal{P}, f) = W(\\mathcal{P}, f')$, where", "$f' : X_{Y'} \\to Y'$ is the base change of $f$.", "\\end{enumerate}", "If $s, t$ have $\\mathcal{R}$ and $g$ has $\\mathcal{Q}$, then", "there exists an open subscheme $W \\subset U'$ such that", "$W \\times_{g, U, t} R = W(\\mathcal{P}, h)$." ], "refs": [], "proofs": [ { "contents": [ "Note that the following diagram is commutative", "$$", "\\xymatrix{", "U' \\times_{g, U, t} R \\times_{t, U, t} R", "\\ar[rr]_-{\\text{pr}_{12}}", "\\ar@<1ex>[d]^-{\\text{pr}_{02}} \\ar@<-1ex>[d]_-{\\text{pr}_{01}} & &", "R \\times_{t, U, t} R", "\\ar@<1ex>[d]^-{\\text{pr}_1} \\ar@<-1ex>[d]_-{\\text{pr}_0}", "\\\\", "U' \\times_{g, U, t} R \\ar[rr]^{\\text{pr}_1} & & R", "}", "$$", "with both squares cartesian (this uses that the two maps", "$t \\circ \\text{pr}_i : R \\times_{t, U, t} R \\to U$ are equal).", "Combining this with the properties of diagram (\\ref{equation-pull})", "we get a commutative diagram", "$$", "\\xymatrix{", "U' \\times_{g, U, t} R \\times_{t, U, t} R", "\\ar[rr]_-{c \\circ (i, 1)}", "\\ar@<1ex>[d]^-{\\text{pr}_{02}} \\ar@<-1ex>[d]_-{\\text{pr}_{01}} & &", "R", "\\ar@<1ex>[d]^-{s} \\ar@<-1ex>[d]_-{t}", "\\\\", "U' \\times_{g, U, t} R \\ar[rr]^h & & U", "}", "$$", "where both squares are cartesian.", "\\medskip\\noindent", "Assume $s, t$ have $\\mathcal{R}$ and $g$ has $\\mathcal{Q}$.", "Then $h$ has $\\mathcal{Q}$ as a composition of $s$ (which has", "$\\mathcal{R}$ hence $\\mathcal{Q}$) and a base change of $g$ (which", "has $\\mathcal{Q}$). Thus $W(\\mathcal{P}, h) \\subset U' \\times_{g, U, t} R$", "exists. By our assumptions we have", "$\\text{pr}_{01}^{-1}(W(\\mathcal{P}, h)) =", "\\text{pr}_{02}^{-1}(W(\\mathcal{P}, h))$", "since both are the largest open on which $c \\circ (i, 1)$ has $\\mathcal{P}$.", "Note that the projection $U' \\times_{g, U, t} R \\to U'$ has a section, namely", "$\\sigma : U' \\to U' \\times_{g, U, t} R$, $u' \\mapsto (u', e(g(u')))$.", "Also via the isomorphism", "$$", "(U' \\times_{g, U, t} R) \\times_{U'} (U' \\times_{g, U, t} R)", "=", "U' \\times_{g, U, t} R \\times_{t, U, t} R", "$$", "the two projections of the left hand side", "to $U' \\times_{g, U, t} R$ agree with the morphisms $\\text{pr}_{01}$", "and $\\text{pr}_{02}$ on the right hand side. Since", "$\\text{pr}_{01}^{-1}(W(\\mathcal{P}, h)) =", "\\text{pr}_{02}^{-1}(W(\\mathcal{P}, h))$", "we conclude that $W(\\mathcal{P}, h)$ is the inverse image of a subset of $U$,", "which is necessarily the open set", "$W = \\sigma^{-1}(W(\\mathcal{P}, h))$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2460, "type": "theorem", "label": "more-groupoids-lemma-property-invariant", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-property-invariant", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid over $S$.", "Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf,", "\\linebreak[0] \\etale, \\linebreak[0]", "smooth, \\linebreak[0] syntomic\\}$\\footnote{The fact that $fpqc$ is missing", "is not a typo.}. Let $\\mathcal{P}$ be a property of morphisms of schemes", "which is $\\tau$-local on the target", "(Descent, Definition \\ref{descent-definition-property-morphisms-local}).", "Assume $\\{s : R \\to U\\}$ and $\\{t : R \\to U\\}$ are coverings for the", "$\\tau$-topology. Let $W \\subset U$ be the maximal open subscheme such that", "$s|_{s^{-1}(W)} : s^{-1}(W) \\to W$ has property $\\mathcal{P}$.", "Then $W$ is $R$-invariant, see", "Groupoids, Definition \\ref{groupoids-definition-invariant-open}." ], "refs": [ "descent-definition-property-morphisms-local", "groupoids-definition-invariant-open" ], "proofs": [ { "contents": [ "The existence and properties of the open $W \\subset U$ are described in", "Descent, Lemma \\ref{descent-lemma-largest-open-of-the-base}.", "In", "Diagram (\\ref{equation-diagram})", "let $W_1 \\subset R$ be the maximal open subscheme over which the morphism", "$\\text{pr}_1 : R \\times_{s, U, t} R \\to R$ has property $\\mathcal{P}$.", "It follows from the aforementioned", "Descent, Lemma \\ref{descent-lemma-largest-open-of-the-base}", "and the assumption that $\\{s : R \\to U\\}$ and $\\{t : R \\to U\\}$ are coverings", "for the $\\tau$-topology that $t^{-1}(W) = W_1 = s^{-1}(W)$ as desired." ], "refs": [ "descent-lemma-largest-open-of-the-base", "descent-lemma-largest-open-of-the-base" ], "ref_ids": [ 14664, 14664 ] } ], "ref_ids": [ 14772, 9685 ] }, { "id": 2461, "type": "theorem", "label": "more-groupoids-lemma-property-G-invariant", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-property-G-invariant", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid over $S$.", "Let $G \\to U$ be its stabilizer group scheme.", "Let $\\tau \\in \\{fppf, \\linebreak[0] \\etale, \\linebreak[0]", "smooth, \\linebreak[0] syntomic\\}$.", "Let $\\mathcal{P}$ be a property of morphisms which is $\\tau$-local", "on the target. Assume $\\{s : R \\to U\\}$ and $\\{t : R \\to U\\}$ are coverings", "for the $\\tau$-topology. Let $W \\subset U$ be the maximal open subscheme", "such that $G_W \\to W$ has property $\\mathcal{P}$. Then $W$ is $R$-invariant", "(see", "Groupoids, Definition", "\\ref{groupoids-definition-invariant-open})." ], "refs": [ "groupoids-definition-invariant-open" ], "proofs": [ { "contents": [ "The existence and properties of the open $W \\subset U$ are described in", "Descent, Lemma \\ref{descent-lemma-largest-open-of-the-base}.", "The morphism", "$$", "G \\times_{U, t} R \\longrightarrow R \\times_{s, U} G, \\quad", "(g, r) \\longmapsto (r, r^{-1} \\circ g \\circ r)", "$$", "is an isomorphism over $R$ (where $\\circ$ denotes", "composition in the groupoid). Hence $s^{-1}(W) = t^{-1}(W)$ by the", "properties of $W$ proved in the aforementioned", "Descent, Lemma \\ref{descent-lemma-largest-open-of-the-base}." ], "refs": [ "descent-lemma-largest-open-of-the-base", "descent-lemma-largest-open-of-the-base" ], "ref_ids": [ 14664, 14664 ] } ], "ref_ids": [ 9685 ] }, { "id": 2462, "type": "theorem", "label": "more-groupoids-lemma-two-fibres", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-two-fibres", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid over $S$.", "Let $r, r' \\in R$ with $t(r) = t(r')$ in $U$.", "Set $u = s(r)$, $u' = s(r')$.", "Denote $F_u = s^{-1}(u)$ and $F_{u'} = s^{-1}(u')$ the scheme", "theoretic fibres.", "\\begin{enumerate}", "\\item There exists a common field extension", "$\\kappa(u) \\subset k$, $\\kappa(u') \\subset k$ and", "an isomorphism $(F_u)_k \\cong (F_{u'})_k$.", "\\item We may choose the isomorphism of (1) such that a point", "lying over $r$ maps to a point lying over $r'$.", "\\item If the morphisms $s$, $t$ are flat then the morphisms of germs", "$s : (R, r) \\to (U, u)$ and $s : (R, r') \\to (U, u')$ are flat", "locally on the base isomorphic.", "\\item If the morphisms $s$, $t$ are \\'etale", "(resp.\\ smooth, syntomic, or flat and locally of finite presentation)", "then the morphisms of germs $s : (R, r) \\to (U, u)$ and", "$s : (R, r') \\to (U, u')$ are locally on the base isomorphic", "in the \\'etale (resp.\\ smooth, syntomic, or fppf) topology.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We repeatedly use the properties and the existence of", "diagram (\\ref{equation-diagram}).", "By the properties of the diagram (and", "Schemes, Lemma \\ref{schemes-lemma-points-fibre-product})", "there exists a point $\\xi$ of $R \\times_{s, U, t} R$", "with $\\text{pr}_0(\\xi) = r$ and $c(\\xi) = r'$.", "Let $\\tilde r = \\text{pr}_1(\\xi) \\in R$.", "\\medskip\\noindent", "Proof of (1). Set $k = \\kappa(\\tilde r)$. Since $t(\\tilde r) = u$", "and $s(\\tilde r) = u'$ we see that $k$ is a common extension", "of both $\\kappa(u)$ and $\\kappa(u')$ and in fact that", "both $(F_u)_k$ and $(F_{u'})_k$ are isomorphic to the fibre of", "$\\text{pr}_1 : R \\times_{s, U, t} R \\to R$ over $\\tilde r$.", "Hence (1) is proved.", "\\medskip\\noindent", "Part (2) follows since the point $\\xi$ maps to $r$, resp.\\ $r'$.", "\\medskip\\noindent", "Part (3) is clear from the above (using the point $\\xi$ for", "$\\tilde u$ and $\\tilde u'$) and the definitions.", "\\medskip\\noindent", "If $s$ and $t$ are flat and of finite presentation, then", "they are open morphisms (Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}).", "Hence the image of some affine open neighbourhood $V''$ of $\\tilde r$ will", "cover an open neighbourhood $V$ of $u$, resp.\\ $V'$ of $u'$.", "These can be used to show that properties (1) and (2) of the", "definition of ``locally on the base isomorphic in the", "$\\tau$-topology''." ], "refs": [ "schemes-lemma-points-fibre-product", "morphisms-lemma-fppf-open" ], "ref_ids": [ 7693, 5267 ] } ], "ref_ids": [] }, { "id": 2463, "type": "theorem", "label": "more-groupoids-lemma-make-CM", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-make-CM", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid over $S$.", "Assume $s$ and $t$ are flat and locally of finite presentation.", "Then there exists an open $U' \\subset U$ such that", "\\begin{enumerate}", "\\item $t^{-1}(U') \\subset R$ is the largest open subscheme of", "$R$ on which the morphism $s$ is Cohen-Macaulay,", "\\item $s^{-1}(U') \\subset R$ is the largest open subscheme of", "$R$ on which the morphism $t$ is Cohen-Macaulay,", "\\item the morphism $t|_{s^{-1}(U')} : s^{-1}(U') \\to U$ is", "surjective,", "\\item the morphism $s|_{t^{-1}(U')} : t^{-1}(U') \\to U$ is", "surjective, and", "\\item the restriction $R' = s^{-1}(U') \\cap t^{-1}(U')$", "of $R$ to $U'$ defines a groupoid $(U', R', s', t', c')$ which has the property", "that the morphisms $s'$ and $t'$ are Cohen-Macaulay and locally of", "finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Apply", "Lemma \\ref{lemma-local-source}", "with", "$g = \\text{id}$ and", "$\\mathcal{Q} =$``locally of finite presentation'',", "$\\mathcal{R} =$``flat and locally of finite presentation'', and", "$\\mathcal{P}=$``Cohen-Macaulay'', see", "Remark \\ref{remark-local-source-apply}.", "This gives us an open $U' \\subset U$ such that", "Let $t^{-1}(U') \\subset R$ is the largest open subscheme of $R$", "on which the morphism $s$ is Cohen-Macaulay.", "This proves (1).", "Let $i : R \\to R$ be the inverse of the groupoid.", "Since $i$ is an isomorphism, and $s \\circ i = t$ and $t \\circ i = s$", "we see that $s^{-1}(U')$ is also the largest open of $R$ on which $t$ is", "Cohen-Macaulay. This proves (2).", "By", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-flat-finite-presentation-CM-open}", "the open subset $t^{-1}(U')$ is dense in every fibre of $s : R \\to U$.", "This proves (3). Same argument for (4).", "Part (5) is a formal consequence of (1) and (2) and the discussion", "of restrictions in", "Groupoids, Section \\ref{groupoids-section-restrict-groupoid}." ], "refs": [ "more-groupoids-lemma-local-source", "more-groupoids-remark-local-source-apply", "more-morphisms-lemma-flat-finite-presentation-CM-open" ], "ref_ids": [ 2459, 2507, 13789 ] } ], "ref_ids": [] }, { "id": 2464, "type": "theorem", "label": "more-groupoids-lemma-restrict-preserves-type", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-restrict-preserves-type", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Let $g : U' \\to U$ be a morphism of schemes.", "Let $(U', R', s', t', c')$ be the restriction of", "$(U, R, s, t, c)$ via $g$.", "\\begin{enumerate}", "\\item If $s, t$ are locally of finite type and $g$ is locally of finite", "type, then $s', t'$ are locally of finite type.", "\\item If $s, t$ are locally of finite presentation and $g$ is locally of finite", "presentation, then $s', t'$ are locally of finite presentation.", "\\item If $s, t$ are flat and $g$ is flat, then $s', t'$ are flat.", "\\item Add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The property of being locally of finite type is stable under composition", "and arbitrary base change, see", "Morphisms, Lemmas \\ref{morphisms-lemma-composition-finite-type} and", "\\ref{morphisms-lemma-base-change-finite-type}.", "Hence (1) is clear from Diagram (\\ref{equation-restriction}).", "For the other cases, see", "Morphisms, Lemmas \\ref{morphisms-lemma-composition-finite-presentation},", "\\ref{morphisms-lemma-base-change-finite-presentation},", "\\ref{morphisms-lemma-composition-flat}, and", "\\ref{morphisms-lemma-base-change-flat}." ], "refs": [ "morphisms-lemma-composition-finite-type", "morphisms-lemma-base-change-finite-type", "morphisms-lemma-composition-finite-presentation", "morphisms-lemma-base-change-finite-presentation", "morphisms-lemma-composition-flat", "morphisms-lemma-base-change-flat" ], "ref_ids": [ 5199, 5200, 5239, 5240, 5263, 5265 ] } ], "ref_ids": [] }, { "id": 2465, "type": "theorem", "label": "more-groupoids-lemma-restrict-property", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-restrict-property", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Let $g : U' \\to U$ be a morphism of schemes.", "Let $(U', R', s', t', c')$ be the restriction of", "$(U, R, s, t, c)$ via $g$, and let", "$h = s \\circ \\text{pr}_1 : U' \\times_{g, U, t} R \\to U$. If", "$\\mathcal{P}$ is a property of morphisms of schemes such that", "\\begin{enumerate}", "\\item $h$ has property $\\mathcal{P}$, and", "\\item $\\mathcal{P}$ is preserved under base change,", "\\end{enumerate}", "then $s', t'$ have property $\\mathcal{P}$." ], "refs": [], "proofs": [ { "contents": [ "This is clear as $s'$ is the base change of $h$ by", "Diagram (\\ref{equation-restriction})", "and $t'$ is isomorphic to $s'$ as a morphism of schemes." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2466, "type": "theorem", "label": "more-groupoids-lemma-double-restrict", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-double-restrict", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Let $g : U' \\to U$ and $g' : U'' \\to U'$ be morphisms of schemes.", "Set $g'' = g \\circ g'$.", "Let $(U', R', s', t', c')$ be the restriction of $R$ to $U'$.", "Let $h = s \\circ \\text{pr}_1 : U' \\times_{g, U, t} R \\to U$,", "let $h' = s' \\circ \\text{pr}_1 : U'' \\times_{g', U', t} R \\to U'$, and", "let $h'' = s \\circ \\text{pr}_1 : U'' \\times_{g'', U, t} R \\to U$.", "The following diagram is commutative", "$$", "\\xymatrix{", "U'' \\times_{g', U', t} R' \\ar[d]^{h'} &", "(U' \\times_{g, U, t} R) \\times_U (U'' \\times_{g'', U, t} R)", "\\ar[l] \\ar[r] \\ar[d] &", "U'' \\times_{g'', U, t} R \\ar[d]_{h''} \\\\", "U' &", "U' \\times_{g, U, t} R \\ar[l]_{\\text{pr}_0} \\ar[r]^h &", "U", "}", "$$", "with both squares cartesian where the left upper horizontal arrow", "is given by the rule", "$$", "\\begin{matrix}", "(U' \\times_{g, U, t} R) \\times_U (U'' \\times_{g'', U, t} R) &", "\\longrightarrow &", "U'' \\times_{g', U', t} R' \\\\", "((u', r_0), (u'', r_1)) &", "\\longmapsto &", "(u'', (c(r_1, i(r_0)), (g'(u''), u')))", "\\end{matrix}", "$$", "with notation as explained in the proof." ], "refs": [], "proofs": [ { "contents": [ "We work this out by exploiting the functorial point of view", "and reducing the lemma to a statement on arrows in restrictions", "of a groupoid category. In the last formula of the lemma the", "notation $((u', r_0), (u'', r_1))$ indicates a $T$-valued point of", "$(U' \\times_{g, U, t} R) \\times_U (U'' \\times_{g'', U, t} R)$.", "This means that $u', u'', r_0, r_1$ are $T$-valued points of $U', U'', R, R$", "and that $g(u') = t(r_0)$, $g(g'(u'')) = g''(u'') = t(r_1)$, and", "$s(r_0) = s(r_1)$. It would be more correct here to write", "$g \\circ u' = t \\circ r_0$ and so on but this makes the notation", "even more unreadable. If we think of $r_1$ and $r_0$ as arrows in", "a groupoid category then we can represent this by the picture", "$$", "\\xymatrix{", "t(r_0) = g(u') &", "s(r_0) = s(r_1) \\ar[l]_{r_0} \\ar[r]^-{r_1} &", "t(r_1) = g(g'(u''))", "}", "$$", "This diagram in particular demonstrates that the composition", "$c(r_1, i(r_0))$ makes sense. Recall that", "$$", "R' = R \\times_{(t, s), U \\times_S U, g \\times g} U' \\times_S U'", "$$", "hence a $T$-valued point of $R'$ looks like $(r, (u'_0, u'_1))$", "with $t(r) = g(u'_0)$ and $s(r) = g(u'_1)$. In particular given", "$((u', r_0), (u'', r_1))$ as above we get the $T$-valued point", "$(c(r_1, i(r_0)), (g'(u''), u'))$ of $R'$ because we have", "$t(c(r_1, i(r_0))) = t(r_1) = g(g'(u''))$ and", "$s(c(r_1, i(r_0))) = s(i(r_0)) = t(r_0) = g(u')$.", "We leave it to the reader to show that the left square commutes", "with this definition.", "\\medskip\\noindent", "To show that the left square is cartesian,", "suppose we are given $(v'', p')$ and $(v', p)$ which are $T$-valued points of", "$U'' \\times_{g', U', t} R'$ and $U' \\times_{g, U, t} R$ with", "$v' = s'(p')$. This also means that $g'(v'') = t'(p')$ and", "$g(v') = t(p)$. By the discussion above we know that we can write", "$p' = (r, (u_0', u_1'))$ with $t(r) = g(u'_0)$ and", "$s(r) = g(u'_1)$. Using this notation we see that", "$v' = s'(p') = u_1'$ and", "$g'(v'') = t'(p') = u_0'$. Here is a picture", "$$", "\\xymatrix{", "s(p) \\ar[r]^-p &", "g(v') = g(u'_1) \\ar[r]^-r &", "g(u'_0) = g(g'(v''))", "}", "$$", "What we have to show is that there exists a unique $T$-valued point", "$((u', r_0), (u'', r_1))$ as above such that", "$v' = u'$, $p = r_0$, $v'' = u''$ and $p' = (c(r_1, i(r_0)), (g'(u''), u'))$.", "Comparing the two diagrams above it is clear that we have no choice", "but to take", "$$", "((u', r_0), (u'', r_1)) = ((v', p), (v'', c(r, p))", "$$", "Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2467, "type": "theorem", "label": "more-groupoids-lemma-double-restrict-property", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-double-restrict-property", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Let $g : U' \\to U$ and $g' : U'' \\to U'$ be morphisms of schemes.", "Set $g'' = g \\circ g'$.", "Let $(U', R', s', t', c')$ be the restriction of $R$ to $U'$.", "Let $h = s \\circ \\text{pr}_1 : U' \\times_{g, U, t} R \\to U$,", "let $h' = s' \\circ \\text{pr}_1 : U'' \\times_{g', U', t} R \\to U'$, and", "let $h'' = s \\circ \\text{pr}_1 : U'' \\times_{g'', U, t} R \\to U$.", "Let $\\tau \\in \\{Zariski, \\linebreak[0] \\etale, \\linebreak[0]", "smooth, \\linebreak[0] syntomic, \\linebreak[0] fppf, \\linebreak[0] fpqc\\}$. Let", "$\\mathcal{P}$ be a property of morphisms of schemes", "which is preserved under base change, and which", "is local on the target for the $\\tau$-topology. If", "\\begin{enumerate}", "\\item $h(U' \\times_U R)$ is open in $U$,", "\\item $\\{h : U' \\times_U R \\to h(U' \\times_U R)\\}$ is a $\\tau$-covering,", "\\item $h'$ has property $\\mathcal{P}$,", "\\end{enumerate}", "then $h''$ has property $\\mathcal{P}$. Conversely, if", "\\begin{enumerate}", "\\item[(a)] $\\{t : R \\to U\\}$ is a $\\tau$-covering,", "\\item[(d)] $h''$ has property $\\mathcal{P}$,", "\\end{enumerate}", "then $h'$ has property $\\mathcal{P}$." ], "refs": [], "proofs": [ { "contents": [ "This follows formally from the properties of the diagram of", "Lemma \\ref{lemma-double-restrict}.", "In the first case, note that the image of the morphism", "$h''$ is contained in the image of $h$, as $g'' = g \\circ g'$.", "Hence we may replace the $U$ in the lower right corner of the", "diagram by $h(U' \\times_U R)$. This explains the significance of", "conditions (1) and (2) in the lemma. In the second case, note that", "$\\{\\text{pr}_0 : U' \\times_{g, U, t} R \\to U'\\}$ is a $\\tau$-covering", "as a base change of $\\tau$ and condition (a)." ], "refs": [ "more-groupoids-lemma-double-restrict" ], "ref_ids": [ 2466 ] } ], "ref_ids": [] }, { "id": 2468, "type": "theorem", "label": "more-groupoids-lemma-groupoid-on-field-open-multiplication", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-groupoid-on-field-open-multiplication", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. If $U$ is the spectrum of a field, then the composition", "morphism $c : R \\times_{s, U, t} R \\to R$ is open." ], "refs": [], "proofs": [ { "contents": [ "The composition is isomorphic to the projection map", "$\\text{pr}_1 : R \\times_{t, U, t} R \\to R$ by", "Diagram (\\ref{equation-pull}).", "The projection is open by", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open}." ], "refs": [ "morphisms-lemma-scheme-over-field-universally-open" ], "ref_ids": [ 5254 ] } ], "ref_ids": [] }, { "id": 2469, "type": "theorem", "label": "more-groupoids-lemma-groupoid-on-field-separated", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-groupoid-on-field-separated", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. If $U$ is the spectrum of a field,", "then $R$ is a separated scheme." ], "refs": [], "proofs": [ { "contents": [ "By", "Groupoids, Lemma \\ref{groupoids-lemma-group-scheme-over-field-separated}", "the stabilizer group scheme $G \\to U$ is separated. By", "Groupoids, Lemma \\ref{groupoids-lemma-diagonal}", "the morphism $j = (t, s) : R \\to U \\times_S U$ is separated.", "As $U$ is the spectrum of a field the scheme", "$U \\times_S U$ is affine (by the construction of fibre products in", "Schemes, Section \\ref{schemes-section-fibre-products}).", "Hence $R$ is a separated scheme, see", "Schemes, Lemma \\ref{schemes-lemma-separated-permanence}." ], "refs": [ "groupoids-lemma-group-scheme-over-field-separated", "groupoids-lemma-diagonal", "schemes-lemma-separated-permanence" ], "ref_ids": [ 9589, 9656, 7714 ] } ], "ref_ids": [] }, { "id": 2470, "type": "theorem", "label": "more-groupoids-lemma-groupoid-on-field-homogeneous", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-groupoid-on-field-homogeneous", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. Assume $U = \\Spec(k)$ with $k$ a field.", "For any points $r, r' \\in R$ there exists a field extension", "$k \\subset k'$ and points", "$r_1, r_2 \\in R \\times_{s, \\Spec(k)} \\Spec(k')$", "and a diagram", "$$", "\\xymatrix{", "R &", "R \\times_{s, \\Spec(k)} \\Spec(k')", "\\ar[l]_-{\\text{pr}_0} \\ar[r]^\\varphi &", "R \\times_{s, \\Spec(k)} \\Spec(k')", "\\ar[r]^-{\\text{pr}_0} &", "R", "}", "$$", "such that $\\varphi$ is an isomorphism of schemes over $\\Spec(k')$,", "we have $\\varphi(r_1) = r_2$, $\\text{pr}_0(r_1) = r$, and", "$\\text{pr}_0(r_2) = r'$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-two-fibres}", "parts (1) and (2)." ], "refs": [ "more-groupoids-lemma-two-fibres" ], "ref_ids": [ 2462 ] } ], "ref_ids": [] }, { "id": 2471, "type": "theorem", "label": "more-groupoids-lemma-restrict-groupoid-on-field", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-restrict-groupoid-on-field", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. Assume $U = \\Spec(k)$ with $k$ a field.", "Let $k \\subset k'$ be a field extension, $U' = \\Spec(k')$", "and let $(U', R', s', t', c')$ be the restriction of", "$(U, R, s, t, c)$ via $U' \\to U$. In the defining diagram", "$$", "\\xymatrix{", "R' \\ar[d] \\ar[r] \\ar@/_3pc/[dd]_{t'} \\ar@/^1pc/[rr]^{s'} \\ar@{..>}[rd] &", "R \\times_{s, U} U' \\ar[r] \\ar[d] &", "U' \\ar[d] \\\\", "U' \\times_{U, t} R \\ar[d] \\ar[r] &", "R \\ar[r]^s \\ar[d]_t &", "U \\\\", "U' \\ar[r] &", "U", "}", "$$", "all the morphisms are surjective, flat, and universally open.", "The dotted arrow $R' \\to R$ is in addition affine." ], "refs": [], "proofs": [ { "contents": [ "The morphism $U' \\to U$ equals $\\Spec(k') \\to \\Spec(k)$,", "hence is affine, surjective and flat. The morphisms $s, t : R \\to U$", "and the morphism $U' \\to U$ are universally open by", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open}.", "Since $R$ is not empty and $U$ is the spectrum of a field the morphisms", "$s, t : R \\to U$ are surjective and flat. Then you conclude by using", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-surjective},", "\\ref{morphisms-lemma-composition-surjective},", "\\ref{morphisms-lemma-composition-open},", "\\ref{morphisms-lemma-base-change-affine},", "\\ref{morphisms-lemma-composition-affine},", "\\ref{morphisms-lemma-base-change-flat}, and", "\\ref{morphisms-lemma-composition-flat}." ], "refs": [ "morphisms-lemma-scheme-over-field-universally-open", "morphisms-lemma-base-change-surjective", "morphisms-lemma-composition-surjective", "morphisms-lemma-composition-open", "morphisms-lemma-base-change-affine", "morphisms-lemma-composition-affine", "morphisms-lemma-base-change-flat", "morphisms-lemma-composition-flat" ], "ref_ids": [ 5254, 5165, 5163, 5253, 5176, 5175, 5265, 5263 ] } ], "ref_ids": [] }, { "id": 2472, "type": "theorem", "label": "more-groupoids-lemma-groupoid-on-field-explain-points", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-groupoid-on-field-explain-points", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. Assume $U = \\Spec(k)$ with $k$ a field.", "For any point $r \\in R$ there exist", "\\begin{enumerate}", "\\item a field extension $k \\subset k'$ with $k'$ algebraically closed,", "\\item a point $r' \\in R'$ where $(U', R', s', t', c')$ is the", "restriction of $(U, R, s, t, c)$ via $\\Spec(k') \\to \\Spec(k)$", "\\end{enumerate}", "such that", "\\begin{enumerate}", "\\item the point $r'$ maps to $r$ under the morphism $R' \\to R$, and", "\\item the maps $s', t' : R' \\to \\Spec(k')$ induce isomorphisms", "$k' \\to \\kappa(r')$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Translating the geometric statement into a statement on fields,", "this means that we can find a diagram", "$$", "\\xymatrix{", "k' & k' \\ar[l]^1 & \\\\", "k' \\ar[u]^\\tau & \\kappa(r) \\ar[lu]^\\sigma & k \\ar[l]^-s \\ar[lu]_i \\\\", "& k \\ar[lu]^i \\ar[u]_t", "}", "$$", "where $i : k \\to k'$ is the embedding of $k$ into $k'$,", "the maps $s, t : k \\to \\kappa(r)$ are induced by $s, t : R \\to U$, and", "the map $\\tau : k' \\to k'$ is an automorphism. To produce such", "a diagram we may proceed in the following way:", "\\begin{enumerate}", "\\item Pick $i : k \\to k'$ a field map with $k'$ algebraically closed of", "very large transcendence degree over $k$.", "\\item Pick an embedding $\\sigma : \\kappa(r) \\to k'$ such that", "$\\sigma \\circ s = i$. Such a $\\sigma$ exists because we can just", "choose a transcendence basis $\\{x_\\alpha\\}_{\\alpha \\in A}$ of $\\kappa(r)$", "over $k$ and find $y_\\alpha \\in k'$, $\\alpha \\in A$ which are algebraically", "independent over $i(k)$, and map $s(k)(\\{x_\\alpha\\})$ into $k'$ by", "the rules $s(\\lambda) \\mapsto i(\\lambda)$ for $\\lambda \\in k$", "and $x_\\alpha \\mapsto y_\\alpha$ for $\\alpha \\in A$.", "Then extend to $\\tau : \\kappa(\\alpha) \\to k'$ using that $k'$ is", "algebraically closed.", "\\item Pick an automorphism $\\tau : k' \\to k'$ such that", "$\\tau \\circ i = \\sigma \\circ t$. To do this pick a transcendence", "basis $\\{x_\\alpha\\}_{\\alpha \\in A}$ of $k$ over its prime field.", "On the one hand, extend $\\{i(x_\\alpha)\\}$ to a transcendence basis of", "$k'$ by adding $\\{y_\\beta\\}_{\\beta \\in B}$ and extend", "$\\{\\sigma(t(x_\\alpha))\\}$ to a transcendence basis of $k'$ by adding", "$\\{z_\\gamma\\}_{\\gamma \\in C}$.", "As $k'$ is algebraically closed we can extend the isomorphism", "$\\sigma \\circ t \\circ i^{-1} : i(k) \\to \\sigma(t(k))$", "to an isomorphism $\\tau' : \\overline{i(k)} \\to \\overline{\\sigma(t(k))}$", "of their algebraic closures in $k'$.", "As $k'$ has large transcendence degree", "we see that the sets $B$ and $C$ have the same cardinality.", "Thus we can use a bijection", "$B \\to C$ to extend $\\tau'$ to an isomorphism", "$$", "\\overline{i(k)}(\\{y_\\beta\\})", "\\longrightarrow", "\\overline{\\sigma(t(k))}(\\{z_\\gamma\\})", "$$", "and then since $k'$ is the algebraic closure of both sides we", "see that this extends to an automorphism $\\tau : k' \\to k'$", "as desired.", "\\end{enumerate}", "This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2473, "type": "theorem", "label": "more-groupoids-lemma-groupoid-on-field-move-point", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-groupoid-on-field-move-point", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. Assume $U = \\Spec(k)$ with $k$ a field.", "If $r \\in R$ is a point such that $s, t$ induce", "isomorphisms $k \\to \\kappa(r)$, then the map", "$$", "R \\longrightarrow R, \\quad", "x \\longmapsto c(r, x)", "$$", "(see proof for precise notation) is an automorphism $R \\to R$", "which maps $e$ to $r$." ], "refs": [], "proofs": [ { "contents": [ "This is completely obvious if you think about groupoids in a", "functorial way. But we will also spell it out completely.", "Denote $a : U \\to R$ the morphism with image $r$ such that", "$s \\circ a = \\text{id}_U$ which exists by the hypothesis", "that $s : k \\to \\kappa(r)$ is an isomorphism. Similarly, denote", "$b : U \\to R$ the morphism with image $r$ such that", "$t \\circ b = \\text{id}_U$. Note that", "$b = a \\circ (t \\circ a)^{-1}$, in particular", "$a \\circ s \\circ b = b$.", "\\medskip\\noindent", "Consider the morphism $\\Psi : R \\to R$ given on $T$-valued points", "by", "$$", "(f : T \\to R) \\longmapsto (c(a \\circ t \\circ f, f) : T \\to R)", "$$", "To see this is defined we have to check that", "$s \\circ a \\circ t \\circ f = t \\circ f$ which is obvious as $s \\circ a = 1$.", "Note that $\\Phi(e) = a$, so that in order to prove the lemma it", "suffices to show that $\\Phi$ is an automorphism of $R$.", "Let $\\Phi : R \\to R$ be the morphism given on $T$-valued points by", "$$", "(g : T \\to R) \\longmapsto (c(i \\circ b \\circ t \\circ g, g) : T \\to R).", "$$", "This is defined because", "$s \\circ i \\circ b \\circ t \\circ g = t \\circ b \\circ t \\circ g =", "t \\circ g$. We claim that $\\Phi$ and $\\Psi$ are inverse to", "each other. To see this we compute", "\\begin{align*}", "& c(a \\circ t \\circ c(i \\circ b \\circ t \\circ g, g),", "c(i \\circ b \\circ t \\circ g, g)) \\\\", "& =", "c(a \\circ t \\circ i \\circ b \\circ t \\circ g,", "c(i \\circ b \\circ t \\circ g, g)) \\\\", "& =", "c(a \\circ s \\circ b \\circ t \\circ g,", "c(i \\circ b \\circ t \\circ g, g)) \\\\", "& =", "c(b \\circ t \\circ g, c(i \\circ b \\circ t \\circ g, g)) \\\\", "& =", "c(c(b \\circ t \\circ g, i \\circ b \\circ t \\circ g), g)) \\\\", "& =", "c(e, g) \\\\", "& = g", "\\end{align*}", "where we have used the relation $a \\circ s \\circ b = b$ shown above.", "In the other direction we have", "\\begin{align*}", "& c(i \\circ b \\circ t \\circ c(a \\circ t \\circ f, f), c(a \\circ t \\circ f, f)) \\\\", "& =", "c(i \\circ b \\circ t \\circ a \\circ t \\circ f, c(a \\circ t \\circ f, f)) \\\\", "& =", "c(i \\circ a \\circ (t \\circ a)^{-1} \\circ t \\circ a \\circ t \\circ f,", "c(a \\circ t \\circ f, f)) \\\\", "& =", "c(i \\circ a \\circ t \\circ f, c(a \\circ t \\circ f, f)) \\\\", "& =", "c(c(i \\circ a \\circ t \\circ f, a \\circ t \\circ f), f) \\\\", "& =", "c(e, f) \\\\", "& = f", "\\end{align*}", "The lemma is proved." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2474, "type": "theorem", "label": "more-groupoids-lemma-groupoid-on-field-translate-open", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-groupoid-on-field-translate-open", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. If $U$ is the spectrum of a field, $W \\subset R$ is open,", "and $Z \\to R$ is a morphism of schemes, then the image of the", "composition $Z \\times_{s, U, t} W \\to R \\times_{s, U, t} R \\to R$ is open." ], "refs": [], "proofs": [ { "contents": [ "Write $U = \\Spec(k)$. Consider a field extension $k \\subset k'$. Denote", "$U' = \\Spec(k')$. Let $R'$ be the restriction of $R$ via $U' \\to U$.", "Set $Z' = Z \\times_R R'$ and $W' = R' \\times_R W$.", "Consider a point $\\xi = (z, w)$ of $Z \\times_{s, U, t} W$.", "Let $r \\in R$ be the image of $z$ under $Z \\to R$.", "Pick $k' \\supset k$ and $r' \\in R'$ as in", "Lemma \\ref{lemma-groupoid-on-field-explain-points}.", "We can choose $z' \\in Z'$ mapping to $z$ and $r'$.", "Then we can find $\\xi' \\in Z' \\times_{s', U', t'} W'$", "mapping to $z'$ and $\\xi$. The open $c(r', W')$", "(Lemma \\ref{lemma-groupoid-on-field-move-point}) is", "contained in the image of $Z' \\times_{s', U', t'} W' \\to R'$.", "Observe that $Z' \\times_{s', U', t'} W' = (Z \\times_{s, U, t} W)", "\\times_{R \\times_{s, U, t} R} (R' \\times_{s', U', t'} R')$.", "Hence the image of $Z' \\times_{s', U', t'} W' \\to R' \\to R$", "is contained in the image of $Z \\times_{s, U, t} W \\to R$.", "As $R' \\to R$ is open (Lemma \\ref{lemma-restrict-groupoid-on-field})", "we conclude the image contains an open neighbourhood of", "the image of $\\xi$ as desired." ], "refs": [ "more-groupoids-lemma-groupoid-on-field-explain-points", "more-groupoids-lemma-groupoid-on-field-move-point", "more-groupoids-lemma-restrict-groupoid-on-field" ], "ref_ids": [ 2472, 2473, 2471 ] } ], "ref_ids": [] }, { "id": 2475, "type": "theorem", "label": "more-groupoids-lemma-groupoid-on-field-geometrically-irreducible", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-groupoid-on-field-geometrically-irreducible", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. Assume $U = \\Spec(k)$ with $k$ a field.", "By abuse of notation denote $e \\in R$ the image of the identity", "morphism $e : U \\to R$. Then", "\\begin{enumerate}", "\\item every local ring $\\mathcal{O}_{R, r}$ of $R$ has a unique", "minimal prime ideal,", "\\item there is exactly one irreducible component $Z$ of $R$", "passing through $e$, and", "\\item $Z$ is geometrically irreducible over $k$ via either", "$s$ or $t$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $r \\in R$ be a point.", "In this proof we will use the correspondence between irreducible components", "of $R$ passing through a point $r$ and minimal primes of the local", "ring $\\mathcal{O}_{R, r}$ without further mention.", "Choose $k \\subset k'$ and $r' \\in R'$ as in", "Lemma \\ref{lemma-groupoid-on-field-explain-points}.", "Note that $\\mathcal{O}_{R, r} \\to \\mathcal{O}_{R', r'}$", "is faithfully flat and local, see", "Lemma \\ref{lemma-restrict-groupoid-on-field}.", "Hence the result for $r' \\in R'$ implies the result for $r \\in R$.", "In other words we may assume that $s, t : k \\to \\kappa(r)$", "are isomorphisms. By", "Lemma \\ref{lemma-groupoid-on-field-move-point}", "there exists an automorphism moving $e$ to $r$.", "Hence we may assume $r = e$, i.e., part (1) follows from part (2).", "\\medskip\\noindent", "We first prove (2) in case $k$ is separably algebraically closed.", "Namely, let $X, Y \\subset R$ be irreducible components", "passing through $e$. Then by", "Varieties, Lemma \\ref{varieties-lemma-bijection-irreducible-components} and", "\\ref{varieties-lemma-separably-closed-irreducible}", "the scheme $X \\times_{s, U, t} Y$ is irreducible as well.", "Hence $c(X \\times_{s, U, t} Y) \\subset R$ is an irreducible subset.", "We claim it contains both $X$ and $Y$ (as subsets of $R$).", "Namely, let $T$ be the spectrum of a field. If $x : T \\to X$ is a $T$-valued", "point of $X$, then $c(x, e \\circ s \\circ x) = x$ and $e \\circ s \\circ x$", "factors through $Y$ as $e \\in Y$. Similarly for points of $Y$.", "This clearly implies that $X = Y$, i.e., there is a unique irreducible", "component of $R$ passing through $e$.", "\\medskip\\noindent", "Proof of (2) and (3) in general.", "Let $k \\subset k'$ be a separable algebraic closure, and", "let $(U', R', s', t', c')$ be the restriction of", "$(U, R, s, t, c)$ via $\\Spec(k') \\to \\Spec(k)$.", "By the previous paragraph there is exactly one irreducible", "component $Z'$ of $R'$ passing through $e'$.", "Denote $e'' \\in R \\times_{s, U} U'$ the base change of $e$.", "As $R' \\to R \\times_{s, U} U'$ is faithfully flat, see", "Lemma \\ref{lemma-restrict-groupoid-on-field},", "and $e' \\mapsto e''$ we see that there is exactly one", "irreducible component $Z''$ of $R \\times_{s, k} k'$ passing", "through $e''$. This implies, as $R \\times_k k' \\to R$ is faithfully", "flat, that there is exactly one irreducible component $Z$ of $R$", "passing through $e$. This proves (2).", "\\medskip\\noindent", "To prove (3) let $Z''' \\subset R \\times_k k'$ be an arbitrary", "irreducible component of $Z \\times_k k'$. By", "Varieties, Lemma \\ref{varieties-lemma-orbit-irreducible-components}", "we see that $Z''' = \\sigma(Z'')$ for some $\\sigma \\in \\text{Gal}(k'/k)$.", "Since $\\sigma(e'') = e''$ we see that $e'' \\in Z'''$ and hence", "$Z''' = Z''$. This means that $Z$ is geometrically irreducible", "over $\\Spec(k)$ via the morphism $s$.", "The same argument implies that $Z$ is geometrically irreducible", "over $\\Spec(k)$ via the morphism $t$." ], "refs": [ "more-groupoids-lemma-groupoid-on-field-explain-points", "more-groupoids-lemma-restrict-groupoid-on-field", "more-groupoids-lemma-groupoid-on-field-move-point", "varieties-lemma-bijection-irreducible-components", "varieties-lemma-separably-closed-irreducible", "more-groupoids-lemma-restrict-groupoid-on-field", "varieties-lemma-orbit-irreducible-components" ], "ref_ids": [ 2472, 2471, 2473, 10934, 10933, 2471, 10943 ] } ], "ref_ids": [] }, { "id": 2476, "type": "theorem", "label": "more-groupoids-lemma-groupoid-on-field-locally-finite-type-dimension", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-groupoid-on-field-locally-finite-type-dimension", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. Assume $U = \\Spec(k)$ with $k$ a field.", "Assume $s, t$ are locally of finite type.", "Then", "\\begin{enumerate}", "\\item $R$ is equidimensional,", "\\item $\\dim(R) = \\dim_r(R)$ for all $r \\in R$,", "\\item for any $r \\in R$ we have", "$\\text{trdeg}_{s(k)}(\\kappa(r)) = \\text{trdeg}_{t(k)}(\\kappa(r))$, and", "\\item for any closed point $r \\in R$ we have", "$\\dim(R) = \\dim(\\mathcal{O}_{R, r})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $r, r' \\in R$.", "Then $\\dim_r(R) = \\dim_{r'}(R)$ by", "Lemma \\ref{lemma-groupoid-on-field-homogeneous}", "and", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}.", "By", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}", "we have", "$$", "\\dim_r(R) =", "\\dim(\\mathcal{O}_{R, r}) + \\text{trdeg}_{s(k)}(\\kappa(r)) =", "\\dim(\\mathcal{O}_{R, r}) + \\text{trdeg}_{t(k)}(\\kappa(r)).", "$$", "On the other hand, the dimension of $R$ (or any open subset of $R$)", "is the supremum of the dimensions of the local rings of $R$, see", "Properties, Lemma \\ref{properties-lemma-codimension-local-ring}.", "Clearly this is maximal for closed points $r$ in which case", "$\\text{trdeg}_k(\\kappa(r)) = 0$ (by the Hilbert Nullstellensatz, see", "Morphisms, Section \\ref{morphisms-section-points-finite-type}).", "Hence the lemma follows." ], "refs": [ "more-groupoids-lemma-groupoid-on-field-homogeneous", "morphisms-lemma-dimension-fibre-after-base-change", "morphisms-lemma-dimension-fibre-at-a-point", "properties-lemma-codimension-local-ring" ], "ref_ids": [ 2470, 5279, 5277, 2979 ] } ], "ref_ids": [] }, { "id": 2477, "type": "theorem", "label": "more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. Assume $U = \\Spec(k)$ with $k$ a field.", "Assume $s, t$ are locally of finite type.", "Then $\\dim(R) = \\dim(G)$ where $G$ is the stabilizer group scheme of $R$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset R$ be the irreducible component passing through $e$ (see", "Lemma \\ref{lemma-groupoid-on-field-geometrically-irreducible})", "thought of as an integral closed subscheme of $R$.", "Let $k'_s$, resp.\\ $k'_t$ be the integral closure of", "$s(k)$, resp.\\ $t(k)$ in $\\Gamma(Z, \\mathcal{O}_Z)$.", "Recall that $k'_s$ and $k'_t$ are fields, see", "Varieties, Lemma \\ref{varieties-lemma-integral-closure-ground-field}.", "By", "Varieties, Proposition \\ref{varieties-proposition-unique-base-field}", "we have $k'_s = k'_t$ as subrings of $\\Gamma(Z, \\mathcal{O}_Z)$.", "As $e$ factors through $Z$ we obtain a commutative diagram", "$$", "\\xymatrix{", "k \\ar[rd]_t \\ar[rrd]^1 \\\\", "& \\Gamma(Z, \\mathcal{O}_Z) \\ar[r]^e & k \\\\", "k \\ar[ru]^s \\ar[rru]_1", "}", "$$", "This on the one hand shows that $k'_s = s(k)$, $k'_t = t(k)$, so", "$s(k) = t(k)$, which combined with the diagram above implies", "that $s = t$! In other words, we conclude that $Z$ is a closed", "subscheme of $G = R \\times_{(t, s), U \\times_S U, \\Delta} U$.", "The lemma follows as both $G$ and $R$ are equidimensional, see", "Lemma \\ref{lemma-groupoid-on-field-locally-finite-type-dimension} and", "Groupoids, Lemma \\ref{groupoids-lemma-group-scheme-finite-type-field}." ], "refs": [ "more-groupoids-lemma-groupoid-on-field-geometrically-irreducible", "varieties-lemma-integral-closure-ground-field", "varieties-proposition-unique-base-field", "more-groupoids-lemma-groupoid-on-field-locally-finite-type-dimension", "groupoids-lemma-group-scheme-finite-type-field" ], "ref_ids": [ 2475, 11021, 11137, 2476, 9599 ] } ], "ref_ids": [] }, { "id": 2478, "type": "theorem", "label": "more-groupoids-lemma-groupoid-characteristic-zero-smooth", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-groupoid-characteristic-zero-smooth", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. Assume", "\\begin{enumerate}", "\\item $U = \\Spec(k)$ with $k$ a field,", "\\item $s, t$ are locally of finite type, and", "\\item the characteristic of $k$ is zero.", "\\end{enumerate}", "Then $s, t : R \\to U$ are smooth." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-sheaf-differentials}", "the sheaf of differentials of $R \\to U$ is free.", "Hence smoothness follows from", "Varieties, Lemma \\ref{varieties-lemma-char-zero-differentials-free-smooth}." ], "refs": [ "more-groupoids-lemma-sheaf-differentials", "varieties-lemma-char-zero-differentials-free-smooth" ], "ref_ids": [ 2457, 11002 ] } ], "ref_ids": [] }, { "id": 2479, "type": "theorem", "label": "more-groupoids-lemma-reduced-group-scheme-perfect-field-characteristic-p-smooth", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-reduced-group-scheme-perfect-field-characteristic-p-smooth", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. Assume", "\\begin{enumerate}", "\\item $U = \\Spec(k)$ with $k$ a field,", "\\item $s, t$ are locally of finite type,", "\\item $R$ is reduced, and", "\\item $k$ is perfect.", "\\end{enumerate}", "Then $s, t : R \\to U$ are smooth." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-sheaf-differentials}", "the sheaf $\\Omega_{R/U}$ is free. Hence the lemma follows from", "Varieties, Lemma \\ref{varieties-lemma-char-p-differentials-free-smooth}." ], "refs": [ "more-groupoids-lemma-sheaf-differentials", "varieties-lemma-char-p-differentials-free-smooth" ], "ref_ids": [ 2457, 11003 ] } ], "ref_ids": [] }, { "id": 2480, "type": "theorem", "label": "more-groupoids-lemma-open-image-is-closed", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-open-image-is-closed", "contents": [ "Notation and assumptions as in", "Situation \\ref{situation-morphism-groupoids-on-field}.", "If $a(R_1)$ is open in $R_2$, then $a(R_1)$ is closed in $R_2$." ], "refs": [], "proofs": [ { "contents": [ "Let $r_2 \\in R_2$ be a point in the closure of $a(R_1)$.", "We want to show $r_2 \\in a(R_1)$. Pick $k \\subset k'$ and", "$r_2' \\in R'_2$ adapted to $(U, R_2, s_2, t_2, c_2)$ and $r_2$ as in", "Lemma \\ref{lemma-groupoid-on-field-explain-points}.", "Let $R_i'$ be the restriction of $R_i$ via the morphism", "$U' = \\Spec(k') \\to U = \\Spec(k)$.", "Let $a' : R'_1 \\to R_2'$ be the base change of $a$. The diagram", "$$", "\\xymatrix{", "R'_1 \\ar[r]_{a'} \\ar[d]_{p_1} & R'_2 \\ar[d]^{p_2} \\\\", "R_1 \\ar[r]^a & R_2", "}", "$$", "is a fibre square. Hence the image of $a'$ is the inverse image of", "the image of $a$ via the morphism $p_2 : R'_2 \\to R_2$. By", "Lemma \\ref{lemma-restrict-groupoid-on-field}", "the map $p_2$ is surjective and open. Hence by", "Topology, Lemma \\ref{topology-lemma-open-morphism-quotient-topology}", "we see that $r_2'$ is in the closure of $a'(R'_1)$.", "This means that we may assume that $r_2 \\in R_2$ has", "the property that the maps $k \\to \\kappa(r_2)$ induced", "by $s_2$ and $t_2$ are isomorphisms.", "\\medskip\\noindent", "In this case we can use", "Lemma \\ref{lemma-groupoid-on-field-move-point}.", "This lemma implies $c(r_2, a(R_1))$ is an open neighbourhood of $r_2$.", "Hence $a(R_1) \\cap c(r_2, a(R_1)) \\not = \\emptyset$ as we assumed", "that $r_2$ was a point of the closure of $a(R_1)$.", "Using the inverse of $R_2$ and $R_1$ we see this means", "$c_2(a(R_1), a(R_1))$ contains $r_2$.", "As $c_2(a(R_1), a(R_1)) \\subset a(c_1(R_1, R_1)) = a(R_1)$", "we conclude $r_2 \\in a(R_1)$ as desired." ], "refs": [ "more-groupoids-lemma-groupoid-on-field-explain-points", "more-groupoids-lemma-restrict-groupoid-on-field", "topology-lemma-open-morphism-quotient-topology", "more-groupoids-lemma-groupoid-on-field-move-point" ], "ref_ids": [ 2472, 2471, 8203, 2473 ] } ], "ref_ids": [] }, { "id": 2481, "type": "theorem", "label": "more-groupoids-lemma-map-groupoids-on-field-image", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-map-groupoids-on-field-image", "contents": [ "Notation and assumptions as in", "Situation \\ref{situation-morphism-groupoids-on-field}.", "Let $Z \\subset R_2$ be the reduced closed subscheme (see", "Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme})", "whose underlying topological space is the closure of the image of", "$a : R_1 \\to R_2$. Then", "$c_2(Z \\times_{s_2, U, t_2} Z) \\subset Z$", "set theoretically." ], "refs": [ "schemes-definition-reduced-induced-scheme" ], "proofs": [ { "contents": [ "Consider the commutative diagram", "$$", "\\xymatrix{", "R_1 \\times_{s_1, U, t_1} R_1 \\ar[r] \\ar[d] & R_1 \\ar[d] \\\\", "R_2 \\times_{s_2, U, t_2} R_2 \\ar[r] & R_2", "}", "$$", "By", "Varieties, Lemma \\ref{varieties-lemma-closure-image-product-map}", "the closure of the image of the left vertical arrow is (set theoretically)", "$Z \\times_{s_2, U, t_2} Z$.", "Hence the result follows." ], "refs": [ "varieties-lemma-closure-image-product-map" ], "ref_ids": [ 11000 ] } ], "ref_ids": [ 7745 ] }, { "id": 2482, "type": "theorem", "label": "more-groupoids-lemma-map-groupoids-on-perfect-field-image", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-map-groupoids-on-perfect-field-image", "contents": [ "Notation and assumptions as in", "Situation \\ref{situation-morphism-groupoids-on-field}.", "Assume that $k$ is perfect.", "Let $Z \\subset R_2$ be the reduced closed subscheme (see", "Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme})", "whose underlying topological space is the closure of the image of", "$a : R_1 \\to R_2$. Then", "$$", "(U, Z, s_2|_Z, t_2|_Z, c_2|_Z)", "$$", "is a groupoid scheme over $S$." ], "refs": [ "schemes-definition-reduced-induced-scheme" ], "proofs": [ { "contents": [ "We first explain why the statement makes sense. Since $U$ is the spectrum", "of a perfect field $k$, the scheme $Z$ is geometrically reduced", "over $k$ (via either projection), see", "Varieties, Lemma \\ref{varieties-lemma-perfect-reduced}.", "Hence the scheme $Z \\times_{s_2, U, t_2} Z \\subset Z$", "is reduced, see", "Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced-any-base-change}.", "Hence by", "Lemma \\ref{lemma-map-groupoids-on-field-image}", "we see that $c$ induces a morphism", "$Z \\times_{s_2, U, t_2} Z \\to Z$.", "Finally, it is clear that $e_2$ factors through $Z$", "and that the map $i_2 : R_2 \\to R_2$ preserves $Z$. Since the morphisms", "of the septuple", "$(U, R_2, s_2, t_2, c_2, e_2, i_2)$", "satisfies the axioms of a groupoid, it follows that after restricting", "to $Z$ they satisfy the axioms." ], "refs": [ "varieties-lemma-perfect-reduced", "varieties-lemma-geometrically-reduced-any-base-change", "more-groupoids-lemma-map-groupoids-on-field-image" ], "ref_ids": [ 10907, 10911, 2481 ] } ], "ref_ids": [ 7745 ] }, { "id": 2483, "type": "theorem", "label": "more-groupoids-lemma-locally-closed-image-is-closed", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-locally-closed-image-is-closed", "contents": [ "Notation and assumptions as in", "Situation \\ref{situation-morphism-groupoids-on-field}.", "If the image $a(R_1)$ is a locally closed subset of $R_2$", "then it is a closed subset." ], "refs": [], "proofs": [ { "contents": [ "Let $k \\subset k'$ be a perfect closure of the field $k$.", "Let $R_i'$ be the restriction of $R_i$ via the morphism", "$U' = \\Spec(k') \\to \\Spec(k)$. Note that the", "morphisms $R_i' \\to R_i$ are universal homeomorphisms as", "compositions of base changes of the universal homeomorphism", "$U' \\to U$ (see diagram in statement of", "Lemma \\ref{lemma-restrict-groupoid-on-field}).", "Hence it suffices to prove that $a'(R_1')$ is closed", "in $R_2'$. In other words, we may assume that $k$ is perfect.", "\\medskip\\noindent", "If $k$ is perfect, then the closure of the image is", "a groupoid scheme $Z \\subset R_2$, by", "Lemma \\ref{lemma-map-groupoids-on-perfect-field-image}.", "By the same lemma applied to", "$\\text{id}_{R_1} : R_1 \\to R_1$", "we see that $(R_2)_{red}$ is a groupoid scheme.", "Thus we may apply", "Lemma \\ref{lemma-open-image-is-closed}", "to the morphism", "$a|_{(R_2)_{red}} : (R_2)_{red} \\to Z$", "to conclude that $Z$ equals the image of $a$." ], "refs": [ "more-groupoids-lemma-restrict-groupoid-on-field", "more-groupoids-lemma-map-groupoids-on-perfect-field-image", "more-groupoids-lemma-open-image-is-closed" ], "ref_ids": [ 2471, 2482, 2480 ] } ], "ref_ids": [] }, { "id": 2484, "type": "theorem", "label": "more-groupoids-lemma-quasi-compact-map-groupoids-on-field-image", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-quasi-compact-map-groupoids-on-field-image", "contents": [ "Notation and assumptions as in", "Situation \\ref{situation-morphism-groupoids-on-field}.", "Assume that $a : R_1 \\to R_2$ is a quasi-compact morphism.", "Let $Z \\subset R_2$ be the scheme theoretic image (see", "Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-image})", "of $a : R_1 \\to R_2$. Then", "$$", "(U, Z, s_2|_Z, t_2|_Z, c_2|_Z)", "$$", "is a groupoid scheme over $S$." ], "refs": [ "morphisms-definition-scheme-theoretic-image" ], "proofs": [ { "contents": [ "The main difficulty is to show that $c_2|_{Z \\times_{s_2, U, t_2} Z}$", "maps into $Z$. Consider the commutative diagram", "$$", "\\xymatrix{", "R_1 \\times_{s_1, U, t_1} R_1 \\ar[r] \\ar[d]^{a \\times a} & R_1 \\ar[d] \\\\", "R_2 \\times_{s_2, U, t_2} R_2 \\ar[r] & R_2", "}", "$$", "By", "Varieties, Lemma \\ref{varieties-lemma-scheme-theoretic-image-product-map}", "we see that the scheme theoretic image of $a \\times a$ is", "$Z \\times_{s_2, U, t_2} Z$. By the commutativity of the diagram we", "conclude that $Z \\times_{s_2, U, t_2} Z$ maps into $Z$ by the bottom", "horizontal arrow. As in the proof of", "Lemma \\ref{lemma-map-groupoids-on-perfect-field-image}", "it is also true that $i_2(Z) \\subset Z$ and that", "$e_2$ factors through $Z$. Hence we conclude as in the", "proof of that lemma." ], "refs": [ "varieties-lemma-scheme-theoretic-image-product-map", "more-groupoids-lemma-map-groupoids-on-perfect-field-image" ], "ref_ids": [ 11001, 2482 ] } ], "ref_ids": [ 5539 ] }, { "id": 2485, "type": "theorem", "label": "more-groupoids-lemma-groupoid-on-field-image", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-groupoid-on-field-image", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. Assume $U$ is the spectrum of a field.", "Let $Z \\subset U \\times_S U$ be the reduced closed subscheme (see", "Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme})", "whose underlying topological space is the closure of the image of", "$j = (t, s) : R \\to U \\times_S U$. Then", "$\\text{pr}_{02}(Z \\times_{\\text{pr}_1, U, \\text{pr}_0} Z) \\subset Z$", "set theoretically." ], "refs": [ "schemes-definition-reduced-induced-scheme" ], "proofs": [ { "contents": [ "As $(U, U \\times_S U, \\text{pr}_1, \\text{pr}_0, \\text{pr}_{02})$", "is a groupoid scheme over $S$ this is a special case of", "Lemma \\ref{lemma-map-groupoids-on-field-image}.", "But we can also prove it directly as follows.", "\\medskip\\noindent", "Write $U = \\Spec(k)$. Denote", "$R_s$ (resp.\\ $Z_s$, resp.\\ $U^2_s$) the scheme", "$R$ (resp.\\ $Z$, resp.\\ $U \\times_S U$) viewed as a scheme over $k$ via", "$s$ (resp.\\ $\\text{pr}_1|_Z$, resp.\\ $\\text{pr}_1$).", "Similarly, denote", "${}_tR$ (resp.\\ ${}_tZ$, resp.\\ ${}_tU^2$) the scheme", "$R$ (resp.\\ $Z$, resp.\\ $U \\times_S U$) viewed as a scheme over $k$ via", "$t$ (resp.\\ $\\text{pr}_0|_Z$, resp.\\ $\\text{pr}_0$).", "The morphism $j$ induces morphisms of schemes", "$j_s : R_s \\to U^2_s$ and ${}_tj : {}_tR \\to {}_tU^2$ over $k$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "R_s \\times_k {}_tR \\ar[r]^c \\ar[d]_{j_s \\times {}_tj} & R \\ar[d]^j \\\\", "U^2_s \\times_k {}_tU^2 \\ar[r] & U \\times_S U", "}", "$$", "By", "Varieties, Lemma \\ref{varieties-lemma-closure-image-product-map}", "we see that the closure of the image of $j_s \\times {}_tj$ is", "$Z_s \\times_k {}_tZ$. By the commutativity of the diagram we", "conclude that $Z_s \\times_k {}_tZ$ maps into $Z$ by the bottom", "horizontal arrow." ], "refs": [ "more-groupoids-lemma-map-groupoids-on-field-image", "varieties-lemma-closure-image-product-map" ], "ref_ids": [ 2481, 11000 ] } ], "ref_ids": [ 7745 ] }, { "id": 2486, "type": "theorem", "label": "more-groupoids-lemma-groupoid-on-perfect-field-image", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-groupoid-on-perfect-field-image", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. Assume $U$ is the spectrum of a perfect field.", "Let $Z \\subset U \\times_S U$ be the reduced closed subscheme (see", "Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme})", "whose underlying topological space is the closure of the image of", "$j = (t, s) : R \\to U \\times_S U$.", "Then", "$$", "(U, Z, \\text{pr}_0|_Z, \\text{pr}_1|_Z,", "\\text{pr}_{02}|_{Z \\times_{\\text{pr}_1, U, \\text{pr}_0} Z})", "$$", "is a groupoid scheme over $S$." ], "refs": [ "schemes-definition-reduced-induced-scheme" ], "proofs": [ { "contents": [ "As $(U, U \\times_S U, \\text{pr}_1, \\text{pr}_0, \\text{pr}_{02})$", "is a groupoid scheme over $S$ this is a special case of", "Lemma \\ref{lemma-map-groupoids-on-perfect-field-image}.", "But we can also prove it directly as follows.", "\\medskip\\noindent", "We first explain why the statement makes sense. Since $U$ is the spectrum", "of a perfect field $k$, the scheme $Z$ is geometrically reduced", "over $k$ (via either projection), see", "Varieties, Lemma \\ref{varieties-lemma-perfect-reduced}.", "Hence the scheme $Z \\times_{\\text{pr}_1, U, \\text{pr}_0} Z \\subset Z$", "is reduced, see", "Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced-any-base-change}.", "Hence by", "Lemma \\ref{lemma-groupoid-on-field-image}", "we see that $\\text{pr}_{02}$ induces a morphism", "$Z \\times_{\\text{pr}_1, U, \\text{pr}_0} Z \\to Z$.", "Finally, it is clear that $\\Delta_{U/S}$ factors through $Z$", "and that the map", "$\\sigma : U \\times_S U \\to U \\times_S U$, $(x, y) \\mapsto (y, x)$", "preserves $Z$. Since", "$(U, U \\times_S U, \\text{pr}_0, \\text{pr}_1, \\text{pr}_{02},", "\\Delta_{U/S}, \\sigma)$", "satisfies the axioms of a groupoid, it follows that after restricting", "to $Z$ they satisfy the axioms." ], "refs": [ "more-groupoids-lemma-map-groupoids-on-perfect-field-image", "varieties-lemma-perfect-reduced", "varieties-lemma-geometrically-reduced-any-base-change", "more-groupoids-lemma-groupoid-on-field-image" ], "ref_ids": [ 2482, 10907, 10911, 2485 ] } ], "ref_ids": [ 7745 ] }, { "id": 2487, "type": "theorem", "label": "more-groupoids-lemma-quasi-compact-groupoid-on-field-image", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-quasi-compact-groupoid-on-field-image", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. Assume $U$ is the spectrum of a field and", "assume $R$ is quasi-compact (equivalently $s, t$ are quasi-compact).", "Let $Z \\subset U \\times_S U$ be the scheme theoretic image (see", "Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-image})", "of $j = (t, s) : R \\to U \\times_S U$.", "Then", "$$", "(U, Z, \\text{pr}_0|_Z, \\text{pr}_1|_Z,", "\\text{pr}_{02}|_{Z \\times_{\\text{pr}_1, U, \\text{pr}_0} Z})", "$$", "is a groupoid scheme over $S$." ], "refs": [ "morphisms-definition-scheme-theoretic-image" ], "proofs": [ { "contents": [ "As $(U, U \\times_S U, \\text{pr}_1, \\text{pr}_0, \\text{pr}_{02})$", "is a groupoid scheme over $S$ this is a special case of", "Lemma \\ref{lemma-quasi-compact-map-groupoids-on-field-image}.", "But we can also prove it directly as follows.", "\\medskip\\noindent", "The main difficulty is to show that", "$\\text{pr}_{02}|_{Z \\times_{\\text{pr}_1, U, \\text{pr}_0} Z}$", "maps into $Z$.", "Write $U = \\Spec(k)$. Denote", "$R_s$ (resp.\\ $Z_s$, resp.\\ $U^2_s$) the scheme", "$R$ (resp.\\ $Z$, resp.\\ $U \\times_S U$) viewed as a scheme over $k$ via", "$s$ (resp.\\ $\\text{pr}_1|_Z$, resp.\\ $\\text{pr}_1$).", "Similarly, denote", "${}_tR$ (resp.\\ ${}_tZ$, resp.\\ ${}_tU^2$) the scheme", "$R$ (resp.\\ $Z$, resp.\\ $U \\times_S U$) viewed as a scheme over $k$ via", "$t$ (resp.\\ $\\text{pr}_0|_Z$, resp.\\ $\\text{pr}_0$).", "The morphism $j$ induces morphisms of schemes", "$j_s : R_s \\to U^2_s$ and ${}_tj : {}_tR \\to {}_tU^2$ over $k$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "R_s \\times_k {}_tR \\ar[r]^c \\ar[d]_{j_s \\times {}_tj} & R \\ar[d]^j \\\\", "U^2_s \\times_k {}_tU^2 \\ar[r] & U \\times_S U", "}", "$$", "By", "Varieties, Lemma \\ref{varieties-lemma-scheme-theoretic-image-product-map}", "we see that the scheme theoretic image of $j_s \\times {}_tj$ is", "$Z_s \\times_k {}_tZ$. By the commutativity of the diagram we", "conclude that $Z_s \\times_k {}_tZ$ maps into $Z$ by the bottom", "horizontal arrow. As in the proof of", "Lemma \\ref{lemma-groupoid-on-perfect-field-image}", "it is also true that $\\sigma(Z) \\subset Z$ and that", "$\\Delta_{U/S}$ factors through $Z$. Hence we conclude as in the", "proof of that lemma." ], "refs": [ "more-groupoids-lemma-quasi-compact-map-groupoids-on-field-image", "varieties-lemma-scheme-theoretic-image-product-map", "more-groupoids-lemma-groupoid-on-perfect-field-image" ], "ref_ids": [ 2484, 11001, 2486 ] } ], "ref_ids": [ 5539 ] }, { "id": 2488, "type": "theorem", "label": "more-groupoids-lemma-slice", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-slice", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$.", "Let $G \\to U$ be the stabilizer group scheme.", "Assume $s$ and $t$ are Cohen-Macaulay and locally of finite presentation.", "Let $u \\in U$ be a finite type point of the scheme $U$, see", "Morphisms, Definition \\ref{morphisms-definition-finite-type-point}.", "With notation as in", "Situation \\ref{situation-slice},", "set", "$$", "d_1 = \\dim(G_u), \\quad", "d_2 = \\dim_{e(u)}(F_u).", "$$", "If $d_2 > d_1$, then there exist an affine scheme $U'$", "and a morphism $g : U' \\to U$ such that (with notation as in", "Situation \\ref{situation-slice})", "\\begin{enumerate}", "\\item $g$ is an immersion", "\\item $u \\in U'$,", "\\item $g$ is locally of finite presentation,", "\\item the morphism $h : U' \\times_{g, U, t} R \\longrightarrow U$", "is Cohen-Macaulay at $(u, e(u))$, and", "\\item we have $\\dim_{e'(u)}(F'_u) = d_2 - 1$.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-finite-type-point" ], "proofs": [ { "contents": [ "Let $\\Spec(A) \\subset U$ be an affine neighbourhood of $u$", "such that $u$ corresponds to a closed point of $U$, see", "Morphisms, Lemma \\ref{morphisms-lemma-identify-finite-type-points}.", "Let $\\Spec(B) \\subset R$ be an affine neighbourhood of $e(u)$", "which maps via $j$ into the open", "$\\Spec(A) \\times_S \\Spec(A) \\subset U \\times_S U$.", "Let $\\mathfrak m \\subset A$ be the maximal ideal corresponding to $u$.", "Let $\\mathfrak q \\subset B$ be the prime ideal corresponding to $e(u)$.", "Pictures:", "$$", "\\vcenter{", "\\xymatrix{", "B & A \\ar[l]^s \\\\", "A \\ar[u]^t", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "B_{\\mathfrak q} & A_{\\mathfrak m} \\ar[l]^s \\\\", "A_{\\mathfrak m} \\ar[u]^t", "}", "}", "$$", "Note that the two induced maps", "$s, t : \\kappa(\\mathfrak m) \\to \\kappa(\\mathfrak q)$", "are equal and isomorphisms as $s \\circ e = t \\circ e = \\text{id}_U$.", "In particular we see that $\\mathfrak q$", "is a maximal ideal as well. The ring maps $s, t : A \\to B$ are", "of finite presentation and flat. By assumption the ring", "$$", "\\mathcal{O}_{F_u, e(u)} = B_{\\mathfrak q}/s(\\mathfrak m)B_{\\mathfrak q}", "$$", "is Cohen-Macaulay of dimension $d_2$. The equality of dimension holds by", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}.", "\\medskip\\noindent", "Let $R''$ be the restriction of $R$ to $u = \\Spec(\\kappa(u))$", "via the morphism $\\Spec(\\kappa(u)) \\to U$.", "As $u \\to U$ is locally of finite type,", "we see that $(\\Spec(\\kappa(u)), R'', s'', t'', c'')$", "is a groupoid scheme with $s'', t''$ locally of finite type, see", "Lemma \\ref{lemma-restrict-preserves-type}.", "By", "Lemma \\ref{lemma-groupoid-on-field-dimension-equal-stabilizer}", "this implies that $\\dim(G'') = \\dim(R'')$. We also have", "$\\dim(R'') = \\dim_{e''}(R'') = \\dim(\\mathcal{O}_{R'', e''})$, see", "Lemma \\ref{lemma-groupoid-on-field-locally-finite-type-dimension}.", "By", "Groupoids, Lemma \\ref{groupoids-lemma-restrict-stabilizer}", "we have $G'' = G_u$. Hence we conclude that", "$\\dim(\\mathcal{O}_{R'', e''}) = d_1$.", "\\medskip\\noindent", "As a scheme $R''$ is", "$$", "R'' =", "R \\times_{(U \\times_S U)}", "\\Big(", "\\Spec(\\kappa(\\mathfrak m)) \\times_S \\Spec(\\kappa(\\mathfrak m))", "\\Big)", "$$", "Hence an affine open neighbourhood of $e''$ is the spectrum of the ring", "$$", "B \\otimes_{(A \\otimes A)} (\\kappa(\\mathfrak m) \\otimes \\kappa(\\mathfrak m))", "=", "B/s(\\mathfrak m)B + t(\\mathfrak m)B", "$$", "We conclude that", "$$", "\\mathcal{O}_{R'', e''} =", "B_{\\mathfrak q}/s(\\mathfrak m)B_{\\mathfrak q} + t(\\mathfrak m)B_{\\mathfrak q}", "$$", "and so now we know that this ring has dimension $d_1$.", "\\medskip\\noindent", "We claim this implies we can find", "an element $f \\in \\mathfrak m$ such that", "$$", "\\dim(B_{\\mathfrak q}/(s(\\mathfrak m)B_{\\mathfrak q} + fB_{\\mathfrak q}) < d_2", "$$", "Namely, suppose $\\mathfrak n_j \\supset s(\\mathfrak m)B_{\\mathfrak q}$,", "$j = 1, \\ldots, m$ correspond to the minimal primes of the local ring", "$B_{\\mathfrak q}/s(\\mathfrak m)B_{\\mathfrak q}$.", "There are finitely many as this ring is Noetherian (since it is essentially", "of finite type over a field -- but also because a Cohen-Macaulay ring is", "Noetherian). By the Cohen-Macaulay condition we have", "$\\dim(B_{\\mathfrak q}/\\mathfrak n_j) = d_2$, for example by", "Algebra, Lemma \\ref{algebra-lemma-CM-dim-formula}.", "Note that", "$\\dim(B_{\\mathfrak q}/(\\mathfrak n_j + t(\\mathfrak m)B_{\\mathfrak q}))", "\\leq d_1$", "as it is a quotient of the ring", "$\\mathcal{O}_{R'', e''} =", "B_{\\mathfrak q}/s(\\mathfrak m)B_{\\mathfrak q} + t(\\mathfrak m)B_{\\mathfrak q}$", "which has dimension $d_1$. As $d_1 < d_2$ this implies that", "$\\mathfrak m \\not \\subset t^{-1}(\\mathfrak n_i)$.", "By prime avoidance, see", "Algebra, Lemma \\ref{algebra-lemma-silly},", "we can find $f \\in \\mathfrak m$ with $t(f) \\not \\in \\mathfrak n_j$ for", "$j = 1, \\ldots, m$. For this choice of $f$ we have", "the displayed inequality above, see", "Algebra, Lemma \\ref{algebra-lemma-one-equation}.", "\\medskip\\noindent", "Set $A' = A/fA$ and $U' = \\Spec(A')$. Then it is clear that", "$U' \\to U$ is an immersion, locally of finite presentation", "and that $u \\in U'$. Thus (1), (2) and (3) of the lemma hold.", "The morphism", "$$", "U' \\times_{g, U, t} R \\longrightarrow U", "$$", "factors through $\\Spec(A)$ and corresponds to the ring map", "$$", "\\xymatrix{", "B/t(f)B \\ar@{=}[r] & A/(f) \\otimes_{A, t} B & A \\ar[l]_-s", "}", "$$", "Now, we see $t(f)$ is not a zerodivisor on", "$B_{\\mathfrak q}/s(\\mathfrak m)B_{\\mathfrak q}$ as this is a", "Cohen-Macaulay ring of positive dimension and $f$ is not contained", "in any minimal prime, see for example", "Algebra, Lemma \\ref{algebra-lemma-reformulate-CM}.", "Hence by", "Algebra, Lemma \\ref{algebra-lemma-grothendieck-general}", "we conclude that $s : A_{\\mathfrak m} \\to B_{\\mathfrak q}/t(f)B_{\\mathfrak q}$", "is flat with fibre ring", "$B_{\\mathfrak q}/(s(\\mathfrak m)B_{\\mathfrak q} + t(f)B_{\\mathfrak q})$", "which is Cohen-Macaulay by", "Algebra, Lemma \\ref{algebra-lemma-reformulate-CM}", "again. This implies part (4) of the lemma.", "To see part (5) note that by Diagram (\\ref{equation-restriction})", "the fibre $F'_u$ is equal to the fibre of $h$ over $u$.", "Hence", "$\\dim_{e'(u)}(F'_u) =", "\\dim(B_{\\mathfrak q}/(s(\\mathfrak m)B_{\\mathfrak q} + t(f)B_{\\mathfrak q}))$", "by", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}", "and the dimension of this ring is $d_2 - 1$ by", "Algebra, Lemma \\ref{algebra-lemma-reformulate-CM}", "once more. This proves the final assertion of the lemma and we win." ], "refs": [ "morphisms-lemma-identify-finite-type-points", "morphisms-lemma-dimension-fibre-at-a-point", "more-groupoids-lemma-restrict-preserves-type", "more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer", "more-groupoids-lemma-groupoid-on-field-locally-finite-type-dimension", "groupoids-lemma-restrict-stabilizer", "algebra-lemma-CM-dim-formula", "algebra-lemma-silly", "algebra-lemma-reformulate-CM", "algebra-lemma-grothendieck-general", "algebra-lemma-reformulate-CM", "morphisms-lemma-dimension-fibre-at-a-point", "algebra-lemma-reformulate-CM" ], "ref_ids": [ 5207, 5277, 2464, 2477, 2476, 9644, 925, 378, 923, 1111, 923, 5277, 923 ] } ], "ref_ids": [ 5550 ] }, { "id": 2489, "type": "theorem", "label": "more-groupoids-lemma-max-slice", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-max-slice", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$.", "Let $G \\to U$ be the stabilizer group scheme.", "Assume $s$ and $t$ are Cohen-Macaulay and locally of finite presentation.", "Let $u \\in U$ be a finite type point of the scheme $U$, see", "Morphisms, Definition \\ref{morphisms-definition-finite-type-point}.", "With notation as in", "Situation \\ref{situation-slice}", "there exist an affine scheme $U'$ and a morphism $g : U' \\to U$ such that", "\\begin{enumerate}", "\\item $g$ is an immersion,", "\\item $u \\in U'$,", "\\item $g$ is locally of finite presentation,", "\\item the morphism $h : U' \\times_{g, U, t} R \\longrightarrow U$", "is Cohen-Macaulay and locally of finite presentation,", "\\item the morphisms $s', t' : R' \\to U'$ are Cohen-Macaulay and", "locally of finite presentation, and", "\\item $\\dim_{e(u)}(F'_u) = \\dim(G'_u)$.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-finite-type-point" ], "proofs": [ { "contents": [ "As $s$ is locally of finite presentation the scheme $F_u$ is", "locally of finite type over $\\kappa(u)$. Hence", "$\\dim_{e(u)}(F_u) < \\infty$ and we may argue by induction on", "$\\dim_{e(u)}(F_u)$.", "\\medskip\\noindent", "If $\\dim_{e(u)}(F_u) = \\dim(G_u)$ there is nothing to prove.", "Assume $\\dim_{e(u)}(F_u) > \\dim(G_u)$. This means that", "Lemma \\ref{lemma-slice}", "applies and we find a morphism $g : U' \\to U$ which has", "properties (1), (2), (3), instead of (6) we have", "$\\dim_{e(u)}(F'_u) < \\dim_{e(u)}(F_u)$,", "and instead of (4) and (5) we have that the composition", "$$", "h = s \\circ \\text{pr}_1 : U' \\times_{g, U, t} R \\longrightarrow U", "$$", "is Cohen-Macaulay at the point $(u, e(u))$. We apply", "Remark \\ref{remark-local-source-apply}", "and we obtain an open subscheme $U'' \\subset U'$ such that", "$U'' \\times_{g, U, t} R \\subset U' \\times_{g, U, t} R$", "is the largest open subscheme on which $h$ is Cohen-Macaulay.", "Since $(u, e(u)) \\in U'' \\times_{g, U, t} R$ we see that $u \\in U''$.", "Hence we may replace $U'$ by $U''$ and assume that in fact $h$ is", "Cohen-Macaulay everywhere! By", "Lemma \\ref{lemma-restrict-property}", "we conclude that $s', t'$ are locally of finite", "presentation and Cohen-Macaulay (use", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-presentation}", "and", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-base-change-CM}).", "\\medskip\\noindent", "By construction $\\dim_{e'(u)}(F'_u) < \\dim_{e(u)}(F_u)$,", "so we may apply the induction hypothesis to $(U', R', s', t', c')$", "and the point $u \\in U'$. Note that $u$ is also a finite type point", "of $U'$ (for example you can see this using the characterization of", "finite type points from", "Morphisms, Lemma \\ref{morphisms-lemma-identify-finite-type-points}).", "Let $g' : U'' \\to U'$ and $(U'', R'', s'', t'', c'')$ be the solution", "of the corresponding problem starting with $(U', R', s', t', c')$", "and the point $u \\in U'$. We claim that the composition", "$$", "g'' = g \\circ g' : U'' \\longrightarrow U", "$$", "is a solution for the original problem. Properties (1), (2), (3), (5),", "and (6) are immediate. To see (4) note that the morphism", "$$", "h'' = s \\circ \\text{pr}_1 : U'' \\times_{g'', U, t} R \\longrightarrow U", "$$", "is locally of finite presentation and Cohen-Macaulay by an application of", "Lemma \\ref{lemma-double-restrict-property}", "(use", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-CM-local-source-and-target}", "to see that Cohen-Macaulay morphisms are fppf local on the target)." ], "refs": [ "more-groupoids-lemma-slice", "more-groupoids-remark-local-source-apply", "more-groupoids-lemma-restrict-property", "morphisms-lemma-base-change-finite-presentation", "more-morphisms-lemma-base-change-CM", "morphisms-lemma-identify-finite-type-points", "more-groupoids-lemma-double-restrict-property", "more-morphisms-lemma-CM-local-source-and-target" ], "ref_ids": [ 2488, 2507, 2465, 5240, 13788, 5207, 2467, 13793 ] } ], "ref_ids": [ 5550 ] }, { "id": 2490, "type": "theorem", "label": "more-groupoids-lemma-max-slice-quasi-finite", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-max-slice-quasi-finite", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$.", "Let $G \\to U$ be the stabilizer group scheme.", "Assume $s$ and $t$ are Cohen-Macaulay and locally of finite presentation.", "Let $u \\in U$ be a finite type point of the scheme $U$, see", "Morphisms, Definition \\ref{morphisms-definition-finite-type-point}.", "Assume that $G \\to U$ is locally quasi-finite.", "With notation as in", "Situation \\ref{situation-slice}", "there exist an affine scheme $U'$ and a morphism $g : U' \\to U$ such that", "\\begin{enumerate}", "\\item $g$ is an immersion,", "\\item $u \\in U'$,", "\\item $g$ is locally of finite presentation,", "\\item the morphism $h : U' \\times_{g, U, t} R \\longrightarrow U$", "is flat, locally of finite presentation, and locally quasi-finite, and", "\\item the morphisms $s', t' : R' \\to U'$ are flat,", "locally of finite presentation, and locally quasi-finite.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-finite-type-point" ], "proofs": [ { "contents": [ "Take $g : U' \\to U$ as in", "Lemma \\ref{lemma-max-slice}.", "Since $h^{-1}(u) = F'_u$ we see that $h$ has relative dimension", "$\\leq 0$ at $(u, e(u))$. Hence, by", "Remark \\ref{remark-local-source-apply},", "we obtain an open subscheme $U'' \\subset U'$ such that", "$u \\in U''$ and $U'' \\times_{g, U, t} R$ is the maximal open subscheme", "of $U' \\times_{g, U, t} R$ on which $h$ has relative dimension $\\leq 0$.", "After replacing $U'$ by $U''$ we see that $h$ has relative dimension $\\leq 0$.", "This implies that $h$ is locally quasi-finite by", "Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}.", "Since it is still locally of finite presentation and Cohen-Macaulay we see", "that it is flat, locally of finite presentation and locally quasi-finite,", "i.e., (4) above holds. This implies that $s'$ is flat, locally of finite", "presentation and locally quasi-finite as a base change of $h$, see", "Lemma \\ref{lemma-restrict-property}." ], "refs": [ "more-groupoids-lemma-max-slice", "more-groupoids-remark-local-source-apply", "morphisms-lemma-locally-quasi-finite-rel-dimension-0", "more-groupoids-lemma-restrict-property" ], "ref_ids": [ 2489, 2507, 5287, 2465 ] } ], "ref_ids": [ 5550 ] }, { "id": 2491, "type": "theorem", "label": "more-groupoids-lemma-quasi-finite-over-base", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-quasi-finite-over-base", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Let $p \\in S$ be a point, and let $u \\in U$ be a point lying over $p$.", "Assume that", "\\begin{enumerate}", "\\item $U \\to S$ is locally of finite type,", "\\item $U \\to S$ is quasi-finite at $u$,", "\\item $U \\to S$ is separated,", "\\item $R \\to S$ is separated,", "\\item $s$, $t$ are flat and locally of finite presentation, and", "\\item $s^{-1}(\\{u\\})$ is finite.", "\\end{enumerate}", "Then there exists an \\'etale neighbourhood $(S', p') \\to (S, p)$ with", "$\\kappa(p) = \\kappa(p')$ and a base change diagram", "$$", "\\xymatrix{", "R' \\amalg W'", "\\ar@{=}[r] &", "S' \\times_S R", "\\ar[r] \\ar@<2ex>[d]^{s'} \\ar@<-2ex>[d]_{t'} &", "R \\ar@<1ex>[d]^s \\ar@<-1ex>[d]_t \\\\", "U' \\amalg W", "\\ar@{=}[r] &", "S' \\times_S U", "\\ar[r] \\ar[d] &", "U \\ar[d] \\\\", " &", "S' \\ar[r] &", "S", "}", "$$", "where the equal signs are decompositions into open and closed", "subschemes such that", "\\begin{enumerate}", "\\item[(a)] there exists a point $u'$ of $U'$ mapping to $u$ in $U$,", "\\item[(b)] the fibre $(U')_{p'}$ equals $t'\\big((s')^{-1}(\\{u'\\})\\big)$", "set theoretically,", "\\item[(c)] the fibre $(R')_{p'}$ equals $(s')^{-1}\\big((U')_{p'}\\big)$", "set theoretically,", "\\item[(d)] the schemes $U'$ and $R'$ are finite over $S'$,", "\\item[(e)] we have $s'(R') \\subset U'$ and $t'(R') \\subset U'$,", "\\item[(f)] we have", "$c'(R' \\times_{s', U', t'} R') \\subset R'$", "where $c'$ is the base change of $c$, and", "\\item[(g)] the morphisms $s', t', c'$ determine a groupoid structure", "by taking the system", "$(U', R', s'|_{R'}, t'|_{R'}, c'|_{R' \\times_{s', U', t'} R'})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us denote $f : U \\to S$ the structure morphism of $U$.", "By assumption (6) we can write $s^{-1}(\\{u\\}) = \\{r_1, \\ldots, r_n\\}$.", "Since this set is finite, we see that $s$ is quasi-finite at each of", "these finitely many inverse images, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-fibre}.", "Hence we see that $f \\circ s : R \\to S$ is quasi-finite at each $r_i$", "(Morphisms, Lemma \\ref{morphisms-lemma-composition-quasi-finite}).", "Hence $r_i$ is isolated in the fibre $R_p$, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}.", "Write $t(\\{r_1, \\ldots, r_n\\}) = \\{u_1, \\ldots, u_m\\}$.", "Note that it may happen that $m < n$ and note that", "$u \\in \\{u_1, \\ldots, u_m\\}$.", "Since $t$ is flat and locally of finite presentation,", "the morphism of fibres $t_p : R_p \\to U_p$ is flat and locally of", "finite presentation (Morphisms,", "Lemmas \\ref{morphisms-lemma-base-change-flat} and", "\\ref{morphisms-lemma-base-change-finite-presentation}),", "hence open (Morphisms,", "Lemma \\ref{morphisms-lemma-fppf-open}).", "The fact that each $r_i$ is isolated in $R_p$ implies that", "each $u_j = t(r_i)$ is isolated in $U_p$. Using", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}", "again, we see that $f$ is quasi-finite at $u_1, \\ldots, u_m$.", "\\medskip\\noindent", "Denote $F_u = s^{-1}(u)$ and $F_{u_j} = s^{-1}(u_j)$ the scheme theoretic", "fibres. Note that $F_u$ is finite over $\\kappa(u)$ as it is locally of finite", "type over $\\kappa(u)$ with finitely many points (for example it follows from", "the much more general", "Morphisms, Lemma", "\\ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded}).", "By", "Lemma \\ref{lemma-two-fibres}", "we see that $F_u$ and $F_{u_j}$ become isomorphic over a common", "field extension of $\\kappa(u)$ and $\\kappa(u_j)$. Hence we see", "that $F_{u_j}$ is finite over $\\kappa(u_j)$. In particular we see", "$s^{-1}(\\{u_j\\})$ is a finite set for each $j = 1, \\ldots, m$.", "Thus we see that assumptions (2) and (6) hold for each $u_j$ also", "(above we saw that $U \\to S$ is quasi-finite at $u_j$).", "Hence the argument of the first paragraph applies to each $u_j$", "and we see that $R \\to U$ is quasi-finite at each of the points of", "$$", "\\{r_1, \\ldots, r_N\\} = s^{-1}(\\{u_1, \\ldots, u_m\\})", "$$", "Note that $t(\\{r_1, \\ldots, r_N\\}) = \\{u_1, \\ldots, u_m\\}$ and", "$t^{-1}(\\{u_1, \\ldots, u_m\\}) = \\{r_1, \\ldots, r_N\\}$", "since $R$ is a groupoid\\footnote{Explanation in groupoid language:", "The original set $\\{r_1, \\ldots, r_n\\}$ was the set of arrows with", "source $u$. The set $\\{u_1, \\ldots, u_m\\}$ was the set of objects", "isomorphic to $u$. And $\\{r_1, \\ldots, r_N\\}$ is the set of all arrows", "between all the objects equivalent to $u$.}. Moreover, we have", "$\\text{pr}_0(c^{-1}(\\{r_1, \\ldots, r_N\\})) = \\{r_1, \\ldots, r_N\\}$", "and", "$\\text{pr}_1(c^{-1}(\\{r_1, \\ldots, r_N\\})) = \\{r_1, \\ldots, r_N\\}$.", "Similarly we get $e(\\{u_1, \\ldots, u_m\\}) \\subset \\{r_1, \\ldots, r_N\\}$", "and $i(\\{r_1, \\ldots, r_N\\}) = \\{r_1, \\ldots, r_N\\}$.", "\\medskip\\noindent", "We may apply", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical}", "to the pairs", "$(U \\to S, \\{u_1, \\ldots, u_m\\})$ and", "$(R \\to S, \\{r_1, \\ldots, r_N\\})$", "to get an \\'etale neighbourhood $(S', p') \\to (S, p)$", "which induces an identification $\\kappa(p) = \\kappa(p')$", "such that $S' \\times_S U$ and $S' \\times_S R$ decompose as", "$$", "S' \\times_S U = U' \\amalg W, \\quad", "S' \\times_S R = R' \\amalg W'", "$$", "with $U' \\to S'$ finite and $(U')_{p'}$ mapping bijectively to", "$\\{u_1, \\ldots, u_m\\}$, and $R' \\to S'$ finite and", "$(R')_{p'}$ mapping bijectively to $\\{r_1, \\ldots, r_N\\}$.", "Moreover, no point of $W_{p'}$ (resp.\\ $(W')_{p'}$) maps to", "any of the points $u_j$ (resp.\\ $r_i$). At this point (a), (b), (c), and (d)", "of the lemma are satisfied. Moreover, the inclusions of (e) and (f) hold", "on fibres over $p'$, i.e., $s'((R')_{p'}) \\subset (U')_{p'}$,", "$t'((R')_{p'}) \\subset (U')_{p'}$, and", "$c'((R' \\times_{s', U', t'} R')_{p'}) \\subset (R')_{p'}$.", "\\medskip\\noindent", "We claim that we can replace $S'$ by a Zariski open neighbourhood", "of $p'$ so that the inclusions of (e) and (f) hold.", "For example, consider the set $E = (s'|_{R'})^{-1}(W)$.", "This is open and closed in $R'$ and does not contain any points", "of $R'$ lying over $p'$. Since $R' \\to S'$ is closed,", "after replacing $S'$ by $S' \\setminus (R' \\to S')(E)$ we reach a", "situation where $E$ is empty. In other words $s'$ maps $R'$ into $U'$.", "Note that this property is preserved under further shrinking $S'$.", "Similarly, we can arrange it so that $t'$ maps $R'$ into $U'$.", "At this point (e) holds. In the same manner, consider the set", "$E = (c'|_{R' \\times_{s', U', t'} R'})^{-1}(W')$.", "It is open and closed in the scheme $R' \\times_{s', U', t'} R'$", "which is finite over $S'$, and does not contain any points lying", "over $p'$. Hence after replacing $S'$ by", "$S' \\setminus (R' \\times_{s', U', t'} R' \\to S')(E)$", "we reach a situation where $E$ is empty. In other words we obtain", "the inclusion in (f). We may repeat the argument also with the identity", "$e' : S' \\times_S U \\to S' \\times_S R$ and the inverse", "$i' : S' \\times_S R \\to S' \\times_S R$ so that we may assume", "(after shrinking $S'$ some more) that $(e'|_{U'})^{-1}(W') = \\emptyset$", "and $(i'|_{R'})^{-1}(W') = \\emptyset$.", "\\medskip\\noindent", "At this point we see that we may consider the structure", "$$", "(U', R', s'|_{R'}, t'|_{R'}, c'|_{R' \\times_{t', U', s'} R'},", "e'|_{U'}, i'|_{R'}).", "$$", "The axioms of a groupoid scheme over $S'$ hold", "because they hold for the groupoid scheme", "$(S' \\times_S U, S' \\times_S R, s', t', c', e', i')$." ], "refs": [ "morphisms-lemma-finite-fibre", "morphisms-lemma-composition-quasi-finite", "morphisms-lemma-quasi-finite-at-point-characterize", "morphisms-lemma-base-change-flat", "morphisms-lemma-base-change-finite-presentation", "morphisms-lemma-fppf-open", "morphisms-lemma-quasi-finite-at-point-characterize", "morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded", "more-groupoids-lemma-two-fibres", "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical" ], "ref_ids": [ 5227, 5232, 5226, 5265, 5240, 5267, 5226, 5531, 2462, 13895 ] } ], "ref_ids": [] }, { "id": 2492, "type": "theorem", "label": "more-groupoids-lemma-quasi-finite-over-base-j-proper", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-quasi-finite-over-base-j-proper", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Let $p \\in S$ be a point, and let $u \\in U$ be a point lying over $p$.", "Assume assumptions (1) -- (6) of", "Lemma \\ref{lemma-quasi-finite-over-base}", "hold as well as", "\\begin{enumerate}", "\\item[(7)] $j : R \\to U \\times_S U$ is universally closed\\footnote{In view of", "the other conditions this is equivalent to requiring $j$ to be proper.}.", "\\end{enumerate}", "Then we can choose $(S', p') \\to (S, p)$ and decompositions", "$S' \\times_S U = U' \\amalg W$ and $S' \\times_S R = R' \\amalg W'$", "and $u' \\in U'$ such that (a) -- (g) of", "Lemma \\ref{lemma-quasi-finite-over-base}", "hold as well as", "\\begin{enumerate}", "\\item[(h)] $R'$ is the restriction of $S' \\times_S R$ to $U'$.", "\\end{enumerate}" ], "refs": [ "more-groupoids-lemma-quasi-finite-over-base", "more-groupoids-lemma-quasi-finite-over-base" ], "proofs": [ { "contents": [ "We apply Lemma \\ref{lemma-quasi-finite-over-base} for the", "groupoid $(U, R, s, t, c)$ over the scheme $S$ with points $p$ and $u$.", "Hence we get an \\'etale neighbourhood", "$(S', p') \\to (S, p)$ and disjoint union decompositions", "$$", "S' \\times_S U = U' \\amalg W, \\quad", "S' \\times_S R = R' \\amalg W'", "$$", "and $u' \\in U'$ satisfying conclusions (a), (b), (c), (d), (e), (f), and (g).", "We may shrink $S'$ to a smaller neighbourhood of $p'$ without", "affecting the conclusions (a) -- (g). We will show that for a suitable", "shrinking conclusion (h) holds as well.", "Let us denote $j'$ the base change of $j$ to $S'$.", "By conclusion (e) it is clear that", "$$", "j'^{-1}(U' \\times_{S'} U') = R' \\amalg Rest", "$$", "for some open and closed $Rest$ piece. Since $U' \\to S'$ is finite", "by conclusion (d) we see that $U' \\times_{S'} U'$ is finite over $S'$.", "Since $j$ is universally closed, also $j'$ is universally closed, and", "hence $j'|_{Rest}$ is universally closed too. By conclusions", "(b) and (c) we see that the fibre of", "$$", "(U' \\times_{S'} U' \\to S') \\circ j'|_{Rest} :", "Rest", "\\longrightarrow", "S'", "$$", "over $p'$ is empty. Hence, since $Rest \\to S'$ is closed as a composition", "of closed morphisms, after replacing $S'$ by", "$S' \\setminus \\Im(Rest \\to S')$, we may assume that", "$Rest = \\emptyset$. And this is exactly the condition that $R'$ is", "the restriction of $S' \\times_S R$ to the open subscheme", "$U' \\subset S' \\times_S U$, see", "Groupoids, Lemma \\ref{groupoids-lemma-restrict-groupoid-relation}", "and its proof." ], "refs": [ "more-groupoids-lemma-quasi-finite-over-base", "groupoids-lemma-restrict-groupoid-relation" ], "ref_ids": [ 2491, 9643 ] } ], "ref_ids": [ 2491, 2491 ] }, { "id": 2493, "type": "theorem", "label": "more-groupoids-lemma-finite-stratify", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-finite-stratify", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$", "are finite. There exists a sequence of $R$-invariant closed subschemes", "$$", "U = Z_0 \\supset Z_1 \\supset Z_2 \\supset \\ldots", "$$", "such that $\\bigcap Z_r = \\emptyset$ and such that", "$s^{-1}(Z_{r - 1}) \\setminus s^{-1}(Z_r) \\to Z_{r - 1} \\setminus Z_r$", "is finite locally free of rank $r$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{Z_r\\}$ be the stratification of $U$ given by the Fitting ideals", "of the finite type quasi-coherent modules $s_*\\mathcal{O}_R$. See", "Divisors, Lemma \\ref{divisors-lemma-locally-free-rank-r-pullback}.", "Since the identity $e : U \\to R$ is a section to $s$ we see that", "$s_*\\mathcal{O}_R$ contains $\\mathcal{O}_S$ as a direct summand.", "Hence $U = Z_{-1} = Z_0$ (details omitted).", "Since formation of Fitting ideals commutes with base change", "(More on Algebra, Lemma \\ref{more-algebra-lemma-fitting-ideal-basics})", "we find that $s^{-1}(Z_r)$ corresponds to the $r$th Fitting ideal", "of $\\text{pr}_{1, *}\\mathcal{O}_{R \\times_{s, U, t} R}$ because", "the lower right square of diagram (\\ref{equation-diagram}) is cartesian.", "Using the fact that the lower left square is also cartesian we conclude", "that $s^{-1}(Z_r) = t^{-1}(Z_r)$, in other words $Z_r$ is $R$-invariant.", "The morphism", "$s^{-1}(Z_{r - 1}) \\setminus s^{-1}(Z_r) \\to Z_{r - 1} \\setminus Z_r$", "is finite locally free of rank $r$ because the module", "$s_*\\mathcal{O}_R$ pulls back to a finite locally free module of rank $r$", "on $Z_{r - 1} \\setminus Z_r$ by", "Divisors, Lemma \\ref{divisors-lemma-locally-free-rank-r-pullback}." ], "refs": [ "divisors-lemma-locally-free-rank-r-pullback", "more-algebra-lemma-fitting-ideal-basics", "divisors-lemma-locally-free-rank-r-pullback" ], "ref_ids": [ 7898, 9834, 7898 ] } ], "ref_ids": [] }, { "id": 2494, "type": "theorem", "label": "more-groupoids-lemma-finite-flat-over-almost-dense-subscheme", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-finite-flat-over-almost-dense-subscheme", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$", "are finite. There exists an open subscheme $W \\subset U$ and a closed", "subscheme $W' \\subset W$ such that", "\\begin{enumerate}", "\\item $W$ and $W'$ are $R$-invariant,", "\\item $U = t(s^{-1}(\\overline{W}))$ set theoretically,", "\\item $W$ is a thickening of $W'$, and", "\\item the maps $s'$, $t'$ of the restriction $(W', R', s', t', c')$", "are finite locally free.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the stratification $U = Z_0 \\supset Z_1 \\supset Z_2 \\supset \\ldots$", "of Lemma \\ref{lemma-finite-stratify}.", "\\medskip\\noindent", "We will construct disjoint unions $W = \\coprod_{r \\geq 1} W_r$ and", "$W' = \\coprod_{r \\geq 1} W'_r$ with each $W'_r \\to W_r$ a thickening", "of $R$-invariant subschemes of $U$ such that the morphisms", "$s_r', t_r'$ of the restrictions $(W_r', R_r', s_r', t_r', c_r')$", "are finite locally free of rank $r$. To begin we set", "$W_1 = W'_1 = U \\setminus Z_1$. This is an $R$-invariant open", "subscheme of $U$, it is true that $W_0$ is a thickening of $W'_0$,", "and the maps $s_1'$, $t_1'$ of the", "restriction $(W_1', R_1', s_1', t_1', c_1')$ are isomorphisms, i.e.,", "finite locally free of rank $1$.", "Moreover, every point of $U \\setminus Z_1$ is in $t(s^{-1}(\\overline{W_1}))$.", "\\medskip\\noindent", "Assume we have found subschemes $W'_r \\subset W_r \\subset U$ for $r \\leq n$", "such that", "\\begin{enumerate}", "\\item $W_1, \\ldots, W_n$ are disjoint,", "\\item $W_r$ and $W_r'$ are $R$-invariant,", "\\item $U \\setminus Z_n \\subset \\bigcup_{r \\leq n} t(s^{-1}(\\overline{W_r}))$", "set theoretically,", "\\item $W_r$ is a thickening of $W'_r$,", "\\item the maps $s_r'$, $t_r'$ of the restriction", "$(W_r', R_r', s_r', t_r', c_r')$ are finite locally free of rank $r$.", "\\end{enumerate}", "Then we set", "$$", "W_{n + 1} = Z_n \\setminus", "\\left(", "Z_{n + 1} \\cup \\bigcup\\nolimits_{r \\leq n} t(s^{-1}(\\overline{W_r}))", "\\right)", "$$", "set theoretically and", "$$", "W'_{n + 1} = Z_n \\setminus", "\\left(", "Z_{n + 1} \\cup \\bigcup\\nolimits_{r \\leq n} t(s^{-1}(\\overline{W_r}))", "\\right)", "$$", "scheme theoretically. Then $W_{n + 1}$ is an $R$-invariant open subscheme", "of $U$ because $Z_{n + 1} \\setminus \\overline{U \\setminus Z_{n + 1}}$", "is open in $U$ and $\\overline{U \\setminus Z_{n + 1}}$ is contained", "in the closed subset $\\bigcup\\nolimits_{r \\leq n} t(s^{-1}(\\overline{W_r}))$", "we are removing by property (3) and the fact that $t$ is a closed morphism.", "It is clear that $W'_{n + 1}$ is a closed subscheme", "of $W_{n + 1}$ with the same underlying topological space.", "Finally, properties (1), (2) and (3) are clear and property (5) follows from", "Lemma \\ref{lemma-finite-stratify}.", "\\medskip\\noindent", "By Lemma \\ref{lemma-finite-stratify} we have $\\bigcap Z_r = \\emptyset$.", "Hence every point of $U$ is contained in $U \\setminus Z_n$", "for some $n$. Thus we see that", "$U = \\bigcup_{r \\geq 1} t(s^{-1}(\\overline{W_r}))$", "set theoretically and we see that (2) holds.", "Thus $W' \\subset W$ satisfy (1), (2), (3), and (4)." ], "refs": [ "more-groupoids-lemma-finite-stratify", "more-groupoids-lemma-finite-stratify", "more-groupoids-lemma-finite-stratify" ], "ref_ids": [ 2493, 2493, 2493 ] } ], "ref_ids": [] }, { "id": 2495, "type": "theorem", "label": "more-groupoids-lemma-finite-flat-over-almost-dense-subscheme-addendum", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-finite-flat-over-almost-dense-subscheme-addendum", "contents": [ "In Lemma \\ref{lemma-finite-flat-over-almost-dense-subscheme}", "assume in addition that $s$ and $t$ are of finite presentation.", "Then", "\\begin{enumerate}", "\\item the morphism $W' \\to W$ is of finite presentation, and", "\\item if $u \\in U$ is a point whose $R$-orbit consists of", "generic points of irreducible components of $U$, then $u \\in W$.", "\\end{enumerate}" ], "refs": [ "more-groupoids-lemma-finite-flat-over-almost-dense-subscheme" ], "proofs": [ { "contents": [ "In this case the stratification", "$U = Z_0 \\supset Z_1 \\supset Z_2 \\supset \\ldots$ of", "Lemma \\ref{lemma-finite-stratify} is given by closed immersions $Z_k \\to U$", "of finite presentation, see", "Divisors, Lemma \\ref{divisors-lemma-locally-free-rank-r-pullback}.", "Part (1) follows immediately from this as $W' \\to W$ is locally given", "by intersecting the open $W$ by $Z_r$. To see part (2)", "let $\\{u_1, \\ldots, u_n\\}$ be the orbit of $u$.", "Since the closed subschemes $Z_k$ are $R$-invariant and", "$\\bigcap Z_k = \\emptyset$, we find an $k$ such that $u_i \\in Z_k$", "and $u_i \\not \\in Z_{k + 1}$ for all $i$.", "The image of $Z_k \\to U$ and $Z_{k + 1} \\to U$ is locally constructible", "(Morphisms, Theorem \\ref{morphisms-theorem-chevalley}).", "Since $u_i \\in U$ is a generic point of an irreducible component", "of $U$, there exists an open neighbourhood $U_i$ of $u_i$ which", "is contained in $Z_k \\setminus Z_{k + 1}$ set theoretically", "(Properties, Lemma \\ref{properties-lemma-generic-point-in-constructible}).", "In the proof of Lemma \\ref{lemma-finite-flat-over-almost-dense-subscheme}", "we have constructed $W$ as a disjoint union $\\coprod W_r$", "with $W_r \\subset Z_{r - 1} \\setminus Z_r$ such that", "$U = \\bigcup t(s^{-1}(\\overline{W_r}))$. As $\\{u_1, \\ldots, u_n\\}$", "is an $R$-orbit we see that $u \\in t(s^{-1}(\\overline{W_r}))$", "implies $u_i \\in \\overline{W_r}$ for some $i$ which implies", "$U_i \\cap W_r \\not = \\emptyset$ which implies $r = k$.", "Thus we conclude that $u$ is in", "$$", "W_{k + 1} = Z_k \\setminus", "\\left(", "Z_{k + 1} \\cup \\bigcup\\nolimits_{r \\leq k} t(s^{-1}(\\overline{W_r}))", "\\right)", "$$", "as desired." ], "refs": [ "more-groupoids-lemma-finite-stratify", "divisors-lemma-locally-free-rank-r-pullback", "morphisms-theorem-chevalley", "properties-lemma-generic-point-in-constructible", "more-groupoids-lemma-finite-flat-over-almost-dense-subscheme" ], "ref_ids": [ 2493, 7898, 5123, 2939, 2494 ] } ], "ref_ids": [ 2494 ] }, { "id": 2496, "type": "theorem", "label": "more-groupoids-lemma-invariant-affine-open-around-generic-point", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-invariant-affine-open-around-generic-point", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$", "are finite and of finite presentation and $U$ quasi-separated. Let", "$u_1, \\ldots, u_m \\in U$ be points whose orbits consist of generic points", "of irreducible components of $U$. Then there exist $R$-invariant subschemes", "$V' \\subset V \\subset U$ such that", "\\begin{enumerate}", "\\item $u_1, \\ldots, u_m \\in V'$,", "\\item $V$ is open in $U$,", "\\item $V'$ and $V$ are affine,", "\\item $V' \\subset V$ is a thickening of finite presentation,", "\\item the morphisms $s', t'$ of the restriction $(V', R', s', t', c')$", "are finite locally free.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $W' \\subset W \\subset U$ be as in", "Lemma \\ref{lemma-finite-flat-over-almost-dense-subscheme}.", "By Lemma \\ref{lemma-finite-flat-over-almost-dense-subscheme-addendum}", "we get $u_j \\in W$ and that $W' \\to W$ is a thickening of finite presentation.", "By Limits, Lemma \\ref{limits-lemma-affines-glued-in-closed-affine}", "it suffices to find an $R$-invariant affine open subscheme", "$V'$ of $W'$ containing $u_j$ (because then we can let $V \\subset W$", "be the corresponding open subscheme which will be affine).", "Thus we may replace $(U, R, s, t, c)$ by the restriction", "$(W', R', s', t', c')$ to $W'$.", "In other words, we may assume we have a groupoid scheme $(U, R, s, t, c)$", "whose morphisms $s$ and $t$ are finite locally free.", "By Properties, Lemma \\ref{properties-lemma-maximal-points-affine}", "we can find an affine open containing the union of the orbits of", "$u_1, \\ldots, u_m$. Finally, we can apply", "Groupoids, Lemma \\ref{groupoids-lemma-find-invariant-affine}", "to conclude." ], "refs": [ "more-groupoids-lemma-finite-flat-over-almost-dense-subscheme", "more-groupoids-lemma-finite-flat-over-almost-dense-subscheme-addendum", "limits-lemma-affines-glued-in-closed-affine", "properties-lemma-maximal-points-affine", "groupoids-lemma-find-invariant-affine" ], "ref_ids": [ 2494, 2495, 15083, 3059, 9664 ] } ], "ref_ids": [] }, { "id": 2497, "type": "theorem", "label": "more-groupoids-lemma-invariant-affine-open-around-generic-point-Noetherian", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-invariant-affine-open-around-generic-point-Noetherian", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$.", "Assume $s, t$ finite, $U$ is locally Noetherian, and $u_1, \\ldots, u_m \\in U$", "points whose orbits consist of generic points of irreducible", "components of $U$. Then there exist $R$-invariant subschemes", "$V' \\subset V \\subset U$ such that", "\\begin{enumerate}", "\\item $u_1, \\ldots, u_m \\in V'$,", "\\item $V$ is open in $U$,", "\\item $V'$ and $V$ are affine,", "\\item $V' \\subset V$ is a thickening,", "\\item the morphisms $s', t'$ of the restriction $(V', R', s', t', c')$", "are finite locally free.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\{u_{j1}, \\ldots, u_{jn_j}\\}$ be the orbit of $u_j$.", "Let $W' \\subset W \\subset U$ be as in", "Lemma \\ref{lemma-finite-flat-over-almost-dense-subscheme}.", "Since $U = t(s^{-1}(\\overline{W}))$ we see that at least", "one $u_{ji} \\in \\overline{W}$. Since $u_{ji}$ is a generic point", "of an irreducible component and $U$ locally Noetherian,", "this implies that $u_{ji} \\in W$. Since $W$ is $R$-invariant, we", "conclude that $u_j \\in W$ and in fact the whole orbit is contained in $W$.", "By Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-image-affine-finite-morphism-affine-Noetherian}", "it suffices to find an $R$-invariant affine open subscheme $V'$", "of $W'$ containing $u_1, \\ldots, u_m$ (because then we can let $V \\subset W$", "be the corresponding open subscheme which will be affine).", "Thus we may replace $(U, R, s, t, c)$", "by the restriction $(W', R', s', t', c')$ to $W'$.", "In other words, we may assume we have a groupoid scheme $(U, R, s, t, c)$", "whose morphisms $s$ and $t$ are finite locally free.", "By Properties, Lemma \\ref{properties-lemma-maximal-points-affine}", "we can find an affine open containing $\\{u_{ij}\\}$", "(a locally Noetherian scheme is quasi-separated by", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian-quasi-separated}).", "Finally, we can apply", "Groupoids, Lemma \\ref{groupoids-lemma-find-invariant-affine}", "to conclude." ], "refs": [ "more-groupoids-lemma-finite-flat-over-almost-dense-subscheme", "coherent-lemma-image-affine-finite-morphism-affine-Noetherian", "properties-lemma-maximal-points-affine", "properties-lemma-locally-Noetherian-quasi-separated", "groupoids-lemma-find-invariant-affine" ], "ref_ids": [ 2494, 3337, 3059, 2953, 9664 ] } ], "ref_ids": [] }, { "id": 2498, "type": "theorem", "label": "more-groupoids-lemma-find-affine-integral", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-find-affine-integral", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$", "with $s, t$ integral. Let $g : U' \\to U$ be an integral morphism", "such that every $R$-orbit in $U$ meets $g(U')$. Let $(U', R', s', t', c')$", "be the restriction of $R$ to $U'$. If $u' \\in U'$ is contained in an", "$R'$-invariant affine open, then the image $u \\in U$ is contained", "in an $R$-invariant affine open of $U$." ], "refs": [], "proofs": [ { "contents": [ "Let $W' \\subset U'$ be an $R'$-invariant affine open.", "Set $\\tilde R = U' \\times_{g, U, t} R$ with maps", "$\\text{pr}_0 : \\tilde R \\to U'$ and $h = s \\circ \\text{pr}_1 : \\tilde R \\to U$.", "Observe that $\\text{pr}_0$ and $h$ are integral.", "It follows that $\\tilde W = \\text{pr}_0^{-1}(W')$ is affine.", "Since $W'$ is $R'$-invariant, the image", "$W = h(\\tilde W)$ is set theoretically $R$-invariant and", "$\\tilde W = h^{-1}(W)$ set theoretically (details omitted).", "Thus, if we can show that $W$ is open, then $W$ is a scheme", "and the morphism $\\tilde W \\to W$ is integral surjective", "which implies that $W$ is affine by", "Limits, Proposition \\ref{limits-proposition-affine}.", "However, our assumption on orbits meeting $U'$ implies", "that $h : \\tilde R \\to U$ is surjective. Since an", "integral surjective morphism is submersive", "(Topology, Lemma \\ref{topology-lemma-closed-morphism-quotient-topology}", "and Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed})", "it follows that $W$ is open." ], "refs": [ "limits-proposition-affine", "topology-lemma-closed-morphism-quotient-topology", "morphisms-lemma-integral-universally-closed" ], "ref_ids": [ 15129, 8204, 5441 ] } ], "ref_ids": [] }, { "id": 2499, "type": "theorem", "label": "more-groupoids-lemma-find-almost-invariant-function", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-find-almost-invariant-function", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme with $s, t$ finite and of", "finite presentation. Let $u_1, \\ldots, u_m \\in U$ be points whose $R$-orbits", "consist of generic points of irreducible components of $U$.", "Let $j : U \\to \\Spec(A)$ be an immersion.", "Let $I \\subset A$ be an ideal such that $j(U) \\cap V(I) = \\emptyset$", "and $V(I) \\cup j(U)$ is closed in $\\Spec(A)$.", "Then there exists an $h \\in I$ such that $j^{-1}D(h)$", "is an $R$-invariant affine open subscheme of $U$ containing", "$u_1, \\ldots, u_m$." ], "refs": [], "proofs": [ { "contents": [ "Let $u_1, \\ldots, u_m \\in V' \\subset V \\subset U$ be as in", "Lemma \\ref{lemma-invariant-affine-open-around-generic-point}.", "Since $U \\setminus V$ is closed in $U$, $j$ an immersion, and $V(I) \\cup j(U)$", "is closed in $\\Spec(A)$, we can find an ideal", "$J \\subset I$ such that $V(J) = V(I) \\cup j(U \\setminus V)$.", "For example we can take the ideal of elements of $I$ which", "vanish on $j(U \\setminus V)$. Thus we can replace", "$(U, R, s, t, c)$, $j : U \\to \\Spec(A)$, and $I$ by", "$(V', R', s', t', c')$, $j|_{V'} : V' \\to \\Spec(A)$, and $J$.", "In other words, we may assume that $U$ is affine and that", "$s$ and $t$ are finite locally free.", "Take any $f \\in I$ which does not vanish at all the", "points in the $R$-orbits of $u_1, \\ldots, u_m$", "(Algebra, Lemma \\ref{algebra-lemma-silly}). Consider", "$$", "g = \\text{Norm}_s(t^\\sharp(j^\\sharp(f))) \\in \\Gamma(U, \\mathcal{O}_U)", "$$", "Since $f \\in I$ and since $V(I) \\cup j(U)$ is closed we see that", "$U \\cap D(f) \\to D(f)$ is a closed immersion. Hence $f^ng$ is the", "image of an element $h \\in I$ for some $n > 0$. We claim that $h$ works.", "Namely, we have seen in", "Groupoids, Lemma \\ref{groupoids-lemma-determinant-trick}", "that $g$ is an $R$-invariant function, hence $D(g) \\subset U$", "is $R$-invariant. Since $f$ does not vanish on the orbit of $u_j$,", "the function $g$ does not vanish at $u_j$. Moreover, we have", "$V(g) \\supset V(j^\\sharp(f))$ and hence $j^{-1}D(h) = D(g)$." ], "refs": [ "more-groupoids-lemma-invariant-affine-open-around-generic-point", "algebra-lemma-silly", "groupoids-lemma-determinant-trick" ], "ref_ids": [ 2496, 378, 9657 ] } ], "ref_ids": [] }, { "id": 2500, "type": "theorem", "label": "more-groupoids-lemma-no-specializations-map-to-same-point", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-no-specializations-map-to-same-point", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme. If $s, t$ are finite,", "and $u, u' \\in R$ are distinct points in the same orbit,", "then $u'$ is not a specialization of $u$." ], "refs": [], "proofs": [ { "contents": [ "Let $r \\in R$ with $s(r) = u$ and $t(r) = u'$. If $u \\leadsto u'$", "then we can find a nontrivial specialization $r \\leadsto r'$ with", "$s(r') = u'$, see ", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-closed}.", "Set $u'' = t(r')$. Note that $u'' \\not = u'$ as there are no", "specializations in the fibres of a finite morphism.", "Hence we can continue and find a nontrivial specialization", "$r' \\leadsto r''$ with $s(r'') = u''$, etc. This shows that the", "orbit of $u$ contains an infinite sequence", "$u \\leadsto u' \\leadsto u'' \\leadsto \\ldots$", "of specializations which is nonsense as the orbit", "$t(s^{-1}(\\{u\\}))$ is finite." ], "refs": [ "schemes-lemma-quasi-compact-closed" ], "ref_ids": [ 7702 ] } ], "ref_ids": [] }, { "id": 2501, "type": "theorem", "label": "more-groupoids-lemma-get-affine", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-get-affine", "contents": [ "Let $j : V \\to \\Spec(A)$ be a quasi-compact immersion of schemes.", "Let $f \\in A$ be such that $j^{-1}D(f)$ is affine and $j(V) \\cap V(f)$", "is closed. Then $V$ is affine." ], "refs": [], "proofs": [ { "contents": [ "This follows from Morphisms, Lemma \\ref{morphisms-lemma-get-affine}", "but we will also give a direct proof.", "Let $A' = \\Gamma(V, \\mathcal{O}_V)$. Then $j' : V \\to \\Spec(A')$ is a", "quasi-compact open immersion, see", "Properties, Lemma \\ref{properties-lemma-quasi-affine}.", "Let $f' \\in A'$ be the image of $f$. Then $(j')^{-1}D(f') = j^{-1}D(f)$", "is affine. On the other hand, $j'(V) \\cap V(f')$ is a subscheme of", "$\\Spec(A')$ which maps isomorphically to the closed subscheme", "$j(V) \\cap V(f)$ of $\\Spec(A)$. Hence it is closed in $\\Spec(A')$", "for example by Schemes, Lemma \\ref{schemes-lemma-section-immersion}.", "Thus we may replace $A$ by $A'$ and assume that $j$ is an open immersion", "and $A = \\Gamma(V, \\mathcal{O}_V)$.", "\\medskip\\noindent", "In this case we claim that $j(V) = \\Spec(A)$ which finishes the proof.", "If not, then we can find a principal affine open $D(g) \\subset \\Spec(A)$", "which meets the complement and avoids the closed subset $j(V) \\cap V(f)$.", "Note that $j$ maps $j^{-1}D(f)$ isomorphically onto $D(f)$, see", "Properties, Lemma \\ref{properties-lemma-invert-f-affine}.", "Hence $D(g)$ meets $V(f)$. On the other hand, $j^{-1}D(g)$", "is a principal open of the affine open $j^{-1}D(f)$ hence affine.", "Hence by", "Properties, Lemma \\ref{properties-lemma-invert-f-affine}", "again we see that $D(g)$ is isomorphic to $j^{-1}D(g) \\subset j^{-1}D(f)$", "which implies that $D(g) \\subset D(f)$. This contradiction finishes", "the proof." ], "refs": [ "morphisms-lemma-get-affine", "properties-lemma-quasi-affine", "properties-lemma-invert-f-affine", "properties-lemma-invert-f-affine" ], "ref_ids": [ 5182, 3009, 3008, 3008 ] } ], "ref_ids": [] }, { "id": 2502, "type": "theorem", "label": "more-groupoids-lemma-find-affine-codimension-1", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-find-affine-codimension-1", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme. Let $u \\in U$. Assume", "\\begin{enumerate}", "\\item $s, t$ are finite morphisms,", "\\item $U$ is separated and locally Noetherian,", "\\item $\\dim(\\mathcal{O}_{U, u'}) \\leq 1$ for every point $u'$", "in the orbit of $u$.", "\\end{enumerate}", "Then $u$ is contained in an $R$-invariant affine open of $U$." ], "refs": [], "proofs": [ { "contents": [ "The $R$-orbit of $u$ is finite. By conditions (2) and (3) it is contained", "in an affine open $U'$ of $U$, see", "Varieties, Proposition", "\\ref{varieties-proposition-finite-set-of-points-of-codim-1-in-affine}.", "Then $t(s^{-1}(U \\setminus U'))$ is an $R$-invariant", "closed subset of $U$ which does not contain $u$. Thus", "$U \\setminus t(s^{-1}(U \\setminus U'))$ is an $R$-invariant open", "of $U'$ containing $u$.", "Replacing $U$ by this open we may assume $U$ is quasi-affine.", "\\medskip\\noindent", "By Lemma \\ref{lemma-find-affine-integral} we may replace $U$ by its reduction", "and assume $U$ is reduced. This means $R$-invariant subschemes", "$W' \\subset W \\subset U$ of", "Lemma \\ref{lemma-finite-flat-over-almost-dense-subscheme}", "are equal $W' = W$. As $U = t(s^{-1}(\\overline{W}))$ some point", "$u'$ of the $R$-orbit of $u$ is contained in $\\overline{W}$", "and by Lemma \\ref{lemma-find-affine-integral}", "we may replace $U$ by $\\overline{W}$ and $u$ by $u'$.", "Hence we may assume there is", "a dense open $R$-invariant subscheme $W \\subset U$ such that", "the morphisms $s_W, t_W$ of the restriction $(W, R_W, s_W, t_W, c_W)$ are", "finite locally free.", "\\medskip\\noindent", "If $u \\in W$ then we are done by", "Groupoids, Lemma \\ref{groupoids-lemma-find-invariant-affine}", "(because $W$ is quasi-affine so any finite set of points", "of $W$ is contained in an affine open, see", "Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-affine}).", "Thus we assume $u \\not \\in W$ and hence none of the points of the", "orbit of $u$ is in $W$. Let $\\xi \\in U$", "be a point with a nontrivial specialization to a point $u'$ in the orbit", "of $u$. Since there are no specializations among the points in the", "orbit of $u$ (Lemma \\ref{lemma-no-specializations-map-to-same-point})", "we see that $\\xi$ is not in the orbit.", "By assumption (3) we see that $\\xi$ is a generic point of $U$", "and hence $\\xi \\in W$.", "As $U$ is Noetherian there are finitely many of these", "points $\\xi_1, \\ldots, \\xi_m \\in W$. Because $s_W, t_W$ are flat the orbit", "of each $\\xi_j$ consists of generic points of irreducible components", "of $W$ (and hence $U$).", "\\medskip\\noindent", "Let $j : U \\to \\Spec(A)$ be an immersion of $U$ into an affine scheme", "(this is possible as $U$ is quasi-affine). Let $J \\subset A$", "be an ideal such that $V(J) \\cap j(W) = \\emptyset$ and $V(J) \\cup j(W)$", "is closed. Apply Lemma \\ref{lemma-find-almost-invariant-function}", "to the groupoid scheme $(W, R_W, s_W, t_W, c_W)$, the morphism", "$j|_W : W \\to \\Spec(A)$, the points $\\xi_j$, and the ideal $J$", "to find an $f \\in J$ such that $(j|_W)^{-1}D(f)$ is an $R_W$-invariant", "affine open containing $\\xi_j$ for all $j$. Since $f \\in J$", "we see that $j^{-1}D(f) \\subset W$, i.e., $j^{-1}D(f)$ is", "an $R$-invariant affine open of $U$ contained in $W$", "containing all $\\xi_j$.", "\\medskip\\noindent", "Let $Z$ be the reduced induced closed subscheme structure on", "$$", "U \\setminus j^{-1}D(f) = j^{-1}V(f).", "$$", "Then $Z$ is set theoretically", "$R$-invariant (but it may not be scheme theoretically $R$-invariant).", "Let $(Z, R_Z, s_Z, t_Z, c_Z)$ be the restriction of $R$ to $Z$.", "Since $Z \\to U$ is finite, it follows that $s_Z$ and $t_Z$ are finite.", "Since $u \\in Z$ the orbit of $u$ is in $Z$ and agrees with the", "$R_Z$-orbit of $u$ viewed as a point of $Z$. Since", "$\\dim(\\mathcal{O}_{U, u'}) \\leq 1$ and since $\\xi_j \\not \\in Z$", "for all $j$, we see that $\\dim(\\mathcal{O}_{Z, u'}) \\leq 0$ for", "all $u'$ in the orbit of $u$. In other words, the $R_Z$-orbit of $u$", "consists of generic points of irreducible components of $Z$.", "\\medskip\\noindent", "Let $I \\subset A$ be an ideal such that $V(I) \\cap j(U) =\\emptyset$", "and $V(I) \\cup j(U)$ is closed. Apply", "Lemma \\ref{lemma-find-almost-invariant-function} to", "the groupoid scheme $(Z, R_Z, s_Z, t_Z, c_Z)$, the restriction $j|_Z$,", "the ideal $I$, and the point $u \\in Z$ to obtain $h \\in I$ such that", "$j^{-1}D(h) \\cap Z$ is an $R_Z$-invariant open affine containing $u$.", "\\medskip\\noindent", "Consider the $R_W$-invariant (Groupoids, Lemma", "\\ref{groupoids-lemma-determinant-trick}) function", "$$", "g = ", "\\text{Norm}_{s_W}(t_W^\\sharp(j^\\sharp(h)|_W)) \\in \\Gamma(W, \\mathcal{O}_W)", "$$", "(In the following we only need the restriction of $g$ to $j^{-1}D(f)$ and", "in this case the norm is along a finite locally free morphism of affines.)", "We claim that", "$$", "V = (W_g \\cap j^{-1}D(f)) \\cup (j^{-1}D(h) \\cap Z)", "$$", "is an $R$-invariant affine open of $U$ which finishes the proof of the lemma.", "It is set theoretically $R$-invariant by construction. As $V$ is a", "constuctible set, to see that it is open it suffices to show it is", "closed under generalization in $U$ (Topology, Lemma", "\\ref{topology-lemma-characterize-closed-Noetherian}", "or the more general", "Topology, Lemma", "\\ref{topology-lemma-constructible-stable-specialization-closed}).", "Since $W_g \\cap j^{-1}D(f)$ is open in $U$, it suffices to consider", "a specialization $u_1 \\leadsto u_2$ of $U$ with", "$u_2 \\in j^{-1}D(h) \\cap Z$.", "This means that $h$ is nonzero in $j(u_2)$ and $u_2 \\in Z$.", "If $u_1 \\in Z$, then $j(u_1) \\leadsto j(u_2)$ and since", "$h$ is nonzero in $j(u_2)$ it is nonzero in $j(u_1)$ which", "implies $u_1 \\in V$. If $u_1 \\not \\in Z$ and", "also not in $W_g \\cap j^{-1}D(f)$, then $u_1 \\in W$, $u_1 \\not \\in W_g$", "because the complement of $Z = j^{-1}V(f)$ is contained in $W \\cap j^{-1}D(f)$.", "Hence there exists a point $r_1 \\in R$ with $s(r_1) = u_1$", "such that $h$ is zero in $t(r_1)$. Since $s$ is finite we", "can find a specialization $r_1 \\leadsto r_2$ with $s(r_2) = u_2$.", "However, then we conclude that $f$ is zero in $u'_2 = t(r_2)$", "which contradicts the fact that $j^{-1}D(h) \\cap Z$", "is $R$-invariant and $u_2$ is in it. Thus $V$ is open.", "\\medskip\\noindent", "Observe that $V \\subset j^{-1}D(h)$ for our function $h \\in I$.", "Thus we obtain an immersion", "$$", "j' : V \\longrightarrow \\Spec(A_h)", "$$", "Let $f' \\in A_h$ be the image of $f$. Then $(j')^{-1}D(f')$", "is the principal open determined by $g$ in the affine", "open $j^{-1}D(f)$ of $U$.", "Hence $(j')^{-1}D(f)$ is affine. Finally,", "$j'(V) \\cap V(f') = j'(j^{-1}D(h) \\cap Z)$", "is closed in $\\Spec(A_h/(f')) = \\Spec((A/f)_h) = D(h) \\cap V(f)$", "by our choice of $h \\in I$ and the ideal $I$. Hence we can apply", "Lemma \\ref{lemma-get-affine}", "to conclude that $V$ is affine as claimed above." ], "refs": [ "varieties-proposition-finite-set-of-points-of-codim-1-in-affine", "more-groupoids-lemma-find-affine-integral", "more-groupoids-lemma-finite-flat-over-almost-dense-subscheme", "more-groupoids-lemma-find-affine-integral", "groupoids-lemma-find-invariant-affine", "properties-lemma-ample-finite-set-in-affine", "more-groupoids-lemma-no-specializations-map-to-same-point", "more-groupoids-lemma-find-almost-invariant-function", "more-groupoids-lemma-find-almost-invariant-function", "groupoids-lemma-determinant-trick", "topology-lemma-characterize-closed-Noetherian", "topology-lemma-constructible-stable-specialization-closed", "more-groupoids-lemma-get-affine" ], "ref_ids": [ 11139, 2498, 2494, 2498, 9664, 3062, 2500, 2499, 2499, 9657, 8290, 8307, 2501 ] } ], "ref_ids": [] }, { "id": 2503, "type": "theorem", "label": "more-groupoids-lemma-sits-in-functions", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-sits-in-functions", "contents": [ "Let $X$ be an ind-quasi-affine scheme. Let $E \\subset X$ be an", "intersection of a nonempty family of quasi-compact opens of $X$.", "Set $A = \\Gamma(E, \\mathcal{O}_X|_E)$ and $Y = \\Spec(A)$.", "Then the canonical morphism", "$$", "j : (E, \\mathcal{O}_X|_E) \\longrightarrow (Y, \\mathcal{O}_Y)", "$$", "of Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}", "determines an isomorphism", "$(E, \\mathcal{O}_X|_E) \\to (E', \\mathcal{O}_Y|_{E'})$", "where $E' \\subset Y$ is an intersection of quasi-compact opens.", "If $W \\subset E$ is open in $X$, then $j(W)$ is open in $Y$." ], "refs": [ "schemes-lemma-morphism-into-affine" ], "proofs": [ { "contents": [ "Note that $(E, \\mathcal{O}_X|_E)$ is a locally ringed space so that", "Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine} applies", "to $A \\to \\Gamma(E, \\mathcal{O}_X|_E)$. Write $E = \\bigcap_{i \\in I} U_i$", "with $I \\not = \\emptyset$ and $U_i \\subset X$ quasi-compact open.", "We may and do assume that for $i, j \\in I$ there exists a $k \\in I$ with", "$U_k \\subset U_i \\cap U_j$. Set $A_i = \\Gamma(U_i, \\mathcal{O}_{U_i})$.", "We obtain commutative diagrams", "$$", "\\xymatrix{", "(E, \\mathcal{O}_X|_E) \\ar[r] \\ar[d] &", "(\\Spec(A), \\mathcal{O}_{\\Spec(A)}) \\ar[d] \\\\", "(U_i, \\mathcal{O}_{U_i}) \\ar[r] &", "(\\Spec(A_i), \\mathcal{O}_{\\Spec(A_i)})", "}", "$$", "Since $U_i$ is quasi-affine, we see that $U_i \\to \\Spec(A_i)$", "is a quasi-compact open immersion. On the other hand", "$A = \\colim A_i$. Hence $\\Spec(A) = \\lim \\Spec(A_i)$ as topological", "spaces (Limits, Lemma \\ref{limits-lemma-topology-limit}). Since", "$E = \\lim U_i$ (by Topology, Lemma \\ref{topology-lemma-make-spectral-space})", "we see that $E \\to \\Spec(A)$ is a homeomorphism onto its", "image $E'$ and that $E'$ is the intersection of the inverse images", "of the opens $U_i \\subset \\Spec(A_i)$ in $\\Spec(A)$. For any", "$e \\in E$ the local ring $\\mathcal{O}_{X, e}$ is the value", "of $\\mathcal{O}_{U_i, e}$ which is the same as the value on $\\Spec(A)$.", "\\medskip\\noindent", "To prove the final assertion of the lemma we argue as follows.", "Pick $i, j \\in I$ with $U_i \\subset U_j$.", "Consider the following commutative diagrams", "$$", "\\xymatrix{", "U_i \\ar[r] \\ar[d] & \\Spec(A_i) \\ar[d] \\\\", "U_i \\ar[r] & \\Spec(A_j)", "}", "\\quad\\quad", "\\xymatrix{", "W \\ar[r] \\ar[d] & \\Spec(A_i) \\ar[d] \\\\", "W \\ar[r] & \\Spec(A_j)", "}", "\\quad\\quad", "\\xymatrix{", "W \\ar[r] \\ar[d] & \\Spec(A) \\ar[d] \\\\", "W \\ar[r] & \\Spec(A_j)", "}", "$$", "By Properties, Lemma", "\\ref{properties-lemma-cartesian-diagram-quasi-affine}", "the first diagram is cartesian. Hence the second is cartesian as well.", "Passing to the limit we find that the third diagram", "is cartesian, so the top horizontal arrow of this diagram is an open immersion." ], "refs": [ "schemes-lemma-morphism-into-affine", "limits-lemma-topology-limit", "topology-lemma-make-spectral-space", "properties-lemma-cartesian-diagram-quasi-affine" ], "ref_ids": [ 7655, 15036, 8323, 3010 ] } ], "ref_ids": [ 7655 ] }, { "id": 2504, "type": "theorem", "label": "more-groupoids-lemma-affine-base-change", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-affine-base-change", "contents": [ "Suppose given a cartesian diagram", "$$", "\\xymatrix{", "X \\ar[d]_f \\ar[r] & \\Spec(B) \\ar[d] \\\\", "Y \\ar[r] & \\Spec(A)", "}", "$$", "of schemes. Let $E \\subset Y$ be an intersection of a nonempty family", "of quasi-compact opens of $Y$. Then", "$$", "\\Gamma(f^{-1}(E), \\mathcal{O}_X|_{f^{-1}(E)}) =", "\\Gamma(E, \\mathcal{O}_Y|_E) \\otimes_A B", "$$", "provided $Y$ is quasi-separated and $A \\to B$ is flat." ], "refs": [], "proofs": [ { "contents": [ "Write $E = \\bigcap_{i \\in I} V_i$ with $V_i \\subset Y$ quasi-compact open.", "We may and do assume that for $i, j \\in I$ there exists a $k \\in I$ with", "$V_k \\subset V_i \\cap V_j$. Then we have similarly that", "$f^{-1}(E) = \\bigcap_{i \\in I} f^{-1}(V_i)$ in $X$.", "Thus the result follows from equation (\\ref{equation-sections-of-intersection})", "and the corresponding result for $V_i$ and $f^{-1}(V_i)$ which is", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}." ], "refs": [ "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 3298 ] } ], "ref_ids": [] }, { "id": 2505, "type": "theorem", "label": "more-groupoids-lemma-ind-quasi-affine", "categories": [ "more-groupoids" ], "title": "more-groupoids-lemma-ind-quasi-affine", "contents": [ "Let $S$ be a scheme. Let $\\{X_i \\to S\\}_{i\\in I}$ be an fpqc covering.", "Let $(V_i/X_i, \\varphi_{ij})$ be a descent datum relative to", "$\\{X_i \\to S\\}$, see Descent, Definition", "\\ref{descent-definition-descent-datum-for-family-of-morphisms}. ", "If each morphism $V_i \\to X_i$ is ind-quasi-affine, then the descent datum", "is effective." ], "refs": [ "descent-definition-descent-datum-for-family-of-morphisms" ], "proofs": [ { "contents": [ "Being ind-quasi-affine is a property of morphisms of schemes", "which is preserved under any base change, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-base-change-ind-quasi-affine}.", "Hence Descent, Lemma \\ref{descent-lemma-descending-types-morphisms} applies", "and it suffices to prove the statement of the lemma", "in case the fpqc-covering is given by a single", "$\\{X \\to S\\}$ flat surjective morphism of affines.", "Say $X = \\Spec(A)$ and $S = \\Spec(R)$ so", "that $R \\to A$ is a faithfully flat ring map.", "Let $(V, \\varphi)$ be a descent datum relative to $X$ over $S$", "and assume that $V \\to X$ is ind-quasi-affine, in other words,", "$V$ is ind-quasi-affine.", "\\medskip\\noindent", "Let $(U, R, s, t, c)$ be the groupoid scheme over $S$ with", "$U = X$ and $R = X \\times_S X$ and $s$, $t$, $c$ as usual.", "By Groupoids, Lemma \\ref{groupoids-lemma-cartesian-equivalent-descent-datum}", "the pair $(V, \\varphi)$ corresponds to a cartesian morphism", "$(U', R', s', t', c') \\to (U, R, s, t, c)$ of groupoid schemes.", "Let $u' \\in U'$ be any point. By", "Groupoids, Lemmas \\ref{groupoids-lemma-constructing-invariant-opens},", "\\ref{groupoids-lemma-first-observation}, and", "\\ref{groupoids-lemma-second-observation}", "we can choose $u' \\in W \\subset E \\subset U'$", "where $W$ is open and $R'$-invariant, and", "$E$ is set-theoretically $R'$-invariant and", "an intersection of a nonempty family of quasi-compact opens.", "\\medskip\\noindent", "Translating back to $(V, \\varphi)$, for any $v \\in V$ we can find", "$v \\in W \\subset E \\subset V$ with the following properties:", "(a) $W$ is open and $\\varphi(W \\times_S X) = X \\times_S W$ and", "(b) $E$ an intersection of quasi-compact opens and", "$\\varphi(E \\times_S X) = X \\times_S E$ set-theoretically.", "Here we use the notation $E \\times_S X$ to mean the", "inverse image of $E$ in $V \\times_S X$ by the projection morphism and", "similarly for $X \\times_S E$. By Lemma \\ref{lemma-affine-base-change}", "this implies that $\\varphi$ defines an isomorphism", "\\begin{align*}", "\\Gamma(E, \\mathcal{O}_V|_E) \\otimes_R A", "& =", "\\Gamma(E \\times_S X, \\mathcal{O}_{V \\times_S X}|_{E \\times_S X}) \\\\", "& \\to", "\\Gamma(X \\times_S E, \\mathcal{O}_{X \\times_S V}|_{X \\times_S E}) \\\\", "& =", "A \\otimes_R \\Gamma(E, \\mathcal{O}_V|_E)", "\\end{align*}", "of $A \\otimes_R A$-algebras which we will call $\\psi$.", "The cocycle condition for $\\varphi$", "translates into the cocycle condition for $\\psi$ as in", "Descent, Definition \\ref{descent-definition-descent-datum-modules}", "(details omitted). By Descent, Proposition", "\\ref{descent-proposition-descent-module}", "we find an $R$-algebra $R'$ and an isomorphism", "$\\chi : R' \\otimes_R A \\to \\Gamma(E, \\mathcal{O}_V|_E)$", "of $A$-algebras, compatible with $\\psi$ and the", "canonical descent datum on $R' \\otimes_R A$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-sits-in-functions} we obtain a canonical ``embedding''", "$$", "j : (E, \\mathcal{O}_V|_E) \\longrightarrow", "\\Spec(\\Gamma(E, \\mathcal{O}_V|_E)) = \\Spec(R' \\otimes_R A)", "$$", "of locally ringed spaces. The construction of this map is canonical", "and we get a commutative diagram", "$$", "\\xymatrix{", "& E \\times_S X \\ar[rr]_\\varphi \\ar[ld] \\ar[rd]^{j'} & &", "X \\times_S E \\ar[rd] \\ar[ld]_{j''} \\\\", "E \\ar[rd]^j & &", "\\Spec(R' \\otimes_R A \\otimes_R A) \\ar[ld] \\ar[rd] & &", "E \\ar[ld]_j \\\\", "& \\Spec(R' \\otimes_R A) \\ar[rd] && \\Spec(R' \\otimes_R A) \\ar[ld] \\\\", "& & \\Spec(R')", "}", "$$", "where $j'$ and $j''$ come from the same construction applied to", "$E \\times_S X \\subset V \\times_S X$ and $X \\times_S E \\subset X \\times_S V$", "via $\\chi$ and the identifications used to construct $\\psi$.", "It follows that $j(W)$ is an open subscheme of $\\Spec(R' \\otimes_R A)$", "whose inverse image under the two projections", "$\\Spec(R' \\otimes_R A \\otimes_R A) \\to \\Spec(R' \\otimes_R A)$", "are equal. By Descent, Lemma \\ref{descent-lemma-open-fpqc-covering}", "we find an open $W_0 \\subset \\Spec(R')$ whose base change", "to $\\Spec(A)$ is $j(W)$. Contemplating the diagram above", "we see that the descent datum $(W, \\varphi|_{W \\times_S X})$", "is effective. By Descent, Lemma", "\\ref{descent-lemma-effective-for-fpqc-is-local-upstairs}", "we see that our descent datum is effective." ], "refs": [ "more-morphisms-lemma-base-change-ind-quasi-affine", "descent-lemma-descending-types-morphisms", "groupoids-lemma-cartesian-equivalent-descent-datum", "groupoids-lemma-constructing-invariant-opens", "groupoids-lemma-first-observation", "groupoids-lemma-second-observation", "more-groupoids-lemma-affine-base-change", "descent-definition-descent-datum-modules", "descent-proposition-descent-module", "more-groupoids-lemma-sits-in-functions", "descent-lemma-open-fpqc-covering", "descent-lemma-effective-for-fpqc-is-local-upstairs" ], "ref_ids": [ 14043, 14747, 9654, 9645, 9646, 9647, 2504, 14759, 14752, 2503, 14637, 14746 ] } ], "ref_ids": [ 14777 ] }, { "id": 2509, "type": "theorem", "label": "examples-lemma-lim-not-quasi-compact", "categories": [ "examples" ], "title": "examples-lemma-lim-not-quasi-compact", "contents": [ "There exists an inverse system of quasi-compact topological spaces", "over $\\mathbf{N}$ whose limit is not quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2510, "type": "theorem", "label": "examples-lemma-noncomplete-completion", "categories": [ "examples" ], "title": "examples-lemma-noncomplete-completion", "contents": [ "There exists a local ring $R$ and a maximal ideal $\\mathfrak m$ such that", "the completion $R^\\wedge$ of $R$ with respect to $\\mathfrak m$ has the", "following properties", "\\begin{enumerate}", "\\item $R^\\wedge$ is local, but its maximal ideal is not equal to", "$\\mathfrak m R^\\wedge$,", "\\item $R^\\wedge$ is not a complete local ring, and", "\\item $R^\\wedge$ is not $\\mathfrak m$-adically complete as an $R$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from the discussion above as (with $R = k[x_1, x_2, x_3, \\ldots]$)", "the completion of the localization $R_{\\mathfrak m}$ is equal to the", "completion of $R$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2511, "type": "theorem", "label": "examples-lemma-noncomplete-quotient", "categories": [ "examples" ], "title": "examples-lemma-noncomplete-quotient", "contents": [ "There exists a ring $R$ complete with respect to a principal ideal", "$I$ and a principal ideal $J$ such that $R/J$ is not $I$-adically", "complete." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2512, "type": "theorem", "label": "examples-lemma-completion-not-exact", "categories": [ "examples" ], "title": "examples-lemma-completion-not-exact", "contents": [ "\\begin{slogan}", "Completion is neither left nor right exact in general.", "\\end{slogan}", "Completion is not an exact functor in general; it is not even", "right exact in general. This holds even when $I$ is finitely", "generated on the category of finitely presented modules." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2513, "type": "theorem", "label": "examples-lemma-complete-modules-not-abelian", "categories": [ "examples" ], "title": "examples-lemma-complete-modules-not-abelian", "contents": [ "Let $R$ be a ring and let $I \\subset R$ be a finitely generated ideal.", "The category of $I$-adically complete $R$-modules has kernels and", "cokernels but is not abelian in general." ], "refs": [], "proofs": [ { "contents": [ "See above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2514, "type": "theorem", "label": "examples-lemma-derived-complete-modules", "categories": [ "examples" ], "title": "examples-lemma-derived-complete-modules", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be an ideal.", "The category $\\mathcal{C}$ of derived complete modules", "is abelian and the inclusion functor $F : \\mathcal{C} \\to \\text{Mod}_A$", "is exact and commutes with arbitrary limits.", "If $I$ is finitely generated, then $\\mathcal{C}$ has", "arbitrary direct sums and colimits, but $F$ does not commute with these", "in general. Finally, filtered colimits are not exact in $\\mathcal{C}$", "in general, hence $\\mathcal{C}$ is not a Grothendieck abelian category." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2515, "type": "theorem", "label": "examples-lemma-countable-fg-tensor", "categories": [ "examples" ], "title": "examples-lemma-countable-fg-tensor", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module which is countable.", "Then $M$ is a finite $R$-module if and only if", "$M \\otimes_R R^\\mathbf{N} \\to M^\\mathbf{N}$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "If $M$ is a finite module, then the map is surjective by Algebra, Proposition", "\\ref{algebra-proposition-fg-tensor}. Conversely, assume the map is surjective.", "Let $m_1, m_2, m_3, \\ldots$ be an enumeration of the elements of $M$.", "Let $\\sum_{j = 1, \\ldots, m} x_j \\otimes a_j$ be an element of the", "tensor product mapping to the element $(m_n) \\in M^\\mathbf{N}$. Then", "we see that $x_1, \\ldots, x_m$ generate $M$ over $R$ as in the proof of", "Algebra, Proposition \\ref{algebra-proposition-fg-tensor}." ], "refs": [ "algebra-proposition-fg-tensor", "algebra-proposition-fg-tensor" ], "ref_ids": [ 1415, 1415 ] } ], "ref_ids": [] }, { "id": 2516, "type": "theorem", "label": "examples-lemma-countable-fp-tensor", "categories": [ "examples" ], "title": "examples-lemma-countable-fp-tensor", "contents": [ "Let $R$ be a countable ring. Let $M$ be a countable $R$-module. Then $M$", "is finitely presented if and only if the canonical map", "$M \\otimes_R R^\\mathbf{N} \\to M^\\mathbf{N}$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "If $M$ is a finitely presented module, then the map is an isomorphism", "by Algebra, Proposition \\ref{algebra-proposition-fp-tensor}. Conversely,", "assume the map is an isomorphism. By Lemma \\ref{lemma-countable-fg-tensor}", "the module $M$ is finite. Choose a surjection $R^{\\oplus m} \\to M$ with", "kernel $K$. Then $K$ is countable as a submodule of $R^{\\oplus m}$.", "Arguing as in the proof of Algebra, Proposition", "\\ref{algebra-proposition-fp-tensor} we see that", "$K \\otimes_R R^\\mathbf{N} \\to K^\\mathbf{N}$ is surjective.", "Hence we conclude that $K$ is a finite $R$-module by", "Lemma \\ref{lemma-countable-fg-tensor}.", "Thus $M$ is finitely presented." ], "refs": [ "algebra-proposition-fp-tensor", "examples-lemma-countable-fg-tensor", "algebra-proposition-fp-tensor", "examples-lemma-countable-fg-tensor" ], "ref_ids": [ 1416, 2515, 1416, 2515 ] } ], "ref_ids": [] }, { "id": 2517, "type": "theorem", "label": "examples-lemma-countable-coherent", "categories": [ "examples" ], "title": "examples-lemma-countable-coherent", "contents": [ "Let $R$ be a countable ring. Then $R$ is coherent if and only if", "$R^\\mathbf{N}$ is a flat $R$-module." ], "refs": [], "proofs": [ { "contents": [ "If $R$ is coherent, then $R^\\mathbf{N}$ is a flat module by", "Algebra, Proposition \\ref{algebra-proposition-characterize-coherent}.", "Assume $R^\\mathbf{N}$ is flat. Let $I \\subset R$ be a finitely", "generated ideal. To prove the lemma we show that $I$ is finitely", "presented as an $R$-module. Namely, the map", "$I \\otimes_R R^\\mathbf{N} \\to R^\\mathbf{N}$ is", "injective as $R^\\mathbf{N}$ is flat and its image is", "$I^\\mathbf{N}$ by Lemma \\ref{lemma-countable-fg-tensor}.", "Thus we conclude by Lemma \\ref{lemma-countable-fp-tensor}." ], "refs": [ "algebra-proposition-characterize-coherent", "examples-lemma-countable-fg-tensor", "examples-lemma-countable-fp-tensor" ], "ref_ids": [ 1418, 2515, 2516 ] } ], "ref_ids": [] }, { "id": 2518, "type": "theorem", "label": "examples-lemma-completion-polynomial-ring-not-flat", "categories": [ "examples" ], "title": "examples-lemma-completion-polynomial-ring-not-flat", "contents": [ "There exists a ring such that the completion $R[[x]]$ of $R[x]$", "at $(x)$ is not flat over $R$ and a fortiori not flat over $R[x]$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2519, "type": "theorem", "label": "examples-lemma-almost-integral-when-powerseries-flat", "categories": [ "examples" ], "title": "examples-lemma-almost-integral-when-powerseries-flat", "contents": [ "Let $R$ be a domain with fraction field $K$.", "If $R[[x]]$ is flat over $R[x]$, then $R$ is normal if and only", "if $R$ is completely normal", "(Algebra, Definition \\ref{algebra-definition-almost-integral})." ], "refs": [ "algebra-definition-almost-integral" ], "proofs": [ { "contents": [ "Suppose we have $\\alpha \\in K$ and a nonzero $r \\in R$ such that", "$r \\alpha^n \\in R$ for all $n \\geq 1$. Then we consider", "$f = \\sum r \\alpha^{n - 1} x^n$ in $R[[x]]$. Write $\\alpha = a/b$", "for $a, b \\in R$ with $b$ nonzero. Then we see that $(a x - b)f = -rb$.", "It follows that $rb$ is in the ideal $(ax - b)R[[x]]$.", "Let $S = \\{h \\in R[x] : h(0) = 1\\}$. This is a multiplicative subset", "and flatness of $R[x] \\to R[[x]]$ implies that $S^{-1}R[x] \\to R[[x]]$", "is faithfully flat (details omitted; hint: use Algebra, Lemma", "\\ref{algebra-lemma-ff-rings}). Hence", "$$", "S^{-1}R/(ax - b)S^{-1}R \\to R[[x]]/(ax - b)R[[x]]", "$$", "is injective. We conclude that", "$h rb = (ax - b) g$ for some $h \\in S$ and $g \\in R[x]$.", "Writing $h = 1 + h_1 x + \\ldots + h_d x^d$ shows that we obtain", "$$", "1 + h_1 x + \\ldots + h_d x^d = (1/r)(\\alpha x - 1)g", "$$", "This factorization in $K[x]$ gives a corresponding factorization", "in $K[x^{-1}]$ which shows that $\\alpha$ is the root of a monic", "polynomial with coefficients in $R$ as desired." ], "refs": [ "algebra-lemma-ff-rings" ], "ref_ids": [ 536 ] } ], "ref_ids": [ 1453 ] }, { "id": 2520, "type": "theorem", "label": "examples-lemma-completion-polynomial-ring-not-flat-bis", "categories": [ "examples" ], "title": "examples-lemma-completion-polynomial-ring-not-flat-bis", "contents": [ "If $R$ is a valuation ring of dimension $> 1$, then $R[[x]]$", "is flat over $R$ but not flat over $R[x]$." ], "refs": [], "proofs": [ { "contents": [ "The arguments above show that this is true if we can show that", "$R$ is not completely normal (valuation rings are", "normal, see Algebra, Lemma \\ref{algebra-lemma-valuation-ring-normal}).", "Let $\\mathfrak p \\subset \\mathfrak m \\subset R$ be a chain of primes.", "Pick nonzero $x \\in \\mathfrak p$ and $y \\in \\mathfrak m \\setminus \\mathfrak p$.", "Then $x y^{-n} \\in R$ for all $n \\geq 1$ (if not then $y^n/x \\in R$", "which is absurd because $y \\not \\in \\mathfrak p$). Hence $1/y$ is", "almost integral over $R$ but not in $R$." ], "refs": [ "algebra-lemma-valuation-ring-normal" ], "ref_ids": [ 616 ] } ], "ref_ids": [] }, { "id": 2521, "type": "theorem", "label": "examples-lemma-nonflat-completion-localization", "categories": [ "examples" ], "title": "examples-lemma-nonflat-completion-localization", "contents": [ "There exists a ring $A$ complete with respect to a principal ideal $I$", "and an element $f \\in A$ such that the $I$-adic completion", "$A_f^\\wedge$ of $A_f$ is not flat over $A$." ], "refs": [], "proofs": [ { "contents": [ "Set $A = R[[x]]$ and $I = (x)$ and observe that $R_f[[x]]$", "is the completion of $R[[x]]_f$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2522, "type": "theorem", "label": "examples-lemma-quasi-coherent-not-abelian", "categories": [ "examples" ], "title": "examples-lemma-quasi-coherent-not-abelian", "contents": [ "The category of quasi-coherent\\footnote{With quasi-coherent modules", "as defined above. Due to how things are setup in the Stacks project,", "this is really the correct definition; as seen above our definition", "agrees with what one would naively have defined to be quasi-coherent modules", "on $\\text{Spf}(A)$, namely complete $A$-modules.}", "modules on a formal algebraic space", "$X$ is not abelian in general, even if $X$ is a Noetherian affine", "formal algebraic space." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2523, "type": "theorem", "label": "examples-lemma-strange-regular-sequence", "categories": [ "examples" ], "title": "examples-lemma-strange-regular-sequence", "contents": [ "There exists a local ring $R$ and a regular sequence $x, y, z$", "(in the maximal ideal) such that there exists a nonzero element", "$\\delta \\in R/zR$ with $x\\delta = y\\delta = 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $R = k[x, y, z] \\oplus E$ where $E$ is the module above considered", "as a square zero ideal. Then it is clear that $x, y, z$ is a regular", "sequence in $R$, and that the element $\\delta \\in E/zE \\subset R/zR$", "gives an element with the desired properties. To get a local example", "we may localize $R$ at the maximal ideal $\\mathfrak m = (x, y, z, E)$.", "The sequence $x, y, z$ remains a regular sequence (as localization is", "exact), and the element $\\delta$ remains nonzero as it is supported", "at $\\mathfrak m$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2524, "type": "theorem", "label": "examples-lemma-base-change-regular-sequence", "categories": [ "examples" ], "title": "examples-lemma-base-change-regular-sequence", "contents": [ "There exists a local homomorphism of local rings $A \\to B$", "and a regular sequence $x, y$ in the maximal ideal of $B$ such that", "$B/(x, y)$ is flat over $A$, but such that the images", "$\\overline{x}, \\overline{y}$ of $x, y$ in $B/\\mathfrak m_AB$ do not", "form a regular sequence, nor even a Koszul-regular sequence." ], "refs": [], "proofs": [ { "contents": [ "Set $A = k[z]_{(z)}$ and let $B = (k[x, y, z] \\oplus E)_{(x, y, z, E)}$.", "Since $x, y, z$ is a regular sequence in $B$, see proof of", "Lemma \\ref{lemma-strange-regular-sequence},", "we see that $x, y$ is a regular sequence in $B$ and that", "$B/(x, y)$ is a torsion free $A$-module, hence flat.", "On the other hand, there exists a nonzero element", "$\\delta \\in B/\\mathfrak m_AB = B/zB$ which is annihilated", "by $\\overline{x}, \\overline{y}$. Hence", "$H_2(K_\\bullet(B/\\mathfrak m_AB, \\overline{x}, \\overline{y})) \\not = 0$.", "Thus $\\overline{x}, \\overline{y}$ is not Koszul-regular, in particular", "it is not a regular sequence, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-regular-koszul-regular}." ], "refs": [ "examples-lemma-strange-regular-sequence", "more-algebra-lemma-regular-koszul-regular" ], "ref_ids": [ 2523, 9973 ] } ], "ref_ids": [] }, { "id": 2525, "type": "theorem", "label": "examples-lemma-nonreduced-recompletion", "categories": [ "examples" ], "title": "examples-lemma-nonreduced-recompletion", "contents": [ "There exists a local Noetherian $2$-dimensional domain $(B, \\mathfrak m)$", "complete with respect to a principal ideal $I = (b)$ and an", "element $f \\in \\mathfrak m$, $f \\not \\in I$ such that", "the $I$-adic completion $C = (B_f)^\\wedge$ of the principal", "localization $B_f$ is nonreduced and even such that", "$C_b = C[1/b] = (B_f)^\\wedge[1/b]$ is nonreduced." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2526, "type": "theorem", "label": "examples-lemma-Noetherian-Jacobson", "categories": [ "examples" ], "title": "examples-lemma-Noetherian-Jacobson", "contents": [ "There exists a Jacobson, universally catenary, Noetherian domain $B$", "with maximal ideals $\\mathfrak m_1, \\mathfrak m_2$ such that", "$\\dim(B_{\\mathfrak m_1}) = 1$ and $\\dim(B_{\\mathfrak m_2}) = 2$." ], "refs": [], "proofs": [ { "contents": [ "The construction of $B$ is given above. We just point out that", "$B$ is universally catenary by", "Algebra, Lemma \\ref{algebra-lemma-localization-catenary} and", "Morphisms, Lemma \\ref{morphisms-lemma-ubiquity-uc}." ], "refs": [ "algebra-lemma-localization-catenary", "morphisms-lemma-ubiquity-uc" ], "ref_ids": [ 932, 5217 ] } ], "ref_ids": [] }, { "id": 2527, "type": "theorem", "label": "examples-lemma-quasi-affine-normalization-not-quasi-affine", "categories": [ "examples" ], "title": "examples-lemma-quasi-affine-normalization-not-quasi-affine", "contents": [ "Let $k$ be a field.", "There exists a variety $X$ whose normalization is quasi-affine but", "which is itself not quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "See discussion above and (insert future reference on normalization here)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2528, "type": "theorem", "label": "examples-lemma-complement-of-affine-does-not-contain-qc-dense-open", "categories": [ "examples" ], "title": "examples-lemma-complement-of-affine-does-not-contain-qc-dense-open", "contents": [ "Nonexistence quasi-compact opens of affines:", "\\begin{enumerate}", "\\item There exist an affine scheme $S$ and affine open $U \\subset S$", "such that there is no quasi-compact open $V \\subset S$ with", "$U \\cap V = \\emptyset$ and $U \\cup V$ dense in $S$.", "\\item There exists an affine scheme $S$ and a closed point $s \\in S$ such that", "$S \\setminus \\{s\\}$ does not contain a quasi-compact dense open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2529, "type": "theorem", "label": "examples-lemma-no-dense-separated-quasi-compact-open-in-qcqs", "categories": [ "examples" ], "title": "examples-lemma-no-dense-separated-quasi-compact-open-in-qcqs", "contents": [ "There exists a quasi-compact and quasi-separated scheme $X$ which does", "not contain a separated quasi-compact dense open." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2530, "type": "theorem", "label": "examples-lemma-nonexistence-qc-dense-open-subscheme", "categories": [ "examples" ], "title": "examples-lemma-nonexistence-qc-dense-open-subscheme", "contents": [ "There exists a quasi-compact and quasi-separated algebraic space", "which does not contain a quasi-compact dense open subscheme." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2531, "type": "theorem", "label": "examples-lemma-cannot-embed-into-affine", "categories": [ "examples" ], "title": "examples-lemma-cannot-embed-into-affine", "contents": [ "There exists a finite type morphism of algebraic spaces $Y \\to X$", "with $Y$ affine and $X$ quasi-separated, such that there does not exist", "an immersion $Y \\to \\mathbf{A}^n_X$ over $X$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2532, "type": "theorem", "label": "examples-lemma-pushforward-quasi-coherent", "categories": [ "examples" ], "title": "examples-lemma-pushforward-quasi-coherent", "contents": [ "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "is sharp in the sense that one can neither drop the assumption", "of quasi-compactness nor the assumption of quasi-separatedness." ], "refs": [ "schemes-lemma-push-forward-quasi-coherent" ], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 7730 ] }, { "id": 2533, "type": "theorem", "label": "examples-lemma-locally-principal-not-invertible", "categories": [ "examples" ], "title": "examples-lemma-locally-principal-not-invertible", "contents": [ "There exists a domain $A$ and a nonzero ideal $I \\subset A$", "such that $I_\\mathfrak q \\subset A_\\mathfrak q$ is a principal", "ideal for all primes $\\mathfrak q \\subset A$ but $I$ is not an invertible", "$A$-module." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2534, "type": "theorem", "label": "examples-lemma-finite-flat-non-projective", "categories": [ "examples" ], "title": "examples-lemma-finite-flat-non-projective", "contents": [ "Strange flat modules.", "\\begin{enumerate}", "\\item There exists a ring $R$ and a finite flat $R$-module $M$ which is", "not projective.", "\\item There exists a closed immersion which is flat but not open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2535, "type": "theorem", "label": "examples-lemma-ideal-generated-by-idempotents-projective", "categories": [ "examples" ], "title": "examples-lemma-ideal-generated-by-idempotents-projective", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal generated by", "a countable collection of idempotents. Then $I$ is projective", "as an $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Say $I = (e_1, e_2, e_3, \\ldots)$ with $e_n$ an idempotent of $R$.", "After inductively replacing $e_{n + 1}$ by $e_n + (1 - e_n)e_{n + 1}$", "we may assume that $(e_1) \\subset (e_2) \\subset (e_3) \\subset \\ldots$", "and hence $I = \\bigcup_{n \\geq 1} (e_n) = \\colim_n e_nR$.", "In this case", "$$", "\\Hom_R(I, M) = \\Hom_R(\\colim_n e_nR, M)", "= \\lim_n \\Hom_R(e_nR, M) = \\lim_n e_nM", "$$", "Note that the transition maps $e_{n + 1}M \\to e_nM$ are given", "by multiplication by $e_n$ and are surjective. Hence by", "Algebra, Lemma \\ref{algebra-lemma-ML-exact-sequence}", "the functor $\\Hom_R(I, M)$ is exact, i.e., $I$ is a projective", "$R$-module." ], "refs": [ "algebra-lemma-ML-exact-sequence" ], "ref_ids": [ 826 ] } ], "ref_ids": [] }, { "id": 2536, "type": "theorem", "label": "examples-lemma-ideal-projective-not-locally-free", "categories": [ "examples" ], "title": "examples-lemma-ideal-projective-not-locally-free", "contents": [ "There exists a ring $R$ and an ideal $I$ such that $I$ is projective as", "an $R$-module but not locally free as an $R$-module." ], "refs": [], "proofs": [ { "contents": [ "See above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2537, "type": "theorem", "label": "examples-lemma-chow-group-product", "categories": [ "examples" ], "title": "examples-lemma-chow-group-product", "contents": [ "Let $K$ be a field.", "Let $C_i$, $i = 1, \\ldots, n$ be smooth, projective, geometrically irreducible", "curves over $K$. Let $P_i \\in C_i(K)$ be a rational point and", "let $Q_i \\in C_i$ be a point such that $[\\kappa(Q_i) : K] = 2$.", "Then $[P_1 \\times \\ldots \\times P_n]$ is nonzero in", "$\\CH_0(U_1 \\times_K \\ldots \\times_K U_n)$ where $U_i = C_i \\setminus \\{Q_i\\}$." ], "refs": [], "proofs": [ { "contents": [ "There is a degree map", "$\\deg : \\CH_0(C_1 \\times_K \\ldots \\times_K C_n) \\to \\mathbf{Z}$", "Because each $Q_i$ has degree $2$ over $K$ we see that", "any zero cycle supported on the ``boundary''", "$$", "C_1 \\times_K \\ldots \\times_K C_n", "\\setminus", "U_1 \\times_K \\ldots \\times_K U_n", "$$", "has degree divisible by $2$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2538, "type": "theorem", "label": "examples-lemma-projective-not-locally-free", "categories": [ "examples" ], "title": "examples-lemma-projective-not-locally-free", "contents": [ "There exists a countable ring $R$ and a projective module $M$", "which is a direct sum of countably many locally free rank $1$", "modules such that $M$ is not locally free." ], "refs": [], "proofs": [ { "contents": [ "See above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2539, "type": "theorem", "label": "examples-lemma-zero-dimensional-flat-ideal", "categories": [ "examples" ], "title": "examples-lemma-zero-dimensional-flat-ideal", "contents": [ "\\begin{slogan}", "Zero dimensional ring with flat ideal.", "\\end{slogan}", "There exists a local ring $R$ with a unique prime ideal", "and a nonzero ideal $I \\subset R$ which is a flat $R$-module" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2540, "type": "theorem", "label": "examples-lemma-epi-not-surjective", "categories": [ "examples" ], "title": "examples-lemma-epi-not-surjective", "contents": [ "There exists an epimorphism of local rings of dimension $0$", "which is not a surjection." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2541, "type": "theorem", "label": "examples-lemma-example-raynaud-gruson", "categories": [ "examples" ], "title": "examples-lemma-example-raynaud-gruson", "contents": [ "There exists a local ring $A$, a finite type ring map $A \\to B$ and a prime", "$\\mathfrak q$ lying over $\\mathfrak m_A$ such that $B_{\\mathfrak q}$ is flat", "over $A$, and for any element $g \\in B$, $g \\not \\in \\mathfrak q$", "the ring $B_g$ is neither finitely presented over $A$ nor flat over $A$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2542, "type": "theorem", "label": "examples-lemma-finite-type-flat-not-finitely-presented", "categories": [ "examples" ], "title": "examples-lemma-finite-type-flat-not-finitely-presented", "contents": [ "There exist examples of", "\\begin{enumerate}", "\\item a flat finite type ring map with geometrically irreducible", "complete intersection fibre rings which is not of finite presentation,", "\\item a flat finite type ring map with geometrically connected,", "geometrically reduced, dimension 1, complete intersection fibre rings", "which is not of finite presentation,", "\\item a proper flat morphism of schemes $X \\to S$ each of whose fibres", "is isomorphic to either $\\mathbf{P}^1_s$ or to the vanishing locus of", "$X_1X_2$ in $\\mathbf{P}^2_s$ which is not of finite presentation, and", "\\item a proper flat morphism of schemes $X \\to S$ each of whose", "fibres is isomorphic to either $\\mathbf{P}^1_s$ or $\\mathbf{P}^2_s$", "which is not of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2543, "type": "theorem", "label": "examples-lemma-topology-finite-type", "categories": [ "examples" ], "title": "examples-lemma-topology-finite-type", "contents": [ "There exists a local homomorphism $A \\to B$ of local domains which is", "essentially of finite type and such that $A/\\mathfrak m_A \\to B/\\mathfrak m_B$", "is finite such that for every prime", "$\\mathfrak q \\not = \\mathfrak m_B$ of $B$ the ring map", "$A \\to B/\\mathfrak q$ is not the localization of a quasi-finite ring map." ], "refs": [], "proofs": [ { "contents": [ "See the discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2544, "type": "theorem", "label": "examples-lemma-pure-not-universally-pure", "categories": [ "examples" ], "title": "examples-lemma-pure-not-universally-pure", "contents": [ "There exists a morphism of affine schemes of finite presentation", "$X \\to S$ and an $\\mathcal{O}_X$-module $\\mathcal{F}$ of finite presentation", "such that $\\mathcal{F}$ is pure relative to $S$, but not universally", "pure relative to $S$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2545, "type": "theorem", "label": "examples-lemma-formally-smooth-nonflat", "categories": [ "examples" ], "title": "examples-lemma-formally-smooth-nonflat", "contents": [ "There exists a formally smooth ring map which is not flat." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2546, "type": "theorem", "label": "examples-lemma-formally-etale-not-presented", "categories": [ "examples" ], "title": "examples-lemma-formally-etale-not-presented", "contents": [ "There exist formally \\'etale nonflat ring maps." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2547, "type": "theorem", "label": "examples-lemma-formally-etale-nontrivial-cotangent-complex", "categories": [ "examples" ], "title": "examples-lemma-formally-etale-nontrivial-cotangent-complex", "contents": [ "There exists a formally \\'etale surjective ring map $A \\to B$", "with $L_{B/A}$ not equal to zero." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2548, "type": "theorem", "label": "examples-lemma-perfect-closure-polynomial-ring", "categories": [ "examples" ], "title": "examples-lemma-perfect-closure-polynomial-ring", "contents": [ "Let $A = \\mathbb{F}_p[T]$ be the polynomial ring in one variable over", "$\\mathbb{F}_p$. Let $A_{perf}$ denote the perfect closure of $A$.", "Then $A \\rightarrow A_{perf}$ is flat and formally unramified,", "but not formally \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Note that under the Frobenius map $F_A : A \\to A$, the target copy of $A$", "is a free-module over the domain with basis $\\{1, T, \\dots, T^{p - 1}\\}$.", "Thus, $F_A$ is faithfully flat, and consequently, so is", "$A \\to A_{perf}$ since it is a colimit of faithfully flat maps.", "Since $A_{perf}$ is a perfect ring, the relative Frobenius", "$F_{A_{perf}/A}$ is a surjection. In other words,", "$A_{perf} = A[A_{perf}^p]$, which readily implies", "$\\Omega_{A_{perf}/A} = 0$. Then", "$A \\rightarrow A_{perf}$ is formally unramified by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-formally-unramified-differentials}", "\\medskip\\noindent", "It suffices to show that $A \\rightarrow A_{perf}$", "is not formally smooth. Note that since $A$ is a", "smooth $\\mathbb{F}_p$-algebra, the cotangent complex", "$L_{A/\\mathbb{F}_P} \\simeq \\Omega_{A/\\mathbb{F}_p}[0]$", "is concentrated in degree $0$, see", "Cotangent, Lemma \\ref{cotangent-lemma-when-projective}. Moreover,", "$L_{A_{perf}/\\mathbb{F}_p} = 0$ in $D(A_{perf})$", "by Cotangent, Lemma \\ref{cotangent-lemma-perfect-zero}.", "Consider the distinguished triangle of cotangent complexes", "$$", "L_{A/\\mathbb{F}_p} \\otimes_A A_{perf} \\to", "L_{A_{perf}/\\mathbb{F}_p} \\to", "L_{A_{perf}/A} \\to", "(L_{A/\\mathbb{F}_p} \\otimes_A A_{perf})[1]", "$$", "in $D(A_{perf})$, see Cotangent, Section \\ref{cotangent-section-triangle}.", "We find $L_{A_{perf}/A} = \\Omega_{A/\\mathbb{F}_p} \\otimes_A A_{perf}[1]$,", "that is, $L_{A_{perf}/A}$ is equal to a free rank $1$ $A_{perf}$", "module placed in degree $-1$. Thus $A \\rightarrow A_{perf}$", "is not formally smooth by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-NL-formally-smooth}", "and", "Cotangent, Lemma \\ref{cotangent-lemma-relation-with-naive-cotangent-complex}." ], "refs": [ "more-morphisms-lemma-formally-unramified-differentials", "cotangent-lemma-when-projective", "cotangent-lemma-perfect-zero", "more-morphisms-lemma-NL-formally-smooth", "cotangent-lemma-relation-with-naive-cotangent-complex" ], "ref_ids": [ 13695, 11199, 11202, 13748, 11204 ] } ], "ref_ids": [] }, { "id": 2549, "type": "theorem", "label": "examples-lemma-completion-etale", "categories": [ "examples" ], "title": "examples-lemma-completion-etale", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring of prime", "characteristic $p > 0$ such that $[\\kappa : \\kappa^p] < \\infty$.", "Then the canonical map $A \\to A^\\wedge$ to the completion of $A$", "is flat and formally unramified. However, if $A$ is regular but not", "excellent, then this map is not formally \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Flatness of the completion is", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}.", "To show that the map is formally unramified, it", "suffices to show that $\\Omega_{A^\\wedge/A} = 0$, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-formally-unramified}.", "\\medskip\\noindent", "We sketch a proof. Choose $x_1, \\ldots, x_r \\in A$ which map to a $p$-basis", "$\\overline{x}_1, \\ldots, \\overline{x}_r$ of $\\kappa$, i.e.,", "such that $\\kappa$ is minimally generated by $\\overline{x}_i$ over $\\kappa^p$.", "Choose a minimal set of generators $y_1, \\ldots, y_s$ of $\\mathfrak m$.", "For each $n$ the elements $x_1, \\ldots, x_r, y_1, \\ldots, y_s$ generate", "$A/\\mathfrak m^n$ over $(A/\\mathfrak m^n)^p$ by Frobenius.", "Some details omitted. We conclude that $F : A^\\wedge \\to A^\\wedge$", "is finite. Hence $\\Omega_{A^\\wedge/A}$ is a finite $A^\\wedge$-module.", "On the other hand, for any $a \\in A^\\wedge$ and $n$ we can find", "$a_0 \\in A$ such that $a - a_0 \\in \\mathfrak m^nA^\\wedge$.", "We conclude that $\\text{d}(a) \\in \\bigcap \\mathfrak m^n \\Omega_{A^\\wedge/A}$", "which implies that $\\text{d}(a)$ is zero by", "Algebra, Lemma \\ref{algebra-lemma-intersect-powers-ideal-module-zero}.", "Thus $\\Omega_{A^\\wedge/A} = 0$.", "\\medskip\\noindent", "Suppose $A$ is regular. Then, using the Cohen structure theorem", "$x_1, \\ldots, x_r, y_1, \\ldots, y_s$ is a $p$-basis for the ring", "$A^\\wedge$, i.e., we have", "$$", "A^\\wedge = \\bigoplus\\nolimits_{I, J} (A^\\wedge)^p", "x_1^{i_1} \\ldots x_r^{i_r} y_1^{j_1} \\ldots y_s^{j_s}", "$$", "with $I = (i_1, \\ldots, i_r)$, $J = (j_1, \\ldots, j_s)$ and", "$0 \\leq i_a, j_b \\leq p - 1$. Details omitted. In particular, we see that", "$\\Omega_{A^\\wedge}$ is a free $A^\\wedge$-module with basis", "$\\text{d}(x_1), \\ldots, \\text{d}(x_r), \\text{d}(y_1), \\ldots, \\text{d}(y_s)$.", "\\medskip\\noindent", "Now if $A \\to A^\\wedge$ is formally \\'etale or even just formally smooth,", "then we see that $\\NL_{A^\\wedge/A}$ has vanishing cohomology in degrees $-1, 0$", "by Algebra, Proposition \\ref{algebra-proposition-characterize-formally-smooth}.", "It follows from the Jacobi-Zariski sequence", "(Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL}) for the ring", "maps $\\mathbf{F}_p \\to A \\to A^\\wedge$ that we get an isomorphism", "$\\Omega_A \\otimes_A A^\\wedge \\cong \\Omega_{A^\\wedge}$.", "Hence we find that $\\Omega_A$ is free on", "$\\text{d}(x_1), \\ldots, \\text{d}(x_r), \\text{d}(y_1), \\ldots, \\text{d}(y_s)$.", "Looking at fraction fields and using that $A$ is normal", "we conclude that $a \\in A$ is a $p$th power if and only if its image in", "$A^\\wedge$ is a $p$th power (details omitted; use", "Algebra, Lemma \\ref{algebra-lemma-derivative-zero-pth-power}).", "A second consequence is that the operators $\\partial/\\partial x_a$ and", "$\\partial/\\partial y_b$ are defined on $A$.", "\\medskip\\noindent", "We will show that the above lead to the conclusion that $A$ is finite", "over $A^p$ with $p$-basis $x_1, \\ldots, x_r, y_1, \\ldots, y_s$.", "This will contradict the non-excellency of $A$ by a result of", "Kunz, see \\cite[Corollary 2.6]{Kun76}.", "Namely, say $a \\in A$ and write", "$$", "a = \\sum\\nolimits_{I, J} (a_{I, J})^p", "x_1^{i_1} \\ldots x_r^{i_r} y_1^{j_1} \\ldots y_s^{j_s}", "$$", "with $a_{I, J} \\in A^\\wedge$. To finish the proof it suffices to", "show that $a_{I, J} \\in A$. Applying the operator", "$$", "(\\partial/\\partial x_1)^{p - 1} \\ldots", "(\\partial/\\partial x_r)^{p - 1}", "(\\partial/\\partial y_1)^{p - 1} \\ldots", "(\\partial/\\partial y_s)^{p - 1}", "$$", "to both sides we conclude that $a_{I, J}^p \\in A$ where", "$I = (p - 1, \\ldots, p - 1)$ and $J = (p - 1, \\ldots, p - 1)$.", "By our remark above, this also implies $a_{I, J} \\in A$.", "After replacing $a$ by $a' = a - a_{I, J}^p x^I y^J$", "we can use a $1$-order lower differential operators to get", "another coefficient $a_{I, J}$ to be in $A$. Etc." ], "refs": [ "algebra-lemma-completion-flat", "algebra-lemma-characterize-formally-unramified", "algebra-lemma-intersect-powers-ideal-module-zero", "algebra-proposition-characterize-formally-smooth", "algebra-lemma-exact-sequence-NL", "algebra-lemma-derivative-zero-pth-power" ], "ref_ids": [ 870, 1254, 627, 1425, 1153, 1315 ] } ], "ref_ids": [] }, { "id": 2550, "type": "theorem", "label": "examples-lemma-excellent-regular-local-rings", "categories": [ "examples" ], "title": "examples-lemma-excellent-regular-local-rings", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of characteristic", "$p > 0$. Suppose $[\\kappa : \\kappa^p] < \\infty$. Then $A$ is excellent", "if and only if $A \\to A^\\wedge$ is formally \\'etale." ], "refs": [], "proofs": [ { "contents": [ "The backward implication follows from Lemma \\ref{lemma-completion-etale}.", "For the forward implication, note that we already know from", "Lemma \\ref{lemma-completion-etale}", "that $A \\to A^\\wedge$ is formally unramified or equivalently", "that $\\Omega_{A^\\wedge/A}$ is zero.", "Thus, it suffices to show that the completion map is formally smooth when", "$A$ is excellent. By N\\'eron-Popescu", "desingularization $A \\to A^\\wedge$ can be written as a filtered", "colimit of smooth $A$-algebras", "(Smoothing Ring Maps, Theorem \\ref{smoothing-theorem-popescu}).", "Hence $\\NL_{A^\\wedge/A}$ has vanishing cohomology in degree $-1$.", "Thus $A \\to A^\\wedge$ is formally smooth by Algebra, Proposition", "\\ref{algebra-proposition-characterize-formally-smooth}." ], "refs": [ "examples-lemma-completion-etale", "examples-lemma-completion-etale", "smoothing-theorem-popescu", "algebra-proposition-characterize-formally-smooth" ], "ref_ids": [ 2549, 2549, 5605, 1425 ] } ], "ref_ids": [] }, { "id": 2551, "type": "theorem", "label": "examples-lemma-not-generated-idempotents", "categories": [ "examples" ], "title": "examples-lemma-not-generated-idempotents", "contents": [ "There exists an affine scheme $X = \\Spec(A)$ and a", "closed subscheme $T \\subset X$ such that $T$ is Zariski locally", "on $X$ cut out by ideals generated by idempotents, but", "$T$ is not cut out by an ideal generated by idempotents." ], "refs": [], "proofs": [ { "contents": [ "See above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2552, "type": "theorem", "label": "examples-lemma-not-ind-etale", "categories": [ "examples" ], "title": "examples-lemma-not-ind-etale", "contents": [ "There is a ring map $A \\to B$ which identifies local rings but", "which is not ind-\\'etale. A fortiori it is not ind-Zariski." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2553, "type": "theorem", "label": "examples-lemma-nonflasque", "categories": [ "examples" ], "title": "examples-lemma-nonflasque", "contents": [ "There exists an affine scheme $X = \\Spec(A)$ and an injective", "$A$-module $J$ such that $\\widetilde{J}$ is not a flasque sheaf on $X$.", "Even the restriction $\\Gamma(X, \\widetilde{J}) \\to \\Gamma(U, \\widetilde{J})$", "with $U$ a standard open need not be surjective." ], "refs": [], "proofs": [ { "contents": [ "See above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2554, "type": "theorem", "label": "examples-lemma-nonvanishing", "categories": [ "examples" ], "title": "examples-lemma-nonvanishing", "contents": [ "There exists an affine scheme $X = \\Spec(A)$ whose underlying", "topological space is Noetherian and an injective", "$A$-module $I$ such that $\\widetilde{I}$ has nonvanishing $H^1$", "on some quasi-compact open $U$ of $X$." ], "refs": [], "proofs": [ { "contents": [ "See above. Note that $\\Spec(A) = \\Spec(k[x, y])$ as topological spaces." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2555, "type": "theorem", "label": "examples-lemma-non-separated-group-scheme", "categories": [ "examples" ], "title": "examples-lemma-non-separated-group-scheme", "contents": [ "There exists a flat group scheme of finite type over the affine line", "which is not separated." ], "refs": [], "proofs": [ { "contents": [ "See the discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2556, "type": "theorem", "label": "examples-lemma-non-quasi-separated-group-scheme", "categories": [ "examples" ], "title": "examples-lemma-non-quasi-separated-group-scheme", "contents": [ "There exists a flat group scheme of finite type over the infinite", "dimensional affine space which is not quasi-separated." ], "refs": [], "proofs": [ { "contents": [ "The same construction as above can be carried out with the infinite dimensional", "affine space $S = \\mathbf{A}^\\infty_k = \\Spec k[x_1, x_2, \\ldots]$ as the base", "and the origin $0 \\in S$ corresponding to the maximal ideal", "$(x_1, x_2, \\ldots)$ as the closed point which is doubled in $G$.", "The resulting group scheme $G \\rightarrow S$ is ", "not quasi-separated as explained in", "Schemes, Example \\ref{schemes-example-not-quasi-separated}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2557, "type": "theorem", "label": "examples-lemma-non-flat-group-scheme", "categories": [ "examples" ], "title": "examples-lemma-non-flat-group-scheme", "contents": [ "There exists a group scheme $G$ over a base $S$ whose identity", "component is flat over $S$ but which is not flat over $S$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2558, "type": "theorem", "label": "examples-lemma-non-separated-group-space", "categories": [ "examples" ], "title": "examples-lemma-non-separated-group-space", "contents": [ "There exists a group algebraic space of finite type over a field", "which is not separated (and not even quasi-separated or locally separated)." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2559, "type": "theorem", "label": "examples-lemma-specializations-fibre-etale", "categories": [ "examples" ], "title": "examples-lemma-specializations-fibre-etale", "contents": [ "There exists an \\'etale morphism of algebraic spaces $f : X \\to Y$", "and a nontrivial specialization of points $x \\leadsto x'$ in $|X|$ with", "$f(x) = f(x')$ in $|Y|$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2560, "type": "theorem", "label": "examples-lemma-torsors-principal-spaces-not-equal", "categories": [ "examples" ], "title": "examples-lemma-torsors-principal-spaces-not-equal", "contents": [ "Let $S$ be a scheme. Let $G$ be a group scheme over $S$.", "The stack $G\\textit{-Principal}$ classifying principal homogeneous $G$-spaces", "(see Examples of Stacks, Subsection", "\\ref{examples-stacks-subsection-principal-homogeneous-spaces})", "and the stack $G\\textit{-Torsors}$ classifying fppf $G$-torsors", "(see Examples of Stacks, Subsection", "\\ref{examples-stacks-subsection-fppf-torsors})", "are not equivalent in general." ], "refs": [], "proofs": [ { "contents": [ "The discussion above shows that the functor", "$G\\textit{-Torsors} \\to G\\textit{-Principal}$ isn't essentially", "surjective in general." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2561, "type": "theorem", "label": "examples-lemma-BG-algebraic", "categories": [ "examples" ], "title": "examples-lemma-BG-algebraic", "contents": [ "Let $k$ be a field. Let $G$ be an affine group scheme over $k$.", "If the stack $[\\Spec(k)/G]$ has a smooth covering by a", "scheme, then $G$ is of finite type over $k$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2562, "type": "theorem", "label": "examples-lemma-limit-preserving-on-objects-not-limit-preserving", "categories": [ "examples" ], "title": "examples-lemma-limit-preserving-on-objects-not-limit-preserving", "contents": [ "Let $S$ be a nonempty scheme. There exists a stack in groupoids", "$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$", "such that $p$ is limit preserving on objects, but $\\mathcal{X}$ is not", "limit preserving." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2563, "type": "theorem", "label": "examples-lemma-not-algebraic", "categories": [ "examples" ], "title": "examples-lemma-not-algebraic", "contents": [ "There exists a functor $F : \\Sch^{opp} \\to \\textit{Sets}$", "which satisfies the sheaf condition for the fpqc topology, has representable", "diagonal $\\Delta : F \\to F \\times F$, and such that there exists a", "surjective, flat, universally open, quasi-compact morphism", "$U \\to F$ where $U$ is a scheme, but such that $F$ is not an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2564, "type": "theorem", "label": "examples-lemma-sheaf-zero-on-low-dimension", "categories": [ "examples" ], "title": "examples-lemma-sheaf-zero-on-low-dimension", "contents": [ "There exists a sheaf of abelian groups $G$ on", "$\\Sch_\\etale$ with the following properties", "\\begin{enumerate}", "\\item $G(X) = 0$ whenever $\\dim(X) < n$,", "\\item $G(X)$ is not zero if $\\dim(X) \\geq n$, and", "\\item if $X \\subset X'$ is a thickening, then $G(X) = G(X')$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See the discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2565, "type": "theorem", "label": "examples-lemma-weird-sheaf", "categories": [ "examples" ], "title": "examples-lemma-weird-sheaf", "contents": [ "There exists a sheaf of abelian groups $G$ on", "$\\Sch_\\etale$ with the following properties", "\\begin{enumerate}", "\\item $G(\\Spec(k)) = 0$ whenever $k$ is a field,", "\\item $G$ is limit preserving,", "\\item if $X \\subset X'$ is a thickening, then $G(X) = G(X')$, and", "\\item $G$ is not zero.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2566, "type": "theorem", "label": "examples-lemma-lisse-etale-not-functorial", "categories": [ "examples" ], "title": "examples-lemma-lisse-etale-not-functorial", "contents": [ "The lisse-\\'etale site is not functorial, even for morphisms of schemes." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2567, "type": "theorem", "label": "examples-lemma-not-a-morphism-of-sites-noetherian-to-all", "categories": [ "examples" ], "title": "examples-lemma-not-a-morphism-of-sites-noetherian-to-all", "contents": [ "With $S = \\Spec(\\mathbf{F}_p)$ the inclusion functor", "$(\\textit{Noetherian}/S)_{fppf} \\to (\\Sch/S)_{fppf}$", "does not define a morphism of sites." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2568, "type": "theorem", "label": "examples-lemma-is-limit", "categories": [ "examples" ], "title": "examples-lemma-is-limit", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $K$ be an object of", "$D(\\mathcal{O}_\\mathcal{X})$ whose cohomology sheaves are locally", "quasi-coherent (Sheaves on Stacks, Definition", "\\ref{stacks-sheaves-definition-locally-quasi-coherent})", "and satisfy the flat base change property (Cohomology of Stacks,", "Definition \\ref{stacks-cohomology-definition-flat-base-change}).", "Then there exists a distinguished triangle", "$$", "K \\to", "\\prod\\nolimits_{n \\geq 0} \\tau_{\\geq -n} K \\to", "\\prod\\nolimits_{n \\geq 0} \\tau_{\\geq -n} K \\to K[1]", "$$", "in $D(\\mathcal{O}_\\mathcal{X})$. In other words, $K$ is the derived", "limit of its canonical truncations." ], "refs": [ "stacks-sheaves-definition-locally-quasi-coherent", "stacks-cohomology-definition-flat-base-change" ], "proofs": [ { "contents": [ "Recall that we work on the ``big fppf site'' $\\mathcal{X}_{fppf}$", "of $\\mathcal{X}$ (by our conventions", "for sheaves of $\\mathcal{O}_\\mathcal{X}$-modules in the chapters", "Sheaves on Stacks and Cohomology on Stacks). Let $\\mathcal{B}$ be the set", "of objects $x$ of $\\mathcal{X}_{fppf}$ which lie over an affine scheme $U$. ", "Combining", "Sheaves on Stacks, Lemmas", "\\ref{stacks-sheaves-lemma-compare-fppf-etale},", "\\ref{stacks-sheaves-lemma-cohomology-restriction},", "Descent, Lemma \\ref{descent-lemma-quasi-coherent-and-flat-base-change},", "and", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}", "we see that $H^p(x, \\mathcal{F}) = 0$ if $\\mathcal{F}$ is", "locally quasi-coherent and $x \\in \\mathcal{B}$.", "Now the claim follows from", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-is-limit-dimension}", "with $d = 0$." ], "refs": [ "stacks-sheaves-lemma-compare-fppf-etale", "stacks-sheaves-lemma-cohomology-restriction", "descent-lemma-quasi-coherent-and-flat-base-change", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "sites-cohomology-lemma-is-limit-dimension" ], "ref_ids": [ 11615, 11591, 14631, 3282, 4273 ] } ], "ref_ids": [ 11630, 4177 ] }, { "id": 2569, "type": "theorem", "label": "examples-lemma-sum-is-product", "categories": [ "examples" ], "title": "examples-lemma-sum-is-product", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. If $\\mathcal{F}_n$ is a collection", "of locally quasi-coherent sheaves with the flat base change property on", "$\\mathcal{X}$, then $\\oplus_n \\mathcal{F}_n[n] \\to \\prod_n \\mathcal{F}_n[n]$", "is an isomorphism in $D(\\mathcal{O}_\\mathcal{X})$." ], "refs": [], "proofs": [ { "contents": [ "This is true because by Lemma \\ref{lemma-is-limit} we see that the direct sum", "is isomorphic to the product." ], "refs": [ "examples-lemma-is-limit" ], "ref_ids": [ 2568 ] } ], "ref_ids": [] }, { "id": 2570, "type": "theorem", "label": "examples-lemma-push-not-OK", "categories": [ "examples" ], "title": "examples-lemma-push-not-OK", "contents": [ "A quasi-compact and quasi-separated morphism", "$f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks", "need not induce a functor", "$Rf_* : D_\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to", "D_\\QCoh(\\mathcal{O}_\\mathcal{Y})$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2571, "type": "theorem", "label": "examples-lemma-big-abelian-category", "categories": [ "examples" ], "title": "examples-lemma-big-abelian-category", "contents": [ "There exists a ``big'' abelian category $\\mathcal{A}$ whose", "$\\Ext$-groups are proper classes." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2572, "type": "theorem", "label": "examples-lemma-example-schematically-dense-missing-weakly-associated-point", "categories": [ "examples" ], "title": "examples-lemma-example-schematically-dense-missing-weakly-associated-point", "contents": [ "There exists a reduced scheme $X$ and a schematically dense open", "$U \\subset X$ such that some weakly associated point $x \\in X$ is not in $U$." ], "refs": [], "proofs": [ { "contents": [ "In the first example we have $\\mathfrak p \\not \\in U$ by construction.", "In Gabber's examples the schemes $\\Spec(R)$ or $\\Spec(R')$ are reduced." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2573, "type": "theorem", "label": "examples-lemma-nonadditivity-of-trace", "categories": [ "examples" ], "title": "examples-lemma-nonadditivity-of-trace", "contents": [ "There exists a ring $R$, a distinguished triangle", "$(K, L, M, \\alpha, \\beta, \\gamma)$ in the homotopy category $K(R)$,", "and an endomorphism $(a, b, c)$ of this distinguished triangle, such that", "$K$, $L$, $M$ are perfect complexes and", "$\\text{Tr}_K(a) + \\text{Tr}_M(c) \\not = \\text{Tr}_L(b)$." ], "refs": [], "proofs": [ { "contents": [ "Consider the example above. The map $\\gamma : C^\\bullet \\to A^\\bullet[1]$", "is given by multiplication by $\\epsilon$ in degree $0$, see", "Derived Categories, Definition \\ref{derived-definition-distinguished-triangle}.", "Hence it is also true that", "$$", "\\xymatrix{", "C^\\bullet \\ar[d] \\ar[r]_\\gamma & A^\\bullet[1] \\ar[d] \\\\", "C^\\bullet \\ar[r]^\\gamma & A^\\bullet[1]", "}", "$$", "commutes in $K(R)$ as $\\epsilon(1 + \\epsilon) = \\epsilon$.", "Thus we indeed have a morphism of distinguished triangles." ], "refs": [ "derived-definition-distinguished-triangle" ], "ref_ids": [ 1981 ] } ], "ref_ids": [] }, { "id": 2574, "type": "theorem", "label": "examples-lemma-non-descending-property-projective", "categories": [ "examples" ], "title": "examples-lemma-non-descending-property-projective", "contents": [ "The properties", "\\begin{enumerate}", "\\item[] $\\mathcal{P}(f) =$``$f$ is projective'', and", "\\item[] $\\mathcal{P}(f) =$``$f$ is quasi-projective''", "\\end{enumerate}", "are not Zariski local on the base. A fortiori, they are not fpqc local", "on the base." ], "refs": [], "proofs": [ { "contents": [ "Following Hironaka \\cite[Example B.3.4.1]{H},", "we define a proper morphism of smooth complex 3-folds", "$f:V_Y\\to Y$", "which is Zariski-locally projective, but not projective. Since $f$ is proper", "and not projective, it is also not quasi-projective.", "\\medskip\\noindent", "Let $Y$ be projective 3-space over the complex numbers ${\\mathbf C}$.", "Let $C$ and $D$ be smooth conics in $Y$ such", "that the closed subscheme $C\\cap D$ is reduced and consists", "of two complex points $P$ and $Q$. (For example,", "let $C=\\{ [x,y,z,w]: xy=z^2, w=0\\}$, $D=\\{ [x,y,z,w]:", "xy=w^2, z=0\\}$, $P=[1,0,0,0]$,", "and $Q=[0,1,0,0]$.) ", "On $Y-Q$, first blow up the curve $C$, and then blow", "up the strict transform of the curve $D$ (Divisors, Definition ", "\\ref{divisors-definition-strict-transform}). On $Y-P$, first blow up", "the curve $D$, and then blow up the strict transform of the curve", "$C$. Over $Y-P-Q$, the two varieties we have constructed are canonically", "isomorphic, and so we can glue them over $Y-P-Q$. The result", "is a smooth proper 3-fold $V_Y$ over ${\\mathbf C}$. The morphism", "$f:V_Y\\to Y$ is proper and Zariski-locally projective (since", "it is a blowup over $Y-P$ and over $Y-Q$), by Divisors,", "Lemma \\ref{divisors-lemma-blowing-up-projective}. We will show that", "$V_Y$ is not projective over ${\\mathbf C}$. That will imply that", "$f$ is not projective.", "\\medskip\\noindent", "To do this, let $L$ be the inverse image in $V_Y$ of a complex point", "of $C-P-Q$, and $M$ the inverse image of a complex point of $D-P-Q$.", "Then $L$ and $M$ are isomorphic to the projective line", "${\\mathbf P}^1_{{\\mathbf C}}$. ", "Next, let $E$ be the inverse image in $V_Y$ of $C\\cup D\\subset Y$ in $V_Y$;", "thus $E\\rightarrow C\\cup D$ is a proper morphism, with fibers", "isomorphic to ${\\mathbf P}^1$ over $(C\\cup D)-\\{P,Q\\}$.", "The inverse", "image of $P$ in $E$ is a union of two lines $L_0$ and $M_0$, and we have", "rational equivalences of cycles $L\\sim L_0+M_0$ and $M\\sim M_0$ on $E$", "(using that $C$ and $D$ are isomorphic to ${\\mathbf P}^1$).", "Note the asymmetry resulting from the order in which we blew", "up the two curves. Near $Q$, the opposite happens. So the inverse image", "of $Q$ is the union of two lines $L_0'$ and $M_0'$, and we have", "rational equivalences $L\\sim L_0'$ and $M\\sim L_0'+M_0'$ on $E$.", "Combining these equivalences, we find that $L_0+M_0'\\sim 0$", "on $E$ and hence on $V_Y$. If $V_Y$ were projective over ${\\mathbf C}$,", "it would have", "an ample line bundle $H$, which would have degree $> 0$ on all curves", "in $V_Y$. In particular $H$ would have positive degree on $L_0+M_0'$,", "contradicting that the degree of a line bundle is well-defined", "on 1-cycles modulo rational equivalence on a proper scheme", "over a field (Chow Homology,", "Lemma \\ref{chow-lemma-proper-pushforward-rational-equivalence}", "and Lemma \\ref{chow-lemma-factors}).", "So $V_Y$ is not projective over ${\\mathbf C}$." ], "refs": [ "divisors-definition-strict-transform", "divisors-lemma-blowing-up-projective", "chow-lemma-proper-pushforward-rational-equivalence", "chow-lemma-factors" ], "ref_ids": [ 8113, 8063, 5694, 5714 ] } ], "ref_ids": [] }, { "id": 2575, "type": "theorem", "label": "examples-lemma-non-effective-descent-projective", "categories": [ "examples" ], "title": "examples-lemma-non-effective-descent-projective", "contents": [ "There is an etale covering $X\\to S$ of schemes and a descent datum", "$(V/X,\\varphi)$ relative to $X\\to S$ such that ", "$V\\to X$ is projective,", "but the descent datum is not effective in the category of schemes." ], "refs": [], "proofs": [ { "contents": [ "We imitate Hironaka's example of a smooth separated complex", "algebraic space of dimension 3", "which is not a scheme \\cite[Example B.3.4.2]{H}.", "\\medskip\\noindent", "Consider the action of the group $G = \\mathbf{Z}/2 = \\{1, g\\}$", "on projective 3-space", "$\\mathbf{P}^3$ over the complex numbers by", "$$", "g[x,y,z,w] = [y,x,w,z].", "$$", "The action is free outside the two disjoint lines", "$L_1=\\{ [x,x,z,z]\\}$ and $L_2=\\{ [x,-x,z,-z]\\}$ in", "${\\mathbf P}^3$. Let $Y={\\mathbf P}^3-(L_1\\cup L_2)$. There is a", "smooth quasi-projective scheme $S=Y/G$ over ${\\mathbf C}$ such that", "$Y\\to S$ is a $G$-torsor (Groupoids,", "Definition \\ref{groupoids-definition-principal-homogeneous-space}).", "Explicitly, we can define $S$ as the image of the open subset $Y$", "in ${\\mathbf P}^3$ under the morphism", "\\begin{align*}", "{\\mathbf P}^3 & \\to \\text{Proj } {\\mathbf C}[x,y,z,w]^G\\\\", " & = \\text{Proj } {\\mathbf C}[u_0,u_1,v_0,v_1,v_2]/(v_0v_1=v_2^2),", "\\end{align*}", "where $u_0=x+y$, $u_1=z+w$, $v_0=(x-y)^2$, $v_1=(z-w)^2$,", "and $v_2=(x-y)(z-w)$, and the ring is graded with $u_0,u_1$", "in degree 1 and $v_0,v_1,v_2$ in degree 2.", "\\medskip\\noindent", "Let $C=\\{ [x,y,z,w]: xy=z^2, w=0\\}$ and $D=\\{ [x,y,z,w]:", "xy=w^2, z=0\\}$. These are smooth conic curves in ${\\mathbf P}^3$, contained", "in the $G$-invariant open subset $Y$, with $g(C)=D$. Also,", "$C\\cap D$ consists of the two points $P:=[1,0,0,0]$", "and $Q:=[0,1,0,0]$, and these two points are switched by the action", "of $G$. ", "\\medskip\\noindent", "Let $V_Y\\to Y$ be the scheme which over $Y-P$", "is defined by blowing up $D$ and then the strict transform", "of $C$, and over $Y-Q$ is defined by blowing up $C$ and then", "the strict transform of $D$. (This is the same construction", "as in the proof of Lemma \\ref{lemma-non-descending-property-projective},", "except that $Y$ here denotes an open subset of ${\\mathbf P}^3$", "rather than all of ${\\mathbf P}^3$.)", "Then the action of $G$ on $Y$ lifts to an action of $G$ on $V_Y$,", "which switches the inverse images of $Y-P$ and $Y-Q$. This action", "of $G$ on $V_Y$ gives a descent datum $(V_Y/Y,\\varphi_Y)$", "on $V_Y$ relative to the $G$-torsor", "$Y\\to S$. The morphism $V_Y\\to Y$", "is proper but not projective, as shown in the proof", "of Lemma \\ref{lemma-non-descending-property-projective}.", "\\medskip\\noindent", "Let $X$ be the disjoint union of the open subsets $Y-P$ and $Y-Q$;", "then we have surjective etale morphisms $X\\to Y\\to S$.", "Let $V$ be the pullback of $V_Y\\to Y$ to $X$; then the morphism", "$V\\to X$ is projective, since $V_Y\\to Y$ is a blowup over each of the open", "subsets $Y-P$ and $Y-Q$. Moreover, the descent datum $(V_Y/Y,\\varphi_Y)$", "pulls back to a descent datum $(V/X,\\varphi)$ relative to the", "etale covering $X\\to S$.", "\\medskip\\noindent", "Suppose that this descent datum is effective in the category", "of schemes. That is, there is a scheme $U\\to S$", "which pulls back to the morphism $V\\to X$ together", "with its descent datum. Then $U$ would be the quotient", "of $V_Y$ by its $G$-action.", "$$", "\\xymatrix{", "V \\ar[r]\\ar[d]& X\\ar[d] \\\\", "V_Y \\ar[r]\\ar[d]& Y\\ar[d] \\\\", "U \\ar[r]& S", "}", "$$", "\\medskip\\noindent", "Let $E$ be the inverse image of $C\\cup D\\subset Y$ in $V_Y$;", "thus $E\\rightarrow C\\cup D$ is a proper morphism, with fibers", "isomorphic to ${\\mathbf P}^1$ over $(C\\cup D)-\\{P,Q\\}$.", "The inverse image of $P$ in $E$ is a union of two lines $L_0$", "and $M_0$. It follows that the inverse image of $Q=g(P)$ in $E$", "is the union of two lines $L_0'=g(M_0)$ and $M_0'=g(L_0)$.", "As shown in the proof", "of Lemma \\ref{lemma-non-descending-property-projective},", "we have a rational equivalence $L_0+M_0'=L_0+g(L_0)\\sim 0$ on $E$.", "\\medskip\\noindent", "By descent of closed subschemes, there is a curve $L_1\\subset U$", "(isomorphic to ${\\mathbf P}^1$)", "whose inverse image in $V_Y$ is $L_0\\cup g(L_0)$. (Use Descent, Lemma", "\\ref{descent-lemma-affine}, noting that a closed immersion is an affine", "morphism.)", "Let $R$ be a complex point of $L_1$. Since", "we assumed that $U$ is a scheme, we can choose a function", "$f$ in the local ring $O_{U,R}$ that vanishes at $R$ but not", "on the whole curve $L_1$. Let $D_{\\text{loc}}$ be an irreducible component", "of the closed subset $\\{f = 0\\}$ in $\\Spec O_{U,R}$; then", "$D_{\\text{loc}}$ has codimension 1.", "The closure of $D_{\\text{loc}}$ in $U$ is an irreducible divisor $D_U$", "in $U$ which contains the point $R$ but not the whole curve $L_1$.", "The inverse image of $D_U$ in $V_Y$ is an effective divisor $D$", "which intersects $L_0\\cup g(L_0)$ but does not contain either", "curve $L_0$ or $g(L_0)$.", "\\medskip\\noindent", "Since the complex 3-fold $V_Y$ is smooth, $O(D)$ is a line", "bundle on $V_Y$. We use here that a regular local ring is factorial,", "or in other words is a UFD, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-regular-local-UFD}.", "The restriction of $O(D)$ to the proper surface", "$E\\subset V_Y$ is a line bundle which has positive degree on the 1-cycle", "$L_0+g(L_0)$, by our information on $D$. Since", "$L_0+g(L_0)\\sim 0$ on $E$, this contradicts ", "that the degree of a line bundle is well-defined", "on 1-cycles modulo rational equivalence on a proper scheme", "over a field (Chow Homology,", "Lemma \\ref{chow-lemma-proper-pushforward-rational-equivalence}", "and Lemma \\ref{chow-lemma-factors}). Therefore the descent datum", "$(V/X,\\varphi)$ is in fact not effective; that is, $U$ does not exist", "as a scheme." ], "refs": [ "groupoids-definition-principal-homogeneous-space", "examples-lemma-non-descending-property-projective", "examples-lemma-non-descending-property-projective", "examples-lemma-non-descending-property-projective", "descent-lemma-affine", "more-algebra-lemma-regular-local-UFD", "chow-lemma-proper-pushforward-rational-equivalence", "chow-lemma-factors" ], "ref_ids": [ 9679, 2574, 2574, 2574, 14748, 10544, 5694, 5714 ] } ], "ref_ids": [] }, { "id": 2576, "type": "theorem", "label": "examples-lemma-family-of-curves-not-scheme", "categories": [ "examples" ], "title": "examples-lemma-family-of-curves-not-scheme", "contents": [ "There exists a field $k$ and a family of curves", "$X \\to \\mathbf{A}^1_k$ such that $X$ is not a scheme." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2577, "type": "theorem", "label": "examples-lemma-no-derived-base-change", "categories": [ "examples" ], "title": "examples-lemma-no-derived-base-change", "contents": [ "Let $R \\to R'$ and $R \\to A$ be ring maps. In general there does not", "exist a functor $T : D(A) \\to D(A \\otimes_R R')$", "of triangulated categories such that an $A$-module $M$ gives an", "object $T(M)$ of $D(A \\otimes_R R')$ which maps to", "$M \\otimes_R^\\mathbf{L} R'$ under the map $D(A \\otimes_R R') \\to D(R')$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2578, "type": "theorem", "label": "examples-lemma-no-good-representatif-compact-object", "categories": [ "examples" ], "title": "examples-lemma-no-good-representatif-compact-object", "contents": [ "There exists a differential graded algebra $(A, \\text{d})$ and", "a compact object $E$ of $D(A, \\text{d})$ such that $E$ cannot", "be represented by a finite and graded projective differential", "graded $A$-module." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2579, "type": "theorem", "label": "examples-lemma-proper-spaces-not-algebraic", "categories": [ "examples" ], "title": "examples-lemma-proper-spaces-not-algebraic", "contents": [ "The stack in groupoids", "$$", "p'_{fp, flat, proper} :", "\\Spacesstack'_{fp, flat, proper}", "\\longrightarrow", "\\Sch_{fppf}", "$$", "whose category of sections over a scheme $S$ is the category of", "flat, proper, finitely presented algebraic spaces over $S$", "(see Quot, Section \\ref{quot-section-stack-of-spaces})", "is not an algebraic stack." ], "refs": [], "proofs": [ { "contents": [ "If it was an algebraic stack, then every formal object would be", "effective, see Artin's Axioms, Lemma \\ref{artin-lemma-effective}.", "The discussion above show this is not the case", "after base change to $\\Spec(\\mathbf{C})$.", "Hence the conclusion." ], "refs": [ "artin-lemma-effective" ], "ref_ids": [ 11362 ] } ], "ref_ids": [] }, { "id": 2580, "type": "theorem", "label": "examples-lemma-torsors-over-two-dimensional-regular", "categories": [ "examples" ], "title": "examples-lemma-torsors-over-two-dimensional-regular", "contents": [ "Let $W$ be a two dimensional regular integral Noetherian scheme", "with function field $K$. Let $G \\to W$ be an abelian scheme.", "Then the map $H^1_{fppf}(W, G) \\to H^1_{fppf}(\\Spec(K), G)$", "is injective." ], "refs": [], "proofs": [ { "contents": [ "[Sketch of proof]", "Let $P \\to W$ be an fppf $G$-torsor which is trivial in the generic point.", "Then we have a morphism $\\Spec(K) \\to P$ over $W$ and we can take", "its scheme theoretic image $Z \\subset P$. Since $P \\to W$ is proper", "(as a torsor for a proper group algebraic space over $W$)", "we see that $Z \\to W$ is a proper birational morphism.", "By Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-finite-in-codim-1}", "the morphism $Z \\to W$ is finite away from finitely many closed points", "of $W$. By (insert future reference on resolving indeterminacies", "of morphisms by blowing quadratic transformations for surfaces)", "the irreducible components of the geometric fibres of $Z \\to W$", "are rational curves. By", "More on Groupoids in Spaces, Lemma", "\\ref{spaces-more-groupoids-lemma-no-nonconstant-morphism-from-P1-to-group}", "there are no nonconstant morphisms from rational curves", "to group schemes or torsors over such.", "Hence $Z \\to W$ is finite, whence $Z$ is a scheme and $Z \\to W$", "is an isomorphism by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-birational-over-normal}.", "In other words, the torsor $P$ is trivial." ], "refs": [ "spaces-over-fields-lemma-finite-in-codim-1", "spaces-more-groupoids-lemma-no-nonconstant-morphism-from-P1-to-group", "morphisms-lemma-finite-birational-over-normal" ], "ref_ids": [ 12822, 13193, 5518 ] } ], "ref_ids": [] }, { "id": 2581, "type": "theorem", "label": "examples-lemma-torsors-over-field-torsion", "categories": [ "examples" ], "title": "examples-lemma-torsors-over-field-torsion", "contents": [ "Let $G$ be a smooth commutative group algebraic space over a field $K$.", "Then $H^1_{fppf}(\\Spec(K), G)$ is torsion." ], "refs": [], "proofs": [ { "contents": [ "Every $G$-torsor $P$ over $\\Spec(K)$ is smooth over $K$ as a form of $G$.", "Hence $P$ has a point over a finite separable extension $K \\subset L$.", "Say $[L : K] = n$. Let $[n](P)$ denote the $G$-torsor whose class is $n$", "times the class of $P$ in $H^1_{fppf}(\\Spec(K), G)$. There is a canonical", "morphism", "$$", "P \\times_{\\Spec(K)} \\ldots \\times_{\\Spec(K)} P \\to [n](P)", "$$", "of algebraic spaces over $K$. This morphism is symmetric as", "$G$ is abelian. Hence it factors through the quotient", "$$", "(P \\times_{\\Spec(K)} \\ldots \\times_{\\Spec(K)} P)/S_n", "$$", "On the other hand, the morphism $\\Spec(L) \\to P$ defines a morphism", "$$", "(\\Spec(L) \\times_{\\Spec(K)} \\ldots \\times_{\\Spec(K)} \\Spec(L))/S_n", "\\longrightarrow (P \\times_{\\Spec(K)} \\ldots \\times_{\\Spec(K)} P)/S_n", "$$", "and the reader can verify that the scheme on the left has a $K$-rational", "point. Thus we see that $[n](P)$ is the trivial torsor." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2582, "type": "theorem", "label": "examples-lemma-not-essentially-surjective", "categories": [ "examples" ], "title": "examples-lemma-not-essentially-surjective", "contents": [ "The canonical map $\\mathcal{X}(S) \\to \\lim \\mathcal{X}(S_n)$", "is not essentially surjective." ], "refs": [], "proofs": [ { "contents": [ "[Sketch of proof]", "Unwinding definitions, it is enough to check that", "$H^1(X, A_0) \\to \\lim H^1(X_n, A_0)$ is not surjective.", "As $X$ is regular and projective, by", "Lemmas \\ref{lemma-torsors-over-field-torsion} and", "\\ref{lemma-torsors-over-two-dimensional-regular}", "each $A_0$-torsor over $X$ is torsion.", "In particular, the group $H^1(X, A_0)$ is torsion.", "It is thus enough to show:", "(a) the group $H^1(X_0, A_0)$ is non-torsion, and", "(b) the maps $H^1(X_{n + 1}, A_0) \\to H^1(X_n, A_0)$ are surjective for all $n$.", "\\medskip\\noindent", "Ad (a). One constructs a nontorsion $A_0$-torsor $P_0$ on the nodal", "curve $X_0$ by glueing trivial $A_0$-torsors on each component", "of $X_0$ using non-torsion points on $A_0$ as the isomorphisms", "over the nodes. More precisely, let $x \\in X_0$ be a node", "which occurs in a loop consisting of rational curves.", "Let $X'_0 \\to X_0$ be the normalization of $X_0$ in $X_0 \\setminus \\{x\\}$.", "Let $x', x'' \\in X'_0$ be the two points mapping to $x_0$.", "Then we take $A_0 \\times_{\\Spec(k)} X'_0$ and we identify", "$A_0 \\times {x'}$ with $A_0 \\times \\{x''\\}$ using translation", "$A_0 \\to A_0$ by a nontorsion point $a_0 \\in A_0(k)$ (there is such", "a nontorsion point as $k$ is algebraically closed and not the algebraic", "closure of a finite field -- this is actually not trivial to prove).", "One can show that the glueing is an algebraic space (in fact one can", "show it is a scheme) and that it is an nontorsion $A_0$-torsor over $X_0$.", "The reason that it is nontorsion is that if $[n](P_0)$ has a section,", "then that section produces a morphism $s : X'_0 \\to A_0$ such that", "$[n](a_0) = s(x') - s(x'')$ in the group law on $A_0(k)$. However,", "since the irreducible components of the loop are rational to", "section $s$ is constant on them (", "More on Groupoids in Spaces, Lemma", "\\ref{spaces-more-groupoids-lemma-no-nonconstant-morphism-from-P1-to-group}).", "Hence $s(x') = s(x'')$ and we obtain a contradiction.", "\\medskip\\noindent", "Ad (b). Deformation theory shows that the obstruction to deforming an", "$A_0$-torsor $P_n \\to X_n$ to an $A_0$-torsor $P_{n + 1} \\to X_{n + 1}$", "lies in $H^2(X_0, \\omega)$ for a suitable vector bundle $\\omega$ on $X_0$.", "The latter vanishes as $X_0$ is a curve, proving the claim." ], "refs": [ "examples-lemma-torsors-over-field-torsion", "examples-lemma-torsors-over-two-dimensional-regular", "spaces-more-groupoids-lemma-no-nonconstant-morphism-from-P1-to-group" ], "ref_ids": [ 2581, 2580, 13193 ] } ], "ref_ids": [] }, { "id": 2583, "type": "theorem", "label": "examples-lemma-non-formal-effectiveness", "categories": [ "examples" ], "title": "examples-lemma-non-formal-effectiveness", "contents": [ "Let $k$ be an algebraically closed field which is not the closure", "of a finite field. Let $A$ be an abelian variety over $k$.", "Let $\\mathcal{X} = [\\Spec(k)/A]$.", "There exists an inverse system of $k$-algebras $R_n$", "with surjective transition maps whose kernels are locally nilpotent", "and a system $(\\xi_n)$ of $\\mathcal{X}$ lying over the system", "$(\\Spec(R_n))$ such that this system is not effective", "in the sense of Artin's Axioms, Remark \\ref{artin-remark-strong-effectiveness}." ], "refs": [ "artin-remark-strong-effectiveness" ], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 11429 ] }, { "id": 2584, "type": "theorem", "label": "examples-lemma-counter-Grothendieck-existence", "categories": [ "examples" ], "title": "examples-lemma-counter-Grothendieck-existence", "contents": [ "Counter examples to algebraization of coherent sheaves.", "\\begin{enumerate}", "\\item Grothendieck's existence theorem as stated in", "Cohomology of Schemes, Theorem \\ref{coherent-theorem-grothendieck-existence}", "is false if we drop the assumption that $X \\to \\Spec(A)$ is separated.", "\\item The stack of coherent sheaves $\\Cohstack_{X/B}$", "of Quot, Theorems \\ref{quot-theorem-coherent-algebraic-general} and", "\\ref{quot-theorem-coherent-algebraic} is in general", "not algebraic if we drop the assumption that $X \\to S$ is separated", "\\item The functor $\\Quotfunctor_{\\mathcal{F}/X/B}$ of", "Quot, Proposition \\ref{quot-proposition-quot}", "is not an algebraic space in general if we drop the assumption", "that $X \\to B$ is separated.", "\\end{enumerate}" ], "refs": [ "coherent-theorem-grothendieck-existence", "quot-theorem-coherent-algebraic-general", "quot-theorem-coherent-algebraic", "quot-proposition-quot" ], "proofs": [ { "contents": [ "Part (1) we saw above. This shows that $\\textit{Coh}_{X/A}$ fails", "axiom [4] of Artin's Axioms, Section \\ref{artin-section-axioms}. Hence it", "cannot be an algebraic stack by Artin's Axioms, Lemma", "\\ref{artin-lemma-effective}.", "In this way we see that (2) is true. To see (3), note that", "there are compatible surjections $\\mathcal{O}_{X_n} \\to \\mathcal{F}_n$", "for all $n$. Thus we see that $\\Quotfunctor_{\\mathcal{O}_X/X/A}$", "fails axiom [4] and we see that (3) is true as before." ], "refs": [ "artin-lemma-effective" ], "ref_ids": [ 11362 ] } ], "ref_ids": [ 3279, 3147, 3146, 3227 ] }, { "id": 2585, "type": "theorem", "label": "examples-lemma-affine-not-mcquillan", "categories": [ "examples" ], "title": "examples-lemma-affine-not-mcquillan", "contents": [ "There exists an affine formal algebraic space which is not McQuillan." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2586, "type": "theorem", "label": "examples-lemma-affine-formal-functions-do-not-separate-points", "categories": [ "examples" ], "title": "examples-lemma-affine-formal-functions-do-not-separate-points", "contents": [ "There exists an affine formal algebraic space $X$", "whose regular functions do not separate points, in the following sense:", "If we write $X = \\colim X_\\lambda$ as in", "Formal Spaces, Definition", "\\ref{formal-spaces-definition-affine-formal-algebraic-space}", "then $\\lim \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})$", "is a field, but $X_{red}$ has infinitely many points." ], "refs": [ "formal-spaces-definition-affine-formal-algebraic-space" ], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 3977 ] }, { "id": 2587, "type": "theorem", "label": "examples-lemma-representable-morphism-affine-formal-not-mcquillan-top", "categories": [ "examples" ], "title": "examples-lemma-representable-morphism-affine-formal-not-mcquillan-top", "contents": [ "There exists a representable morphism $f : X \\to Y$ of", "affine formal algebraic spaces with $Y$ McQuillan, but $X$ not", "McQuillan." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2588, "type": "theorem", "label": "examples-lemma-weird-flat-map", "categories": [ "examples" ], "title": "examples-lemma-weird-flat-map", "contents": [ "There exists a commutative ring $A$ and a flat $A$-algebra $B$", "which cannot be written as a filtered colimit of finitely", "presented flat $A$-algebras. In fact, we may either choose $A$ to", "be a finite type $\\mathbf{F}_p$-algebra or a $1$-dimensional", "Noetherian local ring with residue field of characteristic $0$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2589, "type": "theorem", "label": "examples-lemma-colimit-topology", "categories": [ "examples" ], "title": "examples-lemma-colimit-topology", "contents": [ "There exists a system $G_1 \\to G_2 \\to G_3 \\to \\ldots$ of (abelian)", "topological groups such that $\\colim G_n$ taken in the category of", "topological spaces is different from $\\colim G_n$ taken in the category", "of topological groups." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2590, "type": "theorem", "label": "examples-lemma-universally-submersive-not-V", "categories": [ "examples" ], "title": "examples-lemma-universally-submersive-not-V", "contents": [ "There exists a morphism $X \\to Y$ of affine schemes", "which is universally submersive such that $\\{X \\to Y\\}$", "is not a V covering." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2591, "type": "theorem", "label": "examples-lemma-non-fpqc-descent", "categories": [ "examples" ], "title": "examples-lemma-non-fpqc-descent", "contents": [ "There exists a ring $A$ and an infinite family of flat ring maps", "$\\{A \\to A_i\\}_{i \\in I}$ such that for every $A$-module $M$ ", "$$", "M =", "\\text{Equalizer}\\left(", "\\xymatrix{", "\\prod\\nolimits_{i \\in I} M \\otimes_A A_i \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\prod\\nolimits_{i, j \\in I} M \\otimes_A A_i \\otimes_A A_j", "}", "\\right)", "$$", "but there is no finite subfamily where the same thing is true." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2592, "type": "theorem", "label": "examples-lemma-Z-not-quasi-compact", "categories": [ "examples" ], "title": "examples-lemma-Z-not-quasi-compact", "contents": [ "The scheme $\\Spec(\\mathbf{Z})$ is not quasi-compact", "in the canonical topology on the category of schemes." ], "refs": [], "proofs": [ { "contents": [ "With notation as above consider the family of morphisms", "$$", "\\mathcal{W} = \\{\\Spec(\\mathbf{Z}_A) \\to \\Spec(\\mathbf{Z})\\}_{A \\in U}", "$$", "By Descent, Lemma \\ref{descent-lemma-universal-effective-epimorphism}", "and the two claims proved above", "this is a universally effective epimorphism.", "In any category with fibre products, the universal effective epimorphisms", "give $\\mathcal{C}$ the structure of a site (modulo some set theoretical", "issues which are easy to fix) defining the canonical topology.", "Thus $\\mathcal{W}$ is a covering for the canonical topology.", "On the other hand, we have seen above that any finite subfamily", "$$", "\\{\\Spec(\\mathbf{Z}_{A_i}) \\to \\Spec(\\mathbf{Z})\\}_{i = 1, \\ldots, n},\\quad", "n \\in \\mathbf{N}, A_1, \\ldots, A_n \\in U", "$$", "factors through $\\Spec(\\mathbf{Z}[1/p])$ for some $p$.", "Hence this finite family cannot be a universally effective epimorphism", "and more generally no universally effective epimorphism", "$\\{g_j : T_j \\to \\Spec(\\mathbf{Z})\\}$ can refine", "$\\{\\Spec(\\mathbf{Z}_{A_i}) \\to \\Spec(\\mathbf{Z})\\}_{i = 1, \\ldots, n}$.", "By Sites, Definition \\ref{sites-definition-quasi-compact}", "this means that $\\Spec(\\mathbf{Z})$ is not quasi-compact", "in the canonical topology. To see that our notion of quasi-compactness", "agrees with the usual topos theoretic definition, see", "Sites, Lemma \\ref{sites-lemma-quasi-compact}." ], "refs": [ "descent-lemma-universal-effective-epimorphism", "sites-definition-quasi-compact", "sites-lemma-quasi-compact" ], "ref_ids": [ 14636, 8668, 8530 ] } ], "ref_ids": [] }, { "id": 2593, "type": "theorem", "label": "examples-proposition-nonalghomstack", "categories": [ "examples" ], "title": "examples-proposition-nonalghomstack", "contents": [ "The stack $\\mathcal{X} = \\underline{\\Mor}_S(X, [S/A])$ is not algebraic." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2594, "type": "theorem", "label": "examples-proposition-localization-and-serre-quotients", "categories": [ "examples" ], "title": "examples-proposition-localization-and-serre-quotients", "contents": [ "Let $A$ be a ring. Let $S$ be a multiplicative subset of $A$.", "Let $\\text{Mod}_A$ denote the category of $A$-modules and $\\mathcal{T}$ its", "Serre subcategory of modules for which any element is annihilated by some", "element of $S$. Then there is a canonical equivalence", "$\\text{Mod}_A/\\mathcal{T} \\rightarrow \\text{Mod}_{S^{-1}A}$." ], "refs": [], "proofs": [ { "contents": [ "The functor $\\text{Mod}_A \\to \\text{Mod}_{S^{-1}A}$ given by $M", "\\mapsto M \\otimes_A S^{-1}A$ is exact (by Algebra, Proposition", "\\ref{algebra-proposition-localization-exact})", "and maps modules in $\\mathcal{T}$ to zero.", "Thus, by the universal property given in Homology, Lemma", "\\ref{homology-lemma-serre-subcategory-is-kernel}, the functor descends to a", "functor $\\text{Mod}_A/\\mathcal{T} \\to \\text{Mod}_{S^{-1}A}$.", "\\medskip\\noindent", "Conversely, any $A$-module $M$ with $M \\otimes_A S^{-1}A = 0$", "is an object of $\\mathcal{T}$, since", "$M \\otimes_A S^{-1}A \\cong S^{-1} M$", "(Algebra, Lemma \\ref{algebra-lemma-tensor-localization}). Thus", "Homology, Lemma \\ref{homology-lemma-quotient-by-kernel-exact-functor}", "shows that the functor", "$\\text{Mod}_A/\\mathcal{T} \\to \\text{Mod}_{S^{-1}A}$ is faithful.", "\\medskip\\noindent", "Furthermore, this embedding is essentially surjective: a preimage to an", "$S^{-1}A$-module $N$ is $N_A$, that is $N$ regarded as an $A$-module, since the", "canonical map $N_A \\otimes_A S^{-1}A \\to N$ which maps $x \\otimes a/s$ to", "$(a/s) \\cdot x$ is an isomorphism of $S^{-1}A$-modules." ], "refs": [ "algebra-proposition-localization-exact", "homology-lemma-serre-subcategory-is-kernel", "algebra-lemma-tensor-localization", "homology-lemma-quotient-by-kernel-exact-functor" ], "ref_ids": [ 1402, 12048, 366, 12049 ] } ], "ref_ids": [] }, { "id": 2595, "type": "theorem", "label": "examples-proposition-quotient-by-torsion-modules", "categories": [ "examples" ], "title": "examples-proposition-quotient-by-torsion-modules", "contents": [ "Let $A$ be a ring. Let $Q(A)$ denote its total quotient ring", "(as in Algebra, Example \\ref{algebra-example-localize-at-prime}). Let", "$\\text{Mod}_A$ denote the category of $A$-modules and $\\mathcal{T}$ its", "Serre subcategory of torsion modules. Let $\\text{Mod}_{Q(A)}$", "denote the category", "of $Q(A)$-modules. Then there is a canonical equivalence", "$\\text{Mod}_A/\\mathcal{T} \\rightarrow \\text{Mod}_{Q(A)}$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from applying Proposition", "\\ref{proposition-localization-and-serre-quotients} to the multiplicative subset", "$S = \\{f \\in A \\mid f \\text{ is not a zerodivisor in }A\\}$, since a module is a", "torsion module if and only if all of its elements are each annihilated by some", "element of $S$." ], "refs": [ "examples-proposition-localization-and-serre-quotients" ], "ref_ids": [ 2594 ] } ], "ref_ids": [] }, { "id": 2596, "type": "theorem", "label": "examples-proposition-quotient-by-finitely-generated-torsion-modules", "categories": [ "examples" ], "title": "examples-proposition-quotient-by-finitely-generated-torsion-modules", "contents": [ "Let $A$ be a Noetherian integral domain. Let $K$ denote its field of fractions.", "Let $\\text{Mod}_A^{fg}$ denote the category of finitely generated $A$-modules", "and $\\mathcal{T}^{fg}$ its Serre subcategory of finitely generated torsion", "modules. Then $\\text{Mod}_A^{fg}/\\mathcal{T}^{fg}$ is canonically equivalent", "to the category of finite dimensional $K$-vector spaces." ], "refs": [], "proofs": [ { "contents": [ "The equivalence given in Proposition", "\\ref{proposition-quotient-by-torsion-modules} restricts along the embedding", "$\\text{Mod}_A^{fg}/\\mathcal{T}^{fg} \\to \\text{Mod}_A/\\mathcal{T}$ to an", "equivalence $\\text{Mod}_A^{fg}/\\mathcal{T}^{fg} \\to \\text{Vect}_K^{fd}$.", "The Noetherian assumption guarantees that $\\text{Mod}_A^{fg}$ is an", "abelian category (see", "More on Algebra, Section \\ref{more-algebra-section-abelian-categories-modules})", "and that the canonical functor", "$\\text{Mod}_A^{fg}/\\mathcal{T}^{fg} \\to \\text{Mod}_A/\\mathcal{T}$", "is full (else torsion submodules of finitely", "generated modules might not be objects of $\\mathcal{T}^{fg}$)." ], "refs": [ "examples-proposition-quotient-by-torsion-modules" ], "ref_ids": [ 2595 ] } ], "ref_ids": [] }, { "id": 2597, "type": "theorem", "label": "examples-proposition-quotient-abelian-groups-by-torsion-groups", "categories": [ "examples" ], "title": "examples-proposition-quotient-abelian-groups-by-torsion-groups", "contents": [ "The quotient of the category of abelian groups modulo its", "Serre subcategory of torsion groups is the category of", "$\\mathbf{Q}$-vector spaces." ], "refs": [], "proofs": [ { "contents": [ "The claim follows directly from", "Proposition \\ref{proposition-quotient-by-torsion-modules}." ], "refs": [ "examples-proposition-quotient-by-torsion-modules" ], "ref_ids": [ 2595 ] } ], "ref_ids": [] }, { "id": 2601, "type": "theorem", "label": "bootstrap-theorem-bootstrap", "categories": [ "bootstrap" ], "title": "bootstrap-theorem-bootstrap", "contents": [ "Let $S$ be a scheme.", "Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor.", "Assume that", "\\begin{enumerate}", "\\item the presheaf $F$ is a sheaf,", "\\item the diagonal morphism $F \\to F \\times F$ is representable by", "algebraic spaces, and", "\\item there exists an algebraic space $X$", "and a map $X \\to F$ which is surjective, and \\'etale.", "\\end{enumerate}", "Then $F$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "We will use the remarks directly below", "Definition \\ref{definition-property-transformation}", "without further mention.", "In the situation of the theorem, let $U \\to X$ be a surjective \\'etale morphism", "from a scheme towards $X$.", "By Lemma \\ref{lemma-composition-transformation}", "$U \\to F$ is surjective and \\'etale also.", "Hence the theorem boils down to proving that", "$\\Delta_F$ is representable.", "This follows immediately from", "Lemma \\ref{lemma-bootstrap-diagonal}.", "On the other hand we can circumvent this lemma and show directly $F$", "is an algebraic space as in the next paragraph.", "\\medskip\\noindent", "Let $U$ be a scheme, and let $U \\to F$ be surjective and \\'etale.", "Set $R = U \\times_F U$, which is an algebraic space (see", "Lemma \\ref{lemma-representable-diagonal}).", "The morphism of algebraic spaces $R \\to U \\times_S U$ is a monomorphism,", "hence separated (as the diagonal of a monomorphism is an isomorphism).", "Moreover, since $U \\to F$ is \\'etale, we see that $R \\to U$ is \\'etale, by", "Lemma \\ref{lemma-base-change-transformation-property}.", "In particular, we see that $R \\to U$ is locally quasi-finite, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-etale-locally-quasi-finite}.", "We conclude that also $R \\to U \\times_S U$ is", "locally quasi-finite by", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-permanence-quasi-finite}.", "Hence", "Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}", "applies and $R$ is a scheme. Hence $F = U/R$ is an algebraic", "space according to", "Spaces, Theorem \\ref{spaces-theorem-presentation}." ], "refs": [ "bootstrap-definition-property-transformation", "bootstrap-lemma-composition-transformation", "bootstrap-lemma-bootstrap-diagonal", "bootstrap-lemma-representable-diagonal", "bootstrap-lemma-base-change-transformation-property", "spaces-morphisms-lemma-etale-locally-quasi-finite", "spaces-morphisms-lemma-permanence-quasi-finite", "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "spaces-theorem-presentation" ], "ref_ids": [ 2638, 2609, 2620, 2618, 2613, 4908, 4836, 4983, 8124 ] } ], "ref_ids": [] }, { "id": 2602, "type": "theorem", "label": "bootstrap-theorem-final-bootstrap", "categories": [ "bootstrap" ], "title": "bootstrap-theorem-final-bootstrap", "contents": [ "Let $S$ be a scheme.", "Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor.", "Any one of the following conditions implies that $F$ is an algebraic space:", "\\begin{enumerate}", "\\item $F = U/R$ where $(U, R, s, t, c)$ is a groupoid in algebraic spaces", "over $S$ such that $s, t$ are flat and locally of finite presentation, and", "$j = (t, s) : R \\to U \\times_S U$ is an equivalence relation,", "\\item $F = U/R$ where $(U, R, s, t, c)$ is a groupoid scheme", "over $S$ such that $s, t$ are flat and locally of finite presentation, and", "$j = (t, s) : R \\to U \\times_S U$ is an equivalence relation,", "\\item $F$ is a sheaf and there exists an algebraic space $U$ and a morphism", "$U \\to F$ which is representable by algebraic spaces,", "surjective, flat and locally of finite presentation,", "\\item $F$ is a sheaf and there exists a scheme $U$ and a morphism", "$U \\to F$ which is representable (by algebraic spaces or schemes),", "surjective, flat and locally of finite presentation,", "\\item $F$ is a sheaf, $\\Delta_F$ is representable by algebraic spaces,", "and there exists an algebraic space $U$ and a morphism $U \\to F$ which is", "surjective, flat, and locally of finite presentation, or", "\\item $F$ is a sheaf, $\\Delta_F$ is representable,", "and there exists a scheme $U$ and a morphism $U \\to F$ which is", "surjective, flat, and locally of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Trivial observations: (6) is a special case of (5) and", "(4) is a special case of (3).", "We first prove that cases (5) and (3) reduce to case (1).", "Namely, by bootstrapping the diagonal", "Lemma \\ref{lemma-bootstrap-diagonal}", "we see that (3) implies (5). In case (5) we set $R = U \\times_F U$ which", "is an algebraic space by assumption. Moreover, by assumption both", "projections $s, t : R \\to U$ are surjective, flat and locally of", "finite presentation. The map $j : R \\to U \\times_S U$ is clearly an", "equivalence relation. By", "Lemma \\ref{lemma-surjective-flat-locally-finite-presentation}", "the map $U \\to F$ is a surjection of sheaves. Thus $F = U/R$", "which reduces us to case (1).", "\\medskip\\noindent", "Next, we show that (1) reduces to (2).", "Namely, let $(U, R, s, t, c)$ be a groupoid in algebraic spaces", "over $S$ such that $s, t$ are flat and locally of finite presentation, and", "$j = (t, s) : R \\to U \\times_S U$ is an equivalence relation.", "Choose a scheme $U'$ and a surjective \\'etale morphism $U' \\to U$.", "Let $R' = R|_{U'}$ be the restriction of $R$ to $U'$. By", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-pre-equivalence-relation-restrict}", "we see that $U/R = U'/R'$. Since $s', t' : R' \\to U'$ are also", "flat and locally of finite presentation (see", "More on Groupoids in Spaces,", "Lemma \\ref{spaces-more-groupoids-lemma-restrict-preserves-type})", "this reduces us to the case where $U$ is a scheme.", "As $j$ is an equivalence relation we see that $j$ is a monomorphism.", "As $s : R \\to U$ is locally of finite presentation we see that", "$j : R \\to U \\times_S U$ is locally of finite type, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-permanence-finite-type}.", "By", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}", "we see that $j$ is locally quasi-finite and separated.", "Hence if $U$ is a scheme, then $R$ is a scheme by", "Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}.", "Thus we reduce to proving the theorem in case (2).", "\\medskip\\noindent", "Assume $F = U/R$ where $(U, R, s, t, c)$ is a groupoid scheme", "over $S$ such that $s, t$ are flat and locally of finite presentation, and", "$j = (t, s) : R \\to U \\times_S U$ is an equivalence relation. By", "Lemma \\ref{lemma-slice-equivalence-relation}", "we reduce to that case where $s, t$ are flat,", "locally of finite presentation, and locally quasi-finite.", "Let $U = \\bigcup_{i \\in I} U_i$ be an affine open covering", "(with index set $I$ of cardinality $\\leq$ than the size of $U$ to avoid", "set theoretic problems later -- most readers can safely ignore this remark).", "Let $(U_i, R_i, s_i, t_i, c_i)$ be the restriction of $R$", "to $U_i$. It is clear that $s_i, t_i$ are still flat, locally of finite", "presentation, and locally quasi-finite as $R_i$ is the open subscheme", "$s^{-1}(U_i) \\cap t^{-1}(U_i)$ of $R$", "and $s_i, t_i$ are the restrictions of $s, t$ to this open. By", "Lemma \\ref{lemma-better-finding-opens}", "(or the simpler", "Spaces, Lemma \\ref{spaces-lemma-finding-opens})", "the map $U_i/R_i \\to U/R$ is representable by open immersions.", "Hence if we can show that $F_i = U_i/R_i$ is an algebraic space, then", "$\\coprod_{i \\in I} F_i$ is an algebraic space by", "Spaces, Lemma \\ref{spaces-lemma-coproduct-algebraic-spaces}.", "As $U = \\bigcup U_i$ is an open covering it is clear that", "$\\coprod F_i \\to F$ is surjective. Thus", "it follows that $U/R$ is an algebraic space, by", "Spaces, Lemma \\ref{spaces-lemma-glueing-algebraic-spaces}.", "In this way we reduce to the case where $U$ is affine and $s, t$ are flat,", "locally of finite presentation, and locally quasi-finite and", "$j$ is an equivalence.", "\\medskip\\noindent", "Assume $(U, R, s, t, c)$ is a groupoid scheme over $S$,", "with $U$ affine, such that $s, t$ are flat, locally of finite presentation,", "and locally quasi-finite, and $j$ is an equivalence relation.", "Choose $u \\in U$. We apply", "More on Groupoids in Spaces,", "Lemma \\ref{spaces-more-groupoids-lemma-quasi-splitting-affine-scheme}", "to $u \\in U, R, s, t, c$. We obtain an affine scheme $U'$, an \\'etale", "morphism $g : U' \\to U$, a point $u' \\in U'$ with $\\kappa(u) = \\kappa(u')$", "such that the restriction $R' = R|_{U'}$ is quasi-split over $u'$.", "Note that the image $g(U')$ is open as $g$ is \\'etale and contains $u'$.", "Hence, repeatedly applying the lemma, we can find finitely many", "points $u_i \\in U$, $i = 1, \\ldots, n$,", "affine schemes $U'_i$, \\'etale morphisms $g_i : U_i' \\to U$, points", "$u'_i \\in U'_i$ with $g(u'_i) = u_i$ such that (a) each", "restriction $R'_i$ is quasi-split over some point in $U'_i$ and", "(b) $U = \\bigcup_{i = 1, \\ldots, n} g_i(U'_i)$.", "Now we rerun the last part of the argument in the preceding paragraph:", "Using", "Lemma \\ref{lemma-better-finding-opens}", "(or the simpler", "Spaces, Lemma \\ref{spaces-lemma-finding-opens})", "the map $U'_i/R'_i \\to U/R$ is representable by open immersions.", "If we can show that $F_i = U'_i/R'_i$ is an algebraic space, then", "$\\coprod_{i \\in I} F_i$ is an algebraic space by", "Spaces, Lemma \\ref{spaces-lemma-coproduct-algebraic-spaces}.", "As $\\{g_i : U'_i \\to U\\}$ is an \\'etale covering", "it is clear that $\\coprod F_i \\to F$ is surjective. Thus", "it follows that $U/R$ is an algebraic space, by", "Spaces, Lemma \\ref{spaces-lemma-glueing-algebraic-spaces}.", "In this way we reduce to the case where $U$ is affine and $s, t$ are flat,", "locally of finite presentation, and locally quasi-finite,", "$j$ is an equivalence, and $R$ is quasi-split over $u$ for some", "$u \\in U$.", "\\medskip\\noindent", "Assume $(U, R, s, t, c)$ is a groupoid scheme over $S$,", "with $U$ affine, $u \\in U$ such that $s, t$ are flat, locally", "of finite presentation, and locally quasi-finite and", "$j = (t, s) : R \\to U \\times_S U$ is an equivalence relation", "and $R$ is quasi-split over $u$. Let $P \\subset R$ be a quasi-splitting", "of $R$ over $u$. By", "Lemma \\ref{lemma-divide-subgroupoid}", "we see that $(U, R, s, t, c)$ is the restriction of a groupoid", "$(\\overline{U}, \\overline{R}, \\overline{s}, \\overline{t}, \\overline{c})$", "by a surjective finite locally free morphism $U \\to \\overline{U}$ such that", "$P = U \\times_{\\overline{U}} U$.", "Note that $s, t$ are the base changes of the morphisms", "$\\overline{s}, \\overline{t}$ by $U \\to \\overline{U}$.", "As $\\{U \\to \\overline{U}\\}$ is an fppf covering we conclude", "$\\overline{s}, \\overline{t}$ are flat, locally of finite presentation, and", "locally quasi-finite, see", "Descent, Lemmas \\ref{descent-lemma-descending-property-flat},", "\\ref{descent-lemma-descending-property-locally-finite-presentation}, and", "\\ref{descent-lemma-descending-property-quasi-finite}.", "Consider the commutative diagram", "$$", "\\xymatrix{", "U \\times_{\\overline{U}} U \\ar@{=}[r] \\ar[rd] & P \\ar[r] \\ar[d] & R \\ar[d] \\\\", "& \\overline{U} \\ar[r]^{\\overline{e}} & \\overline{R}", "}", "$$", "It is a general fact about restrictions that the outer four corners", "form a cartesian diagram. By the equality we see the inner square is", "cartesian. Since $P$ is open in $R$ (by definition of a quasi-splitting)", "we conclude that $\\overline{e}$ is an open immersion by", "Descent, Lemma \\ref{descent-lemma-descending-property-open-immersion}.", "An application of", "Groupoids,", "Lemma \\ref{groupoids-lemma-quotient-pre-equivalence-relation-restrict}", "shows that $U/R = \\overline{U}/\\overline{R}$. Hence we have reduced to", "the case where $(U, R, s, t, c)$ is a groupoid scheme over $S$,", "with $U$ affine, $u \\in U$ such that $s, t$ are flat, locally", "of finite presentation, and locally quasi-finite and", "$j = (t, s) : R \\to U \\times_S U$ is an equivalence relation", "and $e : U \\to R$ is an open immersion!", "\\medskip\\noindent", "But of course, if $e$ is an open immersion and", "$s, t$ are flat and locally of finite presentation", "then the morphisms $t, s$ are \\'etale.", "For example you can see this by applying", "More on Groupoids, Lemma \\ref{more-groupoids-lemma-sheaf-differentials}", "which shows that $\\Omega_{R/U} = 0$ which in turn implies", "that $s, t : R \\to U$ is G-unramified (see", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-omega-zero}),", "which in turn implies that $s, t$ are \\'etale (see", "Morphisms, Lemma \\ref{morphisms-lemma-flat-unramified-etale}).", "And if $s, t$ are \\'etale then finally $U/R$ is an algebraic", "space by", "Spaces, Theorem \\ref{spaces-theorem-presentation}." ], "refs": [ "bootstrap-lemma-bootstrap-diagonal", "bootstrap-lemma-surjective-flat-locally-finite-presentation", "spaces-groupoids-lemma-quotient-pre-equivalence-relation-restrict", "spaces-more-groupoids-lemma-restrict-preserves-type", "spaces-morphisms-lemma-permanence-finite-type", "spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "bootstrap-lemma-slice-equivalence-relation", "bootstrap-lemma-better-finding-opens", "spaces-lemma-finding-opens", "spaces-lemma-coproduct-algebraic-spaces", "spaces-lemma-glueing-algebraic-spaces", "spaces-more-groupoids-lemma-quasi-splitting-affine-scheme", "bootstrap-lemma-better-finding-opens", "spaces-lemma-finding-opens", "spaces-lemma-coproduct-algebraic-spaces", "spaces-lemma-glueing-algebraic-spaces", "bootstrap-lemma-divide-subgroupoid", "descent-lemma-descending-property-flat", "descent-lemma-descending-property-locally-finite-presentation", "descent-lemma-descending-property-quasi-finite", "descent-lemma-descending-property-open-immersion", "groupoids-lemma-quotient-pre-equivalence-relation-restrict", "more-groupoids-lemma-sheaf-differentials", "morphisms-lemma-unramified-omega-zero", "morphisms-lemma-flat-unramified-etale", "spaces-theorem-presentation" ], "ref_ids": [ 2620, 2617, 9317, 13179, 4818, 4838, 4983, 2623, 2622, 8151, 8147, 8148, 13213, 2622, 8151, 8147, 8148, 2624, 14680, 14676, 14689, 14681, 9650, 2457, 5343, 5373, 8124 ] } ], "ref_ids": [] }, { "id": 2603, "type": "theorem", "label": "bootstrap-lemma-morphism-spaces-is-representable-by-spaces", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-morphism-spaces-is-representable-by-spaces", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Then $f$ is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "This is formal. It relies on the fact that", "the category of algebraic spaces over $S$ has fibre products, see", "Spaces, Lemma \\ref{spaces-lemma-fibre-product-spaces}." ], "refs": [ "spaces-lemma-fibre-product-spaces" ], "ref_ids": [ 8143 ] } ], "ref_ids": [] }, { "id": 2604, "type": "theorem", "label": "bootstrap-lemma-base-change-transformation", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-base-change-transformation", "contents": [ "\\begin{slogan}", "A base change of a representable by algebraic spaces morphism of", "presheaves is representable by algebraic spaces.", "\\end{slogan}", "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "G' \\times_G F \\ar[r] \\ar[d]^{a'} & F \\ar[d]^a \\\\", "G' \\ar[r] & G", "}", "$$", "be a fibre square of presheaves on $(\\Sch/S)_{fppf}$.", "If $a$ is representable by algebraic spaces so is $a'$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2605, "type": "theorem", "label": "bootstrap-lemma-representable-by-spaces-transformation-to-sheaf", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-representable-by-spaces-transformation-to-sheaf", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$ be representable by algebraic spaces.", "If $G$ is a sheaf, then so is $F$." ], "refs": [], "proofs": [ { "contents": [ "(Same as the proof of", "Spaces, Lemma \\ref{spaces-lemma-representable-transformation-to-sheaf}.)", "Let $\\{\\varphi_i : T_i \\to T\\}$ be a covering of the site", "$(\\Sch/S)_{fppf}$.", "Let $s_i \\in F(T_i)$ which satisfy the sheaf condition.", "Then $\\sigma_i = a(s_i) \\in G(T_i)$ satisfy the sheaf condition", "also. Hence there exists a unique $\\sigma \\in G(T)$ such", "that $\\sigma_i = \\sigma|_{T_i}$. By assumption", "$F' = h_T \\times_{\\sigma, G, a} F$ is a sheaf.", "Note that $(\\varphi_i, s_i) \\in F'(T_i)$ satisfy the", "sheaf condition also, and hence come from some unique", "$(\\text{id}_T, s) \\in F'(T)$. Clearly $s$ is the section of", "$F$ we are looking for." ], "refs": [ "spaces-lemma-representable-transformation-to-sheaf" ], "ref_ids": [ 8129 ] } ], "ref_ids": [] }, { "id": 2606, "type": "theorem", "label": "bootstrap-lemma-representable-by-spaces-transformation-diagonal", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-representable-by-spaces-transformation-diagonal", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$ be representable by algebraic spaces.", "Then $\\Delta_{F/G} : F \\to F \\times_G F$ is representable by", "algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "(Same as the proof of", "Spaces, Lemma \\ref{spaces-lemma-representable-transformation-diagonal}.)", "Let $U$ be a scheme. Let $\\xi = (\\xi_1, \\xi_2) \\in (F \\times_G F)(U)$.", "Set $\\xi' = a(\\xi_1) = a(\\xi_2) \\in G(U)$.", "By assumption there exist an algebraic space $V$ and a morphism $V \\to U$", "representing the fibre product $U \\times_{\\xi', G} F$.", "In particular, the elements $\\xi_1, \\xi_2$ give morphisms", "$f_1, f_2 : U \\to V$ over $U$. Because $V$ represents the", "fibre product $U \\times_{\\xi', G} F$ and because", "$\\xi' = a \\circ \\xi_1 = a \\circ \\xi_2$", "we see that if $g : U' \\to U$ is a morphism then", "$$", "g^*\\xi_1 = g^*\\xi_2", "\\Leftrightarrow", "f_1 \\circ g = f_2 \\circ g.", "$$", "In other words, we see that $U \\times_{\\xi, F \\times_G F} F$", "is represented by $V \\times_{\\Delta, V \\times V, (f_1, f_2)} U$", "which is an algebraic space." ], "refs": [ "spaces-lemma-representable-transformation-diagonal" ], "ref_ids": [ 8130 ] } ], "ref_ids": [] }, { "id": 2607, "type": "theorem", "label": "bootstrap-lemma-representable-by-spaces-over-space", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-representable-by-spaces-over-space", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$ be representable by algebraic spaces.", "If $G$ is an algebraic space, then so is $F$." ], "refs": [], "proofs": [ { "contents": [ "We have seen in", "Lemma \\ref{lemma-representable-by-spaces-transformation-to-sheaf}", "that $F$ is a sheaf.", "\\medskip\\noindent", "Let $U$ be a scheme and let $U \\to G$ be a surjective \\'etale morphism.", "In this case $U \\times_G F$ is an algebraic space. Let $W$ be a scheme", "and let $W \\to U \\times_G F$ be a surjective \\'etale morphism.", "\\medskip\\noindent", "First we claim that $W \\to F$ is representable.", "To see this let $X$ be a scheme and let $X \\to F$ be a morphism.", "Then", "$$", "W \\times_F X = W \\times_{U \\times_G F} U \\times_G F \\times_F X", "= W \\times_{U \\times_G F} (U \\times_G X)", "$$", "Since both $U \\times_G F$ and $G$ are algebraic spaces we see that", "this is a scheme.", "\\medskip\\noindent", "Next, we claim that $W \\to F$ is surjective and \\'etale (this makes", "sense now that we know it is representable). This follows from the", "formula above since both $W \\to U \\times_G F$ and $U \\to G$", "are \\'etale and surjective, hence", "$W \\times_{U \\times_G F} (U \\times_G X) \\to U \\times_G X$ and", "$U \\times_G X \\to X$ are surjective and \\'etale, and the composition of", "surjective \\'etale morphisms is surjective and \\'etale.", "\\medskip\\noindent", "Set $R = W \\times_F W$. By the above $R$ is a scheme and", "the projections $t, s : R \\to W$", "are \\'etale. It is clear that $R$ is an equivalence relation, and", "$W \\to F$ is a surjection of sheaves. Hence $R$ is an \\'etale equivalence", "relation and $F = W/R$. Hence $F$ is an algebraic space by", "Spaces,", "Theorem \\ref{spaces-theorem-presentation}." ], "refs": [ "bootstrap-lemma-representable-by-spaces-transformation-to-sheaf", "spaces-theorem-presentation" ], "ref_ids": [ 2605, 8124 ] } ], "ref_ids": [] }, { "id": 2608, "type": "theorem", "label": "bootstrap-lemma-representable-by-spaces", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-representable-by-spaces", "contents": [ "Let $S$ be a scheme.", "Let $a : F \\to G$ be a map of presheaves on $(\\Sch/S)_{fppf}$.", "Suppose $a : F \\to G$ is representable by algebraic spaces.", "If $X$ is an algebraic space over $S$, and $X \\to G$ is a map of presheaves", "then $X \\times_G F$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-base-change-transformation} the transformation", "$X \\times_G F \\to X$ is representable by algebraic spaces. Hence it is", "an algebraic space by", "Lemma \\ref{lemma-representable-by-spaces-over-space}." ], "refs": [ "bootstrap-lemma-base-change-transformation", "bootstrap-lemma-representable-by-spaces-over-space" ], "ref_ids": [ 2604, 2607 ] } ], "ref_ids": [] }, { "id": 2609, "type": "theorem", "label": "bootstrap-lemma-composition-transformation", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-composition-transformation", "contents": [ "Let $S$ be a scheme.", "Let", "$$", "\\xymatrix{", "F \\ar[r]^a & G \\ar[r]^b & H", "}", "$$", "be maps of presheaves on $(\\Sch/S)_{fppf}$.", "If $a$ and $b$ are representable by algebraic spaces, so is", "$b \\circ a$." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme over $S$, and let $T \\to H$ be a morphism.", "By assumption $T \\times_H G$ is an algebraic space. Hence by", "Lemma \\ref{lemma-representable-by-spaces}", "we see that $T \\times_H F = (T \\times_H G) \\times_G F$ is an", "algebraic space as well." ], "refs": [ "bootstrap-lemma-representable-by-spaces" ], "ref_ids": [ 2608 ] } ], "ref_ids": [] }, { "id": 2610, "type": "theorem", "label": "bootstrap-lemma-product-transformations", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-product-transformations", "contents": [ "Let $S$ be a scheme.", "Let $F_i, G_i : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$, $i = 1, 2$.", "Let $a_i : F_i \\to G_i$, $i = 1, 2$", "be representable by algebraic spaces.", "Then", "$$", "a_1 \\times a_2 : F_1 \\times F_2 \\longrightarrow G_1 \\times G_2", "$$", "is a representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Write $a_1 \\times a_2$ as the composition", "$F_1 \\times F_2 \\to G_1 \\times F_2 \\to G_1 \\times G_2$.", "The first arrow is the base change of $a_1$ by the map", "$G_1 \\times F_2 \\to G_1$, and the second arrow", "is the base change of $a_2$ by the map", "$G_1 \\times G_2 \\to G_2$. Hence this lemma is a formal", "consequence of Lemmas \\ref{lemma-composition-transformation}", "and \\ref{lemma-base-change-transformation}." ], "refs": [ "bootstrap-lemma-composition-transformation", "bootstrap-lemma-base-change-transformation" ], "ref_ids": [ 2609, 2604 ] } ], "ref_ids": [] }, { "id": 2611, "type": "theorem", "label": "bootstrap-lemma-representable-by-spaces-permanence", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-representable-by-spaces-permanence", "contents": [ "Let $S$ be a scheme. Let $a : F \\to G$ and $b : G \\to H$ be", "transformations of functors $(\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Assume", "\\begin{enumerate}", "\\item $\\Delta : G \\to G \\times_H G$ is representable", "by algebraic spaces, and", "\\item $b \\circ a : F \\to H$ is representable by algebraic spaces.", "\\end{enumerate}", "Then $a$ is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme over $S$ and let $\\xi \\in G(U)$. Then", "$$", "U \\times_{\\xi, G, a} F =", "(U \\times_{b(\\xi), H, b \\circ a} F) \\times_{(\\xi, a), (G \\times_H G), \\Delta} G", "$$", "Hence the result using Lemma \\ref{lemma-representable-by-spaces}." ], "refs": [ "bootstrap-lemma-representable-by-spaces" ], "ref_ids": [ 2608 ] } ], "ref_ids": [] }, { "id": 2612, "type": "theorem", "label": "bootstrap-lemma-glueing-sheaves", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-glueing-sheaves", "contents": [ "Let $S \\in \\Ob(\\Sch_{fppf})$. Let $F$ be a presheaf of sets on", "$(\\Sch/S)_{fppf}$. Assume", "\\begin{enumerate}", "\\item $F$ is a sheaf for the Zariski topology on $(\\Sch/S)_{fppf}$,", "\\item there exists an index set $I$ and subfunctors $F_i \\subset F$ such that", "\\begin{enumerate}", "\\item each $F_i$ is an fppf sheaf,", "\\item each $F_i \\to F$ is representable by algebraic spaces,", "\\item $\\coprod F_i \\to F$ becomes surjective after fppf sheafification.", "\\end{enumerate}", "\\end{enumerate}", "Then $F$ is an fppf sheaf." ], "refs": [], "proofs": [ { "contents": [ "Let $T \\in \\Ob((\\Sch/S)_{fppf})$ and let $s \\in F(T)$. By (2)(c)", "there exists an fppf covering $\\{T_j \\to T\\}$ such that", "$s|_{T_j}$ is a section of $F_{\\alpha(j)}$ for some $\\alpha(j) \\in I$.", "Let $W_j \\subset T$ be the image of $T_j \\to T$", "which is an open subscheme Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}.", "By (2)(b) we see", "$F_{\\alpha(j)} \\times_{F, s|_{W_j}} W_j \\to W_j$ is a monomorphism", "of algebraic spaces through which $T_j$ factors. Since $\\{T_j \\to W_j\\}$", "is an fppf covering, we conclude that", "$F_{\\alpha(j)} \\times_{F, s|_{W_j}} W_j = W_j$, in other words", "$s|_{W_j} \\in F_{\\alpha(j)}(W_j)$. Hence we conclude that", "$\\coprod F_i \\to F$ is surjective for the Zariski topology.", "\\medskip\\noindent", "Let $\\{T_j \\to T\\}$ be an fppf covering in $(\\Sch/S)_{fppf}$.", "Let $s, s' \\in F(T)$ with $s|_{T_j} = s'|_{T_j}$ for all $j$.", "We want to show that $s, s'$ are equal. As $F$ is a Zariski sheaf by (1)", "we may work Zariski locally on $T$. By the result of the previous paragraph", "we may assume there exist $i$ such that $s \\in F_i(T)$. Then we see that", "$s'|_{T_j}$ is a section of $F_i$. By (2)(b) we see", "$F_{i} \\times_{F, s'} T \\to T$ is a monomorphism of algebraic spaces", "through which all of the $T_j$ factor. Hence we conclude that", "$s' \\in F_i(T)$. Since $F_i$ is a sheaf for the fppf topology", "we conclude that $s = s'$.", "\\medskip\\noindent", "Let $\\{T_j \\to T\\}$ be an fppf covering in $(\\Sch/S)_{fppf}$ and let", "$s_j \\in F(T_j)$ such that", "$s_j|_{T_j \\times_T T_{j'}} = s_{j'}|_{T_j \\times_T T_{j'}}$. By assumption", "(2)(b) we may refine the covering and assume that $s_j \\in F_{\\alpha(j)}(T_j)$", "for some $\\alpha(j) \\in I$. Let $W_j \\subset T$ be the image of $T_j \\to T$", "which is an open subscheme Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}.", "Then $\\{T_j \\to W_j\\}$ is an fppf covering. Since $F_{\\alpha(j)}$ is a sub", "presheaf of $F$ we see that the two restrictions of $s_j$ to", "$T_j \\times_{W_j} T_j$ agree as elements of", "$F_{\\alpha(j)}(T_j \\times_{W_j} T_j)$. Hence, the sheaf condition for", "$F_{\\alpha(j)}$ implies there exists a $s'_j \\in F_{\\alpha(j)}(W_j)$", "whose restriction to $T_j$ is $s_j$. For a pair of indices", "$j$ and $j'$ the sections $s'_j|_{W_j \\cap W_{j'}}$ and", "$s'_{j'}|_{W_j \\cap W_{j'}}$ of $F$ agree by the result of the", "previous paragraph. This finishes the proof by the fact that", "$F$ is a Zariski sheaf." ], "refs": [ "morphisms-lemma-fppf-open", "morphisms-lemma-fppf-open" ], "ref_ids": [ 5267, 5267 ] } ], "ref_ids": [] }, { "id": 2613, "type": "theorem", "label": "bootstrap-lemma-base-change-transformation-property", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-base-change-transformation-property", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{P}$ be a property as in", "Definition \\ref{definition-property-transformation}.", "Let", "$$", "\\xymatrix{", "G' \\times_G F \\ar[r] \\ar[d]^{a'} & F \\ar[d]^a \\\\", "G' \\ar[r] & G", "}", "$$", "be a fibre square of presheaves on $(\\Sch/S)_{fppf}$.", "If $a$ is representable by algebraic spaces and has $\\mathcal{P}$", "so does $a'$." ], "refs": [ "bootstrap-definition-property-transformation" ], "proofs": [ { "contents": [ "Omitted. Hint: This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 2638 ] }, { "id": 2614, "type": "theorem", "label": "bootstrap-lemma-composition-transformation-property", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-composition-transformation-property", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{P}$ be a property as in", "Definition \\ref{definition-property-transformation},", "and assume $\\mathcal{P}$ is stable under composition.", "Let", "$$", "\\xymatrix{", "F \\ar[r]^a & G \\ar[r]^b & H", "}", "$$", "be maps of presheaves on $(\\Sch/S)_{fppf}$.", "If $a$, $b$ are representable by algebraic spaces and has", "$\\mathcal{P}$ so does $b \\circ a$." ], "refs": [ "bootstrap-definition-property-transformation" ], "proofs": [ { "contents": [ "Omitted. Hint: See", "Lemma \\ref{lemma-composition-transformation}", "and use stability under composition." ], "refs": [ "bootstrap-lemma-composition-transformation" ], "ref_ids": [ 2609 ] } ], "ref_ids": [ 2638 ] }, { "id": 2615, "type": "theorem", "label": "bootstrap-lemma-product-transformations-property", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-product-transformations-property", "contents": [ "Let $S$ be a scheme.", "Let $F_i, G_i : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$,", "$i = 1, 2$.", "Let $a_i : F_i \\to G_i$, $i = 1, 2$ be representable by algebraic spaces.", "Let $\\mathcal{P}$ be a property as in", "Definition \\ref{definition-property-transformation}", "which is stable under composition.", "If $a_1$ and $a_2$ have property $\\mathcal{P}$ so does", "$a_1 \\times a_2 : F_1 \\times F_2 \\longrightarrow G_1 \\times G_2$." ], "refs": [ "bootstrap-definition-property-transformation" ], "proofs": [ { "contents": [ "Note that the lemma makes sense by", "Lemma \\ref{lemma-product-transformations}.", "Proof omitted." ], "refs": [ "bootstrap-lemma-product-transformations" ], "ref_ids": [ 2610 ] } ], "ref_ids": [ 2638 ] }, { "id": 2616, "type": "theorem", "label": "bootstrap-lemma-transformations-property-implication", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-transformations-property-implication", "contents": [ "Let $S$ be a scheme.", "Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$ be a transformation of functors representable by", "algebraic spaces.", "Let $\\mathcal{P}$, $\\mathcal{P}'$ be properties as in", "Definition \\ref{definition-property-transformation}.", "Suppose that for any morphism $f : X \\to Y$ of algebraic spaces over $S$", "we have $\\mathcal{P}(f) \\Rightarrow \\mathcal{P}'(f)$.", "If $a$ has property $\\mathcal{P}$, then", "$a$ has property $\\mathcal{P}'$." ], "refs": [ "bootstrap-definition-property-transformation" ], "proofs": [ { "contents": [ "Formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 2638 ] }, { "id": 2617, "type": "theorem", "label": "bootstrap-lemma-surjective-flat-locally-finite-presentation", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-surjective-flat-locally-finite-presentation", "contents": [ "Let $S$ be a scheme.", "Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be sheaves.", "Let $a : F \\to G$ be representable by algebraic spaces, flat,", "locally of finite presentation, and surjective.", "Then $a : F \\to G$ is surjective as a map of sheaves." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme over $S$ and let $g : T \\to G$ be a $T$-valued point of", "$G$. By assumption $T' = F \\times_G T$ is an algebraic space and", "the morphism $T' \\to T$ is a flat, locally of finite presentation, and", "surjective morphism of algebraic spaces.", "Let $U \\to T'$ be a surjective \\'etale morphism, where $U$ is a scheme.", "Then by the definition of flat morphisms of algebraic spaces", "the morphism of schemes $U \\to T$ is flat. Similarly for", "``locally of finite presentation''. The morphism $U \\to T$ is surjective", "also, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-surjective-local}.", "Hence we see that $\\{U \\to T\\}$ is an fppf covering such", "that $g|_U \\in G(U)$ comes from an element of $F(U)$, namely", "the map $U \\to T' \\to F$. This proves the map is surjective as", "a map of sheaves, see", "Sites, Definition \\ref{sites-definition-sheaves-injective-surjective}." ], "refs": [ "spaces-morphisms-lemma-surjective-local", "sites-definition-sheaves-injective-surjective" ], "ref_ids": [ 4725, 8660 ] } ], "ref_ids": [] }, { "id": 2618, "type": "theorem", "label": "bootstrap-lemma-representable-diagonal", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-representable-diagonal", "contents": [ "\\begin{slogan}", "The diagonal of a presheaf is representable by algebraic spaces if and only if", "every map from a scheme to the presheaf is representable by algebraic spaces.", "\\end{slogan}", "Let $S$ be a scheme.", "If $F$ is a presheaf on $(\\Sch/S)_{fppf}$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $\\Delta_F : F \\to F \\times F$ is representable by algebraic spaces,", "\\item for every scheme $T$ any map $T \\to F$ is representable by algebraic", "spaces, and", "\\item for every algebraic space $X$ any map $X \\to F$ is representable", "by algebraic spaces.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Let $X \\to F$ be as in (3). Let $T$ be a scheme, and let", "$T \\to F$ be a morphism. Then we have", "$$", "T \\times_F X = (T \\times_S X) \\times_{F \\times F, \\Delta} F", "$$", "which is an algebraic space by", "Lemma \\ref{lemma-representable-by-spaces}", "and (1). Hence $X \\to F$ is representable, i.e., (3) holds.", "The implication (3) $\\Rightarrow$ (2) is trivial. Assume (2).", "Let $T$ be a scheme, and let $(a, b) : T \\to F \\times F$ be a morphism.", "Then", "$$", "F \\times_{\\Delta_F, F \\times F} T =", "(T \\times_{a, F, b} T) \\times_{T \\times T, \\Delta_T} T", "$$", "which is an algebraic space by assumption. Hence $\\Delta_F$ is", "representable by algebraic spaces, i.e., (1) holds." ], "refs": [ "bootstrap-lemma-representable-by-spaces" ], "ref_ids": [ 2608 ] } ], "ref_ids": [] }, { "id": 2619, "type": "theorem", "label": "bootstrap-lemma-after-fppf-sep-lqf", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-after-fppf-sep-lqf", "contents": [ "Let $S$ be a scheme.", "Let", "$$", "\\xymatrix{", "E \\ar[r]_a \\ar[d]_f & F \\ar[d]^g \\\\", "H \\ar[r]^b & G", "}", "$$", "be a cartesian diagram of sheaves on $(\\Sch/S)_{fppf}$, so", "$E = H \\times_G F$. If", "\\begin{enumerate}", "\\item $g$ is representable by algebraic spaces, surjective, flat, and", "locally of finite presentation, and", "\\item $a$ is representable by algebraic spaces, separated, and", "locally quasi-finite", "\\end{enumerate}", "then $b$ is representable (by schemes) as well as separated and", "locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme, and let $T \\to G$ be a morphism.", "We have to show that $T \\times_G H$ is a scheme, and that", "the morphism $T \\times_G H \\to T$ is separated and", "locally quasi-finite. Thus we may base change the whole diagram to $T$", "and assume that $G$ is a scheme. In this case $F$ is an algebraic space.", "Let $U$ be a scheme, and let $U \\to F$ be a surjective \\'etale morphism.", "Then $U \\to F$ is representable, surjective, flat and", "locally of finite presentation by", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-etale-flat} and", "\\ref{spaces-morphisms-lemma-etale-locally-finite-presentation}.", "By", "Lemma \\ref{lemma-composition-transformation}", "$U \\to G$ is surjective, flat and locally of finite presentation also.", "Note that the base change $E \\times_F U \\to U$ of $a$ is still", "separated and locally quasi-finite (by", "Lemma \\ref{lemma-base-change-transformation-property}). Hence we", "may replace the upper part of the diagram of the lemma by", "$E \\times_F U \\to U$. In other words, we may assume that", "$F \\to G$ is a surjective, flat morphism of schemes", "which is locally of finite presentation.", "In particular, $\\{F \\to G\\}$ is an fppf covering of schemes.", "By", "Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}", "we conclude that $E$ is a scheme also.", "By", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves}", "the fact that $E = H \\times_G F$ means that we get a descent datum", "on $E$ relative to the fppf covering $\\{F \\to G\\}$.", "By", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend}", "this descent datum is effective.", "By", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves}", "again this implies that $H$ is a scheme.", "By", "Descent, Lemmas \\ref{descent-lemma-descending-property-separated} and", "\\ref{descent-lemma-descending-property-quasi-finite}", "it now follows that $b$ is separated and locally quasi-finite." ], "refs": [ "spaces-morphisms-lemma-etale-flat", "spaces-morphisms-lemma-etale-locally-finite-presentation", "bootstrap-lemma-composition-transformation", "bootstrap-lemma-base-change-transformation-property", "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "descent-lemma-descent-data-sheaves", "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "descent-lemma-descent-data-sheaves", "descent-lemma-descending-property-separated", "descent-lemma-descending-property-quasi-finite" ], "ref_ids": [ 4910, 4911, 2609, 2613, 4983, 14751, 13949, 14751, 14671, 14689 ] } ], "ref_ids": [] }, { "id": 2620, "type": "theorem", "label": "bootstrap-lemma-bootstrap-diagonal", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-bootstrap-diagonal", "contents": [ "Let $S$ be a scheme.", "Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor.", "Assume that", "\\begin{enumerate}", "\\item the presheaf $F$ is a sheaf,", "\\item there exists an algebraic space $X$ and a map $X \\to F$", "which is representable by algebraic spaces, surjective, flat and", "locally of finite presentation.", "\\end{enumerate}", "Then $\\Delta_F$ is representable (by schemes)." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be a surjective \\'etale morphism from a scheme towards $X$.", "Then $U \\to X$ is representable, surjective, flat and", "locally of finite presentation by", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-etale-flat} and", "\\ref{spaces-morphisms-lemma-etale-locally-finite-presentation}.", "By", "Lemma \\ref{lemma-composition-transformation-property}", "the composition $U \\to F$ is representable by algebraic spaces,", "surjective, flat and locally of finite presentation also.", "Thus we see that $R = U \\times_F U$ is an algebraic space, see", "Lemma \\ref{lemma-representable-by-spaces}.", "The morphism of algebraic spaces $R \\to U \\times_S U$ is", "a monomorphism, hence separated (as the diagonal of a monomorphism", "is an isomorphism, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-monomorphism}).", "Since $U \\to F$ is locally of finite presentation, both", "morphisms $R \\to U$ are locally of finite presentation, see", "Lemma \\ref{lemma-base-change-transformation-property}.", "Hence $R \\to U \\times_S U$ is locally of finite type (use", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-finite-presentation-finite-type} and", "\\ref{spaces-morphisms-lemma-permanence-finite-type}).", "Altogether this means that", "$R \\to U \\times_S U$ is a monomorphism which is locally of finite", "type, hence a separated and locally quasi-finite morphism, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}.", "\\medskip\\noindent", "Now we are ready to prove that $\\Delta_F$ is representable.", "Let $T$ be a scheme, and let $(a, b) : T \\to F \\times F$ be a morphism.", "Set", "$$", "T' = (U \\times_S U) \\times_{F \\times F} T.", "$$", "Note that $U \\times_S U \\to F \\times F$ is", "representable by algebraic spaces, surjective, flat and", "locally of finite presentation by", "Lemma \\ref{lemma-product-transformations-property}.", "Hence $T'$ is an algebraic space, and the projection morphism", "$T' \\to T$ is surjective, flat, and locally of finite presentation.", "Consider $Z = T \\times_{F \\times F} F$ (this is a sheaf) and", "$$", "Z' = T' \\times_{U \\times_S U} R", "= T' \\times_T Z.", "$$", "We see that $Z'$ is an algebraic space, and", "$Z' \\to T'$ is separated and locally quasi-finite by the", "discussion in the first paragraph of the proof which showed that $R$ is", "an algebraic space and that the", "morphism $R \\to U \\times_S U$ has those properties.", "Hence we may apply", "Lemma \\ref{lemma-after-fppf-sep-lqf}", "to the diagram", "$$", "\\xymatrix{", "Z' \\ar[r] \\ar[d] & T' \\ar[d] \\\\", "Z \\ar[r] & T", "}", "$$", "and we conclude." ], "refs": [ "spaces-morphisms-lemma-etale-flat", "spaces-morphisms-lemma-etale-locally-finite-presentation", "bootstrap-lemma-composition-transformation-property", "bootstrap-lemma-representable-by-spaces", "spaces-morphisms-lemma-monomorphism", "bootstrap-lemma-base-change-transformation-property", "spaces-morphisms-lemma-finite-presentation-finite-type", "spaces-morphisms-lemma-permanence-finite-type", "spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "bootstrap-lemma-product-transformations-property", "bootstrap-lemma-after-fppf-sep-lqf" ], "ref_ids": [ 4910, 4911, 2614, 2608, 4751, 2613, 4842, 4818, 4838, 2615, 2619 ] } ], "ref_ids": [] }, { "id": 2621, "type": "theorem", "label": "bootstrap-lemma-bootstrap-locally-quasi-finite", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-bootstrap-locally-quasi-finite", "contents": [ "Let $S$ be a scheme. Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a", "functor. Let $X$ be a scheme and let $X \\to F$ be representable by algebraic", "spaces and locally quasi-finite. Then $X \\to F$ is representable", "(by schemes)." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme and let $T \\to F$ be a morphism. We have to show that", "the algebraic space $X \\times_F T$ is representable by a scheme. Consider", "the morphism", "$$", "X \\times_F T \\longrightarrow X \\times_{\\Spec(\\mathbf{Z})} T", "$$", "Since $X \\times_F T \\to T$ is locally quasi-finite, so is the displayed", "arrow (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-permanence-quasi-finite}).", "On the other hand, the displayed arrow is a monomorphism", "and hence separated (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-monomorphism-separated}).", "Thus $X \\times_F T$ is a scheme by Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}." ], "refs": [ "spaces-morphisms-lemma-permanence-quasi-finite", "spaces-morphisms-lemma-monomorphism-separated", "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme" ], "ref_ids": [ 4836, 4752, 4983 ] } ], "ref_ids": [] }, { "id": 2622, "type": "theorem", "label": "bootstrap-lemma-better-finding-opens", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-better-finding-opens", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Let $g : U' \\to U$ be a morphism.", "Assume", "\\begin{enumerate}", "\\item the composition", "$$", "\\xymatrix{", "U' \\times_{g, U, t} R \\ar[r]_-{\\text{pr}_1} \\ar@/^3ex/[rr]^h", "& R \\ar[r]_s & U", "}", "$$", "has an open image $W \\subset U$, and", "\\item the resulting map $h : U' \\times_{g, U, t} R \\to W$", "defines a surjection of sheaves in the fppf topology.", "\\end{enumerate}", "Let $R' = R|_{U'}$ be the restriction of $R$ to $U$. Then the map", "of quotient sheaves", "$$", "U'/R' \\to U/R", "$$", "in the fppf topology is representable, and is an open immersion." ], "refs": [], "proofs": [ { "contents": [ "Note that $W$ is an $R$-invariant open subscheme of $U$.", "This is true because the set of points of $W$ is the set", "of points of $U$ which are equivalent in the sense of", "Groupoids,", "Lemma \\ref{groupoids-lemma-pre-equivalence-equivalence-relation-points}", "to a point of $g(U') \\subset U$ (the lemma applies as $j : R \\to U \\times_S U$", "is a pre-equivalence relation by", "Groupoids, Lemma \\ref{groupoids-lemma-groupoid-pre-equivalence}).", "Also $g : U' \\to U$ factors through $W$.", "Let $R|_W$ be the restriction of $R$ to $W$.", "Then it follows that $R'$ is also the restriction of $R|_W$ to $U'$.", "Hence we can factor the map of sheaves of the lemma as", "$$", "U'/R' \\longrightarrow W/R|_W \\longrightarrow U/R", "$$", "By Groupoids, Lemma \\ref{groupoids-lemma-quotient-groupoid-restrict}", "we see that the first arrow is an isomorphism of sheaves.", "Hence it suffices to show the lemma in case $g$ is the immersion", "of an $R$-invariant open into $U$.", "\\medskip\\noindent", "Assume $U' \\subset U$ is an $R$-invariant open and $g$ is the inclusion", "morphism. Set $F = U/R$ and $F' = U'/R'$. By", "Groupoids,", "Lemma \\ref{groupoids-lemma-quotient-pre-equivalence-relation-restrict}", "or \\ref{groupoids-lemma-quotient-groupoid-restrict}", "the map $F' \\to F$ is injective. Let $\\xi \\in F(T)$.", "We have to show that $T \\times_{\\xi, F} F'$ is representable", "by an open subscheme of $T$.", "There exists an fppf covering $\\{f_i : T_i \\to T\\}$ such that", "$\\xi|_{T_i}$ is the image via $U \\to U/R$ of a morphism $a_i : T_i \\to U$.", "Set $V_i = s_i^{-1}(U')$.", "We claim that $V_i \\times_T T_j = T_i \\times_T V_j$ as open subschemes", "of $T_i \\times_T T_j$.", "\\medskip\\noindent", "As $a_i \\circ \\text{pr}_0$ and $a_j \\circ \\text{pr}_1$ are morphisms", "$T_i \\times_T T_j \\to U$ which both map to the section", "$\\xi|_{T_i \\times_T T_j} \\in F(T_i \\times_T T_j)$ we can find", "an fppf covering $\\{f_{ijk} : T_{ijk} \\to T_i \\times_T T_j\\}$ and morphisms", "$r_{ijk} : T_{ijk} \\to R$ such that", "$$", "a_i \\circ \\text{pr}_0 \\circ f_{ijk} = s \\circ r_{ijk},", "\\quad", "a_j \\circ \\text{pr}_1 \\circ f_{ijk} = t \\circ r_{ijk},", "$$", "see", "Groupoids, Lemma \\ref{groupoids-lemma-quotient-pre-equivalence}.", "Since $U'$ is $R$-invariant we have $s^{-1}(U') = t^{-1}(U')$ and", "hence $f_{ijk}^{-1}(V_i \\times_T T_j) = f_{ijk}^{-1}(T_i \\times_T V_j)$.", "As $\\{f_{ijk}\\}$ is surjective this implies the claim above.", "Hence by", "Descent, Lemma \\ref{descent-lemma-open-fpqc-covering}", "there exists an open subscheme $V \\subset T$ such that", "$f_i^{-1}(V) = V_i$. We claim that $V$ represents $T \\times_{\\xi, F} F'$.", "\\medskip\\noindent", "As a first step, we will show that $\\xi|_V$ lies in $F'(V) \\subset F(V)$.", "Namely, the family of morphisms $\\{V_i \\to V\\}$ is an fppf covering,", "and by construction we have $\\xi|_{V_i} \\in F'(V_i)$.", "Hence by the sheaf property of $F'$ we get $\\xi|_V \\in F'(V)$.", "Finally, let $T' \\to T$ be a morphism of schemes and", "that $\\xi|_{T'} \\in F'(T')$. To finish the proof we have to show that", "$T' \\to T$ factors through $V$.", "We can find a fppf covering $\\{T'_j \\to T'\\}_{j \\in J}$ and morphisms", "$b_j : T'_j \\to U'$ such that $\\xi|_{T'_j}$ is the image via", "$U' \\to U/R$ of $b_j$. Clearly, it is enough to show that the compositions", "$T'_j \\to T$ factor through $V$. Hence we may assume that $\\xi|_{T'}$", "is the image of a morphism $b : T' \\to U'$. Now, it is enough to show", "that $T'\\times_T T_i \\to T_i$ factors through $V_i$. Over the scheme", "$T' \\times_T T_i$ the restriction of $\\xi$ is the image of two", "elements of $(U/R)(T' \\times_T T_i)$, namely $a_i \\circ \\text{pr}_1$, and", "$b \\circ \\text{pr}_0$, the second of which factors through the $R$-invariant", "open $U'$. Hence by", "Groupoids, Lemma \\ref{groupoids-lemma-quotient-pre-equivalence}", "there exists a covering $\\{h_k : Z_k \\to T' \\times_T T_i\\}$ and morphisms", "$r_k : Z_k \\to R$ such that $a_i \\circ \\text{pr}_1 \\circ h_k = s \\circ r_k$", "and $b \\circ \\text{pr}_0 \\circ h_k = t \\circ r_k$. As $U'$ is an $R$-invariant", "open the fact that $b$ has image in $U'$ then implies that each", "$a_i \\circ \\text{pr}_1 \\circ h_k$ has image in $U'$. It follows from this", "that $T' \\times_T T_i \\to T_i$ has image in $V_i$ by definition of $V_i$", "which concludes the proof." ], "refs": [ "groupoids-lemma-pre-equivalence-equivalence-relation-points", "groupoids-lemma-groupoid-pre-equivalence", "groupoids-lemma-quotient-groupoid-restrict", "groupoids-lemma-quotient-pre-equivalence-relation-restrict", "groupoids-lemma-quotient-groupoid-restrict", "groupoids-lemma-quotient-pre-equivalence", "descent-lemma-open-fpqc-covering", "groupoids-lemma-quotient-pre-equivalence" ], "ref_ids": [ 9579, 9620, 9651, 9650, 9651, 9649, 14637, 9649 ] } ], "ref_ids": [] }, { "id": 2623, "type": "theorem", "label": "bootstrap-lemma-slice-equivalence-relation", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-slice-equivalence-relation", "contents": [ "Let $S$ be a scheme.", "Let $j : R \\to U \\times_S U$ be an equivalence relation on schemes over $S$.", "Assume $s, t : R \\to U$ are flat and locally of finite presentation.", "Then there exists an equivalence relation $j' : R' \\to U'\\times_S U'$", "on schemes over $S$, and an isomorphism", "$$", "U'/R' \\longrightarrow U/R", "$$", "induced by a morphism $U' \\to U$ which maps $R'$ into $R$ such that", "$s', t' : R \\to U$ are flat, locally of finite presentation", "and locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "We will prove this lemma in several steps. We will use without further", "mention that an equivalence relation gives rise to a groupoid scheme", "and that the restriction of an equivalence relation is an equivalence", "relation, see", "Groupoids, Lemmas", "\\ref{groupoids-lemma-restrict-relation},", "\\ref{groupoids-lemma-equivalence-groupoid}, and", "\\ref{groupoids-lemma-restrict-groupoid-relation}.", "\\medskip\\noindent", "Step 1: We may assume that $s, t : R \\to U$ are locally of finite presentation", "and Cohen-Macaulay morphisms. Namely, as in", "More on Groupoids, Lemma \\ref{more-groupoids-lemma-make-CM}", "let $g : U' \\to U$ be the open subscheme such that", "$t^{-1}(U') \\subset R$ is the maximal open over which $s : R \\to U$ is", "Cohen-Macaulay, and denote $R'$ the restriction of $R$ to $U'$.", "By the lemma cited above we see that", "$$", "\\xymatrix{", "t^{-1}(U') \\ar@{=}[r] &", "U' \\times_{g, U, t} R \\ar[r]_-{\\text{pr}_1} \\ar@/^3ex/[rr]^h &", "R \\ar[r]_s &", "U", "}", "$$", "is surjective. Since $h$ is flat and locally of finite presentation, we", "see that $\\{h\\}$ is a fppf covering. Hence by", "Groupoids, Lemma \\ref{groupoids-lemma-quotient-groupoid-restrict}", "we see that $U'/R' \\to U/R$ is an isomorphism. By the construction of $U'$", "we see that $s', t'$ are Cohen-Macaulay and locally of finite presentation.", "\\medskip\\noindent", "Step 2. Assume $s, t$ are Cohen-Macaulay and locally of finite presentation.", "Let $u \\in U$ be a point of finite type. By", "More on Groupoids, Lemma \\ref{more-groupoids-lemma-max-slice-quasi-finite}", "there exists an affine scheme $U'$ and a morphism $g : U' \\to U$ such that", "\\begin{enumerate}", "\\item $g$ is an immersion,", "\\item $u \\in U'$,", "\\item $g$ is locally of finite presentation,", "\\item $h$ is flat, locally of finite presentation and locally quasi-finite, and", "\\item the morphisms $s', t' : R' \\to U'$ are flat, locally of finite", "presentation and locally quasi-finite.", "\\end{enumerate}", "Here we have used the notation introduced in", "More on Groupoids, Situation \\ref{more-groupoids-situation-slice}.", "\\medskip\\noindent", "Step 3. For each point $u \\in U$ which is of finite type", "choose a $g_u : U'_u \\to U$ as in", "Step 2 and denote $R'_u$ the restriction of $R$ to $U'_u$.", "Denote $h_u = s \\circ \\text{pr}_1 : U'_u \\times_{g_u, U, t} R \\to U$. Set", "$U' = \\coprod_{u \\in U} U'_u$, and $g = \\coprod g_u$. Let $R'$ be the", "restriction of $R$ to $U$ as above. We claim that", "the pair $(U', g)$ works\\footnote{Here we should check that $U'$ is not", "too large, i.e., that it is isomorphic to an object of the category", "$\\Sch_{fppf}$, see", "Section \\ref{section-conventions}.", "This is a purely set theoretical matter; let us use the notion of size of", "a scheme introduced in", "Sets, Section \\ref{sets-section-categories-schemes}.", "Note that each $U'_u$ has size at most the size of $U$", "and that the cardinality of the index set is at most the cardinality of", "$|U|$ which is bounded by the size of $U$. Hence $U'$ is isomorphic", "to an object of $\\Sch_{fppf}$ by", "Sets, Lemma \\ref{sets-lemma-what-is-in-it} part (6).}.", "Note that", "\\begin{align*}", "R' = &", "\\coprod\\nolimits_{u_1, u_2 \\in U}", "(U'_{u_1} \\times_{g_{u_1}, U, t} R)", "\\times_R", "(R \\times_{s, U, g_{u_2}} U'_{u_2}) \\\\", "= &", "\\coprod\\nolimits_{u_1, u_2 \\in U}", "(U'_{u_1} \\times_{g_{u_1}, U, t} R) \\times_{h_{u_1}, U, g_{u_2}} U'_{u_2}", "\\end{align*}", "Hence the projection $s' : R' \\to U' = \\coprod U'_{u_2}$", "is flat, locally of finite", "presentation and locally quasi-finite as a base change of $\\coprod h_{u_1}$.", "Finally, by construction the morphism", "$h : U' \\times_{g, U, t} R \\to U$ is equal to $\\coprod h_u$ hence", "its image contains all points of finite type of $U$.", "Since each $h_u$ is flat and locally of finite presentation we conclude that", "$h$ is flat and locally of finite presentation.", "In particular, the image of $h$ is open (see", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open})", "and since the set of points of finite type is dense (see", "Morphisms, Lemma \\ref{morphisms-lemma-enough-finite-type-points})", "we conclude that the image of $h$ is $U$. This implies that", "$\\{h\\}$ is an fppf covering. By", "Groupoids, Lemma \\ref{groupoids-lemma-quotient-groupoid-restrict}", "this means that $U'/R' \\to U/R$ is an isomorphism.", "This finishes the proof of the lemma." ], "refs": [ "groupoids-lemma-restrict-relation", "groupoids-lemma-equivalence-groupoid", "groupoids-lemma-restrict-groupoid-relation", "more-groupoids-lemma-make-CM", "groupoids-lemma-quotient-groupoid-restrict", "more-groupoids-lemma-max-slice-quasi-finite", "sets-lemma-what-is-in-it", "morphisms-lemma-fppf-open", "morphisms-lemma-enough-finite-type-points", "groupoids-lemma-quotient-groupoid-restrict" ], "ref_ids": [ 9578, 9621, 9643, 2463, 9651, 2490, 8795, 5267, 5210, 9651 ] } ], "ref_ids": [] }, { "id": 2624, "type": "theorem", "label": "bootstrap-lemma-divide-subgroupoid", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-divide-subgroupoid", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Let $P \\to R$ be monomorphism of schemes. Assume that", "\\begin{enumerate}", "\\item $(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t}P})$ is a groupoid scheme,", "\\item $s|_P, t|_P : P \\to U$ are finite locally free,", "\\item $j|_P : P \\to U \\times_S U$ is a monomorphism.", "\\item $U$ is affine, and", "\\item $j : R \\to U \\times_S U$ is separated and locally quasi-finite,", "\\end{enumerate}", "Then $U/P$ is representable by an affine scheme $\\overline{U}$, the", "quotient morphism $U \\to \\overline{U}$ is finite locally free, and", "$P = U \\times_{\\overline{U}} U$. Moreover, $R$ is the restriction of a", "groupoid scheme", "$(\\overline{U}, \\overline{R}, \\overline{s}, \\overline{t}, \\overline{c})$", "on $\\overline{U}$ via the quotient morphism $U \\to \\overline{U}$." ], "refs": [], "proofs": [ { "contents": [ "Conditions (1), (2), (3), and (4) and", "Groupoids, Proposition \\ref{groupoids-proposition-finite-flat-equivalence}", "imply the affine scheme $\\overline{U}$ representing $U/P$ exists,", "the morphism $U \\to \\overline{U}$ is finite locally free, and", "$P = U \\times_{\\overline{U}} U$. The identification", "$P = U \\times_{\\overline{U}} U$ is such that $t|_P = \\text{pr}_0$ and", "$s|_P = \\text{pr}_1$, and such that composition is equal to", "$\\text{pr}_{02} : U \\times_{\\overline{U}} U \\times_{\\overline{U}} U", "\\to U \\times_{\\overline{U}} U$.", "A product of finite locally free morphisms is finite locally free (see", "Spaces, Lemma \\ref{spaces-lemma-product-representable-transformations-property}", "and", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-finite-locally-free} and", "\\ref{morphisms-lemma-composition-finite-locally-free}).", "To get $\\overline{R}$ we are going to descend", "the scheme $R$ via the finite locally free morphism", "$U \\times_S U \\to \\overline{U} \\times_S \\overline{U}$.", "Namely, note that", "$$", "(U \\times_S U)", "\\times_{(\\overline{U} \\times_S \\overline{U})}", "(U \\times_S U)", "=", "P \\times_S P", "$$", "by the above. Thus giving a descent datum (see", "Descent, Definition \\ref{descent-definition-descent-datum})", "for $R / U \\times_S U / \\overline{U} \\times_S \\overline{U}$", "consists of an isomorphism", "$$", "\\varphi :", "R \\times_{(U \\times_S U), t \\times t} (P \\times_S P)", "\\longrightarrow", "(P \\times_S P) \\times_{s \\times s, (U \\times_S U)} R", "$$", "over $P \\times_S P$ satisfying a cocycle condition. We define $\\varphi$", "on $T$-valued points by the rule", "$$", "\\varphi : (r, (p, p')) \\longmapsto ((p, p'), p^{-1} \\circ r \\circ p')", "$$", "where the composition is taken in the groupoid category", "$(U(T), R(T), s, t, c)$.", "This makes sense because for $(r, (p, p'))$ to be a $T$-valued point", "of the source of $\\varphi$ it needs to be the case that $t(r) = t(p)$", "and $s(r) = t(p')$. Note that this map is an isomorphism", "with inverse given by", "$((p, p'), r') \\mapsto (p \\circ r' \\circ (p')^{-1}, (p, p'))$.", "To check the cocycle condition we have to verify that", "$\\varphi_{02} = \\varphi_{12} \\circ \\varphi_{01}$", "as maps over", "$$", "(U \\times_S U)", "\\times_{(\\overline{U} \\times_S \\overline{U})} (U \\times_S U)", "\\times_{(\\overline{U} \\times_S \\overline{U})} (U \\times_S U) =", "(P \\times_S P) \\times_{s \\times s, (U \\times_S U), t \\times t} (P \\times_S P)", "$$", "By explicit calculation we see that", "$$", "\\begin{matrix}", "\\varphi_{02} & (r, (p_1, p_1'), (p_2, p_2')) & \\mapsto &", "((p_1, p_1'), (p_2, p_2'),", "(p_1 \\circ p_2)^{-1} \\circ r \\circ (p_1' \\circ p_2')) \\\\", "\\varphi_{01} & (r, (p_1, p_1'), (p_2, p_2')) & \\mapsto &", "((p_1, p_1'), p_1^{-1} \\circ r \\circ p_1', (p_2, p_2')) \\\\", "\\varphi_{12} & ((p_1, p_1'), r, (p_2, p_2')) & \\mapsto &", "((p_1, p_1'), (p_2, p_2'), p_2^{-1} \\circ r \\circ p_2')", "\\end{matrix}", "$$", "(with obvious notation) which implies what we want.", "As $j$ is separated and locally quasi-finite by (5) we may apply", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend}", "to get a scheme $\\overline{R} \\to \\overline{U} \\times_S \\overline{U}$", "and an isomorphism", "$$", "R \\to \\overline{R} \\times_{(\\overline{U} \\times_S \\overline{U})} (U \\times_S U)", "$$", "which identifies the descent datum $\\varphi$ with the canonical", "descent datum on", "$\\overline{R} \\times_{(\\overline{U} \\times_S \\overline{U})} (U \\times_S U)$,", "see", "Descent, Definition \\ref{descent-definition-effective}.", "\\medskip\\noindent", "Since $U \\times_S U \\to \\overline{U} \\times_S \\overline{U}$ is finite", "locally free we conclude that $R \\to \\overline{R}$ is finite locally free", "as a base change. Hence $R \\to \\overline{R}$ is surjective as a map of", "sheaves on $(\\Sch/S)_{fppf}$.", "Our choice of $\\varphi$ implies that given $T$-valued points $r, r' \\in R(T)$", "these have the same image in $\\overline{R}$ if and only if", "$p^{-1} \\circ r \\circ p'$ for some $p, p' \\in P(T)$. Thus", "$\\overline{R}$ represents the sheaf", "$$", "T \\longmapsto \\overline{R(T)} = P(T)\\backslash R(T)/P(T)", "$$", "with notation as in the discussion preceding the lemma.", "Hence we can define the groupoid structure on", "$(\\overline{U} = U/P, \\overline{R} = P\\backslash R/P)$ exactly as in", "the discussion of the ``plain'' groupoid case.", "It follows from this that $(U, R, s, t, c)$ is the pullback of", "this groupoid structure via the morphism $U \\to \\overline{U}$.", "This concludes the proof." ], "refs": [ "groupoids-proposition-finite-flat-equivalence", "spaces-lemma-product-representable-transformations-property", "morphisms-lemma-base-change-finite-locally-free", "morphisms-lemma-composition-finite-locally-free", "descent-definition-descent-datum", "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "descent-definition-effective" ], "ref_ids": [ 9669, 8135, 5473, 5472, 14776, 13949, 14780 ] } ], "ref_ids": [] }, { "id": 2625, "type": "theorem", "label": "bootstrap-lemma-locally-algebraic-space", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-locally-algebraic-space", "contents": [ "\\begin{slogan}", "The definition of an algebraic space is fppf local.", "\\end{slogan}", "Let $S$ be a scheme.", "Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor.", "Let $\\{S_i \\to S\\}_{i \\in I}$ be a covering of $(\\Sch/S)_{fppf}$.", "Assume that", "\\begin{enumerate}", "\\item $F$ is a sheaf,", "\\item each $F_i = h_{S_i} \\times F$ is an algebraic space, and", "\\item $\\coprod_{i \\in I} F_i$ is an algebraic space (see", "Spaces, Lemma \\ref{spaces-lemma-coproduct-algebraic-spaces}).", "\\end{enumerate}", "Then $F$ is an algebraic space." ], "refs": [ "spaces-lemma-coproduct-algebraic-spaces" ], "proofs": [ { "contents": [ "Consider the morphism $\\coprod F_i \\to F$. This is the base change", "of $\\coprod S_i \\to S$ via $F \\to S$. Hence it is representable,", "locally of finite presentation, flat and surjective by our definition", "of an fppf covering and", "Lemma \\ref{lemma-base-change-transformation-property}.", "Thus", "Theorem \\ref{theorem-final-bootstrap}", "applies to show that $F$ is an algebraic space." ], "refs": [ "bootstrap-lemma-base-change-transformation-property", "bootstrap-theorem-final-bootstrap" ], "ref_ids": [ 2613, 2602 ] } ], "ref_ids": [ 8147 ] }, { "id": 2626, "type": "theorem", "label": "bootstrap-lemma-locally-algebraic-space-finite-type", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-locally-algebraic-space-finite-type", "contents": [ "Let $S$ be a scheme.", "Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor.", "Let $\\{S_i \\to S\\}_{i \\in I}$ be a covering of $(\\Sch/S)_{fppf}$.", "Assume that", "\\begin{enumerate}", "\\item $F$ is a sheaf,", "\\item each $F_i = h_{S_i} \\times F$ is an algebraic space, and", "\\item the morphisms $F_i \\to S_i$ are of finite type.", "\\end{enumerate}", "Then $F$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-locally-algebraic-space}", "above. To do this we will show that the assumption that", "$F_i$ is of finite type over $S_i$ to prove that the set theoretic", "condition in the lemma is satisfied (after perhaps refining the given", "covering of $S$ a bit).", "We suggest the reader skip the rest of the proof.", "\\medskip\\noindent", "If $S'_i \\to S_i$ is a morphism of schemes then", "$$", "h_{S'_i} \\times F =", "h_{S'_i} \\times_{h_{S_i}} h_{S_i} \\times F =", "h_{S'_i} \\times_{h_{S_i}} F_i", "$$", "is an algebraic space of finite type over $S'_i$, see", "Spaces, Lemma \\ref{spaces-lemma-fibre-product-spaces}", "and", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-base-change-finite-type}.", "Thus we may refine the given covering. After doing this we may assume:", "(a) each $S_i$ is affine, and (b) the cardinality of $I$ is at most", "the cardinality of the set of points of $S$. (Since to cover", "all of $S$ it is enough that each point is in the image of $S_i \\to S$", "for some $i$.)", "\\medskip\\noindent", "Since each $S_i$ is affine and each $F_i$ of finite type over $S_i$", "we conclude that $F_i$ is quasi-compact. Hence by", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-quasi-compact-affine-cover}", "we can find an affine $U_i \\in \\Ob((\\Sch/S)_{fppf})$", "and a surjective \\'etale morphism $U_i \\to F_i$. The fact that", "$F_i \\to S_i$ is locally of finite type then implies that", "$U_i \\to S_i$ is locally of finite type, and in particular", "$U_i \\to S$ is locally of finite type. By", "Sets, Lemma \\ref{sets-lemma-bound-finite-type}", "we conclude that $\\text{size}(U_i) \\leq \\text{size}(S)$.", "Since also $|I| \\leq \\text{size}(S)$ we conclude that", "$\\coprod_{i \\in I} U_i$ is isomorphic to an object of", "$(\\Sch/S)_{fppf}$ by", "Sets, Lemma \\ref{sets-lemma-bound-size}", "and the construction of $\\Sch$. This implies that", "$\\coprod F_i$ is an algebraic space by", "Spaces, Lemma \\ref{spaces-lemma-coproduct-algebraic-spaces}", "and we win." ], "refs": [ "bootstrap-lemma-locally-algebraic-space", "spaces-lemma-fibre-product-spaces", "spaces-morphisms-lemma-base-change-finite-type", "spaces-properties-lemma-quasi-compact-affine-cover", "sets-lemma-bound-finite-type", "sets-lemma-bound-size", "spaces-lemma-coproduct-algebraic-spaces" ], "ref_ids": [ 2625, 8143, 4815, 11832, 8793, 8791, 8147 ] } ], "ref_ids": [] }, { "id": 2627, "type": "theorem", "label": "bootstrap-lemma-descend-algebraic-space", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-descend-algebraic-space", "contents": [ "\\begin{slogan}", "Fppf descent data for algebraic spaces are effective.", "\\end{slogan}", "Let $S$ be a scheme. Let $\\{X_i \\to X\\}_{i \\in I}$ be an fppf", "covering of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item If $I$ is countable\\footnote{The restriction on countablility can be", "ignored by those who do not care about set theoretical issues. We can allow", "larger index sets here if we can bound the size of the algebraic spaces", "which we are descending. See for example", "Lemma \\ref{lemma-locally-algebraic-space-finite-type}.}, then any", "descent datum for algebraic spaces relative to $\\{X_i \\to X\\}$ is effective.", "\\item Any descent datum $(Y_i, \\varphi_{ij})$ relative to", "$\\{X_i \\to X\\}_{i \\in I}$ (Descent on Spaces, Definition", "\\ref{spaces-descent-definition-descent-datum-for-family-of-morphisms})", "with $Y_i \\to X_i$ of finite type", "is effective.", "\\end{enumerate}" ], "refs": [ "bootstrap-lemma-locally-algebraic-space-finite-type", "spaces-descent-definition-descent-datum-for-family-of-morphisms" ], "proofs": [ { "contents": [ "Proof of (1). By", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-descent-data-sheaves}", "this translates into the statement that an fppf sheaf $F$", "endowed with a map $F \\to X$ is an algebraic space provided that", "each $F \\times_X X_i$ is an algebraic space.", "The restriction on the cardinality of $I$ implies that", "coproducts of algebraic spaces indexed by $I$ are algebraic spaces, see", "Spaces, Lemma \\ref{spaces-lemma-coproduct-algebraic-spaces}", "and", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "The morphism", "$$", "\\coprod F \\times_X X_i \\longrightarrow F", "$$", "is representable by algebraic spaces (as the base change of", "$\\coprod X_i \\to X$, see Lemma \\ref{lemma-base-change-transformation}),", "and surjective, flat, and locally of finite presentation", "(as the base change of $\\coprod X_i \\to X$, see", "Lemma \\ref{lemma-base-change-transformation-property}).", "Hence part (1) follows from Theorem \\ref{theorem-final-bootstrap}.", "\\medskip\\noindent", "Proof of (2). First we apply", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-descent-data-sheaves}", "to obtain an fppf sheaf $F$ endowed with a map $F \\to X$", "such that $F \\times_X X_i = Y_i$ for all $i \\in I$.", "Our goal is to show that $F$ is an algebraic space.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "Then $F' = U \\times_X F \\to F$ is representable, surjective, and \\'etale", "as the base change of $U \\to X$.", "By Theorem \\ref{theorem-final-bootstrap} it suffices to show", "that $F' = U \\times_X F$ is an algebraic space.", "We may choose an fppf covering $\\{U_j \\to U\\}_{j \\in J}$", "where $U_j$ is a scheme refining the fppf covering", "$\\{X_i \\times_X U \\to U\\}_{i \\in I}$, see", "Topologies on Spaces, Lemma", "\\ref{spaces-topologies-lemma-refine-fppf-schemes}.", "Thus we get a map $a : J \\to I$ and for each $j$", "a morphism $U_j \\to X_{a(j)}$ over $X$.", "Then we see that $U_j \\times_U F' = U_j \\times_{X_{a(j)}} Y_{a(j)}$", "is of finite type over $U_j$. Hence $F'$ is an algebraic", "space by Lemma \\ref{lemma-locally-algebraic-space-finite-type}." ], "refs": [ "spaces-descent-lemma-descent-data-sheaves", "spaces-lemma-coproduct-algebraic-spaces", "sets-lemma-what-is-in-it", "bootstrap-lemma-base-change-transformation", "bootstrap-lemma-base-change-transformation-property", "bootstrap-theorem-final-bootstrap", "spaces-descent-lemma-descent-data-sheaves", "bootstrap-theorem-final-bootstrap", "spaces-topologies-lemma-refine-fppf-schemes", "bootstrap-lemma-locally-algebraic-space-finite-type" ], "ref_ids": [ 9436, 8147, 8795, 2604, 2613, 2602, 9436, 2602, 3666, 2626 ] } ], "ref_ids": [ 2626, 9445 ] }, { "id": 2628, "type": "theorem", "label": "bootstrap-lemma-representable-by-spaces-cover", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-representable-by-spaces-cover", "contents": [ "Let $S$ be a scheme. Let $a : F \\to G$ and $b : G \\to H$ be", "transformations of functors $(\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Assume", "\\begin{enumerate}", "\\item $F, G, H$ are sheaves,", "\\item $a : F \\to G$ is representable by algebraic spaces, flat,", "locally of finite presentation, and surjective, and", "\\item $b \\circ a : F \\to H$ is representable by algebraic spaces.", "\\end{enumerate}", "Then $b$ is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme over $S$ and let $\\xi \\in H(U)$. We have to show that", "$U \\times_{\\xi, H} G$ is an algebraic space. On the other hand, we know", "that $U \\times_{\\xi, H} F$ is an algebraic space and that", "$U \\times_{\\xi, H} F \\to U \\times_{\\xi, H} G$ is representable by", "algebraic spaces, flat, locally of finite presentation, and surjective", "as a base change of the morphism $a$ (see", "Lemma \\ref{lemma-base-change-transformation-property}).", "Thus the result follows from Theorem \\ref{theorem-final-bootstrap}." ], "refs": [ "bootstrap-lemma-base-change-transformation-property", "bootstrap-theorem-final-bootstrap" ], "ref_ids": [ 2613, 2602 ] } ], "ref_ids": [] }, { "id": 2629, "type": "theorem", "label": "bootstrap-lemma-quotient-stack-isom", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-quotient-stack-isom", "contents": [ "Assume $B \\to S$ and $(U, R, s, t, c)$ are as in", "Groupoids in Spaces,", "Definition \\ref{spaces-groupoids-definition-quotient-stack} (1).", "For any scheme $T$ over $S$ and objects $x, y$ of $[U/R]$ over $T$", "the sheaf $\\mathit{Isom}(x, y)$ on $(\\Sch/T)_{fppf}$", "is an algebraic space." ], "refs": [ "spaces-groupoids-definition-quotient-stack" ], "proofs": [ { "contents": [ "By", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-isom}", "there exists an fppf covering $\\{T_i \\to T\\}_{i \\in I}$", "such that $\\mathit{Isom}(x, y)|_{(\\Sch/T_i)_{fppf}}$", "is an algebraic space for each $i$. By", "Spaces, Lemma \\ref{spaces-lemma-rephrase}", "this means that each $F_i = h_{S_i} \\times \\mathit{Isom}(x, y)$", "is an algebraic space.", "Thus to prove the lemma we only have to verify the set theoretic condition", "that $\\coprod F_i$ is an algebraic space of", "Lemma \\ref{lemma-locally-algebraic-space}", "above to conclude. To do this we use", "Spaces, Lemma \\ref{spaces-lemma-coproduct-algebraic-spaces}", "which requires showing that $I$ and the $F_i$ are not ``too large''.", "We suggest the reader skip the rest of the proof.", "\\medskip\\noindent", "Choose $U' \\in \\Ob(\\Sch/S)_{fppf}$ and a surjective", "\\'etale morphism $U' \\to U$. Let $R'$ be the restriction of $R$ to $U'$.", "Since $[U/R] = [U'/R']$ we may, after replacing $U$ by $U'$,", "assume that $U$ is a scheme. (This step is here so that the", "fibre products below are over a scheme.)", "\\medskip\\noindent", "Note that if we refine the covering $\\{T_i \\to T\\}$ then it remains", "true that each $F_i$ is an algebraic space.", "Hence we may assume that each $T_i$ is affine. Since", "$T_i \\to T$ is locally of finite presentation, this then implies that", "$\\text{size}(T_i) \\leq \\text{size}(T)$, see", "Sets, Lemma \\ref{sets-lemma-bound-finite-type}.", "We may also assume that the cardinality of the index set $I$ is at most the", "cardinality of the set of points of $T$ since to get a", "covering it suffices to check that each point of $T$ is in the image.", "Hence $|I| \\leq \\text{size}(T)$.", "Choose $W \\in \\Ob((\\Sch/S)_{fppf})$", "and a surjective \\'etale morphism $W \\to R$. Note that in the proof of", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-isom}", "we showed that $F_i$ is representable by", "$T_i \\times_{(y_i, x_i), U \\times_B U} R$ for some", "$x_i, y_i : T_i \\to U$. Hence now we see that", "$V_i = T_i \\times_{(y_i, x_i), U \\times_B U} W$ is a", "scheme which comes with an \\'etale surjection $V_i \\to F_i$.", "By", "Sets, Lemma \\ref{sets-lemma-bound-size-fibre-product}", "we see that", "$$", "\\text{size}(V_i) \\leq \\max\\{\\text{size}(T_i), \\text{size}(W)\\}", "\\leq \\max\\{\\text{size}(T), \\text{size}(W)\\}", "$$", "Hence, by", "Sets, Lemma \\ref{sets-lemma-bound-size}", "we conclude that", "$$", "\\text{size}(\\coprod\\nolimits_{i \\in I} V_i)", "\\leq \\max\\{|I|, \\text{size}(T), \\text{size}(W)\\}.", "$$", "Hence we conclude by our construction of $\\Sch$", "that $\\coprod_{i \\in I} V_i$ is isomorphic to an object", "$V$ of $(\\Sch/S)_{fppf}$. This verifies the", "hypothesis of", "Spaces, Lemma \\ref{spaces-lemma-coproduct-algebraic-spaces}", "and we win." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-isom", "spaces-lemma-rephrase", "bootstrap-lemma-locally-algebraic-space", "spaces-lemma-coproduct-algebraic-spaces", "sets-lemma-bound-finite-type", "spaces-groupoids-lemma-quotient-stack-isom", "sets-lemma-bound-size-fibre-product", "sets-lemma-bound-size", "spaces-lemma-coproduct-algebraic-spaces" ], "ref_ids": [ 9325, 8171, 2625, 8147, 8793, 9325, 8792, 8791, 8147 ] } ], "ref_ids": [ 9354 ] }, { "id": 2630, "type": "theorem", "label": "bootstrap-lemma-covering-quotient", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-covering-quotient", "contents": [ "Let $S$ be a scheme. Consider an algebraic space $F$ of the form $F = U/R$", "where $(U, R, s, t, c)$ is a groupoid in algebraic spaces", "over $S$ such that $s, t$ are flat and locally of finite presentation, and", "$j = (t, s) : R \\to U \\times_S U$ is an equivalence relation.", "Then $U \\to F$ is surjective, flat, and locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "This is almost but not quite a triviality. Namely, by", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-pre-equivalence}", "and the fact that $j$ is a monomorphism we see that $R = U \\times_F U$.", "Choose a scheme $W$ and a surjective \\'etale morphism $W \\to F$.", "As $U \\to F$ is a surjection of sheaves we can find an fppf covering", "$\\{W_i \\to W\\}$ and maps $W_i \\to U$ lifting the morphisms $W_i \\to F$.", "Then we see that", "$$", "W_i \\times_F U = W_i \\times_U U \\times_F U = W_i \\times_{U, t} R", "$$", "and the projection $W_i \\times_F U \\to W_i$ is the base change of", "$t : R \\to U$ hence flat and locally of finite presentation, see", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-flat} and", "\\ref{spaces-morphisms-lemma-base-change-finite-presentation}.", "Hence by", "Descent on Spaces, Lemmas", "\\ref{spaces-descent-lemma-descending-property-flat} and", "\\ref{spaces-descent-lemma-descending-property-locally-finite-presentation}", "we see that $U \\to F$ is flat and locally of finite presentation.", "It is surjective by", "Spaces, Remark \\ref{spaces-remark-warning}." ], "refs": [ "spaces-groupoids-lemma-quotient-pre-equivalence", "spaces-morphisms-lemma-base-change-flat", "spaces-morphisms-lemma-base-change-finite-presentation", "spaces-descent-lemma-descending-property-flat", "spaces-descent-lemma-descending-property-locally-finite-presentation", "spaces-remark-warning" ], "ref_ids": [ 9316, 4853, 4840, 9393, 9390, 8187 ] } ], "ref_ids": [] }, { "id": 2631, "type": "theorem", "label": "bootstrap-lemma-quotient-free-action", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-quotient-free-action", "contents": [ "Let $S$ be a scheme. Let $X \\to B$ be a morphism of algebraic spaces over", "$S$. Let $G$ be a group algebraic space over $B$ and let", "$a : G \\times_B X \\to X$ be an action of $G$ on $X$ over $B$.", "If", "\\begin{enumerate}", "\\item $a$ is a free action, and", "\\item $G \\to B$ is flat and locally of finite presentation,", "\\end{enumerate}", "then $X/G$ (see", "Groupoids in Spaces, Definition", "\\ref{spaces-groupoids-definition-quotient-sheaf})", "is an algebraic space and $X \\to X/G$ is surjective, flat, and locally", "of finite presentation." ], "refs": [ "spaces-groupoids-definition-quotient-sheaf" ], "proofs": [ { "contents": [ "The fact that $X/G$ is an algebraic space is immediate from", "Theorem \\ref{theorem-final-bootstrap}", "and the definitions. Namely, $X/G = X/R$ where $R = G \\times_B X$.", "The morphisms $s, t : G \\times_B X \\to X$ are flat and locally of", "finite presentation (clear for $s$ as a base change of $G \\to B$ and", "by symmetry using the inverse it follows for $t$) and the morphism", "$j : G \\times_B X \\to X \\times_B X$ is a monomorphism by", "Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-free-action}", "as the action is free. The assertions about the morphism $X \\to X/G$", "follow from", "Lemma \\ref{lemma-covering-quotient}." ], "refs": [ "bootstrap-theorem-final-bootstrap", "spaces-groupoids-lemma-free-action", "bootstrap-lemma-covering-quotient" ], "ref_ids": [ 2602, 9292, 2630 ] } ], "ref_ids": [ 9352 ] }, { "id": 2632, "type": "theorem", "label": "bootstrap-lemma-descent-torsor", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-descent-torsor", "contents": [ "Let $\\{S_i \\to S\\}_{i \\in I}$ be a covering of $(\\Sch/S)_{fppf}$.", "Let $G$ be a group algebraic space over $S$, and denote", "$G_i = G_{S_i}$ the base changes. Suppose given", "\\begin{enumerate}", "\\item for each $i \\in I$ an fppf $G_i$-torsor $X_i$ over $S_i$,", "and", "\\item for each $i, j \\in I$ a $G_{S_i \\times_S S_j}$-equivariant isomorphism", "$\\varphi_{ij} : X_i \\times_S S_j \\to S_i \\times_S X_j$ satisfying the cocycle", "condition over every $S_i \\times_S S_j \\times_S S_j$.", "\\end{enumerate}", "Then there exists an fppf $G$-torsor $X$ over $S$", "whose base change to $S_i$ is isomorphic to $X_i$ such that we", "recover the descent datum $\\varphi_{ij}$." ], "refs": [], "proofs": [ { "contents": [ "We may think of $X_i$ as a sheaf on $(\\Sch/S_i)_{fppf}$, see", "Spaces, Section \\ref{spaces-section-change-base-scheme}.", "By", "Sites, Section \\ref{sites-section-glueing-sheaves}", "the descent datum $(X_i, \\varphi_{ij})$ is effective in the sense that", "there exists a unique sheaf $X$ on $(\\Sch/S)_{fppf}$ which", "recovers the algebraic spaces $X_i$ after restricting back to", "$(\\Sch/S_i)_{fppf}$. Hence we see that", "$X_i = h_{S_i} \\times X$. By", "Lemma \\ref{lemma-locally-algebraic-space}", "we see that $X$ is an algebraic space, modulo verifying that $\\coprod X_i$", "is an algebraic space which we do at the end of the proof.", "By the equivalence of categories in", "Sites, Lemma \\ref{sites-lemma-mapping-property-glue}", "the action maps $G_i \\times_{S_i} X_i \\to X_i$", "glue to give a map $a : G \\times_S X \\to X$.", "Now we have to show that $a$ is an action and that $X$", "is a pseudo-torsor, and fppf locally trivial (see", "Groupoids in Spaces,", "Definition \\ref{spaces-groupoids-definition-principal-homogeneous-space}).", "These may be checked fppf locally, and", "hence follow from the corresponding properties of the actions", "$G_i \\times_{S_i} X_i \\to X_i$. Hence the lemma is true.", "\\medskip\\noindent", "We suggest the reader skip the rest of the proof, which is purely set", "theoretical. Pick coverings $\\{S_{ij} \\to S_j\\}_{j \\in J_i}$ of", "$(\\Sch/S)_{fppf}$", "which trivialize the $G_i$ torsors $X_i$ (possible by assumption, and", "Topologies, Lemma \\ref{topologies-lemma-fppf-induced} part (1)).", "Then $\\{S_{ij} \\to S\\}_{i \\in I, j \\in J_i}$ is a covering of", "$(\\Sch/S)_{fppf}$ and hence we may assume that each $X_i$", "is the trivial torsor! Of course we may also refine the covering further,", "hence we may assume that each $S_i$ is affine and that the index", "set $I$ has cardinality bounded by the cardinality of the set of points", "of $S$. Choose $U \\in \\Ob((\\Sch/S)_{fppf})$ and a surjective", "\\'etale morphism $U \\to G$. Then we see that $U_i = U \\times_S S_i$ comes", "with an \\'etale surjective morphism to $X_i \\cong G_i$. By", "Sets, Lemma \\ref{sets-lemma-bound-size-fibre-product}", "we see $\\text{size}(U_i) \\leq \\max\\{\\text{size}(U), \\text{size}(S_i)\\}$. By", "Sets, Lemma \\ref{sets-lemma-bound-finite-type}", "we have $\\text{size}(S_i) \\leq \\text{size}(S)$.", "Hence we see that", "$\\text{size}(U_i) \\leq \\max\\{\\text{size}(U), \\text{size}(S)\\}$", "for all $i \\in I$. Together with the bound on $|I|$ we found above we", "conclude from", "Sets, Lemma \\ref{sets-lemma-bound-size}", "that $\\text{size}(\\coprod U_i) \\leq \\max\\{\\text{size}(U), \\text{size}(S)\\}$.", "Hence", "Spaces, Lemma \\ref{spaces-lemma-coproduct-algebraic-spaces}", "applies to show that $\\coprod X_i$ is an algebraic space which is", "what we had to prove." ], "refs": [ "bootstrap-lemma-locally-algebraic-space", "sites-lemma-mapping-property-glue", "spaces-groupoids-definition-principal-homogeneous-space", "topologies-lemma-fppf-induced", "sets-lemma-bound-size-fibre-product", "sets-lemma-bound-finite-type", "sets-lemma-bound-size", "spaces-lemma-coproduct-algebraic-spaces" ], "ref_ids": [ 2625, 8565, 9345, 12474, 8792, 8793, 8791, 8147 ] } ], "ref_ids": [] }, { "id": 2633, "type": "theorem", "label": "bootstrap-lemma-spaces-etale", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-spaces-etale", "contents": [ "Denote the common underlying category of $\\Sch_{fppf}$ and $\\Sch_\\etale$ by", "$\\Sch_\\alpha$ (see Topologies, Remark \\ref{topologies-remark-choice-sites}).", "Let $S$ be an object of $\\Sch_\\alpha$. Let", "$$", "F : (\\Sch_\\alpha/S)^{opp} \\longrightarrow \\textit{Sets}", "$$", "be a presheaf with the following properties:", "\\begin{enumerate}", "\\item $F$ is a sheaf for the \\'etale topology,", "\\item the diagonal $\\Delta : F \\to F \\times F$ is representable, and", "\\item there exists $U \\in \\Ob(\\Sch_\\alpha/S)$", "and $U \\to F$ which is surjective and \\'etale.", "\\end{enumerate}", "Then $F$ is an algebraic space in the sense of", "Algebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}." ], "refs": [ "topologies-remark-choice-sites", "spaces-definition-algebraic-space" ], "proofs": [ { "contents": [ "Note that properties (2) and (3) of the lemma and the corresponding", "properties (2) and (3) of", "Algebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}", "are independent of the topology. This is true because these properties", "involve only the notion of a fibre product of presheaves, maps of", "presheaves, the notion of a representable transformation of functors,", "and what it means for such a transformation to be surjective and \\'etale.", "Thus all we have to prove is that an \\'etale sheaf $F$ with properties", "(2) and (3) is also an fppf sheaf.", "\\medskip\\noindent", "To do this, let $R = U \\times_F U$. By (2) the presheaf $R$ is representable", "by a scheme and by (3) the projections $R \\to U$ are \\'etale. Thus", "$j : R \\to U \\times_S U$ is an \\'etale equivalence relation. Moreover", "$U \\to F$ identifies $F$ as the quotient of $U$ by $R$ for the", "\\'etale topology: (a) if $T \\to F$ is a morphism, then $\\{T \\times_F U \\to T\\}$", "is an \\'etale covering, hence $U \\to F$ is a surjection of sheaves for the", "\\'etale topology, (b) if $a, b : T \\to U$ map to the same section of $F$,", "then $(a, b) : T \\to R$ hence $a$ and $b$ have the same image in the quotient", "of $U$ by $R$ for the \\'etale topology. Next, let $U/R$ denote the quotient", "sheaf in the fppf topology which is an algebraic space by", "Spaces, Theorem \\ref{spaces-theorem-presentation}.", "Thus we have morphisms (transformations of functors)", "$$", "U \\to F \\to U/R.", "$$", "By the aforementioned", "Spaces, Theorem \\ref{spaces-theorem-presentation}", "the composition is representable, surjective, and \\'etale. Hence for any", "scheme $T$ and morphism $T \\to U/R$ the fibre product $V = T \\times_{U/R} U$", "is a scheme surjective and \\'etale over $T$. In other words, $\\{V \\to U\\}$", "is an \\'etale covering. This proves that $U \\to U/R$ is surjective as", "a map of sheaves in the \\'etale topology. It follows that", "$F \\to U/R$ is surjective as a map of sheaves in the \\'etale topology.", "On the other hand, the map $F \\to U/R$ is injective (as a map of presheaves)", "since $R = U \\times_{U/R} U$ again by", "Spaces, Theorem \\ref{spaces-theorem-presentation}.", "It follows that $F \\to U/R$ is an isomorphism of \\'etale sheaves, see", "Sites, Lemma \\ref{sites-lemma-mono-epi-sheaves}", "which concludes the proof." ], "refs": [ "spaces-definition-algebraic-space", "spaces-theorem-presentation", "spaces-theorem-presentation", "spaces-theorem-presentation", "sites-lemma-mono-epi-sheaves" ], "ref_ids": [ 8174, 8124, 8124, 8124, 8517 ] } ], "ref_ids": [ 12553, 8174 ] }, { "id": 2634, "type": "theorem", "label": "bootstrap-lemma-spaces-etale-locally-representable", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-spaces-etale-locally-representable", "contents": [ "Denote the common underlying category of $\\Sch_{fppf}$ and $\\Sch_\\etale$ by", "$\\Sch_\\alpha$ (see Topologies, Remark \\ref{topologies-remark-choice-sites}).", "Let $S$ be an object of $\\Sch_\\alpha$. Let", "$$", "F : (\\Sch_\\alpha/S)^{opp} \\longrightarrow \\textit{Sets}", "$$", "be a presheaf with the following properties:", "\\begin{enumerate}", "\\item $F$ is a sheaf for the \\'etale topology,", "\\item there exists an algebraic space $U$ over $S$", "and a map $U \\to F$ which is representable by", "algebraic spaces, surjective, and \\'etale.", "\\end{enumerate}", "Then $F$ is an algebraic space in the sense of", "Algebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}." ], "refs": [ "topologies-remark-choice-sites", "spaces-definition-algebraic-space" ], "proofs": [ { "contents": [ "Set $R = U \\times_F U$. This is an algebraic space as $U \\to F$ is assumed", "representable by algebraic spaces. The projections $s, t : R \\to U$ are", "\\'etale morphisms of algebraic spaces as $U \\to F$ is assumed \\'etale.", "The map $j = (t, s) : R \\to U \\times_S U$ is a monomorphism and an", "equivalence relation as $R = U \\times_F U$. By", "Theorem \\ref{theorem-final-bootstrap}", "the fppf quotient sheaf $F' = U/R$ is an algebraic space.", "The morphism $U \\to F'$ is surjective, flat, and locally of finite", "presentation by Lemma \\ref{lemma-covering-quotient}.", "The map $R \\to U \\times_{F'} U$ is surjective as a map of fppf", "sheaves by Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-pre-equivalence}", "and since $j$ is a monomorphism it is an isomorphism.", "Hence the base change of $U \\to F'$ by $U \\to F'$ is \\'etale,", "and we conclude that $U \\to F'$ is \\'etale by", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-descending-property-etale}.", "Thus $U \\to F'$ is surjective as a map of \\'etale sheaves.", "This means that $F'$ is equal to the quotient sheaf $U/R$", "in the \\'etale topology (small check omitted). Hence we obtain", "a canonical factorization $U \\to F' \\to F$ and $F' \\to F$ is an injective", "map of sheaves. On the other hand, $U \\to F$ is surjective as a map", "of \\'etale sheaves and hence so is $F' \\to F$. This means that $F' = F$", "and the proof is complete." ], "refs": [ "bootstrap-theorem-final-bootstrap", "bootstrap-lemma-covering-quotient", "spaces-groupoids-lemma-quotient-pre-equivalence", "spaces-descent-lemma-descending-property-etale" ], "ref_ids": [ 2602, 2630, 9316, 9408 ] } ], "ref_ids": [ 12553, 8174 ] }, { "id": 2635, "type": "theorem", "label": "bootstrap-lemma-spaces-etale-smooth-cover", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-spaces-etale-smooth-cover", "contents": [ "Denote the common underlying category of $\\Sch_{fppf}$", "and $\\Sch_\\etale$ by $\\Sch_\\alpha$ (see", "Topologies, Remark \\ref{topologies-remark-choice-sites}). Let $S$ be an object", "of $\\Sch_\\alpha$. ", "$$", "F : (\\Sch_\\alpha/S)^{opp} \\longrightarrow \\textit{Sets}", "$$", "be a presheaf with the following properties:", "\\begin{enumerate}", "\\item $F$ is a sheaf for the \\'etale topology,", "\\item the diagonal $\\Delta : F \\to F \\times F$ is representable", "by algebraic spaces, and", "\\item there exists $U \\in \\Ob(\\Sch_\\alpha/S)$", "and $U \\to F$ which is surjective and smooth.", "\\end{enumerate}", "Then $F$ is an algebraic space in the sense of", "Algebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}." ], "refs": [ "topologies-remark-choice-sites", "spaces-definition-algebraic-space" ], "proofs": [ { "contents": [ "The proof mirrors the proof of Lemma \\ref{lemma-spaces-etale}. Let", "$R = U \\times_F U$. By (2) the presheaf $R$ is an algebraic space and by (3)", "the projections $R \\to U$ are smooth and surjective. Denote $(U, R, s, t, c)$", "the groupoid associated to the equivalence relation $j : R \\to U \\times_S U$", "(see Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-equivalence-groupoid}).", "By Theorem \\ref{theorem-final-bootstrap} we see that $X = U/R$ (quotient", "in the fppf-topology) is an algebraic space. Using that the smooth", "topology and the \\'etale topology have the same sheaves (by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-etale-dominates-smooth})", "we see the map $U \\to F$ identifies $F$ as the quotient of", "$U$ by $R$ for the smooth topology (details omitted).", "Thus we have morphisms (transformations of functors)", "$$", "U \\to F \\to X.", "$$", "By Lemma \\ref{lemma-covering-quotient} we see that $U \\to X$ is", "surjective, flat and locally of finite presentation. By", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-pre-equivalence}", "(and the fact that $j$ is a monomorphism) we have $R = U \\times_X U$. By", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-descending-property-smooth}", "we conclude that $U \\to X$ is smooth and surjective (as the projections", "$R \\to U$ are smooth and surjective and $\\{U \\to X\\}$ is an fppf", "covering). Hence for any scheme $T$ and morphism $T \\to X$ the fibre product", "$T \\times_X U$ is an algebraic space surjective and smooth over $T$.", "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to T \\times_X U$.", "Then $\\{V \\to T\\}$ is a smooth covering such that $V \\to T \\to X$", "lifts to a morphism $V \\to U$. This proves that", "$U \\to X$ is surjective as a map of sheaves in the smooth topology.", "It follows that $F \\to X$ is surjective as a map of sheaves in the smooth", "topology. On the other hand, the map $F \\to X$ is injective (as a map", "of presheaves) since $R = U \\times_X U$.", "It follows that $F \\to X$ is an isomorphism of smooth ($=$ \\'etale)", "sheaves, see Sites, Lemma \\ref{sites-lemma-mono-epi-sheaves}", "which concludes the proof." ], "refs": [ "bootstrap-lemma-spaces-etale", "spaces-groupoids-lemma-equivalence-groupoid", "bootstrap-theorem-final-bootstrap", "more-morphisms-lemma-etale-dominates-smooth", "bootstrap-lemma-covering-quotient", "spaces-groupoids-lemma-quotient-pre-equivalence", "spaces-descent-lemma-descending-property-smooth", "sites-lemma-mono-epi-sheaves" ], "ref_ids": [ 2633, 9298, 2602, 13880, 2630, 9316, 9406, 8517 ] } ], "ref_ids": [ 12553, 8174 ] }, { "id": 2636, "type": "theorem", "label": "bootstrap-lemma-spaces-smooth-locally-representable", "categories": [ "bootstrap" ], "title": "bootstrap-lemma-spaces-smooth-locally-representable", "contents": [ "Denote the common underlying category of $\\Sch_{fppf}$ and $\\Sch_\\etale$ by", "$\\Sch_\\alpha$ (see Topologies, Remark \\ref{topologies-remark-choice-sites}).", "Let $S$ be an object of $\\Sch_\\alpha$. Let", "$$", "F : (\\Sch_\\alpha/S)^{opp} \\longrightarrow \\textit{Sets}", "$$", "be a presheaf with the following properties:", "\\begin{enumerate}", "\\item $F$ is a sheaf for the \\'etale topology,", "\\item there exists an algebraic space $U$ over $S$", "and a map $U \\to F$ which is representable by", "algebraic spaces, surjective, and smooth.", "\\end{enumerate}", "Then $F$ is an algebraic space in the sense of", "Algebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}." ], "refs": [ "topologies-remark-choice-sites", "spaces-definition-algebraic-space" ], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Lemma \\ref{lemma-spaces-etale-locally-representable}.", "Set $R = U \\times_F U$. This is an algebraic space as $U \\to F$ is assumed", "representable by algebraic spaces. The projections $s, t : R \\to U$ are", "smooth morphisms of algebraic spaces as $U \\to F$ is assumed smooth.", "The map $j = (t, s) : R \\to U \\times_S U$ is a monomorphism and an", "equivalence relation as $R = U \\times_F U$. By", "Theorem \\ref{theorem-final-bootstrap}", "the fppf quotient sheaf $F' = U/R$ is an algebraic space.", "The morphism $U \\to F'$ is surjective, flat, and locally of finite", "presentation by Lemma \\ref{lemma-covering-quotient}.", "The map $R \\to U \\times_{F'} U$ is surjective as a map of fppf", "sheaves by Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-pre-equivalence}", "and since $j$ is a monomorphism it is an isomorphism.", "Hence the base change of $U \\to F'$ by $U \\to F'$ is smooth,", "and we conclude that $U \\to F'$ is smooth by", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-descending-property-smooth}.", "Thus $U \\to F'$ is surjective as a map of \\'etale sheaves (as the", "smooth topology is equal to the \\'etale topology by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-etale-dominates-smooth}).", "This means that $F'$ is equal to the quotient sheaf $U/R$", "in the \\'etale topology (small check omitted). Hence we obtain", "a canonical factorization $U \\to F' \\to F$ and $F' \\to F$ is an injective", "map of sheaves. On the other hand, $U \\to F$ is surjective as a map", "of \\'etale sheaves (as the smooth topology is the same as the", "\\'etale topology) and hence so is $F' \\to F$. This means that $F' = F$", "and the proof is complete." ], "refs": [ "bootstrap-lemma-spaces-etale-locally-representable", "bootstrap-theorem-final-bootstrap", "bootstrap-lemma-covering-quotient", "spaces-groupoids-lemma-quotient-pre-equivalence", "spaces-descent-lemma-descending-property-smooth", "more-morphisms-lemma-etale-dominates-smooth" ], "ref_ids": [ 2634, 2602, 2630, 9316, 9406, 13880 ] } ], "ref_ids": [ 12553, 8174 ] }, { "id": 2639, "type": "theorem", "label": "spaces-perfect-theorem-approximation", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-theorem-approximation", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Then approximation by perfect complexes holds on $X$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the induction principle of", "Lemma \\ref{lemma-induction-principle}", "and Lemmas \\ref{lemma-induction-step} and \\ref{lemma-approximation-affine}." ], "refs": [ "spaces-perfect-lemma-induction-principle", "spaces-perfect-lemma-induction-step", "spaces-perfect-lemma-approximation-affine" ], "ref_ids": [ 2670, 2705, 2704 ] } ], "ref_ids": [] }, { "id": 2640, "type": "theorem", "label": "spaces-perfect-theorem-bondal-van-den-Bergh", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-theorem-bondal-van-den-Bergh", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. The category", "$D_\\QCoh(\\mathcal{O}_X)$ can be generated by a single", "perfect object. More precisely, there exists a perfect object", "$P$ of $D(\\mathcal{O}_X)$ such that for ", "$E \\in D_\\QCoh(\\mathcal{O}_X)$ the following are equivalent", "\\begin{enumerate}", "\\item $E = 0$, and", "\\item $\\Hom_{D(\\mathcal{O}_X)}(P[n], E) = 0$ for all $n \\in \\mathbf{Z}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will prove this using the induction principle of", "Lemma \\ref{lemma-induction-principle}.", "\\medskip\\noindent", "If $X$ is affine, then $\\mathcal{O}_X$ is a perfect generator.", "This follows from Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site}", "and", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-affine-compare-bounded}.", "\\medskip\\noindent", "Assume that $(U \\subset X, f : V \\to X)$ is an elementary distinguished", "square with $U$ quasi-compact such that the theorem holds for $U$ and $V$", "is an affine scheme.", "Let $P$ be a perfect object of $D(\\mathcal{O}_U)$ which is a generator", "for $D_\\QCoh(\\mathcal{O}_U)$. Using", "Lemma \\ref{lemma-direct-summand-of-a-restriction} we may", "choose a perfect object", "$Q$ of $D(\\mathcal{O}_X)$ whose restriction to $U$ is a direct sum one", "of whose summands is $P$. Say $V = \\Spec(A)$. Let $Z \\subset V$", "be the reduced closed subscheme which is the inverse image of", "$X \\setminus U$ and maps isomorphically to it", "(see Definition \\ref{definition-elementary-distinguished-square}).", "This is a retrocompact closed subset of $V$.", "Choose $f_1, \\ldots, f_r \\in A$ such that", "$Z = V(f_1, \\ldots, f_r)$. Let $K \\in D(\\mathcal{O}_V)$ be the perfect", "object corresponding to the Koszul complex on $f_1, \\ldots, f_r$ over $A$.", "Note that since $K$ is supported on $Z$, the pushforward", "$K' = Rf_*K$ is a perfect object of $D(\\mathcal{O}_X)$ whose", "restriction to $V$ is $K$ (see Lemmas \\ref{lemma-pushforward-perfect}", "and \\ref{lemma-pushforward-with-support-in-open}).", "We claim that $Q \\oplus K'$ is a generator for", "$D_\\QCoh(\\mathcal{O}_X)$.", "\\medskip\\noindent", "Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$ such that", "there are no nontrivial maps from any shift of $Q \\oplus K'$ into $E$.", "By Lemma \\ref{lemma-pushforward-with-support-in-open}", "we have $K' = f_! K$ and hence", "$$", "\\Hom_{D(\\mathcal{O}_X)}(K'[n], E) = \\Hom_{D(\\mathcal{O}_V)}(K[n], E|_V)", "$$", "Thus by", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-orthogonal-koszul-complex}", "(using also", "Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site})", "the vanishing of these groups implies that $E|_V$ is isomorphic to", "$R(U \\times_X V \\to V)_*E|_{U \\times_X V}$. This implies that", "$E = R(U \\to X)_*E|_U$ (small detail omitted). If this is the case then", "$$", "\\Hom_{D(\\mathcal{O}_X)}(Q[n], E) = \\Hom_{D(\\mathcal{O}_U)}(Q|_U[n], E|_U)", "$$", "which contains $\\Hom_{D(\\mathcal{O}_U)}(P[n], E|_U)$ as a direct summand.", "Thus by our choice of $P$ the vanishing of these groups implies that $E|_U$", "is zero. Whence $E$ is zero." ], "refs": [ "spaces-perfect-lemma-induction-principle", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "perfect-lemma-affine-compare-bounded", "spaces-perfect-lemma-direct-summand-of-a-restriction", "spaces-perfect-definition-elementary-distinguished-square", "spaces-perfect-lemma-pushforward-perfect", "spaces-perfect-lemma-pushforward-with-support-in-open", "spaces-perfect-lemma-pushforward-with-support-in-open", "perfect-lemma-orthogonal-koszul-complex", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site" ], "ref_ids": [ 2670, 2644, 6941, 2707, 2764, 2702, 2679, 2679, 7010, 2644 ] } ], "ref_ids": [] }, { "id": 2641, "type": "theorem", "label": "spaces-perfect-theorem-DQCoh-is-Ddga", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-theorem-DQCoh-is-Ddga", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Then there exist a differential graded algebra $(E, \\text{d})$", "with only a finite number of nonzero cohomology groups $H^i(E)$", "such that $D_\\QCoh(\\mathcal{O}_X)$ is equivalent", "to $D(E, \\text{d})$." ], "refs": [], "proofs": [ { "contents": [ "Let $K^\\bullet$ be a K-injective complex of $\\mathcal{O}$-modules which", "is perfect and generates $D_\\QCoh(\\mathcal{O}_X)$. Such a", "thing exists by Theorem \\ref{theorem-bondal-van-den-Bergh}", "and the existence of K-injective resolutions. We will show the", "theorem holds with", "$$", "(E, \\text{d}) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, K^\\bullet)", "$$", "where $\\text{Comp}^{dg}(\\mathcal{O}_X)$ is the differential graded category", "of complexes of $\\mathcal{O}$-modules. Please see", "Differential Graded Algebra, Section \\ref{dga-section-variant-base-change}.", "Since $K^\\bullet$ is K-injective we", "have", "\\begin{equation}", "\\label{equation-E-is-OK}", "H^n(E) = \\Ext^n_{D(\\mathcal{O}_X)}(K^\\bullet, K^\\bullet)", "\\end{equation}", "for all $n \\in \\mathbf{Z}$. Only a finite number of these Exts", "are nonzero by Lemma \\ref{lemma-ext-from-perfect-into-bounded-QCoh}.", "Consider the functor", "$$", "- \\otimes_E^\\mathbf{L} K^\\bullet :", "D(E, \\text{d}) \\longrightarrow D(\\mathcal{O}_X)", "$$", "of", "Differential Graded Algebra, Lemma", "\\ref{dga-lemma-tensor-with-complex-derived}.", "Since $K^\\bullet$ is perfect, it defines a compact object of", "$D(\\mathcal{O}_X)$, see Proposition \\ref{proposition-compact-is-perfect}.", "Combined with (\\ref{equation-E-is-OK}) the functor above is fully", "faithful as follows from", "Differential Graded Algebra, Lemmas", "\\ref{dga-lemma-fully-faithful-in-compact-case}. It has a right adjoint", "$$", "R\\Hom(K^\\bullet, - ) : D(\\mathcal{O}_X) \\longrightarrow D(E, \\text{d})", "$$", "by Differential Graded Algebra, Lemmas", "\\ref{dga-lemma-tensor-with-complex-hom-adjoint}", "which is a left quasi-inverse functor by generalities on adjoint", "functors. On the other hand, it follows from", "Lemma \\ref{lemma-tensor-with-QCoh-complex} that we obtain", "$$", "- \\otimes_E^\\mathbf{L} K^\\bullet :", "D(E, \\text{d}) \\longrightarrow D_\\QCoh(\\mathcal{O}_X)", "$$", "and by our choice of $K^\\bullet$ as a generator of", "$D_\\QCoh(\\mathcal{O}_X)$ the kernel of the adjoint", "restricted to $D_\\QCoh(\\mathcal{O}_X)$ is zero.", "A formal argument shows that we obtain the desired equivalence, see", "Derived Categories, Lemma", "\\ref{derived-lemma-fully-faithful-adjoint-kernel-zero}." ], "refs": [ "spaces-perfect-theorem-bondal-van-den-Bergh", "spaces-perfect-lemma-ext-from-perfect-into-bounded-QCoh", "dga-lemma-tensor-with-complex-derived", "spaces-perfect-proposition-compact-is-perfect", "dga-lemma-fully-faithful-in-compact-case", "dga-lemma-tensor-with-complex-hom-adjoint", "spaces-perfect-lemma-tensor-with-QCoh-complex", "derived-lemma-fully-faithful-adjoint-kernel-zero" ], "ref_ids": [ 2640, 2712, 13116, 2758, 13119, 13118, 2711, 1793 ] } ], "ref_ids": [] }, { "id": 2642, "type": "theorem", "label": "spaces-perfect-lemma-restrict-direct-image-open", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-restrict-direct-image-open", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Given an \\'etale morphism $V \\to Y$, set $U = V \\times_Y X$", "and denote $g : U \\to V$ the projection morphism. Then", "$(Rf_*E)|_V = Rg_*(E|_U)$ for $E$ in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Represent $E$ by a K-injective complex $\\mathcal{I}^\\bullet$ of", "$\\mathcal{O}_X$-modules. Then $Rf_*(E) = f_*\\mathcal{I}^\\bullet$", "and $Rg_*(E|_U) = g_*(\\mathcal{I}^\\bullet|_U)$ by", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-restrict-K-injective-to-open}.", "Hence the result follows from", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-pushforward-etale-base-change-modules}." ], "refs": [ "sites-cohomology-lemma-restrict-K-injective-to-open", "spaces-properties-lemma-pushforward-etale-base-change-modules" ], "ref_ids": [ 4253, 11898 ] } ], "ref_ids": [] }, { "id": 2643, "type": "theorem", "label": "spaces-perfect-lemma-epsilon-flat", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-epsilon-flat", "contents": [ "The morphism $\\epsilon$ of (\\ref{equation-epsilon})", "is a flat morphism of ringed sites. In particular the functor", "$\\epsilon^* : \\textit{Mod}(\\mathcal{O}_X) \\to", "\\textit{Mod}(\\mathcal{O}_\\etale)$ is exact.", "Moreover, if $\\epsilon^*\\mathcal{F} = 0$, then $\\mathcal{F} = 0$." ], "refs": [], "proofs": [ { "contents": [ "The second assertion follows from the first by", "Modules on Sites, Lemma \\ref{sites-modules-lemma-flat-pullback-exact}.", "To prove the first assertion we have to show that", "$\\mathcal{O}_\\etale$ is a flat $\\epsilon^{-1}\\mathcal{O}_X$-module.", "To do this it suffices to check", "$\\mathcal{O}_{X, x} \\to \\mathcal{O}_{\\etale, \\overline{x}}$", "is flat for any geometric point $\\overline{x}$ of $X$, see", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-check-flat-stalks},", "Sites, Lemma", "\\ref{sites-lemma-point-morphism-sites},", "and", "\\'Etale Cohomology, Remarks", "\\ref{etale-cohomology-remarks-enough-points}.", "By \\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-describe-etale-local-ring}", "we see that $\\mathcal{O}_{\\etale, \\overline{x}}$ is the", "strict henselization of $\\mathcal{O}_{X, x}$. Thus", "$\\mathcal{O}_{X, x} \\to \\mathcal{O}_{\\etale, \\overline{x}}$", "is faithfully flat by More on Algebra,", "Lemma \\ref{more-algebra-lemma-dumb-properties-henselization}.", "The final statement follows also: if $\\epsilon^*\\mathcal{F} = 0$, then", "$$", "0 = \\epsilon^*\\mathcal{F}_{\\overline{x}} =", "\\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\mathcal{O}_\\etale", "$$", "(Modules on Sites, Lemma \\ref{sites-modules-lemma-pullback-stalk})", "for all geometric points $\\overline{x}$. By faithful flatness of", "$\\mathcal{O}_{X, x} \\to \\mathcal{O}_{\\etale, \\overline{x}}$", "we conclude $\\mathcal{F}_x = 0$ for all $x \\in X$." ], "refs": [ "sites-modules-lemma-flat-pullback-exact", "sites-modules-lemma-check-flat-stalks", "sites-lemma-point-morphism-sites", "etale-cohomology-remarks-enough-points", "etale-cohomology-lemma-describe-etale-local-ring", "more-algebra-lemma-dumb-properties-henselization", "sites-modules-lemma-pullback-stalk" ], "ref_ids": [ 14223, 14251, 8603, 6798, 6433, 10055, 14245 ] } ], "ref_ids": [] }, { "id": 2644, "type": "theorem", "label": "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "contents": [ "Let $X$ be a scheme. The functor", "$\\epsilon^* : D_\\QCoh(\\mathcal{O}_X) \\to", "D_\\QCoh(\\mathcal{O}_\\etale)$", "defined above is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by showing the functor", "$R\\epsilon_* : D(\\mathcal{O}_\\etale) \\to D(\\mathcal{O}_X)$", "induces a quasi-inverse. We will use freely that $\\epsilon_*$", "is given by restriction to $X_{Zar} \\subset X_\\etale$ and the description of", "$\\epsilon^* = \\text{id}_{small, \\etale, Zar}^*$", "in Descent, Lemma \\ref{descent-lemma-compare-sites}.", "\\medskip\\noindent", "For a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the adjunction map", "$\\mathcal{F} \\to \\epsilon_*\\epsilon^*\\mathcal{F}$ is an isomorphism by", "the fact that $\\mathcal{F}^a$", "(Descent, Definition \\ref{descent-definition-structure-sheaf})", "is a sheaf as proved in", "Descent, Lemma \\ref{descent-lemma-sheaf-condition-holds}.", "Conversely, every quasi-coherent $\\mathcal{O}_\\etale$-module", "$\\mathcal{H}$ is of the form $\\epsilon^*\\mathcal{F}$ for some quasi-coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$, see", "Descent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent}.", "Then $\\mathcal{F} = \\epsilon_*\\mathcal{H}$ by what we just said and", "we conclude that the adjunction map", "$\\epsilon^*\\epsilon_*\\mathcal{H} \\to \\mathcal{H}$ is an isomorphism for all", "quasi-coherent $\\mathcal{O}_\\etale$-modules $\\mathcal{H}$.", "\\medskip\\noindent", "Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_\\etale)$", "and denote $\\mathcal{H}^q = H^q(E)$ its $q$th cohomology", "sheaf. Let $\\mathcal{B}$ be the set of affine objects of $X_\\etale$.", "Then $H^p(U, \\mathcal{H}^q) = 0$ for all $p > 0$, all $q \\in \\mathbf{Z}$,", "and all $U \\in \\mathcal{B}$, see", "Descent, Proposition \\ref{descent-proposition-same-cohomology-quasi-coherent}", "and", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "By Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-cohomology-over-U-trivial}", "this means that", "$$", "H^q(U, E) = H^0(U, \\mathcal{H}^q)", "$$", "for all $U \\in \\mathcal{B}$. In particular, we find that this holds", "for affine opens $U \\subset X$. It follows that the $q$th cohomology of", "$R\\epsilon_*E$ over $U$ is the value of the sheaf $\\epsilon_*\\mathcal{H}^q$", "over $U$. Applying sheafification we obtain", "$$", "H^q(R\\epsilon_*E) = \\epsilon_*\\mathcal{H}^q", "$$", "which in particular shows that $R\\epsilon_*$ induces a functor", "$D_\\QCoh(\\mathcal{O}_\\etale) \\to D_\\QCoh(\\mathcal{O}_X)$.", "Since $\\epsilon^*$ is exact we then obtain", "$H^q(\\epsilon^*R\\epsilon_*E) = \\epsilon^*\\epsilon_*\\mathcal{H}^q =", "\\mathcal{H}^q$ (by discussion above). Thus the adjunction map", "$\\epsilon^*R\\epsilon_*E \\to E$ is an isomorphism.", "\\medskip\\noindent", "Conversely, for $F \\in D_\\QCoh(\\mathcal{O}_X)$ the", "adjunction map $F \\to R\\epsilon_*\\epsilon^*F$", "is an isomorphism for the same reason, i.e., because", "the cohomology sheaves of $R\\epsilon_*\\epsilon^*F$", "are isomorphic to", "$\\epsilon_*H^m(\\epsilon^*F) = \\epsilon_*\\epsilon^*H^m(F) = H^m(F)$." ], "refs": [ "descent-lemma-compare-sites", "descent-definition-structure-sheaf", "descent-lemma-sheaf-condition-holds", "descent-proposition-equivalence-quasi-coherent", "descent-proposition-same-cohomology-quasi-coherent", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "sites-cohomology-lemma-cohomology-over-U-trivial" ], "ref_ids": [ 14622, 14766, 14621, 14755, 14754, 3282, 4274 ] } ], "ref_ids": [] }, { "id": 2645, "type": "theorem", "label": "spaces-perfect-lemma-check-quasi-coherence-on-covering", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-check-quasi-coherence-on-covering", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $E$ be an object of $D(\\mathcal{O}_X)$. The following are equivalent", "\\begin{enumerate}", "\\item $E$ is in $D_\\QCoh(\\mathcal{O}_X)$,", "\\item for every \\'etale morphism $\\varphi : U \\to X$ where $U$ is an", "affine scheme $\\varphi^*E$ is an object of", "$D_\\QCoh(\\mathcal{O}_U)$,", "\\item for every \\'etale morphism $\\varphi : U \\to X$ where $U$ is a scheme", "$\\varphi^*E$ is an object of", "$D_\\QCoh(\\mathcal{O}_U)$,", "\\item there exists a surjective \\'etale morphism $\\varphi : U \\to X$", "where $U$ is a scheme such that $\\varphi^*E$ is an object of", "$D_\\QCoh(\\mathcal{O}_U)$, and", "\\item there exists a surjective \\'etale morphism of algebraic spaces", "$f : Y \\to X$ such that $Lf^*E$ is an object of", "$D_\\QCoh(\\mathcal{O}_Y)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the discussion preceding the lemma and", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-characterize-quasi-coherent}." ], "refs": [ "spaces-properties-lemma-characterize-quasi-coherent" ], "ref_ids": [ 11911 ] } ], "ref_ids": [] }, { "id": 2646, "type": "theorem", "label": "spaces-perfect-lemma-quasi-coherence-direct-sums", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-quasi-coherence-direct-sums", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Then $D_\\QCoh(\\mathcal{O}_X)$ has direct sums." ], "refs": [], "proofs": [ { "contents": [ "By Injectives, Lemma \\ref{injectives-lemma-derived-products}", "the derived category $D(\\mathcal{O}_X)$ has direct sums and", "they are computed by taking termwise direct sums of any representatives.", "Thus it is clear that the cohomology sheaf of a direct sum is the", "direct sum of the cohomology sheaves as taking direct sums is", "an exact functor (in any Grothendieck abelian category). The lemma", "follows as the direct sum of quasi-coherent sheaves is quasi-coherent, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-properties-quasi-coherent}." ], "refs": [ "injectives-lemma-derived-products", "spaces-properties-lemma-properties-quasi-coherent" ], "ref_ids": [ 7795, 11912 ] } ], "ref_ids": [] }, { "id": 2647, "type": "theorem", "label": "spaces-perfect-lemma-Rlim-quasi-coherent", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-Rlim-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $(K_n)$ be an inverse system of", "$D_\\QCoh(\\mathcal{O}_X)$ with derived limit", "$K = R\\lim K_n$ in $D(\\mathcal{O}_X)$. Assume $H^q(K_{n + 1}) \\to H^q(K_n)$", "is surjective for all $q \\in \\mathbf{Z}$ and $n \\geq 1$.", "Then", "\\begin{enumerate}", "\\item $H^q(K) = \\lim H^q(K_n)$,", "\\item $R\\lim H^q(K_n) = \\lim H^q(K_n)$, and", "\\item for every affine open $U \\subset X$ we have", "$H^p(U, \\lim H^q(K_n)) = 0$ for $p > 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{B} \\subset \\Ob(X_\\etale)$ be the set of affine objects.", "Since $H^q(K_n)$ is quasi-coherent we have $H^p(U, H^q(K_n)) = 0$", "for $U \\in \\mathcal{B}$ by the discussion in", "Cohomology of Spaces, Section", "\\ref{spaces-cohomology-section-higher-direct-image}", "and", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "Moreover, the maps $H^0(U, H^q(K_{n + 1})) \\to H^0(U, H^q(K_n))$", "are surjective for $U \\in \\mathcal{B}$ by similar reasoning.", "Part (1) follows from Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-derived-limit-suitable-system}", "whose conditions we have just verified.", "Parts (2) and (3) follow from", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-inverse-limit-is-derived-limit}." ], "refs": [ "coherent-lemma-quasi-coherent-affine-cohomology-zero", "sites-cohomology-lemma-derived-limit-suitable-system", "sites-cohomology-lemma-inverse-limit-is-derived-limit" ], "ref_ids": [ 3282, 4275, 4268 ] } ], "ref_ids": [] }, { "id": 2648, "type": "theorem", "label": "spaces-perfect-lemma-quasi-coherence-pullback", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-quasi-coherence-pullback", "contents": [ "Let $S$ be a scheme.", "Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$.", "The functor $Lf^*$ sends $D_\\QCoh(\\mathcal{O}_X)$", "into $D_\\QCoh(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$ and $V$ are schemes, the vertical arrows are \\'etale, and", "$a$ is surjective. Since $a^* \\circ Lf^* = Lh^* \\circ b^*$ the result", "follows from", "Lemma \\ref{lemma-check-quasi-coherence-on-covering}", "and the case of schemes which is", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-pullback}." ], "refs": [ "spaces-perfect-lemma-check-quasi-coherence-on-covering", "perfect-lemma-quasi-coherence-pullback" ], "ref_ids": [ 2645, 6944 ] } ], "ref_ids": [] }, { "id": 2649, "type": "theorem", "label": "spaces-perfect-lemma-quasi-coherence-tensor-product", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-quasi-coherence-tensor-product", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "For objects $K, L$ of $D_\\QCoh(\\mathcal{O}_X)$", "the derived tensor product $K \\otimes^\\mathbf{L} L$ is in", "$D_\\QCoh(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi : U \\to X$ be a surjective \\'etale morphism from a scheme $U$.", "Since", "$\\varphi^*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) =", "\\varphi^*K \\otimes_{\\mathcal{O}_U}^\\mathbf{L} \\varphi^*L$", "we see from", "Lemma \\ref{lemma-check-quasi-coherence-on-covering}", "that this follows from the case of schemes which is", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-tensor-product}." ], "refs": [ "spaces-perfect-lemma-check-quasi-coherence-on-covering", "perfect-lemma-quasi-coherence-tensor-product" ], "ref_ids": [ 2645, 6945 ] } ], "ref_ids": [] }, { "id": 2650, "type": "theorem", "label": "spaces-perfect-lemma-nice-K-injective", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-nice-K-injective", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an", "object of $D_\\QCoh(\\mathcal{O}_X)$. Then the canonical map", "$E \\to R\\lim \\tau_{\\geq -n}E$ is an isomorphism\\footnote{In particular,", "$E$ has a K-injective representative as in", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-K-injective}.}." ], "refs": [ "sites-cohomology-lemma-K-injective" ], "proofs": [ { "contents": [ "Denote $\\mathcal{H}^i = H^i(E)$ the $i$th cohomology sheaf of $E$.", "Let $\\mathcal{B}$ be the set of affine objects of $X_\\etale$.", "Then $H^p(U, \\mathcal{H}^i) = 0$ for all $p > 0$, all $i \\in \\mathbf{Z}$,", "and all $U \\in \\mathcal{B}$ as $U$ is an affine scheme.", "See discussion in", "Cohomology of Spaces, Section", "\\ref{spaces-cohomology-section-higher-direct-image}", "and", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "Thus the lemma follows from", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-is-limit-dimension}", "with $d = 0$." ], "refs": [ "coherent-lemma-quasi-coherent-affine-cohomology-zero", "sites-cohomology-lemma-is-limit-dimension" ], "ref_ids": [ 3282, 4273 ] } ], "ref_ids": [ 4276 ] }, { "id": 2651, "type": "theorem", "label": "spaces-perfect-lemma-application-nice-K-injective", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-application-nice-K-injective", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $F : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Ab}$", "be a functor and $N \\geq 0$ an integer. Assume that", "\\begin{enumerate}", "\\item $F$ is left exact,", "\\item $F$ commutes with countable direct products,", "\\item $R^pF(\\mathcal{F}) = 0$ for all $p \\geq N$ and $\\mathcal{F}$", "quasi-coherent.", "\\end{enumerate}", "Then for $E \\in D_\\QCoh(\\mathcal{O}_X)$", "\\begin{enumerate}", "\\item $H^i(RF(\\tau_{\\leq a}E) \\to H^i(RF(E))$ is an isomorphism", "for $i \\leq a$,", "\\item $H^i(RF(E)) \\to H^i(RF(\\tau_{\\geq b - N + 1}E))$ is an isomorphism", "for $i \\geq b$,", "\\item if $H^i(E) = 0$ for $i \\not \\in [a, b]$ for some", "$-\\infty \\leq a \\leq b \\leq \\infty$, then $H^i(RF(E)) = 0$", "for $i \\not \\in [a, b + N - 1]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Statement (1) is", "Derived Categories, Lemma \\ref{derived-lemma-negative-vanishing}.", "\\medskip\\noindent", "Proof of statement (2). Write $E_n = \\tau_{\\geq -n}E$. We have", "$E = R\\lim E_n$, see Lemma \\ref{lemma-nice-K-injective}. Thus", "$RF(E) = R\\lim RF(E_n)$ in $D(\\textit{Ab})$ by Injectives, Lemma", "\\ref{injectives-lemma-RF-commutes-with-Rlim}. Thus for every $i \\in \\mathbf{Z}$", "we have a short exact sequence", "$$", "0 \\to R^1\\lim H^{i - 1}(RF(E_n)) \\to H^i(RF(E)) \\to \\lim H^i(RF(E_n)) \\to 0", "$$", "see More on Algebra, Remark", "\\ref{more-algebra-remark-compare-derived-limit}.", "To prove (2) we will show that the term on the left is zero", "and that the term on the right equals $H^i(RF(E_{-b + N - 1})$", "for any $b$ with $i \\geq b$.", "\\medskip\\noindent", "For every $n$ we have a distinguished triangle", "$$", "H^{-n}(E)[n] \\to E_n \\to E_{n - 1} \\to H^{-n}(E)[n + 1]", "$$", "(Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle})", "in $D(\\mathcal{O}_X)$. Since $H^{-n}(E)$ is quasi-coherent we have", "$$", "H^i(RF(H^{-n}(E)[n])) = R^{i + n}F(H^{-n}(E)) = 0", "$$", "for $i + n \\geq N$ and", "$$", "H^i(RF(H^{-n}(E)[n + 1])) = R^{i + n + 1}F(H^{-n}(E)) = 0", "$$", "for $i + n + 1 \\geq N$. We conclude that", "$$", "H^i(RF(E_n)) \\to H^i(RF(E_{n - 1}))", "$$", "is an isomorphism for $n \\geq N - i$. Thus the systems $H^i(RF(E_n))$ all", "satisfy the ML condition and the $R^1\\lim$ term in our short exact", "sequence is zero (see discussion in", "More on Algebra, Section \\ref{more-algebra-section-Rlim}).", "Moreover, the system $H^i(RF(E_n))$ is constant starting", "with $n = N - i - 1$ as desired.", "\\medskip\\noindent", "Proof of (3). Under the assumption on $E$ we have", "$\\tau_{\\leq a - 1}E = 0$ and we get the vanishing", "of $H^i(RF(E))$ for $i \\leq a - 1$ from (1).", "Similarly, we have $\\tau_{\\geq b + 1}E = 0$ and hence", "we get the vanishing of $H^i(RF(E))$ for $i \\geq b + n$ from", "part (2)." ], "refs": [ "derived-lemma-negative-vanishing", "spaces-perfect-lemma-nice-K-injective", "injectives-lemma-RF-commutes-with-Rlim", "more-algebra-remark-compare-derived-limit", "derived-remark-truncation-distinguished-triangle" ], "ref_ids": [ 1839, 2650, 7796, 10658, 2016 ] } ], "ref_ids": [] }, { "id": 2652, "type": "theorem", "label": "spaces-perfect-lemma-quasi-coherence-direct-image", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-quasi-coherence-direct-image", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-separated and quasi-compact", "morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item The functor $Rf_*$ sends $D_\\QCoh(\\mathcal{O}_X)$", "into $D_\\QCoh(\\mathcal{O}_Y)$.", "\\item If $Y$ is quasi-compact, there exists an integer $N = N(X, Y, f)$", "such that for an object $E$ of $D_\\QCoh(\\mathcal{O}_X)$", "with $H^m(E) = 0$ for $m > 0$ we have", "$H^m(Rf_*E) = 0$ for $m \\geq N$.", "\\item In fact, if $Y$ is quasi-compact we can find $N = N(X, Y, f)$", "such that for every morphism of algebraic spaces $Y' \\to Y$", "the same conclusion holds for the functor $R(f')_*$", "where $f' : X' \\to Y'$ is the base change of $f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$.", "To prove (1) we have to show that $Rf_*E$ has quasi-coherent", "cohomology sheaves. This question is local on $Y$, hence we may", "assume $Y$ is quasi-compact. Pick $N = N(X, Y, f)$ as in", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-vanishing-higher-direct-images}.", "Thus $R^pf_*\\mathcal{F} = 0$ for all quasi-coherent $\\mathcal{O}_X$-modules", "$\\mathcal{F}$ and all $p \\geq N$. Moreover $R^pf_*\\mathcal{F}$", "is quasi-coherent for all $p$ by", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-higher-direct-image}.", "These statements remain true after base change.", "\\medskip\\noindent", "First, assume $E$ is bounded below. We will show (1) and (2) and (3) hold", "for such $E$ with our choice of $N$. In this case we can for example use the", "spectral sequence", "$$", "R^pf_*H^q(E) \\Rightarrow R^{p + q}f_*E", "$$", "(Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}),", "the quasi-coherence of $R^pf_*H^q(E)$, and the vanishing of $R^pf_*H^q(E)$", "for $p \\geq N$ to see that (1), (2), and (3) hold in this case.", "\\medskip\\noindent", "Next we prove (2) and (3). Say $H^m(E) = 0$ for $m > 0$.", "Let $V$ be an affine object of $Y_\\etale$.", "We have $H^p(V \\times_Y X, \\mathcal{F}) = 0$ for $p \\geq N$, see", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-quasi-coherence-higher-direct-images-application}.", "Hence we may apply Lemma \\ref{lemma-application-nice-K-injective}", "to the functor $\\Gamma(V \\times_Y X, -)$ to see that", "$$", "R\\Gamma(V, Rf_*E) = R\\Gamma(V \\times_Y X, E)", "$$", "has vanishing cohomology in degrees $\\geq N$. Since this holds for", "all $V$ affine in $Y_\\etale$ we conclude that $H^m(Rf_*E) = 0$", "for $m \\geq N$.", "\\medskip\\noindent", "Next, we prove (1) in the general case. Recall that there is a", "distinguished triangle", "$$", "\\tau_{\\leq -n - 1}E \\to E \\to \\tau_{\\geq -n}E \\to", "(\\tau_{\\leq -n - 1}E)[1]", "$$", "in $D(\\mathcal{O}_X)$, see Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}.", "By (2) we see that $Rf_*\\tau_{\\leq -n - 1}E$", "has vanishing cohomology sheaves in degrees $\\geq -n + N$.", "Thus, given an integer $q$ we see that $R^qf_*E$ is equal", "to $R^qf_*\\tau_{\\geq -n}E$ for some $n$ and the result", "above applies." ], "refs": [ "spaces-cohomology-lemma-vanishing-higher-direct-images", "spaces-cohomology-lemma-higher-direct-image", "derived-lemma-two-ss-complex-functor", "spaces-cohomology-lemma-quasi-coherence-higher-direct-images-application", "spaces-perfect-lemma-application-nice-K-injective", "derived-remark-truncation-distinguished-triangle" ], "ref_ids": [ 11287, 11271, 1871, 11272, 2651, 2016 ] } ], "ref_ids": [] }, { "id": 2653, "type": "theorem", "label": "spaces-perfect-lemma-quasi-coherence-pushforward-direct-sums", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-quasi-coherence-pushforward-direct-sums", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-separated and", "quasi-compact morphism of algebraic spaces over $S$. Then", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$", "commutes with direct sums." ], "refs": [], "proofs": [ { "contents": [ "Let $E_i$ be a family of objects of $D_\\QCoh(\\mathcal{O}_X)$", "and set $E = \\bigoplus E_i$. We want to show that the map", "$$", "\\bigoplus Rf_*E_i \\longrightarrow Rf_*E", "$$", "is an isomorphism. We will show it induces an isomorphism on", "cohomology sheaves in degree $0$ which will imply the lemma.", "Choose an integer $N$ as in Lemma \\ref{lemma-quasi-coherence-direct-image}.", "Then $R^0f_*E = R^0f_*\\tau_{\\geq -N}E$ and", "$R^0f_*E_i = R^0f_*\\tau_{\\geq -N}E_i$ by the lemma cited. Observe that", "$\\tau_{\\geq -N}E = \\bigoplus \\tau_{\\geq -N}E_i$.", "Thus we may assume all of the $E_i$ have vanishing cohomology", "sheaves in degrees $< -N$. Next we use the spectral sequences", "$$", "R^pf_*H^q(E) \\Rightarrow R^{p + q}f_*E", "\\quad\\text{and}\\quad", "R^pf_*H^q(E_i) \\Rightarrow R^{p + q}f_*E_i", "$$", "(Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor})", "to reduce to the case of a direct sum of quasi-coherent sheaves.", "This case is handled by", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-colimit-cohomology}." ], "refs": [ "spaces-perfect-lemma-quasi-coherence-direct-image", "derived-lemma-two-ss-complex-functor", "spaces-cohomology-lemma-colimit-cohomology" ], "ref_ids": [ 2652, 1871, 11278 ] } ], "ref_ids": [] }, { "id": 2654, "type": "theorem", "label": "spaces-perfect-lemma-affine-morphism", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-affine-morphism", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic", "spaces over $S$. Then", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$", "reflects isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "The statement means that a morphism $\\alpha : E \\to F$ of", "$D_\\QCoh(\\mathcal{O}_X)$ is an isomorphism if", "$Rf_*\\alpha$ is an isomorphism. We may check this on cohomology sheaves.", "In particular, the question is \\'etale local on $Y$. Hence we may assume", "$Y$ and therefore $X$ is affine. In this case the problem reduces to the", "case of schemes", "(Derived Categories of Schemes, Lemma \\ref{perfect-lemma-affine-morphism})", "via Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site} and", "Remark \\ref{remark-match-total-direct-images}." ], "refs": [ "perfect-lemma-affine-morphism", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-remark-match-total-direct-images" ], "ref_ids": [ 6952, 2644, 2768 ] } ], "ref_ids": [] }, { "id": 2655, "type": "theorem", "label": "spaces-perfect-lemma-affine-morphism-pull-push", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-affine-morphism-pull-push", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic", "spaces over $S$. For $E$ in $D_\\QCoh(\\mathcal{O}_Y)$ we have", "$Rf_* Lf^* E = E \\otimes^\\mathbf{L}_{\\mathcal{O}_Y} f_*\\mathcal{O}_X$." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is affine the map $f_*\\mathcal{O}_X \\to Rf_*\\mathcal{O}_X$", "is an isomorphism (Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-affine-vanishing-higher-direct-images}).", "There is a canonical map", "$E \\otimes^\\mathbf{L} f_*\\mathcal{O}_X =", "E \\otimes^\\mathbf{L} Rf_*\\mathcal{O}_X \\to Rf_* Lf^* E$", "adjoint to the map", "$$", "Lf^*(E \\otimes^\\mathbf{L} Rf_*\\mathcal{O}_X) =", "Lf^*E \\otimes^\\mathbf{L} Lf^*Rf_*\\mathcal{O}_X \\longrightarrow", "Lf^* E \\otimes^\\mathbf{L} \\mathcal{O}_X = Lf^* E", "$$", "coming from $1 : Lf^*E \\to Lf^*E$ and the canonical map", "$Lf^*Rf_*\\mathcal{O}_X \\to \\mathcal{O}_X$. To check the map so constructed", "is an isomorphism we may work locally on $Y$. Hence we may assume", "$Y$ and therefore $X$ is affine. In this case the problem reduces to the", "case of schemes", "(Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-morphism-pull-push})", "via Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site} and", "Remark \\ref{remark-match-total-direct-images}." ], "refs": [ "spaces-cohomology-lemma-affine-vanishing-higher-direct-images", "perfect-lemma-affine-morphism-pull-push", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-remark-match-total-direct-images" ], "ref_ids": [ 11288, 6953, 2644, 2768 ] } ], "ref_ids": [] }, { "id": 2656, "type": "theorem", "label": "spaces-perfect-lemma-closed-proper-over-base", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-closed-proper-over-base", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ which is locally of finite type. Let $T \\subset |X|$ be a closed", "subset. The following are equivalent", "\\begin{enumerate}", "\\item the morphism $Z \\to Y$ is proper if $Z$ is the reduced", "induced algebraic space structure on $T$", "(Properties of Spaces, Definition", "\\ref{spaces-properties-definition-reduced-induced-space}),", "\\item for some closed subspace $Z \\subset X$ with $|Z| = T$", "the morphism $Z \\to Y$ is proper, and", "\\item for any closed subspace $Z \\subset X$ with $|Z| = T$ the morphism", "$Z \\to Y$ is proper.", "\\end{enumerate}" ], "refs": [ "spaces-properties-definition-reduced-induced-space" ], "proofs": [ { "contents": [ "The implications (3) $\\Rightarrow$ (1) and (1) $\\Rightarrow$ (2)", "are immediate. Thus it suffices to prove that (2) implies (3).", "We urge the reader to find their own proof of this fact.", "Let $Z'$ and $Z''$ be closed subspaces with $T = |Z'| = |Z''|$", "such that $Z' \\to Y$ is a proper morphism of algebraic spaces.", "We have to show that $Z'' \\to Y$ is proper too.", "Let $Z''' = Z' \\cup Z''$ be the scheme theoretic union, see", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-scheme-theoretic-intersection-union}.", "Then $Z'''$ is another closed subspace with $|Z'''| = T$.", "This follows for example from the description of scheme theoretic unions in", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-scheme-theoretic-union}.", "Since $Z'' \\to Z'''$ is a closed immersion it suffices to prove", "that $Z''' \\to Y$ is proper (see", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-closed-immersion-proper} and", "\\ref{spaces-morphisms-lemma-composition-proper}).", "The morphism $Z' \\to Z'''$ is a bijective closed immersion", "and in particular surjective and universally closed.", "Then the fact that $Z' \\to Y$ is separated implies that", "$Z''' \\to Y$ is separated, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-image-universally-closed-separated}.", "Moreover $Z''' \\to Y$ is locally of finite type", "as $X \\to Y$ is locally of finite type", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-immersion-locally-finite-type} and", "\\ref{spaces-morphisms-lemma-composition-finite-type}).", "Since $Z' \\to Y$ is quasi-compact and $Z' \\to Z'''$ is a", "universal homeomorphism we see that $Z''' \\to Y$ is quasi-compact.", "Finally, since $Z' \\to Y$ is universally closed, we see that", "the same thing is true for $Z''' \\to Y$ by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-image-proper-is-proper}.", "This finishes the proof." ], "refs": [ "spaces-morphisms-definition-scheme-theoretic-intersection-union", "spaces-morphisms-lemma-scheme-theoretic-union", "spaces-morphisms-lemma-closed-immersion-proper", "spaces-morphisms-lemma-composition-proper", "spaces-morphisms-lemma-image-universally-closed-separated", "spaces-morphisms-lemma-immersion-locally-finite-type", "spaces-morphisms-lemma-composition-finite-type", "spaces-morphisms-lemma-image-proper-is-proper" ], "ref_ids": [ 4992, 4775, 4919, 4918, 4750, 4819, 4814, 4921 ] } ], "ref_ids": [ 11932 ] }, { "id": 2657, "type": "theorem", "label": "spaces-perfect-lemma-closed-closed-proper-over-base", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-closed-closed-proper-over-base", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "which is locally of finite type.", "Let $T' \\subset T \\subset |X|$ be closed subsets.", "If $T$ is proper over $Y$, then the same is true for $T'$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2658, "type": "theorem", "label": "spaces-perfect-lemma-base-change-closed-proper-over-base", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-base-change-closed-proper-over-base", "contents": [ "Let $S$ be a scheme.", "Consider a cartesian diagram of algebraic spaces over $S$", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "with $f$ locally of finite type.", "If $T$ is a closed subset of $|X|$ proper over $Y$, then", "$|g'|^{-1}(T)$ is a closed subset of $|X'|$ proper over $Y'$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the statement makes sense as $f'$ is locally of", "finite type by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-finite-type}.", "Let $Z \\subset X$ be the reduced induced closed subspace structure on $T$.", "Denote $Z' = (g')^{-1}(Z)$ the scheme theoretic inverse image.", "Then $Z' = X' \\times_X Z = (Y' \\times_Y X) \\times_X Z = Y' \\times_Y Z$", "is proper over $Y'$ as a base change of $Z$ over $Y$", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-proper}).", "On the other hand, we have $T' = |Z'|$. Hence the lemma holds." ], "refs": [ "spaces-morphisms-lemma-base-change-finite-type", "spaces-morphisms-lemma-base-change-proper" ], "ref_ids": [ 4815, 4917 ] } ], "ref_ids": [] }, { "id": 2659, "type": "theorem", "label": "spaces-perfect-lemma-functoriality-closed-proper-over-base", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-functoriality-closed-proper-over-base", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $f : X \\to Y$ be a morphism of algebraic spaces which", "are locally of finite type over $B$.", "\\begin{enumerate}", "\\item If $Y$ is separated over $B$ and $T \\subset |X|$ is a closed subset", "proper over $B$, then $|f|(T)$ is a closed subset of $|Y|$ proper over $B$.", "\\item If $f$ is universally closed and $T \\subset |X|$ is a", "closed subset proper over $B$, then $|f|(T)$ is a closed subset", "of $Y$ proper over $B$.", "\\item If $f$ is proper and $T \\subset |Y|$ is a closed subset", "proper over $B$, then $|f|^{-1}(T)$ is a closed subset of $|X|$", "proper over $B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Assume $Y$ is separated over $B$ and $T \\subset |X|$", "is a closed subset proper over $B$. Let $Z$ be the reduced induced", "closed subspace structure on $T$ and apply", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-scheme-theoretic-image-is-proper}", "to $Z \\to Y$ over $B$ to conclude.", "\\medskip\\noindent", "Proof of (2). Assume $f$ is universally closed and $T \\subset |X|$ is a", "closed subset proper over $B$. Let $Z$ be the reduced induced", "closed subspace structure on $T$ and let $Z'$ be the reduced", "induced closed subspace structure on $|f|(T)$. We obtain an induced", "morphism $Z \\to Z'$.", "Denote $Z'' = f^{-1}(Z')$ the scheme theoretic inverse image.", "Then $Z'' \\to Z'$ is universally closed as a base change of $f$", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-proper}).", "Hence $Z \\to Z'$ is universally closed as a composition of", "the closed immersion $Z \\to Z''$ and $Z'' \\to Z'$", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-closed-immersion-proper} and", "\\ref{spaces-morphisms-lemma-composition-proper}).", "We conclude that $Z' \\to B$ is separated by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-image-universally-closed-separated}.", "Since $Z \\to B$ is quasi-compact and $Z \\to Z'$ is surjective", "we see that $Z' \\to B$ is quasi-compact.", "Since $Z' \\to B$ is the composition of $Z' \\to Y$ and $Y \\to B$", "we see that $Z' \\to B$ is locally of finite type", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-immersion-locally-finite-type} and", "\\ref{spaces-morphisms-lemma-composition-finite-type}).", "Finally, since $Z \\to B$ is universally closed, we see that", "the same thing is true for $Z' \\to B$ by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-image-proper-is-proper}.", "This finishes the proof.", "\\medskip\\noindent", "Proof of (3). Assume $f$ is proper and $T \\subset |Y|$ is a closed subset", "proper over $B$. Let $Z$ be the reduced induced closed subspace", "structure on $T$. Denote $Z' = f^{-1}(Z)$ the scheme theoretic inverse image.", "Then $Z' \\to Z$ is proper as a base change of $f$", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-proper}).", "Whence $Z' \\to B$ is proper as the composition of $Z' \\to Z$", "and $Z \\to B$", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-composition-proper}).", "This finishes the proof." ], "refs": [ "spaces-morphisms-lemma-scheme-theoretic-image-is-proper", "spaces-morphisms-lemma-base-change-proper", "spaces-morphisms-lemma-closed-immersion-proper", "spaces-morphisms-lemma-composition-proper", "spaces-morphisms-lemma-image-universally-closed-separated", "spaces-morphisms-lemma-immersion-locally-finite-type", "spaces-morphisms-lemma-composition-finite-type", "spaces-morphisms-lemma-image-proper-is-proper", "spaces-morphisms-lemma-base-change-proper", "spaces-morphisms-lemma-composition-proper" ], "ref_ids": [ 4922, 4917, 4919, 4918, 4750, 4819, 4814, 4921, 4917, 4918 ] } ], "ref_ids": [] }, { "id": 2660, "type": "theorem", "label": "spaces-perfect-lemma-union-closed-proper-over-base", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-union-closed-proper-over-base", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "which is locally of finite type.", "Let $T_i \\subset |X|$, $i = 1, \\ldots, n$ be closed subsets.", "If $T_i$, $i = 1, \\ldots, n$ are proper over $Y$, then the same is", "true for $T_1 \\cup \\ldots \\cup T_n$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z_i$ be the reduced induced closed subscheme structure on $T_i$.", "The morphism", "$$", "Z_1 \\amalg \\ldots \\amalg Z_n \\longrightarrow X", "$$", "is finite by Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-closed-immersion-finite} and", "\\ref{spaces-morphisms-lemma-finite-union-finite}.", "As finite morphisms are universally closed", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-proper})", "and since $Z_1 \\amalg \\ldots \\amalg Z_n$ is proper over $S$", "we conclude by", "Lemma \\ref{lemma-functoriality-closed-proper-over-base} part (2)", "that the image $Z_1 \\cup \\ldots \\cup Z_n$ is proper over $S$." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-finite", "spaces-morphisms-lemma-finite-union-finite", "spaces-morphisms-lemma-finite-proper", "spaces-perfect-lemma-functoriality-closed-proper-over-base" ], "ref_ids": [ 4947, 4948, 4946, 2659 ] } ], "ref_ids": [] }, { "id": 2661, "type": "theorem", "label": "spaces-perfect-lemma-module-support-proper-over-base", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-module-support-proper-over-base", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "algebraic spaces over $S$ which is locally of finite type.", "Let $\\mathcal{F}$ be a finite type, quasi-coherent", "$\\mathcal{O}_X$-module. The following are equivalent", "\\begin{enumerate}", "\\item the support of $\\mathcal{F}$ is proper over $Y$,", "\\item the scheme theoretic support of $\\mathcal{F}$", "(Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-scheme-theoretic-support})", "is proper over $Y$, and", "\\item there exists a closed subspace $Z \\subset X$ and", "a finite type, quasi-coherent $\\mathcal{O}_Z$-module", "$\\mathcal{G}$ such that (a) $Z \\to Y$ is proper, and (b)", "$(Z \\to X)_*\\mathcal{G} = \\mathcal{F}$.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-scheme-theoretic-support" ], "proofs": [ { "contents": [ "The support $\\text{Supp}(\\mathcal{F})$ of $\\mathcal{F}$ is a closed subset", "of $|X|$, see Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-support-finite-type}.", "Hence we can apply Definition \\ref{definition-proper-over-base}.", "Since the scheme theoretic support of $\\mathcal{F}$ is a closed", "subspace whose underlying closed subset is $\\text{Supp}(\\mathcal{F})$", "we see that (1) and (2) are equivalent by", "Definition \\ref{definition-proper-over-base}.", "It is clear that (2) implies (3).", "Conversely, if (3) is true, then", "$\\text{Supp}(\\mathcal{F}) \\subset |Z|$", "and hence $\\text{Supp}(\\mathcal{F})$", "is proper over $Y$ for example by", "Lemma \\ref{lemma-closed-closed-proper-over-base}." ], "refs": [ "spaces-morphisms-lemma-support-finite-type", "spaces-perfect-definition-proper-over-base", "spaces-perfect-definition-proper-over-base", "spaces-perfect-lemma-closed-closed-proper-over-base" ], "ref_ids": [ 4777, 2763, 2763, 2657 ] } ], "ref_ids": [ 4993 ] }, { "id": 2662, "type": "theorem", "label": "spaces-perfect-lemma-base-change-module-support-proper-over-base", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-base-change-module-support-proper-over-base", "contents": [ "Let $S$ be a scheme.", "Consider a cartesian diagram of algebraic spaces over $S$", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "with $f$ locally of finite type. Let $\\mathcal{F}$ be a", "finite type, quasi-coherent $\\mathcal{O}_X$-module.", "If the support of $\\mathcal{F}$ is proper over $Y$, then", "the support of $(g')^*\\mathcal{F}$ is proper over $Y'$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the statement makes sense because", "$(g')*\\mathcal{F}$ is of finite type by", "Modules on Sites, Lemma \\ref{sites-modules-lemma-local-pullback}.", "We have $\\text{Supp}((g')^*\\mathcal{F}) = |g'|^{-1}(\\text{Supp}(\\mathcal{F}))$", "by Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-support-finite-type}.", "Thus the lemma follows from", "Lemma \\ref{lemma-base-change-closed-proper-over-base}." ], "refs": [ "sites-modules-lemma-local-pullback", "spaces-morphisms-lemma-support-finite-type", "spaces-perfect-lemma-base-change-closed-proper-over-base" ], "ref_ids": [ 14186, 4777, 2658 ] } ], "ref_ids": [] }, { "id": 2663, "type": "theorem", "label": "spaces-perfect-lemma-cat-module-support-proper-over-base", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-cat-module-support-proper-over-base", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "which is locally of finite type. Let $\\mathcal{F}$, $\\mathcal{G}$", "be finite type, quasi-coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item If the supports of $\\mathcal{F}$, $\\mathcal{G}$", "are proper over $Y$, then the same is true", "for $\\mathcal{F} \\oplus \\mathcal{G}$, for any extension", "of $\\mathcal{G}$ by $\\mathcal{F}$, for $\\Im(u)$ and $\\Coker(u)$", "given any $\\mathcal{O}_X$-module map $u : \\mathcal{F} \\to \\mathcal{G}$,", "and for any quasi-coherent quotient of $\\mathcal{F}$ or $\\mathcal{G}$.", "\\item If $Y$ is locally Noetherian, then the category of", "coherent $\\mathcal{O}_X$-modules with support proper over", "$Y$ is a Serre subcategory (Homology, Definition", "\\ref{homology-definition-serre-subcategory})", "of the abelian category of", "coherent $\\mathcal{O}_X$-modules.", "\\end{enumerate}" ], "refs": [ "homology-definition-serre-subcategory" ], "proofs": [ { "contents": [ "Proof of (1). Let $T$, $T'$ be the support of $\\mathcal{F}$", "and $\\mathcal{G}$. Then all the sheaves mentioned in (1)", "have support contained in $T \\cup T'$. Thus the assertion itself", "is clear from Lemmas \\ref{lemma-closed-closed-proper-over-base} and", "\\ref{lemma-union-closed-proper-over-base}", "provided we check that these sheaves are finite type", "and quasi-coherent.", "For quasi-coherence we refer the reader to", "Properties of Spaces, Section \\ref{spaces-properties-section-quasi-coherent}.", "For ``finite type'' we refer the reader to", "Properties of Spaces, Section", "\\ref{spaces-properties-section-properties-modules}.", "\\medskip\\noindent", "Proof of (2). The proof is the same as the proof of (1). Note that", "the assertions make sense as $X$ is locally Noetherian by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}", "and by the description of the category of coherent modules in", "Cohomology of Spaces, Section \\ref{spaces-cohomology-section-coherent}." ], "refs": [ "spaces-perfect-lemma-closed-closed-proper-over-base", "spaces-perfect-lemma-union-closed-proper-over-base", "spaces-morphisms-lemma-locally-finite-type-locally-noetherian" ], "ref_ids": [ 2657, 2660, 4817 ] } ], "ref_ids": [ 12146 ] }, { "id": 2664, "type": "theorem", "label": "spaces-perfect-lemma-support-proper-over-base-pushforward", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-support-proper-over-base-pushforward", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is locally of finite type and $Y$ locally Noetherian.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module with support", "proper over $Y$. Then $R^pf_*\\mathcal{F}$ is a coherent", "$\\mathcal{O}_Y$-module for all $p \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-module-support-proper-over-base}", "there exists a closed immersion $i : Z \\to X$ with", "$g = f \\circ i : Z \\to Y$ proper and", "$\\mathcal{F} = i_*\\mathcal{G}$ for some coherent module $\\mathcal{G}$", "on $Z$. We see that $R^pg_*\\mathcal{G}$", "is coherent on $S$ by Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-pushforward-coherent}.", "On the other hand, $R^qi_*\\mathcal{G} = 0$ for $q > 0$", "(Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-finite-pushforward-coherent}).", "By Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-relative-Leray}", "we get $R^pf_*\\mathcal{F} = R^pg_*\\mathcal{G}$ and the lemma follows." ], "refs": [ "spaces-perfect-lemma-module-support-proper-over-base", "spaces-cohomology-lemma-proper-pushforward-coherent", "spaces-cohomology-lemma-finite-pushforward-coherent", "sites-cohomology-lemma-relative-Leray" ], "ref_ids": [ 2661, 11331, 11304, 4222 ] } ], "ref_ids": [] }, { "id": 2665, "type": "theorem", "label": "spaces-perfect-lemma-direct-image-coherent", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-direct-image-coherent", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is locally of finite type and $Y$ is Noetherian.", "Let $E$ be an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ such that the", "support of $H^i(E)$ is proper over $Y$ for all $i$.", "Then $Rf_*E$ is an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "Consider the spectral sequence", "$$", "R^pf_*H^q(E) \\Rightarrow R^{p + q}f_*E", "$$", "see Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "By assumption and Lemma \\ref{lemma-support-proper-over-base-pushforward}", "the sheaves $R^pf_*H^q(E)$ are coherent. Hence", "$R^{p + q}f_*E$ is coherent, i.e., $E \\in D_{\\textit{Coh}}(\\mathcal{O}_Y)$.", "Boundedness from below is trivial. Boundedness from above", "follows from", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-vanishing-higher-direct-images}", "or from", "Lemma \\ref{lemma-quasi-coherence-direct-image}." ], "refs": [ "derived-lemma-two-ss-complex-functor", "spaces-perfect-lemma-support-proper-over-base-pushforward", "spaces-cohomology-lemma-vanishing-higher-direct-images", "spaces-perfect-lemma-quasi-coherence-direct-image" ], "ref_ids": [ 1871, 2664, 11287, 2652 ] } ], "ref_ids": [] }, { "id": 2666, "type": "theorem", "label": "spaces-perfect-lemma-direct-image-coherent-bdd-below", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-direct-image-coherent-bdd-below", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is locally of finite type and $Y$ is Noetherian.", "Let $E$ be an object of", "$D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ such that the support of $H^i(E)$", "is proper over $S$ for all $i$.", "Then $Rf_*E$ is an object of $D^+_{\\textit{Coh}}(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "The proof is the same as the proof of", "Lemma \\ref{lemma-direct-image-coherent}.", "You can also deduce it from", "Lemma \\ref{lemma-direct-image-coherent}", "by considering what the exact functor $Rf_*$ does to", "the distinguished triangles", "$\\tau_{\\leq a}E \\to E \\to \\tau_{\\geq a + 1}E \\to \\tau_{\\leq a}E[1]$." ], "refs": [ "spaces-perfect-lemma-direct-image-coherent", "spaces-perfect-lemma-direct-image-coherent" ], "ref_ids": [ 2665, 2665 ] } ], "ref_ids": [] }, { "id": 2667, "type": "theorem", "label": "spaces-perfect-lemma-coherent-internal-hom", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-coherent-internal-hom", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.", "If $L$ is in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$", "and $K$ in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, then", "$R\\SheafHom(K, L)$ is in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "We can check whether an object of $D(\\mathcal{O}_X)$ is in", "$D_{\\textit{Coh}}(\\mathcal{O}_X)$ \\'etale locally on $X$, see", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-coherent-Noetherian}.", "Hence this lemma follows from the case of schemes, see", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-coherent-internal-hom}." ], "refs": [ "spaces-cohomology-lemma-coherent-Noetherian", "perfect-lemma-coherent-internal-hom" ], "ref_ids": [ 11297, 6986 ] } ], "ref_ids": [] }, { "id": 2668, "type": "theorem", "label": "spaces-perfect-lemma-ext-finite", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-ext-finite", "contents": [ "Let $A$ be a Noetherian ring. Let $X$ be a proper algebraic space over $A$.", "For $L$ in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and $K$ in", "$D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, the $A$-modules", "$\\Ext_{\\mathcal{O}_X}^n(K, L)$ are finite." ], "refs": [], "proofs": [ { "contents": [ "Recall that", "$$", "\\Ext_{\\mathcal{O}_X}^n(K, L) =", "H^n(X, R\\SheafHom_{\\mathcal{O}_X}(K, L)) =", "H^n(\\Spec(A), Rf_*R\\SheafHom_{\\mathcal{O}_X}(K, L))", "$$", "see Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-section-RHom-over-U}", "and Cohomology on Sites, Section \\ref{sites-cohomology-section-leray}.", "Thus the result follows from", "Lemmas \\ref{lemma-coherent-internal-hom} and", "\\ref{lemma-direct-image-coherent-bdd-below}." ], "refs": [ "spaces-perfect-lemma-coherent-internal-hom", "spaces-perfect-lemma-direct-image-coherent-bdd-below" ], "ref_ids": [ 2667, 2666 ] } ], "ref_ids": [] }, { "id": 2669, "type": "theorem", "label": "spaces-perfect-lemma-make-more-elementary-distinguished-squares", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-make-more-elementary-distinguished-squares", "contents": [ "Let $S$ be a scheme. Let $(U \\subset W, f : V \\to W)$ be an elementary", "distinguished square of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item If $V' \\subset V$ and", "$U \\subset U' \\subset W$ are open subspaces and $W' = U' \\cup f(V')$", "then $(U' \\subset W', f|_{V'} : V' \\to W')$ is an elementary distinguished", "square.", "\\item If $p : W' \\to W$ is a morphism of algebraic spaces, then", "$(p^{-1}(U) \\subset W', V \\times_W W' \\to W')$ is an elementary distinguished", "square.", "\\item If $S' \\to S$ is a morphism of schemes, then", "$(S' \\times_S U \\subset S' \\times_S W, S' \\times_S V \\to S' \\times_S W)$", "is an elementary distinguished square.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2670, "type": "theorem", "label": "spaces-perfect-lemma-induction-principle", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-induction-principle", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $P$ be a property of the quasi-compact", "and quasi-separated objects of $X_{spaces, \\etale}$. Assume that", "\\begin{enumerate}", "\\item $P$ holds for every affine object of $X_{spaces, \\etale}$,", "\\item for every elementary distinguished square $(U \\subset W, f : V \\to W)$", "such that", "\\begin{enumerate}", "\\item $W$ is a quasi-compact and quasi-separated object of", "$X_{spaces, \\etale}$,", "\\item $U$ is quasi-compact,", "\\item $V$ is affine, and", "\\item $P$ holds for $U$, $V$, and $U \\times_W V$,", "\\end{enumerate}", "then $P$ holds for $W$.", "\\end{enumerate}", "Then $P$ holds for every quasi-compact and quasi-separated object", "of $X_{spaces, \\etale}$ and in particular for $X$." ], "refs": [], "proofs": [ { "contents": [ "We first claim that $P$ holds for every representable", "quasi-compact and quasi-separated object of $X_{spaces, \\etale}$.", "Namely, suppose that $U \\to X$ is \\'etale and $U$ is a", "quasi-compact and quasi-separated scheme. By assumption (1)", "property $P$ holds for every affine open of $U$. Moreover, if", "$W, V \\subset U$ are quasi-compact open with $V$ affine and $P$ holds for ", "$W$, $V$, and $W \\cap V$, then $P$ holds for $W \\cup V$ by (2)", "(as the pair $(W \\subset W \\cup V, V \\to W \\cup V)$ is an elementary", "distinguished square). Thus $P$ holds for $U$ by the induction", "principle for schemes, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}.", "\\medskip\\noindent", "To finish the proof it suffices to prove $P$ holds for $X$", "(because we can simply replace $X$ by any quasi-compact and quasi-separated", "object of $X_{spaces, \\etale}$ we want to prove the result for).", "We will use the filtration", "$$", "\\emptyset = U_{n + 1} \\subset", "U_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X", "$$", "and the morphisms $f_p : V_p \\to U_p$ of", "Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated}.", "We will prove that $P$ holds for $U_p$ by descending induction on $p$.", "Note that $P$ holds for $U_{n + 1}$ by (1)", "as an empty algebraic space is affine. Assume $P$ holds for $U_{p + 1}$.", "Note that $(U_{p + 1} \\subset U_p, f_p : V_p \\to U_p)$ is an elementary", "distinguished square, but (2) may not apply as $V_p$ may not be affine.", "However, as $V_p$ is a quasi-compact scheme we may choose a finite affine open", "covering $V_p = V_{p, 1} \\cup \\ldots \\cup V_{p, m}$.", "Set $W_{p, 0} = U_{p + 1}$ and", "$$", "W_{p, i} = U_{p + 1} \\cup f_p(V_{p, 1} \\cup \\ldots \\cup V_{p, i})", "$$", "for $i = 1, \\ldots, m$. These are quasi-compact open subspaces of $X$.", "Then we have", "$$", "U_{p + 1} = W_{p, 0} \\subset", "W_{p, 1} \\subset \\ldots \\subset", "W_{p, m} = U_p", "$$", "and the pairs", "$$", "(W_{p, 0} \\subset W_{p, 1}, f_p|_{V_{p, 1}}),", "(W_{p, 1} \\subset W_{p, 2}, f_p|_{V_{p, 2}}),\\ldots,", "(W_{p, m - 1} \\subset W_{p, m}, f_p|_{V_{p, m}})", "$$", "are elementary distinguished squares by", "Lemma \\ref{lemma-make-more-elementary-distinguished-squares}.", "Note that $P$ holds for each $V_{p, 1}$ (as affine schemes) and for", "$W_{p, i} \\times_{W_{p, i + 1}} V_{p, i + 1}$ as this is a quasi-compact", "open of $V_{p, i + 1}$ and hence $P$ holds for it by the first paragraph", "of this proof. Thus (2) applies to each of these and we inductively", "conclude $P$ holds for $W_{p, 1}, \\ldots, W_{p, m} = U_p$." ], "refs": [ "coherent-lemma-induction-principle", "decent-spaces-lemma-filter-quasi-compact-quasi-separated", "spaces-perfect-lemma-make-more-elementary-distinguished-squares" ], "ref_ids": [ 3291, 9480, 2669 ] } ], "ref_ids": [] }, { "id": 2671, "type": "theorem", "label": "spaces-perfect-lemma-induction-principle-separated", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-induction-principle-separated", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let", "$\\mathcal{B} \\subset \\Ob(X_{spaces, \\etale})$.", "Let $P$ be a property of the elements of $\\mathcal{B}$.", "Assume that", "\\begin{enumerate}", "\\item every $W \\in \\mathcal{B}$ is quasi-compact and quasi-separated,", "\\item if $W \\in \\mathcal{B}$ and $U \\subset W$ is quasi-compact open, then", "$U \\in \\mathcal{B}$,", "\\item if $V \\in \\Ob(X_{spaces, \\etale})$ is affine, then", "(a) $V \\in \\mathcal{B}$ and (b) $P$ holds for $V$,", "\\item for every elementary distinguished square $(U \\subset W, f : V \\to W)$", "such that", "\\begin{enumerate}", "\\item $W \\in \\mathcal{B}$,", "\\item $U$ is quasi-compact,", "\\item $V$ is affine, and", "\\item $P$ holds for $U$, $V$, and $U \\times_W V$,", "\\end{enumerate}", "then $P$ holds for $W$.", "\\end{enumerate}", "Then $P$ holds for every $W \\in \\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "This is proved in exactly the same manner as the proof of", "Lemma \\ref{lemma-induction-principle}.", "(We remark that (4)(d) makes sense as $U \\times_W V$ is a quasi-compact", "open of $V$ hence an element of $\\mathcal{B}$ by conditions", "(2) and (3).)" ], "refs": [ "spaces-perfect-lemma-induction-principle" ], "ref_ids": [ 2670 ] } ], "ref_ids": [] }, { "id": 2672, "type": "theorem", "label": "spaces-perfect-lemma-induction-principle-enlarge", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-induction-principle-enlarge", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $W \\subset X$ be a quasi-compact open", "subspace. Let $P$ be a property of quasi-compact open subspaces of $X$.", "Assume that", "\\begin{enumerate}", "\\item $P$ holds for $W$, and", "\\item for every elementary distinguished square", "$(W_1 \\subset W_2, f : V \\to W_2)$ where ", "such that", "\\begin{enumerate}", "\\item $W_1$, $W_2$ are quasi-compact open subspaces of $X$,", "\\item $W \\subset W_1$,", "\\item $V$ is affine, and", "\\item $P$ holds for $W_1$,", "\\end{enumerate}", "then $P$ holds for $W_2$.", "\\end{enumerate}", "Then $P$ holds for $X$." ], "refs": [], "proofs": [ { "contents": [ "We can deduce this from Lemma \\ref{lemma-induction-principle-separated},", "but instead we will give a direct argument by explicitly redoing the proof of", "Lemma \\ref{lemma-induction-principle}. We will use the filtration", "$$", "\\emptyset = U_{n + 1} \\subset", "U_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X", "$$", "and the morphisms $f_p : V_p \\to U_p$ of", "Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated}.", "We will prove that $P$ holds for $W_p = W \\cup U_p$ by descending", "induction on $p$. This will finish the proof as $W_1 = X$.", "Note that $P$ holds for $W_{n + 1} = W \\cap U_{n + 1} = W$", "by (1). Assume $P$ holds for $W_{p + 1}$. Observe that", "$W_p \\setminus W_{p + 1}$ (with reduced induced subspace structure)", "is a closed subspace of $U_p \\setminus U_{p + 1}$.", "Since $(U_{p + 1} \\subset U_p, f_p : V_p \\to U_p)$ is an elementary", "distinguished square, the same is true for", "$(W_{p + 1} \\subset W_p, f_p : V_p \\to W_p)$.", "However (2) may not apply as $V_p$ may not be affine.", "However, as $V_p$ is a quasi-compact scheme we may choose", "a finite affine open covering $V_p = V_{p, 1} \\cup \\ldots \\cup V_{p, m}$.", "Set $W_{p, 0} = W_{p + 1}$ and", "$$", "W_{p, i} = W_{p + 1} \\cup f_p(V_{p, 1} \\cup \\ldots \\cup V_{p, i})", "$$", "for $i = 1, \\ldots, m$. These are quasi-compact open subspaces of $X$", "containing $W$. Then we have", "$$", "W_{p + 1} = W_{p, 0} \\subset", "W_{p, 1} \\subset \\ldots \\subset", "W_{p, m} = W_p", "$$", "and the pairs", "$$", "(W_{p, 0} \\subset W_{p, 1}, f_p|_{V_{p, 1}}),", "(W_{p, 1} \\subset W_{p, 2}, f_p|_{V_{p, 2}}),\\ldots,", "(W_{p, m - 1} \\subset W_{p, m}, f_p|_{V_{p, m}})", "$$", "are elementary distinguished squares by", "Lemma \\ref{lemma-make-more-elementary-distinguished-squares}.", "Now (2) applies to each of these and we inductively", "conclude $P$ holds for $W_{p, 1}, \\ldots, W_{p, m} = W_p$." ], "refs": [ "spaces-perfect-lemma-induction-principle-separated", "spaces-perfect-lemma-induction-principle", "decent-spaces-lemma-filter-quasi-compact-quasi-separated", "spaces-perfect-lemma-make-more-elementary-distinguished-squares" ], "ref_ids": [ 2671, 2670, 9480, 2669 ] } ], "ref_ids": [] }, { "id": 2673, "type": "theorem", "label": "spaces-perfect-lemma-exact-sequence-lower-shriek", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-exact-sequence-lower-shriek", "contents": [ "Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary", "distinguished square of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item For a sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$", "we have a short exact sequence", "$$", "0 \\to j_{U \\times_X V!}\\mathcal{F}|_{U \\times_X V} \\to", "j_{U!}\\mathcal{F}|_U \\oplus j_{V!}\\mathcal{F}|_V \\to \\mathcal{F} \\to 0", "$$", "\\item For an object $E$ of $D(\\mathcal{O}_X)$ we have a distinguished", "triangle", "$$", "j_{U \\times_X V!}E|_{U \\times_X V} \\to", "j_{U!}E|_U \\oplus j_{V!}E|_V \\to E \\to ", "j_{U \\times_X V!}E|_{U \\times_X V}[1]", "$$", "in $D(\\mathcal{O}_X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To show the sequence of (1) is exact we may check on stalks at", "geometric points by", "Properties of Spaces, Theorem", "\\ref{spaces-properties-theorem-exactness-stalks}.", "Let $\\overline{x}$ be a geometric point of $X$. By Equations", "(\\ref{equation-stalk-restriction}) and (\\ref{equation-stalk-j-shriek})", "taking stalks at $\\overline{x}$ we obtain the sequence", "$$", "0 \\to", "\\bigoplus\\nolimits_{(\\overline{u}, \\overline{v})} \\mathcal{F}_{\\overline{x}}", "\\to", "\\bigoplus\\nolimits_{\\overline{u}} \\mathcal{F}_{\\overline{x}}", "\\oplus", "\\bigoplus\\nolimits_{\\overline{v}} \\mathcal{F}_{\\overline{x}}", "\\to", "\\mathcal{F}_{\\overline{x}} \\to 0", "$$", "This sequence is exact because for every $\\overline{x}$", "there either is exactly one $\\overline{u}$ mapping to $\\overline{x}$,", "or there is no $\\overline{u}$ and exactly one $\\overline{v}$", "mapping to $\\overline{x}$.", "\\medskip\\noindent", "Proof of (2). We have seen in Cohomology on Sites, Section", "\\ref{sites-cohomology-section-properties-K-injective}", "that the restriction functors and the extension by zero functors", "on derived categories are computed by just applying the functor", "to any complex. Let $\\mathcal{E}^\\bullet$ be a complex", "of $\\mathcal{O}_X$-modules representing $E$.", "The distinguished triangle of the lemma is the", "distinguished triangle associated (by", "Derived Categories, Section", "\\ref{derived-section-canonical-delta-functor} and especially", "Lemma \\ref{derived-lemma-derived-canonical-delta-functor})", "to the short exact sequence of complexes of $\\mathcal{O}_X$-modules", "$$", "0 \\to j_{U \\times_X V!}\\mathcal{E}^\\bullet|_{U \\times_X V} \\to", "j_{U!}\\mathcal{E}^\\bullet|_U \\oplus j_{V!}\\mathcal{E}^\\bullet|_V", "\\to \\mathcal{E}^\\bullet \\to 0", "$$", "which is short exact by (1)." ], "refs": [ "spaces-properties-theorem-exactness-stalks", "derived-lemma-derived-canonical-delta-functor" ], "ref_ids": [ 11813, 1814 ] } ], "ref_ids": [] }, { "id": 2674, "type": "theorem", "label": "spaces-perfect-lemma-exact-sequence-j-star", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-exact-sequence-j-star", "contents": [ "Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary", "distinguished square of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item For every sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$", "we have a short exact sequence", "$$", "0 \\to \\mathcal{F} \\to", "j_{U, *}\\mathcal{F}|_U \\oplus j_{V, *}\\mathcal{F}|_V \\to", "j_{U \\times_X V, *}\\mathcal{F}|_{U \\times_X V} \\to 0", "$$", "\\item For any object $E$ of $D(\\mathcal{O}_X)$ we have a distinguished", "triangle", "$$", "E \\to ", "Rj_{U, *}E|_U \\oplus Rj_{V, *}E|_V \\to", "Rj_{U \\times_X V, *}E|_{U \\times_X V} \\to", "E[1]", "$$", "in $D(\\mathcal{O}_X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $W$ be an object of $X_\\etale$. We claim the sequence", "$$", "0 \\to", "\\mathcal{F}(W) \\to", "\\mathcal{F}(W \\times_X U) \\oplus \\mathcal{F}(W \\times_X V) \\to", "\\mathcal{F}(W \\times_X U \\times_X V)", "$$", "is exact and that an element of the last group can locally on $W$", "be lifted to the middle one.", "By Lemma \\ref{lemma-make-more-elementary-distinguished-squares}", "the pair $(W \\times_X U \\subset W, V \\times_X W \\to W)$ is an elementary", "distinguished square. Thus we may assume $W = X$ and it suffices", "to prove the same thing for", "$$", "0 \\to", "\\mathcal{F}(X) \\to", "\\mathcal{F}(U) \\oplus \\mathcal{F}(V) \\to", "\\mathcal{F}(U \\times_X V)", "$$", "We have seen that", "$$", "0 \\to j_{U \\times_X V!}\\mathcal{O}_{U \\times_X V}", "\\to j_{U!}\\mathcal{O}_U \\oplus", "j_{V!}\\mathcal{O}_V \\to", "\\mathcal{O}_X \\to 0", "$$", "is a exact sequence of $\\mathcal{O}_X$-modules in", "Lemma \\ref{lemma-exact-sequence-lower-shriek} and applying", "the right exact functor $\\Hom_{\\mathcal{O}_X}(- , \\mathcal{F})$", "gives the sequence above. This also means that the obstruction", "to lifting $s \\in \\mathcal{F}(U \\times_X V)$ to", "an element of $\\mathcal{F}(U) \\oplus \\mathcal{F}(V)$ lies in", "$\\Ext^1_{\\mathcal{O}_X}(\\mathcal{O}_X, \\mathcal{F}) =", "H^1(X, \\mathcal{F})$. By locality of cohomology", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-kill-cohomology-class-on-covering})", "this obstruction vanishes \\'etale locally on $X$ and the proof", "of (1) is complete.", "\\medskip\\noindent", "Proof of (2).", "Choose a K-injective complex $\\mathcal{I}^\\bullet$ representing $E$", "whose terms $\\mathcal{I}^n$ are injective objects of", "$\\textit{Mod}(\\mathcal{O}_X)$, see Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "Then $\\mathcal{I}^\\bullet|U$ is a K-injective complex", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-restrict-K-injective-to-open}).", "Hence $Rj_{U, *}E|_U$ is represented by $j_{U, *}\\mathcal{I}^\\bullet|_U$.", "Similarly for $V$ and $U \\times_X V$. Hence the distinguished triangle", "of the lemma is the distinguished triangle associated (by", "Derived Categories, Section", "\\ref{derived-section-canonical-delta-functor} and especially", "Lemma \\ref{derived-lemma-derived-canonical-delta-functor})", "to the short exact sequence of complexes", "$$", "0 \\to", "\\mathcal{I}^\\bullet \\to", "j_{U, *}\\mathcal{I}^\\bullet|_U \\oplus j_{V, *}\\mathcal{I}^\\bullet|_V \\to", "j_{U \\times_X V, *}\\mathcal{I}^\\bullet|_{U \\times_X V} \\to", "0.", "$$", "This sequence is exact by (1)." ], "refs": [ "spaces-perfect-lemma-make-more-elementary-distinguished-squares", "spaces-perfect-lemma-exact-sequence-lower-shriek", "sites-cohomology-lemma-kill-cohomology-class-on-covering", "injectives-theorem-K-injective-embedding-grothendieck", "sites-cohomology-lemma-restrict-K-injective-to-open", "derived-lemma-derived-canonical-delta-functor" ], "ref_ids": [ 2669, 2673, 4188, 7768, 4253, 1814 ] } ], "ref_ids": [] }, { "id": 2675, "type": "theorem", "label": "spaces-perfect-lemma-unbounded-relative-mayer-vietoris", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-unbounded-relative-mayer-vietoris", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $(U \\subset X, V \\to X)$ be an elementary distinguished square.", "Denote $a = f|_U : U \\to Y$, $b = f|_V : V \\to Y$, and", "$c = f|_{U \\times_X V} : U \\times_X V \\to Y$ the restrictions.", "For every object $E$ of $D(\\mathcal{O}_X)$ there exists a", "distinguished triangle", "$$", "Rf_*E \\to", "Ra_*(E|_U) \\oplus Rb_*(E|_V) \\to", "Rc_*(E|_{U \\times_X V}) \\to", "Rf_*E[1]", "$$", "in $D(\\mathcal{O}_Y)$. This triangle is functorial in $E$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $\\mathcal{I}^\\bullet$", "representing $E$. We may assume $\\mathcal{I}^n$ is an injective", "object of $\\textit{Mod}(\\mathcal{O}_X)$ for all $n$, see", "Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "Then $Rf_*E$ is computed by $f_*\\mathcal{I}^\\bullet$.", "Similarly for $U$, $V$, and $U \\cap V$ by", "Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-restrict-K-injective-to-open}.", "Hence the distinguished triangle of the lemma is the distinguished", "triangle associated (by", "Derived Categories, Section", "\\ref{derived-section-canonical-delta-functor} and especially", "Lemma \\ref{derived-lemma-derived-canonical-delta-functor})", "to the short exact sequence of complexes", "$$", "0 \\to", "f_*\\mathcal{I}^\\bullet \\to", "a_*\\mathcal{I}^\\bullet|_U \\oplus b_*\\mathcal{I}^\\bullet|_V \\to", "c_*\\mathcal{I}^\\bullet|_{U \\times_X V} \\to", "0.", "$$", "To see this is a short exact sequence of complexes we argue as", "follows. Pick an injective object $\\mathcal{I}$ of", "$\\textit{Mod}(\\mathcal{O}_X)$. Apply $f_*$ to the short exact sequence", "$$", "0 \\to \\mathcal{I} \\to", "j_{U, *}\\mathcal{I}|_U \\oplus j_{V, *}\\mathcal{I}|_V \\to", "j_{U \\times_X V, *}\\mathcal{I}|_{U \\times_X V} \\to 0", "$$", "of Lemma \\ref{lemma-exact-sequence-j-star}", "and use that $R^1f_*\\mathcal{I} = 0$ to get a short exact sequence", "$$", "0 \\to f_*\\mathcal{I} \\to", "f_*j_{U, *}\\mathcal{I}|_U \\oplus f_*j_{V, *}\\mathcal{I}|_V \\to", "f_*j_{U \\times_X V, *}\\mathcal{I}|_{U \\times_X V} \\to 0", "$$", "The proof is finished by observing that $a_* = f_*j_{U, *}$ and similarly", "for $b_*$ and $c_*$." ], "refs": [ "injectives-theorem-K-injective-embedding-grothendieck", "sites-cohomology-lemma-restrict-K-injective-to-open", "derived-lemma-derived-canonical-delta-functor", "spaces-perfect-lemma-exact-sequence-j-star" ], "ref_ids": [ 7768, 4253, 1814, 2674 ] } ], "ref_ids": [] }, { "id": 2676, "type": "theorem", "label": "spaces-perfect-lemma-mayer-vietoris-hom", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-mayer-vietoris-hom", "contents": [ "Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary", "distinguished square of algebraic spaces over $S$.", "For objects $E$, $F$ of $D(\\mathcal{O}_X)$ we have a", "Mayer-Vietoris sequence", "$$", "\\xymatrix{", "& \\ldots \\ar[r] &", "\\Ext^{-1}(E_{U \\times_X V}, F_{U \\times_X V}) \\ar[lld] \\\\", "\\Hom(E, F) \\ar[r] &", "\\Hom(E_U, F_U) \\oplus", "\\Hom(E_V, F_V) \\ar[r] &", "\\Hom(E_{U \\times_X V}, F_{U \\times_X V})", "}", "$$", "where the subscripts denote restrictions to the relevant opens", "and the $\\Hom$'s are taken in the relevant derived categories." ], "refs": [], "proofs": [ { "contents": [ "Use the distinguished triangle of", "Lemma \\ref{lemma-exact-sequence-lower-shriek}", "to obtain a long exact sequence of $\\Hom$'s", "(from Derived Categories, Lemma \\ref{derived-lemma-representable-homological})", "and use that $\\Hom(j_{U!}E|_U, F) = \\Hom(E|_U, F|_U)$", "by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-adjoint-lower-shriek-restrict}." ], "refs": [ "spaces-perfect-lemma-exact-sequence-lower-shriek", "derived-lemma-representable-homological", "sites-cohomology-lemma-adjoint-lower-shriek-restrict" ], "ref_ids": [ 2673, 1758, 4260 ] } ], "ref_ids": [] }, { "id": 2677, "type": "theorem", "label": "spaces-perfect-lemma-unbounded-mayer-vietoris", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-unbounded-mayer-vietoris", "contents": [ "Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary", "distinguished square of algebraic spaces over $S$. For an object $E$", "of $D(\\mathcal{O}_X)$ we have a distinguished triangle", "$$", "R\\Gamma(X, E) \\to R\\Gamma(U, E) \\oplus R\\Gamma(V, E) \\to", "R\\Gamma(U \\times_X V, E) \\to R\\Gamma(X, E)[1]", "$$", "and in particular a long exact cohomology sequence", "$$", "\\ldots \\to", "H^n(X, E) \\to", "H^n(U, E) \\oplus H^n(V, E) \\to", "H^n(U \\times_X V, E) \\to", "H^{n + 1}(X, E) \\to \\ldots", "$$", "The construction of the distinguished triangle and the", "long exact sequence is functorial in $E$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $\\mathcal{I}^\\bullet$ representing $E$", "whose terms $\\mathcal{I}^n$ are injective objects of", "$\\textit{Mod}(\\mathcal{O}_X)$, see Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "In the proof of Lemma \\ref{lemma-exact-sequence-j-star}", "we found a short exact sequence", "of complexes", "$$", "0 \\to \\mathcal{I}^\\bullet \\to", "j_{U, *}\\mathcal{I}^\\bullet|_U \\oplus j_{V, *}\\mathcal{I}^\\bullet|_V \\to", "j_{U \\times_X V, *}\\mathcal{I}^\\bullet|_{U \\times_X V} \\to 0", "$$", "Since $H^1(X, \\mathcal{I}^n) = 0$, we see that", "taking global sections gives an exact sequence of complexes", "$$", "0 \\to \\Gamma(X, \\mathcal{I}^\\bullet) \\to", "\\Gamma(U, \\mathcal{I}^\\bullet) \\oplus", "\\Gamma(V, \\mathcal{I}^\\bullet) \\to", "\\Gamma(U \\times_X V, \\mathcal{I}^\\bullet) \\to 0", "$$", "Since these complexes represent", "$R\\Gamma(X, E)$, $R\\Gamma(U, E)$, $R\\Gamma(V, E)$, and", "$R\\Gamma(U \\times_X V, E)$ we ", "get a distinguished triangle by", "Derived Categories, Section", "\\ref{derived-section-canonical-delta-functor} and especially", "Lemma \\ref{derived-lemma-derived-canonical-delta-functor}." ], "refs": [ "injectives-theorem-K-injective-embedding-grothendieck", "spaces-perfect-lemma-exact-sequence-j-star", "derived-lemma-derived-canonical-delta-functor" ], "ref_ids": [ 7768, 2674, 1814 ] } ], "ref_ids": [] }, { "id": 2678, "type": "theorem", "label": "spaces-perfect-lemma-restrict-lower-shriek", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-restrict-lower-shriek", "contents": [ "Let $S$ be a scheme. Let $j : U \\to X$ be a \\'etale morphism of algebraic", "spaces over $S$. Given an \\'etale morphism $V \\to Y$, set $W = V \\times_X U$", "and denote $j_W : W \\to V$ the projection morphism. Then", "$(j_!E)|_V = j_{W!}(E|_W)$ for $E$ in $D(\\mathcal{O}_U)$." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$(j_!\\mathcal{F})|_V = j_{W!}(\\mathcal{F}|_W)$", "for an $\\mathcal{O}_X$-module $\\mathcal{F}$ as follows immediately ", "from the construction of the functors $j_!$ and $j_{W!}$, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-extension-by-zero}." ], "refs": [ "sites-modules-lemma-extension-by-zero" ], "ref_ids": [ 14169 ] } ], "ref_ids": [] }, { "id": 2679, "type": "theorem", "label": "spaces-perfect-lemma-pushforward-with-support-in-open", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-pushforward-with-support-in-open", "contents": [ "Let $S$ be a scheme. Let $(U \\subset X, j : V \\to X)$ be an elementary", "distinguished square of algebraic spaces over $S$. Set", "$T = |X| \\setminus |U|$.", "\\begin{enumerate}", "\\item If $E$ is an object of $D(\\mathcal{O}_X)$ supported on $T$, then", "(a) $E \\to Rj_*(E|_V)$ and (b) $j_!(E|_V) \\to E$ are isomorphisms.", "\\item If $F$ is an object of $D(\\mathcal{O}_V)$ supported on $j^{-1}T$, then", "(a) $F \\to (j_!F)|_V$, (b) $(Rj_*F)|_V \\to F$, and (c)", "$j_!F \\to Rj_*F$ are isomorphisms.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $E$ be an object of $D(\\mathcal{O}_X)$ whose cohomology sheaves are", "supported on $T$. Then we see that $E|_U = 0$ and $E|_{U \\times_X V} = 0$", "as $T$ doesn't meet $U$ and $j^{-1}T$ doesn't meet $U \\times_X V$.", "Thus (1)(a) follows from Lemma \\ref{lemma-exact-sequence-j-star}.", "In exactly the same way (1)(b) follows from", "Lemma \\ref{lemma-exact-sequence-lower-shriek}.", "\\medskip\\noindent", "Let $F$ be an object of $D(\\mathcal{O}_V)$ whose cohomology sheaves", "are supported on $j^{-1}T$. By", "Lemma \\ref{lemma-restrict-direct-image-open} we have", "$(Rj_*F)|_U = Rj_{W, *}(F|_W) = 0$ because $F|_W = 0$ by our assumption.", "Similarly $(j_!F)|_U = j_{W!}(F|_W) = 0$ by", "Lemma \\ref{lemma-restrict-lower-shriek}.", "Thus $j_!F$ and $Rj_*F$ are", "supported on $T$ and $(j_!F)|_V$ and $(Rj_*F)|_V$ are supported on", "$j^{-1}(T)$. To check that the maps (2)(a), (b), (c) are isomorphisms", "in the derived category, it suffices to check that these map induce", "isomorphisms on stalks of cohomology sheaves at geometric points of $T$", "and $j^{-1}(T)$ by", "Properties of Spaces, Theorem", "\\ref{spaces-properties-theorem-exactness-stalks}.", "This we may do after replacing $X$ by $V$, $U$ by $U \\times_X V$,", "$V$ by $V \\times_X V$ and $F$ by $F|_{V \\times_X V}$ (restriction via", "first projection), see", "Lemmas \\ref{lemma-restrict-direct-image-open},", "\\ref{lemma-restrict-lower-shriek}, and", "\\ref{lemma-make-more-elementary-distinguished-squares}.", "Since $V \\times_X V \\to V$ has a section this", "reduces (2) to the case that $j : V \\to X$ has a section.", "\\medskip\\noindent", "Assume $j$ has a section $\\sigma : X \\to V$.", "Set $V' = \\sigma(X)$. This is an open subspace of $V$.", "Set $U' = j^{-1}(U)$. This is another open subspace of $V$.", "Then $(U' \\subset V, V' \\to V)$ is an elementary distinguished", "square. Observe that $F|_{U'} = 0$ and $F|_{V' \\cap U'} = 0$", "because $F$ is supported on $j^{-1}(T)$. Denote $j' : V' \\to V$", "the open immersion and $j_{V'} : V' \\to X$ the composition", "$V' \\to V \\to X$ which is the inverse of $\\sigma$.", "Set $F' = \\sigma^*F$. The distinguished triangles of", "Lemmas \\ref{lemma-exact-sequence-lower-shriek} and", "\\ref{lemma-exact-sequence-j-star} show that", "$F = j'_!(F|_{V'})$ and $F = Rj'_*(F|_{V'})$.", "It follows that $j_!F = j_!j'_!(F|_{V'}) = j_{V'!}F = F'$", "because $j_{V'} : V' \\to X$ is an isomorphism and the inverse", "of $\\sigma$. Similarly, $Rj_*F = Rj_*Rj'_*F = Rj_{V', *}F = F'$.", "This proves (2)(c). To prove (2)(a) and (2)(b) it suffices", "to show that $F = F'|_V$. This is clear because both $F$ and $F'|_V$", "restrict to zero on $U'$ and $U' \\cap V'$ and the same object", "on $V'$." ], "refs": [ "spaces-perfect-lemma-exact-sequence-j-star", "spaces-perfect-lemma-exact-sequence-lower-shriek", "spaces-perfect-lemma-restrict-direct-image-open", "spaces-perfect-lemma-restrict-lower-shriek", "spaces-properties-theorem-exactness-stalks", "spaces-perfect-lemma-restrict-direct-image-open", "spaces-perfect-lemma-restrict-lower-shriek", "spaces-perfect-lemma-make-more-elementary-distinguished-squares", "spaces-perfect-lemma-exact-sequence-lower-shriek", "spaces-perfect-lemma-exact-sequence-j-star" ], "ref_ids": [ 2674, 2673, 2642, 2678, 11813, 2642, 2678, 2669, 2673, 2674 ] } ], "ref_ids": [] }, { "id": 2680, "type": "theorem", "label": "spaces-perfect-lemma-glue", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-glue", "contents": [ "Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary", "distinguished square of algebraic spaces over $S$. Suppose given", "\\begin{enumerate}", "\\item an object $A$ of $D(\\mathcal{O}_U)$,", "\\item an object $B$ of $D(\\mathcal{O}_V)$, and", "\\item an isomorphism $c : A|_{U \\times_X V} \\to B|_{U \\times_X V}$.", "\\end{enumerate}", "Then there exists an object $F$ of $D(\\mathcal{O}_X)$", "and isomorphisms $f : F|_U \\to A$, $g : F|_V \\to B$ such", "that $c = g|_{U \\times_X V} \\circ f^{-1}|_{U \\times_X V}$.", "Moreover, given", "\\begin{enumerate}", "\\item an object $E$ of $D(\\mathcal{O}_X)$,", "\\item a morphism $a : A \\to E|_U$ of $D(\\mathcal{O}_U)$,", "\\item a morphism $b : B \\to E|_V$ of $D(\\mathcal{O}_V)$,", "\\end{enumerate}", "such that", "$$", "a|_{U \\times_X V} = b|_{U \\times_X V} \\circ c.", "$$", "Then there exists a morphism $F \\to E$ in $D(\\mathcal{O}_X)$", "whose restriction to $U$ is $a \\circ f$", "and whose restriction to $V$ is $b \\circ g$." ], "refs": [], "proofs": [ { "contents": [ "Denote $j_U$, $j_V$, $j_{U \\times_X V}$ the corresponding morphisms towards", "$X$. Choose a distinguished triangle", "$$", "F \\to Rj_{U, *}A \\oplus Rj_{V, *}B \\to", "Rj_{U \\times_X V, *}(B|_{U \\times_X V}) \\to F[1]", "$$", "Here the map $Rj_{V, *}B \\to Rj_{U \\times_X V, *}(B|_{U \\times_X V})$", "is the obvious one. The map", "$Rj_{U, *}A \\to Rj_{U \\times_X V, *}(B|_{U \\times_X V})$", "is the composition of", "$Rj_{U, *}A \\to Rj_{U \\times_X V, *}(A|_{U \\times_X V})$", "with $Rj_{U \\times_X V, *}c$. Restricting to $U$ we obtain", "$$", "F|_U \\to A \\oplus (Rj_{V, *}B)|_U \\to", "(Rj_{U \\times_X V, *}(B|_{U \\times_X V}))|_U \\to F|_U[1]", "$$", "Denote $j : U \\times_X V \\to U$. Compatibility of restriction and", "total direct image (Lemma \\ref{lemma-restrict-direct-image-open})", "shows that both $(Rj_{V, *}B)|_U$ and", "$(Rj_{U \\times_X V, *}(B|_{U \\times_X V}))|_U$", "are canonically isomorphic to $Rj_*(B|_{U \\times_X V})$.", "Hence the second arrow of the last displayed equation has", "a section, and we conclude that the morphism $F|_U \\to A$ is", "an isomorphism.", "\\medskip\\noindent", "To see that the morphism $F|_V \\to B$ is an isomorphism we will use a trick.", "Namely, choose a distinguished triangle", "$$", "F|_V \\to B \\to B' \\to F[1]|_V", "$$", "in $D(\\mathcal{O}_V)$. Since $F|_U \\to A$ is an isomorphism, and since", "we have the isomorphism $c : A|_{U \\times_X V} \\to B|_{U \\times_X V}$", "the restriction of $F|_V \\to B$ is an isomorphism over $U \\times_X V$.", "Thus $B'$ is supported on $j_V^{-1}(T)$ where $T = |X| \\setminus |U|$.", "On the other hand, there is a morphism of distinguished triangles", "$$", "\\xymatrix{", "F \\ar[r] \\ar[d] &", "Rj_{U, *}F|_U \\oplus Rj_{V, *}F|_V \\ar[r] \\ar[d] &", "Rj_{U \\times_X V, *}F|_{U \\times_X V} \\ar[r] \\ar[d] &", "F[1] \\ar[d] \\\\", "F \\ar[r] &", "Rj_{U, *}A \\oplus Rj_{V, *}B \\ar[r] &", "Rj_{U \\times_X V, *}(B|_{U \\times_X V}) \\ar[r] &", "F[1]", "}", "$$", "The all of the vertical maps in this diagram are isomorphisms, except", "for the map $Rj_{V, *}F|_V \\to Rj_{V, *}B$, hence that is an isomorphism too", "(Derived Categories, Lemma \\ref{derived-lemma-third-isomorphism-triangle}).", "This implies that $Rj_{V, *}B' = 0$. Hence $B' = 0$ by", "Lemma \\ref{lemma-pushforward-with-support-in-open}.", "\\medskip\\noindent", "The existence of the morphism $F \\to E$ follows", "from the Mayer-Vietoris sequence for $\\Hom$, see", "Lemma \\ref{lemma-mayer-vietoris-hom}." ], "refs": [ "spaces-perfect-lemma-restrict-direct-image-open", "derived-lemma-third-isomorphism-triangle", "spaces-perfect-lemma-pushforward-with-support-in-open", "spaces-perfect-lemma-mayer-vietoris-hom" ], "ref_ids": [ 2642, 1759, 2679, 2676 ] } ], "ref_ids": [] }, { "id": 2681, "type": "theorem", "label": "spaces-perfect-lemma-affine-pushforward", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-affine-pushforward", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of", "algebraic spaces over $S$. Then $f_*$ defines a derived functor", "$f_* : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$.", "This functor has the property that", "$$", "\\xymatrix{", "D(\\QCoh(\\mathcal{O}_X)) \\ar[d]_{f_*} \\ar[r] &", "D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\", "D(\\QCoh(\\mathcal{O}_Y)) \\ar[r] &", "D_\\QCoh(\\mathcal{O}_Y)", "}", "$$", "commutes." ], "refs": [], "proofs": [ { "contents": [ "The functor", "$f_* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$", "is exact, see", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-affine-vanishing-higher-direct-images}.", "Hence $f_*$ defines a derived functor", "$f_* : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$", "by simply applying $f_*$ to any representative complex, see", "Derived Categories, Lemma \\ref{derived-lemma-right-derived-exact-functor}.", "For any complex of $\\mathcal{O}_X$-modules", "$\\mathcal{F}^\\bullet$ there is a canonical map", "$f_*\\mathcal{F}^\\bullet \\to Rf_*\\mathcal{F}^\\bullet$.", "To finish the proof we show this is a quasi-isomorphism when", "$\\mathcal{F}^\\bullet$ is a complex with each $\\mathcal{F}^n$", "quasi-coherent. The statement is \\'etale local on $Y$ hence we", "may assume $Y$ affine. As an affine morphism is representable", "we reduce to the case of schemes by the compatibility of", "Remark \\ref{remark-match-total-direct-images}. The case of schemes is", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-affine-pushforward}." ], "refs": [ "spaces-cohomology-lemma-affine-vanishing-higher-direct-images", "derived-lemma-right-derived-exact-functor", "spaces-perfect-remark-match-total-direct-images", "perfect-lemma-affine-pushforward" ], "ref_ids": [ 11288, 1845, 2768, 6963 ] } ], "ref_ids": [] }, { "id": 2682, "type": "theorem", "label": "spaces-perfect-lemma-flat-pushforward-coherator", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-flat-pushforward-coherator", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. Assume $f$ is quasi-compact, quasi-separated, and flat.", "Then, denoting", "$$", "\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))", "$$", "the right derived functor of", "$f_* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$", "we have $RQ_Y \\circ Rf_* = \\Phi \\circ RQ_X$." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by showing that $RQ_Y \\circ Rf_*$ and $\\Phi \\circ RQ_X$", "are right adjoint to the same functor", "$D(\\QCoh(\\mathcal{O}_Y)) \\to D(\\mathcal{O}_X)$.", "\\medskip\\noindent", "Since $f$ is quasi-compact and quasi-separated, we see that", "$f_*$ preserves quasi-coherence, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-pushforward}.", "Recall that $\\QCoh(\\mathcal{O}_X)$ is a Grothendieck abelian category", "(Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-coherator}).", "Hence any $K$ in $D(\\QCoh(\\mathcal{O}_X))$", "can be represented by a K-injective complex $\\mathcal{I}^\\bullet$", "of $\\QCoh(\\mathcal{O}_X)$, see", "Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "Then we can define $\\Phi(K) = f_*\\mathcal{I}^\\bullet$.", "\\medskip\\noindent", "Since $f$ is flat, the functor $f^*$ is exact. Hence $f^*$ defines", "$f^* : D(\\mathcal{O}_Y) \\to D(\\mathcal{O}_X)$ and also", "$f^* : D(\\QCoh(\\mathcal{O}_Y)) \\to D(\\QCoh(\\mathcal{O}_X))$.", "The functor $f^* = Lf^* : D(\\mathcal{O}_Y) \\to D(\\mathcal{O}_X)$", "is left adjoint to", "$Rf_* : D(\\mathcal{O}_X) \\to D(\\mathcal{O}_Y)$,", "see Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-adjoint}.", "Similarly, the functor", "$f^* : D(\\QCoh(\\mathcal{O}_Y)) \\to D(\\QCoh(\\mathcal{O}_X))$", "is left adjoint to", "$\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$", "by Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors}.", "\\medskip\\noindent", "Let $A$ be an object of $D(\\QCoh(\\mathcal{O}_Y))$ and", "$E$ an object of $D(\\mathcal{O}_X)$. Then", "\\begin{align*}", "\\Hom_{D(\\QCoh(\\mathcal{O}_Y))}(A, RQ_Y(Rf_*E))", "& =", "\\Hom_{D(\\mathcal{O}_Y)}(A, Rf_*E) \\\\", "& =", "\\Hom_{D(\\mathcal{O}_X)}(f^*A, E) \\\\", "& =", "\\Hom_{D(\\QCoh(\\mathcal{O}_X))}(f^*A, RQ_X(E)) \\\\", "& =", "\\Hom_{D(\\QCoh(\\mathcal{O}_Y))}(A, \\Phi(RQ_X(E)))", "\\end{align*}", "This implies what we want." ], "refs": [ "spaces-morphisms-lemma-pushforward", "spaces-properties-proposition-coherator", "injectives-theorem-K-injective-embedding-grothendieck", "sites-cohomology-lemma-adjoint", "derived-lemma-derived-adjoint-functors" ], "ref_ids": [ 4760, 11921, 7768, 4249, 1907 ] } ], "ref_ids": [] }, { "id": 2683, "type": "theorem", "label": "spaces-perfect-lemma-affine-coherator", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-affine-coherator", "contents": [ "Let $S$ be a scheme. Let $X$ be an affine algebraic space over $S$.", "Set $A = \\Gamma(X, \\mathcal{O}_X)$. Then", "\\begin{enumerate}", "\\item $Q_X : \\textit{Mod}(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_X)$", "is the functor", "which sends $\\mathcal{F}$ to the quasi-coherent $\\mathcal{O}_X$-module", "associated to the $A$-module $\\Gamma(X, \\mathcal{F})$,", "\\item $RQ_X : D(\\mathcal{O}_X) \\to D(\\QCoh(\\mathcal{O}_X))$", "is the functor which sends $E$ to the complex of quasi-coherent", "$\\mathcal{O}_X$-modules associated to the object $R\\Gamma(X, E)$ of $D(A)$,", "\\item restricted to $D_\\QCoh(\\mathcal{O}_X)$ the functor", "$RQ_X$ defines a quasi-inverse to (\\ref{equation-compare}).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X_0 = \\Spec(A)$ be the affine scheme representing $X$.", "Recall that there is a morphism of ringed sites", "$\\epsilon : X_\\etale \\to X_{0, Zar}$", "which induces equivalences", "$$", "\\xymatrix{", "\\QCoh(\\mathcal{O}_X) \\ar@<1ex>[r]^{{\\epsilon_*}} &", "\\QCoh(\\mathcal{O}_{X_0}) \\ar@<1ex>[l]^{{\\epsilon^*}}", "}", "$$", "see Lemma", "\\ref{lemma-derived-quasi-coherent-small-etale-site}.", "Hence we see that $Q_X = \\epsilon^* \\circ Q_{X_0} \\circ \\epsilon_*$", "by uniqueness of adjoint functors. Hence (1) follows from", "the description of $Q_{X_0}$ in", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-affine-coherator}", "and the fact that", "$\\Gamma(X_0, \\epsilon_*\\mathcal{F}) = \\Gamma(X, \\mathcal{F})$.", "Part (2) follows from (1) and the fact that the functor", "from $A$-modules to quasi-coherent $\\mathcal{O}_X$-modules is exact.", "The third assertion now follows from the result for schemes", "(Derived Categories of Schemes, Lemma \\ref{perfect-lemma-affine-coherator})", "and Lemma", "\\ref{lemma-derived-quasi-coherent-small-etale-site}." ], "refs": [ "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "perfect-lemma-affine-coherator", "perfect-lemma-affine-coherator", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site" ], "ref_ids": [ 2644, 6965, 6965, 2644 ] } ], "ref_ids": [] }, { "id": 2684, "type": "theorem", "label": "spaces-perfect-lemma-argument-proves", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-argument-proves", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Suppose that for every \\'etale morphism", "$j : V \\to W$ with $W \\subset X$ quasi-compact open and $V$ affine", "the right derived functor", "$$", "\\Phi : D(\\QCoh(\\mathcal{O}_U)) \\to D(\\QCoh(\\mathcal{O}_W))", "$$", "of the left exact functor", "$j_* : \\QCoh(\\mathcal{O}_V) \\to \\QCoh(\\mathcal{O}_W)$", "fits into a commutative diagram", "$$", "\\xymatrix{", "D(\\QCoh(\\mathcal{O}_V)) \\ar[d]_\\Phi \\ar[r]_{i_V} &", "D_\\QCoh(\\mathcal{O}_V) \\ar[d]^{Rj_*} \\\\", "D(\\QCoh(\\mathcal{O}_W)) \\ar[r]^{i_W} &", "D_\\QCoh(\\mathcal{O}_W)", "}", "$$", "Then the functor (\\ref{equation-compare})", "$$", "D(\\QCoh(\\mathcal{O}_X))", "\\longrightarrow", "D_\\QCoh(\\mathcal{O}_X)", "$$", "is an equivalence with quasi-inverse given by $RQ_X$." ], "refs": [], "proofs": [ { "contents": [ "We first use the induction principle to prove $i_X$ is fully faithful.", "More precisely, we will use Lemma \\ref{lemma-induction-principle-enlarge}.", "Let $(U \\subset W, V \\to W)$ be an elementary distinguished square", "with $V$ affine and $U, W$ quasi-compact open in $X$. Assume that", "$i_U$ is fully faithful. We have to show that $i_W$ is fully faithful.", "We may replace $X$ by $W$, i.e., we may assume $W = X$ (we do this just", "to simplify the notation -- observe that the condition in the", "statement of the lemma is preserved under this operation).", "\\medskip\\noindent", "Suppose that $A, B$ are objects of $D(\\QCoh(\\mathcal{O}_X))$.", "We want to show that", "$$", "\\Hom_{D(\\QCoh(\\mathcal{O}_X))}(A, B)", "\\longrightarrow", "\\Hom_{D(\\mathcal{O}_X)}(i_X(A), i_X(B))", "$$", "is bijective. Let $T = |X| \\setminus |U|$.", "\\medskip\\noindent", "Assume first $i_X(B)$ is supported on $T$. In this case the map", "$$", "i_X(B) \\to Rj_{V, *}(i_X(B)|_V) = Rj_{V, *}(i_V(B|_V))", "$$", "is a quasi-isomorphism", "(Lemma \\ref{lemma-pushforward-with-support-in-open}).", "By assumption we have an isomorphism", "$i_X(\\Phi(B|_V)) \\to Rj_{V, *}(i_V(B|_V))$ in $D(\\mathcal{O}_X)$.", "Moreover, $\\Phi$ and ${-}|_V$ are adjoint functors on the derived categories of", "quasi-coherent modules (by", "Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors}).", "The adjunction map $B \\to \\Phi(B|_V)$ becomes an isomorphism", "after applying $i_X$, whence is an isomorphism in", "$D(\\QCoh(\\mathcal{O}_X))$.", "Hence", "\\begin{align*}", "\\Mor_{D(\\QCoh(\\mathcal{O}_X))}(A, B)", "& =", "\\Mor_{D(\\QCoh(\\mathcal{O}_X))}(A, \\Phi(B|_V)) \\\\", "& =", "\\Mor_{D(\\QCoh(\\mathcal{O}_V))}(A|_V, B|_V) \\\\", "& =", "\\Mor_{D(\\mathcal{O}_V)}(i_V(A|_V), i_V(B|_V)) \\\\", "& =", "\\Mor_{D(\\mathcal{O}_X)}(i_X(A), Rj_{V, *}(i_V(B|_V))) \\\\", "& =", "\\Mor_{D(\\mathcal{O}_X)}(i_X(A), i_X(B))", "\\end{align*}", "as desired. Here we have used that $i_V$ is fully faithful", "(Lemma \\ref{lemma-affine-coherator}).", "\\medskip\\noindent", "In general, choose any complex $\\mathcal{B}^\\bullet$ of quasi-coherent", "$\\mathcal{O}_X$-modules representing $B$. Next, choose any quasi-isomorphism", "$s : \\mathcal{B}^\\bullet|_U \\to \\mathcal{C}^\\bullet$ of complexes of", "quasi-coherent modules on $U$. As $j_U : U \\to X$ is", "quasi-compact and quasi-separated the functor $j_{U, *}$ transforms", "quasi-coherent modules into quasi-coherent modules", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-pushforward}).", "Thus there is a canonical map", "$\\mathcal{B}^\\bullet \\to j_{U, *}(\\mathcal{B}^\\bullet|_U) \\to", "j_{U, *}\\mathcal{C}^\\bullet$", "of complexes of quasi-coherent modules on $X$.", "Set $B'' = j_{U, *}\\mathcal{C}^\\bullet$ in $D(\\QCoh(\\mathcal{O}_X))$", "and choose a distinguished triangle", "$$", "B \\to B'' \\to B' \\to B[1]", "$$", "in $D(\\QCoh(\\mathcal{O}_X))$. Since the first arrow of the triangle", "restricts to an isomorphism over $U$ we see that $B'$ is supported on $T$.", "Hence in the diagram", "$$", "\\xymatrix{", "\\Hom_{D(\\QCoh(\\mathcal{O}_X))}(A, B'[-1]) \\ar[r] \\ar[d] &", "\\Hom_{D(\\mathcal{O}_X)}(i_X(A), i_X(B')[-1]) \\ar[d] \\\\", "\\Hom_{D(\\QCoh(\\mathcal{O}_X))}(A, B) \\ar[r] \\ar[d] &", "\\Hom_{D(\\mathcal{O}_X)}(i_X(A), i_X(B)) \\ar[d] \\\\", "\\Hom_{D(\\QCoh(\\mathcal{O}_X))}(A, B'') \\ar[r] \\ar[d] &", "\\Hom_{D(\\mathcal{O}_X)}(i_X(A), i_X(B'')) \\ar[d] \\\\", "\\Hom_{D(\\QCoh(\\mathcal{O}_X))}(A, B') \\ar[r] &", "\\Hom_{D(\\mathcal{O}_X)}(i_X(A), i_X(B'))", "}", "$$", "we have exact columns and the top and bottom horizontal arrows are", "bijective. Finally, choose a complex $\\mathcal{A}^\\bullet$", "of quasi-coherent modules representing $A$.", "\\medskip\\noindent", "Let $\\alpha : i_X(A) \\to i_X(B)$ be a morphism between", "in $D(\\mathcal{O}_X)$. The restriction $\\alpha|_U$ comes from a", "morphism in $D(\\QCoh(\\mathcal{O}_U))$ as $i_U$ is fully faithful.", "Hence there exists a choice of", "$s : \\mathcal{B}^\\bullet|_U \\to \\mathcal{C}^\\bullet$ as above", "such that $\\alpha|_U$ is represented by an actual map of complexes", "$\\mathcal{A}^\\bullet|_U \\to \\mathcal{C}^\\bullet$.", "This corresponds to a map of complexes", "$\\mathcal{A} \\to j_{U, *}\\mathcal{C}^\\bullet$.", "In other words, the image of $\\alpha$ in", "$\\Hom_{D(\\mathcal{O}_X)}(i_X(A), i_X(B''))$ comes from", "an element of $\\Hom_{D(\\QCoh(\\mathcal{O}_X))}(A, B'')$.", "A diagram chase then shows that $\\alpha$ comes from a morphism", "$A \\to B$ in $D(\\QCoh(\\mathcal{O}_X))$. Finally, suppose", "that $a : A \\to B$ is a morphism of $D(\\QCoh(\\mathcal{O}_X))$", "which becomes zero in $D(\\mathcal{O}_X)$. After choosing $\\mathcal{B}^\\bullet$", "suitably, we may assume $a$ is represented by a morphism of complexes", "$a^\\bullet : \\mathcal{A}^\\bullet \\to \\mathcal{B}^\\bullet$.", "Since $i_U$ is fully faithul the restriction $a^\\bullet|_U$ is zero", "in $D(\\QCoh(\\mathcal{O}_U))$. Thus we can choose $s$", "such that", "$s \\circ a^\\bullet|_U : \\mathcal{A}^\\bullet|_U \\to \\mathcal{C}^\\bullet$", "is homotopic to zero. Applying the functor $j_{U, *}$ we conclude that", "$\\mathcal{A}^\\bullet \\to j_{U, *}\\mathcal{C}^\\bullet$ is homotopic", "to zero. Thus $a$ maps to zero in", "$\\Hom_{D(\\QCoh(\\mathcal{O}_X))}(A, B'')$.", "Thus we may assume that $a$ is the image of an element", "of $b \\in \\Hom_{D(\\QCoh(\\mathcal{O}_X))}(A, B'[-1])$.", "The image of $b$ in $\\Hom_{D(\\mathcal{O}_X)}(i_X(A), i_X(B')[-1])$", "comes from a $\\gamma \\in \\Hom_{D(\\mathcal{O}_X)}(A, B''[-1])$", "(as $a$ maps to zero in the group on the right). Since we've", "seen above the horizontal arrows are surjective, we see", "that $\\gamma$ comes from a $c$ in", "$\\Hom_{D(\\QCoh(\\mathcal{O}_X))}(A, B''[-1])$", "which implies $a = 0$ as desired.", "\\medskip\\noindent", "At this point we know that $i_X$ is fully faithful for our original $X$.", "Since $RQ_X$ is its right adjoint, we see that", "$RQ_X \\circ i_X = \\text{id}$ (Categories, Lemma", "\\ref{categories-lemma-adjoint-fully-faithful}).", "To finish the proof we show that for any", "$E$ in $D_\\QCoh(\\mathcal{O}_X)$ the map", "$i_X(RQ_X(E)) \\to E$ is an isomorphism. Choose a distinguished triangle", "$$", "i_X(RQ_X(E)) \\to E \\to E' \\to i_X(RQ_X(E))[1]", "$$", "in $D_\\QCoh(\\mathcal{O}_X)$. A formal argument using the", "above shows that $i_X(RQ_X(E')) = 0$. Thus it suffices to prove that", "for $E \\in D_\\QCoh(\\mathcal{O}_X)$ the condition", "$i_X(RQ_X(E)) = 0$ implies that $E = 0$. Consider an \\'etale morphism", "$j : V \\to X$ with $V$ affine. By", "Lemmas \\ref{lemma-affine-coherator} and", "\\ref{lemma-flat-pushforward-coherator}", "and our assumption we have", "$$", "Rj_*(E|_V) = Rj_*(i_V(RQ_V(E|_V))) = i_X(\\Phi(RQ_V(E|_V))) =", "i_X(RQ_X(Rj_*(E|_V)))", "$$", "Choose a distinguished triangle", "$$", "E \\to Rj_*(E|_V) \\to E' \\to E[1]", "$$", "Apply $RQ_X$ to get a distinguished triangle", "$$", "0 \\to RQ_X(Rj_*(E|_V)) \\to RQ_X(E') \\to 0[1]", "$$", "in other words the map in the middle is an isomorphism.", "Combined with the string of equalities above we find", "that our first distinguished triangle becomes a distinguished triangle", "$$", "E \\to i_X(RQ_X(E')) \\to E' \\to E[1]", "$$", "where the middle morphism is the adjunction map. However, the composition", "$E \\to E'$ is zero, hence $E \\to i_X(RQ_X(E'))$ is zero by adjunction!", "Since this morphism is isomorphic to the morphism", "$E \\to Rj_*(E|_V)$ adjoint to $\\text{id} : E|_V \\to E|_V$ we", "conclude that $E|_V$ is zero. Since this holds for all", "affine $V$ \\'etale over $X$ we conclude $E$ is zero as desired." ], "refs": [ "spaces-perfect-lemma-induction-principle-enlarge", "spaces-perfect-lemma-pushforward-with-support-in-open", "derived-lemma-derived-adjoint-functors", "spaces-perfect-lemma-affine-coherator", "spaces-morphisms-lemma-pushforward", "categories-lemma-adjoint-fully-faithful", "spaces-perfect-lemma-affine-coherator", "spaces-perfect-lemma-flat-pushforward-coherator" ], "ref_ids": [ 2672, 2679, 1907, 2683, 4760, 12248, 2683, 2682 ] } ], "ref_ids": [] }, { "id": 2685, "type": "theorem", "label": "spaces-perfect-lemma-direct-image-coherator", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-direct-image-coherator", "contents": [ "Let $S$ be a scheme and let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. Assume $X$ and $Y$ are quasi-compact and have affine diagonal.", "Then, denoting", "$$", "\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))", "$$", "the right derived functor of", "$f_* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$", "the diagram", "$$", "\\xymatrix{", "D(\\QCoh(\\mathcal{O}_X)) \\ar[d]_\\Phi \\ar[r] &", "D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\", "D(\\QCoh(\\mathcal{O}_Y)) \\ar[r] &", "D_\\QCoh(\\mathcal{O}_Y)", "}", "$$", "is commutative." ], "refs": [], "proofs": [ { "contents": [ "Observe that the horizontal arrows in the diagram are", "equivalences of categories by", "Proposition \\ref{proposition-quasi-compact-affine-diagonal}.", "Hence we can identify these categories (and similarly for", "other quasi-compact algebraic spaces with affine diagonal)", "and then the statement of the lemma is that the canonical map", "$\\Phi(K) \\to Rf_*(K)$ is an isomorphism for all $K$ in", "$D(\\QCoh(\\mathcal{O}_X))$. Note that if $K_1 \\to K_2 \\to K_3 \\to K_1[1]$", "is a distinguished triangle in $D(\\QCoh(\\mathcal{O}_X))$ and", "the statement is true for two-out-of-three, then it is true", "for the third.", "\\medskip\\noindent", "Let $\\mathcal{B} \\subset \\Ob(X_{spaces, \\etale})$ be the set of", "objects which are quasi-compact and have affine diagonal.", "For $U \\in \\mathcal{B}$ and any morphism $g : U \\to Z$", "where $Z$ is a quasi-compact algebraic space over $S$ with", "affine diagonal, denote", "$$", "\\Phi_g : D(\\QCoh(\\mathcal{O}_U)) \\to D(\\QCoh(\\mathcal{O}_Z))", "$$", "the derived extension of $g_*$. Let", "$P(U) =$ ``for any $K$ in $D(\\QCoh(\\mathcal{O}_U))$", "and any $g : U \\to Z$ as above the map $\\Phi_g(K) \\to Rg_*K$", "is an isomorphism''.", "By Remark \\ref{remark-how-to} conditions (1), (2), and (3)(a) of", "Lemma \\ref{lemma-induction-principle-separated} hold and we are", "left with proving (3)(b) and (4).", "\\medskip\\noindent", "Checking condition (3)(b). Let $U$ be an affine scheme \\'etale", "over $X$. Let $g : U \\to Z$ be as above. Since the diagonal of $Z$", "is affine the morphism $g : U \\to Z$", "is affine (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-affine-permanence}).", "Hence $P(U)$ holds by Lemma \\ref{lemma-affine-pushforward}.", "\\medskip\\noindent", "Checking condition (4).", "Let $(U \\subset W, V \\to W)$ be an elementary distinguished square", "in $X_{spaces, \\etale}$ with $U, W, V$ in $\\mathcal{B}$ and $V$ affine.", "Assume that $P$ holds for $U$, $V$, and $U \\times_W V$.", "We have to show that $P$ holds for $W$. Let $g : W \\to Z$", "be a morphism to a quasi-compact algebraic space with affine diagonal.", "Let $K$ be an object of $D(\\QCoh(\\mathcal{O}_W))$.", "Consider the distinguished triangle", "$$", "K \\to Rj_{U, *}K|_U \\oplus Rj_{V, *}K|_V \\to", "Rj_{U \\times_W V, *}K|_{U \\times_W V} \\to K[1]", "$$", "in $D(\\mathcal{O}_W)$. By the two-out-of-three property mentioned", "above, it suffices to show that $\\Phi_g(Rj_{U, *}K|_U) \\to Rg_*(Rj_{U, *}K|_U)$", "is an isomorphism and similarly for $V$ and $U \\times_W V$.", "This is discussed in the next paragraph.", "\\medskip\\noindent", "Let $j : U \\to W$ be a morphism $X_{spaces, \\etale}$ with", "$U, W$ in $\\mathcal{B}$ and $P$ holds for $U$. Let $g : W \\to Z$", "be a morphism to a quasi-compact algebraic space with affine diagonal.", "To finish the proof we have to show that", "$\\Phi_g(Rj_*K) \\to Rg_*(Rj_*K)$", "is an isomorphism for any $K$ in $D(\\QCoh(\\mathcal{O}_U))$.", "Let $\\mathcal{I}^\\bullet$ be a K-injective complex in $\\QCoh(\\mathcal{O}_U)$", "representing $K$.", "From $P(U)$ applied to $j$ we see that", "$j_*\\mathcal{I}^\\bullet$ represents $Rj_*K$.", "Since $j_* : \\QCoh(\\mathcal{O}_U) \\to \\QCoh(\\mathcal{O}_X)$", "has an exact left adjoint", "$j^* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_U)$", "we see that $j_*\\mathcal{I}^\\bullet$ is a K-injective complex", "in $\\QCoh(\\mathcal{O}_W)$, see", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}.", "Hence $\\Phi_g(Rj_*K)$ is represented by", "$g_*j_*\\mathcal{I}^\\bullet = (g \\circ j)_*\\mathcal{I}^\\bullet$.", "By $P(U)$ applied to $g \\circ j$ we see that this represents", "$R_{g \\circ j, *}(K) = Rg_*(Rj_*K)$. This finishes the proof." ], "refs": [ "spaces-perfect-proposition-quasi-compact-affine-diagonal", "spaces-perfect-remark-how-to", "spaces-perfect-lemma-induction-principle-separated", "spaces-morphisms-lemma-affine-permanence", "spaces-perfect-lemma-affine-pushforward", "derived-lemma-adjoint-preserve-K-injectives" ], "ref_ids": [ 2756, 2769, 2671, 4804, 2681, 1915 ] } ], "ref_ids": [] }, { "id": 2686, "type": "theorem", "label": "spaces-perfect-lemma-affine-injective-colimit-direct-sum-pushforwards-artin", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-affine-injective-colimit-direct-sum-pushforwards-artin", "contents": [ "Let $S$ be a Noetherian affine scheme. Every injective object of", "$\\QCoh(\\mathcal{O}_S)$ is a filtered colimit $\\colim_i \\mathcal{F}_i$", "of quasi-coherent sheaves of the form", "$$", "\\mathcal{F}_i = (Z_i \\to S)_*\\mathcal{G}_i", "$$", "where $Z_i$ is the spectrum of an Artinian ring and $\\mathcal{G}_i$", "is a coherent module on $Z_i$." ], "refs": [], "proofs": [ { "contents": [ "Let $S = \\Spec(A)$. Let $\\mathcal{J}$ be an injective object of", "$\\QCoh(\\mathcal{O}_S)$. Since $\\QCoh(\\mathcal{O}_S)$ is", "equivalent to the category of $A$-modules we see that $\\mathcal{J}$", "is equal to $\\widetilde{J}$ for some injective $A$-module $J$.", "By Dualizing Complexes, Proposition", "\\ref{dualizing-proposition-structure-injectives-noetherian}", "we can write $J = \\bigoplus E_\\alpha$ with $E_\\alpha$ indecomposable", "and therefore isomorphic to the injective hull of a reside field", "at a point. Thus (because finite disjoint unions of Artinian schemes", "are Artinian) we may assume that $J$ is the injective hull", "of $\\kappa(\\mathfrak p)$ for some prime $\\mathfrak p$ of $A$.", "Then $J = \\bigcup J[\\mathfrak p^n]$ where $J[\\mathfrak p^n]$ is", "the injective hull of $\\kappa(\\mathfrak p)$ over", "$A_\\mathfrak/\\mathfrak p^nA_\\mathfrak p$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-union-artinian}.", "Thus $\\widetilde{J}$ is the colimit of the sheaves", "$(Z_n \\to X)_*\\mathcal{G}_n$ where", "$Z_n = \\Spec(A_\\mathfrak p/\\mathfrak p^nA_\\mathfrak p)$ and", "$\\mathfrak G_n$ the coherent sheaf associated to the", "finite $A_\\mathfrak/\\mathfrak p^nA_\\mathfrak p$-module $J[\\mathfrak p^n]$.", "Finiteness follows from", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-finite}." ], "refs": [ "dualizing-proposition-structure-injectives-noetherian", "dualizing-lemma-union-artinian", "dualizing-lemma-finite" ], "ref_ids": [ 2923, 2806, 2800 ] } ], "ref_ids": [] }, { "id": 2687, "type": "theorem", "label": "spaces-perfect-lemma-injective-colimit-direct-sum-pushforwards-artin", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-injective-colimit-direct-sum-pushforwards-artin", "contents": [ "Let $S$ be an affine scheme. Let $X$ be a Noetherian algebraic space", "over $S$. Every injective object of $\\QCoh(\\mathcal{O}_X)$ is", "a direct summand of a filtered colimit $\\colim_i \\mathcal{F}_i$", "of quasi-coherent sheaves of the form", "$$", "\\mathcal{F}_i = (Z_i \\to X)_*\\mathcal{G}_i", "$$", "where $Z_i$ is the spectrum of an Artinian ring and $\\mathcal{G}_i$", "is a coherent module on $Z_i$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $U$ and a surjective \\'etale morphism", "$j : U \\to X$ (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "Then $U$ is a Noetherian affine scheme. Choose an injective object", "$\\mathcal{J}'$ of $\\QCoh(\\mathcal{O}_U)$ such that there", "exists an injection $\\mathcal{J}|_U \\to \\mathcal{J}'$. Then", "$$", "\\mathcal{J} \\to j_*\\mathcal{J}'", "$$", "is an injective morphism in $\\QCoh(\\mathcal{O}_X)$,", "hence identifies $\\mathcal{J}$ as a direct summand of $j_*\\mathcal{J}'$.", "Thus the result follows from the corresponding result for", "$\\mathcal{J}'$ proved in", "Lemma \\ref{lemma-affine-injective-colimit-direct-sum-pushforwards-artin}." ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-perfect-lemma-affine-injective-colimit-direct-sum-pushforwards-artin" ], "ref_ids": [ 11832, 2686 ] } ], "ref_ids": [] }, { "id": 2688, "type": "theorem", "label": "spaces-perfect-lemma-flat-pullback-injective-quasi-coherent", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-flat-pullback-injective-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a flat, quasi-compact, and", "quasi-separated morphism of algebraic spaces over $S$. If", "$\\mathcal{J}$ is an injective object of $\\QCoh(\\mathcal{O}_X)$,", "then $f_*\\mathcal{J}$ is an injective object of", "$\\QCoh(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is quasi-compact and quasi-separated, the functor", "$f_*$ transforms quasi-coherent sheaves into quasi-coherent sheaves", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-pushforward}).", "The functor $f^*$ is a left adjoint to $f_*$ which", "transforms injections into injections.", "Hence the result follows from", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}" ], "refs": [ "spaces-morphisms-lemma-pushforward", "homology-lemma-adjoint-preserve-injectives" ], "ref_ids": [ 4760, 12116 ] } ], "ref_ids": [] }, { "id": 2689, "type": "theorem", "label": "spaces-perfect-lemma-injective-pushforward", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-injective-pushforward", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. If", "$\\mathcal{J}$ is an injective object of $\\QCoh(\\mathcal{O}_X)$,", "then", "\\begin{enumerate}", "\\item $H^p(U, \\mathcal{J}|_U) = 0$ for $p > 0$ and for", "every quasi-compact and quasi-separated algebraic space $U$ \\'etale over $X$,", "\\item for any morphism $f : X \\to Y$ of algebraic spaces over $S$", "with $Y$ quasi-separated we have $R^pf_*\\mathcal{J} = 0$ for $p > 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Write $\\mathcal{J}$ as a direct summand of", "$\\colim \\mathcal{F}_i$ with $\\mathcal{F}_i = (Z_i \\to X)_*\\mathcal{G}_i$", "as in Lemma \\ref{lemma-injective-colimit-direct-sum-pushforwards-artin}.", "It is clear that it suffices to prove the vanishing for", "$\\colim \\mathcal{F}_i$. Since pullback commutes with colimits", "and since $U$ is quasi-compact and quasi-separated, it suffices", "to prove $H^p(U, \\mathcal{F}_i|_U) = 0$ for $p > 0$, see", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-colimits}.", "Observe that $Z_i \\to X$ is an affine morphism, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-Artinian-affine}.", "Thus", "$$", "\\mathcal{F}_i|_U = (Z_i \\times_X U \\to U)_*\\mathcal{G}'_i =", "R(Z_i \\times_X U \\to U)_*\\mathcal{G}'_i", "$$", "where $\\mathcal{G}'_i$ is the pullback of $\\mathcal{G}_i$", "to $Z_i \\times_X U$, see", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-affine-base-change}.", "Since $Z_i \\times_X U$ is affine we conclude that", "$\\mathcal{G}'_i$ has no higher cohomology on $Z_i \\times_X U$.", "By the Leray spectral sequence we conclude the same", "thing is true for $\\mathcal{F}_i|_U$ (Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-apply-Leray}).", "\\medskip\\noindent", "Proof of (2). Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $V \\to Y$ be an \\'etale morphism with $V$ affine.", "Then $V \\times_Y X \\to X$ is an \\'etale morphism and", "$V \\times_Y X$ is a quasi-compact and quasi-separated algebraic", "space \\'etale over $X$ (details omitted). Hence", "$H^p(V \\times_Y X, \\mathcal{J})$ is zero by part (1).", "Since $R^pf_*\\mathcal{J}$ is the sheaf associated to the presheaf", "$V \\mapsto H^p(V \\times_Y X, \\mathcal{J})$ the result is proved." ], "refs": [ "spaces-perfect-lemma-injective-colimit-direct-sum-pushforwards-artin", "spaces-cohomology-lemma-colimits", "spaces-morphisms-lemma-Artinian-affine", "spaces-cohomology-lemma-affine-base-change", "sites-cohomology-lemma-apply-Leray" ], "ref_ids": [ 2687, 11277, 4805, 11295, 4221 ] } ], "ref_ids": [] }, { "id": 2690, "type": "theorem", "label": "spaces-perfect-lemma-Noetherian-pushforward", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-Noetherian-pushforward", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of Noetherian algebraic spaces over $S$.", "Then $f_*$ on quasi-coherent sheaves has a right derived", "extension", "$\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$", "such that the diagram", "$$", "\\xymatrix{", "D(\\QCoh(\\mathcal{O}_X)) \\ar[d]_{\\Phi} \\ar[r] &", "D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\", "D(\\QCoh(\\mathcal{O}_Y)) \\ar[r] &", "D_\\QCoh(\\mathcal{O}_Y)", "}", "$$", "commutes." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ and $Y$ are Noetherian the morphism is quasi-compact", "and quasi-separated (see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence}).", "Thus $f_*$ preserve quasi-coherence, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-pushforward}.", "Next, let $K$ be an object of $D(\\QCoh(\\mathcal{O}_X))$.", "Since $\\QCoh(\\mathcal{O}_X)$ is a Grothendieck abelian category", "(Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-coherator}), we can", "represent $K$ by a K-injective complex $\\mathcal{I}^\\bullet$", "such that each $\\mathcal{I}^n$ is an injective object of", "$\\QCoh(\\mathcal{O}_X)$, see", "Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "Thus we see that the functor $\\Phi$ is defined by setting", "$$", "\\Phi(K) = f_*\\mathcal{I}^\\bullet", "$$", "where the right hand side is viewed as an object of", "$D(\\QCoh(\\mathcal{O}_Y))$. To finish the proof of the lemma", "it suffices to show that the canonical map", "$$", "f_*\\mathcal{I}^\\bullet \\longrightarrow Rf_*\\mathcal{I}^\\bullet", "$$", "is an isomorphism in $D(\\mathcal{O}_Y)$. To see this it suffices to", "prove the map induces an isomorphism on cohomology sheaves. Pick any", "$m \\in \\mathbf{Z}$. Let $N = N(X, Y, f)$ be as in", "Lemma \\ref{lemma-quasi-coherence-direct-image}.", "Consider the short exact sequence", "$$", "0 \\to \\sigma_{\\geq m - N - 1}\\mathcal{I}^\\bullet \\to", "\\mathcal{I}^\\bullet \\to \\sigma_{\\leq m - N - 2}\\mathcal{I}^\\bullet \\to 0", "$$", "of complexes of quasi-coherent sheaves on $X$. By", "Lemma \\ref{lemma-quasi-coherence-direct-image}", "we see that the cohomology sheaves of", "$Rf_*\\sigma_{\\leq m - N - 2}\\mathcal{I}^\\bullet$", "are zero in degrees $\\geq m - 1$. Thus we see that", "$R^mf_*\\mathcal{I}^\\bullet$ is isomorphic to", "$R^mf_*\\sigma_{\\geq m - N - 1}\\mathcal{I}^\\bullet$.", "In other words, we may assume that $\\mathcal{I}^\\bullet$", "is a bounded below complex of injective objects of", "$\\QCoh(\\mathcal{O}_X)$.", "This case follows from Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "with required vanishing because of Lemma \\ref{lemma-injective-pushforward}." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence", "spaces-morphisms-lemma-pushforward", "spaces-properties-proposition-coherator", "injectives-theorem-K-injective-embedding-grothendieck", "spaces-perfect-lemma-quasi-coherence-direct-image", "spaces-perfect-lemma-quasi-coherence-direct-image", "derived-lemma-leray-acyclicity", "spaces-perfect-lemma-injective-pushforward" ], "ref_ids": [ 4744, 4760, 11921, 7768, 2652, 2652, 1844, 2689 ] } ], "ref_ids": [] }, { "id": 2691, "type": "theorem", "label": "spaces-perfect-lemma-descend-finite-type", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-descend-finite-type", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is of finite type as an $\\mathcal{O}_X$-module, and", "\\item $\\epsilon^*\\mathcal{F}$ is of finite type as an", "$\\mathcal{O}_\\etale$-module on the small \\'etale site of $X$.", "\\end{enumerate}", "Here $\\epsilon$ is as in (\\ref{equation-epsilon})." ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) is a general fact, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-local-pullback}.", "Assume (2). By assumption there exists an \\'etale covering", "$\\{f_i : X_i \\to X\\}$ such that", "$\\epsilon^*\\mathcal{F}|_{(X_i)_\\etale}$ is generated by", "finitely many sections. Let $x \\in X$. We will show that $\\mathcal{F}$", "is generated by finitely many sections in a neighbourhood of $x$.", "Say $x$ is in the image of $X_i \\to X$ and denote $X' = X_i$. Let", "$s_1, \\ldots, s_n \\in", "\\Gamma(X', \\epsilon^*\\mathcal{F}|_{X'_\\etale})$", "be generating sections. As", "$\\epsilon^*\\mathcal{F} =", "\\epsilon^{-1}\\mathcal{F} \\otimes_{\\epsilon^{-1}\\mathcal{O}_X}", "\\mathcal{O}_\\etale$", "we can find an \\'etale morphism $X'' \\to X'$ such that $x$ is", "in the image of $X' \\to X$ and such that", "$s_i|_{X''} = \\sum s_{ij} \\otimes a_{ij}$ for some sections", "$s_{ij} \\in \\epsilon^{-1}\\mathcal{F}(X'')$ and", "$a_{ij} \\in \\mathcal{O}_\\etale(X'')$. Denote $U \\subset X$ the image", "of $X'' \\to X$. This is an open subscheme as $f'' : X'' \\to X$ is \\'etale", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-open}). After possibly", "shrinking $X''$ more we may assume $s_{ij}$ come from elements", "$t_{ij} \\in \\mathcal{F}(U)$ as follows from the construction of", "the inverse image functor $\\epsilon^{-1}$. Now we claim that", "$t_{ij}$ generate $\\mathcal{F}|_U$ which finishes the proof", "of the lemma. Namely, the corresponding map", "$\\mathcal{O}_U^{\\oplus N} \\to \\mathcal{F}|_U$ has the property", "that its pullback by $f''$ to $X''$ is surjective. Since $f'' : X'' \\to U$", "is a surjective flat morphism of schemes, this implies that", "$\\mathcal{O}_U^{\\oplus N} \\to \\mathcal{F}|_U$ is surjective by", "looking at stalks and using that", "$\\mathcal{O}_{U, f''(z)} \\to \\mathcal{O}_{X'', z}$", "is faithfully flat for all $z \\in X''$." ], "refs": [ "sites-modules-lemma-local-pullback", "morphisms-lemma-etale-open" ], "ref_ids": [ 14186, 5370 ] } ], "ref_ids": [] }, { "id": 2692, "type": "theorem", "label": "spaces-perfect-lemma-descend-pseudo-coherent", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-descend-pseudo-coherent", "contents": [ "Let $X$ be a scheme. Let $E$ be an object of $D(\\mathcal{O}_X)$.", "The following are equivalent", "\\begin{enumerate}", "\\item $E$ is $m$-pseudo-coherent, and", "\\item $\\epsilon^*E$ is $m$-pseudo-coherent on the small \\'etale site of $X$.", "\\end{enumerate}", "Here $\\epsilon$ is as in (\\ref{equation-epsilon})." ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) is a general fact, see", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-pseudo-coherent-pullback}.", "Assume $\\epsilon^*E$ is $m$-pseudo-coherent.", "We will use without further mention that $\\epsilon^*$ is", "an exact functor and that therefore", "$$", "\\epsilon^*H^i(E) = H^i(\\epsilon^*E).", "$$", "To show that $E$ is $m$-pseudo-coherent we may work locally on $X$,", "hence we may assume that $X$ is quasi-compact (for example affine).", "Since $X$ is quasi-compact every \\'etale covering $\\{U_i \\to X\\}$", "has a finite refinement. Thus we see that $\\epsilon^*E$ is", "an object of $D^{-}(\\mathcal{O}_\\etale)$, see", "comments following", "Cohomology on Sites, Definition", "\\ref{sites-cohomology-definition-pseudo-coherent}.", "By Lemma \\ref{lemma-epsilon-flat} it follows that $E$ is an object of", "$D^-(\\mathcal{O}_X)$.", "\\medskip\\noindent", "Let $n \\in \\mathbf{Z}$ be the largest integer such that", "$H^n(E)$ is nonzero; then $n$ is also the largest integer", "such that $H^n(\\epsilon^*E)$ is nonzero.", "We will prove the lemma by induction on $n - m$.", "If $n < m$, then the lemma is clearly true.", "If $n \\geq m$, then $H^n(\\epsilon^*E)$ is a finite", "$\\mathcal{O}_\\etale$-module, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-finite-cohomology}.", "Hence $H^n(E)$ is a finite $\\mathcal{O}_X$-module, see", "Lemma \\ref{lemma-descend-finite-type}.", "After replacing $X$ by the members of an open covering, we may", "assume there exists a surjection $\\mathcal{O}_X^{\\oplus t} \\to H^n(E)$.", "We may locally on $X$ lift this to a map of complexes", "$\\alpha : \\mathcal{O}_X^{\\oplus t}[-n] \\to E$ (details omitted).", "Choose a distinguished triangle", "$$", "\\mathcal{O}_X^{\\oplus t}[-n] \\to E \\to C \\to \\mathcal{O}_X^{\\oplus t}[-n + 1]", "$$", "Then $C$ has vanishing cohomology in degrees $\\geq n$. On the other hand, the", "complex $\\epsilon^*C$ is $m$-pseudo-coherent, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cone-pseudo-coherent}.", "Hence by induction we see that $C$ is $m$-pseudo-coherent. Applying", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cone-pseudo-coherent}", "once more we conclude." ], "refs": [ "sites-cohomology-lemma-pseudo-coherent-pullback", "sites-cohomology-definition-pseudo-coherent", "spaces-perfect-lemma-epsilon-flat", "sites-cohomology-lemma-finite-cohomology", "spaces-perfect-lemma-descend-finite-type", "sites-cohomology-lemma-cone-pseudo-coherent", "sites-cohomology-lemma-cone-pseudo-coherent" ], "ref_ids": [ 4367, 4420, 2643, 4371, 2691, 4368, 4368 ] } ], "ref_ids": [] }, { "id": 2693, "type": "theorem", "label": "spaces-perfect-lemma-descend-tor-amplitude", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-descend-tor-amplitude", "contents": [ "Let $X$ be a scheme. Let $E$ be an object of $D(\\mathcal{O}_X)$.", "Then", "\\begin{enumerate}", "\\item $E$ has tor amplitude in $[a, b]$ if and only if", "$\\epsilon^*E$ has tor amplitude in $[a, b]$.", "\\item $E$ has finite tor dimension if and only if $\\epsilon^*E$ has finite", "tor dimension.", "\\end{enumerate}", "Here $\\epsilon$ is as in (\\ref{equation-epsilon})." ], "refs": [], "proofs": [ { "contents": [ "The easy implication follows from", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-tor-amplitude-pullback}.", "For the converse, assume that $\\epsilon^*E$ has tor amplitude in $[a, b]$.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. As $\\epsilon$ is a flat", "morphism of ringed sites (Lemma \\ref{lemma-epsilon-flat})", "we have", "$$", "\\epsilon^*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{F})", "=", "\\epsilon^*E", "\\otimes^\\mathbf{L}_{\\mathcal{O}_\\etale}", "\\epsilon^*\\mathcal{F}", "$$", "Thus the (assumed) vanishing of cohomology sheaves on the right hand side", "implies the desired vanishing of the cohomology sheaves of", "$E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{F}$ via", "Lemma \\ref{lemma-epsilon-flat}." ], "refs": [ "sites-cohomology-lemma-tor-amplitude-pullback", "spaces-perfect-lemma-epsilon-flat", "spaces-perfect-lemma-epsilon-flat" ], "ref_ids": [ 4375, 2643, 2643 ] } ], "ref_ids": [] }, { "id": 2694, "type": "theorem", "label": "spaces-perfect-lemma-tor-dimension-rel", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-tor-dimension-rel", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $E$ be an object", "of $D(\\mathcal{O}_X)$. Then", "\\begin{enumerate}", "\\item $E$ as an object of $D(f^{-1}\\mathcal{O}_Y)$ has tor amplitude in", "$[a, b]$ if and only if $\\epsilon^*E$ has tor amplitude in $[a, b]$", "as an object of $D(f_{small}^{-1}\\mathcal{O}_{Y_\\etale})$.", "\\item $E$ locally has finite tor dimension as an object of", "$D(f^{-1}\\mathcal{O}_Y)$ if and only if $\\epsilon^*E$", "locally has finite tor dimension as an object of", "$D(f_{small}^{-1}\\mathcal{O}_{Y_\\etale})$.", "\\end{enumerate}", "Here $\\epsilon$ is as in (\\ref{equation-epsilon})." ], "refs": [], "proofs": [ { "contents": [ "The easy direction in (1) follows from Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-tor-amplitude-pullback}.", "Let $x \\in X$ be a point and let $\\overline{x}$ be a geometric", "point lying over $x$. Let $y = f(x)$ and denote $\\overline{y}$", "the geometric point of $Y$ coming from $\\overline{x}$.", "Then $(f^{-1}\\mathcal{O}_Y)_x = \\mathcal{O}_{Y, y}$", "(Sheaves, Lemma \\ref{sheaves-lemma-stalk-pullback})", "and", "$$", "(f_{small}^{-1}\\mathcal{O}_{Y_\\etale})_{\\overline{x}} =", "\\mathcal{O}_{Y_\\etale, \\overline{y}} =", "\\mathcal{O}_{Y, y}^{sh}", "$$", "is the strict henselization", "(by \\'Etale Cohomology, Lemmas \\ref{etale-cohomology-lemma-stalk-pullback}", "and \\ref{etale-cohomology-lemma-describe-etale-local-ring}).", "Since the stalk of $\\mathcal{O}_{X_\\etale}$ at $X$ is", "$\\mathcal{O}_{X, x}^{sh}$ we obtain", "$$", "(\\epsilon^*E)_{\\overline{x}} =", "E_x \\otimes_{\\mathcal{O}_{X, x}}^\\mathbf{L} \\mathcal{O}_{X, x}^{sh}", "$$", "by transitivity of pullbacks. If $\\epsilon^*E$ has tor amplitude", "in $[a, b]$ as a complex of $f_{small}^{-1}\\mathcal{O}_{Y_\\etale}$-modules,", "then $(\\epsilon^*E)_{\\overline{x}}$ has tor amplitude in $[a, b]$", "as a complex of $\\mathcal{O}_{Y, y}^{sh}$-modules", "(because taking stalks is a pullback and lemma cited above). By", "More on Flatness, Lemma \\ref{flat-lemma-tor-amplitude-up-down-henselization}", "we find the tor amplitude of", "$(\\epsilon^*E)_{\\overline{x}}$", "as a complex of $\\mathcal{O}_{Y, y}$-modules is in $[a, b]$.", "Since $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X, x}^{sh}$ is faithfully", "flat (More on Algebra, Lemma", "\\ref{more-algebra-lemma-dumb-properties-henselization}) and since", "$(\\epsilon^*E)_{\\overline{x}} =", "E_x \\otimes_{\\mathcal{O}_{X, x}}^\\mathbf{L} \\mathcal{O}_{X, x}^{sh}$", "we may apply", "More on Algebra, Lemma \\ref{more-algebra-lemma-no-change-tor-amplitude}", "to conclude the tor amplitude of $E_x$ as a complex of", "$\\mathcal{O}_{Y, y}$-modules is in $[a, b]$.", "By Cohomology, Lemma \\ref{cohomology-lemma-tor-amplitude-stalk}", "we conclude that $E$ as an object of $D(f^{-1}\\mathcal{O}_Y)$", "has tor amplitude in $[a, b]$. This gives the reverse implication in (1).", "Part (2) follows formally from (1)." ], "refs": [ "sites-cohomology-lemma-tor-amplitude-pullback", "sheaves-lemma-stalk-pullback", "etale-cohomology-lemma-stalk-pullback", "etale-cohomology-lemma-describe-etale-local-ring", "flat-lemma-tor-amplitude-up-down-henselization", "more-algebra-lemma-dumb-properties-henselization", "more-algebra-lemma-no-change-tor-amplitude", "cohomology-lemma-tor-amplitude-stalk" ], "ref_ids": [ 4375, 14507, 6436, 6433, 5983, 10055, 10185, 2217 ] } ], "ref_ids": [] }, { "id": 2695, "type": "theorem", "label": "spaces-perfect-lemma-descend-perfect", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-descend-perfect", "contents": [ "Let $X$ be a scheme. Let $E$ be an object of $D(\\mathcal{O}_X)$.", "Then $E$ is a perfect object of $D(\\mathcal{O}_X)$ if and only if", "$\\epsilon^*E$ is a perfect object of $D(\\mathcal{O}_\\etale)$.", "Here $\\epsilon$ is as in (\\ref{equation-epsilon})." ], "refs": [], "proofs": [ { "contents": [ "The easy implication follows from the general result contained in", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-perfect-pullback}.", "For the converse, we can use the equivalence of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-perfect}", "and the corresponding results for pseudo-coherent and complexes of", "finite tor dimension, namely", "Lemmas \\ref{lemma-descend-pseudo-coherent} and", "\\ref{lemma-descend-tor-amplitude}.", "Some details omitted." ], "refs": [ "sites-cohomology-lemma-perfect-pullback", "sites-cohomology-lemma-perfect", "spaces-perfect-lemma-descend-pseudo-coherent", "spaces-perfect-lemma-descend-tor-amplitude" ], "ref_ids": [ 4384, 4383, 2692, 2693 ] } ], "ref_ids": [] }, { "id": 2696, "type": "theorem", "label": "spaces-perfect-lemma-pseudo-coherent", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-pseudo-coherent", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "If $E$ is an $m$-pseudo-coherent object of $D(\\mathcal{O}_X)$,", "then $H^i(E)$ is a quasi-coherent $\\mathcal{O}_X$-module for $i > m$.", "If $E$ is pseudo-coherent, then $E$ is an object of", "$D_\\QCoh(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Locally $H^i(E)$ is isomorphic to $H^i(\\mathcal{E}^\\bullet)$", "with $\\mathcal{E}^\\bullet$ strictly perfect. The sheaves", "$\\mathcal{E}^i$ are direct summands of finite free modules,", "hence quasi-coherent. The lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2697, "type": "theorem", "label": "spaces-perfect-lemma-identify-pseudo-coherent-noetherian", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-identify-pseudo-coherent-noetherian", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. For", "$m \\in \\mathbf{Z}$ the following are equivalent", "\\begin{enumerate}", "\\item $H^i(E)$ is coherent for $i \\geq m$ and zero for $i \\gg 0$, and", "\\item $E$ is $m$-pseudo-coherent.", "\\end{enumerate}", "In particular, $E$ is pseudo-coherent if and only if $E$ is an object", "of $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "As $X$ is quasi-compact we can find an affine scheme $U$ and a surjective", "\\'etale morphism $U \\to X$ (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "Observe that $U$ is Noetherian.", "Note that $E$ is $m$-pseudo-coherent if and only if $E|_U$ is", "$m$-pseudo-coherent (follows from the definition or from", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-pseudo-coherent-independent-representative}).", "Similarly, $H^i(E)$ is coherent if and only if $H^i(E)|_U = H^i(E|_U)$", "is coherent (see Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-coherent-Noetherian}).", "Thus we may assume that $X$ is representable.", "\\medskip\\noindent", "If $X$ is representable by a scheme $X_0$ then", "(Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site})", "we can write $E = \\epsilon^*E_0$ where $E_0$ is an object of", "$D_\\QCoh(\\mathcal{O}_{X_0})$ and", "$\\epsilon : X_\\etale \\to (X_0)_{Zar}$ is as in", "(\\ref{equation-epsilon}).", "In this case $E$ is $m$-pseudo-coherent", "if and only if $E_0$ is by Lemma \\ref{lemma-descend-pseudo-coherent}.", "Similarly, $H^i(E_0)$ is of finite type (i.e., coherent) if and only if", "$H^i(E)$ is by Lemma \\ref{lemma-descend-finite-type}.", "Finally, $H^i(E_0) = 0$ if and only if $H^i(E) = 0$ by", "Lemma \\ref{lemma-epsilon-flat}.", "Thus we reduce to the case of schemes which is", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-identify-pseudo-coherent-noetherian}." ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "sites-cohomology-lemma-pseudo-coherent-independent-representative", "spaces-cohomology-lemma-coherent-Noetherian", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-descend-pseudo-coherent", "spaces-perfect-lemma-descend-finite-type", "spaces-perfect-lemma-epsilon-flat", "perfect-lemma-identify-pseudo-coherent-noetherian" ], "ref_ids": [ 11832, 4366, 11297, 2644, 2692, 2691, 2643, 6976 ] } ], "ref_ids": [] }, { "id": 2698, "type": "theorem", "label": "spaces-perfect-lemma-tor-qc-qs", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-tor-qc-qs", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$.", "Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. Let $a \\leq b$.", "The following are equivalent", "\\begin{enumerate}", "\\item $E$ has tor amplitude in $[a, b]$, and", "\\item for all $\\mathcal{F}$ in $\\QCoh(\\mathcal{O}_X)$", "we have $H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}) = 0$", "for $i \\not \\in [a, b]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2). Assume (2). Let $j : U \\to X$ be", "an \\'etale morphism with $U$ affine. As $X$ is quasi-separated $j : U \\to X$", "is quasi-compact and separated, hence $j_*$ transforms quasi-coherent", "modules into quasi-coherent modules (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-pushforward}).", "Thus the functor", "$\\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_U)$", "is essentially surjective. It follows that condition (2)", "implies the vanishing of", "$H^i(E|_U \\otimes_{\\mathcal{O}_U}^\\mathbf{L} \\mathcal{G})$", "for $i \\not \\in [a, b]$ for all quasi-coherent $\\mathcal{O}_U$-modules", "$\\mathcal{G}$. Since it suffices to prove that $E|_U$ has tor amplitude", "in $[a, b]$ we reduce to the case where $X$ is representable.", "\\medskip\\noindent", "If $X$ is representable by a scheme $X_0$ then", "(Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site})", "we can write $E = \\epsilon^*E_0$ where $E_0$ is an object of", "$D_\\QCoh(\\mathcal{O}_{X_0})$ and", "$\\epsilon : X_\\etale \\to (X_0)_{Zar}$ is as in", "(\\ref{equation-epsilon}). For every quasi-coherent module", "$\\mathcal{F}_0$ on $X_0$ the module $\\epsilon^*\\mathcal{F}_0$", "is quasi-coherent on $X$ and", "$$", "H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\epsilon^*\\mathcal{F}_0)", "=", "\\epsilon^*H^i(E_0 \\otimes_{\\mathcal{O}_{X_0}}^\\mathbf{L} \\mathcal{F}_0)", "$$", "as $\\epsilon$ is flat (Lemma \\ref{lemma-epsilon-flat}).", "Moreover, the vanishing of these sheaves for $i \\not \\in [a, b]$", "implies the same thing for", "$H^i(E_0 \\otimes_{\\mathcal{O}_{X_0}}^\\mathbf{L} \\mathcal{F}_0)$", "by the same lemma. Thus we've reduced the problem to the case", "of schemes which is treated in", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-tor-qc-qs}." ], "refs": [ "spaces-morphisms-lemma-pushforward", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-epsilon-flat", "perfect-lemma-tor-qc-qs" ], "ref_ids": [ 4760, 2644, 2643, 6979 ] } ], "ref_ids": [] }, { "id": 2699, "type": "theorem", "label": "spaces-perfect-lemma-descend-RHom", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-descend-RHom", "contents": [ "Let $X$ be a scheme. Let $E, F$ be objects of $D(\\mathcal{O}_X)$.", "Assume either", "\\begin{enumerate}", "\\item $E$ is pseudo-coherent and $F$ lies in $D^+(\\mathcal{O}_X)$, or", "\\item $E$ is perfect and $F$ arbitrary,", "\\end{enumerate}", "then there is a canonical isomorphism", "$$", "\\epsilon^*R\\SheafHom(E, F) \\longrightarrow R\\SheafHom(\\epsilon^*E, \\epsilon^*F)", "$$", "Here $\\epsilon$ is as in (\\ref{equation-epsilon})." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\epsilon$ is flat (Lemma \\ref{lemma-epsilon-flat}) and", "hence $\\epsilon^* = L\\epsilon^*$. There is a canonical map", "from left to right by", "Cohomology on Sites, Remark", "\\ref{sites-cohomology-remark-prepare-fancy-base-change}.", "To see this is an isomorphism we can work locally, i.e., we may", "assume $X$ is an affine scheme.", "\\medskip\\noindent", "In case (1) we can represent $E$ by a bounded above complex", "$\\mathcal{E}^\\bullet$ of finite free $\\mathcal{O}_X$-modules, see", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-lift-pseudo-coherent}.", "We may also represent $F$ by a bounded below complex $\\mathcal{F}^\\bullet$", "of $\\mathcal{O}_X$-modules. Applying", "Cohomology, Lemma", "\\ref{cohomology-lemma-Rhom-complex-of-direct-summands-finite-free}", "we see that $R\\SheafHom(E, F)$ is represented by the complex with terms", "$$", "\\bigoplus\\nolimits_{n = - p + q}", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}^p, \\mathcal{F}^q)", "$$", "Applying Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-Rhom-complex-of-direct-summands-finite-free}", "we see that $R\\SheafHom(\\epsilon^*E, \\epsilon^*F)$ is represented by the", "complex with terms", "$$", "\\bigoplus\\nolimits_{n = - p + q}", "\\SheafHom_{\\mathcal{O}_\\etale}", "(\\epsilon^*\\mathcal{E}^p, \\epsilon^*\\mathcal{F}^q)", "$$", "Thus the statement of the lemma boils down to the true fact", "that the canonical map", "$$", "\\epsilon^*\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}, \\mathcal{F})", "\\longrightarrow", "\\SheafHom_{\\mathcal{O}_\\etale}", "(\\epsilon^*\\mathcal{E}, \\epsilon^*\\mathcal{F})", "$$", "is an isomorphism for any $\\mathcal{O}_X$-module $\\mathcal{F}$ and", "finite free $\\mathcal{O}_X$-module $\\mathcal{E}$.", "\\medskip\\noindent", "In case (2) we can represent $E$ by a strictly perfect", "complex $\\mathcal{E}^\\bullet$ of $\\mathcal{O}_X$-modules, use", "Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-affine-compare-bounded} and", "\\ref{perfect-lemma-perfect-affine} and the fact that a perfect", "complex of modules is represented by a finite complex of finite", "projective modules. Thus we can do the exact same proof as", "above, replacing the reference to", "Cohomology, Lemma", "\\ref{cohomology-lemma-Rhom-complex-of-direct-summands-finite-free}", "by a reference to", "Cohomology, Lemma", "\\ref{cohomology-lemma-Rhom-strictly-perfect}." ], "refs": [ "spaces-perfect-lemma-epsilon-flat", "sites-cohomology-remark-prepare-fancy-base-change", "perfect-lemma-lift-pseudo-coherent", "cohomology-lemma-Rhom-complex-of-direct-summands-finite-free", "sites-cohomology-lemma-Rhom-complex-of-direct-summands-finite-free", "perfect-lemma-affine-compare-bounded", "perfect-lemma-perfect-affine", "cohomology-lemma-Rhom-complex-of-direct-summands-finite-free", "cohomology-lemma-Rhom-strictly-perfect" ], "ref_ids": [ 2643, 4431, 6998, 2204, 4365, 6941, 6980, 2204, 2203 ] } ], "ref_ids": [] }, { "id": 2700, "type": "theorem", "label": "spaces-perfect-lemma-quasi-coherence-internal-hom", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-quasi-coherence-internal-hom", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $L, K$ be objects of $D(\\mathcal{O}_X)$.", "If either", "\\begin{enumerate}", "\\item $L$ in $D^+_\\QCoh(\\mathcal{O}_X)$ and $K$ is pseudo-coherent,", "\\item $L$ in $D_\\QCoh(\\mathcal{O}_X)$ and $K$ is perfect,", "\\end{enumerate}", "then $R\\SheafHom(K, L)$ is in $D_\\QCoh(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the analogue for schemes", "(Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-internal-hom})", "via the criterion of Lemma \\ref{lemma-check-quasi-coherence-on-covering},", "the criterion of Lemmas \\ref{lemma-descend-pseudo-coherent} and", "\\ref{lemma-descend-perfect},", "and the result of Lemma \\ref{lemma-descend-RHom}." ], "refs": [ "perfect-lemma-quasi-coherence-internal-hom", "spaces-perfect-lemma-check-quasi-coherence-on-covering", "spaces-perfect-lemma-descend-pseudo-coherent", "spaces-perfect-lemma-descend-perfect", "spaces-perfect-lemma-descend-RHom" ], "ref_ids": [ 6981, 2645, 2692, 2695, 2699 ] } ], "ref_ids": [] }, { "id": 2701, "type": "theorem", "label": "spaces-perfect-lemma-internal-hom-evaluate-tensor-isomorphism", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-internal-hom-evaluate-tensor-isomorphism", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $K, L, M$ be objects of $D_\\QCoh(\\mathcal{O}_X)$.", "The map", "$$", "K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(M, L)", "\\longrightarrow", "R\\SheafHom(M, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)", "$$", "of Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-internal-hom-diagonal-better}", "is an isomorphism in the following cases", "\\begin{enumerate}", "\\item $M$ perfect, or", "\\item $K$ is perfect, or", "\\item $M$ is pseudo-coherent, $L \\in D^+(\\mathcal{O}_X)$, and $K$ has finite", "tor dimension.", "\\end{enumerate}" ], "refs": [ "sites-cohomology-lemma-internal-hom-diagonal-better" ], "proofs": [ { "contents": [ "Checking whether or not the map is an isomorphism can be done", "\\'etale locally hence we may assume $X$ is an affine scheme.", "Then we can write $K, L, M$ as $\\epsilon^*K_0, \\epsilon^*L_0, \\epsilon^*M_0$", "for some $K_0, L_0, M_0$ in $D_\\QCoh(\\mathcal{O}_X)$ by", "Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site}.", "Then we see that Lemma \\ref{lemma-descend-RHom}", "reduces cases (1) and (3) to the case of schemes which", "is Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-internal-hom-evaluate-tensor-isomorphism}.", "If $K$ is perfect but no other assumptions are made, then we", "do not know that either side of the arrow is in $D_\\QCoh(\\mathcal{O}_X)$", "but the result is still true because $K$ will be represented", "(after localizing further) by a finite complex of finite free modules", "in which case it is clear." ], "refs": [ "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-descend-RHom", "perfect-lemma-internal-hom-evaluate-tensor-isomorphism" ], "ref_ids": [ 2644, 2699, 6982 ] } ], "ref_ids": [ 4333 ] }, { "id": 2702, "type": "theorem", "label": "spaces-perfect-lemma-pushforward-perfect", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-pushforward-perfect", "contents": [ "Let $S$ be a scheme. Let $(U \\subset X, j : V \\to X)$ be an", "elementary distinguished square of algebraic space over $S$.", "Let $E$ be a perfect object of $D(\\mathcal{O}_V)$ supported on", "$j^{-1}(T)$ where $T = |X| \\setminus |U|$. Then $Rj_*E$ is a", "perfect object of $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Being perfect is local on $X_\\etale$. Thus it suffices to", "check that $Rj_*E$ is perfect when restricted to $U$ and $V$.", "We have $Rj_*E|_V = E$ by Lemma \\ref{lemma-pushforward-with-support-in-open}", "which is perfect. We have $Rj_*E|_U = 0$ because", "$E|_{V \\setminus j^{-1}(T)} = 0$ (use", "Lemma \\ref{lemma-restrict-direct-image-open})." ], "refs": [ "spaces-perfect-lemma-pushforward-with-support-in-open", "spaces-perfect-lemma-restrict-direct-image-open" ], "ref_ids": [ 2679, 2642 ] } ], "ref_ids": [] }, { "id": 2703, "type": "theorem", "label": "spaces-perfect-lemma-open", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-open", "contents": [ "Let $S$ be a scheme. Let $(U \\subset X, j : V \\to X)$ be an elementary", "distinguished square of algebraic spaces over $S$. Let $T$ be a closed", "subset of $|X| \\setminus |U|$ and let $(T, E, m)$ be a triple as in", "Definition \\ref{definition-approximation-holds}. If", "\\begin{enumerate}", "\\item approximation holds for $(j^{-1}T, E|_V, m)$, and", "\\item the sheaves $H^i(E)$ for $i \\geq m$ are supported on $T$,", "\\end{enumerate}", "then approximation holds for $(T, E, m)$." ], "refs": [ "spaces-perfect-definition-approximation-holds" ], "proofs": [ { "contents": [ "Let $P \\to E|_V$ be an approximation of the triple $(j^{-1}T, E|_V, m)$", "over $V$. Then $Rj_*P$ is a perfect object of $D(\\mathcal{O}_X)$ by", "Lemma \\ref{lemma-pushforward-perfect}. On the other hand,", "$Rj_*P = j_!P$ by Lemma \\ref{lemma-pushforward-with-support-in-open}.", "We see that $j_!P$ is supported on $T$ for example by", "(\\ref{equation-stalk-j-shriek}).", "Hence we obtain an approximation $Rj_*P = j_!P \\to j_!(E|_V) \\to E$." ], "refs": [ "spaces-perfect-lemma-pushforward-perfect", "spaces-perfect-lemma-pushforward-with-support-in-open" ], "ref_ids": [ 2702, 2679 ] } ], "ref_ids": [ 2765 ] }, { "id": 2704, "type": "theorem", "label": "spaces-perfect-lemma-approximation-affine", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-approximation-affine", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ which is", "representable by an affine scheme. Then approximation holds for every", "triple $(T, E, m)$ as in Definition \\ref{definition-approximation-holds}", "such that there exists an integer $r \\geq 0$ with", "\\begin{enumerate}", "\\item $E$ is $m$-pseudo-coherent,", "\\item $H^i(E)$ is supported on $T$ for $i \\geq m - r + 1$,", "\\item $X \\setminus T$ is the union of $r$ affine opens.", "\\end{enumerate}", "In particular, approximation by perfect complexes holds for affine schemes." ], "refs": [ "spaces-perfect-definition-approximation-holds" ], "proofs": [ { "contents": [ "Let $X_0$ be an affine scheme representing $X$. Let $T_0 \\subset X_0$", "by the closed subset corresponding to $T$. Let", "$\\epsilon : X_\\etale \\to X_{0, Zar}$ be the morphism", "(\\ref{equation-epsilon}). We may write $E = \\epsilon^*E_0$ for some object", "$E_0$ of $D_\\QCoh(\\mathcal{O}_{X_0})$, see", "Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site}.", "Then $E_0$ is $m$-pseudo-coherent, see", "Lemma \\ref{lemma-descend-pseudo-coherent}.", "Comparing stalks of cohomology sheaves (see proof of", "Lemma \\ref{lemma-epsilon-flat})", "we see that $H^i(E_0)$ is supported on $T_0$ for $i \\geq m - r + 1$. By", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-approximation-affine}", "there exists an approximation $P_0 \\to E_0$ of", "$(T_0, E_0, m)$. By Lemma \\ref{lemma-descend-perfect}", "we see that $P = \\epsilon^*P_0$ is a perfect object of $D(\\mathcal{O}_X)$.", "Pulling back we obtain an approximation", "$P = \\epsilon^*P_0 \\to \\epsilon^*E_0 = E$ as desired." ], "refs": [ "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-descend-pseudo-coherent", "spaces-perfect-lemma-epsilon-flat", "perfect-lemma-approximation-affine", "spaces-perfect-lemma-descend-perfect" ], "ref_ids": [ 2644, 2692, 2643, 7007, 2695 ] } ], "ref_ids": [ 2765 ] }, { "id": 2705, "type": "theorem", "label": "spaces-perfect-lemma-induction-step", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-induction-step", "contents": [ "Let $S$ be a scheme. Let $(U \\subset X, j : V \\to X)$ be an", "elementary distinguished square of algebraic spaces over $S$.", "Assume $U$ quasi-compact, $V$ affine, and $U \\times_X V$ quasi-compact.", "If approximation by perfect complexes holds on $U$,", "then approximation by perfect complexes holds on $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $T \\subset |X|$ be a closed subset with $X \\setminus T \\to X$", "quasi-compact. Let $r_U$ be the integer of", "Definition \\ref{definition-approximation}", "adapted to the pair $(U, T \\cap |U|)$.", "Set $T' = T \\setminus |U|$. Endow $T'$ with the induced reduced", "subspace structure. Since $|T'|$ is contained in $|X| \\setminus |U|$", "we see that $j^{-1}(T') \\to T'$ is an isomorphism. Moreover,", "$V \\setminus j^{-1}(T')$ is quasi-compact as it is the fibre product", "of $U \\times_X V$ with $X \\setminus T$ over $X$ and we've assumed", "$U \\times_X V$ quasi-compact and $X \\setminus T \\to X$ quasi-compact.", "Let $r'$ be the number of affines needed to cover $V \\setminus j^{-1}(T')$.", "We claim that $r = \\max(r_U, r')$ works for the pair $(X, T)$.", "\\medskip\\noindent", "To see this choose a triple $(T, E, m)$ such that $E$ is", "$(m - r)$-pseudo-coherent and $H^i(E)$ is supported on $T$ for", "$i \\geq m - r$. Let $t$ be the largest integer such that", "$H^t(E)|_U$ is nonzero. (Such an integer exists as $U$ is quasi-compact", "and $E|_U$ is $(m - r)$-pseudo-coherent.)", "We will prove that $E$ can be approximated by induction on $t$.", "\\medskip\\noindent", "Base case: $t \\leq m - r'$. This means that $H^i(E)$ is supported", "on $T'$ for $i \\geq m - r'$. Hence", "Lemma \\ref{lemma-approximation-affine}", "guarantees the existence of an approximation", "$P \\to E|_V$ of $(T', E|_V, m)$ on $V$.", "Applying Lemma \\ref{lemma-open} we see that", "$(T', E, m)$ can be approximated. Such an approximation", "is also an approximation of $(T, E, m)$.", "\\medskip\\noindent", "Induction step. Choose an approximation $P \\to E|_U$ of", "$(T \\cap |U|, E|_U, m)$. This in particular gives a surjection", "$H^t(P) \\to H^t(E|_U)$.", "In the rest of the proof we will use the equivalence of", "Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site}", "(and the compatibilities of Remark \\ref{remark-match-total-direct-images})", "for the representable algebraic spaces $V$ and $U \\times_X V$.", "We will also use the fact that $(m - r)$-pseudo-coherence,", "resp.\\ perfectness on the Zariski site and \\'etale site agree, see", "Lemmas \\ref{lemma-descend-pseudo-coherent} and", "\\ref{lemma-descend-perfect}.", "Thus we can use the results of", "Derived Categories of Schemes, Section \\ref{perfect-section-lift}", "for the open immersion $U \\times_X V \\subset V$. In this way", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-lift-perfect-complex-plus-shift-support}", "implies there exists a perfect object $Q$ in $D(\\mathcal{O}_V)$", "supported on $j^{-1}(T)$ and an isomorphism", "$Q|_{U \\times_X V} \\to (P \\oplus P[1])|_{U \\times_X V}$. By", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-lift-map}", "we can replace $Q$ by $Q \\otimes^\\mathbf{L} I$", "and assume that the map", "$$", "Q|_{U \\times_X V} \\longrightarrow", "(P \\oplus P[1])|_{U \\times_X V} \\longrightarrow", "P|_{U \\times_X V} \\longrightarrow E|_{U \\times_X V}", "$$", "lifts to $Q \\to E|_V$. By Lemma \\ref{lemma-glue}", "we find an morphism $a : R \\to E$ of $D(\\mathcal{O}_X)$", "such that $a|_U$ is isomorphic to $P \\oplus P[1] \\to E|_U$", "and $a|_V$ isomorphic to $Q \\to E|_V$.", "Thus $R$ is perfect and supported on $T$", "and the map $H^t(R) \\to H^t(E)$ is surjective on restriction to $U$.", "Choose a distinguished triangle", "$$", "R \\to E \\to E' \\to R[1]", "$$", "Then $E'$ is $(m - r)$-pseudo-coherent", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cone-pseudo-coherent}),", "$H^i(E')|_U = 0$ for $i \\geq t$, and", "$H^i(E')$ is supported on $T$ for $i \\geq m - r$.", "By induction we find an approximation $R' \\to E'$", "of $(T, E', m)$. Fit the composition $R' \\to E' \\to R[1]$", "into a distinguished triangle $R \\to R'' \\to R' \\to R[1]$", "and extend the morphisms $R' \\to E'$ and $R[1] \\to R[1]$ into", "a morphism of distinguished triangles", "$$", "\\xymatrix{", "R \\ar[r] \\ar[d] & R'' \\ar[d] \\ar[r] & R' \\ar[d] \\ar[r] & R[1] \\ar[d] \\\\", "R \\ar[r] & E \\ar[r] & E' \\ar[r] & R[1]", "}", "$$", "using TR3. Then $R''$ is a perfect complex", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-two-out-of-three-perfect})", "supported on $T$. An easy diagram chase shows that $R'' \\to E$ is the desired", "approximation." ], "refs": [ "spaces-perfect-definition-approximation", "spaces-perfect-lemma-approximation-affine", "spaces-perfect-lemma-open", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-remark-match-total-direct-images", "spaces-perfect-lemma-descend-pseudo-coherent", "spaces-perfect-lemma-descend-perfect", "perfect-lemma-lift-perfect-complex-plus-shift-support", "perfect-lemma-lift-map", "spaces-perfect-lemma-glue", "sites-cohomology-lemma-cone-pseudo-coherent", "sites-cohomology-lemma-two-out-of-three-perfect" ], "ref_ids": [ 2766, 2704, 2703, 2644, 2768, 2692, 2695, 7004, 7001, 2680, 4368, 4385 ] } ], "ref_ids": [] }, { "id": 2706, "type": "theorem", "label": "spaces-perfect-lemma-lift-map-from-perfect-complex-with-support", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-lift-map-from-perfect-complex-with-support", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $W \\subset X$ be a quasi-compact open.", "Let $T \\subset |X|$ be a closed subset such that", "$X \\setminus T \\to X$ is a quasi-compact morphism.", "Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$.", "Let $\\alpha : P \\to E|_W$ be a map where $P$ is a perfect object of", "$D(\\mathcal{O}_W)$ supported on $T \\cap W$. Then there exists a map", "$\\beta : R \\to E$ where $R$ is a perfect object of $D(\\mathcal{O}_X)$", "supported on $T$ such that $P$ is a direct summand of $R|_W$ in", "$D(\\mathcal{O}_W)$ compatible $\\alpha$ and $\\beta|_W$." ], "refs": [], "proofs": [ { "contents": [ "We will use the induction principle of", "Lemma \\ref{lemma-induction-principle-enlarge} to prove this.", "Thus we immediately reduce to the case where we have an", "elementary distinguished square $(W \\subset X, f : V \\to X)$", "with $V$ affine and $P \\to E|_W$ as in the statement of the lemma.", "In the rest of the proof we will use", "Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site}", "(and the compatibilities of Remark \\ref{remark-match-total-direct-images})", "for the representable algebraic spaces $V$ and $W \\times_X V$.", "We will also use the fact that perfectness on the Zariski site", "and \\'etale site agree, see Lemma \\ref{lemma-descend-perfect}.", "\\medskip\\noindent", "By Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-lift-perfect-complex-plus-shift-support}", "we can choose a perfect object $Q$ in $D(\\mathcal{O}_V)$", "supported on $f^{-1}T$ and an isomorphism", "$Q|_{W \\times_X V} \\to (P \\oplus P[1])|_{W \\times_X V}$. By", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-lift-map}", "we can replace $Q$ by $Q \\otimes^\\mathbf{L} I$ (still supported on $f^{-1}T$)", "and assume that the map", "$$", "Q|_{W \\times_X V} \\to (P \\oplus P[1])|_{W \\times V}", "\\longrightarrow P|_{W \\times_X V}", "\\longrightarrow", "E|_{W \\times_X V}", "$$", "lifts to $Q \\to E|_V$. By Lemma \\ref{lemma-glue}", "we find an morphism $a : R \\to E$ of $D(\\mathcal{O}_X)$", "such that $a|_W$ is isomorphic to $P \\oplus P[1] \\to E|_W$", "and $a|_V$ isomorphic to $Q \\to E|_V$.", "Thus $R$ is perfect and supported on $T$ as desired." ], "refs": [ "spaces-perfect-lemma-induction-principle-enlarge", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-remark-match-total-direct-images", "spaces-perfect-lemma-descend-perfect", "perfect-lemma-lift-perfect-complex-plus-shift-support", "perfect-lemma-lift-map", "spaces-perfect-lemma-glue" ], "ref_ids": [ 2672, 2644, 2768, 2695, 7004, 7001, 2680 ] } ], "ref_ids": [] }, { "id": 2707, "type": "theorem", "label": "spaces-perfect-lemma-direct-summand-of-a-restriction", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-direct-summand-of-a-restriction", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $W$ be a quasi-compact open subspace of $X$.", "Let $P$ be a perfect object of $D(\\mathcal{O}_W)$.", "Then $P$ is a direct summand of the restriction of a perfect", "object of $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-lift-map-from-perfect-complex-with-support}." ], "refs": [ "spaces-perfect-lemma-lift-map-from-perfect-complex-with-support" ], "ref_ids": [ 2706 ] } ], "ref_ids": [] }, { "id": 2708, "type": "theorem", "label": "spaces-perfect-lemma-generator-with-support", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-generator-with-support", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $T \\subset |X|$ be a", "closed subset such that $|X| \\setminus T$ is quasi-compact. With notation", "as above, the category $D_{\\QCoh, T}(\\mathcal{O}_X)$ is generated by a", "single perfect object." ], "refs": [], "proofs": [ { "contents": [ "We will prove this using the induction principle of", "Lemma \\ref{lemma-induction-principle}.", "The property is true for representable quasi-compact", "and quasi-separated objects of the site", "$X_{spaces, \\etale}$ by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-generator-with-support}.", "\\medskip\\noindent", "Assume that $(U \\subset X, f : V \\to X)$ is an elementary distinguished", "square such that the lemma holds for $U$ and $V$ is affine. To finish", "the proof we have to show that the result holds for $X$.", "Let $P$ be a perfect object of $D(\\mathcal{O}_U)$ supported on $T \\cap U$", "which is a generator for $D_{\\QCoh, T \\cap U}(\\mathcal{O}_U)$. Using", "Lemma \\ref{lemma-lift-map-from-perfect-complex-with-support}", "we may choose a perfect object $Q$ of $D(\\mathcal{O}_X)$ supported on $T$", "whose restriction to $U$ is a direct sum one of whose summands is $P$.", "Write $V = \\Spec(B)$. Let $Z = X \\setminus U$. Then $f^{-1}Z$ is a closed", "subset of $V$ such that $V \\setminus f^{-1}Z$ is quasi-compact. As $X$", "is quasi-separated, it follows that", "$f^{-1}Z \\cap f^{-1}T = f^{-1}(Z \\cap T)$ is a closed", "subset of $V$ such that $W = V \\setminus f^{-1}(Z \\cap T)$ is quasi-compact.", "Thus we can choose $g_1, \\ldots, g_s \\in B$ such that", "$f^{-1}(Z \\cap T) = V(g_1, \\ldots, g_r)$.", "Let $K \\in D(\\mathcal{O}_V)$ be the perfect object corresponding to the", "Koszul complex on $g_1, \\ldots, g_s$ over $B$. Note that since $K$ is", "supported on $f^{-1}(Z \\cap T) \\subset V$ closed, the pushforward", "$K' = R(V \\to X)_*K$ is a perfect object of $D(\\mathcal{O}_X)$ whose", "restriction to $V$ is $K$ (see Lemmas \\ref{lemma-pushforward-perfect}", "and \\ref{lemma-pushforward-with-support-in-open}).", "We claim that $Q \\oplus K'$ is a generator for", "$D_{\\QCoh, T}(\\mathcal{O}_X)$.", "\\medskip\\noindent", "Let $E$ be an object of $D_{\\QCoh, T}(\\mathcal{O}_X)$ such that", "there are no nontrivial maps from any shift of $Q \\oplus K'$ into $E$.", "By Lemma \\ref{lemma-pushforward-with-support-in-open}", "we have $K' = R(V \\to X)_! K$ and hence", "$$", "\\Hom_{D(\\mathcal{O}_X)}(K'[n], E) = \\Hom_{D(\\mathcal{O}_V)}(K[n], E|_V)", "$$", "Thus by", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-orthogonal-koszul-complex} we have", "$E|_V = Rj_*E|_W$ where $j : W \\to V$ is the inclusion. Picture", "$$", "\\xymatrix{", "W \\ar[r]_j & V & Z \\cap T \\ar[l] \\ar[d] \\\\", "V \\setminus f^{-1}Z \\ar[u]^{j'} \\ar[ru]_{j''} & & Z \\ar[lu]", "}", "$$", "Since $E$ is supported on $T$ we see that $E|_W$ is supported on", "$f^{-1}T \\cap W = f^{-1}T \\cap (V \\setminus f^{-1}Z)$", "which is closed in $W$. We conclude that", "$$", "E|_V = Rj_*(E|_W) = Rj_*(Rj'_*(E|_{U \\cap V})) = Rj''_*(E|_{U \\cap V})", "$$", "Here the second equality is part (1) of", "Cohomology, Lemma \\ref{cohomology-lemma-pushforward-restriction}", "which applies because $V$ is a scheme and $E$ has quasi-coherent", "cohomology sheaves hence pushforward along the quasi-compact", "open immersion $j'$ agrees with pushforward on the underlying schemes, see", "Remark \\ref{remark-match-total-direct-images}.", "This implies that $E = R(U \\to X)_*E|_U$ (small detail omitted). If", "this is the case then", "$$", "\\Hom_{D(\\mathcal{O}_X)}(Q[n], E) = \\Hom_{D(\\mathcal{O}_U)}(Q|_U[n], E|_U)", "$$", "which contains $\\Hom_{D(\\mathcal{O}_U)}(P[n], E|_U)$ as a direct summand.", "Thus by our choice of $P$ the vanishing of these groups implies that $E|_U$", "is zero. Whence $E$ is zero." ], "refs": [ "spaces-perfect-lemma-induction-principle", "perfect-lemma-generator-with-support", "spaces-perfect-lemma-lift-map-from-perfect-complex-with-support", "spaces-perfect-lemma-pushforward-perfect", "spaces-perfect-lemma-pushforward-with-support-in-open", "spaces-perfect-lemma-pushforward-with-support-in-open", "perfect-lemma-orthogonal-koszul-complex", "cohomology-lemma-pushforward-restriction", "spaces-perfect-remark-match-total-direct-images" ], "ref_ids": [ 2670, 7011, 2706, 2702, 2679, 2679, 7010, 2148, 2768 ] } ], "ref_ids": [] }, { "id": 2709, "type": "theorem", "label": "spaces-perfect-lemma-compact-is-perfect-with-support", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-compact-is-perfect-with-support", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $T \\subset |X|$ be a closed subset such that $|X| \\setminus T$", "is quasi-compact. An object of $D_{\\QCoh, T}(\\mathcal{O}_X)$ is compact", "if and only if it is perfect as an object of $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "We observe that $D_{\\QCoh, T}(\\mathcal{O}_X)$ is a triangulated", "category with direct sums by the remark preceding the lemma.", "By Proposition \\ref{proposition-compact-is-perfect}", "the perfect objects define compact objects of $D(\\mathcal{O}_X)$", "hence a fortiori of any subcategory preserved under taking direct", "sums. For the converse we will use there exists a generator", "$E \\in D_{\\QCoh, T}(\\mathcal{O}_X)$ which is a perfect complex", "of $\\mathcal{O}_X$-modules, see", "Lemma \\ref{lemma-generator-with-support}.", "Hence by the above, $E$ is compact. Then it follows from", "Derived Categories, Proposition", "\\ref{derived-proposition-generator-versus-classical-generator}", "that $E$ is a classical generator of the full subcategory", "of compact objects of $D_{\\QCoh, T}(\\mathcal{O}_X)$.", "Thus any compact object can be constructed out of $E$ by", "a finite sequence of operations consisting of", "(a) taking shifts, (b) taking finite direct sums, (c) taking cones, and", "(d) taking direct summands. Each of these operations preserves", "the property of being perfect and the result follows." ], "refs": [ "spaces-perfect-proposition-compact-is-perfect", "spaces-perfect-lemma-generator-with-support", "derived-proposition-generator-versus-classical-generator" ], "ref_ids": [ 2758, 2708, 1965 ] } ], "ref_ids": [] }, { "id": 2710, "type": "theorem", "label": "spaces-perfect-lemma-map-from-pseudo-coherent-to-complex-with-support", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-map-from-pseudo-coherent-to-complex-with-support", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $T \\subset |X|$", "be a closed subset such that the complement $U \\subset X$ is quasi-compact.", "Let $\\alpha : P \\to E$ be a morphism of $D_\\QCoh(\\mathcal{O}_X)$ with", "either", "\\begin{enumerate}", "\\item $P$ is perfect and $E$ supported on $T$, or", "\\item $P$ pseudo-coherent, $E$ supported on $T$, and $E$ bounded below.", "\\end{enumerate}", "Then there exists a perfect complex of $\\mathcal{O}_X$-modules $I$", "and a map $I \\to \\mathcal{O}_X[0]$ such that", "$I \\otimes^\\mathbf{L} P \\to E$ is zero and such that", "$I|_U \\to \\mathcal{O}_U[0]$ is an", "isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Set $\\mathcal{D} = D_{\\QCoh, T}(\\mathcal{O}_X)$. In both cases the complex", "$K = R\\SheafHom(P, E)$ is an object of $\\mathcal{D}$. See", "Lemma \\ref{lemma-quasi-coherence-internal-hom} for quasi-coherence.", "It is clear that $K$ is supported on $T$ as formation of $R\\SheafHom$", "commutes with restriction to opens.", "The map $\\alpha$ defines an element of", "$H^0(K) = \\Hom_{D(\\mathcal{O}_X)}(\\mathcal{O}_X[0], K)$.", "Then it suffices to prove the result for the map", "$\\alpha : \\mathcal{O}_X[0] \\to K$.", "\\medskip\\noindent", "Let $E \\in \\mathcal{D}$ be a perfect generator, see", "Lemma \\ref{lemma-generator-with-support}. Write", "$$", "K = \\text{hocolim} K_n", "$$", "as in Derived Categories, Lemma \\ref{derived-lemma-write-as-colimit}", "using the generator $E$. Since the functor $\\mathcal{D} \\to D(\\mathcal{O}_X)$", "commutes with direct sums, we see that $K = \\text{hocolim} K_n$", "holds in $D(\\mathcal{O}_X)$. Since $\\mathcal{O}_X$ is a compact", "object of $D(\\mathcal{O}_X)$ we find an $n$ and a morphism", "$\\alpha_n : \\mathcal{O}_X \\to K_n$ which gives rise to $\\alpha$, see", "Derived Categories, Lemma \\ref{derived-lemma-commutes-with-countable-sums}.", "By Derived Categories, Lemma \\ref{derived-lemma-factor-through}", "applied to the morphism $\\mathcal{O}_X[0] \\to K_n$ in the ambient", "category $D(\\mathcal{O}_X)$ we see that $\\alpha_n$ factors as", "$\\mathcal{O}_X[0] \\to Q \\to K_n$ where $Q$ is an object", "of $\\langle E \\rangle$. We conclude that $Q$ is a perfect complex", "supported on $T$.", "\\medskip\\noindent", "Choose a distinguished triangle", "$$", "I \\to \\mathcal{O}_X[0] \\to Q \\to I[1]", "$$", "By construction $I$ is perfect, the map $I \\to \\mathcal{O}_X[0]$", "restricts to an isomorphism over $U$, and the composition", "$I \\to K$ is zero as $\\alpha$ factors through $Q$.", "This proves the lemma." ], "refs": [ "spaces-perfect-lemma-quasi-coherence-internal-hom", "spaces-perfect-lemma-generator-with-support", "derived-lemma-write-as-colimit", "derived-lemma-commutes-with-countable-sums", "derived-lemma-factor-through" ], "ref_ids": [ 2700, 2708, 1941, 1924, 1942 ] } ], "ref_ids": [] }, { "id": 2711, "type": "theorem", "label": "spaces-perfect-lemma-tensor-with-QCoh-complex", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-tensor-with-QCoh-complex", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $K^\\bullet$", "be a complex of $\\mathcal{O}_X$-modules whose cohomology sheaves are", "quasi-coherent. Let", "$(E, d) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, K^\\bullet)$", "be the endomorphism differential graded algebra. Then the functor", "$$", "- \\otimes_E^\\mathbf{L} K^\\bullet :", "D(E, \\text{d}) \\longrightarrow D(\\mathcal{O}_X)", "$$", "of", "Differential Graded Algebra, Lemma", "\\ref{dga-lemma-tensor-with-complex-derived}", "has image contained in $D_\\QCoh(\\mathcal{O}_X)$." ], "refs": [ "dga-lemma-tensor-with-complex-derived" ], "proofs": [ { "contents": [ "Let $P$ be a differential graded $E$-module with property $P$.", "Let $F_\\bullet$ be a filtration on $P$ as in", "Differential Graded Algebra, Section \\ref{dga-section-P-resolutions}.", "Then we have", "$$", "P \\otimes_E K^\\bullet = \\text{hocolim}\\ F_iP \\otimes_E K^\\bullet", "$$", "Each of the $F_iP$ has a finite filtration whose graded pieces", "are direct sums of $E[k]$. The result follows easily." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13116 ] }, { "id": 2712, "type": "theorem", "label": "spaces-perfect-lemma-ext-from-perfect-into-bounded-QCoh", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-ext-from-perfect-into-bounded-QCoh", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $K$ be a perfect object of $D(\\mathcal{O}_X)$. Then", "\\begin{enumerate}", "\\item there exist integers $a \\leq b$ such that", "$\\Hom_{D(\\mathcal{O}_X)}(K, L) = 0$ for $L \\in D_\\QCoh(\\mathcal{O}_X)$", "with $H^i(L) = 0$ for $i \\in [a, b]$, and", "\\item if $L$ is bounded, then $\\Ext^n_{D(\\mathcal{O}_X)}(K, L)$", "is zero for all but finitely many $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) follows from (1) as $\\Ext^n_{D(\\mathcal{O}_X)}(K, L) =", "\\Hom_{D(\\mathcal{O}_X)}(K, L[n])$. We prove (1).", "Since $K$ is perfect we have", "$$", "\\Ext^i_{D(\\mathcal{O}_X)}(K, L) =", "H^i(X, K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)", "$$", "where $K^\\vee$ is the ``dual'' perfect complex to $K$, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-dual-perfect-complex}.", "Note that $P = K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$", "is in $D_\\QCoh(X)$ by", "Lemmas \\ref{lemma-quasi-coherence-tensor-product} and", "\\ref{lemma-pseudo-coherent} (to see that a perfect complex", "has quasi-coherent cohomology sheaves). Say $K^\\vee$ has", "tor amplitude in $[a, b]$. Then the spectral sequence", "$$", "E_1^{p, q} = H^p(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} H^q(L))", "\\Rightarrow", "H^{p + q}(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)", "$$", "shows that $H^j(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)$", "is zero if $H^q(L) = 0$ for $q \\in [j - b, j - a]$.", "Let $N$ be the integer $\\max(d_p + p)$ of", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-vanishing-quasi-separated}.", "Then $H^0(X, K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)$", "vanishes if the cohomology sheaves", "$$", "H^{-N}(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L),", "\\ H^{-N + 1}(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L),", "\\ \\ldots,", "\\ H^0(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)", "$$", "are zero. Namely, by the lemma cited and", "Lemma \\ref{lemma-application-nice-K-injective}, we have", "$$", "H^0(X, K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) =", "H^0(X, \\tau_{\\geq -N}(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L))", "$$", "and by the vanishing of cohomology sheaves, this is equal to", "$H^0(X, \\tau_{\\geq 1}(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L))$", "which is zero by Derived Categories, Lemma", "\\ref{derived-lemma-negative-vanishing}.", "It follows that $\\Hom_{D(\\mathcal{O}_X)}(K, L)$ is zero if", "$H^i(L) = 0$ for $i \\in [-b - N, -a]$." ], "refs": [ "sites-cohomology-lemma-dual-perfect-complex", "spaces-perfect-lemma-quasi-coherence-tensor-product", "spaces-perfect-lemma-pseudo-coherent", "spaces-cohomology-lemma-vanishing-quasi-separated", "spaces-perfect-lemma-application-nice-K-injective", "derived-lemma-negative-vanishing" ], "ref_ids": [ 4390, 2649, 2696, 11286, 2651, 1839 ] } ], "ref_ids": [] }, { "id": 2713, "type": "theorem", "label": "spaces-perfect-lemma-pseudo-coherent-hocolim", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-pseudo-coherent-hocolim", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $K \\in D(\\mathcal{O}_X)$. The following are equivalent", "\\begin{enumerate}", "\\item $K$ is pseudo-coherent, and", "\\item $K = \\text{hocolim} K_n$ where", "$K_n$ is perfect and $\\tau_{\\geq -n}K_n \\to \\tau_{\\geq -n}K$", "is an isomorphism for all $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is true on any ringed site.", "Namely, assume (2) holds. Recall that a perfect object of the derived", "category is pseudo-coherent, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-perfect}.", "Then it follows from the definitions that", "$\\tau_{\\geq -n}K_n$ is $(-n + 1)$-pseudo-coherent", "and hence $\\tau_{\\geq -n}K$ is $(-n + 1)$-pseudo-coherent,", "hence $K$ is $(-n + 1)$-pseudo-coherent. This is true for", "all $n$, hence $K$ is pseudo-coherent, see", "Cohomology on Sites, Definition", "\\ref{sites-cohomology-definition-pseudo-coherent}.", "\\medskip\\noindent", "Assume (1). We start by choosing an approximation", "$K_1 \\to K$ of $(X, K, -2)$ by a perfect complex $K_1$, see", "Definitions \\ref{definition-approximation-holds} and", "\\ref{definition-approximation} and", "Theorem \\ref{theorem-approximation}.", "Suppose by induction we have", "$$", "K_1 \\to K_2 \\to \\ldots \\to K_n \\to K", "$$", "with $K_i$ perfect such that", "such that $\\tau_{\\geq -i}K_i \\to \\tau_{\\geq -i}K$ is an isomorphism", "for all $1 \\leq i \\leq n$. Then we pick $a \\leq b$ as in", "Lemma \\ref{lemma-ext-from-perfect-into-bounded-QCoh}", "for the perfect object $K_n$. Choose an approximation", "$K_{n + 1} \\to K$ of $(X, K, \\min(a - 1, -n - 1))$.", "Choose a distinguished triangle", "$$", "K_{n + 1} \\to K \\to C \\to K_{n + 1}[1]", "$$", "Then we see that $C \\in D_\\QCoh(\\mathcal{O}_X)$ has", "$H^i(C) = 0$ for $i \\geq a$. Thus by our choice of $a, b$", "we see that $\\Hom_{D(\\mathcal{O}_X)}(K_n, C) = 0$.", "Hence the composition $K_n \\to K \\to C$ is zero. Hence by", "Derived Categories, Lemma \\ref{derived-lemma-representable-homological}", "we can factor $K_n \\to K$ through $K_{n + 1}$", "proving the induction step.", "\\medskip\\noindent", "We still have to prove that $K = \\text{hocolim} K_n$.", "This follows by an application of", "Derived Categories, Lemma \\ref{derived-lemma-cohomology-of-hocolim}", "to the functors", "$H^i( - ) : D(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X)$", "and our choice of $K_n$." ], "refs": [ "sites-cohomology-lemma-perfect", "sites-cohomology-definition-pseudo-coherent", "spaces-perfect-definition-approximation-holds", "spaces-perfect-definition-approximation", "spaces-perfect-theorem-approximation", "spaces-perfect-lemma-ext-from-perfect-into-bounded-QCoh", "derived-lemma-representable-homological", "derived-lemma-cohomology-of-hocolim" ], "ref_ids": [ 4383, 4420, 2765, 2766, 2639, 2712, 1758, 1923 ] } ], "ref_ids": [] }, { "id": 2714, "type": "theorem", "label": "spaces-perfect-lemma-pseudo-coherent-hocolim-with-support", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-pseudo-coherent-hocolim-with-support", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $T \\subset X$ be a closed subset such that $X \\setminus T$", "is quasi-compact. Let $K \\in D(\\mathcal{O}_X)$ supported on $T$.", "The following are equivalent", "\\begin{enumerate}", "\\item $K$ is pseudo-coherent, and", "\\item $K = \\text{hocolim} K_n$ where", "$K_n$ is perfect, supported on $T$, and", "$\\tau_{\\geq -n}K_n \\to \\tau_{\\geq -n}K$ is an isomorphism for all $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The proof of this lemma is exactly the same as the proof of", "Lemma \\ref{lemma-pseudo-coherent-hocolim}", "except that in the choice of the approximations we use", "the triples $(T, K, m)$." ], "refs": [ "spaces-perfect-lemma-pseudo-coherent-hocolim" ], "ref_ids": [ 2713 ] } ], "ref_ids": [] }, { "id": 2715, "type": "theorem", "label": "spaces-perfect-lemma-better-coherator", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-better-coherator", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "The inclusion functor $D_\\QCoh(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$", "has a right adjoint." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "We will use the induction principle in", "Lemma \\ref{lemma-induction-principle}", "to prove this. If $D(\\QCoh(\\mathcal{O}_X)) \\to D_\\QCoh(\\mathcal{O}_X)$", "is an equivalence, then the lemma is true because the functor", "$RQ_X$ of Section \\ref{section-coherator} is a right adjoint to the functor", "$D(\\QCoh(\\mathcal{O}_X)) \\to D(\\mathcal{O}_X)$.", "In particular, our lemma is true for affine algebraic spaces, see", "Lemma \\ref{lemma-affine-coherator}.", "Thus we see that it suffices to show: if $(U \\subset X, f : V \\to X)$", "is an elementary distinguished square with $U$ quasi-compact", "and $V$ affine and the lemma holds for $U$, $V$, and $U \\times_X V$,", "then the lemma holds for $X$.", "\\medskip\\noindent", "The adjoint exists if and only if for every object $K$ of", "$D(\\mathcal{O}_X)$ we can find a distinguished triangle", "$$", "E' \\to E \\to K \\to E'[1]", "$$", "in $D(\\mathcal{O}_X)$", "such that $E'$ is in $D_\\QCoh(\\mathcal{O}_X)$ and such that", "$\\Hom(M, K) = 0$ for all $M$ in $D_\\QCoh(\\mathcal{O}_X)$. See", "Derived Categories, Lemma \\ref{derived-lemma-right-adjoint}.", "Consider the distinguished triangle", "$$", "E \\to Rj_{U, *}E|_U \\oplus Rj_{V, *}E|_V \\to", "Rj_{U \\times_X V, *}E|_{U \\times_X V} \\to E[1]", "$$", "in $D(\\mathcal{O}_X)$ of Lemma \\ref{lemma-exact-sequence-j-star}.", "By Derived Categories, Lemma \\ref{derived-lemma-prepare-adjoint}", "it suffices to construct the desired distinguished triangles", "for $Rj_{U, *}E|_U$, $Rj_{V, *}E|_V$, and", "$Rj_{U \\times_X V, *}E|_{U \\times_X V}$. This reduces us to the statement", "discussed in the next paragraph.", "\\medskip\\noindent", "Let $j : U \\to X$ be an \\'etale morphism corresponding with", "$U$ quasi-compact and quasi-separated and the lemma is true for $U$.", "Let $L$ be an object of $D(\\mathcal{O}_U)$.", "Then there exists a distinguished triangle", "$$", "E' \\to Rj_*L \\to K \\to E'[1]", "$$", "in $D(\\mathcal{O}_X)$", "such that $E'$ is in $D_\\QCoh(\\mathcal{O}_X)$ and such that", "$\\Hom(M, K) = 0$ for all $M$ in $D_\\QCoh(\\mathcal{O}_X)$.", "To see this we choose a distinguished triangle", "$$", "L' \\to L \\to Q \\to L'[1]", "$$", "in $D(\\mathcal{O}_U)$ such that $L'$ is in $D_\\QCoh(\\mathcal{O}_U)$", "and such that $\\Hom(N, Q) = 0$ for all $N$ in $D_\\QCoh(\\mathcal{O}_U)$.", "This is possible because the statement in", "Derived Categories, Lemma \\ref{derived-lemma-right-adjoint}", "is an if and only if.", "We obtain a distinguished triangle", "$$", "Rj_*L' \\to Rj_*L \\to Rj_*Q \\to Rj_*L'[1]", "$$", "in $D(\\mathcal{O}_X)$. Observe that $Rj_*L'$ is in $D_\\QCoh(\\mathcal{O}_X)$", "by Lemma \\ref{lemma-quasi-coherence-direct-image}.", "On the other hand, if $M$ in $D_\\QCoh(\\mathcal{O}_X)$, then", "$$", "\\Hom(M, Rj_*Q) = \\Hom(Lj^*M, Q) = 0", "$$", "because $Lj^*M$ is in $D_\\QCoh(\\mathcal{O}_U)$ by", "Lemma \\ref{lemma-quasi-coherence-pullback}.", "This finishes the proof." ], "refs": [ "spaces-perfect-lemma-induction-principle", "spaces-perfect-lemma-affine-coherator", "derived-lemma-right-adjoint", "spaces-perfect-lemma-exact-sequence-j-star", "derived-lemma-prepare-adjoint", "derived-lemma-right-adjoint", "spaces-perfect-lemma-quasi-coherence-direct-image", "spaces-perfect-lemma-quasi-coherence-pullback" ], "ref_ids": [ 2670, 2683, 1947, 2674, 1946, 1947, 2652, 2648 ] } ], "ref_ids": [] }, { "id": 2716, "type": "theorem", "label": "spaces-perfect-lemma-pushforward-better-coherator", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-pushforward-better-coherator", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a quasi-compact and quasi-separated", "morphism of algebraic spaces over $S$.", "If the right adjoints $DQ_X$ and $DQ_Y$", "of the inclusion functors $D_\\QCoh \\to D$ exist for $X$ and $Y$, then", "$$", "Rf_* \\circ DQ_X = DQ_Y \\circ Rf_*", "$$" ], "refs": [], "proofs": [ { "contents": [ "The statement makes sense because $Rf_*$ sends", "$D_\\QCoh(\\mathcal{O}_X)$ into $D_\\QCoh(\\mathcal{O}_Y)$ by", "Lemma \\ref{lemma-quasi-coherence-direct-image}.", "The statement is true because $Lf^*$ similarly maps", "$D_\\QCoh(\\mathcal{O}_Y)$ into $D_\\QCoh(\\mathcal{O}_X)$", "(Lemma \\ref{lemma-quasi-coherence-pullback})", "and hence both $Rf_* \\circ DQ_X$ and $DQ_Y \\circ Rf_*$", "are right adjoint to $Lf^* : D_\\QCoh(\\mathcal{O}_Y) \\to D(\\mathcal{O}_X)$." ], "refs": [ "spaces-perfect-lemma-quasi-coherence-direct-image", "spaces-perfect-lemma-quasi-coherence-pullback" ], "ref_ids": [ 2652, 2648 ] } ], "ref_ids": [] }, { "id": 2717, "type": "theorem", "label": "spaces-perfect-lemma-boundedness-better-coherator", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-boundedness-better-coherator", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "The functor $DQ_X$ of Lemma \\ref{lemma-better-coherator}", "has the following boundedness property:", "there exists an integer $N = N(X)$ such that, if", "$K$ in $D(\\mathcal{O}_X)$ with", "$H^i(U, K) = 0$ for $U$ affine \\'etale over $X$ and $i \\not \\in [a, b]$, then", "the cohomology sheaves $H^i(DQ_X(K))$ are zero for", "$i \\not \\in [a, b + N]$." ], "refs": [ "spaces-perfect-lemma-better-coherator" ], "proofs": [ { "contents": [ "We will prove this using the induction principle of", "Lemma \\ref{lemma-induction-principle}.", "\\medskip\\noindent", "If $X$ is affine, then the lemma is true with $N = 0$ because then", "$RQ_X = DQ_X$ is given by taking the complex of", "quasi-coherent sheaves associated to $R\\Gamma(X, K)$.", "See Lemma \\ref{lemma-affine-coherator}.", "\\medskip\\noindent", "Let $(U \\subset W, f : V \\to W)$ be an elementary distinguished square", "with $W$ quasi-compact and quasi-separated, $U \\subset W$", "quasi-compact open, $V$ affine such that", "the lemma holds for $U$, $V$, and $U \\times_W V$.", "Say with integers $N(U)$, $N(V)$, and $N(U \\times_W V)$.", "Now suppose $K$ is in $D(\\mathcal{O}_X)$ with", "$H^i(W, K) = 0$ for all affine $W$ \\'etale over $X$ and all $i \\not \\in [a, b]$.", "Then $K|_U$, $K|_V$, $K|_{U \\times_W V}$ have the same property.", "Hence we see that $RQ_U(K|_U)$ and $RQ_V(K|_V)$ and", "$RQ_{U \\cap V}(K|_{U \\times_W V})$ have vanishing cohomology", "sheaves outside the inverval $[a, b + \\max(N(U), N(V), N(U \\times_W V))$.", "Since the functors $Rj_{U, *}$, $Rj_{V, *}$, $Rj_{U \\times_W V, *}$", "have finite cohomological dimension on $D_\\QCoh$ by", "Lemma \\ref{lemma-quasi-coherence-direct-image}", "we see that there exists an $N$ such that", "$Rj_{U, *}DQ_U(K|_U)$, $Rj_{V, *}DQ_V(K|_V)$, and", "$Rj_{U \\cap V, *}DQ_{U \\times_W V}(K|_{U \\times_W V})$ have vanishing", "cohomology sheaves outside the interval $[a, b + N]$.", "Then finally we conclude by the distinguished triangle", "of Remark \\ref{remark-explain-consequence}." ], "refs": [ "spaces-perfect-lemma-induction-principle", "spaces-perfect-lemma-affine-coherator", "spaces-perfect-lemma-quasi-coherence-direct-image", "spaces-perfect-remark-explain-consequence" ], "ref_ids": [ 2670, 2683, 2652, 2776 ] } ], "ref_ids": [ 2715 ] }, { "id": 2718, "type": "theorem", "label": "spaces-perfect-lemma-cohomology-base-change", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-cohomology-base-change", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and quasi-separated", "morphism of algebraic spaces over $S$. For $E$ in", "$D_\\QCoh(\\mathcal{O}_X)$ and", "$K$ in $D_\\QCoh(\\mathcal{O}_Y)$ we have", "$$", "Rf_*(E) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} K =", "Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*K)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Without any assumptions there is a map", "$Rf_*(E) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} K \\to", "Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*K)$.", "Namely, it is the adjoint to the canonical map", "$$", "Lf^*(Rf_*(E) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} K) =", "Lf^*(Rf_*(E)) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*K", "\\longrightarrow", "E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*K", "$$", "coming from the map $Lf^*Rf_*E \\to E$. See", "Cohomology on Sites, Lemmas", "\\ref{sites-cohomology-lemma-pullback-tensor-product} and", "\\ref{sites-cohomology-lemma-adjoint}.", "To check it is an isomorphism we may work \\'etale locally on $Y$.", "Hence we reduce to the case that $Y$ is an affine scheme.", "\\medskip\\noindent", "Suppose that $K = \\bigoplus K_i$ is a direct", "sum of some complexes $K_i \\in D_\\QCoh(\\mathcal{O}_Y)$.", "If the statement holds for each $K_i$, then it holds for $K$.", "Namely, the functors $Lf^*$ and $\\otimes^\\mathbf{L}$ preserve", "direct sums by construction and $Rf_*$ commutes with direct sums", "(for complexes with quasi-coherent cohomology sheaves) by", "Lemma \\ref{lemma-quasi-coherence-pushforward-direct-sums}.", "Moreover, suppose that $K \\to L \\to M \\to K[1]$ is a distinguished", "triangle in $D_\\QCoh(Y)$. Then if the statement of the", "lemma holds for two of $K, L, M$, then it holds for the third", "(as the functors involved are exact functors of triangulated categories).", "\\medskip\\noindent", "Assume $Y$ affine, say $Y = \\Spec(A)$. The functor", "$\\widetilde{\\ } : D(A) \\to D_\\QCoh(\\mathcal{O}_Y)$ is an equivalence", "by", "Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site} and", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-affine-compare-bounded}.", "Let $T$ be the property for $K \\in D(A)$ that", "the statement of the lemma holds for $\\widetilde{K}$.", "The discussion above and", "More on Algebra, Remark \\ref{more-algebra-remark-P-resolution}", "shows that it suffices to prove $T$ holds for $A[k]$.", "This finishes the proof, as the statement of the lemma", "is clear for shifts of the structure sheaf." ], "refs": [ "sites-cohomology-lemma-pullback-tensor-product", "sites-cohomology-lemma-adjoint", "spaces-perfect-lemma-quasi-coherence-pushforward-direct-sums", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "perfect-lemma-affine-compare-bounded", "more-algebra-remark-P-resolution" ], "ref_ids": [ 4244, 4249, 2653, 2644, 6941, 10653 ] } ], "ref_ids": [] }, { "id": 2719, "type": "theorem", "label": "spaces-perfect-lemma-tor-independent", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-tor-independent", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $X$, $Y$ be algebraic spaces over $B$. The following are equivalent", "\\begin{enumerate}", "\\item $X$ and $Y$ are Tor independent over $B$,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & W \\ar[d] & V \\ar[d] \\ar[l] \\\\", "X \\ar[r] & B & Y \\ar[l]", "}", "$$", "with \\'etale vertical arrows $U$ and $V$ are Tor independent over $W$,", "\\item for some commutative diagram as in (2) with (a) $W \\to B$ \\'etale", "surjective, (b) $U \\to X \\times_B W$ \\'etale surjective, (c)", "$V \\to Y \\times_B W$ \\'etale surjective, the spaces $U$ and $V$ are Tor", "independent over $W$, and", "\\item for some commutative diagram as in (3) with $U$, $V$, $W$ schemes,", "the schemes $U$ and $V$ are Tor independent over $W$ in the sense of", "Derived Categories of Schemes, Definition", "\\ref{perfect-definition-tor-independent}.", "\\end{enumerate}" ], "refs": [ "perfect-definition-tor-independent" ], "proofs": [ { "contents": [ "For an \\'etale morphism $\\varphi : U \\to X$ of algebraic spaces", "and geometric point $\\overline{u}$ the map of local rings", "$\\mathcal{O}_{X, \\varphi(\\overline{u})} \\to \\mathcal{O}_{U, \\overline{u}}$", "is an isomorphism. Hence the equivalence of (1) and (2) follows.", "So does the implication (1) $\\Rightarrow$ (3). Assume (3) and", "pick a diagram of geometric points as in", "Definition \\ref{definition-tor-independent}.", "The assumptions imply that we can first lift $\\overline{b}$ to a geometric", "point $\\overline{w}$ of $W$, then lift the geometric point", "$(\\overline{x}, \\overline{b})$ to a geometric point $\\overline{u}$ of $U$,", "and finally lift the geometric point", "$(\\overline{y}, \\overline{b})$ to a geometric point $\\overline{v}$ of $V$.", "Use Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-geometric-lift-to-usual}", "to find the lifts.", "Using the remark on local rings above we conclude that the condition", "of the definition is satisfied for the given diagram.", "\\medskip\\noindent", "Having made these initial points, it is clear that (4) comes down to the", "statement that", "Definition \\ref{definition-tor-independent}", "agrees with", "Derived Categories of Schemes, Definition", "\\ref{perfect-definition-tor-independent}", "when $X$, $Y$, and $B$ are schemes.", "\\medskip\\noindent", "Let $\\overline{x}, \\overline{b}, \\overline{y}$ be as in", "Definition \\ref{definition-tor-independent} lying over the points", "$x, y, b$. Recall that", "$\\mathcal{O}_{X, \\overline{x}} = \\mathcal{O}_{X, x}^{sh}$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring}) and similarly", "for the other two. By Algebra, Lemma", "\\ref{algebra-lemma-strictly-henselian-functorial-improve} we see that", "$\\mathcal{O}_{X, \\overline{x}}$ is a strict henselization of", "$\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{B, b}} \\mathcal{O}_{B, \\overline{b}}$.", "In particular, the ring map", "$$", "\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{B, b}} \\mathcal{O}_{B, \\overline{b}}", "\\longrightarrow", "\\mathcal{O}_{X, \\overline{x}}", "$$", "is flat (More on Algebra, Lemma", "\\ref{more-algebra-lemma-dumb-properties-henselization}). By", "More on Algebra, Lemma \\ref{more-algebra-lemma-tor-independent-flat}", "we see that", "$$", "\\text{Tor}_i^{\\mathcal{O}_{B, b}}(\\mathcal{O}_{X, x}, \\mathcal{O}_{Y, y})", "\\otimes_{\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{B, b}} \\mathcal{O}_{Y, y}}", "(\\mathcal{O}_{X, \\overline{x}} \\otimes_{\\mathcal{O}_{B, \\overline{b}}}", "\\mathcal{O}_{Y, \\overline y})", "=", "\\text{Tor}_i^{\\mathcal{O}_{B, \\overline{b}}}(", "\\mathcal{O}_{X, \\overline{x}}, \\mathcal{O}_{Y, \\overline{y}})", "$$", "Hence it follows that if $X$ and $Y$ are Tor independent over $B$", "as schemes, then $X$ and $Y$ are Tor independent as algebraic spaces over $B$.", "\\medskip\\noindent", "For the converse, we may assume $X$, $Y$, and $B$ are affine.", "Observe that the ring map", "$$", "\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{B, b}} \\mathcal{O}_{Y, y}", "\\longrightarrow", "\\mathcal{O}_{X, \\overline{x}} \\otimes_{\\mathcal{O}_{B, \\overline{b}}}", "\\mathcal{O}_{Y, \\overline y}", "$$", "is flat by the observations given above. Moreover, the image of the map", "on spectra includes all primes", "$\\mathfrak s \\subset", "\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{B, b}} \\mathcal{O}_{Y, y}$", "lying over $\\mathfrak m_x$ and $\\mathfrak m_y$.", "Hence from this and the displayed formula of Tor's above we see that if", "$X$ and $Y$ are Tor independent over $B$ as algebraic spaces, then", "$$", "\\text{Tor}_i^{\\mathcal{O}_{B, b}}", "(\\mathcal{O}_{X, x}, \\mathcal{O}_{Y, y})_\\mathfrak s = 0", "$$", "for all $i > 0$ and all $\\mathfrak s$ as above. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-tor-independent}", "applied to the ring maps", "$\\Gamma(B, \\mathcal{O}_B) \\to \\Gamma(X, \\mathcal{O}_X)$", "and", "$\\Gamma(B, \\mathcal{O}_B) \\to \\Gamma(X, \\mathcal{O}_X)$", "this implies that $X$ and $Y$ are Tor independent over $B$." ], "refs": [ "spaces-perfect-definition-tor-independent", "spaces-properties-lemma-geometric-lift-to-usual", "spaces-perfect-definition-tor-independent", "perfect-definition-tor-independent", "spaces-perfect-definition-tor-independent", "spaces-properties-lemma-describe-etale-local-ring", "algebra-lemma-strictly-henselian-functorial-improve", "more-algebra-lemma-dumb-properties-henselization", "more-algebra-lemma-tor-independent-flat", "more-algebra-lemma-tor-independent" ], "ref_ids": [ 2767, 11871, 2767, 7118, 2767, 11884, 1305, 10055, 10140, 10143 ] } ], "ref_ids": [ 7118 ] }, { "id": 2720, "type": "theorem", "label": "spaces-perfect-lemma-compare-base-change", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-compare-base-change", "contents": [ "Let $S$ be a scheme. Let $g : Y' \\to Y$ be a morphism of algebraic spaces over", "$S$. Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of", "algebraic spaces over $S$. Consider the base change diagram", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "Y' \\ar[r]^g &", "Y", "}", "$$", "If $X$ and $Y'$ are Tor independent over $Y$, then for all", "$E \\in D_\\QCoh(\\mathcal{O}_X)$ we have", "$Rf'_*L(g')^*E = Lg^*Rf_*E$." ], "refs": [], "proofs": [ { "contents": [ "For any object $E$ of $D(\\mathcal{O}_X)$ we can use", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change}", "to get a canonical base change map $Lg^*Rf_*E \\to Rf'_*L(g')^*E$. To check this", "is an isomorphism we may work \\'etale locally on $Y'$. Hence we may assume", "$g : Y' \\to Y$ is a morphism of affine schemes. In particular, $g$", "is affine and it suffices to show that", "$$", "Rg_*Lg^*Rf_*E \\to Rg_*Rf'_*L(g')^*E = Rf_*(Rg'_* L(g')^* E)", "$$", "is an isomorphism, see Lemma \\ref{lemma-affine-morphism}", "(and use Lemmas \\ref{lemma-quasi-coherence-pullback},", "\\ref{lemma-quasi-coherence-tensor-product}, and", "\\ref{lemma-quasi-coherence-direct-image}", "to see that the objects $Rf'_*L(g')^*E$ and $Lg^*Rf_*E$", "have quasi-coherent cohomology sheaves). Note that $g'$ is", "affine as well (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-affine}).", "By Lemma \\ref{lemma-affine-morphism-pull-push} the map becomes a map", "$$", "Rf_*E \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} g_*\\mathcal{O}_{Y'}", "\\longrightarrow", "Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} g'_*\\mathcal{O}_{X'})", "$$", "Observe that $g'_*\\mathcal{O}_{X'} = f^*g_*\\mathcal{O}_{Y'}$. Thus by", "Lemma \\ref{lemma-cohomology-base-change} it suffices to prove that", "$Lf^*g_*\\mathcal{O}_{Y'} = f^*g_*\\mathcal{O}_{Y'}$. This follows from our", "assumption that $X$ and $Y'$ are Tor independent over $Y$. Namely, to", "check it we may work \\'etale locally on $X$, hence we may also assume $X$", "is affine. Say $X = \\Spec(A)$, $Y = \\Spec(R)$ and $Y' = \\Spec(R')$.", "Our assumption implies that $A$ and $R'$ are Tor independent over $R$", "(see", "Lemma \\ref{lemma-tor-independent}", "and", "More on Algebra, Lemma \\ref{more-algebra-lemma-tor-independent}), i.e.,", "$\\text{Tor}_i^R(A, R') = 0$ for $i > 0$. In other words", "$A \\otimes_R^\\mathbf{L} R' = A \\otimes_R R'$ which exactly means", "that $Lf^*g_*\\mathcal{O}_{Y'} = f^*g_*\\mathcal{O}_{Y'}$." ], "refs": [ "sites-cohomology-remark-base-change", "spaces-perfect-lemma-affine-morphism", "spaces-perfect-lemma-quasi-coherence-pullback", "spaces-perfect-lemma-quasi-coherence-tensor-product", "spaces-perfect-lemma-quasi-coherence-direct-image", "spaces-morphisms-lemma-base-change-affine", "spaces-perfect-lemma-affine-morphism-pull-push", "spaces-perfect-lemma-cohomology-base-change", "spaces-perfect-lemma-tor-independent", "more-algebra-lemma-tor-independent" ], "ref_ids": [ 4424, 2654, 2648, 2649, 2652, 4800, 2655, 2718, 2719, 10143 ] } ], "ref_ids": [] }, { "id": 2721, "type": "theorem", "label": "spaces-perfect-lemma-affine-morphism-and-hom-out-of-perfect", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-affine-morphism-and-hom-out-of-perfect", "contents": [ "Let $g : S' \\to S$ be a morphism of affine schemes.", "Consider a cartesian square", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "of quasi-compact and quasi-separated algebraic spaces. Assume $g$ and $f$", "Tor independent. Write $S = \\Spec(R)$ and $S' = \\Spec(R')$. For", "$M, K \\in D(\\mathcal{O}_X)$ the canonical map", "$$", "R\\Hom_X(M, K) \\otimes^\\mathbf{L}_R R'", "\\longrightarrow", "R\\Hom_{X'}(L(g')^*M, L(g')^*K)", "$$", "in $D(R')$ is an isomorphism in the following two cases", "\\begin{enumerate}", "\\item $M \\in D(\\mathcal{O}_X)$ is perfect and $K \\in D_\\QCoh(X)$, or", "\\item $M \\in D(\\mathcal{O}_X)$ is pseudo-coherent,", "$K \\in D_\\QCoh^+(X)$, and $R'$ has finite tor dimension over $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "There is a canonical map", "$R\\Hom_X(M, K) \\to R\\Hom_{X'}(L(g')^*M, L(g')^*K)$", "in $D(\\Gamma(X, \\mathcal{O}_X))$ of global hom complexes, see", "Cohomology on Sites, Section \\ref{sites-cohomology-section-global-RHom}.", "Restricting scalars we can view this as a map in $D(R)$.", "Then we can use the adjointness of restriction and", "$- \\otimes_R^\\mathbf{L} R'$ to get the displayed map of the lemma.", "Having defined the map it suffices to prove it is an isomorphism", "in the derived category of abelian groups.", "\\medskip\\noindent", "The right hand side is equal to", "$$", "R\\Hom_X(M, R(g')_*L(g')^*K) =", "R\\Hom_X(M, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} g'_*\\mathcal{O}_{X'})", "$$", "by Lemma \\ref{lemma-affine-morphism-pull-push}. In both cases the complex", "$R\\SheafHom(M, K)$ is an object of $D_\\QCoh(\\mathcal{O}_X)$ by", "Lemma \\ref{lemma-quasi-coherence-internal-hom}. There is a natural map", "$$", "R\\SheafHom(M, K) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} g'_*\\mathcal{O}_{X'}", "\\longrightarrow", "R\\SheafHom(M, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} g'_*\\mathcal{O}_{X'})", "$$", "which is an isomorphism in both cases", "Lemma \\ref{lemma-internal-hom-evaluate-tensor-isomorphism}.", "To see that this lemma applies in case (2) we note that", "$g'_*\\mathcal{O}_{X'} = Rg'_*\\mathcal{O}_{X'} =", "Lf^*g_*\\mathcal{O}_X$ the second equality by", "Lemma \\ref{lemma-compare-base-change}.", "Using Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-tor-dimension-affine},", "Lemma \\ref{lemma-descend-tor-amplitude}, and", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-tor-amplitude-pullback}", "we conclude that $g'_*\\mathcal{O}_{X'}$ has finite Tor dimension.", "Hence, in both cases by replacing $K$ by $R\\SheafHom(M, K)$ we reduce", "to proving", "$$", "R\\Gamma(X, K) \\otimes^\\mathbf{L}_A A' \\longrightarrow", "R\\Gamma(X, K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} g'_*\\mathcal{O}_{X'})", "$$", "is an isomorphism.", "Note that the left hand side is equal to $R\\Gamma(X', L(g')^*K)$", "by Lemma \\ref{lemma-affine-morphism-pull-push}.", "Hence the result follows from", "Lemma \\ref{lemma-compare-base-change}." ], "refs": [ "spaces-perfect-lemma-affine-morphism-pull-push", "spaces-perfect-lemma-quasi-coherence-internal-hom", "spaces-perfect-lemma-internal-hom-evaluate-tensor-isomorphism", "spaces-perfect-lemma-compare-base-change", "perfect-lemma-tor-dimension-affine", "spaces-perfect-lemma-descend-tor-amplitude", "sites-cohomology-lemma-tor-amplitude-pullback", "spaces-perfect-lemma-affine-morphism-pull-push", "spaces-perfect-lemma-compare-base-change" ], "ref_ids": [ 2655, 2700, 2701, 2720, 6977, 2693, 4375, 2655, 2720 ] } ], "ref_ids": [] }, { "id": 2722, "type": "theorem", "label": "spaces-perfect-lemma-tor-independence-and-tor-amplitude", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-tor-independence-and-tor-amplitude", "contents": [ "Let $S$ be a scheme. Consider a cartesian square of algebraic spaces", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "over $S$. Assume $g$ and $f$ Tor independent.", "\\begin{enumerate}", "\\item If $E \\in D(\\mathcal{O}_X)$ has tor amplitude", "in $[a, b]$ as a complex of $f^{-1}\\mathcal{O}_Y$-modules,", "then $L(g')^*E$ has tor amplitude", "in $[a, b]$ as a complex of $f^{-1}\\mathcal{O}_{Y'}$-modules.", "\\item If $\\mathcal{G}$ is an $\\mathcal{O}_X$-module flat", "over $Y$, then $L(g')^*\\mathcal{G} = (g')^*\\mathcal{G}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We can compute tor dimension at stalks, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-tor-amplitude-stalk}", "and Properties of Spaces, Theorem", "\\ref{spaces-properties-theorem-exactness-stalks}.", "If $\\overline{x}'$ is a geometric point of $X'$ with image", "$\\overline{x}$ in $X$, then", "$$", "(L(g')^*E)_{\\overline{x}'} =", "E_{\\overline{x}}", "\\otimes_{\\mathcal{O}_{X, \\overline{x}}}^\\mathbf{L}", "\\mathcal{O}_{X', \\overline{x}'}", "$$", "Let $\\overline{y}'$ in $Y'$ and $\\overline{y}$ in $Y$", "be the image of $\\overline{x}'$ and $\\overline{x}$.", "Since $X$ and $Y'$ are tor independent over $Y$, we can apply", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-comparison}", "to see that the right hand side of the displayed formula is equal to", "$E_{\\overline{x}}", "\\otimes_{\\mathcal{O}_{Y, \\overline{y}}}^\\mathbf{L}", "\\mathcal{O}_{Y', \\overline{y}'}$", "in $D(\\mathcal{O}_{Y', \\overline{y}'})$.", "Thus (1) follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-pull-tor-amplitude}.", "To see (2) observe that flatness of $\\mathcal{G}$ is equivalent to", "the condition that $\\mathcal{G}[0]$ has tor amplitude in $[0, 0]$.", "Applying (1) we conclude." ], "refs": [ "sites-cohomology-lemma-tor-amplitude-stalk", "spaces-properties-theorem-exactness-stalks", "more-algebra-lemma-base-change-comparison", "more-algebra-lemma-pull-tor-amplitude" ], "ref_ids": [ 4380, 11813, 10139, 10180 ] } ], "ref_ids": [] }, { "id": 2723, "type": "theorem", "label": "spaces-perfect-lemma-single-complex-base-change-condition", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-single-complex-base-change-condition", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "Y' \\ar[r]^g &", "Y", "}", "$$", "be a cartesian diagram of algebraic spaces over $S$.", "Let $K \\in D_\\QCoh(\\mathcal{O}_X)$ and let $L(g')^*K \\to K'$", "be a map in $D_\\QCoh(\\mathcal{O}_{X'})$. The following are equivalent", "\\begin{enumerate}", "\\item for any $x' \\in X'$ and $i \\in \\mathbf{Z}$ the map (\\ref{equation-bc})", "is an isomorphism,", "\\item for any commutative diagram", "$$", "\\xymatrix{", "& U \\ar[d] \\ar[rd]^a \\\\", "V' \\ar[r] \\ar[rd]^c & V \\ar[rd]^b & X \\ar[d]^f \\\\", "& Y' \\ar[r]^g & Y", "}", "$$", "with $a, b, c$ \\'etale, $U, V, V'$ schemes, and with $U' = V' \\times_V U$", "the equivalent conditions of", "Derived Categories of Schemes, Lemma", "\\ref{lemma-single-complex-base-change-condition}", "hold for $(U \\to X)^*K$ and $(U' \\to X')^*K'$, and", "\\item there is some diagram as in (2) with $U' \\to X'$ surjective.", "\\end{enumerate}" ], "refs": [ "spaces-perfect-lemma-single-complex-base-change-condition" ], "proofs": [ { "contents": [ "Observe that (1) is \\'etale local on $X'$. Working through formal", "implications of what is known, we see that it suffices to prove", "condition (1) of this lemma is equivalent to condition", "(1) of Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-single-complex-base-change-condition}", "if $X, Y, Y', X'$ are representable by schemes", "$X_0, Y_0, Y'_0, X'_0$. Denote $f_0, g_0, g'_0, f'_0$ the", "morphisms between these schemes corresponding to $f, g, g', f'$.", "We may assume $K = \\epsilon^*K_0$ and $K' = \\epsilon^*K'_0$", "for some objects $K_0 \\in D_\\QCoh(\\mathcal{O}_{X_0})$", "and $K'_0 \\in D_\\QCoh(\\mathcal{O}_{X'_0})$, see", "Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site}.", "Moreover, the map $Lg^*K \\to K'$ is the pullback", "of a map $L(g_0)^*K_0 \\to K'_0$ with notation as in", "Remark \\ref{remark-match-total-direct-images}.", "Recall that $\\mathcal{O}_{X, \\overline{x}}$ is the", "strict henselization of $\\mathcal{O}_{X, x}$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring})", "and that we have", "$$", "K_{\\overline{x}} =", "K_{0, x} \\otimes_{\\mathcal{O}_{X, x}}^\\mathbf{L} \\mathcal{O}_{X, \\overline{x}}", "\\quad\\text{and}\\quad", "K'_{\\overline{x}'} =", "K'_{0, x'} \\otimes_{\\mathcal{O}_{X', x'}}^\\mathbf{L}", "\\mathcal{O}_{X', \\overline{x}'}", "$$", "(akin to Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-stalk-quasi-coherent}).", "Consider the commutative diagram", "$$", "\\xymatrix{", "H^i(K_{\\overline{x}}", "\\otimes_{\\mathcal{O}_{Y, \\overline{y}}}^\\mathbf{L}", "\\mathcal{O}_{Y', \\overline{y}'})", "\\otimes_{(\\mathcal{O}_{X, \\overline{x}}", "\\otimes_{\\mathcal{O}_{Y, \\overline{y}}}", "\\mathcal{O}_{Y', \\overline{y}'})}", "\\mathcal{O}_{X', \\overline{x}'}", "\\ar[r] &", "H^i(K'_{\\overline{x}'}) \\\\", "H^i(K_{0, x} \\otimes_{\\mathcal{O}_{Y, y}}^\\mathbf{L} \\mathcal{O}_{Y', y'})", "\\otimes_{(\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{Y, y}} \\mathcal{O}_{Y', y'})}", "\\mathcal{O}_{X', x'}", "\\ar[u] \\ar[r] &", "H^i(K'_{0, x'}) \\ar[u]", "}", "$$", "We have to show that the lower horizontal arrow is an isomorphism if and only", "if the upper horizontal arrow is an isomorphism. Since", "$\\mathcal{O}_{X', x'} \\to \\mathcal{O}_{X', \\overline{x}'}$", "is faithfully flat (More on Algebra, Lemma", "\\ref{more-algebra-lemma-dumb-properties-henselization})", "it suffices to show that the top arrow is the base", "change of the bottom arrow by this map.", "This follows immediately from the relationships between", "stalks given above for the objects on the right.", "For the objects on the left it suffices to show that", "\\begin{align*}", "&", "H^i\\left(", "(K_{0, x}", "\\otimes_{\\mathcal{O}_{X, x}}^\\mathbf{L} \\mathcal{O}_{X, \\overline{x}})", "\\otimes_{\\mathcal{O}_{Y, \\overline{y}}}^\\mathbf{L}", "\\mathcal{O}_{Y', \\overline{y}'}\\right) \\\\", "& =", "H^i(K_{0, x} \\otimes_{\\mathcal{O}_{Y, y}}^\\mathbf{L} \\mathcal{O}_{Y', y'})", "\\otimes_{(\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{Y, y}} \\mathcal{O}_{Y', y'})}", "(\\mathcal{O}_{X, \\overline{x}}", "\\otimes_{\\mathcal{O}_{Y, \\overline{y}}}", "\\mathcal{O}_{Y', \\overline{y}'})", "\\end{align*}", "This follows from More on Algebra, Lemma", "\\ref{more-algebra-lemma-lemma-tor-independent-flat-compare}.", "The flatness assumptions of this lemma hold by what was said", "above as well as Algebra, Lemma", "\\ref{algebra-lemma-strictly-henselian-functorial-improve}", "implying that", "$\\mathcal{O}_{X, \\overline{x}}$ is the strict henselization of", "$\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{Y, y}}", "\\mathcal{O}_{Y, \\overline{y}}$", "and that", "$\\mathcal{O}_{Y', \\overline{y}'}$ is the strict henselization of", "$\\mathcal{O}_{Y', y'} \\otimes_{\\mathcal{O}_{Y, y}}", "\\mathcal{O}_{Y, \\overline{y}}$." ], "refs": [ "perfect-lemma-single-complex-base-change-condition", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-remark-match-total-direct-images", "spaces-properties-lemma-describe-etale-local-ring", "spaces-properties-lemma-stalk-quasi-coherent", "more-algebra-lemma-dumb-properties-henselization", "more-algebra-lemma-lemma-tor-independent-flat-compare", "algebra-lemma-strictly-henselian-functorial-improve" ], "ref_ids": [ 7038, 2644, 2768, 11884, 11909, 10055, 10142, 1305 ] } ], "ref_ids": [ 2723 ] }, { "id": 2724, "type": "theorem", "label": "spaces-perfect-lemma-single-complex-base-change", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-single-complex-base-change", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "Y' \\ar[r]^g &", "Y", "}", "$$", "be a cartesian diagram of algebraic spaces over $S$.", "Let $K \\in D_\\QCoh(\\mathcal{O}_X)$ and let $L(g')^*K \\to K'$", "be a map in $D_\\QCoh(\\mathcal{O}_{X'})$. If", "\\begin{enumerate}", "\\item the equivalent conditions of", "Lemma \\ref{lemma-single-complex-base-change-condition} hold, and", "\\item $f$ is quasi-compact and quasi-separated,", "\\end{enumerate}", "then the composition $Lg^*Rf_*K \\to Rf'_*L(g')^*K \\to Rf'_*K'$", "is an isomorphism." ], "refs": [ "spaces-perfect-lemma-single-complex-base-change-condition" ], "proofs": [ { "contents": [ "To check the map is an isomorphism we may work \\'etale locally on $Y'$.", "Hence we may assume $g : Y' \\to Y$ is a morphism of affine schemes.", "In this case, we will use the induction principle of", "Lemma \\ref{lemma-induction-principle}", "to prove that for a quasi-compact and quasi-separated", "algebraic space $U$ \\'etale over $X$", "the similarly constructed map", "$Lg^*R(U \\to Y)_*K|_U \\to R(U' \\to Y')_*K'|_{U'}$", "is an isomorphism. Here $U' = X' \\times_{g', X} U = Y' \\times_{g, Y} U$.", "\\medskip\\noindent", "If $U$ is a scheme (for example affine), then the result holds.", "Namely, then $Y, Y', U, U'$ are schemes, $K$ and $K'$ come from", "objects of the derived category of the underlying schemes by", "Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site}", "and the condition of", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-single-complex-base-change-condition}", "holds for these complexes by", "Lemma \\ref{lemma-single-complex-base-change-condition}.", "Thus (by the compatibilities explained in", "Remark \\ref{remark-match-total-direct-images})", "we can apply the result in the case of schemes", "which is", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-single-complex-base-change}.", "\\medskip\\noindent", "The induction step. Let $(U \\subset W, V \\to W)$ be an elementary", "distinguished square with $W$ a quasi-compact and quasi-separated", "algebraic space \\'etale over $X$, with $U$ quasi-compact, $V$ affine", "and the result holds for $U$, $V$, and $U \\times_W V$.", "To easy notation we replace $W$ by $X$ (this is permissible at this point).", "Denote $a : U \\to Y$, $b : V \\to Y$, and $c : U \\times_X V \\to Y$", "the obvious morphisms. Let $a' : U' \\to Y'$, $b' : V' \\to Y'$", "and $c' : U' \\times_{X'} V' \\to Y'$ be the base changes of $a$, $b$, and $c$.", "Using the distinguished triangles from relative Mayer-Vietoris", "(Lemma \\ref{lemma-unbounded-relative-mayer-vietoris})", "we obtain a commutative diagram", "$$", "\\xymatrix{", "Lg^*Rf_*K \\ar[r] \\ar[d] & Rf'_*K' \\ar[d] \\\\", "Lg^*Ra_*K|_U \\oplus Lg^*Rb_*K|_V \\ar[r] \\ar[d] &", "Ra'_* K'|_{U'} \\oplus Rb'_* K'|_{V'} \\ar[d] \\\\", "Lg^*Rc_*K|_{U \\times_X V} \\ar[r] \\ar[d] &", "Rc'_*K'|_{U' \\times_{X'} V'} \\ar[d] \\\\", "Lg^*Rf_* K[1] \\ar[r] &", "Rf'_* K'[1]", "}", "$$", "Since the 2nd and 3rd horizontal arrows are isomorphisms so is the first", "(Derived Categories, Lemma \\ref{derived-lemma-third-isomorphism-triangle})", "and the proof of the lemma is finished." ], "refs": [ "spaces-perfect-lemma-induction-principle", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "perfect-lemma-single-complex-base-change-condition", "spaces-perfect-lemma-single-complex-base-change-condition", "spaces-perfect-remark-match-total-direct-images", "perfect-lemma-single-complex-base-change", "spaces-perfect-lemma-unbounded-relative-mayer-vietoris", "derived-lemma-third-isomorphism-triangle" ], "ref_ids": [ 2670, 2644, 7038, 2723, 2768, 7039, 2675, 1759 ] } ], "ref_ids": [ 2723 ] }, { "id": 2725, "type": "theorem", "label": "spaces-perfect-lemma-single-complex-base-change-condition-inherited", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-single-complex-base-change-condition-inherited", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "S' \\ar[r]^g &", "S", "}", "$$", "be a cartesian diagram of algebraic spaces over $S$.", "Let $K \\in D_\\QCoh(\\mathcal{O}_X)$ and let $L(g')^*K \\to K'$", "be a map in $D_\\QCoh(\\mathcal{O}_{X'})$. If the equivalent conditions of", "Lemma \\ref{lemma-single-complex-base-change-condition} hold, then", "\\begin{enumerate}", "\\item for $E \\in D_\\QCoh(\\mathcal{O}_X)$ the equivalent", "conditions of Lemma \\ref{lemma-single-complex-base-change-condition} hold", "for $L(g')^*(E \\otimes^\\mathbf{L} K) \\to L(g')^*E \\otimes^\\mathbf{L} K'$,", "\\item if $E$ in $D(\\mathcal{O}_X)$ is perfect the equivalent conditions of", "Lemma \\ref{lemma-single-complex-base-change-condition} hold for", "$L(g')^*R\\SheafHom(E, K) \\to R\\SheafHom(L(g')^*E, K')$, and", "\\item if $K$ is bounded below and $E$ in $D(\\mathcal{O}_X)$", "pseudo-coherent the equivalent conditions of", "Lemma \\ref{lemma-single-complex-base-change-condition} hold for", "$L(g')^*R\\SheafHom(E, K) \\to R\\SheafHom(L(g')^*E, K')$.", "\\end{enumerate}" ], "refs": [ "spaces-perfect-lemma-single-complex-base-change-condition", "spaces-perfect-lemma-single-complex-base-change-condition", "spaces-perfect-lemma-single-complex-base-change-condition", "spaces-perfect-lemma-single-complex-base-change-condition" ], "proofs": [ { "contents": [ "The statement makes sense as the complexes involved have quasi-coherent", "cohomology sheaves by Lemmas", "\\ref{lemma-quasi-coherence-pullback},", "\\ref{lemma-quasi-coherence-tensor-product}, and", "\\ref{lemma-quasi-coherence-internal-hom} and", "Cohomology on Sites, Lemmas", "\\ref{sites-cohomology-lemma-pseudo-coherent-pullback} and", "\\ref{sites-cohomology-lemma-perfect-pullback}.", "Having said this, we can check the maps (\\ref{equation-bc})", "are isomorphisms in case (1) by computing the source and target", "of (\\ref{equation-bc}) using the transitive property of tensor product, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-triple-tensor-product}.", "The map in (2) and (3) is the composition", "$$", "L(g')^*R\\SheafHom(E, K) \\to R\\SheafHom(L(g')^*E, L(g')^*K)", "\\to R\\SheafHom(L(g')^*E, K')", "$$", "where the first arrow is", "Cohomology on Sites, Remark", "\\ref{sites-cohomology-remark-prepare-fancy-base-change}", "and the second arrow comes from the given map $L(g')^*K \\to K'$.", "To prove the maps (\\ref{equation-bc}) are isomorphisms one represents", "$E_x$ by a bounded complex of finite projective $\\mathcal{O}_{X. x}$-modules", "in case (2) or by a bounded above complex of finite free modules in case (3)", "and computes the source and target of the arrow.", "Some details omitted." ], "refs": [ "spaces-perfect-lemma-quasi-coherence-pullback", "spaces-perfect-lemma-quasi-coherence-tensor-product", "spaces-perfect-lemma-quasi-coherence-internal-hom", "sites-cohomology-lemma-pseudo-coherent-pullback", "sites-cohomology-lemma-perfect-pullback", "more-algebra-lemma-triple-tensor-product", "sites-cohomology-remark-prepare-fancy-base-change" ], "ref_ids": [ 2648, 2649, 2700, 4367, 4384, 10134, 4431 ] } ], "ref_ids": [ 2723, 2723, 2723, 2723 ] }, { "id": 2726, "type": "theorem", "label": "spaces-perfect-lemma-base-change-tensor", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-base-change-tensor", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and", "quasi-separated morphism of algebraic spaces over $S$.", "Let $E \\in D_\\QCoh(\\mathcal{O}_X)$.", "Let $\\mathcal{G}^\\bullet$ be a bounded above complex of", "quasi-coherent $\\mathcal{O}_X$-modules flat over $Y$.", "Then formation of", "$$", "Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)", "$$", "commutes with arbitrary base change (see proof for precise statement)." ], "refs": [], "proofs": [ { "contents": [ "The statement means the following. Let $g : Y' \\to Y$ be a morphism of", "algebraic spaces and consider the base change diagram", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "Y' \\ar[r]^g &", "Y", "}", "$$", "in other words $X' = Y' \\times_Y X$. The lemma asserts that", "$$", "Lg^*Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)", "\\longrightarrow", "Rf'_*(L(g')^*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{X'}} (g')^*\\mathcal{G}^\\bullet)", "$$", "is an isomorphism. Observe that on the right hand side we do {\\bf not}", "use derived pullback on $\\mathcal{G}^\\bullet$.", "To prove this, we apply Lemmas \\ref{lemma-single-complex-base-change} and", "\\ref{lemma-single-complex-base-change-condition-inherited} to see that it", "suffices to prove the canonical map", "$$", "L(g')^*\\mathcal{G}^\\bullet \\to (g')^*\\mathcal{G}^\\bullet", "$$", "satisfies the equivalent conditions of", "Lemma \\ref{lemma-single-complex-base-change-condition}.", "This follows by checking the condition on stalks, where it", "immediately follows from the fact that", "$\\mathcal{G}^\\bullet_{\\overline{x}}", "\\otimes_{\\mathcal{O}_{Y, \\overline{y}}}", "\\mathcal{O}_{Y', \\overline{y}'}$", "computes the derived tensor product by our assumptions on the complex", "$\\mathcal{G}^\\bullet$." ], "refs": [ "spaces-perfect-lemma-single-complex-base-change", "spaces-perfect-lemma-single-complex-base-change-condition-inherited", "spaces-perfect-lemma-single-complex-base-change-condition" ], "ref_ids": [ 2724, 2725, 2723 ] } ], "ref_ids": [] }, { "id": 2727, "type": "theorem", "label": "spaces-perfect-lemma-base-change-RHom", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-base-change-RHom", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and", "quasi-separated morphism of algebraic spaces over $S$. Let $E$", "be an object of $D(\\mathcal{O}_X)$. Let $\\mathcal{G}^\\bullet$", "be a complex of quasi-coherent $\\mathcal{O}_X$-modules. If", "\\begin{enumerate}", "\\item $E$ is perfect, $\\mathcal{G}^\\bullet$ is a bounded above,", "and $\\mathcal{G}^n$ is flat over $Y$, or", "\\item $E$ is pseudo-coherent, $\\mathcal{G}^\\bullet$ is bounded,", "and $\\mathcal{G}^n$ is flat over $Y$,", "\\end{enumerate}", "then formation of", "$$", "Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet)", "$$", "commutes with arbitrary base change (see proof for precise statement)." ], "refs": [], "proofs": [ { "contents": [ "The statement means the following. Let $g : Y' \\to Y$ be a morphism of", "algebraic spaces and consider the base change diagram", "$$", "\\xymatrix{", "X' \\ar[r]_h \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "Y' \\ar[r]^g &", "Y", "}", "$$", "in other words $X' = Y' \\times_Y X$. The lemma asserts that", "$$", "Lg^*Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet)", "\\longrightarrow", "R(f')_*R\\SheafHom(L(g')^*E, (g')^*\\mathcal{G}^\\bullet)", "$$", "is an isomorphism. Observe that on the right hand side we do {\\bf not}", "use the derived pullback on $\\mathcal{G}^\\bullet$. To prove this, we apply", "Lemmas \\ref{lemma-single-complex-base-change} and", "\\ref{lemma-single-complex-base-change-condition-inherited} to see that it", "suffices to prove the canonical map", "$$", "L(g')^*\\mathcal{G}^\\bullet \\to (g')^*\\mathcal{G}^\\bullet", "$$", "satisfies the equivalent conditions of", "Lemma \\ref{lemma-single-complex-base-change-condition}.", "This was shown in the proof of Lemma \\ref{lemma-base-change-tensor}." ], "refs": [ "spaces-perfect-lemma-single-complex-base-change", "spaces-perfect-lemma-single-complex-base-change-condition-inherited", "spaces-perfect-lemma-single-complex-base-change-condition", "spaces-perfect-lemma-base-change-tensor" ], "ref_ids": [ 2724, 2725, 2723, 2726 ] } ], "ref_ids": [] }, { "id": 2728, "type": "theorem", "label": "spaces-perfect-lemma-perfect-direct-image", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-perfect-direct-image", "contents": [ "Let $S$ be a scheme. Let $Y$ be a Noetherian algebraic space over $S$.", "Let $f : X \\to Y$ be a morphism of algebraic spaces which is locally of", "finite type and quasi-separated. Let $E \\in D(\\mathcal{O}_X)$ such that", "\\begin{enumerate}", "\\item $E \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$,", "\\item the support of $H^i(E)$ is proper over $Y$ for all $i$,", "\\item $E$ has finite tor dimension as an object of $D(f^{-1}\\mathcal{O}_Y)$.", "\\end{enumerate}", "Then $Rf_*E$ is a perfect object of $D(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-direct-image-coherent} we see that $Rf_*E$ is an object of", "$D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$. Hence $Rf_*E$ is pseudo-coherent", "(Lemma \\ref{lemma-identify-pseudo-coherent-noetherian}).", "Hence it suffices to show that $Rf_*E$ has finite tor dimension, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-perfect}.", "By Lemma \\ref{lemma-tor-qc-qs} it suffices to check that", "$Rf_*(E) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} \\mathcal{F}$", "has universally bounded cohomology for all quasi-coherent", "sheaves $\\mathcal{F}$ on $Y$. Bounded from above is clear as $Rf_*(E)$", "is bounded from above. Let $T \\subset |X|$ be the union of the supports", "of $H^i(E)$ for all $i$. Then $T$ is proper over $Y$ by assumptions (1)", "and (2) and Lemma \\ref{lemma-union-closed-proper-over-base}.", "In particular there exists a quasi-compact open subspace", "$X' \\subset X$ containing $T$. Setting $f' = f|_{X'}$ we have", "$Rf_*(E) = Rf'_*(E|_{X'})$ because $E$ restricts to zero on $X \\setminus T$.", "Thus we may replace $X$ by $X'$ and assume $f$ is quasi-compact.", "We have assumed $f$ is quasi-separated. Thus", "$$", "Rf_*(E) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} \\mathcal{F} =", "Rf_*\\left(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*\\mathcal{F}\\right) =", "Rf_*\\left(E \\otimes_{f^{-1}\\mathcal{O}_Y}^\\mathbf{L} f^{-1}\\mathcal{F}\\right)", "$$", "by", "Lemma \\ref{lemma-cohomology-base-change}", "and", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-variant-derived-pullback}.", "By assumption (3) the complex", "$E \\otimes_{f^{-1}\\mathcal{O}_Y}^\\mathbf{L} f^{-1}\\mathcal{F}$", "has cohomology sheaves in a", "given finite range, say $[a, b]$. Then $Rf_*$ of it", "has cohomology in the range $[a, \\infty)$ and we win." ], "refs": [ "spaces-perfect-lemma-direct-image-coherent", "spaces-perfect-lemma-identify-pseudo-coherent-noetherian", "sites-cohomology-lemma-perfect", "spaces-perfect-lemma-tor-qc-qs", "spaces-perfect-lemma-union-closed-proper-over-base", "spaces-perfect-lemma-cohomology-base-change", "sites-cohomology-lemma-variant-derived-pullback" ], "ref_ids": [ 2665, 2697, 4383, 2698, 2660, 2718, 4245 ] } ], "ref_ids": [] }, { "id": 2729, "type": "theorem", "label": "spaces-perfect-lemma-tensor-perfect", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-tensor-perfect", "contents": [ "Let $S$ be a scheme. Let $B$ be a Noetherian algebraic space over $S$.", "Let $f : X \\to B$ be a morphism of algebraic spaces which is locally of", "finite type and quasi-separated. Let $E \\in D(\\mathcal{O}_X)$ be perfect.", "Let $\\mathcal{G}^\\bullet$ be a bounded complex of coherent", "$\\mathcal{O}_X$-modules flat over $B$ with support proper over $B$. Then", "$K = Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)$", "is a perfect object of $D(\\mathcal{O}_B)$." ], "refs": [], "proofs": [ { "contents": [ "The object $K$ is perfect by Lemma \\ref{lemma-perfect-direct-image}.", "We check the lemma applies: Locally $E$ is isomorphic to a finite complex", "of finite free $\\mathcal{O}_X$-modules. Hence locally", "$E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet$ is isomorphic", "to a finite complex whose terms are of the form", "$$", "\\bigoplus\\nolimits_{i = a, \\ldots, b} (\\mathcal{G}^i)^{\\oplus r_i}", "$$", "for some integers $a, b, r_a, \\ldots, r_b$. This immediately implies the", "cohomology sheaves $H^i(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G})$", "are coherent. The hypothesis on the tor dimension also follows as", "$\\mathcal{G}^i$ is flat over $f^{-1}\\mathcal{O}_Y$." ], "refs": [ "spaces-perfect-lemma-perfect-direct-image" ], "ref_ids": [ 2728 ] } ], "ref_ids": [] }, { "id": 2730, "type": "theorem", "label": "spaces-perfect-lemma-ext-perfect", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-ext-perfect", "contents": [ "Let $S$ be a scheme. Let $B$ be a Noetherian algebraic space over $S$.", "Let $f : X \\to B$ be a morphism of algebraic spaces which is locally of", "finite type and quasi-separated. Let $E \\in D(\\mathcal{O}_X)$ be perfect.", "Let $\\mathcal{G}^\\bullet$ be a bounded complex of coherent", "$\\mathcal{O}_X$-modules flat over $B$ with support proper over $B$. Then", "$K = Rf_*R\\SheafHom(E, \\mathcal{G})$ is a perfect object of $D(\\mathcal{O}_B)$." ], "refs": [], "proofs": [ { "contents": [ "Since $E$ is a perfect complex there exists a dual perfect complex", "$E^\\vee$, see Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-dual-perfect-complex}.", "Observe that $R\\SheafHom(E, \\mathcal{G}^\\bullet) =", "E^\\vee \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet$.", "Thus the perfectness of $K$ follows from Lemma \\ref{lemma-tensor-perfect}." ], "refs": [ "sites-cohomology-lemma-dual-perfect-complex", "spaces-perfect-lemma-tensor-perfect" ], "ref_ids": [ 4390, 2729 ] } ], "ref_ids": [] }, { "id": 2731, "type": "theorem", "label": "spaces-perfect-lemma-compute-tensor-perfect", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-compute-tensor-perfect", "contents": [ "Assumptions and notation as in Lemma \\ref{lemma-tensor-perfect}.", "Then there are functorial isomorphisms", "$$", "H^i(B, K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F})", "\\longrightarrow", "H^i(X, E \\otimes^\\mathbf{L}_{\\mathcal{O}_X}", "(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}))", "$$", "for $\\mathcal{F}$ quasi-coherent on $B$", "compatible with boundary maps (see proof)." ], "refs": [ "spaces-perfect-lemma-tensor-perfect" ], "proofs": [ { "contents": [ "We have", "$$", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*\\mathcal{F} =", "\\mathcal{G}^\\bullet \\otimes_{f^{-1}\\mathcal{O}_B}^\\mathbf{L} f^{-1}\\mathcal{F} =", "\\mathcal{G}^\\bullet \\otimes_{f^{-1}\\mathcal{O}_B} f^{-1}\\mathcal{F} =", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}", "$$", "the first equality by", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-variant-derived-pullback},", "the second as $\\mathcal{G}^n$ is a flat $f^{-1}\\mathcal{O}_B$-module, and", "the third by definition of pullbacks. Hence we obtain", "\\begin{align*}", "H^i(X, E \\otimes^\\mathbf{L}_{\\mathcal{O}_X}", "(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}))", "& =", "H^i(X, E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*\\mathcal{F}) \\\\", "& =", "H^i(B,", "Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet", "\\otimes^\\mathbf{L}_{\\mathcal{O}_X} Lf^*\\mathcal{F})) \\\\", "& =", "H^i(B,", "Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)", "\\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}) \\\\", "& =", "H^i(B, K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F})", "\\end{align*}", "The first equality by the above, the second by Leray", "(Cohomology on Sites, Remark \\ref{sites-cohomology-remark-before-Leray}), and", "the third equality by Lemma \\ref{lemma-cohomology-base-change}.", "The statement on boundary maps means the following: Given a short", "exact sequence $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "then the isomorphisms fit into commutative diagrams", "$$", "\\xymatrix{", "H^i(B, K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}_3)", "\\ar[r] \\ar[d]_\\delta &", "H^i(X, E \\otimes^\\mathbf{L}_{\\mathcal{O}_X}", "(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_3)) \\ar[d]^\\delta \\\\", "H^{i + 1}(B, K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}_1)", "\\ar[r] &", "H^{i + 1}(X, E \\otimes^\\mathbf{L}_{\\mathcal{O}_X}", "(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_1))", "}", "$$", "where the boundary maps come from the distinguished triangle", "$$", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}_1 \\to", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}_2 \\to", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}_3 \\to", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}_1[1]", "$$", "and the distinguished triangle in $D(\\mathcal{O}_X)$ associated to", "the short exact sequence", "$$", "0 \\to", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_1 \\to", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_2 \\to", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_3 \\to 0", "$$", "of complexes.", "This sequence is exact because $\\mathcal{G}^n$ is flat over $B$.", "We omit the verification of the commutativity of the displayed diagram." ], "refs": [ "sites-cohomology-lemma-variant-derived-pullback", "sites-cohomology-remark-before-Leray", "spaces-perfect-lemma-cohomology-base-change" ], "ref_ids": [ 4245, 4423, 2718 ] } ], "ref_ids": [ 2729 ] }, { "id": 2732, "type": "theorem", "label": "spaces-perfect-lemma-compute-ext-perfect", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-compute-ext-perfect", "contents": [ "Assumption and notation as in Lemma \\ref{lemma-ext-perfect}.", "Then there are functorial isomorphisms", "$$", "H^i(B, K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F})", "\\longrightarrow", "\\Ext^i_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "$$", "for $\\mathcal{F}$ quasi-coherent on $B$", "compatible with boundary maps (see proof)." ], "refs": [ "spaces-perfect-lemma-ext-perfect" ], "proofs": [ { "contents": [ "As in the proof of Lemma \\ref{lemma-ext-perfect} let", "$E^\\vee$ be the dual perfect complex and recall that", "$K = Rf_*(E^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)$.", "Since we also have", "$$", "\\Ext^i_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "=", "H^i(X, E^\\vee \\otimes^\\mathbf{L}_{\\mathcal{O}_X}", "(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}))", "$$", "by construction of $E^\\vee$, the existence of the isomorphisms", "follows from Lemma \\ref{lemma-compute-tensor-perfect} applied to $E^\\vee$", "and $\\mathcal{G}^\\bullet$.", "The statement on boundary maps means the following: Given a short", "exact sequence $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "then the isomorphisms fit into commutative diagrams", "$$", "\\xymatrix{", "H^i(B, K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}_3)", "\\ar[r] \\ar[d]_\\delta &", "\\Ext^i_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_3) \\ar[d]^\\delta \\\\", "H^{i + 1}(B, K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}_1)", "\\ar[r] &", "\\Ext^{i + 1}_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_1)", "}", "$$", "where the boundary maps come from the distinguished triangle", "$$", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}_1 \\to", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}_2 \\to", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}_3 \\to", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}_1[1]", "$$", "and the distinguished triangle in $D(\\mathcal{O}_X)$ associated to", "the short exact sequence", "$$", "0 \\to", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_1 \\to", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_2 \\to", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_3 \\to 0", "$$", "of complexes.", "This sequence is exact because $\\mathcal{G}^n$ is flat over $B$.", "We omit the verification of the commutativity of the displayed diagram." ], "refs": [ "spaces-perfect-lemma-ext-perfect", "spaces-perfect-lemma-compute-tensor-perfect" ], "ref_ids": [ 2730, 2731 ] } ], "ref_ids": [ 2730 ] }, { "id": 2733, "type": "theorem", "label": "spaces-perfect-lemma-compute-ext", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-compute-ext", "contents": [ "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces", "over $S$, $E \\in D(\\mathcal{O}_X)$, and $\\mathcal{F}^\\bullet$ a complex", "of $\\mathcal{O}_X$-modules. Assume", "\\begin{enumerate}", "\\item $B$ is Noetherian,", "\\item $f$ is locally of finite type and quasi-separated,", "\\item $E \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$,", "\\item $\\mathcal{G}^\\bullet$ is a bounded complex of coherent", "$\\mathcal{O}_X$-module flat over $B$ with support proper over $B$.", "\\end{enumerate}", "Then the following two statements are true", "\\begin{enumerate}", "\\item[(A)] for every $m \\in \\mathbf{Z}$ there exists a perfect object $K$", "of $D(\\mathcal{O}_B)$ and functorial maps", "$$", "\\alpha^i_\\mathcal{F} :", "\\Ext^i_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "\\longrightarrow", "H^i(B, K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F})", "$$", "for $\\mathcal{F}$ quasi-coherent on $B$", "compatible with boundary maps (see proof)", "such that $\\alpha^i_\\mathcal{F}$ is an isomorphism for $i \\leq m$, and", "\\item[(B)] there exists a pseudo-coherent $L \\in D(\\mathcal{O}_B)$", "and functorial isomorphisms", "$$", "\\Ext^i_{\\mathcal{O}_B}(L, \\mathcal{F}) \\longrightarrow", "\\Ext^i_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "$$", "for $\\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (A). Suppose $\\mathcal{G}^i$ is nonzero only for $i \\in [a, b]$.", "We may replace $X$ by a quasi-compact open neighbourhood of the union", "of the supports of $\\mathcal{G}^i$. Hence we may assume $X$ is Noetherian.", "In this case $X$ and $f$ are quasi-compact and quasi-separated.", "Choose an approximation $P \\to E$ by a perfect complex $P$ of", "$(X, E, -m - 1 + a)$", "(possible by Theorem \\ref{theorem-approximation}).", "Then the induced map", "$$", "\\Ext^i_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "\\longrightarrow", "\\Ext^i_{\\mathcal{O}_X}(P,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "$$", "is an isomorphism for $i \\leq m$. Namely, the kernel, resp.\\ cokernel of this", "map is a quotient, resp.\\ submodule of", "$$", "\\Ext^i_{\\mathcal{O}_X}(C,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "\\quad\\text{resp.}\\quad", "\\Ext^{i + 1}_{\\mathcal{O}_X}(C,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "$$", "where $C$ is the cone of $P \\to E$. Since $C$ has vanishing cohomology", "sheaves in degrees $\\geq -m - 1 + a$ these $\\Ext$-groups are zero", "for $i \\leq m + 1$ by", "Derived Categories, Lemma \\ref{derived-lemma-negative-exts}.", "This reduces us to the case that", "$E$ is a perfect complex which is Lemma \\ref{lemma-compute-ext-perfect}.", "The statement on boundaries is explained in the proof of", "Lemma \\ref{lemma-compute-ext-perfect}.", "\\medskip\\noindent", "Proof of (B). As in the proof of (A) we may assume $X$ is Noetherian.", "Observe that $E$ is pseudo-coherent by", "Lemma \\ref{lemma-identify-pseudo-coherent-noetherian}.", "By Lemma \\ref{lemma-pseudo-coherent-hocolim} we can write", "$E = \\text{hocolim} E_n$ with $E_n$ perfect and $E_n \\to E$ inducing", "an isomorphism on truncations $\\tau_{\\geq -n}$. Let $E_n^\\vee$", "be the dual perfect complex", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-dual-perfect-complex}).", "We obtain an inverse system $\\ldots \\to E_3^\\vee \\to E_2^\\vee \\to E_1^\\vee$", "of perfect objects. This in turn gives rise to an inverse system", "$$", "\\ldots \\to K_3 \\to K_2 \\to K_1\\quad\\text{with}\\quad", "K_n = Rf_*(E_n^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)", "$$", "perfect on $Y$, see Lemma \\ref{lemma-tensor-perfect}.", "By Lemma \\ref{lemma-compute-ext-perfect} and its proof and", "by the arguments in the previous paragraph (with $P = E_n$)", "for any quasi-coherent $\\mathcal{F}$ on $Y$ we have", "functorial canonical maps", "$$", "\\xymatrix{", "& \\Ext^i_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "\\ar[ld] \\ar[rd] \\\\", "H^i(Y, K_{n + 1} \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} \\mathcal{F})", "\\ar[rr] & &", "H^i(Y, K_n \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} \\mathcal{F})", "}", "$$", "which are isomorphisms for $i \\leq n + a$.", "Let $L_n = K_n^\\vee$ be the dual perfect complex.", "Then we see that $L_1 \\to L_2 \\to L_3 \\to \\ldots$", "is a system of perfect objects in $D(\\mathcal{O}_Y)$", "such that for any quasi-coherent $\\mathcal{F}$ on $Y$", "the maps", "$$", "\\Ext^i_{\\mathcal{O}_Y}(L_{n + 1}, \\mathcal{F})", "\\longrightarrow", "\\Ext^i_{\\mathcal{O}_Y}(L_n, \\mathcal{F})", "$$", "are isomorphisms for $i \\leq n + a - 1$. This implies that", "$L_n \\to L_{n + 1}$ induces an isomorphism on truncations", "$\\tau_{\\geq -n - a + 2}$ (hint: take cone of $L_n \\to L_{n + 1}$", "and look at its last nonvanishing cohomology sheaf).", "Thus $L = \\text{hocolim} L_n$ is pseudo-coherent, see", "Lemma \\ref{lemma-pseudo-coherent-hocolim}. The mapping property", "of homotopy colimits gives that", "$\\Ext^i_{\\mathcal{O}_Y}(L, \\mathcal{F}) =", "\\Ext^i_{\\mathcal{O}_Y}(L_n, \\mathcal{F})$", "for $i \\leq n + a - 3$ which finishes the proof." ], "refs": [ "spaces-perfect-theorem-approximation", "derived-lemma-negative-exts", "spaces-perfect-lemma-compute-ext-perfect", "spaces-perfect-lemma-compute-ext-perfect", "spaces-perfect-lemma-identify-pseudo-coherent-noetherian", "spaces-perfect-lemma-pseudo-coherent-hocolim", "sites-cohomology-lemma-dual-perfect-complex", "spaces-perfect-lemma-tensor-perfect", "spaces-perfect-lemma-compute-ext-perfect", "spaces-perfect-lemma-pseudo-coherent-hocolim" ], "ref_ids": [ 2639, 1893, 2732, 2732, 2697, 2713, 4390, 2729, 2732, 2713 ] } ], "ref_ids": [] }, { "id": 2734, "type": "theorem", "label": "spaces-perfect-lemma-descend-homomorphisms", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-descend-homomorphisms", "contents": [ "In Situation \\ref{situation-descent}. Let $E_0$ and $K_0$ be objects of", "$D(\\mathcal{O}_{X_0})$. Set $E_i = Lf_{i0}^*E_0$ and $K_i = Lf_{i0}^*K_0$", "for $i \\geq 0$ and set $E = Lf_0^*E_0$ and $K = Lf_0^*K_0$. Then the map", "$$", "\\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{X_i})}(E_i, K_i)", "\\longrightarrow", "\\Hom_{D(\\mathcal{O}_X)}(E, K)", "$$", "is an isomorphism if either", "\\begin{enumerate}", "\\item $E_0$ is perfect and $K_0 \\in D_\\QCoh(\\mathcal{O}_{X_0})$, or", "\\item $E_0$ is pseudo-coherent and", "$K_0 \\in D_\\QCoh(\\mathcal{O}_{X_0})$ has finite tor dimension.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For every quasi-compact and quasi-separated object $U_0$ of", "$(X_0)_{spaces, \\etale}$ consider the condition $P$ that the canonical", "map", "$$", "\\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{U_i})}(E_i|_{U_i}, K_i|_{U_i})", "\\longrightarrow", "\\Hom_{D(\\mathcal{O}_U)}(E|_U, K|_U)", "$$", "is an isomorphism, where $U = X \\times_{X_0} U_0$ and", "$U_i = X_i \\times_{X_0} U_0$. We will prove $P$ holds for each $U_0$", "by the induction principle of Lemma \\ref{lemma-induction-principle}.", "Condition (2) of this lemma follows immediately from Mayer-Vietoris", "for hom in the derived category, see Lemma \\ref{lemma-mayer-vietoris-hom}.", "Thus it suffices to prove the lemma when $X_0$ is affine.", "\\medskip\\noindent", "If $X_0$ is affine, then the result follows from the case of schemes, see", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-descend-homomorphisms}.", "To see this use the equivalence of", "Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site}", "and use the translation of properties explained in", "Lemmas \\ref{lemma-descend-pseudo-coherent},", "\\ref{lemma-descend-tor-amplitude}, and", "\\ref{lemma-descend-perfect}." ], "refs": [ "spaces-perfect-lemma-induction-principle", "spaces-perfect-lemma-mayer-vietoris-hom", "perfect-lemma-descend-homomorphisms", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-descend-pseudo-coherent", "spaces-perfect-lemma-descend-tor-amplitude", "spaces-perfect-lemma-descend-perfect" ], "ref_ids": [ 2670, 2676, 7050, 2644, 2692, 2693, 2695 ] } ], "ref_ids": [] }, { "id": 2735, "type": "theorem", "label": "spaces-perfect-lemma-perfect-on-limit", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-perfect-on-limit", "contents": [ "In Situation \\ref{situation-descent} the category of perfect", "objects of $D(\\mathcal{O}_X)$ is the colimit of the categories", "of perfect objects of $D(\\mathcal{O}_{X_i})$." ], "refs": [], "proofs": [ { "contents": [ "For every quasi-compact and quasi-separated object $U_0$ of", "$(X_0)_{spaces, \\etale}$ consider the condition $P$ that", "the functor", "$$", "\\colim_{i \\geq 0} D_{perf}(\\mathcal{O}_{U_i})", "\\longrightarrow", "D_{perf}(\\mathcal{O}_U)", "$$", "is an equivalence where ${}_{perf}$ indicates the full subcategory of", "perfect objects and where $U = X \\times_{X_0} U_0$ and", "$U_i = X_i \\times_{X_0} U_0$. We will prove $P$ holds for every $U_0$", "by the induction principle of Lemma \\ref{lemma-induction-principle}.", "First, we observe that we already know the functor is fully faithful", "by Lemma \\ref{lemma-descend-homomorphisms}. Thus it suffices to prove", "essential surjectivity.", "\\medskip\\noindent", "We first check condition (2) of the induction principle. Thus suppose", "that we have an elementary distinguished square", "$(U_0 \\subset X_0, V_0 \\to X_0)$ and that $P$ holds for", "$U_0$, $V_0$, and $U_0 \\times_{X_0} V_0$. Let $E$ be a perfect object", "of $D(\\mathcal{O}_X)$. We can find $i \\geq 0$ and $E_{U, i}$ perfect on $U_i$", "and $E_{V, i}$ perfect on $V_i$ whose pullback to $U$ and $V$ are isomorphic", "to $E|_U$ and $E|_V$. Denote", "$$", "a : E_{U, i} \\to (R(X \\to X_i)_*E)|_{U_i}", "\\quad\\text{and}\\quad", "b : E_{V, i} \\to (R(X \\to X_i)_*E)|_{V_i}", "$$", "the maps adjoint to the isomorphisms $L(U \\to U_i)^*E_{U, i} \\to E|_U$", "and $L(V \\to V_i)^*E_{V, i} \\to E|_V$. By fully faithfulness, after", "increasing $i$, we can find an isomorphism", "$c : E_{U, i}|_{U_i \\times_{X_i} V_i} \\to E_{V, i}|_{U_i \\times_{X_i} V_i}$", "which pulls back to the identifications ", "$$", "L(U \\to U_i)^*E_{U, i}|_{U \\times_X V} \\to E|_{U \\times_X V} \\to", "L(V \\to V_i)^*E_{V, i}|_{U \\times_X V}.", "$$", "Apply Lemma \\ref{lemma-glue}", "to get an object $E_i$ on $X_i$ and a map $d : E_i \\to R(X \\to X_i)_*E$", "which restricts to the maps $a$ and $b$ over $U_i$ and $V_i$.", "Then it is clear that $E_i$ is perfect and that", "$d$ is adjoint to an isomorphism $L(X \\to X_i)^*E_i \\to E$.", "\\medskip\\noindent", "Finally, we check condition (1) of the induction principle, in other", "words, we check the lemma holds when $X_0$ is affine.", "This follows from the case of schemes, see", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-descend-perfect}.", "To see this use the equivalence of", "Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site}", "and use the translation of Lemma \\ref{lemma-descend-perfect}." ], "refs": [ "spaces-perfect-lemma-induction-principle", "spaces-perfect-lemma-descend-homomorphisms", "spaces-perfect-lemma-glue", "perfect-lemma-descend-perfect", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-descend-perfect" ], "ref_ids": [ 2670, 2734, 2680, 7051, 2644, 2695 ] } ], "ref_ids": [] }, { "id": 2736, "type": "theorem", "label": "spaces-perfect-lemma-base-change-tensor-perfect", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-base-change-tensor-perfect", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of finite presentation", "between algebraic spaces over $S$. Let $E \\in D(\\mathcal{O}_X)$ be a perfect", "object. Let $\\mathcal{G}^\\bullet$ be a bounded complex of finitely presented", "$\\mathcal{O}_X$-modules, flat over $Y$, with support proper over $Y$. Then", "$$", "K = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)", "$$", "is a perfect object of $D(\\mathcal{O}_Y)$ and its formation", "commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "The statement on base change is Lemma \\ref{lemma-base-change-tensor}.", "Thus it suffices to show that $K$ is a perfect object. If $Y$ is", "Noetherian, then this follows from", "Lemma \\ref{lemma-tensor-perfect}.", "We will reduce to this case by Noetherian approximation.", "We encourage the reader to skip the rest of this proof.", "\\medskip\\noindent", "The question is local on $Y$, hence we may assume $Y$ is affine.", "Say $Y = \\Spec(R)$. We write $R = \\colim R_i$ as a filtered colimit", "of Noetherian rings $R_i$. By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-finite-presentation}", "there exists an $i$ and an algebraic space $X_i$ of finite presentation", "over $R_i$ whose base change to $R$ is $X$. By", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-modules-finite-presentation}", "we may assume after increasing $i$, that there exists a", "bounded complex of finitely", "presented $\\mathcal{O}_{X_i}$-modules $\\mathcal{G}_i^\\bullet$ whose", "pullback to $X$ is $\\mathcal{G}^\\bullet$. After increasing $i$", "we may assume $\\mathcal{G}_i^n$ is flat over $R_i$, see", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-flat}.", "After increasing $i$ we may assume the support of $\\mathcal{G}_i^n$", "is proper over $R_i$, see", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-eventually-proper-support}.", "Finally, by Lemma \\ref{lemma-descend-perfect}", "we may, after increasing $i$, assume there exists a perfect", "object $E_i$ of $D(\\mathcal{O}_{X_i})$ whose pullback to", "$X$ is $E$. Applying Lemma \\ref{lemma-compute-tensor-perfect}", "to $X_i \\to \\Spec(R_i)$, $E_i$, $\\mathcal{G}_i^\\bullet$ and using the", "base change property already shown we obtain the result." ], "refs": [ "spaces-perfect-lemma-base-change-tensor", "spaces-perfect-lemma-tensor-perfect", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-modules-finite-presentation", "spaces-limits-lemma-descend-flat", "spaces-limits-lemma-eventually-proper-support", "spaces-perfect-lemma-descend-perfect", "spaces-perfect-lemma-compute-tensor-perfect" ], "ref_ids": [ 2726, 2729, 4598, 4599, 4595, 4618, 2695, 2731 ] } ], "ref_ids": [] }, { "id": 2737, "type": "theorem", "label": "spaces-perfect-lemma-base-change-tensor-pseudo-coherent", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-base-change-tensor-pseudo-coherent", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of finite presentation", "between algebraic spaces over $S$.", "Let $E \\in D(\\mathcal{O}_X)$ be a pseudo-coherent object.", "Let $\\mathcal{G}^\\bullet$ be a bounded above complex of", "finitely presented $\\mathcal{O}_X$-modules,", "flat over $Y$, with support proper over $Y$. Then", "$$", "K = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)", "$$", "is a pseudo-coherent object of $D(\\mathcal{O}_Y)$ and its formation", "commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "The statement on base change is Lemma \\ref{lemma-base-change-tensor}.", "Thus it suffices to show that $K$ is a pseudo-coherent object.", "This will follow from Lemma \\ref{lemma-base-change-tensor-perfect}", "by approximation by perfect complexes. We encourage the reader to", "skip the rest of the proof.", "\\medskip\\noindent", "The question is \\'etale local on $Y$, hence we may assume $Y$ is affine.", "Then $X$ is quasi-compact and quasi-separated. Moreover, there", "exists an integer $N$ such that total direct image", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$", "has cohomological dimension $N$ as explained in", "Lemma \\ref{lemma-quasi-coherence-direct-image}.", "Choose an integer $b$ such that $\\mathcal{G}^i = 0$ for $i > b$.", "It suffices to show that $K$ is $m$-pseudo-coherent for", "every $m$. Choose an approximation $P \\to E$ by a perfect complex $P$", "of $(X, E, m - N - 1 - b)$. This is possible by", "Theorem \\ref{theorem-approximation}.", "Choose a distinguished triangle", "$$", "P \\to E \\to C \\to P[1]", "$$", "in $D_\\QCoh(\\mathcal{O}_X)$. The cohomology sheaves of $C$ are zero", "in degrees $\\geq m - N - 1 - b$. Hence", "the cohomology sheaves of $C \\otimes^\\mathbf{L} \\mathcal{G}^\\bullet$", "are zero in degrees $\\geq m - N - 1$.", "Thus the cohomology sheaves of $Rf_*(C \\otimes^\\mathbf{L} \\mathcal{G})$", "are zero in degrees $\\geq m - 1$. Hence", "$$", "Rf_*(P \\otimes^\\mathbf{L} \\mathcal{G}) \\to", "Rf_*(E \\otimes^\\mathbf{L} \\mathcal{G})", "$$", "is an isomorphism on cohomology sheaves in degrees $\\geq m$.", "Next, suppose that $H^i(P) = 0$ for $i > a$. Then", "$", "P \\otimes^\\mathbf{L} \\sigma_{\\geq m - N - 1 - a}\\mathcal{G}^\\bullet", "\\longrightarrow", "P \\otimes^\\mathbf{L} \\mathcal{G}^\\bullet", "$", "is an isomorphism on cohomology sheaves in degrees $\\geq m - N - 1$.", "Thus again we find that", "$$", "Rf_*(P \\otimes^\\mathbf{L} \\sigma_{\\geq m - N - 1 - a}\\mathcal{G}^\\bullet) \\to", "Rf_*(P \\otimes^\\mathbf{L} \\mathcal{G}^\\bullet)", "$$", "is an isomorphism on cohomology sheaves in degrees $\\geq m$.", "By Lemma \\ref{lemma-base-change-tensor-perfect} the source", "is a perfect complex.", "We conclude that $K$ is $m$-pseudo-coherent as desired." ], "refs": [ "spaces-perfect-lemma-base-change-tensor", "spaces-perfect-lemma-base-change-tensor-perfect", "spaces-perfect-lemma-quasi-coherence-direct-image", "spaces-perfect-theorem-approximation", "spaces-perfect-lemma-base-change-tensor-perfect" ], "ref_ids": [ 2726, 2736, 2652, 2639, 2736 ] } ], "ref_ids": [] }, { "id": 2738, "type": "theorem", "label": "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper", "morphism of finite presentation of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item Let $E \\in D(\\mathcal{O}_X)$ be perfect and $f$ flat. Then", "$Rf_*E$ is a perfect object of $D(\\mathcal{O}_Y)$ and its formation", "commutes with arbitrary base change.", "\\item Let $\\mathcal{G}$ be an $\\mathcal{O}_X$-module of finite presentation,", "flat over $S$. Then $Rf_*\\mathcal{G}$ is a perfect object of", "$D(\\mathcal{O}_Y)$ and its formation commutes with arbitrary base change.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Special cases of", "Lemma \\ref{lemma-base-change-tensor-perfect} applied with", "(1) $\\mathcal{G}^\\bullet$ equal to $\\mathcal{O}_X$ in degree $0$", "and (2) $E = \\mathcal{O}_X$ and $\\mathcal{G}^\\bullet$ consisting", "of $\\mathcal{G}$ sitting in degree $0$." ], "refs": [ "spaces-perfect-lemma-base-change-tensor-perfect" ], "ref_ids": [ 2736 ] } ], "ref_ids": [] }, { "id": 2739, "type": "theorem", "label": "spaces-perfect-lemma-flat-proper-pseudo-coherent-direct-image-general", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-flat-proper-pseudo-coherent-direct-image-general", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a flat proper", "morphism of finite presentation of algebraic spaces over $S$.", "Let $E \\in D(\\mathcal{O}_X)$", "be pseudo-coherent. Then $Rf_*E$ is a pseudo-coherent object of", "$D(\\mathcal{O}_Y)$ and its formation commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "Special case of", "Lemma \\ref{lemma-base-change-tensor-pseudo-coherent} applied with", "$\\mathcal{G} = \\mathcal{O}_X$." ], "refs": [ "spaces-perfect-lemma-base-change-tensor-pseudo-coherent" ], "ref_ids": [ 2737 ] } ], "ref_ids": [] }, { "id": 2740, "type": "theorem", "label": "spaces-perfect-lemma-pullback-and-limits", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-pullback-and-limits", "contents": [ "Let $R$ be a ring. Let $X$ be an algebraic space and let", "$f : X \\to \\Spec(R)$ be proper, flat, and", "of finite presentation. Let $(M_n)$ be an inverse", "system of $R$-modules with surjective transition maps.", "Then the canonical map", "$$", "\\mathcal{O}_X \\otimes_R (\\lim M_n)", "\\longrightarrow", "\\lim \\mathcal{O}_X \\otimes_R M_n", "$$", "induces an isomorphism from the source to $DQ_X$ applied to the target." ], "refs": [], "proofs": [ { "contents": [ "The statement means that for any object $E$ of", "$D_\\QCoh(\\mathcal{O}_X)$ the induced map", "$$", "\\Hom(E, \\mathcal{O}_X \\otimes_R (\\lim M_n))", "\\longrightarrow", "\\Hom(E, \\lim \\mathcal{O}_X \\otimes_R M_n)", "$$", "is an isomorphism. Since $D_\\QCoh(\\mathcal{O}_X)$ has", "a perfect generator (Theorem \\ref{theorem-bondal-van-den-Bergh})", "it suffices to check this for perfect $E$.", "By Lemma \\ref{lemma-Rlim-quasi-coherent} we have", "$\\lim \\mathcal{O}_X \\otimes_R M_n = R\\lim \\mathcal{O}_X \\otimes_R M_n$.", "The exact functor", "$R\\Hom_X(E, -) : D_\\QCoh(\\mathcal{O}_X) \\to D(R)$", "of Cohomology on Sites, Section \\ref{sites-cohomology-section-global-RHom}", "commutes with products and hence with derived limits, whence", "$$", "R\\Hom_X(E, \\lim \\mathcal{O}_X \\otimes_R M_n) =", "R\\lim R\\Hom_X(E, \\mathcal{O}_X \\otimes_R M_n)", "$$", "Let $E^\\vee$ be the dual perfect complex, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-dual-perfect-complex}.", "We have", "$$", "R\\Hom_X(E, \\mathcal{O}_X \\otimes_R M_n) =", "R\\Gamma(X, E^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*M_n) =", "R\\Gamma(X, E^\\vee) \\otimes_R^\\mathbf{L} M_n", "$$", "by Lemma \\ref{lemma-cohomology-base-change}.", "From Lemma \\ref{lemma-flat-proper-perfect-direct-image-general}", "we see $R\\Gamma(X, E^\\vee)$ is a perfect complex of $R$-modules.", "In particular it is a pseudo-coherent complex and by", "More on Algebra, Lemma \\ref{more-algebra-lemma-pseudo-coherent-tensor-limit}", "we obtain", "$$", "R\\lim R\\Gamma(X, E^\\vee) \\otimes_R^\\mathbf{L} M_n =", "R\\Gamma(X, E^\\vee) \\otimes_R^\\mathbf{L} \\lim M_n", "$$", "as desired." ], "refs": [ "spaces-perfect-theorem-bondal-van-den-Bergh", "spaces-perfect-lemma-Rlim-quasi-coherent", "sites-cohomology-lemma-dual-perfect-complex", "spaces-perfect-lemma-cohomology-base-change", "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "more-algebra-lemma-pseudo-coherent-tensor-limit" ], "ref_ids": [ 2640, 2647, 4390, 2718, 2738, 10436 ] } ], "ref_ids": [] }, { "id": 2741, "type": "theorem", "label": "spaces-perfect-lemma-perfect-enough", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-perfect-enough", "contents": [ "Let $A$ be a ring. Let $X$ be an algebraic space over $A$ which is", "quasi-compact and quasi-separated. Let $K \\in D^-_\\QCoh(\\mathcal{O}_X)$.", "If $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent", "in $D(A)$ for every perfect $E$ in $D(\\mathcal{O}_X)$,", "then $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent", "in $D(A)$ for every pseudo-coherent $E$ in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "There exists an integer $N$ such that", "$R\\Gamma(X, -) : D_\\QCoh(\\mathcal{O}_X) \\to D(A)$", "has cohomological dimension $N$ as explained in", "Lemma \\ref{lemma-quasi-coherence-direct-image}.", "Let $b \\in \\mathbf{Z}$ be such that $H^i(K) = 0$ for $i > b$.", "Let $E$ be pseudo-coherent on $X$.", "It suffices to show that $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$", "is $m$-pseudo-coherent for every $m$.", "Choose an approximation $P \\to E$ by a perfect complex $P$", "of $(X, E, m - N - 1 - b)$. This is possible by", "Theorem \\ref{theorem-approximation}.", "Choose a distinguished triangle", "$$", "P \\to E \\to C \\to P[1]", "$$", "in $D_\\QCoh(\\mathcal{O}_X)$. The cohomology sheaves of $C$ are zero", "in degrees $\\geq m - N - 1 - b$. Hence the cohomology", "sheaves of $C \\otimes^\\mathbf{L} K$ are zero in degrees $\\geq m - N - 1$.", "Thus the cohomology of $R\\Gamma(X, C \\otimes^\\mathbf{L} K)$", "are zero in degrees $\\geq m - 1$. Hence", "$$", "R\\Gamma(X, P \\otimes^\\mathbf{L} K) \\to R\\Gamma(X, E \\otimes^\\mathbf{L} K)", "$$", "is an isomorphism on cohomology in degrees $\\geq m$.", "By assumption the source is pseudo-coherent.", "We conclude that $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$", "is $m$-pseudo-coherent as desired." ], "refs": [ "spaces-perfect-lemma-quasi-coherence-direct-image", "spaces-perfect-theorem-approximation" ], "ref_ids": [ 2652, 2639 ] } ], "ref_ids": [] }, { "id": 2742, "type": "theorem", "label": "spaces-perfect-lemma-base-change-RHom-perfect", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-base-change-RHom-perfect", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of finite presentation", "between algebraic spaces over $S$.", "Let $E \\in D(\\mathcal{O}_X)$ be a perfect object. Let $\\mathcal{G}^\\bullet$", "be a bounded complex of finitely presented $\\mathcal{O}_X$-modules,", "flat over $Y$, with support proper over $Y$. Then", "$$", "K = Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet)", "$$", "is a perfect object of $D(\\mathcal{O}_Y)$ and its formation", "commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "The statement on base change is Lemma \\ref{lemma-base-change-RHom}.", "Thus it suffices to show that $K$ is a perfect object. If $Y$ is", "Noetherian, then this follows from Lemma \\ref{lemma-ext-perfect}.", "We will reduce to this case by Noetherian approximation.", "We encourage the reader to skip the rest of this proof.", "\\medskip\\noindent", "The question is local on $Y$, hence we may assume $Y$ is affine.", "Say $Y = \\Spec(R)$. We write $R = \\colim R_i$ as a filtered colimit", "of Noetherian rings $R_i$. By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-finite-presentation}", "there exists an $i$ and an algebraic space $X_i$ of finite presentation", "over $R_i$ whose base change to $R$ is $X$. By", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-modules-finite-presentation}", "we may assume after increasing $i$, that there exists a bounded", "complex of finitely", "presented $\\mathcal{O}_{X_i}$-module $\\mathcal{G}_i^\\bullet$ whose", "pullback to $X$ is $\\mathcal{G}$. After increasing $i$", "we may assume $\\mathcal{G}_i^n$ is flat over $R_i$, see", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-flat}.", "After increasing $i$ we may assume the support of $\\mathcal{G}_i^n$", "is proper over $R_i$, see", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-eventually-proper-support}.", "Finally, by Lemma \\ref{lemma-descend-perfect}", "we may, after increasing $i$, assume there exists a perfect", "object $E_i$ of $D(\\mathcal{O}_{X_i})$ whose pullback to", "$X$ is $E$. Applying Lemma \\ref{lemma-compute-ext-perfect}", "to $X_i \\to \\Spec(R_i)$, $E_i$, $\\mathcal{G}_i^\\bullet$ and using the", "base change property already shown we obtain the result." ], "refs": [ "spaces-perfect-lemma-base-change-RHom", "spaces-perfect-lemma-ext-perfect", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-modules-finite-presentation", "spaces-limits-lemma-descend-flat", "spaces-limits-lemma-eventually-proper-support", "spaces-perfect-lemma-descend-perfect", "spaces-perfect-lemma-compute-ext-perfect" ], "ref_ids": [ 2727, 2730, 4598, 4599, 4595, 4618, 2695, 2732 ] } ], "ref_ids": [] }, { "id": 2743, "type": "theorem", "label": "spaces-perfect-lemma-jump-loci", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-jump-loci", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $E \\in D(\\mathcal{O}_X)$ be pseudo-coherent (for example perfect).", "For any $i \\in \\mathbf{Z}$ consider the function", "$$", "\\beta_i : |X| \\longrightarrow \\{0, 1, 2, \\ldots\\}", "$$", "defined above. Then we have", "\\begin{enumerate}", "\\item formation of $\\beta_i$ commutes with arbitrary base change,", "\\item the functions $\\beta_i$ are upper semi-continuous, and", "\\item the level sets of $\\beta_i$ are \\'etale locally constructible.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$.", "Then $L\\varphi^*E$ is a pseudo-coherent complex on the scheme $U$ (use", "Lemma \\ref{lemma-descend-pseudo-coherent}) and we can apply the result", "for schemes, see", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-jump-loci}.", "The meaning of part (3) is that the inverse image of the level sets", "to $U$ are locally constructible, see", "Properties of Spaces, Definition", "\\ref{spaces-properties-definition-locally-constructible}." ], "refs": [ "spaces-perfect-lemma-descend-pseudo-coherent", "perfect-lemma-jump-loci", "spaces-properties-definition-locally-constructible" ], "ref_ids": [ 2692, 7058, 11928 ] } ], "ref_ids": [] }, { "id": 2744, "type": "theorem", "label": "spaces-perfect-lemma-jump-loci-geometric", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-jump-loci-geometric", "contents": [ "Let $Y$ be a scheme and let $X$ be an algebraic space over $Y$", "such that the structure morphism $f : X \\to Y$", "is flat, proper, and of finite presentation.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation,", "flat over $Y$. For fixed $i \\in \\mathbf{Z}$ consider the function", "$$", "\\beta_i : |Y| \\to \\{0, 1, 2, \\ldots\\},\\quad", "y \\longmapsto \\dim_{\\kappa(y)} H^i(X_y, \\mathcal{F}_y)", "$$", "Then we have", "\\begin{enumerate}", "\\item formation of $\\beta_i$ commutes with arbitrary base change,", "\\item the functions $\\beta_i$ are upper semi-continuous, and", "\\item the level sets of $\\beta_i$ are locally constructible in $Y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By cohomology and base change (more precisely by", "Lemma \\ref{lemma-flat-proper-perfect-direct-image-general})", "the object $K = Rf_*\\mathcal{F}$ is a perfect object of the derived", "category of $Y$ whose formation commutes with arbitrary base change.", "In particular we have", "$$", "H^i(X_y, \\mathcal{F}_y) = H^i(K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} \\kappa(y))", "$$", "Thus the lemma follows from Lemma \\ref{lemma-jump-loci}." ], "refs": [ "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "spaces-perfect-lemma-jump-loci" ], "ref_ids": [ 2738, 2743 ] } ], "ref_ids": [] }, { "id": 2745, "type": "theorem", "label": "spaces-perfect-lemma-chi-locally-constant", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-chi-locally-constant", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $E \\in D(\\mathcal{O}_X)$ be perfect. The function", "$$", "\\chi_E : |X| \\longrightarrow \\mathbf{Z},\\quad", "x \\longmapsto \\sum (-1)^i \\beta_i(x)", "$$", "is locally constant on $X$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints:", "Follows from the case of schemes by \\'etale localization. See", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-chi-locally-constant}." ], "refs": [ "perfect-lemma-chi-locally-constant" ], "ref_ids": [ 7059 ] } ], "ref_ids": [] }, { "id": 2746, "type": "theorem", "label": "spaces-perfect-lemma-open-where-cohomology-in-degree-i-rank-r", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-open-where-cohomology-in-degree-i-rank-r", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $E \\in D(\\mathcal{O}_X)$ be perfect.", "Given $i, r \\in \\mathbf{Z}$, there exists an", "open subspace $U \\subset X$ characterized by the following", "\\begin{enumerate}", "\\item $E|_U \\cong H^i(E|_U)[-i]$ and $H^i(E|_U)$ is a locally free", "$\\mathcal{O}_U$-module of rank $r$,", "\\item a morphism $f : Y \\to X$ factors through $U$ if and only if", "$Lf^*E$ is isomorphic to a locally free module of rank $r$", "placed in degree $i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints:", "Follows from the case of schemes by \\'etale localization. See", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-open-where-cohomology-in-degree-i-rank-r}." ], "refs": [ "perfect-lemma-open-where-cohomology-in-degree-i-rank-r" ], "ref_ids": [ 7060 ] } ], "ref_ids": [] }, { "id": 2747, "type": "theorem", "label": "spaces-perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "which is proper, flat, and of finite presentation.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation,", "flat over $Y$. Fix $i, r \\in \\mathbf{Z}$.", "Then there exists an open subspace", "$V \\subset Y$ with the following property:", "A morphism $T \\to Y$ factors through $V$ if and only if", "$Rf_{T, *}\\mathcal{F}_T$ is isomorphic to a", "finite locally free module of rank $r$ placed in degree $i$." ], "refs": [], "proofs": [ { "contents": [ "By cohomology and base change (", "Lemma \\ref{lemma-flat-proper-perfect-direct-image-general})", "the object $K = Rf_*\\mathcal{F}$ is a perfect object of the derived", "category of $Y$ whose formation commutes with arbitrary base change.", "Thus this lemma follows immediately from", "Lemma \\ref{lemma-open-where-cohomology-in-degree-i-rank-r}." ], "refs": [ "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "spaces-perfect-lemma-open-where-cohomology-in-degree-i-rank-r" ], "ref_ids": [ 2738, 2746 ] } ], "ref_ids": [] }, { "id": 2748, "type": "theorem", "label": "spaces-perfect-lemma-locally-closed-where-H0-locally-free", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-locally-closed-where-H0-locally-free", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $E \\in D(\\mathcal{O}_X)$ be perfect", "of tor-amplitude in $[a, b]$ for some $a, b \\in \\mathbf{Z}$.", "Let $r \\geq 0$.", "Then there exists a locally closed subspace $j : Z \\to X$", "characterized by the following", "\\begin{enumerate}", "\\item $H^a(Lj^*E)$ is a locally free $\\mathcal{O}_Z$-module of rank $r$, and", "\\item a morphism $f : Y \\to X$ factors through $Z$ if and only if", "for all morphisms $g : Y' \\to Y$ the $\\mathcal{O}_{Y'}$-module", "$H^a(L(f \\circ g)^*E)$ is locally free of rank $r$.", "\\end{enumerate}", "Moreover, $j : Z \\to X$ is of finite presentation and we have", "\\begin{enumerate}", "\\item[(3)] if $f : Y \\to X$ factors as $Y \\xrightarrow{g} Z \\to X$, then", "$H^a(Lf^*E) = g^*H^a(Lj^*E)$,", "\\item[(4)] if $\\beta_a(x) \\leq r$ for all $x \\in |X|$, then $j$ is", "a closed immersion and given $f : Y \\to X$ the following are equivalent", "\\begin{enumerate}", "\\item $f : Y \\to X$ factors through $Z$,", "\\item $H^0(Lf^*E)$ is a locally free $\\mathcal{O}_Y$-module of rank $r$,", "\\end{enumerate}", "and if $r = 1$ these are also equivalent to", "\\begin{enumerate}", "\\item[(c)] $\\mathcal{O}_Y \\to \\SheafHom_{\\mathcal{O}_Y}(H^0(Lf^*E), H^0(Lf^*E))$", "is injective.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints:", "Follows from the case of schemes by \\'etale localization. See", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-locally-closed-where-H0-locally-free}." ], "refs": [ "perfect-lemma-locally-closed-where-H0-locally-free" ], "ref_ids": [ 7061 ] } ], "ref_ids": [] }, { "id": 2749, "type": "theorem", "label": "spaces-perfect-lemma-proper-flat-h0", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-proper-flat-h0", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume", "\\begin{enumerate}", "\\item $f$ is proper, flat, and of finite presentation, and", "\\item for a morphism $\\Spec(k) \\to Y$ where $k$ is a field, we have", "$k = H^0(X_k, \\mathcal{O}_{X_k})$.", "\\end{enumerate}", "Then we have", "\\begin{enumerate}", "\\item[(a)] $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and", "this holds after any base change,", "\\item[(b)] \\'etale locally on $Y$ we have", "$$", "Rf_*\\mathcal{O}_X = \\mathcal{O}_Y \\oplus P", "$$", "in $D(\\mathcal{O}_Y)$", "where $P$ is perfect of tor amplitude in $[1, \\infty)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove (a) and (b) \\'etale locally on $Y$, thus we may and do", "assume $Y$ is an affine scheme.", "By cohomology and base change", "(Lemma \\ref{lemma-flat-proper-perfect-direct-image-general})", "the complex $E = Rf_*\\mathcal{O}_X$", "is perfect and its formation commutes with arbitrary base change.", "In particular, for $y \\in Y$ we see that", "$H^0(E \\otimes^\\mathbf{L} \\kappa(y)) =", "H^0(X_y, \\mathcal{O}_{X_y}) = \\kappa(y)$.", "Thus $\\beta_0(y) \\leq 1$ for all $y \\in Y$ with notation as in", "Lemma \\ref{lemma-jump-loci}. Apply", "Lemma \\ref{lemma-locally-closed-where-H0-locally-free}", "with $a = 0$ and $r = 1$. We obtain a universal closed subscheme", "$j : Z \\to Y$ with $H^0(Lj^*E)$ invertible characterized", "by the equivalence of (4)(a), (b), and (c) of the lemma.", "Since formation of $E$ commutes with base change, we have", "$$", "Lf^*E = R\\text{pr}_{1, *}\\mathcal{O}_{X \\times_Y X}", "$$", "The morphism $\\text{pr}_1 : X \\times_Y X$ has a section", "namely the diagonal morphism $\\Delta$ for $X$ over $Y$.", "We obtain maps", "$$", "\\mathcal{O}_X \\longrightarrow R\\text{pr}_{1, *}\\mathcal{O}_{X \\times_Y X}", "\\longrightarrow \\mathcal{O}_X", "$$", "in $D(\\mathcal{O}_X)$ whose composition is the identity. Thus", "$R\\text{pr}_{1, *}\\mathcal{O}_{X \\times_Y X} = \\mathcal{O}_X \\oplus E'$", "in $D(\\mathcal{O}_X)$. Thus $\\mathcal{O}_X$ is a direct summand of", "$H^0(Lf^*E)$ and we conclude that $X \\to Y$ factors through $Z$", "by the equivalence of (4)(c) and (4)(a) of the lemma cited above.", "Since $\\{X \\to Y\\}$ is an fppf covering, we have $Z = Y$.", "Thus $f_*\\mathcal{O}_X$ is an invertible $\\mathcal{O}_Y$-module.", "We conclude $\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$ is an isomorphism", "because a ring map $A \\to B$ such that $B$ is invertible as an $A$-module", "is an isomorphism. Since the assumptions are preserved under base", "change, we see that (a) is true.", "\\medskip\\noindent", "Proof of (b). Above we have seen that for every $y \\in Y$ the map", "$\\mathcal{O}_Y \\to H^0(E \\otimes^\\mathbf{L} \\kappa(y))$ is surjective.", "Thus we may apply", "More on Algebra, Lemma \\ref{more-algebra-lemma-better-cut-complex-in-two}", "to see that in an open neighbourhood of $y$ we have", "a decomposition $Rf_*\\mathcal{O}_X = \\mathcal{O}_Y \\oplus P$" ], "refs": [ "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "spaces-perfect-lemma-jump-loci", "spaces-perfect-lemma-locally-closed-where-H0-locally-free", "more-algebra-lemma-better-cut-complex-in-two" ], "ref_ids": [ 2738, 2743, 2748, 10236 ] } ], "ref_ids": [] }, { "id": 2750, "type": "theorem", "label": "spaces-perfect-lemma-proper-flat-geom-red-connected", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-proper-flat-geom-red-connected", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume", "\\begin{enumerate}", "\\item $f$ is proper, flat, and of finite presentation, and", "\\item the geometric fibres of $f$ are reduced and connected.", "\\end{enumerate}", "Then $f_*\\mathcal{O}_X = \\mathcal{O}_Y$ and this holds", "after any base change." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-proper-flat-h0}", "it suffices to show that $k = H^0(X_k, \\mathcal{O}_{X_k})$", "for all morphisms $\\Spec(k) \\to Y$ where $k$ is a field. This follows from", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-proper-geometrically-reduced-global-sections}", "and the fact that $X_k$ is geometrically connected and geometrically reduced." ], "refs": [ "spaces-perfect-lemma-proper-flat-h0", "spaces-over-fields-lemma-proper-geometrically-reduced-global-sections" ], "ref_ids": [ 2749, 12868 ] } ], "ref_ids": [] }, { "id": 2751, "type": "theorem", "label": "spaces-perfect-lemma-countable-cohomology", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-countable-cohomology", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$", "such that the cohomology sheaves $H^i(K)$ have countable", "sets of sections over affine schemes \\'etale over $X$.", "Then for any quasi-compact and quasi-separated \\'etale morphism $U \\to X$", "and any perfect object $E$ in $D(\\mathcal{O}_X)$", "the sets", "$$", "H^i(U, K \\otimes^\\mathbf{L} E),\\quad \\Ext^i(E|_U, K|_U)", "$$", "are countable." ], "refs": [], "proofs": [ { "contents": [ "Using Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-dual-perfect-complex}", "we see that it suffices to prove the result", "for the groups $H^i(U, K \\otimes^\\mathbf{L} E)$.", "We will use the induction principle to prove the lemma, see", "Lemma \\ref{lemma-induction-principle}.", "\\medskip\\noindent", "When $U = \\Spec(A)$ is affine the result follows from", "the case of schemes, see Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-countable-cohomology}.", "\\medskip\\noindent", "To finish the proof it suffices to show: if $(U \\subset W, V \\to W)$", "is an elementary distinguished triangle", "and the result holds for $U$, $V$, and $U \\times_W V$, then", "the result holds for $W$. This is an immediate consquence", "of the Mayer-Vietoris sequence, see", "Lemma \\ref{lemma-unbounded-mayer-vietoris}." ], "refs": [ "sites-cohomology-lemma-dual-perfect-complex", "spaces-perfect-lemma-induction-principle", "perfect-lemma-countable-cohomology", "spaces-perfect-lemma-unbounded-mayer-vietoris" ], "ref_ids": [ 4390, 2670, 7071, 2677 ] } ], "ref_ids": [] }, { "id": 2752, "type": "theorem", "label": "spaces-perfect-lemma-countable", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-countable", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Assume the sets of sections of $\\mathcal{O}_X$ over affines \\'etale over $X$", "are countable. Let $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. The", "following are equivalent", "\\begin{enumerate}", "\\item $K = \\text{hocolim} E_n$ with $E_n$ a perfect object of", "$D(\\mathcal{O}_X)$, and", "\\item the cohomology sheaves $H^i(K)$ have countable", "sets of sections over affines \\'etale over $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (1) is true, then (2) is true because homotopy colimits commutes", "with taking cohomology sheaves", "(by Derived Categories, Lemma \\ref{derived-lemma-cohomology-of-hocolim})", "and because a perfect complex is", "locally isomorphic to a finite complex of finite free $\\mathcal{O}_X$-modules", "and therefore satisfies (2) by assumption on $X$.", "\\medskip\\noindent", "Assume (2).", "Choose a K-injective complex $\\mathcal{K}^\\bullet$ representing $K$.", "Choose a perfect generator $E$ of $D_\\QCoh(\\mathcal{O}_X)$ and", "represent it by a K-injective complex $\\mathcal{I}^\\bullet$.", "According to Theorem \\ref{theorem-DQCoh-is-Ddga}", "and its proof there is an equivalence", "of triangulated categories $F : D_\\QCoh(\\mathcal{O}_X) \\to D(A, \\text{d})$", "where $(A, \\text{d})$ is the differential graded algebra", "$$", "(A, \\text{d}) =", "\\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}", "(\\mathcal{I}^\\bullet, \\mathcal{I}^\\bullet)", "$$", "which maps $K$ to the differential graded module", "$$", "M = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}", "(\\mathcal{I}^\\bullet, \\mathcal{K}^\\bullet)", "$$", "Note that $H^i(A) = \\Ext^i(E, E)$ and", "$H^i(M) = \\Ext^i(E, K)$.", "Moreover, since $F$ is an equivalence it and its quasi-inverse commute", "with homotopy colimits.", "Therefore, it suffices to write $M$ as a homotopy colimit", "of compact objects of $D(A, \\text{d})$.", "By Differential Graded Algebra, Lemma \\ref{dga-lemma-countable}", "it suffices show that $\\Ext^i(E, E)$ and", "$\\Ext^i(E, K)$ are countable for each $i$.", "This follows from Lemma \\ref{lemma-countable-cohomology}." ], "refs": [ "derived-lemma-cohomology-of-hocolim", "spaces-perfect-theorem-DQCoh-is-Ddga", "dga-lemma-countable", "spaces-perfect-lemma-countable-cohomology" ], "ref_ids": [ 1923, 2641, 13129, 2751 ] } ], "ref_ids": [] }, { "id": 2753, "type": "theorem", "label": "spaces-perfect-lemma-computing-sections-as-colim", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-computing-sections-as-colim", "contents": [ "Let $A$ be a ring. Let $f : U \\to X$ be a flat morphism of algebraic spaces", "of finite presentation over $A$. Then", "\\begin{enumerate}", "\\item there exists an inverse system of perfect objects $L_n$ of", "$D(\\mathcal{O}_X)$ such that", "$$", "R\\Gamma(U, Lf^*K) = \\text{hocolim}\\ R\\Hom_X(L_n, K)", "$$", "in $D(A)$ functorially in $K$ in $D_\\QCoh(\\mathcal{O}_X)$, and", "\\item there exists a system of perfect objects $E_n$ of", "$D(\\mathcal{O}_X)$ such that", "$$", "R\\Gamma(U, Lf^*K) = \\text{hocolim}\\ R\\Gamma(X, E_n \\otimes^\\mathbf{L} K)", "$$", "in $D(A)$ functorially in $K$ in $D_\\QCoh(\\mathcal{O}_X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-cohomology-base-change} we have", "$$", "R\\Gamma(U, Lf^*K) = R\\Gamma(X, Rf_*\\mathcal{O}_U \\otimes^\\mathbf{L} K)", "$$", "functorially in $K$. Observe that $R\\Gamma(X, -)$ commutes with", "homotopy colimits because it commutes with direct sums by", "Lemma \\ref{lemma-quasi-coherence-pushforward-direct-sums}.", "Similarly, $- \\otimes^\\mathbf{L} K$ commutes with derived colimits", "because $- \\otimes^\\mathbf{L} K$ commutes with direct sums", "(because direct sums in $D(\\mathcal{O}_X)$", "are given by direct sums of representing complexes).", "Hence to prove (2) it suffices to write", "$Rf_*\\mathcal{O}_U = \\text{hocolim} E_n$ for a system of", "perfect objects $E_n$ of $D(\\mathcal{O}_X)$. Once this is done", "we obtain (1) by setting $L_n = E_n^\\vee$, see Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-dual-perfect-complex}.", "\\medskip\\noindent", "Write $A = \\colim A_i$ with $A_i$ of finite type over $\\mathbf{Z}$. By", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-descend-finite-presentation}", "we can find an $i$ and morphisms $U_i \\to X_i \\to \\Spec(A_i)$", "of finite presentation whose base change to $\\Spec(A)$ recovers", "$U \\to X \\to \\Spec(A)$.", "After increasing $i$ we may assume that $f_i : U_i \\to X_i$ is", "flat, see Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-flat}.", "By Lemma \\ref{lemma-compare-base-change}", "the derived pullback of $Rf_{i, *}\\mathcal{O}_{U_i}$", "by $g : X \\to X_i$ is equal to $Rf_*\\mathcal{O}_U$.", "Since $Lg^*$ commutes with derived colimits, it suffices", "to prove what we want for $f_i$. Hence we may assume that", "$U$ and $X$ are of finite type over $\\mathbf{Z}$.", "\\medskip\\noindent", "Assume $f : U \\to X$ is a morphism of algebraic spaces", "of finite type over $\\mathbf{Z}$. To finish the proof", "we will show that $Rf_*\\mathcal{O}_U$ is a homotopy", "colimit of perfect complexes. To see this we apply Lemma \\ref{lemma-countable}.", "Thus it suffices to show that $R^if_*\\mathcal{O}_U$", "has countable sets of sections over affines \\'etale over $X$.", "This follows from Lemma \\ref{lemma-countable-cohomology}", "applied to the structure sheaf." ], "refs": [ "spaces-perfect-lemma-cohomology-base-change", "spaces-perfect-lemma-quasi-coherence-pushforward-direct-sums", "sites-cohomology-lemma-dual-perfect-complex", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-flat", "spaces-perfect-lemma-compare-base-change", "spaces-perfect-lemma-countable", "spaces-perfect-lemma-countable-cohomology" ], "ref_ids": [ 2718, 2653, 4390, 4598, 4595, 2720, 2752, 2751 ] } ], "ref_ids": [] }, { "id": 2754, "type": "theorem", "label": "spaces-perfect-lemma-bounded-truncation", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-bounded-truncation", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $P \\in D_{perf}(\\mathcal{O}_X)$ and $E \\in D_{\\QCoh}(\\mathcal{O}_X)$.", "Let $a \\in \\mathbf{Z}$. The following are equivalent", "\\begin{enumerate}", "\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], E) = 0$ for $i \\gg 0$, and", "\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{\\geq a} E) = 0$ for $i \\gg 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Using the triangle $ \\tau_{< a} E \\to E \\to \\tau_{\\geq a} E \\to$", "we see that the equivalence follows if we can show", "$$", "\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{< a} E) =", "\\Hom_{D(\\mathcal{O}_X)}(P, (\\tau_{< a} E)[i]) = 0", "$$", "for $i \\gg 0$. As $P$ is perfect this is true by", "Lemma \\ref{lemma-ext-from-perfect-into-bounded-QCoh}." ], "refs": [ "spaces-perfect-lemma-ext-from-perfect-into-bounded-QCoh" ], "ref_ids": [ 2712 ] } ], "ref_ids": [] }, { "id": 2755, "type": "theorem", "label": "spaces-perfect-lemma-bounded-below-truncation", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-lemma-bounded-below-truncation", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let", "$P \\in D_{perf}(\\mathcal{O}_X)$ and $E \\in D_{\\QCoh}(\\mathcal{O}_X)$.", "Let $a \\in \\mathbf{Z}$. The following are equivalent", "\\begin{enumerate}", "\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], E) = 0$ for $i \\ll 0$, and", "\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{\\leq a} E) = 0$ for $i \\ll 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Using the triangle $ \\tau_{\\leq a} E \\to E \\to \\tau_{> a} E \\to$", "we see that the equivalence follows if we can show", "$$", "\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{> a} E) =", "\\Hom_{D(\\mathcal{O}_X)}(P, (\\tau_{> a} E)[i]) = 0", "$$", "for $i \\ll 0$. As $P$ is perfect this is true by", "Lemma \\ref{lemma-ext-from-perfect-into-bounded-QCoh}." ], "refs": [ "spaces-perfect-lemma-ext-from-perfect-into-bounded-QCoh" ], "ref_ids": [ 2712 ] } ], "ref_ids": [] }, { "id": 2756, "type": "theorem", "label": "spaces-perfect-proposition-quasi-compact-affine-diagonal", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-proposition-quasi-compact-affine-diagonal", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact algebraic space over $S$", "with affine diagonal. Then the functor (\\ref{equation-compare})", "$$", "D(\\QCoh(\\mathcal{O}_X))", "\\longrightarrow", "D_\\QCoh(\\mathcal{O}_X)", "$$", "is an equivalence with quasi-inverse given by $RQ_X$." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\to W$ be an \\'etale morphism with $V$ affine and $W$ a", "quasi-compact open subspace of $X$. Then ", "the morphism $V \\to W$ is affine as $V$ is affine and $W$ has affine diagonal", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-affine-permanence}).", "Lemma \\ref{lemma-affine-pushforward} then guarantees that", "the assumption of Lemma \\ref{lemma-argument-proves} holds.", "Hence we conclude." ], "refs": [ "spaces-morphisms-lemma-affine-permanence", "spaces-perfect-lemma-affine-pushforward", "spaces-perfect-lemma-argument-proves" ], "ref_ids": [ 4804, 2681, 2684 ] } ], "ref_ids": [] }, { "id": 2757, "type": "theorem", "label": "spaces-perfect-proposition-Noetherian", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-proposition-Noetherian", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Then the functor (\\ref{equation-compare})", "$$", "D(\\QCoh(\\mathcal{O}_X))", "\\longrightarrow", "D_\\QCoh(\\mathcal{O}_X)", "$$", "is an equivalence with quasi-inverse given by $RQ_X$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemmas \\ref{lemma-Noetherian-pushforward} and", "\\ref{lemma-argument-proves}." ], "refs": [ "spaces-perfect-lemma-Noetherian-pushforward", "spaces-perfect-lemma-argument-proves" ], "ref_ids": [ 2690, 2684 ] } ], "ref_ids": [] }, { "id": 2758, "type": "theorem", "label": "spaces-perfect-proposition-compact-is-perfect", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-proposition-compact-is-perfect", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "An object of $D_\\QCoh(\\mathcal{O}_X)$ is compact", "if and only if it is perfect." ], "refs": [], "proofs": [ { "contents": [ "If $K$ is a perfect object of $D(\\mathcal{O}_X)$ with dual", "$K^\\vee$ (Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-dual-perfect-complex})", "we have", "$$", "\\Hom_{D(\\mathcal{O}_X)}(K, M) =", "H^0(X, K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)", "$$", "functorially in $M$. Since $K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} -$", "commutes with direct sums and since $H^0(X, -)$ commutes with direct", "sums on $D_\\QCoh(\\mathcal{O}_X)$ by", "Lemma \\ref{lemma-quasi-coherence-pushforward-direct-sums}", "we conclude that $K$ is compact in $D_\\QCoh(\\mathcal{O}_X)$.", "\\medskip\\noindent", "Conversely, let $K$ be a compact object of $D_\\QCoh(\\mathcal{O}_X)$.", "To show that $K$ is perfect, it suffices to show that", "$K|_U$ is perfect for every affine scheme $U$ \\'etale over $X$, see", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-perfect-independent-representative}.", "Observe that $j : U \\to X$ is a quasi-compact and separated morphism.", "Hence", "$Rj_* : D_\\QCoh(\\mathcal{O}_U) \\to D_\\QCoh(\\mathcal{O}_X)$", "commutes with direct sums, see", "Lemma \\ref{lemma-quasi-coherence-pushforward-direct-sums}.", "Thus the adjointness of restriction to $U$ and $Rj_*$ implies that", "$K|_U$ is a perfect object of $D_\\QCoh(\\mathcal{O}_U)$.", "Hence we reduce to the case that $X$ is affine, in particular a", "quasi-compact and quasi-separated scheme. Via", "Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site} and", "\\ref{lemma-descend-perfect}", "we reduce to the case of schemes, i.e., to", "Derived Categories of Schemes, Proposition", "\\ref{perfect-proposition-compact-is-perfect}." ], "refs": [ "sites-cohomology-lemma-dual-perfect-complex", "spaces-perfect-lemma-quasi-coherence-pushforward-direct-sums", "sites-cohomology-lemma-perfect-independent-representative", "spaces-perfect-lemma-quasi-coherence-pushforward-direct-sums", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-descend-perfect", "perfect-proposition-compact-is-perfect" ], "ref_ids": [ 4390, 2653, 4381, 2653, 2644, 2695, 7111 ] } ], "ref_ids": [] }, { "id": 2759, "type": "theorem", "label": "spaces-perfect-proposition-detecting-bounded-above", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-proposition-detecting-bounded-above", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $G \\in D_{perf}(\\mathcal{O}_X)$ be a perfect complex which generates", "$D_\\QCoh (\\mathcal{O}_X)$. Let $E \\in D_\\QCoh (\\mathcal{O}_X)$.", "The following are equivalent", "\\begin{enumerate}", "\\item $E \\in D^-_\\QCoh (\\mathcal{O}_X)$,", "\\item $\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = 0$ for $i \\gg 0$,", "\\item $\\Ext^i_X(G, E) = 0$ for $i \\gg 0$,", "\\item $R\\Hom_X(G, E)$ is in $D^-(\\mathbf{Z})$,", "\\item $H^i(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$", "for $i \\gg 0$,", "\\item $R\\Gamma(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$", "is in $D^-(\\mathbf{Z})$,", "\\item for every perfect object $P$ of $D(\\mathcal{O}_X)$", "\\begin{enumerate}", "\\item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and", "\\item $H^i(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\gg 0$,", "\\item $R\\Gamma(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$", "is in $D^-(\\mathbf{Z})$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Since", "$\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = \\Hom_{D(\\mathcal{O}_X)}(G, E[i])$", "we see that this is zero for $i \\gg 0$ by", "Lemma \\ref{lemma-ext-from-perfect-into-bounded-QCoh}. This proves", "that (1) implies (2).", "\\medskip\\noindent", "Parts (2), (3), (4) are equivalent by the discussion in", "Cohomology on Sites, Section \\ref{sites-cohomology-section-global-RHom}.", "Part (5) and (6) are equivalent as $H^i(X, -) = H^i(R\\Gamma(X, -))$", "by definition. The equivalent conditions (2), (3), (4) are", "equivalent to the equivalent conditions (5), (6) by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-dual-perfect-complex}", "and the fact that $(G[-i])^\\vee = G^\\vee[i]$.", "\\medskip\\noindent", "It is clear that (7) implies (2). Conversely,", "let us prove that the equivalent conditions (2) -- (6) imply (7).", "Recall that $G$ is a classical generator for $D_{perf}(\\mathcal{O}_X)$ by", "Remark \\ref{remark-classical-generator}.", "For $P \\in D_{perf}(\\mathcal{O}_X)$ let $T(P)$ be the assertion that", "$R\\Hom_X(P, E)$ is in $D^-(\\mathbf{Z})$.", "Clearly, $T$ is inherited by direct sums,", "satisfies the 2-out-of-three property for distinguished", "triangles, is inherited by direct summands, and is perserved by shifts.", "Hence by Derived Categories, Remark \\ref{derived-remark-check-on-generator}", "we see that (4) implies $T$ holds on all of $D_{perf}(\\mathcal{O}_X)$.", "The same argument works for all other properties, except that for property", "(7)(b) and (7)(c) we also use that $P \\mapsto P^\\vee$ is a self", "equivalence of $D_{perf}(\\mathcal{O}_X)$. Small detail omitted.", "\\medskip\\noindent", "We will prove the equivalent conditions (2) -- (7) imply (1)", "using the induction principle of", "Lemma \\ref{lemma-induction-principle}.", "\\medskip\\noindent", "First, we prove (2) -- (7) $\\Rightarrow$ (1) if $X$ is affine.", "This follows from the case of schemes, see", "Derived Categories of Schemes, Proposition", "\\ref{perfect-proposition-detecting-bounded-above}.", "\\medskip\\noindent", "Now assume $(U \\subset X, j : V \\to X)$ is an elementary distinguished", "square of quasi-compact and quasi-separated algebraic spaces over $S$", "and assume the implication (2) -- (7) $\\Rightarrow$ (1)", "is known for $U$, $V$, and $U \\times_X V$. To finish the proof", "we have to show the implication (2) -- (7) $\\Rightarrow$ (1) for $X$.", "Suppose $E \\in D_\\QCoh(\\mathcal{O}_X)$ satisfies (2) -- (7).", "By Lemma \\ref{lemma-direct-summand-of-a-restriction} and", "Theorem \\ref{theorem-bondal-van-den-Bergh} there exists a perfect", "complex $Q$ on $X$ such that $Q|_U$ generates $D_\\QCoh (\\mathcal{O}_U)$.", "\\medskip\\noindent", "Say $V = \\Spec(A)$. Let $Z \\subset V$ be the reduced closed subscheme", "which is the inverse image of $X \\setminus U$ and maps isomorphically to it", "(see Definition \\ref{definition-elementary-distinguished-square}).", "This is a retrocompact closed subset of $V$.", "Choose $f_1, \\ldots, f_r \\in A$ such that", "$Z = V(f_1, \\ldots, f_r)$. Let $K \\in D(\\mathcal{O}_V)$ be the perfect", "object corresponding to the Koszul complex on $f_1, \\ldots, f_r$ over $A$.", "Note that since $K$ is supported on $Z$, the pushforward", "$K' = Rj_*K$ is a perfect object of $D(\\mathcal{O}_X)$ whose", "restriction to $V$ is $K$ (see Lemmas \\ref{lemma-pushforward-perfect}", "and \\ref{lemma-pushforward-with-support-in-open}).", "By assumption, we know $R\\Hom_{\\mathcal{O}_X}(Q, E)$ and", "$R\\Hom_{\\mathcal{O}_X}(K', E)$ are bounded above.", "\\medskip\\noindent", "By Lemma \\ref{lemma-pushforward-with-support-in-open}", "we have $K' = j_!K$ and hence", "$$", "\\Hom_{D(\\mathcal{O}_X)}(K'[-i], E) = \\Hom_{D(\\mathcal{O}_V)}(K[-i], E|_V) = 0", "$$", "for $i \\gg 0$. Therefore, we may apply", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-orthogonal-koszul-first-variant} to $E|_V$ to", "obtain an integer $a$ such that", "$\\tau_{\\geq a}(E|_V) =", "\\tau_{\\geq a} R (U \\times_X V \\to V)_* (E|_{U \\times_X V})$.", "Then $\\tau_{\\geq a} E = \\tau_{\\geq a} R (U \\to X)_* (E |_U)$", "(check that the canonical map is an isomorphism after restricting to", "$U$ and to $V$). Hence using Lemma \\ref{lemma-bounded-truncation}", "twice we see that", "$$", "\\Hom_{D(\\mathcal{O}_U)}(Q|_U [-i], E|_U) =", "\\Hom_{D(\\mathcal{O}_X)}(Q[-i], R (U \\to X)_* (E|_U)) = 0", "$$", "for $i \\gg 0$. Since the Proposition holds for $U$ and the generator", "$Q|_U$, we have $E|_U \\in D^-_\\QCoh(\\mathcal{O}_U)$. But then since", "the functor $R (U \\to X)_*$ preserves $D^-_\\QCoh$", "(by Lemma \\ref{lemma-quasi-coherence-direct-image}), we get", "$\\tau_{\\geq a}E \\in D^-_\\QCoh(\\mathcal{O}_X)$. Thus", "$E \\in D^-_\\QCoh (\\mathcal{O}_X)$." ], "refs": [ "spaces-perfect-lemma-ext-from-perfect-into-bounded-QCoh", "sites-cohomology-lemma-dual-perfect-complex", "spaces-perfect-remark-classical-generator", "derived-remark-check-on-generator", "spaces-perfect-lemma-induction-principle", "perfect-proposition-detecting-bounded-above", "spaces-perfect-lemma-direct-summand-of-a-restriction", "spaces-perfect-theorem-bondal-van-den-Bergh", "spaces-perfect-definition-elementary-distinguished-square", "spaces-perfect-lemma-pushforward-perfect", "spaces-perfect-lemma-pushforward-with-support-in-open", "spaces-perfect-lemma-pushforward-with-support-in-open", "perfect-lemma-orthogonal-koszul-first-variant", "spaces-perfect-lemma-bounded-truncation", "spaces-perfect-lemma-quasi-coherence-direct-image" ], "ref_ids": [ 2712, 4390, 2772, 2030, 2670, 7113, 2707, 2640, 2764, 2702, 2679, 2679, 7103, 2754, 2652 ] } ], "ref_ids": [] }, { "id": 2760, "type": "theorem", "label": "spaces-perfect-proposition-detecting-bounded-below", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-proposition-detecting-bounded-below", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $G \\in D_{perf}(\\mathcal{O}_X)$ be a perfect complex which generates", "$D_\\QCoh (\\mathcal{O}_X)$. Let $E \\in D_\\QCoh (\\mathcal{O}_X)$.", "The following are equivalent", "\\begin{enumerate}", "\\item $E \\in D^+_\\QCoh (\\mathcal{O}_X)$,", "\\item $\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = 0$ for $i \\ll 0$,", "\\item $\\Ext^i_X(G, E) = 0$ for $i \\ll 0$,", "\\item $R\\Hom_X(G, E)$ is in $D^+(\\mathbf{Z})$,", "\\item $H^i(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$", "for $i \\ll 0$,", "\\item $R\\Gamma(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$", "is in $D^+(\\mathbf{Z})$,", "\\item for every perfect object $P$ of $D(\\mathcal{O}_X)$", "\\begin{enumerate}", "\\item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and", "\\item $H^i(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\ll 0$,", "\\item $R\\Gamma(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$", "is in $D^+(\\mathbf{Z})$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Since", "$\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = \\Hom_{D(\\mathcal{O}_X)}(G, E[i])$", "we see that this is zero for $i \\ll 0$ by", "Lemma \\ref{lemma-ext-from-perfect-into-bounded-QCoh}. This proves", "that (1) implies (2).", "\\medskip\\noindent", "Parts (2), (3), (4) are equivalent by the discussion in", "Cohomology on Sites, Section \\ref{sites-cohomology-section-global-RHom}.", "Part (5) and (6) are equivalent as $H^i(X, -) = H^i(R\\Gamma(X, -))$", "by definition. The equivalent conditions (2), (3), (4) are", "equivalent to the equivalent conditions (5), (6) by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-dual-perfect-complex}", "and the fact that $(G[-i])^\\vee = G^\\vee[i]$.", "\\medskip\\noindent", "It is clear that (7) implies (2). Conversely,", "let us prove that the equivalent conditions (2) -- (6) imply (7).", "Recall that $G$ is a classical generator for $D_{perf}(\\mathcal{O}_X)$ by", "Remark \\ref{remark-classical-generator}.", "For $P \\in D_{perf}(\\mathcal{O}_X)$ let $T(P)$ be the assertion that", "$R\\Hom_X(P, E)$ is in $D^+(\\mathbf{Z})$.", "Clearly, $T$ is inherited by direct sums,", "satisfies the 2-out-of-three property for distinguished", "triangles, is inherited by direct summands, and is perserved by shifts.", "Hence by Derived Categories, Remark \\ref{derived-remark-check-on-generator}", "we see that (4) implies $T$ holds on all of $D_{perf}(\\mathcal{O}_X)$.", "The same argument works for all other properties, except that for property", "(7)(b) and (7)(c) we also use that $P \\mapsto P^\\vee$ is a self", "equivalence of $D_{perf}(\\mathcal{O}_X)$. Small detail omitted.", "\\medskip\\noindent", "We will prove the equivalent conditions (2) -- (7) imply (1)", "using the induction principle of", "Lemma \\ref{lemma-induction-principle}.", "\\medskip\\noindent", "First, we prove (2) -- (7) $\\Rightarrow$ (1) if $X$ is affine.", "This follows from the case of schemes, see", "Derived Categories of Schemes, Proposition", "\\ref{perfect-proposition-detecting-bounded-below}.", "\\medskip\\noindent", "Now assume $(U \\subset X, j : V \\to X)$ is an elementary distinguished", "square of quasi-compact and quasi-separated algebraic spaces over $S$", "and assume the implication (2) -- (7) $\\Rightarrow$ (1)", "is known for $U$, $V$, and $U \\times_X V$. To finish the proof", "we have to show the implication (2) -- (7) $\\Rightarrow$ (1) for $X$.", "Suppose $E \\in D_\\QCoh(\\mathcal{O}_X)$ satisfies (2) -- (7).", "By Lemma \\ref{lemma-direct-summand-of-a-restriction} and", "Theorem \\ref{theorem-bondal-van-den-Bergh} there exists a perfect", "complex $Q$ on $X$ such that $Q|_U$ generates $D_\\QCoh (\\mathcal{O}_U)$.", "\\medskip\\noindent", "Say $V = \\Spec(A)$. Let $Z \\subset V$ be the reduced closed subscheme", "which is the inverse image of $X \\setminus U$ and maps isomorphically to it", "(see Definition \\ref{definition-elementary-distinguished-square}).", "This is a retrocompact closed subset of $V$.", "Choose $f_1, \\ldots, f_r \\in A$ such that", "$Z = V(f_1, \\ldots, f_r)$. Let $K \\in D(\\mathcal{O}_V)$ be the perfect", "object corresponding to the Koszul complex on $f_1, \\ldots, f_r$ over $A$.", "Note that since $K$ is supported on $Z$, the pushforward", "$K' = Rj_*K$ is a perfect object of $D(\\mathcal{O}_X)$ whose", "restriction to $V$ is $K$ (see Lemmas \\ref{lemma-pushforward-perfect}", "and \\ref{lemma-pushforward-with-support-in-open}).", "By assumption, we know $R\\Hom_{\\mathcal{O}_X}(Q, E)$ and", "$R\\Hom_{\\mathcal{O}_X}(K', E)$ are bounded below.", "\\medskip\\noindent", "By Lemma \\ref{lemma-pushforward-with-support-in-open}", "we have $K' = j_!K$ and hence", "$$", "\\Hom_{D(\\mathcal{O}_X)}(K'[-i], E) = \\Hom_{D(\\mathcal{O}_V)}(K[-i], E|_V) = 0", "$$", "for $i \\ll 0$. Therefore, we may apply", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-orthogonal-koszul-second-variant} to $E|_V$ to", "obtain an integer $a$ such that", "$\\tau_{\\leq a}(E|_V) =", "\\tau_{\\leq a} R (U \\times_X V \\to V)_* (E|_{U \\times_X V})$.", "Then $\\tau_{\\leq a} E = \\tau_{\\leq a} R (U \\to X)_* (E |_U)$", "(check that the canonical map is an isomorphism after restricting to", "$U$ and to $V$). Hence using Lemma \\ref{lemma-bounded-below-truncation}", "twice we see that", "$$", "\\Hom_{D(\\mathcal{O}_U)}(Q|_U [-i], E|_U) =", "\\Hom_{D(\\mathcal{O}_X)}(Q[-i], R (U \\to X)_* (E|_U)) = 0", "$$", "for $i \\ll 0$. Since the Proposition holds for $U$ and the generator", "$Q|_U$, we have $E|_U \\in D^+_\\QCoh(\\mathcal{O}_U)$. But then since", "the functor $R (U \\to X)_*$ preserves $D^+_\\QCoh$", "(by Lemma \\ref{lemma-quasi-coherence-direct-image}), we get", "$\\tau_{\\leq a}E \\in D^+_\\QCoh(\\mathcal{O}_X)$. Thus", "$E \\in D^+_\\QCoh (\\mathcal{O}_X)$." ], "refs": [ "spaces-perfect-lemma-ext-from-perfect-into-bounded-QCoh", "sites-cohomology-lemma-dual-perfect-complex", "spaces-perfect-remark-classical-generator", "derived-remark-check-on-generator", "spaces-perfect-lemma-induction-principle", "perfect-proposition-detecting-bounded-below", "spaces-perfect-lemma-direct-summand-of-a-restriction", "spaces-perfect-theorem-bondal-van-den-Bergh", "spaces-perfect-definition-elementary-distinguished-square", "spaces-perfect-lemma-pushforward-perfect", "spaces-perfect-lemma-pushforward-with-support-in-open", "spaces-perfect-lemma-pushforward-with-support-in-open", "perfect-lemma-orthogonal-koszul-second-variant", "spaces-perfect-lemma-bounded-below-truncation", "spaces-perfect-lemma-quasi-coherence-direct-image" ], "ref_ids": [ 2712, 4390, 2772, 2030, 2670, 7114, 2707, 2640, 2764, 2702, 2679, 2679, 7104, 2755, 2652 ] } ], "ref_ids": [] }, { "id": 2780, "type": "theorem", "label": "dualizing-theorem-local-duality", "categories": [ "dualizing" ], "title": "dualizing-theorem-local-duality", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Let $\\omega_A^\\bullet$ be a normalized dualizing complex.", "Let $E$ be an injective hull of the residue field.", "Let $Z = V(\\mathfrak m) \\subset \\Spec(A)$.", "Denote ${}^\\wedge$ derived completion with respect to $\\mathfrak m$.", "Then", "$$", "R\\Hom_A(K, \\omega_A^\\bullet)^\\wedge \\cong R\\Hom_A(R\\Gamma_Z(K), E[0])", "$$", "for $K$ in $D(A)$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $E[0] \\cong R\\Gamma_Z(\\omega_A^\\bullet)$ by", "Lemma \\ref{lemma-local-cohomology-of-dualizing}.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-completion-RHom}", "completion on the left hand side goes inside.", "Thus we have to prove", "$$", "R\\Hom_A(K^\\wedge, (\\omega_A^\\bullet)^\\wedge)", "=", "R\\Hom_A(R\\Gamma_Z(K), R\\Gamma_Z(\\omega_A^\\bullet))", "$$", "This follows from the equivalence between", "$D_{comp}(A, \\mathfrak m)$ and $D_{\\mathfrak m^\\infty\\text{-torsion}}(A)$", "given in Proposition \\ref{proposition-torsion-complete}.", "More precisely, it is a special case of Lemma \\ref{lemma-compare-RHom}." ], "refs": [ "dualizing-lemma-local-cohomology-of-dualizing", "more-algebra-lemma-completion-RHom", "dualizing-proposition-torsion-complete", "dualizing-lemma-compare-RHom" ], "ref_ids": [ 2872, 10374, 2925, 2833 ] } ], "ref_ids": [] }, { "id": 2781, "type": "theorem", "label": "dualizing-lemma-essential", "categories": [ "dualizing" ], "title": "dualizing-lemma-essential", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item If $A \\subset B$ and $B \\subset C$ are essential extensions, then", "$A \\subset C$ is an essential extension.", "\\item If $A \\subset B$ is an essential extension and $C \\subset B$", "is a subobject, then $A \\cap C \\subset C$ is an essential extension.", "\\item If $A \\to B$ and $B \\to C$ are essential surjections, then", "$A \\to C$ is an essential surjection.", "\\item Given an essential surjection $f : A \\to B$ and a surjection", "$A \\to C$ with kernel $K$, the morphism $C \\to B/f(K)$ is an essential", "surjection.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2782, "type": "theorem", "label": "dualizing-lemma-union-essential-extensions", "categories": [ "dualizing" ], "title": "dualizing-lemma-union-essential-extensions", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module. Let $E = \\colim E_i$", "be a filtered colimit of $R$-modules. Suppose given a compatible", "system of essential injections $M \\to E_i$ of $R$-modules.", "Then $M \\to E$ is an essential injection." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions and the fact that filtered", "colimits are exact (Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact})." ], "refs": [ "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 343 ] } ], "ref_ids": [] }, { "id": 2783, "type": "theorem", "label": "dualizing-lemma-essential-extension", "categories": [ "dualizing" ], "title": "dualizing-lemma-essential-extension", "contents": [ "Let $R$ be a ring. Let $M \\subset N$ be $R$-modules. The following", "are equivalent", "\\begin{enumerate}", "\\item $M \\subset N$ is an essential extension,", "\\item for all $x \\in N$ nonzero there exists an $f \\in R$ such that $fx \\in M$", "and $fx \\not = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and let $x \\in N$ be a nonzero element. By (1) we have", "$Rx \\cap M \\not = 0$. This implies (2).", "\\medskip\\noindent", "Assume (2). Let $N' \\subset N$ be a nonzero submodule. Pick $x \\in N'$", "nonzero. By (2) we can find $f \\in R$ with $fx \\in M$ and $fx \\not = 0$.", "Thus $N' \\cap M \\not = 0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2784, "type": "theorem", "label": "dualizing-lemma-product-injectives", "categories": [ "dualizing" ], "title": "dualizing-lemma-product-injectives", "contents": [ "Let $R$ be a ring. Any product of injective $R$-modules is injective." ], "refs": [], "proofs": [ { "contents": [ "Special case of Homology, Lemma \\ref{homology-lemma-product-injectives}." ], "refs": [ "homology-lemma-product-injectives" ], "ref_ids": [ 12113 ] } ], "ref_ids": [] }, { "id": 2785, "type": "theorem", "label": "dualizing-lemma-injective-flat", "categories": [ "dualizing" ], "title": "dualizing-lemma-injective-flat", "contents": [ "Let $R \\to S$ be a flat ring map. If $E$ is an injective $S$-module,", "then $E$ is injective as an $R$-module." ], "refs": [], "proofs": [ { "contents": [ "This is true because $\\Hom_R(M, E) = \\Hom_S(M \\otimes_R S, E)$", "by Algebra, Lemma \\ref{algebra-lemma-adjoint-tensor-restrict}", "and the fact that tensoring with $S$ is exact." ], "refs": [ "algebra-lemma-adjoint-tensor-restrict" ], "ref_ids": [ 374 ] } ], "ref_ids": [] }, { "id": 2786, "type": "theorem", "label": "dualizing-lemma-injective-epimorphism", "categories": [ "dualizing" ], "title": "dualizing-lemma-injective-epimorphism", "contents": [ "Let $R \\to S$ be an epimorphism of rings. Let $E$ be an $S$-module.", "If $E$ is injective as an $R$-module, then $E$ is an injective $S$-module." ], "refs": [], "proofs": [ { "contents": [ "This is true because $\\Hom_R(N, E) = \\Hom_S(N, E)$ for any $S$-module $N$,", "see Algebra, Lemma \\ref{algebra-lemma-epimorphism-modules}." ], "refs": [ "algebra-lemma-epimorphism-modules" ], "ref_ids": [ 959 ] } ], "ref_ids": [] }, { "id": 2787, "type": "theorem", "label": "dualizing-lemma-hom-injective", "categories": [ "dualizing" ], "title": "dualizing-lemma-hom-injective", "contents": [ "Let $R \\to S$ be a ring map. If $E$ is an injective $R$-module,", "then $\\Hom_R(S, E)$ is an injective $S$-module." ], "refs": [], "proofs": [ { "contents": [ "This is true because $\\Hom_S(N, \\Hom_R(S, E)) = \\Hom_R(N, E)$ by", "Algebra, Lemma \\ref{algebra-lemma-adjoint-hom-restrict}." ], "refs": [ "algebra-lemma-adjoint-hom-restrict" ], "ref_ids": [ 375 ] } ], "ref_ids": [] }, { "id": 2788, "type": "theorem", "label": "dualizing-lemma-essential-extensions-in-injective", "categories": [ "dualizing" ], "title": "dualizing-lemma-essential-extensions-in-injective", "contents": [ "Let $R$ be a ring. Let $I$ be an injective $R$-module. Let $E \\subset I$", "be a submodule. The following are equivalent", "\\begin{enumerate}", "\\item $E$ is injective, and", "\\item for all $E \\subset E' \\subset I$ with $E \\subset E'$ essential", "we have $E = E'$.", "\\end{enumerate}", "In particular, an $R$-module is injective if and only if every essential", "extension is trivial." ], "refs": [], "proofs": [ { "contents": [ "The final assertion follows from the first and the fact that the", "category of $R$-modules has enough injectives", "(More on Algebra, Section \\ref{more-algebra-section-injectives-modules}).", "\\medskip\\noindent", "Assume (1). Let $E \\subset E' \\subset I$ as in (2).", "Then the map $\\text{id}_E : E \\to E$ can be extended", "to a map $\\alpha : E' \\to E$. The kernel of $\\alpha$ has to be", "zero because it intersects $E$ trivially and $E'$ is an essential", "extension. Hence $E = E'$.", "\\medskip\\noindent", "Assume (2). Let $M \\subset N$ be $R$-modules and let $\\varphi : M \\to E$", "be an $R$-module map. In order to prove (1) we have to show that", "$\\varphi$ extends to a morphism $N \\to E$. Consider the set $\\mathcal{S}$", "of pairs", "$(M', \\varphi')$ where $M \\subset M' \\subset N$ and $\\varphi' : M' \\to E$", "is an $R$-module map agreeing with $\\varphi$ on $M$. We define an ordering", "on $\\mathcal{S}$ by the rule $(M', \\varphi') \\leq (M'', \\varphi'')$", "if and only if $M' \\subset M''$ and $\\varphi''|_{M'} = \\varphi'$.", "It is clear that we can take the maximum of a totally ordered subset", "of $\\mathcal{S}$. Hence by Zorn's lemma we may assume $(M, \\varphi)$", "is a maximal element.", "\\medskip\\noindent", "Choose an extension $\\psi : N \\to I$ of $\\varphi$ composed", "with the inclusion $E \\to I$. This is possible as $I$ is injective.", "If $\\psi(N) \\subset E$, then $\\psi$ is the desired extension.", "If $\\psi(N)$ is not contained in $E$, then by (2) the inclusion", "$E \\subset E + \\psi(N)$ is not essential. hence", "we can find a nonzero submodule $K \\subset E + \\psi(N)$ meeting $E$ in $0$.", "This means that $M' = \\psi^{-1}(E + K)$ strictly contains $M$.", "Thus we can extend $\\varphi$ to $M'$ using", "$$", "M' \\xrightarrow{\\psi|_{M'}} E + K \\to (E + K)/K = E", "$$", "This contradicts the maximality of $(M, \\varphi)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2789, "type": "theorem", "label": "dualizing-lemma-sum-injective-modules", "categories": [ "dualizing" ], "title": "dualizing-lemma-sum-injective-modules", "contents": [ "Let $R$ be a Noetherian ring. A direct sum of injective modules", "is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $E_i$ be a family of injective modules parametrized by a set $I$.", "Set $E = \\bigcup E_i$. To show that $E$ is injective we use", "Injectives, Lemma \\ref{injectives-lemma-criterion-baer}.", "Thus let $\\varphi : I \\to E$ be a module map from an ideal of $R$", "into $E$. As $I$ is a finite $R$-module (because $R$ is Noetherian)", "we can find finitely many elements $i_1, \\ldots, i_r \\in I$", "such that $\\varphi$ maps into $\\bigcup_{j = 1, \\ldots, r} E_{i_j}$.", "Then we can extend $\\varphi$ into $\\bigcup_{j = 1, \\ldots, r} E_{i_j}$", "using the injectivity of the modules $E_{i_j}$." ], "refs": [ "injectives-lemma-criterion-baer" ], "ref_ids": [ 7771 ] } ], "ref_ids": [] }, { "id": 2790, "type": "theorem", "label": "dualizing-lemma-localization-injective-modules", "categories": [ "dualizing" ], "title": "dualizing-lemma-localization-injective-modules", "contents": [ "Let $R$ be a Noetherian ring. Let $S \\subset R$ be a multiplicative", "subset. If $E$ is an injective $R$-module, then $S^{-1}E$ is an", "injective $S^{-1}R$-module." ], "refs": [], "proofs": [ { "contents": [ "Since $R \\to S^{-1}R$ is an epimorphism of rings, it suffices", "to show that $S^{-1}E$ is injective as an $R$-module, see", "Lemma \\ref{lemma-injective-epimorphism}.", "To show this we use Injectives, Lemma \\ref{injectives-lemma-criterion-baer}.", "Thus let $I \\subset R$ be an ideal and let", "$\\varphi : I \\to S^{-1} E$ be an $R$-module map.", "As $I$ is a finitely presented $R$-module (because $R$ is Noetherian)", "we can find an $f \\in S$ and an $R$-module map $I \\to E$", "such that $f\\varphi$ is the composition $I \\to E \\to S^{-1}E$", "(Algebra, Lemma \\ref{algebra-lemma-hom-from-finitely-presented}).", "Then we can extend $I \\to E$ to a homomorphism $R \\to E$.", "Then the composition", "$$", "R \\to E \\to S^{-1}E \\xrightarrow{f^{-1}} S^{-1}E", "$$", "is the desired extension of $\\varphi$ to $R$." ], "refs": [ "dualizing-lemma-injective-epimorphism", "injectives-lemma-criterion-baer", "algebra-lemma-hom-from-finitely-presented" ], "ref_ids": [ 2786, 7771, 353 ] } ], "ref_ids": [] }, { "id": 2791, "type": "theorem", "label": "dualizing-lemma-injective-module-divide", "categories": [ "dualizing" ], "title": "dualizing-lemma-injective-module-divide", "contents": [ "Let $R$ be a Noetherian ring. Let $I$ be an injective $R$-module.", "\\begin{enumerate}", "\\item Let $f \\in R$. Then $E = \\bigcup I[f^n] = I[f^\\infty]$", "is an injective submodule of $I$.", "\\item Let $J \\subset R$ be an ideal. Then the $J$-power torsion", "submodule $I[J^\\infty]$ is an injective submodule of $I$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-essential-extensions-in-injective}", "to prove (1).", "Suppose that $E \\subset E' \\subset I$ and that $E'$ is an essential", "extension of $E$. We will show that $E' = E$. If not, then we can", "find $x \\in E'$ and $x \\not \\in E$.", "Let $J = \\{ a \\in R \\mid ax \\in E\\}$. Since $R$ is Noetherian, we may", "choose $x$ so that $J$ is maximal among ideals of this form. Again since", "$R$ is Noetherian, we may write $J = (g_1, \\ldots, g_t)$ for some", "$g_i \\in R$. By definition $E$ is the set of elements of $I$ annihilated", "by powers of $f$, so we may choose integers $n_i$ so that $f^{n_i}g_ix = 0$.", "Set $n = \\mathrm{max}\\{ n_i \\}$. Then $x' = f^n x$ is an element of $E'$", "not in $E$ and is annihilated by $J$. Then by maximality of $J$ we have", "$J = \\{ a \\in R \\mid ax' \\in E \\} = \\text{Ann}(x')$, so $Rx' \\cap E = 0$.", "Hence $E'$ is not an essential extension of $E$, a contradiction.", "\\medskip\\noindent", "To prove (2) write $J = (f_1, \\ldots, f_t)$. Then", "$I[J^\\infty]$ is equal to", "$$", "(\\ldots((I[f_1^\\infty])[f_2^\\infty])\\ldots)[f_t^\\infty]", "$$", "and the result follows from (1) and induction." ], "refs": [ "dualizing-lemma-essential-extensions-in-injective" ], "ref_ids": [ 2788 ] } ], "ref_ids": [] }, { "id": 2792, "type": "theorem", "label": "dualizing-lemma-injective-dimension-over-polynomial-ring", "categories": [ "dualizing" ], "title": "dualizing-lemma-injective-dimension-over-polynomial-ring", "contents": [ "Let $A$ be a Noetherian ring. Let $E$ be an injective $A$-module.", "Then $E \\otimes_A A[x]$ has injective-amplitude $[0, 1]$", "as an object of $D(A[x])$. In particular, $E \\otimes_A A[x]$", "has finite injective dimension as an $A[x]$-module." ], "refs": [], "proofs": [ { "contents": [ "Let us write $E[x] = E \\otimes_A A[x]$. Consider the short exact", "sequence of $A[x]$-modules", "$$", "0 \\to E[x] \\to \\Hom_A(A[x], E[x]) \\to \\Hom_A(A[x], E[x]) \\to 0", "$$", "where the first map sends $p \\in E[x]$ to $f \\mapsto fp$ and the", "second map sends $\\varphi$ to $f \\mapsto \\varphi(xf) - x\\varphi(f)$.", "The second map is surjective because", "$\\Hom_A(A[x], E[x]) = \\prod_{n \\geq 0} E[x]$ as an abelian group and", "the map sends $(e_n)$ to $(e_{n + 1} - xe_n)$ which is surjective.", "As an $A$-module we have $E[x] \\cong \\bigoplus_{n \\geq 0} E$", "which is injective by Lemma \\ref{lemma-sum-injective-modules}.", "Hence the $A[x]$-module $\\Hom_A(A[x], E[x])$ is injective by", "Lemma \\ref{lemma-hom-injective} and the proof is complete." ], "refs": [ "dualizing-lemma-sum-injective-modules", "dualizing-lemma-hom-injective" ], "ref_ids": [ 2789, 2787 ] } ], "ref_ids": [] }, { "id": 2793, "type": "theorem", "label": "dualizing-lemma-projective-cover-unique", "categories": [ "dualizing" ], "title": "dualizing-lemma-projective-cover-unique", "contents": [ "Let $R$ be a ring and let $M$ be an $R$-module. If a projective cover", "of $M$ exists, then it is unique up to isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Let $P \\to M$ and $P' \\to M$ be projective covers. Because $P$ is a", "projective $R$-module and $P' \\to M$ is surjective, we can find an", "$R$-module map $\\alpha : P \\to P'$ compatible with the maps to $M$.", "Since $P' \\to M$ is essential, we see that $\\alpha$ is surjective.", "As $P'$ is a projective $R$-module we can choose a direct sum decomposition", "$P = \\Ker(\\alpha) \\oplus P'$. Since $P' \\to M$ is surjective", "and since $P \\to M$ is essential we conclude that $\\Ker(\\alpha)$", "is zero as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2794, "type": "theorem", "label": "dualizing-lemma-projective-covers-local", "categories": [ "dualizing" ], "title": "dualizing-lemma-projective-covers-local", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Any finite $R$-module has", "a projective cover." ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be a finite $R$-module. Let $r = \\dim_\\kappa(M/\\mathfrak m M)$.", "Choose $x_1, \\ldots, x_r \\in M$ mapping to a basis of $M/\\mathfrak m M$.", "Consider the map $f : R^{\\oplus r} \\to M$. By Nakayama's lemma this is", "a surjection (Algebra, Lemma \\ref{algebra-lemma-NAK}). If", "$N \\subset R^{\\oplus r}$ is a proper submodule, then", "$N/\\mathfrak m N \\to \\kappa^{\\oplus r}$ is not surjective (by", "Nakayama's lemma again) hence $N/\\mathfrak m N \\to M/\\mathfrak m M$", "is not surjective. Thus $f$ is an essential surjection." ], "refs": [ "algebra-lemma-NAK" ], "ref_ids": [ 401 ] } ], "ref_ids": [] }, { "id": 2795, "type": "theorem", "label": "dualizing-lemma-injective-hull", "categories": [ "dualizing" ], "title": "dualizing-lemma-injective-hull", "contents": [ "Let $R$ be a ring. Any $R$-module has an injective hull." ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be an $R$-module. By", "More on Algebra, Section \\ref{more-algebra-section-injectives-modules}", "the category of $R$-modules has enough injectives.", "Choose an injection $M \\to I$ with $I$ an injective $R$-module.", "Consider the set $\\mathcal{S}$ of submodules $M \\subset E \\subset I$", "such that $E$ is an essential extension of $M$. We order $\\mathcal{S}$", "by inclusion. If $\\{E_\\alpha\\}$ is a totally ordered subset", "of $\\mathcal{S}$, then $\\bigcup E_\\alpha$ is an essential extension of $M$", "too (Lemma \\ref{lemma-union-essential-extensions}).", "Thus we can apply Zorn's lemma and find a maximal element", "$E \\in \\mathcal{S}$. We claim $M \\subset E$ is an injective hull, i.e.,", "$E$ is an injective $R$-module. This follows from", "Lemma \\ref{lemma-essential-extensions-in-injective}." ], "refs": [ "dualizing-lemma-union-essential-extensions", "dualizing-lemma-essential-extensions-in-injective" ], "ref_ids": [ 2782, 2788 ] } ], "ref_ids": [] }, { "id": 2796, "type": "theorem", "label": "dualizing-lemma-injective-hull-unique", "categories": [ "dualizing" ], "title": "dualizing-lemma-injective-hull-unique", "contents": [ "Let $R$ be a ring. Let $M$, $N$ be $R$-modules and let $M \\to E$", "and $N \\to E'$ be injective hulls. Then", "\\begin{enumerate}", "\\item for any $R$-module map $\\varphi : M \\to N$ there exists an", "$R$-module map $\\psi : E \\to E'$ such that", "$$", "\\xymatrix{", "M \\ar[r] \\ar[d]_\\varphi & E \\ar[d]^\\psi \\\\", "N \\ar[r] & E'", "}", "$$", "commutes,", "\\item if $\\varphi$ is injective, then $\\psi$ is injective,", "\\item if $\\varphi$ is an essential injection, then $\\psi$ is an isomorphism,", "\\item if $\\varphi$ is an isomorphism, then $\\psi$ is an isomorphism,", "\\item if $M \\to I$ is an embedding of $M$ into an injective $R$-module,", "then there is an isomorphism $I \\cong E \\oplus I'$ compatible with", "the embeddings of $M$,", "\\end{enumerate}", "In particular, the injective hull $E$ of $M$ is unique up to isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from the fact that $E'$ is an injective $R$-module.", "Part (2) follows as $\\Ker(\\psi) \\cap M = 0$", "and $E$ is an essential extension of $M$.", "Assume $\\varphi$ is an essential injection. Then", "$E \\cong \\psi(E) \\subset E'$ by (2) which implies", "$E' = \\psi(E) \\oplus E''$ because $E$ is injective.", "Since $E'$ is an essential extension of", "$M$ (Lemma \\ref{lemma-essential}) we get $E'' = 0$.", "Part (4) is a special case of (3).", "Assume $M \\to I$ as in (5).", "Choose a map $\\alpha : E \\to I$ extending the map $M \\to I$.", "Arguing as before we see that $\\alpha$ is injective.", "Thus as before $\\alpha(E)$ splits off from $I$.", "This proves (5)." ], "refs": [ "dualizing-lemma-essential" ], "ref_ids": [ 2781 ] } ], "ref_ids": [] }, { "id": 2797, "type": "theorem", "label": "dualizing-lemma-indecomposable-injective", "categories": [ "dualizing" ], "title": "dualizing-lemma-indecomposable-injective", "contents": [ "Let $R$ be a ring. Let $E$ be an indecomposable injective $R$-module.", "Then", "\\begin{enumerate}", "\\item $E$ is the injective hull of any nonzero submodule of $E$,", "\\item the intersection of any two nonzero submodules of $E$ is nonzero,", "\\item $\\text{End}_R(E, E)$ is a noncommutative local ring with maximal", "ideal those $\\varphi : E \\to E$ whose kernel is nonzero, and", "\\item the set of zerodivisors on $E$ is a prime ideal $\\mathfrak p$ of $R$", "and $E$ is an injective $R_\\mathfrak p$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from Lemma \\ref{lemma-injective-hull-unique}.", "Part (2) follows from part (1) and the definition of injective hulls.", "\\medskip\\noindent", "Proof of (3). Set $A = \\text{End}_R(E, E)$ and", "$I = \\{\\varphi \\in A \\mid \\Ker(f) \\not = 0\\}$.", "The statement means that $I$ is a two sided ideal and", "that any $\\varphi \\in A$, $\\varphi \\not \\in I$ is invertible.", "Suppose $\\varphi$ and $\\psi$ are not injective.", "Then $\\Ker(\\varphi) \\cap \\Ker(\\psi)$ is nonzero", "by (2). Hence $\\varphi + \\psi \\in I$. It follows that $I$", "is a two sided ideal. If $\\varphi \\in A$, $\\varphi \\not \\in I$,", "then $E \\cong \\varphi(E) \\subset E$ is an injective submodule,", "hence $E = \\varphi(E)$ because $E$ is indecomposable.", "\\medskip\\noindent", "Proof of (4). Consider the ring map $R \\to A$ and let $\\mathfrak p \\subset R$", "be the inverse image of the maximal ideal $I$. Then it is clear", "that $\\mathfrak p$ is a prime ideal and that $R \\to A$ extends to", "$R_\\mathfrak p \\to A$. Thus $E$ is an $R_\\mathfrak p$-module.", "It follows from Lemma \\ref{lemma-injective-epimorphism} that $E$ is injective", "as an $R_\\mathfrak p$-module." ], "refs": [ "dualizing-lemma-injective-hull-unique", "dualizing-lemma-injective-epimorphism" ], "ref_ids": [ 2796, 2786 ] } ], "ref_ids": [] }, { "id": 2798, "type": "theorem", "label": "dualizing-lemma-injective-hull-indecomposable", "categories": [ "dualizing" ], "title": "dualizing-lemma-injective-hull-indecomposable", "contents": [ "Let $\\mathfrak p \\subset R$ be a prime of a ring $R$.", "Let $E$ be the injective hull of $R/\\mathfrak p$. Then", "\\begin{enumerate}", "\\item $E$ is indecomposable,", "\\item $E$ is the injective hull of $\\kappa(\\mathfrak p)$,", "\\item $E$ is the injective hull of $\\kappa(\\mathfrak p)$", "over the ring $R_\\mathfrak p$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-essential-extension} the inclusion", "$R/\\mathfrak p \\subset \\kappa(\\mathfrak p)$ is an essential", "extension. Then Lemma \\ref{lemma-injective-hull-unique}", "shows (2) holds. For $f \\in R$, $f \\not \\in \\mathfrak p$", "the map $f : \\kappa(\\mathfrak p) \\to \\kappa(\\mathfrak p)$ is an isomorphism", "hence the map $f : E \\to E$ is an isomorphism,", "see Lemma \\ref{lemma-injective-hull-unique}.", "Thus $E$ is an $R_\\mathfrak p$-module. It is injective", "as an $R_\\mathfrak p$-module by Lemma \\ref{lemma-injective-epimorphism}.", "Finally, let $E' \\subset E$ be a nonzero injective $R$-submodule.", "Then $J = (R/\\mathfrak p) \\cap E'$ is nonzero. After shrinking $E'$", "we may assume that $E'$ is the injective hull of $J$ (see", "Lemma \\ref{lemma-injective-hull-unique} for example).", "Observe that $R/\\mathfrak p$ is an essential extension of $J$ for example by", "Lemma \\ref{lemma-essential-extension}. Hence $E' \\to E$", "is an isomorphism by Lemma \\ref{lemma-injective-hull-unique} part (3).", "Hence $E$ is indecomposable." ], "refs": [ "dualizing-lemma-essential-extension", "dualizing-lemma-injective-hull-unique", "dualizing-lemma-injective-hull-unique", "dualizing-lemma-injective-epimorphism", "dualizing-lemma-injective-hull-unique", "dualizing-lemma-essential-extension", "dualizing-lemma-injective-hull-unique" ], "ref_ids": [ 2783, 2796, 2796, 2786, 2796, 2783, 2796 ] } ], "ref_ids": [] }, { "id": 2799, "type": "theorem", "label": "dualizing-lemma-indecomposable-injective-noetherian", "categories": [ "dualizing" ], "title": "dualizing-lemma-indecomposable-injective-noetherian", "contents": [ "Let $R$ be a Noetherian ring. Let $E$ be an indecomposable injective", "$R$-module. Then there exists a prime ideal $\\mathfrak p$ of $R$ such that", "$E$ is the injective hull of $\\kappa(\\mathfrak p)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p$ be the prime ideal found in", "Lemma \\ref{lemma-indecomposable-injective}.", "Say $\\mathfrak p = (f_1, \\ldots, f_r)$.", "Pick a nonzero element $x \\in \\bigcap \\Ker(f_i : E \\to E)$,", "see Lemma \\ref{lemma-indecomposable-injective}.", "Then $(R_\\mathfrak p)x$ is a module isomorphic to $\\kappa(\\mathfrak p)$", "inside $E$. We conclude by Lemma \\ref{lemma-indecomposable-injective}." ], "refs": [ "dualizing-lemma-indecomposable-injective", "dualizing-lemma-indecomposable-injective", "dualizing-lemma-indecomposable-injective" ], "ref_ids": [ 2797, 2797, 2797 ] } ], "ref_ids": [] }, { "id": 2800, "type": "theorem", "label": "dualizing-lemma-finite", "categories": [ "dualizing" ], "title": "dualizing-lemma-finite", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be an artinian local ring.", "Let $E$ be an injective hull of $\\kappa$. For every finite", "$R$-module $M$ we have", "$$", "\\text{length}_R(M) = \\text{length}_R(\\Hom_R(M, E))", "$$", "In particular, the injective hull $E$ of $\\kappa$ is a finite $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Because $E$ is an essential extension of $\\kappa$ we have", "$\\kappa = E[\\mathfrak m]$ where $E[\\mathfrak m]$ is the", "$\\mathfrak m$-torsion in $E$ (notation as in More on Algebra, Section", "\\ref{more-algebra-section-formal-glueing}).", "Hence $\\Hom_R(\\kappa, E) \\cong \\kappa$ and the equality of lengths", "holds for $M = \\kappa$. We prove the displayed equality of the lemma", "by induction on the length of $M$. If $M$ is nonzero there exists a surjection", "$M \\to \\kappa$ with kernel $M'$. Since the functor $M \\mapsto \\Hom_R(M, E)$", "is exact we obtain a short exact sequence", "$$", "0 \\to \\Hom_R(\\kappa, E) \\to \\Hom_R(M, E) \\to \\Hom_R(M', E) \\to 0.", "$$", "Additivity of length for this sequence and the sequence", "$0 \\to M' \\to M \\to \\kappa \\to 0$ and the equality for $M'$ (induction", "hypothesis) and $\\kappa$ implies the equality for $M$.", "The final statement of the lemma follows as $E = \\Hom_R(R, E)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2801, "type": "theorem", "label": "dualizing-lemma-evaluate", "categories": [ "dualizing" ], "title": "dualizing-lemma-evaluate", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be an artinian local ring.", "Let $E$ be an injective hull of $\\kappa$.", "For any finite $R$-module $M$ the evaluation map", "$$", "M \\longrightarrow \\Hom_R(\\Hom_R(M, E), E)", "$$", "is an isomorphism. In particular $R = \\Hom_R(E, E)$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the displayed arrow is injective. Namely, if $x \\in M$ is", "a nonzero element, then there is a nonzero map $Rx \\to \\kappa$ which", "we can extend to a map $\\varphi : M \\to E$ that doesn't vanish on $x$.", "Since the source and target of the arrow have the same length by", "Lemma \\ref{lemma-finite}", "we conclude it is an isomorphism. The final statement follows", "on taking $M = R$." ], "refs": [ "dualizing-lemma-finite" ], "ref_ids": [ 2800 ] } ], "ref_ids": [] }, { "id": 2802, "type": "theorem", "label": "dualizing-lemma-duality", "categories": [ "dualizing" ], "title": "dualizing-lemma-duality", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be an artinian local ring.", "Let $E$ be an injective hull of $\\kappa$.", "The functor $D(-) = \\Hom_R(-, E)$ induces an exact anti-equivalence", "$\\text{Mod}^{fg}_R \\to \\text{Mod}^{fg}_R$ and", "$D \\circ D \\cong \\text{id}$." ], "refs": [], "proofs": [ { "contents": [ "We have seen that $D \\circ D = \\text{id}$ on $\\text{Mod}^{fg}_R$", "in Lemma \\ref{lemma-evaluate}. It follows immediately that", "$D$ is an anti-equivalence." ], "refs": [ "dualizing-lemma-evaluate" ], "ref_ids": [ 2801 ] } ], "ref_ids": [] }, { "id": 2803, "type": "theorem", "label": "dualizing-lemma-duality-torsion-cotorsion", "categories": [ "dualizing" ], "title": "dualizing-lemma-duality-torsion-cotorsion", "contents": [ "Assumptions and notation as in Lemma \\ref{lemma-duality}.", "Let $I \\subset R$ be an ideal and $M$ a finite $R$-module.", "Then", "$$", "D(M[I]) = D(M)/ID(M) \\quad\\text{and}\\quad D(M/IM) = D(M)[I]", "$$" ], "refs": [ "dualizing-lemma-duality" ], "proofs": [ { "contents": [ "Say $I = (f_1, \\ldots, f_t)$. Consider the map", "$$", "M^{\\oplus t} \\xrightarrow{f_1, \\ldots, f_t} M", "$$", "with cokernel $M/IM$. Applying the exact functor $D$ we conclude that", "$D(M/IM)$ is $D(M)[I]$. The other case is proved in the same way." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 2802 ] }, { "id": 2804, "type": "theorem", "label": "dualizing-lemma-quotient", "categories": [ "dualizing" ], "title": "dualizing-lemma-quotient", "contents": [ "Let $R \\to S$ be a surjective map of local rings with kernel $I$.", "Let $E$ be the injective hull of the residue field of $R$ over $R$.", "Then $E[I]$ is the injective hull of the residue field of $S$ over $S$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $E[I] = \\Hom_R(S, E)$ as $S = R/I$. Hence $E[I]$ is an injective", "$S$-module by Lemma \\ref{lemma-hom-injective}. Since $E$ is an essential", "extension of $\\kappa = R/\\mathfrak m_R$ it follows that $E[I]$ is an", "essential extension of $\\kappa$ as well. The result follows." ], "refs": [ "dualizing-lemma-hom-injective" ], "ref_ids": [ 2787 ] } ], "ref_ids": [] }, { "id": 2805, "type": "theorem", "label": "dualizing-lemma-torsion-submodule-sum-injective-hulls", "categories": [ "dualizing" ], "title": "dualizing-lemma-torsion-submodule-sum-injective-hulls", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring.", "Let $E$ be the injective hull of $\\kappa$.", "Let $M$ be a $\\mathfrak m$-power torsion $R$-module", "with $n = \\dim_\\kappa(M[\\mathfrak m]) < \\infty$.", "Then $M$ is isomorphic to a submodule of $E^{\\oplus n}$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $E^{\\oplus n}$ is the injective hull of", "$\\kappa^{\\oplus n} = M[\\mathfrak m]$. Thus there is an $R$-module map", "$M \\to E^{\\oplus n}$ which is injective on $M[\\mathfrak m]$.", "Since $M$ is $\\mathfrak m$-power torsion the inclusion", "$M[\\mathfrak m] \\subset M$ is an essential extension", "(for example by Lemma \\ref{lemma-essential-extension})", "we conclude that the kernel of $M \\to E^{\\oplus n}$ is zero." ], "refs": [ "dualizing-lemma-essential-extension" ], "ref_ids": [ 2783 ] } ], "ref_ids": [] }, { "id": 2806, "type": "theorem", "label": "dualizing-lemma-union-artinian", "categories": [ "dualizing" ], "title": "dualizing-lemma-union-artinian", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Let $E$ be an injective hull of $\\kappa$ over $R$.", "Let $E_n$ be an injective hull of $\\kappa$ over $R/\\mathfrak m^n$.", "Then $E = \\bigcup E_n$ and $E_n = E[\\mathfrak m^n]$." ], "refs": [], "proofs": [ { "contents": [ "We have $E_n = E[\\mathfrak m^n]$ by Lemma \\ref{lemma-quotient}.", "We have $E = \\bigcup E_n$ because $\\bigcup E_n = E[\\mathfrak m^\\infty]$", "is an injective $R$-submodule which contains $\\kappa$, see", "Lemma \\ref{lemma-injective-module-divide}." ], "refs": [ "dualizing-lemma-quotient", "dualizing-lemma-injective-module-divide" ], "ref_ids": [ 2804, 2791 ] } ], "ref_ids": [] }, { "id": 2807, "type": "theorem", "label": "dualizing-lemma-compare", "categories": [ "dualizing" ], "title": "dualizing-lemma-compare", "contents": [ "Let $R \\to S$ be a flat local homomorphism of local Noetherian rings", "such that $R/\\mathfrak m_R \\cong S/\\mathfrak m_R S$.", "Then the injective hull of the residue field", "of $R$ is the injective hull of the residue field of $S$." ], "refs": [], "proofs": [ { "contents": [ "Set $\\kappa = R/\\mathfrak m_R = S/\\mathfrak m_S$.", "Let $E_R$ be the injective hull of $\\kappa$ over $R$.", "Let $E_S$ be the injective hull of $\\kappa$ over $S$.", "Observe that $E_S$ is an injective $R$-module by", "Lemma \\ref{lemma-injective-flat}.", "Choose an extension $E_R \\to E_S$ of the identification of", "residue fields. This map is an isomorphism by", "Lemma \\ref{lemma-union-artinian}", "because $R \\to S$ induces an isomorphism", "$R/\\mathfrak m_R^n \\to S/\\mathfrak m_S^n$ for all $n$." ], "refs": [ "dualizing-lemma-injective-flat", "dualizing-lemma-union-artinian" ], "ref_ids": [ 2785, 2806 ] } ], "ref_ids": [] }, { "id": 2808, "type": "theorem", "label": "dualizing-lemma-endos", "categories": [ "dualizing" ], "title": "dualizing-lemma-endos", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Let $E$ be an injective hull of $\\kappa$ over $R$. Then", "$\\Hom_R(E, E)$ is canonically isomorphic to the completion of $R$." ], "refs": [], "proofs": [ { "contents": [ "Write $E = \\bigcup E_n$ with $E_n = E[\\mathfrak m^n]$ as in", "Lemma \\ref{lemma-union-artinian}. Any endomorphism of $E$", "preserves this filtration. Hence", "$$", "\\Hom_R(E, E) = \\lim \\Hom_R(E_n, E_n)", "$$", "The lemma follows as", "$\\Hom_R(E_n, E_n) = \\Hom_{R/\\mathfrak m^n}(E_n, E_n) = R/\\mathfrak m^n$", "by Lemma \\ref{lemma-evaluate}." ], "refs": [ "dualizing-lemma-union-artinian", "dualizing-lemma-evaluate" ], "ref_ids": [ 2806, 2801 ] } ], "ref_ids": [] }, { "id": 2809, "type": "theorem", "label": "dualizing-lemma-injective-hull-has-dcc", "categories": [ "dualizing" ], "title": "dualizing-lemma-injective-hull-has-dcc", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Let $E$ be an injective hull of $\\kappa$ over $R$. Then", "$E$ satisfies the descending chain condition." ], "refs": [], "proofs": [ { "contents": [ "If $E \\supset M_1 \\supset M_2 \\supset \\ldots$ is a sequence of submodules, then", "$$", "\\Hom_R(E, E) \\to \\Hom_R(M_1, E) \\to \\Hom_R(M_2, E) \\to \\ldots", "$$", "is sequence of surjections. By Lemma \\ref{lemma-endos} each of these is a", "module over the completion $R^\\wedge = \\Hom_R(E, E)$.", "Since $R^\\wedge$ is Noetherian", "(Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian-Noetherian})", "the sequence stabilizes: $\\Hom_R(M_n, E) = \\Hom_R(M_{n + 1}, E) = \\ldots$.", "Since $E$ is injective, this can only happen if $\\Hom_R(M_n/M_{n + 1}, E)$", "is zero. However, if $M_n/M_{n + 1}$ is nonzero, then it contains a", "nonzero element annihilated by $\\mathfrak m$, because $E$ is", "$\\mathfrak m$-power torsion by Lemma \\ref{lemma-union-artinian}.", "In this case $M_n/M_{n + 1}$ has a nonzero map into $E$, contradicting", "the assumed vanishing. This finishes the proof." ], "refs": [ "dualizing-lemma-endos", "algebra-lemma-completion-Noetherian-Noetherian", "dualizing-lemma-union-artinian" ], "ref_ids": [ 2808, 874, 2806 ] } ], "ref_ids": [] }, { "id": 2810, "type": "theorem", "label": "dualizing-lemma-describe-categories", "categories": [ "dualizing" ], "title": "dualizing-lemma-describe-categories", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Let $E$ be an injective hull of $\\kappa$.", "\\begin{enumerate}", "\\item For an $R$-module $M$ the following are equivalent:", "\\begin{enumerate}", "\\item $M$ satisfies the ascending chain condition,", "\\item $M$ is a finite $R$-module, and", "\\item there exist $n, m$ and an exact sequence", "$R^{\\oplus m} \\to R^{\\oplus n} \\to M \\to 0$.", "\\end{enumerate}", "\\item For an $R$-module $M$ the following are equivalent:", "\\begin{enumerate}", "\\item $M$ satisfies the descending chain condition,", "\\item $M$ is $\\mathfrak m$-power torsion and", "$\\dim_\\kappa(M[\\mathfrak m]) < \\infty$, and", "\\item there exist $n, m$ and an exact sequence", "$0 \\to M \\to E^{\\oplus n} \\to E^{\\oplus m}$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We omit the proof of (1).", "\\medskip\\noindent", "Let $M$ be an $R$-module with the descending chain condition. Let $x \\in M$.", "Then $\\mathfrak m^n x$ is a descending chain of submodules, hence stabilizes.", "Thus $\\mathfrak m^nx = \\mathfrak m^{n + 1}x$ for some $n$. By Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK}) this implies $\\mathfrak m^n x = 0$,", "i.e., $x$ is $\\mathfrak m$-power torsion. Since $M[\\mathfrak m]$ is a vector", "space over $\\kappa$ it has to be finite dimensional in order to have the", "descending chain condition.", "\\medskip\\noindent", "Assume that $M$ is $\\mathfrak m$-power torsion and has a finite dimensional", "$\\mathfrak m$-torsion submodule $M[\\mathfrak m]$. By", "Lemma \\ref{lemma-torsion-submodule-sum-injective-hulls}", "we see that $M$ is a submodule of $E^{\\oplus n}$ for some $n$.", "Consider the quotient $N = E^{\\oplus n}/M$. By", "Lemma \\ref{lemma-injective-hull-has-dcc} the module $E$ has the", "descending chain condition hence so do $E^{\\oplus n}$ and $N$.", "Therefore $N$ satisfies (2)(a) which implies $N$ satisfies", "(2)(b) by the second paragraph of the proof. Thus by", "Lemma \\ref{lemma-torsion-submodule-sum-injective-hulls}", "again we see that $N$ is a submodule of $E^{\\oplus m}$ for some $m$.", "Thus we have a short exact sequence", "$0 \\to M \\to E^{\\oplus n} \\to E^{\\oplus m}$.", "\\medskip\\noindent", "Assume we have a short exact sequence", "$0 \\to M \\to E^{\\oplus n} \\to E^{\\oplus m}$.", "Since $E$ satisfies the descending chain condition by", "Lemma \\ref{lemma-injective-hull-has-dcc}", "so does $M$." ], "refs": [ "algebra-lemma-NAK", "dualizing-lemma-torsion-submodule-sum-injective-hulls", "dualizing-lemma-injective-hull-has-dcc", "dualizing-lemma-torsion-submodule-sum-injective-hulls", "dualizing-lemma-injective-hull-has-dcc" ], "ref_ids": [ 401, 2805, 2809, 2805, 2809 ] } ], "ref_ids": [] }, { "id": 2811, "type": "theorem", "label": "dualizing-lemma-adjoint", "categories": [ "dualizing" ], "title": "dualizing-lemma-adjoint", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "The functor $R\\Gamma_I$ is right adjoint to the functor", "$D(I^\\infty\\text{-torsion}) \\to D(A)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that taking $I$-power torsion submodules", "is the right adjoint to the inclusion functor", "$I^\\infty\\text{-torsion} \\to \\text{Mod}_A$. See", "Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors}." ], "refs": [ "derived-lemma-derived-adjoint-functors" ], "ref_ids": [ 1907 ] } ], "ref_ids": [] }, { "id": 2812, "type": "theorem", "label": "dualizing-lemma-local-cohomology-ext", "categories": [ "dualizing" ], "title": "dualizing-lemma-local-cohomology-ext", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "For any object $K$ of $D(A)$ we have", "$$", "R\\Gamma_I(K) = \\text{hocolim}\\ R\\Hom_A(A/I^n, K)", "$$", "in $D(A)$ and", "$$", "R^q\\Gamma_I(K) = \\colim_n \\Ext_A^q(A/I^n, K)", "$$", "as modules for all $q \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Let $J^\\bullet$ be a K-injective complex representing $K$.", "Then", "$$", "R\\Gamma_I(K) = J^\\bullet[I^\\infty] = \\colim J^\\bullet[I^n] =", "\\colim \\Hom_A(A/I^n, J^\\bullet)", "$$", "The first equality is the definition.", "By Derived Categories, Lemma \\ref{derived-lemma-colim-hocolim}", "we obtain the second equality. The third equality is clear", "because $H^q(\\Hom_A(A/I^n, J^\\bullet)) = \\Ext^q_A(A/I^n, K)$", "and because filtered colimits are exact in the category of abelian", "groups." ], "refs": [ "derived-lemma-colim-hocolim" ], "ref_ids": [ 1922 ] } ], "ref_ids": [] }, { "id": 2813, "type": "theorem", "label": "dualizing-lemma-bad-local-cohomology-vanishes", "categories": [ "dualizing" ], "title": "dualizing-lemma-bad-local-cohomology-vanishes", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "Let $K^\\bullet$ be a complex of $A$-modules such that", "$f : K^\\bullet \\to K^\\bullet$ is an isomorphism for some", "$f \\in I$, i.e., $K^\\bullet$ is a complex of $A_f$-modules. Then", "$R\\Gamma_I(K^\\bullet) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Namely, in this case the cohomology modules of $R\\Gamma_I(K^\\bullet)$", "are both $f$-power torsion and $f$ acts by automorphisms. Hence the", "cohomology modules are zero and hence the object is zero." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2814, "type": "theorem", "label": "dualizing-lemma-not-equal", "categories": [ "dualizing" ], "title": "dualizing-lemma-not-equal", "contents": [ "Let $A$ be a ring and let $I$ be a finitely generated ideal.", "Let $M$ and $N$ be $I$-power torsion modules.", "\\begin{enumerate}", "\\item $\\Hom_{D(A)}(M, N) = \\Hom_{D({I^\\infty\\text{-torsion}})}(M, N)$,", "\\item $\\Ext^1_{D(A)}(M, N) =", "\\Ext^1_{D({I^\\infty\\text{-torsion}})}(M, N)$,", "\\item $\\Ext^2_{D({I^\\infty\\text{-torsion}})}(M, N) \\to", "\\Ext^2_{D(A)}(M, N)$ is not surjective in general,", "\\item (\\ref{equation-compare-torsion}) is not an equivalence in general.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1) and (2) follow immediately from the fact that $I$-power torsion", "forms a Serre subcategory of $\\text{Mod}_A$. Part (4) follows from", "part (3).", "\\medskip\\noindent", "For part (3) let $A$ be a ring with an element $f \\in A$ such that", "$A[f]$ contains a nonzero element $x$ annihilated by $f$ and", "$A$ contains elements $x_n$ with $f^nx_n = x$. Such a ring $A$", "exists because we can take", "$$", "A = \\mathbf{Z}[f, x, x_n]/(fx, f^nx_n - x)", "$$", "Given $A$ set $I = (f)$. Then the exact sequence", "$$", "0 \\to A[f] \\to A \\xrightarrow{f} A \\to A/fA \\to 0", "$$", "defines an element in $\\Ext^2_A(A/fA, A[f])$. We claim this", "element does not come from an element of", "$\\Ext^2_{D(f^\\infty\\text{-torsion})}(A/fA, A[f])$.", "Namely, if it did, then there would be an exact sequence", "$$", "0 \\to A[f] \\to M \\to N \\to A/fA \\to 0", "$$", "where $M$ and $N$ are $f$-power torsion modules defining the same", "$2$ extension class. Since $A \\to A$ is a complex of free modules", "and since the $2$ extension classes are the same", "we would be able to find a map", "$$", "\\xymatrix{", "0 \\ar[r] &", "A[f] \\ar[r] \\ar[d] &", "A \\ar[r] \\ar[d]_\\varphi &", "A \\ar[r] \\ar[d]_\\psi &", "A/fA \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "A[f] \\ar[r] &", "M \\ar[r] &", "N \\ar[r] &", "A/fA \\ar[r] & 0", "}", "$$", "(some details omitted). Then we could replace $M$ by the image of", "$\\varphi$ and $N$ by the image of $\\psi$. Then $M$ would be a cyclic", "module, hence $f^n M = 0$ for some $n$. Considering $\\varphi(x_{n + 1})$", "we get a contradiction with the fact that $f^{n + 1}x_n = x$ is", "nonzero in $A[f]$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2815, "type": "theorem", "label": "dualizing-lemma-local-cohomology-adjoint", "categories": [ "dualizing" ], "title": "dualizing-lemma-local-cohomology-adjoint", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "There exists a right adjoint $R\\Gamma_Z$ (\\ref{equation-local-cohomology})", "to the inclusion functor $D_{I^\\infty\\text{-torsion}}(A) \\to D(A)$.", "In fact, if $I$ is generated by $f_1, \\ldots, f_r \\in A$, then we have", "$$", "R\\Gamma_Z(K) =", "(A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}}", "\\to \\ldots \\to A_{f_1\\ldots f_r}) \\otimes_A^\\mathbf{L} K", "$$", "functorially in $K \\in D(A)$." ], "refs": [], "proofs": [ { "contents": [ "Say $I = (f_1, \\ldots, f_r)$ is an ideal.", "Let $K^\\bullet$ be a complex of $A$-modules.", "There is a canonical map of complexes", "$$", "(A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to A_{f_1\\ldots f_r}) \\longrightarrow A.", "$$", "from the extended {\\v C}ech complex to $A$.", "Tensoring with $K^\\bullet$, taking associated total complex,", "we get a map", "$$", "\\text{Tot}\\left(", "K^\\bullet \\otimes_A", "(A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to A_{f_1\\ldots f_r})\\right)", "\\longrightarrow", "K^\\bullet", "$$", "in $D(A)$. We claim the cohomology modules of the complex on the left are", "$I$-power torsion, i.e., the LHS is an object of", "$D_{I^\\infty\\text{-torsion}}(A)$. Namely, we have", "$$", "(A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to A_{f_1\\ldots f_r}) = \\colim K(A, f_1^n, \\ldots, f_r^n)", "$$", "by More on Algebra, Lemma", "\\ref{more-algebra-lemma-extended-alternating-Cech-is-colimit-koszul}.", "Moreover, multiplication by $f_i^n$ on the complex", "$K(A, f_1^n, \\ldots, f_r^n)$ is homotopic to zero by", "More on Algebra, Lemma \\ref{more-algebra-lemma-homotopy-koszul}.", "Since", "$$", "H^q\\left( LHS \\right) =", "\\colim H^q(\\text{Tot}(K^\\bullet \\otimes_A K(A, f_1^n, \\ldots, f_r^n)))", "$$", "we obtain our claim. On the other hand, if $K^\\bullet$ is an", "object of $D_{I^\\infty\\text{-torsion}}(A)$, then the complexes", "$K^\\bullet \\otimes_A A_{f_{i_0} \\ldots f_{i_p}}$ have vanishing", "cohomology. Hence in this case the map $LHS \\to K^\\bullet$", "is an isomorphism in $D(A)$. The construction", "$$", "R\\Gamma_Z(K^\\bullet) =", "\\text{Tot}\\left(", "K^\\bullet \\otimes_A", "(A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to A_{f_1\\ldots f_r})\\right)", "$$", "is functorial in $K^\\bullet$ and defines an exact functor", "$D(A) \\to D_{I^\\infty\\text{-torsion}}(A)$ between", "triangulated categories. It follows formally from the", "existence of the natural transformation $R\\Gamma_Z \\to \\text{id}$", "given above and the fact that this evaluates to an isomorphism", "on $K^\\bullet$ in the subcategory, that $R\\Gamma_Z$ is the desired", "right adjoint." ], "refs": [ "more-algebra-lemma-extended-alternating-Cech-is-colimit-koszul", "more-algebra-lemma-homotopy-koszul" ], "ref_ids": [ 9972, 9960 ] } ], "ref_ids": [] }, { "id": 2816, "type": "theorem", "label": "dualizing-lemma-local-cohomology-and-restriction", "categories": [ "dualizing" ], "title": "dualizing-lemma-local-cohomology-and-restriction", "contents": [ "Let $A \\to B$ be a ring homomorphism and let $I \\subset A$", "be a finitely generated ideal. Set $J = IB$. Set $Z = V(I)$", "and $Y = V(J)$. Then", "$$", "R\\Gamma_Z(M_A) = R\\Gamma_Y(M)_A", "$$", "functorially in $M \\in D(B)$. Here $(-)_A$ denotes the restriction", "functors $D(B) \\to D(A)$ and", "$D_{J^\\infty\\text{-torsion}}(B) \\to D_{I^\\infty\\text{-torsion}}(A)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from uniqueness of adjoint functors as both", "$R\\Gamma_Z((-)_A)$ and $R\\Gamma_Y(-)_A$", "are right adjoint to the functor $D_{I^\\infty\\text{-torsion}}(A) \\to D(B)$,", "$K \\mapsto K \\otimes_A^\\mathbf{L} B$.", "Alternatively, one can use the description of $R\\Gamma_Z$ and $R\\Gamma_Y$", "in terms of alternating {\\v C}ech complexes", "(Lemma \\ref{lemma-local-cohomology-adjoint}).", "Namely, if $I = (f_1, \\ldots, f_r)$ then $J$ is generated by the images", "$g_1, \\ldots, g_r \\in B$ of $f_1, \\ldots, f_r$.", "Then the statement of the lemma follows from the existence of", "a canonical isomorphism", "\\begin{align*}", "& M_A \\otimes_A (A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}}", "\\to \\ldots \\to A_{f_1\\ldots f_r}) \\\\", "& = ", "M \\otimes_B (B \\to \\prod\\nolimits_{i_0} B_{g_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} B_{g_{i_0}g_{i_1}}", "\\to \\ldots \\to B_{g_1\\ldots g_r})", "\\end{align*}", "for any $B$-module $M$." ], "refs": [ "dualizing-lemma-local-cohomology-adjoint" ], "ref_ids": [ 2815 ] } ], "ref_ids": [] }, { "id": 2817, "type": "theorem", "label": "dualizing-lemma-torsion-change-rings", "categories": [ "dualizing" ], "title": "dualizing-lemma-torsion-change-rings", "contents": [ "Let $A \\to B$ be a ring homomorphism and let $I \\subset A$", "be a finitely generated ideal. Set $J = IB$. Let $Z = V(I)$ and $Y = V(J)$.", "Then", "$$", "R\\Gamma_Z(K) \\otimes_A^\\mathbf{L} B = R\\Gamma_Y(K \\otimes_A^\\mathbf{L} B)", "$$", "functorially in $K \\in D(A)$." ], "refs": [], "proofs": [ { "contents": [ "Write $I = (f_1, \\ldots, f_r)$. Then $J$ is generated by the images", "$g_1, \\ldots, g_r \\in B$ of $f_1, \\ldots, f_r$. Then we have", "$$", "(A \\to \\prod A_{f_{i_0}} \\to \\ldots \\to A_{f_1\\ldots f_r}) \\otimes_A B =", "(B \\to \\prod B_{g_{i_0}} \\to \\ldots \\to B_{g_1\\ldots g_r})", "$$", "as complexes of $B$-modules. Represent $K$ by a K-flat complex $K^\\bullet$", "of $A$-modules. Since the total complexes associated to", "$$", "K^\\bullet \\otimes_A", "(A \\to \\prod A_{f_{i_0}} \\to \\ldots \\to A_{f_1\\ldots f_r}) \\otimes_A B", "$$", "and", "$$", "K^\\bullet \\otimes_A B \\otimes_B", "(B \\to \\prod B_{g_{i_0}} \\to \\ldots \\to B_{g_1\\ldots g_r})", "$$", "represent the left and right hand side of the displayed formula of the", "lemma (see Lemma \\ref{lemma-local-cohomology-adjoint}) we conclude." ], "refs": [ "dualizing-lemma-local-cohomology-adjoint" ], "ref_ids": [ 2815 ] } ], "ref_ids": [] }, { "id": 2818, "type": "theorem", "label": "dualizing-lemma-local-cohomology-vanishes", "categories": [ "dualizing" ], "title": "dualizing-lemma-local-cohomology-vanishes", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "Let $K^\\bullet$ be a complex of $A$-modules such that", "$f : K^\\bullet \\to K^\\bullet$ is an isomorphism for some", "$f \\in I$, i.e., $K^\\bullet$ is a complex of $A_f$-modules. Then", "$R\\Gamma_Z(K^\\bullet) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Namely, in this case the cohomology modules of $R\\Gamma_Z(K^\\bullet)$", "are both $f$-power torsion and $f$ acts by automorphisms. Hence the", "cohomology modules are zero and hence the object is zero." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2819, "type": "theorem", "label": "dualizing-lemma-torsion-tensor-product", "categories": [ "dualizing" ], "title": "dualizing-lemma-torsion-tensor-product", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "For $K, L \\in D(A)$ we have", "$$", "R\\Gamma_Z(K \\otimes_A^\\mathbf{L} L) =", "K \\otimes_A^\\mathbf{L} R\\Gamma_Z(L) =", "R\\Gamma_Z(K) \\otimes_A^\\mathbf{L} L =", "R\\Gamma_Z(K) \\otimes_A^\\mathbf{L} R\\Gamma_Z(L)", "$$", "If $K$ or $L$ is in $D_{I^\\infty\\text{-torsion}}(A)$ then so is", "$K \\otimes_A^\\mathbf{L} L$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-local-cohomology-adjoint} we know that", "$R\\Gamma_Z$ is given by $C \\otimes^\\mathbf{L} -$ for some $C \\in D(A)$.", "Hence, for $K, L \\in D(A)$ general we have", "$$", "R\\Gamma_Z(K \\otimes_A^\\mathbf{L} L) =", "K \\otimes^\\mathbf{L} L \\otimes_A^\\mathbf{L} C =", "K \\otimes_A^\\mathbf{L} R\\Gamma_Z(L)", "$$", "The other equalities follow formally from this one. This also implies", "the last statement of the lemma." ], "refs": [ "dualizing-lemma-local-cohomology-adjoint" ], "ref_ids": [ 2815 ] } ], "ref_ids": [] }, { "id": 2820, "type": "theorem", "label": "dualizing-lemma-local-cohomology-ss", "categories": [ "dualizing" ], "title": "dualizing-lemma-local-cohomology-ss", "contents": [ "Let $A$ be a ring and let $I, J \\subset A$ be finitely generated", "ideals. Set $Z = V(I)$ and $Y = V(J)$. Then $Z \\cap Y = V(I + J)$", "and $R\\Gamma_Y \\circ R\\Gamma_Z = R\\Gamma_{Y \\cap Z}$ as functors", "$D(A) \\to D_{(I + J)^\\infty\\text{-torsion}}(A)$. For $K \\in D^+(A)$", "there is a spectral sequence", "$$", "E_2^{p, q} = H^p_Y(H^q_Z(K)) \\Rightarrow H^{p + q}_{Y \\cap Z}(K)", "$$", "as in Derived Categories, Lemma", "\\ref{derived-lemma-grothendieck-spectral-sequence}." ], "refs": [ "derived-lemma-grothendieck-spectral-sequence" ], "proofs": [ { "contents": [ "There is a bit of abuse of notation in the lemma as strictly", "speaking we cannot compose $R\\Gamma_Y$ and $R\\Gamma_Z$. The", "meaning of the statement is simply that we are composing", "$R\\Gamma_Z$ with the inclusion $D_{I^\\infty\\text{-torsion}}(A) \\to D(A)$", "and then with $R\\Gamma_Y$. Then the equality", "$R\\Gamma_Y \\circ R\\Gamma_Z = R\\Gamma_{Y \\cap Z}$", "follows from the fact that", "$$", "D_{I^\\infty\\text{-torsion}}(A) \\to D(A) \\xrightarrow{R\\Gamma_Y}", "D_{(I + J)^\\infty\\text{-torsion}}(A)", "$$", "is right adjoint to the inclusion", "$D_{(I + J)^\\infty\\text{-torsion}}(A) \\to D_{I^\\infty\\text{-torsion}}(A)$.", "Alternatively one can prove the formula using", "Lemma \\ref{lemma-local-cohomology-adjoint}", "and the fact that the tensor product of", "extended {\\v C}ech complexes on $f_1, \\ldots, f_r$ and", "$g_1, \\ldots, g_m$ is the extended {\\v C}ech complex on", "$f_1, \\ldots, f_n. g_1, \\ldots, g_m$.", "The final assertion follows from this and the cited lemma." ], "refs": [ "dualizing-lemma-local-cohomology-adjoint" ], "ref_ids": [ 2815 ] } ], "ref_ids": [ 1873 ] }, { "id": 2821, "type": "theorem", "label": "dualizing-lemma-torsion-flat-change-rings", "categories": [ "dualizing" ], "title": "dualizing-lemma-torsion-flat-change-rings", "contents": [ "Let $A \\to B$ be a flat ring map and let $I \\subset A$ be a finitely", "generated ideal such that $A/I = B/IB$. Then base change and", "restriction induce quasi-inverse equivalences", "$D_{I^\\infty\\text{-torsion}}(A) = D_{(IB)^\\infty\\text{-torsion}}(B)$." ], "refs": [], "proofs": [ { "contents": [ "More precisely the functors are $K \\mapsto K \\otimes_A^\\mathbf{L} B$", "for $K$ in $D_{I^\\infty\\text{-torsion}}(A)$ and $M \\mapsto M_A$", "for $M$ in $D_{(IB)^\\infty\\text{-torsion}}(B)$. The reason this works", "is that $H^i(K \\otimes_A^\\mathbf{L} B) = H^i(K) \\otimes_A B = H^i(K)$.", "The first equality holds as $A \\to B$ is flat and the second by", "More on Algebra, Lemma \\ref{more-algebra-lemma-neighbourhood-isomorphism}." ], "refs": [ "more-algebra-lemma-neighbourhood-isomorphism" ], "ref_ids": [ 10340 ] } ], "ref_ids": [] }, { "id": 2822, "type": "theorem", "label": "dualizing-lemma-neighbourhood-extensions", "categories": [ "dualizing" ], "title": "dualizing-lemma-neighbourhood-extensions", "contents": [ "Let $A \\to B$ be a flat ring map and let $I \\subset A$ be a", "finitely generated ideal such that $A/I \\to B/IB$ is an isomorphism.", "For $K \\in D_{I^\\infty\\text{-torsion}}(A)$ and $L \\in D(A)$", "the map", "$$", "R\\Hom_A(K, L) \\longrightarrow R\\Hom_B(K \\otimes_A B, L \\otimes_A B)", "$$", "is a quasi-isomorphism. In particular, if $M$, $N$ are $A$-modules and", "$M$ is $I$-power torsion, then the canonical map", "$$", "\\Ext^i_A(M, N)", "\\longrightarrow", "\\Ext^i_B(M \\otimes_A B, N \\otimes_A B)", "$$", "is an isomorphism for all $i$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z = V(I) \\subset \\Spec(A)$ and $Y = V(IB) \\subset \\Spec(B)$.", "Since the cohomology modules of $K$ are $I$ power torsion, the", "canonical map $R\\Gamma_Z(L) \\to L$ induces an isomorphism", "$$", "R\\Hom_A(K, R\\Gamma_Z(L)) \\to R\\Hom_A(K, L)", "$$", "in $D(A)$. Similarly, the cohomology modules of $K \\otimes_A B$ are", "$IB$ power torsion and we have an isomorphism", "$$", "R\\Hom_B(K \\otimes_A B, R\\Gamma_Y(L \\otimes_A B)) \\to ", "R\\Hom_B(K \\otimes_A B, L \\otimes_A B)", "$$", "in $D(B)$.", "By Lemma \\ref{lemma-torsion-change-rings} we have", "$R\\Gamma_Z(L) \\otimes_A B = R\\Gamma_Y(L \\otimes_A B)$.", "Hence it suffices to show that the map", "$$", "R\\Hom_A(K, R\\Gamma_Z(L)) \\to R\\Hom_B(K \\otimes_A B, R\\Gamma_Z(L) \\otimes_A B)", "$$", "is a quasi-isomorphism. This follows from", "Lemma \\ref{lemma-torsion-flat-change-rings}." ], "refs": [ "dualizing-lemma-torsion-change-rings", "dualizing-lemma-torsion-flat-change-rings" ], "ref_ids": [ 2817, 2821 ] } ], "ref_ids": [] }, { "id": 2823, "type": "theorem", "label": "dualizing-lemma-local-cohomology-noetherian", "categories": [ "dualizing" ], "title": "dualizing-lemma-local-cohomology-noetherian", "contents": [ "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.", "\\begin{enumerate}", "\\item the adjunction $R\\Gamma_I(K) \\to K$ is an isomorphism", "for $K \\in D_{I^\\infty\\text{-torsion}}(A)$,", "\\item the functor", "(\\ref{equation-compare-torsion})", "$D(I^\\infty\\text{-torsion}) \\to D_{I^\\infty\\text{-torsion}}(A)$", "is an equivalence,", "\\item the transformation of functors", "(\\ref{equation-compare-torsion-functors}) is an isomorphism,", "in other words $R\\Gamma_I(K) = R\\Gamma_Z(K)$ for $K \\in D(A)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "A formal argument, which we omit, shows that it suffices to prove (1).", "\\medskip\\noindent", "Let $M$ be an $I$-power torsion $A$-module. Choose an embedding", "$M \\to J$ into an injective $A$-module. Then $J[I^\\infty]$ is", "an injective $A$-module, see Lemma \\ref{lemma-injective-module-divide},", "and we obtain an embedding $M \\to J[I^\\infty]$.", "Thus every $I$-power torsion module has an injective resolution", "$M \\to J^\\bullet$ with $J^n$ also $I$-power torsion. It follows", "that $R\\Gamma_I(M) = M$ (this is not a triviality and this is not", "true in general if $A$ is not Noetherian). Next, suppose that", "$K \\in D_{I^\\infty\\text{-torsion}}^+(A)$. Then the spectral sequence", "$$", "R^q\\Gamma_I(H^p(K)) \\Rightarrow R^{p + q}\\Gamma_I(K)", "$$", "(Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor})", "converges and above we have seen that only the terms with $q = 0$", "are nonzero. Thus we see that $R\\Gamma_I(K) \\to K$ is an isomorphism.", "\\medskip\\noindent", "Suppose $K$ is an arbitrary object of $D_{I^\\infty\\text{-torsion}}(A)$.", "We have", "$$", "R^q\\Gamma_I(K) = \\colim \\Ext^q_A(A/I^n, K)", "$$", "by Lemma \\ref{lemma-local-cohomology-ext}. Choose $f_1, \\ldots, f_r \\in A$", "generating $I$. Let $K_n^\\bullet = K(A, f_1^n, \\ldots, f_r^n)$ be the", "Koszul complex with terms in degrees $-r, \\ldots, 0$. Since the", "pro-objects $\\{A/I^n\\}$ and $\\{K_n^\\bullet\\}$ in $D(A)$ are the same by", "More on Algebra, Lemma \\ref{more-algebra-lemma-sequence-Koszul-complexes},", "we see that", "$$", "R^q\\Gamma_I(K) = \\colim \\Ext^q_A(K_n^\\bullet, K)", "$$", "Pick any complex $K^\\bullet$ of $A$-modules representing $K$.", "Since $K_n^\\bullet$ is a finite complex of finite free modules we see", "that", "$$", "\\Ext^q_A(K_n, K) =", "H^q(\\text{Tot}((K_n^\\bullet)^\\vee \\otimes_A K^\\bullet))", "$$", "where $(K_n^\\bullet)^\\vee$ is the dual of the complex $K_n^\\bullet$.", "See More on Algebra, Lemma \\ref{more-algebra-lemma-RHom-out-of-projective}.", "As $(K_n^\\bullet)^\\vee$ is a complex of finite free $A$-modules sitting", "in degrees $0, \\ldots, r$ we see that the terms of the complex", "$\\text{Tot}((K_n^\\bullet)^\\vee \\otimes_A K^\\bullet)$ are the", "same as the terms of the complex", "$\\text{Tot}((K_n^\\bullet)^\\vee \\otimes_A \\tau_{\\geq q - r - 2} K^\\bullet)$", "in degrees $q - 1$ and higher. Hence we see that", "$$", "\\Ext^q_A(K_n, K) = \\text{Ext}^q_A(K_n, \\tau_{\\geq q - r - 2}K)", "$$", "for all $n$. It follows that", "$$", "R^q\\Gamma_I(K) = R^q\\Gamma_I(\\tau_{\\geq q - r - 2}K) =", "H^q(\\tau_{\\geq q - r - 2}K) = H^q(K)", "$$", "Thus we see that the map $R\\Gamma_I(K) \\to K$ is an isomorphism." ], "refs": [ "dualizing-lemma-injective-module-divide", "derived-lemma-two-ss-complex-functor", "dualizing-lemma-local-cohomology-ext", "more-algebra-lemma-sequence-Koszul-complexes", "more-algebra-lemma-RHom-out-of-projective" ], "ref_ids": [ 2791, 1871, 2812, 10391, 10207 ] } ], "ref_ids": [] }, { "id": 2824, "type": "theorem", "label": "dualizing-lemma-compute-local-cohomology-noetherian", "categories": [ "dualizing" ], "title": "dualizing-lemma-compute-local-cohomology-noetherian", "contents": [ "Let $A$ be a Noetherian ring and let $I = (f_1, \\ldots, f_r)$ be an ideal", "of $A$. Set $Z = V(I) \\subset \\Spec(A)$. There are canonical isomorphisms", "$$", "R\\Gamma_I(A) \\to", "(A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to A_{f_1\\ldots f_r}) \\to R\\Gamma_Z(A)", "$$", "in $D(A)$. If $M$ is an $A$-module, then we have similarly", "$$", "R\\Gamma_I(M) \\cong", "(M \\to \\prod\\nolimits_{i_0} M_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} M_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to M_{f_1\\ldots f_r}) \\cong R\\Gamma_Z(M)", "$$", "in $D(A)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-local-cohomology-noetherian}", "and the computation of the functor $R\\Gamma_Z$ in", "Lemma \\ref{lemma-local-cohomology-adjoint}." ], "refs": [ "dualizing-lemma-local-cohomology-noetherian", "dualizing-lemma-local-cohomology-adjoint" ], "ref_ids": [ 2823, 2815 ] } ], "ref_ids": [] }, { "id": 2825, "type": "theorem", "label": "dualizing-lemma-local-cohomology-change-rings", "categories": [ "dualizing" ], "title": "dualizing-lemma-local-cohomology-change-rings", "contents": [ "If $A \\to B$ is a homomorphism of Noetherian rings and $I \\subset A$", "is an ideal, then in $D(B)$ we have", "$$", "R\\Gamma_I(A) \\otimes_A^\\mathbf{L} B =", "R\\Gamma_Z(A) \\otimes_A^\\mathbf{L} B =", "R\\Gamma_Y(B) = R\\Gamma_{IB}(B)", "$$", "where $Y = V(IB) \\subset \\Spec(B)$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-compute-local-cohomology-noetherian} and", "\\ref{lemma-torsion-change-rings}." ], "refs": [ "dualizing-lemma-compute-local-cohomology-noetherian", "dualizing-lemma-torsion-change-rings" ], "ref_ids": [ 2824, 2817 ] } ], "ref_ids": [] }, { "id": 2826, "type": "theorem", "label": "dualizing-lemma-depth", "categories": [ "dualizing" ], "title": "dualizing-lemma-depth", "contents": [ "Let $A$ be a Noetherian ring, let $I \\subset A$ be an ideal, and", "let $M$ be a finite $A$-module such that $IM \\not = M$. Then", "the following integers are equal:", "\\begin{enumerate}", "\\item $\\text{depth}_I(M)$,", "\\item the smallest integer $i$ such that $\\Ext_A^i(A/I, M)$", "is nonzero, and", "\\item the smallest integer $i$ such that $H^i_I(M)$ is nonzero.", "\\end{enumerate}", "Moreover, we have $\\Ext^i_A(N, M) = 0$ for $i < \\text{depth}_I(M)$", "for any finite $A$-module $N$ annihilated by a power of $I$." ], "refs": [], "proofs": [ { "contents": [ "We prove the equality of (1) and (2) by induction on $\\text{depth}_I(M)$", "which is allowed by", "Algebra, Lemma \\ref{algebra-lemma-depth-finite-noetherian}.", "\\medskip\\noindent", "Base case. If $\\text{depth}_I(M) = 0$, then $I$ is contained in the union", "of the associated primes of $M$", "(Algebra, Lemma \\ref{algebra-lemma-ass-zero-divisors}).", "By prime avoidance (Algebra, Lemma \\ref{algebra-lemma-silly})", "we see that $I \\subset \\mathfrak p$ for some associated prime $\\mathfrak p$.", "Hence $\\Hom_A(A/I, M)$", "is nonzero. Thus equality holds in this case.", "\\medskip\\noindent", "Assume that $\\text{depth}_I(M) > 0$. Let $f \\in I$ be a nonzerodivisor", "on $M$ such that $\\text{depth}_I(M/fM) = \\text{depth}_I(M) - 1$.", "Consider the short exact sequence", "$$", "0 \\to M \\to M \\to M/fM \\to 0", "$$", "and the associated long exact sequence for $\\Ext^*_A(A/I, -)$.", "Note that $\\Ext^i_A(A/I, M)$ is a finite $A/I$-module", "(Algebra, Lemmas \\ref{algebra-lemma-ext-noetherian} and", "\\ref{algebra-lemma-annihilate-ext}). Hence we obtain", "$$", "\\Hom_A(A/I, M/fM) = \\Ext^1_A(A/I, M)", "$$", "and short exact sequences", "$$", "0 \\to \\Ext^i_A(A/I, M) \\to \\text{Ext}^i_A(A/I, M/fM) \\to", "\\Ext^{i + 1}_A(A/I, M) \\to 0", "$$", "Thus the equality of (1) and (2) by induction.", "\\medskip\\noindent", "Observe that $\\text{depth}_I(M) = \\text{depth}_{I^n}(M)$ for all $n \\geq 1$", "for example by Algebra, Lemma \\ref{algebra-lemma-regular-sequence-powers}.", "Hence by the equality of (1) and (2) we see that", "$\\Ext^i_A(A/I^n, M) = 0$ for all $n$ and $i < \\text{depth}_I(M)$.", "Let $N$ be a finite $A$-module annihilated by a power of $I$.", "Then we can choose a short exact sequence", "$$", "0 \\to N' \\to (A/I^n)^{\\oplus m} \\to N \\to 0", "$$", "for some $n, m \\geq 0$. Then", "$\\Hom_A(N, M) \\subset \\Hom_A((A/I^n)^{\\oplus m}, M)$", "and", "$\\Ext^i_A(N, M) \\subset \\text{Ext}^{i - 1}_A(N', M)$", "for $i < \\text{depth}_I(M)$. Thus a simply induction argument", "shows that the final statement of the lemma holds.", "\\medskip\\noindent", "Finally, we prove that (3) is equal to (1) and (2).", "We have $H^p_I(M) = \\colim \\Ext^p_A(A/I^n, M)$ by", "Lemma \\ref{lemma-local-cohomology-ext}.", "Thus we see that $H^i_I(M) = 0$ for $i < \\text{depth}_I(M)$.", "For $i = \\text{depth}_I(M)$, using the vanishing of", "$\\Ext_A^{i - 1}(I/I^n, M)$ we see that the map", "$\\Ext_A^i(A/I, M) \\to H_I^i(M)$ is injective which", "proves nonvanishing in the correct degree." ], "refs": [ "algebra-lemma-depth-finite-noetherian", "algebra-lemma-ass-zero-divisors", "algebra-lemma-silly", "algebra-lemma-ext-noetherian", "algebra-lemma-annihilate-ext", "algebra-lemma-regular-sequence-powers", "dualizing-lemma-local-cohomology-ext" ], "ref_ids": [ 771, 704, 378, 768, 767, 744, 2812 ] } ], "ref_ids": [] }, { "id": 2827, "type": "theorem", "label": "dualizing-lemma-depth-in-ses", "categories": [ "dualizing" ], "title": "dualizing-lemma-depth-in-ses", "contents": [ "Let $A$ be a Noetherian ring. Let $0 \\to N' \\to N \\to N'' \\to 0$", "be a short exact sequence of finite $A$-modules.", "Let $I \\subset A$ be an ideal.", "\\begin{enumerate}", "\\item", "$\\text{depth}_I(N) \\geq \\min\\{\\text{depth}_I(N'), \\text{depth}_I(N'')\\}$", "\\item", "$\\text{depth}_I(N'') \\geq \\min\\{\\text{depth}_I(N), \\text{depth}_I(N') - 1\\}$", "\\item", "$\\text{depth}_I(N') \\geq \\min\\{\\text{depth}_I(N), \\text{depth}_I(N'') + 1\\}$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $IN \\not = N$, $IN' \\not = N'$, and $IN'' \\not = N''$. Then we", "can use the characterization of depth using the Ext groups", "$\\Ext^i(A/I, N)$, see Lemma \\ref{lemma-depth},", "and use the long exact cohomology sequence", "$$", "\\begin{matrix}", "0", "\\to \\Hom_A(A/I, N')", "\\to \\Hom_A(A/I, N)", "\\to \\Hom_A(A/I, N'')", "\\\\", "\\phantom{0\\ }", "\\to \\Ext^1_A(A/I, N')", "\\to \\Ext^1_A(A/I, N)", "\\to \\Ext^1_A(A/I, N'')", "\\to \\ldots", "\\end{matrix}", "$$", "from Algebra, Lemma \\ref{algebra-lemma-long-exact-seq-ext}.", "This argument also works if $IN = N$", "because in this case $\\Ext^i_A(A/I, N) = 0$ for all $i$.", "Similarly in case $IN' \\not = N'$ and/or $IN'' \\not = N''$." ], "refs": [ "dualizing-lemma-depth", "algebra-lemma-long-exact-seq-ext" ], "ref_ids": [ 2826, 765 ] } ], "ref_ids": [] }, { "id": 2828, "type": "theorem", "label": "dualizing-lemma-depth-drops-by-one", "categories": [ "dualizing" ], "title": "dualizing-lemma-depth-drops-by-one", "contents": [ "Let $A$ be a Noetherian ring, let $I \\subset A$ be an ideal, and", "let $M$ a finite $A$-module with $IM \\not = M$.", "\\begin{enumerate}", "\\item If $x \\in I$ is a nonzerodivisor on $M$, then", "$\\text{depth}_I(M/xM) = \\text{depth}_I(M) - 1$.", "\\item Any $M$-regular sequence $x_1, \\ldots, x_r$ in $I$ can be extended to an", "$M$-regular sequence in $I$ of length $\\text{depth}_I(M)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) is a formal consequence of part (1). Let $x \\in I$ be as in (1).", "By the short exact sequence $0 \\to M \\to M \\to M/xM \\to 0$ and", "Lemma \\ref{lemma-depth-in-ses} we see that", "$\\text{depth}_I(M/xM) \\geq \\text{depth}_I(M) - 1$.", "On the other hand, if $x_1, \\ldots, x_r \\in I$", "is a regular sequence for $M/xM$, then $x, x_1, \\ldots, x_r$", "is a regular sequence for $M$. Hence (1) holds." ], "refs": [ "dualizing-lemma-depth-in-ses" ], "ref_ids": [ 2827 ] } ], "ref_ids": [] }, { "id": 2829, "type": "theorem", "label": "dualizing-lemma-depth-CM", "categories": [ "dualizing" ], "title": "dualizing-lemma-depth-CM", "contents": [ "Let $R$ be a Noetherian local ring. If $M$ is a finite Cohen-Macaulay", "$R$-module and $I \\subset R$ a nontrivial ideal. Then", "$$", "\\text{depth}_I(M) = \\dim(\\text{Supp}(M)) - \\dim(\\text{Supp}(M/IM)).", "$$" ], "refs": [], "proofs": [ { "contents": [ "We will prove this by induction on $\\text{depth}_I(M)$.", "\\medskip\\noindent", "If $\\text{depth}_I(M) = 0$, then $I$ is contained in one", "of the associated primes $\\mathfrak p$ of $M$", "(Algebra, Lemma \\ref{algebra-lemma-ideal-nonzerodivisor}).", "Then $\\mathfrak p \\in \\text{Supp}(M/IM)$, hence", "$\\dim(\\text{Supp}(M/IM)) \\geq \\dim(R/\\mathfrak p) = \\dim(\\text{Supp}(M))$", "where equality holds by", "Algebra, Lemma \\ref{algebra-lemma-CM-ass-minimal-support}.", "Thus the lemma holds in this case.", "\\medskip\\noindent", "If $\\text{depth}_I(M) > 0$, we pick $x \\in I$ which is a", "nonzerodivisor on $M$. Note that $(M/xM)/I(M/xM) = M/IM$.", "On the other hand we have", "$\\text{depth}_I(M/xM) = \\text{depth}_I(M) - 1$", "by Lemma \\ref{lemma-depth-drops-by-one}", "and $\\dim(\\text{Supp}(M/xM)) = \\dim(\\text{Supp}(M)) - 1$", "by Algebra, Lemma \\ref{algebra-lemma-one-equation-module}.", "Thus the result by induction hypothesis." ], "refs": [ "algebra-lemma-ideal-nonzerodivisor", "algebra-lemma-CM-ass-minimal-support", "dualizing-lemma-depth-drops-by-one" ], "ref_ids": [ 712, 918, 2828 ] } ], "ref_ids": [] }, { "id": 2830, "type": "theorem", "label": "dualizing-lemma-depth-flat-CM", "categories": [ "dualizing" ], "title": "dualizing-lemma-depth-flat-CM", "contents": [ "Let $R \\to S$ be a flat local ring homomorphism of Noetherian local", "rings. Denote $\\mathfrak m \\subset R$ the maximal ideal.", "Let $I \\subset S$ be an ideal.", "If $S/\\mathfrak mS$ is Cohen-Macaulay, then", "$$", "\\text{depth}_I(S) \\geq \\dim(S/\\mathfrak mS) - \\dim(S/\\mathfrak mS + I)", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-grothendieck-regular-sequence}", "any sequence in $S$ which maps to a regular sequence in $S/\\mathfrak mS$", "is a regular sequence in $S$. Thus it suffices to prove the lemma", "in case $R$ is a field. This is a special case of Lemma \\ref{lemma-depth-CM}." ], "refs": [ "algebra-lemma-grothendieck-regular-sequence", "dualizing-lemma-depth-CM" ], "ref_ids": [ 885, 2829 ] } ], "ref_ids": [] }, { "id": 2831, "type": "theorem", "label": "dualizing-lemma-divide-by-torsion", "categories": [ "dualizing" ], "title": "dualizing-lemma-divide-by-torsion", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "Let $M$ be an $A$-module. Let $Z = V(I)$.", "Then $H^0_I(M) = H^0_Z(M)$. Let $N$ be the common value and", "set $M' = M/N$. Then", "\\begin{enumerate}", "\\item $H^0_I(M') = 0$ and $H^p_I(M) = H^p_I(M')$ and $H^p_I(N) = 0$", "for all $p > 0$,", "\\item $H^0_Z(M') = 0$ and $H^p_Z(M) = H^p_Z(M')$ and $H^p_Z(N) = 0$", "for all $p > 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By definition $H^0_I(M) = M[I^\\infty]$ is $I$-power torsion.", "By Lemma \\ref{lemma-local-cohomology-adjoint} we see that", "$$", "H^0_Z(M) = \\Ker(M \\longrightarrow M_{f_1} \\times \\ldots \\times M_{f_r})", "$$", "if $I = (f_1, \\ldots, f_r)$. Thus $H^0_I(M) \\subset H^0_Z(M)$ and", "conversely, if $x \\in H^0_Z(M)$, then it is annihilated by a $f_i^{e_i}$", "for some $e_i \\geq 1$ hence annihilated by some power of $I$.", "This proves the first equality and moreover $N$ is $I$-power torsion.", "By Lemma \\ref{lemma-adjoint} we see that $R\\Gamma_I(N) = N$.", "By Lemma \\ref{lemma-local-cohomology-adjoint} we see that $R\\Gamma_Z(N) = N$.", "This proves the higher vanishing of $H^p_I(N)$ and $H^p_Z(N)$ in (1) and (2).", "The vanishing of $H^0_I(M')$ and $H^0_Z(M')$ follow from the preceding", "remarks and the fact that $M'$ is $I$-power torsion free by", "More on Algebra, Lemma \\ref{more-algebra-lemma-divide-by-torsion}.", "The equality of higher cohomologies for $M$ and $M'$ follow", "immediately from the long exact cohomology sequence." ], "refs": [ "dualizing-lemma-local-cohomology-adjoint", "dualizing-lemma-adjoint", "dualizing-lemma-local-cohomology-adjoint", "more-algebra-lemma-divide-by-torsion" ], "ref_ids": [ 2815, 2811, 2815, 10335 ] } ], "ref_ids": [] }, { "id": 2832, "type": "theorem", "label": "dualizing-lemma-complete-and-local", "categories": [ "dualizing" ], "title": "dualizing-lemma-complete-and-local", "contents": [ "\\begin{slogan}", "Results of this nature are sometimes referred to as Greenlees-May duality.", "\\end{slogan}", "Let $A$ be a ring and let $I$ be a finitely generated ideal.", "Let $R\\Gamma_Z$ be as in Lemma \\ref{lemma-local-cohomology-adjoint}.", "Let ${\\ }^\\wedge$ denote derived completion as in", "More on Algebra, Lemma \\ref{more-algebra-lemma-derived-completion}.", "For an object $K$ in $D(A)$ we have", "$$", "R\\Gamma_Z(K^\\wedge) = R\\Gamma_Z(K)", "\\quad\\text{and}\\quad", "(R\\Gamma_Z(K))^\\wedge = K^\\wedge", "$$", "in $D(A)$." ], "refs": [ "dualizing-lemma-local-cohomology-adjoint", "more-algebra-lemma-derived-completion" ], "proofs": [ { "contents": [ "Choose $f_1, \\ldots, f_r \\in A$ generating $I$. Recall that", "$$", "K^\\wedge = R\\Hom_A\\left((A \\to \\prod A_{f_{i_0}}", "\\to \\prod A_{f_{i_0i_1}} \\to \\ldots \\to A_{f_1 \\ldots f_r}), K\\right)", "$$", "by More on Algebra, Lemma \\ref{more-algebra-lemma-derived-completion}.", "Hence the cone $C = \\text{Cone}(K \\to K^\\wedge)$", "is given by", "$$", "R\\Hom_A\\left((\\prod A_{f_{i_0}}", "\\to \\prod A_{f_{i_0i_1}} \\to \\ldots \\to A_{f_1 \\ldots f_r}), K\\right)", "$$", "which can be represented by a complex endowed with a finite filtration", "whose successive quotients are isomorphic to", "$$", "R\\Hom_A(A_{f_{i_0} \\ldots f_{i_p}}, K), \\quad p > 0", "$$", "These complexes vanish on applying $R\\Gamma_Z$, see", "Lemma \\ref{lemma-local-cohomology-vanishes}. Applying $R\\Gamma_Z$", "to the distinguished triangle $K \\to K^\\wedge \\to C \\to K[1]$", "we see that the first formula of the lemma is correct.", "\\medskip\\noindent", "Recall that", "$$", "R\\Gamma_Z(K) =", "K \\otimes^\\mathbf{L} (A \\to \\prod A_{f_{i_0}}", "\\to \\prod A_{f_{i_0i_1}} \\to \\ldots \\to A_{f_1 \\ldots f_r})", "$$", "by Lemma \\ref{lemma-local-cohomology-adjoint}.", "Hence the cone $C = \\text{Cone}(R\\Gamma_Z(K) \\to K)$", "can be represented by a complex endowed with a finite filtration", "whose successive quotients are isomorphic to", "$$", "K \\otimes_A A_{f_{i_0} \\ldots f_{i_p}}, \\quad p > 0", "$$", "These complexes vanish on applying ${\\ }^\\wedge$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-derived-completion-vanishes}.", "Applying derived completion to the distinguished triangle", "$R\\Gamma_Z(K) \\to K \\to C \\to R\\Gamma_Z(K)[1]$", "we see that the second formula of the lemma is correct." ], "refs": [ "more-algebra-lemma-derived-completion", "dualizing-lemma-local-cohomology-vanishes", "dualizing-lemma-local-cohomology-adjoint", "more-algebra-lemma-derived-completion-vanishes" ], "ref_ids": [ 10372, 2818, 2815, 10373 ] } ], "ref_ids": [ 2815, 10372 ] }, { "id": 2833, "type": "theorem", "label": "dualizing-lemma-compare-RHom", "categories": [ "dualizing" ], "title": "dualizing-lemma-compare-RHom", "contents": [ "With notation as in Lemma \\ref{lemma-complete-and-local}.", "For objects $K, L$ in $D(A)$ there is a canonical isomorphism", "$$", "R\\Hom_A(K^\\wedge, L^\\wedge) \\longrightarrow R\\Hom_A(R\\Gamma_Z(K), R\\Gamma_Z(L))", "$$", "in $D(A)$." ], "refs": [ "dualizing-lemma-complete-and-local" ], "proofs": [ { "contents": [ "Say $I = (f_1, \\ldots, f_r)$. Denote", "$C = (A \\to \\prod A_{f_i} \\to \\ldots \\to A_{f_1 \\ldots f_r})$ the", "alternating {\\v C}ech complex. Then derived completion is given by", "$R\\Hom_A(C, -)$ (More on Algebra, Lemma", "\\ref{more-algebra-lemma-derived-completion}) and local cohomology by", "$C \\otimes^\\mathbf{L} -$ (Lemma \\ref{lemma-local-cohomology-adjoint}).", "Combining the isomorphism", "$$", "R\\Hom_A(K \\otimes^\\mathbf{L} C, L \\otimes^\\mathbf{L} C) =", "R\\Hom_A(K, R\\Hom_A(C, L \\otimes^\\mathbf{L} C))", "$$", "(More on Algebra, Lemma \\ref{more-algebra-lemma-internal-hom})", "and the map", "$$", "L \\to R\\Hom_A(C, L \\otimes^\\mathbf{L} C)", "$$", "(More on Algebra, Lemma \\ref{more-algebra-lemma-internal-hom-diagonal})", "we obtain a map", "$$", "\\gamma :", "R\\Hom_A(K, L)", "\\longrightarrow", "R\\Hom_A(K \\otimes^\\mathbf{L} C, L \\otimes^\\mathbf{L} C)", "$$", "On the other hand, the right hand side is derived complete as it is", "equal to", "$$", "R\\Hom_A(C, R\\Hom_A(K, L \\otimes^\\mathbf{L} C)).", "$$", "Thus $\\gamma$ factors through the derived completion of", "$R\\Hom_A(K, L)$ by the universal property of derived completion.", "However, the derived completion goes inside the $R\\Hom_A$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-completion-RHom}", "and we obtain the desired map.", "\\medskip\\noindent", "To show that the map of the lemma is an isomorphism", "we may assume that $K$ and $L$ are derived complete, i.e.,", "$K = K^\\wedge$ and $L = L^\\wedge$. In this case we are", "looking at the map", "$$", "\\gamma : R\\Hom_A(K, L) \\longrightarrow R\\Hom_A(R\\Gamma_Z(K), R\\Gamma_Z(L))", "$$", "By Proposition \\ref{proposition-torsion-complete} we know that", "the cohomology groups", "of the left and the right hand side coincide. In other words,", "we have to check that the map $\\gamma$ sends a morphism", "$\\alpha : K \\to L$ in $D(A)$ to the morphism", "$R\\Gamma_Z(\\alpha) : R\\Gamma_Z(K) \\to R\\Gamma_Z(L)$.", "We omit the verification (hint: note that $R\\Gamma_Z(\\alpha)$", "is just the map", "$\\alpha \\otimes \\text{id}_C :", "K \\otimes^\\mathbf{L} C", "\\to", "L \\otimes^\\mathbf{L} C$ which is almost the same as the", "construction of the map in", "More on Algebra, Lemma \\ref{more-algebra-lemma-internal-hom-diagonal})." ], "refs": [ "more-algebra-lemma-derived-completion", "dualizing-lemma-local-cohomology-adjoint", "more-algebra-lemma-internal-hom", "more-algebra-lemma-internal-hom-diagonal", "more-algebra-lemma-completion-RHom", "dualizing-proposition-torsion-complete", "more-algebra-lemma-internal-hom-diagonal" ], "ref_ids": [ 10372, 2815, 10206, 10211, 10374, 2925, 10211 ] } ], "ref_ids": [ 2832 ] }, { "id": 2834, "type": "theorem", "label": "dualizing-lemma-completion-local", "categories": [ "dualizing" ], "title": "dualizing-lemma-completion-local", "contents": [ "Let $I$ and $J$ be ideals in a Noetherian ring $A$. Let $M$ be a finite", "$A$-module. Set $Z =V(J)$. Consider the derived $I$-adic completion", "$R\\Gamma_Z(M)^\\wedge$ of local cohomology. Then", "\\begin{enumerate}", "\\item we have $R\\Gamma_Z(M)^\\wedge = R\\lim R\\Gamma_Z(M/I^nM)$, and", "\\item there are short exact sequences", "$$", "0 \\to R^1\\lim H^{i - 1}_Z(M/I^nM) \\to H^i(R\\Gamma_Z(M)^\\wedge) \\to", "\\lim H^i_Z(M/I^nM) \\to 0", "$$", "\\end{enumerate}", "In particular $R\\Gamma_Z(M)^\\wedge$ has vanishing cohomology", "in negative degrees." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $J = (g_1, \\ldots, g_m)$.", "Then $R\\Gamma_Z(M)$ is computed by the complex", "$$", "M \\to \\prod M_{g_{j_0}} \\to \\prod M_{g_{j_0}g_{j_1}} \\to", "\\ldots \\to M_{g_1g_2\\ldots g_m}", "$$", "by Lemma \\ref{lemma-local-cohomology-adjoint}.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-when-derived-completion-is-completion}", "the derived $I$-adic completion of", "this complex is given by the complex", "$$", "\\lim M/I^nM \\to \\prod \\lim (M/I^nM)_{g_{j_0}} \\to", "\\ldots \\to \\lim (M/I^nM)_{g_1g_2\\ldots g_m}", "$$", "of usual completions. Since $R\\Gamma_Z(M/I^nM)$ is computed by", "the complex $ M/I^nM \\to \\prod (M/I^nM)_{g_{j_0}} \\to", "\\ldots \\to (M/I^nM)_{g_1g_2\\ldots g_m}$ and since the", "transition maps between these complexes are surjective,", "we conclude that (1) holds by", "More on Algebra, Lemma \\ref{more-algebra-lemma-compute-Rlim-modules}.", "Part (2) then follows from More on Algebra, Lemma", "\\ref{more-algebra-lemma-break-long-exact-sequence-modules}." ], "refs": [ "dualizing-lemma-local-cohomology-adjoint", "more-algebra-lemma-when-derived-completion-is-completion", "more-algebra-lemma-compute-Rlim-modules", "more-algebra-lemma-break-long-exact-sequence-modules" ], "ref_ids": [ 2815, 10395, 10324, 10326 ] } ], "ref_ids": [] }, { "id": 2835, "type": "theorem", "label": "dualizing-lemma-completion-local-H0", "categories": [ "dualizing" ], "title": "dualizing-lemma-completion-local-H0", "contents": [ "With notation and hypotheses as in Lemma \\ref{lemma-completion-local}", "assume $A$ is $I$-adically complete. Then", "$$", "H^0(R\\Gamma_Z(M)^\\wedge) = \\colim H^0_{V(J')}(M)", "$$", "where the filtered colimit is over $J' \\subset J$ such that", "$V(J') \\cap V(I) = V(J) \\cap V(I)$." ], "refs": [ "dualizing-lemma-completion-local" ], "proofs": [ { "contents": [ "Since $M$ is a finite $A$-module, we have that $M$ is $I$-adically complete.", "The proof of Lemma \\ref{lemma-completion-local} shows that", "$$", "H^0(R\\Gamma_Z(M)^\\wedge) =", "\\Ker(M^\\wedge \\to \\prod M_{g_j}^\\wedge) =", "\\Ker(M \\to \\prod M_{g_j}^\\wedge)", "$$", "where on the right hand side we have usual $I$-adic completion.", "The kernel $K_j$ of $M_{g_j} \\to M_{g_j}^\\wedge$ is $\\bigcap I^n M_{g_j}$.", "By Algebra, Lemma \\ref{algebra-lemma-intersection-powers-ideal-module}", "for every $\\mathfrak p \\in V(IA_{g_j})$ we find an", "$f \\in A_{g_j}$, $f \\not \\in \\mathfrak p$ such that $(K_j)_f = 0$.", "\\medskip\\noindent", "Let $s \\in H^0(R\\Gamma_Z(M)^\\wedge)$.", "By the above we may think of $s$ as an element of $M$.", "The support $Z'$ of $s$ intersected with $D(g_j)$ is disjoint from", "$D(g_j) \\cap V(I)$ by the arguments above.", "Thus $Z'$ is a closed subset of $\\Spec(A)$ with $Z' \\cap V(I) \\subset V(J)$.", "Then $Z' \\cup V(J) = V(J')$ for some ideal $J' \\subset J$ with", "$V(J') \\cap V(I) \\subset V(J)$ and we have $s \\in H^0_{V(J')}(M)$.", "Conversely, any $s \\in H^0_{V(J')}(M)$ with $J' \\subset J$ and", "$V(J') \\cap V(I) \\subset V(J)$ maps to zero in $M_{g_j}^\\wedge$ for all $j$.", "This proves the lemma." ], "refs": [ "dualizing-lemma-completion-local", "algebra-lemma-intersection-powers-ideal-module" ], "ref_ids": [ 2834, 628 ] } ], "ref_ids": [ 2834 ] }, { "id": 2836, "type": "theorem", "label": "dualizing-lemma-right-adjoint", "categories": [ "dualizing" ], "title": "dualizing-lemma-right-adjoint", "contents": [ "Let $A \\to B$ be a ring homomorphism. The functor $R\\Hom(B, -)$", "constructed above is right adjoint to the restriction functor", "$D(B) \\to D(A)$." ], "refs": [], "proofs": [ { "contents": [ "This is a consequence of the fact that restriction and $\\Hom_A(B, -)$ are", "adjoint functors by Algebra, Lemma \\ref{algebra-lemma-adjoint-hom-restrict}.", "See Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors}." ], "refs": [ "algebra-lemma-adjoint-hom-restrict", "derived-lemma-derived-adjoint-functors" ], "ref_ids": [ 375, 1907 ] } ], "ref_ids": [] }, { "id": 2837, "type": "theorem", "label": "dualizing-lemma-composition-right-adjoints", "categories": [ "dualizing" ], "title": "dualizing-lemma-composition-right-adjoints", "contents": [ "Let $A \\to B \\to C$ be ring maps. Then", "$R\\Hom(C, -) \\circ R\\Hom(B, -) : D(A) \\to D(C)$", "is the functor $R\\Hom(C, -) : D(A) \\to D(C)$." ], "refs": [], "proofs": [ { "contents": [ "Follows from uniqueness of right adjoints and Lemma \\ref{lemma-right-adjoint}." ], "refs": [ "dualizing-lemma-right-adjoint" ], "ref_ids": [ 2836 ] } ], "ref_ids": [] }, { "id": 2838, "type": "theorem", "label": "dualizing-lemma-RHom-ext", "categories": [ "dualizing" ], "title": "dualizing-lemma-RHom-ext", "contents": [ "Let $\\varphi : A \\to B$ be a ring homomorphism. For $K$ in $D(A)$ we have", "$$", "\\varphi_*R\\Hom(B, K) = R\\Hom_A(B, K)", "$$", "where $\\varphi_* : D(B) \\to D(A)$ is restriction. In particular", "$R^q\\Hom(B, K) = \\Ext_A^q(B, K)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $I^\\bullet$ representing $K$.", "Then $R\\Hom(B, K)$ is represented by the complex $\\Hom_A(B, I^\\bullet)$", "of $B$-modules. Since this complex, as a complex of $A$-modules,", "represents $R\\Hom_A(B, K)$ we see that the lemma is true." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2839, "type": "theorem", "label": "dualizing-lemma-exact-support-coherent", "categories": [ "dualizing" ], "title": "dualizing-lemma-exact-support-coherent", "contents": [ "With notation as above, assume $A \\to B$ is a finite ring map of", "Noetherian rings. Then $R\\Hom(B, -)$ maps", "$D^+_{\\textit{Coh}}(A)$ into $D^+_{\\textit{Coh}}(B)$." ], "refs": [], "proofs": [ { "contents": [ "We have to show: if $K \\in D^+(A)$ has finite cohomology modules, then the", "complex $R\\Hom(B, K)$ has finite cohomology modules too.", "This follows for example from Lemma \\ref{lemma-RHom-ext}", "if we can show the ext modules $\\Ext^i_A(B, K)$", "are finite $A$-modules. Since $K$ is bounded below there is a", "convergent spectral sequence", "$$", "\\Ext^p_A(B, H^q(K)) \\Rightarrow \\text{Ext}^{p + q}_A(B, K)", "$$", "This finishes the proof as the modules $\\Ext^p_A(B, H^q(K))$", "are finite by", "Algebra, Lemma \\ref{algebra-lemma-ext-noetherian}." ], "refs": [ "dualizing-lemma-RHom-ext", "algebra-lemma-ext-noetherian" ], "ref_ids": [ 2838, 768 ] } ], "ref_ids": [] }, { "id": 2840, "type": "theorem", "label": "dualizing-lemma-RHom-dga", "categories": [ "dualizing" ], "title": "dualizing-lemma-RHom-dga", "contents": [ "In Situation \\ref{situation-resolution} the functor $R\\Hom(A, -)$", "is equal to the composition of", "$R\\Hom(E, -) : D(R) \\to D(E, \\text{d})$", "and the equivalence $- \\otimes^\\mathbf{L}_E A : D(E, \\text{d}) \\to D(A)$." ], "refs": [], "proofs": [ { "contents": [ "This is true because $R\\Hom(E, -)$ is the right adjoint", "to $- \\otimes^\\mathbf{L}_R E$, see", "Differential Graded Algebra, Lemma \\ref{dga-lemma-tensor-hom-adjoint}.", "Hence this functor plays the same role as the functor", "$R\\Hom(A, -)$ for the map $R \\to A$ (Lemma \\ref{lemma-right-adjoint}),", "whence these functors must correspond via the equivalence", "$- \\otimes^\\mathbf{L}_E A : D(E, \\text{d}) \\to D(A)$." ], "refs": [ "dga-lemma-tensor-hom-adjoint", "dualizing-lemma-right-adjoint" ], "ref_ids": [ 13107, 2836 ] } ], "ref_ids": [] }, { "id": 2841, "type": "theorem", "label": "dualizing-lemma-RHom-is-tensor", "categories": [ "dualizing" ], "title": "dualizing-lemma-RHom-is-tensor", "contents": [ "In Situation \\ref{situation-resolution} assume that", "\\begin{enumerate}", "\\item $E$ viewed as an object of $D(R)$ is compact, and", "\\item $N = \\Hom^\\bullet_R(E^\\bullet, R)$ computes $R\\Hom(E, R)$.", "\\end{enumerate}", "Then $R\\Hom(E, -) : D(R) \\to D(E)$ is isomorphic to", "$K \\mapsto K \\otimes_R^\\mathbf{L} N$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Differential Graded Algebra, Lemma", "\\ref{dga-lemma-RHom-is-tensor}." ], "refs": [ "dga-lemma-RHom-is-tensor" ], "ref_ids": [ 13110 ] } ], "ref_ids": [] }, { "id": 2842, "type": "theorem", "label": "dualizing-lemma-RHom-is-tensor-special", "categories": [ "dualizing" ], "title": "dualizing-lemma-RHom-is-tensor-special", "contents": [ "In Situation \\ref{situation-resolution} assume $A$ is a perfect $R$-module.", "Then", "$$", "R\\Hom(A, -) : D(R) \\to D(A)", "$$", "is given by $K \\mapsto K \\otimes_R^\\mathbf{L} M$", "where $M = R\\Hom(A, R) \\in D(A)$." ], "refs": [], "proofs": [ { "contents": [ "We apply Divided Power Algebra, Lemma", "\\ref{dpa-lemma-tate-resoluton-pseudo-coherent-ring-map}", "to choose a Tate resolution $(E, \\text{d})$ of $A$ over $R$.", "Note that $E^i = 0$ for $i > 0$, $E^0 = R[x_1, \\ldots, x_n]$", "is a polynomial algebra, and $E^i$ is a finite free $E^0$-module", "for $i < 0$. It follows that $E$ viewed as a complex of $R$-modules", "is a bounded above complex of free $R$-modules.", "We check the assumptions of Lemma \\ref{lemma-RHom-is-tensor}.", "The first holds because $A$ is perfect", "(hence compact by More on Algebra, Proposition", "\\ref{more-algebra-proposition-perfect-is-compact})", "and the second by", "More on Algebra, Lemma \\ref{more-algebra-lemma-RHom-out-of-projective}.", "From the lemma conclude that $K \\mapsto R\\Hom(E, K)$ is", "isomorphic to $K \\mapsto K \\otimes_R^\\mathbf{L} N$ for", "some differential graded $E$-module $N$. Observe that", "$$", "(R \\otimes_R E) \\otimes_E^\\mathbf{L} A = R \\otimes_E E \\otimes_E A", "$$", "in $D(A)$. Hence by Differential Graded Algebra, Lemma", "\\ref{dga-lemma-compose-tensor-functors-general-algebra}", "we conclude that the composition of", "$- \\otimes_R^\\mathbf{L} N$ and $- \\otimes_R^\\mathbf{L} A$", "is of the form $- \\otimes_R M$ for some $M \\in D(A)$.", "To finish the proof we apply Lemma \\ref{lemma-RHom-dga}." ], "refs": [ "dpa-lemma-tate-resoluton-pseudo-coherent-ring-map", "dualizing-lemma-RHom-is-tensor", "more-algebra-proposition-perfect-is-compact", "more-algebra-lemma-RHom-out-of-projective", "dga-lemma-compose-tensor-functors-general-algebra", "dualizing-lemma-RHom-dga" ], "ref_ids": [ 1667, 2841, 10585, 10207, 13112, 2840 ] } ], "ref_ids": [] }, { "id": 2843, "type": "theorem", "label": "dualizing-lemma-compute-for-effective-Cartier-algebraic", "categories": [ "dualizing" ], "title": "dualizing-lemma-compute-for-effective-Cartier-algebraic", "contents": [ "Let $R \\to A$ be a surjective ring map whose kernel $I$", "is an invertible $R$-module. The functor", "$R\\Hom(A, -) : D(R) \\to D(A)$", "is isomorphic to $K \\mapsto K \\otimes_R^\\mathbf{L} N[-1]$", "where $N$ is inverse of the invertible $A$-module $I \\otimes_R A$." ], "refs": [], "proofs": [ { "contents": [ "Since $A$ has the finite projective resolution", "$$", "0 \\to I \\to R \\to A \\to 0", "$$", "we see that $A$ is a perfect $R$-module. By", "Lemma \\ref{lemma-RHom-is-tensor-special} it suffices", "to prove that $R\\Hom(A, R)$ is represented by $N[-1]$ in $D(A)$.", "This means $R\\Hom(A, R)$ has a unique nonzero", "cohomology module, namely $N$ in degree $1$. As", "$\\text{Mod}_A \\to \\text{Mod}_R$ is fully faithful it suffice to prove", "this after applying the restriction functor $i_* : D(A) \\to D(R)$.", "By Lemma \\ref{lemma-RHom-ext} we have", "$$", "i_*R\\Hom(A, R) = R\\Hom_R(A, R)", "$$", "Using the finite projective resolution above we find that the latter", "is represented by the complex $R \\to I^{\\otimes -1}$ with $R$", "in degree $0$. The map $R \\to I^{\\otimes -1}$ is injective", "and the cokernel is $N$." ], "refs": [ "dualizing-lemma-RHom-is-tensor-special", "dualizing-lemma-RHom-ext" ], "ref_ids": [ 2842, 2838 ] } ], "ref_ids": [] }, { "id": 2844, "type": "theorem", "label": "dualizing-lemma-check-base-change-is-iso", "categories": [ "dualizing" ], "title": "dualizing-lemma-check-base-change-is-iso", "contents": [ "In the situation above, the map (\\ref{equation-base-change})", "is an isomorphism if and only if the map", "$$", "R\\Hom_R(A, K) \\otimes_R^\\mathbf{L} R'", "\\longrightarrow", "R\\Hom_R(A, K \\otimes_R^\\mathbf{L} R')", "$$", "of More on Algebra, Lemma", "\\ref{more-algebra-lemma-internal-hom-diagonal-better} is an isomorphism." ], "refs": [ "more-algebra-lemma-internal-hom-diagonal-better" ], "proofs": [ { "contents": [ "To see that the map is an isomorphism, it suffices to prove it", "is an isomorphism after applying $\\varphi'_*$.", "Applying the functor $\\varphi'_*$ to (\\ref{equation-base-change})", "and using that $A' = A \\otimes_R^\\mathbf{L} R'$", "we obtain the base change map", "$R\\Hom_R(A, K) \\otimes_R^\\mathbf{L} R' \\to", "R\\Hom_{R'}(A \\otimes_R^\\mathbf{L} R', K \\otimes_R^\\mathbf{L} R')$", "for derived hom of", "More on Algebra, Equation (\\ref{more-algebra-equation-base-change-RHom}).", "Unwinding the left and right hand side exactly as in the proof of", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom}", "and in particular using", "More on Algebra, Lemma \\ref{more-algebra-lemma-upgrade-adjoint-tensor-RHom}", "gives the desired result." ], "refs": [ "more-algebra-lemma-base-change-RHom", "more-algebra-lemma-upgrade-adjoint-tensor-RHom" ], "ref_ids": [ 10418, 10417 ] } ], "ref_ids": [ 10210 ] }, { "id": 2845, "type": "theorem", "label": "dualizing-lemma-flat-bc-surjection", "categories": [ "dualizing" ], "title": "dualizing-lemma-flat-bc-surjection", "contents": [ "Let $R \\to A$ and $R \\to R'$ be ring maps and $A' = A \\otimes_R R'$.", "Assume", "\\begin{enumerate}", "\\item $A$ is pseudo-coherent as an $R$-module,", "\\item $R'$ has finite tor dimension as an $R$-module (for example", "$R \\to R'$ is flat),", "\\item $A$ and $R'$ are tor independent over $R$.", "\\end{enumerate}", "Then (\\ref{equation-base-change}) is an isomorphism for $K \\in D^+(R)$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-check-base-change-is-iso} and", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism} part (4)." ], "refs": [ "dualizing-lemma-check-base-change-is-iso", "more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism" ], "ref_ids": [ 2844, 10415 ] } ], "ref_ids": [] }, { "id": 2846, "type": "theorem", "label": "dualizing-lemma-bc-surjection", "categories": [ "dualizing" ], "title": "dualizing-lemma-bc-surjection", "contents": [ "Let $R \\to A$ and $R \\to R'$ be ring maps and $A' = A \\otimes_R R'$.", "Assume", "\\begin{enumerate}", "\\item $A$ is perfect as an $R$-module,", "\\item $A$ and $R'$ are tor independent over $R$.", "\\end{enumerate}", "Then (\\ref{equation-base-change}) is an isomorphism for all $K \\in D(R)$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-check-base-change-is-iso} and", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism} part (1)." ], "refs": [ "dualizing-lemma-check-base-change-is-iso", "more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism" ], "ref_ids": [ 2844, 10415 ] } ], "ref_ids": [] }, { "id": 2847, "type": "theorem", "label": "dualizing-lemma-finite-ext-into-bounded-injective", "categories": [ "dualizing" ], "title": "dualizing-lemma-finite-ext-into-bounded-injective", "contents": [ "Let $A$ be a Noetherian ring. Let $K, L \\in D_{\\textit{Coh}}(A)$", "and assume $L$ has finite injective dimension. Then", "$R\\Hom_A(K, L)$ is in $D_{\\textit{Coh}}(A)$." ], "refs": [], "proofs": [ { "contents": [ "Pick an integer $n$ and consider the distinguished triangle", "$$", "\\tau_{\\leq n}K \\to K \\to \\tau_{\\geq n + 1}K \\to \\tau_{\\leq n}K[1]", "$$", "see Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}.", "Since $L$ has finite injective dimension we see", "that $R\\Hom_A(\\tau_{\\geq n + 1}K, L)$ has vanishing", "cohomology in degrees $\\geq c - n$ for some constant $c$.", "Hence, given $i$, we see that", "$\\Ext^i_A(K, L) \\to \\Ext^i_A(\\tau_{\\leq n}K, L)$", "is an isomorphism for some $n \\gg - i$. By", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-coherent-internal-hom}", "applied to $\\tau_{\\leq n}K$ and $L$", "we see conclude that $\\Ext^i_A(K, L)$ is", "a finite $A$-module for all $i$. Hence $R\\Hom_A(K, L)$", "is indeed an object of $D_{\\textit{Coh}}(A)$." ], "refs": [ "derived-remark-truncation-distinguished-triangle", "perfect-lemma-coherent-internal-hom" ], "ref_ids": [ 2016, 6986 ] } ], "ref_ids": [] }, { "id": 2848, "type": "theorem", "label": "dualizing-lemma-dualizing", "categories": [ "dualizing" ], "title": "dualizing-lemma-dualizing", "contents": [ "Let $A$ be a Noetherian ring. If $\\omega_A^\\bullet$ is a dualizing", "complex, then the functor", "$$", "D : K \\longmapsto R\\Hom_A(K, \\omega_A^\\bullet)", "$$", "is an anti-equivalence $D_{\\textit{Coh}}(A) \\to D_{\\textit{Coh}}(A)$", "which exchanges $D^+_{\\textit{Coh}}(A)$ and $D^-_{\\textit{Coh}}(A)$", "and induces an anti-equivalence", "$D^b_{\\textit{Coh}}(A) \\to D^b_{\\textit{Coh}}(A)$.", "Moreover $D \\circ D$ is isomorphic to the identity functor." ], "refs": [], "proofs": [ { "contents": [ "Let $K$ be an object of $D_{\\textit{Coh}}(A)$. From", "Lemma \\ref{lemma-finite-ext-into-bounded-injective}", "we see $R\\Hom_A(K, \\omega_A^\\bullet)$ is an object of $D_{\\textit{Coh}}(A)$.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-internal-hom-evaluate-isomorphism-technical}", "and the assumptions on the dualizing complex", "we obtain a canonical isomorphism", "$$", "K = R\\Hom_A(\\omega_A^\\bullet, \\omega_A^\\bullet) \\otimes_A^\\mathbf{L} K", "\\longrightarrow", "R\\Hom_A(R\\Hom_A(K, \\omega_A^\\bullet), \\omega_A^\\bullet)", "$$", "Thus our functor has a quasi-inverse and the proof is complete." ], "refs": [ "dualizing-lemma-finite-ext-into-bounded-injective", "more-algebra-lemma-internal-hom-evaluate-isomorphism-technical" ], "ref_ids": [ 2847, 10414 ] } ], "ref_ids": [] }, { "id": 2849, "type": "theorem", "label": "dualizing-lemma-equivalence-comes-from-invertible", "categories": [ "dualizing" ], "title": "dualizing-lemma-equivalence-comes-from-invertible", "contents": [ "Let $A$ be a Noetherian ring. Let", "$F : D^b_{\\textit{Coh}}(A) \\to D^b_{\\textit{Coh}}(A)$ be an $A$-linear", "equivalence of categories. Then $F(A)$ is an invertible object of $D(A)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak m \\subset A$ be a maximal ideal with residue field $\\kappa$.", "Consider the object $F(\\kappa)$. Since", "$\\kappa = \\Hom_{D(A)}(\\kappa, \\kappa)$ we find that all", "cohomology groups of $F(\\kappa)$ are annihilated by $\\mathfrak m$.", "We also see that", "$$", "\\Ext^i_A(\\kappa, \\kappa) = \\text{Ext}^i_A(F(\\kappa), F(\\kappa))", "= \\Hom_{D(A)}(F(\\kappa), F(\\kappa)[i])", "$$", "is zero for $i < 0$. Say $H^a(F(\\kappa)) \\not = 0$ and", "$H^b(F(\\kappa)) \\not = 0$ with $a$ minimal and $b$ maximal", "(so in particular $a \\leq b$). Then there is a nonzero map", "$$", "F(\\kappa) \\to H^b(F(\\kappa))[-b] \\to H^a(F(\\kappa))[-b]", "\\to F(\\kappa)[a - b]", "$$", "in $D(A)$ (nonzero because it induces a nonzero map on cohomology).", "This proves that $b = a$. We conclude that $F(\\kappa) = \\kappa[-a]$.", "\\medskip\\noindent", "Let $G$ be a quasi-inverse to our functor $F$. Arguing as above", "we find an integer $b$ such that $G(\\kappa) = \\kappa[-b]$.", "On composing we find $a + b = 0$. Let $E$ be a finite $A$-module", "wich is annihilated by a power of $\\mathfrak m$. Arguing by", "induction on the length of $E$ we find that $G(E) = E'[-b]$", "for some finite $A$-module $E'$ annihilated by a power of", "$\\mathfrak m$. Then $E[-a] = F(E')$.", "Next, we consider the groups", "$$", "\\Ext^i_A(A, E') = \\text{Ext}^i_A(F(A), F(E')) =", "\\Hom_{D(A)}(F(A), E[-a + i])", "$$", "The left hand side is nonzero if and only if $i = 0$ and then", "we get $E'$. Applying this with $E = E' = \\kappa$ and using Nakayama's", "lemma this implies that $H^j(F(A))_\\mathfrak m$ is zero for $j > a$ and", "generated by $1$ element for $j = a$. On the other hand, if", "$H^j(F(A))_\\mathfrak m$ is not zero for some $j < a$, then", "there is a map $F(A) \\to E[-a + i]$ for some $i < 0$ and some", "$E$ (More on Algebra, Lemma \\ref{more-algebra-lemma-detect-cohomology})", "which is a contradiction.", "Thus we see that $F(A)_\\mathfrak m = M[-a]$", "for some $A_\\mathfrak m$-module $M$ generated by $1$ element.", "However, since", "$$", "A_\\mathfrak m = \\Hom_{D(A)}(A, A)_\\mathfrak m =", "\\Hom_{D(A)}(F(A), F(A))_\\mathfrak m = \\Hom_{A_\\mathfrak m}(M, M)", "$$", "we see that $M \\cong A_\\mathfrak m$. We conclude that there exists", "an element $f \\in A$, $f \\not \\in \\mathfrak m$ such that", "$F(A)_f$ is isomorphic to $A_f[-a]$. This finishes the proof." ], "refs": [ "more-algebra-lemma-detect-cohomology" ], "ref_ids": [ 10168 ] } ], "ref_ids": [] }, { "id": 2850, "type": "theorem", "label": "dualizing-lemma-dualizing-unique", "categories": [ "dualizing" ], "title": "dualizing-lemma-dualizing-unique", "contents": [ "Let $A$ be a Noetherian ring. If $\\omega_A^\\bullet$ and", "$(\\omega'_A)^\\bullet$ are dualizing complexes, then", "$(\\omega'_A)^\\bullet$ is quasi-isomorphic to", "$\\omega_A^\\bullet \\otimes_A^\\mathbf{L} L$", "for some invertible object $L$ of $D(A)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-dualizing} and", "\\ref{lemma-equivalence-comes-from-invertible} the functor", "$K \\mapsto R\\Hom_A(R\\Hom_A(K, \\omega_A^\\bullet), (\\omega_A')^\\bullet)$", "maps $A$ to an invertible object $L$. In other words, there is", "an isomorphism", "$$", "L \\longrightarrow R\\Hom_A(\\omega_A^\\bullet, (\\omega_A')^\\bullet)", "$$", "Since $L$ has finite tor dimension, this means that we can apply", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-internal-hom-evaluate-isomorphism-technical}", "to see that", "$$", "R\\Hom_A(\\omega_A^\\bullet, (\\omega'_A)^\\bullet) \\otimes_A^\\mathbf{L} K", "\\longrightarrow", "R\\Hom_A(R\\Hom_A(K, \\omega_A^\\bullet), (\\omega_A')^\\bullet)", "$$", "is an isomorphism for $K$ in $D^b_{\\textit{Coh}}(A)$.", "In particular, setting $K = \\omega_A^\\bullet$ finishes the proof." ], "refs": [ "dualizing-lemma-dualizing", "dualizing-lemma-equivalence-comes-from-invertible", "more-algebra-lemma-internal-hom-evaluate-isomorphism-technical" ], "ref_ids": [ 2848, 2849, 10414 ] } ], "ref_ids": [] }, { "id": 2851, "type": "theorem", "label": "dualizing-lemma-dualizing-localize", "categories": [ "dualizing" ], "title": "dualizing-lemma-dualizing-localize", "contents": [ "Let $A$ be a Noetherian ring. Let $B = S^{-1}A$ be a localization.", "If $\\omega_A^\\bullet$ is a dualizing", "complex, then $\\omega_A^\\bullet \\otimes_A B$ is a dualizing", "complex for $B$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\omega_A^\\bullet \\to I^\\bullet$ be a quasi-isomorphism", "with $I^\\bullet$ a bounded complex of injectives.", "Then $S^{-1}I^\\bullet$ is a bounded complex of injective", "$B = S^{-1}A$-modules (Lemma \\ref{lemma-localization-injective-modules})", "representing $\\omega_A^\\bullet \\otimes_A B$.", "Thus $\\omega_A^\\bullet \\otimes_A B$ has finite injective dimension.", "Since $H^i(\\omega_A^\\bullet \\otimes_A B) = H^i(\\omega_A^\\bullet) \\otimes_A B$", "by flatness of $A \\to B$ we see that $\\omega_A^\\bullet \\otimes_A B$", "has finite cohomology modules. Finally, the map", "$$", "B \\longrightarrow", "R\\Hom_A(\\omega_A^\\bullet \\otimes_A B, \\omega_A^\\bullet \\otimes_A B)", "$$", "is a quasi-isomorphism as formation of internal hom commutes with", "flat base change in this case, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom}." ], "refs": [ "dualizing-lemma-localization-injective-modules", "more-algebra-lemma-base-change-RHom" ], "ref_ids": [ 2790, 10418 ] } ], "ref_ids": [] }, { "id": 2852, "type": "theorem", "label": "dualizing-lemma-dualizing-glue", "categories": [ "dualizing" ], "title": "dualizing-lemma-dualizing-glue", "contents": [ "Let $A$ be a Noetherian ring. Let $f_1, \\ldots, f_n \\in A$", "generate the unit ideal. If $\\omega_A^\\bullet$ is a complex", "of $A$-modules such that $(\\omega_A^\\bullet)_{f_i}$ is a dualizing", "complex for $A_{f_i}$ for all $i$, then $\\omega_A^\\bullet$ is a dualizing", "complex for $A$." ], "refs": [], "proofs": [ { "contents": [ "Consider the double complex", "$$", "\\prod\\nolimits_{i_0} (\\omega_A^\\bullet)_{f_{i_0}}", "\\to", "\\prod\\nolimits_{i_0 < i_1} (\\omega_A^\\bullet)_{f_{i_0}f_{i_1}}", "\\to \\ldots", "$$", "The associated total complex is quasi-isomorphic to $\\omega_A^\\bullet$", "for example by Descent, Remark \\ref{descent-remark-standard-covering}", "or by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-alternating-cech-complex-complex-computes-cohomology}.", "By assumption the complexes $(\\omega_A^\\bullet)_{f_i}$ have", "finite injective dimension as complexes of $A_{f_i}$-modules.", "This implies that each of the complexes", "$(\\omega_A^\\bullet)_{f_{i_0} \\ldots f_{i_p}}$, $p > 0$ has", "finite injective dimension over $A_{f_{i_0} \\ldots f_{i_p}}$,", "see Lemma \\ref{lemma-localization-injective-modules}.", "This in turn implies that each of the complexes", "$(\\omega_A^\\bullet)_{f_{i_0} \\ldots f_{i_p}}$, $p > 0$ has", "finite injective dimension over $A$, see", "Lemma \\ref{lemma-injective-flat}. Hence $\\omega_A^\\bullet$", "has finite injective dimension as a complex of $A$-modules", "(as it can be represented by a complex endowed with", "a finite filtration whose graded parts have finite injective", "dimension). Since $H^n(\\omega_A^\\bullet)_{f_i}$ is a finite", "$A_{f_i}$ module for each $i$ we see that $H^i(\\omega_A^\\bullet)$", "is a finite $A$-module, see Algebra, Lemma \\ref{algebra-lemma-cover}.", "Finally, the (derived) base change of the map", "$A \\to R\\Hom_A(\\omega_A^\\bullet, \\omega_A^\\bullet)$ to $A_{f_i}$", "is the map", "$A_{f_i} \\to R\\Hom_A((\\omega_A^\\bullet)_{f_i}, (\\omega_A^\\bullet)_{f_i})$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom}.", "Hence we deduce that", "$A \\to R\\Hom_A(\\omega_A^\\bullet, \\omega_A^\\bullet)$", "is an isomorphism and the proof is complete." ], "refs": [ "descent-remark-standard-covering", "perfect-lemma-alternating-cech-complex-complex-computes-cohomology", "dualizing-lemma-localization-injective-modules", "dualizing-lemma-injective-flat", "algebra-lemma-cover", "more-algebra-lemma-base-change-RHom" ], "ref_ids": [ 14783, 6972, 2790, 2785, 411, 10418 ] } ], "ref_ids": [] }, { "id": 2853, "type": "theorem", "label": "dualizing-lemma-dualizing-finite", "categories": [ "dualizing" ], "title": "dualizing-lemma-dualizing-finite", "contents": [ "Let $A \\to B$ be a finite ring map of Noetherian rings.", "Let $\\omega_A^\\bullet$ be a dualizing complex.", "Then $R\\Hom(B, \\omega_A^\\bullet)$ is a dualizing complex for $B$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\omega_A^\\bullet \\to I^\\bullet$ be a quasi-isomorphism", "with $I^\\bullet$ a bounded complex of injectives.", "Then $\\Hom_A(B, I^\\bullet)$ is a bounded complex of injective", "$B$-modules (Lemma \\ref{lemma-hom-injective}) representing", "$R\\Hom(B, \\omega_A^\\bullet)$.", "Thus $R\\Hom(B, \\omega_A^\\bullet)$ has finite injective dimension.", "By Lemma \\ref{lemma-exact-support-coherent} it is an object of", "$D_{\\textit{Coh}}(B)$. Finally, we compute", "$$", "\\Hom_{D(B)}(R\\Hom(B, \\omega_A^\\bullet), R\\Hom(B, \\omega_A^\\bullet)) =", "\\Hom_{D(A)}(R\\Hom(B, \\omega_A^\\bullet), \\omega_A^\\bullet) = B", "$$", "and for $n \\not = 0$ we compute", "$$", "\\Hom_{D(B)}(R\\Hom(B, \\omega_A^\\bullet), R\\Hom(B, \\omega_A^\\bullet)[n]) =", "\\Hom_{D(A)}(R\\Hom(B, \\omega_A^\\bullet), \\omega_A^\\bullet[n]) = 0", "$$", "which proves the last property of a dualizing complex.", "In the displayed equations, the first", "equality holds by Lemma \\ref{lemma-right-adjoint}", "and the second equality holds by Lemma \\ref{lemma-dualizing}." ], "refs": [ "dualizing-lemma-hom-injective", "dualizing-lemma-exact-support-coherent", "dualizing-lemma-right-adjoint", "dualizing-lemma-dualizing" ], "ref_ids": [ 2787, 2839, 2836, 2848 ] } ], "ref_ids": [] }, { "id": 2854, "type": "theorem", "label": "dualizing-lemma-dualizing-quotient", "categories": [ "dualizing" ], "title": "dualizing-lemma-dualizing-quotient", "contents": [ "Let $A \\to B$ be a surjective homomorphism of Noetherian rings.", "Let $\\omega_A^\\bullet$ be a dualizing complex.", "Then $R\\Hom(B, \\omega_A^\\bullet)$ is a dualizing complex for $B$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-dualizing-finite}." ], "refs": [ "dualizing-lemma-dualizing-finite" ], "ref_ids": [ 2853 ] } ], "ref_ids": [] }, { "id": 2855, "type": "theorem", "label": "dualizing-lemma-dualizing-polynomial-ring", "categories": [ "dualizing" ], "title": "dualizing-lemma-dualizing-polynomial-ring", "contents": [ "Let $A$ be a Noetherian ring. If $\\omega_A^\\bullet$ is a dualizing", "complex, then $\\omega_A^\\bullet \\otimes_A A[x]$ is a dualizing", "complex for $A[x]$." ], "refs": [], "proofs": [ { "contents": [ "Set $B = A[x]$ and $\\omega_B^\\bullet = \\omega_A^\\bullet \\otimes_A B$.", "It follows from Lemma \\ref{lemma-injective-dimension-over-polynomial-ring}", "and More on Algebra, Lemma \\ref{more-algebra-lemma-finite-injective-dimension}", "that $\\omega_B^\\bullet$ has finite injective dimension.", "Since $H^i(\\omega_B^\\bullet) = H^i(\\omega_A^\\bullet) \\otimes_A B$", "by flatness of $A \\to B$ we see that $\\omega_A^\\bullet \\otimes_A B$", "has finite cohomology modules. Finally, the map", "$$", "B \\longrightarrow R\\Hom_B(\\omega_B^\\bullet, \\omega_B^\\bullet)", "$$", "is a quasi-isomorphism as formation of internal hom commutes with", "flat base change in this case, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom}." ], "refs": [ "dualizing-lemma-injective-dimension-over-polynomial-ring", "more-algebra-lemma-finite-injective-dimension", "more-algebra-lemma-base-change-RHom" ], "ref_ids": [ 2792, 10189, 10418 ] } ], "ref_ids": [] }, { "id": 2856, "type": "theorem", "label": "dualizing-lemma-find-function", "categories": [ "dualizing" ], "title": "dualizing-lemma-find-function", "contents": [ "Let $A$ be a Noetherian ring. Let $\\omega_A^\\bullet$ be a dualizing", "complex. Let $\\mathfrak m \\subset A$ be a maximal ideal and set", "$\\kappa = A/\\mathfrak m$. Then", "$R\\Hom_A(\\kappa, \\omega_A^\\bullet) \\cong \\kappa[n]$ for some", "$n \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "This is true because $R\\Hom_A(\\kappa, \\omega_A^\\bullet)$ is a dualizing", "complex over $\\kappa$ (Lemma \\ref{lemma-dualizing-quotient}),", "because dualizing complexes over $\\kappa$ are unique up to shifts", "(Lemma \\ref{lemma-dualizing-unique}), and because $\\kappa$ is a", "dualizing complex over $\\kappa$." ], "refs": [ "dualizing-lemma-dualizing-quotient", "dualizing-lemma-dualizing-unique" ], "ref_ids": [ 2854, 2850 ] } ], "ref_ids": [] }, { "id": 2857, "type": "theorem", "label": "dualizing-lemma-normalized-finite", "categories": [ "dualizing" ], "title": "dualizing-lemma-normalized-finite", "contents": [ "Let $(A, \\mathfrak m, \\kappa) \\to (B, \\mathfrak m', \\kappa')$", "be a finite local map of Noetherian local rings. Let $\\omega_A^\\bullet$", "be a normalized dualizing complex. Then", "$\\omega_B^\\bullet = R\\Hom(B, \\omega_A^\\bullet)$ is a", "normalized dualizing complex for $B$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-dualizing-finite} the complex", "$\\omega_B^\\bullet$ is dualizing for $B$. We have", "$$", "R\\Hom_B(\\kappa', \\omega_B^\\bullet) =", "R\\Hom_B(\\kappa', R\\Hom(B, \\omega_A^\\bullet)) =", "R\\Hom_A(\\kappa', \\omega_A^\\bullet)", "$$", "by Lemma \\ref{lemma-right-adjoint}. Since $\\kappa'$ is isomorphic", "to a finite direct sum of copies of $\\kappa$ as an $A$-module", "and since $\\omega_A^\\bullet$ is normalized, we", "see that this complex only has cohomology placed in degree $0$.", "Thus $\\omega_B^\\bullet$ is a normalized dualizing complex as well." ], "refs": [ "dualizing-lemma-dualizing-finite", "dualizing-lemma-right-adjoint" ], "ref_ids": [ 2853, 2836 ] } ], "ref_ids": [] }, { "id": 2858, "type": "theorem", "label": "dualizing-lemma-normalized-quotient", "categories": [ "dualizing" ], "title": "dualizing-lemma-normalized-quotient", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local", "ring with normalized dualizing complex $\\omega_A^\\bullet$.", "Let $A \\to B$ be surjective. Then", "$\\omega_B^\\bullet = R\\Hom_A(B, \\omega_A^\\bullet)$ is a", "normalized dualizing complex for $B$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-normalized-finite}." ], "refs": [ "dualizing-lemma-normalized-finite" ], "ref_ids": [ 2857 ] } ], "ref_ids": [] }, { "id": 2859, "type": "theorem", "label": "dualizing-lemma-equivalence-finite-length", "categories": [ "dualizing" ], "title": "dualizing-lemma-equivalence-finite-length", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local", "ring. Let $F$ be an $A$-linear self-equivalence of the category of", "finite length $A$-modules. Then $F$ is isomorphic to the identity functor." ], "refs": [], "proofs": [ { "contents": [ "Since $\\kappa$ is the unique simple object of the category we have", "$F(\\kappa) \\cong \\kappa$. Since our category is abelian, we find that", "$F$ is exact. Hence $F(E)$ has the same length as $E$ for all finite", "length modules $E$.", "Since $\\Hom(E, \\kappa) = \\Hom(F(E), F(\\kappa)) \\cong \\Hom(F(E), \\kappa)$", "we conclude from Nakayama's lemma that $E$ and $F(E)$ have the same", "number of generators. Hence $F(A/\\mathfrak m^n)$ is a cyclic $A$-module.", "Pick a generator $e \\in F(A/\\mathfrak m^n)$.", "Since $F$ is $A$-linear we conclude that $\\mathfrak m^n e = 0$.", "The map $A/\\mathfrak m^n \\to F(A/\\mathfrak m^n)$ has to be", "an isomorphism as the lengths are equal. Pick an element", "$$", "e \\in \\lim F(A/\\mathfrak m^n)", "$$", "which maps to a generator for all $n$ (small argument omitted).", "Then we obtain a system of isomorphisms", "$A/\\mathfrak m^n \\to F(A/\\mathfrak m^n)$ compatible with all", "$A$-module maps $A/\\mathfrak m^n \\to A/\\mathfrak m^{n'}$ (by $A$-linearity", "of $F$ again). Since any finite length module is a cokernel", "of a map between direct sums of cyclic modules, we obtain the isomorphism", "of the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2860, "type": "theorem", "label": "dualizing-lemma-dualizing-finite-length", "categories": [ "dualizing" ], "title": "dualizing-lemma-dualizing-finite-length", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local", "ring with normalized dualizing complex $\\omega_A^\\bullet$.", "Let $E$ be an injective hull of $\\kappa$. Then there exists", "a functorial isomorphism", "$$", "R\\Hom_A(N, \\omega_A^\\bullet) = \\Hom_A(N, E)[0]", "$$", "for $N$ running through the finite length $A$-modules." ], "refs": [], "proofs": [ { "contents": [ "By induction on the length of $N$ we see that $R\\Hom_A(N, \\omega_A^\\bullet)$", "is a module of finite length sitting in degree $0$. Thus", "$R\\Hom_A(-, \\omega_A^\\bullet)$ induces an anti-equivalence", "on the category of finite length modules. Since the same is true", "for $\\Hom_A(-, E)$ by Proposition \\ref{proposition-matlis} we see that", "$$", "N \\longmapsto \\Hom_A(R\\Hom_A(N, \\omega_A^\\bullet), E)", "$$", "is an equivalence as in Lemma \\ref{lemma-equivalence-finite-length}.", "Hence it is isomorphic to the identity functor.", "Since $\\Hom_A(-, E)$ applied twice is the identity", "(Proposition \\ref{proposition-matlis}) we obtain", "the statement of the lemma." ], "refs": [ "dualizing-proposition-matlis", "dualizing-lemma-equivalence-finite-length", "dualizing-proposition-matlis" ], "ref_ids": [ 2924, 2859, 2924 ] } ], "ref_ids": [] }, { "id": 2861, "type": "theorem", "label": "dualizing-lemma-sitting-in-degrees", "categories": [ "dualizing" ], "title": "dualizing-lemma-sitting-in-degrees", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with", "normalized dualizing complex $\\omega_A^\\bullet$. Let $M$ be a finite", "$A$-module and let $d = \\dim(\\text{Supp}(M))$. Then", "\\begin{enumerate}", "\\item if $\\Ext^i_A(M, \\omega_A^\\bullet)$ is nonzero, then", "$i \\in \\{-d, \\ldots, 0\\}$,", "\\item the dimension of the support of $\\Ext^i_A(M, \\omega_A^\\bullet)$", "is at most $-i$,", "\\item $\\text{depth}(M)$ is the smallest integer $\\delta \\geq 0$ such that", "$\\Ext^{-\\delta}_A(M, \\omega_A^\\bullet) \\not = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We prove this by induction on $d$. If $d = 0$, this follows from", "Lemma \\ref{lemma-dualizing-finite-length} and Matlis duality", "(Proposition \\ref{proposition-matlis}) which guarantees that", "$\\Hom_A(M, E)$ is nonzero if $M$ is nonzero.", "\\medskip\\noindent", "Assume the result holds for modules with support of dimension $< d$ and that", "$M$ has depth $> 0$. Choose an $f \\in \\mathfrak m$ which is a nonzerodivisor", "on $M$ and consider the short exact sequence", "$$", "0 \\to M \\to M \\to M/fM \\to 0", "$$", "Since $\\dim(\\text{Supp}(M/fM)) = d - 1$", "(Algebra, Lemma \\ref{algebra-lemma-one-equation-module}) we", "may apply the induction hypothesis.", "Writing", "$E^i = \\Ext^i_A(M, \\omega_A^\\bullet)$ and", "$F^i = \\Ext^i_A(M/fM, \\omega_A^\\bullet)$", "we obtain a long exact sequence", "$$", "\\ldots \\to F^i \\to E^i \\xrightarrow{f} E^i \\to F^{i + 1} \\to \\ldots", "$$", "By induction $E^i/fE^i = 0$ for", "$i + 1 \\not \\in \\{-\\dim(\\text{Supp}(M/fM)), \\ldots, -\\text{depth}(M/fM)\\}$.", "By Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "and Algebra, Lemma \\ref{algebra-lemma-depth-drops-by-one}", "we conclude $E^i = 0$ for", "$i \\not \\in \\{-\\dim(\\text{Supp}(M)), \\ldots, -\\text{depth}(M)\\}$.", "Moreover, in the boundary case $i = - \\text{depth}(M)$ we deduce that $E^i$", "is nonzero as $F^{i + 1}$ is nonzero by induction.", "Since $E^i/fE^i \\subset F^{i + 1}$ we get", "$$", "\\dim(\\text{Supp}(F^{i + 1})) \\geq \\dim(\\text{Supp}(E^i/fE^i))", "\\geq \\dim(\\text{Supp}(E^i)) - 1", "$$", "(see lemma used above) we also obtain the dimension estimate (2).", "\\medskip\\noindent", "If $M$ has depth $0$ and $d > 0$ we let $N = M[\\mathfrak m^\\infty]$ and set", "$M' = M/N$ (compare with Lemma \\ref{lemma-divide-by-torsion}).", "Then $M'$ has depth $> 0$ and $\\dim(\\text{Supp}(M')) = d$.", "Thus we know the result for $M'$ and since", "$R\\Hom_A(N, \\omega_A^\\bullet) = \\Hom_A(N, E)$", "(Lemma \\ref{lemma-dualizing-finite-length})", "the long exact cohomology sequence of $\\Ext$'s implies the", "result for $M$." ], "refs": [ "dualizing-lemma-dualizing-finite-length", "dualizing-proposition-matlis", "algebra-lemma-NAK", "algebra-lemma-depth-drops-by-one", "dualizing-lemma-divide-by-torsion", "dualizing-lemma-dualizing-finite-length" ], "ref_ids": [ 2860, 2924, 401, 774, 2831, 2860 ] } ], "ref_ids": [] }, { "id": 2862, "type": "theorem", "label": "dualizing-lemma-local-CM", "categories": [ "dualizing" ], "title": "dualizing-lemma-local-CM", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring", "with normalized dualizing complex $\\omega_A^\\bullet$. Let $M$", "be a finite $A$-module. The following are equivalent", "\\begin{enumerate}", "\\item $M$ is Cohen-Macaulay,", "\\item $\\Ext^i_A(M, \\omega_A^\\bullet)$ is nonzero for a single $i$,", "\\item $\\Ext^{-i}_A(M, \\omega_A^\\bullet)$ is zero for", "$i \\not = \\dim(\\text{Supp}(M))$.", "\\end{enumerate}", "Denote $CM_d$ the category of finite Cohen-Macaulay $A$-modules", "of depth $d$. Then $M \\mapsto \\Ext^{-d}_A(M, \\omega_A^\\bullet)$", "defines an anti-auto-equivalence of $CM_d$." ], "refs": [], "proofs": [ { "contents": [ "We will use the results of Lemma \\ref{lemma-sitting-in-degrees}", "without further mention. Fix a finite module $M$.", "If $M$ is Cohen-Macaulay, then only", "$\\Ext^{-d}_A(M, \\omega_A^\\bullet)$ can be nonzero,", "hence (1) $\\Rightarrow$ (3).", "The implication (3) $\\Rightarrow$ (2) is immediate.", "Assume (2) and let $N = \\Ext^{-\\delta}_A(M, \\omega_A^\\bullet)$", "be the nonzero $\\Ext$ where $\\delta = \\text{depth}(M)$. Then, since", "$$", "M[0] = R\\Hom_A(R\\Hom_A(M, \\omega_A^\\bullet), \\omega_A^\\bullet) =", "R\\Hom_A(N[\\delta], \\omega_A^\\bullet)", "$$", "(Lemma \\ref{lemma-dualizing})", "we conclude that $M = \\Ext_A^{-\\delta}(N, \\omega_A^\\bullet)$.", "Thus $\\delta \\geq \\dim(\\text{Supp}(M))$. However,", "since we also know that $\\delta \\leq \\dim(\\text{Supp}(M))$", "(Algebra, Lemma \\ref{algebra-lemma-bound-depth}) we conclude that $M$ is", "Cohen-Macaulay.", "\\medskip\\noindent", "To prove the final statement, it suffices to show that", "$N = \\Ext^{-d}_A(M, \\omega_A^\\bullet)$ is in $CM_d$", "for $M$ in $CM_d$. Above we have seen that", "$M[0] = R\\Hom_A(N[d], \\omega_A^\\bullet)$ and this proves the", "desired result by the equivalence of (1) and (3)." ], "refs": [ "dualizing-lemma-sitting-in-degrees", "dualizing-lemma-dualizing", "algebra-lemma-bound-depth" ], "ref_ids": [ 2861, 2848, 770 ] } ], "ref_ids": [] }, { "id": 2863, "type": "theorem", "label": "dualizing-lemma-dualizing-artinian", "categories": [ "dualizing" ], "title": "dualizing-lemma-dualizing-artinian", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local", "ring with normalized dualizing complex $\\omega_A^\\bullet$.", "If $\\dim(A) = 0$, then $\\omega_A^\\bullet \\cong E[0]$", "where $E$ is an injective hull of the residue field." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-dualizing-finite-length}." ], "refs": [ "dualizing-lemma-dualizing-finite-length" ], "ref_ids": [ 2860 ] } ], "ref_ids": [] }, { "id": 2864, "type": "theorem", "label": "dualizing-lemma-divide-by-finite-length-ideal", "categories": [ "dualizing" ], "title": "dualizing-lemma-divide-by-finite-length-ideal", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local", "ring with normalized dualizing complex. Let $I \\subset \\mathfrak m$ be an", "ideal of finite length. Set $B = A/I$. Then there is a distinguished", "triangle", "$$", "\\omega_B^\\bullet \\to \\omega_A^\\bullet \\to \\Hom_A(I, E)[0] \\to", "\\omega_B^\\bullet[1]", "$$", "in $D(A)$ where $E$ is an injective hull of $\\kappa$ and", "$\\omega_B^\\bullet$ is a normalized dualizing complex for $B$." ], "refs": [], "proofs": [ { "contents": [ "Use the short exact sequence $0 \\to I \\to A \\to B \\to 0$", "and Lemmas \\ref{lemma-dualizing-finite-length} and", "\\ref{lemma-normalized-quotient}." ], "refs": [ "dualizing-lemma-dualizing-finite-length", "dualizing-lemma-normalized-quotient" ], "ref_ids": [ 2860, 2858 ] } ], "ref_ids": [] }, { "id": 2865, "type": "theorem", "label": "dualizing-lemma-divide-by-nonzerodivisor", "categories": [ "dualizing" ], "title": "dualizing-lemma-divide-by-nonzerodivisor", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local", "ring with normalized dualizing complex $\\omega_A^\\bullet$.", "Let $f \\in \\mathfrak m$ be a", "nonzerodivisor. Set $B = A/(f)$. Then there is a distinguished", "triangle", "$$", "\\omega_B^\\bullet \\to \\omega_A^\\bullet \\to \\omega_A^\\bullet \\to", "\\omega_B^\\bullet[1]", "$$", "in $D(A)$ where $\\omega_B^\\bullet$ is a normalized dualizing complex", "for $B$." ], "refs": [], "proofs": [ { "contents": [ "Use the short exact sequence $0 \\to A \\to A \\to B \\to 0$", "and Lemma \\ref{lemma-normalized-quotient}." ], "refs": [ "dualizing-lemma-normalized-quotient" ], "ref_ids": [ 2858 ] } ], "ref_ids": [] }, { "id": 2866, "type": "theorem", "label": "dualizing-lemma-nonvanishing-generically-local", "categories": [ "dualizing" ], "title": "dualizing-lemma-nonvanishing-generically-local", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with", "normalized dualizing complex $\\omega_A^\\bullet$.", "Let $\\mathfrak p$ be a minimal prime of $A$ with", "$\\dim(A/\\mathfrak p) = e$. Then", "$H^i(\\omega_A^\\bullet)_\\mathfrak p$ is nonzero", "if and only if $i = -e$." ], "refs": [], "proofs": [ { "contents": [ "Since $A_\\mathfrak p$ has dimension zero, there exists an integer", "$n > 0$ such that $\\mathfrak p^nA_\\mathfrak p$ is zero.", "Set $B = A/\\mathfrak p^n$ and", "$\\omega_B^\\bullet = R\\Hom_A(B, \\omega_A^\\bullet)$.", "Since $B_\\mathfrak p = A_\\mathfrak p$ we see that", "$$", "(\\omega_B^\\bullet)_\\mathfrak p =", "R\\Hom_A(B, \\omega_A^\\bullet) \\otimes_A^\\mathbf{L} A_\\mathfrak p =", "R\\Hom_{A_\\mathfrak p}(B_\\mathfrak p, (\\omega_A^\\bullet)_\\mathfrak p) =", "(\\omega_A^\\bullet)_\\mathfrak p", "$$", "The second equality holds by", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom}.", "By Lemma \\ref{lemma-normalized-quotient} we may replace $A$ by $B$.", "After doing so, we see that $\\dim(A) = e$. Then we see that", "$H^i(\\omega_A^\\bullet)_\\mathfrak p$ can only be nonzero if $i = -e$", "by Lemma \\ref{lemma-sitting-in-degrees} parts (1) and (2).", "On the other hand, since $(\\omega_A^\\bullet)_\\mathfrak p$", "is a dualizing complex for the nonzero ring $A_\\mathfrak p$", "(Lemma \\ref{lemma-dualizing-localize})", "we see that the remaining module has to be nonzero." ], "refs": [ "more-algebra-lemma-base-change-RHom", "dualizing-lemma-normalized-quotient", "dualizing-lemma-sitting-in-degrees", "dualizing-lemma-dualizing-localize" ], "ref_ids": [ 10418, 2858, 2861, 2851 ] } ], "ref_ids": [] }, { "id": 2867, "type": "theorem", "label": "dualizing-lemma-nonvanishing-generically", "categories": [ "dualizing" ], "title": "dualizing-lemma-nonvanishing-generically", "contents": [ "Let $A$ be a Noetherian ring. Let $\\mathfrak p$ be a minimal prime", "of $A$. Then $H^i(\\omega_A^\\bullet)_\\mathfrak p$ is nonzero", "for exactly one $i$." ], "refs": [], "proofs": [ { "contents": [ "The complex $\\omega_A^\\bullet \\otimes_A A_\\mathfrak p$", "is a dualizing complex for $A_\\mathfrak p$", "(Lemma \\ref{lemma-dualizing-localize}).", "The dimension of $A_\\mathfrak p$ is zero as $\\mathfrak p$", "is minimal. Hence the result follows from", "Lemma \\ref{lemma-dualizing-artinian}." ], "refs": [ "dualizing-lemma-dualizing-localize", "dualizing-lemma-dualizing-artinian" ], "ref_ids": [ 2851, 2863 ] } ], "ref_ids": [] }, { "id": 2868, "type": "theorem", "label": "dualizing-lemma-quotient-function", "categories": [ "dualizing" ], "title": "dualizing-lemma-quotient-function", "contents": [ "Let $A$ be a Noetherian ring and let $\\omega_A^\\bullet$ be a dualizing", "complex. Let $A \\to B$ be a surjective ring map and let", "$\\omega_B^\\bullet = R\\Hom(B, \\omega_A^\\bullet)$ be the dualizing", "complex for $B$ of Lemma \\ref{lemma-dualizing-quotient}. Then we have", "$$", "\\delta_{\\omega_B^\\bullet} = \\delta_{\\omega_A^\\bullet}|_{\\Spec(B)}", "$$" ], "refs": [ "dualizing-lemma-dualizing-quotient" ], "proofs": [ { "contents": [ "This follows from the definition of the functions and", "Lemma \\ref{lemma-normalized-quotient}." ], "refs": [ "dualizing-lemma-normalized-quotient" ], "ref_ids": [ 2858 ] } ], "ref_ids": [ 2854 ] }, { "id": 2869, "type": "theorem", "label": "dualizing-lemma-dimension-function", "categories": [ "dualizing" ], "title": "dualizing-lemma-dimension-function", "contents": [ "Let $A$ be a Noetherian ring and let $\\omega_A^\\bullet$ be a dualizing", "complex. The function $\\delta = \\delta_{\\omega_A^\\bullet}$", "defined above is a dimension function", "(Topology, Definition \\ref{topology-definition-dimension-function})." ], "refs": [ "topology-definition-dimension-function" ], "proofs": [ { "contents": [ "Let $\\mathfrak p \\subset \\mathfrak q$ be an immediate specialization.", "We have to show that $\\delta(\\mathfrak p) = \\delta(\\mathfrak q) + 1$.", "We may replace $A$ by $A/\\mathfrak p$, the complex $\\omega_A^\\bullet$ by", "$\\omega_{A/\\mathfrak p}^\\bullet = R\\Hom(A/\\mathfrak p, \\omega_A^\\bullet)$,", "the prime $\\mathfrak p$ by $(0)$, and the prime $\\mathfrak q$", "by $\\mathfrak q/\\mathfrak p$,", "see Lemma \\ref{lemma-quotient-function}. Thus we may assume that", "$A$ is a domain, $\\mathfrak p = (0)$, and $\\mathfrak q$ is a prime", "ideal of height $1$.", "\\medskip\\noindent", "Then $H^i(\\omega_A^\\bullet)_{(0)}$ is nonzero", "for exactly one $i$, say $i_0$, by Lemma \\ref{lemma-nonvanishing-generically}.", "In fact $i_0 = -\\delta((0))$ because", "$(\\omega_A^\\bullet)_{(0)}[-\\delta((0))]$", "is a normalized dualizing complex over the field $A_{(0)}$.", "\\medskip\\noindent", "On the other hand $(\\omega_A^\\bullet)_\\mathfrak q[-\\delta(\\mathfrak q)]$", "is a normalized dualizing complex for $A_\\mathfrak q$. By", "Lemma \\ref{lemma-nonvanishing-generically-local}", "we see that", "$$", "H^e((\\omega_A^\\bullet)_\\mathfrak q[-\\delta(\\mathfrak q)])_{(0)} =", "H^{e - \\delta(\\mathfrak q)}(\\omega_A^\\bullet)_{(0)}", "$$", "is nonzero only for $e = -\\dim(A_\\mathfrak q) = -1$.", "We conclude", "$$", "-\\delta((0)) = -1 - \\delta(\\mathfrak q)", "$$", "as desired." ], "refs": [ "dualizing-lemma-quotient-function", "dualizing-lemma-nonvanishing-generically", "dualizing-lemma-nonvanishing-generically-local" ], "ref_ids": [ 2868, 2867, 2866 ] } ], "ref_ids": [ 8367 ] }, { "id": 2870, "type": "theorem", "label": "dualizing-lemma-universally-catenary", "categories": [ "dualizing" ], "title": "dualizing-lemma-universally-catenary", "contents": [ "Let $A$ be a Noetherian ring which has a dualizing", "complex. Then $A$ is universally catenary of finite dimension." ], "refs": [], "proofs": [ { "contents": [ "Because $\\Spec(A)$ has a dimension function by", "Lemma \\ref{lemma-dimension-function}", "it is catenary, see", "Topology, Lemma \\ref{topology-lemma-dimension-function-catenary}.", "Hence $A$ is catenary, see", "Algebra, Lemma \\ref{algebra-lemma-catenary}.", "It follows from", "Proposition \\ref{proposition-dualizing-essentially-finite-type}", "that $A$ is universally catenary.", "\\medskip\\noindent", "Because any dualizing complex $\\omega_A^\\bullet$ is", "in $D^b_{\\textit{Coh}}(A)$ the values of the function", "$\\delta_{\\omega_A^\\bullet}$ in minimal primes are bounded by", "Lemma \\ref{lemma-nonvanishing-generically}.", "On the other hand, for a maximal ideal $\\mathfrak m$ with", "residue field $\\kappa$ the integer $i = -\\delta(\\mathfrak m)$", "is the unique integer such that", "$\\Ext_A^i(\\kappa, \\omega_A^\\bullet)$ is nonzero", "(Lemma \\ref{lemma-find-function}).", "Since $\\omega_A^\\bullet$ has finite injective dimension", "these values are bounded too. Since the dimension of", "$A$ is the maximal value of $\\delta(\\mathfrak p) - \\delta(\\mathfrak m)$", "where $\\mathfrak p \\subset \\mathfrak m$ are a pair", "consisting of a minimal prime and a maximal prime we find that the", "dimension of $\\Spec(A)$ is bounded." ], "refs": [ "dualizing-lemma-dimension-function", "topology-lemma-dimension-function-catenary", "algebra-lemma-catenary", "dualizing-proposition-dualizing-essentially-finite-type", "dualizing-lemma-nonvanishing-generically", "dualizing-lemma-find-function" ], "ref_ids": [ 2869, 8291, 931, 2926, 2867, 2856 ] } ], "ref_ids": [] }, { "id": 2871, "type": "theorem", "label": "dualizing-lemma-depth-dualizing-module", "categories": [ "dualizing" ], "title": "dualizing-lemma-depth-dualizing-module", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with", "normalized dualizing complex $\\omega_A^\\bullet$. Let $d = \\dim(A)$", "and $\\omega_A = H^{-d}(\\omega_A^\\bullet)$. Then", "\\begin{enumerate}", "\\item the support of $\\omega_A$ is the union of the irreducible components", "of $\\Spec(A)$ of dimension $d$,", "\\item $\\omega_A$ satisfies $(S_2)$, see", "Algebra, Definition \\ref{algebra-definition-conditions}.", "\\end{enumerate}" ], "refs": [ "algebra-definition-conditions" ], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-sitting-in-degrees} without further mention.", "By Lemma \\ref{lemma-nonvanishing-generically-local} the support", "of $\\omega_A$ contains the irreducible components of dimension $d$.", "Let $\\mathfrak p \\subset A$ be a prime. By Lemma \\ref{lemma-dimension-function}", "the complex $(\\omega_A^\\bullet)_{\\mathfrak p}[-\\dim(A/\\mathfrak p)]$", "is a normalized dualizing complex for $A_\\mathfrak p$. Hence if", "$\\dim(A/\\mathfrak p) + \\dim(A_\\mathfrak p) < d$, then", "$(\\omega_A)_\\mathfrak p = 0$.", "This proves the support of $\\omega_A$ is the union of the irreducible", "components of dimension $d$, because the complement of this union", "is exactly the primes $\\mathfrak p$ of $A$ for which", "$\\dim(A/\\mathfrak p) + \\dim(A_\\mathfrak p) < d$ as $A$ is catenary", "(Lemma \\ref{lemma-universally-catenary}).", "On the other hand, if $\\dim(A/\\mathfrak p) + \\dim(A_\\mathfrak p) = d$, then", "$$", "(\\omega_A)_\\mathfrak p =", "H^{-\\dim(A_\\mathfrak p)}\\left(", "(\\omega_A^\\bullet)_{\\mathfrak p}[-\\dim(A/\\mathfrak p)] \\right)", "$$", "Hence in order to prove $\\omega_A$ has $(S_2)$ it suffices to show that", "the depth of $\\omega_A$ is at least $\\min(\\dim(A), 2)$.", "We prove this by induction on $\\dim(A)$. The case $\\dim(A) = 0$ is", "trivial.", "\\medskip\\noindent", "Assume $\\text{depth}(A) > 0$. Choose a nonzerodivisor $f \\in \\mathfrak m$", "and set $B = A/fA$. Then $\\dim(B) = \\dim(A) - 1$ and we may apply the", "induction hypothesis to $B$. By Lemma \\ref{lemma-divide-by-nonzerodivisor}", "we see that multiplication by $f$ is injective on $\\omega_A$ and we get", "$\\omega_A/f\\omega_A \\subset \\omega_B$. This proves the depth of $\\omega_A$", "is at least $1$. If $\\dim(A) > 1$, then $\\dim(B) > 0$ and $\\omega_B$", "has depth $ > 0$. Hence $\\omega_A$ has depth $> 1$ and we conclude in", "this case.", "\\medskip\\noindent", "Assume $\\dim(A) > 0$ and $\\text{depth}(A) = 0$. Let", "$I = A[\\mathfrak m^\\infty]$ and set $B = A/I$. Then $B$ has", "depth $\\geq 1$ and $\\omega_A = \\omega_B$ by", "Lemma \\ref{lemma-divide-by-finite-length-ideal}.", "Since we proved the result for $\\omega_B$ above the proof is done." ], "refs": [ "dualizing-lemma-sitting-in-degrees", "dualizing-lemma-nonvanishing-generically-local", "dualizing-lemma-dimension-function", "dualizing-lemma-universally-catenary", "dualizing-lemma-divide-by-nonzerodivisor", "dualizing-lemma-divide-by-finite-length-ideal" ], "ref_ids": [ 2861, 2866, 2869, 2870, 2865, 2864 ] } ], "ref_ids": [ 1547 ] }, { "id": 2872, "type": "theorem", "label": "dualizing-lemma-local-cohomology-of-dualizing", "categories": [ "dualizing" ], "title": "dualizing-lemma-local-cohomology-of-dualizing", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Let $\\omega_A^\\bullet$ be a normalized dualizing complex.", "Let $Z = V(\\mathfrak m) \\subset \\Spec(A)$.", "Then $E = R^0\\Gamma_Z(\\omega_A^\\bullet)$ is an injective hull of", "$\\kappa$ and $R\\Gamma_Z(\\omega_A^\\bullet) = E[0]$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-local-cohomology-noetherian} we have", "$R\\Gamma_{\\mathfrak m} = R\\Gamma_Z$. Thus", "$$", "R\\Gamma_Z(\\omega_A^\\bullet) =", "R\\Gamma_{\\mathfrak m}(\\omega_A^\\bullet) =", "\\text{hocolim}\\ R\\Hom_A(A/\\mathfrak m^n, \\omega_A^\\bullet)", "$$", "by Lemma \\ref{lemma-local-cohomology-ext}. Let $E'$ be an injective", "hull of the residue field.", "By Lemma \\ref{lemma-dualizing-finite-length}", "we can find isomorphisms", "$$", "R\\Hom_A(A/\\mathfrak m^n, \\omega_A^\\bullet) \\cong \\Hom_A(A/\\mathfrak m^n, E')[0]", "$$", "compatible with transition maps. Since", "$E' = \\bigcup E'[\\mathfrak m^n] = \\colim \\Hom_A(A/\\mathfrak m^n, E')$", "by Lemma \\ref{lemma-union-artinian}", "we conclude that $E \\cong E'$ and that all other cohomology", "groups of the complex $R\\Gamma_Z(\\omega_A^\\bullet)$ are zero." ], "refs": [ "dualizing-lemma-local-cohomology-noetherian", "dualizing-lemma-local-cohomology-ext", "dualizing-lemma-dualizing-finite-length", "dualizing-lemma-union-artinian" ], "ref_ids": [ 2823, 2812, 2860, 2806 ] } ], "ref_ids": [] }, { "id": 2873, "type": "theorem", "label": "dualizing-lemma-special-case-local-duality", "categories": [ "dualizing" ], "title": "dualizing-lemma-special-case-local-duality", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Let $\\omega_A^\\bullet$ be a normalized dualizing complex.", "Let $E$ be an injective hull of the residue field.", "Let $K \\in D_{\\textit{Coh}}(A)$. Then", "$$", "\\Ext^{-i}_A(K, \\omega_A^\\bullet)^\\wedge =", "\\Hom_A(H^i_{\\mathfrak m}(K), E)", "$$", "where ${}^\\wedge$ denotes $\\mathfrak m$-adic completion." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-dualizing} we see that $R\\Hom_A(K, \\omega_A^\\bullet)$", "is an object of $D_{\\textit{Coh}}(A)$.", "It follows that the cohomology modules of the derived completion", "of $R\\Hom_A(K, \\omega_A^\\bullet)$ are equal to the usual completions", "$\\Ext^i_A(K, \\omega_A^\\bullet)^\\wedge$ by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-derived-completion-pseudo-coherent}.", "On the other hand, we have $R\\Gamma_{\\mathfrak m} = R\\Gamma_Z$", "for $Z = V(\\mathfrak m)$ by Lemma \\ref{lemma-local-cohomology-noetherian}.", "Moreover, the functor $\\Hom_A(-, E)$ is exact hence", "factors through cohomology.", "Hence the lemma is consequence of", "Theorem \\ref{theorem-local-duality}." ], "refs": [ "dualizing-lemma-dualizing", "more-algebra-lemma-derived-completion-pseudo-coherent", "dualizing-lemma-local-cohomology-noetherian", "dualizing-theorem-local-duality" ], "ref_ids": [ 2848, 10393, 2823, 2780 ] } ], "ref_ids": [] }, { "id": 2874, "type": "theorem", "label": "dualizing-lemma-depth-in-terms-dualizing-complex", "categories": [ "dualizing" ], "title": "dualizing-lemma-depth-in-terms-dualizing-complex", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with", "normalized dualizing complex $\\omega_A^\\bullet$.", "Then $\\text{depth}(A)$ is equal to the smallest integer $\\delta \\geq 0$", "such that $H^{-\\delta}(\\omega_A^\\bullet) \\not = 0$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from", "Lemma \\ref{lemma-sitting-in-degrees}.", "Here are two other ways to see that it is true.", "\\medskip\\noindent", "First alternative. By Nakayama's lemma we see that", "$\\delta$ is the smallest integer such that", "$\\Hom_A(H^{-\\delta}(\\omega_A^\\bullet), \\kappa) \\not = 0$.", "In other words, it is the smallest integer such that", "$\\Ext_A^{-\\delta}(\\omega_A^\\bullet, \\kappa)$", "is nonzero. Using Lemma \\ref{lemma-dualizing} and the fact that", "$\\omega_A^\\bullet$ is normalized this is equal to the", "smallest integer such that $\\Ext_A^\\delta(\\kappa, A)$ is", "nonzero. This is equal to the depth of $A$ by", "Algebra, Lemma \\ref{algebra-lemma-depth-ext}.", "\\medskip\\noindent", "Second alternative. By the local duality theorem", "(in the form of Lemma \\ref{lemma-special-case-local-duality})", "$\\delta$ is the smallest integer such that $H^\\delta_\\mathfrak m(A)$", "is nonzero. This is equal to the depth of $A$ by", "Lemma \\ref{lemma-depth}." ], "refs": [ "dualizing-lemma-sitting-in-degrees", "dualizing-lemma-dualizing", "algebra-lemma-depth-ext", "dualizing-lemma-special-case-local-duality", "dualizing-lemma-depth" ], "ref_ids": [ 2861, 2848, 772, 2873, 2826 ] } ], "ref_ids": [] }, { "id": 2875, "type": "theorem", "label": "dualizing-lemma-apply-CM", "categories": [ "dualizing" ], "title": "dualizing-lemma-apply-CM", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring", "with normalized dualizing complex $\\omega_A^\\bullet$", "and dualizing module $\\omega_A = H^{-\\dim(A)}(\\omega_A^\\bullet)$.", "The following are equivalent", "\\begin{enumerate}", "\\item $A$ is Cohen-Macaulay,", "\\item $\\omega_A^\\bullet$ is concentrated in a single degree, and", "\\item $\\omega_A^\\bullet = \\omega_A[\\dim(A)]$.", "\\end{enumerate}", "In this case $\\omega_A$ is a maximal Cohen-Macaulay module." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-local-CM}." ], "refs": [ "dualizing-lemma-local-CM" ], "ref_ids": [ 2862 ] } ], "ref_ids": [] }, { "id": 2876, "type": "theorem", "label": "dualizing-lemma-has-dualizing-module-CM", "categories": [ "dualizing" ], "title": "dualizing-lemma-has-dualizing-module-CM", "contents": [ "Let $A$ be a Noetherian ring. If there exists a finite $A$-module", "$\\omega_A$ such that $\\omega_A[0]$ is a dualizing complex, then", "$A$ is Cohen-Macaulay." ], "refs": [], "proofs": [ { "contents": [ "We may replace $A$ by the localization at a prime", "(Lemma \\ref{lemma-dualizing-localize} and", "Algebra, Definition \\ref{algebra-definition-ring-CM}).", "In this case the result follows immediately from", "Lemma \\ref{lemma-apply-CM}." ], "refs": [ "dualizing-lemma-dualizing-localize", "algebra-definition-ring-CM", "dualizing-lemma-apply-CM" ], "ref_ids": [ 2851, 1506, 2875 ] } ], "ref_ids": [] }, { "id": 2877, "type": "theorem", "label": "dualizing-lemma-CM-open", "categories": [ "dualizing" ], "title": "dualizing-lemma-CM-open", "contents": [ "Let $A$ be a Noetherian ring with dualizing complex $\\omega_A^\\bullet$.", "Let $M$ be a finite $A$-module. Then", "$$", "U = \\{\\mathfrak p \\in \\Spec(A) \\mid M_\\mathfrak p\\text{ is Cohen-Macaulay}\\}", "$$", "is an open subset of $\\Spec(A)$ whose intersection with", "$\\text{Supp}(M)$ is dense." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathfrak p$ is a generic point of $\\text{Supp}(M)$, then", "$\\text{depth}(M_\\mathfrak p) = \\dim(M_\\mathfrak p) = 0$", "and hence $\\mathfrak p \\in U$. This proves denseness.", "If $\\mathfrak p \\in U$, then we see that", "$$", "R\\Hom_A(M, \\omega_A^\\bullet)_\\mathfrak p =", "R\\Hom_{A_\\mathfrak p}(M_\\mathfrak p, (\\omega_A^\\bullet)_\\mathfrak p)", "$$", "has a unique nonzero cohomology module, say in degree $i_0$, by", "Lemma \\ref{lemma-local-CM}.", "Since $R\\Hom_A(M, \\omega_A^\\bullet)$", "has only a finite number of nonzero cohomology modules $H^i$", "and since each of these is a finite $A$-module, we can", "find an $f \\in A$, $f \\not \\in \\mathfrak p$ such that", "$(H^i)_f = 0$ for $i \\not = i_0$. Then", "$R\\Hom_A(M, \\omega_A^\\bullet)_f$ has a unique nonzero cohomology", "module and reversing the arguments just given we find", "that $D(f) \\subset U$." ], "refs": [ "dualizing-lemma-local-CM" ], "ref_ids": [ 2862 ] } ], "ref_ids": [] }, { "id": 2878, "type": "theorem", "label": "dualizing-lemma-CM", "categories": [ "dualizing" ], "title": "dualizing-lemma-CM", "contents": [ "Let $A$ be a Noetherian ring. If $A$ has a dualizing complex", "$\\omega_A^\\bullet$, then", "$\\{\\mathfrak p \\in \\Spec(A) \\mid A_\\mathfrak p\\text{ is Cohen-Macaulay}\\}$", "is a dense open subset of $\\Spec(A)$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of Lemma \\ref{lemma-CM-open} and the definitions." ], "refs": [ "dualizing-lemma-CM-open" ], "ref_ids": [ 2877 ] } ], "ref_ids": [] }, { "id": 2879, "type": "theorem", "label": "dualizing-lemma-gorenstein-CM", "categories": [ "dualizing" ], "title": "dualizing-lemma-gorenstein-CM", "contents": [ "A Gorenstein ring is Cohen-Macaulay." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-apply-CM}." ], "refs": [ "dualizing-lemma-apply-CM" ], "ref_ids": [ 2875 ] } ], "ref_ids": [] }, { "id": 2880, "type": "theorem", "label": "dualizing-lemma-regular-gorenstein", "categories": [ "dualizing" ], "title": "dualizing-lemma-regular-gorenstein", "contents": [ "A regular local ring is Gorenstein.", "A regular ring is Gorenstein." ], "refs": [], "proofs": [ { "contents": [ "Let $A$ be a regular ring of finite dimension $d$. Then $A$ has finite", "global dimension $d$, see", "Algebra, Lemma \\ref{algebra-lemma-finite-gl-dim-finite-dim-regular}.", "Hence $\\Ext^{d + 1}_A(M, A) = 0$ for all $A$-modules $M$, see", "Algebra, Lemma \\ref{algebra-lemma-projective-dimension-ext}.", "Thus $A$ has finite injective dimension as an $A$-module by", "More on Algebra, Lemma \\ref{more-algebra-lemma-injective-amplitude}.", "It follows that $A[0]$ is a dualizing complex, hence $A$ is", "Gorenstein by the remark following the definition." ], "refs": [ "algebra-lemma-finite-gl-dim-finite-dim-regular", "algebra-lemma-projective-dimension-ext", "more-algebra-lemma-injective-amplitude" ], "ref_ids": [ 980, 971, 10188 ] } ], "ref_ids": [] }, { "id": 2881, "type": "theorem", "label": "dualizing-lemma-gorenstein", "categories": [ "dualizing" ], "title": "dualizing-lemma-gorenstein", "contents": [ "Let $A$ be a Noetherian ring.", "\\begin{enumerate}", "\\item If $A$ has a dualizing complex $\\omega_A^\\bullet$, then", "\\begin{enumerate}", "\\item $A$ is Gorenstein $\\Leftrightarrow$ $\\omega_A^\\bullet$ is an invertible", "object of $D(A)$,", "\\item $A_\\mathfrak p$ is Gorenstein $\\Leftrightarrow$", "$(\\omega_A^\\bullet)_\\mathfrak p$ is an invertible object of", "$D(A_\\mathfrak p)$,", "\\item $\\{\\mathfrak p \\in \\Spec(A) \\mid A_\\mathfrak p\\text{ is Gorenstein}\\}$", "is an open subset.", "\\end{enumerate}", "\\item If $A$ is Gorenstein, then $A$ has a dualizing complex if and", "only if $A[0]$ is a dualizing complex.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For invertible objects of $D(A)$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-invertible-derived}", "and the discussion in Section \\ref{section-dualizing}.", "\\medskip\\noindent", "By Lemma \\ref{lemma-dualizing-localize} for every", "$\\mathfrak p$ the complex $(\\omega_A^\\bullet)_\\mathfrak p$ is a", "dualizing complex over $A_\\mathfrak p$. By definition and uniqueness", "of dualizing complexes (Lemma \\ref{lemma-dualizing-unique})", "we see that (1)(b) holds.", "\\medskip\\noindent", "To see (1)(c) assume that $A_\\mathfrak p$ is Gorenstein.", "Let $n_x$ be the unique integer such that", "$H^{n_{x}}((\\omega_A^\\bullet)_\\mathfrak p)$", "is nonzero and isomorphic to $A_\\mathfrak p$.", "Since $\\omega_A^\\bullet$ is in $D^b_{\\textit{Coh}}(A)$", "there are finitely many nonzero finite $A$-modules", "$H^i(\\omega_A^\\bullet)$. Thus there exists some", "$f \\in A$, $f \\not \\in \\mathfrak p$", "such that only $H^{n_x}((\\omega_A^\\bullet)_f)$", "is nonzero and generated by $1$ element over $A_f$.", "Since dualizing complexes are faithful (by definition)", "we conclude that $A_f \\cong H^{n_x}((\\omega_A^\\bullet)_f)$.", "In this way we see that $A_\\mathfrak q$ is Gorenstein", "for every $\\mathfrak q \\in D(f)$. This proves that the set", "in (1)(c) is open.", "\\medskip\\noindent", "Proof of (1)(a). The implication $\\Leftarrow$ follows from (1)(b).", "The implication $\\Rightarrow$ follows from the discussion", "in the previous paragraph, where we showed that if $A_\\mathfrak p$", "is Gorenstein, then for some $f \\in A$, $f \\not \\in \\mathfrak p$", "the complex $(\\omega_A^\\bullet)_f$ has only one nonzero cohomology module", "which is invertible.", "\\medskip\\noindent", "If $A[0]$ is a dualizing complex then $A$ is Gorenstein by", "part (1). Conversely, we see that part (1) shows that", "$\\omega_A^\\bullet$ is locally isomorphic to a shift of $A$.", "Since being a dualizing complex is local", "(Lemma \\ref{lemma-dualizing-glue})", "the result is clear." ], "refs": [ "more-algebra-lemma-invertible-derived", "dualizing-lemma-dualizing-localize", "dualizing-lemma-dualizing-unique", "dualizing-lemma-dualizing-glue" ], "ref_ids": [ 10575, 2851, 2850, 2852 ] } ], "ref_ids": [] }, { "id": 2882, "type": "theorem", "label": "dualizing-lemma-gorenstein-ext", "categories": [ "dualizing" ], "title": "dualizing-lemma-gorenstein-ext", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Then $A$ is Gorenstein if and only if $\\Ext^i_A(\\kappa, A)$", "is zero for $i \\gg 0$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $A[0]$ is a dualizing complex for $A$ if and only", "if $A$ has finite injective dimension as an $A$-module", "(follows immediately from Definition \\ref{definition-dualizing}).", "Thus the lemma follows from More on Algebra, Lemma", "\\ref{more-algebra-lemma-finite-injective-dimension-Noetherian-local}." ], "refs": [ "dualizing-definition-dualizing", "more-algebra-lemma-finite-injective-dimension-Noetherian-local" ], "ref_ids": [ 2931, 10191 ] } ], "ref_ids": [] }, { "id": 2883, "type": "theorem", "label": "dualizing-lemma-gorenstein-divide-by-nonzerodivisor", "categories": [ "dualizing" ], "title": "dualizing-lemma-gorenstein-divide-by-nonzerodivisor", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$", "be a Noetherian local ring. Let $f \\in \\mathfrak m$ be a", "nonzerodivisor. Set $B = A/(f)$. Then $A$ is Gorenstein if and", "only if $B$ is Gorenstein." ], "refs": [], "proofs": [ { "contents": [ "If $A$ is Gorenstein, then $B$ is Gorenstein by", "Lemma \\ref{lemma-divide-by-nonzerodivisor}.", "Conversely, suppose that $B$ is Gorenstein. Then", "$\\Ext^i_B(\\kappa, B)$ is zero for $i \\gg 0$", "(Lemma \\ref{lemma-gorenstein-ext}).", "Recall that $R\\Hom(B, -) : D(A) \\to D(B)$ is a right adjoint", "to restriction (Lemma \\ref{lemma-right-adjoint}).", "Hence", "$$", "R\\Hom_A(\\kappa, A) = R\\Hom_B(\\kappa, R\\Hom(B, A)) =", "R\\Hom_B(\\kappa, B[1])", "$$", "The final equality by direct computation or by", "Lemma \\ref{lemma-compute-for-effective-Cartier-algebraic}.", "Thus we see that $\\Ext^i_A(\\kappa, A)$ is zero for", "$i \\gg 0$ and $A$ is Gorenstein (Lemma \\ref{lemma-gorenstein-ext})." ], "refs": [ "dualizing-lemma-divide-by-nonzerodivisor", "dualizing-lemma-gorenstein-ext", "dualizing-lemma-right-adjoint", "dualizing-lemma-compute-for-effective-Cartier-algebraic", "dualizing-lemma-gorenstein-ext" ], "ref_ids": [ 2865, 2882, 2836, 2843, 2882 ] } ], "ref_ids": [] }, { "id": 2884, "type": "theorem", "label": "dualizing-lemma-gorenstein-lci", "categories": [ "dualizing" ], "title": "dualizing-lemma-gorenstein-lci", "contents": [ "If $A \\to B$ is a local complete intersection homomorphism of rings and", "$A$ is a Noetherian Gorenstein ring, then $B$ is a Gorenstein ring." ], "refs": [], "proofs": [ { "contents": [ "By More on Algebra, Definition", "\\ref{more-algebra-definition-local-complete-intersection}", "we can write $B = A[x_1, \\ldots, x_n]/I$", "where $I$ is a Koszul-regular ideal. Observe that a polynomial", "ring over a Gorenstein ring $A$ is Gorenstein: reduce to", "$A$ local and then use Lemmas \\ref{lemma-dualizing-polynomial-ring} and", "\\ref{lemma-gorenstein}.", "A Koszul-regular ideal is by definition locally generated", "by a Koszul-regular sequence, see More on Algebra, Section", "\\ref{more-algebra-section-ideals}.", "Looking at local rings of $A[x_1, \\ldots, x_n]$", "we see it suffices to show: if $R$ is a Noetherian local", "Gorenstein ring and $f_1, \\ldots, f_c \\in \\mathfrak m_R$", "is a Koszul regular sequence, then $R/(f_1, \\ldots, f_c)$ is Gorenstein.", "This follows from", "Lemma \\ref{lemma-gorenstein-divide-by-nonzerodivisor} and", "the fact that a Koszul regular sequence in $R$ is just a", "regular sequence (More on Algebra, Lemma", "\\ref{more-algebra-lemma-noetherian-finite-all-equivalent})." ], "refs": [ "more-algebra-definition-local-complete-intersection", "dualizing-lemma-dualizing-polynomial-ring", "dualizing-lemma-gorenstein", "dualizing-lemma-gorenstein-divide-by-nonzerodivisor", "more-algebra-lemma-noetherian-finite-all-equivalent" ], "ref_ids": [ 10609, 2855, 2881, 2883, 9978 ] } ], "ref_ids": [] }, { "id": 2885, "type": "theorem", "label": "dualizing-lemma-flat-under-gorenstein", "categories": [ "dualizing" ], "title": "dualizing-lemma-flat-under-gorenstein", "contents": [ "Let $A \\to B$ be a flat local homomorphism of Noetherian local rings.", "The following are equivalent", "\\begin{enumerate}", "\\item $B$ is Gorenstein, and", "\\item $A$ and $B/\\mathfrak m_A B$ are Gorenstein.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Below we will use without further mention that a local Gorenstein ring", "has finite injective dimension as well as Lemma \\ref{lemma-gorenstein-ext}.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-pseudo-coherence-and-base-change-ext}", "we have", "$$", "\\Ext^i_A(\\kappa_A, A) \\otimes_A B =", "\\Ext^i_B(B/\\mathfrak m_A B, B)", "$$", "for all $i$.", "\\medskip\\noindent", "Assume (2). Using that", "$R\\Hom(B/\\mathfrak m_A B, -) : D(B) \\to D(B/\\mathfrak m_A B)$ is a", "right adjoint to restriction (Lemma \\ref{lemma-right-adjoint}) we obtain", "$$", "R\\Hom_B(\\kappa_B, B) =", "R\\Hom_{B/\\mathfrak m_A B}(\\kappa_B, R\\Hom(B/\\mathfrak m_A B, B))", "$$", "The cohomology modules of $R\\Hom(B/\\mathfrak m_A B, B)$ are the modules", "$\\Ext^i_B(B/\\mathfrak m_A B, B) =", "\\Ext^i_A(\\kappa_A, A) \\otimes_A B$.", "Since $A$ is Gorenstein, we conclude only a finite number of these are nonzero", "and each is isomorphic to a direct sum of copies of $B/\\mathfrak m_A B$.", "Hence since $B/\\mathfrak m_A B$ is Gorenstein we conclude that", "$R\\Hom_B(B/\\mathfrak m_B, B)$ has only a finite number of nonzero", "cohomology modules. Hence $B$ is Gorenstein.", "\\medskip\\noindent", "Assume (1). Since $B$ has finite injective dimension,", "$\\Ext^i_B(B/\\mathfrak m_A B, B)$ is $0$ for $i \\gg 0$.", "Since $A \\to B$ is faithfully flat", "we conclude that $\\Ext^i_A(\\kappa_A, A)$ is $0$", "for $i \\gg 0$. We conclude that $A$ is Gorenstein. This implies that", "$\\Ext^i_A(\\kappa_A, A)$ is nonzero for exactly one $i$,", "namely for $i = \\dim(A)$, and", "$\\Ext^{\\dim(A)}_A(\\kappa_A, A) \\cong \\kappa_A$", "(see Lemmas \\ref{lemma-normalized-finite}, \\ref{lemma-apply-CM}, and", "\\ref{lemma-gorenstein-CM}).", "Thus we see that", "$\\Ext^i_B(B/\\mathfrak m_A B, B)$ is zero except for one $i$,", "namely $i = \\dim(A)$ and", "$\\Ext^{\\dim(A)}_B(B/\\mathfrak m_A B, B) \\cong B/\\mathfrak m_A B$.", "Thus $B/\\mathfrak m_A B$ is Gorenstein by", "Lemma \\ref{lemma-normalized-finite}." ], "refs": [ "dualizing-lemma-gorenstein-ext", "more-algebra-lemma-pseudo-coherence-and-base-change-ext", "dualizing-lemma-right-adjoint", "dualizing-lemma-normalized-finite", "dualizing-lemma-apply-CM", "dualizing-lemma-gorenstein-CM", "dualizing-lemma-normalized-finite" ], "ref_ids": [ 2882, 10165, 2836, 2857, 2875, 2879, 2857 ] } ], "ref_ids": [] }, { "id": 2886, "type": "theorem", "label": "dualizing-lemma-tor-injective-hull", "categories": [ "dualizing" ], "title": "dualizing-lemma-tor-injective-hull", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local Gorenstein ring", "of dimension $d$. Let $E$ be the injective hull of $\\kappa$. Then", "$\\text{Tor}_i^A(E, \\kappa)$ is zero for $i \\not = d$", "and $\\text{Tor}_d^A(E, \\kappa) = \\kappa$." ], "refs": [], "proofs": [ { "contents": [ "Since $A$ is Gorenstein $\\omega_A^\\bullet = A[d]$ is a", "normalized dualizing complex for $A$.", "Also $E$ is the only nonzero cohomology module of", "$R\\Gamma_\\mathfrak m(\\omega_A^\\bullet)$ sitting in degree $0$, see", "Lemma \\ref{lemma-local-cohomology-of-dualizing}.", "By Lemma \\ref{lemma-torsion-tensor-product} we have", "$$", "E \\otimes_A^\\mathbf{L} \\kappa =", "R\\Gamma_\\mathfrak m(\\omega_A^\\bullet) \\otimes_A^\\mathbf{L} \\kappa =", "R\\Gamma_\\mathfrak m(\\omega_A^\\bullet \\otimes_A^\\mathbf{L} \\kappa) =", "R\\Gamma_\\mathfrak m(\\kappa[d]) = \\kappa[d]", "$$", "and the lemma follows." ], "refs": [ "dualizing-lemma-local-cohomology-of-dualizing", "dualizing-lemma-torsion-tensor-product" ], "ref_ids": [ 2872, 2819 ] } ], "ref_ids": [] }, { "id": 2887, "type": "theorem", "label": "dualizing-lemma-flat-unramified", "categories": [ "dualizing" ], "title": "dualizing-lemma-flat-unramified", "contents": [ "Let $A \\to B$ be a local homomorphism of Noetherian local rings.", "Let $\\omega_A^\\bullet$ be a normalized dualizing complex.", "If $A \\to B$ is flat and $\\mathfrak m_A B = \\mathfrak m_B$,", "then $\\omega_A^\\bullet \\otimes_A B$ is a normalized dualizing", "complex for $B$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $\\omega_A^\\bullet \\otimes_A B$ is in $D^b_{\\textit{Coh}}(B)$.", "Let $\\kappa_A$ and $\\kappa_B$ be the residue fields of $A$ and $B$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom}", "we see that", "$$", "R\\Hom_B(\\kappa_B, \\omega_A^\\bullet \\otimes_A B) =", "R\\Hom_A(\\kappa_A, \\omega_A^\\bullet) \\otimes_A B =", "\\kappa_A[0] \\otimes_A B = \\kappa_B[0]", "$$", "Thus $\\omega_A^\\bullet \\otimes_A B$ has finite injective dimension by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-finite-injective-dimension-Noetherian-local}.", "Finally, we can use the same arguments to see that", "$$", "R\\Hom_B(\\omega_A^\\bullet \\otimes_A B, \\omega_A^\\bullet \\otimes_A B) =", "R\\Hom_A(\\omega_A^\\bullet, \\omega_A^\\bullet) \\otimes_A B = A \\otimes_A B = B", "$$", "as desired." ], "refs": [ "more-algebra-lemma-base-change-RHom", "more-algebra-lemma-finite-injective-dimension-Noetherian-local" ], "ref_ids": [ 10418, 10191 ] } ], "ref_ids": [] }, { "id": 2888, "type": "theorem", "label": "dualizing-lemma-flat-iso-mod-I", "categories": [ "dualizing" ], "title": "dualizing-lemma-flat-iso-mod-I", "contents": [ "Let $A \\to B$ be a flat map of Noetherian rings. Let", "$I \\subset A$ be an ideal such that $A/I = B/IB$ and", "such that $IB$ is contained in the Jacobson radical of $B$.", "Let $\\omega_A^\\bullet$ be a dualizing complex.", "Then $\\omega_A^\\bullet \\otimes_A B$ is a dualizing", "complex for $B$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $\\omega_A^\\bullet \\otimes_A B$ is in $D^b_{\\textit{Coh}}(B)$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom}", "we see that", "$$", "R\\Hom_B(K \\otimes_A B, \\omega_A^\\bullet \\otimes_A B) =", "R\\Hom_A(K, \\omega_A^\\bullet) \\otimes_A B", "$$", "for any $K \\in D^b_{\\textit{Coh}}(A)$. For any ideal", "$IB \\subset J \\subset B$ there is a unique ideal $I \\subset J' \\subset A$", "such that $A/J' \\otimes_A B = B/J$. Thus $\\omega_A^\\bullet \\otimes_A B$", "has finite injective dimension by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-finite-injective-dimension-Noetherian-radical}.", "Finally, we also have", "$$", "R\\Hom_B(\\omega_A^\\bullet \\otimes_A B, \\omega_A^\\bullet \\otimes_A B) =", "R\\Hom_A(\\omega_A^\\bullet, \\omega_A^\\bullet) \\otimes_A B = A \\otimes_A B = B", "$$", "as desired." ], "refs": [ "more-algebra-lemma-base-change-RHom", "more-algebra-lemma-finite-injective-dimension-Noetherian-radical" ], "ref_ids": [ 10418, 10190 ] } ], "ref_ids": [] }, { "id": 2889, "type": "theorem", "label": "dualizing-lemma-completion-henselization-dualizing", "categories": [ "dualizing" ], "title": "dualizing-lemma-completion-henselization-dualizing", "contents": [ "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.", "Let $\\omega_A^\\bullet$ be a dualizing complex.", "\\begin{enumerate}", "\\item $\\omega_A^\\bullet \\otimes_A A^h$ is a dualizing complex on the", "henselization $(A^h, I^h)$ of the pair $(A, I)$,", "\\item $\\omega_A^\\bullet \\otimes_A A^\\wedge$ is a dualizing complex on", "the $I$-adic completion $A^\\wedge$, and", "\\item if $A$ is local, then $\\omega_A^\\bullet \\otimes_A A^h$,", "resp.\\ $\\omega_A^\\bullet \\otimes_A A^{sh}$ is a dualzing complex", "on the henselization, resp.\\ strict henselization of $A$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemmas \\ref{lemma-flat-unramified} and", "\\ref{lemma-flat-iso-mod-I}.", "See More on Algebra, Sections \\ref{more-algebra-section-henselian-pairs},", "\\ref{more-algebra-section-permanence-completion}, and", "\\ref{more-algebra-section-permanence-henselization} and", "Algebra, Sections \\ref{algebra-section-completion} and", "\\ref{algebra-section-completion-noetherian}", "for information on completions and henselizations." ], "refs": [ "dualizing-lemma-flat-unramified", "dualizing-lemma-flat-iso-mod-I" ], "ref_ids": [ 2887, 2888 ] } ], "ref_ids": [] }, { "id": 2890, "type": "theorem", "label": "dualizing-lemma-ubiquity-dualizing", "categories": [ "dualizing" ], "title": "dualizing-lemma-ubiquity-dualizing", "contents": [ "The following types of rings have a dualizing complex:", "\\begin{enumerate}", "\\item fields,", "\\item Noetherian complete local rings,", "\\item $\\mathbf{Z}$,", "\\item Dedekind domains,", "\\item any ring which is obtained from one of the rings above by", "taking an algebra essentially of finite type, or by taking an", "ideal-adic completion, or by taking a henselization, ", "or by taking a strict henselization.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (5) follows from Proposition", "\\ref{proposition-dualizing-essentially-finite-type}", "and Lemma \\ref{lemma-completion-henselization-dualizing}.", "By Lemma \\ref{lemma-regular-gorenstein} a regular local ring has a", "dualizing complex.", "A complete Noetherian local ring is the quotient of a regular", "local ring by the Cohen structure theorem", "(Algebra, Theorem \\ref{algebra-theorem-cohen-structure-theorem}).", "Let $A$ be a Dedekind domain. Then every ideal $I$ is a finite", "projective $A$-module (follows from", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}", "and the fact that the local rings of $A$ are discrete valuation ring", "and hence PIDs). Thus every $A$-module has finite injective dimension", "at most $1$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-injective-amplitude}.", "It follows easily that $A[0]$ is a dualizing complex." ], "refs": [ "dualizing-proposition-dualizing-essentially-finite-type", "dualizing-lemma-completion-henselization-dualizing", "dualizing-lemma-regular-gorenstein", "algebra-theorem-cohen-structure-theorem", "algebra-lemma-finite-projective", "more-algebra-lemma-injective-amplitude" ], "ref_ids": [ 2926, 2889, 2880, 327, 795, 10188 ] } ], "ref_ids": [] }, { "id": 2891, "type": "theorem", "label": "dualizing-lemma-formal-fibres-gorenstein", "categories": [ "dualizing" ], "title": "dualizing-lemma-formal-fibres-gorenstein", "contents": [ "Properties (A), (B), (C), (D), and (E) of", "More on Algebra, Section \\ref{more-algebra-section-properties-formal-fibres}", "hold for $P(k \\to R) =$``$R$ is a Gorenstein ring''." ], "refs": [], "proofs": [ { "contents": [ "Since we already know the result holds for Cohen-Macaulay instead", "of Gorenstein, we may in each step assume the ring we have is", "Cohen-Macaulay. This is not particularly helpful for the proof, but", "psychologically may be useful.", "\\medskip\\noindent", "Part (A). Let $k \\subset K$ be a finitely generated field extension.", "Let $R$ be a Gorenstein $k$-algebra.", "We can find a global complete intersection", "$A = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "over $k$ such that $K$ is isomorphic to the fraction field of $A$, see", "Algebra, Lemma \\ref{algebra-lemma-colimit-syntomic}.", "Then $R \\to R \\otimes_k A$ is a relative global complete intersection.", "Hence $R \\otimes_k A$ is Gorenstein by Lemma \\ref{lemma-gorenstein-lci}.", "Thus $R \\otimes_k K$ is too as a localization.", "\\medskip\\noindent", "Proof of (B). This is clear because a ring is Gorenstein", "if and only if all of its local rings are Gorenstein.", "\\medskip\\noindent", "Part (C). Let $A \\to B \\to C$ be flat maps of Noetherian rings.", "Assume the fibres of $A \\to B$ are Gorenstein and $B \\to C$ is regular.", "We have to show the fibres of $A \\to C$ are Gorenstein.", "Clearly, we may assume $A = k$ is a field. Then we may assume that", "$B \\to C$ is a regular local homomorphism of Noetherian local rings.", "Then $B$ is Gorenstein and $C/\\mathfrak m_B C$ is regular, in", "particular Gorenstein (Lemma \\ref{lemma-regular-gorenstein}).", "Then $C$ is Gorenstein by", "Lemma \\ref{lemma-flat-under-gorenstein}.", "\\medskip\\noindent", "Part (D). This follows from Lemma \\ref{lemma-flat-under-gorenstein}.", "Part (E) is immediate as the condition does not refer to the ground field." ], "refs": [ "algebra-lemma-colimit-syntomic", "dualizing-lemma-gorenstein-lci", "dualizing-lemma-regular-gorenstein", "dualizing-lemma-flat-under-gorenstein", "dualizing-lemma-flat-under-gorenstein" ], "ref_ids": [ 1323, 2884, 2880, 2885, 2885 ] } ], "ref_ids": [] }, { "id": 2892, "type": "theorem", "label": "dualizing-lemma-dualizing-gorenstein-formal-fibres", "categories": [ "dualizing" ], "title": "dualizing-lemma-dualizing-gorenstein-formal-fibres", "contents": [ "Let $A$ be a Noetherian local ring. If $A$ has a dualizing complex,", "then the formal fibres of $A$ are Gorenstein." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p$ be a prime of $A$. The formal fibre of $A$ at $\\mathfrak p$", "is isomorphic to the formal fibre of $A/\\mathfrak p$ at $(0)$. The quotient", "$A/\\mathfrak p$ has a dualizing complex", "(Lemma \\ref{lemma-dualizing-quotient}).", "Thus it suffices to check the statement", "when $A$ is a local domain and $\\mathfrak p = (0)$.", "Let $\\omega_A^\\bullet$ be a dualizing complex for $A$. Then", "$\\omega_A^\\bullet \\otimes_A A^\\wedge$ is a dualizing complex", "for the completion $A^\\wedge$", "(Lemma \\ref{lemma-flat-unramified}).", "Then $\\omega_A^\\bullet \\otimes_A K$ is a dualizing", "complex for the fraction field $K$ of $A$", "(Lemma \\ref{lemma-dualizing-localize}).", "Hence $\\omega_A^\\bullet \\otimes_A K$", "is isomorphic ot $K[n]$ for some $n \\in \\mathbf{Z}$.", "Similarly, we conclude a dualizing complex for the formal fibre", "$A^\\wedge \\otimes_A K$ is", "$$", "\\omega_A^\\bullet \\otimes_A A^\\wedge \\otimes_{A^\\wedge} (A^\\wedge \\otimes_A K) =", "(\\omega_A^\\bullet \\otimes_A K) \\otimes_K (A^\\wedge \\otimes_A K) \\cong", "(A^\\wedge \\otimes_A K)[n]", "$$", "as desired." ], "refs": [ "dualizing-lemma-dualizing-quotient", "dualizing-lemma-flat-unramified", "dualizing-lemma-dualizing-localize" ], "ref_ids": [ 2854, 2887, 2851 ] } ], "ref_ids": [] }, { "id": 2893, "type": "theorem", "label": "dualizing-lemma-formal-fibres-lci", "categories": [ "dualizing" ], "title": "dualizing-lemma-formal-fibres-lci", "contents": [ "Properties (A), (B), (C), (D), and (E) of", "More on Algebra, Section \\ref{more-algebra-section-properties-formal-fibres}", "hold for $P(k \\to R) =$``$R$ is a local complete intersection''.", "See Divided Power Algebra, Definition \\ref{dpa-definition-lci}." ], "refs": [ "dpa-definition-lci" ], "proofs": [ { "contents": [ "Part (A). Let $k \\subset K$ be a finitely generated field extension.", "Let $R$ be a $k$-algebra which is a local complete intersection.", "We can find a global complete intersection", "$A = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "over $k$ such that $K$ is isomorphic to the fraction field of $A$, see", "Algebra, Lemma \\ref{algebra-lemma-colimit-syntomic}.", "Then $R \\to R \\otimes_k A$ is a relative global complete intersection.", "It follows that $R \\otimes_k A$ is a local complete intersection", "by Divided Power Algebra, Lemma \\ref{dpa-lemma-avramov}.", "\\medskip\\noindent", "Proof of (B). This is clear", "because a ring is a local complete intersection if and only if all of its", "local rings are complete intersections.", "\\medskip\\noindent", "Part (C). Let $A \\to B \\to C$ be flat maps of Noetherian rings.", "Assume the fibres of $A \\to B$ are local complete intersections", "and $B \\to C$ is regular. We have to show the fibres of $A \\to C$", "are local complete intersections. Clearly, we may assume $A = k$ is a field.", "Then we may assume that $B \\to C$ is a regular local homomorphism", "of Noetherian local rings. Then $B$ is a complete intersection and", "$C/\\mathfrak m_B C$ is regular, in particular a complete intersection", "(by definition). Then $C$ is a complete intersection by", "Divided Power Algebra, Lemma \\ref{dpa-lemma-avramov}.", "\\medskip\\noindent", "Part (D). This follows by the same arguments as in (C) from", "the other implication in", "Divided Power Algebra, Lemma \\ref{dpa-lemma-avramov}.", "Part (E) is immediate as the condition does not refer to the ground", "field." ], "refs": [ "algebra-lemma-colimit-syntomic", "dpa-lemma-avramov", "dpa-lemma-avramov", "dpa-lemma-avramov" ], "ref_ids": [ 1323, 1682, 1682, 1682 ] } ], "ref_ids": [ 1701 ] }, { "id": 2894, "type": "theorem", "label": "dualizing-lemma-well-defined", "categories": [ "dualizing" ], "title": "dualizing-lemma-well-defined", "contents": [ "Let $\\varphi : R \\to A$ be a finite type homomorphism of", "Noetherian rings. The functor $\\varphi^!$ is well defined", "up to isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $\\psi_1 : P_1 = R[x_1, \\ldots, x_n] \\to A$ and", "$\\psi_2 : P_2 = R[y_1, \\ldots, y_m] \\to A$ are two", "surjections from polynomial rings onto $A$. Then we get a", "commutative diagram", "$$", "\\xymatrix{", "R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]", "\\ar[d]^{x_i \\mapsto g_i} \\ar[rr]_-{y_j \\mapsto f_j} & &", "R[x_1, \\ldots, x_n] \\ar[d] \\\\", "R[y_1, \\ldots, y_m] \\ar[rr] & & A", "}", "$$", "where $f_j$ and $g_i$ are chosen such that $\\psi_1(f_j) = \\psi_2(y_j)$", "and $\\psi_2(g_i) = \\psi_1(x_i)$. By symmetry it suffices to prove", "the functors defined using $P \\to A$ and $P[y_1, \\ldots, y_m] \\to A$", "are isomorphic. By induction we may assume $m = 1$. This reduces", "us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Here $\\psi : P \\to A$ is given and $\\chi : P[y] \\to A$ induces", "$\\psi$ on $P$. Write $Q = P[y]$.", "Choose $g \\in P$ with $\\psi(g) = \\chi(y)$.", "Denote $\\pi : Q \\to P$ the $P$-algebra map", "with $\\pi(y) = g$. Then $\\chi = \\psi \\circ \\pi$ and hence", "$\\chi^! = \\psi^! \\circ \\pi^!$ as both are", "adjoint to the restriction functor $D(A) \\to D(Q)$ by the material", "in Section \\ref{section-trivial}. Thus", "$$", "\\chi^!\\left(K \\otimes_R^\\mathbf{L} Q\\right)[n + 1] =", "\\psi^!\\left(\\pi^!\\left(K \\otimes_R^\\mathbf{L} Q\\right)[1]\\right)[n]", "$$", "Hence it suffices to show that", "$\\pi^!(K \\otimes_R^\\mathbf{L} Q[1]) = K \\otimes_R^\\mathbf{L} P$", "Thus it suffices to show that the functor", "$\\pi^!(-) : D(Q) \\to D(P)$", "is isomorphic to $K \\mapsto K \\otimes_Q^\\mathbf{L} P[-1]$.", "This follows from Lemma \\ref{lemma-compute-for-effective-Cartier-algebraic}." ], "refs": [ "dualizing-lemma-compute-for-effective-Cartier-algebraic" ], "ref_ids": [ 2843 ] } ], "ref_ids": [] }, { "id": 2895, "type": "theorem", "label": "dualizing-lemma-shriek-boundedness", "categories": [ "dualizing" ], "title": "dualizing-lemma-shriek-boundedness", "contents": [ "Let $\\varphi : R \\to A$ be a finite type homomorphism of Noetherian rings.", "\\begin{enumerate}", "\\item $\\varphi^!$ maps $D^+(R)$ into $D^+(A)$ and", "$D^+_{\\textit{Coh}}(R)$ into $D^+_{\\textit{Coh}}(A)$.", "\\item if $\\varphi$ is perfect, then $\\varphi^!$ maps", "$D^-(R)$ into $D^-(A)$,", "$D^-_{\\textit{Coh}}(R)$ into $D^-_{\\textit{Coh}}(A)$, and", "$D^b_{\\textit{Coh}}(R)$ into $D^b_{\\textit{Coh}}(A)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a factorization $R \\to P \\to A$ as in the definition of $\\varphi^!$.", "The functor $- \\otimes_R^\\mathbf{L} : D(R) \\to D(P)$ preserves", "the subcategories", "$D^+, D^+_{\\textit{Coh}}, D^-, D^-_{\\textit{Coh}}, D^b_{\\textit{Coh}}$.", "The functor $R\\Hom(A, -) : D(P) \\to D(A)$", "preserves $D^+$ and $D^+_{\\textit{Coh}}$ by", "Lemma \\ref{lemma-exact-support-coherent}.", "If $R \\to A$ is perfect, then $A$ is perfect as a $P$-module, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-perfect-ring-map}.", "Recall that the restriction of $R\\Hom(A, K)$ to $D(P)$ is", "$R\\Hom_P(A, K)$. By More on Algebra, Lemma", "\\ref{more-algebra-lemma-dual-perfect-complex}", "we have $R\\Hom_P(A, K) = E \\otimes_P^\\mathbf{L} K$ for", "some perfect $E \\in D(P)$. Since we can represent $E$ by", "a finite complex of finite projective $P$-modules", "it is clear that $R\\Hom_P(A, K)$ is in", "$D^-(P), D^-_{\\textit{Coh}}(P), D^b_{\\textit{Coh}}(P)$", "as soon as $K$ is. Since the restriction functor", "$D(A) \\to D(P)$ reflects these subcategories, the", "proof is complete." ], "refs": [ "dualizing-lemma-exact-support-coherent", "more-algebra-lemma-perfect-ring-map", "more-algebra-lemma-dual-perfect-complex" ], "ref_ids": [ 2839, 10281, 10224 ] } ], "ref_ids": [] }, { "id": 2896, "type": "theorem", "label": "dualizing-lemma-shriek-dualizing-algebraic", "categories": [ "dualizing" ], "title": "dualizing-lemma-shriek-dualizing-algebraic", "contents": [ "Let $\\varphi$ be a finite type homomorphism of Noetherian rings.", "If $\\omega_R^\\bullet$ is a dualizing complex for $R$, then", "$\\varphi^!(\\omega_R^\\bullet)$ is a dualizing complex for $A$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemmas", "\\ref{lemma-dualizing-polynomial-ring} and", "\\ref{lemma-dualizing-quotient}," ], "refs": [ "dualizing-lemma-dualizing-polynomial-ring", "dualizing-lemma-dualizing-quotient" ], "ref_ids": [ 2855, 2854 ] } ], "ref_ids": [] }, { "id": 2897, "type": "theorem", "label": "dualizing-lemma-flat-bc", "categories": [ "dualizing" ], "title": "dualizing-lemma-flat-bc", "contents": [ "Let $R \\to R'$ be a flat homomorphism of Noetherian rings.", "Let $\\varphi : R \\to A$ be a finite type ring map.", "Let $\\varphi' : R' \\to A' = A \\otimes_R R'$ be the map induced by $\\varphi$.", "Then we have a functorial maps", "$$", "\\varphi^!(K) \\otimes_A^\\mathbf{L} A' \\longrightarrow", "(\\varphi')^!(K \\otimes_R^\\mathbf{L} R')", "$$", "for $K$ in $D(R)$ which are isomorphisms for $K \\in D^+(R)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a factorization $R \\to P \\to A$ where $P$ is a polynomial ring over $R$.", "This gives a corresponding factorization $R' \\to P' \\to A'$ by base change.", "Since we have $(K \\otimes_R^\\mathbf{L} P) \\otimes_P^\\mathbf{L} P' =", "(K \\otimes_R^\\mathbf{L} R') \\otimes_{R'}^\\mathbf{L} P'$", "by More on Algebra, Lemma \\ref{more-algebra-lemma-double-base-change}", "it suffices to construct maps", "$$", "R\\Hom(A, K \\otimes_R^\\mathbf{L} P[n]) \\otimes_A^\\mathbf{L} A'", "\\longrightarrow", "R\\Hom(A', (K \\otimes_R^\\mathbf{L} P[n]) \\otimes_P^\\mathbf{L} P')", "$$", "functorial in $K$. For this we use the map (\\ref{equation-base-change})", "constructed in Section \\ref{section-base-change-trivial-duality}", "for $P, A, P', A'$.", "The map is an isomorphism for $K \\in D^+(R)$ by", "Lemma \\ref{lemma-flat-bc-surjection}." ], "refs": [ "more-algebra-lemma-double-base-change", "dualizing-lemma-flat-bc-surjection" ], "ref_ids": [ 10138, 2845 ] } ], "ref_ids": [] }, { "id": 2898, "type": "theorem", "label": "dualizing-lemma-bc", "categories": [ "dualizing" ], "title": "dualizing-lemma-bc", "contents": [ "Let $R \\to R'$ be a homomorphism of Noetherian rings.", "Let $\\varphi : R \\to A$ be a perfect ring map", "(More on Algebra, Definition", "\\ref{more-algebra-definition-pseudo-coherent-perfect})", "such that $R'$ and $A$ are tor independent over $R$.", "Let $\\varphi' : R' \\to A' = A \\otimes_R R'$ be the map induced by $\\varphi$.", "Then we have a functorial isomorphism", "$$", "\\varphi^!(K) \\otimes_A^\\mathbf{L} A' =", "(\\varphi')^!(K \\otimes_R^\\mathbf{L} R')", "$$", "for $K$ in $D(R)$." ], "refs": [ "more-algebra-definition-pseudo-coherent-perfect" ], "proofs": [ { "contents": [ "We may choose a factorization $R \\to P \\to A$ where $P$", "is a polynomial ring over $R$ such that $A$ is a perfect $P$-module, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-perfect-ring-map}.", "This gives a corresponding factorization $R' \\to P' \\to A'$ by base change.", "Since we have $(K \\otimes_R^\\mathbf{L} P) \\otimes_P^\\mathbf{L} P' =", "(K \\otimes_R^\\mathbf{L} R') \\otimes_{R'}^\\mathbf{L} P'$", "by More on Algebra, Lemma \\ref{more-algebra-lemma-double-base-change}", "it suffices to construct maps", "$$", "R\\Hom(A, K \\otimes_R^\\mathbf{L} P[n]) \\otimes_A^\\mathbf{L} A'", "\\longrightarrow", "R\\Hom(A', (K \\otimes_R^\\mathbf{L} P[n]) \\otimes_P^\\mathbf{L} P')", "$$", "functorial in $K$. We have", "$$", "A \\otimes_P^\\mathbf{L} P' = A \\otimes_R^\\mathbf{L} R' = A'", "$$", "The first equality by", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-comparison}", "applied to $R, R', P, P'$. The second equality because", "$A$ and $R'$ are tor independent over $R$. Hence $A$ and $P'$ are", "tor independent over $P$ and we can use the map (\\ref{equation-base-change})", "constructed in Section \\ref{section-base-change-trivial-duality} for", "$P, A, P', A'$", "get the desired arrow. By Lemma \\ref{lemma-bc-surjection}", "to finish the proof it suffices to prove that $A$ is a perfect $P$-module", "which we saw above." ], "refs": [ "more-algebra-lemma-perfect-ring-map", "more-algebra-lemma-double-base-change", "more-algebra-lemma-base-change-comparison", "dualizing-lemma-bc-surjection" ], "ref_ids": [ 10281, 10138, 10139, 2846 ] } ], "ref_ids": [ 10631 ] }, { "id": 2899, "type": "theorem", "label": "dualizing-lemma-bc-flat", "categories": [ "dualizing" ], "title": "dualizing-lemma-bc-flat", "contents": [ "Let $R \\to R'$ be a homomorphism of Noetherian rings.", "Let $\\varphi : R \\to A$ be flat of finite type.", "Let $\\varphi' : R' \\to A' = A \\otimes_R R'$ be the map induced by $\\varphi$.", "Then we have a functorial isomorphism", "$$", "\\varphi^!(K) \\otimes_A^\\mathbf{L} A' =", "(\\varphi')^!(K \\otimes_R^\\mathbf{L} R')", "$$", "for $K$ in $D(R)$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-bc} by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-flat-finite-presentation-perfect}." ], "refs": [ "dualizing-lemma-bc", "more-algebra-lemma-flat-finite-presentation-perfect" ], "ref_ids": [ 2898, 10283 ] } ], "ref_ids": [] }, { "id": 2900, "type": "theorem", "label": "dualizing-lemma-composition-shriek-algebraic", "categories": [ "dualizing" ], "title": "dualizing-lemma-composition-shriek-algebraic", "contents": [ "Let $A \\xrightarrow{a} B \\xrightarrow{b} C$ be finite type homomorphisms of", "Noetherian rings. Then there is a transformation of functors", "$b^! \\circ a^! \\to (b \\circ a)^!$ which is an isomorphism on $D^+(A)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a polynomial ring $P = A[x_1, \\ldots, x_n]$ over $A$", "and a surjection $P \\to B$. Choose elements $c_1, \\ldots, c_m \\in C$", "generating $C$ over $B$. Set $Q = P[y_1, \\ldots, y_m]$ and", "denote $Q' = Q \\otimes_P B = B[y_1, \\ldots, y_m]$.", "Let $\\chi : Q' \\to C$ be the surjection sending $y_j$ to $c_j$.", "Picture", "$$", "\\xymatrix{", "& Q \\ar[r]_{\\psi'} & Q' \\ar[r]_\\chi & C \\\\", "A \\ar[r] & P \\ar[r]^\\psi \\ar[u] & B \\ar[u]", "}", "$$", "By Lemma \\ref{lemma-flat-bc-surjection} for $M \\in D(P)$ we have an arrow", "$\\psi^!(M) \\otimes_B^\\mathbf{L} Q' \\to (\\psi')^!(M \\otimes_P^\\mathbf{L} Q)$", "which is an isomorphism whenever $M$ is bounded below. Also", "we have $\\chi^! \\circ (\\psi')^! = (\\chi \\circ \\psi')^!$ as both", "functors are adjoint to the restriction functor $D(C) \\to D(Q)$", "by Section \\ref{section-trivial}. Then we see", "\\begin{align*}", "b^!(a^!(K))", "& =", "\\chi^!(\\psi^!(K \\otimes_A^\\mathbf{L} P)[n] \\otimes_B^\\mathbf{L} Q)[m] \\\\", "& \\to", "\\chi^!((\\psi')^!(K \\otimes_A^\\mathbf{L} P \\otimes_P^\\mathbf{L} Q))[n + m] \\\\", "& =", "(\\chi \\circ \\psi')^!(K\\otimes_A^\\mathbf{L} Q)[n + m] \\\\", "& =", "(b \\circ a)^!(K)", "\\end{align*}", "where we have used in addition to the above", "More on Algebra, Lemma \\ref{more-algebra-lemma-double-base-change}." ], "refs": [ "dualizing-lemma-flat-bc-surjection", "more-algebra-lemma-double-base-change" ], "ref_ids": [ 2845, 10138 ] } ], "ref_ids": [] }, { "id": 2901, "type": "theorem", "label": "dualizing-lemma-upper-shriek-finite", "categories": [ "dualizing" ], "title": "dualizing-lemma-upper-shriek-finite", "contents": [ "Let $\\varphi : R \\to A$ be a finite map of Noetherian rings.", "Then $\\varphi^!$ is isomorphic to the functor", "$R\\Hom(A, -) : D(R) \\to D(A)$ from", "Section \\ref{section-trivial}." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $A$ is generated by $n > 1$ elements over $R$.", "Then can factor $R \\to A$ as a composition of two finite ring maps", "where in both steps the number of generators is $< n$.", "Since we have Lemma \\ref{lemma-composition-shriek-algebraic} and", "Lemma \\ref{lemma-composition-right-adjoints}", "we conclude that it suffices", "to prove the lemma when $A$ is generated by one element over $R$.", "Since $A$ is finite over $R$, it follows that $A$ is a quotient", "of $B = R[x]/(f)$ where $f$ is a monic polynomial in $x$", "(Algebra, Lemma \\ref{algebra-lemma-finite-is-integral}).", "Again using the lemmas on composition and the fact that we", "have agreement for surjections by definition, we conclude that", "it suffices to prove the lemma for $R \\to B = R[x]/(f)$.", "In this case, the functor $\\varphi^!$ is isomorphic to", "$K \\mapsto K \\otimes_R^\\mathbf{L} B$; you prove this by", "using Lemma \\ref{lemma-compute-for-effective-Cartier-algebraic}", "for the map $R[x] \\to B$ (note that the shift in the definition", "of $\\varphi^!$ and in the lemma add up to zero).", "For the functor $R\\Hom(B, -) : D(R) \\to D(B)$ we can use", "Lemma \\ref{lemma-RHom-is-tensor-special}", "to see that it suffices to show $\\Hom_R(B, R) \\cong B$", "as $B$-modules. Suppose that $f$ has degree $d$.", "Then an $R$-basis for $B$ is given by $1, x, \\ldots, x^{d - 1}$.", "Let $\\delta_i : B \\to R$, $i = 0, \\ldots, d - 1$", "be the $R$-linear map which picks off the coefficient", "of $x^i$ with respect to the given basis. Then", "$\\delta_0, \\ldots, \\delta_{d - 1}$ is a basis for $\\Hom_R(B, R)$.", "Finally, for $0 \\leq i \\leq d - 1$ a computation shows that", "$$", "x^i \\delta_{d - 1} =", "\\delta_{d - 1 - i} + b_1 \\delta_{d - i} + \\ldots + b_i \\delta_{d - 1}", "$$", "for some $c_1, \\ldots, c_d \\in R$\\footnote{If", "$f = x^d + a_1 x^{d - 1} + \\ldots + a_d$, then", "$c_1 = -a_1$, $c_2 = a_1^2 - a_2$, $c_3 = -a_1^3 + 2a_1a_2 -a_3$, etc.}.", "Hence $\\Hom_R(B, R)$ is a principal $B$-module with generator", "$\\delta_{d - 1}$. By looking", "at ranks we conclude that it is a rank $1$ free $B$-module." ], "refs": [ "dualizing-lemma-composition-shriek-algebraic", "dualizing-lemma-composition-right-adjoints", "algebra-lemma-finite-is-integral", "dualizing-lemma-compute-for-effective-Cartier-algebraic", "dualizing-lemma-RHom-is-tensor-special" ], "ref_ids": [ 2900, 2837, 482, 2843, 2842 ] } ], "ref_ids": [] }, { "id": 2902, "type": "theorem", "label": "dualizing-lemma-upper-shriek-localize", "categories": [ "dualizing" ], "title": "dualizing-lemma-upper-shriek-localize", "contents": [ "Let $R$ be a Noetherian ring and let $f \\in R$.", "If $\\varphi$ denotes the map $R \\to R_f$, then $\\varphi^!$", "is isomorphic to $- \\otimes_R^\\mathbf{L} R_f$.", "More generally, if $\\varphi : R \\to R'$ is a map such that", "$\\Spec(R') \\to \\Spec(R)$ is an open immersion, then", "$\\varphi^!$ is isomorphic to $- \\otimes_R^\\mathbf{L} R'$." ], "refs": [], "proofs": [ { "contents": [ "Choose the presentation $R \\to R[x] \\to R[x]/(fx - 1) = R_f$ and observe", "that $fx - 1$ is a nonzerodivisor in $R[x]$. Thus we can apply", "using Lemma \\ref{lemma-compute-for-effective-Cartier-algebraic}", "to compute the functor $\\varphi^!$. Details omitted;", "note that the shift in the definition", "of $\\varphi^!$ and in the lemma add up to zero.", "\\medskip\\noindent", "In the general case note that $R' \\otimes_R R' = R'$.", "Hence the result follows from the base change results", "above. Either Lemma \\ref{lemma-flat-bc} or", "Lemma \\ref{lemma-bc} will do." ], "refs": [ "dualizing-lemma-compute-for-effective-Cartier-algebraic", "dualizing-lemma-flat-bc", "dualizing-lemma-bc" ], "ref_ids": [ 2843, 2897, 2898 ] } ], "ref_ids": [] }, { "id": 2903, "type": "theorem", "label": "dualizing-lemma-upper-shriek-is-tensor-functor", "categories": [ "dualizing" ], "title": "dualizing-lemma-upper-shriek-is-tensor-functor", "contents": [ "Let $\\varphi : R \\to A$ be a perfect homomorphism of Noetherian rings", "(for example $\\varphi$ is flat of finite type).", "Then $\\varphi^!(K) = K \\otimes_R^\\mathbf{L} \\varphi^!(R)$", "for $K \\in D(R)$." ], "refs": [], "proofs": [ { "contents": [ "(The parenthetical statement follows from", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-flat-finite-presentation-perfect}.)", "We can choose a factorization $R \\to P \\to A$ where $P$ is a polynomial", "ring in $n$ variables over $R$ and then $A$ is a perfect $P$-module, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-perfect-ring-map}.", "Recall that $\\varphi^!(K) = R\\Hom(A, K \\otimes_R^\\mathbf{L} P[n])$.", "Thus the result follows from", "Lemma \\ref{lemma-RHom-is-tensor-special}", "and More on Algebra, Lemma \\ref{more-algebra-lemma-double-base-change}." ], "refs": [ "more-algebra-lemma-flat-finite-presentation-perfect", "more-algebra-lemma-perfect-ring-map", "dualizing-lemma-RHom-is-tensor-special", "more-algebra-lemma-double-base-change" ], "ref_ids": [ 10283, 10281, 2842, 10138 ] } ], "ref_ids": [] }, { "id": 2904, "type": "theorem", "label": "dualizing-lemma-relative-dualizing-if-have-omega", "categories": [ "dualizing" ], "title": "dualizing-lemma-relative-dualizing-if-have-omega", "contents": [ "Let $\\varphi : A \\to B$ be a finite type homomorphism of Noetherian rings.", "Let $\\omega_A^\\bullet$ be a dualizing complex for $A$. Set", "$\\omega_B^\\bullet = \\varphi^!(\\omega_A^\\bullet)$. Denote", "$D_A(K) = R\\Hom_A(K, \\omega_A^\\bullet)$ for $K \\in D_{\\textit{Coh}}(A)$", "and", "$D_B(L) = R\\Hom_B(L, \\omega_B^\\bullet)$ for $L \\in D_{\\textit{Coh}}(B)$.", "Then there is a functorial isomorphism", "$$", "\\varphi^!(K) = D_B(D_A(K) \\otimes_A^\\mathbf{L} B)", "$$", "for $K \\in D_{\\textit{Coh}}(A)$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\omega_B^\\bullet$ is a dualizing complex for $B$ by", "Lemma \\ref{lemma-shriek-dualizing-algebraic}.", "Let $A \\to B \\to C$ be finite type homomorphisms of Noetherian rings.", "If the lemma holds for $A \\to B$ and $B \\to C$, then the lemma holds for", "$A \\to C$. This follows from", "Lemma \\ref{lemma-composition-shriek-algebraic}", "and the fact that $D_B \\circ D_B \\cong \\text{id}$ by", "Lemma \\ref{lemma-dualizing}.", "Thus it suffices to prove the lemma in case $A \\to B$ is", "a surjection and in the case where $B$ is a", "polynomial ring over $A$.", "\\medskip\\noindent", "Assume $B = A[x_1, \\ldots, x_n]$. Since $D_A \\circ D_A \\cong \\text{id}$,", "it suffices to prove", "$D_B(K \\otimes_A B) \\cong D_A(K) \\otimes_A B[n]$ for $K$", "in $D_{\\textit{Coh}}(A)$.", "Choose a bounded complex $I^\\bullet$ of injectives representing", "$\\omega_A^\\bullet$. Choose a quasi-isomorphism", "$I^\\bullet \\otimes_A B \\to J^\\bullet$ where $J^\\bullet$", "is a bounded complex of $B$-modules. Given a complex", "$K^\\bullet$ of $A$-modules, consider the obvious", "map of complexes", "$$", "\\Hom^\\bullet(K^\\bullet, I^\\bullet) \\otimes_A B[n]", "\\longrightarrow", "\\Hom^\\bullet(K^\\bullet \\otimes_A B, J^\\bullet[n])", "$$", "The left hand side represents $D_A(K) \\otimes_A B[n]$ and the right hand", "side represents $D_B(K \\otimes_A B)$. Thus it suffices to prove this", "map is a quasi-isomorphism if the cohomology modules", "of $K^\\bullet$ are finite $A$-modules. Observe that the", "cohomology of the complex in degree $r$ (on either side)", "only depends on finitely many of the $K^i$. Thus we may", "replace $K^\\bullet$ by a truncation, i.e., we may assume", "$K^\\bullet$ represents an object of $D^-_{\\textit{Coh}}(A)$.", "Then $K^\\bullet$ is quasi-isomorphic to a bounded", "above complex of finite free $A$-modules.", "Therefore we may assume $K^\\bullet$ is a bounded", "above complex of finite free $A$-modules.", "In this case it is easy to that the", "displayed map is an isomorphism of complexes which finishes", "the proof in this case.", "\\medskip\\noindent", "Assume that $A \\to B$ is surjective. Denote $i_* : D(B) \\to D(A)$", "the restriction functor and recall that $\\varphi^!(-) = R\\Hom(A, -)$", "is a right adjoint to $i_*$ (Lemma \\ref{lemma-right-adjoint}).", "For $F \\in D(B)$ we have", "\\begin{align*}", "\\Hom_B(F, D_B(D_A(K) \\otimes_A^\\mathbf{L} B))", "& =", "\\Hom_B((D_A(K) \\otimes_A^\\mathbf{L} B) \\otimes_B^\\mathbf{L} F,", "\\omega_B^\\bullet) \\\\", "& =", "\\Hom_A(D_A(K) \\otimes_A^\\mathbf{L} i_*F, \\omega_A^\\bullet) \\\\", "& =", "\\Hom_A(i_*F, D_A(D_A(K))) \\\\", "& =", "\\Hom_A(i_*F, K) \\\\", "& =", "\\Hom_B(F, \\varphi^!(K))", "\\end{align*}", "The first equality follows from More on Algebra, Lemma", "\\ref{more-algebra-lemma-internal-hom} and the definition", "of $D_B$. The second equality by the adjointness mentioned", "above and the equality", "$i_*((D_A(K) \\otimes_A^\\mathbf{L} B) \\otimes_B^\\mathbf{L} F) =", "D_A(K) \\otimes_A^\\mathbf{L} i_*F$", "(More on Algebra, Lemma \\ref{more-algebra-lemma-derived-base-change}).", "The third equality follows from More on Algebra, Lemma", "\\ref{more-algebra-lemma-internal-hom}. The fourth because", "$D_A \\circ D_A = \\text{id}$. The final equality by adjointness again.", "Thus the result holds by the Yoneda lemma." ], "refs": [ "dualizing-lemma-shriek-dualizing-algebraic", "dualizing-lemma-composition-shriek-algebraic", "dualizing-lemma-dualizing", "dualizing-lemma-right-adjoint", "more-algebra-lemma-internal-hom", "more-algebra-lemma-derived-base-change", "more-algebra-lemma-internal-hom" ], "ref_ids": [ 2896, 2900, 2848, 2836, 10206, 10136, 10206 ] } ], "ref_ids": [] }, { "id": 2905, "type": "theorem", "label": "dualizing-lemma-base-change-relative-algebraic", "categories": [ "dualizing" ], "title": "dualizing-lemma-base-change-relative-algebraic", "contents": [ "Let $R \\to R'$ be a homomorphism of Noetherian rings.", "Let $R \\to A$ be of finite type. Set $A' = A \\otimes_R R'$. If", "\\begin{enumerate}", "\\item $R \\to R'$ is flat, or", "\\item $R \\to A$ is flat, or", "\\item $R \\to A$ is perfect", "and $R'$ and $A$ are tor independent over $R$,", "\\end{enumerate}", "then there is an isomorphism", "$\\omega_{A/R}^\\bullet \\otimes_A^\\mathbf{L} A' \\to \\omega^\\bullet_{A'/R'}$", "in $D(A')$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemmas \\ref{lemma-flat-bc}, \\ref{lemma-bc-flat}, and", "\\ref{lemma-bc} and the definitions." ], "refs": [ "dualizing-lemma-flat-bc", "dualizing-lemma-bc-flat", "dualizing-lemma-bc" ], "ref_ids": [ 2897, 2899, 2898 ] } ], "ref_ids": [] }, { "id": 2906, "type": "theorem", "label": "dualizing-lemma-relative-dualizing-algebraic", "categories": [ "dualizing" ], "title": "dualizing-lemma-relative-dualizing-algebraic", "contents": [ "Let $\\varphi : R \\to A$ be a flat finite type map of Noetherian rings.", "Then", "\\begin{enumerate}", "\\item $\\omega_{A/R}^\\bullet$ is in $D^b_{\\textit{Coh}}(A)$", "and $R$-perfect (More on Algebra,", "Definition \\ref{more-algebra-definition-relatively-perfect}),", "\\item $A \\to R\\Hom_A(\\omega_{A/R}^\\bullet, \\omega_{A/R}^\\bullet)$", "is an isomorphism, and", "\\item for every map $R \\to k$ to a field the base change", "$\\omega_{A/R}^\\bullet \\otimes_A^\\mathbf{L} (A \\otimes_R k)$", "is a dualizing complex for $A \\otimes_R k$.", "\\end{enumerate}" ], "refs": [ "more-algebra-definition-relatively-perfect" ], "proofs": [ { "contents": [ "Choose $R \\to P \\to A$ as in the definition of $\\varphi^!$.", "Recall that $R \\to A$ is a perfect ring map", "(More on Algebra, Lemma", "\\ref{more-algebra-lemma-flat-finite-presentation-perfect}) and", "hence $A$ is perfect as a $P$-modue", "(More on Algebra, Lemma \\ref{more-algebra-lemma-perfect-ring-map}).", "This shows that $\\omega_{A/R}^\\bullet$ is in $D^b_{\\textit{Coh}}(A)$", "by Lemma \\ref{lemma-shriek-boundedness}.", "To show $\\omega_{A/R}^\\bullet$ is $R$-perfect it suffices to", "show it has finite tor dimension as a complex of $R$-modules.", "This is true because", "$\\omega_{A/R}^\\bullet = \\varphi^!(R) = R\\Hom(A, P)[n]$", "maps to $R\\Hom_P(A, P)[n]$ in $D(P)$, which is perfect in $D(P)$", "(More on Algebra, Lemma \\ref{more-algebra-lemma-dual-perfect-complex}),", "hence has finite tor dimension in $D(R)$", "as $R \\to P$ is flat. This proves (1).", "\\medskip\\noindent", "Proof of (2). The object", "$R\\Hom_A(\\omega_{A/R}^\\bullet, \\omega_{A/R}^\\bullet)$", "of $D(A)$ maps in $D(P)$ to", "\\begin{align*}", "R\\Hom_P(\\omega_{A/R}^\\bullet, R\\Hom(A, P)[n])", "& =", "R\\Hom_P(R\\Hom_P(A, P)[n], P)[n] \\\\", "& =", "R\\Hom_P(R\\Hom_P(A, P), P)", "\\end{align*}", "This is equal to $A$ by the already used", "More on Algebra, Lemma \\ref{more-algebra-lemma-dual-perfect-complex}.", "\\medskip\\noindent", "Proof of (3). By Lemma \\ref{lemma-base-change-relative-algebraic}", "there is an isomorphism", "$$", "\\omega_{A/R}^\\bullet \\otimes_A^\\mathbf{L} (A \\otimes_R k) \\cong", "\\omega^\\bullet_{A \\otimes_R k/k}", "$$", "and the right hand side is a dualizing complex by", "Lemma \\ref{lemma-shriek-dualizing-algebraic}." ], "refs": [ "more-algebra-lemma-flat-finite-presentation-perfect", "more-algebra-lemma-perfect-ring-map", "dualizing-lemma-shriek-boundedness", "more-algebra-lemma-dual-perfect-complex", "more-algebra-lemma-dual-perfect-complex", "dualizing-lemma-base-change-relative-algebraic", "dualizing-lemma-shriek-dualizing-algebraic" ], "ref_ids": [ 10283, 10281, 2895, 10224, 10224, 2905, 2896 ] } ], "ref_ids": [ 10632 ] }, { "id": 2907, "type": "theorem", "label": "dualizing-lemma-base-change-dualizing-over-field", "categories": [ "dualizing" ], "title": "dualizing-lemma-base-change-dualizing-over-field", "contents": [ "Let $K/k$ be an extension of fields. Let $A$ be a finite type", "$k$-algebra. Let $A_K = A \\otimes_k K$. If", "$\\omega_A^\\bullet$ is a dualizing complex for $A$, then", "$\\omega_A^\\bullet \\otimes_A A_K$ is a dualizing complex for $A_K$." ], "refs": [], "proofs": [ { "contents": [ "By the uniqueness of dualizing complexes, it doesn't matter which", "dualizing complex we pick for $A$; we omit the detailed proof.", "Denote $\\varphi : k \\to A$ the algebra structure.", "We may take $\\omega_A^\\bullet = \\varphi^!(k[0])$ by", "Lemma \\ref{lemma-shriek-dualizing-algebraic}.", "We conclude by", "Lemma \\ref{lemma-relative-dualizing-algebraic}." ], "refs": [ "dualizing-lemma-shriek-dualizing-algebraic", "dualizing-lemma-relative-dualizing-algebraic" ], "ref_ids": [ 2896, 2906 ] } ], "ref_ids": [] }, { "id": 2908, "type": "theorem", "label": "dualizing-lemma-lci-shriek", "categories": [ "dualizing" ], "title": "dualizing-lemma-lci-shriek", "contents": [ "Let $\\varphi : R \\to A$ be a local complete intersection homomorphism of", "Noetherian rings. Then $\\omega_{A/R}^\\bullet$ is an invertible object of", "$D(A)$ and $\\varphi^!(K) = K \\otimes_R^\\mathbf{L} \\omega_{A/R}^\\bullet$", "for all $K \\in D(R)$." ], "refs": [], "proofs": [ { "contents": [ "Recall that a local complete intersection homomorphism is a perfect", "ring map by More on Algebra, Lemma \\ref{more-algebra-lemma-lci-perfect}.", "Hence the final statement holds by", "Lemma \\ref{lemma-upper-shriek-is-tensor-functor}.", "By More on Algebra, Definition", "\\ref{more-algebra-definition-local-complete-intersection}", "we can write $A = R[x_1, \\ldots, x_n]/I$ where $I$ is a", "Koszul-regular ideal.", "The construction of $\\varphi^!$ in", "Section \\ref{section-relative-dualizing-complex-algebraic}", "shows that it suffices to show the lemma in case", "$A = R/I$ where $I \\subset R$ is a Koszul-regular ideal.", "Checking $\\omega_{A/R}^\\bullet$ is invertible in $D(A)$", "is local on $\\Spec(A)$ by More on Algebra, Lemma", "\\ref{more-algebra-lemma-invertible-derived}.", "Moreover, formation of $\\omega_{A/R}^\\bullet$ commutes with", "localization on $R$ by Lemma \\ref{lemma-flat-bc}.", "Combining", "More on Algebra, Definition \\ref{more-algebra-definition-regular-ideal} and", "Lemma \\ref{more-algebra-lemma-noetherian-finite-all-equivalent} and", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-neighbourhood}", "we can find $g_1, \\ldots, g_r \\in R$ generating the unit ideal in $A$", "such that $I_{g_j} \\subset R_{g_j}$ is generated by a regular sequence.", "Thus we may assume $A = R/(f_1, \\ldots, f_c)$ where $f_1, \\ldots, f_c$", "is a regular sequence in $R$. Then we consider the ring maps", "$$", "R \\to R/(f_1) \\to R/(f_1, f_2) \\to \\ldots \\to R/(f_1, \\ldots, f_c) = A", "$$", "and we use Lemma \\ref{lemma-composition-shriek-algebraic}", "(and the final statement already proven)", "to see that it suffices to prove the lemma for each step.", "Finally, in case $A = R/(f)$ for some nonzerodivisor $f$", "we see that the lemma is true since $\\varphi^!(R) = R\\Hom(A, R)$", "is invertible by Lemma \\ref{lemma-compute-for-effective-Cartier-algebraic}." ], "refs": [ "more-algebra-lemma-lci-perfect", "dualizing-lemma-upper-shriek-is-tensor-functor", "more-algebra-definition-local-complete-intersection", "more-algebra-lemma-invertible-derived", "dualizing-lemma-flat-bc", "more-algebra-definition-regular-ideal", "more-algebra-lemma-noetherian-finite-all-equivalent", "algebra-lemma-regular-sequence-in-neighbourhood", "dualizing-lemma-composition-shriek-algebraic", "dualizing-lemma-compute-for-effective-Cartier-algebraic" ], "ref_ids": [ 10285, 2903, 10609, 10575, 2897, 10608, 9978, 741, 2900, 2843 ] } ], "ref_ids": [] }, { "id": 2909, "type": "theorem", "label": "dualizing-lemma-gorenstein-shriek", "categories": [ "dualizing" ], "title": "dualizing-lemma-gorenstein-shriek", "contents": [ "Let $\\varphi : R \\to A$ be a flat finite type homomorphism of Noetherian rings.", "The following are equivalent", "\\begin{enumerate}", "\\item the fibres $A \\otimes_R \\kappa(\\mathfrak p)$ are Gorenstein", "for all primes $\\mathfrak p \\subset R$, and", "\\item $\\omega_{A/R}^\\bullet$ is an invertible object of $D(A)$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-invertible-derived}.", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-invertible-derived" ], "proofs": [ { "contents": [ "If (2) holds, then the fibre rings $A \\otimes_R \\kappa(\\mathfrak p)$", "have invertible dualizing complexes, and hence are Gorenstein.", "See Lemmas \\ref{lemma-relative-dualizing-algebraic} and \\ref{lemma-gorenstein}.", "\\medskip\\noindent", "For the converse, assume (1).", "Observe that $\\omega_{A/R}^\\bullet$ is in $D^b_{\\textit{Coh}}(A)$", "by Lemma \\ref{lemma-shriek-boundedness} (since flat finite type homomorphisms", "of Noetherian rings are perfect, see ", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-flat-finite-presentation-perfect}).", "Take a prime $\\mathfrak q \\subset A$ lying over $\\mathfrak p \\subset R$.", "Then", "$$", "\\omega_{A/R}^\\bullet \\otimes_A^\\mathbf{L} \\kappa(\\mathfrak q) =", "\\omega_{A/R}^\\bullet \\otimes_A^\\mathbf{L}", "(A \\otimes_R \\kappa(\\mathfrak p))", "\\otimes_{(A \\otimes_R \\kappa(\\mathfrak p))}^\\mathbf{L}", "\\kappa(\\mathfrak q)", "$$", "Applying Lemmas \\ref{lemma-relative-dualizing-algebraic} and", "\\ref{lemma-gorenstein} and assumption (1) we find that this complex has $1$", "nonzero cohomology group which is a $1$-dimensional", "$\\kappa(\\mathfrak q)$-vector space. By", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-bounded-pseudo-coherent-to-perfect}", "we conclude that $(\\omega_{A/R}^\\bullet)_f$ is an invertible", "object of $D(A_f)$ for some $f \\in A$, $f \\not \\in \\mathfrak q$.", "This proves (2) holds." ], "refs": [ "dualizing-lemma-relative-dualizing-algebraic", "dualizing-lemma-gorenstein", "dualizing-lemma-shriek-boundedness", "more-algebra-lemma-flat-finite-presentation-perfect", "dualizing-lemma-relative-dualizing-algebraic", "dualizing-lemma-gorenstein", "more-algebra-lemma-lift-bounded-pseudo-coherent-to-perfect" ], "ref_ids": [ 2906, 2881, 2895, 10283, 2906, 2881, 10241 ] } ], "ref_ids": [ 10575 ] }, { "id": 2910, "type": "theorem", "label": "dualizing-lemma-shriek-normalized", "categories": [ "dualizing" ], "title": "dualizing-lemma-shriek-normalized", "contents": [ "Let $\\varphi : R \\to A$ be a finite type homomorphism of Noetherian rings.", "Assume $R$ local and let $\\mathfrak m \\subset A$ be a maximal", "ideal lying over the maximal ideal of $R$. If $\\omega_R^\\bullet$", "is a normalized dualizing complex for $R$, then", "$\\varphi^!(\\omega_R^\\bullet)_\\mathfrak m$ is a normalized", "dualizing complex for $A_\\mathfrak m$." ], "refs": [], "proofs": [ { "contents": [ "We already know that $\\varphi^!(\\omega_R^\\bullet)$ is a dualizing", "complex for $A$, see Lemma \\ref{lemma-shriek-dualizing-algebraic}.", "Choose a factorization $R \\to P \\to A$ with $P = R[x_1, \\ldots, x_n]$", "as in the construction of $\\varphi^!$. If we can prove the", "lemma for $R \\to P$ and the maximal ideal $\\mathfrak m'$ of $P$ corresponding to", "$\\mathfrak m$, then we obtain the result for $R \\to A$ by", "applying Lemma \\ref{lemma-normalized-quotient} to", "$P_{\\mathfrak m'} \\to A_\\mathfrak m$ or by applying", "Lemma \\ref{lemma-quotient-function} to $P \\to A$.", "In the case $A = R[x_1, \\ldots, x_n]$ we see that", "$\\dim(A_\\mathfrak m) = \\dim(R) + n$ for example by", "Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}", "(combined with Algebra, Lemma \\ref{algebra-lemma-dim-affine-space}", "to compute the dimension of the fibre).", "The fact that $\\omega_R^\\bullet$ is normalized means", "that $i = -\\dim(R)$ is the smallest index such that", "$H^i(\\omega_R^\\bullet)$ is nonzero (follows from", "Lemmas \\ref{lemma-sitting-in-degrees} and", "\\ref{lemma-nonvanishing-generically-local}).", "Then $\\varphi^!(\\omega_R^\\bullet)_\\mathfrak m =", "\\omega_R^\\bullet \\otimes_R A_\\mathfrak m[n]$", "has its first nonzero cohomology module in degree $-\\dim(R) - n$", "and therefore is the normalized dualizing complex for $A_\\mathfrak m$." ], "refs": [ "dualizing-lemma-shriek-dualizing-algebraic", "dualizing-lemma-normalized-quotient", "dualizing-lemma-quotient-function", "algebra-lemma-dimension-base-fibre-equals-total", "algebra-lemma-dim-affine-space", "dualizing-lemma-sitting-in-degrees", "dualizing-lemma-nonvanishing-generically-local" ], "ref_ids": [ 2896, 2858, 2868, 987, 992, 2861, 2866 ] } ], "ref_ids": [] }, { "id": 2911, "type": "theorem", "label": "dualizing-lemma-relative-dualizing-trivial-vanishing", "categories": [ "dualizing" ], "title": "dualizing-lemma-relative-dualizing-trivial-vanishing", "contents": [ "Let $R \\to A$ be a finite type homomorphism of Noetherian rings.", "Let $\\mathfrak q \\subset A$ be a prime ideal lying over", "$\\mathfrak p \\subset R$. Then", "$$", "H^i(\\omega_{A/R}^\\bullet)_\\mathfrak q \\not = 0", "\\Rightarrow - d \\leq i", "$$", "where $d$ is the dimension of the fibre of $\\Spec(A) \\to \\Spec(R)$", "over $\\mathfrak p$ at the point $\\mathfrak q$." ], "refs": [], "proofs": [ { "contents": [ "Choose a factorization $R \\to P \\to A$ with $P = R[x_1, \\ldots, x_n]$", "as in Section \\ref{section-relative-dualizing-complex-algebraic}", "so that $\\omega_{A/R}^\\bullet = R\\Hom(A, P)[n]$.", "We have to show that $R\\Hom(A, P)_\\mathfrak q$", "has vanishing cohomology in degrees $< n - d$.", "By Lemma \\ref{lemma-RHom-ext} this means we have to", "show that $\\Ext_P^i(P/I, P)_{\\mathfrak r} = 0$ for $i < n - d$", "where $\\mathfrak r \\subset P$ is the prime corresponding to $\\mathfrak q$", "and $I$ is the kernel of $P \\to A$.", "We may rewrite this as", "$\\Ext_{P_\\mathfrak r}^i(P_\\mathfrak r/IP_\\mathfrak r, P_\\mathfrak r)$", "by More on Algebra, Lemma", "\\ref{more-algebra-lemma-pseudo-coherence-and-base-change-ext}.", "Thus we have to show", "$$", "\\text{depth}_{IP_\\mathfrak r}(P_\\mathfrak r) \\geq n - d", "$$", "by Lemma \\ref{lemma-depth}.", "By Lemma \\ref{lemma-depth-flat-CM} we have", "$$", "\\text{depth}_{IP_\\mathfrak r}(P_\\mathfrak r) \\geq", "\\dim((P \\otimes_R \\kappa(\\mathfrak p))_\\mathfrak r) -", "\\dim((P/I \\otimes_R \\kappa(\\mathfrak p))_\\mathfrak r)", "$$", "The two expressions on the right hand side agree by", "Algebra, Lemma \\ref{algebra-lemma-codimension}." ], "refs": [ "dualizing-lemma-RHom-ext", "more-algebra-lemma-pseudo-coherence-and-base-change-ext", "dualizing-lemma-depth", "dualizing-lemma-depth-flat-CM", "algebra-lemma-codimension" ], "ref_ids": [ 2838, 10165, 2826, 2830, 1008 ] } ], "ref_ids": [] }, { "id": 2912, "type": "theorem", "label": "dualizing-lemma-relative-dualizing-flat-vanishing", "categories": [ "dualizing" ], "title": "dualizing-lemma-relative-dualizing-flat-vanishing", "contents": [ "Let $R \\to A$ be a flat finite type homomorphism of Noetherian rings.", "Let $\\mathfrak q \\subset A$ be a prime ideal lying over", "$\\mathfrak p \\subset R$. Then", "$$", "H^i(\\omega_{A/R}^\\bullet)_\\mathfrak q \\not = 0", "\\Rightarrow - d \\leq i \\leq 0", "$$", "where $d$ is the dimension of the fibre of $\\Spec(A) \\to \\Spec(R)$", "over $\\mathfrak p$ at the point $\\mathfrak q$. If all fibres of", "$\\Spec(A) \\to \\Spec(R)$ have dimension $\\leq d$, then", "$\\omega_{A/R}^\\bullet$ has tor amplitude in $[-d, 0]$", "as a complex of $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "The lower bound has been shown in", "Lemma \\ref{lemma-relative-dualizing-trivial-vanishing}.", "Choose a factorization $R \\to P \\to A$ with $P = R[x_1, \\ldots, x_n]$", "as in Section \\ref{section-relative-dualizing-complex-algebraic}", "so that $\\omega_{A/R}^\\bullet = R\\Hom(A, P)[n]$.", "The upper bound means that $\\Ext^i_P(A, P)$ is zero for $i > n$.", "This follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-perfect-over-polynomial-ring}", "which shows that $A$ is a perfect $P$-module with", "tor amplitude in $[-n, 0]$.", "\\medskip\\noindent", "Proof of the final statement. Let $R \\to R'$ be a ring homomorphism", "of Noetherian rings. Set $A' = A \\otimes_R R'$. Then", "$$", "\\omega_{A'/R'}^\\bullet =", "\\omega_{A/R}^\\bullet \\otimes_A^\\mathbf{L} A' =", "\\omega_{A/R}^\\bullet \\otimes_R^\\mathbf{L} R'", "$$", "The first isomorphism by Lemma \\ref{lemma-base-change-relative-algebraic}", "and the second, which takes place in $D(R')$, by", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-comparison}.", "By the first part of the proof", "(note that the fibres of $\\Spec(A') \\to \\Spec(R')$ have dimension $\\leq d$)", "we conclude that $\\omega_{A/R}^\\bullet \\otimes_R^\\mathbf{L} R'$", "has cohomology only in degrees $[-d, 0]$. Taking $R' = R \\oplus M$", "to be the square zero thickening of $R$ by a finite $R$-module $M$,", "we see that $R\\Hom(A, P) \\otimes_R^\\mathbf{L} M$", "has cohomology only in the interval $[-d, 0]$ for any finite $R$-module $M$.", "Since any $R$-module is a filtered colimit of finite $R$-modules", "and since tensor products commute with colimits we conclude." ], "refs": [ "dualizing-lemma-relative-dualizing-trivial-vanishing", "more-algebra-lemma-perfect-over-polynomial-ring", "dualizing-lemma-base-change-relative-algebraic", "more-algebra-lemma-base-change-comparison" ], "ref_ids": [ 2911, 10245, 2905, 10139 ] } ], "ref_ids": [] }, { "id": 2913, "type": "theorem", "label": "dualizing-lemma-relative-dualizing-CM-vanishing", "categories": [ "dualizing" ], "title": "dualizing-lemma-relative-dualizing-CM-vanishing", "contents": [ "Let $R \\to A$ be a finite type homomorphism of Noetherian rings.", "Let $\\mathfrak p \\subset R$ be a prime ideal. Assume", "\\begin{enumerate}", "\\item $R_\\mathfrak p$ is Cohen-Macaulay, and", "\\item for any minimal prime $\\mathfrak q \\subset A$ we have", "$\\text{trdeg}_{\\kappa(R \\cap \\mathfrak q)} \\kappa(\\mathfrak q) \\leq r$.", "\\end{enumerate}", "Then", "$$", "H^i(\\omega_{A/R}^\\bullet)_\\mathfrak p \\not = 0 \\Rightarrow - r \\leq i", "$$", "and $H^{-r}(\\omega_{A/R}^\\bullet)_\\mathfrak p$ is $(S_2)$", "as an $A_\\mathfrak p$-module." ], "refs": [], "proofs": [ { "contents": [ "We may replace $R$ by $R_\\mathfrak p$ by", "Lemma \\ref{lemma-base-change-relative-algebraic}.", "Thus we may assume $R$ is a Cohen-Macaulay local ring", "and we have to show the assertions of the lemma", "for the $A$-modules $H^i(\\omega_{A/R}^\\bullet)$.", "\\medskip\\noindent", "Let $R^\\wedge$ be the completion of $R$.", "The map $R \\to R^\\wedge$ is flat and $R^\\wedge$ is Cohen-Macaulay", "(More on Algebra, Lemma \\ref{more-algebra-lemma-completion-CM}).", "Observe that the minimal primes of $A \\otimes_R R^\\wedge$", "lie over minimal primes of $A$ by the flatness of", "$A \\to A \\otimes_R R^\\wedge$ (and going down for flatness, see", "Algebra, Lemma \\ref{algebra-lemma-flat-going-down}).", "Thus condition (2) holds for the finite type ring map", "$R^\\wedge \\to A \\otimes_R R^\\wedge$ by", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}.", "Appealing to Lemma \\ref{lemma-base-change-relative-algebraic}", "once again it suffices to prove the lemma for", "$R^\\wedge \\to A \\otimes_R R^\\wedge$. In this way, using", "Lemma \\ref{lemma-ubiquity-dualizing},", "we may assume $R$ is a Noetherian local", "Cohen-Macaulay ring which has a dualizing complex $\\omega_R^\\bullet$.", "\\medskip\\noindent", "Let $\\mathfrak m \\subset A$ be a maximal ideal.", "It suffices to show that the assertions of", "the lemma hold for $H^i(\\omega_{A/R}^\\bullet)_\\mathfrak m$.", "If $\\mathfrak m$ does not lie over the maximal ideal of $R$,", "then we replace $R$ by a localization to reduce to this case", "(small detail omitted). ", "\\medskip\\noindent", "We may assume $\\omega_R^\\bullet$ is normalized.", "Setting $d = \\dim(R)$ we see that $\\omega_R^\\bullet = \\omega_R[d]$", "for some $R$-module $\\omega_R$, see", "Lemma \\ref{lemma-apply-CM}. Set", "$\\omega_A^\\bullet = \\varphi^!(\\omega_R^\\bullet)$.", "By Lemma \\ref{lemma-relative-dualizing-if-have-omega} we have", "$$", "\\omega_{A/R}^\\bullet =", "R\\Hom_A(\\omega_R[d] \\otimes_R^\\mathbf{L} A, \\omega_A^\\bullet)", "$$", "By the dimension formula we have $\\dim(A_\\mathfrak m) \\leq d + r$, see", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-formula-general}", "and use that $\\kappa(\\mathfrak m)$ is finite over the residue field of $R$", "by the Hilbert Nullstellensatz.", "By Lemma \\ref{lemma-shriek-normalized}", "we see that $(\\omega_A^\\bullet)_\\mathfrak m$", "is a normalized dualizing complex for $A_\\mathfrak m$.", "Hence $H^i((\\omega_A^\\bullet)_\\mathfrak m)$ is nonzero", "only for $-d - r \\leq i \\leq 0$, see", "Lemma \\ref{lemma-sitting-in-degrees}.", "Since $\\omega_R[d] \\otimes_R^\\mathbf{L} A$ lives in", "degrees $\\leq -d$ we conclude the vanishing holds.", "Finally, we also see that", "$$", "H^{-r}(\\omega_{A/R}^\\bullet)_\\mathfrak m =", "\\Hom_A(\\omega_R \\otimes_R A, H^{-d - r}(\\omega_A^\\bullet))_\\mathfrak m", "$$", "Since $H^{-d - r}(\\omega_A^\\bullet)_\\mathfrak m$ is $(S_2)$ by", "Lemma \\ref{lemma-depth-dualizing-module}", "we find that the final statement is true by", "More on Algebra, Lemma \\ref{more-algebra-lemma-hom-into-S2}." ], "refs": [ "dualizing-lemma-base-change-relative-algebraic", "more-algebra-lemma-completion-CM", "algebra-lemma-flat-going-down", "morphisms-lemma-dimension-fibre-after-base-change", "dualizing-lemma-base-change-relative-algebraic", "dualizing-lemma-ubiquity-dualizing", "dualizing-lemma-apply-CM", "dualizing-lemma-relative-dualizing-if-have-omega", "morphisms-lemma-dimension-formula-general", "dualizing-lemma-shriek-normalized", "dualizing-lemma-sitting-in-degrees", "dualizing-lemma-depth-dualizing-module", "more-algebra-lemma-hom-into-S2" ], "ref_ids": [ 2905, 10044, 539, 5279, 2905, 2890, 2875, 2904, 5494, 2910, 2861, 2871, 9931 ] } ], "ref_ids": [] }, { "id": 2914, "type": "theorem", "label": "dualizing-lemma-descent", "categories": [ "dualizing" ], "title": "dualizing-lemma-descent", "contents": [ "Let $A \\to B$ be a faithfully flat map of Noetherian rings.", "If $K \\in D(A)$ and $K \\otimes_A^\\mathbf{L} B$", "is a dualizing complex for $B$, then $K$ is a dualizing complex", "for $A$." ], "refs": [], "proofs": [ { "contents": [ "Since $A \\to B$ is flat we have", "$H^i(K) \\otimes_A B = H^i(K \\otimes_A^\\mathbf{L} B)$.", "Since $K \\otimes_A^\\mathbf{L} B$ is in $D^b_{\\textit{Coh}}(B)$", "we first find that $K$ is in $D^b(A)$ and then we see that", "$H^i(K)$ is a finite $A$-module by", "Algebra, Lemma \\ref{algebra-lemma-descend-properties-modules}.", "Let $M$ be a finite $A$-module. Then", "$$", "R\\Hom_A(M, K) \\otimes_A B = R\\Hom_B(M \\otimes_A B, K \\otimes_A^\\mathbf{L} B)", "$$", "by More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom}.", "Since $K \\otimes_A^\\mathbf{L} B$ has finite injective dimension,", "say injective-amplitude in $[a, b]$, we see that the right hand side", "has vanishing cohomology in degrees $> b$.", "Since $A \\to B$ is faithfully flat, we find", "that $R\\Hom_A(M, K)$ has vanishing cohomology in degrees $> b$.", "Thus $K$ has finite injective dimension by", "More on Algebra, Lemma \\ref{more-algebra-lemma-injective-amplitude}.", "To finish the proof we have to show that the map", "$A \\to R\\Hom_A(K, K)$ is an isomorphism.", "For this we again use", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom}", "and the fact that", "$B \\to R\\Hom_B(K \\otimes_A^\\mathbf{L} B, K \\otimes_A^\\mathbf{L} B)$", "is an isomorphism." ], "refs": [ "algebra-lemma-descend-properties-modules", "more-algebra-lemma-base-change-RHom", "more-algebra-lemma-injective-amplitude", "more-algebra-lemma-base-change-RHom" ], "ref_ids": [ 819, 10418, 10188, 10418 ] } ], "ref_ids": [] }, { "id": 2915, "type": "theorem", "label": "dualizing-lemma-descent-ascent", "categories": [ "dualizing" ], "title": "dualizing-lemma-descent-ascent", "contents": [ "Let $\\varphi : A \\to B$ be a homomorphism of Noetherian rings. Assume", "\\begin{enumerate}", "\\item $A \\to B$ is syntomic and induces a surjective map on spectra, or", "\\item $A \\to B$ is a faithfully flat local complete intersection, or", "\\item $A \\to B$ is faithfully flat of finite type with Gorenstein fibres.", "\\end{enumerate}", "Then $K \\in D(A)$ is a dualizing complex for $A$ if and only if", "$K \\otimes_A^\\mathbf{L} B$ is a dualizing complex for $B$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $A \\to B$ satisfies (1) if and only if $A \\to B$", "satisfies (2) by More on Algebra, Lemma \\ref{more-algebra-lemma-syntomic-lci}.", "Observe that in both (2) and (3) the relative dualzing", "complex $\\varphi^!(A) = \\omega_{B/A}^\\bullet$ is an invertible", "object of $D(B)$, see", "Lemmas \\ref{lemma-lci-shriek} and \\ref{lemma-gorenstein-shriek}.", "Moreover we have", "$\\varphi^!(K) = K \\otimes_A^\\mathbf{L} \\omega_{B/A}^\\bullet$", "in both cases, see Lemma \\ref{lemma-upper-shriek-is-tensor-functor}", "for case (3).", "Thus $\\varphi^!(K)$ is the same as $K \\otimes_A^\\mathbf{L} B$", "up to tensoring with an invertible object of $D(B)$.", "Hence $\\varphi^!(K)$ is a dualizing complex for $B$", "if and only if $K \\otimes_A^\\mathbf{L} B$ is", "(as being a dualizing complex is local and invariant under shifts).", "Thus we see that if $K$ is dualizing for $A$, then", "$K \\otimes_A^\\mathbf{L} B$ is dualizing for $B$ by", "Lemma \\ref{lemma-shriek-dualizing-algebraic}.", "To descend the property, see", "Lemma \\ref{lemma-descent}." ], "refs": [ "more-algebra-lemma-syntomic-lci", "dualizing-lemma-lci-shriek", "dualizing-lemma-gorenstein-shriek", "dualizing-lemma-upper-shriek-is-tensor-functor", "dualizing-lemma-shriek-dualizing-algebraic", "dualizing-lemma-descent" ], "ref_ids": [ 10002, 2908, 2909, 2903, 2896, 2914 ] } ], "ref_ids": [] }, { "id": 2916, "type": "theorem", "label": "dualizing-lemma-injective-hull-goes-up", "categories": [ "dualizing" ], "title": "dualizing-lemma-injective-hull-goes-up", "contents": [ "Let $(A, \\mathfrak m, \\kappa) \\to (B, \\mathfrak n, l)$", "be a flat local homorphism of Noetherian rings such that", "$\\mathfrak n = \\mathfrak m B$. If $E$ is the injective", "hull of $\\kappa$, then $E \\otimes_A B$ is the injective", "hull of $l$." ], "refs": [], "proofs": [ { "contents": [ "Write $E = \\bigcup E_n$ as in Lemma \\ref{lemma-union-artinian}.", "It suffices to show that $E_n \\otimes_{A/\\mathfrak m^n} B/\\mathfrak n^n$", "is the injective hull of $l$ over $B/\\mathfrak n$.", "This reduces us to the case where $A$ and $B$ are Artinian local.", "Observe that $\\text{length}_A(A) = \\text{length}_B(B)$ and", "$\\text{length}_A(E) = \\text{length}_B(E \\otimes_A B)$", "by Algebra, Lemma \\ref{algebra-lemma-pullback-module}.", "By Lemma \\ref{lemma-finite} we have", "$\\text{length}_A(E) = \\text{length}_A(A)$ and", "$\\text{length}_B(E') = \\text{length}_B(B)$", "where $E'$ is the injective hull of $l$ over $B$.", "We conclude $\\text{length}_B(E') = \\text{length}_B(E \\otimes_A B)$.", "Observe that", "$$", "\\dim_l((E \\otimes_A B)[\\mathfrak n]) =", "\\dim_l(E[\\mathfrak m] \\otimes_A B) =", "\\dim_\\kappa(E[\\mathfrak m]) = 1", "$$", "where we have used flatness of $A \\to B$ and $\\mathfrak n = \\mathfrak mB$.", "Thus there is an injective $B$-module map $E \\otimes_A B \\to E'$", "by Lemma \\ref{lemma-torsion-submodule-sum-injective-hulls}.", "By equality of lengths shown above this is an isomorphism." ], "refs": [ "dualizing-lemma-union-artinian", "algebra-lemma-pullback-module", "dualizing-lemma-finite", "dualizing-lemma-torsion-submodule-sum-injective-hulls" ], "ref_ids": [ 2806, 640, 2800, 2805 ] } ], "ref_ids": [] }, { "id": 2917, "type": "theorem", "label": "dualizing-lemma-injective-goes-up", "categories": [ "dualizing" ], "title": "dualizing-lemma-injective-goes-up", "contents": [ "Let $\\varphi : A \\to B$ be a flat homorphism of Noetherian rings such", "that for all primes $\\mathfrak q \\subset B$ we have", "$\\mathfrak p B_\\mathfrak q = \\mathfrak qB_\\mathfrak q$", "where $\\mathfrak p = \\varphi^{-1}(\\mathfrak q)$, for example", "if $\\varphi$ is \\'etale.", "If $I$ is an injective $A$-module, then $I \\otimes_A B$ is", "an injective $B$-module." ], "refs": [], "proofs": [ { "contents": [ "\\'Etale maps satisfy the assumption by", "Algebra, Lemma \\ref{algebra-lemma-etale-at-prime}.", "By Lemma \\ref{lemma-sum-injective-modules} and", "Proposition \\ref{proposition-structure-injectives-noetherian}", "we may assume $I$ is the injective hull of $\\kappa(\\mathfrak p)$", "for some prime $\\mathfrak p \\subset A$.", "Then $I$ is a module over $A_\\mathfrak p$.", "It suffices to prove $I \\otimes_A B = I \\otimes_{A_\\mathfrak p} B_\\mathfrak p$", "is injective as a $B_\\mathfrak p$-module, see", "Lemma \\ref{lemma-injective-flat}.", "Thus we may assume $(A, \\mathfrak m, \\kappa)$ is local Noetherian", "and $I = E$ is the injective hull of the residue field $\\kappa$.", "Our assumption implies that the Noetherian ring $B/\\mathfrak m B$", "is a product of fields (details omitted).", "Thus there are finitely many prime ideals", "$\\mathfrak m_1, \\ldots, \\mathfrak m_n$ in $B$", "lying over $\\mathfrak m$ and they are all maximal ideals.", "Write $E = \\bigcup E_n$ as in Lemma \\ref{lemma-union-artinian}.", "Then $E \\otimes_A B = \\bigcup E_n \\otimes_A B$", "and $E_n \\otimes_A B$ is a finite $B$-module with support", "$\\{\\mathfrak m_1, \\ldots, \\mathfrak m_n\\}$ hence decomposes", "as a product over the localizations at $\\mathfrak m_i$.", "Thus $E \\otimes_A B = \\prod (E \\otimes_A B)_{\\mathfrak m_i}$.", "Since $(E \\otimes_A B)_{\\mathfrak m_i} = E \\otimes_A B_{\\mathfrak m_i}$", "is the injective hull of the residue field of $\\mathfrak m_i$", "by Lemma \\ref{lemma-injective-hull-goes-up} we conclude." ], "refs": [ "algebra-lemma-etale-at-prime", "dualizing-lemma-sum-injective-modules", "dualizing-proposition-structure-injectives-noetherian", "dualizing-lemma-injective-flat", "dualizing-lemma-union-artinian", "dualizing-lemma-injective-hull-goes-up" ], "ref_ids": [ 1233, 2789, 2923, 2785, 2806, 2916 ] } ], "ref_ids": [] }, { "id": 2918, "type": "theorem", "label": "dualizing-lemma-uniqueness-relative-dualizing", "categories": [ "dualizing" ], "title": "dualizing-lemma-uniqueness-relative-dualizing", "contents": [ "Let $R \\to A$ be a flat ring map of finite presentation.", "Any two relative dualizing complexes for $R \\to A$ are isomorphic." ], "refs": [], "proofs": [ { "contents": [ "Let $K$ and $L$ be two relative dualizing complexes for $R \\to A$.", "Denote $K_1 = K \\otimes_A^\\mathbf{L} (A \\otimes_R A)$", "and $L_2 = (A \\otimes_R A) \\otimes_A^\\mathbf{L} L$ the", "derived base changes via the first and second coprojections", "$A \\to A \\otimes_R A$. By symmetry the assumption on $L_2$", "implies that $R\\Hom_{A \\otimes_R A}(A, L_2)$ is isomorphic to $A$.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism} part (3)", "applied twice we have", "$$", "A \\otimes_{A \\otimes_R A}^\\mathbf{L} L_2 \\cong", "R\\Hom_{A \\otimes_R A}(A, K_1 \\otimes_{A \\otimes_R A}^\\mathbf{L} L_2) \\cong", "A \\otimes_{A \\otimes_R A}^\\mathbf{L} K_1", "$$", "Applying the restriction functor $D(A \\otimes_R A) \\to D(A)$", "for either coprojection we obtain the desired result." ], "refs": [ "more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism" ], "ref_ids": [ 10415 ] } ], "ref_ids": [] }, { "id": 2919, "type": "theorem", "label": "dualizing-lemma-relative-dualizing-noetherian", "categories": [ "dualizing" ], "title": "dualizing-lemma-relative-dualizing-noetherian", "contents": [ "Let $\\varphi : R \\to A$ be a flat finite type ring map of Noetherian rings.", "Then the relative dualizing complex $\\omega_{A/R}^\\bullet = \\varphi^!(R)$", "of Section \\ref{section-relative-dualizing-complexes-Noetherian}", "is a relative dualizing complex in the sense of", "Definition \\ref{definition-relative-dualizing-complex}." ], "refs": [ "dualizing-definition-relative-dualizing-complex" ], "proofs": [ { "contents": [ "From Lemma \\ref{lemma-relative-dualizing-algebraic} we see that", "$\\varphi^!(R)$ is $R$-perfect.", "Denote $\\delta : A \\otimes_R A \\to A$ the multiplication map", "and $p_1, p_2 : A \\to A \\otimes_R A$ the coprojections.", "Then", "$$", "\\varphi^!(R) \\otimes_A^\\mathbf{L} (A \\otimes_R A) =", "\\varphi^!(R) \\otimes_{A, p_1}^\\mathbf{L} (A \\otimes_R A) =", "p_2^!(A)", "$$", "by Lemma \\ref{lemma-flat-bc}. Recall that", "$", "R\\Hom_{A \\otimes_R A}(A, \\varphi^!(R) \\otimes_A^\\mathbf{L} (A \\otimes_R A))", "$", "is the image of $\\delta^!(\\varphi^!(R) \\otimes_A^\\mathbf{L} (A \\otimes_R A))$", "under the restriction map $\\delta_* : D(A) \\to D(A \\otimes_R A)$.", "Use the definition of $\\delta^!$ from", "Section \\ref{section-relative-dualizing-complex-algebraic}", "and Lemma \\ref{lemma-RHom-ext}.", "Since $\\delta^!(p_2^!(A)) \\cong A$ by", "Lemma \\ref{lemma-composition-shriek-algebraic}", "we conclude." ], "refs": [ "dualizing-lemma-relative-dualizing-algebraic", "dualizing-lemma-flat-bc", "dualizing-lemma-RHom-ext", "dualizing-lemma-composition-shriek-algebraic" ], "ref_ids": [ 2906, 2897, 2838, 2900 ] } ], "ref_ids": [ 2933 ] }, { "id": 2920, "type": "theorem", "label": "dualizing-lemma-base-change-relative-dualizing", "categories": [ "dualizing" ], "title": "dualizing-lemma-base-change-relative-dualizing", "contents": [ "Let $R \\to A$ be a flat ring map of finite presentation. Then", "\\begin{enumerate}", "\\item there exists a relative dualizing complex $K$ in $D(A)$, and", "\\item for any ring map $R \\to R'$ setting $A' = A \\otimes_R R'$", "and $K' = K \\otimes_A^\\mathbf{L} A'$, then $K'$ is a", "relative dualizing complex for $R' \\to A'$.", "\\end{enumerate}", "Moreover, if", "$$", "\\xi : A \\longrightarrow K \\otimes_A^\\mathbf{L} (A \\otimes_R A)", "$$", "is a generator for the cyclic module", "$\\Hom_{D(A \\otimes_R A)}(A, K \\otimes_A^\\mathbf{L} (A \\otimes_R A))$", "then in (2) the derived base change of $\\xi$ by", "$A \\otimes_R A \\to A' \\otimes_{R'} A'$ is a generator for", "the cyclic module", "$\\Hom_{D(A' \\otimes_{R'} A')}(A',", "K' \\otimes_{A'}^\\mathbf{L} (A' \\otimes_{R'} A'))$" ], "refs": [], "proofs": [ { "contents": [ "We first reduce to the Noetherian case. By", "Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "there exists a finite type $\\mathbf{Z}$ subalgebra $R_0 \\subset R$", "and a flat finite type ring map $R_0 \\to A_0$ such that", "$A = A_0 \\otimes_{R_0} R$. By Lemma \\ref{lemma-relative-dualizing-noetherian}", "there exists a relative", "dualizing complex $K_0 \\in D(A_0)$.", "Thus if we show (2) for $K_0$, then we find that", "$K_0 \\otimes_{A_0}^\\mathbf{L} A$ is", "a dualizing complex for $R \\to A$ and that it also satisfies (2)", "by transitivity of derived base change.", "The uniqueness of relative dualizing complexes", "(Lemma \\ref{lemma-uniqueness-relative-dualizing})", "then shows that this holds for", "any relative dualizing complex.", "\\medskip\\noindent", "Assume $R$ Noetherian and let $K$ be a relative dualizing complex", "for $R \\to A$. Given a ring map $R \\to R'$ set $A' = A \\otimes_R R'$", "and $K' = K \\otimes_A^\\mathbf{L} A'$. To finish the proof we have", "to show that $K'$ is a relative dualizing complex for $R' \\to A'$.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-base-change-relatively-perfect}", "we see that $K'$ is $R'$-perfect in all cases.", "By Lemmas \\ref{lemma-base-change-relative-algebraic} and", "\\ref{lemma-relative-dualizing-noetherian}", "if $R'$ is Noetherian, then $K'$ is a relative dualizing complex", "for $R' \\to A'$ (in either sense).", "Transitivity of derived tensor product shows that", "$K \\otimes_A^\\mathbf{L} (A \\otimes_R A)", "\\otimes_{A \\otimes_R A}^\\mathbf{L} (A' \\otimes_{R'} A') =", "K' \\otimes_{A'}^\\mathbf{L} (A' \\otimes_{R'} A')$.", "Flatness of $R \\to A$ guarantees that", "$A \\otimes_{A \\otimes_R A}^\\mathbf{L} (A' \\otimes_{R'} A') = A'$;", "namely $A \\otimes_R A$ and $R'$ are tor independent over $R$", "so we can apply More on Algebra, Lemma", "\\ref{more-algebra-lemma-base-change-comparison}.", "Finally, $A$ is pseudo-coherent as an $A \\otimes_R A$-module", "by More on Algebra, Lemma", "\\ref{more-algebra-lemma-more-relative-pseudo-coherent-is-moot}. Thus", "we have checked all the assumptions of", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-compute-RHom-relatively-perfect}.", "We find there exists a bounded below complex", "$E^\\bullet$ of $R$-flat finitely presented $A \\otimes_R A$-modules", "such that $E^\\bullet \\otimes_R R'$ represents", "$R\\Hom_{A' \\otimes_{R'} A'}(A',", "K' \\otimes_{A'}^\\mathbf{L} (A' \\otimes_{R'} A'))$", "and these identifications are compatible with derived base change.", "Let $n \\in \\mathbf{Z}$, $n \\not = 0$.", "Define $Q^n$ by the sequence", "$$", "E^{n - 1} \\to E^n \\to Q^n \\to 0", "$$", "Since $\\kappa(\\mathfrak p)$ is a Noetherian ring, we know that", "$H^n(E^\\bullet \\otimes_R \\kappa(\\mathfrak p)) = 0$, see remarks above.", "Chasing diagrams this means that", "$$", "Q^n \\otimes_R \\kappa(\\mathfrak p) \\to E^{n + 1} \\otimes_R \\kappa(\\mathfrak p)", "$$", "is injective. Hence for a prime $\\mathfrak q$ of $A \\otimes_R A$", "lying over $\\mathfrak p$ we have $Q^n_\\mathfrak q$ is $R_\\mathfrak p$-flat", "and $Q^n_\\mathfrak p \\to E^{n + 1}_\\mathfrak q$ is", "$R_\\mathfrak p$-universally injective, see", "Algebra, Lemma \\ref{algebra-lemma-mod-injective}.", "Since this holds for all primes,", "we conclude that $Q^n$ is $R$-flat", "and $Q^n \\to E^{n + 1}$ is $R$-universally injective. In particular", "$H^n(E^\\bullet \\otimes_R R') = 0$ for any ring map $R \\to R'$.", "Let $Z^0 = \\Ker(E^0 \\to E^1)$. Since there is an exact sequence", "$0 \\to Z^0 \\to E^0 \\to E^1 \\to Q^1 \\to 0$ we see that $Z^0$", "is $R$-flat and that", "$Z^0 \\otimes_R R' = \\Ker(E^0 \\otimes_R R' \\to E^1 \\otimes_R R')$", "for all $R \\to R'$. Then the short exact sequence", "$0 \\to Q^{-1} \\to Z^0 \\to H^0(E^\\bullet) \\to 0$", "shows that", "$$", "H^0(E^\\bullet \\otimes_R R') = H^0(E^\\bullet) \\otimes_R R'", "= A \\otimes_R R' = A'", "$$", "as desired. This equality furthermore gives the final assertion", "of the lemma." ], "refs": [ "algebra-lemma-flat-finite-presentation-limit-flat", "dualizing-lemma-relative-dualizing-noetherian", "dualizing-lemma-uniqueness-relative-dualizing", "more-algebra-lemma-base-change-relatively-perfect", "dualizing-lemma-base-change-relative-algebraic", "dualizing-lemma-relative-dualizing-noetherian", "more-algebra-lemma-base-change-comparison", "more-algebra-lemma-more-relative-pseudo-coherent-is-moot", "more-algebra-lemma-compute-RHom-relatively-perfect", "algebra-lemma-mod-injective" ], "ref_ids": [ 1389, 2919, 2918, 10291, 2905, 2919, 10139, 10287, 10292, 883 ] } ], "ref_ids": [] }, { "id": 2921, "type": "theorem", "label": "dualizing-lemma-relative-dualizing-RHom", "categories": [ "dualizing" ], "title": "dualizing-lemma-relative-dualizing-RHom", "contents": [ "Let $R \\to A$ be a flat ring map of finite presentation.", "Let $K$ be a relative dualizing complex.", "Then $A \\to R\\Hom_A(K, K)$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By", "Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "there exists a finite type $\\mathbf{Z}$ subalgebra $R_0 \\subset R$", "and a flat finite type ring map $R_0 \\to A_0$ such that", "$A = A_0 \\otimes_{R_0} R$. By Lemmas", "\\ref{lemma-uniqueness-relative-dualizing},", "\\ref{lemma-relative-dualizing-noetherian}, and", "\\ref{lemma-base-change-relative-dualizing}", "there exists a relative dualizing complex $K_0 \\in D(A_0)$", "and its derived base change is $K$.", "This reduces us to the situation discussed in the next paragraph.", "\\medskip\\noindent", "Assume $R$ Noetherian and let $K$ be a relative dualizing complex", "for $R \\to A$. Given a ring map $R \\to R'$ set $A' = A \\otimes_R R'$", "and $K' = K \\otimes_A^\\mathbf{L} A'$. To finish the proof we show", "$R\\Hom_{A'}(K', K') = A'$. By Lemma \\ref{lemma-relative-dualizing-algebraic}", "we know this is true whenever $R'$ is Noetherian.", "Since a general $R'$ is a filtered colimit of Noetherian", "$R$-algebras, we find the result holds by", "More on Algebra, Lemma \\ref{more-algebra-lemma-colimit-relatively-perfect}." ], "refs": [ "algebra-lemma-flat-finite-presentation-limit-flat", "dualizing-lemma-uniqueness-relative-dualizing", "dualizing-lemma-relative-dualizing-noetherian", "dualizing-lemma-base-change-relative-dualizing", "dualizing-lemma-relative-dualizing-algebraic", "more-algebra-lemma-colimit-relatively-perfect" ], "ref_ids": [ 1389, 2918, 2919, 2920, 2906, 10293 ] } ], "ref_ids": [] }, { "id": 2922, "type": "theorem", "label": "dualizing-lemma-relative-dualizing-composition", "categories": [ "dualizing" ], "title": "dualizing-lemma-relative-dualizing-composition", "contents": [ "Let $R \\to A \\to B$ be a ring maps which are flat and of finite presentation.", "Let $K_{A/R}$ and $K_{B/A}$ be relative dualizing complexes for $R \\to A$", "and $A \\to B$. Then $K = K_{A/R} \\otimes_A^\\mathbf{L} K_{B/A}$", "is a relative dualizing complex for $R \\to B$." ], "refs": [], "proofs": [ { "contents": [ "We will use reduction to the Noetherian case.", "Namely, by Algebra, Lemma", "\\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "there exists a finite type $\\mathbf{Z}$ subalgebra $R_0 \\subset R$", "and a flat finite type ring map $R_0 \\to A_0$ such that", "$A = A_0 \\otimes_{R_0} R$. After increasing $R_0$ and correspondingly", "replacing $A_0$ we may assume there is a flat", "finite type ring map $A_0 \\to B_0$ such that $B = B_0 \\otimes_{R_0} R$", "(use the same lemma). If we prove the lemma for $R_0 \\to A_0 \\to B_0$,", "then the lemma follows by Lemmas", "\\ref{lemma-uniqueness-relative-dualizing},", "\\ref{lemma-relative-dualizing-noetherian}, and", "\\ref{lemma-base-change-relative-dualizing}.", "This reduces us to the situation discussed in the next paragraph.", "\\medskip\\noindent", "Assume $R$ is Noetherian and denote $\\varphi : R \\to A$ and", "$\\psi : A \\to B$ the given ring maps. Then $K_{A/R} \\cong \\varphi^!(R)$ and", "$K_{B/A} \\cong \\psi^!(A)$, see references given above.", "Then", "$$", "K = K_{A/R} \\otimes_A^\\mathbf{L} K_{B/A} \\cong", "\\varphi^!(R) \\otimes_A^\\mathbf{L} \\psi^!(A) \\cong", "\\psi^!(\\varphi^!(R)) \\cong (\\psi \\circ \\varphi)^!(R)", "$$", "by Lemmas \\ref{lemma-upper-shriek-is-tensor-functor} and", "\\ref{lemma-composition-shriek-algebraic}. Thus $K$ is a relative", "dualizing complex for $R \\to B$." ], "refs": [ "algebra-lemma-flat-finite-presentation-limit-flat", "dualizing-lemma-uniqueness-relative-dualizing", "dualizing-lemma-relative-dualizing-noetherian", "dualizing-lemma-base-change-relative-dualizing", "dualizing-lemma-upper-shriek-is-tensor-functor", "dualizing-lemma-composition-shriek-algebraic" ], "ref_ids": [ 1389, 2918, 2919, 2920, 2903, 2900 ] } ], "ref_ids": [] }, { "id": 2923, "type": "theorem", "label": "dualizing-proposition-structure-injectives-noetherian", "categories": [ "dualizing" ], "title": "dualizing-proposition-structure-injectives-noetherian", "contents": [ "Let $R$ be a Noetherian ring.", "Every injective module is a direct sum of indecomposable injective modules.", "Every indecomposable injective module is the injective hull of", "the residue field at a prime." ], "refs": [], "proofs": [ { "contents": [ "The second statement is Lemma \\ref{lemma-indecomposable-injective-noetherian}.", "For the first statement, let $I$ be an injective $R$-module.", "We will use transfinite induction to construct $I_\\alpha \\subset I$", "for ordinals $\\alpha$ which are direct sums of indecomposable injective", "$R$-modules $E_{\\beta + 1}$ for $\\beta < \\alpha$.", "For $\\alpha = 0$ we let $I_0 = 0$. Suppose given an ordinal $\\alpha$", "such that $I_\\alpha$ has been constructed. Then $I_\\alpha$ is an", "injective $R$-module by Lemma \\ref{lemma-sum-injective-modules}.", "Hence $I \\cong I_\\alpha \\oplus I'$. If $I' = 0$ we are done.", "If not, then $I'$ has an associated prime by", "Algebra, Lemma \\ref{algebra-lemma-ass-zero}.", "Thus $I'$ contains a copy of $R/\\mathfrak p$ for some prime $\\mathfrak p$.", "Hence $I'$ contains an indecomposable submodule $E$ by", "Lemmas \\ref{lemma-injective-hull-unique} and", "\\ref{lemma-injective-hull-indecomposable}. Set", "$I_{\\alpha + 1} = I_\\alpha \\oplus E_\\alpha$.", "If $\\alpha$ is a limit ordinal and $I_\\beta$ has been constructed", "for $\\beta < \\alpha$, then we set", "$I_\\alpha = \\bigcup_{\\beta < \\alpha} I_\\beta$.", "Observe that $I_\\alpha = \\bigoplus_{\\beta < \\alpha} E_{\\beta + 1}$.", "This concludes the proof." ], "refs": [ "dualizing-lemma-indecomposable-injective-noetherian", "dualizing-lemma-sum-injective-modules", "algebra-lemma-ass-zero", "dualizing-lemma-injective-hull-unique", "dualizing-lemma-injective-hull-indecomposable" ], "ref_ids": [ 2799, 2789, 702, 2796, 2798 ] } ], "ref_ids": [] }, { "id": 2924, "type": "theorem", "label": "dualizing-proposition-matlis", "categories": [ "dualizing" ], "title": "dualizing-proposition-matlis", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a complete local Noetherian ring.", "Let $E$ be an injective hull of $\\kappa$ over $R$. The functor", "$D(-) = \\Hom_R(-, E)$ induces an anti-equivalence", "$$", "\\left\\{", "\\begin{matrix}", "R\\text{-modules with the} \\\\", "\\text{descending chain condition}", "\\end{matrix}", "\\right\\}", "\\longleftrightarrow", "\\left\\{", "\\begin{matrix}", "R\\text{-modules with the} \\\\", "\\text{ascending chain condition}", "\\end{matrix}", "\\right\\}", "$$", "and we have $D \\circ D = \\text{id}$ on either side of the equivalence." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-endos} we have $R = \\Hom_R(E, E) = D(E)$.", "Of course we have $E = \\Hom_R(R, E) = D(R)$. Since $E$ is injective", "the functor $D$ is exact. The result now follows immediately from the", "description of the categories in", "Lemma \\ref{lemma-describe-categories}." ], "refs": [ "dualizing-lemma-endos", "dualizing-lemma-describe-categories" ], "ref_ids": [ 2808, 2810 ] } ], "ref_ids": [] }, { "id": 2925, "type": "theorem", "label": "dualizing-proposition-torsion-complete", "categories": [ "dualizing" ], "title": "dualizing-proposition-torsion-complete", "contents": [ "\\begin{reference}", "This is a special case of \\cite[Theorem 1.1]{Porta-Liran-Yekutieli}.", "\\end{reference}", "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "The functors $R\\Gamma_Z$ and ${\\ }^\\wedge$", "define quasi-inverse equivalences of categories", "$$", "D_{I^\\infty\\text{-torsion}}(A) \\leftrightarrow D_{comp}(A, I)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-complete-and-local}." ], "refs": [ "dualizing-lemma-complete-and-local" ], "ref_ids": [ 2832 ] } ], "ref_ids": [] }, { "id": 2926, "type": "theorem", "label": "dualizing-proposition-dualizing-essentially-finite-type", "categories": [ "dualizing" ], "title": "dualizing-proposition-dualizing-essentially-finite-type", "contents": [ "Let $A$ be a Noetherian ring which has a dualizing complex.", "Then any $A$-algebra essentially of finite type over $A$", "has a dualizing complex." ], "refs": [], "proofs": [ { "contents": [ "This follows from a combination of", "Lemmas \\ref{lemma-dualizing-localize},", "\\ref{lemma-dualizing-quotient}, and \\ref{lemma-dualizing-polynomial-ring}." ], "refs": [ "dualizing-lemma-dualizing-localize", "dualizing-lemma-dualizing-quotient", "dualizing-lemma-dualizing-polynomial-ring" ], "ref_ids": [ 2851, 2854, 2855 ] } ], "ref_ids": [] }, { "id": 2938, "type": "theorem", "label": "properties-lemma-locally-constructible", "categories": [ "properties" ], "title": "properties-lemma-locally-constructible", "contents": [ "Let $X$ be a scheme.", "A subset $E$ of $X$ is locally constructible in $X$ if and only if", "$E \\cap U$ is constructible in $U$ for every affine open $U$ of $X$." ], "refs": [], "proofs": [ { "contents": [ "Assume $E$ is locally constructible. Then there exists an open covering", "$X = \\bigcup U_i$ such that $E \\cap U_i$ is constructible in $U_i$", "for each $i$. Let $V \\subset X$ be any affine open. We can find a finite", "open affine covering $V = V_1 \\cup \\ldots \\cup V_m$ such that for each $j$", "we have $V_j \\subset U_i$ for some $i = i(j)$. By", "Topology, Lemma \\ref{topology-lemma-open-immersion-constructible-inverse-image}", "we see that each $E \\cap V_j$ is constructible in $V_j$. Since the inclusions", "$V_j \\to V$ are quasi-compact (see", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-affine})", "we conclude that $E \\cap V$ is constructible in $V$ by", "Topology, Lemma \\ref{topology-lemma-collate-constructible}.", "The converse implication is immediate." ], "refs": [ "topology-lemma-open-immersion-constructible-inverse-image", "schemes-lemma-quasi-compact-affine", "topology-lemma-collate-constructible" ], "ref_ids": [ 8255, 7697, 8257 ] } ], "ref_ids": [] }, { "id": 2939, "type": "theorem", "label": "properties-lemma-generic-point-in-constructible", "categories": [ "properties" ], "title": "properties-lemma-generic-point-in-constructible", "contents": [ "Let $X$ be a scheme and let $E \\subset X$ be a locally constructible subset.", "Let $\\xi \\in X$ be a generic point of an irreducible component of $X$.", "\\begin{enumerate}", "\\item If $\\xi \\in E$, then an open neighbourhood of", "$\\xi$ is contained in $E$.", "\\item If $\\xi \\not \\in E$, then an open neighbourhood", "of $\\xi$ is disjoint from $E$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As the complement of a locally constructible subset is locally", "constructible it suffices to show (2). We may assume $X$ is", "affine and hence $E$ constructible (Lemma \\ref{lemma-locally-constructible}).", "In this case $X$ is a spectral space", "(Algebra, Lemma \\ref{algebra-lemma-spec-spectral}).", "Then $\\xi \\not \\in E$ implies $\\xi \\not \\in \\overline{E}$ by", "Topology, Lemma \\ref{topology-lemma-constructible-stable-specialization-closed}", "and the fact that there are no points of $X$ different from $\\xi$", "which specialize to $\\xi$." ], "refs": [ "properties-lemma-locally-constructible", "algebra-lemma-spec-spectral", "topology-lemma-constructible-stable-specialization-closed" ], "ref_ids": [ 2938, 423, 8307 ] } ], "ref_ids": [] }, { "id": 2940, "type": "theorem", "label": "properties-lemma-quasi-separated-quasi-compact-open-retrocompact", "categories": [ "properties" ], "title": "properties-lemma-quasi-separated-quasi-compact-open-retrocompact", "contents": [ "Let $X$ be a quasi-separated scheme. The intersection of any two", "quasi-compact opens of $X$ is a quasi-compact open of $X$.", "Every quasi-compact open of $X$ is retrocompact in $X$." ], "refs": [], "proofs": [ { "contents": [ "If $U$ and $V$ are quasi-compact open then", "$U \\cap V = \\Delta^{-1}(U \\times V)$, where $\\Delta : X \\to X \\times X$", "is the diagonal. As $X$ is quasi-separated we see that $\\Delta$ is", "quasi-compact. Hence we see that $U \\cap V$ is quasi-compact as", "$U \\times V$ is quasi-compact (details omitted; use", "Schemes, Lemma \\ref{schemes-lemma-affine-covering-fibre-product}", "to see $U \\times V$ is a finite union of affines).", "The other assertions follow from the first and", "Topology, Lemma \\ref{topology-lemma-topology-quasi-separated-scheme}." ], "refs": [ "schemes-lemma-affine-covering-fibre-product", "topology-lemma-topology-quasi-separated-scheme" ], "ref_ids": [ 7692, 8333 ] } ], "ref_ids": [] }, { "id": 2941, "type": "theorem", "label": "properties-lemma-quasi-compact-quasi-separated-spectral", "categories": [ "properties" ], "title": "properties-lemma-quasi-compact-quasi-separated-spectral", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Then the underlying topological space of $X$ is a spectral space." ], "refs": [], "proofs": [ { "contents": [ "By Topology, Definition \\ref{topology-definition-spectral-space}", "we have to check that $X$ is sober, quasi-compact, has a basis", "of quasi-compact opens, and the intersection of any two", "quasi-compact opens is quasi-compact. This follows from", "Schemes, Lemma \\ref{schemes-lemma-scheme-sober} and", "\\ref{schemes-lemma-basis-affine-opens}", "and", "Lemma \\ref{lemma-quasi-separated-quasi-compact-open-retrocompact}", "above." ], "refs": [ "topology-definition-spectral-space", "schemes-lemma-scheme-sober", "schemes-lemma-basis-affine-opens", "properties-lemma-quasi-separated-quasi-compact-open-retrocompact" ], "ref_ids": [ 8370, 7672, 7673, 2940 ] } ], "ref_ids": [] }, { "id": 2942, "type": "theorem", "label": "properties-lemma-constructible-quasi-compact-quasi-separated", "categories": [ "properties" ], "title": "properties-lemma-constructible-quasi-compact-quasi-separated", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Any locally constructible subset of $X$ is constructible." ], "refs": [], "proofs": [ { "contents": [ "As $X$ is quasi-compact we can choose a finite affine open covering", "$X = V_1 \\cup \\ldots \\cup V_m$. As $X$ is quasi-separated each $V_i$ is", "retrocompact in $X$ by", "Lemma \\ref{lemma-quasi-separated-quasi-compact-open-retrocompact}.", "Hence by", "Topology, Lemma \\ref{topology-lemma-collate-constructible}", "we see that $E \\subset X$ is constructible in $X$ if and only if", "$E \\cap V_j$ is constructible in $V_j$. Thus we win by", "Lemma \\ref{lemma-locally-constructible}." ], "refs": [ "properties-lemma-quasi-separated-quasi-compact-open-retrocompact", "topology-lemma-collate-constructible", "properties-lemma-locally-constructible" ], "ref_ids": [ 2940, 8257, 2938 ] } ], "ref_ids": [] }, { "id": 2943, "type": "theorem", "label": "properties-lemma-retrocompact", "categories": [ "properties" ], "title": "properties-lemma-retrocompact", "contents": [ "Let $X$ be a scheme. A subset $E$ of $X$ is retrocompact in $X$ if and only if", "$E \\cap U$ is quasi-compact for every affine open $U$ of $X$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the fact that every quasi-compact open of $X$ is a finite", "union of affine opens." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2944, "type": "theorem", "label": "properties-lemma-stratification-locally-finite-constructible", "categories": [ "properties" ], "title": "properties-lemma-stratification-locally-finite-constructible", "contents": [ "A partition $X = \\coprod_{i \\in I} X_i$ of a scheme $X$ with", "retrocompact parts is locally finite if and only if the parts", "are locally constructible." ], "refs": [], "proofs": [ { "contents": [ "See Topology, Definitions", "\\ref{topology-definition-quasi-compact},", "\\ref{topology-definition-paritition}, and", "\\ref{topology-definition-locally-finite}", "for the definitions of retrocompact, partition, and locally finite.", "\\medskip\\noindent", "If the partition is locally finite and $U \\subset X$ is an", "affine open, then we see that $U = \\coprod_{i \\in I} U \\cap X_i$", "is a finite partition (more precisely, all but a finite number", "of its parts are empty). Hence $U \\cap X_i$ is quasi-compact", "and its complement is retrocompact in $U$ as a finite union", "of retrocompact parts. Thus $U \\cap X_i$ is constructible", "by Topology, Lemma \\ref{topology-lemma-locally-closed-constructible-image}.", "It follows that $X_i$ is locally constructible by", "Lemma \\ref{lemma-locally-constructible}.", "\\medskip\\noindent", "Assume the parts are locally constructible. Then for any affine", "open $U \\subset X$ we obtain a covering $U = \\coprod X_i \\cap U$", "by constructible subsets. Since the constructible topology is", "quasi-compact, see", "Topology, Lemma \\ref{topology-lemma-constructible-hausdorff-quasi-compact},", "this covering has a finite refinement, i.e.,", "the partition is locally finite." ], "refs": [ "topology-definition-quasi-compact", "topology-definition-paritition", "topology-definition-locally-finite", "topology-lemma-locally-closed-constructible-image", "properties-lemma-locally-constructible", "topology-lemma-constructible-hausdorff-quasi-compact" ], "ref_ids": [ 8360, 8373, 8376, 8264, 2938, 8303 ] } ], "ref_ids": [] }, { "id": 2945, "type": "theorem", "label": "properties-lemma-characterize-reduced", "categories": [ "properties" ], "title": "properties-lemma-characterize-reduced", "contents": [ "Let $X$ be a scheme.", "The following are equivalent.", "\\begin{enumerate}", "\\item The scheme $X$ is reduced, see", "Schemes, Definition \\ref{schemes-definition-reduced}.", "\\item There exists an affine open covering $X = \\bigcup U_i$", "such that each $\\Gamma(U_i, \\mathcal{O}_X)$ is reduced.", "\\item For every affine open $U \\subset X$ the ring", "$\\mathcal{O}_X(U)$ is reduced.", "\\item For every open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is reduced.", "\\end{enumerate}" ], "refs": [ "schemes-definition-reduced" ], "proofs": [ { "contents": [ "See Schemes, Lemmas \\ref{schemes-lemma-reduced} and", "\\ref{schemes-lemma-affine-reduced}." ], "refs": [ "schemes-lemma-reduced", "schemes-lemma-affine-reduced" ], "ref_ids": [ 7679, 7680 ] } ], "ref_ids": [ 7744 ] }, { "id": 2946, "type": "theorem", "label": "properties-lemma-characterize-irreducible", "categories": [ "properties" ], "title": "properties-lemma-characterize-irreducible", "contents": [ "Let $X$ be a scheme.", "The following are equivalent.", "\\begin{enumerate}", "\\item The scheme $X$ is irreducible.", "\\item There exists an affine open covering $X = \\bigcup_{i \\in I} U_i$", "such that $I$ is not empty, $U_i$ is irreducible for all $i \\in I$, and", "$U_i \\cap U_j \\not = \\emptyset$ for all $i, j \\in I$.", "\\item The scheme $X$ is nonempty and every nonempty affine open", "$U \\subset X$ is irreducible.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). By Schemes, Lemma \\ref{schemes-lemma-scheme-sober}", "we see that $X$ has a unique generic point $\\eta$. Then", "$X = \\overline{\\{\\eta\\}}$. Hence $\\eta$ is an element of", "every nonempty affine open $U \\subset X$. This implies", "that $U = \\overline{\\{\\eta\\}}$ and that any two nonempty affines", "meet. Thus (1) implies both (2) and (3).", "\\medskip\\noindent", "Assume (2). Suppose $X = Z_1 \\cup Z_2$ is a union of two closed subsets.", "For every $i$ we see that either $U_i \\subset Z_1$ or $U_i \\subset Z_2$.", "Pick some $i \\in I$ and assume $U_i \\subset Z_1$ (possibly after renumbering", "$Z_1$, $Z_2$). For any $j \\in I$ the open subset $U_i \\cap U_j$ is dense in", "$U_j$ and contained in the closed subset $Z_1 \\cap U_j$. We conclude that", "also $U_j \\subset Z_1$. Thus $X = Z_1$ as desired.", "\\medskip\\noindent", "Assume (3). Choose an affine open covering $X = \\bigcup_{i \\in I} U_i$.", "We may assume that each $U_i$ is nonempty.", "Since $X$ is nonempty we see that $I$ is not empty.", "By assumption each $U_i$ is irreducible.", "Suppose $U_i \\cap U_j = \\emptyset$ for some pair $i, j \\in I$.", "Then the open $U_i \\amalg U_j = U_i \\cup U_j$ is affine, see", "Schemes, Lemma \\ref{schemes-lemma-disjoint-union-affines}.", "Hence it is irreducible by assumption which is absurd. We conclude that (3)", "implies (2). The lemma is proved." ], "refs": [ "schemes-lemma-scheme-sober", "schemes-lemma-disjoint-union-affines" ], "ref_ids": [ 7672, 7659 ] } ], "ref_ids": [] }, { "id": 2947, "type": "theorem", "label": "properties-lemma-characterize-integral", "categories": [ "properties" ], "title": "properties-lemma-characterize-integral", "contents": [ "A scheme $X$ is integral if and only if it is reduced and irreducible." ], "refs": [], "proofs": [ { "contents": [ "If $X$ is irreducible, then every affine open $\\Spec(R) = U \\subset X$", "is irreducible. If $X$ is reduced, then $R$ is reduced, by", "Lemma \\ref{lemma-characterize-reduced} above. Hence $R$ is reduced", "and $(0)$ is a prime ideal, i.e., $R$ is an integral domain.", "\\medskip\\noindent", "If $X$ is integral, then for every nonempty affine open", "$\\Spec(R) = U \\subset X$ the ring $R$ is reduced", "and hence $X$ is reduced by Lemma \\ref{lemma-characterize-reduced}.", "Moreover, every nonempty affine open is irreducible.", "Hence $X$ is irreducible, see Lemma \\ref{lemma-characterize-irreducible}." ], "refs": [ "properties-lemma-characterize-reduced", "properties-lemma-characterize-reduced", "properties-lemma-characterize-irreducible" ], "ref_ids": [ 2945, 2945, 2946 ] } ], "ref_ids": [] }, { "id": 2948, "type": "theorem", "label": "properties-lemma-locally-P", "categories": [ "properties" ], "title": "properties-lemma-locally-P", "contents": [ "Let $X$ be a scheme. Let $P$ be a local property of rings.", "The following are equivalent:", "\\begin{enumerate}", "\\item The scheme $X$ is locally $P$.", "\\item For every affine open $U \\subset X$ the property", "$P(\\mathcal{O}_X(U))$ holds.", "\\item There exists an affine open covering $X = \\bigcup U_i$ such that", "each $\\mathcal{O}_X(U_i)$ satisfies $P$.", "\\item There exists an open covering $X = \\bigcup X_j$", "such that each open subscheme $X_j$ is locally $P$.", "\\end{enumerate}", "Moreover, if $X$ is locally $P$ then every open subscheme", "is locally $P$." ], "refs": [], "proofs": [ { "contents": [ "Of course (1) $\\Leftrightarrow$ (3) and (2) $\\Rightarrow$ (1).", "If (3) $\\Rightarrow$ (2), then the final statement of the lemma", "holds and it follows easily that (4) is also equivalent to (1).", "Thus we show (3) $\\Rightarrow$ (2).", "\\medskip\\noindent", "Let $X = \\bigcup U_i$ be an affine open covering, say", "$U_i = \\Spec(R_i)$. Assume $P(R_i)$.", "Let $\\Spec(R) = U \\subset X$ be an arbitrary affine open.", "By Schemes, Lemma \\ref{schemes-lemma-good-subcover}", "there exists a standard covering of $U = \\Spec(R)$ by", "standard opens $D(f_j)$ such that each ring $R_{f_j}$ is a", "principal localization of one of the rings $R_i$. By", "Definition \\ref{definition-property-local} (1) we get $P(R_{f_j})$.", "Whereupon $P(R)$ by Definition \\ref{definition-property-local} (2)." ], "refs": [ "schemes-lemma-good-subcover", "properties-definition-property-local", "properties-definition-property-local" ], "ref_ids": [ 7676, 3069, 3069 ] } ], "ref_ids": [] }, { "id": 2949, "type": "theorem", "label": "properties-lemma-reduced-is-locally-reduced", "categories": [ "properties" ], "title": "properties-lemma-reduced-is-locally-reduced", "contents": [ "Let $X$ be a scheme. Then $X$ is reduced if and only if $X$ is", "``locally reduced'' in the sense of Definition \\ref{definition-locally-P}." ], "refs": [ "properties-definition-locally-P" ], "proofs": [ { "contents": [ "This is clear from Lemma \\ref{lemma-characterize-reduced}." ], "refs": [ "properties-lemma-characterize-reduced" ], "ref_ids": [ 2945 ] } ], "ref_ids": [ 3070 ] }, { "id": 2950, "type": "theorem", "label": "properties-lemma-properties-local", "categories": [ "properties" ], "title": "properties-lemma-properties-local", "contents": [ "The following properties of a ring $R$ are local.", "\\begin{enumerate}", "\\item (Cohen-Macaulay.)", "The ring $R$ is Noetherian and CM, see", "Algebra, Definition \\ref{algebra-definition-ring-CM}.", "\\item (Regular.)", "The ring $R$ is Noetherian and regular, see", "Algebra, Definition \\ref{algebra-definition-regular}.", "\\item (Absolutely Noetherian.)", "The ring $R$ is of finite type over $Z$.", "\\item Add more here as needed.\\footnote{But we only list those properties", "here which we have not already dealt with separately somewhere else.}", "\\end{enumerate}" ], "refs": [ "algebra-definition-ring-CM", "algebra-definition-regular" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1506, 1512 ] }, { "id": 2951, "type": "theorem", "label": "properties-lemma-locally-Noetherian", "categories": [ "properties" ], "title": "properties-lemma-locally-Noetherian", "contents": [ "Let $X$ be a scheme. The following are equivalent:", "\\begin{enumerate}", "\\item The scheme $X$ is locally Noetherian.", "\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$", "is Noetherian.", "\\item There exists an affine open covering $X = \\bigcup U_i$ such that", "each $\\mathcal{O}_X(U_i)$ is Noetherian.", "\\item There exists an open covering $X = \\bigcup X_j$", "such that each open subscheme $X_j$ is locally Noetherian.", "\\end{enumerate}", "Moreover, if $X$ is locally Noetherian then every open subscheme", "is locally Noetherian." ], "refs": [], "proofs": [ { "contents": [ "To show this it suffices to show that being Noetherian is a local", "property of rings, see Lemma \\ref{lemma-locally-P}.", "Any localization of a Noetherian ring is Noetherian, see", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-permanence}.", "By Algebra, Lemma \\ref{algebra-lemma-cover} we see the second", "property to Definition \\ref{definition-property-local}." ], "refs": [ "properties-lemma-locally-P", "algebra-lemma-Noetherian-permanence", "algebra-lemma-cover", "properties-definition-property-local" ], "ref_ids": [ 2948, 448, 411, 3069 ] } ], "ref_ids": [] }, { "id": 2952, "type": "theorem", "label": "properties-lemma-immersion-into-noetherian", "categories": [ "properties" ], "title": "properties-lemma-immersion-into-noetherian", "contents": [ "Any immersion $Z \\to X$ with $X$ locally Noetherian is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "A closed immersion is clearly quasi-compact.", "A composition of quasi-compact morphisms is quasi-compact,", "see Topology, Lemma \\ref{topology-lemma-composition-quasi-compact}.", "Hence it suffices to show that an open immersion into", "a locally Noetherian scheme is quasi-compact.", "Using Schemes, Lemma \\ref{schemes-lemma-quasi-compact-affine}", "we reduce to the case where $X$ is affine.", "Any open subset of the spectrum of a Noetherian ring", "is quasi-compact (for example", "combine Algebra, Lemma \\ref{algebra-lemma-Noetherian-topology} and", "Topology, Lemmas \\ref{topology-lemma-Noetherian} and", "\\ref{topology-lemma-Noetherian-quasi-compact})." ], "refs": [ "topology-lemma-composition-quasi-compact", "schemes-lemma-quasi-compact-affine", "algebra-lemma-Noetherian-topology", "topology-lemma-Noetherian", "topology-lemma-Noetherian-quasi-compact" ], "ref_ids": [ 8228, 7697, 452, 8220, 8239 ] } ], "ref_ids": [] }, { "id": 2953, "type": "theorem", "label": "properties-lemma-locally-Noetherian-quasi-separated", "categories": [ "properties" ], "title": "properties-lemma-locally-Noetherian-quasi-separated", "contents": [ "A locally Noetherian scheme is quasi-separated." ], "refs": [], "proofs": [ { "contents": [ "By Schemes, Lemma \\ref{schemes-lemma-characterize-quasi-separated}", "we have to show that the intersection $U \\cap V$ of two", "affine opens of $X$ is quasi-compact. This follows from", "Lemma \\ref{lemma-immersion-into-noetherian} above on", "considering the open immersion $U \\cap V \\to U$ for example.", "(But really it is just because any open of the spectrum of a", "Noetherian ring is quasi-compact.)" ], "refs": [ "schemes-lemma-characterize-quasi-separated", "properties-lemma-immersion-into-noetherian" ], "ref_ids": [ 7709, 2952 ] } ], "ref_ids": [] }, { "id": 2954, "type": "theorem", "label": "properties-lemma-Noetherian-topology", "categories": [ "properties" ], "title": "properties-lemma-Noetherian-topology", "contents": [ "A (locally) Noetherian scheme has a (locally)", "Noetherian underlying topological space,", "see Topology, Definition \\ref{topology-definition-noetherian}." ], "refs": [ "topology-definition-noetherian" ], "proofs": [ { "contents": [ "This is because a Noetherian scheme is a finite union of spectra", "of Noetherian rings and", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-topology} and", "Topology, Lemma \\ref{topology-lemma-finite-union-Noetherian}." ], "refs": [ "algebra-lemma-Noetherian-topology", "topology-lemma-finite-union-Noetherian" ], "ref_ids": [ 452, 8222 ] } ], "ref_ids": [ 8355 ] }, { "id": 2955, "type": "theorem", "label": "properties-lemma-locally-closed-in-Noetherian", "categories": [ "properties" ], "title": "properties-lemma-locally-closed-in-Noetherian", "contents": [ "Any locally closed subscheme of a (locally) Noetherian", "scheme is (locally) Noetherian." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Any quotient, and any localization of a Noetherian", "ring is Noetherian. For the Noetherian case use again", "that any subset of a Noetherian space is a Noetherian space", "(with induced topology)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2956, "type": "theorem", "label": "properties-lemma-Noetherian-irreducible-components", "categories": [ "properties" ], "title": "properties-lemma-Noetherian-irreducible-components", "contents": [ "A Noetherian scheme has a finite number of irreducible components." ], "refs": [], "proofs": [ { "contents": [ "The underlying topological space of a Noetherian scheme is Noetherian", "(Lemma \\ref{lemma-Noetherian-topology})", "and we conclude because a Noetherian topological space", "has only finitely many irreducible components", "(Topology, Lemma \\ref{topology-lemma-Noetherian})." ], "refs": [ "properties-lemma-Noetherian-topology", "topology-lemma-Noetherian" ], "ref_ids": [ 2954, 8220 ] } ], "ref_ids": [] }, { "id": 2957, "type": "theorem", "label": "properties-lemma-morphism-Noetherian-schemes-quasi-compact", "categories": [ "properties" ], "title": "properties-lemma-morphism-Noetherian-schemes-quasi-compact", "contents": [ "Any morphism of schemes $f : X \\to Y$ with $X$ Noetherian", "is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Use Lemma \\ref{lemma-Noetherian-topology}", "and use that any subset of a Noetherian topological", "space is quasi-compact (see Topology,", "Lemmas \\ref{topology-lemma-Noetherian} and", "\\ref{topology-lemma-Noetherian-quasi-compact})." ], "refs": [ "properties-lemma-Noetherian-topology", "topology-lemma-Noetherian", "topology-lemma-Noetherian-quasi-compact" ], "ref_ids": [ 2954, 8220, 8239 ] } ], "ref_ids": [] }, { "id": 2958, "type": "theorem", "label": "properties-lemma-locally-Noetherian-closed-point", "categories": [ "properties" ], "title": "properties-lemma-locally-Noetherian-closed-point", "contents": [ "Any nonempty locally Noetherian scheme has a closed point.", "Any nonempty closed subset of a locally Noetherian scheme has a closed point.", "Equivalently, any point of a locally Noetherian scheme specializes", "to a closed point." ], "refs": [], "proofs": [ { "contents": [ "The second assertion follows from the first (using", "Schemes, Lemma \\ref{schemes-lemma-reduced-closed-subscheme}", "and Lemma \\ref{lemma-locally-closed-in-Noetherian}).", "Consider any nonempty affine open $U \\subset X$.", "Let $x \\in U$ be a closed point. If $x$ is a closed point", "of $X$ then we are done. If not, let $X_0 \\subset X$ be the", "reduced induced closed subscheme structure on $\\overline{\\{x\\}}$.", "Then $U_0 = U \\cap X_0$ is an affine open of $X_0$ by", "Schemes, Lemma \\ref{schemes-lemma-closed-subspace-scheme} and", "$U_0 = \\{x\\}$. Let $y \\in X_0$, $y \\not = x$ be a specialization of $x$.", "Consider the local ring $R = \\mathcal{O}_{X_0, y}$.", "This is a Noetherian local ring as $X_0$ is Noetherian", "by Lemma \\ref{lemma-locally-closed-in-Noetherian}. Denote $V \\subset \\Spec(R)$", "the inverse image of $U_0$ in $\\Spec(R)$ by the canonical morphism", "$\\Spec(R) \\to X_0$ (see Schemes, Section \\ref{schemes-section-points}.)", "By construction $V$ is a singleton with unique point corresponding to $x$ (use", "Schemes, Lemma \\ref{schemes-lemma-specialize-points}).", "By", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens}", "we see that $\\dim(R) = 1$.", "In other words, we see that $y$ is an immediate specialization", "of $x$ (see Topology, Definition \\ref{topology-definition-dimension-function}).", "In other words, any", "point $y \\not = x$ such that $x \\leadsto y$ is an immediate", "specialization of $x$. Clearly each of these points is a", "closed point as desired." ], "refs": [ "schemes-lemma-reduced-closed-subscheme", "properties-lemma-locally-closed-in-Noetherian", "schemes-lemma-closed-subspace-scheme", "properties-lemma-locally-closed-in-Noetherian", "schemes-lemma-specialize-points", "algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens", "topology-definition-dimension-function" ], "ref_ids": [ 7681, 2955, 7670, 2955, 7684, 687, 8367 ] } ], "ref_ids": [] }, { "id": 2959, "type": "theorem", "label": "properties-lemma-locally-Noetherian-specialization-dvr", "categories": [ "properties" ], "title": "properties-lemma-locally-Noetherian-specialization-dvr", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $x' \\leadsto x$ be a specialization of points of $X$.", "Then", "\\begin{enumerate}", "\\item there exists a discrete valuation ring $R$ and a morphism", "$f : \\Spec(R) \\to X$ such that the generic point $\\eta$ of", "$\\Spec(R)$ maps to $x'$ and the special point maps to $x$, and", "\\item given a finitely generated field extension $\\kappa(x') \\subset K$", "we may arrange it so that the extension $\\kappa(x') \\subset \\kappa(\\eta)$", "induced by $f$ is isomorphic to the given one.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $x' \\leadsto x$ be a specialization in $X$, and let", "$\\kappa(x') \\subset K$ be a finitely generated extension of fields. By", "Schemes, Lemma \\ref{schemes-lemma-specialize-points}", "and the discussion following", "Schemes, Lemma \\ref{schemes-lemma-characterize-points}", "this leads to ring maps $\\mathcal{O}_{X, x} \\to \\kappa(x') \\to K$.", "Let $R \\subset K$ be any discrete valuation ring whose field of fractions is", "$K$ and which dominates the image of $\\mathcal{O}_{X, x} \\to K$, see", "Algebra, Lemma \\ref{algebra-lemma-exists-dvr}.", "The ring map $\\mathcal{O}_{X, x} \\to R$ induces the morphism", "$f : \\Spec(R) \\to X$, see", "Schemes, Lemma \\ref{schemes-lemma-morphism-from-spec-local-ring}.", "This morphism has all the desired properties by construction." ], "refs": [ "schemes-lemma-specialize-points", "schemes-lemma-characterize-points", "algebra-lemma-exists-dvr", "schemes-lemma-morphism-from-spec-local-ring" ], "ref_ids": [ 7684, 7685, 1028, 7683 ] } ], "ref_ids": [] }, { "id": 2960, "type": "theorem", "label": "properties-lemma-thin-infinite-sequence", "categories": [ "properties" ], "title": "properties-lemma-thin-infinite-sequence", "contents": [ "Let $S$ be a Noetherian scheme. Let $T \\subset S$ be an infinite subset.", "Then there exists an infinite subset $T' \\subset T$", "such that there are no nontrivial specializations among the points $T'$." ], "refs": [], "proofs": [ { "contents": [ "Let $T_0 \\subset T$ be the set of $t \\in T$ which do not specialize", "to another point of $T$. If $T_0$ is infinite, then $T' = T_0$ works.", "Hence we may and do assume $T_0$ is finite.", "Inductively, for $i > 0$, consider the set $T_i \\subset T$", "of $t \\in T$ such that", "\\begin{enumerate}", "\\item $t \\not \\in T_{i - 1} \\cup T_{i - 2} \\cup \\ldots \\cup T_0$,", "\\item there exist a nontrivial specialization $t \\leadsto t'$ with", "$t' \\in T_{i - 1}$, and", "\\item for any nontrivial specialization", "$t \\leadsto t'$ with $t' \\in T$ we have", "$t' \\in T_{i - 1} \\cup T_{i - 2} \\cup \\ldots \\cup T_0$.", "\\end{enumerate}", "Again, if $T_i$ is infinite, then $T' = T_i$ works.", "Let $d$ be the maximum of the dimensions of the local rings", "$\\mathcal{O}_{S, t}$ for $t \\in T_0$; then $d$ is an integer", "because $T_0$ is finite and the dimensions of the local rings", "are finite by Algebra, Proposition \\ref{algebra-proposition-dimension}.", "Then $T_i = \\emptyset$ for $i > d$.", "Namely, if $t \\in T_i$ then we can find a sequence", "of nontrivial specializations", "$t = t_i \\leadsto t_{i - 1} \\leadsto \\ldots \\leadsto t_0$", "with $t_0 \\in T_0$. As", "the points $t = t_i, t_{i - 1}, \\ldots, t_0$ are in", "$\\Spec(\\mathcal{O}_{S, t_0})$", "(Schemes, Lemma \\ref{schemes-lemma-specialize-points}),", "we see that $i \\leq d$.", "Thus $\\bigcup T_i = T_d \\cup \\ldots \\cup T_0$ is a finite subset of $T$.", "\\medskip\\noindent", "Suppose $t \\in T$ is not in $\\bigcup T_i$. Then there must be a specialization", "$t \\leadsto t'$ with $t' \\in T$ and $t' \\not \\in \\bigcup T_i$. (Namely, if", "every specialization of $t$ is in the finite set $T_d \\cup \\ldots \\cup T_0$,", "then there is a maximum $i$ such that there is some specialization", "$t \\leadsto t'$ with $t' \\in T_i$ and then $t \\in T_{i + 1}$ by construction.)", "Hence we get an infinite sequence", "$$", "t \\leadsto t' \\leadsto t'' \\leadsto \\ldots", "$$", "of nontrivial specializations between points of $T \\setminus \\bigcup T_i$.", "This is impossible because the underlying topological space of $S$", "is Noetherian by Lemma \\ref{lemma-locally-Noetherian-quasi-separated}." ], "refs": [ "algebra-proposition-dimension", "schemes-lemma-specialize-points", "properties-lemma-locally-Noetherian-quasi-separated" ], "ref_ids": [ 1411, 7684, 2953 ] } ], "ref_ids": [] }, { "id": 2961, "type": "theorem", "label": "properties-lemma-maximal-points", "categories": [ "properties" ], "title": "properties-lemma-maximal-points", "contents": [ "Let $S$ be a Noetherian scheme. Let $T \\subset S$ be a subset. Let", "$T_0 \\subset T$ be the set of $t \\in T$ such that there is no nontrivial", "specialization $t' \\leadsto t$ with $t' \\in T'$. Then (a) there are", "no specializations among the points of $T_0$, (b) every point of", "$T$ is a specialization of a point of $T_0$, and (c) the closures", "of $T$ and $T_0$ are the same." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\dim(\\mathcal{O}_{S, s}) < \\infty$ for any $s \\in S$, see", "Algebra, Proposition \\ref{algebra-proposition-dimension}. Let $t \\in T$.", "If $t' \\leadsto t$, then by dimension theory", "$\\dim(\\mathcal{O}_{S, t'}) \\leq \\dim(\\mathcal{O}_{S, t})$", "with equality if and only if $t' = t$. Thus if we pick $t' \\leadsto t$", "with $\\dim(\\mathcal{O}_{T, t'})$ minimal, then $t' \\in T_0$.", "In other words, ", "every $t \\in T$ is the specialization of an element of $T_0$." ], "refs": [ "algebra-proposition-dimension" ], "ref_ids": [ 1411 ] } ], "ref_ids": [] }, { "id": 2962, "type": "theorem", "label": "properties-lemma-countable-dense-subset", "categories": [ "properties" ], "title": "properties-lemma-countable-dense-subset", "contents": [ "Let $S$ be a Noetherian scheme. Let $T \\subset S$ be an infinite dense subset.", "Then there exist a countable subset $E \\subset T$ which is dense in $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $T'$ be the set of points $s \\in S$ such that $\\overline{\\{s\\}} \\cap T$", "contains a countable subset whose closure is $\\overline{\\{s\\}}$.", "Since a finite set is countable we have $T \\subset T'$.", "For $s \\in T'$ choose such a countable subset", "$E_s \\subset \\overline{\\{s\\}} \\cap T$.", "Let $E' = \\{s_1, s_2, s_3, \\ldots\\} \\subset T'$", "be a countable subset. Then the closure of $E'$ in $S$ is the", "closure of the countable subset $\\bigcup_n E_{s_n}$ of $T$.", "It follows that if $Z$", "is an irreducible component of the closure of $E'$, then the generic", "point of $Z$ is in $T'$.", "\\medskip\\noindent", "Denote $T'_0 \\subset T'$ the subset of $t \\in T'$ such that", "there is no nontrivial specialization $t' \\leadsto t$ with $t' \\in T'$", "as in Lemma \\ref{lemma-maximal-points} whose results we will use", "without further mention. If $T'_0$ is infinite, then we choose a", "countable subset $E' \\subset T'_0$. By the argument in the first", "paragraph, the generic points of the irreducible components of the", "closure of $E'$ are in $T'$. However, since one of these points specializes to", "infinitely many distinct elements of $E' \\subset T'_0$", "this is a contradiction. Thus $T'_0$ is finite, say", "$T'_0 = \\{s_1, \\ldots, s_m\\}$. Then it follows that $S$, which is", "the closure of $T$, is contained in the closure of", "$\\{s_1, \\ldots, s_m\\}$, which in turn is contained in the closure", "of the countable subset $E_{s_1} \\cup \\ldots \\cup E_{s_m} \\subset T$", "as desired." ], "refs": [ "properties-lemma-maximal-points" ], "ref_ids": [ 2961 ] } ], "ref_ids": [] }, { "id": 2963, "type": "theorem", "label": "properties-lemma-affine-jacobson", "categories": [ "properties" ], "title": "properties-lemma-affine-jacobson", "contents": [ "An affine scheme $\\Spec(R)$ is Jacobson if and only if", "the ring $R$ is Jacobson." ], "refs": [], "proofs": [ { "contents": [ "This is Algebra, Lemma \\ref{algebra-lemma-jacobson}." ], "refs": [ "algebra-lemma-jacobson" ], "ref_ids": [ 469 ] } ], "ref_ids": [] }, { "id": 2964, "type": "theorem", "label": "properties-lemma-locally-jacobson", "categories": [ "properties" ], "title": "properties-lemma-locally-jacobson", "contents": [ "Let $X$ be a scheme. The following are equivalent:", "\\begin{enumerate}", "\\item The scheme $X$ is Jacobson.", "\\item The scheme $X$ is ``locally Jacobson'' in the sense of", "Definition \\ref{definition-locally-P}.", "\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$", "is Jacobson.", "\\item There exists an affine open covering $X = \\bigcup U_i$ such that", "each $\\mathcal{O}_X(U_i)$ is Jacobson.", "\\item There exists an open covering $X = \\bigcup X_j$", "such that each open subscheme $X_j$ is Jacobson.", "\\end{enumerate}", "Moreover, if $X$ is Jacobson then every open subscheme", "is Jacobson." ], "refs": [ "properties-definition-locally-P" ], "proofs": [ { "contents": [ "The final assertion of the lemma holds by", "Topology, Lemma \\ref{topology-lemma-jacobson-inherited}.", "The equivalence of (5) and (1) is", "Topology, Lemma \\ref{topology-lemma-jacobson-local}.", "Hence, using Lemma \\ref{lemma-affine-jacobson},", "we see that (1) $\\Leftrightarrow$ (2).", "To finish proving the lemma it suffices to show that ``Jacobson''", "is a local property of rings, see Lemma \\ref{lemma-locally-P}.", "Any localization of a Jacobson ring at an element is Jacobson, see", "Algebra, Lemma \\ref{algebra-lemma-Jacobson-invert-element}.", "Suppose $R$ is a ring, $f_1, \\ldots, f_n \\in R$ generate the unit", "ideal and each $R_{f_i}$ is Jacobson. Then we see that", "$\\Spec(R) = \\bigcup D(f_i)$ is a union of open subsets", "which are all Jacobson, and hence $\\Spec(R)$ is Jacobson", "by Topology, Lemma \\ref{topology-lemma-jacobson-local} again.", "This proves the second property of Definition \\ref{definition-property-local}." ], "refs": [ "topology-lemma-jacobson-inherited", "topology-lemma-jacobson-local", "properties-lemma-affine-jacobson", "properties-lemma-locally-P", "algebra-lemma-Jacobson-invert-element", "topology-lemma-jacobson-local", "properties-definition-property-local" ], "ref_ids": [ 8279, 8278, 2963, 2948, 475, 8278, 3069 ] } ], "ref_ids": [ 3070 ] }, { "id": 2965, "type": "theorem", "label": "properties-lemma-complement-closed-point-Jacobson", "categories": [ "properties" ], "title": "properties-lemma-complement-closed-point-Jacobson", "contents": [ "Examples of Noetherian Jacobson schemes.", "\\begin{enumerate}", "\\item If $(R, \\mathfrak m)$ is a Noetherian local ring, then", "the punctured spectrum $\\Spec(R) \\setminus \\{\\mathfrak m\\}$", "is a Jacobson scheme.", "\\item If $R$ is a Noetherian ring with Jacobson radical $\\text{rad}(R)$", "then $\\Spec(R) \\setminus V(\\text{rad}(R))$ is a Jacobson scheme.", "\\item If $(R, I)$ is a Zariski pair (More on Algebra, Definition", "\\ref{more-algebra-definition-zariski-pair})", "with $R$ Noetherian, then $\\Spec(R) \\setminus V(I)$ is a", "Jacobson scheme.", "\\end{enumerate}" ], "refs": [ "more-algebra-definition-zariski-pair" ], "proofs": [ { "contents": [ "Proof of (3). Observe that $\\Spec(R) - V(I)$ has a covering by the affine", "opens $\\Spec(R_f)$ for $f \\in I$. The rings $R_f$ are Jacobson by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-noetherian-zariski-jacobson-complement}.", "Hence $\\Spec(R) \\setminus V(I)$ is Jacobson by", "Lemma \\ref{lemma-locally-jacobson}.", "Parts (1) and (2) are special cases of (3).", "\\medskip\\noindent", "Direct proof of case (1).", "Since $\\Spec(R)$ is a Noetherian scheme,", "$S$ is a Noetherian scheme (Lemma \\ref{lemma-locally-closed-in-Noetherian}).", "Hence $S$ is a sober, Noetherian topological space (use", "Schemes, Lemma \\ref{schemes-lemma-scheme-sober}).", "Assume $S$ is not Jacobson to", "get a contradiction. By", "Topology, Lemma \\ref{topology-lemma-non-jacobson-Noetherian-characterize}", "there exists some non-closed point $\\xi \\in S$", "such that $\\{\\xi\\}$ is locally closed. This corresponds", "to a prime $\\mathfrak p \\subset R$ such that (1) there exists", "a prime $\\mathfrak q$, $\\mathfrak p \\subset \\mathfrak q \\subset \\mathfrak m$", "with both inclusions strict, and (2) $\\{\\mathfrak p\\}$ is open in", "$\\Spec(R/\\mathfrak p)$. This is impossible by Algebra,", "Lemma \\ref{algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens}." ], "refs": [ "more-algebra-lemma-noetherian-zariski-jacobson-complement", "properties-lemma-locally-jacobson", "properties-lemma-locally-closed-in-Noetherian", "schemes-lemma-scheme-sober", "topology-lemma-non-jacobson-Noetherian-characterize", "algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens" ], "ref_ids": [ 9856, 2964, 2955, 7672, 8277, 687 ] } ], "ref_ids": [ 10596 ] }, { "id": 2966, "type": "theorem", "label": "properties-lemma-locally-normal", "categories": [ "properties" ], "title": "properties-lemma-locally-normal", "contents": [ "Let $X$ be a scheme. The following are equivalent:", "\\begin{enumerate}", "\\item The scheme $X$ is normal.", "\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$", "is normal.", "\\item There exists an affine open covering $X = \\bigcup U_i$ such that", "each $\\mathcal{O}_X(U_i)$ is normal.", "\\item There exists an open covering $X = \\bigcup X_j$", "such that each open subscheme $X_j$ is normal.", "\\end{enumerate}", "Moreover, if $X$ is normal then every open subscheme", "is normal." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2967, "type": "theorem", "label": "properties-lemma-normal-reduced", "categories": [ "properties" ], "title": "properties-lemma-normal-reduced", "contents": [ "A normal scheme is reduced." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2968, "type": "theorem", "label": "properties-lemma-integral-normal", "categories": [ "properties" ], "title": "properties-lemma-integral-normal", "contents": [ "Let $X$ be an integral scheme.", "Then $X$ is normal if and only if for every affine open", "$U \\subset X$ the ring $\\mathcal{O}_X(U)$ is a normal domain." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Algebra, Lemma \\ref{algebra-lemma-normality-is-local}." ], "refs": [ "algebra-lemma-normality-is-local" ], "ref_ids": [ 510 ] } ], "ref_ids": [] }, { "id": 2969, "type": "theorem", "label": "properties-lemma-normal-locally-finite-nr-irreducibles", "categories": [ "properties" ], "title": "properties-lemma-normal-locally-finite-nr-irreducibles", "contents": [ "Let $X$ be a scheme such that any quasi-compact open has a finite number", "of irreducible components. The following are equivalent:", "\\begin{enumerate}", "\\item $X$ is normal, and", "\\item $X$ is a disjoint union of normal integral schemes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is immediate from the definitions that (2) implies (1).", "Let $X$ be a normal scheme such that every quasi-compact open", "has a finite number of irreducible components.", "If $X$ is affine then $X$ satisfies (2) by", "Algebra, Lemma \\ref{algebra-lemma-characterize-reduced-ring-normal}.", "For a general $X$, let $X = \\bigcup X_i$ be", "an affine open covering. Note that also each $X_i$ has", "but a finite number of irreducible components, and the lemma holds", "for each $X_i$. Let $T \\subset X$ be an irreducible component.", "By the affine case each intersection $T \\cap X_i$ is open in $X_i$", "and an integral normal scheme.", "Hence $T \\subset X$ is open, and an integral normal scheme.", "This proves that $X$ is the disjoint union of its irreducible components,", "which are integral normal schemes." ], "refs": [ "algebra-lemma-characterize-reduced-ring-normal" ], "ref_ids": [ 515 ] } ], "ref_ids": [] }, { "id": 2970, "type": "theorem", "label": "properties-lemma-normal-Noetherian", "categories": [ "properties" ], "title": "properties-lemma-normal-Noetherian", "contents": [ "Let $X$ be a Noetherian scheme.", "The following are equivalent:", "\\begin{enumerate}", "\\item $X$ is normal, and", "\\item $X$ is a finite disjoint union of normal integral schemes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-normal-locally-finite-nr-irreducibles} because a Noetherian", "scheme has a Noetherian underlying topological space", "(Lemma \\ref{lemma-Noetherian-topology}", "and", "Topology, Lemma \\ref{topology-lemma-Noetherian}." ], "refs": [ "properties-lemma-normal-locally-finite-nr-irreducibles", "properties-lemma-Noetherian-topology", "topology-lemma-Noetherian" ], "ref_ids": [ 2969, 2954, 8220 ] } ], "ref_ids": [] }, { "id": 2971, "type": "theorem", "label": "properties-lemma-normal-locally-Noetherian", "categories": [ "properties" ], "title": "properties-lemma-normal-locally-Noetherian", "contents": [ "Let $X$ be a locally Noetherian scheme.", "The following are equivalent:", "\\begin{enumerate}", "\\item $X$ is normal, and", "\\item $X$ is a disjoint union of integral normal schemes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is purely topological from", "Lemma \\ref{lemma-normal-Noetherian}." ], "refs": [ "properties-lemma-normal-Noetherian" ], "ref_ids": [ 2970 ] } ], "ref_ids": [] }, { "id": 2972, "type": "theorem", "label": "properties-lemma-normal-integral-sections", "categories": [ "properties" ], "title": "properties-lemma-normal-integral-sections", "contents": [ "\\begin{slogan}", "The ring of functions on a normal scheme is normal.", "\\end{slogan}", "Let $X$ be an integral normal scheme.", "Then $\\Gamma(X, \\mathcal{O}_X)$ is a normal domain." ], "refs": [], "proofs": [ { "contents": [ "Set $R = \\Gamma(X, \\mathcal{O}_X)$.", "It is clear that $R$ is a domain.", "Suppose $f = a/b$ is an element of its fraction field", "which is integral over $R$. Say we have", "$f^d + \\sum_{i = 0, \\ldots, d - 1} a_i f^i = 0$ with", "$a_i \\in R$. Let $U \\subset X$ be affine open.", "Since $b \\in R$ is not zero and since $X$ is integral we see", "that also $b|_U \\in \\mathcal{O}_X(U)$ is not zero.", "Hence $a/b$ is an element of the fraction field of", "$\\mathcal{O}_X(U)$ which is integral over $\\mathcal{O}_X(U)$", "(because we can use the same polynomial", "$f^d + \\sum_{i = 0, \\ldots, d - 1} a_i|_U f^i = 0$ on $U$).", "Since $\\mathcal{O}_X(U)$ is a normal domain", "(Lemma \\ref{lemma-locally-normal}), we see that", "$f_U = (a|_U)/(b|_U) \\in \\mathcal{O}_X(U)$. It is easy to", "see that $f_U|_V = f_V$ whenever $V \\subset U \\subset X$ are", "affine open. Hence the local sections $f_U$ glue to a global", "section $f$ as desired." ], "refs": [ "properties-lemma-locally-normal" ], "ref_ids": [ 2966 ] } ], "ref_ids": [] }, { "id": 2973, "type": "theorem", "label": "properties-lemma-characterize-Cohen-Macaulay", "categories": [ "properties" ], "title": "properties-lemma-characterize-Cohen-Macaulay", "contents": [ "Let $X$ be a scheme. The following are equivalent:", "\\begin{enumerate}", "\\item $X$ is Cohen-Macaulay,", "\\item $X$ is locally Noetherian and all of its local rings are Cohen-Macaulay,", "and", "\\item $X$ is locally Noetherian and for any closed point $x \\in X$", "the local ring $\\mathcal{O}_{X, x}$ is Cohen-Macaulay.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Algebra, Lemma \\ref{algebra-lemma-localize-CM} says that the localization of", "a Cohen-Macaulay local ring is Cohen-Macaulay. The lemma follows", "by combining this with Lemma \\ref{lemma-locally-Noetherian},", "with the existence of closed", "points on locally Noetherian schemes", "(Lemma \\ref{lemma-locally-Noetherian-closed-point}), and", "the definitions." ], "refs": [ "algebra-lemma-localize-CM", "properties-lemma-locally-Noetherian", "properties-lemma-locally-Noetherian-closed-point" ], "ref_ids": [ 926, 2951, 2958 ] } ], "ref_ids": [] }, { "id": 2974, "type": "theorem", "label": "properties-lemma-locally-Cohen-Macaulay", "categories": [ "properties" ], "title": "properties-lemma-locally-Cohen-Macaulay", "contents": [ "Let $X$ be a scheme. The following are equivalent:", "\\begin{enumerate}", "\\item The scheme $X$ is Cohen-Macaulay.", "\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$", "is Noetherian and Cohen-Macaulay.", "\\item There exists an affine open covering $X = \\bigcup U_i$ such that", "each $\\mathcal{O}_X(U_i)$ is Noetherian and Cohen-Macaulay.", "\\item There exists an open covering $X = \\bigcup X_j$", "such that each open subscheme $X_j$ is Cohen-Macaulay.", "\\end{enumerate}", "Moreover, if $X$ is Cohen-Macaulay then every open subscheme", "is Cohen-Macaulay." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-locally-Noetherian}", "and \\ref{lemma-characterize-Cohen-Macaulay}." ], "refs": [ "properties-lemma-locally-Noetherian", "properties-lemma-characterize-Cohen-Macaulay" ], "ref_ids": [ 2951, 2973 ] } ], "ref_ids": [] }, { "id": 2975, "type": "theorem", "label": "properties-lemma-characterize-regular", "categories": [ "properties" ], "title": "properties-lemma-characterize-regular", "contents": [ "Let $X$ be a scheme. The following are equivalent:", "\\begin{enumerate}", "\\item $X$ is regular,", "\\item $X$ is locally Noetherian and all of its local rings are regular,", "and", "\\item $X$ is locally Noetherian and for any closed point $x \\in X$", "the local ring $\\mathcal{O}_{X, x}$ is regular.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By the discussion in Algebra preceding Algebra, Definition", "\\ref{algebra-definition-regular} we know that the localization of", "a regular local ring is regular. The lemma follows", "by combining this with Lemma \\ref{lemma-locally-Noetherian},", "with the existence of closed", "points on locally Noetherian schemes", "(Lemma \\ref{lemma-locally-Noetherian-closed-point}), and", "the definitions." ], "refs": [ "algebra-definition-regular", "properties-lemma-locally-Noetherian", "properties-lemma-locally-Noetherian-closed-point" ], "ref_ids": [ 1512, 2951, 2958 ] } ], "ref_ids": [] }, { "id": 2976, "type": "theorem", "label": "properties-lemma-locally-regular", "categories": [ "properties" ], "title": "properties-lemma-locally-regular", "contents": [ "Let $X$ be a scheme. The following are equivalent:", "\\begin{enumerate}", "\\item The scheme $X$ is regular.", "\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$", "is Noetherian and regular.", "\\item There exists an affine open covering $X = \\bigcup U_i$ such that", "each $\\mathcal{O}_X(U_i)$ is Noetherian and regular.", "\\item There exists an open covering $X = \\bigcup X_j$", "such that each open subscheme $X_j$ is regular.", "\\end{enumerate}", "Moreover, if $X$ is regular then every open subscheme is regular." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-locally-Noetherian}", "and \\ref{lemma-characterize-regular}." ], "refs": [ "properties-lemma-locally-Noetherian", "properties-lemma-characterize-regular" ], "ref_ids": [ 2951, 2975 ] } ], "ref_ids": [] }, { "id": 2977, "type": "theorem", "label": "properties-lemma-regular-normal", "categories": [ "properties" ], "title": "properties-lemma-regular-normal", "contents": [ "A regular scheme is normal." ], "refs": [], "proofs": [ { "contents": [ "See", "Algebra, Lemma \\ref{algebra-lemma-regular-normal}." ], "refs": [ "algebra-lemma-regular-normal" ], "ref_ids": [ 1312 ] } ], "ref_ids": [] }, { "id": 2978, "type": "theorem", "label": "properties-lemma-dimension", "categories": [ "properties" ], "title": "properties-lemma-dimension", "contents": [ "Let $X$ be a scheme. The following are equal", "\\begin{enumerate}", "\\item The dimension of $X$.", "\\item The supremum of the dimensions of the local rings of $X$.", "\\item The supremum of $\\dim_x(X)$ for $x \\in X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that given a chain of specializations", "$$", "\\xi_n \\leadsto \\xi_{n - 1} \\leadsto \\ldots \\leadsto \\xi_0", "$$", "of points of $X$ all of the points $\\xi_i$ correspond to prime ideals", "of the local ring of $X$ at $\\xi_0$ by", "Schemes, Lemma \\ref{schemes-lemma-specialize-points}.", "Hence we see that the dimension of $X$ is the supremum of the dimensions", "of its local rings. In particular $\\dim_x(X) \\geq \\dim(\\mathcal{O}_{X, x})$", "as $\\dim_x(X)$ is the minimum of the dimensions of open neighbourhoods of", "$x$. Thus $\\sup_{x \\in X} \\dim_x(X) \\geq \\dim(X)$. On the other hand,", "it is clear that $\\sup_{x \\in X} \\dim_x(X) \\leq \\dim(X)$", "as $\\dim(U) \\leq \\dim(X)$ for any open subset of $X$." ], "refs": [ "schemes-lemma-specialize-points" ], "ref_ids": [ 7684 ] } ], "ref_ids": [] }, { "id": 2979, "type": "theorem", "label": "properties-lemma-codimension-local-ring", "categories": [ "properties" ], "title": "properties-lemma-codimension-local-ring", "contents": [ "Let $X$ be a scheme. Let $Y \\subset X$ be an irreducible closed", "subset. Let $\\xi \\in Y$ be the generic point. Then", "$$", "\\text{codim}(Y, X) = \\dim(\\mathcal{O}_{X, \\xi})", "$$", "where the codimension is as defined in", "Topology, Definition \\ref{topology-definition-codimension}." ], "refs": [ "topology-definition-codimension" ], "proofs": [ { "contents": [ "By Topology, Lemma \\ref{topology-lemma-codimension-at-generic-point}", "we may replace $X$ by an affine open neighbourhood of $\\xi$. In this", "case the result follows easily from", "Algebra, Lemma \\ref{algebra-lemma-irreducible-components-containing-x}." ], "refs": [ "topology-lemma-codimension-at-generic-point", "algebra-lemma-irreducible-components-containing-x" ], "ref_ids": [ 8225, 424 ] } ], "ref_ids": [ 8358 ] }, { "id": 2980, "type": "theorem", "label": "properties-lemma-generic-point", "categories": [ "properties" ], "title": "properties-lemma-generic-point", "contents": [ "Let $X$ be a scheme. Let $x \\in X$. Then $x$ is a generic point of", "an irreducible component of $X$ if and only if $\\dim(\\mathcal{O}_{X, x}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-codimension-local-ring} for example." ], "refs": [ "properties-lemma-codimension-local-ring" ], "ref_ids": [ 2979 ] } ], "ref_ids": [] }, { "id": 2981, "type": "theorem", "label": "properties-lemma-locally-Noetherian-dimension-0", "categories": [ "properties" ], "title": "properties-lemma-locally-Noetherian-dimension-0", "contents": [ "A locally Noetherian scheme of dimension $0$ is a disjoint", "union of spectra of Artinian local rings." ], "refs": [], "proofs": [ { "contents": [ "A Noetherian ring of dimension $0$ is a finite product of Artinian local", "rings, see", "Algebra, Proposition \\ref{algebra-proposition-dimension-zero-ring}.", "Hence an affine open of a locally Noetherian scheme $X$ of dimension", "$0$ has discrete underlying topological space. This implies that", "the topology on $X$ is discrete. The lemma follows easily from these", "remarks." ], "refs": [ "algebra-proposition-dimension-zero-ring" ], "ref_ids": [ 1410 ] } ], "ref_ids": [] }, { "id": 2982, "type": "theorem", "label": "properties-lemma-dimension-zero", "categories": [ "properties" ], "title": "properties-lemma-dimension-zero", "contents": [ "\\begin{reference}", "Email from Ofer Gabber dated June 4, 2016", "\\end{reference}", "Let $X$ be a scheme of dimension zero. The following are equivalent", "\\begin{enumerate}", "\\item $X$ is quasi-separated,", "\\item $X$ is separated,", "\\item $X$ is Hausdorff,", "\\item every affine open is closed.", "\\end{enumerate}", "In this case the connected components of $X$ are points." ], "refs": [], "proofs": [ { "contents": [ "As the dimension of $X$ is zero, we see that for any affine open", "$U \\subset X$ the space $U$ is profinite and satisfies", "a bunch of other properties which we will use freely below, see", "Algebra, Lemma \\ref{algebra-lemma-ring-with-only-minimal-primes}.", "We choose an affine open covering $X = \\bigcup U_i$.", "\\medskip\\noindent", "If (4) holds, then $U_i \\cap U_j$ is a closed subset of", "$U_i$, hence quasi-compact, hence $X$ is quasi-separated,", "by Schemes, Lemma \\ref{schemes-lemma-characterize-quasi-separated},", "hence (1) holds.", "\\medskip\\noindent", "If (1) holds, then $U_i \\cap U_j$ is a quasi-compact open", "of $U_i$ hence closed in $U_i$. Then $U_i \\cap U_j \\to U_i$", "is an open immersion whose image is closed, hence it is a", "closed immersion. In particular $U_i \\cap U_j$ is affine", "and $\\mathcal{O}(U_i) \\to \\mathcal{O}_X(U_i \\cap U_j)$ is surjective.", "Thus $X$ is separated", "by Schemes, Lemma \\ref{schemes-lemma-characterize-quasi-separated},", "hence (2) holds.", "\\medskip\\noindent", "Assume (2) and let $x, y \\in X$. Say $x \\in U_i$. If $y \\in U_i$", "too, then we can find disjoint open neighbourhoods of $x$ and $y$", "because $U_i$ is Hausdorff. Say $y \\not \\in U_i$ and $y \\in U_j$.", "Then $y \\not \\in U_i \\cap U_j$ which is an affine open of $U_j$", "and hence closed in $U_j$. Thus we can find an open neighbourhood", "of $y$ not meeting $U_i$ and we conclude that $X$ is Hausdorff,", "hence (3) holds.", "\\medskip\\noindent", "Assume (3). Let $U \\subset X$ be affine open.", "Then $U$ is closed in $X$ by Topology, Lemma", "\\ref{topology-lemma-quasi-compact-in-Hausdorff}.", "This proves (4) holds.", "\\medskip\\noindent", "We omit the proof of the final statement." ], "refs": [ "algebra-lemma-ring-with-only-minimal-primes", "schemes-lemma-characterize-quasi-separated", "schemes-lemma-characterize-quasi-separated", "topology-lemma-quasi-compact-in-Hausdorff" ], "ref_ids": [ 426, 7709, 7709, 8230 ] } ], "ref_ids": [] }, { "id": 2983, "type": "theorem", "label": "properties-lemma-catenary-local", "categories": [ "properties" ], "title": "properties-lemma-catenary-local", "contents": [ "Let $S$ be a scheme. The following are equivalent", "\\begin{enumerate}", "\\item $S$ is catenary,", "\\item there exists an open covering of $S$ all of whose members are", "catenary schemes,", "\\item for every affine open $\\Spec(R) = U \\subset S$ the ring", "$R$ is catenary, and", "\\item there exists an affine open covering $S = \\bigcup U_i$ such", "that each $U_i$ is the spectrum of a catenary ring.", "\\end{enumerate}", "Moreover, in this case any locally closed subscheme of $S$ is catenary", "as well." ], "refs": [], "proofs": [ { "contents": [ "Combine Topology, Lemma \\ref{topology-lemma-catenary}, and", "Algebra, Lemma \\ref{algebra-lemma-catenary}." ], "refs": [ "topology-lemma-catenary", "algebra-lemma-catenary" ], "ref_ids": [ 8226, 931 ] } ], "ref_ids": [] }, { "id": 2984, "type": "theorem", "label": "properties-lemma-catenary-dimension-function", "categories": [ "properties" ], "title": "properties-lemma-catenary-dimension-function", "contents": [ "Let $S$ be a locally Noetherian scheme.", "The following are equivalent:", "\\begin{enumerate}", "\\item $S$ is catenary, and", "\\item locally in the Zariski topology there exists a dimension function", "on $S$ (see Topology, Definition \\ref{topology-definition-dimension-function}).", "\\end{enumerate}" ], "refs": [ "topology-definition-dimension-function" ], "proofs": [ { "contents": [ "This follows from", "Topology, Lemmas", "\\ref{topology-lemma-catenary},", "\\ref{topology-lemma-dimension-function-catenary}, and", "\\ref{topology-lemma-locally-dimension-function},", "Schemes, Lemma \\ref{schemes-lemma-scheme-sober}", "and finally Lemma \\ref{lemma-Noetherian-topology}." ], "refs": [ "topology-lemma-catenary", "topology-lemma-dimension-function-catenary", "topology-lemma-locally-dimension-function", "schemes-lemma-scheme-sober", "properties-lemma-Noetherian-topology" ], "ref_ids": [ 8226, 8291, 8293, 7672, 2954 ] } ], "ref_ids": [ 8367 ] }, { "id": 2985, "type": "theorem", "label": "properties-lemma-catenary-local-rings-catenary", "categories": [ "properties" ], "title": "properties-lemma-catenary-local-rings-catenary", "contents": [ "Let $X$ be a scheme. The following are equivalent", "\\begin{enumerate}", "\\item $X$ is catenary, and", "\\item for any $x \\in X$ the local ring $\\mathcal{O}_{X, x}$ is", "catenary.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $X$ is catenary. Let $x \\in X$. By Lemma \\ref{lemma-catenary-local}", "we may replace $X$ by an affine open neighbourhood of $x$, and", "then $\\Gamma(X, \\mathcal{O}_X)$ is a catenary ring. By", "Algebra, Lemma \\ref{algebra-lemma-localization-catenary} any", "localization of a catenary ring is", "catenary. Whence $\\mathcal{O}_{X, x}$ is catenary.", "\\medskip\\noindent", "Conversely assume all local rings of $X$ are catenary.", "Let $Y \\subset Y'$ be an inclusion of irreducible closed", "subsets of $X$. Let $\\xi \\in Y$ be the generic point.", "Let $\\mathfrak p \\subset \\mathcal{O}_{X, \\xi}$ be the prime", "corresponding to the generic point of $Y'$, see", "Schemes, Lemma \\ref{schemes-lemma-specialize-points}. By that same", "lemma the irreducible closed subsets of $X$ in between $Y$ and $Y'$", "correspond to primes $\\mathfrak q \\subset \\mathcal{O}_{X, \\xi}$", "with $\\mathfrak p \\subset \\mathfrak q \\subset \\mathfrak m_{\\xi}$.", "Hence we see all maximal chains of these are finite and have the", "same length as $\\mathcal{O}_{X, \\xi}$ is a catenary ring." ], "refs": [ "properties-lemma-catenary-local", "algebra-lemma-localization-catenary", "schemes-lemma-specialize-points" ], "ref_ids": [ 2983, 932, 7684 ] } ], "ref_ids": [] }, { "id": 2986, "type": "theorem", "label": "properties-lemma-scheme-regular-iff-all-Rk", "categories": [ "properties" ], "title": "properties-lemma-scheme-regular-iff-all-Rk", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Then $X$ is regular if and only if $X$ has $(R_k)$ for all $k \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-characterize-regular} and the definitions." ], "refs": [ "properties-lemma-characterize-regular" ], "ref_ids": [ 2975 ] } ], "ref_ids": [] }, { "id": 2987, "type": "theorem", "label": "properties-lemma-scheme-CM-iff-all-Sk", "categories": [ "properties" ], "title": "properties-lemma-scheme-CM-iff-all-Sk", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Then $X$ is Cohen-Macaulay if and only if $X$ has $(S_k)$ for all $k \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-characterize-Cohen-Macaulay}", "we reduce to looking at local rings.", "Hence the lemma is true because a Noetherian local ring is Cohen-Macaulay", "if and only if it has depth equal to its dimension." ], "refs": [ "properties-lemma-characterize-Cohen-Macaulay" ], "ref_ids": [ 2973 ] } ], "ref_ids": [] }, { "id": 2988, "type": "theorem", "label": "properties-lemma-criterion-reduced", "categories": [ "properties" ], "title": "properties-lemma-criterion-reduced", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Then $X$ is reduced if and only if $X$ has properties $(S_1)$ and $(R_0)$." ], "refs": [], "proofs": [ { "contents": [ "This is Algebra, Lemma \\ref{algebra-lemma-criterion-reduced}." ], "refs": [ "algebra-lemma-criterion-reduced" ], "ref_ids": [ 1310 ] } ], "ref_ids": [] }, { "id": 2989, "type": "theorem", "label": "properties-lemma-criterion-normal", "categories": [ "properties" ], "title": "properties-lemma-criterion-normal", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Then $X$ is normal if and only if $X$ has properties $(S_2)$ and $(R_1)$." ], "refs": [], "proofs": [ { "contents": [ "This is Algebra, Lemma \\ref{algebra-lemma-criterion-normal}." ], "refs": [ "algebra-lemma-criterion-normal" ], "ref_ids": [ 1311 ] } ], "ref_ids": [] }, { "id": 2990, "type": "theorem", "label": "properties-lemma-normal-dimension-1-regular", "categories": [ "properties" ], "title": "properties-lemma-normal-dimension-1-regular", "contents": [ "Let $X$ be a locally Noetherian scheme which is normal and", "has dimension $\\leq 1$. Then $X$ is regular." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-criterion-normal} and the definitions." ], "refs": [ "properties-lemma-criterion-normal" ], "ref_ids": [ 2989 ] } ], "ref_ids": [] }, { "id": 2991, "type": "theorem", "label": "properties-lemma-normal-dimension-2-Cohen-Macaulay", "categories": [ "properties" ], "title": "properties-lemma-normal-dimension-2-Cohen-Macaulay", "contents": [ "Let $X$ be a locally Noetherian scheme which is normal and", "has dimension $\\leq 2$. Then $X$ is Cohen-Macaulay." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-criterion-normal} and the definitions." ], "refs": [ "properties-lemma-criterion-normal" ], "ref_ids": [ 2989 ] } ], "ref_ids": [] }, { "id": 2992, "type": "theorem", "label": "properties-lemma-nagata-locally-Noetherian", "categories": [ "properties" ], "title": "properties-lemma-nagata-locally-Noetherian", "contents": [ "A Nagata scheme is locally Noetherian." ], "refs": [], "proofs": [ { "contents": [ "This is true because a Nagata ring is Noetherian by definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2993, "type": "theorem", "label": "properties-lemma-locally-Japanese", "categories": [ "properties" ], "title": "properties-lemma-locally-Japanese", "contents": [ "Let $X$ be an integral scheme. The following are equivalent:", "\\begin{enumerate}", "\\item The scheme $X$ is Japanese.", "\\item For every affine open $U \\subset X$ the domain $\\mathcal{O}_X(U)$", "is Japanese.", "\\item There exists an affine open covering $X = \\bigcup U_i$", "such that each $\\mathcal{O}_X(U_i)$ is Japanese.", "\\item There exists an open covering $X = \\bigcup X_j$", "such that each open subscheme $X_j$ is Japanese.", "\\end{enumerate}", "Moreover, if $X$ is Japanese then every open subscheme", "is Japanese." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-locally-P} and", "Algebra, Lemmas \\ref{algebra-lemma-localize-N} and", "\\ref{algebra-lemma-Japanese-local}." ], "refs": [ "properties-lemma-locally-P", "algebra-lemma-localize-N", "algebra-lemma-Japanese-local" ], "ref_ids": [ 2948, 1333, 1334 ] } ], "ref_ids": [] }, { "id": 2994, "type": "theorem", "label": "properties-lemma-locally-universally-Japanese", "categories": [ "properties" ], "title": "properties-lemma-locally-universally-Japanese", "contents": [ "Let $X$ be a scheme. The following are equivalent:", "\\begin{enumerate}", "\\item The scheme $X$ is universally Japanese.", "\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$", "is universally Japanese.", "\\item There exists an affine open covering $X = \\bigcup U_i$", "such that each $\\mathcal{O}_X(U_i)$ is universally Japanese.", "\\item There exists an open covering $X = \\bigcup X_j$", "such that each open subscheme $X_j$ is universally Japanese.", "\\end{enumerate}", "Moreover, if $X$ is universally Japanese then every open subscheme", "is universally Japanese." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-locally-P} and", "Algebra, Lemmas \\ref{algebra-lemma-universally-japanese} and", "\\ref{algebra-lemma-nagata-local}." ], "refs": [ "properties-lemma-locally-P", "algebra-lemma-universally-japanese", "algebra-lemma-nagata-local" ], "ref_ids": [ 2948, 1349, 1352 ] } ], "ref_ids": [] }, { "id": 2995, "type": "theorem", "label": "properties-lemma-locally-nagata", "categories": [ "properties" ], "title": "properties-lemma-locally-nagata", "contents": [ "Let $X$ be a scheme. The following are equivalent:", "\\begin{enumerate}", "\\item The scheme $X$ is Nagata.", "\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$", "is Nagata.", "\\item There exists an affine open covering $X = \\bigcup U_i$", "such that each $\\mathcal{O}_X(U_i)$ is Nagata.", "\\item There exists an open covering $X = \\bigcup X_j$", "such that each open subscheme $X_j$ is Nagata.", "\\end{enumerate}", "Moreover, if $X$ is Nagata then every open subscheme is Nagata." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-locally-P} and", "Algebra, Lemmas \\ref{algebra-lemma-nagata-localize} and", "\\ref{algebra-lemma-nagata-local}." ], "refs": [ "properties-lemma-locally-P", "algebra-lemma-nagata-localize", "algebra-lemma-nagata-local" ], "ref_ids": [ 2948, 1351, 1352 ] } ], "ref_ids": [] }, { "id": 2996, "type": "theorem", "label": "properties-lemma-characterize-nagata", "categories": [ "properties" ], "title": "properties-lemma-characterize-nagata", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Then $X$ is Nagata if and only if every integral closed subscheme", "$Z \\subset X$ is Japanese." ], "refs": [], "proofs": [ { "contents": [ "Assume $X$ is Nagata. Let $Z \\subset X$ be an integral closed subscheme.", "Let $z \\in Z$.", "Let $\\Spec(A) = U \\subset X$ be an affine open containing $z$", "such that $A$ is Nagata. Then", "$Z \\cap U \\cong \\Spec(A/\\mathfrak p)$ for some prime $\\mathfrak p$,", "see Schemes, Lemma \\ref{schemes-lemma-closed-subspace-scheme} (and", "Definition \\ref{definition-integral}). By", "Algebra, Definition \\ref{algebra-definition-nagata} we see", "that $A/\\mathfrak p$ is Japanese. Hence $Z$ is Japanese by definition.", "\\medskip\\noindent", "Assume every integral closed subscheme of $X$ is Japanese.", "Let $\\Spec(A) = U \\subset X$ be any affine open.", "As $X$ is locally Noetherian we see that $A$ is Noetherian", "(Lemma \\ref{lemma-locally-Noetherian}). Let $\\mathfrak p \\subset A$", "be a prime ideal. We have to show that $A/\\mathfrak p$ is Japanese.", "Let $T \\subset U$ be the closed subset $V(\\mathfrak p) \\subset \\Spec(A)$.", "Let $\\overline{T} \\subset X$ be the closure. Then $\\overline{T}$ is", "irreducible as the closure of an irreducible subset. Hence the reduced", "closed subscheme defined by $\\overline{T}$ is an integral closed", "subscheme (called $\\overline{T}$ again), see", "Schemes, Lemma \\ref{schemes-lemma-reduced-closed-subscheme}.", "In other words, $\\Spec(A/\\mathfrak p)$ is an affine", "open of an integral closed subscheme of $X$. This subscheme is Japanese", "by assumption and by Lemma \\ref{lemma-locally-Japanese} we see that", "$A/\\mathfrak p$ is Japanese." ], "refs": [ "schemes-lemma-closed-subspace-scheme", "properties-definition-integral", "algebra-definition-nagata", "properties-lemma-locally-Noetherian", "schemes-lemma-reduced-closed-subscheme", "properties-lemma-locally-Japanese" ], "ref_ids": [ 7670, 3068, 1552, 2951, 7681, 2993 ] } ], "ref_ids": [] }, { "id": 2997, "type": "theorem", "label": "properties-lemma-nagata-universally-Japanese", "categories": [ "properties" ], "title": "properties-lemma-nagata-universally-Japanese", "contents": [ "Let $X$ be a scheme.", "The following are equivalent:", "\\begin{enumerate}", "\\item $X$ is Nagata, and", "\\item $X$ is locally Noetherian and universally Japanese.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is", "Algebra, Proposition \\ref{algebra-proposition-nagata-universally-japanese}." ], "refs": [ "algebra-proposition-nagata-universally-japanese" ], "ref_ids": [ 1430 ] } ], "ref_ids": [] }, { "id": 2998, "type": "theorem", "label": "properties-lemma-normal-geometrically-unibranch", "categories": [ "properties" ], "title": "properties-lemma-normal-geometrically-unibranch", "contents": [ "A normal scheme is geometrically unibranch." ], "refs": [], "proofs": [ { "contents": [ "This follows from the definitions. Namely, a scheme", "is normal if the local rings are normal domains. It is immediate", "from the More on Algebra, Definition \\ref{more-algebra-definition-unibranch}", "that a local normal domain is geometrically unibranch." ], "refs": [ "more-algebra-definition-unibranch" ], "ref_ids": [ 10637 ] } ], "ref_ids": [] }, { "id": 2999, "type": "theorem", "label": "properties-lemma-geometrically-unibranch", "categories": [ "properties" ], "title": "properties-lemma-geometrically-unibranch", "contents": [ "\\begin{reference}", "Compare with \\cite[Proposition 2.3]{Etale-coverings}", "\\end{reference}", "Let $X$ be a Noetherian scheme. The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically unibranch (Definition \\ref{definition-unibranch}),", "\\item for every point $x \\in X$ which is not the generic point of", "an irreducible component of $X$, the punctured spectrum of the", "strict henselization $\\mathcal{O}_{X, x}^{sh}$ is connected.", "\\end{enumerate}" ], "refs": [ "properties-definition-unibranch" ], "proofs": [ { "contents": [ "More on Algebra, Lemma \\ref{more-algebra-lemma-geometrically-unibranch}", "shows that (1) implies that the punctured spectra in (2) are", "irreducible and in particular connected.", "\\medskip\\noindent", "Assume (2). Let $x \\in X$. We have to show that $\\mathcal{O}_{X, x}$", "is geometrically unibranch. By induction on $\\dim(\\mathcal{O}_{X, x})$", "we may assume that the result holds for every nontrivial generalization of $x$.", "We may replace $X$ by $\\Spec(\\mathcal{O}_{X, x})$. In other words,", "we may assume that $X = \\Spec(A)$ with $A$ local and that", "$A_\\mathfrak p$ is geometrically unibranch for each nonmaximal", "prime $\\mathfrak p \\subset A$.", "\\medskip\\noindent", "Let $A^{sh}$ be the strict henselization of $A$. If", "$\\mathfrak q \\subset A^{sh}$ is a prime lying over $\\mathfrak p \\subset A$,", "then $A_\\mathfrak p \\to A^{sh}_\\mathfrak q$ is a", "filtered colimit of \\'etale algebras. Hence the strict henselizations of", "$A_\\mathfrak p$ and $A^{sh}_\\mathfrak q$ are isomorphic.", "Thus by More on Algebra, Lemma \\ref{more-algebra-lemma-geometrically-unibranch}", "we conclude that $A^{sh}_\\mathfrak q$", "has a unique minimal prime ideal for every nonmaximal prime $\\mathfrak q$ of", "$A^{sh}$.", "\\medskip\\noindent", "Let $\\mathfrak q_1, \\ldots, \\mathfrak q_r$ be the minimal primes", "of $A^{sh}$. We have to show that $r = 1$. By the above", "we see that $V(\\mathfrak q_1) \\cap V(\\mathfrak q_j) = \\{\\mathfrak m^{sh}\\}$", "for $j = 2, \\ldots, r$. Hence $V(\\mathfrak q_1) \\setminus \\{\\mathfrak m^{sh}\\}$", "is an open and closed subset of the punctured spectrum of $A^{sh}$", "which is a contradiction with the assumption that this punctured spectrum", "is connected unless $r = 1$." ], "refs": [ "more-algebra-lemma-geometrically-unibranch", "more-algebra-lemma-geometrically-unibranch" ], "ref_ids": [ 10468, 10468 ] } ], "ref_ids": [ 3081 ] }, { "id": 3000, "type": "theorem", "label": "properties-lemma-number-of-branches-irreducible-components", "categories": [ "properties" ], "title": "properties-lemma-number-of-branches-irreducible-components", "contents": [ "Let $X$ be a scheme and $x \\in X$. Let $X_i$, $i \\in I$ be the", "irreducible components of $X$ passing through $x$.", "Then the number of (geometric) branches of $X$ at $x$", "is the sum over $i \\in I$ of the number of (geometric)", "branches of $X_i$ at $x$." ], "refs": [], "proofs": [ { "contents": [ "We view the $X_i$ as integral closed subschemes of $X$, see", "Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme} and", "Lemma \\ref{lemma-characterize-integral}.", "Observe that the number of (geometric) branches of $X_i$ at $x$", "is at least $1$ for all $i$ (essentially by definition).", "Recall that the $X_i$ correspond $1$-to-$1$ with the minimal", "prime ideals $\\mathfrak p_i \\subset \\mathcal{O}_{X, x}$, see", "Algebra, Lemma \\ref{algebra-lemma-irreducible-components-containing-x}.", "Thus, if $I$ is infinite, then $\\mathcal{O}_{X, x}$ has infinitely", "many minimal primes, whence both $\\mathcal{O}_{X, x}^h$", "and $\\mathcal{O}_{X, x}^{sh}$ have infinitely many minimal", "primes (combine Algebra, Lemmas", "\\ref{algebra-lemma-injective-minimal-primes-in-image} and", "\\ref{algebra-lemma-minimal-prime-image-minimal-prime} and", "the injectivity of the maps", "$\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X, x}^h \\to \\mathcal{O}_{X, x}^{sh}$).", "In this case the number of (geometric) branches of $X$ at $x$", "is defined to be $\\infty$ which is also true for the sum.", "Thus we may assume $I$ is finite.", "Let $A'$ be the integral closure of $\\mathcal{O}_{X, x}$", "in the total ring of fractions $Q$ of $(\\mathcal{O}_{X, x})_{red}$.", "Let $A'_i$ be the integral closure of $\\mathcal{O}_{X, x}/\\mathfrak p_i$", "in the total ring of fractions $Q_i$ of $\\mathcal{O}_{X, x}/\\mathfrak p_i$.", "By Algebra, Lemma \\ref{algebra-lemma-total-ring-fractions-no-embedded-points}", "we have $Q = \\prod_{i \\in I} Q_i$. Thus $A' = \\prod A'_i$.", "Then the equality of the lemma follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-number-of-branches-1}", "which expresses the number of (geometric) branches in terms", "of the maximal ideals of $A'$." ], "refs": [ "schemes-definition-reduced-induced-scheme", "properties-lemma-characterize-integral", "algebra-lemma-irreducible-components-containing-x", "algebra-lemma-injective-minimal-primes-in-image", "algebra-lemma-minimal-prime-image-minimal-prime", "algebra-lemma-total-ring-fractions-no-embedded-points", "more-algebra-lemma-number-of-branches-1" ], "ref_ids": [ 7745, 2947, 424, 445, 447, 421, 10469 ] } ], "ref_ids": [] }, { "id": 3001, "type": "theorem", "label": "properties-lemma-number-of-branches-1", "categories": [ "properties" ], "title": "properties-lemma-number-of-branches-1", "contents": [ "Let $X$ be a scheme. Let $x \\in X$.", "\\begin{enumerate}", "\\item The number of branches of $X$ at $x$ is $1$ if and only if", "$X$ is unibranch at $x$.", "\\item The number of geometric branches of $X$ at $x$ is $1$ if and only if", "$X$ is geometrically unibranch at $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma follows immediately from the definitions and the corresponding", "result for rings, see More on Algebra, Lemma", "\\ref{more-algebra-lemma-number-of-branches-1}." ], "refs": [ "more-algebra-lemma-number-of-branches-1" ], "ref_ids": [ 10469 ] } ], "ref_ids": [] }, { "id": 3002, "type": "theorem", "label": "properties-lemma-finite-type-module", "categories": [ "properties" ], "title": "properties-lemma-finite-type-module", "contents": [ "Let $X = \\Spec(R)$ be an affine scheme.", "The quasi-coherent sheaf of $\\mathcal{O}_X$-modules", "$\\widetilde M$ is a finite type $\\mathcal{O}_X$-module", "if and only if $M$ is a finite $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\widetilde M$ is a finite type $\\mathcal{O}_X$-module.", "This means there exists an open covering of $X$ such that", "$\\widetilde M$ restricted to the members of this covering is", "globally generated by finitely many sections.", "Thus there also exists a standard open covering", "$X = \\bigcup_{i = 1, \\ldots, n} D(f_i)$ such that $\\widetilde M|_{D(f_i)}$", "is generated by finitely many sections. Thus $M_{f_i}$ is finitely", "generated for each $i$. Hence we conclude by", "Algebra, Lemma \\ref{algebra-lemma-cover}." ], "refs": [ "algebra-lemma-cover" ], "ref_ids": [ 411 ] } ], "ref_ids": [] }, { "id": 3003, "type": "theorem", "label": "properties-lemma-finite-presentation-module", "categories": [ "properties" ], "title": "properties-lemma-finite-presentation-module", "contents": [ "Let $X = \\Spec(R)$ be an affine scheme. The quasi-coherent sheaf", "of $\\mathcal{O}_X$-modules $\\widetilde M$ is an $\\mathcal{O}_X$-module of", "finite presentation if and only if $M$ is an $R$-module of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\widetilde M$ is an $\\mathcal{O}_X$-module of finite presentation.", "By Lemma \\ref{lemma-finite-type-module} we see that $M$ is a finite $R$-module.", "Choose a surjection $R^n \\to M$ with kernel $K$. By", "Schemes, Lemma \\ref{schemes-lemma-spec-sheaves}", "there is a short exact sequence", "$$", "0 \\to \\widetilde{K} \\to", "\\bigoplus \\mathcal{O}_X^{\\oplus n} \\to", "\\widetilde{M} \\to 0", "$$", "By", "Modules, Lemma", "\\ref{modules-lemma-kernel-surjection-finite-free-onto-finite-presentation}", "we see that $\\widetilde{K}$ is a finite type $\\mathcal{O}_X$-module.", "Hence by Lemma \\ref{lemma-finite-type-module}", "again we see that $K$ is a finite $R$-module.", "Hence $M$ is an $R$-module of finite presentation." ], "refs": [ "properties-lemma-finite-type-module", "schemes-lemma-spec-sheaves", "modules-lemma-kernel-surjection-finite-free-onto-finite-presentation", "properties-lemma-finite-type-module" ], "ref_ids": [ 3002, 7651, 13249, 3002 ] } ], "ref_ids": [] }, { "id": 3004, "type": "theorem", "label": "properties-lemma-invert-f-sections", "categories": [ "properties" ], "title": "properties-lemma-invert-f-sections", "contents": [ "\\begin{slogan}", "Sections of quasi-coherent sheaves have only meromorphic singularities", "at infinity.", "\\end{slogan}", "Let $X$ be a scheme. Let $f \\in \\Gamma(X, \\mathcal{O}_X)$.", "Denote $X_f \\subset X$ the open where $f$ is invertible, see", "Schemes, Lemma \\ref{schemes-lemma-f-open}.", "If $X$ is quasi-compact and quasi-separated, the canonical map", "$$", "\\Gamma(X, \\mathcal{O}_X)_f \\longrightarrow \\Gamma(X_f, \\mathcal{O}_X)", "$$", "is an isomorphism. Moreover, if $\\mathcal{F}$ is a quasi-coherent", "sheaf of $\\mathcal{O}_X$-modules the map", "$$", "\\Gamma(X, \\mathcal{F})_f \\longrightarrow \\Gamma(X_f, \\mathcal{F})", "$$", "is an isomorphism." ], "refs": [ "schemes-lemma-f-open" ], "proofs": [ { "contents": [ "Write $R = \\Gamma(X, \\mathcal{O}_X)$. Consider the canonical morphism", "$$", "\\varphi : X \\longrightarrow \\Spec(R)", "$$", "of schemes, see", "Schemes, Lemma", "\\ref{schemes-lemma-morphism-into-affine}.", "Then the inverse image of the standard open $D(f)$ on the", "right hand side is $X_f$ on the left hand side.", "Moreover, since $X$ is assumed quasi-compact and quasi-separated", "the morphism $\\varphi$ is quasi-compact and quasi-separated,", "see Schemes, Lemma \\ref{schemes-lemma-quasi-compact-affine} and", "\\ref{schemes-lemma-compose-after-separated}. Hence by", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "we see that $\\varphi_*\\mathcal{F}$ is quasi-coherent.", "Hence we see that $\\varphi_*\\mathcal{F} = \\widetilde M$", "with $M = \\Gamma(X, \\mathcal{F})$ as an $R$-module.", "Thus we see that", "$$", "\\Gamma(X_f, \\mathcal{F}) =", "\\Gamma(D(f), \\varphi_*\\mathcal{F}) =", "\\Gamma(D(f), \\widetilde M) = M_f", "$$", "which is exactly the content of the lemma. The first displayed isomorphism", "of the lemma follows by taking $\\mathcal{F} = \\mathcal{O}_X$." ], "refs": [ "schemes-lemma-morphism-into-affine", "schemes-lemma-quasi-compact-affine", "schemes-lemma-compose-after-separated", "schemes-lemma-push-forward-quasi-coherent" ], "ref_ids": [ 7655, 7697, 7715, 7730 ] } ], "ref_ids": [ 7653 ] }, { "id": 3005, "type": "theorem", "label": "properties-lemma-invert-s-sections", "categories": [ "properties" ], "title": "properties-lemma-invert-s-sections", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "Let $s \\in \\Gamma(X, \\mathcal{L})$. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item If $X$ is quasi-compact, then (\\ref{equation-module-invert-s})", "is injective, and", "\\item if $X$ is quasi-compact and quasi-separated, then", "(\\ref{equation-module-invert-s}) is an isomorphism.", "\\end{enumerate}", "In particular, the canonical map", "$$", "\\Gamma_*(X, \\mathcal{L})_{(s)}", "\\longrightarrow", "\\Gamma(X_s, \\mathcal{O}_X),\\quad", "a/s^n \\longmapsto a \\otimes s^{-n}", "$$", "is an isomorphism if $X$ is quasi-compact and quasi-separated." ], "refs": [], "proofs": [ { "contents": [ "Assume $X$ is quasi-compact. Choose a finite affine open covering", "$X = U_1 \\cup \\ldots \\cup U_m$ with $U_j$ affine and", "$\\mathcal{L}|_{U_j} \\cong \\mathcal{O}_{U_j}$. Via this isomorphism,", "the image $s|_{U_j}$ corresponds to some", "$f_j \\in \\Gamma(U_j, \\mathcal{O}_{U_j})$. Then", "$X_s \\cap U_j = D(f_j)$.", "\\medskip\\noindent", "Proof of (1). Let $t/s^n$ be an element in the kernel of", "(\\ref{equation-module-invert-s}). Then $t|_{X_s} = 0$.", "Hence $(t|_{U_j})|_{D(f_j)} = 0$. By", "Lemma \\ref{lemma-invert-f-sections} we conclude that", "$f_j^{e_j} t|_{U_j} = 0$ for some", "$e_j \\geq 0$. Let $e = \\max(e_j)$. Then we see that $t \\otimes s^e$", "restricts to zero on $U_j$ for all $j$, hence is zero. Since $t/s^n$", "is equal to $t \\otimes s^e/s^{n + e}$ in", "$\\Gamma_*(X, \\mathcal{L}, \\mathcal{F})_{(s)}$ we conclude that $t/s^n = 0$", "as desired.", "\\medskip\\noindent", "Proof of (2). Assume $X$ is quasi-compact and quasi-separated.", "Then $U_j \\cap U_{j'}$ is quasi-compact for all pairs $j, j'$, see", "Schemes, Lemma \\ref{schemes-lemma-characterize-quasi-separated}.", "By part (1) we know (\\ref{equation-module-invert-s}) is injective.", "Let $t' \\in \\Gamma(X_s, \\mathcal{F}|_{X_s})$. For every $j$, there exist an", "integer $e_j \\geq 0$ and $t'_j \\in \\Gamma(U_j, \\mathcal{F}|_{U_j})$ such that", "$t'|_{D(f_j)}$ corresponds to $t'_j/f_j^{e_j}$", "via the isomorphism of Lemma \\ref{lemma-invert-f-sections}.", "Set $e = \\max(e_j)$ and", "$$", "t_j = f_j^{e - e_j} t'_j \\otimes q_j^e \\in", "\\Gamma(U_j,", "(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes e})|_{U_j})", "$$", "where $q_j \\in \\Gamma(U_j, \\mathcal{L}|_{U_j})$ is the trivializing", "section coming from the isomorphism", "$\\mathcal{L}|_{U_j} \\cong \\mathcal{O}_{U_j}$. In particular we have", "$s|_{U_j} = f_j q_j$. Using this a calculation shows that", "$t_j|_{U_j \\cap U_{j'}}$ and $t_{j'}|_{U_j \\cap U_{j'}}$", "map to the same section of $\\mathcal{F}$ over $U_j \\cap U_{j'} \\cap X_s$.", "By quasi-compactness of $U_j \\cap U_{j'}$ and part (1) there exists an", "integer $e' \\geq 0$ such that", "$$", "t_j|_{U_j \\cap U_{j'}} \\otimes s^{e'}|_{U_j \\cap U_{j'}} =", "t_{j'}|_{U_j \\cap U_{j'}} \\otimes s^{e'}|_{U_j \\cap U_{j'}}", "$$", "as sections of $\\mathcal{F} \\otimes \\mathcal{L}^{\\otimes e + e'}$ over", "$U_j \\cap U_{j'}$. We may choose the same $e'$ to work for all pairs", "$j, j'$. Then the sheaf conditions implies there is a section", "$t \\in \\Gamma(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes e + e'})$", "whose restriction to $U_j$ is $t_j \\otimes s^{e'}|_{U_j}$.", "A simple computation shows that $t/s^{e + e'}$ maps to $t'$", "as desired." ], "refs": [ "properties-lemma-invert-f-sections", "schemes-lemma-characterize-quasi-separated", "properties-lemma-invert-f-sections" ], "ref_ids": [ 3004, 7709, 3004 ] } ], "ref_ids": [] }, { "id": 3006, "type": "theorem", "label": "properties-lemma-section-maps-backwards", "categories": [ "properties" ], "title": "properties-lemma-section-maps-backwards", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{L})$ be a section.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be quasi-coherent $\\mathcal{O}_X$-modules.", "\\begin{enumerate}", "\\item If $X$ is quasi-compact and $\\mathcal{F}$ is of finite type,", "then (\\ref{equation-hom-invert-s}) is injective, and", "\\item if $X$ is quasi-compact and quasi-separated and $\\mathcal{F}$", "is of finite presentation, then", "(\\ref{equation-hom-invert-s})", "is bijective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We first prove the lemma in case $X = \\Spec(A)$ is affine", "and $\\mathcal{L} = \\mathcal{O}_X$. In this case $s$ corresponds", "to an element $f \\in A$. Say", "$\\mathcal{F} = \\widetilde{M}$ and $\\mathcal{G} = \\widetilde{N}$", "for some $A$-modules $M$ and $N$. Then the lemma translates", "(via Lemmas \\ref{lemma-finite-type-module} and", "\\ref{lemma-finite-presentation-module}) into", "the following algebra statements", "\\begin{enumerate}", "\\item If $M$ is a finite $A$-module and $\\varphi : M \\to N$ is", "an $A$-module map such that the induced map $M_f \\to N_f$ is zero,", "then $f^n\\varphi = 0$ for some $n$.", "\\item If $M$ is a finitely presented $A$-module, then", "$\\Hom_A(M, N)_f = \\Hom_{A_f}(M_f, N_f)$.", "\\end{enumerate}", "The second statement is", "Algebra, Lemma \\ref{algebra-lemma-hom-from-finitely-presented} and we omit", "the proof of the first statement.", "\\medskip\\noindent", "Next, we prove (1) for general $X$.", "Assume $X$ is quasi-compact and hoose a finite affine open covering", "$X = U_1 \\cup \\ldots \\cup U_m$ with $U_j$ affine and", "$\\mathcal{L}|_{U_j} \\cong \\mathcal{O}_{U_j}$. Via this isomorphism,", "the image $s|_{U_j}$ corresponds to some", "$f_j \\in \\Gamma(U_j, \\mathcal{O}_{U_j})$. Then", "$X_s \\cap U_j = D(f_j)$.", "Let $\\alpha/s^n$ be an element in the kernel of", "(\\ref{equation-hom-invert-s}). Then $\\alpha|_{X_s} = 0$.", "Hence $(\\alpha|_{U_j})|_{D(f_j)} = 0$. By the affine case treated above", "we conclude that $f_j^{e_j} \\alpha|_{U_j} = 0$ for some", "$e_j \\geq 0$. Let $e = \\max(e_j)$. Then we see that $\\alpha \\otimes s^e$", "restricts to zero on $U_j$ for all $j$, hence is zero. Since $\\alpha/s^n$", "is equal to $\\alpha \\otimes s^e/s^{n + e}$ in $M_{(s)}$ we conclude that", "$\\alpha/s^n = 0$ as desired.", "\\medskip\\noindent", "Proof of (2). Since $\\mathcal{F}$ is of finite presentation, the", "sheaf $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is", "quasi-coherent, see Schemes, Section \\ref{schemes-section-quasi-coherent}.", "Moreover, it is clear that", "$$", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F},", "\\mathcal{G} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) =", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}", "$$", "for all $n$. Hence in this case the statement follows from", "Lemma \\ref{lemma-invert-s-sections} applied to", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$." ], "refs": [ "properties-lemma-finite-type-module", "properties-lemma-finite-presentation-module", "algebra-lemma-hom-from-finitely-presented", "properties-lemma-invert-s-sections" ], "ref_ids": [ 3002, 3003, 353, 3005 ] } ], "ref_ids": [] }, { "id": 3007, "type": "theorem", "label": "properties-lemma-quasi-coherent-quasi-affine", "categories": [ "properties" ], "title": "properties-lemma-quasi-coherent-quasi-affine", "contents": [ "Let $A$ be a ring and let $U \\subset \\Spec(A)$ be a quasi-compact", "open subscheme. For $\\mathcal{F}$ quasi-coherent on $U$ the canonical map", "$$", "\\widetilde{H^0(U, \\mathcal{F})}|_U \\to \\mathcal{F}", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Denote $j : U \\to \\Spec(A)$ the inclusion morphism. Then", "$H^0(U, \\mathcal{F}) = H^0(\\Spec(A), j_*\\mathcal{F})$ and", "$j_*\\mathcal{F}$ is quasi-coherent by", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}.", "Hence $j_*\\mathcal{F} = \\widetilde{H^0(U, \\mathcal{F})}$ by", "Schemes, Lemma \\ref{schemes-lemma-equivalence-quasi-coherent}.", "Restricting back to $U$ we get the lemma." ], "refs": [ "schemes-lemma-push-forward-quasi-coherent", "schemes-lemma-equivalence-quasi-coherent" ], "ref_ids": [ 7730, 7664 ] } ], "ref_ids": [] }, { "id": 3008, "type": "theorem", "label": "properties-lemma-invert-f-affine", "categories": [ "properties" ], "title": "properties-lemma-invert-f-affine", "contents": [ "Let $X$ be a scheme. Let $f \\in \\Gamma(X, \\mathcal{O}_X)$.", "Assume $X$ is quasi-compact and quasi-separated and assume that", "$X_f$ is affine. Then the canonical morphism", "$$", "j : X \\longrightarrow \\Spec(\\Gamma(X, \\mathcal{O}_X))", "$$", "from Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}", "induces an isomorphism of $X_f = j^{-1}(D(f))$ onto the standard affine", "open $D(f) \\subset \\Spec(\\Gamma(X, \\mathcal{O}_X))$." ], "refs": [ "schemes-lemma-morphism-into-affine" ], "proofs": [ { "contents": [ "This is clear as $j$ induces an isomorphism of rings", "$\\Gamma(X, \\mathcal{O}_X)_f \\to \\mathcal{O}_X(X_f)$ by", "Lemma \\ref{lemma-invert-f-sections} above." ], "refs": [ "properties-lemma-invert-f-sections" ], "ref_ids": [ 3004 ] } ], "ref_ids": [ 7655 ] }, { "id": 3009, "type": "theorem", "label": "properties-lemma-quasi-affine", "categories": [ "properties" ], "title": "properties-lemma-quasi-affine", "contents": [ "Let $X$ be a scheme. Then $X$ is quasi-affine if and only if", "the canonical morphism", "$$", "X \\longrightarrow \\Spec(\\Gamma(X, \\mathcal{O}_X))", "$$", "from Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine} is", "a quasi-compact open immersion." ], "refs": [ "schemes-lemma-morphism-into-affine" ], "proofs": [ { "contents": [ "If the displayed morphism is a quasi-compact open immersion then", "$X$ is isomorphic to a quasi-compact open subscheme of", "$\\Spec(\\Gamma(X, \\mathcal{O}_X))$ and clearly $X$ is quasi-affine.", "\\medskip\\noindent", "Assume $X$ is quasi-affine, say $X \\subset \\Spec(R)$ is", "quasi-compact open. This in particular implies that $X$ is", "separated, see", "Schemes, Lemma \\ref{schemes-lemma-subscheme-of-separated-scheme}.", "Let $A = \\Gamma(X, \\mathcal{O}_X)$.", "Consider the ring map $R \\to A$ coming from", "$R = \\Gamma(\\Spec(R), \\mathcal{O}_{\\Spec(R)})$", "and the restriction mapping of the sheaf $\\mathcal{O}_{\\Spec(R)}$.", "By Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}", "we obtain a factorization:", "$$", "X \\longrightarrow", "\\Spec(A) \\longrightarrow", "\\Spec(R)", "$$", "of the inclusion morphism. Let $x \\in X$. Choose $r \\in R$ such that", "$x \\in D(r)$ and $D(r) \\subset X$. Denote $f \\in A$ the image of $r$", "in $A$. The open $X_f$ of Lemma \\ref{lemma-invert-f-sections}", "above is equal to $D(r) \\subset X$ and hence $A_f \\cong R_r$ by the", "conclusion of that lemma.", "Hence $D(r) \\to \\Spec(A)$ is an isomorphism onto the", "standard affine open $D(f)$ of $\\Spec(A)$. Since $X$", "can be covered by such affine opens $D(f)$ we win." ], "refs": [ "schemes-lemma-subscheme-of-separated-scheme", "schemes-lemma-morphism-into-affine", "properties-lemma-invert-f-sections" ], "ref_ids": [ 7728, 7655, 3004 ] } ], "ref_ids": [ 7655 ] }, { "id": 3010, "type": "theorem", "label": "properties-lemma-cartesian-diagram-quasi-affine", "categories": [ "properties" ], "title": "properties-lemma-cartesian-diagram-quasi-affine", "contents": [ "Let $U \\to V$ be an open immersion of quasi-affine schemes. Then", "$$", "\\xymatrix{", "U \\ar[d] \\ar[rr]_-j & & \\Spec(\\Gamma(U, \\mathcal{O}_U)) \\ar[d] \\\\", "U \\ar[r] & V \\ar[r]^-{j'} & \\Spec(\\Gamma(V, \\mathcal{O}_V))", "}", "$$", "is cartesian." ], "refs": [], "proofs": [ { "contents": [ "The diagram is commutative by Schemes, Lemma", "\\ref{schemes-lemma-morphism-into-affine}.", "Write $A = \\Gamma(U, \\mathcal{O}_U)$ and $B = \\Gamma(V, \\mathcal{O}_V)$. Let", "$g \\in B$ be such that $V_g$ is affine and contained in $U$. This", "means that if $f$ is the image of $g$ in $A$, then $U_f = V_g$. By Lemma", "\\ref{lemma-invert-f-affine} we see that $j'$ induces an isomorphism of", "$V_g$ with the standard open $D(g)$ of $\\Spec(B)$. Thus", "$V_g \\times_{\\Spec(B)} \\Spec(A) \\to \\Spec(A)$ is an", "isomorphism onto $D(f) \\subset \\Spec(A)$. By Lemma \\ref{lemma-invert-f-affine}", "again $j$ maps $U_f$ isomorphically to $D(f)$. Thus we see that", "$U_f = U_f \\times_{\\Spec(B)} \\Spec(A)$. Since by", "Lemma \\ref{lemma-quasi-affine} we can cover $U$ by $V_g = U_f$ as above,", "we see that $U \\to U \\times_{\\Spec(B)} \\Spec(A)$ is an isomorphism." ], "refs": [ "schemes-lemma-morphism-into-affine", "properties-lemma-invert-f-affine", "properties-lemma-invert-f-affine", "properties-lemma-quasi-affine" ], "ref_ids": [ 7655, 3008, 3008, 3009 ] } ], "ref_ids": [] }, { "id": 3011, "type": "theorem", "label": "properties-lemma-quasi-affine-presentation", "categories": [ "properties" ], "title": "properties-lemma-quasi-affine-presentation", "contents": [ "Let $X$ be a quasi-affine scheme. There exists an integer $n \\geq 0$,", "an affine scheme $T$, and a morphism $T \\to X$ such that for every", "morphism $X' \\to X$ with $X'$ affine the fibre product $X' \\times_X T$", "is isomorphic to $\\mathbf{A}^n_{X'}$ over $X'$." ], "refs": [], "proofs": [ { "contents": [ "By definition, there exists a ring $A$ such that $X$ is isomorphic to a", "quasi-compact open subscheme $U \\subset \\Spec(A)$. Recall that the standard", "opens $D(f) \\subset \\Spec(A)$ form a basis for the topology, see", "Algebra, Section \\ref{algebra-section-spectrum-ring}. Since $U$ is", "quasi-compact we can choose $f_1, \\ldots, f_n \\in A$ such that", "$U = D(f_1) \\cup \\ldots \\cup D(f_n)$. Thus we may assume", "$X = \\Spec(A) \\setminus V(I)$ where $I = (f_1, \\ldots, f_n)$. We set", "$$", "T = \\Spec(A[t, x_1, \\ldots, x_n]/(f_1 x_1 + \\ldots + f_n x_n - 1))", "$$", "The structure morphism $T \\to \\Spec(A)$ factors through the open $X$", "to give the morphism $T \\to X$. If $X' = \\Spec(A')$ and the morphism", "$X' \\to X$ corresponds to the ring map $A \\to A'$, then the images", "$f'_1, \\ldots, f'_n \\in A'$ of $f_1, \\ldots, f_n$", "generate the unit ideal in $A'$.", "Say $1 = f'_1 a'_1 + \\ldots + f'_n a'_n$.", "The base change $X' \\times_X T$ is the spectrum of", "$A'[t, x_1, \\ldots, x_n]/(f'_1 x_1 + \\ldots + f'_n x_n - 1)$.", "We claim the $A'$-algebra homomorphism", "$$", "\\varphi :", "A'[y_1, \\ldots, y_n]", "\\longrightarrow", "A'[t, x_1, \\ldots, x_n, x_{n + 1}]/(f'_1 x_1 + \\ldots + f'_n x_n - 1)", "$$", "sending $y_i$ to $a'_i t + x_i$ is an isomorphism. The claim finishes", "the proof of the lemma. The inverse of $\\varphi$ is given by the $A'$-algebra", "homomorphism", "$$", "\\psi :", "A'[t, x_1, \\ldots, x_n, x_{n + 1}]/(f'_1 x_1 + \\ldots + f'_n x_n - 1)", "\\longrightarrow", "A'[y_1, \\ldots, y_n]", "$$", "sending $t$ to $-1 + f'_1 y_1 + \\ldots + f'_n y_n$ and $x_i$ to", "$y_i + a'_i - a'_i(f'_1 y_1 + \\ldots + f'_n y_n)$ for $i = 1, \\ldots, n$.", "This makes sense because $\\sum f'_ix_i$ is mapped to", "$$", "\\begin{matrix}", "\\sum f'_i(y_i + a'_i - a'_i(\\sum f'_j y_j)) =", "(\\sum f'_iy_i) + 1 - (\\sum f'_j y_j) = 1", "\\end{matrix}", "$$", "To see the maps are mutually inverse one computes as follows:", "$$", "\\begin{matrix}", "\\varphi(\\psi(t) = \\varphi(-1 + \\sum f'_i y_i) =", "-1 + \\sum f'_i (a'_i t + x_i) = t \\\\", "\\varphi(\\psi(x_i)) = \\varphi(y_i + a'_i - a'_i(\\sum f'_j y_j)) =", "a'_i t + x_i + a'_i - a'_i(\\sum f'_ja'_jt + f'_jx_j) = x_i \\\\", "\\psi(\\varphi(y_i)) = \\psi(a'_i t + x_i) =", "a'_i(-1 + \\sum f'_j y_j) + y_i + a'_i - a'_i(\\sum f'_j y_j) = y_i", "\\end{matrix}", "$$", "This finishes the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3012, "type": "theorem", "label": "properties-lemma-flat-module", "categories": [ "properties" ], "title": "properties-lemma-flat-module", "contents": [ "\\begin{slogan}", "Flatness is the same for modules and sheaves.", "\\end{slogan}", "Let $X = \\Spec(R)$ be an affine scheme.", "Let $\\mathcal{F} = \\widetilde{M}$ for some $R$-module $M$.", "The quasi-coherent sheaf $\\mathcal{F}$ is a flat", "$\\mathcal{O}_X$-module if and only if $M$ is a flat $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Flatness of $\\mathcal{F}$ may be checked on the stalks, see", "Modules, Lemma \\ref{modules-lemma-flat-stalks-flat}.", "The same is true in the case of modules over a ring, see", "Algebra, Lemma \\ref{algebra-lemma-flat-localization}.", "And since $\\mathcal{F}_x = M_{\\mathfrak p}$ if $x$ corresponds", "to $\\mathfrak p$ the lemma is true." ], "refs": [ "modules-lemma-flat-stalks-flat", "algebra-lemma-flat-localization" ], "ref_ids": [ 13273, 538 ] } ], "ref_ids": [] }, { "id": 3013, "type": "theorem", "label": "properties-lemma-locally-free-module", "categories": [ "properties" ], "title": "properties-lemma-locally-free-module", "contents": [ "Let $X = \\Spec(R)$ be an affine scheme.", "Let $\\mathcal{F} = \\widetilde{M}$ for some $R$-module $M$.", "The quasi-coherent sheaf $\\mathcal{F}$ is a (finite) locally free", "$\\mathcal{O}_X$-module of if and only if $M$ is a (finite)", "locally free $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Follows from the definitions, see", "Modules, Definition \\ref{modules-definition-locally-free}", "and", "Algebra, Definition \\ref{algebra-definition-locally-free}." ], "refs": [ "modules-definition-locally-free", "algebra-definition-locally-free" ], "ref_ids": [ 13342, 1492 ] } ], "ref_ids": [] }, { "id": 3014, "type": "theorem", "label": "properties-lemma-finite-locally-free", "categories": [ "properties" ], "title": "properties-lemma-finite-locally-free", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "The following are equivalent:", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a flat $\\mathcal{O}_X$-module of finite presentation,", "\\item $\\mathcal{F}$ is $\\mathcal{O}_X$-module of finite presentation and", "for all $x \\in X$ the stalk $\\mathcal{F}_x$ is a free", "$\\mathcal{O}_{X, x}$-module,", "\\item $\\mathcal{F}$ is a locally free, finite type $\\mathcal{O}_X$-module,", "\\item $\\mathcal{F}$ is a finite locally free $\\mathcal{O}_X$-module, and", "\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type,", "for every $x \\in X$ the stalk $\\mathcal{F}_x$ is a free", "$\\mathcal{O}_{X, x}$-module, and the function", "$$", "\\rho_\\mathcal{F} : X \\to \\mathbf{Z}, \\quad", "x \\longmapsto", "\\dim_{\\kappa(x)} \\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)", "$$", "is locally constant in the Zariski topology on $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma immediately reduces to the affine case.", "In this case the lemma is a reformulation of", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}.", "The translation uses", "Lemmas \\ref{lemma-finite-type-module},", "\\ref{lemma-finite-presentation-module},", "\\ref{lemma-flat-module}, and", "\\ref{lemma-locally-free-module}." ], "refs": [ "algebra-lemma-finite-projective", "properties-lemma-finite-type-module", "properties-lemma-finite-presentation-module", "properties-lemma-flat-module", "properties-lemma-locally-free-module" ], "ref_ids": [ 795, 3002, 3003, 3012, 3013 ] } ], "ref_ids": [] }, { "id": 3015, "type": "theorem", "label": "properties-lemma-finite-locally-free-reduced", "categories": [ "properties" ], "title": "properties-lemma-finite-locally-free-reduced", "contents": [ "Let $X$ be a reduced scheme. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. Then the equivalent conditions of", "Lemma \\ref{lemma-finite-locally-free} are also equivalent to", "\\begin{enumerate}", "\\item[(6)] $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type and", "the function", "$$", "\\rho_\\mathcal{F} : X \\to \\mathbf{Z}, \\quad", "x \\longmapsto", "\\dim_{\\kappa(x)} \\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)", "$$", "is locally constant in the Zariski topology on $X$.", "\\end{enumerate}" ], "refs": [ "properties-lemma-finite-locally-free" ], "proofs": [ { "contents": [ "This lemma immediately reduces to the affine case.", "In this case the lemma is a reformulation of", "Algebra, Lemma \\ref{algebra-lemma-finite-projective-reduced}." ], "refs": [ "algebra-lemma-finite-projective-reduced" ], "ref_ids": [ 796 ] } ], "ref_ids": [ 3014 ] }, { "id": 3016, "type": "theorem", "label": "properties-lemma-locally-projective", "categories": [ "properties" ], "title": "properties-lemma-locally-projective", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is locally projective, and", "\\item there exists an affine open covering $X = \\bigcup U_i$", "such that the $\\mathcal{O}_X(U_i)$-module", "$\\mathcal{F}(U_i)$ is projective for every $i$.", "\\end{enumerate}", "In particular, if $X = \\Spec(A)$ and $\\mathcal{F} = \\widetilde{M}$", "then $\\mathcal{F}$ is locally projective if and only if $M$ is a projective", "$A$-module." ], "refs": [], "proofs": [ { "contents": [ "First, note that if $M$ is a projective $A$-module and $A \\to B$ is a", "ring map, then $M \\otimes_A B$ is a projective $B$-module, see", "Algebra, Lemma \\ref{algebra-lemma-ascend-properties-modules}.", "Hence if $U$ is an affine open such that $\\mathcal{F}(U)$ is a projective", "$\\mathcal{O}_X(U)$-module, then the standard open $D(f)$ is an", "affine open such that $\\mathcal{F}(D(f))$ is a projective", "$\\mathcal{O}_X(D(f))$-module for all $f \\in \\mathcal{O}_X(U)$.", "Assume (2) holds. Let $U \\subset X$ be an arbitrary affine open.", "We can find an open covering $U = \\bigcup_{j = 1, \\ldots, m} D(f_j)$", "by finitely many standard opens $D(f_j)$ such that for each", "$j$ the open $D(f_j)$ is a standard open of some $U_i$, see", "Schemes, Lemma \\ref{schemes-lemma-standard-open-two-affines}.", "Hence, if we set $A = \\mathcal{O}_X(U)$ and if $M$ is an $A$-module", "such that $\\mathcal{F}|_U$ corresponds to $M$, then we see that", "$M_{f_j}$ is a projective $A_{f_j}$-module. It follows that", "$A \\to B = \\prod A_{f_j}$ is a faithfully flat ring map", "such that $M \\times_A B$ is a projective $B$-module.", "Hence $M$ is projective by", "Algebra, Theorem \\ref{algebra-theorem-ffdescent-projectivity}." ], "refs": [ "algebra-lemma-ascend-properties-modules", "schemes-lemma-standard-open-two-affines", "algebra-theorem-ffdescent-projectivity" ], "ref_ids": [ 853, 7675, 324 ] } ], "ref_ids": [] }, { "id": 3017, "type": "theorem", "label": "properties-lemma-locally-projective-pullback", "categories": [ "properties" ], "title": "properties-lemma-locally-projective-pullback", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.", "If $\\mathcal{G}$ is locally projective on $Y$, then $f^*\\mathcal{G}$", "is locally projective on $X$." ], "refs": [], "proofs": [ { "contents": [ "See", "Algebra, Lemma \\ref{algebra-lemma-ascend-properties-modules}." ], "refs": [ "algebra-lemma-ascend-properties-modules" ], "ref_ids": [ 853 ] } ], "ref_ids": [] }, { "id": 3018, "type": "theorem", "label": "properties-lemma-extend-trivial", "categories": [ "properties" ], "title": "properties-lemma-extend-trivial", "contents": [ "Let $j : U \\to X$ be a quasi-compact open immersion of schemes.", "\\begin{enumerate}", "\\item Any quasi-coherent sheaf on $U$ extends to a quasi-coherent", "sheaf on $X$.", "\\item Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $\\mathcal{G} \\subset \\mathcal{F}|_U$ be a quasi-coherent", "subsheaf. There exists a quasi-coherent subsheaf $\\mathcal{H}$ of", "$\\mathcal{F}$ such that $\\mathcal{H}|_U = \\mathcal{G}$", "as subsheaves of $\\mathcal{F}|_U$.", "\\item Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $\\mathcal{G}$ be a quasi-coherent sheaf on $U$.", "Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}|_U$ be a morphism", "of $\\mathcal{O}_U$-modules. There exists a quasi-coherent sheaf $\\mathcal{H}$", "of $\\mathcal{O}_X$-modules and a map $\\psi : \\mathcal{H} \\to \\mathcal{F}$", "such that $\\mathcal{H}|_U = \\mathcal{G}$ and that", "$\\psi|_U = \\varphi$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "An immersion is separated", "(see Schemes, Lemma \\ref{schemes-lemma-immersions-monomorphisms})", "and $j$ is quasi-compact by assumption.", "Hence for any quasi-coherent sheaf $\\mathcal{G}$ on $U$ the sheaf", "$j_*\\mathcal{G}$ is an extension to $X$. See", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent} and", "Sheaves, Section \\ref{sheaves-section-open-immersions}.", "\\medskip\\noindent", "Assume $\\mathcal{F}$, $\\mathcal{G}$ are as in (2).", "Then $j_*\\mathcal{G}$ is a quasi-coherent sheaf on $X$ (see above).", "It is a subsheaf of $j_*j^*\\mathcal{F}$.", "Hence the kernel", "$$", "\\mathcal{H} =", "\\Ker(\\mathcal{F} \\oplus j_* \\mathcal{G}", "\\longrightarrow j_*j^*\\mathcal{F})", "$$", "is quasi-coherent as well, see", "Schemes, Section \\ref{schemes-section-quasi-coherent}.", "It is formal to check that $\\mathcal{H} \\subset \\mathcal{F}$ and that", "$\\mathcal{H}|_U = \\mathcal{G}$ (using the material in", "Sheaves, Section \\ref{sheaves-section-open-immersions} again).", "\\medskip\\noindent", "Part (3) is proved in the same manner as (2). Just take", "$\\mathcal{H} = \\Ker(\\mathcal{F} \\oplus j_* \\mathcal{G}", "\\to j_*j^*\\mathcal{F})$ with its obvious map to $\\mathcal{F}$", "and its obvious identification with $\\mathcal{G}$ over $U$." ], "refs": [ "schemes-lemma-immersions-monomorphisms", "schemes-lemma-push-forward-quasi-coherent" ], "ref_ids": [ 7727, 7730 ] } ], "ref_ids": [] }, { "id": 3019, "type": "theorem", "label": "properties-lemma-extend", "categories": [ "properties" ], "title": "properties-lemma-extend", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $U \\subset X$ be a quasi-compact open.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\mathcal{G} \\subset \\mathcal{F}|_U$ be a quasi-coherent", "$\\mathcal{O}_U$-submodule which is of finite type. Then", "there exists a quasi-coherent submodule $\\mathcal{G}' \\subset \\mathcal{F}$", "which is of finite type such that $\\mathcal{G}'|_U = \\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Let $n$ be the minimal number of affine opens $U_i \\subset X$,", "$i = 1, \\ldots , n$ such that $X = U \\cup \\bigcup U_i$.", "(Here we use that $X$ is quasi-compact.) Suppose", "we can prove the lemma for the case $n = 1$. Then we can successively", "extend $\\mathcal{G}$", "to a $\\mathcal{G}_1$ over $U \\cup U_1$", "to a $\\mathcal{G}_2$ over $U \\cup U_1 \\cup U_2$", "to a $\\mathcal{G}_3$ over $U \\cup U_1 \\cup U_2 \\cup U_3$,", "and so on.", "Thus we reduce to the case $n = 1$.", "\\medskip\\noindent", "Thus we may assume that $X = U \\cup V$ with $V$ affine.", "Since $X$ is quasi-separated and $U$, $V$ are quasi-compact open,", "we see that $U \\cap V$ is a quasi-compact open. It suffices to prove the", "lemma for the system $(V, U \\cap V, \\mathcal{F}|_V, \\mathcal{G}|_{U \\cap V})$", "since we can glue the resulting sheaf $\\mathcal{G}'$ over $V$", "to the given sheaf $\\mathcal{G}$ over $U$ along the common value", "over $U \\cap V$.", "Thus we reduce to the case where $X$ is affine.", "\\medskip\\noindent", "Assume $X = \\Spec(R)$. Write $\\mathcal{F} = \\widetilde M$", "for some $R$-module $M$. By Lemma \\ref{lemma-extend-trivial} above we may", "find a quasi-coherent subsheaf $\\mathcal{H} \\subset \\mathcal{F}$", "which restricts to $\\mathcal{G}$ over $U$.", "Write $\\mathcal{H} = \\widetilde N$ for some $R$-module $N$.", "For every $u \\in U$ there exists an $f \\in R$ such that", "$u \\in D(f) \\subset U$ and such that $N_f$ is finitely generated,", "see Lemma \\ref{lemma-finite-type-module}.", "Since $U$ is quasi-compact we can cover it by finitely", "many $D(f_i)$ such that $N_{f_i}$ is generated by", "finitely many elements, say $x_{i, 1}/f_i^N, \\ldots, x_{i, r_i}/f_i^N$.", "Let $N' \\subset N$ be the submodule generated by the elements", "$x_{i, j}$. Then the subsheaf", "$\\mathcal{G}' = \\widetilde{N'} \\subset \\mathcal{H} \\subset \\mathcal{F}$", "works." ], "refs": [ "properties-lemma-extend-trivial", "properties-lemma-finite-type-module" ], "ref_ids": [ 3018, 3002 ] } ], "ref_ids": [] }, { "id": 3020, "type": "theorem", "label": "properties-lemma-quasi-coherent-colimit-finite-type", "categories": [ "properties" ], "title": "properties-lemma-quasi-coherent-colimit-finite-type", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Any quasi-coherent sheaf of $\\mathcal{O}_X$-modules", "is the directed colimit of its quasi-coherent", "$\\mathcal{O}_X$-submodules which are of finite type." ], "refs": [], "proofs": [ { "contents": [ "The colimit is directed because if $\\mathcal{G}_1$, $\\mathcal{G}_2$", "are quasi-coherent subsheaves of finite type, then the image of", "$\\mathcal{G}_1 \\oplus \\mathcal{G}_2 \\to \\mathcal{F}$ is", "a quasi-coherent submodule of finite type.", "Let $U \\subset X$ be any affine open, and let", "$s \\in \\Gamma(U, \\mathcal{F})$ be any section.", "Let $\\mathcal{G} \\subset \\mathcal{F}|_U$ be the", "subsheaf generated by $s$. Then clearly $\\mathcal{G}$", "is quasi-coherent and has finite type as an $\\mathcal{O}_U$-module.", "By Lemma \\ref{lemma-extend} we see that $\\mathcal{G}$ is the restriction", "of a quasi-coherent subsheaf $\\mathcal{G}' \\subset \\mathcal{F}$", "which has finite type. Since $X$ has a basis for the topology consisting", "of affine opens we conclude that every local section of", "$\\mathcal{F}$ is locally contained in a quasi-coherent submodule", "of finite type. Thus we win." ], "refs": [ "properties-lemma-extend" ], "ref_ids": [ 3019 ] } ], "ref_ids": [] }, { "id": 3021, "type": "theorem", "label": "properties-lemma-extend-finite-presentation", "categories": [ "properties" ], "title": "properties-lemma-extend-finite-presentation", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $U \\subset X$ be a quasi-compact open.", "Let $\\mathcal{G}$ be an $\\mathcal{O}_U$-module which is of finite presentation.", "Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}|_U$ be a morphism of", "$\\mathcal{O}_U$-modules.", "Then there exists an $\\mathcal{O}_X$-module", "$\\mathcal{G}'$ of finite presentation, and a morphism", "of $\\mathcal{O}_X$-modules $\\varphi' : \\mathcal{G}' \\to \\mathcal{F}$", "such that $\\mathcal{G}'|_U = \\mathcal{G}$ and such that", "$\\varphi'|_U = \\varphi$." ], "refs": [], "proofs": [ { "contents": [ "The beginning of the proof is a repeat of the beginning of the", "proof of Lemma \\ref{lemma-extend}. We write it out carefuly anyway.", "\\medskip\\noindent", "Let $n$ be the minimal number of affine opens $U_i \\subset X$,", "$i = 1, \\ldots , n$ such that $X = U \\cup \\bigcup U_i$.", "(Here we use that $X$ is quasi-compact.) Suppose", "we can prove the lemma for the case $n = 1$. Then we can successively", "extend the pair $(\\mathcal{G}, \\varphi)$", "to a pair $(\\mathcal{G}_1, \\varphi_1)$ over $U \\cup U_1$", "to a pair $(\\mathcal{G}_2, \\varphi_2)$ over $U \\cup U_1 \\cup U_2$", "to a pair $(\\mathcal{G}_3, \\varphi_3)$ over $U \\cup U_1 \\cup U_2 \\cup U_3$,", "and so on.", "Thus we reduce to the case $n = 1$.", "\\medskip\\noindent", "Thus we may assume that $X = U \\cup V$ with $V$ affine.", "Since $X$ is quasi-separated and $U$ quasi-compact,", "we see that $U \\cap V \\subset V$ is quasi-compact.", "Suppose we prove the lemma for the system", "$(V, U \\cap V, \\mathcal{F}|_V, \\mathcal{G}|_{U \\cap V}, \\varphi|_{U \\cap V})$", "thereby producing $(\\mathcal{G}', \\varphi')$ over $V$.", "Then we can glue $\\mathcal{G}'$ over $V$ to the given sheaf $\\mathcal{G}$", "over $U$ along the common value over $U \\cap V$, and similarly we can glue", "the map $\\varphi'$ to the map $\\varphi$ along the common value over", "$U \\cap V$. Thus we reduce to the case where $X$ is affine.", "\\medskip\\noindent", "Assume $X = \\Spec(R)$.", "By Lemma \\ref{lemma-extend-trivial} above we may", "find a quasi-coherent sheaf $\\mathcal{H}$ with", "a map $\\psi : \\mathcal{H} \\to \\mathcal{F}$ over $X$", "which restricts to $\\mathcal{G}$ and $\\varphi$ over $U$.", "By Lemma \\ref{lemma-extend} we can find a finite type", "quasi-coherent $\\mathcal{O}_X$-submodule", "$\\mathcal{H}' \\subset \\mathcal{H}$", "such that $\\mathcal{H}'|_U = \\mathcal{G}$. Thus after", "replacing $\\mathcal{H}$ by $\\mathcal{H}'$", "and $\\psi$ by the restriction of $\\psi$ to $\\mathcal{H}'$", "we may assume that $\\mathcal{H}$ is of finite type.", "By Lemma \\ref{lemma-finite-presentation-module}", "we conclude that $\\mathcal{H} = \\widetilde{N}$ with", "$N$ a finitely generated $R$-module. Hence there exists a surjection", "as in the following short exact sequence of", "quasi-coherent $\\mathcal{O}_X$-modules", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{O}_X^{\\oplus n} \\to \\mathcal{H} \\to 0", "$$", "where $\\mathcal{K}$ is defined as the kernel.", "Since $\\mathcal{G}$ is of finite presentation and", "$\\mathcal{H}|_U = \\mathcal{G}$ by", "Modules, Lemma", "\\ref{modules-lemma-kernel-surjection-finite-free-onto-finite-presentation}", "the restriction $\\mathcal{K}|_U$ is", "an $\\mathcal{O}_U$-module of finite type. Hence by Lemma \\ref{lemma-extend}", "again we see that there exists a finite type quasi-coherent", "$\\mathcal{O}_X$-submodule $\\mathcal{K}' \\subset \\mathcal{K}$ such", "that $\\mathcal{K}'|_U = \\mathcal{K}|_U$. The solution to the problem", "posed in the lemma is to set", "$$", "\\mathcal{G}' = \\mathcal{O}_X^{\\oplus n}/\\mathcal{K}'", "$$", "which is clearly of finite presentation and restricts to give $\\mathcal{G}$", "on $U$ with $\\varphi'$ equal to the composition", "$$", "\\mathcal{G}' = \\mathcal{O}_X^{\\oplus n}/\\mathcal{K}'", "\\to \\mathcal{O}_X^{\\oplus n}/\\mathcal{K} = \\mathcal{H} \\xrightarrow{\\psi}", "\\mathcal{F}.", "$$", "This finishes the proof of the lemma." ], "refs": [ "properties-lemma-extend", "properties-lemma-extend-trivial", "properties-lemma-extend", "properties-lemma-finite-presentation-module", "modules-lemma-kernel-surjection-finite-free-onto-finite-presentation", "properties-lemma-extend" ], "ref_ids": [ 3019, 3018, 3019, 3003, 13249, 3019 ] } ], "ref_ids": [] }, { "id": 3022, "type": "theorem", "label": "properties-lemma-lift-finite-presentation", "categories": [ "properties" ], "title": "properties-lemma-lift-finite-presentation", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \\subset X$", "be a quasi-compact open. Let $\\mathcal{G}$ be an $\\mathcal{O}_U$-module.", "\\begin{enumerate}", "\\item If $\\mathcal{G}$ is quasi-coherent and of finite type, then", "there exists a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{G}'$", "of finite type such that $\\mathcal{G}'|_U = \\mathcal{G}$.", "\\item If $\\mathcal{G}$ is of finite presentation, then", "there exists an $\\mathcal{O}_X$-module $\\mathcal{G}'$", "of finite presentation such that $\\mathcal{G}'|_U = \\mathcal{G}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) is the special case of Lemma \\ref{lemma-extend-finite-presentation}", "where $\\mathcal{F} = 0$. For part (1) we first write", "$\\mathcal{G} = \\mathcal{F}|_U$ for some quasi-coherent $\\mathcal{O}_X$-module", "by Lemma \\ref{lemma-extend-trivial}", "and then we apply Lemma \\ref{lemma-extend} with $\\mathcal{G} = \\mathcal{F}|_U$." ], "refs": [ "properties-lemma-extend-finite-presentation", "properties-lemma-extend-trivial", "properties-lemma-extend" ], "ref_ids": [ 3021, 3018, 3019 ] } ], "ref_ids": [] }, { "id": 3023, "type": "theorem", "label": "properties-lemma-directed-colimit-diagram-finite-presentation", "categories": [ "properties" ], "title": "properties-lemma-directed-colimit-diagram-finite-presentation", "contents": [ "\\begin{slogan}", "Quasi-coherent modules on quasi-compact and quasi-separated schemes", "are filtered colimits of finitely presented modules.", "\\end{slogan}", "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "There exist", "\\begin{enumerate}", "\\item a filtered index category $\\mathcal{I}$ (see", "Categories, Definition \\ref{categories-definition-directed}),", "\\item a diagram $\\mathcal{I} \\to \\textit{Mod}(\\mathcal{O}_X)$ (see", "Categories, Section \\ref{categories-section-limits}),", "$i \\mapsto \\mathcal{F}_i$,", "\\item morphisms of $\\mathcal{O}_X$-modules", "$\\varphi_i : \\mathcal{F}_i \\to \\mathcal{F}$", "\\end{enumerate}", "such that each $\\mathcal{F}_i$ is of finite presentation", "and such that the morphisms $\\varphi_i$ induce an isomorphism", "$$", "\\colim_i \\mathcal{F}_i", "=", "\\mathcal{F}.", "$$" ], "refs": [ "categories-definition-directed" ], "proofs": [ { "contents": [ "Choose a set $I$ and for each $i \\in I$ an $\\mathcal{O}_X$-module", "of finite presentation and a homomorphism of $\\mathcal{O}_X$-modules", "$\\varphi_i : \\mathcal{F}_i \\to \\mathcal{F}$ with the following", "property: For any $\\psi : \\mathcal{G} \\to \\mathcal{F}$ with $\\mathcal{G}$", "of finite presentation there is an $i \\in I$ such that there exists", "an isomorphism $\\alpha : \\mathcal{F}_i \\to \\mathcal{G}$ with", "$\\varphi_i = \\psi \\circ \\alpha$. It is clear from", "Modules, Lemma \\ref{modules-lemma-set-isomorphism-classes-finite-type-modules}", "that such a set exists (see also its proof).", "We denote $\\mathcal{I}$ the category", "with $\\Ob(\\mathcal{I}) = I$ and given $i, i' \\in I$", "we set", "$$", "\\Mor_\\mathcal{I}(i, i') =", "\\{\\alpha : \\mathcal{F}_i \\to \\mathcal{F}_{i'} \\mid", "\\alpha \\circ \\varphi_{i'} = \\varphi_i", "\\}.", "$$", "We claim that $\\mathcal{I}$ is a filtered category and that", "$\\mathcal{F} = \\colim_i \\mathcal{F}_i$.", "\\medskip\\noindent", "Let $i, i' \\in I$. Then we can consider the morphism", "$$", "\\mathcal{F}_i \\oplus \\mathcal{F}_{i'} \\longrightarrow \\mathcal{F}", "$$", "which is the direct sum of $\\varphi_i$ and $\\varphi_{i'}$.", "Since a direct sum of finitely presented $\\mathcal{O}_X$-modules", "is finitely presented we see that there exists some $i'' \\in I$", "such that $\\varphi_{i''} : \\mathcal{F}_{i''} \\to \\mathcal{F}$", "is isomorphic to the displayed arrow towards $\\mathcal{F}$ above.", "Since there are commutative diagrams", "$$", "\\xymatrix{", "\\mathcal{F}_i \\ar[r] \\ar[d] & \\mathcal{F} \\ar@{=}[d] \\\\", "\\mathcal{F}_i \\oplus \\mathcal{F}_{i'} \\ar[r] & \\mathcal{F}", "}", "\\quad", "\\text{and}", "\\quad", "\\xymatrix{", "\\mathcal{F}_{i'} \\ar[r] \\ar[d] & \\mathcal{F} \\ar@{=}[d] \\\\", "\\mathcal{F}_i \\oplus \\mathcal{F}_{i'} \\ar[r] & \\mathcal{F}", "}", "$$", "we see that there are morphisms $i \\to i''$ and $i' \\to i''$", "in $\\mathcal{I}$. Next, suppose that we have $i, i' \\in I$ and", "morphisms $\\alpha, \\beta : i \\to i'$ (corresponding to $\\mathcal{O}_X$-module", "maps $\\alpha, \\beta : \\mathcal{F}_i \\to \\mathcal{F}_{i'}$).", "In this case consider the coequalizer", "$$", "\\mathcal{G} =", "\\Coker(", "\\mathcal{F}_i \\xrightarrow{\\alpha - \\beta} \\mathcal{F}_{i'}", ")", "$$", "Note that $\\mathcal{G}$ is an $\\mathcal{O}_X$-module of finite presentation.", "Since by definition of morphisms in the category $\\mathcal{I}$", "we have $\\varphi_{i'} \\circ \\alpha = \\varphi_{i'} \\circ \\beta$", "we see that we get an induced map $\\psi : \\mathcal{G} \\to \\mathcal{F}$.", "Hence again the pair $(\\mathcal{G}, \\psi)$ is isomorphic to", "the pair $(\\mathcal{F}_{i''}, \\varphi_{i''})$ for some $i''$.", "Hence we see that there exists a morphism $i' \\to i''$ in", "$\\mathcal{I}$ which equalizes $\\alpha$ and $\\beta$. Thus we have", "shown that the category $\\mathcal{I}$ is filtered.", "\\medskip\\noindent", "We still have to show that the colimit of the diagram is $\\mathcal{F}$.", "By definition of the colimit, and by our definition of the category", "$\\mathcal{I}$ there is a canonical map", "$$", "\\varphi :", "\\colim_i \\mathcal{F}_i", "\\longrightarrow", "\\mathcal{F}.", "$$", "Pick $x \\in X$. Let us show that $\\varphi_x$ is an isomorphism.", "Recall that", "$$", "(\\colim_i \\mathcal{F}_i)_x", "=", "\\colim_i \\mathcal{F}_{i, x},", "$$", "see", "Sheaves, Section \\ref{sheaves-section-limits-sheaves}.", "First we show that the map $\\varphi_x$ is injective.", "Suppose that $s \\in \\mathcal{F}_{i, x}$ is an element", "such that $s$ maps to zero in $\\mathcal{F}_x$. Then there exists", "a quasi-compact open $U$ such that $s$ comes from $s \\in \\mathcal{F}_i(U)$", "and such that $\\varphi_i(s) = 0$ in $\\mathcal{F}(U)$.", "By Lemma \\ref{lemma-extend}", "we can find a finite type quasi-coherent subsheaf", "$\\mathcal{K} \\subset \\Ker(\\varphi_i)$ which restricts to", "the quasi-coherent $\\mathcal{O}_U$-submodule of $\\mathcal{F}_i$", "generated by $s$:", "$\\mathcal{K}|_U = \\mathcal{O}_U\\cdot s \\subset \\mathcal{F}_i|_U$.", "Clearly, $\\mathcal{F}_i/\\mathcal{K}$ is of finite presentation and", "the map $\\varphi_i$ factors through the quotient map", "$\\mathcal{F}_i \\to \\mathcal{F}_i/\\mathcal{K}$. Hence we can find", "an $i' \\in I$ and a morphism $\\alpha : \\mathcal{F}_i \\to \\mathcal{F}_{i'}$", "in $\\mathcal{I}$ which can be identified with the quotient map", "$\\mathcal{F}_i \\to \\mathcal{F}_i/\\mathcal{K}$. Then it follows", "that the section $s$ maps to zero in $\\mathcal{F}_{i'}(U)$ and", "in particular in", "$(\\colim_i \\mathcal{F}_i)_x =", "\\colim_i \\mathcal{F}_{i, x}$.", "The injectivity follows.", "Finally, we show that the map $\\varphi_x$ is surjective.", "Pick $s \\in \\mathcal{F}_x$. Choose a quasi-compact open neighbourhood", "$U \\subset X$ of $x$ such that $s$ corresponds to a section", "$s \\in \\mathcal{F}(U)$. Consider the map", "$s : \\mathcal{O}_U \\to \\mathcal{F}$ (multiplication by $s$).", "By Lemma \\ref{lemma-extend-finite-presentation}", "there exists an $\\mathcal{O}_X$-module $\\mathcal{G}$", "of finite presentation and an $\\mathcal{O}_X$-module map", "$\\mathcal{G} \\to \\mathcal{F}$ such that $\\mathcal{G}|_U \\to \\mathcal{F}|_U$", "is identified with", "$s : \\mathcal{O}_U \\to \\mathcal{F}$.", "Again by definition of $\\mathcal{I}$ there exists an $i \\in I$", "such that $\\mathcal{G} \\to \\mathcal{F}$ is isomorphic to", "$\\varphi_i : \\mathcal{F}_i \\to \\mathcal{F}$. Clearly there exists", "a section $s' \\in \\mathcal{F}_i(U)$ mapping to $s \\in \\mathcal{F}(U)$.", "This proves surjectivity and the proof of the lemma is complete." ], "refs": [ "modules-lemma-set-isomorphism-classes-finite-type-modules", "properties-lemma-extend", "properties-lemma-extend-finite-presentation" ], "ref_ids": [ 13242, 3019, 3021 ] } ], "ref_ids": [ 12363 ] }, { "id": 3024, "type": "theorem", "label": "properties-lemma-directed-colimit-finite-presentation", "categories": [ "properties" ], "title": "properties-lemma-directed-colimit-finite-presentation", "contents": [ "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "There exist", "\\begin{enumerate}", "\\item a directed set $I$ (see", "Categories, Definition \\ref{categories-definition-directed-set}),", "\\item a system $(\\mathcal{F}_i, \\varphi_{ii'})$", "over $I$ in $\\textit{Mod}(\\mathcal{O}_X)$ (see", "Categories, Definition \\ref{categories-definition-system-over-poset})", "\\item morphisms of $\\mathcal{O}_X$-modules", "$\\varphi_i : \\mathcal{F}_i \\to \\mathcal{F}$", "\\end{enumerate}", "such that each $\\mathcal{F}_i$ is of finite presentation", "and such that the morphisms $\\varphi_i$ induce an isomorphism", "$$", "\\colim_i \\mathcal{F}_i", "=", "\\mathcal{F}.", "$$" ], "refs": [ "categories-definition-directed-set", "categories-definition-system-over-poset" ], "proofs": [ { "contents": [ "This is a direct consequence of", "Lemma \\ref{lemma-directed-colimit-diagram-finite-presentation} and", "Categories, Lemma \\ref{categories-lemma-directed-category-system}", "(combined with the fact that", "colimits exist in the category of sheaves of $\\mathcal{O}_X$-modules, see", "Sheaves, Section \\ref{sheaves-section-limits-sheaves})." ], "refs": [ "properties-lemma-directed-colimit-diagram-finite-presentation", "categories-lemma-directed-category-system" ], "ref_ids": [ 3023, 12236 ] } ], "ref_ids": [ 12365, 12366 ] }, { "id": 3025, "type": "theorem", "label": "properties-lemma-finite-directed-colimit-surjective-maps", "categories": [ "properties" ], "title": "properties-lemma-finite-directed-colimit-surjective-maps", "contents": [ "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Then we can write $\\mathcal{F} = \\colim \\mathcal{F}_i$ with $\\mathcal{F}_i$", "of finite presentation and all transition maps", "$\\mathcal{F}_i \\to \\mathcal{F}_{i'}$ surjective." ], "refs": [], "proofs": [ { "contents": [ "Write $\\mathcal{F} = \\colim \\mathcal{G}_i$ as a filtered colimit of", "finitely presented $\\mathcal{O}_X$-modules", "(Lemma \\ref{lemma-directed-colimit-finite-presentation}).", "We claim that $\\mathcal{G}_i \\to \\mathcal{F}$ is surjective for some $i$.", "Namely, choose a finite affine open covering $X = U_1 \\cup \\ldots \\cup U_m$.", "Choose sections $s_{jl} \\in \\mathcal{F}(U_j)$ generating", "$\\mathcal{F}|_{U_j}$, see Lemma \\ref{lemma-finite-type-module}.", "By Sheaves, Lemma \\ref{sheaves-lemma-directed-colimits-sections}", "we see that $s_{jl}$ is in the image of $\\mathcal{G}_i \\to \\mathcal{F}$", "for $i$ large enough. Hence $\\mathcal{G}_i \\to \\mathcal{F}$ is surjective", "for $i$ large enough. Choose such an $i$ and let", "$\\mathcal{K} \\subset \\mathcal{G}_i$ be the kernel of the map", "$\\mathcal{G}_i \\to \\mathcal{F}$. Write $\\mathcal{K} = \\colim \\mathcal{K}_a$", "as the filtered colimit of its finite type quasi-coherent submodules", "(Lemma \\ref{lemma-quasi-coherent-colimit-finite-type}). Then", "$\\mathcal{F} = \\colim \\mathcal{G}_i/\\mathcal{K}_a$ is a solution", "to the problem posed by the lemma." ], "refs": [ "properties-lemma-directed-colimit-finite-presentation", "properties-lemma-finite-type-module", "sheaves-lemma-directed-colimits-sections", "properties-lemma-quasi-coherent-colimit-finite-type" ], "ref_ids": [ 3024, 3002, 14526, 3020 ] } ], "ref_ids": [] }, { "id": 3026, "type": "theorem", "label": "properties-lemma-application-directed-colimit", "categories": [ "properties" ], "title": "properties-lemma-application-directed-colimit", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Let $U \\subset X$ be a quasi-compact open such that $\\mathcal{F}|_U$", "is of finite presentation. Then there exists a map of $\\mathcal{O}_X$-modules", "$\\varphi : \\mathcal{G} \\to \\mathcal{F}$ with", "(a) $\\mathcal{G}$ of finite presentation,", "(b) $\\varphi$ is surjective, and", "(c) $\\varphi|_U$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Write $\\mathcal{F} = \\colim \\mathcal{F}_i$ as a directed colimit", "with each $\\mathcal{F}_i$ of finite presentation,", "see Lemma \\ref{lemma-directed-colimit-finite-presentation}.", "Choose a finite affine open covering $X = \\bigcup V_j$ and choose", "finitely many sections $s_{jl} \\in \\mathcal{F}(V_j)$ generating", "$\\mathcal{F}|_{V_j}$, see Lemma \\ref{lemma-finite-type-module}.", "By Sheaves, Lemma \\ref{sheaves-lemma-directed-colimits-sections}", "we see that $s_{jl}$ is in the image of $\\mathcal{F}_i \\to \\mathcal{F}$", "for $i$ large enough. Hence $\\mathcal{F}_i \\to \\mathcal{F}$ is surjective", "for $i$ large enough. Choose such an $i$ and let", "$\\mathcal{K} \\subset \\mathcal{F}_i$ be the kernel of the map", "$\\mathcal{F}_i \\to \\mathcal{F}$. Since $\\mathcal{F}_U$ is of finite", "presentation, we see that $\\mathcal{K}|_U$ is of finite type, see", "Modules, Lemma", "\\ref{modules-lemma-kernel-surjection-finite-free-onto-finite-presentation}.", "Hence we can find a finite type quasi-coherent submodule", "$\\mathcal{K}' \\subset \\mathcal{K}$ with $\\mathcal{K}'|_U = \\mathcal{K}|_U$,", "see Lemma \\ref{lemma-extend}. Then", "$\\mathcal{G} = \\mathcal{F}_i/\\mathcal{K}'$", "with the given map $\\mathcal{G} \\to \\mathcal{F}$ is a solution." ], "refs": [ "properties-lemma-directed-colimit-finite-presentation", "properties-lemma-finite-type-module", "sheaves-lemma-directed-colimits-sections", "modules-lemma-kernel-surjection-finite-free-onto-finite-presentation", "properties-lemma-extend" ], "ref_ids": [ 3024, 3002, 14526, 13249, 3019 ] } ], "ref_ids": [] }, { "id": 3027, "type": "theorem", "label": "properties-lemma-algebra-directed-colimit-finite-presentation", "categories": [ "properties" ], "title": "properties-lemma-algebra-directed-colimit-finite-presentation", "contents": [ "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.", "Let $\\mathcal{A}$ be a quasi-coherent $\\mathcal{O}_X$-algebra.", "There exist", "\\begin{enumerate}", "\\item a directed set $I$ (see", "Categories, Definition \\ref{categories-definition-directed-set}),", "\\item a system $(\\mathcal{A}_i, \\varphi_{ii'})$", "over $I$ in the category of $\\mathcal{O}_X$-algebras,", "\\item morphisms of $\\mathcal{O}_X$-algebras", "$\\varphi_i : \\mathcal{A}_i \\to \\mathcal{A}$", "\\end{enumerate}", "such that each $\\mathcal{A}_i$ is a quasi-coherent $\\mathcal{O}_X$-algebra", "of finite presentation and such that the morphisms $\\varphi_i$", "induce an isomorphism", "$$", "\\colim_i \\mathcal{A}_i", "=", "\\mathcal{A}.", "$$" ], "refs": [ "categories-definition-directed-set" ], "proofs": [ { "contents": [ "First we write $\\mathcal{A} = \\colim_i \\mathcal{F}_i$ as a directed", "colimit of finitely presented quasi-coherent sheaves as in", "Lemma \\ref{lemma-directed-colimit-finite-presentation}.", "For each $i$ let $\\mathcal{B}_i = \\text{Sym}(\\mathcal{F}_i)$ be the", "symmetric algebra on $\\mathcal{F}_i$ over $\\mathcal{O}_X$. Write", "$\\mathcal{I}_i = \\Ker(\\mathcal{B}_i \\to \\mathcal{A})$. Write", "$\\mathcal{I}_i = \\colim_j \\mathcal{F}_{i, j}$ where", "$\\mathcal{F}_{i, j}$ is a finite type quasi-coherent submodule of", "$\\mathcal{I}_i$, see", "Lemma \\ref{lemma-quasi-coherent-colimit-finite-type}.", "Set $\\mathcal{I}_{i, j} \\subset \\mathcal{I}_i$", "equal to the $\\mathcal{B}_i$-ideal generated by $\\mathcal{F}_{i, j}$.", "Set $\\mathcal{A}_{i, j} = \\mathcal{B}_i/\\mathcal{I}_{i, j}$.", "Then $\\mathcal{A}_{i, j}$ is a quasi-coherent finitely presented", "$\\mathcal{O}_X$-algebra. Define $(i, j) \\leq (i', j')$ if", "$i \\leq i'$ and the map $\\mathcal{B}_i \\to \\mathcal{B}_{i'}$", "maps the ideal $\\mathcal{I}_{i, j}$ into the ideal $\\mathcal{I}_{i', j'}$.", "Then it is clear that $\\mathcal{A} = \\colim_{i, j} \\mathcal{A}_{i, j}$." ], "refs": [ "properties-lemma-directed-colimit-finite-presentation", "properties-lemma-quasi-coherent-colimit-finite-type" ], "ref_ids": [ 3024, 3020 ] } ], "ref_ids": [ 12365 ] }, { "id": 3028, "type": "theorem", "label": "properties-lemma-algebra-directed-colimit-finite-type", "categories": [ "properties" ], "title": "properties-lemma-algebra-directed-colimit-finite-type", "contents": [ "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.", "Let $\\mathcal{A}$ be a quasi-coherent $\\mathcal{O}_X$-algebra.", "Then $\\mathcal{A}$ is the directed colimit of its finite type", "quasi-coherent $\\mathcal{O}_X$-subalgebras." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{A}_1, \\mathcal{A}_2 \\subset \\mathcal{F}$ are", "quasi-coherent $\\mathcal{O}_X$-subalgebras of finite type, then the image of", "$\\mathcal{A}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{A}_2 \\to \\mathcal{A}$", "is also a quasi-coherent $\\mathcal{O}_X$-subalgebra of finite type", "(some details omitted) which contains both $\\mathcal{A}_1$ and $\\mathcal{A}_2$.", "In this way we see that the system is directed.", "To show that $\\mathcal{A}$ is the colimit of this system, write", "$\\mathcal{A} = \\colim_i \\mathcal{A}_i$ as a directed", "colimit of finitely presented quasi-coherent $\\mathcal{O}_X$-algebras as in", "Lemma \\ref{lemma-algebra-directed-colimit-finite-presentation}.", "Then the images $\\mathcal{A}'_i = \\Im(\\mathcal{A}_i \\to \\mathcal{A})$ are", "quasi-coherent subalgebras of $\\mathcal{A}$ of finite type. Since", "$\\mathcal{A}$ is the colimit of these the result follows." ], "refs": [ "properties-lemma-algebra-directed-colimit-finite-presentation" ], "ref_ids": [ 3027 ] } ], "ref_ids": [] }, { "id": 3029, "type": "theorem", "label": "properties-lemma-finite-algebra-directed-colimit-finite-finitely-presented", "categories": [ "properties" ], "title": "properties-lemma-finite-algebra-directed-colimit-finite-finitely-presented", "contents": [ "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.", "Let $\\mathcal{A}$ be a finite quasi-coherent $\\mathcal{O}_X$-algebra.", "Then $\\mathcal{A} = \\colim \\mathcal{A}_i$ is a directed colimit of finite", "and finitely presented quasi-coherent $\\mathcal{O}_X$-algebras", "such that all transition maps $\\mathcal{A}_{i'} \\to \\mathcal{A}_i$", "are surjective." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-directed-colimit-surjective-maps}", "there exists a finitely presented $\\mathcal{O}_X$-module", "$\\mathcal{F}$ and a surjection $\\mathcal{F} \\to \\mathcal{A}$.", "Using the algebra structure we obtain a surjection", "$$", "\\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F}) \\longrightarrow \\mathcal{A}", "$$", "Denote $\\mathcal{J}$ the kernel. Write $\\mathcal{J} = \\colim \\mathcal{E}_i$", "as a filtered colimit of finite type $\\mathcal{O}_X$-submodules", "$\\mathcal{E}_i$ (Lemma \\ref{lemma-quasi-coherent-colimit-finite-type}). Set", "$$", "\\mathcal{A}_i = \\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F})/(\\mathcal{E}_i)", "$$", "where $(\\mathcal{E}_i)$ indicates the ideal sheaf generated by", "the image of $\\mathcal{E}_i \\to \\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F})$.", "Then each $\\mathcal{A}_i$ is a finitely presented $\\mathcal{O}_X$-algebra,", "the transition maps are surjections,", "and $\\mathcal{A} = \\colim \\mathcal{A}_i$. To finish the proof we still", "have to show that $\\mathcal{A}_i$ is a finite $\\mathcal{O}_X$-algebra", "for $i$ sufficiently large. To do this we choose an affine open", "covering $X = U_1 \\cup \\ldots \\cup U_m$. Take generators", "$f_{j, 1}, \\ldots, f_{j, N_j} \\in \\Gamma(U_i, \\mathcal{F})$.", "As $\\mathcal{A}(U_j)$ is a finite $\\mathcal{O}_X(U_j)$-algebra we", "see that for each $k$ there exists a monic polynomial", "$P_{j, k} \\in \\mathcal{O}(U_j)[T]$ such that $P_{j, k}(f_{j, k})$", "is zero in $\\mathcal{A}(U_j)$. Since", "$\\mathcal{A} = \\colim \\mathcal{A}_i$ by construction, we", "have $P_{j, k}(f_{j, k}) = 0$ in $\\mathcal{A}_i(U_j)$", "for all sufficiently large $i$. For such $i$ the algebras", "$\\mathcal{A}_i$ are finite." ], "refs": [ "properties-lemma-finite-directed-colimit-surjective-maps", "properties-lemma-quasi-coherent-colimit-finite-type" ], "ref_ids": [ 3025, 3020 ] } ], "ref_ids": [] }, { "id": 3030, "type": "theorem", "label": "properties-lemma-integral-algebra-directed-colimit-finite", "categories": [ "properties" ], "title": "properties-lemma-integral-algebra-directed-colimit-finite", "contents": [ "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.", "Let $\\mathcal{A}$ be an integral quasi-coherent $\\mathcal{O}_X$-algebra.", "Then", "\\begin{enumerate}", "\\item $\\mathcal{A}$ is the directed colimit of its finite", "quasi-coherent $\\mathcal{O}_X$-subalgebras, and", "\\item $\\mathcal{A}$ is a direct colimit of finite and finitely", "presented quasi-coherent $\\mathcal{O}_X$-algebras.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-algebra-directed-colimit-finite-type} we have", "$\\mathcal{A} = \\colim \\mathcal{A}_i$ where", "$\\mathcal{A}_i \\subset \\mathcal{A}$ runs through the", "quasi-coherent $\\mathcal{O}_X$-algebras of finite type.", "Any finite type quasi-coherent $\\mathcal{O}_X$-subalgebra", "of $\\mathcal{A}$ is finite (apply Algebra, Lemma", "\\ref{algebra-lemma-characterize-finite-in-terms-of-integral}", "to $\\mathcal{A}_i(U) \\subset \\mathcal{A}(U)$ for affine opens $U$", "in $X$). This proves (1).", "\\medskip\\noindent", "To prove (2), write $\\mathcal{A} = \\colim \\mathcal{F}_i$", "as a colimit of finitely presented $\\mathcal{O}_X$-modules using", "Lemma \\ref{lemma-directed-colimit-finite-presentation}.", "For each $i$, let $\\mathcal{J}_i$ be the kernel of the map", "$$", "\\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F}_i) \\longrightarrow \\mathcal{A}", "$$", "For $i' \\geq i$ there is an induced map $\\mathcal{J}_i \\to \\mathcal{J}_{i'}$", "and we have $\\mathcal{A} =", "\\colim \\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F}_i)/\\mathcal{J}_i$.", "Moreover, the quasi-coherent $\\mathcal{O}_X$-algebras", "$\\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F}_i)/\\mathcal{J}_i$", "are finite (see above). Write $\\mathcal{J}_i = \\colim \\mathcal{E}_{ik}$", "as a colimit of finitely presented $\\mathcal{O}_X$-modules.", "Given $i' \\geq i$ and $k$ there exists a $k'$ such that we", "have a map $\\mathcal{E}_{ik} \\to \\mathcal{E}_{i'k'}$", "making", "$$", "\\xymatrix{", "\\mathcal{J}_i \\ar[r] & \\mathcal{J}_{i'} \\\\", "\\mathcal{E}_{ik} \\ar[u] \\ar[r] & \\mathcal{E}_{i'k'} \\ar[u]", "}", "$$", "commute. This follows from", "Modules, Lemma \\ref{modules-lemma-finite-presentation-quasi-compact-colimit}.", "This induces a map", "$$", "\\mathcal{A}_{ik} =", "\\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F}_i)/(\\mathcal{E}_{ik})", "\\longrightarrow", "\\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F}_{i'})/(\\mathcal{E}_{i'k'}) =", "\\mathcal{A}_{i'k'}", "$$", "where $(\\mathcal{E}_{ik})$ denotes the ideal generated by $\\mathcal{E}_{ik}$.", "The quasi-coherent $\\mathcal{O}_X$-algebras $\\mathcal{A}_{ki}$", "are of finite presentation and finite for $k$ large enough", "(see proof of", "Lemma \\ref{lemma-finite-algebra-directed-colimit-finite-finitely-presented}).", "Finally, we have", "$$", "\\colim \\mathcal{A}_{ik} = \\colim \\mathcal{A}_i = \\mathcal{A}", "$$", "Namely, the first equality was shown in the proof of", "Lemma \\ref{lemma-finite-algebra-directed-colimit-finite-finitely-presented}", "and the second equality because $\\mathcal{A}$ is the colimit of", "the modules $\\mathcal{F}_i$." ], "refs": [ "properties-lemma-algebra-directed-colimit-finite-type", "algebra-lemma-characterize-finite-in-terms-of-integral", "properties-lemma-directed-colimit-finite-presentation", "modules-lemma-finite-presentation-quasi-compact-colimit", "properties-lemma-finite-algebra-directed-colimit-finite-finitely-presented", "properties-lemma-finite-algebra-directed-colimit-finite-finitely-presented" ], "ref_ids": [ 3028, 484, 3024, 13252, 3029, 3029 ] } ], "ref_ids": [] }, { "id": 3031, "type": "theorem", "label": "properties-lemma-set-of-iso-classes", "categories": [ "properties" ], "title": "properties-lemma-set-of-iso-classes", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\kappa$ be a cardinal.", "There exists a set $T$ and a family $(\\mathcal{F}_t)_{t \\in T}$ of", "$\\kappa$-generated $\\mathcal{O}_X$-modules such that every $\\kappa$-generated", "$\\mathcal{O}_X$-module is isomorphic to one of the $\\mathcal{F}_t$." ], "refs": [], "proofs": [ { "contents": [ "There is a set of coverings of $X$ (provided we disallow repeats).", "Suppose $X = \\bigcup U_i$ is a covering and suppose $\\mathcal{F}_i$", "is an $\\mathcal{O}_{U_i}$-module. Then there is a set of isomorphism", "classes of $\\mathcal{O}_X$-modules $\\mathcal{F}$ with the property", "that $\\mathcal{F}|_{U_i} \\cong \\mathcal{F}_i$ since there is a set of", "glueing maps. This reduces us to proving there is a set of (isomorphism", "classes of) quotients", "$\\oplus_{k \\in \\kappa} \\mathcal{O}_X \\to \\mathcal{F}$", "for any ringed space $X$. This is clear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3032, "type": "theorem", "label": "properties-lemma-colimit-kappa", "categories": [ "properties" ], "title": "properties-lemma-colimit-kappa", "contents": [ "Let $X$ be a scheme. There exists a cardinal $\\kappa$ such that", "every quasi-coherent module $\\mathcal{F}$ is the directed colimit", "of its quasi-coherent $\\kappa$-generated quasi-coherent subsheaves." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine open covering $X = \\bigcup_{i \\in I} U_i$. For each pair", "$i, j$ choose an affine open covering", "$U_i \\cap U_j = \\bigcup_{k \\in I_{ij}} U_{ijk}$.", "Write $U_i = \\Spec(A_i)$ and $U_{ijk} = \\Spec(A_{ijk})$.", "Let $\\kappa$ be any infinite cardinal $\\geq$ than the cardinality", "of any of the sets $I$, $I_{ij}$.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be a quasi-coherent sheaf. Set $M_i = \\mathcal{F}(U_i)$", "and $M_{ijk} = \\mathcal{F}(U_{ijk})$. Note that", "$$", "M_i \\otimes_{A_i} A_{ijk} = M_{ijk} = M_j \\otimes_{A_j} A_{ijk}.", "$$", "see", "Schemes, Lemma \\ref{schemes-lemma-widetilde-pullback}.", "Using the axiom of choice we choose a map", "$$", "(i, j, k, m) \\mapsto S(i, j, k, m)", "$$", "which associates to every $i, j \\in I$, $k \\in I_{ij}$ and $m \\in M_i$", "a finite subset $S(i, j, k, m) \\subset M_j$ such that we have", "$$", "m \\otimes 1 = \\sum\\nolimits_{m' \\in S(i, j, k, m)} m' \\otimes a_{m'}", "$$", "in $M_{ijk}$ for some $a_{m'} \\in A_{ijk}$. Moreover, let's agree", "that $S(i, i, k, m) = \\{m\\}$ for all $i, j = i, k, m$ as above.", "Fix such a map.", "\\medskip\\noindent", "Given a family $\\mathcal{S} = (S_i)_{i \\in I}$ of subsets", "$S_i \\subset M_i$ of cardinality at most $\\kappa$ we set", "$\\mathcal{S}' = (S'_i)$ where", "$$", "S'_j = \\bigcup\\nolimits_{(i, j, k, m)\\text{ such that }m \\in S_i}", "S(i, j, k, m)", "$$", "Note that $S_i \\subset S'_i$. Note that $S'_i$ has cardinality at most", "$\\kappa$ because it is a union over a set of cardinality at most $\\kappa$", "of finite sets. Set $\\mathcal{S}^{(0)} = \\mathcal{S}$,", "$\\mathcal{S}^{(1)} = \\mathcal{S}'$ and by induction", "$\\mathcal{S}^{(n + 1)} = (\\mathcal{S}^{(n)})'$. Then set", "$\\mathcal{S}^{(\\infty)} = \\bigcup_{n \\geq 0} \\mathcal{S}^{(n)}$.", "Writing $\\mathcal{S}^{(\\infty)} = (S^{(\\infty)}_i)$ we see that", "for any element $m \\in S^{(\\infty)}_i$ the image of $m$ in", "$M_{ijk}$ can be written as a finite sum $\\sum m' \\otimes a_{m'}$", "with $m' \\in S_j^{(\\infty)}$. In this way we see that setting", "$$", "N_i = A_i\\text{-submodule of }M_i\\text{ generated by }S^{(\\infty)}_i", "$$", "we have", "$$", "N_i \\otimes_{A_i} A_{ijk} = N_j \\otimes_{A_j} A_{ijk}.", "$$", "as submodules of $M_{ijk}$. Thus there exists a quasi-coherent subsheaf", "$\\mathcal{G} \\subset \\mathcal{F}$ with $\\mathcal{G}(U_i) = N_i$.", "Moreover, by construction the sheaf $\\mathcal{G}$ is $\\kappa$-generated.", "\\medskip\\noindent", "Let $\\{\\mathcal{G}_t\\}_{t \\in T}$ be the set of $\\kappa$-generated", "quasi-coherent subsheaves. If $t, t' \\in T$ then", "$\\mathcal{G}_t + \\mathcal{G}_{t'}$ is also a $\\kappa$-generated", "quasi-coherent subsheaf as it is the image of the map", "$\\mathcal{G}_t \\oplus \\mathcal{G}_{t'} \\to \\mathcal{F}$.", "Hence the system (ordered by inclusion) is directed.", "The arguments above show that every section of $\\mathcal{F}$ over $U_i$", "is in one of the $\\mathcal{G}_t$ (because we can start with $\\mathcal{S}$", "such that the given section is an element of $S_i$). Hence", "$\\colim_t \\mathcal{G}_t \\to \\mathcal{F}$ is both injective and surjective", "as desired." ], "refs": [ "schemes-lemma-widetilde-pullback" ], "ref_ids": [ 7662 ] } ], "ref_ids": [] }, { "id": 3033, "type": "theorem", "label": "properties-lemma-quasi-coherent-finite-type-ideals", "categories": [ "properties" ], "title": "properties-lemma-quasi-coherent-finite-type-ideals", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $U \\subset X$ be an open subscheme. The following are equivalent:", "\\begin{enumerate}", "\\item $U$ is retrocompact in $X$,", "\\item $U$ is quasi-compact,", "\\item $U$ is a finite union of affine opens, and", "\\item there exists a finite type quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_X$ such that $X \\setminus U = V(\\mathcal{I})$", "(set theoretically).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1), (2), and (3) follows from", "Lemma \\ref{lemma-quasi-separated-quasi-compact-open-retrocompact}.", "Assume (1), (2), (3). Let $T = X \\setminus U$. By", "Schemes, Lemma \\ref{schemes-lemma-reduced-closed-subscheme} there exists", "a unique quasi-coherent sheaf of ideals $\\mathcal{J}$ cutting", "out the reduced induced closed subscheme structure on $T$.", "Note that $\\mathcal{J}|_U = \\mathcal{O}_U$ which is an", "$\\mathcal{O}_U$-modules of finite type.", "By Lemma \\ref{lemma-extend} there exists a quasi-coherent subsheaf", "$\\mathcal{I} \\subset \\mathcal{J}$ which is of finite type", "and has the property that $\\mathcal{I}|_U = \\mathcal{J}|_U$.", "Then $X \\setminus U = V(\\mathcal{I})$ and we obtain (4). Conversely,", "if $\\mathcal{I}$ is as in (4) and $W = \\Spec(R) \\subset X$ is an affine", "open, then $\\mathcal{I}|_W = \\widetilde{I}$ for some finitely generated", "ideal $I \\subset R$, see Lemma \\ref{lemma-finite-type-module}.", "It follows that $U \\cap W = \\Spec(R) \\setminus V(I)$ is quasi-compact,", "see Algebra, Lemma \\ref{algebra-lemma-qc-open}. Hence $U \\subset X$", "is retrocompact by Lemma \\ref{lemma-retrocompact}." ], "refs": [ "properties-lemma-quasi-separated-quasi-compact-open-retrocompact", "schemes-lemma-reduced-closed-subscheme", "properties-lemma-extend", "properties-lemma-finite-type-module", "algebra-lemma-qc-open", "properties-lemma-retrocompact" ], "ref_ids": [ 2940, 7681, 3019, 3002, 432, 2943 ] } ], "ref_ids": [] }, { "id": 3034, "type": "theorem", "label": "properties-lemma-sections-annihilated-by-ideal", "categories": [ "properties" ], "title": "properties-lemma-sections-annihilated-by-ideal", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Consider the sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}'$", "which associates to every open $U \\subset X$", "$$", "\\mathcal{F}'(U)", "=", "\\{s \\in \\mathcal{F}(U) \\mid", "\\mathcal{I}s = 0\\}", "$$", "Assume $\\mathcal{I}$ is of finite type. Then", "\\begin{enumerate}", "\\item $\\mathcal{F}'$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules,", "\\item on any affine open $U \\subset X$ we have", "$\\mathcal{F}'(U) = \\{s \\in \\mathcal{F}(U) \\mid \\mathcal{I}(U)s = 0\\}$, and", "\\item $\\mathcal{F}'_x = \\{s \\in \\mathcal{F}_x \\mid \\mathcal{I}_x s = 0\\}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that the rule defining $\\mathcal{F}'$ gives a subsheaf", "of $\\mathcal{F}$ (the sheaf condition is easy to verify). Hence we", "may work locally on $X$ to verify the other statements. In other words", "we may assume that $X = \\Spec(A)$, $\\mathcal{F} = \\widetilde{M}$", "and $\\mathcal{I} = \\widetilde{I}$. It is clear that in this case", "$\\mathcal{F}'(U) = \\{x \\in M \\mid Ix = 0\\} =: M'$ because $\\widetilde{I}$", "is generated by its global sections $I$ which proves (2).", "To show $\\mathcal{F}'$ is quasi-coherent it suffices to show that", "for every $f \\in A$ we have", "$\\{x \\in M_f \\mid I_f x = 0\\} = (M')_f$.", "Write $I = (g_1, \\ldots, g_t)$, which is possible because $\\mathcal{I}$", "is of finite type, see Lemma \\ref{lemma-finite-type-module}.", "If $x = y/f^n$ and $I_fx = 0$, then that means that for every $i$", "there exists an $m \\geq 0$ such that $f^mg_ix = 0$.", "We may choose one $m$ which works for all $i$ (and this is where we", "use that $I$ is finitely generated). Then we see that $f^mx \\in M'$", "and $x/f^n = f^mx/f^{n + m}$ in $(M')_f$ as desired.", "The proof of (3) is similar and omitted." ], "refs": [ "properties-lemma-finite-type-module" ], "ref_ids": [ 3002 ] } ], "ref_ids": [] }, { "id": 3035, "type": "theorem", "label": "properties-lemma-push-sections-annihilated-by-ideal", "categories": [ "properties" ], "title": "properties-lemma-push-sections-annihilated-by-ideal", "contents": [ "Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism", "of schemes. Let $\\mathcal{I} \\subset \\mathcal{O}_Y$ be a quasi-coherent", "sheaf of ideals of finite type. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. Let $\\mathcal{F}' \\subset \\mathcal{F}$", "be the subsheaf of sections annihilated by $f^{-1}\\mathcal{I}\\mathcal{O}_X$.", "Then $f_*\\mathcal{F}' \\subset f_*\\mathcal{F}$ is the subsheaf", "of sections annihilated by $\\mathcal{I}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. (Hint: The assumption that $f$ is quasi-compact and", "quasi-separated implies that $f_*\\mathcal{F}$ is quasi-coherent", "so that Lemma \\ref{lemma-sections-annihilated-by-ideal} applies", "to $\\mathcal{I}$ and $f_*\\mathcal{F}$.)" ], "refs": [ "properties-lemma-sections-annihilated-by-ideal" ], "ref_ids": [ 3034 ] } ], "ref_ids": [] }, { "id": 3036, "type": "theorem", "label": "properties-lemma-sections-supported-on-closed-subset", "categories": [ "properties" ], "title": "properties-lemma-sections-supported-on-closed-subset", "contents": [ "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subset.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Consider the sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}'$", "which associates to every open $U \\subset X$", "$$", "\\mathcal{F}'(U)", "=", "\\{s \\in \\mathcal{F}(U) \\mid", "\\text{the support of }s\\text{ is contained in }Z \\cap U\\}", "$$", "If $X \\setminus Z$ is a retrocompact open of $X$, then", "\\begin{enumerate}", "\\item for an affine open $U \\subset X$ there exist a finitely generated", "ideal $I \\subset \\mathcal{O}_X(U)$ such that $Z \\cap U = V(I)$,", "\\item for $U$ and $I$ as in (1) we have", "$\\mathcal{F}'(U) = \\{x \\in \\mathcal{F}(U) \\mid", "I^nx = 0 \\text{ for some } n\\}$,", "\\item $\\mathcal{F}'$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is Algebra, Lemma \\ref{algebra-lemma-qc-open}.", "Let $U = \\Spec(A)$ and $I$ be as in (1).", "Then $\\mathcal{F}|_U$ is the quasi-coherent sheaf associated to", "some $A$-module $M$. We have", "$$", "\\mathcal{F}'(U) = \\{x \\in M \\mid x = 0\\text{ in }M_\\mathfrak p", "\\text{ for all }\\mathfrak p \\not \\in Z\\}.", "$$", "by Modules, Definition \\ref{modules-definition-support}. Thus", "$x \\in \\mathcal{F}'(U)$ if and only if $V(\\text{Ann}(x)) \\subset V(I)$, see", "Algebra, Lemma \\ref{algebra-lemma-support-element}. Since $I$ is", "finitely generated this is equivalent to $I^n x = 0$ for some $n$.", "This proves (2).", "\\medskip\\noindent", "Proof of (3). Observe that given $U \\subset X$ open there is an exact sequence", "$$", "0 \\to \\mathcal{F}'(U) \\to \\mathcal{F}(U) \\to \\mathcal{F}(U \\setminus Z)", "$$", "If we denote $j : X \\setminus Z \\to X$ the inclusion morphism, then", "we observe that $\\mathcal{F}(U \\setminus Z)$ is the sections of the", "module $j_*(\\mathcal{F}|_{X \\setminus Z})$ over $U$. Thus we have", "an exact sequence", "$$", "0 \\to \\mathcal{F}' \\to \\mathcal{F} \\to j_*(\\mathcal{F}|_{X \\setminus Z})", "$$", "The restriction $\\mathcal{F}|_{X \\setminus Z}$ is quasi-coherent.", "Hence $j_*(\\mathcal{F}|_{X \\setminus Z})$ is quasi-coherent", "by Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "and our assumption that $j$ is quasi-compact (any open immersion", "is separated). Hence $\\mathcal{F}'$ is quasi-coherent", "as a kernel of a map of quasi-coherent modules, see", "Schemes, Section \\ref{schemes-section-quasi-coherent}." ], "refs": [ "algebra-lemma-qc-open", "modules-definition-support", "algebra-lemma-support-element", "schemes-lemma-push-forward-quasi-coherent" ], "ref_ids": [ 432, 13334, 545, 7730 ] } ], "ref_ids": [] }, { "id": 3037, "type": "theorem", "label": "properties-lemma-push-sections-supported-on-closed-subset", "categories": [ "properties" ], "title": "properties-lemma-push-sections-supported-on-closed-subset", "contents": [ "Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism", "of schemes. Let $Z \\subset Y$ be a closed subset such that", "$Y \\setminus Z$ is retrocompact in $Y$. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. Let $\\mathcal{F}' \\subset \\mathcal{F}$", "be the subsheaf of sections supported in $f^{-1}Z$.", "Then $f_*\\mathcal{F}' \\subset f_*\\mathcal{F}$ is the subsheaf", "of sections supported in $Z$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. (Hint: First show that $X \\setminus f^{-1}Z$ is retrocompact", "in $X$ as $Y \\setminus Z$ is retrocompact in $Y$. Hence", "Lemma \\ref{lemma-sections-supported-on-closed-subset}", "applies to $f^{-1}Z$ and $\\mathcal{F}$. As $f$ is quasi-compact and", "quasi-separated we see that $f_*\\mathcal{F}$ is quasi-coherent.", "Hence Lemma \\ref{lemma-sections-supported-on-closed-subset}", "applies to $Z$ and $f_*\\mathcal{F}$. Finally, match the sheaves", "directly.)" ], "refs": [ "properties-lemma-sections-supported-on-closed-subset", "properties-lemma-sections-supported-on-closed-subset" ], "ref_ids": [ 3036, 3036 ] } ], "ref_ids": [] }, { "id": 3038, "type": "theorem", "label": "properties-lemma-sections-over-quasi-compact-open-in-affine", "categories": [ "properties" ], "title": "properties-lemma-sections-over-quasi-compact-open-in-affine", "contents": [ "Let $A$ be a ring.", "Let $I \\subset A$ be a finitely generated ideal.", "Let $M$ be an $A$-module.", "Then there is a canonical map", "$$", "\\colim_n \\Hom_A(I^n, M)", "\\longrightarrow", "\\Gamma(\\Spec(A) \\setminus V(I), \\widetilde{M}).", "$$", "This map is always injective.", "If for all $x \\in M$ we have $Ix = 0 \\Rightarrow x = 0$", "then this map is an isomorphism. In general, set", "$M_n = \\{x \\in M \\mid I^nx = 0\\}$, then there is an", "isomorphism", "$$", "\\colim_n \\Hom_A(I^n, M/M_n)", "\\longrightarrow", "\\Gamma(\\Spec(A) \\setminus V(I), \\widetilde{M}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since $I^{n + 1} \\subset I^n$ and $M_n \\subset M_{n + 1}$ we can", "use composition via these maps to get canonical maps of $A$-modules", "$$", "\\Hom_A(I^n, M)", "\\longrightarrow", "\\Hom_A(I^{n + 1}, M)", "$$", "and", "$$", "\\Hom_A(I^n, M/M_n)", "\\longrightarrow", "\\Hom_A(I^{n + 1}, M/M_{n + 1})", "$$", "which we will use as the transition maps in the systems. Given an", "$A$-module map $\\varphi : I^n \\to M$, then we get a map of", "sheaves $\\widetilde{\\varphi} : \\widetilde{I^n} \\to \\widetilde{M}$", "which we can restrict to the open $\\Spec(A) \\setminus V(I)$.", "Since $\\widetilde{I^n}$ restricted to this open gives the structure", "sheaf we get an element of", "$\\Gamma(\\Spec(A) \\setminus V(I), \\widetilde{M})$.", "We omit the verification that this is compatible with the transition maps", "in the system $\\Hom_A(I^n, M)$. This gives the first arrow.", "To get the second arrow we note that", "$\\widetilde{M}$ and $\\widetilde{M/M_n}$ agree over the open", "$\\Spec(A) \\setminus V(I)$ since the sheaf $\\widetilde{M_n}$", "is clearly supported on $V(I)$. Hence we can use the same mechanism", "as before.", "\\medskip\\noindent", "Next, we work out how to define this arrow in terms of algebra.", "Say $I = (f_1, \\ldots, f_t)$. Then", "$\\Spec(A) \\setminus V(I) = \\bigcup_{i = 1, \\ldots, t} D(f_i)$.", "Hence", "$$", "0 \\to", "\\Gamma(\\Spec(A) \\setminus V(I), \\widetilde{M}) \\to", "\\bigoplus\\nolimits_i M_{f_i} \\to", "\\bigoplus\\nolimits_{i, j} M_{f_if_j}", "$$", "is exact. Suppose that $\\varphi : I^n \\to M$ is an $A$-module map.", "Consider the vector of elements $\\varphi(f_i^n)/f_i^n \\in M_{f_i}$.", "It is easy to see that this vector maps to zero in the", "second direct sum of the exact sequence above. Whence an element", "of $\\Gamma(\\Spec(A) \\setminus V(I), \\widetilde{M})$.", "We omit the verification that this description agrees with the one", "given above.", "\\medskip\\noindent", "Let us show that the first arrow is injective using this description.", "Namely, if $\\varphi$ maps to zero, then for each $i$ the element", "$\\varphi(f_i^n)/f_i^n$ is zero in $M_{f_i}$. In other words we", "see that for each $i$ we have $f_i^m\\varphi(f_i^n) = 0$ for some $m \\geq 0$.", "We may choose a single $m$ which works for all $i$. Then we see that", "$\\varphi(f_i^{n + m}) = 0$ for all $i$. It is easy to see that", "this means that $\\varphi|_{I^{t(n + m - 1) + 1}} = 0$ in other", "words that $\\varphi$ maps to zero in the $t(n + m - 1) + 1$st", "term of the colimit. Hence injectivity follows.", "\\medskip\\noindent", "Note that each $M_n = 0$ in case we have", "$Ix = 0 \\Rightarrow x = 0$ for $x \\in M$. Thus", "to finish the proof of the lemma it suffices to show that", "the second arrow is an isomorphism.", "\\medskip\\noindent", "Let us attempt to construct an inverse of the second map of the lemma.", "Let $s \\in \\Gamma(\\Spec(A) \\setminus V(I), \\widetilde{M})$.", "This corresponds to a vector $x_i/f_i^n$ with $x_i \\in M$ of the", "first direct sum of the exact sequence above.", "Hence for each $i, j$ there exists $m \\geq 0$", "such that $f_i^m f_j^m (f_j^n x_i - f_i^n x_j) = 0$ in $M$.", "We may choose a single $m$ which works for all pairs $i, j$.", "After replacing $x_i$ by $f_i^mx_i$ and $n$ by $n + m$ we see", "that we get $f_j^nx_i = f_i^nx_j$ in $M$ for all $i, j$.", "Let us introduce", "$$", "K_n = \\{x \\in M \\mid f_1^nx = \\ldots = f_t^nx = 0\\}", "$$", "We claim there is an $A$-module map", "$$", "\\varphi :", "I^{t(n - 1) + 1}", "\\longrightarrow", "M/K_n", "$$", "which maps the monomial", "$f_1^{e_1} \\ldots f_t^{e_t}$ with $\\sum e_i = t(n - 1) + 1$", "to the class modulo $K_n$ of the expression", "$f_1^{e_1} \\ldots f_i^{e_i - n} \\ldots f_t^{e_t}x_i$", "where $i$ is chosen such that $e_i \\geq n$ (note that there", "is at least one such $i$).", "To see that this is indeed the case suppose that", "$$", "\\sum\\nolimits_{E = (e_1, \\ldots, e_t), |E| = t(n - 1) + 1}", "a_E f_1^{e_1} \\ldots f_t^{e_t} = 0", "$$", "is a relation between the monomials with coefficients $a_E$ in $A$.", "Then we would map this to", "$$", "z =", "\\sum\\nolimits_{E = (e_1, \\ldots, e_t), |E| = t(n - 1) + 1}", "a_E f_1^{e_1} \\ldots f_{i(E)}^{e_{i(E)} - n} \\ldots f_t^{e_t}x_{i(E)}", "$$", "where for each multiindex $E$ we have chosen a particular $i(E)$", "such that $e_{i(E)} \\geq n$.", "Note that if we multiply this by $f_j^n$ for any $j$, then", "we get zero, since by the relations $f_j^nx_i = f_i^nx_j$ above we get", "\\begin{align*}", "f_j^nz & = \\sum\\nolimits_{E = (e_1, \\ldots, e_t), |E| = t(n - 1) + 1}", "a_E f_1^{e_1} \\ldots f_j^{e_j + n}", "\\ldots f_{i(E)}^{e_{i(E)} - n} \\ldots f_t^{e_t}x_{i(E)} \\\\", "& =", "\\sum\\nolimits_{E = (e_1, \\ldots, e_t), |E| = t(n - 1) + 1}", "a_E f_1^{e_1} \\ldots f_t^{e_t}x_j", "= 0.", "\\end{align*}", "Hence $z \\in K_n$ and we see that every relation gets mapped to zero", "in $M/K_n$. This proves the claim.", "\\medskip\\noindent", "Note that $K_n \\subset M_{t(n - 1) + 1}$. Hence the", "map $\\varphi$ in particular gives rise to a $A$-module map", "$I^{t(n - 1) + 1} \\to M/M_{t(n - 1) + 1}$.", "This proves the second arrow of the lemma is surjective.", "We omit the proof of injectivity." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3039, "type": "theorem", "label": "properties-lemma-sections-over-quasi-compact-open", "categories": [ "properties" ], "title": "properties-lemma-sections-over-quasi-compact-open", "contents": [ "Let $X$ be a quasi-compact scheme.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a", "quasi-coherent sheaf of ideals of finite type.", "Let $Z \\subset X$ be the closed subscheme", "defined by $\\mathcal{I}$ and set $U = X \\setminus Z$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "The canonical map", "$$", "\\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n,", "\\mathcal{F})", "\\longrightarrow", "\\Gamma(U, \\mathcal{F})", "$$", "is injective. Assume further that $X$ is quasi-separated.", "Let $\\mathcal{F}_n \\subset \\mathcal{F}$", "be subsheaf of sections annihilated by $\\mathcal{I}^n$.", "The canonical map", "$$", "\\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n,", "\\mathcal{F}/\\mathcal{F}_n)", "\\longrightarrow", "\\Gamma(U, \\mathcal{F})", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Let $\\Spec(A) = W \\subset X$ be an affine open. Write", "$\\mathcal{F}|_W = \\widetilde{M}$ for some $A$-module $M$", "and $\\mathcal{I}|_W = \\widetilde{I}$ for some finite type", "ideal $I \\subset A$. Restricting the first displayed map", "of the lemma to $W$ we obtain the first displayed map of", "Lemma \\ref{lemma-sections-over-quasi-compact-open-in-affine}.", "Since we can cover $X$ by a finite number of affine opens this", "proves the first displayed map of the lemma is injective.", "\\medskip\\noindent", "We have $\\mathcal{F}_n|_W = \\widetilde{M_n}$ where", "$M_n \\subset M$ is defined as in", "Lemma \\ref{lemma-sections-over-quasi-compact-open-in-affine}", "(details omitted). The lemma guarantees that we have a bijection", "$$", "\\colim_n \\Hom_{\\mathcal{O}_W}(", "\\mathcal{I}^n|_W, (\\mathcal{F}/\\mathcal{F}_n)|_W)", "\\longrightarrow", "\\Gamma(U \\cap W, \\mathcal{F})", "$$", "for any such affine open $W$.", "\\medskip\\noindent", "To see the second displayed arrow of the lemma is bijective,", "we choose a finite affine open covering $X = \\bigcup_{j = 1, \\ldots, m} W_j$.", "The injectivity follows immediately from the above and the finiteness of", "the covering. If $X$ is quasi-separated, then for each pair", "$j, j'$ we choose a finite affine open covering", "$$", "W_j \\cap W_{j'} = \\bigcup\\nolimits_{k = 1, \\ldots, m_{jj'}} W_{jj'k}.", "$$", "Let $s \\in \\Gamma(U, \\mathcal{F})$. As seen above for each $j$ there exists", "an $n_j$ and a map", "$\\varphi_j : \\mathcal{I}^{n_j}|_{W_j} \\to", "(\\mathcal{F}/\\mathcal{F}_{n_j})|_{W_j}$", "which corresponds to $s|_{W_j}$.", "By the same token for each triple $(j, j', k)$ there exists an integer", "$n_{jj'k}$ such that the restriction of $\\varphi_j$ and $\\varphi_{j'}$", "as maps $\\mathcal{I}^{n_{jj'k}} \\to \\mathcal{F}/\\mathcal{F}_{n_{jj'k}}$", "agree over $W_{jj'l}$. Let $n = \\max\\{n_j, n_{jj'k}\\}$ and we see that", "the $\\varphi_j$ glue as maps", "$\\mathcal{I}^n \\to \\mathcal{F}/\\mathcal{F}_n$ over $X$.", "This proves surjectivity of the map." ], "refs": [ "properties-lemma-sections-over-quasi-compact-open-in-affine", "properties-lemma-sections-over-quasi-compact-open-in-affine" ], "ref_ids": [ 3038, 3038 ] } ], "ref_ids": [] }, { "id": 3040, "type": "theorem", "label": "properties-lemma-ample-power-ample", "categories": [ "properties" ], "title": "properties-lemma-ample-power-ample", "contents": [ "\\begin{reference}", "\\cite[II Proposition 4.5.6(i)]{EGA}", "\\end{reference}", "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $n \\geq 1$. Then $\\mathcal{L}$ is ample if and only if", "$\\mathcal{L}^{\\otimes n}$ is ample." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that $X_{s^n} = X_s$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3041, "type": "theorem", "label": "properties-lemma-ample-on-closed", "categories": [ "properties" ], "title": "properties-lemma-ample-on-closed", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.", "For any closed subscheme $Z \\subset X$ the restriction of", "$\\mathcal{L}$ to $Z$ is ample." ], "refs": [], "proofs": [ { "contents": [ "This is clear since a closed subset of a quasi-compact space is quasi-compact", "and a closed subscheme of an affine scheme is affine (see", "Schemes, Lemma \\ref{schemes-lemma-closed-immersion-affine-case})." ], "refs": [ "schemes-lemma-closed-immersion-affine-case" ], "ref_ids": [ 7668 ] } ], "ref_ids": [] }, { "id": 3042, "type": "theorem", "label": "properties-lemma-affine-cap-s-open", "categories": [ "properties" ], "title": "properties-lemma-affine-cap-s-open", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{L})$. For any affine $U \\subset X$", "the intersection $U \\cap X_s$ is affine." ], "refs": [], "proofs": [ { "contents": [ "This translates into the following algebra problem.", "Let $R$ be a ring. Let $N$ be an invertible $R$-module", "(i.e., locally free of rank 1). Let $s \\in N$ be an element.", "Then $U = \\{\\mathfrak p \\mid s \\not \\in \\mathfrak p N\\}$ is", "an affine open subset of $\\Spec(R)$. This you can see", "as follows. Think of $s$ as an $R$-module map $R \\to N$.", "This gives rise to $R$-module maps $N^{\\otimes k} \\to N^{\\otimes k + 1}$.", "Consider", "$$", "R' = \\colim_n N^{\\otimes n}", "$$", "with transition maps as above. Define an $R$-algebra structure on $R'$ by", "the rule $x \\cdot y = x \\otimes y \\in N^{\\otimes n + m}$", "if $x \\in N^{\\otimes n}$ and $y \\in N^{\\otimes m}$. We claim that", "$\\Spec(R') \\to \\Spec(R)$ is an open immersion with", "image $U$.", "\\medskip\\noindent", "To prove this is a local question on $\\Spec(R)$.", "Let $\\mathfrak p \\in \\Spec(R)$. Pick $f \\in R$,", "$f \\not \\in \\mathfrak p$ such that $N_f \\cong R_f$ as a module.", "Replacing $R$ by $R_f$, $N$ by $N_f$ and $R'$ by", "$R'_f = \\colim N_f^{\\otimes n}$ we may assume that $N \\cong R$.", "Say $N = R$. In this case $s$ is an element of $R$ and it is", "easy to see that $R' \\cong R_s$. Thus the lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3043, "type": "theorem", "label": "properties-lemma-ample-tensor-globally-generated", "categories": [ "properties" ], "title": "properties-lemma-ample-tensor-globally-generated", "contents": [ "\\begin{reference}", "\\cite[II Proposition 4.5.6(ii)]{EGA}", "\\end{reference}", "Let $X$ be a scheme. Let $\\mathcal{L}$ and $\\mathcal{M}$", "be invertible $\\mathcal{O}_X$-modules. If", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is ample, and", "\\item the open sets $X_t$ where $t \\in \\Gamma(X, \\mathcal{M}^{\\otimes m})$", "for $m > 0$ cover $X$,", "\\end{enumerate}", "then $\\mathcal{L} \\otimes \\mathcal{M}$ is ample." ], "refs": [], "proofs": [ { "contents": [ "We check the conditions of Definition \\ref{definition-ample}.", "As $\\mathcal{L}$ is ample we see that $X$ is quasi-compact.", "Let $x \\in X$. Choose $n \\geq 1$, $m \\geq 1$,", "$s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$, and", "$t \\in \\Gamma(X, \\mathcal{M}^{\\otimes m})$", "such that $x \\in X_s$, $x \\in X_t$ and $X_s$ is affine.", "Then $s^mt^n \\in \\Gamma(X, (\\mathcal{L} \\otimes \\mathcal{M})^{\\otimes nm})$,", "$x \\in X_{s^mt^n}$, and $X_{s^mt^n}$ is affine by", "Lemma \\ref{lemma-affine-cap-s-open}." ], "refs": [ "properties-definition-ample", "properties-lemma-affine-cap-s-open" ], "ref_ids": [ 3088, 3042 ] } ], "ref_ids": [] }, { "id": 3044, "type": "theorem", "label": "properties-lemma-affine-s-opens", "categories": [ "properties" ], "title": "properties-lemma-affine-s-opens", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Assume the open sets $X_s$, where $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$", "and $n \\geq 1$, form a basis for the topology on $X$.", "Then among those opens, the open sets $X_s$ which are affine", "form a basis for the topology on $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$. Choose an affine open neighbourhood", "$\\Spec(R) = U \\subset X$ of $x$.", "By assumption, there exists", "a $n \\geq 1$ and a $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$", "such that $X_s \\subset U$. By Lemma \\ref{lemma-affine-cap-s-open} above", "the intersection $X_s = U \\cap X_s$ is affine.", "Since $U$ can be chosen arbitrarily small we win." ], "refs": [ "properties-lemma-affine-cap-s-open" ], "ref_ids": [ 3042 ] } ], "ref_ids": [] }, { "id": 3045, "type": "theorem", "label": "properties-lemma-affine-s-opens-cover-quasi-separated", "categories": [ "properties" ], "title": "properties-lemma-affine-s-opens-cover-quasi-separated", "contents": [ "Let $X$ be a scheme and $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Assume for every point $x$ of $X$ there exists $n \\geq 1$ and", "$s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ such that", "$x \\in X_s$ and $X_s$ is affine. Then $X$ is separated." ], "refs": [], "proofs": [ { "contents": [ "We show first that $X$ is quasi-separated. By assumption we", "can find a covering of $X$ by affine opens of the form", "$X_s$. By Lemma \\ref{lemma-affine-cap-s-open}, the", "intersection of any two such sets is affine, so Schemes, Lemma", "\\ref{schemes-lemma-characterize-quasi-separated} implies", "that $X$ is quasi-separated.", "\\medskip\\noindent", "To show that $X$ is separated, we can use the valuative", "criterion, Schemes, Lemma", "\\ref{schemes-lemma-valuative-criterion-separatedness}.", "Thus, let $A$ be a valuation ring with fraction field $K$", "and consider two morphisms $f, g : \\Spec(A) \\to X$ such that", "the two compositions $\\Spec(K) \\to \\Spec(A) \\to X$ agree.", "As $A$ is local, there exists $p, q \\ge 1$, $s \\in \\Gamma(X,", "\\mathcal{L}^{\\otimes p})$, and $t \\in \\Gamma(X,", "\\mathcal{L}^{\\otimes q})$ such that $X_s$ and $X_t$ are", "affine, $f(\\Spec A) \\subseteq X_s$, and $g(\\Spec A)", "\\subseteq X_t$. We now replace $s$ by $s^q$, $t$ by $t^p$,", "and $\\mathcal{L}$ by $\\mathcal{L}^{\\otimes pq}$. This is", "harmless as $X_s = X_{s^q}$ and $X_t = X_{t^p}$, and now $s$", "and $t$ are both sections of the same sheaf $\\mathcal{L}$.", "\\medskip\\noindent", "The quasi-coherent module $f^*\\mathcal{L}$ corresponds to an $A$-module $M$ and", "$g^*\\mathcal{L}$ corresponds to an $A$-module $N$ by our", "classification of quasi-coherent modules over affine schemes", "(Schemes, Lemma \\ref{schemes-lemma-quasi-coherent-affine}).", "The $A$-modules $M$ and $N$ are locally free of rank", "$1$ (Lemma \\ref{lemma-locally-free-module}) and as $A$ is", "local they are free (Algebra, Lemma", "\\ref{algebra-lemma-K0-local}). Therefore we may identify", "$M$ and $N$ with $A$-submodules of $M \\otimes_A K$ and $N", "\\otimes_A K$. The equality $f|_{\\Spec(K)} = g|_{\\Spec(K)}$", "determines an isomorphism $\\phi \\colon M \\otimes_A K \\to N", "\\otimes_A K$.", "\\medskip\\noindent", "Let $x \\in M$ and $y \\in N$ be the elements corresponding to", "the pullback of $s$ along $f$ and $g$, respectively. These", "satisfy $\\phi(x \\otimes 1) = y \\otimes 1$. The image of $f$", "is contained in $X_s$, so $x \\not\\in \\mathfrak{m}_A M$, that", "is, $x$ generates $M$. Hence $\\phi$ determines an", "isomorphism of $M$ with the submodule of $N$ generated by", "$y$. Arguing symmetrically using $t$, $\\phi^{-1}$", "determines an isomorphism of $N$ with a submodule of $M$.", "Consequently $\\phi$ restricts to an isomorphism of $M$ and", "$N$. Since $x$ generates $M$, its image $y$ generates", "$N$, implying $y \\not\\in \\mathfrak{m}_A N$. Therefore", "$g(\\Spec(A)) \\subseteq X_s$. Because $X_s$ is affine, it is", "separated by Schemes, Lemma \\ref{schemes-lemma-affine-separated},", "and we conclude $f = g$." ], "refs": [ "properties-lemma-affine-cap-s-open", "schemes-lemma-characterize-quasi-separated", "schemes-lemma-valuative-criterion-separatedness", "schemes-lemma-quasi-coherent-affine", "properties-lemma-locally-free-module", "algebra-lemma-K0-local", "schemes-lemma-affine-separated" ], "ref_ids": [ 3042, 7709, 7720, 7663, 3013, 654, 7717 ] } ], "ref_ids": [] }, { "id": 3046, "type": "theorem", "label": "properties-lemma-ample-separated", "categories": [ "properties" ], "title": "properties-lemma-ample-separated", "contents": [ "Let $X$ be a scheme. If there exists an ample invertible sheaf on $X$", "then $X$ is separated." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Lemma \\ref{lemma-affine-s-opens-cover-quasi-separated} and", "Definition \\ref{definition-ample}." ], "refs": [ "properties-lemma-affine-s-opens-cover-quasi-separated", "properties-definition-ample" ], "ref_ids": [ 3045, 3088 ] } ], "ref_ids": [] }, { "id": 3047, "type": "theorem", "label": "properties-lemma-map-into-proj", "categories": [ "properties" ], "title": "properties-lemma-map-into-proj", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Set $S = \\Gamma_*(X, \\mathcal{L})$ as a graded ring.", "If every point of $X$ is contained in one of the", "open subschemes $X_s$, for some $s \\in S_{+}$ homogeneous, then", "there is a canonical morphism of schemes", "$$", "f : X \\longrightarrow Y = \\text{Proj}(S),", "$$", "to the homogeneous spectrum of $S$ (see", "Constructions, Section \\ref{constructions-section-proj}).", "This morphism has the following properties", "\\begin{enumerate}", "\\item $f^{-1}(D_{+}(s)) = X_s$ for any $s \\in S_{+}$ homogeneous,", "\\item there are $\\mathcal{O}_X$-module maps", "$f^*\\mathcal{O}_Y(n) \\to \\mathcal{L}^{\\otimes n}$", "compatible with multiplication maps, see", "Constructions, Equation (\\ref{constructions-equation-multiply}),", "\\item the composition", "$S_n \\to \\Gamma(Y, \\mathcal{O}_Y(n)) \\to \\Gamma(X, \\mathcal{L}^{\\otimes n})$", "is the identity map, and", "\\item for every $x \\in X$ there is an integer $d \\geq 1$", "and an open neighbourhood $U \\subset X$ of $x$", "such that $f^*\\mathcal{O}_Y(dn)|_U \\to \\mathcal{L}^{\\otimes dn}|_U$", "is an isomorphism for all $n \\in \\mathbf{Z}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $\\psi : S \\to \\Gamma_*(X, \\mathcal{L})$ the identity map.", "We are going to use the triple", "$(U(\\psi), r_{\\mathcal{L}, \\psi}, \\theta)$ of", "Constructions, Lemma \\ref{constructions-lemma-invertible-map-into-proj}.", "By assumption the open subscheme $U(\\psi)$ of equals $X$. Hence", "$r_{\\mathcal{L}, \\psi} : U(\\psi) \\to Y$ is defined on all of $X$.", "We set $f = r_{\\mathcal{L}, \\psi}$.", "The maps in part (2) are the components of $\\theta$.", "Part (3) follows from condition (2) in the lemma cited above.", "Part (1) follows from (3) combined with condition (1) in the lemma", "cited above. Part (4) follows from the last statement in", "Constructions, Lemma \\ref{constructions-lemma-invertible-map-into-proj}", "since the map $\\alpha$ mentioned there is an isomorphism." ], "refs": [ "constructions-lemma-invertible-map-into-proj", "constructions-lemma-invertible-map-into-proj" ], "ref_ids": [ 12628, 12628 ] } ], "ref_ids": [] }, { "id": 3048, "type": "theorem", "label": "properties-lemma-map-into-proj-quasi-compact", "categories": [ "properties" ], "title": "properties-lemma-map-into-proj-quasi-compact", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Set $S = \\Gamma_*(X, \\mathcal{L})$.", "Assume (a) every point of $X$ is contained in one of the", "open subschemes $X_s$, for some $s \\in S_{+}$ homogeneous,", "and (b) $X$ is quasi-compact. Then the canonical morphism of schemes", "$f : X \\longrightarrow \\text{Proj}(S)$ of Lemma \\ref{lemma-map-into-proj}", "above is quasi-compact with dense image." ], "refs": [ "properties-lemma-map-into-proj" ], "proofs": [ { "contents": [ "To prove $f$ is quasi-compact it suffices to show that $f^{-1}(D_{+}(s))$", "is quasi-compact for any $s \\in S_{+}$ homogeneous. Write", "$X = \\bigcup_{i = 1, \\ldots, n} X_i$ as a finite union of", "affine opens. By Lemma \\ref{lemma-affine-cap-s-open} each intersection", "$X_s \\cap X_i$ is affine. Hence $X_s = \\bigcup_{i = 1, \\ldots, n} X_s \\cap X_i$", "is quasi-compact. Assume that the image of $f$ is not dense to get", "a contradiction. Then, since the opens $D_+(s)$ with $s \\in S_+$ homogeneous", "form a basis for the topology on $\\text{Proj}(S)$, we can find such", "an $s$ with $D_+(s) \\not = \\emptyset$ and $f(X) \\cap D_+(s) = \\emptyset$.", "By Lemma \\ref{lemma-map-into-proj}", "this means $X_s = \\emptyset$. By Lemma \\ref{lemma-invert-s-sections}", "this means that a power $s^n$ is the zero section of", "$\\mathcal{L}^{\\otimes n\\deg(s)}$.", "This in turn means that $D_+(s) = \\emptyset$ which is the", "desired contradiction." ], "refs": [ "properties-lemma-affine-cap-s-open", "properties-lemma-map-into-proj", "properties-lemma-invert-s-sections" ], "ref_ids": [ 3042, 3047, 3005 ] } ], "ref_ids": [ 3047 ] }, { "id": 3049, "type": "theorem", "label": "properties-lemma-ample-immersion-into-proj", "categories": [ "properties" ], "title": "properties-lemma-ample-immersion-into-proj", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Set $S = \\Gamma_*(X, \\mathcal{L})$.", "Assume $\\mathcal{L}$ is ample. Then the canonical morphism of schemes", "$f : X \\longrightarrow \\text{Proj}(S)$ of Lemma \\ref{lemma-map-into-proj}", "is an open immersion with dense image." ], "refs": [ "properties-lemma-map-into-proj" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-affine-s-opens-cover-quasi-separated} we see", "that $X$ is quasi-separated. Choose finitely many", "$s_1, \\ldots, s_n \\in S_{+}$ homogeneous", "such that $X_{s_i}$ are affine, and $X = \\bigcup X_{s_i}$.", "Say $s_i$ has degree $d_i$. The inverse image of", "$D_{+}(s_i)$ under $f$ is $X_{s_i}$, see Lemma \\ref{lemma-map-into-proj}.", "By Lemma \\ref{lemma-invert-s-sections} the ring map", "$$", "(S^{(d_i)})_{(s_i)} = \\Gamma(D_{+}(s_i), \\mathcal{O}_{\\text{Proj}(S)})", "\\longrightarrow", "\\Gamma(X_{s_i}, \\mathcal{O}_X)", "$$", "is an isomorphism. Hence $f$ induces an isomorphism", "$X_{s_i} \\to D_{+}(s_i)$. Thus $f$ is an isomorphism of $X$ onto the open", "subscheme $\\bigcup_{i = 1, \\ldots, n} D_{+}(s_i)$ of $\\text{Proj}(S)$.", "The image is dense by Lemma \\ref{lemma-map-into-proj-quasi-compact}." ], "refs": [ "properties-lemma-affine-s-opens-cover-quasi-separated", "properties-lemma-map-into-proj", "properties-lemma-invert-s-sections", "properties-lemma-map-into-proj-quasi-compact" ], "ref_ids": [ 3045, 3047, 3005, 3048 ] } ], "ref_ids": [ 3047 ] }, { "id": 3050, "type": "theorem", "label": "properties-lemma-open-in-proj-ample", "categories": [ "properties" ], "title": "properties-lemma-open-in-proj-ample", "contents": [ "Let $X$ be a scheme.", "Let $S$ be a graded ring. Assume $X$ is quasi-compact,", "and assume there exists an open immersion", "$$", "j : X \\longrightarrow Y = \\text{Proj}(S).", "$$", "Then $j^*\\mathcal{O}_Y(d)$ is an invertible ample sheaf", "for some $d > 0$." ], "refs": [], "proofs": [ { "contents": [ "This is Constructions, Lemma \\ref{constructions-lemma-ample-on-proj}." ], "refs": [ "constructions-lemma-ample-on-proj" ], "ref_ids": [ 12606 ] } ], "ref_ids": [] }, { "id": 3051, "type": "theorem", "label": "properties-lemma-ample-on-locally-closed", "categories": [ "properties" ], "title": "properties-lemma-ample-on-locally-closed", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{L}$ be an ample invertible", "$\\mathcal{O}_X$-module. Let $i : X' \\to X$ be a morphism of schemes.", "Assume at least one of the following conditions holds", "\\begin{enumerate}", "\\item $i$ is a quasi-compact immersion,", "\\item $X'$ is quasi-compact and $i$ is an immersion,", "\\item $i$ is quasi-compact and induces a homeomorphism", "between $X'$ and $i(X')$,", "\\item $X'$ is quasi-compact and $i$ induces a homeomorphism", "between $X'$ and $i(X')$.", "\\end{enumerate}", "Then $i^*\\mathcal{L}$ is ample on $X'$." ], "refs": [], "proofs": [ { "contents": [ "Observe that in cases (1) and (3) the scheme $X'$ is quasi-compact", "as $X$ is quasi-compact by Definition \\ref{definition-ample}.", "Thus it suffices to prove (2) and (4). Since (2) is a special case", "of (4) it suffices to prove (4).", "\\medskip\\noindent", "Assume condition (4) holds. For $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes d})$", "denote $s' = i^*s$ the pullback of $s$ to $X'$. Note that", "$s'$ is a section of $(i^*\\mathcal{L})^{\\otimes d}$. By", "Proposition \\ref{proposition-characterize-ample}", "the opens $X_s$, for $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes d})$,", "form a basis for the topology on $X$. Since $X'_{s'} = i^{-1}(X_s)$", "and since $X' \\to i(X')$ is a homeomorphism, we conclude", "the opens $X'_{s'}$ form a basis for the topology of $X'$. Hence", "$i^*\\mathcal{L}$ is ample by", "Proposition \\ref{proposition-characterize-ample}." ], "refs": [ "properties-definition-ample", "properties-proposition-characterize-ample", "properties-proposition-characterize-ample" ], "ref_ids": [ 3088, 3067, 3067 ] } ], "ref_ids": [] }, { "id": 3052, "type": "theorem", "label": "properties-lemma-ample-on-product", "categories": [ "properties" ], "title": "properties-lemma-ample-on-product", "contents": [ "Let $S$ be a quasi-separated scheme. Let $X$, $Y$ be schemes over $S$.", "Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module", "and let $\\mathcal{N}$ be an ample invertible $\\mathcal{O}_Y$-module.", "Then $\\mathcal{M} = \\text{pr}_1^*\\mathcal{L}", "\\otimes_{\\mathcal{O}_{X \\times_S Y}} \\text{pr}_2^*\\mathcal{N}$", "is an ample invertible sheaf on $X \\times_S Y$." ], "refs": [], "proofs": [ { "contents": [ "The morphism $i : X \\times_S Y \\to X \\times Y$ is a quasi-compact", "immersion, see Schemes, Lemma \\ref{schemes-lemma-fibre-product-after-map}.", "On the other hand, $\\mathcal{M}$ is the pullback by", "$i$ of the corresponding invertible module on $X \\times Y$.", "By Lemma \\ref{lemma-ample-on-locally-closed} it suffices to prove the", "lemma for $X \\times Y$. We check (1) and (2) of", "Definition \\ref{definition-ample} for $\\mathcal{M}$ on $X \\times Y$.", "\\medskip\\noindent", "Since $X$ and $Y$ are quasi-compact, so is $X \\times Y$.", "Let $z \\in X \\times Y$ be a point. Let $x \\in X$ and $y \\in Y$", "be the projections. Choose $n > 0$ and", "$s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$", "such that $X_s$ is an affine open neighbourhood of $x$.", "Choose $m > 0$ and", "$t \\in \\Gamma(Y, \\mathcal{N}^{\\otimes m})$", "such that $Y_t$ is an affine open neighbourhood of $y$.", "Then $r = \\text{pr}_1^*s \\otimes \\text{pr}_2^*t$ is a section", "of $\\mathcal{M}$ with $(X \\times Y)_r = X_s \\times Y_t$.", "This is an affine open neighbourhood of $z$ and the proof is complete." ], "refs": [ "schemes-lemma-fibre-product-after-map", "properties-lemma-ample-on-locally-closed", "properties-definition-ample" ], "ref_ids": [ 7711, 3051, 3088 ] } ], "ref_ids": [] }, { "id": 3053, "type": "theorem", "label": "properties-lemma-quasi-affine-O-ample", "categories": [ "properties" ], "title": "properties-lemma-quasi-affine-O-ample", "contents": [ "Let $X$ be a scheme.", "Then $X$ is quasi-affine if and only if $\\mathcal{O}_X$ is ample." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $X$ is quasi-affine. Set $A = \\Gamma(X, \\mathcal{O}_X)$.", "Consider the open immersion", "$$", "j : X \\longrightarrow \\Spec(A)", "$$", "from Lemma \\ref{lemma-quasi-affine}. Note that", "$\\Spec(A) = \\text{Proj}(A[T])$, see", "Constructions, Example \\ref{constructions-example-trivial-proj}.", "Hence we can apply Lemma \\ref{lemma-open-in-proj-ample}", "to deduce that $\\mathcal{O}_X$ is ample.", "\\medskip\\noindent", "Suppose that $\\mathcal{O}_X$ is ample.", "Note that $\\Gamma_*(X, \\mathcal{O}_X) \\cong A[T]$", "as graded rings. Hence the result follows from Lemmas", "\\ref{lemma-ample-immersion-into-proj} and \\ref{lemma-quasi-affine}", "taking into account that", "$\\Spec(A) = \\text{Proj}(A[T])$ for any ring $A$", "as seen above." ], "refs": [ "properties-lemma-quasi-affine", "properties-lemma-open-in-proj-ample", "properties-lemma-ample-immersion-into-proj", "properties-lemma-quasi-affine" ], "ref_ids": [ 3009, 3050, 3049, 3009 ] } ], "ref_ids": [] }, { "id": 3054, "type": "theorem", "label": "properties-lemma-quasi-affine-locally-closed", "categories": [ "properties" ], "title": "properties-lemma-quasi-affine-locally-closed", "contents": [ "Let $X$ be a quasi-affine scheme. For any quasi-compact immersion", "$i : X' \\to X$ the scheme $X'$ is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "This can be proved directly without making use of the material on", "ample invertible sheaves; we urge the reader to do this on a napkin.", "Since $X$ is quasi-affine, we have that $\\mathcal{O}_X$ is ample by", "Lemma \\ref{lemma-quasi-affine-O-ample}.", "Then $\\mathcal{O}_{X'}$ is ample by", "Lemma \\ref{lemma-ample-on-locally-closed}. Then $X'$ is quasi-affine by", "Lemma \\ref{lemma-quasi-affine-O-ample}." ], "refs": [ "properties-lemma-quasi-affine-O-ample", "properties-lemma-ample-on-locally-closed", "properties-lemma-quasi-affine-O-ample" ], "ref_ids": [ 3053, 3051, 3053 ] } ], "ref_ids": [] }, { "id": 3055, "type": "theorem", "label": "properties-lemma-characterize-affine", "categories": [ "properties" ], "title": "properties-lemma-characterize-affine", "contents": [ "Let $X$ be a scheme. Suppose that there exist finitely many elements", "$f_1, \\ldots, f_n \\in \\Gamma(X, \\mathcal{O}_X)$ such that", "\\begin{enumerate}", "\\item each $X_{f_i}$ is an affine open of $X$, and", "\\item the ideal generated by $f_1, \\ldots, f_n$ in", "$\\Gamma(X, \\mathcal{O}_X)$ is equal to the unit ideal.", "\\end{enumerate}", "Then $X$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Assume we have $f_1, \\ldots, f_n$ as in the lemma.", "We may write $1 = \\sum g_i f_i$ for some $g_j \\in \\Gamma(X, \\mathcal{O}_X)$", "and hence it is clear that $X = \\bigcup X_{f_i}$. (The $f_i$'s cannot", "all vanish at a point.) Since each $X_{f_i}$", "is quasi-compact (being affine) it follows that $X$ is quasi-compact.", "Hence we see that $X$ is quasi-affine by", "Lemma \\ref{lemma-quasi-affine-O-ample} above.", "Consider the open immersion", "$$", "j : X \\to \\Spec(\\Gamma(X, \\mathcal{O}_X)),", "$$", "see Lemma \\ref{lemma-quasi-affine}. The inverse image of the standard open", "$D(f_i)$ on the right hand side is equal to $X_{f_i}$ on the", "left hand side and the morphism $j$ induces an isomorphism", "$X_{f_i} \\cong D(f_i)$, see", "Lemma \\ref{lemma-invert-f-affine}. Since the $f_i$ generate the unit ideal", "we see that $\\Spec(\\Gamma(X, \\mathcal{O}_X))", "= \\bigcup_{i = 1, \\ldots, n} D(f_i)$. Thus $j$ is an isomorphism." ], "refs": [ "properties-lemma-quasi-affine-O-ample", "properties-lemma-quasi-affine", "properties-lemma-invert-f-affine" ], "ref_ids": [ 3053, 3009, 3008 ] } ], "ref_ids": [] }, { "id": 3056, "type": "theorem", "label": "properties-lemma-ample-gcd-is-one", "categories": [ "properties" ], "title": "properties-lemma-ample-gcd-is-one", "contents": [ "In Situation \\ref{situation-ample}.", "The canonical morphism $f : X \\to Y$", "maps $X$ into the open subscheme $W = W_1 \\subset Y$", "where $\\mathcal{O}_Y(1)$ is invertible and where", "all multiplication maps", "$\\mathcal{O}_Y(n) \\otimes_{\\mathcal{O}_Y} \\mathcal{O}_Y(m) \\to", "\\mathcal{O}_Y(n + m)$", "are isomorphisms (see", "Constructions, Lemma \\ref{constructions-lemma-where-invertible}).", "Moreover, the maps $f^*\\mathcal{O}_Y(n) \\to \\mathcal{L}^{\\otimes n}$", "are all isomorphisms." ], "refs": [ "constructions-lemma-where-invertible" ], "proofs": [ { "contents": [ "By Proposition \\ref{proposition-characterize-ample} there exists an integer", "$n_0$ such that $\\mathcal{L}^{\\otimes n}$ is globally generated for all", "$n \\geq n_0$. Let $x \\in X$ be a point. By the above we can find", "$a \\in S_{n_0}$ and $b \\in S_{n_0 + 1}$ such that", "$a$ and $b$ do not vanish at $x$. Hence", "$f(x) \\in D_{+}(a) \\cap D_{+}(b) = D_{+}(ab)$. By", "Constructions, Lemma \\ref{constructions-lemma-where-invertible}", "we see that $f(x) \\in W_1$ as desired. By", "Constructions, Lemma \\ref{constructions-lemma-invertible-map-into-proj}", "which was used in the construction of the map $f$", "the maps", "$f^*\\mathcal{O}_Y(n_0) \\to \\mathcal{L}^{\\otimes n_0}$ and", "$f^*\\mathcal{O}_Y(n_0 + 1) \\to \\mathcal{L}^{\\otimes n_0 + 1}$", "are isomorphisms in a neighbourhood of $x$. By compatibility with", "the algebra structure and the fact that $f$ maps into $W$", "we conclude all the maps", "$f^*\\mathcal{O}_Y(n) \\to \\mathcal{L}^{\\otimes n}$ are isomorphisms", "in a neighbourhood of $x$. Hence we win." ], "refs": [ "properties-proposition-characterize-ample", "constructions-lemma-where-invertible", "constructions-lemma-invertible-map-into-proj" ], "ref_ids": [ 3067, 12604, 12628 ] } ], "ref_ids": [ 12604 ] }, { "id": 3057, "type": "theorem", "label": "properties-lemma-ample-quasi-coherent", "categories": [ "properties" ], "title": "properties-lemma-ample-quasi-coherent", "contents": [ "In Situation \\ref{situation-ample}.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Set $M = \\Gamma_*(X, \\mathcal{L}, \\mathcal{F})$ as a graded $S$-module.", "There are isomorphisms", "$$", "f^*\\widetilde{M} \\longrightarrow \\mathcal{F}", "$$", "functorial in $\\mathcal{F}$ such that", "$M_0 \\to \\Gamma(\\text{Proj}(S), \\widetilde{M}) \\to \\Gamma(X, \\mathcal{F})$", "is the identity map." ], "refs": [], "proofs": [ { "contents": [ "Let $s \\in S_{+}$ be homogeneous such that $X_s$ is affine open in $X$.", "Recall that $\\widetilde{M}|_{D_{+}(s)}$ corresponds to the", "$S_{(s)}$-module $M_{(s)}$, see", "Constructions, Lemma \\ref{constructions-lemma-proj-sheaves}.", "Recall that $f^{-1}(D_{+}(s)) = X_s$.", "As $X$ carries an ample invertible sheaf it is quasi-compact and", "quasi-separated, see Section \\ref{section-ample}.", "By Lemma \\ref{lemma-invert-s-sections} there is a canonical isomorphism", "$M_{(s)} = \\Gamma_*(X, \\mathcal{L}, \\mathcal{F})_{(s)} \\to", "\\Gamma(X_s, \\mathcal{F})$.", "Since $\\mathcal{F}$ is quasi-coherent this leads to", "a canonical isomorphism", "$$", "f^*\\widetilde{M}|_{X_s} \\to \\mathcal{F}|_{X_s}", "$$", "Since $\\mathcal{L}$ is ample on $X$ we know that $X$ is covered", "by the affine opens of the form $X_s$. Hence it suffices to prove", "that the displayed maps glue on overlaps. Proof of this is", "omitted." ], "refs": [ "constructions-lemma-proj-sheaves", "properties-lemma-invert-s-sections" ], "ref_ids": [ 12594, 3005 ] } ], "ref_ids": [] }, { "id": 3058, "type": "theorem", "label": "properties-lemma-proj-quasi-coherent", "categories": [ "properties" ], "title": "properties-lemma-proj-quasi-coherent", "contents": [ "Let $S$ be a graded ring such that $X = \\text{Proj}(S)$ is quasi-compact.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Set", "$M = \\bigoplus_{n \\in \\mathbf{Z}} \\Gamma(X, \\mathcal{F}(n))$ as", "a graded $S$-module, see", "Constructions, Section \\ref{constructions-section-invertible-on-proj}.", "The map", "$$", "\\widetilde{M} \\longrightarrow \\mathcal{F}", "$$", "of Constructions, Lemma", "\\ref{constructions-lemma-comparison-proj-quasi-coherent}", "is an isomorphism.", "If $X$ is covered by standard opens $D_+(f)$ where $f$ has degree $1$,", "then the induced maps", "$M_n \\to \\Gamma(X, \\mathcal{F}(n))$ are the identity maps." ], "refs": [ "constructions-lemma-comparison-proj-quasi-coherent" ], "proofs": [ { "contents": [ "Since $X$ is quasi-compact we can find homogeneous elements", "$f_1, \\ldots, f_n \\in S$ of positive degrees such that", "$X = D_+(f_1) \\cup \\ldots \\cup D_+(f_n)$. Let $d$ be the", "least common multiple of the degrees of $f_1, \\ldots, f_n$.", "After replacing $f_i$ by a power we may assume that each", "$f_i$ has degree $d$. Then we see that $\\mathcal{L} = \\mathcal{O}_X(d)$ is", "invertible, the multiplication maps", "$\\mathcal{O}_X(ad) \\otimes \\mathcal{O}_X(bd) \\to \\mathcal{O}_X((a + b)d)$", "are isomorphisms, and each $f_i$ determines a global section $s_i$", "of $\\mathcal{L}$ such that $X_{s_i} = D_+(f_i)$, see", "Constructions, Lemmas \\ref{constructions-lemma-where-invertible} and", "\\ref{constructions-lemma-principal-open}.", "Thus $\\Gamma(X, \\mathcal{F}(ad)) =", "\\Gamma(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes a})$.", "Recall that $\\widetilde{M}|_{D_{+}(f_i)}$ corresponds to the", "$S_{(f_i)}$-module $M_{(f_i)}$, see", "Constructions, Lemma \\ref{constructions-lemma-proj-sheaves}.", "Since the degree of $f_i$ is $d$, the isomorphism class of", "$M_{(f_i)}$ depends only on the homogeneous summands of $M$ of", "degree divisible by $d$. More precisely, the isomorphism class of", "$M_{(f_i)}$ depends only on the graded $\\Gamma_*(X, \\mathcal{L})$-module", "$\\Gamma_*(X, \\mathcal{L}, \\mathcal{F})$", "and the image $s_i$ of $f_i$ in $\\Gamma_*(X, \\mathcal{L})$.", "The scheme $X$ is quasi-compact by assumption and", "separated by Constructions, Lemma \\ref{constructions-lemma-proj-separated}.", "By Lemma \\ref{lemma-invert-s-sections} there is a canonical isomorphism", "$$", "M_{(f_i)} = \\Gamma_*(X, \\mathcal{L}, \\mathcal{F})_{(s_i)} \\to", "\\Gamma(X_{s_i}, \\mathcal{F}).", "$$", "The construction of the map in Constructions, Lemma", "\\ref{constructions-lemma-comparison-proj-quasi-coherent}", "then shows that it is an isomorphism over $D_+(f_i)$", "hence an isomorphism as $X$ is covered by these opens.", "We omit the proof of the final statement." ], "refs": [ "constructions-lemma-where-invertible", "constructions-lemma-principal-open", "constructions-lemma-proj-sheaves", "constructions-lemma-proj-separated", "properties-lemma-invert-s-sections", "constructions-lemma-comparison-proj-quasi-coherent" ], "ref_ids": [ 12604, 12605, 12594, 12597, 3005, 12607 ] } ], "ref_ids": [ 12607 ] }, { "id": 3059, "type": "theorem", "label": "properties-lemma-maximal-points-affine", "categories": [ "properties" ], "title": "properties-lemma-maximal-points-affine", "contents": [ "Let $X$ be a quasi-separated scheme.", "Let $Z_1, \\ldots, Z_n$ be pairwise distinct irreducible components of $X$,", "see Topology, Section \\ref{topology-section-irreducible-components}.", "Let $\\eta_i \\in Z_i$ be their generic points, see", "Schemes, Lemma \\ref{schemes-lemma-scheme-sober}.", "There exist affine open neighbourhoods $\\eta_i \\in U_i$", "such that $U_i \\cap U_j = \\emptyset$ for all $i \\not = j$.", "In particular, $U = U_1 \\cup \\ldots \\cup U_n$ is an affine", "open containing all of the points $\\eta_1, \\ldots, \\eta_n$." ], "refs": [ "schemes-lemma-scheme-sober" ], "proofs": [ { "contents": [ "Let $V_i$ be any affine open containing $\\eta_i$", "and disjoint from the closed set $Z_1 \\cup \\ldots \\hat Z_i \\ldots \\cup Z_n$.", "Since $X$ is quasi-separated for each $i$ the union", "$W_i = \\bigcup_{j, j \\not = i} V_i \\cap V_j$ is a quasi-compact", "open of $V_i$ not containing $\\eta_i$.", "We can find open neighbourhoods $U_i \\subset V_i$", "containing $\\eta_i$ and disjoint from $W_i$ by", "Algebra, Lemma \\ref{algebra-lemma-standard-open-containing-maximal-point}.", "Finally, $U$ is affine since it is the spectrum of", "the ring $R_1 \\times \\ldots \\times R_n$ where $R_i = \\mathcal{O}_X(U_i)$,", "see Schemes, Lemma \\ref{schemes-lemma-disjoint-union-affines}." ], "refs": [ "algebra-lemma-standard-open-containing-maximal-point", "schemes-lemma-disjoint-union-affines" ], "ref_ids": [ 425, 7659 ] } ], "ref_ids": [ 7672 ] }, { "id": 3060, "type": "theorem", "label": "properties-lemma-quasi-compact-dense-open-separated", "categories": [ "properties" ], "title": "properties-lemma-quasi-compact-dense-open-separated", "contents": [ "Let $X$ be a quasi-compact scheme.", "There exists a dense open $V \\subset X$ which is separated." ], "refs": [], "proofs": [ { "contents": [ "Say $X = \\bigcup_{i = 1, \\ldots, n} U_i$ is a union of $n$ affine open", "subschemes. We will prove the lemma by induction on $n$. It is trivial for", "$n = 1$. Let $V' \\subset \\bigcup_{i = 1, \\ldots, n - 1} U_i$ be a separated", "dense open subscheme, which exists by induction hypothesis. Consider", "$$", "V = V' \\amalg (U_n \\setminus \\overline{V'}).", "$$", "It is clear that $V$ is separated and a dense open subscheme of $X$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3061, "type": "theorem", "label": "properties-lemma-point-and-maximal-points-affine", "categories": [ "properties" ], "title": "properties-lemma-point-and-maximal-points-affine", "contents": [ "Let $X$ be a quasi-separated scheme. Let $Z_1, \\ldots, Z_n$ be pairwise", "distinct irreducible components of $X$. Let $\\eta_i \\in Z_i$ be their", "generic points. Let $x \\in X$ be arbitrary.", "There exists an affine open $U \\subset X$ containing", "$x$ and all the $\\eta_i$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $x \\in Z_1 \\cap \\ldots \\cap Z_r$ and", "$x \\not \\in Z_{r + 1}, \\ldots, Z_n$. Then we may choose", "an affine open $W \\subset X$ such that", "$x \\in W$ and $W \\cap Z_i = \\emptyset$ for", "$i = r + 1, \\ldots, n$. Note that clearly $\\eta_i \\in W$", "for $i = 1, \\ldots, r$. By Lemma \\ref{lemma-maximal-points-affine}", "we may choose affine opens $U_i \\subset X$ which are pairwise", "disjoint such that $\\eta_i \\in U_i$ for $i = r + 1, \\ldots, n$.", "Since $X$ is quasi-separated the opens $W \\cap U_i$", "are quasi-compact and do not contain $\\eta_i$ for", "$i = r + 1, \\ldots, n$. Hence by", "Algebra, Lemma \\ref{algebra-lemma-standard-open-containing-maximal-point}", "we may shrink $U_i$ such that $W \\cap U_i = \\emptyset$", "for $i = r + 1, \\ldots, n$. Then the union", "$U = W \\cup \\bigcup_{i = r + 1, \\ldots, n} U_i$ is disjoint and hence", "(by Schemes, Lemma \\ref{schemes-lemma-disjoint-union-affines})", "a suitable affine open." ], "refs": [ "properties-lemma-maximal-points-affine", "algebra-lemma-standard-open-containing-maximal-point", "schemes-lemma-disjoint-union-affines" ], "ref_ids": [ 3059, 425, 7659 ] } ], "ref_ids": [] }, { "id": 3062, "type": "theorem", "label": "properties-lemma-ample-finite-set-in-affine", "categories": [ "properties" ], "title": "properties-lemma-ample-finite-set-in-affine", "contents": [ "Let $X$ be a scheme. Assume either", "\\begin{enumerate}", "\\item The scheme $X$ is quasi-affine.", "\\item The scheme $X$ is isomorphic to a locally closed subscheme", "of an affine scheme.", "\\item There exists an ample invertible sheaf on $X$.", "\\item The scheme $X$ is isomorphic to a locally closed subscheme", "of $\\text{Proj}(S)$ for some graded ring $S$.", "\\end{enumerate}", "Then for any finite subset $E \\subset X$ there exists an", "affine open $U \\subset X$ with $E \\subset U$." ], "refs": [], "proofs": [ { "contents": [ "By Properties, Definition \\ref{definition-quasi-affine}", "a quasi-affine scheme is a quasi-compact open subscheme", "of an affine scheme. Any affine scheme $\\Spec(R)$ is isomorphic to", "$\\text{Proj}(R[X])$ where $R[X]$ is graded by setting $\\deg(X) = 1$.", "By Proposition \\ref{proposition-characterize-ample}", "if $X$ has an ample invertible sheaf then $X$ is isomorphic to an open", "subscheme of $\\text{Proj}(S)$ for some graded ring $S$.", "Hence, it suffices to prove the lemma in case (4).", "(We urge the reader to prove case (2) directly for themselves.)", "\\medskip\\noindent", "Thus assume $X \\subset \\text{Proj}(S)$ is a locally closed subscheme", "where $S$ is some graded ring. Let $T = \\overline{X} \\setminus X$.", "Recall that the standard opens $D_{+}(f)$ form a basis of the", "topology on $\\text{Proj}(S)$. Since $E$ is finite we may choose finitely many", "homogeneous elements $f_i \\in S_{+}$ such that", "$$", "E \\subset", "D_{+}(f_1) \\cup \\ldots \\cup D_{+}(f_n) \\subset", "\\text{Proj}(S) \\setminus T", "$$", "Suppose that $E = \\{\\mathfrak p_1, \\ldots, \\mathfrak p_m\\}$", "as a subset of $\\text{Proj}(S)$.", "Consider the ideal $I = (f_1, \\ldots, f_n) \\subset S$.", "Since $I \\not \\subset \\mathfrak p_j$ for all $j = 1, \\ldots, m$", "we see from Algebra, Lemma \\ref{algebra-lemma-graded-silly} that", "there exists a homogeneous element $f \\in I$, $f \\not \\in \\mathfrak p_j$", "for all $j = 1, \\ldots, m$. Then $E \\subset D_{+}(f) \\subset", "D_{+}(f_1) \\cup \\ldots \\cup D_{+}(f_n)$. Since $D_{+}(f)$ does not", "meet $T$ we see that $X \\cap D_{+}(f)$ is a closed subscheme of the", "affine scheme $D_{+}(f)$, hence is an affine open of $X$ as desired." ], "refs": [ "properties-definition-quasi-affine", "properties-proposition-characterize-ample", "algebra-lemma-graded-silly" ], "ref_ids": [ 3083, 3067, 662 ] } ], "ref_ids": [] }, { "id": 3063, "type": "theorem", "label": "properties-lemma-ample-finite-set-in-principal-affine", "categories": [ "properties" ], "title": "properties-lemma-ample-finite-set-in-principal-affine", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{L}$ be an ample invertible sheaf on $X$.", "Let", "$$", "E \\subset W \\subset X", "$$", "with $E$ finite and $W$ open in $X$. Then there exists an $n > 0$", "and a section $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ such that", "$X_s$ is affine and $E \\subset X_s \\subset W$." ], "refs": [], "proofs": [ { "contents": [ "The reader can modify the proof of Lemma \\ref{lemma-ample-finite-set-in-affine}", "to prove this lemma; we will instead deduce the lemma from it.", "By Lemma \\ref{lemma-ample-finite-set-in-affine} we can choose an affine", "open $U \\subset W$ such that $E \\subset U$.", "Consider the graded ring $S = \\Gamma_*(X, \\mathcal{L}) =", "\\bigoplus_{n \\geq 0} \\Gamma(X, \\mathcal{L}^{\\otimes n})$.", "For each $x \\in E$ let $\\mathfrak p_x \\subset S$ be the graded ideal", "of sections vanishing at $x$. It is clear that $\\mathfrak p_x$ is", "a prime ideal and since some power of $\\mathcal{L}$ is globally", "generated, it is clear that $S_{+} \\not \\subset \\mathfrak p_x$.", "Let $I \\subset S$ be the graded ideal of sections vanishing on all", "points of $X \\setminus U$. Since the sets $X_s$ form a basis", "for the topology we see that $I \\not \\subset \\mathfrak p_x$ for", "all $x \\in E$.", "By (graded) prime avoidance (Algebra, Lemma \\ref{algebra-lemma-graded-silly})", "we can find $s \\in I$ homogeneous", "with $s \\not \\in \\mathfrak p_x$ for all $x \\in E$.", "Then $E \\subset X_s \\subset U$ and $X_s$ is affine by", "Lemma \\ref{lemma-affine-cap-s-open}." ], "refs": [ "properties-lemma-ample-finite-set-in-affine", "properties-lemma-ample-finite-set-in-affine", "algebra-lemma-graded-silly", "properties-lemma-affine-cap-s-open" ], "ref_ids": [ 3062, 3062, 662, 3042 ] } ], "ref_ids": [] }, { "id": 3064, "type": "theorem", "label": "properties-lemma-quasi-affine-invertible-nonvanishing-section", "categories": [ "properties" ], "title": "properties-lemma-quasi-affine-invertible-nonvanishing-section", "contents": [ "Let $X$ be a quasi-affine scheme. Let $\\mathcal{L}$ be an invertible", "$\\mathcal{O}_X$-module. Let $E \\subset W \\subset X$ with $E$ finite", "and $W$ open. Then there exists an $s \\in \\Gamma(X, \\mathcal{L})$", "such that $X_s$ is affine and $E \\subset X_s \\subset W$." ], "refs": [], "proofs": [ { "contents": [ "The proof of this lemma has a lot in common with the proof of", "Algebra, Lemma \\ref{algebra-lemma-silly}.", "Say $E = \\{x_1, \\ldots, x_n\\}$. If $E = W = \\emptyset$, then $s = 0$", "works. If $W \\not = \\emptyset$, then we may assume $E \\not = \\emptyset$", "by adding a point if necessary. Thus we may assume $n \\geq 1$.", "We will prove the lemma by induction on $n$.", "\\medskip\\noindent", "Base case: $n = 1$. After replacing $W$ by an affine open neighbourhood", "of $x_1$ in $W$, we may assume $W$ is affine. Combining", "Lemmas \\ref{lemma-quasi-affine-O-ample} and", "Proposition \\ref{proposition-characterize-ample}", "we see that every quasi-coherent", "$\\mathcal{O}_X$-module is globally generated.", "Hence there exists", "a global section $s$ of $\\mathcal{L}$ which does not vanish at $x_1$.", "On the other hand, let $Z \\subset X$ be the", "reduced induced closed subscheme on $X \\setminus W$.", "Applying global generation to the quasi-coherent ideal sheaf", "$\\mathcal{I}$ of $Z$ we find a global section $f$ of $\\mathcal{I}$", "which does not vanish at $x_1$. Then $s' = fs$ is a global section", "of $\\mathcal{L}$ which does not vanish at $x_1$ such that", "$X_{s'} \\subset W$. Then $X_{s'}$ is affine by", "Lemma \\ref{lemma-affine-cap-s-open}.", "\\medskip\\noindent", "Induction step for $n > 1$. If there is a specialization", "$x_i \\leadsto x_j$ for $i \\not = j$, then it suffices", "to prove the lemma for $\\{x_1, \\ldots, x_n\\} \\setminus \\{x_i\\}$", "and we are done by induction. Thus we may assume there are no", "specializations among the $x_i$.", "By either Lemma \\ref{lemma-ample-finite-set-in-affine} or", "Lemma \\ref{lemma-ample-finite-set-in-principal-affine}", "we may assume $W$ is affine.", "By induction we can find a global section", "$s$ of $\\mathcal{L}$ such that $X_s \\subset W$ is affine and contains", "$x_1, \\ldots, x_{n - 1}$. If $x_n \\in X_s$ then we are done.", "Assume $s$ is zero at $x_n$. By the case $n = 1$ we can find", "a global section $s'$ of $\\mathcal{L}$ with", "$\\{x_n\\} \\subset X_{s'} \\subset", "W \\setminus \\overline{\\{x_1, \\ldots, x_{n - 1}\\}}$.", "Here we use that $x_n$ is not a specialization of $x_1, \\ldots, x_{n - 1}$.", "Then $s + s'$", "is a global section of $\\mathcal{L}$ which is nonvanishing", "at $x_1, \\ldots, x_n$ with $X_{s + s'} \\subset W$ and", "we conclude as before." ], "refs": [ "algebra-lemma-silly", "properties-lemma-quasi-affine-O-ample", "properties-proposition-characterize-ample", "properties-lemma-affine-cap-s-open", "properties-lemma-ample-finite-set-in-affine", "properties-lemma-ample-finite-set-in-principal-affine" ], "ref_ids": [ 378, 3053, 3067, 3042, 3062, 3063 ] } ], "ref_ids": [] }, { "id": 3065, "type": "theorem", "label": "properties-lemma-ring-affine-open-injective-local-ring", "categories": [ "properties" ], "title": "properties-lemma-ring-affine-open-injective-local-ring", "contents": [ "Let $X$ be a scheme and $x \\in X$ a point. There exists an affine open", "neighbourhood $U \\subset X$ of $x$ such that the canonical map", "$\\mathcal{O}_X(U) \\to \\mathcal{O}_{X, x}$ is injective in each of", "the following cases:", "\\begin{enumerate}", "\\item $X$ is integral,", "\\item $X$ is locally Noetherian,", "\\item $X$ is reduced and has a finite number of irreducible components.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "After translation into algebra, this follows from", "Algebra, Lemma \\ref{algebra-lemma-subring-of-local-ring}." ], "refs": [ "algebra-lemma-subring-of-local-ring" ], "ref_ids": [ 456 ] } ], "ref_ids": [] }, { "id": 3066, "type": "theorem", "label": "properties-proposition-coherator", "categories": [ "properties" ], "title": "properties-proposition-coherator", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item The category $\\QCoh(\\mathcal{O}_X)$ is a Grothendieck", "abelian category. Consequently, $\\QCoh(\\mathcal{O}_X)$", "has enough injectives and all limits.", "\\item The inclusion functor", "$\\QCoh(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X)$", "has a right adjoint\\footnote{This functor is sometimes called", "the {\\it coherator}.}", "$$", "Q : \\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\QCoh(\\mathcal{O}_X)", "$$", "such that for every quasi-coherent sheaf $\\mathcal{F}$ the adjunction mapping", "$Q(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) means $\\QCoh(\\mathcal{O}_X)$ (a) has all colimits,", "(b) filtered colimits are exact, and (c) has a generator, see", "Injectives, Section \\ref{injectives-section-grothendieck-conditions}.", "By Schemes, Section \\ref{schemes-section-quasi-coherent}", "colimits in $\\QCoh(\\mathcal{O}_X)$ exist and agree", "with colimits in $\\textit{Mod}(\\mathcal{O}_X)$. By", "Modules, Lemma \\ref{modules-lemma-limits-colimits}", "filtered colimits are exact. Hence (a) and (b) hold.", "To construct a generator $U$, pick a cardinal $\\kappa$ as in", "Lemma \\ref{lemma-colimit-kappa}. Pick a collection", "$(\\mathcal{F}_t)_{t \\in T}$ of $\\kappa$-generated quasi-coherent sheaves as in", "Lemma \\ref{lemma-set-of-iso-classes}. Set", "$U = \\bigoplus_{t \\in T} \\mathcal{F}_t$. Since every object of", "$\\QCoh(\\mathcal{O}_X)$ is a filtered colimit of $\\kappa$-generated", "quasi-coherent modules, i.e., of objects isomorphic to $\\mathcal{F}_t$,", "it is clear that $U$ is a generator.", "The assertions on limits and injectives hold in any", "Grothendieck abelian category, see", "Injectives, Theorem", "\\ref{injectives-theorem-injective-embedding-grothendieck} and", "Lemma \\ref{injectives-lemma-grothendieck-products}.", "\\medskip\\noindent", "Proof of (2). To construct $Q$ we use the following general procedure.", "Given an object $\\mathcal{F}$ of $\\textit{Mod}(\\mathcal{O}_X)$", "we consider the functor", "$$", "\\QCoh(\\mathcal{O}_X)^{opp} \\longrightarrow \\textit{Sets},\\quad", "\\mathcal{G} \\longmapsto \\Hom_X(\\mathcal{G}, \\mathcal{F})", "$$", "This functor transforms colimits into limits,", "hence is representable, see", "Injectives, Lemma \\ref{injectives-lemma-grothendieck-brown}.", "Thus there exists a quasi-coherent sheaf $Q(\\mathcal{F})$", "and a functorial isomorphism", "$\\Hom_X(\\mathcal{G}, \\mathcal{F}) = \\Hom_X(\\mathcal{G}, Q(\\mathcal{F}))$", "for $\\mathcal{G}$ in $\\QCoh(\\mathcal{O}_X)$. By the Yoneda lemma", "(Categories, Lemma \\ref{categories-lemma-yoneda})", "the construction $\\mathcal{F} \\leadsto Q(\\mathcal{F})$ is", "functorial in $\\mathcal{F}$. By construction $Q$ is a right", "adjoint to the inclusion functor.", "The fact that $Q(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism", "when $\\mathcal{F}$ is quasi-coherent is a formal consequence of the fact", "that the inclusion functor", "$\\QCoh(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X)$", "is fully faithful." ], "refs": [ "modules-lemma-limits-colimits", "properties-lemma-colimit-kappa", "properties-lemma-set-of-iso-classes", "injectives-theorem-injective-embedding-grothendieck", "injectives-lemma-grothendieck-products", "injectives-lemma-grothendieck-brown", "categories-lemma-yoneda" ], "ref_ids": [ 13222, 3032, 3031, 7767, 7794, 7793, 12203 ] } ], "ref_ids": [] }, { "id": 3067, "type": "theorem", "label": "properties-proposition-characterize-ample", "categories": [ "properties" ], "title": "properties-proposition-characterize-ample", "contents": [ "Let $X$ be a quasi-compact scheme.", "Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "Set $S = \\Gamma_*(X, \\mathcal{L})$.", "The following are equivalent:", "\\begin{enumerate}", "\\item", "\\label{item-ample}", "$\\mathcal{L}$ is ample,", "\\item", "\\label{item-immersion}", "the open sets $X_s$, with $s \\in S_{+}$ homogeneous,", "cover $X$ and the associated morphism $X \\to \\text{Proj}(S)$", "is an open immersion,", "\\item", "\\label{item-s-basis}", "the open sets $X_s$, with $s \\in S_{+}$ homogeneous,", "form a basis for the topology of $X$,", "\\item", "\\label{item-s-affine-basis}", "the open sets $X_s$, with $s \\in S_{+}$ homogeneous,", "which are affine form a basis for the topology of $X$,", "\\item", "\\label{item-qc-gg}", "for every quasi-coherent sheaf $\\mathcal{F}$ on $X$", "the sum of the images of the canonical maps", "$$", "\\Gamma(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n})", "\\otimes_{\\mathbf{Z}} \\mathcal{L}^{\\otimes -n}", "\\longrightarrow", "\\mathcal{F}", "$$", "with $n \\geq 1$ equals $\\mathcal{F}$,", "\\item", "\\label{item-qc-i-gg}", "same property as (\\ref{item-qc-gg}) with $\\mathcal{F}$", "ranging over all quasi-coherent sheaves of ideals,", "\\item", "\\label{item-c-gg}", "$X$ is quasi-separated and", "for every quasi-coherent sheaf $\\mathcal{F}$ of finite type on $X$", "there exists an integer $n_0$ such that", "$\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}$", "is globally generated for all $n \\geq n_0$,", "\\item", "\\label{item-c-q}", "$X$ is quasi-separated and", "for every quasi-coherent sheaf $\\mathcal{F}$ of finite type on $X$", "there exist integers $n > 0$, $k \\geq 0$ such that", "$\\mathcal{F}$ is a quotient of a direct sum of $k$ copies of", "$\\mathcal{L}^{\\otimes - n}$, and", "\\item", "\\label{item-c-i-q}", "same as in (\\ref{item-c-q}) with $\\mathcal{F}$ ranging over all", "sheaves of ideals of finite type on $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Lemma \\ref{lemma-ample-immersion-into-proj} is", "(\\ref{item-ample}) $\\Rightarrow$ (\\ref{item-immersion}).", "Lemmas \\ref{lemma-ample-power-ample} and \\ref{lemma-open-in-proj-ample}", "provide the implication", "(\\ref{item-ample}) $\\Leftarrow$ (\\ref{item-immersion}).", "The implications (\\ref{item-immersion}) $\\Rightarrow$", "(\\ref{item-s-affine-basis}) $\\Rightarrow$ (\\ref{item-s-basis})", "are clear from Constructions, Section \\ref{constructions-section-proj}.", "Lemma \\ref{lemma-affine-s-opens} is", "(\\ref{item-s-basis}) $\\Rightarrow$ (\\ref{item-ample}).", "Thus we see that the first 4 conditions are all equivalent.", "\\medskip\\noindent", "Assume the equivalent conditions (1) -- (4).", "Note that in particular $X$ is separated (as an open", "subscheme of the separated scheme $\\text{Proj}(S)$).", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Choose $s \\in S_{+}$ homogeneous such that $X_s$ is affine.", "We claim that any section $m \\in \\Gamma(X_s, \\mathcal{F})$", "is in the image of one of the maps displayed in", "(\\ref{item-qc-gg}) above. This will imply (\\ref{item-qc-gg})", "since these affines $X_s$ cover $X$.", "Namely, by Lemma \\ref{lemma-invert-s-sections} we may write", "$m$ as the image of $m' \\otimes s^{-n}$ for some", "$n \\geq 1$, some", "$m' \\in \\Gamma(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n})$.", "This proves the claim.", "\\medskip\\noindent", "Clearly (\\ref{item-qc-gg}) $\\Rightarrow$ (\\ref{item-qc-i-gg}).", "Let us assume (\\ref{item-qc-i-gg}) and prove $\\mathcal{L}$ is", "ample. Pick $x \\in X$. Let $U \\subset X$ be an affine open", "which contains $x$. Set $Z = X \\setminus U$. We may think of", "$Z$ as a reduced closed subscheme, see", "Schemes, Section \\ref{schemes-section-reduced}.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent", "sheaf of ideals corresponding to the closed subscheme $Z$.", "By assumption (\\ref{item-qc-i-gg}), there exists an $n \\geq 1$ and a section", "$s \\in \\Gamma(X, \\mathcal{I} \\otimes \\mathcal{L}^{\\otimes n})$", "such that $s$ does not vanish at $x$ (more precisely such that", "$s \\not \\in \\mathfrak m_x \\mathcal{I}_x \\otimes \\mathcal{L}_x^{\\otimes n}$).", "We may think of $s$ as a section of $\\mathcal{L}^{\\otimes n}$.", "Since it clearly vanishes along $Z$ we see that", "$X_s \\subset U$. Hence $X_s$ is affine, see", "Lemma \\ref{lemma-affine-cap-s-open}.", "This proves that $\\mathcal{L}$ is ample.", "At this point we have proved that (1) -- (6) are equivalent.", "\\medskip\\noindent", "Assume the equivalent conditions (1) -- (6). In the following", "we will use the fact that the tensor product of two sheaves of", "modules which are globally generated is globally generated without", "further mention (see", "Modules, Lemma \\ref{modules-lemma-tensor-product-globally-generated}).", "By (1) we can find elements $s_i \\in S_{d_i}$ with $d_i \\geq 1$", "such that $X = \\bigcup_{i = 1, \\ldots, n} X_{s_i}$.", "Set $d = d_1\\ldots d_n$. It follows that $\\mathcal{L}^{\\otimes d}$", "is globally generated by", "$$", "s_1^{d/d_1}, \\ldots, s_n^{d/d_n}.", "$$", "This means that if $\\mathcal{L}^{\\otimes j}$ is globally generated", "then so is $\\mathcal{L}^{\\otimes j + dn}$ for all $n \\geq 0$.", "Fix a $j \\in \\{0, \\ldots, d - 1\\}$. For any point $x \\in X$ there", "exists an $n \\geq 1$ and a global section $s$ of $\\mathcal{L}^{j + dn}$", "which does not vanish at $x$, as follows from (\\ref{item-qc-gg}) applied", "to $\\mathcal{F} = \\mathcal{L}^{\\otimes j}$ and ample invertible", "sheaf $\\mathcal{L}^{\\otimes d}$. Since $X$ is quasi-compact there", "we may find a finite list of integers $n_i$ and global sections", "$s_i$ of $\\mathcal{L}^{\\otimes j + dn_i}$ which do not vanish at any point", "of $X$. Since $\\mathcal{L}^{\\otimes d}$ is globally generated this means that", "$\\mathcal{L}^{\\otimes j + dn}$ is globally generated where $n = \\max\\{n_i\\}$.", "Since we proved this for every congruence class mod $d$ we", "conclude that there exists an $n_0 = n_0(\\mathcal{L})$ such that", "$\\mathcal{L}^{\\otimes n}$ is globally generated for all $n \\geq n_0$.", "At this point we see that if $\\mathcal{F}$ is globally generated then", "so is $\\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}$ for all", "$n \\geq n_0$.", "\\medskip\\noindent", "We continue to assume the equivalent conditions (1) -- (6).", "Let $\\mathcal{F}$ be a quasi-coherent", "sheaf of $\\mathcal{O}_X$-modules of finite type.", "Denote $\\mathcal{F}_n \\subset \\mathcal{F}$ the image of the canonical", "map of (\\ref{item-qc-gg}). By construction", "$\\mathcal{F}_n \\otimes \\mathcal{L}^{\\otimes n}$ is", "globally generated. By (\\ref{item-qc-gg}) we see", "$\\mathcal{F}$ is the sum of the subsheaves $\\mathcal{F}_n$,", "$n \\geq 1$. By", "Modules, Lemma \\ref{modules-lemma-finite-type-quasi-compact-colimit}", "we see that $\\mathcal{F} = \\sum_{n = 1, \\ldots, N} \\mathcal{F}_n$", "for some $N \\geq 1$. It follows that", "$\\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}$ is globally", "generated whenever $n \\geq N + n_0(\\mathcal{L})$ with $n_0(\\mathcal{L})$", "as above. We conclude that (1) -- (6) implies (\\ref{item-c-gg}).", "\\medskip\\noindent", "Assume (\\ref{item-c-gg}). Let $\\mathcal{F}$ be a quasi-coherent", "sheaf of $\\mathcal{O}_X$-modules of finite type.", "By (\\ref{item-c-gg}) there exists an integer $n \\geq 1$ such that", "the canonical map", "$$", "\\Gamma(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n})", "\\otimes_{\\mathbf{Z}} \\mathcal{L}^{\\otimes -n}", "\\longrightarrow", "\\mathcal{F}", "$$", "is surjective. Let $I$ be the set of finite subsets of", "$\\Gamma(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n})$", "partially ordered by inclusion. Then $I$ is a directed partially ordered set.", "For $i = \\{s_1, \\ldots, s_{r(i)}\\}$ let $\\mathcal{F}_i \\subset \\mathcal{F}$", "be the image of the map", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, r(i)} \\mathcal{L}^{\\otimes -n}", "\\longrightarrow", "\\mathcal{F}", "$$", "which is multiplication by $s_j$ on the $j$th factor. The surjectivity above", "implies that $\\mathcal{F} = \\colim_{i \\in I} \\mathcal{F}_i$.", "Hence Modules, Lemma \\ref{modules-lemma-finite-type-quasi-compact-colimit}", "applies and we conclude that", "$\\mathcal{F} = \\mathcal{F}_i$ for some $i$.", "Hence we have proved (\\ref{item-c-q}). In other words,", "(\\ref{item-c-gg}) $\\Rightarrow$ (\\ref{item-c-q}).", "\\medskip\\noindent", "The implication (\\ref{item-c-q}) $\\Rightarrow$ (\\ref{item-c-i-q}) is trivial.", "\\medskip\\noindent", "Finally, assume (\\ref{item-c-i-q}).", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf", "of ideals. By Lemma \\ref{lemma-quasi-coherent-colimit-finite-type}", "(this is where we use the condition that $X$ be quasi-separated)", "we see that $\\mathcal{I} = \\colim_\\alpha I_\\alpha$ with", "each $I_\\alpha$ quasi-coherent of finite type. Since by assumption each of", "the $I_\\alpha$ is a quotient of negative tensor powers of", "$\\mathcal{L}$ we conclude the same for $\\mathcal{I}$ (but of course", "without the finiteness or boundedness of the powers). Hence", "we conclude that (\\ref{item-c-i-q}) implies (\\ref{item-qc-i-gg}).", "This ends the proof of the proposition." ], "refs": [ "properties-lemma-ample-immersion-into-proj", "properties-lemma-ample-power-ample", "properties-lemma-open-in-proj-ample", "properties-lemma-affine-s-opens", "properties-lemma-invert-s-sections", "properties-lemma-affine-cap-s-open", "modules-lemma-tensor-product-globally-generated", "modules-lemma-finite-type-quasi-compact-colimit", "modules-lemma-finite-type-quasi-compact-colimit", "properties-lemma-quasi-coherent-colimit-finite-type" ], "ref_ids": [ 3049, 3040, 3050, 3044, 3005, 3042, 13227, 13241, 13241, 3020 ] } ], "ref_ids": [] }, { "id": 3093, "type": "theorem", "label": "criteria-theorem-bootstrap", "categories": [ "criteria" ], "title": "criteria-theorem-bootstrap", "contents": [ "Let $S$ be a scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$", "be a $1$-morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$. If", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is representable by an algebraic space, and", "\\item $F$ is representable by algebraic spaces, surjective, flat and", "locally of finite presentation,", "\\end{enumerate}", "then $\\mathcal{Y}$ is an algebraic stack." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-flat-finite-presentation-surjective-diagonal}", "we see that the diagonal of $\\mathcal{Y}$ is representable by algebraic", "spaces. Hence we only need to verify the existence of a $1$-morphism", "$f : \\mathcal{V} \\to \\mathcal{Y}$ of stacks in groupoids over", "$(\\Sch/S)_{fppf}$ with $\\mathcal{V}$ representable and", "$f$ surjective and smooth. By", "Lemma \\ref{lemma-hilbert-stack-relative-space}", "we know that", "$$", "\\coprod\\nolimits_{d \\geq 1} \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})", "$$", "is an algebraic stack. It follows from", "Lemma \\ref{lemma-lci-locus-stack-in-groupoids}", "and", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-open-fibred-category-is-algebraic}", "that", "$$", "\\coprod\\nolimits_{d \\geq 1} \\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})", "$$", "is an algebraic stack as well. Choose a representable stack in groupoids", "$\\mathcal{V}$ over $(\\Sch/S)_{fppf}$ and a surjective and smooth", "$1$-morphism", "$$", "\\mathcal{V}", "\\longrightarrow", "\\coprod\\nolimits_{d \\geq 1} \\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y}).", "$$", "We claim that the composition", "$$", "\\mathcal{V}", "\\longrightarrow", "\\coprod\\nolimits_{d \\geq 1} \\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})", "\\longrightarrow", "\\mathcal{Y}", "$$", "is smooth and surjective which finishes the proof of the theorem. In fact,", "the smoothness will be a consequence of", "Lemmas \\ref{lemma-limit-preserving} and \\ref{lemma-lci-formally-smooth}", "and the surjectivity a consequence of", "Lemma \\ref{lemma-lci-surjective}.", "We spell out the details in the following paragraph.", "\\medskip\\noindent", "By construction $\\mathcal{V} \\to", "\\coprod\\nolimits_{d \\geq 1} \\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})$", "is representable by algebraic spaces, surjective, and smooth (and hence", "also locally of finite presentation and formally smooth by the general", "principle", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-representable-transformations-property-implication}", "and", "More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-smooth-formally-smooth}).", "Applying", "Lemmas \\ref{lemma-representable-by-spaces-limit-preserving},", "\\ref{lemma-representable-by-spaces-formally-smooth}, and", "\\ref{lemma-representable-by-spaces-surjective}", "we see that $\\mathcal{V} \\to", "\\coprod\\nolimits_{d \\geq 1} \\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})$", "is limit preserving on objects, formally smooth on objects, and", "surjective on objects. The $1$-morphism", "$\\coprod\\nolimits_{d \\geq 1} \\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})", "\\to \\mathcal{Y}$ is", "\\begin{enumerate}", "\\item limit preserving on objects: this is", "Lemma \\ref{lemma-limit-preserving}", "for $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y}) \\to \\mathcal{Y}$", "and we combine it with Lemmas", "\\ref{lemma-lci-locus-stack-in-groupoids},", "\\ref{lemma-open-immersion-limit-preserving}, and", "\\ref{lemma-composition-limit-preserving}", "to get it for $\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y}) \\to \\mathcal{Y}$,", "\\item formally smooth on objects by", "Lemma \\ref{lemma-lci-formally-smooth},", "and", "\\item surjective on objects by", "Lemma \\ref{lemma-lci-surjective}.", "\\end{enumerate}", "Using", "Lemmas \\ref{lemma-composition-limit-preserving},", "\\ref{lemma-composition-formally-smooth}, and", "\\ref{lemma-composition-surjective}", "we conclude that the composition $\\mathcal{V} \\to \\mathcal{Y}$ is", "limit preserving on objects, formally smooth on objects, and", "surjective on objects.", "Using", "Lemmas \\ref{lemma-representable-by-spaces-limit-preserving},", "\\ref{lemma-representable-by-spaces-formally-smooth}, and", "\\ref{lemma-representable-by-spaces-surjective}", "we see that $\\mathcal{V} \\to \\mathcal{Y}$ is", "locally of finite presentation, formally smooth, and surjective.", "Finally, using (via the general principle", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-representable-transformations-property-implication})", "the infinitesimal lifting criterion", "(More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-smooth-formally-smooth})", "we see that $\\mathcal{V} \\to \\mathcal{Y}$ is smooth and we win." ], "refs": [ "criteria-lemma-flat-finite-presentation-surjective-diagonal", "criteria-lemma-hilbert-stack-relative-space", "criteria-lemma-lci-locus-stack-in-groupoids", "algebraic-lemma-open-fibred-category-is-algebraic", "criteria-lemma-limit-preserving", "criteria-lemma-lci-formally-smooth", "criteria-lemma-lci-surjective", "algebraic-lemma-representable-transformations-property-implication", "spaces-more-morphisms-lemma-smooth-formally-smooth", "criteria-lemma-representable-by-spaces-limit-preserving", "criteria-lemma-representable-by-spaces-formally-smooth", "criteria-lemma-representable-by-spaces-surjective", "criteria-lemma-limit-preserving", "criteria-lemma-lci-locus-stack-in-groupoids", "criteria-lemma-open-immersion-limit-preserving", "criteria-lemma-composition-limit-preserving", "criteria-lemma-lci-formally-smooth", "criteria-lemma-lci-surjective", "criteria-lemma-composition-limit-preserving", "criteria-lemma-composition-formally-smooth", "criteria-lemma-composition-surjective", "criteria-lemma-representable-by-spaces-limit-preserving", "criteria-lemma-representable-by-spaces-formally-smooth", "criteria-lemma-representable-by-spaces-surjective", "algebraic-lemma-representable-transformations-property-implication", "spaces-more-morphisms-lemma-smooth-formally-smooth" ], "ref_ids": [ 3097, 3132, 3133, 8472, 3129, 3135, 3136, 8459, 110, 3101, 3107, 3110, 3129, 3133, 3102, 3100, 3135, 3136, 3100, 3106, 3109, 3101, 3107, 3110, 8459, 110 ] } ], "ref_ids": [] }, { "id": 3094, "type": "theorem", "label": "criteria-theorem-flat-groupoid-gives-algebraic-stack", "categories": [ "criteria" ], "title": "criteria-theorem-flat-groupoid-gives-algebraic-stack", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.", "Assume $s, t$ are flat and locally of finite presentation.", "Then the quotient stack $[U/R]$ is an algebraic stack over $S$." ], "refs": [], "proofs": [ { "contents": [ "We check the two conditions of", "Theorem \\ref{theorem-bootstrap}", "for the morphism", "$$", "(\\Sch/U)_{fppf} \\longrightarrow [U/R].", "$$", "The first is trivial (as $U$ is an algebraic space).", "The second is", "Lemma \\ref{lemma-flat-quotient-flat-presentation}." ], "refs": [ "criteria-theorem-bootstrap", "criteria-lemma-flat-quotient-flat-presentation" ], "ref_ids": [ 3093, 3137 ] } ], "ref_ids": [] }, { "id": 3095, "type": "theorem", "label": "criteria-lemma-etale-permanence", "categories": [ "criteria" ], "title": "criteria-lemma-etale-permanence", "contents": [ "Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$", "be $1$-morphisms of categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$.", "If $\\mathcal{X} \\to \\mathcal{Z}$ and $\\mathcal{Y} \\to \\mathcal{Z}$ are", "representable by algebraic spaces and \\'etale so is", "$\\mathcal{X} \\to \\mathcal{Y}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{U}$ be a representable category fibred in groupoids over $S$.", "Let $f : \\mathcal{U} \\to \\mathcal{Y}$ be a $1$-morphism. We have to show that", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{U}$ is representable by an", "algebraic space and \\'etale over $\\mathcal{U}$.", "Consider the composition $h : \\mathcal{U} \\to \\mathcal{Z}$. Then", "$$", "\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{U}", "\\longrightarrow", "\\mathcal{Y} \\times_\\mathcal{Z} \\mathcal{U}", "$$", "is a $1$-morphism between categories fibres in groupoids which are both", "representable by algebraic spaces and both \\'etale over $\\mathcal{U}$.", "Hence by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-permanence}", "this is represented by an \\'etale morphism of algebraic spaces.", "Finally, we obtain the result we want as the morphism $f$ induces", "a morphism $\\mathcal{U} \\to \\mathcal{Y} \\times_\\mathcal{Z} \\mathcal{U}$", "and we have", "$$", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{U} =", "(\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{U})", "\\times_{(\\mathcal{Y} \\times_\\mathcal{Z} \\mathcal{U})}", "\\mathcal{U}.", "$$" ], "refs": [ "spaces-properties-lemma-etale-permanence" ], "ref_ids": [ 11859 ] } ], "ref_ids": [] }, { "id": 3096, "type": "theorem", "label": "criteria-lemma-stack-in-setoids-descent", "categories": [ "criteria" ], "title": "criteria-lemma-stack-in-setoids-descent", "contents": [ "Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ be stacks in groupoids", "over $(\\Sch/S)_{fppf}$. Suppose that $\\mathcal{X} \\to \\mathcal{Y}$", "and $\\mathcal{Z} \\to \\mathcal{Y}$ are $1$-morphisms.", "If", "\\begin{enumerate}", "\\item $\\mathcal{Y}$, $\\mathcal{Z}$ are representable by algebraic spaces", "$Y$, $Z$ over $S$,", "\\item the associated morphism of algebraic spaces $Y \\to Z$ is surjective,", "flat and locally of finite presentation, and", "\\item $\\mathcal{Y} \\times_\\mathcal{Z} \\mathcal{X}$ is a stack in", "setoids,", "\\end{enumerate}", "then $\\mathcal{X}$ is a stack in setoids." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Stacks, Lemma \\ref{stacks-lemma-stack-in-setoids-descent}." ], "refs": [ "stacks-lemma-stack-in-setoids-descent" ], "ref_ids": [ 8957 ] } ], "ref_ids": [] }, { "id": 3097, "type": "theorem", "label": "criteria-lemma-flat-finite-presentation-surjective-diagonal", "categories": [ "criteria" ], "title": "criteria-lemma-flat-finite-presentation-surjective-diagonal", "contents": [ "Let $S$ be a scheme.", "Let $u : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of", "stacks in groupoids over $(\\Sch/S)_{fppf}$. If", "\\begin{enumerate}", "\\item $\\mathcal{U}$ is representable by an algebraic space, and", "\\item $u$ is representable by algebraic spaces, surjective, flat and", "locally of finite presentation,", "\\end{enumerate}", "then", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$", "representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Given two schemes $T_1$, $T_2$ over $S$ denote", "$\\mathcal{T}_i = (\\Sch/T_i)_{fppf}$ the associated representable", "fibre categories. Suppose given $1$-morphisms", "$f_i : \\mathcal{T}_i \\to \\mathcal{X}$.", "According to", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-representable-diagonal}", "it suffices to prove that the $2$-fibered", "product $\\mathcal{T}_1 \\times_\\mathcal{X} \\mathcal{T}_2$", "is representable by an algebraic space. By", "Stacks, Lemma", "\\ref{stacks-lemma-2-fibre-product-stacks-in-setoids-over-stack-in-groupoids}", "this is in any case a stack in setoids. Thus", "$\\mathcal{T}_1 \\times_\\mathcal{X} \\mathcal{T}_2$ corresponds", "to some sheaf $F$ on $(\\Sch/S)_{fppf}$, see", "Stacks, Lemma \\ref{stacks-lemma-stack-in-setoids-characterize}.", "Let $U$ be the algebraic space which represents $\\mathcal{U}$.", "By assumption", "$$", "\\mathcal{T}_i' = \\mathcal{U} \\times_{u, \\mathcal{X}, f_i} \\mathcal{T}_i", "$$", "is representable by an algebraic space $T'_i$ over $S$. Hence", "$\\mathcal{T}_1' \\times_\\mathcal{U} \\mathcal{T}_2'$ is representable", "by the algebraic space $T'_1 \\times_U T'_2$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "&", "\\mathcal{T}_1 \\times_{\\mathcal X} \\mathcal{T}_2 \\ar[rr]\\ar'[d][dd] & &", "\\mathcal{T}_1 \\ar[dd] \\\\", "\\mathcal{T}_1' \\times_\\mathcal{U} \\mathcal{T}_2' \\ar[ur]\\ar[rr]\\ar[dd] & &", "\\mathcal{T}_1' \\ar[ur]\\ar[dd] \\\\", "&", "\\mathcal{T}_2 \\ar'[r][rr] & &", "\\mathcal X \\\\", "\\mathcal{T}_2' \\ar[rr]\\ar[ur] & &", "\\mathcal{U} \\ar[ur] }", "$$", "In this diagram the bottom square, the right square, the back square, and", "the front square are $2$-fibre products. A formal argument then shows", "that $\\mathcal{T}_1' \\times_\\mathcal{U} \\mathcal{T}_2' \\to", "\\mathcal{T}_1 \\times_{\\mathcal X} \\mathcal{T}_2$", "is the ``base change'' of $\\mathcal{U} \\to \\mathcal{X}$, more precisely", "the diagram", "$$", "\\xymatrix{", "\\mathcal{T}_1' \\times_\\mathcal{U} \\mathcal{T}_2' \\ar[d] \\ar[r] &", "\\mathcal{U} \\ar[d] \\\\", "\\mathcal{T}_1 \\times_{\\mathcal X} \\mathcal{T}_2 \\ar[r] &", "\\mathcal{X}", "}", "$$", "is a $2$-fibre square.", "Hence $T'_1 \\times_U T'_2 \\to F$ is representable by algebraic spaces,", "flat, locally of finite presentation and surjective, see", "Algebraic Stacks, Lemmas", "\\ref{algebraic-lemma-map-fibred-setoids-representable-algebraic-spaces},", "\\ref{algebraic-lemma-base-change-representable-by-spaces},", "\\ref{algebraic-lemma-map-fibred-setoids-property}, and", "\\ref{algebraic-lemma-base-change-representable-transformations-property}.", "Therefore $F$ is an algebraic space by", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}", "and we win." ], "refs": [ "algebraic-lemma-representable-diagonal", "stacks-lemma-2-fibre-product-stacks-in-setoids-over-stack-in-groupoids", "stacks-lemma-stack-in-setoids-characterize", "algebraic-lemma-map-fibred-setoids-representable-algebraic-spaces", "algebraic-lemma-base-change-representable-by-spaces", "algebraic-lemma-map-fibred-setoids-property", "algebraic-lemma-base-change-representable-transformations-property", "bootstrap-theorem-final-bootstrap" ], "ref_ids": [ 8461, 8955, 8951, 8446, 8447, 8454, 8456, 2602 ] } ], "ref_ids": [] }, { "id": 3098, "type": "theorem", "label": "criteria-lemma-second-diagonal", "categories": [ "criteria" ], "title": "criteria-lemma-second-diagonal", "contents": [ "Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\Delta_\\Delta : \\mathcal{X} \\to", "\\mathcal{X} \\times_{\\mathcal{X} \\times \\mathcal{X}} \\mathcal{X}$", "is representable by algebraic spaces,", "\\item for every $1$-morphism $\\mathcal{V} \\to \\mathcal{X} \\times \\mathcal{X}$", "with $\\mathcal{V}$ representable (by a scheme) the fibre product", "$\\mathcal{Y} =", "\\mathcal{X} \\times_{\\Delta, \\mathcal{X} \\times \\mathcal{X}} \\mathcal{V}$", "has diagonal representable by algebraic spaces.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Although this is a bit of a brain twister, it is completely formal.", "Namely, recall that", "$\\mathcal{X} \\times_{\\mathcal{X} \\times \\mathcal{X}} \\mathcal{X} =", "\\mathcal{I}_\\mathcal{X}$ is the inertia of $\\mathcal{X}$ and that", "$\\Delta_\\Delta$ is the identity section of $\\mathcal{I}_\\mathcal{X}$, see", "Categories, Section \\ref{categories-section-inertia}.", "Thus condition (1) says the following: Given a scheme $V$, an object $x$ of", "$\\mathcal{X}$ over $V$, and a morphism $\\alpha : x \\to x$ of $\\mathcal{X}_V$", "the condition ``$\\alpha = \\text{id}_x$'' defines an algebraic space over $V$.", "(In other words, there exists a monomorphism of algebraic spaces $W \\to V$", "such that a morphism of schemes $f : T \\to V$ factors through $W$", "if and only if $f^*\\alpha = \\text{id}_{f^*x}$.)", "\\medskip\\noindent", "On the other hand, let $V$ be a scheme and let $x, y$ be objects of", "$\\mathcal{X}$ over $V$. Then $(x, y)$ define a morphism", "$\\mathcal{V} = (\\Sch/V)_{fppf} \\to \\mathcal{X} \\times \\mathcal{X}$.", "Next, let $h : V' \\to V$ be a morphism of schemes and let", "$\\alpha : h^*x \\to h^*y$ and $\\beta : h^*x \\to h^*y$ be morphisms", "of $\\mathcal{X}_{V'}$. Then $(\\alpha, \\beta)$ define a morphism", "$\\mathcal{V}' = (\\Sch/V)_{fppf} \\to \\mathcal{Y} \\times \\mathcal{Y}$.", "Condition (2) now says that (with any choices as above) the", "condition ``$\\alpha = \\beta$'' defines an algebraic space over $V$.", "\\medskip\\noindent", "To see the equivalence, given $(\\alpha, \\beta)$ as in (2) we see that", "(1) implies that ``$\\alpha^{-1} \\circ \\beta = \\text{id}_{h^*x}$''", "defines an algebraic space. The implication (2) $\\Rightarrow$ (1)", "follows by taking $h = \\text{id}_V$ and $\\beta = \\text{id}_x$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3099, "type": "theorem", "label": "criteria-lemma-base-change-limit-preserving", "categories": [ "criteria" ], "title": "criteria-lemma-base-change-limit-preserving", "contents": [ "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$", "be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "If $p : \\mathcal{X} \\to \\mathcal{Y}$ is limit preserving on objects, then so", "is the base change", "$p' : \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$", "of $p$ by $q$." ], "refs": [], "proofs": [ { "contents": [ "This is formal. Let $U = \\lim_{i \\in I} U_i$ be the directed limit", "of affine schemes $U_i$ over $S$, let $z_i$ be an object of $\\mathcal{Z}$", "over $U_i$ for some $i$, let $w$ be an object of", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ over $U$, and let", "$\\delta : p'(w) \\to z_i|_U$ be an isomorphism.", "We may write", "$w = (U, x, z, \\alpha)$ for some object $x$ of $\\mathcal{X}$ over $U$", "and object $z$ of $\\mathcal{Z}$ over $U$ and isomorphism", "$\\alpha : p(x) \\to q(z)$. Note that $p'(w) = z$ hence", "$\\delta : z \\to z_i|_U$. Set $y_i = q(z_i)$ and", "$\\gamma = q(\\delta) \\circ \\alpha : p(x) \\to y_i|_U$.", "As $p$ is limit preserving on objects there exists an $i' \\geq i$", "and an object $x_{i'}$ of $\\mathcal{X}$ over $U_{i'}$ as well as", "isomorphisms $\\beta : x_{i'}|_U \\to x$ and", "$\\gamma_{i'} : p(x_{i'}) \\to y_i|_{U_{i'}}$ such that", "(\\ref{equation-limit-preserving}) commutes. Then we consider the object", "$w_{i'} = (U_{i'}, x_{i'}, z_i|_{U_{i'}}, \\gamma_{i'})$ of", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ over $U_{i'}$", "and define isomorphisms", "$$", "w_{i'}|_U = (U, x_{i'}|_U, z_i|_U, \\gamma_{i'}|_U)", "\\xrightarrow{(\\beta, \\delta^{-1})}", "(U, x, z, \\alpha) = w", "$$", "and", "$$", "p'(w_{i'}) = z_i|_{U_{i'}} \\xrightarrow{\\text{id}} z_i|_{U_{i'}}.", "$$", "These combine to give a solution to the problem." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3100, "type": "theorem", "label": "criteria-lemma-composition-limit-preserving", "categories": [ "criteria" ], "title": "criteria-lemma-composition-limit-preserving", "contents": [ "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Y} \\to \\mathcal{Z}$", "be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "If $p$ and $q$ are limit preserving on objects, then so is the composition", "$q \\circ p$." ], "refs": [], "proofs": [ { "contents": [ "This is formal. Let $U = \\lim_{i \\in I} U_i$ be the directed limit", "of affine schemes $U_i$ over $S$, let $z_i$ be an object of $\\mathcal{Z}$", "over $U_i$ for some $i$, let $x$ be an object of $\\mathcal{X}$ over $U$,", "and let $\\gamma : q(p(x)) \\to z_i|_U$ be an isomorphism. As $q$ is", "limit preserving on objects there exist an $i' \\geq i$, an object", "$y_{i'}$ of $\\mathcal{Y}$ over $U_{i'}$, an isomorphism", "$\\beta : y_{i'}|_U \\to p(x)$, and an isomorphism", "$\\gamma_{i'} : q(y_{i'}) \\to z_i|_{U_{i'}}$", "such that (\\ref{equation-limit-preserving}) is commutative. As $p$ is", "limit preserving on objects there exist an $i'' \\geq i'$, an object", "$x_{i''}$ of $\\mathcal{X}$ over $U_{i''}$, an isomorphism", "$\\beta' : x_{i''}|_U \\to x$, and an isomorphism", "$\\gamma'_{i''} : p(x_{i''}) \\to y_{i'}|_{U_{i''}}$", "such that (\\ref{equation-limit-preserving}) is commutative.", "The solution is to take $x_{i''}$ over $U_{i''}$ with isomorphism", "$$", "q(p(x_{i''})) \\xrightarrow{q(\\gamma'_{i''})}", "q(y_{i'})|_{U_{i''}} \\xrightarrow{\\gamma_{i'}|_{U_{i''}}}", "z_i|_{U_{i''}}", "$$", "and isomorphism $\\beta' : x_{i''}|_U \\to x$. We omit the verification", "that (\\ref{equation-limit-preserving}) is commutative." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3101, "type": "theorem", "label": "criteria-lemma-representable-by-spaces-limit-preserving", "categories": [ "criteria" ], "title": "criteria-lemma-representable-by-spaces-limit-preserving", "contents": [ "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ is", "representable by algebraic spaces, then the following are equivalent:", "\\begin{enumerate}", "\\item $p$ is limit preserving on objects, and", "\\item $p$ is locally of finite presentation (see", "Algebraic Stacks,", "Definition \\ref{algebraic-definition-relative-representable-property}).", "\\end{enumerate}" ], "refs": [ "algebraic-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Assume (2). Let $U = \\lim_{i \\in I} U_i$ be the directed limit", "of affine schemes $U_i$ over $S$, let $y_i$ be an object of $\\mathcal{Y}$", "over $U_i$ for some $i$, let $x$ be an object of $\\mathcal{X}$ over $U$,", "and let $\\gamma : p(x) \\to y_i|_U$ be an isomorphism. Let", "$X_{y_i}$ denote an algebraic space over $U_i$ representing the $2$-fibre", "product", "$$", "(\\Sch/U_i)_{fppf} \\times_{y_i, \\mathcal{Y}, p} \\mathcal{X}.", "$$", "Note that $\\xi = (U, U \\to U_i, x, \\gamma^{-1})$ defines an object of", "this $2$-fibre product over $U$. Via the $2$-Yoneda lemma $\\xi$ corresponds", "to a morphism $f_\\xi : U \\to X_{y_i}$ over $U_i$. By", "Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-characterize-locally-finite-presentation}", "there exists an $i' \\geq i$ and a morphism $f_{i'} : U_{i'} \\to X_{y_i}$", "such that $f_\\xi$ is the composition of $f_{i'}$ and the projection", "morphism $U \\to U_{i'}$. Also, the $2$-Yoneda lemma tells us that", "$f_{i'}$ corresponds to an object", "$\\xi_{i'} = (U_{i'}, U_{i'} \\to U_i, x_{i'}, \\alpha)$ of", "the displayed $2$-fibre product over $U_{i'}$ whose restriction to", "$U$ recovers $\\xi$. In particular we obtain an isomorphism", "$\\gamma : x_{i'}|U \\to x$. Note that $\\alpha : y_i|_{U_{i'}} \\to p(x_{i'})$.", "Hence we see that taking $x_{i'}$, the isomorphism", "$\\gamma : x_{i'}|U \\to x$, and the isomorphism", "$\\beta = \\alpha^{-1} : p(x_{i'}) \\to y_i|_{U_{i'}}$", "is a solution to the problem.", "\\medskip\\noindent", "Assume (1). Choose a scheme $T$ and a $1$-morphism", "$y : (\\Sch/T)_{fppf} \\to \\mathcal{Y}$. Let", "$X_y$ be an algebraic space over $T$ representing the $2$-fibre product", "$(\\Sch/T)_{fppf} \\times_{y, \\mathcal{Y}, p} \\mathcal{X}$.", "We have to show that $X_y \\to T$ is locally of finite presentation.", "To do this we will use the criterion in", "Limits of Spaces, Remark \\ref{spaces-limits-remark-limit-preserving}.", "Consider an affine scheme $U = \\lim_{i \\in I} U_i$ written as the", "directed limit of affine schemes over $T$.", "Pick any $i \\in I$ and set $y_i = y|_{U_i}$. Also denote $i'$ an element", "of $I$ which is bigger than or equal to $i$. By the $2$-Yoneda lemma", "morphisms $U \\to X_y$ over $T$ correspond bijectively", "to isomorphism classes of pairs $(x, \\alpha)$ where $x$ is an object", "of $\\mathcal{X}$ over $U$ and $\\alpha : y|_U \\to p(x)$ is an isomorphism.", "Of course giving $\\alpha$ is, up to an inverse, the same thing as giving", "an isomorphism $\\gamma : p(x) \\to y_i|_U$.", "Similarly for morphisms $U_{i'} \\to X_y$ over $T$. Hence (1) guarantees", "that the canonical map", "$$", "\\colim_{i' \\geq i} X_y(U_{i'}) \\longrightarrow X_y(U)", "$$", "is surjective in this situation. It follows from", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-surjection-is-enough}", "that $X_y \\to T$ is locally of finite presentation." ], "refs": [ "spaces-limits-proposition-characterize-locally-finite-presentation", "spaces-limits-remark-limit-preserving", "spaces-limits-lemma-surjection-is-enough" ], "ref_ids": [ 4655, 4663, 4564 ] } ], "ref_ids": [ 8483 ] }, { "id": 3102, "type": "theorem", "label": "criteria-lemma-open-immersion-limit-preserving", "categories": [ "criteria" ], "title": "criteria-lemma-open-immersion-limit-preserving", "contents": [ "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $p$ is representable", "by algebraic spaces and an open immersion. Then $p$ is limit preserving", "on objects." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-representable-by-spaces-limit-preserving}", "and (via the general principle", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-representable-transformations-property-implication})", "from the fact that an open immersion of algebraic spaces is", "locally of finite presentation, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-open-immersion-locally-finite-presentation}." ], "refs": [ "criteria-lemma-representable-by-spaces-limit-preserving", "algebraic-lemma-representable-transformations-property-implication", "spaces-morphisms-lemma-open-immersion-locally-finite-presentation" ], "ref_ids": [ 3101, 8459, 4848 ] } ], "ref_ids": [] }, { "id": 3103, "type": "theorem", "label": "criteria-lemma-check-representable-limit-preserving", "categories": [ "criteria" ], "title": "criteria-lemma-check-representable-limit-preserving", "contents": [ "Let $S$ be a scheme.", "Let $\\kappa = \\text{size}(T)$ for some $T \\in \\Ob((\\Sch/S)_{fppf})$.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism", "of categories fibred in groupoids over $(\\Sch/S)_{fppf}$", "such that", "\\begin{enumerate}", "\\item $\\mathcal{Y} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects,", "\\item for an affine scheme $V$ locally of finite presentation over $S$ and", "$y \\in \\Ob(\\mathcal{Y}_V)$ the fibre product", "$(\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}$ is representable", "by an algebraic space of size $\\leq \\kappa$\\footnote{The condition on", "size can be dropped by those ignoring set theoretic issues.},", "\\item $\\mathcal{X}$ and $\\mathcal{Y}$ are stacks for the Zariski topology.", "\\end{enumerate}", "Then $f$ is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Let $V$ be a scheme over $S$ and $y \\in \\mathcal{Y}_V$. We have to prove", "$(\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}$ is representable", "by an algebraic space.", "\\medskip\\noindent", "Case I: $V$ is affine and maps into an affine open $\\Spec(\\Lambda) \\subset S$.", "Then we can write $V = \\lim V_i$ with each $V_i$ affine and of finite", "presentation over $\\Spec(\\Lambda)$, see", "Algebra, Lemma \\ref{algebra-lemma-ring-colimit-fp}.", "Then $y$ comes from an object $y_i$ over $V_i$ for some $i$ by assumption (1).", "By assumption (3) the fibre product", "$(\\Sch/V_i)_{fppf} \\times_{y_i, \\mathcal{Y}} \\mathcal{X}$ is representable", "by an algebraic space $Z_i$. Then ", "$(\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}$ is representable", "by $Z \\times_{V_i} V$.", "\\medskip\\noindent", "Case II: $V$ is general. Choose an affine open covering", "$V = \\bigcup_{i \\in I} V_i$ such that each $V_i$ maps into an affine open", "of $S$. We first claim", "that $\\mathcal{Z} = (\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}$", "is a stack in setoids for the Zariski topology. Namely, it is a stack in", "groupoids for the Zariski topology by", "Stacks, Lemma \\ref{stacks-lemma-2-product-stacks-in-groupoids}.", "Then suppose that $z$ is an object of $\\mathcal{Z}$ over a scheme $T$.", "Denote $g : T \\to V$ the morphism corresponding to the", "projection of $z$ in $(\\Sch/V)_{fppf}$. Consider the Zariski sheaf", "$\\mathit{I} = \\mathit{Isom}_{\\mathcal{Z}}(z, z)$. By Case I we see that", "$\\mathit{I}|_{g^{-1}(V_i)} = *$ (the singleton sheaf). Hence", "$\\mathcal{I} = *$. Thus $\\mathcal{Z}$ is fibred in setoids. To finish", "the proof we have to show that the Zariski sheaf", "$Z : T \\mapsto \\Ob(\\mathcal{Z}_T)/\\cong$ is an algebraic space, see", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-characterize-representable-by-space}.", "There is a map $p : Z \\to V$ (transformation of functors) and by Case I", "we know that $Z_i = p^{-1}(V_i)$ is an algebraic space. The morphisms", "$Z_i \\to Z$ are representable by open immersions and", "$\\coprod Z_i \\to Z$ is surjective (in the Zariski topology).", "Hence $Z$ is a sheaf for the fppf topology by", "Bootstrap, Lemma \\ref{bootstrap-lemma-glueing-sheaves}.", "Thus Spaces, Lemma \\ref{spaces-lemma-glueing-algebraic-spaces}", "applies and we conclude that $Z$ is an algebraic space\\footnote{", "To see that the set theoretic condition of that lemma is satisfied", "we argue as follows: First choose the open covering such that", "$|I| \\leq \\text{size}(V)$. Next, choose schemes $U_i$ of size", "$\\leq \\max(\\kappa, \\text{size}(V))$ and surjective \\'etale morphisms", "$U_i \\to Z_i$; we can do this by assumption (2) and", "Sets, Lemma \\ref{sets-lemma-bound-size-fibre-product}", "(details omitted). Then", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}", "implies that $\\coprod U_i$ is an object of $(\\Sch/S)_{fppf}$.", "Hence $\\coprod Z_i$ is an algebraic space by", "Spaces, Lemma \\ref{spaces-lemma-coproduct-algebraic-spaces}.", "}." ], "refs": [ "algebra-lemma-ring-colimit-fp", "stacks-lemma-2-product-stacks-in-groupoids", "algebraic-lemma-characterize-representable-by-space", "bootstrap-lemma-glueing-sheaves", "spaces-lemma-glueing-algebraic-spaces", "sets-lemma-bound-size-fibre-product", "sets-lemma-what-is-in-it", "spaces-lemma-coproduct-algebraic-spaces" ], "ref_ids": [ 1091, 8949, 8441, 2612, 8148, 8792, 8795, 8147 ] } ], "ref_ids": [] }, { "id": 3104, "type": "theorem", "label": "criteria-lemma-check-property-limit-preserving", "categories": [ "criteria" ], "title": "criteria-lemma-check-property-limit-preserving", "contents": [ "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism", "of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\mathcal{P}$", "be a property of morphisms of algebraic spaces as in", "Algebraic Stacks, Definition", "\\ref{algebraic-definition-relative-representable-property}. If", "\\begin{enumerate}", "\\item $f$ is representable by algebraic spaces,", "\\item $\\mathcal{Y} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects,", "\\item for an affine scheme $V$ locally of finite presentation over $S$ and", "$y \\in \\mathcal{Y}_V$ the resulting morphism of algebraic spaces", "$f_y : F_y \\to V$, see Algebraic Stacks, Equation", "(\\ref{algebraic-equation-representable-by-algebraic-spaces}),", "has property $\\mathcal{P}$.", "\\end{enumerate}", "Then $f$ has property $\\mathcal{P}$." ], "refs": [ "algebraic-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Let $V$ be a scheme over $S$ and $y \\in \\mathcal{Y}_V$. We have to show", "that $F_y \\to V$ has property $\\mathcal{P}$. Since $\\mathcal{P}$ is", "fppf local on the base we may assume that $V$ is an affine scheme which", "maps into an affine open $\\Spec(\\Lambda) \\subset S$. Thus we can write", "$V = \\lim V_i$ with each $V_i$ affine and of finite presentation over", "$\\Spec(\\Lambda)$, see Algebra, Lemma \\ref{algebra-lemma-ring-colimit-fp}.", "Then $y$ comes from an object $y_i$ over $V_i$ for some $i$ by assumption (2).", "By assumption (3) the morphism $F_{y_i} \\to V_i$ has property $\\mathcal{P}$.", "As $\\mathcal{P}$ is stable under arbitrary base change and since", "$F_y = F_{y_i} \\times_{V_i} V$ we conclude that $F_y \\to V$ has property", "$\\mathcal{P}$ as desired." ], "refs": [ "algebra-lemma-ring-colimit-fp" ], "ref_ids": [ 1091 ] } ], "ref_ids": [ 8483 ] }, { "id": 3105, "type": "theorem", "label": "criteria-lemma-base-change-formally-smooth", "categories": [ "criteria" ], "title": "criteria-lemma-base-change-formally-smooth", "contents": [ "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$", "be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "If $p : \\mathcal{X} \\to \\mathcal{Y}$ is formally smooth on objects, then so", "is the base change", "$p' : \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$", "of $p$ by $q$." ], "refs": [], "proofs": [ { "contents": [ "This is formal. Let $U \\subset U'$ be a first order thickening", "of affine schemes over $S$, let $z'$ be an object of $\\mathcal{Z}$", "over $U'$, let $w$ be an object of", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ over $U$, and let", "$\\delta : p'(w) \\to z'|_U$ be an isomorphism.", "We may write", "$w = (U, x, z, \\alpha)$ for some object $x$ of $\\mathcal{X}$ over $U$", "and object $z$ of $\\mathcal{Z}$ over $U$ and isomorphism", "$\\alpha : p(x) \\to q(z)$. Note that $p'(w) = z$ hence", "$\\delta : z \\to z|_U$. Set $y' = q(z')$ and", "$\\gamma = q(\\delta) \\circ \\alpha : p(x) \\to y'|_U$.", "As $p$ is formally smooth on objects there exists an", "object $x'$ of $\\mathcal{X}$ over $U'$ as well as", "isomorphisms $\\beta : x'|_U \\to x$ and $\\gamma' : p(x') \\to y'$ such that", "(\\ref{equation-formally-smooth}) commutes. Then we consider the object", "$w = (U', x', z', \\gamma')$ of $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$", "over $U'$ and define isomorphisms", "$$", "w'|_U = (U, x'|_U, z'|_U, \\gamma'|_U)", "\\xrightarrow{(\\beta, \\delta^{-1})}", "(U, x, z, \\alpha) = w", "$$", "and", "$$", "p'(w') = z' \\xrightarrow{\\text{id}} z'.", "$$", "These combine to give a solution to the problem." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3106, "type": "theorem", "label": "criteria-lemma-composition-formally-smooth", "categories": [ "criteria" ], "title": "criteria-lemma-composition-formally-smooth", "contents": [ "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Y} \\to \\mathcal{Z}$", "be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "If $p$ and $q$ are formally smooth on objects, then so is the composition", "$q \\circ p$." ], "refs": [], "proofs": [ { "contents": [ "This is formal. Let $U \\subset U'$ be a first order thickening", "of affine schemes over $S$, let $z'$ be an object of $\\mathcal{Z}$", "over $U'$, let $x$ be an object of $\\mathcal{X}$ over $U$,", "and let $\\gamma : q(p(x)) \\to z'|_U$ be an isomorphism. As $q$ is", "formally smooth on objects there exist an object", "$y'$ of $\\mathcal{Y}$ over $U'$, an isomorphism", "$\\beta : y'|_U \\to p(x)$, and an isomorphism $\\gamma' : q(y') \\to z'$", "such that (\\ref{equation-formally-smooth}) is commutative. As $p$ is", "formally smooth on objects there exist an object", "$x'$ of $\\mathcal{X}$ over $U'$, an isomorphism", "$\\beta' : x'|_U \\to x$, and an isomorphism $\\gamma'' : p(x') \\to y'$", "such that (\\ref{equation-formally-smooth}) is commutative.", "The solution is to take $x'$ over $U'$ with isomorphism", "$$", "q(p(x')) \\xrightarrow{q(\\gamma'')} q(y') \\xrightarrow{\\gamma'} z'", "$$", "and isomorphism $\\beta' : x'|_U \\to x$. We omit the verification", "that (\\ref{equation-formally-smooth}) is commutative." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3107, "type": "theorem", "label": "criteria-lemma-representable-by-spaces-formally-smooth", "categories": [ "criteria" ], "title": "criteria-lemma-representable-by-spaces-formally-smooth", "contents": [ "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ is", "representable by algebraic spaces, then the following are equivalent:", "\\begin{enumerate}", "\\item $p$ is formally smooth on objects, and", "\\item $p$ is formally smooth (see", "Algebraic Stacks,", "Definition \\ref{algebraic-definition-relative-representable-property}).", "\\end{enumerate}" ], "refs": [ "algebraic-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Assume (2). Let $U \\subset U'$ be a first order thickening", "of affine schemes over $S$, let $y'$ be an object of $\\mathcal{Y}$", "over $U'$, let $x$ be an object of $\\mathcal{X}$ over $U$,", "and let $\\gamma : p(x) \\to y'|_U$ be an isomorphism. Let", "$X_{y'}$ denote an algebraic space over $U'$ representing the $2$-fibre", "product", "$$", "(\\Sch/U')_{fppf} \\times_{y', \\mathcal{Y}, p} \\mathcal{X}.", "$$", "Note that $\\xi = (U, U \\to U', x, \\gamma^{-1})$ defines an object of", "this $2$-fibre product over $U$. Via the $2$-Yoneda lemma $\\xi$ corresponds", "to a morphism $f_\\xi : U \\to X_{y'}$ over $U'$. As $X_{y'} \\to U'$ is", "formally smooth by assumption there exists a morphism", "$f' : U' \\to X_{y'}$ such that $f_\\xi$ is the composition of $f'$", "and the morphism $U \\to U'$. Also, the $2$-Yoneda lemma tells us that", "$f'$ corresponds to an object $\\xi' = (U', U' \\to U', x', \\alpha)$ of", "the displayed $2$-fibre product over $U'$ whose restriction to", "$U$ recovers $\\xi$. In particular we obtain an isomorphism", "$\\gamma : x'|U \\to x$. Note that $\\alpha : y' \\to p(x')$.", "Hence we see that taking $x'$, the isomorphism", "$\\gamma : x'|U \\to x$, and the isomorphism", "$\\beta = \\alpha^{-1} : p(x') \\to y'$", "is a solution to the problem.", "\\medskip\\noindent", "Assume (1). Choose a scheme $T$ and a $1$-morphism", "$y : (\\Sch/T)_{fppf} \\to \\mathcal{Y}$. Let", "$X_y$ be an algebraic space over $T$ representing the $2$-fibre product", "$(\\Sch/T)_{fppf} \\times_{y, \\mathcal{Y}, p} \\mathcal{X}$.", "We have to show that $X_y \\to T$ is formally smooth.", "Hence it suffices to show that given a first order thickening", "$U \\subset U'$ of affine schemes over $T$, then", "$X_y(U') \\to X_y(U')$ is surjective (morphisms in the", "category of algebraic spaces over $T$). Set $y' = y|_{U'}$.", "By the $2$-Yoneda lemma morphisms $U \\to X_y$ over $T$ correspond bijectively", "to isomorphism classes of pairs $(x, \\alpha)$ where $x$ is an object", "of $\\mathcal{X}$ over $U$ and $\\alpha : y|_U \\to p(x)$ is an isomorphism.", "Of course giving $\\alpha$ is, up to an inverse, the same thing as giving", "an isomorphism $\\gamma : p(x) \\to y'|_U$.", "Similarly for morphisms $U' \\to X_y$ over $T$. Hence (1) guarantees", "the surjectivity of $X_y(U') \\to X_y(U')$", "in this situation and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8483 ] }, { "id": 3108, "type": "theorem", "label": "criteria-lemma-base-change-surjective", "categories": [ "criteria" ], "title": "criteria-lemma-base-change-surjective", "contents": [ "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$", "be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "If $p : \\mathcal{X} \\to \\mathcal{Y}$ is surjective on objects, then so", "is the base change", "$p' : \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$", "of $p$ by $q$." ], "refs": [], "proofs": [ { "contents": [ "This is formal. Let $z$ be an object of $\\mathcal{Z}$ over a field $k$.", "As $p$ is surjective on objects there exists an extension $k \\subset K$", "and an object $x$ of $\\mathcal{X}$ over $K$ and an isomorphism", "$\\alpha : p(x) \\to q(z)|_{\\Spec(K)}$. Then", "$w = (\\Spec(K), x, z|_{\\Spec(K)}, \\alpha)$ is an object of", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ over $K$ with", "$p'(w) = z|_{\\Spec(K)}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3109, "type": "theorem", "label": "criteria-lemma-composition-surjective", "categories": [ "criteria" ], "title": "criteria-lemma-composition-surjective", "contents": [ "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Y} \\to \\mathcal{Z}$", "be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "If $p$ and $q$ are surjective on objects, then so is the composition", "$q \\circ p$." ], "refs": [], "proofs": [ { "contents": [ "This is formal. Let $z$ be an object of $\\mathcal{Z}$ over a field $k$.", "As $q$ is surjective on objects there exists a field extension $k \\subset K$", "and an object $y$ of $\\mathcal{Y}$ over $K$ such that", "$q(y) \\cong x|_{\\Spec(K)}$. As $p$ is surjective on objects there", "exists a field extension $K \\subset L$ and an object $x$ of $\\mathcal{X}$", "over $L$ such that $p(x) \\cong y|_{\\Spec(L)}$. Then the field extension", "$k \\subset L$ and the object $x$ of $\\mathcal{X}$ over $L$ satisfy", "$q(p(x)) \\cong z|_{\\Spec(L)}$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3110, "type": "theorem", "label": "criteria-lemma-representable-by-spaces-surjective", "categories": [ "criteria" ], "title": "criteria-lemma-representable-by-spaces-surjective", "contents": [ "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ is", "representable by algebraic spaces, then the following are equivalent:", "\\begin{enumerate}", "\\item $p$ is surjective on objects, and", "\\item $p$ is surjective (see", "Algebraic Stacks,", "Definition \\ref{algebraic-definition-relative-representable-property}).", "\\end{enumerate}" ], "refs": [ "algebraic-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Assume (2). Let $k$ be a field and let $y$ be an object of", "$\\mathcal{Y}$ over $k$. Let $X_y$ denote an algebraic space over $k$", "representing the $2$-fibre product", "$$", "(\\Sch/\\Spec(k))_{fppf} \\times_{y, \\mathcal{Y}, p} \\mathcal{X}.", "$$", "As we've assumed that $p$ is surjective we see that $X_y$ is not empty.", "Hence we can find a field extension $k \\subset K$ and a $K$-valued point", "$x$ of $X_y$. Via the $2$-Yoneda lemma this corresponds to an object", "$x$ of $\\mathcal{X}$ over $K$ together with an isomorphism", "$p(x) \\cong y|_{\\Spec(K)}$ and we see that (1) holds.", "\\medskip\\noindent", "Assume (1). Choose a scheme $T$ and a $1$-morphism", "$y : (\\Sch/T)_{fppf} \\to \\mathcal{Y}$. Let", "$X_y$ be an algebraic space over $T$ representing the $2$-fibre product", "$(\\Sch/T)_{fppf} \\times_{y, \\mathcal{Y}, p} \\mathcal{X}$.", "We have to show that $X_y \\to T$ is surjective. By", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-surjective}", "we have to show that $|X_y| \\to |T|$ is surjective.", "This means exactly that given a field $k$ over $T$ and a", "morphism $t : \\Spec(k) \\to T$ there exists a field extension", "$k \\subset K$ and a morphism $x : \\Spec(K) \\to X_y$ such that", "$$", "\\xymatrix{", "\\Spec(K) \\ar[d] \\ar[r]_x & X_y \\ar[d] \\\\", "\\Spec(k) \\ar[r]^t & T", "}", "$$", "commutes. By the $2$-Yoneda lemma this means exactly that we have to find", "$k \\subset K$ and an object $x$ of $\\mathcal{X}$ over $K$ such that", "$p(x) \\cong t^*y|_{\\Spec(K)}$. Hence (1) guarantees that this is", "the case and we win." ], "refs": [ "spaces-morphisms-definition-surjective" ], "ref_ids": [ 4985 ] } ], "ref_ids": [ 8483 ] }, { "id": 3111, "type": "theorem", "label": "criteria-lemma-algebraic-morphism-to-algebraic", "categories": [ "criteria" ], "title": "criteria-lemma-algebraic-morphism-to-algebraic", "contents": [ "Let $S$ be a scheme.", "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of", "stacks in groupoids over $(\\Sch/S)_{fppf}$. If", "\\begin{enumerate}", "\\item $\\mathcal{Y}$ is an algebraic stack, and", "\\item $F$ is algebraic (see above),", "\\end{enumerate}", "then $\\mathcal{X}$ is an algebraic stack." ], "refs": [], "proofs": [ { "contents": [ "By assumption (1) there exists a scheme $T$ and an object", "$\\xi$ of $\\mathcal{Y}$ over $T$ such that the corresponding", "$1$-morphism $\\xi : (\\Sch/T)_{fppf} \\to \\mathcal{Y}$", "is smooth an surjective. Then", "$\\mathcal{U} = (\\Sch/T)_{fppf} \\times_{\\xi, \\mathcal{Y}} \\mathcal{X}$", "is an algebraic stack by assumption (2).", "Choose a scheme $U$ and a surjective smooth $1$-morphism", "$(\\Sch/U)_{fppf} \\to \\mathcal{U}$.", "The projection $\\mathcal{U} \\longrightarrow \\mathcal{X}$", "is, as the base change of the morphism", "$\\xi : (\\Sch/T)_{fppf} \\to \\mathcal{Y}$,", "surjective and smooth, see", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-base-change-representable-transformations-property}.", "Then the composition", "$(\\Sch/U)_{fppf} \\to \\mathcal{U} \\to \\mathcal{X}$", "is surjective and smooth as a composition of surjective and smooth", "morphisms, see", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-composition-representable-transformations-property}.", "Hence $\\mathcal{X}$ is an algebraic stack by", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-smooth-surjective-morphism-implies-algebraic}." ], "refs": [ "algebraic-lemma-base-change-representable-transformations-property", "algebraic-lemma-composition-representable-transformations-property", "algebraic-lemma-smooth-surjective-morphism-implies-algebraic" ], "ref_ids": [ 8456, 8455, 8470 ] } ], "ref_ids": [] }, { "id": 3112, "type": "theorem", "label": "criteria-lemma-map-from-algebraic", "categories": [ "criteria" ], "title": "criteria-lemma-map-from-algebraic", "contents": [ "Let $S$ be a scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism", "of stacks in groupoids over $(\\Sch/S)_{fppf}$. If $\\mathcal{X}$ is an", "algebraic stack and $\\Delta : \\mathcal{Y} \\to \\mathcal{Y} \\times \\mathcal{Y}$", "is representable by algebraic spaces, then $F$ is algebraic." ], "refs": [], "proofs": [ { "contents": [ "Choose a representable stack in groupoids $\\mathcal{U}$ and a surjective", "smooth $1$-morphism $\\mathcal{U} \\to \\mathcal{X}$. Let $T$ be a scheme and", "let $\\xi$ be an object of $\\mathcal{Y}$ over $T$. The morphism of", "$2$-fibre products", "$$", "(\\Sch/T)_{fppf} \\times_{\\xi, \\mathcal{Y}} \\mathcal{U}", "\\longrightarrow", "(\\Sch/T)_{fppf} \\times_{\\xi, \\mathcal{Y}} \\mathcal{X}", "$$", "is representable by algebraic spaces, surjective, and smooth as a", "base change of $\\mathcal{U} \\to \\mathcal{X}$, see", "Algebraic Stacks,", "Lemmas \\ref{algebraic-lemma-base-change-representable-by-spaces} and", "\\ref{algebraic-lemma-base-change-representable-transformations-property}.", "By our condition on the diagonal of $\\mathcal{Y}$ we see that", "the source of this morphism is representable by an algebraic space, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-representable-diagonal}.", "Hence the target is an algebraic stack by", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-smooth-surjective-morphism-implies-algebraic}." ], "refs": [ "algebraic-lemma-base-change-representable-by-spaces", "algebraic-lemma-base-change-representable-transformations-property", "algebraic-lemma-representable-diagonal", "algebraic-lemma-smooth-surjective-morphism-implies-algebraic" ], "ref_ids": [ 8447, 8456, 8461, 8470 ] } ], "ref_ids": [] }, { "id": 3113, "type": "theorem", "label": "criteria-lemma-diagonals-and-algebraic-morphisms", "categories": [ "criteria" ], "title": "criteria-lemma-diagonals-and-algebraic-morphisms", "contents": [ "Let $S$ be a scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism", "of stacks in groupoids over $(\\Sch/S)_{fppf}$.", "If $F$ is algebraic and", "$\\Delta : \\mathcal{Y} \\to \\mathcal{Y} \\times \\mathcal{Y}$", "is representable by algebraic spaces, then", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$", "is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Assume $F$ is algebraic and", "$\\Delta : \\mathcal{Y} \\to \\mathcal{Y} \\times \\mathcal{Y}$", "is representable by algebraic spaces.", "Take a scheme $U$ over $S$ and two objects $x_1, x_2$ of", "$\\mathcal{X}$ over $U$.", "We have to show that $\\mathit{Isom}(x_1, x_2)$ is an algebraic space", "over $U$, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-representable-diagonal}.", "Set $y_i = F(x_i)$. We have a morphism of sheaves of sets", "$$", "f : \\mathit{Isom}(x_1, x_2) \\to \\mathit{Isom}(y_1, y_2)", "$$", "and the target is an algebraic space by assumption.", "Thus it suffices to show that $f$ is representable by", "algebraic spaces, see Bootstrap, Lemma", "\\ref{bootstrap-lemma-representable-by-spaces-over-space}.", "Thus we can choose a scheme $V$ over $U$ and an", "isomorphism $\\beta : y_{1, V} \\to y_{2, V}$ and", "we have to show the functor", "$$", "(\\Sch/V)_{fppf} \\to \\textit{Sets},\\quad", "T/V \\mapsto \\{\\alpha : x_{1, T} \\to x_{2, T}", "\\text{ in }\\mathcal{X}_T \\mid F(\\alpha) = \\beta|_T\\}", "$$", "is an algebraic space. Consider the objects", "$z_1 = (V, x_{1, V}, \\text{id})$ and ", "$z_2 = (V, x_{2, V}, \\beta)$ of", "$$", "\\mathcal{Z} = (\\Sch/V)_{fppf} \\times_{y_{1, V}, \\mathcal{Y}} \\mathcal{X}", "$$", "Then it is straightforward to verify that", "the functor above is equal to $\\mathit{Isom}(z_1, z_2)$", "on $(\\Sch/V)_{fppf}$. Hence this is an algebraic space", "by our assumption that $F$ is algebraic (and the definition", "of algebraic stacks)." ], "refs": [ "algebraic-lemma-representable-diagonal", "bootstrap-lemma-representable-by-spaces-over-space" ], "ref_ids": [ 8461, 2607 ] } ], "ref_ids": [] }, { "id": 3114, "type": "theorem", "label": "criteria-lemma-surjection-space-of-sections", "categories": [ "criteria" ], "title": "criteria-lemma-surjection-space-of-sections", "contents": [ "Let $Z \\to U$ be a finite morphism of schemes.", "Let $W$ be an algebraic space and let $W \\to Z$ be a", "surjective \\'etale morphism. Then there exists a surjective", "\\'etale morphism $U' \\to U$ and a section", "$$", "\\sigma : Z_{U'} \\to W_{U'}", "$$", "of the morphism $W_{U'} \\to Z_{U'}$." ], "refs": [], "proofs": [ { "contents": [ "We may choose a separated scheme $W'$ and a surjective \\'etale morphism", "$W' \\to W$. Hence after replacing $W$ by $W'$ we may assume that $W$", "is a separated scheme. Write $f : W \\to Z$ and $\\pi : Z \\to U$.", "Note that $f \\circ \\pi : W \\to U$ is separated as", "$W$ is separated (see", "Schemes, Lemma \\ref{schemes-lemma-compose-after-separated}).", "Let $u \\in U$ be a point. Clearly it suffices", "to find an \\'etale neighbourhood $(U', u')$ of $(U, u)$ such that", "a section $\\sigma$ exists over $U'$. Let $z_1, \\ldots, z_r$", "be the points of $Z$ lying above $u$. For each $i$ choose a point", "$w_i \\in W$ which maps to $z_i$. We may pick an \\'etale neighbourhood", "$(U', u') \\to (U, u)$ such that the conclusions of", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant}", "hold for both $Z \\to U$ and the points $z_1, \\ldots, z_r$", "and $W \\to U$ and the points $w_1, \\ldots, w_r$. Hence, after", "replacing $(U, u)$ by $(U', u')$ and relabeling, we may assume that", "all the field extensions $\\kappa(u) \\subset \\kappa(z_i)$ and", "$\\kappa(u) \\subset \\kappa(w_i)$ are purely inseparable, and moreover", "that there exist disjoint union decompositions", "$$", "Z = V_1 \\amalg \\ldots \\amalg V_r \\amalg A, \\quad", "W = W_1 \\amalg \\ldots \\amalg W_r \\amalg B", "$$", "by open and closed subschemes", "with $z_i \\in V_i$, $w_i \\in W_i$ and $V_i \\to U$, $W_i \\to U$ finite.", "After replacing $U$ by $U \\setminus \\pi(A)$ we may assume that", "$A = \\emptyset$, i.e., $Z = V_1 \\amalg \\ldots \\amalg V_r$.", "After replacing $W_i$ by $W_i \\cap f^{-1}(V_i)$ and", "$B$ by $B \\cup \\bigcup W_i \\cap f^{-1}(Z \\setminus V_i)$", "we may assume that $f$ maps $W_i$ into $V_i$.", "Then $f_i = f|_{W_i} : W_i \\to V_i$ is a morphism of schemes finite over $U$,", "hence finite (see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-permanence}).", "It is also \\'etale (by assumption),", "$f_i^{-1}(\\{z_i\\}) = w_i$, and induces an isomorphism of residue", "fields $\\kappa(z_i) = \\kappa(w_i)$ (because both are purely inseparable", "extensions of $\\kappa(u)$ and $\\kappa(z_i) \\subset \\kappa(w_i)$", "is separable as $f$ is \\'etale). Hence by", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-finite-etale-one-point}", "we see that $f_i$ is an isomorphism in a neighbourhood $V_i'$ of", "$z_i$. Since $\\pi : Z \\to U$ is closed, after shrinking $U$, we may assume", "that $W_i \\to V_i$ is an isomorphism. This proves the lemma." ], "refs": [ "schemes-lemma-compose-after-separated", "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant", "morphisms-lemma-finite-permanence", "etale-lemma-finite-etale-one-point" ], "ref_ids": [ 7715, 13896, 5448, 10707 ] } ], "ref_ids": [] }, { "id": 3115, "type": "theorem", "label": "criteria-lemma-space-of-sections", "categories": [ "criteria" ], "title": "criteria-lemma-space-of-sections", "contents": [ "Let $Z \\to U$ be a finite locally free morphism of schemes.", "Let $W$ be an algebraic space and let $W \\to Z$ be an \\'etale morphism.", "Then the functor", "$$", "F : (\\Sch/U)_{fppf}^{opp} \\longrightarrow \\textit{Sets},", "$$", "defined by the rule", "$$", "U' \\longmapsto", "F(U') =", "\\{\\sigma : Z_{U'} \\to W_{U'}\\text{ section of }W_{U'} \\to Z_{U'}\\}", "$$", "is an algebraic space and the morphism $F \\to U$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Assume first that $W \\to Z$ is also separated.", "Let $U'$ be a scheme over $U$ and let $\\sigma \\in F(U')$. By", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-section-immersion}", "the morphism $\\sigma$ is a closed immersion.", "Moreover, $\\sigma$ is \\'etale by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-permanence}.", "Hence $\\sigma$ is also an open immersion, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-etale-universally-injective-open}.", "In other words, $Z_\\sigma = \\sigma(Z_{U'}) \\subset W_{U'}$ is", "an open subspace such that the morphism $Z_\\sigma \\to Z_{U'}$", "is an isomorphism. In particular, the morphism $Z_\\sigma \\to U'$", "is finite. Hence we obtain a transformation of functors", "$$", "F \\longrightarrow (W/U)_{fin}, \\quad", "\\sigma \\longmapsto (U' \\to U, Z_\\sigma)", "$$", "where $(W/U)_{fin}$ is the finite part of the morphism $W \\to U$", "introduced in", "More on Groupoids in Spaces, Section", "\\ref{spaces-more-groupoids-section-finite}.", "It is clear that this transformation of functors is injective (since we can", "recover $\\sigma$ from $Z_\\sigma$ as the inverse of the isomorphism", "$Z_\\sigma \\to Z_{U'}$). By", "More on Groupoids in Spaces, Proposition", "\\ref{spaces-more-groupoids-proposition-finite-algebraic-space}", "we know that $(W/U)_{fin}$ is an algebraic space \\'etale over $U$.", "Hence to finish the proof in this case it suffices to show that", "$F \\to (W/U)_{fin}$ is representable and an open immersion.", "To see this suppose that we are given a morphism of schemes $U' \\to U$", "and an open subspace $Z' \\subset W_{U'}$ such that $Z' \\to U'$", "is finite. Then it suffices to show that there exists an", "open subscheme $U'' \\subset U'$ such that a morphism", "$T \\to U'$ factors through $U''$ if and only if $Z' \\times_{U'} T$", "maps isomorphically to $Z \\times_{U'} T$. This follows from", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-where-isomorphism}", "(here we use that $Z \\to B$ is flat and locally of finite presentation", "as well as finite).", "Hence we have proved the lemma in case $W \\to Z$ is separated", "as well as \\'etale.", "\\medskip\\noindent", "In the general case we choose a separated scheme $W'$ and a surjective", "\\'etale morphism $W' \\to W$. Note that the morphisms $W' \\to W$ and", "$W \\to Z$ are separated as their source is separated. Denote $F'$ the", "functor associated to $W' \\to Z \\to U$ as in the lemma. In the first", "paragraph of the proof we showed that $F'$ is representable by an", "algebraic space \\'etale over $U$. By", "Lemma \\ref{lemma-surjection-space-of-sections}", "the map of functors $F' \\to F$ is surjective for the \\'etale topology", "on $\\Sch/U$. Moreover, if $U'$ and $\\sigma : Z_{U'} \\to W_{U'}$", "define a point $\\xi \\in F(U')$, then the fibre product", "$$", "F'' = F' \\times_{F, \\xi} U'", "$$", "is the functor on $\\Sch/U'$ associated to the morphisms", "$$", "W'_{U'} \\times_{W_{U'}, \\sigma} Z_{U'} \\to Z_{U'} \\to U'.", "$$", "Since the first morphism is separated as a base change of a separated", "morphism, we see that $F''$ is an algebraic space \\'etale over $U'$", "by the result of the first paragraph. It follows that $F' \\to F$ is a", "surjective \\'etale transformation of functors, which is representable", "by algebraic spaces. Hence $F$ is an algebraic space by", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}.", "Since $F' \\to F$ is an \\'etale surjective morphism of algebraic spaces", "it follows that $F \\to U$ is \\'etale because $F' \\to U$ is \\'etale." ], "refs": [ "spaces-properties-lemma-etale-permanence", "spaces-morphisms-lemma-etale-universally-injective-open", "spaces-more-groupoids-proposition-finite-algebraic-space", "spaces-more-morphisms-lemma-where-isomorphism", "criteria-lemma-surjection-space-of-sections", "bootstrap-theorem-final-bootstrap" ], "ref_ids": [ 11859, 4973, 13215, 252, 3114, 2602 ] } ], "ref_ids": [] }, { "id": 3116, "type": "theorem", "label": "criteria-lemma-hom-functor-sheaf", "categories": [ "criteria" ], "title": "criteria-lemma-hom-functor-sheaf", "contents": [ "Let $S$ be a scheme. Let $Z \\to B$ and $X \\to B$ be morphisms of", "algebraic spaces over $S$. Then", "\\begin{enumerate}", "\\item $\\mathit{Mor}_B(Z, X)$ is a sheaf on", "$(\\Sch/S)_{fppf}$.", "\\item If $T$ is an algebraic space over $S$, then there is a", "canonical bijection", "$$", "\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\mathit{Mor}_B(Z, X))", "=", "\\{(a, b)\\text{ as in }(\\ref{equation-hom})\\}", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be an algebraic space over $S$. Let $\\{T_i \\to T\\}$ be an fppf", "covering of $T$ (as in", "Topologies on Spaces, Section \\ref{spaces-topologies-section-fppf}).", "Suppose that $(a_i, b_i) \\in \\mathit{Mor}_B(Z, X)(T_i)$ such", "that $(a_i, b_i)|_{T_i \\times_T T_j} = (a_j, b_j)|_{T_i \\times_T T_j}$", "for all $i, j$. Then by", "Descent on Spaces,", "Lemma \\ref{spaces-descent-lemma-fpqc-universal-effective-epimorphisms}", "there exists a unique morphism $a : T \\to B$ such that $a_i$ is the", "composition of $T_i \\to T$ and $a$. Then", "$\\{T_i \\times_{a_i, B} Z \\to T \\times_{a, B} Z\\}$ is an fppf covering", "too and the same lemma implies there exists a unique morphism", "$b : T \\times_{a, B} Z \\to T \\times_{a, B} X$ such that $b_i$ is the", "composition of $T_i \\times_{a_i, B} Z \\to T \\times_{a, B} Z$ and $b$. Hence", "$(a, b) \\in \\mathit{Mor}_B(Z, X)(T)$ restricts to $(a_i, b_i)$", "over $T_i$ for all $i$.", "\\medskip\\noindent", "Note that the result of the preceding paragraph in particular implies (1).", "\\medskip\\noindent", "Let $T$ be an algebraic space over $S$. In order to prove (2) we will", "construct mutually inverse maps between the displayed sets. In the", "following when we say ``pair'' we mean a pair $(a, b)$ fitting", "into (\\ref{equation-hom}).", "\\medskip\\noindent", "Let $v : T \\to \\mathit{Mor}_B(Z, X)$ be a natural transformation.", "Choose a scheme $U$ and a surjective \\'etale morphism $p : U \\to T$.", "Then $v(p) \\in \\mathit{Mor}_B(Z, X)(U)$ corresponds to a pair $(a_U, b_U)$", "over $U$. Let $R = U \\times_T U$ with projections $t, s : R \\to U$.", "As $v$ is a transformation of functors we see that the pullbacks of", "$(a_U, b_U)$ by $s$ and $t$ agree. Hence, since $\\{U \\to T\\}$ is an", "fppf covering, we may apply the result of the first paragraph that", "deduce that there exists a unique pair $(a, b)$ over $T$.", "\\medskip\\noindent", "Conversely, let $(a, b)$ be a pair over $T$.", "Let $U \\to T$, $R = U \\times_T U$, and $t, s : R \\to U$ be as", "above. Then the restriction $(a, b)|_U$ gives rise to a", "transformation of functors $v : h_U \\to \\mathit{Mor}_B(Z, X)$ by the", "Yoneda lemma", "(Categories, Lemma \\ref{categories-lemma-yoneda}).", "As the two pullbacks $s^*(a, b)|_U$ and $t^*(a, b)|_U$", "are equal, we see that $v$ coequalizes the two maps", "$h_t, h_s : h_R \\to h_U$. Since $T = U/R$ is the fppf quotient sheaf by", "Spaces, Lemma \\ref{spaces-lemma-space-presentation}", "and since $\\mathit{Mor}_B(Z, X)$ is an fppf sheaf by (1) we conclude", "that $v$ factors through a map $T \\to \\mathit{Mor}_B(Z, X)$.", "\\medskip\\noindent", "We omit the verification that the two constructions above are mutually", "inverse." ], "refs": [ "spaces-descent-lemma-fpqc-universal-effective-epimorphisms", "categories-lemma-yoneda", "spaces-lemma-space-presentation" ], "ref_ids": [ 9367, 12203, 8149 ] } ], "ref_ids": [] }, { "id": 3117, "type": "theorem", "label": "criteria-lemma-base-change-hom-functor", "categories": [ "criteria" ], "title": "criteria-lemma-base-change-hom-functor", "contents": [ "Let $S$ be a scheme. Let $Z \\to B$, $X \\to B$, and $B' \\to B$", "be morphisms of algebraic spaces over $S$. Set $Z' = B' \\times_B Z$", "and $X' = B' \\times_B X$. Then", "$$", "\\mathit{Mor}_{B'}(Z', X')", "=", "B' \\times_B \\mathit{Mor}_B(Z, X)", "$$", "in $\\Sh((\\Sch/S)_{fppf})$." ], "refs": [], "proofs": [ { "contents": [ "The equality as functors follows immediately from the definitions.", "The equality as sheaves follows from this because both sides are", "sheaves according to", "Lemma \\ref{lemma-hom-functor-sheaf}", "and the fact that a fibre product of sheaves is the same as the", "corresponding fibre product of pre-sheaves (i.e., functors)." ], "refs": [ "criteria-lemma-hom-functor-sheaf" ], "ref_ids": [ 3116 ] } ], "ref_ids": [] }, { "id": 3118, "type": "theorem", "label": "criteria-lemma-etale-covering-hom-functor", "categories": [ "criteria" ], "title": "criteria-lemma-etale-covering-hom-functor", "contents": [ "Let $S$ be a scheme. Let $Z \\to B$ and $X' \\to X \\to B$ be morphisms of", "algebraic spaces over $S$. Assume", "\\begin{enumerate}", "\\item $X' \\to X$ is \\'etale, and", "\\item $Z \\to B$ is finite locally free.", "\\end{enumerate}", "Then $\\mathit{Mor}_B(Z, X') \\to \\mathit{Mor}_B(Z, X)$ is representable", "by algebraic spaces and \\'etale. If $X' \\to X$ is also surjective,", "then $\\mathit{Mor}_B(Z, X') \\to \\mathit{Mor}_B(Z, X)$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $\\xi = (a, b)$ be an element of", "$\\mathit{Mor}_B(Z, X)(U)$. We have to prove that the functor", "$$", "h_U \\times_{\\xi, \\mathit{Mor}_B(Z, X)} \\mathit{Mor}_B(Z, X')", "$$", "is representable by an algebraic space \\'etale over $U$. Set", "$Z_U = U \\times_{a, B} Z$ and $W = Z_U \\times_{b, X} X'$.", "Then $W \\to Z_U \\to U$ is as in", "Lemma \\ref{lemma-space-of-sections}", "and the sheaf $F$ defined there is identified with the fibre product", "displayed above. Hence the first assertion of the lemma.", "The second assertion follows from this and", "Lemma \\ref{lemma-surjection-space-of-sections}", "which guarantees that $F \\to U$ is surjective in the situation above." ], "refs": [ "criteria-lemma-space-of-sections", "criteria-lemma-surjection-space-of-sections" ], "ref_ids": [ 3115, 3114 ] } ], "ref_ids": [] }, { "id": 3119, "type": "theorem", "label": "criteria-lemma-restriction-of-scalars-sheaf", "categories": [ "criteria" ], "title": "criteria-lemma-restriction-of-scalars-sheaf", "contents": [ "Let $S$ be a scheme. Let $X \\to Z \\to B$ be morphisms of", "algebraic spaces over $S$. Then", "\\begin{enumerate}", "\\item $\\text{Res}_{Z/B}(X)$ is a sheaf on", "$(\\Sch/S)_{fppf}$.", "\\item If $T$ is an algebraic space over $S$, then there is a", "canonical bijection", "$$", "\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\text{Res}_{Z/B}(X))", "=", "\\{(a, b)\\text{ as in }(\\ref{equation-pairs})\\}", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be an algebraic space over $S$. Let $\\{T_i \\to T\\}$ be an fppf", "covering of $T$ (as in", "Topologies on Spaces, Section \\ref{spaces-topologies-section-fppf}).", "Suppose that $(a_i, b_i) \\in \\text{Res}_{Z/B}(X)(T_i)$ such", "that $(a_i, b_i)|_{T_i \\times_T T_j} = (a_j, b_j)|_{T_i \\times_T T_j}$", "for all $i, j$. Then by", "Descent on Spaces,", "Lemma \\ref{spaces-descent-lemma-fpqc-universal-effective-epimorphisms}", "there exists a unique morphism $a : T \\to B$ such that $a_i$ is the", "composition of $T_i \\to T$ and $a$. Then", "$\\{T_i \\times_{a_i, B} Z \\to T \\times_{a, B} Z\\}$ is an fppf covering", "too and the same lemma implies there exists a unique morphism", "$b : T \\times_{a, B} Z \\to X$ such that $b_i$ is the composition", "of $T_i \\times_{a_i, B} Z \\to T \\times_{a, B} Z$ and $b$. Hence", "$(a, b) \\in \\text{Res}_{Z/B}(X)(T)$ restricts to $(a_i, b_i)$", "over $T_i$ for all $i$.", "\\medskip\\noindent", "Note that the result of the preceding paragraph in particular implies (1).", "\\medskip\\noindent", "Let $T$ be an algebraic space over $S$. In order to prove (2) we will", "construct mutually inverse maps between the displayed sets. In the", "following when we say ``pair'' we mean a pair $(a, b)$ fitting", "into (\\ref{equation-pairs}).", "\\medskip\\noindent", "Let $v : T \\to \\text{Res}_{Z/B}(X)$ be a natural transformation.", "Choose a scheme $U$ and a surjective \\'etale morphism $p : U \\to T$.", "Then $v(p) \\in \\text{Res}_{Z/B}(X)(U)$ corresponds to a pair $(a_U, b_U)$", "over $U$. Let $R = U \\times_T U$ with projections $t, s : R \\to U$.", "As $v$ is a transformation of functors we see that the pullbacks of", "$(a_U, b_U)$ by $s$ and $t$ agree. Hence, since $\\{U \\to T\\}$ is an", "fppf covering, we may apply the result of the first paragraph that", "deduce that there exists a unique pair $(a, b)$ over $T$.", "\\medskip\\noindent", "Conversely, let $(a, b)$ be a pair over $T$.", "Let $U \\to T$, $R = U \\times_T U$, and $t, s : R \\to U$ be as", "above. Then the restriction $(a, b)|_U$ gives rise to a", "transformation of functors $v : h_U \\to \\text{Res}_{Z/B}(X)$ by the", "Yoneda lemma", "(Categories, Lemma \\ref{categories-lemma-yoneda}).", "As the two pullbacks $s^*(a, b)|_U$ and $t^*(a, b)|_U$", "are equal, we see that $v$ coequalizes the two maps", "$h_t, h_s : h_R \\to h_U$. Since $T = U/R$ is the fppf quotient sheaf by", "Spaces, Lemma \\ref{spaces-lemma-space-presentation}", "and since $\\text{Res}_{Z/B}(X)$ is an fppf sheaf by (1) we conclude", "that $v$ factors through a map $T \\to \\text{Res}_{Z/B}(X)$.", "\\medskip\\noindent", "We omit the verification that the two constructions above are mutually", "inverse." ], "refs": [ "spaces-descent-lemma-fpqc-universal-effective-epimorphisms", "categories-lemma-yoneda", "spaces-lemma-space-presentation" ], "ref_ids": [ 9367, 12203, 8149 ] } ], "ref_ids": [] }, { "id": 3120, "type": "theorem", "label": "criteria-lemma-etale-base-change-restriction-of-scalars", "categories": [ "criteria" ], "title": "criteria-lemma-etale-base-change-restriction-of-scalars", "contents": [ "Let $S$ be a scheme. Let $X \\to Z \\to B$ and $B' \\to B$", "be morphisms of algebraic spaces over $S$.", "Set $Z' = B' \\times_B Z$ and $X' = B' \\times_B X$. Then", "$$", "\\text{Res}_{Z'/B'}(X')", "=", "B' \\times_B \\text{Res}_{Z/B}(X)", "$$", "in $\\Sh((\\Sch/S)_{fppf})$." ], "refs": [], "proofs": [ { "contents": [ "The equality as functors follows immediately from the definitions.", "The equality as sheaves follows from this because both sides are", "sheaves according to", "Lemma \\ref{lemma-restriction-of-scalars-sheaf}", "and the fact that a fibre product of sheaves is the same as the", "corresponding fibre product of pre-sheaves (i.e., functors)." ], "refs": [ "criteria-lemma-restriction-of-scalars-sheaf" ], "ref_ids": [ 3119 ] } ], "ref_ids": [] }, { "id": 3121, "type": "theorem", "label": "criteria-lemma-etale-covering-restriction-of-scalars", "categories": [ "criteria" ], "title": "criteria-lemma-etale-covering-restriction-of-scalars", "contents": [ "Let $S$ be a scheme. Let $X' \\to X \\to Z \\to B$ be morphisms of", "algebraic spaces over $S$. Assume", "\\begin{enumerate}", "\\item $X' \\to X$ is \\'etale, and", "\\item $Z \\to B$ is finite locally free.", "\\end{enumerate}", "Then $\\text{Res}_{Z/B}(X') \\to \\text{Res}_{Z/B}(X)$ is representable", "by algebraic spaces and \\'etale. If $X' \\to X$ is also surjective,", "then $\\text{Res}_{Z/B}(X') \\to \\text{Res}_{Z/B}(X)$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $\\xi = (a, b)$ be an element of", "$\\text{Res}_{Z/B}(X)(U)$. We have to prove that the functor", "$$", "h_U \\times_{\\xi, \\text{Res}_{Z/B}(X)} \\text{Res}_{Z/B}(X')", "$$", "is representable by an algebraic space \\'etale over $U$. Set", "$Z_U = U \\times_{a, B} Z$ and $W = Z_U \\times_{b, X} X'$.", "Then $W \\to Z_U \\to U$ is as in", "Lemma \\ref{lemma-space-of-sections}", "and the sheaf $F$ defined there is identified with the fibre product", "displayed above. Hence the first assertion of the lemma.", "The second assertion follows from this and", "Lemma \\ref{lemma-surjection-space-of-sections}", "which guarantees that $F \\to U$ is surjective in the situation above." ], "refs": [ "criteria-lemma-space-of-sections", "criteria-lemma-surjection-space-of-sections" ], "ref_ids": [ 3115, 3114 ] } ], "ref_ids": [] }, { "id": 3122, "type": "theorem", "label": "criteria-lemma-fibre-diagram", "categories": [ "criteria" ], "title": "criteria-lemma-fibre-diagram", "contents": [ "Let $S$ be a scheme. Let $X \\to Z \\to B$ be morphisms of", "algebraic spaces over $S$. The following diagram", "$$", "\\xymatrix{", "\\mathit{Mor}_B(Z, X) \\ar[r] & \\mathit{Mor}_B(Z, Z) \\\\", "\\text{Res}_{Z/B}(X) \\ar[r] \\ar[u] & B \\ar[u]_{\\text{id}_Z}", "}", "$$", "is a cartesian diagram of sheaves on $(\\Sch/S)_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Exercise in the functorial point of view in algebraic", "geometry." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3123, "type": "theorem", "label": "criteria-lemma-map-hilbert", "categories": [ "criteria" ], "title": "criteria-lemma-map-hilbert", "contents": [ "Consider a $2$-commutative diagram", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[r]_G \\ar[d]_{F'} & \\mathcal{X} \\ar[d]^F \\\\", "\\mathcal{Y}' \\ar[r]^H & \\mathcal{Y}", "}", "$$", "of stacks in groupoids over $(\\Sch/S)_{fppf}$ with a given", "$2$-isomorphism $\\gamma : H \\circ F' \\to F \\circ G$. In this situation we", "obtain a canonical $1$-morphism", "$\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}') \\to", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$.", "This morphism is compatible with the forgetful $1$-morphisms of", "Examples of Stacks,", "Equation (\\ref{examples-stacks-equation-diagram-hilbert-d-stack})." ], "refs": [], "proofs": [ { "contents": [ "We map the object $(U, Z, y', x', \\alpha')$ to the object", "$(U, Z, H(y'), G(x'), \\gamma \\star \\text{id}_H \\star \\alpha')$", "where $\\star$ denotes horizontal composition of $2$-morphisms, see", "Categories, Definition \\ref{categories-definition-horizontal-composition}.", "To a morphism", "$(f, g, b, a) :", "(U_1, Z_1, y_1', x_1', \\alpha_1') \\to (U_2, Z_2, y_2', x_2', \\alpha_2')$", "we assign", "$(f, g, H(b), G(a))$.", "We omit the verification that this defines a functor between categories over", "$(\\Sch/S)_{fppf}$." ], "refs": [ "categories-definition-horizontal-composition" ], "ref_ids": [ 12377 ] } ], "ref_ids": [] }, { "id": 3124, "type": "theorem", "label": "criteria-lemma-cartesian-map-hilbert", "categories": [ "criteria" ], "title": "criteria-lemma-cartesian-map-hilbert", "contents": [ "In the situation of", "Lemma \\ref{lemma-map-hilbert}", "assume that the given square is $2$-cartesian. Then the diagram", "$$", "\\xymatrix{", "\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}') \\ar[r] \\ar[d] &", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y}) \\ar[d] \\\\", "\\mathcal{Y}' \\ar[r] &", "\\mathcal{Y}", "}", "$$", "is $2$-cartesian." ], "refs": [ "criteria-lemma-map-hilbert" ], "proofs": [ { "contents": [ "We get a $2$-commutative diagram by", "Lemma \\ref{lemma-map-hilbert}", "and hence we get a $1$-morphism (i.e., a functor)", "$$", "\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}')", "\\longrightarrow", "\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})", "$$", "We indicate why this functor is essentially surjective. Namely, an object", "of the category on the right hand side is given by a scheme $U$ over $S$,", "an object $y'$ of $\\mathcal{Y}'_U$, an object $(U, Z, y, x, \\alpha)$", "of $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ over $U$ and an isomorphism", "$H(y') \\to y$ in $\\mathcal{Y}_U$. The assumption means exactly that", "there exists an object $x'$ of $\\mathcal{X}'_Z$ such that there exist", "isomorphisms $G(x') \\cong x$ and $\\alpha' : y'|_Z \\to F'(x')$ compatible", "with $\\alpha$. Then we see that $(U, Z, y', x', \\alpha')$ is an", "object of $\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}')$ over $U$.", "Details omitted." ], "refs": [ "criteria-lemma-map-hilbert" ], "ref_ids": [ 3123 ] } ], "ref_ids": [ 3123 ] }, { "id": 3125, "type": "theorem", "label": "criteria-lemma-etale-covering-hilbert", "categories": [ "criteria" ], "title": "criteria-lemma-etale-covering-hilbert", "contents": [ "In the situation of", "Lemma \\ref{lemma-map-hilbert}", "assume", "\\begin{enumerate}", "\\item $\\mathcal{Y}' = \\mathcal{Y}$ and $H = \\text{id}_\\mathcal{Y}$,", "\\item $G$ is representable by algebraic spaces and \\'etale.", "\\end{enumerate}", "Then $\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}) \\to", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is representable by", "algebraic spaces and \\'etale.", "If $G$ is also surjective, then", "$\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}) \\to", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is surjective." ], "refs": [ "criteria-lemma-map-hilbert" ], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $\\xi = (U, Z, y, x, \\alpha)$ be an object of", "$\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ over $U$.", "We have to prove that the $2$-fibre product", "\\begin{equation}", "\\label{equation-to-show}", "(\\Sch/U)_{fppf}", "\\times_{\\xi, \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})}", "\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y})", "\\end{equation}", "is representable by an algebraic space \\'etale over $U$.", "An object of this over $U'$ corresponds to an object", "$x'$ in the fibre category of $\\mathcal{X}'$ over $Z_{U'}$", "such that $G(x') \\cong x|_{Z_{U'}}$.", "By assumption the $2$-fibre product", "$$", "(\\Sch/Z)_{fppf} \\times_{x, \\mathcal{X}} \\mathcal{X}'", "$$", "is representable by an algebraic space $W$ such that the projection", "$W \\to Z$ is \\'etale. Then (\\ref{equation-to-show})", "is representable by the algebraic space $F$ parametrizing sections of", "$W \\to Z$ over $U$ introduced in", "Lemma \\ref{lemma-space-of-sections}.", "Since $F \\to U$ is \\'etale we conclude that", "$\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}) \\to", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is representable by", "algebraic spaces and \\'etale.", "Finally, if $\\mathcal{X}' \\to \\mathcal{X}$ is surjective also,", "then $W \\to Z$ is surjective, and hence $F \\to U$ is surjective by", "Lemma \\ref{lemma-surjection-space-of-sections}.", "Thus in this case", "$\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}) \\to", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is also surjective." ], "refs": [ "criteria-lemma-space-of-sections", "criteria-lemma-surjection-space-of-sections" ], "ref_ids": [ 3115, 3114 ] } ], "ref_ids": [ 3123 ] }, { "id": 3126, "type": "theorem", "label": "criteria-lemma-etale-map-hilbert", "categories": [ "criteria" ], "title": "criteria-lemma-etale-map-hilbert", "contents": [ "In the situation of", "Lemma \\ref{lemma-map-hilbert}.", "Assume that $G$, $H$ are representable by algebraic spaces and \\'etale.", "Then $\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}') \\to", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is representable by", "algebraic spaces and \\'etale.", "If also $H$ is surjective and the induced functor", "$\\mathcal{X}' \\to \\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X}$", "is surjective, then", "$\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}') \\to", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is surjective." ], "refs": [ "criteria-lemma-map-hilbert" ], "proofs": [ { "contents": [ "Set $\\mathcal{X}'' = \\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X}$. By", "Lemma \\ref{lemma-etale-permanence}", "the $1$-morphism $\\mathcal{X}' \\to \\mathcal{X}''$ is representable by", "algebraic spaces and \\'etale (in particular the condition in the second", "statement of the lemma that $\\mathcal{X}' \\to \\mathcal{X}''$ be surjective", "makes sense). We obtain a $2$-commutative diagram", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[r] \\ar[d] &", "\\mathcal{X}'' \\ar[r] \\ar[d] &", "\\mathcal{X} \\ar[d] \\\\", "\\mathcal{Y}' \\ar[r] &", "\\mathcal{Y}' \\ar[r] &", "\\mathcal{Y}", "}", "$$", "It follows from", "Lemma \\ref{lemma-cartesian-map-hilbert}", "that $\\mathcal{H}_d(\\mathcal{X}''/\\mathcal{Y}')$ is the base change", "of $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ by $\\mathcal{Y}' \\to \\mathcal{Y}$.", "In particular we see that", "$\\mathcal{H}_d(\\mathcal{X}''/\\mathcal{Y}') \\to", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is", "representable by algebraic spaces and \\'etale, see", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-base-change-representable-transformations-property}.", "Moreover, it is also surjective if $H$ is.", " Hence if we can show that", "the result holds for the left square in the diagram, then we're done.", "In this way we reduce to the case where $\\mathcal{Y}' = \\mathcal{Y}$", "which is the content of", "Lemma \\ref{lemma-etale-covering-hilbert}." ], "refs": [ "criteria-lemma-etale-permanence", "criteria-lemma-cartesian-map-hilbert", "algebraic-lemma-base-change-representable-transformations-property", "criteria-lemma-etale-covering-hilbert" ], "ref_ids": [ 3095, 3124, 8456, 3125 ] } ], "ref_ids": [ 3123 ] }, { "id": 3127, "type": "theorem", "label": "criteria-lemma-relative-hilbert", "categories": [ "criteria" ], "title": "criteria-lemma-relative-hilbert", "contents": [ "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids", "over $(\\Sch/S)_{fppf}$. Assume that", "$\\Delta : \\mathcal{Y} \\to \\mathcal{Y} \\times \\mathcal{Y}$", "is representable by algebraic spaces. Then", "$$", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})", "\\longrightarrow", "\\mathcal{H}_d(\\mathcal{X}) \\times \\mathcal{Y}", "$$", "see", "Examples of Stacks, Equation", "(\\ref{examples-stacks-equation-diagram-hilbert-d-stack})", "is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $\\xi = (U, Z, p, x, 1)$ be an object of", "$\\mathcal{H}_d(\\mathcal{X}) = \\mathcal{H}_d(\\mathcal{X}/S)$ over $U$.", "Here $p$ is just the structure morphism of $U$.", "The fifth component $1$ exists and is unique", "since everything is over $S$.", "Also, let $y$ be an object of $\\mathcal{Y}$ over $U$.", "We have to show the $2$-fibre product", "\\begin{equation}", "\\label{equation-res-isom}", "(\\Sch/U)_{fppf}", "\\times_{\\xi \\times y, \\mathcal{H}_d(\\mathcal{X}) \\times \\mathcal{Y}}", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})", "\\end{equation}", "is representable by an algebraic space. To explain why this is so", "we introduce", "$$", "I = \\mathit{Isom}_\\mathcal{Y}(y|_Z, F(x))", "$$", "which is an algebraic space over $Z$ by assumption. Let $a : U' \\to U$", "be a scheme over $U$. What does it mean to give an object of the fibre", "category of (\\ref{equation-res-isom}) over $U'$? Well, it means that we", "have an object $\\xi' = (U', Z', y', x', \\alpha')$ of", "$\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ over $U'$ and isomorphisms", "$(U', Z', p', x', 1) \\cong (U, Z, p, x, 1)|_{U'}$ and", "$y' \\cong y|_{U'}$. Thus $\\xi'$ is isomorphic to", "$(U', U' \\times_{a, U} Z, a^*y, x|_{U' \\times_{a, U} Z}, \\alpha)$", "for some morphism", "$$", "\\alpha :", "a^*y|_{U' \\times_{a, U} Z}", "\\longrightarrow", "F(x|_{U' \\times_{a, U} Z})", "$$", "in the fibre category of $\\mathcal{Y}$ over $U' \\times_{a, U} Z$. Hence", "we can view $\\alpha$ as a morphism $b : U' \\times_{a, U} Z \\to I$.", "In this way we see that (\\ref{equation-res-isom})", "is representable by $\\text{Res}_{Z/U}(I)$ which is an algebraic space by", "Proposition \\ref{proposition-restriction-of-scalars-algebraic-space}." ], "refs": [ "criteria-proposition-restriction-of-scalars-algebraic-space" ], "ref_ids": [ 3143 ] } ], "ref_ids": [] }, { "id": 3128, "type": "theorem", "label": "criteria-lemma-representable-on-top", "categories": [ "criteria" ], "title": "criteria-lemma-representable-on-top", "contents": [ "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ and $G : \\mathcal{X}' \\to \\mathcal{X}$", "be $1$-morphisms of stacks in groupoids over $(\\Sch/S)_{fppf}$.", "If $G$ is representable by algebraic spaces, then the $1$-morphism", "$$", "\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y})", "\\longrightarrow", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})", "$$", "is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $\\xi = (U, Z, y, x, \\alpha)$ be an object of", "$\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ over $U$.", "We have to prove that the $2$-fibre product", "\\begin{equation}", "\\label{equation-to-show-again}", "(\\Sch/U)_{fppf}", "\\times_{\\xi, \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})}", "\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y})", "\\end{equation}", "is representable by an algebraic space \\'etale over $U$.", "An object of this over $a : U' \\to U$ corresponds to an object", "$x'$ of $\\mathcal{X}'$ over $U' \\times_{a, U} Z$ such that", "$G(x') \\cong x|_{U' \\times_{a, U} Z}$. By assumption the $2$-fibre product", "$$", "(\\Sch/Z)_{fppf} \\times_{x, \\mathcal{X}} \\mathcal{X}'", "$$", "is representable by an algebraic space $X$ over $Z$. It follows that", "(\\ref{equation-to-show-again}) is representable by $\\text{Res}_{Z/U}(X)$,", "which is an algebraic space by", "Proposition \\ref{proposition-restriction-of-scalars-algebraic-space}." ], "refs": [ "criteria-proposition-restriction-of-scalars-algebraic-space" ], "ref_ids": [ 3143 ] } ], "ref_ids": [] }, { "id": 3129, "type": "theorem", "label": "criteria-lemma-limit-preserving", "categories": [ "criteria" ], "title": "criteria-lemma-limit-preserving", "contents": [ "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids", "over $(\\Sch/S)_{fppf}$. Assume $F$ is representable by algebraic", "spaces and locally of finite presentation. Then", "$$", "p : \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y}) \\to \\mathcal{Y}", "$$", "is limit preserving on objects." ], "refs": [], "proofs": [ { "contents": [ "This means we have to show the following: Given", "\\begin{enumerate}", "\\item an affine scheme $U = \\lim_i U_i$ which is written as the", "directed limit of affine schemes $U_i$ over $S$,", "\\item an object $y_i$ of $\\mathcal{Y}$ over $U_i$ for some $i$, and", "\\item an object $\\Xi = (U, Z, y, x, \\alpha)$ of", "$\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$", "over $U$ such that $y = y_i|_U$,", "\\end{enumerate}", "then there exists an $i' \\geq i$ and an object", "$\\Xi_{i'} = (U_{i'}, Z_{i'}, y_{i'}, x_{i'}, \\alpha_{i'})$ of", "$\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ over $U_{i'}$ with", "$\\Xi_{i'}|_U = \\Xi$ and $y_{i'} = y_i|_{U_{i'}}$.", "Namely, the last two equalities will take care of the commutativity of", "(\\ref{equation-limit-preserving}).", "\\medskip\\noindent", "Let $X_{y_i} \\to U_i$ be an algebraic space representing the $2$-fibre", "product", "$$", "(\\Sch/U_i)_{fppf} \\times_{y_i, \\mathcal{Y}, F} \\mathcal{X}.", "$$", "Note that $X_{y_i} \\to U_i$ is locally of finite presentation by our", "assumption on $F$. Write $\\Xi $. It is clear that", "$\\xi = (Z, Z \\to U_i, x, \\alpha)$ is an object of the $2$-fibre product", "displayed above, hence $\\xi$ gives rise to a morphism", "$f_\\xi : Z \\to X_{y_i}$ of algebraic spaces over $U_i$", "(since $X_{y_i}$ is the functor of isomorphisms classes of objects of", "$(\\Sch/U_i)_{fppf} \\times_{y, \\mathcal{Y}, F} \\mathcal{X}$, see", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-characterize-representable-by-space}).", "By", "Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation} and", "\\ref{limits-lemma-descend-finite-locally-free}", "there exists an $i' \\geq i$ and a finite locally free morphism", "$Z_{i'} \\to U_{i'}$ of degree $d$ whose base change to $U$ is $Z$. By", "Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-characterize-locally-finite-presentation}", "we may, after replacing $i'$ by a bigger index, assume there exists a", "morphism $f_{i'} : Z_{i'} \\to X_{y_i}$ such that", "$$", "\\xymatrix{", "Z \\ar[d] \\ar[r] \\ar@/^3ex/[rr]^{f_\\xi} &", "Z_{i'} \\ar[d] \\ar[r]_{f_{i'}} & X_{y_i} \\ar[d] \\\\", "U \\ar[r] & U_{i'} \\ar[r] & U_i", "}", "$$", "is commutative. We set", "$\\Xi_{i'} = (U_{i'}, Z_{i'}, y_{i'}, x_{i'}, \\alpha_{i'})$", "where", "\\begin{enumerate}", "\\item $y_{i'}$ is the object of $\\mathcal{Y}$ over $U_{i'}$", "which is the pullback of $y_i$ to $U_{i'}$,", "\\item $x_{i'}$ is the object of $\\mathcal{X}$ over $Z_{i'}$ corresponding", "via the $2$-Yoneda lemma to the $1$-morphism", "$$", "(\\Sch/Z_{i'})_{fppf} \\to", "\\mathcal{S}_{X_{y_i}} \\to", "(\\Sch/U_i)_{fppf} \\times_{y_i, \\mathcal{Y}, F} \\mathcal{X} \\to", "\\mathcal{X}", "$$", "where the middle arrow is the equivalence which defines $X_{y_i}$", "(notation as in", "Algebraic Stacks, Sections", "\\ref{algebraic-section-representable-by-algebraic-spaces} and", "\\ref{algebraic-section-split}).", "\\item $\\alpha_{i'} : y_{i'}|_{Z_{i'}} \\to F(x_{i'})$ is the isomorphism", "coming from the $2$-commutativity of the diagram", "$$", "\\xymatrix{", "(\\Sch/Z_{i'})_{fppf} \\ar[r] \\ar[rd] &", "(\\Sch/U_i)_{fppf} \\times_{y_i, \\mathcal{Y}, F} \\mathcal{X}", "\\ar[r] \\ar[d] &", "\\mathcal{X} \\ar[d]^F \\\\", "& (\\Sch/U_{i'})_{fppf} \\ar[r] & \\mathcal{Y}", "}", "$$", "\\end{enumerate}", "Recall that $f_\\xi : Z \\to X_{y_i}$ was the morphism corresponding to", "the object $\\xi = (Z, Z \\to U_i, x, \\alpha)$ of", "$(\\Sch/U_i)_{fppf} \\times_{y_i, \\mathcal{Y}, F} \\mathcal{X}$", "over $Z$. By construction $f_{i'}$ is the morphism corresponding to", "the object $\\xi_{i'} = (Z_{i'}, Z_{i'} \\to U_i, x_{i'}, \\alpha_{i'})$.", "As $f_\\xi = f_{i'} \\circ (Z \\to Z_{i'})$ we see that", "the object $\\xi_{i'} = (Z_{i'}, Z_{i'} \\to U_i, x_{i'}, \\alpha_{i'})$ pulls", "back to $\\xi$ over $Z$. Thus $x_{i'}$ pulls back to $x$ and $\\alpha_{i'}$", "pulls back to $\\alpha$. This means that $\\Xi_{i'}$ pulls back", "to $\\Xi$ over $U$ and we win." ], "refs": [ "algebraic-lemma-characterize-representable-by-space", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-finite-locally-free", "spaces-limits-proposition-characterize-locally-finite-presentation" ], "ref_ids": [ 8441, 15077, 15063, 4655 ] } ], "ref_ids": [] }, { "id": 3130, "type": "theorem", "label": "criteria-lemma-represent-FAd", "categories": [ "criteria" ], "title": "criteria-lemma-represent-FAd", "contents": [ "The functor in groupoids $FA_d$ defined in (\\ref{equation-FAd})", "is isomorphic (!) to the functor in groupoids which associates", "to a scheme $T$ the category with", "\\begin{enumerate}", "\\item set of objects is $X(T)$,", "\\item set of morphisms is $G(T) \\times X(T)$,", "\\item $s : G(T) \\times X(T) \\to X(T)$ is the projection map,", "\\item $t : G(T) \\times X(T) \\to X(T)$ is $a(T)$, and", "\\item composition $G(T) \\times X(T) \\times_{s, X(T), t} G(T) \\times X(T)", "\\to G(T) \\times X(T)$ is given by $((g, m), (g', m')) \\mapsto (gg', m')$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have seen the rule on objects in (\\ref{equation-objects}).", "We have also seen above that $g \\in G(T)$ can be viewed as", "a morphism from $m$ to $a(g, m)$ for any free $d$-dimensional algebra $m$.", "Conversely, any morphism $m \\to m'$ is given by an invertible linear", "map $\\varphi$ which corresponds to an element $g \\in G(T)$ such", "that $m' = a(g, m)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3131, "type": "theorem", "label": "criteria-lemma-hilbert-stack-of-space", "categories": [ "criteria" ], "title": "criteria-lemma-hilbert-stack-of-space", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Then $\\mathcal{H}_d(X)$ is an algebraic stack." ], "refs": [], "proofs": [ { "contents": [ "The $1$-morphism", "$$", "\\mathcal{H}_d(X) \\longrightarrow \\mathcal{H}_d", "$$", "is representable by algebraic spaces according to", "Lemma \\ref{lemma-representable-on-top}.", "The stack $\\mathcal{H}_d$ is an algebraic stack according to", "Proposition \\ref{proposition-finite-hilbert-point}.", "Hence $\\mathcal{H}_d(X)$ is an algebraic stack by", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-representable-morphism-to-algebraic}." ], "refs": [ "criteria-lemma-representable-on-top", "criteria-proposition-finite-hilbert-point", "algebraic-lemma-representable-morphism-to-algebraic" ], "ref_ids": [ 3128, 3144, 8471 ] } ], "ref_ids": [] }, { "id": 3132, "type": "theorem", "label": "criteria-lemma-hilbert-stack-relative-space", "categories": [ "criteria" ], "title": "criteria-lemma-hilbert-stack-relative-space", "contents": [ "Let $S$ be a scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism", "of stacks in groupoids over $(\\Sch/S)_{fppf}$ such that", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is representable by an algebraic space, and", "\\item $F$ is representable by algebraic spaces, surjective, flat, and", "locally of finite presentation.", "\\end{enumerate}", "Then $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is an algebraic stack." ], "refs": [], "proofs": [ { "contents": [ "Choose a representable stack in groupoids $\\mathcal{U}$ over $S$ and a", "$1$-morphism $f : \\mathcal{U} \\to \\mathcal{H}_d(\\mathcal{X})$", "which is representable by algebraic spaces, smooth, and surjective.", "This is possible because $\\mathcal{H}_d(\\mathcal{X})$ is an algebraic stack by", "Lemma \\ref{lemma-hilbert-stack-of-space}.", "Consider the $2$-fibre product", "$$", "\\mathcal{W} =", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})", "\\times_{\\mathcal{H}_d(\\mathcal{X}), f}", "\\mathcal{U}.", "$$", "Since $\\mathcal{U}$ is representable (in particular a stack in setoids)", "it follows from", "Examples of Stacks, Lemma \\ref{examples-stacks-lemma-faithful-hilbert}", "and", "Stacks, Lemma \\ref{stacks-lemma-2-fibre-product-gives-stack-in-setoids}", "that $\\mathcal{W}$ is a stack in setoids. The $1$-morphism", "$\\mathcal{W} \\to \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is", "representable by algebraic spaces, smooth, and surjective as a base", "change of the morphism $f$ (see", "Algebraic Stacks,", "Lemmas \\ref{algebraic-lemma-base-change-representable-by-spaces} and", "\\ref{algebraic-lemma-base-change-representable-transformations-property}).", "Thus, if we can show that $\\mathcal{W}$ is representable by an algebraic space,", "then the lemma follows from", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-smooth-surjective-morphism-implies-algebraic}.", "\\medskip\\noindent", "The diagonal of $\\mathcal{Y}$ is representable by algebraic spaces according to", "Lemma \\ref{lemma-flat-finite-presentation-surjective-diagonal}.", "We may apply", "Lemma \\ref{lemma-relative-hilbert}", "to see that the $1$-morphism", "$$", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})", "\\longrightarrow", "\\mathcal{H}_d(\\mathcal{X}) \\times \\mathcal{Y}", "$$", "is representable by algebraic spaces. Consider the $2$-fibre product", "$$", "\\mathcal{V} =", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})", "\\times_{(\\mathcal{H}_d(\\mathcal{X}) \\times \\mathcal{Y}), f \\times F}", "(\\mathcal{U} \\times \\mathcal{X}).", "$$", "The projection morphism $\\mathcal{V} \\to \\mathcal{U} \\times \\mathcal{X}$", "is representable by algebraic spaces as a base change of the last", "displayed morphism. Hence $\\mathcal{V}$ is an algebraic space (see", "Bootstrap, Lemma \\ref{bootstrap-lemma-representable-by-spaces-over-space}", "or", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-base-change-by-space-representable-by-space}).", "The $1$-morphism $\\mathcal{V} \\to \\mathcal{U}$ fits into the following", "$2$-cartesian diagram", "$$", "\\xymatrix{", "\\mathcal{V} \\ar[d] \\ar[r] & \\mathcal{X} \\ar[d]^F \\\\", "\\mathcal{W} \\ar[r] & \\mathcal{Y}", "}", "$$", "because", "$$", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})", "\\times_{(\\mathcal{H}_d(\\mathcal{X}) \\times \\mathcal{Y}), f \\times F}", "(\\mathcal{U} \\times \\mathcal{X})", "=", "(\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})", "\\times_{\\mathcal{H}_d(\\mathcal{X}), f}", "\\mathcal{U}) \\times_{\\mathcal{Y}, F} \\mathcal{X}.", "$$", "Hence $\\mathcal{V} \\to \\mathcal{W}$ is representable by algebraic spaces,", "surjective, flat, and locally of finite presentation as a base change", "of $F$. It follows that the same thing is true for the corresponding", "sheaves of sets associated to $\\mathcal{V}$ and $\\mathcal{W}$, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-fibred-setoids-property}.", "Thus we conclude that the sheaf associated to $\\mathcal{W}$ is an", "algebraic space by", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}." ], "refs": [ "criteria-lemma-hilbert-stack-of-space", "examples-stacks-lemma-faithful-hilbert", "stacks-lemma-2-fibre-product-gives-stack-in-setoids", "algebraic-lemma-base-change-representable-by-spaces", "algebraic-lemma-base-change-representable-transformations-property", "algebraic-lemma-smooth-surjective-morphism-implies-algebraic", "criteria-lemma-flat-finite-presentation-surjective-diagonal", "criteria-lemma-relative-hilbert", "bootstrap-lemma-representable-by-spaces-over-space", "algebraic-lemma-base-change-by-space-representable-by-space", "algebraic-lemma-map-fibred-setoids-property", "bootstrap-theorem-final-bootstrap" ], "ref_ids": [ 3131, 9182, 8954, 8447, 8456, 8470, 3097, 3127, 2607, 8448, 8454, 2602 ] } ], "ref_ids": [] }, { "id": 3133, "type": "theorem", "label": "criteria-lemma-lci-locus-stack-in-groupoids", "categories": [ "criteria" ], "title": "criteria-lemma-lci-locus-stack-in-groupoids", "contents": [ "Let $S$ be a scheme. Fix a $1$-morphism", "$F : \\mathcal{X} \\longrightarrow \\mathcal{Y}$", "of stacks in groupoids over $(\\Sch/S)_{fppf}$.", "Assume $F$ is representable by algebraic spaces, flat, and locally", "of finite presentation. Then $\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})$", "is a stack in groupoids and the inclusion functor", "$$", "\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})", "\\longrightarrow", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})", "$$", "is representable and an open immersion." ], "refs": [], "proofs": [ { "contents": [ "Let $\\Xi = (U, Z, y, x, \\alpha)$ be an object of $\\mathcal{H}_d$. It follows", "from the remark following", "(\\ref{equation-relative-map})", "that the pullback of $\\Xi$ by $U' \\to U$ belongs to", "$\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})$ if and only if the base", "change of $x_\\alpha$ is unramified and a local complete intersection morphism.", "Note that $Z \\to U$ is finite locally free (hence flat, locally of", "finite presentation and universally closed) and that $X_y \\to U$ is", "flat and locally of finite presentation by our assumption on $F$. Then", "More on Morphisms of Spaces, Lemmas", "\\ref{spaces-more-morphisms-lemma-where-unramified} and", "\\ref{spaces-more-morphisms-lemma-where-lci}", "imply exists an open subscheme $W \\subset U$ such that a morphism", "$U' \\to U$ factors through $W$ if and only if the base change of", "$x_\\alpha$ via $U' \\to U$ is unramified and a local complete intersection", "morphism. This implies that", "$$", "(\\Sch/U)_{fppf}", "\\times_{\\Xi, \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})}", "\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})", "$$", "is representable by $W$. Hence the final statement of the lemma", "holds. The first statement (that", "$\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})$ is a stack in groupoids)", "follows from this and", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-open-fibred-category-is-algebraic}." ], "refs": [ "spaces-more-morphisms-lemma-where-unramified", "spaces-more-morphisms-lemma-where-lci", "algebraic-lemma-open-fibred-category-is-algebraic" ], "ref_ids": [ 247, 253, 8472 ] } ], "ref_ids": [] }, { "id": 3134, "type": "theorem", "label": "criteria-lemma-lci-unobstructed", "categories": [ "criteria" ], "title": "criteria-lemma-lci-unobstructed", "contents": [ "Let $U \\subset U'$ be a first order thickening of affine schemes.", "Let $X'$ be an algebraic space flat over $U'$. Set $X = U \\times_{U'} X'$.", "Let $Z \\to U$ be finite locally free of degree $d$. Finally, let", "$f : Z \\to X$ be unramified and a local complete intersection morphism.", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "(Z \\subset Z') \\ar[rd] \\ar[rr]_{(f, f')} & & (X \\subset X') \\ar[ld] \\\\", "& (U \\subset U')", "}", "$$", "of algebraic spaces over $U'$ such that $Z' \\to U'$ is finite locally free", "of degree $d$ and $Z = U \\times_{U'} Z'$." ], "refs": [], "proofs": [ { "contents": [ "By", "More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-unramified-lci}", "the conormal sheaf $\\mathcal{C}_{Z/X}$ of the unramified morphism $Z \\to X$", "is a finite locally free $\\mathcal{O}_Z$-module and by", "More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-transitivity-conormal-lci}", "we have an exact sequence", "$$", "0 \\to i^*\\mathcal{C}_{X/X'} \\to", "\\mathcal{C}_{Z/X'} \\to", "\\mathcal{C}_{Z/X} \\to 0", "$$", "of conormal sheaves. Since $Z$ is affine this sequence is split. Choose", "a splitting", "$$", "\\mathcal{C}_{Z/X'} = i^*\\mathcal{C}_{X/X'} \\oplus \\mathcal{C}_{Z/X}", "$$", "Let $Z \\subset Z''$ be the universal first order thickening of $Z$", "over $X'$ (see", "More on Morphisms of Spaces,", "Section \\ref{spaces-more-morphisms-section-universal-thickening}).", "Denote $\\mathcal{I} \\subset \\mathcal{O}_{Z''}$ the quasi-coherent sheaf", "of ideals corresponding to $Z \\subset Z''$. By definition we have", "$\\mathcal{C}_{Z/X'}$ is $\\mathcal{I}$ viewed as a sheaf on $Z$.", "Hence the splitting above determines a splitting", "$$", "\\mathcal{I} = i^*\\mathcal{C}_{X/X'} \\oplus \\mathcal{C}_{Z/X}", "$$", "Let $Z' \\subset Z''$ be the closed subscheme cut out by", "$\\mathcal{C}_{Z/X} \\subset \\mathcal{I}$ viewed as a quasi-coherent sheaf", "of ideals on $Z''$. It is clear that $Z'$ is a first order thickening", "of $Z$ and that we obtain a commutative diagram of first order thickenings", "as in the statement of the lemma.", "\\medskip\\noindent", "Since $X' \\to U'$ is flat and since $X = U \\times_{U'} X'$ we see that", "$\\mathcal{C}_{X/X'}$ is the pullback of $\\mathcal{C}_{U/U'}$ to $X$, see", "More on Morphisms of Spaces, Lemma \\ref{spaces-more-morphisms-lemma-deform}.", "Note that by construction $\\mathcal{C}_{Z/Z'} = i^*\\mathcal{C}_{X/X'}$", "hence we conclude that $\\mathcal{C}_{Z/Z'}$ is isomorphic to the pullback", "of $\\mathcal{C}_{U/U'}$ to $Z$. Applying", "More on Morphisms of Spaces, Lemma \\ref{spaces-more-morphisms-lemma-deform}", "once again (or its analogue for schemes, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-deform})", "we conclude that $Z' \\to U'$ is flat and that $Z = U \\times_{U'} Z'$.", "Finally,", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-deform-property}", "shows that $Z' \\to U'$ is finite locally free of degree $d$." ], "refs": [ "spaces-more-morphisms-lemma-unramified-lci", "spaces-more-morphisms-lemma-transitivity-conormal-lci", "spaces-more-morphisms-lemma-deform", "spaces-more-morphisms-lemma-deform", "more-morphisms-lemma-deform", "more-morphisms-lemma-deform-property" ], "ref_ids": [ 245, 246, 101, 101, 13723, 13725 ] } ], "ref_ids": [] }, { "id": 3135, "type": "theorem", "label": "criteria-lemma-lci-formally-smooth", "categories": [ "criteria" ], "title": "criteria-lemma-lci-formally-smooth", "contents": [ "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids", "over $(\\Sch/S)_{fppf}$. Assume $F$ is representable by algebraic", "spaces, flat, and locally of finite presentation. Then", "$$", "p : \\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y}) \\to \\mathcal{Y}", "$$", "is formally smooth on objects." ], "refs": [], "proofs": [ { "contents": [ "We have to show the following: Given", "\\begin{enumerate}", "\\item an object $(U, Z, y, x, \\alpha)$ of", "$\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})$ over an affine scheme $U$,", "\\item a first order thickening $U \\subset U'$, and", "\\item an object $y'$ of $\\mathcal{Y}$ over $U'$ such that $y'|_U = y$,", "\\end{enumerate}", "then there exists an object $(U', Z', y', x', \\alpha')$ of", "$\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})$ over $U'$ with", "$Z = U \\times_{U'} Z'$, with $x = x'|_Z$, and with", "$\\alpha = \\alpha'|_U$. Namely, the last two equalities will take care", "of the commutativity of (\\ref{equation-formally-smooth}).", "\\medskip\\noindent", "Consider the morphism $x_\\alpha : Z \\to X_y$ constructed in", "Equation (\\ref{equation-relative-map}). Denote similarly $X'_{y'}$", "the algebraic space over $U'$ representing the $2$-fibre product", "$(\\Sch/U')_{fppf} \\times_{y', \\mathcal{Y}, F} \\mathcal{X}$.", "By assumption the morphism $X'_{y'} \\to U'$ is flat (and locally of finite", "presentation). As $y'|_U = y$ we see that $X_y = U \\times_{U'} X'_{y'}$.", "Hence we may apply", "Lemma \\ref{lemma-lci-unobstructed}", "to find $Z' \\to U'$ finite locally free of degree $d$ with", "$Z = U \\times_{U'} Z'$ and with $Z' \\to X'_{y'}$ extending $x_\\alpha$.", "By construction the morphism $Z' \\to X'_{y'}$ corresponds to a pair", "$(x', \\alpha')$. It is clear that $(U', Z', y', x', \\alpha')$", "is an object of $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ over $U'$", "with $Z = U \\times_{U'} Z'$, with $x = x'|_Z$, and with", "$\\alpha = \\alpha'|_U$. As we've seen in", "Lemma \\ref{lemma-lci-locus-stack-in-groupoids}", "that $\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y}) \\subset", "\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is an ``open substack''", "it follows that $(U', Z', y', x', \\alpha')$ is an object of", "$\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})$ as desired." ], "refs": [ "criteria-lemma-lci-unobstructed", "criteria-lemma-lci-locus-stack-in-groupoids" ], "ref_ids": [ 3134, 3133 ] } ], "ref_ids": [] }, { "id": 3136, "type": "theorem", "label": "criteria-lemma-lci-surjective", "categories": [ "criteria" ], "title": "criteria-lemma-lci-surjective", "contents": [ "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids", "over $(\\Sch/S)_{fppf}$. Assume $F$ is representable by algebraic", "spaces, flat, surjective, and locally of finite presentation. Then", "$$", "\\coprod\\nolimits_{d \\geq 1} \\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})", "\\longrightarrow", "\\mathcal{Y}", "$$", "is surjective on objects." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove the following: For any field $k$", "and object $y$ of $\\mathcal{Y}$ over $\\Spec(k)$ there exists", "an integer $d \\geq 1$ and an object $(U, Z, y, x, \\alpha)$ of", "$\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})$ with $U = \\Spec(k)$.", "Namely, in this case we see that $p$ is surjective on objects in the", "strong sense that an extension of the field is not needed.", "\\medskip\\noindent", "Denote $X_y$ the algebraic space over $U = \\Spec(k)$", "representing the $2$-fibre product", "$(\\Sch/U')_{fppf} \\times_{y', \\mathcal{Y}, F} \\mathcal{X}$.", "By assumption the morphism $X_y \\to \\Spec(k)$ is surjective and", "locally of finite presentation (and flat). In particular $X_y$ is", "nonempty. Choose a nonempty affine scheme $V$ and an \\'etale morphism", "$V \\to X_y$. Note that $V \\to \\Spec(k)$ is (flat), surjective,", "and locally of finite presentation (by", "Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-locally-finite-presentation}).", "Pick a closed point $v \\in V$ where $V \\to \\Spec(k)$ is Cohen-Macaulay", "(i.e., $V$ is Cohen-Macaulay at $v$), see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-flat-finite-presentation-CM-open}.", "Applying", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-slice-CM}", "we find a regular immersion $Z \\to V$ with $Z = \\{v\\}$.", "This implies $Z \\to V$ is a closed immersion. Moreover, it follows that", "$Z \\to \\Spec(k)$ is finite (for example by", "Algebra, Lemma \\ref{algebra-lemma-isolated-point}).", "Hence $Z \\to \\Spec(k)$ is finite locally free of some degree $d$.", "Now $Z \\to X_y$ is unramified as the composition", "of a closed immersion followed by an \\'etale morphism", "(see", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-composition-unramified},", "\\ref{spaces-morphisms-lemma-etale-unramified}, and", "\\ref{spaces-morphisms-lemma-immersion-unramified}).", "Finally, $Z \\to X_y$ is a local complete intersection morphism", "as a composition of a regular immersion of schemes and an \\'etale", "morphism of algebraic spaces (see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-regular-immersion-lci}", "and", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-etale-smooth} and", "\\ref{spaces-morphisms-lemma-smooth-syntomic} and", "More on Morphisms of Spaces,", "Lemmas \\ref{spaces-more-morphisms-lemma-flat-lci} and", "\\ref{spaces-more-morphisms-lemma-composition-lci}).", "The morphism $Z \\to X_y$ corresponds to an object $x$ of $\\mathcal{X}$", "over $Z$ together with an isomorphism $\\alpha : y|_Z \\to F(x)$.", "We obtain an object $(U, Z, y, x, \\alpha)$ of", "$\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$. By what was said above about", "the morphism $Z \\to X_y$ we see that it actually is an object of the", "subcategory $\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})$ and we win." ], "refs": [ "spaces-morphisms-definition-locally-finite-presentation", "more-morphisms-lemma-flat-finite-presentation-CM-open", "more-morphisms-lemma-slice-CM", "algebra-lemma-isolated-point", "spaces-morphisms-lemma-composition-unramified", "spaces-morphisms-lemma-etale-unramified", "spaces-morphisms-lemma-immersion-unramified", "more-morphisms-lemma-regular-immersion-lci", "spaces-morphisms-lemma-etale-smooth", "spaces-morphisms-lemma-smooth-syntomic", "spaces-more-morphisms-lemma-flat-lci", "spaces-more-morphisms-lemma-composition-lci" ], "ref_ids": [ 5006, 13789, 13797, 1048, 4896, 4913, 4901, 14007, 4909, 4892, 239, 238 ] } ], "ref_ids": [] }, { "id": 3137, "type": "theorem", "label": "criteria-lemma-flat-quotient-flat-presentation", "categories": [ "criteria" ], "title": "criteria-lemma-flat-quotient-flat-presentation", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.", "Assume $s, t$ are flat and locally of finite presentation.", "Then the morphism $\\mathcal{S}_U \\to [U/R]$ is flat, locally of", "finite presentation, and surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme and let $x : (\\Sch/T)_{fppf} \\to [U/R]$", "be a $1$-morphism. We have to show that the projection", "$$", "\\mathcal{S}_U \\times_{[U/R]} (\\Sch/T)_{fppf}", "\\longrightarrow", "(\\Sch/T)_{fppf}", "$$", "is surjective and smooth. We already know that the left hand side", "is representable by an algebraic space $F$, see", "Algebraic Stacks, Lemmas \\ref{algebraic-lemma-diagonal-quotient-stack} and", "\\ref{algebraic-lemma-representable-diagonal}.", "Hence we have to show the corresponding morphism $F \\to T$ of", "algebraic spaces is surjective, locally of finite presentation, and flat.", "Since we are working with properties of morphisms of algebraic", "spaces which are local on the target in the fppf topology we", "may check this fppf locally on $T$. By construction, there exists", "an fppf covering $\\{T_i \\to T\\}$ of $T$ such that", "$x|_{(\\Sch/T_i)_{fppf}}$ comes from a morphism", "$x_i : T_i \\to U$. (Note that $F \\times_T T_i$ represents the", "$2$-fibre product $\\mathcal{S}_U \\times_{[U/R]} (\\Sch/T_i)_{fppf}$", "so everything is compatible with the base change via $T_i \\to T$.)", "Hence we may assume that $x$ comes from $x : T \\to U$.", "In this case we see that", "$$", "\\mathcal{S}_U \\times_{[U/R]} (\\Sch/T)_{fppf}", "=", "(\\mathcal{S}_U \\times_{[U/R]} \\mathcal{S}_U)", "\\times_{\\mathcal{S}_U} (\\Sch/T)_{fppf}", "=", "\\mathcal{S}_R \\times_{\\mathcal{S}_U} (\\Sch/T)_{fppf}", "$$", "The first equality by", "Categories, Lemma \\ref{categories-lemma-2-fibre-product-erase-factor}", "and the second equality by", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-2-cartesian}.", "Clearly the last $2$-fibre product is represented by the algebraic", "space $F = R \\times_{s, U, x} T$ and the projection", "$R \\times_{s, U, x} T \\to T$ is flat and locally of finite presentation", "as the base change of the flat locally finitely presented", "morphism of algebraic spaces $s : R \\to U$.", "It is also surjective as $s$ has a section (namely the identity", "$e : U \\to R$ of the groupoid).", "This proves the lemma." ], "refs": [ "algebraic-lemma-diagonal-quotient-stack", "algebraic-lemma-representable-diagonal", "categories-lemma-2-fibre-product-erase-factor", "spaces-groupoids-lemma-quotient-stack-2-cartesian" ], "ref_ids": [ 8475, 8461, 12275, 9324 ] } ], "ref_ids": [] }, { "id": 3138, "type": "theorem", "label": "criteria-lemma-quotient-algebraic", "categories": [ "criteria" ], "title": "criteria-lemma-quotient-algebraic", "contents": [ "Let $S$ be a scheme and let $B$ be an algebraic space over $S$.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "The quotient stack $[U/R]$ is an algebraic stack if and only if", "there exists a morphism of algebraic spaces $g : U' \\to U$ such that", "\\begin{enumerate}", "\\item the composition", "$U' \\times_{g, U, t} R \\to R \\xrightarrow{s} U$ is a surjection of", "sheaves, and", "\\item the morphisms $s', t' : R' \\to U'$ are flat and locally of finite", "presentation where $(U', R', s', t', c')$ is the restriction of", "$(U, R, s, t, c)$ via $g$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First, assume that $g : U' \\to U$ satisfies (1) and (2). Property (1)", "implies that $[U'/R'] \\to [U/R]$ is an equivalence, see", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-restrict-equivalence}.", "By", "Theorem \\ref{theorem-flat-groupoid-gives-algebraic-stack}", "the quotient stack $[U'/R']$ is an algebraic stack. Hence", "$[U/R]$ is an algebraic stack too, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-equivalent}.", "\\medskip\\noindent", "Conversely, assume that $[U/R]$ is an algebraic stack. We may choose a", "scheme $W$ and a surjective smooth $1$-morphism", "$$", "f : (\\Sch/W)_{fppf} \\longrightarrow [U/R].", "$$", "By the $2$-Yoneda lemma", "(Algebraic Stacks, Section \\ref{algebraic-section-2-yoneda})", "this corresponds to an object $\\xi$ of $[U/R]$ over $W$.", "By the description of $[U/R]$ in", "Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-quotient-stack-objects}", "we can find a surjective, flat, locally finitely presented morphism", "$b : U' \\to W$ of schemes such that $\\xi' = b^*\\xi$ corresponds to a morphism", "$g : U' \\to U$. Note that the $1$-morphism", "$$", "f' : (\\Sch/U')_{fppf} \\longrightarrow [U/R].", "$$", "corresponding to $\\xi'$ is surjective, flat, and locally of finite", "presentation, see", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-composition-representable-transformations-property}.", "Hence", "$(\\Sch/U')_{fppf} \\times_{[U/R]} (\\Sch/U')_{fppf}$", "which is represented by the algebraic space", "$$", "\\mathit{Isom}_{[U/R]}(\\text{pr}_0^*\\xi', \\text{pr}_1^*\\xi') =", "(U' \\times_S U')", "\\times_{(g \\circ \\text{pr}_0, g \\circ \\text{pr}_1), U \\times_S U} R = R'", "$$", "(see", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-morphisms}", "for the first equality; the second is the definition of restriction)", "is flat and locally of finite presentation over $U'$ via both $s'$ and $t'$", "(by base change, see", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-base-change-representable-transformations-property}).", "By this description of $R'$ and by", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack}", "we obtain a canonical fully faithful $1$-morphism $[U'/R'] \\to [U/R]$.", "This $1$-morphism is essentially surjective because $f'$ is flat,", "locally of finite presentation, and surjective (see", "Stacks, Lemma \\ref{stacks-lemma-characterize-essentially-surjective-when-ff});", "another way to prove this is to use", "Algebraic Stacks, Remark", "\\ref{algebraic-remark-flat-fp-presentation}.", "Finally, we can use", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-restrict-equivalence}", "to conclude that the composition", "$U' \\times_{g, U, t} R \\to R \\xrightarrow{s} U$ is a surjection of sheaves." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-restrict-equivalence", "criteria-theorem-flat-groupoid-gives-algebraic-stack", "algebraic-lemma-equivalent", "spaces-groupoids-lemma-quotient-stack-objects", "algebraic-lemma-composition-representable-transformations-property", "spaces-groupoids-lemma-quotient-stack-morphisms", "algebraic-lemma-base-change-representable-transformations-property", "algebraic-lemma-map-space-into-stack", "stacks-lemma-characterize-essentially-surjective-when-ff", "algebraic-remark-flat-fp-presentation", "spaces-groupoids-lemma-quotient-stack-restrict-equivalence" ], "ref_ids": [ 9330, 3094, 8462, 9328, 8455, 9323, 8456, 8473, 8945, 8491, 9330 ] } ], "ref_ids": [] }, { "id": 3139, "type": "theorem", "label": "criteria-lemma-group-quotient-algebraic", "categories": [ "criteria" ], "title": "criteria-lemma-group-quotient-algebraic", "contents": [ "Let $S$ be a scheme and let $B$ be an algebraic space over $S$.", "Let $G$ be a group algebraic space over $B$.", "Let $X$ be an algebraic space over $B$ and let $a : G \\times_B X \\to X$", "be an action of $G$ on $X$ over $B$.", "The quotient stack $[X/G]$ is an algebraic stack if and only if", "there exists a morphism of algebraic spaces $\\varphi : X' \\to X$ such that", "\\begin{enumerate}", "\\item $G \\times_B X' \\to X$, $(g, x') \\mapsto a(g, \\varphi(x'))$ is a", "surjection of sheaves, and", "\\item the two projections $X'' \\to X'$ of the algebraic space $X''$", "given by the rule", "$$", "T \\longmapsto \\{(x'_1, g, x'_2) \\in (X' \\times_B G \\times_B X')(T)", "\\mid \\varphi(x'_1) = a(g, \\varphi(x'_2))\\}", "$$", "are flat and locally of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma is a special case of", "Lemma \\ref{lemma-quotient-algebraic}.", "Namely, the quotient stack $[X/G]$ is by", "Groupoids in Spaces, Definition \\ref{spaces-groupoids-definition-quotient-stack}", "equal to the quotient stack $[X/G \\times_B X]$ of the groupoid in", "algebraic spaces $(X, G \\times_B X, s, t, c)$ associated to", "the group action in", "Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-groupoid-from-action}.", "There is one small observation that is needed to get condition (1).", "Namely, the morphism $s : G \\times_B X \\to X$ is the second projection", "and the morphism $t : G \\times_B X \\to X$ is the action morphism $a$.", "Hence the morphism $h : U' \\times_{g, U, t} R \\to R \\xrightarrow{s} U$ from", "Lemma \\ref{lemma-quotient-algebraic}", "corresponds to the morphism", "$$", "X' \\times_{\\varphi, X, a} (G \\times_B X) \\xrightarrow{\\text{pr}_1} X", "$$", "in the current setting. However, because of the symmetry given by", "the inverse of $G$ this morphism is isomorphic to the morphism", "$$", "(G \\times_B X) \\times_{\\text{pr}_1, X, \\varphi} X' \\xrightarrow{a} X", "$$", "of the statement of the lemma. Details omitted." ], "refs": [ "criteria-lemma-quotient-algebraic", "spaces-groupoids-definition-quotient-stack", "spaces-groupoids-lemma-groupoid-from-action", "criteria-lemma-quotient-algebraic" ], "ref_ids": [ 3138, 9354, 9308, 3138 ] } ], "ref_ids": [] }, { "id": 3140, "type": "theorem", "label": "criteria-lemma-BG-algebraic", "categories": [ "criteria" ], "title": "criteria-lemma-BG-algebraic", "contents": [ "\\begin{slogan}", "Gerbes are algebraic if and only if the associated groups are flat", "and locally of finite presentation", "\\end{slogan}", "Let $S$ be a scheme and let $B$ be an algebraic space over $S$.", "Let $G$ be a group algebraic space over $B$.", "Endow $B$ with the trivial action of $G$.", "Then the quotient stack $[B/G]$ is an algebraic stack", "if and only if $G$ is flat and locally of finite presentation over $B$." ], "refs": [], "proofs": [ { "contents": [ "If $G$ is flat and locally of finite presentation over $B$, then", "$[B/G]$ is an algebraic stack by", "Theorem \\ref{theorem-flat-groupoid-gives-algebraic-stack}.", "\\medskip\\noindent", "Conversely, assume that $[B/G]$ is an algebraic stack. By", "Lemma \\ref{lemma-group-quotient-algebraic}", "and because the action is trivial, we see", "there exists an algebraic space $B'$ and a morphism", "$B' \\to B$ such that (1) $B' \\to B$ is a surjection", "of sheaves and (2) the projections", "$$", "B' \\times_B G \\times_B B' \\to B'", "$$", "are flat and locally of finite presentation. Note that the base change", "$B' \\times_B G \\times_B B' \\to G \\times_B B'$ of $B' \\to B$", "is a surjection of sheaves also. Thus it follows from", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-curiosity}", "that the projection $G \\times_B B' \\to B'$ is flat and locally", "of finite presentation. By (1) we can find an fppf covering", "$\\{B_i \\to B\\}$ such that $B_i \\to B$ factors through $B' \\to B$.", "Hence $G \\times_B B_i \\to B_i$ is flat and locally of finite presentation", "by base change. By", "Descent on Spaces, Lemmas", "\\ref{spaces-descent-lemma-descending-property-flat} and", "\\ref{spaces-descent-lemma-descending-property-locally-finite-presentation}", "we conclude that $G \\to B$ is flat and locally of finite presentation." ], "refs": [ "criteria-theorem-flat-groupoid-gives-algebraic-stack", "criteria-lemma-group-quotient-algebraic", "spaces-descent-lemma-curiosity", "spaces-descent-lemma-descending-property-flat", "spaces-descent-lemma-descending-property-locally-finite-presentation" ], "ref_ids": [ 3094, 3139, 9368, 9393, 9390 ] } ], "ref_ids": [] }, { "id": 3141, "type": "theorem", "label": "criteria-lemma-stacks-etale", "categories": [ "criteria" ], "title": "criteria-lemma-stacks-etale", "contents": [ "Denote the common underlying category of $\\Sch_{fppf}$", "and $\\Sch_\\etale$ by $\\Sch_\\alpha$ (see", "Sheaves on Stacks, Section \\ref{stacks-sheaves-section-sheaves} and", "Topologies, Remark \\ref{topologies-remark-choice-sites}). Let $S$ be an object", "of $\\Sch_\\alpha$. Let", "$$", "p : \\mathcal{X} \\to \\Sch_\\alpha/S", "$$", "be a category fibred in groupoids with the following properties:", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is a stack in groupoids over $(\\Sch/S)_\\etale$,", "\\item the diagonal $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$", "is representable by algebraic spaces\\footnote{Here we can either mean", "sheaves in the \\'etale topology whose diagonal is representable and which", "have an \\'etale surjective covering by a scheme or algebraic spaces as", "defined in", "Algebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}.", "Namely, by Bootstrap, Lemma \\ref{bootstrap-lemma-spaces-etale}", "there is no difference.}, and", "\\item there exists $U \\in \\Ob(\\Sch_\\alpha/S)$", "and a $1$-morphism $(\\Sch/U)_\\etale \\to \\mathcal{X}$", "which is surjective and smooth.", "\\end{enumerate}", "Then $\\mathcal{X}$ is an algebraic stack in the sense of", "Algebraic Stacks, Definition \\ref{algebraic-definition-algebraic-stack}." ], "refs": [ "topologies-remark-choice-sites", "spaces-definition-algebraic-space", "bootstrap-lemma-spaces-etale", "algebraic-definition-algebraic-stack" ], "proofs": [ { "contents": [ "Note that properties (2) and (3) of the lemma and the corresponding", "properties (2) and (3) of", "Algebraic Stacks, Definition \\ref{algebraic-definition-algebraic-stack}", "are independent of the topology. This is true because these properties", "involve only the notion of a $2$-fibre product of categories fibred in", "groupoids, $1$- and $2$-morphisms of categories fibred in groupoids, the", "notion of a $1$-morphism of categories fibred in groupoids representable", "by algebraic spaces, and what it means for such a $1$-morphism to be", "surjective and smooth.", "Thus all we have to prove is that an \\'etale stack in groupoids", "$\\mathcal{X}$ with properties (2) and (3) is also an fppf stack in groupoids.", "\\medskip\\noindent", "Using (2) let $R$ be an algebraic space representing", "$$", "(\\Sch_\\alpha/U) \\times_\\mathcal{X} (\\Sch_\\alpha/U)", "$$", "By (3) the projections $s, t : R \\to U$ are smooth. Exactly as in the proof of", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack}", "there exists a groupoid in spaces $(U, R, s, t, c)$ and a canonical", "fully faithful $1$-morphism $[U/R]_\\etale \\to \\mathcal{X}$", "where $[U/R]_\\etale$ is the \\'etale stackification of presheaf", "in groupoids", "$$", "T \\longmapsto (U(T), R(T), s(T), t(T), c(T))", "$$", "Claim: If $V \\to T$ is a surjective smooth morphism from an algebraic space", "$V$ to a scheme $T$, then there exists an \\'etale covering $\\{T_i \\to T\\}$", "refining the covering $\\{V \\to T\\}$. This follows from", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-etale-dominates-smooth}", "or the more general", "Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}.", "Using the claim and arguing exactly as in", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-stack-presentation}", "it follows that $[U/R]_\\etale \\to \\mathcal{X}$ is an", "equivalence.", "\\medskip\\noindent", "Next, let $[U/R]$ denote the quotient stack in the fppf topology", "which is an algebraic stack by", "Algebraic Stacks, Theorem", "\\ref{algebraic-theorem-smooth-groupoid-gives-algebraic-stack}.", "Thus we have $1$-morphisms", "$$", "U \\to [U/R]_\\etale \\to [U/R].", "$$", "Both $U \\to [U/R]_\\etale \\cong \\mathcal{X}$ and", "$U \\to [U/R]$ are surjective and smooth (the first by assumption", "and the second by the theorem) and in both cases the", "fibre product $U \\times_\\mathcal{X} U$ and $U \\times_{[U/R]} U$", "is representable by $R$. Hence the $1$-morphism", "$[U/R]_\\etale \\to [U/R]$ is fully faithful (since morphisms", "in the quotient stacks are given by morphisms into $R$, see", "Groupoids in Spaces, Section", "\\ref{spaces-groupoids-section-explicit-quotient-stacks}).", "\\medskip\\noindent", "Finally, for any scheme $T$ and morphism $t : T \\to [U/R]$ the fibre product", "$V = T \\times_{U/R} U$ is an algebraic space surjective and smooth over $T$.", "By the claim above there exists an \\'etale covering $\\{T_i \\to T\\}_{i \\in I}$", "and morphisms $T_i \\to V$ over $T$. This proves that the object", "$t$ of $[U/R]$ over $T$ comes \\'etale locally from $U$. We conclude that", "$[U/R]_\\etale \\to [U/R]$ is an equivalence of stacks in", "groupoids over $(\\Sch/S)_\\etale$ by", "Stacks, Lemma \\ref{stacks-lemma-characterize-essentially-surjective-when-ff}.", "This concludes the proof." ], "refs": [ "algebraic-definition-algebraic-stack", "algebraic-lemma-map-space-into-stack", "more-morphisms-lemma-etale-dominates-smooth", "stacks-sheaves-lemma-surjective-flat-locally-finite-presentation", "algebraic-lemma-stack-presentation", "algebraic-theorem-smooth-groupoid-gives-algebraic-stack", "stacks-lemma-characterize-essentially-surjective-when-ff" ], "ref_ids": [ 8484, 8473, 13880, 11606, 8474, 8435, 8945 ] } ], "ref_ids": [ 12553, 8174, 2633, 8484 ] }, { "id": 3142, "type": "theorem", "label": "criteria-proposition-hom-functor-algebraic-space", "categories": [ "criteria" ], "title": "criteria-proposition-hom-functor-algebraic-space", "contents": [ "Let $S$ be a scheme. Let $Z \\to B$ and $X \\to B$ be morphisms of", "algebraic spaces over $S$. If $Z \\to B$ is finite locally free", "then $\\mathit{Mor}_B(Z, X)$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $B' = \\coprod B'_i$ which is a disjoint union of", "affine schemes $B'_i$ and an \\'etale surjective morphism $B' \\to B$.", "We may also assume that $B'_i \\times_B Z$ is the spectrum of a ring", "which is finite free as a $\\Gamma(B'_i, \\mathcal{O}_{B'_i})$-module.", "By", "Lemma \\ref{lemma-base-change-hom-functor}", "and", "Spaces, Lemma", "\\ref{spaces-lemma-base-change-representable-transformations-property}", "the morphism $\\mathit{Mor}_{B'}(Z', X') \\to \\mathit{Mor}_B(Z, X)$", "is surjective \\'etale. Hence by", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}", "it suffices to prove the proposition when $B = B'$ is a disjoint union of", "affine schemes $B'_i$ so that each $B'_i \\times_B Z$ is finite free", "over $B'_i$. Then it actually suffices to prove the result for the restriction", "to each $B'_i$. Thus we may assume that $B$ is affine and that", "$\\Gamma(Z, \\mathcal{O}_Z)$ is a finite free $\\Gamma(B, \\mathcal{O}_B)$-module.", "\\medskip\\noindent", "Choose a scheme $X'$ which is a disjoint union of affine schemes and", "a surjective \\'etale morphism $X' \\to X$. By", "Lemma \\ref{lemma-etale-covering-hom-functor}", "the morphism $\\mathit{Mor}_B(Z, X') \\to \\mathit{Mor}_B(Z, X)$", "is representable by algebraic spaces, \\'etale, and surjective.", "Hence by", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}", "it suffices to prove the proposition when $X$ is a disjoint union", "of affine schemes. This reduces us to the case discussed in the next", "paragraph.", "\\medskip\\noindent", "Assume $X = \\coprod_{i \\in I} X_i$ is a disjoint union of affine", "schemes, $B$ is affine, and that $\\Gamma(Z, \\mathcal{O}_Z)$ is a finite", "free $\\Gamma(B, \\mathcal{O}_B)$-module. For any finite subset", "$E \\subset I$ set", "$$", "F_E = \\mathit{Mor}_B(Z, \\coprod\\nolimits_{i \\in E} X_i).", "$$", "By More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-hom-from-finite-free-into-affine}", "we see that $F_E$ is an algebraic space. Consider the morphism", "$$", "\\coprod\\nolimits_{E \\subset I\\text{ finite}} F_E", "\\longrightarrow", "\\mathit{Mor}_B(Z, X)", "$$", "Each of the morphisms", "$F_E \\to \\mathit{Mor}_B(Z, X)$ is an open immersion, because it is", "simply the locus parametrizing pairs $(a, b)$ where $b$ maps into", "the open subscheme $\\coprod\\nolimits_{i \\in E} X_i$ of $X$. Moreover,", "if $T$ is quasi-compact, then for any pair $(a, b)$ the image", "of $b$ is contained in $\\coprod\\nolimits_{i \\in E} X_i$ for some", "$E \\subset I$ finite. Hence the displayed arrow is in fact an", "open covering and we win\\footnote{Modulo", "some set theoretic arguments. Namely, we have to show that", "$\\coprod F_E$ is an algebraic space. This follows because", "$|I| \\leq \\text{size}(X)$ and $\\text{size}(F_E) \\leq \\text{size}(X)$", "as follows from the explicit description of $F_E$ in the proof of", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-hom-from-finite-free-into-affine}.", "Some details omitted.} by", "Spaces, Lemma \\ref{spaces-lemma-glueing-algebraic-spaces}." ], "refs": [ "criteria-lemma-base-change-hom-functor", "spaces-lemma-base-change-representable-transformations-property", "bootstrap-theorem-final-bootstrap", "criteria-lemma-etale-covering-hom-functor", "bootstrap-theorem-final-bootstrap", "more-morphisms-lemma-hom-from-finite-free-into-affine", "more-morphisms-lemma-hom-from-finite-free-into-affine", "spaces-lemma-glueing-algebraic-spaces" ], "ref_ids": [ 3117, 8133, 2602, 3118, 2602, 14053, 14053, 8148 ] } ], "ref_ids": [] }, { "id": 3143, "type": "theorem", "label": "criteria-proposition-restriction-of-scalars-algebraic-space", "categories": [ "criteria" ], "title": "criteria-proposition-restriction-of-scalars-algebraic-space", "contents": [ "Let $S$ be a scheme. Let $X \\to Z \\to B$ be morphisms of", "algebraic spaces over $S$. If $Z \\to B$ is finite locally free", "then $\\text{Res}_{Z/B}(X)$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "By", "Proposition \\ref{proposition-hom-functor-algebraic-space}", "the functors $\\mathit{Mor}_B(Z, X)$ and $\\mathit{Mor}_B(Z, Z)$", "are algebraic spaces. Hence this follows from the cartesian", "diagram of", "Lemma \\ref{lemma-fibre-diagram}", "and the fact that fibre products of algebraic spaces exist and", "are given by the fibre product in the underlying category of", "sheaves of sets (see", "Spaces, Lemma", "\\ref{spaces-lemma-fibre-product-spaces-over-sheaf-with-representable-diagonal})." ], "refs": [ "criteria-proposition-hom-functor-algebraic-space", "criteria-lemma-fibre-diagram", "spaces-lemma-fibre-product-spaces-over-sheaf-with-representable-diagonal" ], "ref_ids": [ 3142, 3122, 8142 ] } ], "ref_ids": [] }, { "id": 3144, "type": "theorem", "label": "criteria-proposition-finite-hilbert-point", "categories": [ "criteria" ], "title": "criteria-proposition-finite-hilbert-point", "contents": [ "The stack $\\mathcal{H}_d$ is equivalent to the quotient stack", "$[X/G]$ described above. In particular $\\mathcal{H}_d$ is an", "algebraic stack." ], "refs": [], "proofs": [ { "contents": [ "Note that by", "Groupoids in Spaces, Definition", "\\ref{spaces-groupoids-definition-quotient-stack}", "the quotient stack $[X/G]$ is the stackification of the", "category fibred in groupoids associated to the ``presheaf in groupoids''", "which associates to a scheme $T$ the groupoid", "$$", "(X(T), G(T) \\times X(T), s, t, c).", "$$", "Since this ``presheaf in groupoids'' is isomorphic to $FA_d$ by", "Lemma \\ref{lemma-represent-FAd}", "it suffices to prove that the $\\mathcal{H}_d$ is the stackification", "of (the category fibred in groupoids associated to the", "``presheaf in groupoids'') $FA_d$. To do this we first define a", "functor", "$$", "\\Spec : FA_d \\longrightarrow \\mathcal{H}_d", "$$", "Recall that the fibre category of $\\mathcal{H}_d$ over a scheme $T$", "is the category of finite locally free morphisms $Z \\to T$ of degree $d$.", "Thus given a scheme $T$ and a free $d$-dimensional", "$\\Gamma(T, \\mathcal{O}_T)$-algebra $m$ we may assign to this the object", "$$", "Z = \\underline{\\Spec}_T(\\mathcal{A})", "$$", "of $\\mathcal{H}_{d, T}$", "where $\\mathcal{A} = \\mathcal{O}_T^{\\oplus d}$ endowed with a", "$\\mathcal{O}_T$-algebra structure via $m$. Moreover, if $m'$ is", "a second such free $d$-dimensional $\\Gamma(T, \\mathcal{O}_T)$-algebra", "and if $\\varphi : m \\to m'$ is an isomorphism of these, then", "the induced $\\mathcal{O}_T$-linear map", "$\\varphi : \\mathcal{O}_T^{\\oplus d} \\to \\mathcal{O}_T^{\\oplus d}$", "induces an isomorphism", "$$", "\\varphi : \\mathcal{A}' \\longrightarrow \\mathcal{A}", "$$", "of quasi-coherent $\\mathcal{O}_T$-algebras. Hence", "$$", "\\underline{\\Spec}_T(\\varphi) :", "\\underline{\\Spec}_T(\\mathcal{A})", "\\longrightarrow", "\\underline{\\Spec}_T(\\mathcal{A}')", "$$", "is a morphism in the fibre category $\\mathcal{H}_{d, T}$. We omit the", "verification that this construction is compatible with base change so", "we get indeed a functor $\\Spec : FA_d \\to \\mathcal{H}_d$", "as claimed above.", "\\medskip\\noindent", "To show that $\\Spec : FA_d \\to \\mathcal{H}_d$ induces an equivalence", "between the stackification of $FA_d$ and $\\mathcal{H}_d$ it suffices to", "check that", "\\begin{enumerate}", "\\item $\\mathit{Isom}(m, m') = \\mathit{Isom}(\\Spec(m), \\Spec(m'))$", "for any $m, m' \\in FA_d(T)$.", "\\item for any scheme $T$ and any object $Z \\to T$ of $\\mathcal{H}_{d, T}$", "there exists a covering $\\{T_i \\to T\\}$ such that $Z|_{T_i}$ is", "isomorphic to $\\Spec(m)$ for some $m \\in FA_d(T_i)$, and", "\\end{enumerate}", "see", "Stacks, Lemma \\ref{stacks-lemma-stackify-groupoids}.", "The first statement follows from the observation that any isomorphism", "$$", "\\underline{\\Spec}_T(\\mathcal{A})", "\\longrightarrow", "\\underline{\\Spec}_T(\\mathcal{A}')", "$$", "is necessarily given by a global invertible matrix $g$ when", "$\\mathcal{A} = \\mathcal{A}' = \\mathcal{O}_T^{\\oplus d}$ as modules.", "To prove the second statement let $\\pi : Z \\to T$ be a finite", "locally free morphism of degree $d$. Then $\\mathcal{A}$ is a locally", "free sheaf $\\mathcal{O}_T$-modules of rank $d$.", "Consider the element $1 \\in \\Gamma(T, \\mathcal{A})$. This element is", "nonzero in $\\mathcal{A} \\otimes_{\\mathcal{O}_{T, t}} \\kappa(t)$", "for every $t \\in T$ since the scheme", "$Z_t = \\Spec(\\mathcal{A} \\otimes_{\\mathcal{O}_{T, t}} \\kappa(t))$", "is nonempty being of degree $d > 0$ over $\\kappa(t)$. Thus", "$1 : \\mathcal{O}_T \\to \\mathcal{A}$ can locally be used as the first", "basis element (for example you can use", "Algebra, Lemma \\ref{algebra-lemma-cokernel-flat} parts (1) and (2)", "to see this). Thus, after localizing on", "$T$ we may assume that there exists an isomorphism", "$\\varphi : \\mathcal{A} \\to \\mathcal{O}_T^{\\oplus d}$", "such that $1 \\in \\Gamma(\\mathcal{A})$ corresponds to the first basis element.", "In this situation the multiplication map", "$\\mathcal{A} \\otimes_{\\mathcal{O}_T} \\mathcal{A} \\to \\mathcal{A}$", "translates via $\\varphi$ into a free $d$-dimensional algebra $m$ over", "$\\Gamma(T, \\mathcal{O}_T)$. This finishes the proof." ], "refs": [ "spaces-groupoids-definition-quotient-stack", "criteria-lemma-represent-FAd", "stacks-lemma-stackify-groupoids", "algebra-lemma-cokernel-flat" ], "ref_ids": [ 9354, 3130, 8966, 804 ] } ], "ref_ids": [] }, { "id": 3146, "type": "theorem", "label": "quot-theorem-coherent-algebraic", "categories": [ "quot" ], "title": "quot-theorem-coherent-algebraic", "contents": [ "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces", "over $S$. Assume that $f$ is of finite presentation, separated, and", "flat\\footnote{This assumption is not necessary. See", "Section \\ref{section-not-flat}.}. Then $\\Cohstack_{X/B}$ is", "an algebraic stack over $S$." ], "refs": [], "proofs": [ { "contents": [ "Set $\\mathcal{X} = \\Cohstack_{X/B}$. We have seen that $\\mathcal{X}$", "is a stack in groupoids over $(\\Sch/S)_{fppf}$ with diagonal representable", "by algebraic spaces", "(Lemmas \\ref{lemma-coherent-stack} and \\ref{lemma-coherent-diagonal}).", "Hence it suffices to find a scheme $W$ and a surjective and smooth", "morphism $W \\to \\mathcal{X}$.", "\\medskip\\noindent", "Let $B'$ be a scheme and let $B' \\to B$ be a surjective \\'etale morphism.", "Set $X' = B' \\times_B X$ and denote $f' : X' \\to B'$ the projection.", "Then $\\mathcal{X}' = \\Cohstack_{X'/B'}$ is equal to the $2$-fibre", "product of $\\mathcal{X}$ with the category fibred in sets", "associated to $B'$ over the category fibred in sets associated to $B$", "(Remark \\ref{remark-coherent-base-change}). By the material in", "Algebraic Stacks, Section \\ref{algebraic-section-representable-properties}", "the morphism $\\mathcal{X}' \\to \\mathcal{X}$ is surjective and \\'etale.", "Hence it suffices to prove the result for $\\mathcal{X}'$.", "In other words, we may assume $B$ is a scheme.", "\\medskip\\noindent", "Assume $B$ is a scheme. In this case we may replace $S$ by $B$, see", "Algebraic Stacks, Section \\ref{algebraic-section-change-base-scheme}.", "Thus we may assume $S = B$.", "\\medskip\\noindent", "Assume $S = B$. Choose an affine open covering $S = \\bigcup U_i$.", "Denote $\\mathcal{X}_i$ the restriction of $\\mathcal{X}$ to", "$(\\Sch/U_i)_{fppf}$. If we can find schemes $W_i$ over $U_i$ and", "surjective smooth morphisms $W_i \\to \\mathcal{X}_i$, then we", "set $W = \\coprod W_i$ and we obtain a surjective smooth morphism", "$W \\to \\mathcal{X}$. Thus we may assume $S = B$ is affine.", "\\medskip\\noindent", "Assume $S = B$ is affine, say $S = \\Spec(\\Lambda)$.", "Write $\\Lambda = \\colim \\Lambda_i$ as a filtered colimit with each $\\Lambda_i$", "of finite type over $\\mathbf{Z}$. For some $i$ we can find", "a morphism of algebraic spaces $X_i \\to \\Spec(\\Lambda_i)$", "which is of finite presentation, separated, and flat and whose base change", "to $\\Lambda$ is $X$. See", "Limits of Spaces, Lemmas", "\\ref{spaces-limits-lemma-descend-finite-presentation},", "\\ref{spaces-limits-lemma-descend-separated-morphism}, and", "\\ref{spaces-limits-lemma-descend-flat}.", "If we show that $\\Cohstack_{X_i/\\Spec(\\Lambda_i)}$ is an", "algebraic stack, then it follows by base change", "(Remark \\ref{remark-coherent-base-change} and", "Algebraic Stacks, Section \\ref{algebraic-section-change-base-scheme})", "that $\\mathcal{X}$ is an algebraic stack.", "Thus we may assume that $\\Lambda$ is a finite type $\\mathbf{Z}$-algebra.", "\\medskip\\noindent", "Assume $S = B = \\Spec(\\Lambda)$ is affine of finite type over $\\mathbf{Z}$.", "In this case we will verify conditions (1), (2), (3), (4), and (5) of", "Artin's Axioms, Lemma \\ref{artin-lemma-diagonal-representable}", "to conclude that $\\mathcal{X}$ is an algebraic stack.", "Note that $\\Lambda$ is a G-ring, see", "More on Algebra, Proposition \\ref{more-algebra-proposition-ubiquity-G-ring}.", "Hence all local rings of $S$ are G-rings. Thus (5) holds.", "By Lemma \\ref{lemma-coherent-defo-thy}", "we have that $\\mathcal{X}$ satisfies openness of versality, hence (4) holds.", "To check (2) we have to verify axioms [-1], [0], [1], [2], and [3]", "of Artin's Axioms, Section \\ref{artin-section-axioms}.", "We omit the verification of [-1] and axioms", "[0], [1], [2], [3] correspond respectively to", "Lemmas \\ref{lemma-coherent-stack},", "\\ref{lemma-coherent-limits},", "\\ref{lemma-coherent-RS-star},", "\\ref{lemma-coherent-tangent-space}.", "Condition (3) follows from Lemma \\ref{lemma-coherent-existence}.", "Finally, condition (1) is Lemma \\ref{lemma-coherent-diagonal}.", "This finishes the proof of the theorem." ], "refs": [ "quot-lemma-coherent-stack", "quot-lemma-coherent-diagonal", "quot-remark-coherent-base-change", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-separated-morphism", "spaces-limits-lemma-descend-flat", "quot-remark-coherent-base-change", "artin-lemma-diagonal-representable", "more-algebra-proposition-ubiquity-G-ring", "quot-lemma-coherent-defo-thy", "quot-lemma-coherent-stack", "quot-lemma-coherent-limits", "quot-lemma-coherent-RS-star", "quot-lemma-coherent-tangent-space", "quot-lemma-coherent-existence", "quot-lemma-coherent-diagonal" ], "ref_ids": [ 3162, 3161, 3233, 4598, 4592, 4595, 3233, 11380, 10582, 3168, 3162, 3163, 3164, 3166, 3167, 3161 ] } ], "ref_ids": [] }, { "id": 3147, "type": "theorem", "label": "quot-theorem-coherent-algebraic-general", "categories": [ "quot" ], "title": "quot-theorem-coherent-algebraic-general", "contents": [ "Let $S$ be a scheme. Let $f : X \\to B$ be morphism of algebraic spaces", "over $S$. Assume that $f$ is of finite presentation and separated. Then", "$\\Cohstack_{X/B}$ is an algebraic stack over $S$." ], "refs": [], "proofs": [ { "contents": [ "Only the last step of the proof is different from the proof", "in the flat case, but we repeat all the arguments here to make ", "sure everything works.", "\\medskip\\noindent", "Set $\\mathcal{X} = \\Cohstack_{X/B}$. We have seen that $\\mathcal{X}$", "is a stack in groupoids over $(\\Sch/S)_{fppf}$ with diagonal representable", "by algebraic spaces", "(Lemmas \\ref{lemma-coherent-stack} and \\ref{lemma-coherent-diagonal}).", "Hence it suffices to find a scheme $W$ and a surjective and smooth", "morphism $W \\to \\mathcal{X}$.", "\\medskip\\noindent", "Let $B'$ be a scheme and let $B' \\to B$ be a surjective \\'etale morphism.", "Set $X' = B' \\times_B X$ and denote $f' : X' \\to B'$ the projection.", "Then $\\mathcal{X}' = \\Cohstack_{X'/B'}$ is equal to the $2$-fibre", "product of $\\mathcal{X}$ with the category fibred in sets", "associated to $B'$ over the category fibred in sets associated to $B$", "(Remark \\ref{remark-coherent-base-change}). By the material in", "Algebraic Stacks, Section \\ref{algebraic-section-representable-properties}", "the morphism $\\mathcal{X}' \\to \\mathcal{X}$ is surjective and \\'etale.", "Hence it suffices to prove the result for $\\mathcal{X}'$.", "In other words, we may assume $B$ is a scheme.", "\\medskip\\noindent", "Assume $B$ is a scheme. In this case we may replace $S$ by $B$, see", "Algebraic Stacks, Section \\ref{algebraic-section-change-base-scheme}.", "Thus we may assume $S = B$.", "\\medskip\\noindent", "Assume $S = B$. Choose an affine open covering $S = \\bigcup U_i$.", "Denote $\\mathcal{X}_i$ the restriction of $\\mathcal{X}$ to", "$(\\Sch/U_i)_{fppf}$. If we can find schemes $W_i$ over $U_i$ and", "surjective smooth morphisms $W_i \\to \\mathcal{X}_i$, then we", "set $W = \\coprod W_i$ and we obtain a surjective smooth morphism", "$W \\to \\mathcal{X}$. Thus we may assume $S = B$ is affine.", "\\medskip\\noindent", "Assume $S = B$ is affine, say $S = \\Spec(\\Lambda)$.", "Write $\\Lambda = \\colim \\Lambda_i$ as a filtered colimit with each $\\Lambda_i$", "of finite type over $\\mathbf{Z}$. For some $i$ we can find", "a morphism of algebraic spaces $X_i \\to \\Spec(\\Lambda_i)$", "which is separated and of finite presentation and whose base change", "to $\\Lambda$ is $X$. See Limits of Spaces, Lemmas", "\\ref{spaces-limits-lemma-descend-finite-presentation} and", "\\ref{spaces-limits-lemma-descend-separated-morphism}.", "If we show that $\\Cohstack_{X_i/\\Spec(\\Lambda_i)}$ is an", "algebraic stack, then it follows by base change", "(Remark \\ref{remark-coherent-base-change} and", "Algebraic Stacks, Section \\ref{algebraic-section-change-base-scheme})", "that $\\mathcal{X}$ is an algebraic stack.", "Thus we may assume that $\\Lambda$ is a finite type $\\mathbf{Z}$-algebra.", "\\medskip\\noindent", "Assume $S = B = \\Spec(\\Lambda)$ is affine of finite type over $\\mathbf{Z}$.", "In this case we will verify conditions (1), (2), (3), (4), and (5) of", "Artin's Axioms, Lemma \\ref{artin-lemma-diagonal-representable}", "to conclude that $\\mathcal{X}$ is an algebraic stack.", "Note that $\\Lambda$ is a G-ring, see", "More on Algebra, Proposition \\ref{more-algebra-proposition-ubiquity-G-ring}.", "Hence all local rings of $S$ are G-rings. Thus (5) holds.", "To check (2) we have to verify axioms [-1], [0], [1], [2], and [3]", "of Artin's Axioms, Section \\ref{artin-section-axioms}.", "We omit the verification of [-1] and axioms", "[0], [1], [2], [3] correspond respectively to", "Lemmas \\ref{lemma-coherent-stack},", "\\ref{lemma-coherent-limits},", "\\ref{lemma-coherent-RS-star},", "\\ref{lemma-coherent-tangent-space}.", "Condition (3) is Lemma \\ref{lemma-coherent-existence}.", "Condition (1) is Lemma \\ref{lemma-coherent-diagonal}.", "\\medskip\\noindent", "It remains to show condition (4) which is openness of versality.", "To see this we will use", "Artin's Axioms, Lemma \\ref{artin-lemma-SGE-implies-openness-versality}.", "We have already seen that $\\mathcal{X}$ has diagonal", "representable by algebraic spaces, has (RS*), and is limit preserving", "(see lemmas used above).", "Hence we only need to see that $\\mathcal{X}$ satisfies the strong", "formal effectiveness formulated in", "Artin's Axioms, Lemma \\ref{artin-lemma-SGE-implies-openness-versality}.", "This is Flatness on Spaces, Theorem \\ref{spaces-flat-theorem-existence}", "and the proof is complete." ], "refs": [ "quot-lemma-coherent-stack", "quot-lemma-coherent-diagonal", "quot-remark-coherent-base-change", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-separated-morphism", "quot-remark-coherent-base-change", "artin-lemma-diagonal-representable", "more-algebra-proposition-ubiquity-G-ring", "quot-lemma-coherent-stack", "quot-lemma-coherent-limits", "quot-lemma-coherent-RS-star", "quot-lemma-coherent-tangent-space", "quot-lemma-coherent-existence", "quot-lemma-coherent-diagonal", "artin-lemma-SGE-implies-openness-versality", "artin-lemma-SGE-implies-openness-versality", "spaces-flat-theorem-existence" ], "ref_ids": [ 3162, 3161, 3233, 4598, 4592, 3233, 11380, 10582, 3162, 3163, 3164, 3166, 3167, 3161, 11386, 11386, 7148 ] } ], "ref_ids": [] }, { "id": 3148, "type": "theorem", "label": "quot-theorem-polarized-algebraic", "categories": [ "quot" ], "title": "quot-theorem-polarized-algebraic", "contents": [ "The stack $\\Polarizedstack$ (Situation \\ref{situation-polarized})", "is algebraic. In fact, for any algebraic space $B$ the stack", "$B\\textit{-Polarized}$ (Remark \\ref{remark-polarized-base-change})", "is algebraic." ], "refs": [ "quot-remark-polarized-base-change" ], "proofs": [ { "contents": [ "The absolute case follows from", "Artin's Axioms, Lemma \\ref{artin-lemma-diagonal-representable}", "and Lemmas \\ref{lemma-polarized-diagonal},", "\\ref{lemma-polarized-RS-star},", "\\ref{lemma-polarized-limits},", "\\ref{lemma-polarized-existence}, and", "\\ref{lemma-polarized-defo-thy}.", "The case over $B$ follows from this, the description of", "$B\\textit{-Polarized}$ as a $2$-fibre product in", "Remark \\ref{remark-polarized-base-change}, and the fact", "that algebraic stacks have $2$-fibre products, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-2-fibre-product}." ], "refs": [ "artin-lemma-diagonal-representable", "quot-lemma-polarized-diagonal", "quot-lemma-polarized-RS-star", "quot-lemma-polarized-limits", "quot-lemma-polarized-existence", "quot-lemma-polarized-defo-thy", "quot-remark-polarized-base-change", "algebraic-lemma-2-fibre-product" ], "ref_ids": [ 11380, 3200, 3202, 3201, 3205, 3206, 3241, 8467 ] } ], "ref_ids": [ 3241 ] }, { "id": 3149, "type": "theorem", "label": "quot-theorem-curves-algebraic", "categories": [ "quot" ], "title": "quot-theorem-curves-algebraic", "contents": [ "\\begin{reference}", "See \\cite[Proposition 3.3, page 8]{dJHS} and", "\\cite[Appendix B by Jack Hall, Theorem B.1]{Smyth}.", "\\end{reference}", "The stack $\\Curvesstack$ (Situation \\ref{situation-curves})", "is algebraic. In fact, for any algebraic space $B$ the stack", "$B\\text{-}\\Curvesstack$ (Remark \\ref{remark-curves-base-change})", "is algebraic." ], "refs": [ "quot-remark-curves-base-change" ], "proofs": [ { "contents": [ "The absolute case follows from", "Artin's Axioms, Lemma \\ref{artin-lemma-diagonal-representable}", "and Lemmas \\ref{lemma-curves-diagonal},", "\\ref{lemma-curves-RS-star},", "\\ref{lemma-curves-limits},", "\\ref{lemma-curves-existence}, and", "\\ref{lemma-curves-defo-thy}.", "The case over $B$ follows from this, the description of", "$B\\text{-}\\Curvesstack$ as a $2$-fibre product in", "Remark \\ref{remark-curves-base-change}, and the fact", "that algebraic stacks have $2$-fibre products, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-2-fibre-product}." ], "refs": [ "artin-lemma-diagonal-representable", "quot-lemma-curves-diagonal", "quot-lemma-curves-RS-star", "quot-lemma-curves-limits", "quot-lemma-curves-existence", "quot-lemma-curves-defo-thy", "quot-remark-curves-base-change", "algebraic-lemma-2-fibre-product" ], "ref_ids": [ 11380, 3209, 3211, 3210, 3213, 3214, 3243, 8467 ] } ], "ref_ids": [ 3243 ] }, { "id": 3150, "type": "theorem", "label": "quot-theorem-complexes-algebraic", "categories": [ "quot" ], "title": "quot-theorem-complexes-algebraic", "contents": [ "\\begin{reference}", "\\cite{lieblich-complexes}", "\\end{reference}", "Let $S$ be a scheme. Let $f : X \\to B$ be morphism of algebraic spaces", "over $S$. Assume that $f$ is proper, flat, and of finite presentation.", "Then $\\Complexesstack_{X/B}$ is an algebraic stack over $S$." ], "refs": [], "proofs": [ { "contents": [ "Set $\\mathcal{X} = \\Complexesstack_{X/B}$. We have seen that $\\mathcal{X}$", "is a stack in groupoids over $(\\Sch/S)_{fppf}$ with diagonal representable", "by algebraic spaces", "(Lemmas \\ref{lemma-complexes-stack} and \\ref{lemma-complexes-diagonal}).", "Hence it suffices to find a scheme $W$ and a surjective and smooth", "morphism $W \\to \\mathcal{X}$.", "\\medskip\\noindent", "Let $B'$ be a scheme and let $B' \\to B$ be a surjective \\'etale morphism.", "Set $X' = B' \\times_B X$ and denote $f' : X' \\to B'$ the projection.", "Then $\\mathcal{X}' = \\Complexesstack_{X'/B'}$ is equal to the $2$-fibre", "product of $\\mathcal{X}$ with the category fibred in sets", "associated to $B'$ over the category fibred in sets associated to $B$", "(Remark \\ref{remark-complexes-base-change}). By the material in", "Algebraic Stacks, Section \\ref{algebraic-section-representable-properties}", "the morphism $\\mathcal{X}' \\to \\mathcal{X}$ is surjective and \\'etale.", "Hence it suffices to prove the result for $\\mathcal{X}'$.", "In other words, we may assume $B$ is a scheme.", "\\medskip\\noindent", "Assume $B$ is a scheme. In this case we may replace $S$ by $B$, see", "Algebraic Stacks, Section \\ref{algebraic-section-change-base-scheme}.", "Thus we may assume $S = B$.", "\\medskip\\noindent", "Assume $S = B$. Choose an affine open covering $S = \\bigcup U_i$.", "Denote $\\mathcal{X}_i$ the restriction of $\\mathcal{X}$ to", "$(\\Sch/U_i)_{fppf}$. If we can find schemes $W_i$ over $U_i$ and", "surjective smooth morphisms $W_i \\to \\mathcal{X}_i$, then we", "set $W = \\coprod W_i$ and we obtain a surjective smooth morphism", "$W \\to \\mathcal{X}$. Thus we may assume $S = B$ is affine.", "\\medskip\\noindent", "Assume $S = B$ is affine, say $S = \\Spec(\\Lambda)$.", "Write $\\Lambda = \\colim \\Lambda_i$ as a filtered colimit with each $\\Lambda_i$", "of finite type over $\\mathbf{Z}$. For some $i$ we can find", "a morphism of algebraic spaces $X_i \\to \\Spec(\\Lambda_i)$", "which is proper, flat, of finite presentation and whose base change", "to $\\Lambda$ is $X$. See Limits of Spaces, Lemmas", "\\ref{spaces-limits-lemma-descend-finite-presentation},", "\\ref{spaces-limits-lemma-descend-flat}, and", "\\ref{spaces-limits-lemma-eventually-proper}.", "If we show that $\\Complexesstack_{X_i/\\Spec(\\Lambda_i)}$ is an", "algebraic stack, then it follows by base change", "(Remark \\ref{remark-complexes-base-change} and", "Algebraic Stacks, Section \\ref{algebraic-section-change-base-scheme})", "that $\\mathcal{X}$ is an algebraic stack.", "Thus we may assume that $\\Lambda$ is a finite type $\\mathbf{Z}$-algebra.", "\\medskip\\noindent", "Assume $S = B = \\Spec(\\Lambda)$ is affine of finite type over $\\mathbf{Z}$.", "In this case we will verify conditions (1), (2), (3), (4), and (5) of", "Artin's Axioms, Lemma \\ref{artin-lemma-diagonal-representable}", "to conclude that $\\mathcal{X}$ is an algebraic stack.", "Note that $\\Lambda$ is a G-ring, see", "More on Algebra, Proposition \\ref{more-algebra-proposition-ubiquity-G-ring}.", "Hence all local rings of $S$ are G-rings. Thus (5) holds.", "To check (2) we have to verify axioms [-1], [0], [1], [2], and [3]", "of Artin's Axioms, Section \\ref{artin-section-axioms}.", "We omit the verification of [-1] and axioms", "[0], [1], [2], [3] correspond respectively to", "Lemmas \\ref{lemma-complexes-stack},", "\\ref{lemma-complexes-limits},", "\\ref{lemma-complexes-RS-star},", "\\ref{lemma-complexes-tangent-space}.", "Condition (3) follows from Lemma \\ref{lemma-complexes-strong-effectiveness}.", "Condition (1) is Lemma \\ref{lemma-complexes-diagonal}.", "\\medskip\\noindent", "It remains to show condition (4) which is openness of versality.", "To see this we will use", "Artin's Axioms, Lemma \\ref{artin-lemma-SGE-implies-openness-versality}.", "We have already seen that $\\mathcal{X}$ has diagonal", "representable by algebraic spaces, has (RS*), and is limit preserving", "(see lemmas used above).", "Hence we only need to see that $\\mathcal{X}$ satisfies the strong", "formal effectiveness formulated in", "Artin's Axioms, Lemma \\ref{artin-lemma-SGE-implies-openness-versality}.", "This follows from Lemma \\ref{lemma-complexes-strong-effectiveness}", "and the proof is complete." ], "refs": [ "quot-lemma-complexes-stack", "quot-lemma-complexes-diagonal", "quot-remark-complexes-base-change", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-flat", "spaces-limits-lemma-eventually-proper", "quot-remark-complexes-base-change", "artin-lemma-diagonal-representable", "more-algebra-proposition-ubiquity-G-ring", "quot-lemma-complexes-stack", "quot-lemma-complexes-limits", "quot-lemma-complexes-RS-star", "quot-lemma-complexes-tangent-space", "quot-lemma-complexes-strong-effectiveness", "quot-lemma-complexes-diagonal", "artin-lemma-SGE-implies-openness-versality", "artin-lemma-SGE-implies-openness-versality", "quot-lemma-complexes-strong-effectiveness" ], "ref_ids": [ 3220, 3219, 3245, 4598, 4595, 4596, 3245, 11380, 10582, 3220, 3221, 3222, 3223, 3224, 3219, 11386, 11386, 3224 ] } ], "ref_ids": [] }, { "id": 3151, "type": "theorem", "label": "quot-lemma-hom-sheaf", "categories": [ "quot" ], "title": "quot-lemma-hom-sheaf", "contents": [ "In Situation \\ref{situation-hom} the functor", "$\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ ", "satisfies the sheaf property for the fpqc topology." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{T_i \\to T\\}_{i \\in I}$ be an fpqc covering of schemes over $B$.", "Set $X_i = X_{T_i} = X \\times_S T_i$ and $\\mathcal{F}_i = u_{T_i}$", "and $\\mathcal{G}_i = \\mathcal{G}_{T_i}$.", "Note that $\\{X_i \\to X_T\\}_{i \\in I}$ is an fpqc covering of $X_T$, see", "Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-fpqc}.", "Thus a family of maps $u_i : \\mathcal{F}_i \\to \\mathcal{G}_i$", "such that $u_i$ and $u_j$ restrict to the same map on", "$X_{T_i \\times_T T_j}$ comes from a unique map", "$u : \\mathcal{F}_T \\to \\mathcal{G}_T$ by descent", "(Descent on Spaces, Proposition", "\\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent})." ], "refs": [ "spaces-topologies-lemma-fpqc", "spaces-descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 3678, 9437 ] } ], "ref_ids": [] }, { "id": 3152, "type": "theorem", "label": "quot-lemma-extend-hom-to-spaces", "categories": [ "quot" ], "title": "quot-lemma-extend-hom-to-spaces", "contents": [ "In Situation \\ref{situation-hom}. Let $T$ be an algebraic space over $S$.", "We have", "$$", "\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\mathit{Hom}(\\mathcal{F}, \\mathcal{G})) =", "\\{(h, u) \\mid h : T \\to B, u : \\mathcal{F}_T \\to \\mathcal{G}_T\\}", "$$", "where $\\mathcal{F}_T, \\mathcal{G}_T$ denote the pullbacks of $\\mathcal{F}$", "and $\\mathcal{G}$ to the algebraic space $X \\times_{B, h} T$." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $p : U \\to T$.", "Let $R = U \\times_T U$ with projections $t, s : R \\to U$.", "\\medskip\\noindent", "Let $v : T \\to \\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$", "be a natural transformation. Then $v(p)$ corresponds to a pair", "$(h_U, u_U)$ over $U$. As $v$ is a transformation of functors we see", "that the pullbacks of $(h_U, u_U)$ by $s$ and $t$ agree.", "Since $T = U/R$ (Spaces, Lemma \\ref{spaces-lemma-space-presentation}),", "we obtain a morphism $h : T \\to B$ such that", "$h_U = h \\circ p$. Then $\\mathcal{F}_U$ is the pullback of", "$\\mathcal{F}_T$ to $X_U$ and similarly for $\\mathcal{G}_U$.", "Hence $u_U$ descends to a $\\mathcal{O}_{X_T}$-module map", "$u : \\mathcal{F}_T \\to \\mathcal{G}_T$ by", "Descent on Spaces, Proposition", "\\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}.", "\\medskip\\noindent", "Conversely, let $(h, u)$ be a pair over $T$. Then we get a natural", "transformation $v : T \\to \\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$", "by sending a morphism $a : T' \\to T$ where $T'$ is a scheme", "to $(h \\circ a, a^*u)$. We omit the verification that the construction", "of this and the previous paragraph are mutually inverse." ], "refs": [ "spaces-lemma-space-presentation", "spaces-descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 8149, 9437 ] } ], "ref_ids": [] }, { "id": 3153, "type": "theorem", "label": "quot-lemma-hom-sheaf-in-X", "categories": [ "quot" ], "title": "quot-lemma-hom-sheaf-in-X", "contents": [ "In Situation \\ref{situation-hom} let $\\{X_i \\to X\\}_{i \\in I}$ be an fppf", "covering and for each $i, j \\in I$ let $\\{X_{ijk} \\to X_i \\times_X X_j\\}$", "be an fppf covering. Denote $\\mathcal{F}_i$, resp.\\ $\\mathcal{F}_{ijk}$", "the pullback of $\\mathcal{F}$ to $X_i$, resp.\\ $X_{ijk}$. Similarly", "define $\\mathcal{G}_i$ and $\\mathcal{G}_{ijk}$. For every scheme", "$T$ over $B$ the diagram", "$$", "\\xymatrix{", "\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})(T) \\ar[r] &", "\\prod\\nolimits_i", "\\mathit{Hom}(\\mathcal{F}_i, \\mathcal{G}_i)(T)", "\\ar@<1ex>[r]^-{\\text{pr}_0^*} \\ar@<-1ex>[r]_-{\\text{pr}_1^*}", "&", "\\prod\\nolimits_{i, j, k}", "\\mathit{Hom}(\\mathcal{F}_{ijk}, \\mathcal{G}_{ijk})(T)", "}", "$$", "presents the first arrow as the equalizer of the other two." ], "refs": [], "proofs": [ { "contents": [ "Let $u_i : \\mathcal{F}_{i, T} \\to \\mathcal{G}_{i, T}$ be an element in the", "equalizer of $\\text{pr}_0^*$ and $\\text{pr}_1^*$. Since the base change", "of an fppf covering is an fppf covering", "(Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-fppf})", "we see that $\\{X_{i, T} \\to X_T\\}_{i \\in I}$ and", "$\\{X_{ijk, T} \\to X_{i, T} \\times_{X_T} X_{j, T}\\}$ are fppf coverings.", "Applying Descent on Spaces, Proposition", "\\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}", "we first conclude that $u_i$ and $u_j$ restrict to the same morphism", "over $X_{i, T} \\times_{X_T} X_{j, T}$, whereupon a second application", "shows that there is a unique morphism $u : \\mathcal{F}_T \\to \\mathcal{G}_T$", "restricting to $u_i$ for each $i$. This finishes the proof." ], "refs": [ "spaces-topologies-lemma-fppf", "spaces-descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 3665, 9437 ] } ], "ref_ids": [] }, { "id": 3154, "type": "theorem", "label": "quot-lemma-hom-limits", "categories": [ "quot" ], "title": "quot-lemma-hom-limits", "contents": [ "In Situation \\ref{situation-hom}. If $\\mathcal{F}$ is of finite presentation", "and $f$ is quasi-compact and quasi-separated, then", "$\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ is limit preserving." ], "refs": [], "proofs": [ { "contents": [ "Let $T = \\lim_{i \\in I} T_i$ be a directed limit of affine $B$-schemes.", "We have to show that", "$$", "\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})(T) =", "\\colim \\mathit{Hom}(\\mathcal{F}, \\mathcal{G})(T_i)", "$$", "Pick $0 \\in I$. We may replace $B$ by $T_0$, $X$ by $X_{T_0}$,", "$\\mathcal{F}$ by $\\mathcal{F}_{T_0}$, $\\mathcal{G}$ by", "$\\mathcal{G}_{T_0}$, and $I$ by $\\{i \\in I \\mid i \\geq 0\\}$.", "See Remark \\ref{remark-hom-base-change}.", "Thus we may assume $B = \\Spec(R)$ is affine.", "\\medskip\\noindent", "When $B$ is affine, then $X$ is quasi-compact and quasi-separated.", "Choose a surjective \\'etale morphism $U \\to X$ where $U$ is an", "affine scheme (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "Since $X$ is quasi-separated, the scheme $U \\times_X U$ is quasi-compact", "and we may choose a surjective \\'etale morphism $V \\to U \\times_X U$", "where $V$ is an affine scheme. Applying Lemma \\ref{lemma-hom-sheaf-in-X}", "we see that $\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ is the", "equalizer of two maps between", "$$", "\\mathit{Hom}(\\mathcal{F}|_U, \\mathcal{G}|_U)", "\\quad\\text{and}\\quad", "\\mathit{Hom}(\\mathcal{F}|_V, \\mathcal{G}|_V)", "$$", "This reduces us to the case that $X$ is affine.", "\\medskip\\noindent", "In the affine case the statement of the lemma reduces to", "the following problem: Given a ring map $R \\to A$, two $A$-modules", "$M$, $N$ and a directed system of $R$-algebras $C = \\colim C_i$.", "When is it true that the map", "$$", "\\colim \\Hom_{A \\otimes_R C_i}(M \\otimes_R C_i, N \\otimes_R C_i)", "\\longrightarrow", "\\Hom_{A \\otimes_R C}(M \\otimes_R C, N \\otimes_R C)", "$$", "is bijective? By", "Algebra, Lemma \\ref{algebra-lemma-module-map-property-in-colimit}", "this holds if $M \\otimes_R C$ is of finite presentation over", "$A \\otimes_R C$, i.e., when $M$ is of finite presentation over $A$." ], "refs": [ "quot-remark-hom-base-change", "spaces-properties-lemma-quasi-compact-affine-cover", "quot-lemma-hom-sheaf-in-X", "algebra-lemma-module-map-property-in-colimit" ], "ref_ids": [ 3232, 11832, 3153, 1094 ] } ], "ref_ids": [] }, { "id": 3155, "type": "theorem", "label": "quot-lemma-hom-closed", "categories": [ "quot" ], "title": "quot-lemma-hom-closed", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $i : X' \\to X$ be a closed immersion of algebraic spaces", "over $B$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module", "and let $\\mathcal{G}'$ be a quasi-coherent $\\mathcal{O}_{X'}$-module.", "Then", "$$", "\\mathit{Hom}(\\mathcal{F}, i_*\\mathcal{G}') =", "\\mathit{Hom}(i^*\\mathcal{F}, \\mathcal{G}')", "$$", "as functors on $(\\Sch/B)$." ], "refs": [], "proofs": [ { "contents": [ "Let $g : T \\to B$ be a morphism where $T$ is a scheme.", "Denote $i_T : X'_T \\to X_T$ the base change of $i$.", "Denote $h : X_T \\to X$ and $h' : X'_T \\to X'$ the projections.", "Observe that $(h')^*i^*\\mathcal{F} = i_T^*h^*\\mathcal{F}$.", "As a closed immersion is affine", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-closed-immersion-affine})", "we have $h^*i_*\\mathcal{G} = i_{T, *}(h')^*\\mathcal{G}$ by", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-affine-base-change}.", "Thus we have", "\\begin{align*}", "\\mathit{Hom}(\\mathcal{F}, i_*\\mathcal{G}')(T)", "& =", "\\Hom_{\\mathcal{O}_{X_T}}(h^*\\mathcal{F}, h^*i_*\\mathcal{G}') \\\\", "& =", "\\Hom_{\\mathcal{O}_{X_T}}(h^*\\mathcal{F}, i_{T, *}(h')^*\\mathcal{G}) \\\\", "& =", "\\Hom_{\\mathcal{O}_{X'_T}}(i_T^*h^*\\mathcal{F}, (h')^*\\mathcal{G}) \\\\", "& =", "\\Hom_{\\mathcal{O}_{X'_T}}((h')^*i^*\\mathcal{F}, (h')^*\\mathcal{G}) \\\\", "& =", "\\mathit{Hom}(i^*\\mathcal{F}, \\mathcal{G}')(T)", "\\end{align*}", "as desired. The middle equality follows from the adjointness of the functors", "$i_{T, *}$ and $i_T^*$." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-affine", "spaces-cohomology-lemma-affine-base-change" ], "ref_ids": [ 4801, 11295 ] } ], "ref_ids": [] }, { "id": 3156, "type": "theorem", "label": "quot-lemma-cohomology-perfect-complex", "categories": [ "quot" ], "title": "quot-lemma-cohomology-perfect-complex", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $K$ be a pseudo-coherent object of $D(\\mathcal{O}_B)$.", "\\begin{enumerate}", "\\item If for all $g : T \\to B$ in $(\\Sch/B)$ the cohomology sheaf", "$H^{-1}(Lg^*K)$ is zero, then the functor", "$$", "(\\Sch/B)^{opp} \\longrightarrow \\textit{Sets},\\quad", "(g : T \\to B) \\longmapsto H^0(T, H^0(Lg^*K))", "$$", "is an algebraic space affine and of finite presentation over $B$.", "\\item If for all $g : T \\to B$ in $(\\Sch/B)$ the cohomology sheaves", "$H^i(Lg^*K)$ are zero for $i < 0$, then $K$ is perfect,", "$K$ locally has tor amplitude in $[0, b]$, and the functor", "$$", "(\\Sch/B)^{opp} \\longrightarrow \\textit{Sets},\\quad", "(g : T \\to B) \\longmapsto H^0(T, Lg^*K)", "$$", "is an algebraic space affine and of finite presentation over $B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Under the assumptions of (2) we have $H^0(T, Lg^*K) = H^0(T, H^0(Lg^*K))$.", "Let us prove that the rule $T \\mapsto H^0(T, H^0(Lg^*K))$ satisfies the", "sheaf property for the fppf topology. To do this assume we have an", "fppf covering $\\{h_i : T_i \\to T\\}$ of a scheme $g : T \\to B$ over $B$.", "Set $g_i = g \\circ h_i$. Note that since $h_i$ is flat, we have", "$Lh_i^* = h_i^*$ and $h_i^*$ commutes with taking cohomology. Hence", "$$", "H^0(T_i, H^0(Lg_i^*K)) =", "H^0(T_i, H^0(h_i^*Lg^*K)) =", "H^0(T, h_i^*H^0(Lg^*K))", "$$", "Similarly for the pullback to $T_i \\times_T T_j$.", "Since $Lg^*K$ is a pseudo-coherent complex on $T$", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-pseudo-coherent-pullback})", "the cohomology sheaf $\\mathcal{F} = H^0(Lg^*K)$ is quasi-coherent", "(Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-pseudo-coherent}).", "Hence by Descent on Spaces, Proposition", "\\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}", "we see that", "$$", "H^0(T, \\mathcal{F}) = \\Ker(", "\\prod H^0(T_i, h_i^*\\mathcal{F}) \\to", "\\prod H^0(T_i \\times_T T_j, (T_i \\times_T T_j \\to T)^*\\mathcal{F}))", "$$", "In this way we see that the rules in (1) and (2) satisfy", "the sheaf property for fppf coverings. This means we may apply", "Bootstrap, Lemma \\ref{bootstrap-lemma-locally-algebraic-space-finite-type}", "to see it suffices to prove the representability \\'etale locally on $B$.", "Moreover, we may check whether the end result is affine and", "of finite presentation \\'etale locally on $B$, see", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-affine-local} and", "\\ref{spaces-morphisms-lemma-finite-presentation-local}.", "Hence we may assume that $B$ is an affine scheme.", "\\medskip\\noindent", "Assume $B = \\Spec(A)$ is an affine scheme. By the results of", "Derived Categories of Spaces, Lemmas", "\\ref{spaces-perfect-lemma-pseudo-coherent},", "\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}, and", "\\ref{spaces-perfect-lemma-descend-pseudo-coherent}", "we deduce that in the rest of the proof we may think of $K$ as a perfect", "object of the derived category of complexes of modules on $B$", "in the Zariski topology. By ", "Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-pseudo-coherent},", "\\ref{perfect-lemma-affine-compare-bounded}, and", "\\ref{perfect-lemma-pseudo-coherent-affine} we can find a pseudo-coherent", "complex $M^\\bullet$ of $A$-modules such that $K$ is the corresponding", "object of $D(\\mathcal{O}_B)$. Our assumption on pullbacks implies", "that $M^\\bullet \\otimes^\\mathbf{L}_A \\kappa(\\mathfrak p)$", "has vanishing $H^{-1}$ for all primes $\\mathfrak p \\subset A$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-cut-complex-in-two}", "we can write", "$$", "M^\\bullet =", "\\tau_{\\geq 0}M^\\bullet \\oplus \\tau_{\\leq - 1}M^\\bullet", "$$", "with $\\tau_{\\geq 0}M^\\bullet$ perfect with Tor amplitude in $[0, b]$", "for some $b \\geq 0$ (here we also have used", "More on Algebra, Lemmas \\ref{more-algebra-lemma-glue-perfect} and", "\\ref{more-algebra-lemma-glue-tor-amplitude}).", "Note that in case (2) we also see that $\\tau_{\\leq - 1}M^\\bullet = 0$", "in $D(A)$ whence $M^\\bullet$ and $K$ are perfect with", "tor amplitude in $[0, b]$. For any $B$-scheme $g : T \\to B$ we have", "$$", "H^0(T, H^0(Lg^*K)) = H^0(T, H^0(Lg^*\\tau_{\\geq 0}K))", "$$", "(by the dual of Derived Categories, Lemma", "\\ref{derived-lemma-negative-vanishing})", "hence we may replace $K$ by $\\tau_{\\geq 0}K$ and correspondingly", "$M^\\bullet$ by $\\tau_{\\geq 0}M^\\bullet$. In other words, we may", "assume $M^\\bullet$ has tor amplitude in $[0, b]$.", "\\medskip\\noindent", "Assume $M^\\bullet$ has tor amplitude in $[0, b]$.", "We may assume $M^\\bullet$ is a bounded above complex of finite free", "$A$-modules (by our definition of pseudo-coherent complexes, see", "More on Algebra, Definition \\ref{more-algebra-definition-pseudo-coherent}", "and the discussion following the definition).", "By More on Algebra, Lemma \\ref{more-algebra-lemma-last-one-flat}", "we see that $M = \\Coker(M^{- 1} \\to M^0)$ is flat. By", "Algebra, Lemma \\ref{algebra-lemma-finite-projective} we see that $M$", "is finite locally free. Hence $M^\\bullet$ is quasi-isomorphic to", "$$", "M \\to M^1 \\to M^2 \\to \\ldots \\to M^d \\to 0 \\ldots", "$$", "Note that this is a K-flat complex", "(Cohomology, Lemma \\ref{cohomology-lemma-bounded-flat-K-flat}),", "hence derived pullback of $K$ via a morphism $T \\to B$ is computed", "by the complex", "$$", "g^*\\widetilde{M} \\to g^*\\widetilde{M^1} \\to \\ldots", "$$", "Thus it suffices to show that the functor", "$$", "(g : T \\to B) \\longmapsto", "\\Ker(", "\\Gamma(T,g^*\\widetilde{M})", "\\to", "\\Gamma(T, g^*(\\widetilde{M^1})", ")", "$$", "is representable by an affine scheme of finite presentation over $B$.", "\\medskip\\noindent", "We may still replace $B$ by the members of an affine open covering", "in order to prove this last statement. Hence we may assume that $M$", "is finite free (recall that $M^1$ is finite free to begin with).", "Write $M = A^{\\oplus n}$ and $M^1 = A^{\\oplus m}$. Let the map", "$M \\to M^1$ be given by the $m \\times n$ matrix $(a_{ij})$ with", "coefficients in $A$. Then $\\widetilde{M} = \\mathcal{O}_B^{\\oplus n}$", "and $\\widetilde{M^1} = \\mathcal{O}_B^{\\oplus m}$. Thus the functor", "above is equal to the functor", "$$", "(g : T \\to B) \\longmapsto", "\\{(f_1, \\ldots, f_n) \\in \\Gamma(T, \\mathcal{O}_T) \\mid", "\\sum g^\\sharp(a_{ij})f_i = 0,\\ j = 1, \\ldots, m\\}", "$$", "Clearly this is representable by the affine scheme", "$$", "\\Spec\\left(A[x_1, \\ldots, x_n]/(\\sum a_{ij}x_i; j = 1, \\ldots, m)\\right)", "$$", "and the lemma has been proved." ], "refs": [ "sites-cohomology-lemma-pseudo-coherent-pullback", "spaces-perfect-lemma-pseudo-coherent", "spaces-descent-proposition-fpqc-descent-quasi-coherent", "bootstrap-lemma-locally-algebraic-space-finite-type", "spaces-morphisms-lemma-affine-local", "spaces-morphisms-lemma-finite-presentation-local", "spaces-perfect-lemma-pseudo-coherent", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-descend-pseudo-coherent", "perfect-lemma-pseudo-coherent", "perfect-lemma-affine-compare-bounded", "perfect-lemma-pseudo-coherent-affine", "more-algebra-lemma-cut-complex-in-two", "more-algebra-lemma-glue-perfect", "more-algebra-lemma-glue-tor-amplitude", "derived-lemma-negative-vanishing", "more-algebra-definition-pseudo-coherent", "more-algebra-lemma-last-one-flat", "algebra-lemma-finite-projective", "cohomology-lemma-bounded-flat-K-flat" ], "ref_ids": [ 4367, 2696, 9437, 2626, 4798, 4841, 2696, 2644, 2692, 6974, 6941, 6975, 10238, 10221, 10183, 1839, 10623, 10169, 795, 2109 ] } ], "ref_ids": [] }, { "id": 3157, "type": "theorem", "label": "quot-lemma-noetherian-hom", "categories": [ "quot" ], "title": "quot-lemma-noetherian-hom", "contents": [ "In Situation \\ref{situation-hom} assume that", "\\begin{enumerate}", "\\item $B$ is a Noetherian algebraic space,", "\\item $f$ is locally of finite type and quasi-separated,", "\\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module, and", "\\item $\\mathcal{G}$ is a finite type $\\mathcal{O}_X$-module, flat over $B$,", "with support proper over $B$.", "\\end{enumerate}", "Then the functor $\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ is", "an algebraic space affine and of finite presentation over $B$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $X$ by a quasi-compact open neighbourhood of", "the support of $\\mathcal{G}$, hence we may assume $X$ is Noetherian.", "In this case $X$ and $f$ are quasi-compact and quasi-separated.", "Choose an approximation $P \\to \\mathcal{F}$ by a perfect complex $P$ of", "the triple $(X, \\mathcal{F}, -1)$, see", "Derived Categories of Spaces, Definition", "\\ref{spaces-perfect-definition-approximation-holds} and", "Theorem \\ref{spaces-perfect-theorem-approximation}).", "Then the induced map", "$$", "\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})", "\\longrightarrow", "\\Hom_{D(\\mathcal{O}_X)}(P, \\mathcal{G})", "$$", "is an isomorphism because $P \\to \\mathcal{F}$ induces an isomorphism", "$H^0(P) \\to \\mathcal{F}$ and $H^i(P) = 0$ for $i > 0$.", "Moreover, for any morphism $g : T \\to B$", "denote $h : X_T = T \\times_B X \\to X$ the projection and set", "$P_T = Lh^*P$. Then it is equally true that", "$$", "\\Hom_{\\mathcal{O}_{X_T}}(\\mathcal{F}_T, \\mathcal{G}_T)", "\\longrightarrow", "\\Hom_{D(\\mathcal{O}_{X_T})}(P_T, \\mathcal{G}_T)", "$$", "is an isomorphism, as $P_T = Lh^*P \\to Lh^*\\mathcal{F} \\to \\mathcal{F}_T$", "induces an isomorphism $H^0(P_T) \\to \\mathcal{F}_T$ (because $h^*$ is", "right exact and $H^i(P) = 0$ for $i > 0$). Thus it suffices to prove the", "result for the functor", "$$", "T \\longmapsto \\Hom_{D(\\mathcal{O}_{X_T})}(P_T, \\mathcal{G}_T).", "$$", "By the Leray spectral sequence (see Cohomology on Sites, Remark", "\\ref{sites-cohomology-remark-before-Leray}) we have", "$$", "\\Hom_{D(\\mathcal{O}_{X_T})}(P_T, \\mathcal{G}_T) =", "H^0(X_T, R\\SheafHom(P_T, \\mathcal{G}_T)) =", "H^0(T, Rf_{T, *}R\\SheafHom(P_T, \\mathcal{G}_T))", "$$", "where $f_T : X_T \\to T$ is the base change of $f$. By", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-base-change-RHom}", "we have", "$$", "Rf_{T, *}R\\SheafHom(P_T, \\mathcal{G}_T) = Lg^*Rf_*R\\SheafHom(P, \\mathcal{G}).", "$$", "By", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-ext-perfect}", "the object $K = Rf_*R\\SheafHom(P, \\mathcal{G})$ of $D(\\mathcal{O}_B)$", "is perfect. This means we can apply", "Lemma \\ref{lemma-cohomology-perfect-complex}", "as long as we can prove that the cohomology sheaf", "$H^i(Lg^*K)$ is $0$ for all $i < 0$ and $g : T \\to B$ as above.", "This is clear from the last displayed formula as", "the cohomology sheaves of", "$Rf_{T, *}R\\SheafHom(P_T, \\mathcal{G}_T)$", "are zero in negative degrees", "due to the fact that $R\\SheafHom(P_T, \\mathcal{G}_T)$ has vanishing", "cohomology sheaves in negative degrees as $P_T$ is perfect with", "vanishing cohomology sheaves in positive degrees." ], "refs": [ "spaces-perfect-definition-approximation-holds", "spaces-perfect-theorem-approximation", "sites-cohomology-remark-before-Leray", "spaces-perfect-lemma-base-change-RHom", "spaces-perfect-lemma-ext-perfect", "quot-lemma-cohomology-perfect-complex" ], "ref_ids": [ 2765, 2639, 4423, 2727, 2730, 3156 ] } ], "ref_ids": [] }, { "id": 3158, "type": "theorem", "label": "quot-lemma-isom-sheaf", "categories": [ "quot" ], "title": "quot-lemma-isom-sheaf", "contents": [ "In Situation \\ref{situation-hom} the functor", "$\\mathit{Isom}(\\mathcal{F}, \\mathcal{G})$ ", "satisfies the sheaf property for the fpqc topology." ], "refs": [], "proofs": [ { "contents": [ "We have already seen that $\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$", "satisfies the sheaf property. Hence it remains to show the following:", "Given an fpqc covering $\\{T_i \\to T\\}_{i \\in I}$ of schemes over $B$", "and an $\\mathcal{O}_{X_T}$-linear map", "$u : \\mathcal{F}_T \\to \\mathcal{G}_T$ such that", "$u_{T_i}$ is an isomorphism for all $i$, then $u$ is an isomorphism.", "Since $\\{X_i \\to X_T\\}_{i \\in I}$ is an fpqc covering of $X_T$, see", "Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-fpqc},", "this follows from", "Descent on Spaces, Proposition", "\\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}." ], "refs": [ "spaces-topologies-lemma-fpqc", "spaces-descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 3678, 9437 ] } ], "ref_ids": [] }, { "id": 3159, "type": "theorem", "label": "quot-lemma-extend-isom-to-spaces", "categories": [ "quot" ], "title": "quot-lemma-extend-isom-to-spaces", "contents": [ "In Situation \\ref{situation-hom}. Let $T$ be an algebraic space over $S$.", "We have", "$$", "\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\mathit{Isom}(\\mathcal{F}, \\mathcal{G})) =", "\\{(h, u) \\mid", "h : T \\to B, u : \\mathcal{F}_T \\to \\mathcal{G}_T\\text{ isomorphism}\\}", "$$", "where $\\mathcal{F}_T, \\mathcal{G}_T$ denote the pullbacks of $\\mathcal{F}$", "and $\\mathcal{G}$ to the algebraic space $X \\times_{B, h} T$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the left and right hand side of the equality are", "subsets of the left and right hand side of the equality in", "Lemma \\ref{lemma-extend-hom-to-spaces}.", "We omit the verification that these subsets correspond under", "the identification given in the proof of that lemma." ], "refs": [ "quot-lemma-extend-hom-to-spaces" ], "ref_ids": [ 3152 ] } ], "ref_ids": [] }, { "id": 3160, "type": "theorem", "label": "quot-lemma-coherent-fibred-in-groupoids", "categories": [ "quot" ], "title": "quot-lemma-coherent-fibred-in-groupoids", "contents": [ "In Situation \\ref{situation-coherent} the functor", "$p : \\Cohstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf}$", "is fibred in groupoids." ], "refs": [], "proofs": [ { "contents": [ "We show that $p$ is fibred in groupoids by checking conditions", "(1) and (2) of Categories, Definition", "\\ref{categories-definition-fibred-groupoids}.", "Given an object $(T', g', \\mathcal{F}')$", "of $\\Cohstack_{X/B}$ and a morphism $h : T \\to T'$ of", "schemes over $S$ we can set $g = h \\circ g'$ and", "$\\mathcal{F} = (h')^*\\mathcal{F}'$ where $h' : X_T \\to X_{T'}$", "is the base change of $h$. Then it is clear that we obtain", "a morphism $(T, g, \\mathcal{F}) \\to (T', g', \\mathcal{F}')$", "of $\\Cohstack_{X/B}$ lying over $h$. This proves (1).", "For (2) suppose we are given morphisms", "$$", "(h_1, \\varphi_1) : (T_1, g_1, \\mathcal{F}_1) \\to (T, g, \\mathcal{F})", "\\quad\\text{and}\\quad", "(h_2, \\varphi_2) : (T_2, g_2, \\mathcal{F}_2) \\to (T, g, \\mathcal{F})", "$$", "of $\\Cohstack_{X/B}$ and a morphism $h : T_1 \\to T_2$ such that", "$h_2 \\circ h = h_1$. Then we can let $\\varphi$ be the composition", "$$", "(h')^*\\mathcal{F}_2", "\\xrightarrow{(h')^*\\varphi_2^{-1}}", "(h')^*(h_2)^*\\mathcal{F} = (h_1)^*\\mathcal{F}", "\\xrightarrow{\\varphi_1}", "\\mathcal{F}_1", "$$", "to obtain the morphism", "$(h, \\varphi) : (T_1, g_1, \\mathcal{F}_1) \\to (T_2, g_2, \\mathcal{F}_2)$", "that witnesses the truth of condition (2)." ], "refs": [ "categories-definition-fibred-groupoids" ], "ref_ids": [ 12392 ] } ], "ref_ids": [] }, { "id": 3161, "type": "theorem", "label": "quot-lemma-coherent-diagonal", "categories": [ "quot" ], "title": "quot-lemma-coherent-diagonal", "contents": [ "In Situation \\ref{situation-coherent}. Denote", "$\\mathcal{X} = \\Cohstack_{X/B}$. Then", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is", "representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Consider two objects $x = (T, g, \\mathcal{F})$ and $y = (T, h, \\mathcal{G})$", "of $\\mathcal{X}$ over a scheme $T$. We have to show that", "$\\mathit{Isom}_\\mathcal{X}(x, y)$ is an algebraic space over $T$, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-representable-diagonal}.", "If for $a : T' \\to T$ the restrictions $x|_{T'}$ and $y|_{T'}$ are isomorphic", "in the fibre category $\\mathcal{X}_{T'}$, then $g \\circ a = h \\circ a$.", "Hence there is a transformation of presheaves", "$$", "\\mathit{Isom}_\\mathcal{X}(x, y) \\longrightarrow \\text{Equalizer}(g, h)", "$$", "Since the diagonal of $B$ is representable (by schemes) this equalizer is", "a scheme. Thus we may replace $T$ by this equalizer and the sheaves", "$\\mathcal{F}$ and $\\mathcal{G}$ by their pullbacks. Thus we may assume", "$g = h$. In this case we have", "$\\mathit{Isom}_\\mathcal{X}(x, y) = \\mathit{Isom}(\\mathcal{F}, \\mathcal{G})$", "and the result follows from Proposition \\ref{proposition-isom}." ], "refs": [ "algebraic-lemma-representable-diagonal", "quot-proposition-isom" ], "ref_ids": [ 8461, 3226 ] } ], "ref_ids": [] }, { "id": 3162, "type": "theorem", "label": "quot-lemma-coherent-stack", "categories": [ "quot" ], "title": "quot-lemma-coherent-stack", "contents": [ "In Situation \\ref{situation-coherent} the functor", "$p : \\Cohstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf}$", "is a stack in groupoids." ], "refs": [], "proofs": [ { "contents": [ "To prove that $\\Cohstack_{X/B}$ is a stack in groupoids, we have to show", "that the presheaves $\\mathit{Isom}$ are sheaves and that descent data are", "effective. The statement on $\\mathit{Isom}$ follows from", "Lemma \\ref{lemma-coherent-diagonal}, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-representable-diagonal}.", "Let us prove the statement on descent data.", "Suppose that $\\{a_i : T_i \\to T\\}$ is an fppf covering of schemes over $S$.", "Let $(\\xi_i, \\varphi_{ij})$ be a descent datum for $\\{T_i \\to T\\}$", "with values in $\\Cohstack_{X/B}$.", "For each $i$ we can write $\\xi_i = (T_i, g_i, \\mathcal{F}_i)$.", "Denote $\\text{pr}_0 : T_i \\times_T T_j \\to T_i$ and", "$\\text{pr}_1 : T_i \\times_T T_j \\to T_j$ the projections.", "The condition that $\\xi_i|_{T_i \\times_T T_j} = \\xi_j|_{T_i \\times_T T_j}$", "implies in particular that $g_i \\circ \\text{pr}_0 = g_j \\circ \\text{pr}_1$.", "Thus there exists a unique morphism $g : T \\to B$ such that", "$g_i = g \\circ a_i$, see", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-fpqc-universal-effective-epimorphisms}.", "Denote $X_T = T \\times_{g, B} X$. Set", "$X_i = X_{T_i} = T_i \\times_{g_i, B} X = T_i \\times_{a_i, T} X_T$", "and", "$$", "X_{ij} = X_{T_i} \\times_{X_T} X_{T_j} = X_i \\times_{X_T} X_j", "$$", "with projections $\\text{pr}_i$ and $\\text{pr}_j$ to $X_i$ and $X_j$.", "Observe that the pullback of $(T_i, g_i, \\mathcal{F}_i)$", "by $\\text{pr}_0 : T_i \\times_T T_j \\to T_i$ is given by", "$(T_i \\times_T T_j, g_i \\circ \\text{pr}_0, \\text{pr}_i^*\\mathcal{F}_i)$.", "Hence a descent datum for $\\{T_i \\to T\\}$ in $\\Cohstack_{X/B}$", "is given by the objects $(T_i, g \\circ a_i, \\mathcal{F}_i)$", "and for each pair $i, j$ an isomorphism of $\\mathcal{O}_{X_{ij}}$-modules", "$$", "\\varphi_{ij} :", "\\text{pr}_i^*\\mathcal{F}_i \\longrightarrow \\text{pr}_j^*\\mathcal{F}_j", "$$", "satisfying the cocycle condition over (the pullback of $X$ to)", "$T_i \\times_T T_j \\times_T T_k$.", "Ok, and now we simply use that $\\{X_i \\to X_T\\}$ is an fppf covering", "so that we can view $(\\mathcal{F}_i, \\varphi_{ij})$ as a descent datum", "for this covering. By", "Descent on Spaces, Proposition", "\\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}", "this descent datum is effective and we obtain a quasi-coherent", "sheaf $\\mathcal{F}$ over $X_T$ restricting to $\\mathcal{F}_i$ on $X_i$.", "By Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-permanence}", "we see that $\\mathcal{F}$ is flat over $T$ and", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-finite-presentation-descends}", "guarantees that $\\mathcal{Q}$ is of finite presentation as an", "$\\mathcal{O}_{X_T}$-module. Finally, by", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-descending-property-proper}", "we see that the scheme theoretic support of $\\mathcal{F}$ is proper over", "$T$ as we've assumed the scheme theoretic support of $\\mathcal{F}_i$", "is proper over $T_i$ (note that taking scheme theoretic support commutes", "with flat base change by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-pullback-support}).", "In this way we obtain our desired object over $T$." ], "refs": [ "quot-lemma-coherent-diagonal", "algebraic-lemma-representable-diagonal", "spaces-descent-lemma-fpqc-universal-effective-epimorphisms", "spaces-descent-proposition-fpqc-descent-quasi-coherent", "spaces-morphisms-lemma-flat-permanence", "spaces-descent-lemma-finite-presentation-descends", "spaces-descent-lemma-descending-property-proper", "spaces-morphisms-lemma-flat-pullback-support" ], "ref_ids": [ 3161, 8461, 9367, 9437, 4865, 9360, 9399, 4859 ] } ], "ref_ids": [] }, { "id": 3163, "type": "theorem", "label": "quot-lemma-coherent-limits", "categories": [ "quot" ], "title": "quot-lemma-coherent-limits", "contents": [ "In Situation \\ref{situation-coherent} assume that $B \\to S$", "is locally of finite presentation. Then", "$p : \\Cohstack_{X/B} \\to (\\Sch/S)_{fppf}$ is limit preserving", "(Artin's Axioms, Definition \\ref{artin-definition-limit-preserving})." ], "refs": [ "artin-definition-limit-preserving" ], "proofs": [ { "contents": [ "Write $B(T)$ for the discrete category whose", "objects are the $S$-morphisms $T \\to B$. Let $T = \\lim T_i$ be a filtered", "limit of affine schemes over $S$. Assigning to an object", "$(T, h, \\mathcal{F})$ of $\\Cohstack_{X/B, T}$ the object $h$", "of $B(T)$ gives us a commutative diagram of fibre categories", "$$", "\\xymatrix{", "\\colim \\Cohstack_{X/B, T_i} \\ar[r] \\ar[d] &", "\\Cohstack_{X/B, T} \\ar[d] \\\\", "\\colim B(T_i) \\ar[r] & B(T)", "}", "$$", "We have to show the top horizontal arrow is an equivalence. Since", "we have assumed that $B$ is locally of finite presentation over $S$", "we see from", "Limits of Spaces, Remark \\ref{spaces-limits-remark-limit-preserving}", "that the bottom horizontal arrow is an equivalence. This means that", "we may assume $T = \\lim T_i$ be a filtered limit of affine schemes over", "$B$. Denote $g_i : T_i \\to B$ and $g : T \\to B$ the corresponding", "morphisms. Set $X_i = T_i \\times_{g_i, B} X$ and $X_T = T \\times_{g, B} X$.", "Observe that $X_T = \\colim X_i$ and that the algebraic spaces", "$X_i$ and $X_T$ are quasi-separated and quasi-compact (as they", "are of finite presentation over the affines $T_i$ and $T$).", "By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-modules-finite-presentation}", "we see that", "$$", "\\colim \\textit{FP}(X_i) = \\textit{FP}(X_T).", "$$", "where $\\textit{FP}(W)$ is short hand for the category of finitely", "presented $\\mathcal{O}_W$-modules. The results of", "Limits of Spaces, Lemmas \\ref{spaces-limits-lemma-descend-flat} and", "\\ref{spaces-limits-lemma-eventually-proper-support}", "tell us the same thing is true if we replace $\\textit{FP}(X_i)$", "and $\\textit{FP}(X_T)$ by the full subcategory of objects", "flat over $T_i$ and $T$ with scheme theoretic support proper", "over $T_i$ and $T$. This proves the lemma." ], "refs": [ "spaces-limits-remark-limit-preserving", "spaces-limits-lemma-descend-modules-finite-presentation", "spaces-limits-lemma-descend-flat", "spaces-limits-lemma-eventually-proper-support" ], "ref_ids": [ 4663, 4599, 4595, 4618 ] } ], "ref_ids": [ 11420 ] }, { "id": 3164, "type": "theorem", "label": "quot-lemma-coherent-RS-star", "categories": [ "quot" ], "title": "quot-lemma-coherent-RS-star", "contents": [ "In Situation \\ref{situation-coherent}. Let", "$$", "\\xymatrix{", "Z \\ar[r] \\ar[d] & Z' \\ar[d] \\\\", "Y \\ar[r] & Y'", "}", "$$", "be a pushout in the category of schemes over $S$ where", "$Z \\to Z'$ is a thickening and $Z \\to Y$ is affine, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.", "Then the functor on fibre categories", "$$", "\\Cohstack_{X/B, Y'}", "\\longrightarrow", "\\Cohstack_{X/B, Y} \\times_{\\Cohstack_{X/B, Z}} \\Cohstack_{X/B, Z'}", "$$", "is an equivalence." ], "refs": [ "more-morphisms-lemma-pushout-along-thickening" ], "proofs": [ { "contents": [ "Observe that the corresponding map", "$$", "B(Y') \\longrightarrow B(Y) \\times_{B(Z)} B(Z')", "$$", "is a bijection, see Pushouts of Spaces, Lemma", "\\ref{spaces-pushouts-lemma-pushout-along-thickening-schemes}.", "Thus using the commutative diagram", "$$", "\\xymatrix{", "\\Cohstack_{X/B, Y'} \\ar[r] \\ar[d] &", "\\Cohstack_{X/B, Y} \\times_{\\Cohstack_{X/B, Z}} \\Cohstack_{X/B, Z'}", "\\ar[d] \\\\", "B(Y') \\ar[r] & B(Y) \\times_{B(Z)} B(Z')", "}", "$$", "we see that we may assume that $Y'$ is a scheme over $B'$. By", "Remark \\ref{remark-coherent-base-change}", "we may replace $B$ by $Y'$ and $X$ by $X \\times_B Y'$.", "Thus we may assume $B = Y'$. In this case the statement follows from", "Pushouts of Spaces, Lemma", "\\ref{spaces-pushouts-lemma-space-over-pushout-flat-modules}." ], "refs": [ "spaces-pushouts-lemma-pushout-along-thickening-schemes", "quot-remark-coherent-base-change", "spaces-pushouts-lemma-space-over-pushout-flat-modules" ], "ref_ids": [ 10858, 3233, 10863 ] } ], "ref_ids": [ 13762 ] }, { "id": 3165, "type": "theorem", "label": "quot-lemma-coherent-over-first-order-thickening", "categories": [ "quot" ], "title": "quot-lemma-coherent-over-first-order-thickening", "contents": [ "Let", "$$", "\\xymatrix{", "X \\ar[d] \\ar[r]_i & X' \\ar[d] \\\\", "T \\ar[r] & T'", "}", "$$", "be a cartesian square of algebraic spaces where $T \\to T'$ is a first", "order thickening. Let $\\mathcal{F}'$ be an $\\mathcal{O}_{X'}$-module", "flat over $T'$. Set $\\mathcal{F} = i^*\\mathcal{F}'$. The following", "are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}'$ is a quasi-coherent $\\mathcal{O}_{X'}$-module", "of finite presentation,", "\\item $\\mathcal{F}'$ is an $\\mathcal{O}_{X'}$-module of finite presentation,", "\\item $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module", "of finite presentation,", "\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation,", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that a finitely presented module is quasi-coherent hence the", "equivalence of (1) and (2) and (3) and (4). The equivalence of (2)", "and (4) is a special case of Deformation Theory, Lemma", "\\ref{defos-lemma-deform-fp-module-ringed-topoi}." ], "refs": [ "defos-lemma-deform-fp-module-ringed-topoi" ], "ref_ids": [ 13398 ] } ], "ref_ids": [] }, { "id": 3166, "type": "theorem", "label": "quot-lemma-coherent-tangent-space", "categories": [ "quot" ], "title": "quot-lemma-coherent-tangent-space", "contents": [ "In Situation \\ref{situation-coherent} assume that $S$ is a locally Noetherian", "scheme and $B \\to S$ is locally of finite presentation.", "Let $k$ be a finite type field over $S$ and let", "$x_0 = (\\Spec(k), g_0, \\mathcal{G}_0)$", "be an object of $\\mathcal{X} = \\Cohstack_{X/B}$ over $k$. Then", "the spaces $T\\mathcal{F}_{\\mathcal{X}, k, x_0}$ and", "$\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0})$", "(Artin's Axioms, Section \\ref{artin-section-tangent-spaces})", "are finite dimensional." ], "refs": [], "proofs": [ { "contents": [ "Observe that by Lemma \\ref{lemma-coherent-RS-star}", "our stack in groupoids $\\mathcal{X}$ satisfies property (RS*)", "defined in Artin's Axioms, Section \\ref{artin-section-inf}.", "In particular $\\mathcal{X}$ satisfies (RS).", "Hence all associated predeformation", "categories are deformation categories", "(Artin's Axioms, Lemma \\ref{artin-lemma-deformation-category})", "and the statement makes sense.", "\\medskip\\noindent", "In this paragraph we show that we can reduce to the case $B = \\Spec(k)$.", "Set $X_0 = \\Spec(k) \\times_{g_0, B} X$", "and denote $\\mathcal{X}_0 = \\Cohstack_{X_0/k}$. In", "Remark \\ref{remark-coherent-base-change} we have seen that", "$\\mathcal{X}_0$ is the $2$-fibre product of $\\mathcal{X}$ with", "$\\Spec(k)$ over $B$ as categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$. Thus by", "Artin's Axioms, Lemma \\ref{artin-lemma-fibre-product-tangent-spaces}", "we reduce to proving that $B$, $\\Spec(k)$, and $\\mathcal{X}_0$", "have finite dimensional tangent spaces and infinitesimal automorphism", "spaces. The tangent space of $B$ and $\\Spec(k)$ are finite dimensional by", "Artin's Axioms, Lemma \\ref{artin-lemma-finite-dimension}", "and of course these have vanishing $\\text{Inf}$.", "Thus it suffices to deal with $\\mathcal{X}_0$.", "\\medskip\\noindent", "Let $k[\\epsilon]$ be the dual numbers over $k$.", "Let $\\Spec(k[\\epsilon]) \\to B$ be the composition of $g_0 : \\Spec(k) \\to B$", "and the morphism $\\Spec(k[\\epsilon]) \\to \\Spec(k)$", "coming from the inclusion $k \\to k[\\epsilon]$.", "Set $X_0 = \\Spec(k) \\times_B X$ and", "$X_\\epsilon = \\Spec(k[\\epsilon]) \\times_B X$.", "Observe that $X_\\epsilon$ is a first order thickening of $X_0$", "flat over the first order thickening $\\Spec(k) \\to \\Spec(k[\\epsilon])$.", "Unwinding the definitions and using", "Lemma \\ref{lemma-coherent-over-first-order-thickening}", "we see that $T\\mathcal{F}_{\\mathcal{X}_0, k, x_0}$ is the set of", "lifts of $\\mathcal{G}_0$ to a flat module on $X_\\epsilon$.", "By Deformation Theory, Lemma \\ref{defos-lemma-flat-ringed-topoi}", "we conclude that", "$$", "T\\mathcal{F}_{\\mathcal{X}_0, k, x_0} =", "\\Ext^1_{\\mathcal{O}_{X_0}}(\\mathcal{G}_0, \\mathcal{G}_0)", "$$", "Here we have used the identification $\\epsilon k[\\epsilon] \\cong k$", "of $k[\\epsilon]$-modules. Using", "Deformation Theory, Lemma \\ref{defos-lemma-flat-ringed-topoi}", "once more we see that", "$$", "\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0}) =", "\\Ext^0_{\\mathcal{O}_{X_0}}(\\mathcal{G}_0, \\mathcal{G}_0)", "$$", "These spaces are finite dimensional over $k$ as $\\mathcal{G}_0$", "has support proper over $\\Spec(k)$. Namely, $X_0$ is of finite presentation", "over $\\Spec(k)$, hence Noetherian. Since $\\mathcal{G}_0$ is of finite", "presentation it is a coherent $\\mathcal{O}_{X_0}$-module. Thus we may apply", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-ext-finite}", "to conclude the desired finiteness." ], "refs": [ "quot-lemma-coherent-RS-star", "artin-lemma-deformation-category", "quot-remark-coherent-base-change", "artin-lemma-fibre-product-tangent-spaces", "artin-lemma-finite-dimension", "quot-lemma-coherent-over-first-order-thickening", "defos-lemma-flat-ringed-topoi", "defos-lemma-flat-ringed-topoi", "spaces-perfect-lemma-ext-finite" ], "ref_ids": [ 3164, 11357, 3233, 11360, 11359, 3165, 13404, 13404, 2668 ] } ], "ref_ids": [] }, { "id": 3167, "type": "theorem", "label": "quot-lemma-coherent-existence", "categories": [ "quot" ], "title": "quot-lemma-coherent-existence", "contents": [ "In Situation \\ref{situation-coherent} assume that $S$ is a locally Noetherian", "scheme and that $f : X \\to B$ is separated.", "Let $\\mathcal{X} = \\Cohstack_{X/B}$. Then the functor", "Artin's Axioms, Equation (\\ref{artin-equation-approximation})", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Let $A$ be an $S$-algebra which is a complete local Noetherian ring", "with maximal ideal $\\mathfrak m$", "whose residue field $k$ is of finite type over $S$.", "We have to show that the category of objects over $A$ is", "equivalent to the category of formal objects over $A$.", "Since we know this holds for the category $\\mathcal{S}_B$", "fibred in sets associated to $B$ by Artin's Axioms, ", "Lemma \\ref{artin-lemma-effective}, it suffices to prove this", "for those objects lying over a given morphism $\\Spec(A) \\to B$.", "\\medskip\\noindent", "Set $X_A = \\Spec(A) \\times_B X$ and $X_n = \\Spec(A/\\mathfrak m^n) \\times_B X$.", "By Grothendieck's existence theorem", "(More on Morphisms of Spaces, Theorem", "\\ref{spaces-more-morphisms-theorem-grothendieck-existence})", "we see that the category of coherent modules $\\mathcal{F}$", "on $X_A$ with support proper over $\\Spec(A)$ is equivalent", "to the category of systems $(\\mathcal{F}_n)$ of coherent modules", "$\\mathcal{F}_n$ on $X_n$ with support proper over", "$\\Spec(A/\\mathfrak m^n)$. The equivalence sends $\\mathcal{F}$", "to the system $(\\mathcal{F} \\otimes_A A/\\mathfrak m^n)$. See discussion in", "More on Morphisms of Spaces, Remark", "\\ref{spaces-more-morphisms-remark-reformulate-existence-theorem}.", "To finish the proof of the lemma, it suffices to show that", "$\\mathcal{F}$ is flat over $A$ if and only if all", "$\\mathcal{F} \\otimes_A A/\\mathfrak m^n$ are flat over $A/\\mathfrak m^n$.", "This follows from", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-flatness-over-Noetherian-ring}." ], "refs": [ "artin-lemma-effective", "spaces-more-morphisms-theorem-grothendieck-existence", "spaces-more-morphisms-remark-reformulate-existence-theorem", "spaces-more-morphisms-lemma-flatness-over-Noetherian-ring" ], "ref_ids": [ 11362, 16, 311, 137 ] } ], "ref_ids": [] }, { "id": 3168, "type": "theorem", "label": "quot-lemma-coherent-defo-thy", "categories": [ "quot" ], "title": "quot-lemma-coherent-defo-thy", "contents": [ "In Situation \\ref{situation-coherent} assume that", "$S$ is a locally Noetherian scheme, $S = B$, and $f : X \\to B$ is flat.", "Let $\\mathcal{X} = \\Cohstack_{X/B}$. Then we have openness of", "versality for $\\mathcal{X}$ (see", "Artin's Axioms, Definition \\ref{artin-definition-openness-versality})." ], "refs": [ "artin-definition-openness-versality" ], "proofs": [ { "contents": [ "[First proof]", "This proof is based on the criterion of", "Artin's Axioms, Lemma \\ref{artin-lemma-dual-openness}.", "Let $U \\to S$ be of finite type morphism of schemes, $x$ an object of", "$\\mathcal{X}$ over $U$ and $u_0 \\in U$ a finite type point such that", "$x$ is versal at $u_0$. After shrinking $U$ we may assume that $u_0$", "is a closed point (Morphisms, Lemma \\ref{morphisms-lemma-point-finite-type})", "and $U = \\Spec(A)$ with $U \\to S$ mapping into an", "affine open $\\Spec(\\Lambda)$ of $S$.", "Let $\\mathcal{F}$ be the coherent module on $X_A = \\Spec(A) \\times_S X$", "flat over $A$ corresponding to the given object $x$.", "\\medskip\\noindent", "According to Deformation Theory, Lemma \\ref{defos-lemma-flat-ringed-topoi}", "we have an isomorphism of functors", "$$", "T_x(M) = \\Ext^1_{X_A}(\\mathcal{F}, \\mathcal{F} \\otimes_A M)", "$$", "and given any surjection $A' \\to A$ of $\\Lambda$-algebras with square zero", "kernel $I$ we have an obstruction class", "$$", "\\xi_{A'} \\in \\Ext^2_{X_A}(\\mathcal{F}, \\mathcal{F} \\otimes_A I)", "$$", "This uses that for any $A' \\to A$ as above the base change", "$X_{A'} = \\Spec(A') \\times_B X$ is flat over $A'$.", "Moreover, the construction of the obstruction class is functorial", "in the surjection $A' \\to A$ (for fixed $A$) by", "Deformation Theory, Lemma \\ref{defos-lemma-functorial-ringed-topoi}.", "Apply Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-compute-ext}", "to the computation of the Ext groups", "$\\Ext^i_{X_A}(\\mathcal{F}, \\mathcal{F} \\otimes_A M)$", "for $i \\leq m$ with $m = 2$. We find a perfect object $K \\in D(A)$", "and functorial isomorphisms", "$$", "H^i(K \\otimes_A^\\mathbf{L} M)", "\\longrightarrow", "\\Ext^i_{X_A}(\\mathcal{F}, \\mathcal{F} \\otimes_A M)", "$$", "for $i \\leq m$ compatible with boundary maps. This object $K$, together", "with the displayed identifications above gives us a datum as in", "Artin's Axioms, Situation \\ref{artin-situation-dual}.", "Finally, condition (iv) of", "Artin's Axioms, Lemma \\ref{artin-lemma-dual-obstruction}", "holds by ", "Deformation Theory, Lemma \\ref{defos-lemma-verify-iv-ringed-topoi}.", "Thus Artin's Axioms, Lemma \\ref{artin-lemma-dual-openness}", "does indeed apply and the lemma is proved." ], "refs": [ "artin-lemma-dual-openness", "morphisms-lemma-point-finite-type", "defos-lemma-flat-ringed-topoi", "defos-lemma-functorial-ringed-topoi", "spaces-perfect-lemma-compute-ext", "artin-lemma-dual-obstruction", "defos-lemma-verify-iv-ringed-topoi", "artin-lemma-dual-openness" ], "ref_ids": [ 11398, 5205, 13404, 13405, 2733, 11397, 13406, 11398 ] } ], "ref_ids": [ 11423 ] }, { "id": 3169, "type": "theorem", "label": "quot-lemma-q-sheaf", "categories": [ "quot" ], "title": "quot-lemma-q-sheaf", "contents": [ "In Situation \\ref{situation-q}. The functors", "$\\text{Q}_{\\mathcal{F}/X/B}$ and", "$\\text{Q}^{fp}_{\\mathcal{F}/X/B}$", "satisfy the sheaf property for the fpqc topology." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{T_i \\to T\\}_{i \\in I}$ be an fpqc covering of schemes over $S$.", "Set $X_i = X_{T_i} = X \\times_S T_i$ and $\\mathcal{F}_i = \\mathcal{F}_{T_i}$.", "Note that $\\{X_i \\to X_T\\}_{i \\in I}$ is an fpqc covering of", "$X_T$ (Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-fpqc})", "and that $X_{T_i \\times_T T_{i'}} = X_i \\times_{X_T} X_{i'}$.", "Suppose that $\\mathcal{F}_i \\to \\mathcal{Q}_i$ is a collection of", "elements of $\\text{Q}_{\\mathcal{F}/X/B}(T_i)$ such that $\\mathcal{Q}_i$", "and $\\mathcal{Q}_{i'}$ restrict to the same element of", "$\\text{Q}_{\\mathcal{F}/X/B}(T_i \\times_T T_{i'})$. By", "Remark \\ref{remark-q-sheaf}", "we obtain a surjective map of quasi-coherent $\\mathcal{O}_{X_T}$-modules", "$\\mathcal{F}_T \\to \\mathcal{Q}$ whose restriction to $X_i$ recovers", "the given quotients.", "By Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-permanence}", "we see that $\\mathcal{Q}$ is flat over $T$. Finally, in the case of", "$\\text{Q}^{fp}_{\\mathcal{F}/X/B}$, i.e., if $\\mathcal{Q}_i$ are", "of finite presentation, then", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-finite-presentation-descends}", "guarantees that $\\mathcal{Q}$ is of finite presentation as an", "$\\mathcal{O}_{X_T}$-module." ], "refs": [ "spaces-topologies-lemma-fpqc", "quot-remark-q-sheaf", "spaces-morphisms-lemma-flat-permanence", "spaces-descent-lemma-finite-presentation-descends" ], "ref_ids": [ 3678, 3235, 4865, 9360 ] } ], "ref_ids": [] }, { "id": 3170, "type": "theorem", "label": "quot-lemma-extend-q-to-spaces", "categories": [ "quot" ], "title": "quot-lemma-extend-q-to-spaces", "contents": [ "In Situation \\ref{situation-q}. Let $T$ be an algebraic space over $S$.", "We have", "$$", "\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\text{Q}_{\\mathcal{F}/X/B}) =", "\\left\\{", "\\begin{matrix}", "(h, \\mathcal{F}_T \\to \\mathcal{Q}) \\text{ where }", "h : T \\to B \\text{ and}\\\\", "\\mathcal{Q}\\text{ is quasi-coherent and flat over }T", "\\end{matrix}", "\\right\\}", "$$", "where $\\mathcal{F}_T$ denotes the pullback of $\\mathcal{F}$", "to the algebraic space $X \\times_{B, h} T$. Similarly, we have", "$$", "\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\text{Q}^{fp}_{\\mathcal{F}/X/B}) =", "\\left\\{", "\\begin{matrix}", "(h, \\mathcal{F}_T \\to \\mathcal{Q}) \\text{ where }", "h : T \\to B \\text{ and}\\\\", "\\mathcal{Q}\\text{ is of finite presentation and flat over }T", "\\end{matrix}", "\\right\\}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $p : U \\to T$.", "Let $R = U \\times_T U$ with projections $t, s : R \\to U$.", "\\medskip\\noindent", "Let $v : T \\to \\text{Q}_{\\mathcal{F}/X/B}$", "be a natural transformation. Then $v(p)$ corresponds to a pair", "$(h_U, \\mathcal{F}_U \\to \\mathcal{Q}_U)$ over $U$.", "As $v$ is a transformation of functors we see", "that the pullbacks of $(h_U, \\mathcal{F}_U \\to \\mathcal{Q}_U)$", "by $s$ and $t$ agree.", "Since $T = U/R$ (Spaces, Lemma \\ref{spaces-lemma-space-presentation}),", "we obtain a morphism $h : T \\to B$ such that", "$h_U = h \\circ p$. By Descent on Spaces, Proposition", "\\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}", "the quotient $\\mathcal{Q}_U$ descends to a quotient", "$\\mathcal{F}_T \\to \\mathcal{Q}$ over $X_T$.", "Since $U \\to T$ is surjective and flat, it follows from", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-permanence}", "that $\\mathcal{Q}$ is flat over $T$.", "\\medskip\\noindent", "Conversely, let $(h, \\mathcal{F}_T \\to \\mathcal{Q})$ be a pair over $T$.", "Then we get a natural transformation", "$v : T \\to \\text{Q}_{\\mathcal{F}/X/B}$", "by sending a morphism $a : T' \\to T$ where $T'$ is a scheme", "to $(h \\circ a, \\mathcal{F}_{T'} \\to a^*\\mathcal{Q})$.", "We omit the verification that the construction", "of this and the previous paragraph are mutually inverse.", "\\medskip\\noindent", "In the case of $\\text{Q}^{fp}_{\\mathcal{F}/X/B}$ we", "add: given a morphism $h : T \\to B$, a quasi-coherent sheaf", "on $X_T$ is of finite presentation as an $\\mathcal{O}_{X_T}$-module", "if and only if the pullback to $X_U$ is of finite presentation as an", "$\\mathcal{O}_{X_U}$-module. This follows from the fact that", "$X_U \\to X_T$ is surjective and \\'etale and", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-finite-presentation-descends}." ], "refs": [ "spaces-lemma-space-presentation", "spaces-descent-proposition-fpqc-descent-quasi-coherent", "spaces-morphisms-lemma-flat-permanence", "spaces-descent-lemma-finite-presentation-descends" ], "ref_ids": [ 8149, 9437, 4865, 9360 ] } ], "ref_ids": [] }, { "id": 3171, "type": "theorem", "label": "quot-lemma-q-sheaf-in-X", "categories": [ "quot" ], "title": "quot-lemma-q-sheaf-in-X", "contents": [ "In Situation \\ref{situation-q} let $\\{X_i \\to X\\}_{i \\in I}$ be an fpqc", "covering and for each $i, j \\in I$ let $\\{X_{ijk} \\to X_i \\times_X X_j\\}$", "be an fpqc covering. Denote $\\mathcal{F}_i$, resp.\\ $\\mathcal{F}_{ijk}$", "the pullback of $\\mathcal{F}$ to $X_i$, resp.\\ $X_{ijk}$. For every scheme", "$T$ over $B$ the diagram", "$$", "\\xymatrix{", "Q_{\\mathcal{F}/X/B}(T) \\ar[r] &", "\\prod\\nolimits_i", "Q_{\\mathcal{F}_i/X_i/B}(T)", "\\ar@<1ex>[r]^-{\\text{pr}_0^*} \\ar@<-1ex>[r]_-{\\text{pr}_1^*}", "&", "\\prod\\nolimits_{i, j, k}", "Q_{\\mathcal{F}_{ijk}/X_{ijk}/B}(T)", "}", "$$", "presents the first arrow as the equalizer of the other two.", "The same is true for the functor $\\text{Q}^{fp}_{\\mathcal{F}/X/B}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}_{i, T} \\to \\mathcal{Q}_i$ be an element in the equalizer", "of $\\text{pr}_0^*$ and $\\text{pr}_1^*$. By Remark \\ref{remark-q-sheaf}", "we obtain a surjection $\\mathcal{F}_T \\to \\mathcal{Q}$ of quasi-coherent", "$\\mathcal{O}_{X_T}$-modules whose restriction to $X_{i, T}$ recovers", "$\\mathcal{F}_i \\to \\mathcal{Q}_i$.", "By Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-permanence}", "we see that $\\mathcal{Q}$ is flat over $T$ as desired.", "In the case of the functor $\\text{Q}^{fp}_{\\mathcal{F}/X/B}$, i.e.,", "if $\\mathcal{Q}_i$ is of finite presentation, then", "$\\mathcal{Q}$ is of finite presentation too by", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-finite-presentation-descends}." ], "refs": [ "quot-remark-q-sheaf", "spaces-morphisms-lemma-flat-permanence", "spaces-descent-lemma-finite-presentation-descends" ], "ref_ids": [ 3235, 4865, 9360 ] } ], "ref_ids": [] }, { "id": 3172, "type": "theorem", "label": "quot-lemma-q-limit-preserving", "categories": [ "quot" ], "title": "quot-lemma-q-limit-preserving", "contents": [ "In Situation \\ref{situation-q} assume also that", "(a) $f$ is quasi-compact and quasi-separated and", "(b) $\\mathcal{F}$ is of finite presentation.", "Then the functor $\\text{Q}^{fp}_{\\mathcal{F}/X/B}$", "is limit preserving in the following sense: If $T = \\lim T_i$ is a", "directed limit of affine schemes over $B$, then", "$\\text{Q}^{fp}_{\\mathcal{F}/X/B}(T) =", "\\colim \\text{Q}^{fp}_{\\mathcal{F}/X/B}(T_i)$." ], "refs": [], "proofs": [ { "contents": [ "Let $T = \\lim T_i$ be as in the statement of the lemma.", "Choose $i_0 \\in I$ and replace $I$ by $\\{i \\in I \\mid i \\geq i_0\\}$.", "We may set $B = S = T_{i_0}$ and we may replace $X$ by $X_{T_0}$", "and $\\mathcal{F}$ by the pullback to $X_{T_0}$. Then", "$X_T = \\lim X_{T_i}$, see", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-directed-inverse-system-has-limit}.", "Let $\\mathcal{F}_T \\to \\mathcal{Q}$ be an element of", "$\\text{Q}^{fp}_{\\mathcal{F}/X/B}(T)$. By", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-modules-finite-presentation}", "there exists an $i$ and a map $\\mathcal{F}_{T_i} \\to \\mathcal{Q}_i$", "of $\\mathcal{O}_{X_{T_i}}$-modules of finite presentation whose", "pullback to $X_T$ is the given quotient map.", "\\medskip\\noindent", "We still have to check that, after possibly increasing $i$, the map", "$\\mathcal{F}_{T_i} \\to \\mathcal{Q}_i$ is surjective and $\\mathcal{Q}_i$", "is flat over $T_i$. To do this, choose an affine scheme $U$ and a", "surjective \\'etale morphism $U \\to X$ (see Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "We may check surjectivity and flatness over $T_i$ after pulling", "back to the \\'etale cover $U_{T_i} \\to X_{T_i}$ (by definition).", "This reduces us to the case where $X = \\Spec(B_0)$ is an affine scheme of", "finite presentation over $B = S = T_0 = \\Spec(A_0)$.", "Writing $T_i = \\Spec(A_i)$, then $T = \\Spec(A)$ with $A = \\colim A_i$", "we have reached the following algebra problem. Let $M_i \\to N_i$", "be a map of finitely presented $B_0 \\otimes_{A_0} A_i$-modules", "such that $M_i \\otimes_{A_i} A \\to N_i \\otimes_{A_i} A$ is surjective", "and $N_i \\otimes_{A_i} A$ is flat over $A$. Show that for some $i' \\geq i$", "$M_i \\otimes_{A_i} A_{i'} \\to N_i \\otimes_{A_i} A_{i'}$ is surjective", "and $N_i \\otimes_{A_i} A_{i'}$ is flat over $A$.", "The first follows from", "Algebra, Lemma \\ref{algebra-lemma-module-map-property-in-colimit}", "and the second from", "Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}." ], "refs": [ "spaces-limits-lemma-directed-inverse-system-has-limit", "spaces-limits-lemma-descend-modules-finite-presentation", "spaces-properties-lemma-quasi-compact-affine-cover", "algebra-lemma-module-map-property-in-colimit", "algebra-lemma-flat-finite-presentation-limit-flat" ], "ref_ids": [ 4565, 4599, 11832, 1094, 1389 ] } ], "ref_ids": [] }, { "id": 3173, "type": "theorem", "label": "quot-lemma-q-RS-star", "categories": [ "quot" ], "title": "quot-lemma-q-RS-star", "contents": [ "In Situation \\ref{situation-q}. Let", "$$", "\\xymatrix{", "Z \\ar[r] \\ar[d] & Z' \\ar[d] \\\\", "Y \\ar[r] & Y'", "}", "$$", "be a pushout in the category of schemes over $B$ where", "$Z \\to Z'$ is a thickening and $Z \\to Y$ is affine, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.", "Then the natural map", "$$", "Q_{\\mathcal{F}/X/B}(Y') \\longrightarrow", "Q_{\\mathcal{F}/X/B}(Y) \\times_{Q_{\\mathcal{F}/X/B}(Z)} Q_{\\mathcal{F}/X/B}(Z')", "$$", "is bijective. If $X \\to B$ is locally of finite presentation, then", "the same thing is true for $Q^{fp}_{\\mathcal{F}/X/B}$." ], "refs": [ "more-morphisms-lemma-pushout-along-thickening" ], "proofs": [ { "contents": [ "Let us construct an inverse map. Namely, suppose we have", "$\\mathcal{F}_Y \\to \\mathcal{A}$,", "$\\mathcal{F}_{Z'} \\to \\mathcal{B}'$, and an isomorphism", "$\\mathcal{A}|_{X_Z} \\to \\mathcal{B}'|_{X_Z}$", "compatible with the given surjections.", "Then we apply Pushouts of Spaces, Lemma", "\\ref{spaces-pushouts-lemma-space-over-pushout-flat-modules}", "to get a quasi-coherent module $\\mathcal{A}'$ on $X_{Y'}$", "flat over $Y'$. Since this sheaf is constructed as a fibre product", "(see proof of cited lemma) there is a canonical map", "$\\mathcal{F}_{Y'} \\to \\mathcal{A}'$. That this map is surjective", "can be seen because it factors as", "$$", "\\begin{matrix}", "\\mathcal{F}_{Y'} \\\\", "\\downarrow \\\\", "(X_Y \\to X_{Y'})_*\\mathcal{F}_Y", "\\times_{(X_Z \\to X_{Y'})_*\\mathcal{F}_Z}", "(X_{Z'} \\to X_{Y'})_*\\mathcal{F}_{Z'} \\\\", "\\downarrow \\\\", "\\mathcal{A}' =", "(X_Y \\to X_{Y'})_*\\mathcal{A}", "\\times_{(X_Z \\to X_{Y'})_*\\mathcal{A}|_{X_Z}}", "(X_{Z'} \\to X_{Y'})_*\\mathcal{B}'", "\\end{matrix}", "$$", "and the first arrow is surjective by", "More on Algebra, Lemma \\ref{more-algebra-lemma-module-over-fibre-product-bis}", "and the second by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-surjection-module-over-fibre-product}.", "\\medskip\\noindent", "In the case of $Q^{fp}_{\\mathcal{F}/X/B}$ all we have to show is that", "the construction above produces a finitely presented module.", "This is explained in", "More on Algebra, Remark", "\\ref{more-algebra-remark-relative-modules-over-fibre-product}", "in the commutative algebra setting. The current case of modules", "over algebraic spaces follows from this", "by \\'etale localization." ], "refs": [ "spaces-pushouts-lemma-space-over-pushout-flat-modules", "more-algebra-lemma-module-over-fibre-product-bis", "more-algebra-lemma-surjection-module-over-fibre-product", "more-algebra-remark-relative-modules-over-fibre-product" ], "ref_ids": [ 10863, 9821, 9822, 10649 ] } ], "ref_ids": [ 13762 ] }, { "id": 3174, "type": "theorem", "label": "quot-lemma-quot-sheaf", "categories": [ "quot" ], "title": "quot-lemma-quot-sheaf", "contents": [ "In Situation \\ref{situation-quot}. The functor $\\Quotfunctor_{\\mathcal{F}/X/B}$", "satisfies the sheaf property for the fpqc topology." ], "refs": [], "proofs": [ { "contents": [ "In Lemma \\ref{lemma-q-sheaf} we have seen that the functor", "$\\text{Q}^{fp}_{\\mathcal{F}/X/S}$ is a sheaf. Recall that for a", "scheme $T$ over $S$ the subset", "$\\Quotfunctor_{\\mathcal{F}/X/S}(T) \\subset \\text{Q}_{\\mathcal{F}/X/S}(T)$", "picks out those quotients whose support is proper over $T$.", "This defines a subsheaf by the result of", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-descending-property-proper}", "combined with", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-pullback-support}", "which shows that taking scheme theoretic support commutes", "with flat base change." ], "refs": [ "quot-lemma-q-sheaf", "spaces-descent-lemma-descending-property-proper", "spaces-morphisms-lemma-flat-pullback-support" ], "ref_ids": [ 3169, 9399, 4859 ] } ], "ref_ids": [] }, { "id": 3175, "type": "theorem", "label": "quot-lemma-extend-quot-to-spaces", "categories": [ "quot" ], "title": "quot-lemma-extend-quot-to-spaces", "contents": [ "In Situation \\ref{situation-quot}. Let $T$ be an algebraic space over $S$.", "We have", "$$", "\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\Quotfunctor_{\\mathcal{F}/X/B}) =", "\\left\\{", "\\begin{matrix}", "(h, \\mathcal{F}_T \\to \\mathcal{Q}) \\text{ where }", "h : T \\to B \\text{ and}\\\\", "\\mathcal{Q}\\text{ is of finite presentation and}\\\\", "\\text{flat over }T\\text{ with support proper over }T", "\\end{matrix}", "\\right\\}", "$$", "where $\\mathcal{F}_T$ denotes the pullback of $\\mathcal{F}$", "to the algebraic space $X \\times_{B, h} T$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the left and right hand side of the equality are", "subsets of the left and right hand side of the second equality in", "Lemma \\ref{lemma-extend-q-to-spaces}.", "To see that these subsets correspond under", "the identification given in the proof of that lemma", "it suffices to show: given $h : T \\to B$,", "a surjective \\'etale morphism $U \\to T$,", "a finite type quasi-coherent $\\mathcal{O}_{X_T}$-module $\\mathcal{Q}$", "the following are equivalent", "\\begin{enumerate}", "\\item the scheme theoretic support of $\\mathcal{Q}$ is proper", "over $T$, and", "\\item the scheme theoretic support of $(X_U \\to X_T)^*\\mathcal{Q}$", "is proper over $U$.", "\\end{enumerate}", "This follows from", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-descending-property-proper}", "combined with", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-pullback-support}", "which shows that taking scheme theoretic support commutes", "with flat base change." ], "refs": [ "quot-lemma-extend-q-to-spaces", "spaces-descent-lemma-descending-property-proper", "spaces-morphisms-lemma-flat-pullback-support" ], "ref_ids": [ 3170, 9399, 4859 ] } ], "ref_ids": [] }, { "id": 3176, "type": "theorem", "label": "quot-lemma-hilb-is-quot", "categories": [ "quot" ], "title": "quot-lemma-hilb-is-quot", "contents": [ "In Situation \\ref{situation-hilb} we have", "$\\Hilbfunctor_{X/B} = \\Quotfunctor_{\\mathcal{O}_X/X/B}$." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme over $B$. Given an element", "$Z \\in \\Hilbfunctor_{X/B}(T)$ we can consider the", "quotient $\\mathcal{O}_{X_T} \\to i_*\\mathcal{O}_Z$", "where $i : Z \\to X_T$ is the inclusion morphism.", "Note that $i_*\\mathcal{O}_Z$ is quasi-coherent.", "Since $Z \\to T$ and $X_T \\to T$ are of finite presentation,", "we see that $i$ is of finite presentation (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-permanence}), hence", "$i_*\\mathcal{O}_Z$ is an $\\mathcal{O}_{X_T}$-module of", "finite presentation (Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-finite-finitely-presented-module}).", "Since $Z \\to T$ is proper we see that $i_*\\mathcal{O}_Z$", "has support proper over $T$ (as defined in", "Derived Categories of Spaces, Section", "\\ref{spaces-perfect-section-proper-over-base}).", "Since $\\mathcal{O}_Z$ is flat", "over $T$ and $i$ is affine, we see that $i_*\\mathcal{O}_Z$", "is flat over $T$ (small argument omitted). Hence", "$\\mathcal{O}_{X_T} \\to i_*\\mathcal{O}_Z$", "is an element of $\\Quotfunctor_{\\mathcal{O}_X/X/B}(T)$.", "\\medskip\\noindent", "Conversely, given an element $\\mathcal{O}_{X_T} \\to \\mathcal{Q}$", "of $\\Quotfunctor_{\\mathcal{O}_X/X/B}(T)$, we can consider", "the closed immersion $i : Z \\to X_T$ corresponding to", "the quasi-coherent ideal sheaf", "$\\mathcal{I} = \\Ker(\\mathcal{O}_{X_T} \\to \\mathcal{Q})$", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-closed-immersion-ideals}).", "By construction of $Z$ we see that $\\mathcal{Q} = i_*\\mathcal{O}_Z$.", "Then we can read the arguments given above backwards to see", "that $Z$ defines an element of $\\Hilbfunctor_{X/B}(T)$.", "For example, $\\mathcal{I}$ is quasi-coherent of finite type", "(Modules on Sites, Lemma", "\\ref{sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation})", "hence $i : Z \\to X_T$ is of finite presentation", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-closed-immersion-finite-presentation})", "hence $Z \\to T$ is of finite presentation", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-composition-finite-presentation}).", "Properness of $Z \\to T$ follows from the discussion in", "Derived Categories of Spaces, Section", "\\ref{spaces-perfect-section-proper-over-base}.", "Flatness of $Z \\to T$ follows from flatness of $\\mathcal{Q}$ over $T$.", "\\medskip\\noindent", "We omit the (immediate) verification that the two constructions given", "above are mutually inverse." ], "refs": [ "spaces-morphisms-lemma-finite-presentation-permanence", "spaces-descent-lemma-finite-finitely-presented-module", "spaces-morphisms-lemma-closed-immersion-ideals", "sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation", "spaces-morphisms-lemma-closed-immersion-finite-presentation", "spaces-morphisms-lemma-composition-finite-presentation" ], "ref_ids": [ 4846, 9365, 4765, 14187, 4849, 4839 ] } ], "ref_ids": [] }, { "id": 3177, "type": "theorem", "label": "quot-lemma-extend-hilb-to-spaces", "categories": [ "quot" ], "title": "quot-lemma-extend-hilb-to-spaces", "contents": [ "In Situation \\ref{situation-hilb}. Let $T$ be an algebraic space over $S$.", "We have", "$$", "\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\Hilbfunctor_{X/B}) =", "\\left\\{", "\\begin{matrix}", "(h, Z)\\text{ where }h : T \\to B,\\ Z \\subset X_T \\\\", "\\text{finite presentation, flat, proper over }T", "\\end{matrix}", "\\right\\}", "$$", "where $X_T = X \\times_{B, h} T$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-hilb-is-quot}", "we have $\\Hilbfunctor_{X/B} = \\Quotfunctor_{\\mathcal{O}_X/X/B}$.", "Thus we can apply Lemma \\ref{lemma-extend-quot-to-spaces}", "to see that the left hand side is bijective with the set", "of surjections $\\mathcal{O}_{X_T} \\to \\mathcal{Q}$", "which are finitely presented, flat over $T$, and", "have support proper over $T$. Arguing exactly as in the", "proof of Lemma \\ref{lemma-hilb-is-quot}", "we see that such quotients correspond", "exactly to the closed immersions $Z \\to X_T$ such that", "$Z \\to T$ is proper, flat, and of finite presentation." ], "refs": [ "quot-lemma-hilb-is-quot", "quot-lemma-extend-quot-to-spaces", "quot-lemma-hilb-is-quot" ], "ref_ids": [ 3176, 3175, 3176 ] } ], "ref_ids": [] }, { "id": 3178, "type": "theorem", "label": "quot-lemma-picard-stack-open-in-coh", "categories": [ "quot" ], "title": "quot-lemma-picard-stack-open-in-coh", "contents": [ "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces", "over $S$ which is flat, of finite presentation, and proper.", "The natural map", "$$", "\\Picardstack_{X/B} \\longrightarrow \\Cohstack_{X/B}", "$$", "is representable by open immersions." ], "refs": [], "proofs": [ { "contents": [ "Observe that the map simply sends a triple $(T, g, \\mathcal{L})$", "as in Examples of Stacks, Section \\ref{examples-stacks-section-picard-stack}", "to the same triple $(T, g, \\mathcal{L})$ but where now we view", "this as a triple of the kind described in", "Situation \\ref{situation-coherent}.", "This works because the invertible $\\mathcal{O}_{X_T}$-module", "$\\mathcal{L}$ is certainly a finitely presented $\\mathcal{O}_{X_T}$-module,", "it is flat over $T$ because $X_T \\to T$ is flat, and the support is", "proper over $T$ as $X_T \\to T$ is proper", "(Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-base-change-flat}", "and \\ref{spaces-morphisms-lemma-base-change-proper}).", "Thus the statement makes sense.", "\\medskip\\noindent", "Having said this, it is clear that the content of the lemma is the", "following: given an object $(T, g, \\mathcal{F})$ of", "$\\Cohstack_{X/B}$ there is an open subscheme $U \\subset T$", "such that for a morphism of schemes $T' \\to T$ the following", "are equivalent", "\\begin{enumerate}", "\\item[(a)] $T' \\to T$ factors through $U$,", "\\item[(b)] the pullback $\\mathcal{F}_{T'}$ of", "$\\mathcal{F}$ by $X_{T'} \\to X_T$ is invertible.", "\\end{enumerate}", "Let $W \\subset |X_T|$ be the set of points $x \\in |X_T|$", "such that $\\mathcal{F}$ is locally free in a neighbourhood of $x$. By", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-finite-free-open}.", "$W$ is open and formation", "of $W$ commutes with arbitrary base change.", "Clearly, if $T' \\to T$ satisfies (b), then $|X_{T'}| \\to |X_T|$", "maps into $W$. Hence we may replace $T$ by the open", "$T \\setminus f_T(|X_T| \\setminus W)$ in order", "to construct $U$. After doing so we reach the situation", "where $\\mathcal{F}$ is finite locally free.", "In this case we get a disjoint union decomposition", "$X_T = X_0 \\amalg X_1 \\amalg X_2 \\amalg \\ldots$", "into open and closed subspaces such that the restriction of", "$\\mathcal{F}$ is locally free of rank $i$ on $X_i$. Then clearly", "$$", "U = T \\setminus f_T(|X_0| \\cup |X_2| \\cup |X_3| \\cup \\ldots )", "$$", "works. (Note that if we assume that $T$ is quasi-compact, then", "$X_T$ is quasi-compact hence only a finite number of $X_i$", "are nonempty and so $U$ is indeed open.)" ], "refs": [ "spaces-morphisms-lemma-base-change-flat", "spaces-morphisms-lemma-base-change-proper", "spaces-more-morphisms-lemma-finite-free-open" ], "ref_ids": [ 4853, 4917, 135 ] } ], "ref_ids": [] }, { "id": 3179, "type": "theorem", "label": "quot-lemma-pic-over-pic", "categories": [ "quot" ], "title": "quot-lemma-pic-over-pic", "contents": [ "In Situation \\ref{situation-pic}", "the functor $\\Picardfunctor_{X/B}$ is the sheafification of", "the functor $T \\mapsto \\Ob(\\Picardstack_{X/B, T})/\\cong$." ], "refs": [], "proofs": [ { "contents": [ "Since the fibre category $\\Picardstack_{X/B, T}$ of the Picard stack", "$\\Picardstack_{X/B}$ over $T$ is the category of invertible sheaves on", "$X_T$ (see Section \\ref{section-picard-stack} and", "Examples of Stacks, Section \\ref{examples-stacks-section-picard-stack})", "this is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3180, "type": "theorem", "label": "quot-lemma-flat-geometrically-connected-fibres", "categories": [ "quot" ], "title": "quot-lemma-flat-geometrically-connected-fibres", "contents": [ "In Situation \\ref{situation-pic}.", "If $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism", "for all schemes $T$ over $B$, then", "$$", "0 \\to \\Pic(T) \\to \\Pic(X_T) \\to \\Picardfunctor_{X/B}(T)", "$$", "is an exact sequence for all $T$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $B$ by $T$ and $X$ by $X_T$ and assume that $B = T$", "to simplify the notation. Let $\\mathcal{N}$ be an invertible", "$\\mathcal{O}_B$-module. If $f^*\\mathcal{N} \\cong \\mathcal{O}_X$, then", "we see that $f_*f^*\\mathcal{N} \\cong f_*\\mathcal{O}_X \\cong \\mathcal{O}_B$", "by assumption. Since $\\mathcal{N}$ is locally trivial, we see that", "the canonical map $\\mathcal{N} \\to f_*f^*\\mathcal{N}$ is locally", "an isomorphism (because $\\mathcal{O}_B \\to f_*f^*\\mathcal{O}_B$", "is an isomorphism by assumption). Hence we conclude that", "$\\mathcal{N} \\to f_*f^*\\mathcal{N} \\to \\mathcal{O}_B$ is an isomorphism", "and we see that $\\mathcal{N}$ is trivial. This proves the first arrow", "is injective.", "\\medskip\\noindent", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module which is in", "the kernel of $\\Pic(X) \\to \\Picardfunctor_{X/B}(B)$. Then there exists", "an fppf covering $\\{B_i \\to B\\}$ such that $\\mathcal{L}$ pulls back", "to the trivial invertible sheaf on $X_{B_i}$. Choose a trivializing", "section $s_i$. Then $\\text{pr}_0^*s_i$ and $\\text{pr}_1^*s_j$ are both", "trivialising sections of $\\mathcal{L}$ over $X_{B_i \\times_B B_j}$", "and hence differ by a multiplicative unit", "$$", "f_{ij} \\in", "\\Gamma(X_{S_i \\times_B B_j}, \\mathcal{O}_{X_{B_i \\times_B B_j}}^*) =", "\\Gamma(B_i \\times_B B_j, \\mathcal{O}_{B_i \\times_N B_j}^*)", "$$", "(equality by our assumption on pushforward of structure sheaves).", "Of course these elements satisfy the cocycle condition on", "$B_i \\times_B B_j \\times_B B_k$, hence they define a descent datum", "on invertible sheaves for the fppf covering $\\{B_i \\to B\\}$.", "By Descent, Proposition \\ref{descent-proposition-fpqc-descent-quasi-coherent}", "there is an invertible $\\mathcal{O}_B$-module $\\mathcal{N}$", "with trivializations over $B_i$ whose associated descent datum is", "$\\{f_{ij}\\}$. (The proposition applies because $B$ is a scheme", "by the replacement performed at the start of the proof.)", "Then $f^*\\mathcal{N} \\cong \\mathcal{L}$ as the", "functor from descent data to modules is fully faithful." ], "refs": [ "descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 14753 ] } ], "ref_ids": [] }, { "id": 3181, "type": "theorem", "label": "quot-lemma-flat-geometrically-connected-fibres-with-section", "categories": [ "quot" ], "title": "quot-lemma-flat-geometrically-connected-fibres-with-section", "contents": [ "In Situation \\ref{situation-pic} let $\\sigma : B \\to X$ be a section.", "Assume that $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism", "for all $T$ over $B$. Then", "$$", "0 \\to \\Pic(T) \\to \\Pic(X_T) \\to \\Picardfunctor_{X/B}(T) \\to 0", "$$", "is a split exact sequence with splitting given by", "$\\sigma_T^* : \\Pic(X_T) \\to \\Pic(T)$." ], "refs": [], "proofs": [ { "contents": [ "Denote $K(T) = \\Ker(\\sigma_T^* : \\Pic(X_T) \\to \\Pic(T))$.", "Since $\\sigma$ is a section of $f$ we see that $\\Pic(X_T)$ is the direct", "sum of $\\Pic(T)$ and $K(T)$.", "Thus by Lemma \\ref{lemma-flat-geometrically-connected-fibres} we see that", "$K(T) \\subset \\Picardfunctor_{X/B}(T)$ for all $T$. Moreover, it is clear", "from the construction that $\\Picardfunctor_{X/B}$ is the sheafification", "of the presheaf $K$. To finish the proof it suffices to show that", "$K$ satisfies the sheaf condition for fppf coverings which we do", "in the next paragraph.", "\\medskip\\noindent", "Let $\\{T_i \\to T\\}$ be an fppf covering. Let $\\mathcal{L}_i$ be", "elements of $K(T_i)$ which map to the same elements of $K(T_i \\times_T T_j)$", "for all $i$ and $j$. Choose an isomorphism", "$\\alpha_i : \\mathcal{O}_{T_i} \\to \\sigma_{T_i}^*\\mathcal{L}_i$", "for all $i$. Choose an isomorphism", "$$", "\\varphi_{ij} :", "\\mathcal{L}_i|_{X_{T_i \\times_T T_j}}", "\\longrightarrow", "\\mathcal{L}_j|_{X_{T_i \\times_T T_j}}", "$$", "If the map", "$$", "\\alpha_j|_{T_i \\times_T T_j} \\circ", "\\sigma_{T_i \\times_T T_j}^*\\varphi_{ij} \\circ", "\\alpha_i|_{T_i \\times_T T_j} :", "\\mathcal{O}_{T_i \\times_T T_j} \\to \\mathcal{O}_{T_i \\times_T T_j}", "$$", "is not equal to multiplication by $1$ but some $u_{ij}$, then we can scale", "$\\varphi_{ij}$ by $u_{ij}^{-1}$ to correct this. Having done this, consider", "the self map", "$$", "\\varphi_{ki}|_{X_{T_i \\times_T T_j \\times_T T_k}} \\circ", "\\varphi_{jk}|_{X_{T_i \\times_T T_j \\times_T T_k}} \\circ", "\\varphi_{ij}|_{X_{T_i \\times_T T_j \\times_T T_k}}", "\\quad\\text{on}\\quad", "\\mathcal{L}_i|_{X_{T_i \\times_T T_j \\times_T T_k}}", "$$", "which is given by multiplication by some section $f_{ijk}$", "of the structure sheaf of $X_{T_i \\times_T T_j \\times_T T_k}$.", "By our choice of $\\varphi_{ij}$ we see that the pullback of", "this map by $\\sigma$ is equal to multiplication by $1$. By", "our assumption on functions on $X$, we see that $f_{ijk} = 1$.", "Thus we obtain a descent datum for the fppf covering", "$\\{X_{T_i} \\to X\\}$. By", "Descent on Spaces, Proposition", "\\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}", "there is an invertible $\\mathcal{O}_{X_T}$-module $\\mathcal{L}$", "and an isomorphism $\\alpha : \\mathcal{O}_T \\to \\sigma_T^*\\mathcal{L}$", "whose pullback to $X_{T_i}$ recovers $(\\mathcal{L}_i, \\alpha_i)$", "(small detail omitted). Thus $\\mathcal{L}$ defines an object", "of $K(T)$ as desired." ], "refs": [ "quot-lemma-flat-geometrically-connected-fibres", "spaces-descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 3180, 9437 ] } ], "ref_ids": [] }, { "id": 3182, "type": "theorem", "label": "quot-lemma-pic-with-section-stack", "categories": [ "quot" ], "title": "quot-lemma-pic-with-section-stack", "contents": [ "In Situation \\ref{situation-pic} let $\\sigma : B \\to X$ be a section.", "Then $\\Picardstack_{X/B, \\sigma}$ as defined above is a stack in", "groupoids over $(\\Sch/S)_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "We already know that $\\Picardstack_{X/B}$ is a stack in groupoids", "over $(\\Sch/S)_{fppf}$ by", "Examples of Stacks, Lemma \\ref{examples-stacks-lemma-picard-stack}.", "Let us show descent for objects for $\\Picardstack_{X/B, \\sigma}$.", "Let $\\{T_i \\to T\\}$ be an fppf covering and let", "$\\xi_i = (T_i, h_i, \\mathcal{L}_i, \\alpha_i)$ be an object of", "$\\Picardstack_{X/B, \\sigma}$ lying over $T_i$, and let", "$\\varphi_{ij} : \\text{pr}_0^*\\xi_i \\to \\text{pr}_1^*\\xi_j$", "be a descent datum. Applying the result for $\\Picardstack_{X/B}$", "we see that we may assume we have an object $(T, h, \\mathcal{L})$", "of $\\Picardstack_{X/B}$ over $T$ which pulls back to $\\xi_i$ for all $i$.", "Then we get", "$$", "\\alpha_i : \\mathcal{O}_{T_i} \\to \\sigma_{T_i}^*\\mathcal{L}_i =", "(T_i \\to T)^*\\sigma_T^*\\mathcal{L}", "$$", "Since the maps $\\varphi_{ij}$ are compatible with the $\\alpha_i$", "we see that $\\alpha_i$ and $\\alpha_j$ pullback to the same map", "on $T_i \\times_T T_j$. By descent of quasi-coherent sheaves", "(Descent, Proposition \\ref{descent-proposition-fpqc-descent-quasi-coherent},", "we see that the $\\alpha_i$ are the restriction of a single map", "$\\alpha : \\mathcal{O}_T \\to \\sigma_T^*\\mathcal{L}$ as desired.", "We omit the proof of descent for morphisms." ], "refs": [ "examples-stacks-lemma-picard-stack", "descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 9180, 14753 ] } ], "ref_ids": [] }, { "id": 3183, "type": "theorem", "label": "quot-lemma-compare-pic-with-section", "categories": [ "quot" ], "title": "quot-lemma-compare-pic-with-section", "contents": [ "In Situation \\ref{situation-pic} let $\\sigma : B \\to X$ be a section.", "The morphism $\\Picardstack_{X/B, \\sigma} \\to \\Picardstack_{X/B}$", "is representable, surjective, and smooth." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme and let $(\\Sch/T)_{fppf} \\to \\Picardstack_{X/B}$", "be given by the object $\\xi = (T, h, \\mathcal{L})$ of $\\Picardstack_{X/B}$", "over $T$. We have to show that", "$$", "(\\Sch/T)_{fppf} \\times_{\\xi, \\Picardstack_{X/B}} \\Picardstack_{X/B, \\sigma}", "$$", "is representable by a scheme $V$ and that the corresponding morphism", "$V \\to T$ is surjective and smooth. See", "Algebraic Stacks, Sections \\ref{algebraic-section-representable-morphism},", "\\ref{algebraic-section-morphisms-representable-by-algebraic-spaces}, and", "\\ref{algebraic-section-representable-properties}.", "The forgetful functor $\\Picardstack_{X/B, \\sigma} \\to \\Picardstack_{X/B}$", "is faithful on fibre categories and for $T'/T$ the set of isomorphism", "classes is the set of isomorphisms", "$$", "\\alpha' : \\mathcal{O}_{T'} \\longrightarrow (T' \\to T)^*\\sigma_T^*\\mathcal{L}", "$$", "See Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids}.", "We know this functor is representable by an affine scheme $U$ of finite", "presentation over $T$ by Proposition \\ref{proposition-isom}", "(applied to $\\text{id} : T \\to T$ and $\\mathcal{O}_T$ and", "$\\sigma^*\\mathcal{L}$). Working Zariski locally on $T$ we may", "assume that $\\sigma_T^*\\mathcal{L}$ is isomorphic to $\\mathcal{O}_T$", "and then we see that our functor is representable by", "$\\mathbf{G}_m \\times T$ over $T$. Hence", "$U \\to T$ Zariski locally on $T$ looks like ", "the projection $\\mathbf{G}_m \\times T \\to T$ which", "is indeed smooth and surjective." ], "refs": [ "algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids", "quot-proposition-isom" ], "ref_ids": [ 8442, 3226 ] } ], "ref_ids": [] }, { "id": 3184, "type": "theorem", "label": "quot-lemma-flat-geometrically-connected-fibres-with-section-functor-stack", "categories": [ "quot" ], "title": "quot-lemma-flat-geometrically-connected-fibres-with-section-functor-stack", "contents": [ "In Situation \\ref{situation-pic} let $\\sigma : B \\to X$ be a section.", "If $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism", "for all $T$ over $B$, then", "$\\Picardstack_{X/B, \\sigma} \\to (\\Sch/S)_{fppf}$", "is fibred in setoids with set of isomorphism classes over $T$ given by", "$$", "\\coprod\\nolimits_{h : T \\to B}", "\\Ker(\\sigma_T^* : \\Pic(X \\times_{B, h} T) \\to \\Pic(T))", "$$" ], "refs": [], "proofs": [ { "contents": [ "If $\\xi = (T, h, \\mathcal{L}, \\alpha)$", "is an object of $\\Picardstack_{X/B, \\sigma}$", "over $T$, then an automorphism $\\varphi$ of", "$\\xi$ is given by multiplication with an invertible global section $u$", "of the structure sheaf of $X_T$ such that moreover $\\sigma_T^*u = 1$.", "Then $u = 1$ by our assumption that", "$\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism.", "Hence $\\Picardstack_{X/B, \\sigma}$", "is fibred in setoids over $(\\Sch/S)_{fppf}$.", "Given $T$ and $h : T \\to B$", "the set of isomorphism classes of pairs $(\\mathcal{L}, \\alpha)$", "is the same as the set of isomorphism classes of $\\mathcal{L}$", "with $\\sigma_T^*\\mathcal{L} \\cong \\mathcal{O}_T$ (isomorphism", "not specified). This is clear because any two choices", "of $\\alpha$ differ by a global unit on $T$ and this is the", "same thing as a global unit on $X_T$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3185, "type": "theorem", "label": "quot-lemma-diagonal-pic", "categories": [ "quot" ], "title": "quot-lemma-diagonal-pic", "contents": [ "With assumptions and notation as in Proposition \\ref{proposition-pic-functor}.", "Then the diagonal", "$\\Picardfunctor_{X/B} \\to \\Picardfunctor_{X/B} \\times_B \\Picardfunctor_{X/B}$", "is representable by immersions. In other words, $\\Picardfunctor_{X/B} \\to B$", "is locally separated." ], "refs": [ "quot-proposition-pic-functor" ], "proofs": [ { "contents": [ "Let $T$ be a scheme over $B$ and let $s, t \\in \\Picardfunctor_{X/B}(T)$.", "We want to show that there exists a locally closed subscheme $Z \\subset T$", "such that $s|_Z = t|_Z$ and such that a morphism $T' \\to T$ factors", "through $Z$ if and only if $s|_{T'} = t|_{T'}$.", "\\medskip\\noindent", "We first reduce the general problem to the case where $s$ and $t$ come", "from invertible modules on $X_T$. We suggest the reader skip this step.", "Choose an fppf covering $\\{T_i \\to T\\}_{i \\in I}$ such that", "$s|_{T_i}$ and $t|_{T_i}$ come from $\\Pic(X_{T_i})$ for all $i$.", "Suppose that we can show the result for all the pairs", "$s|_{T_i}, t|_{T_i}$. Then we obtain locally closed subschemes", "$Z_i \\subset T_i$ with the desired universal property.", "It follows that $Z_i$ and $Z_j$ have the same scheme theoretic", "inverse image in $T_i \\times_T T_j$.", "This determines a descend datum on $Z_i/T_i$.", "Since $Z_i \\to T_i$ is locally quasi-finite, it follows from", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend}", "that we obtain a locally quasi-finite morphism $Z \\to T$", "recovering $Z_i \\to T_i$ by base change. Then $Z \\to T$ is an immersion", "by Descent, Lemma \\ref{descent-lemma-descending-fppf-property-immersion}.", "Finally, because $\\Picardfunctor_{X/B}$ is an fppf sheaf, we conclude", "that $s|_Z = t|_Z$ and that $Z$ satisfies the universal property", "mentioned above.", "\\medskip\\noindent", "Assume $s$ and $t$ come from invertible modules $\\mathcal{V}$, $\\mathcal{W}$", "on $X_T$.", "Set $\\mathcal{L} = \\mathcal{V} \\otimes \\mathcal{W}^{\\otimes -1}$", "We are looking for a locally closed subscheme $Z$ of $T$", "such that $T' \\to T$ factors through $Z$ if and only if $\\mathcal{L}_{X_{T'}}$", "is the pullback of an invertible sheaf on $T'$, see", "Lemma \\ref{lemma-flat-geometrically-connected-fibres}.", "Hence the existence of $Z$ follows from", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-diagonal-picard-flat-proper}." ], "refs": [ "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "descent-lemma-descending-fppf-property-immersion", "quot-lemma-flat-geometrically-connected-fibres", "spaces-more-morphisms-lemma-diagonal-picard-flat-proper" ], "ref_ids": [ 13949, 14698, 3180, 269 ] } ], "ref_ids": [ 3230 ] }, { "id": 3186, "type": "theorem", "label": "quot-lemma-Mor-into-Hilb", "categories": [ "quot" ], "title": "quot-lemma-Mor-into-Hilb", "contents": [ "Let $S$ be a scheme. Consider morphisms", "of algebraic spaces $Z \\to B$ and $X \\to B$ over $S$.", "If $X \\to B$ is separated and $Z \\to B$ is", "of finite presentation, flat, and proper,", "then there is a natural", "injective transformation of functors", "$$", "\\mathit{Mor}_B(Z, X) \\longrightarrow \\Hilbfunctor_{Z \\times_B X/B}", "$$", "which maps a morphism $f : Z_T \\to X_T$ to its graph." ], "refs": [], "proofs": [ { "contents": [ "Given a scheme $T$ over $B$ and a morphism $f_T : Z_T \\to X_T$", "over $T$, the graph of $f$ is the morphism", "$\\Gamma_f = (\\text{id}, f) : Z_T \\to Z_T \\times_T X_T = (Z \\times_B X)_T$.", "Recall that being separated, flat, proper, or finite presentation", "are properties of morphisms of algebraic spaces which are stable", "under base change (Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-separated},", "\\ref{spaces-morphisms-lemma-base-change-flat},", "\\ref{spaces-morphisms-lemma-base-change-proper}, and", "\\ref{spaces-morphisms-lemma-base-change-finite-presentation}).", "Hence $\\Gamma_f$ is a closed immersion by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-semi-diagonal}.", "Moreover, $\\Gamma_f(Z_T)$ is flat, proper, and of finite presentation over $T$.", "Thus $\\Gamma_f(Z_T)$ defines an element of $\\Hilbfunctor_{Z \\times_B X/B}(T)$.", "To show the transformation is injective it suffices to show that", "two morphisms with the same graph are the same. This is true because", "if $Y \\subset (Z \\times_B X)_T$ is the graph of a morphism $f$, then", "we can recover $f$ by using the inverse of $\\text{pr}_1|_Y : Y \\to Z_T$", "composed with $\\text{pr}_2|_Y$." ], "refs": [ "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-base-change-flat", "spaces-morphisms-lemma-base-change-proper", "spaces-morphisms-lemma-base-change-finite-presentation", "spaces-morphisms-lemma-semi-diagonal" ], "ref_ids": [ 4714, 4853, 4917, 4840, 4716 ] } ], "ref_ids": [] }, { "id": 3187, "type": "theorem", "label": "quot-lemma-Mor-into-Hilb-open", "categories": [ "quot" ], "title": "quot-lemma-Mor-into-Hilb-open", "contents": [ "Assumption and notation as in Lemma \\ref{lemma-Mor-into-Hilb}.", "The transformation", "$\\mathit{Mor}_B(Z, X) \\longrightarrow \\Hilbfunctor_{Z \\times_B X/B}$", "is representable by open immersions." ], "refs": [ "quot-lemma-Mor-into-Hilb" ], "proofs": [ { "contents": [ "Let $T$ be a scheme over $B$ and let $Y \\subset (Z \\times_B X)_T$", "be an element of $\\Hilbfunctor_{Z \\times_B X/B}(T)$. Then we see that", "$Y$ is the graph of a morphism $Z_T \\to X_T$ over $T$ if and only", "if $k = \\text{pr}_1|_Y : Y \\to Z_T$ is an isomorphism. By", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-where-isomorphism}", "there exists an open subscheme $V \\subset T$ such that", "for any morphism of schemes $T' \\to T$ we have", "$k_{T'} : Y_{T'} \\to Z_{T'}$ is an isomorphism if and", "only if $T' \\to T$ factors through $V$.", "This proves the lemma." ], "refs": [ "spaces-more-morphisms-lemma-where-isomorphism" ], "ref_ids": [ 252 ] } ], "ref_ids": [ 3186 ] }, { "id": 3188, "type": "theorem", "label": "quot-lemma-spaces-fibred-in-groupoids", "categories": [ "quot" ], "title": "quot-lemma-spaces-fibred-in-groupoids", "contents": [ "The category $\\Spacesstack'_{ft}$ is fibred in groupoids", "over $\\Sch_{fppf}$. The same is true for", "$\\Spacesstack'_{fp, flat, proper}$." ], "refs": [], "proofs": [ { "contents": [ "We have seen this in", "Examples of Stacks, Section", "\\ref{examples-stacks-section-stack-in-groupoids-of-finite-type-spaces}", "for the case of $\\Spacesstack'_{ft}$ and this easily implies the", "result for the other case. However, let us also prove", "this directly by checking conditions (1) and (2) of", "Categories, Definition \\ref{categories-definition-fibred-groupoids}.", "\\medskip\\noindent", "Condition (1). Let $X \\to S$ be an object of $\\Spacesstack'_{ft}$", "and let $S' \\to S$ be a morphism of schemes. Then we set", "$X' = S' \\times_S X$. Note that $X' \\to S'$ is of finite type", "by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-finite-type}.", "to obtain a morphism $(X' \\to S') \\to (X \\to S)$", "lying over $S' \\to S$.", "Argue similarly for the other case using", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-finite-presentation},", "\\ref{spaces-morphisms-lemma-base-change-flat}, and", "\\ref{spaces-morphisms-lemma-base-change-proper}.", "\\medskip\\noindent", "Condition (2). Consider morphisms", "$(f, g) : (X' \\to S') \\to (X \\to S)$ and $(a, b) : (Y \\to T) \\to (X \\to S)$", "of $\\Spacesstack'_{ft}$. Given a morphism $h : T \\to S'$ with", "$g \\circ h = b$ we have to show", "there is a unique morphism $(k, h) : (Y \\to T) \\to (X' \\to S')$ of", "$\\Spacesstack'_{ft}$ such that", "$(f, g) \\circ (k, h) = (a, b)$.", "This is clear from the fact that $X' = S' \\times_S X$.", "The same therefore works for any full subcategory of", "$\\Spacesstack'_{ft}$ satisfying (1)." ], "refs": [ "categories-definition-fibred-groupoids", "spaces-morphisms-lemma-base-change-finite-type", "spaces-morphisms-lemma-base-change-finite-presentation", "spaces-morphisms-lemma-base-change-flat", "spaces-morphisms-lemma-base-change-proper" ], "ref_ids": [ 12392, 4815, 4840, 4853, 4917 ] } ], "ref_ids": [] }, { "id": 3189, "type": "theorem", "label": "quot-lemma-spaces-diagonal", "categories": [ "quot" ], "title": "quot-lemma-spaces-diagonal", "contents": [ "The diagonal", "$$", "\\Delta : \\Spacesstack'_{fp, flat, proper} \\longrightarrow", "\\Spacesstack'_{fp, flat, proper} \\times \\Spacesstack'_{fp, flat, proper}", "$$", "is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "We will use criterion (2) of", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-representable-diagonal}.", "Let $S$ be a scheme and let $X$ and $Y$ be algebraic spaces", "of finite presentation over $S$, flat over $S$, and proper over $S$.", "We have to show that the functor", "$$", "\\mathit{Isom}_S(X, Y) : (\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets}, \\quad", "T \\longmapsto \\{f : X_T \\to Y_T \\text{ isomorphism}\\}", "$$", "is an algebraic space. An elementary argument shows that", "$\\mathit{Isom}_S(X, Y)$ sits in a fibre product", "$$", "\\xymatrix{", "\\mathit{Isom}_S(X, Y) \\ar[r] \\ar[d] & S \\ar[d]_{(\\text{id}, \\text{id})} \\\\", "\\mathit{Mor}_S(X, Y) \\times \\mathit{Mor}_S(Y, X) \\ar[r] &", "\\mathit{Mor}_S(X, X) \\times \\mathit{Mor}_S(Y, Y)", "}", "$$", "The bottom arrow sends $(\\varphi, \\psi)$ to", "$(\\psi \\circ \\varphi, \\varphi \\circ \\psi)$.", "By Proposition \\ref{proposition-Mor} the functors on the bottom row", "are algebraic spaces over $S$. ", "Hence the result follows from the fact that the category of", "algebraic spaces over $S$ has fibre products." ], "refs": [ "algebraic-lemma-representable-diagonal", "quot-proposition-Mor" ], "ref_ids": [ 8461, 3231 ] } ], "ref_ids": [] }, { "id": 3190, "type": "theorem", "label": "quot-lemma-spaces-stack", "categories": [ "quot" ], "title": "quot-lemma-spaces-stack", "contents": [ "The category $\\Spacesstack'_{ft}$ is a stack in groupoids", "over $\\Sch_{fppf}$. The same is true for", "$\\Spacesstack'_{fp, flat, proper}$." ], "refs": [], "proofs": [ { "contents": [ "The reason this lemma holds is the slogan: any fppf descent datum for algebraic", "spaces is effective, see Bootstrap, Section", "\\ref{bootstrap-section-applications}.", "More precisely, the lemma for $\\Spacesstack'_{ft}$ follows from", "Examples of Stacks, Lemma", "\\ref{examples-stacks-lemma-stack-of-finite-type-spaces}", "as we saw in Examples of Stacks, Section", "\\ref{examples-stacks-section-stack-in-groupoids-of-finite-type-spaces}.", "However, let us review the proof. We need to check conditions", "(1), (2), and (3) of Stacks, Definition", "\\ref{stacks-definition-stack-in-groupoids}.", "\\medskip\\noindent", "Property (1) we have seen in Lemma \\ref{lemma-spaces-fibred-in-groupoids}.", "\\medskip\\noindent", "Property (2) follows from", "Lemma \\ref{lemma-spaces-diagonal} in the case of", "$\\Spacesstack'_{fp, flat, proper}$.", "In the case of $\\Spacesstack'_{ft}$ it follows", "from Examples of Stacks, Lemma", "\\ref{examples-stacks-lemma-pre-stack-of-spaces}", "(and this is really the ``correct'' reference).", "\\medskip\\noindent", "Condition (3) for $\\Spacesstack'_{ft}$ is checked as follows. Suppose given", "\\begin{enumerate}", "\\item an fppf covering $\\{U_i \\to U\\}_{i \\in I}$ in $\\Sch_{fppf}$,", "\\item for each $i \\in I$ an algebraic space $X_i$ of finite type over", "$U_i$, and", "\\item for each $i, j \\in I$ an isomorphism", "$\\varphi_{ij} : X_i \\times_U U_j \\to U_i \\times_U X_j$ of algebraic spaces", "over $U_i \\times_U U_j$ satisfying the cocycle condition over", "$U_i \\times_U U_j \\times_U U_k$.", "\\end{enumerate}", "We have to show there exists an algebraic space $X$ of finite type over $U$", "and isomorphisms $X_{U_i} \\cong X_i$ over $U_i$ recovering the", "isomorphisms $\\varphi_{ij}$. This follows from", "Bootstrap, Lemma \\ref{bootstrap-lemma-descend-algebraic-space} part (2).", "By Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-finite-type}", "we see that $X \\to U$ is of finite type.", "In the case of $\\Spacesstack'_{fp, flat, proper}$", "one additionally uses", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-finite-presentation},", "\\ref{spaces-descent-lemma-descending-property-flat}, and", "\\ref{spaces-descent-lemma-descending-property-proper}", "in the last step." ], "refs": [ "examples-stacks-lemma-stack-of-finite-type-spaces", "stacks-definition-stack-in-groupoids", "quot-lemma-spaces-fibred-in-groupoids", "quot-lemma-spaces-diagonal", "examples-stacks-lemma-pre-stack-of-spaces", "bootstrap-lemma-descend-algebraic-space", "spaces-descent-lemma-descending-property-finite-type", "spaces-descent-lemma-descending-property-finite-presentation", "spaces-descent-lemma-descending-property-flat", "spaces-descent-lemma-descending-property-proper" ], "ref_ids": [ 9172, 8998, 3188, 3189, 9171, 2627, 9391, 9392, 9393, 9399 ] } ], "ref_ids": [] }, { "id": 3191, "type": "theorem", "label": "quot-lemma-extend-spaces-to-spaces", "categories": [ "quot" ], "title": "quot-lemma-extend-spaces-to-spaces", "contents": [ "Let $T$ be an algebraic space over $\\mathbf{Z}$. Let $\\mathcal{S}_T$", "denote the corresponding algebraic stack (Algebraic Stacks, Sections", "\\ref{algebraic-section-split},", "\\ref{algebraic-section-representable-by-algebraic-spaces}, and", "\\ref{algebraic-section-stacks-spaces}).", "We have an equivalence of categories", "$$", "\\left\\{", "\\begin{matrix}", "\\text{morphisms of algebraic spaces }\\\\", "X \\to T\\text{ of finite type}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\Mor_{\\textit{Cat}/\\Sch_{fppf}}(\\mathcal{S}_T, \\Spacesstack'_{ft})", "$$", "and an equivalence of categories", "$$", "\\left\\{", "\\begin{matrix}", "\\text{morphisms of algebraic spaces }X \\to T\\\\", "\\text{of finite presentation, flat, and proper}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\Mor_{\\textit{Cat}/\\Sch_{fppf}}(\\mathcal{S}_T,", "\\Spacesstack'_{fp, flat, proper})", "$$" ], "refs": [], "proofs": [ { "contents": [ "We are going to deduce this lemma from the fact that it holds for schemes", "(essentially by construction of the stacks) and the fact that fppf descent", "data for algebraic spaces over algerbaic spaces are effective.", "We strongly encourage the reader to skip the proof.", "\\medskip\\noindent", "The construction from left to right in either arrow is straightforward:", "given $X \\to T$ of finite type the functor", "$\\mathcal{S}_T \\to \\Spacesstack'_{ft}$ assigns to $U/T$ the", "base change $X_U \\to U$. We will explain how to construct a quasi-inverse.", "\\medskip\\noindent", "If $T$ is a scheme, then there is a quasi-inverse by the $2$-Yoneda lemma, see", "Categories, Lemma \\ref{categories-lemma-yoneda-2category}.", "Let $p : U \\to T$ be a surjective \\'etale morphism where $U$ is a scheme.", "Let $R = U \\times_T U$ with projections $s, t : R \\to U$.", "Observe that we obtain morphisms", "$$", "\\xymatrix{", "\\mathcal{S}_{U \\times_T U \\times_T U} \\ar@<2ex>[r] \\ar[r] \\ar@<-2ex>[r] &", "\\mathcal{S}_R \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\mathcal{S}_U \\ar[r] &", "\\mathcal{S}_T", "}", "$$", "satisfying various compatibilities (on the nose).", "\\medskip\\noindent", "Let $G : \\mathcal{S}_T \\to \\Spacesstack'_{ft}$ be a functor over $\\Sch_{fppf}$.", "The restriction of $G$ to $\\mathcal{S}_U$ via the map displayed above", "corresponds to a finite type morphism $X_U \\to U$ of algebraic spaces", "via the $2$-Yoneda lemma. Since $p \\circ s = p \\circ t$ we see that", "$R \\times_{s, U} X_U$ and $R \\times_{t, U} X_U$ both correspond to the", "restriction of $G$ to $\\mathcal{S}_R$. Thus we obtain a canonical isomorphism", "$\\varphi : X_U \\times_{U, t} R \\to R \\times_{s, U} X_U$ over $R$.", "This isomorphism satisfies the cocycle condition by the", "various compatibilities of the diagram given above.", "Thus a descent datum which is effective by Bootstrap, Lemma", "\\ref{bootstrap-lemma-descend-algebraic-space} part (2).", "In other words, we obtain an object $X \\to T$ of the right hand side", "category. We omit checking the construction $G \\leadsto X$", "is functorial and that it is quasi-inverse to the other construction.", "In the case of $\\Spacesstack'_{fp, flat, proper}$ one additionally uses", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-finite-presentation},", "\\ref{spaces-descent-lemma-descending-property-flat}, and", "\\ref{spaces-descent-lemma-descending-property-proper}", "in the last step to see that $X \\to T$ is of finite presentation,", "flat, and proper." ], "refs": [ "categories-lemma-yoneda-2category", "bootstrap-lemma-descend-algebraic-space", "spaces-descent-lemma-descending-property-finite-presentation", "spaces-descent-lemma-descending-property-flat", "spaces-descent-lemma-descending-property-proper" ], "ref_ids": [ 12318, 2627, 9392, 9393, 9399 ] } ], "ref_ids": [] }, { "id": 3192, "type": "theorem", "label": "quot-lemma-spaces-limits", "categories": [ "quot" ], "title": "quot-lemma-spaces-limits", "contents": [ "The stack", "$p'_{fp, flat, proper} :", "\\Spacesstack'_{fp, flat, proper} \\to \\Sch_{fppf}$ is limit preserving", "(Artin's Axioms, Definition \\ref{artin-definition-limit-preserving})." ], "refs": [ "artin-definition-limit-preserving" ], "proofs": [ { "contents": [ "Let $T = \\lim T_i$ be the limits of a", "directed inverse system of affine schemes.", "By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-finite-presentation}", "the category of algebraic spaces of finite presentation", "over $T$ is the colimit of the categories of algebraic spaces", "of finite presentation over $T_i$.", "To finish the proof use that flatness and properness", "descends through the limit, see", "Limits of Spaces, Lemmas", "\\ref{spaces-limits-lemma-descend-flat} and", "\\ref{spaces-limits-lemma-eventually-proper}." ], "refs": [ "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-flat", "spaces-limits-lemma-eventually-proper" ], "ref_ids": [ 4598, 4595, 4596 ] } ], "ref_ids": [ 11420 ] }, { "id": 3193, "type": "theorem", "label": "quot-lemma-spaces-RS-star", "categories": [ "quot" ], "title": "quot-lemma-spaces-RS-star", "contents": [ "Let", "$$", "\\xymatrix{", "T \\ar[r] \\ar[d] & T' \\ar[d] \\\\", "S \\ar[r] & S'", "}", "$$", "be a pushout in the category of schemes where", "$T \\to T'$ is a thickening and $T \\to S$ is affine, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.", "Then the functor on fibre categories", "$$", "\\begin{matrix}", "\\Spacesstack'_{fp, flat, proper, S'} \\\\", "\\downarrow \\\\", "\\Spacesstack'_{fp, flat, proper, S}", "\\times_{\\Spacesstack'_{fp, flat, proper, T}}", "\\Spacesstack'_{fp, flat, proper, T'}", "\\end{matrix}", "$$", "is an equivalence." ], "refs": [ "more-morphisms-lemma-pushout-along-thickening" ], "proofs": [ { "contents": [ "The functor is an equivalence if we drop ``proper'' from the list", "of conditions and replace ``of finite presentation'' by", "``locally of finite presentation'', see Pushouts of Spaces, Lemma", "\\ref{spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat}.", "Thus it suffices to show that given a morphism", "$X' \\to S'$ of an algebraic space to $S'$ which is", "flat and locally of finite presentation, then", "$X' \\to S'$ is proper if and only if $S \\times_{S'} X' \\to S$", "and $T' \\times_{S'} X' \\to T'$ are proper.", "One implication follows from the fact that", "properness is preserved under base change", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-proper})", "and the other from the fact that properness of $S \\times_{S'} X' \\to S$", "implies properness of $X' \\to S'$ by", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-thicken-property-morphisms-cartesian}." ], "refs": [ "spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat", "spaces-morphisms-lemma-base-change-proper", "spaces-more-morphisms-lemma-thicken-property-morphisms-cartesian" ], "ref_ids": [ 10864, 4917, 56 ] } ], "ref_ids": [ 13762 ] }, { "id": 3194, "type": "theorem", "label": "quot-lemma-spaces-tangent-space", "categories": [ "quot" ], "title": "quot-lemma-spaces-tangent-space", "contents": [ "Let $k$ be a field and let $x = (X \\to \\Spec(k))$ be an object of", "$\\mathcal{X} = \\Spacesstack'_{fp, flat, proper}$ over $\\Spec(k)$.", "\\begin{enumerate}", "\\item If $k$ is of finite type over $\\mathbf{Z}$, then", "the vector spaces $T\\mathcal{F}_{\\mathcal{X}, k, x}$ and", "$\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x})$", "(see Artin's Axioms, Section \\ref{artin-section-tangent-spaces})", "are finite dimensional, and", "\\item in general the vector spaces $T_x(k)$ and $\\text{Inf}_x(k)$", "(see Artin's Axioms, Section \\ref{artin-section-inf})", "are finite dimensional.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The discussion in Artin's Axioms, Section \\ref{artin-section-tangent-spaces}", "only applies to fields of finite type over the base scheme $\\Spec(\\mathbf{Z})$.", "Our stack satisfies (RS*) by Lemma \\ref{lemma-spaces-RS-star}", "and we may apply", "Artin's Axioms, Lemma \\ref{artin-lemma-properties-lift-RS-star}", "to get the vector spaces $T_x(k)$ and $\\text{Inf}_x(k)$", "mentioned in (2). Moreover, in the finite type case these spaces agree with the", "ones mentioned in (1)", "by Artin's Axioms, Remark \\ref{artin-remark-compare-deformation-spaces}.", "With this out of the way we can start the proof.", "Observe that the first order thickening", "$\\Spec(k) \\to \\Spec(k[\\epsilon]) = \\Spec(k[k])$", "has conormal module $k$. Hence the formula in", "Deformation Theory, Lemma \\ref{defos-lemma-deform-spaces}", "describing infinitesimal deformations of $X$ and infinitesimal", "automorphisms of $X$ become", "$$", "T_x(k) = \\Ext^1_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X)", "\\quad\\text{and}\\quad", "\\text{Inf}_x(k) = \\Ext^0_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X)", "$$", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-netherlander-fp}", "and the fact that $X$ is Noetherian, we see that", "$\\NL_{X/k}$ has coherent cohomology sheaves zero except", "in degrees $0$ and $-1$.", "By Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-ext-finite}", "the displayed $\\Ext$-groups are finite $k$-vector spaces", "and the proof is complete." ], "refs": [ "quot-lemma-spaces-RS-star", "artin-lemma-properties-lift-RS-star", "artin-remark-compare-deformation-spaces", "defos-lemma-deform-spaces", "spaces-more-morphisms-lemma-netherlander-fp", "spaces-perfect-lemma-ext-finite" ], "ref_ids": [ 3193, 11388, 11434, 13412, 125, 2668 ] } ], "ref_ids": [] }, { "id": 3195, "type": "theorem", "label": "quot-lemma-spaces-defo-thy", "categories": [ "quot" ], "title": "quot-lemma-spaces-defo-thy", "contents": [ "The stack in groupoids $\\mathcal{X} = \\Spacesstack'_{fp, flat, proper}$", "satisfies openness of versality over $\\Spec(\\mathbf{Z})$.", "Similarly, after base change (Remark \\ref{remark-spaces-base-change})", "openness of versality holds over any Noetherian base scheme $S$." ], "refs": [ "quot-remark-spaces-base-change" ], "proofs": [ { "contents": [ "For the ``usual'' proof of this fact, please see the discussion", "in the remark following this proof. We will prove this using", "Artin's Axioms, Lemma \\ref{artin-lemma-SGE-implies-openness-versality}.", "We have already seen that $\\mathcal{X}$ has diagonal", "representable by algebraic spaces, has (RS*), and is limit preserving,", "see Lemmas \\ref{lemma-spaces-diagonal},", "\\ref{lemma-spaces-RS-star}, and", "\\ref{lemma-spaces-limits}.", "Hence we only need to see that $\\mathcal{X}$ satisfies the strong", "formal effectiveness formulated in", "Artin's Axioms, Lemma \\ref{artin-lemma-SGE-implies-openness-versality}.", "\\medskip\\noindent", "Let $(R_n)$ be an inverse system of rings such that", "$R_n \\to R_m$ is surjective with square zero kernel for", "all $n \\geq m$. Let $X_n \\to \\Spec(R_n)$ be a finitely presented,", "flat, proper morphism where $X_n$ is an algebraic space and", "let $X_{n + 1} \\to X_n$ be a morphism over $\\Spec(R_{n + 1})$", "inducing an isomorphism $X_n = X_{n + 1} \\times_{\\Spec(R_{n + 1})} \\Spec(R_n)$.", "We have to find a flat, proper, finitely presented morphism", "$X \\to \\Spec(\\lim R_n)$ whose source is an algebraic space", "such that $X_n$ is the base change of $X$ for all $n$.", "\\medskip\\noindent", "Let $I_n = \\Ker(R_n \\to R_1)$. We may think of", "$(X_1 \\subset X_n) \\to (\\Spec(R_1) \\subset \\Spec(R_n))$", "as a morphism of first order thickenings. (Please read some of the material", "on thickenings of algebraic spaces in More on Morphisms of Spaces, Section", "\\ref{spaces-more-morphisms-section-thickenings}", "before continuing.) The structure sheaf of $X_n$ is an extension", "$$", "0 \\to \\mathcal{O}_{X_1} \\otimes_{R_1} I_n \\to", "\\mathcal{O}_{X_n} \\to \\mathcal{O}_{X_1} \\to 0", "$$", "over $0 \\to I_n \\to R_n \\to R_1$, see", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-deform}.", "Let's consider the extension", "$$", "0 \\to \\lim \\mathcal{O}_{X_1} \\otimes_{R_1} I_n \\to", "\\lim \\mathcal{O}_{X_n} \\to \\mathcal{O}_{X_1} \\to 0", "$$", "over $0 \\to \\lim I_n \\to \\lim R_n \\to R_1 \\to 0$.", "The displayed sequence is exact as the $R^1\\lim$ of the system", "of kernels is zero by Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-Rlim-quasi-coherent}.", "Observe that the map", "$$", "\\mathcal{O}_{X_1} \\otimes_{R_1} \\lim I_n \\longrightarrow", "\\lim \\mathcal{O}_{X_1} \\otimes_{R_1} I_n", "$$", "induces an isomorphism upon applying the functor $DQ_X$, see", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-pullback-and-limits}.", "Hence we obtain a unique extension", "$$", "0 \\to \\mathcal{O}_{X_1} \\otimes_{R_1} \\lim I_n \\to", "\\mathcal{O}' \\to \\mathcal{O}_{X_1} \\to 0", "$$", "over $0 \\to \\lim I_n \\to \\lim R_n \\to R_1 \\to 0$", "by the equivalence of categories of", "Deformation Theory, Lemma", "\\ref{defos-lemma-thickening-over-thickening-space-quasi-coherent}.", "The sheaf $\\mathcal{O}'$ determines", "a first order thickening of algebraic spaces $X_1 \\subset X$", "over $\\Spec(R_1) \\subset \\Spec(\\lim R_n)$", "by More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-first-order-thickening}.", "Observe that $X \\to \\Spec(\\lim R_n)$ is flat by the already", "used More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-deform}.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-deform-property}", "we see that $X \\to \\Spec(\\lim R_n)$ is proper", "and of finite presentation.", "This finishes the proof." ], "refs": [ "artin-lemma-SGE-implies-openness-versality", "quot-lemma-spaces-diagonal", "quot-lemma-spaces-RS-star", "quot-lemma-spaces-limits", "artin-lemma-SGE-implies-openness-versality", "spaces-more-morphisms-lemma-deform", "spaces-perfect-lemma-Rlim-quasi-coherent", "spaces-perfect-lemma-pullback-and-limits", "defos-lemma-thickening-over-thickening-space-quasi-coherent", "spaces-more-morphisms-lemma-first-order-thickening", "spaces-more-morphisms-lemma-deform", "spaces-more-morphisms-lemma-deform-property" ], "ref_ids": [ 11386, 3189, 3193, 3192, 11386, 101, 2647, 2740, 13414, 51, 101, 103 ] } ], "ref_ids": [ 3239 ] }, { "id": 3196, "type": "theorem", "label": "quot-lemma-polarized-fibred-in-groupoids", "categories": [ "quot" ], "title": "quot-lemma-polarized-fibred-in-groupoids", "contents": [ "The category $\\Polarizedstack$ is fibred in groupoids over", "$\\Spacesstack'_{fp, flat, proper}$.", "The category $\\Polarizedstack$ is fibred in groupoids over $\\Sch_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "We check conditions (1) and (2) of", "Categories, Definition \\ref{categories-definition-fibred-groupoids}.", "\\medskip\\noindent", "Condition (1). Let $(X \\to S, \\mathcal{L})$ be an object of", "$\\Polarizedstack$ and let $(X' \\to S') \\to (X \\to S)$", "be a morphism of $\\Spacesstack'_{fp, flat, proper}$. Then we", "let $\\mathcal{L}'$ be the pullback of $\\mathcal{L}$ to $X'$.", "Observe that $X, S, S'$ are schemes, hence $X'$ is a scheme", "as well (as the fibre product of schemes). Then", "$\\mathcal{L}'$ is ample on $X'/S'$ by", "Morphisms, Lemma \\ref{morphisms-lemma-ample-base-change}.", "In this way we obtain a morphism", "$(X' \\to S', \\mathcal{L}') \\to (X \\to S, \\mathcal{L})$", "lying over $(X' \\to S') \\to (X \\to S)$.", "\\medskip\\noindent", "Condition (2). Consider morphisms", "$(f, g, \\varphi) : (X' \\to S', \\mathcal{L}') \\to (X \\to S, \\mathcal{L})$ and", "$(a, b, \\psi) : (Y \\to T, \\mathcal{N}) \\to (X \\to S, \\mathcal{L})$", "of $\\Polarizedstack$. Given a morphism $(k, h) : (Y \\to T) \\to (X' \\to S')$", "of $\\Spacesstack'_{fp, flat, proper}$", "with $(f, g) \\circ (k, h) = (a, b)$ we have to show", "there is a unique morphism", "$(k, h, \\chi) : (Y \\to T, \\mathcal{N}) \\to (X' \\to S', \\mathcal{L}')$", "of $\\Polarizedstack$ such that", "$(f, g, \\varphi) \\circ (k, h, \\chi) = (a, b, \\psi)$.", "We can just take", "$$", "\\chi = \\psi \\circ (k^*\\varphi)^{-1}", "$$", "This proves condition (2). A composition of functors defining", "fibred categories defines a fibred category, see", "Categories, Lemma \\ref{categories-lemma-fibred-over-fibred}.", "This we see that $\\Polarizedstack$ is fibred in groupoids over", "$\\Sch_{fppf}$ (strictly speaking we should check the fibre", "categories are groupoids and apply", "Categories, Lemma \\ref{categories-lemma-fibred-groupoids})." ], "refs": [ "categories-definition-fibred-groupoids", "morphisms-lemma-ample-base-change", "categories-lemma-fibred-over-fibred", "categories-lemma-fibred-groupoids" ], "ref_ids": [ 12392, 5385, 12289, 12294 ] } ], "ref_ids": [] }, { "id": 3197, "type": "theorem", "label": "quot-lemma-polarized-stack", "categories": [ "quot" ], "title": "quot-lemma-polarized-stack", "contents": [ "The category $\\Polarizedstack$ is a stack in groupoids over", "$\\Spacesstack'_{fp, flat, proper}$ (endowed with the inherited topology,", "see Stacks, Definition \\ref{stacks-definition-topology-inherited}).", "The category $\\Polarizedstack$ is a stack in groupoids over $\\Sch_{fppf}$." ], "refs": [ "stacks-definition-topology-inherited" ], "proofs": [ { "contents": [ "We prove $\\Polarizedstack$ is a stack in groupoids over", "$\\Spacesstack'_{fp, flat, proper}$", "by checking conditions (1), (2), and (3)", "of Stacks, Definition \\ref{stacks-definition-stack-in-groupoids}.", "We have already seen (1) in", "Lemma \\ref{lemma-polarized-fibred-in-groupoids}.", "\\medskip\\noindent", "A covering of $\\Spacesstack'_{fp, flat, proper}$ comes about", "in the following manner: Let $X \\to S$ be an object of", "$\\Spacesstack'_{fp, flat, proper}$. Suppose that", "$\\{S_i \\to S\\}_{i \\in I}$ is a covering of $\\Sch_{fppf}$.", "Set $X_i = S_i \\times_S X$. Then $\\{(X_i \\to S_i) \\to (X \\to S)\\}_{i \\in I}$", "is a covering of $\\Spacesstack'_{fp, flat, proper}$ and", "every covering of $\\Spacesstack'_{fp, flat, proper}$ is isomorphic", "to one of these. Set $S_{ij} = S_i \\times_S S_j$ and", "$X_{ij} = S_{ij} \\times_S X$ so that $(X_{ij} \\to S_{ij}) =", "(X_i \\to S_i) \\times_{(X \\to S)} (X_j \\to S_j)$.", "Next, suppose that $\\mathcal{L}, \\mathcal{N}$", "are ample invertible sheaves on $X/S$ so that", "$(X \\to S, \\mathcal{L})$ and $(X \\to S, \\mathcal{N})$", "are two objects of $\\Polarizedstack$ over the object $(X \\to S)$.", "To check descent for morphisms, we assume we have morphisms", "$(\\text{id}, \\text{id}, \\varphi_i)$ from", "$(X_i \\to S_i, \\mathcal{L}|_{X_i})$ to", "$(X_i \\to S_i, \\mathcal{N}|_{X_i})$", "whose base changes to morphisms from", "$(X_{ij} \\to S_{ij}, \\mathcal{L}|_{X_{ij}})$ to", "$(X_{ij} \\to S_{ij}, \\mathcal{N}|_{X_{ij}})$", "agree. Then", "$\\varphi_i : \\mathcal{L}|_{X_i} \\to \\mathcal{N}|_{X_i}$", "are isomorphisms of invertible modules over $X_i$ such that", "$\\varphi_i$ and $\\varphi_j$ restrict to the same", "isomorphisms over $X_{ij}$.", "By descent for quasi-coherent sheaves", "(Descent on Spaces, Proposition", "\\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent})", "we obtain a unique isomorphism $\\varphi : \\mathcal{L} \\to \\mathcal{N}$", "whose restriction to $X_i$ recovers $\\varphi_i$.", "\\medskip\\noindent", "Decent for objects is proved in exactly the same manner.", "Namely, suppose that", "$\\{(X_i \\to S_i) \\to (X \\to S)\\}_{i \\in I}$", "is a covering of $\\Spacesstack'_{fp, flat, proper}$", "as above.", "Suppose we have objects $(X_i \\to S_i, \\mathcal{L}_i)$", "of $\\Polarizedstack$ lying over $(X_i \\to S_i)$", "and a descent datum", "$$", "(\\text{id}, \\text{id}, \\varphi_{ij}) :", "(X_{ij} \\to S_{ij}, \\mathcal{L}_i|_{X_{ij}})", "\\to", "(X_{ij} \\to S_{ij}, \\mathcal{L}_j|_{X_{ij}})", "$$", "satisfying the obvious cocycle condition over", "$(X_{ijk} \\to S_{ijk})$ for every triple of indices.", "Then by", " descent for quasi-coherent sheaves", "(Descent on Spaces, Proposition", "\\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent})", "we obtain a unique invertible $\\mathcal{O}_X$-module", "$\\mathcal{L}$ and isomorphisms $\\mathcal{L}|_{X_i} \\to \\mathcal{L}_i$", "recovering the descent datum $\\varphi_{ij}$.", "To show that", "$(X \\to S, \\mathcal{L})$ is an object of", "$\\Polarizedstack$ we have to prove that", "$\\mathcal{L}$ is ample. This follows from", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-descending-property-ample}.", "\\medskip\\noindent", "Since we already have seen that $\\Spacesstack'_{fp, flat, proper}$", "is a stack in groupoids over $\\Sch_{fppf}$", "(Lemma \\ref{lemma-spaces-stack}) it now follows formally", "that $\\Polarizedstack$ is a stack in groupoids over $\\Sch_{fppf}$.", "See Stacks, Lemma \\ref{stacks-lemma-stack-over-stack}." ], "refs": [ "stacks-definition-stack-in-groupoids", "quot-lemma-polarized-fibred-in-groupoids", "spaces-descent-proposition-fpqc-descent-quasi-coherent", "spaces-descent-proposition-fpqc-descent-quasi-coherent", "spaces-descent-lemma-descending-property-ample", "quot-lemma-spaces-stack", "stacks-lemma-stack-over-stack" ], "ref_ids": [ 8998, 3196, 9437, 9437, 9413, 3190, 8973 ] } ], "ref_ids": [ 9002 ] }, { "id": 3198, "type": "theorem", "label": "quot-lemma-extend-polarized-to-spaces", "categories": [ "quot" ], "title": "quot-lemma-extend-polarized-to-spaces", "contents": [ "Let $T$ be an algebraic space over $\\mathbf{Z}$. Let $\\mathcal{S}_T$", "denote the corresponding algebraic stack (Algebraic Stacks, Sections", "\\ref{algebraic-section-split},", "\\ref{algebraic-section-representable-by-algebraic-spaces}, and", "\\ref{algebraic-section-stacks-spaces}).", "We have an equivalence of categories", "$$", "\\left\\{", "\\begin{matrix}", "(X \\to T, \\mathcal{L})\\text{ where }X \\to T\\text{ is a morphism}\\\\", "\\text{of algebraic spaces, is proper, flat, and of}\\\\", "\\text{finite presentation and }\\mathcal{L}\\text{ ample on }X/T", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\Mor_{\\textit{Cat}/\\Sch_{fppf}}(\\mathcal{S}_T, \\Polarizedstack)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints: Argue exactly as in the proof of", "Lemma \\ref{lemma-extend-spaces-to-spaces} and use", "Descent on Spaces, Proposition", "\\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}", "to descent the invertible sheaf in the construction", "of the quasi-inverse functor. The relative ampleness property descends", "by Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-ample}." ], "refs": [ "quot-lemma-extend-spaces-to-spaces", "spaces-descent-proposition-fpqc-descent-quasi-coherent", "spaces-descent-lemma-descending-property-ample" ], "ref_ids": [ 3191, 9437, 9413 ] } ], "ref_ids": [] }, { "id": 3199, "type": "theorem", "label": "quot-lemma-polarized-to-spaces-algebraic", "categories": [ "quot" ], "title": "quot-lemma-polarized-to-spaces-algebraic", "contents": [ "The functor (\\ref{equation-over-proper-spaces}) defines a $1$-morphism", "$$", "\\Polarizedstack \\to \\Spacesstack'_{fp, flat, proper}", "$$", "of stacks in groupoids over $\\Sch_{fppf}$", "which is algebraic in the sense of", "Criteria for Representability, Definition", "\\ref{criteria-definition-algebraic}." ], "refs": [ "criteria-definition-algebraic" ], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-spaces-stack} and \\ref{lemma-polarized-stack}", "the statement makes sense. To prove it, we choose a scheme $S$", "and an object $\\xi = (X \\to S)$ of $\\Spacesstack'_{fp, flat, proper}$", "over $S$. We have to show that", "$$", "\\mathcal{X} = (\\Sch/S)_{fppf} \\times_{\\xi, \\Spacesstack'_{fp, flat, proper}}", "\\Polarizedstack", "$$", "is an algebraic stack over $S$. Observe that an object of $\\mathcal{X}$", "is given by a pair $(T/S, \\mathcal{L})$ where $T$ is a scheme", "over $S$ and $\\mathcal{L}$ is an invertible $\\mathcal{O}_{X_T}$-module", "which is ample on $X_T/T$. Morphisms are defined in the obvious manner.", "In particular, we see immediately that we have an inclusion", "$$", "\\mathcal{X} \\subset \\Picardstack_{X/S}", "$$", "of categories over $(\\Sch/S)_{fppf}$, inducing equality on morphism", "sets. Since $\\Picardstack_{X/S}$ is an algebraic stack by", "Proposition \\ref{proposition-pic} it suffices to show that the inclusion", "above is representable by open immersions. This is exactly the content", "of Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-ample-in-neighbourhood}." ], "refs": [ "quot-lemma-spaces-stack", "quot-lemma-polarized-stack", "quot-proposition-pic", "spaces-descent-lemma-ample-in-neighbourhood" ], "ref_ids": [ 3190, 3197, 3229, 9414 ] } ], "ref_ids": [ 3145 ] }, { "id": 3200, "type": "theorem", "label": "quot-lemma-polarized-diagonal", "categories": [ "quot" ], "title": "quot-lemma-polarized-diagonal", "contents": [ "The diagonal", "$$", "\\Delta : \\Polarizedstack \\longrightarrow", "\\Polarizedstack \\times \\Polarizedstack", "$$", "is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "This is a formal consequence of", "Lemmas \\ref{lemma-polarized-to-spaces-algebraic} and", "\\ref{lemma-spaces-diagonal}.", "See Criteria for Representability, Lemma", "\\ref{criteria-lemma-diagonals-and-algebraic-morphisms}." ], "refs": [ "quot-lemma-polarized-to-spaces-algebraic", "quot-lemma-spaces-diagonal", "criteria-lemma-diagonals-and-algebraic-morphisms" ], "ref_ids": [ 3199, 3189, 3113 ] } ], "ref_ids": [] }, { "id": 3201, "type": "theorem", "label": "quot-lemma-polarized-limits", "categories": [ "quot" ], "title": "quot-lemma-polarized-limits", "contents": [ "The stack in groupoids $\\Polarizedstack$ is limit preserving", "(Artin's Axioms, Definition \\ref{artin-definition-limit-preserving})." ], "refs": [ "artin-definition-limit-preserving" ], "proofs": [ { "contents": [ "Let $I$ be a directed set and let $(A_i, \\varphi_{ii'})$", "be a system of rings over $I$. Set $S = \\Spec(A)$ and", "$S_i = \\Spec(A_i)$. We have to show that on fibre categories we have", "$$", "\\Polarizedstack_S = \\colim \\Polarizedstack_{S_i}", "$$", "We know that the category of schemes of finite presentation over", "$S$ is the colimit of the category of schemes of finite presentation", "over $S_i$, see", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}.", "Moreover, given $X_i \\to S_i$ of finite presentation, with", "limit $X \\to S$, then the category of invertible", "$\\mathcal{O}_X$-modules $\\mathcal{L}$ is the colimit of the categories", "of invertible $\\mathcal{O}_{X_i}$-modules $\\mathcal{L}_i$, see", "Limits, Lemma \\ref{limits-lemma-descend-modules-finite-presentation} and", "\\ref{limits-lemma-descend-invertible-modules}.", "If $X \\to S$ is proper and flat, then for sufficiently large", "$i$ the morphism $X_i \\to S_i$ is proper and flat too, see", "Limits, Lemmas \\ref{limits-lemma-eventually-proper} and", "\\ref{limits-lemma-descend-flat-finite-presentation}.", "Finally, if $\\mathcal{L}$ is ample on $X$", "then $\\mathcal{L}_i$ is ample on $X_i$ for", "$i$ sufficiently large, see", "Limits, Lemma \\ref{limits-lemma-limit-ample}.", "Putting everything together finishes the proof." ], "refs": [ "limits-lemma-descend-finite-presentation", "limits-lemma-descend-modules-finite-presentation", "limits-lemma-descend-invertible-modules", "limits-lemma-eventually-proper", "limits-lemma-descend-flat-finite-presentation", "limits-lemma-limit-ample" ], "ref_ids": [ 15077, 15078, 15079, 15089, 15062, 15045 ] } ], "ref_ids": [ 11420 ] }, { "id": 3202, "type": "theorem", "label": "quot-lemma-polarized-RS-star", "categories": [ "quot" ], "title": "quot-lemma-polarized-RS-star", "contents": [ "In Situation \\ref{situation-coherent}. Let", "$$", "\\xymatrix{", "T \\ar[r] \\ar[d] & T' \\ar[d] \\\\", "S \\ar[r] & S'", "}", "$$", "be a pushout in the category of schemes where", "$T \\to T'$ is a thickening and $T \\to S$ is affine, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.", "Then the functor on fibre categories", "$$", "\\Polarizedstack_{S'}", "\\longrightarrow", "\\Polarizedstack_S \\times_{\\Polarizedstack_T} \\Polarizedstack_{T'}", "$$", "is an equivalence." ], "refs": [ "more-morphisms-lemma-pushout-along-thickening" ], "proofs": [ { "contents": [ "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat}", "there is an equivalence", "$$", "\\textit{flat-lfp}_{S'}", "\\longrightarrow", "\\textit{flat-lfp}_S \\times_{\\textit{flat-lfp}_T} \\textit{flat-lfp}_{T'}", "$$", "where $\\textit{flat-lfp}_S$ signifies the category of schemes flat", "and locally of finite presentation over $S$.", "Let $X'/S'$ on the left hand side correspond to the triple", "$(X/S, Y'/T', \\varphi)$ on the right hand side.", "Set $Y = T \\times_{T'} Y'$ which is isomorphic with", "$T \\times_S X$ via $\\varphi$. Then More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-scheme-over-pushout-flat-modules}", "shows that we have an equivalence", "$$", "\\textit{QCoh-flat}_{X'/S'}", "\\longrightarrow", "\\textit{QCoh-flat}_{X/S}", "\\times_{\\textit{QCoh-flat}_{Y/T}} \\textit{QCoh-flat}_{Y'/T'}", "$$", "where $\\textit{QCoh-flat}_{X/S}$ signifies the category of", "quasi-coherent $\\mathcal{O}_X$-modules flat over $S$.", "Since $X \\to S$, $Y \\to T$, $X' \\to S'$, $Y' \\to T'$ are", "flat, this will in particular apply to invertible modules", "to give an equivalence of categories", "$$", "\\textit{Pic}(X')", "\\longrightarrow", "\\textit{Pic}(X) \\times_{\\textit{Pic}(Y)} \\textit{Pic}(Y')", "$$", "where $\\textit{Pic}(X)$ signifies the category of invertible", "$\\mathcal{O}_X$-modules. There is a small point here:", "one has to show that if an object $\\mathcal{F}'$", "of $\\textit{QCoh-flat}_{X'/S'}$", "pulls back to invertible modules on $X$ and $Y'$, then", "$\\mathcal{F}'$ is an invertible $\\mathcal{O}_{X'}$-module.", "It follows from the cited lemma that $\\mathcal{F}'$", "is an $\\mathcal{O}_{X'}$-module of finite presentation.", "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-flat-and-free-at-point-fibre}", "it suffices to check the restriction of", "$\\mathcal{F}'$ to fibres of $X' \\to S'$ is invertible.", "But the fibres of $X' \\to S'$ are the same as the fibres", "of $X \\to S$ and hence these restrictions are invertible.", "\\medskip\\noindent", "Having said the above we obtain an equivalence of categories if we drop", "the assumption (for the category of objects over $S$) that $X \\to S$ be proper", "and the assumption that $\\mathcal{L}$ be ample.", "Now it is clear that if $X' \\to S'$ is proper, then", "$X \\to S$ and $Y' \\to T'$ are proper (Morphisms, Lemma", "\\ref{morphisms-lemma-base-change-proper}).", "Conversely, if $X \\to S$ and $Y' \\to T'$ are proper, then", "$X' \\to S'$ is proper by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-thicken-property-morphisms-cartesian}.", "Similarly, if $\\mathcal{L}'$ is ample on $X'/S'$, then", "$\\mathcal{L}'|_X$ is ample on $X/S$ and", "$\\mathcal{L}'|_{Y'}$ is ample on $Y'/T'$", "(Morphisms, Lemma \\ref{morphisms-lemma-ample-base-change}).", "Finally, if $\\mathcal{L}'|_X$ is ample on $X/S$ and", "$\\mathcal{L}'|_{Y'}$ is ample on $Y'/T'$, then", "$\\mathcal{L}'$ is ample on $X'/S'$ by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-thicken-property-relatively-ample}." ], "refs": [ "more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat", "more-morphisms-lemma-scheme-over-pushout-flat-modules", "more-morphisms-lemma-flat-and-free-at-point-fibre", "morphisms-lemma-base-change-proper", "more-morphisms-lemma-thicken-property-morphisms-cartesian", "morphisms-lemma-ample-base-change", "more-morphisms-lemma-thicken-property-relatively-ample" ], "ref_ids": [ 13765, 13764, 13771, 5409, 13684, 5385, 13683 ] } ], "ref_ids": [ 13762 ] }, { "id": 3203, "type": "theorem", "label": "quot-lemma-polarized-tangent-space", "categories": [ "quot" ], "title": "quot-lemma-polarized-tangent-space", "contents": [ "Let $k$ be a field and let $x = (X \\to \\Spec(k), \\mathcal{L})$", "be an object of $\\mathcal{X} = \\Polarizedstack$ over $\\Spec(k)$.", "\\begin{enumerate}", "\\item If $k$ is of finite type over $\\mathbf{Z}$, then", "the vector spaces $T\\mathcal{F}_{\\mathcal{X}, k, x}$ and", "$\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x})$", "(see Artin's Axioms, Section \\ref{artin-section-tangent-spaces})", "are finite dimensional, and", "\\item in general the vector spaces $T_x(k)$ and $\\text{Inf}_x(k)$", "(see Artin's Axioms, Section \\ref{artin-section-inf})", "are finite dimensional.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The discussion in Artin's Axioms, Section \\ref{artin-section-tangent-spaces}", "only applies to fields of finite type over the base scheme $\\Spec(\\mathbf{Z})$.", "Our stack satisfies (RS*) by Lemma \\ref{lemma-polarized-RS-star}", "and we may apply", "Artin's Axioms, Lemma \\ref{artin-lemma-properties-lift-RS-star}", "to get the vector spaces $T_x(k)$ and $\\text{Inf}_x(k)$", "mentioned in (2). Moreover, in the finite type case these spaces agree with the", "ones mentioned in part (1)", "by Artin's Axioms, Remark \\ref{artin-remark-compare-deformation-spaces}.", "With this out of the way we can start the proof.", "\\medskip\\noindent", "One proof is to use an argument as in the proof of", "Lemma \\ref{lemma-spaces-tangent-space}; this would", "require us to develop a deformation theory for pairs", "consisting of a scheme and a quasi-coherent module.", "Another proof would be the use the result from", "Lemma \\ref{lemma-spaces-tangent-space},", "the algebraicity of", "$\\Polarizedstack \\to \\Spacesstack'_{fp, flat, proper}$,", "and a computation of the deformation space of an", "invertible module. However, what we will do instead", "is to translate the question into a deformation question", "on graded $k$-algebras and deduce the result that way.", "\\medskip\\noindent", "Let $\\mathcal{C}_k$ be the category of Artinian local $k$-algebras", "$A$ with residue field $k$. We get a predeformation category", "$p : \\mathcal{F} \\to \\mathcal{C}_k$ from our object $x$ of $\\mathcal{X}$", "over $k$, see", "Artin's Axioms, Section \\ref{artin-section-predeformation-categories}.", "Thus $\\mathcal{F}(A)$ is the category of triples", "$(X_A, \\mathcal{L}_A, \\alpha)$, where $(X_A, \\mathcal{L}_A)$", "is an object of $\\Polarizedstack$ over $A$ and $\\alpha$ is an isomorphism", "$(X_A, \\mathcal{L}_A) \\times_{\\Spec(A)} \\Spec(k) \\cong (X, \\mathcal{L})$.", "On the other hand, let $q : \\mathcal{G} \\to \\mathcal{C}_k$", "be the category cofibred in groupoids defined in", "Deformation Problems, Example \\ref{examples-defos-example-graded-algebras}.", "Choose $d_0 \\gg 0$ (we'll see below how large).", "Let $P$ be the graded $k$-algebra", "$$", "P = k \\oplus \\bigoplus\\nolimits_{d \\geq d_0} H^0(X, \\mathcal{L}^{\\otimes d})", "$$", "Then $y = (k, P)$ is an object of $\\mathcal{G}(k)$.", "Let $\\mathcal{G}_y$ be the predeformation category of", "Formal Deformation Theory, Remark", "\\ref{formal-defos-remark-localize-cofibered-groupoid}.", "Given $(X_A, \\mathcal{F}_A, \\alpha)$ as above we set", "$$", "Q = A \\oplus \\bigoplus\\nolimits_{d \\geq d_0} H^0(X_A, \\mathcal{L}_A^{\\otimes d})", "$$", "The isomorphism $\\alpha$ induces a map $\\beta : Q \\to P$.", "By deformation theory of projective schemes", "(More on Morphisms, Lemma \\ref{more-morphisms-lemma-deform-projective})", "we obtain a $1$-morphism", "$$", "\\mathcal{F} \\longrightarrow \\mathcal{G}_y,\\quad", "(X_A, \\mathcal{F}_A, \\alpha) \\longmapsto (Q, \\beta : Q \\to P)", "$$", "of categories cofibred in groupoids over $\\mathcal{C}_k$.", "In fact, this functor is an equivalence with quasi-inverse", "given by $Q \\mapsto \\underline{\\text{Proj}}_A(Q)$.", "Namely, the scheme $X_A = \\underline{\\text{Proj}}_A(Q)$", "is flat over $A$ by Divisors, Lemma \\ref{divisors-lemma-relative-proj-flat}.", "Set $\\mathcal{L}_A = \\mathcal{O}_{X_A}(1)$; this is flat over $A$", "by the same lemma. We get an isomorphism", "$(X_A, \\mathcal{L}_A) \\times_{\\Spec(A)} \\Spec(k) = (X, \\mathcal{L})$", "from $\\beta$. Then we can deduce all the desired properties of", "the pair $(X_A, \\mathcal{L}_A)$ from the corresponding properties", "of $(X, \\mathcal{L})$ using the techniques in", "More on Morphisms, Sections", "\\ref{more-morphisms-section-morphisms-thickenings} and", "\\ref{more-morphisms-section-deform}.", "Some details omitted.", "\\medskip\\noindent", "In conclusion, we see that $T\\mathcal{F} = T\\mathcal{G}_y = T_y\\mathcal{G}$", "and $\\text{Inf}(\\mathcal{F}) = \\text{Inf}_y(\\mathcal{G})$.", "These vector spaces are finite dimensional by Deformation Problems, Lemma", "\\ref{examples-defos-lemma-graded-algebras-TI}", "and the proof is complete." ], "refs": [ "quot-lemma-polarized-RS-star", "artin-lemma-properties-lift-RS-star", "artin-remark-compare-deformation-spaces", "quot-lemma-spaces-tangent-space", "quot-lemma-spaces-tangent-space", "formal-defos-remark-localize-cofibered-groupoid", "more-morphisms-lemma-deform-projective", "divisors-lemma-relative-proj-flat", "examples-defos-lemma-graded-algebras-TI" ], "ref_ids": [ 3202, 11388, 11434, 3194, 3194, 3545, 13728, 8045, 8733 ] } ], "ref_ids": [] }, { "id": 3204, "type": "theorem", "label": "quot-lemma-polarized-strong-effectiveness", "categories": [ "quot" ], "title": "quot-lemma-polarized-strong-effectiveness", "contents": [ "\\begin{slogan}", "Grothendieck's algebraization theorem continues to hold in", "the non-Noetherian setting if one assumes flatness and", "finite presentation.", "\\end{slogan}", "Let $(R_n)$ be an inverse system of rings with surjective transition maps", "whose kernels are locally nilpotent. Set $R = \\lim R_n$.", "Set $S_n = \\Spec(R_n)$ and $S = \\Spec(R)$. Consider a commutative diagram", "$$", "\\xymatrix{", "X_1 \\ar[r]_{i_1} \\ar[d] & X_2 \\ar[r]_{i_2} \\ar[d] & X_3 \\ar[r] \\ar[d] &", "\\ldots \\\\", "S_1 \\ar[r] & S_2 \\ar[r] & S_3 \\ar[r] & \\ldots", "}", "$$", "of schemes with cartesian squares. Suppose given $(\\mathcal{L}_n, \\varphi_n)$", "where each $\\mathcal{L}_n$ is an invertible sheaf on $X_n$ and", "$\\varphi_n : i_n^*\\mathcal{L}_{n + 1} \\to \\mathcal{L}_n$ is an isomorphism.", "If", "\\begin{enumerate}", "\\item $X_n \\to S_n$ is proper, flat, of finite presentation, and", "\\item $\\mathcal{L}_1$ is ample on $X_1$", "\\end{enumerate}", "then there exists a morphism of schemes $X \\to S$", "proper, flat, and of finite presentation", "and an ample invertible $\\mathcal{O}_X$-module $\\mathcal{L}$", "and isomorphisms $X_n \\cong X \\times_S S_n$ and", "$\\mathcal{L}_n \\cong \\mathcal{L}|_{X_n}$ compatible with", "the morphisms $i_n$ and $\\varphi_n$." ], "refs": [], "proofs": [ { "contents": [ "Choose $d_0$ for $X_1 \\to S_1$ and $\\mathcal{L}_1$ as in", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-deform-projective}.", "For any $n \\geq 1$ set", "$$", "A_n = R_n \\oplus", "\\bigoplus\\nolimits_{d \\geq d_0} H^0(X_n, \\mathcal{L}_n^{\\otimes d})", "$$", "By the lemma each $A_n$ is a finitely presented graded $R_n$-algebra", "whose homogeneous parts $(A_n)_d$ are finite projective $R_n$-modules", "such that $X_n = \\text{Proj}(A_n)$ and", "$\\mathcal{L}_n = \\mathcal{O}_{\\text{Proj}(A_n)}(1)$.", "The lemma also guarantees that the maps", "$$", "A_1 \\leftarrow A_2 \\leftarrow A_3 \\leftarrow \\ldots", "$$", "induce isomorphisms $A_n = A_m \\otimes_{R_m} R_n$ for $n \\leq m$.", "We set", "$$", "B = \\bigoplus\\nolimits_{d \\geq 0} B_d", "\\quad\\text{with}\\quad", "B_d = \\lim_n (A_n)_d", "$$", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-lim-finite-projective-gives-finite-projective}", "we see that $B_d$ is a finite projective $R$-module and that", "$B \\otimes_R R_n = A_n$. Thus the scheme", "$$", "X = \\text{Proj}(B)", "\\quad\\text{and}\\quad", "\\mathcal{L} = \\mathcal{O}_X(1)", "$$", "is flat over $S$ and $\\mathcal{L}$ is a quasi-coherent $\\mathcal{O}_X$-module", "flat over $S$, see", "Divisors, Lemma \\ref{divisors-lemma-relative-proj-flat}.", "Because formation of Proj commutes with base change", "(Constructions, Lemma \\ref{constructions-lemma-base-change-map-proj})", "we obtain canonical isomorphisms", "$$", "X \\times_S S_n = X_n", "\\quad\\text{and}\\quad", "\\mathcal{L}|_{X_n} \\cong \\mathcal{L}_n", "$$", "compatible with the transition maps of the system.", "Thus we may think of $X_1 \\subset X$ as a closed subscheme.", "Below we will show that $B$ is of finite presentation over $R$.", "By Divisors, Lemmas \\ref{divisors-lemma-relative-proj-proper} and", "\\ref{divisors-lemma-relative-proj-finite-presentation}", "this implies that $X \\to S$ is of finite presentation", "and proper and that $\\mathcal{L} = \\mathcal{O}_X(1)$", "is of finite presentation as an $\\mathcal{O}_X$-module.", "Since the restriction of $\\mathcal{L}$ to the base change", "$X_1 \\to S_1$ is invertible, we see from", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-finite-free-open}", "that $\\mathcal{L}$ is invertible on an open neighbourhood of $X_1$ in $X$.", "Since $X \\to S$ is closed and since $\\Ker(R \\to R_1)$", "is contained in the Jacobson radical", "(More on Algebra, Lemma \\ref{more-algebra-lemma-limit-henselian})", "we see that any open neighbourhood of $X_1$ in $X$ is equal to $X$.", "Thus $\\mathcal{L}$ is invertible. Finally, the set of points in", "$S$ where $\\mathcal{L}$ is ample on the fibre is open in $S$", "(More on Morphisms, Lemma \\ref{more-morphisms-lemma-ample-in-neighbourhood})", "and contains $S_1$ hence equals $S$. Thus $X \\to S$ and $\\mathcal{L}$", "have all the properties required of them in the statement of the lemma.", "\\medskip\\noindent", "We prove the claim above.", "Choose a presentation $A_1 = R_1[X_1, \\ldots, X_s]/(F_1, \\ldots, F_t)$", "where $X_i$ are variables having degrees $d_i$ and $F_j$", "are homogeneous polynomials in $X_i$ of degree $e_j$.", "Then we can choose a map", "$$", "\\Psi : R[X_1, \\ldots, X_s] \\longrightarrow B", "$$", "lifting the map $R_1[X_1, \\ldots, X_s] \\to A_1$. Since each $B_d$", "is finite projective over $R$ we conclude from ", "Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK}", "using again that $\\Ker(R \\to R_1)$ is contained in the Jacobson radical", "of $R$) that $\\Psi$ is surjective. Since $- \\otimes_R R_1$ is right", "exact we can find $G_1, \\ldots, G_t \\in \\Ker(\\Psi)$", "mapping to $F_1, \\ldots, F_t$ in $R_1[X_1, \\ldots, X_s]$.", "Observe that $\\Ker(\\Psi)_d$ is a finite projective $R$-module", "for all $d \\geq 0$ as the kernel of the surjection", "$R[X_1, \\ldots, X_s]_d \\to B_d$ of finite projective $R$-modules.", "We conclude from Nakayama's lemma once more that ", "$\\Ker(\\Psi)$ is generated by $G_1, \\ldots, G_t$." ], "refs": [ "more-morphisms-lemma-deform-projective", "more-algebra-lemma-lim-finite-projective-gives-finite-projective", "divisors-lemma-relative-proj-flat", "constructions-lemma-base-change-map-proj", "divisors-lemma-relative-proj-proper", "divisors-lemma-relative-proj-finite-presentation", "more-morphisms-lemma-finite-free-open", "more-algebra-lemma-limit-henselian", "more-morphisms-lemma-ample-in-neighbourhood", "algebra-lemma-NAK" ], "ref_ids": [ 13728, 9880, 8045, 12613, 8043, 8046, 13772, 9858, 13933, 401 ] } ], "ref_ids": [] }, { "id": 3205, "type": "theorem", "label": "quot-lemma-polarized-existence", "categories": [ "quot" ], "title": "quot-lemma-polarized-existence", "contents": [ "Consider the stack $\\Polarizedstack$ over the base", "scheme $\\Spec(\\mathbf{Z})$. Then every formal object is effective." ], "refs": [], "proofs": [ { "contents": [ "For definitions of the notions in the lemma, please see", "Artin's Axioms, Section \\ref{artin-section-formal-objects}.", "From the definitions we see the lemma follows immediately", "from the more general Lemma \\ref{lemma-polarized-strong-effectiveness}." ], "refs": [ "quot-lemma-polarized-strong-effectiveness" ], "ref_ids": [ 3204 ] } ], "ref_ids": [] }, { "id": 3206, "type": "theorem", "label": "quot-lemma-polarized-defo-thy", "categories": [ "quot" ], "title": "quot-lemma-polarized-defo-thy", "contents": [ "The stack in groupoids $\\Polarizedstack$", "satisfies openness of versality over $\\Spec(\\mathbf{Z})$.", "Similarly, after base change (Remark \\ref{remark-polarized-base-change})", "openness of versality holds over any Noetherian base scheme $S$." ], "refs": [ "quot-remark-polarized-base-change" ], "proofs": [ { "contents": [ "This follows from", "Artin's Axioms, Lemma \\ref{artin-lemma-SGE-implies-openness-versality}", "and Lemmas \\ref{lemma-polarized-diagonal},", "\\ref{lemma-polarized-RS-star},", "\\ref{lemma-polarized-limits}, and", "\\ref{lemma-polarized-strong-effectiveness}.", "For the ``usual'' proof of this fact, please see the discussion", "in the remark following this proof." ], "refs": [ "artin-lemma-SGE-implies-openness-versality", "quot-lemma-polarized-diagonal", "quot-lemma-polarized-RS-star", "quot-lemma-polarized-limits", "quot-lemma-polarized-strong-effectiveness" ], "ref_ids": [ 11386, 3200, 3202, 3201, 3204 ] } ], "ref_ids": [ 3241 ] }, { "id": 3207, "type": "theorem", "label": "quot-lemma-curves-fibred-in-groupoids", "categories": [ "quot" ], "title": "quot-lemma-curves-fibred-in-groupoids", "contents": [ "The category $\\Curvesstack$ is fibred in groupoids over $\\Sch_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "Using the embedding (\\ref{equation-curves-over-proper-spaces}),", "the description of the image, and", "the corresponding fact for $\\Spacesstack'_{fp, flat, proper}$", "(Lemma \\ref{lemma-spaces-fibred-in-groupoids})", "this reduces to the following statement: Given a morphism", "$$", "\\xymatrix{", "X' \\ar[r] \\ar[d] & X \\ar[d] \\\\", "S' \\ar[r] & S", "}", "$$", "in $\\Spacesstack'_{fp, flat, proper}$ (recall that this implies", "in particular the diagram is cartesian)", "if $X \\to S$ has relative dimension $\\leq 1$, then $X' \\to S'$", "has relative dimension $\\leq 1$.", "This follows from Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}." ], "refs": [ "quot-lemma-spaces-fibred-in-groupoids", "spaces-morphisms-lemma-dimension-fibre-after-base-change" ], "ref_ids": [ 3188, 4872 ] } ], "ref_ids": [] }, { "id": 3208, "type": "theorem", "label": "quot-lemma-curves-stack", "categories": [ "quot" ], "title": "quot-lemma-curves-stack", "contents": [ "The category $\\Curvesstack$ is a stack in groupoids over $\\Sch_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "Using the embedding (\\ref{equation-curves-over-proper-spaces}),", "the description of the image, and", "the corresponding fact for $\\Spacesstack'_{fp, flat, proper}$", "(Lemma \\ref{lemma-spaces-stack})", "this reduces to the following statement: Given an object", "$X \\to S$ of $\\Spacesstack'_{fp, flat, proper}$", "and an fppf covering $\\{S_i \\to S\\}_{i \\in I}$", "the following are equivalent:", "\\begin{enumerate}", "\\item $X \\to S$ has relative dimension $\\leq 1$, and", "\\item for each $i$ the base change $X_i \\to S_i$", "has relative dimension $\\leq 1$.", "\\end{enumerate}", "This follows from Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}." ], "refs": [ "quot-lemma-spaces-stack", "spaces-morphisms-lemma-dimension-fibre-after-base-change" ], "ref_ids": [ 3190, 4872 ] } ], "ref_ids": [] }, { "id": 3209, "type": "theorem", "label": "quot-lemma-curves-diagonal", "categories": [ "quot" ], "title": "quot-lemma-curves-diagonal", "contents": [ "The diagonal", "$$", "\\Delta : \\Curvesstack \\longrightarrow \\Curvesstack \\times \\Curvesstack", "$$", "is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the fully faithful embedding", "(\\ref{equation-curves-over-proper-spaces}) and", "the corresponding fact for $\\Spacesstack'_{fp, flat, proper}$", "(Lemma \\ref{lemma-spaces-diagonal})." ], "refs": [ "quot-lemma-spaces-diagonal" ], "ref_ids": [ 3189 ] } ], "ref_ids": [] }, { "id": 3210, "type": "theorem", "label": "quot-lemma-curves-limits", "categories": [ "quot" ], "title": "quot-lemma-curves-limits", "contents": [ "The stack $\\Curvesstack \\to \\Sch_{fppf}$ is limit preserving", "(Artin's Axioms, Definition \\ref{artin-definition-limit-preserving})." ], "refs": [ "artin-definition-limit-preserving" ], "proofs": [ { "contents": [ "Using the embedding (\\ref{equation-curves-over-proper-spaces}),", "the description of the image, and", "the corresponding fact for $\\Spacesstack'_{fp, flat, proper}$", "(Lemma \\ref{lemma-spaces-limits})", "this reduces to the following statement:", "Let $T = \\lim T_i$ be the limits of a", "directed inverse system of affine schemes.", "Let $i \\in I$ and let $X_i \\to T_i$ be an object of", "$\\Spacesstack'_{fp, flat, proper}$ over $T_i$.", "Assume that $T \\times_{T_i} X_i \\to T$ has", "relative dimension $\\leq 1$.", "Then for some $i' \\geq i$ the morphism", "$T_{i'} \\times_{T_i} X_i \\to T_i$ has", "relative dimension $\\leq 1$. This follows from", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-eventually-relative-dimension}." ], "refs": [ "quot-lemma-spaces-limits", "spaces-limits-lemma-eventually-relative-dimension" ], "ref_ids": [ 3192, 4597 ] } ], "ref_ids": [ 11420 ] }, { "id": 3211, "type": "theorem", "label": "quot-lemma-curves-RS-star", "categories": [ "quot" ], "title": "quot-lemma-curves-RS-star", "contents": [ "Let", "$$", "\\xymatrix{", "T \\ar[r] \\ar[d] & T' \\ar[d] \\\\", "S \\ar[r] & S'", "}", "$$", "be a pushout in the category of schemes where", "$T \\to T'$ is a thickening and $T \\to S$ is affine, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.", "Then the functor on fibre categories", "$$", "\\Curvesstack_{S'}", "\\longrightarrow", "\\Curvesstack_S", "\\times_{\\Curvesstack_T}", "\\Curvesstack_{T'}", "$$", "is an equivalence." ], "refs": [ "more-morphisms-lemma-pushout-along-thickening" ], "proofs": [ { "contents": [ "Using the embedding (\\ref{equation-curves-over-proper-spaces}),", "the description of the image, and", "the corresponding fact for $\\Spacesstack'_{fp, flat, proper}$", "(Lemma \\ref{lemma-spaces-RS-star})", "this reduces to the following statement:", "given a morphism $X' \\to S'$ of an algebraic space to $S'$", "which is of finite presentation, flat, proper then", "$X' \\to S'$ has relative dimension $\\leq 1$", "if and only if $S \\times_{S'} X' \\to S$", "and $T' \\times_{S'} X' \\to T'$ have relative dimension $\\leq 1$.", "One implication follows from the fact that", "having relative dimension $\\leq 1$ is preserved under base change", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}).", "The other follows from the fact that having relative", "dimension $\\leq 1$ is checked on the fibres and that", "the fibres of $X' \\to S'$ (over points of the scheme $S'$)", "are the same as the fibres of $S \\times_{S'} X' \\to S$", "since $S \\to S'$ is a thickening by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}." ], "refs": [ "quot-lemma-spaces-RS-star", "spaces-morphisms-lemma-dimension-fibre-after-base-change", "more-morphisms-lemma-pushout-along-thickening" ], "ref_ids": [ 3193, 4872, 13762 ] } ], "ref_ids": [ 13762 ] }, { "id": 3212, "type": "theorem", "label": "quot-lemma-curves-tangent-space", "categories": [ "quot" ], "title": "quot-lemma-curves-tangent-space", "contents": [ "Let $k$ be a field and let $x = (X \\to \\Spec(k))$ be an object of", "$\\mathcal{X} = \\Curvesstack$ over $\\Spec(k)$.", "\\begin{enumerate}", "\\item If $k$ is of finite type over $\\mathbf{Z}$, then", "the vector spaces $T\\mathcal{F}_{\\mathcal{X}, k, x}$ and", "$\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x})$", "(see Artin's Axioms, Section \\ref{artin-section-tangent-spaces})", "are finite dimensional, and", "\\item in general the vector spaces $T_x(k)$ and $\\text{Inf}_x(k)$", "(see Artin's Axioms, Section \\ref{artin-section-inf})", "are finite dimensional.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the fully faithful embedding", "(\\ref{equation-curves-over-proper-spaces}) and", "the corresponding fact for $\\Spacesstack'_{fp, flat, proper}$", "(Lemma \\ref{lemma-spaces-tangent-space})." ], "refs": [ "quot-lemma-spaces-tangent-space" ], "ref_ids": [ 3194 ] } ], "ref_ids": [] }, { "id": 3213, "type": "theorem", "label": "quot-lemma-curves-existence", "categories": [ "quot" ], "title": "quot-lemma-curves-existence", "contents": [ "Consider the stack $\\Curvesstack$ over the base", "scheme $\\Spec(\\mathbf{Z})$. Then every formal object is effective." ], "refs": [], "proofs": [ { "contents": [ "For definitions of the notions in the lemma, please see", "Artin's Axioms, Section \\ref{artin-section-formal-objects}.", "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian complete", "local ring. Let $(X_n \\to \\Spec(A/\\mathfrak m^n))$", "be a formal object of $\\Curvesstack$ over $A$.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-formal-algebraic-space-proper-reldim-1}", "there exists a projective morphism $X \\to \\Spec(A)$", "and a compatible system of ismomorphisms", "$X \\times_{\\Spec(A)} \\Spec(A/\\mathfrak m^n) \\cong X_n$. By", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-check-flatness-on-infinitesimal-nbhds}", "we see that $X \\to \\Spec(A)$ is flat. By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-dimension-fibres-proper-flat}", "we see that $X \\to \\Spec(A)$ has relative dimension $\\leq 1$.", "This proves the lemma." ], "refs": [ "spaces-more-morphisms-lemma-formal-algebraic-space-proper-reldim-1", "more-morphisms-lemma-check-flatness-on-infinitesimal-nbhds", "more-morphisms-lemma-dimension-fibres-proper-flat" ], "ref_ids": [ 212, 13744, 13844 ] } ], "ref_ids": [] }, { "id": 3214, "type": "theorem", "label": "quot-lemma-curves-defo-thy", "categories": [ "quot" ], "title": "quot-lemma-curves-defo-thy", "contents": [ "The stack in groupoids $\\mathcal{X} = \\Curvesstack$", "satisfies openness of versality over $\\Spec(\\mathbf{Z})$.", "Similarly, after base change (Remark \\ref{remark-curves-base-change})", "openness of versality holds over any Noetherian base scheme $S$." ], "refs": [ "quot-remark-curves-base-change" ], "proofs": [ { "contents": [ "This is immediate from the fully faithful embedding", "(\\ref{equation-curves-over-proper-spaces}) and", "the corresponding fact for $\\Spacesstack'_{fp, flat, proper}$", "(Lemma \\ref{lemma-spaces-defo-thy})." ], "refs": [ "quot-lemma-spaces-defo-thy" ], "ref_ids": [ 3195 ] } ], "ref_ids": [ 3243 ] }, { "id": 3215, "type": "theorem", "label": "quot-lemma-curves-open-and-closed-in-spaces", "categories": [ "quot" ], "title": "quot-lemma-curves-open-and-closed-in-spaces", "contents": [ "The $1$-morphism (\\ref{equation-curves-over-proper-spaces})", "$$", "\\Curvesstack \\longrightarrow \\Spacesstack'_{fp, flat, proper}", "$$", "is representable by open and closed immersions." ], "refs": [], "proofs": [ { "contents": [ "Since (\\ref{equation-curves-over-proper-spaces}) is a fully faithful", "embedding of categories it suffices to show the following:", "given an object $X \\to S$ of $\\Spacesstack'_{fp, flat, proper}$", "there exists an open and closed subscheme $U \\subset S$", "such that a morphism $S' \\to S$ factors through $U$ if and only if the", "base change $X' \\to S'$ of $X \\to S$ has relative dimension $\\leq 1$.", "This follows immediately from", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-dimension-fibres-proper-flat}." ], "refs": [ "spaces-more-morphisms-lemma-dimension-fibres-proper-flat" ], "ref_ids": [ 164 ] } ], "ref_ids": [] }, { "id": 3216, "type": "theorem", "label": "quot-lemma-complexes-open-neg-exts-vanishing", "categories": [ "quot" ], "title": "quot-lemma-complexes-open-neg-exts-vanishing", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume $f$ is proper, flat, and of finite presentation.", "Let $K, E \\in D(\\mathcal{O}_X)$. Assume $K$ is pseudo-coherent", "and $E$ is $Y$-perfect (More on Morphisms of Spaces, Definition", "\\ref{spaces-more-morphisms-definition-relatively-perfect}).", "For a field $k$ and a morphism $y : \\Spec(k) \\to Y$ denote $K_y$, $E_y$", "the pullback to the fibre $X_y$.", "\\begin{enumerate}", "\\item There is an open $W \\subset Y$ characterized by the property", "$$", "y \\in |W|", "\\Leftrightarrow", "\\Ext^i_{\\mathcal{O}_{X_y}}(K_y, E_y) = 0", "\\text{ for }i < 0.", "$$", "\\item For any morphism $V \\to Y$ factoring through $W$ we have", "$$", "\\Ext^i_{\\mathcal{O}_{X_V}}(K_V, E_V) = 0", "\\quad\\text{for}\\quad i < 0", "$$", "where $X_V$ is the base change of $X$ and $K_V$ and $E_V$", "are the derived pullbacks of $K$ and $E$ to $X_V$.", "\\item The functor $V \\mapsto \\Hom_{\\mathcal{O}_{X_V}}(K_V, E_V)$", "is a sheaf on $(\\textit{Spaces}/W)_{fppf}$ representable by an", "algebraic space affine and of finite presentation over $W$.", "\\end{enumerate}" ], "refs": [ "spaces-more-morphisms-definition-relatively-perfect" ], "proofs": [ { "contents": [ "For any morphism $V \\to Y$ the complex $K_V$ is pseudo-coherent", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-pseudo-coherent-pullback})", "and $E_V$ is $V$-perfect (More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-base-change-relatively-perfect}).", "Another observation is that given $y : \\Spec(k) \\to Y$", "and a field extension $k'/k$ with $y' : \\Spec(k') \\to Y$", "the induced morphism, we have", "$$", "\\Ext^i_{\\mathcal{O}_{X_{y'}}}(K_{y'}, E_{y'}) =", "\\Ext^i_{\\mathcal{O}_{X_y}}(K_y, E_y) \\otimes_k k'", "$$", "by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-morphism-and-hom-out-of-perfect}.", "Thus the vanishing in (1) is really a property of the induced", "point $y \\in |Y|$.", "We will use these two observations without further mention in the proof.", "\\medskip\\noindent", "Assume first $Y$ is an affine scheme. Then we may apply", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-compute-ext-rel-perfect}", "and find a pseudo-coherent $L \\in D(\\mathcal{O}_Y)$ which", "``universally computes'' $Rf_*R\\SheafHom(K, E)$ in the sense", "described in that lemma. Unwinding the definitions, we obtain", "for a point $y \\in Y$ the equality", "$$", "\\Ext^i_{\\kappa(y)}(L \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} \\kappa(y),", "\\kappa(y)) = \\Ext^i_{\\mathcal{O}_{X_y}}(K_y, E_y)", "$$", "We conclude that", "$$", "H^i(L \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} \\kappa(y)) = 0", "\\text{ for } i > 0 \\Leftrightarrow", "\\Ext^i_{\\mathcal{O}_{X_y}}(K_y, E_y) = 0 \\text{ for }i < 0.", "$$", "By Derived Categories of Schemes, Lemma \\ref{perfect-lemma-jump-loci}", "the set $W$ of $y \\in Y$ where this happens defines an open of $Y$.", "This open $W$ then satisfies the requirement in (1) for all morphisms", "from spectra of fields, by the ``universality'' of $L$.", "\\medskip\\noindent", "Let's go back to $Y$ a general algebraic space.", "Choose an \\'etale covering $\\{V_i \\to Y\\}$ by affine schemes $V_i$.", "Then we see that the subset $W \\subset |Y|$ pulls back to the corresponding", "subset $W_i \\subset |V_i|$ for $X_{V_i}$, $K_{V_i}$, $E_{V_i}$.", "By the previous paragraph we find that $W_i$ is open, hence $W$ is open.", "This proves (1) in general. Moreover, parts (2) and (3) are entirely formulated", "in terms of the category $\\textit{Spaces}/W$ and the restrictions", "$X_W$, $K_W$, $E_W$. This reduces us to the case $W = Y$.", "\\medskip\\noindent", "Assume $W = Y$. We claim that for any algebraic space $V$ over $Y$", "we have $Rf_{V, *}R\\SheafHom(K_V, E_V)$ has vanishing cohomology", "sheaves in degrees $< 0$. This will prove (2) because", "$$", "\\Ext^i_{\\mathcal{O}_{X_V}}(K_V, E_V) =", "H^i(X_V, R\\SheafHom(K_V, E_V)) = ", "H^i(V, Rf_{V, *}R\\SheafHom(K_V, E_V))", "$$", "by Cohomology on Sites, Lemmas", "\\ref{sites-cohomology-lemma-section-RHom-over-U} and", "\\ref{sites-cohomology-lemma-Leray-unbounded}", "and the vanishing of the cohomology sheaves implies the", "cohomology group $H^i$ is zero for $i < 0$ by", "Derived Categories, Lemma \\ref{derived-lemma-negative-vanishing}.", "\\medskip\\noindent", "To prove the claim, we may work \\'etale locally on $V$.", "In particular, we may assume $Y$ is affine and $W = Y$.", "Let $L \\in D(\\mathcal{O}_Y)$ be as in the second paragraph of the proof.", "For an algebraic space $V$ over $Y$ denote $L_V$ the derived pullback of", "$L$ to $V$. (An important feature we will use is that $L$ ``works'' for all", "algebraic spaces $V$ over $Y$ and not just affine $V$.)", "As $W = Y$ we have $H^i(L) = 0$ for $i > 0$", "(use More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-pseudo-coherent-from-residue-field}", "to go from fibres to stalks). Hence $H^i(L_V) = 0$ for $i > 0$.", "The property defining $L$ is that", "$$", "Rf_{V, *}R\\SheafHom(K_V, E_V) = R\\SheafHom(L_V, \\mathcal{O}_V)", "$$", "Since $L_V$ sits in degrees $\\leq 0$, we conclude that", "$R\\SheafHom(L_V, \\mathcal{O}_V)$ sits in degrees $\\geq 0$", "thereby proving the claim. This finishes the proof of (2).", "\\medskip\\noindent", "Assume $W = Y$ but make no assumptions on the algebraic space $Y$.", "Since we have (2), we see from", "Simplicial Spaces, Lemma \\ref{spaces-simplicial-lemma-fppf-neg-ext-zero-hom}", "that the functor $F$ given by $F(V) = \\Hom_{\\mathcal{O}_{X_V}}(K_V, E_V)$", "is a sheaf\\footnote{To check the sheaf property", "for a covering $\\{V_i \\to V\\}_{i \\in I}$ first consider the", "{\\v C}ech fppf hypercovering $a : V_\\bullet \\to V$ with", "$V_n = \\coprod_{i_0 \\ldots i_n} V_{i_0} \\times_V \\ldots \\times_V V_{i_n}$", "and then set $U_\\bullet = V_\\bullet \\times_{a, V} X_V$. Then", "$U_\\bullet \\to X_V$ is an fppf hypercovering to which we may", "apply Simplicial Spaces, Lemma", "\\ref{spaces-simplicial-lemma-fppf-neg-ext-zero-hom}.}", "on $(\\textit{Spaces}/Y)_{fppf}$. Thus to prove that $F$", "is an algebraic space and that $F \\to Y$ is affine and of", "finite presentation, we may work \\'etale locally on $Y$; see", "Bootstrap, Lemma \\ref{bootstrap-lemma-locally-algebraic-space-finite-type}", "and", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-affine-local} and", "\\ref{spaces-morphisms-lemma-finite-presentation-local}. We conclude", "that it suffices to prove $F$ is an affine algebraic space of", "finite presentation over $Y$ when $Y$ is an affine scheme. In this", "case we go back to our pseudo-coherent complex $L \\in D(\\mathcal{O}_Y)$.", "Since $H^i(L) = 0$ for $i > 0$, we can represent $L$ by a complex", "of the form", "$$", "\\ldots \\to \\mathcal{O}_Y^{\\oplus m_1} \\to \\mathcal{O}_Y^{\\oplus m_0}", "\\to 0 \\to \\ldots", "$$", "with the last term in degree $0$, see More on Algebra, Lemma", "\\ref{more-algebra-lemma-pseudo-coherent}. Combining the two displayed formulas", "earlier in the proof we find that", "$$", "F(V) =", "\\Ker(", "\\Hom_V(\\mathcal{O}_V^{\\oplus m_0}, \\mathcal{O}_V)", "\\to ", "\\Hom_V(\\mathcal{O}_V^{\\oplus m_1}, \\mathcal{O}_V)", ")", "$$", "In other words, there is a fibre product diagram", "$$", "\\xymatrix{", "F \\ar[d] \\ar[r] & Y \\ar[d]^0 \\\\", "\\mathbf{A}_Y^{m_0} \\ar[r] & \\mathbf{A}_Y^{m_1}", "}", "$$", "which proves what we want." ], "refs": [ "sites-cohomology-lemma-pseudo-coherent-pullback", "spaces-more-morphisms-lemma-base-change-relatively-perfect", "perfect-lemma-affine-morphism-and-hom-out-of-perfect", "spaces-more-morphisms-lemma-compute-ext-rel-perfect", "perfect-lemma-jump-loci", "sites-cohomology-lemma-Leray-unbounded", "derived-lemma-negative-vanishing", "more-algebra-lemma-lift-pseudo-coherent-from-residue-field", "spaces-simplicial-lemma-fppf-neg-ext-zero-hom", "spaces-simplicial-lemma-fppf-neg-ext-zero-hom", "bootstrap-lemma-locally-algebraic-space-finite-type", "spaces-morphisms-lemma-affine-local", "spaces-morphisms-lemma-finite-presentation-local", "more-algebra-lemma-pseudo-coherent" ], "ref_ids": [ 4367, 262, 7029, 266, 7058, 4257, 1839, 10231, 9139, 9139, 2626, 4798, 4841, 10148 ] } ], "ref_ids": [ 300 ] }, { "id": 3217, "type": "theorem", "label": "quot-lemma-complexes", "categories": [ "quot" ], "title": "quot-lemma-complexes", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. Assume $f$ is proper, flat, and of finite presentation.", "Let $E \\in D(\\mathcal{O}_X)$.", "Assume", "\\begin{enumerate}", "\\item $E$ is $S$-perfect (More on Morphisms of Spaces, Definition", "\\ref{spaces-more-morphisms-definition-relatively-perfect}), and", "\\item for every point $s \\in S$ we have", "$$", "\\Ext^i_{\\mathcal{O}_{X_s}}(E_s, E_s) = 0", "\\quad\\text{for}\\quad i < 0", "$$", "where $E_s$ is the pullback to the fibre $X_s$.", "\\end{enumerate}", "Then", "\\begin{enumerate}", "\\item[(a)] (1) and (2) are preserved by arbitrary base change $V \\to Y$,", "\\item[(b)] $\\Ext^i_{\\mathcal{O}_{X_V}}(E_V, E_V) = 0$ for $i < 0$", "and all $V$ over $Y$,", "\\item[(c)] $V \\mapsto \\Hom_{\\mathcal{O}_{X_V}}(E_V, E_V)$ is representable", "by an algebraic space affine and of finite presentation over $Y$.", "\\end{enumerate}", "Here $X_V$ is the base change of $X$ and $E_V$ is the derived pullback", "of $E$ to $X_V$." ], "refs": [ "spaces-more-morphisms-definition-relatively-perfect" ], "proofs": [ { "contents": [ "Immediate consequence of Lemma \\ref{lemma-complexes-open-neg-exts-vanishing}." ], "refs": [ "quot-lemma-complexes-open-neg-exts-vanishing" ], "ref_ids": [ 3216 ] } ], "ref_ids": [ 300 ] }, { "id": 3218, "type": "theorem", "label": "quot-lemma-complexes-fibred-in-groupoids", "categories": [ "quot" ], "title": "quot-lemma-complexes-fibred-in-groupoids", "contents": [ "In Situation \\ref{situation-complexes} the functor", "$p : \\Complexesstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf}$", "is fibred in groupoids." ], "refs": [], "proofs": [ { "contents": [ "We show that $p$ is fibred in groupoids by checking conditions", "(1) and (2) of Categories, Definition", "\\ref{categories-definition-fibred-groupoids}.", "Given an object $(T', g', E')$", "of $\\Complexesstack_{X/B}$ and a morphism $h : T \\to T'$ of", "schemes over $S$ we can set $g = h \\circ g'$ and", "$E = L(h')^*E'$ where $h' : X_T \\to X_{T'}$", "is the base change of $h$. Then it is clear that we obtain", "a morphism $(T, g, E) \\to (T', g', E')$", "of $\\Complexesstack_{X/B}$ lying over $h$. This proves (1).", "For (2) suppose we are given morphisms", "$$", "(h_1, \\varphi_1) : (T_1, g_1, E_1) \\to (T, g, E)", "\\quad\\text{and}\\quad", "(h_2, \\varphi_2) : (T_2, g_2, E_2) \\to (T, g, E)", "$$", "of $\\Complexesstack_{X/B}$ and a morphism $h : T_1 \\to T_2$ such that", "$h_2 \\circ h = h_1$. Then we can let $\\varphi$ be the composition", "$$", "L(h')^*E_2", "\\xrightarrow{L(h')^*\\varphi_2^{-1}}", "L(h')^*L(h_2)^*E = L(h_1)^*E", "\\xrightarrow{\\varphi_1}", "E_1", "$$", "to obtain the morphism", "$(h, \\varphi) : (T_1, g_1, E_1) \\to (T_2, g_2, E_2)$", "that witnesses the truth of condition (2)." ], "refs": [ "categories-definition-fibred-groupoids" ], "ref_ids": [ 12392 ] } ], "ref_ids": [] }, { "id": 3219, "type": "theorem", "label": "quot-lemma-complexes-diagonal", "categories": [ "quot" ], "title": "quot-lemma-complexes-diagonal", "contents": [ "In Situation \\ref{situation-complexes}. Denote", "$\\mathcal{X} = \\Complexesstack_{X/B}$. Then", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is", "representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Consider two objects $x = (T, g, E)$ and $y = (T, g', E')$", "of $\\mathcal{X}$ over a scheme $T$. We have to show that", "$\\mathit{Isom}_\\mathcal{X}(x, y)$ is an algebraic space over $T$, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-representable-diagonal}.", "If for $h : T' \\to T$ the restrictions $x|_{T'}$ and $y|_{T'}$ are isomorphic", "in the fibre category $\\mathcal{X}_{T'}$, then $g \\circ h = g' \\circ h$.", "Hence there is a transformation of presheaves", "$$", "\\mathit{Isom}_\\mathcal{X}(x, y) \\longrightarrow \\text{Equalizer}(g, g')", "$$", "Since the diagonal of $B$ is representable (by schemes) this equalizer is", "a scheme. Thus we may replace $T$ by this equalizer and", "$E$ and $E'$ by their pullbacks. Thus we may assume $g = g'$.", "\\medskip\\noindent", "Assume $g = g'$. After replacing $B$ by $T$ and $X$ by $X_T$ we arrive", "at the following problem. Given $E, E' \\in D(\\mathcal{O}_X)$", "satisfying conditions (1), (2) of Lemma \\ref{lemma-complexes}", "we have to show that $\\mathit{Isom}(E, E')$ is an algebraic space.", "Here $\\mathit{Isom}(E, E')$ is the functor", "$$", "(\\Sch/B)^{opp} \\to \\textit{Sets},\\quad", "T \\mapsto \\{\\varphi : E_T \\to E'_T", "\\text{ isomorphism in }D(\\mathcal{O}_{X_T})\\}", "$$", "where $E_T$ and $E'_T$ are the derived pullbacks of $E$ and $E'$", "to $X_T$. Now, let $W \\subset B$, resp.\\ $W' \\subset B$ be the", "open subspace of $B$ associated", "to $E, E'$, resp.\\ to $E', E$ by", "Lemma \\ref{lemma-complexes-open-neg-exts-vanishing}.", "Clearly, if there exists an isomorphism $E_T \\to E'_T$ as in", "the definition of $\\mathit{Isom}(E, E')$, then we see that", "$T \\to B$ factors into both $W$ and $W'$ (because we have", "condition (1) for $E$ and $E'$ and we'll obviously have", "$E_t \\cong E'_t$ so no nonzero maps $E_t[i] \\to E_t$", "or $E'_t[i] \\to E_t$ over the fibre $X_t$ for $i > 0$.", "Thus we may replace $B$ by the open $W \\cap W'$.", "In this case the functor $H = \\SheafHom(E, E')$", "$$", "(\\Sch/B)^{opp} \\to \\textit{Sets},\\quad T \\mapsto", "\\Hom_{\\mathcal{O}_{X_T}}(E_T, E'_T)", "$$", "is an algebraic space affine and of finite presentation over $B$ by", "Lemma \\ref{lemma-complexes-open-neg-exts-vanishing}.", "The same is true for", "$H' = \\SheafHom(E', E)$,", "$I = \\SheafHom(E, E)$, and", "$I' = \\SheafHom(E', E')$.", "Therefore we can repeat the argument of the proof of", "Proposition \\ref{proposition-isom} to see that", "$$", "\\mathit{Isom}(E, E') = (H' \\times_B H) \\times_{c, I \\times_B I', \\sigma} B", "$$", "for some morphisms $c$ and $\\sigma$. Thus", "$\\mathit{Isom}(E, E')$ is an algebraic space." ], "refs": [ "algebraic-lemma-representable-diagonal", "quot-lemma-complexes", "quot-lemma-complexes-open-neg-exts-vanishing", "quot-lemma-complexes-open-neg-exts-vanishing", "quot-proposition-isom" ], "ref_ids": [ 8461, 3217, 3216, 3216, 3226 ] } ], "ref_ids": [] }, { "id": 3220, "type": "theorem", "label": "quot-lemma-complexes-stack", "categories": [ "quot" ], "title": "quot-lemma-complexes-stack", "contents": [ "In Situation \\ref{situation-complexes} the functor", "$p : \\Complexesstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf}$", "is a stack in groupoids." ], "refs": [], "proofs": [ { "contents": [ "To prove that $\\Complexesstack_{X/B}$ is a stack in groupoids,", "we have to show that the presheaves $\\mathit{Isom}$ are sheaves", "and that descent data are effective. The statement on", "$\\mathit{Isom}$ follows from Lemma \\ref{lemma-complexes-diagonal}, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-representable-diagonal}.", "Let us prove the statement on descent data.", "\\medskip\\noindent", "Suppose that $\\{a_i : T_i \\to T\\}$ is an fppf covering of schemes over $S$.", "Let $(\\xi_i, \\varphi_{ij})$ be a descent datum for $\\{T_i \\to T\\}$", "with values in $\\Complexesstack_{X/B}$.", "For each $i$ we can write $\\xi_i = (T_i, g_i, E_i)$.", "Denote $\\text{pr}_0 : T_i \\times_T T_j \\to T_i$ and", "$\\text{pr}_1 : T_i \\times_T T_j \\to T_j$ the projections.", "The condition that $\\xi_i|_{T_i \\times_T T_j} \\cong \\xi_j|_{T_i \\times_T T_j}$", "implies in particular that $g_i \\circ \\text{pr}_0 = g_j \\circ \\text{pr}_1$.", "Thus there exists a unique morphism $g : T \\to B$ such that", "$g_i = g \\circ a_i$, see", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-fpqc-universal-effective-epimorphisms}.", "Denote $X_T = T \\times_{g, B} X$. Set", "$X_i = X_{T_i} = T_i \\times_{g_i, B} X = T_i \\times_{a_i, T} X_T$", "and", "$$", "X_{ij} = X_{T_i} \\times_{X_T} X_{T_j} = X_i \\times_{X_T} X_j", "$$", "with projections $\\text{pr}_i$ and $\\text{pr}_j$ to $X_i$ and $X_j$.", "Observe that the pullback of $(T_i, g_i, E_i)$", "by $\\text{pr}_0 : T_i \\times_T T_j \\to T_i$ is given by", "$(T_i \\times_T T_j, g_i \\circ \\text{pr}_0, L\\text{pr}_i^*E_i)$.", "Hence a descent datum for $\\{T_i \\to T\\}$ in $\\Complexesstack_{X/B}$", "is given by the objects $(T_i, g \\circ a_i, E_i)$", "and for each pair $i, j$ an isomorphism in", "$D\\mathcal{O}_{X_{ij}})$", "$$", "\\varphi_{ij} :", "L\\text{pr}_i^*E_i \\longrightarrow L\\text{pr}_j^*E_j", "$$", "satisfying the cocycle condition over the pullback of $X$ to", "$T_i \\times_T T_j \\times_T T_k$.", "Using the vanishing of negative Exts provided by (b) of", "Lemma \\ref{lemma-complexes}, we may apply", "Simplicial Spaces, Lemma \\ref{spaces-simplicial-lemma-fppf-glue-neg-ext-zero}", "to obtain descent\\footnote{To check this, first consider the", "{\\v C}ech fppf hypercovering $a : T_\\bullet \\to T$ with", "$T_n = \\coprod_{i_0 \\ldots i_n} T_{i_0} \\times_T \\ldots \\times_T T_{i_n}$", "and then set $U_\\bullet = T_\\bullet \\times_{a, T} X_T$. Then", "$U_\\bullet \\to X_T$ is an fppf hypercovering to which we may", "apply Simplicial Spaces, Lemma", "\\ref{spaces-simplicial-lemma-fppf-glue-neg-ext-zero}.}", "for these complexes. In other words, we find there exists an object", "$E$ in $D_\\QCoh(\\mathcal{O}_{X_T})$ restricting to $E_i$ on $X_{T_i}$", "compatible with $\\varphi_{ij}$. Recall that being", "$T$-perfect signifies being pseudo-coherent and having", "locally finite tor dimension over $f^{-1}\\mathcal{O}_T$.", "Thus $E$ is $T$-perfect by an application of", "More on Morphisms of Spaces, Lemmas", "\\ref{spaces-more-morphisms-lemma-pseudo-coherent-descends-fpqc} and", "\\ref{spaces-more-morphisms-lemma-tor-amplitude-descends-fppf}.", "Finally, we have to check condition (2) from", "Lemma \\ref{lemma-complexes} for $E$.", "This immediately follows from the description of the open $W$", "in Lemma \\ref{lemma-complexes-open-neg-exts-vanishing}", "and the fact that (2) holds for $E_i$ on $X_{T_i}/T_i$." ], "refs": [ "quot-lemma-complexes-diagonal", "algebraic-lemma-representable-diagonal", "spaces-descent-lemma-fpqc-universal-effective-epimorphisms", "quot-lemma-complexes", "spaces-simplicial-lemma-fppf-glue-neg-ext-zero", "spaces-simplicial-lemma-fppf-glue-neg-ext-zero", "spaces-more-morphisms-lemma-pseudo-coherent-descends-fpqc", "spaces-more-morphisms-lemma-tor-amplitude-descends-fppf", "quot-lemma-complexes", "quot-lemma-complexes-open-neg-exts-vanishing" ], "ref_ids": [ 3219, 8461, 9367, 3217, 9140, 9140, 270, 271, 3217, 3216 ] } ], "ref_ids": [] }, { "id": 3221, "type": "theorem", "label": "quot-lemma-complexes-limits", "categories": [ "quot" ], "title": "quot-lemma-complexes-limits", "contents": [ "In Situation \\ref{situation-complexes} assume that $B \\to S$", "is locally of finite presentation. Then", "$p : \\Complexesstack_{X/B} \\to (\\Sch/S)_{fppf}$ is limit preserving", "(Artin's Axioms, Definition \\ref{artin-definition-limit-preserving})." ], "refs": [ "artin-definition-limit-preserving" ], "proofs": [ { "contents": [ "Write $B(T)$ for the discrete category whose", "objects are the $S$-morphisms $T \\to B$. Let $T = \\lim T_i$ be a filtered", "limit of affine schemes over $S$. Assigning to an object", "$(T, h, E)$ of $\\Complexesstack_{X/B, T}$ the object $h$", "of $B(T)$ gives us a commutative diagram of fibre categories", "$$", "\\xymatrix{", "\\colim \\Complexesstack_{X/B, T_i} \\ar[r] \\ar[d] &", "\\Complexesstack_{X/B, T} \\ar[d] \\\\", "\\colim B(T_i) \\ar[r] & B(T)", "}", "$$", "We have to show the top horizontal arrow is an equivalence. Since", "we have assume that $B$ is locally of finite presentation over $S$", "we see from", "Limits of Spaces, Remark \\ref{spaces-limits-remark-limit-preserving}", "that the bottom horizontal arrow is an equivalence. This means that", "we may assume $T = \\lim T_i$ be a filtered limit of affine schemes over", "$B$. Denote $g_i : T_i \\to B$ and $g : T \\to B$ the corresponding", "morphisms. Set $X_i = T_i \\times_{g_i, B} X$ and $X_T = T \\times_{g, B} X$.", "Observe that $X_T = \\colim X_i$.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-descend-relatively-perfect}", "the category of $T$-perfect objects of $D(\\mathcal{O}_{X_T})$", "is the colimit of the categories of $T_i$-perfect objects", "of $D(\\mathcal{O}_{X_{T_i}})$.", "Thus all we have to prove is that given an $T_i$-perfect object", "$E_i$ of $D(\\mathcal{O}_{X_{T_i}})$ such that", "the derived pullback $E$ of $E_i$ to $X_T$ satisfies", "condition (2) of Lemma \\ref{lemma-complexes},", "then after increasing $i$ we have that", "$E_i$ satisfies", "condition (2) of Lemma \\ref{lemma-complexes}.", "Let $W \\subset |T_i|$ be the open constructed", "in Lemma \\ref{lemma-complexes-open-neg-exts-vanishing}", "for $E_i$ and $E_i$. By assumption on $E$ we find", "that $T \\to T_i$ factors through $T$.", "Hence there is an $i' \\geq i$", "such that $T_{i'} \\to T_i$ factors through $W$, see", "Limits, Lemma \\ref{limits-lemma-limit-contained-in-constructible}", "Then $i'$ works by construction of $W$." ], "refs": [ "spaces-limits-remark-limit-preserving", "spaces-more-morphisms-lemma-descend-relatively-perfect", "quot-lemma-complexes", "quot-lemma-complexes", "quot-lemma-complexes-open-neg-exts-vanishing", "limits-lemma-limit-contained-in-constructible" ], "ref_ids": [ 4663, 264, 3217, 3217, 3216, 15040 ] } ], "ref_ids": [ 11420 ] }, { "id": 3222, "type": "theorem", "label": "quot-lemma-complexes-RS-star", "categories": [ "quot" ], "title": "quot-lemma-complexes-RS-star", "contents": [ "In Situation \\ref{situation-complexes}. Let", "$$", "\\xymatrix{", "Z \\ar[r] \\ar[d] & Z' \\ar[d] \\\\", "Y \\ar[r] & Y'", "}", "$$", "be a pushout in the category of schemes over $S$ where", "$Z \\to Z'$ is a finite order thickening and $Z \\to Y$ is affine, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.", "Then the functor on fibre categories", "$$", "\\Complexesstack_{X/B, Y'}", "\\longrightarrow", "\\Complexesstack_{X/B, Y}", "\\times_{\\Complexesstack_{X/B, Z}}", "\\Complexesstack_{X/B, Z'}", "$$", "is an equivalence." ], "refs": [ "more-morphisms-lemma-pushout-along-thickening" ], "proofs": [ { "contents": [ "Observe that the corresponding map", "$$", "B(Y') \\longrightarrow B(Y) \\times_{B(Z)} B(Z')", "$$", "is a bijection, see Pushouts of Spaces, Lemma", "\\ref{spaces-pushouts-lemma-pushout-along-thickening-schemes}.", "Thus using the commutative diagram", "$$", "\\xymatrix{", "\\Complexesstack_{X/B, Y'} \\ar[r] \\ar[d] &", "\\Complexesstack_{X/B, Y}", "\\times_{\\Complexesstack_{X/B, Z}}", "\\Complexesstack_{X/B, Z'}", "\\ar[d] \\\\", "B(Y') \\ar[r] & B(Y) \\times_{B(Z)} B(Z')", "}", "$$", "we see that we may assume that $Y'$ is a scheme over $B'$. By", "Remark \\ref{remark-complexes-base-change}", "we may replace $B$ by $Y'$ and $X$ by $X \\times_B Y'$.", "Thus we may assume $B = Y'$.", "\\medskip\\noindent", "Assume $B = Y'$. We first prove fully faithfulness of our functor.", "To do this, let $\\xi_1, \\xi_2$ be two objects of $\\Complexesstack_{X/B}$", "over $Y'$. Then we have to show that", "$$", "\\mathit{Isom}(\\xi_1, \\xi_2)(Y') \\longrightarrow", "\\mathit{Isom}(\\xi_1, \\xi_2)(Y)", "\\times_{\\mathit{Isom}(\\xi_1, \\xi_2)(Z)}", "\\mathit{Isom}(\\xi_1, \\xi_2)(Z')", "$$", "is bijective. However, we already know that $\\mathit{Isom}(\\xi_1, \\xi_2)$", "is an algebraic space over $B = Y'$. Thus this bijectivity follows from", "Artin's Axioms, Lemma \\ref{artin-lemma-pushout} (or the aforementioned", "Pushouts of Spaces, Lemma", "\\ref{spaces-pushouts-lemma-pushout-along-thickening-schemes}).", "\\medskip\\noindent", "Essential surjectivity. Let $(E_Y, E_{Z'}, \\alpha)$ be a triple,", "where $E_Y \\in D(\\mathcal{O}_Y)$ and $E_{Z'} \\in D(\\mathcal{O}_{X_{Z'}})$", "are objects such that $(Y, Y \\to B, E_Y)$ is an object of", "$\\Complexesstack_{X/B}$ over $Y$, such that", "$(Z', Z' \\to B, E_{Z'})$ is an object of $\\Complexesstack_{X/B}$ over $Z'$,", "and $\\alpha : L(X_Z \\to X_Y)^*E_Y \\to L(X_Z \\to X_{Z'})^*E_{Z'}$", "is an isomorphism in $D(\\mathcal{O}_{Z'})$.", "That is to say", "$$", "((Y, Y \\to B, E_Y), (Z', Z' \\to B, E_{Z'}), \\alpha)", "$$", "is an object of the target of the arrow of our lemma.", "Observe that the diagram", "$$", "\\xymatrix{", "X_Z \\ar[r] \\ar[d] & X_{Z'} \\ar[d] \\\\", "X_Y \\ar[r] & X_{Y'}", "}", "$$", "is a pushout with $X_Z \\to X_Y$ affine and $X_Z \\to X_{Z'}$ a thickening", "(see Pushouts of Spaces, Lemma", "\\ref{spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat}).", "Hence by Pushouts of Spaces, Lemma", "\\ref{spaces-pushouts-lemma-pushout-along-thickening-derived}", "we find an object $E_{Y'} \\in D(\\mathcal{O}_{X_{Y'}})$", "together with isomorphisms", "$L(X_Y \\to X_{Y'})^*E_{Y'} \\to E_Y$ and", "$L(X_{Z'} \\to X_{Y'})^*E_{Y'} \\to E_Z$", "compatible with $\\alpha$. Clearly, if we show that", "$E_{Y'}$ is $Y'$-perfect, then we are done, because property (2)", "of Lemma \\ref{lemma-complexes}", "is a property on points (and $Y$ and $Y'$ have the same points).", "This follows from More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-thickening-relatively-perfect}." ], "refs": [ "spaces-pushouts-lemma-pushout-along-thickening-schemes", "quot-remark-complexes-base-change", "artin-lemma-pushout", "spaces-pushouts-lemma-pushout-along-thickening-schemes", "spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat", "spaces-pushouts-lemma-pushout-along-thickening-derived", "quot-lemma-complexes", "spaces-more-morphisms-lemma-thickening-relatively-perfect" ], "ref_ids": [ 10858, 3245, 11354, 10858, 10864, 10866, 3217, 273 ] } ], "ref_ids": [ 13762 ] }, { "id": 3223, "type": "theorem", "label": "quot-lemma-complexes-tangent-space", "categories": [ "quot" ], "title": "quot-lemma-complexes-tangent-space", "contents": [ "In Situation \\ref{situation-complexes} assume that $S$ is a locally Noetherian", "scheme and $B \\to S$ is locally of finite presentation.", "Let $k$ be a finite type field over $S$ and let", "$x_0 = (\\Spec(k), g_0, E_0)$", "be an object of $\\mathcal{X} = \\Complexesstack_{X/B}$ over $k$.", "Then the spaces $T\\mathcal{F}_{\\mathcal{X}, k, x_0}$ and", "$\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0})$", "(Artin's Axioms, Section \\ref{artin-section-tangent-spaces})", "are finite dimensional." ], "refs": [], "proofs": [ { "contents": [ "Observe that by Lemma \\ref{lemma-complexes-RS-star}", "our stack in groupoids $\\mathcal{X}$ satisfies property (RS*)", "defined in Artin's Axioms, Section \\ref{artin-section-RS-star}.", "In particular $\\mathcal{X}$ satisfies (RS).", "Hence all associated predeformation", "categories are deformation categories", "(Artin's Axioms, Lemma \\ref{artin-lemma-deformation-category})", "and the statement makes sense.", "\\medskip\\noindent", "In this paragraph we show that we can reduce to the case $B = \\Spec(k)$.", "Set $X_0 = \\Spec(k) \\times_{g_0, B} X$", "and denote $\\mathcal{X}_0 = \\Complexesstack_{X_0/k}$. In", "Remark \\ref{remark-complexes-base-change} we have seen that", "$\\mathcal{X}_0$ is the $2$-fibre product of $\\mathcal{X}$ with", "$\\Spec(k)$ over $B$ as categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$. Thus by", "Artin's Axioms, Lemma \\ref{artin-lemma-fibre-product-tangent-spaces}", "we reduce to proving that $B$, $\\Spec(k)$, and $\\mathcal{X}_0$", "have finite dimensional tangent spaces and infinitesimal automorphism", "spaces. The tangent space of $B$ and $\\Spec(k)$ are finite dimensional by", "Artin's Axioms, Lemma \\ref{artin-lemma-finite-dimension}", "and of course these have vanishing $\\text{Inf}$.", "Thus it suffices to deal with $\\mathcal{X}_0$.", "\\medskip\\noindent", "Let $k[\\epsilon]$ be the dual numbers over $k$.", "Let $\\Spec(k[\\epsilon]) \\to B$ be the composition of $g_0 : \\Spec(k) \\to B$", "and the morphism $\\Spec(k[\\epsilon]) \\to \\Spec(k)$", "coming from the inclusion $k \\to k[\\epsilon]$.", "Set $X_0 = \\Spec(k) \\times_B X$ and", "$X_\\epsilon = \\Spec(k[\\epsilon]) \\times_B X$.", "Observe that $X_\\epsilon$ is a first order thickening of $X_0$", "flat over the first order thickening $\\Spec(k) \\to \\Spec(k[\\epsilon])$.", "Observe that $X_0$ and $X_\\epsilon$ give rise to canonically equivalent", "small \\'etale topoi, see", "More on Morphisms of Spaces, Section", "\\ref{spaces-more-morphisms-section-thickenings}.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-thickening-relatively-perfect}", "we see that $T\\mathcal{F}_{\\mathcal{X}_0, k, x_0}$ is the set of", "isomorphism classes of lifts of $E_0$ to $X_\\epsilon$ in the sense of", "Deformation Theory, Lemma \\ref{defos-lemma-first-order-defos-complex}.", "We conclude that", "$$", "T\\mathcal{F}_{\\mathcal{X}_0, k, x_0} =", "\\Ext^1_{\\mathcal{O}_{X_0}}(E_0, E_0)", "$$", "Here we have used the identification $\\epsilon k[\\epsilon] \\cong k$", "of $k[\\epsilon]$-modules. Using", "Deformation Theory, Lemma \\ref{defos-lemma-first-order-defos-complex}", "once more we see that there is a surjection", "$$", "\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0})", "\\leftarrow", "\\Ext^0_{\\mathcal{O}_{X_0}}(E_0, E_0)", "$$", "of $k$-vector spaces. As $E_0$ is pseudo-coherent it lies in", "$D^-_{\\textit{Coh}}(\\mathcal{O}_{X_0})$ by", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-identify-pseudo-coherent-noetherian}.", "Since $E_0$ locally has finite tor dimension and $X_0$", "is quasi-compact we see $E_0 \\in D^b_{\\textit{Coh}}(\\mathcal{O}_{X_0})$.", "Thus the $\\Ext$s above are finite dimensional $k$-vector spaces", "by Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-ext-finite}." ], "refs": [ "quot-lemma-complexes-RS-star", "artin-lemma-deformation-category", "quot-remark-complexes-base-change", "artin-lemma-fibre-product-tangent-spaces", "artin-lemma-finite-dimension", "spaces-more-morphisms-lemma-thickening-relatively-perfect", "defos-lemma-first-order-defos-complex", "defos-lemma-first-order-defos-complex", "spaces-perfect-lemma-identify-pseudo-coherent-noetherian", "spaces-perfect-lemma-ext-finite" ], "ref_ids": [ 3222, 11357, 3245, 11360, 11359, 273, 13422, 13422, 2697, 2668 ] } ], "ref_ids": [] }, { "id": 3224, "type": "theorem", "label": "quot-lemma-complexes-strong-effectiveness", "categories": [ "quot" ], "title": "quot-lemma-complexes-strong-effectiveness", "contents": [ "In Situation \\ref{situation-complexes} assume $B = S$ is locally Noetherian.", "Then strong formal effectiveness in the sense of", "Artin's Axioms, Remark \\ref{artin-remark-strong-effectiveness}", "holds for $p : \\Complexesstack_{X/S} \\to (\\Sch/S)_{fppf}$." ], "refs": [ "artin-remark-strong-effectiveness" ], "proofs": [ { "contents": [ "Let $(R_n)$ be an inverse system of $S$-algebras with surjective transition", "maps whose kernels are locally nilpotent. Set $R = \\lim R_n$.", "Let $(\\xi_n)$ be a system of objects of $\\Complexesstack_{X/B}$", "lying over $(\\Spec(R_n))$. We have to show $(\\xi_n)$ is effective, i.e.,", "there exists an object $\\xi$ of $\\Complexesstack_{X/B}$ lying over", "$\\Spec(R)$.", "\\medskip\\noindent", "Write $X_R = \\Spec(R) \\times_S X$ and $X_n = \\Spec(R_n) \\times_S X$.", "Of course $X_n$ is the base change of $X_R$ by $R \\to R_n$. Since $S = B$,", "we see that $\\xi_n$ corresponds simply to an $R_n$-perfect object", "$E_n \\in D(\\mathcal{O}_{X_n})$ satisfying condition (2) of", "Lemma \\ref{lemma-complexes}. In particular $E_n$ is pseudo-coherent.", "The isomorphisms $\\xi_{n + 1}|_{\\Spec(R_n)} \\cong \\xi_n$", "correspond to isomorphisms $L(X_n \\to X_{n + 1})^*E_{n + 1} \\to E_n$.", "Therefore by", "Flatness on Spaces, Theorem \\ref{spaces-flat-theorem-existence-derived}", "we find a pseudo-coherent object $E$ of $D(\\mathcal{O}_{X_R})$", "with $E_n$ equal to the derived pullback of $E$ for all $n$", "compatible with the transition isomorphisms.", "\\medskip\\noindent", "Observe that $(R, \\Ker(R \\to R_1))$ is a henselian pair, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-limit-henselian}.", "In particular, $\\Ker(R \\to R_1)$ is contained in the Jacobson radical of $R$.", "Then we may apply More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-henselian-relatively-perfect}", "to see that $E$ is $R$-perfect.", "\\medskip\\noindent", "Finally, we have to check condition (2) of Lemma \\ref{lemma-complexes}.", "By Lemma \\ref{lemma-complexes-open-neg-exts-vanishing}", "the set of points $t$ of $\\Spec(R)$ where the negative self-exts of $E_t$", "vanish is an open. Since this condition is true in $V(\\Ker(R \\to R_1))$", "and since $\\Ker(R \\to R_1)$ is contained in the Jacobson radical of $R$", "we conclude it holds for all points." ], "refs": [ "quot-lemma-complexes", "spaces-flat-theorem-existence-derived", "more-algebra-lemma-limit-henselian", "spaces-more-morphisms-lemma-henselian-relatively-perfect", "quot-lemma-complexes", "quot-lemma-complexes-open-neg-exts-vanishing" ], "ref_ids": [ 3217, 7149, 9858, 274, 3217, 3216 ] } ], "ref_ids": [ 11429 ] }, { "id": 3225, "type": "theorem", "label": "quot-proposition-hom", "categories": [ "quot" ], "title": "quot-proposition-hom", "contents": [ "In Situation \\ref{situation-hom} assume that", "\\begin{enumerate}", "\\item $f$ is of finite presentation, and", "\\item $\\mathcal{G}$ is a finitely presented $\\mathcal{O}_X$-module,", "flat over $B$, with support proper over $B$.", "\\end{enumerate}", "Then the functor $\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ is", "an algebraic space affine over $B$. If $\\mathcal{F}$", "is of finite presentation, then $\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$", "is of finite presentation over $B$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-hom-sheaf} the functor", "$\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ satisfies", "the sheaf property for fppf coverings. This mean we may\\footnote{We omit", "the verification of the set theoretical condition (3) of the referenced", "lemma.} apply", "Bootstrap, Lemma \\ref{bootstrap-lemma-locally-algebraic-space}", "to check the representability \\'etale locally on $B$. Moreover,", "we may check whether the end result is affine or", "of finite presentation \\'etale locally on $B$, see", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-affine-local} and", "\\ref{spaces-morphisms-lemma-finite-presentation-local}.", "Hence we may assume that $B$ is an affine scheme.", "\\medskip\\noindent", "Assume $B$ is an affine scheme. As $f$ is of finite presentation, it follows", "$X$ is quasi-compact and quasi-separated. Thus we can write", "$\\mathcal{F} = \\colim \\mathcal{F}_i$ as a filtered colimit of", "$\\mathcal{O}_X$-modules of finite presentation", "(Limits of Spaces, Lemma \\ref{spaces-limits-lemma-colimit-finitely-presented}).", "It is clear that", "$$", "\\mathit{Hom}(\\mathcal{F}, \\mathcal{G}) =", "\\lim \\mathit{Hom}(\\mathcal{F}_i, \\mathcal{G})", "$$", "Hence if we can show that each $\\mathit{Hom}(\\mathcal{F}_i, \\mathcal{G})$", "is representable by an affine scheme, then we see that the same thing", "holds for $\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$. Use the material in", "Limits, Section \\ref{limits-section-limits} and", "Limits of Spaces, Section \\ref{spaces-limits-section-limits}.", "Thus we may assume that $\\mathcal{F}$ is of finite presentation.", "\\medskip\\noindent", "Say $B = \\Spec(R)$. Write $R = \\colim R_i$ with each $R_i$ a finite", "type $\\mathbf{Z}$-algebra. Set $B_i = \\Spec(R_i)$. By the results of", "Limits of Spaces, Lemmas", "\\ref{spaces-limits-lemma-descend-finite-presentation} and", "\\ref{spaces-limits-lemma-descend-modules-finite-presentation}", "we can find an $i$, a morphism of algebraic spaces $X_i \\to B_i$,", "and finitely presented $\\mathcal{O}_{X_i}$-modules $\\mathcal{F}_i$ and", "$\\mathcal{G}_i$ such that the base change of", "$(X_i, \\mathcal{F}_i, \\mathcal{G}_i)$ to $B$ recovers", "$(X, \\mathcal{F}, \\mathcal{G})$. By", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-descend-flat}", "we may, after increasing $i$, assume that $\\mathcal{G}_i$", "is flat over $B_i$. By", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-eventually-proper-support}", "we may similarly assume the scheme theoretic support of $\\mathcal{G}_i$", "is proper over $B_i$. At this point we can apply", "Lemma \\ref{lemma-noetherian-hom}", "to see that $H_i = \\mathit{Hom}(\\mathcal{F}_i, \\mathcal{G}_i)$ is", "an algebraic space affine of finite presentation over $B_i$.", "Pulling back to $B$ (using Remark \\ref{remark-hom-base-change})", "we see that $H_i \\times_{B_i} B = \\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ ", "and we win." ], "refs": [ "quot-lemma-hom-sheaf", "bootstrap-lemma-locally-algebraic-space", "spaces-morphisms-lemma-affine-local", "spaces-morphisms-lemma-finite-presentation-local", "spaces-limits-lemma-colimit-finitely-presented", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-modules-finite-presentation", "spaces-limits-lemma-descend-flat", "spaces-limits-lemma-eventually-proper-support", "quot-lemma-noetherian-hom", "quot-remark-hom-base-change" ], "ref_ids": [ 3151, 2625, 4798, 4841, 4601, 4598, 4599, 4595, 4618, 3157, 3232 ] } ], "ref_ids": [] }, { "id": 3226, "type": "theorem", "label": "quot-proposition-isom", "categories": [ "quot" ], "title": "quot-proposition-isom", "contents": [ "In Situation \\ref{situation-hom} assume that", "\\begin{enumerate}", "\\item $f$ is of finite presentation, and", "\\item $\\mathcal{F}$ and $\\mathcal{G}$ are finitely presented", "$\\mathcal{O}_X$-modules, flat over $B$, with support proper over $B$.", "\\end{enumerate}", "Then the functor $\\mathit{Isom}(\\mathcal{F}, \\mathcal{G})$ is", "an algebraic space affine of finite presentation over $B$." ], "refs": [], "proofs": [ { "contents": [ "We will use the abbreviations", "$H = \\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$,", "$I = \\mathit{Hom}(\\mathcal{F}, \\mathcal{F})$,", "$H' = \\mathit{Hom}(\\mathcal{G}, \\mathcal{F})$, and", "$I' = \\mathit{Hom}(\\mathcal{G}, \\mathcal{G})$.", "By Proposition \\ref{proposition-hom} the functors", "$H$, $I$, $H'$, $I'$ are algebraic spaces and the morphisms", "$H \\to B$, $I \\to B$, $H' \\to B$, and $I' \\to B$", "are affine and of finite presentation.", "The composition of maps gives a morphism", "$$", "c : H' \\times_B H \\longrightarrow I \\times_B I',\\quad", "(u', u) \\longmapsto (u \\circ u', u' \\circ u)", "$$", "of algebraic spaces over $B$. Since $I \\times_B I' \\to B$ is separated,", "the section $\\sigma : B \\to I \\times_B I'$ corresponding to", "$(\\text{id}_\\mathcal{F}, \\text{id}_\\mathcal{G})$", "is a closed immersion", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-section-immersion}).", "Moreover, $\\sigma$ is of finite presentation", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-permanence}).", "Hence", "$$", "\\mathit{Isom}(\\mathcal{F}, \\mathcal{G}) =", "(H' \\times_B H) \\times_{c, I \\times_B I', \\sigma} B", "$$", "is an algebraic space affine of finite presentation over $B$ as well.", "Some details omitted." ], "refs": [ "quot-proposition-hom", "spaces-morphisms-lemma-finite-presentation-permanence" ], "ref_ids": [ 3225, 4846 ] } ], "ref_ids": [] }, { "id": 3227, "type": "theorem", "label": "quot-proposition-quot", "categories": [ "quot" ], "title": "quot-proposition-quot", "contents": [ "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic", "spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf", "on $X$. If $f$ is of finite presentation and separated, then", "$\\Quotfunctor_{\\mathcal{F}/X/B}$", "is an algebraic space. If $\\mathcal{F}$ is of finite presentation,", "then $\\Quotfunctor_{\\mathcal{F}/X/B} \\to B$ is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-quot-sheaf}", "we have that $\\Quotfunctor_{\\mathcal{F}/X/B}$ is a sheaf in the", "fppf topology. Let $\\textit{Quot}_{\\mathcal{F}/X/B}$ be the stack in", "groupoids corresponding to $\\Quotfunctor_{\\mathcal{F}/X/S}$, see", "Algebraic Stacks, Section \\ref{algebraic-section-split}.", "By Algebraic Stacks, Proposition", "\\ref{algebraic-proposition-algebraic-stack-no-automorphisms}", "it suffices to show that $\\textit{Quot}_{\\mathcal{F}/X/B}$", "is an algebraic stack.", "Consider the $1$-morphism of stacks in groupoids", "$$", "\\textit{Quot}_{\\mathcal{F}/X/S}", "\\longrightarrow", "\\Cohstack_{X/B}", "$$", "on $(\\Sch/S)_{fppf}$ which associates to the quotient", "$\\mathcal{F}_T \\to \\mathcal{Q}$ the coherent sheaf $\\mathcal{Q}$.", "By Theorem \\ref{theorem-coherent-algebraic-general} we know that", "$\\Cohstack_{X/B}$ is an algebraic stack.", "By Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-representable-morphism-to-algebraic}", "it suffices to show that this $1$-morphism is representable", "by algebraic spaces.", "\\medskip\\noindent", "Let $T$ be a scheme over $S$ and let the object $(h, \\mathcal{G})$ of", "$\\Cohstack_{X/B}$ over $T$ correspond", "to a $1$-morphism $\\xi : (\\Sch/T)_{fppf} \\to \\Cohstack_{X/B}$.", "The $2$-fibre product", "$$", "\\mathcal{Z} =", "(\\Sch/T)_{fppf}", "\\times_{\\xi, \\Cohstack_{X/B}}", "\\textit{Quot}_{\\mathcal{F}/X/S}", "$$", "is a stack in setoids, see", "Stacks, Lemma \\ref{stacks-lemma-2-fibre-product-gives-stack-in-setoids}.", "The corresponding sheaf of sets (i.e., functor, see", "Stacks, Lemmas", "\\ref{stacks-lemma-2-fibre-product-gives-stack-in-setoids} and", "\\ref{stacks-lemma-when-stack-in-sets}) assigns to a scheme", "$T'/T$ the set of surjections $u : \\mathcal{F}_{T'} \\to \\mathcal{G}_{T'}$", "of quasi-coherent modules on $X_{T'}$. Thus we see that", "$\\mathcal{Z}$ is representable by an open subspace", "(by Flatness on Spaces, Lemma \\ref{spaces-flat-lemma-F-surj-open})", "of the algebraic space", "$\\mathit{Hom}(\\mathcal{F}_T, \\mathcal{G})$ from", "Proposition \\ref{proposition-hom}." ], "refs": [ "quot-lemma-quot-sheaf", "algebraic-proposition-algebraic-stack-no-automorphisms", "quot-theorem-coherent-algebraic-general", "algebraic-lemma-representable-morphism-to-algebraic", "stacks-lemma-2-fibre-product-gives-stack-in-setoids", "stacks-lemma-2-fibre-product-gives-stack-in-setoids", "stacks-lemma-when-stack-in-sets", "spaces-flat-lemma-F-surj-open", "quot-proposition-hom" ], "ref_ids": [ 3174, 8480, 3147, 8471, 8954, 8954, 8950, 7184, 3225 ] } ], "ref_ids": [] }, { "id": 3228, "type": "theorem", "label": "quot-proposition-hilb", "categories": [ "quot" ], "title": "quot-proposition-hilb", "contents": [ "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic", "spaces over $S$. If $f$ is of finite presentation and separated, then", "$\\Hilbfunctor_{X/B}$ is an algebraic space locally of finite", "presentation over $B$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Lemma \\ref{lemma-hilb-is-quot}", "and Proposition \\ref{proposition-quot}." ], "refs": [ "quot-lemma-hilb-is-quot", "quot-proposition-quot" ], "ref_ids": [ 3176, 3227 ] } ], "ref_ids": [] }, { "id": 3229, "type": "theorem", "label": "quot-proposition-pic", "categories": [ "quot" ], "title": "quot-proposition-pic", "contents": [ "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic", "spaces over $S$. If $f$ is flat, of finite presentation, and proper, then", "$\\Picardstack_{X/B}$ is an algebraic stack." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Lemma \\ref{lemma-picard-stack-open-in-coh},", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-representable-morphism-to-algebraic}", "and either", "Theorem \\ref{theorem-coherent-algebraic}", "or", "Theorem \\ref{theorem-coherent-algebraic-general}" ], "refs": [ "quot-lemma-picard-stack-open-in-coh", "algebraic-lemma-representable-morphism-to-algebraic", "quot-theorem-coherent-algebraic", "quot-theorem-coherent-algebraic-general" ], "ref_ids": [ 3178, 8471, 3146, 3147 ] } ], "ref_ids": [] }, { "id": 3230, "type": "theorem", "label": "quot-proposition-pic-functor", "categories": [ "quot" ], "title": "quot-proposition-pic-functor", "contents": [ "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic", "spaces over $S$. Assume that", "\\begin{enumerate}", "\\item $f$ is flat, of finite presentation, and proper, and", "\\item $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism", "for all schemes $T$ over $B$.", "\\end{enumerate}", "Then $\\Picardfunctor_{X/B}$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "There exists a surjective, flat, finitely presented morphism", "$B' \\to B$ of algebraic spaces such that the base change $X' = X \\times_B B'$", "over $B'$ has a section: namely, we can take $B' = X$.", "Observe that $\\Picardfunctor_{X'/B'} = B' \\times_B \\Picardfunctor_{X/B}$.", "Hence $\\Picardfunctor_{X'/B'} \\to \\Picardfunctor_{X/B}$ is representable", "by algebraic spaces, surjective, flat, and finitely presented.", "Hence, if we can show that $\\Picardfunctor_{X'/B'}$ is an algebraic space,", "then it follows that $\\Picardfunctor_{X/B}$", "is an algebraic space by Bootstrap, Theorem", "\\ref{bootstrap-theorem-final-bootstrap}.", "In this way we reduce to the case described in the next paragraph.", "\\medskip\\noindent", "In addition to the assumptions of the proposition, assume that", "we have a section $\\sigma : B \\to X$. By", "Proposition \\ref{proposition-pic} we see that", "$\\Picardstack_{X/B}$ is an algebraic stack.", "By Lemma \\ref{lemma-compare-pic-with-section} and", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-representable-morphism-to-algebraic}", "we see that $\\Picardstack_{X/B, \\sigma}$ is an algebraic stack.", "By Lemma", "\\ref{lemma-flat-geometrically-connected-fibres-with-section-functor-stack}", "and Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-characterize-representable-by-space}", "we see that $T \\mapsto \\Ker(\\sigma_T^* : \\Pic(X_T) \\to \\Pic(T))$", "is an algebraic space.", "By Lemma \\ref{lemma-flat-geometrically-connected-fibres-with-section}", "this functor is the same as $\\Picardfunctor_{X/B}$." ], "refs": [ "bootstrap-theorem-final-bootstrap", "quot-proposition-pic", "algebraic-lemma-representable-morphism-to-algebraic", "algebraic-lemma-characterize-representable-by-space" ], "ref_ids": [ 2602, 3229, 8471, 8441 ] } ], "ref_ids": [] }, { "id": 3231, "type": "theorem", "label": "quot-proposition-Mor", "categories": [ "quot" ], "title": "quot-proposition-Mor", "contents": [ "Let $S$ be a scheme. Let $Z \\to B$ and $X \\to B$ be morphisms of algebraic", "spaces over $S$. Assume $X \\to B$ is of finite presentation and separated and", "$Z \\to B$ is of finite presentation, flat, and proper. Then", "$\\mathit{Mor}_B(Z, X)$ is an algebraic space locally of finite", "presentation over $B$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Lemma \\ref{lemma-Mor-into-Hilb-open}", "and Proposition \\ref{proposition-hilb}." ], "refs": [ "quot-lemma-Mor-into-Hilb-open", "quot-proposition-hilb" ], "ref_ids": [ 3187, 3228 ] } ], "ref_ids": [] }, { "id": 3246, "type": "theorem", "label": "spaces-more-cohomology-theorem-proper-base-change", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-theorem-proper-base-change", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "be a cartesian square of algebraic spaces over $S$.", "Assume $f$ is proper.", "Let $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$.", "Then the base change map", "$$", "g^{-1}Rf_*\\mathcal{F} \\longrightarrow Rf'_*(g')^{-1}\\mathcal{F}", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This proof repeats a few of the arguments given in the proof of the", "proper base change theorem for schemes. See", "\\'Etale Cohomology, Section \\ref{etale-cohomology-section-proper-base-change}", "for more details.", "\\medskip\\noindent", "The statement is \\'etale local on $Y'$ and $Y$, hence we may assume", "both $Y$ and $Y'$ are affine schemes. Observe that this in particular", "proves the theorem in case $f$ is representable (we will use this", "below).", "\\medskip\\noindent", "For every $n \\geq 1$ let $\\mathcal{F}[n]$ be the subsheaf of sections", "of $\\mathcal{F}$ annihilated by $n$. Then", "$\\mathcal{F} = \\colim \\mathcal{F}[n]$. By ", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-colimit-cohomology}", "the functors $g^{-1}R^pf_*$ and $R^pf'_*(g')^{-1}$ commute", "with filtered colimits. Hence it suffices to prove the theorem", "if $\\mathcal{F}$ is killed by $n$.", "\\medskip\\noindent", "Let $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ be a resolution by", "injective sheaves of $\\mathbf{Z}/n\\mathbf{Z}$-modules.", "Observe that", "$g^{-1}f_*\\mathcal{I}^\\bullet = f'_*(g')^{-1}\\mathcal{I}^\\bullet$", "by Lemma \\ref{lemma-proper-base-change-f-star}.", "Applying Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "we conclude it suffices to prove", "$R^pf'_*(g')^{-1}\\mathcal{I}^m = 0$ for $p > 0$ and $m \\in \\mathbf{Z}$.", "\\medskip\\noindent", "Choose a surjective proper morphism", "$h : Z \\to X$ where $Z$ is a scheme, see", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-weak-chow}.", "Choose an injective map $h^{-1}\\mathcal{I}^m \\to \\mathcal{J}$", "where $\\mathcal{J}$ is an injective sheaf of", "$\\mathbf{Z}/n\\mathbf{Z}$-modules on $Z_\\etale$.", "Since $h$ is surjective the map $\\mathcal{I}^m \\to h_*\\mathcal{J}$", "is injective (see Lemma \\ref{lemma-surjective-proper}).", "Since $\\mathcal{I}^m$ is injective we see that $\\mathcal{I}^m$", "is a direct summand of $h_*\\mathcal{J}$. Thus it suffices", "to prove the desired vanishing for $h_*\\mathcal{J}$.", "\\medskip\\noindent", "Denote $h'$ the base change by $g$ and denote $g'' : Z' \\to Z$", "the projection. There is a spectral sequence", "$$", "E_2^{p, q} = R^pf'_* R^qh'_* (g'')^{-1}\\mathcal{J}", "$$", "converging to $R^{p + q}(f' \\circ h')_*(g'')^{-1}\\mathcal{J}$.", "Since $h$ and $f \\circ h$ are representable (by schemes)", "we know the result we want holds for them. Thus in the", "spectral sequence we see that $E_2^{p, q} = 0$ for $q > 0$", "and $R^{p + q}(f' \\circ h')_*(g'')^{-1}\\mathcal{J} = 0$", "for $p + q > 0$. It follows that $E_2^{p, 0} = 0$ for $p > 0$.", "Now", "$$", "E_2^{p, 0} = R^pf'_* h'_* (g'')^{-1}\\mathcal{J} =", "R^pf'_* (g')^{-1}h_*\\mathcal{J}", "$$", "by Lemma \\ref{lemma-proper-base-change-f-star}. This finishes the proof." ], "refs": [ "spaces-cohomology-lemma-colimit-cohomology", "spaces-more-cohomology-lemma-proper-base-change-f-star", "derived-lemma-leray-acyclicity", "spaces-cohomology-lemma-weak-chow", "spaces-more-cohomology-lemma-surjective-proper", "spaces-more-cohomology-lemma-proper-base-change-f-star" ], "ref_ids": [ 11278, 3251, 1844, 11327, 3248, 3251 ] } ], "ref_ids": [] }, { "id": 3247, "type": "theorem", "label": "spaces-more-cohomology-lemma-compare-cohomology-other-topologies", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-compare-cohomology-other-topologies", "contents": [ "Let $S$ be a scheme. Let $\\tau \\in \\{\\etale, fppf, ph\\}$ (add more here).", "The inclusion functor", "$$", "(\\Sch/S)_\\tau \\longrightarrow (\\textit{Spaces}/S)_\\tau", "$$", "is a special cocontinuous functor", "(Sites, Definition \\ref{sites-definition-special-cocontinuous-functor})", "and hence identifies topoi." ], "refs": [ "sites-definition-special-cocontinuous-functor" ], "proofs": [ { "contents": [ "The conditions of Sites, Lemma \\ref{sites-lemma-equivalence}", "are immediately verified as our functor is fully faithful", "and as every algebraic space has an \\'etale covering by schemes." ], "refs": [ "sites-lemma-equivalence" ], "ref_ids": [ 8578 ] } ], "ref_ids": [ 8672 ] }, { "id": 3248, "type": "theorem", "label": "spaces-more-cohomology-lemma-surjective-proper", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-surjective-proper", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a surjective proper morphism", "of algebraic spaces over $S$. Let $\\mathcal{F}$ be a sheaf on $X_\\etale$.", "Then $\\mathcal{F} \\to f_*f^{-1}\\mathcal{F}$ is injective with", "image the equalizer of the two maps", "$f_*f^{-1}\\mathcal{F} \\to g_*g^{-1}\\mathcal{F}$ where", "$g$ is the structure morphism $g : Y \\times_X Y \\to X$." ], "refs": [], "proofs": [ { "contents": [ "For any surjective morphism $f : Y \\to X$ of algebraic spaces over $S$,", "the map $\\mathcal{F} \\to f_*f^{-1}\\mathcal{F}$ is injective.", "Namely, if $\\overline{x}$ is a geometric point of $X$, then we", "choose a geometric point $\\overline{y}$ of $Y$ lying over $\\overline{x}$", "and we consider", "$$", "\\mathcal{F}_{\\overline{x}} \\to", "(f_*f^{-1}\\mathcal{F})_{\\overline{x}} \\to", "(f^{-1}\\mathcal{F})_{\\overline{y}} = \\mathcal{F}_{\\overline{x}}", "$$", "See Properties of Spaces, Lemma \\ref{spaces-properties-lemma-stalk-pullback}", "for the last equality.", "\\medskip\\noindent", "The second statement is local on $X$ in the \\'etale topology, hence we may", "and do assume $Y$ is an affine scheme.", "\\medskip\\noindent", "Choose a surjective proper morphism $Z \\to Y$ where $Z$ is a scheme, see", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-weak-chow}.", "The result for $Z \\to X$ implies the result for $Y \\to X$.", "Since $Z \\to X$ is a surjective proper morphism of schemes", "and hence a ph covering", "(Topologies, Lemma \\ref{topologies-lemma-surjective-proper-ph})", "the result for $Z \\to X$ follows from", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-describe-pullback-pi-ph}", "(in fact it is in some sense equivalent to this lemma)." ], "refs": [ "spaces-properties-lemma-stalk-pullback", "spaces-cohomology-lemma-weak-chow", "topologies-lemma-surjective-proper-ph", "etale-cohomology-lemma-describe-pullback-pi-ph" ], "ref_ids": [ 11875, 11327, 12483, 6668 ] } ], "ref_ids": [] }, { "id": 3249, "type": "theorem", "label": "spaces-more-cohomology-lemma-h0-proper-over-henselian-pair", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-h0-proper-over-henselian-pair", "contents": [ "Let $(A, I)$ be a henselian pair. Let $X$ be an algebraic space over $A$", "such that the structure morphism $f : X \\to \\Spec(A)$ is proper.", "Let $i : X_0 \\to X$ be the inclusion of $X \\times_{\\Spec(A)} \\Spec(A/I)$.", "For any sheaf $\\mathcal{F}$ on $X_\\etale$ we", "have $\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, i^{-1}\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjective proper morphism $Y \\to X$ where $Y$ is a scheme, see", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-weak-chow}.", "Consider the diagram", "$$", "\\xymatrix{", "\\Gamma(X_0, \\mathcal{F}_0) \\ar[r] \\ar[d] &", "\\Gamma(Y_0, \\mathcal{G}_0) \\ar@<1ex>[r] \\ar@<-1ex>[r] \\ar[d] &", "\\Gamma((Y \\times_X Y)_0, \\mathcal{H}_0) \\ar[d] \\\\", "\\Gamma(X, \\mathcal{F}) \\ar[r] &", "\\Gamma(Y, \\mathcal{G}) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\Gamma(Y \\times_X Y, \\mathcal{H})", "}", "$$", "Here $\\mathcal{G}$, resp.\\ $\\mathcal{H}$ is the pullbackf or", "$\\mathcal{F}$ to $Y$, resp.\\ $Y \\times_X Y$ and the index $0$", "indicates base change to $\\Spec(A/I)$. By the case of schemes", "(\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-h0-proper-over-henselian-pair})", "we see that the middle and right vertical arrows are bijective.", "By Lemma \\ref{lemma-surjective-proper} it follows that the left one is too." ], "refs": [ "spaces-cohomology-lemma-weak-chow", "etale-cohomology-lemma-h0-proper-over-henselian-pair", "spaces-more-cohomology-lemma-surjective-proper" ], "ref_ids": [ 11327, 6617, 3248 ] } ], "ref_ids": [] }, { "id": 3250, "type": "theorem", "label": "spaces-more-cohomology-lemma-h0-proper-over-henselian-local", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-h0-proper-over-henselian-local", "contents": [ "Let $A$ be a henselian local ring. Let $X$ be an algebraic space", "over $A$ such that $f : X \\to \\Spec(A)$", "be a proper morphism. Let $X_0 \\subset X$ be the fibre of", "$f$ over the closed point. For any sheaf $\\mathcal{F}$ on $X_\\etale$ we", "have $\\Gamma(X, \\mathcal{F}) = \\Gamma(X_0, \\mathcal{F}|_{X_0})$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-h0-proper-over-henselian-pair}." ], "refs": [ "spaces-more-cohomology-lemma-h0-proper-over-henselian-pair" ], "ref_ids": [ 3249 ] } ], "ref_ids": [] }, { "id": 3251, "type": "theorem", "label": "spaces-more-cohomology-lemma-proper-base-change-f-star", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-proper-base-change-f-star", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y' \\to Y$", "be a morphisms of algebraic spaces over $S$. Assume $f$ is proper.", "Set $X' = Y' \\times_Y X$ with projections $f' : X' \\to Y'$ and $g' : X' \\to X$.", "Let $\\mathcal{F}$ be any sheaf on $X_\\etale$. Then", "$g^{-1}f_*\\mathcal{F} = f'_*(g')^{-1}\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "The question is \\'etale local on $Y'$. Choose a scheme $V$ and a surjective", "\\'etale morphism $V \\to Y$. Choose a scheme $V'$ and a surjective \\'etale", "morphism $V' \\to V \\times_Y Y'$. Then we may replace $Y'$ by $V'$ and", "$Y$ by $V$. Hence we may assume $Y$ and $Y'$ are schemes.", "Then we may work Zariski locally on $Y$ and $Y'$ and hence we may", "assume $Y$ and $Y'$ are affine schemes.", "\\medskip\\noindent", "Assume $Y$ and $Y'$ are affine schemes. Choose a surjective proper morphism", "$h_1 : X_1 \\to X$ where $X_1$ is a scheme, see", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-weak-chow}.", "Set $X_2 = X_1 \\times_X X_1$ and denote", "$h_2 : X_2 \\to X$ the structure morphism. Observe this is a scheme.", "By the case of schemes", "(\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-proper-base-change-f-star})", "we know the lemma is true for the cartesian diagrams", "$$", "\\vcenter{", "\\xymatrix{", "X'_1 \\ar[r] \\ar[d] & X_1 \\ar[d] \\\\", "Y' \\ar[r] & Y", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "X'_2 \\ar[r] \\ar[d] & X_2 \\ar[d] \\\\", "Y' \\ar[r] & Y", "}", "}", "$$", "and the sheaves $\\mathcal{F}_i = (X_i \\to X)^{-1}\\mathcal{F}$.", "By Lemma \\ref{lemma-surjective-proper} we have an exact sequence", "$0 \\to \\mathcal{F} \\to h_{1, *}\\mathcal{F}_1 \\to h_{2, *}\\mathcal{F}_2$", "and similarly for $(g')^{-1}\\mathcal{F}$ because", "$X'_2 = X'_1 \\times_{X'} X'_1$. Hence we conlude that the", "lemma is true (some details omitted)." ], "refs": [ "spaces-cohomology-lemma-weak-chow", "etale-cohomology-lemma-proper-base-change-f-star", "spaces-more-cohomology-lemma-surjective-proper" ], "ref_ids": [ 11327, 6620, 3248 ] } ], "ref_ids": [] }, { "id": 3252, "type": "theorem", "label": "spaces-more-cohomology-lemma-proper-pushforward-stalk", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-proper-pushforward-stalk", "contents": [ "Let $S$ be a scheme.", "Let $f : Y \\to X$ be a proper morphism of algebraic spaces over $S$. Let", "$\\overline{x} \\to X$ be a geometric point.", "For any sheaf $\\mathcal{F}$ on $Y_\\etale$", "the canonical map", "$$", "(f_*\\mathcal{F})_{\\overline{x}} \\longrightarrow", "\\Gamma(Y_{\\overline{x}}, \\mathcal{F}_{\\overline{x}})", "$$", "is bijective." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-proper-base-change-f-star}." ], "refs": [ "spaces-more-cohomology-lemma-proper-base-change-f-star" ], "ref_ids": [ 3251 ] } ], "ref_ids": [] }, { "id": 3253, "type": "theorem", "label": "spaces-more-cohomology-lemma-proper-base-change", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-proper-base-change", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "be a cartesian square of algebraic spaces over $S$. Assume $f$ is proper.", "Let $E \\in D^+(X_\\etale)$ have torsion cohomology sheaves.", "Then the base change map $g^{-1}Rf_*E \\to Rf'_*(g')^{-1}E$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is a simple consequence of the proper base change theorem", "(Theorem \\ref{theorem-proper-base-change}) using the spectral", "sequences", "$$", "E_2^{p, q} = R^pf_*H^q(E)", "\\quad\\text{and}\\quad", "{E'}_2^{p, q} = R^pf'_*(g')^{-1}H^q(E)", "$$", "converging to $R^nf_*E$ and $R^nf'_*(g')^{-1}E$.", "The spectral sequences are constructed in", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "Some details omitted." ], "refs": [ "spaces-more-cohomology-theorem-proper-base-change", "derived-lemma-two-ss-complex-functor" ], "ref_ids": [ 3246, 1871 ] } ], "ref_ids": [] }, { "id": 3254, "type": "theorem", "label": "spaces-more-cohomology-lemma-proper-base-change-stalk", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-proper-base-change-stalk", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a proper morphism of algebraic spaces.", "Let $\\overline{y} \\to Y$ be a geometric point.", "\\begin{enumerate}", "\\item For a torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have", "$(R^nf_*\\mathcal{F})_{\\overline{y}} =", "H^n_\\etale(X_{\\overline{y}}, \\mathcal{F}_{\\overline{y}})$.", "\\item For $E \\in D^+(X_\\etale)$ with torsion cohomology sheaves we have", "$(R^nf_*E)_{\\overline{y}} = H^n_\\etale(X_{\\overline{y}}, E_{\\overline{y}})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In the statement, $\\mathcal{F}_{\\overline{y}}$ denotes the pullback", "of $\\mathcal{F}$ to $X_{\\overline{y}} = \\overline{y} \\times_Y X$.", "Since pulling back by $\\overline{y} \\to Y$ produces the", "stalk of $\\mathcal{F}$, the first statement of the lemma", "is a special case of Theorem \\ref{theorem-proper-base-change}.", "The second one is a special case of Lemma \\ref{lemma-proper-base-change}." ], "refs": [ "spaces-more-cohomology-theorem-proper-base-change", "spaces-more-cohomology-lemma-proper-base-change" ], "ref_ids": [ 3246, 3253 ] } ], "ref_ids": [] }, { "id": 3255, "type": "theorem", "label": "spaces-more-cohomology-lemma-base-change-separably-closed", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-base-change-separably-closed", "contents": [ "Let $k \\subset k'$ be an extension of separably closed fields.", "Let $X$ be a proper algebraic space over $k$.", "Let $\\mathcal{F}$ be a torsion abelian sheaf on $X$.", "Then the map $H^q_\\etale(X, \\mathcal{F}) \\to", "H^q_\\etale(X_{k'}, \\mathcal{F}|_{X_{k'}})$ is an isomorphism", "for $q \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Theorem \\ref{theorem-proper-base-change}." ], "refs": [ "spaces-more-cohomology-theorem-proper-base-change" ], "ref_ids": [ 3246 ] } ], "ref_ids": [] }, { "id": 3256, "type": "theorem", "label": "spaces-more-cohomology-lemma-describe-pullback", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-describe-pullback", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a sheaf on $X_\\etale$. Then", "$\\pi_X^{-1}\\mathcal{F}$ is given by the rule", "$$", "(\\pi_X^{-1}\\mathcal{F})(Y) = \\Gamma(Y_\\etale, f_{small}^{-1}\\mathcal{F})", "$$", "for $f : Y \\to X$ in $(\\textit{Spaces}/X)_\\etale$.", "Moreover, $\\pi_Y^{-1}\\mathcal{F}$ satisfies the", "sheaf condition with respect to smooth, syntomic, fppf, fpqc, and ph coverings." ], "refs": [], "proofs": [ { "contents": [ "Since pullback is transitive and $f_{small} = \\pi_X \\circ i_f$", "(see above) we see that", "$i_f^{-1} \\pi_X^{-1}\\mathcal{F} = f_{small}^{-1}\\mathcal{F}$.", "This shows that $\\pi_X^{-1}$ has the description given in the lemma.", "\\medskip\\noindent", "To prove that $\\pi_X^{-1}\\mathcal{F}$ is a sheaf for the ph topology", "it suffices by Topologies on Spaces, Lemma", "\\ref{spaces-topologies-lemma-characterize-sheaf}", "to show that for a surjective proper morphism", "$V \\to U$ of algebraic spaces over $X$ we have", "$(\\pi_X^{-1}\\mathcal{F})(U)$ is the equalizer of the two maps", "$(\\pi_X^{-1}\\mathcal{F})(V) \\to (\\pi_X^{-1}\\mathcal{F})(V \\times_U V)$.", "This we have seen in Lemma \\ref{lemma-surjective-proper}.", "\\medskip\\noindent", "The case of smooth, syntomic, fppf coverings follows from the case", "of ph coverings by Topologies on Spaces, Lemma", "\\ref{spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-ph}.", "\\medskip\\noindent", "Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be an fpqc covering of algebraic", "spaces over $X$. Let $s_i \\in (\\pi_X^{-1}\\mathcal{F})(U_i)$ be sections", "which agree over $U_i \\times_U U_j$. We have to prove there exists a unique", "$s \\in (\\pi_X^{-1}\\mathcal{F})(U)$ restricting to $s_i$ over $U_i$.", "Case I: $U$ and $U_i$ are schemes. This case follows from", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-describe-pullback}.", "Case II: $U$ is a scheme. Here we choose surjective \\'etale morphisms", "$T_i \\to U_i$ where $T_i$ is a scheme. Then $\\mathcal{T} = \\{T_i \\to U\\}$ is an", "fpqc covering by schemes and by case I the result holds for $\\mathcal{T}$.", "We omit the verification that this implies the result for $\\mathcal{U}$.", "Case III: general case. Let $W \\to U$ be a surjective \\'etale", "morphism, where $W$ is a scheme. Then $\\mathcal{W} = \\{U_i \\times_U W \\to W\\}$", "is an fpqc covering (by algebraic spaces) of the scheme $W$.", "By case II the result hold for $\\mathcal{W}$.", "We omit the verification that this implies the result for $\\mathcal{U}$." ], "refs": [ "spaces-topologies-lemma-characterize-sheaf", "spaces-more-cohomology-lemma-surjective-proper", "spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-ph", "etale-cohomology-lemma-describe-pullback" ], "ref_ids": [ 3673, 3248, 3670, 6438 ] } ], "ref_ids": [] }, { "id": 3257, "type": "theorem", "label": "spaces-more-cohomology-lemma-compare-injectives", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-compare-injectives", "contents": [ "Let $S$ be a scheme.", "Let $Y \\to X$ be a morphism of $(\\textit{Spaces}/S)_\\etale$.", "\\begin{enumerate}", "\\item If $\\mathcal{I}$ is injective in", "$\\textit{Ab}((\\textit{Spaces}/X)_\\etale)$, then", "\\begin{enumerate}", "\\item $i_f^{-1}\\mathcal{I}$ is injective in $\\textit{Ab}(Y_\\etale)$,", "\\item $\\mathcal{I}|_{X_\\etale}$ is injective in $\\textit{Ab}(X_\\etale)$,", "\\end{enumerate}", "\\item If $\\mathcal{I}^\\bullet$ is a K-injective complex", "in $\\textit{Ab}((\\textit{Spaces}/X)_\\etale)$, then", "\\begin{enumerate}", "\\item $i_f^{-1}\\mathcal{I}^\\bullet$ is a K-injective complex in", "$\\textit{Ab}(Y_\\etale)$,", "\\item $\\mathcal{I}^\\bullet|_{X_\\etale}$ is a K-injective complex in", "$\\textit{Ab}(X_\\etale)$,", "\\end{enumerate}", "\\end{enumerate}", "The corresponding statements for modules do not hold." ], "refs": [], "proofs": [ { "contents": [ "Parts (1)(b) and (2)(b)", "follow formally from the fact that the restriction functor", "$\\pi_{X, *} = i_X^{-1}$ is a right adjoint of the exact functor", "$\\pi_X^{-1}$, see", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives} and", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}.", "\\medskip\\noindent", "Parts (1)(a) and (2)(a) can be seen in two ways. First proof: We can use", "that $i_f^{-1}$ is a right adjoint of the exact functor $i_{f, !}$.", "This functor is constructed in", "Topologies, Lemma \\ref{topologies-lemma-put-in-T-etale}", "for sheaves of sets and for abelian sheaves in", "Modules on Sites, Lemma \\ref{sites-modules-lemma-g-shriek-adjoint}.", "It is shown in Modules on Sites, Lemma", "\\ref{sites-modules-lemma-exactness-lower-shriek} that it is exact.", "Second proof. We can use that $i_f = i_Y \\circ f_{big}$ as is shown", "in Topologies, Lemma \\ref{topologies-lemma-morphism-big-small-etale}.", "Since $f_{big}$ is a localization, we see that pullback by it", "preserves injectives and K-injectives, see", "Cohomology on Sites, Lemmas \\ref{sites-cohomology-lemma-cohomology-of-open} and", "\\ref{sites-cohomology-lemma-restrict-K-injective-to-open}.", "Then we apply the already proved parts (1)(b) and (2)(b)", "to the functor $i_Y^{-1}$ to conclude.", "\\medskip\\noindent", "To see a counter example for the case of modules we refer to", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-injectives}." ], "refs": [ "homology-lemma-adjoint-preserve-injectives", "derived-lemma-adjoint-preserve-K-injectives", "topologies-lemma-put-in-T-etale", "sites-modules-lemma-g-shriek-adjoint", "sites-modules-lemma-exactness-lower-shriek", "topologies-lemma-morphism-big-small-etale", "sites-cohomology-lemma-cohomology-of-open", "sites-cohomology-lemma-restrict-K-injective-to-open", "etale-cohomology-lemma-compare-injectives" ], "ref_ids": [ 12116, 1915, 12452, 14164, 14165, 12455, 4186, 4253, 6653 ] } ], "ref_ids": [] }, { "id": 3258, "type": "theorem", "label": "spaces-more-cohomology-lemma-compare-higher-direct-image", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-compare-higher-direct-image", "contents": [ "Let $S$ be a scheme.", "Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item For $K$ in $D((\\textit{Spaces}/Y)_\\etale)$ we have", "$", "(Rf_{big, *}K)|_{X_\\etale} = Rf_{small, *}(K|_{Y_\\etale})", "$", "in $D(X_\\etale)$.", "\\item For $K$ in $D((\\textit{Spaces}/Y)_\\etale, \\mathcal{O})$ we have", "$", "(Rf_{big, *}K)|_{X_\\etale} = Rf_{small, *}(K|_{Y_\\etale})", "$", "in $D(\\textit{Mod}(X_\\etale, \\mathcal{O}_X))$.", "\\end{enumerate}", "More generally, let $g : X' \\to X$ be an object of", "$(\\textit{Spaces}/X)_\\etale$. Consider the fibre product", "$$", "\\xymatrix{", "Y' \\ar[r]_{g'} \\ar[d]_{f'} & Y \\ar[d]^f \\\\", "X' \\ar[r]^g & X", "}", "$$", "Then", "\\begin{enumerate}", "\\item[(3)] For $K$ in $D((\\textit{Spaces}/Y)_\\etale)$ we have", "$i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$", "in $D(X'_\\etale)$.", "\\item[(4)] For $K$ in $D((\\textit{Spaces}/Y)_\\etale, \\mathcal{O})$ we have", "$i_g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$", "in $D(\\textit{Mod}(X'_\\etale, \\mathcal{O}_{X'}))$.", "\\item[(5)] For $K$ in $D((\\textit{Spaces}/Y)_\\etale)$ we have", "$g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)$", "in $D((\\textit{Spaces}/X')_\\etale)$.", "\\item[(6)] For $K$ in $D((\\textit{Spaces}/Y)_\\etale, \\mathcal{O})$ we have", "$g_{big}^*(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^*K)$", "in $D(\\textit{Mod}(X'_\\etale, \\mathcal{O}_{X'}))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from", "Lemma \\ref{lemma-compare-injectives}", "and (\\ref{equation-compare-big-small})", "on choosing a K-injective complex of abelian sheaves representing $K$.", "\\medskip\\noindent", "Part (3) follows from Lemma \\ref{lemma-compare-injectives}", "and Topologies, Lemma", "\\ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale}", "on choosing a K-injective complex of abelian sheaves representing $K$.", "\\medskip\\noindent", "Part (5) is Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-localize-cartesian-square}.", "\\medskip\\noindent", "Part (6) is Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-localize-cartesian-square-modules}.", "\\medskip\\noindent", "Part (2) can be proved as follows. Above we have seen", "that $\\pi_X \\circ f_{big} = f_{small} \\circ \\pi_Y$ as morphisms", "of ringed sites. Hence we obtain", "$R\\pi_{X, *} \\circ Rf_{big, *} = Rf_{small, *} \\circ R\\pi_{Y, *}$", "by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-derived-pushforward-composition}.", "Since the restriction functors $\\pi_{X, *}$ and $\\pi_{Y, *}$", "are exact, we conclude.", "\\medskip\\noindent", "Part (4) follows from part (6) and part (2) applied to $f' : Y' \\to X'$." ], "refs": [ "spaces-more-cohomology-lemma-compare-injectives", "spaces-more-cohomology-lemma-compare-injectives", "topologies-lemma-morphism-big-small-cartesian-diagram-etale", "sites-cohomology-lemma-localize-cartesian-square", "sites-cohomology-lemma-localize-cartesian-square-modules", "sites-cohomology-lemma-derived-pushforward-composition" ], "ref_ids": [ 3257, 3257, 12457, 4263, 4264, 4250 ] } ], "ref_ids": [] }, { "id": 3259, "type": "theorem", "label": "spaces-more-cohomology-lemma-compare-cohomology", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-compare-cohomology", "contents": [ "Let $S$ be a scheme.", "Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. Then", "\\begin{enumerate}", "\\item For $K$ in $D(X_\\etale)$ we have", "$H^n_\\etale(X, \\pi_X^{-1}K) = H^n(X_\\etale, K)$.", "\\item For $K$ in $D(X_\\etale, \\mathcal{O}_X)$ we have", "$H^n_\\etale(X, L\\pi_X^*K) = H^n(X_\\etale, K)$.", "\\item For $K$ in $D(X_\\etale)$ we have", "$H^n_\\etale(Y, \\pi_X^{-1}K) = H^n(Y_\\etale, f_{small}^{-1}K)$.", "\\item For $K$ in $D(X_\\etale, \\mathcal{O}_X)$ we have", "$H^n_\\etale(Y, L\\pi_X^*K) = H^n(Y_\\etale, Lf_{small}^*K)$.", "\\item For $M$ in $D((\\textit{Spaces}/X)_\\etale)$ we have", "$H^n_\\etale(Y, M) = H^n(Y_\\etale, i_f^{-1}M)$.", "\\item For $M$ in $D((\\textit{Spaces}/X)_\\etale, \\mathcal{O})$ we have", "$H^n_\\etale(Y, M) = H^n(Y_\\etale, i_f^*M)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove (5) represent $M$ by a K-injective complex of abelian sheaves", "and apply Lemma \\ref{lemma-compare-injectives}", "and work out the definitions. Part (3) follows from", "this as $i_f^{-1}\\pi_X^{-1} = f_{small}^{-1}$. Part (1) is a special", "case of (3).", "\\medskip\\noindent", "Part (6) follows from the very general Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-pullback-same-cohomology}. Then part", "(4) follows because $Lf_{small}^* = i_f^* \\circ L\\pi_X^*$.", "Part (2) is a special case of (4)." ], "refs": [ "spaces-more-cohomology-lemma-compare-injectives", "sites-cohomology-lemma-pullback-same-cohomology" ], "ref_ids": [ 3257, 4339 ] } ], "ref_ids": [] }, { "id": 3260, "type": "theorem", "label": "spaces-more-cohomology-lemma-cohomological-descent-etale", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-cohomological-descent-etale", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "For $K \\in D(X_\\etale)$ the map", "$$", "K \\longrightarrow R\\pi_{X, *}\\pi_X^{-1}K", "$$", "is an isomorphism where", "$\\pi_X : \\Sh((\\textit{Spaces}/X)_\\etale) \\to \\Sh(X_\\etale)$ is as above." ], "refs": [], "proofs": [ { "contents": [ "This is true because both $\\pi_X^{-1}$ and $\\pi_{X, *} = i_X^{-1}$", "are exact functors and the composition $\\pi_{X, *} \\circ \\pi_X^{-1}$", "is the identity functor." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3261, "type": "theorem", "label": "spaces-more-cohomology-lemma-compare-higher-direct-image-proper", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-compare-higher-direct-image-proper", "contents": [ "Let $S$ be a scheme.", "Let $f : Y \\to X$ be a proper morphism of algebraic spaces over $S$.", "Then we have", "\\begin{enumerate}", "\\item $\\pi_X^{-1} \\circ f_{small, *} = f_{big, *} \\circ \\pi_Y^{-1}$", "as functors $\\Sh(Y_\\etale) \\to \\Sh((\\textit{Spaces}/X)_\\etale)$,", "\\item $\\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_Y^{-1}K$", "for $K$ in $D^+(Y_\\etale)$ whose cohomology sheaves are torsion, and", "\\item $\\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_Y^{-1}K$", "for all $K$ in $D(Y_\\etale)$ if $f$ is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $\\mathcal{F}$ be a sheaf on $Y_\\etale$.", "Let $g : X' \\to X$ be an object of $(\\textit{Spaces}/X)_\\etale$.", "Consider the fibre product", "$$", "\\xymatrix{", "Y' \\ar[r]_{f'} \\ar[d]_{g'} & X' \\ar[d]^g \\\\", "Y \\ar[r]^f & X", "}", "$$", "Then we have", "$$", "(f_{big, *}\\pi_Y^{-1}\\mathcal{F})(X') =", "(\\pi_Y^{-1}\\mathcal{F})(Y') =", "((g'_{small})^{-1}\\mathcal{F})(Y') =", "(f'_{small, *}(g'_{small})^{-1}\\mathcal{F})(X')", "$$", "the second equality by Lemma \\ref{lemma-describe-pullback}.", "On the other hand", "$$", "(\\pi_X^{-1}f_{small, *}\\mathcal{F})(X') =", "(g_{small}^{-1}f_{small, *}\\mathcal{F})(X')", "$$", "again by Lemma \\ref{lemma-describe-pullback}.", "Hence by proper base change for sheaves of sets", "(Lemma \\ref{lemma-proper-base-change-f-star})", "we conclude the two sets are canonically isomorphic.", "The isomorphism is compatible with restriction mappings", "and defines an isomorphism", "$\\pi_X^{-1}f_{small, *}\\mathcal{F} = f_{big, *}\\pi_Y^{-1}\\mathcal{F}$.", "Thus an isomorphism of functors", "$\\pi_X^{-1} \\circ f_{small, *} = f_{big, *} \\circ \\pi_Y^{-1}$.", "\\medskip\\noindent", "Proof of (2). There is a canonical base change map", "$\\pi_X^{-1}Rf_{small, *}K \\to Rf_{big, *}\\pi_Y^{-1}K$", "for any $K$ in $D(Y_\\etale)$, see", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change}.", "To prove it is an isomorphism, it suffices to prove the pull back of", "the base change map by $i_g : \\Sh(X'_\\etale) \\to \\Sh((\\Sch/X)_\\etale)$", "is an isomorphism for any object $g : X' \\to X$ of $(\\Sch/X)_\\etale$.", "Let $T', g', f'$ be as in the previous paragraph.", "The pullback of the base change map is", "\\begin{align*}", "g_{small}^{-1}Rf_{small, *}K", "& =", "i_g^{-1}\\pi_X^{-1}Rf_{small, *}K \\\\", "& \\to", "i_g^{-1}Rf_{big, *}\\pi_Y^{-1}K \\\\", "& =", "Rf'_{small, *}(i_{g'}^{-1}\\pi_Y^{-1}K) \\\\", "& =", "Rf'_{small, *}((g'_{small})^{-1}K)", "\\end{align*}", "where we have used $\\pi_X \\circ i_g = g_{small}$,", "$\\pi_Y \\circ i_{g'} = g'_{small}$, and", "Lemma \\ref{lemma-compare-higher-direct-image}.", "This map is an isomorphism by the proper base change theorem", "(Lemma \\ref{lemma-proper-base-change}) provided $K$ is bounded", "below and the cohomology sheaves of $K$ are torsion.", "\\medskip\\noindent", "Proof of (3). If $f$ is finite, then the functors", "$f_{small, *}$ and $f_{big, *}$ are exact. This follows", "from Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-finite-higher-direct-image-zero}", "for $f_{small}$. Since any base change $f'$ of $f$ is finite too,", "we conclude from Lemma \\ref{lemma-compare-higher-direct-image} part (3)", "that $f_{big, *}$ is exact too (as the higher derived functors are zero).", "Thus this case follows from part (1)." ], "refs": [ "spaces-more-cohomology-lemma-describe-pullback", "spaces-more-cohomology-lemma-describe-pullback", "spaces-more-cohomology-lemma-proper-base-change-f-star", "sites-cohomology-remark-base-change", "spaces-more-cohomology-lemma-compare-higher-direct-image", "spaces-more-cohomology-lemma-proper-base-change", "spaces-cohomology-lemma-finite-higher-direct-image-zero", "spaces-more-cohomology-lemma-compare-higher-direct-image" ], "ref_ids": [ 3256, 3256, 3251, 4424, 3258, 3253, 11273, 3258 ] } ], "ref_ids": [] }, { "id": 3262, "type": "theorem", "label": "spaces-more-cohomology-lemma-comparison-fppf-etale", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-comparison-fppf-etale", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item For $\\mathcal{F} \\in \\Sh(X_\\etale)$ we have", "$\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$", "and $a_{X, *}a_X^{-1}\\mathcal{F} = \\mathcal{F}$.", "\\item For $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ we have", "$R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have $a_X^{-1}\\mathcal{F} = \\epsilon_X^{-1} \\pi_X^{-1}\\mathcal{F}$.", "By Lemma \\ref{lemma-describe-pullback} the \\'etale sheaf", "$\\pi_X^{-1}\\mathcal{F}$ is a sheaf for the fppf topology", "and therefore is equal to $a_X^{-1}\\mathcal{F}$ (as pulling", "back by $\\epsilon_X$ is given by fppf sheafification).", "Recall moreover that $\\epsilon_{X, *}$ is the identity", "on underlying presheaves.", "Now part (1) is immediate from the explicit description of $\\pi_X^{-1}$", "in Lemma \\ref{lemma-describe-pullback}.", "\\medskip\\noindent", "We will prove part (2) by reducing it to the case of schemes --", "see part (1) of", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-V-C-all-n-etale-fppf}.", "This will ``clearly work'' as every algebraic space is", "\\'etale locally a scheme. The details are given below but we urge", "the reader to skip the proof.", "\\medskip\\noindent", "For an abelian sheaf $\\mathcal{H}$ on $(\\textit{Spaces}/X)_{fppf}$ the", "higher direct image $R^p\\epsilon_{X, *}\\mathcal{H}$ is the sheaf", "associated to the presheaf $U \\mapsto H^p_{fppf}(U, \\mathcal{H})$", "on $(\\textit{Spaces}/X)_\\etale$. See", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-higher-direct-images}.", "Since every object of $(\\textit{Spaces}/X)_\\etale$ has a covering", "by schemes, it suffices to prove that given $U/X$ a scheme and", "$\\xi \\in H^p_{fppf}(U, a_X^{-1}\\mathcal{F})$ we can find", "an \\'etale covering $\\{U_i \\to U\\}$ such that $\\xi$", "restricts to zero on $U_i$. We have", "\\begin{align*}", "H^p_{fppf}(U, a_X^{-1}\\mathcal{F})", "& =", "H^p((\\textit{Spaces}/U)_{fppf}, (a_X^{-1}\\mathcal{F})|_{\\textit{Spaces}/U}) \\\\", "& =", "H^p((\\Sch/U)_{fppf}, (a_X^{-1}\\mathcal{F})|_{\\Sch/U})", "\\end{align*}", "where the second identification is", "Lemma \\ref{lemma-compare-cohomology-other-topologies}", "and the first is a general fact about restriction", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-of-open}).", "Looking at the first paragraph and the corresponding result in the", "case of schemes (\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-describe-pullback-pi-fppf})", "we conclude that the sheaf $(a_X^{-1}\\mathcal{F})|_{\\Sch/U}$", "matches the pullback by the ``schemes version of $a_U$''.", "Therefore we can find an \\'etale covering", "$\\{U_i \\to U\\}$ such that our class dies in", "$H^p((\\Sch/U_i)_{fppf}, (a_X^{-1}\\mathcal{F})|_{\\Sch/U_i})$", "for each $i$, see", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-V-C-all-n-etale-fppf}", "(the precise statement one should use here is that $V_n$ holds for all $n$", "which is the statement of part (2) for the case of schemes).", "Transporting back (using the same formulas as above but now for", "$U_i$) we conclude $\\xi$ restricts to zero over $U_i$ as desired." ], "refs": [ "spaces-more-cohomology-lemma-describe-pullback", "spaces-more-cohomology-lemma-describe-pullback", "etale-cohomology-lemma-V-C-all-n-etale-fppf", "sites-cohomology-lemma-higher-direct-images", "spaces-more-cohomology-lemma-compare-cohomology-other-topologies", "sites-cohomology-lemma-cohomology-of-open", "etale-cohomology-lemma-describe-pullback-pi-fppf", "etale-cohomology-lemma-V-C-all-n-etale-fppf" ], "ref_ids": [ 3256, 3256, 6663, 4189, 3247, 4186, 6658, 6663 ] } ], "ref_ids": [] }, { "id": 3263, "type": "theorem", "label": "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "For $K \\in D^+(X_\\etale)$ the maps", "$$", "\\pi_X^{-1}K \\longrightarrow R\\epsilon_{X, *}a_X^{-1}K", "\\quad\\text{and}\\quad", "K \\longrightarrow Ra_{X, *}a_X^{-1}K", "$$", "are isomorphisms with", "$a_X : \\Sh((\\textit{Spaces}/X)_{fppf}) \\to \\Sh(X_\\etale)$ as above." ], "refs": [], "proofs": [ { "contents": [ "We only prove the second statement; the first is easier and proved in exactly", "the same manner.", "There is an immediate reduction to the case where", "$K$ is given by a single abelian sheaf. Namely, represent $K$", "by a bounded below complex $\\mathcal{F}^\\bullet$. By the case of a", "sheaf we see that", "$\\mathcal{F}^n = a_{X, *} a_X^{-1} \\mathcal{F}^n$", "and that the sheaves $R^qa_{X, *}a_X^{-1}\\mathcal{F}^n$", "are zero for $q > 0$. By Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "applied to $a_X^{-1}\\mathcal{F}^\\bullet$", "and the functor $a_{X, *}$ we conclude. From now on assume $K = \\mathcal{F}$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-comparison-fppf-etale} we have", "$a_{X, *}a_X^{-1}\\mathcal{F} = \\mathcal{F}$. Thus it suffices to show that", "$R^qa_{X, *}a_X^{-1}\\mathcal{F} = 0$ for $q > 0$.", "For this we can use $a_X = \\epsilon_X \\circ \\pi_X$ and", "the Leray spectral sequence", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-relative-Leray}).", "By Lemma \\ref{lemma-comparison-fppf-etale}", "we have $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$.", "We have", "$\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$", "and by Lemma \\ref{lemma-cohomological-descent-etale} we have", "$R^j\\pi_{X, *}(\\pi_X^{-1}\\mathcal{F}) = 0$ for $j > 0$.", "This concludes the proof." ], "refs": [ "derived-lemma-leray-acyclicity", "spaces-more-cohomology-lemma-comparison-fppf-etale", "sites-cohomology-lemma-relative-Leray", "spaces-more-cohomology-lemma-comparison-fppf-etale", "spaces-more-cohomology-lemma-cohomological-descent-etale" ], "ref_ids": [ 1844, 3262, 4222, 3262, 3260 ] } ], "ref_ids": [] }, { "id": 3264, "type": "theorem", "label": "spaces-more-cohomology-lemma-compare-cohomology-etale-fppf", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-compare-cohomology-etale-fppf", "contents": [ "Let $S$ be a scheme and let $X$ be an algebraic space over $S$.", "With $a_X : \\Sh((\\textit{Spaces}/X)_{fppf}) \\to \\Sh(X_\\etale)$", "as above:", "\\begin{enumerate}", "\\item $H^q(X_\\etale, \\mathcal{F}) = H^q_{fppf}(X, a_X^{-1}\\mathcal{F})$", "for an abelian sheaf $\\mathcal{F}$ on $X_\\etale$,", "\\item $H^q(X_\\etale, K) = H^q_{fppf}(X, a_X^{-1}K)$ for $K \\in D^+(X_\\etale)$.", "\\end{enumerate}", "Example: if $A$ is an abelian group, then", "$H^q_\\etale(X, \\underline{A}) = H^q_{fppf}(X, \\underline{A})$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-cohomological-descent-etale-fppf}", "by Cohomology on Sites, Remark \\ref{sites-cohomology-remark-before-Leray}." ], "refs": [ "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf", "sites-cohomology-remark-before-Leray" ], "ref_ids": [ 3263, 4423 ] } ], "ref_ids": [] }, { "id": 3265, "type": "theorem", "label": "spaces-more-cohomology-lemma-push-pull-fppf-etale", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-push-pull-fppf-etale", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Then there are commutative diagrams of topoi", "$$", "\\xymatrix{", "\\Sh((\\textit{Spaces}/X)_{fppf}) \\ar[rr]_{f_{big, fppf}} \\ar[d]_{\\epsilon_X} & &", "\\Sh((\\textit{Spaces}/Y)_{fppf}) \\ar[d]^{\\epsilon_Y} \\\\", "\\Sh((\\textit{Spaces}/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & &", "\\Sh((\\textit{Spaces}/Y)_\\etale)", "}", "$$", "and", "$$", "\\xymatrix{", "\\Sh((\\textit{Spaces}/X)_{fppf}) \\ar[rr]_{f_{big, fppf}} \\ar[d]_{a_X} & &", "\\Sh((\\textit{Spaces}/Y)_{fppf}) \\ar[d]^{a_Y} \\\\", "\\Sh(X_\\etale) \\ar[rr]^{f_{small}} & &", "\\Sh(Y_\\etale)", "}", "$$", "with $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from working out the definitions of the", "morphisms involved, see", "Topologies on Spaces, Section \\ref{spaces-topologies-section-fppf}", "and Section \\ref{section-compare}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3266, "type": "theorem", "label": "spaces-more-cohomology-lemma-proper-push-pull-fppf-etale", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-proper-push-pull-fppf-etale", "contents": [ "In Lemma \\ref{lemma-push-pull-fppf-etale} if $f$ is proper, then we have", "\\begin{enumerate}", "\\item $a_Y^{-1} \\circ f_{small, *} = f_{big, fppf, *} \\circ a_X^{-1}$, and", "\\item", "$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$", "for $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves.", "\\end{enumerate}" ], "refs": [ "spaces-more-cohomology-lemma-push-pull-fppf-etale" ], "proofs": [ { "contents": [ "Proof of (1). You can prove this by repeating the proof of", "Lemma \\ref{lemma-compare-higher-direct-image-proper} part (1);", "we will instead deduce the result from this.", "As $\\epsilon_{Y, *}$ is the identity functor on underlying presheaves,", "it reflects isomorphisms. Lemma \\ref{lemma-comparison-fppf-etale}", "shows that $\\epsilon_{Y, *} \\circ a_Y^{-1} = \\pi_Y^{-1}$", "and similarly for $X$. To show that the canonical map", "$a_Y^{-1}f_{small, *}\\mathcal{F} \\to f_{big, fppf, *}a_X^{-1}\\mathcal{F}$", "is an isomorphism, it suffices to show that", "\\begin{align*}", "\\pi_Y^{-1}f_{small, *}\\mathcal{F}", "& =", "\\epsilon_{Y, *}a_Y^{-1}f_{small, *}\\mathcal{F} \\\\", "& \\to ", "\\epsilon_{Y, *}f_{big, fppf, *}a_X^{-1}\\mathcal{F} \\\\", "& =", "f_{big, \\etale, *} \\epsilon_{X, *}a_X^{-1}\\mathcal{F} \\\\", "& =", "f_{big, \\etale, *}\\pi_X^{-1}\\mathcal{F}", "\\end{align*}", "is an isomorphism. This is part", "(1) of Lemma \\ref{lemma-compare-higher-direct-image-proper}.", "\\medskip\\noindent", "To see (2) we use that", "\\begin{align*}", "R\\epsilon_{Y, *}Rf_{big, fppf, *}a_X^{-1}K", "& =", "Rf_{big, \\etale, *}R\\epsilon_{X, *}a_X^{-1}K \\\\", "& =", "Rf_{big, \\etale, *}\\pi_X^{-1}K \\\\", "& =", "\\pi_Y^{-1}Rf_{small, *}K \\\\", "& =", "R\\epsilon_{Y, *} a_Y^{-1}Rf_{small, *}K", "\\end{align*}", "The first equality by the commutative diagram in", "Lemma \\ref{lemma-push-pull-fppf-etale}", "and Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-derived-pushforward-composition}.", "Then second equality is Lemma \\ref{lemma-cohomological-descent-etale-fppf}.", "The third is", "Lemma \\ref{lemma-compare-higher-direct-image-proper} part (2).", "The fourth is Lemma \\ref{lemma-cohomological-descent-etale-fppf} again.", "Thus the base change map", "$a_Y^{-1}(Rf_{small, *}K) \\to Rf_{big, fppf, *}(a_X^{-1}K)$", "induces an isomorphism", "$$", "R\\epsilon_{Y, *}a_Y^{-1}Rf_{small, *}K \\to", "R\\epsilon_{Y, *}Rf_{big, fppf, *}a_X^{-1}K", "$$", "The proof is finished by the following remark: a map", "$\\alpha : a_Y^{-1}L \\to M$ with $L$ in $D^+(Y_\\etale)$", "and $M$ in $D^+((\\textit{Spaces}/Y)_{fppf})$ such that", "$R\\epsilon_{Y, *}\\alpha$ is an isomorphism, is an isomorphism. Namely, ", "we show by induction on $i$ that $H^i(\\alpha)$ is an isomorphism.", "This is true for all sufficiently small $i$.", "If it holds for $i \\leq i_0$, then we see that", "$R^j\\epsilon_{Y, *}H^i(M) = 0$ for $j > 0$ and $i \\leq i_0$", "by Lemma \\ref{lemma-comparison-fppf-etale}", "because $H^i(M) = a_Y^{-1}H^i(L)$ in this range.", "Hence $\\epsilon_{Y, *}H^{i_0 + 1}(M) = H^{i_0 + 1}(R\\epsilon_{Y, *}M)$", "by a spectral sequence argument.", "Thus $\\epsilon_{Y, *}H^{i_0 + 1}(M) = \\pi_Y^{-1}H^{i_0 + 1}(L) =", "\\epsilon_{Y, *}a_Y^{-1}H^{i_0 + 1}(L)$.", "This implies $H^{i_0 + 1}(\\alpha)$ is an isomorphism", "(because $\\epsilon_{Y, *}$ reflects isomorphisms as it is the", "identity on underlying presheaves) as desired." ], "refs": [ "spaces-more-cohomology-lemma-compare-higher-direct-image-proper", "spaces-more-cohomology-lemma-comparison-fppf-etale", "spaces-more-cohomology-lemma-compare-higher-direct-image-proper", "spaces-more-cohomology-lemma-push-pull-fppf-etale", "sites-cohomology-lemma-derived-pushforward-composition", "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf", "spaces-more-cohomology-lemma-compare-higher-direct-image-proper", "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf", "spaces-more-cohomology-lemma-comparison-fppf-etale" ], "ref_ids": [ 3261, 3262, 3261, 3265, 4250, 3263, 3261, 3263, 3262 ] } ], "ref_ids": [ 3265 ] }, { "id": 3267, "type": "theorem", "label": "spaces-more-cohomology-lemma-finite-push-pull-fppf-etale", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-finite-push-pull-fppf-etale", "contents": [ "In Lemma \\ref{lemma-push-pull-fppf-etale} if $f$ is finite, then", "$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$", "for $K$ in $D^+(X_\\etale)$." ], "refs": [ "spaces-more-cohomology-lemma-push-pull-fppf-etale" ], "proofs": [ { "contents": [ "Let $V \\to Y$ be a surjective \\'etale morphism where $V$ is a scheme.", "It suffices to prove the base change map is an isomorphism after", "restricting to $V$. Hence we may assume that $Y$ is a scheme.", "As the morphism is finite, hence representable, we conclude", "that we may assume both $X$ and $Y$ are schemes. In this case", "the result follows from the case of schemes", "(\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-V-C-all-n-etale-fppf}", "part (2)) using the comparison of topoi discussed in", "Section \\ref{section-api}", "and in particular given in", "Lemma \\ref{lemma-compare-cohomology-other-topologies}.", "Some details omitted." ], "refs": [ "etale-cohomology-lemma-V-C-all-n-etale-fppf", "spaces-more-cohomology-lemma-compare-cohomology-other-topologies" ], "ref_ids": [ 6663, 3247 ] } ], "ref_ids": [ 3265 ] }, { "id": 3268, "type": "theorem", "label": "spaces-more-cohomology-lemma-descent-sheaf-fppf-etale", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-descent-sheaf-fppf-etale", "contents": [ "In Lemma \\ref{lemma-push-pull-fppf-etale} assume", "$f$ is flat, locally of finite presentation, and surjective.", "Then the functor", "$$", "\\Sh(Y_\\etale) \\longrightarrow", "\\left\\{", "(\\mathcal{G}, \\mathcal{H}, \\alpha)", "\\middle|", "\\begin{matrix}", "\\mathcal{G} \\in \\Sh(X_\\etale),\\ \\mathcal{H} \\in \\Sh((\\Sch/Y)_{fppf}), \\\\", "\\alpha : a_X^{-1}\\mathcal{G} \\to f_{big, fppf}^{-1}\\mathcal{H}", "\\text{ an isomorphism}", "\\end{matrix}", "\\right\\}", "$$", "sending $\\mathcal{F}$ to", "$(f_{small}^{-1}\\mathcal{F}, a_Y^{-1}\\mathcal{F}, can)$ is an equivalence." ], "refs": [ "spaces-more-cohomology-lemma-push-pull-fppf-etale" ], "proofs": [ { "contents": [ "The functor $a_X^{-1}$ is fully faithful (as $a_{X, *}a_X^{-1} = \\text{id}$ by", "Lemma \\ref{lemma-comparison-fppf-etale}). Hence the forgetful functor", "$(\\mathcal{G}, \\mathcal{H}, \\alpha) \\mapsto \\mathcal{H}$ identifies the", "category of triples with a full subcategory of $\\Sh((\\Sch/Y)_{fppf})$.", "Moreover, the functor $a_Y^{-1}$ is fully faithful, hence the functor", "in the lemma is fully faithful as well.", "\\medskip\\noindent", "Suppose that we have an \\'etale covering $\\{Y_i \\to Y\\}$.", "Let $f_i : X_i \\to Y_i$ be the base change of $f$.", "Denote $f_{ij} = f_i \\times f_j : X_i \\times_X X_j \\to Y_i \\times_Y Y_j$.", "Claim: if the lemma is true for $f_i$ and $f_{ij}$ for all $i, j$, then", "the lemma is true for $f$. To see this, note that the given \\'etale covering", "determines an \\'etale covering of the final object in each of", "the four sites $Y_\\etale, X_\\etale, (\\Sch/Y)_{fppf}, (\\Sch/X)_{fppf}$.", "Thus the category of sheaves is equivalent to the category of", "glueing data for this covering", "(Sites, Lemma \\ref{sites-lemma-mapping-property-glue})", "in each of the four cases. A huge commutative diagram of", "categories then finishes the proof of the claim. We omit the details.", "The claim shows that we may work \\'etale locally on $Y$.", "In particular, we may assume $Y$ is a scheme.", "\\medskip\\noindent", "Assume $Y$ is a scheme. Choose a scheme $X'$ and a surjective \\'etale", "morphism $s : X' \\to X$. Set $f' = f \\circ s : X' \\to Y$ and observe", "that $f'$ is surjective, locally of finite presentation, and flat.", "Claim: if the lemma is true for $f'$, then it is true for $f$.", "Namely, given a triple $(\\mathcal{G}, \\mathcal{H}, \\alpha)$", "for $f$, we can pullback by $s$ to get a triple", "$(s_{small}^{-1}\\mathcal{G}, \\mathcal{H}, s_{big, fppf}^{-1}\\alpha)$", "for $f'$. A solution for this triple gives a sheaf", "$\\mathcal{F}$ on $Y_\\etale$ with $a_Y^{-1}\\mathcal{F} = \\mathcal{H}$.", "By the first paragraph of the proof this means the triple is", "in the essential image. This reduces us to the case", "where both $X$ and $Y$ are schemes. This case follows from", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-descent-sheaf-fppf-etale}", "via the discussion in Section \\ref{section-api}", "and in particular Lemma \\ref{lemma-compare-cohomology-other-topologies}." ], "refs": [ "spaces-more-cohomology-lemma-comparison-fppf-etale", "sites-lemma-mapping-property-glue", "etale-cohomology-lemma-descent-sheaf-fppf-etale", "spaces-more-cohomology-lemma-compare-cohomology-other-topologies" ], "ref_ids": [ 3262, 8565, 6661, 3247 ] } ], "ref_ids": [ 3265 ] }, { "id": 3269, "type": "theorem", "label": "spaces-more-cohomology-lemma-review-quasi-coherent", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-review-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item The rule", "$$", "\\mathcal{F}^a : (\\textit{Spaces}/X)_\\etale \\longrightarrow \\textit{Ab},\\quad", "(f : Y \\to X) \\longmapsto \\Gamma(Y, f^*\\mathcal{F})", "$$", "satisfies the sheaf condition for fpqc and a fortiori", "fppf and \\'etale coverings,", "\\item $\\mathcal{F}^a = \\pi_X^*\\mathcal{F}$ on $(\\textit{Spaces}/X)_\\etale$,", "\\item $\\mathcal{F}^a = a_X^*\\mathcal{F}$ on $(\\textit{Spaces}/X)_{fppf}$,", "\\item the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ defines", "an equivalence between quasi-coherent $\\mathcal{O}_X$-modules", "and quasi-coherent modules on", "$((\\textit{Spaces}/X)_\\etale, \\mathcal{O})$,", "\\item the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ defines", "an equivalence between quasi-coherent $\\mathcal{O}_X$-modules", "and quasi-coherent modules on", "$((\\textit{Spaces}/X)_{fppf}, \\mathcal{O})$,", "\\item we have $\\epsilon_{X, *}a_X^*\\mathcal{F} = \\pi_X^*\\mathcal{F}$", "and $a_{X, *}a_X^*\\mathcal{F} = \\mathcal{F}$,", "\\item we have $R^i\\epsilon_{X, *}(a_X^*\\mathcal{F}) = 0$", "and $R^ia_{X, *}(a_X^*\\mathcal{F}) = 0$ for $i > 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is a consequence of fppf descent of quasi-coherent modules.", "Namely, suppose that $\\{f_i : U_i \\to U\\}$ is an fpqc covering", "in $(\\textit{Spaces}/X)_\\etale$. Denote $g : U \\to X$ the structure", "morphism. Suppose that", "we have a family of sections $s_i \\in \\Gamma(U_i , f_i^*g^*\\mathcal{F})$", "such that $s_i|_{U_i \\times_U U_j} = s_j|_{U_i \\times_U U_j}$.", "We have to find the correspond section $s \\in \\Gamma(U, g^*\\mathcal{F})$.", "We can reinterpret the $s_i$ as a family of maps", "$\\varphi_i : f_i^*\\mathcal{O}_U = \\mathcal{O}_{U_i} \\to f_i^*g^*\\mathcal{F}$", "compatible with the canonical descent data associated to the", "quasi-coherent sheaves $\\mathcal{O}_U$ and $g^*\\mathcal{F}$ on $U$.", "Hence by Descent on Spaces, Proposition", "\\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}", "we see that we may (uniquely) descend", "these to a map $\\mathcal{O}_U \\to g^*\\mathcal{F}$ which gives", "us our section $s$.", "\\medskip\\noindent", "We will deduce (2) -- (7) from the corresponding statement for schemes.", "Choose an \\'etale covering $\\{X_i \\to X\\}_{i \\in I}$", "where each $X_i$ is a scheme. Observe that $X_i \\times_X X_j$", "is a scheme too. This covering induces a covering of", "the final object in each of the three sites", "$(\\textit{Spaces}/X)_{fppf}$, $(\\textit{Spaces}/X)_\\etale$, and $X_\\etale$.", "Hence we see that the category of sheaves on these sites", "are equivalent to descent data for these coverings, see", "Sites, Lemma \\ref{sites-lemma-mapping-property-glue}.", "Parts (2), (3) are local (because we have the glueing statement).", "Being quasi-coherent is a local property, hence parts", "(4), (5) are local. Clearly (6) and (7) are local.", "It follows that it suffices to prove parts (2) -- (7)", "of the lemma when $X$ is a scheme.", "\\medskip\\noindent", "Assume $X$ is a scheme. The embeddings", "$(\\Sch/X)_\\etale \\subset (\\textit{Spaces}/X)_\\etale$ and", "$(\\Sch/X)_{fppf} \\subset (\\textit{Spaces}/X)_{fppf}$", "determine equivalences of ringed topoi by", "Lemma \\ref{lemma-compare-cohomology-other-topologies}.", "We conclude that (2) -- (7) follows from the case of schemes.", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-review-quasi-coherent}.", "To transport the property of being quasi-coherent via", "this equivalence use that being quasi-coherent is an", "intrinsic property of modules as explained in", "Modules on Sites, Section \\ref{sites-modules-section-local}.", "Some minor details omitted." ], "refs": [ "spaces-descent-proposition-fpqc-descent-quasi-coherent", "sites-lemma-mapping-property-glue", "spaces-more-cohomology-lemma-compare-cohomology-other-topologies", "etale-cohomology-lemma-review-quasi-coherent" ], "ref_ids": [ 9437, 8565, 3247, 6666 ] } ], "ref_ids": [] }, { "id": 3270, "type": "theorem", "label": "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "For $\\mathcal{F}$ a quasi-coherent $\\mathcal{O}_X$-module the maps", "$$", "\\pi_X^*\\mathcal{F} \\longrightarrow R\\epsilon_{X, *}(a_X^*\\mathcal{F})", "\\quad\\text{and}\\quad", "\\mathcal{F} \\longrightarrow Ra_{X, *}(a_X^*\\mathcal{F})", "$$", "are isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "This is an immediate consequence of parts (6) and (7) of", "Lemma \\ref{lemma-review-quasi-coherent}." ], "refs": [ "spaces-more-cohomology-lemma-review-quasi-coherent" ], "ref_ids": [ 3269 ] } ], "ref_ids": [] }, { "id": 3271, "type": "theorem", "label": "spaces-more-cohomology-lemma-vanishing-adequate", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-vanishing-adequate", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3$", "be a complex of quasi-coherent $\\mathcal{O}_X$-modules.", "Set", "$$", "\\mathcal{H}_\\etale =", "\\Ker(\\pi_X^*\\mathcal{F}_2 \\to \\pi_X^*\\mathcal{F}_3)/", "\\Im(\\pi_X^*\\mathcal{F}_1 \\to \\pi_X^*\\mathcal{F}_2)", "$$", "on $(\\textit{Spaces}/X)_\\etale$ and set", "$$", "\\mathcal{H}_{fppf} =", "\\Ker(a_X^*\\mathcal{F}_2 \\to a_X^*\\mathcal{F}_3)/", "\\Im(a_X^*\\mathcal{F}_1 \\to a_X^*\\mathcal{F}_2)", "$$", "on $(\\textit{Spaces}/X)_{fppf}$.", "Then $\\mathcal{H}_\\etale = \\epsilon_{X, *}\\mathcal{H}_{fppf}$", "and", "$$", "H^p_\\etale(U, \\mathcal{H}_\\etale) = H^p_{fppf}(U, \\mathcal{H}_{fppf}) = 0", "$$", "for $p > 0$ and any affine object $U$ of $(\\textit{Spaces}/X)_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "For any object $f : U \\to X$ of $(\\textit{Spaces}/X)_\\etale$", "consider the restriction", "$\\mathcal{H}_\\etale|_{U_\\etale}$ of $\\mathcal{H}_\\etale$ to $U_\\etale$ via", "the functor $i_f^* = i_f^{-1}$ discussed in Section \\ref{section-compare}.", "The sheaf $\\mathcal{H}_\\etale|_{U_\\etale}$", "is equal to the homology of complex $f^*\\mathcal{F}_\\bullet$ in degree $1$.", "This is true because $i_f \\circ \\pi_X = f$ as morphisms of ringed sites", "$U_\\etale \\to X_\\etale$. In particular we see that", "$\\mathcal{H}_\\etale|_{U_\\etale}$ is a quasi-coherent $\\mathcal{O}_U$-module.", "Next, let $g : V \\to U$ be a flat", "morphism in $(\\textit{Spaces}/X)_\\etale$. Since", "$$", "i_{f \\circ g}^* \\circ \\pi_X^* = (f \\circ g)^* = g^* \\circ f^*", "$$", "as morphisms of sites $V_\\etale \\to X_\\etale$ and since $g$ is flat", "hence $g^*$ is exact, we obtain", "$$", "\\mathcal{H}_\\etale|_{V_\\etale} =", "g^*\\left(\\mathcal{H}_\\etale|_{U_\\etale}\\right)", "$$", "With these preparations we are ready to prove the lemma.", "\\medskip\\noindent", "Let $\\mathcal{U} = \\{g_i : U_i \\to U\\}_{i \\in I}$ be an fppf covering", "with $f : U \\to X$ as above. The sheaf propery holds for", "$\\mathcal{H}_\\etale$ and the covering $\\mathcal{U}$", "by (1) of Lemma \\ref{lemma-review-quasi-coherent}", "applied to $\\mathcal{H}_\\etale|_{U_\\etale}$ and the above.", "Therefore we see that $\\mathcal{H}_\\etale$ is already an fppf", "sheaf and this means that $\\mathcal{H}_{fppf}$ is", "equal to $\\mathcal{H}_\\etale$", "as a presheaf. In particular", "$\\mathcal{H}_\\etale = \\epsilon_{X, *}\\mathcal{H}_{fppf}$.", "\\medskip\\noindent", "Finally, to prove the vanishing, we use", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cech-vanish-collection}.", "We let $\\mathcal{B}$ be the affine objects of", "$(\\textit{Spaces}/X)_{fppf}$ and we let", "$\\text{Cov}$ be the set of finite fppf coverings", "$\\mathcal{U} = \\{U_i \\to U\\}_{i = 1, \\ldots, n}$ with $U$, $U_i$ affine.", "We have", "$$", "{\\check H}^p(\\mathcal{U}, \\mathcal{H}_\\etale) =", "{\\check H}^p(\\mathcal{U}, \\left(\\mathcal{H}_\\etale|_{U_\\etale}\\right)^a)", "$$", "because the values of $\\mathcal{H}_\\etale$ on the", "affine schemes $U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$", "flat over $U$ agree with the values of the pullback", "of the quasi-coherent module $\\mathcal{H}_\\etale|_{U_\\etale}$ by", "the first paragraph. Hence we obtain vanishing by", "Descent, Lemma \\ref{descent-lemma-standard-covering-Cech-quasi-coherent}.", "This finishes the proof." ], "refs": [ "spaces-more-cohomology-lemma-review-quasi-coherent", "sites-cohomology-lemma-cech-vanish-collection", "descent-lemma-standard-covering-Cech-quasi-coherent" ], "ref_ids": [ 3269, 4205, 14625 ] } ], "ref_ids": [] }, { "id": 3272, "type": "theorem", "label": "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules-unbounded", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules-unbounded", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "For $K \\in D_\\QCoh(\\mathcal{O}_X)$ the maps", "$$", "L\\pi_X^*K \\longrightarrow R\\epsilon_{X, *}(La_X^*\\mathcal{F})", "\\quad\\text{and}\\quad", "K \\longrightarrow Ra_{X, *}(La_X^*K)", "$$", "are isomorphisms. Here", "$a_X : \\Sh((\\textit{Spaces}/X)_{fppf}) \\to \\Sh(X_\\etale)$ is as above." ], "refs": [], "proofs": [ { "contents": [ "The question is \\'etale local on $X$ hence we may assume $X$ is affine.", "Say $X = \\Spec(A)$. Then we have $D_\\QCoh(\\mathcal{O}_X) = D(A)$ by", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}", "and", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded}.", "Hence we can choose an K-flat complex of $A$-modules", "$K^\\bullet$ whose corresponding complex", "$\\mathcal{K}^\\bullet$ of quasi-coherent $\\mathcal{O}_X$-modules", "represents $K$.", "We claim that $\\mathcal{K}^\\bullet$ is a K-flat complex", "of $\\mathcal{O}_X$-modules.", "\\medskip\\noindent", "Proof of the claim. By", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-K-flat}", "we see that $\\widetilde{K}^\\bullet$ is K-flat on the scheme", "$(\\Spec(A), \\mathcal{O}_{\\Spec(A)})$.", "Next, note that $\\mathcal{K}^\\bullet = \\epsilon^*\\widetilde{K}^\\bullet$", "where $\\epsilon$ is as in Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}", "whence $\\mathcal{K}^\\bullet$ is K-flat by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-pullback-K-flat-points}", "and the fact that the \\'etale site of a scheme has enough points", "(\\'Etale Cohomology, Remarks \\ref{etale-cohomology-remarks-enough-points}).", "\\medskip\\noindent", "By the claim we see that", "$La_X^*K = a_X^*\\mathcal{K}^\\bullet$ and", "$L\\pi_X^*K = \\pi_X^*\\mathcal{K}^\\bullet$.", "Since the first part of the proof shows that the pullback", "$a_X^*\\mathcal{K}^n$ of the quasi-coherent module", "is acyclic for $\\epsilon_{X, *}$, resp.\\ $a_{X, *}$, surely the proof is done", "by Leray's acyclicity lemma? Actually..., no because Leray's", "acyclicity lemma only applies to bounded below complexes.", "However, in the next paragraph we will show the result does follow", "from the bounded below case because our complex is the derived limit", "of bounded below complexes of quasi-coherent modules.", "\\medskip\\noindent", "The cohomology sheaves of", "$\\pi_X^*\\mathcal{K}^\\bullet$ and $a_X^*\\mathcal{K}^\\bullet$", "have vanishing higher cohomology", "groups over affine objects of $(\\textit{Spaces}/X)_\\etale$ by", "Lemma \\ref{lemma-vanishing-adequate}.", "Therefore we have", "$$", "L\\pi_X^*K = R\\lim \\tau_{\\geq -n}(L\\pi_X^*K)", "\\quad\\text{and}\\quad", "La_X^*K = R\\lim \\tau_{\\geq -n}(La_X^*K)", "$$", "by Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-is-limit-dimension}.", "\\medskip\\noindent", "Proof of $L\\pi_X^*K = R\\epsilon_{X, *}(La_X^*\\mathcal{F})$.", "By the above we have", "$$", "R\\epsilon_{X, *}La_X^*K =", "R\\lim R\\epsilon_{X, *}(\\tau_{\\geq -n}(La_X^*K))", "$$", "by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-Rf-commutes-with-Rlim}.", "Note that $\\tau_{\\geq -n}(La_X^*K)$ is represented by", "$\\tau_{\\geq -n}(a_X^*\\mathcal{K}^\\bullet)$ which may not be the", "same as $a_X^*(\\tau_{\\geq -n}\\mathcal{K}^\\bullet)$.", "But clearly the systems", "$$", "\\{\\tau_{\\geq -n}(a_X^*\\mathcal{K}^\\bullet)\\}_{n \\geq 1}", "\\quad\\text{and}\\quad", "\\{a_X^*(\\tau_{\\geq -n}\\mathcal{K}^\\bullet)\\}_{n \\geq 1}", "$$", "are isomorphic as pro-systems.", "By Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "and the first part of the lemma we see that", "$$", "R\\epsilon_{X, *}(a_X^*(\\tau_{\\geq -n}\\mathcal{K}^\\bullet)) =", "\\pi_X^*(\\tau_{\\geq -n}\\mathcal{K}^\\bullet)", "$$", "Then we can use that the systems", "$$", "\\{\\tau_{\\geq -n}(\\pi_X^*\\mathcal{K}^\\bullet)\\}_{n \\geq 1}", "\\quad\\text{and}\\quad", "\\{\\pi_X^*(\\tau_{\\geq -n}\\mathcal{K}^\\bullet)\\}_{n \\geq 1}", "$$", "are isomorphic as pro-systems. Finally, we put everything together as follows", "\\begin{align*}", "R\\epsilon_{X, *}La_X^*K", "& =", "R\\epsilon_{X, *} (R\\lim \\tau_{\\geq -n}(La_X^*K)) \\\\", "& =", "R\\lim R\\epsilon_{X, *}(\\tau_{\\geq -n}(La_X^*K)) \\\\", "& =", "R\\lim R\\epsilon_{X, *}(\\tau_{\\geq -n}(a_X^*\\mathcal{K}^\\bullet)) \\\\", "& =", "R\\lim R\\epsilon_{X, *}(a_X^*(\\tau_{\\geq -n}\\mathcal{K}^\\bullet)) \\\\", "& =", "R\\lim \\pi_X^*(\\tau_{\\geq -n}\\mathcal{K}^\\bullet) \\\\", "& =", "R\\lim \\tau_{\\geq -n}(\\pi_X^*\\mathcal{K}^\\bullet) \\\\", "& =", "R\\lim \\tau_{\\geq -n}(L\\pi_X^*K) \\\\", "& =", "L\\pi_X^*K", "\\end{align*}", "Here in equalities four and six we have used that isomorphic", "pro-systems have the same $R\\lim$ (small detail omitted).", "You can avoid this step by using more about cohomology of the terms", "of the complex $\\tau_{\\geq -n}a_X^*\\mathcal{K}^\\bullet$ proved", "in Lemma \\ref{lemma-vanishing-adequate} as this will prove", "directly that $R\\epsilon_{X, *}(\\tau_{\\geq -n}(a_X^*\\mathcal{K}^\\bullet)) =", "\\tau_{\\geq -n}(\\pi_X^*\\mathcal{K}^\\bullet)$.", "\\medskip\\noindent", "The equality $K = Ra_{X, *}(La_X^*\\mathcal{F})$ is", "proved in exactly the same way using in the final step that", "$K = R\\lim \\tau_{\\geq -n}K$ by", "Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-nice-K-injective}." ], "refs": [ "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "perfect-lemma-affine-compare-bounded", "perfect-lemma-affine-K-flat", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "sites-cohomology-lemma-pullback-K-flat-points", "etale-cohomology-remarks-enough-points", "spaces-more-cohomology-lemma-vanishing-adequate", "sites-cohomology-lemma-is-limit-dimension", "sites-cohomology-lemma-Rf-commutes-with-Rlim", "derived-lemma-leray-acyclicity", "spaces-more-cohomology-lemma-vanishing-adequate", "spaces-perfect-lemma-nice-K-injective" ], "ref_ids": [ 2644, 6941, 6942, 2644, 4247, 6798, 3271, 4273, 4267, 1844, 3271, 2650 ] } ], "ref_ids": [] }, { "id": 3273, "type": "theorem", "label": "spaces-more-cohomology-lemma-comparison-ph-etale", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-comparison-ph-etale", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item For $\\mathcal{F} \\in \\Sh(X_\\etale)$ we have", "$\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$", "and $a_{X, *}a_X^{-1}\\mathcal{F} = \\mathcal{F}$.", "\\item For $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ torsion we have", "$R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have $a_X^{-1}\\mathcal{F} = \\epsilon_X^{-1} \\pi_X^{-1}\\mathcal{F}$.", "By Lemma \\ref{lemma-describe-pullback} the \\'etale sheaf", "$\\pi_X^{-1}\\mathcal{F}$ is a sheaf for the ph topology", "and therefore is equal to $a_X^{-1}\\mathcal{F}$ (as pulling", "back by $\\epsilon_X$ is given by ph sheafification).", "Recall moreover that $\\epsilon_{X, *}$ is the identity", "on underlying presheaves.", "Now part (1) is immediate from the explicit description of $\\pi_X^{-1}$", "in Lemma \\ref{lemma-describe-pullback}.", "\\medskip\\noindent", "We will prove part (2) by reducing it to the case of schemes --", "see part (1) of", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-V-C-all-n-etale-ph}.", "This will ``clearly work'' as every algebraic space is", "\\'etale locally a scheme. The details are given below but we urge", "the reader to skip the proof.", "\\medskip\\noindent", "For an abelian sheaf $\\mathcal{H}$ on $(\\textit{Spaces}/X)_{ph}$ the", "higher direct image $R^p\\epsilon_{X, *}\\mathcal{H}$ is the sheaf", "associated to the presheaf $U \\mapsto H^p_{ph}(U, \\mathcal{H})$", "on $(\\textit{Spaces}/X)_\\etale$. See", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-higher-direct-images}.", "Since every object of $(\\textit{Spaces}/X)_\\etale$ has a covering", "by schemes, it suffices to prove that given $U/X$ a scheme and", "$\\xi \\in H^p_{ph}(U, a_X^{-1}\\mathcal{F})$ we can find", "an \\'etale covering $\\{U_i \\to U\\}$ such that $\\xi$", "restricts to zero on $U_i$. We have", "\\begin{align*}", "H^p_{ph}(U, a_X^{-1}\\mathcal{F})", "& =", "H^p((\\textit{Spaces}/U)_{ph}, (a_X^{-1}\\mathcal{F})|_{\\textit{Spaces}/U}) \\\\", "& =", "H^p((\\Sch/U)_{ph}, (a_X^{-1}\\mathcal{F})|_{\\Sch/U})", "\\end{align*}", "where the second identification is", "Lemma \\ref{lemma-compare-cohomology-other-topologies}", "and the first is a general fact about restriction", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-of-open}).", "Looking at the first paragraph and the corresponding result in the", "case of schemes (\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-describe-pullback-pi-ph})", "we conclude that the sheaf $(a_X^{-1}\\mathcal{F})|_{\\Sch/U}$", "matches the pullback by the ``schemes version of $a_U$''.", "Therefore we can find an \\'etale covering", "$\\{U_i \\to U\\}$ such that our class dies in", "$H^p((\\Sch/U_i)_{ph}, (a_X^{-1}\\mathcal{F})|_{\\Sch/U_i})$", "for each $i$, see", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-V-C-all-n-etale-ph}", "(the precise statement one should use here is that $V_n$ holds for all $n$", "which is the statement of part (2) for the case of schemes).", "Transporting back (using the same formulas as above but now for", "$U_i$) we conclude $\\xi$ restricts to zero over $U_i$ as desired." ], "refs": [ "spaces-more-cohomology-lemma-describe-pullback", "spaces-more-cohomology-lemma-describe-pullback", "etale-cohomology-lemma-V-C-all-n-etale-ph", "sites-cohomology-lemma-higher-direct-images", "spaces-more-cohomology-lemma-compare-cohomology-other-topologies", "sites-cohomology-lemma-cohomology-of-open", "etale-cohomology-lemma-describe-pullback-pi-ph", "etale-cohomology-lemma-V-C-all-n-etale-ph" ], "ref_ids": [ 3256, 3256, 6672, 4189, 3247, 4186, 6668, 6672 ] } ], "ref_ids": [] }, { "id": 3274, "type": "theorem", "label": "spaces-more-cohomology-lemma-cohomological-descent-etale-ph", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-cohomological-descent-etale-ph", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "For $K \\in D^+(X_\\etale)$ with torsion cohomology sheaves the maps", "$$", "\\pi_X^{-1}K \\longrightarrow R\\epsilon_{X, *}a_X^{-1}K", "\\quad\\text{and}\\quad", "K \\longrightarrow Ra_{X, *}a_X^{-1}K", "$$", "are isomorphisms with", "$a_X : \\Sh((\\textit{Spaces}/X)_{ph}) \\to \\Sh(X_\\etale)$ as above." ], "refs": [], "proofs": [ { "contents": [ "We only prove the second statement; the first is easier and proved in exactly", "the same manner. There is a reduction to the case where", "$K$ is given by a single torsion abelian sheaf. Namely, represent $K$", "by a bounded below complex $\\mathcal{F}^\\bullet$ of torsion", "abelian sheaves. This is possible by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-torsion}.", "By the case of a sheaf we see that", "$\\mathcal{F}^n = a_{X, *} a_X^{-1} \\mathcal{F}^n$", "and that the sheaves $R^qa_{X, *}a_X^{-1}\\mathcal{F}^n$", "are zero for $q > 0$. By Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "applied to $a_X^{-1}\\mathcal{F}^\\bullet$", "and the functor $a_{X, *}$ we conclude. From now on assume", "$K = \\mathcal{F}$ where $\\mathcal{F}$ is a torsion abelian sheaf.", "\\medskip\\noindent", "By Lemma \\ref{lemma-comparison-ph-etale} we have", "$a_{X, *}a_X^{-1}\\mathcal{F} = \\mathcal{F}$. Thus it suffices to show that", "$R^qa_{X, *}a_X^{-1}\\mathcal{F} = 0$ for $q > 0$.", "For this we can use $a_X = \\epsilon_X \\circ \\pi_X$ and", "the Leray spectral sequence", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-relative-Leray}).", "By Lemma \\ref{lemma-comparison-ph-etale}", "we have $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$.", "We have", "$\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$", "and by Lemma \\ref{lemma-cohomological-descent-etale} we have", "$R^j\\pi_{X, *}(\\pi_X^{-1}\\mathcal{F}) = 0$ for $j > 0$.", "This concludes the proof." ], "refs": [ "sites-cohomology-lemma-torsion", "derived-lemma-leray-acyclicity", "spaces-more-cohomology-lemma-comparison-ph-etale", "sites-cohomology-lemma-relative-Leray", "spaces-more-cohomology-lemma-comparison-ph-etale", "spaces-more-cohomology-lemma-cohomological-descent-etale" ], "ref_ids": [ 4252, 1844, 3273, 4222, 3273, 3260 ] } ], "ref_ids": [] }, { "id": 3275, "type": "theorem", "label": "spaces-more-cohomology-lemma-compare-cohomology-etale-ph", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-compare-cohomology-etale-ph", "contents": [ "Let $S$ be a scheme and let $X$ be an algebraic space over $S$.", "With $a_X : \\Sh((\\textit{Spaces}/X)_{ph}) \\to \\Sh(X_\\etale)$", "as above:", "\\begin{enumerate}", "\\item $H^q(X_\\etale, \\mathcal{F}) = H^q_{ph}(X, a_X^{-1}\\mathcal{F})$", "for a torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$,", "\\item $H^q(X_\\etale, K) = H^q_{ph}(X, a_X^{-1}K)$ for $K \\in D^+(X_\\etale)$", "with torsion cohomology sheaves", "\\end{enumerate}", "Example: if $A$ is a torsion abelian group, then", "$H^q_\\etale(X, \\underline{A}) = H^q_{ph}(X, \\underline{A})$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-cohomological-descent-etale-ph}", "by Cohomology on Sites, Remark \\ref{sites-cohomology-remark-before-Leray}." ], "refs": [ "spaces-more-cohomology-lemma-cohomological-descent-etale-ph", "sites-cohomology-remark-before-Leray" ], "ref_ids": [ 3274, 4423 ] } ], "ref_ids": [] }, { "id": 3276, "type": "theorem", "label": "spaces-more-cohomology-lemma-push-pull-ph-etale", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-push-pull-ph-etale", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Then there are commutative diagrams of topoi", "$$", "\\xymatrix{", "\\Sh((\\textit{Spaces}/X)_{ph}) \\ar[rr]_{f_{big, ph}} \\ar[d]_{\\epsilon_X} & &", "\\Sh((\\textit{Spaces}/Y)_{ph}) \\ar[d]^{\\epsilon_Y} \\\\", "\\Sh((\\textit{Spaces}/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & &", "\\Sh((\\textit{Spaces}/Y)_\\etale)", "}", "$$", "and", "$$", "\\xymatrix{", "\\Sh((\\textit{Spaces}/X)_{ph}) \\ar[rr]_{f_{big, ph}} \\ar[d]_{a_X} & &", "\\Sh((\\textit{Spaces}/Y)_{ph}) \\ar[d]^{a_Y} \\\\", "\\Sh(X_\\etale) \\ar[rr]^{f_{small}} & &", "\\Sh(Y_\\etale)", "}", "$$", "with $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from working out the definitions of the", "morphisms involved, see", "Topologies on Spaces, Section \\ref{spaces-topologies-section-ph}", "and Section \\ref{section-compare}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3277, "type": "theorem", "label": "spaces-more-cohomology-lemma-proper-push-pull-ph-etale", "categories": [ "spaces-more-cohomology" ], "title": "spaces-more-cohomology-lemma-proper-push-pull-ph-etale", "contents": [ "In Lemma \\ref{lemma-push-pull-ph-etale} if $f$ is proper, then we have", "\\begin{enumerate}", "\\item $a_Y^{-1} \\circ f_{small, *} = f_{big, ph, *} \\circ a_X^{-1}$, and", "\\item", "$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, ph, *}(a_X^{-1}K)$", "for $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves.", "\\end{enumerate}" ], "refs": [ "spaces-more-cohomology-lemma-push-pull-ph-etale" ], "proofs": [ { "contents": [ "Proof of (1). You can prove this by repeating the proof of", "Lemma \\ref{lemma-compare-higher-direct-image-proper} part (1);", "we will instead deduce the result from this.", "As $\\epsilon_{Y, *}$ is the identity functor on underlying presheaves,", "it reflects isomorphisms. Lemma \\ref{lemma-comparison-ph-etale}", "shows that $\\epsilon_{Y, *} \\circ a_Y^{-1} = \\pi_Y^{-1}$", "and similarly for $X$. To show that the canonical map", "$a_Y^{-1}f_{small, *}\\mathcal{F} \\to f_{big, ph, *}a_X^{-1}\\mathcal{F}$", "is an isomorphism, it suffices to show that", "\\begin{align*}", "\\pi_Y^{-1}f_{small, *}\\mathcal{F}", "& =", "\\epsilon_{Y, *}a_Y^{-1}f_{small, *}\\mathcal{F} \\\\", "& \\to ", "\\epsilon_{Y, *}f_{big, ph, *}a_X^{-1}\\mathcal{F} \\\\", "& =", "f_{big, \\etale, *} \\epsilon_{X, *}a_X^{-1}\\mathcal{F} \\\\", "& =", "f_{big, \\etale, *}\\pi_X^{-1}\\mathcal{F}", "\\end{align*}", "is an isomorphism. This is part", "(1) of Lemma \\ref{lemma-compare-higher-direct-image-proper}.", "\\medskip\\noindent", "To see (2) we use that", "\\begin{align*}", "R\\epsilon_{Y, *}Rf_{big, ph, *}a_X^{-1}K", "& =", "Rf_{big, \\etale, *}R\\epsilon_{X, *}a_X^{-1}K \\\\", "& =", "Rf_{big, \\etale, *}\\pi_X^{-1}K \\\\", "& =", "\\pi_Y^{-1}Rf_{small, *}K \\\\", "& =", "R\\epsilon_{Y, *} a_Y^{-1}Rf_{small, *}K", "\\end{align*}", "The first equality by the commutative diagram in", "Lemma \\ref{lemma-push-pull-ph-etale}", "and Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-derived-pushforward-composition}.", "Then second equality is Lemma \\ref{lemma-cohomological-descent-etale-ph}.", "The third is", "Lemma \\ref{lemma-compare-higher-direct-image-proper} part (2).", "The fourth is Lemma \\ref{lemma-cohomological-descent-etale-ph} again.", "Thus the base change map", "$a_Y^{-1}(Rf_{small, *}K) \\to Rf_{big, ph, *}(a_X^{-1}K)$", "induces an isomorphism", "$$", "R\\epsilon_{Y, *}a_Y^{-1}Rf_{small, *}K \\to", "R\\epsilon_{Y, *}Rf_{big, ph, *}a_X^{-1}K", "$$", "The proof is finished by the following remark: consider a map", "$\\alpha : a_Y^{-1}L \\to M$ with $L$ in $D^+(Y_\\etale)$", "having torsion cohomology sheaves", "and $M$ in $D^+((\\textit{Spaces}/Y)_{ph})$. If", "$R\\epsilon_{Y, *}\\alpha$ is an isomorphism, then $\\alpha$ is an isomorphism.", "Namely, we show by induction on $i$ that $H^i(\\alpha)$ is an isomorphism.", "This is true for all sufficiently small $i$.", "If it holds for $i \\leq i_0$, then we see that", "$R^j\\epsilon_{Y, *}H^i(M) = 0$ for $j > 0$ and $i \\leq i_0$", "by Lemma \\ref{lemma-comparison-ph-etale}", "because $H^i(M) = a_Y^{-1}H^i(L)$ in this range.", "Hence $\\epsilon_{Y, *}H^{i_0 + 1}(M) = H^{i_0 + 1}(R\\epsilon_{Y, *}M)$", "by a spectral sequence argument.", "Thus $\\epsilon_{Y, *}H^{i_0 + 1}(M) = \\pi_Y^{-1}H^{i_0 + 1}(L) =", "\\epsilon_{Y, *}a_Y^{-1}H^{i_0 + 1}(L)$.", "This implies $H^{i_0 + 1}(\\alpha)$ is an isomorphism", "(because $\\epsilon_{Y, *}$ reflects isomorphisms as it is the", "identity on underlying presheaves) as desired." ], "refs": [ "spaces-more-cohomology-lemma-compare-higher-direct-image-proper", "spaces-more-cohomology-lemma-comparison-ph-etale", "spaces-more-cohomology-lemma-compare-higher-direct-image-proper", "spaces-more-cohomology-lemma-push-pull-ph-etale", "sites-cohomology-lemma-derived-pushforward-composition", "spaces-more-cohomology-lemma-cohomological-descent-etale-ph", "spaces-more-cohomology-lemma-compare-higher-direct-image-proper", "spaces-more-cohomology-lemma-cohomological-descent-etale-ph", "spaces-more-cohomology-lemma-comparison-ph-etale" ], "ref_ids": [ 3261, 3273, 3261, 3276, 4250, 3274, 3261, 3274, 3273 ] } ], "ref_ids": [ 3276 ] }, { "id": 3278, "type": "theorem", "label": "coherent-theorem-formal-functions", "categories": [ "coherent" ], "title": "coherent-theorem-formal-functions", "contents": [ "Let $A$ be a Noetherian ring.", "Let $I \\subset A$ be an ideal.", "Let $f : X \\to \\Spec(A)$ be a proper morphism.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Fix $p \\geq 0$.", "The system of maps", "$$", "H^p(X, \\mathcal{F})/I^nH^p(X, \\mathcal{F})", "\\longrightarrow", "H^p(X, \\mathcal{F}/I^n\\mathcal{F})", "$$", "define an isomorphism of limits", "$$", "H^p(X, \\mathcal{F})^\\wedge", "\\longrightarrow", "\\lim_n H^p(X, \\mathcal{F}/I^n\\mathcal{F})", "$$", "where the left hand side is the completion of the $A$-module", "$H^p(X, \\mathcal{F})$ with respect to the ideal $I$, see", "Algebra, Section \\ref{algebra-section-completion}.", "Moreover, this is in fact a homeomorphism for the limit topologies." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-ML-cohomology-powers-ideal} as follows.", "Set $M = H^p(X, \\mathcal{F})$, $M_n = H^p(X, \\mathcal{F}/I^n\\mathcal{F})$,", "and denote $N_n = \\Im(M \\to M_n)$. By", "Lemma \\ref{lemma-ML-cohomology-powers-ideal} parts (2) and (3) we see that", "$(M_n)$ is a Mittag-Leffler system with", "$N_n \\subset M_n$ equal to the image of $M_k$ for all $k \\gg n$.", "It follows that $\\lim M_n = \\lim N_n$ as topological modules (with limit", "topologies). On the other hand, the $N_n$ form an inverse system of quotients", "of the module $M$ and hence $\\lim N_n$ is the completion of $M$", "with respect to the topology given by the kernels $K_n = \\Ker(M \\to N_n)$.", "By Lemma \\ref{lemma-ML-cohomology-powers-ideal} part (1)", "we have $K_n \\subset I^{n - c}M$ and since $N_n \\subset M_n$", "is annihilated by $I^n$ we have $I^n M \\subset K_n$.", "Thus the topology defined using the submodules $K_n$", "as a fundamental system of open neighbourhoods of $0$", "is the same as the $I$-adic topology and we find that the", "induced map $M^\\wedge = \\lim M/I^nM \\to \\lim N_n = \\lim M_n$", "is an isomorphism of topological modules\\footnote{", "To be sure, the limit topology on $M^\\wedge$ is the same as its", "$I$-adic topology as follows from", "Algebra, Lemma \\ref{algebra-lemma-hathat-finitely-generated}.", "See More on Algebra, Section \\ref{more-algebra-section-topological-ring}.}." ], "refs": [ "coherent-lemma-ML-cohomology-powers-ideal", "coherent-lemma-ML-cohomology-powers-ideal", "coherent-lemma-ML-cohomology-powers-ideal", "algebra-lemma-hathat-finitely-generated" ], "ref_ids": [ 3360, 3360, 3360, 859 ] } ], "ref_ids": [] }, { "id": 3279, "type": "theorem", "label": "coherent-theorem-grothendieck-existence", "categories": [ "coherent" ], "title": "coherent-theorem-grothendieck-existence", "contents": [ "\\begin{reference}", "\\cite[III Theorem 5.1.5]{EGA}", "\\end{reference}", "Let $A$ be a Noetherian ring complete with respect to an ideal $I$.", "Let $X$ be a separated, finite type scheme over $A$. Then", "the functor", "(\\ref{equation-completion-functor-proper-over-A})", "$$", "\\textit{Coh}_{\\text{support proper over }A}(\\mathcal{O}_X)", "\\longrightarrow", "\\textit{Coh}_{\\text{support proper over }A}(X, \\mathcal{I})", "$$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "We will use the equivalence of categories of", "Lemma \\ref{lemma-i-star-equivalence}", "without further mention.", "For a closed subscheme $Z \\subset X$ proper over $A$", "in this proof we will say a coherent module on $X$ is", "``supported on $Z$'' if it is annihilated by the ideal", "sheaf of $Z$ or equivalently if it is the pushforward", "of a coherent module on $Z$.", "By Proposition \\ref{proposition-existence-proper} we know", "that the result is true for", "the functor between coherent modules and systems of coherent", "modules supported on $Z$. Hence it suffices to show that", "every object of", "$\\textit{Coh}_{\\text{support proper over }A}(\\mathcal{O}_X)$", "and every object of", "$\\textit{Coh}_{\\text{support proper over }A}(X, \\mathcal{I})$ is", "supported on a closed subscheme $Z \\subset X$ proper over $A$.", "This holds by definition for objects of", "$\\textit{Coh}_{\\text{support proper over }A}(\\mathcal{O}_X)$.", "We will prove this statement for objects of", "$\\textit{Coh}_{\\text{support proper over }A}(X, \\mathcal{I})$", "using the method of proof of Proposition \\ref{proposition-existence-proper}.", "We urge the reader to read that proof first.", "\\medskip\\noindent", "Consider the collection $\\Xi$ of quasi-coherent sheaves of ideals", "$\\mathcal{K} \\subset \\mathcal{O}_X$ such that the statement holds", "for every object $(\\mathcal{F}_n)$ of", "$\\textit{Coh}_{\\text{support proper over }A}(X, \\mathcal{I})$", "annihilated by $\\mathcal{K}$. We want to show $(0)$ is in $\\Xi$.", "If not, then since $X$ is Noetherian there exists a maximal", "quasi-coherent sheaf of ideals $\\mathcal{K}$ not in $\\Xi$, see", "Lemma \\ref{lemma-acc-coherent}.", "After replacing $X$ by the closed subscheme of $X$", "corresponding to $\\mathcal{K}$ we may assume that every nonzero", "$\\mathcal{K}$ is in $\\Xi$. Let $(\\mathcal{F}_n)$ be an object of", "$\\textit{Coh}_{\\text{support proper over }A}(X, \\mathcal{I})$.", "We will show that this object is supported on a closed subscheme", "$Z \\subset X$ proper over $A$, thereby completing the", "proof of the theorem.", "\\medskip\\noindent", "Apply Chow's lemma (Lemma \\ref{lemma-chow-Noetherian}) to find a", "proper surjective morphism $f : Y \\to X$ which is an isomorphism", "over a dense open $U \\subset X$ such that $Y$ is H-quasi-projective", "over $A$. Choose an open immersion $j : Y \\to Y'$ with", "$Y'$ projective over $A$, see", "Morphisms, Lemma \\ref{morphisms-lemma-H-quasi-projective-open-H-projective}.", "Observe that", "$$", "\\text{Supp}(f^*\\mathcal{F}_n) = f^{-1}\\text{Supp}(\\mathcal{F}_n) =", "f^{-1}\\text{Supp}(\\mathcal{F}_1)", "$$", "The first equality by", "Morphisms, Lemma \\ref{morphisms-lemma-support-finite-type}.", "By assumption and", "Lemma \\ref{lemma-functoriality-closed-proper-over-base} part (3)", "we see that $f^{-1}\\text{Supp}(\\mathcal{F}_1)$ is proper over $A$.", "Hence the image of $f^{-1}\\text{Supp}(\\mathcal{F}_1)$", "under $j$ is closed in $Y'$ by", "Lemma \\ref{lemma-functoriality-closed-proper-over-base} part (1).", "Thus $\\mathcal{F}'_n = j_*f^*\\mathcal{F}_n$ is coherent on", "$Y'$ by Lemma \\ref{lemma-pushforward-coherent-on-open}.", "It follows that $(\\mathcal{F}_n')$", "is an object of $\\textit{Coh}(Y', I\\mathcal{O}_{Y'})$.", "By the projective case of Grothendieck's existence theorem", "(Lemma \\ref{lemma-existence-projective})", "there exists a coherent $\\mathcal{O}_{Y'}$-module", "$\\mathcal{F}'$ and an isomorphism", "$(\\mathcal{F}')^\\wedge \\cong (\\mathcal{F}'_n)$ in", "$\\textit{Coh}(Y', I\\mathcal{O}_{Y'})$.", "Since $\\mathcal{F}'/I\\mathcal{F}' = \\mathcal{F}'_1$ we see that", "$$", "\\text{Supp}(\\mathcal{F}') \\cap V(I\\mathcal{O}_{Y'}) =", "\\text{Supp}(\\mathcal{F}'_1) = j(f^{-1}\\text{Supp}(\\mathcal{F}_1))", "$$", "The structure morphism $p' : Y' \\to \\Spec(A)$ is proper, hence", "$p'(\\text{Supp}(\\mathcal{F}') \\setminus j(Y))$", "is closed in $\\Spec(A)$. A nonempty closed subset of $\\Spec(A)$", "contains a point of $V(I)$ as $I$ is contained in the Jacobson radical", "of $A$ by Algebra, Lemma \\ref{algebra-lemma-radical-completion}.", "The displayed equation shows that", "$\\text{Supp}(\\mathcal{F}') \\cap (p')^{-1}V(I) \\subset j(Y)$", "hence we conclude that $\\text{Supp}(\\mathcal{F}') \\subset j(Y)$.", "Thus $\\mathcal{F}'|_Y = j^*\\mathcal{F}'$", "is supported on a closed subscheme $Z'$ of $Y$ proper over $A$", "and $(\\mathcal{F}'|_Y)^\\wedge = (f^*\\mathcal{F}_n)$.", "\\medskip\\noindent", "Let $\\mathcal{K}$ be the quasi-coherent sheaf of ideals cutting", "out the reduced complement $X \\setminus U$. By", "Proposition \\ref{proposition-proper-pushforward-coherent}", "the $\\mathcal{O}_X$-module $\\mathcal{H} = f_*(\\mathcal{F}'|_Y)$ is coherent", "and by Lemma \\ref{lemma-inverse-systems-push-pull}", "there exists a morphism $\\alpha : (\\mathcal{F}_n) \\to \\mathcal{H}^\\wedge$", "of $\\textit{Coh}(X, \\mathcal{I})$ whose kernel and cokernel are", "annihilated by a power $\\mathcal{K}^t$ of $\\mathcal{K}$.", "We obtain an exact sequence", "$$", "0 \\to \\Ker(\\alpha) \\to (\\mathcal{F}_n) \\to", "\\mathcal{H}^\\wedge \\to \\Coker(\\alpha) \\to 0", "$$", "in $\\textit{Coh}(X, \\mathcal{I})$. If $Z_0 \\subset X$ is the scheme theoretic", "support of $\\mathcal{H}$, then it is clear that $Z_0 \\subset f(Z')$", "set-theoretically. Hence $Z_0$ is proper over $A$ by", "Lemma \\ref{lemma-closed-closed-proper-over-base} and", "Lemma \\ref{lemma-functoriality-closed-proper-over-base} part (2).", "Hence $\\mathcal{H}^\\wedge$ is in the subcategory defined in", "Lemma \\ref{lemma-systems-with-proper-support} part (2)", "and a fortiori in", "$\\textit{Coh}_{\\text{support proper over }A}(X, \\mathcal{I})$.", "We conclude that $\\Ker(\\alpha)$ and $\\Coker(\\alpha)$", "are in $\\textit{Coh}_{\\text{support proper over }A}(X, \\mathcal{I})$", "by Lemma \\ref{lemma-systems-with-proper-support} part (1).", "By induction hypothesis, more precisely because $\\mathcal{K}^t$ is in $\\Xi$,", "we see that $\\Ker(\\alpha)$ and $\\Coker(\\alpha)$ are in", "the subcategory defined in", "Lemma \\ref{lemma-systems-with-proper-support} part (2).", "Since this is a Serre subcategory by the lemma, we conclude that the", "same is true for $(\\mathcal{F}_n)$ which is what we wanted to show." ], "refs": [ "coherent-lemma-i-star-equivalence", "coherent-proposition-existence-proper", "coherent-proposition-existence-proper", "coherent-lemma-acc-coherent", "coherent-lemma-chow-Noetherian", "morphisms-lemma-H-quasi-projective-open-H-projective", "morphisms-lemma-support-finite-type", "coherent-lemma-functoriality-closed-proper-over-base", "coherent-lemma-functoriality-closed-proper-over-base", "coherent-lemma-pushforward-coherent-on-open", "coherent-lemma-existence-projective", "algebra-lemma-radical-completion", "coherent-proposition-proper-pushforward-coherent", "coherent-lemma-inverse-systems-push-pull", "coherent-lemma-closed-closed-proper-over-base", "coherent-lemma-functoriality-closed-proper-over-base", "coherent-lemma-systems-with-proper-support", "coherent-lemma-systems-with-proper-support", "coherent-lemma-systems-with-proper-support" ], "ref_ids": [ 3315, 3402, 3402, 3319, 3354, 5428, 5143, 3389, 3389, 3318, 3383, 862, 3401, 3385, 3387, 3389, 3395, 3395, 3395 ] } ], "ref_ids": [] }, { "id": 3280, "type": "theorem", "label": "coherent-theorem-algebraization", "categories": [ "coherent" ], "title": "coherent-theorem-algebraization", "contents": [ "Let $A$ be a Noetherian ring complete with respect to an ideal $I$.", "Set $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Consider a commutative", "diagram", "$$", "\\xymatrix{", "X_1 \\ar[r]_{i_1} \\ar[d] & X_2 \\ar[r]_{i_2} \\ar[d] & X_3 \\ar[r] \\ar[d] &", "\\ldots \\\\", "S_1 \\ar[r] & S_2 \\ar[r] & S_3 \\ar[r] & \\ldots", "}", "$$", "of schemes with cartesian squares. Suppose given $(\\mathcal{L}_n, \\varphi_n)$", "where each $\\mathcal{L}_n$ is an invertible sheaf on $X_n$ and", "$\\varphi_n : i_n^*\\mathcal{L}_{n + 1} \\to \\mathcal{L}_n$", "is an isomorphism. If", "\\begin{enumerate}", "\\item $X_1 \\to S_1$ is proper, and", "\\item $\\mathcal{L}_1$ is ample on $X_1$", "\\end{enumerate}", "then there exists a proper morphism of schemes $X \\to S$", "and an ample invertible $\\mathcal{O}_X$-module $\\mathcal{L}$", "and isomorphisms $X_n \\cong X \\times_S S_n$ and", "$\\mathcal{L}_n \\cong \\mathcal{L}|_{X_n}$ compatible with", "the morphisms $i_n$ and $\\varphi_n$." ], "refs": [], "proofs": [ { "contents": [ "Since the squares in the diagram are cartesian and since the morphisms", "$S_n \\to S_{n + 1}$ are closed immersions, we see that the morphisms", "$i_n$ are closed immersions too. In particular we may think of", "$X_m$ as a closed subscheme of $X_n$ for $m < n$. In fact $X_m$ is", "the closed subscheme cut out by the quasi-coherent sheaf of ideals", "$I^m\\mathcal{O}_{X_n}$. Moreover, the underlying topological spaces", "of the schemes $X_1, X_2, X_3, \\ldots$ are all identified, hence we", "may (and do) think of sheaves $\\mathcal{O}_{X_n}$ as living on the", "same underlying topological space; similarly for coherent", "$\\mathcal{O}_{X_n}$-modules. Set", "$$", "\\mathcal{F}_n =", "\\Ker(\\mathcal{O}_{X_{n + 1}} \\to \\mathcal{O}_{X_n})", "$$", "so that we obtain short exact sequences", "$$", "0 \\to \\mathcal{F}_n \\to \\mathcal{O}_{X_{n + 1}} \\to \\mathcal{O}_{X_n} \\to 0", "$$", "By the above we have $\\mathcal{F}_n = I^n\\mathcal{O}_{X_{n + 1}}$.", "It follows $\\mathcal{F}_n$ is a coherent sheaf on $X_{n + 1}$", "annihilated by $I$, hence we may (and do) think of it as a coherent", "module $\\mathcal{O}_{X_1}$-module. Observe that for $m > n$ the sheaf", "$$", "I^n\\mathcal{O}_{X_m}/I^{n + 1}\\mathcal{O}_{X_m}", "$$", "maps isomorphically to $\\mathcal{F}_n$ under the map", "$\\mathcal{O}_{X_m} \\to \\mathcal{O}_{X_{n + 1}}$. Hence given", "$n_1, n_2 \\geq 0$ we can pick an $m > n_1 + n_2$ and consider the", "multiplication map", "$$", "I^{n_1}\\mathcal{O}_{X_m} \\times I^{n_2}\\mathcal{O}_{X_m}", "\\longrightarrow", "I^{n_1 + n_2}\\mathcal{O}_{X_m} \\to \\mathcal{F}_{n_1 + n_2}", "$$", "This induces an $\\mathcal{O}_{X_1}$-bilinear map", "$$", "\\mathcal{F}_{n_1} \\times \\mathcal{F}_{n_2} \\longrightarrow", "\\mathcal{F}_{n_1 + n_2}", "$$", "which in turn defines the structure of a graded $\\mathcal{O}_{X_1}$-algebra", "on $\\mathcal{F} = \\bigoplus_{n \\geq 0} \\mathcal{F}_n$.", "\\medskip\\noindent", "Set $B = \\bigoplus I^n/I^{n + 1}$; this is a finitely generated", "graded $A/I$-algebra. Set $\\mathcal{B} = (X_1 \\to S_1)^*\\widetilde{B}$.", "The discussion above provides us with a canonical surjection", "$$", "\\mathcal{B} \\longrightarrow \\mathcal{F}", "$$", "of graded $\\mathcal{O}_{X_1}$-algebras. In particular we see that", "$\\mathcal{F}$ is a finite type quasi-coherent graded $\\mathcal{B}$-module.", "By Lemma \\ref{lemma-graded-finiteness} we can find an integer $d_0$", "such that $H^1(X_1, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes d}) = 0$", "for all $d \\geq d_0$. Pick a $d \\geq d_0$ such that there exist sections", "$s_{0, 1}, \\ldots, s_{N, 1} \\in \\Gamma(X_1, \\mathcal{L}_1^{\\otimes d})$", "which induce an immersion", "$$", "\\psi_1 : X_1 \\to \\mathbf{P}^N_{S_1}", "$$", "over $S_1$, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-over-affine-ample-very-ample}.", "As $X_1$ is proper over $S_1$ we see that $\\psi_1$", "is a closed immersion, see", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}", "and", "Schemes, Lemma \\ref{schemes-lemma-immersion-when-closed}.", "We are going to ``lift'' $\\psi_1$ to a compatible system of", "closed immersions of $X_n$ into $\\mathbf{P}^N$.", "\\medskip\\noindent", "Upon tensoring the short exact sequences of the first paragraph", "of the proof", "by $\\mathcal{L}_{n + 1}^{\\otimes d}$ we obtain short exact sequences", "$$", "0 \\to \\mathcal{F}_n \\otimes \\mathcal{L}_{n + 1}^{\\otimes d} \\to", "\\mathcal{L}_{n + 1}^{\\otimes d} \\to \\mathcal{L}_{n + 1}^{\\otimes d} \\to 0", "$$", "Using the isomorphisms $\\varphi_n$ we obtain isomorphisms", "$\\mathcal{L}_{n + 1} \\otimes \\mathcal{O}_{X_l} = \\mathcal{L}_l$", "for $l \\leq n$. Whence the sequence above becomes", "$$", "0 \\to \\mathcal{F}_n \\otimes \\mathcal{L}_1^{\\otimes d} \\to", "\\mathcal{L}_{n + 1}^{\\otimes d} \\to \\mathcal{L}_n^{\\otimes d} \\to 0", "$$", "The vanishing of $H^1(X, \\mathcal{F}_n \\otimes \\mathcal{L}_1^{\\otimes d})$", "implies we can inductively lift", "$s_{0, 1}, \\ldots, s_{N, 1} \\in \\Gamma(X_1, \\mathcal{L}_1^{\\otimes d})$", "to sections", "$s_{0, n}, \\ldots, s_{N, n} \\in \\Gamma(X_n, \\mathcal{L}_n^{\\otimes d})$.", "Thus we obtain a commutative diagram", "$$", "\\xymatrix{", "X_1 \\ar[r]_{i_1} \\ar[d]_{\\psi_1} &", "X_2 \\ar[r]_{i_2} \\ar[d]_{\\psi_2} &", "X_3 \\ar[r] \\ar[d]_{\\psi_3} &", "\\ldots \\\\", "\\mathbf{P}^N_{S_1} \\ar[r] &", "\\mathbf{P}^N_{S_2} \\ar[r] &", "\\mathbf{P}^N_{S_3} \\ar[r] & \\ldots", "}", "$$", "where", "$\\psi_n = \\varphi_{(\\mathcal{L}_n, (s_{0, n}, \\ldots, s_{N, n}))}$", "in the notation of Constructions, Section", "\\ref{constructions-section-projective-space}.", "As the squares in the statement of the theorem are cartesian", "we see that the squares in the above diagram are cartesian.", "We win by applying Lemma \\ref{lemma-algebraize-formal-closed-subscheme}." ], "refs": [ "coherent-lemma-graded-finiteness", "morphisms-lemma-finite-type-over-affine-ample-very-ample", "morphisms-lemma-image-proper-scheme-closed", "schemes-lemma-immersion-when-closed", "coherent-lemma-algebraize-formal-closed-subscheme" ], "ref_ids": [ 3356, 5394, 5411, 7671, 3396 ] } ], "ref_ids": [] }, { "id": 3281, "type": "theorem", "label": "coherent-lemma-cech-cohomology-quasi-coherent-trivial", "categories": [ "coherent" ], "title": "coherent-lemma-cech-cohomology-quasi-coherent-trivial", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\mathcal{U} : U = \\bigcup_{i = 1}^n D(f_i)$ be a standard", "open covering of an affine open of $X$.", "Then $\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0$ for", "all $p > 0$." ], "refs": [], "proofs": [ { "contents": [ "Write $U = \\Spec(A)$ for some ring $A$.", "In other words, $f_1, \\ldots, f_n$ are elements of $A$", "which generate the unit ideal of $A$.", "Write $\\mathcal{F}|_U = \\widetilde{M}$ for some $A$-module $M$.", "Clearly the {\\v C}ech complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "is identified with the complex", "$$", "\\prod\\nolimits_{i_0} M_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0i_1} M_{f_{i_0}f_{i_1}} \\to", "\\prod\\nolimits_{i_0i_1i_2} M_{f_{i_0}f_{i_1}f_{i_2}} \\to", "\\ldots", "$$", "We are asked to show that the extended complex", "\\begin{equation}", "\\label{equation-extended}", "0 \\to", "M \\to", "\\prod\\nolimits_{i_0} M_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0i_1} M_{f_{i_0}f_{i_1}} \\to", "\\prod\\nolimits_{i_0i_1i_2} M_{f_{i_0}f_{i_1}f_{i_2}} \\to", "\\ldots", "\\end{equation}", "(whose truncation we have studied in", "Algebra, Lemma \\ref{algebra-lemma-cover-module}) is exact.", "It suffices to show that (\\ref{equation-extended})", "is exact after localizing at a prime $\\mathfrak p$, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}.", "In fact we will show that the extended complex localized", "at $\\mathfrak p$ is homotopic to zero.", "\\medskip\\noindent", "There exists an index $i$ such that $f_i \\not \\in \\mathfrak p$.", "Choose and fix such an element $i_{\\text{fix}}$. Note that", "$M_{f_{i_{\\text{fix}}}, \\mathfrak p} = M_{\\mathfrak p}$. Similarly", "for a localization at a product $f_{i_0} \\ldots f_{i_p}$ and $\\mathfrak p$", "we can drop any $f_{i_j}$ for which $i_j = i_{\\text{fix}}$.", "Let us define a homotopy", "$$", "h :", "\\prod\\nolimits_{i_0 \\ldots i_{p + 1}}", "M_{f_{i_0} \\ldots f_{i_{p + 1}}, \\mathfrak p}", "\\longrightarrow", "\\prod\\nolimits_{i_0 \\ldots i_p}", "M_{f_{i_0} \\ldots f_{i_p}, \\mathfrak p}", "$$", "by the rule", "$$", "h(s)_{i_0 \\ldots i_p} = s_{i_{\\text{fix}} i_0 \\ldots i_p}", "$$", "(This is ``dual'' to the homotopy in the proof of", "Cohomology, Lemma \\ref{cohomology-lemma-homology-complex}.)", "In other words, $h : \\prod_{i_0} M_{f_{i_0}, \\mathfrak p} \\to M_\\mathfrak p$", "is projection onto the factor", "$M_{f_{i_{\\text{fix}}}, \\mathfrak p} = M_{\\mathfrak p}$ and in general", "the map $h$ equal projection onto the factors", "$M_{f_{i_{\\text{fix}}} f_{i_1} \\ldots f_{i_{p + 1}}, \\mathfrak p}", "= M_{f_{i_1} \\ldots f_{i_{p + 1}}, \\mathfrak p}$. We compute", "\\begin{align*}", "(dh + hd)(s)_{i_0 \\ldots i_p}", "& =", "\\sum\\nolimits_{j = 0}^p", "(-1)^j", "h(s)_{i_0 \\ldots \\hat i_j \\ldots i_p}", "+", "d(s)_{i_{\\text{fix}} i_0 \\ldots i_p}\\\\", "& =", "\\sum\\nolimits_{j = 0}^p", "(-1)^j", "s_{i_{\\text{fix}} i_0 \\ldots \\hat i_j \\ldots i_p}", "+", "s_{i_0 \\ldots i_p}", "+", "\\sum\\nolimits_{j = 0}^p", "(-1)^{j + 1}", "s_{i_{\\text{fix}} i_0 \\ldots \\hat i_j \\ldots i_p} \\\\", "& =", "s_{i_0 \\ldots i_p}", "\\end{align*}", "This proves the identity map is homotopic to zero as desired." ], "refs": [ "algebra-lemma-cover-module", "algebra-lemma-characterize-zero-local", "cohomology-lemma-homology-complex" ], "ref_ids": [ 413, 410, 2049 ] } ], "ref_ids": [] }, { "id": 3282, "type": "theorem", "label": "coherent-lemma-quasi-coherent-affine-cohomology-zero", "categories": [ "coherent" ], "title": "coherent-lemma-quasi-coherent-affine-cohomology-zero", "contents": [ "\\begin{slogan}", "Serre vanishing: Higher cohomology vanishes on affine schemes", "for quasi-coherent modules.", "\\end{slogan}", "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "For any affine open $U \\subset X$ we have", "$H^p(U, \\mathcal{F}) = 0$ for all $p > 0$." ], "refs": [], "proofs": [ { "contents": [ "We are going to apply", "Cohomology, Lemma \\ref{cohomology-lemma-cech-vanish-basis}.", "As our basis $\\mathcal{B}$ for the topology of $X$ we are going to use", "the affine opens of $X$.", "As our set $\\text{Cov}$ of open coverings we are going to use the standard", "open coverings of affine opens of $X$.", "Next we check that conditions (1), (2) and (3) of", "Cohomology, Lemma \\ref{cohomology-lemma-cech-vanish-basis}", "hold. Note that the intersection of standard opens in an affine is", "another standard open. Hence property (1) holds.", "The coverings form a cofinal system of open coverings of any element", "of $\\mathcal{B}$, see", "Schemes, Lemma \\ref{schemes-lemma-standard-open}.", "Hence (2) holds.", "Finally, condition (3) of the lemma follows from", "Lemma \\ref{lemma-cech-cohomology-quasi-coherent-trivial}." ], "refs": [ "cohomology-lemma-cech-vanish-basis", "cohomology-lemma-cech-vanish-basis", "schemes-lemma-standard-open", "coherent-lemma-cech-cohomology-quasi-coherent-trivial" ], "ref_ids": [ 2059, 2059, 7650, 3281 ] } ], "ref_ids": [] }, { "id": 3283, "type": "theorem", "label": "coherent-lemma-relative-affine-vanishing", "categories": [ "coherent" ], "title": "coherent-lemma-relative-affine-vanishing", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "If $f$ is affine then $R^if_*\\mathcal{F} = 0$ for all $i > 0$." ], "refs": [], "proofs": [ { "contents": [ "According to", "Cohomology, Lemma \\ref{cohomology-lemma-describe-higher-direct-images}", "the sheaf", "$R^if_*\\mathcal{F}$ is the sheaf associated to the presheaf", "$V \\mapsto H^i(f^{-1}(V), \\mathcal{F}|_{f^{-1}(V)})$.", "By assumption, whenever $V$ is affine we have that $f^{-1}(V)$ is", "affine, see Morphisms, Definition \\ref{morphisms-definition-affine}.", "By Lemma \\ref{lemma-quasi-coherent-affine-cohomology-zero} we conclude that", "$H^i(f^{-1}(V), \\mathcal{F}|_{f^{-1}(V)}) = 0$", "whenever $V$ is affine. Since $S$ has a basis consisting of affine", "opens we win." ], "refs": [ "cohomology-lemma-describe-higher-direct-images", "morphisms-definition-affine", "coherent-lemma-quasi-coherent-affine-cohomology-zero" ], "ref_ids": [ 2039, 5544, 3282 ] } ], "ref_ids": [] }, { "id": 3284, "type": "theorem", "label": "coherent-lemma-relative-affine-cohomology", "categories": [ "coherent" ], "title": "coherent-lemma-relative-affine-cohomology", "contents": [ "Let $f : X \\to S$ be an affine morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then $H^i(X, \\mathcal{F}) = H^i(S, f_*\\mathcal{F})$ for all $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-relative-affine-vanishing}", "and the Leray spectral sequence. See", "Cohomology, Lemma \\ref{cohomology-lemma-apply-Leray}." ], "refs": [ "coherent-lemma-relative-affine-vanishing", "cohomology-lemma-apply-Leray" ], "ref_ids": [ 3283, 2071 ] } ], "ref_ids": [] }, { "id": 3285, "type": "theorem", "label": "coherent-lemma-affine-diagonal", "categories": [ "coherent" ], "title": "coherent-lemma-affine-diagonal", "contents": [ "Let $X$ be a scheme. The following are equivalent", "\\begin{enumerate}", "\\item $X$ has affine diagonal $\\Delta : X \\to X \\times X$,", "\\item for $U, V \\subset X$ affine open, the intersection", "$U \\cap V$ is affine, and", "\\item there exists an open covering $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$", "such that $U_{i_0 \\ldots i_p}$ is affine open for all $p \\ge 0$ and all", "$i_0, \\ldots, i_p \\in I$.", "\\end{enumerate}", "In particular this holds if $X$ is separated." ], "refs": [], "proofs": [ { "contents": [ "Assume $X$ has affine diagonal. Let $U, V \\subset X$ be affine opens.", "Then $U \\cap V = \\Delta^{-1}(U \\times V)$ is affine. Thus (2) holds.", "It is immediate that (2) implies (3). Conversely, if there is a", "covering of $X$ as in (3), then $X \\times X = \\bigcup U_i \\times U_{i'}$", "is an affine open covering, and we see that", "$\\Delta^{-1}(U_i \\times U_{i'}) = U_i \\cap U_{i'}$", "is affine. Then $\\Delta$ is an affine morphism by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-affine}.", "The final assertion follows from Schemes, Lemma", "\\ref{schemes-lemma-characterize-separated}." ], "refs": [ "morphisms-lemma-characterize-affine", "schemes-lemma-characterize-separated" ], "ref_ids": [ 5172, 7710 ] } ], "ref_ids": [] }, { "id": 3286, "type": "theorem", "label": "coherent-lemma-cech-cohomology-quasi-coherent", "categories": [ "coherent" ], "title": "coherent-lemma-cech-cohomology-quasi-coherent", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be an open covering such that", "$U_{i_0 \\ldots i_p}$ is affine open for all $p \\ge 0$ and all", "$i_0, \\ldots, i_p \\in I$.", "In this case for any quasi-coherent sheaf $\\mathcal{F}$ we have", "$$", "\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = H^p(X, \\mathcal{F})", "$$", "as $\\Gamma(X, \\mathcal{O}_X)$-modules for all $p$." ], "refs": [], "proofs": [ { "contents": [ "In view of", "Lemma \\ref{lemma-quasi-coherent-affine-cohomology-zero}", "this is a special case of", "Cohomology, Lemma", "\\ref{cohomology-lemma-cech-spectral-sequence-application}." ], "refs": [ "coherent-lemma-quasi-coherent-affine-cohomology-zero", "cohomology-lemma-cech-spectral-sequence-application" ], "ref_ids": [ 3282, 2056 ] } ], "ref_ids": [] }, { "id": 3287, "type": "theorem", "label": "coherent-lemma-quasi-compact-h1-zero-covering", "categories": [ "coherent" ], "title": "coherent-lemma-quasi-compact-h1-zero-covering", "contents": [ "\\begin{reference}", "\\cite{Serre-criterion}, \\cite[II, Theorem 5.2.1 (d') and IV (1.7.17)]{EGA}", "\\end{reference}", "\\begin{slogan}", "Serre's criterion for affineness.", "\\end{slogan}", "Let $X$ be a scheme.", "Assume that", "\\begin{enumerate}", "\\item $X$ is quasi-compact,", "\\item for every quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_X$ we have $H^1(X, \\mathcal{I}) = 0$.", "\\end{enumerate}", "Then $X$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$ be a closed point. Let $U \\subset X$ be an affine open", "neighbourhood of $x$. Write $U = \\Spec(A)$ and let", "$\\mathfrak m \\subset A$ be the maximal ideal corresponding to $x$.", "Set $Z = X \\setminus U$ and $Z' = Z \\cup \\{x\\}$.", "By Schemes, Lemma \\ref{schemes-lemma-reduced-closed-subscheme} there", "are quasi-coherent sheaves of ideals", "$\\mathcal{I}$, resp.\\ $\\mathcal{I}'$ cutting out", "the reduced closed subschemes $Z$, resp.\\ $Z'$.", "Consider the short exact sequence", "$$", "0 \\to \\mathcal{I}' \\to \\mathcal{I} \\to \\mathcal{I}/\\mathcal{I}' \\to 0.", "$$", "Since $x$ is a closed point of $X$ and $x \\not \\in Z$ we see that", "$\\mathcal{I}/\\mathcal{I}'$ is supported at $x$. In fact, the restriction", "of $\\mathcal{I}/\\mathcal{I'}$ to $U$ corresponds to the $A$-module", "$A/\\mathfrak m$. Hence we see that $\\Gamma(X, \\mathcal{I}/\\mathcal{I'})", "= A/\\mathfrak m$. Since by assumption $H^1(X, \\mathcal{I}') = 0$", "we see there exists a global section $f \\in \\Gamma(X, \\mathcal{I})$", "which maps to the element $1 \\in A/\\mathfrak m$ as a section of", "$\\mathcal{I}/\\mathcal{I'}$. Clearly we have", "$x \\in X_f \\subset U$. This implies that $X_f = D(f_A)$ where", "$f_A$ is the image of $f$ in $A = \\Gamma(U, \\mathcal{O}_X)$.", "In particular $X_f$ is affine.", "\\medskip\\noindent", "Consider the union $W = \\bigcup X_f$ over all $f \\in \\Gamma(X, \\mathcal{O}_X)$", "such that $X_f$ is affine. Obviously $W$ is open in $X$.", "By the arguments above every closed point of", "$X$ is contained in $W$. The closed subset $X \\setminus W$ of $X$", "is also quasi-compact", "(see Topology, Lemma \\ref{topology-lemma-closed-in-quasi-compact}).", "Hence it has a closed point if it is nonempty (see", "Topology, Lemma \\ref{topology-lemma-quasi-compact-closed-point}).", "This would contradict the fact that all closed points are in", "$W$. Hence we conclude $X = W$.", "\\medskip\\noindent", "Choose finitely many $f_1, \\ldots, f_n \\in \\Gamma(X, \\mathcal{O}_X)$", "such that $X = X_{f_1} \\cup \\ldots \\cup X_{f_n}$ and such that each", "$X_{f_i}$ is affine. This is possible as we've seen above.", "By Properties, Lemma \\ref{properties-lemma-characterize-affine}", "to finish the proof it suffices", "to show that $f_1, \\ldots, f_n$ generate the unit ideal in", "$\\Gamma(X, \\mathcal{O}_X)$. Consider the short exact sequence", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{F} \\ar[r] &", "\\mathcal{O}_X^{\\oplus n} \\ar[rr]^{f_1, \\ldots, f_n} & &", "\\mathcal{O}_X \\ar[r] &", "0", "}", "$$", "The arrow defined by $f_1, \\ldots, f_n$ is surjective since the", "opens $X_{f_i}$ cover $X$. We let $\\mathcal{F}$ be the kernel", "of this surjective map.", "Observe that $\\mathcal{F}$ has a filtration", "$$", "0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset", "\\ldots \\subset \\mathcal{F}_n = \\mathcal{F}", "$$", "so that each subquotient $\\mathcal{F}_i/\\mathcal{F}_{i - 1}$ is", "isomorphic to a quasi-coherent sheaf of ideals.", "Namely we can take $\\mathcal{F}_i$ to be the intersection of", "$\\mathcal{F}$ with the first $i$ direct summands of", "$\\mathcal{O}_X^{\\oplus n}$.", "The assumption", "of the lemma implies that $H^1(X, \\mathcal{F}_i/\\mathcal{F}_{i - 1}) = 0$", "for all $i$. This implies that", "$H^1(X, \\mathcal{F}_2) = 0$ because it is sandwiched between", "$H^1(X, \\mathcal{F}_1)$ and $H^1(X, \\mathcal{F}_2/\\mathcal{F}_1)$.", "Continuing like this we deduce that $H^1(X, \\mathcal{F}) = 0$.", "Therefore we conclude that the map", "$$", "\\xymatrix{", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} \\Gamma(X, \\mathcal{O}_X)", "\\ar[rr]^{f_1, \\ldots, f_n} & &", "\\Gamma(X, \\mathcal{O}_X)", "}", "$$", "is surjective as desired." ], "refs": [ "schemes-lemma-reduced-closed-subscheme", "topology-lemma-closed-in-quasi-compact", "topology-lemma-quasi-compact-closed-point", "properties-lemma-characterize-affine" ], "ref_ids": [ 7681, 8229, 8234, 3055 ] } ], "ref_ids": [] }, { "id": 3288, "type": "theorem", "label": "coherent-lemma-quasi-separated-h1-zero-covering", "categories": [ "coherent" ], "title": "coherent-lemma-quasi-separated-h1-zero-covering", "contents": [ "\\begin{reference}", "\\cite{Serre-criterion}, \\cite[II, Theorem 5.2.1]{EGA}", "\\end{reference}", "\\begin{slogan}", "Serre's criterion for affineness.", "\\end{slogan}", "Let $X$ be a scheme. Assume that", "\\begin{enumerate}", "\\item $X$ is quasi-compact,", "\\item $X$ is quasi-separated, and", "\\item $H^1(X, \\mathcal{I}) = 0$ for every quasi-coherent sheaf", "of ideals $\\mathcal{I}$ of finite type.", "\\end{enumerate}", "Then $X$ is affine." ], "refs": [], "proofs": [ { "contents": [ "By", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-colimit-finite-type}", "every quasi-coherent sheaf of ideals is a directed colimit of", "quasi-coherent sheaves of ideals of finite type.", "By Cohomology, Lemma \\ref{cohomology-lemma-quasi-separated-cohomology-colimit}", "taking cohomology on $X$ commutes with directed colimits.", "Hence we see that $H^1(X, \\mathcal{I}) = 0$", "for every quasi-coherent sheaf of ideals on $X$. In other words", "we see that Lemma \\ref{lemma-quasi-compact-h1-zero-covering} applies." ], "refs": [ "properties-lemma-quasi-coherent-colimit-finite-type", "cohomology-lemma-quasi-separated-cohomology-colimit", "coherent-lemma-quasi-compact-h1-zero-covering" ], "ref_ids": [ 3020, 2082, 3287 ] } ], "ref_ids": [] }, { "id": 3289, "type": "theorem", "label": "coherent-lemma-quasi-compact-h1-zero-invertible", "categories": [ "coherent" ], "title": "coherent-lemma-quasi-compact-h1-zero-invertible", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Assume that", "\\begin{enumerate}", "\\item $X$ is quasi-compact,", "\\item for every quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_X$", "there exists an $n \\geq 1$ such that", "$H^1(X, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) = 0$.", "\\end{enumerate}", "Then $\\mathcal{L}$ is ample." ], "refs": [], "proofs": [ { "contents": [ "This is proved in exactly the same way as", "Lemma \\ref{lemma-quasi-compact-h1-zero-covering}.", "Let $x \\in X$ be a closed point. Let $U \\subset X$ be an affine open", "neighbourhood of $x$ such that $\\mathcal{L}|_U \\cong \\mathcal{O}_U$.", "Write $U = \\Spec(A)$ and let", "$\\mathfrak m \\subset A$ be the maximal ideal corresponding to $x$.", "Set $Z = X \\setminus U$ and $Z' = Z \\cup \\{x\\}$.", "By Schemes, Lemma \\ref{schemes-lemma-reduced-closed-subscheme} there", "are quasi-coherent sheaves of ideals", "$\\mathcal{I}$, resp.\\ $\\mathcal{I}'$ cutting out", "the reduced closed subschemes $Z$, resp.\\ $Z'$.", "Consider the short exact sequence", "$$", "0 \\to \\mathcal{I}' \\to \\mathcal{I} \\to \\mathcal{I}/\\mathcal{I}' \\to 0.", "$$", "For every $n \\geq 1$ we obtain a short exact sequence", "$$", "0 \\to \\mathcal{I}' \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}", "\\to \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n} \\to", "\\mathcal{I}/\\mathcal{I}' \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n} \\to 0.", "$$", "By our assumption we may pick $n$ such that", "$H^1(X, \\mathcal{I}' \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) = 0$.", "Since $x$ is a closed point of $X$ and $x \\not \\in Z$ we see that", "$\\mathcal{I}/\\mathcal{I}'$ is supported at $x$. In fact, the restriction", "of $\\mathcal{I}/\\mathcal{I'}$ to $U$ corresponds to the $A$-module", "$A/\\mathfrak m$. Since $\\mathcal{L}$ is trivial on $U$", "we see that the restriction of", "$\\mathcal{I}/\\mathcal{I}' \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}$", "to $U$ also corresponds to the $A$-module $A/\\mathfrak m$.", "Hence we see that", "$\\Gamma(X, \\mathcal{I}/\\mathcal{I'} \\otimes_{\\mathcal{O}_X}", "\\mathcal{L}^{\\otimes n}) = A/\\mathfrak m$.", "By our choice of $n$ we see there exists a global section", "$s \\in \\Gamma(X, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n})$", "which maps to the element $1 \\in A/\\mathfrak m$. Clearly we have", "$x \\in X_s \\subset U$ because $s$ vanishes at points of $Z$.", "This implies that $X_s = D(f)$ where", "$f \\in A$ is the image of $s$ in $A \\cong \\Gamma(U, \\mathcal{L}^{\\otimes n})$.", "In particular $X_s$ is affine.", "\\medskip\\noindent", "Consider the union $W = \\bigcup X_s$ over all", "$s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ for $n \\geq 1$", "such that $X_s$ is affine. Obviously $W$ is open in $X$.", "By the arguments above every closed point of", "$X$ is contained in $W$. The closed subset $X \\setminus W$ of $X$", "is also quasi-compact", "(see Topology, Lemma \\ref{topology-lemma-closed-in-quasi-compact}).", "Hence it has a closed point if it is nonempty (see", "Topology, Lemma \\ref{topology-lemma-quasi-compact-closed-point}).", "This would contradict the fact that all closed points are in", "$W$. Hence we conclude $X = W$. This means that $\\mathcal{L}$", "is ample by Properties, Definition \\ref{properties-definition-ample}." ], "refs": [ "coherent-lemma-quasi-compact-h1-zero-covering", "schemes-lemma-reduced-closed-subscheme", "topology-lemma-closed-in-quasi-compact", "topology-lemma-quasi-compact-closed-point", "properties-definition-ample" ], "ref_ids": [ 3287, 7681, 8229, 8234, 3088 ] } ], "ref_ids": [] }, { "id": 3290, "type": "theorem", "label": "coherent-lemma-criterion-affine-morphism", "categories": [ "coherent" ], "title": "coherent-lemma-criterion-affine-morphism", "contents": [ "Let $f : X \\to Y$ be a quasi-compact morphism with $X$ and $Y$ quasi-separated.", "If $R^1f_*\\mathcal{I} = 0$ for every quasi-coherent sheaf of ideals", "$\\mathcal{I}$ on $X$, then $f$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset Y$ be an affine open subscheme. We have to show that", "$U = f^{-1}(V)$ is affine. The inclusion morphism $V \\to Y$ is quasi-compact", "by Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}.", "Hence the base change $U \\to X$ is quasi-compact, see", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-preserved-base-change}.", "Thus any quasi-coherent sheaf of ideals $\\mathcal{I}$ on $U$", "extends to a quasi-coherent sheaf of ideals on $X$, see", "Properties, Lemma \\ref{properties-lemma-extend-trivial}.", "Since the formation of $R^1f_*$ is local on $Y$", "(Cohomology, Section \\ref{cohomology-section-locality})", "we conclude that $R^1(U \\to V)_*\\mathcal{I} = 0$ by the assumption", "in the lemma. Hence by the Leray Spectral sequence", "(Cohomology, Lemma \\ref{cohomology-lemma-Leray})", "we conclude that $H^1(U, \\mathcal{I}) = H^1(V, (U \\to V)_*\\mathcal{I})$.", "Since $(U \\to V)_*\\mathcal{I}$ is quasi-coherent by", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}, we have", "$H^1(V, (U \\to V)_*\\mathcal{I}) = 0$ by", "Lemma \\ref{lemma-quasi-coherent-affine-cohomology-zero}.", "Thus we find that $U$ is affine by", "Lemma \\ref{lemma-quasi-compact-h1-zero-covering}." ], "refs": [ "schemes-lemma-quasi-compact-permanence", "schemes-lemma-quasi-compact-preserved-base-change", "properties-lemma-extend-trivial", "cohomology-lemma-Leray", "schemes-lemma-push-forward-quasi-coherent", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "coherent-lemma-quasi-compact-h1-zero-covering" ], "ref_ids": [ 7716, 7698, 3018, 2070, 7730, 3282, 3287 ] } ], "ref_ids": [] }, { "id": 3291, "type": "theorem", "label": "coherent-lemma-induction-principle", "categories": [ "coherent" ], "title": "coherent-lemma-induction-principle", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Let $P$ be a property", "of the quasi-compact opens of $X$. Assume that", "\\begin{enumerate}", "\\item $P$ holds for every affine open of $X$,", "\\item if $U$ is quasi-compact open, $V$ affine open,", "$P$ holds for $U$, $V$, and $U \\cap V$, then", "$P$ holds for $U \\cup V$.", "\\end{enumerate}", "Then $P$ holds for every quasi-compact open of $X$", "and in particular for $X$." ], "refs": [], "proofs": [ { "contents": [ "First we argue by induction that $P$ holds for {\\it separated} quasi-compact", "opens $W \\subset X$. Namely, such an open can be written as", "$W = U_1 \\cup \\ldots \\cup U_n$ and we can do induction on $n$ using", "property (2) with $U = U_1 \\cup \\ldots \\cup U_{n - 1}$ and $V = U_n$.", "This is allowed because", "$U \\cap V = (U_1 \\cap U_n) \\cup \\ldots \\cup (U_{n - 1} \\cap U_n)$", "is also a union of $n - 1$ affine open subschemes by", "Schemes, Lemma \\ref{schemes-lemma-characterize-separated}", "applied to the affine opens $U_i$ and $U_n$ of $W$.", "Having said this, for any quasi-compact open $W \\subset X$ we can", "do induction on the number of affine opens needed to cover $W$", "using the same trick as before and using that the quasi-compact open", "$U_i \\cap U_n$ is separated as an open subscheme of the affine scheme $U_n$." ], "refs": [ "schemes-lemma-characterize-separated" ], "ref_ids": [ 7710 ] } ], "ref_ids": [] }, { "id": 3292, "type": "theorem", "label": "coherent-lemma-vanishing-nr-affines", "categories": [ "coherent" ], "title": "coherent-lemma-vanishing-nr-affines", "contents": [ "\\begin{slogan}", "For schemes with affine diagonal, the cohomology of quasi-coherent", "modules vanishes in degrees bigger than the number of affine", "opens needed in a covering.", "\\end{slogan}", "Let $X$ be a quasi-compact scheme with affine diagonal (for example", "if $X$ is separated).", "Let $t = t(X)$ be the minimal number of affine opens needed to", "cover $X$. Then $H^n(X, \\mathcal{F}) = 0$ for all $n \\geq t$ and all", "quasi-coherent sheaves $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "First proof.", "By induction on $t$.", "If $t = 1$ the result follows from", "Lemma \\ref{lemma-quasi-coherent-affine-cohomology-zero}.", "If $t > 1$ write $X = U \\cup V$ with $V$ affine open and", "$U = U_1 \\cup \\ldots \\cup U_{t - 1}$ a union of $t - 1$ open affines.", "Note that in this case", "$U \\cap V = (U_1 \\cap V) \\cup \\ldots (U_{t - 1} \\cap V)$", "is also a union of $t - 1$ affine open subschemes.", "Namely, since the diagonal is affine, the intersection of two", "affine opens is affine, see Lemma \\ref{lemma-affine-diagonal}.", "We apply the Mayer-Vietoris long exact sequence", "$$", "0 \\to", "H^0(X, \\mathcal{F}) \\to", "H^0(U, \\mathcal{F}) \\oplus H^0(V, \\mathcal{F}) \\to", "H^0(U \\cap V, \\mathcal{F}) \\to", "H^1(X, \\mathcal{F}) \\to \\ldots", "$$", "see Cohomology, Lemma \\ref{cohomology-lemma-mayer-vietoris}.", "By induction we see that the groups $H^i(U, \\mathcal{F})$,", "$H^i(V, \\mathcal{F})$, $H^i(U \\cap V, \\mathcal{F})$ are zero for", "$i \\geq t - 1$. It follows immediately that $H^i(X, \\mathcal{F})$", "is zero for $i \\geq t$.", "\\medskip\\noindent", "Second proof.", "Let $\\mathcal{U} : X = \\bigcup_{i = 1}^t U_i$ be a finite affine open", "covering. Since $X$ is has affine diagonal the multiple intersections", "$U_{i_0 \\ldots i_p}$ are all affine, see", "Lemma \\ref{lemma-affine-diagonal}.", "By Lemma \\ref{lemma-cech-cohomology-quasi-coherent} the {\\v C}ech", "cohomology groups $\\check{H}^p(\\mathcal{U}, \\mathcal{F})$", "agree with the cohomology groups. By", "Cohomology, Lemma \\ref{cohomology-lemma-alternating-usual}", "the {\\v C}ech cohomology groups may be computed using the alternating", "{\\v C}ech complex $\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})$.", "As the covering consists of $t$ elements we see immediately", "that $\\check{\\mathcal{C}}_{alt}^p(\\mathcal{U}, \\mathcal{F}) = 0$", "for all $p \\geq t$. Hence the result follows." ], "refs": [ "coherent-lemma-quasi-coherent-affine-cohomology-zero", "coherent-lemma-affine-diagonal", "cohomology-lemma-mayer-vietoris", "coherent-lemma-affine-diagonal", "coherent-lemma-cech-cohomology-quasi-coherent", "cohomology-lemma-alternating-usual" ], "ref_ids": [ 3282, 3285, 2042, 3285, 3286, 2095 ] } ], "ref_ids": [] }, { "id": 3293, "type": "theorem", "label": "coherent-lemma-affine-diagonal-universal-delta-functor", "categories": [ "coherent" ], "title": "coherent-lemma-affine-diagonal-universal-delta-functor", "contents": [ "Let $X$ be a quasi-compact scheme with affine diagonal", "(for example if $X$ is separated). Then", "\\begin{enumerate}", "\\item given a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "there exists an embedding $\\mathcal{F} \\to \\mathcal{F}'$ of", "quasi-coherent $\\mathcal{O}_X$-modules", "such that $H^p(X, \\mathcal{F}') = 0$ for all $p \\geq 1$, and", "\\item $\\{H^n(X, -)\\}_{n \\geq 0}$", "is a universal $\\delta$-functor from $\\QCoh(\\mathcal{O}_X)$ to", "$\\textit{Ab}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X = \\bigcup U_i$ be a finite affine open covering.", "Set $U = \\coprod U_i$ and denote $j : U \\to X$", "the morphism inducing the given open immersions $U_i \\to X$.", "Since $U$ is an affine scheme and $X$ has affine diagonal,", "the morphism $j$ is affine, see", "Morphisms, Lemma \\ref{morphisms-lemma-affine-permanence}.", "For every $\\mathcal{O}_X$-module $\\mathcal{F}$ there is", "a canonical map $\\mathcal{F} \\to j_*j^*\\mathcal{F}$.", "This map is injective as can be seen by checking on stalks:", "if $x \\in U_i$, then we have a factorization", "$$", "\\mathcal{F}_x \\to (j_*j^*\\mathcal{F})_x", "\\to (j^*\\mathcal{F})_{x'} = \\mathcal{F}_x", "$$", "where $x' \\in U$ is the point $x$ viewed as a point of $U_i \\subset U$.", "Now if $\\mathcal{F}$ is quasi-coherent, then $j^*\\mathcal{F}$", "is quasi-coherent on the affine scheme $U$ hence has vanishing", "higher cohomology by", "Lemma \\ref{lemma-quasi-coherent-affine-cohomology-zero}.", "Then $H^p(X, j_*j^*\\mathcal{F}) = 0$ for", "$p > 0$ by Lemma \\ref{lemma-relative-affine-cohomology}", "as $j$ is affine. This proves (1).", "Finally, we see that the map", "$H^p(X, \\mathcal{F}) \\to H^p(X, j_*j^*\\mathcal{F})$", "is zero and part (2) follows from", "Homology, Lemma \\ref{homology-lemma-efface-implies-universal}." ], "refs": [ "morphisms-lemma-affine-permanence", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "coherent-lemma-relative-affine-cohomology", "homology-lemma-efface-implies-universal" ], "ref_ids": [ 5179, 3282, 3284, 12052 ] } ], "ref_ids": [] }, { "id": 3294, "type": "theorem", "label": "coherent-lemma-vanishing-nr-affines-quasi-separated", "categories": [ "coherent" ], "title": "coherent-lemma-vanishing-nr-affines-quasi-separated", "contents": [ "Let $X$ be a quasi-compact quasi-separated scheme.", "Let $X = U_1 \\cup \\ldots \\cup U_t$ be an affine open covering.", "Set", "$$", "d = \\max\\nolimits_{I \\subset \\{1, \\ldots, t\\}}", "\\left(|I| + t(\\bigcap\\nolimits_{i \\in I} U_i)\\right)", "$$", "where $t(U)$ is the minimal number of affines needed to cover", "the scheme $U$. Then $H^n(X, \\mathcal{F}) = 0$ for all $n \\geq d$ and all", "quasi-coherent sheaves $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Note that since $X$ is quasi-separated the numbers", "$t(\\bigcap_{i \\in I} U_i)$ are finite.", "Let $\\mathcal{U} : X = \\bigcup_{i = 1}^t U_i$.", "By", "Cohomology, Lemma \\ref{cohomology-lemma-cech-spectral-sequence}", "there is a spectral sequence", "$$", "E_2^{p, q} = \\check{H}^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F}))", "$$", "converging to $H^{p + q}(U, \\mathcal{F})$. By", "Cohomology, Lemma \\ref{cohomology-lemma-alternating-usual}", "we have", "$$", "E_2^{p, q} =", "H^p(\\check{\\mathcal{C}}_{alt}^\\bullet(", "\\mathcal{U}, \\underline{H}^q(\\mathcal{F}))", "$$", "The alternating {\\v C}ech complex with values in the presheaf", "$\\underline{H}^q(\\mathcal{F})$ vanishes in high degrees by", "Lemma \\ref{lemma-vanishing-nr-affines},", "more precisely $E_2^{p, q} = 0$ for $p + q \\geq d$.", "Hence the result follows." ], "refs": [ "cohomology-lemma-cech-spectral-sequence", "cohomology-lemma-alternating-usual", "coherent-lemma-vanishing-nr-affines" ], "ref_ids": [ 2055, 2095, 3292 ] } ], "ref_ids": [] }, { "id": 3295, "type": "theorem", "label": "coherent-lemma-quasi-coherence-higher-direct-images", "categories": [ "coherent" ], "title": "coherent-lemma-quasi-coherence-higher-direct-images", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume that $f$ is quasi-separated and quasi-compact.", "\\begin{enumerate}", "\\item For any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the", "higher direct images $R^pf_*\\mathcal{F}$ are quasi-coherent on $S$.", "\\item If $S$ is quasi-compact, there exists an integer $n = n(X, S, f)$", "such that $R^pf_*\\mathcal{F} = 0$ for all $p \\geq n$ and any", "quasi-coherent sheaf $\\mathcal{F}$ on $X$.", "\\item In fact, if $S$ is quasi-compact we can find $n = n(X, S, f)$", "such that for every", "morphism of schemes $S' \\to S$ we have $R^p(f')_*\\mathcal{F}' = 0$", "for $p \\geq n$ and any quasi-coherent sheaf $\\mathcal{F}'$", "on $X'$. Here $f' : X' = S' \\times_S X \\to S'$ is the base change of $f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We first prove (1). Note that under the hypotheses of the lemma the sheaf", "$R^0f_*\\mathcal{F} = f_*\\mathcal{F}$ is quasi-coherent by", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}.", "Using", "Cohomology, Lemma \\ref{cohomology-lemma-localize-higher-direct-images}", "we see that forming higher direct images commutes with restriction", "to open subschemes. Since being quasi-coherent is local on $S$ we", "may assume $S$ is affine.", "\\medskip\\noindent", "Assume $S$ is affine and $f$ quasi-compact and separated.", "Let $t \\geq 1$ be the minimal number of affine opens needed to cover $X$.", "We will prove this case of (1) by induction on $t$.", "If $t = 1$ then the morphism $f$ is affine by", "Morphisms, Lemma \\ref{morphisms-lemma-morphism-affines-affine}", "and (1) follows from", "Lemma \\ref{lemma-relative-affine-vanishing}.", "If $t > 1$ write $X = U \\cup V$ with $V$ affine open and", "$U = U_1 \\cup \\ldots \\cup U_{t - 1}$ a union of $t - 1$ open affines.", "Note that in this case", "$U \\cap V = (U_1 \\cap V) \\cup \\ldots (U_{t - 1} \\cap V)$", "is also a union of $t - 1$ affine open subschemes, see", "Schemes, Lemma \\ref{schemes-lemma-characterize-separated}.", "We will apply the relative Mayer-Vietoris sequence", "$$", "0 \\to", "f_*\\mathcal{F} \\to", "a_*(\\mathcal{F}|_U) \\oplus b_*(\\mathcal{F}|_V) \\to", "c_*(\\mathcal{F}|_{U \\cap V}) \\to", "R^1f_*\\mathcal{F} \\to \\ldots", "$$", "see Cohomology, Lemma \\ref{cohomology-lemma-relative-mayer-vietoris}.", "By induction we see that", "$R^pa_*\\mathcal{F}$, $R^pb_*\\mathcal{F}$ and $R^pc_*\\mathcal{F}$", "are all quasi-coherent. This implies that each of the sheaves", "$R^pf_*\\mathcal{F}$ is quasi-coherent since it sits in the middle of a short", "exact sequence with a cokernel of a map between quasi-coherent sheaves", "on the left and a kernel of a map between quasi-coherent sheaves on the right.", "Using the results on quasi-coherent sheaves in", "Schemes, Section \\ref{schemes-section-quasi-coherent} we see", "conclude $R^pf_*\\mathcal{F}$ is quasi-coherent.", "\\medskip\\noindent", "Assume $S$ is affine and $f$ quasi-compact and quasi-separated.", "Let $t \\geq 1$ be the minimal number of affine opens needed to cover $X$.", "We will prove (1) by induction on $t$.", "In case $t = 1$ the morphism $f$ is separated and we are back", "in the previous case (see previous paragraph).", "If $t > 1$ write $X = U \\cup V$ with $V$ affine open and", "$U$ a union of $t - 1$ open affines.", "Note that in this case $U \\cap V$ is an open subscheme of an affine", "scheme and hence separated (see", "Schemes, Lemma \\ref{schemes-lemma-affine-separated}).", "We will apply the relative Mayer-Vietoris sequence", "$$", "0 \\to", "f_*\\mathcal{F} \\to", "a_*(\\mathcal{F}|_U) \\oplus b_*(\\mathcal{F}|_V) \\to", "c_*(\\mathcal{F}|_{U \\cap V}) \\to", "R^1f_*\\mathcal{F} \\to \\ldots", "$$", "see Cohomology, Lemma \\ref{cohomology-lemma-relative-mayer-vietoris}.", "By induction and the result of the previous paragraph we see that", "$R^pa_*\\mathcal{F}$, $R^pb_*\\mathcal{F}$ and $R^pc_*\\mathcal{F}$", "are quasi-coherent. As in the previous paragraph this implies each of", "sheaves $R^pf_*\\mathcal{F}$ is quasi-coherent.", "\\medskip\\noindent", "Next, we prove (3) and a fortiori (2). Choose a finite affine open", "covering $S = \\bigcup_{j = 1, \\ldots m} S_j$. For each $i$ choose", "a finite affine open covering", "$f^{-1}(S_j) = \\bigcup_{i = 1, \\ldots t_j} U_{ji} $.", "Let", "$$", "d_j = \\max\\nolimits_{I \\subset \\{1, \\ldots, t_j\\}}", "\\left(|I| + t(\\bigcap\\nolimits_{i \\in I} U_{ji})\\right)", "$$", "be the integer found in", "Lemma \\ref{lemma-vanishing-nr-affines-quasi-separated}.", "We claim that $n(X, S, f) = \\max d_j$ works.", "\\medskip\\noindent", "Namely, let $S' \\to S$ be a morphism of schemes and let", "$\\mathcal{F}'$ be a quasi-coherent sheaf on $X' = S' \\times_S X$.", "We want to show that $R^pf'_*\\mathcal{F}' = 0$ for $p \\geq n(X, S, f)$.", "Since this question is local on $S'$ we may assume that $S'$ is affine", "and maps into $S_j$ for some $j$. Then $X' = S' \\times_{S_j} f^{-1}(S_j)$", "is covered by the open affines $S' \\times_{S_j} U_{ji}$, $i = 1, \\ldots t_j$", "and the intersections", "$$", "\\bigcap\\nolimits_{i \\in I} S' \\times_{S_j} U_{ji} =", "S' \\times_{S_j} \\bigcap\\nolimits_{i \\in I} U_{ji}", "$$", "are covered by the same number of affines as before the base change.", "Applying", "Lemma \\ref{lemma-vanishing-nr-affines-quasi-separated}", "we get $H^p(X', \\mathcal{F}') = 0$. By the first part of the proof", "we already know that each $R^qf'_*\\mathcal{F}'$ is quasi-coherent", "hence has vanishing higher cohomology groups on our affine scheme $S'$,", "thus we see that $H^0(S', R^pf'_*\\mathcal{F}') = H^p(X', \\mathcal{F}') = 0$", "by Cohomology, Lemma \\ref{cohomology-lemma-apply-Leray}.", "Since $R^pf'_*\\mathcal{F}'$ is quasi-coherent", "we conclude that $R^pf'_*\\mathcal{F}' = 0$." ], "refs": [ "schemes-lemma-push-forward-quasi-coherent", "cohomology-lemma-localize-higher-direct-images", "morphisms-lemma-morphism-affines-affine", "coherent-lemma-relative-affine-vanishing", "schemes-lemma-characterize-separated", "cohomology-lemma-relative-mayer-vietoris", "schemes-lemma-affine-separated", "cohomology-lemma-relative-mayer-vietoris", "coherent-lemma-vanishing-nr-affines-quasi-separated", "coherent-lemma-vanishing-nr-affines-quasi-separated", "cohomology-lemma-apply-Leray" ], "ref_ids": [ 7730, 2040, 5180, 3283, 7710, 2043, 7717, 2043, 3294, 3294, 2071 ] } ], "ref_ids": [] }, { "id": 3296, "type": "theorem", "label": "coherent-lemma-quasi-coherence-higher-direct-images-application", "categories": [ "coherent" ], "title": "coherent-lemma-quasi-coherence-higher-direct-images-application", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume that $f$ is quasi-separated and quasi-compact.", "Assume $S$ is affine.", "For any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "we have", "$$", "H^q(X, \\mathcal{F}) = H^0(S, R^qf_*\\mathcal{F})", "$$", "for all $q \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the Leray spectral sequence $E_2^{p, q} = H^p(S, R^qf_*\\mathcal{F})$", "converging to $H^{p + q}(X, \\mathcal{F})$, see", "Cohomology, Lemma \\ref{cohomology-lemma-Leray}.", "By Lemma \\ref{lemma-quasi-coherence-higher-direct-images}", "we see that the sheaves $R^qf_*\\mathcal{F}$ are quasi-coherent.", "By Lemma \\ref{lemma-quasi-coherent-affine-cohomology-zero}", "we see that $E_2^{p, q} = 0$ when $p > 0$.", "Hence the spectral sequence degenerates at $E_2$ and we win.", "See also", "Cohomology, Lemma \\ref{cohomology-lemma-apply-Leray} (2)", "for the general principle." ], "refs": [ "cohomology-lemma-Leray", "coherent-lemma-quasi-coherence-higher-direct-images", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "cohomology-lemma-apply-Leray" ], "ref_ids": [ 2070, 3295, 3282, 2071 ] } ], "ref_ids": [] }, { "id": 3297, "type": "theorem", "label": "coherent-lemma-affine-base-change", "categories": [ "coherent" ], "title": "coherent-lemma-affine-base-change", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume $f$ is affine.", "In this case $f_*\\mathcal{F} \\cong Rf_*\\mathcal{F}$ is", "a quasi-coherent sheaf, and for every base change diagram", "(\\ref{equation-base-change-diagram})", "we have", "$$", "g^*f_*\\mathcal{F} = f'_*(g')^*\\mathcal{F}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "The vanishing of higher direct images is", "Lemma \\ref{lemma-relative-affine-vanishing}.", "The statement is local on $S$ and $S'$. Hence we may", "assume $X = \\Spec(A)$, $S = \\Spec(R)$,", "$S' = \\Spec(R')$ and $\\mathcal{F} = \\widetilde{M}$", "for some $A$-module $M$.", "We use Schemes, Lemma \\ref{schemes-lemma-widetilde-pullback}", "to describe pullbacks and pushforwards of $\\mathcal{F}$.", "Namely, $X' = \\Spec(R' \\otimes_R A)$ and", "$\\mathcal{F}'$ is the quasi-coherent sheaf associated", "to $(R' \\otimes_R A) \\otimes_A M$.", "Thus we see that the lemma boils down to the", "equality", "$$", "(R' \\otimes_R A) \\otimes_A M = R' \\otimes_R M", "$$", "as $R'$-modules." ], "refs": [ "coherent-lemma-relative-affine-vanishing", "schemes-lemma-widetilde-pullback" ], "ref_ids": [ 3283, 7662 ] } ], "ref_ids": [] }, { "id": 3298, "type": "theorem", "label": "coherent-lemma-flat-base-change-cohomology", "categories": [ "coherent" ], "title": "coherent-lemma-flat-base-change-cohomology", "contents": [ "Consider a cartesian diagram of schemes", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module", "with pullback $\\mathcal{F}' = (g')^*\\mathcal{F}$.", "Assume that $g$ is flat and that $f$ is quasi-compact and quasi-separated.", "For any $i \\geq 0$", "\\begin{enumerate}", "\\item the base change map of", "Cohomology, Lemma \\ref{cohomology-lemma-base-change-map-flat-case}", "is an isomorphism", "$$", "g^*R^if_*\\mathcal{F} \\longrightarrow R^if'_*\\mathcal{F}',", "$$", "\\item if $S = \\Spec(A)$ and $S' = \\Spec(B)$, then", "$H^i(X, \\mathcal{F}) \\otimes_A B = H^i(X', \\mathcal{F}')$.", "\\end{enumerate}" ], "refs": [ "cohomology-lemma-base-change-map-flat-case" ], "proofs": [ { "contents": [ "Using Cohomology, Lemma \\ref{cohomology-lemma-base-change-map-flat-case} in (1)", "is allowed since $g'$ is flat by", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat}.", "Having said this, part (1) follows from part (2). Namely,", "part (1) is local on $S'$ and hence we may assume $S$", "and $S'$ are affine. In other words, we have $S = \\Spec(A)$", "and $S' = \\Spec(B)$ as in (2).", "Then since $R^if_*\\mathcal{F}$ is quasi-coherent", "(Lemma \\ref{lemma-quasi-coherence-higher-direct-images}),", "it is the quasi-coherent $\\mathcal{O}_S$-module associated to the", "$A$-module $H^0(S, R^if_*\\mathcal{F}) = H^i(X, \\mathcal{F})$", "(equality by", "Lemma \\ref{lemma-quasi-coherence-higher-direct-images-application}).", "Similarly, $R^if'_*\\mathcal{F}'$ is the quasi-coherent", "$\\mathcal{O}_{S'}$-module associated to the $B$-module", "$H^i(X', \\mathcal{F}')$. Since pullback by $g$ corresponds", "to $- \\otimes_A B$ on modules", "(Schemes, Lemma \\ref{schemes-lemma-widetilde-pullback})", "we see that it suffices to prove (2).", "\\medskip\\noindent", "Let $A \\to B$ be a flat ring homomorphism.", "Let $X$ be a quasi-compact and quasi-separated scheme over $A$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Set $X_B = X \\times_{\\Spec(A)} \\Spec(B)$ and denote", "$\\mathcal{F}_B$ the pullback of $\\mathcal{F}$.", "We are trying to show that the map", "$$", "H^i(X, \\mathcal{F}) \\otimes_A B \\longrightarrow H^i(X_B, \\mathcal{F}_B)", "$$", "(given by the reference in the statement of the lemma)", "is an isomorphism.", "\\medskip\\noindent", "In case $X$ is separated, choose an affine open covering", "$\\mathcal{U} : X = U_1 \\cup \\ldots \\cup U_t$ and recall that", "$$", "\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = H^p(X, \\mathcal{F}),", "$$", "see", "Lemma \\ref{lemma-cech-cohomology-quasi-coherent}.", "If $\\mathcal{U}_B : X_B = (U_1)_B \\cup \\ldots \\cup (U_t)_B$ we obtain", "by base change, then it is still the case that each $(U_i)_B$ is affine", "and that $X_B$ is separated. Thus we obtain", "$$", "\\check{H}^p(\\mathcal{U}_B, \\mathcal{F}_B) = H^p(X_B, \\mathcal{F}_B).", "$$", "We have the following relation between the {\\v C}ech complexes", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}_B, \\mathcal{F}_B) =", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\otimes_A B", "$$", "as follows from", "Lemma \\ref{lemma-affine-base-change}.", "Since $A \\to B$ is flat, the same thing remains true on taking cohomology.", "\\medskip\\noindent", "In case $X$ is quasi-separated, choose an affine open covering", "$\\mathcal{U} : X = U_1 \\cup \\ldots \\cup U_t$. We will use the", "{\\v C}ech-to-cohomology spectral sequence", "Cohomology, Lemma \\ref{cohomology-lemma-cech-spectral-sequence}.", "The reader who wishes to avoid this spectral sequence", "can use Mayer-Vietoris and induction on $t$ as in the proof of", "Lemma \\ref{lemma-quasi-coherence-higher-direct-images}.", "The spectral sequence has $E_2$-page", "$E_2^{p, q} = \\check{H}^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F}))$", "and converges to $H^{p + q}(X, \\mathcal{F})$.", "Similarly, we have a spectral sequence with $E_2$-page", "$E_2^{p, q} = \\check{H}^p(\\mathcal{U}_B, \\underline{H}^q(\\mathcal{F}_B))$", "which converges to $H^{p + q}(X_B, \\mathcal{F}_B)$.", "Since the intersections $U_{i_0 \\ldots i_p}$ are quasi-compact", "and separated, the result of the second paragraph of the proof gives", "$\\check{H}^p(\\mathcal{U}_B, \\underline{H}^q(\\mathcal{F}_B)) =", "\\check{H}^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F})) \\otimes_A B$.", "Using that $A \\to B$ is flat we conclude that", "$H^i(X, \\mathcal{F}) \\otimes_A B \\to H^i(X_B, \\mathcal{F}_B)$", "is an isomorphism for all $i$ and we win." ], "refs": [ "cohomology-lemma-base-change-map-flat-case", "morphisms-lemma-base-change-flat", "coherent-lemma-quasi-coherence-higher-direct-images", "coherent-lemma-quasi-coherence-higher-direct-images-application", "schemes-lemma-widetilde-pullback", "coherent-lemma-cech-cohomology-quasi-coherent", "coherent-lemma-affine-base-change", "cohomology-lemma-cech-spectral-sequence", "coherent-lemma-quasi-coherence-higher-direct-images" ], "ref_ids": [ 2079, 5265, 3295, 3296, 7662, 3286, 3297, 2055, 3295 ] } ], "ref_ids": [ 2079 ] }, { "id": 3299, "type": "theorem", "label": "coherent-lemma-finite-locally-free-base-change-cohomology", "categories": [ "coherent" ], "title": "coherent-lemma-finite-locally-free-base-change-cohomology", "contents": [ "Consider a cartesian diagram of schemes", "$$", "\\xymatrix{", "Y \\ar[d]_{g} \\ar[r]_h & X \\ar[d]^f \\\\", "\\Spec(B) \\ar[r] & \\Spec(A)", "}", "$$", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module", "with pullback $\\mathcal{G} = h^*\\mathcal{F}$.", "If $B$ is a finite locally free $A$-module, then", "$H^i(X, \\mathcal{F}) \\otimes_A B = H^i(Y, \\mathcal{G})$." ], "refs": [], "proofs": [ { "contents": [ "In case $X$ is separated, choose an affine open covering", "$\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ and recall that", "$$", "\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = H^p(X, \\mathcal{F}),", "$$", "see", "Lemma \\ref{lemma-cech-cohomology-quasi-coherent}.", "Let $\\mathcal{V} : Y = \\bigcup_{i \\in I} g^{-1}(U_i)$", "be the corresponding affine open covering of $Y$.", "The opens $V_i = g^{-1}(U_i) = U_i \\times_{\\Spec(A)} \\Spec(B)$", "are affine and $Y$ is separated. Thus we obtain", "$$", "\\check{H}^p(\\mathcal{V}, \\mathcal{G}) = H^p(Y, \\mathcal{G}).", "$$", "We claim the map of {\\v C}ech complexes", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\otimes_A B", "\\longrightarrow", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G})", "$$", "is an isomorphism. Namely, as $B$ is finitely presented as an $A$-module", "we see that tensoring with $B$ over $A$ commutes with products, see", "Algebra, Proposition \\ref{algebra-proposition-fp-tensor}.", "Thus it suffices to show that the maps", "$\\Gamma(U_{i_0 \\ldots i_p}, \\mathcal{F}) \\otimes_A B \\to", "\\Gamma(V_{i_0 \\ldots i_p}, \\mathcal{G})$ ", "are isomorphisms which follows from", "Lemma \\ref{lemma-affine-base-change}.", "Since $A \\to B$ is flat, the same thing remains true on taking cohomology.", "\\medskip\\noindent", "In the general case we argue in exactly the same way using affine", "open covering $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ and the", "corresponding covering $\\mathcal{V} : Y = \\bigcup_{i \\in I} V_i$", "with $V_i = g^{-1}(U_i)$ as above. We will use the", "{\\v C}ech-to-cohomology spectral sequence", "Cohomology, Lemma \\ref{cohomology-lemma-cech-spectral-sequence}.", "The spectral sequence has $E_2$-page", "$E_2^{p, q} = \\check{H}^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F}))$", "and converges to $H^{p + q}(X, \\mathcal{F})$.", "Similarly, we have a spectral sequence with $E_2$-page", "$E_2^{p, q} = \\check{H}^p(\\mathcal{V}, \\underline{H}^q(\\mathcal{G}))$", "which converges to $H^{p + q}(Y, \\mathcal{G})$.", "Since the intersections $U_{i_0 \\ldots i_p}$ are separated, the result", "of the previous paragraph gives isomorphisms", "$\\Gamma(U_{i_0 \\ldots i_p}, \\underline{H}^q(\\mathcal{F})) \\otimes_A B", "\\to \\Gamma(V_{i_0 \\ldots i_p}, \\underline{H}^q(\\mathcal{G}))$.", "Using that $- \\otimes_A B$ commutes with products and is exact, we conclude", "that", "$\\check{H}^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F})) \\otimes_A B", "\\to \\check{H}^p(\\mathcal{V}, \\underline{H}^q(\\mathcal{G}))$", "is an isomorphism. Using that $A \\to B$ is flat we conclude that", "$H^i(X, \\mathcal{F}) \\otimes_A B \\to H^i(Y, \\mathcal{G})$", "is an isomorphism for all $i$ and we win." ], "refs": [ "coherent-lemma-cech-cohomology-quasi-coherent", "algebra-proposition-fp-tensor", "coherent-lemma-affine-base-change", "cohomology-lemma-cech-spectral-sequence" ], "ref_ids": [ 3286, 1416, 3297, 2055 ] } ], "ref_ids": [] }, { "id": 3300, "type": "theorem", "label": "coherent-lemma-colimit-cohomology", "categories": [ "coherent" ], "title": "coherent-lemma-colimit-cohomology", "contents": [ "Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes.", "Let $\\mathcal{F} = \\colim \\mathcal{F}_i$ be a filtered colimit", "of quasi-coherent sheaves on $X$.", "Then for any $p \\geq 0$ we have", "$$", "R^pf_*\\mathcal{F} = \\colim R^pf_*\\mathcal{F}_i.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Recall that $R^pf_*\\mathcal{F}$ is the sheaf associated to", "$U \\mapsto H^p(f^{-1}U, \\mathcal{F})$, see", "Cohomology, Lemma \\ref{cohomology-lemma-describe-higher-direct-images}.", "Recall that the colimit is the sheaf associated to the presheaf colimit", "(taking colimits over opens). Hence we can apply", "Cohomology, Lemma \\ref{cohomology-lemma-quasi-separated-cohomology-colimit}", "to $H^p(f^{-1}U, -)$ where $U$ is affine to conclude. (Because the", "basis of affine opens in $f^{-1}U$ satisfies the assumptions of that", "lemma.)" ], "refs": [ "cohomology-lemma-describe-higher-direct-images", "cohomology-lemma-quasi-separated-cohomology-colimit" ], "ref_ids": [ 2039, 2082 ] } ], "ref_ids": [] }, { "id": 3301, "type": "theorem", "label": "coherent-lemma-separated-case-relative-cech", "categories": [ "coherent" ], "title": "coherent-lemma-separated-case-relative-cech", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume $X$ is quasi-compact and $X$ and $S$ have affine diagonal", "(e.g., if $X$ and $S$ are separated).", "In this case we can compute $Rf_*\\mathcal{F}$ as follows:", "\\begin{enumerate}", "\\item Choose a finite affine open covering", "$\\mathcal{U} : X = \\bigcup_{i = 1, \\ldots, n} U_i$.", "\\item For $i_0, \\ldots, i_p \\in \\{1, \\ldots, n\\}$ denote", "$f_{i_0 \\ldots i_p} : U_{i_0 \\ldots i_p} \\to S$ the restriction of $f$", "to the intersection $U_{i_0 \\ldots i_p} = U_{i_0} \\cap \\ldots \\cap U_{i_p}$.", "\\item Set $\\mathcal{F}_{i_0 \\ldots i_p}$ equal to the restriction", "of $\\mathcal{F}$ to $U_{i_0 \\ldots i_p}$.", "\\item Set", "$$", "\\check{\\mathcal{C}}^p(\\mathcal{U}, f, \\mathcal{F}) =", "\\bigoplus\\nolimits_{i_0 \\ldots i_p}", "f_{i_0 \\ldots i_p *} \\mathcal{F}_{i_0 \\ldots i_p}", "$$", "and define differentials", "$d : \\check{\\mathcal{C}}^p(\\mathcal{U}, f, \\mathcal{F})", "\\to \\check{\\mathcal{C}}^{p + 1}(\\mathcal{U}, f, \\mathcal{F})$", "as in Cohomology, Equation (\\ref{cohomology-equation-d-cech}).", "\\end{enumerate}", "Then the complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F})$", "is a complex of quasi-coherent sheaves on $S$ which comes equipped with an", "isomorphism", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F})", "\\longrightarrow", "Rf_*\\mathcal{F}", "$$", "in $D^{+}(S)$. This isomorphism is functorial in the quasi-coherent", "sheaf $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the resolution", "$\\mathcal{F} \\to {\\mathfrak C}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "of Cohomology, Lemma \\ref{cohomology-lemma-covering-resolution}.", "We have an equality of complexes", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F}) =", "f_*{\\mathfrak C}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "of quasi-coherent $\\mathcal{O}_S$-modules.", "The morphisms $j_{i_0 \\ldots i_p} : U_{i_0 \\ldots i_p} \\to X$", "and the morphisms $f_{i_0 \\ldots i_p} : U_{i_0 \\ldots i_p} \\to S$", "are affine by Morphisms, Lemma \\ref{morphisms-lemma-affine-permanence}", "and Lemma \\ref{lemma-affine-diagonal}.", "Hence $R^qj_{i_0 \\ldots i_p *}\\mathcal{F}_{i_0 \\ldots i_p}$", "as well as $R^qf_{i_0 \\ldots i_p *}\\mathcal{F}_{i_0 \\ldots i_p}$", "are zero for $q > 0$ (Lemma \\ref{lemma-relative-affine-vanishing}).", "Using $f \\circ j_{i_0 \\ldots i_p} = f_{i_0 \\ldots i_p}$ and", "the spectral sequence of", "Cohomology, Lemma \\ref{cohomology-lemma-relative-Leray}", "we conclude that", "$R^qf_*(j_{i_0 \\ldots i_p *}\\mathcal{F}_{i_0 \\ldots i_p}) = 0$", "for $q > 0$.", "Since the terms of the complex", "${\\mathfrak C}^\\bullet(\\mathcal{U}, \\mathcal{F})$ are finite direct", "sums of the sheaves $j_{i_0 \\ldots i_p *}\\mathcal{F}_{i_0 \\ldots i_p}$", "we conclude using Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "that", "$$", "Rf_* \\mathcal{F} = f_*{\\mathfrak C}^\\bullet(\\mathcal{U}, \\mathcal{F}) =", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F})", "$$", "as desired." ], "refs": [ "cohomology-lemma-covering-resolution", "morphisms-lemma-affine-permanence", "coherent-lemma-affine-diagonal", "coherent-lemma-relative-affine-vanishing", "cohomology-lemma-relative-Leray", "derived-lemma-leray-acyclicity" ], "ref_ids": [ 2097, 5179, 3285, 3283, 2073, 1844 ] } ], "ref_ids": [] }, { "id": 3302, "type": "theorem", "label": "coherent-lemma-base-change-complex", "categories": [ "coherent" ], "title": "coherent-lemma-base-change-complex", "contents": [ "With notation as in diagram (\\ref{equation-base-change-diagram}).", "Assume $f : X \\to S$ and $\\mathcal{F}$ satisfy the hypotheses of", "Lemma \\ref{lemma-separated-case-relative-cech}. Choose a finite", "affine open covering $\\mathcal{U} : X = \\bigcup U_i$ of $X$.", "There is a canonical isomorphism", "$$", "g^*\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F})", "\\longrightarrow", "Rf'_*\\mathcal{F}'", "$$", "in $D^{+}(S')$. Moreover, if $S' \\to S$ is affine, then in fact", "$$", "g^*\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F})", "=", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}', f', \\mathcal{F}')", "$$", "with $\\mathcal{U}' : X' = \\bigcup U_i'$ where", "$U_i' = (g')^{-1}(U_i) = U_{i, S'}$ is also affine." ], "refs": [ "coherent-lemma-separated-case-relative-cech" ], "proofs": [ { "contents": [ "In fact we may define $U_i' = (g')^{-1}(U_i) = U_{i, S'}$ no matter", "whether $S'$ is affine over $S$ or not.", "Let $\\mathcal{U}' : X' = \\bigcup U_i'$ be the induced covering of $X'$.", "In this case we claim that", "$$", "g^*\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F})", "=", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}', f', \\mathcal{F}')", "$$", "with $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}', f', \\mathcal{F}')$", "defined in exactly the same manner as in", "Lemma \\ref{lemma-separated-case-relative-cech}.", "This is clear from the case of affine morphisms", "(Lemma \\ref{lemma-affine-base-change}) by working locally on $S'$.", "Moreover, exactly as in the proof of", "Lemma \\ref{lemma-separated-case-relative-cech}", "one sees that there is an isomorphism", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}', f', \\mathcal{F}')", "\\longrightarrow", "Rf'_*\\mathcal{F}'", "$$", "in $D^{+}(S')$ since the morphisms $U_i' \\to X'$ and $U_i' \\to S'$", "are still affine (being base changes of affine morphisms).", "Details omitted." ], "refs": [ "coherent-lemma-separated-case-relative-cech", "coherent-lemma-affine-base-change", "coherent-lemma-separated-case-relative-cech" ], "ref_ids": [ 3301, 3297, 3301 ] } ], "ref_ids": [ 3301 ] }, { "id": 3303, "type": "theorem", "label": "coherent-lemma-hypercoverings", "categories": [ "coherent" ], "title": "coherent-lemma-hypercoverings", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Assume that $f$ is quasi-compact and quasi-separated and", "that $S$ is quasi-compact and separated.", "There exists a bounded below complex $\\mathcal{K}^\\bullet$", "of quasi-coherent $\\mathcal{O}_S$-modules with the", "following property: For every morphism", "$g : S' \\to S$ the complex $g^*\\mathcal{K}^\\bullet$ is", "a representative for $Rf'_*\\mathcal{F}'$ with notation as in", "diagram (\\ref{equation-base-change-diagram})." ], "refs": [], "proofs": [ { "contents": [ "(If $f$ is separated as well, please see", "Lemma \\ref{lemma-base-change-complex}.)", "The assumptions imply in particular that $X$", "is quasi-compact and quasi-separated as a scheme.", "Let $\\mathcal{B}$ be the set of affine opens of $X$. By", "Hypercoverings,", "Lemma \\ref{hypercovering-lemma-quasi-separated-quasi-compact-hypercovering}", "we can find a hypercovering $K = (I, \\{U_i\\})$ such that each", "$I_n$ is finite and each $U_i$ is an affine open of $X$. By", "Hypercoverings, Lemma \\ref{hypercovering-lemma-cech-spectral-sequence}", "there is a spectral sequence with $E_2$-page", "$$", "E_2^{p, q} = \\check{H}^p(K, \\underline{H}^q(\\mathcal{F}))", "$$", "converging to $H^{p + q}(X, \\mathcal{F})$. Note that", "$\\check{H}^p(K, \\underline{H}^q(\\mathcal{F}))$ is the $p$th cohomology", "group of the complex", "$$", "\\prod\\nolimits_{i \\in I_0} H^q(U_i, \\mathcal{F})", "\\to", "\\prod\\nolimits_{i \\in I_1} H^q(U_i, \\mathcal{F})", "\\to", "\\prod\\nolimits_{i \\in I_2} H^q(U_i, \\mathcal{F})", "\\to \\ldots", "$$", "Since each $U_i$ is affine we see that this is zero unless $q = 0$", "in which case we obtain", "$$", "\\prod\\nolimits_{i \\in I_0} \\mathcal{F}(U_i)", "\\to", "\\prod\\nolimits_{i \\in I_1} \\mathcal{F}(U_i)", "\\to", "\\prod\\nolimits_{i \\in I_2} \\mathcal{F}(U_i)", "\\to \\ldots", "$$", "Thus we conclude that $R\\Gamma(X, \\mathcal{F})$ is computed by", "this complex.", "\\medskip\\noindent", "For any $n$ and $i \\in I_n$ denote $f_i : U_i \\to S$ the restriction of", "$f$ to $U_i$. As $S$ is separated and $U_i$ is affine this morphism", "is affine. Consider the complex of quasi-coherent sheaves", "$$", "\\mathcal{K}^\\bullet = (", "\\prod\\nolimits_{i \\in I_0} f_{i, *}\\mathcal{F}|_{U_i}", "\\to", "\\prod\\nolimits_{i \\in I_1} f_{i, *}\\mathcal{F}|_{U_i}", "\\to", "\\prod\\nolimits_{i \\in I_2} f_{i, *}\\mathcal{F}|_{U_i}", "\\to \\ldots )", "$$", "on $S$. As in", "Hypercoverings, Lemma \\ref{hypercovering-lemma-cech-spectral-sequence}", "we obtain a map $\\mathcal{K}^\\bullet \\to Rf_*\\mathcal{F}$ in", "$D(\\mathcal{O}_S)$ by choosing an injective resolution of $\\mathcal{F}$", "(details omitted). Consider any affine scheme $V$ and a morphism", "$g : V \\to S$. Then the base change $X_V$ has a hypercovering", "$K_V = (I, \\{U_{i, V}\\})$ obtained by base change. Moreover,", "$g^*f_{i, *}\\mathcal{F} = f_{i, V, *}(g')^*\\mathcal{F}|_{U_{i, V}}$.", "Thus the arguments above prove that $\\Gamma(V, g^*\\mathcal{K}^\\bullet)$", "computes $R\\Gamma(X_V, (g')^*\\mathcal{F})$.", "This finishes the proof of the lemma as it suffices to prove", "the equality of complexes Zariski locally on $S'$." ], "refs": [ "coherent-lemma-base-change-complex", "hypercovering-lemma-quasi-separated-quasi-compact-hypercovering", "hypercovering-lemma-cech-spectral-sequence", "hypercovering-lemma-cech-spectral-sequence" ], "ref_ids": [ 3302, 8413, 8398, 8398 ] } ], "ref_ids": [] }, { "id": 3304, "type": "theorem", "label": "coherent-lemma-cohomology-projective-space-over-ring", "categories": [ "coherent" ], "title": "coherent-lemma-cohomology-projective-space-over-ring", "contents": [ "\\begin{reference}", "\\cite[III Proposition 2.1.12]{EGA}", "\\end{reference}", "Let $R$ be a ring.", "Let $n \\geq 0$ be an integer.", "We have", "$$", "H^q(\\mathbf{P}^n, \\mathcal{O}_{\\mathbf{P}^n_R}(d)) =", "\\left\\{", "\\begin{matrix}", "(R[T_0, \\ldots, T_n])_d & \\text{if} & q = 0 \\\\", "0 & \\text{if} & q \\not = 0, n \\\\", "\\left(\\frac{1}{T_0 \\ldots T_n} R[\\frac{1}{T_0}, \\ldots, \\frac{1}{T_n}]\\right)_d", "& \\text{if} & q = n", "\\end{matrix}", "\\right.", "$$", "as $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "We will use the standard affine open covering", "$$", "\\mathcal{U} : \\mathbf{P}^n_R = \\bigcup\\nolimits_{i = 0}^n D_{+}(T_i)", "$$", "to compute the cohomology using the {\\v C}ech complex.", "This is permissible by Lemma \\ref{lemma-cech-cohomology-quasi-coherent}", "since any intersection of finitely many affine $D_{+}(T_i)$ is also a", "standard affine open (see", "Constructions, Section \\ref{constructions-section-proj}).", "In fact, we can use the alternating or ordered {\\v C}ech complex according to", "Cohomology, Lemmas \\ref{cohomology-lemma-ordered-alternating} and", "\\ref{cohomology-lemma-alternating-usual}.", "\\medskip\\noindent", "The ordering we will use on $\\{0, \\ldots, n\\}$ is the usual one.", "Hence the complex we are looking at has terms", "$$", "\\check{\\mathcal{C}}_{ord}^p(\\mathcal{U}, \\mathcal{O}_{\\mathbf{P}_R}(d))", "=", "\\bigoplus\\nolimits_{i_0 < \\ldots < i_p}", "(R[T_0, \\ldots, T_n, \\frac{1}{T_{i_0} \\ldots T_{i_p}}])_d", "$$", "Moreover, the maps are given by the usual formula", "$$", "d(s)_{i_0 \\ldots i_{p + 1}} =", "\\sum\\nolimits_{j = 0}^{p + 1} (-1)^j s_{i_0 \\ldots \\hat i_j \\ldots i_{p + 1}}", "$$", "see Cohomology, Section \\ref{cohomology-section-alternating-cech}.", "Note that each term of this complex has a natural", "$\\mathbf{Z}^{n + 1}$-grading. Namely, we get this by declaring a monomial", "$T_0^{e_0} \\ldots T_n^{e_n}$ to be homogeneous with weight", "$(e_0, \\ldots, e_n) \\in \\mathbf{Z}^{n + 1}$. It is clear that the differential", "given above respects the grading. In a formula we have", "$$", "\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{O}_{\\mathbf{P}_R}(d))", "=", "\\bigoplus\\nolimits_{\\vec{e} \\in \\mathbf{Z}^{n + 1}}", "\\check{\\mathcal{C}}^\\bullet(\\vec{e})", "$$", "where not all summands on the right hand side occur (see below).", "Hence in order to compute the cohomology", "modules of the complex it suffices to compute the cohomology of the graded", "pieces and take the direct sum at the end.", "\\medskip\\noindent", "Fix $\\vec{e} = (e_0, \\ldots, e_n) \\in \\mathbf{Z}^{n + 1}$. In order for this", "weight to occur in the complex above we need to assume", "$e_0 + \\ldots + e_n = d$ (if not then it occurs for a different twist of", "the structure sheaf of course). Assuming this, set", "$$", "NEG(\\vec{e}) = \\{i \\in \\{0, \\ldots, n\\} \\mid e_i < 0\\}.", "$$", "With this notation the weight $\\vec{e}$ summand", "$\\check{\\mathcal{C}}^\\bullet(\\vec{e})$ of the {\\v C}ech complex above has", "the following terms", "$$", "\\check{\\mathcal{C}}^p(\\vec{e})", "=", "\\bigoplus\\nolimits_{i_0 < \\ldots < i_p,", "\\ NEG(\\vec{e}) \\subset \\{i_0, \\ldots, i_p\\}}", "R \\cdot T_0^{e_0} \\ldots T_n^{e_n}", "$$", "In other words, the terms corresponding to $i_0 < \\ldots < i_p$ such", "that $NEG(\\vec{e})$ is not contained in $\\{i_0 \\ldots i_p\\}$ are zero.", "The differential of the complex $\\check{\\mathcal{C}}^\\bullet(\\vec{e})$", "is still given by the exact same formula as above.", "\\medskip\\noindent", "Suppose that $NEG(\\vec{e}) = \\{0, \\ldots, n\\}$, i.e., that all", "exponents $e_i$ are negative.", "In this case the complex $\\check{\\mathcal{C}}^\\bullet(\\vec{e})$ has", "only one term, namely $\\check{\\mathcal{C}}^n(\\vec{e}) =", "R \\cdot \\frac{1}{T_0^{-e_0} \\ldots T_n^{-e_n}}$. Hence in this", "case", "$$", "H^q(\\check{\\mathcal{C}}^\\bullet(\\vec{e})) =", "\\left\\{", "\\begin{matrix}", "R \\cdot \\frac{1}{T_0^{-e_0} \\ldots T_n^{-e_n}} & \\text{if} & q = n \\\\", "0 & \\text{if} & \\text{else}", "\\end{matrix}", "\\right.", "$$", "The direct sum of all of these terms clearly gives the value", "$$", "\\left(\\frac{1}{T_0 \\ldots T_n} R[\\frac{1}{T_0}, \\ldots, \\frac{1}{T_n}]\\right)_d", "$$", "in degree $n$ as stated in the lemma. Moreover these terms do not contribute", "to cohomology in other degrees (also in accordance with the statement of the", "lemma).", "\\medskip\\noindent", "Assume $NEG(\\vec{e}) = \\emptyset$. In this case the complex", "$\\check{\\mathcal{C}}^\\bullet(\\vec{e})$ has a summand $R$ corresponding", "to all $i_0 < \\ldots < i_p$.", "Let us compare the complex $\\check{\\mathcal{C}}^\\bullet(\\vec{e})$", "to another complex. Namely, consider the affine open covering", "$$", "\\mathcal{V} : \\Spec(R) = \\bigcup\\nolimits_{i \\in \\{0, \\ldots, n\\}} V_i", "$$", "where $V_i = \\Spec(R)$ for all $i$. Consider the alternating", "{\\v C}ech complex", "$$", "\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{V}, \\mathcal{O}_{\\Spec(R)})", "$$", "By the same reasoning as above this computes the cohomology of the", "structure sheaf on $\\Spec(R)$. Hence we see that", "$H^p(", "\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{V}, \\mathcal{O}_{\\Spec(R)})", ") = R$ if $p = 0$ and is $0$ whenever $p > 0$.", "For these facts, see", "Lemma \\ref{lemma-cech-cohomology-quasi-coherent-trivial} and its proof.", "Note that also", "$\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{V}, \\mathcal{O}_{\\Spec(R)})$", "has a summand $R$ for every $i_0 < \\ldots < i_p$ and has exactly the same", "differential as $\\check{\\mathcal{C}}^\\bullet(\\vec{e})$. In other words", "these complexes are isomorphic complexes and hence have the same cohomology.", "We conclude that", "$$", "H^q(\\check{\\mathcal{C}}^\\bullet(\\vec{e})) =", "\\left\\{", "\\begin{matrix}", "R \\cdot T_0^{e_0} \\ldots T_n^{e_n} & \\text{if} & q = 0 \\\\", "0 & \\text{if} & \\text{else}", "\\end{matrix}", "\\right.", "$$", "in the case that $NEG(\\vec{e}) = \\emptyset$.", "The direct sum of all of these terms clearly gives the value", "$$", "(R[T_0, \\ldots, T_n])_d", "$$", "in degree $0$ as stated in the lemma. Moreover these terms do not contribute", "to cohomology in other degrees (also in accordance with the statement of the", "lemma).", "\\medskip\\noindent", "To finish the proof of the lemma we have to show that the complexes", "$\\check{\\mathcal{C}}^\\bullet(\\vec{e})$ are acyclic when", "$NEG(\\vec{e})$ is neither empty nor equal to $\\{0, \\ldots, n\\}$.", "Pick an index $i_{\\text{fix}} \\not \\in NEG(\\vec{e})$ (such an index exists).", "Consider the map", "$$", "h :", "\\check{\\mathcal{C}}^{p + 1}(\\vec{e})", "\\to", "\\check{\\mathcal{C}}^p(\\vec{e})", "$$", "given by the rule", "$$", "h(s)_{i_0 \\ldots i_p} = s_{i_{\\text{fix}} i_0 \\ldots i_p}", "$$", "(compare with the proof of", "Lemma \\ref{lemma-cech-cohomology-quasi-coherent-trivial}).", "It is clear that this is well defined since", "$$", "NEG(\\vec{e}) \\subset \\{i_0, \\ldots, i_p\\}", "\\Leftrightarrow", "NEG(\\vec{e}) \\subset \\{i_{\\text{fix}}, i_0, \\ldots, i_p\\}", "$$", "Also $\\check{\\mathcal{C}}^0(\\vec{e}) = 0$ so that this", "formula does work for all $p$ including $p = - 1$.", "The exact same (combinatorial) computation as in the", "proof of Lemma \\ref{lemma-cech-cohomology-quasi-coherent-trivial}", "shows that", "$$", "(hd + dh)(s)_{i_0 \\ldots i_p}", "=", "s_{i_0 \\ldots i_p}", "$$", "Hence we see that the identity map of the complex", "$\\check{\\mathcal{C}}^\\bullet(\\vec{e})$ is homotopic to zero", "which implies that it is acyclic." ], "refs": [ "coherent-lemma-cech-cohomology-quasi-coherent", "cohomology-lemma-ordered-alternating", "cohomology-lemma-alternating-usual", "coherent-lemma-cech-cohomology-quasi-coherent-trivial", "coherent-lemma-cech-cohomology-quasi-coherent-trivial", "coherent-lemma-cech-cohomology-quasi-coherent-trivial" ], "ref_ids": [ 3286, 2093, 2095, 3281, 3281, 3281 ] } ], "ref_ids": [] }, { "id": 3305, "type": "theorem", "label": "coherent-lemma-identify-functorially", "categories": [ "coherent" ], "title": "coherent-lemma-identify-functorially", "contents": [ "The identifications of Equation (\\ref{equation-identify}) are", "compatible with base change w.r.t.\\ ring maps $R \\to R'$.", "Moreover, for any $f \\in R[T_0, \\ldots, T_n]$ homogeneous", "of degree $m$ the map multiplication by $f$", "$$", "\\mathcal{O}_{\\mathbf{P}^n_R}(d)", "\\longrightarrow", "\\mathcal{O}_{\\mathbf{P}^n_R}(d + m)", "$$", "induces the map on the cohomology group via the identifications", "of Equation (\\ref{equation-identify}) which is multiplication by", "$f$ for $H^0$ and the contragredient of multiplication by $f$", "$$", "(R[T_0, \\ldots, T_n])_{-n - 1 - (d + m)}", "\\longrightarrow", "(R[T_0, \\ldots, T_n])_{-n - 1 - d}", "$$", "on $H^n$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $R \\to R'$ is a ring map.", "Let $\\mathcal{U}$ be the standard affine open covering of $\\mathbf{P}^n_R$,", "and let $\\mathcal{U}'$ be the standard affine open covering of", "$\\mathbf{P}^n_{R'}$. Note that $\\mathcal{U}'$ is the pullback of the covering", "$\\mathcal{U}$ under the canonical morphism", "$\\mathbf{P}^n_{R'} \\to \\mathbf{P}^n_R$. Hence there", "is a map of {\\v C}ech complexes", "$$", "\\gamma :", "\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U},", "\\mathcal{O}_{\\mathbf{P}_R}(d))", "\\longrightarrow", "\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}',", "\\mathcal{O}_{\\mathbf{P}_{R'}}(d))", "$$", "which is compatible with the map on cohomology by", "Cohomology, Lemma \\ref{cohomology-lemma-functoriality-cech}.", "It is clear from the computations in the proof of", "Lemma \\ref{lemma-cohomology-projective-space-over-ring}", "that this map of {\\v C}ech complexes is compatible with the identifications", "of the cohomology groups in question. (Namely the basis elements for", "the {\\v C}ech complex over $R$ simply map to the corresponding basis elements", "for the {\\v C}ech complex over $R'$.) Whence the first statement of the lemma.", "\\medskip\\noindent", "Now fix the ring $R$ and consider two homogeneous polynomials", "$f, g \\in R[T_0, \\ldots, T_n]$ both of the same degree $m$.", "Since cohomology is an additive functor, it is clear that the", "map induced by multiplication by $f + g$ is the same as the sum", "of the maps induced by multiplication by $f$ and the map induced", "by multiplication by $g$. Moreover, since cohomology is a functor,", "a similar result holds for multiplication by a product $fg$ where", "$f, g$ are both homogeneous (but not necessarily of the same degree).", "Hence to verify the second statement of the lemma it suffices to", "prove this when $f = x \\in R$ or when $f = T_i$.", "In the case of multiplication by an element $x \\in R$ the result", "follows since every cohomology groups or complex in sight has the", "structure of an $R$-module or complex of $R$-modules.", "Finally, we consider the case of multiplication by $T_i$", "as a $\\mathcal{O}_{\\mathbf{P}^n_R}$-linear map", "$$", "\\mathcal{O}_{\\mathbf{P}^n_R}(d)", "\\longrightarrow", "\\mathcal{O}_{\\mathbf{P}^n_R}(d + 1)", "$$", "The statement on $H^0$ is clear. For the statement on $H^n$", "consider multiplication by $T_i$ as a map on {\\v C}ech complexes", "$$", "\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U},", "\\mathcal{O}_{\\mathbf{P}_R}(d))", "\\longrightarrow", "\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U},", "\\mathcal{O}_{\\mathbf{P}_R}(d + 1))", "$$", "We are going to use the notation introduced in the proof of", "Lemma \\ref{lemma-cohomology-projective-space-over-ring}.", "We consider the effect of multiplication by $T_i$", "in terms of the decompositions", "$$", "\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{O}_{\\mathbf{P}_R}(d))", "=", "\\bigoplus\\nolimits_{\\vec{e} \\in \\mathbf{Z}^{n + 1}, \\ \\sum e_i = d}", "\\check{\\mathcal{C}}^\\bullet(\\vec{e})", "$$", "and", "$$", "\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U},", "\\mathcal{O}_{\\mathbf{P}_R}(d + 1))", "=", "\\bigoplus\\nolimits_{\\vec{e} \\in \\mathbf{Z}^{n + 1}, \\ \\sum e_i = d + 1}", "\\check{\\mathcal{C}}^\\bullet(\\vec{e})", "$$", "It is clear that it maps the subcomplex", "$\\check{\\mathcal{C}}^\\bullet(\\vec{e})$ to the subcomplex", "$\\check{\\mathcal{C}}^\\bullet(\\vec{e} + \\vec{b}_i)$ where", "$\\vec{b}_i = (0, \\ldots, 0, 1, 0, \\ldots, 0))$ the $i$th basis vector.", "In other words, it maps the summand of $H^n$ corresponding to", "$\\vec{e}$ with $e_i < 0$ and $\\sum e_i = d$", "to the summand of $H^n$ corresponding to", "$\\vec{e} + \\vec{b}_i$ (which is zero if $e_i + b_i \\geq 0$).", "It is easy to see that this corresponds exactly to the action", "of the contragredient of multiplication by $T_i$ as a map", "$$", "(R[T_0, \\ldots, T_n])_{-n - 1 - (d + 1)}", "\\longrightarrow", "(R[T_0, \\ldots, T_n])_{-n - 1 - d}", "$$", "This proves the lemma." ], "refs": [ "cohomology-lemma-functoriality-cech", "coherent-lemma-cohomology-projective-space-over-ring", "coherent-lemma-cohomology-projective-space-over-ring" ], "ref_ids": [ 2075, 3304, 3304 ] } ], "ref_ids": [] }, { "id": 3306, "type": "theorem", "label": "coherent-lemma-cohomology-projective-space-over-base", "categories": [ "coherent" ], "title": "coherent-lemma-cohomology-projective-space-over-base", "contents": [ "Let $S$ be a scheme.", "Let $n \\geq 0$ be an integer.", "Consider the structure morphism", "$$", "f : \\mathbf{P}^n_S \\longrightarrow S.", "$$", "We have", "$$", "R^qf_*(\\mathcal{O}_{\\mathbf{P}^n_S}(d)) =", "\\left\\{", "\\begin{matrix}", "(\\mathcal{O}_S[T_0, \\ldots, T_n])_d & \\text{if} & q = 0 \\\\", "0 & \\text{if} & q \\not = 0, n \\\\", "\\SheafHom_{\\mathcal{O}_S}(", "(\\mathcal{O}_S[T_0, \\ldots, T_n])_{- n - 1 - d}, \\mathcal{O}_S)", "& \\text{if} & q = n", "\\end{matrix}", "\\right.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This follows since the identifications in", "(\\ref{equation-identify}) are compatible with affine base change", "by Lemma \\ref{lemma-identify-functorially}." ], "refs": [ "coherent-lemma-identify-functorially" ], "ref_ids": [ 3305 ] } ], "ref_ids": [] }, { "id": 3307, "type": "theorem", "label": "coherent-lemma-cohomology-projective-bundle", "categories": [ "coherent" ], "title": "coherent-lemma-cohomology-projective-bundle", "contents": [ "Let $S$ be a scheme. Let $n \\geq 1$.", "Let $\\mathcal{E}$ be a finite locally", "free $\\mathcal{O}_S$-module of constant rank $n + 1$.", "Consider the structure morphism", "$$", "\\pi : \\mathbf{P}(\\mathcal{E}) \\longrightarrow S.", "$$", "We have", "$$", "R^q\\pi_*(\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(d)) =", "\\left\\{", "\\begin{matrix}", "\\text{Sym}^d(\\mathcal{E}) & \\text{if} & q = 0 \\\\", "0 & \\text{if} & q \\not = 0, n \\\\", "\\SheafHom_{\\mathcal{O}_S}(", "\\text{Sym}^{- n - 1 - d}(\\mathcal{E})", "\\otimes_{\\mathcal{O}_S}", "\\wedge^{n + 1}\\mathcal{E},", "\\mathcal{O}_S)", "& \\text{if} & q = n", "\\end{matrix}", "\\right.", "$$", "These identifications are compatible with base change and", "isomorphism between locally free sheaves." ], "refs": [], "proofs": [ { "contents": [ "Consider the canonical map", "$$", "\\pi^*\\mathcal{E} \\longrightarrow \\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(1)", "$$", "and twist down by $1$ to get", "$$", "\\pi^*(\\mathcal{E})(-1) \\longrightarrow \\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}", "$$", "This is a surjective map from a locally free rank $n + 1$ sheaf onto", "the structure sheaf. Hence the corresponding Koszul complex is", "exact (More on Algebra, Lemma", "\\ref{more-algebra-lemma-homotopy-koszul-abstract}).", "In other words there is an exact complex", "$$", "0 \\to", "\\pi^*(\\wedge^{n + 1}\\mathcal{E})(-n - 1) \\to", "\\ldots \\to", "\\pi^*(\\wedge^i\\mathcal{E})(-i) \\to", "\\ldots \\to", "\\pi^*\\mathcal{E}(-1) \\to", "\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})} \\to 0", "$$", "We will think of the term $\\pi^*(\\wedge^i\\mathcal{E})(-i)$ as being", "in degree $-i$.", "We are going to compute the higher direct images", "of this acyclic complex using the first spectral sequence of", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "Namely, we see that there is a spectral sequence with terms", "$$", "E_1^{p, q} = R^q\\pi_*\\left(\\pi^*(\\wedge^{-p}\\mathcal{E})(p)\\right)", "$$", "converging to zero! By the projection formula", "(Cohomology, Lemma \\ref{cohomology-lemma-projection-formula})", "we have", "$$", "E_1^{p, q} = \\wedge^{-p} \\mathcal{E} \\otimes_{\\mathcal{O}_S}", "R^q\\pi_*\\left(\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(p)\\right).", "$$", "Note that locally on $S$ the sheaf $\\mathcal{E}$ is trivial,", "i.e., isomorphic to $\\mathcal{O}_S^{\\oplus n + 1}$, hence locally on", "$S$ the morphism $\\mathbf{P}(\\mathcal{E}) \\to S$ can be identified", "with $\\mathbf{P}^n_S \\to S$. Hence", "locally on $S$ we can use the result of Lemmas", "\\ref{lemma-cohomology-projective-space-over-ring},", "\\ref{lemma-identify-functorially}, or", "\\ref{lemma-cohomology-projective-space-over-base}.", "It follows that $E_1^{p, q} = 0$ unless $(p, q)$ is $(0, 0)$", "or $(-n - 1, n)$. The nonzero terms are", "\\begin{align*}", "E_1^{0, 0} & = \\pi_*\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})} = \\mathcal{O}_S \\\\", "E_1^{-n - 1, n} & =", "R^n\\pi_*\\left(\\pi^*(\\wedge^{n + 1}\\mathcal{E})(-n - 1)\\right) =", "\\wedge^{n + 1}\\mathcal{E} \\otimes_{\\mathcal{O}_S}", "R^n\\pi_*\\left(\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(-n - 1)\\right)", "\\end{align*}", "Hence there can only be one nonzero", "differential in the spectral sequence namely the map", "$d_{n + 1}^{-n - 1, n} : E_{n + 1}^{-n - 1, n} \\to E_{n + 1}^{0, 0}$", "which has to be an isomorphism (because the spectral sequence converges", "to the $0$ sheaf). Thus $E_1^{p, q} = E_{n + 1}^{p, q}$ and", "we obtain a canonical isomorphism", "$$", "\\wedge^{n + 1}\\mathcal{E} \\otimes_{\\mathcal{O}_S}", "R^n\\pi_*\\left(\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(-n - 1)\\right) =", "R^n\\pi_*\\left(\\pi^*(\\wedge^{n + 1}\\mathcal{E})(-n - 1)\\right)", "\\xrightarrow{d_{n + 1}^{-n - 1, n}}", "\\mathcal{O}_S", "$$", "Since $\\wedge^{n + 1}\\mathcal{E}$ is an invertible", "sheaf, this implies that", "$R^n\\pi_*\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(-n - 1)$ is invertible", "as well and canonically isomorphic to the inverse of", "$\\wedge^{n + 1}\\mathcal{E}$. In other words we have proved the case", "$d = - n - 1$ of the lemma.", "\\medskip\\noindent", "Working locally on $S$ we see immediately from the computation of", "cohomology in Lemmas \\ref{lemma-cohomology-projective-space-over-ring},", "\\ref{lemma-identify-functorially}, or", "\\ref{lemma-cohomology-projective-space-over-base} the statements on", "vanishing of the lemma. Moreover the result on $R^0\\pi_*$ is clear", "as well, since there are canonical maps", "$\\text{Sym}^d(\\mathcal{E}) \\to \\pi_* \\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(d)$", "for all $d$. It remains to show that the description of", "$R^n\\pi_*\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(d)$ is correct", "for $d < -n - 1$. In order to do this we consider the map", "$$", "\\pi^*(\\text{Sym}^{-d - n - 1}(\\mathcal{E}))", "\\otimes_{\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}}", "\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(d)", "\\longrightarrow", "\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(-n - 1)", "$$", "Applying $R^n\\pi_*$ and the projection formula (see above) we get a map", "$$", "\\text{Sym}^{-d - n - 1}(\\mathcal{E})", "\\otimes_{\\mathcal{O}_S}", "R^n\\pi_*(\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(d))", "\\longrightarrow", "R^n\\pi_*\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(-n - 1) =", "(\\wedge^{n + 1}\\mathcal{E})^{\\otimes -1}", "$$", "(the last equality we have shown above).", "Again by the local calculations of Lemmas", "\\ref{lemma-cohomology-projective-space-over-ring},", "\\ref{lemma-identify-functorially}, or", "\\ref{lemma-cohomology-projective-space-over-base}", "it follows that this map induces a perfect pairing between", "$R^n\\pi_*(\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(d))$ and", "$\\text{Sym}^{-d - n - 1}(\\mathcal{E}) \\otimes \\wedge^{n + 1}(\\mathcal{E})$", "as desired." ], "refs": [ "more-algebra-lemma-homotopy-koszul-abstract", "derived-lemma-two-ss-complex-functor", "cohomology-lemma-projection-formula", "coherent-lemma-cohomology-projective-space-over-ring", "coherent-lemma-identify-functorially", "coherent-lemma-cohomology-projective-space-over-base", "coherent-lemma-cohomology-projective-space-over-ring", "coherent-lemma-identify-functorially", "coherent-lemma-cohomology-projective-space-over-base", "coherent-lemma-cohomology-projective-space-over-ring", "coherent-lemma-identify-functorially", "coherent-lemma-cohomology-projective-space-over-base" ], "ref_ids": [ 9959, 1871, 2243, 3304, 3305, 3306, 3304, 3305, 3306, 3304, 3305, 3306 ] } ], "ref_ids": [] }, { "id": 3308, "type": "theorem", "label": "coherent-lemma-coherent-Noetherian", "categories": [ "coherent" ], "title": "coherent-lemma-coherent-Noetherian", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is coherent,", "\\item $\\mathcal{F}$ is a quasi-coherent, finite type $\\mathcal{O}_X$-module,", "\\item $\\mathcal{F}$ is a finitely presented $\\mathcal{O}_X$-module,", "\\item for any affine open $\\Spec(A) = U \\subset X$ we have", "$\\mathcal{F}|_U = \\widetilde M$ with $M$ a finite $A$-module, and", "\\item there exists an affine open covering $X = \\bigcup U_i$,", "$U_i = \\Spec(A_i)$ such that each", "$\\mathcal{F}|_{U_i} = \\widetilde M_i$ with $M_i$ a finite $A_i$-module.", "\\end{enumerate}", "In particular $\\mathcal{O}_X$ is coherent, any invertible", "$\\mathcal{O}_X$-module is coherent, and more generally any", "finite locally free $\\mathcal{O}_X$-module is coherent." ], "refs": [], "proofs": [ { "contents": [ "The implications (1) $\\Rightarrow$ (2) and (1) $\\Rightarrow$ (3) hold", "in general, see", "Modules, Lemma \\ref{modules-lemma-coherent-finite-presentation}.", "If $\\mathcal{F}$ is finitely presented then $\\mathcal{F}$ is", "quasi-coherent, see", "Modules, Lemma \\ref{modules-lemma-finite-presentation-quasi-coherent}.", "Hence also (3) $\\Rightarrow$ (2).", "\\medskip\\noindent", "Assume $\\mathcal{F}$ is a quasi-coherent, finite type $\\mathcal{O}_X$-module.", "By", "Properties, Lemma \\ref{properties-lemma-finite-type-module}", "we see that on any affine open", "$\\Spec(A) = U \\subset X$ we have $\\mathcal{F}|_U = \\widetilde M$", "with $M$ a finite $A$-module. Since $A$ is Noetherian we see that", "$M$ has a finite resolution", "$$", "A^{\\oplus m} \\to A^{\\oplus n} \\to M \\to 0.", "$$", "Hence $\\mathcal{F}$ is of finite presentation by", "Properties, Lemma \\ref{properties-lemma-finite-presentation-module}.", "In other words (2) $\\Rightarrow$ (3).", "\\medskip\\noindent", "By Modules, Lemma \\ref{modules-lemma-coherent-structure-sheaf} it suffices", "to show that $\\mathcal{O}_X$ is coherent in order to show that (3)", "implies (1). Thus we have to show: given any open $U \\subset X$ and", "any finite collection of sections $f_i \\in \\mathcal{O}_X(U)$,", "$i = 1, \\ldots, n$ the kernel of the map", "$\\bigoplus_{i = 1, \\ldots, n} \\mathcal{O}_U \\to \\mathcal{O}_U$", "is of finite type. Since being of finite type is a local property", "it suffices to check this in a neighbourhood of any $x \\in U$.", "Thus we may assume $U = \\Spec(A)$ is affine. In this case", "$f_1, \\ldots, f_n \\in A$ are elements of $A$. Since $A$ is", "Noetherian, see", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian}", "the kernel $K$ of the map $\\bigoplus_{i = 1, \\ldots, n} A \\to A$", "is a finite $A$-module. See for example", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-basic}.", "As the functor\\ $\\widetilde{ }$\\ is exact, see", "Schemes, Lemma \\ref{schemes-lemma-spec-sheaves}", "we get an exact sequence", "$$", "\\widetilde K \\to", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} \\mathcal{O}_U \\to", "\\mathcal{O}_U", "$$", "and by", "Properties, Lemma \\ref{properties-lemma-finite-type-module}", "again we see that $\\widetilde K$ is of finite type. We conclude", "that (1), (2) and (3) are all equivalent.", "\\medskip\\noindent", "It follows from", "Properties, Lemma \\ref{properties-lemma-finite-type-module}", "that (2) implies (4). It is trivial that (4) implies (5).", "The discussion in", "Schemes, Section \\ref{schemes-section-quasi-coherent}", "show that (5) implies", "that $\\mathcal{F}$ is quasi-coherent and it is clear that (5)", "implies that $\\mathcal{F}$ is of finite type. Hence (5) implies", "(2) and we win." ], "refs": [ "modules-lemma-coherent-finite-presentation", "modules-lemma-finite-presentation-quasi-coherent", "properties-lemma-finite-type-module", "properties-lemma-finite-presentation-module", "modules-lemma-coherent-structure-sheaf", "properties-lemma-locally-Noetherian", "algebra-lemma-Noetherian-basic", "schemes-lemma-spec-sheaves", "properties-lemma-finite-type-module", "properties-lemma-finite-type-module" ], "ref_ids": [ 13254, 13248, 3002, 3003, 13256, 2951, 624, 7651, 3002, 3002 ] } ], "ref_ids": [] }, { "id": 3309, "type": "theorem", "label": "coherent-lemma-coherent-abelian-Noetherian", "categories": [ "coherent" ], "title": "coherent-lemma-coherent-abelian-Noetherian", "contents": [ "Let $X$ be a locally Noetherian scheme.", "The category of coherent $\\mathcal{O}_X$-modules is abelian.", "More precisely, the kernel and cokernel of a map of coherent", "$\\mathcal{O}_X$-modules are coherent. Any extension", "of coherent sheaves is coherent." ], "refs": [], "proofs": [ { "contents": [ "This is a restatement of", "Modules, Lemma \\ref{modules-lemma-coherent-abelian}", "in a particular case." ], "refs": [ "modules-lemma-coherent-abelian" ], "ref_ids": [ 13255 ] } ], "ref_ids": [] }, { "id": 3310, "type": "theorem", "label": "coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient", "categories": [ "coherent" ], "title": "coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Any quasi-coherent submodule of $\\mathcal{F}$ is coherent.", "Any quasi-coherent quotient module of $\\mathcal{F}$ is coherent." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $X$ is affine, say $X = \\Spec(A)$.", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian}", "implies that $A$ is Noetherian. Lemma \\ref{lemma-coherent-Noetherian}", "turns this into algebra. The algebraic counter part of", "the lemma is that a quotient, or a submodule of a finite $A$-module", "is a finite $A$-module, see for example", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-basic}." ], "refs": [ "properties-lemma-locally-Noetherian", "coherent-lemma-coherent-Noetherian", "algebra-lemma-Noetherian-basic" ], "ref_ids": [ 2951, 3308, 624 ] } ], "ref_ids": [] }, { "id": 3311, "type": "theorem", "label": "coherent-lemma-tensor-hom-coherent", "categories": [ "coherent" ], "title": "coherent-lemma-tensor-hom-coherent", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.", "The $\\mathcal{O}_X$-modules $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$", "and $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ are", "coherent." ], "refs": [], "proofs": [ { "contents": [ "It is shown in", "Modules, Lemma \\ref{modules-lemma-internal-hom-locally-kernel-direct-sum} that", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is coherent.", "The result for tensor products is", "Modules, Lemma \\ref{modules-lemma-tensor-product-permanence}" ], "refs": [ "modules-lemma-internal-hom-locally-kernel-direct-sum", "modules-lemma-tensor-product-permanence" ], "ref_ids": [ 13298, 13271 ] } ], "ref_ids": [] }, { "id": 3312, "type": "theorem", "label": "coherent-lemma-local-isomorphism", "categories": [ "coherent" ], "title": "coherent-lemma-local-isomorphism", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.", "Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a homomorphism", "of $\\mathcal{O}_X$-modules. Let $x \\in X$.", "\\begin{enumerate}", "\\item If $\\mathcal{F}_x = 0$ then there exists an open neighbourhood", "$U \\subset X$ of $x$ such that $\\mathcal{F}|_U = 0$.", "\\item If $\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ is injective,", "then there exists an open neighbourhood $U \\subset X$ of $x$ such that", "$\\varphi|_U$ is injective.", "\\item If $\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ is surjective,", "then there exists an open neighbourhood $U \\subset X$ of $x$ such that", "$\\varphi|_U$ is surjective.", "\\item If $\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ is bijective,", "then there exists an open neighbourhood $U \\subset X$ of $x$ such that", "$\\varphi|_U$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See Modules, Lemmas", "\\ref{modules-lemma-finite-type-surjective-on-stalk},", "\\ref{modules-lemma-finite-type-stalk-zero}, and", "\\ref{modules-lemma-finite-type-to-coherent-injective-on-stalk}." ], "refs": [ "modules-lemma-finite-type-surjective-on-stalk", "modules-lemma-finite-type-stalk-zero", "modules-lemma-finite-type-to-coherent-injective-on-stalk" ], "ref_ids": [ 13238, 13239, 13257 ] } ], "ref_ids": [] }, { "id": 3313, "type": "theorem", "label": "coherent-lemma-map-stalks-local-map", "categories": [ "coherent" ], "title": "coherent-lemma-map-stalks-local-map", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.", "Let $x \\in X$.", "Suppose $\\psi : \\mathcal{G}_x \\to \\mathcal{F}_x$ is a map of", "$\\mathcal{O}_{X, x}$-modules.", "Then there exists an open neighbourhood $U \\subset X$ of $x$ and a map", "$\\varphi : \\mathcal{G}|_U \\to \\mathcal{F}|_U$ such that", "$\\varphi_x = \\psi$." ], "refs": [], "proofs": [ { "contents": [ "In view of Lemma \\ref{lemma-coherent-Noetherian}", "this is a reformulation of", "Modules, Lemma \\ref{modules-lemma-stalk-internal-hom}." ], "refs": [ "coherent-lemma-coherent-Noetherian", "modules-lemma-stalk-internal-hom" ], "ref_ids": [ 3308, 13296 ] } ], "ref_ids": [] }, { "id": 3314, "type": "theorem", "label": "coherent-lemma-coherent-support-closed", "categories": [ "coherent" ], "title": "coherent-lemma-coherent-support-closed", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent", "$\\mathcal{O}_X$-module. Then $\\text{Supp}(\\mathcal{F})$ is closed, and", "$\\mathcal{F}$ comes from a coherent sheaf on the scheme theoretic support", "of $\\mathcal{F}$, see", "Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-support}." ], "refs": [ "morphisms-definition-scheme-theoretic-support" ], "proofs": [ { "contents": [ "Let $i : Z \\to X$ be the scheme theoretic support of $\\mathcal{F}$ and", "let $\\mathcal{G}$ be the finite type quasi-coherent sheaf on $Z$", "such that $i_*\\mathcal{G} \\cong \\mathcal{F}$.", "Since $Z = \\text{Supp}(\\mathcal{F})$ we see that the support is closed.", "The scheme $Z$ is locally Noetherian by", "Morphisms, Lemmas \\ref{morphisms-lemma-immersion-locally-finite-type}", "and \\ref{morphisms-lemma-finite-type-noetherian}.", "Finally, $\\mathcal{G}$ is a coherent $\\mathcal{O}_Z$-module by", "Lemma \\ref{lemma-coherent-Noetherian}" ], "refs": [ "morphisms-lemma-immersion-locally-finite-type", "morphisms-lemma-finite-type-noetherian", "coherent-lemma-coherent-Noetherian" ], "ref_ids": [ 5201, 5202, 3308 ] } ], "ref_ids": [ 5538 ] }, { "id": 3315, "type": "theorem", "label": "coherent-lemma-i-star-equivalence", "categories": [ "coherent" ], "title": "coherent-lemma-i-star-equivalence", "contents": [ "Let $i : Z \\to X$ be a closed immersion of locally Noetherian schemes.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent sheaf of ideals", "cutting out $Z$. The functor $i_*$ induces an equivalence between the", "category of coherent $\\mathcal{O}_X$-modules annihilated by $\\mathcal{I}$", "and the category of coherent $\\mathcal{O}_Z$-modules." ], "refs": [], "proofs": [ { "contents": [ "The functor is fully faithful by", "Morphisms, Lemma \\ref{morphisms-lemma-i-star-equivalence}.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module", "annihilated by $\\mathcal{I}$. By", "Morphisms, Lemma \\ref{morphisms-lemma-i-star-equivalence}", "we can write $\\mathcal{F} = i_*\\mathcal{G}$ for some quasi-coherent", "sheaf $\\mathcal{G}$ on $Z$. By", "Modules, Lemma \\ref{modules-lemma-i-star-reflects-finite-type}", "we see that $\\mathcal{G}$ is of finite type.", "Hence $\\mathcal{G}$ is coherent by", "Lemma \\ref{lemma-coherent-Noetherian}.", "Thus the functor is also essentially surjective as desired." ], "refs": [ "morphisms-lemma-i-star-equivalence", "morphisms-lemma-i-star-equivalence", "modules-lemma-i-star-reflects-finite-type", "coherent-lemma-coherent-Noetherian" ], "ref_ids": [ 5136, 5136, 13259, 3308 ] } ], "ref_ids": [] }, { "id": 3316, "type": "theorem", "label": "coherent-lemma-finite-pushforward-coherent", "categories": [ "coherent" ], "title": "coherent-lemma-finite-pushforward-coherent", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume $f$ is finite and $Y$ locally Noetherian.", "Then $R^pf_*\\mathcal{F} = 0$ for $p > 0$ and", "$f_*\\mathcal{F}$ is coherent if $\\mathcal{F}$ is coherent." ], "refs": [], "proofs": [ { "contents": [ "The higher direct images vanish by", "Lemma \\ref{lemma-relative-affine-vanishing} and because", "a finite morphism is affine (by definition).", "Note that the assumptions imply that also $X$ is locally Noetherian", "(see Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian})", "and hence the statement makes sense.", "Let $\\Spec(A) = V \\subset Y$ be an affine open subset.", "By Morphisms, Definition \\ref{morphisms-definition-integral}", "we see that $f^{-1}(V) = \\Spec(B)$ with $A \\to B$ finite.", "Lemma \\ref{lemma-coherent-Noetherian}", "turns the statement of the lemma into the following algebra", "fact: If $M$ is a finite $B$-module, then $M$ is also finite", "viewed as a $A$-module, see", "Algebra, Lemma \\ref{algebra-lemma-finite-module-over-finite-extension}." ], "refs": [ "coherent-lemma-relative-affine-vanishing", "morphisms-lemma-finite-type-noetherian", "morphisms-definition-integral", "coherent-lemma-coherent-Noetherian", "algebra-lemma-finite-module-over-finite-extension" ], "ref_ids": [ 3283, 5202, 5573, 3308, 336 ] } ], "ref_ids": [] }, { "id": 3317, "type": "theorem", "label": "coherent-lemma-coherent-support-dimension-0", "categories": [ "coherent" ], "title": "coherent-lemma-coherent-support-dimension-0", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$", "be a coherent sheaf with $\\dim(\\text{Supp}(\\mathcal{F})) \\leq 0$.", "Then $\\mathcal{F}$ is generated by global sections and", "$H^i(X, \\mathcal{F}) = 0$ for $i > 0$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-coherent-support-closed} we see that", "$\\mathcal{F} = i_*\\mathcal{G}$ where $i : Z \\to X$ is the inclusion", "of the scheme theoretic support of $\\mathcal{F}$ and where $\\mathcal{G}$", "is a coherent $\\mathcal{O}_Z$-module. Since the dimension of $Z$ is", "$0$, we see $Z$ is a disjoint union of affines (Properties, Lemma", "\\ref{properties-lemma-locally-Noetherian-dimension-0}).", "Hence $\\mathcal{G}$ is globally generated and the higher", "cohomology groups of $\\mathcal{G}$ are zero", "(Lemma \\ref{lemma-quasi-coherent-affine-cohomology-zero}).", "Hence $\\mathcal{F} = i_*\\mathcal{G}$ is globally generated.", "Since the cohomologies of $\\mathcal{F}$ and $\\mathcal{G}$ agree", "(Lemma \\ref{lemma-relative-affine-cohomology} applies as a", "closed immersion is affine)", "we conclude that the higher cohomology groups of $\\mathcal{F}$ are zero." ], "refs": [ "coherent-lemma-coherent-support-closed", "properties-lemma-locally-Noetherian-dimension-0", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "coherent-lemma-relative-affine-cohomology" ], "ref_ids": [ 3314, 2981, 3282, 3284 ] } ], "ref_ids": [] }, { "id": 3318, "type": "theorem", "label": "coherent-lemma-pushforward-coherent-on-open", "categories": [ "coherent" ], "title": "coherent-lemma-pushforward-coherent-on-open", "contents": [ "Let $X$ be a scheme. Let $j : U \\to X$ be the inclusion of an open.", "Let $T \\subset X$ be a closed subset contained in $U$.", "If $\\mathcal{F}$ is a coherent $\\mathcal{O}_U$-module", "with $\\text{Supp}(\\mathcal{F}) \\subset T$, then", "$j_*\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "Consider the open covering $X = U \\cup (X \\setminus T)$.", "Then $j_*\\mathcal{F}|_U = \\mathcal{F}$ is coherent and", "$j_*\\mathcal{F}|_{X \\setminus T} = 0$ is also coherent.", "Hence $j_*\\mathcal{F}$ is coherent." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3319, "type": "theorem", "label": "coherent-lemma-acc-coherent", "categories": [ "coherent" ], "title": "coherent-lemma-acc-coherent", "contents": [ "Let $X$ be a Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "The ascending chain condition holds for quasi-coherent submodules", "of $\\mathcal{F}$. In other words, given any sequence", "$$", "\\mathcal{F}_1 \\subset \\mathcal{F}_2 \\subset \\ldots \\subset \\mathcal{F}", "$$", "of quasi-coherent submodules, then", "$\\mathcal{F}_n = \\mathcal{F}_{n + 1} = \\ldots $ for some $n \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite affine open covering.", "On each member of the covering we get stabilization by", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-basic}.", "Hence the lemma follows." ], "refs": [ "algebra-lemma-Noetherian-basic" ], "ref_ids": [ 624 ] } ], "ref_ids": [] }, { "id": 3320, "type": "theorem", "label": "coherent-lemma-power-ideal-kills-sheaf", "categories": [ "coherent" ], "title": "coherent-lemma-power-ideal-kills-sheaf", "contents": [ "Let $X$ be a Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent", "sheaf of ideals corresponding to a closed subscheme $Z \\subset X$.", "Then there is some $n \\geq 0$ such that $\\mathcal{I}^n\\mathcal{F} = 0$", "if and only if $\\text{Supp}(\\mathcal{F}) \\subset Z$ (set theoretically)." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-power-ideal-kills-module}", "because $X$ has a finite covering by spectra of Noetherian rings." ], "refs": [ "algebra-lemma-Noetherian-power-ideal-kills-module" ], "ref_ids": [ 694 ] } ], "ref_ids": [] }, { "id": 3321, "type": "theorem", "label": "coherent-lemma-Artin-Rees", "categories": [ "coherent" ], "title": "coherent-lemma-Artin-Rees", "contents": [ "Let $X$ be a Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Let $\\mathcal{G} \\subset \\mathcal{F}$ be a quasi-coherent subsheaf.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of", "ideals.", "Then there exists a $c \\geq 0$ such that for all $n \\geq c$ we", "have", "$$", "\\mathcal{I}^{n - c}(\\mathcal{I}^c\\mathcal{F} \\cap \\mathcal{G})", "=", "\\mathcal{I}^n\\mathcal{F} \\cap \\mathcal{G}", "$$" ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from", "Algebra, Lemma \\ref{algebra-lemma-Artin-Rees}", "because $X$ has a finite covering by spectra of Noetherian rings." ], "refs": [ "algebra-lemma-Artin-Rees" ], "ref_ids": [ 625 ] } ], "ref_ids": [] }, { "id": 3322, "type": "theorem", "label": "coherent-lemma-homs-over-open", "categories": [ "coherent" ], "title": "coherent-lemma-homs-over-open", "contents": [ "Let $X$ be a Noetherian scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-module.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of", "ideals. Denote $Z \\subset X$ the corresponding closed subscheme and", "set $U = X \\setminus Z$.", "There is a canonical isomorphism", "$$", "\\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n\\mathcal{G}, \\mathcal{F})", "\\longrightarrow", "\\Hom_{\\mathcal{O}_U}(\\mathcal{G}|_U, \\mathcal{F}|_U).", "$$", "In particular we have an isomorphism", "$$", "\\colim_n \\Hom_{\\mathcal{O}_X}(", "\\mathcal{I}^n, \\mathcal{F})", "\\longrightarrow", "\\Gamma(U, \\mathcal{F}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "We first prove the second map is an isomorphism. It is injective by ", "Properties, Lemma \\ref{properties-lemma-sections-over-quasi-compact-open}.", "Since $\\mathcal{F}$ is the union of its coherent submodules, see", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-colimit-finite-type}", "(and Lemma \\ref{lemma-coherent-Noetherian})", "we may and do assume that $\\mathcal{F}$ is coherent to prove surjectivity.", "Let $\\mathcal{F}_n$ denote the quasi-coherent subsheaf of $\\mathcal{F}$", "consisting of sections annihilated by $\\mathcal{I}^n$,", "see Properties, Lemma \\ref{properties-lemma-sections-over-quasi-compact-open}.", "Since $\\mathcal{F}_1 \\subset \\mathcal{F}_2 \\subset \\ldots$ we see that", "$\\mathcal{F}_n = \\mathcal{F}_{n + 1} = \\ldots $ for some $n \\geq 0$", "by Lemma \\ref{lemma-acc-coherent}. Set $\\mathcal{H} = \\mathcal{F}_n$", "for this $n$. By Artin-Rees (Lemma \\ref{lemma-Artin-Rees})", "there exists an $c \\geq 0$ such that", "$\\mathcal{I}^m\\mathcal{F} \\cap \\mathcal{H}", "\\subset \\mathcal{I}^{m - c}\\mathcal{H}$. Picking $m = n + c$ we get", "$\\mathcal{I}^m\\mathcal{F} \\cap \\mathcal{H} \\subset \\mathcal{I}^n\\mathcal{H}", "= 0$. Thus if we set $\\mathcal{F}' = \\mathcal{I}^m\\mathcal{F}$ then we", "see that $\\mathcal{F}' \\cap \\mathcal{F}_n = 0$ and", "$\\mathcal{F}'|_U = \\mathcal{F}|_U$. Note in particular that the subsheaf", "$(\\mathcal{F}')_N$ of sections annihilated by $\\mathcal{I}^N$ is zero", "for all $N \\geq 0$. Hence by", "Properties, Lemma \\ref{properties-lemma-sections-over-quasi-compact-open}", "we deduce that", "the top horizontal arrow in the following commutative", "diagram is a bijection:", "$$", "\\xymatrix{", "\\colim_n \\Hom_{\\mathcal{O}_X}(", "\\mathcal{I}^n, \\mathcal{F}')", "\\ar[r] \\ar[d] &", "\\Gamma(U, \\mathcal{F}') \\ar[d] \\\\", "\\colim_n \\Hom_{\\mathcal{O}_X}(", "\\mathcal{I}^n, \\mathcal{F})", "\\ar[r] &", "\\Gamma(U, \\mathcal{F})", "}", "$$", "Since also the right vertical arrow is a bijection we conclude that", "the bottom horizontal arrow is surjective as desired.", "\\medskip\\noindent", "Next, we prove the first arrow of the lemma is a bijection.", "By Lemma \\ref{lemma-coherent-Noetherian} the sheaf $\\mathcal{G}$", "is of finite presentation and hence the sheaf", "$\\mathcal{H} = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{F})$", "is quasi-coherent, see", "Schemes, Section \\ref{schemes-section-quasi-coherent}.", "By definition we have", "$$", "\\mathcal{H}(U)", "=", "\\Hom_{\\mathcal{O}_U}(\\mathcal{G}|_U, \\mathcal{F}|_U)", "$$", "Pick a $\\psi$ in the right hand side of the first arrow of the", "lemma, i.e., $\\psi \\in \\mathcal{H}(U)$. The result just proved applies", "to $\\mathcal{H}$ and hence there exists an $n \\geq 0$ and an", "$\\varphi : \\mathcal{I}^n \\to \\mathcal{H}$ which recovers", "$\\psi$ on restriction to $U$. By", "Modules, Lemma \\ref{modules-lemma-internal-hom}", "$\\varphi$ corresponds to a map", "$$", "\\varphi :", "\\mathcal{I}^{\\otimes n} \\otimes_{\\mathcal{O}_X} \\mathcal{G}", "\\longrightarrow", "\\mathcal{F}.", "$$", "This is almost what we want except that the source of the arrow", "is the tensor product of $\\mathcal{I}^n$ and $\\mathcal{G}$", "and not the product. We will show that, at the cost of increasing $n$,", "the difference is irrelevant. Consider the short exact sequence", "$$", "0 \\to \\mathcal{K} \\to", "\\mathcal{I}^n \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to", "\\mathcal{I}^n\\mathcal{G} \\to 0", "$$", "where $\\mathcal{K}$ is defined as the kernel. Note that", "$\\mathcal{I}^n\\mathcal{K} = 0$ (proof omitted). By Artin-Rees", "again we see that", "$$", "\\mathcal{K}", "\\cap", "\\mathcal{I}^m(\\mathcal{I}^n \\otimes_{\\mathcal{O}_X} \\mathcal{G})", "=", "0", "$$", "for some $m$ large enough. In other words we see that", "$$", "\\mathcal{I}^m(\\mathcal{I}^n \\otimes_{\\mathcal{O}_X} \\mathcal{G})", "\\longrightarrow", "\\mathcal{I}^{n + m}\\mathcal{G}", "$$", "is an isomorphism. Let $\\varphi'$ be the restriction of", "$\\varphi$ to this submodule thought of as a map", "$\\mathcal{I}^{m + n}\\mathcal{G} \\to \\mathcal{F}$.", "Then $\\varphi'$ gives an element", "of the left hand side of the first arrow of the lemma which", "maps to $\\psi$ via the arrow. In other words we have proved surjectivity", "of the arrow. We omit the proof of injectivity." ], "refs": [ "properties-lemma-sections-over-quasi-compact-open", "properties-lemma-quasi-coherent-colimit-finite-type", "coherent-lemma-coherent-Noetherian", "properties-lemma-sections-over-quasi-compact-open", "coherent-lemma-acc-coherent", "coherent-lemma-Artin-Rees", "properties-lemma-sections-over-quasi-compact-open", "coherent-lemma-coherent-Noetherian", "modules-lemma-internal-hom" ], "ref_ids": [ 3039, 3020, 3308, 3039, 3319, 3321, 3039, 3308, 13294 ] } ], "ref_ids": [] }, { "id": 3323, "type": "theorem", "label": "coherent-lemma-extend-coherent", "categories": [ "coherent" ], "title": "coherent-lemma-extend-coherent", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$, $\\mathcal{G}$", "be coherent $\\mathcal{O}_X$-modules. Let $U \\subset X$ be open and", "let $\\varphi : \\mathcal{F}|_U \\to \\mathcal{G}|_U$ be an", "$\\mathcal{O}_U$-module map. Then there exists a coherent", "submodule $\\mathcal{F}' \\subset \\mathcal{F}$ agreeing with", "$\\mathcal{F}$ over $U$ such that $\\varphi$ extends to", "$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the coherent sheaf of ideals", "cutting out the reduced induced scheme structure on $X \\setminus U$.", "If $X$ is Noetherian, then Lemma \\ref{lemma-homs-over-open} tells us that", "we can take $\\mathcal{F}' = \\mathcal{I}^n\\mathcal{F}$", "for some $n$. The general case will follow from this using Zorn's lemma.", "\\medskip\\noindent", "Consider the set of triples $(U', \\mathcal{F}', \\varphi')$ where", "$U \\subset U' \\subset X$ is open, $\\mathcal{F}' \\subset \\mathcal{F}|_{U'}$", "is a coherent subsheaf agreeing with $\\mathcal{F}$ over $U$, and", "$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}|_{U'}$ restricts to", "$\\varphi$ over $U$. We say", "$(U'', \\mathcal{F}'', \\varphi'') \\geq (U', \\mathcal{F}', \\varphi')$", "if and only if $U'' \\supset U'$, $\\mathcal{F}''|_{U'} = \\mathcal{F}'$,", "and $\\varphi''|_{U'} = \\varphi'$.", "It is clear that if we have a totally ordered collection", "of triples $(U_i, \\mathcal{F}_i, \\varphi_i)$, then we", "can glue the $\\mathcal{F}_i$ to a subsheaf $\\mathcal{F}'$ of $\\mathcal{F}$", "over $U' = \\bigcup U_i$ and extend $\\varphi$ to a map", "$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}|_{U'}$.", "Hence any totally ordered subset of triples has an upper bound.", "Finally, suppose that $(U', \\mathcal{F}', \\varphi')$", "is any triple but $U' \\not = X$. Then we can choose an", "affine open $W \\subset X$ which is not contained in $U'$.", "By the result of the first paragraph we can extend", "the subsheaf $\\mathcal{F}'|_{W \\cap U'}$ and the restriction", "$\\varphi'|_{W \\cap U'}$ to some subsheaf $\\mathcal{F}'' \\subset \\mathcal{F}|_W$", "and map $\\varphi'' : \\mathcal{F}'' \\to \\mathcal{G}|_W$.", "Of course the agreement between $(\\mathcal{F}', \\varphi')$ and", "$(\\mathcal{F}'', \\varphi'')$ over $W \\cap U'$ exactly means that", "we can extend this to a triple $(U' \\cup W, \\mathcal{F}''', \\varphi''')$.", "Hence any maximal triple $(U', \\mathcal{F}', \\varphi')$", "(which exist by Zorn's lemma) must have $U' = X$ and the", "proof is complete." ], "refs": [ "coherent-lemma-homs-over-open" ], "ref_ids": [ 3322 ] } ], "ref_ids": [] }, { "id": 3324, "type": "theorem", "label": "coherent-lemma-hom-into-depth", "categories": [ "coherent" ], "title": "coherent-lemma-hom-into-depth", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$, $\\mathcal{G}$", "be coherent $\\mathcal{O}_X$-modules and $x \\in X$.", "\\begin{enumerate}", "\\item If $\\mathcal{G}_x$ has depth $\\geq 1$, then", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})_x$", "has depth $\\geq 1$.", "\\item If $\\mathcal{G}_x$ has depth $\\geq 2$, then", "$\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})_x$ has depth $\\geq 2$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is", "a coherent $\\mathcal{O}_X$-module by Lemma \\ref{lemma-tensor-hom-coherent}.", "Coherent modules are of finite presentation", "(Lemma \\ref{lemma-coherent-Noetherian}) hence taking stalks commutes", "with taking $\\SheafHom$ and $\\Hom$, see", "Modules, Lemma \\ref{modules-lemma-stalk-internal-hom}.", "Thus we reduce to the case of finite modules over local", "rings which is More on Algebra, Lemma \\ref{more-algebra-lemma-hom-into-depth}." ], "refs": [ "coherent-lemma-tensor-hom-coherent", "coherent-lemma-coherent-Noetherian", "modules-lemma-stalk-internal-hom", "more-algebra-lemma-hom-into-depth" ], "ref_ids": [ 3311, 3308, 13296, 9930 ] } ], "ref_ids": [] }, { "id": 3325, "type": "theorem", "label": "coherent-lemma-hom-into-S2", "categories": [ "coherent" ], "title": "coherent-lemma-hom-into-S2", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$, $\\mathcal{G}$", "be coherent $\\mathcal{O}_X$-modules.", "\\begin{enumerate}", "\\item If $\\mathcal{G}$ has property $(S_1)$, then", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ has property $(S_1)$.", "\\item If $\\mathcal{G}$ has property $(S_2)$, then", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ has property $(S_2)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-hom-into-depth}", "and the definitions." ], "refs": [ "coherent-lemma-hom-into-depth" ], "ref_ids": [ 3324 ] } ], "ref_ids": [] }, { "id": 3326, "type": "theorem", "label": "coherent-lemma-Cohen-Macaulay-over-regular", "categories": [ "coherent" ], "title": "coherent-lemma-Cohen-Macaulay-over-regular", "contents": [ "Let $X$ be a regular scheme. Let $\\mathcal{F}$ be a coherent", "$\\mathcal{O}_X$-module. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is Cohen-Macaulay and $\\text{Supp}(\\mathcal{F}) = X$,", "\\item $\\mathcal{F}$ is finite locally free of rank $> 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$. If (2) holds, then $\\mathcal{F}_x$ is a free", "$\\mathcal{O}_{X, x}$-module of rank $> 0$. Hence", "$\\text{depth}(\\mathcal{F}_x) = \\dim(\\mathcal{O}_{X, x})$", "because a regular local ring is Cohen-Macaulay", "(Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}).", "Conversely, if (1) holds, then $\\mathcal{F}_x$ is a", "maximal Cohen-Macaulay module over $\\mathcal{O}_{X, x}$", "(Algebra, Definition \\ref{algebra-definition-maximal-CM}).", "Hence $\\mathcal{F}_x$ is free by", "Algebra, Lemma \\ref{algebra-lemma-regular-mcm-free}." ], "refs": [ "algebra-lemma-regular-ring-CM", "algebra-definition-maximal-CM", "algebra-lemma-regular-mcm-free" ], "ref_ids": [ 941, 1503, 944 ] } ], "ref_ids": [] }, { "id": 3327, "type": "theorem", "label": "coherent-lemma-prepare-filter-support", "categories": [ "coherent" ], "title": "coherent-lemma-prepare-filter-support", "contents": [ "Let $X$ be a Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Suppose that $\\text{Supp}(\\mathcal{F}) = Z \\cup Z'$ with $Z$, $Z'$ closed.", "Then there exists a short exact sequence of coherent sheaves", "$$", "0 \\to \\mathcal{G}' \\to \\mathcal{F} \\to \\mathcal{G} \\to 0", "$$", "with $\\text{Supp}(\\mathcal{G}') \\subset Z'$ and", "$\\text{Supp}(\\mathcal{G}) \\subset Z$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the sheaf of ideals", "defining the reduced induced closed subscheme structure on $Z$, see", "Schemes, Lemma \\ref{schemes-lemma-reduced-closed-subscheme}.", "Consider the subsheaves", "$\\mathcal{G}'_n = \\mathcal{I}^n\\mathcal{F}$ and the", "quotients $\\mathcal{G}_n = \\mathcal{F}/\\mathcal{I}^n\\mathcal{F}$.", "For each $n$ we have a short exact sequence", "$$", "0 \\to \\mathcal{G}'_n \\to \\mathcal{F} \\to \\mathcal{G}_n \\to 0", "$$", "For every point $x$ of $Z' \\setminus Z$ we have", "$\\mathcal{I}_x = \\mathcal{O}_{X, x}$", "and hence $\\mathcal{G}_{n, x} = 0$. Thus we see that", "$\\text{Supp}(\\mathcal{G}_n) \\subset Z$. Note that $X \\setminus Z'$", "is a Noetherian scheme. Hence by Lemma \\ref{lemma-power-ideal-kills-sheaf}", "there exists an $n$ such that", "$\\mathcal{G}'_n|_{X \\setminus Z'} =", "\\mathcal{I}^n\\mathcal{F}|_{X \\setminus Z'} = 0$.", "For such an $n$ we see that $\\text{Supp}(\\mathcal{G}'_n) \\subset Z'$.", "Thus setting", "$\\mathcal{G}' = \\mathcal{G}'_n$ and $\\mathcal{G} = \\mathcal{G}_n$", "works." ], "refs": [ "schemes-lemma-reduced-closed-subscheme", "coherent-lemma-power-ideal-kills-sheaf" ], "ref_ids": [ 7681, 3320 ] } ], "ref_ids": [] }, { "id": 3328, "type": "theorem", "label": "coherent-lemma-prepare-filter-irreducible", "categories": [ "coherent" ], "title": "coherent-lemma-prepare-filter-irreducible", "contents": [ "Let $X$ be a Noetherian scheme.", "Let $i : Z \\to X$ be an integral closed subscheme.", "Let $\\xi \\in Z$ be the generic point.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Assume that $\\mathcal{F}_\\xi$ is annihilated by", "$\\mathfrak m_\\xi$. Then there exists an integer", "$r \\geq 0$ and a sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$", "and an injective map of coherent sheaves", "$$", "i_*\\left(\\mathcal{I}^{\\oplus r}\\right) \\to \\mathcal{F}", "$$", "which is an isomorphism in a neighbourhood of $\\xi$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{J} \\subset \\mathcal{O}_X$ be the ideal sheaf of $Z$.", "Let $\\mathcal{F}' \\subset \\mathcal{F}$ be the subsheaf of", "local sections of $\\mathcal{F}$ which are annihilated by", "$\\mathcal{J}$. It is a quasi-coherent sheaf by", "Properties, Lemma \\ref{properties-lemma-sections-annihilated-by-ideal}.", "Moreover, $\\mathcal{F}'_\\xi = \\mathcal{F}_\\xi$ because", "$\\mathcal{J}_\\xi = \\mathfrak m_\\xi$ and part (3) of", "Properties, Lemma \\ref{properties-lemma-sections-annihilated-by-ideal}.", "By Lemma \\ref{lemma-local-isomorphism} we see that", "$\\mathcal{F}' \\to \\mathcal{F}$", "induces an isomorphism in a neighbourhood of $\\xi$.", "Hence we may replace $\\mathcal{F}$ by $\\mathcal{F}'$ and assume", "that $\\mathcal{F}$ is annihilated by $\\mathcal{J}$.", "\\medskip\\noindent", "Assume $\\mathcal{J}\\mathcal{F} = 0$. By", "Lemma \\ref{lemma-i-star-equivalence} we can write", "$\\mathcal{F} = i_*\\mathcal{G}$ for some coherent", "sheaf $\\mathcal{G}$ on $Z$. Suppose we can find a morphism", "$\\mathcal{I}^{\\oplus r} \\to \\mathcal{G}$ which is an isomorphism", "in a neighbourhood of the generic point $\\xi$ of $Z$.", "Then applying $i_*$ (which is left exact) we get the result of the lemma.", "Hence we have reduced to the case $X = Z$.", "\\medskip\\noindent", "Suppose $Z = X$ is an integral Noetherian scheme with generic point $\\xi$.", "Note that $\\mathcal{O}_{X, \\xi} = \\kappa(\\xi)$ is the function field of $X$", "in this case.", "Since $\\mathcal{F}_\\xi$ is a finite $\\mathcal{O}_\\xi$-module we see", "that $r = \\dim_{\\kappa(\\xi)} \\mathcal{F}_\\xi$ is finite.", "Hence the sheaves $\\mathcal{O}_X^{\\oplus r}$ and $\\mathcal{F}$", "have isomorphic stalks at $\\xi$.", "By Lemma \\ref{lemma-map-stalks-local-map} there exists a nonempty", "open $U \\subset X$ and a morphism", "$\\psi : \\mathcal{O}_X^{\\oplus r}|_U \\to \\mathcal{F}|_U$", "which is an isomorphism", "at $\\xi$, and hence an isomorphism in a neighbourhood of $\\xi$ by", "Lemma \\ref{lemma-local-isomorphism}.", "By Schemes, Lemma \\ref{schemes-lemma-reduced-closed-subscheme}", "there exists a quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_X$", "whose associated closed subscheme $Z \\subset X$ is the complement", "of $U$.", "By Lemma \\ref{lemma-homs-over-open} there exists an $n \\geq 0$ and a morphism", "$\\mathcal{I}^n(\\mathcal{O}_X^{\\oplus r}) \\to \\mathcal{F}$", "which recovers our $\\psi$ over $U$. Since", "$\\mathcal{I}^n(\\mathcal{O}_X^{\\oplus r}) = (\\mathcal{I}^n)^{\\oplus r}$", "we get a map as in the lemma. It is injective because $X$ is", "integral and it is injective at the generic point of $X$", "(easy proof omitted)." ], "refs": [ "properties-lemma-sections-annihilated-by-ideal", "properties-lemma-sections-annihilated-by-ideal", "coherent-lemma-local-isomorphism", "coherent-lemma-i-star-equivalence", "coherent-lemma-map-stalks-local-map", "coherent-lemma-local-isomorphism", "schemes-lemma-reduced-closed-subscheme", "coherent-lemma-homs-over-open" ], "ref_ids": [ 3034, 3034, 3312, 3315, 3313, 3312, 7681, 3322 ] } ], "ref_ids": [] }, { "id": 3329, "type": "theorem", "label": "coherent-lemma-coherent-filter", "categories": [ "coherent" ], "title": "coherent-lemma-coherent-filter", "contents": [ "Let $X$ be a Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "There exists a filtration", "$$", "0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset", "\\ldots \\subset \\mathcal{F}_m = \\mathcal{F}", "$$", "by coherent subsheaves such that for each $j = 1, \\ldots, m$", "there exists an integral closed subscheme $Z_j \\subset X$", "and a sheaf of ideals $\\mathcal{I}_j \\subset \\mathcal{O}_{Z_j}$", "such that", "$$", "\\mathcal{F}_j/\\mathcal{F}_{j - 1}", "\\cong (Z_j \\to X)_* \\mathcal{I}_j", "$$" ], "refs": [], "proofs": [ { "contents": [ "Consider the collection", "$$", "\\mathcal{T} =", "\\left\\{", "\\begin{matrix}", "Z \\subset X", "\\text{ closed such that there exists a coherent sheaf }", "\\mathcal{F} \\\\", "\\text{ with }", "\\text{Supp}(\\mathcal{F}) = Z", "\\text{ for which the lemma is wrong}", "\\end{matrix}", "\\right\\}", "$$", "We are trying to show that $\\mathcal{T}$ is empty. If not, then", "because $X$ is Noetherian we can choose a minimal element", "$Z \\in \\mathcal{T}$. This means that there exists a coherent", "sheaf $\\mathcal{F}$ on $X$ whose support is $Z$ and for which the", "lemma does not hold. Clearly $Z \\not = \\emptyset$ since the only", "sheaf whose support is empty is the zero sheaf for which the", "lemma does hold (with $m = 0$).", "\\medskip\\noindent", "If $Z$ is not irreducible, then we can write $Z = Z_1 \\cup Z_2$", "with $Z_1, Z_2$ closed and strictly smaller than $Z$.", "Then we can apply Lemma \\ref{lemma-prepare-filter-support}", "to get a short exact sequence of coherent sheaves", "$$", "0 \\to", "\\mathcal{G}_1 \\to", "\\mathcal{F} \\to", "\\mathcal{G}_2 \\to 0", "$$", "with $\\text{Supp}(\\mathcal{G}_i) \\subset Z_i$. By minimality of", "$Z$ each of $\\mathcal{G}_i$ has a filtration as in the statement", "of the lemma. By considering the induced filtration on $\\mathcal{F}$", "we arrive at a contradiction. Hence we conclude", "that $Z$ is irreducible.", "\\medskip\\noindent", "Suppose $Z$ is irreducible. Let $\\mathcal{J}$ be the sheaf of ideals", "cutting out the reduced induced closed subscheme structure of $Z$,", "see Schemes, Lemma \\ref{schemes-lemma-reduced-closed-subscheme}.", "By Lemma \\ref{lemma-power-ideal-kills-sheaf} we see there exists", "an $n \\geq 0$ such that $\\mathcal{J}^n\\mathcal{F} = 0$. Hence we obtain", "a filtration", "$$", "0 = \\mathcal{J}^n\\mathcal{F} \\subset \\mathcal{J}^{n - 1}\\mathcal{F}", "\\subset \\ldots \\subset \\mathcal{J}\\mathcal{F} \\subset \\mathcal{F}", "$$", "each of whose successive subquotients is annihilated by $\\mathcal{J}$.", "Hence if each of these subquotients has a filtration as in the statement", "of the lemma then also $\\mathcal{F}$ does. In other words we may", "assume that $\\mathcal{J}$ does annihilate $\\mathcal{F}$.", "\\medskip\\noindent", "In the case where $Z$ is irreducible and $\\mathcal{J}\\mathcal{F} = 0$", "we can apply Lemma \\ref{lemma-prepare-filter-irreducible}.", "This gives a short exact sequence", "$$", "0 \\to", "i_*(\\mathcal{I}^{\\oplus r}) \\to", "\\mathcal{F} \\to", "\\mathcal{Q} \\to 0", "$$", "where $\\mathcal{Q}$ is defined as the quotient.", "Since $\\mathcal{Q}$ is zero in a neighbourhood of $\\xi$ by", "the lemma just cited we see that the support of $\\mathcal{Q}$", "is strictly smaller than $Z$. Hence we see that $\\mathcal{Q}$", "has a filtration of the desired type by minimality of $Z$.", "But then clearly $\\mathcal{F}$ does too, which is our final contradiction." ], "refs": [ "coherent-lemma-prepare-filter-support", "schemes-lemma-reduced-closed-subscheme", "coherent-lemma-power-ideal-kills-sheaf", "coherent-lemma-prepare-filter-irreducible" ], "ref_ids": [ 3327, 7681, 3320, 3328 ] } ], "ref_ids": [] }, { "id": 3330, "type": "theorem", "label": "coherent-lemma-property-initial", "categories": [ "coherent" ], "title": "coherent-lemma-property-initial", "contents": [ "Let $X$ be a Noetherian scheme.", "Let $\\mathcal{P}$ be a property of coherent sheaves on $X$. Assume", "\\begin{enumerate}", "\\item For any short exact sequence of coherent sheaves", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0", "$$", "if $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$", "then so does $\\mathcal{F}$.", "\\item For every integral closed subscheme $Z \\subset X$", "and every quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_Z$ we have", "$\\mathcal{P}$ for $i_*\\mathcal{I}$.", "\\end{enumerate}", "Then property $\\mathcal{P}$ holds for every coherent sheaf", "on $X$." ], "refs": [], "proofs": [ { "contents": [ "First note that if $\\mathcal{F}$ is a coherent sheaf with a filtration", "$$", "0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset", "\\ldots \\subset \\mathcal{F}_m = \\mathcal{F}", "$$", "by coherent subsheaves such that each of $\\mathcal{F}_i/\\mathcal{F}_{i - 1}$", "has property $\\mathcal{P}$, then so does $\\mathcal{F}$.", "This follows from the property (1) for $\\mathcal{P}$.", "On the other hand, by Lemma \\ref{lemma-coherent-filter}", "we can filter any $\\mathcal{F}$", "with successive subquotients as in (2).", "Hence the lemma follows." ], "refs": [ "coherent-lemma-coherent-filter" ], "ref_ids": [ 3329 ] } ], "ref_ids": [] }, { "id": 3331, "type": "theorem", "label": "coherent-lemma-property-irreducible", "categories": [ "coherent" ], "title": "coherent-lemma-property-irreducible", "contents": [ "Let $X$ be a Noetherian scheme. Let $Z_0 \\subset X$ be an irreducible closed", "subset with generic point $\\xi$. Let $\\mathcal{P}$ be a property of coherent", "sheaves on $X$ with support contained in $Z_0$ such that", "\\begin{enumerate}", "\\item For any short exact sequence of coherent sheaves if two", "out of three of them have property $\\mathcal{P}$ then so does the", "third.", "\\item For every integral closed subscheme $Z \\subset Z_0 \\subset X$,", "$Z \\not = Z_0$ and every quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_Z$ we have", "$\\mathcal{P}$ for $(Z \\to X)_*\\mathcal{I}$.", "\\item There exists some coherent sheaf $\\mathcal{G}$ on $X$ such that", "\\begin{enumerate}", "\\item $\\text{Supp}(\\mathcal{G}) = Z_0$,", "\\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$,", "\\item $\\dim_{\\kappa(\\xi)} \\mathcal{G}_\\xi = 1$, and", "\\item property $\\mathcal{P}$ holds for $\\mathcal{G}$.", "\\end{enumerate}", "\\end{enumerate}", "Then property $\\mathcal{P}$ holds for every coherent sheaf", "$\\mathcal{F}$ on $X$ whose support is contained in $Z_0$." ], "refs": [], "proofs": [ { "contents": [ "First note that if $\\mathcal{F}$ is a coherent sheaf with support", "contained in $Z_0$ with a filtration", "$$", "0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset", "\\ldots \\subset \\mathcal{F}_m = \\mathcal{F}", "$$", "by coherent subsheaves such that each of $\\mathcal{F}_i/\\mathcal{F}_{i - 1}$", "has property $\\mathcal{P}$, then so does $\\mathcal{F}$. Or, if $\\mathcal{F}$", "has property $\\mathcal{P}$ and all but one of the", "$\\mathcal{F}_i/\\mathcal{F}_{i - 1}$ has property $\\mathcal{P}$ then", "so does the last one. This follows from assumption (1).", "\\medskip\\noindent", "As a first application we conclude that any coherent sheaf whose support", "is strictly contained in $Z_0$ has property $\\mathcal{P}$. Namely, such a", "sheaf has a filtration (see Lemma \\ref{lemma-coherent-filter})", "whose subquotients have property $\\mathcal{P}$ according to (2).", "\\medskip\\noindent", "Let $\\mathcal{G}$ be as in (3). By Lemma \\ref{lemma-prepare-filter-irreducible}", "there exist a sheaf of ideals $\\mathcal{I}$ on $Z_0$, an", "integer $r \\geq 1$, and a short exact sequence", "$$", "0 \\to", "\\left((Z_0 \\to X)_*\\mathcal{I}\\right)^{\\oplus r} \\to", "\\mathcal{G} \\to", "\\mathcal{Q} \\to 0", "$$", "where the support of $\\mathcal{Q}$ is strictly contained in $Z_0$.", "By (3)(c) we see that $r = 1$. Since $\\mathcal{Q}$ has property $\\mathcal{P}$", "too we conclude that $(Z_0 \\to X)_*\\mathcal{I}$ has property", "$\\mathcal{P}$.", "\\medskip\\noindent", "Next, suppose that $\\mathcal{I}' \\not = 0$ is another quasi-coherent", "sheaf of ideals on $Z_0$. Then we can consider the intersection", "$\\mathcal{I}'' = \\mathcal{I}' \\cap \\mathcal{I}$ and we get", "two short exact sequences", "$$", "0 \\to", "(Z_0 \\to X)_*\\mathcal{I}'' \\to", "(Z_0 \\to X)_*\\mathcal{I} \\to", "\\mathcal{Q} \\to 0", "$$", "and", "$$", "0 \\to", "(Z_0 \\to X)_*\\mathcal{I}'' \\to", "(Z_0 \\to X)_*\\mathcal{I}' \\to", "\\mathcal{Q}' \\to 0.", "$$", "Note that the support of the coherent sheaves $\\mathcal{Q}$ and", "$\\mathcal{Q}'$ are strictly contained in $Z_0$.", "Hence $\\mathcal{Q}$ and $\\mathcal{Q}'$ have property $\\mathcal{P}$", "(see above). Hence we conclude using (1)", "that $(Z_0 \\to X)_*\\mathcal{I}''$ and $(Z_0 \\to X)_*\\mathcal{I}'$", "both have $\\mathcal{P}$ as well.", "\\medskip\\noindent", "The final step of the proof is to note that any coherent sheaf", "$\\mathcal{F}$ on $X$ whose support is contained in $Z_0$ has a filtration", "(see Lemma \\ref{lemma-coherent-filter} again) whose subquotients", "all have property $\\mathcal{P}$ by what we just said." ], "refs": [ "coherent-lemma-coherent-filter", "coherent-lemma-prepare-filter-irreducible", "coherent-lemma-coherent-filter" ], "ref_ids": [ 3329, 3328, 3329 ] } ], "ref_ids": [] }, { "id": 3332, "type": "theorem", "label": "coherent-lemma-property", "categories": [ "coherent" ], "title": "coherent-lemma-property", "contents": [ "Let $X$ be a Noetherian scheme.", "Let $\\mathcal{P}$ be a property of coherent sheaves on $X$ such that", "\\begin{enumerate}", "\\item For any short exact sequence of coherent sheaves if two", "out of three of them have property $\\mathcal{P}$ then so does the", "third.", "\\item For every integral closed subscheme $Z \\subset X$", "with generic point $\\xi$ there exists", "some coherent sheaf $\\mathcal{G}$ such that", "\\begin{enumerate}", "\\item $\\text{Supp}(\\mathcal{G}) = Z$,", "\\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$,", "\\item $\\dim_{\\kappa(\\xi)} \\mathcal{G}_\\xi = 1$, and", "\\item property $\\mathcal{P}$ holds for $\\mathcal{G}$.", "\\end{enumerate}", "\\end{enumerate}", "Then property $\\mathcal{P}$ holds for every coherent sheaf", "on $X$." ], "refs": [], "proofs": [ { "contents": [ "According to Lemma \\ref{lemma-property-initial} it suffices to show that", "for all integral closed subschemes $Z \\subset X$ and all quasi-coherent", "ideal sheaves $\\mathcal{I} \\subset \\mathcal{O}_Z$ we have $\\mathcal{P}$", "for $(Z \\to X)_*\\mathcal{I}$. If this fails, then since $X$ is Noetherian", "there is a minimal integral closed subscheme $Z_0 \\subset X$ such that", "$\\mathcal{P}$ fails for $(Z_0 \\to X)_*\\mathcal{I}_0$ for some", "quasi-coherent sheaf of ideals $\\mathcal{I}_0 \\subset \\mathcal{O}_{Z_0}$,", "but $\\mathcal{P}$ does hold for $(Z \\to X)_*\\mathcal{I}$ for all integral", "closed subschemes $Z \\subset Z_0$, $Z \\not = Z_0$ and quasi-coherent", "ideal sheaves $\\mathcal{I} \\subset \\mathcal{O}_Z$. Since we have the", "existence of $\\mathcal{G}$ for $Z_0$ by part (2), according to", "Lemma \\ref{lemma-property-irreducible} this cannot happen." ], "refs": [ "coherent-lemma-property-initial", "coherent-lemma-property-irreducible" ], "ref_ids": [ 3330, 3331 ] } ], "ref_ids": [] }, { "id": 3333, "type": "theorem", "label": "coherent-lemma-property-irreducible-higher-rank-cohomological", "categories": [ "coherent" ], "title": "coherent-lemma-property-irreducible-higher-rank-cohomological", "contents": [ "Let $X$ be a Noetherian scheme. Let $Z_0 \\subset X$ be an irreducible", "closed subset with generic point $\\xi$. Let $\\mathcal{P}$ be a property", "of coherent sheaves on $X$ such that", "\\begin{enumerate}", "\\item For any short exact sequence of coherent sheaves", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0", "$$", "if $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$", "then so does $\\mathcal{F}$.", "\\item If $\\mathcal{P}$ holds for $\\mathcal{F}^{\\oplus r}$ for", "some $r \\geq 1$, then it holds for $\\mathcal{F}$.", "\\item For every integral closed subscheme $Z \\subset Z_0 \\subset X$,", "$Z \\not = Z_0$ and every quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_Z$ we have", "$\\mathcal{P}$ for $(Z \\to X)_*\\mathcal{I}$.", "\\item There exists some coherent sheaf $\\mathcal{G}$ such that", "\\begin{enumerate}", "\\item $\\text{Supp}(\\mathcal{G}) = Z_0$,", "\\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$, and", "\\item for every quasi-coherent sheaf of ideals", "$\\mathcal{J} \\subset \\mathcal{O}_X$ such that", "$\\mathcal{J}_\\xi = \\mathcal{O}_{X, \\xi}$ there exists a quasi-coherent", "subsheaf $\\mathcal{G}' \\subset \\mathcal{J}\\mathcal{G}$ with", "$\\mathcal{G}'_\\xi = \\mathcal{G}_\\xi$ and such that", "$\\mathcal{P}$ holds for $\\mathcal{G}'$.", "\\end{enumerate}", "\\end{enumerate}", "Then property $\\mathcal{P}$ holds for every coherent sheaf", "$\\mathcal{F}$ on $X$ whose support is contained in $Z_0$." ], "refs": [], "proofs": [ { "contents": [ "Note that if $\\mathcal{F}$ is a coherent sheaf with a filtration", "$$", "0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset", "\\ldots \\subset \\mathcal{F}_m = \\mathcal{F}", "$$", "by coherent subsheaves such that each of $\\mathcal{F}_i/\\mathcal{F}_{i - 1}$", "has property $\\mathcal{P}$, then so does $\\mathcal{F}$.", "This follows from assumption (1).", "\\medskip\\noindent", "As a first application we conclude that any coherent sheaf whose support", "is strictly contained in $Z_0$ has property $\\mathcal{P}$. Namely, such a", "sheaf has a filtration (see Lemma \\ref{lemma-coherent-filter})", "whose subquotients have property $\\mathcal{P}$ according to (3).", "\\medskip\\noindent", "Let us denote $i : Z_0 \\to X$ the closed immersion.", "Consider a coherent sheaf $\\mathcal{G}$ as in (4).", "By Lemma \\ref{lemma-prepare-filter-irreducible}", "there exists a sheaf of ideals $\\mathcal{I}$ on $Z_0$ and", "a short exact sequence", "$$", "0 \\to", "i_*\\mathcal{I}^{\\oplus r} \\to", "\\mathcal{G} \\to", "\\mathcal{Q} \\to 0", "$$", "where the support of $\\mathcal{Q}$ is strictly contained in $Z_0$.", "In particular $r > 0$ and $\\mathcal{I}$ is nonzero", "because the support of $\\mathcal{G}$ is equal to $Z_0$.", "Let $\\mathcal{I}' \\subset \\mathcal{I}$ be any nonzero quasi-coherent", "sheaf of ideals on $Z_0$ contained in $\\mathcal{I}$.", "Then we also get a short exact sequence", "$$", "0 \\to", "i_*(\\mathcal{I}')^{\\oplus r} \\to", "\\mathcal{G} \\to", "\\mathcal{Q}' \\to 0", "$$", "where $\\mathcal{Q}'$ has support properly contained in $Z_0$.", "Let $\\mathcal{J} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf", "of ideals cutting out the support of $\\mathcal{Q}'$ (for example", "the ideal corresponding to the reduced induced closed subscheme", "structure on the support of $\\mathcal{Q}'$). Then", "$\\mathcal{J}_\\xi = \\mathcal{O}_{X, \\xi}$. By", "Lemma \\ref{lemma-power-ideal-kills-sheaf}", "we see that $\\mathcal{J}^n\\mathcal{Q}' = 0$ for some $n$.", "Hence $\\mathcal{J}^n\\mathcal{G} \\subset i_*(\\mathcal{I}')^{\\oplus r}$.", "By assumption (4)(c) of the lemma we see there exists", "a quasi-coherent subsheaf $\\mathcal{G}' \\subset \\mathcal{J}^n\\mathcal{G}$", "with $\\mathcal{G}'_\\xi = \\mathcal{G}_\\xi$", "for which property $\\mathcal{P}$ holds.", "Hence we get a short exact sequence", "$$", "0 \\to \\mathcal{G}' \\to", "i_*(\\mathcal{I}')^{\\oplus r} \\to", "\\mathcal{Q}'' \\to 0", "$$", "where $\\mathcal{Q}''$ has support properly contained in $Z_0$.", "Thus by our initial remarks and property (1) of the lemma", "we conclude that $i_*(\\mathcal{I}')^{\\oplus r}$ satisfies", "$\\mathcal{P}$. Hence we see that $i_*\\mathcal{I}'$ satisfies", "$\\mathcal{P}$ by (2). Finally, for an arbitrary quasi-coherent", "sheaf of ideals $\\mathcal{I}'' \\subset \\mathcal{O}_{Z_0}$ we can set", "$\\mathcal{I}' = \\mathcal{I}'' \\cap \\mathcal{I}$ and we get", "a short exact sequence", "$$", "0 \\to", "i_*(\\mathcal{I}') \\to", "i_*(\\mathcal{I}'') \\to", "\\mathcal{Q}''' \\to 0", "$$", "where $\\mathcal{Q}'''$ has support properly contained in $Z_0$.", "Hence we conclude that property $\\mathcal{P}$ holds for", "$i_*\\mathcal{I}''$ as well.", "\\medskip\\noindent", "The final step of the proof is to note that any coherent sheaf", "$\\mathcal{F}$ on $X$ whose support is contained in $Z_0$ has a filtration", "(see Lemma \\ref{lemma-coherent-filter} again) whose subquotients", "all have property $\\mathcal{P}$ by what we just said." ], "refs": [ "coherent-lemma-coherent-filter", "coherent-lemma-prepare-filter-irreducible", "coherent-lemma-power-ideal-kills-sheaf", "coherent-lemma-coherent-filter" ], "ref_ids": [ 3329, 3328, 3320, 3329 ] } ], "ref_ids": [] }, { "id": 3334, "type": "theorem", "label": "coherent-lemma-property-higher-rank-cohomological", "categories": [ "coherent" ], "title": "coherent-lemma-property-higher-rank-cohomological", "contents": [ "Let $X$ be a Noetherian scheme.", "Let $\\mathcal{P}$ be a property of coherent sheaves on $X$ such that", "\\begin{enumerate}", "\\item For any short exact sequence of coherent sheaves", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0", "$$", "if $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$", "then so does $\\mathcal{F}$.", "\\item If $\\mathcal{P}$ holds for $\\mathcal{F}^{\\oplus r}$ for", "some $r \\geq 1$, then it holds for $\\mathcal{F}$.", "\\item For every integral closed subscheme $Z \\subset X$", "with generic point $\\xi$ there exists", "some coherent sheaf $\\mathcal{G}$ such that", "\\begin{enumerate}", "\\item $\\text{Supp}(\\mathcal{G}) = Z$,", "\\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$, and", "\\item for every quasi-coherent sheaf of ideals", "$\\mathcal{J} \\subset \\mathcal{O}_X$ such that", "$\\mathcal{J}_\\xi = \\mathcal{O}_{X, \\xi}$ there exists a quasi-coherent", "subsheaf $\\mathcal{G}' \\subset \\mathcal{J}\\mathcal{G}$ with", "$\\mathcal{G}'_\\xi = \\mathcal{G}_\\xi$ and such that", "$\\mathcal{P}$ holds for $\\mathcal{G}'$.", "\\end{enumerate}", "\\end{enumerate}", "Then property $\\mathcal{P}$ holds for every coherent sheaf", "on $X$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-property-irreducible-higher-rank-cohomological}", "in exactly the same way that Lemma \\ref{lemma-property} follows from", "Lemma \\ref{lemma-property-irreducible}." ], "refs": [ "coherent-lemma-property-irreducible-higher-rank-cohomological", "coherent-lemma-property", "coherent-lemma-property-irreducible" ], "ref_ids": [ 3333, 3332, 3331 ] } ], "ref_ids": [] }, { "id": 3335, "type": "theorem", "label": "coherent-lemma-finite-morphism-Noetherian", "categories": [ "coherent" ], "title": "coherent-lemma-finite-morphism-Noetherian", "contents": [ "Let $f : Y \\to X$ be a morphism of schemes.", "Assume $f$ is finite, surjective and $X$ locally Noetherian.", "Let $Z \\subset X$ be an integral closed subscheme with", "generic point $\\xi$. Then", "there exists a coherent sheaf $\\mathcal{F}$ on $Y$", "such that the support of $f_*\\mathcal{F}$ is equal to $Z$", "and $(f_*\\mathcal{F})_\\xi$ is annihilated by $\\mathfrak m_\\xi$." ], "refs": [], "proofs": [ { "contents": [ "Note that $Y$ is locally Noetherian by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}.", "Because $f$ is surjective the fibre $Y_\\xi$ is not empty.", "Pick $\\xi' \\in Y$ mapping to $\\xi$. Let $Z' = \\overline{\\{\\xi'\\}}$.", "We may think of $Z' \\subset Y$ as a reduced closed subscheme,", "see Schemes, Lemma \\ref{schemes-lemma-reduced-closed-subscheme}.", "Hence the sheaf $\\mathcal{F} = (Z' \\to Y)_*\\mathcal{O}_{Z'}$", "is a coherent sheaf on $Y$ (see", "Lemma \\ref{lemma-finite-pushforward-coherent}).", "Look at the commutative diagram", "$$", "\\xymatrix{", "Z' \\ar[r]_{i'} \\ar[d]_{f'} &", "Y \\ar[d]^f \\\\", "Z \\ar[r]^i &", "X", "}", "$$", "We see that $f_*\\mathcal{F} = i_*f'_*\\mathcal{O}_{Z'}$.", "Hence the stalk of $f_*\\mathcal{F}$ at $\\xi$ is the stalk", "of $f'_*\\mathcal{O}_{Z'}$ at $\\xi$. Note that since $Z'$ is", "integral with generic point $\\xi'$ we have that", "$\\xi'$ is the only point of $Z'$ lying over $\\xi$, see", "Algebra, Lemmas \\ref{algebra-lemma-finite-is-integral} and", "\\ref{algebra-lemma-integral-no-inclusion}.", "Hence the stalk of $f'_*\\mathcal{O}_{Z'}$ at $\\xi$", "equal $\\mathcal{O}_{Z', \\xi'} = \\kappa(\\xi')$. In particular", "the stalk of $f_*\\mathcal{F}$ at $\\xi$ is not zero.", "This combined with the fact that $f_*\\mathcal{F}$ is", "of the form $i_*f'_*(\\text{something})$ implies the lemma." ], "refs": [ "morphisms-lemma-finite-type-noetherian", "schemes-lemma-reduced-closed-subscheme", "coherent-lemma-finite-pushforward-coherent", "algebra-lemma-finite-is-integral", "algebra-lemma-integral-no-inclusion" ], "ref_ids": [ 5202, 7681, 3316, 482, 498 ] } ], "ref_ids": [] }, { "id": 3336, "type": "theorem", "label": "coherent-lemma-affine-morphism-projection-ideal", "categories": [ "coherent" ], "title": "coherent-lemma-affine-morphism-projection-ideal", "contents": [ "Let $f : Y \\to X$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $Y$.", "Let $\\mathcal{I}$ be a quasi-coherent sheaf of ideals on $X$.", "If the morphism $f$ is affine then", "$\\mathcal{I}f_*\\mathcal{F} = f_*(f^{-1}\\mathcal{I}\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "The notation means the following. Since $f^{-1}$ is an exact functor", "we see that $f^{-1}\\mathcal{I}$ is a sheaf", "of ideals of $f^{-1}\\mathcal{O}_X$. Via the map", "$f^\\sharp : f^{-1}\\mathcal{O}_X \\to \\mathcal{O}_Y$ this acts on", "$\\mathcal{F}$. Then $f^{-1}\\mathcal{I}\\mathcal{F}$ is the subsheaf", "generated by sums of local sections of the form $as$ where $a$", "is a local section of $f^{-1}\\mathcal{I}$ and $s$ is a local section", "of $\\mathcal{F}$. It is a quasi-coherent $\\mathcal{O}_Y$-submodule", "of $\\mathcal{F}$ because it is also the image of a natural map", "$f^*\\mathcal{I} \\otimes_{\\mathcal{O}_Y} \\mathcal{F} \\to \\mathcal{F}$.", "\\medskip\\noindent", "Having said this the proof is straightforward. Namely, the question is local", "and hence we may assume $X$ is affine. Since $f$ is affine we see that", "$Y$ is affine too. Thus we may write", "$Y = \\Spec(B)$, $X = \\Spec(A)$, $\\mathcal{F} = \\widetilde{M}$,", "and $\\mathcal{I} = \\widetilde{I}$. The assertion of the lemma in this", "case boils down to the statement that", "$$", "I(M_A) = ((IB)M)_A", "$$", "where $M_A$ indicates the $A$-module associated to the $B$-module $M$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3337, "type": "theorem", "label": "coherent-lemma-image-affine-finite-morphism-affine-Noetherian", "categories": [ "coherent" ], "title": "coherent-lemma-image-affine-finite-morphism-affine-Noetherian", "contents": [ "Let $f : Y \\to X$ be a morphism of schemes.", "Assume", "\\begin{enumerate}", "\\item $f$ finite,", "\\item $f$ surjective,", "\\item $Y$ affine, and", "\\item $X$ Noetherian.", "\\end{enumerate}", "Then $X$ is affine." ], "refs": [], "proofs": [ { "contents": [ "We will prove that under the assumptions of the lemma for any coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ we have $H^1(X, \\mathcal{F}) = 0$.", "This will in particular imply that $H^1(X, \\mathcal{I}) = 0$", "for every quasi-coherent sheaf of ideals of $\\mathcal{O}_X$. Then it", "follows that $X$ is affine from either", "Lemma \\ref{lemma-quasi-compact-h1-zero-covering} or", "Lemma \\ref{lemma-quasi-separated-h1-zero-covering}.", "\\medskip\\noindent", "Let $\\mathcal{P}$ be the property of coherent sheaves", "$\\mathcal{F}$ on $X$ defined by the rule", "$$", "\\mathcal{P}(\\mathcal{F}) \\Leftrightarrow H^1(X, \\mathcal{F}) = 0.", "$$", "We are going to apply Lemma \\ref{lemma-property-higher-rank-cohomological}.", "Thus we have to verify (1), (2) and (3) of that lemma for $\\mathcal{P}$.", "Property (1) follows from the long exact cohomology sequence associated", "to a short exact sequence of sheaves. Property (2) follows since", "$H^1(X, -)$ is an additive functor. To see (3) let $Z \\subset X$ be", "an integral closed subscheme with generic point $\\xi$.", "Let $\\mathcal{F}$ be a coherent sheaf on $Y$ such that", "the support of $f_*\\mathcal{F}$ is equal to $Z$", "and $(f_*\\mathcal{F})_\\xi$ is annihilated by $\\mathfrak m_\\xi$,", "see Lemma \\ref{lemma-finite-morphism-Noetherian}. We claim that", "taking $\\mathcal{G} = f_*\\mathcal{F}$ works. We only have to verify", "part (3)(c) of Lemma \\ref{lemma-property-higher-rank-cohomological}.", "Hence assume that $\\mathcal{J} \\subset \\mathcal{O}_X$ is a", "quasi-coherent sheaf of ideals such that", "$\\mathcal{J}_\\xi = \\mathcal{O}_{X, \\xi}$.", "A finite morphism is affine hence by", "Lemma \\ref{lemma-affine-morphism-projection-ideal} we see that", "$\\mathcal{J}\\mathcal{G} = f_*(f^{-1}\\mathcal{J}\\mathcal{F})$.", "Also, as pointed out in the proof of", "Lemma \\ref{lemma-affine-morphism-projection-ideal} the sheaf", "$f^{-1}\\mathcal{J}\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_Y$-module.", "Since $Y$ is affine we see that $H^1(Y, f^{-1}\\mathcal{J}\\mathcal{F}) = 0$,", "see Lemma \\ref{lemma-quasi-coherent-affine-cohomology-zero}.", "Since $f$ is finite, hence affine, we see that", "$$", "H^1(X, \\mathcal{J}\\mathcal{G}) =", "H^1(X, f_*(f^{-1}\\mathcal{J}\\mathcal{F})) =", "H^1(Y, f^{-1}\\mathcal{J}\\mathcal{F}) = 0", "$$", "by Lemma \\ref{lemma-relative-affine-cohomology}.", "Hence the quasi-coherent subsheaf $\\mathcal{G}' = \\mathcal{J}\\mathcal{G}$", "satisfies $\\mathcal{P}$. This verifies property (3)(c) of", "Lemma \\ref{lemma-property-higher-rank-cohomological} as desired." ], "refs": [ "coherent-lemma-quasi-compact-h1-zero-covering", "coherent-lemma-quasi-separated-h1-zero-covering", "coherent-lemma-property-higher-rank-cohomological", "coherent-lemma-finite-morphism-Noetherian", "coherent-lemma-property-higher-rank-cohomological", "coherent-lemma-affine-morphism-projection-ideal", "coherent-lemma-affine-morphism-projection-ideal", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "coherent-lemma-relative-affine-cohomology", "coherent-lemma-property-higher-rank-cohomological" ], "ref_ids": [ 3287, 3288, 3334, 3335, 3334, 3336, 3336, 3282, 3284, 3334 ] } ], "ref_ids": [] }, { "id": 3338, "type": "theorem", "label": "coherent-lemma-coherent-projective", "categories": [ "coherent" ], "title": "coherent-lemma-coherent-projective", "contents": [ "Let $R$ be a Noetherian ring.", "Let $n \\geq 0$ be an integer.", "For every coherent sheaf $\\mathcal{F}$ on $\\mathbf{P}^n_R$", "we have the following:", "\\begin{enumerate}", "\\item There exists an $r \\geq 0$ and", "$d_1, \\ldots, d_r \\in \\mathbf{Z}$ and a surjection", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, r}", "\\mathcal{O}_{\\mathbf{P}^n_R}(d_j)", "\\longrightarrow", "\\mathcal{F}.", "$$", "\\item We have $H^i(\\mathbf{P}^n_R, \\mathcal{F}) = 0$ unless", "$0 \\leq i \\leq n$.", "\\item For any $i$ the cohomology group $H^i(\\mathbf{P}^n_R, \\mathcal{F})$", "is a finite $R$-module.", "\\item If $i > 0$, then", "$H^i(\\mathbf{P}^n_R, \\mathcal{F}(d)) = 0$ for all $d$ large enough.", "\\item For any $k \\in \\mathbf{Z}$ the graded $R[T_0, \\ldots, T_n]$-module", "$$", "\\bigoplus\\nolimits_{d \\geq k} H^0(\\mathbf{P}^n_R, \\mathcal{F}(d))", "$$", "is a finite $R[T_0, \\ldots, T_n]$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use that $\\mathcal{O}_{\\mathbf{P}^n_R}(1)$ is an ample invertible", "sheaf on", "the scheme $\\mathbf{P}^n_R$. This follows directly from the definition", "since $\\mathbf{P}^n_R$ covered by the standard affine opens $D_{+}(T_i)$.", "Hence by", "Properties, Proposition \\ref{properties-proposition-characterize-ample}", "every finite type quasi-coherent $\\mathcal{O}_{\\mathbf{P}^n_R}$-module", "is a quotient of a finite direct sum of tensor powers of", "$\\mathcal{O}_{\\mathbf{P}^n_R}(1)$. On the other hand coherent sheaves", "and finite type quasi-coherent sheaves are the same thing on projective", "space over $R$ by Lemma \\ref{lemma-coherent-Noetherian}. Thus we see (1).", "\\medskip\\noindent", "Projective $n$-space $\\mathbf{P}^n_R$ is covered by $n + 1$ affines,", "namely the standard opens $D_{+}(T_i)$, $i = 0, \\ldots, n$, see Constructions,", "Lemma \\ref{constructions-lemma-standard-covering-projective-space}.", "Hence we see that for any quasi-coherent", "sheaf $\\mathcal{F}$ on $\\mathbf{P}^n_R$", "we have $H^i(\\mathbf{P}^n_R, \\mathcal{F}) = 0$ for $i \\geq n + 1$,", "see Lemma \\ref{lemma-vanishing-nr-affines}. Hence (2) holds.", "\\medskip\\noindent", "Let us prove (3) and (4) simultaneously for all coherent sheaves", "on $\\mathbf{P}^n_R$ by descending induction on $i$. Clearly the result", "holds for $i \\geq n + 1$ by (2). Suppose we know the result for", "$i + 1$ and we want to show the result for $i$. (If $i = 0$, then", "part (4) is vacuous.) Let $\\mathcal{F}$ be a coherent sheaf on", "$\\mathbf{P}^n_R$. Choose a surjection as in (1) and denote", "$\\mathcal{G}$ the kernel so that we have a short exact sequence", "$$", "0 \\to \\mathcal{G} \\to", "\\bigoplus\\nolimits_{j = 1, \\ldots, r}", "\\mathcal{O}_{\\mathbf{P}^n_R}(d_j)", "\\to", "\\mathcal{F} \\to 0", "$$", "By Lemma \\ref{lemma-coherent-abelian-Noetherian}", "we see that $\\mathcal{G}$ is coherent. The long exact", "cohomology sequence gives an exact sequence", "$$", "H^i(\\mathbf{P}^n_R, \\bigoplus\\nolimits_{j = 1, \\ldots, r}", "\\mathcal{O}_{\\mathbf{P}^n_R}(d_j))", "\\to", "H^i(\\mathbf{P}^n_R, \\mathcal{F})", "\\to", "H^{i + 1}(\\mathbf{P}^n_R, \\mathcal{G}).", "$$", "By induction assumption the right $R$-module is finite and by", "Lemma \\ref{lemma-cohomology-projective-space-over-ring} the left", "$R$-module is finite. Since $R$ is Noetherian it follows immediately", "that $H^i(\\mathbf{P}^n_R, \\mathcal{F})$ is a finite $R$-module.", "This proves the induction step for assertion (3).", "Since $\\mathcal{O}_{\\mathbf{P}^n_R}(d)$ is invertible", "we see that twisting on $\\mathbf{P}^n_R$ is an exact functor (since", "you get it by tensoring with an invertible sheaf, see", "Constructions, Definition \\ref{constructions-definition-twist}).", "This means that for all $d \\in \\mathbf{Z}$ the sequence", "$$", "0 \\to \\mathcal{G}(d) \\to", "\\bigoplus\\nolimits_{j = 1, \\ldots, r}", "\\mathcal{O}_{\\mathbf{P}^n_R}(d_j + d)", "\\to", "\\mathcal{F}(d) \\to 0", "$$", "is short exact. The resulting cohomology sequence is", "$$", "H^i(\\mathbf{P}^n_R, \\bigoplus\\nolimits_{j = 1, \\ldots, r}", "\\mathcal{O}_{\\mathbf{P}^n_R}(d_j + d))", "\\to", "H^i(\\mathbf{P}^n_R, \\mathcal{F}(d))", "\\to", "H^{i + 1}(\\mathbf{P}^n_R, \\mathcal{G}(d)).", "$$", "By induction assumption we see the module on the right is zero", "for $d \\gg 0$ and by the computation in", "Lemma \\ref{lemma-cohomology-projective-space-over-ring}", "the module on the left is zero as soon as $d \\geq -\\min\\{d_j\\}$", "and $i \\geq 1$. Hence the induction step for assertion (4).", "This concludes the proof of (3) and (4).", "\\medskip\\noindent", "In order to prove (5) note that for all sufficiently large $d$", "the map", "$$", "H^0(\\mathbf{P}^n_R, \\bigoplus\\nolimits_{j = 1, \\ldots, r}", "\\mathcal{O}_{\\mathbf{P}^n_R}(d_j + d))", "\\to", "H^0(\\mathbf{P}^n_R, \\mathcal{F}(d))", "$$", "is surjective by the vanishing of $H^1(\\mathbf{P}^n_R, \\mathcal{G}(d))$", "we just proved. In other words, the module", "$$", "M_k", "=", "\\bigoplus\\nolimits_{d \\geq k} H^0(\\mathbf{P}^n_R, \\mathcal{F}(d))", "$$", "is for $k$ large enough a quotient of the corresponding module", "$$", "N_k", "=", "\\bigoplus\\nolimits_{d \\geq k} H^0(\\mathbf{P}^n_R,", "\\bigoplus\\nolimits_{j = 1, \\ldots, r}", "\\mathcal{O}_{\\mathbf{P}^n_R}(d_j + d)", ")", "$$", "When $k$ is sufficiently small (e.g.\\ $k < -d_j$ for all $j$) then", "$$", "N_k = \\bigoplus\\nolimits_{j = 1, \\ldots, r}", "R[T_0, \\ldots, T_n](d_j)", "$$", "by our computations in Section \\ref{section-cohomology-projective-space}.", "In particular it is finitely generated.", "Suppose $k \\in \\mathbf{Z}$ is arbitrary.", "Choose $k_{-} \\ll k \\ll k_{+}$.", "Consider the diagram", "$$", "\\xymatrix{", "N_{k_{-}} & N_{k_{+}} \\ar[d] \\ar[l] \\\\", "M_k & M_{k_{+}} \\ar[l]", "}", "$$", "where the vertical arrow is the surjective map above and", "the horizontal arrows are the obvious inclusion maps.", "By what was said above we see that $N_{k_{-}}$ is a finitely", "generated $R[T_0, \\ldots, T_n]$-module. Hence $N_{k_{+}}$ is", "a finitely generated $R[T_0, \\ldots, T_n]$-module because it", "is a submodule of a finitely generated module and the ring", "$R[T_0, \\ldots, T_n]$ is Noetherian. Since the vertical arrow", "is surjective we conclude that $M_{k_{+}}$ is a finitely", "generated $R[T_0, \\ldots, T_n]$-module. The quotient", "$M_k/M_{k_{+}}$ is finite as an $R$-module since it is a", "finite direct sum of the finite $R$-modules", "$H^0(\\mathbf{P}^n_R, \\mathcal{F}(d))$ for $k \\leq d < k_{+}$.", "Note that we use part (3) for $i = 0$ here. Hence", "$M_k/M_{k_{+}}$ is a fortiori a finite $R[T_0, \\ldots, T_n]$-module.", "In other words, we have sandwiched $M_k$ between two finite", "$R[T_0, \\ldots, T_n]$-modules and we win." ], "refs": [ "properties-proposition-characterize-ample", "coherent-lemma-coherent-Noetherian", "constructions-lemma-standard-covering-projective-space", "coherent-lemma-vanishing-nr-affines", "coherent-lemma-coherent-abelian-Noetherian", "coherent-lemma-cohomology-projective-space-over-ring", "constructions-definition-twist", "coherent-lemma-cohomology-projective-space-over-ring" ], "ref_ids": [ 3067, 3308, 12622, 3292, 3309, 3304, 12663, 3304 ] } ], "ref_ids": [] }, { "id": 3339, "type": "theorem", "label": "coherent-lemma-coherent-on-proj", "categories": [ "coherent" ], "title": "coherent-lemma-coherent-on-proj", "contents": [ "Let $A$ be a graded ring such that $A_0$ is Noetherian and", "$A$ is generated by finitely many elements of $A_1$ over $A_0$.", "Set $X = \\text{Proj}(A)$. Then $X$ is a Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item There exists an $r \\geq 0$ and", "$d_1, \\ldots, d_r \\in \\mathbf{Z}$ and a surjection", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, r} \\mathcal{O}_X(d_j)", "\\longrightarrow \\mathcal{F}.", "$$", "\\item For any $p$ the cohomology group $H^p(X, \\mathcal{F})$ is a finite", "$A_0$-module.", "\\item If $p > 0$, then $H^p(X, \\mathcal{F}(d)) = 0$ for all $d$ large enough.", "\\item For any $k \\in \\mathbf{Z}$ the graded $A$-module", "$$", "\\bigoplus\\nolimits_{d \\geq k} H^0(X, \\mathcal{F}(d))", "$$", "is a finite $A$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By assumption there exists a surjection of graded $A_0$-algebras", "$$", "A_0[T_0, \\ldots, T_n] \\longrightarrow A", "$$", "where $\\deg(T_j) = 1$ for $j = 0, \\ldots, n$. By Constructions, Lemma", "\\ref{constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj}", "this defines a closed immersion $i : X \\to \\mathbf{P}^n_{A_0}$", "such that $i^*\\mathcal{O}_{\\mathbf{P}^n_{A_0}}(1) = \\mathcal{O}_X(1)$.", "In particular, $X$ is Noetherian as a closed subscheme of the Noetherian", "scheme $\\mathbf{P}^n_{A_0}$. We claim that the results of the lemma for", "$\\mathcal{F}$ follow from the corresponding", "results of Lemma \\ref{lemma-coherent-projective} for the coherent sheaf", "$i_*\\mathcal{F}$ (Lemma \\ref{lemma-i-star-equivalence}) on", "$\\mathbf{P}^n_{A_0}$. For example, by this lemma there", "exists a surjection", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, r} \\mathcal{O}_{\\mathbf{P}^n_{A_0}}(d_j)", "\\longrightarrow i_*\\mathcal{F}.", "$$", "By adjunction this corresponds to a map", "$\\bigoplus_{j = 1, \\ldots, r} \\mathcal{O}_X(d_j) \\longrightarrow \\mathcal{F}$", "which is surjective as well. The statements on cohomology follow from the", "fact that", "$H^p(X, \\mathcal{F}(d)) = H^p(\\mathbf{P}^n_{A_0}, i_*\\mathcal{F}(d))$", "by Lemma \\ref{lemma-relative-affine-cohomology}." ], "refs": [ "constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj", "coherent-lemma-coherent-projective", "coherent-lemma-i-star-equivalence", "coherent-lemma-relative-affine-cohomology" ], "ref_ids": [ 12612, 3338, 3315, 3284 ] } ], "ref_ids": [] }, { "id": 3340, "type": "theorem", "label": "coherent-lemma-recover-tail-graded-module", "categories": [ "coherent" ], "title": "coherent-lemma-recover-tail-graded-module", "contents": [ "Let $A$ be a graded ring such that $A_0$ is Noetherian and $A$ is generated", "by finitely many elements of $A_1$ over $A_0$. Let $M$ be a", "finite graded $A$-module. Set $X = \\text{Proj}(A)$ and let $\\widetilde{M}$", "be the quasi-coherent $\\mathcal{O}_X$-module on $X$ associated to $M$.", "The maps", "$$", "M_n \\longrightarrow \\Gamma(X, \\widetilde{M}(n))", "$$", "from Constructions, Lemma \\ref{constructions-lemma-apply-modules}", "are isomorphisms for all sufficiently large $n$." ], "refs": [ "constructions-lemma-apply-modules" ], "proofs": [ { "contents": [ "Because $M$ is a finite $A$-module we see that", "$\\widetilde{M}$ is a finite type $\\mathcal{O}_X$-module,", "i.e., a coherent $\\mathcal{O}_X$-module.", "Set $N = \\bigoplus_{n \\geq 0} \\Gamma(X, \\widetilde{M}(n))$.", "We have to show that the map $M \\to N$ of graded $A$-modules", "is an isomorphism in all sufficiently large degrees.", "By Properties, Lemma \\ref{properties-lemma-proj-quasi-coherent}", "we have a canonical isomorphism $\\widetilde{N} \\to \\widetilde{M}$", "such that $M_n \\to N_n = \\Gamma(X, \\widetilde{M}(n))$", "is the canonical map. Let $K = \\Ker(M \\to N)$ and $Q = \\Coker(M \\to N)$.", "Recall that the functor", "$M \\mapsto \\widetilde{M}$ is exact, see", "Constructions, Lemma \\ref{constructions-lemma-proj-sheaves}.", "Hence we see that $\\widetilde{K} = 0$ and $\\widetilde{Q} = 0$.", "On the other hand, $A$ is a Noetherian ring and $M$ and $N$", "are finitely generated $A$-modules (for $N$ this follows from", "the last part of Lemma \\ref{lemma-coherent-on-proj}).", "Hence $K$ and $Q$ are finite $A$-modules. Thus it suffices to show", "that a finite $A$-module $K$ with $\\widetilde{K} = 0$", "has only finitely many nonzero homogeneous parts $K_d$.", "To do this, let $x_1, \\ldots, x_r \\in K$ be homogeneous generators", "say sitting in degrees $d_1, \\ldots, d_r$.", "Let $f_1, \\ldots, f_n \\in A_1$ be elements generating $A$ over $A_0$.", "For each $i$ and $j$ there exists an $n_{ij} \\geq 0$ such that", "$f_i^{n_{ij}} x_j = 0$ in $K_{d_j + n_{ij}}$: if not then", "$x_i/f_i^{d_i} \\in K_{(f_i)}$ would not be zero, i.e., $\\widetilde{K}$", "would not be zero.", "Then we see that $K_d$ is zero for $d > \\max_j(d_j + \\sum_i n_{ij})$", "as every element of $K_d$ is a sum of terms where each term is a", "monomials in the $f_i$ times one of the $x_j$ of total degree $d$." ], "refs": [ "properties-lemma-proj-quasi-coherent", "constructions-lemma-proj-sheaves", "coherent-lemma-coherent-on-proj" ], "ref_ids": [ 3058, 12594, 3339 ] } ], "ref_ids": [ 12603 ] }, { "id": 3341, "type": "theorem", "label": "coherent-lemma-coherent-on-proj-general", "categories": [ "coherent" ], "title": "coherent-lemma-coherent-on-proj-general", "contents": [ "Let $A$ be a Noetherian graded ring. Set $X = \\text{Proj}(A)$. Then $X$", "is a Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item There exists an $r \\geq 0$ and", "$d_1, \\ldots, d_r \\in \\mathbf{Z}$ and a surjection", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, r} \\mathcal{O}_X(d_j)", "\\longrightarrow \\mathcal{F}.", "$$", "\\item For any $p$ the cohomology group $H^p(X, \\mathcal{F})$ is a finite", "$A_0$-module.", "\\item If $p > 0$, then $H^p(X, \\mathcal{F}(d)) = 0$ for all $d$ large enough.", "\\item For any $k \\in \\mathbf{Z}$ the graded $A$-module", "$$", "\\bigoplus\\nolimits_{d \\geq k} H^0(X, \\mathcal{F}(d))", "$$", "is a finite $A$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will prove this by reducing the statement to", "Lemma \\ref{lemma-coherent-on-proj}.", "By Algebra, Lemmas \\ref{algebra-lemma-graded-Noetherian} and", "\\ref{algebra-lemma-S-plus-generated} the ring $A_0$ is Noetherian", "and $A$ is generated over $A_0$ by finitely many elements", "$f_1, \\ldots, f_r$ homogeneous of positive degree.", "Let $d$ be a sufficiently divisible integer. Set $A' = A^{(d)}$ with notation", "as in Algebra, Section \\ref{algebra-section-graded}.", "Then $A'$ is generated over $A'_0 = A_0$ by elements of", "degree $1$, see Algebra, Lemma \\ref{algebra-lemma-uple-generated-degree-1}.", "Thus Lemma \\ref{lemma-coherent-on-proj} applies to $X' = \\text{Proj}(A')$.", "\\medskip\\noindent", "By Constructions, Lemma \\ref{constructions-lemma-d-uple} there exist", "an isomorphism of schemes $i : X \\to X'$ and", "isomorphisms $\\mathcal{O}_X(nd) \\to i^*\\mathcal{O}_{X'}(n)$", "compatible with the map $A' \\to A$ and the maps", "$A_n \\to H^0(X, \\mathcal{O}_X(n))$ and $A'_n \\to H^0(X', \\mathcal{O}_{X'}(n))$.", "Thus Lemma \\ref{lemma-coherent-on-proj} implies $X$ is Noetherian and that", "(1) and (2) hold. To see (3) and (4)", "we can use that for any fixed $k$, $p$, and $q$ we have", "$$", "\\bigoplus\\nolimits_{dn + q \\geq k} H^p(X, \\mathcal{F}(dn + q)) =", "\\bigoplus\\nolimits_{dn + q \\geq k} H^p(X', (i_*\\mathcal{F}(q))(n)", "$$", "by the compatibilities above. If $p > 0$, we have the vanishing of the right", "hand side for $k$ depending on $q$ large enough by", "Lemma \\ref{lemma-coherent-on-proj}. Since there are only a finite number", "of congruence classes of integers modulo $d$, we see that (3) holds for", "$\\mathcal{F}$ on $X$. If $p = 0$, then we have that the right hand side", "is a finite $A'$-module by Lemma \\ref{lemma-coherent-on-proj}. Using", "the finiteness of congruence classes once more, we find that", "$\\bigoplus_{n \\geq k} H^0(X, \\mathcal{F}(n))$ is a finite $A'$-module too.", "Since the $A'$-module structure comes from the $A$-module structure", "(by the compatibilities mentioned above), we conclude it is finite", "as an $A$-module as well." ], "refs": [ "coherent-lemma-coherent-on-proj", "algebra-lemma-graded-Noetherian", "algebra-lemma-S-plus-generated", "algebra-lemma-uple-generated-degree-1", "coherent-lemma-coherent-on-proj", "constructions-lemma-d-uple", "coherent-lemma-coherent-on-proj", "coherent-lemma-coherent-on-proj", "coherent-lemma-coherent-on-proj" ], "ref_ids": [ 3339, 668, 667, 657, 3339, 12615, 3339, 3339, 3339 ] } ], "ref_ids": [] }, { "id": 3342, "type": "theorem", "label": "coherent-lemma-recover-tail-graded-module-general", "categories": [ "coherent" ], "title": "coherent-lemma-recover-tail-graded-module-general", "contents": [ "Let $A$ be a Noetherian graded ring and let $d$ be the lcm of generators", "of $A$ over $A_0$. Let $M$ be a finite graded $A$-module.", "Set $X = \\text{Proj}(A)$ and let $\\widetilde{M}$ be", "the quasi-coherent $\\mathcal{O}_X$-module on $X$ associated to $M$.", "Let $k \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item $N' = \\bigoplus_{n \\geq k} H^0(X, \\widetilde{M(n)})$", "is a finite $A$-module,", "\\item $N = \\bigoplus_{n \\geq k} H^0(X, \\widetilde{M}(n))$", "is a finite $A$-module,", "\\item there is a canonical map $N \\to N'$,", "\\item if $k$ is small enough there is a canonical map $M \\to N'$,", "\\item the map $M_n \\to N'_n$ is an isomorphism for $n \\gg 0$,", "\\item $N_n \\to N'_n$ is an isomorphism for $d | n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The map $N \\to N'$ in (3) comes from", "Constructions, Equation (\\ref{constructions-equation-multiply-more-generally})", "by taking global sections.", "\\medskip\\noindent", "By", "Constructions, Equation", "(\\ref{constructions-equation-global-sections-more-generally})", "there is a map of graded $A$-modules", "$M \\to \\bigoplus_{n \\in \\mathbf{Z}} H^0(X, \\widetilde{M(n)})$.", "If the generators of $M$ sit in degrees $\\geq k$, then the image", "is contained in the submodule", "$N' \\subset \\bigoplus_{n \\in \\mathbf{Z}} H^0(X, \\widetilde{M(n)})$", "and we get the map in (4).", "\\medskip\\noindent", "By Algebra, Lemmas \\ref{algebra-lemma-graded-Noetherian} and", "\\ref{algebra-lemma-S-plus-generated} the ring $A_0$ is Noetherian", "and $A$ is generated over $A_0$ by finitely many elements", "$f_1, \\ldots, f_r$ homogeneous of positive degree.", "Let $d = \\text{lcm}(\\deg(f_i))$. Then we see that (6) holds", "for example by", "Constructions, Lemma \\ref{constructions-lemma-where-invertible}.", "\\medskip\\noindent", "Because $M$ is a finite $A$-module we see that", "$\\widetilde{M}$ is a finite type $\\mathcal{O}_X$-module,", "i.e., a coherent $\\mathcal{O}_X$-module.", "Thus part (2) follows from Lemma \\ref{lemma-coherent-on-proj-general}.", "\\medskip\\noindent", "We will deduce (1) from (2) using a trick. For $q \\in \\{0, \\ldots, d - 1\\}$", "write", "$$", "{}^qN = \\bigoplus\\nolimits_{n + q \\geq k} H^0(X, \\widetilde{M(q)}(n))", "$$", "By part (2) these are finite $A$-modules. The Noetherian ring $A$", "is finite over $A^{(d)} = \\bigoplus_{n \\geq 0} A_{dn}$, because", "it is generated by $f_i$ over $A^{(d)}$ and $f_i^d \\in A^{(d)}$.", "Hence ${}^qN$ is a finite $A^{(d)}$-module.", "Moreover, $A^{(d)}$ is Noetherian (follows from", "Algebra, Lemma \\ref{algebra-lemma-dehomogenize-finite-type}).", "It follows that the $A^{(d)}$-submodule", "${}^qN^{(d)} = \\bigoplus_{n \\in \\mathbf{Z}} {}^qN_{dn}$", "is a finite module over $A^{(d)}$. Using the isomorphisms", "$\\widetilde{M(dn + q)} = \\widetilde{M(q)}(dn)$ we can write", "$$", "N' =", "\\bigoplus\\nolimits_{q \\in \\{0, \\ldots, d - 1\\}}", "\\bigoplus\\nolimits_{dn + q \\geq k}", "H^0(X, \\widetilde{M(q)}(dn)) =", "\\bigoplus\\nolimits_{q \\in \\{0, \\ldots, d - 1\\}} {}^qN^{(d)}", "$$", "Thus $N'$ is finite over $A^{(d)}$ and a fortiori finite over $A$.", "Thus (1) is true.", "\\medskip\\noindent", "Let $K$ be a finite $A$-module such that $\\widetilde{K} = 0$. We claim", "that $K_n = 0$ for $d|n$ and $n \\gg 0$. Arguing as above we see that", "$K^{(d)}$ is a finite $A^{(d)}$-module. Let", "$x_1, \\ldots, x_m \\in K$ be homogeneous generators of $K^{(d)}$", "over $A^{(d)}$, say sitting in degrees $d_1, \\ldots, d_m$ with $d | d_j$.", "For each $i$ and $j$ there exists an $n_{ij} \\geq 0$ such that", "$f_i^{n_{ij}} x_j = 0$ in $K_{d_j + n_{ij}}$: if not then", "$x_j/f_i^{d_i/\\deg(f_i)} \\in K_{(f_i)}$ would not be zero, i.e.,", "$\\widetilde{K}$ would not be zero. Here we use that $\\deg(f_i) | d | d_j$", "for all $i, j$. We conclude that $K_n$ is zero for", "$n$ with $d | n$ and $n > \\max_j (d_j + \\sum_i n_{ij} \\deg(f_i))$", "as every element of $K_n$ is a sum of terms where each term is a", "monomials in the $f_i$ times one of the $x_j$ of total degree $n$.", "\\medskip\\noindent", "To finish the proof, we have to show that $M \\to N'$ is an isomorphism", "in all sufficiently large degrees. The map $N \\to N'$ induces an", "isomorphism $\\widetilde{N} \\to \\widetilde{N'}$ because on the affine", "opens $D_+(f_i) = D_+(f_i^d)$ the corresponding modules are isomorphic:", "$N_{(f_i)} \\cong N_{(f_i^d)} \\cong N'_{(f_i^d)} \\cong N'_{(f_i)}$", "by property (6).", "By Properties, Lemma \\ref{properties-lemma-proj-quasi-coherent}", "we have a canonical isomorphism $\\widetilde{N} \\to \\widetilde{M}$.", "The composition $\\widetilde{N} \\to \\widetilde{M} \\to \\widetilde{N'}$", "is the isomorphism above (proof omitted; hint: look on standard", "affine opens to check this). Thus the map $M \\to N'$ induces an isomorphism", "$\\widetilde{M} \\to \\widetilde{N'}$.", "Let $K = \\Ker(M \\to N')$ and $Q = \\Coker(M \\to N')$.", "Recall that the functor", "$M \\mapsto \\widetilde{M}$ is exact, see", "Constructions, Lemma \\ref{constructions-lemma-proj-sheaves}.", "Hence we see that $\\widetilde{K} = 0$ and $\\widetilde{Q} = 0$.", "By the result of the previous paragraph we see that $K_n = 0$ and", "$Q_n = 0$ for $d | n$ and $n \\gg 0$. At this point we finally see", "the advantage of using $N'$ over $N$: the functor $M \\leadsto N'$", "is compatible with shifts (immediate from the construction).", "Thus, repeating the whole argument with $M$ replaced by $M(q)$", "we find that $K_n = 0$ and $Q_n = 0$ for $n \\equiv q \\bmod d$", "and $n \\gg 0$. Since there are", "only finitely many congruence classes modulo $n$ the proof is finished." ], "refs": [ "algebra-lemma-graded-Noetherian", "algebra-lemma-S-plus-generated", "constructions-lemma-where-invertible", "coherent-lemma-coherent-on-proj-general", "algebra-lemma-dehomogenize-finite-type", "properties-lemma-proj-quasi-coherent", "constructions-lemma-proj-sheaves" ], "ref_ids": [ 668, 667, 12604, 3341, 665, 3058, 12594 ] } ], "ref_ids": [] }, { "id": 3343, "type": "theorem", "label": "coherent-lemma-coherent-proper-ample", "categories": [ "coherent" ], "title": "coherent-lemma-coherent-proper-ample", "contents": [ "Let $R$ be a Noetherian ring. Let $X \\to \\Spec(R)$ be a proper morphism.", "Let $\\mathcal{L}$ be an ample invertible sheaf on $X$. Let $\\mathcal{F}$", "be a coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item The graded ring", "$A = \\bigoplus_{d \\geq 0} H^0(X, \\mathcal{L}^{\\otimes d})$", "is a finitely generated $R$-algebra.", "\\item There exists an $r \\geq 0$ and", "$d_1, \\ldots, d_r \\in \\mathbf{Z}$ and a surjection", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, r} \\mathcal{L}^{\\otimes d_j}", "\\longrightarrow \\mathcal{F}.", "$$", "\\item For any $p$ the cohomology group $H^p(X, \\mathcal{F})$ is a finite", "$R$-module.", "\\item If $p > 0$, then", "$H^p(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) = 0$", "for all $d$ large enough.", "\\item For any $k \\in \\mathbf{Z}$ the graded $A$-module", "$$", "\\bigoplus\\nolimits_{d \\geq k}", "H^0(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})", "$$", "is a finite $A$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-over-affine-ample-very-ample}", "there exists a $d > 0$ and an immersion $i : X \\to \\mathbf{P}^n_R$", "such that $\\mathcal{L}^{\\otimes d} \\cong i^*\\mathcal{O}_{\\mathbf{P}^n_R}(1)$.", "Since $X$ is proper over $R$ the morphism $i$ is a closed immersion", "(Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}).", "Thus we have $H^i(X, \\mathcal{G}) = H^i(\\mathbf{P}^n_R, i_*\\mathcal{G})$", "for any quasi-coherent sheaf $\\mathcal{G}$ on $X$", "(by Lemma \\ref{lemma-relative-affine-cohomology} and the fact that", "closed immersions are affine, see", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-affine}).", "Moreover, if $\\mathcal{G}$ is coherent, then $i_*\\mathcal{G}$", "is coherent as well (Lemma \\ref{lemma-i-star-equivalence}).", "We will use these facts without further mention.", "\\medskip\\noindent", "Proof of (1). Set $S = R[T_0, \\ldots, T_n]$ so that", "$\\mathbf{P}^n_R = \\text{Proj}(S)$.", "Observe that $A$ is an $S$-algebra (but the ring map $S \\to A$ is not", "a homomorphism of graded rings because $S_n$ maps into $A_{dn}$).", "By the projection formula", "(Cohomology, Lemma \\ref{cohomology-lemma-projection-formula})", "we have", "$$", "i_*(\\mathcal{L}^{\\otimes nd + q}) =", "i_*(\\mathcal{L}^{\\otimes q})", "\\otimes_{\\mathcal{O}_{\\mathbf{P}^n_R}}", "\\mathcal{O}_{\\mathbf{P}^n_R}(n)", "$$", "for all $n \\in \\mathbf{Z}$. We conclude that $\\bigoplus_{n \\geq 0} A_{nd + q}$", "is a finite graded $S$-module by Lemma \\ref{lemma-coherent-projective}.", "Since", "$A = \\bigoplus_{q \\in \\{0, \\ldots, d - 1} \\bigoplus_{n \\geq 0} A_{nd + q}$", "we see that $A$ is finite as an $S$-algebra, hence (1) is true.", "\\medskip\\noindent", "Proof of (2). This follows from", "Properties, Proposition \\ref{properties-proposition-characterize-ample}.", "\\medskip\\noindent", "Proof of (3). Apply Lemma \\ref{lemma-coherent-projective}", "and use $H^p(X, \\mathcal{F}) = H^p(\\mathbf{P}^n_R, i_*\\mathcal{F})$.", "\\medskip\\noindent", "Proof of (4). Fix $p > 0$. By the projection formula we have", "$$", "i_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes nd + q}) =", "i_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes q})", "\\otimes_{\\mathcal{O}_{\\mathbf{P}^n_R}}", "\\mathcal{O}_{\\mathbf{P}^n_R}(n)", "$$", "for all $n \\in \\mathbf{Z}$. By Lemma \\ref{lemma-coherent-projective}", "we conclude that $H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{nd + q}) = 0$", "for $n \\gg 0$. Since there are only finitely many congruence classes", "of integers modulo $d$ this proves (4).", "\\medskip\\noindent", "Proof of (5). Fix an integer $k$. Set", "$M = \\bigoplus_{n \\geq k} H^0(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n})$.", "Arguing as above we conclude that $\\bigoplus_{nd + q \\geq k} A_{nd + q}$", "is a finite graded $S$-module. Since", "$M = \\bigoplus_{q \\in \\{0, \\ldots, d - 1\\}}", "\\bigoplus_{nd + q \\geq k} M_{nd + q}$", "we see that $M$ is finite as an $S$-module. Since the $S$-module structure", "factors through the ring map $S \\to A$, we conclude that $M$ is finite", "as an $A$-module." ], "refs": [ "morphisms-lemma-finite-type-over-affine-ample-very-ample", "morphisms-lemma-image-proper-scheme-closed", "coherent-lemma-relative-affine-cohomology", "morphisms-lemma-closed-immersion-affine", "coherent-lemma-i-star-equivalence", "cohomology-lemma-projection-formula", "coherent-lemma-coherent-projective", "properties-proposition-characterize-ample", "coherent-lemma-coherent-projective", "coherent-lemma-coherent-projective" ], "ref_ids": [ 5394, 5411, 3284, 5177, 3315, 2243, 3338, 3067, 3338, 3338 ] } ], "ref_ids": [] }, { "id": 3344, "type": "theorem", "label": "coherent-lemma-kill-by-twisting", "categories": [ "coherent" ], "title": "coherent-lemma-kill-by-twisting", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "Assume that", "\\begin{enumerate}", "\\item $S$ is Noetherian,", "\\item $f$ is proper,", "\\item $\\mathcal{F}$ is coherent, and", "\\item $\\mathcal{L}$ is relatively ample on $X/S$.", "\\end{enumerate}", "Then there exists an $n_0$ such that for all $n \\geq n_0$", "we have", "$$", "R^pf_*\\left(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}\\right)", "=", "0", "$$", "for all $p > 0$." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite affine open covering $S = \\bigcup V_j$ and", "set $X_j = f^{-1}(V_j)$.", "Clearly, if we solve the question for each of the finitely many", "systems $(X_j \\to V_j, \\mathcal{L}|_{X_j}, \\mathcal{F}|_{V_j})$", "then the result follows. Thus we may assume $S$ is affine.", "In this case the vanishing of", "$R^pf_*(\\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n})$", "is equivalent to the vanishing of", "$H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n})$, see", "Lemma \\ref{lemma-quasi-coherence-higher-direct-images-application}.", "Thus the required vanishing follows", "from Lemma \\ref{lemma-coherent-proper-ample} (which applies because", "$\\mathcal{L}$ is ample on $X$ by Morphisms, Lemma", "\\ref{morphisms-lemma-finite-type-over-affine-ample-very-ample})." ], "refs": [ "coherent-lemma-quasi-coherence-higher-direct-images-application", "coherent-lemma-coherent-proper-ample", "morphisms-lemma-finite-type-over-affine-ample-very-ample" ], "ref_ids": [ 3296, 3343, 5394 ] } ], "ref_ids": [] }, { "id": 3345, "type": "theorem", "label": "coherent-lemma-locally-projective-pushforward", "categories": [ "coherent" ], "title": "coherent-lemma-locally-projective-pushforward", "contents": [ "Let $S$ be a locally Noetherian scheme.", "Let $f : X \\to S$ be a locally projective morphism.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Then $R^if_*\\mathcal{F}$ is a coherent $\\mathcal{O}_S$-module", "for all $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "We first remark that a locally projective morphism is proper", "(Morphisms, Lemma \\ref{morphisms-lemma-locally-projective-proper})", "and hence of finite type.", "In particular $X$ is locally Noetherian", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian})", "and hence the statement makes sense.", "Moreover, by Lemma \\ref{lemma-quasi-coherence-higher-direct-images}", "the sheaves $R^pf_*\\mathcal{F}$ are quasi-coherent.", "\\medskip\\noindent", "Having said this the statement is local on $S$ (for example by", "Cohomology, Lemma \\ref{cohomology-lemma-localize-higher-direct-images}).", "Hence we may assume $S = \\Spec(R)$ is the spectrum of", "a Noetherian ring, and $X$ is a closed subscheme of", "$\\mathbf{P}^n_R$ for some $n$, see", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-locally-projective}.", "In this case, the sheaves $R^pf_*\\mathcal{F}$ are the quasi-coherent", "sheaves associated to the $R$-modules $H^p(X, \\mathcal{F})$, see", "Lemma \\ref{lemma-quasi-coherence-higher-direct-images-application}.", "Hence it suffices to show that $R$-modules $H^p(X, \\mathcal{F})$", "are finite $R$-modules (Lemma \\ref{lemma-coherent-Noetherian}).", "This follows from Lemma \\ref{lemma-coherent-proper-ample}", "(because the restriction of $\\mathcal{O}_{\\mathbf{P}^n_R}(1)$", "to $X$ is ample on $X$)." ], "refs": [ "morphisms-lemma-locally-projective-proper", "morphisms-lemma-finite-type-noetherian", "coherent-lemma-quasi-coherence-higher-direct-images", "cohomology-lemma-localize-higher-direct-images", "morphisms-lemma-characterize-locally-projective", "coherent-lemma-quasi-coherence-higher-direct-images-application", "coherent-lemma-coherent-Noetherian", "coherent-lemma-coherent-proper-ample" ], "ref_ids": [ 5422, 5202, 3295, 2040, 5421, 3296, 3308, 3343 ] } ], "ref_ids": [] }, { "id": 3346, "type": "theorem", "label": "coherent-lemma-vanshing-gives-ample", "categories": [ "coherent" ], "title": "coherent-lemma-vanshing-gives-ample", "contents": [ "\\begin{reference}", "\\cite[III Proposition 2.6.1]{EGA}", "\\end{reference}", "Let $R$ be a Noetherian ring. Let $f : X \\to \\Spec(R)$ be a proper morphism.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is ample on $X$ (this is equivalent to many other", "things, see", "Properties, Proposition \\ref{properties-proposition-characterize-ample} and", "Morphisms, Lemma", "\\ref{morphisms-lemma-finite-type-over-affine-ample-very-ample}),", "\\item for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ there exists", "an $n_0 \\geq 0$ such that", "$H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}) = 0$ for all $n \\geq n_0$", "and $p > 0$, and", "\\item for every quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_X$, there exists an $n \\geq 1$", "such that $H^1(X, \\mathcal{I} \\otimes \\mathcal{L}^{\\otimes n}) = 0$.", "\\end{enumerate}" ], "refs": [ "properties-proposition-characterize-ample", "morphisms-lemma-finite-type-over-affine-ample-very-ample" ], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) follows from", "Lemma \\ref{lemma-coherent-proper-ample}.", "The implication (2) $\\Rightarrow$ (3) is trivial.", "The implication (3) $\\Rightarrow$ (1) is", "Lemma \\ref{lemma-quasi-compact-h1-zero-invertible}." ], "refs": [ "coherent-lemma-coherent-proper-ample", "coherent-lemma-quasi-compact-h1-zero-invertible" ], "ref_ids": [ 3343, 3289 ] } ], "ref_ids": [ 3067, 5394 ] }, { "id": 3347, "type": "theorem", "label": "coherent-lemma-surjective-finite-morphism-ample", "categories": [ "coherent" ], "title": "coherent-lemma-surjective-finite-morphism-ample", "contents": [ "Let $R$ be a Noetherian ring. Let $f : Y \\to X$ be a morphism of", "schemes proper over $R$. Let $\\mathcal{L}$ be an", "invertible $\\mathcal{O}_X$-module. Assume $f$ is finite and surjective.", "Then $\\mathcal{L}$ is ample if and only if $f^*\\mathcal{L}$ is ample." ], "refs": [], "proofs": [ { "contents": [ "The pullback of an ample invertible sheaf by a quasi-affine morphism", "is ample, see Morphisms, Lemma", "\\ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}.", "This proves one of the implications as a finite morphism is affine", "by definition.", "\\medskip\\noindent", "Assume that $f^*\\mathcal{L}$ is ample. Let $P$ be the following property on", "coherent $\\mathcal{O}_X$-modules $\\mathcal{F}$: there exists an $n_0$", "such that $H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}) = 0$", "for all $n \\geq n_0$ and $p > 0$. We will prove that $P$ holds", "for any coherent $\\mathcal{O}_X$-module $\\mathcal{F}$, which implies", "$\\mathcal{L}$ is ample by Lemma \\ref{lemma-vanshing-gives-ample}.", "We are going to apply Lemma \\ref{lemma-property-higher-rank-cohomological}.", "Thus we have to verify (1), (2) and (3) of that lemma for $P$.", "Property (1) follows from the long exact cohomology sequence associated", "to a short exact sequence of sheaves and the fact that tensoring with", "an invertible sheaf is an exact functor. Property (2) follows since", "$H^p(X, -)$ is an additive functor. To see (3) let $Z \\subset X$ be", "an integral closed subscheme with generic point $\\xi$.", "Let $\\mathcal{F}$ be a coherent sheaf on $Y$ such that", "the support of $f_*\\mathcal{F}$ is equal to $Z$", "and $(f_*\\mathcal{F})_\\xi$ is annihilated by $\\mathfrak m_\\xi$,", "see Lemma \\ref{lemma-finite-morphism-Noetherian}. We claim that", "taking $\\mathcal{G} = f_*\\mathcal{F}$ works. We only have to verify", "part (3)(c) of Lemma \\ref{lemma-property-higher-rank-cohomological}.", "Hence assume that $\\mathcal{J} \\subset \\mathcal{O}_X$ is a", "quasi-coherent sheaf of ideals such that", "$\\mathcal{J}_\\xi = \\mathcal{O}_{X, \\xi}$.", "A finite morphism is affine hence by", "Lemma \\ref{lemma-affine-morphism-projection-ideal} we see that", "$\\mathcal{J}\\mathcal{G} = f_*(f^{-1}\\mathcal{J}\\mathcal{F})$.", "Also, as pointed out in the proof of", "Lemma \\ref{lemma-affine-morphism-projection-ideal} the sheaf", "$f^{-1}\\mathcal{J}\\mathcal{F}$ is a coherent $\\mathcal{O}_Y$-module.", "As $\\mathcal{L}$ is ample we see from Lemma \\ref{lemma-vanshing-gives-ample}", "that there exists an $n_0$ such that", "$$", "H^p(Y, f^{-1}\\mathcal{J}\\mathcal{F}", "\\otimes_{\\mathcal{O}_Y} f^*\\mathcal{L}^{\\otimes n}) = 0,", "$$", "for $n \\geq n_0$ and $p > 0$. Since $f$ is finite, hence affine, we see that", "\\begin{align*}", "H^p(X, \\mathcal{J}\\mathcal{G} \\otimes_{\\mathcal{O}_X}", "\\mathcal{L}^{\\otimes n})", "& =", "H^p(X, f_*(f^{-1}\\mathcal{J}\\mathcal{F}) \\otimes_{\\mathcal{O}_X}", "\\mathcal{L}^{\\otimes n}) \\\\", "& =", "H^p(X, f_*(f^{-1}\\mathcal{J}\\mathcal{F} \\otimes_{\\mathcal{O}_Y}", "f^*\\mathcal{L}^{\\otimes n})) \\\\", "& =", "H^p(Y, f^{-1}\\mathcal{J}\\mathcal{F} \\otimes_{\\mathcal{O}_Y}", "f^*\\mathcal{L}^{\\otimes n}) = 0", "\\end{align*}", "Here we have used the projection formula", "(Cohomology, Lemma \\ref{cohomology-lemma-projection-formula}) and", "Lemma \\ref{lemma-relative-affine-cohomology}.", "Hence the quasi-coherent subsheaf $\\mathcal{G}' = \\mathcal{J}\\mathcal{G}$", "satisfies $P$. This verifies property (3)(c) of", "Lemma \\ref{lemma-property-higher-rank-cohomological} as desired." ], "refs": [ "morphisms-lemma-pullback-ample-tensor-relatively-ample", "coherent-lemma-vanshing-gives-ample", "coherent-lemma-property-higher-rank-cohomological", "coherent-lemma-finite-morphism-Noetherian", "coherent-lemma-property-higher-rank-cohomological", "coherent-lemma-affine-morphism-projection-ideal", "coherent-lemma-affine-morphism-projection-ideal", "coherent-lemma-vanshing-gives-ample", "cohomology-lemma-projection-formula", "coherent-lemma-relative-affine-cohomology", "coherent-lemma-property-higher-rank-cohomological" ], "ref_ids": [ 5383, 3346, 3334, 3335, 3334, 3336, 3336, 3346, 2243, 3284, 3334 ] } ], "ref_ids": [] }, { "id": 3348, "type": "theorem", "label": "coherent-lemma-invert-s-cohomology", "categories": [ "coherent" ], "title": "coherent-lemma-invert-s-cohomology", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "Let $s \\in \\Gamma(X, \\mathcal{L})$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "If $X$ is quasi-compact and quasi-separated, the canonical map", "$$", "H^p_*(X, \\mathcal{L}, \\mathcal{F})_{(s)}", "\\longrightarrow", "H^p(X_s, \\mathcal{F})", "$$", "which maps $\\xi/s^n$ to $s^{-n}\\xi$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Note that for $p = 0$ this is", "Properties, Lemma \\ref{properties-lemma-invert-s-sections}.", "We will prove the statement using the induction", "principle (Lemma \\ref{lemma-induction-principle}) where for", "$U \\subset X$ quasi-compact open we let $P(U)$ be the property:", "for all $p \\geq 0$ the map", "$$", "H^p_*(U, \\mathcal{L}, \\mathcal{F})_{(s)}", "\\longrightarrow", "H^p(U_s, \\mathcal{F})", "$$", "is an isomorphism.", "\\medskip\\noindent", "If $U$ is affine, then both sides of the arrow displayed above", "are zero for $p > 0$ by", "Lemma \\ref{lemma-quasi-coherent-affine-cohomology-zero}", "and", "Properties, Lemma \\ref{properties-lemma-affine-cap-s-open}", "and the statement is true. If $P$ is true for $U$, $V$, and $U \\cap V$,", "then we can use the Mayer-Vietoris sequences", "(Cohomology, Lemma \\ref{cohomology-lemma-mayer-vietoris}) to obtain", "a map of long exact sequences", "$$", "\\xymatrix{", "H^{p - 1}_*(U \\cap V, \\mathcal{L}, \\mathcal{F})_{(s)} \\ar[r] \\ar[d] &", "H^p_*(U \\cup V, \\mathcal{L}, \\mathcal{F})_{(s)} \\ar[r] \\ar[d] &", "H^p_*(U, \\mathcal{L}, \\mathcal{F})_{(s)}", "\\oplus", "H^p_*(V, \\mathcal{L}, \\mathcal{F})_{(s)} \\ar[d] \\\\", "H^{p - 1}(U_s \\cap V_s, \\mathcal{F}) \\ar[r]&", "H^p(U_s \\cup V_s, \\mathcal{F}) \\ar[r] &", "H^p(U_s, \\mathcal{F})", "\\oplus", "H^p(V_s, \\mathcal{F})", "}", "$$", "(only a snippet shown). Observe that $U_s \\cap V_s = (U \\cap V)_s$ and", "that $U_s \\cup V_s = (U \\cup V)_s$. Thus the left and right vertical", "maps are isomorphisms (as well as one more to the right and one more", "to the left which are not shown in the diagram).", "We conclude that $P(U \\cup V)$ holds by", "the 5-lemma (Homology, Lemma \\ref{homology-lemma-five-lemma}).", "This finishes the proof." ], "refs": [ "properties-lemma-invert-s-sections", "coherent-lemma-induction-principle", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "properties-lemma-affine-cap-s-open", "cohomology-lemma-mayer-vietoris", "homology-lemma-five-lemma" ], "ref_ids": [ 3005, 3291, 3282, 3042, 2042, 12030 ] } ], "ref_ids": [] }, { "id": 3349, "type": "theorem", "label": "coherent-lemma-section-affine-open-kills-classes", "categories": [ "coherent" ], "title": "coherent-lemma-section-affine-open-kills-classes", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{L})$ be a section.", "Assume that", "\\begin{enumerate}", "\\item $X$ is quasi-compact and quasi-separated, and", "\\item $X_s$ is affine.", "\\end{enumerate}", "Then for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ and", "every $p > 0$ and all $\\xi \\in H^p(X, \\mathcal{F})$ there exists", "an $n \\geq 0$ such that $s^n\\xi = 0$ in", "$H^p(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n})$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $H^p(X_s, \\mathcal{G})$ is zero for every quasi-coherent", "module $\\mathcal{G}$ by", "Lemma \\ref{lemma-quasi-coherent-affine-cohomology-zero}.", "Hence the lemma follows from", "Lemma \\ref{lemma-invert-s-cohomology}." ], "refs": [ "coherent-lemma-quasi-coherent-affine-cohomology-zero", "coherent-lemma-invert-s-cohomology" ], "ref_ids": [ 3282, 3348 ] } ], "ref_ids": [] }, { "id": 3350, "type": "theorem", "label": "coherent-lemma-ample-on-reduction", "categories": [ "coherent" ], "title": "coherent-lemma-ample-on-reduction", "contents": [ "Let $i : Z \\to X$ be a closed immersion of Noetherian schemes", "inducing a homeomorphism of underlying topological spaces.", "Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "Then $i^*\\mathcal{L}$ is ample on $Z$, if and only if", "$\\mathcal{L}$ is ample on $X$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{L}$ is ample, then $i^*\\mathcal{L}$ is ample for", "example by Morphisms, Lemma", "\\ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}.", "Assume $i^*\\mathcal{L}$ is ample. We have to show that $\\mathcal{L}$", "is ample on $X$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the coherent sheaf of ideals", "cutting out the closed subscheme $Z$. Since $i(Z) = X$ set theoretically", "we see that $\\mathcal{I}^n = 0$ for some $n$ by", "Lemma \\ref{lemma-power-ideal-kills-sheaf}.", "Consider the sequence", "$$", "X = Z_n \\supset Z_{n - 1} \\supset Z_{n - 2} \\supset \\ldots \\supset Z_1 = Z", "$$", "of closed subschemes cut out by", "$0 = \\mathcal{I}^n \\subset \\mathcal{I}^{n - 1} \\subset \\ldots \\subset", "\\mathcal{I}$. Then each of the closed immersions $Z_i \\to Z_{i - 1}$", "is defined by a coherent sheaf of ideals of square zero. In this way", "we reduce to the case that $\\mathcal{I}^2 = 0$.", "\\medskip\\noindent", "Consider the short exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_X \\to i_*\\mathcal{O}_Z \\to 0", "$$", "of quasi-coherent $\\mathcal{O}_X$-modules. Tensoring with", "$\\mathcal{L}^{\\otimes n}$ we obtain short exact sequences", "\\begin{equation}", "\\label{equation-ses}", "0 \\to \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}", "\\to \\mathcal{L}^{\\otimes n} \\to i_*i^*\\mathcal{L}^{\\otimes n} \\to 0", "\\end{equation}", "As $\\mathcal{I}^2 = 0$, we can use", "Morphisms, Lemma \\ref{morphisms-lemma-i-star-equivalence}", "to think of $\\mathcal{I}$ as a quasi-coherent $\\mathcal{O}_Z$-module", "and then $\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n} =", "\\mathcal{I} \\otimes_{\\mathcal{O}_Z} i^*\\mathcal{L}^{\\otimes n}$ with", "obvious abuse of notation.", "Moreover, the cohomology of this sheaf over $Z$ is canonically", "the same as the cohomology of this sheaf over $X$ (as $i$ is a", "homeomorphism).", "\\medskip\\noindent", "Let $x \\in X$ be a point and denote $z \\in Z$ the corresponding point.", "Because $i^*\\mathcal{L}$ is ample there exists an $n$ and a section", "$s \\in \\Gamma(Z, i^*\\mathcal{L}^{\\otimes n})$ with $z \\in Z_s$", "and with $Z_s$ affine. The obstruction to lifting $s$ to a section", "of $\\mathcal{L}^{\\otimes n}$ over $X$ is the boundary", "$$", "\\xi = \\partial s \\in", "H^1(X, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) =", "H^1(Z, \\mathcal{I} \\otimes_{\\mathcal{O}_Z} i^*\\mathcal{L}^{\\otimes n})", "$$", "coming from the short exact sequence of sheaves (\\ref{equation-ses}).", "If we replace $s$ by $s^{e + 1}$ then $\\xi$ is replaced by", "$\\partial(s^{e + 1}) = (e + 1) s^e \\xi$ in", "$H^1(Z, \\mathcal{I} \\otimes_{\\mathcal{O}_Z} i^*\\mathcal{L}^{\\otimes (e + 1)n})$", "because the boundary map for", "$$", "0 \\to", "\\bigoplus\\nolimits_{m \\geq 0}", "\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes m} \\to", "\\bigoplus\\nolimits_{m \\geq 0}", "\\mathcal{L}^{\\otimes m} \\to", "\\bigoplus\\nolimits_{m \\geq 0}", "i_*i^*\\mathcal{L}^{\\otimes m} \\to 0", "$$", "is a derivation by Cohomology, Lemma", "\\ref{cohomology-lemma-boundary-derivation-over-cup-product}. By", "Lemma \\ref{lemma-section-affine-open-kills-classes}", "we see that $s^e \\xi$ is zero for $e$ large enough.", "Hence, after replacing $s$ by a power, we can assume $s$ is the image", "of a section $s' \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$.", "Then $X_{s'}$ is an open subscheme and $Z_s \\to X_{s'}$ is a surjective", "closed immersion of Noetherian schemes with $Z_s$ affine. Hence", "$X_s$ is affine by", "Lemma \\ref{lemma-image-affine-finite-morphism-affine-Noetherian} and", "we conclude that $\\mathcal{L}$ is ample." ], "refs": [ "morphisms-lemma-pullback-ample-tensor-relatively-ample", "coherent-lemma-power-ideal-kills-sheaf", "morphisms-lemma-i-star-equivalence", "cohomology-lemma-boundary-derivation-over-cup-product", "coherent-lemma-image-affine-finite-morphism-affine-Noetherian" ], "ref_ids": [ 5383, 3320, 5136, 2101, 3337 ] } ], "ref_ids": [] }, { "id": 3351, "type": "theorem", "label": "coherent-lemma-thickening-quasi-affine", "categories": [ "coherent" ], "title": "coherent-lemma-thickening-quasi-affine", "contents": [ "Let $i : Z \\to X$ be a closed immersion of Noetherian schemes", "inducing a homeomorphism of underlying topological spaces.", "Then $X$ is quasi-affine if and only if $Z$ is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "Recall that a scheme is quasi-affine", "if and only if the structure sheaf is ample, see", "Properties, Lemma \\ref{properties-lemma-quasi-affine-O-ample}.", "Hence if $Z$ is quasi-affine, then $\\mathcal{O}_Z$ is ample,", "hence $\\mathcal{O}_X$ is ample by", "Lemma \\ref{lemma-ample-on-reduction}, hence", "$X$ is quasi-affine. A proof of the converse, which", "can also be seen in an elementary way, is gotten by", "reading the argument just given backwards." ], "refs": [ "properties-lemma-quasi-affine-O-ample", "coherent-lemma-ample-on-reduction" ], "ref_ids": [ 3053, 3350 ] } ], "ref_ids": [] }, { "id": 3352, "type": "theorem", "label": "coherent-lemma-affine-in-presence-ample", "categories": [ "coherent" ], "title": "coherent-lemma-affine-in-presence-ample", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{L}$ be an ample invertible", "$\\mathcal{O}_X$-module. Let $n_0$ be an integer.", "If $H^p(X, \\mathcal{L}^{\\otimes -n}) = 0$ for $n \\geq n_0$ and $p > 0$,", "then $X$ is affine." ], "refs": [], "proofs": [ { "contents": [ "We claim $H^p(X, \\mathcal{F}) = 0$ for every quasi-coherent", "$\\mathcal{O}_X$-module and $p > 0$. Since $X$ is quasi-compact", "by Properties, Definition \\ref{properties-definition-ample}", "the claim finishes the proof", "by Lemma \\ref{lemma-quasi-compact-h1-zero-covering}.", "The scheme $X$ is separated by", "Properties, Lemma \\ref{properties-lemma-ample-separated}.", "Say $X$ is covered by $e + 1$ affine opens. Then", "$H^p(X, \\mathcal{F}) = 0$ for $p > e$, see", "Lemma \\ref{lemma-vanishing-nr-affines}. Thus we may use descending", "induction on $p$ to prove the claim. Writing $\\mathcal{F}$", "as a filtered colimit of finite type quasi-coherent", "modules (Properties, Lemma", "\\ref{properties-lemma-quasi-coherent-colimit-finite-type})", "and using Cohomology, Lemma", "\\ref{cohomology-lemma-quasi-separated-cohomology-colimit}", "we may assume $\\mathcal{F}$ is of finite type.", "Then we can choose $n > n_0$ such that", "$\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}$", "is globally generated, see Properties, Proposition", "\\ref{properties-proposition-characterize-ample}.", "This means there is a short exact sequence", "$$", "0 \\to \\mathcal{F}' \\to", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{L}^{\\otimes -n}", "\\to \\mathcal{F} \\to 0", "$$", "for some set $I$ (in fact we can choose $I$ finite). By", "induction hypothesis we have $H^{p + 1}(X, \\mathcal{F}') = 0$", "and by assumption (combined with the already used", "commutation of cohomology with colimits)", "we have $H^p(X, \\bigoplus_{i \\in I} \\mathcal{L}^{\\otimes -n}) = 0$.", "From the long exact cohomology sequence we conclude that", "$H^p(X, \\mathcal{F}) = 0$ as desired." ], "refs": [ "properties-definition-ample", "coherent-lemma-quasi-compact-h1-zero-covering", "properties-lemma-ample-separated", "coherent-lemma-vanishing-nr-affines", "properties-lemma-quasi-coherent-colimit-finite-type", "cohomology-lemma-quasi-separated-cohomology-colimit", "properties-proposition-characterize-ample" ], "ref_ids": [ 3088, 3287, 3046, 3292, 3020, 2082, 3067 ] } ], "ref_ids": [] }, { "id": 3353, "type": "theorem", "label": "coherent-lemma-affine-if-quasi-affine", "categories": [ "coherent" ], "title": "coherent-lemma-affine-if-quasi-affine", "contents": [ "Let $X$ be a quasi-affine scheme.", "If $H^p(X, \\mathcal{O}_X) = 0$ for $p > 0$,", "then $X$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathcal{O}_X$ is ample by", "Properties, Lemma \\ref{properties-lemma-quasi-affine-O-ample}", "this follows from Lemma \\ref{lemma-affine-in-presence-ample}." ], "refs": [ "properties-lemma-quasi-affine-O-ample", "coherent-lemma-affine-in-presence-ample" ], "ref_ids": [ 3053, 3352 ] } ], "ref_ids": [] }, { "id": 3354, "type": "theorem", "label": "coherent-lemma-chow-Noetherian", "categories": [ "coherent" ], "title": "coherent-lemma-chow-Noetherian", "contents": [ "\\begin{reference}", "\\cite[II Theorem 5.6.1(a)]{EGA}", "\\end{reference}", "Let $S$ be a Noetherian scheme.", "Let $f : X \\to S$ be a separated morphism of finite type.", "Then there exists an $n \\geq 0$ and a diagram", "$$", "\\xymatrix{", "X \\ar[rd] & X' \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_S \\ar[dl] \\\\", "& S &", "}", "$$", "where $X' \\to \\mathbf{P}^n_S$ is an immersion, and", "$\\pi : X' \\to X$ is proper and surjective. Moreover, we may", "arrange it such that there exists a dense open subscheme", "$U \\subset X$ such that $\\pi^{-1}(U) \\to U$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "All of the schemes we will encounter during the rest of the proof", "are going to be of finite type over the Noetherian scheme $S$ and", "hence Noetherian", "(see Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}).", "All morphisms between them will automatically be quasi-compact, locally of", "finite type and quasi-separated, see", "Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type} and", "Properties,", "Lemmas \\ref{properties-lemma-locally-Noetherian-quasi-separated} and", "\\ref{properties-lemma-morphism-Noetherian-schemes-quasi-compact}.", "\\medskip\\noindent", "The scheme $X$ has only finitely many irreducible components", "(Properties, Lemma \\ref{properties-lemma-Noetherian-irreducible-components}).", "Say $X = X_1 \\cup \\ldots \\cup X_r$ is the decomposition", "of $X$ into irreducible components.", "Let $\\eta_i \\in X_i$ be the generic point.", "For every point $x \\in X$ there exists an affine open", "$U_x \\subset X$ which contains $x$ and each of the generic", "points $\\eta_i$. See", "Properties, Lemma \\ref{properties-lemma-point-and-maximal-points-affine}.", "Since $X$ is quasi-compact, we can find a finite affine open", "covering $X = U_1 \\cup \\ldots \\cup U_m$ such that", "each $U_i$ contains $\\eta_1, \\ldots, \\eta_r$.", "In particular we conclude that the open", "$U = U_1 \\cap \\ldots \\cap U_m \\subset X$ is", "a dense open. This and the fact that the $U_i$ are affine opens", "covering $X$ is all that we will use below.", "\\medskip\\noindent", "Let $X^* \\subset X$ be the scheme theoretic closure of $U \\to X$, see", "Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-image}.", "Let $U_i^* = X^* \\cap U_i$. Note that $U_i^*$ is a closed subscheme", "of $U_i$. Hence $U_i^*$ is affine. Since $U$ is dense in $X$ the", "morphism $X^* \\to X$ is a surjective closed immersion. It is an", "isomorphism over $U$. Hence we may replace $X$ by $X^*$ and", "$U_i$ by $U_i^*$ and assume that $U$ is scheme theoretically dense", "in $X$, see", "Morphisms, Definition \\ref{morphisms-definition-scheme-theoretically-dense}.", "\\medskip\\noindent", "By Morphisms, Lemma \\ref{morphisms-lemma-quasi-projective-finite-type-over-S}", "we can find an immersion $j_i : U_i \\to \\mathbf{P}_S^{n_i}$", "for each $i$. By", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-immersion} we can find", "closed subschemes $Z_i \\subset \\mathbf{P}_S^{n_i}$", "such that $j_i : U_i \\to Z_i$ is a scheme theoretically", "dense open immersion. Note that $Z_i \\to S$ is proper, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-projective-proper}.", "Consider the morphism", "$$", "j = (j_1|_U, \\ldots, j_m|_U) : U \\longrightarrow", "\\mathbf{P}_S^{n_1} \\times_S \\ldots \\times_S \\mathbf{P}_S^{n_m}.", "$$", "By the lemma cited above we can find a closed subscheme", "$Z$ of $\\mathbf{P}_S^{n_1} \\times_S \\ldots \\times_S \\mathbf{P}_S^{n_m}$", "such that $j : U \\to Z$ is an open immersion and such that $U$", "is scheme theoretically dense in $Z$. The morphism $Z \\to S$", "is proper. Consider the $i$th projection", "$$", "\\text{pr}_i|_Z : Z \\longrightarrow \\mathbf{P}^{n_i}_S.", "$$", "This morphism factors through $Z_i$ (see Morphisms,", "Lemma \\ref{morphisms-lemma-factor-factor}). Denote $p_i : Z \\to Z_i$", "the induced morphism. This is a proper morphism, see", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}", "for example. At this point we have that", "$U \\subset U_i \\subset Z_i$ are scheme theoretically", "dense open immersions. Moreover, we can think of $Z$ as the", "scheme theoretic image of the ``diagonal'' morphism", "$U \\to Z_1 \\times_S \\ldots \\times_S Z_m$.", "\\medskip\\noindent", "Set $V_i = p_i^{-1}(U_i)$. Note that $p_i|_{V_i} : V_i \\to U_i$ is proper.", "Set $X' = V_1 \\cup \\ldots \\cup V_m$. By construction $X'$ has an immersion", "into the scheme", "$\\mathbf{P}^{n_1}_S \\times_S \\ldots \\times_S \\mathbf{P}^{n_m}_S$.", "Thus by the Segre embedding (see", "Constructions, Lemma \\ref{constructions-lemma-segre-embedding})", "we see that $X'$ has", "an immersion into a projective space over $S$.", "\\medskip\\noindent", "We claim that the morphisms $p_i|_{V_i}: V_i \\to U_i$ glue to a morphism", "$X' \\to X$. Namely, it is clear that $p_i|_U$ is the identity map", "from $U$ to $U$. Since $U \\subset X'$ is scheme theoretically", "dense by construction, it is also scheme theoretically dense", "in the open subscheme $V_i \\cap V_j$. Thus we see that", "$p_i|_{V_i \\cap V_j} = p_j|_{V_i \\cap V_j}$ as morphisms into the", "separated $S$-scheme $X$, see", "Morphisms, Lemma \\ref{morphisms-lemma-equality-of-morphisms}.", "We denote the resulting morphism $\\pi : X' \\to X$.", "\\medskip\\noindent", "We claim that $\\pi^{-1}(U_i) = V_i$.", "Since $\\pi|_{V_i} = p_i|_{V_i}$ it follows that", "$V_i \\subset \\pi^{-1}(U_i)$. Consider the diagram", "$$", "\\xymatrix{", "V_i \\ar[r] \\ar[rd]_{p_i|_{V_i}} & \\pi^{-1}(U_i) \\ar[d] \\\\", "& U_i", "}", "$$", "Since $V_i \\to U_i$ is proper we see that the image of", "the horizontal arrow is closed, see", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}.", "Since $V_i \\subset \\pi^{-1}(U_i)$ is scheme", "theoretically dense (as it contains $U$)", "we conclude that $V_i = \\pi^{-1}(U_i)$ as claimed.", "\\medskip\\noindent", "This shows that $\\pi^{-1}(U_i) \\to U_i$ is identified with the proper", "morphism $p_i|_{V_i} : V_i \\to U_i$. Hence we see that $X$ has a", "finite affine covering $X = \\bigcup U_i$ such that the restriction", "of $\\pi$ is proper on each member of the covering.", "Thus by Morphisms, Lemma \\ref{morphisms-lemma-proper-local-on-the-base}", "we see that $\\pi$ is proper.", "\\medskip\\noindent", "Finally we have to show that $\\pi^{-1}(U) = U$. To see this we argue in the", "same way as above using the diagram", "$$", "\\xymatrix{", "U \\ar[r] \\ar[rd] & \\pi^{-1}(U) \\ar[d] \\\\", "& U", "}", "$$", "and using that $\\text{id}_U : U \\to U$ is proper and that", "$U$ is scheme theoretically dense in $\\pi^{-1}(U)$." ], "refs": [ "morphisms-lemma-finite-type-noetherian", "morphisms-lemma-permanence-finite-type", "properties-lemma-locally-Noetherian-quasi-separated", "properties-lemma-morphism-Noetherian-schemes-quasi-compact", "properties-lemma-Noetherian-irreducible-components", "properties-lemma-point-and-maximal-points-affine", "morphisms-definition-scheme-theoretic-image", "morphisms-definition-scheme-theoretically-dense", "morphisms-lemma-quasi-projective-finite-type-over-S", "morphisms-lemma-quasi-compact-immersion", "morphisms-lemma-locally-projective-proper", "morphisms-lemma-factor-factor", "morphisms-lemma-image-proper-scheme-closed", "constructions-lemma-segre-embedding", "morphisms-lemma-equality-of-morphisms", "morphisms-lemma-image-proper-scheme-closed", "morphisms-lemma-proper-local-on-the-base" ], "ref_ids": [ 5202, 5204, 2953, 2957, 2956, 3061, 5539, 5540, 5393, 5154, 5422, 5148, 5411, 12624, 5157, 5411, 5407 ] } ], "ref_ids": [] }, { "id": 3355, "type": "theorem", "label": "coherent-lemma-proper-over-affine-cohomology-finite", "categories": [ "coherent" ], "title": "coherent-lemma-proper-over-affine-cohomology-finite", "contents": [ "Let $S = \\Spec(A)$ with $A$ a Noetherian ring.", "Let $f : X \\to S$ be a proper morphism.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Then $H^i(X, \\mathcal{F})$ is finite $A$-module for all $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "This is just the affine case of", "Proposition \\ref{proposition-proper-pushforward-coherent}.", "Namely, by Lemmas \\ref{lemma-quasi-coherence-higher-direct-images} and", "\\ref{lemma-quasi-coherence-higher-direct-images-application} we know that", "$R^if_*\\mathcal{F}$ is the quasi-coherent sheaf associated", "to the $A$-module $H^i(X, \\mathcal{F})$", "and by Lemma \\ref{lemma-coherent-Noetherian} this is", "a coherent sheaf if and only if $H^i(X, \\mathcal{F})$", "is an $A$-module of finite type." ], "refs": [ "coherent-proposition-proper-pushforward-coherent", "coherent-lemma-quasi-coherence-higher-direct-images", "coherent-lemma-quasi-coherence-higher-direct-images-application", "coherent-lemma-coherent-Noetherian" ], "ref_ids": [ 3401, 3295, 3296, 3308 ] } ], "ref_ids": [] }, { "id": 3356, "type": "theorem", "label": "coherent-lemma-graded-finiteness", "categories": [ "coherent" ], "title": "coherent-lemma-graded-finiteness", "contents": [ "Let $A$ be a Noetherian ring.", "Let $B$ be a finitely generated graded $A$-algebra.", "Let $f : X \\to \\Spec(A)$ be a proper morphism.", "Set $\\mathcal{B} = f^*\\widetilde B$.", "Let $\\mathcal{F}$ be a quasi-coherent", "graded $\\mathcal{B}$-module of finite type.", "\\begin{enumerate}", "\\item For every $p \\geq 0$ the graded $B$-module $H^p(X, \\mathcal{F})$", "is a finite $B$-module.", "\\item If $\\mathcal{L}$ is an ample invertible $\\mathcal{O}_X$-module,", "then there exists an integer $d_0$ such that", "$H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes d}) = 0$", "for all $p > 0$ and $d \\geq d_0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove this we consider the fibre product diagram", "$$", "\\xymatrix{", "X' = \\Spec(B) \\times_{\\Spec(A)} X", "\\ar[r]_-\\pi \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "\\Spec(B) \\ar[r] &", "\\Spec(A)", "}", "$$", "Note that $f'$ is a proper morphism, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-proper}.", "Also, $B$ is a finitely generated $A$-algebra, and hence", "Noetherian (Algebra, Lemma \\ref{algebra-lemma-Noetherian-permanence}).", "This implies that $X'$ is a Noetherian scheme", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}).", "Note that $X'$ is the relative spectrum of the quasi-coherent", "$\\mathcal{O}_X$-algebra $\\mathcal{B}$ by", "Constructions, Lemma \\ref{constructions-lemma-spec-properties}.", "Since $\\mathcal{F}$ is a quasi-coherent $\\mathcal{B}$-module", "we see that there is a unique quasi-coherent", "$\\mathcal{O}_{X'}$-module $\\mathcal{F}'$ such that", "$\\pi_*\\mathcal{F}' = \\mathcal{F}$, see", "Morphisms, Lemma \\ref{morphisms-lemma-affine-equivalence-modules}", "Since $\\mathcal{F}$ is finite type as a $\\mathcal{B}$-module we", "conclude that $\\mathcal{F}'$ is a finite type", "$\\mathcal{O}_{X'}$-module (details omitted). In other words,", "$\\mathcal{F}'$ is a coherent $\\mathcal{O}_{X'}$-module", "(Lemma \\ref{lemma-coherent-Noetherian}).", "Since the morphism $\\pi : X' \\to X$ is affine we have", "$$", "H^p(X, \\mathcal{F}) = H^p(X', \\mathcal{F}')", "$$", "by Lemma \\ref{lemma-relative-affine-cohomology}.", "Thus (1) follows from", "Lemma \\ref{lemma-proper-over-affine-cohomology-finite}.", "Given $\\mathcal{L}$ as in (2) we set", "$\\mathcal{L}' = \\pi^*\\mathcal{L}$. Note that $\\mathcal{L}'$ is", "ample on $X'$ by", "Morphisms, Lemma \\ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}.", "By the projection formula", "(Cohomology, Lemma \\ref{cohomology-lemma-projection-formula}) we have", "$\\pi_*(\\mathcal{F}' \\otimes \\mathcal{L}') = \\mathcal{F} \\otimes \\mathcal{L}$.", "Thus part (2) follows by the same reasoning as above from", "Lemma \\ref{lemma-kill-by-twisting}." ], "refs": [ "morphisms-lemma-base-change-proper", "algebra-lemma-Noetherian-permanence", "morphisms-lemma-finite-type-noetherian", "constructions-lemma-spec-properties", "morphisms-lemma-affine-equivalence-modules", "coherent-lemma-coherent-Noetherian", "coherent-lemma-relative-affine-cohomology", "coherent-lemma-proper-over-affine-cohomology-finite", "morphisms-lemma-pullback-ample-tensor-relatively-ample", "cohomology-lemma-projection-formula", "coherent-lemma-kill-by-twisting" ], "ref_ids": [ 5409, 448, 5202, 12590, 5174, 3308, 3284, 3355, 5383, 2243, 3344 ] } ], "ref_ids": [] }, { "id": 3357, "type": "theorem", "label": "coherent-lemma-cohomology-powers-ideal-times-F", "categories": [ "coherent" ], "title": "coherent-lemma-cohomology-powers-ideal-times-F", "contents": [ "Let $A$ be a Noetherian ring.", "Let $I \\subset A$ be an ideal.", "Set $B = \\bigoplus_{n \\geq 0} I^n$.", "Let $f : X \\to \\Spec(A)$ be a proper morphism.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Then for every $p \\geq 0$ the graded $B$-module", "$\\bigoplus_{n \\geq 0} H^p(X, I^n\\mathcal{F})$ is", "a finite $B$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{B} = \\bigoplus I^n\\mathcal{O}_X = f^*\\widetilde{B}$.", "Then $\\bigoplus I^n\\mathcal{F}$ is a finite type", "graded $\\mathcal{B}$-module. Hence the result follows", "from Lemma \\ref{lemma-graded-finiteness} part (1)." ], "refs": [ "coherent-lemma-graded-finiteness" ], "ref_ids": [ 3356 ] } ], "ref_ids": [] }, { "id": 3358, "type": "theorem", "label": "coherent-lemma-cohomology-powers-ideal-times-sheaf", "categories": [ "coherent" ], "title": "coherent-lemma-cohomology-powers-ideal-times-sheaf", "contents": [ "Given a morphism of schemes $f : X \\to Y$, a quasi-coherent sheaf", "$\\mathcal{F}$ on $X$, and a quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_Y$. Assume $Y$ locally", "Noetherian, $f$ proper, and $\\mathcal{F}$ coherent.", "Then", "$$", "\\mathcal{M} =", "\\bigoplus\\nolimits_{n \\geq 0} R^pf_*(\\mathcal{I}^n\\mathcal{F})", "$$", "is a graded $\\mathcal{A} = \\bigoplus_{n \\geq 0} \\mathcal{I}^n$-module", "which is quasi-coherent and of finite type." ], "refs": [], "proofs": [ { "contents": [ "The statement is local on $Y$, hence this reduces to the", "case where $Y$ is affine. In the affine case the result follows", "from Lemma \\ref{lemma-cohomology-powers-ideal-times-F}.", "Details omitted." ], "refs": [ "coherent-lemma-cohomology-powers-ideal-times-F" ], "ref_ids": [ 3357 ] } ], "ref_ids": [] }, { "id": 3359, "type": "theorem", "label": "coherent-lemma-cohomology-powers-ideal-application", "categories": [ "coherent" ], "title": "coherent-lemma-cohomology-powers-ideal-application", "contents": [ "Let $A$ be a Noetherian ring.", "Let $I \\subset A$ be an ideal.", "Let $f : X \\to \\Spec(A)$ be a proper morphism.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Then for every $p \\geq 0$ there exists an integer $c \\geq 0$", "such that", "\\begin{enumerate}", "\\item the multiplication map", "$I^{n - c} \\otimes H^p(X, I^c\\mathcal{F}) \\to H^p(X, I^n\\mathcal{F})$", "is surjective for all $n \\geq c$,", "\\item the image of $H^p(X, I^{n + m}\\mathcal{F}) \\to H^p(X, I^n\\mathcal{F})$", "is contained in the submodule $I^{m - e} H^p(X, I^n\\mathcal{F})$", "where $e = \\max(0, c - n)$ for $n + m \\geq c$, $n, m \\geq 0$,", "\\item we have", "$$", "\\Ker(H^p(X, I^n\\mathcal{F}) \\to H^p(X, \\mathcal{F})) =", "\\Ker(H^p(X, I^n\\mathcal{F}) \\to H^p(X, I^{n - c}\\mathcal{F}))", "$$", "for $n \\geq c$,", "\\item there are maps $I^nH^p(X, \\mathcal{F}) \\to H^p(X, I^{n - c}\\mathcal{F})$", "for $n \\geq c$ such that the compositions", "$$", "H^p(X, I^n\\mathcal{F}) \\to ", "I^{n - c}H^p(X, \\mathcal{F}) \\to", "H^p(X, I^{n - 2c}\\mathcal{F})", "$$", "and", "$$", "I^nH^p(X, \\mathcal{F}) \\to", "H^p(X, I^{n - c}\\mathcal{F}) \\to", "I^{n - 2c}H^p(X, \\mathcal{F})", "$$", "for $n \\geq 2c$ are the canonical ones, and", "\\item the inverse systems $(H^p(X, I^n\\mathcal{F}))$ and", "$(I^nH^p(X, \\mathcal{F}))$ are pro-isomorphic.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $M_n = H^p(X, I^n\\mathcal{F})$ for $n \\geq 1$ and", "$M_0 = H^p(X, \\mathcal{F})$ so that we have maps", "$\\ldots \\to M_3 \\to M_2 \\to M_1 \\to M_0$. Setting", "$B = \\bigoplus_{n \\geq 0} I^n$, then $M = \\bigoplus_{n \\geq 0} M_n$", "is a finite graded $B$-module, see", "Lemma \\ref{lemma-cohomology-powers-ideal-times-F}.", "Observe that the products", "$B_n \\otimes M_m \\to M_{m + n}$, $a \\otimes m \\mapsto a \\cdot m$", "are compatible with the maps in our inverse system in the sense", "that the diagrams", "$$", "\\xymatrix{", "B_n \\otimes_A M_m \\ar[r] \\ar[d] & M_{n + m} \\ar[d] \\\\", "B_n \\otimes_A M_{m'} \\ar[r] & M_{n + m'}", "}", "$$", "commute for $n, m' \\geq 0$ and $m \\geq m'$.", "\\medskip\\noindent", "Proof of (1). Choose $d_1, \\ldots, d_t \\geq 0$ and $x_i \\in M_{d_i}$", "such that $M$ is generated by $x_1, \\ldots, x_t$ over $B$.", "For any $c \\geq \\max\\{d_i\\}$ we conclude that", "$B_{n - c} \\cdot M_c = M_n$ for $n \\geq c$ and we conclude (1) is true.", "\\medskip\\noindent", "Proof of (2). Let $c$ be as in the proof of (1). Let $n + m \\geq c$.", "We have $M_{n + m} = B_{n + m - c} \\cdot M_c$.", "If $c > n$ then we use $M_c \\to M_n$ and the compatibility of products", "with transition maps pointed out above to conclude that", "the image of $M_{n + m} \\to M_n$ is contained in $I^{n + m - c}M_n$.", "If $c \\leq n$, then we write $M_{n + m} = B_m \\cdot B_{n - c} \\cdot M_c =", "B_m \\cdot M_n$ to see that the image is contained in $I^m M_n$.", "This proves (2).", "\\medskip\\noindent", "Let $K_n \\subset M_n$ be the kernel of the map $M_n \\to M_0$. The", "compatibility of products with transition maps pointed out above", "shows that $K = \\bigoplus K_n \\subset M$ is a graded $B$-submodule.", "As $B$ is Noetherian and $M$ is a finitely generated graded $B$-module,", "this shows that $K$ is a finitely generated graded $B$-module.", "Choose $d'_1, \\ldots, d'_{t'} \\geq 0$ and $y_i \\in K_{d'_i}$", "such that $K$ is generated by $y_1, \\ldots, y_{t'}$ over $B$.", "Set $c = \\max(d_i, d'_j)$. Since $y_i \\in \\Ker(M_{d'_i} \\to M_0)$", "we see that $B_n \\cdot y_i \\subset \\Ker(M_{n + d_i} \\to M_n)$.", "In this way we see that $K_n = \\Ker(M_n \\to M_{n - c})$ for $n \\geq c$.", "This proves (3).", "\\medskip\\noindent", "Consider the following commutative solid diagram", "$$", "\\xymatrix{", "I^n \\otimes_A M_0 \\ar[r] \\ar[d] &", "I^nM_0 \\ar[r] \\ar@{..>}[d] &", "M_0 \\ar[d] \\\\", "M_n \\ar[r] &", "M_{n - c} \\ar[r] &", "M_0", "}", "$$", "Since the kernel of the surjective arrow $I^n \\otimes_A M_0 \\to I^nM_0$", "maps into $K_n$ by the above we obtain the dotted arrow and the", "composition $I^nM_0 \\to M_{n - c} \\to M_0$ is the canonical map.", "Then clearly the composition $I^nM_0 \\to M_{n - c} \\to I^{n - 2c}M_0$", "is the canonical map for $n \\geq 2c$. Consider the composition", "$M_n \\to I^{n - c}M_0 \\to M_{n - 2c}$. The first map", "sends an element of the form $a \\cdot m$", "with $a \\in I^{n - c}$ and $m \\in M_c$", "to $a m'$ where $m'$ is the image of $m$ in $M_0$.", "Then the second map sends this to $a \\cdot m'$ in $M_{n - 2c}$ and", "we see (4) is true.", "\\medskip\\noindent", "Part (5) is an immediate consequence of (4) and the definition of", "morphisms of pro-objects." ], "refs": [ "coherent-lemma-cohomology-powers-ideal-times-F" ], "ref_ids": [ 3357 ] } ], "ref_ids": [] }, { "id": 3360, "type": "theorem", "label": "coherent-lemma-ML-cohomology-powers-ideal", "categories": [ "coherent" ], "title": "coherent-lemma-ML-cohomology-powers-ideal", "contents": [ "Let $A$ be a Noetherian ring.", "Let $I \\subset A$ be an ideal.", "Let $f : X \\to \\Spec(A)$ be a proper morphism.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Fix $p \\geq 0$. There exists a $c \\geq 0$ such that", "\\begin{enumerate}", "\\item for all $n \\geq c$ we have", "$$", "\\Ker(H^p(X, \\mathcal{F}) \\to H^p(X, \\mathcal{F}/I^n\\mathcal{F})) \\subset", "I^{n - c}H^p(X, \\mathcal{F}).", "$$", "\\item the inverse system", "$$", "\\left(H^p(X, \\mathcal{F}/I^n\\mathcal{F})\\right)_{n \\in \\mathbf{N}}", "$$", "satisfies the Mittag-Leffler condition (see", "Homology, Definition \\ref{homology-definition-Mittag-Leffler}), and", "\\item we have", "$$", "\\Im(H^p(X, \\mathcal{F}/I^k\\mathcal{F})", "\\to H^p(X, \\mathcal{F}/I^n\\mathcal{F}))", "=", "\\Im(H^p(X, \\mathcal{F})", "\\to H^p(X, \\mathcal{F}/I^n\\mathcal{F}))", "$$", "for all $k \\geq n + c$.", "\\end{enumerate}" ], "refs": [ "homology-definition-Mittag-Leffler" ], "proofs": [ { "contents": [ "Let $c = \\max\\{c_p, c_{p + 1}\\}$, where $c_p, c_{p + 1}$ are the integers", "found in Lemma \\ref{lemma-cohomology-powers-ideal-application} for", "$H^p$ and $H^{p + 1}$.", "\\medskip\\noindent", "Let us prove part (1). Consider the short exact sequence", "$$", "0 \\to I^n\\mathcal{F} \\to \\mathcal{F} \\to \\mathcal{F}/I^n\\mathcal{F} \\to 0", "$$", "From the long exact cohomology sequence we see that", "$$", "\\Ker(", "H^p(X, \\mathcal{F}) \\to H^p(X, \\mathcal{F}/I^n\\mathcal{F})", ")", "=", "\\Im(", "H^p(X, I^n\\mathcal{F}) \\to H^p(X, \\mathcal{F})", ")", "$$", "Hence by Lemma \\ref{lemma-cohomology-powers-ideal-application} part (2)", "we see that this is contained in $I^{n - c}H^p(X, \\mathcal{F})$ for $n \\geq c$.", "\\medskip\\noindent", "Note that part (3) implies part (2) by definition of the Mittag-Leffler", "systems.", "\\medskip\\noindent", "Let us prove part (3). Fix an $n$. Consider the commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "I^n\\mathcal{F} \\ar[r] &", "\\mathcal{F} \\ar[r] &", "\\mathcal{F}/I^n\\mathcal{F} \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "I^{n + m}\\mathcal{F} \\ar[r] \\ar[u] &", "\\mathcal{F} \\ar[r] \\ar[u] &", "\\mathcal{F}/I^{n + m}\\mathcal{F} \\ar[r] \\ar[u] &", "0", "}", "$$", "This gives rise to the following commutative diagram", "$$", "\\xymatrix{", "H^p(X, \\mathcal{F}) \\ar[r] &", "H^p(X, \\mathcal{F}/I^n\\mathcal{F}) \\ar[r]_\\delta &", "H^{p + 1}(X, I^n\\mathcal{F}) \\ar[r] &", "H^{p + 1}(X, \\mathcal{F}) \\\\", "H^p(X, \\mathcal{F}) \\ar[r] \\ar[u]^1 &", "H^p(X, \\mathcal{F}/I^{n + m}\\mathcal{F}) \\ar[r] \\ar[u]^\\gamma &", "H^{p + 1}(X, I^{n + m}\\mathcal{F}) \\ar[u]^\\alpha \\ar[r]^-\\beta &", "H^{p + 1}(X, \\mathcal{F}) \\ar[u]_1", "}", "$$", "with exact rows. By", "Lemma \\ref{lemma-cohomology-powers-ideal-application} part (4) the kernel", "of $\\beta$ is equal to the kernel of $\\alpha$ for $m \\geq c$.", "By a diagram chase this shows that the image of $\\gamma$ is contained", "in the kernel of $\\delta$ which shows that part (3) is true", "(set $k = n + m$ to get it)." ], "refs": [ "coherent-lemma-cohomology-powers-ideal-application", "coherent-lemma-cohomology-powers-ideal-application", "coherent-lemma-cohomology-powers-ideal-application" ], "ref_ids": [ 3359, 3359, 3359 ] } ], "ref_ids": [ 12188 ] }, { "id": 3361, "type": "theorem", "label": "coherent-lemma-spell-out-theorem-formal-functions", "categories": [ "coherent" ], "title": "coherent-lemma-spell-out-theorem-formal-functions", "contents": [ "Let $A$ be a ring. Let $I \\subset A$ be an ideal. Assume $A$ is", "Noetherian and complete with respect to $I$.", "Let $f : X \\to \\Spec(A)$ be a proper morphism.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Then", "$$", "H^p(X, \\mathcal{F}) = \\lim_n H^p(X, \\mathcal{F}/I^n\\mathcal{F})", "$$", "for all $p \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of the theorem on formal functions", "(Theorem \\ref{theorem-formal-functions}) in the", "case of a complete Noetherian base ring. Namely, in this case the", "$A$-module $H^p(X, \\mathcal{F})$ is finite", "(Lemma \\ref{lemma-proper-over-affine-cohomology-finite}) hence", "$I$-adically complete (Algebra, Lemma \\ref{algebra-lemma-completion-tensor})", "and we see that completion on the left hand side is not necessary." ], "refs": [ "coherent-theorem-formal-functions", "coherent-lemma-proper-over-affine-cohomology-finite", "algebra-lemma-completion-tensor" ], "ref_ids": [ 3278, 3355, 869 ] } ], "ref_ids": [] }, { "id": 3362, "type": "theorem", "label": "coherent-lemma-formal-functions-stalk", "categories": [ "coherent" ], "title": "coherent-lemma-formal-functions-stalk", "contents": [ "Given a morphism of schemes $f : X \\to Y$ and a quasi-coherent sheaf", "$\\mathcal{F}$ on $X$. Assume", "\\begin{enumerate}", "\\item $Y$ locally Noetherian,", "\\item $f$ proper, and", "\\item $\\mathcal{F}$ coherent.", "\\end{enumerate}", "Let $y \\in Y$ be a point. Consider the infinitesimal neighbourhoods", "$$", "\\xymatrix{", "X_n =", "\\Spec(\\mathcal{O}_{Y, y}/\\mathfrak m_y^n) \\times_Y X", "\\ar[r]_-{i_n} \\ar[d]_{f_n} &", "X \\ar[d]^f \\\\", "\\Spec(\\mathcal{O}_{Y, y}/\\mathfrak m_y^n) \\ar[r]^-{c_n} & Y", "}", "$$", "of the fibre $X_1 = X_y$ and set $\\mathcal{F}_n = i_n^*\\mathcal{F}$.", "Then we have", "$$", "\\left(R^pf_*\\mathcal{F}\\right)_y^\\wedge", "\\cong", "\\lim_n H^p(X_n, \\mathcal{F}_n)", "$$", "as $\\mathcal{O}_{Y, y}^\\wedge$-modules." ], "refs": [], "proofs": [ { "contents": [ "This is just a reformulation of a special case of the theorem", "on formal functions, Theorem \\ref{theorem-formal-functions}.", "Let us spell it out. Note that $\\mathcal{O}_{Y, y}$ is a Noetherian", "local ring. Consider the canonical morphism", "$c : \\Spec(\\mathcal{O}_{Y, y}) \\to Y$, see", "Schemes, Equation (\\ref{schemes-equation-canonical-morphism}).", "This is a flat morphism as it identifies local rings.", "Denote momentarily $f' : X' \\to \\Spec(\\mathcal{O}_{Y, y})$", "the base change of $f$ to this local ring. We see that", "$c^*R^pf_*\\mathcal{F} = R^pf'_*\\mathcal{F}'$ by", "Lemma \\ref{lemma-flat-base-change-cohomology}.", "Moreover, the infinitesimal neighbourhoods of", "the fibre $X_y$ and $X'_y$ are identified (verification omitted; hint:", "the morphisms $c_n$ factor through $c$).", "\\medskip\\noindent", "Hence we may assume that $Y = \\Spec(A)$ is the spectrum of", "a Noetherian local ring $A$ with maximal ideal $\\mathfrak m$", "and that $y \\in Y$ corresponds to the closed point (i.e., to $\\mathfrak m$).", "In particular it follows that", "$$", "\\left(R^pf_*\\mathcal{F}\\right)_y =", "\\Gamma(Y, R^pf_*\\mathcal{F}) =", "H^p(X, \\mathcal{F}).", "$$", "\\medskip\\noindent", "In this case also, the morphisms $c_n$ are each closed immersions.", "Hence their base changes $i_n$ are closed immersions as well.", "Note that $i_{n, *}\\mathcal{F}_n = i_{n, *}i_n^*\\mathcal{F}", "= \\mathcal{F}/\\mathfrak m^n\\mathcal{F}$. By the Leray spectral sequence", "for $i_n$, and Lemma \\ref{lemma-finite-pushforward-coherent} we see that", "$$", "H^p(X_n, \\mathcal{F}_n) =", "H^p(X, i_{n, *}\\mathcal{F}) =", "H^p(X, \\mathcal{F}/\\mathfrak m^n\\mathcal{F})", "$$", "Hence we may indeed apply the theorem on formal functions to compute", "the limit in the statement of the lemma and we win." ], "refs": [ "coherent-theorem-formal-functions", "coherent-lemma-flat-base-change-cohomology", "coherent-lemma-finite-pushforward-coherent" ], "ref_ids": [ 3278, 3298, 3316 ] } ], "ref_ids": [] }, { "id": 3363, "type": "theorem", "label": "coherent-lemma-higher-direct-images-zero-finite-fibre", "categories": [ "coherent" ], "title": "coherent-lemma-higher-direct-images-zero-finite-fibre", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $y \\in Y$.", "Assume", "\\begin{enumerate}", "\\item $Y$ locally Noetherian,", "\\item $f$ is proper, and", "\\item $f^{-1}(\\{y\\})$ is finite.", "\\end{enumerate}", "Then for any coherent sheaf $\\mathcal{F}$ on $X$ we have", "$(R^pf_*\\mathcal{F})_y = 0$ for all $p > 0$." ], "refs": [], "proofs": [ { "contents": [ "The fibre $X_y$ is finite, and by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-fibre} it", "is a finite discrete space. Moreover, the underlying topological", "space of each infinitesimal neighbourhood $X_n$ is the same.", "Hence each of the schemes $X_n$ is affine according to", "Schemes, Lemma \\ref{schemes-lemma-scheme-finite-discrete-affine}.", "Hence it follows that $H^p(X_n, \\mathcal{F}_n) = 0$ for all", "$p > 0$. Hence we see that $(R^pf_*\\mathcal{F})_y^\\wedge = 0$", "by Lemma \\ref{lemma-formal-functions-stalk}.", "Note that $R^pf_*\\mathcal{F}$ is coherent by", "Proposition \\ref{proposition-proper-pushforward-coherent} and", "hence $R^pf_*\\mathcal{F}_y$ is a finite", "$\\mathcal{O}_{Y, y}$-module. By Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK})", "if the completion of a finite module over a local ring", "is zero, then the module is zero. Whence", "$(R^pf_*\\mathcal{F})_y = 0$." ], "refs": [ "morphisms-lemma-finite-fibre", "schemes-lemma-scheme-finite-discrete-affine", "coherent-lemma-formal-functions-stalk", "coherent-proposition-proper-pushforward-coherent", "algebra-lemma-NAK" ], "ref_ids": [ 5227, 7678, 3362, 3401, 401 ] } ], "ref_ids": [] }, { "id": 3364, "type": "theorem", "label": "coherent-lemma-higher-direct-images-zero-above-dimension-fibre", "categories": [ "coherent" ], "title": "coherent-lemma-higher-direct-images-zero-above-dimension-fibre", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $y \\in Y$.", "Assume", "\\begin{enumerate}", "\\item $Y$ locally Noetherian,", "\\item $f$ is proper, and", "\\item $\\dim(X_y) = d$.", "\\end{enumerate}", "Then for any coherent sheaf $\\mathcal{F}$ on $X$ we have", "$(R^pf_*\\mathcal{F})_y = 0$ for all $p > d$." ], "refs": [], "proofs": [ { "contents": [ "The fibre $X_y$ is of finite type over $\\Spec(\\kappa(y))$.", "Hence $X_y$ is a Noetherian scheme by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}.", "Hence the underlying topological space of $X_y$ is Noetherian, see", "Properties, Lemma \\ref{properties-lemma-Noetherian-topology}.", "Moreover, the underlying topological space of each infinitesimal", "neighbourhood $X_n$ is the same as that of $X_y$.", "Hence $H^p(X_n, \\mathcal{F}_n) = 0$ for all $p > d$ by", "Cohomology, Proposition \\ref{cohomology-proposition-vanishing-Noetherian}.", "Hence we see that $(R^pf_*\\mathcal{F})_y^\\wedge = 0$", "by Lemma \\ref{lemma-formal-functions-stalk} for $p > d$.", "Note that $R^pf_*\\mathcal{F}$ is coherent by", "Proposition \\ref{proposition-proper-pushforward-coherent} and", "hence $R^pf_*\\mathcal{F}_y$ is a finite", "$\\mathcal{O}_{Y, y}$-module. By Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK})", "if the completion of a finite module over a local ring", "is zero, then the module is zero. Whence", "$(R^pf_*\\mathcal{F})_y = 0$." ], "refs": [ "morphisms-lemma-finite-type-noetherian", "properties-lemma-Noetherian-topology", "cohomology-proposition-vanishing-Noetherian", "coherent-lemma-formal-functions-stalk", "coherent-proposition-proper-pushforward-coherent", "algebra-lemma-NAK" ], "ref_ids": [ 5202, 2954, 2246, 3362, 3401, 401 ] } ], "ref_ids": [] }, { "id": 3365, "type": "theorem", "label": "coherent-lemma-characterize-finite", "categories": [ "coherent" ], "title": "coherent-lemma-characterize-finite", "contents": [ "(For a more general version see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-characterize-finite}.)", "Let $f : X \\to S$ be a morphism of schemes.", "Assume $S$ is locally Noetherian.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is finite, and", "\\item $f$ is proper with finite fibres.", "\\end{enumerate}" ], "refs": [ "more-morphisms-lemma-characterize-finite" ], "proofs": [ { "contents": [ "A finite morphism is proper according to", "Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}.", "A finite morphism is quasi-finite according to", "Morphisms, Lemma \\ref{morphisms-lemma-finite-quasi-finite}.", "A quasi-finite morphism has finite fibres, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite}.", "Hence a finite morphism is proper and has finite fibres.", "\\medskip\\noindent", "Assume $f$ is proper with finite fibres.", "We want to show $f$ is finite.", "In fact it suffices to prove $f$ is affine.", "Namely, if $f$ is affine, then it follows that", "$f$ is integral by", "Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed}", "whereupon it follows from", "Morphisms, Lemma \\ref{morphisms-lemma-finite-integral}", "that $f$ is finite.", "\\medskip\\noindent", "To show that $f$ is affine we may assume that $S$ is affine, and our", "goal is to show that $X$ is affine too.", "Since $f$ is proper we see that $X$ is separated and quasi-compact.", "Hence we may use the criterion of", "Lemma \\ref{lemma-quasi-separated-h1-zero-covering} to prove that $X$", "is affine. To see this let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be a finite type ideal sheaf. In particular $\\mathcal{I}$ is", "a coherent sheaf on $X$. By", "Lemma \\ref{lemma-higher-direct-images-zero-finite-fibre} we conclude that", "$R^1f_*\\mathcal{I}_s = 0$ for all $s \\in S$.", "In other words, $R^1f_*\\mathcal{I} = 0$. Hence we see from", "the Leray Spectral Sequence for $f$ that", "$H^1(X , \\mathcal{I}) = H^1(S, f_*\\mathcal{I})$.", "Since $S$ is affine, and $f_*\\mathcal{I}$ is quasi-coherent", "(Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent})", "we conclude $H^1(S, f_*\\mathcal{I}) = 0$", "from Lemma \\ref{lemma-quasi-coherent-affine-cohomology-zero}", "as desired. Hence $H^1(X, \\mathcal{I}) = 0$ as desired." ], "refs": [ "morphisms-lemma-finite-proper", "morphisms-lemma-finite-quasi-finite", "morphisms-lemma-quasi-finite", "morphisms-lemma-integral-universally-closed", "morphisms-lemma-finite-integral", "coherent-lemma-quasi-separated-h1-zero-covering", "coherent-lemma-higher-direct-images-zero-finite-fibre", "schemes-lemma-push-forward-quasi-coherent", "coherent-lemma-quasi-coherent-affine-cohomology-zero" ], "ref_ids": [ 5445, 5444, 5230, 5441, 5438, 3288, 3363, 7730, 3282 ] } ], "ref_ids": [ 13903 ] }, { "id": 3366, "type": "theorem", "label": "coherent-lemma-proper-finite-fibre-finite-in-neighbourhood", "categories": [ "coherent" ], "title": "coherent-lemma-proper-finite-fibre-finite-in-neighbourhood", "contents": [ "\\begin{slogan}", "A proper morphism is finite in a neighbourhood of a finite fiber.", "\\end{slogan}", "(For a more general version see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood}.)", "Let $f : X \\to S$ be a morphism of schemes.", "Let $s \\in S$.", "Assume", "\\begin{enumerate}", "\\item $S$ is locally Noetherian,", "\\item $f$ is proper, and", "\\item $f^{-1}(\\{s\\})$ is a finite set.", "\\end{enumerate}", "Then there exists an open neighbourhood $V \\subset S$ of $s$", "such that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite." ], "refs": [ "more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood" ], "proofs": [ { "contents": [ "The morphism $f$ is quasi-finite at all the points of $f^{-1}(\\{s\\})$", "by Morphisms, Lemma \\ref{morphisms-lemma-finite-fibre}.", "By Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-points-open} the", "set of points at which $f$ is quasi-finite is an open $U \\subset X$.", "Let $Z = X \\setminus U$. Then $s \\not \\in f(Z)$. Since $f$ is proper", "the set $f(Z) \\subset S$ is closed. Choose any open neighbourhood", "$V \\subset S$ of $s$ with $Z \\cap V = \\emptyset$. Then", "$f^{-1}(V) \\to V$ is locally quasi-finite and proper.", "Hence it is quasi-finite", "(Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-locally-quasi-compact}),", "hence has finite fibres", "(Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite}), hence", "is finite by Lemma \\ref{lemma-characterize-finite}." ], "refs": [ "morphisms-lemma-finite-fibre", "morphisms-lemma-quasi-finite-points-open", "morphisms-lemma-quasi-finite-locally-quasi-compact", "morphisms-lemma-quasi-finite", "coherent-lemma-characterize-finite" ], "ref_ids": [ 5227, 5521, 5229, 5230, 3365 ] } ], "ref_ids": [ 13904 ] }, { "id": 3367, "type": "theorem", "label": "coherent-lemma-ample-on-fibre", "categories": [ "coherent" ], "title": "coherent-lemma-ample-on-fibre", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes with $Y$ Noetherian.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Let $y \\in Y$ be a point such that $\\mathcal{L}_y$ is ample on $X_y$.", "Then there exists a $d_0$ such that for all $d \\geq d_0$ we have", "$$", "R^pf_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})_y = 0", "\\text{ for }p > 0", "$$", "and the map", "$$", "f_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})_y", "\\longrightarrow", "H^0(X_y, \\mathcal{F}_y \\otimes_{\\mathcal{O}_{X_y}} \\mathcal{L}_y^{\\otimes d})", "$$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Note that $\\mathcal{O}_{Y, y}$ is a Noetherian local ring.", "Consider the canonical morphism", "$c : \\Spec(\\mathcal{O}_{Y, y}) \\to Y$, see", "Schemes, Equation (\\ref{schemes-equation-canonical-morphism}).", "This is a flat morphism as it identifies local rings.", "Denote momentarily $f' : X' \\to \\Spec(\\mathcal{O}_{Y, y})$", "the base change of $f$ to this local ring. We see that", "$c^*R^pf_*\\mathcal{F} = R^pf'_*\\mathcal{F}'$ by", "Lemma \\ref{lemma-flat-base-change-cohomology}.", "Moreover, the fibres $X_y$ and $X'_y$ are identified.", "Hence we may assume that $Y = \\Spec(A)$ is the spectrum of", "a Noetherian local ring $(A, \\mathfrak m, \\kappa)$ and $y \\in Y$", "corresponds to $\\mathfrak m$. In this case", "$R^pf_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})_y =", "H^p(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})$", "for all $p \\geq 0$. Denote $f_y : X_y \\to \\Spec(\\kappa)$ the projection.", "\\medskip\\noindent", "Let $B = \\text{Gr}_\\mathfrak m(A) =", "\\bigoplus_{n \\geq 0} \\mathfrak m^n/\\mathfrak m^{n + 1}$.", "Consider the sheaf $\\mathcal{B} = f_y^*\\widetilde{B}$", "of quasi-coherent graded $\\mathcal{O}_{X_y}$-algebras.", "We will use notation as in Section \\ref{section-theorem-formal-functions}", "with $I$ replaced by $\\mathfrak m$.", "Since $X_y$ is the closed subscheme of $X$ cut out by", "$\\mathfrak m\\mathcal{O}_X$ we may think of", "$\\mathfrak m^n\\mathcal{F}/\\mathfrak m^{n + 1}\\mathcal{F}$", "as a coherent $\\mathcal{O}_{X_y}$-module, see", "Lemma \\ref{lemma-i-star-equivalence}. Then", "$\\bigoplus_{n \\geq 0} \\mathfrak m^n\\mathcal{F}/\\mathfrak m^{n + 1}\\mathcal{F}$", "is a quasi-coherent graded $\\mathcal{B}$-module of finite type", "because it is generated in degree zero over $\\mathcal{B}$", "abd because the degree zero part is", "$\\mathcal{F}_y = \\mathcal{F}/\\mathfrak m \\mathcal{F}$", "which is a coherent $\\mathcal{O}_{X_y}$-module.", "Hence by Lemma \\ref{lemma-graded-finiteness} part (2)", "we see that", "$$", "H^p(X_y, \\mathfrak m^n \\mathcal{F}/ \\mathfrak m^{n + 1}\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X_y}} \\mathcal{L}_y^{\\otimes d}) = 0", "$$", "for all $p > 0$, $d \\geq d_0$, and $n \\geq 0$. By", "Lemma \\ref{lemma-relative-affine-cohomology}", "this is the same as the statement that", "$", "H^p(X, \\mathfrak m^n \\mathcal{F}/ \\mathfrak m^{n + 1}\\mathcal{F}", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) = 0", "$", "for all $p > 0$, $d \\geq d_0$, and $n \\geq 0$.", "\\medskip\\noindent", "Consider the short exact sequences", "$$", "0 \\to \\mathfrak m^n\\mathcal{F}/\\mathfrak m^{n + 1} \\mathcal{F}", "\\to \\mathcal{F}/\\mathfrak m^{n + 1} \\mathcal{F}", "\\to \\mathcal{F}/\\mathfrak m^n \\mathcal{F} \\to 0", "$$", "of coherent $\\mathcal{O}_X$-modules. Tensoring with $\\mathcal{L}^{\\otimes d}$", "is an exact functor and we obtain short exact sequences", "$$", "0 \\to", "\\mathfrak m^n\\mathcal{F}/\\mathfrak m^{n + 1} \\mathcal{F}", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}", "\\to \\mathcal{F}/\\mathfrak m^{n + 1} \\mathcal{F}", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}", "\\to \\mathcal{F}/\\mathfrak m^n \\mathcal{F}", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d} \\to 0", "$$", "Using the long exact cohomology sequence and the vanishing above", "we conclude (using induction) that", "\\begin{enumerate}", "\\item $H^p(X, \\mathcal{F}/\\mathfrak m^n \\mathcal{F}", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) = 0$", "for all $p > 0$, $d \\geq d_0$, and $n \\geq 0$, and", "\\item $H^0(X, \\mathcal{F}/\\mathfrak m^n \\mathcal{F}", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) \\to", "H^0(X_y, \\mathcal{F}_y \\otimes_{\\mathcal{O}_{X_y}} \\mathcal{L}_y^{\\otimes d})$", "is surjective for all $d \\geq d_0$ and $n \\geq 1$.", "\\end{enumerate}", "By the theorem on formal functions (Theorem \\ref{theorem-formal-functions})", "we find that the $\\mathfrak m$-adic completion of", "$H^p(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})$", "is zero for all $d \\geq d_0$ and $p > 0$.", "Since $H^p(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})$", "is a finite $A$-module by", "Lemma \\ref{lemma-proper-over-affine-cohomology-finite}", "it follows from Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "that $H^p(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})$", "is zero for all $d \\geq d_0$ and $p > 0$.", "For $p = 0$ we deduce from", "Lemma \\ref{lemma-ML-cohomology-powers-ideal} part (3)", "that $H^0(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) \\to", "H^0(X_y, \\mathcal{F}_y \\otimes_{\\mathcal{O}_{X_y}} \\mathcal{L}_y^{\\otimes d})$", "is surjective, which gives the final statement of the lemma." ], "refs": [ "coherent-lemma-flat-base-change-cohomology", "coherent-lemma-i-star-equivalence", "coherent-lemma-graded-finiteness", "coherent-lemma-relative-affine-cohomology", "coherent-theorem-formal-functions", "coherent-lemma-proper-over-affine-cohomology-finite", "algebra-lemma-NAK", "coherent-lemma-ML-cohomology-powers-ideal" ], "ref_ids": [ 3298, 3315, 3356, 3284, 3278, 3355, 401, 3360 ] } ], "ref_ids": [] }, { "id": 3368, "type": "theorem", "label": "coherent-lemma-ample-in-neighbourhood", "categories": [ "coherent" ], "title": "coherent-lemma-ample-in-neighbourhood", "contents": [ "(For a more general version see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-ample-in-neighbourhood}.)", "Let $f : X \\to Y$ be a proper morphism of schemes with $Y$ Noetherian.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $y \\in Y$ be a point such that $\\mathcal{L}_y$ is ample", "on $X_y$. Then there is an open neighbourhood $V \\subset Y$", "of $y$ such that $\\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$." ], "refs": [ "more-morphisms-lemma-ample-in-neighbourhood" ], "proofs": [ { "contents": [ "Pick $d_0$ as in Lemma \\ref{lemma-ample-on-fibre} for", "$\\mathcal{F} = \\mathcal{O}_X$. Pick $d \\geq d_0$", "so that we can find $r \\geq 0$ and sections", "$s_{y, 0}, \\ldots, s_{y, r} \\in H^0(X_y, \\mathcal{L}_y^{\\otimes d})$", "which define a closed immersion", "$$", "\\varphi_y =", "\\varphi_{\\mathcal{L}_y^{\\otimes d}, (s_{y, 0}, \\ldots, s_{y, r})} :", "X_y \\to \\mathbf{P}^r_{\\kappa(y)}.", "$$", "This is possible by Morphisms, Lemma", "\\ref{morphisms-lemma-finite-type-over-affine-ample-very-ample}", "but we also use", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}", "to see that $\\varphi_y$ is a closed immersion and", "Constructions, Section \\ref{constructions-section-projective-space}", "for the description of morphisms into projective", "space in terms of invertible sheaves and sections.", "By our choice of $d_0$, after replacing $Y$ by an open neighbourhood", "of $y$, we can choose", "$s_0, \\ldots, s_r \\in H^0(X, \\mathcal{L}^{\\otimes d})$", "mapping to $s_{y, 0}, \\ldots, s_{y, r}$.", "Let $X_{s_i} \\subset X$ be the open subset where $s_i$", "is a generator of $\\mathcal{L}^{\\otimes d}$. Since", "the $s_{y, i}$ generate $\\mathcal{L}_y^{\\otimes d}$ we see that", "$X_y \\subset U = \\bigcup X_{s_i}$.", "Since $X \\to Y$ is closed, we see that", "there is an open neighbourhood $y \\in V \\subset Y$", "such that $f^{-1}(V) \\subset U$.", "After replacing $Y$ by $V$ we may assume that", "the $s_i$ generate $\\mathcal{L}^{\\otimes d}$. Thus we", "obtain a morphism", "$$", "\\varphi = \\varphi_{\\mathcal{L}^{\\otimes d}, (s_0, \\ldots, s_r)} :", "X \\longrightarrow \\mathbf{P}^r_Y", "$$", "with $\\mathcal{L}^{\\otimes d} \\cong \\varphi^*\\mathcal{O}_{\\mathbf{P}^r_Y}(1)$", "whose base change to $y$ gives $\\varphi_y$.", "\\medskip\\noindent", "We will finish the proof by a sleight of hand; the ``correct'' proof", "proceeds by directly showing that $\\varphi$ is a closed", "immersion after base changing to an open neighbourhood of $y$.", "Namely, by Lemma \\ref{lemma-proper-finite-fibre-finite-in-neighbourhood}", "we see that $\\varphi$ is a finite over an open neighbourhood", "of the fibre $\\mathbf{P}^r_{\\kappa(y)}$ of $\\mathbf{P}^r_Y \\to Y$", "above $y$. Using that $\\mathbf{P}^r_Y \\to Y$ is closed, after", "shrinking $Y$ we may assume that $\\varphi$ is finite.", "Then $\\mathcal{L}^{\\otimes d} \\cong \\varphi^*\\mathcal{O}_{\\mathbf{P}^r_Y}(1)$", "is ample by the very general", "Morphisms, Lemma \\ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}." ], "refs": [ "coherent-lemma-ample-on-fibre", "morphisms-lemma-finite-type-over-affine-ample-very-ample", "morphisms-lemma-image-proper-scheme-closed", "coherent-lemma-proper-finite-fibre-finite-in-neighbourhood", "morphisms-lemma-pullback-ample-tensor-relatively-ample" ], "ref_ids": [ 3367, 5394, 5411, 3366, 5383 ] } ], "ref_ids": [ 13933 ] }, { "id": 3369, "type": "theorem", "label": "coherent-lemma-perfect-direct-image", "categories": [ "coherent" ], "title": "coherent-lemma-perfect-direct-image", "contents": [ "Let $A$ be a Noetherian ring and set $S = \\Spec(A)$. Let $f : X \\to S$ be a", "proper morphism of schemes. Let $\\mathcal{F}$ be a coherent", "$\\mathcal{O}_X$-module flat over $S$. Then", "\\begin{enumerate}", "\\item $R\\Gamma(X, \\mathcal{F})$ is a perfect object of $D(A)$, and", "\\item for any ring map $A \\to A'$ the base change map", "$$", "R\\Gamma(X, \\mathcal{F}) \\otimes_A^{\\mathbf{L}} A'", "\\longrightarrow", "R\\Gamma(X_{A'}, \\mathcal{F}_{A'})", "$$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a finite affine open covering $X = \\bigcup_{i = 1, \\ldots, n} U_i$.", "By Lemmas \\ref{lemma-separated-case-relative-cech} and", "\\ref{lemma-base-change-complex} the {\\v C}ech complex", "$K^\\bullet = {\\check C}^\\bullet(\\mathcal{U}, \\mathcal{F})$ satisfies", "$$", "K^\\bullet \\otimes_A A' = R\\Gamma(X_{A'}, \\mathcal{F}_{A'})", "$$", "for all ring maps $A \\to A'$. Let", "$K_{alt}^\\bullet = {\\check C}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "be the alternating {\\v C}ech complex. By", "Cohomology, Lemma \\ref{cohomology-lemma-alternating-usual}", "there is a homotopy equivalence $K_{alt}^\\bullet \\to K^\\bullet$", "of $A$-modules. In particular, we have", "$$", "K_{alt}^\\bullet \\otimes_A A' = R\\Gamma(X_{A'}, \\mathcal{F}_{A'})", "$$", "as well. Since $\\mathcal{F}$ is flat over $A$ we see that each $K_{alt}^n$", "is flat over $A$ (see", "Morphisms, Lemma \\ref{morphisms-lemma-flat-module-characterize}).", "Since moreover $K_{alt}^\\bullet$ is bounded above (this is why we switched", "to the alternating {\\v C}ech complex)", "$K_{alt}^\\bullet \\otimes_A A' = K_{alt}^\\bullet \\otimes_A^{\\mathbf{L}} A'$", "by the definition of derived tensor products (see", "More on Algebra, Section \\ref{more-algebra-section-derived-tensor-product}).", "By", "Lemma \\ref{lemma-proper-over-affine-cohomology-finite}", "the cohomology groups $H^i(K_{alt}^\\bullet)$ are finite $A$-modules.", "As $K_{alt}^\\bullet$ is bounded, we conclude that $K_{alt}^\\bullet$", "is pseudo-coherent, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-Noetherian-pseudo-coherent}.", "Given any $A$-module $M$ set $A' = A \\oplus M$ where $M$ is a square zero", "ideal, i.e., $(a, m) \\cdot (a', m') = (aa', am' + a'm)$. By the", "above we see that $K_{alt}^\\bullet \\otimes_A^\\mathbf{L} A'$ has cohomology", "in degrees $0, \\ldots, n$. Hence $K_{alt}^\\bullet \\otimes_A^\\mathbf{L} M$", "has cohomology in degrees $0, \\ldots, n$. Hence $K_{alt}^\\bullet$ has", "finite Tor dimension, see", "More on Algebra, Definition \\ref{more-algebra-definition-tor-amplitude}.", "We win by More on Algebra, Lemma \\ref{more-algebra-lemma-perfect}." ], "refs": [ "coherent-lemma-separated-case-relative-cech", "coherent-lemma-base-change-complex", "cohomology-lemma-alternating-usual", "morphisms-lemma-flat-module-characterize", "coherent-lemma-proper-over-affine-cohomology-finite", "more-algebra-lemma-Noetherian-pseudo-coherent", "more-algebra-definition-tor-amplitude", "more-algebra-lemma-perfect" ], "ref_ids": [ 3301, 3302, 2095, 5259, 3355, 10160, 10624, 10212 ] } ], "ref_ids": [] }, { "id": 3370, "type": "theorem", "label": "coherent-lemma-inverse-systems-affine", "categories": [ "coherent" ], "title": "coherent-lemma-inverse-systems-affine", "contents": [ "If $X = \\Spec(A)$ is the spectrum of a Noetherian ring and", "$\\mathcal{I}$ is the quasi-coherent sheaf of ideals associated to the ideal", "$I \\subset A$, then $\\textit{Coh}(X, \\mathcal{I})$ is equivalent to the", "category of finite $A^\\wedge$-modules where $A^\\wedge$ is the completion", "of $A$ with respect to $I$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\text{Mod}^{fg}_{A, I}$ be the category of inverse systems $(M_n)$", "of finite $A$-modules satisfying: (1) $M_n$ is annihilated by $I^n$ and (2)", "$M_{n + 1}/I^nM_{n + 1} = M_n$. By the correspondence between coherent", "sheaves on $X$ and finite $A$-modules (Lemma \\ref{lemma-coherent-Noetherian})", "it suffices to show $\\text{Mod}^{fg}_{A, I}$ is equivalent to the category of", "finite $A^\\wedge$-modules. To see this it suffices to prove that given", "an object $(M_n)$ of $\\text{Mod}^{fg}_{A, I}$ the module", "$$", "M = \\lim M_n", "$$", "is a finite $A^\\wedge$-module and that $M/I^nM = M_n$. As the transition", "maps are surjective, we see that $M \\to M_1$ is surjective.", "Pick $x_1, \\ldots, x_t \\in M$ which map to generators of $M_1$.", "This induces a map of systems $(A/I^n)^{\\oplus t} \\to M_n$.", "By Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK}) these maps are", "surjective. Let $K_n \\subset (A/I^n)^{\\oplus t}$ be the kernel.", "Property (2) implies that $K_{n + 1} \\to K_n$ is surjective, in particular", "the system $(K_n)$ satisfies the Mittag-Leffler condition.", "By Homology, Lemma \\ref{homology-lemma-Mittag-Leffler}", "we obtain an exact sequence", "$0 \\to K \\to (A^\\wedge)^{\\oplus t} \\to M \\to 0$", "with $K = \\lim K_n$.", "Hence $M$ is a finite $A^\\wedge$-module.", "As $K \\to K_n$ is surjective it follows that", "$$", "M/I^nM = \\Coker(K \\to (A/I^n)^{\\oplus t}) =", "(A/I^n)^{\\oplus t}/K_n = M_n", "$$", "as desired." ], "refs": [ "coherent-lemma-coherent-Noetherian", "algebra-lemma-NAK", "homology-lemma-Mittag-Leffler" ], "ref_ids": [ 3308, 401, 12124 ] } ], "ref_ids": [] }, { "id": 3371, "type": "theorem", "label": "coherent-lemma-inverse-systems-abelian", "categories": [ "coherent" ], "title": "coherent-lemma-inverse-systems-abelian", "contents": [ "Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be a quasi-coherent sheaf of ideals.", "\\begin{enumerate}", "\\item The category $\\textit{Coh}(X, \\mathcal{I})$ is abelian.", "\\item For $U \\subset X$ open the restriction functor", "$\\textit{Coh}(X, \\mathcal{I}) \\to \\textit{Coh}(U, \\mathcal{I}|_U)$", "is exact.", "\\item Exactness in $\\textit{Coh}(X, \\mathcal{I})$ may be checked by", "restricting to the members of an open covering of $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha =(\\alpha_n) : (\\mathcal{F}_n) \\to (\\mathcal{G}_n)$ be a morphism of", "$\\textit{Coh}(X, \\mathcal{I})$. The cokernel of $\\alpha$ is the inverse system", "$(\\Coker(\\alpha_n))$ (details omitted). To describe the kernel let", "$$", "\\mathcal{K}'_{l, m} = \\Im(\\Ker(\\alpha_l) \\to \\mathcal{F}_m)", "$$", "for $l \\geq m$.", "We claim:", "\\begin{enumerate}", "\\item[(a)] the inverse system $(\\mathcal{K}'_{l, m})_{l \\geq m}$ is", "eventually constant, say with value $\\mathcal{K}'_m$,", "\\item[(b)] the system $(\\mathcal{K}'_m/\\mathcal{I}^n\\mathcal{K}'_m)_{m \\geq n}$", "is eventually constant, say with value $\\mathcal{K}_n$,", "\\item[(c)] the system $(\\mathcal{K}_n)$ forms an object of", "$\\textit{Coh}(X, \\mathcal{I})$, and", "\\item[(d)] this object is the kernel of $\\alpha$.", "\\end{enumerate}", "To see (a), (b), and (c) we may work affine locally, say $X = \\Spec(A)$", "and $\\mathcal{I}$ corresponds to the ideal $I \\subset A$. By", "Lemma \\ref{lemma-inverse-systems-affine}", "$\\alpha$ corresponds to a map $f : M \\to N$ of finite $A^\\wedge$-modules.", "Denote $K = \\Ker(f)$. Note that $A^\\wedge$ is a Noetherian", "ring (Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian-Noetherian}).", "Choose an integer $c \\geq 0$ such that", "$K \\cap I^n M \\subset I^{n - c}K$ for $n \\geq c$", "(Algebra, Lemma \\ref{algebra-lemma-Artin-Rees})", "and which satisfies Algebra, Lemma \\ref{algebra-lemma-map-AR}", "for the map $f$ and the ideal $I^\\wedge = IA^\\wedge$. Then", "$\\mathcal{K}'_{l, m}$ corresponds to the $A$-module", "$$", "K'_{l, m} = \\frac{a^{-1}(I^lN) + I^mM}{I^mM} =", "\\frac{K + I^{l - c}f^{-1}(I^cN) + I^mM}{I^mM} =", "\\frac{K + I^mM}{I^mM}", "$$", "where the last equality holds if $l \\geq m + c$. So $\\mathcal{K}'_m$", "corresponds to the $A$-module $K/K \\cap I^mM$ and", "$\\mathcal{K}'_m/\\mathcal{I}^n\\mathcal{K}'_m$ corresponds to", "$$", "\\frac{K}{K \\cap I^mM + I^nK} = \\frac{K}{I^nK}", "$$", "for $m \\geq n + c$ by our choice of $c$ above. Hence $\\mathcal{K}_n$", "corresponds to $K/I^nK$.", "\\medskip\\noindent", "We prove (d). It is clear from the description on affines above that", "the composition $(\\mathcal{K}_n) \\to (\\mathcal{F}_n) \\to (\\mathcal{G}_n)$", "is zero. Let $\\beta : (\\mathcal{H}_n) \\to (\\mathcal{F}_n)$", "be a morphism such that $\\alpha \\circ \\beta = 0$. Then", "$\\mathcal{H}_l \\to \\mathcal{F}_l$ maps into $\\Ker(\\alpha_l)$.", "Since $\\mathcal{H}_m = \\mathcal{H}_l/\\mathcal{I}^m\\mathcal{H}_l$", "for $l \\geq m$ we obtain a system of maps", "$\\mathcal{H}_m \\to \\mathcal{K}'_{l, m}$. Thus a map", "$\\mathcal{H}_m \\to \\mathcal{K}_m'$. Since", "$\\mathcal{H}_n = \\mathcal{H}_m/\\mathcal{I}^n\\mathcal{H}_m$ we obtain", "a system of maps $\\mathcal{H}_n \\to \\mathcal{K}'_m/\\mathcal{I}^n\\mathcal{K}'_m$", "and hence a map $\\mathcal{H}_n \\to \\mathcal{K}_n$ as desired.", "\\medskip\\noindent", "To finish the proof of (1) we still have to show that $\\Coim = \\Im$", "in $\\textit{Coh}(X, \\mathcal{I})$. We have seen above that taking", "kernels and cokernels commutes, over affines, with the description", "of $\\textit{Coh}(X, \\mathcal{I})$ as a category of modules. Since", "$\\Im = \\Coim$ holds in the category of modules", "this gives $\\Coim = \\Im$ in $\\textit{Coh}(X, \\mathcal{I})$.", "Parts (2) and (3) of the lemma are immediate from our construction", "of kernels and cokernels." ], "refs": [ "coherent-lemma-inverse-systems-affine", "algebra-lemma-completion-Noetherian-Noetherian", "algebra-lemma-Artin-Rees", "algebra-lemma-map-AR" ], "ref_ids": [ 3370, 874, 625, 626 ] } ], "ref_ids": [] }, { "id": 3372, "type": "theorem", "label": "coherent-lemma-inverse-systems-surjective", "categories": [ "coherent" ], "title": "coherent-lemma-inverse-systems-surjective", "contents": [ "Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be a quasi-coherent sheaf of ideals. A map", "$(\\mathcal{F}_n) \\to (\\mathcal{G}_n)$ is surjective in", "$\\textit{Coh}(X, \\mathcal{I})$", "if and only if $\\mathcal{F}_1 \\to \\mathcal{G}_1$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Look on affine opens, use", "Lemma \\ref{lemma-inverse-systems-affine}, and use", "Algebra, Lemma \\ref{algebra-lemma-NAK}." ], "refs": [ "coherent-lemma-inverse-systems-affine", "algebra-lemma-NAK" ], "ref_ids": [ 3370, 401 ] } ], "ref_ids": [] }, { "id": 3373, "type": "theorem", "label": "coherent-lemma-exact", "categories": [ "coherent" ], "title": "coherent-lemma-exact", "contents": [ "The functor (\\ref{equation-completion-functor}) is exact." ], "refs": [], "proofs": [ { "contents": [ "It suffices to check this locally on $X$. Hence we may assume $X$ is", "affine, i.e., we have a situation as in", "Lemma \\ref{lemma-inverse-systems-affine}.", "The functor is the functor $\\text{Mod}^{fg}_A \\to \\text{Mod}^{fg}_{A^\\wedge}$", "which associates to a finite $A$-module $M$ the completion $M^\\wedge$.", "Thus the result follows from", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}." ], "refs": [ "coherent-lemma-inverse-systems-affine", "algebra-lemma-completion-flat" ], "ref_ids": [ 3370, 870 ] } ], "ref_ids": [] }, { "id": 3374, "type": "theorem", "label": "coherent-lemma-completion-internal-hom", "categories": [ "coherent" ], "title": "coherent-lemma-completion-internal-hom", "contents": [ "Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be a quasi-coherent sheaf of ideals. Let $\\mathcal{F}$, $\\mathcal{G}$ be", "coherent $\\mathcal{O}_X$-modules. Set", "$\\mathcal{H} = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{F})$.", "Then", "$$", "\\lim H^0(X, \\mathcal{H}/\\mathcal{I}^n\\mathcal{H}) =", "\\Mor_{\\textit{Coh}(X, \\mathcal{I})}", "(\\mathcal{G}^\\wedge, \\mathcal{F}^\\wedge).", "$$" ], "refs": [], "proofs": [ { "contents": [ "To prove this we may work affine locally on $X$.", "Hence we may assume $X = \\Spec(A)$ and $\\mathcal{F}$, $\\mathcal{G}$", "given by finite $A$-module $M$ and $N$. Then $\\mathcal{H}$", "corresponds to the finite $A$-module $H = \\Hom_A(M, N)$.", "The statement of the lemma becomes the statement", "$$", "H^\\wedge = \\Hom_{A^\\wedge}(M^\\wedge, N^\\wedge)", "$$", "via the equivalence of Lemma \\ref{lemma-inverse-systems-affine}.", "By Algebra, Lemma \\ref{algebra-lemma-completion-flat}", "(used 3 times) we have", "$$", "H^\\wedge = \\Hom_A(M, N) \\otimes_A A^\\wedge =", "\\Hom_{A^\\wedge}(M \\otimes_A A^\\wedge, N \\otimes_A A^\\wedge) =", "\\Hom_{A^\\wedge}(M^\\wedge, N^\\wedge)", "$$", "where the second equality uses that $A^\\wedge$ is flat over $A$", "(see More on Algebra, Lemma", "\\ref{more-algebra-lemma-pseudo-coherence-and-base-change-ext}).", "The lemma follows." ], "refs": [ "coherent-lemma-inverse-systems-affine", "algebra-lemma-completion-flat", "more-algebra-lemma-pseudo-coherence-and-base-change-ext" ], "ref_ids": [ 3370, 870, 10165 ] } ], "ref_ids": [] }, { "id": 3375, "type": "theorem", "label": "coherent-lemma-existence-easy", "categories": [ "coherent" ], "title": "coherent-lemma-existence-easy", "contents": [ "Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be a quasi-coherent sheaf of ideals. Let $\\mathcal{G}$ be a coherent", "$\\mathcal{O}_X$-module. Let $(\\mathcal{F}_n)$ an object of", "$\\textit{Coh}(X, \\mathcal{I})$.", "\\begin{enumerate}", "\\item If $\\alpha : (\\mathcal{F}_n) \\to \\mathcal{G}^\\wedge$ is", "a map whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$,", "then there exists a unique (up to unique isomorphism) triple", "$(\\mathcal{F}, a, \\beta)$ where", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module,", "\\item $a : \\mathcal{F} \\to \\mathcal{G}$ is an $\\mathcal{O}_X$-module map", "whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$,", "\\item $\\beta : (\\mathcal{F}_n) \\to \\mathcal{F}^\\wedge$ is an isomorphism, and", "\\item $\\alpha = a^\\wedge \\circ \\beta$.", "\\end{enumerate}", "\\item If $\\alpha : \\mathcal{G}^\\wedge \\to (\\mathcal{F}_n)$ is", "a map whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$,", "then there exists a unique (up to unique isomorphism) triple", "$(\\mathcal{F}, a, \\beta)$ where", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module,", "\\item $a : \\mathcal{G} \\to \\mathcal{F}$ is an $\\mathcal{O}_X$-module map", "whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$,", "\\item $\\beta : \\mathcal{F}^\\wedge \\to (\\mathcal{F}_n)$ is an isomorphism, and", "\\item $\\alpha = \\beta \\circ a^\\wedge$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). The uniqueness implies it suffices to construct", "$(\\mathcal{F}, a, \\beta)$", "Zariski locally on $X$. Thus we may assume $X = \\Spec(A)$ and $\\mathcal{I}$", "corresponds to the ideal $I \\subset A$. In this situation", "Lemma \\ref{lemma-inverse-systems-affine} applies.", "Let $M'$ be the finite $A^\\wedge$-module corresponding", "to $(\\mathcal{F}_n)$. Let $N$ be the finite $A$-module corresponding to", "$\\mathcal{G}$. Then $\\alpha$ corresponds to a map", "$$", "\\varphi : M' \\longrightarrow N^\\wedge", "$$", "whose kernel and cokernel are annihilated by $I^t$ for some $t$. Recall that", "$N^\\wedge = N \\otimes_A A^\\wedge$", "(Algebra, Lemma \\ref{algebra-lemma-completion-tensor}).", "By More on Algebra, Lemma \\ref{more-algebra-lemma-application-formal-glueing}", "there is an $A$-module map $\\psi : M \\to N$ whose kernel and cokernel are", "$I$-power torsion and an isomorphism", "$M \\otimes_A A^\\wedge = M'$ compatible with $\\varphi$.", "As $N$ and $M'$ are finite modules, we conclude that $M$", "is a finite $A$-module, see", "More on Algebra, Remark \\ref{more-algebra-remark-formal-glueing-algebras}.", "Hence $M \\otimes_A A^\\wedge = M^\\wedge$. We omit the verification", "that the triple $(M, N \\to M, M^\\wedge \\to M')$ so obtained", "is unique up to unique isomorphism.", "\\medskip\\noindent", "The proof of (2) is exactly the same and we omit it." ], "refs": [ "coherent-lemma-inverse-systems-affine", "algebra-lemma-completion-tensor", "more-algebra-lemma-application-formal-glueing", "more-algebra-remark-formal-glueing-algebras" ], "ref_ids": [ 3370, 869, 10352, 10663 ] } ], "ref_ids": [] }, { "id": 3376, "type": "theorem", "label": "coherent-lemma-torsion-hom-ext", "categories": [ "coherent" ], "title": "coherent-lemma-torsion-hom-ext", "contents": [ "Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be a quasi-coherent sheaf of ideals. Any object of", "$\\textit{Coh}(X, \\mathcal{I})$ which is annihilated", "by a power of $\\mathcal{I}$ is in the essential image of", "(\\ref{equation-completion-functor}).", "Moreover, if $\\mathcal{F}$, $\\mathcal{G}$ are in $\\textit{Coh}(\\mathcal{O}_X)$", "and either $\\mathcal{F}$ or $\\mathcal{G}$ is annihilated by a power of", "$\\mathcal{I}$, then the maps", "$$", "\\xymatrix{", "\\Hom_X(\\mathcal{F}, \\mathcal{G}) \\ar[d] &", "\\Ext_X(\\mathcal{F}, \\mathcal{G}) \\ar[d] \\\\", "\\Hom_{\\textit{Coh}(X, \\mathcal{I})}(\\mathcal{F}^\\wedge, \\mathcal{G}^\\wedge) &", "\\Ext_{\\textit{Coh}(X, \\mathcal{I})}(\\mathcal{F}^\\wedge, \\mathcal{G}^\\wedge)", "}", "$$", "are isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "Suppose $(\\mathcal{F}_n)$ is an object of $\\textit{Coh}(X, \\mathcal{I})$", "which is annihilated by $\\mathcal{I}^c$ for some $c \\geq 1$. Then", "$\\mathcal{F}_n \\to \\mathcal{F}_c$ is an isomorphism for $n \\geq c$.", "Hence if we set $\\mathcal{F} = \\mathcal{F}_c$, then we see that", "$\\mathcal{F}^\\wedge \\cong (\\mathcal{F}_n)$. This proves the first assertion.", "\\medskip\\noindent", "Let $\\mathcal{F}$, $\\mathcal{G}$ be objects of $\\textit{Coh}(\\mathcal{O}_X)$", "such that either $\\mathcal{F}$ or $\\mathcal{G}$ is annihilated by", "$\\mathcal{I}^c$ for some $c \\geq 1$. Then", "$\\mathcal{H} = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{F})$", "is a coherent $\\mathcal{O}_X$-module annihilated by $\\mathcal{I}^c$.", "Hence we see that", "$$", "\\Hom_X(\\mathcal{G}, \\mathcal{F}) =", "H^0(X, \\mathcal{H}) =", "\\lim H^0(X, \\mathcal{H}/\\mathcal{I}^n\\mathcal{H}) =", "\\Mor_{\\textit{Coh}(X, \\mathcal{I})}", "(\\mathcal{G}^\\wedge, \\mathcal{F}^\\wedge).", "$$", "see Lemma \\ref{lemma-completion-internal-hom}.", "This proves the statement on homomorphisms.", "\\medskip\\noindent", "The notation $\\Ext$ refers to extensions as defined in", "Homology, Section \\ref{homology-section-extensions}.", "The injectivity of the map on $\\Ext$'s follows immediately", "from the bijectivity of the map on $\\Hom$'s.", "For surjectivity, assume $\\mathcal{F}$ is annihilated", "by a power of $I$. Then part (1) of Lemma \\ref{lemma-existence-easy}", "shows that given an extension", "$$", "0 \\to \\mathcal{G}^\\wedge \\to (\\mathcal{E}_n) \\to \\mathcal{F}^\\wedge \\to 0", "$$", "in $\\textit{Coh}(U, I\\mathcal{O}_U)$", "the morphism $\\mathcal{G}^\\wedge \\to (\\mathcal{E}_n)$ is", "isomorphic to $\\mathcal{G} \\to \\mathcal{E}^\\wedge$", "for some $\\mathcal{G} \\to \\mathcal{E}$ in $\\textit{Coh}(\\mathcal{O}_U)$.", "Similarly in the other case." ], "refs": [ "coherent-lemma-completion-internal-hom", "coherent-lemma-existence-easy" ], "ref_ids": [ 3374, 3375 ] } ], "ref_ids": [] }, { "id": 3377, "type": "theorem", "label": "coherent-lemma-finite-over-rees-algebra", "categories": [ "coherent" ], "title": "coherent-lemma-finite-over-rees-algebra", "contents": [ "Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be a quasi-coherent sheaf of ideals. If $(\\mathcal{F}_n)$ is an object of", "$\\textit{Coh}(X, \\mathcal{I})$ then", "$\\bigoplus \\Ker(\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n)$ is", "a finite type, graded, quasi-coherent", "$\\bigoplus \\mathcal{I}^n/\\mathcal{I}^{n + 1}$-module." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$ hence we may assume $X$ is affine, i.e.,", "we have a situation as in Lemma \\ref{lemma-inverse-systems-affine}.", "In this case, if $(\\mathcal{F}_n)$ corresponds to the finite $A^\\wedge$", "module $M$, then $\\bigoplus \\Ker(\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n)$", "corresponds to $\\bigoplus I^nM/I^{n + 1}M$ which is clearly a finite", "module over $\\bigoplus I^n/I^{n + 1}$." ], "refs": [ "coherent-lemma-inverse-systems-affine" ], "ref_ids": [ 3370 ] } ], "ref_ids": [] }, { "id": 3378, "type": "theorem", "label": "coherent-lemma-inverse-systems-pullback", "categories": [ "coherent" ], "title": "coherent-lemma-inverse-systems-pullback", "contents": [ "Let $f : X \\to Y$ be a morphism of Noetherian schemes.", "Let $\\mathcal{J} \\subset \\mathcal{O}_Y$ be a quasi-coherent sheaf", "of ideals and set $\\mathcal{I} = f^{-1}\\mathcal{J} \\mathcal{O}_X$.", "Then there is a right exact functor", "$$", "f^* : \\textit{Coh}(Y, \\mathcal{J}) \\longrightarrow \\textit{Coh}(X, \\mathcal{I})", "$$", "which sends $(\\mathcal{G}_n)$ to $(f^*\\mathcal{G}_n)$. If $f$ is flat,", "then $f^*$ is an exact functor." ], "refs": [], "proofs": [ { "contents": [ "Since $f^* : \\textit{Coh}(\\mathcal{O}_Y) \\to \\textit{Coh}(\\mathcal{O}_X)$", "is right exact we have", "$$", "f^*\\mathcal{G}_n =", "f^*(\\mathcal{G}_{n + 1}/\\mathcal{I}^n\\mathcal{G}_{n + 1}) =", "f^*\\mathcal{G}_{n + 1}/f^{-1}\\mathcal{I}^nf^*\\mathcal{G}_{n + 1} =", "f^*\\mathcal{G}_{n + 1}/\\mathcal{J}^nf^*\\mathcal{G}_{n + 1}", "$$", "hence the pullback of a system is a system. The construction of", "cokernels in the proof of Lemma \\ref{lemma-inverse-systems-abelian}", "shows that", "$f^* : \\textit{Coh}(Y, \\mathcal{J}) \\to \\textit{Coh}(X, \\mathcal{I})$", "is always right exact. If $f$ is flat, then", "$f^* : \\textit{Coh}(\\mathcal{O}_Y) \\to \\textit{Coh}(\\mathcal{O}_X)$", "is an exact functor. It follows from the construction of kernels", "in the proof of Lemma \\ref{lemma-inverse-systems-abelian}", "that in this case", "$f^* : \\textit{Coh}(Y, \\mathcal{J}) \\to \\textit{Coh}(X, \\mathcal{I})$", "also transforms kernels into kernels." ], "refs": [ "coherent-lemma-inverse-systems-abelian", "coherent-lemma-inverse-systems-abelian" ], "ref_ids": [ 3371, 3371 ] } ], "ref_ids": [] }, { "id": 3379, "type": "theorem", "label": "coherent-lemma-inverse-systems-pullback-equivalence", "categories": [ "coherent" ], "title": "coherent-lemma-inverse-systems-pullback-equivalence", "contents": [ "Let $f : X' \\to X$ be a morphism of Noetherian schemes. Let $Z \\subset X$", "be a closed subscheme and denote $Z' = f^{-1}Z$ the scheme theoretic", "inverse image. Let $\\mathcal{I} \\subset \\mathcal{O}_X$,", "$\\mathcal{I}' \\subset \\mathcal{O}_{X'}$ be the corresponding", "quasi-coherent sheaves of ideals.", "If $f$ is flat and the induced morphism $Z' \\to Z$", "is an isomorphism, then the pullback functor", "$f^* : \\textit{Coh}(X, \\mathcal{I}) \\to \\textit{Coh}(X', \\mathcal{I}')$", "(Lemma \\ref{lemma-inverse-systems-pullback})", "is an equivalence." ], "refs": [ "coherent-lemma-inverse-systems-pullback" ], "proofs": [ { "contents": [ "If $X$ and $X'$ are affine, then this follows immediately from", "More on Algebra, Lemma \\ref{more-algebra-lemma-neighbourhood-equivalence}.", "To prove it in general we let $Z_n \\subset X$, $Z'_n \\subset X'$", "be the $n$th infinitesimal neighbourhoods of $Z$, $Z'$.", "The induced morphism $Z_n \\to Z'_n$ is a homeomorphism on", "underlying topological spaces. On the other hand, if $z' \\in Z'$", "maps to $z \\in Z$, then the ring map", "$\\mathcal{O}_{X, z} \\to \\mathcal{O}_{X', z'}$ is flat", "and induces an isomorphism", "$\\mathcal{O}_{X, z}/\\mathcal{I}_z \\to \\mathcal{O}_{X', z'}/\\mathcal{I}'_{z'}$.", "Hence it induces an isomorphism", "$\\mathcal{O}_{X, z}/\\mathcal{I}_z^n \\to", "\\mathcal{O}_{X', z'}/(\\mathcal{I}'_{z'})^n$", "for all $n \\geq 1$ for example by", "More on Algebra, Lemma \\ref{more-algebra-lemma-neighbourhood-isomorphism}.", "Thus $Z'_n \\to Z_n$ is an isomorphism of schemes.", "Thus $f^*$ induces an equivalence between the", "category of coherent $\\mathcal{O}_X$-modules annihilated by $\\mathcal{I}^n$", "and the", "category of coherent $\\mathcal{O}_{X'}$-modules annihilated by", "$(\\mathcal{I}')^n$, see", "Lemma \\ref{lemma-i-star-equivalence}.", "This clearly implies the lemma." ], "refs": [ "more-algebra-lemma-neighbourhood-equivalence", "more-algebra-lemma-neighbourhood-isomorphism", "coherent-lemma-i-star-equivalence" ], "ref_ids": [ 10341, 10340, 3315 ] } ], "ref_ids": [ 3378 ] }, { "id": 3380, "type": "theorem", "label": "coherent-lemma-inverse-systems-ideals-equivalence", "categories": [ "coherent" ], "title": "coherent-lemma-inverse-systems-ideals-equivalence", "contents": [ "Let $X$ be a Noetherian scheme. Let", "$\\mathcal{I}, \\mathcal{J} \\subset \\mathcal{O}_X$", "be quasi-coherent sheaves of ideals.", "If $V(\\mathcal{I}) = V(\\mathcal{J})$ is the same closed subset", "of $X$, then $\\textit{Coh}(X, \\mathcal{I})$ and $\\textit{Coh}(X, \\mathcal{J})$", "are equivalent." ], "refs": [], "proofs": [ { "contents": [ "First, assume $X = \\Spec(A)$ is affine. Let $I, J \\subset A$ be the ideals", "corresponding to $\\mathcal{I}, \\mathcal{J}$. Then $V(I) = V(J)$", "implies we have $I^c \\subset J$ and $J^d \\subset I$ for some $c, d \\geq 1$", "by elementary properties of the Zariski topology", "(see Algebra, Section \\ref{algebra-section-spectrum-ring} and", "Lemma \\ref{algebra-lemma-Noetherian-power}).", "Hence the $I$-adic and $J$-adic completions of $A$ agree, see", "Algebra, Lemma \\ref{algebra-lemma-change-ideal-completion}.", "Thus the equivalence follows from Lemma \\ref{lemma-inverse-systems-affine}", "in this case.", "\\medskip\\noindent", "In general, using what we said above and the fact that", "$X$ is quasi-compact, to choose $c, d \\geq 1$ such that", "$\\mathcal{I}^c \\subset \\mathcal{J}$ and $\\mathcal{J}^d \\subset \\mathcal{I}$.", "Then given an object $(\\mathcal{F}_n)$ in", "$\\textit{Coh}(X, \\mathcal{I})$ we claim that the", "inverse system", "$$", "(\\mathcal{F}_{cn}/\\mathcal{J}^n\\mathcal{F}_{cn})", "$$", "is in $\\textit{Coh}(X, \\mathcal{J})$. This may be checked on", "the members of an affine covering; we omit the details.", "In the same manner we can construct an object of", "$\\textit{Coh}(X, \\mathcal{I})$ starting with an object of", "$\\textit{Coh}(X, \\mathcal{J})$. We omit the verification", "that these constructions define mutually quasi-inverse functors." ], "refs": [ "algebra-lemma-Noetherian-power", "algebra-lemma-change-ideal-completion", "coherent-lemma-inverse-systems-affine" ], "ref_ids": [ 460, 865, 3370 ] } ], "ref_ids": [] }, { "id": 3381, "type": "theorem", "label": "coherent-lemma-fully-faithful", "categories": [ "coherent" ], "title": "coherent-lemma-fully-faithful", "contents": [ "Let $A$ be Noetherian ring complete with respect to an ideal $I$.", "Let $f : X \\to \\Spec(A)$ be a proper morphism. Let", "$\\mathcal{I} = I\\mathcal{O}_X$.", "Then the functor (\\ref{equation-completion-functor}) is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.", "Then $\\mathcal{H} = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{F})$", "is a coherent $\\mathcal{O}_X$-module, see", "Modules, Lemma \\ref{modules-lemma-internal-hom-locally-kernel-direct-sum}.", "By Lemma \\ref{lemma-completion-internal-hom} the map", "$$", "\\lim_n H^0(X, \\mathcal{H}/\\mathcal{I}^n\\mathcal{H})", "\\to", "\\Mor_{\\textit{Coh}(X, \\mathcal{I})}", "(\\mathcal{G}^\\wedge, \\mathcal{F}^\\wedge)", "$$", "is bijective. Hence fully faithfulness of", "(\\ref{equation-completion-functor}) follows from the theorem on formal", "functions (Lemma \\ref{lemma-spell-out-theorem-formal-functions})", "for the coherent sheaf $\\mathcal{H}$." ], "refs": [ "modules-lemma-internal-hom-locally-kernel-direct-sum", "coherent-lemma-completion-internal-hom", "coherent-lemma-spell-out-theorem-formal-functions" ], "ref_ids": [ 13298, 3374, 3361 ] } ], "ref_ids": [] }, { "id": 3382, "type": "theorem", "label": "coherent-lemma-vanishing-projective", "categories": [ "coherent" ], "title": "coherent-lemma-vanishing-projective", "contents": [ "Let $A$ be Noetherian ring and $I \\subset A$ and ideal.", "Let $f : X \\to \\Spec(A)$ be a proper morphism and let", "$\\mathcal{L}$ be an $f$-ample invertible sheaf. Let", "$\\mathcal{I} = I\\mathcal{O}_X$. Let $(\\mathcal{F}_n)$ be an", "object of $\\textit{Coh}(X, \\mathcal{I})$. Then there exists an", "integer $d_0$ such that", "$$", "H^1(X, \\Ker(\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n)", "\\otimes \\mathcal{L}^{\\otimes d} )", "= 0", "$$", "for all $n \\geq 0$ and all $d \\geq d_0$." ], "refs": [], "proofs": [ { "contents": [ "Set $B = \\bigoplus I^n/I^{n + 1}$ and", "$\\mathcal{B} = \\bigoplus \\mathcal{I}^n/\\mathcal{I}^{n + 1} = f^*\\widetilde{B}$.", "By Lemma \\ref{lemma-finite-over-rees-algebra} the graded quasi-coherent", "$\\mathcal{B}$-module", "$\\mathcal{G} = \\bigoplus \\Ker(\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n)$", "is of finite type. Hence the lemma follows from", "Lemma \\ref{lemma-graded-finiteness} part (2)." ], "refs": [ "coherent-lemma-finite-over-rees-algebra", "coherent-lemma-graded-finiteness" ], "ref_ids": [ 3377, 3356 ] } ], "ref_ids": [] }, { "id": 3383, "type": "theorem", "label": "coherent-lemma-existence-projective", "categories": [ "coherent" ], "title": "coherent-lemma-existence-projective", "contents": [ "Let $A$ be Noetherian ring complete with respect to an ideal $I$.", "Let $f : X \\to \\Spec(A)$ be a projective morphism. Let", "$\\mathcal{I} = I\\mathcal{O}_X$.", "Then the functor (\\ref{equation-completion-functor}) is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "We have already seen that (\\ref{equation-completion-functor}) is", "fully faithful in Lemma \\ref{lemma-fully-faithful}. Thus it suffices", "to show that the functor is essentially surjective.", "\\medskip\\noindent", "We first show that every object $(\\mathcal{F}_n)$ of", "$\\textit{Coh}(X, \\mathcal{I})$ is the quotient of an object", "in the image of (\\ref{equation-completion-functor}). ", "Let $\\mathcal{L}$ be an $f$-ample invertible sheaf on $X$.", "Choose $d_0$ as in Lemma \\ref{lemma-vanishing-projective}.", "Choose a $d \\geq d_0$ such that", "$\\mathcal{F}_1 \\otimes \\mathcal{L}^{\\otimes d}$", "is globally generated by some sections $s_{1, 1}, \\ldots, s_{t, 1}$.", "Since the transition maps of the system", "$$", "H^0(X, \\mathcal{F}_{n + 1} \\otimes \\mathcal{L}^{\\otimes d})", "\\longrightarrow", "H^0(X, \\mathcal{F}_n \\otimes \\mathcal{L}^{\\otimes d})", "$$", "are surjective by the vanishing of $H^1$ we can lift", "$s_{1, 1}, \\ldots, s_{t, 1}$ to a compatible system of global sections", "$s_{1, n}, \\ldots, s_{t, n}$ of", "$\\mathcal{F}_n \\otimes \\mathcal{L}^{\\otimes d}$.", "These determine a compatible system of maps", "$$", "(s_{1, n}, \\ldots, s_{t, n}) :", "(\\mathcal{L}^{\\otimes -d})^{\\oplus t} \\longrightarrow \\mathcal{F}_n", "$$", "Using Lemma \\ref{lemma-inverse-systems-surjective}", "we deduce that we have a surjective map", "$$", "\\left((\\mathcal{L}^{\\otimes -d})^{\\oplus t}\\right)^\\wedge", "\\longrightarrow", "(\\mathcal{F}_n)", "$$", "as desired.", "\\medskip\\noindent", "The result of the previous paragraph and the fact that", "$\\textit{Coh}(X, \\mathcal{I})$ is abelian", "(Lemma \\ref{lemma-inverse-systems-abelian})", "implies that", "every object of $\\textit{Coh}(X, \\mathcal{I})$ is a cokernel", "of a map between objects coming from $\\textit{Coh}(\\mathcal{O}_X)$.", "As (\\ref{equation-completion-functor}) is fully faithful and exact by", "Lemmas \\ref{lemma-fully-faithful} and \\ref{lemma-exact}", "we conclude." ], "refs": [ "coherent-lemma-fully-faithful", "coherent-lemma-vanishing-projective", "coherent-lemma-inverse-systems-surjective", "coherent-lemma-inverse-systems-abelian", "coherent-lemma-fully-faithful", "coherent-lemma-exact" ], "ref_ids": [ 3381, 3382, 3372, 3371, 3381, 3373 ] } ], "ref_ids": [] }, { "id": 3384, "type": "theorem", "label": "coherent-lemma-existence-tricky", "categories": [ "coherent" ], "title": "coherent-lemma-existence-tricky", "contents": [ "Let $X$ be a Noetherian scheme. Let", "$\\mathcal{I}, \\mathcal{K} \\subset \\mathcal{O}_X$", "be quasi-coherent sheaves of ideals.", "Let $X_e \\subset X$ be the closed subscheme cut out by $\\mathcal{K}^e$.", "Let $\\mathcal{I}_e = \\mathcal{I}\\mathcal{O}_{X_e}$.", "Let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(X, \\mathcal{I})$.", "Assume", "\\begin{enumerate}", "\\item the functor", "$\\textit{Coh}(\\mathcal{O}_{X_e}) \\to \\textit{Coh}(X_e, \\mathcal{I}_e)$", "is an equivalence for all $e \\geq 1$, and", "\\item there exists a coherent sheaf $\\mathcal{H}$ on $X$ and a map", "$\\alpha : (\\mathcal{F}_n) \\to \\mathcal{H}^\\wedge$ whose", "kernel and cokernel are annihilated by a power of $\\mathcal{K}$.", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ is in the essential image of", "(\\ref{equation-completion-functor})." ], "refs": [], "proofs": [ { "contents": [ "During this proof we will use without further mention that for a closed", "immersion $i : Z \\to X$ the functor $i_*$ gives an equivalence between the", "category of coherent modules on $Z$ and coherent modules on $X$ annihilated", "by the ideal sheaf of $Z$, see Lemma \\ref{lemma-i-star-equivalence}.", "In particular we may identify $\\textit{Coh}(\\mathcal{O}_{X_e})$", "with the category of coherent $\\mathcal{O}_X$-modules annihilated by", "$\\mathcal{K}^e$ and $\\textit{Coh}(X_e, \\mathcal{I}_e)$ as the full subcategory", "of $\\textit{Coh}(X, \\mathcal{I})$ of objects annihilated by $\\mathcal{K}^e$.", "Moreover (1) tells us these two categories are equivalent under the", "completion functor (\\ref{equation-completion-functor}).", "\\medskip\\noindent", "Applying this equivalence we get a coherent $\\mathcal{O}_X$-module", "$\\mathcal{G}_e$ annihilated by $\\mathcal{K}^e$ corresponding to the system", "$(\\mathcal{F}_n/\\mathcal{K}^e\\mathcal{F}_n)$ of", "$\\textit{Coh}(X, \\mathcal{I})$. The maps", "$\\mathcal{F}_n/\\mathcal{K}^{e + 1}\\mathcal{F}_n \\to", "\\mathcal{F}_n/\\mathcal{K}^e\\mathcal{F}_n$ correspond to canonical maps", "$\\mathcal{G}_{e + 1} \\to \\mathcal{G}_e$ which induce isomorphisms", "$\\mathcal{G}_{e + 1}/\\mathcal{K}^e\\mathcal{G}_{e + 1} \\to \\mathcal{G}_e$.", "Hence $(\\mathcal{G}_e)$ is an object of $\\textit{Coh}(X, \\mathcal{K})$.", "The map $\\alpha$ induces a system of maps", "$$", "\\mathcal{F}_n/\\mathcal{K}^e\\mathcal{F}_n", "\\longrightarrow", "\\mathcal{H}/(\\mathcal{I}^n + \\mathcal{K}^e)\\mathcal{H}", "$$", "whence maps $\\mathcal{G}_e \\to \\mathcal{H}/\\mathcal{K}^e\\mathcal{H}$", "(by the equivalence of categories again).", "Let $t \\geq 1$ be an integer, which exists by assumption (2),", "such that $\\mathcal{K}^t$ annihilates the kernel and cokernel of all the maps", "$\\mathcal{F}_n \\to \\mathcal{H}/\\mathcal{I}^n\\mathcal{H}$.", "Then $\\mathcal{K}^{2t}$ annihilates the kernel and cokernel of the maps", "$\\mathcal{F}_n/\\mathcal{K}^e\\mathcal{F}_n \\to", "\\mathcal{H}/(\\mathcal{I}^n + \\mathcal{K}^e)\\mathcal{H}$, see", "Remark \\ref{remark-inverse-systems-kernel-cokernel-annihilated-by}.", "Whereupon we conclude that $\\mathcal{K}^{4t}$ annihilates the kernel and", "the cokernel of the maps", "$$", "\\mathcal{G}_e", "\\longrightarrow", "\\mathcal{H}/\\mathcal{K}^e\\mathcal{H},", "$$", "see Remark \\ref{remark-inverse-systems-kernel-cokernel-annihilated-by}.", "We apply Lemma \\ref{lemma-existence-easy} to obtain a coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$, a map", "$a : \\mathcal{F} \\to \\mathcal{H}$ and an isomorphism", "$\\beta : (\\mathcal{G}_e) \\to (\\mathcal{F}/\\mathcal{K}^e\\mathcal{F})$", "in $\\textit{Coh}(X, \\mathcal{K})$. Working backwards, for a given $n$", "the triple", "$(\\mathcal{F}/\\mathcal{I}^n\\mathcal{F}, a \\bmod \\mathcal{I}^n, \\beta", "\\bmod \\mathcal{I}^n)$ is a triple as in the lemma for the morphism", "$\\alpha_n \\bmod \\mathcal{K}^e :", "(\\mathcal{F}_n/\\mathcal{K}^e\\mathcal{F}_n) \\to", "(\\mathcal{H}/(\\mathcal{I}^n + \\mathcal{K}^e)\\mathcal{H})$", "of $\\textit{Coh}(X, \\mathcal{K})$. Thus the uniqueness in", "Lemma \\ref{lemma-existence-easy}", "gives a canonical isomorphism", "$\\mathcal{F}/\\mathcal{I}^n\\mathcal{F} \\to \\mathcal{F}_n$", "compatible with all the morphisms in sight. This finishes the proof", "of the lemma." ], "refs": [ "coherent-lemma-i-star-equivalence", "coherent-remark-inverse-systems-kernel-cokernel-annihilated-by", "coherent-remark-inverse-systems-kernel-cokernel-annihilated-by", "coherent-lemma-existence-easy", "coherent-lemma-existence-easy" ], "ref_ids": [ 3315, 3408, 3408, 3375, 3375 ] } ], "ref_ids": [] }, { "id": 3385, "type": "theorem", "label": "coherent-lemma-inverse-systems-push-pull", "categories": [ "coherent" ], "title": "coherent-lemma-inverse-systems-push-pull", "contents": [ "Let $Y$ be a Noetherian scheme. Let", "$\\mathcal{J}, \\mathcal{K} \\subset \\mathcal{O}_Y$", "be quasi-coherent sheaves of ideals.", "Let $f : X \\to Y$ be a proper morphism which is an isomorphism", "over $V = Y \\setminus V(\\mathcal{K})$.", "Set $\\mathcal{I} = f^{-1}\\mathcal{J} \\mathcal{O}_X$.", "Let $(\\mathcal{G}_n)$ be an object of $\\textit{Coh}(Y, \\mathcal{J})$,", "let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module, and let", "$\\beta : (f^*\\mathcal{G}_n) \\to \\mathcal{F}^\\wedge$ be an isomorphism in", "$\\textit{Coh}(X, \\mathcal{I})$. Then there exists a map", "$$", "\\alpha :", "(\\mathcal{G}_n)", "\\longrightarrow", "(f_*\\mathcal{F})^\\wedge", "$$", "in $\\textit{Coh}(Y, \\mathcal{J})$ whose kernel and cokernel", "are annihilated by a power of $\\mathcal{K}$." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is a proper morphism we see that $f_*\\mathcal{F}$", "is a coherent $\\mathcal{O}_Y$-module", "(Proposition \\ref{proposition-proper-pushforward-coherent}).", "Thus the statement of the lemma makes sense.", "Consider the compositions", "$$", "\\gamma_n : \\mathcal{G}_n \\to", "f_*f^*\\mathcal{G}_n \\to", "f_*(\\mathcal{F}/\\mathcal{I}^n\\mathcal{F}).", "$$", "Here the first map is the adjunction map and the second is $f_*\\beta_n$.", "We claim that there exists a unique $\\alpha$ as in the lemma", "such that the compositions", "$$", "\\mathcal{G}_n \\xrightarrow{\\alpha_n}", "f_*\\mathcal{F}/\\mathcal{J}^nf_*\\mathcal{F} \\to", "f_*(\\mathcal{F}/\\mathcal{I}^n\\mathcal{F})", "$$", "equal $\\gamma_n$ for all $n$. Because of the uniqueness we may assume", "that $Y = \\Spec(B)$ is affine. Let $J \\subset B$ corresponds to the", "ideal $\\mathcal{J}$. Set", "$$", "M_n = H^0(X, \\mathcal{F}/\\mathcal{I}^n\\mathcal{F})", "\\quad\\text{and}\\quad", "M = H^0(X, \\mathcal{F})", "$$", "By Lemma \\ref{lemma-ML-cohomology-powers-ideal} and", "Theorem \\ref{theorem-formal-functions}", "the inverse limit of the modules", "$M_n$ equals the completion $M^\\wedge = \\lim M/J^nM$.", "Set $N_n = H^0(Y, \\mathcal{G}_n)$ and $N = \\lim N_n$.", "Via the equivalence of categories of", "Lemma \\ref{lemma-inverse-systems-affine}", "the finite $B^\\wedge$ modules $N$ and $M^\\wedge$ correspond", "to $(\\mathcal{G}_n)$ and $f_*\\mathcal{F}^\\wedge$.", "It follows from this that $\\alpha$ has to be the morphism of", "$\\textit{Coh}(Y, \\mathcal{J})$ corresponding to the homomorphism", "$$", "\\lim \\gamma_n : N = \\lim_n N_n \\longrightarrow \\lim M_n = M^\\wedge", "$$", "of finite $B^\\wedge$-modules.", "\\medskip\\noindent", "We still have to show that the kernel and cokernel of $\\alpha$ are", "annihilated by a power of $\\mathcal{K}$. Set $Y' = \\Spec(B^\\wedge)$", "and $X' = Y' \\times_Y X$. Let $\\mathcal{K}'$, $\\mathcal{J}'$, $\\mathcal{G}'_n$", "and $\\mathcal{I}'$, $\\mathcal{F}'$ be the pullback of", "$\\mathcal{K}$, $\\mathcal{J}$, $\\mathcal{G}_n$ and", "$\\mathcal{I}$, $\\mathcal{F}$, to $Y'$ and $X'$.", "The projection morphism $f' : X' \\to Y'$ is the base change of", "$f$ by $Y' \\to Y$. Note that $Y' \\to Y$ is a flat morphism of schemes", "as $B \\to B^\\wedge$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}.", "Hence $f'_*\\mathcal{F}'$, resp.\\ $f'_*(f')^*\\mathcal{G}_n'$", "is the pullback of $f_*\\mathcal{F}$, resp.\\ $f_*f^*\\mathcal{G}_n$", "to $Y'$ by Lemma \\ref{lemma-flat-base-change-cohomology}.", "The uniqueness of our construction shows the pullback of $\\alpha$ to $Y'$", "is the corresponding map $\\alpha'$ constructed for the situation on $Y'$.", "Moreover, to check that the kernel and cokernel of $\\alpha$ are", "annihilated by $\\mathcal{K}^t$ it suffices to check that the", "kernel and cokernel of $\\alpha'$ are annihilated by", "$(\\mathcal{K}')^t$. Namely, to see this we need to check this for", "kernels and cokernels of the maps $\\alpha_n$ and $\\alpha'_n$", "(see Remark \\ref{remark-inverse-systems-kernel-cokernel-annihilated-by})", "and the ring map $B \\to B^\\wedge$ induces", "an equivalence of categories between modules annihilated by", "$J^n$ and $(J')^n$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-neighbourhood-equivalence}.", "Thus we may assume $B$ is complete with respect to $J$.", "\\medskip\\noindent", "Assume $Y = \\Spec(B)$ is affine, $\\mathcal{J}$ corresponds to the ideal", "$J \\subset B$, and $B$ is complete with respect to $J$.", "In this case $(\\mathcal{G}_n)$ is in the essential image of the functor", "$\\textit{Coh}(\\mathcal{O}_Y) \\to \\textit{Coh}(Y, \\mathcal{J})$.", "Say $\\mathcal{G}$ is a coherent $\\mathcal{O}_Y$-module such that", "$(\\mathcal{G}_n) = \\mathcal{G}^\\wedge$. Note that", "$f^*(\\mathcal{G}^\\wedge) = (f^*\\mathcal{G})^\\wedge$. Hence", "Lemma \\ref{lemma-fully-faithful}", "tells us that $\\beta$ comes from an isomorphism", "$b : f^*\\mathcal{G} \\to \\mathcal{F}$", "and $\\alpha$ is the completion functor applied to", "$$", "\\mathcal{G} \\to f_*f^*\\mathcal{G} \\cong f_*\\mathcal{F}", "$$", "Hence we are trying to verify that the kernel and cokernel of the", "adjunction map $c : \\mathcal{G} \\to f_*f^*\\mathcal{G}$ are annihilated by", "a power of $\\mathcal{K}$. However, since the restriction", "$f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is an isomorphism", "we see that $c|_V$ is an isomorphism. Thus the coherent sheaves", "$\\Ker(c)$ and $\\Coker(c)$ are supported on $V(\\mathcal{K})$", "hence are annihilated by a power of $\\mathcal{K}$", "(Lemma \\ref{lemma-power-ideal-kills-sheaf}) as desired." ], "refs": [ "coherent-proposition-proper-pushforward-coherent", "coherent-lemma-ML-cohomology-powers-ideal", "coherent-theorem-formal-functions", "coherent-lemma-inverse-systems-affine", "algebra-lemma-completion-flat", "coherent-lemma-flat-base-change-cohomology", "coherent-remark-inverse-systems-kernel-cokernel-annihilated-by", "more-algebra-lemma-neighbourhood-equivalence", "coherent-lemma-fully-faithful", "coherent-lemma-power-ideal-kills-sheaf" ], "ref_ids": [ 3401, 3360, 3278, 3370, 870, 3298, 3408, 10341, 3381, 3320 ] } ], "ref_ids": [] }, { "id": 3386, "type": "theorem", "label": "coherent-lemma-closed-proper-over-base", "categories": [ "coherent" ], "title": "coherent-lemma-closed-proper-over-base", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $Z \\subset X$ be a closed subset. The following are equivalent", "\\begin{enumerate}", "\\item the morphism $Z \\to S$ is proper if $Z$ is endowed with the reduced", "induced closed subscheme structure", "(Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme}),", "\\item for some closed subscheme structure on $Z$ the morphism $Z \\to S$", "is proper,", "\\item for any closed subscheme structure on $Z$ the morphism", "$Z \\to S$ is proper.", "\\end{enumerate}" ], "refs": [ "schemes-definition-reduced-induced-scheme" ], "proofs": [ { "contents": [ "The implications (3) $\\Rightarrow$ (2) and (1) $\\Rightarrow$ (2)", "are immediate. Thus it suffices to prove that (2) implies (3).", "We urge the reader to find their own proof of this fact.", "Let $Z'$ and $Z''$ be closed subscheme structures on $Z$", "such that $Z' \\to S$ is proper. We have to show that $Z'' \\to S$ is proper.", "Let $Z''' = Z' \\cup Z''$ be the scheme theoretic union, see", "Morphisms, Definition", "\\ref{morphisms-definition-scheme-theoretic-intersection-union}.", "Then $Z'''$ is another closed subscheme structure on $Z$.", "This follows for example from the description of scheme theoretic unions in", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-union}.", "Since $Z'' \\to Z'''$ is a closed immersion it suffices to prove", "that $Z''' \\to S$ is proper (see", "Morphisms, Lemmas \\ref{morphisms-lemma-closed-immersion-proper} and", "\\ref{morphisms-lemma-composition-proper}).", "The morphism $Z' \\to Z'''$ is a bijective closed immersion", "and in particular surjective and universally closed.", "Then the fact that $Z' \\to S$ is separated implies that", "$Z''' \\to S$ is separated, see", "Morphisms, Lemma \\ref{morphisms-lemma-image-universally-closed-separated}.", "Moreover $Z''' \\to S$ is locally of finite type", "as $X \\to S$ is locally of finite type", "(Morphisms, Lemmas \\ref{morphisms-lemma-immersion-locally-finite-type} and", "\\ref{morphisms-lemma-composition-finite-type}).", "Since $Z' \\to S$ is quasi-compact and $Z' \\to Z'''$ is a homeomorphism", "we see that $Z''' \\to S$ is quasi-compact.", "Finally, since $Z' \\to S$ is universally closed, we see that", "the same thing is true for $Z''' \\to S$ by", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-is-proper}.", "This finishes the proof." ], "refs": [ "morphisms-definition-scheme-theoretic-intersection-union", "morphisms-lemma-scheme-theoretic-union", "morphisms-lemma-closed-immersion-proper", "morphisms-lemma-composition-proper", "morphisms-lemma-image-universally-closed-separated", "morphisms-lemma-immersion-locally-finite-type", "morphisms-lemma-composition-finite-type", "morphisms-lemma-image-proper-is-proper" ], "ref_ids": [ 5537, 5140, 5410, 5408, 5415, 5201, 5199, 5413 ] } ], "ref_ids": [ 7745 ] }, { "id": 3387, "type": "theorem", "label": "coherent-lemma-closed-closed-proper-over-base", "categories": [ "coherent" ], "title": "coherent-lemma-closed-closed-proper-over-base", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $Y \\subset Z \\subset X$ be closed subsets.", "If $Z$ is proper over $S$, then the same is true for $Y$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3388, "type": "theorem", "label": "coherent-lemma-base-change-closed-proper-over-base", "categories": [ "coherent" ], "title": "coherent-lemma-base-change-closed-proper-over-base", "contents": [ "Consider a cartesian diagram of schemes", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "with $f$ locally of finite type.", "If $Z$ is a closed subset of $X$ proper over $S$, then", "$(g')^{-1}(Z)$ is a closed subset of $X'$ proper over $S'$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the statement makes sense as $f'$ is locally of", "finite type by Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-type}.", "Endow $Z$ with the reduced induced closed subscheme structure.", "Denote $Z' = (g')^{-1}(Z)$ the scheme theoretic inverse image", "(Schemes, Definition \\ref{schemes-definition-inverse-image-closed-subscheme}).", "Then $Z' = X' \\times_X Z = (S' \\times_S X) \\times_X Z = S' \\times_S Z$", "is proper over $S'$ as a base change of $Z$ over $S$", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-proper})." ], "refs": [ "morphisms-lemma-base-change-finite-type", "schemes-definition-inverse-image-closed-subscheme", "morphisms-lemma-base-change-proper" ], "ref_ids": [ 5200, 7749, 5409 ] } ], "ref_ids": [] }, { "id": 3389, "type": "theorem", "label": "coherent-lemma-functoriality-closed-proper-over-base", "categories": [ "coherent" ], "title": "coherent-lemma-functoriality-closed-proper-over-base", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes which", "are locally of finite type over $S$.", "\\begin{enumerate}", "\\item If $Y$ is separated over $S$ and $Z \\subset X$ is a closed subset", "proper over $S$, then $f(Z)$ is a closed subset of $Y$ proper over $S$.", "\\item If $f$ is universally closed and $Z \\subset X$ is a", "closed subset proper over $S$, then $f(Z)$ is a closed subset", "of $Y$ proper over $S$.", "\\item If $f$ is proper and $Z \\subset Y$ is a closed subset", "proper over $S$, then $f^{-1}(Z)$ is a closed subset of $X$ proper over $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Assume $Y$ is separated over $S$ and $Z \\subset X$", "is a closed subset proper over $S$. Endow $Z$ with the reduced induced", "closed subscheme structure and apply", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-image-is-proper}", "to $Z \\to Y$ over $S$ to conclude.", "\\medskip\\noindent", "Proof of (2). Assume $f$ is universally closed and $Z \\subset X$ is a", "closed subset proper over $S$. Endow $Z$ and $Z' = f(Z)$ with their reduced", "induced closed subscheme structures. We obtain an induced", "morphism $Z \\to Z'$.", "Denote $Z'' = f^{-1}(Z')$ the scheme theoretic inverse image", "(Schemes, Definition \\ref{schemes-definition-inverse-image-closed-subscheme}).", "Then $Z'' \\to Z'$ is universally closed as a base change of $f$", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-proper}).", "Hence $Z \\to Z'$ is universally closed as a composition of", "the closed immersion $Z \\to Z''$ and $Z'' \\to Z'$", "(Morphisms, Lemmas", "\\ref{morphisms-lemma-closed-immersion-proper} and", "\\ref{morphisms-lemma-composition-proper}).", "We conclude that $Z' \\to S$ is separated by", "Morphisms, Lemma \\ref{morphisms-lemma-image-universally-closed-separated}.", "Since $Z \\to S$ is quasi-compact and $Z \\to Z'$ is surjective", "we see that $Z' \\to S$ is quasi-compact.", "Since $Z' \\to S$ is the composition of $Z' \\to Y$ and $Y \\to S$", "we see that $Z' \\to S$ is locally of finite type", "(Morphisms, Lemmas \\ref{morphisms-lemma-immersion-locally-finite-type} and", "\\ref{morphisms-lemma-composition-finite-type}).", "Finally, since $Z \\to S$ is universally closed, we see that", "the same thing is true for $Z' \\to S$ by", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-is-proper}.", "This finishes the proof.", "\\medskip\\noindent", "Proof of (3). Assume $f$ is proper and $Z \\subset Y$ is a closed subset", "proper over $S$. Endow $Z$ with the reduced induced closed subscheme", "structure. Denote $Z' = f^{-1}(Z)$ the scheme theoretic inverse image", "(Schemes, Definition \\ref{schemes-definition-inverse-image-closed-subscheme}).", "Then $Z' \\to Z$ is proper as a base change of $f$", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-proper}).", "Whence $Z' \\to S$ is proper as the composition of $Z' \\to Z$", "and $Z \\to S$", "(Morphisms, Lemma \\ref{morphisms-lemma-composition-proper}).", "This finishes the proof." ], "refs": [ "morphisms-lemma-scheme-theoretic-image-is-proper", "schemes-definition-inverse-image-closed-subscheme", "morphisms-lemma-base-change-proper", "morphisms-lemma-closed-immersion-proper", "morphisms-lemma-composition-proper", "morphisms-lemma-image-universally-closed-separated", "morphisms-lemma-immersion-locally-finite-type", "morphisms-lemma-composition-finite-type", "morphisms-lemma-image-proper-is-proper", "schemes-definition-inverse-image-closed-subscheme", "morphisms-lemma-base-change-proper", "morphisms-lemma-composition-proper" ], "ref_ids": [ 5414, 7749, 5409, 5410, 5408, 5415, 5201, 5199, 5413, 7749, 5409, 5408 ] } ], "ref_ids": [] }, { "id": 3390, "type": "theorem", "label": "coherent-lemma-union-closed-proper-over-base", "categories": [ "coherent" ], "title": "coherent-lemma-union-closed-proper-over-base", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $Z_i \\subset X$, $i = 1, \\ldots, n$ be closed subsets.", "If $Z_i$, $i = 1, \\ldots, n$ are proper over $S$, then the same is", "true for $Z_1 \\cup \\ldots \\cup Z_n$." ], "refs": [], "proofs": [ { "contents": [ "Endow $Z_i$ with their reduced induced closed subscheme structures.", "The morphism", "$$", "Z_1 \\amalg \\ldots \\amalg Z_n \\longrightarrow X", "$$", "is finite by Morphisms, Lemmas", "\\ref{morphisms-lemma-closed-immersion-finite} and", "\\ref{morphisms-lemma-finite-union-finite}.", "As finite morphisms are universally closed", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-proper})", "and since $Z_1 \\amalg \\ldots \\amalg Z_n$ is proper over $S$", "we conclude by", "Lemma \\ref{lemma-functoriality-closed-proper-over-base} part (2)", "that the image $Z_1 \\cup \\ldots \\cup Z_n$ is proper over $S$." ], "refs": [ "morphisms-lemma-closed-immersion-finite", "morphisms-lemma-finite-union-finite", "morphisms-lemma-finite-proper", "coherent-lemma-functoriality-closed-proper-over-base" ], "ref_ids": [ 5446, 5447, 5445, 3389 ] } ], "ref_ids": [] }, { "id": 3391, "type": "theorem", "label": "coherent-lemma-module-support-proper-over-base", "categories": [ "coherent" ], "title": "coherent-lemma-module-support-proper-over-base", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally", "of finite type. Let $\\mathcal{F}$ be a finite type, quasi-coherent", "$\\mathcal{O}_X$-module. The following are equivalent", "\\begin{enumerate}", "\\item the support of $\\mathcal{F}$ is proper over $S$,", "\\item the scheme theoretic support of $\\mathcal{F}$", "(Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-support})", "is proper over $S$, and", "\\item there exists a closed subscheme $Z \\subset X$ and", "a finite type, quasi-coherent $\\mathcal{O}_Z$-module", "$\\mathcal{G}$ such that (a) $Z \\to S$ is proper, and (b)", "$(Z \\to X)_*\\mathcal{G} = \\mathcal{F}$.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-scheme-theoretic-support" ], "proofs": [ { "contents": [ "The support $\\text{Supp}(\\mathcal{F})$ of $\\mathcal{F}$ is a closed subset", "of $X$, see Morphisms, Lemma \\ref{morphisms-lemma-support-finite-type}.", "Hence we can apply Definition \\ref{definition-proper-over-base}.", "Since the scheme theoretic support of $\\mathcal{F}$ is a closed", "subscheme whose underlying closed subset is $\\text{Supp}(\\mathcal{F})$", "we see that (1) and (2) are equivalent by", "Definition \\ref{definition-proper-over-base}.", "It is clear that (2) implies (3).", "Conversely, if (3) is true, then", "$\\text{Supp}(\\mathcal{F}) \\subset Z$", "(an inclusion of closed subsets of $X$)", "and hence $\\text{Supp}(\\mathcal{F})$", "is proper over $S$ for example by", "Lemma \\ref{lemma-closed-closed-proper-over-base}." ], "refs": [ "morphisms-lemma-support-finite-type", "coherent-definition-proper-over-base", "coherent-definition-proper-over-base", "coherent-lemma-closed-closed-proper-over-base" ], "ref_ids": [ 5143, 3405, 3405, 3387 ] } ], "ref_ids": [ 5538 ] }, { "id": 3392, "type": "theorem", "label": "coherent-lemma-base-change-module-support-proper-over-base", "categories": [ "coherent" ], "title": "coherent-lemma-base-change-module-support-proper-over-base", "contents": [ "Consider a cartesian diagram of schemes", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "with $f$ locally of finite type. Let $\\mathcal{F}$ be a", "finite type, quasi-coherent $\\mathcal{O}_X$-module.", "If the support of $\\mathcal{F}$ is proper over $S$, then", "the support of $(g')^*\\mathcal{F}$ is proper over $S'$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the statement makes sense because", "$(g')*\\mathcal{F}$ is of finite type by", "Modules, Lemma \\ref{modules-lemma-pullback-finite-type}.", "We have $\\text{Supp}((g')^*\\mathcal{F}) = (g')^{-1}(\\text{Supp}(\\mathcal{F}))$", "by Morphisms, Lemma \\ref{morphisms-lemma-support-finite-type}.", "Thus the lemma follows from", "Lemma \\ref{lemma-base-change-closed-proper-over-base}." ], "refs": [ "modules-lemma-pullback-finite-type", "morphisms-lemma-support-finite-type", "coherent-lemma-base-change-closed-proper-over-base" ], "ref_ids": [ 13236, 5143, 3388 ] } ], "ref_ids": [] }, { "id": 3393, "type": "theorem", "label": "coherent-lemma-cat-module-support-proper-over-base", "categories": [ "coherent" ], "title": "coherent-lemma-cat-module-support-proper-over-base", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally", "of finite type. Let $\\mathcal{F}$, $\\mathcal{G}$", "be finite type, quasi-coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item If the supports of $\\mathcal{F}$, $\\mathcal{G}$", "are proper over $S$, then the same is true", "for $\\mathcal{F} \\oplus \\mathcal{G}$, for any extension", "of $\\mathcal{G}$ by $\\mathcal{F}$, for $\\Im(u)$ and $\\Coker(u)$", "given any $\\mathcal{O}_X$-module map $u : \\mathcal{F} \\to \\mathcal{G}$,", "and for any quasi-coherent quotient of $\\mathcal{F}$ or $\\mathcal{G}$.", "\\item If $S$ is locally Noetherian, then the category of", "coherent $\\mathcal{O}_X$-modules with support proper over", "$S$ is a Serre subcategory (Homology, Definition", "\\ref{homology-definition-serre-subcategory})", "of the abelian category of", "coherent $\\mathcal{O}_X$-modules.", "\\end{enumerate}" ], "refs": [ "homology-definition-serre-subcategory" ], "proofs": [ { "contents": [ "Proof of (1). Let $Z$, $Z'$ be the support of $\\mathcal{F}$", "and $\\mathcal{G}$. Then all the sheaves mentioned in (1)", "have support contained in $Z \\cup Z'$. Thus the assertion itself", "is clear from Lemmas \\ref{lemma-closed-closed-proper-over-base} and", "\\ref{lemma-union-closed-proper-over-base}", "provided we check that these sheaves are finite type", "and quasi-coherent. For quasi-coherence we refer the reader to", "Schemes, Section \\ref{schemes-section-quasi-coherent}.", "For ``finite type'' we suggest the reader take a look at", "Modules, Section \\ref{modules-section-finite-type}.", "\\medskip\\noindent", "Proof of (2). The proof is the same as the proof of (1). Note that", "the assertions make sense as $X$ is locally Noetherian by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}", "and by the description of the category of coherent", "modules in Section \\ref{section-coherent-sheaves}." ], "refs": [ "coherent-lemma-closed-closed-proper-over-base", "coherent-lemma-union-closed-proper-over-base", "morphisms-lemma-finite-type-noetherian" ], "ref_ids": [ 3387, 3390, 5202 ] } ], "ref_ids": [ 12146 ] }, { "id": 3394, "type": "theorem", "label": "coherent-lemma-support-proper-over-base-pushforward", "categories": [ "coherent" ], "title": "coherent-lemma-support-proper-over-base-pushforward", "contents": [ "Let $S$ be a locally Noetherian scheme.", "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module", "with support proper over $S$. Then $R^pf_*\\mathcal{F}$", "is a coherent $\\mathcal{O}_S$-module for all $p \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-module-support-proper-over-base}", "there exists a closed immersion $i : Z \\to X$ and", "a finite type, quasi-coherent $\\mathcal{O}_Z$-module", "$\\mathcal{G}$ such that (a) $g = f \\circ i : Z \\to S$ is proper, and (b)", "$i_*\\mathcal{G} = \\mathcal{F}$.", "We see that $R^pg_*\\mathcal{G}$ is coherent on $S$ by", "Proposition \\ref{proposition-proper-pushforward-coherent}.", "On the other hand, $R^qi_*\\mathcal{G} = 0$ for $q > 0$", "(Lemma \\ref{lemma-finite-pushforward-coherent}).", "By Cohomology, Lemma \\ref{cohomology-lemma-relative-Leray}", "we get $R^pf_*\\mathcal{F} = R^pg_*\\mathcal{G}$ which concludes the proof." ], "refs": [ "coherent-lemma-module-support-proper-over-base", "coherent-proposition-proper-pushforward-coherent", "coherent-lemma-finite-pushforward-coherent", "cohomology-lemma-relative-Leray" ], "ref_ids": [ 3391, 3401, 3316, 2073 ] } ], "ref_ids": [] }, { "id": 3395, "type": "theorem", "label": "coherent-lemma-systems-with-proper-support", "categories": [ "coherent" ], "title": "coherent-lemma-systems-with-proper-support", "contents": [ "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a finite type morphism.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be", "a quasi-coherent sheaf of ideals. The following are Serre subcategories", "of $\\textit{Coh}(X, \\mathcal{I})$", "\\begin{enumerate}", "\\item the full subcategory of $\\textit{Coh}(X, \\mathcal{I})$", "consisting of those objects $(\\mathcal{F}_n)$ such that", "the support of $\\mathcal{F}_1$ is proper over $S$,", "\\item the full subcategory of $\\textit{Coh}(X, \\mathcal{I})$", "consisting of those objects $(\\mathcal{F}_n)$ such that", "there exists a closed subscheme $Z \\subset X$ proper over $S$", "with $\\mathcal{I}_Z \\mathcal{F}_n = 0$ for all $n \\geq 1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion of", "Homology, Lemma \\ref{homology-lemma-characterize-serre-subcategory}.", "Moreover, we will use that if", "$0 \\to (\\mathcal{G}_n) \\to (\\mathcal{F}_n) \\to (\\mathcal{H}_n) \\to 0$", "is a short exact sequence of $\\textit{Coh}(X, \\mathcal{I})$, then", "(a) $\\mathcal{G}_n \\to \\mathcal{F}_n \\to \\mathcal{H}_n \\to 0$", "is exact for all $n \\geq 1$ and", "(b) $\\mathcal{G}_n$ is a quotient of $\\Ker(\\mathcal{F}_m \\to \\mathcal{H}_m)$", "for some $m \\geq n$. See proof of Lemma \\ref{lemma-inverse-systems-abelian}.", "\\medskip\\noindent", "Proof of (1). Let $(\\mathcal{F}_n)$ be an object of", "$\\textit{Coh}(X, \\mathcal{I})$. Then", "$\\text{Supp}(\\mathcal{F}_n) = \\text{Supp}(\\mathcal{F}_1)$ for all $n \\geq 1$.", "Hence by remarks (a) and (b) above we see that", "for any short exact sequence", "$0 \\to (\\mathcal{G}_n) \\to (\\mathcal{F}_n) \\to (\\mathcal{H}_n) \\to 0$", "of $\\textit{Coh}(X, \\mathcal{I})$ we have", "$\\text{Supp}(\\mathcal{G}_1) \\cup \\text{Supp}(\\mathcal{H}_1) =", "\\text{Supp}(\\mathcal{F}_1)$.", "This proves that the category defined in (1)", "is a Serre subcategory of $\\textit{Coh}(X, \\mathcal{I})$.", "\\medskip\\noindent", "Proof of (2). Here we argue the same way. Let", "$0 \\to (\\mathcal{G}_n) \\to (\\mathcal{F}_n) \\to (\\mathcal{H}_n) \\to 0$", "be a short exact sequence of $\\textit{Coh}(X, \\mathcal{I})$.", "If $Z \\subset X$ is a closed subscheme and $\\mathcal{I}_Z$", "annihilates $\\mathcal{F}_n$ for all $n$, then", "$\\mathcal{I}_Z$ annihilates $\\mathcal{G}_n$ and $\\mathcal{H}_n$", "for all $n$ by (a) and (b) above.", "Hence if $Z \\to S$ is proper, then we conclude that the category", "defined in (2) is closed under taking sub and quotient objects", "inside of $\\textit{Coh}(X, \\mathcal{I})$.", "Finally, suppose that $Z \\subset X$ and $Y \\subset X$ are", "closed subschemes proper over $S$ such that", "$\\mathcal{I}_Z \\mathcal{G}_n = 0$ and", "$\\mathcal{I}_Y \\mathcal{H}_n = 0$ for all $n \\geq 1$.", "Then it follows from (a) above that", "$\\mathcal{I}_{Z \\cup Y} = \\mathcal{I}_Z \\cdot \\mathcal{I}_Y$", "annihilates $\\mathcal{F}_n$ for all $n$.", "By Lemma \\ref{lemma-union-closed-proper-over-base}", "(and via Definition \\ref{definition-proper-over-base} which", "tells us we may choose an arbitrary scheme structure used on the union)", "we see that $Z \\cup Y \\to S$ is proper and the proof is complete." ], "refs": [ "homology-lemma-characterize-serre-subcategory", "coherent-lemma-inverse-systems-abelian", "coherent-lemma-union-closed-proper-over-base", "coherent-definition-proper-over-base" ], "ref_ids": [ 12045, 3371, 3390, 3405 ] } ], "ref_ids": [] }, { "id": 3396, "type": "theorem", "label": "coherent-lemma-algebraize-formal-closed-subscheme", "categories": [ "coherent" ], "title": "coherent-lemma-algebraize-formal-closed-subscheme", "contents": [ "Let $A$ be a Noetherian ring complete with respect to an ideal $I$.", "Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$.", "Let $X \\to S$ be a separated morphism of finite type.", "For $n \\geq 1$ we set $X_n = X \\times_S S_n$.", "Suppose given a commutative diagram", "$$", "\\xymatrix{", "Z_1 \\ar[r] \\ar[d] & Z_2 \\ar[r] \\ar[d] & Z_3 \\ar[r] \\ar[d] & \\ldots \\\\", "X_1 \\ar[r]^{i_1} & X_2 \\ar[r]^{i_2} & X_3 \\ar[r] & \\ldots", "}", "$$", "of schemes with cartesian squares. Assume that", "\\begin{enumerate}", "\\item $Z_1 \\to X_1$ is a closed immersion, and", "\\item $Z_1 \\to S_1$ is proper.", "\\end{enumerate}", "Then there exists a closed immersion of schemes $Z \\to X$ such that", "$Z_n = Z \\times_S S_n$. Moreover, $Z$ is proper over $S$." ], "refs": [], "proofs": [ { "contents": [ "Let's write $j_n : Z_n \\to X_n$ for the vertical morphisms.", "As the squares in the statement are cartesian", "we see that the base change of $j_n$ to $X_1$ is $j_1$.", "Thus Morphisms, Lemma \\ref{morphisms-lemma-check-closed-infinitesimally}", "shows that $j_n$ is a closed immersion.", "Set $\\mathcal{F}_n = j_{n, *}\\mathcal{O}_{Z_n}$, so that", "$j_n^\\sharp$ is a surjection $\\mathcal{O}_{X_n} \\to \\mathcal{F}_n$.", "Again using that the squares are cartesian we see that", "the pullback of $\\mathcal{F}_{n + 1}$ to $X_n$ is $\\mathcal{F}_n$.", "Hence Grothendieck's existence theorem, as reformulated in", "Remark \\ref{remark-reformulate-existence-theorem},", "tells us there exists a map", "$\\mathcal{O}_X \\to \\mathcal{F}$", "of coherent $\\mathcal{O}_X$-modules whose restriction to", "$X_n$ recovers $\\mathcal{O}_{X_n} \\to \\mathcal{F}_n$.", "Moreover, the support of $\\mathcal{F}$ is proper over $S$.", "As the completion functor is exact (Lemma \\ref{lemma-exact})", "we see that the cokernel $\\mathcal{Q}$ of $\\mathcal{O}_X \\to \\mathcal{F}$", "has vanishing completion. Since $\\mathcal{F}$ has support", "proper over $S$ and so does $\\mathcal{Q}$ this implies that", "$\\mathcal{Q} = 0$ for example because the functor", "(\\ref{equation-completion-functor-proper-over-A}) is an equivalence", "by Grothendieck's existence theorem.", "Thus $\\mathcal{F} = \\mathcal{O}_X/\\mathcal{J}$", "for some quasi-coherent sheaf of ideals $\\mathcal{J}$.", "Setting $Z = V(\\mathcal{J})$ finishes the proof." ], "refs": [ "morphisms-lemma-check-closed-infinitesimally", "coherent-remark-reformulate-existence-theorem", "coherent-lemma-exact" ], "ref_ids": [ 5456, 3409, 3373 ] } ], "ref_ids": [] }, { "id": 3397, "type": "theorem", "label": "coherent-lemma-algebraize-formal-scheme-finite-over-proper", "categories": [ "coherent" ], "title": "coherent-lemma-algebraize-formal-scheme-finite-over-proper", "contents": [ "Let $A$ be a Noetherian ring complete with respect to an ideal $I$.", "Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$.", "Let $X \\to S$ be a separated morphism of finite type.", "For $n \\geq 1$ we set $X_n = X \\times_S S_n$.", "Suppose given a commutative diagram", "$$", "\\xymatrix{", "Y_1 \\ar[r] \\ar[d] & Y_2 \\ar[r] \\ar[d] & Y_3 \\ar[r] \\ar[d] & \\ldots \\\\", "X_1 \\ar[r]^{i_1} & X_2 \\ar[r]^{i_2} & X_3 \\ar[r] & \\ldots", "}", "$$", "of schemes with cartesian squares. Assume that", "\\begin{enumerate}", "\\item $Y_n \\to X_n$ is a finite morphism, and", "\\item $Y_1 \\to S_1$ is proper.", "\\end{enumerate}", "Then there exists a finite morphism of schemes $Y \\to X$ such that", "$Y_n = Y \\times_S S_n$. Moreover, $Y$ is proper over $S$." ], "refs": [], "proofs": [ { "contents": [ "Let's write $f_n : Y_n \\to X_n$ for the vertical morphisms.", "Set $\\mathcal{F}_n = f_{n, *}\\mathcal{O}_{Y_n}$. This is", "a coherent $\\mathcal{O}_{X_n}$-module as $f_n$ is finite", "(Lemma \\ref{lemma-finite-pushforward-coherent}).", "Using that the squares are cartesian we see that", "the pullback of $\\mathcal{F}_{n + 1}$ to $X_n$ is $\\mathcal{F}_n$.", "Hence Grothendieck's existence theorem, as reformulated in", "Remark \\ref{remark-reformulate-existence-theorem},", "tells us there exists a coherent $\\mathcal{O}_X$-module", "$\\mathcal{F}$ whose restriction to $X_n$ recovers $\\mathcal{F}_n$.", "Moreover, the support of $\\mathcal{F}$ is proper over $S$.", "As the completion functor is fully faithful", "(Theorem \\ref{theorem-grothendieck-existence})", "we see that the multiplication maps", "$\\mathcal{F}_n \\otimes_{\\mathcal{O}_{X_n}} \\mathcal{F}_n \\to", "\\mathcal{F}_n$ fit together to give an algebra structure on $\\mathcal{F}$.", "Setting $Y = \\underline{\\Spec}_X(\\mathcal{F})$ finishes the proof." ], "refs": [ "coherent-lemma-finite-pushforward-coherent", "coherent-remark-reformulate-existence-theorem", "coherent-theorem-grothendieck-existence" ], "ref_ids": [ 3316, 3409, 3279 ] } ], "ref_ids": [] }, { "id": 3398, "type": "theorem", "label": "coherent-lemma-algebraize-morphism", "categories": [ "coherent" ], "title": "coherent-lemma-algebraize-morphism", "contents": [ "Let $A$ be a Noetherian ring complete with respect to an ideal $I$.", "Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Let $X$, $Y$ be schemes", "over $S$. For $n \\geq 1$ we set $X_n = X \\times_S S_n$ and", "$Y_n = Y \\times_S S_n$. Suppose given a compatible system of", "commutative diagrams", "$$", "\\xymatrix{", "& & X_{n + 1} \\ar[rd] \\ar[rr]_{g_{n + 1}} & & Y_{n + 1} \\ar[ld] \\\\", "X_n \\ar[rru] \\ar[rd] \\ar[rr]_{g_n} & & Y_n \\ar[rru] \\ar[ld] & S_{n + 1} \\\\", "& S_n \\ar[rru]", "}", "$$", "Assume that", "\\begin{enumerate}", "\\item $X \\to S$ is proper, and", "\\item $Y \\to S$ is separated of finite type.", "\\end{enumerate}", "Then there exists a unique morphism of schemes $g : X \\to Y$", "over $S$ such that $g_n$ is the base change of $g$ to $S_n$." ], "refs": [], "proofs": [ { "contents": [ "The morphisms $(1, g_n) : X_n \\to X_n \\times_S Y_n$ are closed immersions", "because $Y_n \\to S_n$ is separated", "(Schemes, Lemma \\ref{schemes-lemma-section-immersion}). Thus by", "Lemma \\ref{lemma-algebraize-formal-closed-subscheme}", "there exists a closed subscheme $Z \\subset X \\times_S Y$", "proper over $S$ whose base change to $S_n$ recovers", "$X_n \\subset X_n \\times_S Y_n$. The first projection $p : Z \\to X$", "is a proper morphism (as $Z$ is proper over $S$, see", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed})", "whose base change to $S_n$ is an isomorphism", "for all $n$. In particular, $p : Z \\to X$ is finite over an open", "neighbourhood of $X_0$ by", "Lemma \\ref{lemma-proper-finite-fibre-finite-in-neighbourhood}.", "As $X$ is proper over $S$ this open neighbourhood is all of $X$", "and we conclude $p : Z \\to X$ is finite.", "Applying the equivalence of Proposition \\ref{proposition-existence-proper}", "we see that $p_*\\mathcal{O}_Z = \\mathcal{O}_X$ as this is true", "modulo $I^n$ for all $n$. Hence $p$ is an isomorphism and we obtain", "the morphism $g$ as the composition $X \\cong Z \\to Y$.", "We omit the proof of uniqueness." ], "refs": [ "coherent-lemma-algebraize-formal-closed-subscheme", "morphisms-lemma-image-proper-scheme-closed", "coherent-lemma-proper-finite-fibre-finite-in-neighbourhood", "coherent-proposition-existence-proper" ], "ref_ids": [ 3396, 5411, 3366, 3402 ] } ], "ref_ids": [] }, { "id": 3399, "type": "theorem", "label": "coherent-proposition-coherent-modules-on-proj", "categories": [ "coherent" ], "title": "coherent-proposition-coherent-modules-on-proj", "contents": [ "Let $A$ be a graded ring such that $A_0$ is Noetherian and $A$ is generated", "by finitely many elements of $A_1$ over $A_0$.", "Set $X = \\text{Proj}(A)$. The functor $M \\mapsto \\widetilde M$", "induces an equivalence", "$$", "\\text{Mod}^{fg}_A/\\text{Mod}^{fg}_{A, torsion}", "\\longrightarrow", "\\textit{Coh}(\\mathcal{O}_X)", "$$", "whose quasi-inverse is given by", "$\\mathcal{F} \\longmapsto \\bigoplus_{n \\geq 0} \\Gamma(X, \\mathcal{F}(n))$." ], "refs": [], "proofs": [ { "contents": [ "The subcategory $\\text{Mod}^{fg}_{A, torsion}$ is a Serre subcategory", "of $\\text{Mod}^{fg}_A$, see", "Homology, Definition \\ref{homology-definition-serre-subcategory}.", "This is clear from the description of objects given above but it also follows", "from More on Algebra, Lemma \\ref{more-algebra-lemma-I-power-torsion}.", "Hence the quotient category on the left of the arrow is defined", "in Homology, Lemma \\ref{homology-lemma-serre-subcategory-is-kernel}.", "To define the functor of the proposition, it suffices to show that", "the functor $M \\mapsto \\widetilde M$ sends torsion modules to $0$.", "This is clear because for any $f \\in A_+$ homogeneous the", "module $M_f$ is zero and hence the value $M_{(f)}$ of $\\widetilde M$", "on $D_+(f)$ is zero too.", "\\medskip\\noindent", "By Lemma \\ref{lemma-coherent-on-proj} the proposed quasi-inverse", "makes sense. Namely, the lemma shows that", "$\\mathcal{F} \\longmapsto \\bigoplus_{n \\geq 0} \\Gamma(X, \\mathcal{F}(n))$", "is a functor $\\textit{Coh}(\\mathcal{O}_X) \\to \\text{Mod}^{fg}_A$", "which we can compose with the quotient functor", "$\\text{Mod}^{fg}_A \\to \\text{Mod}^{fg}_A/\\text{Mod}^{fg}_{A, torsion}$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-recover-tail-graded-module}", "the composite left to right to left is isomorphic to the identity functor.", "\\medskip\\noindent", "Finally, let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Set $M = \\bigoplus_{n \\in \\mathbf{Z}} \\Gamma(X, \\mathcal{F}(n))$", "viewed as a graded $A$-module, so that our functor sends $\\mathcal{F}$ to", "$M_{\\geq 0} = \\bigoplus_{n \\geq 0} M_n$.", "By Properties, Lemma \\ref{properties-lemma-proj-quasi-coherent}", "the canonical map $\\widetilde M \\to \\mathcal{F}$", "is an isomorphism. Since the inclusion map", "$M_{\\geq 0} \\to M$ defines an isomorphism", "$\\widetilde{M_{\\geq 0}} \\to \\widetilde M$ we conclude that", "the composite right to left to right is isomorphic to the identity", "functor as well." ], "refs": [ "homology-definition-serre-subcategory", "more-algebra-lemma-I-power-torsion", "homology-lemma-serre-subcategory-is-kernel", "coherent-lemma-coherent-on-proj", "coherent-lemma-recover-tail-graded-module", "properties-lemma-proj-quasi-coherent" ], "ref_ids": [ 12146, 10336, 12048, 3339, 3340, 3058 ] } ], "ref_ids": [] }, { "id": 3400, "type": "theorem", "label": "coherent-proposition-coherent-modules-on-proj-general", "categories": [ "coherent" ], "title": "coherent-proposition-coherent-modules-on-proj-general", "contents": [ "Let $A$ be a Noetherian graded ring. Set $X = \\text{Proj}(A)$. The functor", "$M \\mapsto \\widetilde M$ induces an equivalence", "$$", "\\text{Mod}^{fg}_A/\\text{Mod}^{fg}_{A, irrelevant}", "\\longrightarrow", "\\textit{Coh}(\\mathcal{O}_X)", "$$", "whose quasi-inverse is given by", "$\\mathcal{F} \\longmapsto \\bigoplus_{n \\geq 0} \\Gamma(X, \\mathcal{F}(n))$." ], "refs": [], "proofs": [ { "contents": [ "We urge the reader to read the proof in the case where $A$", "is generated in degree $1$ first, see", "Proposition \\ref{proposition-coherent-modules-on-proj}.", "Let $f_1, \\ldots, f_r \\in A$ be homogeneous elements of positive degree", "which generate $A$ over $A_0$. Let $d$ be the lcm of the degrees", "$d_i$ of $f_i$. Let $M$ be a finite $A$-module.", "Let us show that $\\widetilde{M}$ is zero if and", "only if $M$ is an irrelevant graded $A$-module (as defined above", "the statement of the proposition). Namely, let $x \\in M$ be a", "homogeneous element.", "Choose $k \\in \\mathbf{Z}$ sufficiently small and let", "$N \\to N'$ and $M \\to N'$ be as in", "Lemma \\ref{lemma-recover-tail-graded-module-general}.", "We may also pick $l$ sufficiently large such that", "$M_n \\to N_n$ is an isomorphism for $n \\geq l$.", "If $\\widetilde{M}$ is zero, then $N = 0$.", "Thus for any $f \\in A_+$ homogeneous with", "$\\deg(f) + \\deg(x) = nd$ and $nd > l$ we see that $fx$ is zero", "because $N_{nd} \\to N'_{nd}$ and $M_{nd} \\to N'_{nd}$ are isomorphisms.", "Hence $x$ is irrelevant.", "Conversely, assume $M$ is irrelevant. Then $M_{nd}$ is zero", "for $n \\gg 0$ (see discussion above proposition).", "Clearly this implies that $M_{(f_i)} = M_{(f_i^{d/\\deg(f_i)})} = 0$,", "whence $\\widetilde{M} = 0$ by construction.", "\\medskip\\noindent", "It follows that the subcategory $\\text{Mod}^{fg}_{A, irrelevant}$", "is a Serre subcategory of $\\text{Mod}^{fg}_A$ as the kernel", "of the exact functor $M \\mapsto \\widetilde M$, see", "Homology, Lemma \\ref{homology-lemma-kernel-exact-functor}", "and Constructions, Lemma \\ref{constructions-lemma-proj-sheaves}.", "Hence the quotient category on the left of the arrow is defined", "in Homology, Lemma \\ref{homology-lemma-serre-subcategory-is-kernel}.", "To define the functor of the proposition, it suffices to show that", "the functor $M \\mapsto \\widetilde M$ sends irrelevant modules to $0$", "which we have shown above.", "\\medskip\\noindent", "By Lemma \\ref{lemma-coherent-on-proj-general} the proposed quasi-inverse", "makes sense. Namely, the lemma shows that", "$\\mathcal{F} \\longmapsto \\bigoplus_{n \\geq 0} \\Gamma(X, \\mathcal{F}(n))$", "is a functor $\\textit{Coh}(\\mathcal{O}_X) \\to \\text{Mod}^{fg}_A$", "which we can compose with the quotient functor", "$\\text{Mod}^{fg}_A \\to \\text{Mod}^{fg}_A/\\text{Mod}^{fg}_{A, irrelevant}$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-recover-tail-graded-module-general}", "the composite left to right to left is isomorphic to the identity functor.", "Namely, let $M$ be a finite graded $A$-module and let", "$k \\in \\mathbf{Z}$ sufficiently small and let", "$N \\to N'$ and $M \\to N'$ be as in", "Lemma \\ref{lemma-recover-tail-graded-module-general}.", "Then the kernel and cokernel of $M \\to N'$ are nonzero in", "only finitely many degrees, hence are irrelevant. Moreover, the", "kernel and cokernel of the map $N \\to N'$ are zero in all sufficiently", "large degrees divisible by $d$, hence these are irrelevant modules too.", "Thus $M \\to N'$ and $N \\to N'$ are both isomorphisms in the quotient", "category, as desired.", "\\medskip\\noindent", "Finally, let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Set $M = \\bigoplus_{n \\in \\mathbf{Z}} \\Gamma(X, \\mathcal{F}(n))$", "viewed as a graded $A$-module, so that our functor sends $\\mathcal{F}$ to", "$M_{\\geq 0} = \\bigoplus_{n \\geq 0} M_n$.", "By Properties, Lemma \\ref{properties-lemma-proj-quasi-coherent}", "the canonical map $\\widetilde M \\to \\mathcal{F}$", "is an isomorphism. Since the inclusion map", "$M_{\\geq 0} \\to M$ defines an isomorphism", "$\\widetilde{M_{\\geq 0}} \\to \\widetilde M$ we conclude that", "the composite right to left to right is isomorphic to the identity", "functor as well." ], "refs": [ "coherent-proposition-coherent-modules-on-proj", "coherent-lemma-recover-tail-graded-module-general", "homology-lemma-kernel-exact-functor", "constructions-lemma-proj-sheaves", "homology-lemma-serre-subcategory-is-kernel", "coherent-lemma-coherent-on-proj-general", "coherent-lemma-recover-tail-graded-module-general", "coherent-lemma-recover-tail-graded-module-general", "properties-lemma-proj-quasi-coherent" ], "ref_ids": [ 3399, 3342, 12047, 12594, 12048, 3341, 3342, 3342, 3058 ] } ], "ref_ids": [] }, { "id": 3401, "type": "theorem", "label": "coherent-proposition-proper-pushforward-coherent", "categories": [ "coherent" ], "title": "coherent-proposition-proper-pushforward-coherent", "contents": [ "\\begin{reference}", "\\cite[III Theorem 3.2.1]{EGA}", "\\end{reference}", "Let $S$ be a locally Noetherian scheme.", "Let $f : X \\to S$ be a proper morphism.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Then $R^if_*\\mathcal{F}$ is a coherent $\\mathcal{O}_S$-module", "for all $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Since the problem is local on $S$ we may assume that $S$ is", "a Noetherian scheme. Since a proper morphism is of finite type", "we see that in this case $X$ is a Noetherian scheme also.", "Consider the property $\\mathcal{P}$ of coherent sheaves", "on $X$ defined by the rule", "$$", "\\mathcal{P}(\\mathcal{F}) \\Leftrightarrow", "R^pf_*\\mathcal{F}\\text{ is coherent for all }p \\geq 0", "$$", "We are going to use the result of", "Lemma \\ref{lemma-property} to prove that", "$\\mathcal{P}$ holds for every coherent sheaf on $X$.", "\\medskip\\noindent", "Let", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0", "$$", "be a short exact sequence of coherent sheaves on $X$.", "Consider the long exact sequence of higher direct images", "$$", "R^{p - 1}f_*\\mathcal{F}_3 \\to", "R^pf_*\\mathcal{F}_1 \\to", "R^pf_*\\mathcal{F}_2 \\to", "R^pf_*\\mathcal{F}_3 \\to", "R^{p + 1}f_*\\mathcal{F}_1", "$$", "Then it is clear that if 2-out-of-3 of the sheaves $\\mathcal{F}_i$", "have property $\\mathcal{P}$, then the higher direct images of the", "third are sandwiched in this exact complex between two coherent", "sheaves. Hence these higher direct images are also coherent by", "Lemma \\ref{lemma-coherent-abelian-Noetherian} and", "\\ref{lemma-coherent-Noetherian-quasi-coherent-sub-quotient}.", "Hence property $\\mathcal{P}$ holds for the third as well.", "\\medskip\\noindent", "Let $Z \\subset X$ be an integral closed subscheme.", "We have to find a coherent sheaf $\\mathcal{F}$ on $X$ whose support is", "contained in $Z$, whose stalk at the generic point $\\xi$ of $Z$ is a", "$1$-dimensional vector space over $\\kappa(\\xi)$ such that $\\mathcal{P}$", "holds for $\\mathcal{F}$. Denote $g = f|_Z : Z \\to S$ the restriction of $f$.", "Suppose we can find a coherent sheaf $\\mathcal{G}$ on $Z$ such", "that", "(a) $\\mathcal{G}_\\xi$ is a $1$-dimensional vector space over $\\kappa(\\xi)$,", "(b) $R^pg_*\\mathcal{G} = 0$ for $p > 0$, and", "(c) $g_*\\mathcal{G}$ is coherent. Then we can consider", "$\\mathcal{F} = (Z \\to X)_*\\mathcal{G}$. As $Z \\to X$ is a closed immersion", "we see that $(Z \\to X)_*\\mathcal{G}$ is coherent on $X$", "and $R^p(Z \\to X)_*\\mathcal{G} = 0$ for $p > 0$", "(Lemma \\ref{lemma-finite-pushforward-coherent}).", "Hence by the relative Leray spectral sequence", "(Cohomology, Lemma \\ref{cohomology-lemma-relative-Leray})", "we will have $R^pf_*\\mathcal{F} = R^pg_*\\mathcal{G} = 0$ for $p > 0$", "and $f_*\\mathcal{F} = g_*\\mathcal{G}$ is coherent.", "Finally $\\mathcal{F}_\\xi = ((Z \\to X)_*\\mathcal{G})_\\xi = \\mathcal{G}_\\xi$", "which verifies the condition on the stalk at $\\xi$.", "Hence everything depends on finding a coherent sheaf $\\mathcal{G}$", "on $Z$ which has properties (a), (b), and (c).", "\\medskip\\noindent", "We can apply Chow's Lemma \\ref{lemma-chow-Noetherian}", "to the morphism $Z \\to S$. Thus we get a diagram", "$$", "\\xymatrix{", "Z \\ar[rd]_g & Z' \\ar[d]^-{g'} \\ar[l]^\\pi \\ar[r]_i & \\mathbf{P}^n_S \\ar[dl] \\\\", "& S &", "}", "$$", "as in the statement of Chow's lemma. Also, let $U \\subset Z$ be", "the dense open subscheme such that $\\pi^{-1}(U) \\to U$ is an isomorphism.", "By the discussion in Remark \\ref{remark-chow-Noetherian} we see that", "$i' = (i, \\pi) : Z' \\to \\mathbf{P}^n_Z$ is", "a closed immersion. Hence", "$$", "\\mathcal{L} = i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1) \\cong", "(i')^*\\mathcal{O}_{\\mathbf{P}^n_Z}(1)", "$$", "is $g'$-relatively ample and $\\pi$-relatively ample (for example by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-ample-on-finite-type}).", "Hence by Lemma \\ref{lemma-kill-by-twisting}", "there exists an $n \\geq 0$ such that", "both $R^p\\pi_*\\mathcal{L}^{\\otimes n} = 0$ for all $p > 0$ and", "$R^p(g')_*\\mathcal{L}^{\\otimes n} = 0$ for all $p > 0$.", "Set $\\mathcal{G} = \\pi_*\\mathcal{L}^{\\otimes n}$.", "Property (a) holds because $\\pi_*\\mathcal{L}^{\\otimes}|_U$ is", "an invertible sheaf (as $\\pi^{-1}(U) \\to U$ is an isomorphism).", "Properties (b) and (c) hold because by the relative Leray", "spectral sequence", "(Cohomology, Lemma \\ref{cohomology-lemma-relative-Leray})", "we have", "$$", "E_2^{p, q} = R^pg_* R^q\\pi_*\\mathcal{L}^{\\otimes n}", "\\Rightarrow", "R^{p + q}(g')_*\\mathcal{L}^{\\otimes n}", "$$", "and by choice of $n$ the only nonzero terms in $E_2^{p, q}$ are", "those with $q = 0$ and the only nonzero terms of", "$R^{p + q}(g')_*\\mathcal{L}^{\\otimes n}$ are those with $p = q = 0$.", "This implies that $R^pg_*\\mathcal{G} = 0$ for $p > 0$ and", "that $g_*\\mathcal{G} = (g')_*\\mathcal{L}^{\\otimes n}$.", "Finally, applying the previous", "Lemma \\ref{lemma-locally-projective-pushforward}", "we see that $g_*\\mathcal{G} = (g')_*\\mathcal{L}^{\\otimes n}$ is", "coherent as desired." ], "refs": [ "coherent-lemma-property", "coherent-lemma-coherent-abelian-Noetherian", "coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient", "coherent-lemma-finite-pushforward-coherent", "cohomology-lemma-relative-Leray", "coherent-lemma-chow-Noetherian", "coherent-remark-chow-Noetherian", "morphisms-lemma-characterize-ample-on-finite-type", "coherent-lemma-kill-by-twisting", "cohomology-lemma-relative-Leray", "coherent-lemma-locally-projective-pushforward" ], "ref_ids": [ 3332, 3309, 3310, 3316, 2073, 3354, 3406, 5397, 3344, 2073, 3345 ] } ], "ref_ids": [] }, { "id": 3402, "type": "theorem", "label": "coherent-proposition-existence-proper", "categories": [ "coherent" ], "title": "coherent-proposition-existence-proper", "contents": [ "Let $A$ be a Noetherian ring complete with respect to an ideal $I$.", "Let $f : X \\to \\Spec(A)$ be a proper morphism of schemes.", "Set $\\mathcal{I} = I\\mathcal{O}_X$.", "Then the functor (\\ref{equation-completion-functor}) is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "We have already seen that (\\ref{equation-completion-functor}) is", "fully faithful in Lemma \\ref{lemma-fully-faithful}. Thus it suffices", "to show that the functor is essentially surjective.", "\\medskip\\noindent", "Consider the collection $\\Xi$ of quasi-coherent sheaves of ideals", "$\\mathcal{K} \\subset \\mathcal{O}_X$ such that every object", "$(\\mathcal{F}_n)$ annihilated by $\\mathcal{K}$ is in the essential image.", "We want to show $(0)$ is in $\\Xi$. If not, then since $X$ is Noetherian", "there exists a maximal quasi-coherent sheaf of ideals $\\mathcal{K}$", "not in $\\Xi$, see", "Lemma \\ref{lemma-acc-coherent}.", "After replacing $X$ by the closed subscheme of $X$", "corresponding to $\\mathcal{K}$ we may assume that every nonzero", "$\\mathcal{K}$ is in $\\Xi$. (This uses the correspondence by", "coherent modules annihilated by $\\mathcal{K}$ and coherent modules", "on the closed subscheme corresponding to $\\mathcal{K}$, see", "Lemma \\ref{lemma-i-star-equivalence}.)", "Let $(\\mathcal{F}_n)$ be an object of", "$\\textit{Coh}(X, \\mathcal{I})$.", "We will show that this object is in the essential image of the", "functor (\\ref{equation-completion-functor}), thereby completion the", "proof of the proposition.", "\\medskip\\noindent", "Apply Chow's lemma (Lemma \\ref{lemma-chow-Noetherian}) to find a", "proper surjective morphism $f : X' \\to X$ which is an isomorphism", "over a dense open $U \\subset X$ such that $X'$ is projective over $A$.", "Let $\\mathcal{K}$ be the quasi-coherent sheaf of ideals cutting", "out the reduced complement $X \\setminus U$. By the projective", "case of Grothendieck's existence theorem", "(Lemma \\ref{lemma-existence-projective})", "there exists a coherent module $\\mathcal{F}'$ on $X'$ such", "that $(\\mathcal{F}')^\\wedge \\cong (f^*\\mathcal{F}_n)$. By", "Proposition \\ref{proposition-proper-pushforward-coherent}", "the $\\mathcal{O}_X$-module $\\mathcal{H} = f_*\\mathcal{F}'$ is coherent", "and by Lemma \\ref{lemma-inverse-systems-push-pull}", "there exists a morphism $(\\mathcal{F}_n) \\to \\mathcal{H}^\\wedge$", "of $\\textit{Coh}(X, \\mathcal{I})$ whose kernel and cokernel are", "annihilated by a power of $\\mathcal{K}$. The powers $\\mathcal{K}^e$", "are all in $\\Xi$ so that (\\ref{equation-completion-functor})", "is an equivalence for the closed subschemes $X_e = V(\\mathcal{K}^e)$.", "We conclude by Lemma \\ref{lemma-existence-tricky}." ], "refs": [ "coherent-lemma-fully-faithful", "coherent-lemma-acc-coherent", "coherent-lemma-i-star-equivalence", "coherent-lemma-chow-Noetherian", "coherent-lemma-existence-projective", "coherent-proposition-proper-pushforward-coherent", "coherent-lemma-inverse-systems-push-pull", "coherent-lemma-existence-tricky" ], "ref_ids": [ 3381, 3319, 3315, 3354, 3383, 3401, 3385, 3384 ] } ], "ref_ids": [] }, { "id": 3410, "type": "theorem", "label": "formal-defos-theorem-miniversal-object-existence", "categories": [ "formal-defos" ], "title": "formal-defos-theorem-miniversal-object-existence", "contents": [ "Let $\\mathcal{F}$ be a predeformation category.", "Consider the following conditions", "\\begin{enumerate}", "\\item $\\mathcal{F}$ has a minimal versal formal object satisfying", "(\\ref{equation-bijective}),", "\\item $\\mathcal{F}$ has a minimal versal formal object satisfying", "(\\ref{equation-bijective-orbits}),", "\\item the following conditions hold:", "\\begin{enumerate}", "\\item $\\mathcal{F}$ satisfies (S1).", "\\item $\\mathcal{F}$ satisfies (S2).", "\\item $\\dim_k T\\mathcal{F}$ is finite.", "\\end{enumerate}", "\\end{enumerate}", "We always have", "$$", "(1) \\Rightarrow (3) \\Rightarrow (2).", "$$", "If $k' \\subset k$ is separable, then all three are equivalent." ], "refs": [], "proofs": [ { "contents": [ "Lemma \\ref{lemma-miniversal-object-existence-1}", "shows that (1) $\\Rightarrow$ (3).", "Lemmas \\ref{lemma-versal-object-existence} and", "\\ref{lemma-construct-bijective-orbits}", "show that (3) $\\Rightarrow$ (2). If $k' \\subset k$ is separable", "then $\\text{Der}_\\Lambda(k, k) = 0$ and we see that", "(\\ref{equation-bijective}) $=$ (\\ref{equation-bijective-orbits}), i.e.,", "(1) is the same as (2).", "\\medskip\\noindent", "An alternative proof of (3) $\\Rightarrow$ (1) in the classical case", "is to add a few words to the proof of", "Lemma \\ref{lemma-versal-object-existence}", "to see that one can right away construct a versal object which", "satisfies (\\ref{equation-bijective}) in this case. This avoids the use of", "Lemma \\ref{lemma-versal-object-existence}", "in the classical case. Details omitted." ], "refs": [ "formal-defos-lemma-miniversal-object-existence-1", "formal-defos-lemma-versal-object-existence", "formal-defos-lemma-construct-bijective-orbits", "formal-defos-lemma-versal-object-existence", "formal-defos-lemma-versal-object-existence" ], "ref_ids": [ 3464, 3458, 3465, 3458, 3458 ] } ], "ref_ids": [] }, { "id": 3411, "type": "theorem", "label": "formal-defos-theorem-Schlessinger-prorepresentability", "categories": [ "formal-defos" ], "title": "formal-defos-theorem-Schlessinger-prorepresentability", "contents": [ "Let $F: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a functor.", "Then $F$ is prorepresentable if and only if", "(a) $F$ is a deformation functor,", "(b) $\\dim_k TF$ is finite, and (c) $\\gamma : \\text{Der}_\\Lambda(k, k) \\to TF$", "is injective." ], "refs": [], "proofs": [ { "contents": [ "Assume $F$ is prorepresentable by $R \\in \\widehat{\\mathcal{C}}_\\Lambda$.", "We see $F$ is a deformation functor by", "Example \\ref{example-prorepresentable-deformation-functor}.", "We see $\\dim_k TF$ is finite by", "Example \\ref{example-tangent-space-prorepresentable-functor}.", "Finally, $\\text{Der}_\\Lambda(k, k) \\to TF$ is identified with", "$\\text{Der}_\\Lambda(k, k) \\to \\text{Der}_\\Lambda(R, k)$ by", "Example \\ref{example-tangent-space-map-prorepresentable-functor}", "which is injective because $R \\to k$ is surjective.", "\\medskip\\noindent", "Conversely, assume (a), (b), and (c) hold. By", "Lemma \\ref{lemma-RS-implies-S1-S2}", "we see that (S1) and (S2) hold. Hence by", "Theorem \\ref{theorem-miniversal-object-existence}", "there exists a minimal versal formal object $\\xi$ of $F$ such that", "(\\ref{equation-bijective-orbits}) holds. Say $\\xi$ lies over $R$.", "The map", "$$", "d\\underline{\\xi} : \\text{Der}_\\Lambda(R, k) \\to T\\mathcal{F}", "$$", "is bijective on $\\text{Der}_\\Lambda(k, k)$-orbits. Since the action", "of $\\text{Der}_\\Lambda(k, k)$ on the left hand side is free by (c) and", "Lemma \\ref{lemma-action-linear}", "we see that the map is bijective. Thus we see that $\\underline{\\xi}$", "is an isomorphism by", "Lemma \\ref{lemma-minimal-smooth-morphism-functors}." ], "refs": [ "formal-defos-lemma-RS-implies-S1-S2", "formal-defos-theorem-miniversal-object-existence", "formal-defos-lemma-action-linear", "formal-defos-lemma-minimal-smooth-morphism-functors" ], "ref_ids": [ 3469, 3410, 3454, 3474 ] } ], "ref_ids": [] }, { "id": 3412, "type": "theorem", "label": "formal-defos-theorem-presentation-deformation-groupoid", "categories": [ "formal-defos" ], "title": "formal-defos-theorem-presentation-deformation-groupoid", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$. Then $\\mathcal{F}$ admits a presentation by a", "smooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$", "if and only if the following conditions hold:", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a deformation category.", "\\item $\\dim_k T\\mathcal{F}$ is finite.", "\\item $\\dim_k \\text{Inf}(\\mathcal{F})$ is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that a prorepresentable functor is a deformation functor, see", "Example \\ref{example-prorepresentable-deformation-functor}. Thus", "if $\\mathcal{F}$ is equivalent to a smooth prorepresentable groupoid in", "functors, then conditions (1), (2), and (3) follow from", "Lemma \\ref{lemma-deformation-groupoid-quotient} (1), (2), and (3).", "\\medskip\\noindent", "Conversely, assume conditions (1), (2), and (3) hold. Condition (1)", "implies that (S1) and (S2) are satisfied, see", "Lemma \\ref{lemma-RS-implies-S1-S2}.", "By", "Lemma \\ref{lemma-versal-object-existence}", "there exists a versal formal object $\\xi$. Setting", "$U = \\underline{R}|_{\\mathcal{C}_\\Lambda}$ the", "associated map $\\underline{\\xi} : U \\to \\mathcal{F}$ is smooth (this is", "the definition of a versal formal object).", "Let $(U, R, s, t, c)$ be the groupoid in functors constructed in", "Lemma \\ref{lemma-presentation-construction}", "from the map $\\underline{\\xi}$. By", "Lemma \\ref{lemma-smooth-groupoid-in-functors-construction}", "we see that $(U, R, s, t, c)$ is a smooth groupoid in functors and that", "$[U/R] \\to \\mathcal{F}$ is an equivalence. By", "Lemma \\ref{lemma-prorepresentable-groupoid-in-functors-construction}", "we see that $(U, R, s, t, c)$ is prorepresentable.", "Hence $[U/R] \\to \\mathcal{F}$ is the desired presentation of $\\mathcal{F}$." ], "refs": [ "formal-defos-lemma-deformation-groupoid-quotient", "formal-defos-lemma-RS-implies-S1-S2", "formal-defos-lemma-versal-object-existence", "formal-defos-lemma-presentation-construction", "formal-defos-lemma-smooth-groupoid-in-functors-construction", "formal-defos-lemma-prorepresentable-groupoid-in-functors-construction" ], "ref_ids": [ 3489, 3469, 3458, 3490, 3492, 3494 ] } ], "ref_ids": [] }, { "id": 3413, "type": "theorem", "label": "formal-defos-theorem-minimal-smooth-prorepresentable-presentations", "categories": [ "formal-defos" ], "title": "formal-defos-theorem-minimal-smooth-prorepresentable-presentations", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$. Consider the following conditions", "\\begin{enumerate}", "\\item $\\mathcal{F}$ admits a presentation by a normalized", "smooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$,", "\\item $\\mathcal{F}$ admits a presentation by a", "smooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$,", "\\item $\\mathcal{F}$ admits a presentation by a minimal", "smooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$, and", "\\item $\\mathcal{F}$ satisfies the following conditions", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a deformation category.", "\\item $\\dim_k T\\mathcal{F}$ is finite.", "\\item $\\dim_k \\text{Inf}(\\mathcal{F})$ is finite.", "\\end{enumerate}", "\\end{enumerate}", "Then (2), (3), (4) are equivalent and are implied by (1).", "If $k' \\subset k$ is separable, then (1), (2), (3), (4) are all equivalent.", "Furthermore, the minimal smooth prorepresentable groupoids in functors", "which provide a presentation of $\\mathcal{F}$ are unique up to isomorphism." ], "refs": [], "proofs": [ { "contents": [ "We see that (1) implies (3) and is equivalent to (3) if", "$k' \\subset k$ is separable from", "Lemma \\ref{lemma-characterize-minimal-groupoid-in-functors}.", "It is clear that (3) implies (2). We see that (2) implies (4) by", "Theorem \\ref{theorem-presentation-deformation-groupoid}.", "We see that (4) implies (3) by", "Lemma \\ref{lemma-minimal-groupoid-in-functors-construction}.", "This proves all the implications.", "The final uniqueness statement follows from", "Lemma \\ref{lemma-minimal-presentations-equivalent}." ], "refs": [ "formal-defos-lemma-characterize-minimal-groupoid-in-functors", "formal-defos-theorem-presentation-deformation-groupoid", "formal-defos-lemma-minimal-groupoid-in-functors-construction", "formal-defos-lemma-minimal-presentations-equivalent" ], "ref_ids": [ 3495, 3412, 3499, 3500 ] } ], "ref_ids": [] }, { "id": 3414, "type": "theorem", "label": "formal-defos-lemma-factor-small-extension", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-factor-small-extension", "contents": [ "Let $f: B \\to A$ be a surjective ring map in $\\mathcal{C}_\\Lambda$.", "Then $f$ can be factored as a composition of small extensions." ], "refs": [], "proofs": [ { "contents": [ "Let $I$ be the kernel of $f$. The maximal ideal $\\mathfrak{m}_B$ is", "nilpotent since $B$ is Artinian, say $\\mathfrak{m}_B^n = 0$. Hence we get a", "factorization", "$$", "B = B/I\\mathfrak{m}_B^{n-1} \\to B/I\\mathfrak{m}_B^{n-2} \\to", "\\ldots \\to B/I \\cong A", "$$", "of $f$ into a composition of surjective maps whose kernels are annihilated by", "the maximal ideal. Thus it suffices to prove the lemma when $f$ itself is such", "a map, i.e.\\ when $I$ is annihilated by $\\mathfrak{m}_B$. In this case", "$I$ is a $k$-vector space, which has finite dimension, see", "Algebra, Lemma \\ref{algebra-lemma-artinian-finite-length}.", "Take a basis $x_1, \\ldots, x_n$ of $I$ as a $k$-vector space to get a", "factorization", "$$", "B \\to B/(x_1) \\to \\ldots \\to B/(x_1, \\ldots, x_n) \\cong A", "$$", "of $f$ into a composition of small extensions." ], "refs": [ "algebra-lemma-artinian-finite-length" ], "ref_ids": [ 646 ] } ], "ref_ids": [] }, { "id": 3415, "type": "theorem", "label": "formal-defos-lemma-length", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-length", "contents": [ "Let $A$ be a local $\\Lambda$-algebra with residue field $k$.", "Let $M$ be an $A$-module. Then", "$[k : k'] \\text{length}_A(M) = \\text{length}_\\Lambda(M)$.", "In the classical case we have", "$\\text{length}_A(M) = \\text{length}_\\Lambda(M)$." ], "refs": [], "proofs": [ { "contents": [ "If $M$ is a simple $A$-module then $M \\cong k$ as an $A$-module, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-length-1}.", "In this case $\\text{length}_A(M) = 1$ and", "$\\text{length}_\\Lambda(M) = [k' : k]$, see", "Algebra, Lemma \\ref{algebra-lemma-dimension-is-length}.", "If $\\text{length}_A(M)$ is finite, then the result follows on", "choosing a filtration of $M$ by $A$-submodules with simple quotients", "using additivity, see", "Algebra, Lemma \\ref{algebra-lemma-length-additive}.", "If $\\text{length}_A(M)$ is infinite, the result follows from the obvious", "inequality $\\text{length}_A(M) \\leq \\text{length}_\\Lambda(M)$." ], "refs": [ "algebra-lemma-characterize-length-1", "algebra-lemma-dimension-is-length", "algebra-lemma-length-additive" ], "ref_ids": [ 637, 634, 631 ] } ], "ref_ids": [] }, { "id": 3416, "type": "theorem", "label": "formal-defos-lemma-surjective", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-surjective", "contents": [ "Let $A \\to B$ be a ring map in $\\mathcal{C}_\\Lambda$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is surjective,", "\\item $\\mathfrak m_A/\\mathfrak m_A^2 \\to \\mathfrak m_B/\\mathfrak m_B^2$", "is surjective, and", "\\item $\\mathfrak m_A/(\\mathfrak m_\\Lambda A + \\mathfrak m_A^2)", "\\to \\mathfrak m_B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2)$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For any ring map $f : A \\to B$ in $\\mathcal{C}_\\Lambda$ we have", "$f(\\mathfrak m_A) \\subset \\mathfrak m_B$ for example because", "$\\mathfrak m_A$, $\\mathfrak m_B$ is the set of nilpotent elements of", "$A$, $B$. Suppose $f$ is surjective. Let $y \\in \\mathfrak m_B$.", "Choose $x \\in A$ with $f(x) = y$. Since $f$ induces an isomorphism", "$A/\\mathfrak m_A \\to B/\\mathfrak m_B$ we see that $x \\in \\mathfrak m_A$.", "Hence the induced map", "$\\mathfrak m_A/\\mathfrak m_A^2 \\to \\mathfrak m_B/\\mathfrak m_B^2$", "is surjective. In this way we see that (1) implies (2).", "\\medskip\\noindent", "It is clear that (2) implies (3). The map $A \\to B$ gives rise", "to a canonical commutative diagram", "$$", "\\xymatrix{", "\\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 \\otimes_{k'} k \\ar[r] \\ar[d] &", "\\mathfrak m_A/\\mathfrak m_A^2 \\ar[r] \\ar[d] &", "\\mathfrak m_A/(\\mathfrak m_\\Lambda A + \\mathfrak m_A^2) \\ar[r] \\ar[d] & 0 \\\\", "\\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 \\otimes_{k'} k \\ar[r] &", "\\mathfrak m_B/\\mathfrak m_B^2 \\ar[r] &", "\\mathfrak m_B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2) \\ar[r] & 0", "}", "$$", "with exact rows. Hence if (3) holds, then so does (2).", "\\medskip\\noindent", "Assume (2). To show that $A \\to B$ is surjective it suffices by", "Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "to show that $A/\\mathfrak m_A \\to B/\\mathfrak m_AB$ is surjective.", "(Note that $\\mathfrak m_A$ is a nilpotent ideal.)", "As $k = A/\\mathfrak m_A = B/\\mathfrak m_B$ it suffices to show that", "$\\mathfrak m_AB \\to \\mathfrak m_B$ is surjective. Applying", "Nakayama's lemma once more we see that it suffices to see that", "$\\mathfrak m_AB/\\mathfrak m_A\\mathfrak m_B \\to \\mathfrak m_B/\\mathfrak m_B^2$", "is surjective which is what we assumed." ], "refs": [ "algebra-lemma-NAK" ], "ref_ids": [ 401 ] } ], "ref_ids": [] }, { "id": 3417, "type": "theorem", "label": "formal-defos-lemma-fiber-product-CLambda", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-fiber-product-CLambda", "contents": [ "Let $f_1 : A_1 \\to A$ and $f_2 : A_2 \\to A$ be ring maps in", "$\\mathcal{C}_\\Lambda$. Then:", "\\begin{enumerate}", "\\item If $f_1$ or $f_2$ is surjective, then", "$A_1 \\times_A A_2$ is in $\\mathcal{C}_\\Lambda$.", "\\item If $f_2$ is a small extension, then so is", "$A_1 \\times_A A_2 \\to A_1$.", "\\item If the field extension $k' \\subset k$ is separable, then", "$A_1 \\times_A A_2$ is in $\\mathcal{C}_\\Lambda$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The ring $A_1 \\times_A A_2$ is a $\\Lambda$-algebra via the map", "$\\Lambda \\to A_1 \\times_A A_2$ induced by the maps", "$\\Lambda \\to A_1$ and $\\Lambda \\to A_2$. It is a local ring with unique", "maximal ideal", "$$", "\\mathfrak m_{A_1} \\times_{\\mathfrak m_A} \\mathfrak m_{A_2} =", "\\Ker(A_1 \\times_A A_2 \\longrightarrow k)", "$$", "A ring is Artinian if and only if it has finite length as a module", "over itself, see", "Algebra, Lemma \\ref{algebra-lemma-artinian-finite-length}.", "Since $A_1$ and $A_2$ are Artinian, Lemma \\ref{lemma-length} implies", "$\\text{length}_\\Lambda(A_1)$ and $\\text{length}_\\Lambda(A_2)$,", "and hence $\\text{length}_\\Lambda(A_1 \\times A_2)$, are all finite. As", "$A_1 \\times_A A_2 \\subset A_1 \\times A_2$ is a $\\Lambda$-submodule, this", "implies", "$\\text{length}_{A_1 \\times_A A_2}(A_1 \\times_A A_2) \\leq", "\\text{length}_\\Lambda(A_1 \\times_A A_2)$ is finite. So $A_1", "\\times_A A_2$ is Artinian. Thus the only thing that is keeping", "$A_1 \\times_A A_2$ from being an object of $\\mathcal{C}_\\Lambda$ is", "the possibility that its residue field maps to a proper subfield of $k$", "via the map $A_1 \\times_A A_2 \\to A \\to A/\\mathfrak m_A = k$ above.", "\\medskip\\noindent", "Proof of (1). If $f_2$ is surjective, then the projection", "$A_1 \\times_A A_2 \\to A_1$ is surjective. Hence the composition", "$A_1 \\times_A A_2 \\to A_1 \\to A_1/\\mathfrak m_{A_1} = k$ is surjective", "and we conclude that $A_1 \\times_A A_2$ is an object of $\\mathcal{C}_\\Lambda$.", "\\medskip\\noindent", "Proof of (2). If $f_2$ is a small extension then $A_2 \\to A$ and", "$A_1 \\times_A A_2 \\to A_1$ are both surjective with the same kernel.", "Hence the kernel of $A_1 \\times_A A_2 \\to A_1$ is a $1$-dimensional", "$k$-vector space and we see that $A_1 \\times_A A_2 \\to A_1$ is a small", "extension.", "\\medskip\\noindent", "Proof of (3). Choose $\\overline{x} \\in k$ such that", "$k = k'(\\overline{x})$ (see", "Fields, Lemma \\ref{fields-lemma-primitive-element}).", "Let $P'(T) \\in k'[T]$ be the minimal polynomial of $\\overline{x}$ over $k'$.", "Since $k/k'$ is separable we see that", "$\\text{d}P/\\text{d}T(\\overline{x}) \\not = 0$.", "Choose a monic $P \\in \\Lambda[T]$ which maps to $P'$ under the surjective map", "$\\Lambda[T] \\to k'[T]$. Because $A, A_1, A_2$ are henselian, see", "Algebra, Lemma \\ref{algebra-lemma-local-dimension-zero-henselian},", "we can find $x, x_1, x_2 \\in A, A_1, A_2$ with $P(x) = 0, P(x_1) = 0,", "P(x_2) = 0$ and such that the image of $x, x_1, x_2$ in $k$ is $\\overline{x}$.", "Then $(x_1, x_2) \\in A_1 \\times_A A_2$ because $x_1, x_2$", "map to $x \\in A$ by uniqueness, see", "Algebra, Lemma \\ref{algebra-lemma-uniqueness}.", "Hence the residue field of", "$A_1 \\times_A A_2$ contains a generator of $k$ over $k'$ and we win." ], "refs": [ "algebra-lemma-artinian-finite-length", "formal-defos-lemma-length", "fields-lemma-primitive-element", "algebra-lemma-local-dimension-zero-henselian", "algebra-lemma-uniqueness" ], "ref_ids": [ 646, 3415, 4498, 1283, 1275 ] } ], "ref_ids": [] }, { "id": 3418, "type": "theorem", "label": "formal-defos-lemma-essential-surjection-mod-squares", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-essential-surjection-mod-squares", "contents": [ "Let $f: B \\to A$ be a ring map in $\\mathcal{C}_\\Lambda$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is an essential surjection,", "\\item the map $B/\\mathfrak m_B^2 \\to A/\\mathfrak m_A^2$ is an essential", "surjection, and", "\\item the map", "$B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2) \\to", "A/(\\mathfrak m_\\Lambda A + \\mathfrak m_A^2)$ is an essential surjection.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (3). Let $C \\to B$ be a ring map in $\\mathcal{C}_\\Lambda$ such", "that $C \\to A$ is surjective. Then", "$C \\to A/(\\mathfrak m_\\Lambda A + \\mathfrak m_A^2)$ is surjective", "too. We conclude that $C \\to B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2)$", "is surjective by our assumption. Hence $C \\to B$ is surjective by applying", "Lemma \\ref{lemma-surjective} (2 times).", "\\medskip\\noindent", "Assume (1). Let $C \\to B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2)$", "be a morphism of $\\mathcal{C}_\\Lambda$ such that", "$C \\to A/(\\mathfrak m_\\Lambda A + \\mathfrak m_A^2)$ is surjective. Set", "$C' = C \\times_{B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2)} B$", "which is an object of $\\mathcal{C}_\\Lambda$ by", "Lemma \\ref{lemma-fiber-product-CLambda}.", "Note that $C' \\to A/(\\mathfrak m_\\Lambda A + \\mathfrak m_A^2)$", "is still surjective, hence $C' \\to A$ is surjective by", "Lemma \\ref{lemma-surjective}.", "Thus $C' \\to B$ is surjective by our assumption. This implies", "that $C' \\to B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2)$ is", "surjective, which implies by the construction of $C'$ that", "$C \\to B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2)$ is surjective.", "\\medskip\\noindent", "In the first paragraph we proved (3) $\\Rightarrow$ (1) and in the second", "paragraph we proved (1) $\\Rightarrow$ (3). The equivalence of", "(2) and (3) is a special case of the equivalence of (1) and (3), hence", "we are done." ], "refs": [ "formal-defos-lemma-surjective", "formal-defos-lemma-fiber-product-CLambda", "formal-defos-lemma-surjective" ], "ref_ids": [ 3416, 3417, 3416 ] } ], "ref_ids": [] }, { "id": 3419, "type": "theorem", "label": "formal-defos-lemma-H1-separable-case", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-H1-separable-case", "contents": [ "There is a canonical map", "$$", "\\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 \\longrightarrow H_1(L_{k/\\Lambda}).", "$$", "If $k' \\subset k$ is separable (for example if the characteristic", "of $k$ is zero), then this map induces an isomorphism", "$\\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 \\otimes_{k'} k = H_1(L_{k/\\Lambda})$.", "If $k = k'$ (for example in the classical case), then", "$\\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 = H_1(L_{k/\\Lambda})$.", "The composition", "$$", "\\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 \\longrightarrow", "H_1(L_{k/\\Lambda}) \\longrightarrow \\mathfrak m_A/\\mathfrak m_A^2", "$$", "comes from the canonical map $\\mathfrak m_\\Lambda \\to \\mathfrak m_A$." ], "refs": [], "proofs": [ { "contents": [ "Note that $H_1(L_{k'/\\Lambda}) = \\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2$", "as $\\Lambda \\to k'$ is surjective with kernel $\\mathfrak m_\\Lambda$.", "The map arises from functoriality of the naive cotangent complex.", "If $k' \\subset k$ is separable, then $k' \\to k$ is an \\'etale ring map, see", "Algebra, Lemma \\ref{algebra-lemma-etale-over-field}.", "Thus its naive cotangent complex has trivial homology groups, see", "Algebra, Definition \\ref{algebra-definition-etale}.", "Then", "Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL}", "applied to the ring maps $\\Lambda \\to k' \\to k$ implies that", "$\\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 \\otimes_{k'} k = H_1(L_{k/\\Lambda})$.", "We omit the proof of the final statement." ], "refs": [ "algebra-lemma-etale-over-field", "algebra-definition-etale", "algebra-lemma-exact-sequence-NL" ], "ref_ids": [ 1232, 1539, 1153 ] } ], "ref_ids": [] }, { "id": 3420, "type": "theorem", "label": "formal-defos-lemma-essential-surjection", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-essential-surjection", "contents": [ "Let $f: B \\to A$ be a ring map in $\\mathcal{C}_\\Lambda$.", "Notation as in (\\ref{equation-sequence-functorial}).", "\\begin{enumerate}", "\\item The equivalent conditions of", "Lemma \\ref{lemma-essential-surjection-mod-squares}", "characterizing when $f$ is surjective are also equivalent to", "\\begin{enumerate}", "\\item $\\Im(\\text{d}_B) \\to \\Im(\\text{d}_A)$ is surjective, and", "\\item the map $\\Omega_{B/\\Lambda} \\otimes_B k \\to", "\\Omega_{A/\\Lambda} \\otimes_A k$ is surjective.", "\\end{enumerate}", "\\item The following are equivalent", "\\begin{enumerate}", "\\item $f$ is an essential surjection,", "\\item the map $\\Im(\\text{d}_B) \\to \\Im(\\text{d}_A)$ is an", "isomorphism, and", "\\item the map $\\Omega_{B/\\Lambda} \\otimes_B k \\to", "\\Omega_{A/\\Lambda} \\otimes_A k$ is an isomorphism.", "\\end{enumerate}", "\\item If $k/k'$ is separable, then $f$ is an essential surjection if", "and only if the map", "$\\mathfrak m_B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2) \\to", "\\mathfrak m_A/(\\mathfrak m_\\Lambda A + \\mathfrak m_A^2)$", "is an isomorphism.", "\\item If $f$ is a small extension, then $f$ is not essential if and only if", "$f$ has a section $s: A \\to B$ in $\\mathcal{C}_\\Lambda$", "with $f \\circ s = \\text{id}_A$.", "\\end{enumerate}" ], "refs": [ "formal-defos-lemma-essential-surjection-mod-squares" ], "proofs": [ { "contents": [ "Proof of (1). It follows from (\\ref{equation-sequence-functorial})", "that (1)(a) and (1)(b) are equivalent. Also, if", "$A \\to B$ is surjective, then (1)(a) and (1)(b) hold. Assume (1)(a).", "Since the kernel of $\\text{d}_A$ is the image of", "$H_1(L_{k/\\Lambda})$ which also maps to", "$\\mathfrak m_B/\\mathfrak m_B^2$ we conclude that", "$\\mathfrak m_B/\\mathfrak m_B^2 \\to \\mathfrak m_A/\\mathfrak m_A^2$", "is surjective. Hence $B \\to A$ is surjective by", "Lemma \\ref{lemma-surjective}. This finishes the proof of (1).", "\\medskip\\noindent", "Proof of (2). The equivalence of (2)(b) and (2)(c) is immediate from", "(\\ref{equation-sequence-functorial}).", "\\medskip\\noindent", "Assume (2)(b). Let $g : C \\to B$ be a ring map in $\\mathcal{C}_\\Lambda$", "such that $f \\circ g$ is surjective. We conclude that", "$\\mathfrak m_C/\\mathfrak m_C^2 \\to \\mathfrak m_A/\\mathfrak m_A^2$", "is surjective by", "Lemma \\ref{lemma-surjective}.", "Hence", "$\\Im(\\text{d}_C) \\to \\Im(\\text{d}_A)$ is surjective", "and by the assumption we see that", "$\\Im(\\text{d}_C) \\to \\Im(\\text{d}_B)$ is surjective.", "It follows that $C \\to B$ is surjective by (1).", "\\medskip\\noindent", "Assume (2)(a). Then $f$ is surjective and we see that", "$\\Omega_{B/\\Lambda} \\otimes_B k \\to \\Omega_{A/\\Lambda} \\otimes_A k$", "is surjective. Let $K$ be the kernel. Note that", "$K = \\text{d}_B(\\Ker(\\mathfrak m_B/\\mathfrak m_B^2 \\to", "\\mathfrak m_A/\\mathfrak m_A^2))$ by (\\ref{equation-sequence-functorial}).", "Choose a splitting", "$$", "\\Omega_{B/\\Lambda} \\otimes_B k =", "\\Omega_{A/\\Lambda} \\otimes_A k \\oplus K", "$$", "of $k$-vector space. The map $\\text{d} : B \\to \\Omega_{B/\\Lambda}$", "induces via the projection onto $K$ a map $D : B \\to K$. Set", "$C = \\{b \\in B \\mid D(b) = 0\\}$. The Leibniz rule shows that this is", "a $\\Lambda$-subalgebra of $B$. Let $\\overline{x} \\in k$. Choose $x \\in B$", "mapping to $\\overline{x}$. If $D(x) \\not = 0$, then we can find an element", "$y \\in \\mathfrak m_B$ such that $D(y) = D(x)$. Hence $x - y \\in C$ is", "an element which maps to $\\overline{x}$. Thus $C \\to k$ is surjective", "and $C$ is an object of $\\mathcal{C}_\\Lambda$. Similarly, pick", "$\\omega \\in \\Im(\\text{d}_A)$. We can find $x \\in \\mathfrak m_B$", "such that $\\text{d}_B(x)$ maps to $\\omega$ by (1). If $D(x) \\not = 0$, then", "we can find an element $y \\in \\mathfrak m_B$ which maps to zero", "in $\\mathfrak m_A/\\mathfrak m_A^2$ such that $D(y) = D(x)$.", "Hence $z = x - y$ is an element of $\\mathfrak m_C$ whose", "image $\\text{d}_C(z) \\in \\Omega_{C/k} \\otimes_C k$ maps to $\\omega$.", "Hence $\\Im(\\text{d}_C) \\to \\Im(\\text{d}_A)$ is surjective.", "We conclude that $C \\to A$ is surjective by (1). Hence $C \\to B$ is", "surjective by assumption. Hence $D = 0$, i.e., $K = 0$, i.e., (2)(c) holds.", "This finishes the proof of (2).", "\\medskip\\noindent", "Proof of (3). If $k'/k$ is separable, then", "$H_1(L_{k/\\Lambda}) =", "\\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 \\otimes_{k'} k$, see", "Lemma \\ref{lemma-H1-separable-case}.", "Hence $\\Im(\\text{d}_A) =", "\\mathfrak m_A/(\\mathfrak m_\\Lambda A + \\mathfrak m_A^2)$", "and similarly for $B$. Thus (3) follows from (2).", "\\medskip\\noindent", "Proof of (4). A section $s$ of $f$ is not surjective (by definition a", "small extension has nontrivial kernel), hence $f$ is not essentially", "surjective. Conversely, assume $f$ is a small extension but not an", "essential surjection. Choose a ring map $C \\to B$ in $\\mathcal{C}_\\Lambda$", "which is not surjective, such that $C \\to A$ is surjective. Let", "$C' \\subset B$ be the image of $C \\to B$. Then $C' \\not = B$ but", "$C'$ surjects onto $A$. Since $f : B \\to A$ is a small extension,", "$\\text{length}_C(B) = \\text{length}_C(A) + 1$. Thus", "$\\text{length}_C(C') \\leq \\text{length}_C(A)$ since", "$C'$ is a proper subring of $B$. But $C' \\to A$ is surjective, so in", "fact we must have $\\text{length}_C(C') = \\text{length}_C(A)$ and", "$C' \\to A$ is an isomorphism which gives us our section." ], "refs": [ "formal-defos-lemma-surjective", "formal-defos-lemma-surjective", "formal-defos-lemma-H1-separable-case" ], "ref_ids": [ 3416, 3416, 3419 ] } ], "ref_ids": [ 3418 ] }, { "id": 3421, "type": "theorem", "label": "formal-defos-lemma-surjective-cotangent-space", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-surjective-cotangent-space", "contents": [ "Let $f: R \\to S$ be a ring map in $\\widehat{\\mathcal{C}}_\\Lambda$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is surjective,", "\\item the map", "$\\mathfrak m_R/\\mathfrak m_R^2 \\to \\mathfrak m_S/\\mathfrak m_S^2$", "is surjective, and", "\\item the map", "$\\mathfrak m_R/(\\mathfrak m_\\Lambda R + \\mathfrak m_R^2) \\to", "\\mathfrak m_S/(\\mathfrak m_\\Lambda S + \\mathfrak m_S^2)$", "is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that for $n \\geq 2$ we have the equality of relative cotangent spaces", "$$", "\\mathfrak m_R/(\\mathfrak m_\\Lambda R + \\mathfrak m_R^2)", "=", "\\mathfrak m_{R_n}/(\\mathfrak m_\\Lambda R_n + \\mathfrak m_{R_n}^2)", "$$", "and similarly for $S$. Hence by", "Lemma \\ref{lemma-surjective}", "we see that $R_n \\to S_n$ is surjective for all $n$.", "Now let $K_n$ be the kernel of $R_n \\to S_n$. Then the sequences", "$$", "0 \\to K_n \\to R_n \\to S_n \\to 0", "$$", "form an exact sequence of directed inverse systems. The system $(K_n)$ is", "Mittag-Leffler since each $K_n$ is Artinian. Hence by", "Algebra, Lemma \\ref{algebra-lemma-ML-exact-sequence}", "taking limits preserves exactness. So", "$\\lim R_n \\to \\lim S_n$ is surjective, i.e., $f$ is surjective." ], "refs": [ "formal-defos-lemma-surjective", "algebra-lemma-ML-exact-sequence" ], "ref_ids": [ 3416, 826 ] } ], "ref_ids": [] }, { "id": 3422, "type": "theorem", "label": "formal-defos-lemma-CLambdahat-pushouts", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-CLambdahat-pushouts", "contents": [ "The category $\\widehat{\\mathcal{C}}_\\Lambda$ admits pushouts." ], "refs": [], "proofs": [ { "contents": [ "Let $R \\to S_1$ and $R \\to S_2$ be morphisms of", "$\\widehat{\\mathcal{C}}_\\Lambda$. Consider the ring", "$C = S_1 \\otimes_R S_2$.", "This ring has a finitely generated maximal ideal", "$\\mathfrak m = \\mathfrak m_{S_1} \\otimes S_2 +", "S_1 \\otimes \\mathfrak m_{S_2}$ with residue field $k$.", "Set $C^\\wedge$ equal to the completion of $C$ with respect to $\\mathfrak m$.", "Then $C^\\wedge$ is a Noetherian ring complete with respect to", "the maximal ideal $\\mathfrak m^\\wedge = \\mathfrak mC^\\wedge$", "whose residue field is identified with $k$, see", "Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian}.", "Hence $C^\\wedge$ is an object of $\\widehat{\\mathcal{C}}_\\Lambda$.", "Then $S_1 \\to C^\\wedge$ and $S_2 \\to C^\\wedge$ turn $C^\\wedge$", "into a pushout over $R$ in $\\widehat{\\mathcal{C}}_\\Lambda$ (details omitted)." ], "refs": [ "algebra-lemma-completion-Noetherian" ], "ref_ids": [ 873 ] } ], "ref_ids": [] }, { "id": 3423, "type": "theorem", "label": "formal-defos-lemma-CLambdahat-coproducts", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-CLambdahat-coproducts", "contents": [ "The category $\\widehat{\\mathcal{C}}_\\Lambda$ admits coproducts", "of pairs of objects." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ and $S$ be objects of $\\widehat{\\mathcal{C}}_\\Lambda$.", "Consider the ring $C = R \\otimes_\\Lambda S$. There is a canonical", "surjective map $C \\to R \\otimes_\\Lambda S \\to k \\otimes_\\Lambda k \\to k$", "where the last map is the multiplication map. The kernel of", "$C \\to k$ is a maximal ideal $\\mathfrak m$. Note that $\\mathfrak m$", "is generated by $\\mathfrak m_R C$, $\\mathfrak m_S C$ and finitely many", "elements of $C$ which map to generators of the kernel of", "$k \\otimes_\\Lambda k \\to k$. Hence $\\mathfrak m$ is a finitely", "generated ideal. Set", "$C^\\wedge$ equal to the completion of $C$ with respect to $\\mathfrak m$.", "Then $C^\\wedge$ is a Noetherian ring complete with respect to", "the maximal ideal $\\mathfrak m^\\wedge = \\mathfrak mC^\\wedge$", "with residue field $k$, see", "Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian}.", "Hence $C^\\wedge$ is an object of $\\widehat{\\mathcal{C}}_\\Lambda$.", "Then $R \\to C^\\wedge$ and $S \\to C^\\wedge$ turn $C^\\wedge$", "into a coproduct in $\\widehat{\\mathcal{C}}_\\Lambda$ (details omitted)." ], "refs": [ "algebra-lemma-completion-Noetherian" ], "ref_ids": [ 873 ] } ], "ref_ids": [] }, { "id": 3424, "type": "theorem", "label": "formal-defos-lemma-derivations-finite", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-derivations-finite", "contents": [ "Let $S$ be an object of $\\widehat{\\mathcal{C}}_\\Lambda$.", "Then $\\dim_k \\text{Der}_\\Lambda(S, k) < \\infty$." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1, \\ldots, x_n \\in \\mathfrak m_S$ map to a $k$-basis", "for the relative cotangent space", "$\\mathfrak m_S/(\\mathfrak m_\\Lambda S + \\mathfrak m_S^2)$.", "Choose $y_1, \\ldots, y_m \\in S$ whose images in $k$ generate $k$", "over $k'$. We claim that $\\dim_k \\text{Der}_\\Lambda(S, k) \\leq n + m$.", "To see this it suffices to prove that if $D(x_i) = 0$ and", "$D(y_j) = 0$, then $D = 0$. Let $a \\in S$. We can find a", "polynomial $P = \\sum \\lambda_J y^J$ with $\\lambda_J \\in \\Lambda$", "whose image in $k$ is the same as the image of $a$ in $k$.", "Then we see that $D(a - P) = D(a) - D(P) = D(a)$ by our assumption", "that $D(y_j) = 0$ for all $j$. Thus we may assume $a \\in \\mathfrak m_S$.", "Write $a = \\sum a_i x_i$ with $a_i \\in S$. By the Leibniz rule", "$$", "D(a) = \\sum x_iD(a_i) + \\sum a_iD(x_i) = \\sum x_iD(a_i)", "$$", "as we assumed $D(x_i) = 0$. We have $\\sum x_iD(a_i) = 0$", "as multiplication by $x_i$ is zero on $k$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3425, "type": "theorem", "label": "formal-defos-lemma-derivations-surjective", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-derivations-surjective", "contents": [ "Let $f : R \\to S$ be a morphism of $\\widehat{\\mathcal{C}}_\\Lambda$.", "If $\\text{Der}_\\Lambda(S, k) \\to \\text{Der}_\\Lambda(R, k)$ is injective,", "then $f$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "If $f$ is not surjective, then", "$\\mathfrak m_S/(\\mathfrak m_R S + \\mathfrak m_S^2)$ is nonzero by", "Lemma \\ref{lemma-surjective-cotangent-space}.", "Then also $Q = S/(f(R) + \\mathfrak m_R S + \\mathfrak m_S^2)$ is nonzero.", "Note that $Q$ is a $k = R/\\mathfrak m_R$-vector space via $f$. We turn", "$Q$ into an $S$-module via $S \\to k$. The quotient", "map $D : S \\to Q$ is an $R$-derivation: if $a_1, a_2 \\in S$, we can write", "$a_1 = f(b_1) + a_1'$ and $a_2 = f(b_2) + a_2'$ for some", "$b_1, b_2 \\in R$ and $a_1', a_2' \\in \\mathfrak m_S$. Then", "$b_i$ and $a_i$ have the same image in $k$ for $i = 1, 2$ and", "\\begin{align*}", "a_1a_2 & = (f(b_1) + a_1')(f(b_2) + a_2') \\\\", "& = f(b_1)a_2' + f(b_2)a_1' \\\\", "& = f(b_1)(f(b_2) + a_2') + f(b_2)(f(b_1) + a_1') \\\\", "& = f(b_1)a_2 + f(b_2)a_1", "\\end{align*}", "in $Q$ which proves the Leibniz rule. Hence $D : S \\to Q$ is a", "$\\Lambda$-derivation which is zero on composing with $R \\to S$.", "Since $Q \\not = 0$ there also exist derivations $D : S \\to k$ which", "are zero on composing with $R \\to S$, i.e.,", "$\\text{Der}_\\Lambda(S, k) \\to \\text{Der}_\\Lambda(R, k)$ is not injective." ], "refs": [ "formal-defos-lemma-surjective-cotangent-space" ], "ref_ids": [ 3421 ] } ], "ref_ids": [] }, { "id": 3426, "type": "theorem", "label": "formal-defos-lemma-m-adic-topology", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-m-adic-topology", "contents": [ "Let $R$ be an object of $\\widehat{\\mathcal{C}}_\\Lambda$. Let $(J_n)$ be a", "decreasing sequence of ideals such that $\\mathfrak m_R^n \\subset J_n$.", "Set $J = \\bigcap J_n$. Then the sequence $(J_n/J)$ defines the", "$\\mathfrak m_{R/J}$-adic topology on $R/J$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $\\mathfrak m_{R/J}^n \\subset J_n/J$. Thus it suffices", "to show that for every $n$ there exists an $N$ such that", "$J_N/J \\subset \\mathfrak m_{R/J}^n$. This is equivalent to", "$J_N \\subset \\mathfrak m_R^n + J$. For each $n$ the ring $R/\\mathfrak m_R^n$", "is Artinian, hence there exists a $N_n$ such that", "$$", "J_{N_n} + \\mathfrak m_R^n = J_{N_n + 1} + \\mathfrak m_R^n = \\ldots", "$$", "Set $E_n = (J_{N_n} + \\mathfrak m_R^n)/\\mathfrak m_R^n$.", "Set $E = \\lim E_n \\subset \\lim R/\\mathfrak m_R^n = R$.", "Note that $E \\subset J$ as for any $f \\in E$ and any $m$", "we have $f \\in J_m + \\mathfrak m_R^n$ for all $n \\gg 0$, so", "$f \\in J_m$ by Artin-Rees, see", "Algebra, Lemma \\ref{algebra-lemma-intersect-powers-ideal-module-zero}.", "Since the transition maps $E_n \\to E_{n - 1}$ are all surjective,", "we see that $J$ surjects onto $E_n$. Hence for $N = N_n$ works." ], "refs": [ "algebra-lemma-intersect-powers-ideal-module-zero" ], "ref_ids": [ 627 ] } ], "ref_ids": [] }, { "id": 3427, "type": "theorem", "label": "formal-defos-lemma-limit-artinian", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-limit-artinian", "contents": [ "Let $\\ldots \\to A_3 \\to A_2 \\to A_1$ be a sequence of surjective", "ring maps in $\\mathcal{C}_\\Lambda$. If", "$\\dim_k (\\mathfrak m_{A_n}/\\mathfrak m_{A_n}^2)$ is bounded, then", "$S = \\lim A_n$ is an object in $\\widehat{\\mathcal{C}}_\\Lambda$", "and the ideals $I_n = \\Ker(S \\to A_n)$ define the", "$\\mathfrak m_S$-adic topology on $S$." ], "refs": [], "proofs": [ { "contents": [ "We will use freely that the maps $S \\to A_n$ are surjective for all $n$.", "Note that the maps", "$\\mathfrak m_{A_{n + 1}}/\\mathfrak m_{A_{n + 1}}^2 \\to", "\\mathfrak m_{A_n}/\\mathfrak m_{A_n}^2$ are surjective, see", "Lemma \\ref{lemma-surjective-cotangent-space}.", "Hence for $n$ sufficiently large the dimension", "$\\dim_k (\\mathfrak m_{A_n}/\\mathfrak m_{A_n}^2)$ stabilizes to an", "integer, say $r$.", "Thus we can find $x_1, \\ldots, x_r \\in \\mathfrak m_S$ whose images in", "$A_n$ generate $\\mathfrak m_{A_n}$. Moreover, pick $y_1, \\ldots, y_t \\in S$", "whose images in $k$ generate $k$ over $\\Lambda$. Then we get a ring map", "$P = \\Lambda[z_1, \\ldots, z_{r + t}] \\to S$, $z_i \\mapsto x_i$ and", "$z_{r + j} \\mapsto y_j$ such that the composition", "$P \\to S \\to A_n$ is surjective for all $n$. Let $\\mathfrak m \\subset P$", "be the kernel of $P \\to k$. Let $R = P^\\wedge$ be the $\\mathfrak m$-adic", "completion of $P$; this is an object of $\\widehat{\\mathcal{C}}_\\Lambda$.", "Since we still have the compatible system of (surjective) maps $R \\to A_n$", "we get a map $R \\to S$. Set $J_n = \\Ker(R \\to A_n)$.", "Set $J = \\bigcap J_n$. By", "Lemma \\ref{lemma-m-adic-topology}", "we see that $R/J = \\lim R/J_n = \\lim A_n = S$", "and that the ideals $J_n/J = I_n$ define the $\\mathfrak m$-adic topology.", "(Note that for each $n$ we have $\\mathfrak m_R^{N_n} \\subset J_n$ for", "some $N_n$ and not necessarily $N_n = n$, so a renumbering of the ideals", "$J_n$ may be necessary before applying the lemma.)" ], "refs": [ "formal-defos-lemma-surjective-cotangent-space", "formal-defos-lemma-m-adic-topology" ], "ref_ids": [ 3421, 3426 ] } ], "ref_ids": [] }, { "id": 3428, "type": "theorem", "label": "formal-defos-lemma-power-series", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-power-series", "contents": [ "Let $R', R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$. Suppose that", "$R = R' \\oplus I$ for some ideal $I$ of $R$. Let $x_1, \\ldots, x_r \\in I$", "map to a basis of $I/\\mathfrak m_R I$. Set $S = R'[[X_1, \\ldots, X_r]]$", "and consider the $R'$-algebra map $S \\to R$ mapping $X_i$ to $x_i$.", "Assume that for every $n \\gg 0$ the map", "$S/\\mathfrak m_S^n \\to R/\\mathfrak m_R^n$ has a left inverse in", "$\\mathcal{C}_\\Lambda$. Then $S \\to R$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "As $R = R' \\oplus I$ we have", "$$", "\\mathfrak m_R/\\mathfrak m_R^2 =", "\\mathfrak m_{R'}/\\mathfrak m_{R'}^2 \\oplus I/\\mathfrak m_RI", "$$", "and similarly", "$$", "\\mathfrak m_S/\\mathfrak m_S^2 =", "\\mathfrak m_{R'}/\\mathfrak m_{R'}^2 \\oplus \\bigoplus kX_i", "$$", "Hence for $n > 1$ the map $S/\\mathfrak m_S^n \\to R/\\mathfrak m_R^n$", "induces an isomorphism on cotangent spaces. Thus a left inverse", "$h_n : R/\\mathfrak m_R^n \\to S/\\mathfrak m_S^n$ is surjective by", "Lemma \\ref{lemma-surjective-cotangent-space}.", "Since $h_n$ is injective as a left inverse it is an isomorphism.", "Thus the canonical surjections $S/\\mathfrak m_S^n \\to R/\\mathfrak m_R^n$", "are all isomorphisms and we win." ], "refs": [ "formal-defos-lemma-surjective-cotangent-space" ], "ref_ids": [ 3421 ] } ], "ref_ids": [] }, { "id": 3429, "type": "theorem", "label": "formal-defos-lemma-completion-cofibred", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-completion-cofibred", "contents": [ "Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in", "groupoids. Then", "$\\widehat{p} : \\widehat{\\mathcal{F}} \\to \\widehat{\\mathcal{C}}_\\Lambda$", "is a category cofibered in groupoids." ], "refs": [], "proofs": [ { "contents": [ "Let $R \\to S$ be a ring map in $\\widehat{\\mathcal{C}}_\\Lambda$.", "Let $(R, \\xi_n, f_n)$ be an object of $\\widehat{\\mathcal{F}}$.", "For each $n$ choose a pushforward $\\xi_n \\to \\eta_n$ of $\\xi_n$", "along $R/\\mathfrak m_R^n \\to S/\\mathfrak m_S^n$. For each $n$ there", "exists a unique morphism $g_n : \\eta_{n + 1} \\to \\eta_n$ in $\\mathcal{F}$", "lying over $S/\\mathfrak m_S^{n + 1} \\to S/\\mathfrak m_S^n$ such that", "$$", "\\xymatrix{", "\\xi_{n + 1} \\ar[d] \\ar[r]_{f_n} & \\xi_n \\ar[d] \\\\", "\\eta_{n + 1} \\ar[r]^{g_n} & \\eta_n", "}", "$$", "commutes (by the first axiom of a category cofibred in groupoids).", "Hence we obtain a morphism $(R, \\xi_n, f_n) \\to (S, \\eta_n, g_n)$", "lying over $R \\to S$, i.e., the first axiom of a category cofibred in", "groupoids holds for $\\widehat{\\mathcal{F}}$. To see the second axiom", "suppose that we have morphisms", "$a : (R, \\xi_n, f_n) \\to (S, \\eta_n, g_n)$ and", "$b : (R, \\xi_n, f_n) \\to (T, \\theta_n, h_n)$ in $\\widehat{\\mathcal{F}}$", "and a morphism $c_0 : S \\to T$ in $\\widehat{\\mathcal{C}}_\\Lambda$ such that", "$c_0 \\circ a_0 = b_0$. By the second axiom of a category cofibred in groupoids", "for $\\mathcal{F}$ we obtain unique maps $c_n : \\eta_n \\to \\theta_n$", "lying over $S/\\mathfrak m_S^n \\to T/\\mathfrak m_T^n$ such that", "$c_n \\circ a_n = b_n$. Setting $c = (c_n)_{n \\geq 0}$ gives the desired", "morphism $c : (S, \\eta_n, g_n) \\to (T, \\theta_n, h_n)$ in", "$\\widehat{\\mathcal{F}}$ (we omit the verification that", "$h_n \\circ c_{n + 1} = c_n \\circ g_n$)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3430, "type": "theorem", "label": "formal-defos-lemma-formal-objects-different-filtration", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-formal-objects-different-filtration", "contents": [ "In the situation above, $\\widehat{\\mathcal{F}}_\\mathcal{I}(R)$ is equivalent", "to the category $\\widehat{\\mathcal{F}}(R)$." ], "refs": [], "proofs": [ { "contents": [ "An equivalence", "$\\widehat{\\mathcal{F}}_\\mathcal{I}(R) \\to \\widehat{\\mathcal{F}}(R)$", "can be defined as follows. For each $n$, let $m(n)$ be the least $m$", "that $I_m \\subset \\mathfrak m_R^n$. Given an object", "$(\\xi_n, f_n)$ of $\\widehat{\\mathcal{F}}_\\mathcal{I}(R)$, let", "$\\eta_n$ be the pushforward of $\\xi_{m(n)}$ along", "$R/I_{m(n)} \\to R/\\mathfrak m_R^n$. Let $g_n : \\eta_{n + 1} \\to \\eta_n$", "be the unique morphism of $\\mathcal{F}$ lying over", "$R/\\mathfrak m_R^{n + 1} \\to R/\\mathfrak m_R^n$ such that", "$$", "\\xymatrix{", "\\xi_{m(n + 1)} \\ar[rrr]_{f_{m(n)} \\circ \\ldots \\circ f_{m(n + 1) - 1}} \\ar[d]", "& & & \\xi_{m(n)} \\ar[d] \\\\", "\\eta_{n + 1} \\ar[rrr]^{g_n} & & & \\eta_n", "}", "$$", "commutes (existence and uniqueness is guaranteed by the axioms of a", "cofibred category). The functor", "$\\widehat{\\mathcal{F}}_\\mathcal{I}(R) \\to \\widehat{\\mathcal{F}}(R)$", "sends $(\\xi_n, f_n)$ to $(R, \\eta_n, g_n)$. We omit the", "verification that this is indeed an equivalence of categories." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3431, "type": "theorem", "label": "formal-defos-lemma-smoothness-small-extensions", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-smoothness-small-extensions", "contents": [ "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of categories", "cofibered in groupoids over $\\mathcal{C}_\\Lambda$. Then $\\varphi$ is smooth", "if the condition in Definition \\ref{definition-smooth-morphism} is assumed to", "hold only for small extensions $B \\to A$." ], "refs": [ "formal-defos-definition-smooth-morphism" ], "proofs": [ { "contents": [ "Let $B \\to A$ be a surjective ring map in $\\mathcal{C}_\\Lambda$.", "Let $y \\in \\Ob(\\mathcal{G}(B))$, $x \\in \\Ob(\\mathcal{F}(A))$,", "and $y \\to \\varphi(x)$ be a morphism lying over $B \\to A$. By", "Lemma \\ref{lemma-factor-small-extension} we can factor $B \\to A$ into", "small extensions $B = B_n \\to B_{n-1} \\to \\ldots \\to B_0 = A$.", "We argue by induction on $n$. If $n = 1$ the result is true by assumption.", "If $n > 1$, then denote $f : B = B_n \\to B_{n - 1}$ and denote", "$g : B_{n - 1} \\to B_0 = A$. Choose a pushforward", "$y \\to f_* y$ of $y$ along $f$, so that the morphism $y \\to \\varphi(x)$", "factors as $y \\to f_* y \\to \\varphi(x)$. By the induction hypothesis", "we can find $x_{n - 1} \\to x$ lying over $g : B_{n - 1} \\to A$ and", "$a : \\varphi(x_{n - 1}) \\to f_*y$ lying over", "$\\text{id} : B_{n - 1} \\to B_{n - 1}$ such that", "$$", "\\xymatrix{", "\\varphi(x_{n - 1}) \\ar[r]_-a \\ar[dr] & f_*y \\ar[d] \\\\", "& \\varphi(x)", "}", "$$", "commutes. We can apply the assumption to the composition", "$y \\to \\varphi(x_{n - 1})$ of", "$y \\to f_*y$ with $a^{-1} : f_*y \\to \\varphi(x_{n - 1})$. We obtain", "$x_n \\to x_{n - 1}$ lying over $B_n \\to B_{n - 1}$ and", "$\\varphi(x_n) \\to y$ lying over $\\text{id} : B_n \\to B_n$ so that the diagram", "$$", "\\xymatrix{", "\\varphi(x_n) \\ar[r] \\ar[d] & y \\ar[d] \\\\", "\\varphi(x_{n - 1}) \\ar[r]^-a \\ar[dr] & f_*y \\ar[d] \\\\", "& \\varphi(x)", "}", "$$", "commutes. Then the composition $x_n \\to x_{n - 1} \\to x$ and", "$\\varphi(x_n) \\to y$ are the morphisms required by the definition of", "smoothness." ], "refs": [ "formal-defos-lemma-factor-small-extension" ], "ref_ids": [ 3414 ] } ], "ref_ids": [ 3520 ] }, { "id": 3432, "type": "theorem", "label": "formal-defos-lemma-smooth-morphism-power-series", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-smooth-morphism-power-series", "contents": [ "Let $R \\to S$ be a ring map in $\\widehat{\\mathcal{C}}_\\Lambda$. Then", "the induced morphism", "$\\underline{S}|_{\\mathcal{C}_\\Lambda} \\to \\underline{R}|_{\\mathcal{C}_\\Lambda}$", "is smooth if and only if $S$ is a power series ring over $R$." ], "refs": [], "proofs": [ { "contents": [ "Assume $S$ is a power series ring over $R$. Say $S = R[[x_1, \\ldots, x_n]]$.", "Smoothness of", "$\\underline{S}|_{\\mathcal{C}_\\Lambda} \\to \\underline{R}|_{\\mathcal{C}_\\Lambda}$", "means the following (see Remark \\ref{remark-compare-smooth-schlessinger}):", "Given a surjective ring map $B \\to A$ in", "$\\mathcal{C}_\\Lambda$, a ring map $R \\to B$, a ring map $S \\to A$ such that", "the solid diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar@{..>}[rd] & A \\\\", "R \\ar[u] \\ar[r] & B \\ar[u]", "}", "$$", "is commutative then a dotted arrow exists making the diagram commute.", "(Note the similarity with", "Algebra, Definition \\ref{algebra-definition-formally-smooth}.)", "To construct the dotted arrow choose elements $b_i \\in B$ whose images", "in $A$ are equal to the images of $x_i$ in $A$. Note that", "$b_i \\in \\mathfrak m_B$ as $x_i$ maps to an element of $\\mathfrak m_A$.", "Hence there is a unique $R$-algebra map $R[[x_1, \\ldots, x_n]] \\to B$", "which maps $x_i$ to $b_i$ and which can serve as our dotted arrow.", "\\medskip\\noindent", "Conversely, assume", "$\\underline{S}|_{\\mathcal{C}_\\Lambda} \\to \\underline{R}|_{\\mathcal{C}_\\Lambda}$", "is smooth. Let $x_1, \\ldots, x_n \\in S$ be elements whose images", "form a basis in the relative cotangent space", "$\\mathfrak m_S/(\\mathfrak m_R S + \\mathfrak m_S^2)$ of $S$ over $R$.", "Set $T = R[[X_1, \\ldots, X_n]]$. Note that both", "$$", "S/(\\mathfrak m_R S + \\mathfrak m_S^2) \\cong", "R/\\mathfrak m_R[x_1, \\ldots, x_n]/(x_ix_j)", "$$", "and", "$$", "T/(\\mathfrak m_R T + \\mathfrak m_T^2) \\cong", "R/\\mathfrak m_R[X_1, \\ldots, X_n]/(X_iX_j).", "$$", "Let", "$S/(\\mathfrak m_R S + \\mathfrak m_S^2) \\to", "T/(\\mathfrak m_R T + \\mathfrak m_T^2)$", "be the local $R$-algebra isomorphism given by mapping", "the class of $x_i$ to the class of $X_i$. Let", "$f_1 : S \\to T/(\\mathfrak m_R T + \\mathfrak m_T^2)$ be the", "composition", "$S \\to S/(\\mathfrak m_R S + \\mathfrak m_S^2)", "\\to T/(\\mathfrak m_R T + \\mathfrak m_T^2)$.", "The assumption that", "$\\underline{S}|_{\\mathcal{C}_\\Lambda} \\to \\underline{R}|_{\\mathcal{C}_\\Lambda}$", "is smooth means we can lift $f_1$ to a map", "$f_2 : S \\to T/\\mathfrak{m}_T^2$, then to a map", "$f_3 : S \\to T/\\mathfrak{m}_T^3$, and so on, for all $n \\geq 1$. Thus", "we get an induced map $f : S \\to T = \\lim T/\\mathfrak m_T^n$", "of local $R$-algebras. By our choice of $f_1$, the map $f$ induces an", "isomorphism", "$\\mathfrak m_S/(\\mathfrak m_R S + \\mathfrak m_S^2) \\to", "\\mathfrak m_T/(\\mathfrak m_R T + \\mathfrak m_T^2)$", "of relative cotangent spaces.", "Hence $f$ is surjective by", "Lemma \\ref{lemma-surjective-cotangent-space}", "(where we think of $f$ as a map in $\\widehat{\\mathcal{C}}_R$).", "Choose preimages $y_i \\in S$ of $X_i \\in T$ under $f$. As $T$ is a", "power series ring over $R$ there exists a local", "$R$-algebra homomorphism $s : T \\to S$ mapping $X_i$ to $y_i$.", "By construction $f \\circ s = \\text{id}$. Then $s$ is injective.", "But $s$ induces an isomorphism on relative cotangent spaces since", "$f$ does, so it is also surjective by", "Lemma \\ref{lemma-surjective-cotangent-space}", "again. Hence $s$ and $f$ are isomorphisms." ], "refs": [ "formal-defos-remark-compare-smooth-schlessinger", "algebra-definition-formally-smooth", "formal-defos-lemma-surjective-cotangent-space", "formal-defos-lemma-surjective-cotangent-space" ], "ref_ids": [ 3557, 1537, 3421, 3421 ] } ], "ref_ids": [] }, { "id": 3433, "type": "theorem", "label": "formal-defos-lemma-smooth-properties", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-smooth-properties", "contents": [ "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and $\\psi : \\mathcal{G}", "\\to \\mathcal{H}$ be morphisms of categories cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$.", "\\begin{enumerate}", "\\item If $\\varphi$ and $\\psi$ are smooth, then $\\psi \\circ \\varphi$ is smooth.", "\\item If $\\varphi$ is essentially surjective and $\\psi \\circ \\varphi$ is", "smooth, then $\\psi$ is smooth.", "\\item If $\\mathcal{G}' \\to \\mathcal{G}$ is a morphism of categories", "cofibered in groupoids and $\\varphi$ is smooth, then", "$\\mathcal{F} \\times_\\mathcal{G} \\mathcal{G}' \\to \\mathcal{G}'$ is smooth.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Statements (1) and (2) follow immediately from the definitions.", "Proof of (3) omitted. Hints: use the formulation of smoothness given in", "Remark \\ref{remark-smoothness-2-categorical}", "and use that $\\mathcal{F} \\times_\\mathcal{G} \\mathcal{G}'$", "is the $2$-fibre product, see", "Remarks \\ref{remarks-cofibered-groupoids} (\\ref{item-fibre-product})." ], "refs": [ "formal-defos-remark-smoothness-2-categorical", "formal-defos-remarks-cofibered-groupoids" ], "ref_ids": [ 3556, 3585 ] } ], "ref_ids": [] }, { "id": 3434, "type": "theorem", "label": "formal-defos-lemma-smooth-morphism-essentially-surjective", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-smooth-morphism-essentially-surjective", "contents": [ "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a smooth morphism of", "categories cofibered in groupoids over $\\mathcal{C}_\\Lambda$. Assume", "$\\varphi : \\mathcal{F}(k) \\to \\mathcal{G}(k)$ is essentially surjective.", "Then $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and", "$\\widehat{\\varphi} : \\widehat{\\mathcal{F}} \\to \\widehat{\\mathcal{G}}$", "are essentially surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $y$ be an object of $\\mathcal{G}$ lying over", "$A \\in \\Ob(\\mathcal{C}_\\Lambda)$. Let $y \\to y_0$ be a pushforward", "of $y$ along $A \\to k$. By the assumption on essential surjectivity of", "$\\varphi : \\mathcal{F}(k) \\to \\mathcal{G}(k)$ there exist an object", "$x_0$ of $\\mathcal{F}$ lying over $k$ and an isomorphism", "$y_0 \\to \\varphi(x_0)$. Smoothness of $\\varphi$ implies there exists", "an object $x$ of $\\mathcal{F}$ over $A$ whose image $\\varphi(x)$", "is isomorphic to $y$. Thus $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "is essentially surjective.", "\\medskip\\noindent", "Let $\\eta = (R, \\eta_n, g_n)$ be an object of $\\widehat{\\mathcal{G}}$. We", "construct an object $\\xi$ of $\\widehat{\\mathcal{F}}$ with an isomorphism", "$\\eta \\to \\varphi(\\xi)$. By the assumption on essential surjectivity of", "$\\varphi : \\mathcal{F}(k) \\to \\mathcal{G}(k)$, there exists a morphism", "$\\eta_1 \\to \\varphi(\\xi_1)$ in $\\mathcal{G}(k)$ for some", "$\\xi_1 \\in \\Ob(\\mathcal{F}(k))$. The morphism", "$\\eta_2 \\xrightarrow{g_1} \\eta_1 \\to \\varphi(\\xi_1)$", "lies over the surjective ring map $R/\\mathfrak m_R^2 \\to k$, hence", "by smoothness of $\\varphi$ there exists", "$\\xi_2 \\in \\Ob(\\mathcal{F}(R/\\mathfrak m_R^2))$, a", "morphism $f_1: \\xi_2 \\to \\xi_1$ lying over", "$R/\\mathfrak m_R^2 \\to k$, and a morphism", "$\\eta_2 \\to \\varphi(\\xi_2)$ such that", "$$", "\\xymatrix{", "\\varphi(\\xi_2) \\ar[r]^{\\varphi(f_1)} & \\varphi(\\xi_{1}) \\\\", "\\eta_2 \\ar[u] \\ar[r]^{g_1} & \\eta_1 \\ar[u] \\\\", "}", "$$", "commutes. Continuing in this way we construct an object", "$\\xi = (R, \\xi_n, f_n)$ of $\\widehat{\\mathcal{F}}$ and a morphism", "$\\eta \\to \\varphi(\\xi) = (R, \\varphi(\\xi_n), \\varphi(f_n))$", "in $\\widehat{\\mathcal{G}}(R)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3435, "type": "theorem", "label": "formal-defos-lemma-versal-object-quasi-initial", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-versal-object-quasi-initial", "contents": [ "Let $\\mathcal{F}$ be a predeformation category.", "Let $\\xi$ be a versal formal object of $\\mathcal{F}$.", "For any formal object $\\eta$ of $\\widehat{\\mathcal{F}}$,", "there exists a morphism $\\xi \\to \\eta$." ], "refs": [], "proofs": [ { "contents": [ "By assumption the morphism", "$\\underline{\\xi} : \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$", "is smooth. Then", "$\\iota(\\xi) : \\underline{R} \\to \\widehat{\\mathcal{F}}$", "is the completion of $\\underline{\\xi}$, see", "Remark \\ref{remark-formal-objects-yoneda}.", "By", "Lemma \\ref{lemma-smooth-morphism-essentially-surjective}", "there exists an object $f$ of $\\underline{R}$ such that", "$\\iota(\\xi)(f) = \\eta$. Then $f$ is", "a ring map $f : R \\to S$ in $\\widehat{\\mathcal{C}}_\\Lambda$. And", "$\\iota(\\xi)(f) = \\eta$ means that", "$f_*\\xi \\cong \\eta$ which means exactly that there is a morphism", "$\\xi \\to \\eta$ lying over $f$." ], "refs": [ "formal-defos-remark-formal-objects-yoneda", "formal-defos-lemma-smooth-morphism-essentially-surjective" ], "ref_ids": [ 3553, 3434 ] } ], "ref_ids": [] }, { "id": 3436, "type": "theorem", "label": "formal-defos-lemma-smooth", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-smooth", "contents": [ "Let $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$. The following are", "equivalent", "\\begin{enumerate}", "\\item $\\underline{R}|_{\\mathcal{C}_\\Lambda}$ is smooth,", "\\item $\\Lambda \\to R$ is formally smooth in the $\\mathfrak m_R$-adic topology,", "\\item $\\Lambda \\to R$ is flat and $R \\otimes_\\Lambda k'$ is", "geometrically regular over $k'$, and", "\\item $\\Lambda \\to R$ is flat and $k' \\to R \\otimes_\\Lambda k'$ is", "formally smooth in the $\\mathfrak m_R$-adic topology.", "\\end{enumerate}", "In the classical case, these are also equivalent to", "\\begin{enumerate}", "\\item[(5)] $R$ is isomorphic to $\\Lambda[[x_1, \\ldots, x_n]]$", "for some $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Smoothness of", "$p : \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{C}_\\Lambda$", "means that given $B \\to A$ surjective in $\\mathcal{C}_\\Lambda$", "and given $R \\to A$ we can find the dotted arrow in the", "diagram", "$$", "\\xymatrix{", "R \\ar[r] \\ar@{-->}[rd] & A \\\\", "\\Lambda \\ar[r] \\ar[u] & B \\ar[u]", "}", "$$", "This is certainly true if $\\Lambda \\to R$ is formally smooth in the", "$\\mathfrak m_R$-adic topology, see", "More on Algebra, Definitions", "\\ref{more-algebra-definition-formally-smooth-adic} and", "\\ref{more-algebra-definition-formally-smooth}.", "Conversely, if this holds, then we see that", "$\\Lambda \\to R$ is formally smooth in the $\\mathfrak m_R$-adic", "topology by More on Algebra, Lemma \\ref{more-algebra-lemma-fs-local}.", "Thus (1) and (2) are equivalent.", "\\medskip\\noindent", "The equivalence of (2), (3), and (4) is", "More on Algebra, Proposition \\ref{more-algebra-proposition-fs-flat-fibre-fs}.", "The equivalence with (5) follows for example from", "Lemma \\ref{lemma-smooth-morphism-power-series}", "and the fact that $\\mathcal{C}_\\Lambda$ is the same", "as $\\underline{\\Lambda}|_{\\mathcal{C}_\\Lambda}$ in the classical case." ], "refs": [ "more-algebra-definition-formally-smooth-adic", "more-algebra-definition-formally-smooth", "more-algebra-lemma-fs-local", "more-algebra-proposition-fs-flat-fibre-fs", "formal-defos-lemma-smooth-morphism-power-series" ], "ref_ids": [ 10612, 10611, 10021, 10577, 3432 ] } ], "ref_ids": [] }, { "id": 3437, "type": "theorem", "label": "formal-defos-lemma-smooth-power-series-classical", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-smooth-power-series-classical", "contents": [ "Let $\\mathcal{F}$ be a predeformation category.", "Let $\\xi$ be a versal formal object of $\\mathcal{F}$ lying over", "$R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$. The following are", "equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is unobstructed, and", "\\item $\\Lambda \\to R$ is formally smooth in the $\\mathfrak m_R$-adic topology.", "\\end{enumerate}", "In the classical case these are also equivalent to", "\\begin{enumerate}", "\\item[(3)] $R \\cong \\Lambda[[x_1, \\ldots, x_n]]$ for some $n$.\\", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (1) holds, i.e., if $\\mathcal{F}$ is unobstructed, then the composition", "$$", "\\underline{R}|_{\\mathcal{C}_\\Lambda}", "\\xrightarrow{\\underline{\\xi}}", "\\mathcal{F}", "\\to", "\\mathcal{C}_\\Lambda", "$$", "is smooth, see Lemma \\ref{lemma-smooth-properties}.", "Hence we see that (2) holds by Lemma \\ref{lemma-smooth}.", "Conversely, if (2) holds, then the composition is smooth", "and moreover the first arrow is essentially surjective by", "Lemma \\ref{lemma-versal-object-quasi-initial}. Hence we", "find that the second arrow is smooth by Lemma \\ref{lemma-smooth-properties}", "which means that $\\mathcal{F}$ is unobstructed by definition.", "The equivalence with (3) in the classical case follows", "from Lemma \\ref{lemma-smooth}." ], "refs": [ "formal-defos-lemma-smooth-properties", "formal-defos-lemma-smooth", "formal-defos-lemma-versal-object-quasi-initial", "formal-defos-lemma-smooth-properties", "formal-defos-lemma-smooth" ], "ref_ids": [ 3433, 3436, 3435, 3433, 3436 ] } ], "ref_ids": [] }, { "id": 3438, "type": "theorem", "label": "formal-defos-lemma-exists-smooth", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-exists-smooth", "contents": [ "There exists an $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$", "such that the equivalent conditions of Lemma \\ref{lemma-smooth}", "hold and moreover $H_1(L_{k/\\Lambda}) = \\mathfrak m_R/\\mathfrak m_R^2$", "and $\\Omega_{R/\\Lambda} \\otimes_R k = \\Omega_{k/\\Lambda}$." ], "refs": [ "formal-defos-lemma-smooth" ], "proofs": [ { "contents": [ "In the classical case we choose $R = \\Lambda$. More generally, if", "the residue field extension $k/k'$ is separable, then there exists", "a unique finite \\'etale extension $\\Lambda^\\wedge \\to R$", "(Algebra, Lemmas \\ref{algebra-lemma-complete-henselian} and", "\\ref{algebra-lemma-henselian-cat-finite-etale})", "of the completion $\\Lambda^\\wedge$ of $\\Lambda$", "inducing the extension $k/k'$ on residue fields.", "\\medskip\\noindent", "In the general case we proceed as follows. Choose a smooth", "$\\Lambda$-algebra $P$ and a $\\Lambda$-algebra surjection $P \\to k$.", "(For example, let $P$ be a polynomial algebra.)", "Denote $\\mathfrak m_P$ the kernel of $P \\to k$. The Jacobi-Zariski sequence,", "see (\\ref{equation-sequence-extended}) and ", "Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL}, is an", "exact sequence", "$$", "0 \\to H_1(\\NL_{k/\\Lambda}) \\to", "\\mathfrak m_P/\\mathfrak m_P^2 \\to", "\\Omega_{P/\\Lambda} \\otimes_P k \\to", "\\Omega_{k/\\Lambda} \\to 0", "$$", "We have the $0$ on the left because $P/k$ is smooth, hence", "$\\NL_{P/\\Lambda}$ is quasi-isomorphic to a finite projective module", "placed in degree $0$, hence $H_1(\\NL_{P/\\Lambda} \\otimes_P k) = 0$.", "Suppose $f \\in \\mathfrak m_P$ maps to a nonzero element", "of $\\Omega_{P/\\Lambda} \\otimes_P k$. Setting $P' = P/(f)$ we have a", "$\\Lambda$-algebra surjection $P' \\to k$.", "Observe that $P'$ is smooth at $\\mathfrak m_{P'}$:", "this follows from More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-slice-smooth-given-element}.", "Thus after replacing $P$ by a principal localization", "of $P'$, we see that $\\dim(\\mathfrak m_P/\\mathfrak m_P^2)$", "decreases. Repeating finitely many times, we may assume", "the map $\\mathfrak m_P/\\mathfrak m_P^2 \\to", "\\Omega_{P/\\Lambda} \\otimes_P k$ is zero so that", "the exact sequence breaks into isomorphisms", "$H_1(L_{k/\\Lambda}) = \\mathfrak m_P/\\mathfrak m_P^2$ and", "$\\Omega_{P/\\Lambda} \\otimes_P k = \\Omega_{k/\\Lambda}$.", "\\medskip\\noindent", "Let $R$ be the $\\mathfrak m_P$-adic completion of $P$.", "Then $R$ is an object of $\\widehat{\\mathcal{C}}_\\Lambda$.", "Namely, it is a complete local Noetherian ring (see", "Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian-Noetherian})", "and its residue field is identified with $k$.", "We claim that $R$ works.", "\\medskip\\noindent", "First observe that the map $P \\to R$ induces isomorphisms", "$\\mathfrak m_P/\\mathfrak m_P^2 = \\mathfrak m_R/\\mathfrak m_R^2$", "and $\\Omega_{P/\\Lambda} \\otimes_P k = \\Omega_{R/\\Lambda} \\otimes_R k$.", "This is true because both $\\mathfrak m_P/\\mathfrak m_P^2$", "and $\\Omega_{P/\\Lambda} \\otimes_P k$ only depend on the", "$\\Lambda$-algebra $P/\\mathfrak m_P^2$, see", "Algebra, Lemma \\ref{algebra-lemma-differential-mod-power-ideal},", "the same holds for $R$ and we have $P/\\mathfrak m_P^2 = R/\\mathfrak m_R^2$.", "Using the functoriality of the Jacobi-Zariski sequence", "(\\ref{equation-sequence-functorial})", "we deduce that $H_1(L_{k/\\Lambda}) = \\mathfrak m_R/\\mathfrak m_R^2$ and", "$\\Omega_{R/\\Lambda} \\otimes_R k = \\Omega_{k/\\Lambda}$", "as the same is true for $P$.", "\\medskip\\noindent", "Finally, since $\\Lambda \\to P$ is smooth we see that", "$\\Lambda \\to P$ is formally smooth by", "Algebra, Proposition \\ref{algebra-proposition-smooth-formally-smooth}.", "Then $\\Lambda \\to P$ is formally smooth for the $\\mathfrak m_P$-adic", "topology by More on Algebra, Lemma \\ref{more-algebra-lemma-formally-smooth}.", "This property is inherited by the completion $R$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-formally-smooth-completion}", "and the proof is complete.", "In fact, it turns out that whenever $\\underline{R}|_{\\mathcal{C}_\\Lambda}$", "is smooth, then $R$ is isomorphic to a completion of a smooth", "algebra over $\\Lambda$, but we won't use this." ], "refs": [ "algebra-lemma-complete-henselian", "algebra-lemma-henselian-cat-finite-etale", "algebra-lemma-exact-sequence-NL", "more-morphisms-lemma-slice-smooth-given-element", "algebra-lemma-completion-Noetherian-Noetherian", "algebra-lemma-differential-mod-power-ideal", "algebra-proposition-smooth-formally-smooth", "more-algebra-lemma-formally-smooth", "more-algebra-lemma-formally-smooth-completion" ], "ref_ids": [ 1282, 1280, 1153, 13875, 874, 1137, 1426, 10014, 10015 ] } ], "ref_ids": [ 3436 ] }, { "id": 3439, "type": "theorem", "label": "formal-defos-lemma-S1-small-extensions", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-S1-small-extensions", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal", "C_\\Lambda$. Then $\\mathcal{F}$ satisfies (S1) if the condition of (S1)", "is assumed to hold only when $A_2 \\to A$ is a small extension." ], "refs": [], "proofs": [ { "contents": [ "Proof omitted. Hints: apply Lemma \\ref{lemma-factor-small-extension}", "and use induction similar to the proof of", "Lemma \\ref{lemma-smoothness-small-extensions}." ], "refs": [ "formal-defos-lemma-factor-small-extension", "formal-defos-lemma-smoothness-small-extensions" ], "ref_ids": [ 3414, 3431 ] } ], "ref_ids": [] }, { "id": 3440, "type": "theorem", "label": "formal-defos-lemma-S2-extensions", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-S2-extensions", "contents": [ "Let $\\mathcal{F}$ be a category cofibred in groupoids over", "$\\mathcal{C}_\\Lambda$. If $\\mathcal{F}$ satisfies (S2), then the", "condition of (S2) also holds when $k[\\epsilon]$ is replaced by $k[V]$", "for any finite dimensional $k$-vector space $V$." ], "refs": [], "proofs": [ { "contents": [ "In the case that $\\mathcal{F}$ is cofibred in sets, i.e., corresponds", "to a functor $F : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ this follows", "from the description of (S2) for $F$ in", "Remark \\ref{remark-compare-S1-S2-schlessinger}", "and the fact that", "$k[V] \\cong k[\\epsilon] \\times_k \\ldots \\times_k k[\\epsilon]$", "with $\\dim_k V$ factors. The case of functors is what we will use in", "the rest of this chapter.", "\\medskip\\noindent", "We prove the general case by induction on $\\dim(V)$. If $\\dim(V) = 1$, then", "$k[V] \\cong k[\\epsilon]$ and the result holds by assumption.", "If $\\dim(V) > 1$ we write $V = V' \\oplus k\\epsilon$. Pick a diagram", "$$", "\\vcenter{", "\\xymatrix{", "& x_V \\ar[d] \\\\", "x \\ar[r] & x_0", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "& k[V] \\ar[d] \\\\", "A \\ar[r] & k", "}", "}", "$$", "Choose a morphism $x_V \\to x_{V'}$ lying over $k[V] \\to k[V']$", "and a morphism $x_V \\to x_\\epsilon$ lying over $k[V] \\to k[\\epsilon]$.", "Note that the morphism $x_V \\to x_0$ factors as", "$x_V \\to x_{V'} \\to x_0$ and as $x_V \\to x_\\epsilon \\to x_0$.", "By induction hypothesis we can find a diagram", "$$", "\\vcenter{", "\\xymatrix{", "y' \\ar[d] \\ar[r] & x_{V'} \\ar[d] \\\\", "x \\ar[r] & x_0", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "A \\times_k k[V'] \\ar[d] \\ar[r] & k[V'] \\ar[d] \\\\", "A \\ar[r] & k", "}", "}", "$$", "This gives us a commutative diagram", "$$", "\\vcenter{", "\\xymatrix{", "& x_\\epsilon \\ar[d] \\\\", "y' \\ar[r] & x_0", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "& k[\\epsilon] \\ar[d] \\\\", "A \\times_k k[V'] \\ar[r] & k", "}", "}", "$$", "Hence by (S2) we get a commutative diagram", "$$", "\\vcenter{", "\\xymatrix{", "y \\ar[d] \\ar[r] & x_\\epsilon \\ar[d] \\\\", "y' \\ar[r] & x_0", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "(A \\times_k k[V']) \\times_k k[\\epsilon] \\ar[d] \\ar[r] & k[\\epsilon] \\ar[d] \\\\", "A \\times_k k[V'] \\ar[r] & k", "}", "}", "$$", "Note that", "$(A \\times_k k[V']) \\times_k k[\\epsilon] = A \\times_k k[V' \\oplus k\\epsilon]", "= A \\times_k k[V]$. We claim that $y$ fits into the correct commutative", "diagram. To see this we let $y \\to y_V$ be a morphism lying over", "$A \\times_k k[V] \\to k[V]$. We can factor the morphisms", "$y \\to y' \\to x_{V'}$ and $y \\to x_\\epsilon$ through the morphism", "$y \\to y_V$ (by the axioms of categories cofibred in groupoids).", "Hence we see that both $y_V$ and $x_V$ fit into commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "y_V \\ar[r] \\ar[d] & x_\\epsilon \\ar[d] \\\\", "x_{V'} \\ar[r] & x_0", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "x_V \\ar[r] \\ar[d] & x_\\epsilon \\ar[d] \\\\", "x_{V'} \\ar[r] & x_0", "}", "}", "$$", "and hence by the second part of (S2) there exists an isomorphism", "$y_V \\to x_V$ compatible with $y_V \\to x_{V'}$ and $x_V \\to x_{V'}$", "and in particular compatible with the maps to $x_0$.", "The composition $y \\to y_V \\to x_V$ then fits into the required commutative", "diagram", "$$", "\\vcenter{", "\\xymatrix{", "y \\ar[r] \\ar[d] & x_V \\ar[d] \\\\", "x \\ar[r] & x_0", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "A \\times_k k[V] \\ar[d] \\ar[r] & k[V] \\ar[d] \\\\", "A \\ar[r] & k", "}", "}", "$$", "In this way we see that the first part of $(S2)$ holds with $k[\\epsilon]$", "replaced by $k[V]$.", "\\medskip\\noindent", "To prove the second part suppose given two commutative", "diagrams", "$$", "\\vcenter{", "\\xymatrix{", "y \\ar[r] \\ar[d] & x_V \\ar[d] \\\\", "x \\ar[r] & x_0", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "y' \\ar[r] \\ar[d] & x_V \\ar[d] \\\\", "x \\ar[r] & x_0", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "A \\times_k k[V] \\ar[d] \\ar[r] & k[V] \\ar[d] \\\\", "A \\ar[r] & k", "}", "}", "$$", "We will use the morphisms $x_V \\to x_{V'} \\to x_0$ and", "$x_V \\to x_\\epsilon \\to x_0$ introduced in the first paragraph of the proof.", "Choose morphisms $y \\to y_{V'}$ and $y' \\to y'_{V'}$", "lying over $A \\times_k k[V] \\to A \\times_k k[V']$. The axioms of a", "cofibred category imply we can find commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "y_{V'} \\ar[r] \\ar[d] & x_{V'} \\ar[d] \\\\", "x \\ar[r] & x_0", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "y'_{V'} \\ar[r] \\ar[d] & x_{V'} \\ar[d] \\\\", "x \\ar[r] & x_0", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "A \\times_k k[V'] \\ar[d] \\ar[r] & k[V'] \\ar[d] \\\\", "A \\ar[r] & k", "}", "}", "$$", "By induction hypothesis we obtain an isomorphism", "$b : y_{V'} \\to y'_{V'}$", "compatible with the morphisms $y_{V'} \\to x$ and $y'_{V'} \\to x$,", "in particular compatible with the morphisms to $x_0$.", "Then we have commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "y \\ar[r] \\ar[d] & x_\\epsilon \\ar[d] \\\\", "y'_{V'} \\ar[r] & x_0", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "y' \\ar[r] \\ar[d] & x_\\epsilon \\ar[d] \\\\", "y'_{V'} \\ar[r] & x_0", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "A \\times_k k[\\epsilon] \\ar[d] \\ar[r] & k[\\epsilon] \\ar[d] \\\\", "A \\ar[r] & k", "}", "}", "$$", "where the morphism $y \\to y'_{V'}$ is the composition", "$y \\to y_{V'} \\xrightarrow{b} y'_{V'}$ and where the morphisms", "$y \\to x_\\epsilon$ and $y' \\to x_\\epsilon$ are the compositions of", "the maps $y \\to x_V$ and $y' \\to x_V$ with the morphism $x_V \\to x_\\epsilon$.", "Then the second part of (S2) guarantees the existence of an isomorphism", "$y \\to y'$ compatible with the maps to $y'_{V'}$, in particular compatible", "with the maps to $x$ (because $b$ was compatible with the maps to $x$)." ], "refs": [ "formal-defos-remark-compare-S1-S2-schlessinger" ], "ref_ids": [ 3561 ] } ], "ref_ids": [] }, { "id": 3441, "type": "theorem", "label": "formal-defos-lemma-S1-S2-associated-functor", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-S1-S2-associated-functor", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ satisfies (S1), then so does", "$\\overline{\\mathcal{F}}$.", "\\item If $\\mathcal{F}$ satisfies (S2), then so does", "$\\overline{\\mathcal{F}}$ provided at least one of the following conditions is", "satisfied", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a predeformation category,", "\\item the category $\\mathcal{F}(k)$ is a set or a setoid, or", "\\item for any morphism $x_\\epsilon \\to x_0$ of $\\mathcal{F}$", "lying over $k[\\epsilon] \\to k$ the pushforward map", "$\\text{Aut}_{k[\\epsilon]}(x_\\epsilon) \\to \\text{Aut}_k(x_0)$", "is surjective.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{F}$ has (S1).", "Suppose we have ring maps $f_i : A_i \\to A$ in $\\mathcal{C}_\\Lambda$", "with $f_2$ surjective. Let $x_i \\in \\mathcal{F}(A_i)$ such that", "the pushforwards $f_{1, *}(x_1)$ and $f_{2, *}(x_2)$ are isomorphic.", "Then we can denote $x$ an object of $\\mathcal{F}$ over $A$ isomorphic", "to both of these and we obtain a diagram as in (S1). Hence we find", "an object $y$ of $\\mathcal{F}$ over $A_1 \\times_A A_2$ whose pushforward", "to $A_1$, resp.\\ $A_2$ is isomorphic to $x_1$, resp.\\ $x_2$. In this way", "we see that (S1) holds for $\\overline{\\mathcal{F}}$.", "\\medskip\\noindent", "Assume $\\mathcal{F}$ has (S2).", "The first part of (S2) for $\\overline{\\mathcal{F}}$ follows as in", "the argument above. The second part of (S2) for", "$\\overline{\\mathcal{F}}$ signifies that the map", "$$", "\\overline{\\mathcal{F}}(A \\times_k k[\\epsilon]) \\to", "\\overline{\\mathcal{F}}(A)", "\\times_{\\overline{\\mathcal{F}}(k)} \\overline{\\mathcal{F}}(k[\\epsilon])", "$$", "is injective for any ring $A$ in $\\mathcal{C}_\\Lambda$. Suppose that", "$y, y' \\in \\mathcal{F}(A \\times_k k[\\epsilon])$. Using the axioms", "of cofibred categories we can choose commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "y \\ar[r]_c \\ar[d]_a & x_\\epsilon \\ar[d]^e \\\\", "x \\ar[r]^d & x_0", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "y' \\ar[r]_{c'} \\ar[d]_{a'} & x'_\\epsilon \\ar[d]^{e'} \\\\", "x' \\ar[r]^{d'} & x'_0", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "A \\times_k k[\\epsilon] \\ar[d] \\ar[r] & k[\\epsilon] \\ar[d] \\\\", "A \\ar[r] & k", "}", "}", "$$", "Assume that there exist isomorphisms", "$\\alpha : x \\to x'$ in $\\mathcal{F}(A)$ and", "$\\beta : x_\\epsilon \\to x'_\\epsilon$ in $\\mathcal{F}(k[\\epsilon])$.", "This also means there exists an isomorphism $\\gamma : x_0 \\to x'_0$", "compatible with $\\alpha$. To prove (S2) for $\\overline{\\mathcal{F}}$", "we have to show that there exists an isomorphism $y \\to y'$ in", "$\\mathcal{F}(A \\times_k k[\\epsilon])$.", "By (S2) for $\\mathcal{F}$ such a morphism will exist if we can", "choose the isomorphisms $\\alpha$ and $\\beta$ and $\\gamma$ such that", "$$", "\\xymatrix{", "x \\ar[d]^\\alpha \\ar[r] & x_0 \\ar[d]^\\gamma &", "x_\\epsilon \\ar[d]^\\beta \\ar[l]^e \\\\", "x' \\ar[r] & x'_0 & x'_\\epsilon \\ar[l]_{e'}", "}", "$$", "is commutative (because then we can replace $x$ by $x'$ and $x_\\epsilon$", "by $x'_\\epsilon$ in the previous displayed diagram). The left hand square", "commutes by our choice of $\\gamma$. We can factor $e' \\circ \\beta$ as", "$\\gamma' \\circ e$ for some second map", "$\\gamma' : x_0 \\to x'_0$. Now the question is whether we can arrange it so", "that $\\gamma = \\gamma'$? This is clear if $\\mathcal{F}(k)$ is a set, or a", "setoid. Moreover, if", "$\\text{Aut}_{k[\\epsilon]}(x_\\epsilon) \\to \\text{Aut}_k(x_0)$", "is surjective, then we can adjust the choice of $\\beta$ by precomposing", "with an automorphism of $x_\\epsilon$ whose image is", "$\\gamma^{-1} \\circ \\gamma'$ to make things work." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3442, "type": "theorem", "label": "formal-defos-lemma-S1-S2-localize", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-S1-S2-localize", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$. Let $x_0 \\in \\Ob(\\mathcal{F}(k))$.", "Let $\\mathcal{F}_{x_0}$ be the category cofibred in groupoids over", "$\\mathcal{C}_\\Lambda$ constructed in", "Remark \\ref{remark-localize-cofibered-groupoid}.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ satisfies (S1), then so does $\\mathcal{F}_{x_0}$.", "\\item If $\\mathcal{F}$ satisfies (S2), then so does $\\mathcal{F}_{x_0}$.", "\\end{enumerate}" ], "refs": [ "formal-defos-remark-localize-cofibered-groupoid" ], "proofs": [ { "contents": [ "Any diagram as in", "Definition \\ref{definition-S1-S2}", "in $\\mathcal{F}_{x_0}$ gives rise to a diagram in $\\mathcal{F}$", "and the output of condition (S1) or (S2) for this diagram in $\\mathcal{F}$", "can be viewed as an output for $\\mathcal{F}_{x_0}$ as well." ], "refs": [ "formal-defos-definition-S1-S2" ], "ref_ids": [ 3523 ] } ], "ref_ids": [ 3545 ] }, { "id": 3443, "type": "theorem", "label": "formal-defos-lemma-lifting-section", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-lifting-section", "contents": [ "Let $p: \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in", "groupoids. Consider a diagram of $\\mathcal{F}$", "$$", "\\vcenter{", "\\xymatrix{", "y \\ar[r] \\ar[d]_a & x_\\epsilon \\ar[d]_e \\\\", "x \\ar[r]^d & x_0", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "A \\times_k k[\\epsilon] \\ar[r] \\ar[d] & k[\\epsilon] \\ar[d] \\\\", "A \\ar[r] & k.", "}", "}", "$$", "in $\\mathcal{C}_\\Lambda$. Assume $\\mathcal{F}$ satisfies (S2).", "Then there exists a morphism $s : x \\to y$ with $a \\circ s = \\text{id}_x$", "if and only if there exists a morphism $s_\\epsilon : x \\to x_\\epsilon$", "with $e \\circ s_\\epsilon = d$." ], "refs": [], "proofs": [ { "contents": [ "The ``only if'' direction is clear. Conversely, assume there exists a", "morphism $s_\\epsilon : x \\to x_\\epsilon$ with $e \\circ s_\\epsilon = d$.", "Note that $p(s_\\epsilon) : A \\to k[\\epsilon]$ is a ring map compatible", "with the map $A \\to k$. Hence we obtain", "$$", "\\sigma = (\\text{id}_A, p(s_\\epsilon)) : A \\to A \\times_k k[\\epsilon].", "$$", "Choose a pushforward $x \\to \\sigma_*x$. By construction we can factor", "$s_\\epsilon$ as $x \\to \\sigma_*x \\to x_\\epsilon$. Moreover, as $\\sigma$", "is a section of $A \\times_k k[\\epsilon] \\to A$, we get a morphism", "$\\sigma_*x \\to x$ such that $x \\to \\sigma_*x \\to x$ is $\\text{id}_x$.", "Because $e \\circ s_\\epsilon = d$ we find that the diagram", "$$", "\\xymatrix{", "\\sigma_*x \\ar[r] \\ar[d] & x_\\epsilon \\ar[d]_e \\\\", "x \\ar[r]^d & x_0", "}", "$$", "is commutative. Hence by (S2) we obtain a morphism $\\sigma_*x \\to y$", "such that $\\sigma_*x \\to y \\to x$ is the given map $\\sigma_*x \\to x$.", "The solution to the problem is now to take $a : x \\to y$ equal to", "the composition $x \\to \\sigma_*x \\to y$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3444, "type": "theorem", "label": "formal-defos-lemma-lifting-along-small-extension", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-lifting-along-small-extension", "contents": [ "Consider a commutative diagram in a predeformation category $\\mathcal{F}$", "$$", "\\vcenter{", "\\xymatrix{", "y \\ar[r] \\ar[d] & x_2 \\ar[d]^{a_2} \\\\", "x_1 \\ar[r]^{a_1} & x", "}", "}", "\\quad\\text{lying over}", "\\vcenter{", "\\xymatrix{", "A_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d]^{f_2} \\\\", "A_1 \\ar[r]^{f_1} & A", "}", "}", "$$", "in $\\mathcal{C}_\\Lambda$ where", "$f_2 : A_2 \\to A$ is a small extension.", "Assume there is a map $h : A_1 \\to A_2$ such that $f_2 = f_1 \\circ h$.", "Let $I = \\Ker(f_2)$. Consider the ring map", "$$", "g : A_1 \\times_A A_2 \\longrightarrow k[I] = k \\oplus I, \\quad", "(u, v) \\longmapsto \\overline{u} \\oplus (v - h(u))", "$$", "Choose a pushforward $y \\to g_*y$. Assume $\\mathcal{F}$ satisfies (S2).", "If there exists a morphism $x_1 \\to g_*y$, then there exists a", "morphism $b: x_1 \\to x_2$ such that $a_1 = a_2 \\circ b$." ], "refs": [], "proofs": [ { "contents": [ "Note that", "$\\text{id}_{A_1} \\times g : A_1 \\times_A A_2 \\to A_1 \\times_k k[I]$", "is an isomorphism and that $k[I] \\cong k[\\epsilon]$. Hence we have a diagram", "$$", "\\vcenter{", "\\xymatrix{", "y \\ar[r] \\ar[d] & g_*y \\ar[d] \\\\", "x_1 \\ar[r] & x_0", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "A_1 \\times_k k[\\epsilon] \\ar[r] \\ar[d] & k[\\epsilon] \\ar[d] \\\\", "A_1 \\ar[r] & k.", "}", "}", "$$", "where $x_0$ is an object of $\\mathcal{F}$ lying over $k$ (every object", "of $\\mathcal{F}$ has a unique morphism to $x_0$, see", "discussion following Definition \\ref{definition-predeformation-category}).", "If we have a morphism $x_1 \\to g_*y$ then", "Lemma \\ref{lemma-lifting-section}", "provides us with a section $s : x_1 \\to y$ of the map $y \\to x_1$.", "Composing this with the map $y \\to x_2$ we obtain $b : x_1 \\to x_2$", "which has the property that $a_1 = a_2 \\circ b$ because", "the diagram of the lemma commutes and because $s$ is a section." ], "refs": [ "formal-defos-definition-predeformation-category" ], "ref_ids": [ 3517 ] } ], "ref_ids": [] }, { "id": 3445, "type": "theorem", "label": "formal-defos-lemma-linear-functor", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-linear-functor", "contents": [ "Let $L: \\text{Mod}^{fg}_R \\to \\textit{Sets}$,", "resp.\\ $L: \\text{Mod}_R \\to \\textit{Sets}$ be a", "functor. Suppose $L(0)$ is a one element set and $L$ preserves finite", "products. Then there exists a unique $R$-linear functor", "$\\widetilde{L} : \\text{Mod}^{fg}_R \\to \\text{Mod}_R$,", "resp.\\ $\\widetilde{L} : \\text{Mod}^{fg}_R \\to \\text{Mod}_R$,", "such that", "$$", "\\vcenter{", "\\xymatrix{", "& \\text{Mod}_R \\ar[dr]^{\\text{forget}} & \\\\", "\\text{Mod}^{fg}_R \\ar[ur]^{\\widetilde{L}} \\ar[rr]^{L} & &", "\\textit{Sets}", "}", "}", "\\quad\\text{resp.}\\quad", "\\vcenter{", "\\xymatrix{", "& \\text{Mod}_R \\ar[dr]^{\\text{forget}} & \\\\", "\\text{Mod}_R \\ar[ur]^{\\widetilde{L}} \\ar[rr]^{L} & &", "\\textit{Sets}", "}", "}", "$$", "commutes." ], "refs": [], "proofs": [ { "contents": [ "We only prove this in case $L: \\text{Mod}^{fg}_R \\to \\textit{Sets}$.", "Let $M$ be a finitely generated $R$-module. We define $\\widetilde{L}(M)$ to be", "the set $L(M)$ with the following $R$-module structure.", "\\medskip\\noindent", "Multiplication: If $r \\in R$, multiplication by $r$ on $L(M)$ is defined to be", "the map $L(M) \\to L(M)$ induced by the multiplication map", "$r \\cdot : M \\to M$.", "\\medskip\\noindent", "Addition: The sum map $M \\times M \\to M: (m_1, m_2) \\mapsto m_1 + m_2$", "induces a map $L(M \\times M) \\to L(M)$. By assumption $L(M \\times M)$", "is canonically isomorphic to $L(M) \\times L(M)$. Addition on $L(M)$ is defined", "by the map $L(M) \\times L(M) \\cong L(M \\times M) \\to L(M)$.", "\\medskip\\noindent", "Zero: There is a unique map $0 \\to M$. The zero element of $L(M)$ is", "the image of $L(0) \\to L(M)$.", "\\medskip\\noindent", "We omit the verification that this defines an $R$-module $\\widetilde{L}(M)$,", "the unique such that is $R$-linearly functorial in $M$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3446, "type": "theorem", "label": "formal-defos-lemma-morphism-linear-functors", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-morphism-linear-functors", "contents": [ "Let $L_1, L_2: \\text{Mod}^{fg}_R \\to \\textit{Sets}$ be", "functors that take $0$ to a one element set and preserve finite products.", "Let $t : L_1 \\to L_2$ be a morphism of functors. Then $t$ induces a morphism", "$\\widetilde{t} : \\widetilde{L}_1 \\to \\widetilde{L}_2$ between the", "functors guaranteed by Lemma \\ref{lemma-linear-functor}, which is given simply", "by $\\widetilde{t}_M = t_M: \\widetilde{L}_1(M) \\to \\widetilde{L}_2(M)$", "for each $M \\in \\Ob(\\text{Mod}^{fg}_R)$. In other words,", "$t_M: \\widetilde{L}_1(M) \\to \\widetilde{L}_2(M)$ is a map of $R$-modules." ], "refs": [ "formal-defos-lemma-linear-functor" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 3445 ] }, { "id": 3447, "type": "theorem", "label": "formal-defos-lemma-linear-functor-over-field", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-linear-functor-over-field", "contents": [ "Let $K$ be a field. Let $L: \\text{Mod}^{fg}_K \\to", "\\text{Mod}_K$ be a $K$-linear functor. Then $L$ is isomorphic to the", "functor $L(K) \\otimes_K - : \\text{Mod}^{fg}_K \\to", "\\text{Mod}_K$." ], "refs": [], "proofs": [ { "contents": [ "For $V \\in \\Ob(\\text{Mod}^{fg}_K)$, the isomorphism", "$L(K) \\otimes_K V \\to L(V)$ is given on pure tensors by", "$x \\otimes v \\mapsto L(f_v)(x)$, where $f_v: K \\to V$ is the $K$-linear map", "sending $1 \\mapsto v$. When $V = K$, this is the isomorphism", "$L(K) \\otimes_K K \\to L(K)$ given by multiplication by $K$.", "For general $V$, it is an isomorphism by the case $V = K$ and the", "fact that $L$ commutes with finite", "products (Remark \\ref{remark-linear-functor})." ], "refs": [ "formal-defos-remark-linear-functor" ], "ref_ids": [ 3563 ] } ], "ref_ids": [] }, { "id": 3448, "type": "theorem", "label": "formal-defos-lemma-preserves-products", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-preserves-products", "contents": [ "Let $R$ be an $S$-algebra. Then the functor", "$\\text{Mod}_R \\to S\\text{-Alg}/R$ described above preserves finite products." ], "refs": [], "proofs": [ { "contents": [ "This is merely the statement that if $M$ and $N$ are $R$-modules, then the map", "$R[M \\times N] \\to R[M] \\times_R R[N]$ is an isomorphism in", "$S\\text{-Alg}/R$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3449, "type": "theorem", "label": "formal-defos-lemma-tangent-space-functor", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-tangent-space-functor", "contents": [ "Let $R$ be an $S$-algebra, and let $\\mathcal{C}$ be a strictly full", "subcategory of $S\\text{-Alg}/R$ containing $R[M]$ for all", "$M \\in \\Ob(\\text{Mod}^{fg}_R)$.", "Let $F: \\mathcal{C} \\to \\textit{Sets}$ be a functor. Suppose that", "$F(R)$ is a one element set and that for any $M, N \\in", "\\Ob(\\text{Mod}^{fg}_R)$, the induced map", "$$", "F(R[M] \\times_R R[N]) \\to F(R[M]) \\times F(R[N])", "$$", "is a bijection. Then $F(R[M])$ has a natural $R$-module structure for any $M", "\\in \\Ob(\\text{Mod}^{fg}_R)$." ], "refs": [], "proofs": [ { "contents": [ "Note that $R \\cong R[0]$ and $R[M] \\times_R R[N] \\cong R[M \\times N]$ hence", "$R$ and $R[M] \\times_R R[N]$ are objects of $\\mathcal{C}$ by our assumptions on", "$\\mathcal{C}$. Thus the conditions on $F$ make sense.", "The functor $\\text{Mod}_R \\to S\\text{-Alg}/R$ of", "Lemma \\ref{lemma-preserves-products}", "restricts to a functor $\\text{Mod}^{fg}_R \\to \\mathcal{C}$", "by the assumption on $\\mathcal{C}$. Let $L$ be the composition", "$\\text{Mod}^{fg}_R \\to \\mathcal{C} \\to \\textit{Sets}$, i.e.,", "$L(M) = F(R[M])$.", "Then $L$ preserves finite products by", "Lemma \\ref{lemma-preserves-products}", "and the assumption on $F$. Hence", "Lemma \\ref{lemma-linear-functor}", "shows that $L(M) = F(R[M])$ has a natural $R$-module structure for any", "$M \\in \\Ob(\\text{Mod}^{fg}_R)$." ], "refs": [ "formal-defos-lemma-preserves-products", "formal-defos-lemma-preserves-products", "formal-defos-lemma-linear-functor" ], "ref_ids": [ 3448, 3448, 3445 ] } ], "ref_ids": [] }, { "id": 3450, "type": "theorem", "label": "formal-defos-lemma-morphism-tangent-spaces", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-morphism-tangent-spaces", "contents": [ "Let $F, G: \\mathcal{C} \\to \\textit{Sets}$ be functors satisfying", "the hypotheses of", "Lemma \\ref{lemma-tangent-space-functor}.", "Let $t : F \\to G$ be a morphism of functors. For any", "$M \\in \\Ob(\\text{Mod}^{fg}_R)$, the map", "$t_{R[M]}: F(R[M]) \\to G(R[M])$ is a map of $R$-modules, where", "$F(R[M])$ and $G(R[M])$ are given the $R$-module structure from", "Lemma \\ref{lemma-tangent-space-functor}.", "In particular, $t_{R[\\epsilon]} : TF \\to TG$ is a map of $R$-modules." ], "refs": [ "formal-defos-lemma-tangent-space-functor", "formal-defos-lemma-tangent-space-functor" ], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-morphism-linear-functors}." ], "refs": [ "formal-defos-lemma-morphism-linear-functors" ], "ref_ids": [ 3446 ] } ], "ref_ids": [ 3449, 3449 ] }, { "id": 3451, "type": "theorem", "label": "formal-defos-lemma-tangent-space-tensor", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-tangent-space-tensor", "contents": [ "Let $F: \\mathcal{C} \\to \\textit{Sets}$ be a functor satisfying the", "hypotheses of", "Lemma \\ref{lemma-tangent-space-functor}.", "Assume $R = K$ is a field. Then $F(K[V]) \\cong TF \\otimes_K V$", "for any finite dimensional $K$-vector space $V$." ], "refs": [ "formal-defos-lemma-tangent-space-functor" ], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-linear-functor-over-field}." ], "refs": [ "formal-defos-lemma-linear-functor-over-field" ], "ref_ids": [ 3447 ] } ], "ref_ids": [ 3449 ] }, { "id": 3452, "type": "theorem", "label": "formal-defos-lemma-tangent-space-vector-space", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-tangent-space-vector-space", "contents": [ "Let $\\mathcal{F}$ be a predeformation category such that", "$\\overline{\\mathcal{F}}$ satisfies (S2)\\footnote{For example", "if $\\mathcal{F}$ satisfies (S2), see", "Lemma \\ref{lemma-S1-S2-associated-functor}.}. Then $T \\mathcal{F}$ has a", "natural $k$-vector space structure. For any finite dimensional", "vector space $V$ we have", "$\\overline{\\mathcal{F}}(k[V]) = T\\mathcal{F} \\otimes_k V$", "functorially in $V$." ], "refs": [ "formal-defos-lemma-S1-S2-associated-functor" ], "proofs": [ { "contents": [ "Let us write", "$F = \\overline{\\mathcal{F}} : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$.", "This is a predeformation functor and $F$ satisfies (S2). By", "Lemma \\ref{lemma-S2-extensions}", "(and the translation of", "Remark \\ref{remark-compare-S1-S2-schlessinger})", "we see that", "$$", "F(A \\times_k k[V]) \\longrightarrow F(A) \\times F(k[V])", "$$", "is a bijection for every finite dimensional vector space $V$ and every", "$A \\in \\Ob(\\mathcal{C}_\\Lambda)$. In particular, if $A = k[W]$", "then we see that $F(k[W] \\times_k k[V]) = F(k[W]) \\times F(k[V])$.", "In other words, the hypotheses of", "Lemma \\ref{lemma-tangent-space-functor}", "hold and we see that $TF = T \\mathcal{F}$", "has a natural $k$-vector space structure.", "The final assertion follows from", "Lemma \\ref{lemma-tangent-space-tensor}." ], "refs": [ "formal-defos-lemma-S2-extensions", "formal-defos-remark-compare-S1-S2-schlessinger", "formal-defos-lemma-tangent-space-functor", "formal-defos-lemma-tangent-space-tensor" ], "ref_ids": [ 3440, 3561, 3449, 3451 ] } ], "ref_ids": [ 3441 ] }, { "id": 3453, "type": "theorem", "label": "formal-defos-lemma-k-linear-differential", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-k-linear-differential", "contents": [ "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of predeformation", "categories. Assume $\\overline{\\mathcal{F}}$ and $\\overline{\\mathcal{G}}$ both", "satisfy (S2). Then $d \\varphi : T \\mathcal{F} \\to T \\mathcal{G}$ is $k$-linear." ], "refs": [], "proofs": [ { "contents": [ "In the proof of", "Lemma \\ref{lemma-tangent-space-vector-space}", "we have seen that $\\overline{\\mathcal{F}}$ and $\\overline{\\mathcal{G}}$", "satisfy the hypotheses of", "Lemma \\ref{lemma-tangent-space-functor}.", "Hence the lemma follows from", "Lemma \\ref{lemma-morphism-tangent-spaces}." ], "refs": [ "formal-defos-lemma-tangent-space-vector-space", "formal-defos-lemma-tangent-space-functor", "formal-defos-lemma-morphism-tangent-spaces" ], "ref_ids": [ 3452, 3449, 3450 ] } ], "ref_ids": [] }, { "id": 3454, "type": "theorem", "label": "formal-defos-lemma-action-linear", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-action-linear", "contents": [ "Let $\\mathcal{F}$ be a predeformation category over $\\mathcal{C}_\\Lambda$.", "If $\\overline{\\mathcal{F}}$ has (S2) then the maps $\\gamma_V$ are", "$k$-linear and we have $a_V(D, x) = x + \\gamma_V(D)$." ], "refs": [], "proofs": [ { "contents": [ "In the proof of", "Lemma \\ref{lemma-tangent-space-vector-space}", "we have seen that the functor $V \\mapsto \\overline{\\mathcal{F}}(k[V])$", "transforms $0$ to a singleton and products to products. The same is", "true of the functor $V \\mapsto \\text{Der}_\\Lambda(k, V)$.", "Hence $\\gamma_V$ is linear by", "Lemma \\ref{lemma-morphism-linear-functors}.", "Let $D : k \\to V$ be a $\\Lambda$-derivation.", "Set $D_1 : k \\to V^{\\oplus 2}$ equal to $a \\mapsto (D(a), 0)$.", "Then", "$$", "\\xymatrix{", "k[V \\times V] \\ar[r]_{+} \\ar[d]^{f_{1, D_1}} & k[V] \\ar[d]^{f_{1, D}} \\\\", "k[V \\times V] \\ar[r]^{+} & k[V]", "}", "$$", "commutes. Unwinding the definitions and using that", "$\\overline{F}(V \\times V) = \\overline{F}(V) \\times \\overline{F}(V)$", "this means that $a_D(x_1) + x_2 = a_D(x_1 + x_2)$ for all", "$x_1, x_2 \\in \\overline{F}(V)$. Thus it suffices to show that", "$a_V(D, 0) = 0 + \\gamma_V(D)$ where $0 \\in \\overline{F}(V)$ is", "the zero vector. By definition this is the element $f_{0, *}(x_0)$.", "Since $f_D = f_{1, D} \\circ f_0$ the desired result follows." ], "refs": [ "formal-defos-lemma-tangent-space-vector-space", "formal-defos-lemma-morphism-linear-functors" ], "ref_ids": [ 3452, 3446 ] } ], "ref_ids": [] }, { "id": 3455, "type": "theorem", "label": "formal-defos-lemma-versal-object-S1", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-versal-object-S1", "contents": [ "Let $\\mathcal{F}$ be a predeformation category.", "Assume $\\mathcal{F}$ has a versal formal object.", "Then $\\mathcal{F}$ satisfies (S1)." ], "refs": [], "proofs": [ { "contents": [ "Let $\\xi$ be a versal formal object of $\\mathcal{F}$. Let", "$$", "\\xymatrix{", " & x_2 \\ar[d] \\\\", "x_1 \\ar[r] & x", "}", "$$", "be a diagram in $\\mathcal{F}$ such that $x_2 \\to x$ lies over a", "surjective ring map. Since the natural morphism", "$\\widehat{\\mathcal{F}}|_{\\mathcal{C}_\\Lambda} \\xrightarrow{\\sim} \\mathcal{F}$", "is an equivalence (see", "Remark \\ref{remark-restrict-completion}), we can consider this", "diagram also as a diagram in $\\widehat{\\mathcal{F}}$. By", "Lemma \\ref{lemma-versal-object-quasi-initial} there exists a morphism", "$\\xi \\to x_1$, so by", "Remark \\ref{remark-versal-object} we also get a", "morphism $\\xi \\to x_2$ making the diagram", "$$", "\\xymatrix{", "\\xi \\ar[r] \\ar[d] & x_2 \\ar[d] \\\\", "x_1 \\ar[r] & x", "}", "$$", "commute. If $x_1 \\to x$ and $x_2 \\to x$ lie above ring maps", "$A_1 \\to A$ and $A_2 \\to A$ then taking the pushforward of", "$\\xi$ to $A_1 \\times_A A_2$ gives an object $y$ as required by (S1)." ], "refs": [ "formal-defos-remark-restrict-completion", "formal-defos-lemma-versal-object-quasi-initial", "formal-defos-remark-versal-object" ], "ref_ids": [ 3548, 3435, 3559 ] } ], "ref_ids": [] }, { "id": 3456, "type": "theorem", "label": "formal-defos-lemma-versal-criterion", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-versal-criterion", "contents": [ "Let $\\mathcal{F}$ be a predeformation category satisfying (S1) and", "(S2). Let $\\xi$ be a formal object of $\\mathcal{F}$ corresponding to", "$\\underline{\\xi} : \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$, see", "Remark \\ref{remark-formal-objects-yoneda}.", "Then $\\xi$ is versal if and only if the following two conditions hold:", "\\begin{enumerate}", "\\item the map", "$d\\underline{\\xi} : T\\underline{R}|_{\\mathcal{C}_\\Lambda} \\to T\\mathcal{F}$", "on tangent spaces is surjective, and", "\\item given a diagram in $\\widehat{\\mathcal{F}}$", "$$", "\\vcenter{", "\\xymatrix{", " & y \\ar[d] \\\\", "\\xi \\ar[r] & x", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", " & B \\ar[d]^{f} \\\\", "R \\ar[r] & A", "}", "}", "$$", "in $\\widehat{\\mathcal{C}}_\\Lambda$ with $B \\to A$ a small extension of", "Artinian rings, then there exists a ring map $R \\to B$ such that", "$$", "\\xymatrix{", " & B \\ar[d]^{f} \\\\", "R \\ar[ur] \\ar[r] & A", "}", "$$", "commutes.", "\\end{enumerate}" ], "refs": [ "formal-defos-remark-formal-objects-yoneda" ], "proofs": [ { "contents": [ "If $\\xi$ is versal then (1) holds by", "Lemma \\ref{lemma-smooth-morphism-essentially-surjective}", "and (2) holds by", "Remark \\ref{remark-versal-object}.", "Assume (1) and (2) hold. By", "Remark \\ref{remark-versal-object}", "we must show that given a diagram in $\\widehat{\\mathcal{F}}$ as in (2),", "there exists $\\xi \\to y$ such that", "$$", "\\xymatrix{", " & y \\ar[d] \\\\", "\\xi \\ar[ur] \\ar[r] & x", "}", "$$", "commutes. Let $b : R \\to B$ be the map guaranteed by (2). Denote", "$y' = b_*\\xi$ and choose a factorization $\\xi \\to y' \\to x$", "lying over $R \\to B \\to A$ of the given morphism $\\xi \\to x$.", "By (S1) we obtain a commutative diagram", "$$", "\\vcenter{", "\\xymatrix{", "z \\ar[r] \\ar[d] & y \\ar[d] \\\\", "y' \\ar[r] & x", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "B \\times_A B \\ar[d] \\ar[r] & B \\ar[d]^{f} \\\\", "B \\ar[r]^{f} & A .", "}", "}", "$$", "Set $I = \\Ker(f)$. Let $\\overline{g} : B \\times_A B \\to k[I]$", "be the ring map $(u, v) \\mapsto \\overline{u} \\oplus (v - u)$,", "cf.\\ Lemma \\ref{lemma-lifting-along-small-extension}.", "By (1) there exists a morphism $\\xi \\to \\overline{g}_*z$ which lies over a ring", "map $i : R \\to k[\\epsilon]$. Choose an Artinian quotient", "$b_1 : R \\to B_1$ such that both $b : R \\to B$ and $i : R \\to k[\\epsilon]$", "factor through $R \\to B_1$, i.e., giving", "$h : B_1 \\to B$ and $i' : B_1 \\to k[\\epsilon]$.", "Choose a pushforward $y_1 = b_{1, *}\\xi$, a factorization", "$\\xi \\to y_1 \\to y'$ lying over $R \\to B_1 \\to B$ of $\\xi \\to y'$, and a", "factorization $\\xi \\to y_1 \\to \\overline{g}_*z$ lying over", "$R \\to B_1 \\to k[\\epsilon]$ of $\\xi \\to \\overline{g}_*z$.", "Applying (S1) once more we obtain", "$$", "\\vcenter{", "\\xymatrix{", "z_1 \\ar[r] \\ar[d] & z \\ar[r] \\ar[d] & y \\ar[d] \\\\", "y_1 \\ar[r] & y' \\ar[r] & x", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "B_1 \\times_A B \\ar[d] \\ar[r] & B \\times_A B \\ar[r] \\ar[d] & B \\ar[d]^{f} \\\\", "B_1 \\ar[r] & B \\ar[r] & A .", "}", "}", "$$", "Note that the map $g : B_1 \\times_A B \\to k[I]$ of", "Lemma \\ref{lemma-lifting-along-small-extension}", "(defined using $h$)", "is the composition of $B_1 \\times_A B \\to B \\times_A B$ and the map", "$\\overline{g}$ above. By construction there exists a morphism", "$y_1 \\to g_*z_1 \\cong \\overline{g}_*z$! Hence", "Lemma \\ref{lemma-lifting-along-small-extension}", "applies (to the outer rectangles in the diagrams above)", "to give a morphism $y_1 \\to y$ and precomposing", "with $\\xi \\to y_1$ gives the desired morphism $\\xi \\to y$." ], "refs": [ "formal-defos-lemma-smooth-morphism-essentially-surjective", "formal-defos-remark-versal-object", "formal-defos-remark-versal-object", "formal-defos-lemma-lifting-along-small-extension", "formal-defos-lemma-lifting-along-small-extension", "formal-defos-lemma-lifting-along-small-extension" ], "ref_ids": [ 3434, 3559, 3559, 3444, 3444, 3444 ] } ], "ref_ids": [ 3553 ] }, { "id": 3457, "type": "theorem", "label": "formal-defos-lemma-largest-closed-where-lift", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-largest-closed-where-lift", "contents": [ "Let $\\mathcal{F}$ be a category cofibred in groupoids over", "$\\mathcal{C}_\\Lambda$ which has (S1). Let $B \\to A$ be a surjection", "in $\\mathcal{C}_\\Lambda$ with kernel $I$ annihilated by $\\mathfrak m_B$.", "Let $x \\in \\mathcal{F}(A)$. The set of ideals", "$$", "\\mathcal{J} = \\{ J \\subset I \\mid", "\\text{there exists an }y \\to x\\text{ lying over }B/J \\to A\\}", "$$", "has a smallest element." ], "refs": [], "proofs": [ { "contents": [ "Note that $\\mathcal{J}$ is nonempty as $I \\in \\mathcal{J}$.", "Also, if $J \\in \\mathcal{J}$ and $J \\subset J' \\subset I$ then", "$J' \\in \\mathcal{J}$ because we can pushforward the object $y$ to an", "object $y'$ over $B/J'$. Let $J$ and $K$ be elements of the displayed set.", "We claim that $J \\cap K \\in \\mathcal{J}$ which will prove the lemma.", "Since $I$ is a $k$-vector space we can find an ideal $J \\subset J' \\subset I$", "such that $J \\cap K = J' \\cap K$ and such that $J' + K = I$. By the above", "we may replace $J$ by $J'$ and assume that $J + K = I$. In this case", "$$", "A/(J \\cap K) = A/J \\times_{A/I} A/K.", "$$", "Hence the existence of an element $z \\in \\mathcal{F}(A/(J \\cap K))$", "mapping to $x$ follows, via (S1), from the existence of the elements we have", "assumed exist over $A/J$ and $A/K$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3458, "type": "theorem", "label": "formal-defos-lemma-versal-object-existence", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-versal-object-existence", "contents": [ "Let $\\mathcal{F}$ be a category cofibred in groupoids over", "$\\mathcal{C}_\\Lambda$. Assume the following conditions hold:", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a predeformation category.", "\\item $\\mathcal{F}$ satisfies (S1).", "\\item $\\mathcal{F}$ satisfies (S2).", "\\item $\\dim_k T\\mathcal{F}$ is finite.", "\\end{enumerate}", "Then $\\mathcal{F}$ has a versal formal object." ], "refs": [], "proofs": [ { "contents": [ "Assume (1), (2), (3), and (4) hold. Choose an object", "$R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$", "such that $\\underline{R}|_{\\mathcal{C}_\\Lambda}$ is smooth.", "See Lemma \\ref{lemma-exists-smooth}.", "Let $r = \\dim_k T\\mathcal{F}$ and put $S = R[[X_1, \\ldots, X_r]]$.", "\\medskip\\noindent", "We are going to inductively construct for $n \\geq 2$ pairs", "$(J_n, f_{n - 1} : \\xi_n \\to \\xi_{n - 1})$", "where $J_n \\subset S$ is an decreasing sequence of ideals and", "$f_{n - 1} : \\xi_n \\to \\xi_{n - 1}$ is a morphism of", "$\\mathcal{F}$ lying over the projection $S/J_n \\to S/J_{n - 1}$.", "\\medskip\\noindent", "Step 1. Let $J_1 = \\mathfrak m_S$. Let $\\xi_1$ be the unique", "(up to unique isomorphism) object of $\\mathcal{F}$ over", "$k = S/J_1 = S/\\mathfrak m_S$", "\\medskip\\noindent", "Step 2. Let", "$J_2 = \\mathfrak m_S^2 + \\mathfrak{m}_R S$. Then", "$S/J_2 = k[V]$ with $V = kX_1 \\oplus \\ldots \\oplus kX_r$", "By (S2) for $\\overline{\\mathcal{F}}$ we get a bijection", "$$", "\\overline{\\mathcal{F}}(S/J_2)", "\\longrightarrow", "T\\mathcal{F} \\otimes_k V,", "$$", "see", "Lemmas \\ref{lemma-S1-S2-associated-functor} and", "\\ref{lemma-tangent-space-vector-space}.", "Choose a basis $\\theta_1, \\ldots, \\theta_r$ for $T\\mathcal{F}$ and set", "$\\xi_2 = \\sum \\theta_i \\otimes X_i \\in \\Ob(\\mathcal{F}(S/J_2))$.", "The point of this choice is that", "$$", "d\\xi_2 :", "\\Mor_{\\mathcal{C}_\\Lambda}(S/J_2, k[\\epsilon])", "\\longrightarrow", "T\\mathcal{F}", "$$", "is surjective. Let $f_1 : \\xi_2 \\to \\xi_1$ be the unique morphism.", "\\medskip\\noindent", "Induction step. Assume $(J_n, f_{n - 1} : \\xi_n \\to \\xi_{n - 1})$ has been", "constructed for some $n \\geq 2$. There is a minimal element $J_{n + 1}$ of", "the set of ideals $J \\subset S$ satisfying:", "(a) $\\mathfrak m_S J_n \\subset J \\subset J_n$ and", "(b) there exists a morphism $\\xi_{n + 1} \\to \\xi_n$ lying over", "$S/J \\to S/J_n$, see", "Lemma \\ref{lemma-largest-closed-where-lift}.", "Let $f_n : \\xi_{n + 1} \\to \\xi_n$ be any morphism of $\\mathcal{F}$", "lying over $S/J_{n + 1} \\to S/J_n$.", "\\medskip\\noindent", "Set $J = \\bigcap J_n$. Set $\\overline{S} = S/J$. Set $\\overline{J}_n = J_n/J$.", "By", "Lemma \\ref{lemma-m-adic-topology}", "the sequence of ideals $(\\overline{J}_n)$ induces the", "$\\mathfrak m_{\\overline{S}}$-adic topology on $\\overline{S}$.", "Since $(\\xi_n, f_n)$ is an object of", "$\\widehat{\\mathcal{F}}_\\mathcal{I}(\\overline{S})$, where $\\mathcal{I}$", "is the filtration $(\\overline{J}_n)$ of $\\overline{S}$,", "we see that $(\\xi_n, f_n)$", "induces an object $\\xi$ of $\\widehat{\\mathcal{F}}(\\overline{S})$.", "see", "Lemma \\ref{lemma-formal-objects-different-filtration}.", "\\medskip\\noindent", "We prove $\\xi$ is versal. For versality it suffices to check", "conditions (1) and (2) of", "Lemma \\ref{lemma-versal-criterion}.", "Condition (1) follows from our choice of $\\xi_2$ in Step 2 above.", "Suppose given a diagram in $\\widehat{\\mathcal{F}}$", "$$", "\\vcenter{", "\\xymatrix{", " & y \\ar[d] \\\\", "\\eta \\ar[r] & x", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", " & B \\ar[d]^{f} \\\\", "\\overline{S} \\ar[r] & A", "}", "}", "$$", "in $\\widehat{\\mathcal{C}}_\\Lambda$ with $f: B \\to A$ a small extension", "of Artinian rings. We have to show there is a map $\\overline{S} \\to B$ fitting", "into the diagram on the right. Choose $n$ such that", "$\\overline{S} \\to A$ factors through $\\overline{S} \\to S/J_n$. This is", "possible as the sequence $(\\overline{J}_n)$ induces the", "$\\mathfrak m_{\\overline{S}}$-adic topology as we saw above.", "The pushforward of $\\xi$ along $\\overline{S} \\to S/J_n$ is $\\xi_n$.", "We may factor $\\xi \\to x$ as $\\xi \\to \\xi_n \\to x$ hence we get a diagram", "in $\\mathcal{F}$", "$$", "\\vcenter{", "\\xymatrix{", " & y \\ar[d] \\\\", "\\xi_n \\ar[r] & x", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", " & B \\ar[d]^{f} \\\\", "S/J_n \\ar[r] & A .", "}", "}", "$$", "To check condition (2) of", "Lemma \\ref{lemma-versal-criterion}", "it suffices to complete the diagram", "$$", "\\xymatrix{", "S/J_{n + 1} \\ar[d] \\ar@{-->}[r] & B \\ar[d]^{f} \\\\", "S/J_n \\ar[r] & A", "}", "$$", "or equivalently, to complete the diagram", "$$", "\\xymatrix{", " & S/J_n \\times_A B \\ar[d]^{p_1} \\\\", "S/J_{n + 1} \\ar@{-->}[ur] \\ar[r] & S/J_n.", "}", "$$", "If $p_1$ has a section we are done. If not, by", "Lemma \\ref{lemma-fiber-product-CLambda} (2)", "$p_1$ is a small extension, so by", "Lemma \\ref{lemma-essential-surjection} (4)", "$p_1$ is an essential surjection. Recall that $S = R[[X_1, \\ldots, X_r]]$", "and that we chose $R$ such that $\\underline{R}|_{\\mathcal{C}_\\Lambda}$", "is smooth. Hence there exists a map $h : R \\to B$ lifting the map", "$R \\to S \\to S/J_n \\to A$. By the universal property of a power series", "ring there is an $R$-algebra map $h : S = R[[X_1, \\ldots, X_2]] \\to B$", "lifting the given map $S \\to S/J_n \\to A$. This induces a map", "$g: S \\to S/J_n \\times_A B$ making the solid square in the diagram", "$$", "\\xymatrix{", "S \\ar[d] \\ar[r]_-g & S/J_n \\times_A B \\ar[d]^{p_1} \\\\", "S/J_{n + 1} \\ar@{-->}[ur] \\ar[r] & S/J_n", "}", "$$", "commute. Then $g$ is a surjection since $p_1$ is an essential surjection.", "We claim the ideal $K = \\Ker(g)$ of $S$ satisfies conditions (a) and", "(b) of the construction of $J_{n + 1}$ in the induction step above.", "Namely, $K \\subset J_n$ is clear and $\\mathfrak m_SJ_n \\subset K$ as $p_1$", "is a small extension; this proves (a). By (S1) applied to", "$$", "\\xymatrix{", " & y \\ar[d] \\\\", "\\xi_n \\ar[r] & x,", "}", "$$", "there exists a lifting of $\\xi_n$ to $S/K \\cong S/J_n \\times_A B$, so (b)", "holds. Since $J_{n + 1}$ was the minimal ideal with properties (a) and (b)", "this implies $J_{n + 1} \\subset K$. Thus the desired map", "$S/J_{n+1} \\to S/K \\cong S/J_n \\times_A B$ exists." ], "refs": [ "formal-defos-lemma-exists-smooth", "formal-defos-lemma-S1-S2-associated-functor", "formal-defos-lemma-tangent-space-vector-space", "formal-defos-lemma-largest-closed-where-lift", "formal-defos-lemma-m-adic-topology", "formal-defos-lemma-formal-objects-different-filtration", "formal-defos-lemma-versal-criterion", "formal-defos-lemma-versal-criterion", "formal-defos-lemma-fiber-product-CLambda", "formal-defos-lemma-essential-surjection" ], "ref_ids": [ 3438, 3441, 3452, 3457, 3426, 3430, 3456, 3456, 3417, 3420 ] } ], "ref_ids": [] }, { "id": 3459, "type": "theorem", "label": "formal-defos-lemma-smallest-where-descends", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-smallest-where-descends", "contents": [ "Let $\\mathcal{F}$ be a category cofibred in groupoids over", "$\\mathcal{C}_\\Lambda$ which has (S1).", "\\begin{enumerate}", "\\item For $y \\to x$ in $\\mathcal{F}$ a minimal", "object in $\\mathcal{S}_y$ maps to a minimal object of $\\mathcal{S}_x$.", "\\item For $y \\to x$ in $\\mathcal{F}$ lying over a surjection", "$f : B \\to A$ in $\\mathcal{C}_\\Lambda$ every minimal object", "of $\\mathcal{S}_x$ is the image of a minimal object of", "$\\mathcal{S}_y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Say $y \\to x$ lies over $f : B \\to A$. Let $y' \\to y$", "lying over $B' \\subset B$ be a minimal object of $\\mathcal{S}_y$. Let", "$$", "\\vcenter{", "\\xymatrix{", "y' \\ar[d] \\ar[r] & x' \\ar[d] \\\\", "y \\ar[r] & x", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "B' \\ar[d] \\ar[r] & f(B') \\ar[d] \\\\", "B \\ar[r] & A", "}", "}", "$$", "be as in the construction of $f_*$ above. Suppose that", "$(x'' \\to x) \\to (x' \\to x)$ is a morphism of $\\mathcal{S}_x$", "with $x'' \\to x'$ lying over $A'' \\subset f(B')$. By (S1)", "there exists $y'' \\to y'$ lying over $B' \\times_{f(B')} A'' \\to B'$.", "Since $y' \\to y$ is minimal we conclude that", "$B' \\times_{f(B')} A'' \\to B'$ is an isomorphism, which implies that", "$A'' = f(B')$, i.e., $x' \\to x$ is minimal.", "\\medskip\\noindent", "Proof of (2). Suppose $f : B \\to A$ is surjective and $y \\to x$ lies over $f$.", "Let $x' \\to x$ be a minimal object of $\\mathcal{S}_x$ lying over $A' \\subset A$.", "By (S1) there exists $y' \\to y$ lying over", "$B' = f^{-1}(A') = B \\times_A A' \\to B$ whose image in $\\mathcal{S}_x$ is", "$x' \\to x$. So $f_*(y' \\to y) = x' \\to x$.", "Choose a morphism $(y'' \\to y) \\to (y' \\to y)$ in", "$\\mathcal{S}_y$ with $y'' \\to y$ a minimal object (this is possible by", "the remark on lengths above the lemma). Then $f_*(y'' \\to y)$ is an", "object of $\\mathcal{S}_x$ which maps to $x' \\to x$ (by functoriality of", "$f_*$) hence is isomorphic to $x' \\to x$ by minimality of $x' \\to x$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3460, "type": "theorem", "label": "formal-defos-lemma-smallest-where-descends-versal", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-smallest-where-descends-versal", "contents": [ "Let $\\mathcal{F}$ be a category cofibred in groupoids over", "$\\mathcal{C}_\\Lambda$ which has (S1). Let $\\xi$ be a versal formal object", "of $\\mathcal{F}$ lying over $R$. There exists a morphism $\\xi' \\to \\xi$", "lying over $R' \\subset R$ with the following minimality properties", "\\begin{enumerate}", "\\item for every $f : R \\to A$ with $A \\in \\Ob(\\mathcal{C}_\\Lambda)$", "the pushforwards", "$$", "\\vcenter{", "\\xymatrix{", "\\xi' \\ar[d] \\ar[r] & x' \\ar[d] \\\\", "\\xi \\ar[r] & x", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "R' \\ar[d] \\ar[r] & f(R') \\ar[d] \\\\", "R \\ar[r] & A", "}", "}", "$$", "produce a minimal object $x' \\to x$ of $\\mathcal{S}_x$, and", "\\item for any morphism of formal objects $\\xi'' \\to \\xi'$", "the corresponding morphism $R'' \\to R'$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $\\xi = (R, \\xi_n, f_n)$. Set $R'_1 = k$ and", "$\\xi'_1 = \\xi_1$. Suppose that we have constructed", "minimal objects $\\xi'_m \\to \\xi_m$ of $\\mathcal{S}_{\\xi_m}$", "lying over $R'_m \\subset R/\\mathfrak m_R^m$ for $m \\leq n$", "and morphisms $f'_m : \\xi'_{m + 1} \\to \\xi'_m$ compatible with $f_m$", "for $m \\leq n - 1$. By", "Lemma \\ref{lemma-smallest-where-descends} (2)", "there exists a minimal object $\\xi'_{n + 1} \\to \\xi_{n + 1}$ lying over", "$R'_{n + 1} \\subset R/\\mathfrak m_R^{n + 1}$ whose image", "is $\\xi'_n \\to \\xi_n$ over $R'_n \\subset R/\\mathfrak m_R^n$.", "This produces the commutative diagram", "$$", "\\xymatrix{", "\\xi'_{n + 1} \\ar[r]_{f'_n} \\ar[d] & \\xi'_n \\ar[d] \\\\", "\\xi_{n + 1} \\ar[r]^{f_n} & \\xi_n", "}", "$$", "by construction. Moreover the ring map $R'_{n + 1} \\to R'_n$", "is surjective. Set $R' = \\lim_n R'_n$. Then $R' \\to R$ is injective.", "\\medskip\\noindent", "However, it isn't a priori clear that $R'$ is Noetherian. To prove this", "we use that $\\xi$ is versal. Namely, versality implies that there exists", "a morphism $\\xi \\to \\xi'_n$ in $\\widehat{\\mathcal{F}}$, see", "Lemma \\ref{lemma-versal-object-quasi-initial}.", "The corresponding map $R \\to R'_n$ has to be surjective (as", "$\\xi'_n \\to \\xi_n$ is minimal in $\\mathcal{S}_{\\xi_n}$).", "Thus the dimensions of the cotangent spaces are bounded and", "Lemma \\ref{lemma-limit-artinian}", "implies $R'$ is Noetherian, i.e., an object of $\\widehat{\\mathcal{C}}_\\Lambda$.", "By", "Lemma \\ref{lemma-formal-objects-different-filtration}", "(plus the result on filtrations of", "Lemma \\ref{lemma-limit-artinian})", "the sequence of elements $\\xi'_n$ defines a formal object $\\xi'$ over $R'$", "and we have a map $\\xi' \\to \\xi$.", "\\medskip\\noindent", "By construction (1) holds for $R \\to R/\\mathfrak m_R^n$ for each $n$.", "Since each $R \\to A$ as in (1) factors through $R \\to R/\\mathfrak m_R^n \\to A$", "we see that (1) for $x' \\to x$ over $f(R) \\subset A$ follows from the", "minimality of $\\xi'_n \\to \\xi_n$ over $R'_n \\to R/\\mathfrak m_R^n$ by", "Lemma \\ref{lemma-smallest-where-descends} (1).", "\\medskip\\noindent", "If $R'' \\to R'$ as in (2) is not surjective, then $R'' \\to R' \\to R'_n$", "would not be surjective for some $n$ and $\\xi'_n \\to \\xi_n$ wouldn't", "be minimal, a contradiction. This contradiction proves (2)." ], "refs": [ "formal-defos-lemma-smallest-where-descends", "formal-defos-lemma-versal-object-quasi-initial", "formal-defos-lemma-limit-artinian", "formal-defos-lemma-formal-objects-different-filtration", "formal-defos-lemma-limit-artinian", "formal-defos-lemma-smallest-where-descends" ], "ref_ids": [ 3459, 3435, 3427, 3430, 3427, 3459 ] } ], "ref_ids": [] }, { "id": 3461, "type": "theorem", "label": "formal-defos-lemma-descends-versal", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-descends-versal", "contents": [ "Let $\\mathcal{F}$ be a category cofibred in groupoids over", "$\\mathcal{C}_\\Lambda$ which has (S1). Let $\\xi$ be a versal formal object", "of $\\mathcal{F}$ lying over $R$. Let $\\xi' \\to \\xi$ be a morphism", "of formal objects lying over $R' \\subset R$ as constructed in", "Lemma \\ref{lemma-smallest-where-descends-versal}.", "Then", "$$", "R \\cong R'[[x_1, \\ldots, x_r]]", "$$", "is a power series ring over $R'$.", "Moreover, $\\xi'$ is a versal formal object too." ], "refs": [ "formal-defos-lemma-smallest-where-descends-versal" ], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-versal-object-quasi-initial}", "there exists a morphism $\\xi \\to \\xi'$. By", "Lemma \\ref{lemma-smallest-where-descends-versal}", "the corresponding map $f : R \\to R'$ induces a surjection", "$f|_{R'} : R' \\to R'$. This is an isomorphism by", "Algebra, Lemma \\ref{algebra-lemma-surjective-endo-noetherian-ring-is-iso}.", "Hence $I = \\Ker(f)$ is an ideal of $R$ such that $R = R' \\oplus I$.", "Let $x_1, \\ldots, x_n \\in I$ be elements which form a basis for", "$I/\\mathfrak m_RI$. Consider the map", "$S = R'[[X_1, \\ldots, X_r]] \\to R$ mapping $X_i$ to $x_i$.", "For every $n \\geq 1$ we get a surjection of Artinian $R'$-algebras", "$B = S/\\mathfrak m_S^n \\to R/\\mathfrak m_R^n = A$. Denote", "$y \\in \\Ob(\\mathcal{F}(B)$, resp.\\ $x \\in \\Ob(\\mathcal{F}(A))$", "the pushforward of $\\xi'$ along $R' \\to S \\to B$, resp.\\ $R' \\to S \\to A$.", "Note that $x$ is also the pushforward of $\\xi$ along $R \\to A$ as", "$\\xi$ is the pushforward of $\\xi'$ along $R' \\to R$.", "Thus we have a solid diagram", "$$", "\\vcenter{", "\\xymatrix{", "& y \\ar[d] \\\\", "\\xi \\ar[r] \\ar@{..>}[ru] & x", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "& S/\\mathfrak m_S^n \\ar[d] \\\\", "R \\ar[r] \\ar@{..>}[ru] & R/\\mathfrak m_R^n", "}", "}", "$$", "Because $\\xi$ is versal, using", "Remark \\ref{remark-versal-object}", "we obtain the dotted arrows fitting into these diagrams.", "In particular, the maps $S/\\mathfrak m_S^n \\to R/\\mathfrak m_R^n$", "have sections $h_n : R/\\mathfrak m_R^n \\to S/\\mathfrak m_S^n$.", "It follows from", "Lemma \\ref{lemma-power-series}", "that $S \\to R$ is an isomorphism.", "\\medskip\\noindent", "As $\\xi$ is a pushforward of $\\xi'$ along $R' \\to R$ we obtain from", "Remark \\ref{remark-formal-objects-yoneda-map}", "a commutative diagram", "$$", "\\xymatrix{", "\\underline{R}|_{\\mathcal{C}_\\Lambda} \\ar[rr] \\ar[rd]_{\\underline{\\xi}} & &", "\\underline{R'}|_{\\mathcal{C}_\\Lambda} \\ar[ld]^{\\underline{\\xi'}} \\\\", "& \\mathcal{F}", "}", "$$", "Since $R' \\to R$ has a left inverse (namely $R \\to R/I = R'$) we see that", "$\\underline{R}|_{\\mathcal{C}_\\Lambda} \\to", "\\underline{R'}|_{\\mathcal{C}_\\Lambda}$ is essentially surjective.", "Hence by", "Lemma \\ref{lemma-smooth-properties}", "we see that $\\underline{\\xi'}$ is smooth, i.e., $\\xi'$ is a versal", "formal object." ], "refs": [ "formal-defos-lemma-versal-object-quasi-initial", "formal-defos-lemma-smallest-where-descends-versal", "algebra-lemma-surjective-endo-noetherian-ring-is-iso", "formal-defos-remark-versal-object", "formal-defos-lemma-power-series", "formal-defos-remark-formal-objects-yoneda-map", "formal-defos-lemma-smooth-properties" ], "ref_ids": [ 3435, 3460, 457, 3559, 3428, 3554, 3433 ] } ], "ref_ids": [ 3460 ] }, { "id": 3462, "type": "theorem", "label": "formal-defos-lemma-minimal-versal", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-minimal-versal", "contents": [ "Let $\\mathcal{F}$ be a predeformation category which", "has a versal formal object. Then", "\\begin{enumerate}", "\\item $\\mathcal{F}$ has a minimal versal formal object,", "\\item minimal versal objects are unique up to isomorphism, and", "\\item any versal object is the pushforward of a minimal versal", "object along a power series ring extension.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose $\\mathcal{F}$ has a versal formal object $\\xi$ over $R$.", "Then it satisfies (S1), see", "Lemma \\ref{lemma-versal-object-S1}.", "Let $\\xi' \\to \\xi$ over $R' \\subset R$ be any of the morphisms constructed in", "Lemma \\ref{lemma-smallest-where-descends-versal}.", "By", "Lemma \\ref{lemma-descends-versal}", "we see that $\\xi'$ is versal, hence it is a minimal versal formal", "object (by construction). This proves (1).", "Also, $R \\cong R'[[x_1, \\ldots, x_n]]$ which proves (3).", "\\medskip\\noindent", "Suppose that $\\xi_i/R_i$ are two minimal versal formal objects. By", "Lemma \\ref{lemma-versal-object-quasi-initial}", "there exist morphisms $\\xi_1 \\to \\xi_2$ and $\\xi_2 \\to \\xi_1$.", "The corresponding ring maps $f : R_1 \\to R_2$ and $g : R_2 \\to R_1$", "are surjective by minimality. Hence the compositions", "$g \\circ f : R_1 \\to R_1$ and $f \\circ g : R_2 \\to R_2$ are", "isomorphisms by", "Algebra, Lemma \\ref{algebra-lemma-surjective-endo-noetherian-ring-is-iso}.", "Thus $f$ and $g$ are isomorphisms whence the maps", "$\\xi_1 \\to \\xi_2$ and $\\xi_2 \\to \\xi_1$ are isomorphisms", "(because $\\widehat{\\mathcal{F}}$ is cofibred in groupoids by", "Lemma \\ref{lemma-completion-cofibred}). This proves (2) and", "finishes the proof of the lemma." ], "refs": [ "formal-defos-lemma-versal-object-S1", "formal-defos-lemma-smallest-where-descends-versal", "formal-defos-lemma-descends-versal", "formal-defos-lemma-versal-object-quasi-initial", "algebra-lemma-surjective-endo-noetherian-ring-is-iso", "formal-defos-lemma-completion-cofibred" ], "ref_ids": [ 3455, 3460, 3461, 3435, 457, 3429 ] } ], "ref_ids": [] }, { "id": 3463, "type": "theorem", "label": "formal-defos-lemma-miniversal-object-unique", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-miniversal-object-unique", "contents": [ "Let $\\mathcal{F}$ be a predeformation category.", "Let $\\xi$ be a versal formal object of $\\mathcal{F}$ such that", "(\\ref{equation-bijective-orbits}) holds.", "Then $\\xi$ is a minimal versal formal object.", "In particular, such $\\xi$ are unique up to isomorphism." ], "refs": [], "proofs": [ { "contents": [ "If $\\xi$ is not minimal, then there exists a morphism", "$\\xi' \\to \\xi$ lying over $R' \\to R$ such that", "$R = R'[[x_1, \\ldots, x_n]]$ with $n > 0$, see", "Lemma \\ref{lemma-minimal-versal}.", "Thus $d\\underline{\\xi}$ factors as", "$$", "\\text{Der}_\\Lambda(R, k) \\to", "\\text{Der}_\\Lambda(R', k) \\to T\\mathcal{F}", "$$", "and we see that (\\ref{equation-bijective-orbits}) cannot hold", "because $D : f \\mapsto \\partial/\\partial x_1(f) \\bmod \\mathfrak m_R$", "is an element of the kernel of the first arrow which is not in the image of", "$\\text{Der}_\\Lambda(k, k) \\to \\text{Der}_\\Lambda(R, k)$." ], "refs": [ "formal-defos-lemma-minimal-versal" ], "ref_ids": [ 3462 ] } ], "ref_ids": [] }, { "id": 3464, "type": "theorem", "label": "formal-defos-lemma-miniversal-object-existence-1", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-miniversal-object-existence-1", "contents": [ "Let $\\mathcal{F}$ be a predeformation category.", "Let $\\xi$ be a versal formal object of $\\mathcal{F}$ such that", "(\\ref{equation-bijective}) holds. Then", "\\begin{enumerate}", "\\item $\\mathcal{F}$ satisfies (S1).", "\\item $\\mathcal{F}$ satisfies (S2).", "\\item $\\dim_k T\\mathcal{F}$ is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Condition (S1) holds by", "Lemma \\ref{lemma-versal-object-S1}.", "The first part of (S2) holds since (S1) holds. Let", "$$", "\\vcenter{", "\\xymatrix{", "y \\ar[r]_c \\ar[d]_a & x_\\epsilon \\ar[d]^e \\\\", "x \\ar[r]^d & x_0", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "y' \\ar[r]_{c'} \\ar[d]_{a'} & x_\\epsilon \\ar[d]^e \\\\", "x \\ar[r]^d & x_0", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "A \\times_k k[\\epsilon] \\ar[r] \\ar[d] & k[\\epsilon] \\ar[d] \\\\", "A \\ar[r] & k", "}", "}", "$$", "be diagrams as in the second part of (S2). As above we can find", "morphisms $b : \\xi \\to y$ and $b' : \\xi \\to y'$ such that", "$$", "\\xymatrix{", "\\xi \\ar[r]^{b'} \\ar[d]_b & y' \\ar[d]^{a'} \\\\", "y \\ar[r]^{a} & x", "}", "$$", "commutes. Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ denote the", "structure morphism. Say $\\widehat{p}(\\xi) = R$, i.e., $\\xi$ lies over", "$R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$. We see that the", "pushforward of $\\xi$ via $p(c) \\circ p(b)$ is $x_\\epsilon$", "and that the pushforward of $\\xi$ via $p(c') \\circ p(b')$ is $x_\\epsilon$.", "Since $\\xi$ satisfies (\\ref{equation-bijective}), we see that", "$p(c) \\circ p(b) = p(c') \\circ p(b')$", "as maps $R \\to k[\\epsilon]$. Hence $p(b) = p(b')$ as maps from", "$R \\to A \\times_k k[\\epsilon]$. Thus we see that $y$ and $y'$ are", "isomorphic to the pushforward of $\\xi$ along this map and we get", "a unique morphism $y \\to y'$ over $A \\times_k k[\\epsilon]$", "compatible with $b$ and $b'$ as desired.", "\\medskip\\noindent", "Finally, by", "Example \\ref{example-tangent-space-prorepresentable-functor}", "we see", "$\\dim_k T\\mathcal{F} = \\dim_k T\\underline{R}|_{\\mathcal{C}_\\Lambda}$", "is finite." ], "refs": [ "formal-defos-lemma-versal-object-S1" ], "ref_ids": [ 3455 ] } ], "ref_ids": [] }, { "id": 3465, "type": "theorem", "label": "formal-defos-lemma-construct-bijective-orbits", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-construct-bijective-orbits", "contents": [ "Let $\\mathcal{F}$ be a predeformation category satisfying", "(S2) which has a versal formal object. Then its minimal versal", "formal object satisfies (\\ref{equation-bijective-orbits})." ], "refs": [], "proofs": [ { "contents": [ "Let $\\xi$ be a minimal versal formal object for $\\mathcal{F}$, see", "Lemma \\ref{lemma-minimal-versal}.", "Say $\\xi$ lies over $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$.", "In order to parse (\\ref{equation-bijective-orbits}) we point out", "that $T\\mathcal{F}$ has a natural $k$-vector space structure", "(see", "Lemma \\ref{lemma-tangent-space-vector-space}),", "that $d\\underline{\\xi} : \\text{Der}_\\Lambda(R, k) \\to T\\mathcal{F}$", "is linear (see", "Lemma \\ref{lemma-k-linear-differential}),", "and that the action of $\\text{Der}_\\Lambda(k, k)$ is", "given by addition (see", "Lemma \\ref{lemma-action-linear}).", "Consider the diagram", "$$", "\\xymatrix{", "& \\Hom_k(\\mathfrak m_R/\\mathfrak m_R^2, k) \\\\", "K \\ar[r] & \\text{Der}_\\Lambda(R, k) \\ar[r]^{d\\underline{\\xi}} \\ar[u] &", "T\\mathcal{F} \\\\", "& \\text{Der}_\\Lambda(k, k) \\ar[u] \\ar[ru]", "}", "$$", "The vector space $K$ is the kernel of $d\\underline{\\xi}$.", "Note that the middle column is exact in the middle as it is dual to the", "sequence (\\ref{equation-sequence}). If (\\ref{equation-bijective-orbits})", "fails, then we can find a nonzero element $D \\in K$ which", "does not map to zero in $\\Hom_k(\\mathfrak m_R/\\mathfrak m_R^2, k)$.", "This means there exists an $t \\in \\mathfrak m_R$ such that", "$D(t) = 1$. Set $R' = \\{a \\in R \\mid D(a) = 0\\}$. As $D$ is a derivation", "this is a subring of $R$. Since $D(t) = 1$ we see that $R' \\to k$", "is surjective (compare with the proof of", "Lemma \\ref{lemma-essential-surjection}).", "Note that $\\mathfrak m_{R'} = \\Ker(D : \\mathfrak m_R \\to k)$", "is an ideal of $R$ and $\\mathfrak m_R^2 \\subset \\mathfrak m_{R'}$. Hence", "$$", "\\mathfrak m_R/\\mathfrak m_R^2 =", "\\mathfrak m_{R'}/\\mathfrak m_R^2 + k\\overline{t}", "$$", "which implies that the map", "$$", "R'/\\mathfrak m_R^2 \\times_k k[\\epsilon] \\to R/\\mathfrak m_R^2", "$$", "sending $\\epsilon$ to $\\overline{t}$ is an isomorphism. In particular", "there is a map $R/\\mathfrak m_R^2 \\to R'/\\mathfrak m_R^2$.", "\\medskip\\noindent", "Let $\\xi \\to y$ be a morphism lying over $R \\to R/\\mathfrak m_R^2$.", "Let $y \\to x$ be a morphism lying over", "$R/\\mathfrak m_R^2 \\to R'/\\mathfrak m_R^2$.", "Let $y \\to x_\\epsilon$ be a morphism lying over", "$R/\\mathfrak m_R^2 \\to k[\\epsilon]$. Let $x_0$ be the unique (up to unique", "isomorphism) object of $\\mathcal{F}$ over $k$.", "By the axioms of a category cofibred in groupoids we obtain", "a commutative diagram", "$$", "\\vcenter{", "\\xymatrix{", "y \\ar[r] \\ar[d] & x_\\epsilon \\ar[d] \\\\", "x \\ar[r] & x_0", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "R'/\\mathfrak m_R^2 \\times_k k[\\epsilon] \\ar[r] \\ar[d] & k[\\epsilon] \\ar[d] \\\\", "R'/\\mathfrak m_R^2 \\ar[r] & k.", "}", "}", "$$", "Because $D \\in K$ we see that $x_\\epsilon$ is isomorphic", "to $0 \\in \\mathcal{F}(k[\\epsilon])$, i.e., $x_\\epsilon$ is", "the pushforward of $x_0$ via $k \\to k[\\epsilon], a \\mapsto a$.", "Hence by", "Lemma \\ref{lemma-lifting-section}", "we see that there exists a morphism $x \\to y$. Since", "$\\text{length}_\\Lambda(R'/\\mathfrak m_R^2) <", "\\text{length}_\\Lambda(R/\\mathfrak m_R^2)$", "the corresponding ring map $R'/\\mathfrak m_R^2 \\to R/\\mathfrak m_R^2$", "is not surjective. This contradicts the minimality of", "$\\xi/R$, see part (1) of", "Lemma \\ref{lemma-smallest-where-descends-versal}.", "This contradiction shows that such a $D$ cannot exist, hence", "we win." ], "refs": [ "formal-defos-lemma-minimal-versal", "formal-defos-lemma-tangent-space-vector-space", "formal-defos-lemma-k-linear-differential", "formal-defos-lemma-action-linear", "formal-defos-lemma-essential-surjection", "formal-defos-lemma-smallest-where-descends-versal" ], "ref_ids": [ 3462, 3452, 3453, 3454, 3420, 3460 ] } ], "ref_ids": [] }, { "id": 3466, "type": "theorem", "label": "formal-defos-lemma-RS-fiber-square", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-RS-fiber-square", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$ satisfying (RS). Given a commutative diagram", "in $\\mathcal{F}$", "$$", "\\vcenter{", "\\xymatrix{", "y \\ar[r] \\ar[d] & x_2 \\ar[d] \\\\", "x_1 \\ar[r] & x", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "A_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d] \\\\", "A_1 \\ar[r] & A.", "}", "}", "$$", "with $A_2 \\to A$ surjective, then it is a fiber square." ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathcal{F}$ satisfies (RS), there exists a fiber product diagram", "$$", "\\vcenter{", "\\xymatrix{", "x_1 \\times_x x_2 \\ar[r] \\ar[d] & x_2 \\ar[d] \\\\", "x_1 \\ar[r] & x", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "A_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d] \\\\", "A_1 \\ar[r] & A.", "}", "}", "$$", "The induced map $y \\to x_1 \\times_x x_2$ lies over", "$\\text{id} : A_1 \\times_A A_1 \\to A_1 \\times_A A_1$, hence it is an", "isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3467, "type": "theorem", "label": "formal-defos-lemma-RS-small-extension", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-RS-small-extension", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal", "C_\\Lambda$. Then $\\mathcal{F}$ satisfies (RS) if the condition in", "Definition \\ref{definition-RS} is assumed to hold only when $A_2 \\to A$", "is a small extension." ], "refs": [ "formal-defos-definition-RS" ], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-factor-small-extension}. The proof is similar to that", "of Lemma \\ref{lemma-smoothness-small-extensions}." ], "refs": [ "formal-defos-lemma-factor-small-extension", "formal-defos-lemma-smoothness-small-extensions" ], "ref_ids": [ 3414, 3431 ] } ], "ref_ids": [ 3529 ] }, { "id": 3468, "type": "theorem", "label": "formal-defos-lemma-RS-2-categorical", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-RS-2-categorical", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ satisfies (RS),", "\\item the functor", "$\\mathcal{F}(A_1 \\times_A A_2) \\to", "\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$", "see (\\ref{equation-compare}) is an equivalence of", "categories whenever $A_2 \\to A$ is surjective, and", "\\item same as in (2) whenever $A_2 \\to A$ is a small extension.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). By", "Lemma \\ref{lemma-RS-fiber-square}", "we see that every object of $\\mathcal{F}(A_1 \\times_A A_2)$", "is of the form $x_1 \\times_x x_2$. Moreover", "$$", "\\Mor_{A_1 \\times_A A_2}(x_1 \\times_x x_2, y_1 \\times_y y_2) =", "\\Mor_{A_1}(x_1, y_1) \\times_{\\Mor_A(x, y)}", "\\Mor_{A_2}(x_2, y_2).", "$$", "Hence we see that $\\mathcal{F}(A_1 \\times_A A_2)$ is a $2$-fibre product", "of $\\mathcal{F}(A_1)$ with $\\mathcal{F}(A_2)$ over $\\mathcal{F}(A)$ by", "Categories, Remark \\ref{categories-remark-other-description-2-fibre-product}.", "In other words, we see that (2) holds.", "\\medskip\\noindent", "The implication (2) $\\Rightarrow$ (3) is immediate.", "\\medskip\\noindent", "Assume (3). Let $q_1 : A_1 \\to A$ and $q_2 : A_2 \\to A$ be given with", "$q_2$ a small extension. We will use the description of the $2$-fibre product", "$\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$ from", "Categories, Remark \\ref{categories-remark-other-description-2-fibre-product}.", "Hence let $y \\in \\mathcal{F}(A_1 \\times_A A_2)$ correspond to", "$(x_1, x_2, x, a_1 : x_1 \\to x, a_2 : x_2 \\to x)$.", "Let $z$ be an object of $\\mathcal{F}$ lying over $C$. Then", "\\begin{align*}", "\\Mor_\\mathcal{F}(z, y) & =", "\\{(f, \\alpha) \\mid f : C \\to A_1 \\times_A A_2,", "\\alpha : f_*z \\to y\\} \\\\", "& = \\{(f_1, f_2, \\alpha_1, \\alpha_2) \\mid", "f_i : C \\to A_i, \\ \\alpha_i : f_{i, *}z \\to x_i, \\\\", "& \\quad\\quad\\quad\\quad", "q_1 \\circ f_1 = q_2 \\circ f_2, \\ q_{1, *} \\alpha_1 = q_{2, *}\\alpha_2\\} \\\\", "& =", "\\Mor_\\mathcal{F}(z, x_1) \\times_{\\Mor_\\mathcal{F}(z, x)}", "\\Mor_\\mathcal{F}(z, x_2)", "\\end{align*}", "whence $y$ is a fibre product of $x_1$ and $x_2$ over $x$. Thus we see", "that $\\mathcal{F}$ satisfies (RS) in case $A_2 \\to A$ is a small extension.", "Hence (RS) holds by", "Lemma \\ref{lemma-RS-small-extension}." ], "refs": [ "formal-defos-lemma-RS-fiber-square", "categories-remark-other-description-2-fibre-product", "categories-remark-other-description-2-fibre-product", "formal-defos-lemma-RS-small-extension" ], "ref_ids": [ 3466, 12428, 12428, 3467 ] } ], "ref_ids": [] }, { "id": 3469, "type": "theorem", "label": "formal-defos-lemma-RS-implies-S1-S2", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-RS-implies-S1-S2", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$. The condition (RS) for $\\mathcal{F}$", "implies both (S1) and (S2) for $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Using the reformulation of", "Lemma \\ref{lemma-RS-2-categorical}", "and the explanation of (S1) following", "Definition \\ref{definition-S1-S2}", "it is immediate that (RS) implies (S1).", "This proves the first part of (S2). The second part of (S2)", "follows because", "Lemma \\ref{lemma-RS-fiber-square}", "tells us that $y = x_1 \\times_{d, x_0, e} x_2 = y'$ if", "$y, y'$ are as in the second part of the definition of (S2) in", "Definition \\ref{definition-S1-S2}. (In fact the morphism", "$y \\to y'$ is compatible with both $a, a'$ and $c, c'$!)" ], "refs": [ "formal-defos-lemma-RS-2-categorical", "formal-defos-definition-S1-S2", "formal-defos-lemma-RS-fiber-square", "formal-defos-definition-S1-S2" ], "ref_ids": [ 3468, 3523, 3466, 3523 ] } ], "ref_ids": [] }, { "id": 3470, "type": "theorem", "label": "formal-defos-lemma-RS-associated-functor", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-RS-associated-functor", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$ satisfying (RS).", "The following conditions are equivalent:", "\\begin{enumerate}", "\\item $\\overline{\\mathcal{F}}$ satisfies (RS).", "\\item Let $f_1: A_1 \\to A$ and $f_2: A_2 \\to A$ be ring maps in", "$\\mathcal{C}_\\Lambda$ with $f_2$ surjective. The induced map", "of sets of isomorphism classes", "$$", "\\overline{\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)}", "\\to \\overline{\\mathcal{F}}(A_1) \\times_{\\overline{\\mathcal{F}}(A)}", "\\overline{\\mathcal{F}}(A_2)", "$$", "is injective.", "\\item For every morphism $x' \\to x$ in $\\mathcal{F}$ lying over a", "surjective ring map $A' \\to A$, the map", "$\\text{Aut}_{A'}(x') \\to \\text{Aut}_A(x)$ is surjective.", "\\item For every morphism $x' \\to x$ in $\\mathcal{F}$ lying over a small", "extension $A' \\to A$, the map", "$\\text{Aut}_{A'}(x') \\to \\text{Aut}_A(x)$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We prove that (1) is equivalent to (2) and (2) is equivalent to (3). The", "equivalence of (3) and (4) follows from Lemma", "\\ref{lemma-factor-small-extension}.", "\\medskip\\noindent", "Let $f_1: A_1 \\to A$ and $f_2: A_2 \\to A$ be ring maps in", "$\\mathcal{C}_\\Lambda$ with $f_2$ surjective. By", "Remark \\ref{remark-compare-schlessinger-H4}", "we see $\\overline{\\mathcal{F}}$ satisfies (RS) if and", "only if the map", "$$", "\\overline{\\mathcal{F}}(A_1 \\times_A A_2) \\to \\overline{\\mathcal", "F}(A_1) \\times_{\\overline{\\mathcal{F}}(A)} \\overline{\\mathcal{F}}(A_2)", "$$", "is bijective for any such $f_1, f_2$.", "This map is at least surjective since that", "is the condition of (S1) and $\\overline{\\mathcal{F}}$ satisfies (S1) by", "Lemmas \\ref{lemma-RS-implies-S1-S2} and", "\\ref{lemma-S1-S2-associated-functor}.", "Moreover, this map factors as", "$$", "\\overline{\\mathcal{F}}(A_1 \\times_A A_2)", "\\longrightarrow", "\\overline{\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)}", "\\longrightarrow", "\\overline{\\mathcal{F}}(A_1) \\times_{\\overline{\\mathcal{F}}(A)}", "\\overline{\\mathcal{F}}(A_2),", "$$", "where the first map is a bijection since", "$$", "\\mathcal{F}(A_1 \\times_A A_2)", "\\longrightarrow", "\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)", "$$", "is an equivalence by (RS) for $\\mathcal{F}$. Hence (1) is equivalent to (2).", "\\medskip\\noindent", "Assume (2) holds. Let $x' \\to x$ be a morphism in $\\mathcal{F}$ lying", "over a surjective ring map $f: A' \\to A$. Let", "$a \\in \\text{Aut}_A(x)$. The", "objects", "$$", "(x', x', a : x \\to x), \\ (x', x', \\text{id} : x \\to x)", "$$", "of", "$\\mathcal{F}(A') \\times_{\\mathcal{F}(A)} \\mathcal{F}(A')$", "have the same image in", "$\\overline{\\mathcal{F}}(A') \\times_{\\overline{\\mathcal{F}}(A)}", "\\overline{\\mathcal{F}}(A')$. By (2) there exists maps", "$b_1, b_2 : x' \\to x'$ such that", "$$", "\\xymatrix{", "x \\ar[r]_a \\ar[d]_{f_*b_1} & x \\ar[d]^{f_*b_2} \\\\", "x \\ar[r]^{\\text{id}} & x", "}", "$$", "commutes. Hence $b_2^{-1} \\circ b_1 \\in \\text{Aut}_{A'}(x')$ has image", "$a \\in \\text{Aut}_A(x)$. Hence (3) holds.", "\\medskip\\noindent", "Assume (3) holds. Suppose", "$$", "(x_1, x_2, a : (f_1)_*x_1 \\to (f_2)_*x_2),", "\\ (x'_1, x'_2, a' : (f_1)_*x'_1 \\to (f_2)_*x'_2)", "$$", "are objects of", "$\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$", "with the same image in", "$\\overline{\\mathcal{F}}(A_1) \\times_{\\overline{\\mathcal{F}}(A)}", "\\overline{\\mathcal{F}}(A_2)$. Then there are morphisms $b_1: x_1 \\to", "x'_1$ in $\\mathcal{F}(A_1)$ and $b_2: x_2 \\to x'_2$ in $\\mathcal", "F(A_2)$. By (3) we can modify $b_2$ by an automorphism of $x_2$ over $A_2$ so", "that the diagram", "$$", "\\xymatrix{", "(f_1)_*x_1 \\ar[r]_a \\ar[d]_{(f_1)_*b_1} & (f_2)_*x_2 \\ar[d]^{(f_2)_*b_2} \\\\", "(f_1)_*x'_1 \\ar[r]^{a'} & (f_2)_*x'_2.", "}", "$$", "commutes. This proves $(x_1, x_2, a) \\cong (x'_1, x'_2, a')$ in", "$\\overline{\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)}$.", "Hence (2) holds." ], "refs": [ "formal-defos-lemma-factor-small-extension", "formal-defos-remark-compare-schlessinger-H4", "formal-defos-lemma-RS-implies-S1-S2", "formal-defos-lemma-S1-S2-associated-functor" ], "ref_ids": [ 3414, 3568, 3469, 3441 ] } ], "ref_ids": [] }, { "id": 3471, "type": "theorem", "label": "formal-defos-lemma-localize-RS", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-localize-RS", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$. Let $x_0 \\in \\Ob(\\mathcal{F}(k))$.", "Let $\\mathcal{F}_{x_0}$ be the category cofibred in groupoids over", "$\\mathcal{C}_\\Lambda$ constructed in", "Remark \\ref{remark-localize-cofibered-groupoid}.", "If $\\mathcal{F}$ satisfies (RS), then so does $\\mathcal{F}_{x_0}$.", "In particular, $\\mathcal{F}_{x_0}$ is a deformation category." ], "refs": [ "formal-defos-remark-localize-cofibered-groupoid" ], "proofs": [ { "contents": [ "Any diagram as in", "Definition \\ref{definition-RS}", "in $\\mathcal{F}_{x_0}$ gives rise to a diagram in $\\mathcal{F}$", "and the output of (RS) for this diagram in $\\mathcal{F}$", "can be viewed as an output for $\\mathcal{F}_{x_0}$ as well." ], "refs": [ "formal-defos-definition-RS" ], "ref_ids": [ 3529 ] } ], "ref_ids": [ 3545 ] }, { "id": 3472, "type": "theorem", "label": "formal-defos-lemma-RS-fiber-product-morphisms", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-RS-fiber-product-morphisms", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{H} \\times_\\mathcal{F} \\mathcal{G} \\ar[r] \\ar[d] &", "\\mathcal{G} \\ar[d]^g \\\\", "\\mathcal{H} \\ar[r]^f & \\mathcal{F}", "}", "$$", "be $2$-fibre product of categories cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$. If $\\mathcal{F}, \\mathcal{G}, \\mathcal{H}$", "all satisfy (RS), then $\\mathcal{H} \\times_\\mathcal{F} \\mathcal{G}$", "satisfies (RS)." ], "refs": [], "proofs": [ { "contents": [ "If $A$ is an object of $\\mathcal{C}_\\Lambda$, then an object of the fiber", "category of $\\mathcal{H} \\times_\\mathcal{F} \\mathcal{G}$ over $A$", "is a triple $(u, v, a)$ where $u \\in \\mathcal{H}(A)$, $v \\in \\mathcal{G}(A)$,", "and $a : f(u) \\to g(v)$ is a morphism in $\\mathcal{F}(A)$. Consider", "a diagram in $\\mathcal{H} \\times_\\mathcal{F} \\mathcal{G}$", "$$", "\\vcenter{", "\\xymatrix{", " & (u_2, v_2, a_2) \\ar[d] \\\\", "(u_1, v_1, a_1) \\ar[r] & (u, v, a)", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", " & A_2 \\ar[d] \\\\", "A_1 \\ar[r] & A", "}", "}", "$$", "in $\\mathcal{C}_\\Lambda$ with $A_2 \\to A$ surjective. Since", "$\\mathcal{H}$ and $\\mathcal{G}$ satisfy (RS), there are fiber", "products $u_1 \\times_u u_2$ and $v_1 \\times_v v_2$ lying over", "$A_1 \\times_A A_2$. Since $\\mathcal{F}$ satisfies (RS),", "Lemma \\ref{lemma-RS-fiber-square} shows", "$$", "\\vcenter{", "\\xymatrix{", "f(u_1 \\times_u u_2) \\ar[r] \\ar[d] & f(u_2) \\ar[d] \\\\", "f(u_1) \\ar[r] & f(u)", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "g(v_1 \\times_v v_2) \\ar[r] \\ar[d] & g(v_2) \\ar[d] \\\\", "g(v_1) \\ar[r] & g(v)", "}", "}", "$$", "are both fiber squares in $\\mathcal{F}$. Thus we can view", "$a_1 \\times_a a_2$ as a morphism from $f(u_1 \\times_u u_2)$ to", "$g(v_1 \\times_v v_2)$ over $A_1 \\times_A A_2$.", "It follows that", "$$", "\\xymatrix{", " (u_1 \\times_u u_2, v_1 \\times_v v_2, a_{1} \\times_a a_2) \\ar[d] \\ar[r] &", "(u_2, v_2, a_2) \\ar[d] \\\\", "(u_1, v_1, a_1) \\ar[r] & (u, v, a)", "}", "$$", "is a fiber square in $\\mathcal{H} \\times_\\mathcal{F} \\mathcal{G}$", "as desired." ], "refs": [ "formal-defos-lemma-RS-fiber-square" ], "ref_ids": [ 3466 ] } ], "ref_ids": [] }, { "id": 3473, "type": "theorem", "label": "formal-defos-lemma-free-transitive-action", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-free-transitive-action", "contents": [ "Let $\\mathcal{F}$ be a deformation category.", "Let $A' \\to A$ be a surjective ring map in", "$\\mathcal{C}_\\Lambda$ whose kernel $I$ is annihilated", "by $\\mathfrak m_{A'}$. Let $x \\in \\Ob(\\mathcal{F}(A))$.", "If $\\text{Lift}(x, A')$ is nonempty,", "then there is a free and transitive action of", "$T\\mathcal{F} \\otimes_k I$ on $\\text{Lift}(x, A')$." ], "refs": [], "proofs": [ { "contents": [ "Consider the ring map $g : A' \\times_A A' \\to k[I]$ defined by the", "rule $g(a_1, a_2) = \\overline{a_1} \\oplus a_2 - a_1$ (compare with", "Lemma \\ref{lemma-lifting-along-small-extension}).", "There is an isomorphism", "$$", "A' \\times_A A' \\xrightarrow{\\sim} A' \\times_k k[I]", "$$", "given by $(a_1, a_2) \\mapsto (a_1, g(a_1, a_2))$.", "This isomorphism commutes with the projections to $A'$ on the first", "factor, and hence with the projections of", "$A' \\times_A A'$ and $A' \\times_k k[I]$ to $A$. Thus there is a bijection", "\\begin{equation}", "\\label{equation-one}", "\\text{Lift}(x, A' \\times_A A')", "\\longrightarrow", "\\text{Lift}(x, A' \\times_k k[I])", "\\end{equation}", "By Remark \\ref{remark-lift-bijections} there is a bijection", "\\begin{equation}", "\\label{equation-two}", "\\text{Lift}(x, A') \\times \\text{Lift}(x, A')", "\\longrightarrow", "\\text{Lift}(x, A' \\times_A A')", "\\end{equation}", "There is a commutative diagram", "$$", "\\xymatrix{", "A' \\times_k k[I] \\ar[r] \\ar[d] & A \\times_k k[I] \\ar[r] \\ar[d] & k[I] \\ar[d] \\\\", "A' \\ar[r] & A \\ar[r] & k.", "}", "$$", "Thus if we choose a pushforward $x \\to x_0$ of $x$ along", "$A \\to k$, we obtain by the end of", "Remark \\ref{remark-lift-bijections}", "a bijection", "\\begin{equation}", "\\label{equation-three}", "\\text{Lift}(x, A' \\times_k k[I])", "\\longrightarrow", "\\text{Lift}(x, A') \\times \\text{Lift}(x_0, k[I])", "\\end{equation}", "Composing (\\ref{equation-two}), (\\ref{equation-one}), and", "(\\ref{equation-three})", "we get a bijection", "$$", "\\Phi :", "\\text{Lift}(x, A') \\times \\text{Lift}(x, A')", "\\longrightarrow", "\\text{Lift}(x, A') \\times \\text{Lift}(x_0, k[I]).", "$$", "This bijection commutes with the projections on the first factors.", "By Remark \\ref{remark-tangent-space-lifting}", "we see that $\\text{Lift}(x_0, k[I]) = T\\mathcal{F} \\otimes_k I$.", "If $\\text{pr}_2$ is the second projection of", "$\\text{Lift}(x, A') \\times \\text{Lift}(x, A')$, then we get a map", "$$", "a = \\text{pr}_2 \\circ \\Phi^{-1} :", "\\text{Lift}(x, A') \\times (T\\mathcal{F} \\otimes_k I)", "\\longrightarrow", "\\text{Lift}(x, A').", "$$", "Unwinding all the above we see that $a(x' \\to x, \\theta)$", "is the unique lift $x'' \\to x$ such that $g_*(x', x'') = \\theta$", "in $\\text{Lift}(x_0, k[I]) = T\\mathcal{F} \\otimes_k I$.", "To see this is an action of $T\\mathcal{F} \\otimes_k I$ on $\\text{Lift}(x, A')$", "we have to show the following: if $x', x'', x'''$ are lifts of $x$ and", "$g_*(x', x'') = \\theta$, $g_*(x'', x''') = \\theta'$, then", "$g_*(x', x''') = \\theta + \\theta'$. This follows from the commutative", "diagram", "$$", "\\xymatrix{", "A' \\times_A A' \\times_A A'", "\\ar[rrrrr]_-{(a_1, a_2, a_3) \\mapsto (g(a_1, a_2), g(a_2, a_3))}", "\\ar[rrrrrd]_{(a_1, a_2, a_3) \\mapsto g(a_1, a_3)} & & & & &", "k[I] \\times_k k[I] = k[I \\times I] \\ar[d]^{+} \\\\", "& & & & & k[I]", "}", "$$", "The action is free and transitive because $\\Phi$ is bijective." ], "refs": [ "formal-defos-lemma-lifting-along-small-extension", "formal-defos-remark-lift-bijections", "formal-defos-remark-lift-bijections", "formal-defos-remark-tangent-space-lifting" ], "ref_ids": [ 3444, 3572, 3572, 3571 ] } ], "ref_ids": [] }, { "id": 3474, "type": "theorem", "label": "formal-defos-lemma-minimal-smooth-morphism-functors", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-minimal-smooth-morphism-functors", "contents": [ "Let $F, G: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be deformation", "functors. Let $\\varphi : F \\to G$ be a smooth morphism which induces", "an isomorphism $d\\varphi : TF \\to TG$ of tangent", "spaces. Then $\\varphi$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "We prove $F(A) \\to G(A)$ is a bijection for all $A \\in", "\\Ob(\\mathcal{C}_\\Lambda)$ by induction on", "$\\text{length}_A(A)$. For $A = k$ the statement follows from the", "assumption that $F$ and $G$ are deformation functors. Suppose that the", "statement holds for rings of length less than $n$ and let $A'$ be a ring of", "length $n$. Choose a small extension $f : A' \\to A$. We have a", "commutative diagram", "$$", "\\xymatrix{", "F(A') \\ar[r] \\ar[d]_{F(f)} & G(A') \\ar[d]^{G(f)} \\\\", "F(A) \\ar[r]^{\\sim} & G(A)", "}", "$$", "where the map $F(A) \\to G(A)$ is a bijection. By smoothness of $F", "\\to G$, $F(A') \\to G(A')$ is surjective (Lemma", "\\ref{lemma-smooth-morphism-essentially-surjective}). Thus we can check", "bijectivity by checking it on fibers $F(f)^{-1}(x) \\to", "G(f)^{-1}(\\varphi(x))$ for $x \\in F(A)$ such that $F(f)^{-1}(x)$ is nonempty.", "These fibers are precisely $\\text{Lift}(x, A')$ and", "$\\text{Lift}(\\varphi(x), A')$ and by assumption we have an isomorphism", "$d\\varphi \\otimes \\text{id} :", "TF \\otimes_k \\Ker(f) \\to TG \\otimes_k \\Ker(f)$.", "Thus, by", "Lemma \\ref{lemma-free-transitive-action} and", "Remark \\ref{remark-free-transitive-action-functorial},", "for $x \\in F(A)$ such that $F(f)^{-1}(x)$ is nonempty the map", "$F(f)^{-1}(x) \\to G(f)^{-1}(\\varphi(x))$ is a map of sets commuting", "with free transitive actions by $TF \\otimes_k \\Ker(f)$.", "Hence it is bijective." ], "refs": [ "formal-defos-lemma-smooth-morphism-essentially-surjective", "formal-defos-lemma-free-transitive-action", "formal-defos-remark-free-transitive-action-functorial" ], "ref_ids": [ 3434, 3473, 3573 ] } ], "ref_ids": [] }, { "id": 3475, "type": "theorem", "label": "formal-defos-lemma-Aut-functor-RS", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-Aut-functor-RS", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$ satisfying (RS). Let", "$x \\in \\Ob(\\mathcal{F}(A))$. Then", "$\\mathit{Aut}(x): \\mathcal{C}_A \\to \\textit{Sets}$ satisfies (RS)." ], "refs": [], "proofs": [ { "contents": [ "It follows that $\\mathit{Aut}(x)$ satisfies (RS) from the fully", "faithfulness of the functor", "$\\mathcal{F}(A_1 \\times_A A_2) \\to", "\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$ in", "Lemma \\ref{lemma-RS-2-categorical}." ], "refs": [ "formal-defos-lemma-RS-2-categorical" ], "ref_ids": [ 3468 ] } ], "ref_ids": [] }, { "id": 3476, "type": "theorem", "label": "formal-defos-lemma-Aut-functor-tangent-space", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-Aut-functor-tangent-space", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$ satisfying (RS). Let", "$x \\in \\Ob(\\mathcal{F}(A))$. Let $x_0$ be a pushforward of $x$ to", "$\\mathcal{F}(k)$.", "\\begin{enumerate}", "\\item $T_{\\text{id}_{x_0}} \\mathit{Aut}(x)$ has a natural $k$-vector", "space structure such that addition agrees with composition in", "$T_{\\text{id}_{x_0}} \\mathit{Aut}(x)$. In particular, composition in", "$T_{\\text{id}_{x_0}} \\mathit{Aut}(x)$ is commutative.", "\\item There is a canonical isomorphism", "$T_{\\text{id}_{x_0}} \\mathit{Aut}(x) \\to", "T_{\\text{id}_{x_0}} \\mathit{Aut}(x_0)$", "of $k$-vector spaces.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We apply", "Remark \\ref{remark-localize-cofibered-groupoid}", "to the functor $\\mathit{Aut}(x) : \\mathcal{C}_A \\to \\textit{Sets}$", "and the element $\\text{id}_{x_0} \\in \\mathit{Aut}(x)(k)$ to get", "a predeformation functor $F = \\mathit{Aut}(x)_{\\text{id}_{x_0}}$. By", "Lemmas \\ref{lemma-Aut-functor-RS} and \\ref{lemma-localize-RS}", "$F$ is a deformation functor. By definition", "$T_{\\text{id}_{x_0}} \\mathit{Aut}(x) = TF = F(k[\\epsilon])$", "which has a natural", "$k$-vector space structure specified by Lemma", "\\ref{lemma-tangent-space-functor}.", "\\medskip\\noindent", "Addition is defined as the composition", "$$", "F(k[\\epsilon]) \\times F(k[\\epsilon]) \\longrightarrow", "F(k[\\epsilon] \\times_k k[\\epsilon]) \\longrightarrow", "F(k[\\epsilon])", "$$", "where the first map is the inverse of the bijection guaranteed by (RS) and the", "second is induced by the $k$-algebra map", "$k[\\epsilon] \\times_k k[\\epsilon] \\to k[\\epsilon]$", "which maps $(\\epsilon, 0)$ and $(0, \\epsilon)$ to $\\epsilon$.", "If $A \\to B$ is a ring map in $\\mathcal{C}_\\Lambda$, then $F(A) \\to F(B)$", "is a homomorphism where $F(A) = \\mathit{Aut}(x)_{\\text{id}_{x_0}}(A)$ and", "$F(B) = \\mathit{Aut}(x)_{\\text{id}_{x_0}}(B)$ are groups under", "composition. We conclude that", "$+ : F(k[\\epsilon]) \\times F(k[\\epsilon])\\to F(k[\\epsilon])$", "is a homomorphism where $F(k[\\epsilon])$ is regarded as a", "group under composition. With $\\text{id} \\in F(k[\\epsilon])$ the", "unit element we see that $+(v, \\text{id}) =", "+(\\text{id}, v) = v$ for any $v \\in F(k[\\epsilon])$ because", "$(\\text{id}, v)$ is the pushforward of $v$ along the ring map", "$k[\\epsilon] \\to k[\\epsilon] \\times_k k[\\epsilon]$ with", "$\\epsilon \\mapsto (\\epsilon, 0)$.", "In general, given a group $G$ with multiplication $\\circ$", "and $+ : G \\times G \\to G$ is a homomorphism such that", "$+(g, 1) = +(1, g) = g$, where $1$ is the identity of $G$, then $+ = \\circ$.", "This shows addition in the $k$-vector space structure on $F(k[\\epsilon])$", "agrees with composition.", "\\medskip\\noindent", "Finally, (2) is a matter of unwinding the definitions. Namely", "$T_{\\text{id}_{x_0}} \\mathit{Aut}(x)$ is the set of", "automorphisms $\\alpha$ of the pushforward of $x$ along", "$A \\to k \\to k[\\epsilon]$ which are trivial modulo $\\epsilon$.", "On the other hand $T_{\\text{id}_{x_0}} \\mathit{Aut}(x_0)$ is the set of", "automorphisms of the pushforward of $x_0$ along", "$k \\to k[\\epsilon]$ which are trivial modulo $\\epsilon$.", "Since $x_0$ is the pushforward of $x$ along $A \\to k$ the result", "is clear." ], "refs": [ "formal-defos-remark-localize-cofibered-groupoid", "formal-defos-lemma-Aut-functor-RS", "formal-defos-lemma-localize-RS", "formal-defos-lemma-tangent-space-functor" ], "ref_ids": [ 3545, 3475, 3471, 3449 ] } ], "ref_ids": [] }, { "id": 3477, "type": "theorem", "label": "formal-defos-lemma-infaut-vector-space", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-infaut-vector-space", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$ satisfying (RS). Let $x_0 \\in \\Ob(\\mathcal{F}(k))$.", "Then $\\text{Inf}_{x_0}(\\mathcal{F})$ is equal as a set to", "$T_{\\text{id}_{x_0}} \\mathit{Aut}(x_0)$, and so has a natural $k$-vector", "space structure such that addition agrees with composition of automorphisms." ], "refs": [], "proofs": [ { "contents": [ "The equality of sets is as in the end of", "Remark \\ref{remark-infaut-lifting-equalities}", "and the statement about the vector space structure follows from", "Lemma \\ref{lemma-Aut-functor-tangent-space}." ], "refs": [ "formal-defos-remark-infaut-lifting-equalities", "formal-defos-lemma-Aut-functor-tangent-space" ], "ref_ids": [ 3576, 3476 ] } ], "ref_ids": [] }, { "id": 3478, "type": "theorem", "label": "formal-defos-lemma-k-linear-infaut", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-k-linear-infaut", "contents": [ "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of categories", "cofibred in groupoids over $\\mathcal{C}_\\Lambda$ satisfying (RS).", "Let $x_0 \\in \\Ob(\\mathcal{F}(k))$. Then $\\varphi$ induces a $k$-linear", "map $\\text{Inf}_{x_0}(\\mathcal{F}) \\to \\text{Inf}_{\\varphi(x_0)}(\\mathcal{G})$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $\\varphi$ induces a morphism from", "$\\mathit{Aut}(x_0) \\to \\mathit{Aut}(\\varphi(x_0))$", "which maps the identity to the identity. Hence this follows from", "the result for tangent spaces, see", "Lemma \\ref{lemma-k-linear-differential}." ], "refs": [ "formal-defos-lemma-k-linear-differential" ], "ref_ids": [ 3453 ] } ], "ref_ids": [] }, { "id": 3479, "type": "theorem", "label": "formal-defos-lemma-lifted-automorphisms-torsor", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-lifted-automorphisms-torsor", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$ satisfying (RS). Let $x' \\to x$ be a", "morphism lying over a surjective ring map $A' \\to A$ with kernel $I$", "annihilated by $\\mathfrak m_{A'}$. Let $x_0$ be a pushforward of $x$ to", "$\\mathcal{F}(k)$. Then $\\text{Inf}(x'/x)$ has a free and transitive action by", "$T_{\\text{id}_{x_0}} \\mathit{Aut}(x') \\otimes_k I", "= \\text{Inf}_{x_0}(\\mathcal{F}) \\otimes_k I$." ], "refs": [], "proofs": [ { "contents": [ "This is just the analogue of", "Lemma \\ref{lemma-free-transitive-action}", "in the setting of automorphism sheaves.", "To be precise, we apply", "Remark \\ref{remark-localize-cofibered-groupoid}", "to the functor $\\mathit{Aut}(x') : \\mathcal{C}_{A'} \\to \\textit{Sets}$", "and the element $\\text{id}_{x_0} \\in \\mathit{Aut}(x)(k)$ to get", "a predeformation functor $F = \\mathit{Aut}(x')_{\\text{id}_{x_0}}$. By", "Lemmas \\ref{lemma-Aut-functor-RS} and \\ref{lemma-localize-RS}", "$F$ is a deformation functor. Hence", "Lemma \\ref{lemma-free-transitive-action}", "gives a free and transitive action", "of $TF \\otimes_k I$ on $\\text{Lift}(\\text{id}_x, A')$, because as", "$\\text{Lift}(\\text{id}_x, A')$ is a group it is always nonempty.", "Note that we have equalities of vector spaces", "$$", "TF = T_{\\text{id}_{x_0}} \\mathit{Aut}(x') \\otimes_k I =", "\\text{Inf}_{x_0}(\\mathcal{F}) \\otimes_k I", "$$", "by", "Lemma \\ref{lemma-Aut-functor-tangent-space}.", "The equality $\\text{Inf}(x'/x) = \\text{Lift}(\\text{id}_x, A')$ of", "Remark \\ref{remark-infaut-lifting-equalities}", "finishes the proof." ], "refs": [ "formal-defos-lemma-free-transitive-action", "formal-defos-remark-localize-cofibered-groupoid", "formal-defos-lemma-Aut-functor-RS", "formal-defos-lemma-localize-RS", "formal-defos-lemma-free-transitive-action", "formal-defos-lemma-Aut-functor-tangent-space", "formal-defos-remark-infaut-lifting-equalities" ], "ref_ids": [ 3473, 3545, 3475, 3471, 3473, 3476, 3576 ] } ], "ref_ids": [] }, { "id": 3480, "type": "theorem", "label": "formal-defos-lemma-infaut-trivial", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-infaut-trivial", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$ satisfying (RS). Let $x' \\to x$ be a morphism", "in $\\mathcal{F}$ lying over a surjective ring map. Let $x_0$ be a pushforward", "of $x$ to $\\mathcal{F}(k)$. If $\\text{Inf}_{x_0}(\\mathcal{F}) = 0$ then", "$\\text{Inf}(x'/x) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemmas \\ref{lemma-factor-small-extension} and", "\\ref{lemma-lifted-automorphisms-torsor}." ], "refs": [ "formal-defos-lemma-factor-small-extension", "formal-defos-lemma-lifted-automorphisms-torsor" ], "ref_ids": [ 3414, 3479 ] } ], "ref_ids": [] }, { "id": 3481, "type": "theorem", "label": "formal-defos-lemma-infdef-trivial", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-infdef-trivial", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$ satisfying (RS). Let", "$x_0 \\in \\Ob(\\mathcal{F}(k))$. Then $\\text{Inf}_{x_0}(\\mathcal{F}) = 0$", "if and only if the natural morphism", "$\\mathcal{F}_{x_0} \\to \\overline{\\mathcal{F}_{x_0}}$ of", "categories cofibered in groupoids is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "The morphism $\\mathcal{F}_{x_0} \\to \\overline{\\mathcal{F}_{x_0}}$ is an", "equivalence if and only if $\\mathcal{F}_{x_0}$ is fibered in setoids,", "cf.\\ Categories, Section \\ref{categories-section-fibred-in-setoids}", "(a setoid is by definition a groupoid in", "which the only automorphism of any object is the identity). We prove that", "$\\text{Inf}_{x_0}(\\mathcal{F}) = 0$ if and only if this condition holds", "for $\\mathcal{F}_{x_0}$. Obviously if $\\mathcal{F}_{x_0}$ is fibered in", "setoids then $\\text{Inf}_{x_0}(\\mathcal{F}) = 0$. Conversely assume", "$\\text{Inf}_{x_0}(\\mathcal{F}) = 0$. Let $A$ be an object of", "$\\mathcal{C}_\\Lambda$. Then by", "Lemma \\ref{lemma-infaut-trivial},", "$\\text{Inf}(x/x_0) = 0$ for any object $x \\to x_0$ of", "$\\mathcal{F}_{x_0}(A)$. Since by definition $\\text{Inf}(x/x_0)$", "equals the group of automorphisms of $x \\to x_0$ in $\\mathcal{F}_{x_0}(A)$,", "this proves $\\mathcal{F}_{x_0}(A)$ is a setoid." ], "refs": [ "formal-defos-lemma-infaut-trivial" ], "ref_ids": [ 3480 ] } ], "ref_ids": [] }, { "id": 3482, "type": "theorem", "label": "formal-defos-lemma-deformation-categories-fiber-product-morphisms", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-deformation-categories-fiber-product-morphisms", "contents": [ "Let $f : \\mathcal{H} \\to \\mathcal{F}$ and $g : \\mathcal{G} \\to \\mathcal{F}$", "be $1$-morphisms of deformation categories. Then", "\\begin{enumerate}", "\\item $\\mathcal{W} = \\mathcal{H} \\times_\\mathcal{F} \\mathcal{G}$ is a", "deformation category, and", "\\item we have a $6$-term exact sequence of vector spaces", "$$", "0 \\to \\text{Inf}(\\mathcal{W})", "\\to \\text{Inf}(\\mathcal{H}) \\oplus \\text{Inf}(\\mathcal{G})", "\\to \\text{Inf}(\\mathcal{F}) \\to", "T\\mathcal{W} \\to T\\mathcal{H} \\oplus T\\mathcal{G} \\to T\\mathcal{F}", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from Lemma \\ref{lemma-RS-fiber-product-morphisms}", "and the fact that $\\mathcal{W}(k)$ is the fibre product of", "two setoids with a unique isomorphism class over a setoid with", "a unique isomorphism class.", "\\medskip\\noindent", "Part (2). Let $w_0 \\in \\Ob(\\mathcal{W}(k))$ and let $x_0, y_0, z_0$ be", "the image of $w_0$ in $\\mathcal{F}, \\mathcal{H}, \\mathcal{G}$.", "Then $\\text{Inf}(\\mathcal{W}) = \\text{Inf}_{w_0}(\\mathcal{W})$", "and simlarly for $\\mathcal{H}$, $\\mathcal{G}$, and $\\mathcal{F}$, see", "Remark \\ref{remark-trivial-aut-point}.", "We apply Lemmas \\ref{lemma-k-linear-differential} and", "\\ref{lemma-k-linear-infaut} to get all the linear maps", "except for the ``boundary map''", "$\\delta : \\text{Inf}_{x_0}(\\mathcal{F}) \\to T\\mathcal{W}$.", "We will insert suitable signs later.", "\\medskip\\noindent", "Construction of $\\delta$. Choose a pushforward $w_0 \\to w'_0$ along", "$k \\to k[\\epsilon]$. Denote $x'_0, y'_0, z'_0$ the images of $w'_0$ in", "$\\mathcal{F}, \\mathcal{H}, \\mathcal{G}$. In particular", "we obtain isomorphisms $b' : f(y'_0) \\to x'_0$ and $c' : x'_0 \\to g(z'_0)$.", "Denote $b : f(y_0) \\to x_0$ and $c : x_0 \\to g(z_0)$ the pushforwards", "along $k[\\epsilon] \\to k$. Observe that this means", "$w'_0 = (k[\\epsilon], y'_0, z'_0, c' \\circ b')$ and", "$w_0 = (k, y_0, z_0, c \\circ b)$ in terms of the explicit form", "of the fibre product of categories,", "see Remarks \\ref{remarks-cofibered-groupoids} (\\ref{item-fibre-product}).", "Given $\\alpha : x'_0 \\to x'_0$ we set", "$\\delta(\\alpha) = (k[\\epsilon], y'_0, z'_0, c' \\circ \\alpha \\circ b')$", "which is indeed an object of $\\mathcal{W}$ over $k[\\epsilon]$ and comes", "with a morphism $(k[\\epsilon], y'_0, z'_0, c' \\circ \\alpha \\circ b') \\to w_0$", "over $k[\\epsilon] \\to k$ as $\\alpha$ pushes forward to the identity over $k$.", "More generally, for any $k$-vector space $V$ we can define a map", "$$", "\\text{Lift}(\\text{id}_{x_0}, k[V])", "\\longrightarrow", "\\text{Lift}(w_0, k[V])", "$$", "using exactly the same formulae. This construction is functorial", "in the vector space $V$ (details omitted). Hence $\\delta$ is $k$-linear", "by an application of", "Lemma \\ref{lemma-morphism-linear-functors}.", "\\medskip\\noindent", "Having constructed these maps it is straightforward to show the sequence", "is exact. Injectivity of the first map comes from the fact that", "$f \\times g : \\mathcal{W} \\to \\mathcal{H} \\times \\mathcal{G}$", "is faithful. If", "$(\\beta, \\gamma) \\in", "\\text{Inf}_{y_0}(\\mathcal{H}) \\oplus \\text{Inf}_{z_0}(\\mathcal{G})$", "map to the same element of $\\text{Inf}_{x_0}(\\mathcal{F})$ then", "$(\\beta, \\gamma)$ defines an automorphism of", "$w'_0 = (k[\\epsilon], y'_0, z'_0, c' \\circ b')$ whence exactness", "at the second spot. If $\\alpha$ as above gives the trivial deformation", "$(k[\\epsilon], y'_0, z'_0, c' \\circ \\alpha \\circ b')$", "of $w_0$, then the isomorphism", "$w'_0 = (k[\\epsilon], y'_0, z'_0, c' \\circ b') \\to", "(k[\\epsilon], y'_0, z'_0, c' \\circ \\alpha \\circ b')$", "produces a pair $(\\beta, \\gamma)$ which is a preimage of $\\alpha$.", "If $w = (k[\\epsilon], y, z, \\phi)$ is a deformation of $w_0$", "such that $y'_0 \\cong y$ and $z \\cong z'_0$ then the map", "$$", "f(y'_0) \\to f(y) \\xrightarrow{\\phi} g(z) \\to g(z'_0)", "$$", "is an $\\alpha$ which maps to $w$ under $\\delta$.", "Finally, if $y$ and $z$ are deformations of $y_0$ and $z_0$", "and there exists an isomorphism $\\phi : f(y) \\to g(z)$ of deformations", "of $f(y_0) = x_0 = g(z_0)$ then we get a preimage", "$w = (k[\\epsilon], y, z, \\phi)$ of $(x, y)$ in $T\\mathcal{W}$.", "This finishes the proof." ], "refs": [ "formal-defos-lemma-RS-fiber-product-morphisms", "formal-defos-remark-trivial-aut-point", "formal-defos-lemma-k-linear-differential", "formal-defos-lemma-k-linear-infaut", "formal-defos-remarks-cofibered-groupoids", "formal-defos-lemma-morphism-linear-functors" ], "ref_ids": [ 3472, 3575, 3453, 3478, 3585, 3446 ] } ], "ref_ids": [] }, { "id": 3483, "type": "theorem", "label": "formal-defos-lemma-map-fibre-products-smooth", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-map-fibre-products-smooth", "contents": [ "Let $\\mathcal{H}_1 \\to \\mathcal{G}$, $\\mathcal{H}_2 \\to \\mathcal{G}$, and", "$\\mathcal{G} \\to \\mathcal{F}$ be maps of categories cofibred in groupoids", "over $\\mathcal{C}_\\Lambda$. Assume", "\\begin{enumerate}", "\\item $\\mathcal{F}$ and $\\mathcal{G}$ are deformation categories,", "\\item $T\\mathcal{G} \\to T\\mathcal{F}$ is injective, and", "\\item $\\text{Inf}(\\mathcal{G}) \\to \\text{Inf}(\\mathcal{F})$ is surjective.", "\\end{enumerate}", "Then $\\mathcal{H}_1 \\times_\\mathcal{G} \\mathcal{H}_2 \\to", "\\mathcal{H}_1 \\times_\\mathcal{F} \\mathcal{H}_2$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "Denote $p_i : \\mathcal{H}_i \\to \\mathcal{G}$ and", "$q : \\mathcal{G} \\to \\mathcal{F}$ be the given maps.", "Let $A' \\to A$ be a small extension in $\\mathcal{C}_\\Lambda$.", "An object of $\\mathcal{H}_1 \\times_\\mathcal{F} \\mathcal{H}_2$", "over $A'$ is a triple $(x'_1, x'_2, a')$ where", "$x'_i$ is an object of $\\mathcal{H}_i$ over $A'$ and", "$a' : q(p_1(x'_1)) \\to q(p_2(x'_2))$ is a morphism", "of the fibre category of $\\mathcal{F}$ over $A'$.", "By pushforward along $A' \\to A$ we get", "$(x_1, x_2, a)$. Lifting this to an object of", "$\\mathcal{H}_1 \\times_\\mathcal{G} \\mathcal{H}_2$", "over $A$ means finding a morphism", "$b : p_1(x_1) \\to p_2(x_2)$ over $A$ with $q(b) = a$.", "Thus we have to show that we can lift $b$ to a morphism", "$b' : p_1(x'_1) \\to p_2(x'_2)$ whose image under $q$ is $a'$.", "\\medskip\\noindent", "Observe that we can think of", "$$", "p_1(x'_1) \\to p_1(x_1) \\xrightarrow{b} p_2(x_2)", "\\quad\\text{and}\\quad", "p_2(x'_2) \\to p_2(x_2)", "$$", "as two objects of $\\textit{Lift}(p_2(x_2), A' \\to A)$.", "The functor $q$ sends these objects to the two objects", "$$", "q(p_1(x'_1)) \\to q(p_1(x_1)) \\xrightarrow{b} q(p_2(x_2))", "\\quad\\text{and}\\quad", "q(p_2(x'_2)) \\to q(p_2(x_2))", "$$", "of $\\textit{Lift}(q(p_2(x_2)), A' \\to A)$", "which are isomorphic using the map $a' : q(p_1(x'_1)) \\to q(p_2(x'_2))$.", "On the other hand, the functor", "$$", "q : \\textit{Lift}(p_2(x_2), A' \\to A) \\to", "\\textit{Lift}(q(p_2(x_2)), A' \\to A)", "$$", "defines a injection on isomorphism classes by", "Lemma \\ref{lemma-free-transitive-action}", "and our assumption on tangent spaces. Thus we see that there is a morphism", "$b' : p_1(x_1') \\to p_2(x'_2)$ whose pushforward to $A$ is $b$.", "However, we may need to adjust our choice of $b'$", "to achieve $q(b') = a'$. For this it suffices to see that", "$q : \\text{Inf}(p_2(x'_2)/p_2(x_2)) \\to \\text{Inf}(q(p_2(x'_2))/q(p_2(x_2)))$", "is surjective. This follows from our assumption on", "infinitesimal automorphisms and Lemma \\ref{lemma-lifted-automorphisms-torsor}." ], "refs": [ "formal-defos-lemma-free-transitive-action", "formal-defos-lemma-lifted-automorphisms-torsor" ], "ref_ids": [ 3473, 3479 ] } ], "ref_ids": [] }, { "id": 3484, "type": "theorem", "label": "formal-defos-lemma-easy-check-smooth", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-easy-check-smooth", "contents": [ "Let $f : \\mathcal{F} \\to \\mathcal{G}$ be a map of deformation", "categories. Let $x_0 \\in \\Ob(\\mathcal{F}(k))$ with image", "$y_0 \\in \\Ob(\\mathcal{G}(k))$. If", "\\begin{enumerate}", "\\item the map $T\\mathcal{F} \\to T\\mathcal{G}$ is surjective, and", "\\item for every small extension $A' \\to A$ in $\\mathcal{C}_\\Lambda$", "and $x \\in \\mathcal{F}(A)$ with image $y \\in \\mathcal{G}(A)$", "if there is a lift of $y$ to $A'$, then there is a lift", "of $x$ to $A'$,", "\\end{enumerate}", "then $\\mathcal{F} \\to \\mathcal{G}$ is smooth (and vice versa)." ], "refs": [], "proofs": [ { "contents": [ "Let $A' \\to A$ be a small extension. Let $x \\in \\mathcal{F}(A)$.", "Let $y' \\to f(x)$ be a morphism in $\\mathcal{G}$ over $A' \\to A$.", "Consider the functor $\\text{Lift}(A', x) \\to \\text{Lift}(A', f(x))$", "induced by $f$.", "We have to show that there exists an object $x' \\to x$", "of $\\text{Lift}(A', x)$ mapping to $y' \\to f(x)$, see", "Lemma \\ref{lemma-smoothness-small-extensions}.", "By condition (2) we know that $\\text{Lift}(A', x)$", "is not the empty category.", "By condition (2) and", "Lemma \\ref{lemma-free-transitive-action}", "we conlude that the map on isomorphism classes", "is surjective as desired." ], "refs": [ "formal-defos-lemma-smoothness-small-extensions", "formal-defos-lemma-free-transitive-action" ], "ref_ids": [ 3431, 3473 ] } ], "ref_ids": [] }, { "id": 3485, "type": "theorem", "label": "formal-defos-lemma-map-between-smooth", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-map-between-smooth", "contents": [ "Let $\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}$ be maps of categories", "cofibred in groupoids over $\\mathcal{C}_\\Lambda$. If", "\\begin{enumerate}", "\\item $\\mathcal{F}$, $\\mathcal{G}$ are deformation categories", "\\item the map $T\\mathcal{F} \\to T\\mathcal{G}$ is surjective, and", "\\item $\\mathcal{F} \\to \\mathcal{H}$ is smooth.", "\\end{enumerate}", "Then $\\mathcal{F} \\to \\mathcal{G}$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "Let $A' \\to A$ be a small extension in $\\mathcal{C}_\\Lambda$", "and let $x \\in \\mathcal{F}(A)$ with image $y \\in \\mathcal{G}(A)$.", "Assume there is a lift $y' \\in \\mathcal{G}(A')$. According to", "Lemma \\ref{lemma-easy-check-smooth} all we have to do is check", "that $x$ has a lift too. Take the image $z' \\in \\mathcal{H}(A')$", "of $y'$. Since $\\mathcal{F} \\to \\mathcal{H}$ is smooth, there is", "an $x' \\in \\mathcal{F}(A')$ mapping to both $x \\in \\mathcal{F}(A)$", "and $z' \\in \\mathcal{H}(A')$, see", "Definition \\ref{definition-smooth-morphism}. This finishes the proof." ], "refs": [ "formal-defos-lemma-easy-check-smooth", "formal-defos-definition-smooth-morphism" ], "ref_ids": [ 3484, 3520 ] } ], "ref_ids": [] }, { "id": 3486, "type": "theorem", "label": "formal-defos-lemma-groupoid-in-functors-prorep-equivalences", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-groupoid-in-functors-prorep-equivalences", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}_\\Lambda$.", "\\begin{enumerate}", "\\item $(U, R, s, t, c)$ is prorepresentable if and only if its completion is", "representable as a groupoid in functors on $\\widehat{\\mathcal{C}}_\\Lambda$.", "\\item $(U, R, s, t, c)$ is prorepresentable if and only if $U$ and $R$ are", "prorepresentable.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from", "Remark \\ref{remark-groupoid-in-functors-complete-restrict}.", "For (2), the ``only if'' direction is clear from the definition", "of a prorepresentable groupoid in functors. Conversely, assume $U$ and $R$", "are prorepresentable, say $U \\cong \\underline{R_0}|_{\\mathcal{C}_\\Lambda}$", "and $R \\cong \\underline{R_1}|_{\\mathcal{C}_\\Lambda}$ for objects $R_0$ and", "$R_1$ of $\\widehat{\\mathcal{C}}_\\Lambda$.", "Since $\\underline{R_0} \\cong \\widehat{\\underline{R_0}|_{\\mathcal{C}_\\Lambda}}$", "and $\\underline{R_1} \\cong \\widehat{\\underline{R_1}|_{\\mathcal{C}_\\Lambda}}$", "by", "Remark \\ref{remark-restrict-complete-continuous-functor}", "we see that the completion $(U, R, s, t, c)^\\wedge$ is a groupoid in", "functors of the form", "$(\\underline{R_0}, \\underline{R_1}, \\widehat{s}, \\widehat{t}, \\widehat{c})$.", "By", "Lemma \\ref{lemma-CLambdahat-pushouts}", "the pushout", "$\\underline{R_1} \\times_{\\widehat{s}, \\underline{R_1}, \\widehat{t}}", "\\underline{R_1}$ exists. Hence", "$(\\underline{R_0}, \\underline{R_1}, \\widehat{s}, \\widehat{t}, \\widehat{c})$", "is a representable groupoid in functors on $\\widehat{\\mathcal{C}}_\\Lambda$.", "Finally, the restriction", "$(\\underline{R_0}, \\underline{R_1}, s, t, c)|_{\\mathcal{C}_\\Lambda}$", "gives back $(U, R, s, t, c)$ by", "Remark \\ref{remark-groupoid-in-functors-complete-restrict}", "hence $(U, R, s, t, c)$ is prorepresentable by definition." ], "refs": [ "formal-defos-remark-groupoid-in-functors-complete-restrict", "formal-defos-remark-restrict-complete-continuous-functor", "formal-defos-lemma-CLambdahat-pushouts", "formal-defos-remark-groupoid-in-functors-complete-restrict" ], "ref_ids": [ 3582, 3552, 3422, 3582 ] } ], "ref_ids": [] }, { "id": 3487, "type": "theorem", "label": "formal-defos-lemma-smooth-quotient-morphism", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-smooth-quotient-morphism", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}_\\Lambda$.", "The following are equivalent:", "\\begin{enumerate}", "\\item The groupoid in functors $(U, R, s, t, c)$ is smooth.", "\\item The morphism $s : R \\to U$ is smooth.", "\\item The morphism $t : R \\to U$ is smooth.", "\\item The quotient morphism $U \\to [U/R]$ is smooth.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Statement (2) is equivalent to (3) since the inverse $i: R \\to R$ of", "$(U, R, s, t, c)$ is an isomorphism and $t = s \\circ i$. By definition (1) is", "equivalent to (2) and (3) together, hence it is equivalent to either of them", "individually.", "\\medskip\\noindent", "Finally we prove (2) is equivalent to (4). Unwinding the definitions:", "\\begin{enumerate}", "\\item[(2)] Smoothness of $s: R \\to U$ amounts to the following", "condition: If $f: B \\to A$ is a surjective ring map in", "$\\mathcal{C}_\\Lambda$, $a \\in R(A)$, and $y \\in U(B)$ such that", "$s(a) = U(f)(y)$, then there exists $a' \\in R(B)$ such that", "$R(f)(a') = a$ and $s(a') = y$.", "\\item[(4)] Smoothness of $U \\to [U/R]$ amounts to the following", "condition: If $f: B \\to A$ is a surjective ring map in", "$\\mathcal{C}_\\Lambda$ and $(f, a) : (B, y) \\to (A, x)$", "is a morphism of $[U/R]$, then there exists $x' \\in U(B)$ and", "$b \\in R(B)$ with $s(b) = x'$, $t(b) = y$", "such that $c(a, R(f)(b)) = e(x)$. Here $e : U \\to R$ denotes the", "identity and the notation $(f, a)$ is as in Remarks", "\\ref{remarks-cofibered-groupoids}", "(\\ref{item-construction-associated-cofibered-groupoid});", "in particular $a \\in R(A)$ with $s(a) = U(f)(y)$ and $t(a) = x$.", "\\end{enumerate}", "If (4) holds and $f, a, y$ as in (2) are given, let $x = t(a)$ so that", "we have a morphism $(f, a): (B, y) \\to (A, x)$.", "Then (4) produces $x'$ and $b$, and $a' = i(b)$ satisfies the requirements", "of (2). Conversely, assume (2) holds and let $(f, a): (B, y) \\to (A, x)$", "as in (4) be given. Then (2) produces $a' \\in R(B)$, and $x' = t(a')$ and", "$b = i(a')$ satisfy the requirements of (4)." ], "refs": [ "formal-defos-remarks-cofibered-groupoids" ], "ref_ids": [ 3585 ] } ], "ref_ids": [] }, { "id": 3488, "type": "theorem", "label": "formal-defos-lemma-smooth-RS-groupoid-in-functors-quotient", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-smooth-RS-groupoid-in-functors-quotient", "contents": [ "Let $(U, R, s, t, c)$ be a smooth groupoid in functors on $\\mathcal{C}_\\Lambda$.", "Assume $U$ and $R$ satisfy (RS). Then $[U/R]$ satisfies (RS)." ], "refs": [], "proofs": [ { "contents": [ "Let", "$$", "\\xymatrix{", " & (A_2, x_2) \\ar[d]^{(f_2, a_2)} \\\\", "(A_1, x_1) \\ar[r]^{(f_1, a_1)} & (A, x)", "}", "$$", "be a diagram in $[U/R]$ such that $f_2: A_2 \\to A$ is surjective. The", "notation is as in", "Remarks \\ref{remarks-cofibered-groupoids}", "(\\ref{item-construction-associated-cofibered-groupoid}).", "Hence $f_1: A_1 \\to A, f_2: A_2 \\to A$ are maps in", "$\\mathcal{C}_\\Lambda$, $x \\in U(A)$, $x_1 \\in U(A_1)$, $x_2 \\in U(A_2)$,", "and $a_1, a_2 \\in R(A)$ with $s(a_1) = U(f_1)(x_1)$,", "$t(a_1) = x$ and $s(a_2) = U(f_2)(x_2)$, $t(a_2) = x$.", "We construct a fiber product lying over $A_1 \\times_A A_2$", "for this diagram in $[U/R]$ as follows.", "\\medskip\\noindent", "Let $a = c(i(a_1), a_2)$, where $i: R \\to R$ is the inverse morphism.", "Then $a \\in R(A)$, $x_2 \\in U(A_2)$ and $s(a) = U(f_2)(x_2)$.", "Hence an element $(a, x_2) \\in R(A) \\times_{s, U(A), U(f_2)} U(A_2)$.", "By smoothness of $s : R \\to U$ there is an element", "$\\widetilde{a} \\in R(A_2)$ with $R(f_2)(\\widetilde{a}) = a$ and", "$s(\\widetilde{a}) = x_2$. In particular", "$U(f_2)(t(\\widetilde{a})) = t(a) = U(f_1)(x_1)$. Thus $x_1$ and", "$t(\\widetilde{a})$ define an element", "$$", "(x_1, t(\\widetilde{a})) \\in U(A_1) \\times_{U(A)} U(A_2).", "$$", "By the assumption that $U$ satisfies (RS), we have an identification", "$U(A_1) \\times_{U(A)} U(A_2) = U(A_1 \\times_A A_2)$. Let us denote", "$x_1 \\times t(\\widetilde{a}) \\in U(A_1 \\times_A A_2)$ the element", "corresponding to $(x_1, t(\\widetilde{a})) \\in U(A_1) \\times_{U(A)} U(A_2)$.", "Let $p_1, p_2$ be the projections of $A_1 \\times_A A_2$. We claim", "$$", "\\xymatrix{", "(A_1 \\times_A A_2, x_1 \\times t(\\widetilde{a})) \\ar[d]_{(p_1, e(x_1))}", "\\ar[rr]_-{(p_2, i(\\widetilde{a}))} & & (A_2, x_2) \\ar[d]^{(f_2, a_2)} \\\\", "(A_1, x_1) \\ar[rr]^{(f_1, a_1)} & & (A, x)", "}", "$$", "is a fiber square in $[U/R]$. (Note $e: U \\to R$ denotes the identity.)", "\\medskip\\noindent", "The diagram is commutative because", "$c(a_2, R(f_2)(i(\\widetilde{a}))) = c(a_2, i(a)) = a_1$.", "To check it is a fiber square, let", "$$", "\\xymatrix{", "(B, z) \\ar[d]_{(g_1, b_1)} \\ar[rr]_{(g_2, b_2)} & & (A_2, x_2)", "\\ar[d]^{(f_2, a_2)} \\\\", "(A_1, x_1) \\ar[rr]^{(f_1, a_1)} & & (A, x)", "}", "$$", "be a commutative diagram in $[U/R]$. We will show there is a unique morphism", "$(g, b) : (B, z) \\to (A_1 \\times_A A_2, x_1 \\times t(\\widetilde{a}))$", "compatible with the morphisms to $(A_1, x_1)$ and $(A_2, x_2)$.", "We must take $g = (g_1, g_2) : B \\to A_1 \\times_A A_2$.", "Since by assumption $R$ satisfies (RS), we have an identification", "$R(A_1 \\times_A A_2) = R(A_1) \\times_{R(A)} R(A_2)$.", "Hence we can write $b = (b'_1, b'_2)$ for some", "$b'_1 \\in R(A_1)$, $b'_2 \\in R(A_2)$ which agree in $R(A)$.", "Then", "$((g_1, g_2), (b'_1, b'_2)) : (B, z) \\to", "(A_1 \\times_A A_2, x_1 \\times t(\\widetilde{a}))$", "will commute with the projections if and only if", "$b'_1 = b_1$ and $b'_2 = c(\\widetilde{a}, b_2)$ proving unicity and", "existence." ], "refs": [ "formal-defos-remarks-cofibered-groupoids" ], "ref_ids": [ 3585 ] } ], "ref_ids": [] }, { "id": 3489, "type": "theorem", "label": "formal-defos-lemma-deformation-groupoid-quotient", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-deformation-groupoid-quotient", "contents": [ "Let $(U, R, s, t, c)$ be a smooth groupoid in functors on $\\mathcal{C}_\\Lambda$.", "Assume $U$ and $R$ are deformation functors. Then:", "\\begin{enumerate}", "\\item The quotient $[U/R]$ is a deformation category.", "\\item The tangent space of $[U/R]$ is", "$$", "T[U/R] = \\Coker(ds-dt: TR \\to TU).", "$$", "\\item The space of infinitesimal automorphisms of $[U/R]$ is", "$$", "\\text{Inf}([U/R]) =", "\\Ker(ds \\oplus dt : TR \\to TU \\oplus TU).", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $U$ and $R$ are deformation functors $[U/R]$ is a predeformation", "category. Since (RS) holds for deformation functors by", "definition we see that (RS) holds for [U/R] by", "Lemma \\ref{lemma-smooth-RS-groupoid-in-functors-quotient}.", "Hence $[U/R]$ is a deformation category. Statements (2) and (3)", "follow directly from the definitions." ], "refs": [ "formal-defos-lemma-smooth-RS-groupoid-in-functors-quotient" ], "ref_ids": [ 3488 ] } ], "ref_ids": [] }, { "id": 3490, "type": "theorem", "label": "formal-defos-lemma-presentation-construction", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-presentation-construction", "contents": [ "Let $\\mathcal{F}$ be category cofibered in groupoids over a category", "$\\mathcal{C}$. Let $U : \\mathcal{C} \\to \\textit{Sets}$ be a functor.", "Let $f : U \\to \\mathcal{F}$ be a morphism of categories cofibered in groupoids", "over $\\mathcal{C}$. Define $R, s, t, c$ as follows:", "\\begin{enumerate}", "\\item $R : \\mathcal{C} \\to \\textit{Sets}$ is the functor", "$U \\times_{f, \\mathcal{F}, f} U$.", "\\item $t, s : R \\to U$ are the first and second projections,", "respectively.", "\\item $c : R \\times_{s, U, t} R \\to R$ is the morphism given by projection", "onto the first and last factors of", "$U \\times_{f, \\mathcal{F}, f} U \\times_{f, \\mathcal{F}, f} U$", "under the canonical isomorphism", "$R \\times_{s, U, t} R \\to", "U \\times_{f, \\mathcal{F}, f} U \\times_{f, \\mathcal{F}, f} U$.", "\\end{enumerate}", "Then $(U, R, s, t, c)$ is a groupoid in functors on $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3491, "type": "theorem", "label": "formal-defos-lemma-presentation-morphism", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-presentation-morphism", "contents": [ "Let $\\mathcal{F}$ be category cofibered in groupoids over a category", "$\\mathcal{C}$. Let $U : \\mathcal{C} \\to \\textit{Sets}$ be a functor.", "Let $f : U \\to \\mathcal{F}$ be a morphism of categories cofibered in groupoids", "over $\\mathcal{C}$. Let $(U, R, s, t, c)$ be the groupoid in functors on", "$\\mathcal{C}$ constructed from $f : U \\to \\mathcal{F}$ in", "Lemma \\ref{lemma-presentation-construction}.", "Then there is a natural morphism $[f] : [U/R] \\to \\mathcal{F}$ such that:", "\\begin{enumerate}", "\\item $[f]: [U/R] \\to \\mathcal{F}$ is fully faithful.", "\\item $[f]: [U/R] \\to \\mathcal{F}$ is an equivalence if and only if", "$f : U \\to \\mathcal{F}$ is essentially surjective.", "\\end{enumerate}" ], "refs": [ "formal-defos-lemma-presentation-construction" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 3490 ] }, { "id": 3492, "type": "theorem", "label": "formal-defos-lemma-smooth-groupoid-in-functors-construction", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-smooth-groupoid-in-functors-construction", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$. Let $U : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$", "be a functor. Let $f : U \\to \\mathcal{F}$ be a smooth morphism of", "categories cofibered in groupoids. Then:", "\\begin{enumerate}", "\\item If $(U, R, s, t, c)$ is the groupoid in functors on", "$\\mathcal{C}_\\Lambda$ constructed from $f : U \\to \\mathcal{F}$ in", "Lemma \\ref{lemma-presentation-construction}, then $(U, R, s, t, c)$", "is smooth.", "\\item If $f : U(k) \\to \\mathcal{F}(k)$ is essentially surjective,", "then the morphism $[f] : [U/R] \\to \\mathcal{F}$ of", "Lemma \\ref{lemma-presentation-morphism}", "is an equivalence.", "\\end{enumerate}" ], "refs": [ "formal-defos-lemma-presentation-construction", "formal-defos-lemma-presentation-morphism" ], "proofs": [ { "contents": [ "From the construction of", "Lemma \\ref{lemma-presentation-construction}", "we have a commutative diagram", "$$", "\\xymatrix{", "R = U \\times_{f, \\mathcal{F}, f} U \\ar[r]_-s \\ar[d]_t & U", "\\ar[d]^f \\\\", "U \\ar[r]^f & \\mathcal{F}", "}", "$$", "where $t, s$ are the first and second projections. So $t, s$ are smooth by", "Lemma \\ref{lemma-smooth-properties}. Hence (1) holds.", "\\medskip\\noindent", "If the assumption of (2) holds, then by", "Lemma \\ref{lemma-smooth-morphism-essentially-surjective}", "the morphism $f : U \\to \\mathcal{F}$ is essentially surjective. Hence by", "Lemma \\ref{lemma-presentation-morphism}", "the morphism $[f] : [U/R] \\to \\mathcal{F}$ is an equivalence." ], "refs": [ "formal-defos-lemma-presentation-construction", "formal-defos-lemma-smooth-properties", "formal-defos-lemma-smooth-morphism-essentially-surjective", "formal-defos-lemma-presentation-morphism" ], "ref_ids": [ 3490, 3433, 3434, 3491 ] } ], "ref_ids": [ 3490, 3491 ] }, { "id": 3493, "type": "theorem", "label": "formal-defos-lemma-deformation-functor-diagonal", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-deformation-functor-diagonal", "contents": [ "Let $\\mathcal{F}$ be a deformation category.", "Let $U : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a deformation functor.", "Let $f: U \\to \\mathcal{F}$ be a morphism of categories cofibered in groupoids.", "Then $U \\times_{f, \\mathcal{F}, f} U$ is a deformation functor", "with tangent space fitting into an exact sequence of $k$-vector spaces", "$$", "0 \\to \\text{Inf}(\\mathcal{F}) \\to", "T(U \\times_{f, \\mathcal{F}, f} U) \\to TU \\oplus TU", "$$" ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-deformation-categories-fiber-product-morphisms}", "and the fact that $\\text{Inf}(U) = (0)$." ], "refs": [ "formal-defos-lemma-deformation-categories-fiber-product-morphisms" ], "ref_ids": [ 3482 ] } ], "ref_ids": [] }, { "id": 3494, "type": "theorem", "label": "formal-defos-lemma-prorepresentable-groupoid-in-functors-construction", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-prorepresentable-groupoid-in-functors-construction", "contents": [ "Let $\\mathcal{F}$ be a deformation category.", "Let $U : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a prorepresentable functor.", "Let $f : U \\to \\mathcal{F}$ be a morphism of categories cofibered in groupoids.", "Let $(U, R, s, t, c)$ be the groupoid in functors on $\\mathcal{C}_\\Lambda$", "constructed from $f : U \\to \\mathcal{F}$ in", "Lemma \\ref{lemma-presentation-construction}. If", "$\\dim_k \\text{Inf}(\\mathcal{F}) < \\infty$, then", "$(U, R, s, t, c)$ is prorepresentable." ], "refs": [ "formal-defos-lemma-presentation-construction" ], "proofs": [ { "contents": [ "Note that $U$ is a deformation functor by", "Example \\ref{example-prorepresentable-deformation-functor}.", "By", "Lemma \\ref{lemma-deformation-functor-diagonal}", "we see that $R = U \\times_{f, \\mathcal{F}, f} U$", "is a deformation functor whose tangent space", "$TR = T(U \\times_{f, \\mathcal{F}, f} U)$ sits in an exact sequence", "$0 \\to \\text{Inf}(\\mathcal{F}) \\to TR \\to TU \\oplus TU$.", "Since we have assumed the first space has finite dimension and since", "$TU$ has finite dimension by", "Example \\ref{example-tangent-space-prorepresentable-functor}", "we see that $\\dim TR < \\infty$. The map", "$\\gamma : \\text{Der}_\\Lambda(k, k) \\to TR$ see (\\ref{equation-map})", "is injective because its composition with $TR \\to TU$ is injective by", "Theorem \\ref{theorem-Schlessinger-prorepresentability}", "for the prorepresentable functor $U$. Thus $R$ is prorepresentable by", "Theorem \\ref{theorem-Schlessinger-prorepresentability}.", "It follows from", "Lemma \\ref{lemma-groupoid-in-functors-prorep-equivalences}", "that $(U, R, s, t, c)$ is prorepresentable." ], "refs": [ "formal-defos-lemma-deformation-functor-diagonal", "formal-defos-theorem-Schlessinger-prorepresentability", "formal-defos-theorem-Schlessinger-prorepresentability", "formal-defos-lemma-groupoid-in-functors-prorep-equivalences" ], "ref_ids": [ 3493, 3411, 3411, 3486 ] } ], "ref_ids": [ 3490 ] }, { "id": 3495, "type": "theorem", "label": "formal-defos-lemma-characterize-minimal-groupoid-in-functors", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-characterize-minimal-groupoid-in-functors", "contents": [ "Let $(U, R, s, t, c)$ be a smooth prorepresentable groupoid in", "functors on $\\mathcal{C}_\\Lambda$.", "\\begin{enumerate}", "\\item $(U, R, s, t, c)$ is normalized if and only if the morphism", "$U \\to [U/R]$ induces an isomorphism on tangent spaces, and", "\\item $(U, R, s, t, c)$ is minimal if and only if the kernel of", "$TU \\to T[U/R]$ is contained in the image of", "$\\text{Der}_\\Lambda(k, k) \\to TU$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows immediately from the definitions.", "To see part (2) set $\\mathcal{F} = [U/R]$. Since $\\mathcal{F}$", "has a presentation it is a deformation category, see", "Theorem \\ref{theorem-presentation-deformation-groupoid}.", "In particular it satisfies (RS), (S1), and (S2), see", "Lemma \\ref{lemma-RS-implies-S1-S2}.", "Recall that minimal versal formal objects are unique up to isomorphism, see", "Lemma \\ref{lemma-minimal-versal}.", "By", "Theorem \\ref{theorem-miniversal-object-existence}", "a minimal versal object induces a map", "$\\underline{\\xi} : \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$", "satisfying (\\ref{equation-bijective-orbits}). Since", "$U \\cong \\underline{R}|_{\\mathcal{C}_\\Lambda}$ over $\\mathcal{F}$", "we see that $TU \\to T\\mathcal{F} = T[U/R]$ satisfies the property", "as stated in the lemma." ], "refs": [ "formal-defos-theorem-presentation-deformation-groupoid", "formal-defos-lemma-RS-implies-S1-S2", "formal-defos-lemma-minimal-versal", "formal-defos-theorem-miniversal-object-existence" ], "ref_ids": [ 3412, 3469, 3462, 3410 ] } ], "ref_ids": [] }, { "id": 3496, "type": "theorem", "label": "formal-defos-lemma-surjective-morphism-prorepresentable-functor", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-surjective-morphism-prorepresentable-functor", "contents": [ "Let $U: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a", "prorepresentable functor. Let $\\varphi : U \\to U$ be a morphism such", "that $d\\varphi : TU \\to TU$ is an isomorphism. Then $\\varphi$ is an", "isomorphism." ], "refs": [], "proofs": [ { "contents": [ "If $U \\cong \\underline{R}|_{\\mathcal{C}_\\Lambda}$ for some", "$R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$,", "then completing $\\varphi$ gives a morphism $\\underline{R} \\to \\underline{R}$.", "If $f: R \\to R$ is the corresponding morphism in", "$\\widehat{\\mathcal{C}}_\\Lambda$, then $f$ induces an isomorphism", "$\\text{Der}_\\Lambda(R, k) \\to \\text{Der}_\\Lambda(R, k)$, see", "Example \\ref{example-tangent-space-map-prorepresentable-functor}.", "In particular $f$ is a surjection by", "Lemma \\ref{lemma-derivations-surjective}.", "As a surjective endomorphism of a Noetherian ring is an isomorphism (see", "Algebra, Lemma \\ref{algebra-lemma-surjective-endo-noetherian-ring-is-iso})", "we conclude $f$, hence $\\underline{R}", "\\to \\underline{R}$, hence $\\varphi : U \\to U$", "is an isomorphism." ], "refs": [ "formal-defos-lemma-derivations-surjective", "algebra-lemma-surjective-endo-noetherian-ring-is-iso" ], "ref_ids": [ 3425, 457 ] } ], "ref_ids": [] }, { "id": 3497, "type": "theorem", "label": "formal-defos-lemma-minimal-prorepresentable-groupoid-autoequivalence", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-minimal-prorepresentable-groupoid-autoequivalence", "contents": [ "Let $(U, R, s, t, c)$ be a minimal smooth prorepresentable groupoid in", "functors on $\\mathcal{C}_\\Lambda$. If $\\varphi : [U/R] \\to [U/R]$ is an", "equivalence of categories cofibered in groupoids, then $\\varphi$ is an", "isomorphism." ], "refs": [], "proofs": [ { "contents": [ "A morphism $\\varphi : [U/R] \\to [U/R]$ is the same thing as a", "morphism $\\varphi : (U, R, s, t, c) \\to (U, R, s, t, c)$ of", "groupoids in functors over $\\mathcal{C}_\\Lambda$ as defined in", "Definition \\ref{definition-groupoid-in-functors}.", "Denote $\\phi : U \\to U$ and $\\psi : R \\to R$ the corresponding morphisms.", "Because the diagram", "$$", "\\xymatrix{", "& \\text{Der}_\\Lambda(k, k) \\ar[dr]_\\gamma \\ar[dl]^\\gamma \\\\", "TU \\ar[rr]_{d\\phi} \\ar[d] & & TU \\ar[d] \\\\", "T[U/R] \\ar[rr]^{d\\varphi} & & T[U/R]", "}", "$$", "is commutative, since $d\\varphi$ is bijective, and since we have", "the characterization of minimality in", "Lemma \\ref{lemma-characterize-minimal-groupoid-in-functors}", "we conclude that $d\\phi$ is injective (hence bijective by dimension reasons).", "Thus $\\phi : U \\to U$ is an isomorphism by", "Lemma \\ref{lemma-surjective-morphism-prorepresentable-functor}.", "We can use a similar argument, using the exact sequence", "$$", "0 \\to \\text{Inf}([U/R]) \\to TR \\to TU \\oplus TU", "$$", "of", "Lemma \\ref{lemma-deformation-functor-diagonal}", "to prove that $\\psi : R \\to R$ is an isomorphism. But is also a consequence", "of the fact that $R = U \\times_{[U/R]} U$ and that $\\varphi$ and $\\phi$", "are isomorphisms." ], "refs": [ "formal-defos-definition-groupoid-in-functors", "formal-defos-lemma-characterize-minimal-groupoid-in-functors", "formal-defos-lemma-surjective-morphism-prorepresentable-functor", "formal-defos-lemma-deformation-functor-diagonal" ], "ref_ids": [ 3535, 3495, 3496, 3493 ] } ], "ref_ids": [] }, { "id": 3498, "type": "theorem", "label": "formal-defos-lemma-minimal-prorepresentable-groupoid-equivalence", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-minimal-prorepresentable-groupoid-equivalence", "contents": [ "Let $(U, R, s, t, c)$ and $(U', R', s', t', c')$ be minimal smooth", "prorepresentable groupoids in functors on $\\mathcal{C}_\\Lambda$. If", "$\\varphi : [U/R] \\to [U'/R']$ is an equivalence of categories cofibered", "in groupoids, then $\\varphi$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Let $\\psi : [U'/R'] \\to [U/R]$ be a quasi-inverse to $\\varphi$.", "Then $\\psi \\circ \\varphi$ and $\\varphi \\circ \\psi$ are isomorphisms by", "Lemma \\ref{lemma-minimal-prorepresentable-groupoid-autoequivalence},", "hence $\\varphi$ and $\\psi$ are isomorphisms." ], "refs": [ "formal-defos-lemma-minimal-prorepresentable-groupoid-autoequivalence" ], "ref_ids": [ 3497 ] } ], "ref_ids": [] }, { "id": 3499, "type": "theorem", "label": "formal-defos-lemma-minimal-groupoid-in-functors-construction", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-minimal-groupoid-in-functors-construction", "contents": [ "Let $\\mathcal{F}$ be a deformation category such that", "$\\dim_k T\\mathcal{F} <\\infty$ and", "$\\dim_k \\text{Inf}(\\mathcal{F}) < \\infty$.", "Then there exists a minimal versal formal object $\\xi$ of $\\mathcal{F}$.", "Say $\\xi$ lies over $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$.", "Let $U = \\underline{R}|_{\\mathcal{C}_\\Lambda}$.", "Let $f = \\underline{\\xi} : U \\to \\mathcal{F}$ be the associated", "morphism. Let $(U, R, s, t, c)$ be the groupoid in functors on", "$\\mathcal{C}_\\Lambda$ constructed from $f : U \\to \\mathcal{F}$ in", "Lemma \\ref{lemma-presentation-construction}.", "Then $(U, R, s, t, c)$ is a minimal smooth prorepresentable", "groupoid in functors on $\\mathcal{C}_\\Lambda$ and there", "is an equivalence $[U/R] \\to \\mathcal{F}$." ], "refs": [ "formal-defos-lemma-presentation-construction" ], "proofs": [ { "contents": [ "As $\\mathcal{F}$ is a deformation category it satisfies (S1) and (S2), see", "Lemma \\ref{lemma-RS-implies-S1-S2}.", "By", "Lemma \\ref{lemma-versal-object-existence}", "there exists a versal formal object. By", "Lemma \\ref{lemma-minimal-versal}", "there exists a minimal versal formal object $\\xi/R$ as in the statement", "of the lemma. Setting", "$U = \\underline{R}|_{\\mathcal{C}_\\Lambda}$ the", "associated map $\\underline{\\xi} : U \\to \\mathcal{F}$ is smooth (this is", "the definition of a versal formal object).", "Let $(U, R, s, t, c)$ be the groupoid in functors constructed in", "Lemma \\ref{lemma-presentation-construction}", "from the map $\\underline{\\xi}$. By", "Lemma \\ref{lemma-smooth-groupoid-in-functors-construction}", "we see that $(U, R, s, t, c)$ is a smooth groupoid in functors and that", "$[U/R] \\to \\mathcal{F}$ is an equivalence. By", "Lemma \\ref{lemma-prorepresentable-groupoid-in-functors-construction}", "we see that $(U, R, s, t, c)$ is prorepresentable.", "Finally, $(U, R, s, t, c)$ is minimal because $U \\to [U/R] = \\mathcal{F}$", "corresponds to the minimal versal formal object $\\xi$." ], "refs": [ "formal-defos-lemma-RS-implies-S1-S2", "formal-defos-lemma-versal-object-existence", "formal-defos-lemma-minimal-versal", "formal-defos-lemma-presentation-construction", "formal-defos-lemma-smooth-groupoid-in-functors-construction", "formal-defos-lemma-prorepresentable-groupoid-in-functors-construction" ], "ref_ids": [ 3469, 3458, 3462, 3490, 3492, 3494 ] } ], "ref_ids": [ 3490 ] }, { "id": 3500, "type": "theorem", "label": "formal-defos-lemma-minimal-presentations-equivalent", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-minimal-presentations-equivalent", "contents": [ "Let $\\mathcal{F}$ be category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$. Assume there exist presentations of", "$\\mathcal{F}$ by minimal smooth prorepresentable groupoids", "in functors $(U, R, s, t, c)$ and $(U', R', s', t', c')$.", "Then $(U, R, s, t, c)$ and $(U', R', s', t', c')$ are isomorphic." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-minimal-prorepresentable-groupoid-equivalence}", "and the observation that a morphism", "$[U/R] \\to [U'/R']$ is the same thing as a morphism", "of groupoids in functors (by our explicit construction of $[U/R]$ in", "Definition \\ref{definition-quotient})." ], "refs": [ "formal-defos-lemma-minimal-prorepresentable-groupoid-equivalence", "formal-defos-definition-quotient" ], "ref_ids": [ 3498, 3538 ] } ], "ref_ids": [] }, { "id": 3501, "type": "theorem", "label": "formal-defos-lemma-homotopy", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-homotopy", "contents": [ "Being formally homotopic is an equivalence relation on", "sets of morphisms in $\\widehat{\\mathcal{C}}_\\Lambda$." ], "refs": [], "proofs": [ { "contents": [ "Suppose we have any $r \\geq 1$ and two maps", "$h_1, h_2 : R \\to R[[t_1, \\ldots, t_r]]$ such that", "$h_1(a) \\bmod (t_1, \\ldots, t_r) = h_2(a) \\bmod (t_1, \\ldots, t_r) = a$", "for all $a \\in R$ and a map $k : R[[t_1, \\ldots, t_r]] \\to S$.", "Then we claim $k \\circ h_1$ is formally homotopic to $k \\circ h_2$.", "The symmetric inherent in this claim will show that our notion", "of formally homotopic is symmetric. Namely, the map", "$$", "\\Psi :", "R[[t_1, \\ldots, t_r]]", "\\longrightarrow", "R[[t_1, \\ldots, t_r]],\\quad", "\\sum a_I t^I \\longmapsto \\sum h_1(a_I)t^I", "$$", "is an isomorphism. Set $h(a) = \\Psi^{-1}(h_2(a))$ for $a \\in R$ and", "$k' = k \\circ \\Psi$, then we see that $(r, h, k')$ is a formal", "homotopy between $k \\circ h_1$ and $k \\circ h_2$, proving the claim", "\\medskip\\noindent", "Say we have three maps $f_1, f_2, f_3 : R \\to S$ as above", "and a formal homotopy $(r_1, h_1, k_1)$ between $f_1$ and $f_2$", "and a formal homotopy $(r_2, h_2, k_2)$ between $f_3$ and $f_2$ (!).", "After relabeling the coordinates we may assume", "$h_2 : R \\to R[[t_{r_1 + 1}, \\ldots, t_{r_1 + r_2}]]$", "and", "$k_2 : R[[t_{r_1 + 1}, \\ldots, t_{r_1 + r_2}]] \\to S$.", "By choosing a suitable isomorphism", "$$", "R[[t_1, \\ldots, t_{r_1 + r_2}]]", "\\longrightarrow", "R[[t_{r_1 + 1}, \\ldots, t_{r_1 + r_2}]]", "\\widehat{\\otimes}_{h_2, R, h_1}", "R[[t_1, \\ldots, t_{r_1}]]", "$$", "we may fit these maps into a commutative diagram", "$$", "\\xymatrix{", "R \\ar[r]_{h_1} \\ar[d]^{h_2} &", "R[[t_1, \\ldots, t_{r_1}]] \\ar[d]^{h_2'} \\\\", "R[[t_{r_1 + 1}, \\ldots, t_{r_1 + r_2}]] \\ar[r]^{h_1'} &", "R[[t_1, \\ldots, t_{r_1 + r_2}]]", "}", "$$", "with $h_2'(t_i) = t_i$ for $1 \\leq i \\leq r_1$ and", "$h_1'(t_i) = t_i$ for $r_1 + 1 \\leq i \\leq r_2$.", "Some details omitted.", "Since this diagram is a pushout in the category", "$\\widehat{\\mathcal{C}}_\\Lambda$ (see proof of", "Lemma \\ref{lemma-CLambdahat-pushouts})", "and since $k_1 \\circ h_1 = f_2 = k_2 \\circ h_2$ we conclude", "there exists a map", "$$", "k : R[[t_1, \\ldots, t_{r_1 + r_2}]] \\to S", "$$", "with $k_1 = k \\circ h_2'$ and $k_2 = k \\circ h_1'$.", "Denote $h = h_1' \\circ h_2 = h_2' \\circ h_1$.", "Then we have", "\\begin{enumerate}", "\\item $k(h_1'(a)) = k_2(a) = f_3(a)$, and", "\\item $k(h_2'(a)) = k_1(a) = f_1(a)$.", "\\end{enumerate}", "By the claim in the first paragraph of the proof this shows that", "$f_1$ and $f_3$ are formally homotopic." ], "refs": [ "formal-defos-lemma-CLambdahat-pushouts" ], "ref_ids": [ 3422 ] } ], "ref_ids": [] }, { "id": 3502, "type": "theorem", "label": "formal-defos-lemma-composition-homotopic", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-composition-homotopic", "contents": [ "In the category $\\widehat{\\mathcal{C}}_\\Lambda$, if $f_1, f_2 : R \\to S$", "are formally homotopic and $g : S \\to S'$ is a morphism, then", "$g \\circ f_1$ and $g \\circ f_2$ are formally homotopic." ], "refs": [], "proofs": [ { "contents": [ "Namely, if $(r, h, k)$ is a formal homotopy between $f_1$ and $f_2$, then", "$(r, h, g \\circ k)$ is a formal homotopy between $g \\circ f_1$ and", "$g \\circ f_2$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3503, "type": "theorem", "label": "formal-defos-lemma-versal-unique-up-to-homotopy", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-versal-unique-up-to-homotopy", "contents": [ "Let $\\mathcal{F}$ be a deformation category over $\\mathcal{C}_\\Lambda$", "with $\\dim_k T\\mathcal{F} < \\infty$ and", "$\\dim_k \\text{Inf}(\\mathcal{F}) < \\infty$. Let $\\xi$ be a versal formal", "object lying over $R$. Let $\\eta$ be a formal object lying over $S$.", "Then any two maps", "$$", "f, g : R \\to S", "$$", "such that $f_*\\xi \\cong \\eta \\cong g_*\\xi$ are formally homotopic." ], "refs": [], "proofs": [ { "contents": [ "By Theorem \\ref{theorem-presentation-deformation-groupoid}", "and its proof, $\\mathcal{F}$ has a presentation by", "a smooth prorepresentable groupoid", "$$", "(\\underline{R}, \\underline{R_1}, s, t, c, e, i)|_{\\mathcal{C}_\\Lambda}", "$$", "in functors on $\\mathcal{C}_\\lambda$ such that $\\mathcal{F}$.", "Then the maps $s : R \\to R_1$ and $t : R \\to R_1$ are formally", "smooth ring maps and $e : R_1 \\to R$ is a section.", "In particular, we can choose an isomorphism", "$R_1 = R[[t_1, \\ldots, t_r]]$ for some $r \\geq 0$ such", "that $s$ is the embedding $R \\subset R[[t_1, \\ldots, t_r]]$", "and $t$ corresponds to a map $h : R \\to R[[t_1, \\ldots, t_r]]$", "with $h(a) \\bmod (t_1, \\ldots, t_r) = a$ for all $a \\in R$.", "The existence of the isomorphism $\\alpha : f_*\\xi \\to g_*\\xi$", "means exactly that there is a map $k : R_1 \\to S$", "such that $f = k \\circ s$ and $g = k \\circ t$.", "This exactly means that $(r, h, k)$ is a formal homotopy between", "$f$ and $g$." ], "refs": [ "formal-defos-theorem-presentation-deformation-groupoid" ], "ref_ids": [ 3412 ] } ], "ref_ids": [] }, { "id": 3504, "type": "theorem", "label": "formal-defos-lemma-homotopic-minimal-prime", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-homotopic-minimal-prime", "contents": [ "In the category $\\widehat{\\mathcal{C}}_\\Lambda$, if $f_1, f_2 : R \\to S$", "are formally homotopic and $\\mathfrak p \\subset R$ is a minimal", "prime ideal, then $f_1(\\mathfrak p)S = f_2(\\mathfrak p)S$ as ideals." ], "refs": [], "proofs": [ { "contents": [ "Suppose $(r, h, k)$ is a formal homotopy between $f_1$ and $f_2$.", "We claim that", "$\\mathfrak pR[[t_1, \\ldots, t_r]] = h(\\mathfrak p)R[[t_1, \\ldots, t_r]]$.", "The claim implies the lemma by further composing with $k$.", "To prove the claim, observe that the map", "$\\mathfrak p \\mapsto \\mathfrak pR[[t_1, \\ldots, t_r]]$", "is a bijection between the minimal prime ideals of $R$ and the", "minimal prime ideals of $R[[t_1, \\ldots, t_r]]$.", "Finally, $h(\\mathfrak p)R[[t_1, \\ldots, t_r]]$ is a minimal", "prime as $h$ is flat, and hence of the form", "$\\mathfrak q R[[t_1, \\ldots, t_r]]$ for some minimal", "prime $\\mathfrak q \\subset R$ by what we just said.", "But since $h \\bmod (t_1, \\ldots, t_r) = \\text{id}_R$ by", "definition of a formal homotopy, we conclude that", "$\\mathfrak q = \\mathfrak p$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3505, "type": "theorem", "label": "formal-defos-lemma-elementary-properties-change-of-field", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-elementary-properties-change-of-field", "contents": [ "With notation and assumptions as in Situation \\ref{situation-change-of-fields}.", "\\begin{enumerate}", "\\item We have $\\overline{\\mathcal{F}_{l/k}} = (\\overline{\\mathcal{F}})_{l/k}$.", "\\item If $\\mathcal{F}$ is a predeformation category, then $\\mathcal{F}_{l/k}$", "is a predeformation category.", "\\item If $\\mathcal{F}$ satisfies (S1), then $\\mathcal{F}_{l/k}$", "satisfies (S1).", "\\item If $\\mathcal{F}$ satisfies (S2), then $\\mathcal{F}_{l/k}$", "satisfies (S2).", "\\item If $\\mathcal{F}$ satisfies (RS), then $\\mathcal{F}_{l/k}$", "satisfies (RS).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is immediate from the definitions.", "\\medskip\\noindent", "Since $\\mathcal{F}_{l/k}(l) = \\mathcal{F}(k)$ part (2) follows from the", "definition, see Definition \\ref{definition-predeformation-category}.", "\\medskip\\noindent", "Part (3) follows as the functor (\\ref{equation-comparison}) commutes with", "fibre products and transforms surjective maps into surjective maps, see", "Definition \\ref{definition-S1-S2}.", "\\medskip\\noindent", "Part (4). To see this consider a diagram", "$$", "\\xymatrix{", " & l[\\epsilon] \\ar[d] \\\\", "B \\ar[r] & l", "}", "$$", "in $\\mathcal{C}_{\\Lambda, l}$ as in Definition \\ref{definition-S1-S2}.", "Applying the functor (\\ref{equation-comparison}) we obtain", "$$", "\\xymatrix{", " & k[l\\epsilon] \\ar[d] \\\\", "B \\times_l k \\ar[r] & k", "}", "$$", "where $l\\epsilon$ denotes the finite dimensional $k$-vector space", "$l\\epsilon \\subset l[\\epsilon]$. According to Lemma \\ref{lemma-S2-extensions}", "the condition of (S2) for $\\mathcal{F}$ also holds for this diagram.", "Hence (S2) holds for $\\mathcal{F}_{l/k}$.", "\\medskip\\noindent", "Part (5) follows from the characterization of (RS) in", "Lemma \\ref{lemma-RS-2-categorical} part (2) and the fact that", "(\\ref{equation-comparison}) commutes with fibre products." ], "refs": [ "formal-defos-definition-predeformation-category", "formal-defos-definition-S1-S2", "formal-defos-definition-S1-S2", "formal-defos-lemma-S2-extensions", "formal-defos-lemma-RS-2-categorical" ], "ref_ids": [ 3517, 3523, 3523, 3440, 3468 ] } ], "ref_ids": [] }, { "id": 3506, "type": "theorem", "label": "formal-defos-lemma-tangent-space-change-of-field", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-tangent-space-change-of-field", "contents": [ "With notation and assumptions as in Situation \\ref{situation-change-of-fields}.", "Assume $\\mathcal{F}$ is a predeformation category and", "$\\overline{\\mathcal{F}}$ satisfies (S2). Then there is a", "canonical $l$-vector space isomorphism", "$$", "T\\mathcal{F} \\otimes_k l \\longrightarrow T\\mathcal{F}_{l/k}", "$$", "of tangent spaces." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-elementary-properties-change-of-field} we may replace", "$\\mathcal{F}$ by $\\overline{\\mathcal{F}}$. Moreover we see that", "$T\\mathcal{F}$, resp.\\ $T\\mathcal{F}_{l/k}$ has a canonical", "$k$-vector space structure, resp.\\ $l$-vector space structure, see", "Lemma \\ref{lemma-tangent-space-vector-space}. Then", "$$", "T\\mathcal{F}_{l/k} = \\mathcal{F}_{l/k}(l[\\epsilon])", "= \\mathcal{F}(k[l\\epsilon]) = T\\mathcal{F} \\otimes_k l", "$$", "the last equality by Lemma \\ref{lemma-tangent-space-vector-space}.", "More generally, given a finite dimensional $l$-vector space $V$ we have", "$$", "\\mathcal{F}_{l/k}(l[V]) = \\mathcal{F}(k[V_k]) = T\\mathcal{F} \\otimes_k V_k", "$$", "where $V_k$ denotes $V$ seen as a $k$-vector space. We conclude that", "the functors $V \\mapsto \\mathcal{F}_{l/k}(l[V])$ and", "$V \\mapsto T\\mathcal{F} \\otimes_k V_k$ are canonically identified", "as functors to the category of sets. By Lemma \\ref{lemma-linear-functor}", "we see there is at most one way to turn either functor into an $l$-linear", "functor. Hence the isomorphisms are compatible with the $l$-vector space", "structures and we win." ], "refs": [ "formal-defos-lemma-elementary-properties-change-of-field", "formal-defos-lemma-tangent-space-vector-space", "formal-defos-lemma-tangent-space-vector-space", "formal-defos-lemma-linear-functor" ], "ref_ids": [ 3505, 3452, 3452, 3445 ] } ], "ref_ids": [] }, { "id": 3507, "type": "theorem", "label": "formal-defos-lemma-inf-aut-change-of-field", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-inf-aut-change-of-field", "contents": [ "With notation and assumptions as in Situation \\ref{situation-change-of-fields}.", "Assume $\\mathcal{F}$ is a deformation category.", "Then there is a", "canonical $l$-vector space isomorphism", "$$", "\\text{Inf}(\\mathcal{F}) \\otimes_k l", "\\longrightarrow", "\\text{Inf}(\\mathcal{F}_{l/k})", "$$", "of infinitesimal automorphism spaces." ], "refs": [], "proofs": [ { "contents": [ "Let $x_0 \\in \\Ob(\\mathcal{F}(k))$ and denote $x_{l, 0}$ the corresponding", "object of $\\mathcal{F}_{l/k}$ over $l$. Recall that", "$\\text{Inf}(\\mathcal{F}) = \\text{Inf}_{x_0}(\\mathcal{F})$ and", "$\\text{Inf}(\\mathcal{F}_{l/k}) = \\text{Inf}_{x_{l, 0}}(\\mathcal{F}_{l/k})$,", "see Remark \\ref{remark-trivial-aut-point}.", "Recall that the vector space structure on", "$\\text{Inf}_{x_0}(\\mathcal{F})$ comes from identifying it with the tangent", "space of the functor $\\mathit{Aut}(x_0)$ which is defined on", "the category $\\mathcal{C}_{k, k}$ of Artinian local $k$-algebras with", "residue field $k$. Similarly, $\\text{Inf}_{x_{l, 0}}(\\mathcal{F}_{l/k})$", "is the tangent space of $\\mathit{Aut}(x_{l, 0})$ which is defined on the", "category $\\mathcal{C}_{l, l}$ of Artinian local $l$-algebras with residue", "field $l$. Unwinding the definitions we see that", "$\\mathit{Aut}(x_{l, 0})$ is the restriction of $\\mathit{Aut}(x_0)_{l/k}$", "(which lives on $\\mathcal{C}_{k, l}$) to $\\mathcal{C}_{l, l}$. Since", "there is no difference between the tangent space of", "$\\mathit{Aut}(x_0)_{l/k}$ seen as a functor on $\\mathcal{C}_{k, l}$ or", "$\\mathcal{C}_{l, l}$, the lemma follows from ", "Lemma \\ref{lemma-tangent-space-change-of-field}", "and the fact that $\\mathit{Aut}(x_0)$ satisfies (RS) by", "Lemma \\ref{lemma-Aut-functor-RS} (whence we have (S2) by", "Lemma \\ref{lemma-RS-implies-S1-S2})." ], "refs": [ "formal-defos-remark-trivial-aut-point", "formal-defos-lemma-tangent-space-change-of-field", "formal-defos-lemma-Aut-functor-RS", "formal-defos-lemma-RS-implies-S1-S2" ], "ref_ids": [ 3575, 3506, 3475, 3469 ] } ], "ref_ids": [] }, { "id": 3508, "type": "theorem", "label": "formal-defos-lemma-change-of-fields-smooth", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-change-of-fields-smooth", "contents": [ "With notation and assumptions as in Situation \\ref{situation-change-of-fields}.", "If $\\mathcal{F} \\to \\mathcal{G}$ is a smooth morphism of categories cofibred", "in groupoids over $\\mathcal{C}_{\\Lambda, k}$, then", "$\\mathcal{F}_{l/k} \\to \\mathcal{G}_{l/k}$ is a smooth morphism of categories", "cofibred in groupoids over $\\mathcal{C}_{\\Lambda, l}$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the definitions and the fact that", "(\\ref{equation-comparison}) preserves surjections." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3509, "type": "theorem", "label": "formal-defos-lemma-change-of-field-versal-ring", "categories": [ "formal-defos" ], "title": "formal-defos-lemma-change-of-field-versal-ring", "contents": [ "With notation and assumptions as in Situation \\ref{situation-change-of-fields}.", "Let $\\xi$ be a versal formal object for $\\mathcal{F}$ lying over", "$R \\in \\Ob(\\widehat{\\mathcal{C}}_{\\Lambda, k})$. Then there exist", "\\begin{enumerate}", "\\item an $S \\in \\Ob(\\widehat{\\mathcal{C}}_{\\Lambda, l})$", "and a local $\\Lambda$-algebra homomorphism $R \\to S$ which is", "formally smooth in the $\\mathfrak m_S$-adic topology and induces", "the given field extension $l/k$ on residue fieds, and", "\\item a versal formal object of $\\mathcal{F}_{l/k}$ lying over $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Construction of $S$. Choose a surjection $R[x_1, \\ldots, x_n] \\to l$", "of $R$-algebras. The kernel is a maximal ideal $\\mathfrak m$.", "Set $S$ equal to the $\\mathfrak m$-adic completion of the Noetherian ring", "$R[x_1, \\ldots, x_n]$. Then $S$ is in $\\widehat{\\mathcal{C}}_{\\Lambda, l}$", "by Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian-Noetherian}.", "The map $R \\to S$ is formally smooth in the $\\mathfrak m_S$-adic topology by", "More on Algebra, Lemmas", "\\ref{more-algebra-lemma-formally-smooth} and", "\\ref{more-algebra-lemma-formally-smooth-completion}", "and the fact that $R \\to R[x_1, \\ldots, x_n]$ is formally smooth.", "(Compare with the proof Lemma \\ref{lemma-exists-smooth}.)", "\\medskip\\noindent", "Since $\\xi$ is versal, the transformation", "$\\underline{\\xi} : \\underline{R}|_{\\mathcal{C}_{\\Lambda, k}} \\to \\mathcal{F}$", "is smooth. By Lemma \\ref{lemma-change-of-fields-smooth} the induced map", "$$", "(\\underline{R}|_{\\mathcal{C}_{\\Lambda, k}})_{l/k}", "\\longrightarrow", "\\mathcal{F}_{l/k}", "$$", "is smooth. Thus it suffices to construct a smooth morphism", "$\\underline{S}|_{\\mathcal{C}_{\\Lambda, l}} \\to", "(\\underline{R}|_{\\mathcal{C}_{\\Lambda, k}})_{l/k}$.", "To give such a map means for every object $B$ of $\\mathcal{C}_{\\Lambda, l}$", "a map of sets", "$$", "\\Mor_{\\widehat{\\mathcal{C}}_{\\Lambda, l}}(S, B)", "\\longrightarrow", "\\Mor_{\\widehat{\\mathcal{C}}_{\\Lambda, k}}(R, B \\times_l k)", "$$", "functorial in $B$. Given an element $\\varphi : S \\to B$", "on the left hand side we send it to the composition $R \\to S \\to B$", "whose image is contained in the sub $\\Lambda$-algebra $B \\times_l k$.", "Smoothness of the map means that given a surjection", "$B' \\to B$ and a commutative diagram", "$$", "\\xymatrix{", "S \\ar[r] & B \\ar@{=}[r] & B \\\\", "R \\ar[u] \\ar[r] & B' \\times_l k \\ar[u] \\ar[r] & B' \\ar[u]", "}", "$$", "we have to find a ring map $S \\to B'$ fitting into the outer rectangle.", "The existence of this map is guaranteed as we chose $R \\to S$ to be", "formally smooth in the $\\mathfrak m_S$-adic topology, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-lift-continuous}." ], "refs": [ "algebra-lemma-completion-Noetherian-Noetherian", "more-algebra-lemma-formally-smooth", "more-algebra-lemma-formally-smooth-completion", "formal-defos-lemma-exists-smooth", "formal-defos-lemma-change-of-fields-smooth", "more-algebra-lemma-lift-continuous" ], "ref_ids": [ 874, 10014, 10015, 3438, 3508, 10016 ] } ], "ref_ids": [] }, { "id": 3586, "type": "theorem", "label": "adequate-lemma-adequate-finite-presentation", "categories": [ "adequate" ], "title": "adequate-lemma-adequate-finite-presentation", "contents": [ "Let $A$ be a ring.", "Let $F$ be an adequate functor on $\\textit{Alg}_A$.", "If $B = \\colim B_i$ is a filtered", "colimit of $A$-algebras, then $F(B) = \\colim F(B_i)$." ], "refs": [], "proofs": [ { "contents": [ "This holds because for any $A$-module $M$ we have", "$M \\otimes_A B = \\colim M \\otimes_A B_i$ (see", "Algebra, Lemma \\ref{algebra-lemma-tensor-products-commute-with-limits})", "and because filtered colimits commute with exact sequences, see", "Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}." ], "refs": [ "algebra-lemma-tensor-products-commute-with-limits", "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 363, 343 ] } ], "ref_ids": [] }, { "id": 3587, "type": "theorem", "label": "adequate-lemma-adequate-flat", "categories": [ "adequate" ], "title": "adequate-lemma-adequate-flat", "contents": [ "Let $A$ be a ring.", "Let $F$ be an adequate functor on $\\textit{Alg}_A$.", "If $B \\to B'$ is flat, then $F(B) \\otimes_B B' \\to F(B')$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Choose an exact sequence $0 \\to F \\to \\underline{M} \\to \\underline{N}$.", "This gives the diagram", "$$", "\\xymatrix{", "0 \\ar[r] & F(B) \\otimes_B B' \\ar[r] \\ar[d] &", "(M \\otimes_A B)\\otimes_B B' \\ar[r] \\ar[d] &", "(N \\otimes_A B)\\otimes_B B' \\ar[d] \\\\", "0 \\ar[r] & F(B') \\ar[r] &", "M \\otimes_A B' \\ar[r] &", "N \\otimes_A B'", "}", "$$", "where the rows are exact (the top one because $B \\to B'$ is flat).", "Since the right two vertical arrows are isomorphisms, so is the", "left one." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3588, "type": "theorem", "label": "adequate-lemma-adequate-surjection-from-linear", "categories": [ "adequate" ], "title": "adequate-lemma-adequate-surjection-from-linear", "contents": [ "Let $A$ be a ring.", "Let $F$ be an adequate functor on $\\textit{Alg}_A$. Then there exists a", "surjection $L \\to F$ with $L$ a direct sum of linearly adequate functors." ], "refs": [], "proofs": [ { "contents": [ "Choose an exact sequence $0 \\to F \\to \\underline{M} \\to \\underline{N}$", "where $\\underline{M} \\to \\underline{N}$ is given by", "$\\varphi : M \\to N$. By", "Lemma \\ref{lemma-adequate-finite-presentation}", "it suffices to construct $L \\to F$ such that $L(B) \\to F(B)$ is surjective", "for every finitely presented $A$-algebra $B$. Hence it suffices to construct,", "given a finitely presented $A$-algebra $B$ and an element $\\xi \\in F(B)$", "a map $L \\to F$ with $L$ linearly adequate such that $\\xi$ is in the image", "of $L(B) \\to F(B)$.", "(Because there is a set worth of such pairs $(B, \\xi)$ up to isomorphism.)", "\\medskip\\noindent", "To do this write $\\sum_{i = 1, \\ldots, n} m_i \\otimes b_i$ the image of", "$\\xi$ in $\\underline{M}(B) = M \\otimes_A B$. We know that", "$\\sum \\varphi(m_i) \\otimes b_i = 0$ in $N \\otimes_A B$.", "As $N$ is a filtered colimit of finitely presented $A$-modules, we can", "find a finitely presented $A$-module $N'$, a commutative diagram", "of $A$-modules", "$$", "\\xymatrix{", "A^{\\oplus n} \\ar[r] \\ar[d]_{m_1, \\ldots, m_n} & N' \\ar[d] \\\\", "M \\ar[r] & N", "}", "$$", "such that $(b_1, \\ldots, b_n)$ maps to zero in $N' \\otimes_A B$.", "Choose a presentation $A^{\\oplus l} \\to A^{\\oplus k} \\to N' \\to 0$.", "Choose a lift $A^{\\oplus n} \\to A^{\\oplus k}$ of the map", "$A^{\\oplus n} \\to N'$ of the diagram. Then we see that there exist", "$(c_1, \\ldots, c_l) \\in B^{\\oplus l}$ such that", "$(b_1, \\ldots, b_n, c_1, \\ldots, c_l)$ maps to zero in $B^{\\oplus k}$", "under the map $B^{\\oplus n} \\oplus B^{\\oplus l} \\to B^{\\oplus k}$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "A^{\\oplus n} \\oplus A^{\\oplus l} \\ar[r] \\ar[d] & A^{\\oplus k} \\ar[d] \\\\", "M \\ar[r] & N", "}", "$$", "where the left vertical arrow is zero on the summand $A^{\\oplus l}$.", "Then we see that $L$ equal to the kernel of $\\underline{A^{\\oplus n + l}}", "\\to \\underline{A^{\\oplus k}}$ works because the element", "$(b_1, \\ldots, b_n, c_1, \\ldots, c_l) \\in L(B)$ maps to $\\xi$." ], "refs": [ "adequate-lemma-adequate-finite-presentation" ], "ref_ids": [ 3586 ] } ], "ref_ids": [] }, { "id": 3589, "type": "theorem", "label": "adequate-lemma-flat-functor-split", "categories": [ "adequate" ], "title": "adequate-lemma-flat-functor-split", "contents": [ "Let $A$ be a ring.", "Let $F$ be a module-valued functor on $\\textit{Alg}_A$.", "Assume that for $B \\to B'$ flat the map", "$F(B) \\otimes_B B' \\to F(B')$ is an isomorphism.", "Let $B$ be a graded $A$-algebra. Then", "\\begin{enumerate}", "\\item $F(B) = \\bigoplus_{k \\in \\mathbf{Z}} F(B)^{(k)}$, and", "\\item the map $B \\to B_0 \\to B$ induces map $F(B) \\to F(B)$", "whose image is contained in $F(B)^{(0)}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in F(B)$. The map $p : B \\to B[t, t^{-1}]$ is free", "hence we know that", "$$", "F(B[t, t^{-1}]) =", "\\bigoplus\\nolimits_{k \\in \\mathbf{Z}} F(p)(F(B)) \\cdot t^k =", "\\bigoplus\\nolimits_{k \\in \\mathbf{Z}} F(B) \\cdot t^k", "$$", "as indicated we drop the $F(p)$ in the rest of the proof.", "Write $F(a)(x) = \\sum t^k x_k$ for some $x_k \\in F(B)$.", "Denote $\\epsilon : B[t, t^{-1}] \\to B$", "the $B$-algebra map $t \\mapsto 1$. Note that the compositions", "$\\epsilon \\circ p, \\epsilon \\circ a : B \\to B[t, t^{-1}] \\to B$ are", "the identity. Hence we see that", "$$", "x = F(\\epsilon)(F(a)(x)) = F(\\epsilon)(\\sum t^k x_k) = \\sum x_k.", "$$", "On the other hand, we claim that $x_k \\in F(B)^{(k)}$. Namely, consider", "the commutative diagram", "$$", "\\xymatrix{", "B \\ar[r]_a \\ar[d]_{a'} &", "B[t, t^{-1}] \\ar[d]^f \\\\", "B[s, s^{-1}] \\ar[r]^-g &", "B[t, s, t^{-1}, s^{-1}]", "}", "$$", "where $a'(b) = s^{\\deg(b)}b$, $f(b) = b$, $f(t) = st$ and", "$g(b) = t^{\\deg(b)}b$ and $g(s) = s$. Then", "$$", "F(g)(F(a'))(x) = F(g)(\\sum s^k x_k) =", "\\sum s^k F(a)(x_k)", "$$", "and going the other way we see", "$$", "F(f)(F(a))(x) = F(f)(\\sum t^k x_k) = \\sum (st)^k x_k.", "$$", "Since $B \\to B[s, t, s^{-1}, t^{-1}]$ is free we see that", "$F(B[t, s, t^{-1}, s^{-1}]) =", "\\bigoplus_{k, l \\in \\mathbf{Z}} F(B) \\cdot t^ks^l$ and", "comparing coefficients in the expressions above we find", "$F(a)(x_k) = t^k x_k$ as desired.", "\\medskip\\noindent", "Finally, the image of $F(B_0) \\to F(B)$ is contained in $F(B)^{(0)}$", "because $B_0 \\to B \\xrightarrow{a} B[t, t^{-1}]$ is equal to", "$B_0 \\to B \\xrightarrow{p} B[t, t^{-1}]$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3590, "type": "theorem", "label": "adequate-lemma-lift-map", "categories": [ "adequate" ], "title": "adequate-lemma-lift-map", "contents": [ "Let $A$ be a ring. Given a solid diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "L \\ar[d]_\\varphi \\ar[r] &", "\\underline{A^{\\oplus n}} \\ar[r] \\ar@{..>}[ld] &", "\\underline{A^{\\oplus m}} \\\\", "& \\underline{M}", "}", "$$", "of module-valued functors on $\\textit{Alg}_A$", "with exact row there exists a dotted arrow making the diagram commute." ], "refs": [], "proofs": [ { "contents": [ "Suppose that the map $A^{\\oplus n} \\to A^{\\oplus m}$ is given by the", "$m \\times n$-matrix $(a_{ij})$. Consider the ring", "$B = A[x_1, \\ldots, x_n]/(\\sum a_{ij}x_j)$. The element", "$(x_1, \\ldots, x_n) \\in \\underline{A^{\\oplus n}}(B)$ maps to zero in", "$\\underline{A^{\\oplus m}}(B)$ hence is the image of a unique element", "$\\xi \\in L(B)$. Note that $\\xi$ has the following universal property:", "for any $A$-algebra $C$ and any $\\xi' \\in L(C)$ there exists an $A$-algebra", "map $B \\to C$ such that $\\xi$ maps to $\\xi'$ via the map $L(B) \\to L(C)$.", "\\medskip\\noindent", "Note that $B$ is a graded $A$-algebra, hence we can use", "Lemmas \\ref{lemma-flat-functor-split} and \\ref{lemma-adequate-flat}", "to decompose the values of our functors on $B$ into graded pieces.", "Note that $\\xi \\in L(B)^{(1)}$ as $(x_1, \\ldots, x_n)$ is an element", "of degree one in $\\underline{A^{\\oplus n}}(B)$. Hence we see that", "$\\varphi(\\xi) \\in \\underline{M}(B)^{(1)} = M \\otimes_A B_1$.", "Since $B_1$ is generated by $x_1, \\ldots, x_n$ as an $A$-module we", "can write $\\varphi(\\xi) = \\sum m_i \\otimes x_i$. Consider the map", "$A^{\\oplus n} \\to M$ which maps the $i$th basis vector to $m_i$.", "By construction the associated map", "$\\underline{A^{\\oplus n}} \\to \\underline{M}$", "maps the element $\\xi$ to $\\varphi(\\xi)$. It follows from the", "universal property mentioned above that the diagram commutes." ], "refs": [ "adequate-lemma-flat-functor-split", "adequate-lemma-adequate-flat" ], "ref_ids": [ 3589, 3587 ] } ], "ref_ids": [] }, { "id": 3591, "type": "theorem", "label": "adequate-lemma-cokernel-into-module", "categories": [ "adequate" ], "title": "adequate-lemma-cokernel-into-module", "contents": [ "Let $A$ be a ring.", "Let $\\varphi : F \\to \\underline{M}$ be a map of module-valued functors", "on $\\textit{Alg}_A$ with $F$ adequate.", "Then $\\Coker(\\varphi)$ is adequate." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-adequate-surjection-from-linear}", "we may assume that $F = \\bigoplus L_i$ is a direct sum of linearly adequate", "functors. Choose exact sequences", "$0 \\to L_i \\to \\underline{A^{\\oplus n_i}} \\to \\underline{A^{\\oplus m_i}}$.", "For each $i$ choose a map $A^{\\oplus n_i} \\to M$ as in", "Lemma \\ref{lemma-lift-map}.", "Consider the diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\bigoplus L_i \\ar[r] \\ar[d] &", "\\bigoplus \\underline{A^{\\oplus n_i}} \\ar[r] \\ar[ld] &", "\\bigoplus \\underline{A^{\\oplus m_i}} \\\\", "& \\underline{M}", "}", "$$", "Consider the $A$-modules", "$$", "Q =", "\\Coker(\\bigoplus A^{\\oplus n_i} \\to M \\oplus \\bigoplus A^{\\oplus m_i})", "\\quad\\text{and}\\quad", "P = \\Coker(\\bigoplus A^{\\oplus n_i} \\to \\bigoplus A^{\\oplus m_i}).", "$$", "Then we see that $\\Coker(\\varphi)$ is isomorphic to the", "kernel of $\\underline{Q} \\to \\underline{P}$." ], "refs": [ "adequate-lemma-adequate-surjection-from-linear", "adequate-lemma-lift-map" ], "ref_ids": [ 3588, 3590 ] } ], "ref_ids": [] }, { "id": 3592, "type": "theorem", "label": "adequate-lemma-cokernel-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-cokernel-adequate", "contents": [ "\\begin{slogan}", "The cokernel of a map of adequate functors on the category of algebras", "over a ring is adequate.", "\\end{slogan}", "Let $A$ be a ring.", "Let $\\varphi : F \\to G$ be a map of adequate functors on $\\textit{Alg}_A$.", "Then $\\Coker(\\varphi)$ is adequate." ], "refs": [], "proofs": [ { "contents": [ "Choose an injection $G \\to \\underline{M}$.", "Then we have an injection $G/F \\to \\underline{M}/F$. By", "Lemma \\ref{lemma-cokernel-into-module}", "we see that $\\underline{M}/F$ is adequate, hence we can find an injection", "$\\underline{M}/F \\to \\underline{N}$.", "Composing we obtain an injection $G/F \\to \\underline{N}$. By", "Lemma \\ref{lemma-cokernel-into-module}", "the cokernel of the induced map $G \\to \\underline{N}$ is adequate", "hence we can find an injection $\\underline{N}/G \\to \\underline{K}$.", "Then $0 \\to G/F \\to \\underline{N} \\to \\underline{K}$ is exact and", "we win." ], "refs": [ "adequate-lemma-cokernel-into-module", "adequate-lemma-cokernel-into-module" ], "ref_ids": [ 3591, 3591 ] } ], "ref_ids": [] }, { "id": 3593, "type": "theorem", "label": "adequate-lemma-kernel-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-kernel-adequate", "contents": [ "Let $A$ be a ring.", "Let $\\varphi : F \\to G$ be a map of adequate functors on $\\textit{Alg}_A$.", "Then $\\Ker(\\varphi)$ is adequate." ], "refs": [], "proofs": [ { "contents": [ "Choose an injection $F \\to \\underline{M}$ and an injection", "$G \\to \\underline{N}$. Denote $F \\to \\underline{M \\oplus N}$", "the diagonal map so that", "$$", "\\xymatrix{", "F \\ar[d] \\ar[r] & G \\ar[d] \\\\", "\\underline{M \\oplus N} \\ar[r] & \\underline{N}", "}", "$$", "commutes. By", "Lemma \\ref{lemma-cokernel-adequate}", "we can find a module map $M \\oplus N \\to K$ such that", "$F$ is the kernel of $\\underline{M \\oplus N} \\to \\underline{K}$.", "Then $\\Ker(\\varphi)$ is the kernel of", "$\\underline{M \\oplus N} \\to \\underline{K \\oplus N}$." ], "refs": [ "adequate-lemma-cokernel-adequate" ], "ref_ids": [ 3592 ] } ], "ref_ids": [] }, { "id": 3594, "type": "theorem", "label": "adequate-lemma-colimit-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-colimit-adequate", "contents": [ "Let $A$ be a ring.", "An arbitrary direct sum of adequate functors on $\\textit{Alg}_A$", "is adequate. A colimit of adequate functors is adequate." ], "refs": [], "proofs": [ { "contents": [ "The statement on direct sums is immediate.", "A general colimit can be written as a kernel of a map between", "direct sums, see", "Categories, Lemma \\ref{categories-lemma-colimits-coproducts-coequalizers}.", "Hence this follows from", "Lemma \\ref{lemma-kernel-adequate}." ], "refs": [ "categories-lemma-colimits-coproducts-coequalizers", "adequate-lemma-kernel-adequate" ], "ref_ids": [ 12214, 3593 ] } ], "ref_ids": [] }, { "id": 3595, "type": "theorem", "label": "adequate-lemma-flat-linear-functor", "categories": [ "adequate" ], "title": "adequate-lemma-flat-linear-functor", "contents": [ "Let $A$ be a ring.", "Let $F, G$ be module-valued functors on $\\textit{Alg}_A$.", "Let $\\varphi : F \\to G$ be a transformation of functors. Assume", "\\begin{enumerate}", "\\item $\\varphi$ is additive,", "\\item for every $A$-algebra $B$ and $\\xi \\in F(B)$ and unit", "$u \\in B^*$ we have $\\varphi(u\\xi) = u\\varphi(\\xi)$ in $G(B)$, and", "\\item for any flat ring map $B \\to B'$ we have", "$G(B) \\otimes_B B' = G(B')$.", "\\end{enumerate}", "Then $\\varphi$ is a morphism of module-valued functors." ], "refs": [], "proofs": [ { "contents": [ "Let $B$ be an $A$-algebra, $\\xi \\in F(B)$, and $b \\in B$. We have to show", "that $\\varphi(b \\xi) = b \\varphi(\\xi)$. Consider the ring map", "$$", "B \\to B' = B[x, y, x^{-1}, y^{-1}]/(x + y - b).", "$$", "This ring map is faithfully flat, hence $G(B) \\subset G(B')$. On the", "other hand", "$$", "\\varphi(b\\xi) = \\varphi((x + y)\\xi) =", "\\varphi(x\\xi) + \\varphi(y\\xi) = x\\varphi(\\xi) + y\\varphi(\\xi)", "= (x + y)\\varphi(\\xi) = b\\varphi(\\xi)", "$$", "because $x, y$ are units in $B'$. Hence we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3596, "type": "theorem", "label": "adequate-lemma-extension-adequate-key", "categories": [ "adequate" ], "title": "adequate-lemma-extension-adequate-key", "contents": [ "Let $A$ be a ring.", "Let $0 \\to \\underline{M} \\to G \\to L \\to 0$ be a short exact sequence", "of module-valued functors on $\\textit{Alg}_A$ with $L$ linearly adequate.", "Then $G$ is adequate." ], "refs": [], "proofs": [ { "contents": [ "We first point out that for any flat $A$-algebra map", "$B \\to B'$ the map $G(B) \\otimes_B B' \\to G(B')$ is an isomorphism.", "Namely, this holds for $\\underline{M}$ and $L$, see", "Lemma \\ref{lemma-adequate-flat}", "and hence follows for $G$ by the five lemma. In particular, by", "Lemma \\ref{lemma-flat-functor-split}", "we see that $G(B) = \\bigoplus_{k \\in \\mathbf{Z}} G(B)^{(k)}$", "for any graded $A$-algebra $B$.", "\\medskip\\noindent", "Choose an exact sequence", "$0 \\to L \\to \\underline{A^{\\oplus n}} \\to \\underline{A^{\\oplus m}}$.", "Suppose that the map $A^{\\oplus n} \\to A^{\\oplus m}$ is given by the", "$m \\times n$-matrix $(a_{ij})$. Consider the graded $A$-algebra", "$B = A[x_1, \\ldots, x_n]/(\\sum a_{ij}x_j)$. The element", "$(x_1, \\ldots, x_n) \\in \\underline{A^{\\oplus n}}(B)$ maps to zero in", "$\\underline{A^{\\oplus m}}(B)$ hence is the image of a unique element", "$\\xi \\in L(B)$. Observe that $\\xi \\in L(B)^{(1)}$. The map", "$$", "\\Hom_A(B, C) \\longrightarrow L(C), \\quad", "f \\longmapsto L(f)(\\xi)", "$$", "defines an isomorphism of functors. The reason is that $f$ is", "determined by the images $c_i = f(x_i) \\in C$ which have to", "satisfy the relations $\\sum a_{ij}c_j = 0$. And $L(C)$ is the", "set of $n$-tuples $(c_1, \\ldots, c_n)$ satisfying the relations", "$\\sum a_{ij} c_j = 0$.", "\\medskip\\noindent", "Since the value of each of the functors $\\underline{M}$, $G$, $L$", "on $B$ is a direct sum of its weight spaces (by the lemma mentioned", "above) exactness of $0 \\to \\underline{M} \\to G \\to L \\to 0$ implies", "the sequence $0 \\to \\underline{M}(B)^{(1)} \\to G(B)^{(1)} \\to L(B)^{(1)} \\to 0$", "is exact. Thus we may choose an element $\\theta \\in G(B)^{(1)}$ mapping", "to $\\xi$.", "\\medskip\\noindent", "Consider the graded $A$-algebra", "$$", "C = A[x_1, \\ldots, x_n, y_1, \\ldots, y_n]/", "(\\sum a_{ij}x_j, \\sum a_{ij}y_j)", "$$", "There are three graded $A$-algebra homomorphisms $p_1, p_2, m : B \\to C$", "defined by the rules", "$$", "p_1(x_i) = x_i, \\quad", "p_1(x_i) = y_i, \\quad", "m(x_i) = x_i + y_i.", "$$", "We will show that the element", "$$", "\\tau = G(m)(\\theta) - G(p_1)(\\theta) - G(p_2)(\\theta) \\in G(C)", "$$", "is zero. First, $\\tau$ maps to zero in $L(C)$ by a direct calculation.", "Hence $\\tau$ is an element of $\\underline{M}(C)$.", "Moreover, since $m$, $p_1$, $p_2$ are graded algebra maps we see", "that $\\tau \\in G(C)^{(1)}$ and since $\\underline{M} \\subset G$", "we conclude", "$$", "\\tau \\in \\underline{M}(C)^{(1)} = M \\otimes_A C_1.", "$$", "We may write uniquely", "$\\tau = \\underline{M}(p_1)(\\tau_1) + \\underline{M}(p_2)(\\tau_2)$ with", "$\\tau_i \\in M \\otimes_A B_1 = \\underline{M}(B)^{(1)}$ because", "$C_1 = p_1(B_1) \\oplus p_2(B_1)$.", "Consider the ring map $q_1 : C \\to B$ defined by $x_i \\mapsto x_i$ and", "$y_i \\mapsto 0$. Then", "$\\underline{M}(q_1)(\\tau) =", "\\underline{M}(q_1)(\\underline{M}(p_1)(\\tau_1) + \\underline{M}(p_2)(\\tau_2)) =", "\\tau_1$.", "On the other hand, because", "$q_1 \\circ m = q_1 \\circ p_1$ we see that", "$G(q_1)(\\tau) = - G(q_1 \\circ p_2)(\\tau)$. Since $q_1 \\circ p_2$ factors as", "$B \\to A \\to B$ we see that $G(q_1 \\circ p_2)(\\tau)$ is in", "$G(B)^{(0)}$, see", "Lemma \\ref{lemma-flat-functor-split}.", "Hence $\\tau_1 = 0$ because it is in", "$G(B)^{(0)} \\cap \\underline{M}(B)^{(1)} \\subset", "G(B)^{(0)} \\cap G(B)^{(1)} = 0$.", "Similarly $\\tau_2 = 0$, whence $\\tau = 0$.", "\\medskip\\noindent", "Since $\\theta \\in G(B)$ we obtain a transformation of functors", "$$", "\\psi : L(-) = \\Hom_A(B, - ) \\longrightarrow G(-)", "$$", "by mapping $f : B \\to C$ to $G(f)(\\theta)$. Since $\\theta$ is a lift of", "$\\xi$ the map $\\psi$ is a right inverse of $G \\to L$. In terms of", "$\\psi$ the statements proved above have the following meaning:", "$\\tau = 0$ means that $\\psi$ is additive and", "$\\theta \\in G(B)^{(1)}$ implies that for any $A$-algebra $D$ we have", "$\\psi(ul) = u\\psi(l)$ in $G(D)$ for $l \\in L(D)$ and $u \\in D^*$ a unit.", "This implies that $\\psi$ is a morphism of module-valued functors, see", "Lemma \\ref{lemma-flat-linear-functor}.", "Clearly this implies that $G \\cong \\underline{M} \\oplus L$ and we win." ], "refs": [ "adequate-lemma-adequate-flat", "adequate-lemma-flat-functor-split", "adequate-lemma-flat-functor-split", "adequate-lemma-flat-linear-functor" ], "ref_ids": [ 3587, 3589, 3589, 3595 ] } ], "ref_ids": [] }, { "id": 3597, "type": "theorem", "label": "adequate-lemma-extension-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-extension-adequate", "contents": [ "Let $A$ be a ring.", "Let $0 \\to F \\to G \\to H \\to 0$ be a short exact sequence of", "module-valued functors on $\\textit{Alg}_A$.", "If $F$ and $H$ are adequate, so is $G$." ], "refs": [], "proofs": [ { "contents": [ "Choose an exact sequence $0 \\to F \\to \\underline{M} \\to \\underline{N}$.", "If we can show that $(\\underline{M} \\oplus G)/F$ is adequate, then", "$G$ is the kernel of the map of adequate functors", "$(\\underline{M} \\oplus G)/F \\to \\underline{N}$, hence", "adequate by", "Lemma \\ref{lemma-kernel-adequate}.", "Thus we may assume $F = \\underline{M}$.", "\\medskip\\noindent", "We can choose a surjection $L \\to H$ where $L$ is a direct sum of", "linearly adequate functors, see", "Lemma \\ref{lemma-adequate-surjection-from-linear}.", "If we can show that the pullback $G \\times_H L$ is adequate, then", "$G$ is the cokernel of the map $\\Ker(L \\to H) \\to G \\times_H L$", "hence adequate by", "Lemma \\ref{lemma-cokernel-adequate}.", "Thus we may assume that $H = \\bigoplus L_i$ is a direct sum of", "linearly adequate functors. By", "Lemma \\ref{lemma-extension-adequate-key}", "each of the pullbacks $G \\times_H L_i$ is adequate. By", "Lemma \\ref{lemma-colimit-adequate}", "we see that $\\bigoplus G \\times_H L_i$ is adequate.", "Then $G$ is the cokernel of", "$$", "\\bigoplus\\nolimits_{i \\not = i'} F \\longrightarrow", "\\bigoplus G \\times_H L_i", "$$", "where $\\xi$ in the summand $(i, i')$ maps to", "$(0, \\ldots, 0, \\xi, 0, \\ldots, 0, -\\xi, 0, \\ldots, 0)$", "with nonzero entries in the summands $i$ and $i'$.", "Thus $G$ is adequate by", "Lemma \\ref{lemma-cokernel-adequate}." ], "refs": [ "adequate-lemma-kernel-adequate", "adequate-lemma-adequate-surjection-from-linear", "adequate-lemma-cokernel-adequate", "adequate-lemma-extension-adequate-key", "adequate-lemma-colimit-adequate", "adequate-lemma-cokernel-adequate" ], "ref_ids": [ 3593, 3588, 3592, 3596, 3594, 3592 ] } ], "ref_ids": [] }, { "id": 3598, "type": "theorem", "label": "adequate-lemma-base-change-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-base-change-adequate", "contents": [ "Let $A \\to A'$ be a ring map. If $F$ is an adequate functor on", "$\\textit{Alg}_A$, then its restriction $F'$ to", "$\\textit{Alg}_{A'}$ is adequate too." ], "refs": [], "proofs": [ { "contents": [ "Choose an exact sequence $0 \\to F \\to \\underline{M} \\to \\underline{N}$.", "Then $F'(B') = F(B') = \\Ker(M \\otimes_A B' \\to N \\otimes_A B')$.", "Since $M \\otimes_A B' = M \\otimes_A A' \\otimes_{A'} B'$ and similarly", "for $N$ we see that $F'$ is the kernel of", "$\\underline{M \\otimes_A A'} \\to \\underline{N \\otimes_A A'}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3599, "type": "theorem", "label": "adequate-lemma-pushforward-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-pushforward-adequate", "contents": [ "Let $A \\to A'$ be a ring map. If $F'$ is an adequate functor on", "$\\textit{Alg}_{A'}$, then the module-valued functor", "$F : B \\mapsto F'(A' \\otimes_A B)$ on $\\textit{Alg}_A$ is adequate too." ], "refs": [], "proofs": [ { "contents": [ "Choose an exact sequence $0 \\to F' \\to \\underline{M'} \\to \\underline{N'}$.", "Then", "\\begin{align*}", "F(B) & = F'(A' \\otimes_A B) \\\\", "& = \\Ker(M' \\otimes_{A'} (", "A' \\otimes_A B) \\to N' \\otimes_{A'} (A' \\otimes_A B)) \\\\", "& = \\Ker(M' \\otimes_A B \\to N' \\otimes_A B)", "\\end{align*}", "Thus $F$ is the kernel of", "$\\underline{M} \\to \\underline{N}$", "where $M = M'$ and $N = N'$ viewed as $A$-modules." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3600, "type": "theorem", "label": "adequate-lemma-adequate-product", "categories": [ "adequate" ], "title": "adequate-lemma-adequate-product", "contents": [ "Let $A = A_1 \\times \\ldots \\times A_n$ be a product of rings.", "An adequate functor over $A$ is the same thing as a sequence", "$F_1, \\ldots, F_n$ of adequate functors $F_i$ over $A_i$." ], "refs": [], "proofs": [ { "contents": [ "This is true because an $A$-algebra $B$ is canonically a product", "$B_1 \\times \\ldots \\times B_n$ and the same thing holds for $A$-modules.", "Setting $F(B) = \\coprod F_i(B_i)$ gives the correspondence.", "Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3601, "type": "theorem", "label": "adequate-lemma-adequate-descent", "categories": [ "adequate" ], "title": "adequate-lemma-adequate-descent", "contents": [ "Let $A \\to A'$ be a ring map and let $F$ be a module-valued functor on", "$\\textit{Alg}_A$ such that", "\\begin{enumerate}", "\\item the restriction $F'$ of $F$ to the category of $A'$-algebras is", "adequate, and", "\\item for any $A$-algebra $B$ the sequence", "$$", "0 \\to F(B) \\to F(B \\otimes_A A') \\to F(B \\otimes_A A' \\otimes_A A')", "$$", "is exact.", "\\end{enumerate}", "Then $F$ is adequate." ], "refs": [], "proofs": [ { "contents": [ "The functors $B \\to F(B \\otimes_A A')$ and", "$B \\mapsto F(B \\otimes_A A' \\otimes_A A')$ are adequate, see", "Lemmas \\ref{lemma-pushforward-adequate} and", "\\ref{lemma-base-change-adequate}.", "Hence $F$ as a kernel of a map of adequate functors is adequate, see", "Lemma \\ref{lemma-kernel-adequate}." ], "refs": [ "adequate-lemma-pushforward-adequate", "adequate-lemma-base-change-adequate", "adequate-lemma-kernel-adequate" ], "ref_ids": [ 3599, 3598, 3593 ] } ], "ref_ids": [] }, { "id": 3602, "type": "theorem", "label": "adequate-lemma-adjoint", "categories": [ "adequate" ], "title": "adequate-lemma-adjoint", "contents": [ "Let $A$ be a ring.", "For every module-valued functor $F$ on $\\textit{Alg}_A$", "there exists a morphism $Q(F) \\to F$ of module-valued functors on", "$\\textit{Alg}_A$ such that (1) $Q(F)$ is adequate and (2) for every", "adequate functor $G$ the map $\\Hom(G, Q(F)) \\to \\Hom(G, F)$", "is a bijection." ], "refs": [], "proofs": [ { "contents": [ "Choose a set $\\{L_i\\}_{i \\in I}$ of linearly adequate functors such that", "every linearly adequate functor is isomorphic to one of the $L_i$.", "This is possible. Suppose that we can find $Q(F) \\to F$ with (1) and", "(2)' or every $i \\in I$ the map $\\Hom(L_i, Q(F)) \\to \\Hom(L_i, F)$", "is a bijection. Then (2) holds. Namely, combining", "Lemmas \\ref{lemma-adequate-surjection-from-linear} and", "\\ref{lemma-kernel-adequate}", "we see that every adequate functor $G$ sits in an exact sequence", "$$", "K \\to L \\to G \\to 0", "$$", "with $K$ and $L$ direct sums of linearly adequate functors. Hence (2)'", "implies that", "$\\Hom(L, Q(F)) \\to \\Hom(L, F)$", "and", "$\\Hom(K, Q(F)) \\to \\Hom(K, F)$", "are bijections, whence the same thing for $G$.", "\\medskip\\noindent", "Consider the category $\\mathcal{I}$ whose objects are pairs", "$(i, \\varphi)$ where $i \\in I$ and $\\varphi : L_i \\to F$ is a morphism.", "A morphism $(i, \\varphi) \\to (i', \\varphi')$ is a map", "$\\psi : L_i \\to L_{i'}$ such that $\\varphi' \\circ \\psi = \\varphi$.", "Set", "$$", "Q(F) = \\colim_{(i, \\varphi) \\in \\Ob(\\mathcal{I})} L_i", "$$", "There is a natural map $Q(F) \\to F$, by", "Lemma \\ref{lemma-colimit-adequate}", "it is adequate, and by construction it has property (2)'." ], "refs": [ "adequate-lemma-adequate-surjection-from-linear", "adequate-lemma-kernel-adequate", "adequate-lemma-colimit-adequate" ], "ref_ids": [ 3588, 3593, 3594 ] } ], "ref_ids": [] }, { "id": 3603, "type": "theorem", "label": "adequate-lemma-enough-injectives", "categories": [ "adequate" ], "title": "adequate-lemma-enough-injectives", "contents": [ "Let $A$ be a ring. Denote $\\mathcal{P}$ the category of module-valued", "functors on $\\textit{Alg}_A$ and $\\mathcal{A}$ the category of adequate", "functors on $\\textit{Alg}_A$. Denote $i : \\mathcal{A} \\to \\mathcal{P}$", "the inclusion functor. Denote $Q : \\mathcal{P} \\to \\mathcal{A}$", "the construction of Lemma \\ref{lemma-adjoint}.", "Then", "\\begin{enumerate}", "\\item $i$ is fully faithful, exact, and its image is a weak Serre subcategory,", "\\item $\\mathcal{P}$ has enough injectives,", "\\item the functor $Q$ is a right adjoint to $i$ hence left exact,", "\\item $Q$ transforms injectives into injectives,", "\\item $\\mathcal{A}$ has enough injectives.", "\\end{enumerate}" ], "refs": [ "adequate-lemma-adjoint" ], "proofs": [ { "contents": [ "This lemma just collects some facts we have already seen so far.", "Part (1) is clear from the definitions, the characterization of", "weak Serre subcategories (see", "Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}),", "and", "Lemmas \\ref{lemma-cokernel-adequate}, \\ref{lemma-kernel-adequate},", "and \\ref{lemma-extension-adequate}.", "Recall that $\\mathcal{P}$ is equivalent to the category", "$\\textit{PMod}((\\textit{Aff}/\\Spec(A))_\\tau, \\mathcal{O})$.", "Hence (2) by", "Injectives, Proposition \\ref{injectives-proposition-presheaves-modules}.", "Part (3) follows from", "Lemma \\ref{lemma-adjoint}", "and", "Categories, Lemma \\ref{categories-lemma-adjoint-exact}.", "Parts (4) and (5) follow from", "Homology, Lemmas \\ref{homology-lemma-adjoint-preserve-injectives} and", "\\ref{homology-lemma-adjoint-enough-injectives}." ], "refs": [ "homology-lemma-characterize-weak-serre-subcategory", "adequate-lemma-cokernel-adequate", "adequate-lemma-kernel-adequate", "adequate-lemma-extension-adequate", "injectives-proposition-presheaves-modules", "adequate-lemma-adjoint", "categories-lemma-adjoint-exact", "homology-lemma-adjoint-preserve-injectives", "homology-lemma-adjoint-enough-injectives" ], "ref_ids": [ 12046, 3592, 3593, 3597, 7806, 3602, 12249, 12116, 12117 ] } ], "ref_ids": [ 3602 ] }, { "id": 3604, "type": "theorem", "label": "adequate-lemma-tangent-functor", "categories": [ "adequate" ], "title": "adequate-lemma-tangent-functor", "contents": [ "Let $A$ be a ring. Let $F$ be a module valued functor.", "For every $B \\in \\Ob(\\textit{Alg}_A)$ and $B$-module $N$", "there is a canonical decomposition", "$$", "F(B[N]) = F(B) \\oplus TF(B, N)", "$$", "characterized by the following properties", "\\begin{enumerate}", "\\item $TF(B, N) = \\Ker(F(B[N]) \\to F(B))$,", "\\item there is a $B$-module structure $TF(B, N)$", "compatible with $B[N]$-module structure on $F(B[N])$,", "\\item $TF$ is a functor from the category of pairs $(B, N)$,", "\\item", "\\label{item-mult-map}", "there are canonical maps $N \\otimes_B F(B) \\to TF(B, N)$", "inducing a transformation between functors defined on the category", "of pairs $(B, N)$,", "\\item $TF(B, 0) = 0$ and the map $TF(B, N) \\to TF(B, N')$ is", "zero when $N \\to N'$ is the zero map.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $B \\to B[N] \\to B$ is the identity we see that $F(B) \\to F(B[N])$", "is a direct summand whose complement is $TF(N, B)$ as defined in (1).", "This construction is functorial in the pair $(B, N)$ simply because", "given a morphism of pairs $(B, N) \\to (B', N')$ we obtain a commutative", "diagram", "$$", "\\xymatrix{", "B' \\ar[r] & B'[N'] \\ar[r] & B' \\\\", "B \\ar[r] \\ar[u] & B[N] \\ar[r] \\ar[u] & B \\ar[u]", "}", "$$", "in $\\textit{Alg}_A$. The $B$-module structure comes from the $B[N]$-module", "structure and the ring map $B \\to B[N]$. The map in (4) is the", "composition", "$$", "N \\otimes_B F(B) \\longrightarrow", "B[N] \\otimes_{B[N]} F(B[N]) \\longrightarrow F(B[N])", "$$", "whose image is contained in $TF(B, N)$. (The first arrow uses the inclusions", "$N \\to B[N]$ and $F(B) \\to F(B[N])$ and the second arrow is the multiplication", "map.) If $N = 0$, then $B = B[N]$", "hence $TF(B, 0) = 0$. If $N \\to N'$ is zero then it factors as", "$N \\to 0 \\to N'$ hence the induced map is zero since $TF(B, 0) = 0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3605, "type": "theorem", "label": "adequate-lemma-tangent-injective", "categories": [ "adequate" ], "title": "adequate-lemma-tangent-injective", "contents": [ "Let $A$ be a ring. Let $I$ be an injective object of the category", "of module-valued functors. Then for any $B \\in \\Ob(\\textit{Alg}_A)$", "and short exact sequence", "$0 \\to N_1 \\to N \\to N_2 \\to 0$", "of $B$-modules the sequence", "$$", "TI(B, N_1) \\to TI(B, N) \\to TI(B, N_2) \\to 0", "$$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "We will use the results of", "Lemma \\ref{lemma-tangent-functor}", "without further mention.", "Denote $h : \\textit{Alg}_A \\to \\textit{Sets}$ the functor given by", "$h(C) = \\Mor_A(B[N], C)$. Similarly for $h_1$ and $h_2$.", "The map $B[N] \\to B[N_2]$ corresponding to the surjection $N \\to N_2$", "is surjective. It corresponds to a map $h_2 \\to h$ such that", "$h_2(C) \\to h(C)$ is injective for all $A$-algebras $C$. On the other", "hand, there are two maps $p, q : h \\to h_1$, corresponding to the", "zero map $N_1 \\to N$ and the injection $N_1 \\to N$. Note that", "$$", "\\xymatrix{", "h_2 \\ar[r] & h \\ar@<1ex>[r] \\ar@<-1ex>[r] & h_1", "}", "$$", "is an equalizer diagram. Denote $\\mathcal{O}_h$ the module-valued functor", "$C \\mapsto \\bigoplus_{h(C)} C$. Similarly for $\\mathcal{O}_{h_1}$ and", "$\\mathcal{O}_{h_2}$. Note that", "$$", "\\Hom_\\mathcal{P}(\\mathcal{O}_h, F) = F(B[N])", "$$", "where $\\mathcal{P}$ is the category of module-valued functors on", "$\\textit{Alg}_A$. We claim there is an equalizer diagram", "$$", "\\xymatrix{", "\\mathcal{O}_{h_2} \\ar[r] &", "\\mathcal{O}_h \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\mathcal{O}_{h_1}", "}", "$$", "in $\\mathcal{P}$. Namely, suppose that $C \\in \\Ob(\\textit{Alg}_A)$", "and $\\xi = \\sum_{i = 1, \\ldots, n} c_i \\cdot f_i$ where $c_i \\in C$ and", "$f_i : B[N] \\to C$ is an element of", "$\\mathcal{O}_h(C)$. If $p(\\xi) = q(\\xi)$, then", "we see that", "$$", "\\sum c_i \\cdot f_i \\circ z = \\sum c_i \\cdot f_i \\circ y", "$$", "where $z, y : B[N_1] \\to B[N]$ are the maps $z : (b, m_1) \\mapsto (b, 0)$", "and $y : (b, m_1) \\mapsto (b, m_1)$. This means that for every $i$", "there exists a $j$ such that $f_j \\circ z = f_i \\circ y$. Clearly, this", "implies that $f_i(N_1) = 0$, i.e., $f_i$ factors through a unique map", "$\\overline{f}_i : B[N_2] \\to C$. Hence $\\xi$ is the image of", "$\\overline{\\xi} = \\sum c_i \\cdot \\overline{f}_i$.", "Since $I$ is injective, it transforms this equalizer diagram", "into a coequalizer diagram", "$$", "\\xymatrix{", "I(B[N_1]) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "I(B[N]) \\ar[r] &", "I(B[N_2])", "}", "$$", "This diagram is compatible with the direct sum decompositions", "$I(B[N]) = I(B) \\oplus TI(B, N)$ and $I(B[N_i]) = I(B) \\oplus TI(B, N_i)$.", "The zero map $N \\to N_1$ induces the zero map $TI(B, N) \\to TI(B, N_1)$.", "Thus we see that the coequalizer property", "above means we have an exact sequence", "$TI(B, N_1) \\to TI(B, N) \\to TI(B, N_2) \\to 0$", "as desired." ], "refs": [ "adequate-lemma-tangent-functor" ], "ref_ids": [ 3604 ] } ], "ref_ids": [] }, { "id": 3606, "type": "theorem", "label": "adequate-lemma-exactness-implies", "categories": [ "adequate" ], "title": "adequate-lemma-exactness-implies", "contents": [ "Let $A$ be a ring. Let $F$ be a module-valued functor", "such that for any $B \\in \\Ob(\\textit{Alg}_A)$ the", "functor $TF(B, -)$ on $B$-modules transforms a short exact sequence", "of $B$-modules into a right exact sequence. Then", "\\begin{enumerate}", "\\item $TF(B, N_1 \\oplus N_2) = TF(B, N_1) \\oplus TF(B, N_2)$,", "\\item there is a second functorial $B$-module structure on $TF(B, N)$", "defined by setting $x \\cdot b = TF(B, b\\cdot 1_N)(x)$ for $x \\in TF(B, N)$", "and $b \\in B$,", "\\item", "\\label{item-mult-map-linear}", "the canonical map $N \\otimes_B F(B) \\to TF(B, N)$ of", "Lemma \\ref{lemma-tangent-functor}", "is $B$-linear also with respect to the second $B$-module structure,", "\\item", "\\label{item-tangent-right-exact}", "given a finitely presented $B$-module $N$ there is a canonical", "isomorphism $TF(B, B) \\otimes_B N \\to TF(B, N)$ where the tensor", "product uses the second $B$-module structure on $TF(B, B)$.", "\\end{enumerate}" ], "refs": [ "adequate-lemma-tangent-functor" ], "proofs": [ { "contents": [ "We will use the results of", "Lemma \\ref{lemma-tangent-functor}", "without further mention.", "The maps $N_1 \\to N_1 \\oplus N_2$ and $N_2 \\to N_1 \\oplus N_2$ give", "a map $TF(B, N_1) \\oplus TF(B, N_2) \\to TF(B, N_1 \\oplus N_2)$", "which is injective since the maps $N_1 \\oplus N_2 \\to N_1$ and", "$N_1 \\oplus N_2 \\to N_2$ induce an inverse.", "Since $TF$ is right exact we see that", "$TF(B, N_1) \\to TF(B, N_1 \\oplus N_2) \\to TF(B, N_2) \\to 0$ is exact.", "Hence $TF(B, N_1) \\oplus TF(B, N_2) \\to TF(B, N_1 \\oplus N_2)$ is an", "isomorphism. This proves (1).", "\\medskip\\noindent", "To see (2) the only thing we need to show is that", "$x \\cdot (b_1 + b_2) = x \\cdot b_1 + x \\cdot b_2$.", "(Associativity and additivity are clear.) To see this consider", "$$", "N \\xrightarrow{(b_1, b_2)} N \\oplus N \\xrightarrow{+} N", "$$", "and apply $TF(B, -)$.", "\\medskip\\noindent", "Part (3) follows immediately from the fact that", "$N \\otimes_B F(B) \\to TF(B, N)$ is functorial in the pair $(B, N)$.", "\\medskip\\noindent", "Suppose $N$ is a finitely presented $B$-module. Choose a presentation", "$B^{\\oplus m} \\to B^{\\oplus n} \\to N \\to 0$. This gives an exact", "sequence", "$$", "TF(B, B^{\\oplus m}) \\to TF(B, B^{\\oplus n}) \\to TF(B, N) \\to 0", "$$", "by right exactness of $TF(B, -)$. By part (1) we can write", "$TF(B, B^{\\oplus m}) = TF(B, B)^{\\oplus m}$ and", "$TF(B, B^{\\oplus n}) = TF(B, B)^{\\oplus n}$. Next, suppose that", "$B^{\\oplus m} \\to B^{\\oplus n}$ is given by the matrix $T = (b_{ij})$.", "Then the induced map $TF(B, B)^{\\oplus m} \\to TF(B, B)^{\\oplus n}$", "is given by the matrix with entries $TF(B, b_{ij} \\cdot 1_B)$.", "This combined with right exactness of $\\otimes$ proves (4)." ], "refs": [ "adequate-lemma-tangent-functor" ], "ref_ids": [ 3604 ] } ], "ref_ids": [ 3604 ] }, { "id": 3607, "type": "theorem", "label": "adequate-lemma-exactness-permanence", "categories": [ "adequate" ], "title": "adequate-lemma-exactness-permanence", "contents": [ "Let $A$ be a ring. For $F$ a module-valued functor on $\\textit{Alg}_A$", "say $(*)$ holds if for all $B \\in \\Ob(\\textit{Alg}_A)$ the", "functor $TF(B, -)$ on $B$-modules transforms a short exact sequence", "of $B$-modules into a right exact sequence. Let", "$0 \\to F \\to G \\to H \\to 0$ be a short exact sequence of", "module-valued functors on $\\textit{Alg}_A$.", "\\begin{enumerate}", "\\item If $(*)$ holds for $F, G$ then $(*)$ holds for $H$.", "\\item If $(*)$ holds for $F, H$ then $(*)$ holds for $G$.", "\\item If $H' \\to H$ is morphism of module-valued functors on $\\textit{Alg}_A$", "and $(*)$ holds for $F$, $G$, $H$, and $H'$, then $(*)$ holds for", "$G \\times_H H'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $B$ be given. Let $0 \\to N_1 \\to N_2 \\to N_3 \\to 0$ be a short exact", "sequence of $B$-modules. Part (1) follows from a diagram chase in", "the diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "TF(B, N_1) \\ar[r] \\ar[d] &", "TG(B, N_1) \\ar[r] \\ar[d] &", "TH(B, N_1) \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "TF(B, N_2) \\ar[r] \\ar[d] &", "TG(B, N_2) \\ar[r] \\ar[d] &", "TH(B, N_2) \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "TF(B, N_3) \\ar[r] \\ar[d] &", "TG(B, N_3) \\ar[r] \\ar[d] &", "TH(B, N_3) \\ar[r] & 0 \\\\", "& 0 & 0", "}", "$$", "with exact horizontal rows and exact columns involving $TF$ and $TG$.", "To prove part (2) we do a diagram chase in the diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "TF(B, N_1) \\ar[r] \\ar[d] &", "TG(B, N_1) \\ar[r] \\ar[d] &", "TH(B, N_1) \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "TF(B, N_2) \\ar[r] \\ar[d] &", "TG(B, N_2) \\ar[r] \\ar[d] &", "TH(B, N_2) \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "TF(B, N_3) \\ar[r] \\ar[d] &", "TG(B, N_3) \\ar[r] &", "TH(B, N_3) \\ar[r] \\ar[d] & 0 \\\\", "& 0 & & 0", "}", "$$", "with exact horizontal rows and exact columns involving $TF$ and $TH$.", "Part (3) follows from part (2) as $G \\times_H H'$ sits in the exact", "sequence $0 \\to F \\to G \\times_H H' \\to H' \\to 0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3608, "type": "theorem", "label": "adequate-lemma-ext-group-zero-key", "categories": [ "adequate" ], "title": "adequate-lemma-ext-group-zero-key", "contents": [ "Let $A$ be a ring. Let $M$, $P$ be $A$-modules with $P$ of finite", "presentation. Then", "$\\Ext^i_\\mathcal{P}(\\underline{P}, \\underline{M}) = 0$", "for $i > 0$ where $\\mathcal{P}$ is the category of module-valued", "functors on $\\textit{Alg}_A$." ], "refs": [], "proofs": [ { "contents": [ "Choose an injective resolution $\\underline{M} \\to I^\\bullet$ in", "$\\mathcal{P}$, see", "Lemma \\ref{lemma-enough-injectives}.", "By", "Derived Categories, Lemma \\ref{derived-lemma-compute-ext-resolutions}", "any element of $\\Ext^i_\\mathcal{P}(\\underline{P}, \\underline{M})$", "comes from a morphism $\\varphi : \\underline{P} \\to I^i$ with", "$d^i \\circ \\varphi = 0$. We will prove that the", "Yoneda extension", "$$", "E : 0 \\to \\underline{M} \\to I^0 \\to \\ldots \\to", "I^{i - 1} \\times_{\\Ker(d^i)} \\underline{P} \\to \\underline{P} \\to 0", "$$", "of $\\underline{P}$ by $\\underline{M}$", "associated to $\\varphi$ is trivial, which will prove the lemma by", "Derived Categories, Lemma \\ref{derived-lemma-yoneda-extension}.", "\\medskip\\noindent", "For $F$ a module-valued functor on $\\textit{Alg}_A$", "say $(*)$ holds if for all $B \\in \\Ob(\\textit{Alg}_A)$ the", "functor $TF(B, -)$ on $B$-modules transforms a short exact sequence", "of $B$-modules into a right exact sequence.", "Recall that the module-valued functors $\\underline{M}, I^n, \\underline{P}$", "each have property $(*)$, see", "Lemma \\ref{lemma-tangent-injective}", "and the remarks preceding it.", "By splitting $0 \\to \\underline{M} \\to I^\\bullet$ into short", "exact sequences we find that each of the functors", "$\\Im(d^{n - 1}) = \\Ker(d^n) \\subset I^n$ has property $(*)$ by", "Lemma \\ref{lemma-exactness-permanence}", "and also that $I^{i - 1} \\times_{\\Ker(d^i)} \\underline{P}$ has property", "$(*)$.", "\\medskip\\noindent", "Thus we may assume the Yoneda extension is given as", "$$", "E : 0 \\to \\underline{M} \\to F_{i - 1} \\to \\ldots \\to", "F_0 \\to \\underline{P} \\to 0", "$$", "where each of the module-valued functors $F_j$ has property $(*)$.", "Set $G_j(B) = TF_j(B, B)$ viewed as a $B$-module via the {\\it second}", "$B$-module structure defined in", "Lemma \\ref{lemma-exactness-implies}.", "Since $TF_j$ is a functor on pairs we see that $G_j$ is a module-valued", "functor on $\\textit{Alg}_A$. Moreover, since $E$ is an exact sequence", "the sequence $G_{j + 1} \\to G_j \\to G_{j - 1}$ is exact (see remark", "preceding", "Lemma \\ref{lemma-exactness-permanence}).", "Observe that $T\\underline{M}(B, B) = M \\otimes_A B = \\underline{M}(B)$", "and that the two $B$-module structures agree on this.", "Thus we obtain a Yoneda extension", "$$", "E' : 0 \\to \\underline{M} \\to G_{i - 1} \\to \\ldots \\to", "G_0 \\to \\underline{P} \\to 0", "$$", "Moreover, the canonical maps", "$$", "F_j(B) = B \\otimes_B F_j(B) \\longrightarrow TF_j(B, B) = G_j(B)", "$$", "of", "Lemma \\ref{lemma-tangent-functor} (\\ref{item-mult-map})", "are $B$-linear by", "Lemma \\ref{lemma-exactness-implies} (\\ref{item-mult-map-linear})", "and functorial in $B$. Hence a map", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\underline{M} \\ar[r] \\ar[d]^1 &", "F_{i - 1} \\ar[r] \\ar[d] &", "\\ldots \\ar[r] &", "F_0 \\ar[r] \\ar[d] &", "\\underline{P} \\ar[r] \\ar[d]^1 & 0 \\\\", "0 \\ar[r] &", "\\underline{M} \\ar[r] &", "G_{i - 1} \\ar[r] &", "\\ldots \\ar[r] &", "G_0 \\ar[r] &", "\\underline{P} \\ar[r] & 0", "}", "$$", "of Yoneda extensions. In particular we see that $E$ and $E'$ have the", "same class in $\\Ext^i_\\mathcal{P}(\\underline{P}, \\underline{M})$", "by the lemma on Yoneda Exts mentioned above. Finally, let $N$ be a", "$A$-module of finite presentation. Then we see that", "$$", "0 \\to T\\underline{M}(A, N) \\to TF_{i - 1}(A, N) \\to \\ldots \\to", "TF_0(A, N) \\to T\\underline{P}(A, N) \\to 0", "$$", "is exact. By", "Lemma \\ref{lemma-exactness-implies} (\\ref{item-tangent-right-exact})", "with $B = A$ this translates into the exactness of the sequence of", "$A$-modules", "$$", "0 \\to M \\otimes_A N \\to G_{i - 1}(A) \\otimes_A N \\to \\ldots \\to", "G_0(A) \\otimes_A N \\to P \\otimes_A N \\to 0", "$$", "Hence the sequence of $A$-modules", "$0 \\to M \\to G_{i - 1}(A) \\to \\ldots \\to G_0(A) \\to P \\to 0$", "is universally exact, in the sense that it remains exact on tensoring", "with any finitely presented $A$-module $N$. Let", "$K = \\Ker(G_0(A) \\to P)$ so that we have exact sequences", "$$", "0 \\to K \\to G_0(A) \\to P \\to 0", "\\quad\\text{and}\\quad", "G_2(A) \\to G_1(A) \\to K \\to 0", "$$", "Tensoring the second sequence with $N$ we obtain that", "$K \\otimes_A N = \\Coker(G_2(A) \\otimes_A N \\to G_1(A) \\otimes_A N)$.", "Exactness of $G_2(A) \\otimes_A N \\to G_1(A) \\otimes_A N \\to G_0(A) \\otimes_A N$", "then implies that $K \\otimes_A N \\to G_0(A) \\otimes_A N$ is injective.", "By", "Algebra, Theorem \\ref{algebra-theorem-universally-exact-criteria}", "this means that the $A$-module extension $0 \\to K \\to G_0(A) \\to P \\to 0$", "is exact, and because $P$ is assumed of finite presentation this means", "the sequence is split, see", "Algebra, Lemma \\ref{algebra-lemma-universally-exact-split}.", "Any splitting $P \\to G_0(A)$ defines a map $\\underline{P} \\to G_0$", "which splits the surjection $G_0 \\to \\underline{P}$. Thus the", "Yoneda extension $E'$ is equivalent to the trivial Yoneda extension", "and we win." ], "refs": [ "adequate-lemma-enough-injectives", "derived-lemma-compute-ext-resolutions", "derived-lemma-yoneda-extension", "adequate-lemma-tangent-injective", "adequate-lemma-exactness-permanence", "adequate-lemma-exactness-implies", "adequate-lemma-exactness-permanence", "adequate-lemma-tangent-functor", "adequate-lemma-exactness-implies", "adequate-lemma-exactness-implies", "algebra-theorem-universally-exact-criteria", "algebra-lemma-universally-exact-split" ], "ref_ids": [ 3603, 1892, 1894, 3605, 3607, 3606, 3607, 3604, 3606, 3606, 319, 808 ] } ], "ref_ids": [] }, { "id": 3609, "type": "theorem", "label": "adequate-lemma-ext-group-zero", "categories": [ "adequate" ], "title": "adequate-lemma-ext-group-zero", "contents": [ "Let $A$ be a ring. Let $M$ be an $A$-module. Let $L$ be a linearly", "adequate functor on $\\textit{Alg}_A$. Then", "$\\Ext^i_\\mathcal{P}(L, \\underline{M}) = 0$", "for $i > 0$ where $\\mathcal{P}$ is the category of module-valued", "functors on $\\textit{Alg}_A$." ], "refs": [], "proofs": [ { "contents": [ "Since $L$ is linearly adequate there exists an exact sequence", "$$", "0 \\to L \\to \\underline{A^{\\oplus m}} \\to \\underline{A^{\\oplus n}} \\to", "\\underline{P} \\to 0", "$$", "Here $P = \\Coker(A^{\\oplus m} \\to A^{\\oplus n})$ is the cokernel", "of the map of finite free $A$-modules which is given by the definition", "of linearly adequate functors. By", "Lemma \\ref{lemma-ext-group-zero-key}", "we have the vanishing of", "$\\Ext^i_\\mathcal{P}(\\underline{P}, \\underline{M})$", "and", "$\\Ext^i_\\mathcal{P}(\\underline{A}, \\underline{M})$", "for $i > 0$.", "Let $K = \\Ker(\\underline{A^{\\oplus n}} \\to \\underline{P})$.", "By the long exact sequence of Ext groups associated to the exact sequence", "$0 \\to K \\to \\underline{A^{\\oplus n}} \\to \\underline{P} \\to 0$", "we conclude that", "$\\Ext^i_\\mathcal{P}(K, \\underline{M}) = 0$ for $i > 0$.", "Repeating with the sequence", "$0 \\to L \\to \\underline{A^{\\oplus m}} \\to K \\to 0$", "we win." ], "refs": [ "adequate-lemma-ext-group-zero-key" ], "ref_ids": [ 3608 ] } ], "ref_ids": [] }, { "id": 3610, "type": "theorem", "label": "adequate-lemma-RQ-zero", "categories": [ "adequate" ], "title": "adequate-lemma-RQ-zero", "contents": [ "With notation as in", "Lemma \\ref{lemma-enough-injectives}", "we have $R^pQ(F) = 0$ for all $p > 0$ and any adequate functor $F$." ], "refs": [ "adequate-lemma-enough-injectives" ], "proofs": [ { "contents": [ "Choose an exact sequence $0 \\to F \\to \\underline{M^0} \\to \\underline{M^1}$.", "Set $M^2 = \\Coker(M^0 \\to M^1)$ so that", "$0 \\to F \\to \\underline{M^0} \\to \\underline{M^1}", "\\to \\underline{M^2} \\to 0$ is a resolution. By", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}", "we obtain a spectral sequence", "$$", "R^pQ(\\underline{M^q}) \\Rightarrow R^{p + q}Q(F)", "$$", "Since $Q(\\underline{M^q}) = \\underline{M^q}$ it suffices to prove", "$R^pQ(\\underline{M}) = 0$, $p > 0$ for any $A$-module $M$.", "\\medskip\\noindent", "Choose an injective resolution $\\underline{M} \\to I^\\bullet$ in", "the category $\\mathcal{P}$. Suppose that $R^iQ(\\underline{M})$ is nonzero.", "Then $\\Ker(Q(I^i) \\to Q(I^{i + 1}))$ is strictly bigger", "than the image of $Q(I^{i - 1}) \\to Q(I^i)$. Hence by", "Lemma \\ref{lemma-adequate-surjection-from-linear}", "there exists a linearly adequate functor $L$ and a map", "$\\varphi : L \\to Q(I^i)$ mapping into the kernel of $Q(I^i) \\to Q(I^{i + 1})$", "which does not factor through the image of $Q(I^{i - 1}) \\to Q(I^i)$.", "Because $Q$ is a left adjoint to the inclusion functor the map", "$\\varphi$ corresponds to a map $\\varphi' : L \\to I^i$ with the same properties.", "Thus $\\varphi'$ gives a nonzero element of", "$\\Ext^i_\\mathcal{P}(L, \\underline{M})$ contradicting", "Lemma \\ref{lemma-ext-group-zero}." ], "refs": [ "derived-lemma-two-ss-complex-functor", "adequate-lemma-adequate-surjection-from-linear", "adequate-lemma-ext-group-zero" ], "ref_ids": [ 1871, 3588, 3609 ] } ], "ref_ids": [ 3603 ] }, { "id": 3611, "type": "theorem", "label": "adequate-lemma-adequate-local", "categories": [ "adequate" ], "title": "adequate-lemma-adequate-local", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ be an adequate $\\mathcal{O}$-module on", "$(\\Sch/S)_\\tau$. For any affine scheme $\\Spec(A)$ over $S$", "the functor $F_{\\mathcal{F}, A}$ is adequate." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{\\Spec(A_i) \\to S\\}_{i \\in I}$ be a $\\tau$-covering", "such that $F_{\\mathcal{F}, A_i}$ is adequate for all $i \\in I$.", "We can find a standard affine $\\tau$-covering", "$\\{\\Spec(A'_j) \\to \\Spec(A)\\}_{j = 1, \\ldots, m}$", "such that $\\Spec(A'_j) \\to \\Spec(A) \\to S$ factors", "through $\\Spec(A_{i(j)})$ for some $i(j) \\in I$. Then we see that", "$F_{\\mathcal{F}, A'_j}$ is the restriction of", "$F_{\\mathcal{F}, A_{i(j)}}$ to the category of $A'_j$-algebras.", "Hence $F_{\\mathcal{F}, A'_j}$ is adequate by", "Lemma \\ref{lemma-base-change-adequate}.", "By", "Lemma \\ref{lemma-adequate-product}", "the sequence", "$F_{\\mathcal{F}, A'_j}$ corresponds to an adequate ``product'' functor", "$F'$ over $A' = A'_1 \\times \\ldots \\times A'_m$. As $\\mathcal{F}$ is a", "sheaf (for the Zariski topology) this product functor $F'$ is equal", "to $F_{\\mathcal{F}, A'}$, i.e., is the restriction of $F$ to $A'$-algebras.", "Finally, $\\{\\Spec(A') \\to \\Spec(A)\\}$ is a $\\tau$-covering.", "It follows from", "Lemma \\ref{lemma-adequate-descent}", "that $F_{\\mathcal{F}, A}$ is adequate." ], "refs": [ "adequate-lemma-base-change-adequate", "adequate-lemma-adequate-product", "adequate-lemma-adequate-descent" ], "ref_ids": [ 3598, 3600, 3601 ] } ], "ref_ids": [] }, { "id": 3612, "type": "theorem", "label": "adequate-lemma-adequate-affine", "categories": [ "adequate" ], "title": "adequate-lemma-adequate-affine", "contents": [ "Let $S = \\Spec(A)$ be an affine scheme. The category of adequate", "$\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$ is equivalent to the", "category of adequate module-valued functors on $\\textit{Alg}_A$." ], "refs": [], "proofs": [ { "contents": [ "Given an adequate module $\\mathcal{F}$ the functor $F_{\\mathcal{F}, A}$", "is adequate by Lemma \\ref{lemma-adequate-local}.", "Given an adequate functor $F$ we choose an exact sequence", "$0 \\to F \\to \\underline{M} \\to \\underline{N}$ and we consider", "the $\\mathcal{O}$-module $\\mathcal{F} = \\Ker(M^a \\to N^a)$ where", "$M^a$ denotes the quasi-coherent $\\mathcal{O}$-module on", "$(\\Sch/S)_\\tau$ associated to the quasi-coherent sheaf", "$\\widetilde{M}$ on $S$. Note that $F = F_{\\mathcal{F}, A}$, in particular", "the module $\\mathcal{F}$ is adequate by definition.", "We omit the proof that the constructions define mutually inverse", "equivalences of categories." ], "refs": [ "adequate-lemma-adequate-local" ], "ref_ids": [ 3611 ] } ], "ref_ids": [] }, { "id": 3613, "type": "theorem", "label": "adequate-lemma-pullback-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-pullback-adequate", "contents": [ "Let $f : T \\to S$ be a morphism of schemes.", "The pullback $f^*\\mathcal{F}$ of an adequate $\\mathcal{O}$-module", "$\\mathcal{F}$ on $(\\Sch/S)_\\tau$ is an adequate", "$\\mathcal{O}$-module on $(\\Sch/T)_\\tau$." ], "refs": [], "proofs": [ { "contents": [ "The pullback map", "$f^* : \\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O}) \\to", "\\textit{Mod}((\\Sch/T)_\\tau, \\mathcal{O})$", "is given by restriction, i.e., $f^*\\mathcal{F}(V) = \\mathcal{F}(V)$", "for any scheme $V$ over $T$. Hence this lemma follows immediately from", "Lemma \\ref{lemma-adequate-local}", "and the definition." ], "refs": [ "adequate-lemma-adequate-local" ], "ref_ids": [ 3611 ] } ], "ref_ids": [] }, { "id": 3614, "type": "theorem", "label": "adequate-lemma-adequate-characterize", "categories": [ "adequate" ], "title": "adequate-lemma-adequate-characterize", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module on", "$(\\Sch/S)_\\tau$. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is adequate,", "\\item there exists an affine open covering $S = \\bigcup S_i$ and", "maps of quasi-coherent $\\mathcal{O}_{S_i}$-modules", "$\\mathcal{G}_i \\to \\mathcal{H}_i$", "such that $\\mathcal{F}|_{(\\Sch/S_i)_\\tau}$ is the", "kernel of $\\mathcal{G}_i^a \\to \\mathcal{H}_i^a$", "\\item there exists a $\\tau$-covering $\\{S_i \\to S\\}_{i \\in I}$ and", "maps of $\\mathcal{O}_{S_i}$-quasi-coherent modules", "$\\mathcal{G}_i \\to \\mathcal{H}_i$", "such that $\\mathcal{F}|_{(\\Sch/S_i)_\\tau}$ is the", "kernel of $\\mathcal{G}_i^a \\to \\mathcal{H}_i^a$,", "\\item there exists a $\\tau$-covering $\\{f_i : S_i \\to S\\}_{i \\in I}$", "such that each $f_i^*\\mathcal{F}$ is adequate,", "\\item for any affine scheme $U$ over $S$ the restriction", "$\\mathcal{F}|_{(\\Sch/U)_\\tau}$ is the kernel", "of a map $\\mathcal{G}^a \\to \\mathcal{H}^a$ of quasi-coherent", "$\\mathcal{O}_U$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $U = \\Spec(A)$ be an affine scheme over $S$.", "Set $F = F_{\\mathcal{F}, A}$. By definition, the functor", "$F$ is adequate if and only if there exists a map of $A$-modules", "$M \\to N$ such that $F = \\Ker(\\underline{M} \\to \\underline{N})$.", "Combining with", "Lemmas \\ref{lemma-adequate-local} and", "\\ref{lemma-adequate-affine}", "we see that (1) and (5) are equivalent.", "\\medskip\\noindent", "It is clear that (5) implies (2) and (2) implies (3).", "If (3) holds then we can refine the covering", "$\\{S_i \\to S\\}$ such that each $S_i = \\Spec(A_i)$ is affine.", "Then we see, by the preliminary remarks of the proof, that", "$F_{\\mathcal{F}, A_i}$ is adequate. Thus $\\mathcal{F}$", "is adequate by definition. Hence (3) implies (1).", "\\medskip\\noindent", "Finally, (4) is equivalent to (1) using", "Lemma \\ref{lemma-pullback-adequate}", "for one direction and that", "a composition of $\\tau$-coverings is a $\\tau$-covering for the other." ], "refs": [ "adequate-lemma-adequate-local", "adequate-lemma-adequate-affine", "adequate-lemma-pullback-adequate" ], "ref_ids": [ 3611, 3612, 3613 ] } ], "ref_ids": [] }, { "id": 3615, "type": "theorem", "label": "adequate-lemma-adequate-fpqc", "categories": [ "adequate" ], "title": "adequate-lemma-adequate-fpqc", "contents": [ "Let $\\mathcal{F}$ be an adequate $\\mathcal{O}$-module on", "$(\\Sch/S)_\\tau$. For any surjective flat morphism", "$\\Spec(B) \\to \\Spec(A)$ of affines over $S$", "the extended {\\v C}ech complex", "$$", "0 \\to \\mathcal{F}(\\Spec(A)) \\to", "\\mathcal{F}(\\Spec(B)) \\to", "\\mathcal{F}(\\Spec(B \\otimes_A B)) \\to \\ldots", "$$", "is exact. In particular $\\mathcal{F}$ satisfies the sheaf condition", "for fpqc coverings, and is a sheaf of $\\mathcal{O}$-modules", "on $(\\Sch/S)_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "With $A \\to B$ as in the lemma let $F = F_{\\mathcal{F}, A}$. This functor", "is adequate by", "Lemma \\ref{lemma-adequate-local}.", "By", "Lemma \\ref{lemma-adequate-flat}", "since $A \\to B$, $A \\to B \\otimes_A B$, etc are flat we see that", "$F(B) = F(A) \\otimes_A B$,", "$F(B \\otimes_A B) = F(A) \\otimes_A B \\otimes_A B$, etc.", "Exactness follows from", "Descent, Lemma \\ref{descent-lemma-ff-exact}.", "\\medskip\\noindent", "Thus $\\mathcal{F}$ satisfies the sheaf condition for", "$\\tau$-coverings (in particular Zariski coverings) and any faithfully", "flat covering of an affine by an affine. Arguing as in the proofs of", "Descent, Lemma \\ref{descent-lemma-standard-fpqc-covering}", "and", "Descent, Proposition \\ref{descent-proposition-fpqc-descent-quasi-coherent}", "we conclude that $\\mathcal{F}$ satisfies the sheaf condition for all", "fpqc coverings (made out of objects of $(\\Sch/S)_\\tau$).", "Details omitted." ], "refs": [ "adequate-lemma-adequate-local", "adequate-lemma-adequate-flat", "descent-lemma-ff-exact", "descent-lemma-standard-fpqc-covering", "descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 3611, 3587, 14598, 14609, 14753 ] } ], "ref_ids": [] }, { "id": 3616, "type": "theorem", "label": "adequate-lemma-same-cohomology-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-same-cohomology-adequate", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ be an adequate", "$\\mathcal{O}$-module on $(\\Sch/S)_\\tau$.", "\\begin{enumerate}", "\\item The restriction $\\mathcal{F}|_{S_{Zar}}$ is a quasi-coherent", "$\\mathcal{O}_S$-module on the scheme $S$.", "\\item The restriction $\\mathcal{F}|_{S_\\etale}$ is the", "quasi-coherent module associated to $\\mathcal{F}|_{S_{Zar}}$.", "\\item For any affine scheme $U$ over $S$ we have $H^q(U, \\mathcal{F}) = 0$", "for all $q > 0$.", "\\item There is a canonical isomorphism", "$$", "H^q(S, \\mathcal{F}|_{S_{Zar}}) =", "H^q((\\Sch/S)_\\tau, \\mathcal{F}).", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-adequate-flat}", "and", "Lemma \\ref{lemma-adequate-local}", "we see that for any flat morphism of affines $U \\to V$ over $S$", "we have", "$\\mathcal{F}(U) = \\mathcal{F}(V) \\otimes_{\\mathcal{O}(V)} \\mathcal{O}(U)$.", "This works in particular if $U \\subset V \\subset S$ are affine opens of", "$S$, hence $\\mathcal{F}|_{S_{Zar}}$ is quasi-coherent.", "Thus (1) holds.", "\\medskip\\noindent", "Let $S' \\to S$ be an \\'etale morphism of schemes.", "Then for $U \\subset S'$ affine open mapping into an affine open", "$V \\subset S$ we see that", "$\\mathcal{F}(U) = \\mathcal{F}(V) \\otimes_{\\mathcal{O}(V)} \\mathcal{O}(U)$", "because $U \\to V$ is \\'etale, hence flat. Therefore", "$\\mathcal{F}|_{S'_{Zar}}$ is the pullback of $\\mathcal{F}|_{S_{Zar}}$.", "This proves (2).", "\\medskip\\noindent", "We are going to apply", "Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-cech-vanish-collection}", "to the site $(\\Sch/S)_\\tau$ with", "$\\mathcal{B}$ the set of affine schemes over $S$ and", "$\\text{Cov}$ the set of standard affine $\\tau$-coverings.", "Assumption (3) of the lemma is satisfied by", "Descent, Lemma \\ref{descent-lemma-standard-covering-Cech}", "and", "Lemma \\ref{lemma-adequate-fpqc}", "for the case of a covering by a single affine.", "Hence we conclude that $H^p(U, \\mathcal{F}) = 0$ for every", "affine scheme $U$ over $S$. This proves (3).", "In exactly the same way as in the proof of", "Descent, Proposition \\ref{descent-proposition-same-cohomology-quasi-coherent}", "this implies the equality of cohomologies (4)." ], "refs": [ "adequate-lemma-adequate-flat", "adequate-lemma-adequate-local", "sites-cohomology-lemma-cech-vanish-collection", "descent-lemma-standard-covering-Cech", "adequate-lemma-adequate-fpqc", "descent-proposition-same-cohomology-quasi-coherent" ], "ref_ids": [ 3587, 3611, 4205, 14624, 3615, 14754 ] } ], "ref_ids": [] }, { "id": 3617, "type": "theorem", "label": "adequate-lemma-sheafification-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-sheafification-adequate", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules", "on $(\\Sch/S)_\\tau$. If for every affine scheme", "$\\Spec(A)$ over $S$ the functor $F_{\\mathcal{F}, A}$ is", "adequate, then the sheafification of $\\mathcal{F}$ is an adequate", "$\\mathcal{O}$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $U = \\Spec(A)$ be an affine scheme over $S$.", "Set $F = F_{\\mathcal{F}, A}$.", "The sheafification $\\mathcal{F}^\\# = (\\mathcal{F}^+)^+$, see", "Sites, Section \\ref{sites-section-sheafification}.", "By construction", "$$", "(\\mathcal{F})^+(U) =", "\\colim_\\mathcal{U} \\check{H}^0(\\mathcal{U}, \\mathcal{F})", "$$", "where the colimit is over coverings in the site $(\\Sch/S)_\\tau$.", "Since $U$ is affine it suffices to take the limit over standard", "affine $\\tau$-coverings", "$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I} =", "\\{\\Spec(A_i) \\to \\Spec(A)\\}_{i \\in I}$ of $U$.", "Since each $A \\to A_i$ and $A \\to A_i \\otimes_A A_j$ is flat we see that", "$$", "\\check{H}^0(\\mathcal{U}, \\mathcal{F}) =", "\\Ker(\\prod F(A) \\otimes_A A_i \\to \\prod F(A) \\otimes_A A_i \\otimes_A A_j)", "$$", "by", "Lemma \\ref{lemma-adequate-flat}.", "Since $A \\to \\prod A_i$ is faithfully flat we see that this always", "is canonically isomorphic to $F(A)$ by", "Descent, Lemma \\ref{descent-lemma-ff-exact}.", "Thus the presheaf $(\\mathcal{F})^+$ has the same value as", "$\\mathcal{F}$ on all affine schemes over $S$. Repeating the argument", "once more we deduce the same thing for $\\mathcal{F}^\\# = ((\\mathcal{F})^+)^+$.", "Thus $F_{\\mathcal{F}, A} = F_{\\mathcal{F}^\\#, A}$ and we conclude", "that $\\mathcal{F}^\\#$ is adequate." ], "refs": [ "adequate-lemma-adequate-flat", "descent-lemma-ff-exact" ], "ref_ids": [ 3587, 14598 ] } ], "ref_ids": [] }, { "id": 3618, "type": "theorem", "label": "adequate-lemma-abelian-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-abelian-adequate", "contents": [ "Let $S$ be a scheme.", "\\begin{enumerate}", "\\item The category $\\textit{Adeq}(\\mathcal{O})$ is abelian.", "\\item The functor", "$\\textit{Adeq}(\\mathcal{O}) \\to", "\\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O})$", "is exact.", "\\item If $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "is a short exact sequence of $\\mathcal{O}$-modules and", "$\\mathcal{F}_1$ and $\\mathcal{F}_3$ are adequate, then", "$\\mathcal{F}_2$ is adequate.", "\\item The category $\\textit{Adeq}(\\mathcal{O})$ has colimits and", "$\\textit{Adeq}(\\mathcal{O}) \\to", "\\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O})$", "commutes with them.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of adequate", "$\\mathcal{O}$-modules. To prove (1) and (2) it suffices to show that", "$\\mathcal{K} = \\Ker(\\varphi)$ and", "$\\mathcal{Q} = \\Coker(\\varphi)$ computed in", "$\\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O})$ are adequate.", "Let $U = \\Spec(A)$ be an affine scheme over $S$.", "Let $F = F_{\\mathcal{F}, A}$ and $G = F_{\\mathcal{G}, A}$.", "By", "Lemmas \\ref{lemma-kernel-adequate} and", "\\ref{lemma-cokernel-adequate}", "the kernel $K$ and cokernel $Q$ of the induced map", "$F \\to G$ are adequate functors.", "Because the kernel is computed on the level of presheaves, we see", "that $K = F_{\\mathcal{K}, A}$ and we conclude $\\mathcal{K}$ is adequate.", "To prove the result for the cokernel, denote $\\mathcal{Q}'$ the presheaf", "cokernel of $\\varphi$. Then $Q = F_{\\mathcal{Q}', A}$ and", "$\\mathcal{Q} = (\\mathcal{Q}')^\\#$. Hence $\\mathcal{Q}$", "is adequate by", "Lemma \\ref{lemma-sheafification-adequate}.", "\\medskip\\noindent", "Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "is a short exact sequence of $\\mathcal{O}$-modules and", "$\\mathcal{F}_1$ and $\\mathcal{F}_3$ are adequate.", "Let $U = \\Spec(A)$ be an affine scheme over $S$.", "Let $F_i = F_{\\mathcal{F}_i, A}$. The sequence of functors", "$$", "0 \\to F_1 \\to F_2 \\to F_3 \\to 0", "$$", "is exact, because for $V = \\Spec(B)$ affine over $U$ we have", "$H^1(V, \\mathcal{F}_1) = 0$ by", "Lemma \\ref{lemma-same-cohomology-adequate}.", "Since $F_1$ and $F_3$ are adequate functors by", "Lemma \\ref{lemma-adequate-local}", "we see that $F_2$ is adequate by", "Lemma \\ref{lemma-extension-adequate}.", "Thus $\\mathcal{F}_2$ is adequate.", "\\medskip\\noindent", "Let $\\mathcal{I} \\to \\textit{Adeq}(\\mathcal{O})$, $i \\mapsto \\mathcal{F}_i$", "be a diagram. Denote $\\mathcal{F} = \\colim_i \\mathcal{F}_i$", "the colimit computed in", "$\\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O})$.", "To prove (4) it suffices to show that $\\mathcal{F}$ is adequate.", "Let $\\mathcal{F}' = \\colim_i \\mathcal{F}_i$ be the colimit computed", "in presheaves of $\\mathcal{O}$-modules. Then", "$\\mathcal{F} = (\\mathcal{F}')^\\#$.", "Let $U = \\Spec(A)$ be an affine scheme over $S$.", "Let $F_i = F_{\\mathcal{F}_i, A}$. By", "Lemma \\ref{lemma-colimit-adequate}", "the functor $\\colim_i F_i = F_{\\mathcal{F}', A}$ is adequate.", "Lemma \\ref{lemma-sheafification-adequate}", "shows that $\\mathcal{F}$ is adequate." ], "refs": [ "adequate-lemma-kernel-adequate", "adequate-lemma-cokernel-adequate", "adequate-lemma-sheafification-adequate", "adequate-lemma-same-cohomology-adequate", "adequate-lemma-adequate-local", "adequate-lemma-extension-adequate", "adequate-lemma-colimit-adequate", "adequate-lemma-sheafification-adequate" ], "ref_ids": [ 3593, 3592, 3617, 3616, 3611, 3597, 3594, 3617 ] } ], "ref_ids": [] }, { "id": 3619, "type": "theorem", "label": "adequate-lemma-direct-image-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-direct-image-adequate", "contents": [ "Let $f : T \\to S$ be a quasi-compact and quasi-separated morphism", "of schemes. For any adequate $\\mathcal{O}_T$-module on", "$(\\Sch/T)_\\tau$ the pushforward", "$f_*\\mathcal{F}$ and the higher direct images $R^if_*\\mathcal{F}$", "are adequate $\\mathcal{O}_S$-modules on $(\\Sch/S)_\\tau$." ], "refs": [], "proofs": [ { "contents": [ "First we explain how to compute the higher direct images.", "Choose an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$.", "Then $R^if_*\\mathcal{F}$ is the $i$th cohomology sheaf of the", "complex $f_*\\mathcal{I}^\\bullet$.", "Hence $R^if_*\\mathcal{F}$ is the sheaf associated to the presheaf", "which associates to an object $U/S$ of $(\\Sch/S)_\\tau$", "the module", "\\begin{align*}", "\\frac{\\Ker(f_*\\mathcal{I}^i(U) \\to f_*\\mathcal{I}^{i + 1}(U))}", "{\\Im(f_*\\mathcal{I}^{i - 1}(U) \\to f_*\\mathcal{I}^i(U))}", "& =", "\\frac{\\Ker(\\mathcal{I}^i(U \\times_S T) \\to", "\\mathcal{I}^{i + 1}(U \\times_S T))}", "{\\Im(\\mathcal{I}^{i - 1}(U \\times_S T) \\to \\mathcal{I}^i(U \\times_S T))}", "\\\\", "& =", "H^i(U \\times_S T, \\mathcal{F}) \\\\", "& = H^i((\\Sch/U \\times_S T)_\\tau,", "\\mathcal{F}|_{(\\Sch/U \\times_S T)_\\tau}) \\\\", "& = H^i(U \\times_S T, \\mathcal{F}|_{(U \\times_S T)_{Zar}})", "\\end{align*}", "The first equality by", "Topologies, Lemma \\ref{topologies-lemma-morphism-big-fppf}", "(and its analogues for other topologies),", "the second equality by definition of cohomology of $\\mathcal{F}$", "over an object of $(\\Sch/T)_\\tau$,", "the third equality by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-of-open},", "and the last equality by", "Lemma \\ref{lemma-same-cohomology-adequate}.", "Thus by", "Lemma \\ref{lemma-sheafification-adequate}", "it suffices to prove the claim stated in the following paragraph.", "\\medskip\\noindent", "Let $A$ be a ring. Let $T$ be a scheme quasi-compact and quasi-separated", "over $A$. Let $\\mathcal{F}$ be an adequate $\\mathcal{O}_T$-module on", "$(\\Sch/T)_\\tau$. For an $A$-algebra $B$ set", "$T_B = T \\times_{\\Spec(A)} \\Spec(B)$ and denote", "$\\mathcal{F}_B = \\mathcal{F}|_{(T_B)_{Zar}}$ the restriction of", "$\\mathcal{F}$ to the small Zariski site of $T_B$.", "(Recall that this is a ``usual'' quasi-coherent sheaf on the scheme", "$T_B$, see", "Lemma \\ref{lemma-same-cohomology-adequate}.)", "Claim: The functor", "$$", "B \\longmapsto H^q(T_B, \\mathcal{F}_B)", "$$", "is adequate. We will prove the lemma by the usual", "procedure of cutting $T$ into pieces.", "\\medskip\\noindent", "Case I: $T$ is affine. In this case the schemes $T_B$ are all affine", "and $H^q(T_B, \\mathcal{F}_B) = 0$ for all $q \\geq 1$.", "The functor $B \\mapsto H^0(T_B, \\mathcal{F}_B)$ is adequate by", "Lemma \\ref{lemma-pushforward-adequate}.", "\\medskip\\noindent", "Case II: $T$ is separated. Let $n$ be the minimal number of affines needed", "to cover $T$. We argue by induction on $n$. The base case is Case I.", "Choose an affine open covering $T = V_1 \\cup \\ldots \\cup V_n$.", "Set $V = V_1 \\cup \\ldots \\cup V_{n - 1}$ and $U = V_n$. Observe that", "$$", "U \\cap V = (V_1 \\cap V_n) \\cup \\ldots \\cup (V_{n - 1} \\cap V_n)", "$$", "is also a union of $n - 1$ affine opens as $T$ is separated, see", "Schemes, Lemma \\ref{schemes-lemma-characterize-separated}.", "Note that for each $B$ the base changes $U_B$, $V_B$ and", "$(U \\cap V)_B = U_B \\cap V_B$ behave in the same way. Hence we see that", "for each $B$ we have a long exact sequence", "$$", "0 \\to", "H^0(T_B, \\mathcal{F}_B) \\to", "H^0(U_B, \\mathcal{F}_B) \\oplus H^0(V_B, \\mathcal{F}_B) \\to", "H^0((U \\cap V)_B, \\mathcal{F}_B) \\to", "H^1(T_B, \\mathcal{F}_B) \\to \\ldots", "$$", "functorial in $B$, see", "Cohomology, Lemma \\ref{cohomology-lemma-mayer-vietoris}.", "By induction hypothesis the functors", "$B \\mapsto H^q(U_B, \\mathcal{F}_B)$,", "$B \\mapsto H^q(V_B, \\mathcal{F}_B)$, and", "$B \\mapsto H^q((U \\cap V)_B, \\mathcal{F}_B)$", "are adequate. Using", "Lemmas \\ref{lemma-kernel-adequate} and", "\\ref{lemma-cokernel-adequate}", "we see that our functor $B \\mapsto H^q(T_B, \\mathcal{F}_B)$ sits in the", "middle of a short exact sequence whose outer terms are adequate.", "Thus the claim follows from", "Lemma \\ref{lemma-extension-adequate}.", "\\medskip\\noindent", "Case III: General quasi-compact and quasi-separated case.", "The proof is again by induction on the number $n$ of affines needed to", "cover $T$. The base case $n = 1$ is Case I.", "Choose an affine open covering $T = V_1 \\cup \\ldots \\cup V_n$.", "Set $V = V_1 \\cup \\ldots \\cup V_{n - 1}$ and $U = V_n$. Note that", "since $T$ is quasi-separated $U \\cap V$ is a quasi-compact open of an", "affine scheme, hence Case II applies to it. The rest of the argument", "proceeds in exactly the same manner as in the paragraph above and is", "omitted." ], "refs": [ "topologies-lemma-morphism-big-fppf", "sites-cohomology-lemma-cohomology-of-open", "adequate-lemma-same-cohomology-adequate", "adequate-lemma-sheafification-adequate", "adequate-lemma-same-cohomology-adequate", "adequate-lemma-pushforward-adequate", "schemes-lemma-characterize-separated", "cohomology-lemma-mayer-vietoris", "adequate-lemma-kernel-adequate", "adequate-lemma-cokernel-adequate", "adequate-lemma-extension-adequate" ], "ref_ids": [ 12478, 4186, 3616, 3617, 3616, 3599, 7710, 2042, 3593, 3592, 3597 ] } ], "ref_ids": [] }, { "id": 3620, "type": "theorem", "label": "adequate-lemma-parasitic-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-parasitic-adequate", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{F}$ be an adequate $\\mathcal{O}$-module on", "$(\\Sch/S)_\\tau$. The following are equivalent:", "\\begin{enumerate}", "\\item $v\\mathcal{F} = 0$,", "\\item $\\mathcal{F}$ is parasitic,", "\\item $\\mathcal{F}$ is parasitic for the $\\tau$-topology,", "\\item $\\mathcal{F}(U) = 0$ for all $U \\subset S$ open, and", "\\item there exists an affine open covering $S = \\bigcup U_i$", "such that $\\mathcal{F}(U_i) = 0$ for all $i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implications (2) $\\Rightarrow$ (3) $\\Rightarrow$ (4) $\\Rightarrow$ (5)", "are immediate from the definitions. Assume (5). Suppose that", "$S = \\bigcup U_i$ is an affine open covering such that $\\mathcal{F}(U_i) = 0$", "for all $i$. Let $V \\to S$ be a flat morphism. There exists an affine", "open covering $V = \\bigcup V_j$ such that each $V_j$ maps into some", "$U_i$. As the morphism $V_j \\to S$ is flat, also $V_j \\to U_i$ is flat. ", "Hence the corresponding ring map", "$A_i = \\mathcal{O}(U_i) \\to \\mathcal{O}(V_j) = B_j$ is flat. Thus by", "Lemma \\ref{lemma-adequate-local}", "and", "Lemma \\ref{lemma-adequate-flat}", "we see that $\\mathcal{F}(U_i) \\otimes_{A_i} B_j \\to \\mathcal{F}(V_j)$", "is an isomorphism. Hence $\\mathcal{F}(V_j) = 0$. Since $\\mathcal{F}$ is", "a sheaf for the Zariski topology we conclude that $\\mathcal{F}(V) = 0$.", "In this way we see that (5) implies (2).", "\\medskip\\noindent", "This proves the equivalence of (2), (3), (4), and (5).", "As (1) is equivalent to (3) (see", "Remark \\ref{remark-compare})", "we conclude that all five conditions are equivalent." ], "refs": [ "adequate-lemma-adequate-local", "adequate-lemma-adequate-flat", "adequate-remark-compare" ], "ref_ids": [ 3611, 3587, 3646 ] } ], "ref_ids": [] }, { "id": 3621, "type": "theorem", "label": "adequate-lemma-adequate-by-parasitic", "categories": [ "adequate" ], "title": "adequate-lemma-adequate-by-parasitic", "contents": [ "Let $S$ be a scheme. The subcategory", "$\\mathcal{C} \\subset \\textit{Adeq}(\\mathcal{O})$ of parasitic adequate", "modules is a Serre subcategory. Moreover, the functor $v$ induces", "an equivalence of categories", "$$", "\\textit{Adeq}(\\mathcal{O}) / \\mathcal{C} = \\QCoh(\\mathcal{O}_S).", "$$" ], "refs": [], "proofs": [ { "contents": [ "The category $\\mathcal{C}$ is the kernel of the exact functor", "$v : \\textit{Adeq}(\\mathcal{O}) \\to \\QCoh(\\mathcal{O}_S)$, see", "Lemma \\ref{lemma-parasitic-adequate}.", "Hence it is a Serre subcategory by", "Homology, Lemma \\ref{homology-lemma-kernel-exact-functor}.", "By", "Homology, Lemma \\ref{homology-lemma-serre-subcategory-is-kernel}", "we obtain an induced exact functor", "$\\overline{v} :", "\\textit{Adeq}(\\mathcal{O}) / \\mathcal{C}", "\\to", "\\QCoh(\\mathcal{O}_S)$.", "Because $u$ is a right inverse to $v$ we see right away that", "$\\overline{v}$ is essentially surjective.", "We see that $\\overline{v}$ is faithful by", "Homology, Lemma \\ref{homology-lemma-quotient-by-kernel-exact-functor}.", "Because $u$ is a right inverse to $v$ we finally conclude that", "$\\overline{v}$ is fully faithful." ], "refs": [ "adequate-lemma-parasitic-adequate", "homology-lemma-kernel-exact-functor", "homology-lemma-serre-subcategory-is-kernel", "homology-lemma-quotient-by-kernel-exact-functor" ], "ref_ids": [ 3620, 12047, 12048, 12049 ] } ], "ref_ids": [] }, { "id": 3622, "type": "theorem", "label": "adequate-lemma-direct-image-parasitic-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-direct-image-parasitic-adequate", "contents": [ "Let $f : T \\to S$ be a quasi-compact and quasi-separated morphism", "of schemes. For any parasitic adequate $\\mathcal{O}_T$-module on", "$(\\Sch/T)_\\tau$ the pushforward", "$f_*\\mathcal{F}$ and the higher direct images $R^if_*\\mathcal{F}$", "are parasitic adequate $\\mathcal{O}_S$-modules on $(\\Sch/S)_\\tau$." ], "refs": [], "proofs": [ { "contents": [ "We have already seen in", "Lemma \\ref{lemma-direct-image-adequate}", "that these higher direct images are adequate.", "Hence it suffices to show that", "$(R^if_*\\mathcal{F})(U_i) = 0$ for any $\\tau$-covering", "$\\{U_i \\to S\\}$ open. And $R^if_*\\mathcal{F}$", "is parasitic by", "Descent, Lemma \\ref{descent-lemma-direct-image-parasitic}." ], "refs": [ "adequate-lemma-direct-image-adequate", "descent-lemma-direct-image-parasitic" ], "ref_ids": [ 3619, 14630 ] } ], "ref_ids": [] }, { "id": 3623, "type": "theorem", "label": "adequate-lemma-quotient-easy", "categories": [ "adequate" ], "title": "adequate-lemma-quotient-easy", "contents": [ "Let $S$ be a scheme. Let", "$\\mathcal{C} \\subset \\textit{Adeq}(\\mathcal{O})$ denote the", "full subcategory consisting of parasitic adequate modules.", "Then", "$$", "D(\\textit{Adeq}(\\mathcal{O}))/D_\\mathcal{C}(\\textit{Adeq}(\\mathcal{O}))", "= D(\\QCoh(\\mathcal{O}_S))", "$$", "and similarly for the bounded versions." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Derived Categories, Lemma \\ref{derived-lemma-quotient-by-serre-easy}." ], "refs": [ "derived-lemma-quotient-by-serre-easy" ], "ref_ids": [ 1848 ] } ], "ref_ids": [] }, { "id": 3624, "type": "theorem", "label": "adequate-lemma-describe-Dplus-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-describe-Dplus-adequate", "contents": [ "Let $U = \\Spec(A)$ be an affine scheme.", "The bounded below derived category", "$D^+(\\textit{Adeq}(\\mathcal{O}))$ is the localization", "of $K^+(\\QCoh(\\mathcal{O}_U))$ at the multiplicative subset of", "universal quasi-isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "If $\\varphi : \\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$", "is a morphism of complexes of quasi-coherent", "$\\mathcal{O}_U$-modules, then $u\\varphi : u\\mathcal{F}^\\bullet \\to", "u\\mathcal{G}^\\bullet$ is a quasi-isomorphism if and only if $\\varphi$ is", "a universal quasi-isomorphism. Hence the collection $S$", "of universal quasi-isomorphisms is a saturated multiplicative", "system compatible with the triangulated structure by", "Derived Categories, Lemma \\ref{derived-lemma-triangle-functor-localize}.", "Hence $S^{-1}K^+(\\QCoh(\\mathcal{O}_U))$ exists and is a", "triangulated category, see", "Derived Categories, Proposition", "\\ref{derived-proposition-construct-localization}.", "We obtain a canonical functor", "$can : S^{-1}K^+(\\QCoh(\\mathcal{O}_U)) \\to", "D^{+}(\\textit{Adeq}(\\mathcal{O}))$ by", "Derived Categories, Lemma \\ref{derived-lemma-universal-property-localization}.", "\\medskip\\noindent", "Note that, almost by definition, every adequate module on $U$ has an", "embedding into a quasi-coherent sheaf, see", "Lemma \\ref{lemma-adequate-characterize}. Hence by", "Derived Categories, Lemma \\ref{derived-lemma-subcategory-right-resolution}", "given $\\mathcal{F}^\\bullet \\in \\Ob(K^+(\\textit{Adeq}(\\mathcal{O})))$", "there exists a quasi-isomorphism", "$\\mathcal{F}^\\bullet \\to u\\mathcal{G}^\\bullet$", "where $\\mathcal{G}^\\bullet \\in \\Ob(K^+(\\QCoh(\\mathcal{O}_U)))$.", "This proves that $can$ is essentially surjective.", "\\medskip\\noindent", "Similarly, suppose that $\\mathcal{F}^\\bullet$ and $\\mathcal{G}^\\bullet$", "are bounded below complexes of quasi-coherent $\\mathcal{O}_U$-modules.", "A morphism in $D^+(\\textit{Adeq}(\\mathcal{O}))$ between these", "consists of a pair $f : u\\mathcal{F}^\\bullet \\to \\mathcal{H}^\\bullet$", "and $s : u\\mathcal{G}^\\bullet \\to \\mathcal{H}^\\bullet$ where $s$", "is a quasi-isomorphism. Pick a quasi-isomorphism", "$s' : \\mathcal{H}^\\bullet \\to u\\mathcal{E}^\\bullet$. Then we see that", "$s' \\circ f : \\mathcal{F} \\to \\mathcal{E}^\\bullet$ and the", "universal quasi-isomorphism", "$s' \\circ s : \\mathcal{G}^\\bullet \\to \\mathcal{E}^\\bullet$ give", "a morphism in $S^{-1}K^{+}(\\QCoh(\\mathcal{O}_U))$ mapping", "to the given morphism. This proves the \"fully\" part of full faithfulness.", "Faithfulness is proved similarly." ], "refs": [ "derived-lemma-triangle-functor-localize", "derived-proposition-construct-localization", "derived-lemma-universal-property-localization", "adequate-lemma-adequate-characterize", "derived-lemma-subcategory-right-resolution" ], "ref_ids": [ 1779, 1959, 1781, 3614, 1836 ] } ], "ref_ids": [] }, { "id": 3625, "type": "theorem", "label": "adequate-lemma-right-adjoint-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-right-adjoint-adequate", "contents": [ "Let $U = \\Spec(A)$ be an affine scheme.", "The inclusion functor", "$$", "\\textit{Adeq}(\\mathcal{O}) \\to", "\\textit{Mod}((\\Sch/U)_\\tau, \\mathcal{O})", "$$", "has a right adjoint $A$\\footnote{This is the ``adequator''.}.", "Moreover, the adjunction mapping", "$A(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism for every", "adequate module $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "By", "Topologies, Lemma \\ref{topologies-lemma-affine-big-site-fppf}", "(and similarly for the other topologies)", "we may work with $\\mathcal{O}$-modules on $(\\textit{Aff}/U)_\\tau$.", "Denote $\\mathcal{P}$ the category of module-valued", "functors on $\\textit{Alg}_A$ and $\\mathcal{A}$ the category of adequate", "functors on $\\textit{Alg}_A$. Denote $i : \\mathcal{A} \\to \\mathcal{P}$", "the inclusion functor. Denote $Q : \\mathcal{P} \\to \\mathcal{A}$", "the construction of Lemma \\ref{lemma-adjoint}.", "We have the commutative diagram", "\\begin{equation}", "\\label{equation-categories}", "\\vcenter{", "\\xymatrix{", "\\textit{Adeq}(\\mathcal{O}) \\ar[r]_-k \\ar@{=}[d] &", "\\textit{Mod}((\\textit{Aff}/U)_\\tau, \\mathcal{O}) \\ar[r]_-j &", "\\textit{PMod}((\\textit{Aff}/U)_\\tau, \\mathcal{O}) \\ar@{=}[d] \\\\", "\\mathcal{A} \\ar[rr]^-i & & \\mathcal{P}", "}", "}", "\\end{equation}", "The left vertical equality is", "Lemma \\ref{lemma-adequate-affine}", "and the right vertical equality was explained in", "Section \\ref{section-quasi-coherent}.", "Define $A(\\mathcal{F}) = Q(j(\\mathcal{F}))$.", "Since $j$ is fully faithful it follows immediately that $A$", "is a right adjoint of the inclusion functor $k$. Also, since", "$k$ is fully faithful too, the final assertion follows formally." ], "refs": [ "topologies-lemma-affine-big-site-fppf", "adequate-lemma-adjoint", "adequate-lemma-adequate-affine" ], "ref_ids": [ 12477, 3602, 3612 ] } ], "ref_ids": [] }, { "id": 3626, "type": "theorem", "label": "adequate-lemma-RA-zero", "categories": [ "adequate" ], "title": "adequate-lemma-RA-zero", "contents": [ "Let $U = \\Spec(A)$ be an affine scheme.", "For any object $\\mathcal{F}$ of $\\textit{Adeq}(\\mathcal{O})$", "we have $R^pA(\\mathcal{F}) = 0$ for all $p > 0$ where $A$ is", "as in", "Lemma \\ref{lemma-right-adjoint-adequate}." ], "refs": [ "adequate-lemma-right-adjoint-adequate" ], "proofs": [ { "contents": [ "With notation as in the proof of", "Lemma \\ref{lemma-right-adjoint-adequate}", "choose an injective resolution $k(\\mathcal{F}) \\to \\mathcal{I}^\\bullet$", "in the category of $\\mathcal{O}$-modules on $(\\textit{Aff}/U)_\\tau$.", "By", "Cohomology on Sites, Lemmas \\ref{sites-cohomology-lemma-include-modules}", "and", "Lemma \\ref{lemma-same-cohomology-adequate}", "the complex $j(\\mathcal{I}^\\bullet)$ is exact.", "On the other hand, each $j(\\mathcal{I}^n)$ is an injective object", "of the category of presheaves of modules by", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-injective-module-injective-presheaf}.", "It follows that $R^pA(\\mathcal{F}) = R^pQ(j(k(\\mathcal{F})))$.", "Hence the result now follows from", "Lemma \\ref{lemma-RQ-zero}." ], "refs": [ "adequate-lemma-right-adjoint-adequate", "sites-cohomology-lemma-include-modules", "adequate-lemma-same-cohomology-adequate", "sites-cohomology-lemma-injective-module-injective-presheaf", "adequate-lemma-RQ-zero" ], "ref_ids": [ 3625, 4208, 3616, 4207, 3610 ] } ], "ref_ids": [ 3625 ] }, { "id": 3627, "type": "theorem", "label": "adequate-lemma-bounded-below", "categories": [ "adequate" ], "title": "adequate-lemma-bounded-below", "contents": [ "If $U = \\Spec(A)$ is an affine scheme, then the bounded", "below version", "\\begin{equation}", "\\label{equation-compare-bounded-adequate}", "D^+(\\textit{Adeq}(\\mathcal{O}))", "\\longrightarrow", "D^+_{\\textit{Adeq}}(\\mathcal{O})", "\\end{equation}", "of the functor above is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Let $A : \\textit{Mod}(\\mathcal{O}) \\to \\textit{Adeq}(\\mathcal{O})$", "be the right adjoint to the inclusion functor constructed in", "Lemma \\ref{lemma-right-adjoint-adequate}.", "Since $A$ is left exact and since $\\textit{Mod}(\\mathcal{O})$", "has enough injectives, $A$ has a right derived functor", "$RA : D^+_{\\textit{Adeq}}(\\mathcal{O}) \\to D^+(\\textit{Adeq}(\\mathcal{O}))$.", "We claim that $RA$ is a quasi-inverse to", "(\\ref{equation-compare-bounded-adequate}).", "To see this the key fact is that if $\\mathcal{F}$ is an adequate module, then", "the adjunction map $\\mathcal{F} \\to RA(\\mathcal{F})$ is a", "quasi-isomorphism by Lemma \\ref{lemma-RA-zero}.", "\\medskip\\noindent", "Namely, to prove the lemma in full it suffices to show:", "\\begin{enumerate}", "\\item Given $\\mathcal{F}^\\bullet \\in K^+(\\textit{Adeq}(\\mathcal{O}))$", "the canonical map $\\mathcal{F}^\\bullet \\to RA(\\mathcal{F}^\\bullet)$", "is a quasi-isomorphism, and", "\\item given $\\mathcal{G}^\\bullet \\in K^+(\\textit{Mod}(\\mathcal{O}))$", "the canonical map $RA(\\mathcal{G}^\\bullet) \\to \\mathcal{G}^\\bullet$", "is a quasi-isomorphism.", "\\end{enumerate}", "Both (1) and (2) follow from the key fact via a spectral sequence", "argument using one of the spectral sequences of", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "Some details omitted." ], "refs": [ "adequate-lemma-right-adjoint-adequate", "adequate-lemma-RA-zero", "derived-lemma-two-ss-complex-functor" ], "ref_ids": [ 3625, 3626, 1871 ] } ], "ref_ids": [] }, { "id": 3628, "type": "theorem", "label": "adequate-lemma-ext-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-ext-adequate", "contents": [ "Let $U = \\Spec(A)$ be an affine scheme.", "Let $\\mathcal{F}$ and $\\mathcal{G}$ be adequate $\\mathcal{O}$-modules.", "For any $i \\geq 0$ the natural map", "$$", "\\Ext^i_{\\textit{Adeq}(\\mathcal{O})}(\\mathcal{F}, \\mathcal{G})", "\\longrightarrow", "\\Ext^i_{\\textit{Mod}(\\mathcal{O})}(\\mathcal{F}, \\mathcal{G})", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By definition these ext groups are computed as hom sets in the", "derived category. Hence this follows immediately from", "Lemma \\ref{lemma-bounded-below}." ], "refs": [ "adequate-lemma-bounded-below" ], "ref_ids": [ 3627 ] } ], "ref_ids": [] }, { "id": 3629, "type": "theorem", "label": "adequate-lemma-pure-projective", "categories": [ "adequate" ], "title": "adequate-lemma-pure-projective", "contents": [ "Let $A$ be a ring.", "\\begin{enumerate}", "\\item A module is pure projective if and only if", "it is a direct summand of a direct sum of finitely presented $A$-modules.", "\\item For any module $M$ there exists a universally exact sequence", "$0 \\to N \\to P \\to M \\to 0$ with $P$ pure projective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First note that a finitely presented $A$-module is pure projective by", "Algebra, Theorem \\ref{algebra-theorem-universally-exact-criteria}.", "Hence a direct summand of a direct sum of finitely presented $A$-modules", "is indeed pure projective. Let $M$ be any $A$-module.", "Write $M = \\colim_{i \\in I} P_i$ as a filtered colimit of", "finitely presented $A$-modules. Consider the sequence", "$$", "0 \\to N \\to \\bigoplus P_i \\to M \\to 0.", "$$", "For any finitely presented $A$-module $P$ the map", "$\\Hom_A(P, \\bigoplus P_i) \\to \\Hom_A(P, M)$", "is surjective, as any map $P \\to M$ factors through some $P_i$.", "Hence by", "Algebra, Theorem \\ref{algebra-theorem-universally-exact-criteria}", "this sequence is universally exact. This proves (2).", "If now $M$ is pure projective, then the sequence is split and", "we see that $M$ is a direct summand of $\\bigoplus P_i$." ], "refs": [ "algebra-theorem-universally-exact-criteria", "algebra-theorem-universally-exact-criteria" ], "ref_ids": [ 319, 319 ] } ], "ref_ids": [] }, { "id": 3630, "type": "theorem", "label": "adequate-lemma-pure-injective", "categories": [ "adequate" ], "title": "adequate-lemma-pure-injective", "contents": [ "Let $A$ be a ring. For any $A$-module $M$ set", "$M^\\vee = \\Hom_\\mathbf{Z}(M, \\mathbf{Q}/\\mathbf{Z})$.", "\\begin{enumerate}", "\\item For any $A$-module $M$ the $A$-module $M^\\vee$ is pure injective.", "\\item An $A$-module $I$ is pure injective if and only if the map", "$I \\to (I^\\vee)^\\vee$ splits.", "\\item For any module $M$ there exists a universally exact sequence", "$0 \\to M \\to I \\to N \\to 0$ with $I$ pure injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use the properties of the functor $M \\mapsto M^\\vee$ found in", "More on Algebra, Section \\ref{more-algebra-section-injectives-modules}", "without further mention. Part (1) holds because", "$\\Hom_A(N, M^\\vee) =", "\\Hom_\\mathbf{Z}(N \\otimes_A M, \\mathbf{Q}/\\mathbf{Z})$", "and because $\\mathbf{Q}/\\mathbf{Z}$ is injective in the category of", "abelian groups. Hence if $I \\to (I^\\vee)^\\vee$ is split, then", "$I$ is pure injective. We claim that for any $A$-module $M$ the", "evaluation map $ev : M \\to (M^\\vee)^\\vee$ is universally injective.", "To see this note that $ev^\\vee : ((M^\\vee)^\\vee)^\\vee \\to M^\\vee$", "has a right inverse, namely $ev' : M^\\vee \\to ((M^\\vee)^\\vee)^\\vee$.", "Then for any $A$-module $N$ applying the exact faithful functor", "${}^\\vee$ to the map $N \\otimes_A M \\to N \\otimes_A (M^\\vee)^\\vee$", "gives", "$$", "\\Hom_A(N, ((M^\\vee)^\\vee)^\\vee) =", "\\Big(N \\otimes_A (M^\\vee)^\\vee\\Big)^\\vee", "\\to", "\\Big(N \\otimes_A M\\Big)^\\vee =", "\\Hom_A(N, M^\\vee)", "$$", "which is surjective by the existence of the right inverse. The claim follows.", "The claim implies (3) and the necessity of the condition in (2)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3631, "type": "theorem", "label": "adequate-lemma-split-universally-exact-sequence", "categories": [ "adequate" ], "title": "adequate-lemma-split-universally-exact-sequence", "contents": [ "Let $A$ be a ring.", "\\begin{enumerate}", "\\item Let $L \\to M \\to N$ be a universally exact sequence", "of $A$-modules. Let $K = \\Im(M \\to N)$.", "Then $K \\to N$ is universally injective.", "\\item Any universally exact complex", "can be split into universally exact short exact sequences.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). For any $A$-module $T$ the sequence", "$L \\otimes_A T \\to M \\otimes_A T \\to K \\otimes_A T \\to 0$ is exact", "by right exactness of $\\otimes$. By assumption the sequence", "$L \\otimes_A T \\to M \\otimes_A T \\to N \\otimes_A T$ is exact.", "Combined this shows that $K \\otimes_A T \\to N \\otimes_A T$ is injective.", "\\medskip\\noindent", "Part (2) means the following: Suppose that $M^\\bullet$ is a universally", "exact complex of $A$-modules. Set $K^i = \\Ker(d^i) \\subset M^i$.", "Then the short exact sequences $0 \\to K^i \\to M^i \\to K^{i + 1} \\to 0$", "are universally exact. This follows immediately from part (1)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3632, "type": "theorem", "label": "adequate-lemma-pure-projective-resolutions", "categories": [ "adequate" ], "title": "adequate-lemma-pure-projective-resolutions", "contents": [ "Let $A$ be a ring.", "\\begin{enumerate}", "\\item Any $A$-module has a pure projective resolution.", "\\end{enumerate}", "Let $M \\to N$ be a map of $A$-modules.", "Let $P_\\bullet \\to M$ be a pure projective resolution and", "let $N_\\bullet \\to N$ be a universally exact resolution.", "\\begin{enumerate}", "\\item[(2)] There exists a map of complexes $P_\\bullet \\to N_\\bullet$", "inducing the given map", "$$", "M = \\Coker(P_1 \\to P_0) \\to \\Coker(N_1 \\to N_0) = N", "$$", "\\item[(3)] two maps $\\alpha, \\beta : P_\\bullet \\to N_\\bullet$", "inducing the same map $M \\to N$ are homotopic.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows immediately from", "Lemma \\ref{lemma-pure-projective}.", "Before we prove (2) and (3) note that by", "Lemma \\ref{lemma-split-universally-exact-sequence}", "we can split the universally exact complex $N_\\bullet \\to N \\to 0$", "into universally exact short exact sequences $0 \\to K_0 \\to N_0 \\to N \\to 0$", "and $0 \\to K_i \\to N_i \\to K_{i - 1} \\to 0$.", "\\medskip\\noindent", "Proof of (2). Because $P_0$ is pure projective", "we can find a map $P_0 \\to N_0$ lifting the map $P_0 \\to M \\to N$.", "We obtain an induced map $P_1 \\to F_0 \\to N_0$ wich ends up in $K_0$.", "Since $P_1$ is pure projective we may lift this", "to a map $P_1 \\to N_1$. This in turn induces a map", "$P_2 \\to P_1 \\to N_1$ which maps to zero into", "$N_0$, i.e., into $K_1$. Hence we may lift to get a map", "$P_2 \\to N_2$. Repeat.", "\\medskip\\noindent", "Proof of (3). To show that $\\alpha, \\beta$ are homotopic it suffices", "to show the difference $\\gamma = \\alpha - \\beta$ is homotopic", "to zero. Note that the image of $\\gamma_0 : P_0 \\to N_0$", "is contained in $K_0$. Hence we may lift", "$\\gamma_0$ to a map $h_0 : P_0 \\to N_1$. Consider the map", "$\\gamma_1' = \\gamma_1 - h_0 \\circ d_{P, 1} : P_1 \\to N_1$.", "By our choice of $h_0$ we see that the image of $\\gamma_1'$", "is contained in $K_1$. Since $P_1$ is pure projective may lift", "$\\gamma_1'$ to a map $h_1 : P_1 \\to N_2$. At this point we have", "$\\gamma_1 = h_0 \\circ d_{F, 1} + d_{N, 2} \\circ h_1$. Repeat." ], "refs": [ "adequate-lemma-pure-projective", "adequate-lemma-split-universally-exact-sequence" ], "ref_ids": [ 3629, 3631 ] } ], "ref_ids": [] }, { "id": 3633, "type": "theorem", "label": "adequate-lemma-pure-injective-resolutions", "categories": [ "adequate" ], "title": "adequate-lemma-pure-injective-resolutions", "contents": [ "Let $A$ be a ring.", "\\begin{enumerate}", "\\item Any $A$-module has a pure injective resolution.", "\\end{enumerate}", "Let $M \\to N$ be a map of $A$-modules.", "Let $M \\to M^\\bullet$ be a universally exact resolution and", "let $N \\to I^\\bullet$ be a pure injective resolution.", "\\begin{enumerate}", "\\item[(2)] There exists a map of complexes $M^\\bullet \\to I^\\bullet$", "inducing the given map", "$$", "M = \\Ker(M^0 \\to M^1) \\to \\Ker(I^0 \\to I^1) = N", "$$", "\\item[(3)] two maps $\\alpha, \\beta : M^\\bullet \\to I^\\bullet$", "inducing the same map $M \\to N$ are homotopic.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma is dual to", "Lemma \\ref{lemma-pure-projective-resolutions}.", "The proof is identical, except one has to reverse all the arrows." ], "refs": [ "adequate-lemma-pure-projective-resolutions" ], "ref_ids": [ 3632 ] } ], "ref_ids": [] }, { "id": 3634, "type": "theorem", "label": "adequate-lemma-facts-pext", "categories": [ "adequate" ], "title": "adequate-lemma-facts-pext", "contents": [ "Let $A$ be a ring.", "\\begin{enumerate}", "\\item $\\text{Pext}^i_A(M, N) = 0$ for $i > 0$ whenever $N$ is pure injective,", "\\item $\\text{Pext}^i_A(M, N) = 0$ for $i > 0$ whenever $M$ is pure projective,", "in particular if $M$ is an $A$-module of finite presentation,", "\\item $\\text{Pext}^i_A(M, N)$ is also the $i$th cohomology module", "of the complex $\\Hom_A(P_\\bullet, N)$ where $P_\\bullet$", "is a pure projective resolution of $M$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To see (3) consider the double complex", "$$", "A^{\\bullet, \\bullet} = \\Hom_A(P_\\bullet, I^\\bullet)", "$$", "Each of its rows is exact except in degree $0$ where its cohomology", "is $\\Hom_A(M, I^q)$. Each of its columns is exact except in degree $0$", "where its cohomology is $\\Hom_A(P_p, N)$. Hence the two spectral", "sequences associated to this complex in", "Homology, Section \\ref{homology-section-double-complex}", "degenerate, giving the equality." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3635, "type": "theorem", "label": "adequate-lemma-pure-injective-injective-adequate", "categories": [ "adequate" ], "title": "adequate-lemma-pure-injective-injective-adequate", "contents": [ "Let $A$ be a ring.", "Let $\\mathcal{A}$ be the category of adequate functors on $\\textit{Alg}_A$.", "The injective objects of $\\mathcal{A}$ are exactly the functors", "$\\underline{I}$ where $I$ is a pure injective $A$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $I$ be an injective object of $\\mathcal{A}$.", "Choose an embedding $I \\to \\underline{M}$ for some $A$-module $M$.", "As $I$ is injective we see that $\\underline{M} = I \\oplus F$ for some", "module-valued functor $F$. Then $M = I(A) \\oplus F(A)$ and it follows", "that $I = \\underline{I(A)}$. Thus we see that any injective object", "is of the form $\\underline{I}$ for some $A$-module $I$.", "It is clear that the module $I$ has to be pure injective", "since any universally exact sequence $0 \\to M \\to N \\to L \\to 0$", "gives rise to an exact sequence", "$0 \\to \\underline{M} \\to \\underline{N} \\to \\underline{L} \\to 0$", "of $\\mathcal{A}$.", "\\medskip\\noindent", "Finally, suppose that $I$ is a pure injective", "$A$-module. Choose an embedding $\\underline{I} \\to J$", "into an injective object of $\\mathcal{A}$ (see", "Lemma \\ref{lemma-enough-injectives}).", "We have seen above that $J = \\underline{I'}$", "for some $A$-module $I'$ which is pure injective. As", "$\\underline{I} \\to \\underline{I'}$ is injective", "the map $I \\to I'$ is universally injective. By assumption on $I$", "it splits. Hence $\\underline{I}$ is a summand of $J = \\underline{I'}$", "whence an injective object of the category $\\mathcal{A}$." ], "refs": [ "adequate-lemma-enough-injectives" ], "ref_ids": [ 3603 ] } ], "ref_ids": [] }, { "id": 3636, "type": "theorem", "label": "adequate-lemma-big-ext", "categories": [ "adequate" ], "title": "adequate-lemma-big-ext", "contents": [ "Let $U = \\Spec(A)$ be an affine scheme. Let $M$, $N$ be $A$-modules.", "For all $i$ we have a canonical isomorphism", "$$", "\\Ext^i_{\\textit{Mod}(\\mathcal{O})}(M^a, N^a) = \\text{Pext}^i_A(M, N)", "$$", "functorial in $M$ and $N$." ], "refs": [], "proofs": [ { "contents": [ "Let us construct a canonical arrow from right to left. Namely, if", "$N \\to I^\\bullet$ is a pure injective resolution, then", "$M^a \\to (I^\\bullet)^a$ is an exact complex of (adequate)", "$\\mathcal{O}$-modules. Hence any element of $\\text{Pext}^i_A(M, N)$", "gives rise to a map $N^a \\to M^a[i]$ in $D(\\mathcal{O})$, i.e.,", "an element of the group on the left.", "\\medskip\\noindent", "To prove this map is an isomorphism, note that we may replace", "$\\Ext^i_{\\textit{Mod}(\\mathcal{O})}(M^a, N^a)$ by", "$\\Ext^i_{\\textit{Adeq}(\\mathcal{O})}(M^a, N^a)$, see", "Lemma \\ref{lemma-ext-adequate}.", "Let $\\mathcal{A}$ be the category of adequate functors", "on $\\textit{Alg}_A$. We have seen that $\\mathcal{A}$ is", "equivalent to $\\textit{Adeq}(\\mathcal{O})$, see", "Lemma \\ref{lemma-adequate-affine}; see also the proof of", "Lemma \\ref{lemma-right-adjoint-adequate}.", "Hence now it suffices to prove that", "$$", "\\Ext^i_\\mathcal{A}(\\underline{M}, \\underline{N}) =", "\\text{Pext}^i_A(M, N)", "$$", "However, this is clear from", "Lemma \\ref{lemma-pure-injective-injective-adequate}", "as a pure injective resolution $N \\to I^\\bullet$ exactly corresponds", "to an injective resolution of $\\underline{N}$ in $\\mathcal{A}$." ], "refs": [ "adequate-lemma-ext-adequate", "adequate-lemma-adequate-affine", "adequate-lemma-right-adjoint-adequate", "adequate-lemma-pure-injective-injective-adequate" ], "ref_ids": [ 3628, 3612, 3625, 3635 ] } ], "ref_ids": [] }, { "id": 3650, "type": "theorem", "label": "spaces-topologies-lemma-zariski", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-zariski", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$", "is a Zariski covering of $X$.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a Zariski covering and for each", "$i$ we have a Zariski covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then", "$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a Zariski covering.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a Zariski covering", "and $X' \\to X$ is a morphism of algebraic spaces then", "$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a Zariski covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3651, "type": "theorem", "label": "spaces-topologies-lemma-zariski-etale", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-zariski-etale", "contents": [ "Any Zariski covering is an \\'etale covering." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions and the fact that an open immersion", "is an \\'etale morphism (this follows from", "Morphisms, Lemma \\ref{morphisms-lemma-open-immersion-etale} via", "Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}", "as immersions are representable)." ], "refs": [ "morphisms-lemma-open-immersion-etale", "spaces-lemma-representable-transformations-property-implication" ], "ref_ids": [ 5366, 8136 ] } ], "ref_ids": [] }, { "id": 3652, "type": "theorem", "label": "spaces-topologies-lemma-etale", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-etale", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$", "is a \\'etale covering of $X$.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a \\'etale covering and for each", "$i$ we have a \\'etale covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then", "$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a \\'etale covering.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a \\'etale covering", "and $X' \\to X$ is a morphism of algebraic spaces then", "$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a \\'etale covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3653, "type": "theorem", "label": "spaces-topologies-lemma-etale-dominates-smooth", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-etale-dominates-smooth", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\{X_i \\to X\\}_{i \\in I}$ be a smooth covering of $X$.", "Then there exists an \\'etale covering $\\{U_j \\to X\\}_{j \\in J}$", "of $X$ which refines $\\{X_i \\to X\\}_{i \\in I}$." ], "refs": [], "proofs": [ { "contents": [ "First choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "For each $i$ choose a scheme $W_i$ and a surjective \\'etale morphism", "$W_i \\to X_i$. Then $\\{W_i \\to X\\}_{i \\in I}$ is a smooth covering", "which refines $\\{X_i \\to X\\}_{i \\in I}$. Hence", "$\\{W_i \\times_X U \\to U\\}_{i \\in I}$ is a smooth covering of schemes.", "By More on Morphisms, Lemma \\ref{more-morphisms-lemma-etale-dominates-smooth}", "we can choose an \\'etale covering $\\{U_j \\to U\\}$ which refines", "$\\{W_i \\times_X U \\to U\\}$. Then $\\{U_j \\to X\\}_{j \\in J}$", "is an \\'etale covering refining $\\{X_i \\to X\\}_{i \\in I}$." ], "refs": [ "more-morphisms-lemma-etale-dominates-smooth" ], "ref_ids": [ 13880 ] } ], "ref_ids": [] }, { "id": 3654, "type": "theorem", "label": "spaces-topologies-lemma-put-in-T-etale", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-put-in-T-etale", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of", "$(\\textit{Spaces}/S)_\\etale$. The inclusion functor", "$Y_{spaces, \\etale} \\to (\\textit{Spaces}/X)_\\etale$", "is cocontinuous and induces a morphism of topoi", "$$", "i_f :", "\\Sh(Y_\\etale)", "\\longrightarrow", "\\Sh((\\textit{Spaces}/X)_\\etale)", "$$", "For a sheaf $\\mathcal{G}$ on $(\\textit{Spaces}/X)_\\etale$", "we have the formula $(i_f^{-1}\\mathcal{G})(U/Y) = \\mathcal{G}(U/X)$.", "The functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes", "with fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "Denote the functor $u : Y_{spaces, \\etale} \\to (\\textit{Spaces}/X)_\\etale$.", "In other words, given an \\'etale morphism $j : U \\to Y$ corresponding", "to an object of $Y_{spaces, \\etale}$ we set", "$u(U \\to T) = (f \\circ j : U \\to S)$.", "The category $Y_{spaces, \\etale}$", "has fibre products and equalizers and $u$ commutes with them.", "It is immediate that $u$ cocontinuous.", "The functor $u$ is also continuous as $u$ transforms coverings to coverings and", "commutes with fibre products. Hence the Lemma follows from", "Sites, Lemmas \\ref{sites-lemma-when-shriek}", "and \\ref{sites-lemma-preserve-equalizers}." ], "refs": [ "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers" ], "ref_ids": [ 8545, 8546 ] } ], "ref_ids": [] }, { "id": 3655, "type": "theorem", "label": "spaces-topologies-lemma-at-the-bottom-etale", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-at-the-bottom-etale", "contents": [ "Let $S$ be a scheme. Let $X$ be an object of $(\\textit{Spaces}/S)_\\etale$.", "The inclusion functor $X_{spaces, \\etale} \\to (\\textit{Spaces}/X)_\\etale$", "satisfies the hypotheses of Sites, Lemma \\ref{sites-lemma-bigger-site}", "and hence induces a morphism of sites", "$$", "\\pi_X :", "(\\textit{Spaces}/X)_\\etale", "\\longrightarrow", "X_{spaces, \\etale}", "$$", "and a morphism of topoi", "$$", "i_X :", "\\Sh(X_\\etale)", "\\longrightarrow", "\\Sh((\\textit{Spaces}/X)_\\etale)", "$$", "such that $\\pi_X \\circ i_X = \\text{id}$. Moreover, $i_X = i_{\\text{id}_X}$", "with $i_{\\text{id}_X}$ as in Lemma \\ref{lemma-put-in-T-etale}.", "In particular the functor $i_X^{-1} = \\pi_{X, *}$ is described by the rule", "$i_X^{-1}(\\mathcal{G})(U/X) = \\mathcal{G}(U/X)$." ], "refs": [ "sites-lemma-bigger-site", "spaces-topologies-lemma-put-in-T-etale" ], "proofs": [ { "contents": [ "In this case the functor", "$u : X_{spaces, \\etale} \\to (\\textit{Spaces}/X)_\\etale$,", "in addition to the properties seen in the proof of", "Lemma \\ref{lemma-put-in-T-etale} above, also is fully faithful", "and transforms the final object into the final object.", "The lemma follows from Sites, Lemma \\ref{sites-lemma-bigger-site}." ], "refs": [ "spaces-topologies-lemma-put-in-T-etale", "sites-lemma-bigger-site" ], "ref_ids": [ 3654, 8548 ] } ], "ref_ids": [ 8548, 3654 ] }, { "id": 3656, "type": "theorem", "label": "spaces-topologies-lemma-morphism-big-etale", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-morphism-big-etale", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism in", "$(\\textit{Spaces}/S)_\\etale$. The functor", "$$", "u :", "(\\textit{Spaces}/Y)_\\etale", "\\longrightarrow", "(\\textit{Spaces}/X)_\\etale,", "\\quad", "V/Y \\longmapsto V/X", "$$", "is cocontinuous, and has a continuous right adjoint", "$$", "v :", "(\\textit{Spaces}/X)_\\etale", "\\longrightarrow", "(\\textit{Spaces}/Y)_\\etale,", "\\quad", "(U \\to X) \\longmapsto (U \\times_X Y \\to Y).", "$$", "They induce the same morphism of topoi", "$$", "f_{big} :", "\\Sh((\\textit{Spaces}/Y)_\\etale)", "\\longrightarrow", "\\Sh((\\textit{Spaces}/X)_\\etale)", "$$", "We have $f_{big}^{-1}(\\mathcal{G})(U/Y) = \\mathcal{G}(U/X)$.", "We have $f_{big, *}(\\mathcal{F})(U/X) = \\mathcal{F}(U \\times_X Y/Y)$.", "Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with", "fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "The functor $u$ is cocontinuous, continuous and commutes with fibre products", "and equalizers (details omitted; compare with the proof of", "Lemma \\ref{lemma-put-in-T-etale}).", "Hence", "Sites, Lemmas \\ref{sites-lemma-when-shriek} and", "\\ref{sites-lemma-preserve-equalizers}", "apply and we deduce the formula", "for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover,", "the functor $v$ is a right adjoint because given $U/Y$ and $V/X$", "we have $\\Mor_X(u(U), V) = \\Mor_Y(U, V \\times_X Y)$", "as desired. Thus we may apply", "Sites, Lemmas \\ref{sites-lemma-have-functor-other-way} and", "\\ref{sites-lemma-have-functor-other-way-morphism} to get the", "formula for $f_{big, *}$." ], "refs": [ "spaces-topologies-lemma-put-in-T-etale", "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers", "sites-lemma-have-functor-other-way", "sites-lemma-have-functor-other-way-morphism" ], "ref_ids": [ 3654, 8545, 8546, 8549, 8550 ] } ], "ref_ids": [] }, { "id": 3657, "type": "theorem", "label": "spaces-topologies-lemma-morphism-big-small-etale", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-morphism-big-small-etale", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism in", "$(\\textit{Spaces}/S)_\\etale$.", "\\begin{enumerate}", "\\item We have $i_f = f_{big} \\circ i_T$ with $i_f$ as in", "Lemma \\ref{lemma-put-in-T-etale} and $i_T$ as in", "Lemma \\ref{lemma-at-the-bottom-etale}.", "\\item The functor $X_{spaces, \\etale} \\to T_{spaces, \\etale}$,", "$(U \\to X) \\mapsto (U \\times_X Y \\to Y)$ is continuous and induces", "a morphism of sites", "$$", "f_{spaces, \\etale} : Y_{spaces, \\etale} \\longrightarrow X_{spaces, \\etale}", "$$", "The corresponding morphism of small \\'etale topoi is denoted", "$$", "f_{small} : \\Sh(Y_\\etale) \\to \\Sh(X_\\etale)", "$$", "We have $f_{small, *}(\\mathcal{F})(U/X) = \\mathcal{F}(U \\times_X Y/Y)$.", "\\item We have a commutative diagram of morphisms of sites", "$$", "\\xymatrix{", "Y_{spaces, \\etale} \\ar[d]_{f_{spaces, \\etale}} &", "(\\textit{Spaces}/Y)_\\etale \\ar[d]^{f_{big}} \\ar[l]^-{\\pi_Y}\\\\", "X_{spaces, \\etale} &", "(\\textit{Spaces}/X)_\\etale \\ar[l]_-{\\pi_X}", "}", "$$", "so that $f_{small} \\circ \\pi_Y = \\pi_X \\circ f_{big}$ as morphisms of topoi.", "\\item We have $f_{small} = \\pi_X \\circ f_{big} \\circ i_Y = \\pi_X \\circ i_f$.", "\\end{enumerate}" ], "refs": [ "spaces-topologies-lemma-put-in-T-etale", "spaces-topologies-lemma-at-the-bottom-etale" ], "proofs": [ { "contents": [ "The equality $i_f = f_{big} \\circ i_Y$ follows from the", "equality $i_f^{-1} = i_T^{-1} \\circ f_{big}^{-1}$ which is", "clear from the descriptions of these functors above.", "Thus we see (1).", "\\medskip\\noindent", "The functor $u : X_{spaces, \\etale} \\to Y_{spaces, \\etale}$,", "$u(U \\to X) = (U \\times_X Y \\to Y)$ was shown to give", "rise to a morphism of sites and correspong morphism of", "small \\'etale topoi in", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-functoriality-etale-site}. The description", "of the pushforward is clear.", "\\medskip\\noindent", "Part (3) follows because $\\pi_X$ and $\\pi_Y$ are given by the", "inclusion functors and $f_{spaces, \\etale}$ and $f_{big}$ by the", "base change functors $U \\mapsto U \\times_X Y$.", "\\medskip\\noindent", "Statement (4) follows from (3) by precomposing with $i_Y$." ], "refs": [ "spaces-properties-lemma-functoriality-etale-site" ], "ref_ids": [ 11864 ] } ], "ref_ids": [ 3654, 3655 ] }, { "id": 3658, "type": "theorem", "label": "spaces-topologies-lemma-composition-etale", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-composition-etale", "contents": [ "Let $S$ be a scheme. Given morphisms $f : X \\to Y$, $g : Y \\to Z$", "in $(\\textit{Spaces}/S)_\\etale$ we have", "$g_{big} \\circ f_{big} = (g \\circ f)_{big}$ and", "$g_{small} \\circ f_{small} = (g \\circ f)_{small}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the simple description of pushforward", "and pullback for the functors on the big sites from", "Lemma \\ref{lemma-morphism-big-etale}. For the functors", "on the small sites this follows from the description of", "the pushforward functors in Lemma \\ref{lemma-morphism-big-small-etale}." ], "refs": [ "spaces-topologies-lemma-morphism-big-etale", "spaces-topologies-lemma-morphism-big-small-etale" ], "ref_ids": [ 3656, 3657 ] } ], "ref_ids": [] }, { "id": 3659, "type": "theorem", "label": "spaces-topologies-lemma-morphism-big-small-cartesian-diagram-etale", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-morphism-big-small-cartesian-diagram-etale", "contents": [ "Let $S$ be a scheme. Consider a cartesian diagram", "$$", "\\xymatrix{", "Y' \\ar[r]_{g'} \\ar[d]_{f'} & Y \\ar[d]^f \\\\", "X' \\ar[r]^g & X", "}", "$$", "in $(\\textit{Spaces}/S)_\\etale$. Then", "$i_g^{-1} \\circ f_{big, *} = f'_{small, *} \\circ (i_{g'})^{-1}$", "and $g_{big}^{-1} \\circ f_{big, *} = f'_{big, *} \\circ (g'_{big})^{-1}$." ], "refs": [], "proofs": [ { "contents": [ "Since the diagram is cartesian, we have for $U'/X'$", "that $U' \\times_{X'} Y' = U' \\times_X Y$. Hence both", "$i_g^{-1} \\circ f_{big, *}$ and $f'_{small, *} \\circ (i_{g'})^{-1}$", "send a sheaf $\\mathcal{F}$ on $(\\textit{Spaces}/Y)_\\etale$ to the sheaf", "$U' \\mapsto \\mathcal{F}(U' \\times_{X'} Y')$ on $X'_\\etale$", "(use Lemmas \\ref{lemma-put-in-T-etale} and \\ref{lemma-morphism-big-etale}).", "The second equality can be proved in the same manner or can be", "deduced from the very general", "Sites, Lemma \\ref{sites-lemma-localize-morphism}." ], "refs": [ "spaces-topologies-lemma-put-in-T-etale", "spaces-topologies-lemma-morphism-big-etale", "sites-lemma-localize-morphism" ], "ref_ids": [ 3654, 3656, 8571 ] } ], "ref_ids": [] }, { "id": 3660, "type": "theorem", "label": "spaces-topologies-lemma-zariski-etale-smooth", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-zariski-etale-smooth", "contents": [ "Any \\'etale covering is a smooth covering, and a fortiori,", "any Zariski covering is a smooth covering." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions, the fact that an", "\\'etale morphism is smooth", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-etale-smooth}), and", "Lemma \\ref{lemma-zariski-etale}." ], "refs": [ "spaces-morphisms-lemma-etale-smooth", "spaces-topologies-lemma-zariski-etale" ], "ref_ids": [ 4909, 3651 ] } ], "ref_ids": [] }, { "id": 3661, "type": "theorem", "label": "spaces-topologies-lemma-smooth", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-smooth", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$", "is a smooth covering of $X$.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a smooth covering and for each", "$i$ we have a smooth covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then", "$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a smooth covering.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a smooth covering", "and $X' \\to X$ is a morphism of algebraic spaces then", "$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a smooth covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3662, "type": "theorem", "label": "spaces-topologies-lemma-zariski-etale-smooth-syntomic", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-zariski-etale-smooth-syntomic", "contents": [ "Any smooth covering is a syntomic covering, and a fortiori,", "any \\'etale or Zariski covering is a syntomic covering." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions and the fact that a smooth", "morphism is syntomic", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-smooth-syntomic}),", "and Lemma \\ref{lemma-zariski-etale-smooth}." ], "refs": [ "spaces-morphisms-lemma-smooth-syntomic", "spaces-topologies-lemma-zariski-etale-smooth" ], "ref_ids": [ 4892, 3660 ] } ], "ref_ids": [] }, { "id": 3663, "type": "theorem", "label": "spaces-topologies-lemma-syntomic", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-syntomic", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$", "is a syntomic covering of $X$.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a syntomic covering and for each", "$i$ we have a syntomic covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then", "$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a syntomic covering.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a syntomic covering", "and $X' \\to X$ is a morphism of algebraic spaces then", "$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a syntomic covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3664, "type": "theorem", "label": "spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf", "contents": [ "Any syntomic covering is an fppf covering, and a fortiori,", "any smooth, \\'etale, or Zariski covering is an fppf covering." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions, the fact that a syntomic morphism", "is flat and locally of finite presentation", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-syntomic-locally-finite-presentation} and", "\\ref{spaces-morphisms-lemma-syntomic-flat}) and", "Lemma \\ref{lemma-zariski-etale-smooth-syntomic}." ], "refs": [ "spaces-morphisms-lemma-syntomic-locally-finite-presentation", "spaces-morphisms-lemma-syntomic-flat", "spaces-topologies-lemma-zariski-etale-smooth-syntomic" ], "ref_ids": [ 4883, 4884, 3662 ] } ], "ref_ids": [] }, { "id": 3665, "type": "theorem", "label": "spaces-topologies-lemma-fppf", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-fppf", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$", "is an fppf covering of $X$.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is an fppf covering and for each", "$i$ we have an fppf covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then", "$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is an fppf covering.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is an fppf covering", "and $X' \\to X$ is a morphism of algebraic spaces then", "$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is an fppf covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3666, "type": "theorem", "label": "spaces-topologies-lemma-refine-fppf-schemes", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-refine-fppf-schemes", "contents": [ "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.", "Suppose that $\\mathcal{U} = \\{f_i : X_i \\to X\\}_{i \\in I}$ is an", "fppf covering of $X$. Then there exists a refinement", "$\\mathcal{V} = \\{g_i : T_i \\to X\\}$ of $\\mathcal{U}$ which is an", "fppf covering such that each $T_i$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: For each $i$ choose a scheme $T_i$ and a surjective \\'etale", "morphism $T_i \\to X_i$. Then check that $\\{T_i \\to X\\}$ is an fppf covering." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3667, "type": "theorem", "label": "spaces-topologies-lemma-fppf-covering-surjective", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-fppf-covering-surjective", "contents": [ "Let $S$ be a scheme.", "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fppf covering of algebraic", "spaces over $S$. Then the map of sheaves", "$$", "\\coprod X_i \\longrightarrow X", "$$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Spaces, Lemma \\ref{spaces-lemma-surjective-flat-locally-finite-presentation}.", "See also", "Spaces, Remark \\ref{spaces-remark-warning}", "in case you are confused about the meaning of this lemma." ], "refs": [ "spaces-lemma-surjective-flat-locally-finite-presentation", "spaces-remark-warning" ], "ref_ids": [ 8137, 8187 ] } ], "ref_ids": [] }, { "id": 3668, "type": "theorem", "label": "spaces-topologies-lemma-morphism-big-fppf", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-morphism-big-fppf", "contents": [ "Let $S$ be a scheme.", "Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$.", "The functor", "$$", "u : (\\textit{Spaces}/Y)_{fppf} \\longrightarrow (\\textit{Spaces}/X)_{fppf},", "\\quad", "V/Y \\longmapsto V/X", "$$", "is cocontinuous, and has a continuous right adjoint", "$$", "v : (\\textit{Spaces}/X)_{fppf} \\longrightarrow (\\textit{Spaces}/Y)_{fppf},", "\\quad", "(U \\to Y) \\longmapsto (U \\times_X Y \\to Y).", "$$", "They induce the same morphism of topoi", "$$", "f_{big} :", "\\Sh((\\textit{Spaces}/Y)_{fppf})", "\\longrightarrow", "\\Sh((\\textit{Spaces}/X)_{fppf})", "$$", "We have $f_{big}^{-1}(\\mathcal{G})(U/Y) = \\mathcal{G}(U/X)$.", "We have $f_{big, *}(\\mathcal{F})(U/X) = \\mathcal{F}(U \\times_X Y/Y)$.", "Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with", "fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "The functor $u$ is cocontinuous, continuous, and commutes with fibre products", "and equalizers. Hence", "Sites, Lemmas \\ref{sites-lemma-when-shriek} and", "\\ref{sites-lemma-preserve-equalizers}", "apply and we deduce the formula", "for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover,", "the functor $v$ is a right adjoint because given $U/T$ and $V/X$", "we have $\\Mor_X(u(U), V) = \\Mor_Y(U, V \\times_X Y)$", "as desired. Thus we may apply", "Sites, Lemmas \\ref{sites-lemma-have-functor-other-way} and", "\\ref{sites-lemma-have-functor-other-way-morphism} to get the", "formula for $f_{big, *}$." ], "refs": [ "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers", "sites-lemma-have-functor-other-way", "sites-lemma-have-functor-other-way-morphism" ], "ref_ids": [ 8545, 8546, 8549, 8550 ] } ], "ref_ids": [] }, { "id": 3669, "type": "theorem", "label": "spaces-topologies-lemma-composition-fppf", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-composition-fppf", "contents": [ "Let $S$ be a scheme. Given morphisms $f : X \\to Y$, $g : Y \\to Z$", "of algebraic spaces over $S$ we have", "$g_{big} \\circ f_{big} = (g \\circ f)_{big}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the simple description of pushforward", "and pullback for the functors on the big sites from", "Lemma \\ref{lemma-morphism-big-fppf}." ], "refs": [ "spaces-topologies-lemma-morphism-big-fppf" ], "ref_ids": [ 3668 ] } ], "ref_ids": [] }, { "id": 3670, "type": "theorem", "label": "spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-ph", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-ph", "contents": [ "Any fppf covering is a ph covering, and a fortiori,", "any syntomic, smooth, \\'etale or Zariski covering is a ph covering." ], "refs": [], "proofs": [ { "contents": [ "We will show that an fppf covering is a ph covering, and then the", "rest follows from Lemma \\ref{lemma-zariski-etale-smooth-syntomic-fppf}.", "Let $\\{X_i \\to X\\}_{i \\in I}$ be an fppf covering of algebraic spaces", "over a base scheme $S$. Let $U$ be an affine scheme and let", "$U \\to X$ be a morphism. We can refine the fppf covering", "$\\{X_i \\times_U U \\to U\\}_{i \\in I}$ by an fppf covering", "$\\{T_i \\to U\\}_{i \\in I}$ where $T_i$ is a scheme", "(Lemma \\ref{lemma-refine-fppf-schemes}).", "Then we can find a standard ph covering $\\{U_j \\to U\\}_{j = 1, \\ldots, m}$", "refining $\\{T_i \\to U\\}_{i \\in I}$ by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-fppf-ph}", "(and the definition of ph coverings for schemes).", "Thus $\\{X_i \\to X\\}_{i \\in I}$ is a ph covering by definition." ], "refs": [ "spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf", "spaces-topologies-lemma-refine-fppf-schemes", "more-morphisms-lemma-fppf-ph" ], "ref_ids": [ 3664, 3666, 13927 ] } ], "ref_ids": [] }, { "id": 3671, "type": "theorem", "label": "spaces-topologies-lemma-surjective-proper-ph", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-surjective-proper-ph", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a surjective proper morphism", "of algebraic spaces over $S$. Then $\\{Y \\to X\\}$ is a ph covering." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be a morphism with $U$ affine.", "By Chow's lemma (in the weak form given as", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-weak-chow})", "we see that there is a surjective proper morphism of schemes", "$V \\to U$ which factors through $Y \\times_X U \\to U$.", "Taking any finite affine open cover of $V$ we obtain a", "standard ph covering of $U$ refining $\\{X \\times_Y U \\to U\\}$", "as desired." ], "refs": [ "spaces-cohomology-lemma-weak-chow" ], "ref_ids": [ 11327 ] } ], "ref_ids": [] }, { "id": 3672, "type": "theorem", "label": "spaces-topologies-lemma-ph", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-ph", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$", "is a ph covering of $X$.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a ph covering and for each", "$i$ we have a ph covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then", "$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a ph covering.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a ph covering", "and $X' \\to X$ is a morphism of algebraic spaces then", "$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a ph covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is clear. Consider $g : X' \\to X$ and", "$\\{X_i \\to X\\}_{i\\in I}$ a ph covering as in (3). By", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-finite-type}", "the morphisms $X' \\times_X X_i \\to X'$ are locally of finite type.", "If $h' : Z \\to X'$ is a morphism from an affine scheme", "towards $X'$, then set $h = g \\circ h' : Z \\to X$. The assumption", "on $\\{X_i \\to X\\}_{i\\in I}$ means there exists a standard ph covering", "$\\{Z_j \\to Z\\}_{j = 1, \\ldots, n}$ and morphisms $Z_j \\to X_{i(j)}$ covering", "$h$ for certain $i(j) \\in I$. By the universal property of the fibre product", "we obtain morphisms $Z_j \\to X' \\times_X X_{i(j)}$ over $h'$ also.", "Hence $\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a ph covering.", "This proves (3).", "\\medskip\\noindent", "Let $\\{X_i \\to X\\}_{i\\in I}$ and $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$ be as", "in (2). Let $h : Z \\to X$ be a morphism from an affine scheme towards $X$.", "By assumption there exists a standard ph covering", "$\\{Z_j \\to Z\\}_{j = 1, \\ldots, n}$ and morphisms $h_j : Z_j \\to X_{i(j)}$", "covering $h$ for some indices $i(j) \\in I$. By assumption there exist", "standard ph coverings", "$\\{Z_{j, l} \\to Z_j\\}_{l = 1, \\ldots, n(j)}$", "and morphisms $Z_{j, l} \\to X_{i(j)j(l)}$ covering", "$h_j$ for some indices $j(l) \\in J_{i(j)}$. By", "Topologies, Lemma \\ref{topologies-lemma-refine-by-standard-ph}", "the family $\\{Z_{j, l} \\to Z\\}$ can be refined by a standard ph covering.", "Hence we conclude that $\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$", "is a ph covering." ], "refs": [ "spaces-morphisms-lemma-base-change-finite-type", "topologies-lemma-refine-by-standard-ph" ], "ref_ids": [ 4815, 12481 ] } ], "ref_ids": [] }, { "id": 3673, "type": "theorem", "label": "spaces-topologies-lemma-characterize-sheaf", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-characterize-sheaf", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a presheaf on $(\\textit{Spaces}/X)_{ph}$.", "Then $\\mathcal{F}$ is a sheaf if and only if", "\\begin{enumerate}", "\\item $\\mathcal{F}$ satisfies the sheaf condition for \\'etale coverings, and", "\\item if $f : V \\to U$ is a proper surjective morphism of", "$(\\textit{Spaces}/X)_{ph}$, then", "$\\mathcal{F}(U)$ maps bijectively to the equalizer", "of the two maps $\\mathcal{F}(V) \\to \\mathcal{F}(V \\times_U V)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will show that if (1) and (2) hold, then $\\mathcal{F}$ is sheaf.", "Let $\\{T_i \\to T\\}$ be a ph covering, i.e., a covering in", "$(\\textit{Spaces}/X)_{ph}$.", "We will verify the sheaf condition for this covering.", "Let $s_i \\in \\mathcal{F}(T_i)$ be sections which restrict to the same", "section over $T_i \\times_T T_{i'}$. We will show that there exists a", "unique section $s \\in \\mathcal{F}$ restricting to $s_i$ over $T_i$.", "Let $\\{U_j \\to T\\}$ be an \\'etale covering with $U_j$ affine.", "By property (1) it suffices to produce sections $s_j \\in \\mathcal{F}(U_j)$", "which agree on $U_j \\cap U_{j'}$ in order to produce $s$.", "Consider the ph coverings $\\{T_i \\times_T U_j \\to U_j\\}$.", "Then $s_{ji} = s_i|_{T_i \\times_T U_j}$ are sections agreeing", "over $(T_i \\times_T U_j) \\times_{U_j} (T_{i'} \\times_T U_j)$.", "Choose a proper surjective morphism $V_j \\to U_j$ and a finite affine", "open covering $V_j = \\bigcup V_{jk}$ such that the standard ph covering", "$\\{V_{jk} \\to U_j\\}$ refines $\\{T_i \\times_T U_j \\to U_j\\}$.", "If $s_{jk} \\in \\mathcal{F}(V_{jk})$", "denotes the pullback of $s_{ji}$ to $V_{jk}$ by the", "implied morphisms, then we find that $s_{jk}$ glue to a section", "$s'_j \\in \\mathcal{F}(V_j)$. Using the agreement on overlaps", "once more, we find that $s'_j$ is in the equalizer of the two", "maps $\\mathcal{F}(V_j) \\to \\mathcal{F}(V_j \\times_{U_j} V_j)$.", "Hence by (2) we find that $s'_j$ comes from a unique section", "$s_j \\in \\mathcal{F}(U_j)$. We omit the verification that these", "sections $s_j$ have all the desired properties." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3674, "type": "theorem", "label": "spaces-topologies-lemma-morphism-big-ph", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-morphism-big-ph", "contents": [ "Let $S$ be a scheme.", "Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$.", "The functor", "$$", "u : (\\textit{Spaces}/Y)_{ph} \\longrightarrow (\\textit{Spaces}/X)_{ph},", "\\quad", "V/Y \\longmapsto V/X", "$$", "is cocontinuous, and has a continuous right adjoint", "$$", "v : (\\textit{Spaces}/X)_{ph} \\longrightarrow (\\textit{Spaces}/Y)_{ph},", "\\quad", "(U \\to Y) \\longmapsto (U \\times_X Y \\to Y).", "$$", "They induce the same morphism of topoi", "$$", "f_{big} :", "\\Sh((\\textit{Spaces}/Y)_{ph})", "\\longrightarrow", "\\Sh((\\textit{Spaces}/X)_{ph})", "$$", "We have $f_{big}^{-1}(\\mathcal{G})(U/Y) = \\mathcal{G}(U/X)$.", "We have $f_{big, *}(\\mathcal{F})(U/X) = \\mathcal{F}(U \\times_X Y/Y)$.", "Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with", "fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "The functor $u$ is cocontinuous, continuous, and commutes with fibre products", "and equalizers. Hence", "Sites, Lemmas \\ref{sites-lemma-when-shriek} and", "\\ref{sites-lemma-preserve-equalizers}", "apply and we deduce the formula", "for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover,", "the functor $v$ is a right adjoint because given $U/T$ and $V/X$", "we have $\\Mor_X(u(U), V) = \\Mor_Y(U, V \\times_X Y)$", "as desired. Thus we may apply", "Sites, Lemmas \\ref{sites-lemma-have-functor-other-way} and", "\\ref{sites-lemma-have-functor-other-way-morphism} to get the", "formula for $f_{big, *}$." ], "refs": [ "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers", "sites-lemma-have-functor-other-way", "sites-lemma-have-functor-other-way-morphism" ], "ref_ids": [ 8545, 8546, 8549, 8550 ] } ], "ref_ids": [] }, { "id": 3675, "type": "theorem", "label": "spaces-topologies-lemma-composition-ph", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-composition-ph", "contents": [ "Let $S$ be a scheme. Given morphisms $f : X \\to Y$, $g : Y \\to Z$", "of algebraic spaces over $S$ we have", "$g_{big} \\circ f_{big} = (g \\circ f)_{big}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the simple description of pushforward", "and pullback for the functors on the big sites from", "Lemma \\ref{lemma-morphism-big-ph}." ], "refs": [ "spaces-topologies-lemma-morphism-big-ph" ], "ref_ids": [ 3674 ] } ], "ref_ids": [] }, { "id": 3676, "type": "theorem", "label": "spaces-topologies-lemma-cech-enough", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-cech-enough", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $P$ be a property of objects in $(\\textit{Spaces}/X)_{fppf}$", "such that whenever $\\{U_i \\to U\\}$ is a covering in", "$(\\textit{Spaces}/X)_{fppf}$, then", "$$", "P(U_{i_0} \\times_U \\ldots \\times_U U_{i_p})", "\\text{ for all }", "p \\geq 0,\\ i_0, \\ldots, i_p \\in I", "\\Rightarrow P(U)", "$$", "If $P(U)$ for all $U$ affine and flat, locally of finite presentation over $X$,", "then $P(X)$." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a separated algebraic space locally of finite presentation over $X$.", "Then we can choose an \\'etale covering $\\{U_i \\to U\\}_{i \\in I}$ with $V_i$", "affine. Since $U$ is separated, we conclude that", "$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ is always affine.", "Whence $P(U_{i_0} \\times_U \\ldots \\times_U U_{i_p})$ always.", "Hence $P(U)$ holds. Choose a scheme $U$ which is a disjoint union of", "affines and a surjective \\'etale morphism $U \\to X$.", "Then $U \\times_X \\ldots \\times_X U$ (with $p + 1$ factors)", "is a separated algebraic space", "\\'etale over $X$. Hence $P(U \\times_X \\ldots \\times_X U)$ by the above.", "We conclude that $P(X)$ is true." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3677, "type": "theorem", "label": "spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc", "contents": [ "Any fppf covering is an fpqc covering, and a fortiori,", "any syntomic, smooth, \\'etale or Zariski covering is an fpqc covering." ], "refs": [], "proofs": [ { "contents": [ "We will show that an fppf covering is an fpqc covering, and then the", "rest follows from", "Lemma \\ref{lemma-zariski-etale-smooth-syntomic-fppf}.", "Let $\\{f_i : U_i \\to U\\}_{i \\in I}$ be an", "fppf covering of algebraic spaces over $S$.", "By definition this means that the $f_i$ are flat which checks the first", "condition of Definition \\ref{definition-fpqc-covering}. To check the", "second, let $V \\to U$ be a morphism with $V$ affine.", "We may choose an \\'etale covering $\\{V_{ij} \\to V \\times_U U_i\\}$", "with $V_{ij}$ affine. Then the compositions", "$f_{ij} : V_{ij} \\to V \\times_U U_i \\to V$ are flat and locally of", "finite presentation as compositions of such", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-composition-finite-presentation},", "\\ref{spaces-morphisms-lemma-composition-flat},", "\\ref{spaces-morphisms-lemma-etale-flat}, and", "\\ref{spaces-morphisms-lemma-etale-locally-finite-presentation}).", "Hence these morphisms are open", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-fppf-open})", "and we see that", "$|V| = \\bigcup_{i \\in I} \\bigcup_{j \\in J_i} f_{ij}(|V_{ij}|)$", "is an open covering of $|V|$.", "Since $|V|$ is quasi-compact, this covering has a finite", "refinement.", "Say $V_{i_1j_1}, \\ldots, V_{i_Nj_N}$ do the job.", "Then $\\{V_{i_kj_k} \\to V\\}_{k = 1, \\ldots, N}$ is", "a standard fpqc covering of $V$ refinining the family", "$\\{U_i \\times_U V \\to V\\}$.", "This finishes the proof." ], "refs": [ "spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf", "spaces-topologies-definition-fpqc-covering", "spaces-morphisms-lemma-composition-finite-presentation", "spaces-morphisms-lemma-composition-flat", "spaces-morphisms-lemma-etale-flat", "spaces-morphisms-lemma-etale-locally-finite-presentation", "spaces-morphisms-lemma-fppf-open" ], "ref_ids": [ 3664, 3694, 4839, 4852, 4910, 4911, 4855 ] } ], "ref_ids": [] }, { "id": 3678, "type": "theorem", "label": "spaces-topologies-lemma-fpqc", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-fpqc", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$", "is an fpqc covering of $X$.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is an fpqc covering and for each", "$i$ we have an fpqc covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then", "$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is an fpqc covering.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is an fpqc covering", "and $X' \\to X$ is a morphism of algebraic spaces then", "$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is an fpqc covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is clear. Consider $g : X' \\to X$ and", "$\\{X_i \\to X\\}_{i\\in I}$ an fpqc covering as in (3). By", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-flat}", "the morphisms $X' \\times_X X_i \\to X'$", "are flat. If $h' : Z \\to X'$ is a morphism from an affine scheme", "towards $X'$, then set $h = g \\circ h' : Z \\to X$. The assumption", "on $\\{X_i \\to X\\}_{i\\in I}$ means there exists a standard fpqc covering", "$\\{Z_j \\to Z\\}_{j = 1, \\ldots, n}$ and morphisms $Z_j \\to X_{i(j)}$ covering", "$h$ for certain $i(j) \\in I$. By the universal property of the fibre product", "we obtain morphisms $Z_j \\to X' \\times_X X_{i(j)}$ over $h'$ also.", "Hence $\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is an fpqc covering.", "This proves (3).", "\\medskip\\noindent", "Let $\\{X_i \\to X\\}_{i\\in I}$ and $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$ be as", "in (2). Let $h : Z \\to X$ be a morphism from an affine scheme towards $X$.", "By assumption there exists a standard fpqc covering", "$\\{Z_j \\to Z\\}_{j = 1, \\ldots, n}$ and morphisms $h_j : Z_j \\to X_{i(j)}$", "covering $h$ for some indices $i(j) \\in I$. By assumption there exist", "standard fpqc coverings", "$\\{Z_{j, l} \\to Z_j\\}_{l = 1, \\ldots, n(j)}$", "and morphisms $Z_{j, l} \\to X_{i(j)j(l)}$ covering", "$h_j$ for some indices $j(l) \\in J_{i(j)}$. By", "Topologies, Lemma \\ref{topologies-lemma-fpqc-affine-axioms}", "the family $\\{Z_{j, l} \\to Z\\}$ is a standard fpqc covering.", "Hence we conclude that $\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$", "is an fpqc covering." ], "refs": [ "spaces-morphisms-lemma-base-change-flat", "topologies-lemma-fpqc-affine-axioms" ], "ref_ids": [ 4853, 12500 ] } ], "ref_ids": [] }, { "id": 3679, "type": "theorem", "label": "spaces-topologies-lemma-recognize-fpqc-covering", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-recognize-fpqc-covering", "contents": [ "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.", "Suppose that $\\{f_i : X_i \\to X\\}_{i \\in I}$ is a family of morphisms of", "algebraic spaces with target $X$. Let $U \\to X$ be a surjective", "\\'etale morphism from a scheme towards $X$. Then", "$\\{f_i : X_i \\to X\\}_{i \\in I}$ is an fpqc covering of $X$ if and only", "if $\\{U \\times_X X_i \\to U\\}_{i \\in I}$ is an fpqc covering of $U$." ], "refs": [], "proofs": [ { "contents": [ "If $\\{X_i \\to X\\}_{i \\in I}$ is an fpqc covering, then so is", "$\\{U \\times_X X_i \\to U\\}_{i \\in I}$ by Lemma \\ref{lemma-fpqc}.", "Assume that $\\{U \\times_X X_i \\to U\\}_{i \\in I}$ is an fpqc covering.", "Let $h : Z \\to X$ be a morphism from an affine scheme towards $X$.", "Then we see that $U \\times_X Z \\to Z$ is a surjective \\'etale morphism", "of schemes, in particular open. Hence we can find finitely many affine opens", "$W_1, \\ldots, W_t$ of $U \\times_X Z$ whose images cover $Z$.", "For each $j$ we may apply the condition that", "$\\{U \\times_X X_i \\to U\\}_{i \\in I}$ is an fpqc covering", "to the morphism $W_j \\to U$, and obtain a standard fpqc covering", "$\\{W_{jl} \\to W_j\\}$ which refines $\\{W_j \\times_X X_i \\to W_j\\}_{i \\in I}$.", "Hence $\\{W_{jl} \\to Z\\}$ is a standard fpqc covering of $Z$", "(see", "Topologies, Lemma \\ref{topologies-lemma-fpqc-affine-axioms})", "which refines $\\{Z \\times_X X_i \\to X\\}$ and we win." ], "refs": [ "spaces-topologies-lemma-fpqc", "topologies-lemma-fpqc-affine-axioms" ], "ref_ids": [ 3678, 12500 ] } ], "ref_ids": [] }, { "id": 3680, "type": "theorem", "label": "spaces-topologies-lemma-refine-fpqc-schemes", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-lemma-refine-fpqc-schemes", "contents": [ "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.", "Suppose that $\\mathcal{U} = \\{f_i : X_i \\to X\\}_{i \\in I}$ is an", "fpqc covering of $X$. Then there exists a refinement", "$\\mathcal{V} = \\{g_i : T_i \\to X\\}$ of $\\mathcal{U}$ which is an", "fpqc covering such that each $T_i$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: For each $i$ choose a scheme $T_i$ and a surjective \\'etale", "morphism $T_i \\to X_i$. Then check that $\\{T_i \\to X\\}$ is an fpqc covering." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3696, "type": "theorem", "label": "proetale-theorem-proper-base-change", "categories": [ "proetale" ], "title": "proetale-theorem-proper-base-change", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$ be", "a morphism of schemes giving rise to the base change diagram", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal", "such that $\\Lambda/I$ is torsion. Let $K$ be an object", "of $D(X_\\proetale)$ such that", "\\begin{enumerate}", "\\item $K$ is derived complete, and", "\\item $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$ is", "bounded below with cohomology sheaves coming from $X_\\etale$,", "\\item $\\Lambda/I^n$ is a perfect $\\Lambda$-module\\footnote{This assumption", "can be removed if $K$ is a constructible complex, see \\cite{BS}.}.", "\\end{enumerate}", "Then the base change map", "$$", "Lg_{comp}^*Rf_*K \\longrightarrow Rf'_*L(g')^*_{comp}K", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "We omit the construction of the base change map (this uses only", "formal properties of derived pushforward and completed derived pullback,", "compare with", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change}).", "Write $K_n = K \\otimes^\\mathbf{L}_\\Lambda \\underline{\\Lambda/I^n}$.", "By Lemma \\ref{lemma-naive-completion} we have $K = R\\lim K_n$", "because $K$ is derived complete.", "By Lemmas \\ref{lemma-pushforward-Noetherian-case} and", "\\ref{lemma-naive-completion} we can unwind the left hand side", "$$", "Lg_{comp}^* Rf_* K =", "R\\lim Lg^*(Rf_*K)\\otimes^\\mathbf{L}_\\Lambda \\underline{\\Lambda/I^n} =", "R\\lim Lg^* Rf_* K_n", "$$", "the last equality because $\\Lambda/I^n$ is a perfect module and", "the projection formula (Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-projection-formula}).", "Using Lemma \\ref{lemma-pushforward-Noetherian-case} we can unwind the right", "hand side", "$$", "Rf'_* L(g')^*_{comp} K =", "Rf'_* R\\lim L(g')^* K_n =", "R\\lim Rf'_* L(g')^* K_n", "$$", "the last equality because $Rf'_*$ commutes with $R\\lim$", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-Rf-commutes-with-Rlim}).", "Thus it suffices to show the maps", "$$", "Lg^* Rf_* K_n \\longrightarrow Rf'_* L(g')^* K_n", "$$", "are isomorphisms. By Lemma \\ref{lemma-compare-derived} and our second", "condition we can write $K_n = \\epsilon^{-1}L_n$ for some", "$L_n \\in D^+(X_\\etale, \\Lambda/I^n)$. By Lemma \\ref{lemma-morphism-comparison}", "and the fact that $\\epsilon^{-1}$ commutes with pullbacks", "we obtain", "$$", "Lg^* Rf_* K_n =", "Lg^* Rf_* \\epsilon^*L_n =", "Lg^* \\epsilon^{-1} Rf_* L_n =", "\\epsilon^{-1} Lg^* Rf_* L_n", "$$", "and", "$$", "Rf'_* L(g')^* K_n =", "Rf'_* L(g')^* \\epsilon^{-1} L_n =", "Rf'_* \\epsilon^{-1} L(g')^* L_n =", "\\epsilon^{-1} Rf'_* L(g')^* L_n", "$$", "(this also uses that $L_n$ is bounded below).", "Finally, by the proper base change theorem for \\'etale cohomology", "(\\'Etale Cohomology, Theorem", "\\ref{etale-cohomology-theorem-proper-base-change}) we have", "$$", "Lg^* Rf_* L_n = Rf'_* L(g')^* L_n", "$$", "(again using that $L_n$ is bounded below)", "and the theorem is proved." ], "refs": [ "sites-cohomology-remark-base-change", "proetale-lemma-naive-completion", "proetale-lemma-pushforward-Noetherian-case", "proetale-lemma-naive-completion", "sites-cohomology-lemma-projection-formula", "proetale-lemma-pushforward-Noetherian-case", "sites-cohomology-lemma-Rf-commutes-with-Rlim", "proetale-lemma-compare-derived", "proetale-lemma-morphism-comparison", "etale-cohomology-theorem-proper-base-change" ], "ref_ids": [ 4424, 3790, 3791, 3790, 4396, 3791, 4267, 3786, 3793, 6397 ] } ], "ref_ids": [] }, { "id": 3697, "type": "theorem", "label": "proetale-lemma-spectral-split", "categories": [ "proetale" ], "title": "proetale-lemma-spectral-split", "contents": [ "Let $X$ be a spectral space. Let $X_0 \\subset X$ be the set of closed points.", "The following are equivalent", "\\begin{enumerate}", "\\item Every open covering of $X$ can be refined by a finite", "disjoint union decomposition $X = \\coprod U_i$ with $U_i$", "open and closed in $X$.", "\\item The composition $X_0 \\to X \\to \\pi_0(X)$ is bijective.", "\\end{enumerate}", "Moreover, if $X_0$ is closed in $X$ and every point of $X$ specializes", "to a unique point of $X_0$, then these conditions are satisfied." ], "refs": [], "proofs": [ { "contents": [ "We will use without further mention that", "$X_0$ is quasi-compact", "(Topology, Lemma \\ref{topology-lemma-closed-points-quasi-compact})", "and $\\pi_0(X)$ is profinite", "(Topology, Lemma \\ref{topology-lemma-spectral-pi0}).", "Picture", "$$", "\\xymatrix{", "X_0 \\ar[rd]_f \\ar[r] & X \\ar[d]^\\pi \\\\", "& \\pi_0(X)", "}", "$$", "If (2) holds, the continuous bijective map $f : X_0 \\to \\pi_0(X)$ is", "a homeomorphism by", "Topology, Lemma \\ref{topology-lemma-bijective-map}.", "Given an open covering $X = \\bigcup U_i$, we get an open covering", "$\\pi_0(X) = \\bigcup f(X_0 \\cap U_i)$. By", "Topology, Lemma \\ref{topology-lemma-profinite-refine-open-covering}", "we can find a finite open covering of the form $\\pi_0(X) = \\coprod V_j$", "which refines this covering.", "Since $X_0 \\to \\pi_0(X)$ is bijective each connected component of", "$X$ has a unique closed point, whence is equal to the set of points", "specializing to this closed point. Hence $\\pi^{-1}(V_j)$ is the", "set of points specializing to the points of $f^{-1}(V_j)$.", "Now, if $f^{-1}(V_j) \\subset X_0 \\cap U_i \\subset U_i$, then", "it follows that $\\pi^{-1}(V_j) \\subset U_i$ (because the open set", "$U_i$ is closed under generalizations). In this way we see", "that the open covering $X = \\coprod \\pi^{-1}(V_j)$ refines", "the covering we started out with. In this way we see that", "(2) implies (1).", "\\medskip\\noindent", "Assume (1). Let $x, y \\in X$ be closed points. Then we have the open covering", "$X = (X \\setminus \\{x\\}) \\cup (X \\setminus \\{y\\})$.", "It follows from (1) that there exists a disjoint union decomposition", "$X = U \\amalg V$ with $U$ and $V$ open (and closed) and $x \\in U$ and", "$y \\in V$. In particular we see that every connected component of $X$", "has at most one closed point. By", "Topology, Lemma \\ref{topology-lemma-quasi-compact-closed-point}", "every connected component (being closed) also does have a closed point.", "Thus $X_0 \\to \\pi_0(X)$ is bijective. In this way we see that (1) implies (2).", "\\medskip\\noindent", "Assume $X_0$ is closed in $X$ and every point specializes to a unique", "point of $X_0$. Then $X_0$ is a spectral space", "(Topology, Lemma \\ref{topology-lemma-spectral-sub})", "consisting of closed points, hence profinite", "(Topology, Lemma \\ref{topology-lemma-characterize-profinite-spectral}).", "Let $x, y \\in X_0$ be distinct. By", "Topology, Lemma \\ref{topology-lemma-profinite-refine-open-covering}", "we can find a disjoint union decomposition", "$X_0 = U_0 \\amalg V_0$ with $U_0$ and $V_0$ open and closed", "and $x \\in U_0$ and $y \\in V_0$.", "Let $U \\subset X$, resp.\\ $V \\subset X$", "be the set of points specializing to $U_0$, resp.\\ $V_0$.", "Observe that $X = U \\amalg V$.", "By Topology, Lemma \\ref{topology-lemma-make-spectral-space}", "we see that $U$ is an intersection of quasi-compact open subsets.", "Hence $U$ is closed in the constructible topology.", "Since $U$ is closed under specialization, we see that", "$U$ is closed by Topology, Lemma", "\\ref{topology-lemma-constructible-stable-specialization-closed}.", "By symmetry $V$ is closed and hence $U$ and $V$ are both", "open and closed.", "This proves that $x, y$ are not in the same connected component of $X$.", "In other words, $X_0 \\to \\pi_0(X)$ is injective. The map is also", "surjective by", "Topology, Lemma \\ref{topology-lemma-quasi-compact-closed-point}", "and the fact that connected components are closed.", "In this way we see that the final condition implies (2)." ], "refs": [ "topology-lemma-closed-points-quasi-compact", "topology-lemma-spectral-pi0", "topology-lemma-bijective-map", "topology-lemma-profinite-refine-open-covering", "topology-lemma-quasi-compact-closed-point", "topology-lemma-spectral-sub", "topology-lemma-characterize-profinite-spectral", "topology-lemma-profinite-refine-open-covering", "topology-lemma-make-spectral-space", "topology-lemma-constructible-stable-specialization-closed", "topology-lemma-quasi-compact-closed-point" ], "ref_ids": [ 8235, 8310, 8275, 8301, 8234, 8306, 8309, 8301, 8323, 8307, 8234 ] } ], "ref_ids": [] }, { "id": 3698, "type": "theorem", "label": "proetale-lemma-closed-subspace-w-local", "categories": [ "proetale" ], "title": "proetale-lemma-closed-subspace-w-local", "contents": [ "Let $X$ be a w-local spectral space. If $Y \\subset X$ is closed,", "then $Y$ is w-local." ], "refs": [], "proofs": [ { "contents": [ "The subset $Y_0 \\subset Y$ of closed points is closed because", "$Y_0 = X_0 \\cap Y$. Since $X$ is $w$-local, every $y \\in Y$ specializes", "to a unique point of $X_0$. This specialization is in $Y$, and hence", "also in $Y_0$, because $\\overline{\\{y\\}}\\subset Y$. In conclusion, $Y$", "is $w$-local." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3699, "type": "theorem", "label": "proetale-lemma-silly", "categories": [ "proetale" ], "title": "proetale-lemma-silly", "contents": [ "Let $X$ be a spectral space. Let", "$$", "\\xymatrix{", "Y \\ar[r] \\ar[d] & T \\ar[d] \\\\", "X \\ar[r] & \\pi_0(X)", "}", "$$", "be a cartesian diagram in the category of topological spaces", "with $T$ profinite. Then $Y$ is spectral and $T = \\pi_0(Y)$.", "If moreover $X$ is w-local, then $Y$ is w-local, $Y \\to X$ is w-local,", "and the set of closed points of $Y$ is the inverse image of the", "set of closed points of $X$." ], "refs": [], "proofs": [ { "contents": [ "Note that $Y$ is a closed subspace of $X \\times T$ as $\\pi_0(X)$", "is a profinite space hence Hausdorff", "(use Topology, Lemmas \\ref{topology-lemma-spectral-pi0} and", "\\ref{topology-lemma-fibre-product-closed}).", "Since $X \\times T$ is spectral", "(Topology, Lemma \\ref{topology-lemma-product-spectral-spaces})", "it follows that $Y$ is spectral", "(Topology, Lemma \\ref{topology-lemma-spectral-sub}).", "Let $Y \\to \\pi_0(Y) \\to T$ be the canonical factorization", "(Topology, Lemma \\ref{topology-lemma-space-connected-components}).", "It is clear that $\\pi_0(Y) \\to T$ is surjective.", "The fibres of $Y \\to T$ are homeomorphic to the fibres of", "$X \\to \\pi_0(X)$. Hence these fibres are connected. It follows", "that $\\pi_0(Y) \\to T$ is injective. We conclude that $\\pi_0(Y) \\to T$", "is a homeomorphism by", "Topology, Lemma \\ref{topology-lemma-bijective-map}.", "\\medskip\\noindent", "Next, assume that $X$ is w-local and let $X_0 \\subset X$ be the", "set of closed points. The inverse image $Y_0 \\subset Y$ of $X_0$ in", "$Y$ maps bijectively onto $T$ as $X_0 \\to \\pi_0(X)$ is a bijection", "by Lemma \\ref{lemma-spectral-split}. Moreover, $Y_0$ is quasi-compact", "as a closed subset of the spectral space $Y$. Hence", "$Y_0 \\to \\pi_0(Y) = T$ is a homeomorphism by", "Topology, Lemma \\ref{topology-lemma-bijective-map}.", "It follows that all points of $Y_0$ are closed in $Y$.", "Conversely, if $y \\in Y$ is a closed point, then", "it is closed in the fibre of $Y \\to \\pi_0(Y) = T$", "and hence its image $x$ in $X$ is closed in the (homeomorphic) fibre of", "$X \\to \\pi_0(X)$. This implies $x \\in X_0$ and hence $y \\in Y_0$.", "Thus $Y_0$ is the collection of closed points of $Y$", "and for each $y \\in Y_0$ the set of generalizations of $y$ is", "the fibre of $Y \\to \\pi_0(Y)$. The lemma follows." ], "refs": [ "topology-lemma-spectral-pi0", "topology-lemma-fibre-product-closed", "topology-lemma-product-spectral-spaces", "topology-lemma-spectral-sub", "topology-lemma-space-connected-components", "topology-lemma-bijective-map", "proetale-lemma-spectral-split", "topology-lemma-bijective-map" ], "ref_ids": [ 8310, 8193, 8311, 8306, 8210, 8275, 3697, 8275 ] } ], "ref_ids": [] }, { "id": 3700, "type": "theorem", "label": "proetale-lemma-base-change-local-isomorphism", "categories": [ "proetale" ], "title": "proetale-lemma-base-change-local-isomorphism", "contents": [ "Let $A \\to B$ and $A \\to A'$ be ring maps. Let $B' = B \\otimes_A A'$", "be the base change of $B$.", "\\begin{enumerate}", "\\item If $A \\to B$ is a local isomorphism, then $A' \\to B'$ is a", "local isomorphism.", "\\item If $A \\to B$ identifies local rings, then $A' \\to B'$", "identifies local rings.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3701, "type": "theorem", "label": "proetale-lemma-composition-local-isomorphism", "categories": [ "proetale" ], "title": "proetale-lemma-composition-local-isomorphism", "contents": [ "Let $A \\to B$ and $B \\to C$ be ring maps.", "\\begin{enumerate}", "\\item If $A \\to B$ and $B \\to C$ are local isomorphisms, then $A \\to C$", "is a local isomorphism.", "\\item If $A \\to B$ and $B \\to C$ identify local rings, then $A \\to C$", "identifies local rings.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3702, "type": "theorem", "label": "proetale-lemma-local-isomorphism-permanence", "categories": [ "proetale" ], "title": "proetale-lemma-local-isomorphism-permanence", "contents": [ "Let $A$ be a ring. Let $B \\to C$ be an $A$-algebra homomorphism.", "\\begin{enumerate}", "\\item If $A \\to B$ and $A \\to C$ are local isomorphisms, then $B \\to C$", "is a local isomorphism.", "\\item If $A \\to B$ and $A \\to C$ identify local rings, then $B \\to C$", "identifies local rings.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3703, "type": "theorem", "label": "proetale-lemma-local-isomorphism-implies", "categories": [ "proetale" ], "title": "proetale-lemma-local-isomorphism-implies", "contents": [ "Let $A \\to B$ be a local isomorphism. Then", "\\begin{enumerate}", "\\item $A \\to B$ is \\'etale,", "\\item $A \\to B$ identifies local rings,", "\\item $A \\to B$ is quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3704, "type": "theorem", "label": "proetale-lemma-structure-local-isomorphism", "categories": [ "proetale" ], "title": "proetale-lemma-structure-local-isomorphism", "contents": [ "Let $A \\to B$ be a local isomorphism. Then there exist $n \\geq 0$,", "$g_1, \\ldots, g_n \\in B$, $f_1, \\ldots, f_n \\in A$ such that", "$(g_1, \\ldots, g_n) = B$ and $A_{f_i} \\cong B_{g_i}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3705, "type": "theorem", "label": "proetale-lemma-fully-faithful-spaces-over-X", "categories": [ "proetale" ], "title": "proetale-lemma-fully-faithful-spaces-over-X", "contents": [ "Let $p : (Y, \\mathcal{O}_Y) \\to (X, \\mathcal{O}_X)$ and", "$q : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$", "be morphisms of locally ringed spaces.", "If $\\mathcal{O}_Y = p^{-1}\\mathcal{O}_X$, then", "$$", "\\Mor_{\\text{LRS}/(X, \\mathcal{O}_X)}((Z, \\mathcal{O}_Z), (Y, \\mathcal{O}_Y))", "\\longrightarrow", "\\Mor_{\\textit{Top}/X}(Z, Y),\\quad", "(f, f^\\sharp) \\longmapsto f", "$$", "is bijective. Here $\\text{LRS}/(X, \\mathcal{O}_X)$ is the category of", "locally ringed spaces over $X$ and $\\textit{Top}/X$ is the category", "of topological spaces over $X$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3706, "type": "theorem", "label": "proetale-lemma-local-isomorphism-fully-faithful", "categories": [ "proetale" ], "title": "proetale-lemma-local-isomorphism-fully-faithful", "contents": [ "Let $A$ be a ring. Set $X = \\Spec(A)$. The functor", "$$", "B \\longmapsto \\Spec(B)", "$$", "from the category of $A$-algebras $B$ such that $A \\to B$ identifies", "local rings to the category of", "topological spaces over $X$ is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-fully-faithful-spaces-over-X}", "and the fact that if $A \\to B$ identifies local rings, then the pullback", "of the structure sheaf of $\\Spec(A)$ via $p : \\Spec(B) \\to \\Spec(A)$", "is equal to the structure sheaf of $\\Spec(B)$." ], "refs": [ "proetale-lemma-fully-faithful-spaces-over-X" ], "ref_ids": [ 3705 ] } ], "ref_ids": [] }, { "id": 3707, "type": "theorem", "label": "proetale-lemma-base-change-ind-zariski", "categories": [ "proetale" ], "title": "proetale-lemma-base-change-ind-zariski", "contents": [ "Let $A \\to B$ and $A \\to A'$ be ring maps. Let $B' = B \\otimes_A A'$", "be the base change of $B$.", "If $A \\to B$ is ind-Zariski, then $A' \\to B'$ is ind-Zariski." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3708, "type": "theorem", "label": "proetale-lemma-composition-ind-zariski", "categories": [ "proetale" ], "title": "proetale-lemma-composition-ind-zariski", "contents": [ "Let $A \\to B$ and $B \\to C$ be ring maps. If $A \\to B$ and $B \\to C$", "are ind-Zariski, then $A \\to C$ is ind-Zariski." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3709, "type": "theorem", "label": "proetale-lemma-ind-zariski-permanence", "categories": [ "proetale" ], "title": "proetale-lemma-ind-zariski-permanence", "contents": [ "Let $A$ be a ring. Let $B \\to C$ be an $A$-algebra homomorphism.", "If $A \\to B$ and $A \\to C$ are ind-Zariski, then $B \\to C$", "is ind-Zariski." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3710, "type": "theorem", "label": "proetale-lemma-ind-ind-zariski", "categories": [ "proetale" ], "title": "proetale-lemma-ind-ind-zariski", "contents": [ "A filtered colimit of ind-Zariski $A$-algebras is ind-Zariski over $A$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3711, "type": "theorem", "label": "proetale-lemma-ind-zariski-implies", "categories": [ "proetale" ], "title": "proetale-lemma-ind-zariski-implies", "contents": [ "Let $A \\to B$ be ind-Zariski. Then $A \\to B$ identifies local rings," ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3712, "type": "theorem", "label": "proetale-lemma-localization", "categories": [ "proetale" ], "title": "proetale-lemma-localization", "contents": [ "Let $A$ be a ring. Set $X = \\Spec(A)$. Let $Z \\subset X$ be a locally closed", "subscheme which is of the form $D(f) \\cap V(I)$ for some $f \\in A$ and", "ideal $I \\subset A$. Then", "\\begin{enumerate}", "\\item there exists a multiplicative subset $S \\subset A$ such that", "$\\Spec(S^{-1}A)$ maps by a homeomorphism to the set of points of $X$", "specializing to $Z$,", "\\item the $A$-algebra $A_Z^\\sim = S^{-1}A$ depends only on", "the underlying locally closed subset $Z \\subset X$,", "\\item $Z$ is a closed subscheme of $\\Spec(A_Z^\\sim)$,", "\\end{enumerate}", "If $A \\to A'$ is a ring map and $Z' \\subset X' = \\Spec(A')$ is a", "locally closed subscheme of the same form which maps into $Z$,", "then there is a unique $A$-algebra map", "$A_Z^\\sim \\to (A')_{Z'}^\\sim$." ], "refs": [], "proofs": [ { "contents": [ "Let $S \\subset A$ be the multiplicative set of elements which map", "to invertible elements of $\\Gamma(Z, \\mathcal{O}_Z) = (A/I)_f$.", "If $\\mathfrak p$ is a prime of $A$ which does not specialize to $Z$,", "then $\\mathfrak p$ generates the unit ideal in $(A/I)_f$. Hence", "we can write $f^n = g + h$ for some $n \\geq 0$, $g \\in \\mathfrak p$,", "$h \\in I$. Then $g \\in S$ and we see that $\\mathfrak p$ is not in", "the spectrum of $S^{-1}A$. Conversely, if $\\mathfrak p$ does specialize", "to $Z$, say $\\mathfrak p \\subset \\mathfrak q \\supset I$ with", "$f \\not \\in \\mathfrak q$, then we see that $S^{-1}A$ maps to", "$A_\\mathfrak q$ and hence $\\mathfrak p$ is in the spectrum of $S^{-1}A$.", "This proves (1).", "\\medskip\\noindent", "The isomorphism class of the localization $S^{-1}A$ depends only", "on the corresponding subset $\\Spec(S^{-1}A) \\subset \\Spec(A)$, whence", "(2) holds. By construction $S^{-1}A$ maps surjectively onto", "$(A/I)_f$, hence (3). The final statement follows as the multiplicative subset", "$S' \\subset A'$ corresponding to $Z'$ contains the image of the", "multiplicative subset $S$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3713, "type": "theorem", "label": "proetale-lemma-refine", "categories": [ "proetale" ], "title": "proetale-lemma-refine", "contents": [ "Let $X = \\Spec(A)$ as above. Given any finite stratification", "$X = \\coprod T_i$ by constructible subsets, there exists a finite", "subset $E \\subset A$ such that the stratification (\\ref{equation-stratify})", "refines $X = \\coprod T_i$." ], "refs": [], "proofs": [ { "contents": [ "We may write $T_i = \\bigcup_j U_{i, j} \\cap V_{i, j}^c$ as a finite union", "for some $U_{i, j}$ and $V_{i, j}$ quasi-compact open in $X$.", "Then we may write $U_{i, j} = \\bigcup D(f_{i, j, k})$ and", "$V_{i, j} = \\bigcup D(g_{i, j, l})$. Then we set", "$E = \\{f_{i, j, k}\\} \\cup \\{g_{i, j, l}\\}$. This does the job, because", "the stratification (\\ref{equation-stratify}) is the one whose strata are", "labeled by the vanishing pattern of the elements of $E$ which", "clearly refines the given stratification." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3714, "type": "theorem", "label": "proetale-lemma-make-w-local", "categories": [ "proetale" ], "title": "proetale-lemma-make-w-local", "contents": [ "Let $X = \\Spec(A)$ be an affine scheme. With $A \\to A_w$, $X_w = \\Spec(A_w)$,", "and $Z \\subset X_w$ as above.", "\\begin{enumerate}", "\\item $A \\to A_w$ is ind-Zariski and faithfully flat,", "\\item $X_w \\to X$ induces a bijection $Z \\to X$,", "\\item $Z$ is the set of closed points of $X_w$,", "\\item $Z$ is a reduced scheme, and", "\\item every point of $X_w$ specializes to a unique point of $Z$.", "\\end{enumerate}", "In particular, $X_w$ is w-local (Definition \\ref{definition-w-local})." ], "refs": [ "proetale-definition-w-local" ], "proofs": [ { "contents": [ "The map $A \\to A_w$ is ind-Zariski by construction.", "For every $E$ the morphism $Z_E \\to X$ is a bijection, hence (2).", "As $Z \\subset X_w$ we conclude $X_w \\to X$ is surjective and", "$A \\to A_w$ is faithfully flat by", "Algebra, Lemma \\ref{algebra-lemma-ff-rings}. This proves (1).", "\\medskip\\noindent", "Suppose that $y \\in X_w$, $y \\not \\in Z$. Then there", "exists an $E$ such that the image of $y$ in $X_E$ is not contained in", "$Z_E$. Then for all $E \\subset E'$ also $y$ maps to an element of $X_{E'}$", "not contained in $Z_{E'}$. Let $T_{E'} \\subset X_{E'}$ be the reduced", "closed subscheme which is the closure of the image of $y$. It is", "clear that $T = \\lim_{E \\subset E'} T_{E'}$ is the closure of $y$ in $X_w$.", "For every $E \\subset E'$ the scheme $T_{E'} \\cap Z_{E'}$ is nonempty", "by construction of $X_{E'}$. Hence $\\lim T_{E'} \\cap Z_{E'}$ is nonempty", "and we conclude that $T \\cap Z$ is nonempty. Thus $y$ is not a closed point.", "It follows that every closed point of $X_w$ is in $Z$.", "\\medskip\\noindent", "Suppose that $y \\in X_w$ specializes to $z, z' \\in Z$. We will show that", "$z = z'$ which will finish the proof of (3) and will imply (5).", "Let $x, x' \\in X$ be the images of $z$ and $z'$. Since $Z \\to X$ is", "bijective it suffices to show that $x = x'$. If $x \\not = x'$, then", "there exists an $f \\in A$ such that $x \\in D(f)$ and $x' \\in V(f)$", "(or vice versa). Set $E = \\{f\\}$ so that", "$$", "X_E = \\Spec(A_f) \\amalg \\Spec(A_{V(f)}^\\sim)", "$$", "Then we see that $z$ and $z'$ map $x_E$ and $x'_E$ which are in different", "parts of the given decomposition of $X_E$ above. But then it impossible", "for $x_E$ and $x'_E$ to be specializations of a common point.", "This is the desired contradiction.", "\\medskip\\noindent", "Recall that given a finite subset $E \\subset A$ we have $Z_E$", "is a disjoint union of the locally closed subschemes $Z(E', E'')$", "each isomorphic to the spectrum of $(A/I)_f$ where $I$ is the ideal", "generated by $E''$ and $f$ the product of the elements of $E'$.", "Any nilpotent element $b$ of $(A/I)_f$ is the class of $g/f^n$", "for some $g \\in A$. Then setting $E' = E \\cup \\{g\\}$ the reader", "verifies that $b$ is pulls back to zero under the transition map", "$Z_{E'} \\to Z_E$ of the system. This proves (4)." ], "refs": [ "algebra-lemma-ff-rings" ], "ref_ids": [ 536 ] } ], "ref_ids": [ 3830 ] }, { "id": 3715, "type": "theorem", "label": "proetale-lemma-universal", "categories": [ "proetale" ], "title": "proetale-lemma-universal", "contents": [ "Let $A$ be a ring. Let $A \\to A_w$ be the ring map constructed in", "Lemma \\ref{lemma-make-w-local}. For any ring map $A \\to B$ such that", "$\\Spec(B)$ is w-local, there is a unique factorization $A \\to A_w \\to B$", "such that $\\Spec(B) \\to \\Spec(A_w)$ is w-local." ], "refs": [ "proetale-lemma-make-w-local" ], "proofs": [ { "contents": [ "Denote $Y = \\Spec(B)$ and $Y_0 \\subset Y$ the set of closed points.", "Denote $f : Y \\to X$ the given morphism.", "Recall that $Y_0$ is profinite, in particular every constructible", "subset of $Y_0$ is open and closed. Let $E \\subset A$ be a finite subset.", "Recall that $A_w = \\colim A_E$ and that the set of closed points of", "$\\Spec(A_w)$ is the limit of the closed subsets $Z_E \\subset X_E = \\Spec(A_E)$.", "Thus it suffices to show there is a unique factorization $A \\to A_E \\to B$", "such that $Y \\to X_E$ maps $Y_0$ into $Z_E$.", "Since $Z_E \\to X = \\Spec(A)$ is bijective, and since the strata", "$Z(E', E'')$ are constructible we see that", "$$", "Y_0 = \\coprod f^{-1}(Z(E', E'')) \\cap Y_0", "$$", "is a disjoint union decomposition into open and closed subsets.", "As $Y_0 = \\pi_0(Y)$ we obtain a corresponding decomposition of", "$Y$ into open and closed pieces. Thus it suffices to construct", "the factorization in case $f(Y_0) \\subset Z(E', E'')$ for", "some decomposition $E = E' \\amalg E''$.", "In this case $f(Y)$ is contained in the set of points of $X$", "specializing to $Z(E', E'')$ which is homeomorphic to $X_{E', E''}$.", "Thus we obtain a unique continuous map $Y \\to X_{E', E''}$ over $X$. By", "Lemma \\ref{lemma-fully-faithful-spaces-over-X}", "this corresponds to a unique morphism of schemes", "$Y \\to X_{E', E''}$ over $X$. This finishes the proof." ], "refs": [ "proetale-lemma-fully-faithful-spaces-over-X" ], "ref_ids": [ 3705 ] } ], "ref_ids": [ 3714 ] }, { "id": 3716, "type": "theorem", "label": "proetale-lemma-profinite-goes-up", "categories": [ "proetale" ], "title": "proetale-lemma-profinite-goes-up", "contents": [ "Let $A$ be a ring such that $\\Spec(A)$ is profinite. Let $A \\to B$ be a", "ring map. Then $\\Spec(B)$ is profinite in each of the following cases:", "\\begin{enumerate}", "\\item if $\\mathfrak q,\\mathfrak q' \\subset B$ lie over the same", "prime of $A$, then neither $\\mathfrak q \\subset \\mathfrak q'$, nor", "$\\mathfrak q' \\subset \\mathfrak q$,", "\\item $A \\to B$ induces algebraic extensions of residue fields,", "\\item $A \\to B$ is a local isomorphism,", "\\item $A \\to B$ identifies local rings,", "\\item $A \\to B$ is weakly \\'etale,", "\\item $A \\to B$ is quasi-finite,", "\\item $A \\to B$ is unramified,", "\\item $A \\to B$ is \\'etale,", "\\item $B$ is a filtered colimit of $A$-algebras as in (1) -- (8),", "\\item etc.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By the references mentioned above", "(Algebra, Lemma \\ref{algebra-lemma-ring-with-only-minimal-primes} or", "Topology, Lemma \\ref{topology-lemma-characterize-profinite-spectral})", "there are no specializations between distinct points of $\\Spec(A)$ and", "$\\Spec(B)$ is profinite if and only if there are no specializations", "between distinct points of $\\Spec(B)$. These specializations can only", "happen in the fibres of $\\Spec(B) \\to \\Spec(A)$. In this way we see", "that (1) is true.", "\\medskip\\noindent", "The assumption in (2) implies all primes of $B$ are maximal by", "Algebra, Lemma \\ref{algebra-lemma-finite-residue-extension-closed}.", "Thus (2) holds.", "If $A \\to B$ is a local isomorphism or identifies local rings,", "then the residue field extensions are trivial, so (3) and (4)", "follow from (2).", "If $A \\to B$ is weakly \\'etale, then More on Algebra, Lemma", "\\ref{more-algebra-lemma-weakly-etale-residue-field-extensions}", "tells us it induces separable algebraic residue field extensions, so", "(5) follows from (2).", "If $A \\to B$ is quasi-finite, then the fibres are finite discrete", "topological spaces. Hence (6) follows from (1).", "Hence (3) follows from (1). Cases (7) and (8)", "follow from this as unramified and \\'etale ring map are quasi-finite", "(Algebra, Lemmas", "\\ref{algebra-lemma-unramified-quasi-finite} and", "\\ref{algebra-lemma-etale-quasi-finite}).", "If $B = \\colim B_i$ is a filtered colimit of $A$-algebras, then", "$\\Spec(B) = \\lim \\Spec(B_i)$ in the category of topological spaces by", "Limits, Lemma \\ref{limits-lemma-inverse-limit-top}.", "Hence if each $\\Spec(B_i)$ is profinite, so is $\\Spec(B)$ by", "Topology, Lemma \\ref{topology-lemma-directed-inverse-limit-profinite}.", "This proves (9)." ], "refs": [ "algebra-lemma-ring-with-only-minimal-primes", "topology-lemma-characterize-profinite-spectral", "algebra-lemma-finite-residue-extension-closed", "more-algebra-lemma-weakly-etale-residue-field-extensions", "algebra-lemma-unramified-quasi-finite", "algebra-lemma-etale-quasi-finite", "limits-lemma-inverse-limit-top", "topology-lemma-directed-inverse-limit-profinite" ], "ref_ids": [ 426, 8309, 472, 10453, 1269, 1234, 15033, 8300 ] } ], "ref_ids": [] }, { "id": 3717, "type": "theorem", "label": "proetale-lemma-localize-along-closed-profinite", "categories": [ "proetale" ], "title": "proetale-lemma-localize-along-closed-profinite", "contents": [ "Let $A$ be a ring. Let $V(I) \\subset \\Spec(A)$ be a closed subset", "which is a profinite topological space. Then there exists an", "ind-Zariski ring map $A \\to B$ such that $\\Spec(B)$ is w-local,", "the set of closed points is $V(IB)$, and $A/I \\cong B/IB$." ], "refs": [], "proofs": [ { "contents": [ "Let $A \\to A_w$ and $Z \\subset Y = \\Spec(A_w)$ as in", "Lemma \\ref{lemma-make-w-local}.", "Let $T \\subset Z$ be the inverse image of $V(I)$.", "Then $T \\to V(I)$ is a homeomorphism by", "Topology, Lemma \\ref{topology-lemma-bijective-map}.", "Let $B = (A_w)_T^\\sim$, see Lemma \\ref{lemma-localization}.", "It is clear that $B$ is w-local with closed points $V(IB)$.", "The ring map $A/I \\to B/IB$ is ind-Zariski", "and induces a homeomorphism on underlying", "topological spaces. Hence it is an isomorphism by", "Lemma \\ref{lemma-local-isomorphism-fully-faithful}." ], "refs": [ "proetale-lemma-make-w-local", "topology-lemma-bijective-map", "proetale-lemma-localization", "proetale-lemma-local-isomorphism-fully-faithful" ], "ref_ids": [ 3714, 8275, 3712, 3706 ] } ], "ref_ids": [] }, { "id": 3718, "type": "theorem", "label": "proetale-lemma-w-local-algebraic-residue-field-extensions", "categories": [ "proetale" ], "title": "proetale-lemma-w-local-algebraic-residue-field-extensions", "contents": [ "Let $A$ be a ring such that $X = \\Spec(A)$ is w-local. Let $I \\subset A$", "be the radical ideal cutting out the set $X_0$ of closed points in $X$.", "Let $A \\to B$ be a ring map inducing algebraic extensions on residue", "fields at primes. Then", "\\begin{enumerate}", "\\item every point of $Z = V(IB)$ is a closed point of $\\Spec(B)$,", "\\item there exists an ind-Zariski ring map $B \\to C$ such that", "\\begin{enumerate}", "\\item $B/IB \\to C/IC$ is an isomorphism,", "\\item the space $Y = \\Spec(C)$ is w-local,", "\\item the induced map $p : Y \\to X$ is w-local, and", "\\item $p^{-1}(X_0)$ is the set of closed points of $Y$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-profinite-goes-up} applied to $A/I \\to B/IB$", "all points of $Z = V(IB) = \\Spec(B/IB)$ are closed, in fact $\\Spec(B/IB)$", "is a profinite space.", "To finish the proof we apply Lemma \\ref{lemma-localize-along-closed-profinite}", "to $IB \\subset B$." ], "refs": [ "proetale-lemma-profinite-goes-up", "proetale-lemma-localize-along-closed-profinite" ], "ref_ids": [ 3716, 3717 ] } ], "ref_ids": [] }, { "id": 3719, "type": "theorem", "label": "proetale-lemma-construct", "categories": [ "proetale" ], "title": "proetale-lemma-construct", "contents": [ "Let $A$ be a ring. Let $X = \\Spec(A)$. Let $T \\subset \\pi_0(X)$ be a", "closed subset. There exists a surjective ind-Zariski ring map $A \\to B$", "such that $\\Spec(B) \\to \\Spec(A)$ induces a homeomorphism of $\\Spec(B)$", "with the inverse image of $T$ in $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset X$ be the inverse image of $T$. Then $Z$ is the intersection", "$Z = \\bigcap Z_\\alpha$ of the open and closed subsets of $X$ containing $Z$,", "see Topology, Lemma \\ref{topology-lemma-closed-union-connected-components}.", "For each $\\alpha$ we have $Z_\\alpha = \\Spec(A_\\alpha)$ where", "$A \\to A_\\alpha$ is a local isomorphism (a localization at an idempotent).", "Setting $B = \\colim A_\\alpha$ proves the lemma." ], "refs": [ "topology-lemma-closed-union-connected-components" ], "ref_ids": [ 8238 ] } ], "ref_ids": [] }, { "id": 3720, "type": "theorem", "label": "proetale-lemma-construct-profinite", "categories": [ "proetale" ], "title": "proetale-lemma-construct-profinite", "contents": [ "Let $A$ be a ring and let $X = \\Spec(A)$. Let $T$ be a profinite space and", "let $T \\to \\pi_0(X)$ be a continuous map. There exists an", "ind-Zariski ring map $A \\to B$ such that with $Y = \\Spec(B)$ the diagram", "$$", "\\xymatrix{", "Y \\ar[r] \\ar[d] & \\pi_0(Y) \\ar[d] \\\\", "X \\ar[r] & \\pi_0(X)", "}", "$$", "is cartesian in the category of topological spaces and such that", "$\\pi_0(Y) = T$ as spaces over $\\pi_0(X)$." ], "refs": [], "proofs": [ { "contents": [ "Namely, write $T = \\lim T_i$ as the limit of an inverse system finite", "discrete spaces over a directed set (see", "Topology, Lemma \\ref{topology-lemma-profinite}). For each $i$ let", "$Z_i = \\Im(T \\to \\pi_0(X) \\times T_i)$. This is a closed subset.", "Observe that $X \\times T_i$ is the spectrum of $A_i = \\prod_{t \\in T_i} A$", "and that $A \\to A_i$ is a local isomorphism. By Lemma \\ref{lemma-construct}", "we see that $Z_i \\subset \\pi_0(X \\times T_i) = \\pi_0(X) \\times T_i$", "corresponds to a surjection $A_i \\to B_i$ which is ind-Zariski", "such that $\\Spec(B_i) = X \\times_{\\pi_0(X)} Z_i$ as subsets of", "$X \\times T_i$. The transition maps $T_i \\to T_{i'}$ induce maps", "$Z_i \\to Z_{i'}$ and $X \\times_{\\pi_0(X)} Z_i \\to X \\times_{\\pi_0(X)} Z_{i'}$.", "Hence ring maps $B_{i'} \\to B_i$", "(Lemmas \\ref{lemma-local-isomorphism-fully-faithful} and", "\\ref{lemma-ind-zariski-implies}).", "Set $B = \\colim B_i$. Because $T = \\lim Z_i$ we have", "$X \\times_{\\pi_0(X)} T = \\lim X \\times_{\\pi_0(X)} Z_i$", "and hence $Y = \\Spec(B) = \\lim \\Spec(B_i)$", "fits into the cartesian diagram", "$$", "\\xymatrix{", "Y \\ar[r] \\ar[d] & T \\ar[d] \\\\", "X \\ar[r] & \\pi_0(X)", "}", "$$", "of topological spaces. By Lemma \\ref{lemma-silly}", "we conclude that $T = \\pi_0(Y)$." ], "refs": [ "topology-lemma-profinite", "proetale-lemma-construct", "proetale-lemma-local-isomorphism-fully-faithful", "proetale-lemma-ind-zariski-implies", "proetale-lemma-silly" ], "ref_ids": [ 8299, 3719, 3706, 3711, 3699 ] } ], "ref_ids": [] }, { "id": 3721, "type": "theorem", "label": "proetale-lemma-w-local-morphism-equal-points-stalks-is-iso", "categories": [ "proetale" ], "title": "proetale-lemma-w-local-morphism-equal-points-stalks-is-iso", "contents": [ "Let $A \\to B$ be ring map such that", "\\begin{enumerate}", "\\item $A \\to B$ identifies local rings,", "\\item the topological spaces $\\Spec(B)$, $\\Spec(A)$ are w-local,", "\\item $\\Spec(B) \\to \\Spec(A)$ is w-local, and", "\\item $\\pi_0(\\Spec(B)) \\to \\pi_0(\\Spec(A))$ is bijective.", "\\end{enumerate}", "Then $A \\to B$ is an isomorphism" ], "refs": [], "proofs": [ { "contents": [ "Let $X_0 \\subset X = \\Spec(A)$ and $Y_0 \\subset Y = \\Spec(B)$ be the", "sets of closed points. By assumption $Y_0$ maps into $X_0$ and", "the induced map $Y_0 \\to X_0$ is a bijection.", "As a space $\\Spec(A)$ is the disjoint union of the spectra", "of the local rings of $A$ at closed points.", "Similarly for $B$. Hence $X \\to Y$ is a bijection.", "Since $A \\to B$ is flat we have going down", "(Algebra, Lemma \\ref{algebra-lemma-flat-going-down}).", "Thus Algebra, Lemma \\ref{algebra-lemma-unique-prime-over-localize-below}", "shows for any prime $\\mathfrak q \\subset B$ lying over", "$\\mathfrak p \\subset A$ we have $B_\\mathfrak q = B_\\mathfrak p$.", "Since $B_\\mathfrak q = A_\\mathfrak p$ by assumption, we", "see that $A_\\mathfrak p = B_\\mathfrak p$ for all primes $\\mathfrak p$", "of $A$. Thus $A = B$ by", "Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}." ], "refs": [ "algebra-lemma-flat-going-down", "algebra-lemma-unique-prime-over-localize-below", "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 539, 556, 410 ] } ], "ref_ids": [] }, { "id": 3722, "type": "theorem", "label": "proetale-lemma-w-local-morphism-equal-stalks-is-ind-zariski", "categories": [ "proetale" ], "title": "proetale-lemma-w-local-morphism-equal-stalks-is-ind-zariski", "contents": [ "Let $A \\to B$ be ring map such that", "\\begin{enumerate}", "\\item $A \\to B$ identifies local rings,", "\\item the topological spaces $\\Spec(B)$, $\\Spec(A)$ are w-local, and", "\\item $\\Spec(B) \\to \\Spec(A)$ is w-local.", "\\end{enumerate}", "Then $A \\to B$ is ind-Zariski." ], "refs": [], "proofs": [ { "contents": [ "Set $X = \\Spec(A)$ and $Y = \\Spec(B)$. Let $X_0 \\subset X$ and", "$Y_0 \\subset Y$ be the set of closed points. Let $A \\to A'$ be the ind-Zariski", "morphism of affine schemes such that with $X' = \\Spec(A')$ the diagram", "$$", "\\xymatrix{", "X' \\ar[r] \\ar[d] & \\pi_0(X') \\ar[d] \\\\", "X \\ar[r] & \\pi_0(X)", "}", "$$", "is cartesian in the category of topological spaces and such that", "$\\pi_0(X') = \\pi_0(Y)$ as spaces over $\\pi_0(X)$, see", "Lemma \\ref{lemma-construct-profinite}. By", "Lemma \\ref{lemma-silly} we see that $X'$ is w-local and", "the set of closed points $X'_0 \\subset X'$ is the inverse image of $X_0$.", "\\medskip\\noindent", "We obtain a continuous map $Y \\to X'$ of underlying topological spaces", "over $X$ identifying $\\pi_0(Y)$ with $\\pi_0(X')$. By", "Lemma \\ref{lemma-local-isomorphism-fully-faithful}", "(and Lemma \\ref{lemma-ind-zariski-implies})", "this is corresponds to a morphism of affine schemes $Y \\to X'$", "over $X$. Since $Y \\to X$ maps $Y_0$ into $X_0$ we see that", "$Y \\to X'$ maps $Y_0$ into $X'_0$, i.e., $Y \\to X'$ is w-local.", "By Lemma \\ref{lemma-w-local-morphism-equal-points-stalks-is-iso}", "we see that $Y \\cong X'$ and we win." ], "refs": [ "proetale-lemma-construct-profinite", "proetale-lemma-silly", "proetale-lemma-local-isomorphism-fully-faithful", "proetale-lemma-ind-zariski-implies", "proetale-lemma-w-local-morphism-equal-points-stalks-is-iso" ], "ref_ids": [ 3720, 3699, 3706, 3711, 3721 ] } ], "ref_ids": [] }, { "id": 3723, "type": "theorem", "label": "proetale-lemma-w-local-extremally-disconnected", "categories": [ "proetale" ], "title": "proetale-lemma-w-local-extremally-disconnected", "contents": [ "Let $A$ be a ring. The following are equivalent", "\\begin{enumerate}", "\\item every faithfully flat ring map $A \\to B$ identifying local rings", "has a section,", "\\item every faithfully flat ind-Zariski ring map $A \\to B$ has a section, and", "\\item $A$ satisfies", "\\begin{enumerate}", "\\item $\\Spec(A)$ is w-local, and", "\\item $\\pi_0(\\Spec(A))$ is extremally disconnected.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows immediately from", "Proposition \\ref{proposition-maps-wich-identify-local-rings}.", "\\medskip\\noindent", "Assume (3)(a) and (3)(b). Let $A \\to B$ be faithfully flat and ind-Zariski.", "We will use without further mention the fact that a flat map", "$A \\to B$ is faithfully flat if and only if every closed point", "of $\\Spec(A)$ is in the image of $\\Spec(B) \\to \\Spec(A)$", "We will show that $A \\to B$ has a section.", "\\medskip\\noindent", "Let $I \\subset A$ be an ideal such that $V(I) \\subset \\Spec(A)$ is", "the set of closed points of $\\Spec(A)$. ", "We may replace $B$ by the ring $C$ constructed in", "Lemma \\ref{lemma-w-local-algebraic-residue-field-extensions}", "for $A \\to B$ and $I \\subset A$.", "Thus we may assume $\\Spec(B)$ is w-local such that the set of", "closed points of $\\Spec(B)$ is $V(IB)$.", "\\medskip\\noindent", "Assume $\\Spec(B)$ is w-local and the set of closed points of $\\Spec(B)$", "is $V(IB)$. Choose a continuous section to the surjective", "continuous map $V(IB) \\to V(I)$. This is possible as", "$V(I) \\cong \\pi_0(\\Spec(A))$ is extremally disconnected, see", "Topology, Proposition", "\\ref{topology-proposition-projective-in-category-hausdorff-qc}.", "The image is a closed subspace $T \\subset \\pi_0(\\Spec(B)) \\cong V(JB)$", "mapping homeomorphically onto $\\pi_0(A)$. Replacing $B$ by the ind-Zariski", "quotient ring constructed in Lemma \\ref{lemma-construct}", "we see that we may assume $\\pi_0(\\Spec(B)) \\to \\pi_0(\\Spec(A))$", "is bijective. At this point $A \\to B$ is an isomorphism by", "Lemma \\ref{lemma-w-local-morphism-equal-points-stalks-is-iso}.", "\\medskip\\noindent", "Assume (1) or equivalently (2). Let $A \\to A_w$ be the ring map constructed in", "Lemma \\ref{lemma-make-w-local}. By (1) there is a section $A_w \\to A$.", "Thus $\\Spec(A)$ is homeomorphic to a closed subset of $\\Spec(A_w)$. By", "Lemma \\ref{lemma-closed-subspace-w-local} we see (3)(a) holds.", "Finally, let $T \\to \\pi_0(A)$ be a surjective map with $T$ an", "extremally disconnected, quasi-compact, Hausdorff topological space", "(Topology, Lemma \\ref{topology-lemma-existence-projective-cover}).", "Choose $A \\to B$ as in Lemma \\ref{lemma-construct-profinite}", "adapted to $T \\to \\pi_0(\\Spec(A))$. By (1) there is a section", "$B \\to A$. Thus we see that $T = \\pi_0(\\Spec(B)) \\to \\pi_0(\\Spec(A))$", "has a section. A formal categorical argument, using", "Topology, Proposition", "\\ref{topology-proposition-projective-in-category-hausdorff-qc},", "implies that $\\pi_0(\\Spec(A))$ is extremally disconnected." ], "refs": [ "proetale-proposition-maps-wich-identify-local-rings", "proetale-lemma-w-local-algebraic-residue-field-extensions", "topology-proposition-projective-in-category-hausdorff-qc", "proetale-lemma-construct", "proetale-lemma-w-local-morphism-equal-points-stalks-is-iso", "proetale-lemma-make-w-local", "proetale-lemma-closed-subspace-w-local", "topology-lemma-existence-projective-cover", "proetale-lemma-construct-profinite", "topology-proposition-projective-in-category-hausdorff-qc" ], "ref_ids": [ 3825, 3718, 8345, 3719, 3721, 3714, 3698, 8332, 3720, 8345 ] } ], "ref_ids": [] }, { "id": 3724, "type": "theorem", "label": "proetale-lemma-find-Zariski-w-contractible", "categories": [ "proetale" ], "title": "proetale-lemma-find-Zariski-w-contractible", "contents": [ "Let $A$ be a ring. There exists a faithfully flat, ind-Zariski ring", "map $A \\to B$ such that $B$ satisfies the equivalent conditions", "of Lemma \\ref{lemma-w-local-extremally-disconnected}." ], "refs": [ "proetale-lemma-w-local-extremally-disconnected" ], "proofs": [ { "contents": [ "We first apply Lemma \\ref{lemma-make-w-local} to see that we may assume that", "$\\Spec(A)$ is w-local.", "Choose an extremally disconnected space $T$ and a surjective", "continuous map $T \\to \\pi_0(\\Spec(A))$, see", "Topology, Lemma \\ref{topology-lemma-existence-projective-cover}.", "Note that $T$ is profinite. Apply Lemma \\ref{lemma-construct-profinite}", "to find an ind-Zariski ring map $A \\to B$ such that", "$\\pi_0(\\Spec(B)) \\to \\pi_0(\\Spec(A))$ realizes $T \\to \\pi_0(\\Spec(A))$", "and such that", "$$", "\\xymatrix{", "\\Spec(B) \\ar[r] \\ar[d] & \\pi_0(\\Spec(B)) \\ar[d] \\\\", "\\Spec(A) \\ar[r] & \\pi_0(\\Spec(A))", "}", "$$", "is cartesian in the category of topological spaces. Note that $\\Spec(B)$", "is w-local, that $\\Spec(B) \\to \\Spec(A)$ is w-local, and that the", "set of closed points of $\\Spec(B)$ is the inverse image of the", "set of closed points of $\\Spec(A)$, see Lemma \\ref{lemma-silly}.", "Thus condition (3) of", "Lemma \\ref{lemma-w-local-extremally-disconnected}", "holds for $B$." ], "refs": [ "proetale-lemma-make-w-local", "topology-lemma-existence-projective-cover", "proetale-lemma-construct-profinite", "proetale-lemma-silly", "proetale-lemma-w-local-extremally-disconnected" ], "ref_ids": [ 3714, 8332, 3720, 3699, 3723 ] } ], "ref_ids": [ 3723 ] }, { "id": 3725, "type": "theorem", "label": "proetale-lemma-base-change-ind-etale", "categories": [ "proetale" ], "title": "proetale-lemma-base-change-ind-etale", "contents": [ "Let $A \\to B$ and $A \\to A'$ be ring maps. Let $B' = B \\otimes_A A'$", "be the base change of $B$.", "If $A \\to B$ is ind-\\'etale, then $A' \\to B'$ is ind-\\'etale." ], "refs": [], "proofs": [ { "contents": [ "This is Algebra, Lemma \\ref{algebra-lemma-base-change-colimit-etale}." ], "refs": [ "algebra-lemma-base-change-colimit-etale" ], "ref_ids": [ 1287 ] } ], "ref_ids": [] }, { "id": 3726, "type": "theorem", "label": "proetale-lemma-composition-ind-etale", "categories": [ "proetale" ], "title": "proetale-lemma-composition-ind-etale", "contents": [ "Let $A \\to B$ and $B \\to C$ be ring maps. If $A \\to B$ and $B \\to C$", "are ind-\\'etale, then $A \\to C$ is ind-\\'etale." ], "refs": [], "proofs": [ { "contents": [ "This is Algebra, Lemma \\ref{algebra-lemma-composition-colimit-etale}." ], "refs": [ "algebra-lemma-composition-colimit-etale" ], "ref_ids": [ 1288 ] } ], "ref_ids": [] }, { "id": 3727, "type": "theorem", "label": "proetale-lemma-ind-ind-etale", "categories": [ "proetale" ], "title": "proetale-lemma-ind-ind-etale", "contents": [ "A filtered colimit of ind-\\'etale $A$-algebras is ind-\\'etale over $A$." ], "refs": [], "proofs": [ { "contents": [ "This is Algebra, Lemma \\ref{algebra-lemma-colimit-colimit-etale}." ], "refs": [ "algebra-lemma-colimit-colimit-etale" ], "ref_ids": [ 1289 ] } ], "ref_ids": [] }, { "id": 3728, "type": "theorem", "label": "proetale-lemma-ind-etale-permanence", "categories": [ "proetale" ], "title": "proetale-lemma-ind-etale-permanence", "contents": [ "Let $A$ be a ring. Let $B \\to C$ be an $A$-algebra map of ind-\\'etale", "$A$-algebras. Then $C$ is an ind-\\'etale $B$-algebra." ], "refs": [], "proofs": [ { "contents": [ "This is Algebra, Lemma \\ref{algebra-lemma-colimits-of-etale}." ], "refs": [ "algebra-lemma-colimits-of-etale" ], "ref_ids": [ 1290 ] } ], "ref_ids": [] }, { "id": 3729, "type": "theorem", "label": "proetale-lemma-ind-etale-implies", "categories": [ "proetale" ], "title": "proetale-lemma-ind-etale-implies", "contents": [ "Let $A \\to B$ be ind-\\'etale. Then $A \\to B$ is weakly \\'etale", "(More on Algebra, Definition \\ref{more-algebra-definition-weakly-etale})." ], "refs": [ "more-algebra-definition-weakly-etale" ], "proofs": [ { "contents": [ "This follows from More on Algebra, Lemma", "\\ref{more-algebra-lemma-when-weakly-etale}." ], "refs": [ "more-algebra-lemma-when-weakly-etale" ], "ref_ids": [ 10450 ] } ], "ref_ids": [ 10635 ] }, { "id": 3730, "type": "theorem", "label": "proetale-lemma-lift-ind-etale", "categories": [ "proetale" ], "title": "proetale-lemma-lift-ind-etale", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be an ideal. The base change functor", "$$", "\\text{ind-\\'etale }A\\text{-algebras}", "\\longrightarrow", "\\text{ind-\\'etale }A/I\\text{-algebras},\\quad", "C \\longmapsto C/IC", "$$", "has a fully faithful right adjoint $v$. In particular, given", "an ind-\\'etale $A/I$-algebra $\\overline{C}$ there exists", "an ind-\\'etale $A$-algebra $C = v(\\overline{C})$ such that", "$\\overline{C} = C/IC$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\overline{C}$ be an ind-\\'etale $A/I$-algebra.", "Consider the category $\\mathcal{C}$ of factorizations", "$A \\to B \\to \\overline{C}$ where $A \\to B$ is \\'etale.", "(We ignore some set theoretical issues in this proof.)", "We will show that this category is directed and that", "$C = \\colim_\\mathcal{C} B$ is an ind-\\'etale $A$-algebra", "such that $\\overline{C} = C/IC$.", "\\medskip\\noindent", "We first prove that $\\mathcal{C}$ is directed", "(Categories, Definition \\ref{categories-definition-directed}).", "The category is nonempty as $A \\to A \\to \\overline{C}$ is an object.", "Suppose that $A \\to B \\to \\overline{C}$ and $A \\to B' \\to \\overline{C}$", "are two objects of $\\mathcal{C}$. Then $A \\to B \\otimes_A B' \\to \\overline{C}$", "is another (use Algebra, Lemma \\ref{algebra-lemma-etale}).", "Suppose that $f, g : B \\to B'$ are two maps between", "objects $A \\to B \\to \\overline{C}$ and $A \\to B' \\to \\overline{C}$", "of $\\mathcal{C}$. Then a coequalizer is", "$A \\to B' \\otimes_{f, B, g} B' \\to \\overline{C}$.", "This is an object of $\\mathcal{C}$ by", "Algebra, Lemmas \\ref{algebra-lemma-etale} and", "\\ref{algebra-lemma-map-between-etale}.", "Thus the category $\\mathcal{C}$ is directed.", "\\medskip\\noindent", "Write $\\overline{C} = \\colim \\overline{B_i}$ as a filtered colimit with", "$\\overline{B_i}$ \\'etale over $A/I$. For every $i$", "there exists $A \\to B_i$ \\'etale with $\\overline{B_i} = B_i/IB_i$, see", "Algebra, Lemma \\ref{algebra-lemma-lift-etale}.", "Thus $C \\to \\overline{C}$ is surjective.", "Since $C/IC \\to \\overline{C}$ is ind-\\'etale", "(Lemma \\ref{lemma-ind-etale-permanence})", "we see that it is flat. Hence $\\overline{C}$ is a localization of", "$C/IC$ at some multiplicative subset $S \\subset C/IC$", "(Algebra, Lemma \\ref{algebra-lemma-pure}).", "Take an $f \\in C$ mapping to an element of $S \\subset C/IC$.", "Choose $A \\to B \\to \\overline{C}$ in $\\mathcal{C}$ and $g \\in B$", "mapping to $f$ in the colimit. Then we see that $A \\to B_g \\to \\overline{C}$", "is an object of $\\mathcal{C}$ as well. Thus $f$ is an invertible", "element of $C$. It follows that $C/IC = \\overline{C}$.", "\\medskip\\noindent", "Next, we claim that for an ind-\\'etale algebra $D$ over $A$ we have", "$$", "\\Mor_A(D, C) = \\Mor_{A/I}(D/ID, \\overline{C})", "$$", "Namely, let $D/ID \\to \\overline{C}$ be an $A/I$-algebra map.", "Write $D = \\colim_{i \\in I} D_i$ as a colimit over a directed set $I$", "with $D_i$ \\'etale over $A$. By choice of $\\mathcal{C}$", "we obtain a transformation $I \\to \\mathcal{C}$ and hence a map", "$D \\to C$ compatible with maps to $\\overline{C}$. Whence the claim.", "\\medskip\\noindent", "It follows that the functor $v$ defined by the rule", "$$", "\\overline{C} ", "\\longmapsto", "v(\\overline{C}) = \\colim_{A \\to B \\to \\overline{C}} B", "$$", "is a right adjoint to the base change functor $u$ as required by the lemma.", "The functor $v$ is fully faithful because", "$u \\circ v = \\text{id}$ by construction, see", "Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}." ], "refs": [ "categories-definition-directed", "algebra-lemma-etale", "algebra-lemma-etale", "algebra-lemma-map-between-etale", "algebra-lemma-lift-etale", "proetale-lemma-ind-etale-permanence", "algebra-lemma-pure", "categories-lemma-adjoint-fully-faithful" ], "ref_ids": [ 12363, 1231, 1231, 1236, 1238, 3728, 960, 12248 ] } ], "ref_ids": [] }, { "id": 3731, "type": "theorem", "label": "proetale-lemma-first-construction", "categories": [ "proetale" ], "title": "proetale-lemma-first-construction", "contents": [ "Given a ring $A$ there exists a faithfully flat ind-\\'etale $A$-algebra $C$", "such that every faithfully flat \\'etale ring map $C \\to B$ has a section." ], "refs": [], "proofs": [ { "contents": [ "Set $T^1(A) = T(A)$ and $T^{n + 1}(A) = T(T^n(A))$. Let", "$$", "C = \\colim T^n(A)", "$$", "This algebra is faithfully flat over each $T^n(A)$ and in particular", "over $A$, see", "Algebra, Lemma \\ref{algebra-lemma-colimit-faithfully-flat}.", "Moreover, $C$ is ind-\\'etale over $A$ by Lemma \\ref{lemma-ind-ind-etale}.", "If $C \\to B$ is \\'etale, then there exists an $n$ and an \\'etale", "ring map $T^n(A) \\to B'$ such that $B = C \\otimes_{T^n(A)} B'$, see", "Algebra, Lemma \\ref{algebra-lemma-etale}.", "If $C \\to B$ is faithfully flat, then $\\Spec(B) \\to \\Spec(C) \\to \\Spec(T^n(A))$", "is surjective, hence $\\Spec(B') \\to \\Spec(T^n(A))$ is surjective.", "In other words, $T^n(A) \\to B'$ is faithfully flat.", "By our construction, there is a $T^n(A)$-algebra map", "$B' \\to T^{n + 1}(A)$. This induces a $C$-algebra map $B \\to C$", "which finishes the proof." ], "refs": [ "algebra-lemma-colimit-faithfully-flat", "proetale-lemma-ind-ind-etale", "algebra-lemma-etale" ], "ref_ids": [ 540, 3727, 1231 ] } ], "ref_ids": [] }, { "id": 3732, "type": "theorem", "label": "proetale-lemma-have-sections-quotient", "categories": [ "proetale" ], "title": "proetale-lemma-have-sections-quotient", "contents": [ "Let $A$ be a ring such that every faithfully flat \\'etale ring map", "$A \\to B$ has a section. Then the same is true for every quotient ring", "$A/I$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3733, "type": "theorem", "label": "proetale-lemma-have-sections-strictly-henselian", "categories": [ "proetale" ], "title": "proetale-lemma-have-sections-strictly-henselian", "contents": [ "Let $A$ be a ring such that every faithfully flat \\'etale ring map", "$A \\to B$ has a section. Then every local ring of $A$ at a maximal", "ideal is strictly henselian." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak m$ be a maximal ideal of $A$. Let $A \\to B$ be an", "\\'etale ring map and let $\\mathfrak q \\subset B$ be a prime", "lying over $\\mathfrak m$. By the description of the strict henselization", "$A_\\mathfrak m^{sh}$ in", "Algebra, Lemma \\ref{algebra-lemma-strict-henselization-different}", "it suffices to show that $A_\\mathfrak m = B_\\mathfrak q$.", "Note that there are finitely many primes", "$\\mathfrak q = \\mathfrak q_1, \\mathfrak q_2, \\ldots, \\mathfrak q_n$", "lying over $\\mathfrak m$ and there are no specializations", "between them as an \\'etale ring map is quasi-finite, see", "Algebra, Lemma \\ref{algebra-lemma-etale-quasi-finite}.", "Thus $\\mathfrak q_i$ is a maximal ideal and we can find", "$g \\in \\mathfrak q_2 \\cap \\ldots \\cap \\mathfrak q_n$, $g \\not \\in \\mathfrak q$", "(Algebra, Lemma \\ref{algebra-lemma-silly}).", "After replacing $B$ by $B_g$ we see that $\\mathfrak q$", "is the only prime of $B$ lying over $\\mathfrak m$.", "The image $U \\subset \\Spec(A)$ of $\\Spec(B) \\to \\Spec(A)$ is", "open (Algebra, Proposition \\ref{algebra-proposition-fppf-open}).", "Thus the complement $\\Spec(A) \\setminus U$ is closed", "and we can find $f \\in A$, $f \\not \\in \\mathfrak p$ such that", "$\\Spec(A) = U \\cup D(f)$. The ring map $A \\to B \\times A_f$", "is faithfully flat and \\'etale, hence has a section", "$\\sigma : B \\times A_f \\to A$ by assumption on $A$.", "Observe that $\\sigma$ is \\'etale, hence flat as a map between \\'etale", "$A$-algebras (Algebra, Lemma \\ref{algebra-lemma-map-between-etale}).", "Since $\\mathfrak q$ is the only prime of $B \\times A_f$ lying", "over $A$ we find that $A_\\mathfrak p \\to B_\\mathfrak q$ has", "a section which is also flat. Thus", "$A_\\mathfrak p \\to B_\\mathfrak q \\to A_\\mathfrak p$", "are flat local ring maps whose composition is the identity. Since", "a flat local homomorphism of local rings is injective we conclude these", "maps are isomorphisms as desired." ], "refs": [ "algebra-lemma-strict-henselization-different", "algebra-lemma-etale-quasi-finite", "algebra-lemma-silly", "algebra-proposition-fppf-open", "algebra-lemma-map-between-etale" ], "ref_ids": [ 1304, 1234, 378, 1407, 1236 ] } ], "ref_ids": [] }, { "id": 3734, "type": "theorem", "label": "proetale-lemma-have-sections-localize", "categories": [ "proetale" ], "title": "proetale-lemma-have-sections-localize", "contents": [ "Let $A$ be a ring such that every faithfully flat \\'etale ring map", "$A \\to B$ has a section. Let $Z \\subset \\Spec(A)$ be a closed subscheme", "of the form $D(f) \\cap V(I)$ and let $A \\to A_Z^\\sim$ be as constructed", "in Lemma \\ref{lemma-localization}.", "Then every faithfully flat \\'etale ring map $A_Z^\\sim \\to C$ has", "a section." ], "refs": [ "proetale-lemma-localization" ], "proofs": [ { "contents": [ "There exists an \\'etale ring map $A \\to B'$ such that", "$C = B' \\otimes_A A_Z^\\sim$ as $A_Z^\\sim$-algebras.", "The image $U' \\subset \\Spec(A)$ of $\\Spec(B') \\to \\Spec(A)$", "is open and contains $V(I)$, hence we can find $f \\in I$ such", "that $\\Spec(A) = U' \\cup D(f)$. Then $A \\to B' \\times A_f$", "is \\'etale and faithfully flat. By assumption there is a section", "$B' \\times A_f \\to A$. Localizing we obtain the desired section", "$C \\to A_Z^\\sim$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 3712 ] }, { "id": 3735, "type": "theorem", "label": "proetale-lemma-get-w-local-algebraic-residue-field-extensions", "categories": [ "proetale" ], "title": "proetale-lemma-get-w-local-algebraic-residue-field-extensions", "contents": [ "Let $A \\to B$ be a ring map inducing algebraic extensions on residue fields.", "There exists a commutative diagram", "$$", "\\xymatrix{", "B \\ar[r] & D \\\\", "A \\ar[r] \\ar[u] & C \\ar[u]", "}", "$$", "with the following properties:", "\\begin{enumerate}", "\\item $A \\to C$ is faithfully flat and ind-\\'etale,", "\\item $B \\to D$ is faithfully flat and ind-\\'etale,", "\\item $\\Spec(C)$ is w-local,", "\\item $\\Spec(D)$ is w-local,", "\\item $\\Spec(D) \\to \\Spec(C)$ is w-local,", "\\item the set of closed points of $\\Spec(D)$ is the inverse image", "of the set of closed points of $\\Spec(C)$,", "\\item the set of closed points of $\\Spec(C)$ surjects onto $\\Spec(A)$,", "\\item the set of closed points of $\\Spec(D)$ surjects onto $\\Spec(B)$,", "\\item for $\\mathfrak m \\subset C$ maximal the local ring", "$C_\\mathfrak m$ is strictly henselian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "There is a faithfully flat, ind-Zariski ring map $A \\to A'$ such that", "$\\Spec(A')$ is w-local and such that the set of closed points of", "$\\Spec(A')$ maps onto $\\Spec(A)$, see Lemma \\ref{lemma-make-w-local}.", "Let $I \\subset A'$ be the ideal such that $V(I)$ is the set", "of closed points of $\\Spec(A')$.", "Choose $A' \\to C'$ as in Lemma \\ref{lemma-first-construction}.", "Note that the local rings $C'_{\\mathfrak m'}$ at maximal ideals", "$\\mathfrak m' \\subset C'$ are strictly henselian by", "Lemma \\ref{lemma-have-sections-strictly-henselian}.", "We apply Lemma \\ref{lemma-w-local-algebraic-residue-field-extensions}", "to $A' \\to C'$ and $I \\subset A'$ to get $C' \\to C$ with $C'/IC' \\cong C/IC$.", "Note that since $A' \\to C'$ is faithfully flat, $\\Spec(C'/IC')$", "surjects onto the set of closed points of $A'$ and in particular", "onto $\\Spec(A)$. Moreover, as $V(IC) \\subset \\Spec(C)$", "is the set of closed points of $C$ and $C' \\to C$ is ind-Zariski", "(and identifies local rings) we obtain properties (1), (3), (7), and (9).", "\\medskip\\noindent", "Denote $J \\subset C$ the ideal such that $V(J)$ is the set of closed", "points of $\\Spec(C)$. Set $D' = B \\otimes_A C$. The ring", "map $C \\to D'$ induces algebraic residue field extensions. Keep in mind that", "since $V(J) \\to \\Spec(A)$ is surjective the map $T = V(JD) \\to \\Spec(B)$", "is surjective too. Apply", "Lemma \\ref{lemma-w-local-algebraic-residue-field-extensions}", "to $C \\to D'$ and $J \\subset C$ to get ", "$D' \\to D$ with $D'/JD' \\cong D/JD$.", "All of the remaining properties given in the lemma are", "immediate from the results of", "Lemma \\ref{lemma-w-local-algebraic-residue-field-extensions}." ], "refs": [ "proetale-lemma-make-w-local", "proetale-lemma-first-construction", "proetale-lemma-have-sections-strictly-henselian", "proetale-lemma-w-local-algebraic-residue-field-extensions", "proetale-lemma-w-local-algebraic-residue-field-extensions", "proetale-lemma-w-local-algebraic-residue-field-extensions" ], "ref_ids": [ 3714, 3731, 3733, 3718, 3718, 3718 ] } ], "ref_ids": [] }, { "id": 3736, "type": "theorem", "label": "proetale-lemma-pro-V-V", "categories": [ "proetale" ], "title": "proetale-lemma-pro-V-V", "contents": [ "Let $Y$ be an affine scheme. Let $X = \\lim X_i$ be a directed limit", "of affine schemes over $Y$. The following are equivalent", "\\begin{enumerate}", "\\item $\\{X \\to Y\\}$ is a standard V covering", "(Topologies, Definition \\ref{topologies-definition-standard-V-covering}), and", "\\item $\\{X_i \\to Y\\}$ is a standard V covering for all $i$.", "\\end{enumerate}" ], "refs": [ "topologies-definition-standard-V-covering" ], "proofs": [ { "contents": [ "A singleton $\\{X \\to Y\\}$ is a standard V covering if and only if", "given a morphism $g : \\Spec(V) \\to Y$ there is an extension of", "valuation rings $V \\subset W$ and a commutative diagram", "$$", "\\xymatrix{", "\\Spec(W) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(V) \\ar[r]^g & Y", "}", "$$", "Thus (1) $\\Rightarrow$ (2) is immediate from the definition.", "Conversely, assume (2) and let $g : \\Spec(V) \\to Y$ as above", "be given. Write $\\Spec(V) \\times_Y X_i = \\Spec(A_i)$.", "Since $\\{X_i \\to Y\\}$ is a standard V covering, we may choose", "a valuation ring $W_i$ and a ring map $A_i \\to W_i$ such that", "the composition $V \\to A_i \\to W_i$ is an extension of valuation", "rings. In particular, the quotient $A'_i$ of $A_i$ by its", "$V$-torsion is a faitfhully flat $V$-algebra. Flatness by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-valuation-ring-torsion-free-flat} and", "surjectivity on spectra because $A_i \\to W_i$ factors through $A'_i$.", "Thus", "$$", "A = \\colim A'_i", "$$", "is a faithfully flat $V$-algebra", "(Algebra, Lemma \\ref{algebra-lemma-colimit-faithfully-flat}).", "Since $\\{\\Spec(A) \\to \\Spec(V)\\}$ is a standard fpqc cover, it", "is a standard V cover", "(Topologies, Lemma \\ref{topologies-lemma-standard-fpqc-standard-V})", "and hence we can choose $\\Spec(W) \\to \\Spec(A)$ such that", "$V \\to W$ is an extension of valuation rings. Since we can compose", "with the morphism $\\Spec(A) \\to X = \\Spec(\\colim A_i)$", "the proof is complete." ], "refs": [ "more-algebra-lemma-valuation-ring-torsion-free-flat", "algebra-lemma-colimit-faithfully-flat", "topologies-lemma-standard-fpqc-standard-V" ], "ref_ids": [ 9920, 540, 12504 ] } ], "ref_ids": [ 12550 ] }, { "id": 3737, "type": "theorem", "label": "proetale-lemma-pro-h-V", "categories": [ "proetale" ], "title": "proetale-lemma-pro-h-V", "contents": [ "Let $X \\to Y$ be a morphism of affine schemes. The following are equivalent", "\\begin{enumerate}", "\\item $\\{X \\to Y\\}$ is a standard V covering", "(Topologies, Definition \\ref{topologies-definition-standard-V-covering}),", "\\item $X = \\lim X_i$ is a directed limit of affine schemes over $Y$", "such that $\\{X_i \\to Y\\}$ is a ph covering for each $i$, and", "\\item $X = \\lim X_i$ is a directed limit of affine schemes over $Y$", "such that $\\{X_i \\to Y\\}$ is an h covering for each $i$.", "\\end{enumerate}" ], "refs": [ "topologies-definition-standard-V-covering" ], "proofs": [ { "contents": [ "Proof of (2) $\\Rightarrow$ (1). Recall that a V covering given by a", "single arrow between affines is a standard V covering, see", "Topologies, Definition \\ref{topologies-definition-V-covering} and", "Lemma \\ref{topologies-lemma-refine-standard-V}.", "Recall that any ph covering is a V covering, see", "Topologies, Lemma", "\\ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc-ph-V}.", "Hence if $X = \\lim X_i$ as in (2), then $\\{X_i \\to Y\\}$", "is a standard V covering for each $i$. Thus by", "Lemma \\ref{lemma-pro-V-V} we see that (1) is true.", "\\medskip\\noindent", "Proof of (3) $\\Rightarrow$ (2). This is clear because an h covering", "is always a ph covering, see", "More on Flatness, Definition \\ref{flat-definition-h-covering}.", "\\medskip\\noindent", "Proof of (1) $\\Rightarrow$ (3). This is the interesting direction, but", "the interesting content in this proof is hidden in More on Flatness, Lemma", "\\ref{flat-lemma-equivalence-h-v-locally-finite-presentation}.", "Write $X = \\Spec(A)$ and $Y = \\Spec(R)$. We can write", "$A = \\colim A_i$ with $A_i$ of finite presentation over $R$, see", "Algebra, Lemma \\ref{algebra-lemma-ring-colimit-fp}.", "Set $X_i = \\Spec(A_i)$. Then $\\{X_i \\to Y\\}$ is a standard V covering", "for all $i$ by (1) and", "Topologies, Lemma \\ref{topologies-lemma-refine-standard-V}.", "Hence $\\{X_i \\to Y\\}$ is an h covering by", "More on Flatness, Definition \\ref{flat-definition-h-covering}.", "This finishes the proof." ], "refs": [ "topologies-definition-V-covering", "topologies-lemma-refine-standard-V", "topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc-ph-V", "proetale-lemma-pro-V-V", "flat-definition-h-covering", "flat-lemma-equivalence-h-v-locally-finite-presentation", "algebra-lemma-ring-colimit-fp", "topologies-lemma-refine-standard-V", "flat-definition-h-covering" ], "ref_ids": [ 12551, 12508, 12511, 3736, 6220, 6139, 1091, 12508, 6220 ] } ], "ref_ids": [ 12550 ] }, { "id": 3738, "type": "theorem", "label": "proetale-lemma-h-limit-preserving", "categories": [ "proetale" ], "title": "proetale-lemma-h-limit-preserving", "contents": [ "Let $S$ be a scheme. Let $F$ be a contravariant functor defined", "on the category of all schemes over $S$. If", "\\begin{enumerate}", "\\item $F$ satisfies the sheaf property for the h topology, and", "\\item $F$ is limit preserving", "(Limits, Remark \\ref{limits-remark-limit-preserving}),", "\\end{enumerate}", "then $F$ satisfies the sheaf property for the V topology." ], "refs": [ "limits-remark-limit-preserving" ], "proofs": [ { "contents": [ "We will prove this by verifying (1) and (2') of", "Topologies, Lemma \\ref{topologies-lemma-sheaf-property-V}.", "The sheaf property for Zariski coverings follows from", "the fact that $F$ has the sheaf property for all h coverings.", "Finally, suppose that $X \\to Y$ is a morphism of affine schemes", "over $S$ such that $\\{X \\to Y\\}$ is a V covering.", "By Lemma \\ref{lemma-pro-h-V} we can write $X = \\lim X_i$", "as a directed limit of affine schemes", "over $Y$ such that $\\{X_i \\to Y\\}$ is an h covering for each $i$.", "We obtain", "\\begin{align*}", "&", "\\text{Equalizer}(", "\\xymatrix{", "F(X) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "F(X \\times_Y X)", "}", ")", "\\\\", "& =", "\\text{Equalizer}(", "\\xymatrix{", "\\colim F(X_i) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\colim F(X_i \\times_Y X_i)", "}", ")", "\\\\", "& =", "\\colim", "\\text{Equalizer}(", "\\xymatrix{", "F(X_i) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "F(X_i \\times_Y X_i)", "}", ")", "\\\\", "& =", "\\colim F(Y) = F(Y)", "\\end{align*}", "which is what we wanted to show.", "The first equality because $F$ is limit preserving and $X = \\lim X_i$ and", "$X \\times_Y X = \\lim X_i \\times_Y X_i$.", "The second equality because filtered colimits are exact.", "The third equality because $F$ satisfies the sheaf property", "for h coverings." ], "refs": [ "topologies-lemma-sheaf-property-V", "proetale-lemma-pro-h-V" ], "ref_ids": [ 12512, 3737 ] } ], "ref_ids": [ 15130 ] }, { "id": 3739, "type": "theorem", "label": "proetale-lemma-w-local-strictly-henselian-extremally-disconnected", "categories": [ "proetale" ], "title": "proetale-lemma-w-local-strictly-henselian-extremally-disconnected", "contents": [ "Let $A$ be a ring. The following are equivalent", "\\begin{enumerate}", "\\item $A$ is w-contractible,", "\\item every faithfully flat, ind-\\'etale ring map $A \\to B$ has", "a section, and", "\\item $A$ satisfies", "\\begin{enumerate}", "\\item $\\Spec(A)$ is w-local,", "\\item $\\pi_0(\\Spec(A))$ is extremally disconnected, and", "\\item for every maximal ideal $\\mathfrak m \\subset A$ the", "local ring $A_\\mathfrak m$ is strictly henselian.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows immediately from", "Proposition \\ref{proposition-weakly-etale}.", "\\medskip\\noindent", "Assume (3)(a), (3)(b), and (3)(c). Let $A \\to B$ be faithfully flat", "and ind-\\'etale. We will use without further mention the fact that a flat map", "$A \\to B$ is faithfully flat if and only if every closed point", "of $\\Spec(A)$ is in the image of $\\Spec(B) \\to \\Spec(A)$", "We will show that $A \\to B$ has a section.", "\\medskip\\noindent", "Let $I \\subset A$ be an ideal such that $V(I) \\subset \\Spec(A)$ is", "the set of closed points of $\\Spec(A)$. ", "We may replace $B$ by the ring $C$ constructed in", "Lemma \\ref{lemma-w-local-algebraic-residue-field-extensions}", "for $A \\to B$ and $I \\subset A$.", "Thus we may assume $\\Spec(B)$ is w-local such that the set of", "closed points of $\\Spec(B)$ is $V(IB)$. In this case $A \\to B$", "identifies local rings by condition (3)(c) as it suffices to check", "this at maximal ideals of $B$ which lie over maximal ideals of $A$.", "Thus $A \\to B$ has a section by", "Lemma \\ref{lemma-w-local-extremally-disconnected}.", "\\medskip\\noindent", "Assume (1) or equivalently (2). We have (3)(c) by", "Lemma \\ref{lemma-have-sections-strictly-henselian}.", "Properties (3)(a) and (3)(b) follow from", "Lemma \\ref{lemma-w-local-extremally-disconnected}." ], "refs": [ "proetale-proposition-weakly-etale", "proetale-lemma-w-local-algebraic-residue-field-extensions", "proetale-lemma-w-local-extremally-disconnected", "proetale-lemma-have-sections-strictly-henselian", "proetale-lemma-w-local-extremally-disconnected" ], "ref_ids": [ 3826, 3718, 3723, 3733, 3723 ] } ], "ref_ids": [] }, { "id": 3740, "type": "theorem", "label": "proetale-lemma-finite-finitely-presented-over-extremally-disconnected", "categories": [ "proetale" ], "title": "proetale-lemma-finite-finitely-presented-over-extremally-disconnected", "contents": [ "Let $A \\to B$ be a quasi-finite and finitely presented ring map.", "If the residue fields of $A$ are separably algebraically closed", "and $\\Spec(A)$ is extremally disconnected, then $\\Spec(B)$ is", "extremally disconnected." ], "refs": [], "proofs": [ { "contents": [ "Set $X = \\Spec(A)$ and $Y = \\Spec(B)$. Choose a finite partition", "$X = \\coprod X_i$ and $X'_i \\to X_i$ as in", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-decompose-quasi-finite-morphism}.", "Because $X$ is extremally disconnected, every constructible", "locally closed subset is open and closed, hence we see that $X$", "is topologically the disjoint union of the strata $X_i$. Thus we may", "replace $X$ by the $X_i$ and assume there exists a surjective finite", "locally free morphism $X' \\to X$ such that $(X' \\times_X Y)_{red}$", "is isomorphic to a finite disjoint union of copies of $X'_{red}$.", "Picture", "$$", "\\xymatrix{", "\\coprod_{i = 1, \\ldots, r} X' \\ar[r] \\ar[d] & Y \\ar[d] \\\\", "X' \\ar[r] & X", "}", "$$", "The assumption on the residue fields of $A$ implies that", "this diagram is a fibre product diagram on underlying sets", "of points (details omitted).", "Since $X$ is extremally disconnected and $X'$ is Hausdorff", "(Lemma \\ref{lemma-profinite-goes-up}), the continuous map", "$X' \\to X$ has a continuous section $\\sigma$. Then", "$\\coprod_{i = 1, \\ldots, r} \\sigma(X) \\to Y$ is a bijective", "continuous map. By", "Topology, Lemma \\ref{topology-lemma-bijective-map}", "we see that it is a homeomorphism and the proof is done." ], "refs": [ "etale-cohomology-lemma-decompose-quasi-finite-morphism", "proetale-lemma-profinite-goes-up", "topology-lemma-bijective-map" ], "ref_ids": [ 6537, 3716, 8275 ] } ], "ref_ids": [] }, { "id": 3741, "type": "theorem", "label": "proetale-lemma-finite-finitely-presented-over-w-contractible", "categories": [ "proetale" ], "title": "proetale-lemma-finite-finitely-presented-over-w-contractible", "contents": [ "Let $A \\to B$ be a finite and finitely presented ring map.", "If $A$ is w-contractible, so is $B$." ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion of", "Lemma \\ref{lemma-w-local-strictly-henselian-extremally-disconnected}.", "Set $X = \\Spec(A)$ and $Y = \\Spec(B)$.", "As $Y \\to X$ is a finite morphism, we see that the set of closed", "points $Y_0$ of $Y$ is the inverse image of the set of closed points", "$X_0$ of $X$. Moreover, every point of $Y$ specializes to a unique", "point of $Y_0$ as (a) this is true for $X$ and (b) the map", "$X \\to Y$ is separated. For every $y \\in Y_0$ with image $x \\in X_0$", "we see that $\\mathcal{O}_{Y, y}$ is strictly henselian by", "Algebra, Lemma \\ref{algebra-lemma-finite-over-henselian}", "applied to $\\mathcal{O}_{X, x} \\to B \\otimes_A \\mathcal{O}_{X, x}$.", "It remains to show that $Y_0$ is extremally disconnected.", "To do this we look at $X_0 \\times_X Y \\to X_0$", "where $X_0 \\subset X$ is the reduced induced scheme structure.", "Note that the underlying topological space of", "$X_0 \\times_X Y$ agrees with $Y_0$. Now the desired result follows from", "Lemma \\ref{lemma-finite-finitely-presented-over-extremally-disconnected}." ], "refs": [ "proetale-lemma-w-local-strictly-henselian-extremally-disconnected", "algebra-lemma-finite-over-henselian", "proetale-lemma-finite-finitely-presented-over-extremally-disconnected" ], "ref_ids": [ 3739, 1277, 3740 ] } ], "ref_ids": [] }, { "id": 3742, "type": "theorem", "label": "proetale-lemma-localization-w-contractible", "categories": [ "proetale" ], "title": "proetale-lemma-localization-w-contractible", "contents": [ "Let $A$ be a ring. Let $Z \\subset \\Spec(A)$ be a closed subset", "of the form $Z = V(f_1, \\ldots, f_r)$. Set $B = A_Z^\\sim$, see", "Lemma \\ref{lemma-localization}. If $A$ is w-contractible, so is $B$." ], "refs": [ "proetale-lemma-localization" ], "proofs": [ { "contents": [ "Let $A_Z^\\sim \\to B$ be a weakly \\'etale faithfully flat ring map.", "Consider the ring map", "$$", "A \\longrightarrow A_{f_1} \\times \\ldots \\times A_{f_r} \\times B", "$$", "this is faithful flat and weakly \\'etale. If $A$ is w-contractible,", "then there is a section $\\sigma$. Consider the morphism", "$$", "\\Spec(A_Z^\\sim) \\to \\Spec(A) \\xrightarrow{\\Spec(\\sigma)}", "\\coprod \\Spec(A_{f_i}) \\amalg \\Spec(B)", "$$", "Every point of $Z \\subset \\Spec(A_Z^\\sim)$ maps into the component", "$\\Spec(B)$. Since every point of $\\Spec(A_Z^\\sim)$ specializes to a", "point of $Z$ we find a morphism $\\Spec(A_Z^\\sim) \\to \\Spec(B)$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 3712 ] }, { "id": 3743, "type": "theorem", "label": "proetale-lemma-recognize-proetale-covering", "categories": [ "proetale" ], "title": "proetale-lemma-recognize-proetale-covering", "contents": [ "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of", "morphisms of schemes with target $T$. The following are equivalent", "\\begin{enumerate}", "\\item $\\{f_i : T_i \\to T\\}_{i \\in I}$ is a pro-\\'etale covering,", "\\item each $f_i$ is weakly \\'etale and $\\{f_i : T_i \\to T\\}_{i \\in I}$", "is an fpqc covering,", "\\item each $f_i$ is weakly \\'etale and for every affine open $U \\subset T$", "there exist quasi-compact opens $U_i \\subset T_i$ which are almost all empty,", "such that $U = \\bigcup f_i(U_i)$,", "\\item each $f_i$ is weakly \\'etale and there exists an affine open covering", "$T = \\bigcup_{\\alpha \\in A} U_\\alpha$ and for each $\\alpha \\in A$", "there exist $i_{\\alpha, 1}, \\ldots, i_{\\alpha, n(\\alpha)} \\in I$", "and quasi-compact opens $U_{\\alpha, j} \\subset T_{i_{\\alpha, j}}$ such that", "$U_\\alpha =", "\\bigcup_{j = 1, \\ldots, n(\\alpha)} f_{i_{\\alpha, j}}(U_{\\alpha, j})$.", "\\end{enumerate}", "If $T$ is quasi-separated, these are also equivalent to", "\\begin{enumerate}", "\\item[(5)] each $f_i$ is weakly \\'etale, and for every $t \\in T$ there exist", "$i_1, \\ldots, i_n \\in I$ and quasi-compact opens $U_j \\subset T_{i_j}$", "such that $\\bigcup_{j = 1, \\ldots, n} f_{i_j}(U_j)$ is a", "(not necessarily open) neighbourhood of $t$ in $T$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is immediate from the definitions.", "Hence the lemma follows from", "Topologies, Lemma \\ref{topologies-lemma-recognize-fpqc-covering}." ], "refs": [ "topologies-lemma-recognize-fpqc-covering" ], "ref_ids": [ 12493 ] } ], "ref_ids": [] }, { "id": 3744, "type": "theorem", "label": "proetale-lemma-etale-proetale", "categories": [ "proetale" ], "title": "proetale-lemma-etale-proetale", "contents": [ "Any \\'etale covering and any Zariski covering is a pro-\\'etale covering." ], "refs": [], "proofs": [ { "contents": [ "This follows from the corresponding result for fpqc coverings", "(Topologies, Lemma", "\\ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc}),", "Lemma \\ref{lemma-recognize-proetale-covering}, and", "the fact that an \\'etale morphism is a weakly \\'etale morphism, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-when-weakly-etale}." ], "refs": [ "topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc", "proetale-lemma-recognize-proetale-covering", "more-morphisms-lemma-when-weakly-etale" ], "ref_ids": [ 12497, 3743, 14029 ] } ], "ref_ids": [] }, { "id": 3745, "type": "theorem", "label": "proetale-lemma-proetale", "categories": [ "proetale" ], "title": "proetale-lemma-proetale", "contents": [ "Let $T$ be a scheme.", "\\begin{enumerate}", "\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$", "is a pro-\\'etale covering of $T$.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a pro-\\'etale covering and for each", "$i$ we have a pro-\\'etale covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then", "$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a pro-\\'etale covering.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a pro-\\'etale covering", "and $T' \\to T$ is a morphism of schemes then", "$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a pro-\\'etale covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that composition and base changes", "of weakly \\'etale morphisms are weakly \\'etale", "(More on Morphisms, Lemmas", "\\ref{more-morphisms-lemma-composition-weakly-etale} and", "\\ref{more-morphisms-lemma-base-change-weakly-etale}),", "Lemma \\ref{lemma-recognize-proetale-covering}, and", "the corresponding results for fpqc coverings, see", "Topologies, Lemma \\ref{topologies-lemma-fpqc}." ], "refs": [ "more-morphisms-lemma-composition-weakly-etale", "more-morphisms-lemma-base-change-weakly-etale", "proetale-lemma-recognize-proetale-covering", "topologies-lemma-fpqc" ], "ref_ids": [ 14025, 14026, 3743, 12498 ] } ], "ref_ids": [] }, { "id": 3746, "type": "theorem", "label": "proetale-lemma-proetale-affine", "categories": [ "proetale" ], "title": "proetale-lemma-proetale-affine", "contents": [ "Let $T$ be an affine scheme. Let $\\{T_i \\to T\\}_{i \\in I}$ be a pro-\\'etale", "covering of $T$. Then there exists a pro-\\'etale covering", "$\\{U_j \\to T\\}_{j = 1, \\ldots, n}$ which is a refinement", "of $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine", "scheme. Moreover, we may choose each $U_j$ to be open affine", "in one of the $T_i$." ], "refs": [], "proofs": [ { "contents": [ "This follows directly from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3747, "type": "theorem", "label": "proetale-lemma-verify-site-proetale", "categories": [ "proetale" ], "title": "proetale-lemma-verify-site-proetale", "contents": [ "Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale site", "containing $S$. Both $S_\\proetale$ and $(\\textit{Aff}/S)_\\proetale$ are sites." ], "refs": [], "proofs": [ { "contents": [ "Let us show that $S_\\proetale$ is a site. It is a category with a", "given set of families of morphisms with fixed target. Thus we", "have to show properties (1), (2) and (3) of", "Sites, Definition \\ref{sites-definition-site}.", "Since $(\\Sch/S)_\\proetale$ is a site, it suffices to prove", "that given any covering $\\{U_i \\to U\\}$ of $(\\Sch/S)_\\proetale$", "with $U \\in \\Ob(S_\\proetale)$ we also have $U_i \\in \\Ob(S_\\proetale)$.", "This follows from the definitions", "as the composition of weakly \\'etale morphisms is weakly \\'etale.", "\\medskip\\noindent", "To show that $(\\textit{Aff}/S)_\\proetale$ is a site, reasoning as above,", "it suffices to show that the collection of standard pro-\\'etale coverings", "of affines satisfies properties (1), (2) and (3) of", "Sites, Definition \\ref{sites-definition-site}.", "This follows from Lemma \\ref{lemma-recognize-proetale-covering}", "and the corresponding result for standard fpqc coverings", "(Topologies, Lemma \\ref{topologies-lemma-fpqc-affine-axioms})." ], "refs": [ "sites-definition-site", "sites-definition-site", "proetale-lemma-recognize-proetale-covering", "topologies-lemma-fpqc-affine-axioms" ], "ref_ids": [ 8652, 8652, 3743, 12500 ] } ], "ref_ids": [] }, { "id": 3748, "type": "theorem", "label": "proetale-lemma-fibre-products-proetale", "categories": [ "proetale" ], "title": "proetale-lemma-fibre-products-proetale", "contents": [ "Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale", "site containing $S$. Let $\\Sch$ be the category of all schemes.", "\\begin{enumerate}", "\\item The categories $\\Sch_\\proetale$, $(\\Sch/S)_\\proetale$,", "$S_\\proetale$, and $(\\textit{Aff}/S)_\\proetale$ have fibre products", "agreeing with fibre products in $\\Sch$.", "\\item The categories $\\Sch_\\proetale$, $(\\Sch/S)_\\proetale$,", "$S_\\proetale$ have equalizers agreeing with equalizers in $\\Sch$.", "\\item The categories $(\\Sch/S)_\\proetale$, and $S_\\proetale$ both have", "a final object, namely $S/S$.", "\\item The category $\\Sch_\\proetale$ has a final object agreeing", "with the final object of $\\Sch$, namely $\\Spec(\\mathbf{Z})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The category $\\Sch_\\proetale$ contains $\\Spec(\\mathbf{Z})$ and", "is closed under products and fibre products by construction, see", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "Suppose we have $U \\to S$, $V \\to U$, $W \\to U$ morphisms", "of schemes with $U, V, W \\in \\Ob(\\Sch_\\proetale)$.", "The fibre product $V \\times_U W$ in $\\Sch_\\proetale$", "is a fibre product in $\\Sch$ and", "is the fibre product of $V/S$ with $W/S$ over $U/S$ in", "the category of all schemes over $S$, and hence also a", "fibre product in $(\\Sch/S)_\\proetale$.", "This proves the result for $(\\Sch/S)_\\proetale$.", "If $U \\to S$, $V \\to U$ and $W \\to U$ are weakly \\'etale then so is", "$V \\times_U W \\to S$ (see", "More on Morphisms, Section \\ref{more-morphisms-section-weakly-etale})", "and hence we get fibre products for $S_\\proetale$.", "If $U, V, W$ are affine, so is $V \\times_U W$ and hence we", "get fibre products for $(\\textit{Aff}/S)_\\proetale$.", "\\medskip\\noindent", "Let $a, b : U \\to V$ be two morphisms in $\\Sch_\\proetale$.", "In this case the equalizer of $a$ and $b$ (in the category of schemes) is", "$$", "V", "\\times_{\\Delta_{V/\\Spec(\\mathbf{Z})}, V \\times_{\\Spec(\\mathbf{Z})} V, (a, b)}", "(U \\times_{\\Spec(\\mathbf{Z})} U)", "$$", "which is an object of $\\Sch_\\proetale$ by what we saw above.", "Thus $\\Sch_\\proetale$ has equalizers. If $a$ and $b$ are morphisms over $S$,", "then the equalizer (in the category of schemes) is also given by", "$$", "V \\times_{\\Delta_{V/S}, V \\times_S V, (a, b)} (U \\times_S U)", "$$", "hence we see that $(\\Sch/S)_\\proetale$ has equalizers. Moreover, if", "$U$ and $V$ are weakly-\\'etale over $S$, then so is the equalizer", "above as a fibre product of schemes weakly \\'etale over $S$.", "Thus $S_\\proetale$ has equalizers. The statements on final objects", "is clear." ], "refs": [ "sets-lemma-what-is-in-it" ], "ref_ids": [ 8795 ] } ], "ref_ids": [] }, { "id": 3749, "type": "theorem", "label": "proetale-lemma-affine-big-site-proetale", "categories": [ "proetale" ], "title": "proetale-lemma-affine-big-site-proetale", "contents": [ "Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale", "site containing $S$.", "The functor $(\\textit{Aff}/S)_\\proetale \\to (\\Sch/S)_\\proetale$", "is a special cocontinuous functor. Hence it induces an equivalence", "of topoi from $\\Sh((\\textit{Aff}/S)_\\proetale)$ to", "$\\Sh((\\Sch/S)_\\proetale)$." ], "refs": [], "proofs": [ { "contents": [ "The notion of a special cocontinuous functor is introduced in", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}.", "Thus we have to verify assumptions (1) -- (5) of", "Sites, Lemma \\ref{sites-lemma-equivalence}.", "Denote the inclusion functor", "$u : (\\textit{Aff}/S)_\\proetale \\to (\\Sch/S)_\\proetale$.", "Being cocontinuous just means that any pro-\\'etale covering of", "$T/S$, $T$ affine, can be refined by a standard pro-\\'etale", "covering of $T$. This is the content of", "Lemma \\ref{lemma-proetale-affine}.", "Hence (1) holds. We see $u$ is continuous simply because a standard", "pro-\\'etale covering is a pro-\\'etale covering. Hence (2) holds.", "Parts (3) and (4) follow immediately from the fact that $u$ is", "fully faithful. And finally condition (5) follows from the", "fact that every scheme has an affine open covering." ], "refs": [ "sites-definition-special-cocontinuous-functor", "sites-lemma-equivalence", "proetale-lemma-proetale-affine" ], "ref_ids": [ 8672, 8578, 3746 ] } ], "ref_ids": [] }, { "id": 3750, "type": "theorem", "label": "proetale-lemma-put-in-T", "categories": [ "proetale" ], "title": "proetale-lemma-put-in-T", "contents": [ "Let $\\Sch_\\proetale$ be a big pro-\\'etale site.", "Let $f : T \\to S$ be a morphism in $\\Sch_\\proetale$.", "The functor $T_\\proetale \\to (\\Sch/S)_\\proetale$", "is cocontinuous and induces a morphism of topoi", "$$", "i_f :", "\\Sh(T_\\proetale)", "\\longrightarrow", "\\Sh((\\Sch/S)_\\proetale)", "$$", "For a sheaf $\\mathcal{G}$ on $(\\Sch/S)_\\proetale$", "we have the formula $(i_f^{-1}\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.", "The functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes", "with fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "Denote the functor $u : T_\\proetale \\to (\\Sch/S)_\\proetale$.", "In other words, given a weakly \\'etale morphism $j : U \\to T$ corresponding", "to an object of $T_\\proetale$ we set $u(U \\to T) = (f \\circ j : U \\to S)$.", "This functor commutes with fibre products, see", "Lemma \\ref{lemma-fibre-products-proetale}.", "Moreover, $T_\\proetale$ has equalizers and $u$ commutes with them", "by Lemma \\ref{lemma-fibre-products-proetale}.", "It is clearly cocontinuous.", "It is also continuous as $u$ transforms coverings to coverings and", "commutes with fibre products. Hence the lemma follows from", "Sites, Lemmas \\ref{sites-lemma-when-shriek}", "and \\ref{sites-lemma-preserve-equalizers}." ], "refs": [ "proetale-lemma-fibre-products-proetale", "proetale-lemma-fibre-products-proetale", "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers" ], "ref_ids": [ 3748, 3748, 8545, 8546 ] } ], "ref_ids": [] }, { "id": 3751, "type": "theorem", "label": "proetale-lemma-at-the-bottom", "categories": [ "proetale" ], "title": "proetale-lemma-at-the-bottom", "contents": [ "Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale", "site containing $S$.", "The inclusion functor $S_\\proetale \\to (\\Sch/S)_\\proetale$", "satisfies the hypotheses of Sites, Lemma \\ref{sites-lemma-bigger-site}", "and hence induces a morphism of sites", "$$", "\\pi_S : (\\Sch/S)_\\proetale \\longrightarrow S_\\proetale", "$$", "and a morphism of topoi", "$$", "i_S : \\Sh(S_\\proetale) \\longrightarrow \\Sh((\\Sch/S)_\\proetale)", "$$", "such that $\\pi_S \\circ i_S = \\text{id}$. Moreover, $i_S = i_{\\text{id}_S}$", "with $i_{\\text{id}_S}$ as in Lemma \\ref{lemma-put-in-T}. In particular the", "functor $i_S^{-1} = \\pi_{S, *}$ is described by the rule", "$i_S^{-1}(\\mathcal{G})(U/S) = \\mathcal{G}(U/S)$." ], "refs": [ "sites-lemma-bigger-site", "proetale-lemma-put-in-T" ], "proofs": [ { "contents": [ "In this case the functor $u : S_\\proetale \\to (\\Sch/S)_\\proetale$,", "in addition to the properties seen in the proof of", "Lemma \\ref{lemma-put-in-T} above, also is fully faithful", "and transforms the final object into the final object.", "The lemma follows from Sites, Lemma \\ref{sites-lemma-bigger-site}." ], "refs": [ "proetale-lemma-put-in-T", "sites-lemma-bigger-site" ], "ref_ids": [ 3750, 8548 ] } ], "ref_ids": [ 8548, 3750 ] }, { "id": 3752, "type": "theorem", "label": "proetale-lemma-morphism-big", "categories": [ "proetale" ], "title": "proetale-lemma-morphism-big", "contents": [ "Let $\\Sch_\\proetale$ be a big pro-\\'etale site.", "Let $f : T \\to S$ be a morphism in $\\Sch_\\proetale$.", "The functor", "$$", "u : (\\Sch/T)_\\proetale \\longrightarrow (\\Sch/S)_\\proetale, \\quad", "V/T \\longmapsto V/S", "$$", "is cocontinuous, and has a continuous right adjoint", "$$", "v : (\\Sch/S)_\\proetale \\longrightarrow (\\Sch/T)_\\proetale, \\quad", "(U \\to S) \\longmapsto (U \\times_S T \\to T).", "$$", "They induce the same morphism of topoi", "$$", "f_{big} :", "\\Sh((\\Sch/T)_\\proetale)", "\\longrightarrow", "\\Sh((\\Sch/S)_\\proetale)", "$$", "We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.", "We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.", "Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with", "fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "The functor $u$ is cocontinuous, continuous, and commutes with fibre products", "and equalizers (details omitted; compare with proof of", "Lemma \\ref{lemma-put-in-T}). Hence", "Sites, Lemmas \\ref{sites-lemma-when-shriek} and", "\\ref{sites-lemma-preserve-equalizers}", "apply and we deduce the formula", "for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover,", "the functor $v$ is a right adjoint because given $U/T$ and $V/S$", "we have $\\Mor_S(u(U), V) = \\Mor_T(U, V \\times_S T)$", "as desired. Thus we may apply", "Sites, Lemmas \\ref{sites-lemma-have-functor-other-way} and", "\\ref{sites-lemma-have-functor-other-way-morphism}", "to get the formula for $f_{big, *}$." ], "refs": [ "proetale-lemma-put-in-T", "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers", "sites-lemma-have-functor-other-way", "sites-lemma-have-functor-other-way-morphism" ], "ref_ids": [ 3750, 8545, 8546, 8549, 8550 ] } ], "ref_ids": [] }, { "id": 3753, "type": "theorem", "label": "proetale-lemma-morphism-big-small", "categories": [ "proetale" ], "title": "proetale-lemma-morphism-big-small", "contents": [ "Let $\\Sch_\\proetale$ be a big pro-\\'etale site.", "Let $f : T \\to S$ be a morphism in $\\Sch_\\proetale$.", "\\begin{enumerate}", "\\item We have $i_f = f_{big} \\circ i_T$ with $i_f$ as in", "Lemma \\ref{lemma-put-in-T} and $i_T$ as in", "Lemma \\ref{lemma-at-the-bottom}.", "\\item The functor $S_\\proetale \\to T_\\proetale$,", "$(U \\to S) \\mapsto (U \\times_S T \\to T)$ is continuous and induces", "a morphism of topoi", "$$", "f_{small} : \\Sh(T_\\proetale) \\longrightarrow \\Sh(S_\\proetale).", "$$", "We have $f_{small, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.", "\\item We have a commutative diagram of morphisms of sites", "$$", "\\xymatrix{", "T_\\proetale \\ar[d]_{f_{small}} &", "(\\Sch/T)_\\proetale \\ar[d]^{f_{big}} \\ar[l]^{\\pi_T}\\\\", "S_\\proetale &", "(\\Sch/S)_\\proetale \\ar[l]_{\\pi_S}", "}", "$$", "so that $f_{small} \\circ \\pi_T = \\pi_S \\circ f_{big}$ as morphisms of topoi.", "\\item We have $f_{small} = \\pi_S \\circ f_{big} \\circ i_T = \\pi_S \\circ i_f$.", "\\end{enumerate}" ], "refs": [ "proetale-lemma-put-in-T", "proetale-lemma-at-the-bottom" ], "proofs": [ { "contents": [ "The equality $i_f = f_{big} \\circ i_T$ follows from the", "equality $i_f^{-1} = i_T^{-1} \\circ f_{big}^{-1}$ which is", "clear from the descriptions of these functors above.", "Thus we see (1).", "\\medskip\\noindent", "The functor $u : S_\\proetale \\to T_\\proetale$,", "$u(U \\to S) = (U \\times_S T \\to T)$", "transforms coverings into coverings and commutes with fibre products,", "see Lemmas \\ref{lemma-proetale} and \\ref{lemma-fibre-products-proetale}.", "Moreover, both $S_\\proetale$, $T_\\proetale$ have final objects,", "namely $S/S$ and $T/T$ and $u(S/S) = T/T$. Hence by", "Sites, Proposition \\ref{sites-proposition-get-morphism}", "the functor $u$ corresponds to a morphism of sites", "$T_\\proetale \\to S_\\proetale$. This in turn gives rise to the", "morphism of topoi, see", "Sites, Lemma \\ref{sites-lemma-morphism-sites-topoi}. The description", "of the pushforward is clear from these references.", "\\medskip\\noindent", "Part (3) follows because $\\pi_S$ and $\\pi_T$ are given by the", "inclusion functors and $f_{small}$ and $f_{big}$ by the", "base change functors $U \\mapsto U \\times_S T$.", "\\medskip\\noindent", "Statement (4) follows from (3) by precomposing with $i_T$." ], "refs": [ "proetale-lemma-proetale", "proetale-lemma-fibre-products-proetale", "sites-proposition-get-morphism", "sites-lemma-morphism-sites-topoi" ], "ref_ids": [ 3745, 3748, 8641, 8528 ] } ], "ref_ids": [ 3750, 3751 ] }, { "id": 3754, "type": "theorem", "label": "proetale-lemma-composition-proetale", "categories": [ "proetale" ], "title": "proetale-lemma-composition-proetale", "contents": [ "Given schemes $X$, $Y$, $Y$ in $\\Sch_\\proetale$", "and morphisms $f : X \\to Y$, $g : Y \\to Z$ we have", "$g_{big} \\circ f_{big} = (g \\circ f)_{big}$ and", "$g_{small} \\circ f_{small} = (g \\circ f)_{small}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the simple description of pushforward", "and pullback for the functors on the big sites from", "Lemma \\ref{lemma-morphism-big}. For the functors", "on the small sites this follows from the description of", "the pushforward functors in Lemma \\ref{lemma-morphism-big-small}." ], "refs": [ "proetale-lemma-morphism-big", "proetale-lemma-morphism-big-small" ], "ref_ids": [ 3752, 3753 ] } ], "ref_ids": [] }, { "id": 3755, "type": "theorem", "label": "proetale-lemma-morphism-big-small-cartesian-diagram", "categories": [ "proetale" ], "title": "proetale-lemma-morphism-big-small-cartesian-diagram", "contents": [ "Let $\\Sch_\\proetale$ be a big pro-\\'etale site. Consider a cartesian diagram", "$$", "\\xymatrix{", "T' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "in $\\Sch_\\proetale$. Then", "$i_g^{-1} \\circ f_{big, *} = f'_{small, *} \\circ (i_{g'})^{-1}$", "and $g_{big}^{-1} \\circ f_{big, *} = f'_{big, *} \\circ (g'_{big})^{-1}$." ], "refs": [], "proofs": [ { "contents": [ "Since the diagram is cartesian, we have for $U'/S'$", "that $U' \\times_{S'} T' = U' \\times_S T$. Hence both", "$i_g^{-1} \\circ f_{big, *}$ and $f'_{small, *} \\circ (i_{g'})^{-1}$", "send a sheaf $\\mathcal{F}$ on $(\\Sch/T)_\\proetale$ to the sheaf", "$U' \\mapsto \\mathcal{F}(U' \\times_{S'} T')$ on $S'_\\proetale$", "(use Lemmas \\ref{lemma-put-in-T} and \\ref{lemma-morphism-big}).", "The second equality can be proved in the same manner or can be", "deduced from the very general", "Sites, Lemma \\ref{sites-lemma-localize-morphism}." ], "refs": [ "proetale-lemma-put-in-T", "proetale-lemma-morphism-big", "sites-lemma-localize-morphism" ], "ref_ids": [ 3750, 3752, 8571 ] } ], "ref_ids": [] }, { "id": 3756, "type": "theorem", "label": "proetale-lemma-characterize-sheaf-big", "categories": [ "proetale" ], "title": "proetale-lemma-characterize-sheaf-big", "contents": [ "Let $S$ be a scheme contained in a big pro-\\'etale site $\\Sch_\\proetale$.", "A sheaf $\\mathcal{F}$ on the big pro-\\'etale site $(\\Sch/S)_\\proetale$", "is given by the following data:", "\\begin{enumerate}", "\\item for every $T/S \\in \\Ob((\\Sch/S)_\\proetale)$ a sheaf", "$\\mathcal{F}_T$ on $T_\\proetale$,", "\\item for every $f : T' \\to T$ in", "$(\\Sch/S)_\\proetale$ a map", "$c_f : f_{small}^{-1}\\mathcal{F}_T \\to \\mathcal{F}_{T'}$.", "\\end{enumerate}", "These data are subject to the following conditions:", "\\begin{enumerate}", "\\item[(a)] given any $f : T' \\to T$ and $g : T'' \\to T'$ in", "$(\\Sch/S)_\\proetale$ the composition", "$g_{small}^{-1}c_f \\circ c_g$ is equal to $c_{f \\circ g}$, and", "\\item[(b)] if $f : T' \\to T$ in $(\\Sch/S)_\\proetale$", "is weakly \\'etale then $c_f$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Identical to the proof of", "Topologies, Lemma \\ref{topologies-lemma-characterize-sheaf-big-etale}." ], "refs": [ "topologies-lemma-characterize-sheaf-big-etale" ], "ref_ids": [ 12458 ] } ], "ref_ids": [] }, { "id": 3757, "type": "theorem", "label": "proetale-lemma-alternative", "categories": [ "proetale" ], "title": "proetale-lemma-alternative", "contents": [ "Let $S$ be a scheme. Let $S_{affine, \\proetale}$ denote the full subcategory", "of $S_\\proetale$ consisting of affine objects. A covering of", "$S_{affine, \\proetale}$ will be a standard pro-\\'etale covering, see", "Definition \\ref{definition-standard-proetale}.", "Then restriction", "$$", "\\mathcal{F} \\longmapsto \\mathcal{F}|_{S_{affine, \\etale}}", "$$", "defines an equivalence of topoi", "$\\Sh(S_\\proetale) \\cong \\Sh(S_{affine, \\proetale})$." ], "refs": [ "proetale-definition-standard-proetale" ], "proofs": [ { "contents": [ "This you can show directly from the definitions, and is a good exercise.", "But it also follows immediately from", "Sites, Lemma \\ref{sites-lemma-equivalence}", "by checking that the inclusion functor", "$S_{affine, \\proetale} \\to S_\\proetale$", "is a special cocontinuous functor (see", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor})." ], "refs": [ "sites-lemma-equivalence", "sites-definition-special-cocontinuous-functor" ], "ref_ids": [ 8578, 8672 ] } ], "ref_ids": [ 3836 ] }, { "id": 3758, "type": "theorem", "label": "proetale-lemma-affine-alternative", "categories": [ "proetale" ], "title": "proetale-lemma-affine-alternative", "contents": [ "Let $S$ be an affine scheme. Let $S_{app}$ denote the full subcategory", "of $S_\\proetale$ consisting of affine objects $U$ such that", "$\\mathcal{O}(S) \\to \\mathcal{O}(U)$ is ind-\\'etale. A covering of", "$S_{app}$ will be a standard pro-\\'etale covering, see", "Definition \\ref{definition-standard-proetale}.", "Then restriction", "$$", "\\mathcal{F} \\longmapsto \\mathcal{F}|_{S_{app}}", "$$", "defines an equivalence of topoi $\\Sh(S_\\proetale) \\cong \\Sh(S_{app})$." ], "refs": [ "proetale-definition-standard-proetale" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-alternative} we may replace $S_\\proetale$ by", "$S_{affine, \\proetale}$.", "The lemma follows from Sites, Lemma \\ref{sites-lemma-equivalence}", "by checking that the inclusion functor $S_{app} \\to S_{affine, \\proetale}$", "is a special cocontinuous functor, see", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}.", "The conditions of Sites, Lemma \\ref{sites-lemma-equivalence}", "follow immediately from the definition and the facts", "(a) any object $U$ of $S_{affine, \\proetale}$ has a covering", "$\\{V \\to U\\}$ with $V$ ind-\\'etale over $X$", "(Proposition \\ref{proposition-weakly-etale})", "and (b) the functor $u$ is fully faithful." ], "refs": [ "proetale-lemma-alternative", "sites-lemma-equivalence", "sites-definition-special-cocontinuous-functor", "sites-lemma-equivalence", "proetale-proposition-weakly-etale" ], "ref_ids": [ 3757, 8578, 8672, 8578, 3826 ] } ], "ref_ids": [ 3836 ] }, { "id": 3759, "type": "theorem", "label": "proetale-lemma-proetale-subcanonical", "categories": [ "proetale" ], "title": "proetale-lemma-proetale-subcanonical", "contents": [ "Let $S$ be a scheme. The topology on each of the pro-\\'etale sites", "$\\Sch_\\proetale$, $S_\\proetale$, $(\\Sch/S)_\\proetale$,", "$S_{affine, \\proetale}$, and $(\\textit{Aff}/S)_\\proetale$ is subcanonical." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-recognize-proetale-covering} and", "Descent, Lemma \\ref{descent-lemma-fpqc-universal-effective-epimorphisms}." ], "refs": [ "proetale-lemma-recognize-proetale-covering", "descent-lemma-fpqc-universal-effective-epimorphisms" ], "ref_ids": [ 3743, 14638 ] } ], "ref_ids": [] }, { "id": 3760, "type": "theorem", "label": "proetale-lemma-w-contractible-proetale-cover", "categories": [ "proetale" ], "title": "proetale-lemma-w-contractible-proetale-cover", "contents": [ "Let $T = \\Spec(A)$ be an affine scheme. The following are equivalent", "\\begin{enumerate}", "\\item $A$ is w-contractible, and", "\\item every pro-\\'etale covering of $T$ can be refined by", "a Zariski covering of the form $T = \\coprod_{i = 1, \\ldots, n} U_i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $A$ is w-contractible. By Lemma \\ref{lemma-proetale-affine}", "it suffices to prove we can refine every standard pro-\\'etale covering", "$\\{f_i : T_i \\to T\\}_{i = 1, \\ldots, n}$ by a Zariski covering of $T$.", "The morphism $\\coprod T_i \\to T$ is a surjective weakly \\'etale morphism", "of affine schemes. Hence by Definition \\ref{definition-w-contractible}", "there exists a morphism $\\sigma : T \\to \\coprod T_i$ over $T$.", "Then the Zariski covering $T = \\coprod \\sigma^{-1}(T_i)$", "refines $\\{f_i : T_i \\to T\\}$.", "\\medskip\\noindent", "Conversely, assume (2). If $A \\to B$ is faithfully flat and weakly \\'etale,", "then $\\{\\Spec(B) \\to T\\}$ is a pro-\\'etale covering.", "Hence there exists a Zariski covering $T = \\coprod U_i$", "and morphisms $U_i \\to \\Spec(B)$ over $T$. Since $T = \\coprod U_i$", "we obtain $T \\to \\Spec(B)$, i.e., an $A$-algebra map $B \\to A$.", "This means $A$ is w-contractible." ], "refs": [ "proetale-lemma-proetale-affine", "proetale-definition-w-contractible" ], "ref_ids": [ 3746, 3834 ] } ], "ref_ids": [] }, { "id": 3761, "type": "theorem", "label": "proetale-lemma-w-contractible-is-weakly-contractible", "categories": [ "proetale" ], "title": "proetale-lemma-w-contractible-is-weakly-contractible", "contents": [ "Let $\\Sch_\\proetale$ be a big pro-\\'etale site as in", "Definition \\ref{definition-big-proetale-site}.", "Let $T = \\Spec(A)$ be an affine object of $\\Sch_\\proetale$.", "The following are equivalent", "\\begin{enumerate}", "\\item $A$ is w-contractible,", "\\item $T$ is a weakly contractible", "(Sites, Definition \\ref{sites-definition-w-contractible})", "object of $\\Sch_\\proetale$, and", "\\item every pro-\\'etale covering of $T$ can be refined by", "a Zariski covering of the form $T = \\coprod_{i = 1, \\ldots, n} U_i$.", "\\end{enumerate}" ], "refs": [ "proetale-definition-big-proetale-site", "sites-definition-w-contractible" ], "proofs": [ { "contents": [ "We have seen the equivalence of (1) and (3) in", "Lemma \\ref{lemma-w-contractible-proetale-cover}.", "\\medskip\\noindent", "Assume (3) and let $\\mathcal{F} \\to \\mathcal{G}$ be a surjection of sheaves on", "$\\Sch_\\proetale$. Let $s \\in \\mathcal{G}(T)$. To prove (2) we will show that", "$s$ is in the image of $\\mathcal{F}(T) \\to \\mathcal{G}(T)$. We can find a", "covering $\\{T_i \\to T\\}$ of $\\Sch_\\proetale$ such that $s$ lifts", "to a section of $\\mathcal{F}$ over $T_i$", "(Sites, Definition \\ref{sites-definition-sheaves-injective-surjective}).", "By (3) we may assume we have a finite covering", "$T = \\coprod_{j = 1, \\ldots, m} U_j$ by open and closed subsets", "and we have $t_j \\in \\mathcal{F}(U_j)$ mapping to $s|_{U_j}$.", "Since Zariski coverings are coverings in $\\Sch_\\proetale$", "(Lemma \\ref{lemma-etale-proetale}) we conclude that", "$\\mathcal{F}(T) = \\prod \\mathcal{F}(U_j)$.", "Thus $t = (t_1, \\ldots, t_m) \\in \\mathcal{F}(T)$", "is a section mapping to $s$.", "\\medskip\\noindent", "Assume (2). Let $A \\to D$ be as in", "Proposition \\ref{proposition-find-w-contractible}.", "Then $\\{V \\to T\\}$ is a covering of $\\Sch_\\proetale$.", "(Note that $V = \\Spec(D)$ is an object of $\\Sch_\\proetale$", "by Remark \\ref{remark-size-w-contractible}", "combined with our choice of the function", "$Bound$ in Definition \\ref{definition-big-proetale-site}", "and the computation of the size of affine schemes in", "Sets, Lemma \\ref{sets-lemma-bound-size}.)", "Since the topology on $\\Sch_\\proetale$ is subcanonical", "(Lemma \\ref{lemma-proetale-subcanonical})", "we see that $h_V \\to h_T$ is a surjective map of sheaves", "(Sites, Lemma \\ref{sites-lemma-covering-surjective-after-sheafification}).", "Since $T$ is assumed weakly contractible, we see that there is an element", "$f \\in h_V(T) = \\Mor(T, V)$ whose image in $h_T(T)$ is $\\text{id}_T$.", "Thus $A \\to D$ has a section $\\sigma : D \\to A$.", "Now if $A \\to B$ is faithfully flat and weakly \\'etale, then", "$D \\to D \\otimes_A B$ has the same properties, hence", "there is a section $D \\otimes_A B \\to D$ and combined", "with $\\sigma$ we get a section $B \\to D \\otimes_A B \\to D \\to A$", "of $A \\to B$. Thus $A$ is w-contractible and (1) holds." ], "refs": [ "proetale-lemma-w-contractible-proetale-cover", "sites-definition-sheaves-injective-surjective", "proetale-lemma-etale-proetale", "proetale-proposition-find-w-contractible", "proetale-remark-size-w-contractible", "proetale-definition-big-proetale-site", "sets-lemma-bound-size", "proetale-lemma-proetale-subcanonical", "sites-lemma-covering-surjective-after-sheafification" ], "ref_ids": [ 3760, 8660, 3744, 3827, 3850, 3837, 8791, 3759, 8519 ] } ], "ref_ids": [ 3837, 8680 ] }, { "id": 3762, "type": "theorem", "label": "proetale-lemma-get-many-weakly-contractible", "categories": [ "proetale" ], "title": "proetale-lemma-get-many-weakly-contractible", "contents": [ "Let $\\Sch_\\proetale$ be a big pro-\\'etale site as in", "Definition \\ref{definition-big-proetale-site}.", "For every object $T$ of $\\Sch_\\proetale$ there exists", "a covering $\\{T_i \\to T\\}$ in $\\Sch_\\proetale$", "with each $T_i$ affine and the spectrum of a w-contractible", "ring. In particular, $T_i$ is weakly contractible in $\\Sch_\\proetale$." ], "refs": [ "proetale-definition-big-proetale-site" ], "proofs": [ { "contents": [ "For those readers who do not care about set-theoretical issues", "this lemma is a trivial consequence of", "Lemma \\ref{lemma-w-contractible-is-weakly-contractible} and", "Proposition \\ref{proposition-find-w-contractible}.", "Here are the details.", "Choose an affine open covering $T = \\bigcup U_i$. Write $U_i = \\Spec(A_i)$.", "Choose faithfully flat, ind-\\'etale ring maps $A_i \\to D_i$", "such that $D_i$ is w-contractible as in", "Proposition \\ref{proposition-find-w-contractible}.", "The family of morphisms $\\{\\Spec(D_i) \\to T\\}$ is a", "pro-\\'etale covering.", "If we can show that $\\Spec(D_i)$ is isomorphic to an object, say $T_i$,", "of $\\Sch_\\proetale$, then $\\{T_i \\to T\\}$ will be combinatorially", "equivalent to a covering of $\\Sch_\\proetale$ by the construction", "of $\\Sch_\\proetale$ in Definition \\ref{definition-big-proetale-site}", "and more precisely the application of", "Sets, Lemma \\ref{sets-lemma-coverings-site} in the last step.", "To prove $\\Spec(D_i)$ is isomorphic to an object of", "$\\Sch_\\proetale$, it suffices to prove that", "$|D_i| \\leq Bound(\\text{size}(T))$ by the construction", "of $\\Sch_\\proetale$ in Definition \\ref{definition-big-proetale-site}", "and more precisely the application of", "Sets, Lemma \\ref{sets-lemma-construct-category} in step (3).", "Since $|A_i| \\leq \\text{size}(U_i) \\leq \\text{size}(T)$", "by Sets, Lemmas \\ref{sets-lemma-bound-affine} and", "\\ref{sets-lemma-bound-finite-type} we get", "$|D_i| \\leq \\kappa^{2^{2^{2^\\kappa}}}$ where $\\kappa = \\text{size}(T)$", "by Remark \\ref{remark-size-w-contractible}.", "Thus by our choice of the function $Bound$ in", "Definition \\ref{definition-big-proetale-site} we win." ], "refs": [ "proetale-lemma-w-contractible-is-weakly-contractible", "proetale-proposition-find-w-contractible", "proetale-proposition-find-w-contractible", "proetale-definition-big-proetale-site", "sets-lemma-coverings-site", "proetale-definition-big-proetale-site", "sets-lemma-construct-category", "sets-lemma-bound-affine", "sets-lemma-bound-finite-type", "proetale-remark-size-w-contractible", "proetale-definition-big-proetale-site" ], "ref_ids": [ 3761, 3827, 3827, 3837, 8800, 3837, 8789, 8790, 8793, 3850, 3837 ] } ], "ref_ids": [ 3837 ] }, { "id": 3763, "type": "theorem", "label": "proetale-lemma-proetale-enough-w-contractible", "categories": [ "proetale" ], "title": "proetale-lemma-proetale-enough-w-contractible", "contents": [ "Let $S$ be a scheme. The pro-\\'etale sites", "$S_\\proetale$, $(\\Sch/S)_\\proetale$, $S_{affine, \\proetale}$, and", "$(\\textit{Aff}/S)_\\proetale$ and if $S$ is affine $S_{app}$", "have enough (affine) quasi-compact, weakly contractible", "objects, see Sites, Definition \\ref{sites-definition-w-contractible}." ], "refs": [ "sites-definition-w-contractible" ], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-get-many-weakly-contractible}." ], "refs": [ "proetale-lemma-get-many-weakly-contractible" ], "ref_ids": [ 3762 ] } ], "ref_ids": [ 8680 ] }, { "id": 3764, "type": "theorem", "label": "proetale-lemma-weakly-contractible-cover", "categories": [ "proetale" ], "title": "proetale-lemma-weakly-contractible-cover", "contents": [ "Let $S$ be a scheme. The pro-\\'etale sites", "$\\Sch_\\proetale$, $S_\\proetale$, $(\\Sch/S)_\\proetale$", "have the following property: for any object", "$U$ there exists a covering $\\{V \\to U\\}$ with $V$ a", "weakly contractible object. If $U$ is quasi-compact, then we", "may choose $V$ affine and weakly contractible." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $V = \\coprod_{j \\in J} V_j$ is an object of $(\\Sch/S)_\\proetale$", "which is the disjoint union of weakly contractible objects $V_j$.", "Since a disjoint union decomposition is a pro-\\'etale covering we", "see that $\\mathcal{F}(V) = \\prod_{j \\in J} \\mathcal{F}(V_j)$ for any", "pro-\\'etale sheaf $\\mathcal{F}$. Let $\\mathcal{F} \\to \\mathcal{G}$", "be a surjective map of sheaves of sets. Since $V_j$ is weakly contractible,", "the map $\\mathcal{F}(V_j) \\to \\mathcal{G}(V_j)$ is surjective, see", "Sites, Definition \\ref{sites-definition-w-contractible}.", "Thus $\\mathcal{F}(V) \\to \\mathcal{G}(V)$ is surjective as a product", "of surjective maps of sets and we conclude that $V$ is weakly contractible.", "\\medskip\\noindent", "Choose a covering $\\{U_i \\to U\\}_{i \\in I}$ with $U_i$ affine and", "weakly contractible as in Lemma \\ref{lemma-get-many-weakly-contractible}.", "Take $V = \\coprod_{i \\in I} U_i$ (there is a set theoretic issue here", "which we will address below). Then $\\{V \\to U\\}$ is the desired", "pro-\\'etale covering by a weakly contractible object", "(to check it is a covering use", "Lemma \\ref{lemma-recognize-proetale-covering}).", "If $U$ is quasi-compact, then it follows immediately from", "Lemma \\ref{lemma-recognize-proetale-covering}", "that we can choose a finite subset $I' \\subset I$ such that", "$\\{U_i \\to U\\}_{i \\in I'}$ is still a covering", "and then $\\{\\coprod_{i \\in I'} U_i \\to U\\}$ is the desired covering", "by an affine and weakly contractible object.", "\\medskip\\noindent", "In this paragraph, which we urge the reader to skip, we address", "set theoretic problems. In order to know that the disjoint union", "lies in our partial universe, we need to bound the cardinality of", "the index set $I$. It is seen immediately from the construction of", "the covering $\\{U_i \\to U\\}_{i \\in I}$ in the proof of", "Lemma \\ref{lemma-get-many-weakly-contractible}", "that $|I| \\leq \\text{size}(U)$ where the size of a scheme is as", "defined in Sets, Section \\ref{sets-section-categories-schemes}.", "Moreover, for each $i$ we have", "$\\text{size}(U_i) \\leq Bound(\\text{size}(U))$;", "this follows for the bound of the cardinality of", "$\\Gamma(U_i, \\mathcal{O}_{U_i})$ in", "the proof of Lemma \\ref{lemma-get-many-weakly-contractible}", "and Sets, Lemma \\ref{sets-lemma-bound-affine}.", "Thus $\\text{size}(\\coprod_{i \\in I} U_i)) \\leq Bound(\\text{size}(U))$", "by Sets, Lemma \\ref{sets-lemma-bound-size}.", "Hence by construction of the big pro-\\'etale site through", "Sets, Lemma \\ref{sets-lemma-construct-category}", "we see that $\\coprod_{i \\in I} U_i$ is isomorphic to an object", "of our site and the proof is complete." ], "refs": [ "sites-definition-w-contractible", "proetale-lemma-get-many-weakly-contractible", "proetale-lemma-recognize-proetale-covering", "proetale-lemma-recognize-proetale-covering", "proetale-lemma-get-many-weakly-contractible", "proetale-lemma-get-many-weakly-contractible", "sets-lemma-bound-affine", "sets-lemma-bound-size", "sets-lemma-construct-category" ], "ref_ids": [ 8680, 3762, 3743, 3743, 3762, 3762, 8790, 8791, 8789 ] } ], "ref_ids": [] }, { "id": 3765, "type": "theorem", "label": "proetale-lemma-w-contractible-hypercovering", "categories": [ "proetale" ], "title": "proetale-lemma-w-contractible-hypercovering", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item For every object $U$ of $X_\\proetale$ there exists a hypercovering", "$K$ of $U$ in $X_\\proetale$ such that each term $K_n$ consists of a", "single weakly contractible object of $X_\\proetale$ covering $U$.", "\\item For every quasi-compact and quasi-separated object $U$ of $X_\\proetale$", "there exists a hypercovering $K$ of $U$ in $X_\\proetale$ such that each", "term $K_n$ consists of a single affine and weakly contractible object of", "$X_\\proetale$ covering $U$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{B} \\subset \\Ob(X_\\proetale)$ be the set of weakly contractible", "objects of $X_\\proetale$. Every object $T$ of $X_\\proetale$ has a", "covering $\\{T_i \\to T\\}_{i \\in I}$ with $I$ finite and $T_i \\in \\mathcal{B}$ by", "Lemma \\ref{lemma-weakly-contractible-cover}.", "By Hypercoverings, Lemma \\ref{hypercovering-lemma-w-contractible}", "we get a hypercovering $K$ of $U$ such that $K_n = \\{U_{n, i}\\}_{i \\in I_n}$", "with $I_n$ finite and $U_{n, i}$ weakly contractible.", "Then we can replace $K$ by the hypercovering of $U$ given by $\\{U_n\\}$", "in degree $n$ where $U_n = \\coprod_{i \\in I_n} U_{n, i}$", "This is allowed by", "Hypercoverings, Remark \\ref{hypercovering-remark-take-unions-hypercovering-X}.", "\\medskip\\noindent", "Let $X_{qcqs, \\proetale} \\subset X_\\proetale$ be the full subcategory", "consisting of quasi-compact and quasi-separated objects.", "A covering of $X_{qcqs, \\proetale}$ will be a finite pro-\\'etale covering.", "Then $X_{qcqs, \\proetale}$ is a site, has fibre products, and", "the inclusion functor $X_{qcqs, \\proetale} \\to X_\\proetale$ is continuous", "and commutes with fibre products.", "In particular, if $K$ is a hypercovering of an object $U$ in", "$X_{qcqs, \\proetale}$ then $K$ is a hypercovering of $U$ in $X_\\proetale$", "by Hypercoverings, Lemma", "\\ref{hypercovering-lemma-hypercovering-continuous-functor}.", "Let $\\mathcal{B} \\subset \\Ob(X_{qcqs, \\proetale})$ be the set of", "affine and weakly contractible objects. By ", "Lemma \\ref{lemma-get-many-weakly-contractible}", "and the fact that finite unions of affines are affine,", "for every object $U$ of $X_{qcqs, \\proetale}$ there exists a covering", "$\\{V \\to U\\}$ of $X_{qcqs, \\proetale}$ with $V \\in \\mathcal{B}$.", "By Hypercoverings, Lemma \\ref{hypercovering-lemma-w-contractible}", "we get a hypercovering $K$ of $U$ such that $K_n = \\{U_{n, i}\\}_{i \\in I_n}$", "with $I_n$ finite and $U_{n, i}$ affine and weakly contractible.", "Then we can replace $K$ by the hypercovering of $U$", "given by $\\{U_n\\}$ in degree $n$ where", "$U_n = \\coprod_{i \\in I_n} U_{n, i}$. This is allowed by", "Hypercoverings, Remark \\ref{hypercovering-remark-take-unions-hypercovering-X}." ], "refs": [ "proetale-lemma-weakly-contractible-cover", "hypercovering-lemma-w-contractible", "hypercovering-remark-take-unions-hypercovering-X", "hypercovering-lemma-hypercovering-continuous-functor", "proetale-lemma-get-many-weakly-contractible", "hypercovering-lemma-w-contractible", "hypercovering-remark-take-unions-hypercovering-X" ], "ref_ids": [ 3764, 8419, 8434, 8418, 3762, 8419, 8434 ] } ], "ref_ids": [] }, { "id": 3766, "type": "theorem", "label": "proetale-lemma-compute-cohomology", "categories": [ "proetale" ], "title": "proetale-lemma-compute-cohomology", "contents": [ "Let $X$ be a scheme. Let $E \\in D^+(X_\\proetale)$ be represented by", "a bounded below complex $\\mathcal{E}^\\bullet$ of abelian sheaves.", "Let $K$ be a hypercovering of $U \\in \\Ob(X_\\proetale)$ with", "$K_n = \\{U_n \\to U\\}$ where $U_n$ is a weakly contractible object of", "$X_\\proetale$. Then", "$$", "R\\Gamma(U, E) = \\text{Tot}(s(\\mathcal{E}^\\bullet(K)))", "$$", "in $D(\\textit{Ab})$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{E}$ is an abelian", "sheaf on $X_\\proetale$, then the spectral sequence of", "Hypercoverings, Lemma \\ref{hypercovering-lemma-cech-spectral-sequence}", "implies that", "$$", "R\\Gamma(X_\\proetale, \\mathcal{E}) = s(\\mathcal{E}(K))", "$$", "because the higher cohomology groups of any sheaf over $U_n$ vanish, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-w-contractible}.", "\\medskip\\noindent", "If $\\mathcal{E}^\\bullet$ is bounded below, then we can choose an injective", "resolution $\\mathcal{E}^\\bullet \\to \\mathcal{I}^\\bullet$ and consider", "the map of complexes", "$$", "\\text{Tot}(s(\\mathcal{E}^\\bullet(K)))", "\\longrightarrow", "\\text{Tot}(s(\\mathcal{I}^\\bullet(K)))", "$$", "For every $n$ the map $\\mathcal{E}^\\bullet(U_n) \\to \\mathcal{I}^\\bullet(U_n)$", "is a quasi-isomorphism because taking sections over $U_n$ is exact.", "Hence the displayed map is a quasi-isomorphism by one of the spectral", "sequences of", "Homology, Lemma \\ref{homology-lemma-first-quadrant-ss}.", "Using the result of the first paragraph we see that for every $p$", "the complex $s(\\mathcal{I}^p(K))$ is acyclic in degrees $n > 0$ and", "computes $\\mathcal{I}^p(U)$ in degree $0$. Thus the other spectral", "sequence of", "Homology, Lemma \\ref{homology-lemma-first-quadrant-ss}", "shows $\\text{Tot}(s(\\mathcal{I}^\\bullet(K)))$ computes", "$R\\Gamma(U, E) = \\mathcal{I}^\\bullet(U)$." ], "refs": [ "hypercovering-lemma-cech-spectral-sequence", "sites-cohomology-lemma-w-contractible", "homology-lemma-first-quadrant-ss", "homology-lemma-first-quadrant-ss" ], "ref_ids": [ 8398, 4397, 12105, 12105 ] } ], "ref_ids": [] }, { "id": 3767, "type": "theorem", "label": "proetale-lemma-quasi-compact-quasi-separated-commutes-direct-sums", "categories": [ "proetale" ], "title": "proetale-lemma-quasi-compact-quasi-separated-commutes-direct-sums", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "The functor $R\\Gamma(X, -) : D^+(X_\\proetale) \\to D(\\textit{Ab})$", "commutes with direct sums and homotopy colimits." ], "refs": [], "proofs": [ { "contents": [ "The statement means the following: Suppose we have a family of objects", "$E_i$ of $D^+(X_\\proetale)$ such that $\\bigoplus E_i$ is an object", "of $D^+(X_\\proetale)$. Then", "$R\\Gamma(X, \\bigoplus E_i) = \\bigoplus R\\Gamma(X, E_i)$.", "To see this choose a hypercovering $K$ of $X$ with $K_n = \\{U_n \\to X\\}$", "where $U_n$ is an affine and weakly contractible scheme, see", "Lemma \\ref{lemma-w-contractible-hypercovering}.", "Let $N$ be an integer such that $H^p(E_i) = 0$ for $p < N$.", "Choose a complex of abelian sheaves $\\mathcal{E}_i^\\bullet$", "representing $E_i$ with $\\mathcal{E}_i^p = 0$ for $p < N$.", "The termwise direct sum $\\bigoplus \\mathcal{E}_i^\\bullet$ represents", "$\\bigoplus E_i$ in $D(X_\\proetale)$, see", "Injectives, Lemma \\ref{injectives-lemma-derived-products}.", "By Lemma \\ref{lemma-compute-cohomology} we have", "$$", "R\\Gamma(X, \\bigoplus E_i) =", "\\text{Tot}(s((\\bigoplus \\mathcal{E}^\\bullet_i)(K)))", "$$", "and", "$$", "R\\Gamma(X, E_i) = \\text{Tot}(s(\\mathcal{E}^\\bullet_i(K)))", "$$", "Since each $U_n$ is quasi-compact we see that", "$$", "\\text{Tot}(s((\\bigoplus \\mathcal{E}^\\bullet_i)(K))) =", "\\bigoplus \\text{Tot}(s(\\mathcal{E}^\\bullet_i(K)))", "$$", "by Modules on Sites, Lemma", "\\ref{sites-modules-lemma-sections-over-quasi-compact}.", "The statement on homotopy colimits is a formal consequence of the fact", "that $R\\Gamma$ is an exact functor of triangulated categories and the", "fact (just proved) that it commutes with direct sums." ], "refs": [ "proetale-lemma-w-contractible-hypercovering", "injectives-lemma-derived-products", "proetale-lemma-compute-cohomology", "sites-modules-lemma-sections-over-quasi-compact" ], "ref_ids": [ 3765, 7795, 3766, 14215 ] } ], "ref_ids": [] }, { "id": 3768, "type": "theorem", "label": "proetale-lemma-enough-compact-proetale", "categories": [ "proetale" ], "title": "proetale-lemma-enough-compact-proetale", "contents": [ "Let $S$ be a scheme. Let $\\Lambda$ be a ring.", "\\begin{enumerate}", "\\item $D(S_\\proetale)$ is compactly generated,", "\\item $D(S_\\proetale, \\Lambda)$ is compactly generated,", "\\item $D(S_\\proetale, \\mathcal{A})$ is compactly generated", "for any sheaf of rings $\\mathcal{A}$ on $S_\\proetale$,", "\\item $D((\\Sch/S)_\\proetale)$ is compactly generated,", "\\item $D((\\Sch/S)_\\proetale, \\Lambda)$ is compactly generated, and", "\\item $D((\\Sch/S)_\\proetale, \\mathcal{A})$ is compactly generated", "for any sheaf of rings $\\mathcal{A}$ on $(\\Sch/S)_\\proetale$,", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (3). Let $U$ be an affine object of $S_\\proetale$ which is", "weakly contractible. Then $j_{U!}\\mathcal{A}_U$ is a compact object", "of the derived category $D(S_\\proetale, \\mathcal{A})$, see", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-quasi-compact-weakly-contractible-compact}.", "Choose a set $I$ and for each $i \\in I$ an affine weakly contractible", "object $U_i$ of $S_\\proetale$ such that every affine weakly contractible", "object of $S_\\proetale$ is isomorphic to one of the $U_i$. This is possible", "because $\\Ob(S_\\proetale)$ is a set. To finish the proof of (3) it suffices", "to show that $\\bigoplus j_{U_i, !}\\mathcal{A}_{U_i}$ is a generator", "of $D(S_\\proetale, \\mathcal{A})$, see", "Derived Categories, Definition \\ref{derived-definition-generators}.", "To see this, let $K$ be a nonzero object", "of $D(S_\\proetale, \\mathcal{A})$. Then there exists an object $T$", "of our site $S_\\proetale$ and a nonzero element $\\xi$ of $H^n(K)(T)$.", "In other words, $\\xi$ is a nonzero section of the $n$th cohomology", "sheaf of $K$.", "We may assume $K$ is represented by a complex $\\mathcal{K}^\\bullet$", "of sheaves of $\\mathcal{A}$-modules and $\\xi$ is the class of a", "section $s \\in \\mathcal{K}^n(T)$ with $\\text{d}(s) = 0$.", "Namely, $\\xi$ is locally represented as the class of a section", "(so you get the result after replacing $T$ by a member of a covering of $T$).", "Next, we choose a covering $\\{T_j \\to T\\}_{j \\in J}$", "as in Lemma \\ref{lemma-get-many-weakly-contractible}.", "Since $H^n(K)$ is a sheaf, we see that for some $j$ the restriction", "$\\xi|_{T_j}$ remains nonzero. Thus $s|_{T_j}$ defines a nonzero map", "$j_{T_j, !}\\mathcal{A}_{T_j} \\to K$ in $D(S_\\proetale, \\mathcal{A})$.", "Since $T_j \\cong U_i$ for some $i \\in I$ we conclude.", "\\medskip\\noindent", "The exact same argument works for the big pro-\\'etale site of $S$." ], "refs": [ "sites-cohomology-lemma-quasi-compact-weakly-contractible-compact", "derived-definition-generators", "proetale-lemma-get-many-weakly-contractible" ], "ref_ids": [ 4403, 2003, 3762 ] } ], "ref_ids": [] }, { "id": 3769, "type": "theorem", "label": "proetale-lemma-presheaf-value-weakly-contractible", "categories": [ "proetale" ], "title": "proetale-lemma-presheaf-value-weakly-contractible", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{F}$ be a presheaf of sets on $X_\\proetale$", "which sends finite disjoint unions to products. Then", "$\\mathcal{F}^\\#(W) = \\mathcal{F}(W)$ if $W$ is an affine weakly contractible", "object of $X_\\proetale$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\mathcal{F}^\\#$ is equal to $(\\mathcal{F}^+)^+$, see", "Sites, Theorem \\ref{sites-theorem-plus}, where $\\mathcal{F}^+$", "is the presheaf which sends an object $U$ of $X_\\proetale$ to", "$\\colim H^0(\\mathcal{U}, \\mathcal{F})$ where the colimit is", "over all pro-\\'etale coverings $\\mathcal{U}$ of $U$.", "Thus it suffices to prove that (a) $\\mathcal{F}^+$ sends finite", "disjoint unions to products and (b) sends $W$ to $\\mathcal{F}(W)$.", "If $U = U_1 \\amalg U_2$, then given a pro-\\'etale covering", "$\\mathcal{U} = \\{f_j : V_j \\to U\\}$ of $U$ we obtain pro-\\'etale coverings", "$\\mathcal{U}_i = \\{f_j^{-1}(U_i) \\to U_i\\}$ and we clearly have", "$$", "H^0(\\mathcal{U}, \\mathcal{F}) =", "H^0(\\mathcal{U}_1, \\mathcal{F}) \\times", "H^0(\\mathcal{U}_2, \\mathcal{F})", "$$", "because $\\mathcal{F}$ sends finite disjoint unions to products (this includes", "the condition that $\\mathcal{F}$ sends the empty scheme to the singleton).", "This proves (a).", "Finally, any pro-\\'etale covering of $W$ can be refined by a finite disjoint", "union decomposition $W = W_1 \\amalg \\ldots W_n$ by", "Lemma \\ref{lemma-w-contractible-is-weakly-contractible}.", "Hence $\\mathcal{F}^+(W) = \\mathcal{F}(W)$ exactly because the value of", "$\\mathcal{F}$ on $W$ is the product of the values of $\\mathcal{F}$", "on the $W_j$. This proves (b)." ], "refs": [ "sites-theorem-plus", "proetale-lemma-w-contractible-is-weakly-contractible" ], "ref_ids": [ 8492, 3761 ] } ], "ref_ids": [] }, { "id": 3770, "type": "theorem", "label": "proetale-lemma-small-pullback-weakly-contractible", "categories": [ "proetale" ], "title": "proetale-lemma-small-pullback-weakly-contractible", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{F}$ be a sheaf", "of sets on $X_\\proetale$. If $W$ is an affine weakly contractible", "object of $X_\\proetale$, then", "$$", "f_{small}^{-1}\\mathcal{F}(W) = \\colim_{W \\to V} \\mathcal{F}(V)", "$$", "where the colimit is over morphisms $W \\to V$ over $Y$", "with $V \\in Y_\\proetale$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $f_{small}^{-1}\\mathcal{F}$ is the sheaf associated to the", "presheaf", "$$", "u_p\\mathcal{F} : U \\mapsto \\colim_{U \\to V} \\mathcal{F}(V)", "$$", "on $X_\\etale$, see Sites, Sections \\ref{sites-section-morphism-sites} and", "\\ref{sites-section-continuous-functors}; we've surpressed from the notation", "that the colimit is over the opposite of the category", "$\\{U \\to V, V \\in Y_\\proetale\\}$. By", "Lemma \\ref{lemma-presheaf-value-weakly-contractible}", "it suffices to prove that $u_p\\mathcal{F}$ sends finite disjoint unions", "to products.", "Suppose that $U = U_1 \\amalg U_2$ is a disjoint union of open and closed", "subschemes. There is a functor", "$$", "\\{U_1 \\to V_1\\} \\times \\{U_2 \\to V_2\\} \\longrightarrow", "\\{U \\to V\\},\\quad", "(U_1 \\to V_1, U_2 \\to V_2) \\longmapsto (U \\to V_1 \\amalg V_2)", "$$", "which is initial (Categories, Definition \\ref{categories-definition-initial}).", "Hence the corresponding functor on opposite categories is cofinal and", "by Categories, Lemma \\ref{categories-lemma-cofinal} we see", "that $u_p\\mathcal{F}$ on $U$ is the colimit of the values", "$\\mathcal{F}(V_1 \\amalg V_2)$ over the product category.", "Since $\\mathcal{F}$ is a sheaf it sends disjoint unions to products and", "we conclude $u_p\\mathcal{F}$ does too." ], "refs": [ "proetale-lemma-presheaf-value-weakly-contractible", "categories-definition-initial", "categories-lemma-cofinal" ], "ref_ids": [ 3769, 12362, 12217 ] } ], "ref_ids": [] }, { "id": 3771, "type": "theorem", "label": "proetale-lemma-describe-pullback", "categories": [ "proetale" ], "title": "proetale-lemma-describe-pullback", "contents": [ "Let $S$ be a scheme. Consider the morphism", "$$", "\\pi_S : (\\Sch/S)_\\proetale \\longrightarrow S_\\proetale", "$$", "of Lemma \\ref{lemma-at-the-bottom}. Let $\\mathcal{F}$ be a sheaf on", "$S_\\proetale$. Then $\\pi_S^{-1}\\mathcal{F}$ is given by the rule", "$$", "(\\pi_S^{-1}\\mathcal{F})(T) = \\Gamma(T_\\proetale, f_{small}^{-1}\\mathcal{F})", "$$", "where $f : T \\to S$. Moreover, $\\pi_S^{-1}\\mathcal{F}$ satisfies the", "sheaf condition with respect to fpqc coverings." ], "refs": [ "proetale-lemma-at-the-bottom" ], "proofs": [ { "contents": [ "Observe that we have a morphism", "$i_f : \\Sh(T_\\proetale) \\to \\Sh(\\Sch/S)_\\proetale)$", "such that $\\pi_S \\circ i_f = f_{small}$ as morphisms", "$T_\\proetale \\to S_\\proetale$, see Lemma \\ref{lemma-put-in-T}.", "Since pullback is transitive we see that", "$i_f^{-1} \\pi_S^{-1}\\mathcal{F} = f_{small}^{-1}\\mathcal{F}$ as desired.", "\\medskip\\noindent", "Let $\\{g_i : T_i \\to T\\}_{i \\in I}$ be an fpqc covering. The final statement", "means the following: Given a sheaf $\\mathcal{G}$ on $T_\\proetale$ and given", "sections $s_i \\in \\Gamma(T_i, g_{i, small}^{-1}\\mathcal{G})$ whose pullbacks", "to $T_i \\times_T T_j$ agree, there is a unique section $s$ of $\\mathcal{G}$", "over $T$ whose pullback to $T_i$ agrees with $s_i$. We will prove this", "statement when $T$ is affine and the covering is given by a single", "surjective flat morphism $T' \\to T$ of affines and omit the reduction of", "the general case to this case.", "\\medskip\\noindent", "Let $g : T' \\to T$ be a surjective flat morphism of affines and let", "$s' \\in g_{small}^{-1}\\mathcal{G}(T')$ be a section with", "$\\text{pr}_0^*s' = \\text{pr}_1^*s'$ on $T' \\times_T T'$.", "Choose a surjective weakly \\'etale morphism $W \\to T'$ with", "$W$ affine and weakly contractible, see", "Lemma \\ref{lemma-weakly-contractible-cover}. By", "Lemma \\ref{lemma-small-pullback-weakly-contractible}", "the restriction $s'|_W$ is an element of $\\colim_{W \\to U} \\mathcal{G}(U)$.", "Choose $\\phi : W \\to U_0$ and $s_0 \\in \\mathcal{G}(U_0)$ corresponding to $s'$.", "Choose a surjective weakly \\'etale morphism $V \\to W \\times_T W$", "with $V$ affine and weakly contractible.", "Denote $a, b : V \\to W$ the induced morphisms.", "Since $a^*(s'|_W) = b^*(s'|_W)$ and since the category", "$\\{V \\to U, U \\in T_\\proetale\\}$ is cofiltered", "(this is clear but see", "Sites, Lemma \\ref{sites-lemma-directed-morphism} if in doubt),", "we see that the two morphisms $\\phi \\circ a , \\phi \\circ b : V \\to U_0$", "have to be equal. By the results in", "Descent, Section \\ref{descent-section-fpqc-universal-effective-epimorphisms}", "(especially", "Descent, Lemma \\ref{descent-lemma-fpqc-universal-effective-epimorphisms})", "it follows there is a unique morphism $T \\to U_0$ such that $\\phi$", "is the composition of this morphism with the structure morphism $W \\to T$", "(small detail omitted). Then we can let $s$ be the pullback", "of $s_0$ by this morphism. We omit the verification that", "$s$ pulls back to $s'$ on $T'$." ], "refs": [ "proetale-lemma-put-in-T", "proetale-lemma-weakly-contractible-cover", "proetale-lemma-small-pullback-weakly-contractible", "sites-lemma-directed-morphism", "descent-lemma-fpqc-universal-effective-epimorphisms" ], "ref_ids": [ 3750, 3764, 3770, 8526, 14638 ] } ], "ref_ids": [ 3751 ] }, { "id": 3772, "type": "theorem", "label": "proetale-lemma-compare-injectives", "categories": [ "proetale" ], "title": "proetale-lemma-compare-injectives", "contents": [ "Let $S$ be a scheme. Let $T$ be an object of $(\\Sch/S)_\\proetale$.", "\\begin{enumerate}", "\\item If $\\mathcal{I}$ is injective in $\\textit{Ab}((\\Sch/S)_\\proetale)$, then", "\\begin{enumerate}", "\\item $i_f^{-1}\\mathcal{I}$ is injective in $\\textit{Ab}(T_\\proetale)$,", "\\item $\\mathcal{I}|_{S_\\proetale}$ is injective in $\\textit{Ab}(S_\\proetale)$,", "\\end{enumerate}", "\\item If $\\mathcal{I}^\\bullet$ is a K-injective complex", "in $\\textit{Ab}((\\Sch/S)_\\proetale)$, then", "\\begin{enumerate}", "\\item $i_f^{-1}\\mathcal{I}^\\bullet$ is a K-injective complex in", "$\\textit{Ab}(T_\\proetale)$,", "\\item $\\mathcal{I}^\\bullet|_{S_\\proetale}$ is a K-injective complex in", "$\\textit{Ab}(S_\\proetale)$,", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1)(a) and (2)(a): $i_f^{-1}$ is a right adjoint of an", "exact functor $i_{f, !}$. Namely, recall that $i_f$ corresponds", "to a cocontinuous functor $u : T_\\proetale \\to (\\Sch/S)_\\proetale$", "which is continuous and commutes with fibre products and equalizers, see", "Lemma \\ref{lemma-put-in-T} and its proof. Hence we obtain $i_{f, !}$ by", "Modules on Sites, Lemma \\ref{sites-modules-lemma-g-shriek-adjoint}.", "It is shown in Modules on Sites, Lemma", "\\ref{sites-modules-lemma-exactness-lower-shriek} that it is exact.", "Then we conclude (1)(a) and (2)(a) hold by", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives} and", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}.", "\\medskip\\noindent", "Parts (1)(b) and (2)(b) are special cases of (1)(a) and (2)(a)", "as $i_S = i_{\\text{id}_S}$." ], "refs": [ "proetale-lemma-put-in-T", "sites-modules-lemma-g-shriek-adjoint", "sites-modules-lemma-exactness-lower-shriek", "homology-lemma-adjoint-preserve-injectives", "derived-lemma-adjoint-preserve-K-injectives" ], "ref_ids": [ 3750, 14164, 14165, 12116, 1915 ] } ], "ref_ids": [] }, { "id": 3773, "type": "theorem", "label": "proetale-lemma-compare-higher-direct-image", "categories": [ "proetale" ], "title": "proetale-lemma-compare-higher-direct-image", "contents": [ "Let $f : T \\to S$ be a morphism of schemes.", "For $K$ in $D((\\Sch/T)_\\proetale)$ we have", "$$", "(Rf_{big, *}K)|_{S_\\proetale} = Rf_{small, *}(K|_{T_\\proetale})", "$$", "in $D(S_\\proetale)$. More generally, let $S' \\in \\Ob((\\Sch/S)_\\proetale)$", "with structure morphism $g : S' \\to S$. Consider the fibre product", "$$", "\\xymatrix{", "T' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "Then for $K$ in $D((\\Sch/T)_\\proetale)$ we have", "$$", "i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)", "$$", "in $D(S'_\\proetale)$ and", "$$", "g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)", "$$", "in $D((\\Sch/S')_\\proetale)$." ], "refs": [], "proofs": [ { "contents": [ "The first equality follows from Lemma \\ref{lemma-compare-injectives}", "and (\\ref{equation-compare-big-small})", "on choosing a K-injective complex of abelian sheaves representing $K$.", "The second equality follows from Lemma \\ref{lemma-compare-injectives}", "and Lemma", "\\ref{lemma-morphism-big-small-cartesian-diagram}", "on choosing a K-injective complex of abelian sheaves representing $K$.", "The third equality follows similarly from", "Cohomology on Sites, Lemmas \\ref{sites-cohomology-lemma-cohomology-of-open} and", "\\ref{sites-cohomology-lemma-restrict-K-injective-to-open}", "and Lemma \\ref{lemma-morphism-big-small-cartesian-diagram}", "on choosing a K-injective complex of abelian sheaves representing $K$." ], "refs": [ "proetale-lemma-compare-injectives", "proetale-lemma-compare-injectives", "proetale-lemma-morphism-big-small-cartesian-diagram", "sites-cohomology-lemma-cohomology-of-open", "sites-cohomology-lemma-restrict-K-injective-to-open", "proetale-lemma-morphism-big-small-cartesian-diagram" ], "ref_ids": [ 3772, 3772, 3755, 4186, 4253, 3755 ] } ], "ref_ids": [] }, { "id": 3774, "type": "theorem", "label": "proetale-lemma-compare-cohomology-big-small", "categories": [ "proetale" ], "title": "proetale-lemma-compare-cohomology-big-small", "contents": [ "Let $f : T \\to S$ be a morphism of schemes. For $K$ in $D(S_\\proetale)$", "we have", "$$", "H^n_\\proetale(S, \\pi_S^{-1}K) = H^n(S_\\proetale, K)", "$$", "and", "$$", "H^n_\\proetale(T, \\pi_S^{-1}K) = H^n(T_\\proetale, f_{small}^{-1}K).", "$$", "For $M$ in $D((\\Sch/S)_\\proetale)$ we have", "$$", "H^n_\\proetale(T, M) = H^n(T_\\proetale, i_f^{-1}M).", "$$" ], "refs": [], "proofs": [ { "contents": [ "To prove the last equality represent $M$", "by a K-injective complex of abelian sheaves", "and apply Lemma \\ref{lemma-compare-injectives}", "and work out the definitions. The second equality follows from", "this as $i_f^{-1} \\circ \\pi_S^{-1} = f_{small}^{-1}$. The first", "equality is a special", "case of the second one." ], "refs": [ "proetale-lemma-compare-injectives" ], "ref_ids": [ 3772 ] } ], "ref_ids": [] }, { "id": 3775, "type": "theorem", "label": "proetale-lemma-cohomological-descent-proetale", "categories": [ "proetale" ], "title": "proetale-lemma-cohomological-descent-proetale", "contents": [ "Let $S$ be a scheme. For $K \\in D(S_\\proetale)$ the map", "$$", "K \\longrightarrow R\\pi_{S, *}\\pi_S^{-1}K", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is true because both $\\pi_S^{-1}$ and $\\pi_{S, *} = i_S^{-1}$", "are exact functors and the composition $\\pi_{S, *} \\circ \\pi_S^{-1}$", "is the identity functor." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3776, "type": "theorem", "label": "proetale-lemma-points-proetale", "categories": [ "proetale" ], "title": "proetale-lemma-points-proetale", "contents": [ "Let $S$ be a scheme. The pro-\\'etale sites $\\Sch_\\proetale$,", "$S_\\proetale$, $(\\Sch/S)_\\proetale$, $S_{affine, \\proetale}$, and", "$(\\textit{Aff}/S)_\\proetale$ have enough points." ], "refs": [], "proofs": [ { "contents": [ "The big pro-\\'etale topos of $S$ is equivalent to the topos defined by", "$(\\textit{Aff}/S)_\\proetale$, see", "Lemma \\ref{lemma-affine-big-site-proetale}.", "The topos of sheaves on $S_\\proetale$ is equivalent to the topos", "associated to $S_{affine, \\proetale}$, see", "Lemma \\ref{lemma-alternative}.", "The result for the sites $(\\textit{Aff}/S)_\\proetale$ and", "$S_{affine, \\proetale}$ follows immediately from Deligne's result", "Sites, Lemma \\ref{sites-lemma-criterion-points}.", "The case $\\Sch_\\proetale$ is handled because it is equal to", "$(\\Sch/\\Spec(\\mathbf{Z}))_\\proetale$." ], "refs": [ "proetale-lemma-affine-big-site-proetale", "proetale-lemma-alternative", "sites-lemma-criterion-points" ], "ref_ids": [ 3749, 3757, 8616 ] } ], "ref_ids": [] }, { "id": 3777, "type": "theorem", "label": "proetale-lemma-cofinal-etale", "categories": [ "proetale" ], "title": "proetale-lemma-cofinal-etale", "contents": [ "Let $S$ be a scheme and let $\\overline{s} : \\Spec(k) \\to S$ be a", "geometric point. The category of pro-\\'etale neighbourhoods of", "$\\overline{s}$ is cofiltered." ], "refs": [], "proofs": [ { "contents": [ "The proof is identitical to the proof of", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-cofinal-etale}", "but using the corresponding facts about weakly \\'etale morphisms", "proven in", "More on Morphisms, Lemmas", "\\ref{more-morphisms-lemma-composition-weakly-etale},", "\\ref{more-morphisms-lemma-base-change-weakly-etale}, and", "\\ref{more-morphisms-lemma-weakly-etale-permanence}." ], "refs": [ "etale-cohomology-lemma-cofinal-etale", "more-morphisms-lemma-composition-weakly-etale", "more-morphisms-lemma-base-change-weakly-etale", "more-morphisms-lemma-weakly-etale-permanence" ], "ref_ids": [ 6422, 14025, 14026, 14033 ] } ], "ref_ids": [] }, { "id": 3778, "type": "theorem", "label": "proetale-lemma-geometric-lift-to-cover", "categories": [ "proetale" ], "title": "proetale-lemma-geometric-lift-to-cover", "contents": [ "Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point of $S$.", "Let $\\mathcal{U} = \\{\\varphi_i : S_i \\to S\\}_{i\\in I}$ be a", "pro-\\'etale covering. Then there exist $i \\in I$ and geometric", "point $\\overline{s}_i$ of $S_i$ mapping to $\\overline{s}$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the fact that $\\coprod \\varphi_i$ is surjective", "and that residue field extensions induced by weakly \\'etale", "morphisms are separable algebraic (see for example", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-weakly-etale-strictly-henselian-local-rings}." ], "refs": [ "more-morphisms-lemma-weakly-etale-strictly-henselian-local-rings" ], "ref_ids": [ 14031 ] } ], "ref_ids": [] }, { "id": 3779, "type": "theorem", "label": "proetale-lemma-classical-point", "categories": [ "proetale" ], "title": "proetale-lemma-classical-point", "contents": [ "In the situation above the scheme $\\Spec(\\mathcal{O}_{S, \\overline{s}}^{sh})$", "is an object of $X_\\proetale$ and there is a canonical isomorphism", "$$", "\\mathcal{F}(\\Spec(\\mathcal{O}_{S, \\overline{s}}^{sh})) =", "\\mathcal{F}_{\\overline{s}}", "$$", "functorial in $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "The first statement is clear from the construction of the strict henselization", "as a filtered colimit of \\'etale algebras over $S$, or by the characterization", "of weakly \\'etale morphisms of", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-weakly-etale-strictly-henselian-local-rings}.", "The second statement follows as by Olivier's theorem", "(More on Algebra, Theorem \\ref{more-algebra-theorem-olivier})", "the scheme $\\Spec(\\mathcal{O}_{S, \\overline{s}}^{sh})$", "is an initial object of the category of pro-\\'etale neighbourhoods", "of $\\overline{s}$." ], "refs": [ "more-morphisms-lemma-weakly-etale-strictly-henselian-local-rings", "more-algebra-theorem-olivier" ], "ref_ids": [ 14031, 9805 ] } ], "ref_ids": [] }, { "id": 3780, "type": "theorem", "label": "proetale-lemma-limit-pullback", "categories": [ "proetale" ], "title": "proetale-lemma-limit-pullback", "contents": [ "Let $X$ be a scheme. Let $Y = \\lim Y_i$ be the limit of a directed inverse", "system of quasi-compact and quasi-separated objects of $X_\\proetale$", "with affine transition morphisms. For any sheaf $\\mathcal{F}$", "on $X_\\etale$ we have", "$\\epsilon^{-1}\\mathcal{F}(Y) = \\colim \\mathcal{F}(Y_i)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F} = h_U$ be a representable sheaf on $X_\\etale$", "with $U$ an object of $X_\\etale$. In this case", "$\\epsilon^{-1}h_U = h_{u(U)}$ where $u(U)$ is $U$ viewed as an object of", "$X_\\proetale$ (Sites, Lemma \\ref{sites-lemma-pullback-representable-sheaf}).", "Then", "\\begin{align*}", "(\\epsilon^{-1}h_U)(Y)", "& = h_{u(U)}(Y) \\\\", "& = \\Mor_X(Y, U) \\\\", "& = \\colim \\Mor_X(Y_i, U) \\\\", "& = \\colim h_U(Y_i)", "\\end{align*}", "Here the only nonformal equality is the $3$rd which holds", "by Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}.", "Hence the lemma holds for every representable sheaf. Since every sheaf", "is a coequalizer of a map of coproducts of representable sheaves", "(Sites, Lemma \\ref{sites-lemma-sheaf-coequalizer-representable})", "we obtain the result in general." ], "refs": [ "sites-lemma-pullback-representable-sheaf", "limits-proposition-characterize-locally-finite-presentation", "sites-lemma-sheaf-coequalizer-representable" ], "ref_ids": [ 8524, 15127, 8520 ] } ], "ref_ids": [] }, { "id": 3781, "type": "theorem", "label": "proetale-lemma-fully-faithful", "categories": [ "proetale" ], "title": "proetale-lemma-fully-faithful", "contents": [ "Let $X$ be a scheme. For every sheaf $\\mathcal{F}$ on $X_\\etale$", "the adjunction map $\\mathcal{F} \\to \\epsilon_*\\epsilon^{-1}\\mathcal{F}$ is an", "isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $U$ is a quasi-compact and quasi-separated scheme \\'etale", "over $X$. Then", "$$", "\\epsilon_*\\epsilon^{-1}\\mathcal{F}(U) =", "\\epsilon^{-1}\\mathcal{F}(U) =", "\\mathcal{F}(U)", "$$", "the second equality by (a special case of) Lemma \\ref{lemma-limit-pullback}.", "Since every object of $X_\\etale$", "has a covering by quasi-compact and quasi-separated objects we conclude." ], "refs": [ "proetale-lemma-limit-pullback" ], "ref_ids": [ 3780 ] } ], "ref_ids": [] }, { "id": 3782, "type": "theorem", "label": "proetale-lemma-affine-vanishing", "categories": [ "proetale" ], "title": "proetale-lemma-affine-vanishing", "contents": [ "Let $X$ be an affine scheme. For injective abelian sheaf $\\mathcal{I}$ on", "$X_\\etale$ we have $H^p(X_\\proetale, \\epsilon^{-1}\\mathcal{I}) = 0$", "for $p > 0$." ], "refs": [], "proofs": [ { "contents": [ "We are going to use", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cech-vanish-collection}", "to prove this. Let $\\mathcal{B} \\subset \\Ob(X_\\proetale)$ be the set of affine", "objects $U$ of $X_\\proetale$ such that $\\mathcal{O}(X) \\to \\mathcal{O}(U)$ is", "ind-\\'etale. Let $\\text{Cov}$ be the set of pro-\\'etale coverings", "$\\{U_i \\to U\\}_{i = 1, \\ldots, n}$ with $U \\in \\mathcal{B}$ such that", "$\\mathcal{O}(U) \\to \\mathcal{O}(U_i)$ is ind-\\'etale for $i = 1, \\ldots, n$.", "Properties (1) and (2) of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cech-vanish-collection}", "hold for $\\mathcal{B}$ and $\\text{Cov}$ by", "Lemmas \\ref{lemma-composition-ind-etale},", "\\ref{lemma-base-change-ind-etale}, and", "\\ref{lemma-proetale-affine} and", "Proposition \\ref{proposition-weakly-etale}.", "\\medskip\\noindent", "To check condition (3) suppose that", "$\\mathcal{U} = \\{U_i \\to U\\}_{i = 1, \\ldots, n}$", "is an element of $\\text{Cov}$. We have to show that the higher", "Cech cohomology groups of $\\epsilon^{-1}\\mathcal{I}$", "with respect to $\\mathcal{U}$ are zero.", "First we write $U_i = \\lim_{a \\in A_i} U_{i, a}$ as a directed inverse limit", "with $U_{i, a} \\to U$ \\'etale and $U_{i, a}$ affine.", "We think of $A_1 \\times \\ldots \\times A_n$ as a direct set", "with ordering $(a_1, \\ldots, a_n) \\geq (a_1', \\ldots, a_n')$", "if and only if $a_i \\geq a_i'$ for $i = 1, \\ldots, n$.", "Observe that", "$\\mathcal{U}_{(a_1, \\ldots, a_n)} = \\{U_{i, a_i} \\to U\\}_{i = 1, \\ldots, n}$", "is an \\'etale covering for all", "$a_1, \\ldots, a_n \\in A_1 \\times \\ldots \\times A_n$.", "Observe that", "$$", "U_{i_0} \\times_U U_{i_1} \\times_U \\ldots \\times_U U_{i_p}", "=", "\\lim_{(a_1, \\ldots, a_n) \\in A_1 \\times \\ldots \\times A_n}", "U_{i_0, a_{i_0}} \\times_U", "U_{i_1, a_{i_1}} \\times_U \\ldots \\times_U U_{i_p, a_{i_p}}", "$$", "for all $i_0, \\ldots, i_p \\in \\{1, \\ldots, n\\}$ because limits", "commute with fibred products. Hence by Lemma \\ref{lemma-limit-pullback}", "and exactness of filtered colimits we have", "$$", "\\check{H}^p(\\mathcal{U}, \\epsilon^{-1}\\mathcal{I}) =", "\\colim \\check{H}^p(\\mathcal{U}_{(a_1, \\ldots, a_n)}, \\epsilon^{-1}\\mathcal{I}) ", "$$", "Thus it suffices to prove the vanishing for \\'etale coverings", "of $U$!", "\\medskip\\noindent", "Let $\\mathcal{U} = \\{U_i \\to U\\}_{i = 1, \\ldots, n}$ be an \\'etale covering", "with $U_i$ affine. Write $U = \\lim_{b \\in B} U_b$ as a directed inverse limit", "with $U_b$ affine and $U_b \\to X$ \\'etale.", "By Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation},", "\\ref{limits-lemma-limit-affine}, and", "\\ref{limits-lemma-descend-etale}", "we can choose a $b_0 \\in B$ such that for $i = 1, \\ldots, n$ there is an", "\\'etale morphism $U_{i, b_0} \\to U_{b_0}$ of affines", "such that $U_i = U \\times_{U_{b_0}} U_{i, b_0}$.", "Set $U_{i, b} = U_b \\times_{U_{b_0}} U_{i, b_0}$ for $b \\geq b_0$.", "For $b$ large enough the family", "$\\mathcal{U}_b = \\{U_{i, b} \\to U_b\\}_{i = 1, \\ldots, n}$", "is an \\'etale covering, see", "Limits, Lemma \\ref{limits-lemma-descend-surjective}.", "Exactly as before we find that", "$$", "\\check{H}^p(\\mathcal{U}, \\epsilon^{-1}\\mathcal{I}) =", "\\colim \\check{H}^p(\\mathcal{U}_b, \\epsilon^{-1}\\mathcal{I}) =", "\\colim \\check{H}^p(\\mathcal{U}_b, \\mathcal{I})", "$$", "the final equality by Lemma \\ref{lemma-fully-faithful}.", "Since each of the {\\v C}ech complexes on the right hand side is", "acyclic in positive degrees", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-injective-trivial-cech})", "it follows that the one on the left is too. This proves condition (3) of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cech-vanish-collection}.", "Since $X \\in \\mathcal{B}$ the lemma follows." ], "refs": [ "sites-cohomology-lemma-cech-vanish-collection", "sites-cohomology-lemma-cech-vanish-collection", "proetale-lemma-composition-ind-etale", "proetale-lemma-base-change-ind-etale", "proetale-lemma-proetale-affine", "proetale-proposition-weakly-etale", "proetale-lemma-limit-pullback", "limits-lemma-descend-finite-presentation", "limits-lemma-limit-affine", "limits-lemma-descend-etale", "limits-lemma-descend-surjective", "proetale-lemma-fully-faithful", "sites-cohomology-lemma-injective-trivial-cech", "sites-cohomology-lemma-cech-vanish-collection" ], "ref_ids": [ 4205, 4205, 3726, 3725, 3746, 3826, 3780, 15077, 15043, 15065, 15069, 3781, 4198, 4205 ] } ], "ref_ids": [] }, { "id": 3783, "type": "theorem", "label": "proetale-lemma-relative-comparison", "categories": [ "proetale" ], "title": "proetale-lemma-relative-comparison", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item For an abelian sheaf $\\mathcal{F}$ on $X_\\etale$", "we have $R\\epsilon_*(\\epsilon^{-1}\\mathcal{F}) = \\mathcal{F}$.", "\\item For $K \\in D^+(X_\\etale)$ the map $K \\to R\\epsilon_*\\epsilon^{-1}K$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I}$ be an injective abelian sheaf on $X_\\etale$.", "Recall that $R^q\\epsilon_*(\\epsilon^{-1}\\mathcal{I})$ is the sheaf associated", "to $U \\mapsto H^q(U_\\proetale, \\epsilon^{-1}\\mathcal{I})$, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-higher-direct-images}.", "By Lemma \\ref{lemma-affine-vanishing} we see that this is zero for $q > 0$", "and $U$ affine and \\'etale over $X$. Since every object of", "$X_\\etale$ has a covering by affine objects, it follows that", "$R^q\\epsilon_*(\\epsilon^{-1}\\mathcal{I}) = 0$ for $q > 0$.", "\\medskip\\noindent", "Let $K \\in D^+(X_\\etale)$. Choose a bounded below complex $\\mathcal{I}^\\bullet$", "of injective abelian sheaves on $X_\\etale$ representing $K$.", "Then $\\epsilon^{-1}K$ is represented by $\\epsilon^{-1}\\mathcal{I}^\\bullet$.", "By Leray's acyclicity lemma (Derived Categories, Lemma", "\\ref{derived-lemma-leray-acyclicity}) we see that $R\\epsilon_*\\epsilon^{-1}K$", "is represented by $\\epsilon_*\\epsilon^{-1}\\mathcal{I}^\\bullet$.", "By Lemma \\ref{lemma-fully-faithful} we conclude that", "$R\\epsilon_*\\epsilon^{-1}\\mathcal{I}^\\bullet = \\mathcal{I}^\\bullet$", "and the proof of (2) is complete. Part (1) is a special case of (2)." ], "refs": [ "sites-cohomology-lemma-higher-direct-images", "proetale-lemma-affine-vanishing", "derived-lemma-leray-acyclicity", "proetale-lemma-fully-faithful" ], "ref_ids": [ 4189, 3782, 1844, 3781 ] } ], "ref_ids": [] }, { "id": 3784, "type": "theorem", "label": "proetale-lemma-compare-cohomology", "categories": [ "proetale" ], "title": "proetale-lemma-compare-cohomology", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item For an abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have", "$$", "H^i(X_\\etale, \\mathcal{F}) =", "H^i(X_\\proetale, \\epsilon^{-1}\\mathcal{F})", "$$", "for all $i$.", "\\item For $K \\in D^+(X_\\etale)$ we have", "$$", "R\\Gamma(X_\\etale, K) = R\\Gamma(X_\\proetale, \\epsilon^{-1}K)", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of Lemma \\ref{lemma-relative-comparison} and", "the Leray spectral sequence (Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-apply-Leray})." ], "refs": [ "proetale-lemma-relative-comparison", "sites-cohomology-lemma-apply-Leray" ], "ref_ids": [ 3783, 4221 ] } ], "ref_ids": [] }, { "id": 3785, "type": "theorem", "label": "proetale-lemma-compare-cohomology-nonabelian", "categories": [ "proetale" ], "title": "proetale-lemma-compare-cohomology-nonabelian", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{G}$ be a sheaf of (possibly", "noncommutative) groups on $X_\\etale$. We have", "$$", "H^1(X_\\etale, \\mathcal{G}) =", "H^1(X_\\proetale, \\epsilon^{-1}\\mathcal{G})", "$$", "where $H^1$ is defined as the set of isomorphism classes of", "torsors (see", "Cohomology on Sites, Section \\ref{sites-cohomology-section-h1-torsors})." ], "refs": [], "proofs": [ { "contents": [ "Since the functor $\\epsilon^{-1}$ is fully faithful by", "Lemma \\ref{lemma-fully-faithful}", "it is clear that the map", "$H^1(X_\\etale, \\mathcal{G}) \\to H^1(X_\\proetale, \\epsilon^{-1}\\mathcal{G})$", "is injective. To show surjectivity it suffices to show that", "any $\\epsilon^{-1}\\mathcal{G}$-torsor $\\mathcal{F}$ is \\'etale", "locally trivial. To do this we may assume that $X$ is affine.", "Thus we reduce to proving surjectivity for $X$ affine.", "\\medskip\\noindent", "Choose a covering $\\{U \\to X\\}$ with (a) $U$ affine, (b)", "$\\mathcal{O}(X) \\to \\mathcal{O}(U)$ ind-\\'etale, and (c) $\\mathcal{F}(U)$", "nonempty. We can do this by Proposition \\ref{proposition-weakly-etale}", "and the fact that", "standard pro-\\'etale coverings of $X$ are cofinal among all pro-\\'etale", "coverings of $X$ (Lemma \\ref{lemma-proetale-affine}).", "Write $U = \\lim U_i$ as a limit of affine schemes \\'etale over $X$.", "Pick $s \\in \\mathcal{F}(U)$. Let", "$g \\in \\epsilon^{-1}\\mathcal{G}(U \\times_X U)$", "be the unique section such that $g \\cdot \\text{pr}_1^*s = \\text{pr}_2^*s$ in", "$\\mathcal{F}(U \\times_X U)$. Then $g$ satisfies the cocycle condition", "$$", "\\text{pr}_{12}^*g \\cdot \\text{pr}_{23}^*g = \\text{pr}_{13}^*g", "$$", "in $\\epsilon^{-1}\\mathcal{G}(U \\times_X U \\times_X U)$. By", "Lemma \\ref{lemma-limit-pullback}", "we have", "$$", "\\epsilon^{-1}\\mathcal{G}(U \\times_X U) =", "\\colim \\mathcal{G}(U_i \\times_X U_i)", "$$", "and", "$$", "\\epsilon^{-1}\\mathcal{G}(U \\times_X U \\times_X U) =", "\\colim \\mathcal{G}(U_i \\times_X U_i \\times_X U_i)", "$$", "hence we can find an $i$ and an element $g_i \\in \\mathcal{G}(U_i)$", "mapping to $g$ satisfying the cocycle condition.", "The cocycle $g_i$ then defines a torsor for $\\mathcal{G}$ on", "$X_\\etale$ whose pullback is isomorphic to $\\mathcal{F}$", "by construction. Some details omitted (namely, the relationship", "between torsors and 1-cocycles which should be added to the chapter", "on cohomology on sites)." ], "refs": [ "proetale-lemma-fully-faithful", "proetale-proposition-weakly-etale", "proetale-lemma-proetale-affine", "proetale-lemma-limit-pullback" ], "ref_ids": [ 3781, 3826, 3746, 3780 ] } ], "ref_ids": [] }, { "id": 3786, "type": "theorem", "label": "proetale-lemma-compare-derived", "categories": [ "proetale" ], "title": "proetale-lemma-compare-derived", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a ring.", "\\begin{enumerate}", "\\item The essential image of the fully faithful functor", "$\\epsilon^{-1} : \\textit{Mod}(X_\\etale, \\Lambda) \\to", "\\textit{Mod}(X_\\proetale, \\Lambda)$", "is a weak Serre subcategory $\\mathcal{C}$.", "\\item The functor $\\epsilon^{-1}$ defines an equivalence of categories", "of $D^+(X_\\etale, \\Lambda)$ with $D^+_\\mathcal{C}(X_\\proetale, \\Lambda)$", "with question inverse given by $R\\epsilon_*$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove (1) we will prove conditions (1) -- (4) of", "Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}.", "Since $\\epsilon^{-1}$ is fully faithful (Lemma \\ref{lemma-fully-faithful})", "and exact, everything is clear except for condition (4).", "However, if", "$$", "0 \\to \\epsilon^{-1}\\mathcal{F}_1 \\to \\mathcal{G} \\to", "\\epsilon^{-1}\\mathcal{F}_2 \\to 0", "$$", "is a short exact sequence of sheaves of $\\Lambda$-modules on $X_\\proetale$,", "then we get", "$$", "0 \\to \\epsilon_*\\epsilon^{-1}\\mathcal{F}_1 \\to \\epsilon_*\\mathcal{G} \\to", "\\epsilon_*\\epsilon^{-1}\\mathcal{F}_2 \\to", "R^1\\epsilon_*\\epsilon^{-1}\\mathcal{F}_1", "$$", "which by Lemma \\ref{lemma-relative-comparison}", "is the same as a short exact sequence", "$$", "0 \\to \\mathcal{F}_1 \\to \\epsilon_*\\mathcal{G} \\to \\mathcal{F}_2 \\to 0", "$$", "Pulling pack we find that $\\mathcal{G} = \\epsilon^{-1}\\epsilon_*\\mathcal{G}$.", "This proves (1).", "\\medskip\\noindent", "Part (2) follows from part (1) and Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-equivalence-bounded}." ], "refs": [ "homology-lemma-characterize-weak-serre-subcategory", "proetale-lemma-fully-faithful", "proetale-lemma-relative-comparison", "sites-cohomology-lemma-equivalence-bounded" ], "ref_ids": [ 12046, 3781, 3783, 4290 ] } ], "ref_ids": [] }, { "id": 3787, "type": "theorem", "label": "proetale-lemma-compare-locally-constant", "categories": [ "proetale" ], "title": "proetale-lemma-compare-locally-constant", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a ring.", "The functor $\\epsilon^{-1}$ defines an equivalence of categories", "$$", "\\left\\{", "\\begin{matrix}", "\\text{locally constant sheaves}\\\\", "\\text{of }\\Lambda\\text{-modules on }X_\\etale\\\\", "\\text{of finite presentation}", "\\end{matrix}", "\\right\\}", "\\longleftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{locally constant sheaves}\\\\", "\\text{of }\\Lambda\\text{-modules on }X_\\proetale\\\\", "\\text{of finite presentation}", "\\end{matrix}", "\\right\\}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a locally constant sheaf of $\\Lambda$-modules", "on $X_\\proetale$ of finite presentation. Choose a pro-\\'etale covering", "$\\{U_i \\to X\\}$ such that $\\mathcal{F}|_{U_i}$ is constant, say", "$\\mathcal{F}|_{U_i} \\cong \\underline{M_i}_{U_i}$.", "Observe that $U_i \\times_X U_j$ is empty if $M_i$ is not isomorphic", "to $M_j$.", "For each $\\Lambda$-module $M$ let $I_M = \\{i \\in I \\mid M_i \\cong M\\}$.", "As pro-\\'etale coverings are fpqc coverings and by", "Descent, Lemma \\ref{descent-lemma-open-fpqc-covering}", "we see that $U_M = \\bigcup_{i \\in I_M} \\Im(U_i \\to X)$", "is an open subset of $X$. Then $X = \\coprod U_M$ is a disjoint", "open covering of $X$. We may replace $X$ by $U_M$ for some $M$ and", "assume that $M_i = M$ for all $i$.", "\\medskip\\noindent", "Consider the sheaf $\\mathcal{I} = \\mathit{Isom}(\\underline{M}, \\mathcal{F})$.", "This sheaf is a torsor for", "$\\mathcal{G} = \\mathit{Isom}(\\underline{M}, \\underline{M})$.", "By Modules on Sites, Lemma \\ref{sites-modules-lemma-locally-constant}", "we have $\\mathcal{G} = \\underline{G}$", "where $G = \\mathit{Isom}_\\Lambda(M, M)$.", "Since torsors for the \\'etale topology", "and the pro-\\'etale topology agree by", "Lemma \\ref{lemma-compare-cohomology-nonabelian}", "it follows that $\\mathcal{I}$ has sections \\'etale locally on $X$.", "Thus $\\mathcal{F}$ is \\'etale locally a constant sheaf which is", "what we had to show." ], "refs": [ "descent-lemma-open-fpqc-covering", "sites-modules-lemma-locally-constant", "proetale-lemma-compare-cohomology-nonabelian" ], "ref_ids": [ 14637, 14272, 3785 ] } ], "ref_ids": [] }, { "id": 3788, "type": "theorem", "label": "proetale-lemma-compare-locally-constant-derived", "categories": [ "proetale" ], "title": "proetale-lemma-compare-locally-constant-derived", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.", "Let $D_{flc}(X_\\etale, \\Lambda)$, resp.\\ $D_{flc}(X_\\proetale, \\Lambda)$", "be the full subcategory of", "$D(X_\\etale, \\Lambda)$, resp.\\ $D(X_\\proetale, \\Lambda)$", "consisting of those complexes whose cohomology sheaves are locally", "constant sheaves of $\\Lambda$-modules of finite type. Then", "$$", "\\epsilon^{-1} :", "D_{flc}^+(X_\\etale, \\Lambda)", "\\longrightarrow", "D_{flc}^+(X_\\proetale, \\Lambda)", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "The categories $D_{flc}(X_\\etale, \\Lambda)$ and $D_{flc}(X_\\proetale, \\Lambda)$", "are strictly full, saturated, triangulated subcategories of", "$D(X_\\etale, \\Lambda)$ and $D(X_\\proetale, \\Lambda)$", "by Modules on Sites, Lemma", "\\ref{sites-modules-lemma-kernel-finite-locally-constant}", "and", "Derived Categories, Section \\ref{derived-section-triangulated-sub}", "The statement of the lemma follows by combining", "Lemmas \\ref{lemma-compare-derived} and", "\\ref{lemma-compare-locally-constant}." ], "refs": [ "sites-modules-lemma-kernel-finite-locally-constant", "proetale-lemma-compare-derived", "proetale-lemma-compare-locally-constant" ], "ref_ids": [ 14273, 3786, 3787 ] } ], "ref_ids": [] }, { "id": 3789, "type": "theorem", "label": "proetale-lemma-compare-truncations", "categories": [ "proetale" ], "title": "proetale-lemma-compare-truncations", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.", "Let $K$ be an object of $D(X_\\proetale, \\Lambda)$.", "Set $K_n = K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$.", "If $K_1$ is", "\\begin{enumerate}", "\\item in the essential image of", "$\\epsilon^{-1} :D(X_\\etale, \\Lambda/I) \\to D(X_\\proetale, \\Lambda/I)$, and", "\\item has tor amplitude in $[a,\\infty)$ for some $a \\in \\mathbf{Z}$,", "\\end{enumerate}", "then (1) and (2) hold for $K_n$ as an object of $D(X_\\proetale, \\Lambda/I^n)$." ], "refs": [], "proofs": [ { "contents": [ "For assertion (2) this follows from the more general", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-bounded}.", "The second assertion follows from the fact that", "the essential image of $\\epsilon^{-1}$ is a triangulated", "subcategory of $D^+(X_\\proetale, \\Lambda/I^n)$", "(Lemma \\ref{lemma-compare-derived}),", "the distinguished triangles", "$$", "K \\otimes_\\Lambda^\\mathbf{L} \\underline{I^n/I^{n + 1}} \\to", "K_{n + 1} \\to", "K_n \\to", "K \\otimes_\\Lambda^\\mathbf{L} \\underline{I^n/I^{n + 1}}[1]", "$$", "and the isomorphism", "$$", "K \\otimes_\\Lambda^\\mathbf{L} \\underline{I^n/I^{n + 1}} =", "K_1 \\otimes_{\\Lambda/I}^\\mathbf{L} \\underline{I^n/I^{n + 1}}", "$$" ], "refs": [ "sites-cohomology-lemma-bounded", "proetale-lemma-compare-derived" ], "ref_ids": [ 4379, 3786 ] } ], "ref_ids": [] }, { "id": 3790, "type": "theorem", "label": "proetale-lemma-naive-completion", "categories": [ "proetale" ], "title": "proetale-lemma-naive-completion", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a Noetherian ring", "and let $I \\subset \\Lambda$ be an ideal. The left adjoint", "to the inclusion functor", "$D_{comp}(\\mathcal{C}, \\Lambda) \\to D(\\mathcal{C}, \\Lambda)$", "of Algebraic and Formal Geometry, Proposition", "\\ref{algebraization-proposition-derived-completion} sends $K$ to", "$$", "K^\\wedge = R\\lim(K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n})", "$$", "In particular, $K$ is derived complete if and only if", "$K = R\\lim(K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n})$." ], "refs": [ "algebraization-proposition-derived-completion" ], "proofs": [ { "contents": [ "Choose generators $f_1, \\ldots, f_r$ of $I$. By", "Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-derived-completion-koszul}", "we have", "$$", "K^\\wedge = ", "R\\lim (K \\otimes_\\Lambda^\\mathbf{L} \\underline{K_n})", "$$", "where $K_n = K(\\Lambda, f_1^n, \\ldots, f_r^n)$.", "In More on Algebra, Lemma \\ref{more-algebra-lemma-sequence-Koszul-complexes}", "we have seen that the pro-systems $\\{K_n\\}$ and", "$\\{\\Lambda/I^n\\}$ of $D(\\Lambda)$", "are isomorphic. Thus the lemma follows." ], "refs": [ "algebraization-lemma-derived-completion-koszul", "more-algebra-lemma-sequence-Koszul-complexes" ], "ref_ids": [ 12701, 10391 ] } ], "ref_ids": [ 12790 ] }, { "id": 3791, "type": "theorem", "label": "proetale-lemma-pushforward-Noetherian-case", "categories": [ "proetale" ], "title": "proetale-lemma-pushforward-Noetherian-case", "contents": [ "Let $\\Lambda$ be a Noetherian ring. Let $I \\subset \\Lambda$ be an ideal.", "Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi.", "Then", "\\begin{enumerate}", "\\item $Rf_*$ sends $D_{comp}(\\mathcal{D}, \\Lambda)$", "into $D_{comp}(\\mathcal{C}, \\Lambda)$,", "\\item the map $Rf_* : D_{comp}(\\mathcal{D}, \\Lambda) \\to", "D_{comp}(\\mathcal{C}, \\Lambda)$ has a left adjoint", "$Lf_{comp}^* : D_{comp}(\\mathcal{C}, \\Lambda) \\to", "D_{comp}(\\mathcal{D}, \\Lambda)$ which is $Lf^*$ followed by", "derived completion,", "\\item $Rf_*$ commutes with derived completion,", "\\item for $K$ in $D_{comp}(\\mathcal{D}, \\Lambda)$ we have", "$Rf_*K = R\\lim Rf_*(K \\otimes^\\mathbf{L}_\\Lambda \\underline{\\Lambda/I^n})$.", "\\item for $M$ in $D_{comp}(\\mathcal{C}, \\Lambda)$ we have", "$Lf^*_{comp}M =", "R\\lim Lf^*(M \\otimes^\\mathbf{L}_\\Lambda \\underline{\\Lambda/I^n})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have seen (1) and (2) in", "Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-pushforward-derived-complete-adjoint}.", "Part (3) follows from", "Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-pushforward-commutes-with-derived-completion}.", "For (4) let $K$ be derived complete. Then", "$$", "Rf_*K = Rf_*( R\\lim K \\otimes^\\mathbf{L}_\\Lambda \\underline{\\Lambda/I^n}) =", "R\\lim Rf_*(K \\otimes^\\mathbf{L}_\\Lambda \\underline{\\Lambda/I^n})", "$$", "the first equality by Lemma \\ref{lemma-naive-completion}", "and the second because $Rf_*$ commutes with $R\\lim$", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-Rf-commutes-with-Rlim}). This proves (4).", "To prove (5), by Lemma \\ref{lemma-naive-completion} we have", "$$", "Lf_{comp}^*M =", "R\\lim ( Lf^*M \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n})", "$$", "Since $Lf^*$ commutes with derived tensor product by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-pullback-tensor-product}", "and since $Lf^*\\underline{\\Lambda/I^n} = \\underline{\\Lambda/I^n}$", "we get (5)." ], "refs": [ "algebraization-lemma-pushforward-derived-complete-adjoint", "algebraization-lemma-pushforward-commutes-with-derived-completion", "proetale-lemma-naive-completion", "sites-cohomology-lemma-Rf-commutes-with-Rlim", "proetale-lemma-naive-completion", "sites-cohomology-lemma-pullback-tensor-product" ], "ref_ids": [ 12706, 12707, 3790, 4267, 3790, 4244 ] } ], "ref_ids": [] }, { "id": 3792, "type": "theorem", "label": "proetale-lemma-more-general-point", "categories": [ "proetale" ], "title": "proetale-lemma-more-general-point", "contents": [ "Let $k$ be a field. Let $G = \\text{Gal}(k^{sep}/k)$ be its absolute", "Galois group. Further, ", "\\begin{enumerate}", "\\item let $M$ be a profinite abelian group with a continuous", "$G$-action, or", "\\item let $\\Lambda$ be a Noetherian ring and $I \\subset \\Lambda$ an ideal", "an let $M$ be an $I$-adically complete $\\Lambda$-module with continuous", "$G$-action.", "\\end{enumerate}", "Then there is a canonical sheaf $\\underline{M}^\\wedge$ on", "$\\Spec(k)_\\proetale$ associated to $M$ such that", "$$", "H^i(\\Spec(k), \\underline{M}^\\wedge) = H^i_{cont}(G, M)", "$$", "as abelian groups or $\\Lambda$-modules." ], "refs": [], "proofs": [ { "contents": [ "Proof in case (2). Set $M_n = M/I^nM$. Then $M = \\lim M_n$ as $M$ is", "assumed $I$-adically complete. Since the action of $G$ is continuous", "we get continuous actions of $G$ on $M_n$. By \\'Etale Cohomology, Theorem", "\\ref{etale-cohomology-theorem-equivalence-sheaves-point}", "this action corresponds to a (locally constant) sheaf", "$\\underline{M_n}$ of $\\Lambda/I^n$-modules on $\\Spec(k)_\\etale$.", "Pull back to $\\Spec(k)_\\proetale$ by the comparison morphism", "$\\epsilon$ and take the limit", "$$", "\\underline{M}^\\wedge = \\lim \\epsilon^{-1}\\underline{M_n}", "$$", "to get the sheaf promised in the lemma. Exactly the same argument as", "given in the introduction of this section gives the comparison", "with Tate's continuous Galois cohomology." ], "refs": [ "etale-cohomology-theorem-equivalence-sheaves-point" ], "ref_ids": [ 6386 ] } ], "ref_ids": [] }, { "id": 3793, "type": "theorem", "label": "proetale-lemma-morphism-comparison", "categories": [ "proetale" ], "title": "proetale-lemma-morphism-comparison", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$. Then we have", "$f_{\\proetale, *}\\epsilon^{-1}\\mathcal{F} =", "\\epsilon^{-1}f_{\\etale, *}\\mathcal{F}$.", "\\item Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Then we have", "$Rf_{\\proetale, *}\\epsilon^{-1}\\mathcal{F} =", "\\epsilon^{-1}Rf_{\\etale, *}\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$. There", "is a canonical map $\\epsilon^{-1}f_{\\etale, *}\\mathcal{F} \\to", "f_{\\proetale, *}\\epsilon^{-1}\\mathcal{F}$, see", "Sites, Section \\ref{sites-section-pullback}.", "To show it is an isomorphism we may work (Zariski) locally on $Y$, hence", "we may assume $Y$ is affine. In this case", "every object of $Y_\\proetale$ has a covering by objects $V = \\lim V_i$", "which are limits of affine schemes $V_i$ \\'etale over $Y$ (by", "Proposition \\ref{proposition-weakly-etale}", "for example). Evaluating the map", "$\\epsilon^{-1}f_{\\etale, *}\\mathcal{F} \\to", "f_{\\proetale, *}\\epsilon^{-1}\\mathcal{F}$", "on $V$ we obtain a map", "$$", "\\colim \\Gamma(X \\times_Y V_i, \\mathcal{F})", "\\longrightarrow", "\\Gamma(X \\times_Y V, \\epsilon^*\\mathcal{F}).", "$$", "see Lemma \\ref{lemma-limit-pullback} for the left hand side.", "By Lemma \\ref{lemma-limit-pullback} we have", "$$", "\\Gamma(X \\times_Y V, \\epsilon^*\\mathcal{F}) =", "\\Gamma(X \\times_Y V, \\mathcal{F})", "$$", "Hence the result holds by", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-directed-colimit-cohomology}.", "\\medskip\\noindent", "Proof of (2). Arguing in exactly the same manner as above", "we see that it suffices to show that", "$$", "\\colim H^i_\\etale(X \\times_Y V_i, \\mathcal{F})", "\\longrightarrow", "H^i_\\etale(X \\times_Y V, \\mathcal{F})", "$$", "which follows once more from \\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-directed-colimit-cohomology}." ], "refs": [ "proetale-proposition-weakly-etale", "proetale-lemma-limit-pullback", "proetale-lemma-limit-pullback", "etale-cohomology-lemma-directed-colimit-cohomology", "etale-cohomology-lemma-directed-colimit-cohomology" ], "ref_ids": [ 3826, 3780, 3780, 6473, 6473 ] } ], "ref_ids": [] }, { "id": 3794, "type": "theorem", "label": "proetale-lemma-finite", "categories": [ "proetale" ], "title": "proetale-lemma-finite", "contents": [ "Let $f : Z \\to X$ be a finite morphism of schemes which is", "locally of finite presentation. Then", "$f_{\\proetale, *} : \\textit{Ab}(Z_\\proetale) \\to \\textit{Ab}(X_\\proetale)$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "The prove this we may work (Zariski) locally on $X$ and assume that $X$", "is affine, say $X = \\Spec(A)$. Then $Z = \\Spec(B)$ for some finite", "$A$-algebra $B$ of finite presentation. The construction in the proof of", "Proposition \\ref{proposition-find-w-contractible}", "produces a faithfully flat, ind-\\'etale ring map $A \\to D$", "with $D$ w-contractible. We may check exactness of a sequence of", "sheaves by evaluating on $U = \\Spec(D)$ be such an object. Then", "$f_{\\proetale, *}\\mathcal{F}$", "evaluated at $U$ is equal to $\\mathcal{F}$ evaluated at", "$V = \\Spec(D \\otimes_A B)$. Since $D \\otimes_A B$ is w-contractible", "by Lemma \\ref{lemma-finite-finitely-presented-over-w-contractible}", "evaluation at $V$ is exact." ], "refs": [ "proetale-proposition-find-w-contractible", "proetale-lemma-finite-finitely-presented-over-w-contractible" ], "ref_ids": [ 3827, 3741 ] } ], "ref_ids": [] }, { "id": 3795, "type": "theorem", "label": "proetale-lemma-closed-immersion-affines", "categories": [ "proetale" ], "title": "proetale-lemma-closed-immersion-affines", "contents": [ "Let $i : Z \\to X$ be a closed immersion morphism of affine schemes.", "Denote $X_{app}$ and $Z_{app}$ the sites introduced in", "Lemma \\ref{lemma-affine-alternative}.", "The base change functor", "$$", "u : X_{app} \\to Z_{app},\\quad U \\longmapsto u(U) = U \\times_X Z", "$$", "is continuous and has a fully faithful left adjoint $v$.", "For $V$ in $Z_{app}$ the morphism $V \\to v(V)$ is a closed immersion", "identifying $V$ with $u(v(V)) = v(V) \\times_X Z$ and every point of", "$v(V)$ specializes to a point of $V$.", "The functor $v$ is cocontinuous and sends coverings to coverings." ], "refs": [ "proetale-lemma-affine-alternative" ], "proofs": [ { "contents": [ "The existence of the adjoint follows immediately from", "Lemma \\ref{lemma-lift-ind-etale} and the definitions.", "It is clear that $u$ is continuous from the definition of", "coverings in $X_{app}$.", "\\medskip\\noindent", "Write $X = \\Spec(A)$ and $Z = \\Spec(A/I)$. Let $V = \\Spec(\\overline{C})$", "be an object of $Z_{app}$ and let $v(V) = \\Spec(C)$.", "We have seen in the statement of Lemma \\ref{lemma-lift-ind-etale}", "that $V$ equals $v(V) \\times_X Z = \\Spec(C/IC)$.", "Any $g \\in C$ which maps to an invertible element of", "$C/IC = \\overline{C}$ is invertible in $C$. Namely, we have the", "$A$-algebra maps $C \\to C_g \\to C/IC$ and by adjointness", "we obtain an $C$-algebra map $C_g \\to C$.", "Thus every point of $v(V)$ specializes to a point of $V$.", "\\medskip\\noindent", "Suppose that $\\{V_i \\to V\\}$ is a covering in $Z_{app}$.", "Then $\\{v(V_i) \\to v(V)\\}$ is a finite family of morphisms of", "$Z_{app}$ such that every point of $V \\subset v(V)$ is", "in the image of one of the maps $v(V_i) \\to v(V)$. As the", "morphisms $v(V_i) \\to v(V)$ are flat (since they are weakly \\'etale)", "we conclude that $\\{v(V_i) \\to v(V)\\}$ is jointly surjective.", "This proves that $v$ sends coverings to coverings.", "\\medskip\\noindent", "Let $V$ be an object of $Z_{app}$ and let $\\{U_i \\to v(V)\\}$", "be a covering in $X_{app}$. Then we see that", "$\\{u(U_i) \\to u(v(V)) = V\\}$ is a covering of $Z_{app}$.", "By adjointness we obtain morphisms $v(u(U_i)) \\to U_i$.", "Thus the family $\\{v(u(U_i)) \\to v(V)\\}$ refines the given", "covering and we conclude that $v$ is cocontinuous." ], "refs": [ "proetale-lemma-lift-ind-etale", "proetale-lemma-lift-ind-etale" ], "ref_ids": [ 3730, 3730 ] } ], "ref_ids": [ 3758 ] }, { "id": 3796, "type": "theorem", "label": "proetale-lemma-closed-immersion-affines-apply", "categories": [ "proetale" ], "title": "proetale-lemma-closed-immersion-affines-apply", "contents": [ "Let $Z \\to X$ be a closed immersion morphism of affine schemes.", "The corresponding morphism of topoi $i = i_\\proetale$", "is equal to the morphism of topoi", "associated to the fully faithful cocontinuous functor", "$v : Z_{app} \\to X_{app}$ of Lemma \\ref{lemma-closed-immersion-affines}.", "It follows that", "\\begin{enumerate}", "\\item $i^{-1}\\mathcal{F}$ is the sheaf associated to the presheaf", "$V \\mapsto \\mathcal{F}(v(V))$,", "\\item for a weakly contractible object $V$ of $Z_{app}$ we have", "$i^{-1}\\mathcal{F}(V) = \\mathcal{F}(v(V))$,", "\\item $i^{-1} : \\Sh(X_\\proetale) \\to \\Sh(Z_\\proetale)$", "has a left adjoint $i^{Sh}_!$,", "\\item $i^{-1} : \\textit{Ab}(X_\\proetale) \\to \\textit{Ab}(Z_\\proetale)$", "has a left adjoint $i_!$,", "\\item $\\text{id} \\to i^{-1}i^{Sh}_!$, $\\text{id} \\to i^{-1}i_!$, and", "$i^{-1}i_* \\to \\text{id}$ are isomorphisms, and", "\\item $i_*$, $i^{Sh}_!$ and $i_!$ are fully faithful.", "\\end{enumerate}" ], "refs": [ "proetale-lemma-closed-immersion-affines" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-affine-alternative} we may describe $i_\\proetale$ in terms", "of the morphism of sites $u : X_{app} \\to Z_{app}$, $V \\mapsto V \\times_X Z$.", "The first statement of the lemma follows from", "Sites, Lemma \\ref{sites-lemma-have-functor-other-way-morphism}", "(but with the roles of $u$ and $v$ reversed).", "\\medskip\\noindent", "Proof of (1). By the description of $i$ as the morphism of topoi associated", "to $v$ this holds by the construction, see", "Sites, Lemma \\ref{sites-lemma-cocontinuous-morphism-topoi}.", "\\medskip\\noindent", "Proof of (2). Since the functor $v$ sends coverings to coverings by", "Lemma \\ref{lemma-closed-immersion-affines} we see that the presheaf", "$\\mathcal{G} : V \\mapsto \\mathcal{F}(v(V))$ is a separated presheaf", "(Sites, Definition \\ref{sites-definition-separated}). Hence", "the sheafification of $\\mathcal{G}$ is $\\mathcal{G}^+$, see", "Sites, Theorem \\ref{sites-theorem-plus}. Next, let $V$ be a weakly", "contractible object of $Z_{app}$. Let", "$\\mathcal{V} = \\{V_i \\to V\\}_{i = 1, \\ldots, n}$", "be any covering in $Z_{app}$. Set $\\mathcal{V}' = \\{\\coprod V_i \\to V\\}$.", "Since $v$ commutes with finite disjoint unions (as a left adjoint or by", "the construction) and since $\\mathcal{F}$ sends finite disjoint", "unions into products, we see that", "$$", "H^0(\\mathcal{V}, \\mathcal{G}) = H^0(\\mathcal{V}', \\mathcal{G})", "$$", "(notation as in Sites, Section \\ref{sites-section-sheafification};", "compare with", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-cech-complex}).", "Thus we may assume the covering is given by a single morphism, like", "so $\\{V' \\to V\\}$. Since $V$ is weakly contractible, this covering", "can be refined by the trivial covering $\\{V \\to V\\}$.", "It therefore follows that the value of $\\mathcal{G}^+ = i^{-1}\\mathcal{F}$", "on $V$ is simply $\\mathcal{F}(v(V))$ and (2) is proved.", "\\medskip\\noindent", "Proof of (3). Every object of $Z_{app}$ has a covering by weakly", "contractible objects (Lemma \\ref{lemma-proetale-enough-w-contractible}).", "By the above we see that we would have $i^{Sh}_!h_V = h_{v(V)}$ for $V$", "weakly contractible if $i^{Sh}_!$ existed. The existence of", "$i^{Sh}_!$ then follows from", "Sites, Lemma \\ref{sites-lemma-existence-lower-shriek}.", "\\medskip\\noindent", "Proof of (4). Existence of $i_!$ follows in the same way by setting", "$i_!\\mathbf{Z}_V = \\mathbf{Z}_{v(V)}$ for $V$ weakly contractible in $Z_{app}$,", "using similar for direct sums, and applying", "Homology, Lemma \\ref{homology-lemma-partially-defined-adjoint}.", "Details omitted.", "\\medskip\\noindent", "Proof of (5). Let $V$ be a contractible object of $Z_{app}$.", "Then $i^{-1}i^{Sh}_!h_V = i^{-1}h_{v(V)} = h_{u(v(V))} = h_V$.", "(It is a general fact that $i^{-1}h_U = h_{u(U)}$.) Since the", "sheaves $h_V$ for $V$ contractible generate $\\Sh(Z_{app})$", "(Sites, Lemma \\ref{sites-lemma-sheaf-coequalizer-representable})", "we conclude $\\text{id} \\to i^{-1}i^{Sh}_!$ is an isomorphism.", "Similarly for the map $\\text{id} \\to i^{-1}i_!$. Then", "$(i^{-1}i_*\\mathcal{H})(V) = i_*\\mathcal{H}(v(V)) =", "\\mathcal{H}(u(v(V))) = \\mathcal{H}(V)$ and we find that", "$i^{-1}i_* \\to \\text{id}$ is an isomorphism.", "\\medskip\\noindent", "The fully faithfulness statements of (6) now follow from", "Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}." ], "refs": [ "proetale-lemma-affine-alternative", "sites-lemma-have-functor-other-way-morphism", "sites-lemma-cocontinuous-morphism-topoi", "proetale-lemma-closed-immersion-affines", "sites-definition-separated", "sites-theorem-plus", "etale-cohomology-lemma-cech-complex", "proetale-lemma-proetale-enough-w-contractible", "sites-lemma-existence-lower-shriek", "homology-lemma-partially-defined-adjoint", "sites-lemma-sheaf-coequalizer-representable", "categories-lemma-adjoint-fully-faithful" ], "ref_ids": [ 3758, 8550, 8543, 3795, 8658, 8492, 6416, 3763, 8552, 12119, 8520, 12248 ] } ], "ref_ids": [ 3795 ] }, { "id": 3797, "type": "theorem", "label": "proetale-lemma-closed-immersion", "categories": [ "proetale" ], "title": "proetale-lemma-closed-immersion", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes. Then", "\\begin{enumerate}", "\\item $i_\\proetale^{-1}$ commutes with limits,", "\\item $i_{\\proetale, *}$ is fully faithful, and", "\\item $i_\\proetale^{-1}i_{\\proetale, *} \\cong \\text{id}_{\\Sh(Z_\\proetale)}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assertions (2) and (3) are equivalent by", "Sites, Lemma \\ref{sites-lemma-exactness-properties}.", "Parts (1) and (3) are (Zariski) local on $X$, hence we may assume that", "$X$ is affine. In this case the result follows from", "Lemma \\ref{lemma-closed-immersion-affines-apply}." ], "refs": [ "sites-lemma-exactness-properties", "proetale-lemma-closed-immersion-affines-apply" ], "ref_ids": [ 8618, 3796 ] } ], "ref_ids": [] }, { "id": 3798, "type": "theorem", "label": "proetale-lemma-thickening", "categories": [ "proetale" ], "title": "proetale-lemma-thickening", "contents": [ "Let $i : Z \\to X$ be an integral universally injective and surjective morphism", "of schemes. Then", "$i_{\\proetale, *}$ and $i_\\proetale^{-1}$ are quasi-inverse", "equivalences of categories of pro-\\'etale topoi." ], "refs": [], "proofs": [ { "contents": [ "There is an immediate reduction to the case that $X$ is affine.", "Then $Z$ is affine too. Set $A = \\mathcal{O}(X)$ and $B = \\mathcal{O}(Z)$.", "Then the categories of \\'etale algebras over", "$A$ and $B$ are equivalent, see ", "\\'Etale Cohomology, Theorem", "\\ref{etale-cohomology-theorem-topological-invariance} and", "Remark \\ref{etale-cohomology-remark-affine-inside-equivalence}.", "Thus the categories of ind-\\'etale algebras over $A$ and $B$ are", "equivalent. In other words the categories $X_{app}$ and $Z_{app}$", "of Lemma \\ref{lemma-affine-alternative} are equivalent.", "We omit the verification", "that this equivalence sends coverings to coverings and vice versa.", "Thus the result as Lemma \\ref{lemma-affine-alternative}", "tells us the pro-\\'etale topos is the topos of sheaves on $X_{app}$." ], "refs": [ "etale-cohomology-theorem-topological-invariance", "etale-cohomology-remark-affine-inside-equivalence", "proetale-lemma-affine-alternative", "proetale-lemma-affine-alternative" ], "ref_ids": [ 6383, 6782, 3758, 3758 ] } ], "ref_ids": [] }, { "id": 3799, "type": "theorem", "label": "proetale-lemma-compute-i-star", "categories": [ "proetale" ], "title": "proetale-lemma-compute-i-star", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes.", "Let $U \\to X$ be an object of $X_\\proetale$ such that", "\\begin{enumerate}", "\\item $U$ is affine and weakly contractible, and", "\\item every point of $U$ specializes to a point of $U \\times_X Z$.", "\\end{enumerate}", "Then $i_\\proetale^{-1}\\mathcal{F}(U \\times_X Z) = \\mathcal{F}(U)$", "for all abelian sheaves on $X_\\proetale$." ], "refs": [], "proofs": [ { "contents": [ "Since pullback commutes with restriction, we may replace $X$ by $U$.", "Thus we may assume that $X$ is affine and weakly contractible", "and that every point of $X$ specializes to a point of $Z$.", "By Lemma \\ref{lemma-closed-immersion-affines-apply} part (1)", "it suffices to show that $v(Z) = X$ in this case.", "Thus we have to show: If $A$ is a w-contractible ring, $I \\subset A$", "an ideal contained in the Jacobson radical of $A$ and $A \\to B \\to A/I$", "is a factorization with $A \\to B$ ind-\\'etale, then there is", "a unique section $B \\to A$ compatible with maps to $A/I$.", "Observe that $B/IB = A/I \\times R$ as $A/I$-algebras.", "After replacing $B$ by a localization we may assume $B/IB = A/I$.", "Note that $\\Spec(B) \\to \\Spec(A)$ is surjective as the image", "contains $V(I)$ and hence all closed points and is closed under", "specialization. Since $A$ is w-contractible there is a section $B \\to A$.", "Since $B/IB = A/I$ this section is compatible with the map to $A/I$.", "We omit the proof of uniqueness (hint: use that $A$ and $B$ have", "isomorphic local rings at maximal ideals of $A$)." ], "refs": [ "proetale-lemma-closed-immersion-affines-apply" ], "ref_ids": [ 3796 ] } ], "ref_ids": [] }, { "id": 3800, "type": "theorem", "label": "proetale-lemma-closed-immersion-complement-retrocompact-exact", "categories": [ "proetale" ], "title": "proetale-lemma-closed-immersion-complement-retrocompact-exact", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes.", "If $X \\setminus i(Z)$ is a retrocompact open of $X$, then", "$i_{\\proetale, *}$ is exact." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$ hence we may assume $X$ is affine.", "Say $X = \\Spec(A)$ and $Z = \\Spec(A/I)$. There exist", "$f_1, \\ldots, f_r \\in I$ such that $Z = V(f_1, \\ldots, f_r)$", "set theoretically, see Algebra, Lemma \\ref{algebra-lemma-qc-open}.", "By Lemma \\ref{lemma-thickening} we may assume that", "$Z = \\Spec(A/(f_1, \\ldots, f_r))$. In this case", "the functor $i_{\\proetale, *}$ is exact by", "Lemma \\ref{lemma-finite}." ], "refs": [ "algebra-lemma-qc-open", "proetale-lemma-thickening", "proetale-lemma-finite" ], "ref_ids": [ 432, 3798, 3794 ] } ], "ref_ids": [] }, { "id": 3801, "type": "theorem", "label": "proetale-lemma-jshriek-comparison", "categories": [ "proetale" ], "title": "proetale-lemma-jshriek-comparison", "contents": [ "Let $j : U \\to X$ be an \\'etale morphism of schemes.", "Let $\\mathcal{G}$ be an abelian sheaf on $U_\\etale$.", "Then $\\epsilon^{-1} j_!\\mathcal{G} = j_!\\epsilon^{-1}\\mathcal{G}$", "as sheaves on $X_\\proetale$." ], "refs": [], "proofs": [ { "contents": [ "This is true because both are left adjoints to", "$j_{\\proetale, *}\\epsilon^{-1} = \\epsilon^{-1}j_{\\etale, *}$, see", "Lemma \\ref{lemma-morphism-comparison}." ], "refs": [ "proetale-lemma-morphism-comparison" ], "ref_ids": [ 3793 ] } ], "ref_ids": [] }, { "id": 3802, "type": "theorem", "label": "proetale-lemma-jshriek-zero", "categories": [ "proetale" ], "title": "proetale-lemma-jshriek-zero", "contents": [ "Let $j : U \\to X$ be a weakly \\'etale morphism of schemes.", "Let $i : Z \\to X$ be a closed immersion such that $U \\times_X Z = \\emptyset$.", "Let $V \\to X$ be an affine object of $X_\\proetale$ such that every point", "of $V$ specializes to a point of $V_Z = Z \\times_X V$.", "Then $j_!\\mathcal{F}(V) = 0$ for all abelian sheaves on $U_\\proetale$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{V_i \\to V\\}$ be a pro-\\'etale covering. The lemma follows if we", "can refine this covering to a covering where the members have no", "morphisms into $U$ over $X$ (see construction of $j_!$ in", "Modules on Sites, Section \\ref{sites-modules-section-localize}).", "First refine the covering to get a finite covering with $V_i$ affine.", "For each $i$ let $V_i = \\Spec(A_i)$ and let $Z_i \\subset V_i$ be the", "inverse image of $Z$.", "Set $W_i = \\Spec(A_{i, Z_i}^\\sim)$ with notation as in", "Lemma \\ref{lemma-localization}.", "Then $\\coprod W_i \\to V$ is weakly \\'etale and the image contains all", "points of $V_Z$. Hence the image contains all points of $V$ by", "our assumption on specializations. Thus $\\{W_i \\to V\\}$ is a", "pro-\\'etale covering refining the given one. But each point in $W_i$", "specializes to a point lying over $Z$, hence there are no morphisms", "$W_i \\to U$ over $X$." ], "refs": [ "proetale-lemma-localization" ], "ref_ids": [ 3712 ] } ], "ref_ids": [] }, { "id": 3803, "type": "theorem", "label": "proetale-lemma-open-immersion", "categories": [ "proetale" ], "title": "proetale-lemma-open-immersion", "contents": [ "Let $j : U \\to X$ be an open immersion of schemes.", "Then $\\text{id} \\cong j^{-1}j_!$ and $j^{-1}j_* \\cong \\text{id}$", "and the functors $j_!$ and $j_*$ are fully faithful." ], "refs": [], "proofs": [ { "contents": [ "See Modules on Sites, Lemma \\ref{sites-modules-lemma-restrict-back}", "(and Sites, Lemma \\ref{sites-lemma-restrict-back} for the case", "of sheaves of sets) and", "Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}." ], "refs": [ "sites-modules-lemma-restrict-back", "sites-lemma-restrict-back", "categories-lemma-adjoint-fully-faithful" ], "ref_ids": [ 14173, 8569, 12248 ] } ], "ref_ids": [] }, { "id": 3804, "type": "theorem", "label": "proetale-lemma-ses-associated-to-open", "categories": [ "proetale" ], "title": "proetale-lemma-ses-associated-to-open", "contents": [ "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme and let", "$U \\subset X$ be the complement. Denote $i : Z \\to X$ and $j : U \\to X$", "the inclusion morphisms. Assume that $j$ is a quasi-compact morphism.", "For every abelian sheaf on $X_\\proetale$ there is a canonical short exact", "sequence", "$$", "0 \\to j_!j^{-1}\\mathcal{F} \\to \\mathcal{F} \\to i_*i^{-1}\\mathcal{F} \\to 0", "$$", "on $X_\\proetale$ where all the functors are for the pro-\\'etale topology." ], "refs": [], "proofs": [ { "contents": [ "We obtain the maps by the adjointness properties of the functors involved.", "It suffices to show that $X_\\proetale$ has enough objects", "(Sites, Definition \\ref{sites-definition-w-contractible}) on which", "the sequence evaluates to a short exact sequence.", "Let $V = \\Spec(A)$ be an affine object of $X_\\proetale$", "such that $A$ is w-contractible (there are enough objects of this type).", "Then $V \\times_X Z$ is cut out by an ideal $I \\subset A$.", "The assumption that $j$ is quasi-compact implies there exist", "$f_1, \\ldots, f_r \\in I$ such that $V(I) = V(f_1, \\ldots, f_r)$.", "We obtain a faithfully flat, ind-Zariski ring map", "$$", "A \\longrightarrow A_{f_1} \\times \\ldots \\times A_{f_r} \\times", "A_{V(I)}^\\sim", "$$", "with $A_{V(I)}^\\sim$ as in Lemma \\ref{lemma-localization}.", "Since $V_i = \\Spec(A_{f_i}) \\to X$ factors through $U$ we have", "$$", "j_!j^{-1}\\mathcal{F}(V_i) = \\mathcal{F}(V_i)", "\\quad\\text{and}\\quad", "i_*i^{-1}\\mathcal{F}(V_i) = 0", "$$", "On the other hand, for the scheme $V^\\sim = \\Spec(A_{V(I)}^\\sim)$", "we have", "$$", "j_!j^{-1}\\mathcal{F}(V^\\sim) = 0", "\\quad\\text{and}\\quad", "\\mathcal{F}(V^\\sim) = i_*i^{-1}\\mathcal{F}(V^\\sim)", "$$", "the first equality by Lemma \\ref{lemma-jshriek-zero}", "and the second by", "Lemmas \\ref{lemma-compute-i-star} and \\ref{lemma-localization-w-contractible}.", "Thus the sequence evaluates to an exact sequence on", "$\\Spec(A_{f_1} \\times \\ldots \\times A_{f_r} \\times A_{V(I)}^\\sim)$", "and the lemma is proved." ], "refs": [ "sites-definition-w-contractible", "proetale-lemma-localization", "proetale-lemma-jshriek-zero", "proetale-lemma-compute-i-star", "proetale-lemma-localization-w-contractible" ], "ref_ids": [ 8680, 3712, 3802, 3799, 3742 ] } ], "ref_ids": [] }, { "id": 3805, "type": "theorem", "label": "proetale-lemma-j-shriek-limits", "categories": [ "proetale" ], "title": "proetale-lemma-j-shriek-limits", "contents": [ "Let $j : U \\to X$ be a quasi-compact open immersion", "morphism of schemes. The functor", "$j_! : \\textit{Ab}(U_\\proetale) \\to \\textit{Ab}(X_\\proetale)$", "commutes with limits." ], "refs": [], "proofs": [ { "contents": [ "Since $j_!$ is exact it suffices to show that $j_!$ commutes with products.", "The question is local on $X$, hence we may assume $X$ affine.", "Let $\\mathcal{G}$ be an abelian sheaf on $U_\\proetale$.", "We have $j^{-1}j_*\\mathcal{G} = \\mathcal{G}$. Hence applying", "the exact sequence of Lemma \\ref{lemma-ses-associated-to-open}", "we get", "$$", "0 \\to j_!\\mathcal{G} \\to j_*\\mathcal{G} \\to i_*i^{-1}j_*\\mathcal{G} \\to 0", "$$", "where $i : Z \\to X$ is the inclusion of the reduced induced scheme", "structure on the complement $Z = X \\setminus U$.", "The functors $j_*$ and $i_*$ commute with products as right adjoints.", "The functor $i^{-1}$ commutes with products by", "Lemma \\ref{lemma-closed-immersion}.", "Hence $j_!$ does because on the pro-\\'etale site products", "are exact", "(Cohomology on Sites, Proposition", "\\ref{sites-cohomology-proposition-enough-weakly-contractibles})." ], "refs": [ "proetale-lemma-ses-associated-to-open", "proetale-lemma-closed-immersion", "sites-cohomology-proposition-enough-weakly-contractibles" ], "ref_ids": [ 3804, 3797, 4410 ] } ], "ref_ids": [] }, { "id": 3806, "type": "theorem", "label": "proetale-lemma-compare-constructible", "categories": [ "proetale" ], "title": "proetale-lemma-compare-constructible", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.", "The functor $\\epsilon^{-1}$ defines an equivalence of categories", "$$", "\\left\\{", "\\begin{matrix}", "\\text{constructible sheaves of}\\\\", "\\Lambda\\text{-modules on }X_\\etale\\\\", "\\end{matrix}", "\\right\\}", "\\longleftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{constructible sheaves of}\\\\", "\\Lambda\\text{-modules on }X_\\proetale\\\\", "\\end{matrix}", "\\right\\}", "$$", "between constructible sheaves of $\\Lambda$-modules on $X_\\etale$", "and constructible sheaves of $\\Lambda$-modules on $X_\\proetale$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-fully-faithful} the functor $\\epsilon^{-1}$", "is fully faithful and commutes with pullback (restriction) to", "the strata. Hence $\\epsilon^{-1}$ of a constructible \\'etale", "sheaf is a constructible pro-\\'etale sheaf. To finish the", "proof let $\\mathcal{F}$ be a constructible sheaf of $\\Lambda$-modules", "on $X_\\proetale$ as in Definition \\ref{definition-constructible}.", "There is a canonical map", "$$", "\\epsilon^{-1}\\epsilon_*\\mathcal{F} \\longrightarrow \\mathcal{F}", "$$", "We will show this map is an isomorphism. This will prove that", "$\\mathcal{F}$ is in the essential image of $\\epsilon^{-1}$", "and finish the proof (details omitted).", "\\medskip\\noindent", "To prove this we may assume that $X$ is affine. In this case we have", "a finite partition $X = \\coprod_i X_i$ by constructible locally closed", "strata such that $\\mathcal{F}|_{X_i}$ is locally constant of finite", "type. Let $U \\subset X$ be one of the open strata in the partition", "and let $Z \\subset X$ be the reduced induced structure on the complement.", "By Lemma \\ref{lemma-ses-associated-to-open}", "we have a short exact sequence", "$$", "0 \\to j_!j^{-1}\\mathcal{F} \\to \\mathcal{F} \\to i_*i^{-1}\\mathcal{F} \\to 0", "$$", "on $X_\\proetale$. Functoriality gives a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\epsilon^{-1}\\epsilon_*j_!j^{-1}\\mathcal{F} \\ar[r] \\ar[d] &", "\\epsilon^{-1}\\epsilon_*\\mathcal{F} \\ar[r] \\ar[d] &", "\\epsilon^{-1}\\epsilon_*i_*i^{-1}\\mathcal{F} \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "j_!j^{-1}\\mathcal{F} \\ar[r] &", "\\mathcal{F} \\ar[r] &", "i_*i^{-1}\\mathcal{F} \\ar[r] & 0", "}", "$$", "By induction on the length of the partition we know that", "on the one hand", "$\\epsilon^{-1}\\epsilon_*i^{-1}\\mathcal{F} \\to i^{-1}\\mathcal{F}$", "and", "$\\epsilon^{-1}\\epsilon_*j^{-1}\\mathcal{F} \\to j^{-1}\\mathcal{F}$", "are isomorphisms and on the other that", "$i^{-1}\\mathcal{F} = \\epsilon^{-1}\\mathcal{A}$", "and", "$j^{-1}\\mathcal{F} = \\epsilon^{-1}\\mathcal{B}$", "for some constructible sheaves of $\\Lambda$-modules", "$\\mathcal{A}$ on $Z_\\etale$ and $\\mathcal{B}$ on $U_\\etale$.", "Then", "$$", "\\epsilon^{-1}\\epsilon_*j_!j^{-1}\\mathcal{F} =", "\\epsilon^{-1}\\epsilon_*j_!\\epsilon^{-1}\\mathcal{B} =", "\\epsilon^{-1}\\epsilon_*\\epsilon^{-1}j_!\\mathcal{B} =", "\\epsilon^{-1}j_!\\mathcal{B} =", "j_!\\epsilon^{-1}\\mathcal{B} =", "j_!j^{-1}\\mathcal{F}", "$$", "the second equality by Lemma \\ref{lemma-jshriek-comparison},", "the third equality by Lemma \\ref{lemma-fully-faithful}, and the", "fourth equality by Lemma \\ref{lemma-jshriek-comparison} again.", "Similarly, we have", "$$", "\\epsilon^{-1}\\epsilon_*i_*i^{-1}\\mathcal{F} =", "\\epsilon^{-1}\\epsilon_*i_*\\epsilon^{-1}\\mathcal{A} =", "\\epsilon^{-1}\\epsilon_*\\epsilon^{-1}i_*\\mathcal{A} =", "\\epsilon^{-1}i_*\\mathcal{A} =", "i_*\\epsilon^{-1}\\mathcal{A} =", "i_*i^{-1}\\mathcal{F}", "$$", "this time using Lemma \\ref{lemma-morphism-comparison}.", "By the five lemma we conclude the", "vertical map in the middle of the big diagram is an isomorphism." ], "refs": [ "proetale-lemma-fully-faithful", "proetale-definition-constructible", "proetale-lemma-ses-associated-to-open", "proetale-lemma-jshriek-comparison", "proetale-lemma-fully-faithful", "proetale-lemma-jshriek-comparison", "proetale-lemma-morphism-comparison" ], "ref_ids": [ 3781, 3841, 3804, 3801, 3781, 3801, 3793 ] } ], "ref_ids": [] }, { "id": 3807, "type": "theorem", "label": "proetale-lemma-constructible-serre", "categories": [ "proetale" ], "title": "proetale-lemma-constructible-serre", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.", "The category of constructible sheaves of $\\Lambda$-modules on $X_\\proetale$", "is a weak Serre subcategory of $\\textit{Mod}(X_\\proetale, \\Lambda)$." ], "refs": [], "proofs": [ { "contents": [ "This is a formal consequence of", "Lemmas \\ref{lemma-compare-constructible} and \\ref{lemma-compare-derived}", "and the result for the \\'etale site", "(\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-constructible-abelian})." ], "refs": [ "proetale-lemma-compare-constructible", "proetale-lemma-compare-derived", "etale-cohomology-lemma-constructible-abelian" ], "ref_ids": [ 3806, 3786, 6531 ] } ], "ref_ids": [] }, { "id": 3808, "type": "theorem", "label": "proetale-lemma-compare-constructible-derived", "categories": [ "proetale" ], "title": "proetale-lemma-compare-constructible-derived", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.", "Let $D_c(X_\\etale, \\Lambda)$, resp.\\ $D_c(X_\\proetale, \\Lambda)$", "be the full subcategory of", "$D(X_\\etale, \\Lambda)$, resp.\\ $D(X_\\proetale, \\Lambda)$", "consisting of those complexes whose cohomology sheaves are", "constructible sheaves of $\\Lambda$-modules. Then", "$$", "\\epsilon^{-1} :", "D_c^+(X_\\etale, \\Lambda)", "\\longrightarrow", "D_c^+(X_\\proetale, \\Lambda)", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "The categories $D_c(X_\\etale, \\Lambda)$ and $D_c(X_\\proetale, \\Lambda)$", "are strictly full, saturated, triangulated subcategories of", "$D(X_\\etale, \\Lambda)$ and $D(X_\\proetale, \\Lambda)$ by", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-constructible-abelian}", "and", "Lemma \\ref{lemma-constructible-serre}", "and", "Derived Categories, Section \\ref{derived-section-triangulated-sub}.", "The statement of the lemma follows by combining", "Lemmas \\ref{lemma-compare-derived} and", "\\ref{lemma-compare-constructible}." ], "refs": [ "etale-cohomology-lemma-constructible-abelian", "proetale-lemma-constructible-serre", "proetale-lemma-compare-derived", "proetale-lemma-compare-constructible" ], "ref_ids": [ 6531, 3807, 3786, 3806 ] } ], "ref_ids": [] }, { "id": 3809, "type": "theorem", "label": "proetale-lemma-tensor-c", "categories": [ "proetale" ], "title": "proetale-lemma-tensor-c", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.", "Let $K, L \\in D_c^-(X_\\proetale, \\Lambda)$. Then", "$K \\otimes_\\Lambda^\\mathbf{L} L$ is in $D_c^-(X_\\proetale, \\Lambda)$." ], "refs": [], "proofs": [ { "contents": [ "Note that $H^i(K \\otimes_\\Lambda^\\mathbf{L} L)$ is the same as", "$H^i(\\tau_{\\geq i - 1}K \\otimes_\\Lambda^\\mathbf{L} \\tau_{\\geq i - 1}L)$.", "Thus we may assume $K$ and $L$ are bounded.", "In this case we can apply Lemma \\ref{lemma-compare-constructible-derived} to", "reduce to the case of the \\'etale site, see", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-tensor-c}." ], "refs": [ "proetale-lemma-compare-constructible-derived", "etale-cohomology-lemma-tensor-c" ], "ref_ids": [ 3808, 6559 ] } ], "ref_ids": [] }, { "id": 3810, "type": "theorem", "label": "proetale-lemma-compare-truncations-constructible", "categories": [ "proetale" ], "title": "proetale-lemma-compare-truncations-constructible", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.", "Let $K$ be an object of $D(X_\\proetale, \\Lambda)$.", "Set $K_n = K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$.", "If $K_1$ is in $D^-_c(X_\\proetale, \\Lambda/I)$, then", "$K_n$ is in $D^-_c(X_\\proetale, \\Lambda/I^n)$ for all $n$." ], "refs": [], "proofs": [ { "contents": [ "Consider the distinguished triangles", "$$", "K \\otimes_\\Lambda^\\mathbf{L} \\underline{I^n/I^{n + 1}} \\to", "K_{n + 1} \\to", "K_n \\to", "K \\otimes_\\Lambda^\\mathbf{L} \\underline{I^n/I^{n + 1}}[1]", "$$", "and the isomorphisms", "$$", "K \\otimes_\\Lambda^\\mathbf{L} \\underline{I^n/I^{n + 1}} =", "K_1 \\otimes_{\\Lambda/I}^\\mathbf{L} \\underline{I^n/I^{n + 1}}", "$$", "By Lemma \\ref{lemma-tensor-c} we see that this tensor product has", "constructible cohomology sheaves (and vanishing when $K_1$ has", "vanishing cohomology). Hence by induction on $n$ using", "Lemma \\ref{lemma-constructible-serre}", "we see that each $K_n$ has constructible cohomology sheaves." ], "refs": [ "proetale-lemma-tensor-c", "proetale-lemma-constructible-serre" ], "ref_ids": [ 3809, 3807 ] } ], "ref_ids": [] }, { "id": 3811, "type": "theorem", "label": "proetale-lemma-Noetherian-constructible", "categories": [ "proetale" ], "title": "proetale-lemma-Noetherian-constructible", "contents": [ "Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring and", "let $I \\subset \\Lambda$ be an ideal. Let $\\mathcal{F}$ be a", "constructible $\\Lambda$-sheaf on $X_\\proetale$.", "Then there exists a finite partition $X = \\coprod X_i$ by", "locally closed subschemes such that the restriction $\\mathcal{F}|_{X_i}$", "is lisse." ], "refs": [], "proofs": [ { "contents": [ "Let $R = \\bigoplus I^n/I^{n + 1}$. Observe that $R$ is a Noetherian ring.", "Since each of the sheaves", "$\\mathcal{F}/I^n\\mathcal{F}$ is a constructible sheaf of", "$\\Lambda/I^n\\Lambda$-modules also $I^n\\mathcal{F}/I^{n + 1}\\mathcal{F}$", "is a constructible sheaf of $\\Lambda/I$-modules and hence the pullback", "of a constructible sheaf $\\mathcal{G}_n$ on $X_\\etale$ by", "Lemma \\ref{lemma-compare-constructible}.", "Set $\\mathcal{G} = \\bigoplus \\mathcal{G}_n$. This is a sheaf", "of $R$-modules on $X_\\etale$ and the map", "$$", "\\mathcal{G}_0 \\otimes_{\\Lambda/I} \\underline{R}", "\\longrightarrow", "\\mathcal{G}", "$$", "is surjective because the maps", "$$", "\\mathcal{F}/I\\mathcal{F} \\otimes \\underline{I^n/I^{n + 1}} \\to", "I^n\\mathcal{F}/I^{n + 1}\\mathcal{F}", "$$", "are surjective. Hence $\\mathcal{G}$ is a constructible sheaf of", "$R$-modules by \\'Etale Cohomology, Proposition", "\\ref{etale-cohomology-proposition-constructible-over-noetherian}.", "Choose a partition $X = \\coprod X_i$ such that", "$\\mathcal{G}|_{X_i}$ is a locally constant sheaf of $R$-modules", "of finite type (\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-constructible-quasi-compact-quasi-separated}).", "We claim this is a partition as in the lemma.", "Namely, replacing $X$ by $X_i$ we may assume $\\mathcal{G}$ is locally", "constant. It follows that each of the sheaves", " $I^n\\mathcal{F}/I^{n + 1}\\mathcal{F}$", "is locally constant. Using the short exact sequences", "$$", "0 \\to I^n\\mathcal{F}/I^{n + 1}\\mathcal{F} \\to", "\\mathcal{F}/I^{n + 1}\\mathcal{F} \\to \\mathcal{F}/I^n\\mathcal{F} \\to 0", "$$", "induction and Modules on Sites, Lemma", "\\ref{sites-modules-lemma-kernel-finite-locally-constant}", "the lemma follows." ], "refs": [ "proetale-lemma-compare-constructible", "etale-cohomology-proposition-constructible-over-noetherian", "etale-cohomology-lemma-constructible-quasi-compact-quasi-separated", "sites-modules-lemma-kernel-finite-locally-constant" ], "ref_ids": [ 3806, 6706, 6527, 14273 ] } ], "ref_ids": [] }, { "id": 3812, "type": "theorem", "label": "proetale-lemma-weird", "categories": [ "proetale" ], "title": "proetale-lemma-weird", "contents": [ "Let $X$ be a weakly contractible affine scheme. Let $\\Lambda$ be a Noetherian", "ring and $I \\subset \\Lambda$ be an ideal. Let $\\mathcal{F}$ be a sheaf of", "$\\Lambda$-modules on $X_\\proetale$ such that", "\\begin{enumerate}", "\\item $\\mathcal{F} = \\lim \\mathcal{F}/I^n\\mathcal{F}$,", "\\item $\\mathcal{F}/I^n\\mathcal{F}$ is a constant sheaf of", "$\\Lambda/I^n$-modules,", "\\item $\\mathcal{F}/I\\mathcal{F}$ is of finite type.", "\\end{enumerate}", "Then $\\mathcal{F} \\cong \\underline{M}^\\wedge$ where $M$ is", "a finite $\\Lambda^\\wedge$-module." ], "refs": [], "proofs": [ { "contents": [ "Pick a $\\Lambda/I^n$-module $M_n$ such that", "$\\mathcal{F}/I^n\\mathcal{F} \\cong \\underline{M_n}$.", "Since we have the surjections", "$\\mathcal{F}/I^{n + 1}\\mathcal{F} \\to \\mathcal{F}/I^n\\mathcal{F}$", "we conclude that there", "exist surjections $M_{n + 1} \\to M_n$ inducing isomorphisms", "$M_{n + 1}/I^nM_{n + 1} \\to M_n$. Fix a choice of such surjections", "and set $M = \\lim M_n$. Then $M$ is an $I$-adically complete", "$\\Lambda$-module with $M/I^nM = M_n$, see", "Algebra, Lemma \\ref{algebra-lemma-limit-complete}.", "Since $M_1$ is a finite type $\\Lambda$-module", "(Modules on Sites, Lemma", "\\ref{sites-modules-lemma-locally-constant-finite-type})", "we see that $M$ is a finite $\\Lambda^\\wedge$-module.", "Consider the sheaves", "$$", "\\mathcal{I}_n = \\mathit{Isom}(\\underline{M_n}, \\mathcal{F}/I^n\\mathcal{F})", "$$", "on $X_\\proetale$. Modding out by $I^n$ defines a transition map", "$$", "\\mathcal{I}_{n + 1} \\longrightarrow \\mathcal{I}_n", "$$", "By our choice of $M_n$ the sheaf $\\mathcal{I}_n$ is a torsor under", "$$", "\\mathit{Isom}(\\underline{M_n}, \\underline{M_n}) =", "\\underline{\\text{Isom}_\\Lambda(M_n, M_n)}", "$$", "(Modules on Sites, Lemma \\ref{sites-modules-lemma-locally-constant})", "since $\\mathcal{F}/I^n\\mathcal{F}$ is (\\'etale) locally isomorphic", "to $\\underline{M_n}$. It follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-hom-systems-ML}", "that the system of sheaves $(\\mathcal{I}_n)$ is Mittag-Leffler.", "For each $n$ let $\\mathcal{I}'_n \\subset \\mathcal{I}_n$ be the", "image of $\\mathcal{I}_N \\to \\mathcal{I}_n$ for all $N \\gg n$.", "Then", "$$", "\\ldots \\to \\mathcal{I}'_3 \\to \\mathcal{I}'_2 \\to \\mathcal{I}'_1 \\to *", "$$", "is a sequence of sheaves of sets on $X_\\proetale$ with surjective", "transition maps. Since $*(X)$ is a singleton (not empty)", "and since evaluating at $X$ transforms surjective maps of sheaves of sets", "into surjections of sets, we can pick", "$s \\in \\lim \\mathcal{I}'_n(X)$. The sections define isomorphisms", "$\\underline{M}^\\wedge \\to \\lim \\mathcal{F}/I^n\\mathcal{F} = \\mathcal{F}$", "and the proof is done." ], "refs": [ "algebra-lemma-limit-complete", "sites-modules-lemma-locally-constant-finite-type", "sites-modules-lemma-locally-constant", "more-algebra-lemma-hom-systems-ML" ], "ref_ids": [ 880, 14269, 14272, 10422 ] } ], "ref_ids": [] }, { "id": 3813, "type": "theorem", "label": "proetale-lemma-connected-lisse", "categories": [ "proetale" ], "title": "proetale-lemma-connected-lisse", "contents": [ "Let $X$ be a connected scheme. Let $\\Lambda$ be a Noetherian ring and let", "$I \\subset \\Lambda$ be an ideal. If $\\mathcal{F}$ is a lisse", "constructible $\\Lambda$-sheaf on $X_\\proetale$, then $\\mathcal{F}$", "is adic lisse." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-compare-locally-constant} we have", "$\\mathcal{F}/I^n\\mathcal{F} = \\epsilon^{-1}\\mathcal{G}_n$", "for some locally constant sheaf $\\mathcal{G}_n$ of $\\Lambda/I^n$-modules. By", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-connected-locally-constant}", "there exists a finite $\\Lambda/I^n$-module $M_n$ such that", "$\\mathcal{G}_n$ is locally isomorphic to $\\underline{M_n}$.", "Choose a covering $\\{W_t \\to X\\}_{t \\in T}$ with each $W_t$", "affine and weakly contractible.", "Then $\\mathcal{F}|_{W_t}$ satisfies the assumptions of", "Lemma \\ref{lemma-weird}", "and hence $\\mathcal{F}|_{W_t} \\cong \\underline{N_t}^\\wedge$", "for some finite $\\Lambda^\\wedge$-module $N_t$. Note that", "$N_t/I^nN_t \\cong M_n$ for all $t$ and $n$. Hence", "$N_t \\cong N_{t'}$ for all $t, t' \\in T$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-isomorphic-completions}.", "This proves that $\\mathcal{F}$ is adic lisse." ], "refs": [ "proetale-lemma-compare-locally-constant", "etale-cohomology-lemma-connected-locally-constant", "proetale-lemma-weird", "more-algebra-lemma-isomorphic-completions" ], "ref_ids": [ 3787, 6507, 3812, 10423 ] } ], "ref_ids": [] }, { "id": 3814, "type": "theorem", "label": "proetale-lemma-Noetherian-adic-constructible", "categories": [ "proetale" ], "title": "proetale-lemma-Noetherian-adic-constructible", "contents": [ "Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring and", "let $I \\subset \\Lambda$ be an ideal. Let $\\mathcal{F}$ be a", "constructible $\\Lambda$-sheaf on $X_\\proetale$. Then $\\mathcal{F}$", "is adic constructible." ], "refs": [], "proofs": [ { "contents": [ "This is a consequence of Lemmas \\ref{lemma-Noetherian-constructible} and", "\\ref{lemma-connected-lisse}, the fact that a Noetherian scheme", "is locally connected", "(Topology, Lemma \\ref{topology-lemma-locally-Noetherian-locally-connected}),", "and the definitions." ], "refs": [ "proetale-lemma-Noetherian-constructible", "proetale-lemma-connected-lisse", "topology-lemma-locally-Noetherian-locally-connected" ], "ref_ids": [ 3811, 3813, 8223 ] } ], "ref_ids": [] }, { "id": 3815, "type": "theorem", "label": "proetale-lemma-derived-complete-zero", "categories": [ "proetale" ], "title": "proetale-lemma-derived-complete-zero", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a ring and let", "$I \\subset \\Lambda$ be a finitely generated ideal.", "Let $\\mathcal{F}$ be a sheaf of $\\Lambda$-modules on $X_\\proetale$.", "If $\\mathcal{F}$ is derived complete and $\\mathcal{F}/I\\mathcal{F} = 0$,", "then $\\mathcal{F} = 0$." ], "refs": [], "proofs": [ { "contents": [ "Assume that $\\mathcal{F}/I\\mathcal{F}$ is zero.", "Let $I = (f_1, \\ldots, f_r)$. Let $i < r$ be the largest", "integer such that $\\mathcal{G} = \\mathcal{F}/(f_1, \\ldots, f_i)\\mathcal{F}$", "is nonzero. If $i$ does not exist, then $\\mathcal{F} = 0$ which is what we", "want to show. Then $\\mathcal{G}$ is derived complete as a cokernel", "of a map between derived complete modules, see", "Proposition \\ref{proposition-enough-weakly-contractibles}.", "By our choice of $i$ we have that $f_{i + 1} : \\mathcal{G} \\to \\mathcal{G}$", "is surjective. Hence", "$$", "\\lim (\\ldots \\to \\mathcal{G} \\xrightarrow{f_{i + 1}} \\mathcal{G}", "\\xrightarrow{f_{i + 1}} \\mathcal{G})", "$$", "is nonzero, contradicting the derived completeness of $\\mathcal{G}$." ], "refs": [ "proetale-proposition-enough-weakly-contractibles" ], "ref_ids": [ 3828 ] } ], "ref_ids": [] }, { "id": 3816, "type": "theorem", "label": "proetale-lemma-derived-complete-limit", "categories": [ "proetale" ], "title": "proetale-lemma-derived-complete-limit", "contents": [ "Let $X$ be a weakly contractible affine scheme.", "Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal.", "Let $\\mathcal{F}$ be a derived complete sheaf of $\\Lambda$-modules", "on $X_\\proetale$ with $\\mathcal{F}/I\\mathcal{F}$ a locally", "constant sheaf of $\\Lambda/I$-modules of finite type.", "Then there exists an integer $t$ and a surjective map", "$$", "(\\underline{\\Lambda}^\\wedge)^{\\oplus t} \\to \\mathcal{F}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is weakly contractible, there exists a finite disjoint open", "covering $X = \\coprod U_i$ such that $\\mathcal{F}/I\\mathcal{F}|_{U_i}$", "is isomorphic to the constant sheaf associated to a finite $\\Lambda/I$-module", "$M_i$. Choose finitely many generators $m_{ij}$ of $M_i$. We", "can find sections $s_{ij} \\in \\mathcal{F}(X)$ restricting to", "$m_{ij}$ viewed as a section of $\\mathcal{F}/I\\mathcal{F}$ over $U_i$. ", "Let $t$ be the total number of $s_{ij}$. Then we obtain a map", "$$", "\\alpha : \\underline{\\Lambda}^{\\oplus t} \\longrightarrow \\mathcal{F}", "$$", "which is surjective modulo $I$ by construction. By", "Lemma \\ref{lemma-naive-completion}", "the derived completion of $\\underline{\\Lambda}^{\\oplus t}$ is the", "sheaf $(\\underline{\\Lambda}^\\wedge)^{\\oplus t}$. Since $\\mathcal{F}$", "is derived complete we see that $\\alpha$ factors through a map", "$$", "\\alpha^\\wedge :", "(\\underline{\\Lambda}^\\wedge)^{\\oplus t}", "\\longrightarrow", "\\mathcal{F}", "$$", "Then $\\mathcal{Q} = \\Coker(\\alpha^\\wedge)$ is a derived complete", "sheaf of $\\Lambda$-modules by", "Proposition \\ref{proposition-enough-weakly-contractibles}.", "By construction $\\mathcal{Q}/I\\mathcal{Q} = 0$. It follows from", "Lemma \\ref{lemma-derived-complete-zero}", "that $\\mathcal{Q} = 0$ which is what we wanted to show." ], "refs": [ "proetale-lemma-naive-completion", "proetale-proposition-enough-weakly-contractibles", "proetale-lemma-derived-complete-zero" ], "ref_ids": [ 3790, 3828, 3815 ] } ], "ref_ids": [] }, { "id": 3817, "type": "theorem", "label": "proetale-lemma-describe-constructible-complexes", "categories": [ "proetale" ], "title": "proetale-lemma-describe-constructible-complexes", "contents": [ "In the situation above suppose $K$ is in $D_{cons}(X, \\Lambda)$", "and $X$ is quasi-compact. Set", "$K_n = K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$.", "There exist $a, b$ such that", "\\begin{enumerate}", "\\item $K = R\\lim K_n$ and $H^i(K) = 0$ for $i \\not \\in [a, b]$,", "\\item each $K_n$ has tor amplitude in $[a, b]$,", "\\item each $K_n$ has constructible cohomology sheaves,", "\\item each $K_n = \\epsilon^{-1}L_n$ for some", "$L_n \\in D_{ctf}(X_\\etale, \\Lambda/I^n)$", "(\\'Etale Cohomology, Definition \\ref{etale-cohomology-definition-ctf}).", "\\end{enumerate}" ], "refs": [ "etale-cohomology-definition-ctf" ], "proofs": [ { "contents": [ "By definition of local having finite tor dimension, we can find", "$a, b$ such that $K_1$ has tor amplitude in $[a, b]$.", "Part (2) follows from ", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-bounded}.", "Then (1) follows as $K$ is derived complete by the description", "of limits in", "Cohomology on Sites, Proposition", "\\ref{sites-cohomology-proposition-enough-weakly-contractibles}", "and the fact that $H^b(K_{n + 1}) \\to H^b(K_n)$ is surjective", "as $K_n = K_{n + 1} \\otimes^\\mathbf{L}_\\Lambda \\underline{\\Lambda/I^n}$.", "Part (3) follows from", "Lemma \\ref{lemma-compare-truncations-constructible},", "Part (4) follows from", "Lemma \\ref{lemma-compare-constructible-derived}", "and the fact that $L_n$ has finite tor dimension because $K_n$ does", "(small argument omitted)." ], "refs": [ "sites-cohomology-lemma-bounded", "sites-cohomology-proposition-enough-weakly-contractibles", "proetale-lemma-compare-truncations-constructible", "proetale-lemma-compare-constructible-derived" ], "ref_ids": [ 4379, 4410, 3810, 3808 ] } ], "ref_ids": [ 6758 ] }, { "id": 3818, "type": "theorem", "label": "proetale-lemma-local-structure-constructible-complex", "categories": [ "proetale" ], "title": "proetale-lemma-local-structure-constructible-complex", "contents": [ "Let $X$ be a weakly contractible affine scheme. Let $\\Lambda$ be a Noetherian", "ring and let $I \\subset \\Lambda$ be an ideal. Let $K$ be an object of", "$D_{cons}(X, \\Lambda)$ such that the cohomology sheaves of", "$K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$ are locally", "constant. Then there exists a finite disjoint open covering", "$X = \\coprod U_i$ and for each $i$ a finite collection of", "finite projective $\\Lambda^\\wedge$-modules $M_a, \\ldots, M_b$", "such that $K|_{U_i}$ is represented by a complex", "$$", "(\\underline{M^a})^\\wedge \\to \\ldots \\to (\\underline{M^b})^\\wedge", "$$", "in $D(U_{i, \\proetale}, \\Lambda)$ for some maps of sheaves of", "$\\Lambda$-modules $(\\underline{M^i})^\\wedge \\to (\\underline{M^{i + 1}})^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "We freely use the results of", "Lemma \\ref{lemma-describe-constructible-complexes}.", "Choose $a, b$ as in that lemma. We will prove the lemma by", "induction on $b - a$. Let $\\mathcal{F} = H^b(K)$.", "Note that $\\mathcal{F}$ is a derived complete sheaf of", "$\\Lambda$-modules by", "Proposition \\ref{proposition-enough-weakly-contractibles}.", "Moreover $\\mathcal{F}/I\\mathcal{F}$ is a locally", "constant sheaf of $\\Lambda/I$-modules of finite type.", "Apply Lemma \\ref{lemma-derived-complete-limit} to get a surjection", "$\\rho : (\\underline{\\Lambda}^\\wedge)^{\\oplus t} \\to \\mathcal{F}$.", "\\medskip\\noindent", "If $a = b$, then $K = \\mathcal{F}[-b]$. In this case we see that", "$$", "\\mathcal{F} \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I} =", "\\mathcal{F}/I\\mathcal{F}", "$$", "As $X$ is weakly contractible and $\\mathcal{F}/I\\mathcal{F}$", "locally constant, we can find a finite disjoint", "union decomposition $X = \\coprod U_i$ by affine opens $U_i$", "and $\\Lambda/I$-modules $\\overline{M}_i$", "such that $\\mathcal{F}/I\\mathcal{F}$ restricts to", "$\\underline{\\overline{M}_i}$ on $U_i$. After refining the covering", "we may assume the map", "$$", "\\rho|_{U_i} \\bmod I :", "\\underline{\\Lambda/I}^{\\oplus t}", "\\longrightarrow", "\\underline{\\overline{M}_i}", "$$", "is equal to $\\underline{\\alpha_i}$ for some surjective module map", "$\\alpha_i : \\Lambda/I^{\\oplus t} \\to \\overline{M}_i$, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-morphism-locally-constant}.", "Note that each $\\overline{M}_i$ is a finite $\\Lambda/I$-module.", "Since $\\mathcal{F}/I\\mathcal{F}$ has tor amplitude in $[0, 0]$", "we conclude that $\\overline{M}_i$ is a flat $\\Lambda/I$-module.", "Hence $\\overline{M}_i$ is finite projective", "(Algebra, Lemma \\ref{algebra-lemma-finite-projective}).", "Hence we can find a projector", "$\\overline{p}_i : (\\Lambda/I)^{\\oplus t} \\to (\\Lambda/I)^{\\oplus t}$", "whose image maps isomorphically to $\\overline{M}_i$ under the map $\\alpha_i$.", "We can lift $\\overline{p}_i$ to a projector", "$p_i : (\\Lambda^\\wedge)^{\\oplus t} \\to", "(\\Lambda^\\wedge)^{\\oplus t}$\\footnote{Proof: by", "Algebra, Lemma \\ref{algebra-lemma-lift-idempotents-noncommutative}", "we can lift $\\overline{p}_i$ to a compatible system of", "projectors $p_{i, n} : (\\Lambda/I^n)^{\\oplus t} \\to (\\Lambda/I^n)^{\\oplus t}$", "and then we set $p_i = \\lim p_{i, n}$ which works because", "$\\Lambda^\\wedge = \\lim \\Lambda/I^n$.}.", "Then $M_i = \\Im(p_i)$ is a finite $I$-adically complete", "$\\Lambda^\\wedge$-module with $M_i/IM_i = \\overline{M}_i$.", "Over $U_i$ consider the maps", "$$", "\\underline{M_i}^\\wedge \\to", "(\\underline{\\Lambda}^\\wedge)^{\\oplus t} \\to", "\\mathcal{F}|_{U_i}", "$$", "By construction the composition induces an isomorphism modulo $I$.", "The source and target are derived complete, hence so are the cokernel", "$\\mathcal{Q}$ and the kernel $\\mathcal{K}$. We have", "$\\mathcal{Q}/I\\mathcal{Q} = 0$ by construction hence $\\mathcal{Q}$", "is zero by Lemma \\ref{lemma-derived-complete-zero}.", "Then", "$$", "0 \\to \\mathcal{K}/I\\mathcal{K} \\to", "\\underline{\\overline{M}_i}", "\\to \\mathcal{F}/I\\mathcal{F} \\to 0", "$$", "is exact by the vanishing of $\\text{Tor}_1$ see at the start of this", "paragraph; also use that", "$\\underline{\\Lambda}^\\wedge/I\\overline{\\Lambda}^\\wedge$ by", "Modules on Sites, Lemma \\ref{sites-modules-lemma-completion-flat}", "to see that", "$\\underline{M_i}^\\wedge/I\\underline{M_i}^\\wedge = \\underline{\\overline{M}_i}$.", "Hence $\\mathcal{K}/I\\mathcal{K} = 0$ by construction and we conclude", "that $\\mathcal{K} = 0$ as before. This proves the result in case $a = b$.", "\\medskip\\noindent", "If $b > a$, then we lift the map $\\rho$ to a map", "$$", "\\tilde \\rho : (\\underline{\\Lambda}^\\wedge)^{\\oplus t}[-b] \\longrightarrow K", "$$", "in $D(X_\\proetale, \\Lambda)$. This is possible as we can think of", "$K$ as a complex of $\\underline{\\Lambda}^\\wedge$-modules by", "discussion in the introduction to", "Section \\ref{section-derived-completion-noetherian}", "and because $X_\\proetale$ is weakly contractible", "hence there is no obstruction to lifting the elements", "$\\rho(e_s) \\in H^0(X, \\mathcal{F})$ to elements of $H^b(X, K)$.", "Fitting $\\tilde \\rho$ into a distinguished triangle", "$$", "(\\underline{\\Lambda}^\\wedge)^{\\oplus t}[-b] \\to K \\to L \\to", "(\\underline{\\Lambda}^\\wedge)^{\\oplus t}[-b + 1]", "$$", "we see that $L$ is an object of $D_{cons}(X, \\Lambda)$ such", "that $L \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$", "has tor amplitude contained in $[a, b - 1]$ (details omitted).", "By induction we can describe $L$ locally as stated in the lemma, say", "$L$ is isomorphic to", "$$", "(\\underline{M^a})^\\wedge \\to \\ldots \\to (\\underline{M^{b - 1}})^\\wedge", "$$", "The map", "$L \\to (\\underline{\\Lambda}^\\wedge)^{\\oplus t}[-b + 1]$", "corresponds to a map", "$(\\underline{M^{b - 1}})^\\wedge \\to (\\underline{\\Lambda}^\\wedge)^{\\oplus t}$", "which allows us to extend the complex by one. The corresponding", "complex is isomorphic to $K$ in the derived category by the properties", "of triangulated categories. This finishes the proof." ], "refs": [ "proetale-lemma-describe-constructible-complexes", "proetale-proposition-enough-weakly-contractibles", "proetale-lemma-derived-complete-limit", "sites-modules-lemma-morphism-locally-constant", "algebra-lemma-finite-projective", "algebra-lemma-lift-idempotents-noncommutative", "proetale-lemma-derived-complete-zero", "sites-modules-lemma-completion-flat" ], "ref_ids": [ 3817, 3828, 3816, 14271, 795, 462, 3815, 14268 ] } ], "ref_ids": [] }, { "id": 3819, "type": "theorem", "label": "proetale-lemma-weakly-contractible-locally-constant-ML", "categories": [ "proetale" ], "title": "proetale-lemma-weakly-contractible-locally-constant-ML", "contents": [ "Let $X$ be a weakly contractible affine scheme. Let $\\Lambda$ be a Noetherian", "ring and let $I \\subset \\Lambda$ be an ideal. Let $K$ be an object of", "$D_{cons}(X, \\Lambda)$ such that", "$K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$", "is isomorphic in $D(X_\\proetale, \\Lambda/I^n)$ to a", "complex of constant sheaves of $\\Lambda/I^n$-modules. Then", "$$", "H^0(X, K \\otimes_\\Lambda^\\mathbf{L} \\Lambda/I^n)", "$$", "has the Mittag-Leffler condition." ], "refs": [], "proofs": [ { "contents": [ "Say $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$ is isomorphic", "to $\\underline{E_n}$ for some object $E_n$ of $D(\\Lambda/I^n)$.", "Since $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$ has", "finite tor dimension and has finite type cohomology sheaves", "we see that $E_1$ is perfect (see", "More on Algebra, Lemma \\ref{more-algebra-lemma-perfect}). The transition maps", "$$", "K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^{n + 1}}", "\\to", "K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}", "$$", "locally come from (possibly many distinct) maps of complexes", "$E_{n + 1} \\to E_n$ in $D(\\Lambda/I^{n + 1})$ see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-locally-constant-map}.", "For each $n$ choose one such map and observe that it induces", "an isomorphism", "$E_{n + 1} \\otimes_{\\Lambda/I^{n + 1}}^\\mathbf{L} \\Lambda/I^n \\to E_n$", "in $D(\\Lambda/I^n)$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-Rlim-perfect-gives-complete}", "we can find a finite complex $M^\\bullet$ of finite projective", "$\\Lambda^\\wedge$-modules and isomorphisms $M^\\bullet/I^nM^\\bullet \\to E_n$", "in $D(\\Lambda/I^n)$ compatible with the transition maps.", "\\medskip\\noindent", "Now observe that for each finite collection of indices", "$n > m > k$ the triple of maps", "$$", "H^0(X, K \\otimes_\\Lambda^\\mathbf{L} \\Lambda/I^n)", "\\to", "H^0(X, K \\otimes_\\Lambda^\\mathbf{L} \\Lambda/I^m)", "\\to", "H^0(X, K \\otimes_\\Lambda^\\mathbf{L} \\Lambda/I^k)", "$$", "is isomorphic to", "$$", "H^0(X, \\underline{M^\\bullet/I^nM^\\bullet})", "\\to", "H^0(X, \\underline{M^\\bullet/I^mM^\\bullet})", "\\to", "H^0(X, \\underline{M^\\bullet/I^kM^\\bullet})", "$$", "Namely, choose any isomorphism", "$$", "\\underline{M^\\bullet/I^nM^\\bullet} \\to", "K \\otimes_\\Lambda^\\mathbf{L} \\Lambda/I^n", "$$", "induces similar isomorphisms module $I^m$ and $I^k$ and we see that", "the assertion is true. Thus to prove the lemma it suffices to show that", "the system", "$H^0(X, \\underline{M^\\bullet/I^nM^\\bullet})$ has Mittag-Leffler.", "Since taking sections over $X$ is exact, it suffices to prove that", "the system of $\\Lambda$-modules", "$$", "H^0(M^\\bullet/I^nM^\\bullet)", "$$", "has Mittag-Leffler. Set $A = \\Lambda^\\wedge$ and consider the spectral", "sequence", "$$", "\\text{Tor}_{-p}^A(H^q(M^\\bullet), A/I^nA) \\Rightarrow", "H^{p + q}(M^\\bullet/I^nM^\\bullet)", "$$", "By More on Algebra, Lemma \\ref{more-algebra-lemma-tor-strictly-pro-zero}", "the pro-systems $\\{\\text{Tor}_{-p}^A(H^q(M^\\bullet), A/I^nA)\\}$ are zero", "for $p > 0$. Thus the pro-system $\\{H^0(M^\\bullet/I^nM^\\bullet)\\}$", "is equal to the pro-system $\\{H^0(M^\\bullet)/I^nH^0(M^\\bullet)\\}$", "and the lemma is proved." ], "refs": [ "more-algebra-lemma-perfect", "sites-cohomology-lemma-locally-constant-map", "more-algebra-lemma-Rlim-perfect-gives-complete", "more-algebra-lemma-tor-strictly-pro-zero" ], "ref_ids": [ 10212, 4407, 10410, 9954 ] } ], "ref_ids": [] }, { "id": 3820, "type": "theorem", "label": "proetale-lemma-connected-adic-lisse", "categories": [ "proetale" ], "title": "proetale-lemma-connected-adic-lisse", "contents": [ "Let $X$ be a connected scheme. Let $\\Lambda$ be a Noetherian ring and let", "$I \\subset \\Lambda$ be an ideal. If $K$ is in $D_{cons}(X, \\Lambda)$", "such that $K \\otimes_\\Lambda \\underline{\\Lambda/I}$", "has locally constant cohomology sheaves, then $K$ is adic lisse", "(Definition \\ref{definition-adic-constructible})." ], "refs": [ "proetale-definition-adic-constructible" ], "proofs": [ { "contents": [ "Write $K_n = K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$.", "We will use the results of Lemma \\ref{lemma-describe-constructible-complexes}", "without further mention. By Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-locally-constant-bounded}", "we see that $K_n$ has locally constant cohomology sheaves for all $n$.", "We have $K_n = \\epsilon^{-1}L_n$ some $L_n$ in", "$D_{ctf}(X_\\etale, \\Lambda/I^n)$ with locally constant cohomology sheaves.", "By \\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-connected-ctf-locally-constant}", "there exist perfect $M_n \\in D(\\Lambda/I^n)$", "such that $L_n$ is \\'etale locally isomorphic to $\\underline{M_n}$.", "The maps $L_{n + 1} \\to L_n$ corresponding to $K_{n + 1} \\to K_n$", "induces isomorphisms", "$L_{n + 1} \\otimes_{\\Lambda/I^{n + 1}}^\\mathbf{L} \\underline{\\Lambda/I^n}", "\\to L_n$. Looking locally on $X$ we conclude that there", "exist maps $M_{n + 1} \\to M_n$ in $D(\\Lambda/I^{n + 1})$", "inducing isomorphisms", "$M_{n + 1} \\otimes_{\\Lambda/I^{n + 1}} \\Lambda/I^n \\to M_n$, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-locally-constant-map}.", "Fix a choice of such maps. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-Rlim-perfect-gives-complete}", "we can find a finite complex $M^\\bullet$ of finite projective", "$\\Lambda^\\wedge$-modules and isomorphisms $M^\\bullet/I^nM^\\bullet \\to M_n$", "in $D(\\Lambda/I^n)$ compatible with the transition maps.", "To finish the proof we will show that $K$ is locally isomorphic", "to", "$$", "\\underline{M^\\bullet}^\\wedge =", "\\lim \\underline{M^\\bullet/I^nM^\\bullet} =", "R\\lim \\underline{M^\\bullet/I^nM^\\bullet}", "$$", "Let $E^\\bullet$ be the dual complex to $M^\\bullet$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-dual-perfect-complex}", "and its proof. Consider the objects", "$$", "H_n =", "R\\SheafHom_{\\Lambda/I^n}(\\underline{M^\\bullet/I^nM^\\bullet}, K_n) =", "\\underline{E^\\bullet/I^nE^\\bullet} \\otimes_{\\Lambda/I^n}^\\mathbf{L} K_n", "$$", "of $D(X_\\proetale, \\Lambda/I^n)$. Modding out by $I^n$ defines a", "transition map $H_{n + 1} \\to H_n$. Set $H = R\\lim H_n$. Then $H$ is an", "object of $D_{cons}(X, \\Lambda)$ (details omitted) with", "$H \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n} = H_n$.", "Choose a covering $\\{W_t \\to X\\}_{t \\in T}$ with each $W_t$", "affine and weakly contractible. By our choice of $M^\\bullet$", "we see that", "\\begin{align*}", "H_n|_{W_t} & \\cong ", "R\\SheafHom_{\\Lambda/I^n}(\\underline{M^\\bullet/I^nM^\\bullet},", "\\underline{M^\\bullet/I^nM^\\bullet}) \\\\", "& =", "\\underline{", "\\text{Tot}(E^\\bullet/I^nE^\\bullet \\otimes_{\\Lambda/I^n} M^\\bullet/I^nM^\\bullet)", "}", "\\end{align*}", "Thus we may apply Lemma \\ref{lemma-weakly-contractible-locally-constant-ML}", "to $H = R\\lim H_n$. We conclude the system $H^0(W_t, H_n)$ satisfies", "Mittag-Leffler. Since for all $n \\gg 1$ there is an element of $H^0(W_t, H_n)$", "which maps to an isomorphism in", "$$", "H^0(W_t, H_1) = \\Hom(\\underline{M^\\bullet/IM^\\bullet}, K_1)", "$$", "we find an element $(\\varphi_{t, n})$", "in the inverse limit which produces an isomorphism mod $I$. Then", "$$", "R\\lim \\varphi_{t, n} :", "\\underline{M^\\bullet}^\\wedge|_{W_t} =", "R\\lim \\underline{M^\\bullet/I^nM^\\bullet}|_{W_t}", "\\longrightarrow", "R\\lim K_n|_{W_t} = K|_{W_t}", "$$", "is an isomorphism. This finishes the proof." ], "refs": [ "proetale-lemma-describe-constructible-complexes", "sites-cohomology-lemma-locally-constant-bounded", "etale-cohomology-lemma-connected-ctf-locally-constant", "sites-cohomology-lemma-locally-constant-map", "more-algebra-lemma-Rlim-perfect-gives-complete", "more-algebra-lemma-dual-perfect-complex", "proetale-lemma-weakly-contractible-locally-constant-ML" ], "ref_ids": [ 3817, 4409, 6563, 4407, 10410, 10224, 3819 ] } ], "ref_ids": [ 3844 ] }, { "id": 3821, "type": "theorem", "label": "proetale-lemma-proetale-induced", "categories": [ "proetale" ], "title": "proetale-lemma-proetale-induced", "contents": [ "Let $\\Sch_\\proetale$ be a big pro-\\'etale site as in", "Definition \\ref{definition-big-proetale-site}.", "Let $T \\in \\Ob(\\Sch_\\proetale)$.", "Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary pro-\\'etale covering of $T$.", "There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site", "$\\Sch_\\proetale$ which refines $\\{T_i \\to T\\}_{i \\in I}$." ], "refs": [ "proetale-definition-big-proetale-site" ], "proofs": [ { "contents": [ "Namely, we first let $\\{V_k \\to T\\}$ be a covering as in", "Lemma \\ref{lemma-get-many-weakly-contractible}.", "Then the pro-\\'etale coverings $\\{T_i \\times_T V_k \\to V_k\\}$", "can be refined by a finite disjoint open covering", "$V_k = V_{k, 1} \\amalg \\ldots \\amalg V_{k, n_k}$, see", "Lemma \\ref{lemma-w-contractible-proetale-cover}.", "Then $\\{V_{k, i} \\to T\\}$ is a covering of $\\Sch_\\proetale$", "which refines $\\{T_i \\to T\\}_{i \\in I}$." ], "refs": [ "proetale-lemma-get-many-weakly-contractible", "proetale-lemma-w-contractible-proetale-cover" ], "ref_ids": [ 3762, 3760 ] } ], "ref_ids": [ 3837 ] }, { "id": 3822, "type": "theorem", "label": "proetale-lemma-proetale-cohomology-independent-partial-universe", "categories": [ "proetale" ], "title": "proetale-lemma-proetale-cohomology-independent-partial-universe", "contents": [ "Let $S$ be a scheme. Let $S_\\proetale \\subset S_\\proetale'$ be", "two small pro-\\'etale sites of $S$ as constructed in", "Definition \\ref{definition-big-small-proetale}. Then the inclusion functor", "satisfies the assumptions of ", "Sites, Lemma \\ref{sites-lemma-bigger-site}.", "Hence there exist morphisms of topoi", "$$", "\\xymatrix{", "\\Sh(S_\\proetale) \\ar[r]^g &", "\\Sh(S_\\proetale') \\ar[r]^f &", "\\Sh(S_\\proetale)", "}", "$$", "whose composition is isomorphic to the identity and with $f_* = g^{-1}$.", "Moreover,", "\\begin{enumerate}", "\\item for $\\mathcal{F}' \\in \\textit{Ab}(S_\\proetale')$ we have", "$H^p(S_\\proetale', \\mathcal{F}') = H^p(S_\\proetale, g^{-1}\\mathcal{F}')$,", "\\item for $\\mathcal{F} \\in \\textit{Ab}(S_\\proetale)$ we have", "$$", "H^p(S_\\proetale, \\mathcal{F}) =", "H^p(S_\\proetale', g_*\\mathcal{F}) =", "H^p(S_\\proetale', f^{-1}\\mathcal{F}).", "$$", "\\end{enumerate}" ], "refs": [ "proetale-definition-big-small-proetale", "sites-lemma-bigger-site" ], "proofs": [ { "contents": [ "The inclusion functor is fully faithful and continuous.", "We have seen that $S_\\proetale$ and $S_\\proetale'$ have fibre products", "and final objects and that our functor commutes with these", "(Lemma \\ref{lemma-fibre-products-proetale}).", "It follows from Lemma \\ref{lemma-proetale-induced}", "that the inclusion functor is cocontinuous.", "Hence the existence of $f$ and $g$ follows from", "Sites, Lemma \\ref{sites-lemma-bigger-site}.", "The equality in (1) is", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-bigger-site}.", "Part (2) follows from (1) as", "$\\mathcal{F} = g^{-1}g_*\\mathcal{F} = g^{-1}f^{-1}\\mathcal{F}$." ], "refs": [ "proetale-lemma-fibre-products-proetale", "proetale-lemma-proetale-induced", "sites-lemma-bigger-site", "sites-cohomology-lemma-cohomology-bigger-site" ], "ref_ids": [ 3748, 3821, 8548, 4187 ] } ], "ref_ids": [ 3838, 8548 ] }, { "id": 3823, "type": "theorem", "label": "proetale-lemma-change-alpha", "categories": [ "proetale" ], "title": "proetale-lemma-change-alpha", "contents": [ "Suppose given big sites $\\Sch_\\proetale$ and $\\Sch'_\\proetale$ as in", "Definition \\ref{definition-big-proetale-site}.", "Assume that $\\Sch_\\proetale$ is contained in $\\Sch'_\\proetale$.", "The inclusion functor $\\Sch_\\proetale \\to \\Sch'_\\proetale$ satisfies", "the assumptions of Sites, Lemma \\ref{sites-lemma-bigger-site}.", "There are morphisms of topoi", "\\begin{eqnarray*}", "g : \\Sh(\\Sch_\\proetale) &", "\\longrightarrow &", "\\Sh(\\Sch'_\\proetale) \\\\", "f : \\Sh(\\Sch'_\\proetale) &", "\\longrightarrow &", "\\Sh(\\Sch_\\proetale)", "\\end{eqnarray*}", "such that $f \\circ g \\cong \\text{id}$. For any object $S$", "of $\\Sch_\\proetale$ the inclusion functor", "$(\\Sch/S)_\\proetale \\to (\\Sch'/S)_\\proetale$ satisfies", "the assumptions of Sites, Lemma \\ref{sites-lemma-bigger-site}", "also. Hence similarly we obtain morphisms", "\\begin{eqnarray*}", "g : \\Sh((\\Sch/S)_\\proetale) &", "\\longrightarrow &", "\\Sh((\\Sch'/S)_\\proetale) \\\\", "f : \\Sh((\\Sch'/S)_\\proetale) &", "\\longrightarrow &", "\\Sh((\\Sch/S)_\\proetale)", "\\end{eqnarray*}", "with $f \\circ g \\cong \\text{id}$." ], "refs": [ "proetale-definition-big-proetale-site", "sites-lemma-bigger-site", "sites-lemma-bigger-site" ], "proofs": [ { "contents": [ "Assumptions (b), (c), and (e) of", "Sites, Lemma \\ref{sites-lemma-bigger-site}", "are immediate for the functors", "$\\Sch_\\proetale \\to \\Sch'_\\proetale$ and", "$(\\Sch/S)_\\proetale \\to (\\Sch'/S)_\\proetale$. Property (a) holds by", "Lemma \\ref{lemma-proetale-induced}.", "Property (d) holds because", "fibre products in the categories $\\Sch_\\proetale$, $\\Sch'_\\proetale$", "exist and are compatible with fibre products in the category of schemes." ], "refs": [ "sites-lemma-bigger-site", "proetale-lemma-proetale-induced" ], "ref_ids": [ 8548, 3821 ] } ], "ref_ids": [ 3837, 8548, 8548 ] }, { "id": 3824, "type": "theorem", "label": "proetale-lemma-cohomology-enlarge-partial-universe", "categories": [ "proetale" ], "title": "proetale-lemma-cohomology-enlarge-partial-universe", "contents": [ "Let $S$ be a scheme. Let $(\\Sch/S)_\\proetale$ and $(\\Sch'/S)_\\proetale$ be two", "big pro-\\'etale sites of $S$ as in", "Definition \\ref{definition-big-small-proetale}.", "Assume that the first is contained in", "the second. In this case", "\\begin{enumerate}", "\\item for any abelian sheaf $\\mathcal{F}'$ defined on $(\\Sch'/S)_\\proetale$", "and any object $U$ of $(\\Sch/S)_\\proetale$ we have", "$$", "H^p(U, \\mathcal{F}'|_{(\\Sch/S)_\\proetale}) =", "H^p(U, \\mathcal{F}')", "$$", "In words: the cohomology of $\\mathcal{F}'$ over $U$ computed in the bigger site", "agrees with the cohomology of $\\mathcal{F}'$ restricted to the smaller site", "over $U$.", "\\item for any abelian sheaf $\\mathcal{F}$ on $(\\Sch/S)_\\proetale$ there is an", "abelian sheaf $\\mathcal{F}'$ on $(\\Sch/S)_\\proetale'$ whose restriction to", "$(\\Sch/S)_\\proetale$ is isomorphic to $\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [ "proetale-definition-big-small-proetale" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-change-alpha} the inclusion functor", "$(\\Sch/S)_\\proetale \\to (\\Sch'/S)_\\proetale$ satisfies the assumptions of", "Sites, Lemma \\ref{sites-lemma-bigger-site}. This implies (2) and (1)", "follows from", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-bigger-site}." ], "refs": [ "proetale-lemma-change-alpha", "sites-lemma-bigger-site", "sites-cohomology-lemma-cohomology-bigger-site" ], "ref_ids": [ 3823, 8548, 4187 ] } ], "ref_ids": [ 3838 ] }, { "id": 3825, "type": "theorem", "label": "proetale-proposition-maps-wich-identify-local-rings", "categories": [ "proetale" ], "title": "proetale-proposition-maps-wich-identify-local-rings", "contents": [ "Let $A \\to B$ be a ring map which identifies local rings.", "Then there exists a faithfully flat, ind-Zariski ring map", "$B \\to B'$ such that $A \\to B'$ is ind-Zariski." ], "refs": [], "proofs": [ { "contents": [ "Let $A \\to A_w$, resp. $B \\to B_w$ be the faithfully flat, ind-Zariski ring", "map constructed in Lemma \\ref{lemma-make-w-local} for $A$, resp.\\ $B$.", "Since $\\Spec(B_w)$ is w-local, there exists a unique factorization", "$A \\to A_w \\to B_w$ such that $\\Spec(B_w) \\to \\Spec(A_w)$ is w-local", "by Lemma \\ref{lemma-universal}. Note that $A_w \\to B_w$ identifies", "local rings, see", "Lemma \\ref{lemma-local-isomorphism-permanence}.", "By Lemma \\ref{lemma-w-local-morphism-equal-stalks-is-ind-zariski}", "this means $A_w \\to B_w$ is ind-Zariski. Since $B \\to B_w$ is", "faithfully flat, ind-Zariski (Lemma \\ref{lemma-make-w-local})", "and the composition $A \\to B \\to B_w$ is ind-Zariski", "(Lemma \\ref{lemma-composition-ind-zariski})", "the proposition is proved." ], "refs": [ "proetale-lemma-make-w-local", "proetale-lemma-universal", "proetale-lemma-local-isomorphism-permanence", "proetale-lemma-w-local-morphism-equal-stalks-is-ind-zariski", "proetale-lemma-make-w-local", "proetale-lemma-composition-ind-zariski" ], "ref_ids": [ 3714, 3715, 3702, 3722, 3714, 3708 ] } ], "ref_ids": [] }, { "id": 3826, "type": "theorem", "label": "proetale-proposition-weakly-etale", "categories": [ "proetale" ], "title": "proetale-proposition-weakly-etale", "contents": [ "Let $A \\to B$ be a weakly \\'etale ring map.", "Then there exists a faithfully flat, ind-\\'etale ring map", "$B \\to B'$ such that $A \\to B'$ is ind-\\'etale." ], "refs": [], "proofs": [ { "contents": [ "The ring map $A \\to B$ induces (separable) algebraic extensions of", "residue fields, see More on Algebra, Lemma", "\\ref{more-algebra-lemma-weakly-etale-residue-field-extensions}.", "Thus we may apply", "Lemma \\ref{lemma-get-w-local-algebraic-residue-field-extensions}", "and choose a diagram", "$$", "\\xymatrix{", "B \\ar[r] & D \\\\", "A \\ar[r] \\ar[u] & C \\ar[u]", "}", "$$", "with the properties as listed in the lemma. Note that $C \\to D$", "is weakly \\'etale by", "More on Algebra, Lemma \\ref{more-algebra-lemma-weakly-etale-permanence}.", "Pick a maximal ideal $\\mathfrak m \\subset D$. By construction", "this lies over a maximal ideal $\\mathfrak m' \\subset C$.", "By More on Algebra, Theorem \\ref{more-algebra-theorem-olivier}", "the ring map $C_{\\mathfrak m'} \\to D_\\mathfrak m$ is an isomorphism.", "As every point of $\\Spec(C)$ specializes to a closed point we conclude that", "$C \\to D$ identifies local rings.", "Thus Proposition \\ref{proposition-maps-wich-identify-local-rings}", "applies to the ring map $C \\to D$. Pick $D \\to D'$ faithfully flat", "and ind-Zariski such that $C \\to D'$ is ind-Zariski. Then", "$B \\to D'$ is a solution to the problem posed in the proposition." ], "refs": [ "more-algebra-lemma-weakly-etale-residue-field-extensions", "proetale-lemma-get-w-local-algebraic-residue-field-extensions", "more-algebra-lemma-weakly-etale-permanence", "more-algebra-theorem-olivier", "proetale-proposition-maps-wich-identify-local-rings" ], "ref_ids": [ 10453, 3735, 10447, 9805, 3825 ] } ], "ref_ids": [] }, { "id": 3827, "type": "theorem", "label": "proetale-proposition-find-w-contractible", "categories": [ "proetale" ], "title": "proetale-proposition-find-w-contractible", "contents": [ "For every ring $A$ there exists a faithfully flat, ind-\\'etale ring", "map $A \\to D$ such that $D$ is w-contractible." ], "refs": [], "proofs": [ { "contents": [ "Applying Lemma \\ref{lemma-get-w-local-algebraic-residue-field-extensions}", "to $\\text{id}_A : A \\to A$ we find a faithfully flat, ind-\\'etale ring map", "$A \\to C$ such that $C$ is w-local and such that every local ring at a", "maximal ideal of $C$ is strictly henselian.", "Choose an extremally disconnected space $T$ and a surjective", "continuous map $T \\to \\pi_0(\\Spec(C))$, see", "Topology, Lemma \\ref{topology-lemma-existence-projective-cover}.", "Note that $T$ is profinite. Apply Lemma \\ref{lemma-construct-profinite}", "to find an ind-Zariski ring map $C \\to D$ such that", "$\\pi_0(\\Spec(D)) \\to \\pi_0(\\Spec(C))$ realizes $T \\to \\pi_0(\\Spec(C))$", "and such that", "$$", "\\xymatrix{", "\\Spec(D) \\ar[r] \\ar[d] & \\pi_0(\\Spec(D)) \\ar[d] \\\\", "\\Spec(C) \\ar[r] & \\pi_0(\\Spec(C))", "}", "$$", "is cartesian in the category of topological spaces. Note that $\\Spec(D)$", "is w-local, that $\\Spec(D) \\to \\Spec(C)$ is w-local, and that the", "set of closed points of $\\Spec(D)$ is the inverse image of the", "set of closed points of $\\Spec(C)$, see Lemma \\ref{lemma-silly}.", "Thus it is still true that the local rings of $D$ at its maximal", "ideals are strictly henselian (as they are isomorphic to the", "local rings at the corresponding maximal ideals of $C$).", "It follows from", "Lemma \\ref{lemma-w-local-strictly-henselian-extremally-disconnected}", "that $D$ is w-contractible." ], "refs": [ "proetale-lemma-get-w-local-algebraic-residue-field-extensions", "topology-lemma-existence-projective-cover", "proetale-lemma-construct-profinite", "proetale-lemma-silly", "proetale-lemma-w-local-strictly-henselian-extremally-disconnected" ], "ref_ids": [ 3735, 8332, 3720, 3699, 3739 ] } ], "ref_ids": [] }, { "id": 3828, "type": "theorem", "label": "proetale-proposition-enough-weakly-contractibles", "categories": [ "proetale" ], "title": "proetale-proposition-enough-weakly-contractibles", "contents": [ "Let $\\mathcal{C}$ be a site. Assume $\\mathcal{C}$ has enough", "weakly contractible objects.", "Let $\\Lambda$ be a Noetherian ring. Let $I \\subset \\Lambda$ be an ideal.", "\\begin{enumerate}", "\\item The category of derived complete sheaves $\\Lambda$-modules is a", "weak Serre subcategory of $\\textit{Mod}(\\mathcal{C}, \\Lambda)$.", "\\item A sheaf $\\mathcal{F}$ of $\\Lambda$-modules satisfies", "$\\mathcal{F} = \\lim \\mathcal{F}/I^n\\mathcal{F}$ if and only if", "$\\mathcal{F}$ is derived complete and $\\bigcap I^n\\mathcal{F} = 0$.", "\\item The sheaf $\\underline{\\Lambda}^\\wedge$ is derived complete.", "\\item If $\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1$", "is an inverse system of derived complete sheaves of $\\Lambda$-modules,", "then $\\lim \\mathcal{F}_n$ is derived complete.", "\\item An object $K \\in D(\\mathcal{C}, \\Lambda)$ is derived complete if", "and only if each cohomology sheaf $H^p(K)$ is derived complete.", "\\item An object $K \\in D_{comp}(\\mathcal{C}, \\Lambda)$ is bounded above", "if and only if $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$", "is bounded above.", "\\item An object $K \\in D_{comp}(\\mathcal{C}, \\Lambda)$ is bounded", "if $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$ has finite", "tor dimension.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset such that every", "$U \\in \\mathcal{B}$ is weakly contractible and every object of $\\mathcal{C}$", "has a covering by elements of $\\mathcal{B}$.", "We will use the results of Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-w-contractible} and", "Proposition \\ref{sites-cohomology-proposition-enough-weakly-contractibles}", "without further mention.", "\\medskip\\noindent", "Recall that $R\\lim$ commutes with $R\\Gamma(U, -)$,", "see Injectives, Lemma \\ref{injectives-lemma-RF-commutes-with-Rlim}.", "Let $f \\in I$. Recall that $T(K, f)$ is the homotopy limit", "of the system", "$$", "\\ldots \\xrightarrow{f} K \\xrightarrow{f} K \\xrightarrow{f} K", "$$", "in $D(\\mathcal{C}, \\Lambda)$. Thus", "$$", "R\\Gamma(U, T(K, f)) = T(R\\Gamma(U, K), f).", "$$", "Since we can test isomorphisms of maps between objects of", "$D(\\mathcal{C}, \\Lambda)$ by evaluating at $U \\in \\mathcal{B}$", "we conclude an object $K$ of $D(\\mathcal{C}, \\Lambda)$", "is derived complete if and only if for every $U \\in \\mathcal{B}$ the", "object $R\\Gamma(U, K)$ is derived complete as an object of $D(\\Lambda)$.", "\\medskip\\noindent", "The remark above implies that items (1), (5) follow from the corresponding", "results for modules over rings, see", "More on Algebra, Lemmas \\ref{more-algebra-lemma-hom-from-Af} and", "\\ref{more-algebra-lemma-serre-subcategory}.", "In the same way (2) can be deduced from", "More on Algebra, Proposition", "\\ref{more-algebra-proposition-derived-complete-modules}", "as $(I^n\\mathcal{F})(U) = I^n \\cdot \\mathcal{F}(U)$", "for $U \\in \\mathcal{B}$ (by exactness of evaluating at $U$).", "\\medskip\\noindent", "Proof of (4). The homotopy limit $R\\lim \\mathcal{F}_n$ is in", "$D_{comp}(X, \\Lambda)$ (see discussion following", "Algebraic and Formal Geometry, Definition", "\\ref{algebraization-definition-derived-complete}).", "By part (5) just proved we conclude that", "$\\lim \\mathcal{F}_n = H^0(R\\lim \\mathcal{F}_n)$", "is derived complete.", "Part (3) is a special case of (4).", "\\medskip\\noindent", "Proof of (6) and (7). Follows from", "Lemma \\ref{lemma-naive-completion}", "and", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-bounded}", "and the computation of homotopy limits in Cohomology on Sites,", "Proposition \\ref{sites-cohomology-proposition-enough-weakly-contractibles}." ], "refs": [ "sites-cohomology-lemma-w-contractible", "sites-cohomology-proposition-enough-weakly-contractibles", "injectives-lemma-RF-commutes-with-Rlim", "more-algebra-lemma-hom-from-Af", "more-algebra-lemma-serre-subcategory", "more-algebra-proposition-derived-complete-modules", "algebraization-definition-derived-complete", "proetale-lemma-naive-completion", "sites-cohomology-lemma-bounded", "sites-cohomology-proposition-enough-weakly-contractibles" ], "ref_ids": [ 4397, 4410, 7796, 10365, 10368, 10589, 12803, 3790, 4379, 4410 ] } ], "ref_ids": [] }, { "id": 3829, "type": "theorem", "label": "proetale-proposition-Noetherian-adic-constructible", "categories": [ "proetale" ], "title": "proetale-proposition-Noetherian-adic-constructible", "contents": [ "Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring and", "let $I \\subset \\Lambda$ be an ideal. Let $K$ be an object of", "$D_{cons}(X, \\Lambda)$. Then $K$ is adic constructible", "(Definition \\ref{definition-adic-constructible})." ], "refs": [ "proetale-definition-adic-constructible" ], "proofs": [ { "contents": [ "This is a consequence of Lemma \\ref{lemma-connected-adic-lisse}", "and the fact that a Noetherian scheme is locally connected", "(Topology, Lemma \\ref{topology-lemma-locally-Noetherian-locally-connected}),", "and the definitions." ], "refs": [ "proetale-lemma-connected-adic-lisse", "topology-lemma-locally-Noetherian-locally-connected" ], "ref_ids": [ 3820, 8223 ] } ], "ref_ids": [ 3844 ] }, { "id": 3852, "type": "theorem", "label": "formal-spaces-lemma-fully-faithful", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-fully-faithful", "contents": [ "Choose a category of schemes $\\Sch_\\alpha$", "as in Sets, Lemma \\ref{sets-lemma-construct-category}.", "Given a formal scheme $\\mathfrak X$ let", "$$", "h_\\mathfrak X : (\\Sch_\\alpha)^{opp} \\longrightarrow \\textit{Sets},\\quad", "h_\\mathfrak X(S) = \\Mor_{\\textit{Formal Schemes}}(S, \\mathfrak X)", "$$", "be its functor of points. Then we have", "$$", "\\Mor_{\\textit{Formal Schemes}}(\\mathfrak X, \\mathfrak Y) =", "\\Mor_{\\textit{PSh}(\\Sch_\\alpha)}(h_\\mathfrak X, h_\\mathfrak Y)", "$$", "provided the size of $\\mathfrak X$ is not too large." ], "refs": [ "sets-lemma-construct-category" ], "proofs": [ { "contents": [ "First we observe that $h_\\mathfrak X$ satisfies the sheaf property for", "the Zariski topology for any formal scheme $\\mathfrak X$ (see", "Schemes, Definition \\ref{schemes-definition-representable-by-open-immersions}).", "This follows from the local nature of morphisms in the category", "of formal schemes. Also, for an open immersion", "$\\mathfrak V \\to \\mathfrak W$ of formal schemes,", "the corresponding transformation of functors $h_\\mathfrak V \\to h_\\mathfrak W$", "is injective and representable by open immersions (see", "Schemes, Definition \\ref{schemes-definition-representable-by-open-immersions}).", "Choose an open covering $\\mathfrak X = \\bigcup \\mathfrak U_i$", "of a formal scheme by affine formal schemes $\\mathfrak U_i$.", "Then the collection of functors", "$h_{\\mathfrak U_i}$ covers $h_\\mathfrak X$ (see", "Schemes, Definition \\ref{schemes-definition-representable-by-open-immersions}).", "Finally, note that", "$$", "h_{\\mathfrak U_i} \\times_{h_\\mathfrak X} h_{\\mathfrak U_j} =", "h_{\\mathfrak U_i \\cap \\mathfrak U_j}", "$$", "Hence in order to give a map $h_\\mathfrak X \\to h_\\mathfrak Y$", "is equivalent to giving a family of maps", "$h_{\\mathfrak U_i} \\to h_\\mathfrak Y$ which agree on overlaps.", "Thus we can reduce the bijectivity (resp.\\ injectivity) of the map", "of the lemma to bijectivity (resp.\\ injectivity) for the pairs", "$(\\mathfrak U_i, \\mathfrak Y)$", "and injectivity (resp.\\ nothing)", "for $(\\mathfrak U_i \\cap \\mathfrak U_j, \\mathfrak Y)$.", "In this way we reduce to the case where $\\mathfrak X$ is an", "affine formal scheme. Say $\\mathfrak X = \\text{Spf}(A)$", "for some admissible topological ring $A$. Also, choose a", "fundamental system of ideals of definition $I_\\lambda \\subset A$.", "\\medskip\\noindent", "We can also localize on $\\mathfrak Y$.", "Namely, suppose that $\\mathfrak V \\subset \\mathfrak Y$ is an", "open formal subscheme and $\\varphi : h_\\mathfrak X \\to h_\\mathfrak Y$.", "Then", "$$", "h_\\mathfrak V \\times_{h_\\mathfrak Y, \\varphi} h_\\mathfrak X \\to h_\\mathfrak X", "$$", "is representable by open immersions. Pulling back to", "$\\Spec(A/I_\\lambda)$ for all $\\lambda$ we find an open subscheme", "$U_\\lambda \\subset \\Spec(A/I_\\lambda)$. However, for", "$I_\\lambda \\subset I_\\mu$ the morphism $\\Spec(A/I_\\lambda) \\to \\Spec(A/I_\\mu)$", "pulls back $U_\\mu$ to $U_\\lambda$. Thus these glue to give", "an open formal subscheme $\\mathfrak U \\subset \\mathfrak X$.", "A straightforward argument (omitted) shows that", "$$", "h_\\mathfrak U = h_\\mathfrak V \\times_{h_\\mathfrak Y} h_\\mathfrak X", "$$", "In this way we see that given an open covering", "$\\mathfrak Y = \\bigcup \\mathfrak V_j$ and a transformation", "of functors $\\varphi : h_\\mathfrak X \\to h_\\mathfrak Y$", "we obtain a corresponding open covering of $\\mathfrak X$.", "Since $\\mathfrak X$ is affine, we can refine this covering by", "a finite open covering", "$\\mathfrak X = \\mathfrak U_1 \\cup \\ldots \\cup \\mathfrak U_n$", "by affine formal subschemes. In other words, for each $i$ there", "is a $j$ and a map $\\varphi_i : h_{\\mathfrak U_i} \\to h_{\\mathfrak V_j}$", "such that", "$$", "\\xymatrix{", "h_{\\mathfrak U_i} \\ar[r]_{\\varphi_i} \\ar[d] & h_{\\mathfrak V_j} \\ar[d] \\\\", "h_{\\mathfrak X} \\ar[r]^\\varphi & h_\\mathfrak Y", "}", "$$", "commutes. With a few additional arguments (which we omit) this implies", "that it suffices to prove the bijectivity of the lemma in case", "both $\\mathfrak X$ and $\\mathfrak Y$ are affine formal schemes.", "\\medskip\\noindent", "Assume $\\mathfrak X$ and $\\mathfrak Y$ are affine formal schemes.", "Say $\\mathfrak X = \\text{Spf}(A)$ and $\\mathfrak Y = \\text{Spf}(B)$.", "Let $\\varphi : h_\\mathfrak X \\to h_\\mathfrak Y$ be a transformation", "of functors. Let $I_\\lambda \\subset A$ be a fundamental system of", "ideals of definition. The canonical inclusion morphism", "$i_\\lambda : \\Spec(A/I_\\lambda) \\to \\mathfrak X$ maps to a morphism", "$\\varphi(i_\\lambda) : \\Spec(A/I_\\lambda) \\to \\mathfrak Y$.", "By (\\ref{equation-morphisms-affine-formal-schemes}) this corresponds", "to a continuous map $\\chi_\\lambda : B \\to A/I_\\lambda$.", "Since $\\varphi$ is a transformation of functors it follows", "that for $I_\\lambda \\subset I_\\mu$ the composition", "$B \\to A/I_\\lambda \\to A/I_\\mu$ is equal to $\\chi_\\mu$.", "In other words we obtain a ring map", "$$", "\\chi = \\lim \\chi_\\lambda : B \\longrightarrow \\lim A/I_\\lambda = A", "$$", "This is a continuous homomorphism because the inverse image", "of $I_\\lambda$ is open for all $\\lambda$ (as $A/I_\\lambda$ has the discrete", "topology and $\\chi_\\lambda$ is continuous). Thus we obtain", "a morphism $\\text{Spf}(\\chi) : \\mathfrak X \\to \\mathfrak Y$ by", "(\\ref{equation-morphisms-affine-formal-schemes}).", "We omit the verification that this construction is the inverse", "to the map of the lemma in this case.", "\\medskip\\noindent", "Set theoretic remarks. To make this work on the given category", "of schemes $\\Sch_\\alpha$ we just have to make sure all the", "schemes used in the proof above are isomorphic to objects of $\\Sch_\\alpha$.", "In fact, a careful analysis shows that it suffices if the", "schemes $\\Spec(A/I_\\lambda)$ occurring above are isomorphic to", "objects of $\\Sch_\\alpha$. For this it certainly suffices to assume", "the size of $\\mathfrak X$ is at most the size of", "a scheme contained in $\\Sch_\\alpha$." ], "refs": [ "schemes-definition-representable-by-open-immersions", "schemes-definition-representable-by-open-immersions", "schemes-definition-representable-by-open-immersions" ], "ref_ids": [ 7747, 7747, 7747 ] } ], "ref_ids": [ 8789 ] }, { "id": 3853, "type": "theorem", "label": "formal-spaces-lemma-formal-scheme-sheaf-fppf", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-formal-scheme-sheaf-fppf", "contents": [ "\\begin{slogan}", "Formal schemes are fpqc sheaves", "\\end{slogan}", "Let $\\mathfrak X$ be a formal scheme. The functor of points", "$h_\\mathfrak X$ (see Lemma \\ref{lemma-fully-faithful})", "satisfies the sheaf condition for fpqc coverings." ], "refs": [ "formal-spaces-lemma-fully-faithful" ], "proofs": [ { "contents": [ "Topologies, Lemma \\ref{topologies-lemma-sheaf-property-fpqc}", "reduces us to the case of a Zariski covering and a covering", "$\\{\\Spec(S) \\to \\Spec(R)\\}$ with $R \\to S$ faithfully flat.", "We observed in the proof of Lemma \\ref{lemma-fully-faithful} ", "that $h_\\mathfrak X$ satisfies the sheaf condition for Zariski coverings.", "\\medskip\\noindent", "Suppose that $R \\to S$ is a faithfully flat ring map.", "Denote $\\pi : \\Spec(S) \\to \\Spec(R)$ the", "corresponding morphism of schemes. It is surjective and flat.", "Let $f : \\Spec(S) \\to \\mathfrak X$ be a morphism", "such that $f \\circ \\text{pr}_1 = f \\circ \\text{pr}_2$", "as maps $\\Spec(S \\otimes_R S) \\to \\mathfrak X$.", "By Descent, Lemma \\ref{descent-lemma-equiv-fibre-product}", "we see that as a map on the underlying", "sets $f$ is of the form $f = g \\circ \\pi$ for some", "(set theoretic) map $g : \\Spec(R) \\to \\mathfrak X$.", "By Morphisms, Lemma \\ref{morphisms-lemma-fpqc-quotient-topology}", "and the fact that $f$ is continuous we see that $g$", "is continuous.", "\\medskip\\noindent", "Pick $y \\in \\Spec(R)$. Choose $\\mathfrak U \\subset \\mathfrak X$", "an affine formal open subscheme containing $g(y)$.", "Say $\\mathfrak U = \\text{Spf}(A)$ for some admissible topological", "ring $A$. By the above we may choose an $r \\in R$ such that", "$y \\in D(r) \\subset g^{-1}(\\mathfrak U)$.", "The restriction of $f$ to $\\pi^{-1}(D(r))$ into $\\mathfrak U$", "corresponds to a continuous ring map $A \\to S_r$ by", "(\\ref{equation-morphisms-affine-formal-schemes}). The two induced ring maps", "$A \\to S_r \\otimes_{R_r} S_r = (S \\otimes_R S)_r$ are equal", "by assumption on $f$.", "Note that $R_r \\to S_r$ is faithfully flat.", "By Descent, Lemma \\ref{descent-lemma-ff-exact} the equalizer of", "the two arrows $S_r \\to S_r \\otimes_{R_r} S_r$ is $R_r$.", "We conclude that $A \\to S_r$ factors uniquely through a map $A \\to R_r$", "which is also continuous as it has the same (open) kernel as the", "map $A \\to S_r$. This map in turn gives a morphism $D(r) \\to \\mathfrak U$ by", "(\\ref{equation-morphisms-affine-formal-schemes}).", "\\medskip\\noindent", "What have we proved so far? We have shown that for any $y \\in \\Spec(R)$", "there exists a standard affine open", "$y \\in D(r) \\subset \\Spec(R)$ such that the morphism", "$f|_{\\pi^{-1}(D(r))} : \\pi^{-1}(D(r)) \\to \\mathfrak X$ factors uniquely", "though some morphism $D(r) \\to \\mathfrak X$. We omit the", "verification that these morphisms glue to the desired", "morphism $\\Spec(R) \\to \\mathfrak X$." ], "refs": [ "topologies-lemma-sheaf-property-fpqc", "formal-spaces-lemma-fully-faithful", "descent-lemma-equiv-fibre-product", "morphisms-lemma-fpqc-quotient-topology", "descent-lemma-ff-exact" ], "ref_ids": [ 12502, 3852, 14632, 5269, 14598 ] } ], "ref_ids": [ 3852 ] }, { "id": 3854, "type": "theorem", "label": "formal-spaces-lemma-closed", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-closed", "contents": [ "Let $R$ be a topological ring. Let $M$ be a linearly topologized", "$R$-module and let $M_\\lambda$, $\\lambda \\in \\Lambda$ be a fundamental", "system of open submodules. Let $N \\subset M$ be a submodule.", "The closure of $N$ is $\\bigcap_{\\lambda \\in \\Lambda} (N + M_\\lambda)$." ], "refs": [], "proofs": [ { "contents": [ "Since each $N + M_\\lambda$ is open, it is also closed. Hence the", "intersection is closed. If $x \\in M$ is not in the closure of $N$,", "then $(x + M_\\lambda) \\cap N = 0$ for some $\\lambda$. Hence", "$x \\not \\in N + M_\\lambda$. This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3855, "type": "theorem", "label": "formal-spaces-lemma-closure", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-closure", "contents": [ "Let $R$ be a topological ring. Let $M$ be a linearly topologized", "$R$-module. Let $N \\subset M$ be a submodule. Then", "\\begin{enumerate}", "\\item $0 \\to N^\\wedge \\to M^\\wedge \\to (M/N)^\\wedge$ is exact, and", "\\item $N^\\wedge$ is the closure of the image of $N \\to M^\\wedge$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $M_\\lambda$, $\\lambda \\in \\Lambda$ be a fundamental system of", "open submodules. Then $N \\cap M_\\lambda$ is a fundamental system", "of open submodules of $N$ and $M_\\lambda + N/N$ is a fundamental system", "of open submodules of $M/N$. Thus we see that (1) follows from", "the exactness of the sequences", "$$", "0 \\to N/N \\cap M_\\lambda \\to M/M_\\lambda \\to M/(M_\\lambda + N) \\to 0", "$$", "and the fact that taking limits commutes with limits. The second", "statement follows from this and the fact that $N \\to N^\\wedge$", "has dense image and that the kernel of $M^\\wedge \\to (M/N)^\\wedge$ is closed." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3856, "type": "theorem", "label": "formal-spaces-lemma-quotient-by-closed", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-quotient-by-closed", "contents": [ "Let $R$ be a topological ring. Let $M$ be a complete, linearly topologized", "$R$-module. Let $N \\subset M$ be a closed submodule. If $M$ has a", "countable fundamental system of neighbourhoods of $0$, then", "$M/N$ is complete and the map $M \\to M/N$ is open." ], "refs": [], "proofs": [ { "contents": [ "Let $M_n$, $n \\in \\mathbf{N}$ be a fundamental system of open submodules of $M$.", "We may assume $M_{n + 1} \\subset M_n$", "for all $n$. The $(M_n + N)/N$ is a fundamental system in $M/N$.", "Hence we have to show that $M/N = \\lim M/(M_n + N)$. Consider", "the short exact sequences", "$$", "0 \\to N/N \\cap M_n \\to M/M_n \\to M/(M_n + N) \\to 0", "$$", "Since the transition maps of the system $\\{N/N\\cap M_n\\}$ are surjective", "we see that $M = \\lim M/M_n$ (by completeness of $M$) surjects onto", "$\\lim M/(M_n + N)$ by", "Algebra, Lemma \\ref{algebra-lemma-ML-exact-sequence}.", "As $N$ is closed we see that the kernel of $M \\to \\lim M/(M_n + N)$", "is $N$ (see Lemma \\ref{lemma-closed}). Finally, $M \\to M/N$", "is open by definition of the quotient topology." ], "refs": [ "algebra-lemma-ML-exact-sequence", "formal-spaces-lemma-closed" ], "ref_ids": [ 826, 3854 ] } ], "ref_ids": [] }, { "id": 3857, "type": "theorem", "label": "formal-spaces-lemma-ses", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-ses", "contents": [ "\\begin{reference}", "\\cite[Theorem 8.1]{Ma}", "\\end{reference}", "Let $R$ be a topological ring. Let $M$ be a linearly topologized", "$R$-module. Let $N \\subset M$ be a submodule. Assume $M$ has a", "countable fundamental system of neighbourhoods of $0$. Then", "\\begin{enumerate}", "\\item $0 \\to N^\\wedge \\to M^\\wedge \\to (M/N)^\\wedge \\to 0$ is exact,", "\\item $N^\\wedge$ is the closure of the image of $N \\to M^\\wedge$,", "\\item $M^\\wedge \\to (M/N)^\\wedge$ is open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have $0 \\to N^\\wedge \\to M^\\wedge \\to (M/N)^\\wedge$ is exact", "and statement (2) by Lemma \\ref{lemma-closure}.", "This produces a canonical map $c : M^\\wedge/N^\\wedge \\to (M/N)^\\wedge$.", "The module $M^\\wedge/N^\\wedge$ is complete and", "$M^\\wedge \\to M^\\wedge/N^\\wedge$ is open by", "Lemma \\ref{lemma-quotient-by-closed}.", "By the universal property of completion we obtain a canonical", "map $b : (M/N)^\\wedge \\to M^\\wedge/N^\\wedge$.", "Then $b$ and $c$ are mutually inverse as they are on a dense subset." ], "refs": [ "formal-spaces-lemma-closure", "formal-spaces-lemma-quotient-by-closed" ], "ref_ids": [ 3855, 3856 ] } ], "ref_ids": [] }, { "id": 3858, "type": "theorem", "label": "formal-spaces-lemma-completion-adic-star", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-completion-adic-star", "contents": [ "Let $R$ be a topological ring. Let $M$ be a topological $R$-module.", "Let $I \\subset R$ be a finitely generated ideal. Assume $M$", "has an open submodule whose topology is $I$-adic. Then $M^\\wedge$", "has an open submodule whose topology is $I$-adic and we have", "$M^\\wedge/I^n M^\\wedge = M/I^nM$ for all $n \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "Let $M' \\subset M$ be an open submodule whose topology is $I$-adic.", "Then $\\{I^nM'\\}_{n \\geq 1}$ is a fundamental system of open submodules", "of $M$. Thus $M^\\wedge = \\lim M/I^nM'$ contains", "$(M')^\\wedge = \\lim M'/I^nM'$", "as an open submodule and the topology on $(M')^\\wedge$ is", "$I$-adic by Algebra, Lemma \\ref{algebra-lemma-hathat-finitely-generated}.", "Since $I$ is finitely generated, $I^n$ is finitely generated,", "say by $f_1, \\ldots, f_r$. Observe that the surjection", "$(f_1, \\ldots, f_r) : M^{\\oplus r} \\to I^n M$ is continuous", "and open by our description of the topology on $M$ above.", "By Lemma \\ref{lemma-ses} applied to this surjection and to the", "short exact sequence $0 \\to I^nM \\to M \\to M/I^nM \\to 0$", "we conclude that", "$$", "(f_1, \\ldots, f_r) :", "(M^\\wedge)^{\\oplus r} \\longrightarrow M^\\wedge", "$$", "surjects onto the kernel of the surjection $M^\\wedge \\to M/I^nM$.", "Since $f_1, \\ldots, f_r$ generate $I^n$ we conclude." ], "refs": [ "algebra-lemma-hathat-finitely-generated", "formal-spaces-lemma-ses" ], "ref_ids": [ 859, 3857 ] } ], "ref_ids": [] }, { "id": 3859, "type": "theorem", "label": "formal-spaces-lemma-weakly-admissible-henselian", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-weakly-admissible-henselian", "contents": [ "Let $A$ be a weakly admissible topological ring. Let $I \\subset A$", "be a weak ideal of definition. Then $(A, I)$ is a henselian pair." ], "refs": [], "proofs": [ { "contents": [ "Let $A \\to A'$ be an \\'etale ring map and let $\\sigma : A' \\to A/I$", "be an $A$-algebra map. By More on Algebra, Lemma", "\\ref{more-algebra-lemma-characterize-henselian-pair} it suffices", "to lift $\\sigma$ to an $A$-algebra map $A' \\to A$.", "To do this, as $A$ is complete, it suffices to find,", "for every open ideal $J \\subset I$, a unique $A$-algebra map $A' \\to A/J$", "lifting $\\sigma$. Since $I$ is a weak ideal of definition,", "the ideal $I/J$ is locally nilpotent. We conclude by", "More on Algebra, Lemma \\ref{more-algebra-lemma-locally-nilpotent-henselian}." ], "refs": [ "more-algebra-lemma-characterize-henselian-pair", "more-algebra-lemma-locally-nilpotent-henselian" ], "ref_ids": [ 9861, 9857 ] } ], "ref_ids": [] }, { "id": 3860, "type": "theorem", "label": "formal-spaces-lemma-topologically-nilpotent", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-topologically-nilpotent", "contents": [ "Let $B$ be a linearly topologized ring. The set of topologically nilpotent", "elements of $B$ is a closed, radical ideal of $B$.", "Let $\\varphi : A \\to B$ be a continuous map of linearly topologized rings.", "\\begin{enumerate}", "\\item If $f \\in A$ is topologically nilpotent, then $\\varphi(f)$ is", "topologically nilpotent.", "\\item If $I \\subset A$ consists of topologically nilpotent elements,", "then the closure of $\\varphi(I)B$ consists of topologically nilpotent", "elements.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak b \\subset B$ be the set of topologically nilpotent elements.", "We omit the proof of the fact that $\\mathfrak b$ is a radical ideal", "(good exercise in the definitions). Let $g$ be an element of the closure", "of $\\mathfrak b$. Our goal is to show that $g$ is topologically nilpotent.", "Let $J \\subset B$ be an open ideal. We have to show", "$g^e \\in J$ for some $e \\geq 1$. We have $g \\in \\mathfrak b + J$", "by Lemma \\ref{lemma-closed}. Hence $g = f + h$", "for some $f \\in \\mathfrak b$ and $h \\in J$. Pick $m \\geq 1$ such that", "$f^m \\in J$. Then $g^{m + 1} \\in J$ as desired.", "\\medskip\\noindent", "Let $\\varphi : A \\to B$ be as in the statement of the lemma.", "Assertion (1) is clear and assertion (2) follows from this and", "the fact that $\\mathfrak b$ is a closed ideal." ], "refs": [ "formal-spaces-lemma-closed" ], "ref_ids": [ 3854 ] } ], "ref_ids": [] }, { "id": 3861, "type": "theorem", "label": "formal-spaces-lemma-taut-weakly-admissible", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-taut-weakly-admissible", "contents": [ "Let $\\varphi : A \\to B$ be a continuous map of weakly admissible topological", "rings. The following are equivalent", "\\begin{enumerate}", "\\item $\\varphi$ is taut,", "\\item for every weak ideal of definition $I \\subset A$ the closure of", "$\\varphi(I)B$ is a weak ideal of definition of $B$ and these form a", "fundamental system of weak ideals of definition of $B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (2) implies (1). The other implication follows", "from Lemma \\ref{lemma-topologically-nilpotent}." ], "refs": [ "formal-spaces-lemma-topologically-nilpotent" ], "ref_ids": [ 3860 ] } ], "ref_ids": [] }, { "id": 3862, "type": "theorem", "label": "formal-spaces-lemma-closure-image-ideal", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-closure-image-ideal", "contents": [ "Let $A \\to B$ be a continuous map of linearly topologized rings.", "Let $I \\subset A$ be an ideal. The closure of $IB$", "is the kernel of $B \\to B \\widehat{\\otimes}_A A/I$." ], "refs": [], "proofs": [ { "contents": [ "Let $J_\\mu$ be a fundamental system of open ideals of $B$.", "The closure of $IB$ is $\\bigcap (IB + J_\\lambda)$ by Lemma \\ref{lemma-closed}.", "Let $I_\\mu$ be a fundamental system of open ideals in $A$.", "Then", "$$", "B \\widehat{\\otimes}_A A/I = \\lim (B/J_\\lambda \\otimes_A A/(I_\\mu + I)) =", "\\lim B/(J_\\lambda + I_\\mu B + I B)", "$$", "Since $A \\to B$ is continuous, for every $\\lambda$ there", "is a $\\mu$ such that $I_\\mu B \\subset J_\\lambda$, see discussion in", "Example \\ref{example-what-does-it-mean}. Hence the limit", "can be written as $\\lim B/(J_\\lambda + IB)$ and the result is clear." ], "refs": [ "formal-spaces-lemma-closed" ], "ref_ids": [ 3854 ] } ], "ref_ids": [] }, { "id": 3863, "type": "theorem", "label": "formal-spaces-lemma-dense-image-surjective", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-dense-image-surjective", "contents": [ "Let $\\varphi : A \\to B$ be a continuous homomorphism of", "linearly topologized rings. If", "\\begin{enumerate}", "\\item $\\varphi$ is taut,", "\\item $\\varphi$ has dense image,", "\\item $A$ is complete,", "\\item $B$ is separated, and", "\\item $A$ has a countable fundamental system of neighbourhoods of $0$.", "\\end{enumerate}", "Then $\\varphi$ is surjective and open, $B$ is complete, and $B = A/K$ for", "some closed ideal $K \\subset A$." ], "refs": [], "proofs": [ { "contents": [ "We may choose a sequence of open ideals", "$A \\supset I_1 \\supset I_2 \\supset I_3 \\supset \\ldots$", "which form a fundamental system of neighbourhoods of $0$.", "For each $i$ let $J_i \\subset B$ be the closure of $\\varphi(I_i)B$.", "As $\\varphi$ is taut we see that these form a fundamental system", "of open ideals of $B$. Set $I_0 = A$ and $J_0 = B$.", "Let $n \\geq 0$ and let $y_n \\in J_n$. Since $J_n$ is the", "closure of $\\varphi(I_n)B$ we can write", "$$", "y_n = \\sum\\nolimits_t \\varphi(f_t)b_t + y'_{n + 1}", "$$", "for some $f_t \\in I_n$, $b_t \\in B$, and $y'_{n + 1} \\in J_{n + 1}$.", "Since $\\varphi$ has dense image we can choose $a_t \\in A$ with", "$\\varphi(a_t) = b_t \\bmod J_{n + 1}$. Thus", "$$", "y_n = \\varphi(f_n) + y_{n + 1}", "$$", "with $f_n = \\sum f_ta_t \\in I_n$ and", "$y_{n + 1} = y'_{n + 1} + \\sum f_t(b_t - \\varphi(a_t)) \\in J_{n + 1}$.", "Thus, starting with any $y = y_0 \\in B$, we can find by induction", "a sequence $f_m \\in I_m$, $m \\geq 0$ such that ", "$$", "y = y_0 = \\varphi(f_0 + f_1 + \\ldots + f_n) + y_{n + 1}", "$$", "with $y_{n + 1} \\in J_{n + 1}$. Since $A$ is complete we see that", "$$", "x = x_0 = f_0 + f_1 + f_2 + \\ldots", "$$", "exists. Since the partial sums approximate $x$ in $A$, since $\\varphi$", "is continuous, and since $B$ is separated we find that $\\varphi(x) = y$", "because above we've shown that the images of the partial sums approximate $y$", "in $B$. Thus $\\varphi$ is surjective. In exactly the same manner we", "find that $\\varphi(I_n) = J_n$ for all $n \\geq 1$. This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3864, "type": "theorem", "label": "formal-spaces-lemma-taut-is-adic", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-taut-is-adic", "contents": [ "Let $\\varphi : A \\to B$ be a continuous map of linearly topologized rings.", "Let $I \\subset A$ be an ideal. Assume", "\\begin{enumerate}", "\\item $I$ is finitely generated,", "\\item $A$ has the $I$-adic topology,", "\\item $B$ is complete, and", "\\item $\\varphi$ is taut.", "\\end{enumerate}", "Then the topology on $B$ is the $I$-adic topology." ], "refs": [], "proofs": [ { "contents": [ "Let $J_n$ be the closure of $\\varphi(I^n)B$ in $B$.", "Since $B$ is complete we have $B = \\lim B/J_n$.", "Let $B' = \\lim B/I^nB$ be the $I$-adic completion of $B$.", "By Algebra, Lemma \\ref{algebra-lemma-hathat-finitely-generated},", "the $I$-adic topology on $B'$ is complete and", "$B'/I^nB' = B/I^nB$. Thus the ring map $B' \\to B$ is continuous", "and has dense image as $B' \\to B/I^nB \\to B/J_n$ is surjective", "for all $n$. Finally, the map $B' \\to B$ is taut", "because $(I^nB')B = I^nB$ and $A \\to B$ is taut.", "By Lemma \\ref{lemma-dense-image-surjective} we see that $B' \\to B$ is open", "and surjective which implies the lemma." ], "refs": [ "algebra-lemma-hathat-finitely-generated", "formal-spaces-lemma-dense-image-surjective" ], "ref_ids": [ 859, 3863 ] } ], "ref_ids": [] }, { "id": 3865, "type": "theorem", "label": "formal-spaces-lemma-completed-tensor-product", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-completed-tensor-product", "contents": [ "Let $B \\to A$ and $B \\to C$ be continuous homomorphisms of", "linearly topologized rings.", "\\begin{enumerate}", "\\item If $A$ and $C$ are weakly pre-admissible, then", "$A \\widehat{\\otimes}_B C$ is weakly admissible.", "\\item If $A$ and $C$ are pre-admissible, then", "$A \\widehat{\\otimes}_B C$ is admissible.", "\\item If $A$ and $C$ have a countable fundamental system of open", "ideals, then $A \\widehat{\\otimes}_B C$ has a countable fundamental", "system of open ideals.", "\\item If $A$ and $C$ are pre-adic and have finitely generated ideals", "of definition, then $A \\widehat{\\otimes}_B C$ is adic and has", "a finitely generated ideal of definition.", "\\item If $A$ and $C$ are pre-adic Noetherian rings and", "$B/\\mathfrak b \\to A/\\mathfrak a$ is of finite type", "where $\\mathfrak a \\subset A$ and $\\mathfrak b \\subset B$", "are the ideals of topologically nilpotent elements, then", "$A \\widehat{\\otimes}_B C$ is adic Noetherian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $I_\\lambda \\subset A$, $\\lambda \\in \\Lambda$ and", "$J_\\mu \\subset C$, $\\mu \\in M$", "be fundamental systems of open ideals, then by definition", "$$", "A \\widehat{\\otimes}_B C =", "\\lim_{\\lambda, \\mu} A/I_\\lambda \\otimes_B C/J_\\mu", "$$", "with the limit topology. Thus a fundamental system of open ideals", "is given by the kernels $K_{\\lambda, \\mu}$ of the maps", "$A \\widehat{\\otimes}_B C \\to A/I_\\lambda \\otimes_B C/J_\\mu$.", "Note that $K_{\\lambda, \\mu}$ is the closure of the ideal", "$I_\\lambda(A \\widehat{\\otimes}_B C) + J_\\mu(A \\widehat{\\otimes}_B C)$.", "Finally, we have a ring homomorphism", "$\\tau : A \\otimes_B C \\to A \\widehat{\\otimes}_B C$ with dense image.", "\\medskip\\noindent", "Proof of (1). If $I_\\lambda$ and $J_\\mu$ consist of topologically", "nilpotent elements, then so does $K_{\\lambda, \\mu}$ by", "Lemma \\ref{lemma-topologically-nilpotent}. Hence ", "$A \\widehat{\\otimes}_B C$ is weakly admissible by definition.", "\\medskip\\noindent", "Proof of (2). Assume for some $\\lambda_0$ and $\\mu_0$ the ideals", "$I = I_{\\lambda_0} \\subset A$ and $J_{\\mu_0} \\subset C$ are ideals of", "definition. Thus for every $\\lambda$ there exists an $n$ such that", "$I^n \\subset I_\\lambda$. For every $\\mu$ there exists an $m$ such that", "$J^m \\subset J_\\mu$. Then", "$$", "\\left(I(A \\widehat{\\otimes}_B C) + J(A \\widehat{\\otimes}_B C)\\right)^{n + m}", "\\subset", "I_\\lambda(A \\widehat{\\otimes}_B C) + J_\\mu(A \\widehat{\\otimes}_B C)", "$$", "It follows that the open ideal $K = K_{\\lambda_0, \\mu_0}$", "satisfies $K^{n + m} \\subset K_{\\lambda, \\mu}$. Hence $K$", "is an ideal of definition of $A \\widehat{\\otimes}_B C$", "and $A \\widehat{\\otimes}_B C$ is admissible by definition.", "\\medskip\\noindent", "Proof of (3). If $\\Lambda$ and $M$ are countable, so is", "$\\Lambda \\times M$.", "\\medskip\\noindent", "Proof of (4). Assume $\\Lambda = \\mathbf{N}$ and $M = \\mathbf{N}$", "and we have finitely generated ideals $I \\subset A$ and $J \\subset C$", "such that $I_n = I^n$ and $J_n = J^n$. Then", "$$", "I(A \\widehat{\\otimes}_B C) + J(A \\widehat{\\otimes}_B C)", "$$", "is a finitely generated ideal and it is easily seen that", "$A \\widehat{\\otimes}_B C$ is the completion of", "$A \\otimes_B C$ with respect to this ideal. Hence (4)", "follows from Algebra, Lemma \\ref{algebra-lemma-hathat-finitely-generated}.", "\\medskip\\noindent", "Proof of (5).", "Let $\\mathfrak c \\subset C$ be the ideal of topologically nilpotent elements.", "Since $A$ and $C$ are adic Noetherian, we see that", "$\\mathfrak a$ and $\\mathfrak c$ are ideals of definition (details omitted).", "From part (4) we already know that", "$A \\widehat{\\otimes}_B C$ is adic and that", "$\\mathfrak a(A \\widehat{\\otimes}_B C) + \\mathfrak c(A \\widehat{\\otimes}_B C)$", "is a finitely generated ideal of definition. Since", "$$", "A \\widehat{\\otimes}_B C /", "\\left(\\mathfrak a(A \\widehat{\\otimes}_B C) +", "\\mathfrak c(A \\widehat{\\otimes}_B C)\\right)", "=", "A/\\mathfrak a \\otimes_{B/\\mathfrak b} C/\\mathfrak c", "$$", "is Noetherian as a finite type algebra over the Noetherian ring", "$C/\\mathfrak c$ we conclude by", "Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian}." ], "refs": [ "formal-spaces-lemma-topologically-nilpotent", "algebra-lemma-hathat-finitely-generated", "algebra-lemma-completion-Noetherian" ], "ref_ids": [ 3860, 859, 873 ] } ], "ref_ids": [] }, { "id": 3866, "type": "theorem", "label": "formal-spaces-lemma-diagonal-affine-formal-algebraic-space", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-diagonal-affine-formal-algebraic-space", "contents": [ "Let $S$ be a scheme. If $X$ is an affine formal algebraic space over", "$S$, then the diagonal morphism $\\Delta : X \\to X \\times_S X$", "is representable and a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Suppose given $U \\to X$ and $V \\to X$ where $U, V$ are schemes over $S$.", "Let us show that $U \\times_X V$ is representable. Write $X = \\colim X_\\lambda$", "as in Definition \\ref{definition-affine-formal-algebraic-space}.", "The discussion above shows that Zariski locally on $U$ and $V$ the morphisms", "factors through some $X_\\lambda$. In this case", "$U \\times_X V = U \\times_{X_\\lambda} V$ which is a scheme.", "Thus the diagonal is representable, see", "Spaces, Lemma \\ref{spaces-lemma-representable-diagonal}.", "Given $(a, b) : W \\to X \\times_S X$ where $W$ is a scheme over $S$", "consider the map $X \\times_{\\Delta, X \\times_S X, (a, b)} W \\to W$.", "As before locally on $W$ the morphisms $a$ and $b$ map into", "the affine scheme $X_\\lambda$ for some $\\lambda$ and then", "we get the morphism", "$X_\\lambda", "\\times_{\\Delta_\\lambda, X_\\lambda \\times_S X_\\lambda, (a, b)} W \\to W$.", "This is the base change of", "$\\Delta_\\lambda : X_\\lambda \\to X_\\lambda \\times_S X_\\lambda$", "which is a closed immersion as $X_\\lambda \\to S$ is separated", "(because $X_\\lambda$ is affine).", "Thus $X \\to X \\times_S X$ is a closed immersion." ], "refs": [ "formal-spaces-definition-affine-formal-algebraic-space", "spaces-lemma-representable-diagonal" ], "ref_ids": [ 3977, 8138 ] } ], "ref_ids": [] }, { "id": 3867, "type": "theorem", "label": "formal-spaces-lemma-covering-by-thickenings", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-covering-by-thickenings", "contents": [ "Let $X_\\lambda, \\lambda \\in \\Lambda$ and $X = \\colim X_\\lambda$", "be as in Definition \\ref{definition-affine-formal-algebraic-space}.", "Then $X_\\lambda \\to X$ is representable and a thickening." ], "refs": [ "formal-spaces-definition-affine-formal-algebraic-space" ], "proofs": [ { "contents": [ "The statement makes sense by the discussion in", "Spaces, Section \\ref{spaces-section-representable} and", "\\ref{spaces-section-representable-properties}.", "By Lemma \\ref{lemma-diagonal-affine-formal-algebraic-space}", "the morphisms $X_\\lambda \\to X$ are representable.", "Given $U \\to X$ where $U$ is a scheme,", "then the discussion following", "Definition \\ref{definition-affine-formal-algebraic-space}", "shows that Zariski locally on $U$ the", "morphism factors through some $X_\\mu$ with $\\lambda \\leq \\mu$.", "In this case $U \\times_X X_\\lambda = U \\times_{X_\\mu} X_\\lambda$", "so that $U \\times_X X_\\lambda \\to U$ is a base change of", "the thickening $X_\\lambda \\to X_\\mu$." ], "refs": [ "formal-spaces-lemma-diagonal-affine-formal-algebraic-space", "formal-spaces-definition-affine-formal-algebraic-space" ], "ref_ids": [ 3866, 3977 ] } ], "ref_ids": [ 3977 ] }, { "id": 3868, "type": "theorem", "label": "formal-spaces-lemma-factor-through-thickening", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-factor-through-thickening", "contents": [ "Let $X_\\lambda, \\lambda \\in \\Lambda$ and $X = \\colim X_\\lambda$", "be as in Definition \\ref{definition-affine-formal-algebraic-space}.", "If $Y$ is a quasi-compact algebraic space over $S$, then any", "morphism $Y \\to X$ factors through an $X_\\lambda$." ], "refs": [ "formal-spaces-definition-affine-formal-algebraic-space" ], "proofs": [ { "contents": [ "Choose an affine scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$. The composition $V \\to Y \\to X$ factors through", "$X_\\lambda$ for some $\\lambda$ by the discussion following", "Definition \\ref{definition-affine-formal-algebraic-space}.", "Since $V \\to Y$ is a surjection of sheaves, we conclude." ], "refs": [ "formal-spaces-definition-affine-formal-algebraic-space" ], "ref_ids": [ 3977 ] } ], "ref_ids": [ 3977 ] }, { "id": 3869, "type": "theorem", "label": "formal-spaces-lemma-characterize-affine-formal-algebraic-space", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-characterize-affine-formal-algebraic-space", "contents": [ "Let $S$ be a scheme. Let $X$ be a sheaf on $(\\Sch/S)_{fppf}$.", "Then $X$ is an affine formal algebraic space if and only if", "the following hold", "\\begin{enumerate}", "\\item any morphism $U \\to X$ where $U$ is an affine scheme over $S$", "factors through a morphism $T \\to X$ which is representable and a", "thickening with $T$ an affine scheme over $S$, and", "\\item a set theoretic condition as in Remark \\ref{remark-set-theoretic}.", "\\end{enumerate}" ], "refs": [ "formal-spaces-remark-set-theoretic" ], "proofs": [ { "contents": [ "It follows from Lemmas \\ref{lemma-covering-by-thickenings} and", "\\ref{lemma-factor-through-thickening} that an affine formal algebraic space", "satisfies (1) and (2). In order to prove the converse we may", "assume $X$ is not empty.", "Let $\\Lambda$ be the category of representable morphisms $T \\to X$ which are", "thickenings where $T$ is an affine scheme over $S$. This category", "is directed. Since $X$ is not empty, $\\Lambda$ contains at least one", "object. If $T \\to X$ and $T' \\to X$ are in $\\Lambda$, then we can", "factor $T \\amalg T' \\to X$ through $T'' \\to X$ in $\\Lambda$. Between", "any two objects of $\\Lambda$ there is a unique arrow or none. Thus", "$\\Lambda$ is a directed set and by assumption", "$X = \\colim_{T \\to X\\text{ in }\\Lambda} T$. To finish the proof", "we need to show that any arrow $T \\to T'$ in $\\Lambda$ is a thickening.", "This is true because $T' \\to X$ is a monomorphism of sheaves, so that", "$T = T \\times_{T'} T' = T \\times_X T'$ and hence the morphism", "$T \\to T'$ equals the projection $T \\times_X T' \\to T'$ which is", "a thickening because $T \\to X$ is a thickening." ], "refs": [ "formal-spaces-lemma-covering-by-thickenings", "formal-spaces-lemma-factor-through-thickening" ], "ref_ids": [ 3867, 3868 ] } ], "ref_ids": [ 4001 ] }, { "id": 3870, "type": "theorem", "label": "formal-spaces-lemma-mcquillan-affine-formal-algebraic-space", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-mcquillan-affine-formal-algebraic-space", "contents": [ "Let $S$ be a scheme. Let $X$ be an fppf sheaf on $(\\Sch/S)_{fppf}$", "which satisfies the set theoretic condition of", "Remark \\ref{remark-set-theoretic}.", "The following are equivalent:", "\\begin{enumerate}", "\\item there exists a weakly admissible topological ring $A$ over $S$", "(see Remark \\ref{remark-mcquillan}) such that", "$X = \\colim_{I \\subset A\\text{ weak ideal of definition}} \\Spec(A/I)$,", "\\item $X$ is an affine formal algebraic space and", "there exists an $S$-algebra $A$ and a map $X \\to \\Spec(A)$", "such that for a closed immersion $T \\to X$ with $T$ an affine scheme", "the composition $T \\to \\Spec(A)$ is a closed immersion,", "\\item $X$ is an affine formal algebraic space and", "there exists an $S$-algebra $A$ and a map $X \\to \\Spec(A)$", "such that for a closed immersion $T \\to X$ with $T$ a scheme", "the composition $T \\to \\Spec(A)$ is a closed immersion,", "\\item $X$ is an affine formal algebraic space and", "for some choice of $X = \\colim X_\\lambda$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}", "the projections $\\lim \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})", "\\to \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})$ are surjective,", "\\item $X$ is an affine formal algebraic space and for any choice", "of $X = \\colim X_\\lambda$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}", "the projections $\\lim \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})", "\\to \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})$ are surjective.", "\\end{enumerate}", "Moreover, the weakly admissible topological ring is", "$A = \\lim \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})$", "endowed with its limit topology and the weak ideals of definition", "classify exactly the morphisms $T \\to X$ which are representable", "and thickenings." ], "refs": [ "formal-spaces-remark-set-theoretic", "formal-spaces-remark-mcquillan", "formal-spaces-definition-affine-formal-algebraic-space", "formal-spaces-definition-affine-formal-algebraic-space" ], "proofs": [ { "contents": [ "It is clear that (5) implies (4).", "\\medskip\\noindent", "Assume (4) for $X = \\colim X_\\lambda$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}.", "Set $A = \\lim \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})$.", "Let $T \\to X$ be a closed immersion with $T$ a scheme", "(note that $T \\to X$ is representable by", "Lemma \\ref{lemma-diagonal-affine-formal-algebraic-space}).", "Since $X_\\lambda \\to X$ is a thickening, so is", "$X_\\lambda \\times_X T \\to T$. On the other hand,", "$X_\\lambda \\times_X T \\to X_\\lambda$ is a closed immersion,", "hence $X_\\lambda \\times_X T$ is affine. Hence $T$ is affine", "by Limits, Proposition \\ref{limits-proposition-affine}.", "Then $T \\to X$ factors through $X_\\lambda$ for some $\\lambda$", "by Lemma \\ref{lemma-factor-through-thickening}.", "Thus $A \\to \\Gamma(X_\\lambda, \\mathcal{O}) \\to \\Gamma(T, \\mathcal{O})$", "is surjective. In this way we see that (3) holds.", "\\medskip\\noindent", "It is clear that (3) implies (2).", "\\medskip\\noindent", "Assume (2) for $A$ and $X \\to \\Spec(A)$. Write $X = \\colim X_\\lambda$", "as in Definition \\ref{definition-affine-formal-algebraic-space}.", "Then $A_\\lambda = \\Gamma(X_\\lambda, \\mathcal{O})$ is a quotient", "of $A$ by assumption (2). Hence $A^\\wedge = \\lim A_\\lambda$", "is a complete topological ring, see discussion in", "More on Algebra, Section \\ref{more-algebra-section-topological-ring}.", "The maps $A^\\wedge \\to A_\\lambda$ are surjective as $A \\to A_\\lambda$ is.", "We claim that for any $\\lambda$ the kernel $I_\\lambda \\subset A^\\wedge$ of", "$A^\\wedge \\to A_\\lambda$ is a weak ideal of definition.", "Namely, it is open by definition of the limit topology.", "If $f \\in I_\\lambda$, then for any $\\mu \\in \\Lambda$", "the image of $f$ in $A_\\mu$ is zero in all the residue fields", "of the points of $X_\\mu$. Hence it is a nilpotent element", "of $A_\\mu$. Hence some power $f^n \\in I_\\mu$. Thus $f^n \\to 0$", "as $n \\to 0$. Thus $A^\\wedge$ is weakly admissible.", "Finally, suppose that $I \\subset A^\\wedge$ is a weak ideal", "of definition. Then $I \\subset A^\\wedge$ is open and hence there exists", "some $\\lambda$ such that $I \\supset I_\\lambda$. Thus we obtain a morphism", "$\\Spec(A^\\wedge/I) \\to \\Spec(A_\\lambda) \\to X$.", "Then it follows that $X = \\colim \\Spec(A^\\wedge/I)$ where now", "the colimit is over all weak ideals of definition.", "Thus (1) holds.", "\\medskip\\noindent", "Assume (1). In this case it is clear that $X$ is an affine formal", "algebraic space. Let $X = \\colim X_\\lambda$ be any presentation as in", "Definition \\ref{definition-affine-formal-algebraic-space}.", "For each $\\lambda$ we can find a weak ideal of definition", "$I \\subset A$ such that $X_\\lambda \\to X$ factors through", "$\\Spec(A/I) \\to X$, see Lemma \\ref{lemma-factor-through-thickening}.", "Then $X_\\lambda = \\Spec(A/I_\\lambda)$ with $I \\subset I_\\lambda$.", "Conversely, for any weak ideal of definition $I \\subset A$", "the morphism $\\Spec(A/I) \\to X$ factors through $X_\\lambda$", "for some $\\lambda$, i.e., $I_\\lambda \\subset I$.", "It follows that each $I_\\lambda$ is a weak ideal of definition", "and that they form a cofinal subset of the set of weak ideals", "of definition. Hence $A = \\lim A/I = \\lim A/I_\\lambda$", "and we see that (5) is true and moreover that", "$A = \\lim \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})$." ], "refs": [ "formal-spaces-definition-affine-formal-algebraic-space", "formal-spaces-lemma-diagonal-affine-formal-algebraic-space", "limits-proposition-affine", "formal-spaces-lemma-factor-through-thickening", "formal-spaces-definition-affine-formal-algebraic-space", "formal-spaces-definition-affine-formal-algebraic-space", "formal-spaces-lemma-factor-through-thickening" ], "ref_ids": [ 3977, 3866, 15129, 3868, 3977, 3977, 3868 ] } ], "ref_ids": [ 4001, 3997, 3977, 3977 ] }, { "id": 3871, "type": "theorem", "label": "formal-spaces-lemma-morphism-between-formal-spectra", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-morphism-between-formal-spectra", "contents": [ "Let $S$ be a scheme. Let $A$, $B$ be weakly admissible", "topological rings over $S$. Any morphism $f : \\text{Spf}(B) \\to \\text{Spf}(A)$", "of affine formal algebraic spaces over $S$", "is equal to $\\text{Spf}(f^\\sharp)$ for a unique continuous", "$S$-algebra map $f^\\sharp : A \\to B$." ], "refs": [], "proofs": [ { "contents": [ "Let $f : \\text{Spf}(B) \\to \\text{Spf}(A)$ be as in the lemma.", "Let $J \\subset B$ be a weak ideal of definition. By", "Lemma \\ref{lemma-factor-through-thickening}", "there exists a weak ideal of definition $I \\subset A$ such that", "$\\Spec(B/J) \\to \\text{Spf}(B) \\to \\text{Spf}(A)$", "factors through $\\Spec(A/I)$. By", "Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}", "we obtain an $S$-algebra map $A/I \\to B/J$.", "These maps are compatible for varying $J$ and define the", "map $f^\\sharp : A \\to B$. This map is continuous because", "for every weak ideal of definition $J \\subset B$ there is a", "weak ideal of definition $I \\subset A$ such that", "$f^\\sharp(I) \\subset J$. The equality $f = \\text{Spf}(f^\\sharp)$", "holds by our choice of the ring maps $A/I \\to B/J$ which make up $f^\\sharp$." ], "refs": [ "formal-spaces-lemma-factor-through-thickening", "schemes-lemma-morphism-into-affine" ], "ref_ids": [ 3868, 7655 ] } ], "ref_ids": [] }, { "id": 3872, "type": "theorem", "label": "formal-spaces-lemma-presentation-representable", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-presentation-representable", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a map", "of presheaves on $(\\Sch/S)_{fppf}$. If $X$ is an affine formal algebraic", "space and $f$ is representable by algebraic spaces and locally quasi-finite,", "then $f$ is representable (by schemes)." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme over $S$ and $T \\to Y$ a map. We have to show that", "the algebraic space $X \\times_Y T$ is a scheme. Write $X = \\colim X_\\lambda$", "as in Definition", "\\ref{definition-affine-formal-algebraic-space}.", "Let $W \\subset X \\times_Y T$", "be a quasi-compact open subspace. The restriction of the projection", "$X \\times_Y T \\to X$ to $W$ factors through $X_\\lambda$ for some $\\lambda$.", "Then", "$$", "W \\to X_\\lambda \\times_S T", "$$", "is a monomorphism (hence separated) and locally quasi-finite (because", "$W \\to X \\times_Y T \\to T$ is locally quasi-finite by our assumption", "on $X \\to Y$, see Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-permanence-quasi-finite}).", "Hence $W$ is a scheme by", "Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}.", "Thus $X \\times_Y T$ is a scheme by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme}." ], "refs": [ "formal-spaces-definition-affine-formal-algebraic-space", "spaces-morphisms-lemma-permanence-quasi-finite", "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "spaces-properties-lemma-subscheme" ], "ref_ids": [ 3977, 4836, 4983, 11848 ] } ], "ref_ids": [] }, { "id": 3873, "type": "theorem", "label": "formal-spaces-lemma-countable-affine-formal-algebraic-space", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-countable-affine-formal-algebraic-space", "contents": [ "Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item there exists a system $X_1 \\to X_2 \\to X_3 \\to \\ldots$", "of thickenings of affine schemes over $S$ such that $X = \\colim X_n$,", "\\item there exists a choice $X = \\colim X_\\lambda$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}", "such that $\\Lambda$ is countable.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-affine-formal-algebraic-space" ], "proofs": [ { "contents": [ "This follows from the observation that a countable directed set", "has a cofinal subset isomorphic to $(\\mathbf{N}, \\geq)$.", "See proof of Algebra, Lemma \\ref{algebra-lemma-ML-limit-nonempty}." ], "refs": [ "algebra-lemma-ML-limit-nonempty" ], "ref_ids": [ 825 ] } ], "ref_ids": [ 3977 ] }, { "id": 3874, "type": "theorem", "label": "formal-spaces-lemma-implications-between-types", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-implications-between-types", "contents": [ "Let $X$ be an affine formal algebraic space over a scheme $S$.", "\\begin{enumerate}", "\\item If $X$ is Noetherian, then $X$ is adic*.", "\\item If $X$ is adic*, then $X$ is adic.", "\\item If $X$ is adic, then $X$ is countably indexed.", "\\item If $X$ is countably indexed, then $X$ is McQuillan.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1) and (2) are immediate from the definitions.", "\\medskip\\noindent", "Proof of (3). By definition there exists an adic topological ring $A$", "such that $X = \\colim \\Spec(A/I)$ where the colimit is over the ideals", "of definition of $A$. As $A$ is adic, there exits an ideal $I$", "such that $\\{I^n\\}$ forms a fundamental system of neighbourhoods of $0$.", "Then each $I^n$ is an ideal of definition and $X = \\colim \\Spec(A/I^n)$.", "Thus $X$ is countably indexed.", "\\medskip\\noindent", "Proof of (4). Write $X = \\colim X_n$", "for some system $X_1 \\to X_2 \\to X_3 \\to \\ldots$ of thickenings of affine", "schemes over $S$. Then", "$$", "A = \\lim \\Gamma(X_n, \\mathcal{O}_{X_n})", "$$", "surjects onto each $\\Gamma(X_n, \\mathcal{O}_{X_n})$ because the transition", "maps are surjections as the morphisms $X_n \\to X_{n + 1}$ are closed", "immersions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3875, "type": "theorem", "label": "formal-spaces-lemma-countably-indexed", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-countably-indexed", "contents": [ "Let $S$ be a scheme. Let $X$ be a presheaf on $(\\Sch/S)_{fppf}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is a countably indexed affine formal algebraic space,", "\\item $X = \\text{Spf}(A)$ where $A$ is a weakly admissible topological", "$S$-algebra which has a countable fundamental system of neighbourhoods of $0$,", "\\item $X = \\text{Spf}(A)$ where $A$ is a weakly admissible topological", "$S$-algebra which has a fundamental system", "$A \\supset I_1 \\supset I_2 \\supset I_3 \\supset \\ldots$", "of weak ideals of definition,", "\\item $X = \\text{Spf}(A)$ where $A$ is a complete topological $S$-algebra", "with a fundamental system of open neighbourhoods of $0$ given by a", "countable sequence $A \\supset I_1 \\supset I_2 \\supset I_3 \\supset \\ldots$", "of ideals such that $I_n/I_{n + 1}$ is locally nilpotent, and", "\\item $X = \\text{Spf}(A)$ where $A = \\lim B/J_n$ with the limit topology", "where $B \\supset J_1 \\supset J_2 \\supset J_3 \\supset \\ldots$ is a", "sequence of ideals in an $S$-algebra $B$ with $J_n/J_{n + 1}$", "locally nilpotent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). By Lemma \\ref{lemma-implications-between-types}", "we can write $X = \\text{Spf}(A)$ where $A$ is a weakly admissible", "topological $S$-algebra. For any presentation $X = \\colim X_n$ as in", "Lemma \\ref{lemma-countable-affine-formal-algebraic-space} part (1)", "we see that $A = \\lim A_n$ with $X_n = \\Spec(A_n)$ and", "$A_n = A/I_n$ for some weak ideal of definition $I_n \\subset A$.", "This follows from the final statement of", "Lemma \\ref{lemma-mcquillan-affine-formal-algebraic-space}", "which moreover implies that $\\{I_n\\}$ is a fundamental system", "of open neighbourhoods of $0$. Thus we have a sequence", "$$", "A \\supset I_1 \\supset I_2 \\supset I_3 \\supset \\ldots", "$$", "of weak ideals of definition with $A = \\lim A/I_n$. In this way", "we see that condition (1) implies each of the conditions (2) -- (5).", "\\medskip\\noindent", "Assume (5). First note that the limit topology on", "$A = \\lim B/J_n$ is a linearly topologized, complete topology, see", "More on Algebra, Section \\ref{more-algebra-section-topological-ring}.", "If $f \\in A$ maps to zero in $B/J_1$, then some power maps to zero", "in $B/J_2$ as its image in $J_1/J_2$ is nilpotent, then a further", "power maps to zero in $J_2/J_3$, etc, etc. In this way we see", "the open ideal $\\Ker(A \\to B/J_1)$ is a weak ideal of definition.", "Thus $A$ is weakly admissible. In this way we see that (5) implies (2).", "\\medskip\\noindent", "It is clear that (4) is a special case of (5) by taking $B = A$.", "It is clear that (3) is a special case of (2).", "\\medskip\\noindent", "Assume $A$ is as in (2). Let $E_n$ be a countable fundamental", "system of neighbourhoods of $0$ in $A$. Since $A$ is a weakly", "admissible topological ring we can find open ideals $I_n \\subset E_n$.", "We can also choose a weak ideal of definition $J \\subset A$.", "Then $J \\cap I_n$ is a fundamental system of weak ideals of definition", "of $A$ and we get", "$X = \\text{Spf}(A) = \\colim \\Spec(A/(J \\cap I_n))$", "which shows that $X$ is a countably indexed affine formal algebraic space." ], "refs": [ "formal-spaces-lemma-implications-between-types", "formal-spaces-lemma-countable-affine-formal-algebraic-space", "formal-spaces-lemma-mcquillan-affine-formal-algebraic-space" ], "ref_ids": [ 3874, 3873, 3870 ] } ], "ref_ids": [] }, { "id": 3876, "type": "theorem", "label": "formal-spaces-lemma-characterize-noetherian-affine", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-characterize-noetherian-affine", "contents": [ "Let $S$ be a scheme. Let $X$ be an affine formal algebraic space.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is Noetherian,", "\\item $X$ is adic* and for some choice of $X = \\colim X_\\lambda$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}", "the schemes $X_\\lambda$ are Noetherian,", "\\item $X$ is adic* and for any closed immersion $T \\to X$ with $T$", "a scheme, $T$ is Noetherian.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-affine-formal-algebraic-space" ], "proofs": [ { "contents": [ "This follows from the fact that if $A$ is a ring complete with", "respect to a finitely generated ideal $I$, then $A$ is Noetherian", "if and only if $A/I$ is Noetherian, see", "Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian}.", "Details omitted." ], "refs": [ "algebra-lemma-completion-Noetherian" ], "ref_ids": [ 873 ] } ], "ref_ids": [ 3977 ] }, { "id": 3877, "type": "theorem", "label": "formal-spaces-lemma-diagonal-formal-algebraic-space", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-diagonal-formal-algebraic-space", "contents": [ "Let $S$ be a scheme. If $X$ is a formal algebraic space over", "$S$, then the diagonal morphism $\\Delta : X \\to X \\times_S X$", "is representable, a monomorphism, locally quasi-finite,", "locally of finite type, and separated." ], "refs": [], "proofs": [ { "contents": [ "Suppose given $U \\to X$ and $V \\to X$ with $U, V$ schemes over $S$.", "Then $U \\times_X V$ is a sheaf. Choose $\\{X_i \\to X\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}.", "For every $i$ the morphism", "$$", "(U \\times_X X_i) \\times_{X_i} (V \\times_X X_i)", "= (U \\times_X V) \\times_X X_i \\to U \\times_X V", "$$", "is representable and \\'etale as a base change of $X_i \\to X$", "and its source is a scheme (use", "Lemmas \\ref{lemma-diagonal-affine-formal-algebraic-space} and", "\\ref{lemma-presentation-representable}). These maps are jointly surjective", "hence $U \\times_X V$ is an algebraic space by", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}.", "The morphism $U \\times_X V \\to U \\times_S V$ is a monomorphism.", "It is also locally quasi-finite, because on precomposing with", "the morphism displayed above we obtain the composition", "$$", "(U \\times_X X_i) \\times_{X_i} (V \\times_X X_i)", "\\to (U \\times_X X_i) \\times_S (V \\times_X X_i)", "\\to U \\times_S V", "$$", "which is locally quasi-finite as a composition of a closed", "immersion (Lemma \\ref{lemma-diagonal-affine-formal-algebraic-space})", "and an \\'etale morphism, see", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-locally-quasi-finite-etale-local-source}.", "Hence we conclude that $U \\times_X V$ is a scheme by", "Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}.", "Thus $\\Delta$ is representable, see", "Spaces, Lemma \\ref{spaces-lemma-representable-diagonal}.", "\\medskip\\noindent", "In fact, since we've shown above that the morphisms of schemes", "$U \\times_X V \\to U \\times_S V$ are aways monomorphisms and", "locally quasi-finite we conclude that $\\Delta : X \\to X \\times_S X$", "is a monomorphism and locally quasi-finite, see", "Spaces, Lemma \\ref{spaces-lemma-transformation-diagonal-properties}.", "Then we can use the principle of", "Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}", "to see that $\\Delta$ is separated and locally of finite type.", "Namely, a monomorphism of schemes is separated", "(Schemes, Lemma \\ref{schemes-lemma-monomorphism-separated})", "and a locally quasi-finite morphism of schemes is", "locally of finite type", "(follows from the definition in", "Morphisms, Section \\ref{morphisms-section-quasi-finite})." ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-lemma-diagonal-affine-formal-algebraic-space", "formal-spaces-lemma-presentation-representable", "bootstrap-theorem-final-bootstrap", "formal-spaces-lemma-diagonal-affine-formal-algebraic-space", "spaces-descent-lemma-locally-quasi-finite-etale-local-source", "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "spaces-lemma-representable-diagonal", "spaces-lemma-transformation-diagonal-properties", "spaces-lemma-representable-transformations-property-implication", "schemes-lemma-monomorphism-separated" ], "ref_ids": [ 3981, 3866, 3872, 2602, 3866, 9425, 4983, 8138, 8139, 8136, 7722 ] } ], "ref_ids": [] }, { "id": 3878, "type": "theorem", "label": "formal-spaces-lemma-space-to-formal-space", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-space-to-formal-space", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism from an", "algebraic space over $S$ to a formal algebraic space over $S$.", "Then $f$ is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\to Y$ be a morphism where $Z$ is a scheme over $S$.", "We have to show that $X \\times_Y Z$ is an algebraic space.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "Then $U \\times_Y Z \\to X \\times_Y Z$ is representable surjective \\'etale", "(Spaces, Lemma", "\\ref{spaces-lemma-base-change-representable-transformations-property})", "and $U \\times_Y Z$ is a scheme by", "Lemma \\ref{lemma-diagonal-formal-algebraic-space}.", "Hence the result by", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}." ], "refs": [ "spaces-lemma-base-change-representable-transformations-property", "formal-spaces-lemma-diagonal-formal-algebraic-space", "bootstrap-theorem-final-bootstrap" ], "ref_ids": [ 8133, 3877, 2602 ] } ], "ref_ids": [] }, { "id": 3879, "type": "theorem", "label": "formal-spaces-lemma-reduction-formal-algebraic-space", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-reduction-formal-algebraic-space", "contents": [ "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.", "There exists a reduced algebraic space $X_{red}$ and a representable", "morphism $X_{red} \\to X$ which is a thickening. A morphism $U \\to X$", "with $U$ a reduced algebraic space factors uniquely through $X_{red}$." ], "refs": [], "proofs": [ { "contents": [ "First assume that $X$ is an affine formal algebraic space.", "Say $X = \\colim X_\\lambda$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}.", "Since the transition morphisms are thickenings, the affine", "schemes $X_\\lambda$ all have isomorphic reductions $X_{red}$.", "The morphism $X_{red} \\to X$ is representable and a thickening", "by Lemma \\ref{lemma-covering-by-thickenings} and the fact that", "compositions of thickenings are thickenings. We omit the", "verification of the universal", "property (use Schemes, Definition", "\\ref{schemes-definition-reduced-induced-scheme},", "Schemes, Lemma \\ref{schemes-lemma-map-into-reduction},", "Properties of Spaces, Definition", "\\ref{spaces-properties-definition-reduced-induced-space}, and", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-map-into-reduction}).", "\\medskip\\noindent", "Let $X$ and $\\{X_i \\to X\\}_{i \\in I}$ be as in", "Definition \\ref{definition-formal-algebraic-space}.", "For each $i$ let $X_{i, red} \\to X_i$ be the reduction as", "constructed above. For $i, j \\in I$ the projection", "$X_{i, red} \\times_X X_j \\to X_{i, red}$ is an \\'etale (by assumption)", "morphism of schemes (by Lemma \\ref{lemma-presentation-representable}).", "Hence $X_{i, red} \\times_X X_j$ is reduced (see", "Descent, Lemma \\ref{descent-lemma-reduced-local-smooth}).", "Thus the projection $X_{i, red} \\times_X X_j \\to X_j$ factors", "through $X_{j, red}$ by the universal property. We conclude that", "$$", "R_{ij} = X_{i, red} \\times_X X_j = X_{i, red} \\times_X X_{j, red} =", "X_i \\times_X X_{j, red}", "$$", "because the morphisms $X_{i, red} \\to X_i$ are injections of sheaves.", "Set $U = \\coprod X_{i, red}$, set", "$R = \\coprod R_{ij}$, and denote $s, t : R \\to U$ the two", "projections. As a sheaf $R = U \\times_X U$ and $s$ and $t$", "are \\'etale. Then $(t, s) : R \\to U$ defines an \\'etale equivalence", "relation by our observations above. Thus $X_{red} = U/R$ is an", "algebraic space by Spaces, Theorem \\ref{spaces-theorem-presentation}.", "By construction the diagram", "$$", "\\xymatrix{", "\\coprod X_{i, red} \\ar[r] \\ar[d] & \\coprod X_i \\ar[d] \\\\", "X_{red} \\ar[r] & X", "}", "$$", "is cartesian. Since the right vertical arrow is \\'etale surjective", "and the top horizontal arrow is representable and a thickening", "we conclude that $X_{red} \\to X$ is representable by", "Bootstrap, Lemma \\ref{bootstrap-lemma-after-fppf-sep-lqf}", "(to verify the assumptions of the lemma use that a surjective", "\\'etale morphism is surjective, flat, and locally of finite", "presentation and use that thickenings are separated and locally quasi-finite).", "Then we can use Spaces, Lemma", "\\ref{spaces-lemma-descent-representable-transformations-property}", "to conclude that $X_{red} \\to X$ is a thickening", "(use that being a thickening is equivalent to being", "a surjective closed immersion).", "\\medskip\\noindent", "Finally, suppose that $U \\to X$ is a morphism with", "$U$ a reduced algebraic space over $S$. Then each $X_i \\times_X U$", "is \\'etale over $U$ and therefore reduced (by our definition of", "reduced algebraic spaces in Properties of Spaces, Section", "\\ref{spaces-properties-section-types-properties}).", "Then $X_i \\times_X U \\to X_i$ factors through $X_{i, red}$.", "Hence $U \\to X$ factors through $X_{red}$ because", "$\\{X_i \\times_X U \\to U\\}$ is an \\'etale covering." ], "refs": [ "formal-spaces-definition-affine-formal-algebraic-space", "formal-spaces-lemma-covering-by-thickenings", "schemes-definition-reduced-induced-scheme", "schemes-lemma-map-into-reduction", "spaces-properties-definition-reduced-induced-space", "spaces-properties-lemma-map-into-reduction", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-lemma-presentation-representable", "descent-lemma-reduced-local-smooth", "spaces-theorem-presentation", "bootstrap-lemma-after-fppf-sep-lqf", "spaces-lemma-descent-representable-transformations-property" ], "ref_ids": [ 3977, 3867, 7745, 7682, 11932, 11847, 3981, 3872, 14653, 8124, 2619, 8134 ] } ], "ref_ids": [] }, { "id": 3880, "type": "theorem", "label": "formal-spaces-lemma-reduction-smooth", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-reduction-smooth", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "formal algebraic spaces over $S$ which is representable by", "algebraic spaces and smooth (for example \\'etale).", "Then $X_{red} = X \\times_Y Y_{red}$." ], "refs": [], "proofs": [ { "contents": [ "(The \\'etale case follows directly from the construction of", "the underlying reduced algebraic space in the proof of", "Lemma \\ref{lemma-reduction-formal-algebraic-space}.)", "Assume $f$ is smooth. Observe that $X \\times_Y Y_{red} \\to Y_{red}$", "is a smooth morphism of algebraic spaces. Hence $X \\times_Y Y_{red}$", "is a reduced algebraic space by Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-reduced-local-smooth}.", "Then the univeral property of reduction shows that the canonical morphism", "$X_{red} \\to X \\times_Y Y_{red}$ is an isomorphism." ], "refs": [ "formal-spaces-lemma-reduction-formal-algebraic-space", "spaces-descent-lemma-reduced-local-smooth" ], "ref_ids": [ 3879, 9377 ] } ], "ref_ids": [] }, { "id": 3881, "type": "theorem", "label": "formal-spaces-lemma-reduction-surjective", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-reduction-surjective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "formal algebraic spaces over $S$ which is representable by", "algebraic spaces. Then $f$ is surjective in the sense of", "Bootstrap, Definition \\ref{bootstrap-definition-property-transformation}", "if and only if $f_{red} : X_{red} \\to Y_{red}$ is a", "surjective morphism of algebraic spaces." ], "refs": [ "bootstrap-definition-property-transformation" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 2638 ] }, { "id": 3882, "type": "theorem", "label": "formal-spaces-lemma-colimit-is-formal", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-colimit-is-formal", "contents": [ "Let $S$ be a scheme. Suppose given a directed set", "$\\Lambda$ and a system of algebraic spaces $(X_\\lambda, f_{\\lambda \\mu})$", "over $\\Lambda$ where each $f_{\\lambda \\mu} : X_\\lambda \\to X_\\mu$ is a", "thickening. Then $X = \\colim_{\\lambda \\in \\Lambda} X_\\lambda$", "is a formal algebraic space over $S$." ], "refs": [], "proofs": [ { "contents": [ "Since we take the colimit in the category of fppf sheaves, we", "see that $X$ is a sheaf. Choose and fix $\\lambda \\in \\Lambda$. Choose an", "\\'etale covering $\\{X_{i, \\lambda} \\to X_\\lambda\\}$ where $X_i$ is an affine", "scheme over $S$, see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-cover-by-union-affines}.", "For each $\\mu \\geq \\lambda$ there exists a cartesian diagram", "$$", "\\xymatrix{", "X_{i, \\lambda} \\ar[r] \\ar[d] & X_{i, \\mu} \\ar[d] \\\\", "X_\\lambda \\ar[r] & X_\\mu", "}", "$$", "with \\'etale vertical arrows, see", "More on Morphisms of Spaces, Theorem", "\\ref{spaces-more-morphisms-theorem-topological-invariance}", "(this also uses that a thickening is a surjective closed immersion which", "satisfies the conditions of the theorem). Moreover, these diagrams are", "unique up to unique isomorphism and hence", "$X_{i, \\mu} = X_\\mu \\times_{X_{\\mu'}} X_{i, \\mu'}$ for", "$\\mu' \\geq \\mu$. The morphisms $X_{i, \\mu} \\to X_{i, \\mu'}$", "is a thickening as a base change of a thickening. Each $X_{i, \\mu}$", "is an affine scheme by Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-affine} and the fact that", "$X_{i, \\lambda}$ is affine.", "Set $X_i = \\colim_{\\mu \\geq \\lambda} X_{i, \\mu}$. Then $X_i$ is", "an affine formal algebraic space. The morphism $X_i \\to X$", "is \\'etale because given an affine scheme $U$ any $U \\to X$", "factors through $X_\\mu$ for some $\\mu \\geq \\lambda$ (details omitted).", "In this way we see that $X$ is a formal algebraic space." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-more-morphisms-theorem-topological-invariance", "spaces-limits-proposition-affine" ], "ref_ids": [ 11830, 9, 4658 ] } ], "ref_ids": [] }, { "id": 3883, "type": "theorem", "label": "formal-spaces-lemma-completion-affine-is-affine-formal-algebraic-space", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-completion-affine-is-affine-formal-algebraic-space", "contents": [ "Let $S$ be a scheme. Let $X$ be an affine scheme over $S$.", "Let $T \\subset |X|$ be a closed subset. Then the functor", "$$", "(\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets},\\quad", "U \\longmapsto \\{f : U \\to X \\mid f(|U|) \\subset T\\}", "$$", "is a McQuillan affine formal algebraic space." ], "refs": [], "proofs": [ { "contents": [ "Say $X = \\Spec(A)$ and $T$ corresponds to the radical ideal $I \\subset A$.", "Let $U = \\Spec(B)$ be an affine scheme over $S$ and let", "$f : U \\to X$ be an element of $F(U)$. Then $f$ corresponds to a", "ring map $\\varphi : A \\to B$ such that every prime of $B$ contains", "$\\varphi(I) B$. Thus every element of $\\varphi(I)$ is nilpotent in $B$, see", "Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}.", "Setting $J = \\Ker(\\varphi)$ we conclude that $I/J$ is a locally nilpotent", "ideal in $A/J$. Equivalently, $V(J) = V(I) = T$.", "In other words, the functor of the lemma equals", "$\\colim \\Spec(A/J)$ where the colimit is over the", "collection of ideals $J$ with $V(J) = T$.", "Thus our functor is an affine formal algebraic space. It is McQuillan", "(Definition \\ref{definition-types-affine-formal-algebraic-space})", "because the maps $A \\to A/J$ are surjective", "and hence $A^\\wedge = \\lim A/J \\to A/J$ is surjective, see", "Lemma \\ref{lemma-mcquillan-affine-formal-algebraic-space}." ], "refs": [ "algebra-lemma-Zariski-topology", "formal-spaces-definition-types-affine-formal-algebraic-space", "formal-spaces-lemma-mcquillan-affine-formal-algebraic-space" ], "ref_ids": [ 389, 3978, 3870 ] } ], "ref_ids": [] }, { "id": 3884, "type": "theorem", "label": "formal-spaces-lemma-completion-is-formal-algebraic-space", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-completion-is-formal-algebraic-space", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $T \\subset |X|$ be a closed subset. Then the functor", "$$", "(\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets},\\quad", "U \\longmapsto \\{f : U \\to X \\mid f(|U|) \\subset T\\}", "$$", "is a formal algebraic space." ], "refs": [], "proofs": [ { "contents": [ "Denote $F$ the functor. Let $\\{U_i \\to U\\}$ be an fppf covering.", "Then $\\coprod |U_i| \\to |U|$ is surjective. Since $X$ is an fppf", "sheaf, it follows that $F$ is an fppf sheaf.", "\\medskip\\noindent", "Let $\\{g_i : X_i \\to X\\}$ be an \\'etale covering such that $X_i$ is affine", "for all $i$, see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-cover-by-union-affines}.", "The morphisms $F \\times_X X_i \\to F$ are \\'etale", "(see Spaces, Lemma", "\\ref{spaces-lemma-base-change-representable-transformations-property})", "and the map $\\coprod F \\times_X X_i \\to F$ is a surjection of sheaves.", "Thus it suffices to prove that $F \\times_X X_i$ is an affine formal", "algebraic space. A $U$-valued point of $F \\times_X X_i$ is a", "morphism $U \\to X_i$ whose image is contained in the closed subset", "$g_i^{-1}(T) \\subset |X_i|$. Thus this follows from", "Lemma \\ref{lemma-completion-affine-is-affine-formal-algebraic-space}." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-lemma-base-change-representable-transformations-property", "formal-spaces-lemma-completion-affine-is-affine-formal-algebraic-space" ], "ref_ids": [ 11830, 8133, 3883 ] } ], "ref_ids": [] }, { "id": 3885, "type": "theorem", "label": "formal-spaces-lemma-map-completions-representable", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-map-completions-representable", "contents": [ "Let $S$ be a scheme. Let $f : X' \\to X$ be a morphism", "of algebraic spaces over $S$. Let $T \\subset |X|$", "be a closed subset and let $T' = |f|^{-1}(T) \\subset |X'|$.", "Then", "$$", "\\xymatrix{", "X'_{/T'} \\ar[r] \\ar[d] & X' \\ar[d]^f \\\\", "X_{/T} \\ar[r] & X", "}", "$$", "is a cartesian diagram of sheaves. In particular, the morphism", "$X'_{/T'} \\to X_{/T}$ is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Namely, suppose that $Y \\to X$ is a morphism from a scheme into $X$ such", "that $|Y|$ maps into $T$. Then $Y \\times_X X' \\to X$ is a morphism of", "algebraic spaces such that $|Y \\times_X X'|$ maps into $T'$. Hence the", "functor $Y \\times_{X_{/T}} X'_{/T'}$ is represented by $Y \\times_X X'$", "and we see that the lemma holds." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3886, "type": "theorem", "label": "formal-spaces-lemma-reduction-completion", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-reduction-completion", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $T \\subset |X|$ be a closed subset. The reduction $(X_{/T})_{red}$", "of the completion $X_{/T}$ of $X$ along $T$ is", "the reduced induced closed subspace $Z$ of $X$ corresponding to $T$." ], "refs": [], "proofs": [ { "contents": [ "It follows from Lemma \\ref{lemma-reduction-formal-algebraic-space},", "Properties of Spaces, Definition", "\\ref{spaces-properties-definition-reduced-induced-space}", "(which uses Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-reduced-closed-subspace} to construct $Z$),", "and the definition of $X_{/T}$ that", "$Z$ and $(X_{/T})_{red}$ are reduced algebraic spaces", "characterized the same mapping property:", "a morphism $g : Y \\to X$ whose source is a reduced algebraic space", "factors through them if and only if $|Y|$ maps into $T \\subset |X|$." ], "refs": [ "formal-spaces-lemma-reduction-formal-algebraic-space", "spaces-properties-definition-reduced-induced-space", "spaces-properties-lemma-reduced-closed-subspace" ], "ref_ids": [ 3879, 11932, 11846 ] } ], "ref_ids": [] }, { "id": 3887, "type": "theorem", "label": "formal-spaces-lemma-affine-formal-completion-types", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-affine-formal-completion-types", "contents": [ "Let $S$ be a scheme. Let $X = \\Spec(A)$ be an affine scheme over $S$.", "Let $T \\subset X$ be a closed subset. Let $X_{/T}$ be the", "formal completion of $X$ along $T$.", "\\begin{enumerate}", "\\item If $X \\setminus T$ is quasi-compact, i.e., $T$ is constructible,", "then $X_{/T}$ is adic*.", "\\item If $T = V(I)$ for some finitely generated ideal $I \\subset A$,", "then $X_{/T} = \\text{Spf}(A^\\wedge)$ where $A^\\wedge$ is the", "$I$-adic completion of $A$.", "\\item If $X$ is Noetherian, then $X_{/T}$ is Noetherian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-qc-open} if (1) holds, then", "we can find an ideal $I \\subset A$ as in (2). If (3) holds then", "we can find an ideal $I \\subset A$ as in (2). Moreover, completions", "of Noetherian rings are Noetherian by", "Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian-Noetherian}.", "All in all we see that it suffices to prove (2).", "\\medskip\\noindent", "Proof of (2).", "Let $I = (f_1, \\ldots, f_r) \\subset A$ cut out $T$.", "If $Z = \\Spec(B)$ is an affine scheme and $g : Z \\to X$ is", "a morphism with $g(Z) \\subset T$ (set theoretically), then", "$g^\\sharp(f_i)$ is nilpotent in $B$ for each $i$. Thus", "$I^n$ maps to zero in $B$ for some $n$. Hence we see that", "$X_{/T} = \\colim \\Spec(A/I^n) = \\text{Spf}(A^\\wedge)$." ], "refs": [ "algebra-lemma-qc-open", "algebra-lemma-completion-Noetherian-Noetherian" ], "ref_ids": [ 432, 874 ] } ], "ref_ids": [] }, { "id": 3888, "type": "theorem", "label": "formal-spaces-lemma-completion-countably-indexed", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-completion-countably-indexed", "contents": [ "\\begin{reference}", "Email by Ofer Gabber of September 11, 2014.", "\\end{reference}", "Let $S$ be a scheme. Let $X = \\Spec(A)$ be an affine scheme over $S$.", "Let $T \\subset X$ be a closed subscheme.", "\\begin{enumerate}", "\\item If the formal completion $X_{/T}$ is countably indexed", "and there exist countably many $f_1, f_2, f_3, \\ldots \\in A$ such that", "$T = V(f_1, f_2, f_3, \\ldots)$, then $X_{/T}$ is adic*.", "\\item The conclusion of (1) is wrong if we omit the assumption that", "$T$ can be cut out by countably many functions in $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The assumption that $X_{/T}$ is countably indexed means that there exists a", "sequence of ideals", "$$", "A \\supset J_1 \\supset J_2 \\supset J_3 \\supset \\ldots", "$$", "with $V(J_n) = T$ such that every ideal $J \\subset A$ with $V(J) = T$", "there exists an $n$ such that $J \\supset J_n$.", "\\medskip\\noindent", "To construct an example for (2) let $\\omega_1$ be the first uncountable", "ordinal. Let $k$ be a field and let", "$A$ be the $k$-algebra generated by $x_\\alpha$, $\\alpha \\in \\omega_1$", "and $y_{\\alpha \\beta}$ with $\\alpha \\in \\beta \\in \\omega_1$", "subject to the relations $x_\\alpha = y_{\\alpha \\beta} x_\\beta$.", "Let $T = V(x_\\alpha)$. Let $J_n = (x_\\alpha^n)$.", "If $J \\subset A$ is an ideal such that", "$V(J) = T$, then $x_\\alpha^{n_\\alpha} \\in J$ for some $n_\\alpha \\geq 1$.", "One of the sets $\\{\\alpha \\mid n_\\alpha = n\\}$ must be unbounded in", "$\\omega_1$. Then the relations imply that $J_n \\subset J$.", "\\medskip\\noindent", "To see that (2) holds it now suffices to show that $A^\\wedge = \\lim A/J_n$", "is not a ring complete with respect to a finitely generated ideal.", "For $\\gamma \\in \\omega_1$ let $A_\\gamma$ be the quotient of $A$", "by the ideal generated by $x_\\alpha$, $\\alpha \\in \\gamma$ and", "$y_{\\alpha \\beta}$, $\\alpha \\in \\gamma$. As $A/J_1$ is reduced,", "every topologically nilpotent element $f$ of $\\lim A/J_n$ is in", "$J_1^\\wedge = \\lim J_1/J_n$. This means $f$ is an infinite series", "involving only a countable number of generators. Hence $f$ dies in", "$A_\\gamma^\\wedge = \\lim A_\\gamma/J_nA_\\gamma$ for some $\\gamma$.", "Note that $A^\\wedge \\to A_\\gamma^\\wedge$ is continuous and open by", "Lemma \\ref{lemma-ses}.", "If the topology on $A^\\wedge$ was $I$-adic for some finitely generated ideal", "$I \\subset A^\\wedge$, then $I$ would go to zero in some", "$A_\\gamma^\\wedge$. This would mean that $A_\\gamma^\\wedge$ is discrete,", "which is not the case as there is a surjective continuous and open", "(by Lemma \\ref{lemma-ses}) map", "$A_\\gamma^\\wedge \\to k[[t]]$ given by", "$x_\\alpha \\mapsto t$, $y_{\\alpha \\beta} \\mapsto 1$ for", "$\\gamma = \\alpha$ or $\\gamma \\in \\alpha$.", "\\medskip\\noindent", "Before we prove (1) we first prove the following: If $I \\subset A^\\wedge$ is", "a finitely generated ideal whose closure $\\bar I$ is open, then $I = \\bar I$.", "Since $V(J_n^2) = T$ there exists an $m$ such that $J_n^2 \\supset J_m$.", "Thus, we may assume that $J_n^2 \\supset J_{n + 1}$ for all $n$ by passing", "to a subsequence. Set $J_n^\\wedge = \\lim_{k \\geq n} J_n/J_k \\subset A^\\wedge$.", "Since the closure $\\bar I = \\bigcap (I + J_n^\\wedge)$", "(Lemma \\ref{lemma-closed}) is open we see that there exists an $m$ such that", "$I + J_n^\\wedge \\supset J_m^\\wedge$ for all $n \\geq m$. Fix such an $m$.", "We have", "$$", "J_{n - 1}^\\wedge I + J_{n + 1}^\\wedge \\supset", "J_{n - 1}^\\wedge (I + J_{n + 1}^\\wedge) \\supset", "J_{n - 1}^\\wedge J_m^\\wedge", "$$", "for all $n \\geq m + 1$. Namely, the first inclusion is trivial and the", "second was shown above. Because $J_{n - 1}J_m \\supset J_{n - 1}^2 \\supset J_n$", "these inclusions show that the image of $J_n$ in $A^\\wedge$", "is contained in the ideal $J_{n - 1}^\\wedge I + J_{n + 1}^\\wedge$.", "Because this ideal is open we conclude that", "$$", "J_{n - 1}^\\wedge I + J_{n + 1}^\\wedge \\supset J_n^\\wedge.", "$$", "Say $I = (g_1, \\ldots, g_t)$. Pick $f \\in J_{m + 1}^\\wedge$.", "Using the last displayed inclusion, valid for all $n \\geq m + 1$,", "we can write by induction on $c \\geq 0$", "$$", "f = \\sum f_{i, c} g_i \\mod J_{m + 1+ c}^\\wedge", "$$", "with $f_{i, c} \\in J_m^\\wedge$ and", "$f_{i, c} \\equiv f_{i, c - 1} \\bmod J_{m + c}^\\wedge$.", "It follows that $IJ_m^\\wedge \\supset J_{m + 1}^\\wedge$.", "Combined with $I + J_{m + 1}^\\wedge \\supset J_m^\\wedge$", "we conclude that $I$ is open.", "\\medskip\\noindent", "Proof of (1). Assume $T = V(f_1, f_2, f_3, \\ldots)$.", "Let $I_m \\subset A^\\wedge$ be the ideal generated by $f_1, \\ldots, f_m$.", "We distinguish two cases.", "\\medskip\\noindent", "Case I: For some $m$ the closure of $I_m$ is open.", "Then $I_m$ is open by the result of the previous paragraph.", "For any $n$ we have $(J_n)^2 \\supset J_{n+1}$ by design, so", "the closure of $(J_n^\\wedge)^2$ contains $J_{n+1}^\\wedge$", "and thus is open. Taking $n$ large, it follows that the closure", "of the product of any two open ideals in $A^\\wedge$ is open.", "Let us prove $I_m^k$ is open for $k \\ge 1$ by induction on $k$.", "The case $k = 1$ is our hypothesis on $m$ in Case I.", "For $k > 1$, suppose $I_m^{k - 1}$ is open. Then", "$I_m^k = I_m^{k - 1} \\cdot I_m$ is the product of two open ideals", "and hence has open closure. But then since $I_m^k$", "is finitely generated it follows that $I_m^k$", "is open by the previous paragraph (applied to $I = I_m^k$),", "so we can continue the induction on $k$.", "As each element of $I_m$ is topologically nilpotent, we conclude", "that $I_m$ is an ideal of definition which proves that $A^\\wedge$", "is adic with a finitely generated ideal of definition, i.e.,", "$X_{/T}$ is adic*.", "\\medskip\\noindent", "Case II. For all $m$ the closure $\\bar I_m$ of $I_m$ is not open.", "Then the topology on $A^\\wedge/\\bar I_m$ is not discrete. This means", "we can pick $\\phi(m) \\geq m$ such that", "$$", "\\Im(J_{\\phi(m)} \\to A/(f_1, \\ldots, f_m)) \\not =", "\\Im(J_{\\phi(m) + 1} \\to A/(f_1, \\ldots, f_m))", "$$", "To see this we have used that", "$A^\\wedge/(\\bar I_m + J_n^\\wedge) = A/((f_1, \\ldots, f_m) + J_n)$.", "Choose exponents $e_i > 0$ such that $f_i^{e_i} \\in J_{\\phi(m) + 1}$", "for $0 < m < i$. Let $J = (f_1^{e_1}, f_2^{e_2}, f_3^{e_3}, \\ldots)$.", "Then $V(J) = T$. We claim that $J \\not \\supset J_n$ for all $n$", "which is a contradiction proving Case II does not occur.", "Namely, the image of $J$ in $A/(f_1, \\ldots, f_m)$ is contained", "in the image of $J_{\\phi(m) + 1}$ which is properly contained in the", "image of $J_m$." ], "refs": [ "formal-spaces-lemma-ses", "formal-spaces-lemma-ses", "formal-spaces-lemma-closed" ], "ref_ids": [ 3857, 3857, 3854 ] } ], "ref_ids": [] }, { "id": 3889, "type": "theorem", "label": "formal-spaces-lemma-etale-covering-by-formal-algebraic-spaces", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-etale-covering-by-formal-algebraic-spaces", "contents": [ "Let $S$ be a scheme. Let $\\{X_i \\to X\\}_{i \\in I}$ be a family of maps", "of sheaves on $(\\Sch/S)_{fppf}$. Assume (a) $X_i$ is a", "formal algebraic space over $S$, (b) $X_i \\to X$ is representable", "by algebraic spaces and \\'etale, and (c) $\\coprod X_i \\to X$", "is a surjection of sheaves. Then $X$ is a formal algebraic space", "over $S$." ], "refs": [], "proofs": [ { "contents": [ "For each $i$ pick $\\{X_{ij} \\to X_i\\}_{j \\in J_i}$ as in", "Definition \\ref{definition-formal-algebraic-space}.", "Then $\\{X_{ij} \\to X\\}_{i \\in I, j \\in J_i}$ is a family", "as in Definition \\ref{definition-formal-algebraic-space}", "for $X$." ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "ref_ids": [ 3981, 3981 ] } ], "ref_ids": [] }, { "id": 3890, "type": "theorem", "label": "formal-spaces-lemma-fibre-products-general", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-fibre-products-general", "contents": [ "Let $S$ be a scheme. Let $X, Y$ be formal algebraic spaces over $S$", "and let $Z$ be a sheaf whose diagonal is representable by", "algebraic spaces. Let $X \\to Z$ and $Y \\to Z$ be maps of sheaves.", "Then $X \\times_Z Y$ is a formal algebraic space." ], "refs": [], "proofs": [ { "contents": [ "Choose $\\{X_i \\to X\\}$ and $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}.", "Then $\\{X_i \\times_Z Y_j \\to X \\times_Z Y\\}$ is a family", "of maps which are representable by algebraic spaces and \\'etale.", "Thus Lemma \\ref{lemma-etale-covering-by-formal-algebraic-spaces}", "tells us it suffices to show that $X \\times_Z Y$ is a formal", "algebraic space when $X$ and $Y$ are affine formal algebraic spaces.", "\\medskip\\noindent", "Assume $X$ and $Y$ are affine formal algebraic spaces.", "Write $X = \\colim X_\\lambda$ and $Y = \\colim Y_\\mu$ as", "in Definition \\ref{definition-affine-formal-algebraic-space}.", "Then $X \\times_Z Y = \\colim X_\\lambda \\times_Z Y_\\mu$.", "Each $X_\\lambda \\times_Z Y_\\mu$ is an algebraic space.", "For $\\lambda \\leq \\lambda'$ and $\\mu \\leq \\mu'$ the morphism", "$$", "X_\\lambda \\times_Z Y_\\mu \\to", "X_\\lambda \\times_Z Y_{\\mu'} \\to", "X_{\\lambda'} \\times_Z Y_{\\mu'}", "$$", "is a thickening as a composition of base changes of thickenings.", "Thus we conclude by applying Lemma \\ref{lemma-colimit-is-formal}." ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-lemma-etale-covering-by-formal-algebraic-spaces", "formal-spaces-definition-affine-formal-algebraic-space", "formal-spaces-lemma-colimit-is-formal" ], "ref_ids": [ 3981, 3889, 3977, 3882 ] } ], "ref_ids": [] }, { "id": 3891, "type": "theorem", "label": "formal-spaces-lemma-fibre-products", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-fibre-products", "contents": [ "Let $S$ be a scheme. The category of formal algebraic spaces over $S$", "has fibre products." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-fibre-products-general}", "because formal algebraic spaces have representable diagonals, see", "Lemma \\ref{lemma-diagonal-formal-algebraic-space}." ], "refs": [ "formal-spaces-lemma-fibre-products-general", "formal-spaces-lemma-diagonal-formal-algebraic-space" ], "ref_ids": [ 3890, 3877 ] } ], "ref_ids": [] }, { "id": 3892, "type": "theorem", "label": "formal-spaces-lemma-reduction-fibre-products", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-reduction-fibre-products", "contents": [ "Let $S$ be a scheme. Let $X \\to Z$ and $Y \\to Z$ be morphisms of", "formal algebraic spaces over $S$. Then", "$(X \\times_Z Y)_{red} = (X_{red} \\times_{Z_{red}} Y_{red})_{red}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the universal property of the reduction", "in Lemma \\ref{lemma-reduction-formal-algebraic-space}." ], "refs": [ "formal-spaces-lemma-reduction-formal-algebraic-space" ], "ref_ids": [ 3879 ] } ], "ref_ids": [] }, { "id": 3893, "type": "theorem", "label": "formal-spaces-lemma-diagonal-morphism-formal-algebraic-spaces", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-diagonal-morphism-formal-algebraic-spaces", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces", "over $S$. The diagonal morphism $\\Delta : X \\to X \\times_Y X$", "is representable (by schemes), a monomorphism, locally quasi-finite,", "locally of finite type, and separated." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme and let $T \\to X \\times_Y X$ be a morphism.", "Then", "$$", "T \\times_{(X \\times_Y X)} X = T \\times_{(X \\times_S X)} X", "$$", "Hence the result follows immediately from", "Lemma \\ref{lemma-diagonal-formal-algebraic-space}." ], "refs": [ "formal-spaces-lemma-diagonal-formal-algebraic-space" ], "ref_ids": [ 3877 ] } ], "ref_ids": [] }, { "id": 3894, "type": "theorem", "label": "formal-spaces-lemma-characterize-quasi-separated", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-characterize-quasi-separated", "contents": [ "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item the reduction of $X$", "(Lemma \\ref{lemma-reduction-formal-algebraic-space}) is a", "quasi-separated algebraic space,", "\\item for $U \\to X$, $V \\to X$ with $U$, $V$ quasi-compact schemes", "the fibre product $U \\times_X V$ is quasi-compact,", "\\item for $U \\to X$, $V \\to X$ with $U$, $V$ affine", "the fibre product $U \\times_X V$ is quasi-compact.", "\\end{enumerate}" ], "refs": [ "formal-spaces-lemma-reduction-formal-algebraic-space" ], "proofs": [ { "contents": [ "Observe that $U \\times_X V$ is a scheme by", "Lemma \\ref{lemma-diagonal-formal-algebraic-space}.", "Let $U_{red}, V_{red}, X_{red}$ be the reduction of $U, V, X$.", "Then", "$$", "U_{red} \\times_{X_{red}} V_{red} = U_{red} \\times_X V_{red} \\to U \\times_X V", "$$", "is a thickening of schemes. From this the equivalence of (1) and (2)", "is clear, keeping in mind the analogous lemma for algebraic spaces, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-characterize-quasi-separated}.", "We omit the proof of the equivalence of (2) and (3)." ], "refs": [ "formal-spaces-lemma-diagonal-formal-algebraic-space", "spaces-properties-lemma-characterize-quasi-separated" ], "ref_ids": [ 3877, 11816 ] } ], "ref_ids": [ 3879 ] }, { "id": 3895, "type": "theorem", "label": "formal-spaces-lemma-characterize-separated", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-characterize-separated", "contents": [ "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item the reduction of $X$", "(Lemma \\ref{lemma-reduction-formal-algebraic-space}) is a separated", "algebraic space,", "\\item for $U \\to X$, $V \\to X$ with $U$, $V$ affine", "the fibre product $U \\times_X V$ is affine and", "$$", "\\mathcal{O}(U) \\otimes_\\mathbf{Z} \\mathcal{O}(V)", "\\longrightarrow", "\\mathcal{O}(U \\times_X V)", "$$", "is surjective.", "\\end{enumerate}" ], "refs": [ "formal-spaces-lemma-reduction-formal-algebraic-space" ], "proofs": [ { "contents": [ "If (2) holds, then $X_{red}$ is a separated algebraic space", "by applying Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-characterize-quasi-separated}", "to morphisms $U \\to X_{red}$ and $V \\to X_{red}$", "with $U, V$ affine and using that $U \\times_{X_{red}} V = U \\times_X V$.", "\\medskip\\noindent", "Assume (1). Let $U \\to X$ and $V \\to X$ be as in (2).", "Observe that $U \\times_X V$ is a scheme by", "Lemma \\ref{lemma-diagonal-formal-algebraic-space}.", "Let $U_{red}, V_{red}, X_{red}$ be the reduction of $U, V, X$.", "Then", "$$", "U_{red} \\times_{X_{red}} V_{red} = U_{red} \\times_X V_{red} \\to U \\times_X V", "$$", "is a thickening of schemes. It follows that", "$(U \\times_X V)_{red} = (U_{red} \\times_{X_{red}} V_{red})_{red}$.", "In particular, we see that $(U \\times_X V)_{red}$ is an affine scheme", "and that", "$$", "\\mathcal{O}(U) \\otimes_\\mathbf{Z} \\mathcal{O}(V)", "\\longrightarrow", "\\mathcal{O}((U \\times_X V)_{red})", "$$", "is surjective, see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-characterize-quasi-separated}.", "Then $U \\times_X V$ is affine by", "Limits of Spaces, Proposition \\ref{spaces-limits-proposition-affine}.", "On the other hand, the morphism $U \\times_X V \\to U \\times V$", "of affine schemes is the composition", "$$", "U \\times_X V = X \\times_{(X \\times_S X)} (U \\times_S V)", "\\to U \\times_S V \\to U \\times V", "$$", "The first morphism is a monomorphism and locally of finite type", "(Lemma \\ref{lemma-diagonal-formal-algebraic-space}).", "The second morphism is an immersion", "(Schemes, Lemma \\ref{schemes-lemma-fibre-product-after-map}).", "Hence the composition is a monomorphism which is locally of finite type.", "On the other hand, the composition is integral as the map on", "underlying reduced affine schemes is a closed immersion", "by the above and hence universally closed (use", "Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed}).", "Thus the ring map", "$$", "\\mathcal{O}(U) \\otimes_\\mathbf{Z} \\mathcal{O}(V)", "\\longrightarrow", "\\mathcal{O}(U \\times_X V)", "$$", "is an epimorphism which is integral of finite type", "hence finite hence surjective (use", "Morphisms, Lemma \\ref{morphisms-lemma-finite-integral}", "and", "Algebra, Lemma \\ref{algebra-lemma-finite-epimorphism-surjective})." ], "refs": [ "spaces-properties-lemma-characterize-quasi-separated", "formal-spaces-lemma-diagonal-formal-algebraic-space", "spaces-properties-lemma-characterize-quasi-separated", "spaces-limits-proposition-affine", "formal-spaces-lemma-diagonal-formal-algebraic-space", "schemes-lemma-fibre-product-after-map", "morphisms-lemma-integral-universally-closed", "morphisms-lemma-finite-integral", "algebra-lemma-finite-epimorphism-surjective" ], "ref_ids": [ 11816, 3877, 11816, 4658, 3877, 7711, 5441, 5438, 952 ] } ], "ref_ids": [ 3879 ] }, { "id": 3896, "type": "theorem", "label": "formal-spaces-lemma-fibre-product-affines-over-separated", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-fibre-product-affines-over-separated", "contents": [ "Let $S$ be a scheme. Let $X \\to Z$ and $Y \\to Z$ be morphisms", "of formal algebraic spaces over $S$. Assume $Z$ separated.", "\\begin{enumerate}", "\\item If $X$ and $Y$ are affine formal algebraic spaces, then", "so is $X \\times_Z Y$.", "\\item If $X$ and $Y$ are McQuillan affine formal algebraic spaces, then", "so is $X \\times_Z Y$.", "\\item If $X$, $Y$, and $Z$ are McQuillan affine formal algebraic spaces", "corresponding to the weakly admissible topological $S$-algebras", "$A$, $B$, and $C$, then $X \\times_Z Y$ corresponds to", "$A \\widehat{\\otimes}_C B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $X = \\colim X_\\lambda$ and $Y = \\colim Y_\\mu$ as", "in Definition \\ref{definition-affine-formal-algebraic-space}.", "Then $X \\times_Z Y = \\colim X_\\lambda \\times_Z Y_\\mu$.", "Since $Z$ is separated the fibre products are affine, hence", "we see that (1) holds. Assume $X$ and $Y$ corresponds to", "the weakly admissible topological $S$-algebras $A$ and $B$", "and $X_\\lambda = \\Spec(A/I_\\lambda)$ and $Y_\\mu = \\Spec(B/J_\\mu)$.", "Then", "$$", "X_\\lambda \\times_Z Y_\\mu \\to", "X_\\lambda \\times Y_\\mu \\to \\Spec(A \\otimes B)", "$$", "is a closed immersion. Thus one of the conditions of", "Lemma \\ref{lemma-mcquillan-affine-formal-algebraic-space}", "holds and we conclude that $X \\times_Z Y$ is McQuillan.", "If also $Z$ is McQuillan corresponding to $C$, then", "$$", "X_\\lambda \\times_Z Y_\\mu = \\Spec(A/I_\\lambda \\otimes_C B/J_\\mu)", "$$", "hence we see that the weakly admissible topological ring", "corresponding to $X \\times_Z Y$ is the completed tensor product", "(see Definition \\ref{definition-toplogy-tensor-product})." ], "refs": [ "formal-spaces-definition-affine-formal-algebraic-space", "formal-spaces-lemma-mcquillan-affine-formal-algebraic-space", "formal-spaces-definition-toplogy-tensor-product" ], "ref_ids": [ 3977, 3870, 3974 ] } ], "ref_ids": [] }, { "id": 3897, "type": "theorem", "label": "formal-spaces-lemma-separated-from-separated", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-separated-from-separated", "contents": [ "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.", "Let $U \\to X$ be a morphism where $U$ is a separated algebraic", "space over $S$. Then $U \\to X$ is separated." ], "refs": [], "proofs": [ { "contents": [ "The statement makes sense because $U \\to X$ is representable by", "algebraic spaces (Lemma \\ref{lemma-space-to-formal-space}).", "Let $T$ be a scheme and $T \\to X$ a morphism. We have to show", "that $U \\times_X T \\to T$ is separated. Since $U \\times_X T \\to U \\times_S T$", "is a monomorphism, it suffices to show that $U \\times_S T \\to T$", "is separated. As this is the base change of $U \\to S$ this", "follows. We used in the argument above:", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-separated},", "\\ref{spaces-morphisms-lemma-composition-separated},", "\\ref{spaces-morphisms-lemma-monomorphism-separated}, and", "\\ref{spaces-morphisms-lemma-separated-implies-morphism-separated}." ], "refs": [ "formal-spaces-lemma-space-to-formal-space", "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-composition-separated", "spaces-morphisms-lemma-monomorphism-separated", "spaces-morphisms-lemma-separated-implies-morphism-separated" ], "ref_ids": [ 3878, 4714, 4718, 4752, 4721 ] } ], "ref_ids": [] }, { "id": 3898, "type": "theorem", "label": "formal-spaces-lemma-characterize-quasi-compact", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-characterize-quasi-compact", "contents": [ "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item the reduction of $X$", "(Lemma \\ref{lemma-reduction-formal-algebraic-space}) is a quasi-compact", "algebraic space,", "\\item we can find $\\{X_i \\to X\\}_{i \\in I}$ as in", "Definition \\ref{definition-formal-algebraic-space} with $I$ finite,", "\\item there exists a morphism $Y \\to X$ representable by algebraic", "spaces which is \\'etale and surjective and where", "$Y$ is an affine formal algebraic space.", "\\end{enumerate}" ], "refs": [ "formal-spaces-lemma-reduction-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 3879, 3981 ] }, { "id": 3899, "type": "theorem", "label": "formal-spaces-lemma-characterize-quasi-compact-morphism", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-characterize-quasi-compact-morphism", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces over $S$. The following are equivalent", "\\begin{enumerate}", "\\item the induced map $f_{red} : X_{red} \\to Y_{red}$ between reductions", "(Lemma \\ref{lemma-reduction-formal-algebraic-space}) is a quasi-compact", "morphism of algebraic spaces,", "\\item for every quasi-compact scheme $T$ and morphism $T \\to Y$", "the fibre product $X \\times_Y T$ is a quasi-compact formal", "algebraic space,", "\\item for every affine scheme $T$ and morphism $T \\to Y$", "the fibre product $X \\times_Y T$ is a quasi-compact formal", "algebraic space, and", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "such that each $X \\times_Y Y_j$ is a quasi-compact formal algebraic space.", "\\end{enumerate}" ], "refs": [ "formal-spaces-lemma-reduction-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 3879, 3981 ] }, { "id": 3900, "type": "theorem", "label": "formal-spaces-lemma-quasi-compact-representable", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-quasi-compact-representable", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces over $S$ which is representable by algebraic spaces.", "Then $f$ is quasi-compact in the sense of", "Definition \\ref{definition-quasi-compact-morphism}", "if and only if $f$ is quasi-compact in the sense of", "Bootstrap, Definition \\ref{bootstrap-definition-property-transformation}." ], "refs": [ "formal-spaces-definition-quasi-compact-morphism", "bootstrap-definition-property-transformation" ], "proofs": [ { "contents": [ "This is immediate from the definitions and", "Lemma \\ref{lemma-characterize-quasi-compact-morphism}." ], "refs": [ "formal-spaces-lemma-characterize-quasi-compact-morphism" ], "ref_ids": [ 3899 ] } ], "ref_ids": [ 3985, 2638 ] }, { "id": 3901, "type": "theorem", "label": "formal-spaces-lemma-structure-quasi-compact-quasi-separated", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-structure-quasi-compact-quasi-separated", "contents": [ "\\begin{reference}", "\\cite[Proposition 3.32]{Yasuda}", "\\end{reference}", "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "formal algebraic space over $S$. Then $X = \\colim X_\\lambda$", "for a system of algebraic spaces $(X_\\lambda, f_{\\lambda \\mu})$", "over a directed set $\\Lambda$ where each", "$f_{\\lambda \\mu} : X_\\lambda \\to X_\\mu$ is a thickening." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-characterize-quasi-compact} we may choose an", "affine formal algebraic space $Y$ and a representable surjective", "\\'etale morphism $Y \\to X$. Write $Y = \\colim Y_\\lambda$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}.", "\\medskip\\noindent", "Pick $\\lambda \\in \\Lambda$. Then $Y_\\lambda \\times_X Y$ is a scheme by", "Lemma \\ref{lemma-presentation-representable}. The reduction", "(Lemma \\ref{lemma-reduction-formal-algebraic-space})", "of $Y_\\lambda \\times_X Y$ is equal to the reduction of", "$Y_{red} \\times_{X_{red}} Y_{red}$ which is quasi-compact as $X$", "is quasi-separated and $Y_{red}$ is affine.", "Therefore $Y_\\lambda \\times_X Y$ is a quasi-compact scheme.", "Hence there exists a $\\mu \\geq \\lambda$ such that", "$\\text{pr}_2 : Y_\\lambda \\times_X Y \\to Y$ factors", "through $Y_\\mu$, see Lemma \\ref{lemma-factor-through-thickening}.", "Let $Z_\\lambda$ be the scheme theoretic image of the morphism", "$\\text{pr}_2 : Y_\\lambda \\times_X Y \\to Y_\\mu$.", "This is independent of the choice of $\\mu$ and we can and", "will think of $Z_\\lambda \\subset Y$ as the scheme theoretic", "image of the morphism $\\text{pr}_2 : Y_\\lambda \\times_X Y \\to Y$.", "Observe that $Z_\\lambda$ is also equal to the scheme theoretic image", "of the morphism $\\text{pr}_1 : Y \\times_X Y_\\lambda \\to Y$ since", "this is isomorphic to the morphism used to define $Z_\\lambda$.", "We claim that $Z_\\lambda \\times_X Y = Y \\times_X Z_\\lambda$ as subfunctors", "of $Y \\times_X Y$. Namely, since $Y \\to X$ is \\'etale we see that", "$Z_\\lambda \\times_X Y$ is the scheme theoretic image of the morphism", "$$", "\\text{pr}_{13} = \\text{pr}_1 \\times \\text{id}_Y :", "Y \\times_X Y_\\lambda \\times_X Y \\longrightarrow Y \\times_X Y", "$$", "by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image}.", "By the same token, $Y \\times_X Z_\\lambda$ is the scheme theoretic image", "of the morphism", "$$", "\\text{pr}_{13} = \\text{id}_Y \\times \\text{pr}_2 : ", "Y \\times_X Y_\\lambda \\times_X Y \\longrightarrow Y \\times_X Y", "$$", "The claim follows. Then", "$R_\\lambda = Z_\\lambda \\times_X Y = Y \\times_X Z_\\lambda$", "together with the morphism $R_\\lambda \\to Z_\\lambda \\times_S Z_\\lambda$", "defines an \\'etale equivalence relation. In this way we obtain an algebraic", "space $X_\\lambda = Z_\\lambda/R_\\lambda$. By construction the diagram", "$$", "\\xymatrix{", "Z_\\lambda \\ar[r] \\ar[d] & Y \\ar[d] \\\\", "X_\\lambda \\ar[r] & X", "}", "$$", "is cartesian (because $X$ is the coequalizer of the two projections", "$R = Y \\times_X Y \\to Y$, because $Z_\\lambda \\subset Y$ is $R$-invariant,", "and because $R_\\lambda$ is the restriction of $R$ to $Z_\\lambda$).", "Hence $X_\\lambda \\to X$ is representable and a closed immersion, see", "Spaces, Lemma", "\\ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale}.", "On the other hand, since $Y_\\lambda \\subset Z_\\lambda$ we see that", "$(X_\\lambda)_{red} = X_{red}$, in other words, $X_\\lambda \\to X$", "is a thickening. Finally, we claim that", "$$", "X = \\colim X_\\lambda", "$$", "We have $Y \\times_X X_\\lambda = Z_\\lambda \\supset Y_\\lambda$. Every", "morphism $T \\to X$ where $T$ is a scheme over $S$ lifts \\'etale locally", "to a morphism into $Y$ which lifts \\'etale locally into a morphism", "into some $Y_\\lambda$. Hence $T \\to X$ lifts \\'etale locally on", "$T$ to a morphism into $X_\\lambda$. This finishes the proof." ], "refs": [ "formal-spaces-lemma-characterize-quasi-compact", "formal-spaces-definition-affine-formal-algebraic-space", "formal-spaces-lemma-presentation-representable", "formal-spaces-lemma-reduction-formal-algebraic-space", "formal-spaces-lemma-factor-through-thickening", "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image", "spaces-lemma-morphism-sheaves-with-P-effective-descent-etale" ], "ref_ids": [ 3898, 3977, 3872, 3879, 3868, 4780, 8158 ] } ], "ref_ids": [] }, { "id": 3902, "type": "theorem", "label": "formal-spaces-lemma-characterize-affine", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-characterize-affine", "contents": [ "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.", "Then $X$ is an affine formal algebraic space if and only if", "its reduction $X_{red}$ (Lemma \\ref{lemma-reduction-formal-algebraic-space})", "is affine." ], "refs": [ "formal-spaces-lemma-reduction-formal-algebraic-space" ], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-characterize-quasi-separated} and", "\\ref{lemma-characterize-quasi-compact} and", "Definitions \\ref{definition-separated} and \\ref{definition-quasi-compact}", "we see that $X$ is quasi-compact and quasi-separated.", "By Yasuda's lemma (Lemma \\ref{lemma-structure-quasi-compact-quasi-separated})", "we can write $X = \\colim X_\\lambda$ as a filtered colimit", "of thickenings of algebraic spaces. However, each $X_\\lambda$", "is affine by Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-reduction-scheme}", "because $(X_\\lambda)_{red} = X_{red}$.", "Hence $X$ is an affine formal algebraic space by definition." ], "refs": [ "formal-spaces-lemma-characterize-quasi-separated", "formal-spaces-lemma-characterize-quasi-compact", "formal-spaces-definition-separated", "formal-spaces-definition-quasi-compact", "formal-spaces-lemma-structure-quasi-compact-quasi-separated", "spaces-limits-lemma-reduction-scheme" ], "ref_ids": [ 3894, 3898, 3983, 3984, 3901, 4627 ] } ], "ref_ids": [ 3879 ] }, { "id": 3903, "type": "theorem", "label": "formal-spaces-lemma-composition-representable", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-composition-representable", "contents": [ "The composition of morphisms representable by algebraic spaces is", "representable by algebraic spaces. The same holds for representable", "(by schemes)." ], "refs": [], "proofs": [ { "contents": [ "See Bootstrap, Lemma \\ref{bootstrap-lemma-composition-transformation}." ], "refs": [ "bootstrap-lemma-composition-transformation" ], "ref_ids": [ 2609 ] } ], "ref_ids": [] }, { "id": 3904, "type": "theorem", "label": "formal-spaces-lemma-base-change-representable", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-base-change-representable", "contents": [ "A base change of a morphism representable by algebraic spaces is", "representable by algebraic spaces. The same holds for representable", "(by schemes)." ], "refs": [], "proofs": [ { "contents": [ "See Bootstrap, Lemma \\ref{bootstrap-lemma-base-change-transformation}." ], "refs": [ "bootstrap-lemma-base-change-transformation" ], "ref_ids": [ 2604 ] } ], "ref_ids": [] }, { "id": 3905, "type": "theorem", "label": "formal-spaces-lemma-permanence-representable", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-permanence-representable", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of", "formal algebraic spaces over $S$. If $g \\circ f : X \\to Z$ is representable", "by algebraic spaces, then $f : X \\to Y$ is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Note that the diagonal of $Y \\to Z$ is representable by", "Lemma \\ref{lemma-diagonal-morphism-formal-algebraic-spaces}.", "Thus $X \\to Y$ is representable by algebraic spaces by", "Bootstrap, Lemma \\ref{bootstrap-lemma-representable-by-spaces-permanence}." ], "refs": [ "formal-spaces-lemma-diagonal-morphism-formal-algebraic-spaces", "bootstrap-lemma-representable-by-spaces-permanence" ], "ref_ids": [ 3893, 2611 ] } ], "ref_ids": [] }, { "id": 3906, "type": "theorem", "label": "formal-spaces-lemma-representable-by-algebraic-spaces-local", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-representable-by-algebraic-spaces-local", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces over $S$. The following are equivalent:", "\\begin{enumerate}", "\\item the morphism $f$ is representable by algebraic spaces,", "\\item there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are formal algebraic spaces, the vertical arrows are", "representable by algebraic spaces, $U \\to X$", "is surjective \\'etale, and $U \\to V$ is representable by algebraic spaces,", "\\item for any commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are formal algebraic spaces and the vertical arrows are", "representable by algebraic spaces, the morphism $U \\to V$ is", "representable by algebraic spaces,", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "and for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in", "Definition \\ref{definition-formal-algebraic-space} such that", "$X_{ji} \\to Y_j$ is representable by algebraic spaces for each $j$ and $i$,", "\\item there exist a covering $\\{X_i \\to X\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$", "is an affine formal algebraic space, $Y_i \\to Y$ is representable", "by algebraic spaces, such that $X_i \\to Y_i$ is representable by algebraic", "spaces, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "It is clear that (1) implies (2) because we can take $U = X$ and $V = Y$.", "Conversely, (2) implies (1) by", "Bootstrap, Lemma \\ref{bootstrap-lemma-representable-by-spaces-cover}", "applied to $U \\to X \\to Y$.", "\\medskip\\noindent", "Assume (1) is true and consider a diagram as in (3).", "Then $U \\to Y$ is representable by algebraic spaces", "(as the composition $U \\to X \\to Y$, see", "Bootstrap, Lemma \\ref{bootstrap-lemma-composition-transformation})", "and factors through $V$. Thus $U \\to V$ is representable by", "algebraic spaces by Lemma \\ref{lemma-permanence-representable}.", "\\medskip\\noindent", "It is clear that (3) implies (2). Thus now (1) -- (3) are equivalent.", "\\medskip\\noindent", "Observe that the condition in (4) makes sense as the fibre product", "$Y_j \\times_Y X$ is a formal algebraic space by", "Lemma \\ref{lemma-fibre-products}.", "It is clear that (4) implies (5).", "\\medskip\\noindent", "Assume $X_i \\to Y_i \\to Y$ as in (5). Then we set", "$V = \\coprod Y_i$ and $U = \\coprod X_i$ to see that", "(5) implies (2).", "\\medskip\\noindent", "Finally, assume (1) -- (3) are true.", "Thus we can choose any covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "and for each $j$ any covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}.", "Then $X_{ij} \\to Y_j$ is representable by algebraic spaces by (3)", "and we see that (4) is true. This concludes the proof." ], "refs": [ "bootstrap-lemma-representable-by-spaces-cover", "bootstrap-lemma-composition-transformation", "formal-spaces-lemma-permanence-representable", "formal-spaces-lemma-fibre-products", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "ref_ids": [ 2628, 2609, 3905, 3891, 3981, 3981 ] } ], "ref_ids": [ 3981, 3981, 3981 ] }, { "id": 3907, "type": "theorem", "label": "formal-spaces-lemma-algebraic-space-over-affine-formal", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-algebraic-space-over-affine-formal", "contents": [ "Let $S$ be a scheme. Let $Y$ be an affine formal algebraic space over $S$.", "Let $f : X \\to Y$ be a map of sheaves on $(\\Sch/S)_{fppf}$ which is", "representable by algebraic spaces. Then $X$ is a formal", "algebraic space." ], "refs": [], "proofs": [ { "contents": [ "Write $Y = \\colim Y_\\lambda$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}.", "For each $\\lambda$ the fibre product", "$X \\times_Y Y_\\lambda$ is an algebraic space.", "Hence $X = \\colim X \\times_Y Y_\\lambda$ is a formal", "algebraic space by Lemma \\ref{lemma-colimit-is-formal}." ], "refs": [ "formal-spaces-definition-affine-formal-algebraic-space", "formal-spaces-lemma-colimit-is-formal" ], "ref_ids": [ 3977, 3882 ] } ], "ref_ids": [] }, { "id": 3908, "type": "theorem", "label": "formal-spaces-lemma-representable-by-algebraic-spaces", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-representable-by-algebraic-spaces", "contents": [ "Let $S$ be a scheme. Let $Y$ be a formal algebraic space over $S$.", "Let $f : X \\to Y$ be a map of sheaves on $(\\Sch/S)_{fppf}$ which is", "representable by algebraic spaces. Then $X$ is a formal", "algebraic space." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{Y_i \\to Y\\}$ be as in", "Definition \\ref{definition-formal-algebraic-space}.", "Then $X \\times_Y Y_i \\to X$ is a family of morphisms", "representable by algebraic spaces, \\'etale, and jointly", "surjective. Thus it suffices to show that", "$X \\times_Y Y_i$ is a formal algebraic space, see", "Lemma \\ref{lemma-etale-covering-by-formal-algebraic-spaces}.", "This follows from Lemma \\ref{lemma-algebraic-space-over-affine-formal}." ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-lemma-etale-covering-by-formal-algebraic-spaces", "formal-spaces-lemma-algebraic-space-over-affine-formal" ], "ref_ids": [ 3981, 3889, 3907 ] } ], "ref_ids": [] }, { "id": 3909, "type": "theorem", "label": "formal-spaces-lemma-affine-representable-by-algebraic-spaces", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-affine-representable-by-algebraic-spaces", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "affine formal algebraic spaces which is representable by", "algebraic spaces. Then $f$ is representable (by schemes) and affine." ], "refs": [], "proofs": [ { "contents": [ "We will show that $f$ is affine; it will then follow that", "$f$ is representable and affine by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-affine-local}.", "Write $Y = \\colim Y_\\mu$ and $X = \\colim X_\\lambda$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}.", "Let $T \\to Y$ be a morphism where $T$ is a scheme", "over $S$. We have to show that $X \\times_Y T \\to T$ is affine, see", "Bootstrap, Definition", "\\ref{bootstrap-definition-property-transformation}.", "To do this we may assume that $T$ is affine and we have to prove", "that $X \\times_Y T$ is affine. In this case $T \\to Y$ factors", "through $Y_\\mu \\to Y$ for some $\\mu$, see", "Lemma \\ref{lemma-factor-through-thickening}.", "Since $f$ is quasi-compact we see that $X \\times_Y T$ is", "quasi-compact (Lemma \\ref{lemma-characterize-quasi-compact-morphism}).", "Hence $X \\times_Y T \\to X$ factors through $X_\\lambda$ for some", "$\\lambda$. Similarly $X_\\lambda \\to Y$ factors through $Y_\\mu$", "after increasing $\\mu$. Then", "$X \\times_Y T = X_\\lambda \\times_{Y_\\mu} T$.", "We conclude as fibre products of affine schemes are affine." ], "refs": [ "spaces-morphisms-lemma-affine-local", "formal-spaces-definition-affine-formal-algebraic-space", "bootstrap-definition-property-transformation", "formal-spaces-lemma-factor-through-thickening", "formal-spaces-lemma-characterize-quasi-compact-morphism" ], "ref_ids": [ 4798, 3977, 2638, 3868, 3899 ] } ], "ref_ids": [] }, { "id": 3910, "type": "theorem", "label": "formal-spaces-lemma-property-goes-up-affine-morphism", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-property-goes-up-affine-morphism", "contents": [ "Let $S$ be a scheme. Let $Y$ be an affine formal algebraic space.", "Let $f : X \\to Y$ be a map of sheaves on $(\\Sch/S)_{fppf}$ which", "is representable and affine. Then", "\\begin{enumerate}", "\\item $X$ is an affine formal algebraic space.", "\\item if $Y$ is countably indexed, then $X$ is countably indexed.", "\\item if $Y$ is adic*, then $X$ is adic*,", "\\item if $Y$ is Noetherian and $f$ is (locally) of finite type, then", "$X$ is Noetherian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Write $Y = \\colim_{\\lambda \\in \\Lambda} Y_\\lambda$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}.", "Since $f$ is representable and affine, the fibre products", "$X_\\lambda = Y_\\lambda \\times_Y X$ are affine. And", "$X = \\colim Y_\\lambda \\times_Y X$.", "Thus $X$ is an affine formal algebraic space.", "\\medskip\\noindent", "Proof of (2). If $Y$ is countably indexed, then in the argument above", "we may assume $\\Lambda$ is countable.", "Then we immediately see that $X$ is countably indexed too.", "\\medskip\\noindent", "Proof of (3). Assume $Y$ is adic*. Then $Y = \\text{Spf}(B)$ for some adic", "topological ring $B$ which has a finitely generated", "ideal $J$ such that $\\{J^n\\}$ is a fundamental system of open ideals.", "Of course, then $Y = \\colim \\Spec(B/J^n)$.", "The schemes $X \\times_Y \\Spec(B/J^n)$ are affine", "and we can write $X \\times_Y \\Spec(B/J^n) = \\Spec(A_n)$.", "Then $X = \\colim \\Spec(A_n)$. The $B$-algebra maps $A_{n + 1} \\to A_n$", "are surjective and induce isomorphisms $A_{n + 1}/J^nA_{n + 1} \\to A_n$.", "By Algebra, Lemma \\ref{algebra-lemma-limit-complete} the ring", "$A = \\lim A_n$ is $J$-adically complete and $A/J^n A = A_n$.", "Hence $X = \\text{Spf}(A^\\wedge)$ is adic*.", "\\medskip\\noindent", "Proof of (4). Combining (3) with Lemma \\ref{lemma-implications-between-types}", "we see that $X$ is adic*. Thus we can use the criterion of", "Lemma \\ref{lemma-characterize-noetherian-affine}.", "First, it tells us the affine schemes $Y_\\lambda$ are Noetherian.", "Then $X_\\lambda \\to Y_\\lambda$ is of finite type, hence $X_\\lambda$", "is Noetherian too (Morphisms, Lemma", "\\ref{morphisms-lemma-finite-type-noetherian}).", "Then the criterion tells us $X$ is Noetherian and the proof is", "complete." ], "refs": [ "formal-spaces-definition-affine-formal-algebraic-space", "algebra-lemma-limit-complete", "formal-spaces-lemma-implications-between-types", "formal-spaces-lemma-characterize-noetherian-affine", "morphisms-lemma-finite-type-noetherian" ], "ref_ids": [ 3977, 880, 3874, 3876, 5202 ] } ], "ref_ids": [] }, { "id": 3911, "type": "theorem", "label": "formal-spaces-lemma-property-goes-up-affine", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-property-goes-up-affine", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of affine formal", "algebraic spaces which is representable by algebraic spaces. Then", "\\begin{enumerate}", "\\item if $Y$ is countably indexed, then $X$ is countably indexed.", "\\item if $Y$ is adic*, then $X$ is adic*,", "\\item if $Y$ is Noetherian and $f$ is (locally) of finite type, then", "$X$ is Noetherian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-affine-representable-by-algebraic-spaces} and", "\\ref{lemma-property-goes-up-affine-morphism}." ], "refs": [ "formal-spaces-lemma-affine-representable-by-algebraic-spaces", "formal-spaces-lemma-property-goes-up-affine-morphism" ], "ref_ids": [ 3909, 3910 ] } ], "ref_ids": [] }, { "id": 3912, "type": "theorem", "label": "formal-spaces-lemma-representable-affine", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-representable-affine", "contents": [ "Let $S$ be a scheme. Let $\\varphi : A \\to B$ be a continuous map of", "weakly admissible topological rings over $S$. The following", "are equivalent", "\\begin{enumerate}", "\\item $\\text{Spf}(\\varphi) : \\text{Spf}(B) \\to \\text{Spf}(A)$", "is representable by algebraic spaces,", "\\item $\\text{Spf}(\\varphi) : \\text{Spf}(B) \\to \\text{Spf}(A)$", "is representable (by schemes),", "\\item $\\varphi$ is taut, see Definition \\ref{definition-taut}.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-taut" ], "proofs": [ { "contents": [ "Parts (1) and (2) are equivalent by", "Lemma \\ref{lemma-affine-representable-by-algebraic-spaces}.", "\\medskip\\noindent", "Assume the equivalent conditions (1) and (2) hold.", "If $I \\subset A$ is a weak ideal of definition, then", "$\\Spec(A/I) \\to \\text{Spf}(A)$ is representable and a thickening", "(this is clear from the construction of the formal spectrum", "but it also follows from", "Lemma \\ref{lemma-mcquillan-affine-formal-algebraic-space}).", "Then $\\Spec(A/I) \\times_{\\text{Spf}(A)} \\text{Spf}(B) \\to \\text{Spf}(B)$", "is representable and a thickening as a base change.", "Hence by", "Lemma \\ref{lemma-mcquillan-affine-formal-algebraic-space}", "there is a weak ideal of definition $J(I) \\subset B$ such that", "$\\Spec(A/I) \\times_{\\text{Spf}(A)} \\text{Spf}(B) = \\Spec(B/J(I))$", "as subfunctors of $\\text{Spf}(B)$. We obtain a cartesian diagram", "$$", "\\xymatrix{", "\\Spec(B/J(I)) \\ar[d] \\ar[r] & \\Spec(A/I) \\ar[d] \\\\", "\\text{Spf}(B) \\ar[r] & \\text{Spf}(A)", "}", "$$", "By Lemma \\ref{lemma-fibre-product-affines-over-separated}", "we see that $B/J(I) = B \\widehat{\\otimes}_A A/I$.", "It follows that $J(I)$ is the closure of the ideal $\\varphi(I)B$, see", "Lemma \\ref{lemma-closure-image-ideal}.", "Since $\\text{Spf}(A) = \\colim \\Spec(A/I)$ with $I$ as above,", "we find that $\\text{Spf}(B) = \\colim \\Spec(B/J(I))$.", "Thus the ideals $J(I)$ form a fundamental system of weak", "ideals of definition (see", "Lemma \\ref{lemma-mcquillan-affine-formal-algebraic-space}).", "Hence (3) holds.", "\\medskip\\noindent", "Assume (3) holds. We are essentially just going to reverse the", "arguments given in the previous paragraph.", "Let $I \\subset A$ be a weak ideal of definition.", "By Lemma \\ref{lemma-fibre-product-affines-over-separated}", "we get a cartesian diagram", "$$", "\\xymatrix{", "\\text{Spf}(B \\widehat{\\otimes}_A A/I) \\ar[d] \\ar[r] & \\Spec(A/I) \\ar[d] \\\\", "\\text{Spf}(B) \\ar[r] & \\text{Spf}(A)", "}", "$$", "If $J(I)$ is the closure of $IB$, then $J(I)$ is open in $B$", "by tautness of $\\varphi$. Hence if $J$ is open in $B$ and $J \\subset J(B)$,", "then $B/J \\otimes_A A/I = B/(IB + J) = B/J(I)$ because", "$J(I) = \\bigcap_{J \\subset B\\text{ open}} (IB + J)$ by Lemma \\ref{lemma-closed}.", "Hence the limit defining the completed tensor product collapses to give", "$B \\widehat{\\otimes}_A A/I = B/J(I)$.", "Thus $\\text{Spf}(B \\widehat{\\otimes}_A A/I) = \\Spec(B/J(I))$.", "This proves that $\\text{Spf}(B) \\times_{\\text{Spf}(A)} \\Spec(A/I)$", "is representable for every weak ideal of definition $I \\subset A$.", "Since every morphism $T \\to \\text{Spf}(A)$ with $T$ quasi-compact", "factors through $\\Spec(A/I)$ for some weak ideal of definition $I$", "(Lemma \\ref{lemma-factor-through-thickening})", "we conclude that $\\text{Spf}(\\varphi)$ is representable, i.e.,", "(2) holds. This finishes the proof." ], "refs": [ "formal-spaces-lemma-affine-representable-by-algebraic-spaces", "formal-spaces-lemma-mcquillan-affine-formal-algebraic-space", "formal-spaces-lemma-mcquillan-affine-formal-algebraic-space", "formal-spaces-lemma-fibre-product-affines-over-separated", "formal-spaces-lemma-closure-image-ideal", "formal-spaces-lemma-mcquillan-affine-formal-algebraic-space", "formal-spaces-lemma-fibre-product-affines-over-separated", "formal-spaces-lemma-closed", "formal-spaces-lemma-factor-through-thickening" ], "ref_ids": [ 3909, 3870, 3870, 3896, 3862, 3870, 3896, 3854, 3868 ] } ], "ref_ids": [ 3976 ] }, { "id": 3913, "type": "theorem", "label": "formal-spaces-lemma-etale", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-etale", "contents": [ "Let $S$ be a scheme. Let $Y$ be a McQuillan affine formal algebraic space", "over $S$, i.e., $Y = \\text{Spf}(B)$ for some weakly admissible topological", "$S$-algebra $B$. Then there is an equivalence of categories between", "\\begin{enumerate}", "\\item the category of morphisms $f : X \\to Y$", "of affine formal algebraic spaces which are representable", "by algebraic spaces and \\'etale, and", "\\item the category of topological $B$-algebras of the form", "$A^\\wedge$ where $A$ is an \\'etale $B$-algebra and", "$A^\\wedge = \\lim A/JA$ with $J \\subset B$ running over the", "weak ideals of definition of $B$.", "\\end{enumerate}", "The equivalence is given by sending $A^\\wedge$ to $X = \\text{Spf}(A^\\wedge)$.", "In particular, any $X$ as in (1) is McQuillan." ], "refs": [], "proofs": [ { "contents": [ "Let $A$ be an \\'etale $B$-algebra. Then $B/J \\to A/JA$ is", "\\'etale for every open ideal $J \\subset B$. Hence", "the morphism $\\text{Spf}(A^\\wedge) \\to Y$ is representable and \\'etale.", "The functor $\\text{Spf}$ is fully faithful by", "Lemma \\ref{lemma-morphism-between-formal-spectra}.", "To finish the proof we will show in the next paragraph", "that any $X \\to Y$ as in (1) is in the essential image.", "\\medskip\\noindent", "Choose a weak ideal of definition $J_0 \\subset B$. Set", "$Y_0 = \\Spec(B/J_0)$ and $X_0 = Y_0 \\times_Y X$. Then $X_0 \\to Y_0$", "is an \\'etale morphism of affine schemes (see", "Lemma \\ref{lemma-affine-representable-by-algebraic-spaces}).", "Say $X_0 = \\Spec(A_0)$. By Algebra, Lemma \\ref{algebra-lemma-lift-etale}", "we can find an \\'etale algebra map $B \\to A$ such that", "$A_0 \\cong A/J_0A$. Consider an ideal of definition $J \\subset J_0$.", "As above we may write $\\Spec(B/J) \\times_Y X = \\Spec(\\bar A)$", "for some \\'etale ring map $B/J \\to \\bar A$. Then both", "$B/J \\to \\bar A$ and $B/J \\to A/JA$ are \\'etale ring maps", "lifting the \\'etale ring map $B/J_0 \\to A_0$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-locally-nilpotent-henselian}", "there is a unique $B/J$-algebra isomorphism", "$\\varphi_J : A/JA \\to \\bar A$ lifting the identification modulo $J_0$.", "Since the maps $\\varphi_J$ are unique they are compatible for varying $J$.", "Thus", "$$", "X = \\colim \\Spec(B/J) \\times_Y X = \\colim \\Spec(A/JA) = \\text{Spf}(A)", "$$", "and we see that the lemma holds." ], "refs": [ "formal-spaces-lemma-morphism-between-formal-spectra", "formal-spaces-lemma-affine-representable-by-algebraic-spaces", "algebra-lemma-lift-etale", "more-algebra-lemma-locally-nilpotent-henselian" ], "ref_ids": [ 3871, 3909, 1238, 9857 ] } ], "ref_ids": [] }, { "id": 3914, "type": "theorem", "label": "formal-spaces-lemma-etale-surjective", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-etale-surjective", "contents": [ "With notation and assumptions as in Lemma \\ref{lemma-etale} let", "$f : X \\to Y$ correspond to $B \\to A^\\wedge$. The following are equivalent", "\\begin{enumerate}", "\\item $f : X \\to Y$ is surjective,", "\\item $B \\to A$ is faithfully flat,", "\\item for every weak ideal of definition $J \\subset B$", "the ring map $B/J \\to A/JA$ is faithfully flat, and", "\\item for some weak ideal of definition $J \\subset B$", "the ring map $B/J \\to A/JA$ is faithfully flat.", "\\end{enumerate}" ], "refs": [ "formal-spaces-lemma-etale" ], "proofs": [ { "contents": [ "Let $J \\subset B$ be a weak ideal of definition. As every element of $J$", "is topologically nilpotent, we see that every element of $1 + J$ is", "a unit. It follows that $J$ is contained in the Jacobson radical of $B$", "(Algebra, Lemma \\ref{algebra-lemma-contained-in-radical}).", "Hence a flat ring map $B \\to A$ is faithfully flat if and only if", "$B/J \\to A/JA$ is faithfully flat", "(Algebra, Lemma \\ref{algebra-lemma-ff-rings}).", "In this way we see that (2) -- (4) are equivalent.", "If (1) holds, then for every weak ideal of definition $J \\subset B$", "the morphism", "$\\Spec(A/JA) = \\Spec(B/J) \\times_Y X \\to \\Spec(B/J)$ is surjective", "which implies (3). Conversely, assume (3).", "A morphism $T \\to Y$ with $T$ quasi-compact", "factors through $\\Spec(B/J)$ for some ideal of definition $J$ of $B$", "(Lemma \\ref{lemma-factor-through-thickening}).", "Hence $X \\times_Y T = \\Spec(A/JA) \\times_{\\Spec(B/J)} T \\to T$", "is surjective as a base change of the surjective morphism", "$\\Spec(A/JA) \\to \\Spec(B/J)$. Thus (1) holds." ], "refs": [ "algebra-lemma-contained-in-radical", "algebra-lemma-ff-rings", "formal-spaces-lemma-factor-through-thickening" ], "ref_ids": [ 399, 536, 3868 ] } ], "ref_ids": [ 3913 ] }, { "id": 3915, "type": "theorem", "label": "formal-spaces-lemma-iff-countably-indexed", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-iff-countably-indexed", "contents": [ "Let $S$ be a scheme. Let $X \\to Y$ be a morphism of affine", "formal algebraic spaces which is representable by algebraic spaces,", "surjective, and flat. Then $X$ is countably indexed if and only", "if $Y$ is countably indexed." ], "refs": [], "proofs": [ { "contents": [ "Assume $X$ is countably indexed. We write $X = \\colim X_n$ as in", "Lemma \\ref{lemma-countable-affine-formal-algebraic-space}.", "Write $Y = \\colim Y_\\lambda$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}.", "For every $n$ we can pick a $\\lambda_n$ such that", "$X_n \\to Y$ factors through $Y_{\\lambda_n}$, see", "Lemma \\ref{lemma-factor-through-thickening}.", "On the other hand, for every $\\lambda$ the scheme", "$Y_\\lambda \\times_Y X$ is affine", "(Lemma \\ref{lemma-affine-representable-by-algebraic-spaces})", "and hence $Y_\\lambda \\times_Y X \\to X$ factors through", "$X_n$ for some $n$ (Lemma \\ref{lemma-factor-through-thickening}).", "Picture", "$$", "\\xymatrix{", "Y_\\lambda \\times_Y X \\ar[r] \\ar[d] & X_n \\ar[r] \\ar[d] & X \\ar[d] \\\\", "Y_\\lambda \\ar@{..>}[r] \\ar@/_1pc/[rr] & Y_{\\lambda_n} \\ar[r] & Y", "}", "$$", "If we can show the dotted arrow exists, then we conclude that", "$Y = \\colim Y_{\\lambda_n}$ and $Y$ is countably indexed. To do this we", "pick a $\\mu$ with $\\mu \\geq \\lambda$ and $\\mu \\geq \\lambda_n$.", "Thus both $Y_\\lambda \\to Y$ and $Y_{\\lambda_n} \\to Y$ factor", "through $Y_\\mu \\to Y$.", "Say $Y_\\mu = \\Spec(B_\\mu)$, the closed subscheme $Y_\\lambda$ corresponds to", "$J \\subset B_\\mu$, and the closed subscheme $Y_{\\lambda_n}$ corresponds to", "$J' \\subset B_\\mu$. We are trying to show that $J' \\subset J$.", "By the diagram above we know $J'A_\\mu \\subset JA_\\mu$", "where $Y_\\mu \\times_Y X = \\Spec(A_\\mu)$.", "Since $X \\to Y$ is surjective and flat the morphism", "$Y_\\lambda \\times_Y X \\to Y_\\lambda$ is a faithfully flat morphism", "of affine schemes, hence $B_\\mu \\to A_\\mu$ is", "faithfully flat. Thus $J' \\subset J$ as desired.", "\\medskip\\noindent", "Assume $Y$ is countably indexed. Then $X$ is countably indexed", "by Lemma \\ref{lemma-property-goes-up-affine}." ], "refs": [ "formal-spaces-lemma-countable-affine-formal-algebraic-space", "formal-spaces-definition-affine-formal-algebraic-space", "formal-spaces-lemma-factor-through-thickening", "formal-spaces-lemma-affine-representable-by-algebraic-spaces", "formal-spaces-lemma-factor-through-thickening", "formal-spaces-lemma-property-goes-up-affine" ], "ref_ids": [ 3873, 3977, 3868, 3909, 3868, 3911 ] } ], "ref_ids": [] }, { "id": 3916, "type": "theorem", "label": "formal-spaces-lemma-iff-adic-star", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-iff-adic-star", "contents": [ "Let $S$ be a scheme. Let $X \\to Y$ be a morphism of affine", "formal algebraic spaces which is representable by algebraic spaces,", "surjective, and flat. Then $X$ is adic* if and only if $Y$ is adic*." ], "refs": [], "proofs": [ { "contents": [ "Assume $Y$ is adic*. Then $X$ is adic* by", "Lemma \\ref{lemma-property-goes-up-affine}.", "\\medskip\\noindent", "Assume $X$ is adic*. Write $X = \\text{Spf}(A)$ for some adic ring $A$", "which has a finitely generated ideal $I$ such that $\\{I^n\\}$ is a", "fundamental system of open ideals. By Lemmas \\ref{lemma-iff-countably-indexed}", "we see that $Y$ is countably indexed. Thus, by", "Lemma \\ref{lemma-countably-indexed},", "we can write $Y = \\text{Spf}(B)$ where $B$ is a weakly admissible", "topological ring with a countable fundamental system $\\{J_m\\}$ of", "weak ideals of definition. By Lemma \\ref{lemma-morphism-between-formal-spectra}", "the morphism $X \\to Y$ corresponds to a continuous ring map", "$B \\to A$ which is taut by Lemma \\ref{lemma-representable-affine}.", "Our first goal is to reduce to the case where $J_m A$ is equal to $I^m$.", "\\medskip\\noindent", "Set $Y_m = \\Spec(B/J_m)$ so that", "$Y = \\colim Y_m$. The scheme $Y_m \\times_Y X$ is affine", "(Lemma \\ref{lemma-affine-representable-by-algebraic-spaces})", "and we have $X = \\colim Y_m \\times_Y X$. Say $Y_m \\times_Y X = \\Spec(A_m)$", "so that $B/J_m \\to A_m$ is a faithfully flat ring map.", "It follows from Lemma \\ref{lemma-fibre-product-affines-over-separated}", "that $\\Ker(A \\to A_m)$ is the closure of $J_mA$.", "\\medskip\\noindent", "Choose $n \\geq 1$. There exists an $m$ such that $\\Spec(A/I^n) \\to Y$", "factors through $Y_m$. In terms of ideals", "\\begin{equation}", "\\label{equation-first}", "\\forall n\\ \\exists m,\\ J_m A \\subset I^n.", "\\end{equation}", "Choose $m \\geq 1$. We can find an $n$ such that the morphism", "$\\Spec(A_m) \\to X$ factors through $\\Spec(A/I^n)$. In terms of ideals", "\\begin{equation}", "\\label{equation-second}", "\\forall m\\ \\exists n,\\ I^n \\subset \\Ker(A \\to A_m).", "\\end{equation}", "Given an $m$ we can pick an $n = n(m)$ such that", "$I^n \\subset \\Ker(A \\to A_m)$ by (\\ref{equation-second}).", "Choose generators $f_1, \\ldots, f_r$ of $I$.", "For any $E = (e_1, \\ldots, e_r)$ with $|E| = \\sum e_i = n$ write", "$$", "f_1^{e_1} \\ldots f_r^{e_r} =", "\\sum g_{E, j} a_{E, j} + \\delta_E", "$$", "with $g_{E, j} \\in J_m$, $a_{E, j} \\in A$, and $\\delta_E \\in I^{n + 1}$", "(possible by the above). Let $J = (g_{E, j}) \\subset J_m \\subset B$.", "Then we see that", "$$", "I^n \\subset J A + I^{n + 1}", "$$", "As $I$ is contained in the Jacobson radical of $A$ and", "$I^n$ is finitely generated", "we see that $I^n \\subset JA$ by Nakayama's lemma. More precisely", "apply part (2) of Algebra, Lemma \\ref{algebra-lemma-NAK}", "to see that $M = (I^n + JA)/JA$ is zero.", "\\medskip\\noindent", "We first apply what we just proved as follows: since for every $m$ there", "is an $n(m)$ with $I^{n(m)} \\subset J_mA$ we see that", "$J_mA$ is open in $A$, hence closed, hence $\\Ker(A \\to A_m) = J_mA$,", "in other words, $A_m = A/J_mA$. This holds for every $m$.", "\\medskip\\noindent", "Next, we pick $m$ with $J_mA \\subset I$ (\\ref{equation-first}).", "Then choose $n = n(m)$ and finitely generated ideal $J \\subset J_m$", "with $I^n \\subset JA \\subset I$ as above.", "For every $k \\geq 1$ we define $\\mathfrak b_k = \\Ker(B \\to A/J^kA)$.", "Observe that $\\mathfrak b_k \\supset \\mathfrak b_{k + 1}$.", "For every $k$ there exists", "an $m'$ with $J_{m'} \\subset \\mathfrak b_k$ as we have", "$I^{nk} \\subset J^kA$ and we can apply (\\ref{equation-first}).", "On the other hand, for every $m'$ there exists a $k$ such that", "$I^k \\subset J_{m'}A$ because $J_{m'}A$ is open. Then", "$\\mathfrak b_k$ maps to zero in $A/J_{m'}A$ which is faithfully", "flat over $B/J_{m'}$. Hence $\\mathfrak b_k \\subset J_{m'}$.", "In other words, we see that the topology on $B$ is defined by the", "sequence of ideals $\\mathfrak b_k$. Note that $J^k \\subset \\mathfrak b_k$", "which implies that $\\mathfrak b_k A = J^kA$.", "In other words, we have reduced the problem to the", "situation discussed in the following paragraph.", "\\medskip\\noindent", "We are given a ring map $B \\to A$ where", "\\begin{enumerate}", "\\item $B$ is a weakly admissible topological ring with a fundamental system", "$J_1 \\supset J_2 \\supset J_3 \\supset \\ldots$ of ideals of definition,", "\\item $A$ is a ring complete with respect to a finitely generated", "ideal $I$,", "\\item we have $J_k A = I^k$ for all $k$, and", "\\item $B/J_k \\to A/I^k$ is faithfully flat.", "\\end{enumerate}", "We want to deduce that $B$ is adic*.", "Pick $g_1, \\ldots, g_r \\in J_1$", "whose images in $A/I^2$ generate $I/I^2$; this is possible because", "$J_1A/J_2A = I/I^2$. Then for all $k \\geq 1$ we see that the elements", "$g^E = g_1^{e_1} \\ldots g_r^{e_r}$ with $|E| = k$ are in $J_k$", "because $B/J_k \\to A/I^k$ is faithfully flat and $J_1A = I$.", "Also we have $J_1 J_k \\subset J_{k + 1}$ by similar reasoning.", "The classes of $g^E$ with $|E| = k$ in $J_k/J_{k + 1}$ map to", "generators of $I^k/I^{k + 1}$ because the images of $g_1, \\ldots, g_r$", "generate $I/I^2$. Since $B/J_{k + 1} \\to A/I^{k + 1}$", "is flat we see that", "$$", "J_k/J_{k + 1} \\otimes_{B/J_1} A/I =", "J_k/J_{k + 1} \\otimes_{B/J_{k + 1}} A/I^{k + 1} \\to I^k/I^{k + 1}", "$$", "is an isomorphism (see More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-deform}). Since $B/J_1 \\to A/I$ is", "faithfully flat, we conclude that the classes of the elements", "$g^E$, $|E| = k$ generate $J_k/J_{k + 1}$. We claim that", "$J_k = (g^E, |E| = k)$. Namely, suppose that $x_k \\in J_k$.", "By the above we can write", "$$", "x_k = \\sum\\nolimits_{|E| = k} b_{E, 0} g^E + x_{k + 1}", "$$", "with $x_{k + 1} \\in J_{k + 1}$ and some $b_{E, 0} \\in B$. Now we can write", "$x_{k + 1}$ as follows", "$$", "x_{k + 1} =", "\\sum\\nolimits_{|E| = k}", "\\left(\\sum\\nolimits_{|E'| = 1} b_{E, E'}g^{E'}\\right) g^E + x_{k + 2}", "$$", "because every multi-index of degree $k + 1$ is a sum of a multi-index", "of degree $k$ and a multi-index of degree $1$. Continuing in this", "manner we can find $b_{E, E'} \\in B$ such that for every $l > 1$", "we have", "$$", "x_k = \\sum\\nolimits_{|E| = k}", "\\left(\\sum\\nolimits_{0 \\leq |E'| < l} b_{E, E'} g^{E'}\\right) g^E + x_{k + l}", "$$", "with some $x_{k + l} \\in J_{k + l}$. Then we can finally define", "$$", "b_E = \\sum\\nolimits_{E'} b_{E, E'} g^{E'}", "$$", "as an element in $B$ and we see that $x_k = \\sum b_E g^E$ as desired.", "This finishes the proof as now $J_1$ is finitely generated and $J_k = J_1^k$", "for all $k \\geq 1$." ], "refs": [ "formal-spaces-lemma-property-goes-up-affine", "formal-spaces-lemma-iff-countably-indexed", "formal-spaces-lemma-countably-indexed", "formal-spaces-lemma-morphism-between-formal-spectra", "formal-spaces-lemma-representable-affine", "formal-spaces-lemma-affine-representable-by-algebraic-spaces", "formal-spaces-lemma-fibre-product-affines-over-separated", "algebra-lemma-NAK", "more-morphisms-lemma-deform" ], "ref_ids": [ 3911, 3915, 3875, 3871, 3912, 3909, 3896, 401, 13723 ] } ], "ref_ids": [] }, { "id": 3917, "type": "theorem", "label": "formal-spaces-lemma-iff-noetherian", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-iff-noetherian", "contents": [ "Let $S$ be a scheme. Let $X \\to Y$ be a morphism of affine", "formal algebraic spaces which is representable by algebraic spaces,", "surjective, flat, and (locally) of finite type. Then $X$ is Noetherian", "if and only if $Y$ is Noetherian." ], "refs": [], "proofs": [ { "contents": [ "Observe that a Noetherian affine formal algebraic space is adic*, see", "Lemma \\ref{lemma-implications-between-types}. Thus by", "Lemma \\ref{lemma-iff-adic-star} we may assume that both $X$ and $Y$", "are adic*. We will use the criterion of", "Lemma \\ref{lemma-characterize-noetherian-affine}", "to see that the lemma holds. Namely, write $Y = \\colim Y_n$", "as in Lemma \\ref{lemma-countable-affine-formal-algebraic-space}.", "For each $n$ set $X_n = Y_n \\times_Y X$. Then $X_n$ is an", "affine scheme (Lemma \\ref{lemma-affine-representable-by-algebraic-spaces})", "and $X = \\colim X_n$. Each of the morphisms $X_n \\to Y_n$ is", "faithfully flat and of finite type. Thus the lemma follows from the", "fact that in this situation $X_n$ is Noetherian if and only if $Y_n$", "is Noetherian, see", "Algebra, Lemma \\ref{algebra-lemma-descent-Noetherian} (to go down)", "and", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-permanence} (to go up)." ], "refs": [ "formal-spaces-lemma-implications-between-types", "formal-spaces-lemma-iff-adic-star", "formal-spaces-lemma-characterize-noetherian-affine", "formal-spaces-lemma-countable-affine-formal-algebraic-space", "formal-spaces-lemma-affine-representable-by-algebraic-spaces", "algebra-lemma-descent-Noetherian", "algebra-lemma-Noetherian-permanence" ], "ref_ids": [ 3874, 3916, 3876, 3873, 3909, 1370, 448 ] } ], "ref_ids": [] }, { "id": 3918, "type": "theorem", "label": "formal-spaces-lemma-type-local", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-type-local", "contents": [ "Let $S$ be a scheme. Let $P \\in \\{countably\\ indexed, adic*, Noetherian\\}$.", "Let $X$ be a formal algebraic space over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item if $Y$ is an affine formal algebraic space and", "$f : Y \\to X$ is representable by algebraic spaces and \\'etale,", "then $Y$ has property $P$,", "\\item for some $\\{X_i \\to X\\}_{i \\in I}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "each $X_i$ has property $P$.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "It is clear that (1) implies (2). Assume (2) and let", "$Y \\to X$ be as in (1). Since the fibre products $X_i \\times_X Y$", "are formal algebraic spaces (Lemma \\ref{lemma-fibre-products-general})", "we can pick coverings $\\{X_{ij} \\to X_i \\times_X Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}.", "Since $Y$ is quasi-compact, there exist", "$(i_1, j_1), \\ldots, (i_n, j_n)$ such that", "$$", "X_{i_1 j_1} \\amalg \\ldots \\amalg X_{i_n j_n} \\longrightarrow Y", "$$", "is surjective and \\'etale. Then $X_{i_kj_k} \\to X_{i_k}$ is representable", "by algebraic spaces and \\'etale hence $X_{i_kj_k}$ has property $P$ by", "Lemma \\ref{lemma-property-goes-up-affine}.", "Then $X_{i_1 j_1} \\amalg \\ldots \\amalg X_{i_n j_n}$ is an", "affine formal algebraic space with property $P$ (small detail", "omitted on finite disjoint unions of affine formal algebraic spaces).", "Hence we conclude by applying one of", "Lemmas \\ref{lemma-iff-countably-indexed},", "\\ref{lemma-iff-adic-star}, and", "\\ref{lemma-iff-noetherian}." ], "refs": [ "formal-spaces-lemma-fibre-products-general", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-lemma-property-goes-up-affine", "formal-spaces-lemma-iff-countably-indexed", "formal-spaces-lemma-iff-adic-star", "formal-spaces-lemma-iff-noetherian" ], "ref_ids": [ 3890, 3981, 3911, 3915, 3916, 3917 ] } ], "ref_ids": [ 3981 ] }, { "id": 3919, "type": "theorem", "label": "formal-spaces-lemma-formal-completion-types", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-formal-completion-types", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $T \\subset |X|$ be a closed subset. Let $X_{/T}$ be the", "formal completion of $X$ along $T$.", "\\begin{enumerate}", "\\item If $X \\setminus T \\to X$ is quasi-compact,", "then $X_{/T}$ is locally adic*.", "\\item If $X$ is locally Noetherian, then $X_{/T}$ is locally", "Noetherian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a surjective \\'etale morphism $U \\to X$ with $U = \\coprod U_i$", "a disjoint union of affine schemes, see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-cover-by-union-affines}.", "Let $T_i \\subset U_i$ be the inverse image of $T$.", "We have $X_{/T} \\times_X U_i = (U_i)_{/T_i}$", "(Lemma \\ref{lemma-map-completions-representable}).", "Hence $\\{(U_i)_{/T_i} \\to X_{/T}\\}$ is a covering as in", "Definition \\ref{definition-formal-algebraic-space}.", "Moreover, if $X \\setminus T \\to X$ is quasi-compact, so is", "$U_i \\setminus T_i \\to U_i$ and if $X$ is locally Noetherian, so is", "$U_i$. Thus the lemma follows from the affine case which is", "Lemma \\ref{lemma-affine-formal-completion-types}." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "formal-spaces-lemma-map-completions-representable", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-lemma-affine-formal-completion-types" ], "ref_ids": [ 11830, 3885, 3981, 3887 ] } ], "ref_ids": [] }, { "id": 3920, "type": "theorem", "label": "formal-spaces-lemma-types-fibre-products", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-types-fibre-products", "contents": [ "Let $S$ be a scheme. Let $X \\to Y$ and $Z \\to Y$ be", "morphisms of formal algebraic space over $S$. Then", "\\begin{enumerate}", "\\item If $X$ and $Z$ are locally countably indexed, then $X \\times_Y Z$", "is locally countably indexed.", "\\item If $X$ and $Z$ are locally adic*, then $X \\times_Y Z$ is", "locally adic*.", "\\item If $X$ and $Z$ are locally Noetherian and $X_{red} \\to Y_{red}$", "is locally of finite type, then $X \\times_Y Z$ is locally Noetherian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}.", "For each $j$ choose a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$", "as in Definition \\ref{definition-formal-algebraic-space}.", "For each $j$ choose a covering $\\{Z_{jk} \\to Y_j \\times_Y Z\\}$", "as in Definition \\ref{definition-formal-algebraic-space}.", "Observe that $X_{ji} \\times_{Y_j} Z_{jk}$ is an", "affine formal algebraic space by", "Lemma \\ref{lemma-fibre-product-affines-over-separated}.", "Hence", "$$", "\\{X_{ji} \\times_{Y_j} Z_{jk} \\to X \\times_Y Z\\}", "$$", "is a covering as in Definition \\ref{definition-formal-algebraic-space}.", "Thus it suffices to prove (1) and (2) in case $X$, $Y$, and $Z$", "are affine formal algebraic spaces.", "\\medskip\\noindent", "Assume $X$, $Y$, and $Z$ are affine formal algebraic spaces and", "McQuillan. Then we can write", "$X = \\text{Spf}(A)$, $Y = \\text{Spf}(B)$, $Z = \\text{Spf}(C)$", "for some weakly admissible topological rings $A$, $B$, and $C$", "and the morphsms $X \\to Y$ and $Z \\to Y$ are given by", "continuous ring maps $B \\to A$ and $B \\to C$, see", "Definition \\ref{definition-types-affine-formal-algebraic-space}", "and Lemma \\ref{lemma-morphism-between-formal-spectra}", "By Lemma \\ref{lemma-fibre-product-affines-over-separated}", "we see that $X \\times_Y Z = \\text{Spf}(A \\widehat{\\otimes}_B C)$", "and that $A \\widehat{\\otimes}_B C$ is a weakly admissible topological ring.", "This reduces cases (1) and (2) of our lemma to parts (3) and (4) of", "Lemma \\ref{lemma-completed-tensor-product}.", "\\medskip\\noindent", "To deduce case (3) from Lemma \\ref{lemma-completed-tensor-product} part (5)", "we need to match the hypotheses. First, we observe that taking the", "reduction corresponds to dividing by the ideal of topologically", "nilpotent elements (Example \\ref{example-reduction-affine-formal-spectrum}).", "Second, if $X_{red} \\to Y_{red}$ is locally of finite type, then", "$(X_{ji})_{red} \\to (Y_j)_{red}$ is locally of finite type", "(and hence correspond to finite type homomorphisms of rings).", "This follows", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-type-local}", "and the fact that in the commutative diagram", "$$", "\\xymatrix{", "(X_{ji})_{red} \\ar[d] \\ar[r] & (Y_j)_{red} \\ar[d] \\\\", "X_{red} \\ar[r] & Y_{red}", "}", "$$", "the vertical morphisms are \\'etale. Namely, we have", "$(X_{ji})_{red} = X_{ij} \\times_X X_{red}$ and", "$(Y_j)_{red} = Y_j \\times_Y Y_{red}$", "by Lemma \\ref{lemma-reduction-smooth}." ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-lemma-fibre-product-affines-over-separated", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-types-affine-formal-algebraic-space", "formal-spaces-lemma-morphism-between-formal-spectra", "formal-spaces-lemma-fibre-product-affines-over-separated", "formal-spaces-lemma-completed-tensor-product", "formal-spaces-lemma-completed-tensor-product", "spaces-morphisms-lemma-finite-type-local", "formal-spaces-lemma-reduction-smooth" ], "ref_ids": [ 3981, 3981, 3981, 3896, 3981, 3978, 3871, 3896, 3865, 3865, 4816, 3880 ] } ], "ref_ids": [] }, { "id": 3921, "type": "theorem", "label": "formal-spaces-lemma-structure-locally-noetherian", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-structure-locally-noetherian", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian formal algebraic space", "over $S$. Then $X = \\colim X_n$ for a system $X_1 \\to X_2 \\to X_3 \\to \\ldots$", "of finite order thickenings of locally Noetherian algebraic spaces over $S$", "where $X_1 = X_{red}$ and $X_n$ is the $n$th infinitesimal neighbourhood of", "$X_1$ in $X_m$ for all $m \\geq n$." ], "refs": [], "proofs": [ { "contents": [ "We only sketch the proof and omit some of the details.", "Set $X_1 = X_{red}$. Define $X_n \\subset X$ as the subfunctor", "defined by the rule: a morphism $f : T \\to X$ where $T$ is a scheme factors", "through $X_n$ if and only if the $n$th power of the ideal sheaf", "of the closed immersion $X_1 \\times_X T \\to T$ is zero. Then $X_n \\subset X$", "is a subsheaf as vanishing of quasi-coherent modules can be checked", "fppf locally. We claim that $X_n \\to X$ is representable by schemes,", "a closed immersion, and that $X = \\colim X_n$ (as fppf sheaves).", "To check this we may work \\'etale locally on $X$. Hence we may assume", "$X = \\text{Spf}(A)$ is a Noetherian affine formal algebraic space.", "Then $X_1 = \\Spec(A/\\mathfrak a)$ where $\\mathfrak a \\subset A$", "is the ideal of topologically nilpotent elements of the Noetherian", "adic topological ring $A$. Then $X_n = \\Spec(A/\\mathfrak a^n)$", "and we obtain what we want." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3922, "type": "theorem", "label": "formal-spaces-lemma-completion-in-sub", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-completion-in-sub", "contents": [ "Let $A \\in \\Ob(\\textit{WAdm})$. Let $A \\to A'$ be a ring", "map (no topology). Let $(A')^\\wedge = \\lim_{I \\subset A\\text{ w.i.d}} A'/IA'$", "be the object of $\\textit{WAdm}$ constructed in", "Example \\ref{example-representable-morphism-from-completion}.", "\\begin{enumerate}", "\\item If $A$ is in $\\textit{WAdm}^{count}$, so is $(A')^\\wedge$.", "\\item If $A$ is in $\\textit{WAdm}^{adic*}$, so is $(A')^\\wedge$.", "\\item If $A$ is in $\\textit{WAdm}^{Noeth}$ and $A'$ is Noetherian, then", "$(A')^\\wedge$ is in $\\textit{WAdm}^{Noeth}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is clear from the construction.", "Assume $A$ has a finitely generated ideal of definition $I \\subset A$.", "Then $I^n(A')^\\wedge = \\Ker((A')^\\wedge \\to A'/I^nA')$ by Algebra, Lemma", "\\ref{algebra-lemma-hathat-finitely-generated}.", "Thus $I(A')^\\wedge$ is a finitely generated ideal of definition", "and we see that (2) holds.", "Finally, assume that $A$ is Noetherian and adic.", "By (2) we know that $(A')^\\wedge$ is adic.", "By Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian-Noetherian}", "we see that $(A')^\\wedge$ is Noetherian. Hence (3) holds." ], "refs": [ "algebra-lemma-hathat-finitely-generated", "algebra-lemma-completion-Noetherian-Noetherian" ], "ref_ids": [ 859, 874 ] } ], "ref_ids": [] }, { "id": 3923, "type": "theorem", "label": "formal-spaces-lemma-property-defines-property-morphisms", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-property-defines-property-morphisms", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "locally countably indexed formal algebraic spaces over $S$.", "Let $P$ be a local property of morphisms of $\\textit{WAdm}^{count}$.", "The following are equivalent", "\\begin{enumerate}", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$", "representable by algebraic spaces and \\'etale, the morphism $U \\to V$", "corresponds to a morphism of $\\textit{WAdm}^{count}$ with property $P$,", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space} and for each $j$", "a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "such that each $X_{ji} \\to Y_j$ corresponds", "to a morphism of $\\textit{WAdm}^{count}$ with property $P$, and", "\\item there exist a covering $\\{X_i \\to X\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$", "is an affine formal algebraic space, $Y_i \\to Y$ is representable", "by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds", "to a morphism of $\\textit{WAdm}^{count}$ with property $P$.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "It is clear that (1) implies (2) and that (2) implies (3).", "Assume $\\{X_i \\to X\\}$ and $X_i \\to Y_i \\to Y$ as in (3)", "and let a diagram as in (1) be given.", "Since $Y_i \\times_Y V$ is a formal algebraic space", "(Lemma \\ref{lemma-fibre-products-general}) we may pick", "coverings $\\{Y_{ij} \\to Y_i \\times_Y V\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}.", "For each $(i, j)$ we may similarly choose coverings", "$\\{X_{ijk} \\to Y_{ij} \\times_{Y_i} X_i \\times_X U\\}$", "as in Definition \\ref{definition-formal-algebraic-space}.", "Since $U$ is quasi-compact we can choose", "$(i_1, j_1, k_1), \\ldots, (i_n, j_n, k_n)$ such that", "$$", "X_{i_1 j_1 k_1} \\amalg \\ldots \\amalg X_{i_n j_n k_n} \\longrightarrow U", "$$", "is surjective. For $s = 1, \\ldots, n$ consider the commutative diagram", "$$", "\\xymatrix{", "& & & X_{i_s j_s k_s} \\ar[ld] \\ar[d] \\ar[rd] \\\\", "X \\ar[d] & X_{i_s} \\ar[l] \\ar[d] &", "X_{i_s} \\times_X U \\ar[l] \\ar[d] & Y_{i_s j_s} \\ar[ld] \\ar[rd] &", "X_{i_s} \\times_X U \\ar[d] \\ar[r] &", "U \\ar[d] \\ar[r] & X \\ar[d] \\\\", "Y & Y_{i_s} \\ar[l] &", "Y_{i_s} \\times_Y V \\ar[l] & &", "Y_{i_s} \\times_Y V \\ar[r] &", "V \\ar[r] & Y", "}", "$$", "Let us say that $P$ holds for a morphism of countably indexed", "affine formal algebraic spaces if it holds for the corresponding", "morphism of $\\textit{WAdm}^{count}$. Observe that the maps", "$X_{i_s j_s k_s} \\to X_{i_s}$, $Y_{i_s j_s} \\to Y_{i_s}$", "are given by completions of \\'etale ring maps, see Lemma \\ref{lemma-etale}.", "Hence we see that $P(X_{i_s} \\to Y_{i_s})$ implies", "$P(X_{i_s j_s k_s} \\to Y_{i_s j_s})$ by axiom (\\ref{item-axiom-1}).", "Observe that the maps $Y_{i_s j_s} \\to V$ are given by completions of", "\\'etale rings maps (same lemma as before).", "By axiom (\\ref{item-axiom-2}) applied to the diagram", "$$", "\\xymatrix{", "X_{i_s j_s k_s} \\ar@{=}[r] \\ar[d] & X_{i_s j_s k_s} \\ar[d] \\\\", "Y_{i_s j_s} \\ar[r] & V", "}", "$$", "(this is permissible as identities are faithfully flat ring maps)", "we conclude that $P(X_{i_s j_s k_s} \\to V)$ holds.", "By axiom (\\ref{item-axiom-3}) we find that", "$P(\\coprod_{s = 1, \\ldots, n} X_{i_s j_s k_s} \\to V)$ holds.", "Since the morphism $\\coprod X_{i_s j_s k_s} \\to U$ is surjective", "by construction, the corresponding morphism of $\\textit{WAdm}^{count}$", "is the completion of a faithfully flat \\'etale ring map, see", "Lemma \\ref{lemma-etale-surjective}.", "One more application of axiom (\\ref{item-axiom-2})", "(with $B' = B$) implies that $P(U \\to V)$ is true as desired." ], "refs": [ "formal-spaces-lemma-fibre-products-general", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-lemma-etale", "formal-spaces-lemma-etale-surjective" ], "ref_ids": [ 3890, 3981, 3981, 3913, 3914 ] } ], "ref_ids": [ 3981, 3981, 3981 ] }, { "id": 3924, "type": "theorem", "label": "formal-spaces-lemma-base-change-property-morphisms", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-base-change-property-morphisms", "contents": [ "Let $S$ be a scheme. Let $P$ be a local property of morphisms of", "$\\textit{WAdm}^{count}$ which is stable under base change.", "Let $f : X \\to Y$ and $g : Z \\to Y$ be morphisms of locally countably indexed", "formal algebraic spaces over $S$. If $f$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-property-defines-property-morphisms}", "then so does $\\text{pr}_2 : X \\times_Y Z \\to Z$." ], "refs": [ "formal-spaces-lemma-property-defines-property-morphisms" ], "proofs": [ { "contents": [ "Choose a covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}.", "For each $j$ choose a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$", "as in Definition \\ref{definition-formal-algebraic-space}.", "For each $j$ choose a covering $\\{Z_{jk} \\to Y_j \\times_Y Z\\}$", "as in Definition \\ref{definition-formal-algebraic-space}.", "Observe that $X_{ji} \\times_{Y_j} Z_{jk}$ is an", "affine formal algebraic space which is countably indexed, see", "Lemma \\ref{lemma-types-fibre-products}.", "Then we see that", "$$", "\\{X_{ji} \\times_{Y_j} Z_{jk} \\to X \\times_Y Z\\}", "$$", "is a covering as in Definition \\ref{definition-formal-algebraic-space}.", "Moreover, the morphisms $X_{ji} \\times_{Y_j} Z_{jk} \\to Z$", "factor through $Z_{jk}$. By assumption we know that", "$X_{ji} \\to Y_j$ corresponds to a morphism $B_j \\to A_{ji}$ of", "$\\text{WAdm}^{count}$ having property $P$.", "The morphisms $Z_{jk} \\to Y_j$ correspond to morphisms $B_j \\to C_{jk}$ in", "$\\text{WAdm}^{count}$. Since", "$X_{ji} \\times_{Y_j} Z_{jk} =", "\\text{Spf}(A_{ji} \\widehat{\\otimes}_{B_j} C_{jk})$", "by Lemma \\ref{lemma-fibre-product-affines-over-separated}", "we see that it suffices to show that", "$C_{jk} \\to A_{ji} \\widehat{\\otimes}_{B_j} C_{jk}$", "has property $P$ which is exactly what", "the condition that $P$ is stable under base change guarantees." ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-lemma-types-fibre-products", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-lemma-fibre-product-affines-over-separated" ], "ref_ids": [ 3981, 3981, 3981, 3920, 3981, 3896 ] } ], "ref_ids": [ 3923 ] }, { "id": 3925, "type": "theorem", "label": "formal-spaces-lemma-composition-property-morphisms", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-composition-property-morphisms", "contents": [ "Let $S$ be a scheme. Let $P$ be a local property of morphisms of", "$\\textit{WAdm}^{count}$ which is stable under composition.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of locally countably indexed", "formal algebraic spaces over $S$. If $f$ and $g$", "satisfies the equivalent conditions of", "Lemma \\ref{lemma-property-defines-property-morphisms}", "then so does $g \\circ f : X \\to Z$." ], "refs": [ "formal-spaces-lemma-property-defines-property-morphisms" ], "proofs": [ { "contents": [ "Choose a covering $\\{Z_k \\to Z\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}.", "For each $k$ choose a covering $\\{Y_{kj} \\to Z_k \\times_Z Y\\}$", "as in Definition \\ref{definition-formal-algebraic-space}.", "For each $k$ and $j$ choose a covering $\\{X_{kji} \\to Y_{kj} \\times_Y X\\}$", "as in Definition \\ref{definition-formal-algebraic-space}.", "If $f$ and $g$", "satisfies the equivalent conditions of", "Lemma \\ref{lemma-property-defines-property-morphisms}", "then $X_{kji} \\to Y_{jk}$ and $Y_{jk} \\to Z_k$", "correspond to arrows", "$B_{kj} \\to A_{kji}$ and $C_k \\to B_{kj}$ of", "$\\text{WAdm}^{count}$ having property $P$.", "Hence the compositions do too and we conclude." ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-lemma-property-defines-property-morphisms" ], "ref_ids": [ 3981, 3981, 3981, 3923 ] } ], "ref_ids": [ 3923 ] }, { "id": 3926, "type": "theorem", "label": "formal-spaces-lemma-permanence-property-morphisms", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-permanence-property-morphisms", "contents": [ "Let $S$ be a scheme. Let $P$ be a local property of morphisms of", "$\\textit{WAdm}^{count}$ which has the cancellation property.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of locally countably indexed", "formal algebraic spaces over $S$. If $g \\circ f$ and $g$", "satisfies the equivalent conditions of", "Lemma \\ref{lemma-property-defines-property-morphisms}", "then so does $f : X \\to Y$." ], "refs": [ "formal-spaces-lemma-property-defines-property-morphisms" ], "proofs": [ { "contents": [ "Choose a covering $\\{Z_k \\to Z\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}.", "For each $k$ choose a covering $\\{Y_{kj} \\to Z_k \\times_Z Y\\}$", "as in Definition \\ref{definition-formal-algebraic-space}.", "For each $k$ and $j$ choose a covering $\\{X_{kji} \\to Y_{kj} \\times_Y X\\}$", "as in Definition \\ref{definition-formal-algebraic-space}.", "Let $X_{kji} \\to Y_{jk}$ and $Y_{jk} \\to Z_k$", "correspond to arrows", "$B_{kj} \\to A_{kji}$ and $C_k \\to B_{kj}$ of", "$\\text{WAdm}^{count}$.", "If $g \\circ f$ and $g$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-property-defines-property-morphisms}", "then $C_k \\to B_{kj}$ and $C_k \\to A_{kji}$ satisfy $P$.", "Hence $B_{kj} \\to A_{kji}$ does too and we conclude." ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-lemma-property-defines-property-morphisms" ], "ref_ids": [ 3981, 3981, 3981, 3923 ] } ], "ref_ids": [ 3923 ] }, { "id": 3927, "type": "theorem", "label": "formal-spaces-lemma-representable-property-rings", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-representable-property-rings", "contents": [ "Let $B \\to A$ be an arrow of $\\textit{WAdm}^{count}$.", "The following are equivalent", "\\begin{enumerate}", "\\item[(a)] $B \\to A$ is taut (Definition \\ref{definition-taut}),", "\\item[(b)] for $B \\supset J_1 \\supset J_2 \\supset J_3 \\supset \\ldots$", "a fundamental system of weak ideals of definitions there exist", "a commutative diagram", "$$", "\\xymatrix{", "A \\ar[r] & \\ldots \\ar[r] & A_3 \\ar[r] & A_2 \\ar[r] & A_1 \\\\", "B \\ar[r] \\ar[u] & \\ldots \\ar[r] & B/J_3 \\ar[r] \\ar[u] &", "B/J_2 \\ar[r] \\ar[u] & B/J_1 \\ar[u]", "}", "$$", "such that $A_{n + 1}/J_nA_{n + 1} = A_n$ and $A = \\lim A_n$", "as topological ring.", "\\end{enumerate}", "Moreover, these equivalent conditions define a local property,", "i.e., they satisfy axioms (\\ref{item-axiom-1}), (\\ref{item-axiom-2}),", "(\\ref{item-axiom-3})." ], "refs": [ "formal-spaces-definition-taut" ], "proofs": [ { "contents": [ "The equivalence of (a) and (b) is immediate. Below we will give an", "algebraic proof of the axioms, but it turns out we've already proven", "them. Namely, using Lemma \\ref{lemma-representable-affine}", "the equivalent conditions (a) and (b) translate to saying the", "corresponding morphism of affine formal algebraic spaces is representable.", "Since this condition is ``\\'etale local on the source and target'' by", "Lemma \\ref{lemma-representable-by-algebraic-spaces-local}", "we immediately get axioms (\\ref{item-axiom-1}), (\\ref{item-axiom-2}), and", "(\\ref{item-axiom-3}).", "\\medskip\\noindent", "Direct algebraic proof of (\\ref{item-axiom-1}), (\\ref{item-axiom-2}),", "(\\ref{item-axiom-3}). Let a diagram (\\ref{equation-localize}) as in", "Situation \\ref{situation-local-property} be given.", "By Example \\ref{example-representable-morphism-from-completion}", "the maps $A \\to (A')^\\wedge$ and $B \\to (B')^\\wedge$", "satisfy (a) and (b).", "\\medskip\\noindent", "Assume (a) and (b) hold for $\\varphi$. Let $J \\subset B$ be a weak ideal", "of definition. Then the closure of $JA$, resp.\\ $J(B')^\\wedge$", "is a weak ideal of definition $I \\subset A$, resp.\\ $J' \\subset (B')^\\wedge$.", "Then the closure of $I(A')^\\wedge$ is a weak ideal of definition", "$I' \\subset (A')^\\wedge$. A topological argument shows that $I'$ is also", "the closure of $J(A')^\\wedge$ and of $J'(A')^\\wedge$.", "Finally, as $J$ runs over a fundamental system of weak ideals of definition", "of $B$ so do the ideals $I$ and $I'$ in $A$ and $(A')^\\wedge$.", "It follows that (a) holds for $\\varphi'$. This proves (\\ref{item-axiom-1}).", "\\medskip\\noindent", "Assume $A \\to A'$ is faithfully flat and that (a) and (b) hold for $\\varphi'$.", "Let $J \\subset B$ be a weak ideal of definition. Using (a) and (b)", "for the maps $B \\to (B')^\\wedge \\to (A')^\\wedge$ we find that the", "closure $I'$ of $J(A')^\\wedge$ is a weak ideal of definition.", "In particular, $I'$ is open and hence the inverse image of $I'$", "in $A$ is open. Now we have (explanation below)", "\\begin{align*}", "A \\cap I'", "& =", "A \\cap \\bigcap (J(A')^\\wedge + \\Ker((A')^\\wedge \\to A'/I_0A')) \\\\", "& =", "A \\cap \\bigcap \\Ker((A')^\\wedge \\to A'/JA' + I_0 A') \\\\", "& = \\bigcap (JA + I_0)", "\\end{align*}", "which is the closure of $JA$ by Lemma \\ref{lemma-closed}.", "The intersections are over weak ideals of definition $I_0 \\subset A$.", "The first equality because a fundamental system of neighbourhoods of", "$0$ in $(A')^\\wedge$ are the kernels of the maps $(A')^\\wedge \\to A'/I_0A'$.", "The second equality is trivial. The third equality because $A \\to A'$", "is faithfully flat, see", "Algebra, Lemma \\ref{algebra-lemma-faithfully-flat-universally-injective}.", "Thus the closure of $JA$ is open. By Lemma \\ref{lemma-topologically-nilpotent}", "the closure of $JA$", "is a weak ideal of definition of $A$. Finally, given a weak", "ideal of definition $I \\subset A$ we can find $J$ such that", "$J(A')^\\wedge$ is contained in the closure of $I(A')^\\wedge$", "by property (a) for $B \\to (B')^\\wedge$ and $\\varphi'$.", "Thus we see that (a) holds for $\\varphi$. This proves (\\ref{item-axiom-2}).", "\\medskip\\noindent", "We omit the proof of (\\ref{item-axiom-3})." ], "refs": [ "formal-spaces-lemma-representable-affine", "formal-spaces-lemma-representable-by-algebraic-spaces-local", "formal-spaces-lemma-closed", "algebra-lemma-faithfully-flat-universally-injective", "formal-spaces-lemma-topologically-nilpotent" ], "ref_ids": [ 3912, 3906, 3854, 814, 3860 ] } ], "ref_ids": [ 3976 ] }, { "id": 3928, "type": "theorem", "label": "formal-spaces-lemma-representable-local-property", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-representable-local-property", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "locally countably indexed formal algebraic spaces over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$", "representable by algebraic spaces and \\'etale, the morphism $U \\to V$", "corresponds to a taut map $B \\to A$ of $\\textit{WAdm}^{count}$,", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space} and for each $j$", "a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "such that each $X_{ji} \\to Y_j$ corresponds", "to a taut ring map in $\\textit{WAdm}^{count}$,", "\\item there exist a covering $\\{X_i \\to X\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$", "is an affine formal algebraic space, $Y_i \\to Y$ is representable", "by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds", "to a taut ring map in $\\textit{WAdm}^{count}$, and", "\\item $f$ is representable by algebraic spaces.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "The property of a map in $\\textit{WAdm}^{count}$ being", "``taut'' is a local property by", "Lemma \\ref{lemma-representable-property-rings}.", "Thus Lemma \\ref{lemma-property-defines-property-morphisms}", "exactly tells us that (1), (2), and (3) are equivalent.", "On the other hand, by Lemma \\ref{lemma-representable-affine}", "being ``taut'' on maps in $\\textit{WAdm}^{count}$ corresponds exactly to being", "``representable by algebraic spaces'' for the corresponding morphisms of", "countably indexed affine formal algebraic spaces.", "Thus the implication (1) $\\Rightarrow$ (2) of", "Lemma \\ref{lemma-representable-by-algebraic-spaces-local}", "shows that (4) implies (1) of the current lemma.", "Similarly, the implication (4) $\\Rightarrow$ (1) of", "Lemma \\ref{lemma-representable-by-algebraic-spaces-local}", "shows that (2) implies (4) of the current lemma." ], "refs": [ "formal-spaces-lemma-representable-property-rings", "formal-spaces-lemma-property-defines-property-morphisms", "formal-spaces-lemma-representable-affine", "formal-spaces-lemma-representable-by-algebraic-spaces-local", "formal-spaces-lemma-representable-by-algebraic-spaces-local" ], "ref_ids": [ 3927, 3923, 3912, 3906, 3906 ] } ], "ref_ids": [ 3981, 3981, 3981 ] }, { "id": 3929, "type": "theorem", "label": "formal-spaces-lemma-adic-homomorphism", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-adic-homomorphism", "contents": [ "Let $A$ and $B$ be pre-adic topological rings. Let", "$\\varphi : A \\to B$ be a continuous ring homomorphism.", "\\begin{enumerate}", "\\item If $\\varphi$ is adic, then $\\varphi$ is taut.", "\\item If $B$ is complete, $A$ has a finitely generated", "ideal of definition, and $\\varphi$ is taut, then $\\varphi$ is adic.", "\\end{enumerate}", "In particular the conditions ``$\\varphi$ is adic'' and ``$\\varphi$ is taut''", "are equivalent on the category $\\textit{WAdm}^{adic*}$." ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows immediately from the definitions, please see", "Definition \\ref{definition-taut} for the definition of a taut ring maps.", "Conversely, assume $B$ is complete, $I \\subset A$ is a finitely generated", "ideal of definition, and $\\varphi$ is taut. Then", "Lemma \\ref{lemma-taut-is-adic} tells us the topology on $B$", "is the $I$-adic topology as desired. This proves (2). The final", "statement is a trivial consequence of (1) and (2)." ], "refs": [ "formal-spaces-definition-taut", "formal-spaces-lemma-taut-is-adic" ], "ref_ids": [ 3976, 3864 ] } ], "ref_ids": [] }, { "id": 3930, "type": "theorem", "label": "formal-spaces-lemma-characterize-finite-type", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-characterize-finite-type", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces over $S$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is of finite type,", "\\item $f$ is representable by algebraic spaces and is of finite type in", "the sense of", "Bootstrap, Definition \\ref{bootstrap-definition-property-transformation}.", "\\end{enumerate}" ], "refs": [ "bootstrap-definition-property-transformation" ], "proofs": [ { "contents": [ "This follows from Bootstrap, Lemma", "\\ref{bootstrap-lemma-transformations-property-implication},", "the implication ``quasi-compact $+$ locally of finite type", "$\\Rightarrow$ finite type'' for morphisms of algebraic spaces, and", "Lemma \\ref{lemma-quasi-compact-representable}." ], "refs": [ "bootstrap-lemma-transformations-property-implication", "formal-spaces-lemma-quasi-compact-representable" ], "ref_ids": [ 2616, 3900 ] } ], "ref_ids": [ 2638 ] }, { "id": 3931, "type": "theorem", "label": "formal-spaces-lemma-composition-finite-type", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-composition-finite-type", "contents": [ "The composition of finite type morphisms is of finite type.", "The same holds for locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "See Bootstrap, Lemma \\ref{bootstrap-lemma-composition-transformation-property}", "and use Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-composition-finite-type}." ], "refs": [ "bootstrap-lemma-composition-transformation-property", "spaces-morphisms-lemma-composition-finite-type" ], "ref_ids": [ 2614, 4814 ] } ], "ref_ids": [] }, { "id": 3932, "type": "theorem", "label": "formal-spaces-lemma-base-change-finite-type", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-base-change-finite-type", "contents": [ "A base change of a finite type morphism is finite type.", "The same holds for locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "See Bootstrap, Lemma \\ref{bootstrap-lemma-base-change-transformation-property}", "and use Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-finite-type}." ], "refs": [ "bootstrap-lemma-base-change-transformation-property", "spaces-morphisms-lemma-base-change-finite-type" ], "ref_ids": [ 2613, 4815 ] } ], "ref_ids": [] }, { "id": 3933, "type": "theorem", "label": "formal-spaces-lemma-permanence-finite-type", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-permanence-finite-type", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of", "formal algebraic spaces over $S$. If $g \\circ f : X \\to Z$ is locally of", "finite type, then $f : X \\to Y$ is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-permanence-representable} we see that $f$ is", "representable by algebraic spaces. Let $T$ be a scheme and let", "$T \\to Z$ be a morphism. Then we can apply", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-permanence-finite-type}", "to the morphisms", "$T \\times_Z X \\to T \\times_Z Y \\to T$ of algebraic spaces to conclude." ], "refs": [ "formal-spaces-lemma-permanence-representable", "spaces-morphisms-lemma-permanence-finite-type" ], "ref_ids": [ 3905, 4818 ] } ], "ref_ids": [] }, { "id": 3934, "type": "theorem", "label": "formal-spaces-lemma-finite-type-local", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-finite-type-local", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces over $S$. The following are equivalent:", "\\begin{enumerate}", "\\item the morphism $f$ is locally of finite type,", "\\item there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are formal algebraic spaces, the vertical arrows are", "representable by algebraic spaces and \\'etale, $U \\to X$", "is surjective, and $U \\to V$ is locally of finite type,", "\\item for any commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are formal algebraic spaces and vertical arrows", "representable by algebraic spaces and \\'etale, the morphism", "$U \\to V$ is locally of finite type,", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "and for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in", "Definition \\ref{definition-formal-algebraic-space} such that", "$X_{ji} \\to Y_j$ is locally of finite type for each $j$ and $i$,", "\\item there exist a covering $\\{X_i \\to X\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$", "is an affine formal algebraic space, $Y_i \\to Y$ is representable", "by algebraic spaces and \\'etale, such that $X_i \\to Y_i$ is", "locally of finite type, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "In each of the 5 cases the morphism $f : X \\to Y$ is representable", "by algebraic spaces, see", "Lemma \\ref{lemma-representable-by-algebraic-spaces-local}.", "We will use this below without further mention.", "\\medskip\\noindent", "It is clear that (1) implies (2) because we can take $U = X$ and $V = Y$.", "Conversely, assume given a diagram as in (2). Let $T$ be a scheme and", "let $T \\to Y$ be a morphism. Then we can consider", "$$", "\\xymatrix{", "U \\times_Y T \\ar[d] \\ar[r] & V \\times_Y T \\ar[d] \\\\", "X \\times_Y T \\ar[r] & T", "}", "$$", "The vertical arrows are \\'etale and the top horizontal arrow", "is locally of finite type as base changes of such morphisms.", "Hence by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-type-local} we conclude that", "$X \\times_Y T \\to T$ is locally of finite type. In other words", "(1) holds.", "\\medskip\\noindent", "Assume (1) is true and consider a diagram as in (3).", "Then $U \\to Y$ is locally of finite type", "(as the composition $U \\to X \\to Y$, see", "Bootstrap, Lemma \\ref{bootstrap-lemma-composition-transformation-property}).", "Let $T$ be a scheme and let $T \\to V$ be a morphism.", "Then the projection $T \\times_V U \\to T$ factors as", "$$", "T \\times_V U = (T \\times_Y U) \\times_{(V \\times_Y V)} V", "\\to T \\times_Y U \\to T", "$$", "The second arrow is locally of finite type (as a base change", "of the composition $U \\to X \\to Y$) and", "the first is the base change of the diagonal $V \\to V \\times_Y V$", "which is locally of finite type by", "Lemma \\ref{lemma-diagonal-morphism-formal-algebraic-spaces}.", "\\medskip\\noindent", "It is clear that (3) implies (2). Thus now (1) -- (3) are equivalent.", "\\medskip\\noindent", "Observe that the condition in (4) makes sense as the fibre product", "$Y_j \\times_Y X$ is a formal algebraic space by", "Lemma \\ref{lemma-fibre-products}.", "It is clear that (4) implies (5).", "\\medskip\\noindent", "Assume $X_i \\to Y_i \\to Y$ as in (5). Then we set", "$V = \\coprod Y_i$ and $U = \\coprod X_i$ to see that", "(5) implies (2).", "\\medskip\\noindent", "Finally, assume (1) -- (3) are true.", "Thus we can choose any covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "and for each $j$ any covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}.", "Then $X_{ij} \\to Y_j$ is locally of finite type by (3)", "and we see that (4) is true. This concludes the proof." ], "refs": [ "formal-spaces-lemma-representable-by-algebraic-spaces-local", "spaces-morphisms-lemma-finite-type-local", "bootstrap-lemma-composition-transformation-property", "formal-spaces-lemma-diagonal-morphism-formal-algebraic-spaces", "formal-spaces-lemma-fibre-products", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "ref_ids": [ 3906, 4816, 2614, 3893, 3891, 3981, 3981 ] } ], "ref_ids": [ 3981, 3981, 3981 ] }, { "id": 3935, "type": "theorem", "label": "formal-spaces-lemma-locally-finite-type-locally-noetherian", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-locally-finite-type-locally-noetherian", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "formal algebraic spaces over $S$. If $Y$ is locally Noetherian and", "$f$ locally of finite type, then $X$ is locally Noetherian." ], "refs": [], "proofs": [ { "contents": [ "Pick $\\{Y_j \\to Y\\}$ and $\\{X_{ij} \\to Y_j \\times_Y X\\}$", "as in Lemma \\ref{lemma-finite-type-local}. Then it follows", "from Lemma \\ref{lemma-property-goes-up-affine-morphism}", "that each $X_{ij}$ is Noetherian. This proves the lemma." ], "refs": [ "formal-spaces-lemma-finite-type-local", "formal-spaces-lemma-property-goes-up-affine-morphism" ], "ref_ids": [ 3934, 3910 ] } ], "ref_ids": [] }, { "id": 3936, "type": "theorem", "label": "formal-spaces-lemma-fibre-product-Noetherian", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-fibre-product-Noetherian", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $Z \\to Y$ be morphisms", "of formal algebraic spaces over $S$. If $Z$ is locally", "Noetherian and $f$ locally of finite type, then", "$Z \\times_Y X$ is locally Noetherian." ], "refs": [], "proofs": [ { "contents": [ "The morphism $Z \\times_Y X \\to Z$ is locally of finite type by", "Lemma \\ref{lemma-base-change-finite-type}. Hence this follows", "from Lemma \\ref{lemma-locally-finite-type-locally-noetherian}." ], "refs": [ "formal-spaces-lemma-base-change-finite-type", "formal-spaces-lemma-locally-finite-type-locally-noetherian" ], "ref_ids": [ 3932, 3935 ] } ], "ref_ids": [] }, { "id": 3937, "type": "theorem", "label": "formal-spaces-lemma-composition-surjective", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-composition-surjective", "contents": [ "The composition of two surjective morphisms is a surjective morphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3938, "type": "theorem", "label": "formal-spaces-lemma-base-change-surjective", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-base-change-surjective", "contents": [ "A base change of a surjective morphism is a surjective morphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3939, "type": "theorem", "label": "formal-spaces-lemma-characterize-surjective", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-characterize-surjective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces over $S$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is surjective,", "\\item for every scheme $T$ and morphism $T \\to Y$", "the projection $X \\times_Y T \\to T$ is a surjective morphism", "of formal algebraic spaces,", "\\item for every affine scheme $T$ and morphism $T \\to Y$", "the projection $X \\times_Y T \\to T$ is a surjective morphism of formal", "algebraic spaces,", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "such that each $X \\times_Y Y_j \\to Y_j$ is a surjective morphism of", "formal algebraic spaces,", "\\item there exists a surjective morphism $Z \\to Y$", "of formal algebraic spaces such that $X \\times_Y Z \\to Z$ is surjective, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 3981 ] }, { "id": 3940, "type": "theorem", "label": "formal-spaces-lemma-composition-monomorphism", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-composition-monomorphism", "contents": [ "The composition of two monomorphisms is a monomorphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3941, "type": "theorem", "label": "formal-spaces-lemma-base-change-monomorphism", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-base-change-monomorphism", "contents": [ "A base change of a monomorphism is a monomorphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3942, "type": "theorem", "label": "formal-spaces-lemma-characterize-monomorphisms", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-characterize-monomorphisms", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces over $S$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is a monomorphism,", "\\item for every scheme $T$ and morphism $T \\to Y$", "the projection $X \\times_Y T \\to T$ is a monomorphism", "of formal algebraic spaces,", "\\item for every affine scheme $T$ and morphism $T \\to Y$", "the projection $X \\times_Y T \\to T$ is a monomorphism of formal", "algebraic spaces,", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "such that each $X \\times_Y Y_j \\to Y_j$ is a monomorphism of", "formal algebraic spaces, and", "\\item there exists a family of morphisms $\\{Y_j \\to Y\\}$ such", "that $\\coprod Y_j \\to Y$ is a surjection of sheaves on", "$(\\Sch/S)_{fppf}$ such that each $X \\times_Y Y_j \\to Y_j$ is a", "monomorphism for all $j$,", "\\item there exists a morphism $Z \\to Y$ of formal algebraic spaces", "which is representable by algebraic spaces, surjective, flat, and locally", "of finite presentation such that $X \\times_Y Z \\to X$ is a monomorphism, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 3981 ] }, { "id": 3943, "type": "theorem", "label": "formal-spaces-lemma-composition-closed-immersion", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-composition-closed-immersion", "contents": [ "The composition of two closed immersions is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3944, "type": "theorem", "label": "formal-spaces-lemma-base-change-closed-immersion", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-base-change-closed-immersion", "contents": [ "A base change of a closed immersion is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3945, "type": "theorem", "label": "formal-spaces-lemma-characterize-closed-immersion", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-characterize-closed-immersion", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces over $S$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is a closed immersion,", "\\item for every scheme $T$ and morphism $T \\to Y$ the projection", "$X \\times_Y T \\to T$ is a closed immersion,", "\\item for every affine scheme $T$ and morphism $T \\to Y$", "the projection $X \\times_Y T \\to T$ is a closed immersion,", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "such that each $X \\times_Y Y_j \\to Y_j$ is a closed immersion, and", "\\item there exists a morphism $Z \\to Y$ of formal algebraic spaces", "which is representable by algebraic spaces, surjective, flat, and locally", "of finite presentation such that $X \\times_Y Z \\to X$ is a", "closed immersion, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 3981 ] }, { "id": 3946, "type": "theorem", "label": "formal-spaces-lemma-closed-immersion-into-McQuillan", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-closed-immersion-into-McQuillan", "contents": [ "Let $S$ be a scheme. Let $X$ be a McQuillan affine formal algebraic space", "over $S$. Let $f : Y \\to X$ be a closed immersion of formal algebraic spaces", "over $S$. Then $Y$ is a McQuillan affine formal algebraic space and", "$f$ corresponds to a continuous homomorphism $A \\to B$ of weakly admissible", "topological $S$-algebras which is taut, has closed kernel, and has", "dense image." ], "refs": [], "proofs": [ { "contents": [ "Write $X = \\text{Spf}(A)$ where $A$ is a weakly admissible topological ring.", "Let $I_\\lambda$ be a fundamental system of weakly admissible ideals", "of definition in $A$. Then $Y \\times_X \\Spec(A/I_\\lambda)$ is", "a closed subscheme of $\\Spec(A/I_\\lambda)$", "and hence affine (Definition \\ref{definition-closed-immersion}).", "Say $Y \\times_X \\Spec(A/I_\\lambda) = \\Spec(B_\\lambda)$.", "The ring map $A/I_\\lambda \\to B_\\lambda$", "is surjective. Hence the projections", "$$", "B = \\lim B_\\lambda \\longrightarrow B_\\lambda", "$$", "are surjective as the compositions $A \\to B \\to B_\\lambda$ are surjective.", "It follows that $Y$ is McQuillan by", "Lemma \\ref{lemma-mcquillan-affine-formal-algebraic-space}.", "The ring map $A \\to B$ is taut by Lemma \\ref{lemma-representable-affine}.", "The kernel is closed because $B$ is complete and $A \\to B$ is", "continuous. Finally, as $A \\to B_\\lambda$ is surjective for all $\\lambda$", "we see that the image of $A$ in $B$ is dense." ], "refs": [ "formal-spaces-definition-closed-immersion", "formal-spaces-lemma-mcquillan-affine-formal-algebraic-space", "formal-spaces-lemma-representable-affine" ], "ref_ids": [ 3992, 3870, 3912 ] } ], "ref_ids": [] }, { "id": 3947, "type": "theorem", "label": "formal-spaces-lemma-monomorphism-iso-over-red", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-monomorphism-iso-over-red", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces. Assume", "\\begin{enumerate}", "\\item $f$ is representable by algebraic spaces,", "\\item $f$ is a monomorphism,", "\\item the inclusion $Y_{red} \\to Y$ factors through $f$, and", "\\item $f$ is locally of finite type or $Y$ is locally Noetherian.", "\\end{enumerate}", "Then $f$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Assumptions (2) and (3) imply that", "$X_{red} = X \\times_Y Y_{red} = Y_{red}$.", "We will use this without further mention.", "\\medskip\\noindent", "If $Y' \\to Y$ is an \\'etale morphism of formal algebraic spaces over $S$,", "then the base change $f' : X \\times_Y Y' \\to Y'$ satisfies conditions", "(1) -- (4). Hence by Lemma \\ref{lemma-characterize-closed-immersion}", "we may assume $Y$ is an affine formal algebraic space.", "\\medskip\\noindent", "Say $Y = \\colim_{\\lambda \\in \\Lambda} Y_\\lambda$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}.", "Then $X_\\lambda = X \\times_Y Y_\\lambda$ is an algebraic space", "endowed with a monomorphism $f_\\lambda : X_\\lambda \\to Y_\\lambda$", "which induces an isomorphism $X_{\\lambda, red} \\to Y_{\\lambda, red}$.", "Thus $X_\\lambda$ is an affine scheme by", "Limits of Spaces, Proposition \\ref{spaces-limits-proposition-affine}", "(as $X_{\\lambda, red} \\to X_\\lambda$ is surjective and integral).", "To finish the proof it suffices to show that", "$X_\\lambda \\to Y_\\lambda$ is a closed immersion", "which we will do in the next paragraph.", "\\medskip\\noindent", "Let $X \\to Y$ be a monomorphism of affine schemes such that", "$X_{red} = X \\times_Y Y_{red} = Y_{red}$. In general, this does", "not imply that $X \\to Y$ is a closed immersion, see", "Examples, Section \\ref{examples-section-epimorphism-not-surjective}.", "However, under our assumption (4) we know that in the previous parapgrah", "either $X_\\lambda \\to Y_\\lambda$ is of finite type or $Y_\\lambda$", "is Noetherian. This means that", "$X \\to Y$ corresponds to a ring map $R \\to A$ such that", "$R/I \\to A/IA$ is an isomorphism where $I \\subset R$ is the", "nil radical (ie., the maximal locally nilpotent ideal of $R$)", "and either $R \\to A$ is of finite type or $R$ is Noetherian.", "In the first case $R \\to A$ is surjective by", "Algebra, Lemma \\ref{algebra-lemma-surjective-mod-locally-nilpotent}", "and in the second case $I$ is finitely generated, hence", "nilpotent, hence $R \\to A$ is surjective by Nakayama's lemma, see", "Algebra, Lemma \\ref{algebra-lemma-NAK} part (11)." ], "refs": [ "formal-spaces-lemma-characterize-closed-immersion", "formal-spaces-definition-affine-formal-algebraic-space", "spaces-limits-proposition-affine", "algebra-lemma-surjective-mod-locally-nilpotent", "algebra-lemma-NAK" ], "ref_ids": [ 3945, 3977, 4658, 1087, 401 ] } ], "ref_ids": [] }, { "id": 3948, "type": "theorem", "label": "formal-spaces-lemma-topologically-finite-type-finite-type", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-topologically-finite-type-finite-type", "contents": [ "Let $S$ be a scheme. Let $\\varphi : A \\to B$ be a continuous map of", "weakly admissible topological rings over $S$. The following", "are equivalent", "\\begin{enumerate}", "\\item $\\text{Spf}(\\varphi) : Y = \\text{Spf}(B) \\to \\text{Spf}(A) = X$", "is of finite type,", "\\item $\\varphi$ is taut and $B$ is topologically of finite type over $A$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We can use Lemma \\ref{lemma-representable-affine} to relate tautness of", "$\\varphi$ to representability of $\\text{Spf}(\\varphi)$. We will use this", "without further mention below. It follows that $X = \\colim \\Spec(A/I)$", "and $Y = \\colim \\Spec(B/J(I))$ where $I \\subset A$ runs over the weak", "ideals of definition of $A$ and $J(I)$ is the closure of $IB$ in $B$.", "\\medskip\\noindent", "Assume (2).", "Choose a ring map $A[x_1, \\ldots, x_r] \\to B$ whose image is dense.", "Then $A[x_1, \\ldots, x_r] \\to B \\to B/J(I)$ has dense image too", "which means that it is surjective. Therefore $B/J(I)$ is of", "finite type over $A/I$. Let $T \\to X$ be a morphism with", "$T$ a quasi-compact scheme. Then $T \\to X$ factors through", "$\\Spec(A/I)$ for some $I$ (Lemma \\ref{lemma-factor-through-thickening}).", "Then $T \\times_X Y = T \\times_{\\Spec(A/I)} \\Spec(B/J(I))$, see proof of", "Lemma \\ref{lemma-representable-affine}.", "Hence $T \\times_Y X \\to T$ is of finite type as the base change of", "the morphism $\\Spec(B/J(I)) \\to \\Spec(A/I)$ which is of finite", "type. Thus (1) is true.", "\\medskip\\noindent", "Assume (1). Pick any $I \\subset A$ as above. Since", "$\\Spec(A/I) \\times_X Y = \\Spec(B/J(I))$ we see that $A/I \\to B/J(I)$", "is of finite type. Choose $b_1, \\ldots, b_r \\in B$", "mapping to generators of $B/J(I)$ over $A/I$. We claim that the image", "of the ring map $A[x_1, \\ldots, x_r] \\to B$ sending $x_i$ to $b_i$", "is dense. To prove this, let $I' \\subset I$ be a second weak ideal", "of definition. Then we have", "$$", "B/(J(I') + IB) = B/J(I)", "$$", "because $J(I)$ is the closure of $IB$ and because $J(I')$ is open.", "Hence we may apply Algebra, Lemma", "\\ref{algebra-lemma-surjective-mod-locally-nilpotent}", "to see that $(A/I')[x_1, \\ldots, x_r] \\to B/J(I')$ is surjective.", "Thus (2) is true, concluding the proof." ], "refs": [ "formal-spaces-lemma-representable-affine", "formal-spaces-lemma-factor-through-thickening", "formal-spaces-lemma-representable-affine", "algebra-lemma-surjective-mod-locally-nilpotent" ], "ref_ids": [ 3912, 3868, 3912, 1087 ] } ], "ref_ids": [] }, { "id": 3949, "type": "theorem", "label": "formal-spaces-lemma-category-affine-over", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-category-affine-over", "contents": [ "Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$.", "Assume $X$ is McQuillan and let $A$ be the weakly admissible topological", "ring associated to $X$. Then there is an anti-equivalence of categories", "between", "\\begin{enumerate}", "\\item the category $\\mathcal{C}$ introduced above, and", "\\item the category of maps $Y \\to X$ of finite type of", "affine formal algebraic spaces.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $(I_\\lambda)$ be a fundamental system of weakly admissible ideals", "of definition in $A$. Consider $Y$ as in (2). Then", "$Y \\times_X \\Spec(A/I_\\lambda)$ is affine", "(Definition \\ref{definition-finite-type}", "and Lemma \\ref{lemma-affine-representable-by-algebraic-spaces}).", "Say $Y \\times_X \\Spec(A/I_\\lambda) = \\Spec(B_\\lambda)$.", "The ring map $A/I_\\lambda \\to B_\\lambda$ is of finite type", "because $\\Spec(B_\\lambda) \\to \\Spec(A/I_\\lambda)$ is of", "finite type", "(by Definition \\ref{definition-finite-type}).", "Then $(B_\\lambda)$ is an object of $\\mathcal{C}$.", "\\medskip\\noindent", "Conversely, given an object $(B_\\lambda)$ of $\\mathcal{C}$ we can set", "$Y = \\colim \\Spec(B_\\lambda)$. This is an affine formal algebraic", "space. We claim that", "$$", "Y \\times_X \\Spec(A/I_\\lambda) =", "\\left(\\colim_\\mu \\Spec(B_\\mu)\\right) \\times_X \\Spec(A/I_\\lambda) =", "\\Spec(B_\\lambda)", "$$", "To show this it suffices we get the same values if we evaluate", "on a quasi-compact scheme $U$. A morphism", "$U \\to \\left(\\colim_\\mu \\Spec(B_\\mu)\\right) \\times_X \\Spec(A/I_\\lambda)$", "comes from a morphism", "$U \\to \\Spec(B_\\mu) \\times_{\\Spec(A/I_\\mu)} \\Spec(A/I_\\lambda)$", "for some $\\mu \\geq \\lambda$ (use", "Lemma \\ref{lemma-factor-through-thickening} two times). Since", "$\\Spec(B_\\mu) \\times_{\\Spec(A/I_\\mu)} \\Spec(A/I_\\lambda) = \\Spec(B_\\lambda)$", "by our second assumption on objects of $\\mathcal{C}$", "this proves what we want. Using this we can show the morphism", "$Y \\to X$ is of finite type. Namely, we note that", "for any morphism $U \\to X$ with $U$ a quasi-compact scheme, we get", "a factorization $U \\to \\Spec(A/I_\\lambda) \\to X$ for some $\\lambda$", "(see lemma cited above). Hence", "$$", "Y \\times_X U =", "Y \\times_X \\Spec(A/I_\\lambda)) \\times_{\\Spec(A/I_\\lambda)} U =", "\\Spec(B_\\lambda) \\times_{\\Spec(A/I_\\lambda)} U", "$$", "is a scheme of finite type over $U$ as desired. Thus the construction", "$(B_\\lambda) \\mapsto \\colim \\Spec(B_\\lambda)$ does give a functor", "from category (1) to category (2).", "\\medskip\\noindent", "To finish the proof we show that the above constructions", "define quasi-inverse functors between the categories (1) and (2).", "In one direction you have to show that", "$$", "\\left(\\colim_\\mu \\Spec(B_\\mu)\\right) \\times_X \\Spec(A/I_\\lambda) =", "\\Spec(B_\\lambda)", "$$", "for any object $(B_\\lambda)$ in the category $\\mathcal{C}$.", "This we proved above. For", "the other direction you have to show that", "$$", "Y = \\colim (Y \\times_X \\Spec(A/I_\\lambda))", "$$", "given $Y$ in the category (2). Again this is true by evaluating on", "quasi-compact test objects and because $X = \\colim \\Spec(A/I_\\lambda)$." ], "refs": [ "formal-spaces-definition-finite-type", "formal-spaces-lemma-affine-representable-by-algebraic-spaces", "formal-spaces-definition-finite-type", "formal-spaces-lemma-factor-through-thickening" ], "ref_ids": [ 3989, 3909, 3989, 3868 ] } ], "ref_ids": [] }, { "id": 3950, "type": "theorem", "label": "formal-spaces-lemma-closed-immersion-into-countably-indexed", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-closed-immersion-into-countably-indexed", "contents": [ "Let $S$ be a scheme. Let $X$ be a countably indexed affine formal algebraic", "space over $S$. Let $f : Y \\to X$ be a closed immersion of formal algebraic", "spaces over $S$. Then $Y$ is a countably indexed affine formal algebraic space", "and $f$ corresponds to $A \\to A/K$ where $A$ is an object of", "$\\textit{WAdm}^{count}$", "(Section \\ref{section-morphisms-rings})", "and $K \\subset A$ is a closed ideal." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-countably-indexed}", "we see that $X = \\text{Spf}(A)$ where $A$ is an object of", "$\\textit{WAdm}^{count}$. Since a closed immersion is representable", "and affine, we conclude by Lemma \\ref{lemma-property-goes-up-affine-morphism}", "that $Y$ is an affine formal algebraic space and countably index.", "Thus applying Lemma \\ref{lemma-countably-indexed}", "again we see that $Y = \\text{Spf}(B)$ with $B$ an object of", "$\\textit{WAdm}^{count}$. By Lemma \\ref{lemma-closed-immersion-into-McQuillan}", "we conclude that $f$ is given by a morphism $A \\to B$ of", "$\\textit{WAdm}^{count}$ which is taut and has dense image.", "To finish the proof we apply", "Lemma \\ref{lemma-dense-image-surjective}." ], "refs": [ "formal-spaces-lemma-countably-indexed", "formal-spaces-lemma-property-goes-up-affine-morphism", "formal-spaces-lemma-countably-indexed", "formal-spaces-lemma-closed-immersion-into-McQuillan", "formal-spaces-lemma-dense-image-surjective" ], "ref_ids": [ 3875, 3910, 3875, 3946, 3863 ] } ], "ref_ids": [] }, { "id": 3951, "type": "theorem", "label": "formal-spaces-lemma-quotient-restricted-power-series", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-quotient-restricted-power-series", "contents": [ "Let $B \\to A$ be an arrow of $\\textit{WAdm}^{count}$, see", "Section \\ref{section-morphisms-rings}.", "The following are equivalent", "\\begin{enumerate}", "\\item[(a)] $B \\to A$ is taut and $B/J \\to A/I$ is of finite type for", "every weak ideal of definition $J \\subset B$ where $I \\subset A$ is the", "closure of $JA$,", "\\item[(b)] $B \\to A$ is taut and $B/J_\\lambda \\to A/I_\\lambda$", "is of finite type for a cofinal system $(J_\\lambda)$", "of weak ideals of definition of $B$ where", "$I_\\lambda \\subset A$ is the closure of $J_\\lambda A$,", "\\item[(c)] $B \\to A$ is taut and $A$ is topologically of finite", "type over $B$,", "\\item[(d)] $A$ is isomorphic as a topological $B$-algebra to a quotient of", "$B\\{x_1, \\ldots, x_n\\}$ by a closed ideal.", "\\end{enumerate}", "Moreover, these equivalent conditions define a local property,", "i.e., they satisfy", "Axioms (\\ref{item-axiom-1}), (\\ref{item-axiom-2}), (\\ref{item-axiom-3})." ], "refs": [], "proofs": [ { "contents": [ "The implications (a) $\\Rightarrow$ (b), (c) $\\Rightarrow$ (a),", "(d) $\\Rightarrow$ (c) are straightforward from the definitions.", "Assume (b) holds and let $J \\subset B$ and $I \\subset A$ be as in (a).", "Choose a commutative diagram", "$$", "\\xymatrix{", "A \\ar[r] & \\ldots \\ar[r] & A_3 \\ar[r] & A_2 \\ar[r] & A_1 \\\\", "B \\ar[r] \\ar[u] & \\ldots \\ar[r] & B/J_3 \\ar[r] \\ar[u] &", "B/J_2 \\ar[r] \\ar[u] & B/J_1 \\ar[u]", "}", "$$", "such that $A_{n + 1}/J_nA_{n + 1} = A_n$ and such that $A = \\lim A_n$ as in", "Lemma \\ref{lemma-representable-property-rings}.", "For every $m$ there exists a $\\lambda$ such that $J_\\lambda \\subset J_m$.", "Since $B/J_\\lambda \\to A/I_\\lambda$ is of finite type, this implies", "that $B/J_m \\to A/I_m$ is of finite type.", "Let $\\alpha_1, \\ldots, \\alpha_n \\in A_1$ be generators of $A_1$ over", "$B/J_1$. Since $A$ is a countable limit of a system with surjective", "transition maps, we can find $a_1, \\ldots, a_n \\in A$ mapping to", "$\\alpha_1, \\ldots, \\alpha_n$ in $A_1$. By", "Remark \\ref{remark-universal-property} we find a continuous map", "$B\\{x_1, \\ldots, x_n\\} \\to A$ mapping $x_i$ to $a_i$. This map", "induces surjections $(B/J_m)[x_1, \\ldots, x_n] \\to A_m$ by", "Algebra, Lemma \\ref{algebra-lemma-surjective-mod-locally-nilpotent}.", "For $m \\geq 1$ we obtain a short exact sequence", "$$", "0 \\to K_m \\to (B/J_m)[x_1, \\ldots, x_n] \\to A_m \\to 0", "$$", "The induced transition maps $K_{m + 1} \\to K_m$ are surjective because", "$A_{m + 1}/J_mA_{m + 1} = A_m$. Hence the inverse limit of these", "short exact sequences is exact, see", "Algebra, Lemma \\ref{algebra-lemma-ML-exact-sequence}.", "Since $B\\{x_1, \\ldots, x_n\\} = \\lim (B/J_m)[x_1, \\ldots, x_n]$", "and $A = \\lim A_m$", "we conclude that $B\\{x_1, \\ldots, x_n\\} \\to A$ is surjective and open.", "As $A$ is complete the kernel is a closed ideal. In this way we see that", "(a), (b), (c), and (d) are equivalent.", "\\medskip\\noindent", "Let a diagram (\\ref{equation-localize}) as in", "Situation \\ref{situation-local-property}", "be given. By Example \\ref{example-finite-type-from-finite-type-ring-map}", "the maps $A \\to (A')^\\wedge$ and $B \\to (B')^\\wedge$", "satisfy (a), (b), (c), and (d). Moreover, by", "Lemma \\ref{lemma-representable-property-rings}", "in order to prove Axioms (\\ref{item-axiom-1}) and (\\ref{item-axiom-2})", "we may assume both $B \\to A$ and $(B')^\\wedge \\to (A')^\\wedge$", "are taut. Now pick a weak ideal of definition $J \\subset B$. Let", "$J' \\subset (B')^\\wedge$, $I \\subset A$, $I' \\subset (A')^\\wedge$", "be the closure of $J(B')^\\wedge$, $JA$, $J(A')^\\wedge$.", "By what was said above, it suffices to consider the commutative", "diagram", "$$", "\\xymatrix{", "A/I \\ar[r] & (A')^\\wedge/I' \\\\", "B/J \\ar[r] \\ar[u]^{\\overline{\\varphi}} &", "(B')^\\wedge/J' \\ar[u]_{\\overline{\\varphi}'}", "}", "$$", "and to show (1) $\\overline{\\varphi}$ finite type", "$\\Rightarrow \\overline{\\varphi}'$", "finite type, and (2) if $A \\to A'$ is faithfully flat, then", "$\\overline{\\varphi}'$ finite type $\\Rightarrow \\overline{\\varphi}$", "finite type. Note that $(B')^\\wedge/J' = B'/JB'$ and", "$(A')^\\wedge/I' = A'/IA'$ by the construction of the topologies on", "$(B')^\\wedge$ and $(A')^\\wedge$. In particular the horizontal", "maps in the diagram are \\'etale. Part (1) now follows from", "Algebra, Lemma \\ref{algebra-lemma-compose-finite-type}", "and part (2) from", "Descent, Lemma \\ref{descent-lemma-finite-type-local-source-fppf-algebra}", "as the ring map $A/I \\to (A')^\\wedge/I' = A'/IA'$ is faithfully flat", "and \\'etale.", "\\medskip\\noindent", "We omit the proof of Axiom (\\ref{item-axiom-3})." ], "refs": [ "formal-spaces-lemma-representable-property-rings", "formal-spaces-remark-universal-property", "algebra-lemma-surjective-mod-locally-nilpotent", "algebra-lemma-ML-exact-sequence", "formal-spaces-lemma-representable-property-rings", "algebra-lemma-compose-finite-type", "descent-lemma-finite-type-local-source-fppf-algebra" ], "ref_ids": [ 3927, 4016, 1087, 826, 3927, 333, 14641 ] } ], "ref_ids": [] }, { "id": 3952, "type": "theorem", "label": "formal-spaces-lemma-quotient-restricted-power-series-admissible", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-quotient-restricted-power-series-admissible", "contents": [ "In Lemma \\ref{lemma-quotient-restricted-power-series}", "if $B$ is admissible (for example adic), then the equivalent conditions", "(a) -- (d) are also equivalent to", "\\begin{enumerate}", "\\item[(e)] $B \\to A$ is taut and $B/J \\to A/I$ is of finite type for", "some ideal of definition $J \\subset B$ where $I \\subset A$ is", "the closure of $JA$.", "\\end{enumerate}" ], "refs": [ "formal-spaces-lemma-quotient-restricted-power-series" ], "proofs": [ { "contents": [ "It is enough to show that (e) implies (a). Let $J' \\subset B$ be a weak ideal", "of definition and let $I' \\subset A$ be the closure of $J'A$. We have", "to show that $B/J' \\to A/I'$ is of finite type. If the corresponding statement", "holds for the smaller weak ideal of definition $J'' = J' \\cap J$, then it", "holds for $J'$. Thus we may assume $J' \\subset J$. As $J$ is an ideal", "of definition (and not just a weak ideal of definition), we get", "$J^n \\subset J'$ for some $n \\geq 1$. Thus we can consider the", "diagram", "$$", "\\xymatrix{", "0 \\ar[r] & I/I' \\ar[r] & A/I' \\ar[r] & A/I \\ar[r] & 0 \\\\", "0 \\ar[r] & J/J' \\ar[r] \\ar[u] & B/J' \\ar[r] \\ar[u] & B/J \\ar[r] \\ar[u] & 0", "}", "$$", "with exact rows. Since $I' \\subset A$ is open and since", "$I$ is the closure of $J A$ we see that $I/I' = (J/J') \\cdot A/I'$.", "Because $J/J'$ is a nilpotent ideal and as $B/J \\to A/I$ is of finite type,", "we conclude from Algebra, Lemma \\ref{algebra-lemma-finite-type-mod-nilpotent}", "that $A/I'$ is of finite type over $B/J'$ as desired." ], "refs": [ "algebra-lemma-finite-type-mod-nilpotent" ], "ref_ids": [ 1086 ] } ], "ref_ids": [ 3951 ] }, { "id": 3953, "type": "theorem", "label": "formal-spaces-lemma-representable-affine-finite-type", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-representable-affine-finite-type", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "affine formal algebraic spaces. Assume $Y$ countably indexed.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is locally of finite type,", "\\item $f$ is of finite type,", "\\item $f$ corresponds to a morphism $B \\to A$ of $\\textit{WAdm}^{count}$", "(Section \\ref{section-morphisms-rings})", "satisfying the equivalent conditions of", "Lemma \\ref{lemma-quotient-restricted-power-series}.", "\\end{enumerate}" ], "refs": [ "formal-spaces-lemma-quotient-restricted-power-series" ], "proofs": [ { "contents": [ "Since $X$ and $Y$ are affine it is clear that conditions (1)", "and (2) are equivalent. In cases (1) and (2) the morphism $f$", "is representable by algebraic spaces by definition, hence", "affine by Lemma \\ref{lemma-affine-representable-by-algebraic-spaces}.", "Thus if (1) or (2) holds we see that", "$X$ is countably indexed by", "Lemma \\ref{lemma-property-goes-up-affine-morphism}.", "Write $X = \\text{Spf}(A)$ and $Y = \\text{Spf}(B)$", "for topological $S$-algebras $A$ and $B$ in $\\textit{WAdm}^{count}$, see", "Lemma \\ref{lemma-countably-indexed}. By", "Lemma \\ref{lemma-morphism-between-formal-spectra}", "we see that $f$ corresponds to a continuous map $B \\to A$.", "Hence now the result follows from", "Lemma \\ref{lemma-topologically-finite-type-finite-type}." ], "refs": [ "formal-spaces-lemma-affine-representable-by-algebraic-spaces", "formal-spaces-lemma-property-goes-up-affine-morphism", "formal-spaces-lemma-countably-indexed", "formal-spaces-lemma-morphism-between-formal-spectra", "formal-spaces-lemma-topologically-finite-type-finite-type" ], "ref_ids": [ 3909, 3910, 3875, 3871, 3948 ] } ], "ref_ids": [ 3951 ] }, { "id": 3954, "type": "theorem", "label": "formal-spaces-lemma-finite-type-local-property", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-finite-type-local-property", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "locally countably indexed formal algebraic spaces over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$", "representable by algebraic spaces and \\'etale, the morphism $U \\to V$", "corresponds to a morphism of $\\textit{WAdm}^{count}$ which is", "taut and topologically of finite type,", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space} and for each $j$", "a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "such that each $X_{ji} \\to Y_j$ corresponds", "to a morphism of $\\textit{WAdm}^{count}$ which is", "taut and topologically of finite type,", "\\item there exist a covering $\\{X_i \\to X\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$", "is an affine formal algebraic space, $Y_i \\to Y$ is representable", "by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds", "to a morphism of $\\textit{WAdm}^{count}$ which is,", "taut and topologically of finite type, and", "\\item $f$ is locally of finite type.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-quotient-restricted-power-series}", "the property", "$P(\\varphi)=$``$\\varphi$ is taut and topologically of finite type''", "is local on $\\text{WAdm}^{count}$. Hence by", "Lemma \\ref{lemma-property-defines-property-morphisms}", "we see that conditions (1), (2), and (3) are equivalent.", "On the other hand, by Lemma \\ref{lemma-representable-affine-finite-type}", "the condition $P$ on morphisms of $\\textit{WAdm}^{count}$", "corresponds exactly to morphisms of countably indexed, affine", "formal algebraic spaces being locally of finite type.", "Thus the implication (1) $\\Rightarrow$ (3) of", "Lemma \\ref{lemma-finite-type-local}", "shows that (4) implies (1) of the current lemma.", "Similarly, the implication (4) $\\Rightarrow$ (1) of", "Lemma \\ref{lemma-finite-type-local}", "shows that (2) implies (4) of the current lemma." ], "refs": [ "formal-spaces-lemma-quotient-restricted-power-series", "formal-spaces-lemma-property-defines-property-morphisms", "formal-spaces-lemma-representable-affine-finite-type", "formal-spaces-lemma-finite-type-local", "formal-spaces-lemma-finite-type-local" ], "ref_ids": [ 3951, 3923, 3953, 3934, 3934 ] } ], "ref_ids": [ 3981, 3981, 3981 ] }, { "id": 3955, "type": "theorem", "label": "formal-spaces-lemma-base-change-separated", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-base-change-separated", "contents": [ "All of the separation axioms listed in", "Definition \\ref{definition-separated-morphism}", "are stable under base change." ], "refs": [ "formal-spaces-definition-separated-morphism" ], "proofs": [ { "contents": [ "Let $f : X \\to Y$ and $Y' \\to Y$ be morphisms of formal algebraic spaces.", "Let $f' : X' \\to Y'$ be the base change of $f$ by $Y' \\to Y$. Then", "$\\Delta_{X'/Y'}$ is the base change of $\\Delta_{X/Y}$ by", "the morphism $X' \\times_{Y'} X' \\to X \\times_Y X$. Each of the properties", "of the diagonal used in Definition \\ref{definition-separated-morphism}", "is stable under base change. Hence the lemma is true." ], "refs": [ "formal-spaces-definition-separated-morphism" ], "ref_ids": [ 3994 ] } ], "ref_ids": [ 3994 ] }, { "id": 3956, "type": "theorem", "label": "formal-spaces-lemma-fibre-product-after-map", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-fibre-product-after-map", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Z$, $g : Y \\to Z$ and $Z \\to T$", "be morphisms of formal algebraic spaces over $S$. Consider the induced", "morphism $i : X \\times_Z Y \\to X \\times_T Y$. Then", "\\begin{enumerate}", "\\item $i$ is representable (by schemes), locally of finite type,", "locally quasi-finite, separated, and a monomorphism,", "\\item if $Z \\to T$ is separated, then $i$ is a closed immersion, and", "\\item if $Z \\to T$ is quasi-separated, then $i$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By general category theory the following diagram", "$$", "\\xymatrix{", "X \\times_Z Y \\ar[r]_i \\ar[d] & X \\times_T Y \\ar[d] \\\\", "Z \\ar[r]^-{\\Delta_{Z/T}} \\ar[r] & Z \\times_T Z", "}", "$$", "is a fibre product diagram. Hence $i$ is the base change of the", "diagonal morphism $\\Delta_{Z/T}$. Thus the lemma follows", "from Lemma \\ref{lemma-diagonal-morphism-formal-algebraic-spaces}." ], "refs": [ "formal-spaces-lemma-diagonal-morphism-formal-algebraic-spaces" ], "ref_ids": [ 3893 ] } ], "ref_ids": [] }, { "id": 3957, "type": "theorem", "label": "formal-spaces-lemma-composition-separated", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-composition-separated", "contents": [ "All of the separation axioms listed in", "Definition \\ref{definition-separated-morphism}", "are stable under composition of morphisms." ], "refs": [ "formal-spaces-definition-separated-morphism" ], "proofs": [ { "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of formal algebraic spaces", "to which the axiom in question applies.", "The diagonal $\\Delta_{X/Z}$ is the composition", "$$", "X \\longrightarrow X \\times_Y X \\longrightarrow X \\times_Z X.", "$$", "Our separation axiom is defined by requiring the diagonal", "to have some property $\\mathcal{P}$. By", "Lemma \\ref{lemma-fibre-product-after-map} above we see that", "the second arrow also has this property. Hence the lemma follows", "since the composition of (representable) morphisms with property", "$\\mathcal{P}$ also is a morphism with property $\\mathcal{P}$." ], "refs": [ "formal-spaces-lemma-fibre-product-after-map" ], "ref_ids": [ 3956 ] } ], "ref_ids": [ 3994 ] }, { "id": 3958, "type": "theorem", "label": "formal-spaces-lemma-separated-local", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-separated-local", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces", "over $S$. Let $\\mathcal{P}$ be any of the separation axioms of", "Definition \\ref{definition-separated-morphism}.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is $\\mathcal{P}$,", "\\item for every scheme $Z$ and morphism $Z \\to Y$ the", "base change $Z \\times_Y X \\to Z$ of $f$ is $\\mathcal{P}$,", "\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the", "base change $Z \\times_Y X \\to Z$ of $f$ is $\\mathcal{P}$,", "\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the", "formal algebraic space $Z \\times_Y X$ is $\\mathcal{P}$ (see", "Definition \\ref{definition-separated}),", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "such that the base change $Y_j \\times_Y X \\to Y_j$ has", "$\\mathcal{P}$ for all $j$.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-separated-morphism", "formal-spaces-definition-separated", "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "We will repeatedly use", "Lemma \\ref{lemma-base-change-separated}", "without further mention. In particular, it is clear that", "(1) implies (2) and (2) implies (3).", "\\medskip\\noindent", "Assume (3) and let $Z \\to Y$ be a morphism where $Z$ is an affine scheme.", "Let $U$, $V$ be affine schemes and let $a : U \\to Z \\times_Y X$", "and $b : V \\to Z \\times_Y X$ be morphisms. Then", "$$", "U \\times_{Z \\times_Y X} V =", "(Z \\times_Y X) \\times_{\\Delta, (Z \\times_Y X) \\times_Z (Z \\times_Y X)}", "(U \\times_Z V)", "$$", "and we see that this is quasi-compact if $\\mathcal{P} =$``quasi-separated''", "or an affine scheme equipped with a closed immersion into", "$U \\times_Z V$ if $\\mathcal{P} =$``separated''. Thus (4) holds.", "\\medskip\\noindent", "Assume (4) and let $Z \\to Y$ be a morphism where $Z$ is an affine scheme.", "Let $U$, $V$ be affine schemes and let $a : U \\to Z \\times_Y X$", "and $b : V \\to Z \\times_Y X$ be morphisms. Reading the argument above", "backwards, we see that $U \\times_{Z \\times_Y X} V \\to U \\times_Z V$", "is quasi-compact if $\\mathcal{P} =$``quasi-separated'' or a closed", "immersion if $\\mathcal{P} =$``separated''. Since we can choose $U$ and", "$V$ as above such that $U$ varies through an", "\\'etale covering of $Z \\times_Y X$, we find", "that the corresponding morphisms", "$$", "U \\times_Z V \\to (Z \\times_Y X) \\times_Z (Z \\times_Y X)", "$$", "form an \\'etale covering by affines. Hence we conclude that", "$\\Delta : (Z \\times_Y X) \\to (Z \\times_Y X) \\times_Z (Z \\times_Y X)$", "is quasi-compact, resp.\\ a closed immersion. Thus (3) holds.", "\\medskip\\noindent", "Let us prove that (3) implies (5). Assume (3) and let", "$\\{Y_j \\to Y\\}$ be as in", "Definition \\ref{definition-formal-algebraic-space}.", "We have to show that the morphisms", "$$", "\\Delta_j :", "Y_j \\times_Y X", "\\longrightarrow", "(Y_j \\times_Y X) \\times_{Y_j} (Y_j \\times_Y X) =", "Y_j \\times_Y X \\times_Y X", "$$", "has the corresponding property (i.e., is quasi-compact or a closed immersion).", "Write $Y_j = \\colim Y_{j, \\lambda}$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}.", "Replacing $Y_j$ by $Y_{j, \\lambda}$ in the formula above, we have the", "property by our assumption that (3) holds. Since the displayed arrow", "is the colimit of the arrows $\\Delta_{j, \\lambda}$ and since we", "can test whether $\\Delta_j$ has the corresponding property by", "testing after base change by affine schemes mapping into", "$Y_j \\times_Y X \\times_Y X$, we conclude by", "Lemma \\ref{lemma-factor-through-thickening}.", "\\medskip\\noindent", "Let us prove that (5) implies (1). Let $\\{Y_j \\to Y\\}$ be as in (5).", "Then we have the fibre product diagram", "$$", "\\xymatrix{", "\\coprod Y_j \\times_Y X \\ar[r] \\ar[d] &", "X \\ar[d] \\\\", "\\coprod Y_j \\times_Y X \\times_Y X \\ar[r] &", "X \\times_Y X", "}", "$$", "By assumption the left vertical arrow is quasi-compact or a closed immersion.", "It follows from", "Spaces, Lemma \\ref{spaces-lemma-descent-representable-transformations-property}", "that also the right vertical arrow is quasi-compact or a", "closed immersion." ], "refs": [ "formal-spaces-lemma-base-change-separated", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-affine-formal-algebraic-space", "formal-spaces-lemma-factor-through-thickening", "spaces-lemma-descent-representable-transformations-property" ], "ref_ids": [ 3955, 3981, 3977, 3868, 8134 ] } ], "ref_ids": [ 3994, 3983, 3981 ] }, { "id": 3959, "type": "theorem", "label": "formal-spaces-lemma-proper-local", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-proper-local", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces", "over $S$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is proper,", "\\item for every scheme $Z$ and morphism $Z \\to Y$ the", "base change $Z \\times_Y X \\to Z$ of $f$ is proper,", "\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the", "base change $Z \\times_Y X \\to Z$ of $f$ is proper,", "\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the", "formal algebraic space $Z \\times_Y X$ is an algebraic space proper over $Z$,", "\\item there exists a covering $\\{Y_j \\to Y\\}$ as in", "Definition \\ref{definition-formal-algebraic-space}", "such that the base change $Y_j \\times_Y X \\to Y_j$ is proper for all $j$.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-formal-algebraic-space" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 3981 ] }, { "id": 3960, "type": "theorem", "label": "formal-spaces-lemma-base-change-proper", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-base-change-proper", "contents": [ "Proper morphisms of formal algebraic spaces are preserved by base change." ], "refs": [], "proofs": [ { "contents": [ "This is an immediate consequence of Lemma \\ref{lemma-proper-local}", "and transitivity of base change." ], "refs": [ "formal-spaces-lemma-proper-local" ], "ref_ids": [ 3959 ] } ], "ref_ids": [] }, { "id": 3961, "type": "theorem", "label": "formal-spaces-lemma-sheaf-fpqc", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-sheaf-fpqc", "contents": [ "\\begin{slogan}", "Formal algebraic spaces are fpqc sheaves", "\\end{slogan}", "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Then", "$X$ satisfies the sheaf property for the fpqc topology." ], "refs": [], "proofs": [ { "contents": [ "The proof is {\\bf identical} to the proof of", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-sheaf-fpqc}.", "Since $X$ is a sheaf for the Zariski topology it", "suffices to show the following. Given a surjective", "flat morphism of affines $f : T' \\to T$ we have:", "$X(T)$ is the equalizer of the two maps $X(T') \\to X(T' \\times_T T')$.", "See Topologies, Lemma \\ref{topologies-lemma-sheaf-property-fpqc}.", "\\medskip\\noindent", "Let $a, b : T \\to X$ be two morphisms such that $a \\circ f = b \\circ f$.", "We have to show $a = b$. Consider the fibre product", "$$", "E = X \\times_{\\Delta_{X/S}, X \\times_S X, (a, b)} T.", "$$", "By Lemma \\ref{lemma-diagonal-formal-algebraic-space}", "the morphism $\\Delta_{X/S}$ is a representable monomorphism. Hence", "$E \\to T$ is a monomorphism of schemes. Our assumption that", "$a \\circ f = b \\circ f$ implies that $T' \\to T$ factors (uniquely) through $E$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "T' \\times_T E \\ar[r] \\ar[d] & E \\ar[d] \\\\", "T' \\ar[r] \\ar@/^5ex/[u] \\ar[ru] & T", "}", "$$", "Since the projection $T' \\times_T E \\to T'$ is a monomorphism", "with a section we conclude it is an isomorphism. Hence we conclude that", "$E \\to T$ is an isomorphism by", "Descent, Lemma \\ref{descent-lemma-descending-property-isomorphism}.", "This means $a = b$ as desired.", "\\medskip\\noindent", "Next, let $c : T' \\to X$ be a morphism such that the two compositions", "$T' \\times_T T' \\to T' \\to X$ are the same. We have to find a morphism", "$a : T \\to X$ whose composition with $T' \\to T$ is $c$. Choose a", "formal affine scheme $U$ and an \\'etale morphism $U \\to X$ such that the image", "of $|U| \\to |X_{red}|$ contains the image of $|c| : |T'| \\to |X_{red}|$.", "This is possible by", "Definition \\ref{definition-formal-algebraic-space},", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-topology-points},", "the fact that a finite union of formal affine algebraic spaces is a", "formal affine algebraic space, and the fact that $|T'|$ is quasi-compact", "(small argument omitted). The morphism $U \\to X$ is representable", "by schemes (Lemma \\ref{lemma-presentation-representable}) and", "separated (Lemma \\ref{lemma-separated-from-separated}). Thus", "$$", "V = U \\times_{X, c} T' \\longrightarrow T'", "$$", "is an \\'etale and separated morphism of schemes. It is also surjective", "by our choice of $U \\to X$ (if you do not want to argue this you can", "replace $U$ by a disjoint union of formal affine algebraic spaces so that", "$U \\to X$ is surjective everything else still works as well). The fact that", "$c \\circ \\text{pr}_0 = c \\circ \\text{pr}_1$ means that we obtain a", "descent datum on $V/T'/T$", "(Descent, Definition \\ref{descent-definition-descent-datum})", "because", "\\begin{align*}", "V \\times_{T'} (T' \\times_T T')", "& =", "U \\times_{X, c \\circ \\text{pr}_0} (T' \\times_T T') \\\\", "& =", "(T' \\times_T T') \\times_{c \\circ \\text{pr}_1, X} U \\\\", "& =", "(T' \\times_T T') \\times_{T'} V", "\\end{align*}", "The morphism $V \\to T'$ is ind-quasi-affine by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-separated-ind-quasi-affine}", "(because \\'etale morphisms are locally quasi-finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-locally-quasi-finite}).", "By More on Groupoids, Lemma \\ref{more-groupoids-lemma-ind-quasi-affine}", "the descent datum is effective. Say $W \\to T$ is a morphism", "such that there is an isomorphism $\\alpha : T' \\times_T W \\to V$", "compatible with the given descent datum on $V$ and the canonical descent", "datum on $T' \\times_T W$. Then $W \\to T$ is surjective and \\'etale", "(Descent, Lemmas \\ref{descent-lemma-descending-property-surjective} and", "\\ref{descent-lemma-descending-property-etale}).", "Consider the composition", "$$", "b' : T' \\times_T W \\longrightarrow V = U \\times_{X, c} T' \\longrightarrow U", "$$", "The two compositions", "$b' \\circ (\\text{pr}_0, 1), ", "b' \\circ (\\text{pr}_1, 1) :", "(T' \\times_T T') \\times_T W \\to T' \\times_T W \\to U$", "agree by our choice of $\\alpha$ and the corresponding property of $c$", "(computation omitted). Hence $b'$ descends to a morphism $b : W \\to U$ by", "Descent, Lemma \\ref{descent-lemma-fpqc-universal-effective-epimorphisms}.", "The diagram", "$$", "\\xymatrix{", "T' \\times_T W \\ar[r] \\ar[d] & W \\ar[r]_b & U \\ar[d] \\\\", "T' \\ar[rr]^c & & X", "}", "$$", "is commutative. What this means is that we have proved the existence", "of $a$ \\'etale locally on $T$, i.e., we have an $a' : W \\to X$.", "However, since we have proved uniqueness", "in the first paragraph, we find that this \\'etale local solution", "satisfies the glueing condition, i.e., we have", "$\\text{pr}_0^*a' = \\text{pr}_1^*a'$ as elements of $X(W \\times_T W)$.", "Since $X$ is an \\'etale sheaf we find an unique $a \\in X(T)$ restricting", "to $a'$ on $W$." ], "refs": [ "spaces-properties-proposition-sheaf-fpqc", "topologies-lemma-sheaf-property-fpqc", "formal-spaces-lemma-diagonal-formal-algebraic-space", "descent-lemma-descending-property-isomorphism", "formal-spaces-definition-formal-algebraic-space", "spaces-properties-lemma-topology-points", "formal-spaces-lemma-presentation-representable", "formal-spaces-lemma-separated-from-separated", "descent-definition-descent-datum", "more-morphisms-lemma-etale-separated-ind-quasi-affine", "morphisms-lemma-etale-locally-quasi-finite", "more-groupoids-lemma-ind-quasi-affine", "descent-lemma-descending-property-surjective", "descent-lemma-descending-property-etale", "descent-lemma-fpqc-universal-effective-epimorphisms" ], "ref_ids": [ 11919, 12502, 3877, 14682, 3981, 11822, 3872, 3897, 14776, 14045, 5363, 2505, 14672, 14694, 14638 ] } ], "ref_ids": [] }, { "id": 3962, "type": "theorem", "label": "formal-spaces-lemma-map-into-affine", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-map-into-affine", "contents": [ "Let $S$ be a scheme. Let $A$ be a weakly admissible topological", "$S$-algebra. Let $X$ be an affine scheme over $S$. Then", "the natural map", "$$", "\\Mor_S(\\Spec(A), X)", "\\longrightarrow", "\\Mor_S(\\text{Spf}(A), X)", "$$", "is bijective." ], "refs": [], "proofs": [ { "contents": [ "If $X$ is affine, say $X = \\Spec(B)$, then we see from", "Lemma \\ref{lemma-morphism-between-formal-spectra}", "that morphisms $\\text{Spf}(A) \\to \\Spec(B)$ correspond to continuous", "$S$-algebra maps $B \\to A$ where $B$ has the discrete topology.", "These are just $S$-algebra maps, which correspond to morphisms", "$\\Spec(A) \\to \\Spec(B)$." ], "refs": [ "formal-spaces-lemma-morphism-between-formal-spectra" ], "ref_ids": [ 3871 ] } ], "ref_ids": [] }, { "id": 3963, "type": "theorem", "label": "formal-spaces-lemma-map-into-scheme", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-map-into-scheme", "contents": [ "Let $S$ be a scheme. Let $A$ be a weakly admissible topological", "$S$-algebra such that $A/I$ is a local ring for some weak ideal", "of definition $I \\subset A$. Let $X$ be a scheme over $S$. Then", "the natural map", "$$", "\\Mor_S(\\Spec(A), X)", "\\longrightarrow", "\\Mor_S(\\text{Spf}(A), X)", "$$", "is bijective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi : \\text{Spf}(A) \\to X$ be a morphism. Since $\\Spec(A/I)$", "is local we see that $\\varphi$ maps $\\Spec(A/I)$ into an affine", "open $U \\subset X$. However, this then implies that $\\Spec(A/J)$", "maps into $U$ for every ideal of definition $J$. Hence we may", "apply Lemma \\ref{lemma-map-into-affine} to see that $\\varphi$ comes", "from a morphism $\\Spec(A) \\to X$. This proves surjectivity of the map.", "We omit the proof of injectivity." ], "refs": [ "formal-spaces-lemma-map-into-affine" ], "ref_ids": [ 3962 ] } ], "ref_ids": [] }, { "id": 3964, "type": "theorem", "label": "formal-spaces-lemma-map-into-algebraic-space", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-map-into-algebraic-space", "contents": [ "Let $S$ be a scheme. Let $R$ be a complete local Noetherian $S$-algebra.", "Let $X$ be an algebraic space over $S$. Then the natural map", "$$", "\\Mor_S(\\Spec(R), X)", "\\longrightarrow", "\\Mor_S(\\text{Spf}(R), X)", "$$", "is bijective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak m$ be the maximal ideal of $R$. We have to show that", "$$", "\\Mor_S(\\Spec(R), X) \\longrightarrow \\lim \\Mor_S(\\Spec(R/\\mathfrak m^n), X)", "$$", "is bijective for $R$ as above.", "\\medskip\\noindent", "Injectivity: Let $x, x' : \\Spec(R) \\to X$", "be two morphisms mapping to the same element in the right hand side.", "Consider the fibre product", "$$", "T = \\Spec(R) \\times_{(x, x'), X \\times_S X, \\Delta} X", "$$", "Then $T$ is a scheme and $T \\to \\Spec(R)$ is locally of finite type,", "monomorphism, separated, and locally quasi-finite, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-properties-diagonal}.", "In particular $T$ is locally Noetherian, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}.", "Let $t \\in T$ be the unique point mapping to the closed point of $\\Spec(R)$", "which exists as $x$ and $x'$ agree over $R/\\mathfrak m$. Then", "$R \\to \\mathcal{O}_{T, t}$ is a local ring map of Noetherian rings such that", "$R/\\mathfrak m^n \\to \\mathcal{O}_{T, t}/\\mathfrak m^n\\mathcal{O}_{T, t}$", "is an isomorphism for all $n$ (because $x$ and $x'$ agree over", "$\\Spec(R/\\mathfrak m^n)$ for all $n$). Since $\\mathcal{O}_{T, t}$", "maps injectively into its completion (see", "Algebra, Lemma \\ref{algebra-lemma-intersect-powers-ideal-module-zero})", "we conclude that $R = \\mathcal{O}_{T, t}$. Hence $x$ and $x'$ agree", "over $R$.", "\\medskip\\noindent", "Surjectivity: Let $(x_n)$ be an element of the right hand side.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$. ", "Denote $x_0 : \\Spec(k) \\to X$ the morphism induced on the residue field", "$k = R/\\mathfrak m$. The morphism of schemes", "$U \\times_{X, x_0} \\Spec(k) \\to \\Spec(k)$ is surjective \\'etale.", "Thus $U \\times_{X, x_0} \\Spec(k)$ is a nonempty disjoint union of spectra", "of finite separable field extensions of $k$, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}.", "Hence we can find a finite separable field extension $k \\subset k'$", "and a $k'$-point $u_0 : \\Spec(k') \\to U$ such that", "$$", "\\xymatrix{", "\\Spec(k') \\ar[d] \\ar[r]_-{u_0} & U \\ar[d] \\\\", "\\Spec(k) \\ar[r]^-{x_0} & X", "}", "$$", "commutes. Let $R \\subset R'$ be the finite \\'etale extension of Noetherian", "complete local rings which induces $k \\subset k'$ on residue fields", "(see Algebra, Lemmas \\ref{algebra-lemma-henselian-cat-finite-etale} and", "\\ref{algebra-lemma-complete-henselian}). Denote $x'_n$ the restriction", "of $x_n$ to $\\Spec(R'/\\mathfrak m^nR')$. By", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-etale-formally-etale}", "we can find an element", "$(u'_n) \\in \\lim \\Mor_S(\\Spec(R'/\\mathfrak m^nR'), U)$", "mapping to $(x'_n)$. By Lemma \\ref{lemma-map-into-scheme}", "the family $(u'_n)$ comes from a unique", "morphism $u' : \\Spec(R') \\to U$. Denote $x' : \\Spec(R') \\to X$ the", "composition. Note that $R' \\otimes_R R'$ is a finite product of spectra of", "Noetherian complete local rings to which our current discussion applies.", "Hence the diagram", "$$", "\\xymatrix{", "\\Spec(R' \\otimes_R R') \\ar[r] \\ar[d] & \\Spec(R') \\ar[d]^{x'} \\\\", "\\Spec(R') \\ar[r]^{x'} & X", "}", "$$", "is commutative by the injectivity shown above and the fact that", "$x'_n$ is the restriction of $x_n$ which is defined over $R/\\mathfrak m^n$.", "Since $\\{\\Spec(R') \\to \\Spec(R)\\}$ is an fppf covering we conclude", "that $x'$ descends to a morphism $x : \\Spec(R) \\to X$.", "We omit the proof that $x_n$ is the restriction of $x$ to", "$\\Spec(R/\\mathfrak m^n)$." ], "refs": [ "spaces-morphisms-lemma-properties-diagonal", "morphisms-lemma-finite-type-noetherian", "algebra-lemma-intersect-powers-ideal-module-zero", "morphisms-lemma-etale-over-field", "algebra-lemma-henselian-cat-finite-etale", "algebra-lemma-complete-henselian", "spaces-more-morphisms-lemma-etale-formally-etale", "formal-spaces-lemma-map-into-scheme" ], "ref_ids": [ 4712, 5202, 627, 5364, 1280, 1282, 95, 3963 ] } ], "ref_ids": [] }, { "id": 3965, "type": "theorem", "label": "formal-spaces-lemma-adic-into-completion", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-adic-into-completion", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $T \\subset |X|$ be a closed subset such that", "$X \\setminus T \\to X$ is quasi-compact. Let $R$ be a complete local", "Noetherian $S$-algebra. Then an adic morphism $p : \\text{Spf}(R) \\to X_{/T}$", "corresponds to a unique morphism $g : \\Spec(R) \\to X$ such", "that $g^{-1}(T) = \\{\\mathfrak m_R\\}$." ], "refs": [], "proofs": [ { "contents": [ "The statement makes sense because $X_{/T}$ is adic* by", "Lemma \\ref{lemma-formal-completion-types} (and hence we're", "allowed to use the terminology adic for morphisms, see", "Definition \\ref{definition-adic-morphism}).", "Let $p$ be given. By Lemma \\ref{lemma-map-into-algebraic-space}", "we get a unique morphism $g : \\Spec(R) \\to X$ corresponding to", "the composition $\\text{Spf}(R) \\to X_{/T} \\to X$.", "Let $Z \\subset X$ be the reduced induced closed subspace structure", "on $T$. The incusion morphism $Z \\to X$ corresponds to a morphism", "$Z \\to X_{/T}$. Since $p$ is adic it is representable by algebraic", "spaces and we find", "$$", "\\text{Spf}(R) \\times_{X_{/T}} Z = \\text{Spf}(R) \\times_X Z", "$$", "is an algebraic space endowed with a closed immersion to $\\text{Spf}(R)$.", "(Equality holds because $X_{/T} \\to X$ is a monomorphism.)", "Thus this fibre product is equal to $\\Spec(R/J)$ for some", "ideal $J \\subset R$ wich contains $\\mathfrak m_R^{n_0}$ for some", "$n_0 \\geq 1$. This implies that $\\Spec(R) \\times_X Z$", "is a closed subscheme of $\\Spec(R)$,", "say $\\Spec(R) \\times_X Z = \\Spec(R/I)$, whose intersection with", "$\\Spec(R/\\mathfrak m_R^n)$ for $n \\geq n_0$ is equal to $\\Spec(R/J)$.", "In algebraic terms this says", "$I + \\mathfrak m_R^n = J + \\mathfrak m_R^n = J$ for all $n \\geq n_0$.", "By Krull's intersection theorem", "this implies $I = J$ and we conclude." ], "refs": [ "formal-spaces-lemma-formal-completion-types", "formal-spaces-definition-adic-morphism", "formal-spaces-lemma-map-into-algebraic-space" ], "ref_ids": [ 3919, 3988, 3964 ] } ], "ref_ids": [] }, { "id": 3966, "type": "theorem", "label": "formal-spaces-lemma-functoriality-etale-site", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-functoriality-etale-site", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$.", "\\begin{enumerate}", "\\item There is a continuous functor", "$Y_{spaces, \\etale} \\to X_{spaces, \\etale}$", "which induces a morphism of sites", "$$", "f_{spaces, \\etale} : X_{spaces, \\etale} \\to Y_{spaces, \\etale}.", "$$", "\\item The rule $f \\mapsto f_{spaces, \\etale}$ is compatible with", "compositions, in other words $(f \\circ g)_{spaces, \\etale}", "= f_{spaces, \\etale} \\circ g_{spaces, \\etale}$ (see", "Sites, Definition \\ref{sites-definition-composition-morphisms-sites}).", "\\item The morphism of topoi associated to $f_{spaces, \\etale}$", "induces, via (\\ref{equation-etale-topos}), a morphism of topoi", "$f_{small} : \\Sh(X_\\etale) \\to \\Sh(Y_\\etale)$", "whose construction is compatible with compositions.", "\\end{enumerate}" ], "refs": [ "sites-definition-composition-morphisms-sites" ], "proofs": [ { "contents": [ "The only point here is that $f$ induces a morphism of reductions", "$X_{red} \\to Y_{red}$ by Lemma \\ref{lemma-reduction-formal-algebraic-space}.", "Hence this lemma is immediate from the corresponding lemma for", "morphisms of algebraic spaces (Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-functoriality-etale-site})." ], "refs": [ "formal-spaces-lemma-reduction-formal-algebraic-space", "spaces-properties-lemma-functoriality-etale-site" ], "ref_ids": [ 3879, 11864 ] } ], "ref_ids": [ 8666 ] }, { "id": 3967, "type": "theorem", "label": "formal-spaces-lemma-etale-morphism-topoi", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-etale-morphism-topoi", "contents": [ "Let $S$ be a scheme, and let $f : X \\to Y$ be a morphism of", "formal algebraic spaces over $S$. Assume $f$ is representable", "by algebraic spaces and \\'etale. In this case there is a", "cocontinuous functor $j : X_\\etale \\to Y_\\etale$.", "The morphism of topoi $f_{small}$ is the", "morphism of topoi associated to $j$, see", "Sites, Lemma \\ref{sites-lemma-cocontinuous-morphism-topoi}.", "Moreover, $j$ is continuous as well, hence", "Sites, Lemma \\ref{sites-lemma-when-shriek} applies." ], "refs": [ "sites-lemma-cocontinuous-morphism-topoi", "sites-lemma-when-shriek" ], "proofs": [ { "contents": [ "This will follow immediately from the case of algebraic spaces", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-etale-morphism-topoi})", "if we can show that the induced morphism $X_{red} \\to Y_{red}$", "is \\'etale. Observe that $X \\times_Y Y_{red}$ is an", "algebraic space, \\'etale over the reduced algebraic space $Y_{red}$,", "and hence reduced itself (by our definition of reduced algebraic", "spaces in Properties of Spaces, Section", "\\ref{spaces-properties-section-types-properties}.", "Hence $X_{red} = X \\times_Y Y_{red}$ as desired." ], "refs": [ "spaces-properties-lemma-etale-morphism-topoi" ], "ref_ids": [ 11866 ] } ], "ref_ids": [ 8543, 8545 ] }, { "id": 3968, "type": "theorem", "label": "formal-spaces-lemma-affine-identify-affine-etale", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-affine-identify-affine-etale", "contents": [ "Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$.", "Then $X_{affine, \\etale}$ is equivalent to the category whose objects", "are morphisms $\\varphi : U \\to X$ of formal algebraic spaces such that", "\\begin{enumerate}", "\\item $U$ is an affine formal algebraic space,", "\\item $\\varphi$ is representable by algebraic spaces and \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $\\mathcal{C}$ the category introduced in the lemma.", "Observe that for $\\varphi : U \\to X$ in $\\mathcal{C}$ the", "morphism $\\varphi$ is representable (by schemes) and affine, see", "Lemma \\ref{lemma-affine-representable-by-algebraic-spaces}.", "Recall that $X_{affine, \\etale} = X_{red, affine, \\etale}$.", "Hence we can define a functor", "$$", "\\mathcal{C} \\longrightarrow X_{affine, \\etale},\\quad", "(U \\to X) \\longmapsto U \\times_X X_{red}", "$$", "because $U \\times_X X_{red}$ is an affine scheme.", "\\medskip\\noindent", "To finish the proof we will construct a quasi-inverse.", "Namely, write $X = \\colim X_\\lambda$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}.", "For each $\\lambda$ we have $X_{red} \\subset X_\\lambda$", "is a thickening. Thus for every $\\lambda$ we have an", "equivalence", "$$", "X_{red, affine, \\etale} = X_{\\lambda, affine, \\etale}", "$$", "for example by", "More on Algebra, Lemma \\ref{more-algebra-lemma-locally-nilpotent-henselian}.", "Hence if $U_{red} \\to X_{red}$ is an \\'etale morphism with", "$U_{red}$ affine, then we obtain a system of \\'etale morphisms", "$U_\\lambda \\to X_\\lambda$ of affine schemes compatible with the", "transition morphisms in the system defining $X$. Hence we can take", "$$", "U = \\colim U_\\lambda", "$$", "as our affine formal algebraic space over $X$. The construction gives that", "$U \\times_X X_\\lambda = U_\\lambda$. This shows that $U \\to X$ is", "representable and \\'etale. We omit the verification that the constructions", "are mutually inverse to each other." ], "refs": [ "formal-spaces-lemma-affine-representable-by-algebraic-spaces", "formal-spaces-definition-affine-formal-algebraic-space", "more-algebra-lemma-locally-nilpotent-henselian" ], "ref_ids": [ 3909, 3977, 9857 ] } ], "ref_ids": [] }, { "id": 3969, "type": "theorem", "label": "formal-spaces-lemma-affine-etale-mcquillan", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-affine-etale-mcquillan", "contents": [ "Let $S$ be a scheme. Let $X$ be an affine formal", "algebraic space over $S$. Assume $X$ is McQuillan, i.e.,", "equal to $\\text{Spf}(A)$ for some", "weakly admissible topological $S$-algebra $A$.", "Then $(X_{affine, \\etale})^{opp}$ is equivalent to", "the category whose", "\\begin{enumerate}", "\\item objects are $A$-algebras of the form", "$B^\\wedge = \\lim B/JB$ where $A \\to B$ is an \\'etale ring map", "and $J$ runs over the weak ideals of definition of $A$, and", "\\item morphisms are continuous $A$-algebra homomorphisms.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-affine-identify-affine-etale} and \\ref{lemma-etale}." ], "refs": [ "formal-spaces-lemma-affine-identify-affine-etale", "formal-spaces-lemma-etale" ], "ref_ids": [ 3968, 3913 ] } ], "ref_ids": [] }, { "id": 3970, "type": "theorem", "label": "formal-spaces-lemma-identify-spaces-etale", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-identify-spaces-etale", "contents": [ "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.", "Then $X_{spaces, \\etale}$ is equivalent to the category whose objects", "are morphisms $\\varphi : U \\to X$ of formal algebraic spaces such that", "$\\varphi$ is representable by algebraic spaces and \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Denote $\\mathcal{C}$ the category introduced in the lemma.", "Recall that $X_{spaces, \\etale} = X_{red, spaces, \\etale}$.", "Hence we can define a functor", "$$", "\\mathcal{C} \\longrightarrow X_{spaces, \\etale},\\quad", "(U \\to X) \\longmapsto U \\times_X X_{red}", "$$", "because $U \\times_X X_{red}$ is an algebraic space \\'etale over $X_{red}$.", "\\medskip\\noindent", "To finish the proof we will construct a quasi-inverse.", "Choose an object $\\psi : V \\to X_{red}$ of $X_{red, spaces, \\etale}$.", "Consider the functor", "$U_{V, \\psi} : (\\Sch/S)_{fppf} \\to \\textit{Sets}$ given by", "$$", "U_{V, \\psi}(T) = \\{(a, b) \\mid", "a : T \\to X,", "\\ b : T \\times_{a, X} X_{red} \\to V,", "\\ \\psi \\circ b = a|_{T \\times_{a, X} X_{red}}\\}", "$$", "We claim that the transformation $U_{V, \\psi} \\to X$, $(a, b) \\mapsto a$", "defines an object of the category $\\mathcal{C}$.", "First, let's prove that $U_{V, \\psi}$ is a formal algebraic space.", "Observe that $U_{V, \\psi}$ is a sheaf for the fppf topology (some details", "omitted). Next, suppose that $X_i \\to X$ is an \\'etale covering by", "affine formal algebraic spaces as in", "Definition \\ref{definition-formal-algebraic-space}.", "Set $V_i = V \\times_{X_{red}} X_{i, red}$ and denote", "$\\psi_i : V_i \\to X_{i, red}$ the projection. Then ", "we have", "$$", "U_{V, \\psi} \\times_X X_i = U_{V_i, \\psi_i}", "$$", "by a formal argument because $X_{i, red} = X_i \\times_X X_{red}$", "(as $X_i \\to X$ is representable by algebraic spaces and \\'etale).", "Hence it suffices to show that $U_{V_i, \\psi_i}$ is an", "affine formal algebraic space, because then we will have", "a covering $U_{V_i, \\psi_i} \\to U_{V, \\psi}$ as in", "Definition \\ref{definition-formal-algebraic-space}.", "On the other hand, we have seen in the proof of", "Lemma \\ref{lemma-etale-morphism-topoi}", "that $\\psi_i : V_i \\to X_i$ is the base", "change of a representable and \\'etale morphism", "$U_i \\to X_i$ of affine formal algebraic spaces.", "Then it is not hard to see that $U_i = U_{V_i, \\psi_i}$", "as desired.", "\\medskip\\noindent", "We omit the verification that $U_{V, \\psi} \\to X$", "is representable by algebraic spaces and \\'etale.", "Thus we obtain our functor $(V, \\psi) \\mapsto (U_{V, \\psi} \\to X)$", "in the other direction.", "We omit the verification that the constructions", "are mutually inverse to each other." ], "refs": [ "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-lemma-etale-morphism-topoi" ], "ref_ids": [ 3981, 3981, 3967 ] } ], "ref_ids": [] }, { "id": 3971, "type": "theorem", "label": "formal-spaces-lemma-identify-affine-etale", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-identify-affine-etale", "contents": [ "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.", "Then $X_{affine, \\etale}$ is equivalent to the category whose objects", "are morphisms $\\varphi : U \\to X$ of formal algebraic spaces such that", "\\begin{enumerate}", "\\item $U$ is an affine formal algebraic space,", "\\item $\\varphi$ is representable by algebraic spaces and \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows by combining Lemmas \\ref{lemma-identify-spaces-etale} and", "\\ref{lemma-characterize-affine}." ], "refs": [ "formal-spaces-lemma-identify-spaces-etale", "formal-spaces-lemma-characterize-affine" ], "ref_ids": [ 3970, 3902 ] } ], "ref_ids": [] }, { "id": 3972, "type": "theorem", "label": "formal-spaces-lemma-structure-sheaf", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-structure-sheaf", "contents": [ "Every formal algebraic space has a structure sheaf." ], "refs": [], "proofs": [ { "contents": [ "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.", "By (\\ref{equation-etale-topos}) it suffices to construct $\\mathcal{O}_X$", "as a sheaf of topological rings on $X_{affine, \\etale}$.", "Denote $\\mathcal{C}$ the category whose objects", "are morphisms $\\varphi : U \\to X$ of formal algebraic spaces such that", "$U$ is an affine formal algebraic space and", "$\\varphi$ is representable by algebraic spaces and \\'etale.", "By Lemma \\ref{lemma-identify-affine-etale}", "the functor $U \\mapsto U_{red}$ is an equivalence of categories", "$\\mathcal{C} \\to X_{affine, \\etale}$. Hence by the rule given", "above the lemma, we already have $\\mathcal{O}_X$ as a presheaf of", "topological rings on $X_{affine, \\etale}$. Thus it suffices to check", "the sheaf condition.", "\\medskip\\noindent", "By definition of $X_{affine, \\etale}$ a covering corresponds to a finite", "family $\\{g_i : U_i \\to U\\}_{i = 1, \\ldots, n}$ of morphisms of $\\mathcal{C}$", "such that $\\{U_{i, red} \\to U_{red}\\}$ is an \\'etale covering.", "The morphisms $g_i$ are representably by algebraic spaces", "(Lemma \\ref{lemma-permanence-representable}) hence affine", "(Lemma \\ref{lemma-affine-representable-by-algebraic-spaces}).", "Then $g_i$ is \\'etale (follows formally from", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-permanence}", "as $U_i$ and $U$ are \\'etale over $X$ in the sense of", "Bootstrap, Section \\ref{bootstrap-section-representable-by-spaces-properties}).", "Finally, write $U = \\colim U_\\lambda$ as in", "Definition \\ref{definition-affine-formal-algebraic-space}.", "\\medskip\\noindent", "With these preparations out of the way, we can prove the sheaf", "property as follows. For each $\\lambda$ we set", "$U_{i, \\lambda} = U_i \\times_U U_\\lambda$ and", "$U_{ij, \\lambda} = (U_i \\times_U U_j) \\times_U U_\\lambda$.", "By the above, these are affine schemes, $\\{U_{i, \\lambda} \\to U_\\lambda\\}$", "is an \\'etale covering, and", "$U_{ij, \\lambda} = U_{i, \\lambda} \\times_{U_\\lambda} U_{j, \\lambda}$.", "Also we have $U_i = \\colim U_{i, \\lambda}$ and", "$U_i \\times_U U_j = \\colim U_{ij, \\lambda}$.", "For each $\\lambda$ we have an exact sequence", "$$", "0 \\to", "\\Gamma(U_\\lambda, \\mathcal{O}_{U_\\lambda}) \\to", "\\prod\\nolimits_i \\Gamma(U_{i, \\lambda}, \\mathcal{O}_{U_{i, \\lambda}}) \\to", "\\prod\\nolimits_{i, j} \\Gamma(U_{ij, \\lambda}, \\mathcal{O}_{U_{ij, \\lambda}})", "$$", "as we have the sheaf condition for the structure sheaf", "on $U_\\lambda$ and the \\'etale topology", "(see \\'Etale Cohomology, Proposition", "\\ref{etale-cohomology-proposition-quasi-coherent-sheaf-fpqc}).", "Since limits commute with limits, the inverse limit of these", "exact sequences is an exact sequence", "$$", "0 \\to", "\\lim \\Gamma(U_\\lambda, \\mathcal{O}_{U_\\lambda}) \\to", "\\prod\\nolimits_i \\lim \\Gamma(U_{i, \\lambda}, \\mathcal{O}_{U_{i, \\lambda}}) \\to", "\\prod\\nolimits_{i, j} \\lim", "\\Gamma(U_{ij, \\lambda}, \\mathcal{O}_{U_{ij, \\lambda}})", "$$", "which exactly means that", "$$", "0 \\to", "\\mathcal{O}_X(U_{red}) \\to", "\\prod\\nolimits_i \\mathcal{O}_X(U_{i, red}) \\to", "\\prod\\nolimits_{i, j} \\mathcal{O}_X((U_i \\times_U U_j)_{red})", "$$", "is exact and hence the sheaf propery holds as desired." ], "refs": [ "formal-spaces-lemma-identify-affine-etale", "formal-spaces-lemma-permanence-representable", "formal-spaces-lemma-affine-representable-by-algebraic-spaces", "spaces-properties-lemma-etale-permanence", "formal-spaces-definition-affine-formal-algebraic-space", "etale-cohomology-proposition-quasi-coherent-sheaf-fpqc" ], "ref_ids": [ 3971, 3905, 3909, 11859, 3977, 6696 ] } ], "ref_ids": [] }, { "id": 3973, "type": "theorem", "label": "formal-spaces-lemma-higher-vanishing-structure-sheaf", "categories": [ "formal-spaces" ], "title": "formal-spaces-lemma-higher-vanishing-structure-sheaf", "contents": [ "If $X$ is a countably indexed affine formal algebraic space, then", "we have $H^n(X_\\etale, \\mathcal{O}_X) = 0$ for $n > 0$." ], "refs": [], "proofs": [ { "contents": [ "We may work with $X_{affine, \\etale}$ as this gives the same topos.", "We will apply Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-cech-vanish-collection}", "to show we have vanishing. Since $X_{affine, \\etale}$", "has finite disjoint unions, this reduces us to the {\\v C}ech", "complex of a covering given by a single arrow $\\{U_{red} \\to V_{red}\\}$", "in $X_{affine, \\etale} = X_{red, affine, \\etale}$", "(see \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-cech-complex}).", "Thus we have to show that", "$$", "0 \\to \\mathcal{O}_X(V_{red}) \\to \\mathcal{O}_X(U_{red}) \\to", "\\mathcal{O}_X(U_{red} \\times_{V_{red}} U_{red}) \\to \\ldots", "$$", "is exact. We will do this below in the case $V_{red} = X_{red}$.", "The general case is proven in exactly the same way.", "\\medskip\\noindent", "Recall that $X = \\text{Spf}(A)$ where $A$ is a", "weakly admissible topological ring having a countable", "fundamental system of weak ideals of definition.", "We have seen in Lemmas", "\\ref{lemma-affine-identify-affine-etale} and", "\\ref{lemma-affine-etale-mcquillan}", "that the object $U_{red}$ in $X_{affine, \\etale}$", "corresponds to a morphism $U \\to X$ of affine formal", "algebraic spaces which is representable by algebraic space and \\'etale", "and $U = \\text{Spf}(B^\\wedge)$ where $B$ is an \\'etale $A$-algebra.", "By our rule for the structure sheaf we see", "$$", "\\mathcal{O}_X(U_{red}) = B^\\wedge", "$$", "We recall that $B^\\wedge = \\lim B/JB$ where the limit is over weak", "ideals of definition $J \\subset A$.", "Working through the definitions we obtain", "$$", "\\mathcal{O}_X(U_{red} \\times_{X_{red}} U_{red}) = (B \\otimes_A B)^\\wedge", "$$", "and so on. Since $U \\to X$ is a covering the map $A \\to B$", "is faithfully flat, see Lemma \\ref{lemma-etale-surjective}.", "Hence the complex", "$$", "0 \\to A \\to B \\to B \\otimes_A B \\to B \\otimes_A B \\otimes_A B \\to \\ldots", "$$", "is {\\bf universally} exact, see Descent, Lemma \\ref{descent-lemma-ff-exact}.", "Our goal is to show that", "$$", "H^n(0 \\to A^\\wedge \\to B^\\wedge \\to (B \\otimes_A B)^\\wedge", "\\to (B \\otimes_A B \\otimes_A B)^\\wedge \\to \\ldots)", "$$", "is zero for $p > 0$. To see what is going on, let's split our", "exact complex (before completion) into short exact sequences", "$$", "0 \\to A \\to B \\to M_1 \\to 0,\\quad", "0 \\to M_i \\to B^{\\otimes_A i + 1} \\to M_{i + 1} \\to 0", "$$", "By what we said above, these are universally exact short exact", "sequences. Hence $JM_i = M_i \\cap J(B^{\\otimes_A i + 1})$ for", "every ideal $J$ of $A$. In particular, the", "topology on $M_i$ as a submodule of $B^{\\otimes_A i + 1}$", "is the same as the topology on $M_i$ as a quotient module of", "$B^{\\otimes_A i}$. Therefore, since there exists a countable fundamental system", "of weak ideals of definition in $A$, the sequences", "$$", "0 \\to A^\\wedge \\to B^\\wedge \\to M_1^\\wedge \\to 0,\\quad", "0 \\to M_i^\\wedge \\to (B^{\\otimes_A i + 1})^\\wedge \\to M_{i + 1}^\\wedge \\to 0", "$$", "remain exact by Lemma \\ref{lemma-ses}. This proves the lemma." ], "refs": [ "sites-cohomology-lemma-cech-vanish-collection", "etale-cohomology-lemma-cech-complex", "formal-spaces-lemma-affine-identify-affine-etale", "formal-spaces-lemma-affine-etale-mcquillan", "formal-spaces-lemma-etale-surjective", "descent-lemma-ff-exact", "formal-spaces-lemma-ses" ], "ref_ids": [ 4205, 6416, 3968, 3969, 3914, 14598, 3857 ] } ], "ref_ids": [] }, { "id": 4021, "type": "theorem", "label": "pione-theorem-fundamental-group", "categories": [ "pione" ], "title": "pione-theorem-fundamental-group", "contents": [ "Let $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point", "of $X$.", "\\begin{enumerate}", "\\item The fibre functor $F_{\\overline{x}}$ defines an equivalence of", "categories", "$$", "\\textit{F\\'Et}_X \\longrightarrow", "\\textit{Finite-}\\pi_1(X, \\overline{x})\\textit{-Sets}", "$$", "\\item Given a second geometric point $\\overline{x}'$ of $X$ there", "exists an isomorphism $t : F_{\\overline{x}} \\to F_{\\overline{x}'}$.", "This gives an isomorphism $\\pi_1(X, \\overline{x}) \\to \\pi_1(X, \\overline{x}')$", "compatible with the equivalences in (1). This isomorphism is", "independent of $t$ up to inner conjugation.", "\\item Given a morphism $f : X \\to Y$ of connected schemes denote", "$\\overline{y} = f \\circ \\overline{x}$. There is a canonical", "continuous homomorphism", "$$", "f_* : \\pi_1(X, \\overline{x}) \\to \\pi_1(Y, \\overline{y})", "$$", "such that the diagram", "$$", "\\xymatrix{", "\\textit{F\\'Et}_Y \\ar[r]_{\\text{base change}} \\ar[d]_{F_{\\overline{y}}} &", "\\textit{F\\'Et}_X \\ar[d]^{F_{\\overline{x}}} \\\\", "\\textit{Finite-}\\pi_1(Y, \\overline{y})\\textit{-Sets} \\ar[r]^{f_*} &", "\\textit{Finite-}\\pi_1(X, \\overline{x})\\textit{-Sets}", "}", "$$", "is commutative.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from Lemma \\ref{lemma-finite-etale-connected-galois-category}", "and Proposition \\ref{proposition-galois}.", "Part (2) is a special case of Lemma \\ref{lemma-functoriality-galois}.", "For part (3) observe that the diagram", "$$", "\\xymatrix{", "\\textit{F\\'Et}_Y \\ar[r] \\ar[d]_{F_{\\overline{y}}} &", "\\textit{F\\'Et}_X \\ar[d]^{F_{\\overline{x}}} \\\\", "\\textit{Sets} \\ar@{=}[r] & \\textit{Sets}", "}", "$$", "is commutative (actually commutative, not just $2$-commutative) because", "$\\overline{y} = f \\circ \\overline{x}$. Hence", "we can apply Lemma \\ref{lemma-functoriality-galois} with the implied", "transformation of functors to get (3)." ], "refs": [ "pione-lemma-finite-etale-connected-galois-category", "pione-proposition-galois", "pione-lemma-functoriality-galois", "pione-lemma-functoriality-galois" ], "ref_ids": [ 4039, 4130, 4031, 4031 ] } ], "ref_ids": [] }, { "id": 4022, "type": "theorem", "label": "pione-theorem-global", "categories": [ "pione" ], "title": "pione-theorem-global", "contents": [ "Let $Y$ be an excellent regular scheme over a field. Let $f : X \\to Y$", "be a finite type morphism of schemes with $X$ normal. Let $V \\subset X$", "be the maximal open subscheme where $f$ is \\'etale. Then the inclusion", "morphism $V \\to X$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$ with image $y \\in Y$. It suffices to prove that", "$V \\cap W$ is affine for some affine open neighbourhood $W$ of $x$.", "Since $\\Spec(\\mathcal{O}_{X, x})$ is the limit of the schemes $W$,", "this holds if and only if", "$$", "V_x = V \\times_X \\Spec(\\mathcal{O}_{X, x})", "$$", "is affine (Limits, Lemma \\ref{limits-lemma-limit-affine}).", "Thus, if the theorem holds for the morphism", "$X \\times_Y \\Spec(\\mathcal{O}_{Y, y}) \\to \\Spec(\\mathcal{O}_{Y, y})$,", "then the theorem holds. In particular, we may assume $Y$", "is regular of finite dimension, which allows us to do induction", "on the dimension $d = \\dim(Y)$. Combining this with the same argument again,", "we may assume that $Y$ is local with closed point $y$ and that", "$V \\cap (X \\setminus f^{-1}(\\{y\\}) \\to X \\setminus f^{-1}(\\{y\\})$", "is affine.", "\\medskip\\noindent", "Let $x \\in X$ be a point lying over $y$. If $x \\in V$, then", "there is nothing to prove. Observe that $f^{-1}(\\{y\\}) \\cap V$", "is a finite set of closed points (the fibres of an \\'etale morphism", "are discrete). Thus after replacing $X$", "by an affine open neighbourhood of $x$ we may assume", "$y \\not \\in f(V)$. We have to prove that $V$ is affine.", "\\medskip\\noindent", "Let $e(V)$ be the maximum $i$ with $H^i(V, \\mathcal{O}_V) \\not = 0$.", "As $X$ is affine the integer $e(V)$ is the maximum of the numbers $e(V_x)$", "where $x \\in X \\setminus V$, see", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-cd-local}", "and the characterization of cohomological dimension", "in Local Cohomology, Lemma \\ref{local-cohomology-lemma-cd}.", "We have $e(V_x) \\leq \\dim(\\mathcal{O}_{X, x}) - 1$ by", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-cd-dimension}.", "If $\\dim(\\mathcal{O}_{X, x}) \\geq 2$ then purity of", "ramification locus (Lemma \\ref{lemma-purity-ramification})", "shows that $V_x$ is strictly smaller than the punctured spectrum of", "$\\mathcal{O}_{X, x}$. Since $\\mathcal{O}_{X, x}$ is", "normal and excellent, this implies", "$e(V_x) \\leq \\dim(\\mathcal{O}_{X, x}) - 2$ by", "Hartshorne-Lichtenbaum vanishing", "(Local Cohomology, Lemma \\ref{local-cohomology-lemma-affine-complement}).", "On the other hand, since $X \\to Y$ is of finite type", "and $V \\subset X$ is dense (after possibly replacing $X$", "by the closure of $V$), we see that $\\dim(\\mathcal{O}_{X, x}) \\leq d$", "by the dimension formula", "(Morphisms, Lemma \\ref{morphisms-lemma-dimension-formula}).", "Whence $e(V) \\leq \\max(0, d - 2)$.", "Thus $V$ is affine by Lemma \\ref{lemma-conclude}", "if $d \\geq 2$. If $d = 1$ or $d = 0$, then the punctured spectrum of", "$\\mathcal{O}_{Y, y}$ is affine and hence $V$ is affine." ], "refs": [ "limits-lemma-limit-affine", "local-cohomology-lemma-cd-local", "local-cohomology-lemma-cd", "local-cohomology-lemma-cd-dimension", "pione-lemma-purity-ramification", "local-cohomology-lemma-affine-complement", "morphisms-lemma-dimension-formula", "pione-lemma-conclude" ], "ref_ids": [ 15043, 9707, 9703, 9708, 4116, 9757, 5493, 4118 ] } ], "ref_ids": [] }, { "id": 4023, "type": "theorem", "label": "pione-theorem-specialization-map-isomorphism-prime-to-p", "categories": [ "pione" ], "title": "pione-theorem-specialization-map-isomorphism-prime-to-p", "contents": [ "Let $f : X \\to S$ be a smooth proper morphism with geometrically", "connected fibres. Let $s' \\leadsto s$ be a specialization.", "If the characteristic of $\\kappa(s)$ is $p$, then the specialization", "map", "$$", "sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})", "$$", "is surjective and induces an isomorphism", "$$", "\\pi'_1(X_{\\overline{s}'}) \\cong \\pi'_1(X_{\\overline{s}})", "$$", "of the maximal prime-to-p quotients" ], "refs": [], "proofs": [ { "contents": [ "This is proved in exactly the same manner as", "Proposition \\ref{proposition-specialization-map-isomorphism}", "with the following differences", "\\begin{enumerate}", "\\item Given $X/A$ we no longer show that the functor", "$\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_{X_{\\overline{\\eta}}}$", "is essentially surjective. We show only that Galois objects", "whose Galois group has order prime to $p$ are in the essential", "image. This will be enough to conclude the injectivity of", "$\\pi'_1(X_{\\overline{s}'}) \\to \\pi'_1(X_{\\overline{s}})$ by", "exactly the same argument.", "\\item The extensions", "$\\mathcal{O}_{X_B, \\xi_B} \\subset \\mathcal{O}_{Z, \\xi_i}$", "are tamely ramified as the associated extension of fraction", "fields is Galois with group of order prime to $p$. See", "More on Algebra, Lemma \\ref{more-algebra-lemma-galois-conclusion}.", "\\item The extension $\\kappa_A \\subset \\kappa_B$ is no longer", "necessarily trivial, but it is purely inseparable.", "Hence the morphism $X_{\\kappa_B} \\to X_{\\kappa_A}$", "is a universal homeomorphism and induces an isomorphism", "of fundamental groups by Proposition \\ref{proposition-universal-homeomorphism}.", "\\end{enumerate}" ], "refs": [ "pione-proposition-specialization-map-isomorphism", "more-algebra-lemma-galois-conclusion", "pione-proposition-universal-homeomorphism" ], "ref_ids": [ 4139, 10499, 4131 ] } ], "ref_ids": [] }, { "id": 4024, "type": "theorem", "label": "pione-lemma-sheaves-point", "categories": [ "pione" ], "title": "pione-lemma-sheaves-point", "contents": [ "Let $K$ be a field. Let $K^{sep}$ be a separable closure of $K$.", "Consider the profinite group $G = \\text{Gal}(K^{sep}/K)$.", "The functor", "$$", "\\begin{matrix}", "\\text{schemes \\'etale over }K &", "\\longrightarrow &", "G\\textit{-Sets} \\\\", "X/K & \\longmapsto &", "\\Mor_{\\Spec(K)}(\\Spec(K^{sep}), X)", "\\end{matrix}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "A scheme $X$ over $K$ is \\'etale over $K$ if and only if", "$X \\cong \\coprod_{i\\in I} \\Spec(K_i)$ with", "each $K_i$ a finite separable extension of $K$", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}).", "The functor of the lemma associates to $X$ the $G$-set", "$$", "\\coprod\\nolimits_i \\Hom_K(K_i, K^{sep})", "$$", "with its natural left $G$-action. Each element has an open stabilizer", "by definition of the topology on $G$. Conversely, any $G$-set $S$", "is a disjoint union of its orbits. Say $S = \\coprod S_i$. Pick $s_i \\in S_i$", "and denote $G_i \\subset G$ its open stabilizer. By Galois theory", "(Fields, Theorem \\ref{fields-theorem-inifinite-galois-theory})", "the fields $(K^{sep})^{G_i}$ are finite separable field extensions of $K$, and", "hence the scheme", "$$", "\\coprod\\nolimits_i \\Spec((K^{sep})^{G_i})", "$$", "is \\'etale over $K$. This gives an inverse to the functor of the lemma.", "Some details omitted." ], "refs": [ "morphisms-lemma-etale-over-field", "fields-theorem-inifinite-galois-theory" ], "ref_ids": [ 5364, 4443 ] } ], "ref_ids": [] }, { "id": 4025, "type": "theorem", "label": "pione-lemma-aut-inverse-limit", "categories": [ "pione" ], "title": "pione-lemma-aut-inverse-limit", "contents": [ "Let $\\mathcal{C}$ be a category and let $F : \\mathcal{C} \\to \\textit{Sets}$", "be a functor. The map (\\ref{equation-embedding-product}) identifies", "$\\text{Aut}(F)$ with a closed subgroup of", "$\\prod_{X \\in \\Ob(\\mathcal{C})} \\text{Aut}(F(X))$.", "In particular, if $F(X)$ is finite for all $X$, then", "$\\text{Aut}(F)$ is a profinite group." ], "refs": [], "proofs": [ { "contents": [ "Let $\\xi = (\\gamma_X) \\in \\prod \\text{Aut}(F(X))$ be an element not in", "$\\text{Aut}(F)$. Then there exists a morphism $f : X \\to X'$ of $\\mathcal{C}$", "and an element $x \\in F(X)$ such that", "$F(f)(\\gamma_X(x)) \\not = \\gamma_{X'}(F(f)(x))$.", "Consider the open neighbourhood", "$U = \\{\\gamma \\in \\text{Aut}(F(X)) \\mid \\gamma(x) = \\gamma_X(x)\\}$", "of $\\gamma_X$ and the open neighbourhood", "$U' = \\{\\gamma' \\in \\text{Aut}(F(X')) \\mid \\gamma'(F(f)(x)) =", "\\gamma_{X'}(F(f)(x))\\}$.", "Then", "$U \\times U' \\times \\prod_{X'' \\not = X, X'} \\text{Aut}(F(X''))$", "is an open neighbourhood of $\\xi$ not meeting $\\text{Aut}(F)$.", "The final statement is follows from the fact that", "$\\prod \\text{Aut}(F(X))$ is a profinite space if each $F(X)$ is finite." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4026, "type": "theorem", "label": "pione-lemma-single-out-profinite", "categories": [ "pione" ], "title": "pione-lemma-single-out-profinite", "contents": [ "Let $G$ be a topological group. The automorphism group of the functor", "(\\ref{equation-forgetful}) endowed with its profinite topology from", "Lemma \\ref{lemma-aut-inverse-limit} is the profinite completion of $G$." ], "refs": [ "pione-lemma-aut-inverse-limit" ], "proofs": [ { "contents": [ "Denote $F_G$ the functor (\\ref{equation-forgetful}). Any morphism", "$X \\to Y$ in $\\textit{Finite-}G\\textit{-Sets}$ commutes with the action", "of $G$. Thus any $g \\in G$ defines an automorphism of $F_G$ and", "we obtain a canonical homomorphism $G \\to \\text{Aut}(F_G)$ of groups.", "Observe that any finite $G$-set $X$ is a finite disjoint union of", "$G$-sets of the form $G/H_i$ with canonical $G$-action where", "$H_i \\subset G$ is an open subgroup of finite index. Then", "$U_i = \\bigcap gH_ig^{-1}$ is open, normal, and has finite index.", "Moreover $U_i$ acts trivially on $G/H_i$ hence", "$U = \\bigcap U_i$ acts trivially on $F_G(X)$. Hence the action", "$G \\times F_G(X) \\to F_G(X)$ is continuous. By the universal", "property of the topology on $\\text{Aut}(F_G)$ the map", "$G \\to \\text{Aut}(F_G)$ is continuous.", "By Lemma \\ref{lemma-aut-inverse-limit} and the universal property", "of profinite completion there is an induced", "continuous group homomorphism", "$$", "G^\\wedge \\longrightarrow \\text{Aut}(F_G)", "$$", "Moreover, since $G/U$ acts faithfully on $G/U$ this map is", "injective. If the image is dense, then the map is surjective and hence a", "homeomorphism by Topology, Lemma \\ref{topology-lemma-bijective-map}.", "\\medskip\\noindent", "Let $\\gamma \\in \\text{Aut}(F_G)$ and let $X \\in \\Ob(\\mathcal{C})$.", "We will show there is a $g \\in G$ such that $\\gamma$ and $g$", "induce the same action on $F_G(X)$. This will finish the proof.", "As before we see that $X$ is a finite disjoint union of $G/H_i$.", "With $U_i$ and $U$ as above, the finite $G$-set $Y = G/U$", "surjects onto $G/H_i$ for all $i$ and hence it suffices to", "find $g \\in G$ such that $\\gamma$ and $g$ induce the same action", "on $F_G(G/U) = G/U$. Let $e \\in G$ be the neutral element and", "say that $\\gamma(eU) = g_0U$ for some $g_0 \\in G$. For any", "$g_1 \\in G$ the morphism", "$$", "R_{g_1} : G/U \\longrightarrow G/U,\\quad gU \\longmapsto gg_1U", "$$", "of $\\textit{Finite-}G\\textit{-Sets}$ commutes with the action of", "$\\gamma$. Hence", "$$", "\\gamma(g_1U) = \\gamma(R_{g_1}(eU)) = R_{g_1}(\\gamma(eU)) =", "R_{g_1}(g_0U) = g_0g_1U", "$$", "Thus we see that $g = g_0$ works." ], "refs": [ "pione-lemma-aut-inverse-limit", "topology-lemma-bijective-map" ], "ref_ids": [ 4025, 8275 ] } ], "ref_ids": [ 4025 ] }, { "id": 4027, "type": "theorem", "label": "pione-lemma-second-fundamental-functor", "categories": [ "pione" ], "title": "pione-lemma-second-fundamental-functor", "contents": [ "Let $G$ be a topological group. Let", "$F : \\textit{Finite-}G\\textit{-Sets} \\to \\textit{Sets}$", "be an exact functor with $F(X)$ finite for all $X$.", "Then $F$ is isomorphic to the functor (\\ref{equation-forgetful})." ], "refs": [], "proofs": [ { "contents": [ "Let $X$ be a nonempty object of $\\textit{Finite-}G\\textit{-Sets}$.", "The diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d] & \\{*\\} \\ar[d] \\\\", "\\{*\\} \\ar[r] & \\{*\\}", "}", "$$", "is cocartesian. Hence we conclude that $F(X)$ is nonempty.", "Let $U \\subset G$ be an open, normal subgroup with finite index.", "Observe that", "$$", "G/U \\times G/U = \\coprod\\nolimits_{gU \\in G/U} G/U", "$$", "where the summand corresponding to $gU$ corresponds to the orbit of", "$(eU, gU)$ on the left hand side. Then we see that", "$$", "F(G/U) \\times F(G/U) = F(G/U \\times G/U) = \\coprod\\nolimits_{gU \\in G/U} F(G/U)", "$$", "Hence $|F(G/U)| = |G/U|$ as $F(G/U)$ is nonempty. Thus we see that", "$$", "\\lim_{U \\subset G\\text{ open, normal, finite idex}} F(G/U)", "$$", "is nonempty (Categories, Lemma \\ref{categories-lemma-nonempty-limit}).", "Pick $\\gamma = (\\gamma_U)$ an element in this limit.", "Denote $F_G$ the functor (\\ref{equation-forgetful}). We can identify", "$F_G$ with the functor", "$$", "X \\longmapsto \\colim_U \\Mor(G/U, X)", "$$", "where $f : G/U \\to X$ corresponds to $f(eU) \\in X = F_G(X)$", "(details omitted). Hence the element $\\gamma$ determines", "a well defined map", "$$", "t : F_G \\longrightarrow F", "$$", "Namely, given $x \\in X$ choose $U$ and $f : G/U \\to X$ sending", "$eU$ to $x$ and then set $t_X(x) = F(f)(\\gamma_U)$.", "We will show that $t$ induces a bijective map", "$t_{G/U} : F_G(G/U) \\to F(G/U)$ for any $U$.", "This implies in a straightforward manner that $t$", "is an isomorphism (details omitted).", "Since $|F_G(G/U)| = |F(G/U)|$ it suffices to show", "that $t_{G/U}$ is surjective. The image contains at least", "one element, namely", "$t_{G/U}(eU) = F(\\text{id}_{G/U})(\\gamma_U) = \\gamma_U$.", "For $g \\in G$ denote $R_g : G/U \\to G/U$ right multiplication.", "Then set of fixed points of $F(R_g) : F(G/U) \\to F(G/U)$", "is equal to $F(\\emptyset) = \\emptyset$ if $g \\not \\in U$ because $F$", "commutes with equalizers. It follows that if", "$g_1, \\ldots, g_{|G/U|}$ is a system of representatives", "for $G/U$, then the elements $F(R_{g_i})(\\gamma_U)$ are pairwise distinct", "and hence fill out $F(G/U)$. Then", "$$", "t_{G/U}(g_iU) = F(R_{g_i})(\\gamma_U)", "$$", "and the proof is complete." ], "refs": [ "categories-lemma-nonempty-limit" ], "ref_ids": [ 12237 ] } ], "ref_ids": [] }, { "id": 4028, "type": "theorem", "label": "pione-lemma-epi-mono", "categories": [ "pione" ], "title": "pione-lemma-epi-mono", "contents": [ "Let $(\\mathcal{C}, F)$ be a Galois category. Let", "$X \\to Y \\in \\text{Arrows}(\\mathcal{C})$. Then", "\\begin{enumerate}", "\\item $F$ is faithful,", "\\item $X \\to Y$ is a monomorphism", "$\\Leftrightarrow F(X) \\to F(Y)$ is injective,", "\\item $X \\to Y$ is an epimorphism", "$\\Leftrightarrow F(X) \\to F(Y)$ is surjective,", "\\item an object $A$ of $\\mathcal{C}$ is initial if and only if", "$F(A) = \\emptyset$,", "\\item an object $Z$ of $\\mathcal{C}$ is final if and only if", "$F(Z)$ is a singleton,", "\\item if $X$ and $Y$ are connected, then $X \\to Y$ is an epimorphism,", "\\item", "\\label{item-one-element}", "if $X$ is connected and $a, b : X \\to Y$ are two morphisms", "then $a = b$ as soon as $F(a)$ and $F(b)$ agree on one element of $F(X)$,", "\\item if $X = \\coprod_{i = 1, \\ldots, n} X_i$ and", "$Y = \\coprod_{j = 1, \\ldots, m} Y_j$ where $X_i$, $Y_j$ are connected,", "then there is map $\\alpha : \\{1, \\ldots, n\\} \\to \\{1, \\ldots, m\\}$", "such that $X \\to Y$ comes from a collection of morphisms", "$X_i \\to Y_{\\alpha(i)}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Suppose $a, b : X \\to Y$ with $F(a) = F(b)$.", "Let $E$ be the equalizer of $a$ and $b$. Then $F(E) = F(X)$", "and we see that $E = X$ because $F$ reflects isomorphisms.", "\\medskip\\noindent", "Proof of (2). This is true because $F$ turns the morphism $X \\to X \\times_Y X$", "into the map $F(X) \\to F(X) \\times_{F(Y)} F(X)$ and $F$ reflects isomorphisms.", "\\medskip\\noindent", "Proof of (3). This is true because $F$ turns the morphism $Y \\amalg_X Y \\to Y$", "into the map $F(Y) \\amalg_{F(X)} F(Y) \\to F(Y)$ and $F$ reflects isomorphisms.", "\\medskip\\noindent", "Proof of (4). There exists an initial object $A$ and certainly", "$F(A) = \\emptyset$. On the other hand, if $X$ is an object with", "$F(X) = \\emptyset$, then the unique map $A \\to X$ induces a bijection", "$F(A) \\to F(X)$ and hence $A \\to X$ is an isomorphism.", "\\medskip\\noindent", "Proof of (5). There exists a final object $Z$ and certainly", "$F(Z)$ is a singleton. On the other hand, if $X$ is an object with", "$F(X)$ a singleton, then the unique map $X \\to Z$ induces a bijection", "$F(X) \\to F(Z)$ and hence $X \\to Z$ is an isomorphism.", "\\medskip\\noindent", "Proof of (6). The equalizer $E$ of the two maps $Y \\to Y \\amalg_X Y$ is not", "an initial object of $\\mathcal{C}$ because $X \\to Y$ factors through $E$", "and $F(X) \\not = \\emptyset$. Hence $E = Y$ and we conclude.", "\\medskip\\noindent", "Proof of (\\ref{item-one-element}).", "The equalizer $E$ of $a$ and $b$ comes with a monomorphism", "$E \\to X$ and $F(E) \\subset F(X)$ is the set of elements where", "$F(a)$ and $F(b)$ agree. To finish use that either $E$ is initial", "or $E = X$.", "\\medskip\\noindent", "Proof of (8). For each $i, j$ we see that $E_{ij} = X_i \\times_Y Y_j$", "is either initial or equal to $X_i$. Picking $s \\in F(X_i)$", "we see that $E_{ij} = X_i$ if and only if $s$ maps to an element", "of $F(Y_j) \\subset F(Y)$, hence this happens for a unique $j = \\alpha(i)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4029, "type": "theorem", "label": "pione-lemma-galois", "categories": [ "pione" ], "title": "pione-lemma-galois", "contents": [ "Let $(\\mathcal{C}, F)$ be a Galois category. For any connected object $X$", "of $\\mathcal{C}$ there exists a Galois object $Y$ and a morphism $Y \\to X$." ], "refs": [], "proofs": [ { "contents": [ "We will use the results of Lemma \\ref{lemma-epi-mono} without further mention.", "Let $n = |F(X)|$. Consider $X^n$ endowed with its natural action of", "$S_n$. Let", "$$", "X^n = \\coprod\\nolimits_{t \\in T} Z_t", "$$", "be the decomposition into connected objects. Pick a $t$ such that", "$F(Z_t)$ contains $(s_1, \\ldots, s_n)$ with $s_i$ pairwise distinct.", "If $(s'_1, \\ldots, s'_n) \\in F(Z_t)$ is another element, then we", "claim $s'_i$ are pairwise distinct as well. Namely, if not, say", "$s'_i = s'_j$, then $Z_t$ is the image of an connected component of", "$X^{n - 1}$ under the diagonal morphism", "$$", "\\Delta_{ij} : X^{n - 1} \\longrightarrow X^n", "$$", "Since morphisms of connected objects are epimorphisms and induce", "surjections after applying $F$ it would follow that $s_i = s_j$", "which is not the case.", "\\medskip\\noindent", "Let $G \\subset S_n$ be the subgroup of elements with $g(Z_t) = Z_t$.", "Looking at the action of $S_n$ on", "$$", "F(X)^n = F(X^n) = \\coprod\\nolimits_{t' \\in T} F(Z_{t'})", "$$", "we see that $G = \\{g \\in S_n \\mid g(s_1, \\ldots, s_n) \\in F(Z_t)\\}$.", "Now pick a second element $(s'_1, \\ldots, s'_n) \\in F(Z_t)$.", "Above we have seen that $s'_i$ are pairwise distinct. Thus we can", "find a $g \\in S_n$ with $g(s_1, \\ldots, s_n) = (s'_1, \\ldots, s'_n)$.", "In other words, the action of $G$ on $F(Z_t)$ is transitive and", "the proof is complete." ], "refs": [ "pione-lemma-epi-mono" ], "ref_ids": [ 4028 ] } ], "ref_ids": [] }, { "id": 4030, "type": "theorem", "label": "pione-lemma-tame", "categories": [ "pione" ], "title": "pione-lemma-tame", "contents": [ "\\begin{reference}", "Compare with \\cite[Definition 7.2.4]{BS}.", "\\end{reference}", "Let $(\\mathcal{C}, F)$ be a Galois category. Let $G = \\text{Aut}(F)$", "be as in Example \\ref{example-from-C-F-to-G-sets}. For any connected", "$X$ in $\\mathcal{C}$ the action of $G$ on $F(X)$ is transitive." ], "refs": [], "proofs": [ { "contents": [ "We will use the results of Lemma \\ref{lemma-epi-mono} without further mention.", "Let $I$ be the set of isomorphism classes of Galois objects in $\\mathcal{C}$.", "For each $i \\in I$ let $X_i$ be a representative of the isomorphism class.", "Choose $\\gamma_i \\in F(X_i)$ for each $i \\in I$.", "We define a partial ordering on $I$ by setting $i \\geq i'$ if", "and only if there is a morphism $f_{ii'} : X_i \\to X_{i'}$.", "Given such a morphism we can post-compose by an automorphism", "$X_{i'} \\to X_{i'}$ to assure that $F(f_{ii'})(\\gamma_i) = \\gamma_{i'}$.", "With this normalization the morphism $f_{ii'}$ is unique.", "Observe that $I$ is a directed partially ordered set:", "(Categories, Definition \\ref{categories-definition-directed-set})", "if $i_1, i_2 \\in I$ there exists a Galois object $Y$ and a morphism", "$Y \\to X_{i_1} \\times X_{i_2}$ by Lemma \\ref{lemma-galois} applied", "to a connected component of $X_{i_1} \\times X_{i_2}$.", "Then $Y \\cong X_i$ for some $i \\in I$ and $i \\geq i_1$, $i \\geq I_2$.", "\\medskip\\noindent", "We claim that the functor $F$ is isomorphic to the functor $F'$", "which sends $X$ to", "$$", "F'(X) = \\colim_I \\Mor_\\mathcal{C}(X_i, X)", "$$", "via the transformation of functors $t : F' \\to F$ defined as follows:", "given $f : X_i \\to X$ we set $t_X(f) = F(f)(\\gamma_i)$.", "Using (\\ref{item-one-element}) we find that $t_X$ is injective.", "To show surjectivity, let $\\gamma \\in F(X)$. Then we can immediately", "reduce to the case where $X$ is connected by the definition of", "a Galois category. Then we may assume $X$ is Galois by", "Lemma \\ref{lemma-galois}. In this case $X$ is isomorphic to $X_i$", "for some $i$ and we can choose the isomorphism $X_i \\to X$ such", "that $\\gamma_i$ maps to $\\gamma$ (by definition of Galois objects).", "We conclude that $t$ is an isomorphism.", "\\medskip\\noindent", "Set $A_i = \\text{Aut}(X_i)$.", "We claim that for $i \\geq i'$ there is a canonical map", "$h_{ii'} : A_i \\to A_{i'}$ such that for all $a \\in A_i$", "the diagram", "$$", "\\xymatrix{", "X_i \\ar[d]_a \\ar[r]_{f_{ii'}} & X_{i'} \\ar[d]^{h_{ii'}(a)} \\\\", "X_i \\ar[r]^{f_{ii'}} & X_{i'}", "}", "$$", "commutes. Namely, just let $h_{ii'}(a) = a' : X_{i'} \\to X_{i'}$", "be the unique automorphism such that", "$F(a')(\\gamma_{i'}) = F(f_{ii'} \\circ a)(\\gamma_i)$.", "As before this makes the diagram commute and moreover the choice", "is unique.", "It follows that", "$h_{i'i''} \\circ h_{ii'} = h_{ii''}$", "if $i \\geq i' \\geq i''$.", "Since $F(X_i) \\to F(X_{i'})$ is surjective we see that", "$A_i \\to A_{i'}$ is surjective.", "Taking the inverse limit we obtain a group", "$$", "A = \\lim_I A_i", "$$", "This is a profinite group since the automorphism groups are finite.", "The map $A \\to A_i$ is surjective for all $i$ by", "Categories, Lemma \\ref{categories-lemma-nonempty-limit}.", "\\medskip\\noindent", "Since elements of $A$ act on the inverse system $X_i$ we get an action of", "$A$ (on the right) on $F'$ by pre-composing. In other words, we get", "a homomorphism $A^{opp} \\to G$. Since $A \\to A_i$ is surjective we conclude", "that $G$ acts transitively on $F(X_i)$ for all $i$. Since every connected", "object is dominated by one of the $X_i$ we conclude the lemma is true." ], "refs": [ "pione-lemma-epi-mono", "categories-definition-directed-set", "pione-lemma-galois", "pione-lemma-galois", "categories-lemma-nonempty-limit" ], "ref_ids": [ 4028, 12365, 4029, 4029, 12237 ] } ], "ref_ids": [] }, { "id": 4031, "type": "theorem", "label": "pione-lemma-functoriality-galois", "categories": [ "pione" ], "title": "pione-lemma-functoriality-galois", "contents": [ "Let $(\\mathcal{C}, F)$ and $(\\mathcal{C}', F')$ be Galois categories.", "Let $H : \\mathcal{C} \\to \\mathcal{C}'$ be an exact functor.", "There exists an isomorphism $t : F' \\circ H \\to F$.", "The choice of $t$ determines a continuous homomorphism", "$h : G' = \\text{Aut}(F') \\to \\text{Aut}(F) = G$ and", "a $2$-commutative diagram", "$$", "\\xymatrix{", "\\mathcal{C} \\ar[r]_H \\ar[d] & \\mathcal{C}' \\ar[d] \\\\", "\\textit{Finite-}G\\textit{-Sets} \\ar[r]^h &", "\\textit{Finite-}G'\\textit{-Sets}", "}", "$$", "The map $h$ is independent of $t$ up", "to an inner automorphism of $G$.", "Conversely, given a continuous homomorphism $h : G' \\to G$ there", "is an exact functor $H : \\mathcal{C} \\to \\mathcal{C}'$ and an", "isomorphism $t$ recovering $h$ as above." ], "refs": [], "proofs": [ { "contents": [ "By Proposition \\ref{proposition-galois} and", "Lemma \\ref{lemma-single-out-profinite} we may assume", "$\\mathcal{C} = \\textit{Finite-}G\\textit{-Sets}$ and $F$ is the", "forgetful functor and similarly for $\\mathcal{C}'$. Thus the existence of", "$t$ follows from Lemma \\ref{lemma-second-fundamental-functor}. The map $h$", "comes from transport of structure via $t$. The commutativity of the", "diagram is obvious. Uniqueness of $h$ up to inner conjugation by", "an element of $G$ comes from the fact that the choice of $t$ is", "unique up to an element of $G$. The final statement is straightforward." ], "refs": [ "pione-proposition-galois", "pione-lemma-single-out-profinite", "pione-lemma-second-fundamental-functor" ], "ref_ids": [ 4130, 4026, 4027 ] } ], "ref_ids": [] }, { "id": 4032, "type": "theorem", "label": "pione-lemma-functoriality-galois-surjective", "categories": [ "pione" ], "title": "pione-lemma-functoriality-galois-surjective", "contents": [ "In diagram (\\ref{equation-translation}) the following are equivalent", "\\begin{enumerate}", "\\item $h : G' \\to G$ is surjective,", "\\item $H : \\mathcal{C} \\to \\mathcal{C}'$ is fully faithful,", "\\item if $X \\in \\Ob(\\mathcal{C})$ is connected, then $H(X)$ is connected,", "\\item if $X \\in \\Ob(\\mathcal{C})$ is connected and there is", "a morphism $*' \\to H(X)$ in $\\mathcal{C}'$, then", "there is a morphism $* \\to X$, and", "\\item for any object $X$ of $\\mathcal{C}$ the map", "$\\Mor_\\mathcal{C}(*, X) \\to \\Mor_{\\mathcal{C}'}(*', H(X))$", "is bijective.", "\\end{enumerate}", "Here $*$ and $*'$ are final objects of $\\mathcal{C}$ and $\\mathcal{C}'$." ], "refs": [], "proofs": [ { "contents": [ "The implications (5) $\\Rightarrow$ (4) and (2) $\\Rightarrow$ (5) are clear.", "\\medskip\\noindent", "Assume (3). Let $X$ be a connected object of $\\mathcal{C}$ and let", "$*' \\to H(X)$ be a morphism. Since $H(X)$ is connected by (3)", "we see that $*' \\to H(X)$ is an isomorphism. Hence the $G'$-set", "corresponding to $H(X)$ has exactly one element, which means the", "$G$-set corresponding to $X$ has one element which means $X$ is", "isomorphic to the final object of $\\mathcal{C}$, in particular", "there is a map $* \\to X$. In this way we see that (3) $\\Rightarrow$ (4).", "\\medskip\\noindent", "If (1) is true, then the functor", "$\\textit{Finite-}G\\textit{-Sets} \\to \\textit{Finite-}G'\\textit{-Sets}$", "is fully faithful: in this case a map of $G$-sets commutes with the", "action of $G$ if and only if it commutes with the action of $G'$.", "Thus (1) $\\Rightarrow$ (2).", "\\medskip\\noindent", "If (1) is true, then for a $G$-set $X$ the $G$-orbits and $G'$-orbits", "agree. Thus (1) $\\Rightarrow$ (3).", "\\medskip\\noindent", "To finish the proof it suffices to show that (4) implies (1).", "If (1) is false, i.e., if $h$ is not surjective, then there is", "an open subgroup $U \\subset G$ containing $h(G')$ which is not", "equal to $G$. Then the finite $G$-set $M = G/U$ has a transitive", "action but $G'$ has a fixed point. The object $X$ of $\\mathcal{C}$", "corresponding to $M$ would contradict (3). In this way we see that", "(3) $\\Rightarrow$ (1) and the proof is complete." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4033, "type": "theorem", "label": "pione-lemma-composition-trivial", "categories": [ "pione" ], "title": "pione-lemma-composition-trivial", "contents": [ "In diagram (\\ref{equation-translation}) the following are equivalent", "\\begin{enumerate}", "\\item $h \\circ h'$ is trivial, and", "\\item the image of $H' \\circ H$ consists of objects isomorphic to finite", "coproducts of final objects.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may replace $H$ and $H'$ by the canonical functors", "$\\textit{Finite-}G\\textit{-Sets} \\to \\textit{Finite-}G'\\textit{-Sets}", "\\to \\textit{Finite-}G''\\textit{-Sets}$ determined by $h$ and $h'$.", "Then we are saying that the action of $G''$ on every $G$-set is trivial", "if and only if the homomorphism $G'' \\to G$ is trivial. This is clear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4034, "type": "theorem", "label": "pione-lemma-functoriality-galois-ses", "categories": [ "pione" ], "title": "pione-lemma-functoriality-galois-ses", "contents": [ "In diagram (\\ref{equation-translation}) the following are equivalent", "\\begin{enumerate}", "\\item the sequence $G'' \\xrightarrow{h'} G' \\xrightarrow{h} G \\to 1$", "is exact in the following sense: $h$ is surjective, $h \\circ h'$ is trivial,", "and $\\Ker(h)$ is the smallest closed normal subgroup containing $\\Im(h')$,", "\\item $H$ is fully faithful and an object $X'$ of $\\mathcal{C}'$ is in", "the essential image of $H$ if and only if $H'(X')$ is isomorphic to a", "finite coproduct of final objects, and", "\\item $H$ is fully faithful, $H \\circ H'$ sends every object to a finite", "coproduct of final objects, and for an object $X'$ of $\\mathcal{C}'$", "such that $H'(X')$ is a finite coproduct of final objects there exists", "an object $X$ of $\\mathcal{C}$ and an epimorphism $H(X) \\to X'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-functoriality-galois-surjective} and", "\\ref{lemma-composition-trivial} we may assume that", "$H$ is fully faithful, $h$ is surjective, $H' \\circ H$ maps", "objects to disjoint unions of the final object, and $h \\circ h'$", "is trivial. Let $N \\subset G'$ be the smallest closed normal", "subgroup containing the image of $h'$. It is clear that", "$N \\subset \\Ker(h)$.", "We may assume the functors $H$ and $H'$ are the canonical functors", "$\\textit{Finite-}G\\textit{-Sets} \\to \\textit{Finite-}G'\\textit{-Sets}", "\\to \\textit{Finite-}G''\\textit{-Sets}$ determined by $h$ and $h'$.", "\\medskip\\noindent", "Suppose that (2) holds. This means that for a finite $G'$-set $X'$", "such that $G''$ acts trivially, the action of $G'$ factors through $G$.", "Apply this to $X' = G'/U'N$ where $U'$ is a small open subgroup of $G'$.", "Then we see that $\\Ker(h) \\subset U'N$ for all $U'$. Since $N$ is closed", "this implies $\\Ker(h) \\subset N$, i.e., (1) holds.", "\\medskip\\noindent", "Suppose that (1) holds. This means that $N = \\Ker(h)$. Let $X'$ be a", "finite $G'$-set such that $G''$ acts trivially. This means that", "$\\Ker(G' \\to \\text{Aut}(X'))$ is a closed normal subgroup containing", "$\\Im(h')$. Hence $N = \\Ker(h)$ is contained in it and the $G'$-action", "on $X'$ factors through $G$, i.e., (2) holds.", "\\medskip\\noindent", "Suppose that (3) holds. This means that for a finite $G'$-set $X'$", "such that $G''$ acts trivially, there is a surjection of $G'$-sets", "$X \\to X'$ where $X$ is a $G$-set. Clearly this means the action of", "$G'$ on $X'$ factors through $G$, i.e., (2) holds.", "\\medskip\\noindent", "The implication (2) $\\Rightarrow$ (3) is immediate. This finishes the proof." ], "refs": [ "pione-lemma-functoriality-galois-surjective", "pione-lemma-composition-trivial" ], "ref_ids": [ 4032, 4033 ] } ], "ref_ids": [] }, { "id": 4035, "type": "theorem", "label": "pione-lemma-functoriality-galois-injective", "categories": [ "pione" ], "title": "pione-lemma-functoriality-galois-injective", "contents": [ "In diagram (\\ref{equation-translation}) the following are equivalent", "\\begin{enumerate}", "\\item $h'$ is injective, and", "\\item for every connected object $X''$ of $\\mathcal{C}''$", "there exists an object $X'$ of $\\mathcal{C}'$ and a diagram", "$$", "X'' \\leftarrow Y'' \\rightarrow H(X')", "$$", "in $\\mathcal{C}''$ where $Y'' \\to X''$ is an epimorphism and", "$Y'' \\to H(X')$ is a monomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may replace $H'$ by the corresponding functor between the categories", "of finite $G'$-sets and finite $G''$-sets.", "\\medskip\\noindent", "Assume $h' : G'' \\to G'$ is injective. Let $H'' \\subset G''$", "be an open subgroup. Since the topology on $G''$ is the induced", "topology from $G'$ there exists an open subgroup $H' \\subset G'$", "such that $(h')^{-1}(H') \\subset H''$.", "Then the desired diagram is", "$$", "G''/H'' \\leftarrow G''/(h')^{-1}(H') \\rightarrow G'/H'", "$$", "Conversely, assume (2) holds for the functor", "$\\textit{Finite-}G'\\textit{-Sets} \\to \\textit{Finite-}G''\\textit{-Sets}$.", "Let $g'' \\in \\Ker(h')$. Pick any open subgroup $H'' \\subset G''$.", "By assumption there exists a finite $G'$-set $X'$ and a diagram", "$$", "G''/H'' \\leftarrow Y'' \\rightarrow X'", "$$", "of $G''$-sets with the left arrow surjective and the right arrow injective.", "Since $g''$ is in the kernel of $h'$ we see that $g''$ acts trivially on $X'$.", "Hence $g''$ acts trivially on $Y''$ and hence trivially on $G''/H''$.", "Thus $g'' \\in H''$. As this holds for all open subgroups we conclude", "that $g''$ is the identity element as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4036, "type": "theorem", "label": "pione-lemma-functoriality-galois-normal", "categories": [ "pione" ], "title": "pione-lemma-functoriality-galois-normal", "contents": [ "In diagram (\\ref{equation-translation}) the following are equivalent", "\\begin{enumerate}", "\\item the image of $h'$ is normal, and", "\\item for every connected object $X'$ of $\\mathcal{C}'$ such that", "there is a morphism from the final object of $\\mathcal{C}''$", "to $H'(X')$ we have that $H'(X')$ is isomorphic to a finite coproduct", "of final objects.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This translates into the following statement for the continuous", "group homomorphism $h' : G'' \\to G'$: the image of $h'$ is normal", "if and only if every open subgroup $U' \\subset G'$ which", "contains $h'(G'')$ also contains every conjugate of $h'(G'')$.", "The result follows easily from this; some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4037, "type": "theorem", "label": "pione-lemma-finite-etale-covers-limits-colimits", "categories": [ "pione" ], "title": "pione-lemma-finite-etale-covers-limits-colimits", "contents": [ "Let $X$ be a scheme. The category $\\textit{F\\'Et}_X$ has finite limits and", "finite colimits and for any morphism $X' \\to X$ the base change functor", "$\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_{X'}$ is exact." ], "refs": [], "proofs": [ { "contents": [ "Finite limits and left exactness. By", "Categories, Lemma \\ref{categories-lemma-finite-limits-exist}", "it suffices to show that $\\textit{F\\'Et}_X$ has a final object", "and fibred products. This is clear because the category of", "all schemes over $X$ has a final object (namely $X$) and fibred products", "and fibred products of schemes finite \\'etale over $X$ are", "finite \\'etale over $X$. Moreover, it is clear that base", "change commutes with these operations and hence base change", "is left exact (Categories, Lemma", "\\ref{categories-lemma-characterize-left-exact}).", "\\medskip\\noindent", "Finite colimits and right exactness. By", "Categories, Lemma \\ref{categories-lemma-colimits-exist}", "it suffices to show that $\\textit{F\\'Et}_X$ has finite", "coproducts and coequalizers. Finite coproducts are given", "by disjoint unions (the empty coproduct is the empty scheme).", "Let $a, b : Z \\to Y$ be two morphisms of $\\textit{F\\'Et}_X$.", "Since $Z \\to X$ and $Y \\to X$ are finite \\'etale we can write", "$Z = \\underline{\\Spec}(\\mathcal{C})$ and $Y = \\underline{\\Spec}(\\mathcal{B})$", "for some finite locally free $\\mathcal{O}_X$-algebras $\\mathcal{C}$", "and $\\mathcal{B}$. The morphisms $a, b$ induce two maps", "$a^\\sharp, b^\\sharp : \\mathcal{B} \\to \\mathcal{C}$.", "Let $\\mathcal{A} = \\text{Eq}(a^\\sharp, b^\\sharp)$ be their", "equalizer. If", "$$", "\\underline{\\Spec}(\\mathcal{A}) \\longrightarrow X", "$$", "is finite \\'etale, then it is clear that this is the coequalizer", "(after all we can write any object of $\\textit{F\\'Et}_X$", "as the relative spectrum of a sheaf of $\\mathcal{O}_X$-algebras).", "This we may do after replacing $X$ by the members of an \\'etale", "covering (Descent, Lemmas \\ref{descent-lemma-descending-property-finite}", "and \\ref{descent-lemma-descending-property-separated}).", "Thus by \\'Etale Morphisms, Lemma \\ref{etale-lemma-finite-etale-etale-local}", "we may assume that", "$Y = \\coprod_{i = 1, \\ldots, n} X$ and $Z = \\coprod_{j = 1, \\ldots, m} X$.", "Then", "$$", "\\mathcal{C} = \\prod\\nolimits_{1 \\leq j \\leq m} \\mathcal{O}_X", "\\quad\\text{and}\\quad", "\\mathcal{B} = \\prod\\nolimits_{1 \\leq i \\leq n} \\mathcal{O}_X", "$$", "After a further replacement by the members of an open covering", "we may assume that $a, b$ correspond to", "maps $a_s, b_s : \\{1, \\ldots, m\\} \\to \\{1, \\ldots, n\\}$, i.e.,", "the summand $X$ of $Z$ corresponding to the index $j$ maps into", "the summand $X$ of $Y$ corresponding to the index $a_s(j)$, resp.\\ $b_s(j)$", "under the morphism $a$, resp.\\ $b$.", "Let $\\{1, \\ldots, n\\} \\to T$ be the coequalizer of $a_s, b_s$.", "Then we see that", "$$", "\\mathcal{A} = \\prod\\nolimits_{t \\in T} \\mathcal{O}_X", "$$", "whose spectrum is certainly finite \\'etale over $X$. We", "omit the verification that this is compatible with base change.", "Thus base change is a right exact functor." ], "refs": [ "categories-lemma-finite-limits-exist", "categories-lemma-characterize-left-exact", "categories-lemma-colimits-exist", "descent-lemma-descending-property-finite", "descent-lemma-descending-property-separated", "etale-lemma-finite-etale-etale-local" ], "ref_ids": [ 12224, 12245, 12227, 14688, 14671, 10714 ] } ], "ref_ids": [] }, { "id": 4038, "type": "theorem", "label": "pione-lemma-internal-hom-finite-etale", "categories": [ "pione" ], "title": "pione-lemma-internal-hom-finite-etale", "contents": [ "Let $X$ be a scheme. Given $U, V$ finite \\'etale over $X$ there", "exists a scheme $W$ finite \\'etale over $X$ such that", "$$", "\\Mor_X(X, W) = \\Mor_X(U, V)", "$$", "and such that the same remains true after any base change." ], "refs": [], "proofs": [ { "contents": [ "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-hom-from-finite-locally-free-separated-lqf}", "there exists a scheme $W$ representing $\\mathit{Mor}_X(U, V)$.", "(Use that an \\'etale morphism is locally quasi-finite by", "Morphisms, Lemmas \\ref{morphisms-lemma-etale-locally-quasi-finite}", "and that a finite morphism is separated.)", "This scheme clearly satisfies the formula after any base change.", "To finish the proof we have to show that $W \\to X$ is finite \\'etale.", "This we may do after replacing $X$ by the members of an \\'etale", "covering (Descent, Lemmas \\ref{descent-lemma-descending-property-finite}", "and \\ref{descent-lemma-descending-property-separated}).", "Thus by \\'Etale Morphisms, Lemma \\ref{etale-lemma-finite-etale-etale-local}", "we may assume that $U = \\coprod_{i = 1, \\ldots, n} X$", "and $V = \\coprod_{j = 1, \\ldots, m} X$.", "In this case", "$W = \\coprod_{\\alpha : \\{1, \\ldots, n\\} \\to \\{1, \\ldots, m\\}} X$", "by inspection (details omitted) and the proof is complete." ], "refs": [ "more-morphisms-lemma-hom-from-finite-locally-free-separated-lqf", "morphisms-lemma-etale-locally-quasi-finite", "descent-lemma-descending-property-finite", "descent-lemma-descending-property-separated", "etale-lemma-finite-etale-etale-local" ], "ref_ids": [ 14056, 5363, 14688, 14671, 10714 ] } ], "ref_ids": [] }, { "id": 4039, "type": "theorem", "label": "pione-lemma-finite-etale-connected-galois-category", "categories": [ "pione" ], "title": "pione-lemma-finite-etale-connected-galois-category", "contents": [ "Let $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point.", "The functor", "$$", "F_{\\overline{x}} : \\textit{F\\'Et}_X \\longrightarrow \\textit{Sets},\\quad", "Y \\longmapsto |Y_{\\overline{x}}|", "$$", "defines a Galois category (Definition \\ref{definition-galois-category})." ], "refs": [ "pione-definition-galois-category" ], "proofs": [ { "contents": [ "After identifying $\\textit{F\\'Et}_{\\overline{x}}$ with the category of", "finite sets (Example \\ref{example-finite-etale-geometric-point})", "we see that our functor $F_{\\overline{x}}$", "is nothing but the base change functor for the morphism $\\overline{x} \\to X$.", "Thus we see that $\\textit{F\\'Et}_X$ has finite limits and finite colimits", "and that $F_{\\overline{x}}$ is exact by", "Lemma \\ref{lemma-finite-etale-covers-limits-colimits}.", "We will also use that finite limits in $\\textit{F\\'Et}_X$", "agree with the corresponding finite limits in the category", "of schemes over $X$, see Remark \\ref{remark-colimits-commute-forgetful}.", "\\medskip\\noindent", "If $Y' \\to Y$ is a monomorphism in $\\textit{F\\'Et}_X$", "then we see that $Y' \\to Y' \\times_Y Y'$ is an isomorphism, and", "hence $Y' \\to Y$ is a monomorphism of schemes. It follows that", "$Y' \\to Y$ is an open immersion", "(\\'Etale Morphisms, Theorem \\ref{etale-theorem-etale-radicial-open}). Since", "$Y'$ is finite over $X$ and $Y$ separated over $X$,", "the morphism $Y' \\to Y$ is finite", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-permanence}), hence closed", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}),", "hence it is the inclusion of an open and closed subscheme of $Y$.", "It follows that $Y$ is a connected objects of the category", "$\\textit{F\\'Et}_X$ (as in Definition \\ref{definition-galois-category})", "if and only if $Y$ is connected as a scheme. Then it follows from", "Topology, Lemma \\ref{topology-lemma-finite-fibre-connected-components}", "that $Y$ is a finite coproduct of its connected components", "both as a scheme and in the sense of", "Definition \\ref{definition-galois-category}.", "\\medskip\\noindent", "Let $Y \\to Z$ be a morphism in $\\textit{F\\'Et}_X$ which induces a", "bijection $F_{\\overline{x}}(Y) \\to F_{\\overline{x}}(Z)$. We have to", "show that $Y \\to Z$ is an isomorphism. By the above we may assume", "$Z$ is connected. Since $Y \\to Z$ is finite \\'etale and hence finite", "locally free it suffices to show that $Y \\to Z$ is finite locally", "free of degree $1$. This is true in a neighbourhood of any point of", "$Z$ lying over $\\overline{x}$ and since $Z$ is connected and", "the degree is locally constant we conclude." ], "refs": [ "pione-lemma-finite-etale-covers-limits-colimits", "pione-remark-colimits-commute-forgetful", "etale-theorem-etale-radicial-open", "morphisms-lemma-finite-permanence", "morphisms-lemma-finite-proper", "pione-definition-galois-category", "topology-lemma-finite-fibre-connected-components", "pione-definition-galois-category" ], "ref_ids": [ 4037, 4145, 10692, 5448, 5445, 4142, 8209, 4142 ] } ], "ref_ids": [ 4142 ] }, { "id": 4040, "type": "theorem", "label": "pione-lemma-fundamental-group-Galois-group", "categories": [ "pione" ], "title": "pione-lemma-fundamental-group-Galois-group", "contents": [ "Let $K$ be a field and set $X = \\Spec(K)$. Let $\\overline{K}$ be an", "algebraic closure and denote $\\overline{x} : \\Spec(\\overline{K}) \\to X$", "the corresponding geometric point. Let $K^{sep} \\subset \\overline{K}$", "be the separable algebraic closure.", "\\begin{enumerate}", "\\item The functor of Lemma \\ref{lemma-sheaves-point} induces an equivalence", "$$", "\\textit{F\\'Et}_X \\longrightarrow", "\\textit{Finite-}\\text{Gal}(K^{sep}/K)\\textit{-Sets}.", "$$", "compatible with $F_{\\overline{x}}$ and the functor", "$\\textit{Finite-}\\text{Gal}(K^{sep}/K)\\textit{-Sets} \\to \\textit{Sets}$.", "\\item This induces a canonical isomorphism", "$$", "\\text{Gal}(K^{sep}/K) \\longrightarrow \\pi_1(X, \\overline{x})", "$$", "of profinite topological groups.", "\\end{enumerate}" ], "refs": [ "pione-lemma-sheaves-point" ], "proofs": [ { "contents": [ "The functor of Lemma \\ref{lemma-sheaves-point} is the same as the functor", "$F_{\\overline{x}}$ because for any $Y$ \\'etale over $X$ we have", "$$", "\\Mor_X(\\Spec(\\overline{K}), Y) = \\Mor_X(\\Spec(K^{sep}), Y)", "$$", "Namely, as seen in the proof of Lemma \\ref{lemma-sheaves-point} we have", "$Y = \\coprod_{i \\in I} \\Spec(L_i)$ with $L_i/K$ finite separable over $K$.", "Hence any $K$-algebra homomorphism $L_i \\to \\overline{K}$ factors", "through $K^{sep}$. Also, note that $F_{\\overline{x}}(Y)$ is finite", "if and only if $I$ is finite if and only if $Y \\to X$ is finite \\'etale.", "This proves (1).", "\\medskip\\noindent", "Part (2) is a formal consequence of (1),", "Lemma \\ref{lemma-functoriality-galois}, and", "Lemma \\ref{lemma-single-out-profinite}.", "(Please also see the remark below.)" ], "refs": [ "pione-lemma-sheaves-point", "pione-lemma-sheaves-point", "pione-lemma-functoriality-galois", "pione-lemma-single-out-profinite" ], "ref_ids": [ 4024, 4024, 4031, 4026 ] } ], "ref_ids": [ 4024 ] }, { "id": 4041, "type": "theorem", "label": "pione-lemma-what-equivalence-gives", "categories": [ "pione" ], "title": "pione-lemma-what-equivalence-gives", "contents": [ "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated schemes", "such that the base change functor $\\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_X$", "is an equivalence of categories. In this case", "\\begin{enumerate}", "\\item $f$ induces a homeomorphism $\\pi_0(X) \\to \\pi_0(Y)$,", "\\item if $X$ or equivalently $Y$ is connected, then", "$\\pi_1(X, \\overline{x}) = \\pi_1(Y, \\overline{y})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $Y = Y_0 \\amalg Y_1$ be a decomposition into nonempty open and closed", "subschemes. We claim that $f(X)$ meets both $Y_i$. Namely, if not,", "say $f(X) \\subset Y_1$, then we can consider the finite \\'etale", "morphism $V = Y_1 \\to Y$. This is not an", "isomorphism but $V \\times_Y X \\to X$ is an isomorphism, which is", "a contradiction.", "\\medskip\\noindent", "Suppose that $X = X_0 \\amalg X_1$ is a decomposition into open and closed", "subschemes. Consider the finite \\'etale morphism $U = X_1 \\to X$. Then", "$U = X \\times_Y V$ for some finite \\'etale morphism $V \\to Y$. The degree", "of the morphism $V \\to Y$ is locally constant, hence we obtain a decomposition", "$Y = \\coprod_{d \\geq 0} Y_d$ into open and closed subschemes", "such that $V \\to Y$ has degree $d$ over $Y_d$. Since", "$f^{-1}(Y_d) = \\emptyset$ for $d > 1$ we conclude that $Y_d = \\emptyset$", "for $d > 1$ by the above. And we conclude that $f^{-1}(Y_i) = X_i$", "for $i = 0, 1$.", "\\medskip\\noindent", "It follows that $f^{-1}$ induces a bijection between the set of", "open and closed subsets of $Y$ and the set of open and closed subsets of $X$.", "Note that $X$ and $Y$ are spectral spaces, see Properties, Lemma", "\\ref{properties-lemma-quasi-compact-quasi-separated-spectral}.", "By Topology, Lemma \\ref{topology-lemma-connected-component-intersection}", "the lattice of open and closed subsets of a spectral space", "determines the set of connected components.", "Hence $\\pi_0(X) \\to \\pi_0(Y)$ is bijective. Since $\\pi_0(X)$ and", "$\\pi_0(Y)$ are profinite spaces", "(Topology, Lemma \\ref{topology-lemma-pi0-profinite})", "we conclude that $\\pi_0(X) \\to \\pi_0(Y)$ is a homeomorphism by", "Topology, Lemma \\ref{topology-lemma-bijective-map}. This proves (1).", "Part (2) is immediate." ], "refs": [ "properties-lemma-quasi-compact-quasi-separated-spectral", "topology-lemma-connected-component-intersection", "topology-lemma-pi0-profinite", "topology-lemma-bijective-map" ], "ref_ids": [ 2941, 8236, 8302, 8275 ] } ], "ref_ids": [] }, { "id": 4042, "type": "theorem", "label": "pione-lemma-gabber", "categories": [ "pione" ], "title": "pione-lemma-gabber", "contents": [ "Let $(A, I)$ be a henselian pair. Set $X = \\Spec(A)$ and $Z = \\Spec(A/I)$.", "The functor", "$$", "\\textit{F\\'Et}_X \\longrightarrow \\textit{F\\'Et}_Z,\\quad", "U \\longmapsto U \\times_X Z", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "This is a translation of", "More on Algebra, Lemma \\ref{more-algebra-lemma-finite-etale-equivalence}." ], "refs": [ "more-algebra-lemma-finite-etale-equivalence" ], "ref_ids": [ 9879 ] } ], "ref_ids": [] }, { "id": 4043, "type": "theorem", "label": "pione-lemma-thickening", "categories": [ "pione" ], "title": "pione-lemma-thickening", "contents": [ "Let $X \\subset X'$ be a thickening of schemes. The functor", "$$", "\\textit{F\\'Et}_{X'} \\longrightarrow \\textit{F\\'Et}_X,\\quad", "U' \\longmapsto U' \\times_{X'} X", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "For a discussion of thickenings see", "More on Morphisms, Section \\ref{more-morphisms-section-thickenings}.", "Let $U' \\to X'$ be an \\'etale morphism such that $U = U' \\times_{X'} X \\to X$", "is finite \\'etale. Then $U' \\to X'$ is finite \\'etale as well.", "This follows for example from More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-properties-that-extend-over-thickenings}.", "Now, if $X \\subset X'$ is a finite order thickening then this remark", "combined with \\'Etale Morphisms, Theorem", "\\ref{etale-theorem-remarkable-equivalence}", "proves the lemma. Below we will prove the lemma for general thickenings, but", "we suggest the reader skip the proof.", "\\medskip\\noindent", "Let $X' = \\bigcup X_i'$ be an affine open covering. Set", "$X_i = X \\times_{X'} X_i'$, $X_{ij}' = X'_i \\cap X'_j$,", "$X_{ij} = X \\times_{X'} X_{ij}'$, $X_{ijk}' = X'_i \\cap X'_j \\cap X'_k$,", "$X_{ijk} = X \\times_{X'} X_{ijk}'$.", "Suppose that we can prove", "the theorem for each of the thickenings", "$X_i \\subset X'_i$, $X_{ij} \\subset X_{ij}'$, and $X_{ijk} \\subset X_{ijk}'$.", "Then the result follows for $X \\subset X'$ by relative glueing of", "schemes, see", "Constructions, Section \\ref{constructions-section-relative-glueing}.", "Observe that the schemes $X_i'$, $X_{ij}'$, $X_{ijk}'$ are", "each separated as open subschemes of affine schemes. Repeating the", "argument one more time we reduce to the case where the schemes", "$X'_i$, $X_{ij}'$, $X_{ijk}'$ are affine.", "\\medskip\\noindent", "In the affine case we have $X' = \\Spec(A')$ and $X = \\Spec(A'/I')$", "where $I'$ is a locally nilpotent ideal. Then $(A', I')$ is a", "henselian pair (More on Algebra, Lemma", "\\ref{more-algebra-lemma-locally-nilpotent-henselian})", "and the result follows from Lemma \\ref{lemma-gabber} (which is", "much easier in this case)." ], "refs": [ "more-morphisms-lemma-properties-that-extend-over-thickenings", "etale-theorem-remarkable-equivalence", "more-algebra-lemma-locally-nilpotent-henselian", "pione-lemma-gabber" ], "ref_ids": [ 13685, 10694, 9857, 4042 ] } ], "ref_ids": [] }, { "id": 4044, "type": "theorem", "label": "pione-lemma-finite-etale-on-proper-over-henselian", "categories": [ "pione" ], "title": "pione-lemma-finite-etale-on-proper-over-henselian", "contents": [ "Let $A$ be a henselian local ring. Let $X$ be a proper scheme over $A$", "with closed fibre $X_0$. Then the functor", "$$", "\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_{X_0},\\quad", "U \\longmapsto U_0 = U \\times_X X_0", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "The proof given here is an example of applying algebraization and", "approximation. We proceed in a number of stages.", "\\medskip\\noindent", "Essential surjectivity when $A$ is a complete local Noetherian ring.", "Let $X_n = X \\times_{\\Spec(A)} \\Spec(A/\\mathfrak m^{n + 1})$.", "By \\'Etale Morphisms, Theorem \\ref{etale-theorem-remarkable-equivalence}", "the inclusions", "$$", "X_0 \\to X_1 \\to X_2 \\to \\ldots", "$$", "induce equivalence of categories between the category", "of schemes \\'etale over $X_0$ and the category of schemes", "\\'etale over $X_n$.", "Moreover, if $U_n \\to X_n$ corresponds to a finite \\'etale", "morphism $U_0 \\to X_0$, then $U_n \\to X_n$ is finite too, for example", "by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-thicken-property-morphisms-cartesian}.", "In this case the morphism $U_0 \\to \\Spec(A/\\mathfrak m)$", "is proper as $X_0$ is proper over $A/\\mathfrak m$. Thus we may apply", "Grothendieck's algebraization theorem", "(in the form of", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-algebraize-formal-scheme-finite-over-proper})", "to see that there is a finite morphism $U \\to X$ whose restriction", "to $X_0$ recovers $U_0$. By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds}", "we see that $U \\to X$ is \\'etale at every point of $U_0$.", "However, since every point of $U$ specializes to a point of $U_0$", "(as $U$ is proper over $A$), we conclude that $U \\to X$ is \\'etale.", "In this way we conclude the functor is essentially surjective.", "\\medskip\\noindent", "Fully faithfulness when $A$ is a complete local Noetherian ring.", "Let $U \\to X$ and $V \\to X$ be finite \\'etale morphisms and", "let $\\varphi_0 : U_0 \\to V_0$ be a morphism over $X_0$. Look at", "the morphism", "$$", "\\Gamma_{\\varphi_0} : U_0 \\longrightarrow U_0 \\times_{X_0} V_0", "$$", "This morphism is both finite \\'etale and a closed immersion.", "By essential surjectivity applied to $X = U \\times_X V$ we find", "a finite \\'etale morphism $W \\to U \\times_X V$ whose special", "fibre is isomorphic to $\\Gamma_{\\varphi_0}$. Consider the projection", "$W \\to U$. It is finite \\'etale and an isomorphism over $U_0$ by", "construction. By \\'Etale Morphisms, Lemma", "\\ref{etale-lemma-finite-etale-one-point}", "$W \\to U$ is an isomorphism in an open neighbourhood of $U_0$.", "Thus it is an isomorphism and the composition $\\varphi : U \\cong W \\to V$", "is the desired lift of $\\varphi_0$.", "\\medskip\\noindent", "Essential surjectivity when $A$ is a henselian local Noetherian G-ring.", "Let $U_0 \\to X_0$ be a finite \\'etale morphism.", "Let $A^\\wedge$ be the completion of $A$ with respect to the maximal ideal.", "Let $X^\\wedge$ be the base change of $X$ to $A^\\wedge$.", "By the result above there exists a finite \\'etale morphism", "$V \\to X^\\wedge$ whose special fibre is $U_0$.", "Write $A^\\wedge = \\colim A_i$ with $A \\to A_i$ of finite type.", "By Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "there exists an $i$ and a finitely presented morphism $U_i \\to X_{A_i}$", "whose base change to $X^\\wedge$ is $V$. After increasing $i$", "we may assume that $U_i \\to X_{A_i}$ is finite and \\'etale", "(Limits, Lemmas \\ref{limits-lemma-descend-finite-finite-presentation} and", "\\ref{limits-lemma-descend-etale}). Writing", "$$", "A_i = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)", "$$", "the ring map $A_i \\to A^\\wedge$ can be reinterpreted as a solution", "$(a_1, \\ldots, a_n)$ in $A^\\wedge$ for the system of equations $f_j = 0$.", "By Smoothing Ring Maps, Theorem \\ref{smoothing-theorem-approximation-property}", "we can approximate this solution (to order $11$ for example) by a solution", "$(b_1, \\ldots, b_n)$ in $A$. Translating back we find an $A$-algebra map", "$A_i \\to A$ which gives the same closed point as the original map", "$A_i \\to A^\\wedge$ (as $11 > 1$). The base change $U \\to X$ of $V \\to X_{A_i}$", "by this ring map will therefore be a finite \\'etale morphism whose", "special fibre is isomorphic to $U_0$.", "\\medskip\\noindent", "Fully faithfulness when $A$ is a henselian local Noetherian G-ring.", "This can be deduced from essential surjectivity in exactly the same", "manner as was done in the case that $A$ is complete Noetherian.", "\\medskip\\noindent", "General case. Let $(A, \\mathfrak m)$ be a henselian local ring.", "Set $S = \\Spec(A)$ and denote $s \\in S$ the closed point. By Limits, Lemma", "\\ref{limits-lemma-proper-limit-of-proper-finite-presentation-noetherian}", "we can write $X \\to \\Spec(A)$ as a cofiltered limit of", "proper morphisms $X_i \\to S_i$ with $S_i$ of finite type over $\\mathbf{Z}$.", "For each $i$ let $s_i \\in S_i$ be the image of $s$.", "Since $S = \\lim S_i$ and $A = \\mathcal{O}_{S, s}$ we have", "$A = \\colim \\mathcal{O}_{S_i, s_i}$. The ring $A_i = \\mathcal{O}_{S_i, s_i}$", "is a Noetherian local G-ring (More on Algebra, Proposition", "\\ref{more-algebra-proposition-ubiquity-G-ring}).", "By More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-colimit}", "we see that $A = \\colim A_i^h$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-G-ring}", "the rings $A_i^h$ are G-rings. Thus we see that $A = \\colim A_i^h$ and", "$$", "X = \\lim (X_i \\times_{S_i} \\Spec(A_i^h))", "$$", "as schemes. The category of schemes finite \\'etale over $X$ is the limit", "of the category of schemes finite \\'etale over", "$X_i \\times_{S_i} \\Spec(A_i^h)$ (by", "Limits, Lemmas", "\\ref{limits-lemma-descend-finite-presentation},", "\\ref{limits-lemma-descend-finite-finite-presentation}, and", "\\ref{limits-lemma-descend-etale})", "The same thing is true for schemes finite \\'etale over", "$X_0 = \\lim (X_i \\times_{S_i} s_i)$.", "Thus we formally deduce the result for $X / \\Spec(A)$", "from the result for the $(X_i \\times_{S_i} \\Spec(A_i^h)) / \\Spec(A_i^h)$", "which we dealt with above." ], "refs": [ "etale-theorem-remarkable-equivalence", "more-morphisms-lemma-thicken-property-morphisms-cartesian", "coherent-lemma-algebraize-formal-scheme-finite-over-proper", "more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds", "etale-lemma-finite-etale-one-point", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-finite-finite-presentation", "limits-lemma-descend-etale", "smoothing-theorem-approximation-property", "limits-lemma-proper-limit-of-proper-finite-presentation-noetherian", "more-algebra-proposition-ubiquity-G-ring", "more-algebra-lemma-henselization-colimit", "more-algebra-lemma-henselization-G-ring", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-finite-finite-presentation", "limits-lemma-descend-etale" ], "ref_ids": [ 10694, 13684, 3397, 13743, 10707, 15077, 15058, 15065, 5606, 15091, 10582, 9875, 10089, 15077, 15058, 15065 ] } ], "ref_ids": [] }, { "id": 4045, "type": "theorem", "label": "pione-lemma-finite-etale-invariant-over-proper", "categories": [ "pione" ], "title": "pione-lemma-finite-etale-invariant-over-proper", "contents": [ "Let $k \\subset k'$ be an extension of algebraically closed fields.", "Let $X$ be a proper scheme over $k$. Then the functor", "$$", "U \\longmapsto U_{k'}", "$$", "is an equivalence of categories between schemes finite \\'etale over", "$X$ and schemes finite \\'etale over $X_{k'}$." ], "refs": [], "proofs": [ { "contents": [ "Let us prove the functor is essentially surjective.", "Let $U' \\to X_{k'}$ be a finite \\'etale morphism.", "Write $k' = \\colim A_i$ as a filtered colimit of finite type $k$-algebras.", "By Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "there exists an $i$ and a finitely presented morphism $U_i \\to X_{A_i}$", "whose base change to $X_{k'}$ is $U'$. After increasing $i$", "we may assume that $U_i \\to X_{A_i}$ is finite and \\'etale", "(Limits, Lemmas \\ref{limits-lemma-descend-finite-finite-presentation} and", "\\ref{limits-lemma-descend-etale}).", "Since $k$ is algebraically closed we can find a", "$k$-valued point $t$ in $\\Spec(A_i)$. Let $U = (U_i)_t$ be the", "fibre of $U_i$ over $t$. Let $A_i^h$ be the", "henselization of $(A_i)_{\\mathfrak m}$ where $\\mathfrak m$ is", "the maximal ideal corresponding to the point $t$. By", "Lemma \\ref{lemma-finite-etale-on-proper-over-henselian}", "we see that $(U_i)_{A_i^h} = U \\times \\Spec(A_i^h)$ as schemes", "over $X_{A_i^h}$. Now since", "$A_i^h$ is algebraic over $A_i$ (see for example discussion in", "Smoothing Ring Maps, Example \\ref{smoothing-example-describe-henselian})", "and since $k'$ is algebraically closed", "we can find a ring map $A_i^h \\to k'$ extending the given", "inclusion $A_i \\subset k'$. Hence we conclude that $U'$", "is isomorphic to the base change of $U$.", "The proof of fully faithfulness is exactly the same." ], "refs": [ "limits-lemma-descend-finite-presentation", "limits-lemma-descend-finite-finite-presentation", "limits-lemma-descend-etale", "pione-lemma-finite-etale-on-proper-over-henselian" ], "ref_ids": [ 15077, 15058, 15065, 4044 ] } ], "ref_ids": [] }, { "id": 4046, "type": "theorem", "label": "pione-lemma-dense-faithful", "categories": [ "pione" ], "title": "pione-lemma-dense-faithful", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. If $f(X)$ is dense in $Y$", "then the base change functor $\\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_X$", "is faithful." ], "refs": [], "proofs": [ { "contents": [ "Since the category of finite \\'etale coverings has an", "internal hom (Lemma \\ref{lemma-internal-hom-finite-etale})", "it suffices to prove the following: Given $W$ finite \\'etale over $Y$", "and a morphism $s : X \\to W$ over $X$ there is at most one section", "$t : Y \\to W$ such that $s = t \\circ f$. Consider two sections", "$t_1, t_2 : Y \\to W$ such that $s = t_1 \\circ f = t_2 \\circ f$.", "Since the equalizer of $t_1$ and $t_2$ is closed in $Y$", "(Schemes, Lemma \\ref{schemes-lemma-where-are-they-equal})", "and since $f(X)$ is dense in $Y$ we see that $t_1$ and $t_2$", "agree on $Y_{red}$. Then it follows that $t_1$ and $t_2$ have", "the same image which is an open and closed subscheme of $W$ mapping", "isomorphically to $Y$", "(\\'Etale Morphisms, Proposition \\ref{etale-proposition-properties-sections})", "hence they are equal." ], "refs": [ "pione-lemma-internal-hom-finite-etale", "schemes-lemma-where-are-they-equal", "etale-proposition-properties-sections" ], "ref_ids": [ 4038, 7708, 10726 ] } ], "ref_ids": [] }, { "id": 4047, "type": "theorem", "label": "pione-lemma-same-etale-extensions", "categories": [ "pione" ], "title": "pione-lemma-same-etale-extensions", "contents": [ "Let $(A, \\mathfrak m)$ be a local ring. Set $X = \\Spec(A)$", "and let $U = X \\setminus \\{\\mathfrak m\\}$. If the punctured spectrum", "of the strict henselization of $A$ is connected, then", "$$", "\\textit{F\\'Et}_X \\longrightarrow \\textit{F\\'Et}_U,\\quad", "Y \\longmapsto Y \\times_X U", "$$", "is a fully faithful functor." ], "refs": [], "proofs": [ { "contents": [ "Assume $A$ is strictly henselian. In this case any finite \\'etale", "cover $Y$ of $X$ is isomorphic to a finite disjoint union of", "copies of $X$. Thus it suffices to prove that any morphism", "$U \\to U \\amalg \\ldots \\amalg U$ over $U$, extends uniquely to a morphism", "$X \\to X \\amalg \\ldots \\amalg X$ over $X$.", "If $U$ is connected (in particular nonempty), then this is true.", "\\medskip\\noindent", "The general case. Since the category of finite \\'etale coverings has an", "internal hom (Lemma \\ref{lemma-internal-hom-finite-etale})", "it suffices to prove the following: Given $Y$ finite \\'etale over $X$", "any morphism $s : U \\to Y$ over $X$ extends to a morphism $t : X \\to Y$", "over $X$. Let $A^{sh}$ be the strict henselization of $A$ and denote", "$X^{sh} = \\Spec(A^{sh})$, $U^{sh} = U \\times_X X^{sh}$,", "$Y^{sh} = Y \\times_X X^{sh}$. By the first paragraph and our assumption", "on $A$, we can extend the base change $s^{sh} : U^{sh} \\to Y^{sh}$ of $s$ to", "$t^{sh} : X^{sh} \\to Y^{sh}$. Set $A' = A^{sh} \\otimes_A A^{sh}$.", "Then the two pullbacks $t'_1, t'_2$ of $t^{sh}$ to $X' = \\Spec(A')$", "are extensions of the pullback $s'$ of $s$ to $U' = U \\times_X X'$.", "As $A \\to A'$ is flat we see that $U' \\subset X'$ is (topologically) dense", "by going down for $A \\to A'$", "(Algebra, Lemma \\ref{algebra-lemma-flat-going-down}). Thus", "$t'_1 = t'_2$ by Lemma \\ref{lemma-dense-faithful}.", "Hence $t^{sh}$ descends to a morphism $t : X \\to Y$", "for example by", "Descent, Lemma \\ref{descent-lemma-fpqc-universal-effective-epimorphisms}." ], "refs": [ "pione-lemma-internal-hom-finite-etale", "algebra-lemma-flat-going-down", "pione-lemma-dense-faithful", "descent-lemma-fpqc-universal-effective-epimorphisms" ], "ref_ids": [ 4038, 539, 4046, 14638 ] } ], "ref_ids": [] }, { "id": 4048, "type": "theorem", "label": "pione-lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian", "categories": [ "pione" ], "title": "pione-lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian", "contents": [ "Let $X$ be a scheme. Let $U \\subset X$ be a dense open. Assume", "\\begin{enumerate}", "\\item the underlying topological space of $X$ is Noetherian, and", "\\item for every $x \\in X \\setminus U$ the punctured spectrum of the", "strict henselization of $\\mathcal{O}_{X, x}$ is connected.", "\\end{enumerate}", "Then $\\textit{F\\'Et}_X \\to \\textit{F\\'et}_U$ is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "Let $Y_1, Y_2$ be finite \\'etale over $X$ and let", "$\\varphi : (Y_1)_U \\to (Y_2)_U$ be a morphism over $U$. We have to show that", "$\\varphi$ lifts uniquely to a morphism $Y_1 \\to Y_2$ over $X$.", "Uniqueness follows from Lemma \\ref{lemma-dense-faithful}.", "\\medskip\\noindent", "Let $x \\in X \\setminus U$ be a generic point of an irreducible component", "of $X \\setminus U$. Set $V = U \\times_X \\Spec(\\mathcal{O}_{X, x})$.", "By our choice of $x$ this is the punctured spectrum of", "$\\Spec(\\mathcal{O}_{X, x})$. By", "Lemma \\ref{lemma-same-etale-extensions}", "we can extend the morphism $\\varphi_V : (Y_1)_V \\to (Y_2)_V$", "uniquely to a morphism", "$(Y_1)_{\\Spec(\\mathcal{O}_{X, x})} \\to (Y_2)_{\\Spec(\\mathcal{O}_{X, x})}$.", "By Limits, Lemma \\ref{limits-lemma-glueing-near-point}", "we find an open $U \\subset U'$ containing $x$ and an extension", "$\\varphi' : (Y_1)_{U'} \\to (Y_2)_{U'}$ of $\\varphi$.", "Since the underlying topological space of $X$ is Noetherian", "this finishes the proof by Noetherian induction on the complement", "of the open over which $\\varphi$ is defined." ], "refs": [ "pione-lemma-dense-faithful", "pione-lemma-same-etale-extensions", "limits-lemma-glueing-near-point" ], "ref_ids": [ 4046, 4047, 15114 ] } ], "ref_ids": [] }, { "id": 4049, "type": "theorem", "label": "pione-lemma-retrocompact-dense-open-connected-at-infinity-closed", "categories": [ "pione" ], "title": "pione-lemma-retrocompact-dense-open-connected-at-infinity-closed", "contents": [ "Let $X$ be a scheme. Let $U \\subset X$ be a dense open. Assume", "\\begin{enumerate}", "\\item $U \\to X$ is quasi-compact,", "\\item every point of $X \\setminus U$ is closed, and", "\\item for every $x \\in X \\setminus U$ the punctured spectrum of the", "strict henselization of $\\mathcal{O}_{X, x}$ is connected.", "\\end{enumerate}", "Then $\\textit{F\\'Et}_X \\to \\textit{F\\'et}_U$ is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "Let $Y_1, Y_2$ be finite \\'etale over $X$ and let", "$\\varphi : (Y_1)_U \\to (Y_2)_U$ be a morphism over $U$. We have to show that", "$\\varphi$ lifts uniquely to a morphism $Y_1 \\to Y_2$ over $X$.", "Uniqueness follows from Lemma \\ref{lemma-dense-faithful}.", "\\medskip\\noindent", "Let $x \\in X \\setminus U$. Set $V = U \\times_X \\Spec(\\mathcal{O}_{X, x})$.", "Since every point of $X \\setminus U$ is closed $V$ is the punctured spectrum", "of $\\Spec(\\mathcal{O}_{X, x})$. By", "Lemma \\ref{lemma-same-etale-extensions}", "we can extend the morphism $\\varphi_V : (Y_1)_V \\to (Y_2)_V$", "uniquely to a morphism", "$(Y_1)_{\\Spec(\\mathcal{O}_{X, x})} \\to (Y_2)_{\\Spec(\\mathcal{O}_{X, x})}$.", "By Limits, Lemma \\ref{limits-lemma-glueing-near-point}", "(this uses that $U$ is retrocompact in $X$)", "we find an open $U \\subset U'_x$ containing $x$ and an extension", "$\\varphi'_x : (Y_1)_{U'_x} \\to (Y_2)_{U'_x}$ of $\\varphi$.", "Note that given two points $x, x' \\in X \\setminus U$ the", "morphisms $\\varphi'_x$ and $\\varphi'_{x'}$ agree over", "$U'_x \\cap U'_{x'}$ as $U$ is dense in that open", "(Lemma \\ref{lemma-dense-faithful}). Thus we can extend $\\varphi$", "to $\\bigcup U'_x = X$ as desired." ], "refs": [ "pione-lemma-dense-faithful", "pione-lemma-same-etale-extensions", "limits-lemma-glueing-near-point", "pione-lemma-dense-faithful" ], "ref_ids": [ 4046, 4047, 15114, 4046 ] } ], "ref_ids": [] }, { "id": 4050, "type": "theorem", "label": "pione-lemma-quasi-compact-dense-open-connected-at-infinity", "categories": [ "pione" ], "title": "pione-lemma-quasi-compact-dense-open-connected-at-infinity", "contents": [ "Let $X$ be a scheme. Let $U \\subset X$ be a dense open. Assume", "\\begin{enumerate}", "\\item every quasi-compact open of $X$ has finitely many", "irreducible components,", "\\item for every $x \\in X \\setminus U$ the punctured spectrum of the", "strict henselization of $\\mathcal{O}_{X, x}$ is connected.", "\\end{enumerate}", "Then $\\textit{F\\'Et}_X \\to \\textit{F\\'et}_U$ is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "Let $Y_1, Y_2$ be finite \\'etale over $X$ and let", "$\\varphi : (Y_1)_U \\to (Y_2)_U$ be a morphism over $U$. We have to show that", "$\\varphi$ lifts uniquely to a morphism $Y_1 \\to Y_2$ over $X$.", "Uniqueness follows from Lemma \\ref{lemma-dense-faithful}.", "We will prove existence by showing that we can enlarge $U$", "if $U \\not = X$ and using Zorn's lemma to finish the proof.", "\\medskip\\noindent", "Let $x \\in X \\setminus U$ be a generic point of an irreducible component", "of $X \\setminus U$. Set $V = U \\times_X \\Spec(\\mathcal{O}_{X, x})$.", "By our choice of $x$ this is the punctured spectrum of", "$\\Spec(\\mathcal{O}_{X, x})$. By", "Lemma \\ref{lemma-same-etale-extensions}", "we can extend the morphism $\\varphi_V : (Y_1)_V \\to (Y_2)_V$", "(uniquely) to a morphism", "$(Y_1)_{\\Spec(\\mathcal{O}_{X, x})} \\to (Y_2)_{\\Spec(\\mathcal{O}_{X, x})}$.", "Choose an affine neighbourhood $W \\subset X$ of $x$.", "Since $U \\cap W$ is dense in $W$ it contains the generic points", "$\\eta_1, \\ldots, \\eta_n$ of $W$. Choose an affine open", "$W' \\subset W \\cap U$ containing $\\eta_1, \\ldots, \\eta_n$.", "Set $V' = W' \\times_X \\Spec(\\mathcal{O}_{X, x})$.", "By Limits, Lemma \\ref{limits-lemma-glueing-near-point}", "applied to $x \\in W \\supset W'$", "we find an open $W' \\subset W'' \\subset W$ with $x \\in W''$", "and a morphism $\\varphi'' : (Y_1)_{W''} \\to (Y_2)_{W''}$", "agreeing with $\\varphi$ over $W'$. Since $W'$ is dense in", "$W'' \\cap U$, we see by Lemma \\ref{lemma-dense-faithful}", "that $\\varphi$ and $\\varphi''$ agree over $U \\cap W'$.", "Thus $\\varphi$ and $\\varphi''$ glue to a morphism", "$\\varphi'$ over $U' = U \\cup W''$ agreeing with $\\varphi$ over $U$.", "Observe that $x \\in U'$ so that we've extended $\\varphi$", "to a strictly larger open.", "\\medskip\\noindent", "Consider the set $\\mathcal{S}$ of pairs $(U', \\varphi')$ where $U \\subset U'$", "and $\\varphi'$ is an extension of $\\varphi$. We endow $\\mathcal{S}$", "with a partial ordering in the obvious manner. If $(U'_i, \\varphi'_i)$", "is a totally ordered subset, then it has a maximum $(U', \\varphi')$.", "Just take $U' = \\bigcup U'_i$ and let", "$\\varphi' : (Y_1)_{U'} \\to (Y_2)_{U'}$ be the morphism", "agreeing with $\\varphi'_i$ over $U'_i$. Thus Zorn's lemma applies", "and $\\mathcal{S}$ has a maximal element. By the argument above", "we see that this maximal element is an extension of $\\varphi$", "over all of $X$." ], "refs": [ "pione-lemma-dense-faithful", "pione-lemma-same-etale-extensions", "limits-lemma-glueing-near-point", "pione-lemma-dense-faithful" ], "ref_ids": [ 4046, 4047, 15114, 4046 ] } ], "ref_ids": [] }, { "id": 4051, "type": "theorem", "label": "pione-lemma-local-exact-sequence", "categories": [ "pione" ], "title": "pione-lemma-local-exact-sequence", "contents": [ "Let $(A, \\mathfrak m)$ be a local ring. Set $X = \\Spec(A)$ and", "$U = X \\setminus \\{\\mathfrak m\\}$. Let $U^{sh}$ be the punctured spectrum", "of the strict henselization $A^{sh}$ of $A$.", "Assume $U$ is quasi-compact and $U^{sh}$ is connected. Then the sequence", "$$", "\\pi_1(U^{sh}, \\overline{u}) \\to \\pi_1(U, \\overline{u}) \\to", "\\pi_1(X, \\overline{u}) \\to 1", "$$", "is exact in the sense of Lemma \\ref{lemma-functoriality-galois-ses} part (1)." ], "refs": [ "pione-lemma-functoriality-galois-ses" ], "proofs": [ { "contents": [ "The map $\\pi_1(U) \\to \\pi_1(X)$ is surjective by", "Lemmas \\ref{lemma-same-etale-extensions} and", "\\ref{lemma-functoriality-galois-surjective}.", "\\medskip\\noindent", "Write $X^{sh} = \\Spec(A^{sh})$. Let $Y \\to X$ be a finite \\'etale morphism.", "Then $Y^{sh} = Y \\times_X X^{sh} \\to X^{sh}$ is a finite \\'etale morphism.", "Since $A^{sh}$ is strictly henselian we see that $Y^{sh}$ is isomorphic", "to a disjoint union of copies of $X^{sh}$. Thus the same is true for", "$Y \\times_X U^{sh}$. It follows that the composition", "$\\pi_1(U^{sh}) \\to \\pi_1(U) \\to \\pi_1(X)$ is trivial, see", "Lemma \\ref{lemma-composition-trivial}.", "\\medskip\\noindent", "To finish the proof, it suffices according to", "Lemma \\ref{lemma-functoriality-galois-ses}", "to show the following: Given a finite \\'etale morphism", "$V \\to U$ such that $V \\times_U U^{sh}$ is a disjoint", "union of copies of $U^{sh}$, we can find a finite \\'etale", "morphism $Y \\to X$ with $V \\cong Y \\times_X U$ over $U$.", "The assumption implies that there exists a finite \\'etale", "morphism $Y^{sh} \\to X^{sh}$ and an isomorphism", "$V \\times_U U^{sh} \\cong Y^{sh} \\times_{X^{sh}} U^{sh}$.", "Consider the following diagram", "$$", "\\xymatrix{", "U \\ar[d] & U^{sh} \\ar[d] \\ar[l] &", "U^{sh} \\times_U U^{sh} \\ar[d] \\ar@<1ex>[l] \\ar@<-1ex>[l] &", "U^{sh} \\times_U U^{sh} \\times_U U^{sh}", "\\ar[d] \\ar@<1ex>[l] \\ar[l] \\ar@<-1ex>[l] \\\\", "X & X^{sh} \\ar[l] &", "X^{sh} \\times_X X^{sh} \\ar@<1ex>[l] \\ar@<-1ex>[l] &", "X^{sh} \\times_X X^{sh} \\times_X X^{sh} \\ar@<1ex>[l] \\ar[l] \\ar@<-1ex>[l]", "}", "$$", "Since $U \\subset X$ is quasi-compact by assumption, all the", "downward arrows are quasi-compact open immersions.", "Let $\\xi \\in X^{sh} \\times_X X^{sh}$ be a point not", "in $U^{sh} \\times_U U^{sh}$. Then $\\xi$ lies over the closed", "point $x^{sh}$ of $X^{sh}$.", "Consider the local ring homomorphism", "$$", "A^{sh} = \\mathcal{O}_{X^{sh}, x^{sh}} \\to", "\\mathcal{O}_{X^{sh} \\times_X X^{sh}, \\xi}", "$$", "determined by the first projection $X^{sh} \\times_X X^{sh}$.", "This is a filtered colimit of local homomorphisms which are", "localizations \\'etale ring maps.", "Since $A^{sh}$ is strictly henselian, we conclude that it is an", "isomorphism. Since this holds for every $\\xi$ in the complement", "it follows there are no specializations among these points and", "hence every such $\\xi$ is a closed point (you can also prove", "this directly). As the local ring at $\\xi$ is isomorphic", "to $A^{sh}$, it is strictly henselian and has connected punctured spectrum.", "Similarly for points $\\xi$ of $X^{sh} \\times_X X^{sh} \\times_X X^{sh}$ not", "in $U^{sh} \\times_U U^{sh} \\times_U U^{sh}$. It follows from", "Lemma \\ref{lemma-retrocompact-dense-open-connected-at-infinity-closed}", "that pullback along the vertical arrows induce fully faithful functors on", "the categories of finite \\'etale schemes. Thus the", "canonical descent datum on $V \\times_U U^{sh}$ relative to", "the fpqc covering $\\{U^{sh} \\to U\\}$ translates into a", "descent datum for $Y^{sh}$ relative to the fpqc covering $\\{X^{sh} \\to X\\}$.", "Since $Y^{sh} \\to X^{sh}$ is finite hence affine, this descent datum is", "effective (Descent, Lemma \\ref{descent-lemma-affine}).", "Thus we get an affine morphism $Y \\to X$ and an isomorphism", "$Y \\times_X X^{sh} \\to Y^{sh}$ compatible with descent data.", "By fully faithfulness of descent data", "(as in Descent, Lemma \\ref{descent-lemma-refine-coverings-fully-faithful})", "we get an isomorphism $V \\to U \\times_X Y$.", "Finally, $Y \\to X$ is finite \\'etale as $Y^{sh} \\to X^{sh}$ is, see", "Descent, Lemmas \\ref{descent-lemma-descending-property-etale} and", "\\ref{descent-lemma-descending-property-finite}." ], "refs": [ "pione-lemma-same-etale-extensions", "pione-lemma-functoriality-galois-surjective", "pione-lemma-composition-trivial", "pione-lemma-functoriality-galois-ses", "pione-lemma-retrocompact-dense-open-connected-at-infinity-closed", "descent-lemma-affine", "descent-lemma-refine-coverings-fully-faithful", "descent-lemma-descending-property-etale", "descent-lemma-descending-property-finite" ], "ref_ids": [ 4047, 4032, 4033, 4034, 4049, 14748, 14745, 14694, 14688 ] } ], "ref_ids": [ 4034 ] }, { "id": 4052, "type": "theorem", "label": "pione-lemma-irreducible-geometrically-unibranch", "categories": [ "pione" ], "title": "pione-lemma-irreducible-geometrically-unibranch", "contents": [ "Let $X$ be an irreducible, geometrically unibranch scheme.", "For any nonempty open $U \\subset X$ the canonical map", "$$", "\\pi_1(U, \\overline{u}) \\longrightarrow \\pi_1(X, \\overline{u})", "$$", "is surjective. The map (\\ref{equation-inclusion-generic-point})", "$\\pi_1(\\eta, \\overline{\\eta}) \\to \\pi_1(X, \\overline{\\eta})$", "is surjective as well." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-thickening} we may replace $X$ by its reduction.", "Thus we may assume that $X$ is an integral scheme. By", "Lemma \\ref{lemma-functoriality-galois-surjective}", "the assertion of the lemma translates into the statement that", "the functors $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_U$ and", "$\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_\\eta$ are fully faithful.", "\\medskip\\noindent", "The result for $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_U$ follows", "from Lemma \\ref{lemma-quasi-compact-dense-open-connected-at-infinity}", "and the fact that for a local ring $A$ which is", "geometrically unibranch its strict henselization has an", "irreducible spectrum. See", "More on Algebra, Lemma \\ref{more-algebra-lemma-geometrically-unibranch}.", "\\medskip\\noindent", "Observe that the residue field $\\kappa(\\eta) = \\mathcal{O}_{X, \\eta}$", "is the filtered colimit of $\\mathcal{O}_X(U)$ over $U \\subset X$", "nonempty open affine. Hence $\\textit{F\\'Et}_\\eta$ is the colimit of the", "categories $\\textit{F\\'Et}_U$ over such $U$, see", "Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation},", "\\ref{limits-lemma-descend-finite-finite-presentation}, and", "\\ref{limits-lemma-descend-etale}.", "A formal argument then shows that fully faithfulness for", "$\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_\\eta$ follows from the", "fully faithfulness of the functors $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_U$." ], "refs": [ "pione-lemma-thickening", "pione-lemma-functoriality-galois-surjective", "pione-lemma-quasi-compact-dense-open-connected-at-infinity", "more-algebra-lemma-geometrically-unibranch", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-finite-finite-presentation", "limits-lemma-descend-etale" ], "ref_ids": [ 4043, 4032, 4050, 10468, 15077, 15058, 15065 ] } ], "ref_ids": [] }, { "id": 4053, "type": "theorem", "label": "pione-lemma-exact-sequence-finite-nr-closed-pts", "categories": [ "pione" ], "title": "pione-lemma-exact-sequence-finite-nr-closed-pts", "contents": [ "Let $X$ be a scheme. Let $x_1, \\ldots, x_n \\in X$ be a finite", "number of closed points such that", "\\begin{enumerate}", "\\item $U = X \\setminus \\{x_1, \\ldots, x_n\\}$ is connected and is", "a retrocompact open of $X$, and", "\\item for each $i$ the punctured spectrum $U_i^{sh}$ of the", "strict henselization of $\\mathcal{O}_{X, x_i}$ is connected.", "\\end{enumerate}", "Then the map $\\pi_1(U) \\to \\pi_1(X)$ is surjective and the kernel", "is the smallest closed normal subgroup of $\\pi_1(U)$ containing", "the image of $\\pi_1(U_i^{sh}) \\to \\pi_1(U)$ for $i = 1, \\ldots, n$." ], "refs": [], "proofs": [ { "contents": [ "Surjectivity follows from", "Lemmas \\ref{lemma-retrocompact-dense-open-connected-at-infinity-closed} and", "\\ref{lemma-functoriality-galois-surjective}.", "We can consider the sequence of maps", "$$", "\\pi_1(U) \\to \\ldots \\to", "\\pi_1(X \\setminus \\{x_1, x_2\\}) \\to \\pi_1(X \\setminus \\{x_1\\}) \\to \\pi_1(X)", "$$", "A group theory argument then shows it suffices to prove the statement on the", "kernel in the case $n = 1$ (details omitted). Write", "$x = x_1$, $U^{sh} = U_1^{sh}$,", "set $A = \\mathcal{O}_{X, x}$, and let $A^{sh}$ be the strict henselization.", "Consider the diagram", "$$", "\\xymatrix{", "U \\ar[d] &", "\\Spec(A) \\setminus \\{\\mathfrak m\\} \\ar[l] \\ar[d] &", "U^{sh} \\ar[d] \\ar[l] \\\\", "X & \\Spec(A) \\ar[l] & \\Spec(A^{sh}) \\ar[l]", "}", "$$", "By Lemma \\ref{lemma-functoriality-galois-ses}", "we have to show finite \\'etale morphisms", "$V \\to U$ which pull back to trivial coverings of $U^{sh}$", "extend to finite \\'etale schemes over $X$.", "By Lemma \\ref{lemma-local-exact-sequence}", "we know the corresponding statement", "for finite \\'etale schemes over the punctured spectrum of $A$.", "However, by Limits, Lemma \\ref{limits-lemma-glueing-near-closed-point}", "schemes of finite presentation over $X$ are the same thing as", "schemes of finite presentation over $U$ and $A$ glued over", "the punctured spectrum of $A$. This finishes the proof." ], "refs": [ "pione-lemma-retrocompact-dense-open-connected-at-infinity-closed", "pione-lemma-functoriality-galois-surjective", "pione-lemma-functoriality-galois-ses", "pione-lemma-local-exact-sequence", "limits-lemma-glueing-near-closed-point" ], "ref_ids": [ 4049, 4032, 4034, 4051, 15112 ] } ], "ref_ids": [] }, { "id": 4054, "type": "theorem", "label": "pione-lemma-unramified-in-L", "categories": [ "pione" ], "title": "pione-lemma-unramified-in-L", "contents": [ "In the situation above the following are equivalent", "\\begin{enumerate}", "\\item $X$ is unramified in $L$,", "\\item $Y \\to X$ is \\'etale, and", "\\item $Y \\to X$ is finite \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $Y \\to X$ is an integral morphism.", "In each case the morphism $Y \\to X$ is locally of finite type", "by definition.", "Hence we find that in each case the lemma is finite by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-integral}.", "In particular we see that (2) is equivalent to (3).", "An \\'etale morphism is unramified, hence (2) implies (1).", "\\medskip\\noindent", "Conversely, assume $Y \\to X$ is unramified. Let $x \\in X$.", "We can choose an \\'etale neighbourhood $(U, u) \\to (X, x)$ such that", "$$", "Y \\times_X U = \\coprod V_j \\longrightarrow U", "$$", "is a disjoint union of closed immersions, see", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-finite-unramified-etale-local}.", "Shrinking we may assume $U$ is quasi-compact.", "Then $U$ has finitely many irreducible components", "(Descent, Lemma \\ref{descent-lemma-locally-finite-nr-irred-local-fppf}).", "Since $U$ is normal", "(Descent, Lemma \\ref{descent-lemma-normal-local-smooth}) the", "irreducible components of $U$ are open and closed", "(Properties, Lemma \\ref{properties-lemma-normal-locally-finite-nr-irreducibles})", "and we may assume $U$ is irreducible. Then $U$ is an integral", "scheme whose generic point $\\xi$ maps to the generic point of $X$.", "On the other hand, we know that $Y \\times_X U$", "is the normalization of $U$ in $\\Spec(L) \\times_X U$", "by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-normalization-smooth-localization}.", "Every point of $\\Spec(L) \\times_X U$ maps to $\\xi$.", "Thus every $V_j$ contains a point mapping to $\\xi$ by", "Morphisms, Lemma \\ref{morphisms-lemma-normalization-generic}.", "Thus $V_j \\to U$ is an isomorphism as $U = \\overline{\\{\\xi\\}}$.", "Thus $Y \\times_X U \\to U$ is \\'etale. By", "Descent, Lemma \\ref{descent-lemma-descending-property-etale}", "we conclude that $Y \\to X$ is \\'etale over the", "image of $U \\to X$ (an open neighbourhood of $x$)." ], "refs": [ "morphisms-lemma-finite-integral", "etale-lemma-finite-unramified-etale-local", "descent-lemma-locally-finite-nr-irred-local-fppf", "descent-lemma-normal-local-smooth", "properties-lemma-normal-locally-finite-nr-irreducibles", "more-morphisms-lemma-normalization-smooth-localization", "morphisms-lemma-normalization-generic", "descent-lemma-descending-property-etale" ], "ref_ids": [ 5438, 10711, 14650, 14654, 2969, 13774, 5504, 14694 ] } ], "ref_ids": [] }, { "id": 4055, "type": "theorem", "label": "pione-lemma-finite-etale-covering-normal-unramified", "categories": [ "pione" ], "title": "pione-lemma-finite-etale-covering-normal-unramified", "contents": [ "Let $X$ be a normal integral scheme with function field $K$.", "Let $Y \\to X$ be a finite \\'etale morphism. If $Y$ is connected,", "then $Y$ is an integral normal scheme and $Y$ is the normalization", "of $X$ in the function field of $Y$." ], "refs": [], "proofs": [ { "contents": [ "The scheme $Y$ is normal by", "Descent, Lemma \\ref{descent-lemma-normal-local-smooth}.", "Since $Y \\to X$ is flat every generic point of $Y$ maps", "to the generic point of $X$ by", "Morphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat}.", "Since $Y \\to X$ is finite we see that $Y$ has a finite number", "of irreducible components. Thus $Y$ is the disjoint union of", "a finite number of integral normal schemes by", "Properties, Lemma \\ref{properties-lemma-normal-locally-finite-nr-irreducibles}.", "Thus if $Y$ is connected, then $Y$ is an integral normal scheme.", "\\medskip\\noindent", "Let $L$ be the function field of $Y$ and let $Y' \\to X$ be the normalization", "of $X$ in $L$. By", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-normalization}", "we obtain a factorization $Y' \\to Y \\to X$ and $Y' \\to Y$ is", "the normalization of $Y$ in $L$. Since $Y$ is normal it is clear", "that $Y' = Y$ (this can also be deduced from", "Morphisms, Lemma \\ref{morphisms-lemma-finite-birational-over-normal})." ], "refs": [ "descent-lemma-normal-local-smooth", "morphisms-lemma-generalizations-lift-flat", "properties-lemma-normal-locally-finite-nr-irreducibles", "morphisms-lemma-characterize-normalization", "morphisms-lemma-finite-birational-over-normal" ], "ref_ids": [ 14654, 5266, 2969, 5499, 5518 ] } ], "ref_ids": [] }, { "id": 4056, "type": "theorem", "label": "pione-lemma-local-exact-sequence-normal", "categories": [ "pione" ], "title": "pione-lemma-local-exact-sequence-normal", "contents": [ "Let $(A, \\mathfrak m)$ be a normal local ring.", "Set $X = \\Spec(A)$. Let $A^{sh}$ be the strict henselization of $A$.", "Let $K$ and $K^{sh}$ be the fraction fields of $A$ and $A^{sh}$.", "Then the sequence", "$$", "\\pi_1(\\Spec(K^{sh})) \\to \\pi_1(\\Spec(K)) \\to \\pi_1(X) \\to 1", "$$", "is exact in the sense of Lemma \\ref{lemma-functoriality-galois-ses} part (1)." ], "refs": [ "pione-lemma-functoriality-galois-ses" ], "proofs": [ { "contents": [ "Note that $A^{sh}$ is a normal domain, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-normal}.", "The map $\\pi_1(\\Spec(K)) \\to \\pi_1(X)$ is surjective by", "Proposition \\ref{proposition-normal}.", "\\medskip\\noindent", "Write $X^{sh} = \\Spec(A^{sh})$. Let $Y \\to X$ be a finite \\'etale morphism.", "Then $Y^{sh} = Y \\times_X X^{sh} \\to X^{sh}$ is a finite \\'etale morphism.", "Since $A^{sh}$ is strictly henselian we see that $Y^{sh}$ is isomorphic", "to a disjoint union of copies of $X^{sh}$. Thus the same is true for", "$Y \\times_X \\Spec(K^{sh})$. It follows that the composition", "$\\pi_1(\\Spec(K^{sh})) \\to \\pi_1(X)$ is trivial, see", "Lemma \\ref{lemma-composition-trivial}.", "\\medskip\\noindent", "To finish the proof, it suffices according to", "Lemma \\ref{lemma-functoriality-galois-ses}", "to show the following: Given a finite \\'etale morphism", "$V \\to \\Spec(K)$ such that $V \\times_{\\Spec(K)} \\Spec(K^{sh})$", "is a disjoint union of copies of $\\Spec(K^{sh})$, we can find a", "finite \\'etale morphism", "$Y \\to X$ with $V \\cong Y \\times_X \\Spec(K)$ over $\\Spec(K)$.", "Write $V = \\Spec(L)$, so $L$ is a finite product of", "finite separable extensions of $K$.", "Let $B \\subset L$ be the integral closure of $A$ in $L$.", "If $A \\to B$ is \\'etale, then we can take $Y = \\Spec(B)$", "and the proof is complete. By", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-commutes-smooth}", "(and a limit argument we omit)", "we see that $B \\otimes_A A^{sh}$ is the integral closure of", "$A^{sh}$ in $L^{sh} = L \\otimes_K K^{sh}$.", "Our assumption is that $L^{sh}$ is a product of copies of", "$K^{sh}$ and hence $B^{sh}$ is a product of copies of $A^{sh}$.", "Thus $A^{sh} \\to B^{sh}$ is \\'etale. As $A \\to A^{sh}$ is", "faithfully flat it follows that $A \\to B$ is \\'etale", "(Descent, Lemma \\ref{descent-lemma-descending-property-etale})", "as desired." ], "refs": [ "more-algebra-lemma-henselization-normal", "pione-proposition-normal", "pione-lemma-composition-trivial", "pione-lemma-functoriality-galois-ses", "algebra-lemma-integral-closure-commutes-smooth", "descent-lemma-descending-property-etale" ], "ref_ids": [ 10060, 4132, 4033, 4034, 1252, 14694 ] } ], "ref_ids": [ 4034 ] }, { "id": 4057, "type": "theorem", "label": "pione-lemma-get-algebraic-closure", "categories": [ "pione" ], "title": "pione-lemma-get-algebraic-closure", "contents": [ "Let $A$ be a normal domain whose fraction field $K$ is separably algebraically", "closed. Let $\\mathfrak p \\subset A$ be a nonzero prime ideal.", "Then the residue field $\\kappa(\\mathfrak p)$ is algebraically closed." ], "refs": [], "proofs": [ { "contents": [ "Assume the lemma is not true to get a contradiction. Then there exists a", "monic irreducible polynomial $P(T) \\in \\kappa(\\mathfrak p)[T]$ of", "degree $d > 1$. After replacing $P$ by $a^d P(a^{-1}T)$ for suitable $a \\in A$", "(to clear denominators) we may assume that $P$ is the image of a", "monic polynomial $Q$ in $A[T]$. Observe that $Q$ is irreducible in", "$K[T]$. Namely a factorization over $K$ leads to a factorization", "over $A$ by Algebra, Lemma \\ref{algebra-lemma-polynomials-divide}", "which we could reduce modulo $\\mathfrak p$ to get a factorization of $P$.", "As $K$ is separably closed, $Q$ is not a separable polynomial", "(Fields, Definition \\ref{fields-definition-separable}).", "Then the characteristic of $K$ is $p > 0$ and $Q$ has", "vanishing linear term (Fields, Definition \\ref{fields-definition-separable}).", "However, then we can replace $Q$ by", "$Q + a T$ where $a \\in \\mathfrak p$ is nonzero to get a contradiction." ], "refs": [ "algebra-lemma-polynomials-divide", "fields-definition-separable", "fields-definition-separable" ], "ref_ids": [ 520, 4537, 4537 ] } ], "ref_ids": [] }, { "id": 4058, "type": "theorem", "label": "pione-lemma-normal-local-domain-separablly-closed-fraction-field", "categories": [ "pione" ], "title": "pione-lemma-normal-local-domain-separablly-closed-fraction-field", "contents": [ "A normal local ring with separably closed fraction field is", "strictly henselian." ], "refs": [], "proofs": [ { "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be normal local with separably", "closed fraction field $K$. If $A = K$, then we are done. If not,", "then the residue field $\\kappa$ is algebraically closed", "by Lemma \\ref{lemma-get-algebraic-closure} and it suffices to", "check that $A$ is henselian.", "Let $f \\in A[T]$ be monic and let $a_0 \\in \\kappa$ be a root", "of multiplicity $1$ of the reduction $\\overline{f} \\in \\kappa[T]$.", "Let $f = \\prod f_i$ be the factorization in $K[T]$.", "By Algebra, Lemma \\ref{algebra-lemma-polynomials-divide} we have", "$f_i \\in A[T]$. Thus $a_0$ is a root of $f_i$ for some $i$.", "After replacing $f$ by $f_i$ we may assume $f$ is irreducible.", "Then, since the derivative $f'$ cannot be zero in $A[T]$", "as $a_0$ is a single root, we conclude that $f$ is linear", "due to the fact that $K$ is separably algebraically closed.", "Thus $A$ is henselian, see", "Algebra, Definition \\ref{algebra-definition-henselian}." ], "refs": [ "pione-lemma-get-algebraic-closure", "algebra-lemma-polynomials-divide", "algebra-definition-henselian" ], "ref_ids": [ 4057, 520, 1545 ] } ], "ref_ids": [] }, { "id": 4059, "type": "theorem", "label": "pione-lemma-inertia-base-change", "categories": [ "pione" ], "title": "pione-lemma-inertia-base-change", "contents": [ "Let $G$ be a finite group acting on a ring $R$. Let $R^G \\to A$ be a ring", "map. Let $\\mathfrak q' \\subset A \\otimes_{R^G} R$ be a prime lying", "over the prime $\\mathfrak q \\subset R$. Then", "$$", "I_\\mathfrak q = \\{\\sigma \\in G \\mid", "\\sigma(\\mathfrak q) = \\mathfrak q\\text{ and }", "\\sigma \\bmod \\mathfrak q = \\text{id}_{\\kappa(\\mathfrak q)}\\}", "$$", "is equal to", "$$", "I_{\\mathfrak q'} = \\{\\sigma \\in G \\mid", "\\sigma(\\mathfrak q') = \\mathfrak q'\\text{ and }", "\\sigma \\bmod \\mathfrak q' = \\text{id}_{\\kappa(\\mathfrak q')}\\}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathfrak q$ is the inverse image of $\\mathfrak q'$", "and since $\\kappa(\\mathfrak q) \\subset \\kappa(\\mathfrak q')$,", "we get $I_{\\mathfrak q'} \\subset I_\\mathfrak q$.", "Conversely, if $\\sigma \\in I_\\mathfrak q$, the $\\sigma$", "acts trivially on the fibre ring $A \\otimes_{R^G} \\kappa(\\mathfrak q)$.", "Thus $\\sigma$ fixes all the primes lying over $\\mathfrak q$", "and induces the identity on their residue fields." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4060, "type": "theorem", "label": "pione-lemma-inertia-invariants-etale", "categories": [ "pione" ], "title": "pione-lemma-inertia-invariants-etale", "contents": [ "Let $G$ be a finite group acting on a ring $R$. Let $\\mathfrak q \\subset R$", "be a prime. Set", "$$", "I = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q) = \\mathfrak q", "\\text{ and } \\sigma \\bmod \\mathfrak q = \\text{id}_\\mathfrak q\\}", "$$", "Then $R^G \\to R^I$ is \\'etale at $R^I \\cap \\mathfrak q$." ], "refs": [], "proofs": [ { "contents": [ "The strategy of the proof is to use \\'etale localization to", "reduce to the case where $R \\to R^I$ is a local isomorphism at", "$R^I \\cap \\mathfrak p$.", "Let $R^G \\to A$ be an \\'etale ring map. We claim that if the result", "holds for the action of $G$ on $A \\otimes_{R^G} R$ and some prime", "$\\mathfrak q'$ of $A \\otimes_{R^G} R$ lying over $\\mathfrak q$, then", "the result is true.", "\\medskip\\noindent", "To check this, note that since $R^G \\to A$ is flat we have", "$A = (A \\otimes_{R^G} R)^G$, see More on Algebra,", "Lemma \\ref{more-algebra-lemma-base-change-invariants}.", "By Lemma \\ref{lemma-inertia-base-change} the group $I$ does not change.", "Then a second application of More on Algebra,", "Lemma \\ref{more-algebra-lemma-base-change-invariants}", "shows that $A \\otimes_{R^G} R^I = (A \\otimes_{R^G} R)^I$", "(because $R^I \\to A \\otimes_{R^G} R^I$ is flat).", "Thus", "$$", "\\xymatrix{", "\\Spec((A \\otimes_{R^G} R)^I) \\ar[d] \\ar[r] & \\Spec(R^I) \\ar[d] \\\\", "\\Spec(A) \\ar[r] & \\Spec(R^G)", "}", "$$", "is cartesian and the horizontal arrows are \\'etale. Thus if the", "left vertical arrow is \\'etale in some open neighbourhood $W$ of", "$(A \\otimes_{R^G} R)^I \\cap \\mathfrak q'$, then the right vertical", "arrow is \\'etale at the points of the (open) image of $W$ in", "$\\Spec(R^I)$, see", "Descent, Lemma \\ref{descent-lemma-smooth-permanence}. In particular", "the morphism $\\Spec(R^I) \\to \\Spec(R^G)$ is \\'etale at $R^I \\cap \\mathfrak q$.", "\\medskip\\noindent", "Let $\\mathfrak p = R^G \\cap \\mathfrak q$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-one-orbit}", "the fibre of $\\Spec(R) \\to \\Spec(R^G)$ over $\\mathfrak p$ is", "finite. Moreover the residue field extensions at these points", "are algebraic, normal, with finite automorphism groups by", "More on Algebra, Lemma \\ref{more-algebra-lemma-one-orbit-geometric}.", "Thus we may apply", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-etale-makes-integral-split}", "to the integral ring map $R^G \\to R$ and the prime $\\mathfrak p$.", "Combined with the claim above we reduce to the case where", "$R = A_1 \\times \\ldots \\times A_n$ with each $A_i$ having a single", "prime $\\mathfrak q_i$ lying over $\\mathfrak p$ such that the", "residue field extensions $\\kappa(\\mathfrak q_i)/\\kappa(\\mathfrak p)$", "are purely inseparable. Of course $\\mathfrak q$ is one of", "these primes, say $\\mathfrak q = \\mathfrak q_1$.", "\\medskip\\noindent", "It may not be the case that $G$ permutes the factors $A_i$", "(this would be true if the spectrum of $A_i$ were connected,", "for example if $R^G$ was local). This we can fix as follows;", "we suggest the reader think this through for themselves, perhaps", "using idempotents instead of topology.", "Recall that the product decomposition gives a corresponding", "disjoint union decomposition of $\\Spec(R)$ by open and closed", "subsets $U_i$. Since $G$ is finite, we can refine this covering", "by a finite disjoint union decomposition", "$\\Spec(R) = \\coprod_{j \\in J} W_j$ by open", "and closed subsets $W_j$, such that for all $j \\in J$ there exists", "a $j' \\in J$ with $\\sigma(W_j) = W_{j'}$. The union of the", "$W_j$ not meeting $\\{\\mathfrak q_1, \\ldots, \\mathfrak q_n\\}$", "is a closed subset not meeting the fibre over $\\mathfrak p$", "hence maps to a closed subset of $\\Spec(R^G)$ not meeting", "$\\mathfrak p$ as $\\Spec(R) \\to \\Spec(R^G)$ is closed.", "Hence after replacing $R^G$ by a principal localization", "(permissible by the claim) we may assume each $W_j$ meets", "one of the points $\\mathfrak q_i$. Then we set $U_i = W_j$", "if $\\mathfrak q_i \\in W_j$. The corresponding product decomposition", "$R = A_1 \\times \\ldots \\times A_n$ is one", "where $G$ permutes the factors $A_i$.", "\\medskip\\noindent", "Thus we may assume we have a product decomposition", "$R = A_1 \\times \\ldots \\times A_n$ compatible with $G$-action,", "where each $A_i$ has a single prime $\\mathfrak q_i$ lying", "over $\\mathfrak p$ and the field extensions", "$\\kappa(\\mathfrak q_i)/\\kappa(\\mathfrak p)$ are purely inseparable.", "Write $A' = A_2 \\times \\ldots \\times A_n$ so that", "$$", "R = A_1 \\times A'", "$$", "Since $\\mathfrak q = \\mathfrak q_1$ we find that every", "$\\sigma \\in I$ preserves the product decomposition above.", "Hence", "$$", "R^I = (A_1)^I \\times (A')^I", "$$", "Observe that $I = D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q) = \\mathfrak q\\}$", "because $\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$ is purely inseparable.", "Since the action of $G$ on primes over $\\mathfrak p$ is transitive", "(More on Algebra, Lemma \\ref{more-algebra-lemma-one-orbit})", "we conclude that, the index of $I$ in $G$ is $n$ and we can write", "$G = eI \\amalg \\sigma_2I \\amalg \\ldots \\amalg \\sigma_nI$ so that", "$A_i = \\sigma_i(A_1)$ for $i = 2, \\ldots, n$. It follows that", "$$", "R^G = (A_1)^I.", "$$", "Thus the map $R^G \\to R^I$ is \\'etale at $R^I \\cap \\mathfrak q$", "and the proof is complete." ], "refs": [ "more-algebra-lemma-base-change-invariants", "pione-lemma-inertia-base-change", "more-algebra-lemma-base-change-invariants", "descent-lemma-smooth-permanence", "more-algebra-lemma-one-orbit", "more-algebra-lemma-one-orbit-geometric", "more-morphisms-lemma-etale-makes-integral-split", "more-algebra-lemma-one-orbit" ], "ref_ids": [ 10487, 4059, 10487, 14644, 10488, 10489, 13898, 10488 ] } ], "ref_ids": [] }, { "id": 4061, "type": "theorem", "label": "pione-lemma-inertial-invariants-unramified", "categories": [ "pione" ], "title": "pione-lemma-inertial-invariants-unramified", "contents": [ "Let $A$ be a normal domain with fraction field $K$.", "Let $L/K$ be a (possibly infinite) Galois extension.", "Let $G = \\text{Gal}(L/K)$ and let", "$B$ be the integral closure of $A$ in $L$.", "Let $\\mathfrak q \\subset B$. Set", "$$", "I = \\{\\sigma \\in G \\mid", "\\sigma(\\mathfrak q) = \\mathfrak q \\text{ and }", "\\sigma \\bmod \\mathfrak q = \\text{id}_{\\kappa(\\mathfrak q)}\\}", "$$", "Then $(B^I)_{B^I \\cap \\mathfrak q}$ is a filtered colimit", "of \\'etale $A$-algebras." ], "refs": [], "proofs": [ { "contents": [ "We can write $L$ as the filtered colimit of finite Galois extensions", "of $K$. Hence it suffices to prove this lemma in case $L/K$ is", "a finite Galois extension, see", "Algebra, Lemma \\ref{algebra-lemma-colimit-colimit-etale}.", "Since $A = B^G$ as $A$ is integrally", "closed in $K = L^G$ the result follows from", "Lemma \\ref{lemma-inertia-invariants-etale}." ], "refs": [ "algebra-lemma-colimit-colimit-etale", "pione-lemma-inertia-invariants-etale" ], "ref_ids": [ 1289, 4060 ] } ], "ref_ids": [] }, { "id": 4062, "type": "theorem", "label": "pione-lemma-identify-inertia", "categories": [ "pione" ], "title": "pione-lemma-identify-inertia", "contents": [ "In the situation described above, via the isomorphism", "$\\pi_1(U) = \\text{Gal}(K^{sep}/K)$ the diagram", "(\\ref{equation-inertia-diagram-pione})", "translates into the diagram", "$$", "\\xymatrix{", "I \\ar[r] \\ar[rd]_1 & D \\ar[d] \\ar[r] & \\text{Gal}(K^{sep}/K) \\ar[d] \\\\", "& \\text{Gal}(\\kappa(\\mathfrak m^{sh})/\\kappa) \\ar[r] & \\text{Gal}(M/K)", "}", "$$", "where $K^{sep}/M/K$ is the maximal subextension unramified", "with respect to $A$. Moreover, the vertical arrows are surjective,", "the kernel of the left vertical arrow is $I$ and the kernel of the", "right vertical arrow is", "the smallest closed normal subgroup of $\\text{Gal}(K^{sep}/K)$", "containing $I$." ], "refs": [], "proofs": [ { "contents": [ "By construction the group $D$ acts on $(A^{sep})_{\\mathfrak m^{sep}}$", "over $A$. By the uniqueness of $A^{sh} \\to (A^{sep})_{\\mathfrak m^{sep}}$", "given the map on residue fields", "(Algebra, Lemma \\ref{algebra-lemma-strictly-henselian-functorial})", "we see that the image of $A^{sh} \\to (A^{sep})_{\\mathfrak m^{sep}}$", "is contained in $((A^{sep})_{\\mathfrak m^{sep}})^I$.", "On the other hand,", "Lemma \\ref{lemma-inertial-invariants-unramified}", "shows that $((A^{sep})_{\\mathfrak m^{sep}})^I$", "is a filtered colimit of \\'etale extensions of $A$.", "Since $A^{sh}$ is the maximal such extension, we conclude", "that $A^{sh} = ((A^{sep})_{\\mathfrak m^{sep}})^I$.", "Hence $K^{sh} = (K^{sep})^I$.", "\\medskip\\noindent", "Recall that $I$ is the kernel of a surjective map", "$D \\to \\text{Aut}(\\kappa(\\mathfrak m^{sep})/\\kappa)$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-one-orbit-geometric-galois}.", "We have $\\text{Aut}(\\kappa(\\mathfrak m^{sep})/\\kappa) =", "\\text{Gal}(\\kappa(\\mathfrak m^{sh})/\\kappa)$", "as we have seen above that these fields are the algebraic", "and separable algebraic closures of $\\kappa$.", "On the other hand, any automorphism of $A^{sh}$ over $A$", "is an automorphism of $A^{sh}$ over $A^h$ by the uniqueness", "in Algebra, Lemma \\ref{algebra-lemma-henselian-functorial}.", "Furthermore, $A^{sh}$ is the colimit of finite \\'etale", "extensions $A^h \\subset A'$ which correspond $1$-to-$1$", "with finite separable extension $\\kappa'/\\kappa$, see", "Algebra, Remark \\ref{algebra-remark-construct-sh-from-h}.", "Thus", "$$", "\\text{Aut}(A^{sh}/A) = \\text{Aut}(A^{sh}/A^h) =", "\\text{Gal}(\\kappa(\\mathfrak m^{sh})/\\kappa)", "$$", "Let $\\kappa \\subset \\kappa'$ be a finite Galois extension with", "Galois group $G$. Let $A^h \\subset A'$ be the finite \\'etale extension", "corresponding to $\\kappa \\subset \\kappa'$ by", "Algebra, Lemma \\ref{algebra-lemma-henselian-cat-finite-etale}.", "Then it follows that", "$(A')^G = A^h$ by looking at fraction fields and degrees", "(small detail omitted). Taking the colimit we conclude that", "$(A^{sh})^{\\text{Gal}(\\kappa(\\mathfrak m^{sh})/\\kappa)} = A^h$.", "Combining all of the above, we find $A^h = ((A^{sep})_{\\mathfrak m^{sep}})^D$.", "Hence $K^h = (K^{sep})^D$.", "\\medskip\\noindent", "Since $U$, $U^h$, $U^{sh}$ are the spectra of the fields", "$K$, $K^h$, $K^{sh}$ we see that the top lines of the diagrams", "correspond via", "Lemma \\ref{lemma-fundamental-group-Galois-group}.", "By Lemma \\ref{lemma-gabber} we have", "$\\pi_1(X^h) = \\text{Gal}(\\kappa(\\mathfrak m^{sh})/\\kappa)$.", "The exactness of the sequence", "$1 \\to I \\to D \\to \\text{Gal}(\\kappa(\\mathfrak m^{sh})/\\kappa) \\to 1$", "was pointed out above.", "By Proposition \\ref{proposition-normal}", "we see that $\\pi_1(X) = \\text{Gal}(M/K)$.", "Finally, the statement on the kernel of", "$\\text{Gal}(K^{sep}/K) \\to \\text{Gal}(M/K) = \\pi_1(X)$", "follows from Lemma \\ref{lemma-local-exact-sequence-normal}.", "This finishes the proof." ], "refs": [ "algebra-lemma-strictly-henselian-functorial", "pione-lemma-inertial-invariants-unramified", "more-algebra-lemma-one-orbit-geometric-galois", "algebra-lemma-henselian-functorial", "algebra-remark-construct-sh-from-h", "algebra-lemma-henselian-cat-finite-etale", "pione-lemma-fundamental-group-Galois-group", "pione-lemma-gabber", "pione-proposition-normal", "pione-lemma-local-exact-sequence-normal" ], "ref_ids": [ 1303, 4061, 10490, 1297, 1581, 1280, 4040, 4042, 4132, 4056 ] } ], "ref_ids": [] }, { "id": 4063, "type": "theorem", "label": "pione-lemma-normal-pione-quotient-inertia", "categories": [ "pione" ], "title": "pione-lemma-normal-pione-quotient-inertia", "contents": [ "Let $X$ be a normal integral scheme with function field $K$.", "With notation as above, the following three subgroups of", "$\\text{Gal}(K^{sep}/K) = \\pi_1(\\Spec(K))$", "are equal", "\\begin{enumerate}", "\\item the kernel of the surjection", "$\\text{Gal}(K^{sep}/K) \\longrightarrow \\pi_1(X)$,", "\\item the smallest normal closed subgroup containing $I_y$", "for all $y \\in X^{sep}$, and", "\\item the smallest normal closed subgroup containing", "$\\text{Gal}(K^{sep}/K_x^{sh})$ for all $x \\in X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (2) and (3) follows from", "Lemma \\ref{lemma-identify-inertia}", "which tells us that $I_y$ is conjugate to $\\text{Gal}(K^{sep}/K_x^{sh})$", "if $y$ lies over $x$. By Lemma \\ref{lemma-local-exact-sequence-normal}", "we see that $\\text{Gal}(K^{sep}/K_x^{sh})$ maps trivially to", "$\\pi_1(\\Spec(\\mathcal{O}_{X, x}))$ and therefore the subgroup", "$N \\subset G = \\text{Gal}(K^{sep}/K)$", "of (2) and (3) is contained in the kernel of", "$G \\longrightarrow \\pi_1(X)$.", "\\medskip\\noindent", "To prove the other inclusion, since $N$ is normal, it suffices to prove:", "given $N \\subset U \\subset G$ with $U$ open normal,", "the quotient map $G \\to G/U$ factors through $\\pi_1(X)$.", "In other words, if $L/K$ is the Galois extension corresponding", "to $U$, then we have to show that $X$ is unramified in $L$", "(Section \\ref{section-normal}, especially", "Proposition \\ref{proposition-normal}).", "It suffices to do this when $X$ is affine (we do this", "so we can refer to algebra results in the rest of the proof).", "Let $Y \\to X$ be the normalization of $X$ in $L$.", "The inclusion $L \\subset K^{sep}$ induces a morphism", "$\\pi : X^{sep} \\to Y$. For $y \\in X^{sep}$", "the inertia group of $\\pi(y)$ in $\\text{Gal}(L/K)$", "is the image of $I_y$ in $\\text{Gal}(L/K)$; this follows", "from More on Algebra, Lemma", "\\ref{more-algebra-lemma-one-orbit-geometric-galois-compare}.", "Since $N \\subset U$ all these inertia groups are trivial.", "We conclude that $Y \\to X$ is \\'etale by applying", "Lemma \\ref{lemma-inertia-invariants-etale}.", "(Alternative: you can use Lemma \\ref{lemma-local-exact-sequence-normal}", "to see that the pullback of $Y$ to $\\Spec(\\mathcal{O}_{X, x})$ is", "\\'etale for all $x \\in X$ and then conclude from there", "with a bit more work.)" ], "refs": [ "pione-lemma-identify-inertia", "pione-lemma-local-exact-sequence-normal", "pione-proposition-normal", "more-algebra-lemma-one-orbit-geometric-galois-compare", "pione-lemma-inertia-invariants-etale", "pione-lemma-local-exact-sequence-normal" ], "ref_ids": [ 4062, 4056, 4132, 10491, 4060, 4056 ] } ], "ref_ids": [] }, { "id": 4064, "type": "theorem", "label": "pione-lemma-unramified", "categories": [ "pione" ], "title": "pione-lemma-unramified", "contents": [ "Let $X$ be an integral normal scheme with function field $K$.", "Let $L/K$ be a finite extension. Let $Y \\to X$ be the normalization", "of $X$ in $L$. The following are equivalent", "\\begin{enumerate}", "\\item $X$ is unramified in $L$ as defined in Section \\ref{section-normal},", "\\item $Y \\to X$ is an unramified morphism of schemes,", "\\item $Y \\to X$ is an \\'etale morphism of schemes,", "\\item $Y \\to X$ is a finite \\'etale morphism of schemes,", "\\item for $x \\in X$ the projection", "$Y \\times_X \\Spec(\\mathcal{O}_{X, x}) \\to \\Spec(\\mathcal{O}_{X, x})$", "is unramified,", "\\item same as in (5) but with $\\mathcal{O}_{X, x}^h$,", "\\item same as in (5) but with $\\mathcal{O}_{X, x}^{sh}$,", "\\item for $x \\in X$ the scheme theoretic fibre $Y_x$", "is \\'etale over $x$ of degree $\\geq [L : K]$.", "\\end{enumerate}", "If $L/K$ is Galois with Galois group $G$, then these are also", "equivalent to", "\\begin{enumerate}", "\\item[(9)] for $y \\in Y$ the group", "$I_y = \\{g \\in G \\mid g(y) = y\\text{ and }", "g \\bmod \\mathfrak m_y = \\text{id}_{\\kappa(y)}\\}$ is trivial.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is the definition of (1).", "The equivalence of (2), (3), and (4) is Lemma \\ref{lemma-unramified-in-L}.", "It is straightforward to prove that (4) $\\Rightarrow$ (5),", "(5) $\\Rightarrow$ (6), (6) $\\Rightarrow$ (7).", "\\medskip\\noindent", "Assume (7). Observe that $\\mathcal{O}_{X, x}^{sh}$ is a normal local domain", "(More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-normal}).", "Let $L^{sh} = L \\otimes_K K_x^{sh}$ where $K_x^{sh}$ is the fraction field", "of $\\mathcal{O}_{X, x}^{sh}$. Then $L^{sh} = \\prod_{i = 1, \\ldots, n} L_i$", "with $L_i/K_x^{sh}$ finite separable. By", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-commutes-smooth}", "(and a limit argument we omit)", "we see that $Y \\times_X \\Spec(\\mathcal{O}_{X, x}^{sh})$", "is the integral closure of $\\Spec(\\mathcal{O}_{X, x}^{sh})$ in $L^{sh}$.", "Hence by Lemma \\ref{lemma-unramified-in-L} (applied to the factors", "$L_i$ of $L^{sh}$) we see that", "$Y \\times_X \\Spec(\\mathcal{O}_{X, x}^{sh}) \\to \\Spec(\\mathcal{O}_{X, x}^{sh})$", "is finite \\'etale. Looking at the generic point we see that", "the degree is equal to $[L : K]$ and hence we see that (8) is true.", "\\medskip\\noindent", "Assume (8). Assume that $x \\in X$ and that the scheme theoretic fibre $Y_x$", "is \\'etale over $x$ of degree $\\geq [L : K]$. Observe that this means", "that $Y$ has $\\geq [L : K]$ geometric points lying over $x$.", "We will show that $Y \\to X$ is finite \\'etale over a neighbourhood of $x$.", "This will prove (1) holds.", "To prove this we may assume $X = \\Spec(R)$, the point $x$ corresponds to", "the prime $\\mathfrak p \\subset R$, and $Y = \\Spec(S)$. We apply", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-etale-makes-integral-split} and we find an", "\\'etale neighbourhood $(U, u) \\to (X, x)$ such that", "$Y \\times_X U = V_1 \\amalg \\ldots \\amalg V_m$ such that $V_i$", "has a unique point $v_i$ lying over $u$ with $\\kappa(v_i)/\\kappa(u)$", "purely inseparable. Shrinking $U$ if necessary we may assume $U$ is", "a normal integral scheme with generic point $\\xi$ (use", "Descent, Lemmas \\ref{descent-lemma-locally-finite-nr-irred-local-fppf} and", "\\ref{descent-lemma-normal-local-smooth} and", "Properties, Lemma \\ref{properties-lemma-normal-locally-finite-nr-irreducibles}).", "By our remark on geometric points we see that $m \\geq [L : K]$.", "On the other hand, by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-normalization-smooth-localization}", "we see that $\\coprod V_i \\to U$ is the normalization of $U$ in", "$\\Spec(L) \\times_X U$. As $K \\subset \\kappa(\\xi)$ is finite separable,", "we can write $\\Spec(L) \\times_X U = \\Spec(\\prod_{i = 1, \\ldots, n} L_i)$", "with $L_i/\\kappa(\\xi)$ finite and $[L : K] = \\sum [L_i : \\kappa(\\xi)]$.", "Since $V_j$ is nonempty for each $j$ and $m \\geq [L : K]$", "we conclude that $m = n$ and $[L_i : \\kappa(\\xi)] = 1$", "for all $i$. Then $V_j \\to U$ is an isomorphism in particular", "\\'etale, hence $Y \\times_X U \\to U$ is \\'etale. By", "Descent, Lemma \\ref{descent-lemma-descending-property-etale}", "we conclude that $Y \\to X$ is \\'etale over the", "image of $U \\to X$ (an open neighbourhood of $x$).", "\\medskip\\noindent", "Assume $L/K$ is Galois and (9) holds. Then $Y \\to X$ is \\'etale", "by Lemma \\ref{lemma-inertial-invariants-unramified}.", "We omit the proof that (1) implies (9)." ], "refs": [ "pione-lemma-unramified-in-L", "more-algebra-lemma-henselization-normal", "algebra-lemma-integral-closure-commutes-smooth", "pione-lemma-unramified-in-L", "more-morphisms-lemma-etale-makes-integral-split", "descent-lemma-locally-finite-nr-irred-local-fppf", "descent-lemma-normal-local-smooth", "properties-lemma-normal-locally-finite-nr-irreducibles", "more-morphisms-lemma-normalization-smooth-localization", "descent-lemma-descending-property-etale", "pione-lemma-inertial-invariants-unramified" ], "ref_ids": [ 4054, 10060, 1252, 4054, 13898, 14650, 14654, 2969, 13774, 14694, 4061 ] } ], "ref_ids": [] }, { "id": 4065, "type": "theorem", "label": "pione-lemma-structure-decomposition", "categories": [ "pione" ], "title": "pione-lemma-structure-decomposition", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "Let $L/K$ be a (possibly infinite) Galois extension.", "Let $B$ be the integral closure of $A$ in $L$.", "Let $\\mathfrak m$ be a maximal ideal of $B$.", "Let $G = \\text{Gal}(L/K)$,", "$D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak m) = \\mathfrak m\\}$, and", "$I = \\{\\sigma \\in D \\mid \\sigma \\bmod \\mathfrak m =", "\\text{id}_{\\kappa(\\mathfrak m)}\\}$.", "The decomposition group $D$ fits into a canonical exact sequence", "$$", "1 \\to I \\to D \\to \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa_A) \\to 1", "$$", "The inertia group $I$ fits into a canonical exact sequence", "$$", "1 \\to P \\to I \\to I_t \\to 1", "$$", "such that", "\\begin{enumerate}", "\\item $P$ is a normal subgroup of $D$,", "\\item $P$ is a pro-$p$-group if the characteristic of", "$\\kappa_A$ is $p > 1$ and $P = \\{1\\}$ if the characteristic of $\\kappa_A$", "is zero,", "\\item there is a multiplicatively directed $S \\subset \\mathbf{N}$", "such that $\\kappa(\\mathfrak m)$ contains a primitive $n$th root of unity", "for each $n \\in S$ (elements of $S$ are prime to $p$),", "\\item there exists a canonical surjective map", "$$", "\\theta_{can} : I \\to \\lim_{n \\in S} \\mu_n(\\kappa(\\mathfrak m))", "$$", "whose kernel is $P$, which satisfies", "$\\theta_{can}(\\tau \\sigma \\tau^{-1}) = \\tau(\\theta_{can}(\\sigma))$", "for $\\tau \\in D$, $\\sigma \\in I$, and which induces an isomorphism", "$I_t \\to \\lim_{n \\in S} \\mu_n(\\kappa(\\mathfrak m))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is mostly a reformulation of the results on finite Galois extensions", "proved in More on Algebra, Section \\ref{more-algebra-section-ramification}.", "The surjectivity of the map $D \\to \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa)$ is", "More on Algebra, Lemma \\ref{more-algebra-lemma-one-orbit-geometric-galois}.", "This gives the first exact sequence.", "\\medskip\\noindent", "To construct the second short exact sequence let $\\Lambda$ be the set", "of finite Galois subextensions, i.e., $\\lambda \\in \\Lambda$ corresponds", "to $L/L_\\lambda/K$. Set $G_\\lambda = \\text{Gal}(L_\\lambda/K)$.", "Recall that $G_\\lambda$ is an inverse system of finite groups with surjective", "transition maps and that $G = \\lim_{\\lambda \\in \\Lambda} G_\\lambda$, see", "Fields, Lemma \\ref{fields-lemma-infinite-galois-limit}.", "We let $B_\\lambda$ be the integral closure of $A$ in $L_\\lambda$.", "Then we set $\\mathfrak m_\\lambda = \\mathfrak m \\cap B_\\lambda$", "and we denote $P_\\lambda, I_\\lambda, D_\\lambda$ the", "wild inertia, inertia, and decomposition group of", "$\\mathfrak m_\\lambda$, see More on Algebra, Lemma", "\\ref{more-algebra-lemma-galois-inertia}.", "For $\\lambda \\geq \\lambda'$ the restriction defines", "a commutative diagram", "$$", "\\xymatrix{", "P_\\lambda \\ar[d] \\ar[r] &", "I_\\lambda \\ar[d] \\ar[r] &", "D_\\lambda \\ar[d] \\ar[r] &", "G_\\lambda \\ar[d] \\\\", "P_{\\lambda'} \\ar[r] &", "I_{\\lambda'} \\ar[r] &", "D_{\\lambda'} \\ar[r] &", "G_{\\lambda'}", "}", "$$", "with surjective vertical maps, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-compare-inertia}.", "\\medskip\\noindent", "From the definitions it follows immediately", "that $I = \\lim I_\\lambda$ and $D = \\lim D_\\lambda$", "under the isomorphism $G = \\lim G_\\lambda$ above.", "Since $L = \\colim L_\\lambda$ we have $B = \\colim B_\\lambda$", "and $\\kappa(\\mathfrak m) = \\colim \\kappa(\\mathfrak m_\\lambda)$.", "Since the transition maps of the system $D_\\lambda$", "are compatible with the maps", "$D_\\lambda \\to \\text{Aut}(\\kappa(\\mathfrak m_\\lambda)/\\kappa)$", "(see More on Algebra, Lemma \\ref{more-algebra-lemma-compare-inertia})", "we see that the map $D \\to \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa)$", "is the limit of the maps", "$D_\\lambda \\to \\text{Aut}(\\kappa(\\mathfrak m_\\lambda)/\\kappa)$.", "\\medskip\\noindent", "There exist canonical maps", "$$", "\\theta_{\\lambda, can} :", "I_\\lambda", "\\longrightarrow", "\\mu_{n_\\lambda}(\\kappa(\\mathfrak m_\\lambda))", "$$", "where $n_\\lambda = |I_\\lambda|/|P_\\lambda|$, where", "$\\mu_{n_\\lambda}(\\kappa(\\mathfrak m_\\lambda))$ has", "order $n_\\lambda$, such that", "$\\theta_{\\lambda, can}(\\tau \\sigma \\tau^{-1}) =", "\\tau(\\theta_{\\lambda, can}(\\sigma))$ for", "$\\tau \\in D_\\lambda$ and $\\sigma \\in I_\\lambda$, and such that", "we get commutative diagrams", "$$", "\\xymatrix{", "I_\\lambda \\ar[r]_-{\\theta_{\\lambda, can}} \\ar[d] &", "\\mu_{n_\\lambda}(\\kappa(\\mathfrak m_\\lambda))", "\\ar[d]^{(-)^{n_\\lambda/n_{\\lambda'}}} \\\\", "I_{\\lambda'} \\ar[r]^-{\\theta_{\\lambda', can}} &", "\\mu_{n_{\\lambda'}}(\\kappa(\\mathfrak m_{\\lambda'}))", "}", "$$", "see", "More on Algebra, Remark \\ref{more-algebra-remark-canonical-inertia-character}.", "\\medskip\\noindent", "Let $S \\subset \\mathbf{N}$ be the collection of integers $n_\\lambda$.", "Since $\\Lambda$ is directed, we see that $S$ is multiplicatively directed.", "By the displayed commutative diagrams above we can take the limits of", "the maps $\\theta_{\\lambda, can}$ to obtain", "$$", "\\theta_{can} : I \\to \\lim_{n \\in S} \\mu_n(\\kappa(\\mathfrak m)).", "$$", "This map is continuous (small detail omitted). Since the transition maps", "of the system of $I_\\lambda$ are surjective", "and $\\Lambda$ is directed, the projections $I \\to I_\\lambda$", "are surjective. For every $\\lambda$ the diagram", "$$", "\\xymatrix{", "I \\ar[d] \\ar[r]_-{\\theta_{can}} &", "\\lim_{n \\in S} \\mu_n(\\kappa(\\mathfrak m)) \\ar[d] \\\\", "I_{\\lambda} \\ar[r]^-{\\theta_{\\lambda, can}} &", "\\mu_{n_\\lambda}(\\kappa(\\mathfrak m_\\lambda))", "}", "$$", "commutes. Hence the image of $\\theta_{can}$ surjects onto the finite group", "$\\mu_{n_\\lambda}(\\kappa(\\mathfrak m)) =", "\\mu_{n_\\lambda}(\\kappa(\\mathfrak m_\\lambda))$ of order $n_\\lambda$", "(see above). It follows that the image of $\\theta_{can}$ is dense.", "On the other hand $\\theta_{can}$ is continuous and the", "source is a profinite group. Hence $\\theta_{can}$ is surjective", "by a topological argument.", "\\medskip\\noindent", "The property $\\theta_{can}(\\tau \\sigma \\tau^{-1}) = \\tau(\\theta_{can}(\\sigma))$", "for $\\tau \\in D$, $\\sigma \\in I$ follows from the corresponding properties", "of the maps $\\theta_{\\lambda, can}$ and the compatibility of the map", "$D \\to \\text{Aut}(\\kappa(\\mathfrak m))$ with the maps", "$D_\\lambda \\to \\text{Aut}(\\kappa(\\mathfrak m_\\lambda))$.", "Setting $P = \\Ker(\\theta_{can})$ this implies", "that $P$ is a normal subgroup of $D$. Setting $I_t = I/P$", "we obtain the isomorphism $I_t \\to \\lim_{n \\in S} \\mu_n(\\kappa(\\mathfrak m))$", "from the surjectivity of $\\theta_{can}$.", "\\medskip\\noindent", "To finish the proof we show that $P = \\lim P_\\lambda$ which proves", "that $P$ is a pro-$p$-group. Recall that the tame inertia group", "$I_{\\lambda, t} = I_\\lambda/P_\\lambda$ has order $n_\\lambda$.", "Since the transition maps $P_\\lambda \\to P_{\\lambda'}$ are surjective", "and $\\Lambda$ is directed, we obtain a short exact sequence", "$$", "1 \\to \\lim P_\\lambda \\to I \\to \\lim I_{\\lambda, t} \\to 1", "$$", "(details omitted). Since for each $\\lambda$ the map $\\theta_{\\lambda, can}$", "induces an isomorphism", "$I_{\\lambda, t} \\cong \\mu_{n_\\lambda}(\\kappa(\\mathfrak m))$", "the desired result follows." ], "refs": [ "more-algebra-lemma-one-orbit-geometric-galois", "fields-lemma-infinite-galois-limit", "more-algebra-lemma-galois-inertia", "more-algebra-lemma-compare-inertia", "more-algebra-lemma-compare-inertia", "more-algebra-remark-canonical-inertia-character" ], "ref_ids": [ 10490, 4511, 10501, 10504, 10504, 10678 ] } ], "ref_ids": [] }, { "id": 4066, "type": "theorem", "label": "pione-lemma-structure-decomposition-separable-closure", "categories": [ "pione" ], "title": "pione-lemma-structure-decomposition-separable-closure", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "Let $K^{sep}$ be a separable closure of $K$.", "Let $A^{sep}$ be the integral closure of $A$ in $K^{sep}$.", "Let $\\mathfrak m^{sep}$ be a maximal ideal of $A^{sep}$.", "Let $\\mathfrak m = \\mathfrak m^{sep} \\cap A$, let", "$\\kappa = A/\\mathfrak m$, and let", "$\\overline{\\kappa} = A^{sep}/\\mathfrak m^{sep}$.", "Then $\\overline{\\kappa}$ is an algebraic closure of $\\kappa$.", "Let $G = \\text{Gal}(K^{sep}/K)$,", "$D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak m^{sep}) = \\mathfrak m^{sep}\\}$, and", "$I = \\{\\sigma \\in D \\mid \\sigma \\bmod \\mathfrak m^{sep} =", "\\text{id}_{\\kappa(\\mathfrak m^{sep})}\\}$.", "The decomposition group $D$ fits into a canonical exact sequence", "$$", "1 \\to I \\to D \\to \\text{Gal}(\\kappa^{sep}/\\kappa) \\to 1", "$$", "where $\\kappa^{sep} \\subset \\overline{\\kappa}$ is the separable", "closure of $\\kappa$.", "The inertia group $I$ fits into a canonical exact sequence", "$$", "1 \\to P \\to I \\to I_t \\to 1", "$$", "such that", "\\begin{enumerate}", "\\item $P$ is a normal subgroup of $D$,", "\\item $P$ is a pro-$p$-group if the characteristic of", "$\\kappa_A$ is $p > 1$ and $P = \\{1\\}$ if the characteristic of $\\kappa_A$", "is zero,", "\\item there exists a canonical surjective map", "$$", "\\theta_{can} : I \\to \\lim_{n\\text{ prime to }p} \\mu_n(\\kappa^{sep})", "$$", "whose kernel is $P$, which satisfies", "$\\theta_{can}(\\tau \\sigma \\tau^{-1}) = \\tau(\\theta_{can}(\\sigma))$", "for $\\tau \\in D$, $\\sigma \\in I$, and which induces an isomorphism", "$I_t \\to \\lim_{n\\text{ prime to }p} \\mu_n(\\kappa^{sep})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The field $\\overline{\\kappa}$ is the algebraic closure of $\\kappa$ by", "Lemma \\ref{lemma-get-algebraic-closure}.", "Most of the statements immediately follow from the corresponding", "parts of Lemma \\ref{lemma-structure-decomposition}. For example because", "$\\text{Aut}(\\overline{\\kappa}/\\kappa) = \\text{Gal}(\\kappa^{sep}/\\kappa)$", "we obtain the first sequence.", "Then the only other assertion that needs a proof is the fact that", "with $S$ as in Lemma \\ref{lemma-structure-decomposition} the", "limit $\\lim_{n \\in S} \\mu_n(\\overline{\\kappa})$ is equal to", "$\\lim_{n\\text{ prime to }p} \\mu_n(\\kappa^{sep})$. To see this", "it suffices to show that every integer $n$ prime to $p$", "divides an element of $S$.", "Let $\\pi \\in A$ be a uniformizer and consider the splitting", "field $L$ of the polynomial $X^n - \\pi$. Since the polynomial", "is separable we see that $L$ is a finite Galois extension of $K$.", "Choose an embedding $L \\to K^{sep}$.", "Observe that if $B$ is the integral closure of $A$ in $L$,", "then the ramification index of $A \\to B_{\\mathfrak m^{sep} \\cap B}$", "is divisible by $n$ (because $\\pi$ has an $n$th root in $B$; in fact", "the ramification index equals $n$ but we do not need this).", "Then it follows from the construction of the $S$ in the proof of", "Lemma \\ref{lemma-structure-decomposition}", "that $n$ divides an element of $S$." ], "refs": [ "pione-lemma-get-algebraic-closure", "pione-lemma-structure-decomposition", "pione-lemma-structure-decomposition", "pione-lemma-structure-decomposition" ], "ref_ids": [ 4057, 4065, 4065, 4065 ] } ], "ref_ids": [] }, { "id": 4067, "type": "theorem", "label": "pione-lemma-limit", "categories": [ "pione" ], "title": "pione-lemma-limit", "contents": [ "Let $I$ be a directed set. Let $X_i$ be an", "inverse system of quasi-compact and quasi-separated schemes", "over $I$ with affine transition morphisms.", "Let $X = \\lim X_i$ as in Limits, Section \\ref{limits-section-limits}.", "Then there is an equivalence of categories", "$$", "\\colim \\textit{F\\'Et}_{X_i} = \\textit{F\\'Et}_X", "$$", "If $X_i$ is connected for all sufficiently large $i$ and $\\overline{x}$", "is a geometric point of $X$, then", "$$", "\\pi_1(X, \\overline{x}) = \\lim \\pi_1(X_i, \\overline{x})", "$$" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of categories follows from Limits, Lemmas", "\\ref{limits-lemma-descend-finite-presentation},", "\\ref{limits-lemma-descend-finite-finite-presentation}, and", "\\ref{limits-lemma-descend-etale}.", "The second statement is formal given the statement on", "categories." ], "refs": [ "limits-lemma-descend-finite-presentation", "limits-lemma-descend-finite-finite-presentation", "limits-lemma-descend-etale" ], "ref_ids": [ 15077, 15058, 15065 ] } ], "ref_ids": [] }, { "id": 4068, "type": "theorem", "label": "pione-lemma-perfection", "categories": [ "pione" ], "title": "pione-lemma-perfection", "contents": [ "Let $k$ be a field with perfection $k^{perf}$. Let $X$ be a connected scheme", "over $k$. Then $X_{k^{perf}}$ is connected and", "$\\pi_1(X_{k^{perf}}) \\to \\pi_1(X)$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Special case of topological invariance of the fundamental group.", "See Proposition \\ref{proposition-universal-homeomorphism}.", "To see that $\\Spec(k^{perf}) \\to \\Spec(k)$ is a universal", "homeomorphism you can use", "Algebra, Lemma \\ref{algebra-lemma-radicial-integral-bijective}." ], "refs": [ "pione-proposition-universal-homeomorphism", "algebra-lemma-radicial-integral-bijective" ], "ref_ids": [ 4131, 585 ] } ], "ref_ids": [] }, { "id": 4069, "type": "theorem", "label": "pione-lemma-ses-field", "categories": [ "pione" ], "title": "pione-lemma-ses-field", "contents": [ "Let $k$ be a field with algebraic closure $\\overline{k}$.", "Let $X$ be a quasi-compact and quasi-separated scheme over $k$.", "If the base change $X_{\\overline{k}}$ is connected, then", "there is a short exact sequence", "$$", "1 \\to \\pi_1(X_{\\overline{k}}) \\to \\pi_1(X) \\to \\pi_1(\\Spec(k)) \\to 1", "$$", "of profinite topological groups." ], "refs": [], "proofs": [ { "contents": [ "Connected objects of $\\textit{F\\'Et}_{\\Spec(k)}$ are of the form", "$\\Spec(k') \\to \\Spec(k)$ with $k'/k$ a finite separable extension.", "Then $X_{\\Spec{k'}}$ is connected, as the morphism", "$X_{\\overline{k}} \\to X_{\\Spec(k')}$ is surjective and", "$X_{\\overline{k}}$ is connected by assumption. Thus", "$\\pi_1(X) \\to \\pi_1(\\Spec(k))$ is surjective by", "Lemma \\ref{lemma-functoriality-galois-surjective}.", "\\medskip\\noindent", "Before we go on, note that we may assume that $k$ is a perfect field.", "Namely, we have $\\pi_1(X_{k^{perf}}) = \\pi_1(X)$ and", "$\\pi_1(\\Spec(k^{perf})) = \\pi_1(\\Spec(k))$ by Lemma \\ref{lemma-perfection}.", "\\medskip\\noindent", "It is clear that the composition of the functors", "$\\textit{F\\'Et}_{\\Spec(k)} \\to \\textit{F\\'Et}_X \\to", "\\textit{F\\'Et}_{X_{\\overline{k}}}$ sends objects to disjoint unions", "of copies of $X_{\\Spec(\\overline{k})}$. Therefore the composition", "$\\pi_1(X_{\\overline{k}}) \\to \\pi_1(X) \\to \\pi_1(\\Spec(k))$", "is the trivial homomorphism by Lemma \\ref{lemma-composition-trivial}.", "\\medskip\\noindent", "Let $U \\to X$ be a finite \\'etale morphism with $U$ connected.", "Observe that $U \\times_X X_{\\overline{k}} = U_{\\overline{k}}$.", "Suppose that $U_{\\overline{k}} \\to X_{\\overline{k}}$", "has a section $s : X_{\\overline{k}} \\to U_{\\overline{k}}$.", "Then $s(X_{\\overline{k}})$ is an open connected component of", "$U_{\\overline{k}}$. For $\\sigma \\in \\text{Gal}(\\overline{k}/k)$", "denote $s^\\sigma$ the base change of $s$ by $\\Spec(\\sigma)$.", "Since $U_{\\overline{k}} \\to X_{\\overline{k}}$ is finite \\'etale", "it has only a finite number of sections. Thus", "$$", "\\overline{T} = \\bigcup s^\\sigma(X_{\\overline{k}})", "$$", "is a finite union and we see that $\\overline{T}$ is a", "$\\text{Gal}(\\overline{k}/k)$-stable open and closed subset.", "By Varieties, Lemma \\ref{varieties-lemma-closed-fixed-by-Galois}", "we see that $\\overline{T}$ is the inverse image of a closed", "subset $T \\subset U$. Since $U_{\\overline{k}} \\to U$ is open", "(Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open})", "we conclude that $T$ is open as well. As $U$ is connected we", "see that $T = U$. Hence $U_{\\overline{k}}$ is a (finite) disjoint", "union of copies of $X_{\\overline{k}}$. By", "Lemma \\ref{lemma-functoriality-galois-normal} we conclude that the image of", "$\\pi_1(X_{\\overline{k}}) \\to \\pi_1(X)$ is normal.", "\\medskip\\noindent", "Let $V \\to X_{\\overline{k}}$ be a finite \\'etale cover. Recall that", "$\\overline{k}$ is the union of finite separable extensions of $k$.", "By Lemma \\ref{lemma-limit} we find a finite separable extension $k'/k$", "and a finite \\'etale morphism $U \\to X_{k'}$ such that", "$V = X_{\\overline{k}} \\times_{X_{k'}} U =", "U \\times_{\\Spec(k')} \\Spec(\\overline{k})$.", "Then the composition $U \\to X_{k'} \\to X$ is finite \\'etale", "and $U \\times_{\\Spec(k)} \\Spec(\\overline{k})$", "contains $V = U \\times_{\\Spec(k')} \\Spec(\\overline{k})$", "as an open and closed subscheme. (Because $\\Spec(\\overline{k})$", "is an open and closed subscheme of", "$\\Spec(k') \\times_{\\Spec(k)} \\Spec(\\overline{k})$ via", "the multiplication map $k' \\otimes_k \\overline{k} \\to \\overline{k}$.) By", "Lemma \\ref{lemma-functoriality-galois-injective}", "we conclude that $\\pi_1(X_{\\overline{k}}) \\to \\pi_1(X)$ is injective.", "\\medskip\\noindent", "Finally, we have to show that for any finite \\'etale morphism", "$U \\to X$ such that $U_{\\overline{k}}$ is a disjoint union", "of copies of $X_{\\overline{k}}$ there is a finite \\'etale", "morphism $V \\to \\Spec(k)$ and a surjection $V \\times_{\\Spec(k)} X \\to U$.", "See Lemma \\ref{lemma-functoriality-galois-ses}.", "Arguing as above using Lemma \\ref{lemma-limit}", "we find a finite separable extension $k'/k$", "such that there is an isomorphism", "$U_{k'} \\cong \\coprod_{i = 1, \\ldots, n} X_{k'}$.", "Thus setting $V = \\coprod_{i = 1, \\ldots, n} \\Spec(k')$", "we conclude." ], "refs": [ "pione-lemma-functoriality-galois-surjective", "pione-lemma-perfection", "pione-lemma-composition-trivial", "varieties-lemma-closed-fixed-by-Galois", "morphisms-lemma-scheme-over-field-universally-open", "pione-lemma-functoriality-galois-normal", "pione-lemma-limit", "pione-lemma-functoriality-galois-injective", "pione-lemma-functoriality-galois-ses", "pione-lemma-limit" ], "ref_ids": [ 4032, 4068, 4033, 10922, 5254, 4036, 4067, 4035, 4034, 4067 ] } ], "ref_ids": [] }, { "id": 4070, "type": "theorem", "label": "pione-lemma-stein-factorization-etale", "categories": [ "pione" ], "title": "pione-lemma-stein-factorization-etale", "contents": [ "\\begin{reference}", "\\cite[Expose X, Proposition 1.2, p. 262]{SGA1}.", "\\end{reference}", "Let $f : X \\to S$ be a proper morphism of schemes.", "Let $X \\to S' \\to S$ be the Stein factorization of $f$, see", "More on Morphisms, Theorem", "\\ref{more-morphisms-theorem-stein-factorization-general}.", "If $f$ is of finite presentation, flat, with geometrically", "reduced fibres, then $S' \\to S$ is finite \\'etale." ], "refs": [ "more-morphisms-theorem-stein-factorization-general" ], "proofs": [ { "contents": [ "This follows from Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-proper-flat-geom-red}", "and the information contained in", "More on Morphisms, Theorem", "\\ref{more-morphisms-theorem-stein-factorization-general}." ], "refs": [ "perfect-lemma-proper-flat-geom-red", "more-morphisms-theorem-stein-factorization-general" ], "ref_ids": [ 7069, 13675 ] } ], "ref_ids": [ 13675 ] }, { "id": 4071, "type": "theorem", "label": "pione-lemma-specialization-map-base-change", "categories": [ "pione" ], "title": "pione-lemma-specialization-map-base-change", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "Y \\ar[d]_g \\ar[r] & X \\ar[d]^f \\\\", "T \\ar[r] & S", "}", "$$", "of schemes where $f$ and $g$ are proper with geometrically connected", "fibres. Let $t' \\leadsto t$ be a specialization of points in $T$", "and consider a specialization map", "$sp : \\pi_1(Y_{\\overline{t}'}) \\to \\pi_1(Y_{\\overline{t}})$ as above.", "Then there is a commutative diagram", "$$", "\\xymatrix{", "\\pi_1(Y_{\\overline{t}'}) \\ar[r]_{sp} \\ar[d] & \\pi_1(Y_{\\overline{t}}) \\ar[d] \\\\", "\\pi_1(X_{\\overline{s}'}) \\ar[r]^{sp} & \\pi_1(X_{\\overline{s}})", "}", "$$", "of specialization maps where $\\overline{s}$ and $\\overline{s}'$", "are the images of $\\overline{t}$ and $\\overline{t}'$." ], "refs": [], "proofs": [ { "contents": [ "Let $B$ be the strict henselization of $\\mathcal{O}_{T, t}$ with respect to", "$\\kappa(t) \\subset \\kappa(t)^{sep} \\subset \\kappa(\\overline{t})$.", "Pick $\\psi : \\overline{t}' \\to \\Spec(B)$ lifting $\\overline{t}' \\to T$", "as in the construction of the specialization map.", "Let $s$ and $s'$ denote the images of $t$ and $t'$ in $S$.", "Let $A$ be the strict henselization of $\\mathcal{O}_{S, s}$", "with respect to", "$\\kappa(s) \\subset \\kappa(s)^{sep} \\subset \\kappa(\\overline{s})$.", "Since $\\kappa(\\overline{s}) = \\kappa(\\overline{t})$,", "by the functoriality of strict henselization", "(Algebra, Lemma \\ref{algebra-lemma-strictly-henselian-functorial})", "we obtain a ring map $A \\to B$ fitting into the commutative diagram", "$$", "\\xymatrix{", "\\overline{t}' \\ar[r]_-\\psi \\ar[d] & \\Spec(B) \\ar[d] \\ar[r] & T \\ar[d] \\\\", "\\overline{s}' \\ar[r]^-\\varphi & \\Spec(A) \\ar[r] & S", "}", "$$", "Here the morphism $\\varphi : \\overline{s}' \\to \\Spec(A)$ is simply taken", "to be the composition $\\overline{t}' \\to \\Spec(B) \\to \\Spec(A)$.", "Applying base change we obtain a commutative diagram", "$$", "\\xymatrix{", "Y_{\\overline{t}'} \\ar[r] \\ar[d] & Y_B \\ar[d] \\\\", "X_{\\overline{s}'} \\ar[r] & X_A", "}", "$$", "and from the construction of the specialization map the commutativity", "of this diagram implies the commutativity of the diagram of the lemma." ], "refs": [ "algebra-lemma-strictly-henselian-functorial" ], "ref_ids": [ 1303 ] } ], "ref_ids": [] }, { "id": 4072, "type": "theorem", "label": "pione-lemma-specialization-map-composition", "categories": [ "pione" ], "title": "pione-lemma-specialization-map-composition", "contents": [ "Let $f : X \\to S$ be a proper morphism with geometrically connected fibres.", "Let $s'' \\leadsto s' \\leadsto s$ be specializations of points of $S$.", "A composition of specialization maps", "$\\pi_1(X_{\\overline{s}''}) \\to \\pi_1(X_{\\overline{s}'}) \\to", "\\pi_1(X_{\\overline{s}})$ is a specialization map", "$\\pi_1(X_{\\overline{s}''}) \\to \\pi_1(X_{\\overline{s}})$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{O}_{S, s} \\to A$ be the strict henselization", "constructed using $\\kappa(s) \\to \\kappa(\\overline{s})$.", "Let $A \\to \\kappa(\\overline{s}')$ be the map used to construct", "the first specialization map. Let $\\mathcal{O}_{S, s'} \\to A'$", "be the strict henselization constructed using", "$\\kappa(s') \\subset \\kappa(\\overline{s}')$.", "By functoriality of strict henselization, there is a map", "$A \\to A'$ such that the composition with $A' \\to \\kappa(\\overline{s}')$", "is the given map", "(Algebra, Lemma \\ref{algebra-lemma-map-into-henselian-colimit}).", "Next, let $A' \\to \\kappa(\\overline{s}'')$ be the map used to", "construct the second specialization map. Then it is clear that", "the composition of the first and second specialization maps", "is the specialization map", "$\\pi_1(X_{\\overline{s}''}) \\to \\pi_1(X_{\\overline{s}})$", "constructed using $A \\to A' \\to \\kappa(\\overline{s}'')$." ], "refs": [ "algebra-lemma-map-into-henselian-colimit" ], "ref_ids": [ 1291 ] } ], "ref_ids": [] }, { "id": 4073, "type": "theorem", "label": "pione-lemma-specialization-map-valuation-ring", "categories": [ "pione" ], "title": "pione-lemma-specialization-map-valuation-ring", "contents": [ "Let $f : X \\to S$ be a proper morphism with geometrically connected fibres.", "Let $s' \\leadsto s$ be a specialization of points of $S$ and let", "$sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})$", "be a specialization map. Then there exists a strictly henselian", "valuation ring $R$ over $S$ with algebraically closed fraction field", "such that $sp$ is isomorphic to $sp_R$ defined above." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{O}_{S, s} \\to A$ be the strict henselization", "constructed using $\\kappa(s) \\to \\kappa(\\overline{s})$.", "Let $A \\to \\kappa(\\overline{s}')$ be the map used to construct $sp$.", "Let $R \\subset \\kappa(\\overline{s}')$ be a valuation ring with", "fraction field $\\kappa(\\overline{s}')$ dominating the image of $A$.", "See Algebra, Lemma \\ref{algebra-lemma-dominate}.", "Observe that $R$ is strictly henselian for example by", "Lemma \\ref{lemma-normal-local-domain-separablly-closed-fraction-field}", "and Algebra, Lemma \\ref{algebra-lemma-valuation-ring-normal}.", "Then the lemma is clear." ], "refs": [ "algebra-lemma-dominate", "pione-lemma-normal-local-domain-separablly-closed-fraction-field", "algebra-lemma-valuation-ring-normal" ], "ref_ids": [ 608, 4058, 616 ] } ], "ref_ids": [] }, { "id": 4074, "type": "theorem", "label": "pione-lemma-specialization-map-discrete-valuation-ring", "categories": [ "pione" ], "title": "pione-lemma-specialization-map-discrete-valuation-ring", "contents": [ "Let $f : X \\to S$ be a proper morphism with geometrically connected fibres.", "Let $s' \\leadsto s$ be a specialization of points of $S$ and let", "$sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})$", "be a specialization map. If $S$ is Noetherian, then", "there exists a strictly henselian", "discrete valuation ring $R$ over $S$ such that $sp$ is isomorphic to $sp_R$", "defined above." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{O}_{S, s} \\to A$ be the strict henselization", "constructed using $\\kappa(s) \\to \\kappa(\\overline{s})$.", "Let $A \\to \\kappa(\\overline{s}')$ be the map used to construct $sp$.", "Let $R \\subset \\kappa(\\overline{s}')$ be a discrete valuation ring", "dominating the image of $A$, see Algebra, Lemma \\ref{algebra-lemma-exists-dvr}.", "Choose a diagram of fields", "$$", "\\xymatrix{", "\\kappa(\\overline{s}) \\ar[r] & k \\\\", "A/\\mathfrak m_A \\ar[r] \\ar[u] & R/\\mathfrak m_R \\ar[u]", "}", "$$", "with $k$ algebraically closed. Let $R^{sh}$ be the strict", "henselization of $R$ constructed using $R \\to k$. Then", "$R^{sh}$ is a discrete valuation ring by", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-dvr}.", "Denote $\\eta, o$ the generic and closed point of $\\Spec(R^{sh})$.", "Since the diagram of schemes", "$$", "\\xymatrix{", "\\overline{\\eta} \\ar[d] \\ar[r] & \\Spec(R^{sh}) \\ar[d] &", "\\Spec(k) \\ar[d] \\ar[l] \\\\", "\\overline{s}' \\ar[r] & \\Spec(A) & \\overline{s} \\ar[l]", "}", "$$", "commutes, we obtain a commutative diagram", "$$", "\\xymatrix{", "\\pi_1(X_{\\overline{\\eta}}) \\ar[d] \\ar[r]_{sp_{R^{sh}}} & \\pi_1(X_o) \\ar[d] \\\\", "\\pi_1(X_{\\overline{s}'}) \\ar[r]^{sp} & X_{\\overline{s}}", "}", "$$", "of specialization maps by the construction of these maps.", "Since the vertical arrows are isomorphisms", "(Lemma \\ref{lemma-finite-etale-invariant-over-proper}), this proves the lemma." ], "refs": [ "algebra-lemma-exists-dvr", "more-algebra-lemma-henselization-dvr", "pione-lemma-finite-etale-invariant-over-proper" ], "ref_ids": [ 1028, 10065, 4045 ] } ], "ref_ids": [] }, { "id": 4075, "type": "theorem", "label": "pione-lemma-restriction-fully-faithful", "categories": [ "pione" ], "title": "pione-lemma-restriction-fully-faithful", "contents": [ "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme", "with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.", "Assume the completion functor", "$$", "\\textit{Coh}(\\mathcal{O}_X)", "\\longrightarrow ", "\\textit{Coh}(X, \\mathcal{I}),\\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge", "$$", "is fully faithful on the full subcategory of finite locally free objects", "(see above).", "Then the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$", "is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "Since the category of finite \\'etale coverings has an", "internal hom (Lemma \\ref{lemma-internal-hom-finite-etale})", "it suffices to prove the following: Given $U$ finite \\'etale over $X$", "and a morphism $t : Y \\to U$ over $X$ there exists a unique section", "$s : X \\to U$ such that $t = s|_Y$. Picture", "$$", "\\xymatrix{", "& U \\ar[d]^f \\\\", "Y \\ar[r] \\ar[ru] & X \\ar@{..>}@/^1em/[u]", "}", "$$", "Finding the dotted arrow $s$ is the same thing as finding an", "$\\mathcal{O}_X$-algebra map", "$$", "s^\\sharp : f_*\\mathcal{O}_U \\longrightarrow \\mathcal{O}_X", "$$", "which reduces modulo the ideal sheaf of $Y$ to the given algebra map", "$t^\\sharp : f_*\\mathcal{O}_U \\to \\mathcal{O}_Y$.", "By Lemma \\ref{lemma-thickening} we can lift $t$ uniquely to a compatible", "system of maps $t_n : Y_n \\to U$ and hence a map", "$$", "\\lim t_n^\\sharp : f_*\\mathcal{O}_U \\longrightarrow \\lim \\mathcal{O}_{Y_n}", "$$", "of sheaves of algebras on $X$.", "Since $f_*\\mathcal{O}_U$ is a finite locally free $\\mathcal{O}_X$-module,", "we conclude that we get a unique $\\mathcal{O}_X$-module map", "$\\sigma : f_*\\mathcal{O}_U \\to \\mathcal{O}_X$ whose completion", "is $\\lim t_n^\\sharp$. To see that $\\sigma$ is an algebra homomorphism,", "we need to check that the diagram", "$$", "\\xymatrix{", "f_*\\mathcal{O}_U \\otimes_{\\mathcal{O}_X} f_*\\mathcal{O}_U", "\\ar[r] \\ar[d]_{\\sigma \\otimes \\sigma} &", "f_*\\mathcal{O}_U \\ar[d]^\\sigma \\\\", "\\mathcal{O}_X \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X \\ar[r] &", "\\mathcal{O}_X", "}", "$$", "commutes. For every $n$ we know this diagram commutes after restricting", "to $Y_n$, i.e., the diagram commutes after applying the completion functor.", "Hence by faithfulness of the completion functor we conclude." ], "refs": [ "pione-lemma-internal-hom-finite-etale", "pione-lemma-thickening" ], "ref_ids": [ 4038, 4043 ] } ], "ref_ids": [] }, { "id": 4076, "type": "theorem", "label": "pione-lemma-restriction-equivalence", "categories": [ "pione" ], "title": "pione-lemma-restriction-equivalence", "contents": [ "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme", "with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.", "Assume the completion functor", "$$", "\\textit{Coh}(\\mathcal{O}_X)", "\\longrightarrow ", "\\textit{Coh}(X, \\mathcal{I}),\\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge", "$$", "is an equivalence on full subcategories of finite locally free objects", "(see above).", "Then the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "The restriction functor is fully faithful by", "Lemma \\ref{lemma-restriction-fully-faithful}.", "\\medskip\\noindent", "Let $U_1 \\to Y$ be a finite \\'etale morphism. To finish the proof", "we will show that $U_1$ is in the essential image of the", "restriction functor.", "\\medskip\\noindent", "For $n \\geq 1$ let $Y_n$ be the $n$th infinitesimal neighbourhood of $Y$.", "By Lemma \\ref{lemma-thickening}", "there is a unique finite \\'etale morphism", "$\\pi_n : U_n \\to Y_n$ whose base change to $Y = Y_1$", "recovers $U_1 \\to Y_1$.", "Consider the sheaves $\\mathcal{F}_n = \\pi_{n, *}\\mathcal{O}_{U_n}$.", "We may and do view $\\mathcal{F}_n$ as an $\\mathcal{O}_X$-module on $X$", "wich is locally isomorphic to", "$(\\mathcal{O}_X/f^{n + 1}\\mathcal{O}_X)^{\\oplus r}$.", "This $(\\mathcal{F}_n)$ is a finite locally free object of", "$\\textit{Coh}(X, \\mathcal{I})$.", "By assumption there exists a finite locally free $\\mathcal{O}_X$-module", "$\\mathcal{F}$ and a compatible system of isomorphisms", "$$", "\\mathcal{F}/\\mathcal{I}^n\\mathcal{F} \\to \\mathcal{F}_n", "$$", "of $\\mathcal{O}_X$-modules.", "\\medskip\\noindent", "To construct an algebra structure on $\\mathcal{F}$ consider the multiplication", "maps", "$\\mathcal{F}_n \\otimes_{\\mathcal{O}_X} \\mathcal{F}_n \\to \\mathcal{F}_n$", "coming from the fact that $\\mathcal{F}_n = \\pi_{n, *}\\mathcal{O}_{U_n}$", "are sheaves of algebras. These define a map", "$$", "(\\mathcal{F}\\otimes_{\\mathcal{O}_X} \\mathcal{F})^\\wedge", "\\longrightarrow", "\\mathcal{F}^\\wedge", "$$", "in the category $\\textit{Coh}(X, \\mathcal{I})$. Hence by assumption", "we may assume there is a map", "$\\mu : \\mathcal{F}\\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{F}$", "whose restriction to $Y_n$ gives the multiplication maps above.", "After possibly shrinking further we may assume $\\mu$", "defines a commutative $\\mathcal{O}_X$-algebra", "structure on $\\mathcal{F}$ compatible with the given algebra", "structures on $\\mathcal{F}_n$.", "Setting", "$$", "U = \\underline{\\Spec}_X((\\mathcal{F}, \\mu))", "$$", "we obtain a finite locally free scheme $\\pi : U \\to X$ whose restriction", "to $Y$ is isomorphic to $U_1$. The the discriminant of $\\pi$ is the zero", "set of the section", "$$", "\\det(Q_\\pi) :", "\\mathcal{O}_X", "\\longrightarrow", "\\wedge^{top}(\\pi_*\\mathcal{O}_U)^{\\otimes -2}", "$$", "constructed in", "Discriminants, Section \\ref{discriminant-section-discriminant}.", "Since the restriction of this to $Y_n$ is an isomorphism for all $n$", "by Discriminants, Lemma \\ref{discriminant-lemma-discriminant}", "we conclude that it is an isomorphism. Thus $\\pi$ is \\'etale by", "Discriminants, Lemma \\ref{discriminant-lemma-discriminant}." ], "refs": [ "pione-lemma-restriction-fully-faithful", "pione-lemma-thickening", "discriminant-lemma-discriminant", "discriminant-lemma-discriminant" ], "ref_ids": [ 4075, 4043, 14955, 14955 ] } ], "ref_ids": [] }, { "id": 4077, "type": "theorem", "label": "pione-lemma-restriction-fully-faithful-general", "categories": [ "pione" ], "title": "pione-lemma-restriction-fully-faithful-general", "contents": [ "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme", "with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.", "Let $\\mathcal{V}$ be the set of open subschemes $V \\subset X$ containing $Y$", "ordered by reverse inclusion. Assume the completion functor", "$$", "\\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V)", "\\longrightarrow", "\\textit{Coh}(X, \\mathcal{I}),", "\\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge", "$$", "defines is fully faithful on the full subcategory of", "finite locally free objects (see above).", "Then the restriction functor", "$\\colim_\\mathcal{V} \\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y$", "is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\mathcal{V}$ is a directed set, so the colimits are", "as in Categories, Section \\ref{categories-section-directed-colimits}.", "The rest of the argument is almost exactly the same as the argument", "in the proof of Lemma \\ref{lemma-restriction-fully-faithful}; we urge", "the reader to skip it.", "\\medskip\\noindent", "Since the category of finite \\'etale coverings has an", "internal hom (Lemma \\ref{lemma-internal-hom-finite-etale})", "it suffices to prove the following: Given $U$ finite \\'etale over", "$V \\in \\mathcal{V}$", "and a morphism $t : Y \\to U$ over $V$ there exists a $V' \\geq V$", "and a morphism $s : V' \\to U$ over $V$ such that $t = s|_Y$. Picture", "$$", "\\xymatrix{", "& & U \\ar[d]^f \\\\", "Y \\ar[r] \\ar[rru] & V' \\ar@{..>}[ru] \\ar[r] & V", "}", "$$", "Finding the dotted arrow $s$ is the same thing as finding an", "$\\mathcal{O}_{V'}$-algebra map", "$$", "s^\\sharp : f_*\\mathcal{O}_U|_{V'} \\longrightarrow \\mathcal{O}_{V'}", "$$", "which reduces modulo the ideal sheaf of $Y$ to the given algebra map", "$t^\\sharp : f_*\\mathcal{O}_U \\to \\mathcal{O}_Y$.", "By Lemma \\ref{lemma-thickening} we can lift $t$ uniquely to a compatible", "system of maps $t_n : Y_n \\to U$ and hence a map", "$$", "\\lim t_n^\\sharp : f_*\\mathcal{O}_U \\longrightarrow \\lim \\mathcal{O}_{Y_n}", "$$", "of sheaves of algebras on $V$.", "Observe that $f_*\\mathcal{O}_U$ is a finite locally free", "$\\mathcal{O}_V$-module. Hence we get a $V' \\geq V$ a map", "$\\sigma : f_*\\mathcal{O}_U|_{V'} \\to \\mathcal{O}_{V'}$", "whose completion is $\\lim t_n^\\sharp$.", "To see that $\\sigma$ is an algebra homomorphism, we need to check", "that the diagram", "$$", "\\xymatrix{", "(f_*\\mathcal{O}_U \\otimes_{\\mathcal{O}_V} f_*\\mathcal{O}_U)|_{V'}", "\\ar[r] \\ar[d]_{\\sigma \\otimes \\sigma} &", "f_*\\mathcal{O}_U|_{V'} \\ar[d]^\\sigma \\\\", "\\mathcal{O}_{V'} \\otimes_{\\mathcal{O}_{V'}} \\mathcal{O}_{V'} \\ar[r] &", "\\mathcal{O}_{V'}", "}", "$$", "commutes. For every $n$ we know this diagram commutes after restricting", "to $Y_n$, i.e., the diagram commutes after applying the completion functor.", "Hence by faithfulness of the completion functor", "we deduce that there exists a $V'' \\geq V'$ such that", "$\\sigma|_{V''}$ is an algebra homomorphism as desired." ], "refs": [ "pione-lemma-restriction-fully-faithful", "pione-lemma-internal-hom-finite-etale", "pione-lemma-thickening" ], "ref_ids": [ 4075, 4038, 4043 ] } ], "ref_ids": [] }, { "id": 4078, "type": "theorem", "label": "pione-lemma-restriction-equivalence-general", "categories": [ "pione" ], "title": "pione-lemma-restriction-equivalence-general", "contents": [ "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme", "with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.", "Let $\\mathcal{V}$ be the set of open subschemes $V \\subset X$ containing $Y$", "ordered by reverse inclusion. Assume the completion functor", "$$", "\\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V)", "\\longrightarrow", "\\textit{Coh}(X, \\mathcal{I}),", "\\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge", "$$", "defines an equivalence of the full subcategories of", "finite locally free objects (see explanation above).", "Then the restriction functor", "$$", "\\colim_\\mathcal{V} \\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y", "$$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\mathcal{V}$ is a directed set, so the colimits are", "as in Categories, Section \\ref{categories-section-directed-colimits}.", "The rest of the argument is almost exactly the same as the argument", "in the proof of Lemma \\ref{lemma-restriction-equivalence}; we urge", "the reader to skip it.", "\\medskip\\noindent", "The restriction functor is fully faithful by", "Lemma \\ref{lemma-restriction-fully-faithful-general}.", "\\medskip\\noindent", "Let $U_1 \\to Y$ be a finite \\'etale morphism. To finish the proof", "we will show that $U_1$ is in the essential image of the", "restriction functor.", "\\medskip\\noindent", "For $n \\geq 1$ let $Y_n$ be the $n$th infinitesimal neighbourhood of $Y$.", "By Lemma \\ref{lemma-thickening}", "there is a unique finite \\'etale morphism", "$\\pi_n : U_n \\to Y_n$ whose base change to $Y = Y_1$", "recovers $U_1 \\to Y_1$.", "Consider the sheaves $\\mathcal{F}_n = \\pi_{n, *}\\mathcal{O}_{U_n}$.", "We may and do view $\\mathcal{F}_n$ as an $\\mathcal{O}_X$-module on $X$", "wich is locally isomorphic to", "$(\\mathcal{O}_X/f^{n + 1}\\mathcal{O}_X)^{\\oplus r}$.", "This $(\\mathcal{F}_n)$ is a finite locally free object of", "$\\textit{Coh}(X, \\mathcal{I})$.", "By assumption there exists a $V \\in \\mathcal{V}$", "and a finite locally free $\\mathcal{O}_V$-module $\\mathcal{F}$", "and a compatible system of isomorphisms", "$$", "\\mathcal{F}/\\mathcal{I}^n\\mathcal{F} \\to \\mathcal{F}_n", "$$", "of $\\mathcal{O}_V$-modules.", "\\medskip\\noindent", "To construct an algebra structure on $\\mathcal{F}$ consider the multiplication", "maps", "$\\mathcal{F}_n \\otimes_{\\mathcal{O}_V} \\mathcal{F}_n \\to \\mathcal{F}_n$", "coming from the fact that $\\mathcal{F}_n = \\pi_{n, *}\\mathcal{O}_{U_n}$", "are sheaves of algebras. These define a map", "$$", "(\\mathcal{F}\\otimes_{\\mathcal{O}_V} \\mathcal{F})^\\wedge", "\\longrightarrow", "\\mathcal{F}^\\wedge", "$$", "in the category $\\textit{Coh}(X, \\mathcal{I})$. Hence by assumption", "after shrinking $V$ we may assume there is a map", "$\\mu : \\mathcal{F}\\otimes_{\\mathcal{O}_V} \\mathcal{F} \\to \\mathcal{F}$", "whose restriction to $Y_n$ gives the multiplication maps above.", "After possibly shrinking further we may assume $\\mu$", "defines a commutative $\\mathcal{O}_V$-algebra", "structure on $\\mathcal{F}$ compatible with the given algebra", "structures on $\\mathcal{F}_n$.", "Setting", "$$", "U = \\underline{\\Spec}_V((\\mathcal{F}, \\mu))", "$$", "we obtain a finite locally free scheme over $V$ whose restriction", "to $Y$ is isomorphic to $U_1$. It follows that $U \\to V$", "is \\'etale at all points lying over $Y$, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds}.", "Thus after shrinking $V$ once more we may assume $U \\to V$ is", "finite \\'etale. This finishes the proof." ], "refs": [ "pione-lemma-restriction-equivalence", "pione-lemma-restriction-fully-faithful-general", "pione-lemma-thickening", "more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds" ], "ref_ids": [ 4076, 4077, 4043, 13743 ] } ], "ref_ids": [] }, { "id": 4079, "type": "theorem", "label": "pione-lemma-restriction-faithful", "categories": [ "pione" ], "title": "pione-lemma-restriction-faithful", "contents": [ "Let $X$ be a scheme and let $Y \\subset X$ be a closed subscheme.", "If every connected component of $X$ meets $Y$, then", "the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$", "is faithful." ], "refs": [], "proofs": [ { "contents": [ "Let $a, b : U \\to U'$ be two morphisms of schemes finite \\'etale over $X$", "whose restriction to $Y$ are the same. The image of a connected component", "of $U$ is an connected component of $X$; this follows from", "Topology, Lemma \\ref{topology-lemma-finite-fibre-connected-components}", "applied to the restriction of $U \\to X$ to a connected component of $X$.", "Hence the image of every connected component of $U$ meets $Y$", "by assumption. We conclude that $a = b$ after restriction to each", "connected component of $U$ by \\'Etale Morphisms, Proposition", "\\ref{etale-proposition-equality}. Since the equalizer of $a$ and $b$", "is an open subscheme of $U$ (as the diagonal of $U'$ over $X$ is open)", "we conclude." ], "refs": [ "topology-lemma-finite-fibre-connected-components", "etale-proposition-equality" ], "ref_ids": [ 8209, 10727 ] } ], "ref_ids": [] }, { "id": 4080, "type": "theorem", "label": "pione-lemma-restriction-fully-faithful-special", "categories": [ "pione" ], "title": "pione-lemma-restriction-fully-faithful-special", "contents": [ "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme.", "Let $Y_n \\subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$.", "Assume one of the following holds", "\\begin{enumerate}", "\\item $X$ is quasi-affine and", "$\\Gamma(X, \\mathcal{O}_X) \\to \\lim \\Gamma(Y_n, \\mathcal{O}_{Y_n})$", "is an isomorphism, or", "\\item $X$ has an ample invertible module $\\mathcal{L}$ and", "$\\Gamma(X, \\mathcal{L}^{\\otimes m}) \\to", "\\lim \\Gamma(Y_n, \\mathcal{L}^{\\otimes m}|_{Y_n})$", "is an isomorphism for all $m \\gg 0$, or", "\\item for every finite locally free $\\mathcal{O}_X$-module", "$\\mathcal{E}$ the map", "$\\Gamma(X, \\mathcal{E}) \\to \\lim \\Gamma(Y_n, \\mathcal{E}|_{Y_n})$", "is an isomorphism.", "\\end{enumerate}", "Then the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$", "is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "This lemma follows formally from", "Lemma \\ref{lemma-restriction-fully-faithful} and", "Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-completion-fully-faithful}." ], "refs": [ "pione-lemma-restriction-fully-faithful", "algebraization-lemma-completion-fully-faithful" ], "ref_ids": [ 4075, 12738 ] } ], "ref_ids": [] }, { "id": 4081, "type": "theorem", "label": "pione-lemma-restriction-fully-faithful-general-special", "categories": [ "pione" ], "title": "pione-lemma-restriction-fully-faithful-general-special", "contents": [ "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme.", "Let $Y_n \\subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$.", "Let $\\mathcal{V}$ be the set of open subschemes $V \\subset X$ containing $Y$", "ordered by reverse inclusion. Assume one of the following holds", "\\begin{enumerate}", "\\item $X$ is quasi-affine and", "$$", "\\colim_\\mathcal{V} \\Gamma(V, \\mathcal{O}_V)", "\\longrightarrow", "\\lim \\Gamma(Y_n, \\mathcal{O}_{Y_n})", "$$", "is an isomorphism, or", "\\item $X$ has an ample invertible module $\\mathcal{L}$ and", "$$", "\\colim_\\mathcal{V} \\Gamma(V, \\mathcal{L}^{\\otimes m})", "\\longrightarrow", "\\lim \\Gamma(Y_n, \\mathcal{L}^{\\otimes m}|_{Y_n})", "$$", "is an isomorphism for all $m \\gg 0$, or", "\\item for every $V \\in \\mathcal{V}$ and every finite locally free", "$\\mathcal{O}_V$-module $\\mathcal{E}$ the map", "$$", "\\colim_{V' \\geq V} \\Gamma(V', \\mathcal{E}|_{V'})", "\\longrightarrow", "\\lim \\Gamma(Y_n, \\mathcal{E}|_{Y_n})", "$$", "is an isomorphism.", "\\end{enumerate}", "Then the functor", "$$", "\\colim_\\mathcal{V} \\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y", "$$", "is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "This lemma follows formally from", "Lemma \\ref{lemma-restriction-fully-faithful-general} and", "Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-completion-fully-faithful-general}." ], "refs": [ "pione-lemma-restriction-fully-faithful-general", "algebraization-lemma-completion-fully-faithful-general" ], "ref_ids": [ 4077, 12739 ] } ], "ref_ids": [] }, { "id": 4082, "type": "theorem", "label": "pione-lemma-pushout-along-closed-immersion-and-integral", "categories": [ "pione" ], "title": "pione-lemma-pushout-along-closed-immersion-and-integral", "contents": [ "In More on Morphisms, Situation", "\\ref{more-morphisms-situation-pushout-along-closed-immersion-and-integral},", "for example if $Z \\to Y$ and $Z \\to X$ are closed immersions of schemes,", "there is an equivalence of categories", "$$", "\\textit{F\\'Et}_{Y \\amalg_Z X}", "\\longrightarrow", "\\textit{F\\'Et}_Y", "\\times_{\\textit{F\\'Et}_Z}", "\\textit{F\\'Et}_X", "$$" ], "refs": [], "proofs": [ { "contents": [ "The pushout exists by", "More on Morphisms, Proposition", "\\ref{more-morphisms-proposition-pushout-along-closed-immersion-and-integral}.", "The functor is given by sending a scheme $U$ finite \\'etale over the", "pushout to the base changes $Y' = U \\times_{Y \\amalg_Z X} Y$", "and $X' = U \\times_{Y \\amalg_Z X} X$ and the natural isomorphism", "$Y' \\times_Y Z \\to X' \\times_X Z$ over $Z$. To prove this functor", "is an equivalence we use", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-pushout-functor-equivalence-flat}", "to construct a quasi-inverse functor.", "The only thing left to prove is to show that given a morphism", "$U \\to Y \\amalg_Z X$ which is separated, quasi-finite and \\'etale", "such that $X' \\to X$ and $Y' \\to Y$ are finite,", "then $U \\to Y \\amalg_Z X$ is finite.", "This can either be deduced from the corresponding algebra fact", "(More on Algebra, Lemma", "\\ref{more-algebra-lemma-finite-module-over-fibre-product})", "or it can be seen because", "$$", "X' \\amalg Y' \\to U", "$$", "is surjective and $X'$ and $Y'$ are proper over $Y \\amalg_Z X$", "(this uses the description of the pushout in More on Morphisms, Proposition", "\\ref{more-morphisms-proposition-pushout-along-closed-immersion-and-integral})", "and then we can apply", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-image-is-proper}", "to conclude that $U$ is proper over $Y \\amalg_Z X$.", "Since a quasi-finite and proper morphism is finite", "(More on Morphisms, Lemma \\ref{more-morphisms-lemma-characterize-finite})", "we win." ], "refs": [ "more-morphisms-proposition-pushout-along-closed-immersion-and-integral", "more-morphisms-lemma-pushout-functor-equivalence-flat", "more-algebra-lemma-finite-module-over-fibre-product", "more-morphisms-proposition-pushout-along-closed-immersion-and-integral", "morphisms-lemma-scheme-theoretic-image-is-proper", "more-morphisms-lemma-characterize-finite" ], "ref_ids": [ 14103, 14050, 9823, 14103, 5414, 13903 ] } ], "ref_ids": [] }, { "id": 4083, "type": "theorem", "label": "pione-lemma-faithful", "categories": [ "pione" ], "title": "pione-lemma-faithful", "contents": [ "In Situation \\ref{situation-local-lefschetz}.", "Assume one of the following holds", "\\begin{enumerate}", "\\item $\\dim(A/\\mathfrak p) \\geq 2$ for every minimal prime", "$\\mathfrak p \\subset A$ with $f \\not \\in \\mathfrak p$, or", "\\item every connected component of $U$ meets $U_0$.", "\\end{enumerate}", "Then", "$$", "\\textit{F\\'Et}_U \\longrightarrow \\textit{F\\'Et}_{U_0},\\quad", "V \\longmapsto V_0 = V \\times_U U_0", "$$", "is a faithful functor." ], "refs": [], "proofs": [ { "contents": [ "Case (2) is immediate from Lemma \\ref{lemma-restriction-faithful}.", "Assumption (1) implies every irreducible component of $U$ meets $U_0$, see", "Algebra, Lemma \\ref{algebra-lemma-one-equation}.", "Hence (1) follows from (2)." ], "refs": [ "pione-lemma-restriction-faithful" ], "ref_ids": [ 4079 ] } ], "ref_ids": [] }, { "id": 4084, "type": "theorem", "label": "pione-lemma-fill-in-missing", "categories": [ "pione" ], "title": "pione-lemma-fill-in-missing", "contents": [ "In Situation \\ref{situation-local-lefschetz}. Let $V \\to U$ be a finite", "morphism. Let $A^\\wedge$ be the $\\mathfrak m$-adic completion of $A$,", "let $X' = \\Spec(A^\\wedge)$ and let $U'$ and $V'$ be the base changes of", "$U$ and $V$ to $X'$. If $Y' \\to X'$ is a finite morphism such that", "$V' = Y' \\times_{X'} U'$, then there exists a finite morphism $Y \\to X$", "such that $V = Y \\times_X U$ and $Y' = Y \\times_X X'$." ], "refs": [], "proofs": [ { "contents": [ "This is a straightforward application of", "More on Algebra, Proposition \\ref{more-algebra-proposition-equivalence}.", "Namely, choose generators $f_1, \\ldots, f_t$ of $\\mathfrak m$.", "For each $i$ write $V \\times_U D(f_i) = \\Spec(B_i)$.", "For $1 \\leq i, j \\leq n$ we obtain an isomorphism", "$\\alpha_{ij} : (B_i)_{f_j} \\to (B_j)_{f_i}$ of $A_{f_if_j}$-algebras", "because the spectrum of both represent $V \\times_U D(f_if_j)$.", "Write $Y' = \\Spec(B')$. Since $V \\times_U U' = Y \\times_{X'} U'$", "we get isomorphisms $\\alpha_i : B'_{f_i} \\to B_i \\otimes_A A^\\wedge$.", "A straightforward argument shows that $(B', B_i, \\alpha_i, \\alpha_{ij})$", "is an object of $\\text{Glue}(A \\to A^\\wedge, f_1, \\ldots, f_t)$, see", "More on Algebra, Remark \\ref{more-algebra-remark-glueing-data}.", "Applying the proposition cited above (and using", "More on Algebra, Remark \\ref{more-algebra-remark-formal-glueing-algebras}", "to obtain the algebra structure) we find an $A$-algebra $B$ such that", "$\\text{Can}(B)$ is isomorphic to $(B', B_i, \\alpha_i, \\alpha_{ij})$.", "Setting $Y = \\Spec(B)$ we see that $Y \\to X$ is a morphism", "which comes equipped with compatible isomorphisms", "$V \\cong Y \\times_X U$ and $Y' = Y \\times_X X'$ as desired." ], "refs": [ "more-algebra-proposition-equivalence", "more-algebra-remark-glueing-data", "more-algebra-remark-formal-glueing-algebras" ], "ref_ids": [ 10587, 10662, 10663 ] } ], "ref_ids": [] }, { "id": 4085, "type": "theorem", "label": "pione-lemma-fully-faithful-henselian-completion", "categories": [ "pione" ], "title": "pione-lemma-fully-faithful-henselian-completion", "contents": [ "In Situation \\ref{situation-local-lefschetz} assume $A$ is henselian", "or more generally that $(A, (f))$ is a henselian pair.", "Let $A^\\wedge$ be the $\\mathfrak m$-adic completion of $A$,", "let $X' = \\Spec(A^\\wedge)$ and let $U'$ and $U'_0$ be the base changes of", "$U$ and $U_0$ to $X'$. If $\\textit{F\\'Et}_{U'} \\to \\textit{F\\'Et}_{U'_0}$", "is fully faithful, then $\\textit{F\\'Et}_U \\to \\textit{F\\'Et}_{U_0}$", "is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\textit{F\\'Et}_{U'} \\longrightarrow \\textit{F\\'Et}_{U'_0}$", "is a fully faithful. Since $X' \\to X$ is faithfully flat, it is", "immediate that the functor $V \\to V_0 = V \\times_U U_0$ is faithful.", "Since the category of finite \\'etale coverings has an internal hom", "(Lemma \\ref{lemma-internal-hom-finite-etale})", "it suffices to prove the following: Given $V$ finite \\'etale over $U$", "we have", "$$", "\\Mor_U(U, V) = \\Mor_{U_0}(U_0, V_0)", "$$", "The we assume we have a morphism $s_0 : U_0 \\to V_0$ over $U_0$ and we will", "produce a morphism $s : U \\to V$ over $U$.", "\\medskip\\noindent", "By our assumption there does exist a morphism $s' : U' \\to V'$", "whose restriction to $V'_0$ is the base change $s'_0$ of $s_0$.", "Since $V' \\to U'$ is finite \\'etale this means that $V' = s'(U') \\amalg W'$", "for some $W' \\to U'$ finite and \\'etale.", "Choose a finite morphism $Z' \\to X'$ such that $W' = Z' \\times_{X'} U'$.", "This is possible by Zariski's main theorem in the form stated in", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite}", "(small detail omitted).", "Then", "$$", "V' = s'(U') \\amalg W' \\longrightarrow X' \\amalg Z' = Y'", "$$", "is an open immersion such that $V' = Y' \\times_{X'} U'$.", "By Lemma \\ref{lemma-fill-in-missing} we can find $Y \\to X$ finite", "such that $V = Y \\times_X U$ and $Y' = Y \\times_X X'$.", "Write $Y = \\Spec(B)$ so that $Y' = \\Spec(B \\otimes_A A^\\wedge)$.", "Then $B \\otimes_A A^\\wedge$ has an idempotent $e'$", "corresponding to the open and closed subscheme $X'$ of $Y' = X' \\amalg Z'$.", "\\medskip\\noindent", "The case $A$ is henselian (slightly easier). The image $\\overline{e}$", "of $e'$ in $B \\otimes_A \\kappa(\\mathfrak m) = B/\\mathfrak mB$ lifts to an", "idempotent $e$ of $B$ as $A$ is henselian (because $B$ is a product of", "local rings by Algebra, Lemma \\ref{algebra-lemma-characterize-henselian}).", "Then we see that $e$ maps to $e'$ by uniqueness of lifts of idempotents", "(using that $B \\otimes_A A^\\wedge$ is a product of local rings).", "Let $Y_1 \\subset Y$ be the open and closed subscheme corresponding to $e$.", "Then $Y_1 \\times_X X' = s'(X')$ which implies that $Y_1 \\to X$ is", "an isomorphism (by faithfully flat descent) and gives the desired section.", "\\medskip\\noindent", "The case where $(A, (f))$ is a henselian pair. Here we use that $s'$ is", "a lift of $s'_0$. Namely, let $Y_{0, 1} \\subset Y_0 = Y \\times_X X_0$", "be the closure of $s_0(U_0) \\subset V_0 = Y_0 \\times_{X_0} U_0$.", "As $X' \\to X$ is flat, the base change $Y'_{0, 1} \\subset Y'_0$", "is the closure of $s'_0(U'_0)$ which is equal to $X'_0 \\subset Y'_0$", "(see Morphisms, Lemma", "\\ref{morphisms-lemma-flat-base-change-scheme-theoretic-image}).", "Since $Y'_0 \\to Y_0$ is submersive", "(Morphisms, Lemma \\ref{morphisms-lemma-fpqc-quotient-topology})", "we conclude that $Y_{0, 1}$ is open and closed in $Y_0$.", "Let $e_0 \\in B/fB$ be the corresponding idempotent.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-characterize-henselian-pair}", "we can lift $e_0$ to an idempotent $e \\in B$.", "Then we conclude as before." ], "refs": [ "pione-lemma-internal-hom-finite-etale", "more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "pione-lemma-fill-in-missing", "algebra-lemma-characterize-henselian", "morphisms-lemma-flat-base-change-scheme-theoretic-image", "morphisms-lemma-fpqc-quotient-topology", "more-algebra-lemma-characterize-henselian-pair" ], "ref_ids": [ 4038, 13901, 4084, 1276, 5273, 5269, 9861 ] } ], "ref_ids": [] }, { "id": 4086, "type": "theorem", "label": "pione-lemma-fully-faithful-simple", "categories": [ "pione" ], "title": "pione-lemma-fully-faithful-simple", "contents": [ "In Situation \\ref{situation-local-lefschetz}. Assume", "\\begin{enumerate}", "\\item[(a)] $A$ has a dualizing complex,", "\\item[(b)] the pair $(A, (f))$ is henselian,", "\\item[(c)] one of the following is true", "\\begin{enumerate}", "\\item[(i)] $A_f$ is $(S_2)$ and every irreducible component of $X$", "not contained in $X_0$ has dimension $\\geq 3$, or", "\\item[(ii)] for every prime", "$\\mathfrak p \\subset A$, $f \\not \\in \\mathfrak p$ we have", "$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 2$.", "\\end{enumerate}", "\\end{enumerate}", "Then the restriction functor", "$\\textit{F\\'Et}_U \\longrightarrow \\textit{F\\'Et}_{U_0}$", "is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "Let $A'$ be the $\\mathfrak m$-adic completion of $A$. We will show that", "the hypotheses remain true for $A'$. This is clear for conditions", "(a) and (b). Condition (c)(ii) is preserved by", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-change-completion}.", "Next, assume (c)(i) holds. Since $A$ is universally catenary", "(Dualizing Complexes, Lemma \\ref{dualizing-lemma-universally-catenary})", "we see that every irreducible component of $\\Spec(A')$ not contained in $V(f)$", "has dimension $\\geq 3$, see", "More on Algebra, Proposition \\ref{more-algebra-proposition-ratliff}.", "Since $A \\to A'$ is flat with Gorenstein fibres,", "the condition that $A_f$ is $(S_2)$ implies that $A'_f$ is $(S_2)$.", "References used:", "Dualizing Complexes, Section \\ref{dualizing-section-formal-fibres},", "More on Algebra, Section \\ref{more-algebra-section-properties-formal-fibres},", "and Algebra, Lemma \\ref{algebra-lemma-Sk-goes-up}.", "Thus by Lemma \\ref{lemma-fully-faithful-henselian-completion}", "we may assume that $A$ is a Noetherian complete local ring.", "\\medskip\\noindent", "Assume $A$ is a complete local ring in addition to the other assumptions.", "By Lemma \\ref{lemma-restriction-fully-faithful} the result follows from", "Algebraic and Formal Geometry, Lemmas", "\\ref{algebraization-lemma-fully-faithful-simple-one} and", "\\ref{algebraization-lemma-fully-faithful-simple-two}." ], "refs": [ "local-cohomology-lemma-change-completion", "dualizing-lemma-universally-catenary", "more-algebra-proposition-ratliff", "algebra-lemma-Sk-goes-up", "pione-lemma-fully-faithful-henselian-completion", "pione-lemma-restriction-fully-faithful", "algebraization-lemma-fully-faithful-simple-one", "algebraization-lemma-fully-faithful-simple-two" ], "ref_ids": [ 9740, 2870, 10591, 1363, 4085, 4075, 12742, 12744 ] } ], "ref_ids": [] }, { "id": 4087, "type": "theorem", "label": "pione-lemma-fully-faithful-minimal", "categories": [ "pione" ], "title": "pione-lemma-fully-faithful-minimal", "contents": [ "\\begin{reference}", "\\cite[Corollary 1.11]{Bhatt-local}", "\\end{reference}", "In Situation \\ref{situation-local-lefschetz}. Assume", "\\begin{enumerate}", "\\item $H^1_\\mathfrak m(A)$ and $H^2_\\mathfrak m(A)$ are", "annihilated by a power of $f$, and", "\\item $A$ is henselian or more generally $(A, (f))$ is a henselian pair.", "\\end{enumerate}", "Then the restriction functor", "$\\textit{F\\'Et}_U \\longrightarrow \\textit{F\\'Et}_{U_0}$", "is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-fully-faithful-henselian-completion}", "we may assume that $A$ is a Noetherian complete local ring.", "(The assumptions carry over; use", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-torsion-change-rings}.)", "By Lemma \\ref{lemma-restriction-fully-faithful}", "the result follows from", "Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-fully-faithful-alternative}." ], "refs": [ "pione-lemma-fully-faithful-henselian-completion", "dualizing-lemma-torsion-change-rings", "pione-lemma-restriction-fully-faithful", "algebraization-lemma-fully-faithful-alternative" ], "ref_ids": [ 4085, 2817, 4075, 12743 ] } ], "ref_ids": [] }, { "id": 4088, "type": "theorem", "label": "pione-lemma-fully-faithful", "categories": [ "pione" ], "title": "pione-lemma-fully-faithful", "contents": [ "In Situation \\ref{situation-local-lefschetz} assume $A$ has depth $\\geq 3$", "and $A$ is henselian or more generally $(A, (f))$ is a henselian pair. Then", "the restriction functor", "$\\textit{F\\'Et}_U \\to \\textit{F\\'Et}_{U_0}$", "is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "The assumption of depth forces", "$H^1_\\mathfrak m(A) = H^2_\\mathfrak m(A) = 0$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth}.", "Hence Lemma \\ref{lemma-fully-faithful-minimal} applies." ], "refs": [ "dualizing-lemma-depth", "pione-lemma-fully-faithful-minimal" ], "ref_ids": [ 2826, 4087 ] } ], "ref_ids": [] }, { "id": 4089, "type": "theorem", "label": "pione-lemma-sections-over-punctured-spec", "categories": [ "pione" ], "title": "pione-lemma-sections-over-punctured-spec", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$", "and let $U = X \\setminus \\{\\mathfrak m\\}$.", "Let $\\pi : Y \\to X$ be a finite morphism such that", "$\\text{depth}(\\mathcal{O}_{Y, y}) \\geq 2$ for all closed points", "$y \\in Y$.", "Then $Y$ is the spectrum of $B = \\mathcal{O}_Y(\\pi^{-1}(U))$." ], "refs": [], "proofs": [ { "contents": [ "Set $V = \\pi^{-1}(U)$ and denote $\\pi' : V \\to U$ the restriction of $\\pi$.", "Consider the $\\mathcal{O}_X$-module map", "$$", "\\pi_*\\mathcal{O}_Y \\longrightarrow j_*\\pi'_*\\mathcal{O}_V", "$$", "where $j : U \\to X$ is the inclusion morphism. We claim", "Divisors, Lemma \\ref{divisors-lemma-depth-2-hartog}", "applies to this map. If so, then $B = \\Gamma(Y, \\mathcal{O}_Y)$", "and we see that the lemma holds. Let $x \\in X$ be the closed point.", "It suffices to show that", "$\\text{depth}((\\pi_*\\mathcal{O}_Y)_x) \\geq 2$.", "Let $y_1, \\ldots, y_n \\in Y$ be the points mapping to $x$.", "By Algebra, Lemma \\ref{algebra-lemma-depth-goes-down-finite}", "it suffices to show that", "$\\text{depth}(\\mathcal{O}_{Y, y_i}) \\geq 2$ for $i = 1, \\ldots, n$.", "Since this is the assumption of the lemma the proof is complete." ], "refs": [ "divisors-lemma-depth-2-hartog", "algebra-lemma-depth-goes-down-finite" ], "ref_ids": [ 7881, 778 ] } ], "ref_ids": [] }, { "id": 4090, "type": "theorem", "label": "pione-lemma-reformulate-purity", "categories": [ "pione" ], "title": "pione-lemma-reformulate-purity", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$", "and let $U = X \\setminus \\{\\mathfrak m\\}$.", "Let $V$ be finite \\'etale", "over $U$. Assume $A$ has depth $\\geq 2$. The following are equivalent", "\\begin{enumerate}", "\\item $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale,", "\\item $B = \\Gamma(V, \\mathcal{O}_V)$ is finite \\'etale over $A$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $\\pi : V \\to U$ the given finite \\'etale morphism.", "Assume $Y$ as in (1) exists. Let $x \\in X$ be the point", "corresponding to $\\mathfrak m$.", "Let $y \\in Y$ be a point mapping to $x$. We claim that", "$\\text{depth}(\\mathcal{O}_{Y, y}) \\geq 2$.", "This is true because $Y \\to X$ is \\'etale and hence", "$A = \\mathcal{O}_{X, x}$ and $\\mathcal{O}_{Y, y}$ have", "the same depth (Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck}).", "Hence Lemma \\ref{lemma-sections-over-punctured-spec}", "applies and $Y = \\Spec(B)$.", "\\medskip\\noindent", "The implication (2) $\\Rightarrow$ (1) is easier and the", "details are omitted." ], "refs": [ "algebra-lemma-apply-grothendieck", "pione-lemma-sections-over-punctured-spec" ], "ref_ids": [ 1361, 4089 ] } ], "ref_ids": [] }, { "id": 4091, "type": "theorem", "label": "pione-lemma-reformulate-purity-normal", "categories": [ "pione" ], "title": "pione-lemma-reformulate-purity-normal", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$", "and let $U = X \\setminus \\{\\mathfrak m\\}$. Assume $A$ is normal", "of dimension $\\geq 2$. The functor", "$$", "\\textit{F\\'Et}_U \\longrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{finite normal }A\\text{-algebras }B\\text{ such} \\\\", "\\text{that }\\Spec(B) \\to X\\text{ is \\'etale over }U", "\\end{matrix}", "\\right\\},", "\\quad", "V \\longmapsto \\Gamma(V, \\mathcal{O}_V)", "$$", "is an equivalence. Moreover, $V = Y \\times_X U$ for some $Y \\to X$", "finite \\'etale if and only if $B = \\Gamma(V, \\mathcal{O}_V)$", "is finite \\'etale over $A$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\text{depth}(A) \\geq 2$ because $A$ is normal", "(Serre's criterion for normality, Algebra, Lemma", "\\ref{algebra-lemma-criterion-normal}).", "Thus the final statement follows from Lemma \\ref{lemma-reformulate-purity}.", "Given $\\pi : V \\to U$ finite \\'etale, set $B = \\Gamma(V, \\mathcal{O}_V)$.", "If we can show that $B$ is normal and finite over $A$, then", "we obtain the displayed functor. Since there is an obvious", "quasi-inverse functor, this is also all that we have to show.", "\\medskip\\noindent", "Since $A$ is normal, the scheme $V$ is normal", "(Descent, Lemma \\ref{descent-lemma-normal-local-smooth}).", "Hence $V$ is a finite disjoint union of integral schemes", "(Properties, Lemma \\ref{properties-lemma-normal-Noetherian}).", "Thus we may assume $V$ is integral.", "In this case the function field $L$ of $V$", "(Morphisms, Section \\ref{morphisms-section-rational-maps})", "is a finite separable extension of the fraction field of $A$", "(because we get it by looking at the generic fibre", "of $V \\to U$ and using Morphisms, Lemma", "\\ref{morphisms-lemma-etale-over-field}).", "By Algebra, Lemma", "\\ref{algebra-lemma-Noetherian-normal-domain-finite-separable-extension}", "the integral closure $B' \\subset L$ of $A$ in $L$ is finite over $A$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-integral-closure-reflexive}", "we see that $B'$ is a reflexive $A$-module, which in turn implies", "that $\\text{depth}_A(B') \\geq 2$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-reflexive-over-normal}.", "\\medskip\\noindent", "Let $f \\in \\mathfrak m$. Then $B_f = \\Gamma(V \\times_U D(f), \\mathcal{O}_V)$", "(Properties, Lemma \\ref{properties-lemma-invert-f-sections}).", "Hence $B'_f = B_f$ because $B_f$ is normal (see above),", "finite over $A_f$ with fraction field $L$.", "It follows that $V = \\Spec(B') \\times_X U$.", "Then we conclude that $B = B'$ from", "Lemma \\ref{lemma-sections-over-punctured-spec}", "applied to $\\Spec(B') \\to X$.", "This lemma applies because the localizations $B'_{\\mathfrak m'}$", "of $B'$ at maximal ideals $\\mathfrak m' \\subset B'$ lying over", "$\\mathfrak m$ have depth $\\geq 2$ by", "Algebra, Lemma \\ref{algebra-lemma-depth-goes-down-finite}", "and the remark on depth in the preceding paragraph." ], "refs": [ "algebra-lemma-criterion-normal", "pione-lemma-reformulate-purity", "descent-lemma-normal-local-smooth", "properties-lemma-normal-Noetherian", "morphisms-lemma-etale-over-field", "algebra-lemma-Noetherian-normal-domain-finite-separable-extension", "more-algebra-lemma-integral-closure-reflexive", "more-algebra-lemma-reflexive-over-normal", "properties-lemma-invert-f-sections", "pione-lemma-sections-over-punctured-spec", "algebra-lemma-depth-goes-down-finite" ], "ref_ids": [ 1311, 4090, 14654, 2970, 5364, 1338, 9939, 9937, 3004, 4089, 778 ] } ], "ref_ids": [] }, { "id": 4092, "type": "theorem", "label": "pione-lemma-purity-and-completion", "categories": [ "pione" ], "title": "pione-lemma-purity-and-completion", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$", "and let $U = X \\setminus \\{\\mathfrak m\\}$.", "Let $V$ be finite \\'etale over $U$.", "Let $A^\\wedge$ be the $\\mathfrak m$-adic completion of $A$,", "let $X' = \\Spec(A^\\wedge)$ and let $U'$ and $V'$ be the base changes of", "$U$ and $V$ to $X'$. The following are equivalent", "\\begin{enumerate}", "\\item $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale, and", "\\item $V' = Y' \\times_{X'} U'$ for some $Y' \\to X'$ finite \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) follows from taking the base change", "of a solution $Y \\to X$. Let $Y' \\to X'$ be as in (2).", "By Lemma \\ref{lemma-fill-in-missing} we can find $Y \\to X$ finite", "such that $V = Y \\times_X U$ and $Y' = Y \\times_X X'$.", "By descent we see that $Y \\to X$ is finite \\'etale", "(Algebra, Lemmas \\ref{algebra-lemma-descend-properties-modules} and", "\\ref{algebra-lemma-etale}). This finishes the proof." ], "refs": [ "pione-lemma-fill-in-missing", "algebra-lemma-descend-properties-modules", "algebra-lemma-etale" ], "ref_ids": [ 4084, 819, 1231 ] } ], "ref_ids": [] }, { "id": 4093, "type": "theorem", "label": "pione-lemma-lift-simple", "categories": [ "pione" ], "title": "pione-lemma-lift-simple", "contents": [ "In Situation \\ref{situation-local-lefschetz}. Let $V$ be finite", "\\'etale over $U$. Assume", "\\begin{enumerate}", "\\item[(a)] $A$ has a dualizing complex,", "\\item[(b)] the pair $(A, (f))$ is henselian,", "\\item[(c)] one of the following is true", "\\begin{enumerate}", "\\item[(i)] $A_f$ is $(S_2)$ and every irreducible component of $X$", "not contained in $X_0$ has dimension $\\geq 3$, or", "\\item[(ii)] for every prime $\\mathfrak p \\subset A$, $f \\not \\in \\mathfrak p$", "we have $\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 2$.", "\\end{enumerate}", "\\item[(d)] $V_0 = V \\times_U U_0$ is equal to $Y_0 \\times_{X_0} U_0$", "for some $Y_0 \\to X_0$ finite \\'etale.", "\\end{enumerate}", "Then $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale." ], "refs": [], "proofs": [ { "contents": [ "We reduce to the complete case using Lemma \\ref{lemma-purity-and-completion}.", "(The assumptions carry over; see proof of", "Lemma \\ref{lemma-fully-faithful-simple}.)", "\\medskip\\noindent", "In the complete case we can lift $Y_0 \\to X_0$ to a finite \\'etale", "morphism $Y \\to X$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-finite-etale-equivalence};", "observe that $(A, fA)$ is a henselian pair by", "More on Algebra, Lemma \\ref{more-algebra-lemma-complete-henselian}.", "Then we can use Lemma \\ref{lemma-fully-faithful-simple}", "to see that $V$ is isomorphic to $Y \\times_X U$ and", "the proof is complete." ], "refs": [ "pione-lemma-purity-and-completion", "pione-lemma-fully-faithful-simple", "more-algebra-lemma-finite-etale-equivalence", "more-algebra-lemma-complete-henselian", "pione-lemma-fully-faithful-simple" ], "ref_ids": [ 4092, 4086, 9879, 9859, 4086 ] } ], "ref_ids": [] }, { "id": 4094, "type": "theorem", "label": "pione-lemma-lift-purity-general", "categories": [ "pione" ], "title": "pione-lemma-lift-purity-general", "contents": [ "In Situation \\ref{situation-local-lefschetz}.", "Let $V$ be finite \\'etale over $U$. Assume", "\\begin{enumerate}", "\\item $H^1_\\mathfrak m(A)$ and $H^2_\\mathfrak m(A)$", "are annihilated by a power of $f$,", "\\item $V_0 = V \\times_U U_0$ is equal to $Y_0 \\times_{X_0} U_0$", "for some $Y_0 \\to X_0$ finite \\'etale.", "\\end{enumerate}", "Then $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale." ], "refs": [], "proofs": [ { "contents": [ "We reduce to the complete case using Lemma \\ref{lemma-purity-and-completion}.", "(The assumptions carry over; use Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-torsion-change-rings}.)", "\\medskip\\noindent", "In the complete case we can lift $Y_0 \\to X_0$ to a finite \\'etale", "morphism $Y \\to X$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-finite-etale-equivalence};", "observe that $(A, fA)$ is a henselian pair by", "More on Algebra, Lemma \\ref{more-algebra-lemma-complete-henselian}.", "Then we can use Lemma \\ref{lemma-fully-faithful-minimal}", "to see that $V$ is isomorphic to $Y \\times_X U$ and", "the proof is complete." ], "refs": [ "pione-lemma-purity-and-completion", "dualizing-lemma-torsion-change-rings", "more-algebra-lemma-finite-etale-equivalence", "more-algebra-lemma-complete-henselian", "pione-lemma-fully-faithful-minimal" ], "ref_ids": [ 4092, 2817, 9879, 9859, 4087 ] } ], "ref_ids": [] }, { "id": 4095, "type": "theorem", "label": "pione-lemma-lift-purity", "categories": [ "pione" ], "title": "pione-lemma-lift-purity", "contents": [ "In Situation \\ref{situation-local-lefschetz}.", "Let $V$ be finite \\'etale over $U$. Assume", "\\begin{enumerate}", "\\item $A$ has depth $\\geq 3$,", "\\item $V_0 = V \\times_U U_0$ is equal to $Y_0 \\times_{X_0} U_0$", "for some $Y_0 \\to X_0$ finite \\'etale.", "\\end{enumerate}", "Then $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale." ], "refs": [], "proofs": [ { "contents": [ "The assumption of depth forces", "$H^1_\\mathfrak m(A) = H^2_\\mathfrak m(A) = 0$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth}.", "Hence Lemma \\ref{lemma-lift-purity-general} applies." ], "refs": [ "dualizing-lemma-depth", "pione-lemma-lift-purity-general" ], "ref_ids": [ 2826, 4094 ] } ], "ref_ids": [] }, { "id": 4096, "type": "theorem", "label": "pione-lemma-find-point-codim-1", "categories": [ "pione" ], "title": "pione-lemma-find-point-codim-1", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring with $\\dim(A) \\geq 1$.", "Let $f \\in \\mathfrak m$. Then there exist a $\\mathfrak p \\in V(f)$ with", "$\\dim(A_\\mathfrak p) = 1$." ], "refs": [], "proofs": [ { "contents": [ "By induction on $\\dim(A)$. If $\\dim(A) = 1$, then $\\mathfrak p = \\mathfrak m$", "works. If $\\dim(A) > 1$, then let $Z \\subset \\Spec(A)$ be an irreducible", "component of dimension $> 1$. Then $V(f) \\cap Z$ has dimension $> 0$", "(Algebra, Lemma \\ref{algebra-lemma-one-equation}). Pick a prime", "$\\mathfrak q \\in V(f) \\cap Z$, $\\mathfrak q \\not = \\mathfrak m$", "corresponding to a closed point of the punctured spectrum of $A$;", "this is possible by", "Properties, Lemma \\ref{properties-lemma-complement-closed-point-Jacobson}.", "Then $\\mathfrak q$ is not the generic point of $Z$. Hence", "$0 < \\dim(A_\\mathfrak q) < \\dim(A)$ and $f \\in \\mathfrak q A_\\mathfrak q$.", "By induction on the dimension we can find", "$f \\in \\mathfrak p \\subset A_\\mathfrak q$ with", "$\\dim((A_\\mathfrak q)_\\mathfrak p) = 1$.", "Then $\\mathfrak p \\cap A$ works." ], "refs": [ "properties-lemma-complement-closed-point-Jacobson" ], "ref_ids": [ 2965 ] } ], "ref_ids": [] }, { "id": 4097, "type": "theorem", "label": "pione-lemma-ramification-quasi-finite-flat", "categories": [ "pione" ], "title": "pione-lemma-ramification-quasi-finite-flat", "contents": [ "Let $f : X \\to Y$ be a morphism of locally Noetherian schemes.", "Let $x \\in X$. Assume", "\\begin{enumerate}", "\\item $f$ is flat,", "\\item $f$ is quasi-finite at $x$,", "\\item $x$ is not a generic point of an irreducible component of $X$,", "\\item for specializations $x' \\leadsto x$ with", "$\\dim(\\mathcal{O}_{X, x'}) = 1$ our $f$ is unramified at $x'$.", "\\end{enumerate}", "Then $f$ is \\'etale at $x$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the set of points where $f$ is unramified is the same as", "the set of points where $f$ is \\'etale and that this set is open.", "See Morphisms, Definitions \\ref{morphisms-definition-unramified}", "and \\ref{morphisms-definition-etale} and", "Lemma \\ref{morphisms-lemma-flat-unramified-etale}.", "To check $f$ is \\'etale at $x$ we may work \\'etale", "locally on the base and on the", "target (Descent, Lemmas \\ref{descent-lemma-descending-property-etale} and", "\\ref{descent-lemma-etale-etale-local-source}).", "Thus we can apply More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point}", "and assume that $f : X \\to Y$ is finite and that $x$ is the unique", "point of $X$ lying over $y = f(x)$.", "Then it follows that $f$ is finite locally free", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}).", "\\medskip\\noindent", "Assume $f$ is finite locally free and that $x$ is the unique point of", "$X$ lying over $y = f(x)$. By", "Discriminants, Lemma \\ref{discriminant-lemma-discriminant}", "we find a locally principal closed subscheme $D_\\pi \\subset Y$", "such that $y' \\in D_\\pi$ if and only if there exists an $x' \\in X$", "with $f(x') = y'$ and $f$ ramified at $x'$. Thus we have to prove", "that $y \\not \\in D_\\pi$. Assume $y \\in D_\\pi$ to get a contradiction.", "\\medskip\\noindent", "By condition (3) we have $\\dim(\\mathcal{O}_{X, x}) \\geq 1$.", "We have $\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y})$ by", "Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}.", "By Lemma \\ref{lemma-find-point-codim-1}", "we can find $y' \\in D_\\pi$ specializing to $y$", "with $\\dim(\\mathcal{O}_{Y, y'}) = 1$.", "Choose $x' \\in X$ with $f(x') = y'$ where $f$ is ramified. Since $f$", "is finite it is closed, and hence $x' \\leadsto x$.", "We have $\\dim(\\mathcal{O}_{X, x'}) = \\dim(\\mathcal{O}_{Y, y'}) = 1$", "as before. This contradicts property (4)." ], "refs": [ "morphisms-definition-unramified", "morphisms-definition-etale", "morphisms-lemma-flat-unramified-etale", "descent-lemma-descending-property-etale", "descent-lemma-etale-etale-local-source", "more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point", "morphisms-lemma-finite-flat", "discriminant-lemma-discriminant", "algebra-lemma-dimension-base-fibre-equals-total", "pione-lemma-find-point-codim-1" ], "ref_ids": [ 5566, 5567, 5373, 14694, 14716, 13892, 5471, 14955, 987, 4096 ] } ], "ref_ids": [] }, { "id": 4098, "type": "theorem", "label": "pione-lemma-local-purity", "categories": [ "pione" ], "title": "pione-lemma-local-purity", "contents": [ "Let $(A, \\mathfrak m)$ be a regular local ring of dimension $d \\geq 2$.", "Set $X = \\Spec(A)$ and $U = X \\setminus \\{\\mathfrak m\\}$. Then", "\\begin{enumerate}", "\\item the functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_U$", "is essentially surjective, i.e., purity holds for $A$,", "\\item any finite $A \\to B$ with $B$ normal which", "induces a finite \\'etale morphism on punctured spectra is \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that a regular local ring is normal by", "Algebra, Lemma \\ref{algebra-lemma-regular-normal}.", "Hence (1) and (2) are equivalent by", "Lemma \\ref{lemma-reformulate-purity-normal}.", "We prove the lemma by induction on $d$.", "\\medskip\\noindent", "The case $d = 2$. In this case $A \\to B$ is flat.", "Namely, we have going down for $A \\to B$ by", "Algebra, Proposition \\ref{algebra-proposition-going-down-normal-integral}.", "Then $\\dim(B_{\\mathfrak m'}) = 2$ for all maximal ideals", "$\\mathfrak m' \\subset B$ by", "Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}.", "Then $B_{\\mathfrak m'}$ is Cohen-Macaulay by", "Algebra, Lemma \\ref{algebra-lemma-criterion-normal}.", "Hence and this is the important step", "Algebra, Lemma \\ref{algebra-lemma-CM-over-regular-flat}", "applies to show $A \\to B_{\\mathfrak m'}$ is flat.", "Then Algebra, Lemma \\ref{algebra-lemma-flat-localization}", "shows $A \\to B$ is flat. Thus we can apply", "Lemma \\ref{lemma-ramification-quasi-finite-flat}", "(or you can directly argue using the easier", "Discriminants, Lemma \\ref{discriminant-lemma-discriminant})", "to see that $A \\to B$ is \\'etale.", "\\medskip\\noindent", "The case $d \\geq 3$. Let $V \\to U$ be finite \\'etale.", "Let $f \\in \\mathfrak m_A$, $f \\not \\in \\mathfrak m_A^2$.", "Then $A/fA$ is a regular local ring of dimension $d - 1 \\geq 2$, see", "Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}.", "Let $U_0$ be the punctured spectrum of $A/fA$ and let", "$V_0 = V \\times_U U_0$.", "By Lemma \\ref{lemma-lift-purity}", "it suffices to show that $V_0$ is in the essential", "image of $\\textit{F\\'Et}_{\\Spec(A/fA)} \\to \\textit{F\\'Et}_{U_0}$.", "This follows from the induction hypothesis." ], "refs": [ "algebra-lemma-regular-normal", "pione-lemma-reformulate-purity-normal", "algebra-proposition-going-down-normal-integral", "algebra-lemma-dimension-base-fibre-equals-total", "algebra-lemma-criterion-normal", "algebra-lemma-CM-over-regular-flat", "algebra-lemma-flat-localization", "pione-lemma-ramification-quasi-finite-flat", "discriminant-lemma-discriminant", "algebra-lemma-regular-ring-CM", "pione-lemma-lift-purity" ], "ref_ids": [ 1312, 4091, 1406, 987, 1311, 1107, 538, 4097, 14955, 941, 4095 ] } ], "ref_ids": [] }, { "id": 4099, "type": "theorem", "label": "pione-lemma-purity", "categories": [ "pione" ], "title": "pione-lemma-purity", "contents": [ "\\begin{reference}", "\\cite{Nagata-Purity} and \\cite[Exp. X, Thm. 3.1]{SGA1}", "\\end{reference}", "\\begin{history}", "This result was first stated and proved by Zariski in", "geometric form in \\cite{Zariski-Purity}.", "The generalization to nonperfect ground fields by Nagata", "was published as the next article in the same volume of the", "Proceedings of the National Academy of Sciences of the United States of America", "in \\cite{Nagata-Remarks-Purity}. In the following year Nagata", "proved the result for Noetherian local rings in \\cite{Nagata-Purity}.", "His proof uses a result of Chow which is a Bertini theorem for", "complete local domains, see \\cite{Chow-Bertini};", "the history of Bertini's theorems is discussed in", "Kleiman's historical article \\cite{Kleiman-Bertini}.", "A few years later a completely different proof was found by", "Auslander, see \\cite{Auslander-Purity}.", "\\end{history}", "Let $f : X \\to Y$ be a morphism of locally Noetherian schemes.", "Let $x \\in X$ and set $y = f(x)$. Assume", "\\begin{enumerate}", "\\item $\\mathcal{O}_{X, x}$ is normal,", "\\item $\\mathcal{O}_{Y, y}$ is regular,", "\\item $f$ is quasi-finite at $x$,", "\\item $\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y}) \\geq 1$", "\\item for specializations $x' \\leadsto x$ with", "$\\dim(\\mathcal{O}_{X, x'}) = 1$ our $f$ is unramified at $x'$.", "\\end{enumerate}", "Then $f$ is \\'etale at $x$." ], "refs": [], "proofs": [ { "contents": [ "We will prove the lemma by induction on", "$d = \\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y})$.", "\\medskip\\noindent", "An uninteresting case is when $d = 1$.", "In that case we are assuming that $f$ is unramified at $x$", "and that $\\mathcal{O}_{Y, y}$ is a discrete valuation ring", "(Algebra, Lemma \\ref{algebra-lemma-characterize-dvr}).", "Then $\\mathcal{O}_{X, x}$ is flat over $\\mathcal{O}_{Y, y}$", "(otherwise the map would not be quasi-finite at $x$)", "and we see that $f$ is flat at $x$. Since flat $+$", "unramified is \\'etale we conclude (some details omitted).", "\\medskip\\noindent", "The case $d \\geq 2$. We will use induction on $d$ to reduce", "to the case discussed in Lemma \\ref{lemma-local-purity}.", "To check $f$ is \\'etale at $x$ we may work \\'etale locally", "on the base and on the target", "(Descent, Lemmas \\ref{descent-lemma-descending-property-etale} and", "\\ref{descent-lemma-etale-etale-local-source}).", "Thus we can apply More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point}", "and assume that $f : X \\to Y$ is finite and that $x$ is the unique", "point of $X$ lying over $y$. Here we use that \\'etale extensions of", "local rings do not change dimension, normality, and regularity, see", "More on Algebra, Section \\ref{more-algebra-section-permanence-etale}", "and", "\\'Etale Morphisms, Section \\ref{etale-section-properties-permanence}.", "\\medskip\\noindent", "Next, we can base change by $\\Spec(\\mathcal{O}_{Y, y})$", "and assume that $Y$ is the spectrum of a regular local ring.", "It follows that $X = \\Spec(\\mathcal{O}_{X, x})$ as", "every point of $X$ necessarily specializes to $x$.", "\\medskip\\noindent", "The ring map $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ is", "finite and necessarily injective (by equality of dimensions).", "We conclude we have going down for", "$\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ by", "Algebra, Proposition \\ref{algebra-proposition-going-down-normal-integral}", "(and the fact that a regular ring is a normal ring by", "Algebra, Lemma \\ref{algebra-lemma-regular-normal}).", "Pick $x' \\in X$, $x' \\not = x$ with image $y' = f(x')$.", "Then $\\mathcal{O}_{X, x'}$ is normal as a localization", "of a normal domain. Similarly, $\\mathcal{O}_{Y, y'}$ is", "regular (see Algebra, Lemma", "\\ref{algebra-lemma-localization-of-regular-local-is-regular}).", "We have $\\dim(\\mathcal{O}_{X, x'}) = \\dim(\\mathcal{O}_{Y, y'})$ by", "Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}", "(we checked going down above).", "Of course these dimensions are strictly less than $d$ as $x' \\not = x$", "and by induction on $d$ we conclude that $f$ is \\'etale at $x'$.", "\\medskip\\noindent", "Thus we arrive at the following situation: We have a finite", "local homomorphism $A \\to B$ of Noetherian local rings", "of dimension $d \\geq 2$, with $A$ regular, $B$ normal, which", "induces a finite \\'etale morphism $V \\to U$ on punctured spectra.", "Our goal is to show that $A \\to B$ is \\'etale.", "This follows from Lemma \\ref{lemma-local-purity}", "and the proof is complete." ], "refs": [ "algebra-lemma-characterize-dvr", "pione-lemma-local-purity", "descent-lemma-descending-property-etale", "descent-lemma-etale-etale-local-source", "more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point", "algebra-proposition-going-down-normal-integral", "algebra-lemma-regular-normal", "algebra-lemma-localization-of-regular-local-is-regular", "algebra-lemma-dimension-base-fibre-equals-total", "pione-lemma-local-purity" ], "ref_ids": [ 1023, 4098, 14694, 14716, 13892, 1406, 1312, 979, 987, 4098 ] } ], "ref_ids": [] }, { "id": 4100, "type": "theorem", "label": "pione-lemma-extend-S2", "categories": [ "pione" ], "title": "pione-lemma-extend-S2", "contents": [ "Let $j : U \\to X$ be an open immersion of locally Noetherian schemes", "such that $\\text{depth}(\\mathcal{O}_{X, x}) \\geq 2$ for $x \\not \\in U$.", "Let $\\pi : V \\to U$ be finite \\'etale. Then", "\\begin{enumerate}", "\\item $\\mathcal{B} = j_*\\pi_*\\mathcal{O}_V$ is a reflexive coherent", "$\\mathcal{O}_X$-algebra, set $Y = \\underline{\\Spec}_X(\\mathcal{B})$,", "\\item $Y \\to X$ is the unique finite morphism such that", "$V = Y \\times_X U$ and $\\text{depth}(\\mathcal{O}_{Y, y}) \\geq 2$", "for $y \\in Y \\setminus V$,", "\\item $Y \\to X$ is \\'etale at $y$ if and only if $Y \\to X$ is flat at $y$, and", "\\item $Y \\to X$ is \\'etale if and only if $\\mathcal{B}$", "is finite locally free as an $\\mathcal{O}_X$-module.", "\\end{enumerate}", "Moreover, (a) the construction of $\\mathcal{B}$ and $Y \\to X$ commutes", "with base change by flat morphisms $X' \\to X$ of locally Noetherian", "schemes, and (b) if $V' \\to U'$ is a finite \\'etale morphism with", "$U \\subset U' \\subset X$ open which restricts to $V \\to U$ over $U$,", "then there is a unique isomorphism $Y' \\times_X U' = V'$ over $U'$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\pi_*\\mathcal{O}_V$ is a finite locally free", "$\\mathcal{O}_U$-module, in particular reflexive.", "By Divisors, Lemma \\ref{divisors-lemma-reflexive-S2-extend}", "the module $j_*\\pi_*\\mathcal{O}_V$ is the unique", "reflexive coherent module on $X$ restricting to", "$\\pi_*\\mathcal{O}_V$ over $U$. This proves (1).", "\\medskip\\noindent", "By construction $Y \\times_X U = V$.", "Since $\\mathcal{B}$ is coherent, we see that $Y \\to X$ is finite.", "We have $\\text{depth}(\\mathcal{B}_x) \\geq 2$ for $x \\in X \\setminus U$", "by Divisors, Lemma \\ref{divisors-lemma-reflexive-S2}.", "Hence $\\text{depth}(\\mathcal{O}_{Y, y}) \\geq 2$ for $y \\in Y \\setminus V$", "by Algebra, Lemma \\ref{algebra-lemma-depth-goes-down-finite}.", "Conversely, suppose that $\\pi' : Y' \\to X$ is a finite morphism such that", "$V = Y' \\times_X U$ and $\\text{depth}(\\mathcal{O}_{Y', y'}) \\geq 2$", "for $y' \\in Y' \\setminus V$. Then $\\pi'_*\\mathcal{O}_{Y'}$", "restricts to $\\pi_*\\mathcal{O}_V$ over $U$ and satisfies", "$\\text{depth}((\\pi'_*\\mathcal{O}_{Y'})_x) \\geq 2$ for", "$x \\in X \\setminus U$ by", "Algebra, Lemma \\ref{algebra-lemma-depth-goes-down-finite}.", "Then $\\pi'_*\\mathcal{O}_{Y'}$ is canonically isomorphic", "to $j_*\\pi_*\\mathcal{O}_V$ for example by", "Divisors, Lemma \\ref{divisors-lemma-depth-2-hartog}.", "This proves (2).", "\\medskip\\noindent", "If $Y \\to X$ is \\'etale at $y$, then $Y \\to X$ is flat at $y$.", "Conversely, suppose that $Y \\to X$ is flat at $y$.", "If $y \\in V$, then $Y \\to X$ is \\'etale at $y$.", "If $y \\not \\in V$, then we check (1), (2), (3), and (4) of", "Lemma \\ref{lemma-ramification-quasi-finite-flat} hold", "to see that $Y \\to X$ is \\'etale at $y$. Parts (1) and (2)", "are clear and so is (3) since $\\text{depth}(\\mathcal{O}_{Y, y}) \\geq 2$.", "If $y' \\leadsto y$ is a specialization and $\\dim(\\mathcal{O}_{Y, y'}) = 1$,", "then $y' \\in V$ since otherwise the depth of this local ring", "would be $2$ a contradiction by", "Algebra, Lemma \\ref{algebra-lemma-bound-depth}.", "Hence $Y \\to X$ is \\'etale at $y'$ and we conclude (4) of", "Lemma \\ref{lemma-ramification-quasi-finite-flat} holds too.", "This finishes the proof of (3).", "\\medskip\\noindent", "Part (4) follows from (3) and the fact that $((Y \\to X)_*\\mathcal{O}_Y)_x$", "is a flat $\\mathcal{O}_{X, x}$-module if and only if $\\mathcal{O}_{Y, y}$", "is a flat $\\mathcal{O}_{X, x}$-module for all $y \\in Y$ mapping to $x$, see", "Algebra, Lemma \\ref{algebra-lemma-flat-localization}. Here we also", "use that a finite flat module over a Noetherian ring is finite locally", "free, see Algebra, Lemma \\ref{algebra-lemma-finite-projective}", "(and", "Algebra, Lemma", "\\ref{algebra-lemma-Noetherian-finite-type-is-finite-presentation}).", "\\medskip\\noindent", "As to the final assertions of the lemma, part (a) follows from", "flat base change, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}", "and part (b) follows from the uniqueness in (2) applied to the restriction", "$Y \\times_X U'$." ], "refs": [ "divisors-lemma-reflexive-S2-extend", "divisors-lemma-reflexive-S2", "algebra-lemma-depth-goes-down-finite", "algebra-lemma-depth-goes-down-finite", "divisors-lemma-depth-2-hartog", "pione-lemma-ramification-quasi-finite-flat", "algebra-lemma-bound-depth", "pione-lemma-ramification-quasi-finite-flat", "algebra-lemma-flat-localization", "algebra-lemma-finite-projective", "algebra-lemma-Noetherian-finite-type-is-finite-presentation", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 7923, 7922, 778, 778, 7881, 4097, 770, 4097, 538, 795, 451, 3298 ] } ], "ref_ids": [] }, { "id": 4101, "type": "theorem", "label": "pione-lemma-extend-pure", "categories": [ "pione" ], "title": "pione-lemma-extend-pure", "contents": [ "Let $j : U \\to X$ be an open immersion of Noetherian schemes", "such that purity holds for $\\mathcal{O}_{X, x}$ for all $x \\not \\in U$.", "Then", "$$", "\\textit{F\\'Et}_X \\longrightarrow \\textit{F\\'Et}_U", "$$", "is essentially surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\to U$ be a finite \\'etale morphism. By Noetherian", "induction it suffices to extend $V \\to U$ to a finite \\'etale", "morphism to a strictly larger open subset of $X$.", "Let $x \\in X \\setminus U$ be the generic point of", "an irreducible component of $X \\setminus U$.", "Then the inverse image $U_x$ of $U$ in $\\Spec(\\mathcal{O}_{X, x})$", "is the punctured spectrum of $\\mathcal{O}_{X, x}$.", "By assumption $V_x = V \\times_U U_x$ is the restriction", "of a finite \\'etale morphism $Y_x \\to \\Spec(\\mathcal{O}_{X, x})$", "to $U_x$.", "By Limits, Lemma \\ref{limits-lemma-glueing-near-point}", "we find an open subscheme $U \\subset U' \\subset X$", "containing $x$ and a morphism $V' \\to U'$ of finite presentation", "whose restriction to $U$ recovers $V \\to U$ and", "whose restriction to $\\Spec(\\mathcal{O}_{X, x})$ recovering $Y_x$.", "Finally, the morphism $V' \\to U'$ is finite \\'etale", "after possible shrinking $U'$ to a smaller open by", "Limits, Lemma \\ref{limits-lemma-glueing-near-point-properties}." ], "refs": [ "limits-lemma-glueing-near-point", "limits-lemma-glueing-near-point-properties" ], "ref_ids": [ 15114, 15115 ] } ], "ref_ids": [] }, { "id": 4102, "type": "theorem", "label": "pione-lemma-faithful-general", "categories": [ "pione" ], "title": "pione-lemma-faithful-general", "contents": [ "In Situation \\ref{situation-local-lefschetz}. Let $U' \\subset U$", "be open and contain $U_0$. Assume for $\\mathfrak p \\subset A$ minimal", "with $\\mathfrak p \\in U'$, $\\mathfrak p \\not \\in U_0$ we have", "$\\dim(A/\\mathfrak p) \\geq 2$. Then", "$$", "\\textit{F\\'Et}_{U'} \\longrightarrow \\textit{F\\'Et}_{U_0},\\quad", "V' \\longmapsto V_0 = V' \\times_{U'} U_0", "$$", "is a faithful functor. Moreover, there exists a $U'$ satisfying", "the assumption and any smaller open $U'' \\subset U'$ containing", "$U_0$ also satisfies this assumption. In particular, the restriction", "functor", "$$", "\\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'}", "\\longrightarrow", "\\textit{F\\'Et}_{U_0}", "$$", "is faithful." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-one-equation}", "we see that $V(\\mathfrak p)$ meets $U_0$ for", "every prime $\\mathfrak p$ of $A$ with $\\dim(A/\\mathfrak p) \\geq 2$.", "Thus the displayed functor is faithful for a $U$ as in the statement", "by Lemma \\ref{lemma-restriction-faithful}.", "To see the existence of such a $U'$ note that for", "$\\mathfrak p \\subset A$ with $\\mathfrak p \\in U$,", "$\\mathfrak p \\not \\in U_0$ with $\\dim(A/\\mathfrak p) = 1$", "then $\\mathfrak p$ corresponds to a closed point of $U$", "and hence $V(\\mathfrak p) \\cap U_0 = \\emptyset$.", "Thus we can take $U'$ to be the complement of the irreducible", "components of $X$ which do not meet $U_0$ and have dimension $1$." ], "refs": [ "pione-lemma-restriction-faithful" ], "ref_ids": [ 4079 ] } ], "ref_ids": [] }, { "id": 4103, "type": "theorem", "label": "pione-lemma-fully-faithful-general-better", "categories": [ "pione" ], "title": "pione-lemma-fully-faithful-general-better", "contents": [ "In Situation \\ref{situation-local-lefschetz} assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex and is $f$-adically complete,", "\\item every irreducible component of $X$ not contained in $X_0$", "has dimension $\\geq 3$.", "\\end{enumerate}", "Then the restriction functor", "$$", "\\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'}", "\\longrightarrow", "\\textit{F\\'Et}_{U_0}", "$$", "is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "To prove this we may replace $A$ by its reduction by the topological", "invariance of the fundamental group, see Lemma \\ref{lemma-thickening}.", "Then the result follows from", "Lemma \\ref{lemma-restriction-fully-faithful-general}", "and Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-fully-faithful-general}." ], "refs": [ "pione-lemma-thickening", "pione-lemma-restriction-fully-faithful-general", "algebraization-lemma-fully-faithful-general" ], "ref_ids": [ 4043, 4077, 12745 ] } ], "ref_ids": [] }, { "id": 4104, "type": "theorem", "label": "pione-lemma-fully-faithful-general", "categories": [ "pione" ], "title": "pione-lemma-fully-faithful-general", "contents": [ "In Situation \\ref{situation-local-lefschetz} assume", "\\begin{enumerate}", "\\item $A$ is $f$-adically complete,", "\\item $f$ is a nonzerodivisor.", "\\item $H^1_\\mathfrak m(A/fA)$ is a finite $A$-module.", "\\end{enumerate}", "Then the restriction functor", "$$", "\\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'}", "\\longrightarrow", "\\textit{F\\'Et}_{U_0}", "$$", "is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-restriction-fully-faithful-general} and", "Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-fully-faithful-general-alternative}." ], "refs": [ "pione-lemma-restriction-fully-faithful-general", "algebraization-lemma-fully-faithful-general-alternative" ], "ref_ids": [ 4077, 12746 ] } ], "ref_ids": [] }, { "id": 4105, "type": "theorem", "label": "pione-lemma-essentially-surjective-general-better", "categories": [ "pione" ], "title": "pione-lemma-essentially-surjective-general-better", "contents": [ "In Situation \\ref{situation-local-lefschetz} assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex and is $f$-adically complete,", "\\item one of the following is true", "\\begin{enumerate}", "\\item $A_f$ is $(S_2)$ and every irreducible component of $X$", "not contained in $X_0$ has dimension $\\geq 4$, or", "\\item if $\\mathfrak p \\not \\in V(f)$ and", "$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then", "$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.", "\\end{enumerate}", "\\end{enumerate}", "Then the restriction functor", "$$", "\\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'}", "\\longrightarrow", "\\textit{F\\'Et}_{U_0}", "$$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-restriction-equivalence-general}", "and", "Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-equivalence-better}." ], "refs": [ "pione-lemma-restriction-equivalence-general", "algebraization-lemma-equivalence-better" ], "ref_ids": [ 4078, 12777 ] } ], "ref_ids": [] }, { "id": 4106, "type": "theorem", "label": "pione-lemma-essentially-surjective-general", "categories": [ "pione" ], "title": "pione-lemma-essentially-surjective-general", "contents": [ "In Situation \\ref{situation-local-lefschetz} assume", "\\begin{enumerate}", "\\item $A$ is $f$-adically complete,", "\\item $f$ is a nonzerodivisor,", "\\item $H^1_\\mathfrak m(A/fA)$ and $H^2_\\mathfrak m(A/fA)$", "are finite $A$-modules.", "\\end{enumerate}", "Then the restriction functor", "$$", "\\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'}", "\\longrightarrow", "\\textit{F\\'Et}_{U_0}", "$$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-restriction-equivalence-general}", "and", "Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-equivalence}." ], "refs": [ "pione-lemma-restriction-equivalence-general", "algebraization-lemma-equivalence" ], "ref_ids": [ 4078, 12778 ] } ], "ref_ids": [] }, { "id": 4107, "type": "theorem", "label": "pione-lemma-equivalence-better", "categories": [ "pione" ], "title": "pione-lemma-equivalence-better", "contents": [ "In Situation \\ref{situation-local-lefschetz} assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex and is $f$-adically complete,", "\\item one of the following is true", "\\begin{enumerate}", "\\item $A_f$ is $(S_2)$ and every irreducible component of $X$", "not contained in $X_0$ has dimension $\\geq 4$, or", "\\item if $\\mathfrak p \\not \\in V(f)$ and", "$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then", "$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.", "\\end{enumerate}", "\\item for every maximal ideal $\\mathfrak p \\subset A_f$", "purity holds for $(A_f)_\\mathfrak p$.", "\\end{enumerate}", "Then the restriction functor $\\textit{F\\'Et}_U \\to \\textit{F\\'Et}_{U_0}$", "is essentially surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $V_0 \\to U_0$ be a finite \\'etale morphism. By", "Lemma \\ref{lemma-essentially-surjective-general-better}", "there exists an open $U' \\subset U$ containing $U_0$ and", "a finite \\'etale morphism $V' \\to U$ whose base change to $U_0$", "is isomorphic to $V_0 \\to U_0$. Since $U' \\supset U_0$", "we see that $U \\setminus U'$ consists of points corresponding", "to prime ideals $\\mathfrak p_1, \\ldots, \\mathfrak p_n$ as in (3).", "By assumption we can find finite \\'etale morphisms", "$V'_i \\to \\Spec(A_{\\mathfrak p_i})$ agreeing with", "$V' \\to U'$ over $U' \\times_U \\Spec(A_{\\mathfrak p_i})$.", "By Limits, Lemma \\ref{limits-lemma-glueing-near-closed-point}", "applied $n$ times we see that $V' \\to U'$ extends to a finite \\'etale", "morphism $V \\to U$." ], "refs": [ "pione-lemma-essentially-surjective-general-better", "limits-lemma-glueing-near-closed-point" ], "ref_ids": [ 4105, 15112 ] } ], "ref_ids": [] }, { "id": 4108, "type": "theorem", "label": "pione-lemma-equivalence", "categories": [ "pione" ], "title": "pione-lemma-equivalence", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "Let $f \\in \\mathfrak m$. Assume", "\\begin{enumerate}", "\\item $A$ is $f$-adically complete,", "\\item $f$ is a nonzerodivisor,", "\\item $H^1_\\mathfrak m(A/fA)$ and $H^2_\\mathfrak m(A/fA)$ are finite", "$A$-modules,", "\\item for every maximal ideal $\\mathfrak p \\subset A_f$", "purity holds for $(A_f)_\\mathfrak p$.", "\\end{enumerate}", "Then the restriction functor $\\textit{F\\'Et}_U \\to \\textit{F\\'Et}_{U_0}$", "is essentially surjective." ], "refs": [], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Lemma \\ref{lemma-equivalence-better}", "using", "Lemma \\ref{lemma-essentially-surjective-general}", "in stead of", "Lemma \\ref{lemma-essentially-surjective-general-better}." ], "refs": [ "pione-lemma-equivalence-better", "pione-lemma-essentially-surjective-general", "pione-lemma-essentially-surjective-general-better" ], "ref_ids": [ 4107, 4106, 4105 ] } ], "ref_ids": [] }, { "id": 4109, "type": "theorem", "label": "pione-lemma-purity-inherited-by-hypersurface-better", "categories": [ "pione" ], "title": "pione-lemma-purity-inherited-by-hypersurface-better", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "Let $f \\in \\mathfrak m$. Assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex and is $f$-adically complete,", "\\item one of the following is true", "\\begin{enumerate}", "\\item $A_f$ is $(S_2)$ and every irreducible component of $X$", "not contained in $X_0$ has dimension $\\geq 4$, or", "\\item if $\\mathfrak p \\not \\in V(f)$ and", "$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then", "$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.", "\\end{enumerate}", "\\item for every maximal ideal $\\mathfrak p \\subset A_f$", "purity holds for $(A_f)_\\mathfrak p$, and", "\\item purity holds for $A$.", "\\end{enumerate}", "Then purity holds for $A/fA$." ], "refs": [], "proofs": [ { "contents": [ "Denote $X = \\Spec(A)$ and $U = X \\setminus \\{\\mathfrak m\\}$", "the punctured spectrum. Similarly we have $X_0 = \\Spec(A/fA)$", "and $U_0 = X_0 \\setminus \\{\\mathfrak m\\}$.", "Let $V_0 \\to U_0$ be a finite \\'etale morphism. By", "Lemma \\ref{lemma-equivalence-better}", "we find a finite \\'etale morphism $V \\to U$", "whose base change to $U_0$", "is isomorphic to $V_0 \\to U_0$.", "By assumption (5) we find that $V \\to U$ extends", "to a finite \\'etale morphism $Y \\to X$. Then the restriction of", "$Y$ to $X_0$ is the desired extension of $V_0 \\to U_0$." ], "refs": [ "pione-lemma-equivalence-better" ], "ref_ids": [ 4107 ] } ], "ref_ids": [] }, { "id": 4110, "type": "theorem", "label": "pione-lemma-purity-inherited-by-hypersurface", "categories": [ "pione" ], "title": "pione-lemma-purity-inherited-by-hypersurface", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "Let $f \\in \\mathfrak m$. Assume", "\\begin{enumerate}", "\\item $A$ is $f$-adically complete,", "\\item $f$ is a nonzerodivisor,", "\\item $H^1_\\mathfrak m(A/fA)$ and $H^2_\\mathfrak m(A/fA)$ are finite", "$A$-modules,", "\\item for every maximal ideal $\\mathfrak p \\subset A_f$", "purity holds for $(A_f)_\\mathfrak p$,", "\\item purity holds for $A$.", "\\end{enumerate}", "Then purity holds for $A/fA$." ], "refs": [], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Lemma \\ref{lemma-purity-inherited-by-hypersurface-better}", "using", "Lemma \\ref{lemma-equivalence}", "in stead of", "Lemma \\ref{lemma-equivalence-better}." ], "refs": [ "pione-lemma-purity-inherited-by-hypersurface-better", "pione-lemma-equivalence", "pione-lemma-equivalence-better" ], "ref_ids": [ 4109, 4108, 4107 ] } ], "ref_ids": [] }, { "id": 4111, "type": "theorem", "label": "pione-lemma-fully-faithful-power-series-over-depth2", "categories": [ "pione" ], "title": "pione-lemma-fully-faithful-power-series-over-depth2", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring of depth $\\geq 2$.", "Let $B = A[[x_1, \\ldots, x_d]]$ with $d \\geq 1$.", "Set $Y = \\Spec(B)$ and $Y_0 = V(x_1, \\ldots, x_d)$.", "For any open subscheme $V \\subset Y$ with", "$V_0 = V \\cap Y_0$ equal to $Y_0 \\setminus \\{\\mathfrak m_B\\}$", "the restriction functor", "$$", "\\textit{F\\'Et}_V \\longrightarrow \\textit{F\\'Et}_{V_0}", "$$", "is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "Set $I = (x_1, \\ldots, x_d)$. Set $X = \\Spec(A)$.", "If we use the map $Y \\to X$ to identify $Y_0$ with $X$,", "then $V_0$ is identified with the punctured spectrum $U$ of $A$.", "Pushing forward modules by this affine morphism we get", "\\begin{align*}", "\\lim_n \\Gamma(V_0, \\mathcal{O}_V/I^n\\mathcal{O}_V)", "& =", "\\lim_n \\Gamma(V_0, \\mathcal{O}_Y/I^n\\mathcal{O}_Y) \\\\", "& =", "\\lim_n \\Gamma(U, \\mathcal{O}_U[x_1, \\ldots, x_d]/(x_1, \\ldots, x_d)^n) \\\\", "& =", "\\lim_n A[x_1, \\ldots, x_d]/(x_1, \\ldots, x_d)^n \\\\", "& =", "B", "\\end{align*}", "Namely, as the depth of $A$ is $\\geq 2$ we have $\\Gamma(U, \\mathcal{O}_U) = A$,", "see Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}.", "Thus for any $V \\subset Y$ open as in the lemma we get", "$$", "B = \\Gamma(Y, \\mathcal{O}_Y) \\to \\Gamma(V, \\mathcal{O}_V) \\to", "\\lim_n \\Gamma(V_0, \\mathcal{O}_Y/I^n\\mathcal{O}_Y) = B", "$$", "which implies both arrows are isomorphisms (small detail omitted).", "By Algebraic and Formal Geometry, Lemma ", "\\ref{algebraization-lemma-completion-fully-faithful}", "we conclude that", "$\\textit{Coh}(\\mathcal{O}_V) \\to \\textit{Coh}(V, I\\mathcal{O}_V)$", "is fully faithful on the full subcategory of finite locally free objects.", "Thus we conclude by", "Lemma \\ref{lemma-restriction-fully-faithful}." ], "refs": [ "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "algebraization-lemma-completion-fully-faithful", "pione-lemma-restriction-fully-faithful" ], "ref_ids": [ 9729, 12738, 4075 ] } ], "ref_ids": [] }, { "id": 4112, "type": "theorem", "label": "pione-lemma-purity-power-series-over-depth2", "categories": [ "pione" ], "title": "pione-lemma-purity-power-series-over-depth2", "contents": [ "\\begin{slogan}", "Ramanujam-Samuel for finite \\'etale covers", "\\end{slogan}", "Let $(A, \\mathfrak m)$ be a Noetherian local ring of depth $\\geq 2$. Let", "$B = A[[x_1, \\ldots, x_d]]$ with $d \\geq 1$. For any open", "$V \\subset Y = \\Spec(B)$ which contains", "\\begin{enumerate}", "\\item any prime $\\mathfrak q \\subset B$ such that", "$\\mathfrak q \\cap A \\not = \\mathfrak m$,", "\\item the prime $\\mathfrak m B$", "\\end{enumerate}", "the functor", "$", "\\textit{F\\'Et}_Y", "\\to", "\\textit{F\\'Et}_V", "$", "is an equivalence. In particular purity holds for $B$." ], "refs": [], "proofs": [ { "contents": [ "A prime $\\mathfrak q \\subset B$ which is not contained in $V$", "lies over $\\mathfrak m$. In this case $A \\to B_\\mathfrak q$", "is a flat local homomorphism and hence $\\text{depth}(B_\\mathfrak q) \\geq 2$", "(Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck}).", "Thus the functor is fully faithful by", "Lemma \\ref{lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian}", "combined with Local Cohomology,", "Lemma \\ref{local-cohomology-lemma-depth-2-connected-punctured-spectrum}.", "\\medskip\\noindent", "Let $W \\to V$ be a finite \\'etale morphism. Let $B \\to C$ be the unique finite", "ring map such that $\\Spec(C) \\to Y$ is the finite morphism extending", "$W \\to V$ constructed in Lemma \\ref{lemma-extend-S2}.", "Observe that $C = \\Gamma(W, \\mathcal{O}_W)$.", "\\medskip\\noindent", "Set $Y_0 = V(x_1, \\ldots, x_d)$ and $V_0 = V \\cap Y_0$. Set $X = \\Spec(A)$.", "If we use the map $Y \\to X$ to identify $Y_0$ with $X$,", "then $V_0$ is identified with the punctured spectrum $U$ of $A$.", "Thus we may view $W_0 = W \\times_Y Y_0$ as a finite \\'etale scheme", "over $U$. Then", "$$", "W_0 \\times_U (U \\times_X Y)", "\\quad\\text{and}\\quad", "W \\times_V (U \\times_X Y)", "$$", "are schemes finite \\'etale over $U \\times_X Y$ which restrict to", "isomorphic finite \\'etale schemes over $V_0$. By", "Lemma \\ref{lemma-fully-faithful-power-series-over-depth2}", "applied to the open $U \\times_X Y$ we obtain an isomorphism", "$$", "W_0 \\times_U (U \\times_X Y) \\longrightarrow W \\times_V (U \\times_X Y)", "$$", "over $U \\times_X Y$.", "\\medskip\\noindent", "Observe that $C_0 = \\Gamma(W_0, \\mathcal{O}_{W_0})$ is a finite $A$-algebra", "by Lemma \\ref{lemma-extend-S2} applied to $W_0 \\to U \\subset X$ (exactly", "as we did for $B \\to C$ above). Since the construction in", "Lemma \\ref{lemma-extend-S2} is compatible with flat base change", "and with change of opens, the isomorphism above induces", "an isomorphism", "$$", "\\Psi : C \\longrightarrow C_0 \\otimes_A B", "$$", "of finite $B$-algebras. However, we know that $\\Spec(C) \\to Y$", "is \\'etale at all points above at least one point of $Y$ lying", "over $\\mathfrak m \\in X$. Since $\\Psi$ is an isomorphism, we", "conclude that $\\Spec(C_0) \\to X$", "is \\'etale above $\\mathfrak m$ (small detail omitted).", "Of course this means that $A \\to C_0$ is finite", "\\'etale and hence $B \\to C$ is finite \\'etale." ], "refs": [ "algebra-lemma-apply-grothendieck", "pione-lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian", "local-cohomology-lemma-depth-2-connected-punctured-spectrum", "pione-lemma-extend-S2", "pione-lemma-fully-faithful-power-series-over-depth2", "pione-lemma-extend-S2", "pione-lemma-extend-S2" ], "ref_ids": [ 1361, 4048, 9701, 4100, 4111, 4100, 4100 ] } ], "ref_ids": [] }, { "id": 4113, "type": "theorem", "label": "pione-lemma-purity-smooth-over-depth2", "categories": [ "pione" ], "title": "pione-lemma-purity-smooth-over-depth2", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $U \\subset X$", "be an open subscheme. Assume", "\\begin{enumerate}", "\\item $f$ is smooth,", "\\item $S$ is Noetherian,", "\\item for $s \\in S$ with $\\text{depth}(\\mathcal{O}_{S, s}) \\leq 1$", "we have $X_s = U_s$,", "\\item $U_s \\subset X_s$ is dense for all $s \\in S$.", "\\end{enumerate}", "Then $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_U$ is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "The functor is fully faithful by", "Lemma \\ref{lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian}", "combined with Local Cohomology,", "Lemma \\ref{local-cohomology-lemma-depth-2-connected-punctured-spectrum}", "(plus an application of", "Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck}", "to check the depth condition).", "\\medskip\\noindent", "Let $\\pi : V \\to U$ be a finite \\'etale morphism. Let $Y \\to X$", "be the finite morphism constructed in Lemma \\ref{lemma-extend-S2}.", "We have to show that $Y \\to X$ is finite \\'etale.", "To show that this is true for all points $x \\in X$ mapping to a", "given point $s \\in S$ we may perform a base change by a flat", "morphism $S' \\to S$ of Noetherian schemes such that $s$ is", "in the image. This follows from the compatibility of the", "construction in Lemma \\ref{lemma-extend-S2} with flat base change.", "\\medskip\\noindent", "After enlarging $U$ we may assume $U \\subset X$ is", "the maximal open over which $Y \\to X$ is finite \\'etale.", "Let $Z \\subset X$ be the complement of $U$.", "To get a contradiction, assume $Z \\not = \\emptyset$.", "Let $s \\in S$ be a point in the image of $Z \\to S$", "such that no strict generalization of $s$ is in the image.", "Then after base change to $\\Spec(\\mathcal{O}_{S, s})$", "we see that $S = \\Spec(A)$ with $(A, \\mathfrak m, \\kappa)$", "a local Noetherian ring of depth $\\geq 2$ and $Z$", "contained in the closed fibre $X_s$", "and nowhere dense in $X_s$. Choose a closed point $z \\in Z$.", "Then $\\kappa(z)/\\kappa$ is finite (by the Hilbert Nullstellensatz, see", "Algebra, Theorem \\ref{algebra-theorem-nullstellensatz}).", "Choose a finite flat morphism $(S', s') \\to (S, s)$ of local schemes", "realizing the residue field extension $\\kappa(z)/\\kappa$, see", "Algebra, Lemma \\ref{algebra-lemma-finite-free-given-residue-field-extension}.", "After doing a base change by $S' \\to S$ we reduce to the case", "where $\\kappa(z) = \\kappa$.", "\\medskip\\noindent", "By More on Morphisms, Lemma \\ref{more-morphisms-lemma-slice-smooth}", "there exists a locally closed subscheme $S' \\subset X$ passing through $z$", "such that $S' \\to S$ is \\'etale at $z$. After performing the base change", "by $S' \\to S$, we may assume there is a section $\\sigma : S \\to X$", "such that $\\sigma(s) = z$. Choose an affine neighbourhood", "$\\Spec(B) \\subset X$ of $s$. Then $A \\to B$ is a smooth ring", "map which has a section $\\sigma : B \\to A$. Denote $I = \\Ker(\\sigma)$", "and denote $B^\\wedge$ the $I$-adic completion of $B$.", "Then $B^\\wedge \\cong A[[x_1, \\ldots, x_d]]$ for some $d \\geq 0$, see", "Algebra, Lemma \\ref{algebra-lemma-section-smooth}.", "Observe that $d > 0$ since otherwise we see that $X \\to S$", "is \\'etale at $z$ which would imply that $z$ is a generic point of", "$X_s$ and hence $z \\in U$ by assumption (4).", "Similarly, if $d > 0$, then $\\mathfrak m B^\\wedge$ maps into", "$U$ via the morphism $\\Spec(B^\\wedge) \\to X$.", "It suffices prove $Y \\to X$ is finite \\'etale after base change", "to $\\Spec(B^\\wedge)$. Since $B \\to B^\\wedge$ is flat", "(Algebra, Lemma \\ref{algebra-lemma-completion-flat})", "this follows from Lemma \\ref{lemma-purity-power-series-over-depth2}", "and the uniqueness in the construction of $Y \\to X$." ], "refs": [ "pione-lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian", "local-cohomology-lemma-depth-2-connected-punctured-spectrum", "algebra-lemma-apply-grothendieck", "pione-lemma-extend-S2", "pione-lemma-extend-S2", "algebra-theorem-nullstellensatz", "algebra-lemma-finite-free-given-residue-field-extension", "more-morphisms-lemma-slice-smooth", "algebra-lemma-completion-flat", "pione-lemma-purity-power-series-over-depth2" ], "ref_ids": [ 4048, 9701, 1361, 4100, 4100, 316, 1326, 13878, 870, 4112 ] } ], "ref_ids": [] }, { "id": 4114, "type": "theorem", "label": "pione-lemma-characterize-rational-singularity", "categories": [ "pione" ], "title": "pione-lemma-characterize-rational-singularity", "contents": [ "Let $A$ be a Noetherian normal local domain of dimension $2$.", "Assume $A$ is Nagata, has a dualizing module $\\omega_A$, and has a", "resolution of singularities $f : X \\to \\Spec(A)$.", "Let $\\omega_X$ be as in Resolution of Surfaces,", "Remark \\ref{resolve-remark-dualizing-setup}.", "If $\\omega_X \\cong \\mathcal{O}_X(E)$ for some effective", "Cartier divisor $E \\subset X$ supported on the exceptional", "fibre, then $A$ defines a rational singularity.", "If $f$ is a minimal resolution, then $E = 0$." ], "refs": [ "resolve-remark-dualizing-setup" ], "proofs": [ { "contents": [ "There is a trace map $Rf_*\\omega_X \\to \\omega_A$, see", "Duality for Schemes, Section \\ref{duality-section-trace}.", "By Grauert-Riemenschneider", "(Resolution of Surfaces,", "Proposition \\ref{resolve-proposition-Grauert-Riemenschneider})", "we have $R^1f_*\\omega_X = 0$.", "Thus the trace map is a map $f_*\\omega_X \\to \\omega_A$.", "Then we can consider", "$$", "\\mathcal{O}_{\\Spec(A)} = f_*\\mathcal{O}_X \\to f_*\\omega_X \\to \\omega_A", "$$", "where the first map comes from the map", "$\\mathcal{O}_X \\to \\mathcal{O}_X(E) = \\omega_X$ which is", "assumed to exist in the statement of the lemma.", "The composition is an isomorphism by Divisors, Lemma", "\\ref{divisors-lemma-check-isomorphism-via-depth-and-ass}", "as it is an isomorphism over the punctured spectrum of $A$", "(by the assumption in the lemma and the fact that $f$ is an isomorphism", "over the punctured spectrum) and $A$ and $\\omega_A$", "are $A$-modules of depth $2$ (by", "Algebra, Lemma \\ref{algebra-lemma-criterion-normal} and", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth-dualizing-module}).", "Hence $f_*\\omega_X \\to \\omega_A$ is surjective whence an isomorphism.", "Thus $Rf_*\\omega_X = \\omega_A$ which by duality implies", "$Rf_*\\mathcal{O}_X = \\mathcal{O}_{\\Spec(A)}$.", "Whence $H^1(X, \\mathcal{O}_X) = 0$ which implies that $A$", "defines a rational singularity (see discussion in", "Resolution of Surfaces, Section", "\\ref{resolve-section-bounded} in particular", "Lemmas \\ref{resolve-lemma-regular-rational} and", "\\ref{resolve-lemma-exact-sequence}).", "If $f$ is minimal, then $E = 0$ because the map", "$f^*\\omega_A \\to \\omega_X$ is surjective by", "a repeated application of Resolution of Surfaces, Lemma", "\\ref{resolve-lemma-dualizing-blow-up-rational}", "and $\\omega_A \\cong A$ as we've seen above." ], "refs": [ "resolve-proposition-Grauert-Riemenschneider", "divisors-lemma-check-isomorphism-via-depth-and-ass", "algebra-lemma-criterion-normal", "dualizing-lemma-depth-dualizing-module", "resolve-lemma-regular-rational", "resolve-lemma-exact-sequence", "resolve-lemma-dualizing-blow-up-rational" ], "ref_ids": [ 11711, 7864, 1311, 2871, 11666, 11661, 11675 ] } ], "ref_ids": [ 11717 ] }, { "id": 4115, "type": "theorem", "label": "pione-lemma-key-purity-ramification", "categories": [ "pione" ], "title": "pione-lemma-key-purity-ramification", "contents": [ "Let $f : X \\to \\Spec(A)$ be a finite type morphism.", "Let $x \\in X$ be a point. Assume", "\\begin{enumerate}", "\\item $A$ is an excellent regular local ring,", "\\item $\\mathcal{O}_{X, x}$ is normal of dimension $2$,", "\\item $f$ is \\'etale outside of $\\overline{\\{x\\}}$.", "\\end{enumerate}", "Then $f$ is \\'etale at $x$." ], "refs": [], "proofs": [ { "contents": [ "We first replace $X$ by an affine open neighbourhood of $x$.", "Observe that $\\mathcal{O}_{X, x}$ is an excellent local ring", "(More on Algebra, Lemma \\ref{more-algebra-lemma-finite-type-over-excellent}).", "Thus we can choose a minimal resolution of singularities", "$W \\to \\Spec(\\mathcal{O}_{X, x})$, see", "Resolution of Surfaces, Theorem \\ref{resolve-theorem-resolve}.", "After possibly replacing $X$ by an affine open neighbourhood of $x$", "we can find a proper morphism $b : X' \\to X$ such that", "$X' \\times_X \\Spec(\\mathcal{O}_{X, x}) = W$, see", "Limits, Lemma \\ref{limits-lemma-glueing-near-closed-point}.", "After shrinking $X$ further, we may assume $X'$ is regular.", "Namely, we know $W$ is regular and $X'$ is excellent", "and the regular locus of the spectrum of an excellent ring is open.", "Since $W \\to \\Spec(\\mathcal{O}_{X, x})$ is projective", "(as a sequence of normalized blowing ups), we may assume", "after shrinking $X$ that $b$ is projective (details omitted).", "Let $U = X \\setminus \\overline{\\{x\\}}$.", "Since $W \\to \\Spec(\\mathcal{O}_{X, x})$ is an isomorphism over the", "punctured spectrum, we may assume $b : X' \\to X$ is an isomorphism over $U$.", "Thus we may and will think of $U$ as an open subscheme of $X'$ as well.", "Set $f' = f \\circ b : X' \\to \\Spec(A)$.", "\\medskip\\noindent", "Since $A$ is regular we see that $\\mathcal{O}_Y$ is a dualizing complex for $Y$.", "Hence $f^!\\mathcal{O}_Y$ is a dualzing complex on $X$", "(Duality for Schemes, Lemma \\ref{duality-lemma-shriek-dualizing}).", "The Cohen-Macaulay locus of $X$ is open by", "Duality for Schemes, Lemma \\ref{duality-lemma-dualizing-module-CM-scheme}", "(this can also be proven using excellency).", "Since $\\mathcal{O}_{X, x}$ is Cohen-Macaulay, after shrinking", "$X$ we may assume $X$ is Cohen-Macaulay.", "Observe that an \\'etale morphism is a local complete intersection.", "Thus", "Duality for Schemes, Lemma \\ref{duality-lemma-fundamental-class-almost-lci}", "applies with $r = 0$ and we get a map", "$$", "\\mathcal{O}_X \\longrightarrow \\omega_{X/Y} = H^0(f^!\\mathcal{O}_Y)", "$$", "which is an isomorphism over $X \\setminus \\overline{\\{x\\}}$.", "Since $\\omega_{X/Y}$ is $(S_2)$ by", "Duality for Schemes, Lemma \\ref{duality-lemma-shriek-over-CM}", "we find this map is an isomorphism by", "Divisors, Lemma \\ref{divisors-lemma-check-isomorphism-via-depth-and-ass}.", "This already shows that $X$ and in particular $\\mathcal{O}_{X, x}$ is", "Gorenstein.", "\\medskip\\noindent", "Set $\\omega_{X'/Y} = H^0((f')^!\\mathcal{O}_Y)$. Arguing in exactly the", "same manner as above we find that $(f')^!\\mathcal{O}_Y = \\omega_{X'/Y}[0]$", "is a dualizing complex for $X'$. Since $X'$ is regular", "the morphism $X' \\to Y$ is a local complete intersection morphism, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-morphism-regular-schemes-is-lci}.", "By Duality for Schemes, Lemma \\ref{duality-lemma-fundamental-class-lci}", "there exists a map", "$$", "\\mathcal{O}_{X'} \\longrightarrow \\omega_{X'/Y}", "$$", "which is an isomorphism over $U$. We conclude", "$\\omega_{X'/Y} = \\mathcal{O}_{X'}(E)$ for some effective", "Cartier divisor $E \\subset X'$ disjoint from $U$.", "\\medskip\\noindent", "Since $\\omega_{X/Y} = \\mathcal{O}_Y$ we see that", "$\\omega_{X'/Y} = b^! f^!\\mathcal{O}_Y = b^!\\mathcal{O}_X$.", "Returning to $W \\to \\Spec(\\mathcal{O}_{X, x})$", "we see that $\\omega_W = \\mathcal{O}_W(E|_W)$.", "By Lemma \\ref{lemma-characterize-rational-singularity}", "we find $E|_W = 0$.", "This means that $f' : X' \\to Y$ is \\'etale by (the already used)", "Duality for Schemes, Lemma \\ref{duality-lemma-fundamental-class-lci}.", "This immediately finishes the proof, as \\'etaleness", "of $f'$ forces $b$ to be an isomorphism." ], "refs": [ "more-algebra-lemma-finite-type-over-excellent", "resolve-theorem-resolve", "limits-lemma-glueing-near-closed-point", "duality-lemma-shriek-dualizing", "duality-lemma-dualizing-module-CM-scheme", "duality-lemma-fundamental-class-almost-lci", "duality-lemma-shriek-over-CM", "divisors-lemma-check-isomorphism-via-depth-and-ass", "more-morphisms-lemma-morphism-regular-schemes-is-lci", "duality-lemma-fundamental-class-lci", "pione-lemma-characterize-rational-singularity", "duality-lemma-fundamental-class-lci" ], "ref_ids": [ 10106, 11635, 15112, 13560, 13586, 13620, 13580, 7864, 14009, 13619, 4114, 13619 ] } ], "ref_ids": [] }, { "id": 4116, "type": "theorem", "label": "pione-lemma-purity-ramification", "categories": [ "pione" ], "title": "pione-lemma-purity-ramification", "contents": [ "\\begin{reference}", "This result for complex spaces can be found on page 170 of \\cite{Fischer}.", "In general this is \\cite[Theorem 2.4]{Zong} attributed to Gabber.", "\\end{reference}", "Let $f : X \\to Y$ be a morphism of locally Noetherian schemes.", "Let $x \\in X$ and set $y = f(x)$. Assume", "\\begin{enumerate}", "\\item $\\mathcal{O}_{X, x}$ is normal of dimension $\\geq 1$,", "\\item $\\mathcal{O}_{Y, y}$ is regular,", "\\item $f$ is locally of finite type, and", "\\item for specializations $x' \\leadsto x$ with", "$\\dim(\\mathcal{O}_{X, x'}) = 1$ our $f$ is \\'etale at $x'$.", "\\end{enumerate}", "Then $f$ is \\'etale at $x$." ], "refs": [], "proofs": [ { "contents": [ "We will prove the lemma by induction on $d = \\dim(\\mathcal{O}_{X, x})$.", "\\medskip\\noindent", "An uninteresting case is $d = 1$ since in that case the morphism", "$f$ is \\'etale at $x$ by assumption. Assume $d \\geq 2$.", "\\medskip\\noindent", "We can base change by $\\Spec(\\mathcal{O}_{Y, y}) \\to Y$", "without affecting the conclusion of the lemma, see", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-etale}.", "Thus we may assume $Y = \\Spec(A)$ where $A$ is a regular local", "ring and $y$ corresponds to the maximal ideal $\\mathfrak m$ of $A$.", "\\medskip\\noindent", "Let $x' \\leadsto x$ be a specialization with $x' \\not = x$.", "Then $\\mathcal{O}_{X, x'}$ is normal as a localization of", "$\\mathcal{O}_{X, x}$. If $x'$ is not a generic point of $X$,", "then $1 \\leq \\dim(\\mathcal{O}_{X, x'}) < d$ and we conclude that", "$f$ is \\'etale at $x'$ by induction hypothesis.", "Thus we may assume that $f$ is \\'etale at all points specializing to", "$x$. Since the set of points where $f$ is \\'etale is open in $X$", "(by definition) we may after replacing $X$ by an open neighbourhood of $x$", "assume that $f$ is \\'etale away from $\\overline{\\{x\\}}$.", "In particular, we see that $f$ is \\'etale except at points", "lying over the closed point $y \\in Y = \\Spec(A)$.", "\\medskip\\noindent", "Let $X' = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$. Let $x' \\in X'$", "be the unique point lying over $x$. By the above we see that", "$X'$ is \\'etale over $\\Spec(A^\\wedge)$ away from the closed fibre and", "hence $X'$ is normal away from the closed fibre. Since $X$ is normal", "we conclude that $X'$ is normal by", "Resolution of Surfaces, Lemma \\ref{resolve-lemma-normalization-completion}.", "Then if we can show $X' \\to \\Spec(A^\\wedge)$ is \\'etale at $x'$,", "then $f$ is \\'etale at $x$ (by the aforementioned", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-etale}).", "Thus we may and do assume $A$ is a regular complete local ring.", "\\medskip\\noindent", "The case $d = 2$ now follows from Lemma \\ref{lemma-key-purity-ramification}.", "\\medskip\\noindent", "Assume $d > 2$. Let $t \\in \\mathfrak m$, $t \\not \\in \\mathfrak m^2$.", "Set $Y_0 = \\Spec(A/tA)$ and $X_0 = X \\times_Y Y_0$.", "Then $X_0 \\to Y_0$ is \\'etale away from the fibre over the closed point.", "Since $d > 2$ we have $\\dim(\\mathcal{O}_{X_0, x}) = d - 1$ is $\\geq 2$.", "The normalization $X_0' \\to X_0$ is surjective and finite", "(as we're working over a complete local ring and such rings are Nagata).", "Let $x' \\in X_0'$ be a point mapping to $x$. By induction hypothesis the", "morphism $X'_0 \\to Y$ is \\'etale at $x'$. From the inclusions", "$\\kappa(y) \\subset \\kappa(x) \\subset \\kappa(x')$", "we conclude that $\\kappa(x)$ is finite over $\\kappa(y)$.", "Hence $x$ is a closed point of the fibre of $X \\to Y$", "over $y$. But since $x$ is also a generic point of", "this fibre, we conclude that $f$ is quasi-finite at $x$", "and we reduce to the case of purity of branch locus, see", "Lemma \\ref{lemma-purity}." ], "refs": [ "morphisms-lemma-set-points-where-fibres-etale", "resolve-lemma-normalization-completion", "morphisms-lemma-set-points-where-fibres-etale", "pione-lemma-key-purity-ramification", "pione-lemma-purity" ], "ref_ids": [ 5374, 11684, 5374, 4115, 4099 ] } ], "ref_ids": [] }, { "id": 4117, "type": "theorem", "label": "pione-lemma-structure-cohomology", "categories": [ "pione" ], "title": "pione-lemma-structure-cohomology", "contents": [ "Let $(A, \\mathfrak m)$ be a regular local ring which contains a field.", "Let $f : V \\to \\Spec(A)$ be \\'etale and quasi-compact.", "Assume that $\\mathfrak m \\not \\in f(V)$ and assume that", "$g : V \\to \\Spec(A) \\setminus \\{\\mathfrak m\\}$ is affine.", "Then $H^i(V, \\mathcal{O}_V)$, $i > 0$ is isomorphic to a direct", "sum of copies of the injective hull of the residue field of $A$." ], "refs": [], "proofs": [ { "contents": [ "Denote $U = \\Spec(A) \\setminus \\{\\mathfrak m\\}$ the punctured spectrum.", "Thus $g : V \\to U$ is affine.", "We have $H^i(V, \\mathcal{O}_V) = H^i(U, g_*\\mathcal{O}_V)$ by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology}.", "The $\\mathcal{O}_U$-module $g_*\\mathcal{O}_V$ is quasi-coherent by", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}.", "For any quasi-coherent $\\mathcal{O}_U$-module $\\mathcal{F}$", "the cohomology $H^i(U, \\mathcal{F})$, $i > 0$", "is $\\mathfrak m$-power torsion, see for example", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-local-cohomology}.", "In particular, the $A$-modules $H^i(V, \\mathcal{O}_V)$, $i > 0$", "are $\\mathfrak m$-power torsion.", "For any flat ring map $A \\to A'$ we have", "$H^i(V, \\mathcal{O}_V) \\otimes_A A' = H^i(V', \\mathcal{O}_{V'})$", "where $V' = V \\times_{\\Spec(A)} \\Spec(A')$ by flat base change", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}.", "If we take $A'$ to be the completion of $A$ (flat by", "More on Algebra, Section \\ref{more-algebra-section-permanence-completion}),", "then we see that", "$$", "H^i(V, \\mathcal{O}_V) = H^i(V, \\mathcal{O}_V) \\otimes_A A' =", "H^i(V', \\mathcal{O}_{V'}),\\quad\\text{for } i > 0", "$$", "The first equality by the torsion property we just proved and", "More on Algebra, Lemma \\ref{more-algebra-lemma-neighbourhood-equivalence}.", "Moreover, the injective hull of the residue field $k$", "is the same for $A$ and $A'$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-compare}.", "In this way we reduce to the case $A = k[[x_1, \\ldots, x_d]]$, see", "Algebra, Section \\ref{algebra-section-cohen-structure-theorem}.", "\\medskip\\noindent", "Assume the characteristic of $k$ is $p > 0$. Since $F : A \\to A$,", "$a \\mapsto a^p$ is flat (Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-frobenius-flat-regular})", "and since $V \\times_{\\Spec(A), \\Spec(F)} \\Spec(A) \\cong V$ as schemes", "over $\\Spec(A)$ by", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-relative-frobenius-etale}", "the above gives", "$H^i(V, \\mathcal{O}_V) \\otimes_{A, F} A \\cong H^i(V, \\mathcal{O}_V)$.", "Thus we get the result by", "Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-structure-torsion-Frobenius-regular}.", "\\medskip\\noindent", "Assume the characteristic of $k$ is $0$. By", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-etale-derivation}", "there are additive operators $D_j$, $j = 1, \\ldots, d$ on", "$H^i(V, \\mathcal{O}_V)$ satisfying the Leibniz rule with", "respect to $\\partial_j = \\partial/\\partial x_j$.", "Thus we get the result by", "Local Cohomology,", "Lemma \\ref{local-cohomology-lemma-structure-torsion-D-module-regular}." ], "refs": [ "coherent-lemma-relative-affine-cohomology", "schemes-lemma-push-forward-quasi-coherent", "local-cohomology-lemma-local-cohomology", "coherent-lemma-flat-base-change-cohomology", "more-algebra-lemma-neighbourhood-equivalence", "dualizing-lemma-compare", "local-cohomology-lemma-frobenius-flat-regular", "etale-lemma-relative-frobenius-etale", "local-cohomology-lemma-structure-torsion-Frobenius-regular", "local-cohomology-lemma-etale-derivation", "local-cohomology-lemma-structure-torsion-D-module-regular" ], "ref_ids": [ 3284, 7730, 9696, 3298, 10341, 2807, 9763, 10708, 9765, 9768, 9764 ] } ], "ref_ids": [] }, { "id": 4118, "type": "theorem", "label": "pione-lemma-conclude", "categories": [ "pione" ], "title": "pione-lemma-conclude", "contents": [ "In the situation of Lemma \\ref{lemma-structure-cohomology}", "assume that $H^i(V, \\mathcal{O}_V) = 0$ for $i \\geq \\dim(A) - 1$.", "Then $V$ is affine." ], "refs": [ "pione-lemma-structure-cohomology" ], "proofs": [ { "contents": [ "Let $k = A/\\mathfrak m$. Since $V \\times_{\\Spec(A)} \\Spec(k) = \\emptyset$,", "by cohomology and base change we have", "$$", "R\\Gamma(V, \\mathcal{O}_V) \\otimes_A^\\mathbf{L} k = 0", "$$", "See", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-compare-base-change}.", "Thus there is a spectral sequence", "(More on Algebra, Example \\ref{more-algebra-example-tor})", "$$", "E_2^{p, q} = \\text{Tor}_{-p}(k, H^q(V, \\mathcal{O}_V)),\\quad", "d_2^{p, q} : E_2^{p, q} \\to E_2^{p + 2, q - 1}", "$$", "and $d_r^{p, q} : E_r^{p, q} \\to E_r^{p + r, q - r + 1}$", "converging to zero. By Lemma \\ref{lemma-structure-cohomology},", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-tor-injective-hull},", "and our assumption $H^i(V, \\mathcal{O}_V) = 0$ for $i \\geq \\dim(A) - 1$", "we conclude that there is no nonzero differential", "entering or leaving the $(p, q) = (0, 0)$ spot. Thus", "$H^0(V, \\mathcal{O}_V) \\otimes_A k = 0$. This means that", "if $\\mathfrak m = (x_1, \\ldots, x_d)$ then we have", "an open covering $V = \\bigcup V \\times_{\\Spec(A)} \\Spec(A_{x_i})$", "by affine open subschemes $V \\times_{\\Spec(A)} \\Spec(A_{x_i})$", "(because $V$ is affine over the punctured spectrum of $A$)", "such that $x_1, \\ldots, x_d$ generate", "the unit ideal in $\\Gamma(V, \\mathcal{O}_V)$.", "This implies $V$ is affine by", "Properties, Lemma \\ref{properties-lemma-characterize-affine}." ], "refs": [ "perfect-lemma-compare-base-change", "pione-lemma-structure-cohomology", "dualizing-lemma-tor-injective-hull", "properties-lemma-characterize-affine" ], "ref_ids": [ 7028, 4117, 2886, 3055 ] } ], "ref_ids": [ 4117 ] }, { "id": 4119, "type": "theorem", "label": "pione-lemma-specialization-map-surjective", "categories": [ "pione" ], "title": "pione-lemma-specialization-map-surjective", "contents": [ "Let $f : X \\to S$ be a flat proper morphism with geometrically", "connected fibres. Let $s' \\leadsto s$ be a specialization.", "If $X_s$ is geometrically reduced, then the specialization", "map $sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Since $X_s$ is geometrically reduced, we may assume all", "fibres are geometrically reduced after possibly shrinking $S$, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-geometrically-reduced-open}.", "Let $\\mathcal{O}_{S, s} \\to A \\to \\kappa(\\overline{s}')$ be as", "in the construction of the specialization map, see", "Section \\ref{section-specialization-map}.", "Thus it suffices to show that", "$$", "\\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_A)", "$$", "is surjective. This follows from", "Proposition \\ref{proposition-first-homotopy-sequence}", "and $\\pi_1(\\Spec(A)) = \\{1\\}$." ], "refs": [ "more-morphisms-lemma-geometrically-reduced-open", "pione-proposition-first-homotopy-sequence" ], "ref_ids": [ 13819, 4133 ] } ], "ref_ids": [] }, { "id": 4120, "type": "theorem", "label": "pione-lemma-pullback-tame-codim1", "categories": [ "pione" ], "title": "pione-lemma-pullback-tame-codim1", "contents": [ "Let $X' \\to X$ be a morphism of locally Noetherian schemes.", "Let $U \\subset X$ be a dense open. Assume", "\\begin{enumerate}", "\\item $U' = f^{-1}(U)$ is dense open in $X'$,", "\\item for every prime divisor $Z \\subset X$ with $Z \\cap U = \\emptyset$", "the local ring $\\mathcal{O}_{X, \\xi}$ of $X$ at the generic point $\\xi$", "of $Z$ is a discrete valuation ring,", "\\item for every prime divisor $Z' \\subset X'$", "with $Z' \\cap U' = \\emptyset$ the local ring $\\mathcal{O}_{X', \\xi'}$", "of $X'$ at the generic point $\\xi'$ of $Z'$ is a discrete valuation ring,", "\\item if $\\xi' \\in X'$ is as in (3), then $\\xi = f(\\xi')$ is as in (2).", "\\end{enumerate}", "Then if $f : Y \\to U$ is finite \\'etale and", "$Y$ is unramified, resp.\\ tamely ramified over $X$", "in codimension $1$, then $Y' = Y \\times_X X' \\to U'$ is finite \\'etale", "and $Y'$ is unramified, resp.\\ tamely ramified over $X'$ in codimension $1$." ], "refs": [], "proofs": [ { "contents": [ "The only interesting fact in this lemma is the commutative algebra", "result given in More on Algebra, Lemma \\ref{more-algebra-lemma-tame-goes-up}." ], "refs": [ "more-algebra-lemma-tame-goes-up" ], "ref_ids": [ 10514 ] } ], "ref_ids": [] }, { "id": 4121, "type": "theorem", "label": "pione-lemma-purity-one-divisor", "categories": [ "pione" ], "title": "pione-lemma-purity-one-divisor", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$", "be an effective Cartier divisor such that $D$ is a regular scheme.", "Let $Y \\to X \\setminus D$ be a finite \\'etale morphism.", "If $Y$ is unramified over $X$ in codimension $1$, then", "there exists a finite \\'etale morphism $Y' \\to X$", "whose restriction to $X \\setminus D$ is $Y$." ], "refs": [], "proofs": [ { "contents": [ "Before we start we note that $\\mathcal{O}_{X, x}$ is a regular", "local ring for all $x \\in D$. This follows from", "Algebra, Lemma \\ref{algebra-lemma-regular-mod-x}", "and our assumption that $\\mathcal{O}_{D, x}$ is regular.", "Let $\\xi \\in D$ be a generic point of an irreducible component of $D$.", "By the above $\\mathcal{O}_{X, \\xi}$ is a discrete valuation ring.", "Hence the statement of the lemma makes sense.", "As in the discussion above, write", "$Y \\times_U \\Spec(K_\\xi) = \\Spec(L_\\xi)$.", "Denote $B_\\xi$ the integral closure of $\\mathcal{O}_{X, \\xi}$ in", "$L_\\xi$. Our assumption that $Y$ is unramified over $X$ in codimension $1$", "signifies that $\\mathcal{O}_{X, \\xi} \\to B_\\xi$ is finite \\'etale.", "Thus we get $Y_\\xi \\to \\Spec(\\mathcal{O}_{X, \\xi})$ finite", "\\'etale and an isomorphism", "$$", "Y \\times_U \\Spec(K_\\xi) \\cong", "Y_\\xi \\times_{\\Spec(\\mathcal{O}_{X, \\xi})} \\Spec(K_\\xi)", "$$", "over $\\Spec(K_\\xi)$.", "By Limits, Lemma \\ref{limits-lemma-glueing-near-point}", "we find an open subscheme $X \\setminus D \\subset U' \\subset X$", "containing $\\xi$ and a morphism $Y' \\to U'$ of finite presentation", "whose restriction to $X \\setminus D$ recovers $Y$ and", "whose restriction to $\\Spec(\\mathcal{O}_{X, \\xi})$ recovers $Y_\\xi$.", "Finally, the morphism $Y' \\to U'$ is finite \\'etale", "after possible shrinking $U'$ to a smaller open by", "Limits, Lemma \\ref{limits-lemma-glueing-near-point-properties}.", "Repeating the argument with the other generic points of", "$D$ we may assume that we have a finite \\'etale morphism $Y' \\to U'$", "extending $Y \\to X\\setminus D$ to an open subscheme", "containing $U' \\subset X$ containing $X \\setminus D$", "and all generic points of $D$.", "We finish by applying Lemma \\ref{lemma-extend-pure}", "to $Y' \\to U'$. Namely, all local rings $\\mathcal{O}_{X, x}$", "for $x \\in D$ are regular (see above) and if $x \\not \\in U'$", "we have $\\dim(\\mathcal{O}_{X, x}) \\geq 2$. Hence we have", "purity for $\\mathcal{O}_{X, x}$ by", "Lemma \\ref{lemma-local-purity}." ], "refs": [ "algebra-lemma-regular-mod-x", "limits-lemma-glueing-near-point", "limits-lemma-glueing-near-point-properties", "pione-lemma-extend-pure", "pione-lemma-local-purity" ], "ref_ids": [ 945, 15114, 15115, 4101, 4098 ] } ], "ref_ids": [] }, { "id": 4122, "type": "theorem", "label": "pione-lemma-abhyankar-one-divisor", "categories": [ "pione" ], "title": "pione-lemma-abhyankar-one-divisor", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$", "be an effective Cartier divisor such that $D$ is a regular scheme.", "Let $Y \\to X \\setminus D$ be a finite \\'etale morphism.", "If $Y$ is tamely ramified over $X$ in codimension $1$, then", "\\'etale locally on $X$ the morphism $Y \\to X$ is as given", "as a finite disjoint union of standard tamely ramified", "morphisms as described in Example \\ref{example-tamely-ramified}." ], "refs": [], "proofs": [ { "contents": [ "Before we start we note that $\\mathcal{O}_{X, x}$ is a regular", "local ring for all $x \\in D$. This follows from", "Algebra, Lemma \\ref{algebra-lemma-regular-mod-x}", "and our assumption that $\\mathcal{O}_{D, x}$ is regular.", "Below we will also use that regular rings are normal, see", "Algebra, Lemma \\ref{algebra-lemma-regular-normal}.", "\\medskip\\noindent", "To prove the lemma we may work locally on $X$.", "Thus we may assume $X = \\Spec(A)$ and $D \\subset X$", "is given by a nonzerodivisor $f \\in A$.", "Then $Y = \\Spec(B)$ as a finite \\'etale scheme over $A_f$.", "Let $\\mathfrak p_1, \\ldots, \\mathfrak p_r$ be the minimal", "primes of $A$ over $f$. Then $A_i = A_{\\mathfrak p_i}$", "is a discrete valuation ring; denote its fraction field $K_i$.", "By assumption", "$$", "K_i \\otimes_{A_f} B = \\prod L_{ij}", "$$", "is a finite product of fields each tamely ramified with respect to $A_i$.", "Choose $e \\geq 1$ sufficiently divisible (namely, divisible by", "all ramification indices for $L_{ij}$ over $A_i$ as in", "More on Algebra, Remark \\ref{more-algebra-remark-finite-separable-extension}).", "Warning: at this point we do not know that $e$ is invertible on $A$.", "\\medskip\\noindent", "Consider the finite free $A$-algebra", "$$", "A' = A[x]/(x^e - f)", "$$", "Observe that $f' = x$ is a nonzerodivisor in $A'$ and that", "$A'/f'A' \\cong A/fA$ is a regular ring. Set", "$B' = B \\otimes_A A' = B \\otimes_{A_f} A'_{f'}$.", "By Abhyankar's lemma", "(More on Algebra, Lemma \\ref{more-algebra-lemma-abhyankar})", "we see that $\\Spec(B')$ is unramified over $\\Spec(A')$", "in codimension $1$. Namely, by Lemma \\ref{lemma-pullback-tame-codim1}", "we see that $\\Spec(B')$ is still at least tamely ramified", "over $\\Spec(A')$ in codimension $1$. But Abhyankar's lemma", "tells us that the ramification indices have all become equal to $1$.", "By Lemma \\ref{lemma-purity-one-divisor} we conclude that", "$\\Spec(B') \\to \\Spec(A'_{f'})$ extends to a finite \\'etale morphism", "$\\Spec(C) \\to \\Spec(A')$.", "\\medskip\\noindent", "For a point $x \\in D$ corresponding to $\\mathfrak p \\in V(f)$", "denote $A^{sh}$ a strict henselization of $A_\\mathfrak p = \\mathcal{O}_{X, x}$.", "Observe that $A^{sh}$ and $A^{sh}/fA^{sh} = (A/fA)^{sh}$", "(Algebra, Lemma \\ref{algebra-lemma-quotient-strict-henselization})", "are regular local rings, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-regular}.", "Observe that $A'$ has a unique prime $\\mathfrak p'$ lying over", "$\\mathfrak p$ with identical residue field. Thus", "$$", "(A')^{sh} = A^{sh} \\otimes_A A' = A^{sh}[x]/(x^e - f)", "$$", "is a strictly henselian local ring finite over $A^{sh}$", "(Algebra, Lemma \\ref{algebra-lemma-quasi-finite-strict-henselization}).", "Since $f'$ is a nonzerodivisor in $(A')^{sh}$ and since", "$(A')^{sh}/f'(A')^{sh} = A^{sh}/fA^{sh}$ is regular, we conclude", "that $(A')^{sh}$ is a regular local ring (see above).", "Observe that the induced extension", "$$", "Q(A^{sh}) \\subset Q((A')^{sh}) = Q(A^{sh})[x]/(x^e - f)", "$$", "of fraction fields has degree $e$ (and not less).", "Since $A' \\to C$ is finite \\'etale we see that", "$A^{sh} \\otimes_A C$ is a finite product of copies of $(A')^{sh}$", "(Algebra, Lemma \\ref{algebra-lemma-mop-up-strictly-henselian}).", "We have the inclusions", "$$", "A^{sh}_f \\subset", "A^{sh} \\otimes_A B \\subset", "A^{sh} \\otimes_A B' = A^{sh} \\otimes_A C_{f'}", "$$", "and each of these rings is Noetherian and normal; this follows from", "Algebra, Lemma \\ref{algebra-lemma-normal-goes-up} for the ring", "in the middle. Taking total quotient rings, using the product", "decomposition of $A^{sh} \\otimes_A C$ and using", "Fields, Lemma \\ref{fields-lemma-subfields-kummer} we conclude that", "there is an isomorphism", "$$", "Q(A^{sh}) \\otimes_A B \\cong \\prod\\nolimits_{i \\in I} F_i,\\quad", "F_i \\cong Q(A^{sh})[x]/(x^{e_i} - f)", "$$", "of $Q(A^{sh})$-algebras for some finite set $I$ and integers $e_i | e$.", "Since $A^{sh} \\otimes_A B$ is a normal ring, it must be the", "integral closure of $A^{sh}$ in its total quotient ring.", "We conclude that we have an isomorphism", "$$", "A^{sh} \\otimes_A B \\cong \\prod A^{sh}_f[x]/(x^{e_i} - f)", "$$", "over $A^{sh}_f$ because the algebras $A^{sh}[x]/(x^{e_i} - f)$", "are regular and hence normal. The discriminant of", "$A^{sh}[x]/(x^{e_i} - f)$ over $A^{sh}$ is $e_i^{e_i}f^{e_i - 1}$", "(up to sign; calculation omitted). Since $A_f \\to B$ is finite", "\\'etale we see that $e_i$ must be invertible in $A^{sh}_f$.", "On the other hand, since $A_f \\to B$ is tamely ramified", "over $\\Spec(A)$ in codimension $1$, by Lemma \\ref{lemma-pullback-tame-codim1}", "the ring map $A^{sh}_f \\to A^{sh} \\otimes_A B$", "is tamely ramified over $\\Spec(A^{sh})$ in codimension $1$.", "This implies $e_i$ is nonzero in $A^{sh}/fA^{sh}$", "(as it must map to an invertible element of the fraction field of", "this domain by definition of tamely ramified extensions).", "We conclude that $V(e_i) \\subset \\Spec(A^{sh})$", "has codimension $\\geq 2$ which is absurd unless it is empty.", "In other words, $e_i$ is an invertible element of $A^{sh}$.", "We conclude that the pullback of $Y$ to $\\Spec(A^{sh})$", "is indeed a finite disjoint union of standard tamely ramified morphisms.", "\\medskip\\noindent", "To finish the proof, we write $A^{sh} = \\colim A_\\lambda$", "as a filtered colimit of \\'etale $A$-algebras $A_\\lambda$.", "The isomorphism", "$$", "A^{sh} \\otimes_A B \\cong", "\\prod\\nolimits_{i \\in I} A^{sh}_f[x]/(x^{e_i} - f)", "$$", "descends to an isomorphism", "$$", "A_\\lambda \\otimes_A B \\cong \\prod\\nolimits_{i \\in I}", "(A_\\lambda)_f[x]/(x^{e_i} - f)", "$$", "for suitably large $\\lambda$. After increasing $\\lambda$ a bit", "more we may assume $e_i$ is invertible in $A_\\lambda$. Then", "$\\Spec(A_\\lambda) \\to \\Spec(A)$ is the desired \\'etale neighbourhood", "of $x$ and the proof is complete." ], "refs": [ "algebra-lemma-regular-mod-x", "algebra-lemma-regular-normal", "more-algebra-remark-finite-separable-extension", "more-algebra-lemma-abhyankar", "pione-lemma-pullback-tame-codim1", "pione-lemma-purity-one-divisor", "algebra-lemma-quotient-strict-henselization", "more-algebra-lemma-henselization-regular", "algebra-lemma-quasi-finite-strict-henselization", "algebra-lemma-mop-up-strictly-henselian", "algebra-lemma-normal-goes-up", "fields-lemma-subfields-kummer", "pione-lemma-pullback-tame-codim1" ], "ref_ids": [ 945, 1312, 10676, 10509, 4120, 4121, 1307, 10064, 1306, 1279, 1368, 4516, 4120 ] } ], "ref_ids": [] }, { "id": 4123, "type": "theorem", "label": "pione-lemma-extend-tame-covering-normal", "categories": [ "pione" ], "title": "pione-lemma-extend-tame-covering-normal", "contents": [ "In the situation of Lemma \\ref{lemma-abhyankar-one-divisor}", "the normalization of $X$ in $Y$ is a finite locally free morphism", "$\\pi : Y' \\to X$ such that", "\\begin{enumerate}", "\\item the restriction of $Y'$ to $X \\setminus D$ is isomorphic to $Y$,", "\\item $D' = \\pi^{-1}(D)_{red}$ is an effective Cartier divisor on $Y'$, and", "\\item $D'$ is a regular scheme.", "\\end{enumerate}", "Moreover, \\'etale locally on $X$ the morphism $Y' \\to X$ is a finite disjoint", "union of morphisms", "$$", "\\Spec(A[x]/(x^e - f)) \\to \\Spec(A)", "$$", "where $A$ is a Noetherian ring, $f \\in A$ is a nonzerodivisor with", "$A/fA$ regular, and $e \\geq 1$ is invertible in $A$." ], "refs": [ "pione-lemma-abhyankar-one-divisor" ], "proofs": [ { "contents": [ "This is just an addendum to Lemma \\ref{lemma-abhyankar-one-divisor}", "and in fact the truth of this lemma follows almost immediately if", "you've read the proof of that lemma. But we can also deduce", "the lemma from the result of Lemma \\ref{lemma-abhyankar-one-divisor}.", "Namely, taking the normalization of $X$ in $Y$ commutes with", "\\'etale base change, see More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-normalization-smooth-localization}.", "Hence we see that we may prove the statements on the local structure", "of $Y' \\to X$ \\'etale locally on $X$. Thus, by", "Lemma \\ref{lemma-abhyankar-one-divisor} we may assume that", "$X = \\Spec(A)$ where $A$ is a Noetherian ring, that we have a", "nonzerodivisor $f\\in A$ such that $A/fA$ is regular, and that $Y$", "is a finite disjoint union of spectra of rings $A_f[x]/(x^e - f)$", "where $e$ is invertible in $A$. We omit the verification that", "the integral closure of $A$ in $A_f[x]/(x^e - f)$ is", "equal to $A' = A[x]/(x^e - f)$. (To see this argue that", "the localizations of $A'$ at primes lying over $(f)$ are regular.)", "We omit the details." ], "refs": [ "pione-lemma-abhyankar-one-divisor", "pione-lemma-abhyankar-one-divisor", "more-morphisms-lemma-normalization-smooth-localization", "pione-lemma-abhyankar-one-divisor" ], "ref_ids": [ 4122, 4122, 13774, 4122 ] } ], "ref_ids": [ 4122 ] }, { "id": 4124, "type": "theorem", "label": "pione-lemma-tame-covering-split", "categories": [ "pione" ], "title": "pione-lemma-tame-covering-split", "contents": [ "In the situation of Lemma \\ref{lemma-abhyankar-one-divisor}", "let $Y' \\to X$ be as in Lemma \\ref{lemma-extend-tame-covering-normal}.", "Let $R$ be a discrete valuation ring with fraction field $K$.", "Let", "$$", "t : \\Spec(R) \\to X", "$$", "be a morphism such that the scheme theoretic inverse image", "$t^{-1}D$ is the reduced closed point of $\\Spec(R)$.", "\\begin{enumerate}", "\\item If $t|_{\\Spec(K)}$ lifts to a point of $Y$, then", "we get a lift $t' : \\Spec(R) \\to Y'$ such that $Y' \\to X$", "is \\'etale along $t'(\\Spec(R))$.", "\\item If $\\Spec(K) \\times_X Y$ is isomorphic to a disjoint union", "of copies of $\\Spec(K)$, then $Y' \\to X$ is finite \\'etale", "over an open neighbourhood of $t(\\Spec(R))$.", "\\end{enumerate}" ], "refs": [ "pione-lemma-abhyankar-one-divisor", "pione-lemma-extend-tame-covering-normal" ], "proofs": [ { "contents": [ "By the valuative criterion of properness applied to", "the finite morphism $Y' \\to X$ we see that $\\Spec(K)$-valued", "points of $Y$ matching $t|_{\\Spec(K)}$ as maps into $X$", "lift uniquely to morphisms $t' : \\Spec(R) \\to Y'$.", "Thus statement (1) make sense.", "\\medskip\\noindent", "Choose an \\'etale neighbourhood $(U, u) \\to (X, t(\\mathfrak m_R))$", "such that $U = \\Spec(A)$ and such that $Y' \\times_X U \\to U$", "has a description as in Lemma \\ref{lemma-extend-tame-covering-normal}", "for some $f \\in A$. Then $\\Spec(R) \\times_X U \\to \\Spec(R)$ is \\'etale", "and surjective. If $R'$ denotes the local ring of", "$\\Spec(R) \\times_X U$ lying over the closed point of $\\Spec(R)$,", "then $R'$ is a discrete valuation ring and $R \\subset R'$", "is an unramified extension of discrete valuation rings", "(More on Algebra, Lemma \\ref{more-algebra-lemma-Dedekind-etale-extension}).", "The assumption on $t$ signifies that the map $A \\to R'$", "corresponding to", "$$", "\\Spec(R') \\to \\Spec(R) \\times_X U \\to U", "$$", "maps $f$ to a uniformizer $\\pi \\in R'$. Now suppose that", "$$", "Y' \\times_X U =", "\\coprod\\nolimits_{i \\in I} \\Spec(A[x]/(x^{e_i} - f))", "$$", "for some $e_i \\geq 1$. Then we see that", "$$", "\\Spec(R') \\times_U (Y' \\times_X U) =", "\\coprod\\nolimits_{i \\in I} \\Spec(R'[x]/(x^{e_i} - \\pi))", "$$", "The rings $R'[x]/(x^{e_i} - f)$ are discrete valuation rings", "(More on Algebra, Lemma \\ref{more-algebra-lemma-pull-root-uniformizer})", "and hence have no map into the fraction field of $R'$ unless $e_i = 1$.", "\\medskip\\noindent", "Proof of (1). In this case the map $t' : \\Spec(R) \\to Y'$ base changes", "to determine a corresponding map $t'' : \\Spec(R') \\to Y' \\times_X U$", "which must map into a summand corresponding to $i \\in I$ with $e_i = 1$", "by the discussion above. Thus clearly we see that $Y' \\times_X U \\to U$", "is \\'etale along the image of $t''$. Since being \\'etale is a property", "one can check after \\'etale base chamge, this proves (1).", "\\medskip\\noindent", "Proof of (2). In this case the assumption implies that $e_i = 1$", "for all $i \\in I$. Thus $Y' \\times_X U \\to U$ is finite \\'etale", "and we conclude as before." ], "refs": [ "pione-lemma-extend-tame-covering-normal", "more-algebra-lemma-Dedekind-etale-extension", "more-algebra-lemma-pull-root-uniformizer" ], "ref_ids": [ 4123, 10054, 10507 ] } ], "ref_ids": [ 4122, 4123 ] }, { "id": 4125, "type": "theorem", "label": "pione-lemma-extend-covering", "categories": [ "pione" ], "title": "pione-lemma-extend-covering", "contents": [ "Let $S$ be an integral normal Noetherian scheme with generic point $\\eta$.", "Let $f : X \\to S$ be a smooth morphism with geometrically connected fibres.", "Let $\\sigma : S \\to X$ be a section of $f$. Let $Z \\to X_\\eta$ be a", "finite \\'etale Galois cover (Section \\ref{section-finite-etale-under-galois})", "with group $G$ of order invertible on $S$ such that", "$Z$ has a $\\kappa(\\eta)$-rational point mapping to $\\sigma(\\eta)$.", "Then there exists a finite \\'etale Galois cover $Y \\to X$ with group $G$", "whose restriction to $X_\\eta$ is $Z$." ], "refs": [], "proofs": [ { "contents": [ "First assume $S = \\Spec(R)$ is the spectrum of a discrete valuation ring $R$", "with closed point $s \\in S$. Then $X_s$ is an effective Cartier", "divisor in $X$ and $X_s$ is regular as a scheme smooth over a field.", "Moreover the generic fibre $X_\\eta$ is the open subscheme $X \\setminus X_s$.", "It follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-galois-conclusion}", "and the assumption on $G$ that $Z$ is tamely ramified", "over $X$ in codimension $1$. Let $Z' \\to X$ be as in", "Lemma \\ref{lemma-extend-tame-covering-normal}. Observe that", "the action of $G$ on $Z$ extends to an action of $G$ on $Z'$.", "By Lemma \\ref{lemma-tame-covering-split}", "we see that $Z' \\to X$ is finite \\'etale over an open", "neighbourhood of $\\sigma(y)$.", "Since $X_s$ is irreducible, this implies $Z \\to X_\\eta$", "is unramified over $X$ in codimension $1$.", "Then we get a finite \\'etale morphism $Y \\to X$", "whose restriction to $X_\\eta$ is $Z$ by", "Lemma \\ref{lemma-purity-one-divisor}.", "Of course $Y \\cong Z'$ (details omitted; hint: compute \\'etale locally)", "and hence $Y$ is a Galois cover with group $G$.", "\\medskip\\noindent", "General case. Let $U \\subset S$ be a maximal open subscheme", "such that there exists a finite \\'etale Galois cover", "$Y \\to X \\times_S U$ with group $G$", "whose restriction to $X_\\eta$ is isomorphic to $Z$.", "Assume $U \\not = S$ to get a contradiction.", "Let $s \\in S \\setminus U$ be a generic point of an irreducible", "component of $S \\setminus U$. Then the inverse image $U_s$", "of $U$ in $\\Spec(\\mathcal{O}_{S, s})$", "is the punctured spectrum of $\\mathcal{O}_{S, s}$.", "We claim $Y \\times_S U_s \\to X \\times_S U_s$", "is the restriction of a finite \\'etale Galois cover", "$Y'_s \\to X \\times_S \\Spec(\\mathcal{O}_{S, s})$", "with group $G$.", "\\medskip\\noindent", "Let us first prove the claim produces the desired contradiction.", "By Limits, Lemma \\ref{limits-lemma-glueing-near-point}", "we find an open subscheme $U \\subset U' \\subset S$", "containing $s$ and a morphism $Y'' \\to U'$ of finite presentation", "whose restriction to $U$ recovers $Y' \\to U$ and", "whose restriction to $\\Spec(\\mathcal{O}_{S, s})$ recovers $Y'_s$.", "Moreover, by the equivalence of categories given in the", "lemma, we may assume after shrinking $U'$", "there is a morphism $Y'' \\to U' \\times_S X$", "and there is an action of $G$ on $Y''$ over $U' \\times_S X$", "compatible with the given morphisms and actions after base", "change to $U$ and $\\Spec(\\mathcal{O}_{S, s})$.", "After shrinking $U'$ further if necessary, we may", "assume $Y'' \\to U \\times_S X$ is finite \\'etale, see", "Limits, Lemma \\ref{limits-lemma-glueing-near-point-properties}.", "This means we have found a strictly larger open of $S$ over which", "$Y$ extends to a finite \\'etale Galois cover with group $G$", "which gives the contradiction we were looking for.", "\\medskip\\noindent", "Proof of the claim. We may and do replace $S$ by $\\Spec(\\mathcal{O}_{S, s})$.", "Then $S = \\Spec(A)$ where $(A, \\mathfrak m)$ is a local normal domain.", "Also $U \\subset S$ is the punctured spectrum and we have a finite", "\\'etale Galois cover $Y \\to X \\times_S U$ with group $G$.", "If $\\dim(A) = 1$, then we can construct the extension of $Y$ to", "a Galois covering of $X$ by the first paragraph of the proof.", "Thus we may assume $\\dim(A) \\geq 2$ and hence $\\text{depth}(A) \\geq 2$", "as $S$ is normal, see Algebra, Lemma \\ref{algebra-lemma-criterion-normal}.", "Since $X \\to S$ is flat, we conclude that", "$\\text{depth}(\\mathcal{O}_{X, x}) \\geq 2$ for every point $x \\in X$", "mapping to $s$, see", "Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck}.", "Let", "$$", "Y' \\longrightarrow X", "$$", "be the finite morphism constructed in Lemma \\ref{lemma-extend-S2}", "using $Y \\to X \\times_S U$. Observe that we obtain a canonical", "$G$-action on $Y$. Thus all that remains is to show that $Y'$", "is \\'etale over $X$. In fact, by", "Lemma \\ref{lemma-purity-smooth-over-depth2} (for example)", "it even suffices to show that $Y' \\to X$ is \\'etale over the", "(unique) generic point of the fibre $X_s$. This we do by", "a local calculation in a (formal) neighbourhood of $\\sigma(s)$.", "\\medskip\\noindent", "Choose an affine open $\\Spec(B) \\subset X$ containing $\\sigma(s)$.", "Then $A \\to B$ is a smooth ring map which has a section $\\sigma : B \\to A$.", "Denote $I = \\Ker(\\sigma)$ and denote $B^\\wedge$ the", "$I$-adic completion of $B$. Then $B^\\wedge \\cong A[[x_1, \\ldots, x_d]]$", "for some $d \\geq 0$, see", "Algebra, Lemma \\ref{algebra-lemma-section-smooth}.", "Of course $B \\to B^\\wedge$ is flat", "(Algebra, Lemma \\ref{algebra-lemma-completion-flat})", "and the image of $\\Spec(B^\\wedge) \\to X$ contains the generic point of $X_s$.", "Let $V \\subset \\Spec(B^\\wedge)$ be the inverse image of $U$.", "Consider the finite \\'etale morphism", "$$", "W = Y \\times_{(X \\times_S U)} V \\longrightarrow V", "$$", "By the compatibility of the construction of $Y'$", "with flat base change in Lemma \\ref{lemma-extend-S2}", "we find that the base chang", "$Y' \\times_X \\Spec(B^\\wedge) \\to \\Spec(B^\\wedge)$", "is constructed from $W \\to V$ over $\\Spec(B^\\wedge)$", "by the procedure in Lemma \\ref{lemma-extend-S2}.", "Set $V_0 = V \\cap V(x_1, \\ldots, x_d) \\subset V$ and $W_0 = W \\times_V V_0$.", "This is a normal integral scheme which maps into $\\sigma(S)$", "by the morphism $\\Spec(B^\\wedge) \\to X$ and in fact is identified", "with $\\sigma(U)$. Hence we know that $W_0 \\to V_0 = U$ completely decomposes", "as this is true for its generic fibre by our assumption", "on $Z \\to X_\\eta$ having a $\\kappa(\\eta)$-rational point lying over", "$\\sigma(\\eta)$ (and of course the $G$-action then implies the whole", "fibre $Z_{\\sigma(\\eta)}$ is a disjoint union of copies of", "the scheme $\\eta = \\Spec(\\kappa(\\eta))$).", "Finally, by", "Lemma \\ref{lemma-fully-faithful-power-series-over-depth2}", "we have", "$$", "W_0 \\times_U V \\cong W", "$$", "This shows that $W$ is a disjoint union of copies of $V$", "and hence $Y' \\times_X \\Spec(B^\\wedge)$ is a disjoint", "union of copies of $\\Spec(B^\\wedge)$ and the proof is complete." ], "refs": [ "more-algebra-lemma-galois-conclusion", "pione-lemma-extend-tame-covering-normal", "pione-lemma-tame-covering-split", "pione-lemma-purity-one-divisor", "limits-lemma-glueing-near-point", "limits-lemma-glueing-near-point-properties", "algebra-lemma-criterion-normal", "algebra-lemma-apply-grothendieck", "pione-lemma-extend-S2", "pione-lemma-purity-smooth-over-depth2", "algebra-lemma-completion-flat", "pione-lemma-extend-S2", "pione-lemma-extend-S2", "pione-lemma-fully-faithful-power-series-over-depth2" ], "ref_ids": [ 10499, 4123, 4124, 4121, 15114, 15115, 1311, 1361, 4100, 4113, 870, 4100, 4100, 4111 ] } ], "ref_ids": [] }, { "id": 4126, "type": "theorem", "label": "pione-lemma-extend-covering-general", "categories": [ "pione" ], "title": "pione-lemma-extend-covering-general", "contents": [ "Let $S$ be a quasi-compact and quasi-separated integral normal scheme", "with generic point $\\eta$. Let $f : X \\to S$ be a quasi-compact and", "quasi-separated smooth morphism with geometrically connected fibres.", "Let $\\sigma : S \\to X$ be a section of $f$. Let $Z \\to X_\\eta$ be a", "finite \\'etale Galois cover (Section \\ref{section-finite-etale-under-galois})", "with group $G$ of order invertible on $S$ such that", "$Z$ has a $\\kappa(\\eta)$-rational point mapping to $\\sigma(\\eta)$.", "Then there exists a finite \\'etale Galois cover $Y \\to X$ with group $G$", "whose restriction to $X_\\eta$ is $Z$." ], "refs": [], "proofs": [ { "contents": [ "If $S$ is Noetherian, then this is the result of", "Lemma \\ref{lemma-extend-covering}. The general case", "follows from this by a standard limit argument.", "We strongly urge the reader to skip the proof.", "\\medskip\\noindent", "We can write $S = \\lim S_i$ as a directed limit of a system", "of schemes with affine transition morphisms and with $S_i$ of", "finite type over $\\mathbf{Z}$, see", "Limits, Proposition \\ref{limits-proposition-approximate}.", "For each $i$ let $S \\to S'_i \\to S_i$ be the normalization", "of $S_i$ in $S$, see", "Morphisms, Section \\ref{morphisms-section-normalization-X-in-Y}.", "Combining Algebra, Proposition \\ref{algebra-proposition-ubiquity-nagata}", "Morphisms, Lemmas \\ref{morphisms-lemma-nagata-normalization-finite} and", "\\ref{morphisms-lemma-normal-normalization}", "we conclude that $S'_i$ is of finite type over $\\mathbf{Z}$,", "finite over $S_i$, and that $S'_i$ is an integral normal", "scheme such that $S \\to S'_i$ is dominant.", "By Morphisms, Lemma \\ref{morphisms-lemma-functoriality-normalization}", "we obtain transition morphisms $S'_{i'} \\to S'_i$ compatible", "with the transition morphisms $S_{i'} \\to S_i$ and with", "the morphisms with source $S$.", "We claim that $S = \\lim S'_i$. Proof of claim omitted (hint:", "look on affine opens over a chosen affine open in $S_i$", "for some $i$ to translate this into a straightforward algebra", "problem). We conclude that we may write $S = \\lim S_i$", "as a directed limit of a system of normal integral schemes $S_i$", "with affine transition morphisms and with $S_i$ of", "finite type over $\\mathbf{Z}$.", "\\medskip\\noindent", "For some $i$ we can find a smooth morphism $X_i \\to S_i$", "of finite presentation whose base change to $S$ is $X \\to S$.", "See Limits, Lemmas", "\\ref{limits-lemma-descend-finite-presentation} and", "\\ref{limits-lemma-descend-smooth}.", "After increasing $i$ we may assume the section", "$\\sigma$ lifts to a section $\\sigma_i : S_i \\to X_i$", "(by the equivalence of categories in", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}).", "We may replace $X_i$ by the open subscheme $X_i^0$", "of it studied in More on Morphisms, Section", "\\ref{more-morphisms-section-connected-components}", "since the image of $X \\to X_i$ clearly maps into it", "(openness by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-connected-along-section-open}).", "Thus we may assume the fibres of $X_i \\to S_i$ are", "geometrically connected.", "After increasing $i$ we may assume $|G|$ is invertible", "on $S_i$.", "Let $\\eta_i \\in S_i$ be the generic point.", "Since $X_\\eta$ is the limit of the schemes", "$X_{i, \\eta_i}$ we can use the exact same arguments", "to descent $Z \\to X_\\eta$ to some finite \\'etale Galois", "cover $Z_i \\to X_{i, \\eta_i}$ after possibly increasing $i$.", "See Lemma \\ref{lemma-limit}.", "After possibly increasing $i$ once more we may", "assume $Z_i$ has a $\\kappa(\\eta_i)$-rational point", "mapping to $\\sigma_i(\\eta_i)$.", "Then we apply the lemma in the Noetherian case", "and we pullback to $X$ to conclude." ], "refs": [ "pione-lemma-extend-covering", "limits-proposition-approximate", "algebra-proposition-ubiquity-nagata", "morphisms-lemma-nagata-normalization-finite", "morphisms-lemma-normal-normalization", "morphisms-lemma-functoriality-normalization", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-smooth", "limits-lemma-descend-finite-presentation", "pione-lemma-limit" ], "ref_ids": [ 4125, 15126, 1431, 5510, 5508, 5500, 15077, 15064, 15077, 4067 ] } ], "ref_ids": [] }, { "id": 4127, "type": "theorem", "label": "pione-lemma-lower-invariant", "categories": [ "pione" ], "title": "pione-lemma-lower-invariant", "contents": [ "In the situation above, if $NF(C, \\varphi, \\tau) > 0$, then there exist", "an \\'etale $k$-algebra map $\\varphi'$ and a surjective $k$-algebra map", "$\\tau'$ fitting into the commutative diagram", "$$", "\\xymatrix{", "& B \\\\", "C \\ar[r] & C/\\varphi'(I)C \\ar[u]_{\\tau'} \\\\", "k[x_1, \\ldots, x_n] \\ar[u]^{\\varphi'} \\ar[r] &", "A \\ar[u] \\ar@/_3em/[uu]_\\pi", "}", "$$", "with $NF(C, \\varphi', \\tau') < NF(C, \\varphi, \\tau)$." ], "refs": [], "proofs": [ { "contents": [ "Choose $r \\geq 0$ and $y_1, \\ldots, y_r \\in C$ which generate", "$C$ over $\\Im(\\varphi)$ and let $0 \\leq s \\leq r$ be such that", "$y_1, \\ldots, y_s$ are integral over $\\Im(\\varphi)$", "such that $r - s = NF(C, \\varphi, \\tau) > 0$.", "Since $B$ is finite over $A$, the image of $y_{s + 1}$ in $B$", "satisfies a monic polynomial over $A$. Hence we can find", "$d \\geq 1$ and $f_1, \\ldots, f_d \\in k[x_1, \\ldots, x_n]$", "such that", "$$", "z = y_{s + 1}^d + \\varphi(f_1) y_{s + 1}^{d - 1} + \\ldots + \\varphi(f_d) \\in", "J = \\Ker(C \\to C/\\varphi(I)C \\xrightarrow{\\tau} B)", "$$", "Since $\\varphi : k[x_1, \\ldots, x_n] \\to C$ is \\'etale, we can find a", "nonzero and nonconstant polynomial $g \\in k[T_1, \\ldots, T_{n + 1}]$", "such that", "$$", "g(\\varphi(x_1), \\ldots, \\varphi(x_n), z) = 0", "\\quad\\text{in}\\quad", "C", "$$", "To see this you can use for example that", "$C \\otimes_{\\varphi, k[x_1, \\ldots, x_n]} k(x_1, \\ldots, x_n)$", "is a finite product of finite separable field extensions", "of $k(x_1, \\ldots, x_n)$", "(see Algebra, Lemmas \\ref{algebra-lemma-etale-over-field})", "and hence $z$ satisfies a monic", "polynomial over $k(x_1, \\ldots, x_n)$. Clearing denominators", "we obtain $g$.", "\\medskip\\noindent", "The existence of $g$ and", "Algebra, Lemma \\ref{algebra-lemma-helper-polynomial}", "produce integers $e_1, e_2, \\ldots, e_n \\geq 1$ such that", "$z$ is integral over the subring $C'$ of $C$ generated by", "$t_1 = \\varphi(x_1) + z^{pe_1}, \\ldots, t_n = \\varphi(x_n) + z^{pe_n}$.", "Of course, the elements $\\varphi(x_1), \\ldots, \\varphi(x_n)$", "are also integral over $C'$ as are the elements", "$y_1, \\ldots, y_s$. Finally, by our choice of $z$ the element", "$y_{s + 1}$ is integral over $C'$ too.", "\\medskip\\noindent", "Consider the ring map", "$$", "\\varphi' : k[x_1, \\ldots, x_n] \\longrightarrow C, \\quad", "x_i \\longmapsto t_i", "$$", "with image $C'$. Since", "$\\text{d}(\\varphi(x_i)) = \\text{d}(t_i) = \\text{d}(\\varphi'(x_i))$", "in $\\Omega_{C/k}$ (and this is where we use the characteristic of $k$", "is $p > 0$) we conclude that $\\varphi'$ is \\'etale because $\\varphi$ is", "\\'etale, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-etale-over-polynomial-ring}.", "Observe that $\\varphi'(x_i) - \\varphi(x_i) = t_i - \\varphi(x_i) = z^{pe_i}$", "is in the kernel $J$ of the map $C \\to C/\\varphi(I)C \\to B$ by our", "choice of $z$ as an element of $J$.", "Hence for $f \\in I$ the element", "$$", "\\varphi'(f) =", "f(t_1, \\ldots, t_n) =", "f(\\varphi(x_1) + z^{pe_1}, \\ldots, \\varphi(x_n) + z^{pe_n}) =", "\\varphi(f) + \\text{element of }(z)", "$$", "is in $J$ as well. In other words, $\\varphi'(I)C \\subset J$ and", "we obtain a surjection", "$$", "\\tau' : C/\\varphi'(I)C \\longrightarrow C/J \\cong B", "$$", "of algebras \\'etale over $A$. Finally, the algebra", "$C$ is generated by the elements", "$\\varphi(x_1), \\ldots, \\varphi(x_n), y_1, \\ldots, y_r$", "over $C' = \\Im(\\varphi')$ with", "$\\varphi(x_1), \\ldots, \\varphi(x_n), y_1, \\ldots, y_{s + 1}$", "integral over $C' = \\Im(\\varphi')$. Hence", "$NF(C, \\varphi', \\tau') < r - s = NF(C, \\varphi, \\tau)$.", "This finishes the proof." ], "refs": [ "algebra-lemma-etale-over-field", "algebra-lemma-helper-polynomial", "algebra-lemma-characterize-etale-over-polynomial-ring" ], "ref_ids": [ 1232, 999, 1272 ] } ], "ref_ids": [] }, { "id": 4128, "type": "theorem", "label": "pione-lemma-affine-etale-over-affine-space", "categories": [ "pione" ], "title": "pione-lemma-affine-etale-over-affine-space", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $X \\to \\mathbf{A}^n_k$ be an", "\\'etale morphism with $X$ affine. Then there exists a finite \\'etale", "morphism $X \\to \\mathbf{A}^n_k$." ], "refs": [], "proofs": [ { "contents": [ "Write $X = \\Spec(C)$. Set $A = 0$ and denote $I = k[x_1, \\ldots, x_n]$.", "By assumption there exists some \\'etale $k$-algebra map", "$\\varphi : k[x_1, \\ldots, x_n] \\to C$. Denote $\\tau : C/\\varphi(I)C \\to 0$", "the unique surjection. We may choose $\\varphi$ and $\\tau$ such", "that $N(C, \\varphi, \\tau)$ is minimal. By Lemma \\ref{lemma-lower-invariant}", "we get $N(C, \\varphi, \\tau) = 0$. Hence $\\varphi$ is finite \\'etale." ], "refs": [ "pione-lemma-lower-invariant" ], "ref_ids": [ 4127 ] } ], "ref_ids": [] }, { "id": 4129, "type": "theorem", "label": "pione-lemma-dominate-affine-space", "categories": [ "pione" ], "title": "pione-lemma-dominate-affine-space", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $Z \\subset \\mathbf{A}^n_k$", "be a closed subscheme. Let $Y \\to Z$ be finite \\'etale. There exists a", "finite \\'etale morphism $f : U \\to \\mathbf{A}^n_k$ such that", "there is an open and closed immersion $Y \\to f^{-1}(Z)$ over $Z$." ], "refs": [], "proofs": [ { "contents": [ "Let us turn the problem into algebra. Write", "$\\mathbf{A}^n_k = \\Spec(k[x_1, \\ldots, x_n])$.", "Then $Z = \\Spec(A)$ where $A = k[x_1, \\ldots, x_n]/I$", "for some ideal $I \\subset k[x_1, \\ldots, x_n]$.", "Write $Y = \\Spec(B)$ so that $Y \\to Z$ corresponds to", "the finite \\'etale $k$-algebra map $A \\to B$.", "\\medskip\\noindent", "By Algebra, Lemma \\ref{algebra-lemma-lift-etale}", "there exists an \\'etale ring map", "$$", "\\varphi : k[x_1, \\ldots, x_n] \\to C", "$$", "and a surjective $A$-algebra map $\\tau : C/\\varphi(I)C \\to B$.", "(We can even choose $C, \\varphi, \\tau$ such that $\\tau$ is an isomorphism,", "but we won't use this). We may choose $\\varphi$ and $\\tau$ such", "that $N(C, \\varphi, \\tau)$ is minimal. By Lemma \\ref{lemma-lower-invariant}", "we get $N(C, \\varphi, \\tau) = 0$. Hence $\\varphi$ is finite \\'etale.", "\\medskip\\noindent", "Let $f : U = \\Spec(C) \\to \\mathbf{A}^n_k$ be the finite \\'etale", "morphism corresponding to $\\varphi$. The morphism", "$Y \\to f^{-1}(Z) = \\Spec(C/\\varphi(I)C)$ induced by $\\tau$", "is a closed immersion as $\\tau$ is surjective and open as", "it is an \\'etale morphism by Morphisms, Lemma", "\\ref{morphisms-lemma-etale-permanence}. This finishes the proof." ], "refs": [ "algebra-lemma-lift-etale", "pione-lemma-lower-invariant", "morphisms-lemma-etale-permanence" ], "ref_ids": [ 1238, 4127, 5375 ] } ], "ref_ids": [] }, { "id": 4130, "type": "theorem", "label": "pione-proposition-galois", "categories": [ "pione" ], "title": "pione-proposition-galois", "contents": [ "\\begin{reference}", "This is a weak version of \\cite[Expos\\'e V]{SGA1}.", "The proof is borrowed from \\cite[Theorem 7.2.5]{BS}.", "\\end{reference}", "Let $(\\mathcal{C}, F)$ be a Galois category. Let $G = \\text{Aut}(F)$", "be as in Example \\ref{example-from-C-F-to-G-sets}. The functor", "$F : \\mathcal{C} \\to \\textit{Finite-}G\\textit{-Sets}$", "(\\ref{equation-remember}) an equivalence." ], "refs": [], "proofs": [ { "contents": [ "We will use the results of Lemma \\ref{lemma-epi-mono} without further mention.", "In particular we know the functor is faithful.", "By Lemma \\ref{lemma-tame} we know that for any connected $X$ the", "action of $G$ on $F(X)$ is transitive. Hence $F$ preserves", "the decomposition into connected components (existence of which is", "an axiom of a Galois category). Let $X$ and $Y$ be objects and let", "$s : F(X) \\to F(Y)$ be a map. Then the graph", "$\\Gamma_s \\subset F(X) \\times F(Y)$ of $s$", "is a union of connected components. Hence there exists a", "union of connected components $Z$ of $X \\times Y$,", "which comes equipped with a monomorphism $Z \\to X \\times Y$,", "with $F(Z) = \\Gamma_s$. Since $F(Z) \\to F(X)$ is bijective", "we see that $Z \\to X$ is an isomorphism and we conclude", "that $s = F(f)$ where $f : X \\cong Z \\to Y$ is the composition.", "Hence $F$ is fully faithful.", "\\medskip\\noindent", "To finish the proof we show that $F$ is essentially surjective.", "It suffices to show that $G/H$ is in the essential image for", "any open subgroup $H \\subset G$ of finite index.", "By definition of the topology on $G$ there exists a finite", "collection of objects $X_i$ such that", "$$", "\\Ker(G \\longrightarrow \\prod\\nolimits_i \\text{Aut}(F(X_i)))", "$$", "is contained in $H$. We may assume $X_i$ is connected", "for all $i$. We can choose a Galois object $Y$ mapping", "to a connected component of $\\prod X_i$ using", "Lemma \\ref{lemma-galois}. Choose an isomorphism $F(Y) = G/U$", "in $G\\textit{-sets}$ for some open subgroup $U \\subset G$.", "As $Y$ is Galois, the group", "$\\text{Aut}(Y) = \\text{Aut}_{G\\textit{-Sets}}(G/U)$ acts transitively", "on $F(Y) = G/U$. This implies that $U$ is normal. Since", "$F(Y)$ surjects onto $F(X_i)$ for each $i$ we see that", "$U \\subset H$. Let $M \\subset \\text{Aut}(Y)$ be the finite subgroup", "corresponding to", "$$", "(H/U)^{opp} \\subset (G/U)^{opp} = \\text{Aut}_{G\\textit{-Sets}}(G/U)", "= \\text{Aut}(Y).", "$$", "Set $X = Y/M$, i.e., $X$ is the coequalizer", "of the arrows $m : Y \\to Y$, $m \\in M$.", "Since $F$ is exact we see that $F(X) = G/H$ and the", "proof is complete." ], "refs": [ "pione-lemma-epi-mono", "pione-lemma-tame", "pione-lemma-galois" ], "ref_ids": [ 4028, 4030, 4029 ] } ], "ref_ids": [] }, { "id": 4131, "type": "theorem", "label": "pione-proposition-universal-homeomorphism", "categories": [ "pione" ], "title": "pione-proposition-universal-homeomorphism", "contents": [ "Let $f : X \\to Y$ be a universal homeomorphism of schemes. Then", "$$", "\\textit{F\\'Et}_Y \\longrightarrow \\textit{F\\'Et}_X,\\quad", "V \\longmapsto V \\times_Y X", "$$", "is an equivalence. Thus if $X$ and $Y$ are connected, then", "$f$ induces an isomorphism $\\pi_1(X, \\overline{x}) \\to \\pi_1(Y, \\overline{y})$", "of fundamental groups." ], "refs": [], "proofs": [ { "contents": [ "Recall that a universal homeomorphism is the same thing as an", "integral, universally injective, surjective morphism, see", "Morphisms, Lemma \\ref{morphisms-lemma-universal-homeomorphism}.", "In particular, the diagonal $\\Delta : X \\to X \\times_Y X$ is a thickening", "by Morphisms, Lemma \\ref{morphisms-lemma-universally-injective}.", "Thus by Lemma \\ref{lemma-thickening}", "we see that given a finite \\'etale morphism $U \\to X$", "there is a unique isomorphism", "$$", "\\varphi : U \\times_Y X \\to X \\times_Y U", "$$", "of schemes finite \\'etale over $X \\times_Y X$ which pulls back under", "$\\Delta$ to $\\text{id} : U \\to U$ over $X$.", "Since $X \\to X \\times_Y X \\times_Y X$", "is a thickening as well (it is bijective and a closed immersion)", "we conclude that $(U, \\varphi)$ is a descent datum relative to $X/Y$.", "By \\'Etale Morphisms, Proposition \\ref{etale-proposition-effective}", "we conclude that $U = X \\times_Y V$ for some $V \\to Y$", "quasi-compact, separated, and \\'etale.", "We omit the proof that $V \\to Y$ is finite (hints:", "the morphism $U \\to V$ is surjective and $U \\to Y$ is integral).", "We conclude that $\\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_X$", "is essentially surjective.", "\\medskip\\noindent", "Arguing in the same manner as above we see that given", "$V_1 \\to Y$ and $V_2 \\to Y$ in $\\textit{F\\'Et}_Y$ any", "morphism $a : X \\times_Y V_1 \\to X \\times_Y V_2$ over $X$", "is compatible with the canonical descent data. Thus $a$", "descends to a morphism $V_1 \\to V_2$ over $Y$ by", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-fully-faithful-cases}." ], "refs": [ "morphisms-lemma-universal-homeomorphism", "morphisms-lemma-universally-injective", "pione-lemma-thickening", "etale-proposition-effective", "etale-lemma-fully-faithful-cases" ], "ref_ids": [ 5454, 5167, 4043, 10733, 10718 ] } ], "ref_ids": [] }, { "id": 4132, "type": "theorem", "label": "pione-proposition-normal", "categories": [ "pione" ], "title": "pione-proposition-normal", "contents": [ "Let $X$ be a normal integral scheme with function field $K$.", "Then the canonical map (\\ref{equation-inclusion-generic-point})", "$$", "\\text{Gal}(K^{sep}/K) = \\pi_1(\\eta, \\overline{\\eta})", "\\longrightarrow \\pi_1(X, \\overline{\\eta})", "$$", "is identified with the quotient map", "$\\text{Gal}(K^{sep}/K) \\to \\text{Gal}(M/K)$ where $M \\subset K^{sep}$", "is the union of the finite subextensions $L$", "such that $X$ is unramified in $L$." ], "refs": [], "proofs": [ { "contents": [ "The normal scheme $X$ is geometrically unibranch", "(Properties, Lemma \\ref{properties-lemma-normal-geometrically-unibranch}).", "Hence Lemma \\ref{lemma-irreducible-geometrically-unibranch} applies to $X$.", "Thus $\\pi_1(\\eta, \\overline{\\eta}) \\to \\pi_1(X, \\overline{\\eta})$", "is surjective and top horizontal arrow of the commutative diagram", "$$", "\\xymatrix{", "\\textit{F\\'Et}_X \\ar[r] \\ar[d] \\ar[rd]_c & \\textit{F\\'Et}_\\eta \\ar[d] \\\\", "\\textit{Finite-}\\pi_1(X, \\overline{\\eta})\\textit{-sets} \\ar[r] &", "\\textit{Finite-}\\text{Gal}(K^{sep}/K)\\textit{-sets}", "}", "$$", "is fully faithful. The left vertical arrow is the equivalence of", "Theorem \\ref{theorem-fundamental-group}", "and the right vertical arrow is the equivalence of", "Lemma \\ref{lemma-fundamental-group-Galois-group}. The lower", "horizontal arrow is induced by the map of the proposition.", "By Lemmas \\ref{lemma-unramified-in-L} and", "\\ref{lemma-finite-etale-covering-normal-unramified}", "we see that the essential image of $c$", "consists of $\\text{Gal}(K^{sep}/K)\\textit{-Sets}$ isomorphic", "to sets of the form", "$$", "S = \\Hom_K(\\prod\\nolimits_{i = 1, \\ldots, n} L_i, K^{sep}) =", "\\coprod\\nolimits_{i = 1, \\ldots, n} \\Hom_K(L_i, K^{sep})", "$$", "with $L_i/K$ finite separable such that $X$ is unramified in $L_i$.", "Thus if $M \\subset K^{sep}$ is as in the statement of the lemma,", "then $\\text{Gal}(K^{sep}/M)$ is exactly the subgroup of", "$\\text{Gal}(K^{sep}/K)$ acting trivially on every object", "in the essential image of $c$. On the other hand, the essential image of $c$", "is exactly the category of $S$ such that the $\\text{Gal}(K^{sep}/K)$-action", "factors through the surjection", "$\\text{Gal}(K^{sep}/K) \\to \\pi_1(X, \\overline{\\eta})$.", "We conclude that $\\text{Gal}(K^{sep}/M)$ is the kernel.", "Hence $\\text{Gal}(K^{sep}/M)$ is a normal subgroup, $M/K$ is Galois,", "and we have a short exact sequence", "$$", "1 \\to \\text{Gal}(K^{sep}/M) \\to", "\\text{Gal}(K^{sep}/K) \\to", "\\text{Gal}(M/K) \\to 1", "$$", "by Galois theory (Fields, Theorem", "\\ref{fields-theorem-inifinite-galois-theory} and", "Lemma \\ref{fields-lemma-ses-infinite-galois}). The proof is done." ], "refs": [ "properties-lemma-normal-geometrically-unibranch", "pione-lemma-irreducible-geometrically-unibranch", "pione-theorem-fundamental-group", "pione-lemma-fundamental-group-Galois-group", "pione-lemma-unramified-in-L", "pione-lemma-finite-etale-covering-normal-unramified", "fields-theorem-inifinite-galois-theory", "fields-lemma-ses-infinite-galois" ], "ref_ids": [ 2998, 4052, 4021, 4040, 4054, 4055, 4443, 4512 ] } ], "ref_ids": [] }, { "id": 4133, "type": "theorem", "label": "pione-proposition-first-homotopy-sequence", "categories": [ "pione" ], "title": "pione-proposition-first-homotopy-sequence", "contents": [ "Let $f : X \\to S$ be a flat proper morphism of finite presentation whose", "geometric fibres are connected and reduced. Assume $S$ is connected and", "let $\\overline{s}$ be a geometric point of $S$. Then there is an exact", "sequence", "$$", "\\pi_1(X_{\\overline{s}}) \\to \\pi_1(X) \\to \\pi_1(S) \\to 1", "$$", "of fundamental groups." ], "refs": [], "proofs": [ { "contents": [ "Let $Y \\to X$ be a finite \\'etale morphism. Consider the Stein factorization", "$$", "\\xymatrix{", "Y \\ar[d] \\ar[r] & X \\ar[d] \\\\", "T \\ar[r] & S", "}", "$$", "of $Y \\to S$. By Lemma \\ref{lemma-stein-factorization-etale}", "the morphism $T \\to S$ is finite \\'etale. In this way we obtain", "a functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_S$.", "For any finite \\'etale morphism $U \\to S$ a morphism", "$Y \\to U \\times_S X$ over $X$ is the same thing as a morphism", "$Y \\to U$ over $S$ and such a morphism factors uniquely through", "the Stein factorization, i.e., corresponds to a unique", "morphism $T \\to U$", "(by the construction of the Stein factorization as a relative", "normalization in More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-stein-universally-closed}", "and factorization by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-normalization}).", "Thus we see that the functors", "$\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_S$ and", "$\\textit{F\\'Et}_S \\to \\textit{F\\'Et}_X$ are adjoints.", "Note that the Stein factorization of $U \\times_S X \\to S$ is", "$U$, because the fibres of $U \\times_S X \\to U$ are geometrically connected.", "\\medskip\\noindent", "By the discussion above and", "Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}", "we conclude that", "$\\textit{F\\'Et}_S \\to \\textit{F\\'Et}_X$", "is fully faithful, i.e., $\\pi_1(X) \\to \\pi_1(S)$ is surjective", "(Lemma \\ref{lemma-functoriality-galois-surjective}).", "\\medskip\\noindent", "It is immediate that the composition", "$\\textit{F\\'Et}_S \\to \\textit{F\\'Et}_X \\to \\textit{F\\'Et}_{X_{\\overline{s}}}$", "sends any $U$ to a disjoint union of copies of $X_{\\overline{s}}$.", "Hence $\\pi_1(X_{\\overline{s}}) \\to \\pi_1(X) \\to \\pi_1(S)$ is trivial", "by Lemma \\ref{lemma-composition-trivial}.", "\\medskip\\noindent", "Let $Y \\to X$ be a finite \\'etale morphism with $Y$ connected such that", "$Y \\times_X X_{\\overline{s}}$ contains a connected component $Z$", "isomorphic to $X_{\\overline{s}}$. Consider the Stein factorization $T$", "as above. Let $\\overline{t} \\in T_{\\overline{s}}$ be the point corresponding", "to the fibre $Z$. Observe that $T$ is connected (as the image of a connected", "scheme) and by the surjectivity above $T \\times_S X$ is connected.", "Now consider the factorization", "$$", "\\pi : Y \\longrightarrow T \\times_S X", "$$", "Let $\\overline{x} \\in X_{\\overline{s}}$ be any closed point. Note that", "$\\kappa(\\overline{t}) = \\kappa(\\overline{s}) = \\kappa(\\overline{x})$", "is an algebraically closed field.", "Then the fibre of $\\pi$ over $(\\overline{t}, \\overline{x})$ consists", "of a unique point, namely the unique point $\\overline{z} \\in Z$", "corresponding to $\\overline{x} \\in X_{\\overline{s}}$ via the", "isomorphism $Z \\to X_{\\overline{s}}$. We conclude that the finite", "\\'etale morphism $\\pi$ has degree $1$ in a neighbourhood of", "$(\\overline{t}, \\overline{x})$. Since $T \\times_S X$ is connected", "it has degree $1$ everywhere and we find that $Y \\cong T \\times_S X$.", "Thus $Y \\times_X X_{\\overline{s}}$ splits completely.", "Combining all of the above we see that", "Lemmas \\ref{lemma-functoriality-galois-ses} and", "\\ref{lemma-functoriality-galois-normal}", "both apply and the proof is complete." ], "refs": [ "pione-lemma-stein-factorization-etale", "more-morphisms-lemma-stein-universally-closed", "morphisms-lemma-characterize-normalization", "categories-lemma-adjoint-fully-faithful", "pione-lemma-functoriality-galois-surjective", "pione-lemma-composition-trivial", "pione-lemma-functoriality-galois-ses", "pione-lemma-functoriality-galois-normal" ], "ref_ids": [ 4070, 13942, 5499, 12248, 4032, 4033, 4034, 4036 ] } ], "ref_ids": [] }, { "id": 4134, "type": "theorem", "label": "pione-proposition-purity-complete-intersection", "categories": [ "pione" ], "title": "pione-proposition-purity-complete-intersection", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring. If $A$ is a", "complete intersection of dimension $\\geq 3$, then purity", "holds for $A$ in the sense that any finite \\'etale cover of", "the punctured spectrum extends." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-purity-and-completion} we may assume that $A$ is", "a complete local ring. By assumption we can write", "$A = B/(f_1, \\ldots, f_r)$ where $B$ is a complete regular local", "ring and $f_1, \\ldots, f_r$ is a regular sequence.", "We will finish the proof by induction on $r$.", "The base case is $r = 0$ which follows from", "Lemma \\ref{lemma-local-purity} which applies to", "regular rings of dimension $\\geq 2$.", "\\medskip\\noindent", "Assume that $A = B/(f_1, \\ldots, f_r)$ and that the proposition", "holds for $r - 1$. Set $A' = B/(f_1, \\ldots, f_{r - 1})$ and apply", "Lemma \\ref{lemma-purity-inherited-by-hypersurface} to $f_r \\in A'$.", "This is permissible:", "condition (1) holds as $f_1, \\ldots, f_r$ is a regular sequence,", "condition (2) holds as $B$ and hence $A'$ is complete,", "condition (3) holds as $A = A'/f_r A'$ is Cohen-Macaulay of dimension", "$\\dim(A) \\geq 3$, see Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth},", "condition (4) holds by induction hypothesis as", "$\\dim((A'_{f_r})_\\mathfrak p) \\geq 3$ for a maximal", "prime $\\mathfrak p$ of $A'_{f_r}$ and as", "$(A'_{f_r})_\\mathfrak p = B_\\mathfrak q/(f_1, \\ldots, f_{r - 1})$", "for some $\\mathfrak q \\subset B$,", "condition (5) holds by induction hypothesis." ], "refs": [ "pione-lemma-purity-and-completion", "pione-lemma-local-purity", "pione-lemma-purity-inherited-by-hypersurface", "dualizing-lemma-depth" ], "ref_ids": [ 4092, 4098, 4110, 2826 ] } ], "ref_ids": [] }, { "id": 4135, "type": "theorem", "label": "pione-proposition-purity-smooth-over-depth2", "categories": [ "pione" ], "title": "pione-proposition-purity-smooth-over-depth2", "contents": [ "Let $A \\to B$ be a local homomorphism of local Noetherian rings.", "Assume $A$ has depth $\\geq 2$, $A \\to B$ is formally smooth for the", "$\\mathfrak m_B$-adic topology, and $\\dim(B) > \\dim(A)$. For any open", "$V \\subset Y = \\Spec(B)$ which contains", "\\begin{enumerate}", "\\item any prime $\\mathfrak q \\subset B$ such that", "$\\mathfrak q \\cap A \\not = \\mathfrak m_A$,", "\\item the prime $\\mathfrak m_A B$", "\\end{enumerate}", "the functor $\\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_V$", "is an equivalence. In particular purity holds for $B$." ], "refs": [], "proofs": [ { "contents": [ "A prime $\\mathfrak q \\subset B$ which is not contained in $V$", "lies over $\\mathfrak m_A$. In this case $A \\to B_\\mathfrak q$", "is a flat local homomorphism and hence $\\text{depth}(B_\\mathfrak q) \\geq 2$", "(Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck}).", "Thus the functor is fully faithful by", "Lemma \\ref{lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian}", "combined with Local Cohomology,", "Lemma \\ref{local-cohomology-lemma-depth-2-connected-punctured-spectrum}.", "\\medskip\\noindent", "Denote $A^\\wedge$ and $B^\\wedge$ the completions of $A$ and $B$", "with respect to their maximal ideals. Observe that the assumptions", "of the proposition hold for $A^\\wedge \\to B^\\wedge$, see", "More on Algebra, Lemmas", "\\ref{more-algebra-lemma-completion-dimension},", "\\ref{more-algebra-lemma-completion-depth}, and", "\\ref{more-algebra-lemma-formally-smooth-completion}.", "By the uniqueness and compatibility with flat base change", "of the construction of Lemma \\ref{lemma-extend-S2}", "it suffices to prove the essential surjectivity for", "$A^\\wedge \\to B^\\wedge$ and the inverse image of $V$", "(details omitted; compare with Lemma \\ref{lemma-purity-and-completion}", "for the case where $V$ is the punctured spectrum).", "By More on Algebra, Proposition \\ref{more-algebra-proposition-fs-regular}", "this means we may assume $A \\to B$ is regular.", "\\medskip\\noindent", "Let $W \\to V$ be a finite \\'etale morphism.", "By Popescu's theorem", "(Smoothing Ring Maps, Theorem \\ref{smoothing-theorem-popescu})", "we can write $B = \\colim B_i$ as a filtered colimit", "of smooth $A$-algebras. We can pick an $i$ and an", "open $V_i \\subset \\Spec(B_i)$ whose inverse image is $V$", "(Limits, Lemma \\ref{limits-lemma-descend-opens}).", "After increasing $i$ we may assume there is a finite", "\\'etale morphism $W_i \\to V_i$ whose base change to $V$", "is $W \\to V$, see", "Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation},", "\\ref{limits-lemma-descend-finite-finite-presentation}, and", "\\ref{limits-lemma-descend-etale}.", "We may assume the complement of $V_i$ is contained", "in the closed fibre of $\\Spec(B_i) \\to \\Spec(A)$ as this", "is true for $V$ (either choose $V_i$ this way or use", "the lemma above to show this is true for $i$ large enough).", "Let $\\eta$ be the generic point of the closed fibre", "of $\\Spec(B) \\to \\Spec(A)$. Since $\\eta \\in V$, the image of", "$\\eta$ is in $V_i$. Hence after replacing $V_i$ by an", "affine open neighbourhood of the image of the closed point", "of $\\Spec(B)$, we may assume that the closed fibre", "of $\\Spec(B_i) \\to \\Spec(A)$ is irreducible and that", "its generic point is contained in $V_i$ (details omitted; use that", "a scheme smooth over a field is a disjoint union of irreducible schemes).", "At this point we may apply Lemma \\ref{lemma-purity-smooth-over-depth2}", "to see that $W_i \\to V_i$ extends to a finite \\'etale", "morphism $\\Spec(C_i) \\to \\Spec(B_i)$ and pulling", "back to $\\Spec(B)$ we conclude that $W$ is in", "the essential image of the functor", "$\\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_V$", "as desired." ], "refs": [ "algebra-lemma-apply-grothendieck", "pione-lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian", "local-cohomology-lemma-depth-2-connected-punctured-spectrum", "more-algebra-lemma-completion-dimension", "more-algebra-lemma-completion-depth", "more-algebra-lemma-formally-smooth-completion", "pione-lemma-extend-S2", "pione-lemma-purity-and-completion", "more-algebra-proposition-fs-regular", "smoothing-theorem-popescu", "limits-lemma-descend-opens", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-finite-finite-presentation", "limits-lemma-descend-etale", "pione-lemma-purity-smooth-over-depth2" ], "ref_ids": [ 1361, 4048, 9701, 10042, 10043, 10015, 4100, 4092, 10579, 5605, 15041, 15077, 15058, 15065, 4113 ] } ], "ref_ids": [] }, { "id": 4136, "type": "theorem", "label": "pione-proposition-lefschetz-fully-faithful", "categories": [ "pione" ], "title": "pione-proposition-lefschetz-fully-faithful", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$.", "Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{L})$. Let $Y = Z(s)$ be the", "zero scheme of $s$. Assume that for all $x \\in X \\setminus Y$", "we have", "$$", "\\text{depth}(\\mathcal{O}_{X, x}) + \\dim(\\overline{\\{x\\}}) > 1", "$$", "Then the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$", "is fully faithful. In fact, for any open subscheme $V \\subset X$", "containing $Y$ the restriction functor", "$\\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y$", "is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "The first statement is a formal consequence of", "Lemma \\ref{lemma-restriction-fully-faithful-special}", "and", "Algebraic and Formal Geometry, Proposition", "\\ref{algebraization-proposition-lefschetz}.", "The second statement follows from", "Lemma \\ref{lemma-restriction-fully-faithful-special}", "and", "Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-lefschetz-addendum}." ], "refs": [ "pione-lemma-restriction-fully-faithful-special", "algebraization-proposition-lefschetz", "pione-lemma-restriction-fully-faithful-special", "algebraization-lemma-lefschetz-addendum" ], "ref_ids": [ 4080, 12800, 4080, 12788 ] } ], "ref_ids": [] }, { "id": 4137, "type": "theorem", "label": "pione-proposition-lefschetz-equivalence-general", "categories": [ "pione" ], "title": "pione-proposition-lefschetz-equivalence-general", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$.", "Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{L})$. Let $Y = Z(s)$ be the", "zero scheme of $s$. Let $\\mathcal{V}$ be the set of open", "subschemes of $X$ containing $Y$ ordered by reverse inclusion.", "Assume that for all $x \\in X \\setminus Y$ we have", "$$", "\\text{depth}(\\mathcal{O}_{X, x}) + \\dim(\\overline{\\{x\\}}) > 2", "$$", "Then the restriction functor", "$$", "\\colim_\\mathcal{V} \\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y", "$$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "This is a formal consequence of", "Lemma \\ref{lemma-restriction-equivalence-general} and", "Algebraic and Formal Geometry, Proposition", "\\ref{algebraization-proposition-lefschetz-equivalence}." ], "refs": [ "pione-lemma-restriction-equivalence-general", "algebraization-proposition-lefschetz-equivalence" ], "ref_ids": [ 4078, 12802 ] } ], "ref_ids": [] }, { "id": 4138, "type": "theorem", "label": "pione-proposition-lefschetz-equivalence", "categories": [ "pione" ], "title": "pione-proposition-lefschetz-equivalence", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$.", "Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{L})$. Let $Y = Z(s)$ be the", "zero scheme of $s$.", "Assume that for all $x \\in X \\setminus Y$ we have", "$$", "\\text{depth}(\\mathcal{O}_{X, x}) + \\dim(\\overline{\\{x\\}}) > 2", "$$", "and that for $x \\in X \\setminus Y$ closed purity holds for", "$\\mathcal{O}_{X, x}$. Then the restriction functor", "$\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$", "is an equivalence. If $X$ or equivalently $Y$ is connected, then", "$$", "\\pi_1(Y, \\overline{y}) \\to \\pi_1(X, \\overline{y})", "$$", "is an isomorphism for any geometric point $\\overline{y}$ of $Y$." ], "refs": [], "proofs": [ { "contents": [ "Fully faithfulness holds by", "Proposition \\ref{proposition-lefschetz-fully-faithful}.", "By Proposition \\ref{proposition-lefschetz-equivalence-general}", "any object of $\\textit{F\\'Et}_Y$", "is isomorphic to the fibre product $U \\times_V Y$ for some", "finite \\'etale morphism $U \\to V$ where $V \\subset X$", "is an open subscheme containing $Y$.", "The complement $T = X \\setminus V$", "is\\footnote{Namely, $T$ is proper over $k$ (being closed in $X$)", "and affine (being closed in the affine scheme $X \\setminus Y$, see", "Morphisms, Lemma \\ref{morphisms-lemma-proper-ample-delete-affine})", "and hence finite over $k$", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}).", "Thus $T$ is a finite set of closed points.}", "a finite set of closed points of $X \\setminus Y$.", "Say $T = \\{x_1, \\ldots, x_n\\}$.", "By assumption we can find finite \\'etale morphisms", "$V'_i \\to \\Spec(\\mathcal{O}_{X, x_i})$ agreeing with", "$U \\to V$ over $V \\times_X \\Spec(\\mathcal{O}_{X, x_i})$.", "By Limits, Lemma \\ref{limits-lemma-glueing-near-closed-point}", "applied $n$ times we see that $U \\to V$ extends to a finite \\'etale", "morphism $U' \\to X$ as desired.", "See Lemma \\ref{lemma-what-equivalence-gives} for the final statement." ], "refs": [ "pione-proposition-lefschetz-fully-faithful", "pione-proposition-lefschetz-equivalence-general", "morphisms-lemma-proper-ample-delete-affine", "morphisms-lemma-finite-proper", "limits-lemma-glueing-near-closed-point", "pione-lemma-what-equivalence-gives" ], "ref_ids": [ 4136, 4137, 5435, 5445, 15112, 4041 ] } ], "ref_ids": [] }, { "id": 4139, "type": "theorem", "label": "pione-proposition-specialization-map-isomorphism", "categories": [ "pione" ], "title": "pione-proposition-specialization-map-isomorphism", "contents": [ "Let $f : X \\to S$ be a smooth proper morphism with geometrically", "connected fibres. Let $s' \\leadsto s$ be a specialization.", "If the characteristic to $\\kappa(s)$ is zero, then the specialization", "map", "$$", "sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The map is surjective by", "Lemma \\ref{lemma-specialization-map-surjective}.", "Thus we have to show it is injective.", "\\medskip\\noindent", "We may assume $S$ is affine. Then $S$ is a cofiltered limit of affine", "schemes of finite type over $\\mathbf{Z}$.", "Hence we can assume $X \\to S$ is the", "base change of $X_0 \\to S_0$ where $S_0$ is the spectrum of a finite", "type $\\mathbf{Z}$-algebra and $X_0 \\to S_0$ is smooth and proper.", "See Limits, Lemma \\ref{limits-lemma-descend-finite-presentation},", "\\ref{limits-lemma-descend-smooth}, and", "\\ref{limits-lemma-eventually-proper}. By", "Lemma \\ref{lemma-specialization-map-base-change}", "we reduce to the case where the base is Noetherian.", "\\medskip\\noindent", "Applying Lemma \\ref{lemma-specialization-map-discrete-valuation-ring}", "we reduce to the case where the base $S$ is the spectrum of a", "strictly henselian discrete valuation ring $A$ and we are", "looking at the specialization map over $A$.", "Let $K$ be the fraction field of $A$.", "Choose an algebraic closure $\\overline{K}$ which", "corresponds to a geometric generic point $\\overline{\\eta}$ of $\\Spec(A)$.", "For $\\overline{K}/L/K$ finite separable, let $B \\subset L$ be the", "integral closure of $A$ in $L$. This is a discrete", "valuation ring by", "More on Algebra, Remark \\ref{more-algebra-remark-finite-separable-extension}.", "\\medskip\\noindent", "Let $X \\to \\Spec(A)$ be as in the previous paragraph.", "To show injectivity of the specialization map", "it suffices to prove that every finite", "\\'etale cover $V$ of $X_{\\overline{\\eta}}$ is the base", "change of a finite \\'etale cover $Y \\to X$.", "Namely, then $\\pi_1(X_{\\overline{\\eta}}) \\to \\pi_1(X) = \\pi_1(X_s)$", "is injective by Lemma \\ref{lemma-functoriality-galois-injective}.", "\\medskip\\noindent", "Given $V$ we can first descend $V$ to $V' \\to X_{K^{sep}}$ by", "Lemma \\ref{lemma-perfection} and then to", "$V'' \\to X_L$ by Lemma \\ref{lemma-limit}.", "Let $Z \\to X_B$ be the normalization of $X_B$ in $V''$.", "Observe that $Z$ is normal and that $Z_L = V''$ as schemes", "over $X_L$. Hence $Z \\to X_B$ is finite \\'etale over", "the generic fibre. The problem is that we do not know that", "$Z \\to X_B$ is everywhere \\'etale. Since $X \\to \\Spec(A)$", "has geometrically connected smooth fibres, we see that", "the special fibre $X_s$ is geometrically irreducible.", "Hence the special fibre of $X_B \\to \\Spec(B)$ is irreducible;", "let $\\xi_B$ be its generic point. Let", "$\\xi_1, \\ldots, \\xi_r$ be the points of $Z$ mapping to", "$\\xi_B$. Our first (and it will turn out only) problem", "is now that the extensions", "$$", "\\mathcal{O}_{X_B, \\xi_B} \\subset \\mathcal{O}_{Z, \\xi_i}", "$$", "of discrete valuation rings may be ramified. Let $e_i$ be", "the ramification index of this extension. Note that since the", "characteristic of $\\kappa(s)$ is zero, the ramification is tame!", "\\medskip\\noindent", "To get rid of the ramification we are going to choose a further finite", "separable extension $K^{sep}/L'/L/K$ such that the ramification", "index $e$ of the induced extensions $B'/B$ is divisible by $e_i$.", "Consider the normalized base change $Z'$ of $Z$ with respect to", "$\\Spec(B') \\to \\Spec(B)$, see discussion in", "More on Morphisms, Section \\ref{more-morphisms-section-reduced-fibre-theorem}.", "Let $\\xi_{i, j}$ be the points of $Z'$ mapping to $\\xi_{B'}$", "and to $\\xi_i$ in $Z$. Then the local rings", "$$", "\\mathcal{O}_{Z', \\xi_{i, j}}", "$$", "are localizations of the integral closure of $\\mathcal{O}_{Z, \\xi_i}$", "in $L' \\otimes_L F_i$ where $F_i$ is the fraction field of", "$\\mathcal{O}_{Z, \\xi_i}$; details omitted. Hence Abhyankar's lemma", "(More on Algebra, Lemma \\ref{more-algebra-lemma-abhyankar})", "tells us that", "$$", "\\mathcal{O}_{X_{B'}, \\xi_{B'}} \\subset \\mathcal{O}_{Z', \\xi_{i, j}}", "$$", "is unramified. We conclude that the morphism $Z' \\to X_{B'}$", "is \\'etale away from codimension $1$. Hence by purity of", "branch locus (Lemma \\ref{lemma-purity})", "we see that $Z' \\to X_{B'}$ is finite \\'etale!", "\\medskip\\noindent", "However, since the residue field extension induced by $A \\to B'$", "is trivial (as the residue field of $A$ is algebraically closed", "being separably closed of characteristic zero)", "we conclude that $Z'$ is the base change of a finite \\'etale", "cover $Y \\to X$ by applying", "Lemma \\ref{lemma-finite-etale-on-proper-over-henselian}", "twice (first to get $Y$ over $A$, then to prove that", "the pullback to $B$ is isomorphic to $Z'$).", "This finishes the proof." ], "refs": [ "pione-lemma-specialization-map-surjective", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-smooth", "limits-lemma-eventually-proper", "pione-lemma-specialization-map-base-change", "pione-lemma-specialization-map-discrete-valuation-ring", "more-algebra-remark-finite-separable-extension", "pione-lemma-functoriality-galois-injective", "pione-lemma-perfection", "pione-lemma-limit", "more-algebra-lemma-abhyankar", "pione-lemma-purity", "pione-lemma-finite-etale-on-proper-over-henselian" ], "ref_ids": [ 4119, 15077, 15064, 15089, 4071, 4074, 10676, 4035, 4068, 4067, 10509, 4099, 4044 ] } ], "ref_ids": [] }, { "id": 4140, "type": "theorem", "label": "pione-proposition-injective", "categories": [ "pione" ], "title": "pione-proposition-injective", "contents": [ "Let $p$ be a prime number. Let $i : Z \\to X$ be a closed immersion", "of connected affine schemes over $\\mathbf{F}_p$. For any geometric", "point $\\overline{z}$ of $Z$ the map", "$$", "\\pi_1(Z, \\overline{z}) \\to \\pi_1(X, \\overline{z})", "$$", "is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $Y \\to Z$ be a finite \\'etale morphism. It suffices to construct", "a finite \\'etale morphism $f : U \\to X$ such that $Y$ is isomorphic to", "an open and closed subscheme of $f^{-1}(Z)$, see", "Lemma \\ref{lemma-functoriality-galois-injective}.", "Write $Y = \\Spec(A)$ and $X = \\Spec(R)$ so the closed immersion", "$Y \\to X$ is given by a surjection $R \\to A$. We may write", "$A = \\colim A_i$ as the filtered colimit of its $\\mathbf{F}_p$-subalgebras", "of finite type. By Lemma \\ref{lemma-limit}", "we can find an $i$ and a finite \\'etale morphism $Y_i \\to Z_i = \\Spec(A_i)$", "such that $Y = Z \\times_{Z_i} Y_i$.", "\\medskip\\noindent", "Choose a surjection $\\mathbf{F}_p[x_1, \\ldots, x_n] \\to A_i$.", "This determines a closed immersion", "$$", "Z_i = \\Spec(A_i)", "\\longrightarrow", "X_i = \\mathbf{A}^n_{\\mathbf{F}_p} = \\Spec(\\mathbf{F}_p[x_1, \\ldots, x_n])", "$$", "By the universal property of polynomial algebras and since", "$R \\to A$ is surjective, we can find a commutative diagram", "$$", "\\xymatrix{", "\\mathbf{F}_p[x_1, \\ldots, x_n] \\ar[r] \\ar[d] &", "A_i \\ar[d] \\\\", "R \\ar[r] &", "A", "}", "$$", "of $\\mathbf{F}_p$-algebras. Thus we have a commutative diagram", "$$", "\\xymatrix{", "Y_i \\ar[r] &", "Z_i \\ar[r] &", "X_i \\\\", "Y \\ar[u] \\ar[r] &", "Z \\ar[u] \\ar[r] &", "X \\ar[u]", "}", "$$", "whose right square is cartesian. Clearly, if we can", "find $f_i : U_i \\to X_i$ finite \\'etale such that $Y_i$ is", "isomorphic to an open and closed subscheme of $f_i^{-1}(Z_i)$,", "then the base change $f : U \\to X$ of $f_i$ by $X \\to X_i$", "is a solution to our problem. Thus we conclude by applying", "Lemma \\ref{lemma-dominate-affine-space} to", "$Y_i \\to Z_i \\to X_i = \\mathbf{A}^n_{\\mathbf{F}_p}$." ], "refs": [ "pione-lemma-functoriality-galois-injective", "pione-lemma-limit", "pione-lemma-dominate-affine-space" ], "ref_ids": [ 4035, 4067, 4129 ] } ], "ref_ids": [] }, { "id": 4148, "type": "theorem", "label": "stacks-cohomology-lemma-flat-pullback-quasi-coherent", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-flat-pullback-quasi-coherent", "contents": [ "If $f : \\mathcal{X} \\to \\mathcal{Y}$ is a flat morphism of algebraic stacks", "then $f^* : \\QCoh(\\mathcal{O}_\\mathcal{Y}) \\to", "\\QCoh(\\mathcal{O}_\\mathcal{X})$ is an exact functor." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective smooth morphism $V \\to \\mathcal{Y}$.", "Choose a scheme $U$ and a surjective smooth morphism", "$U \\to V \\times_\\mathcal{Y} \\mathcal{X}$. Then $U \\to \\mathcal{X}$ is", "still smooth and surjective as a composition of two such morphisms.", "From the commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_{f'} & V \\ar[d] \\\\", "\\mathcal{X} \\ar[r]^f & \\mathcal{Y}", "}", "$$", "we obtain a commutative diagram", "$$", "\\xymatrix{", "\\QCoh(\\mathcal{O}_U) & \\QCoh(\\mathcal{O}_V) \\ar[l] \\\\", "\\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar[u] &", "\\QCoh(\\mathcal{O}_\\mathcal{Y}) \\ar[l] \\ar[u]", "}", "$$", "of abelian categories. Our proof that the bottom two categories in this", "diagram are abelian showed that the vertical functors are faithful", "exact functors (see proof of", "Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-quasi-coherent-algebraic-stack}).", "Since $f'$ is a flat morphism of schemes (by our definition of", "flat morphisms of algebraic stacks) we see that $(f')^*$ is an", "exact functor on quasi-coherent sheaves on $V$. Thus we win." ], "refs": [ "stacks-sheaves-lemma-quasi-coherent-algebraic-stack" ], "ref_ids": [ 11590 ] } ], "ref_ids": [] }, { "id": 4149, "type": "theorem", "label": "stacks-cohomology-lemma-general-pushforward", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-general-pushforward", "contents": [ "Let $\\mathcal{M}$ be a rule which associates to every algebraic stack", "$\\mathcal{X}$ a subcategory $\\mathcal{M}_\\mathcal{X}$ of", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "such that", "\\begin{enumerate}", "\\item $\\mathcal{M}_\\mathcal{X}$ is a weak Serre subcategory", "of $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "(see Homology, Definition \\ref{homology-definition-serre-subcategory})", "for all algebraic stacks $\\mathcal{X}$,", "\\item for a smooth morphism of algebraic stacks", "$f : \\mathcal{Y} \\to \\mathcal{X}$ the functor $f^*$ maps", "$\\mathcal{M}_\\mathcal{X}$ into $\\mathcal{M}_\\mathcal{Y}$,", "\\item if $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ is a family of smooth", "morphisms of algebraic stacks with", "$|\\mathcal{X}| = \\bigcup |f_i|(|\\mathcal{X}_i|)$, then an object", "$\\mathcal{F}$ of", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "is in $\\mathcal{M}_\\mathcal{X}$ if and only if", "$f_i^*\\mathcal{F}$ is in $\\mathcal{M}_{\\mathcal{X}_i}$ for all $i$, and", "\\item if $f : \\mathcal{Y} \\to \\mathcal{X}$ is a morphism of algebraic", "stacks such that $\\mathcal{X}$ and $\\mathcal{Y}$ are representable", "by affine schemes, then $R^if_*$ maps $\\mathcal{M}_\\mathcal{Y}$", "into $\\mathcal{M}_\\mathcal{X}$.", "\\end{enumerate}", "Then for any quasi-compact and quasi-separated morphism ", "$f : \\mathcal{Y} \\to \\mathcal{X}$ of algebraic stacks", "$R^if_*$ maps $\\mathcal{M}_\\mathcal{Y}$", "into $\\mathcal{M}_\\mathcal{X}$. (Higher direct images computed in \\'etale", "topology.)" ], "refs": [ "homology-definition-serre-subcategory" ], "proofs": [ { "contents": [ "Let $f : \\mathcal{Y} \\to \\mathcal{X}$ be a quasi-compact and quasi-separated", "morphism of algebraic stacks and let $\\mathcal{F}$ be an object of", "$\\mathcal{M}_\\mathcal{Y}$. Choose a surjective smooth morphism", "$\\mathcal{U} \\to \\mathcal{X}$ where $\\mathcal{U}$ is representable by", "a scheme. By", "Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-base-change-higher-direct-images}", "taking higher direct images commutes with base change.", "Assumption (2) shows that the pullback of $\\mathcal{F}$", "to $\\mathcal{U} \\times_\\mathcal{X} \\mathcal{Y}$ is in", "$\\mathcal{M}_{\\mathcal{U} \\times_\\mathcal{X} \\mathcal{Y}}$", "because the projection", "$\\mathcal{U} \\times_\\mathcal{X} \\mathcal{Y} \\to \\mathcal{Y}$", "is smooth as a base change of a smooth morphism. Hence (3) shows we may", "replace $\\mathcal{Y} \\to \\mathcal{X}$ by the projection", "$\\mathcal{U} \\times_\\mathcal{X} \\mathcal{Y} \\to \\mathcal{U}$.", "In other words, we may assume that $\\mathcal{X}$", "is representable by a scheme.", "Using (3) once more, we see that the question is Zariski local on", "$\\mathcal{X}$, hence we may assume that $\\mathcal{X}$ is representable by", "an affine scheme. Since $f$ is quasi-compact this implies that also", "$\\mathcal{Y}$ is quasi-compact. Thus we may choose a surjective smooth", "morphism $g : \\mathcal{V} \\to \\mathcal{Y}$ where $\\mathcal{V}$ is representable", "by an affine scheme.", "\\medskip\\noindent", "In this situation we have the spectral sequence", "$$", "E_2^{p, q} = R^q(f \\circ g_p)_*g_p^*\\mathcal{F}", "\\Rightarrow", "R^{p + q}f_*\\mathcal{F}", "$$", "of", "Sheaves on Stacks, Proposition", "\\ref{stacks-sheaves-proposition-smooth-covering-compute-direct-image}.", "Recall that this is a first quadrant spectral sequence hence we may", "use the last part of Homology, Lemma \\ref{homology-lemma-first-quadrant-ss}.", "Note that the morphisms", "$$", "g_p : \\mathcal{V}_p =", "\\mathcal{V} \\times_\\mathcal{Y} \\ldots \\times_\\mathcal{Y} \\mathcal{V}", "\\longrightarrow", "\\mathcal{Y}", "$$", "are smooth as compositions of base changes of the smooth morphism $g$.", "Thus the sheaves $g_p^*\\mathcal{F}$ are in", "$\\mathcal{M}_{\\mathcal{V}_p}$ by (2). Hence it suffices to prove that the", "higher direct images of objects of $\\mathcal{M}_{\\mathcal{V}_p}$ under", "the morphisms", "$$", "\\mathcal{V}_p =", "\\mathcal{V} \\times_\\mathcal{Y} \\ldots \\times_\\mathcal{Y} \\mathcal{V}", "\\longrightarrow", "\\mathcal{X}", "$$", "are in $\\mathcal{M}_\\mathcal{X}$. The algebraic stacks $\\mathcal{V}_p$", "are quasi-compact and quasi-separated by", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-quasi-compact-quasi-separated-permanence}.", "Of course each $\\mathcal{V}_p$ is representable by an algebraic space", "(the diagonal of the algebraic stack $\\mathcal{Y}$ is representable", "by algebraic spaces). This reduces us to the case where", "$\\mathcal{Y}$ is representable by an algebraic space and $\\mathcal{X}$", "is representable by an affine scheme.", "\\medskip\\noindent", "In the situation where $\\mathcal{Y}$ is representable by an algebraic", "space and $\\mathcal{X}$ is representable by an affine scheme, we choose", "anew a surjective smooth morphism $\\mathcal{V} \\to \\mathcal{Y}$ where", "$\\mathcal{V}$ is representable by an affine scheme. Going through the", "argument above once again we once again reduce to the morphisms", "$\\mathcal{V}_p \\to \\mathcal{X}$. But in the current situation the algebraic", "stacks $\\mathcal{V}_p$ are representable by quasi-compact and quasi-separated", "schemes (because the diagonal of an algebraic space is representable by", "schemes).", "\\medskip\\noindent", "Thus we may assume $\\mathcal{Y}$ is representable by a scheme and", "$\\mathcal{X}$ is representable by an affine scheme. Choose (again)", "a surjective smooth morphism $\\mathcal{V} \\to \\mathcal{Y}$ where", "$\\mathcal{V}$ is representable by an affine scheme. In this case all", "the algebraic stacks $\\mathcal{V}_p$ are representable by separated", "schemes (because the diagonal of a scheme is separated).", "\\medskip\\noindent", "Thus we may assume $\\mathcal{Y}$ is representable by a separated scheme and", "$\\mathcal{X}$ is representable by an affine scheme. Choose (yet again)", "a surjective smooth morphism $\\mathcal{V} \\to \\mathcal{Y}$ where", "$\\mathcal{V}$ is representable by an affine scheme. In this case all", "the algebraic stacks $\\mathcal{V}_p$ are representable by affine schemes", "(because the diagonal of a separated scheme is a closed immersion hence affine)", "and this case is handled by assumption (4).", "This finishes the proof." ], "refs": [ "stacks-sheaves-lemma-base-change-higher-direct-images", "stacks-sheaves-proposition-smooth-covering-compute-direct-image", "homology-lemma-first-quadrant-ss", "stacks-morphisms-lemma-quasi-compact-quasi-separated-permanence" ], "ref_ids": [ 11610, 11621, 12105, 7428 ] } ], "ref_ids": [ 12146 ] }, { "id": 4150, "type": "theorem", "label": "stacks-cohomology-lemma-general-pushforward-fppf", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-general-pushforward-fppf", "contents": [ "Let $\\mathcal{M}$ be a rule which associates to every algebraic stack", "$\\mathcal{X}$ a subcategory $\\mathcal{M}_\\mathcal{X}$ of", "$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$", "such that", "\\begin{enumerate}", "\\item $\\mathcal{M}_\\mathcal{X}$ is a weak Serre subcategory", "of $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$", "for all algebraic stacks $\\mathcal{X}$,", "\\item for a smooth morphism of algebraic stacks", "$f : \\mathcal{Y} \\to \\mathcal{X}$ the functor $f^*$ maps", "$\\mathcal{M}_\\mathcal{X}$ into $\\mathcal{M}_\\mathcal{Y}$,", "\\item if $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ is a family of smooth", "morphisms of algebraic stacks with", "$|\\mathcal{X}| = \\bigcup |f_i|(|\\mathcal{X}_i|)$, then an object", "$\\mathcal{F}$ of $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$", "is in $\\mathcal{M}_\\mathcal{X}$ if and only if", "$f_i^*\\mathcal{F}$ is in $\\mathcal{M}_{\\mathcal{X}_i}$ for all $i$, and", "\\item if $f : \\mathcal{Y} \\to \\mathcal{X}$ is a morphism of algebraic", "stacks and $\\mathcal{X}$ and $\\mathcal{Y}$ are representable", "by affine schemes, then $R^if_*$ maps $\\mathcal{M}_\\mathcal{Y}$", "into $\\mathcal{M}_\\mathcal{X}$.", "\\end{enumerate}", "Then for any quasi-compact and quasi-separated morphism ", "$f : \\mathcal{Y} \\to \\mathcal{X}$ of algebraic stacks", "$R^if_*$ maps $\\mathcal{M}_\\mathcal{Y}$", "into $\\mathcal{M}_\\mathcal{X}$. (Higher direct images computed in fppf", "topology.)" ], "refs": [], "proofs": [ { "contents": [ "Identical to the proof of Lemma \\ref{lemma-general-pushforward}." ], "refs": [ "stacks-cohomology-lemma-general-pushforward" ], "ref_ids": [ 4149 ] } ], "ref_ids": [] }, { "id": 4151, "type": "theorem", "label": "stacks-cohomology-lemma-check-lqc-on-etale-covering", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-check-lqc-on-etale-covering", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let", "$f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of smooth", "morphisms of algebraic stacks with", "$|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules", "on $\\mathcal{X}_\\etale$. If each $f_j^{-1}\\mathcal{F}$", "is locally quasi-coherent, then so is $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "We may replace each of the algebraic stacks $\\mathcal{X}_j$ by", "a scheme $U_j$ (using that any algebraic stack has a smooth covering by", "a scheme and that compositions of smooth morphisms are smooth, see", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-composition-smooth}).", "The pullback of $\\mathcal{F}$ to $(\\Sch/U_j)_\\etale$ is still", "locally quasi-coherent, see", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-pullback-lqc}.", "Then $f = \\coprod f_j : U = \\coprod U_j \\to \\mathcal{X}$ is a surjective", "smooth morphism. Let $x$ be an object of $\\mathcal{X}$. By", "Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}", "there exists an \\'etale covering $\\{x_i \\to x\\}_{i \\in I}$", "such that each $x_i$ lifts to an object $u_i$ of $(\\Sch/U)_\\etale$.", "This just means that $x$, $x_i$ live over schemes $V$, $V_i$, that", "$\\{V_i \\to V\\}$ is an \\'etale covering, and that $x_i$ comes from", "a morphism $u_i : V_i \\to U$. The restriction", "$x_i^*\\mathcal{F}|_{V_{i, \\etale}}$ is equal to the restriction", "of $f^*\\mathcal{F}$ to $V_{i, \\etale}$, see", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-comparison}.", "Hence $x^*\\mathcal{F}|_{V_\\etale}$", "is a sheaf on the small \\'etale site of $V$ which is quasi-coherent", "when restricted to $V_{i, \\etale}$ for each $i$.", "This implies that it is quasi-coherent (as desired), for example by", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-characterize-quasi-coherent}." ], "refs": [ "stacks-morphisms-lemma-composition-smooth", "stacks-sheaves-lemma-pullback-lqc", "stacks-sheaves-lemma-surjective-flat-locally-finite-presentation", "stacks-sheaves-lemma-comparison", "spaces-properties-lemma-characterize-quasi-coherent" ], "ref_ids": [ 7539, 11584, 11606, 11575, 11911 ] } ], "ref_ids": [] }, { "id": 4152, "type": "theorem", "label": "stacks-cohomology-lemma-pushforward-locally-quasi-coherent", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-pushforward-locally-quasi-coherent", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and", "quasi-separated morphism of algebraic stacks. Let ", "$\\mathcal{F}$ be a locally quasi-coherent", "$\\mathcal{O}_\\mathcal{X}$-module on $\\mathcal{X}_\\etale$.", "Then $R^if_*\\mathcal{F}$ (computed in the \\'etale topology) is", "locally quasi-coherent on $\\mathcal{Y}_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-general-pushforward}", "to prove this. We will check its assumptions (1) -- (4).", "Parts (1) and (2) follows from", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-lqc-colimits}.", "Part (3) follows from", "Lemma \\ref{lemma-check-lqc-on-etale-covering}.", "Thus it suffices to show (4).", "\\medskip\\noindent", "Suppose $f : \\mathcal{X} \\to \\mathcal{Y}$ is a morphism of algebraic stacks", "such that $\\mathcal{X}$ and $\\mathcal{Y}$ are representable by affine", "schemes $X$ and $Y$. Choose any object $y$ of $\\mathcal{Y}$ lying over a", "scheme $V$. For clarity, denote $\\mathcal{V} = (\\Sch/V)_{fppf}$ the", "algebraic stack corresponding to $V$. Consider the cartesian diagram", "$$", "\\xymatrix{", "\\mathcal{Z} \\ar[d] \\ar[r]_g \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\", "\\mathcal{V} \\ar[r]^y & \\mathcal{Y}", "}", "$$", "Thus $\\mathcal{Z}$ is representable by the scheme $Z = V \\times_Y X$", "and $f'$ is quasi-compact and separated (even affine). By", "Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-compare-representable-morphism-cohomology}", "we have", "$$", "R^if_*\\mathcal{F}|_{V_\\etale} =", "R^if'_{small, *}\\big(g^*\\mathcal{F}|_{Z_\\etale}\\big)", "$$", "The right hand side is a quasi-coherent sheaf on $V_\\etale$ by", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-higher-direct-image}.", "This implies the left hand side is quasi-coherent which is what", "we had to prove." ], "refs": [ "stacks-cohomology-lemma-general-pushforward", "stacks-sheaves-lemma-lqc-colimits", "stacks-cohomology-lemma-check-lqc-on-etale-covering", "stacks-sheaves-lemma-compare-representable-morphism-cohomology", "spaces-cohomology-lemma-higher-direct-image" ], "ref_ids": [ 4149, 11585, 4151, 11613, 11271 ] } ], "ref_ids": [] }, { "id": 4153, "type": "theorem", "label": "stacks-cohomology-lemma-check-lqc-on-flat-covering", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-check-lqc-on-flat-covering", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let", "$f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of flat", "and locally finitely presented morphisms of algebraic stacks with", "$|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules", "on $\\mathcal{X}_{fppf}$. If each $f_j^{-1}\\mathcal{F}$", "is locally quasi-coherent, then so is $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "First, suppose there is a morphism $a : \\mathcal{U} \\to \\mathcal{X}$", "which is surjective, flat, locally of finite presentation, quasi-compact,", "and quasi-separated such that $a^*\\mathcal{F}$ is locally quasi-coherent.", "Then there is an exact sequence", "$$", "0 \\to \\mathcal{F} \\to a_*a^*\\mathcal{F} \\to b_*b^*\\mathcal{F}", "$$", "where $b$ is the morphism", "$b : \\mathcal{U} \\times_\\mathcal{X} \\mathcal{U} \\to \\mathcal{X}$, see", "Sheaves on Stacks, Proposition", "\\ref{stacks-sheaves-proposition-exactness-cech-complex} and", "Lemma \\ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}.", "Moreover, the pullback $b^*\\mathcal{F}$ is the pullback of $a^*\\mathcal{F}$", "via one of the projection morphisms, hence is locally quasi-coherent", "(Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-pullback-lqc}).", "The modules $a_*a^*\\mathcal{F}$ and $b_*b^*\\mathcal{F}$ are locally", "quasi-coherent by Lemma \\ref{lemma-pushforward-locally-quasi-coherent}.", "(Note that $a_*$ and $b_*$ don't care about which topology is", "used to calculate them.)", "We conclude that $\\mathcal{F}$ is locally quasi-coherent, see", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-lqc-colimits}.", "\\medskip\\noindent", "We are going to reduce the proof of the general case the", "situation in the first paragraph. Let $x$ be an object of $\\mathcal{X}$", "lying over the scheme $U$. We have to show that", "$\\mathcal{F}|_{U_\\etale}$ is a quasi-coherent $\\mathcal{O}_U$-module.", "It suffices to do this (Zariski) locally on $U$, hence we may", "assume that $U$ is affine. By", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-surjective-family-flat-locally-finite-presentation}", "there exists an fppf covering $\\{a_i : U_i \\to U\\}$ such that", "each $x \\circ a_i$ factors through some $f_j$. Hence $a_i^*\\mathcal{F}$", "is locally quasi-coherent on $(\\Sch/U_i)_{fppf}$. After refining", "the covering we may assume $\\{U_i \\to U\\}_{i = 1, \\ldots, n}$", "is a standard fppf covering. Then $x^*\\mathcal{F}$ is an fppf", "module on $(\\Sch/U)_{fppf}$ whose pullback by the morphism", "$a : U_1 \\amalg \\ldots \\amalg U_n \\to U$ is locally quasi-coherent.", "Hence by the first paragraph we see that $x^*\\mathcal{F}$ is locally", "quasi-coherent, which certainly implies that $\\mathcal{F}|_{U_\\etale}$", "is quasi-coherent." ], "refs": [ "stacks-sheaves-proposition-exactness-cech-complex", "stacks-sheaves-lemma-surjective-flat-locally-finite-presentation", "stacks-sheaves-lemma-pullback-lqc", "stacks-cohomology-lemma-pushforward-locally-quasi-coherent", "stacks-sheaves-lemma-lqc-colimits", "stacks-morphisms-lemma-surjective-family-flat-locally-finite-presentation" ], "ref_ids": [ 11619, 11606, 11584, 4152, 11585, 7512 ] } ], "ref_ids": [] }, { "id": 4154, "type": "theorem", "label": "stacks-cohomology-lemma-check-flat-comparison-on-etale-covering", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-check-flat-comparison-on-etale-covering", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{F}$", "be an $\\mathcal{O}_\\mathcal{X}$-module on $\\mathcal{X}_\\etale$.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ has the flat base change property then for any morphism", "$g : \\mathcal{Y} \\to \\mathcal{X}$ of algebraic stacks, the", "pullback $g^*\\mathcal{F}$ does too.", "\\item The full subcategory of", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "consisting of modules with the flat base change property", "is a weak Serre subcategory.", "\\item Let $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ be a family of", "smooth morphisms of algebraic stacks such that", "$|\\mathcal{X}| = \\bigcup_i |f_i|(|\\mathcal{X}_i|)$. If each", "$f_i^*\\mathcal{F}$ has the flat base change property then so does", "$\\mathcal{F}$.", "\\item The category of $\\mathcal{O}_\\mathcal{X}$-modules", "on $\\mathcal{X}_\\etale$ with the flat base change property", "has colimits and they agree with colimits in", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $g : \\mathcal{Y} \\to \\mathcal{X}$ be as in (1).", "Let $y$ be an object of $\\mathcal{Y}$ lying over a scheme $V$. By", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-comparison}", "we have", "$(g^*\\mathcal{F})|_{V_\\etale} = \\mathcal{F}|_{V_\\etale}$.", "Moreover a comparison mapping for the sheaf $g^*\\mathcal{F}$ on $\\mathcal{Y}$", "is a special case of a comparison map for the sheaf $\\mathcal{F}$ on", "$\\mathcal{X}$, see", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-comparison}.", "In this way (1) is clear.", "\\medskip\\noindent", "Proof of (2). We use the characterization of weak Serre subcategories of", "Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}.", "Kernels and cokernels of", "maps between sheaves having the flat base change property", "also have the flat base change property. This is clear because", "$f_{small}^*$ is exact for a flat morphism of schemes and since the", "restriction functors $(-)|_{U_\\etale}$ are exact (because we", "are working in the \\'etale topology). Finally, if", "$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "is a short exact sequence of", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "and the outer two sheaves have the flat base change property then", "the middle one does as well, again because of the exactness of", "$f_{small}^*$ and the restriction functors (and the 5 lemma).", "\\medskip\\noindent", "Proof of (3).", "Let $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ be a jointly surjective family of", "smooth morphisms of algebraic stacks and assume each $f_i^*\\mathcal{F}$", "has the flat base change property. By part (1), the definition of", "an algebraic stack, and the fact that compositions of smooth morphisms", "are smooth (see", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-composition-smooth})", "we may assume that each $\\mathcal{X}_i$ is representable by a scheme.", "Let $\\varphi : x \\to x'$ be a morphism of $\\mathcal{X}$ lying over", "a flat morphism $a : U \\to U'$ of schemes. By", "Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}", "there exists a jointly surjective family of \\'etale morphisms", "$U'_i \\to U'$ such that $U'_i \\to U' \\to \\mathcal{X}$ factors through", "$\\mathcal{X}_i$. Thus we obtain commutative diagrams", "$$", "\\xymatrix{", "U_i = U \\times_{U'} U_i' \\ar[r]_-{a_i} \\ar[d] &", "U_i' \\ar[r]_{x_i'} \\ar[d] & \\mathcal{X}_i \\ar[d]^{f_i} \\\\", "U \\ar[r]^a & U' \\ar[r]^{x'} & \\mathcal{X}", "}", "$$", "Note that each $a_i$ is a flat morphism of schemes as a base change of $a$.", "Denote $\\psi_i : x_i \\to x'_i$ the morphism of $\\mathcal{X}_i$ lying over", "$a_i$ with target $x_i'$. By assumption the comparison maps", "$c_{\\psi_i} :", "(a_i)_{small}^*\\big(f_i^*\\mathcal{F}|_{(U'_i)_\\etale}\\big)", "\\to f_i^*\\mathcal{F}|_{(U_i)_\\etale}$ is an isomorphism.", "Because the vertical arrows $U_i' \\to U'$ and $U_i \\to U$ are \\'etale,", "the sheaves $f_i^*\\mathcal{F}|_{(U_i')_\\etale}$ and", "$f_i^*\\mathcal{F}|_{(U_i)_\\etale}$ are the restrictions of", "$\\mathcal{F}|_{U'_\\etale}$ and $\\mathcal{F}|_{U_\\etale}$", "and the map $c_{\\psi_i}$ is the restriction of $c_\\varphi$ to", "$(U_i)_\\etale$, see", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-comparison}.", "Since $\\{U_i \\to U\\}$ is an \\'etale covering, this implies", "that the comparison map $c_\\varphi$ is an isomorphism which is what", "we wanted to prove.", "\\medskip\\noindent", "Proof of (4). Let", "$\\mathcal{I} \\to", "\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$,", "$i \\mapsto \\mathcal{F}_i$ be a diagram and assume each $\\mathcal{F}_i$", "has the flat base change property. Recall that $\\colim_i \\mathcal{F}_i$", "is the sheafification of the presheaf colimit. As we are using the", "\\'etale topology, it is clear that", "$$", "(\\colim_i \\mathcal{F}_i)|_{U_\\etale} =", "\\colim_i {\\mathcal{F}_i}|_{U_\\etale}", "$$", "As $f_{small}^*$ commutes with colimits (as a left adjoint)", "we see that (4) holds." ], "refs": [ "stacks-sheaves-lemma-comparison", "stacks-sheaves-lemma-comparison", "homology-lemma-characterize-weak-serre-subcategory", "stacks-morphisms-lemma-composition-smooth", "stacks-sheaves-lemma-surjective-flat-locally-finite-presentation", "stacks-sheaves-lemma-comparison" ], "ref_ids": [ 11575, 11575, 12046, 7539, 11606, 11575 ] } ], "ref_ids": [] }, { "id": 4155, "type": "theorem", "label": "stacks-cohomology-lemma-flat-comparison", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-flat-comparison", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and", "quasi-separated morphism of algebraic stacks. Let ", "$\\mathcal{F}$ be an object of", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "which is locally quasi-coherent and has the flat base change property.", "Then each $R^if_*\\mathcal{F}$ (computed in the \\'etale topology)", "has the flat base change property." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-general-pushforward}", "to prove this. For every algebraic stack $\\mathcal{X}$ let", "$\\mathcal{M}_\\mathcal{X}$ denote the full subcategory of", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "consisting of locally quasi-coherent sheaves with the flat base", "change property. Once we verify conditions (1) -- (4) of", "Lemma \\ref{lemma-general-pushforward}", "the lemma will follow. Properties (1), (2), and (3) follow from", "Sheaves on Stacks, Lemmas \\ref{stacks-sheaves-lemma-pullback-lqc} and", "\\ref{stacks-sheaves-lemma-lqc-colimits}", "and", "Lemmas \\ref{lemma-check-lqc-on-etale-covering} and", "\\ref{lemma-check-flat-comparison-on-etale-covering}.", "Thus it suffices to show part (4).", "\\medskip\\noindent", "Suppose $f : \\mathcal{X} \\to \\mathcal{Y}$ is a morphism of algebraic stacks", "such that $\\mathcal{X}$ and $\\mathcal{Y}$ are representable by affine", "schemes $X$ and $Y$. In this case, suppose that", "$\\psi : y \\to y'$ is a morphism of $\\mathcal{Y}$ lying over", "a flat morphism $b : V \\to V'$ of schemes. For clarity denote", "$\\mathcal{V} = (\\Sch/V)_{fppf}$ and $\\mathcal{V}' = (\\Sch/V')_{fppf}$", "the corresponding algebraic stacks. Consider the diagram", "of algebraic stacks", "$$", "\\xymatrix{", "\\mathcal{Z} \\ar[d]_{f''} \\ar[r]_a &", "\\mathcal{Z}' \\ar[r]_{x'} \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\", "\\mathcal{V} \\ar[r]^b & \\mathcal{V}' \\ar[r]^{y'} & \\mathcal{Y}", "}", "$$", "with both squares cartesian. As $f$ is representable by schemes", "(and quasi-compact and separated -- even affine) we see that $\\mathcal{Z}$ and", "$\\mathcal{Z}'$ are representable by schemes $Z$ and $Z'$ and in", "fact $Z = V \\times_{V'} Z'$. Since $\\mathcal{F}$ has the flat", "base change property we see that", "$$", "a_{small}^*\\big(\\mathcal{F}|_{Z'_\\etale}\\big)", "\\longrightarrow", "\\mathcal{F}|_{Z_\\etale}", "$$", "is an isomorphism. Moreover,", "$$", "R^if_*\\mathcal{F}|_{V'_\\etale} =", "R^i(f')_{small, *}\\big(\\mathcal{F}|_{Z'_\\etale}\\big)", "$$", "and", "$$", "R^if_*\\mathcal{F}|_{V_\\etale} =", "R^i(f'')_{small, *}\\big(\\mathcal{F}|_{Z_\\etale}\\big)", "$$", "by", "Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-compare-representable-morphism-cohomology}.", "Hence we see that the comparison map", "$$", "c_\\psi :", "b_{small}^*(R^if_*\\mathcal{F}|_{V'_\\etale})", "\\longrightarrow", "R^if_*\\mathcal{F}|_{V_\\etale}", "$$", "is an isomorphism by", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-flat-base-change-cohomology}.", "Thus $R^if_*\\mathcal{F}$ has the flat base change property.", "Since $R^if_*\\mathcal{F}$ is locally quasi-coherent by", "Lemma \\ref{lemma-pushforward-locally-quasi-coherent}", "we win." ], "refs": [ "stacks-cohomology-lemma-general-pushforward", "stacks-cohomology-lemma-general-pushforward", "stacks-sheaves-lemma-pullback-lqc", "stacks-sheaves-lemma-lqc-colimits", "stacks-cohomology-lemma-check-lqc-on-etale-covering", "stacks-cohomology-lemma-check-flat-comparison-on-etale-covering", "stacks-sheaves-lemma-compare-representable-morphism-cohomology", "spaces-cohomology-lemma-flat-base-change-cohomology", "stacks-cohomology-lemma-pushforward-locally-quasi-coherent" ], "ref_ids": [ 4149, 4149, 11584, 11585, 4151, 4154, 11613, 11296, 4152 ] } ], "ref_ids": [] }, { "id": 4156, "type": "theorem", "label": "stacks-cohomology-lemma-check-lqc-fbc-on-covering", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-check-lqc-fbc-on-covering", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. With $\\mathcal{M}_\\mathcal{X}$", "the category of locally quasi-coherent modules with the flat base change", "property.", "\\begin{enumerate}", "\\item Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of smooth", "morphisms of algebraic stacks with", "$|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules", "on $\\mathcal{X}_\\etale$. If each $f_j^{-1}\\mathcal{F}$", "is in $\\mathcal{M}_{\\mathcal{X}_i}$, then $\\mathcal{F}$ is in", "$\\mathcal{M}_\\mathcal{X}$.", "\\item Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of flat", "and locally finitely presented morphisms of algebraic stacks with", "$|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules", "on $\\mathcal{X}_{fppf}$. If each $f_j^{-1}\\mathcal{F}$", "is in $\\mathcal{M}_{\\mathcal{X}_i}$, then $\\mathcal{F}$ is in", "$\\mathcal{M}_\\mathcal{X}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from a combination of", "Lemmas \\ref{lemma-check-lqc-on-etale-covering} and", "\\ref{lemma-check-flat-comparison-on-etale-covering}.", "The proof of (2) is analogous to the proof of", "Lemma \\ref{lemma-check-lqc-on-flat-covering}.", "Let $\\mathcal{F}$ of a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules", "on $\\mathcal{X}_{fppf}$.", "\\medskip\\noindent", "First, suppose there is a morphism $a : \\mathcal{U} \\to \\mathcal{X}$", "which is surjective, flat, locally of finite presentation, quasi-compact,", "and quasi-separated such that $a^*\\mathcal{F}$ is locally quasi-coherent", "and has the flat base change property.", "Then there is an exact sequence", "$$", "0 \\to \\mathcal{F} \\to a_*a^*\\mathcal{F} \\to b_*b^*\\mathcal{F}", "$$", "where $b$ is the morphism", "$b : \\mathcal{U} \\times_\\mathcal{X} \\mathcal{U} \\to \\mathcal{X}$, see", "Sheaves on Stacks, Proposition", "\\ref{stacks-sheaves-proposition-exactness-cech-complex} and", "Lemma \\ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}.", "Moreover, the pullback $b^*\\mathcal{F}$ is the pullback of $a^*\\mathcal{F}$", "via one of the projection morphisms, hence is locally quasi-coherent", "and has the flat base change property, see", "Proposition \\ref{proposition-lcq-flat-base-change}.", "The modules $a_*a^*\\mathcal{F}$ and $b_*b^*\\mathcal{F}$ are locally", "quasi-coherent and have the flat base change property by", "Proposition \\ref{proposition-lcq-flat-base-change}.", "We conclude that $\\mathcal{F}$ is locally quasi-coherent and", "has the flat base change property by", "Proposition \\ref{proposition-lcq-flat-base-change}.", "\\medskip\\noindent", "Choose a scheme $U$ and a surjective smooth morphism $x : U \\to \\mathcal{X}$.", "By part (1) it suffices to show that $x^*\\mathcal{F}$ is locally", "quasi-coherent and has the flat base change property.", "Again by part (1) it suffices to do this (Zariski) locally on $U$,", "hence we may assume that $U$ is affine. By", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-surjective-family-flat-locally-finite-presentation}", "there exists an fppf covering $\\{a_i : U_i \\to U\\}$ such that", "each $x \\circ a_i$ factors through some $f_j$. Hence the module", "$a_i^*\\mathcal{F}$ on $(\\Sch/U_i)_{fppf}$", "is locally quasi-coherent and has the flat base change property.", "After refining the covering we may assume $\\{U_i \\to U\\}_{i = 1, \\ldots, n}$", "is a standard fppf covering. Then $x^*\\mathcal{F}$ is an fppf", "module on $(\\Sch/U)_{fppf}$ whose pullback by the morphism", "$a : U_1 \\amalg \\ldots \\amalg U_n \\to U$ is locally quasi-coherent", "and has the flat base change property.", "Hence by the previous paragraph we see that $x^*\\mathcal{F}$ is locally", "quasi-coherent and has the flat base change property as desired." ], "refs": [ "stacks-cohomology-lemma-check-lqc-on-etale-covering", "stacks-cohomology-lemma-check-flat-comparison-on-etale-covering", "stacks-cohomology-lemma-check-lqc-on-flat-covering", "stacks-sheaves-proposition-exactness-cech-complex", "stacks-sheaves-lemma-surjective-flat-locally-finite-presentation", "stacks-cohomology-proposition-lcq-flat-base-change", "stacks-cohomology-proposition-lcq-flat-base-change", "stacks-cohomology-proposition-lcq-flat-base-change", "stacks-morphisms-lemma-surjective-family-flat-locally-finite-presentation" ], "ref_ids": [ 4151, 4154, 4153, 11619, 11606, 4173, 4173, 4173, 7512 ] } ], "ref_ids": [] }, { "id": 4157, "type": "theorem", "label": "stacks-cohomology-lemma-parasitic", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-parasitic", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{F}$", "be a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is parasitic and", "$g : \\mathcal{Y} \\to \\mathcal{X}$ is a flat morphism of algebraic stacks,", "then $g^*\\mathcal{F}$ is parasitic.", "\\item For $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$", "we have", "\\begin{enumerate}", "\\item the $\\tau$ sheafification of a parasitic presheaf of modules is", "parasitic, and", "\\item the full subcategory of", "$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$", "consisting of parasitic modules is a Serre subcategory.", "\\end{enumerate}", "\\item Suppose $\\mathcal{F}$ is a sheaf for the \\'etale topology.", "Let $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ be a family of", "smooth morphisms of algebraic stacks such that", "$|\\mathcal{X}| = \\bigcup_i |f_i|(|\\mathcal{X}_i|)$. If each", "$f_i^*\\mathcal{F}$ is parasitic then so is $\\mathcal{F}$.", "\\item Suppose $\\mathcal{F}$ is a sheaf for the fppf topology.", "Let $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ be a family of", "flat and locally finitely presented morphisms of algebraic stacks such that", "$|\\mathcal{X}| = \\bigcup_i |f_i|(|\\mathcal{X}_i|)$. If each", "$f_i^*\\mathcal{F}$ is parasitic then so is $\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To see part (1) let $y$ be an object of $\\mathcal{Y}$ which lies", "over a scheme $V$ such that the corresponding morphism $y : V \\to \\mathcal{Y}$", "is flat. Then $g(y) : V \\to \\mathcal{Y} \\to \\mathcal{X}$ is flat", "as a composition of flat morphisms (see", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-composition-flat})", "hence $\\mathcal{F}(g(y))$ is zero by assumption. Since", "$g^*\\mathcal{F} = g^{-1}\\mathcal{F}(y) = \\mathcal{F}(g(y))$ we conclude", "$g^*\\mathcal{F}$ is parasitic.", "\\medskip\\noindent", "To see part (2)(a) note that if $\\{x_i \\to x\\}$ is a $\\tau$-covering", "of $\\mathcal{X}$, then each of the morphisms $x_i \\to x$ lies", "over a flat morphism of schemes. Hence if $x$ lies over a scheme", "$U$ such that $x : U \\to \\mathcal{X}$ is flat, so do all of the", "objects $x_i$. Hence the presheaf $\\mathcal{F}^+$ (see", "Sites, Section \\ref{sites-section-sheafification})", "is parasitic if the presheaf $\\mathcal{F}$ is", "parasitic. This proves (2)(a) as the sheafification of $\\mathcal{F}$", "is $(\\mathcal{F}^+)^+$.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be a parasitic $\\tau$-module. It is immediate from the", "definitions that any submodule of $\\mathcal{F}$ is parasitic. On the other", "hand, if $\\mathcal{F}' \\subset \\mathcal{F}$ is a submodule, then it is", "equally clear that the presheaf", "$x \\mapsto \\mathcal{F}(x)/\\mathcal{F}'(x)$", "is parasitic. Hence the quotient $\\mathcal{F}/\\mathcal{F}'$ is a parasitic", "module by (2)(a). Finally, we have to show that given a short exact sequence", "$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "with $\\mathcal{F}_1$ and $\\mathcal{F}_3$ parasitic, then $\\mathcal{F}_2$", "is parasitic. This follows immediately on evaluating on $x$ lying", "over a scheme flat over $\\mathcal{X}$. This proves (2)(b), see", "Homology, Lemma \\ref{homology-lemma-characterize-serre-subcategory}.", "\\medskip\\noindent", "Let $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ be a jointly surjective family of", "smooth morphisms of algebraic stacks and assume each $f_i^*\\mathcal{F}$", "is parasitic. Let $x$ be an object of $\\mathcal{X}$ which lies over a", "scheme $U$ such that $x : U \\to \\mathcal{X}$ is flat. Consider a surjective", "smooth covering $W_i \\to U \\times_{x, \\mathcal{X}} \\mathcal{X}_i$.", "Denote $y_i : W_i \\to \\mathcal{X}_i$ the projection. It follows", "that $\\{f_i(y_i) \\to x\\}$ is a covering for the smooth topology", "on $\\mathcal{X}$. Since a composition of flat morphisms is flat we see that", "$f_i^*\\mathcal{F}(y_i) = 0$. On the other hand, as we saw in the proof", "of (1), we have $f_i^*\\mathcal{F}(y_i) = \\mathcal{F}(f_i(y_i))$.", "Hence we see that for some smooth covering $\\{x_i \\to x\\}_{i \\in I}$", "in $\\mathcal{X}$ we have $\\mathcal{F}(x_i) = 0$. This implies", "$\\mathcal{F}(x) = 0$ because the smooth topology is the same", "as the \\'etale topology, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-etale-dominates-smooth}.", "Namely, $\\{x_i \\to x\\}_{i \\in I}$ lies over a smooth covering", "$\\{U_i \\to U\\}_{i \\in I}$ of schemes. By the lemma just referenced", "there exists an \\'etale covering $\\{V_j \\to U\\}_{j \\in J}$ which", "refines $\\{U_i \\to U\\}_{i \\in I}$. Denote $x'_j = x|_{V_j}$.", "Then $\\{x'_j \\to x\\}$ is an \\'etale covering in $\\mathcal{X}$", "refining $\\{x_i \\to x\\}_{i \\in I}$. This means the map", "$\\mathcal{F}(x) \\to \\prod_{j \\in J} \\mathcal{F}(x'_j)$, which is", "injective as $\\mathcal{F}$ is a sheaf in the \\'etale topology,", "factors through $\\mathcal{F}(x) \\to \\prod_{i \\in I} \\mathcal{F}(x_i)$", "which is zero. Hence $\\mathcal{F}(x) = 0$ as desired.", "\\medskip\\noindent", "Proof of (4): omitted. Hint: similar, but simpler, than the proof of (3)." ], "refs": [ "stacks-morphisms-lemma-composition-flat", "homology-lemma-characterize-serre-subcategory", "more-morphisms-lemma-etale-dominates-smooth" ], "ref_ids": [ 7494, 12045, 13880 ] } ], "ref_ids": [] }, { "id": 4158, "type": "theorem", "label": "stacks-cohomology-lemma-pushforward-parasitic", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-pushforward-parasitic", "contents": [ "Let $\\tau \\in \\{\\etale, fppf\\}$.", "Let $\\mathcal{X}$ be an algebraic stack.", "Let $\\mathcal{F}$ be a parasitic object of", "$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$.", "\\begin{enumerate}", "\\item $H^i_\\tau(\\mathcal{X}, \\mathcal{F}) = 0$ for all $i$.", "\\item Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Then $R^if_*\\mathcal{F}$ (computed in $\\tau$-topology) is a", "parasitic object of $\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We first reduce (2) to (1).", "By Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-pushforward-restriction}", "we see that $R^if_*\\mathcal{F}$ is the sheaf associated to the presheaf", "$$", "y \\longmapsto", "H^i_\\tau\\Big(V \\times_{y, \\mathcal{Y}} \\mathcal{X},", "\\ \\text{pr}^{-1}\\mathcal{F}\\Big)", "$$", "Here $y$ is a typical object of $\\mathcal{Y}$ lying over the scheme $V$.", "By Lemma \\ref{lemma-parasitic} it suffices to show that", "these cohomology groups are zero when $y : V \\to \\mathcal{Y}$ is flat.", "Note that $\\text{pr} : V \\times_{y, \\mathcal{Y}} \\mathcal{X} \\to \\mathcal{X}$", "is flat as a base change of $y$. Hence by", "Lemma \\ref{lemma-parasitic} we see that $\\text{pr}^{-1}\\mathcal{F}$", "is parasitic. Thus it suffices to prove (1).", "\\medskip\\noindent", "To see (1) we can use the spectral sequence of", "Sheaves on Stacks, Proposition", "\\ref{stacks-sheaves-proposition-smooth-covering-compute-cohomology}", "to reduce this to the case where $\\mathcal{X}$", "is an algebraic stack representable by an algebraic space.", "Note that in the spectral sequence each", "$f_p^{-1}\\mathcal{F} = f_p^*\\mathcal{F}$ is a parasitic module by", "Lemma \\ref{lemma-parasitic} because the morphisms", "$f_p : \\mathcal{U}_p =", "\\mathcal{U} \\times_\\mathcal{X} \\ldots", "\\times_\\mathcal{X} \\mathcal{U} \\to \\mathcal{X}$ are flat.", "Reusing this spectral sequence one more time (as in the", "proof of the key Lemma \\ref{lemma-general-pushforward})", "we reduce to the case where the", "algebraic stack $\\mathcal{X}$ is representable by a scheme $X$.", "Then $H^i_\\tau(\\mathcal{X}, \\mathcal{F}) = H^i((\\Sch/X)_\\tau, \\mathcal{F})$.", "In this case the vanishing follows easily from an argument", "with {\\v C}ech coverings, see", "Descent, Lemma \\ref{descent-lemma-cohomology-parasitic}." ], "refs": [ "stacks-sheaves-lemma-pushforward-restriction", "stacks-cohomology-lemma-parasitic", "stacks-cohomology-lemma-parasitic", "stacks-sheaves-proposition-smooth-covering-compute-cohomology", "stacks-cohomology-lemma-parasitic", "stacks-cohomology-lemma-general-pushforward", "descent-lemma-cohomology-parasitic" ], "ref_ids": [ 11609, 4157, 4157, 11620, 4157, 4149, 14629 ] } ], "ref_ids": [] }, { "id": 4159, "type": "theorem", "label": "stacks-cohomology-lemma-exact-sequence-quasi-coherent-parasitic-cohomology", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-exact-sequence-quasi-coherent-parasitic-cohomology", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let", "$\\mathcal{F}^\\bullet$ be an", "exact complex in $\\QCoh(\\mathcal{O}_\\mathcal{X})$.", "Then the cohomology sheaves of $\\mathcal{F}^\\bullet$", "in either the \\'etale or the fppf topology", "are parasitic $\\mathcal{O}_\\mathcal{X}$-modules." ], "refs": [], "proofs": [ { "contents": [ "Let $x : U \\to \\mathcal{X}$ be a flat morphism where $U$ is a scheme.", "Then $x^*\\mathcal{F}^\\bullet$ is exact by", "Lemma \\ref{lemma-flat-pullback-quasi-coherent}.", "Hence the restriction $x^*\\mathcal{F}^\\bullet|_{U_\\etale}$", "is exact which is what we had to prove." ], "refs": [ "stacks-cohomology-lemma-flat-pullback-quasi-coherent" ], "ref_ids": [ 4148 ] } ], "ref_ids": [] }, { "id": 4160, "type": "theorem", "label": "stacks-cohomology-lemma-adjoint", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-adjoint", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{M}_\\mathcal{X}$", "be the category of locally quasi-coherent modules with the", "flat base change property, see", "Proposition \\ref{proposition-lcq-flat-base-change}.", "The inclusion functor", "$i : \\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\mathcal{M}_\\mathcal{X}$", "has a right adjoint", "$$", "Q : \\mathcal{M}_\\mathcal{X} \\to \\QCoh(\\mathcal{O}_\\mathcal{X})", "$$", "such that $Q \\circ i$ is the identity functor." ], "refs": [ "stacks-cohomology-proposition-lcq-flat-base-change" ], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective smooth morphism $f : U \\to \\mathcal{X}$.", "Set $R = U \\times_\\mathcal{X} U$ so that we obtain a smooth groupoid", "$(U, R, s, t, c)$ in algebraic spaces with the property that", "$\\mathcal{X} = [U/R]$, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-stack-presentation}.", "We may and do replace $\\mathcal{X}$ by $[U/R]$.", "In the proof of", "Sheaves on Stacks, Proposition \\ref{stacks-sheaves-proposition-quasi-coherent}", "we constructed a functor", "$$", "q_1 :", "\\QCoh(U, R, s, t, c)", "\\longrightarrow", "\\QCoh(\\mathcal{O}_\\mathcal{X}).", "$$", "The construction of the inverse functor in the proof of", "Sheaves on Stacks, Proposition \\ref{stacks-sheaves-proposition-quasi-coherent}", "works for objects of $\\mathcal{M}_\\mathcal{X}$ and induces a functor", "$$", "q_2 :", "\\mathcal{M}_\\mathcal{X}", "\\longrightarrow", "\\QCoh(U, R, s, t, c).", "$$", "Namely, if $\\mathcal{F}$ is an object of $\\mathcal{M}_\\mathcal{X}$", "the we set", "$$", "q_2(\\mathcal{F}) = (f^*\\mathcal{F}|_{U_\\etale}, \\alpha)", "$$", "where $\\alpha$ is the isomorphism", "$$", "t_{small}^*(f^*\\mathcal{F}|_{U_\\etale})", "\\to", "t^*f^*\\mathcal{F}|_{R_\\etale} \\to", "s^*f^*\\mathcal{F}|_{R_\\etale} \\to", "s_{small}^*(f^*\\mathcal{F}|_{U_\\etale})", "$$", "where the outer two morphisms are the comparison maps. Note that", "$q_2(\\mathcal{F})$ is quasi-coherent precisely because $\\mathcal{F}$ is", "locally quasi-coherent (and we used the flat base change property", "in the construction of the descent datum $\\alpha$). We omit the", "verification that the cocycle condition (see", "Groupoids in Spaces, Definition", "\\ref{spaces-groupoids-definition-groupoid-module})", "holds. We define $Q = q_1 \\circ q_2$.", "Let $\\mathcal{F}$ be an object of $\\mathcal{M}_\\mathcal{X}$ and", "let $\\mathcal{G}$ be an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$.", "We have", "\\begin{align*}", "\\Mor_{\\mathcal{M}_\\mathcal{X}}(i(\\mathcal{G}), \\mathcal{F})", "& =", "\\Mor_{\\QCoh(U, R, s, t, c)}(q_2(\\mathcal{G}), q_2(\\mathcal{F})) \\\\", "& =", "\\Mor_{\\QCoh(\\mathcal{O}_\\mathcal{X})}(\\mathcal{G}, Q(\\mathcal{F}))", "\\end{align*}", "where the first equality is", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-map-from-quasi-coherent}", "and the second equality holds because $q_1$ and $q_2$ are inverse", "equivalences of categories. The assertion $Q \\circ i \\cong \\text{id}$", "is a formal consequence of the fact that $i$ is fully faithful." ], "refs": [ "algebraic-lemma-stack-presentation", "stacks-sheaves-proposition-quasi-coherent", "stacks-sheaves-proposition-quasi-coherent", "spaces-groupoids-definition-groupoid-module", "stacks-sheaves-lemma-map-from-quasi-coherent" ], "ref_ids": [ 8474, 11617, 11617, 9348, 11589 ] } ], "ref_ids": [ 4173 ] }, { "id": 4161, "type": "theorem", "label": "stacks-cohomology-lemma-adjoint-kernel-parasitic", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-adjoint-kernel-parasitic", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Let $Q : \\mathcal{M}_\\mathcal{X} \\to \\QCoh(\\mathcal{O}_\\mathcal{X})$", "be the functor constructed in Lemma \\ref{lemma-adjoint}.", "\\begin{enumerate}", "\\item The kernel of $Q$ is exactly the collection of parasitic objects", "of $\\mathcal{M}_\\mathcal{X}$.", "\\item For any object $\\mathcal{F}$", "of $\\mathcal{M}_\\mathcal{X}$ both the kernel and the cokernel of the", "adjunction map $Q(\\mathcal{F}) \\to \\mathcal{F}$ are parasitic.", "\\item The functor $Q$ is exact.", "\\end{enumerate}" ], "refs": [ "stacks-cohomology-lemma-adjoint" ], "proofs": [ { "contents": [ "Write $\\mathcal{X} = [U/R]$ as in the proof of Lemma \\ref{lemma-adjoint}.", "Let $\\mathcal{F}$ be an object of $\\mathcal{M}_\\mathcal{X}$.", "It is clear from the proof of Lemma \\ref{lemma-adjoint}", "that $\\mathcal{F}$ is in the kernel of $Q$ if and only if", "$\\mathcal{F}|_{U_\\etale} = 0$.", "In particular, if $\\mathcal{F}$ is parasitic then $\\mathcal{F}$ is in", "the kernel. Next, let $x : V \\to \\mathcal{X}$ be a flat morphism, where", "$V$ is a scheme. Set $W = V \\times_\\mathcal{X} U$ and consider the diagram", "$$", "\\xymatrix{", "W \\ar[d]_p \\ar[r]_q & V \\ar[d] \\\\", "U \\ar[r] & \\mathcal{X}", "}", "$$", "Note that the projection $p : W \\to U$ is flat and the projection", "$q : W \\to V$ is smooth and surjective. This implies that $q_{small}^*$", "is a faithful functor on quasi-coherent modules. By assumption $\\mathcal{F}$", "has the flat base change property so that we obtain", "$p_{small}^*\\mathcal{F}|_{U_\\etale} \\cong", "q_{small}^*\\mathcal{F}|_{V_\\etale}$. Thus if $\\mathcal{F}$", "is in the kernel of $Q$, then $\\mathcal{F}|_{V_\\etale} = 0$", "which completes the proof of (1).", "\\medskip\\noindent", "Part (2) follows from the discussion above and the fact", "that the map $Q(\\mathcal{F}) \\to \\mathcal{F}$ becomes an isomorphism after", "restricting to $U_\\etale$.", "\\medskip\\noindent", "To see part (3) note that $Q$", "is left exact as a right adjoint. Suppose that", "$0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0$", "is a short exact sequence in $\\mathcal{M}_\\mathcal{X}$. Let", "$\\mathcal{E} = \\Coker(Q(\\mathcal{G}) \\to Q(\\mathcal{H}))$ in", "$\\QCoh(\\mathcal{O}_\\mathcal{X})$. Since", "$\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\mathcal{M}_\\mathcal{X}$", "is a left adjoint it is right exact. Hence we see that", "$Q(\\mathcal{G}) \\to Q(\\mathcal{H}) \\to \\mathcal{E} \\to 0$", "is exact in $\\mathcal{M}_\\mathcal{X}$. Using", "Lemma \\ref{lemma-exact-sequence-quasi-coherent-parasitic-cohomology}", "we find that the top row of the following commutative diagram", "has parasitic cohomology sheaves at $Q(\\mathcal{F})$ and $Q(\\mathcal{G})$:", "$$", "\\xymatrix{", "0 \\ar[r] &", "Q(\\mathcal{F}) \\ar[r] \\ar[d]_a &", "Q(\\mathcal{G}) \\ar[r] \\ar[d]_b &", "Q(\\mathcal{H}) \\ar[r] \\ar[d]_c &", "\\mathcal{E} \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "\\mathcal{F} \\ar[r] &", "\\mathcal{G} \\ar[r] &", "\\mathcal{H} \\ar[r] & 0", "}", "$$", "The bottom row is exact and the vertical arrows $a, b, c$", "have parasitic kernel and cokernels by part (2). It follows that", "$\\mathcal{E}$ is parasitic: in the quotient category of", "$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})/\\text{Parasitic}$", "(see Homology, Lemma \\ref{homology-lemma-serre-subcategory-is-kernel} and ", "Lemma \\ref{lemma-parasitic})", "we see that $a, b, c$ are isomorphisms and that the top row becomes", "exact. As it is also quasi-coherent, we conclude", "that $\\mathcal{E}$ is zero because $\\mathcal{E} = Q(\\mathcal{E}) = 0$ by", "part (1)." ], "refs": [ "stacks-cohomology-lemma-adjoint", "stacks-cohomology-lemma-adjoint", "stacks-cohomology-lemma-exact-sequence-quasi-coherent-parasitic-cohomology", "homology-lemma-serre-subcategory-is-kernel", "stacks-cohomology-lemma-parasitic" ], "ref_ids": [ 4160, 4160, 4159, 12048, 4157 ] } ], "ref_ids": [ 4160 ] }, { "id": 4162, "type": "theorem", "label": "stacks-cohomology-lemma-leray", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-leray", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$", "be a quasi-compact and quasi-separated morphism of algebraic stacks.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $\\mathcal{X}$. Then", "there exists a spectral sequence with $E_2$-page", "$$", "E_2^{p, q} = H^p(\\mathcal{Y}, R^qf_{\\QCoh, *}\\mathcal{F})", "$$", "converging to $H^{p + q}(\\mathcal{X}, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "By Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-Leray}", "the Leray spectral sequence with", "$$", "E_2^{p, q} = H^p(\\mathcal{Y}, R^qf_*\\mathcal{F})", "$$", "converges to $H^{p + q}(\\mathcal{X}, \\mathcal{F})$. The kernel and cokernel", "of the adjunction map", "$$", "R^qf_{\\QCoh, *}\\mathcal{F} \\longrightarrow R^qf_*\\mathcal{F}", "$$", "are parasitic modules on $\\mathcal{Y}$", "(Lemma \\ref{lemma-adjoint-kernel-parasitic})", "hence have vanishing cohomology", "(Lemma \\ref{lemma-pushforward-parasitic}).", "It follows formally that", "$H^p(\\mathcal{Y}, R^qf_{\\QCoh, *}\\mathcal{F}) =", "H^p(\\mathcal{Y}, R^qf_*\\mathcal{F})$ and we win." ], "refs": [ "sites-cohomology-lemma-Leray", "stacks-cohomology-lemma-adjoint-kernel-parasitic", "stacks-cohomology-lemma-pushforward-parasitic" ], "ref_ids": [ 4220, 4161, 4158 ] } ], "ref_ids": [] }, { "id": 4163, "type": "theorem", "label": "stacks-cohomology-lemma-relative-leray", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-relative-leray", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Y} \\to \\mathcal{Z}$", "be quasi-compact and quasi-separated morphisms of algebraic stacks.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $\\mathcal{X}$. Then", "there exists a spectral sequence with $E_2$-page", "$$", "E_2^{p, q} = R^pg_{\\QCoh, *}(R^qf_{\\QCoh, *}\\mathcal{F})", "$$", "converging to $R^{p + q}(g \\circ f)_{\\QCoh, *}\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "By Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-relative-Leray}", "the Leray spectral sequence with", "$$", "E_2^{p, q} = R^pg_*(R^qf_*\\mathcal{F})", "$$", "converges to $R^{p + q}(g \\circ f)_*\\mathcal{F}$. By the results of", "Proposition \\ref{proposition-lcq-flat-base-change}", "all the terms of this spectral sequence are objects of", "$\\mathcal{M}_\\mathcal{Z}$. Applying the exact functor", "$Q_\\mathcal{Z} : \\mathcal{M}_\\mathcal{Z} \\to", "\\QCoh(\\mathcal{O}_\\mathcal{Z})$ we obtain a spectral sequence in", "$\\QCoh(\\mathcal{O}_\\mathcal{Z})$ covering to", "$R^{p + q}(g \\circ f)_{\\QCoh, *}\\mathcal{F}$. Hence", "the result follows if we can show that", "$$", "Q_\\mathcal{Z}(R^pg_*(R^qf_*\\mathcal{F})) =", "Q_\\mathcal{Z}(R^pg_*(Q_\\mathcal{X}(R^qf_*\\mathcal{F}))", "$$", "This follows from the fact that the kernel and cokernel of the map", "$$", "Q_\\mathcal{X}(R^qf_*\\mathcal{F}) \\longrightarrow R^qf_*\\mathcal{F}", "$$", "are parasitic (Lemma \\ref{lemma-adjoint-kernel-parasitic}) and that", "$R^pg_*$ transforms parasitic modules into parasitic modules", "(Lemma \\ref{lemma-pushforward-parasitic})." ], "refs": [ "sites-cohomology-lemma-relative-Leray", "stacks-cohomology-proposition-lcq-flat-base-change", "stacks-cohomology-lemma-adjoint-kernel-parasitic", "stacks-cohomology-lemma-pushforward-parasitic" ], "ref_ids": [ 4222, 4173, 4161, 4158 ] } ], "ref_ids": [] }, { "id": 4164, "type": "theorem", "label": "stacks-cohomology-lemma-lisse-etale", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-lisse-etale", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "\\begin{enumerate}", "\\item The inclusion functor", "$\\mathcal{X}_{lisse,\\etale} \\to \\mathcal{X}_\\etale$", "is fully faithful, continuous and cocontinuous. It follows that", "\\begin{enumerate}", "\\item there is a morphism of topoi", "$$", "g :", "\\Sh(\\mathcal{X}_{lisse,\\etale})", "\\longrightarrow", "\\Sh(\\mathcal{X}_\\etale)", "$$", "with $g^{-1}$ given by restriction,", "\\item the functor $g^{-1}$ has a left adjoint $g_!^{Sh}$ on sheaves of sets,", "\\item the adjunction maps $g^{-1}g_* \\to \\text{id}$ and", "$\\text{id} \\to g^{-1}g_!^{Sh}$ are isomorphisms,", "\\item the functor $g^{-1}$ has a left adjoint $g_!$ on abelian sheaves,", "\\item the adjunction map $\\text{id} \\to g^{-1}g_!$ is an isomorphism, and", "\\item we have $g^{-1}\\mathcal{O}_\\mathcal{X} =", "\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}$ hence $g$ induces a flat", "morphism of ringed topoi such that $g^{-1} = g^*$.", "\\end{enumerate}", "\\item The inclusion functor", "$\\mathcal{X}_{flat,fppf} \\to \\mathcal{X}_{fppf}$", "is fully faithful, continuous and cocontinuous. It follows that", "\\begin{enumerate}", "\\item there is a morphism of topoi", "$$", "g :", "\\Sh(\\mathcal{X}_{flat,fppf})", "\\longrightarrow", "\\Sh(\\mathcal{X}_{fppf})", "$$", "with $g^{-1}$ given by restriction,", "\\item the functor $g^{-1}$ has a left adjoint $g_!^{Sh}$ on sheaves of sets,", "\\item the adjunction maps $g^{-1}g_* \\to \\text{id}$ and", "$\\text{id} \\to g^{-1}g_!^{Sh}$ are isomorphisms,", "\\item the functor $g^{-1}$ has a left adjoint $g_!$ on abelian sheaves,", "\\item the adjunction map $\\text{id} \\to g^{-1}g_!$ is an isomorphism, and", "\\item we have $g^{-1}\\mathcal{O}_\\mathcal{X} =", "\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}$ hence $g$ induces a flat", "morphism of ringed topoi such that $g^{-1} = g^*$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In both cases it is immediate that the functor is fully faithful,", "continuous, and cocontinuous (see", "Sites, Definitions \\ref{sites-definition-continuous} and", "\\ref{sites-definition-cocontinuous}).", "Hence properties (a), (b), (c) follow from", "Sites, Lemmas \\ref{sites-lemma-when-shriek} and", "\\ref{sites-lemma-back-and-forth}.", "Parts (d), (e) follow from", "Modules on Sites, Lemmas \\ref{sites-modules-lemma-g-shriek-adjoint} and", "\\ref{sites-modules-lemma-back-and-forth}.", "Part (f) is immediate." ], "refs": [ "sites-definition-continuous", "sites-definition-cocontinuous", "sites-lemma-when-shriek", "sites-lemma-back-and-forth", "sites-modules-lemma-g-shriek-adjoint", "sites-modules-lemma-back-and-forth" ], "ref_ids": [ 8664, 8670, 8545, 8547, 14164, 14166 ] } ], "ref_ids": [] }, { "id": 4165, "type": "theorem", "label": "stacks-cohomology-lemma-lisse-etale-modules", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-lisse-etale-modules", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Notation as in", "Lemma \\ref{lemma-lisse-etale}.", "\\begin{enumerate}", "\\item There exists a functor", "$$", "g_! :", "\\textit{Mod}(\\mathcal{X}_{lisse,\\etale},", "\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})", "\\longrightarrow", "\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_{\\mathcal{X}})", "$$", "which is left adjoint to $g^*$. Moreover it agrees with the functor $g_!$", "on abelian sheaves and $g^*g_! = \\text{id}$.", "\\item There exists a functor", "$$", "g_! :", "\\textit{Mod}(\\mathcal{X}_{flat,fppf},", "\\mathcal{O}_{\\mathcal{X}_{flat,fppf}})", "\\longrightarrow", "\\textit{Mod}(\\mathcal{X}_{fppf}, \\mathcal{O}_{\\mathcal{X}})", "$$", "which is left adjoint to $g^*$. Moreover it agrees with the functor $g_!$", "on abelian sheaves and $g^*g_! = \\text{id}$.", "\\end{enumerate}" ], "refs": [ "stacks-cohomology-lemma-lisse-etale" ], "proofs": [ { "contents": [ "In both cases, the existence of the functor $g_!$ follows from", "Modules on Sites, Lemma \\ref{sites-modules-lemma-lower-shriek-modules}.", "To see that $g_!$ agrees with the functor on abelian sheaves we will", "show the maps Modules on Sites, Equation", "(\\ref{sites-modules-equation-compare-on-localizations})", "are isomorphisms.", "\\medskip\\noindent", "Lisse-\\'etale case. Let $x \\in \\Ob(\\mathcal{X}_{lisse,\\etale})$", "lying over a scheme $U$ with $x : U \\to \\mathcal{X}$ smooth.", "Consider the induced fully faithful functor", "$$", "g' :", "\\mathcal{X}_{lisse,\\etale}/x", "\\longrightarrow", "\\mathcal{X}_\\etale/x", "$$", "The right hand side is identified with $(\\Sch/U)_\\etale$ and the", "left hand side with the full subcategory of schemes $U'/U$ such that", "the composition $U' \\to U \\to \\mathcal{X}$ is smooth. Thus", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-compare-structure-sheaves}", "applies.", "\\medskip\\noindent", "Flat-fppf case. Let $x \\in \\Ob(\\mathcal{X}_{flat,fppf})$", "lying over a scheme $U$ with $x : U \\to \\mathcal{X}$ flat.", "Consider the induced fully faithful functor", "$$", "g' :", "\\mathcal{X}_{flat,fppf}/x", "\\longrightarrow", "\\mathcal{X}_{fppf}/x", "$$", "The right hand side is identified with $(\\Sch/U)_{fppf}$ and the", "left hand side with the full subcategory of schemes $U'/U$ such that", "the composition $U' \\to U \\to \\mathcal{X}$ is flat. Thus", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-compare-structure-sheaves}", "applies.", "\\medskip\\noindent", "In both cases the equality $g^*g_! = \\text{id}$ follows from", "$g^* = g^{-1}$ and the", "equality for abelian sheaves in Lemma \\ref{lemma-lisse-etale}." ], "refs": [ "sites-modules-lemma-lower-shriek-modules", "etale-cohomology-lemma-compare-structure-sheaves", "etale-cohomology-lemma-compare-structure-sheaves", "stacks-cohomology-lemma-lisse-etale" ], "ref_ids": [ 14262, 6468, 6468, 4164 ] } ], "ref_ids": [ 4164 ] }, { "id": 4166, "type": "theorem", "label": "stacks-cohomology-lemma-lisse-etale-structure-sheaf", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-lisse-etale-structure-sheaf", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Notation as in", "Lemmas \\ref{lemma-lisse-etale} and \\ref{lemma-lisse-etale-modules}.", "\\begin{enumerate}", "\\item We have $g_!\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}} =", "\\mathcal{O}_\\mathcal{X}$.", "\\item We have $g_!\\mathcal{O}_{\\mathcal{X}_{flat, fppf}} =", "\\mathcal{O}_\\mathcal{X}$.", "\\end{enumerate}" ], "refs": [ "stacks-cohomology-lemma-lisse-etale", "stacks-cohomology-lemma-lisse-etale-modules" ], "proofs": [ { "contents": [ "In this proof we write", "$\\mathcal{C} = \\mathcal{X}_\\etale$", "(resp.\\ $\\mathcal{C} = \\mathcal{X}_{fppf}$)", "and we denote", "$\\mathcal{C}' = \\mathcal{X}_{lisse,\\etale}$", "(resp.\\ $\\mathcal{C}' = \\mathcal{X}_{flat, fppf}$).", "Then $\\mathcal{C}'$ is a full subcategory of $\\mathcal{C}$.", "In this proof we will think of objects $V$ of $\\mathcal{C}$", "as schemes over $\\mathcal{X}$ and objects $U$ of $\\mathcal{C}'$", "as schemes smooth (resp.\\ flat) over $\\mathcal{X}$.", "Finally, we write $\\mathcal{O} = \\mathcal{O}_\\mathcal{X}$", "and $\\mathcal{O}' = \\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}$", "(resp.\\ $\\mathcal{O}' = \\mathcal{O}_{\\mathcal{X}_{flat,fppf}}$).", "In the notation above we have $\\mathcal{O}(V) = \\Gamma(V, \\mathcal{O}_V)$", "and $\\mathcal{O}'(U) = \\Gamma(U, \\mathcal{O}_U)$.", "Consider the $\\mathcal{O}$-module homomorphism", "$g_!\\mathcal{O}' \\to \\mathcal{O}$", "adjoint to the identification $\\mathcal{O}' = g^{-1}\\mathcal{O}$.", "\\medskip\\noindent", "Recall that $g_!\\mathcal{O}'$ is the sheaf associated to the presheaf", "$g_{p!}\\mathcal{O}'$ given by the rule", "$$", "V \\longmapsto \\colim_{V \\to U} \\mathcal{O}'(U)", "$$", "where the colimit is taken in the category of abelian groups", "(Modules on Sites, Definition \\ref{sites-modules-definition-g-shriek}).", "Below we will use frequently that if", "$$", "V \\to U \\to U'", "$$", "are morphisms and if $f' \\in \\mathcal{O}'(U')$ restricts to", "$f \\in \\mathcal{O}'(U)$, then $(V \\to U, f)$ and $(V \\to U', f')$", "define the same element of the colimit. Also,", "$g_!\\mathcal{O}' \\to \\mathcal{O}$ maps the element", "$(V \\to U, f)$ simply to the pullback of $f$ to $V$.", "\\medskip\\noindent", "To see that $g_!\\mathcal{O}' \\to \\mathcal{O}$ is surjective it", "suffices to show that $1 \\in \\Gamma(\\mathcal{C}, \\mathcal{O})$ is", "locally in the image. Choose an object $U$ of $\\mathcal{C}'$", "corresponding to a surjective smooth morphism $U \\to \\mathcal{X}$.", "Then viewing $U$ both as an object of $\\mathcal{C}'$ and $\\mathcal{C}$", "we see that $(U \\to U, 1)$ is an element of the colimit above which", "maps to $1 \\in \\mathcal{O}(U)$. Since $U$ surjects onto the final", "object of $\\Sh(\\mathcal{C})$ we conclude $g_!\\mathcal{O}' \\to \\mathcal{O}$", "is surjective.", "\\medskip\\noindent", "Suppose that $s \\in g_!\\mathcal{O}'(V)$ is a section", "mapping to zero in $\\mathcal{O}(V)$. To finish the proof we have to show", "that $s$ is zero. After replacing $V$ by the members", "of a covering we may assume $s$ is an element of the colimit", "$$", "\\colim_{V \\to U} \\mathcal{O}'(U)", "$$", "Say $s = \\sum (\\varphi_i, s_i)$ is a finite sum with", "$\\varphi_i : V \\to U_i$, $U_i$ smooth (resp.\\ flat) over $\\mathcal{X}$, and", "$s_i \\in \\Gamma(U_i, \\mathcal{O}_{U_i})$. Choose a scheme $W$ surjective", "\\'etale over the algebraic space", "$U = U_1 \\times_\\mathcal{X} \\ldots \\times_\\mathcal{X} U_n$.", "Note that $W$ is still smooth (resp.\\ flat) over $\\mathcal{X}$, i.e.,", "defines an object of $\\mathcal{C}'$. The fibre product", "$$", "V' = V \\times_{(\\varphi_1, \\ldots, \\varphi_n), U} W", "$$", "is surjective \\'etale over $V$, hence it suffices to show that $s$ maps", "to zero in $g_!\\mathcal{O}'(V')$. Note that the restriction", "$\\sum (\\varphi_i, s_i)|_{V'}$ corresponds to the sum of the pullbacks", "of the functions $s_i$ to $W$. In other words, we have reduced to the case", "of $(\\varphi, s)$ where $\\varphi : V \\to U$ is a morphism with $U$ in", "$\\mathcal{C}'$ and $s \\in \\mathcal{O}'(U)$ restricts to zero in", "$\\mathcal{O}(V)$. By the commutative diagram", "$$", "\\xymatrix{", "V \\ar[rr]_-{(\\varphi, 0)} \\ar[rrd]_\\varphi & & U \\times \\mathbf{A}^1 \\\\", "& & U \\ar[u]_{(\\text{id}, 0)}", "}", "$$", "we see that", "$((\\varphi, 0) : V \\to U \\times \\mathbf{A}^1, \\text{pr}_2^*x)$", "represents zero in the colimit above. Hence we may", "replace $U$ by $U \\times \\mathbf{A}^1$, $\\varphi$ by $(\\varphi, 0)$", "and $s$ by $\\text{pr}_1^*s + \\text{pr}_2^*x$. Thus we may assume that", "the vanishing locus $Z : s = 0$ in $U$ of $s$ is smooth (resp.\\ flat)", "over $\\mathcal{X}$. Then we see that $(V \\to Z, 0)$ and $(\\varphi, s)$", "have the same value in the colimit, i.e., we see that the element $s$", "is zero as desired." ], "refs": [ "sites-modules-definition-g-shriek" ], "ref_ids": [ 14285 ] } ], "ref_ids": [ 4164, 4165 ] }, { "id": 4167, "type": "theorem", "label": "stacks-cohomology-lemma-parasitic-in-terms-flat-fppf", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-parasitic-in-terms-flat-fppf", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{X}$-module", "with the flat base change property on $\\mathcal{X}_\\etale$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is parasitic, and", "\\item $g^*\\mathcal{F} = 0$ where", "$g : \\Sh(\\mathcal{X}_{lisse,\\etale}) \\to", "\\Sh(\\mathcal{X}_\\etale)$ is as in Lemma \\ref{lemma-lisse-etale}.", "\\end{enumerate}", "\\item Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{X}$-module on", "$\\mathcal{X}_{fppf}$. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is parasitic, and", "\\item $g^*\\mathcal{F} = 0$ where", "$g : \\Sh(\\mathcal{X}_{flat,fppf}) \\to \\Sh(\\mathcal{X}_{fppf})$", "is as in Lemma \\ref{lemma-lisse-etale}.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [ "stacks-cohomology-lemma-lisse-etale", "stacks-cohomology-lemma-lisse-etale" ], "proofs": [ { "contents": [ "Part (2) is immediate from the definitions (this is one of the advantages", "of the flat-fppf site over the lisse-\\'etale site). The implication", "(1)(a) $\\Rightarrow$ (1)(b) is immediate as well. To see (1)(b)", "$\\Rightarrow$ (1)(a) let $U$ be a scheme and let $x : U \\to \\mathcal{X}$", "be a surjective smooth morphism. Then $x$ is an object of the", "lisse-\\'etale site of $\\mathcal{X}$. Hence we see that (1)(b)", "implies that $\\mathcal{F}|_{U_\\etale} = 0$. Let $V \\to \\mathcal{X}$", "be an flat morphism where $V$ is a scheme. Set $W = U \\times_\\mathcal{X} V$", "and consider the diagram", "$$", "\\xymatrix{", "W \\ar[d]_p \\ar[r]_q & V \\ar[d] \\\\", "U \\ar[r] & \\mathcal{X}", "}", "$$", "Note that the projection $p : W \\to U$ is flat and the projection", "$q : W \\to V$ is smooth and surjective. This implies that $q_{small}^*$", "is a faithful functor on quasi-coherent modules. By assumption $\\mathcal{F}$", "has the flat base change property so that we obtain", "$p_{small}^*\\mathcal{F}|_{U_\\etale} \\cong", "q_{small}^*\\mathcal{F}|_{V_\\etale}$. Thus if $\\mathcal{F}$", "is in the kernel of $g^*$, then $\\mathcal{F}|_{V_\\etale} = 0$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 4164, 4164 ] }, { "id": 4168, "type": "theorem", "label": "stacks-cohomology-lemma-lisse-etale-functorial", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-lisse-etale-functorial", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "\\begin{enumerate}", "\\item If $f$ is smooth, then $f$ restricts to a continuous and cocontinuous", "functor", "$\\mathcal{X}_{lisse,\\etale} \\to \\mathcal{Y}_{lisse,\\etale}$", "which gives a morphism of ringed topoi fitting into the following", "commutative diagram", "$$", "\\xymatrix{", "\\Sh(\\mathcal{X}_{lisse,\\etale}) \\ar[r]_{g'} \\ar[d]_{f'} &", "\\Sh(\\mathcal{X}_\\etale) \\ar[d]^f \\\\", "\\Sh(\\mathcal{Y}_{lisse,\\etale}) \\ar[r]^g &", "\\Sh(\\mathcal{Y}_\\etale)", "}", "$$", "We have $f'_*(g')^{-1} = g^{-1}f_*$ and $g'_!(f')^{-1} = f^{-1}g_!$.", "\\item If $f$ is flat, then $f$ restricts to a continuous and cocontinuous", "functor", "$\\mathcal{X}_{flat,fppf} \\to \\mathcal{Y}_{flat,fppf}$", "which gives a morphism of ringed topoi fitting into the following", "commutative diagram", "$$", "\\xymatrix{", "\\Sh(\\mathcal{X}_{flat,fppf}) \\ar[r]_{g'} \\ar[d]_{f'} &", "\\Sh(\\mathcal{X}_{fppf}) \\ar[d]^f \\\\", "\\Sh(\\mathcal{Y}_{flat,fppf}) \\ar[r]^g &", "\\Sh(\\mathcal{Y}_{fppf})", "}", "$$", "We have $f'_*(g')^{-1} = g^{-1}f_*$ and $g'_!(f')^{-1} = f^{-1}g_!$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The initial statement comes from the fact that if $x \\in \\Ob(\\mathcal{X})$", "lies over a scheme $U$ such that $x : U \\to \\mathcal{X}$ is smooth", "(resp.\\ flat) and if $f$ is smooth (resp.\\ flat) then", "$f(x) : U \\to \\mathcal{Y}$ is smooth (resp.\\ flat), see", "Morphisms of Stacks, Lemmas \\ref{stacks-morphisms-lemma-composition-smooth} and", "\\ref{stacks-morphisms-lemma-composition-flat}. The induced functor", "$\\mathcal{X}_{lisse,\\etale} \\to \\mathcal{Y}_{lisse,\\etale}$", "(resp.\\ $\\mathcal{X}_{flat,fppf} \\to \\mathcal{Y}_{flat,fppf}$) is", "continuous and cocontinuous by our definition of coverings in these", "categories. Finally, the commutativity of the diagram is a consequence of the", "fact that the horizontal morphisms are given by the inclusion functors (see", "Lemma \\ref{lemma-lisse-etale}) and", "Sites, Lemma \\ref{sites-lemma-composition-cocontinuous}.", "\\medskip\\noindent", "To show that $f'_*(g')^{-1} = g^{-1}f_*$ let $\\mathcal{F}$ be a sheaf", "on $\\mathcal{X}_\\etale$ (resp.\\ $\\mathcal{X}_{fppf}$).", "There is a canonical pullback map", "$$", "g^{-1}f_*\\mathcal{F} \\longrightarrow f'_*(g')^{-1}\\mathcal{F}", "$$", "see Sites, Section \\ref{sites-section-pullback}.", "We claim this map is an isomorphism.", "To prove this pick an object $y$ of $\\mathcal{Y}_{lisse,\\etale}$", "(resp.\\ $\\mathcal{Y}_{flat,fppf}$). Say $y$ lies over the scheme $V$", "such that $y : V \\to \\mathcal{Y}$ is smooth (resp.\\ flat). Since", "$g^{-1}$ is the restriction we find that", "$$", "\\left(g^{-1}f_*\\mathcal{F}\\right)(y) =", "\\Gamma(V \\times_{y, \\mathcal{Y}} \\mathcal{X},\\ \\text{pr}^{-1}\\mathcal{F})", "$$", "by Sheaves on Stacks, Equation (\\ref{stacks-sheaves-equation-pushforward}).", "Let", "$(V \\times_{y, \\mathcal{Y}} \\mathcal{X})' \\subset", "V \\times_{y, \\mathcal{Y}} \\mathcal{X}$", "be the full subcategory consisting of objects", "$z : W \\to V \\times_{y, \\mathcal{Y}} \\mathcal{X}$ such that the induced", "morphism $W \\to \\mathcal{X}$ is smooth (resp.\\ flat). Denote", "$$", "\\text{pr}' :", "(V \\times_{y, \\mathcal{Y}} \\mathcal{X})'", "\\longrightarrow", "\\mathcal{X}_{lisse,\\etale}", "\\ (\\text{resp. }\\mathcal{X}_{flat,fppf})", "$$", "the restriction of the functor $\\text{pr}$ used in the formula above.", "Exactly the same argument that proves", "Sheaves on Stacks, Equation (\\ref{stacks-sheaves-equation-pushforward})", "shows that for any sheaf $\\mathcal{H}$ on $\\mathcal{X}_{lisse,\\etale}$", "(resp.\\ $\\mathcal{X}_{flat,fppf}$) we have", "\\begin{equation}", "\\label{equation-pushforward-lisse-etale}", "f'_*\\mathcal{H}(y) =", "\\Gamma((V \\times_{y, \\mathcal{Y}} \\mathcal{X})',", "\\ (\\text{pr}')^{-1}\\mathcal{H})", "\\end{equation}", "Since $(g')^{-1}$ is restriction we see that", "$$", "\\left(f'_*(g')^{-1}\\mathcal{F}\\right)(y) =", "\\Gamma((V \\times_{y, \\mathcal{Y}} \\mathcal{X})',", "\\ \\text{pr}^{-1}\\mathcal{F}|_{(V \\times_{y, \\mathcal{Y}} \\mathcal{X})'})", "$$", "By", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-cohomology-on-subcategory}", "we see that", "$$", "\\Gamma((V \\times_{y, \\mathcal{Y}} \\mathcal{X})',", "\\ \\text{pr}^{-1}\\mathcal{F}|_{(V \\times_{y, \\mathcal{Y}} \\mathcal{X})'})", "=", "\\Gamma(V \\times_{y, \\mathcal{Y}} \\mathcal{X},\\ \\text{pr}^{-1}\\mathcal{F})", "$$", "are equal as desired; although we omit the verification of the assumptions", "of the lemma we note that the fact that $V \\to \\mathcal{Y}$ is smooth", "(resp.\\ flat) is used to verify the second condition.", "\\medskip\\noindent", "Finally, the equality $g'_!(f')^{-1} = f^{-1}g_!$ follows formally from", "the equality $f'_*(g')^{-1} = g^{-1}f_*$ by the adjointness of", "$f^{-1}$ and $f_*$, the adjointness of $g_!$ and $g^{-1}$, and their", "``primed'' versions." ], "refs": [ "stacks-morphisms-lemma-composition-smooth", "stacks-morphisms-lemma-composition-flat", "stacks-cohomology-lemma-lisse-etale", "sites-lemma-composition-cocontinuous", "stacks-sheaves-lemma-cohomology-on-subcategory" ], "ref_ids": [ 7539, 7494, 4164, 8544, 11616 ] } ], "ref_ids": [] }, { "id": 4169, "type": "theorem", "label": "stacks-cohomology-lemma-check-qc-on-etale-covering", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-check-qc-on-etale-covering", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "\\begin{enumerate}", "\\item Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of smooth", "morphisms of algebraic stacks with", "$|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules", "on $\\mathcal{X}_\\etale$. If each $f_j^{-1}\\mathcal{F}$", "is quasi-coherent, then so is $\\mathcal{F}$.", "\\item Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of flat and", "locally finitely presented morphisms of algebraic stacks with", "$|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules", "on $\\mathcal{X}_{fppf}$. If each $f_j^{-1}\\mathcal{F}$", "is quasi-coherent, then so is $\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). We may replace each of the algebraic stacks $\\mathcal{X}_j$", "by a scheme $U_j$ (using that any algebraic stack has a smooth covering by", "a scheme and that compositions of smooth morphisms are smooth, see", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-composition-smooth}).", "The pullback of $\\mathcal{F}$ to $(\\Sch/U_j)_\\etale$ is still", "quasi-coherent, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-local-pullback}.", "Then $f = \\coprod f_j : U = \\coprod U_j \\to \\mathcal{X}$ is a smooth surjective", "morphism. Let $x : V \\to \\mathcal{X}$ be an object of $\\mathcal{X}$. By", "Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}", "there exists an \\'etale covering $\\{x_i \\to x\\}_{i \\in I}$", "such that each $x_i$ lifts to an object $u_i$ of $(\\Sch/U)_\\etale$.", "This just means that $x_i$ lives over a scheme $V_i$, that", "$\\{V_i \\to V\\}$ is an \\'etale covering, and that $x_i$ comes from", "a morphism $u_i : V_i \\to U$. Then", "$x_i^*\\mathcal{F} = u_i^*f^*\\mathcal{F}$ is quasi-coherent.", "This implies that $x^*\\mathcal{F}$ on $(\\Sch/V)_\\etale$", "is quasi-coherent, for example by", "Modules on Sites, Lemma \\ref{sites-modules-lemma-local-final-object}.", "By Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-characterize-quasi-coherent-bis}", "we see that $x^*\\mathcal{F}$ is an fppf sheaf and since $x$", "was arbitrary we see that $\\mathcal{F}$ is a sheaf in the", "fppf topology. Applying Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-characterize-quasi-coherent}", "we see that $\\mathcal{F}$ is quasi-coherent.", "\\medskip\\noindent", "Proof of (2). This is proved using exactly the same argument, which we fully", "write out here. We may replace each of the algebraic stacks $\\mathcal{X}_j$", "by a scheme $U_j$ (using that any algebraic stack has a smooth covering by", "a scheme and that flat and locally finite presented morphisms are preserved", "by composition, see Morphisms of Stacks, Lemmas", "\\ref{stacks-morphisms-lemma-composition-flat} and", "\\ref{stacks-morphisms-lemma-composition-finite-presentation}).", "The pullback of $\\mathcal{F}$ to $(\\Sch/U_j)_\\etale$ is still", "locally quasi-coherent, see", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-pullback-quasi-coherent}.", "Then $f = \\coprod f_j : U = \\coprod U_j \\to \\mathcal{X}$ is a surjective,", "flat, and locally finitely presented morphism. Let", "$x : V \\to \\mathcal{X}$ be an object of $\\mathcal{X}$. By", "Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}", "there exists an fppf covering $\\{x_i \\to x\\}_{i \\in I}$", "such that each $x_i$ lifts to an object $u_i$ of $(\\Sch/U)_\\etale$.", "This just means that $x_i$ lives over a scheme $V_i$, that", "$\\{V_i \\to V\\}$ is an fppf covering, and that $x_i$ comes from", "a morphism $u_i : V_i \\to U$. Then", "$x_i^*\\mathcal{F} = u_i^*f^*\\mathcal{F}$ is quasi-coherent.", "This implies that $x^*\\mathcal{F}$ on $(\\Sch/V)_\\etale$", "is quasi-coherent, for example by", "Modules on Sites, Lemma \\ref{sites-modules-lemma-local-final-object}.", "By Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-characterize-quasi-coherent}", "we see that $\\mathcal{F}$ is quasi-coherent." ], "refs": [ "stacks-morphisms-lemma-composition-smooth", "sites-modules-lemma-local-pullback", "stacks-sheaves-lemma-surjective-flat-locally-finite-presentation", "sites-modules-lemma-local-final-object", "stacks-sheaves-lemma-characterize-quasi-coherent-bis", "stacks-sheaves-lemma-characterize-quasi-coherent", "stacks-morphisms-lemma-composition-flat", "stacks-morphisms-lemma-composition-finite-presentation", "stacks-sheaves-lemma-pullback-quasi-coherent", "stacks-sheaves-lemma-surjective-flat-locally-finite-presentation", "sites-modules-lemma-local-final-object", "stacks-sheaves-lemma-characterize-quasi-coherent" ], "ref_ids": [ 7539, 14186, 11606, 14185, 11582, 11581, 7494, 7500, 11580, 11606, 14185, 11581 ] } ], "ref_ids": [] }, { "id": 4170, "type": "theorem", "label": "stacks-cohomology-lemma-shriek-quasi-coherent", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-shriek-quasi-coherent", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Notation as in", "Lemma \\ref{lemma-lisse-etale}.", "\\begin{enumerate}", "\\item Let $\\mathcal{H}$ be a quasi-coherent", "$\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}$-module ", "on the lisse-\\'etale site of $\\mathcal{X}$. Then $g_!\\mathcal{H}$ is a", "quasi-coherent module on $\\mathcal{X}$.", "\\item Let $\\mathcal{H}$ be a quasi-coherent", "$\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}$-module ", "on the flat-fppf site of $\\mathcal{X}$. Then $g_!\\mathcal{H}$ is a", "quasi-coherent module on $\\mathcal{X}$.", "\\end{enumerate}" ], "refs": [ "stacks-cohomology-lemma-lisse-etale" ], "proofs": [ { "contents": [ "Pick a scheme $U$ and a surjective smooth morphism $x : U \\to \\mathcal{X}$.", "By", "Modules on Sites, Definition \\ref{sites-modules-definition-site-local}", "there exists an \\'etale (resp.\\ fppf) covering", "$\\{U_i \\to U\\}_{i \\in I}$ such that each pullback $f_i^{-1}\\mathcal{H}$", "has a global presentation (see", "Modules on Sites, Definition \\ref{sites-modules-definition-global}).", "Here $f_i : U_i \\to \\mathcal{X}$ is the composition", "$U_i \\to U \\to \\mathcal{X}$ which is a morphism of algebraic stacks.", "(Recall that the pullback ``is'' the restriction to $\\mathcal{X}/f_i$, see", "Sheaves on Stacks, Definition \\ref{stacks-sheaves-definition-pullback}", "and the discussion following.) Since each $f_i$ is smooth (resp.\\ flat) by", "Lemma \\ref{lemma-lisse-etale-functorial}", "we see that $f_i^{-1}g_!\\mathcal{H} = g_{i, !}(f'_i)^{-1}\\mathcal{H}$.", "Using Lemma \\ref{lemma-check-qc-on-etale-covering}", "we reduce the statement of the lemma to the case where $\\mathcal{H}$", "has a global presentation. Say we have", "$$", "\\bigoplus\\nolimits_{j \\in J} \\mathcal{O} \\longrightarrow", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{O} \\longrightarrow", "\\mathcal{H} \\longrightarrow 0", "$$", "of $\\mathcal{O}$-modules where", "$\\mathcal{O} = \\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}$", "(resp.\\ $\\mathcal{O} = \\mathcal{O}_{\\mathcal{X}_{flat,fppf}}$).", "Since $g_!$ commutes with arbitrary colimits (as a left adjoint functor, see", "Lemma \\ref{lemma-lisse-etale-modules} and", "Categories, Lemma \\ref{categories-lemma-adjoint-exact})", "we conclude that there exists an exact sequence", "$$", "\\bigoplus\\nolimits_{j \\in J} g_!\\mathcal{O} \\longrightarrow", "\\bigoplus\\nolimits_{i \\in I} g_!\\mathcal{O} \\longrightarrow", "g_!\\mathcal{H} \\longrightarrow 0", "$$", "Lemma \\ref{lemma-lisse-etale-structure-sheaf}", "shows that $g_!\\mathcal{O} = \\mathcal{O}_\\mathcal{X}$.", "In case (2) we are done. In case (1) we apply", "Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-characterize-quasi-coherent-bis}", "to conclude." ], "refs": [ "sites-modules-definition-site-local", "sites-modules-definition-global", "stacks-sheaves-definition-pullback", "stacks-cohomology-lemma-lisse-etale-functorial", "stacks-cohomology-lemma-check-qc-on-etale-covering", "stacks-cohomology-lemma-lisse-etale-modules", "categories-lemma-adjoint-exact", "stacks-cohomology-lemma-lisse-etale-structure-sheaf", "stacks-sheaves-lemma-characterize-quasi-coherent-bis" ], "ref_ids": [ 14289, 14286, 11628, 4168, 4169, 4165, 12249, 4166, 11582 ] } ], "ref_ids": [ 4164 ] }, { "id": 4171, "type": "theorem", "label": "stacks-cohomology-lemma-quasi-coherent", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-quasi-coherent", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{M}_\\mathcal{X}$", "be the category of locally quasi-coherent $\\mathcal{O}_\\mathcal{X}$-modules", "with the flat base change property.", "\\begin{enumerate}", "\\item With $g$ as in Lemma \\ref{lemma-lisse-etale}", "for the lisse-\\'etale site we have", "\\begin{enumerate}", "\\item the functors $g^{-1}$ and $g_!$ define mutually inverse functors", "$$", "\\xymatrix{", "\\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar@<1ex>[r]^-{g^{-1}} &", "\\QCoh(\\mathcal{X}_{lisse,\\etale},", "\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}) \\ar@<1ex>[l]^-{g_!}", "}", "$$", "\\item if $\\mathcal{F}$ is in $\\mathcal{M}_\\mathcal{X}$", "then $g^{-1}\\mathcal{F}$ is in", "$\\QCoh(\\mathcal{X}_{lisse,\\etale},", "\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$ and", "\\item $Q(\\mathcal{F}) = g_!g^{-1}\\mathcal{F}$ where $Q$ is as in", "Lemma \\ref{lemma-adjoint}.", "\\end{enumerate}", "\\item With $g$ as in Lemma \\ref{lemma-lisse-etale}", "for the flat-fppf site we have", "\\begin{enumerate}", "\\item the functors $g^{-1}$ and $g_!$ define mutually inverse functors", "$$", "\\xymatrix{", "\\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar@<1ex>[r]^-{g^{-1}} &", "\\QCoh(\\mathcal{X}_{flat,fppf},", "\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}) \\ar@<1ex>[l]^-{g_!}", "}", "$$", "\\item if $\\mathcal{F}$ is in $\\mathcal{M}_\\mathcal{X}$", "then $g^{-1}\\mathcal{F}$ is in", "$\\QCoh(\\mathcal{X}_{flat,fppf}, \\mathcal{O}_{\\mathcal{X}_{flat,fppf}})$", "and", "\\item $Q(\\mathcal{F}) = g_!g^{-1}\\mathcal{F}$ where $Q$ is as in", "Lemma \\ref{lemma-adjoint}.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [ "stacks-cohomology-lemma-lisse-etale", "stacks-cohomology-lemma-adjoint", "stacks-cohomology-lemma-lisse-etale", "stacks-cohomology-lemma-adjoint" ], "proofs": [ { "contents": [ "Pullback by any morphism of ringed topoi preserves categories of quasi-coherent", "modules, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-local-pullback}.", "Hence $g^{-1}$ preserves the categories of quasi-coherent modules;", "here we use that", "$\\QCoh(\\mathcal{O}_\\mathcal{X}) =", "\\QCoh(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "by Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-characterize-quasi-coherent-bis}.", "The same is true for $g_!$ by", "Lemma \\ref{lemma-shriek-quasi-coherent}.", "We know that $\\mathcal{H} \\to g^{-1}g_!\\mathcal{H}$ is an isomorphism by", "Lemma \\ref{lemma-lisse-etale}.", "Conversely, if $\\mathcal{F}$ is in $\\QCoh(\\mathcal{O}_\\mathcal{X})$", "then the map $g_!g^{-1}\\mathcal{F} \\to \\mathcal{F}$ is a map of quasi-coherent", "modules on $\\mathcal{X}$ whose restriction to any scheme smooth over", "$\\mathcal{X}$ is an isomorphism. Then the discussion in", "Sheaves on Stacks, Sections", "\\ref{stacks-sheaves-section-quasi-coherent-presentation} and", "\\ref{stacks-sheaves-section-quasi-coherent-algebraic-stacks}", "(comparing with quasi-coherent modules on presentations)", "shows it is an isomorphism. This proves (1)(a) and (2)(a).", "\\medskip\\noindent", "Let $\\mathcal{F}$ be an object of $\\mathcal{M}_\\mathcal{X}$. By", "Lemma \\ref{lemma-adjoint-kernel-parasitic}", "the kernel and cokernel of the map", "$Q(\\mathcal{F}) \\to \\mathcal{F}$ are parasitic. Hence by", "Lemma \\ref{lemma-parasitic-in-terms-flat-fppf}", "and since $g^* = g^{-1}$ is exact, we conclude", "$g^*Q(\\mathcal{F}) \\to g^*\\mathcal{F}$ is an isomorphism. Thus", "$g^*\\mathcal{F}$ is quasi-coherent. This proves (1)(b) and (2)(b).", "Finally, (1)(c) and (2)(c) follow because", "$g_!g^*Q(\\mathcal{F}) \\to Q(\\mathcal{F})$ is an isomorphism by", "our arguments above." ], "refs": [ "sites-modules-lemma-local-pullback", "stacks-sheaves-lemma-characterize-quasi-coherent-bis", "stacks-cohomology-lemma-shriek-quasi-coherent", "stacks-cohomology-lemma-lisse-etale", "stacks-cohomology-lemma-adjoint-kernel-parasitic", "stacks-cohomology-lemma-parasitic-in-terms-flat-fppf" ], "ref_ids": [ 14186, 11582, 4170, 4164, 4161, 4167 ] } ], "ref_ids": [ 4164, 4160, 4164, 4160 ] }, { "id": 4172, "type": "theorem", "label": "stacks-cohomology-lemma-quasi-coherent-weak-serre", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-lemma-quasi-coherent-weak-serre", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "\\begin{enumerate}", "\\item $\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$", "is a weak Serre subcategory of", "$\\textit{Mod}(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$.", "\\item $\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat,fppf}})$", "is a weak Serre subcategory of", "$\\textit{Mod}(\\mathcal{O}_{\\mathcal{X}_{flat,fppf}})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will verify conditions (1), (2), (3), (4) of", "Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}.", "Since $0$ is a quasi-coherent module on any ringed site we see that (1)", "holds. By definition $\\QCoh(\\mathcal{O})$", "is a strictly full subcategory $\\textit{Mod}(\\mathcal{O})$, so (2) holds.", "Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a morphism of quasi-coherent", "modules on $\\mathcal{X}_{lisse,\\etale}$ or $\\mathcal{X}_{flat,fppf}$.", "We have $g^*g_!\\mathcal{F} = \\mathcal{F}$ and similarly for", "$\\mathcal{G}$ and $\\varphi$, see Lemma \\ref{lemma-lisse-etale-modules}.", "By Lemma \\ref{lemma-shriek-quasi-coherent}", "we see that $g_!\\mathcal{F}$ and $g_!\\mathcal{G}$ are quasi-coherent", "$\\mathcal{O}_\\mathcal{X}$-modules. Hence we see that", "$\\Ker(g_!\\varphi)$ and $\\Coker(g_!\\varphi)$ are quasi-coherent", "modules on $\\mathcal{X}$. Since $g^*$ is exact (see", "Lemma \\ref{lemma-lisse-etale}) we see that", "$g^*\\Ker(g_!\\varphi) = \\Ker(g^*g_!\\varphi) = \\Ker(\\varphi)$", "and", "$g^*\\Coker(g_!\\varphi) = \\Coker(g^*g_!\\varphi) =", "\\Coker(\\varphi)$", "are quasi-coherent too (see Lemma \\ref{lemma-quasi-coherent}).", "This proves (3).", "\\medskip\\noindent", "Finally, suppose that", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{E} \\to \\mathcal{G} \\to 0", "$$", "is an extension of $\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}$-modules", "(resp.\\ $\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}$-modules) with $\\mathcal{F}$", "and $\\mathcal{G}$ quasi-coherent. We have to show that $\\mathcal{E}$", "is quasi-coherent on $\\mathcal{X}_{lisse,\\etale}$", "(resp.\\ $\\mathcal{X}_{flat,fppf}$). We strongly urge the reader to write", "out what this means on a napkin and prove it him/herself rather than", "reading the somewhat labyrinthine proof that follows.", "By Lemma \\ref{lemma-quasi-coherent}", "this is true if and only if $g_!\\mathcal{E}$ is quasi-coherent.", "By Lemmas \\ref{lemma-check-qc-on-etale-covering} and", "Lemma \\ref{lemma-lisse-etale-functorial}", "we may check this after replacing $\\mathcal{X}$ by a smooth", "(resp.\\ fppf) cover. Choose a scheme $U$ and a smooth surjective", "morphism $U \\to \\mathcal{X}$. By definition there exists an \\'etale", "(resp.\\ fppf) covering $\\{U_i \\to U\\}_i$ such that $\\mathcal{G}$", "has a global presentation over each $U_i$. Replacing $\\mathcal{X}$", "by $U_i$ (which is permissible by the discussion above)", "we may assume that the site $\\mathcal{X}_{lisse,\\etale}$", "(resp.\\ $\\mathcal{X}_{flat,fppf}$) has a final object $U$", "(in other words $\\mathcal{X}$ is representable by the scheme $U$)", "and that $\\mathcal{G}$ has a global presentation", "$$", "\\bigoplus\\nolimits_{j \\in J} \\mathcal{O} \\longrightarrow", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{O} \\longrightarrow", "\\mathcal{G} \\longrightarrow 0", "$$", "of $\\mathcal{O}$-modules where", "$\\mathcal{O} = \\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}$", "(resp.\\ $\\mathcal{O} = \\mathcal{O}_{\\mathcal{X}_{flat,fppf}}$).", "Let $\\mathcal{E}'$ be the pullback of $\\mathcal{E}$ via the map", "$\\bigoplus\\nolimits_{i \\in I} \\mathcal{O} \\to \\mathcal{G}$.", "Then we see that $\\mathcal{E}$ is the cokernel of a map", "$\\bigoplus\\nolimits_{j \\in J} \\mathcal{O} \\to \\mathcal{E}'$", "hence by property (3) which we proved above, it suffices to prove", "that $\\mathcal{E}'$ is quasi-coherent. Consider the exact sequence", "$$", "L_1g_!\\left(\\bigoplus\\nolimits_{i \\in I}\\mathcal{O}\\right) \\to", "g_!\\mathcal{F} \\to", "g_!\\mathcal{E}' \\to", "g_!\\left(\\bigoplus\\nolimits_{i \\in I}\\mathcal{O}\\right) \\to 0", "$$", "where $L_1g_!$ is the first left derived functor of", "$g_! : \\textit{Mod}(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}) \\to", "\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "(resp.\\ $g_! :", "\\textit{Mod}(\\mathcal{X}_{flat,fppf}, \\mathcal{O}_{\\mathcal{X}_{flat,fppf}})", "\\to \\textit{Mod}(\\mathcal{X}_{fppf}, \\mathcal{O}_{\\mathcal{X}})$).", "This derived functor exists by Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-existence-derived-lower-shriek}.", "Moreover, since $\\mathcal{O} = j_{U!}\\mathcal{O}_U$ we have", "$Lg_!\\mathcal{O} = g_!\\mathcal{O} = \\mathcal{O}_\\mathcal{X}$", "also by Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-existence-derived-lower-shriek}.", "Thus the left hand term vanishes and we obtain a short exact sequence", "$$", "0 \\to", "g_!\\mathcal{F} \\to", "g_!\\mathcal{E}' \\to", "\\bigoplus\\nolimits_{i \\in I}\\mathcal{O}_\\mathcal{X} \\to 0", "$$", "By Proposition \\ref{proposition-lcq-flat-base-change}", "it follows that $g_!\\mathcal{E}'$ is locally quasi-coherent with the", "flat base change property. Finally, Lemma \\ref{lemma-quasi-coherent}", "implies that $\\mathcal{E}' = g^{-1}g_!\\mathcal{E}'$ is quasi-coherent", "as desired." ], "refs": [ "homology-lemma-characterize-weak-serre-subcategory", "stacks-cohomology-lemma-lisse-etale-modules", "stacks-cohomology-lemma-shriek-quasi-coherent", "stacks-cohomology-lemma-lisse-etale", "stacks-cohomology-lemma-quasi-coherent", "stacks-cohomology-lemma-quasi-coherent", "stacks-cohomology-lemma-check-qc-on-etale-covering", "stacks-cohomology-lemma-lisse-etale-functorial", "sites-cohomology-lemma-existence-derived-lower-shriek", "sites-cohomology-lemma-existence-derived-lower-shriek", "stacks-cohomology-proposition-lcq-flat-base-change", "stacks-cohomology-lemma-quasi-coherent" ], "ref_ids": [ 12046, 4165, 4170, 4164, 4171, 4171, 4169, 4168, 4337, 4337, 4173, 4171 ] } ], "ref_ids": [] }, { "id": 4173, "type": "theorem", "label": "stacks-cohomology-proposition-lcq-flat-base-change", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-proposition-lcq-flat-base-change", "contents": [ "Summary of results on locally quasi-coherent modules having the flat", "base change property.", "\\begin{enumerate}", "\\item Let $\\mathcal{X}$ be an algebraic stack.", "If $\\mathcal{F}$ is an object of", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "which is locally quasi-coherent and has the flat base change property,", "then $\\mathcal{F}$ is a sheaf for the fppf topology, i.e., it is", "an object of $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.", "\\item The category of modules which are locally quasi-coherent", "and have the flat base change property is a weak Serre subcategory", "$\\mathcal{M}_\\mathcal{X}$ of both $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$", "and $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.", "\\item Pullback $f^*$ along any morphism of algebraic stacks", "$f : \\mathcal{X} \\to \\mathcal{Y}$ induces a functor", "$f^* : \\mathcal{M}_\\mathcal{Y} \\to \\mathcal{M}_\\mathcal{X}$.", "\\item If $f : \\mathcal{X} \\to \\mathcal{Y}$ is a", "quasi-compact and quasi-separated morphism of algebraic stacks", "and $\\mathcal{F}$ is an object of $\\mathcal{M}_\\mathcal{X}$, then", "\\begin{enumerate}", "\\item the derived direct image $Rf_*\\mathcal{F}$ and the higher direct", "images $R^if_*\\mathcal{F}$ can be computed in either the \\'etale or the", "fppf topology with the same result, and", "\\item each $R^if_*\\mathcal{F}$ is an object of $\\mathcal{M}_\\mathcal{Y}$.", "\\end{enumerate}", "\\item The category $\\mathcal{M}_\\mathcal{X}$ has colimits and they agree", "with colimits in", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "as well as in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is", "Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-lqc-flat-base-change-fppf-sheaf}.", "\\medskip\\noindent", "Part (2) for the embedding $\\mathcal{M}_\\mathcal{X} \\subset", "\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "we have seen in the proof of", "Lemma \\ref{lemma-flat-comparison}.", "Let us prove (2) for the embedding", "$\\mathcal{M}_\\mathcal{X} \\subset \\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism between", "objects of $\\mathcal{M}_\\mathcal{X}$. Since $\\Ker(\\varphi)$", "is the same whether computed in the \\'etale or the fppf", "topology, we see that $\\Ker(\\varphi)$ is in", "$\\mathcal{M}_\\mathcal{X}$ by the \\'etale case. On the other hand,", "the cokernel computed in the fppf topology is the fppf sheafification", "of the cokernel computed in the \\'etale topology. However, this", "\\'etale cokernel is in $\\mathcal{M}_\\mathcal{X}$ hence an fppf sheaf", "by (1) and we see that the cokernel is in $\\mathcal{M}_\\mathcal{X}$.", "Finally, suppose that", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0", "$$", "is an exact sequence in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$", "(i.e., using the fppf topology) with $\\mathcal{F}_1$, $\\mathcal{F}_2$", "in $\\mathcal{M}_\\mathcal{X}$. In order to show that $\\mathcal{F}_2$", "is an object of $\\mathcal{M}_\\mathcal{X}$ it suffices to show that", "the sequence is also exact in the \\'etale topology. To do this it", "suffices to show that any element of $H^1_{fppf}(x, \\mathcal{F}_1)$", "becomes zero on the members of an \\'etale covering of $x$ (for any", "object $x$ of $\\mathcal{X}$). This is true because", "$H^1_{fppf}(x, \\mathcal{F}_1) = H^1_\\etale(x, \\mathcal{F}_1)$ by", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-compare-fppf-etale}", "and because of locality of cohomology, see", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-kill-cohomology-class-on-covering}.", "This proves (2).", "\\medskip\\noindent", "Part (3) follows from", "Lemma \\ref{lemma-check-flat-comparison-on-etale-covering}", "and", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-pullback-lqc}.", "\\medskip\\noindent", "Part (4)(b) for $R^if_*\\mathcal{F}$ computed in the \\'etale cohomology", "follows from Lemma \\ref{lemma-flat-comparison}.", "Whereupon part (4)(a) follows from", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-compare-fppf-etale}", "combined with (1) above.", "\\medskip\\noindent", "Part (5) for the \\'etale topology follows from", "Sheaves on Stacks, Lemma \\ref{stacks-sheaves-lemma-lqc-colimits} and", "Lemma \\ref{lemma-check-flat-comparison-on-etale-covering}.", "The fppf version then follows as the colimit in the \\'etale", "topology is already an fppf sheaf by part (1)." ], "refs": [ "stacks-sheaves-lemma-lqc-flat-base-change-fppf-sheaf", "stacks-cohomology-lemma-flat-comparison", "stacks-sheaves-lemma-compare-fppf-etale", "sites-cohomology-lemma-kill-cohomology-class-on-covering", "stacks-cohomology-lemma-check-flat-comparison-on-etale-covering", "stacks-sheaves-lemma-pullback-lqc", "stacks-cohomology-lemma-flat-comparison", "stacks-sheaves-lemma-compare-fppf-etale", "stacks-sheaves-lemma-lqc-colimits", "stacks-cohomology-lemma-check-flat-comparison-on-etale-covering" ], "ref_ids": [ 11614, 4155, 11615, 4188, 4154, 11584, 4155, 11615, 11585, 4154 ] } ], "ref_ids": [] }, { "id": 4174, "type": "theorem", "label": "stacks-cohomology-proposition-direct-image-quasi-coherent", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-proposition-direct-image-quasi-coherent", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and quasi-separated", "morphism of algebraic stacks. The functor", "$f^* : \\QCoh(\\mathcal{O}_\\mathcal{Y}) \\to", "\\QCoh(\\mathcal{O}_\\mathcal{X})$", "has a right adjoint", "$$", "f_{\\QCoh, *} :", "\\QCoh(\\mathcal{O}_\\mathcal{X})", "\\longrightarrow", "\\QCoh(\\mathcal{O}_\\mathcal{Y})", "$$", "which can be defined as the composition", "$$", "\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\mathcal{M}_\\mathcal{X}", "\\xrightarrow{f_*} \\mathcal{M}_\\mathcal{Y}", "\\xrightarrow{Q} \\QCoh(\\mathcal{O}_\\mathcal{Y})", "$$", "where the functors $f_*$ and $Q$ are as in", "Proposition \\ref{proposition-lcq-flat-base-change}", "and", "Lemma \\ref{lemma-adjoint}.", "Moreover, if we define $R^if_{\\QCoh, *}$ as the composition", "$$", "\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\mathcal{M}_\\mathcal{X}", "\\xrightarrow{R^if_*} \\mathcal{M}_\\mathcal{Y}", "\\xrightarrow{Q} \\QCoh(\\mathcal{O}_\\mathcal{Y})", "$$", "then the sequence of functors $\\{R^if_{\\QCoh, *}\\}_{i \\geq 0}$", "forms a cohomological $\\delta$-functor." ], "refs": [ "stacks-cohomology-proposition-lcq-flat-base-change", "stacks-cohomology-lemma-adjoint" ], "proofs": [ { "contents": [ "This is a combination of the results mentioned in the statement.", "The adjointness can be shown as follows: Let $\\mathcal{F}$", "be a quasi-coherent $\\mathcal{O}_\\mathcal{X}$-module and let", "$\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_\\mathcal{Y}$-module.", "Then we have", "\\begin{align*}", "\\Mor_{\\QCoh(\\mathcal{O}_\\mathcal{X})}(f^*\\mathcal{G}, \\mathcal{F})", "& =", "\\Mor_{\\mathcal{M}_\\mathcal{Y}}(\\mathcal{G}, f_*\\mathcal{F}) \\\\", "& =", "\\Mor_{\\QCoh(\\mathcal{O}_\\mathcal{Y})}(\\mathcal{G}, Q(f_*\\mathcal{F}))", "\\\\", "& =", "\\Mor_{\\QCoh(\\mathcal{O}_\\mathcal{Y})}(\\mathcal{G},", "f_{\\QCoh, *}\\mathcal{F})", "\\end{align*}", "the first equality by adjointness of $f_*$ and $f^*$ (for arbitrary sheaves", "of modules). By", "Proposition \\ref{proposition-lcq-flat-base-change}", "we see that $f_*\\mathcal{F}$ is an object of $\\mathcal{M}_\\mathcal{Y}$", "(and can be computed in either the fppf or \\'etale topology) and we", "obtain the second equality by Lemma \\ref{lemma-adjoint}. The third", "equality is the definition of $f_{\\QCoh, *}$.", "\\medskip\\noindent", "To see that $\\{R^if_{\\QCoh, *}\\}_{i \\geq 0}$ is a cohomological", "$\\delta$-functor as defined in", "Homology, Definition \\ref{homology-definition-cohomological-delta-functor}", "let", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0", "$$", "be a short exact sequence of $\\QCoh(\\mathcal{O}_\\mathcal{X})$.", "This sequence may not be an exact sequence in", "$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ but we know that it is", "up to parasitic modules, see", "Lemma \\ref{lemma-exact-sequence-quasi-coherent-parasitic-cohomology}.", "Thus we may break up the sequence into short exact sequences", "$$", "\\begin{matrix}", "0 \\to \\mathcal{P}_1 \\to \\mathcal{F}_1 \\to \\mathcal{I}_2 \\to 0 \\\\", "0 \\to \\mathcal{I}_2 \\to \\mathcal{F}_2 \\to \\mathcal{Q}_2 \\to 0 \\\\", "0 \\to \\mathcal{P}_2 \\to \\mathcal{Q}_2 \\to \\mathcal{I}_3 \\to 0 \\\\", "0 \\to \\mathcal{I}_3 \\to \\mathcal{F}_3 \\to \\mathcal{P}_3 \\to 0", "\\end{matrix}", "$$", "of $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ with $\\mathcal{P}_i$ parasitic.", "Note that each of the sheaves", "$\\mathcal{P}_j$, $\\mathcal{I}_j$, $\\mathcal{Q}_j$ is an object of", "$\\mathcal{M}_\\mathcal{X}$, see", "Proposition \\ref{proposition-lcq-flat-base-change}.", "Applying $R^if_*$ we obtain long exact sequences ", "$$", "\\begin{matrix}", "0 \\to f_*\\mathcal{P}_1 \\to f_*\\mathcal{F}_1 \\to f_*\\mathcal{I}_2 \\to", "R^1f_*\\mathcal{P}_1 \\to \\ldots \\\\", "0 \\to f_*\\mathcal{I}_2 \\to f_*\\mathcal{F}_2 \\to f_*\\mathcal{Q}_2 \\to", "R^1f_*\\mathcal{I}_2 \\to \\ldots \\\\", "0 \\to f_*\\mathcal{P}_2 \\to f_*\\mathcal{Q}_2 \\to f_*\\mathcal{I}_3 \\to", "R^1f_*\\mathcal{P}_2 \\to \\ldots \\\\", "0 \\to f_*\\mathcal{I}_3 \\to f_*\\mathcal{F}_3 \\to f_*\\mathcal{P}_3 \\to", "R^1f_*\\mathcal{I}_3 \\to \\ldots", "\\end{matrix}", "$$", "where are the terms are objects of $\\mathcal{M}_\\mathcal{Y}$ by", "Proposition \\ref{proposition-lcq-flat-base-change}.", "By", "Lemma \\ref{lemma-pushforward-parasitic}", "the sheaves $R^if_*\\mathcal{P}_j$ are parasitic, hence vanish on applying", "the functor $Q$, see", "Lemma \\ref{lemma-adjoint-kernel-parasitic}.", "Since $Q$ is exact the maps", "$$", "Q(R^if_*\\mathcal{F}_3)", "\\cong", "Q(R^if_*\\mathcal{I}_3)", "\\cong", "Q(R^if_*\\mathcal{Q}_2)", "\\rightarrow", "Q(R^{i + 1}f_*\\mathcal{I}_2)", "\\cong", "Q(R^{i + 1}f_*\\mathcal{F}_1)", "$$", "can serve as the connecting map which turns the family of functors", "$\\{R^if_{\\QCoh, *}\\}_{i \\geq 0}$", "into a cohomological $\\delta$-functor." ], "refs": [ "stacks-cohomology-proposition-lcq-flat-base-change", "stacks-cohomology-lemma-adjoint", "homology-definition-cohomological-delta-functor", "stacks-cohomology-lemma-exact-sequence-quasi-coherent-parasitic-cohomology", "stacks-cohomology-proposition-lcq-flat-base-change", "stacks-cohomology-proposition-lcq-flat-base-change", "stacks-cohomology-lemma-pushforward-parasitic", "stacks-cohomology-lemma-adjoint-kernel-parasitic" ], "ref_ids": [ 4173, 4160, 12149, 4159, 4173, 4173, 4158, 4161 ] } ], "ref_ids": [ 4173, 4160 ] }, { "id": 4175, "type": "theorem", "label": "stacks-cohomology-proposition-smooth-covering-compute-cohomology", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-proposition-smooth-covering-compute-cohomology", "contents": [ "Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a morphism of algebraic stacks.", "Assume $f$ is representable by algebraic spaces, surjective, flat, and", "locally of finite presentation. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_\\mathcal{X}$-module. Then there is a spectral sequence", "$$", "E_2^{p, q} = H^q(\\mathcal{U}_p, f_p^*\\mathcal{F})", "\\Rightarrow", "H^{p + q}(\\mathcal{X}, \\mathcal{F})", "$$", "where $f_p$ is the morphism", "$\\mathcal{U} \\times_\\mathcal{X} \\ldots \\times_\\mathcal{X} \\mathcal{U} \\to", "\\mathcal{X}$ ($p + 1$ factors)." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Sheaves on Stacks, Proposition", "\\ref{stacks-sheaves-proposition-smooth-covering-compute-cohomology}." ], "refs": [ "stacks-sheaves-proposition-smooth-covering-compute-cohomology" ], "ref_ids": [ 11620 ] } ], "ref_ids": [] }, { "id": 4176, "type": "theorem", "label": "stacks-cohomology-proposition-smooth-covering-compute-direct-image", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-proposition-smooth-covering-compute-direct-image", "contents": [ "Let $f : \\mathcal{U} \\to \\mathcal{X}$ and $g : \\mathcal{X} \\to \\mathcal{Y}$", "be composable morphisms of algebraic stacks.", "Assume that", "\\begin{enumerate}", "\\item $f$ is representable by algebraic spaces, surjective,", "flat, locally of finite presentation, quasi-compact, and quasi-separated, and", "\\item $g$ is quasi-compact and quasi-separated.", "\\end{enumerate}", "If $\\mathcal{F}$ is in $\\QCoh(\\mathcal{O}_\\mathcal{X})$ then", "there is a spectral sequence", "$$", "E_2^{p, q} = R^q(g \\circ f_p)_{\\QCoh, *}f_p^*\\mathcal{F}", "\\Rightarrow", "R^{p + q}g_{\\QCoh, *}\\mathcal{F}", "$$", "in $\\QCoh(\\mathcal{O}_\\mathcal{Y})$." ], "refs": [], "proofs": [ { "contents": [ "Note that each of the morphisms", "$f_p : \\mathcal{U} \\times_\\mathcal{X} \\ldots \\times_\\mathcal{X} \\mathcal{U} \\to", "\\mathcal{X}$ is quasi-compact and quasi-separated, hence $g \\circ f_p$", "is quasi-compact and quasi-separated, hence the assertion makes sense", "(i.e., the functors $R^q(g \\circ f_p)_{\\QCoh, *}$ are defined).", "There is a spectral sequence", "$$", "E_2^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}", "\\Rightarrow", "R^{p + q}g_*\\mathcal{F}", "$$", "by Sheaves on Stacks, Proposition", "\\ref{stacks-sheaves-proposition-smooth-covering-compute-direct-image}.", "Applying the exact functor", "$Q_\\mathcal{Y} : \\mathcal{M}_\\mathcal{Y} \\to", "\\QCoh(\\mathcal{O}_\\mathcal{Y})$ gives the desired spectral sequence in", "$\\QCoh(\\mathcal{O}_\\mathcal{Y})$." ], "refs": [ "stacks-sheaves-proposition-smooth-covering-compute-direct-image" ], "ref_ids": [ 11621 ] } ], "ref_ids": [] }, { "id": 4181, "type": "theorem", "label": "sites-cohomology-lemma-trivial-torsor", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-trivial-torsor", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{G}$ be a sheaf of (possibly non-commutative)", "groups on $\\mathcal{C}$.", "A $\\mathcal{G}$-torsor $\\mathcal{F}$ is trivial if and only if", "$\\Gamma(\\mathcal{C}, \\mathcal{F}) \\not = \\emptyset$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4182, "type": "theorem", "label": "sites-cohomology-lemma-torsors-h1", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-torsors-h1", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{H}$ be an abelian sheaf on $\\mathcal{C}$.", "There is a canonical bijection between the set of isomorphism", "classes of $\\mathcal{H}$-torsors and $H^1(\\mathcal{C}, \\mathcal{H})$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a $\\mathcal{H}$-torsor.", "Consider the free abelian sheaf $\\mathbf{Z}[\\mathcal{F}]$", "on $\\mathcal{F}$. It is the sheafification of the rule", "which associates to $U \\in \\Ob(\\mathcal{C})$ the collection of finite", "formal sums $\\sum n_i[s_i]$ with $n_i \\in \\mathbf{Z}$", "and $s_i \\in \\mathcal{F}(U)$. There is a natural map", "$$", "\\sigma : \\mathbf{Z}[\\mathcal{F}] \\longrightarrow \\underline{\\mathbf{Z}}", "$$", "which to a local section $\\sum n_i[s_i]$ associates $\\sum n_i$.", "The kernel of $\\sigma$ is generated by sections of the form", "$[s] - [s']$. There is a canonical map", "$a : \\Ker(\\sigma) \\to \\mathcal{H}$", "which maps $[s] - [s'] \\mapsto h$ where $h$ is the local section of", "$\\mathcal{H}$ such that $h \\cdot s = s'$. Consider the pushout diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Ker(\\sigma) \\ar[r] \\ar[d]^a &", "\\mathbf{Z}[\\mathcal{F}] \\ar[r] \\ar[d] &", "\\underline{\\mathbf{Z}} \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{H} \\ar[r] &", "\\mathcal{E} \\ar[r] &", "\\underline{\\mathbf{Z}} \\ar[r] &", "0", "}", "$$", "Here $\\mathcal{E}$ is the extension obtained by pushout.", "From the long exact cohomology sequence associated to the lower", "short exact sequence we obtain an element", "$\\xi = \\xi_\\mathcal{F} \\in H^1(\\mathcal{C}, \\mathcal{H})$", "by applying the boundary operator to", "$1 \\in H^0(\\mathcal{C}, \\underline{\\mathbf{Z}})$.", "\\medskip\\noindent", "Conversely, given $\\xi \\in H^1(\\mathcal{C}, \\mathcal{H})$ we can associate to", "$\\xi$ a torsor as follows. Choose an embedding $\\mathcal{H} \\to \\mathcal{I}$", "of $\\mathcal{H}$ into an injective abelian sheaf $\\mathcal{I}$. We set", "$\\mathcal{Q} = \\mathcal{I}/\\mathcal{H}$ so that we have a short exact", "sequence", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{H} \\ar[r] &", "\\mathcal{I} \\ar[r] &", "\\mathcal{Q} \\ar[r] &", "0", "}", "$$", "The element $\\xi$ is the image of a global section", "$q \\in H^0(\\mathcal{C}, \\mathcal{Q})$", "because $H^1(\\mathcal{C}, \\mathcal{I}) = 0$ (see", "Derived Categories, Lemma \\ref{derived-lemma-higher-derived-functors}).", "Let $\\mathcal{F} \\subset \\mathcal{I}$ be the subsheaf (of sets) of sections", "that map to $q$ in the sheaf $\\mathcal{Q}$. It is easy to verify that", "$\\mathcal{F}$ is a $\\mathcal{H}$-torsor.", "\\medskip\\noindent", "We omit the verification that the two constructions given", "above are mutually inverse." ], "refs": [ "derived-lemma-higher-derived-functors" ], "ref_ids": [ 1869 ] } ], "ref_ids": [] }, { "id": 4183, "type": "theorem", "label": "sites-cohomology-lemma-h1-extensions", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-h1-extensions", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules on $\\mathcal{C}$.", "There is a canonical bijection", "$$", "\\Ext^1_{\\textit{Mod}(\\mathcal{O})}(\\mathcal{O}, \\mathcal{F})", "\\longrightarrow", "H^1(\\mathcal{C}, \\mathcal{F})", "$$", "which associates to the extension", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{E} \\to \\mathcal{O} \\to 0", "$$", "the image of $1 \\in \\Gamma(\\mathcal{C}, \\mathcal{O})$ in", "$H^1(\\mathcal{C}, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Let us construct the inverse of the map given in the lemma.", "Let $\\xi \\in H^1(\\mathcal{C}, \\mathcal{F})$.", "Choose an injection $\\mathcal{F} \\subset \\mathcal{I}$ with", "$\\mathcal{I}$ injective in $\\textit{Mod}(\\mathcal{O})$.", "Set $\\mathcal{Q} = \\mathcal{I}/\\mathcal{F}$.", "By the long exact sequence of cohomology, we see that", "$\\xi$ is the image of a section", "$\\tilde \\xi \\in \\Gamma(\\mathcal{C}, \\mathcal{Q}) =", "\\Hom_\\mathcal{O}(\\mathcal{O}, \\mathcal{Q})$.", "Now, we just form the pullback", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{F} \\ar[r] \\ar@{=}[d] &", "\\mathcal{E} \\ar[r] \\ar[d] &", "\\mathcal{O} \\ar[r] \\ar[d]^{\\tilde \\xi} &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{F} \\ar[r] &", "\\mathcal{I} \\ar[r] &", "\\mathcal{Q} \\ar[r] &", "0", "}", "$$", "see Homology, Section \\ref{homology-section-extensions}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4184, "type": "theorem", "label": "sites-cohomology-lemma-h1-mod-ab-agree", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-h1-mod-ab-agree", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules on $\\mathcal{C}$.", "Let $\\mathcal{F}_{ab}$ denote the underlying sheaf of abelian", "groups. Then there is a functorial isomorphism", "$$", "H^1(\\mathcal{C}, \\mathcal{F}_{ab})", "=", "H^1(\\mathcal{C}, \\mathcal{F})", "$$", "where the left hand side is cohomology computed in", "$\\textit{Ab}(\\mathcal{C})$ and the right hand side", "is cohomology computed in $\\textit{Mod}(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\underline{\\mathbf{Z}}$ denote the constant sheaf", "$\\mathbf{Z}$. As", "$\\textit{Ab}(\\mathcal{C}) = \\textit{Mod}(\\underline{\\mathbf{Z}})$", "we may apply", "Lemma \\ref{lemma-h1-extensions}", "twice, and it follows that we have to show", "$$", "\\Ext^1_{\\textit{Mod}(\\mathcal{O})}(\\mathcal{O}, \\mathcal{F})", "=", "\\Ext^1_{\\textit{Mod}(\\underline{\\mathbf{Z}})}(", "\\underline{\\mathbf{Z}}, \\mathcal{F}_{ab}).", "$$", "Suppose that $0 \\to \\mathcal{F} \\to \\mathcal{E} \\to \\mathcal{O} \\to 0$", "is an extension in $\\textit{Mod}(\\mathcal{O})$. Then we can use", "the obvious map of abelian sheaves", "$1 : \\underline{\\mathbf{Z}} \\to \\mathcal{O}$", "and pullback to obtain an extension $\\mathcal{E}_{ab}$, like so:", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{F}_{ab} \\ar[r] \\ar@{=}[d] &", "\\mathcal{E}_{ab} \\ar[r] \\ar[d] &", "\\underline{\\mathbf{Z}} \\ar[r] \\ar[d]^{1} &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{F} \\ar[r] &", "\\mathcal{E} \\ar[r] &", "\\mathcal{O} \\ar[r] &", "0", "}", "$$", "The converse is a little more fun. Suppose that", "$0 \\to \\mathcal{F}_{ab} \\to \\mathcal{E}_{ab} \\to \\underline{\\mathbf{Z}} \\to 0$", "is an extension in $\\textit{Mod}(\\underline{\\mathbf{Z}})$.", "Since $\\underline{\\mathbf{Z}}$ is a flat $\\underline{\\mathbf{Z}}$-module", "we see that the sequence", "$$", "0 \\to \\mathcal{F}_{ab} \\otimes_{\\underline{\\mathbf{Z}}} \\mathcal{O}", "\\to \\mathcal{E}_{ab} \\otimes_{\\underline{\\mathbf{Z}}} \\mathcal{O}", "\\to \\underline{\\mathbf{Z}} \\otimes_{\\underline{\\mathbf{Z}}} \\mathcal{O}", "\\to 0", "$$", "is exact, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-flat-tor-zero}.", "Of course", "$\\underline{\\mathbf{Z}} \\otimes_{\\underline{\\mathbf{Z}}} \\mathcal{O}", "= \\mathcal{O}$.", "Hence we can form the pushout via the ($\\mathcal{O}$-linear) multiplication map", "$\\mu : \\mathcal{F} \\otimes_{\\underline{\\mathbf{Z}}} \\mathcal{O}", "\\to \\mathcal{F}$ to get an extension of $\\mathcal{O}$ by", "$\\mathcal{F}$, like this", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{F}_{ab} \\otimes_{\\underline{\\mathbf{Z}}} \\mathcal{O}", "\\ar[r] \\ar[d]^\\mu &", "\\mathcal{E}_{ab} \\otimes_{\\underline{\\mathbf{Z}}} \\mathcal{O}", "\\ar[r] \\ar[d] &", "\\mathcal{O} \\ar[r] \\ar@{=}[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{F} \\ar[r] &", "\\mathcal{E} \\ar[r] &", "\\mathcal{O} \\ar[r] &", "0", "}", "$$", "which is the desired extension. We omit the verification that these", "constructions are mutually inverse." ], "refs": [ "sites-cohomology-lemma-h1-extensions", "sites-modules-lemma-flat-tor-zero" ], "ref_ids": [ 4183, 14204 ] } ], "ref_ids": [] }, { "id": 4185, "type": "theorem", "label": "sites-cohomology-lemma-h1-invertible", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-h1-invertible", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a locally ringed site.", "There is a canonical isomorphism", "$$", "H^1(\\mathcal{C}, \\mathcal{O}^*) = \\Pic(\\mathcal{O}).", "$$", "of abelian groups." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}$-module.", "Consider the presheaf $\\mathcal{L}^*$ defined by the rule", "$$", "U \\longmapsto \\{s \\in \\mathcal{L}(U)", "\\text{ such that } \\mathcal{O}_U \\xrightarrow{s \\cdot -} \\mathcal{L}_U", "\\text{ is an isomorphism}\\}", "$$", "This presheaf satisfies the sheaf condition. Moreover, if", "$f \\in \\mathcal{O}^*(U)$ and $s \\in \\mathcal{L}^*(U)$, then clearly", "$fs \\in \\mathcal{L}^*(U)$. By the same token, if $s, s' \\in \\mathcal{L}^*(U)$", "then there exists a unique $f \\in \\mathcal{O}^*(U)$ such that", "$fs = s'$. Moreover, the sheaf $\\mathcal{L}^*$ has sections locally", "by Modules on Sites, Lemma", "\\ref{sites-modules-lemma-invertible-is-locally-free-rank-1}.", "In other words we", "see that $\\mathcal{L}^*$ is a $\\mathcal{O}^*$-torsor. Thus we get", "a map", "$$", "\\begin{matrix}", "\\text{set of invertible sheaves on }(\\mathcal{C}, \\mathcal{O}) \\\\", "\\text{ up to isomorphism}", "\\end{matrix}", "\\longrightarrow", "\\begin{matrix}", "\\text{set of }\\mathcal{O}^*\\text{-torsors} \\\\", "\\text{ up to isomorphism}", "\\end{matrix}", "$$", "We omit the verification that this is a homomorphism of abelian groups.", "By", "Lemma \\ref{lemma-torsors-h1}", "the right hand side is canonically", "bijective to $H^1(\\mathcal{C}, \\mathcal{O}^*)$.", "Thus we have to show this map is injective and surjective.", "\\medskip\\noindent", "Injective. If the torsor $\\mathcal{L}^*$ is trivial, this means by", "Lemma \\ref{lemma-trivial-torsor}", "that $\\mathcal{L}^*$ has a global section.", "Hence this means exactly that $\\mathcal{L} \\cong \\mathcal{O}$ is", "the neutral element in $\\Pic(\\mathcal{O})$.", "\\medskip\\noindent", "Surjective. Let $\\mathcal{F}$ be an $\\mathcal{O}^*$-torsor.", "Consider the presheaf of sets", "$$", "\\mathcal{L}_1 : U \\longmapsto", "(\\mathcal{F}(U) \\times \\mathcal{O}(U))/\\mathcal{O}^*(U)", "$$", "where the action of $f \\in \\mathcal{O}^*(U)$ on", "$(s, g)$ is $(fs, f^{-1}g)$. Then $\\mathcal{L}_1$ is a presheaf", "of $\\mathcal{O}$-modules by setting", "$(s, g) + (s', g') = (s, g + (s'/s)g')$ where $s'/s$ is the local", "section $f$ of $\\mathcal{O}^*$ such that $fs = s'$, and", "$h(s, g) = (s, hg)$ for $h$ a local section of $\\mathcal{O}$.", "We omit the verification that the sheafification", "$\\mathcal{L} = \\mathcal{L}_1^\\#$ is an invertible $\\mathcal{O}$-module", "whose associated $\\mathcal{O}^*$-torsor $\\mathcal{L}^*$ is isomorphic", "to $\\mathcal{F}$." ], "refs": [ "sites-modules-lemma-invertible-is-locally-free-rank-1", "sites-cohomology-lemma-torsors-h1", "sites-cohomology-lemma-trivial-torsor" ], "ref_ids": [ 14257, 4182, 4181 ] } ], "ref_ids": [] }, { "id": 4186, "type": "theorem", "label": "sites-cohomology-lemma-cohomology-of-open", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cohomology-of-open", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $U$ be an object of $\\mathcal{C}$.", "\\begin{enumerate}", "\\item If $\\mathcal{I}$ is an injective $\\mathcal{O}$-module", "then $\\mathcal{I}|_U$ is an injective $\\mathcal{O}_U$-module.", "\\item For any sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ we have", "$H^p(U, \\mathcal{F}) = H^p(\\mathcal{C}/U, \\mathcal{F}|_U)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that the functor $j_U^{-1}$ of restriction to $U$ is a right adjoint", "to the functor $j_{U!}$ of extension by $0$, see", "Modules on Sites, Section", "\\ref{sites-modules-section-localize}.", "Moreover, $j_{U!}$ is exact. Hence (1) follows from", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}.", "\\medskip\\noindent", "By definition $H^p(U, \\mathcal{F}) = H^p(\\mathcal{I}^\\bullet(U))$", "where $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ is an injective resolution", "in $\\textit{Mod}(\\mathcal{O})$.", "By the above we see that $\\mathcal{F}|_U \\to \\mathcal{I}^\\bullet|_U$", "is an injective resolution in $\\textit{Mod}(\\mathcal{O}_U)$.", "Hence $H^p(U, \\mathcal{F}|_U)$ is equal to", "$H^p(\\mathcal{I}^\\bullet|_U(U))$.", "Of course $\\mathcal{F}(U) = \\mathcal{F}|_U(U)$ for", "any sheaf $\\mathcal{F}$ on $\\mathcal{C}$.", "Hence the equality in (2)." ], "refs": [ "homology-lemma-adjoint-preserve-injectives" ], "ref_ids": [ 12116 ] } ], "ref_ids": [] }, { "id": 4187, "type": "theorem", "label": "sites-cohomology-lemma-cohomology-bigger-site", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cohomology-bigger-site", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "Assume $u$ satisfies the hypotheses of", "Sites, Lemma \\ref{sites-lemma-bigger-site}.", "Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "be the associated morphism of topoi.", "For any abelian sheaf $\\mathcal{F}$ on $\\mathcal{D}$ we have", "isomorphisms", "$$", "R\\Gamma(\\mathcal{C}, g^{-1}\\mathcal{F}) = R\\Gamma(\\mathcal{D}, \\mathcal{F}),", "$$", "in particular", "$H^p(\\mathcal{C}, g^{-1}\\mathcal{F}) = H^p(\\mathcal{D}, \\mathcal{F})$", "and for any $U \\in \\Ob(\\mathcal{C})$ we have isomorphisms", "$$", "R\\Gamma(U, g^{-1}\\mathcal{F}) = R\\Gamma(u(U), \\mathcal{F}),", "$$", "in particular", "$H^p(U, g^{-1}\\mathcal{F}) = H^p(u(U), \\mathcal{F})$. All of these", "isomorphisms are functorial in $\\mathcal{F}$." ], "refs": [ "sites-lemma-bigger-site" ], "proofs": [ { "contents": [ "Since it is clear that", "$\\Gamma(\\mathcal{C}, g^{-1}\\mathcal{F}) = \\Gamma(\\mathcal{D}, \\mathcal{F})$", "by hypothesis (e), it suffices to show that $g^{-1}$ transforms injective", "abelian sheaves into injective abelian sheaves. As usual we use", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}", "to see this. The left adjoint to $g^{-1}$ is $g_! = f^{-1}$ with the", "notation of", "Sites, Lemma \\ref{sites-lemma-bigger-site}", "which is an exact functor. Hence the lemma does indeed apply." ], "refs": [ "homology-lemma-adjoint-preserve-injectives", "sites-lemma-bigger-site" ], "ref_ids": [ 12116, 8548 ] } ], "ref_ids": [ 8548 ] }, { "id": 4188, "type": "theorem", "label": "sites-cohomology-lemma-kill-cohomology-class-on-covering", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-kill-cohomology-class-on-covering", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.", "Let $U$ be an object of $\\mathcal{C}$.", "Let $n > 0$ and let $\\xi \\in H^n(U, \\mathcal{F})$.", "Then there exists a covering $\\{U_i \\to U\\}$ of $\\mathcal{C}$", "such that $\\xi|_{U_i} = 0$ for all $i \\in I$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ be an injective resolution.", "Then", "$$", "H^n(U, \\mathcal{F}) =", "\\frac{\\Ker(\\mathcal{I}^n(U) \\to \\mathcal{I}^{n + 1}(U))}", "{\\Im(\\mathcal{I}^{n - 1}(U) \\to \\mathcal{I}^n(U))}.", "$$", "Pick an element $\\tilde \\xi \\in \\mathcal{I}^n(U)$ representing the", "cohomology class in the presentation above. Since $\\mathcal{I}^\\bullet$", "is an injective resolution of $\\mathcal{F}$ and $n > 0$ we see that", "the complex $\\mathcal{I}^\\bullet$ is exact in degree $n$. Hence", "$\\Im(\\mathcal{I}^{n - 1} \\to \\mathcal{I}^n) =", "\\Ker(\\mathcal{I}^n \\to \\mathcal{I}^{n + 1})$ as sheaves.", "Since $\\tilde \\xi$ is a section of the kernel sheaf over $U$", "we conclude there exists a covering $\\{U_i \\to U\\}$ of the site", "such that $\\tilde \\xi|_{U_i}$ is the image under $d$ of a section", "$\\xi_i \\in \\mathcal{I}^{n - 1}(U_i)$. By our definition of the", "restriction $\\xi|_{U_i}$ as corresponding to the class of", "$\\tilde \\xi|_{U_i}$ we conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4189, "type": "theorem", "label": "sites-cohomology-lemma-higher-direct-images", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-higher-direct-images", "contents": [ "Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to", "(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites", "corresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.", "For any $\\mathcal{F} \\in \\Ob(\\textit{Mod}(\\mathcal{O}_\\mathcal{C}))$", "the sheaf $R^if_*\\mathcal{F}$ is the sheaf associated to the", "presheaf", "$$", "V \\longmapsto H^i(u(V), \\mathcal{F})", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ be an injective resolution.", "Then $R^if_*\\mathcal{F}$ is by definition the $i$th cohomology sheaf", "of the complex", "$$", "f_*\\mathcal{I}^0 \\to f_*\\mathcal{I}^1 \\to f_*\\mathcal{I}^2 \\to \\ldots", "$$", "By definition of the abelian category structure on", "$\\mathcal{O}_\\mathcal{D}$-modules", "this cohomology sheaf is the sheaf associated to the presheaf", "$$", "V", "\\longmapsto", "\\frac{\\Ker(f_*\\mathcal{I}^i(V) \\to f_*\\mathcal{I}^{i + 1}(V))}", "{\\Im(f_*\\mathcal{I}^{i - 1}(V) \\to f_*\\mathcal{I}^i(V))}", "$$", "and this is obviously equal to", "$$", "\\frac{\\Ker(\\mathcal{I}^i(u(V)) \\to \\mathcal{I}^{i + 1}(u(V)))}", "{\\Im(\\mathcal{I}^{i - 1}(u(V)) \\to \\mathcal{I}^i(u(V)))}", "$$", "which is equal to $H^i(u(V), \\mathcal{F})$", "and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4190, "type": "theorem", "label": "sites-cohomology-lemma-cech-h0", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cech-h0", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{F}$ be an abelian presheaf on $\\mathcal{C}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{C}$ and", "\\item for every covering $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$", "of the site $\\mathcal{C}$ the natural map", "$$", "\\mathcal{F}(U) \\to \\check{H}^0(\\mathcal{U}, \\mathcal{F})", "$$", "(see Sites, Section \\ref{sites-section-sheafification}) is bijective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is true since the sheaf condition is exactly that", "$\\mathcal{F}(U) \\to \\check{H}^0(\\mathcal{U}, \\mathcal{F})$", "is bijective for every covering of $\\mathcal{C}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4191, "type": "theorem", "label": "sites-cohomology-lemma-cech-exact-presheaves", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cech-exact-presheaves", "contents": [ "The functor given by Equation (\\ref{equation-cech-functor})", "is an exact functor (see Homology, Lemma \\ref{homology-lemma-exact-functor})." ], "refs": [ "homology-lemma-exact-functor" ], "proofs": [ { "contents": [ "For any object $W$ of $\\mathcal{C}$ the functor", "$\\mathcal{F} \\mapsto \\mathcal{F}(W)$ is an additive exact functor", "from $\\textit{PAb}(\\mathcal{C})$ to $\\textit{Ab}$.", "The terms $\\check{\\mathcal{C}}^p(\\mathcal{U}, \\mathcal{F})$", "of the complex are products of these exact functors and hence exact.", "Moreover a sequence of complexes is exact if and only if the sequence", "of terms in a given degree is exact. Hence the lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12034 ] }, { "id": 4192, "type": "theorem", "label": "sites-cohomology-lemma-cech-cohomology-delta-functor-presheaves", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cech-cohomology-delta-functor-presheaves", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a family of morphisms", "with fixed target such that all fibre products", "$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$.", "The functors $\\mathcal{F} \\mapsto \\check{H}^n(\\mathcal{U}, \\mathcal{F})$", "form a $\\delta$-functor from the abelian category $\\textit{PAb}(\\mathcal{C})$", "to the category of $\\mathbf{Z}$-modules (see", "Homology, Definition \\ref{homology-definition-cohomological-delta-functor})." ], "refs": [ "homology-definition-cohomological-delta-functor" ], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-cech-exact-presheaves}", "a short exact sequence of abelian presheaves", "$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "is turned into a short exact sequence of complexes of", "$\\mathbf{Z}$-modules. Hence we can use", "Homology, Lemma \\ref{homology-lemma-long-exact-sequence-cochain}", "to get the boundary maps", "$\\delta_{\\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3} :", "\\check{H}^n(\\mathcal{U}, \\mathcal{F}_3) \\to", "\\check{H}^{n + 1}(\\mathcal{U}, \\mathcal{F}_1)$", "and a corresponding long exact sequence. We omit the verification", "that these maps are compatible with maps between short exact", "sequences of presheaves." ], "refs": [ "sites-cohomology-lemma-cech-exact-presheaves", "homology-lemma-long-exact-sequence-cochain" ], "ref_ids": [ 4191, 12061 ] } ], "ref_ids": [ 12149 ] }, { "id": 4193, "type": "theorem", "label": "sites-cohomology-lemma-cech-map-into", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cech-map-into", "contents": [ "Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$", "be a family of morphisms with fixed target such that all fibre products", "$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$.", "Consider the chain complex $\\mathbf{Z}_{\\mathcal{U}, \\bullet}$", "of abelian presheaves", "$$", "\\ldots", "\\to", "\\bigoplus_{i_0i_1i_2} \\mathbf{Z}_{U_{i_0} \\times_U U_{i_1} \\times_U U_{i_2}}", "\\to", "\\bigoplus_{i_0i_1} \\mathbf{Z}_{U_{i_0} \\times_U U_{i_1}}", "\\to", "\\bigoplus_{i_0} \\mathbf{Z}_{U_{i_0}}", "\\to 0 \\to \\ldots", "$$", "where the last nonzero term is placed in degree $0$", "and where the map", "$$", "\\mathbf{Z}_{U_{i_0} \\times_U \\ldots \\times_u U_{i_{p + 1}}}", "\\longrightarrow", "\\mathbf{Z}_{U_{i_0} \\times_U", "\\ldots \\widehat{U_{i_j}} \\ldots \\times_U U_{i_{p + 1}}}", "$$", "is given by $(-1)^j$ times the canonical map.", "Then there is an isomorphism", "$$", "\\Hom_{\\textit{PAb}(\\mathcal{C})}(\\mathbf{Z}_{\\mathcal{U}, \\bullet}, \\mathcal{F})", "=", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "$$", "functorial in $\\mathcal{F} \\in \\Ob(\\textit{PAb}(\\mathcal{C}))$." ], "refs": [], "proofs": [ { "contents": [ "This is a tautology based on the fact that", "\\begin{align*}", "\\Hom_{\\textit{PAb}(\\mathcal{C})}(", "\\bigoplus_{i_0 \\ldots i_p}", "\\mathbf{Z}_{U_{i_0} \\times_U \\ldots \\times_U U_{i_p}},", "\\mathcal{F})", "& =", "\\prod_{i_0 \\ldots i_p}", "\\Hom_{\\textit{PAb}(\\mathcal{C})}(", "\\mathbf{Z}_{U_{i_0} \\times_U \\ldots \\times_U U_{i_p}},", "\\mathcal{F}) \\\\", "& =", "\\prod_{i_0 \\ldots i_p}", "\\mathcal{F}(U_{i_0} \\times_U \\ldots \\times_U U_{i_p})", "\\end{align*}", "see Modules on Sites, Lemma \\ref{sites-modules-lemma-obvious-adjointness}." ], "refs": [ "sites-modules-lemma-obvious-adjointness" ], "ref_ids": [ 14141 ] } ], "ref_ids": [] }, { "id": 4194, "type": "theorem", "label": "sites-cohomology-lemma-homology-complex", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-homology-complex", "contents": [ "Let $\\mathcal{C}$ be a category. Let", "$\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ be a family of morphisms", "with fixed target such that all fibre products", "$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$.", "The chain complex $\\mathbf{Z}_{\\mathcal{U}, \\bullet}$ of presheaves", "of Lemma \\ref{lemma-cech-map-into} above is exact in positive", "degrees, i.e., the homology presheaves", "$H_i(\\mathbf{Z}_{\\mathcal{U}, \\bullet})$ are zero for $i > 0$." ], "refs": [ "sites-cohomology-lemma-cech-map-into" ], "proofs": [ { "contents": [ "Let $V$ be an object of $\\mathcal{C}$. We have to show that the chain complex", "of abelian groups $\\mathbf{Z}_{\\mathcal{U}, \\bullet}(V)$ is exact in", "degrees $> 0$. This is the complex", "$$", "\\xymatrix{", "\\ldots \\ar[d] \\\\", "\\bigoplus_{i_0i_1i_2}", "\\mathbf{Z}[", "\\Mor_\\mathcal{C}(V, U_{i_0} \\times_U U_{i_1} \\times_U U_{i_2})", "]", "\\ar[d] \\\\", "\\bigoplus_{i_0i_1}", "\\mathbf{Z}[", "\\Mor_\\mathcal{C}(V, U_{i_0} \\times_U U_{i_1})", "]", "\\ar[d] \\\\", "\\bigoplus_{i_0}", "\\mathbf{Z}[", "\\Mor_\\mathcal{C}(V, U_{i_0})", "] \\ar[d] \\\\", "0", "}", "$$", "For any morphism $\\varphi : V \\to U$ denote", "$\\Mor_\\varphi(V, U_i) = \\{\\varphi_i : V \\to U_i \\mid", "f_i \\circ \\varphi_i = \\varphi\\}$. We will use a similar notation", "for $\\Mor_\\varphi(V, U_{i_0} \\times_U \\ldots \\times_U U_{i_p})$.", "Note that composing with the various projection maps between the", "fibred products $U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ preserves", "these morphism sets. Hence we see that the complex above", "is the same as the complex", "$$", "\\xymatrix{", "\\ldots \\ar[d] \\\\", "\\bigoplus_\\varphi", "\\bigoplus_{i_0i_1i_2}", "\\mathbf{Z}[", "\\Mor_\\varphi(V, U_{i_0} \\times_U U_{i_1} \\times_U U_{i_2})", "]", "\\ar[d] \\\\", "\\bigoplus_\\varphi", "\\bigoplus_{i_0i_1}", "\\mathbf{Z}[", "\\Mor_\\varphi(V, U_{i_0} \\times_U U_{i_1})", "]", "\\ar[d] \\\\", "\\bigoplus_\\varphi", "\\bigoplus_{i_0}", "\\mathbf{Z}[", "\\Mor_\\varphi(V, U_{i_0})", "] \\ar[d] \\\\", "0", "}", "$$", "Next, we make the remark that we have", "$$", "\\Mor_\\varphi(V, U_{i_0} \\times_U \\ldots \\times_U U_{i_p})", "=", "\\Mor_\\varphi(V, U_{i_0}) \\times \\ldots", "\\times \\Mor_\\varphi(V, U_{i_p})", "$$", "Using this and the fact that $\\mathbf{Z}[A] \\oplus \\mathbf{Z}[B] =", "\\mathbf{Z}[A \\amalg B]$ we see that the complex becomes", "$$", "\\xymatrix{", "\\ldots \\ar[d] \\\\", "\\bigoplus_\\varphi", "\\mathbf{Z}\\left[", "\\coprod_{i_0i_1i_2}", "\\Mor_\\varphi(V, U_{i_0}) \\times \\Mor_\\varphi(V, U_{i_1}) \\times", "\\Mor_\\varphi(V, U_{i_2})", "\\right]", "\\ar[d] \\\\", "\\bigoplus_\\varphi", "\\mathbf{Z}\\left[", "\\coprod_{i_0i_1}", "\\Mor_\\varphi(V, U_{i_0}) \\times \\Mor_\\varphi(V, U_{i_1})", "\\right]", "\\ar[d] \\\\", "\\bigoplus_\\varphi", "\\mathbf{Z}\\left[", "\\coprod_{i_0}", "\\Mor_\\varphi(V, U_{i_0})", "\\right] \\ar[d] \\\\", "0", "}", "$$", "Finally, on setting $S_\\varphi = \\coprod_{i \\in I} \\Mor_\\varphi(V, U_i)$", "we see that we get", "$$", "\\bigoplus\\nolimits_\\varphi \\left(\\ldots \\to", "\\mathbf{Z}[S_\\varphi \\times S_\\varphi \\times S_\\varphi] \\to", "\\mathbf{Z}[S_\\varphi \\times S_\\varphi] \\to", "\\mathbf{Z}[S_\\varphi] \\to 0 \\to \\ldots", "\\right)", "$$", "Thus we have simplified our task. Namely, it suffices to show that", "for any nonempty set $S$ the (extended) complex of free abelian groups", "$$", "\\ldots \\to", "\\mathbf{Z}[S \\times S \\times S] \\to", "\\mathbf{Z}[S \\times S] \\to", "\\mathbf{Z}[S] \\xrightarrow{\\Sigma} \\mathbf{Z} \\to 0 \\to \\ldots", "$$", "is exact in all degrees. To see this fix an element $s \\in S$, and", "use the homotopy", "$$", "n_{(s_0, \\ldots, s_p)} \\longmapsto n_{(s, s_0, \\ldots, s_p)}", "$$", "with obvious notations." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 4193 ] }, { "id": 4195, "type": "theorem", "label": "sites-cohomology-lemma-complex-tensored-still-exact", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-complex-tensored-still-exact", "contents": [ "\\begin{slogan}", "The integral presheaf {\\v C}ech complex is a flat resolution of the", "constant presheaf of integers.", "\\end{slogan}", "Let $\\mathcal{C}$ be a category. Let", "$\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ be a family of morphisms", "with fixed target such that all fibre products", "$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$.", "Let $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$.", "The chain complex", "$$", "\\mathbf{Z}_{\\mathcal{U}, \\bullet}", "\\otimes_{p, \\mathbf{Z}}", "\\mathcal{O}", "$$", "is exact in positive degrees. Here $\\mathbf{Z}_{\\mathcal{U}, \\bullet}$", "is the chain complex of Lemma \\ref{lemma-cech-map-into}, and", "the tensor product is over the constant presheaf of rings", "with value $\\mathbf{Z}$." ], "refs": [ "sites-cohomology-lemma-cech-map-into" ], "proofs": [ { "contents": [ "Let $V$ be an object of $\\mathcal{C}$.", "In the proof of Lemma \\ref{lemma-homology-complex} we saw that", "$\\mathbf{Z}_{\\mathcal{U}, \\bullet}(V)$ is isomorphic as a complex", "to a direct sum of complexes which are homotopic to $\\mathbf{Z}$", "placed in degree zero. Hence also", "$\\mathbf{Z}_{\\mathcal{U}, \\bullet}(V) \\otimes_\\mathbf{Z} \\mathcal{O}(V)$", "is isomorphic as a complex to a direct sum of complexes which are homotopic", "to $\\mathcal{O}(V)$ placed in degree zero.", "Or you can use", "Modules on Sites, Lemma \\ref{sites-modules-lemma-flat-resolution-of-flat},", "which applies since the presheaves $\\mathbf{Z}_{\\mathcal{U}, i}$ are flat,", "and the proof of Lemma \\ref{lemma-homology-complex} shows that", "$H_0(\\mathbf{Z}_{\\mathcal{U}, \\bullet})$ is a flat presheaf also." ], "refs": [ "sites-cohomology-lemma-homology-complex", "sites-modules-lemma-flat-resolution-of-flat", "sites-cohomology-lemma-homology-complex" ], "ref_ids": [ 4194, 14206, 4194 ] } ], "ref_ids": [ 4193 ] }, { "id": 4196, "type": "theorem", "label": "sites-cohomology-lemma-cech-cohomology-derived-presheaves", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cech-cohomology-derived-presheaves", "contents": [ "Let $\\mathcal{C}$ be a category. Let", "$\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ be a family of morphisms", "with fixed target such that all fibre products", "$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$.", "The {\\v C}ech cohomology functors $\\check{H}^p(\\mathcal{U}, -)$", "are canonically isomorphic as a $\\delta$-functor to", "the right derived functors of the functor", "$$", "\\check{H}^0(\\mathcal{U}, -) :", "\\textit{PAb}(\\mathcal{C})", "\\longrightarrow", "\\textit{Ab}.", "$$", "Moreover, there is a functorial quasi-isomorphism", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\longrightarrow", "R\\check{H}^0(\\mathcal{U}, \\mathcal{F})", "$$", "where the right hand side indicates the derived functor", "$$", "R\\check{H}^0(\\mathcal{U}, -) :", "D^{+}(\\textit{PAb}(\\mathcal{C}))", "\\longrightarrow", "D^{+}(\\mathbf{Z})", "$$", "of the left exact functor $\\check{H}^0(\\mathcal{U}, -)$." ], "refs": [], "proofs": [ { "contents": [ "Note that the category of abelian presheaves has enough injectives, see", "Injectives, Proposition \\ref{injectives-proposition-presheaves-injectives}.", "Note that $\\check{H}^0(\\mathcal{U}, -)$ is a left exact functor", "from the category of abelian presheaves", "to the category of $\\mathbf{Z}$-modules.", "Hence the derived functor and the right derived functor exist, see", "Derived Categories, Section \\ref{derived-section-right-derived-functor}.", "\\medskip\\noindent", "Let $\\mathcal{I}$ be a injective abelian presheaf.", "In this case the functor", "$\\Hom_{\\textit{PAb}(\\mathcal{C})}(-, \\mathcal{I})$", "is exact on $\\textit{PAb}(\\mathcal{C})$. By", "Lemma \\ref{lemma-cech-map-into} we have", "$$", "\\Hom_{\\textit{PAb}(\\mathcal{C})}(", "\\mathbf{Z}_{\\mathcal{U}, \\bullet}, \\mathcal{I})", "=", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}).", "$$", "By Lemma \\ref{lemma-homology-complex} we have that", "$\\mathbf{Z}_{\\mathcal{U}, \\bullet}$ is exact in positive degrees.", "Hence by the exactness of Hom into $\\mathcal{I}$ mentioned above we see", "that $\\check{H}^i(\\mathcal{U}, \\mathcal{I}) = 0$ for all", "$i > 0$. Thus the $\\delta$-functor $(\\check{H}^n, \\delta)$", "(see Lemma \\ref{lemma-cech-cohomology-delta-functor-presheaves})", "satisfies the assumptions of", "Homology, Lemma \\ref{homology-lemma-efface-implies-universal},", "and hence is a universal $\\delta$-functor.", "\\medskip\\noindent", "By", "Derived Categories, Lemma \\ref{derived-lemma-higher-derived-functors}", "also the sequence $R^i\\check{H}^0(\\mathcal{U}, -)$", "forms a universal $\\delta$-functor. By the uniqueness of universal", "$\\delta$-functors, see", "Homology, Lemma \\ref{homology-lemma-uniqueness-universal-delta-functor}", "we conclude that", "$R^i\\check{H}^0(\\mathcal{U}, -) = \\check{H}^i(\\mathcal{U}, -)$.", "This is enough for most applications", "and the reader is suggested to skip the rest of the proof.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be any abelian presheaf on $\\mathcal{C}$.", "Choose an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$", "in the category $\\textit{PAb}(\\mathcal{C})$.", "Consider the double complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet)$", "with terms $\\check{\\mathcal{C}}^p(\\mathcal{U}, \\mathcal{I}^q)$.", "Next, consider the total complex", "$\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet))$", "associated to this double complex, see", "Homology, Section \\ref{homology-section-double-complexes}.", "There is a map of complexes", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\longrightarrow", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet))", "$$", "coming from the maps", "$\\check{\\mathcal{C}}^p(\\mathcal{U}, \\mathcal{F})", "\\to \\check{\\mathcal{C}}^p(\\mathcal{U}, \\mathcal{I}^0)$", "and there is a map of complexes", "$$", "\\check{H}^0(\\mathcal{U}, \\mathcal{I}^\\bullet)", "\\longrightarrow", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet))", "$$", "coming from the maps", "$\\check{H}^0(\\mathcal{U}, \\mathcal{I}^q) \\to", "\\check{\\mathcal{C}}^0(\\mathcal{U}, \\mathcal{I}^q)$.", "Both of these maps are quasi-isomorphisms by an application of", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}.", "Namely, the columns of the double complex are exact in positive degrees", "because the {\\v C}ech complex as a functor is exact", "(Lemma \\ref{lemma-cech-exact-presheaves})", "and the rows of the double complex are exact in positive degrees", "since as we just saw the higher {\\v C}ech cohomology groups of the injective", "presheaves $\\mathcal{I}^q$ are zero.", "Since quasi-isomorphisms become invertible", "in $D^{+}(\\mathbf{Z})$ this gives the last displayed morphism", "of the lemma. We omit the verification that this morphism is", "functorial." ], "refs": [ "injectives-proposition-presheaves-injectives", "sites-cohomology-lemma-cech-map-into", "sites-cohomology-lemma-homology-complex", "sites-cohomology-lemma-cech-cohomology-delta-functor-presheaves", "homology-lemma-efface-implies-universal", "derived-lemma-higher-derived-functors", "homology-lemma-uniqueness-universal-delta-functor", "homology-lemma-double-complex-gives-resolution", "sites-cohomology-lemma-cech-exact-presheaves" ], "ref_ids": [ 7805, 4193, 4194, 4192, 12052, 1869, 12053, 12106, 4191 ] } ], "ref_ids": [] }, { "id": 4197, "type": "theorem", "label": "sites-cohomology-lemma-injective-abelian-sheaf-injective-presheaf", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-injective-abelian-sheaf-injective-presheaf", "contents": [ "Let $\\mathcal{C}$ be a site. An injective abelian sheaf is also injective as an", "object in the category $\\textit{PAb}(\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "Apply Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}", "to the categories $\\mathcal{A} = \\textit{Ab}(\\mathcal{C})$,", "$\\mathcal{B} = \\textit{PAb}(\\mathcal{C})$, the inclusion functor", "and sheafification. (See", "Modules on Sites, Section \\ref{sites-modules-section-abelian-sheaves} to see", "that all assumptions of the lemma are satisfied.)" ], "refs": [ "homology-lemma-adjoint-preserve-injectives" ], "ref_ids": [ 12116 ] } ], "ref_ids": [] }, { "id": 4198, "type": "theorem", "label": "sites-cohomology-lemma-injective-trivial-cech", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-injective-trivial-cech", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$.", "Let $\\mathcal{I}$ be an injective abelian sheaf, i.e., an injective", "object of $\\textit{Ab}(\\mathcal{C})$.", "Then", "$$", "\\check{H}^p(\\mathcal{U}, \\mathcal{I}) =", "\\left\\{", "\\begin{matrix}", "\\mathcal{I}(U) & \\text{if} & p = 0 \\\\", "0 & \\text{if} & p > 0", "\\end{matrix}", "\\right.", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-injective-abelian-sheaf-injective-presheaf}", "we see that $\\mathcal{I}$ is an injective object in", "$\\textit{PAb}(\\mathcal{C})$.", "Hence we can apply Lemma \\ref{lemma-cech-cohomology-derived-presheaves}", "(or its proof) to see the vanishing of higher {\\v C}ech cohomology group.", "For the zeroth see Lemma \\ref{lemma-cech-h0}." ], "refs": [ "sites-cohomology-lemma-injective-abelian-sheaf-injective-presheaf", "sites-cohomology-lemma-cech-cohomology-derived-presheaves", "sites-cohomology-lemma-cech-h0" ], "ref_ids": [ 4197, 4196, 4190 ] } ], "ref_ids": [] }, { "id": 4199, "type": "theorem", "label": "sites-cohomology-lemma-cech-cohomology", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cech-cohomology", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$.", "There is a transformation", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, -)", "\\longrightarrow", "R\\Gamma(U, -)", "$$", "of functors", "$\\textit{Ab}(\\mathcal{C}) \\to D^{+}(\\mathbf{Z})$.", "In particular this gives a transformation of functors", "$\\check{H}^p(U, \\mathcal{F}) \\to H^p(U, \\mathcal{F})$ for", "$\\mathcal{F}$ ranging over $\\textit{Ab}(\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be an abelian sheaf. Choose an injective resolution", "$\\mathcal{F} \\to \\mathcal{I}^\\bullet$. Consider the double complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet)$", "with terms $\\check{\\mathcal{C}}^p(\\mathcal{U}, \\mathcal{I}^q)$.", "Next, consider the associated total complex", "$\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet))$,", "see Homology, Definition \\ref{homology-definition-associated-simple-complex}.", "There is a map of complexes", "$$", "\\alpha :", "\\Gamma(U, \\mathcal{I}^\\bullet)", "\\longrightarrow", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet))", "$$", "coming from the maps", "$\\mathcal{I}^q(U) \\to \\check{H}^0(\\mathcal{U}, \\mathcal{I}^q)$", "and a map of complexes", "$$", "\\beta :", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\longrightarrow", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet))", "$$", "coming from the map $\\mathcal{F} \\to \\mathcal{I}^0$.", "We can apply", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}", "to see that $\\alpha$ is a quasi-isomorphism.", "Namely, Lemma \\ref{lemma-injective-trivial-cech} implies that", "the $q$th row of the double complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet)$ is a", "resolution of $\\Gamma(U, \\mathcal{I}^q)$.", "Hence $\\alpha$ becomes invertible in $D^{+}(\\mathbf{Z})$ and", "the transformation of the lemma is the composition of $\\beta$", "followed by the inverse of $\\alpha$. We omit the verification", "that this is functorial." ], "refs": [ "homology-definition-associated-simple-complex", "homology-lemma-double-complex-gives-resolution", "sites-cohomology-lemma-injective-trivial-cech" ], "ref_ids": [ 12164, 12106, 4198 ] } ], "ref_ids": [] }, { "id": 4200, "type": "theorem", "label": "sites-cohomology-lemma-cech-h1", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cech-h1", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{G}$ be an abelian sheaf", "on $\\mathcal{C}$. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a", "covering of $\\mathcal{C}$. The map", "$$", "\\check{H}^1(\\mathcal{U}, \\mathcal{G})", "\\longrightarrow", "H^1(U, \\mathcal{G})", "$$", "is injective and identifies $\\check{H}^1(\\mathcal{U}, \\mathcal{G})$ via", "the bijection of Lemma \\ref{lemma-torsors-h1}", "with the set of isomorphism classes of $\\mathcal{G}|_U$-torsors", "which restrict to trivial torsors over each $U_i$." ], "refs": [ "sites-cohomology-lemma-torsors-h1" ], "proofs": [ { "contents": [ "To see this we construct an inverse map. Namely, let $\\mathcal{F}$ be a", "$\\mathcal{G}|_U$-torsor on $\\mathcal{C}/U$ whose restriction to", "$\\mathcal{C}/U_i$ is trivial. By Lemma \\ref{lemma-trivial-torsor}", "this means there", "exists a section $s_i \\in \\mathcal{F}(U_i)$. On $U_{i_0} \\times_U U_{i_1}$", "there is a unique section $s_{i_0i_1}$ of $\\mathcal{G}$ such that", "$s_{i_0i_1} \\cdot s_{i_0}|_{U_{i_0} \\times_U U_{i_1}} =", "s_{i_1}|_{U_{i_0} \\times_U U_{i_1}}$. An easy computation shows", "that $s_{i_0i_1}$ is a {\\v C}ech cocycle and that its class is well", "defined (i.e., does not depend on the choice of the sections $s_i$).", "The inverse maps the isomorphism class of $\\mathcal{F}$ to the cohomology", "class of the cocycle $(s_{i_0i_1})$.", "We omit the verification that this map is indeed an inverse." ], "refs": [ "sites-cohomology-lemma-trivial-torsor" ], "ref_ids": [ 4181 ] } ], "ref_ids": [ 4182 ] }, { "id": 4201, "type": "theorem", "label": "sites-cohomology-lemma-include", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-include", "contents": [ "Let $\\mathcal{C}$ be a site.", "Consider the functor", "$i : \\textit{Ab}(\\mathcal{C}) \\to \\textit{PAb}(\\mathcal{C})$.", "It is a left exact functor with right derived functors given by", "$$", "R^pi(\\mathcal{F}) = \\underline{H}^p(\\mathcal{F}) :", "U \\longmapsto H^p(U, \\mathcal{F})", "$$", "see discussion in Section \\ref{section-locality}." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $i$ is left exact.", "Choose an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$.", "By definition $R^pi$ is the $p$th cohomology {\\it presheaf}", "of the complex $\\mathcal{I}^\\bullet$. In other words, the", "sections of $R^pi(\\mathcal{F})$ over an object $U$ of $\\mathcal{C}$", "are given by", "$$", "\\frac{\\Ker(\\mathcal{I}^n(U) \\to \\mathcal{I}^{n + 1}(U))}", "{\\Im(\\mathcal{I}^{n - 1}(U) \\to \\mathcal{I}^n(U))}.", "$$", "which is the definition of $H^p(U, \\mathcal{F})$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4202, "type": "theorem", "label": "sites-cohomology-lemma-cech-spectral-sequence", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cech-spectral-sequence", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$", "be a covering of $\\mathcal{C}$. For any abelian sheaf $\\mathcal{F}$ there", "is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with", "$$", "E_2^{p, q} = \\check{H}^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F}))", "$$", "converging to $H^{p + q}(U, \\mathcal{F})$.", "This spectral sequence is functorial in $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "This is a Grothendieck spectral sequence (see", "Derived Categories, Lemma \\ref{derived-lemma-grothendieck-spectral-sequence})", "for the functors", "$$", "i : \\textit{Ab}(\\mathcal{C}) \\to \\textit{PAb}(\\mathcal{C})", "\\quad\\text{and}\\quad", "\\check{H}^0(\\mathcal{U}, - ) : \\textit{PAb}(\\mathcal{C})", "\\to \\textit{Ab}.", "$$", "Namely, we have $\\check{H}^0(\\mathcal{U}, i(\\mathcal{F})) = \\mathcal{F}(U)$", "by Lemma \\ref{lemma-cech-h0}. We have that $i(\\mathcal{I})$ is", "{\\v C}ech acyclic by Lemma \\ref{lemma-injective-trivial-cech}. And we", "have that $\\check{H}^p(\\mathcal{U}, -) = R^p\\check{H}^0(\\mathcal{U}, -)$", "as functors on $\\textit{PAb}(\\mathcal{C})$", "by Lemma \\ref{lemma-cech-cohomology-derived-presheaves}.", "Putting everything together gives the lemma." ], "refs": [ "derived-lemma-grothendieck-spectral-sequence", "sites-cohomology-lemma-cech-h0", "sites-cohomology-lemma-injective-trivial-cech", "sites-cohomology-lemma-cech-cohomology-derived-presheaves" ], "ref_ids": [ 1873, 4190, 4198, 4196 ] } ], "ref_ids": [] }, { "id": 4203, "type": "theorem", "label": "sites-cohomology-lemma-cech-spectral-sequence-application", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cech-spectral-sequence-application", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a covering.", "Let $\\mathcal{F} \\in \\Ob(\\textit{Ab}(\\mathcal{C}))$.", "Assume that $H^i(U_{i_0} \\times_U \\ldots \\times_U U_{i_p}, \\mathcal{F}) = 0$", "for all $i > 0$, all $p \\geq 0$ and all $i_0, \\ldots, i_p \\in I$.", "Then $\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = H^p(U, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "We will use the spectral sequence of", "Lemma \\ref{lemma-cech-spectral-sequence}.", "The assumptions mean that $E_2^{p, q} = 0$ for all $(p, q)$ with", "$q \\not = 0$. Hence the spectral sequence degenerates at $E_2$", "and the result follows." ], "refs": [ "sites-cohomology-lemma-cech-spectral-sequence" ], "ref_ids": [ 4202 ] } ], "ref_ids": [] }, { "id": 4204, "type": "theorem", "label": "sites-cohomology-lemma-ses-cech-h1", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-ses-cech-h1", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0", "$$", "be a short exact sequence of abelian sheaves on $\\mathcal{C}$.", "Let $U$ be an object of $\\mathcal{C}$. If there exists a cofinal system", "of coverings $\\mathcal{U}$ of $U$ such that", "$\\check{H}^1(\\mathcal{U}, \\mathcal{F}) = 0$,", "then the map $\\mathcal{G}(U) \\to \\mathcal{H}(U)$ is", "surjective." ], "refs": [], "proofs": [ { "contents": [ "Take an element $s \\in \\mathcal{H}(U)$. Choose a covering", "$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ such that", "(a) $\\check{H}^1(\\mathcal{U}, \\mathcal{F}) = 0$ and (b)", "$s|_{U_i}$ is the image of a section $s_i \\in \\mathcal{G}(U_i)$.", "Since we can certainly find a covering such that (b) holds", "it follows from the assumptions of the lemma that we can find", "a covering such that (a) and (b) both hold.", "Consider the sections", "$$", "s_{i_0i_1} =", "s_{i_1}|_{U_{i_0} \\times_U U_{i_1}} - s_{i_0}|_{U_{i_0} \\times_U U_{i_1}}.", "$$", "Since $s_i$ lifts $s$ we see that", "$s_{i_0i_1} \\in \\mathcal{F}(U_{i_0} \\times_U U_{i_1})$.", "By the vanishing of $\\check{H}^1(\\mathcal{U}, \\mathcal{F})$ we can", "find sections $t_i \\in \\mathcal{F}(U_i)$ such that", "$$", "s_{i_0i_1} =", "t_{i_1}|_{U_{i_0} \\times_U U_{i_1}} - t_{i_0}|_{U_{i_0} \\times_U U_{i_1}}.", "$$", "Then clearly the sections $s_i - t_i$ satisfy the sheaf condition", "and glue to a section of $\\mathcal{G}$ over $U$ which maps to $s$.", "Hence we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4205, "type": "theorem", "label": "sites-cohomology-lemma-cech-vanish-collection", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cech-vanish-collection", "contents": [ "(Variant of Cohomology, Lemma \\ref{cohomology-lemma-cech-vanish}.)", "Let $\\mathcal{C}$ be a site. Let $\\text{Cov}_\\mathcal{C}$ be the set", "of coverings of $\\mathcal{C}$ (see", "Sites, Definition \\ref{sites-definition-site}). Let", "$\\mathcal{B} \\subset \\Ob(\\mathcal{C})$, and", "$\\text{Cov} \\subset \\text{Cov}_\\mathcal{C}$", "be subsets. Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}$.", "Assume that", "\\begin{enumerate}", "\\item For every $\\mathcal{U} \\in \\text{Cov}$,", "$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ we have", "$U, U_i \\in \\mathcal{B}$ and every", "$U_{i_0} \\times_U \\ldots \\times_U U_{i_p} \\in \\mathcal{B}$.", "\\item For every $U \\in \\mathcal{B}$ the coverings of $U$", "occurring in $\\text{Cov}$ is a cofinal system of coverings of $U$.", "\\item For every $\\mathcal{U} \\in \\text{Cov}$ we have", "$\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0$ for all $p > 0$.", "\\end{enumerate}", "Then $H^p(U, \\mathcal{F}) = 0$ for all $p > 0$ and any $U \\in \\mathcal{B}$." ], "refs": [ "cohomology-lemma-cech-vanish", "sites-definition-site" ], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ and $\\text{Cov}$ be as in the lemma.", "We will indicate this by saying ``$\\mathcal{F}$ has vanishing higher", "{\\v C}ech cohomology for any $\\mathcal{U} \\in \\text{Cov}$''.", "Choose an embedding $\\mathcal{F} \\to \\mathcal{I}$ into an", "injective abelian sheaf.", "By Lemma \\ref{lemma-injective-trivial-cech} $\\mathcal{I}$", "has vanishing higher {\\v C}ech cohomology for any $\\mathcal{U} \\in \\text{Cov}$.", "Let $\\mathcal{Q} = \\mathcal{I}/\\mathcal{F}$", "so that we have a short exact sequence", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{I} \\to \\mathcal{Q} \\to 0.", "$$", "By Lemma \\ref{lemma-ses-cech-h1} and our assumption (2)", "this sequence gives rise to an exact sequence", "$$", "0 \\to \\mathcal{F}(U) \\to \\mathcal{I}(U) \\to \\mathcal{Q}(U) \\to 0.", "$$", "for every $U \\in \\mathcal{B}$. Hence for any $\\mathcal{U} \\in \\text{Cov}$", "we get a short exact sequence of {\\v C}ech complexes", "$$", "0 \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}) \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{Q}) \\to 0", "$$", "since each term in the {\\v C}ech complex is made up out of a product of", "values over elements of $\\mathcal{B}$ by assumption (1).", "In particular we have a long exact sequence of {\\v C}ech cohomology", "groups for any covering $\\mathcal{U} \\in \\text{Cov}$.", "This implies that $\\mathcal{Q}$ is also an abelian sheaf", "with vanishing higher {\\v C}ech cohomology for all", "$\\mathcal{U} \\in \\text{Cov}$.", "\\medskip\\noindent", "Next, we look at the long exact cohomology sequence", "$$", "\\xymatrix{", "0 \\ar[r] &", "H^0(U, \\mathcal{F}) \\ar[r] &", "H^0(U, \\mathcal{I}) \\ar[r] &", "H^0(U, \\mathcal{Q}) \\ar[lld] \\\\", "&", "H^1(U, \\mathcal{F}) \\ar[r] &", "H^1(U, \\mathcal{I}) \\ar[r] &", "H^1(U, \\mathcal{Q}) \\ar[lld] \\\\", "&", "\\ldots & \\ldots & \\ldots \\\\", "}", "$$", "for any $U \\in \\mathcal{B}$. Since $\\mathcal{I}$ is injective we", "have $H^n(U, \\mathcal{I}) = 0$ for $n > 0$ (see", "Derived Categories, Lemma \\ref{derived-lemma-higher-derived-functors}).", "By the above we see that $H^0(U, \\mathcal{I}) \\to H^0(U, \\mathcal{Q})$", "is surjective and hence $H^1(U, \\mathcal{F}) = 0$.", "Since $\\mathcal{F}$ was an arbitrary abelian sheaf with", "vanishing higher {\\v C}ech cohomology for all $\\mathcal{U} \\in \\text{Cov}$", "we conclude that also $H^1(U, \\mathcal{Q}) = 0$ since $\\mathcal{Q}$ is", "another of these sheaves (see above). By the long exact sequence this in", "turn implies that $H^2(U, \\mathcal{F}) = 0$. And so on and so forth." ], "refs": [ "sites-cohomology-lemma-injective-trivial-cech", "sites-cohomology-lemma-ses-cech-h1", "derived-lemma-higher-derived-functors" ], "ref_ids": [ 4198, 4204, 1869 ] } ], "ref_ids": [ 2058, 8652 ] }, { "id": 4206, "type": "theorem", "label": "sites-cohomology-lemma-existence", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-existence", "contents": [ "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$", "be a gerbe over a site whose automorphism sheaves are abelian.", "Let $\\mathcal{G}$ be the sheaf of abelian groups constructed", "in Stacks, Lemma \\ref{stacks-lemma-gerbe-abelian-auts}.", "Let $U$ be an object of $\\mathcal{C}$ such that", "\\begin{enumerate}", "\\item there exists a cofinal system of coverings $\\{U_i \\to U\\}$", "of $U$ in $\\mathcal{C}$ such that $H^1(U_i, \\mathcal{G}) = 0$ and", "$H^1(U_i \\times_U U_j, \\mathcal{G}) = 0$", "for all $i, j$, and", "\\item $H^2(U, \\mathcal{G}) = 0$.", "\\end{enumerate}", "Then there exists an object of $\\mathcal{S}$ lying over $U$." ], "refs": [ "stacks-lemma-gerbe-abelian-auts" ], "proofs": [ { "contents": [ "By Stacks, Definition \\ref{stacks-definition-gerbe}", "there exists a covering $\\mathcal{U} = \\{U_i \\to U\\}$", "and $x_i$ in $\\mathcal{S}$ lying over $U_i$.", "Write $U_{ij} = U_i \\times_U U_j$. By (1) after refining the covering", "we may assume that $H^1(U_i, \\mathcal{G}) = 0$ and", "$H^1(U_{ij}, \\mathcal{G}) = 0$.", "Consider the sheaf", "$$", "\\mathcal{F}_{ij} =", "\\mathit{Isom}(x_i|_{U_{ij}}, x_j|_{U_{ij}})", "$$", "on $\\mathcal{C}/U_{ij}$. Since", "$\\mathcal{G}|_{U_{ij}} = \\mathit{Aut}(x_i|_{U_{ij}})$ we see that there", "is an action", "$$", "\\mathcal{G}|_{U_{ij}} \\times \\mathcal{F}_{ij} \\to \\mathcal{F}_{ij}", "$$", "by precomposition. It is clear that $\\mathcal{F}_{ij}$ is a", "pseudo $\\mathcal{G}|_{U_{ij}}$-torsor and in fact a torsor because", "any two objects of a gerbe are locally isomorphic.", "By our choice of the covering and by", "Lemma \\ref{lemma-torsors-h1}", "these torsors are trivial (and hence have global sections by", "Lemma \\ref{lemma-trivial-torsor}).", "In other words, we can choose isomorphisms", "$$", "\\varphi_{ij} : x_i|_{U_{ij}} \\longrightarrow x_j|_{U_{ij}}", "$$", "To find an object $x$ over $U$ we are going to massage our choice", "of these $\\varphi_{ij}$ to get a descent datum (which is necessarily", "effective as $p : \\mathcal{S} \\to \\mathcal{C}$ is a stack).", "Namely, the obstruction to being a descent datum is that the cocycle", "condition may not hold. Namely, set $U_{ijk} = U_i \\times_U U_j \\times_U U_k$.", "Then we can consider", "$$", "g_{ijk} = \\varphi_{ik}^{-1}|_{U_{ijk}} \\circ \\varphi_{jk}|_{U_{ijk}} \\circ", "\\varphi_{ij}|_{U_{ijk}}", "$$", "which is an automorphism of $x_i$ over $U_{ijk}$. Thus we may and do", "consider $g_{ijk}$ as a section of $\\mathcal{G}$ over $U_{ijk}$.", "A computation (omitted) shows that $(g_{i_0i_1i_2})$ is a $2$-cocycle", "in the {\\v C}ech complex ${\\check C}^\\bullet(\\mathcal{U}, \\mathcal{G})$", "of $\\mathcal{G}$ with respect to the covering $\\mathcal{U}$.", "By the spectral sequence of", "Lemma \\ref{lemma-cech-spectral-sequence}", "and since $H^1(U_i, \\mathcal{G}) = 0$ for all $i$", "we see that ${\\check H}^2(\\mathcal{U}, \\mathcal{G}) \\to H^2(U, \\mathcal{G})$", "is injective. Hence $(g_{i_0i_1i_2})$ is a coboundary", "by our assumption that $H^2(U, \\mathcal{G}) = 0$.", "Thus we can find sections $g_{ij} \\in \\mathcal{G}(U_{ij})$ such that", "$g_{ik}^{-1}|_{U_{ijk}} g_{jk}|_{U_{ijk}} g_{ij}|_{U_{ijk}} = g_{ijk}$", "for all $i, j, k$.", "After replacing $\\varphi_{ij}$ by $\\varphi_{ij}g_{ij}^{-1}$", "we see that $\\varphi_{ij}$ gives a descent datum on the objects", "$x_i$ over $U_i$ and the proof is complete." ], "refs": [ "stacks-definition-gerbe", "sites-cohomology-lemma-torsors-h1", "sites-cohomology-lemma-trivial-torsor", "sites-cohomology-lemma-cech-spectral-sequence" ], "ref_ids": [ 9003, 4182, 4181, 4202 ] } ], "ref_ids": [ 8979 ] }, { "id": 4207, "type": "theorem", "label": "sites-cohomology-lemma-injective-module-injective-presheaf", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-injective-module-injective-presheaf", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "An injective sheaf of modules is also injective as an", "object in the category $\\textit{PMod}(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "Apply Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}", "to the categories $\\mathcal{A} = \\textit{Mod}(\\mathcal{O})$,", "$\\mathcal{B} = \\textit{PMod}(\\mathcal{O})$, the inclusion functor", "and sheafification. (See", "Modules on Sites,", "Section \\ref{sites-modules-section-sheafification-presheaves-modules}", "to see that all assumptions of the lemma are satisfied.)" ], "refs": [ "homology-lemma-adjoint-preserve-injectives" ], "ref_ids": [ 12116 ] } ], "ref_ids": [] }, { "id": 4208, "type": "theorem", "label": "sites-cohomology-lemma-include-modules", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-include-modules", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Consider the functor", "$i : \\textit{Mod}(\\mathcal{C}) \\to \\textit{PMod}(\\mathcal{C})$.", "It is a left exact functor with right derived functors given by", "$$", "R^pi(\\mathcal{F}) = \\underline{H}^p(\\mathcal{F}) :", "U \\longmapsto H^p(U, \\mathcal{F})", "$$", "see discussion in", "Section \\ref{section-locality}." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $i$ is left exact.", "Choose an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$", "in $\\textit{Mod}(\\mathcal{O})$.", "By definition $R^pi$ is the $p$th cohomology {\\it presheaf}", "of the complex $\\mathcal{I}^\\bullet$. In other words, the", "sections of $R^pi(\\mathcal{F})$ over an object $U$ of $\\mathcal{C}$", "are given by", "$$", "\\frac{\\Ker(\\mathcal{I}^n(U) \\to \\mathcal{I}^{n + 1}(U))}", "{\\Im(\\mathcal{I}^{n - 1}(U) \\to \\mathcal{I}^n(U))}.", "$$", "which is the definition of $H^p(U, \\mathcal{F})$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4209, "type": "theorem", "label": "sites-cohomology-lemma-injective-module-trivial-cech", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-injective-module-trivial-cech", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$.", "Let $\\mathcal{I}$ be an injective $\\mathcal{O}$-module, i.e., an injective", "object of $\\textit{Mod}(\\mathcal{O})$. Then", "$$", "\\check{H}^p(\\mathcal{U}, \\mathcal{I}) =", "\\left\\{", "\\begin{matrix}", "\\mathcal{I}(U) & \\text{if} & p = 0 \\\\", "0 & \\text{if} & p > 0", "\\end{matrix}", "\\right.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Lemma \\ref{lemma-cech-map-into} gives the first equality in the following", "sequence of equalities", "\\begin{align*}", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I})", "& =", "\\Mor_{\\textit{PAb}(\\mathcal{C})}(", "\\mathbf{Z}_{\\mathcal{U}, \\bullet}, \\mathcal{I}) \\\\", "& =", "\\Mor_{\\textit{PMod}(\\mathbf{Z})}(", "\\mathbf{Z}_{\\mathcal{U}, \\bullet}, \\mathcal{I}) \\\\", "& =", "\\Mor_{\\textit{PMod}(\\mathcal{O})}(", "\\mathbf{Z}_{\\mathcal{U}, \\bullet} \\otimes_{p, \\mathbf{Z}} \\mathcal{O},", "\\mathcal{I})", "\\end{align*}", "The third equality by", "Modules on Sites,", "Lemma \\ref{sites-modules-lemma-adjointness-tensor-restrict-presheaves}.", "By Lemma \\ref{lemma-injective-module-injective-presheaf}", "we see that $\\mathcal{I}$ is an injective object in", "$\\textit{PMod}(\\mathcal{O})$.", "Hence $\\Hom_{\\textit{PMod}(\\mathcal{O})}(-, \\mathcal{I})$", "is an exact functor. By", "Lemma \\ref{lemma-complex-tensored-still-exact} we see the vanishing of", "higher {\\v C}ech cohomology groups.", "For the zeroth see Lemma \\ref{lemma-cech-h0}." ], "refs": [ "sites-cohomology-lemma-cech-map-into", "sites-modules-lemma-adjointness-tensor-restrict-presheaves", "sites-cohomology-lemma-injective-module-injective-presheaf", "sites-cohomology-lemma-complex-tensored-still-exact", "sites-cohomology-lemma-cech-h0" ], "ref_ids": [ 4193, 14146, 4207, 4195, 4190 ] } ], "ref_ids": [] }, { "id": 4210, "type": "theorem", "label": "sites-cohomology-lemma-cohomology-modules-abelian-agree", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cohomology-modules-abelian-agree", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$.", "Let $\\mathcal{F}$ be an $\\mathcal{O}$-module, and denote", "$\\mathcal{F}_{ab}$ the underlying sheaf of abelian groups.", "Then we have", "$$", "H^i(\\mathcal{C}, \\mathcal{F}_{ab})", "=", "H^i(\\mathcal{C}, \\mathcal{F})", "$$", "and for any object $U$ of $\\mathcal{C}$ we also have", "$$", "H^i(U, \\mathcal{F}_{ab})", "=", "H^i(U, \\mathcal{F}).", "$$", "Here the left hand side is cohomology computed in", "$\\textit{Ab}(\\mathcal{C})$ and the right hand side", "is cohomology computed in $\\textit{Mod}(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "By", "Derived Categories, Lemma \\ref{derived-lemma-higher-derived-functors}", "the $\\delta$-functor $(\\mathcal{F} \\mapsto H^p(U, \\mathcal{F}))_{p \\geq 0}$", "is universal. The functor", "$\\textit{Mod}(\\mathcal{O}) \\to \\textit{Ab}(\\mathcal{C})$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_{ab}$ is exact. Hence", "$(\\mathcal{F} \\mapsto H^p(U, \\mathcal{F}_{ab}))_{p \\geq 0}$", "is a $\\delta$-functor also. Suppose we show that", "$(\\mathcal{F} \\mapsto H^p(U, \\mathcal{F}_{ab}))_{p \\geq 0}$", "is also universal. This will imply the second statement of the lemma", "by uniqueness of universal $\\delta$-functors, see", "Homology, Lemma \\ref{homology-lemma-uniqueness-universal-delta-functor}.", "Since $\\textit{Mod}(\\mathcal{O})$ has enough injectives,", "it suffices to show that $H^i(U, \\mathcal{I}_{ab}) = 0$", "for any injective object $\\mathcal{I}$ in $\\textit{Mod}(\\mathcal{O})$, see", "Homology, Lemma \\ref{homology-lemma-efface-implies-universal}.", "\\medskip\\noindent", "Let $\\mathcal{I}$ be an injective object of $\\textit{Mod}(\\mathcal{O})$.", "Apply Lemma \\ref{lemma-cech-vanish-collection}", "with $\\mathcal{F} = \\mathcal{I}$, $\\mathcal{B} = \\mathcal{C}$", "and $\\text{Cov} = \\text{Cov}_\\mathcal{C}$.", "Assumption (3) of that lemma holds by", "Lemma \\ref{lemma-injective-module-trivial-cech}.", "Hence we see that $H^i(U, \\mathcal{I}_{ab}) = 0$", "for every object $U$ of $\\mathcal{C}$.", "\\medskip\\noindent", "If $\\mathcal{C}$ has a final", "object then this also implies the first equality. If not, then", "according to Sites, Lemma \\ref{sites-lemma-topos-good-site} we see that", "the ringed topos $(\\Sh(\\mathcal{C}), \\mathcal{O})$ is equivalent to a", "ringed topos where the underlying site does have a final object.", "Hence the lemma follows." ], "refs": [ "derived-lemma-higher-derived-functors", "homology-lemma-uniqueness-universal-delta-functor", "homology-lemma-efface-implies-universal", "sites-cohomology-lemma-cech-vanish-collection", "sites-cohomology-lemma-injective-module-trivial-cech", "sites-lemma-topos-good-site" ], "ref_ids": [ 1869, 12053, 12052, 4205, 4209, 8581 ] } ], "ref_ids": [] }, { "id": 4211, "type": "theorem", "label": "sites-cohomology-lemma-cohomology-products", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cohomology-products", "contents": [ "Let $\\mathcal{C}$ be a site. Let $I$ be a set. For $i \\in I$ let ", "$\\mathcal{F}_i$ be an abelian sheaf on $\\mathcal{C}$. Let", "$U \\in \\Ob(\\mathcal{C})$. The canonical map", "$$", "H^p(U, \\prod\\nolimits_{i \\in I} \\mathcal{F}_i)", "\\longrightarrow", "\\prod\\nolimits_{i \\in I} H^p(U, \\mathcal{F}_i)", "$$", "is an isomorphism for $p = 0$ and injective for $p = 1$." ], "refs": [], "proofs": [ { "contents": [ "The statement for $p = 0$ is true because the product of sheaves", "is equal to the product of the underlying presheaves, see", "Sites, Lemma \\ref{sites-lemma-limit-sheaf}.", "Proof for $p = 1$. Set $\\mathcal{F} = \\prod \\mathcal{F}_i$.", "Let $\\xi \\in H^1(U, \\mathcal{F})$ map to zero in", "$\\prod H^1(U, \\mathcal{F}_i)$. By locality of cohomology, see", "Lemma \\ref{lemma-kill-cohomology-class-on-covering},", "there exists a covering $\\mathcal{U} = \\{U_j \\to U\\}$ such that", "$\\xi|_{U_j} = 0$ for all $j$. By Lemma \\ref{lemma-cech-h1} this means", "$\\xi$ comes from an element", "$\\check \\xi \\in \\check H^1(\\mathcal{U}, \\mathcal{F})$.", "Since the maps", "$\\check H^1(\\mathcal{U}, \\mathcal{F}_i) \\to H^1(U, \\mathcal{F}_i)$", "are injective for all $i$ (by Lemma \\ref{lemma-cech-h1}), and since", "the image of $\\xi$ is zero in $\\prod H^1(U, \\mathcal{F}_i)$ we see", "that the image", "$\\check \\xi_i = 0$ in $\\check H^1(\\mathcal{U}, \\mathcal{F}_i)$.", "However, since $\\mathcal{F} = \\prod \\mathcal{F}_i$ we see that", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$ is the", "product of the complexes", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}_i)$,", "hence by", "Homology, Lemma \\ref{homology-lemma-product-abelian-groups-exact}", "we conclude that $\\check \\xi = 0$ as desired." ], "refs": [ "sites-lemma-limit-sheaf", "sites-cohomology-lemma-kill-cohomology-class-on-covering", "sites-cohomology-lemma-cech-h1", "sites-cohomology-lemma-cech-h1", "homology-lemma-product-abelian-groups-exact" ], "ref_ids": [ 8508, 4188, 4200, 4200, 12130 ] } ], "ref_ids": [] }, { "id": 4212, "type": "theorem", "label": "sites-cohomology-lemma-restriction-along-monomorphism-surjective", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-restriction-along-monomorphism-surjective", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $a : U' \\to U$ be a", "monomorphism in $\\mathcal{C}$. Then for any injective $\\mathcal{O}$-module", "$\\mathcal{I}$ the restriction mapping $\\mathcal{I}(U) \\to \\mathcal{I}(U')$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $j : \\mathcal{C}/U \\to \\mathcal{C}$ and", "$j' : \\mathcal{C}/U' \\to \\mathcal{C}$ be the localization morphisms", "(Modules on Sites, Section \\ref{sites-modules-section-localize}).", "Since $j_!$ is a left adjoint to restriction we see that", "for any sheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules", "$$", "\\Hom_\\mathcal{O}(j_!\\mathcal{O}_U, \\mathcal{F})", "=", "\\Hom_{\\mathcal{O}_U}(\\mathcal{O}_U, \\mathcal{F}|_U)", "=", "\\mathcal{F}(U)", "$$", "Similarly, the sheaf $j'_!\\mathcal{O}_{U'}$ represents the", "functor $\\mathcal{F} \\mapsto \\mathcal{F}(U')$.", "Moreover below we describe a canonical map of $\\mathcal{O}$-modules", "$$", "j'_!\\mathcal{O}_{U'} \\longrightarrow j_!\\mathcal{O}_U", "$$", "which corresponds to the restriction mapping", "$\\mathcal{F}(U) \\to \\mathcal{F}(U')$ via Yoneda's lemma", "(Categories, Lemma \\ref{categories-lemma-yoneda}).", "It suffices to prove the displayed map of modules is injective, see", "Homology, Lemma \\ref{homology-lemma-characterize-injectives}.", "\\medskip\\noindent", "To construct our map it suffices to construct a map between the", "presheaves which assign to an object $V$ of $\\mathcal{C}$ the", "$\\mathcal{O}(V)$-module", "$$", "\\bigoplus\\nolimits_{\\varphi' \\in \\Mor_\\mathcal{C}(V, U')} \\mathcal{O}(V)", "\\quad\\text{and}\\quad", "\\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)} \\mathcal{O}(V)", "$$", "see Modules on Sites, Lemma \\ref{sites-modules-lemma-extension-by-zero}.", "We take the map which maps the summand corresponding to $\\varphi'$", "to the summand corresponding to $\\varphi = a \\circ \\varphi'$", "by the identity map on $\\mathcal{O}(V)$. As $a$ is a monomorphism,", "this map is injective. As sheafification is exact, the result", "follows." ], "refs": [ "categories-lemma-yoneda", "homology-lemma-characterize-injectives", "sites-modules-lemma-extension-by-zero" ], "ref_ids": [ 12203, 12112, 14169 ] } ], "ref_ids": [] }, { "id": 4213, "type": "theorem", "label": "sites-cohomology-lemma-compute-cohomology-on-sheaf-sets", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-compute-cohomology-on-sheaf-sets", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $K$ be a presheaf of sets on $\\mathcal{C}$.", "Let $\\mathcal{F}$ be an $\\mathcal{O}$-module and denote", "$\\mathcal{F}_{ab}$ the underlying sheaf of abelian groups.", "Then $H^p(K, \\mathcal{F}) = H^p(K, \\mathcal{F}_{ab})$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $K$ by its sheafification and assume $K$ is a sheaf.", "Note that both $H^p(K, \\mathcal{F})$ and $H^p(K, \\mathcal{F}_{ab})$", "depend only on the topos, not on the underlying site. Hence by", "Sites, Lemma \\ref{sites-lemma-topos-good-site}", "we may replace $\\mathcal{C}$ by a ``larger'' site such", "that $K = h_U$ for some object $U$ of $\\mathcal{C}$.", "In this case the result follows from", "Lemma \\ref{lemma-cohomology-modules-abelian-agree}." ], "refs": [ "sites-lemma-topos-good-site", "sites-cohomology-lemma-cohomology-modules-abelian-agree" ], "ref_ids": [ 8581, 4210 ] } ], "ref_ids": [] }, { "id": 4214, "type": "theorem", "label": "sites-cohomology-lemma-cech-to-cohomology-sheaf-sets", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cech-to-cohomology-sheaf-sets", "contents": [ "Let $\\mathcal{C}$ be a site. Let $K' \\to K$ be a map of presheaves", "of sets on $\\mathcal{C}$ whose sheafification is surjective. Set", "$K'_p = K' \\times_K \\ldots \\times_K K'$ ($p + 1$-factors).", "For every abelian sheaf $\\mathcal{F}$ there is a spectral sequence", "with $E_1^{p, q} = H^q(K'_p, \\mathcal{F})$ converging to", "$H^{p + q}(K, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Since sheafification is exact, we see that", "$(K_p')^\\#$ is equal to", "$(K')^\\# \\times_{K^\\#} \\ldots \\times_{K^\\#} (K')^\\#$", "($p + 1$-factors). Thus we may replace $K$ and $K'$ by", "their sheafifications and assume $K \\to K'$ is a surjective", "map of sheaves. After replacing $\\mathcal{C}$ by a ``larger'' site as in", "Sites, Lemma \\ref{sites-lemma-topos-good-site} ", "we may assume that $K, K'$ are objects of $\\mathcal{C}$ and that", "$\\mathcal{U} = \\{K' \\to K\\}$ is a covering. Then we have the {\\v C}ech", "to cohomology spectral sequence of Lemma \\ref{lemma-cech-spectral-sequence}", "whose $E_1$ page is as indicated in the statement of the lemma." ], "refs": [ "sites-lemma-topos-good-site", "sites-cohomology-lemma-cech-spectral-sequence" ], "ref_ids": [ 8581, 4202 ] } ], "ref_ids": [] }, { "id": 4215, "type": "theorem", "label": "sites-cohomology-lemma-cohomology-on-sheaf-sets", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cohomology-on-sheaf-sets", "contents": [ "Let $\\mathcal{C}$ be a site. Let $K$ be a sheaf of sets on $\\mathcal{C}$.", "Consider the morphism of topoi", "$j : \\Sh(\\mathcal{C}/K) \\to \\Sh(\\mathcal{C})$, see", "Sites, Lemma \\ref{sites-lemma-localize-topos-site}.", "Then $j^{-1}$ preserves injectives and", "$H^p(K, \\mathcal{F}) = H^p(\\mathcal{C}/K, j^{-1}\\mathcal{F})$", "for any abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}$." ], "refs": [ "sites-lemma-localize-topos-site" ], "proofs": [ { "contents": [ "By", "Sites, Lemmas \\ref{sites-lemma-localize-topos} and", "\\ref{sites-lemma-localize-topos-site}", "the morphism of topoi $j$ is", "equivalent to a localization. Hence this follows from", "Lemma \\ref{lemma-cohomology-of-open}." ], "refs": [ "sites-lemma-localize-topos", "sites-lemma-localize-topos-site", "sites-cohomology-lemma-cohomology-of-open" ], "ref_ids": [ 8583, 8585, 4186 ] } ], "ref_ids": [ 8585 ] }, { "id": 4216, "type": "theorem", "label": "sites-cohomology-lemma-characterize-limp", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-characterize-limp", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be an abelian sheaf. If", "\\begin{enumerate}", "\\item $H^p(U, \\mathcal{F}) = 0$ for $p > 0$ and $U \\in \\Ob(\\mathcal{C})$, and", "\\item for every surjection $K' \\to K$ of sheaves of sets the", "extended {\\v C}ech complex", "$$", "0 \\to H^0(K, \\mathcal{F}) \\to H^0(K', \\mathcal{F}) \\to", "H^0(K' \\times_K K', \\mathcal{F}) \\to \\ldots", "$$", "is exact,", "\\end{enumerate}", "then $\\mathcal{F}$ is totally acyclic (and the converse holds too)." ], "refs": [], "proofs": [ { "contents": [ "By assumption (1) we have $H^p(h_U^\\#, g^{-1}\\mathcal{I}) = 0$ for all", "$p > 0$ and all objects $U$ of $\\mathcal{C}$. Note that if", "$K = \\coprod K_i$ is a coproduct of sheaves of sets on $\\mathcal{C}$", "then $H^p(K, g^{-1}\\mathcal{I}) = \\prod H^p(K_i, g^{-1}\\mathcal{I})$.", "For any sheaf of sets $K$ there exists a surjection", "$$", "K' = \\coprod h_{U_i}^\\# \\longrightarrow K", "$$", "see Sites, Lemma \\ref{sites-lemma-sheaf-coequalizer-representable}.", "Thus we conclude that: (*) for every sheaf of sets $K$ there exists a", "surjection $K' \\to K$ of sheaves of sets such that $H^p(K', \\mathcal{F}) = 0$", "for $p > 0$. We claim that (*) and condition (2) imply that $\\mathcal{F}$", "is totally acyclic.", "Note that conditions (*) and (2) only depend on $\\mathcal{F}$ as an", "object of the topos $\\Sh(\\mathcal{C})$ and not on the underlying site.", "(We will not use property (1) in the rest of the proof.)", "\\medskip\\noindent", "We are going to prove by induction on $n \\geq 0$ that (*) and (2)", "imply the following induction hypothesis $IH_n$:", "$H^p(K, \\mathcal{F}) = 0$ for all $0 < p \\leq n$ and", "all sheaves of sets $K$. Note that $IH_0$ holds. Assume $IH_n$. Pick", "a sheaf of sets $K$. Pick a surjection $K' \\to K$ such that", "$H^p(K', \\mathcal{F}) = 0$ for all $p > 0$. We have a", "spectral sequence with", "$$", "E_1^{p, q} = H^q(K'_p, \\mathcal{F})", "$$", "covering to $H^{p + q}(K, \\mathcal{F})$, see", "Lemma \\ref{lemma-cech-to-cohomology-sheaf-sets}.", "By $IH_n$ we see that $E_1^{p, q} = 0$ for $0 < q \\leq n$ and by", "assumption (2) we see that $E_2^{p, 0} = 0$ for $p > 0$. Finally, we have", "$E_1^{0, q} = 0$ for $q > 0$ because $H^q(K', \\mathcal{F}) = 0$ by", "choice of $K'$. Hence we conclude that $H^{n + 1}(K, \\mathcal{F}) = 0$", "because all the terms $E_2^{p, q}$ with $p + q = n + 1$ are zero." ], "refs": [ "sites-lemma-sheaf-coequalizer-representable", "sites-cohomology-lemma-cech-to-cohomology-sheaf-sets" ], "ref_ids": [ 8520, 4214 ] } ], "ref_ids": [] }, { "id": 4217, "type": "theorem", "label": "sites-cohomology-lemma-direct-image-injective-sheaf", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-direct-image-injective-sheaf", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.", "Then for any injective object $\\mathcal{I}$ in", "$\\textit{Mod}(\\mathcal{O}_\\mathcal{C})$", "the pushforward $f_*\\mathcal{I}$ is totally acyclic." ], "refs": [], "proofs": [ { "contents": [ "Let $K$ be a sheaf of sets on $\\mathcal{D}$.", "By", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites}", "we may replace $\\mathcal{C}$, $\\mathcal{D}$ by ``larger'' sites such", "that $f$ comes from a morphism of ringed sites induced by a continuous", "functor $u : \\mathcal{D} \\to \\mathcal{C}$ such that", "$K = h_V$ for some object $V$ of $\\mathcal{D}$.", "\\medskip\\noindent", "Thus we have to show that $H^q(V, f_*\\mathcal{I})$ is zero", "for $q > 0$ and all objects $V$ of $\\mathcal{D}$ when $f$ is given", "by a morphism of ringed sites. Let $\\mathcal{V} = \\{V_j \\to V\\}$", "be any covering of $\\mathcal{D}$. Since $u$ is continuous we see that", "$\\mathcal{U} = \\{u(V_j) \\to u(V)\\}$ is a covering of $\\mathcal{C}$.", "Then we have an equality of {\\v C}ech complexes", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, f_*\\mathcal{I})", "=", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I})", "$$", "by the definition of $f_*$. By", "Lemma \\ref{lemma-injective-module-trivial-cech}", "we see that the cohomology of this complex is zero in positive degrees.", "We win by", "Lemma \\ref{lemma-cech-vanish-collection}." ], "refs": [ "sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites", "sites-cohomology-lemma-injective-module-trivial-cech", "sites-cohomology-lemma-cech-vanish-collection" ], "ref_ids": [ 14145, 4209, 4205 ] } ], "ref_ids": [] }, { "id": 4218, "type": "theorem", "label": "sites-cohomology-lemma-pushforward-injective-flat", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-pushforward-injective-flat", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.", "If $f$ is flat, then $f_*\\mathcal{I}$ is an injective", "$\\mathcal{O}_\\mathcal{D}$-module", "for any injective $\\mathcal{O}_\\mathcal{C}$-module $\\mathcal{I}$." ], "refs": [], "proofs": [ { "contents": [ "In this case the functor $f^*$ is exact, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-flat-pullback-exact}.", "Hence the result follows from", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}." ], "refs": [ "sites-modules-lemma-flat-pullback-exact", "homology-lemma-adjoint-preserve-injectives" ], "ref_ids": [ 14223, 12116 ] } ], "ref_ids": [] }, { "id": 4219, "type": "theorem", "label": "sites-cohomology-lemma-limp-acyclic", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-limp-acyclic", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$ be a ringed topos.", "A totally acyclic sheaf is right acyclic for the following functors:", "\\begin{enumerate}", "\\item the functor $H^0(U, -)$ for any object $U$ of $\\mathcal{C}$,", "\\item the functor $\\mathcal{F} \\mapsto \\mathcal{F}(K)$ for any", "presheaf of sets $K$,", "\\item the functor $\\Gamma(\\mathcal{C}, -)$ of global sections,", "\\item the functor $f_*$ for any morphism", "$f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ of ringed topoi.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) is the definition of a totally acyclic sheaf.", "Part (1) is a consequence of (2) as pointed out in the discussion following the", "definition of totally acyclic sheaves.", "Part (3) is a special case of (2) where $K = e$ is the final object", "of $\\Sh(\\mathcal{C})$.", "\\medskip\\noindent", "To prove (4) we may assume, by", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites}", "that $f$ is given by a morphism of sites. In this case we see that", "$R^if_*$, $i > 0$ of a totally acyclic sheaf are zero by the description of", "higher direct images in", "Lemma \\ref{lemma-higher-direct-images}." ], "refs": [ "sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites", "sites-cohomology-lemma-higher-direct-images" ], "ref_ids": [ 14145, 4189 ] } ], "ref_ids": [] }, { "id": 4220, "type": "theorem", "label": "sites-cohomology-lemma-Leray", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-Leray", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.", "Let $\\mathcal{F}^\\bullet$ be a bounded below complex of", "$\\mathcal{O}_\\mathcal{C}$-modules. There is a spectral sequence", "$$", "E_2^{p, q} = H^p(\\mathcal{D}, R^qf_*(\\mathcal{F}^\\bullet))", "$$", "converging to $H^{p + q}(\\mathcal{C}, \\mathcal{F}^\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "This is just the Grothendieck spectral sequence", "Derived Categories, Lemma \\ref{derived-lemma-grothendieck-spectral-sequence}", "coming from the composition of functors", "$\\Gamma(\\mathcal{C}, -) = \\Gamma(\\mathcal{D}, -) \\circ f_*$.", "To see that the assumptions of", "Derived Categories, Lemma \\ref{derived-lemma-grothendieck-spectral-sequence}", "are satisfied, see", "Lemmas \\ref{lemma-direct-image-injective-sheaf} and", "\\ref{lemma-limp-acyclic}." ], "refs": [ "derived-lemma-grothendieck-spectral-sequence", "derived-lemma-grothendieck-spectral-sequence", "sites-cohomology-lemma-direct-image-injective-sheaf", "sites-cohomology-lemma-limp-acyclic" ], "ref_ids": [ 1873, 1873, 4217, 4219 ] } ], "ref_ids": [] }, { "id": 4221, "type": "theorem", "label": "sites-cohomology-lemma-apply-Leray", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-apply-Leray", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{C}$-module.", "\\begin{enumerate}", "\\item If $R^qf_*\\mathcal{F} = 0$ for $q > 0$, then", "$H^p(\\mathcal{C}, \\mathcal{F}) = H^p(\\mathcal{D}, f_*\\mathcal{F})$ for all $p$.", "\\item If $H^p(\\mathcal{D}, R^qf_*\\mathcal{F}) = 0$ for all $q$ and $p > 0$,", "then $H^q(\\mathcal{C}, \\mathcal{F}) = H^0(\\mathcal{D}, R^qf_*\\mathcal{F})$", "for all $q$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "These are two simple conditions that force the Leray spectral sequence to", "converge. You can also prove these facts directly (without using the", "spectral sequence) which is a good exercise in cohomology of sheaves." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4222, "type": "theorem", "label": "sites-cohomology-lemma-relative-Leray", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-relative-Leray", "contents": [ "Let", "$f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "and", "$g : (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\to", "(\\Sh(\\mathcal{E}), \\mathcal{O}_\\mathcal{E})$", "be morphisms of ringed topoi.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{C}$-module.", "There is a spectral sequence with", "$$", "E_2^{p, q} = R^pg_*(R^qf_*\\mathcal{F})", "$$", "converging to $R^{p + q}(g \\circ f)_*\\mathcal{F}$.", "This spectral sequence is functorial in $\\mathcal{F}$, and there", "is a version for bounded below complexes of $\\mathcal{O}_\\mathcal{C}$-modules." ], "refs": [], "proofs": [ { "contents": [ "This is a Grothendieck spectral sequence for composition of functors, see", "Derived Categories, Lemma \\ref{derived-lemma-grothendieck-spectral-sequence}", "and", "Lemmas \\ref{lemma-direct-image-injective-sheaf} and", "\\ref{lemma-limp-acyclic}." ], "refs": [ "derived-lemma-grothendieck-spectral-sequence", "sites-cohomology-lemma-direct-image-injective-sheaf", "sites-cohomology-lemma-limp-acyclic" ], "ref_ids": [ 1873, 4217, 4219 ] } ], "ref_ids": [] }, { "id": 4223, "type": "theorem", "label": "sites-cohomology-lemma-base-change-map-flat-case", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-base-change-map-flat-case", "contents": [ "Let", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})", "\\ar[r]_{g'} \\ar[d]_{f'} &", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^f \\\\", "(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'})", "\\ar[r]^g &", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})", "}", "$$", "be a commutative diagram of ringed topoi.", "Let $\\mathcal{F}^\\bullet$ be a bounded below complex of", "$\\mathcal{O}_\\mathcal{C}$-modules.", "Assume both $g$ and $g'$ are flat.", "Then there exists a canonical base change map", "$$", "g^*Rf_*\\mathcal{F}^\\bullet", "\\longrightarrow", "R(f')_*(g')^*\\mathcal{F}^\\bullet", "$$", "in $D^{+}(\\mathcal{O}_{\\mathcal{D}'})$." ], "refs": [], "proofs": [ { "contents": [ "Choose injective resolutions $\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$", "and $(g')^*\\mathcal{F}^\\bullet \\to \\mathcal{J}^\\bullet$.", "By Lemma \\ref{lemma-pushforward-injective-flat} we see that", "$(g')_*\\mathcal{J}^\\bullet$ is a complex of injectives representing", "$R(g')_*(g')^*\\mathcal{F}^\\bullet$. Hence by", "Derived Categories, Lemmas \\ref{derived-lemma-morphisms-lift}", "and \\ref{derived-lemma-morphisms-equal-up-to-homotopy}", "the arrow $\\beta$ in the diagram", "$$", "\\xymatrix{", "(g')_*(g')^*\\mathcal{F}^\\bullet \\ar[r] &", "(g')_*\\mathcal{J}^\\bullet \\\\", "\\mathcal{F}^\\bullet \\ar[u]^{adjunction} \\ar[r] &", "\\mathcal{I}^\\bullet \\ar[u]_\\beta", "}", "$$", "exists and is unique up to homotopy.", "Pushing down to $\\mathcal{D}$ we get", "$$", "f_*\\beta :", "f_*\\mathcal{I}^\\bullet", "\\longrightarrow", "f_*(g')_*\\mathcal{J}^\\bullet", "=", "g_*(f')_*\\mathcal{J}^\\bullet", "$$", "By adjunction of $g^*$ and $g_*$ we get a map of complexes", "$g^*f_*\\mathcal{I}^\\bullet \\to (f')_*\\mathcal{J}^\\bullet$.", "Note that this map is unique up to homotopy since the only", "choice in the whole process was the choice of the map $\\beta$", "and everything was done on the level of complexes." ], "refs": [ "sites-cohomology-lemma-pushforward-injective-flat", "derived-lemma-morphisms-lift", "derived-lemma-morphisms-equal-up-to-homotopy" ], "ref_ids": [ 4218, 1853, 1854 ] } ], "ref_ids": [] }, { "id": 4224, "type": "theorem", "label": "sites-cohomology-lemma-colim-works-over-collection", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-colim-works-over-collection", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\text{Cov}_\\mathcal{C}$ be the set", "of coverings of $\\mathcal{C}$ (see", "Sites, Definition \\ref{sites-definition-site}). Let", "$\\mathcal{B} \\subset \\Ob(\\mathcal{C})$, and", "$\\text{Cov} \\subset \\text{Cov}_\\mathcal{C}$", "be subsets. Assume that", "\\begin{enumerate}", "\\item For every $\\mathcal{U} \\in \\text{Cov}$ we have", "$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ with $I$ finite,", "$U, U_i \\in \\mathcal{B}$ and every", "$U_{i_0} \\times_U \\ldots \\times_U U_{i_p} \\in \\mathcal{B}$.", "\\item For every $U \\in \\mathcal{B}$ the coverings of $U$", "occurring in $\\text{Cov}$ is a cofinal system of coverings of $U$.", "\\end{enumerate}", "Then the map", "$$", "\\colim_i H^p(U, \\mathcal{F}_i)", "\\longrightarrow", "H^p(U, \\colim_i \\mathcal{F}_i)", "$$", "is an isomorphism for every $p \\geq 0$, every $U \\in \\mathcal{B}$, and", "every filtered diagram $\\mathcal{I} \\to \\textit{Ab}(\\mathcal{C})$." ], "refs": [ "sites-definition-site" ], "proofs": [ { "contents": [ "To prove the lemma we will argue by induction on $p$.", "Note that we require in (1) the coverings $\\mathcal{U} \\in \\text{Cov}$", "to be finite, so that all the elements of $\\mathcal{B}$ are quasi-compact.", "Hence (2) and (1) imply that any $U \\in \\mathcal{B}$ satisfies the hypothesis", "of Sites, Lemma \\ref{sites-lemma-directed-colimits-sections} (4).", "Thus we see that the result holds for $p = 0$.", "Now we assume the lemma holds for $p$ and prove it for $p + 1$.", "\\medskip\\noindent", "Choose a filtered diagram", "$\\mathcal{F} : \\mathcal{I} \\to \\textit{Ab}(\\mathcal{C})$,", "$i \\mapsto \\mathcal{F}_i$.", "Since $\\textit{Ab}(\\mathcal{C})$ has functorial injective embeddings, see", "Injectives, Theorem \\ref{injectives-theorem-sheaves-injectives},", "we can find a morphism of filtered diagrams", "$\\mathcal{F} \\to \\mathcal{I}$", "such that each $\\mathcal{F}_i \\to \\mathcal{I}_i$ is an injective map of", "abelian sheaves into an injective abelian sheaf. Denote $\\mathcal{Q}_i$", "the cokernel so that we have short exact sequences", "$$", "0 \\to", "\\mathcal{F}_i \\to", "\\mathcal{I}_i \\to", "\\mathcal{Q}_i \\to 0.", "$$", "Since colimits of sheaves are the sheafification of colimits on the level", "of presheaves, since sheafification is exact, and since filtered", "colimits of abelian groups are exact", "(see Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}),", "we see the sequence", "$$", "0 \\to", "\\colim_i \\mathcal{F}_i \\to", "\\colim_i \\mathcal{I}_i \\to", "\\colim_i \\mathcal{Q}_i \\to 0.", "$$", "is also a short exact sequence. We claim that", "$H^q(U, \\colim_i \\mathcal{I}_i) = 0$ for all $U \\in \\mathcal{B}$", "and all $q \\geq 1$. Accepting this claim", "for the moment consider the diagram", "$$", "\\xymatrix{", "\\colim_i H^p(U, \\mathcal{I}_i) \\ar[d] \\ar[r] &", "\\colim_i H^p(U, \\mathcal{Q}_i) \\ar[d] \\ar[r] &", "\\colim_i H^{p + 1}(U, \\mathcal{F}_i) \\ar[d] \\ar[r] &", "0 \\ar[d] \\\\", "H^p(U, \\colim_i \\mathcal{I}_i) \\ar[r] &", "H^p(U, \\colim_i \\mathcal{Q}_i) \\ar[r] &", "H^{p + 1}(U, \\colim_i \\mathcal{F}_i) \\ar[r] &", "0", "}", "$$", "The zero at the lower right corner comes from the claim and the", "zero at the upper right corner comes from the fact that the sheaves", "$\\mathcal{I}_i$ are injective.", "The top row is exact by an application of", "Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}.", "Hence by the snake lemma we deduce the", "result for $p + 1$.", "\\medskip\\noindent", "It remains to show that the claim is true. We will use", "Lemma \\ref{lemma-cech-vanish-collection}.", "By the result for $p = 0$ we see that for $\\mathcal{U} \\in \\text{Cov}$", "we have", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\colim_i \\mathcal{I}_i)", "=", "\\colim_i \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}_i)", "$$", "because all the $U_{j_0} \\times_U \\ldots \\times_U U_{j_p}$", "are in $\\mathcal{B}$. By", "Lemma \\ref{lemma-injective-trivial-cech}", "each of the complexes in the colimit of {\\v C}ech complexes is", "acyclic in degree $\\geq 1$. Hence by", "Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}", "we see that also the {\\v C}ech complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\colim_i \\mathcal{I}_i)$", "is acyclic in degrees $\\geq 1$. In other words we see that", "$\\check{H}^p(\\mathcal{U}, \\colim_i \\mathcal{I}_i) = 0$", "for all $p \\geq 1$. Thus the assumptions of", "Lemma \\ref{lemma-cech-vanish-collection}.", "are satisfied and the claim follows." ], "refs": [ "sites-lemma-directed-colimits-sections", "injectives-theorem-sheaves-injectives", "algebra-lemma-directed-colimit-exact", "algebra-lemma-directed-colimit-exact", "sites-cohomology-lemma-cech-vanish-collection", "sites-cohomology-lemma-injective-trivial-cech", "algebra-lemma-directed-colimit-exact", "sites-cohomology-lemma-cech-vanish-collection" ], "ref_ids": [ 8531, 7765, 343, 343, 4205, 4198, 343, 4205 ] } ], "ref_ids": [ 8652 ] }, { "id": 4225, "type": "theorem", "label": "sites-cohomology-lemma-colim-sites-injective", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-colim-sites-injective", "contents": [ "Let $\\mathcal{I}$ be a cofiltered index category and let", "$(\\mathcal{C}_i, f_a)$ be an inverse system of sites over $\\mathcal{I}$", "as in Sites, Situation \\ref{sites-situation-inverse-limit-sites}.", "Set $\\mathcal{C} = \\colim \\mathcal{C}_i$ as in Sites,", "Lemmas \\ref{sites-lemma-colimit-sites} and", "\\ref{sites-lemma-compute-pullback-to-limit}.", "Moreover, assume given", "\\begin{enumerate}", "\\item an abelian sheaf $\\mathcal{F}_i$ on $\\mathcal{C}_i$ for all", "$i \\in \\Ob(\\mathcal{I})$,", "\\item for $a : j \\to i$ a map", "$\\varphi_a : f_a^{-1}\\mathcal{F}_i \\to \\mathcal{F}_j$", "of abelian sheaves on $\\mathcal{C}_j$", "\\end{enumerate}", "such that $\\varphi_c = \\varphi_b \\circ f_b^{-1}\\varphi_a$", "whenever $c = a \\circ b$. Then there exists a map of systems", "$(\\mathcal{F}_i, \\varphi_a) \\to (\\mathcal{G}_i, \\psi_a)$", "such that $\\mathcal{F}_i \\to \\mathcal{G}_i$ is injective and", "$\\mathcal{G}_i$ is an injective abelian sheaf." ], "refs": [ "sites-lemma-colimit-sites", "sites-lemma-compute-pullback-to-limit" ], "proofs": [ { "contents": [ "For each $i$ we pick an injection $\\mathcal{F}_i \\to \\mathcal{A}_i$", "where $\\mathcal{A}_i$ is an injective abelian sheaf on $\\mathcal{C}_i$.", "Then we can consider the family of maps", "$$", "\\gamma_i :", "\\mathcal{F}_i", "\\longrightarrow", "\\prod\\nolimits_{b : k \\to i} f_{b, *}\\mathcal{A}_k = \\mathcal{G}_i", "$$", "where the component maps are the maps adjoint to the maps", "$f_b^{-1}\\mathcal{F}_i \\to \\mathcal{F}_k \\to \\mathcal{A}_k$.", "For $a : j \\to i$ in $\\mathcal{I}$ there is a canonical map", "$$", "\\psi_a : f_a^{-1}\\mathcal{G}_i \\to \\mathcal{G}_j", "$$", "whose components are the canonical maps", "$f_b^{-1}f_{a \\circ b, *}\\mathcal{A}_k \\to f_{b, *}\\mathcal{A}_k$", "for $b : k \\to j$. Thus we find an injection", "$(\\gamma_i) : (\\mathcal{F}_i, \\varphi_a) \\to (\\mathcal{G}_i, \\psi_a)$", "of systems of abelian sheaves. Note that $\\mathcal{G}_i$ is an injective", "sheaf of abelian groups on $\\mathcal{C}_i$, see", "Lemma \\ref{lemma-pushforward-injective-flat} and", "Homology, Lemma \\ref{homology-lemma-product-injectives}.", "This finishes the construction." ], "refs": [ "sites-cohomology-lemma-pushforward-injective-flat", "homology-lemma-product-injectives" ], "ref_ids": [ 4218, 12113 ] } ], "ref_ids": [ 8532, 8533 ] }, { "id": 4226, "type": "theorem", "label": "sites-cohomology-lemma-colimit", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-colimit", "contents": [ "In the situation of Lemma \\ref{lemma-colim-sites-injective} set", "$\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$.", "Let $i \\in \\Ob(\\mathcal{I})$, $X_i \\in \\text{Ob}(\\mathcal{C}_i)$. Then", "$$", "\\colim_{a : j \\to i} H^p(u_a(X_i), \\mathcal{F}_j) =", "H^p(u_i(X_i), \\mathcal{F})", "$$", "for all $p \\geq 0$." ], "refs": [ "sites-cohomology-lemma-colim-sites-injective" ], "proofs": [ { "contents": [ "The case $p = 0$ is Sites, Lemma \\ref{sites-lemma-colimit}.", "\\medskip\\noindent", "Choose $(\\mathcal{F}_i, \\varphi_a) \\to (\\mathcal{G}_i, \\psi_a)$", "as in Lemma \\ref{lemma-colim-sites-injective}.", "Arguing exactly as in the proof of", "Lemma \\ref{lemma-colim-works-over-collection}", "we see that it suffices to prove that", "$H^p(X, \\colim f_i^{-1}\\mathcal{G}_i) = 0$ for $p > 0$.", "\\medskip\\noindent", "Set $\\mathcal{G} = \\colim f_i^{-1}\\mathcal{G}_i$.", "To show vanishing of cohomology of $\\mathcal{G}$ on every object", "of $\\mathcal{C}$ we show that the {\\v C}ech cohomology of $\\mathcal{G}$", "for any covering $\\mathcal{U}$ of $\\mathcal{C}$ is zero", "(Lemma \\ref{lemma-cech-vanish-collection}).", "The covering $\\mathcal{U}$ comes from a covering", "$\\mathcal{U}_i$ of $\\mathcal{C}_i$ for some $i$. We have", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{G}) =", "\\colim_{a : j \\to i}", "\\check{\\mathcal{C}}^\\bullet(u_a(\\mathcal{U}_i), \\mathcal{G}_j)", "$$", "by the case $p = 0$. The right hand side is acyclic in positive degrees", "as a filtered colimit of acyclic complexes by", "Lemma \\ref{lemma-injective-trivial-cech}. See", "Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}." ], "refs": [ "sites-lemma-colimit", "sites-cohomology-lemma-colim-sites-injective", "sites-cohomology-lemma-colim-works-over-collection", "sites-cohomology-lemma-cech-vanish-collection", "sites-cohomology-lemma-injective-trivial-cech", "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 8534, 4225, 4224, 4205, 4198, 343 ] } ], "ref_ids": [ 4225 ] }, { "id": 4227, "type": "theorem", "label": "sites-cohomology-lemma-derived-tor-exact", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-derived-tor-exact", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{G}^\\bullet$ be a complex of $\\mathcal{O}$-modules.", "The functors", "$$", "K(\\textit{Mod}(\\mathcal{O}))", "\\longrightarrow", "K(\\textit{Mod}(\\mathcal{O})),", "\\quad", "\\mathcal{F}^\\bullet \\longmapsto", "\\text{Tot}(\\mathcal{G}^\\bullet \\otimes_\\mathcal{O} \\mathcal{F}^\\bullet)", "$$", "and", "$$", "K(\\textit{Mod}(\\mathcal{O}))", "\\longrightarrow", "K(\\textit{Mod}(\\mathcal{O})),", "\\quad", "\\mathcal{F}^\\bullet \\longmapsto", "\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{G}^\\bullet)", "$$", "are exact functors of triangulated categories." ], "refs": [], "proofs": [ { "contents": [ "This follows from Derived Categories, Remark", "\\ref{derived-remark-double-complex-as-tensor-product-of}." ], "refs": [ "derived-remark-double-complex-as-tensor-product-of" ], "ref_ids": [ 2014 ] } ], "ref_ids": [] }, { "id": 4228, "type": "theorem", "label": "sites-cohomology-lemma-K-flat-quasi-isomorphism", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-K-flat-quasi-isomorphism", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{K}^\\bullet$ be a K-flat complex.", "Then the functor", "$$", "K(\\textit{Mod}(\\mathcal{O}))", "\\longrightarrow", "K(\\textit{Mod}(\\mathcal{O})), \\quad", "\\mathcal{F}^\\bullet", "\\longmapsto", "\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{K}^\\bullet)", "$$", "transforms quasi-isomorphisms into quasi-isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-derived-tor-exact}", "and the fact that quasi-isomorphisms are characterized by having", "acyclic cones." ], "refs": [ "sites-cohomology-lemma-derived-tor-exact" ], "ref_ids": [ 4227 ] } ], "ref_ids": [] }, { "id": 4229, "type": "theorem", "label": "sites-cohomology-lemma-restriction-K-flat", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-restriction-K-flat", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $U$ be an object of $\\mathcal{C}$.", "If $\\mathcal{K}^\\bullet$ is a K-flat complex of $\\mathcal{O}$-modules, then", "$\\mathcal{K}^\\bullet|_U$ is a K-flat complex of $\\mathcal{O}_U$-modules." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{G}^\\bullet$ be an exact complex of $\\mathcal{O}_U$-modules.", "Since $j_{U!}$ is exact", "(Modules on Sites, Lemma \\ref{sites-modules-lemma-extension-by-zero-exact})", "and $\\mathcal{K}^\\bullet$ is a K-flat complex of $\\mathcal{O}$-modules", "we see that the complex", "$$", "j_{U!}(\\text{Tot}(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_U}", "\\mathcal{K}^\\bullet|_U)) =", "\\text{Tot}(j_{U!}\\mathcal{G}^\\bullet \\otimes_\\mathcal{O} \\mathcal{K}^\\bullet)", "$$", "is exact. Here the equality comes from", "Modules on Sites, Lemma \\ref{sites-modules-lemma-j-shriek-and-tensor}", "and the fact that $j_{U!}$ commutes with direct sums (as a left adjoint).", "We conclude because $j_{U!}$ reflects exactness by", "Modules on Sites, Lemma \\ref{sites-modules-lemma-j-shriek-reflects-exactness}." ], "refs": [ "sites-modules-lemma-extension-by-zero-exact", "sites-modules-lemma-j-shriek-and-tensor", "sites-modules-lemma-j-shriek-reflects-exactness" ], "ref_ids": [ 14170, 14197, 14171 ] } ], "ref_ids": [] }, { "id": 4230, "type": "theorem", "label": "sites-cohomology-lemma-tensor-product-K-flat", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-tensor-product-K-flat", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "If $\\mathcal{K}^\\bullet$, $\\mathcal{L}^\\bullet$ are K-flat complexes", "of $\\mathcal{O}$-modules, then", "$\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet)$", "is a K-flat complex of $\\mathcal{O}$-modules." ], "refs": [], "proofs": [ { "contents": [ "Follows from the isomorphism", "$$", "\\text{Tot}(\\mathcal{M}^\\bullet \\otimes_\\mathcal{O}", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet))", "=", "\\text{Tot}(\\text{Tot}(\\mathcal{M}^\\bullet \\otimes_\\mathcal{O}", "\\mathcal{K}^\\bullet) \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet)", "$$", "and the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4231, "type": "theorem", "label": "sites-cohomology-lemma-K-flat-two-out-of-three", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-K-flat-two-out-of-three", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{K}_1^\\bullet, \\mathcal{K}_2^\\bullet, \\mathcal{K}_3^\\bullet)$", "be a distinguished triangle in $K(\\textit{Mod}(\\mathcal{O}))$.", "If two out of three of $\\mathcal{K}_i^\\bullet$ are K-flat, so is the third." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-derived-tor-exact}", "and the fact that in a distinguished triangle in", "$K(\\textit{Mod}(\\mathcal{O}))$", "if two out of three are acyclic, so is the third." ], "refs": [ "sites-cohomology-lemma-derived-tor-exact" ], "ref_ids": [ 4227 ] } ], "ref_ids": [] }, { "id": 4232, "type": "theorem", "label": "sites-cohomology-lemma-K-flat-two-out-of-three-ses", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-K-flat-two-out-of-three-ses", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$0 \\to \\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet \\to", "\\mathcal{K}_3^\\bullet \\to 0$ be a short exact sequence of complexes", "such that the terms of $\\mathcal{K}_3^\\bullet$ are flat $\\mathcal{O}$-modules.", "If two out of three of $\\mathcal{K}_i^\\bullet$ are K-flat, so is the third." ], "refs": [], "proofs": [ { "contents": [ "By Modules on Sites, Lemma \\ref{sites-modules-lemma-flat-tor-zero}", "for every complex $\\mathcal{L}^\\bullet$", "we obtain a short exact sequence", "$$", "0 \\to", "\\text{Tot}(\\mathcal{L}^\\bullet \\otimes_\\mathcal{O} \\mathcal{K}_1^\\bullet) \\to", "\\text{Tot}(\\mathcal{L}^\\bullet \\otimes_\\mathcal{O} \\mathcal{K}_1^\\bullet) \\to", "\\text{Tot}(\\mathcal{L}^\\bullet \\otimes_\\mathcal{O} \\mathcal{K}_1^\\bullet) \\to 0", "$$", "of complexes. Hence the lemma follows from the long exact sequence of", "cohomology sheaves and the definition of K-flat complexes." ], "refs": [ "sites-modules-lemma-flat-tor-zero" ], "ref_ids": [ 14204 ] } ], "ref_ids": [] }, { "id": 4233, "type": "theorem", "label": "sites-cohomology-lemma-bounded-flat-K-flat", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-bounded-flat-K-flat", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. A bounded above complex", "of flat $\\mathcal{O}$-modules is K-flat." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{K}^\\bullet$ be a bounded above complex of flat", "$\\mathcal{O}$-modules. Let $\\mathcal{L}^\\bullet$ be an acyclic complex", "of $\\mathcal{O}$-modules. Note that", "$\\mathcal{L}^\\bullet = \\colim_m \\tau_{\\leq m}\\mathcal{L}^\\bullet$", "where we take termwise colimits. Hence also", "$$", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet)", "=", "\\colim_m \\text{Tot}(", "\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\tau_{\\leq m}\\mathcal{L}^\\bullet)", "$$", "termwise. Hence to prove the complex on the left is acyclic it suffices", "to show each of the complexes on the right is acyclic. Since", "$\\tau_{\\leq m}\\mathcal{L}^\\bullet$ is acyclic this reduces us to the", "case where $\\mathcal{L}^\\bullet$ is bounded above.", "In this case the spectral sequence of", "Homology, Lemma \\ref{homology-lemma-first-quadrant-ss}", "has", "$$", "{}'E_1^{p, q} = H^p(\\mathcal{L}^\\bullet \\otimes_R \\mathcal{K}^q)", "$$", "which is zero as $\\mathcal{K}^q$ is flat and $\\mathcal{L}^\\bullet$ acyclic.", "Hence we win." ], "refs": [ "homology-lemma-first-quadrant-ss" ], "ref_ids": [ 12105 ] } ], "ref_ids": [] }, { "id": 4234, "type": "theorem", "label": "sites-cohomology-lemma-colimit-K-flat", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-colimit-K-flat", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet \\to \\ldots$", "be a system of K-flat complexes.", "Then $\\colim_i \\mathcal{K}_i^\\bullet$ is K-flat." ], "refs": [], "proofs": [ { "contents": [ "Because we are taking termwise colimits it is clear that", "$$", "\\colim_i \\text{Tot}(", "\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{K}_i^\\bullet)", "=", "\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O}", "\\colim_i \\mathcal{K}_i^\\bullet)", "$$", "Hence the lemma follows from the fact that filtered colimits are", "exact." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4235, "type": "theorem", "label": "sites-cohomology-lemma-resolution-by-direct-sums-extensions-by-zero", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-resolution-by-direct-sums-extensions-by-zero", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "For any complex $\\mathcal{G}^\\bullet$ of $\\mathcal{O}$-modules", "there exists a commutative diagram of complexes of $\\mathcal{O}$-modules", "$$", "\\xymatrix{", "\\mathcal{K}_1^\\bullet \\ar[d] \\ar[r] &", "\\mathcal{K}_2^\\bullet \\ar[d] \\ar[r] & \\ldots \\\\", "\\tau_{\\leq 1}\\mathcal{G}^\\bullet \\ar[r] &", "\\tau_{\\leq 2}\\mathcal{G}^\\bullet \\ar[r] & \\ldots", "}", "$$", "with the following properties: (1) the vertical arrows are quasi-isomorphisms", "and termwise surjective,", "(2) each $\\mathcal{K}_n^\\bullet$ is a bounded above complex whose terms", "are direct sums of $\\mathcal{O}$-modules of the form $j_{U!}\\mathcal{O}_U$, and", "(3) the maps $\\mathcal{K}_n^\\bullet \\to \\mathcal{K}_{n + 1}^\\bullet$ are", "termwise split injections whose cokernels are direct sums of", "$\\mathcal{O}$-modules of the form $j_{U!}\\mathcal{O}_U$. Moreover, the map", "$\\colim \\mathcal{K}_n^\\bullet \\to \\mathcal{G}^\\bullet$ is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The existence of the diagram and properties (1), (2), (3) follows immediately", "from", "Modules on Sites, Lemma \\ref{sites-modules-lemma-module-quotient-flat}", "and", "Derived Categories, Lemma \\ref{derived-lemma-special-direct-system}.", "The induced map", "$\\colim \\mathcal{K}_n^\\bullet \\to \\mathcal{G}^\\bullet$", "is a quasi-isomorphism because filtered colimits are exact." ], "refs": [ "sites-modules-lemma-module-quotient-flat", "derived-lemma-special-direct-system" ], "ref_ids": [ 14203, 1903 ] } ], "ref_ids": [] }, { "id": 4236, "type": "theorem", "label": "sites-cohomology-lemma-K-flat-resolution", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-K-flat-resolution", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. For any complex", "$\\mathcal{G}^\\bullet$ there exists a $K$-flat complex $\\mathcal{K}^\\bullet$", "whose terms are flat $\\mathcal{O}$-modules and a quasi-isomorphism", "$\\mathcal{K}^\\bullet \\to \\mathcal{G}^\\bullet$ which is termwise surjective." ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram as in", "Lemma \\ref{lemma-resolution-by-direct-sums-extensions-by-zero}.", "Each complex $\\mathcal{K}_n^\\bullet$ is a bounded", "above complex of flat modules, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-j-shriek-flat}.", "Hence $\\mathcal{K}_n^\\bullet$ is K-flat by", "Lemma \\ref{lemma-bounded-flat-K-flat}.", "Thus $\\colim \\mathcal{K}_n^\\bullet$ is K-flat by", "Lemma \\ref{lemma-colimit-K-flat}.", "The induced map", "$\\colim \\mathcal{K}_n^\\bullet \\to \\mathcal{G}^\\bullet$", "is a quasi-isomorphism and termwise surjective by construction.", "Property (3) of Lemma \\ref{lemma-resolution-by-direct-sums-extensions-by-zero}", "shows that $\\colim \\mathcal{K}_n^m$ is a direct sum of", "flat modules and hence flat which proves the final assertion." ], "refs": [ "sites-cohomology-lemma-resolution-by-direct-sums-extensions-by-zero", "sites-modules-lemma-j-shriek-flat", "sites-cohomology-lemma-bounded-flat-K-flat", "sites-cohomology-lemma-colimit-K-flat", "sites-cohomology-lemma-resolution-by-direct-sums-extensions-by-zero" ], "ref_ids": [ 4235, 14202, 4233, 4234, 4235 ] } ], "ref_ids": [] }, { "id": 4237, "type": "theorem", "label": "sites-cohomology-lemma-derived-tor-quasi-isomorphism-other-side", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-derived-tor-quasi-isomorphism-other-side", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$\\alpha : \\mathcal{P}^\\bullet \\to \\mathcal{Q}^\\bullet$ be a", "quasi-isomorphism of K-flat complexes of $\\mathcal{O}$-modules.", "For every complex $\\mathcal{F}^\\bullet$ of $\\mathcal{O}$-modules", "the induced map", "$$", "\\text{Tot}(\\text{id}_{\\mathcal{F}^\\bullet} \\otimes \\alpha) :", "\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{P}^\\bullet)", "\\longrightarrow", "\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{Q}^\\bullet)", "$$", "is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Choose a quasi-isomorphism $\\mathcal{K}^\\bullet \\to \\mathcal{F}^\\bullet$", "with $\\mathcal{K}^\\bullet$ a K-flat complex, see", "Lemma \\ref{lemma-K-flat-resolution}.", "Consider the commutative diagram", "$$", "\\xymatrix{", "\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_\\mathcal{O} \\mathcal{P}^\\bullet) \\ar[r] \\ar[d] &", "\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_\\mathcal{O} \\mathcal{Q}^\\bullet) \\ar[d] \\\\", "\\text{Tot}(\\mathcal{F}^\\bullet", "\\otimes_\\mathcal{O} \\mathcal{P}^\\bullet) \\ar[r] &", "\\text{Tot}(\\mathcal{F}^\\bullet", "\\otimes_\\mathcal{O} \\mathcal{Q}^\\bullet)", "}", "$$", "The result follows as by", "Lemma \\ref{lemma-K-flat-quasi-isomorphism}", "the vertical arrows and the top horizontal arrow are quasi-isomorphisms." ], "refs": [ "sites-cohomology-lemma-K-flat-resolution", "sites-cohomology-lemma-K-flat-quasi-isomorphism" ], "ref_ids": [ 4236, 4228 ] } ], "ref_ids": [] }, { "id": 4238, "type": "theorem", "label": "sites-cohomology-lemma-flat-tor-zero", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-flat-tor-zero", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{F}$ be an $\\mathcal{O}$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a flat $\\mathcal{O}$-module, and", "\\item $\\text{Tor}_1^\\mathcal{O}(\\mathcal{F}, \\mathcal{G}) = 0$", "for every $\\mathcal{O}$-module $\\mathcal{G}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{F}$ is flat, then $\\mathcal{F} \\otimes_\\mathcal{O} -$", "is an exact functor and the satellites vanish. Conversely assume (2)", "holds. Then if $\\mathcal{G} \\to \\mathcal{H}$ is injective with cokernel", "$\\mathcal{Q}$, the long exact sequence of $\\text{Tor}$ shows that", "the kernel of", "$\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G} \\to", "\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{H}$", "is a quotient of", "$\\text{Tor}_1^\\mathcal{O}(\\mathcal{F}, \\mathcal{Q})$", "which is zero by assumption. Hence $\\mathcal{F}$ is flat." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4239, "type": "theorem", "label": "sites-cohomology-lemma-K-flat-flat-acyclic", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-K-flat-flat-acyclic", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{K}^\\bullet$", "be a K-flat, acyclic complex with flat terms. Then", "$\\mathcal{F} = \\Ker(\\mathcal{K}^n \\to \\mathcal{K}^{n + 1})$", "is a flat $\\mathcal{O}$-module." ], "refs": [], "proofs": [ { "contents": [ "Observe that", "$$", "\\ldots \\to \\mathcal{K}^{n - 2} \\to \\mathcal{K}^{n - 1} \\to", "\\mathcal{F} \\to 0", "$$", "is a flat resolution of our module $\\mathcal{F}$. Since a bounded above complex", "of flat modules is K-flat (Lemma \\ref{lemma-bounded-flat-K-flat})", "we may use this resolution to compute", "$\\text{Tor}_i(\\mathcal{F}, \\mathcal{G})$ for any", "$\\mathcal{O}$-module $\\mathcal{G}$. On the one hand", "$\\mathcal{K}^\\bullet \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{G}$", "is zero in $D(\\mathcal{O})$ because $\\mathcal{K}^\\bullet$ is acyclic", "and on the other hand it is represented by", "$\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{G}$.", "Hence we see that", "$$", "\\mathcal{K}^{n - 3} \\otimes_\\mathcal{O} \\mathcal{G} \\to", "\\mathcal{K}^{n - 2} \\otimes_\\mathcal{O} \\mathcal{G} \\to", "\\mathcal{K}^{n - 1} \\otimes_\\mathcal{O} \\mathcal{G}", "$$", "is exact. Thus $\\text{Tor}_1(\\mathcal{F}, \\mathcal{G}) = 0$", "and we conclude by Lemma \\ref{lemma-flat-tor-zero}." ], "refs": [ "sites-cohomology-lemma-bounded-flat-K-flat", "sites-cohomology-lemma-flat-tor-zero" ], "ref_ids": [ 4233, 4238 ] } ], "ref_ids": [] }, { "id": 4240, "type": "theorem", "label": "sites-cohomology-lemma-factor-through-K-flat", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-factor-through-K-flat", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed space.", "Let $a : \\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet$ be a map of complexes", "of $\\mathcal{O}$-modules. If $\\mathcal{K}^\\bullet$ is K-flat, then", "there exist a complex $\\mathcal{N}^\\bullet$ and maps of complexes", "$b : \\mathcal{K}^\\bullet \\to \\mathcal{N}^\\bullet$", "and $c : \\mathcal{N}^\\bullet \\to \\mathcal{L}^\\bullet$ such that", "\\begin{enumerate}", "\\item $\\mathcal{N}^\\bullet$ is K-flat,", "\\item $c$ is a quasi-isomorphism,", "\\item $a$ is homotopic to $c \\circ b$.", "\\end{enumerate}", "If the terms of $\\mathcal{K}^\\bullet$ are flat, then we may choose", "$\\mathcal{N}^\\bullet$, $b$, and $c$", "such that the same is true for $\\mathcal{N}^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "We will use that the homotopy category $K(\\textit{Mod}(\\mathcal{O}))$", "is a triangulated category, see Derived Categories, Proposition", "\\ref{derived-proposition-homotopy-category-triangulated}.", "Choose a distinguished triangle", "$\\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet \\to", "\\mathcal{C}^\\bullet \\to \\mathcal{K}^\\bullet[1]$.", "Choose a quasi-isomorphism $\\mathcal{M}^\\bullet \\to \\mathcal{C}^\\bullet$ with", "$\\mathcal{M}^\\bullet$ K-flat with flat terms, see", "Lemma \\ref{lemma-K-flat-resolution}.", "By the axioms of triangulated categories,", "we may fit the composition", "$\\mathcal{M}^\\bullet \\to \\mathcal{C}^\\bullet \\to \\mathcal{K}^\\bullet[1]$", "into a distinguished triangle", "$\\mathcal{K}^\\bullet \\to \\mathcal{N}^\\bullet \\to", "\\mathcal{M}^\\bullet \\to \\mathcal{K}^\\bullet[1]$.", "By Lemma \\ref{lemma-K-flat-two-out-of-three} we see that", "$\\mathcal{N}^\\bullet$ is K-flat.", "Again using the axioms of triangulated categories,", "we can choose a map $\\mathcal{N}^\\bullet \\to \\mathcal{L}^\\bullet$ fitting into", "the following morphism of distinghuised triangles", "$$", "\\xymatrix{", "\\mathcal{K}^\\bullet \\ar[r] \\ar[d] &", "\\mathcal{N}^\\bullet \\ar[r] \\ar[d] &", "\\mathcal{M}^\\bullet \\ar[r] \\ar[d] &", "\\mathcal{K}^\\bullet[1] \\ar[d] \\\\", "\\mathcal{K}^\\bullet \\ar[r] &", "\\mathcal{L}^\\bullet \\ar[r] &", "\\mathcal{C}^\\bullet \\ar[r] &", "\\mathcal{K}^\\bullet[1]", "}", "$$", "Since two out of three of the arrows are quasi-isomorphisms, so is", "the third arrow $\\mathcal{N}^\\bullet \\to \\mathcal{L}^\\bullet$", "by the long exact sequences", "of cohomology associated to these distinguished triangles", "(or you can look at the image of this diagram in $D(\\mathcal{O})$ and use", "Derived Categories, Lemma \\ref{derived-lemma-third-isomorphism-triangle}", "if you like). This finishes the proof of (1), (2), and (3).", "To prove the final assertion, we may choose $\\mathcal{N}^\\bullet$ such that", "$\\mathcal{N}^n \\cong \\mathcal{M}^n \\oplus \\mathcal{K}^n$, see", "Derived Categories, Lemma", "\\ref{derived-lemma-improve-distinguished-triangle-homotopy}.", "Hence we get the desired flatness", "if the terms of $\\mathcal{K}^\\bullet$ are flat." ], "refs": [ "derived-proposition-homotopy-category-triangulated", "sites-cohomology-lemma-K-flat-resolution", "sites-cohomology-lemma-K-flat-two-out-of-three", "derived-lemma-third-isomorphism-triangle", "derived-lemma-improve-distinguished-triangle-homotopy" ], "ref_ids": [ 1960, 4236, 4231, 1759, 1809 ] } ], "ref_ids": [] }, { "id": 4241, "type": "theorem", "label": "sites-cohomology-lemma-pullback-K-flat", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-pullback-K-flat", "contents": [ "Let $f : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$", "be a morphism of ringed topoi. Let $\\mathcal{K}^\\bullet$ be a K-flat complex", "of $\\mathcal{O}$-modules whose terms are flat $\\mathcal{O}$-modules. Then", "$f^*\\mathcal{K}^\\bullet$ is a K-flat complex of $\\mathcal{O}'$-modules whose", "terms are flat $\\mathcal{O}'$-modules." ], "refs": [], "proofs": [ { "contents": [ "The terms $f^*\\mathcal{K}^n$ are flat $\\mathcal{O}'$-modules by", "Modules on Sites, Lemma \\ref{sites-modules-lemma-pullback-flat}.", "Choose a diagram", "$$", "\\xymatrix{", "\\mathcal{K}_1^\\bullet \\ar[d] \\ar[r] &", "\\mathcal{K}_2^\\bullet \\ar[d] \\ar[r] & \\ldots \\\\", "\\tau_{\\leq 1}\\mathcal{K}^\\bullet \\ar[r] &", "\\tau_{\\leq 2}\\mathcal{K}^\\bullet \\ar[r] & \\ldots", "}", "$$", "as in Lemma \\ref{lemma-resolution-by-direct-sums-extensions-by-zero}.", "We will use all of the properties stated in the", "lemma without further mention. Each $\\mathcal{K}_n^\\bullet$ is a bounded", "above complex of flat modules, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-j-shriek-flat}.", "Consider the short exact sequence of complexes", "$$", "0 \\to \\mathcal{M}^\\bullet \\to", "\\colim \\mathcal{K}_n^\\bullet \\to", "\\mathcal{K}^\\bullet \\to 0", "$$", "defining $\\mathcal{M}^\\bullet$. By Lemmas \\ref{lemma-bounded-flat-K-flat} and", "\\ref{lemma-colimit-K-flat} the complex $\\colim \\mathcal{K}_n^\\bullet$", "is K-flat and by Modules on Sites, Lemma \\ref{sites-modules-lemma-colimits-flat}", "it has flat terms. By Modules on Sites, Lemma \\ref{sites-modules-lemma-flat-ses}", "$\\mathcal{M}^\\bullet$ has flat terms, by", "Lemma \\ref{lemma-K-flat-two-out-of-three-ses}", "$\\mathcal{M}^\\bullet$ is K-flat, and by the long exact", "cohomology sequence $\\mathcal{M}^\\bullet$ is acyclic (because", "the second arrow is a quasi-isomorphism). The pullback", "$f^*(\\colim \\mathcal{K}_n^\\bullet) = \\colim f^*\\mathcal{K}_n^\\bullet$", "is a colimit of bounded below complexes of flat $\\mathcal{O}'$-modules", "and hence is K-flat (by the same lemmas as above).", "The pullback of our short exact sequence", "$$", "0 \\to f^*\\mathcal{M}^\\bullet \\to", "f^*(\\colim \\mathcal{K}_n^\\bullet) \\to", "f^*\\mathcal{K}^\\bullet \\to 0", "$$", "is a short exact sequence of complexes by", "Modules on Sites, Lemma \\ref{sites-modules-lemma-pullback-ses}.", "Hence by Lemma \\ref{lemma-K-flat-two-out-of-three-ses}", "it suffices to show that $f^*\\mathcal{M}^\\bullet$", "is K-flat. This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $\\mathcal{K}^\\bullet$ is acyclic as well as K-flat and", "with flat terms. Then Lemma \\ref{lemma-K-flat-flat-acyclic}", "guarantees that all terms of $\\tau_{\\leq n}\\mathcal{K}^\\bullet$", "are flat $\\mathcal{O}$-modules. We choose a diagram as above and", "we will use all the properties proven above for this diagram.", "Denote $\\mathcal{M}_n^\\bullet$ the kernel of the map of complexes", "$\\mathcal{K}_n^\\bullet \\to \\tau_{\\leq n}\\mathcal{K}^\\bullet$", "so that we have short exact sequences of complexes", "$$", "0 \\to \\mathcal{M}_n^\\bullet \\to \\mathcal{K}_n^\\bullet \\to", "\\tau_{\\leq n}\\mathcal{K}^\\bullet \\to 0", "$$", "By Modules on Sites, Lemma \\ref{sites-modules-lemma-flat-ses}", "we see that the terms of the complex $\\mathcal{M}_n^\\bullet$ are flat.", "Hence we see that $\\mathcal{M} = \\colim \\mathcal{M}_n^\\bullet$", "is a filtered colimit of bounded below complexes of flat modules", "in this case. Thus $f^*\\mathcal{M}^\\bullet$ is K-flat", "(same argument as above) and we win." ], "refs": [ "sites-modules-lemma-pullback-flat", "sites-cohomology-lemma-resolution-by-direct-sums-extensions-by-zero", "sites-modules-lemma-j-shriek-flat", "sites-cohomology-lemma-bounded-flat-K-flat", "sites-cohomology-lemma-colimit-K-flat", "sites-modules-lemma-colimits-flat", "sites-modules-lemma-flat-ses", "sites-cohomology-lemma-K-flat-two-out-of-three-ses", "sites-modules-lemma-pullback-ses", "sites-cohomology-lemma-K-flat-two-out-of-three-ses", "sites-cohomology-lemma-K-flat-flat-acyclic", "sites-modules-lemma-flat-ses" ], "ref_ids": [ 14249, 4235, 14202, 4233, 4234, 14200, 14205, 4232, 14252, 4232, 4239, 14205 ] } ], "ref_ids": [] }, { "id": 4242, "type": "theorem", "label": "sites-cohomology-lemma-derived-base-change", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-derived-base-change", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$", "be a morphism of ringed topoi. There exists an exact functor", "$$", "Lf^* : D(\\mathcal{O}') \\longrightarrow D(\\mathcal{O})", "$$", "of triangulated categories so that", "$Lf^*\\mathcal{K}^\\bullet = f^*\\mathcal{K}^\\bullet$ for any", "K-flat complex $\\mathcal{K}^\\bullet$ with flat terms and", "in particular for any bounded above complex of flat $\\mathcal{O}'$-modules." ], "refs": [], "proofs": [ { "contents": [ "To see this we use the general theory developed in", "Derived Categories, Section \\ref{derived-section-derived-functors}.", "Set $\\mathcal{D} = K(\\mathcal{O}')$ and $\\mathcal{D}' = D(\\mathcal{O})$.", "Let us write $F : \\mathcal{D} \\to \\mathcal{D}'$ the exact functor", "of triangulated categories defined by the rule", "$F(\\mathcal{G}^\\bullet) = f^*\\mathcal{G}^\\bullet$.", "We let $S$ be the set of quasi-isomorphisms in", "$\\mathcal{D} = K(\\mathcal{O}')$.", "This gives a situation as in", "Derived Categories, Situation \\ref{derived-situation-derived-functor}", "so that", "Derived Categories, Definition", "\\ref{derived-definition-right-derived-functor-defined}", "applies. We claim that $LF$ is everywhere defined.", "This follows from", "Derived Categories, Lemma \\ref{derived-lemma-find-existence-computes}", "with $\\mathcal{P} \\subset \\Ob(\\mathcal{D})$ the collection", "of K-flat complexes $\\mathcal{K}^\\bullet$ with flat terms.", "Namely, (1) follows from Lemma \\ref{lemma-K-flat-resolution}", "and to see (2) we have to show that for a quasi-isomorphism", "$\\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet$ between", "elements of $\\mathcal{P}$ the map", "$f^*\\mathcal{K}_1^\\bullet \\to f^*\\mathcal{K}_2^\\bullet$ is a", "quasi-isomorphism. To see this write this as", "$$", "f^{-1}\\mathcal{K}_1^\\bullet \\otimes_{f^{-1}\\mathcal{O}'} \\mathcal{O}", "\\longrightarrow", "f^{-1}\\mathcal{K}_2^\\bullet \\otimes_{f^{-1}\\mathcal{O}'} \\mathcal{O}", "$$", "The functor $f^{-1}$ is exact, hence the map", "$f^{-1}\\mathcal{K}_1^\\bullet \\to f^{-1}\\mathcal{K}_2^\\bullet$ is a", "quasi-isomorphism. The complexes", "$f^{-1}\\mathcal{K}_1^\\bullet$ and $f^{-1}\\mathcal{K}_2^\\bullet$", "are K-flat complexes of $f^{-1}\\mathcal{O}'$-modules by", "Lemma \\ref{lemma-pullback-K-flat}", "because we can consider the morphism of ringed topoi", "$(\\Sh(\\mathcal{C}), f^{-1}\\mathcal{O}') \\to", "(\\Sh(\\mathcal{C}'), \\mathcal{O}')$. Hence", "Lemma \\ref{lemma-derived-tor-quasi-isomorphism-other-side}", "guarantees that the displayed map is a quasi-isomorphism.", "Thus we obtain a derived functor", "$$", "LF :", "D(\\mathcal{O}') = S^{-1}\\mathcal{D}", "\\longrightarrow", "\\mathcal{D}' = D(\\mathcal{O})", "$$", "see", "Derived Categories, Equation (\\ref{derived-equation-everywhere}).", "Finally,", "Derived Categories, Lemma \\ref{derived-lemma-find-existence-computes}", "also guarantees that", "$LF(\\mathcal{K}^\\bullet) = F(\\mathcal{K}^\\bullet) = f^*\\mathcal{K}^\\bullet$", "when $\\mathcal{K}^\\bullet$ is in $\\mathcal{P}$.", "The proof is finished by observing that", "bounded above complexes of flat modules are in $\\mathcal{P}$", "by Lemma \\ref{lemma-bounded-flat-K-flat}." ], "refs": [ "derived-definition-right-derived-functor-defined", "derived-lemma-find-existence-computes", "sites-cohomology-lemma-K-flat-resolution", "sites-cohomology-lemma-pullback-K-flat", "sites-cohomology-lemma-derived-tor-quasi-isomorphism-other-side", "derived-lemma-find-existence-computes", "sites-cohomology-lemma-bounded-flat-K-flat" ], "ref_ids": [ 1987, 1832, 4236, 4241, 4237, 1832, 4233 ] } ], "ref_ids": [] }, { "id": 4243, "type": "theorem", "label": "sites-cohomology-lemma-derived-pullback-composition", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-derived-pullback-composition", "contents": [ "Consider morphisms of ringed topoi", "$f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "and", "$g : (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\to", "(\\Sh(\\mathcal{E}), \\mathcal{O}_\\mathcal{E})$.", "Then $Lf^* \\circ Lg^* = L(g \\circ f)^*$ as functors", "$D(\\mathcal{O}_\\mathcal{E}) \\to D(\\mathcal{O}_\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "Let $E$ be an object of $D(\\mathcal{O}_\\mathcal{E})$.", "We may represent $E$ by a K-flat complex $\\mathcal{K}^\\bullet$", "with flat terms, see Lemma \\ref{lemma-K-flat-resolution}.", "By construction $Lg^*E$ is computed by $g^*\\mathcal{K}^\\bullet$, see", "Lemma \\ref{lemma-derived-base-change}.", "By Lemma \\ref{lemma-pullback-K-flat} the complex", "$g^*\\mathcal{K}^\\bullet$ is K-flat with flat terms.", "Hence $Lf^*Lg^*E$ is represented by $f^*g^*\\mathcal{K}^\\bullet$.", "Since also $L(g \\circ f)^*E$ is represented by", "$(g \\circ f)^*\\mathcal{K}^\\bullet = f^*g^*\\mathcal{K}^\\bullet$", "we conclude." ], "refs": [ "sites-cohomology-lemma-K-flat-resolution", "sites-cohomology-lemma-derived-base-change", "sites-cohomology-lemma-pullback-K-flat" ], "ref_ids": [ 4236, 4242, 4241 ] } ], "ref_ids": [] }, { "id": 4244, "type": "theorem", "label": "sites-cohomology-lemma-pullback-tensor-product", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-pullback-tensor-product", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$", "be a morphism of ringed topoi.", "There is a canonical bifunctorial isomorphism", "$$", "Lf^*(", "\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}'}^{\\mathbf{L}} \\mathcal{G}^\\bullet", ") =", "Lf^*\\mathcal{F}^\\bullet ", "\\otimes_{\\mathcal{O}}^{\\mathbf{L}}", "Lf^*\\mathcal{G}^\\bullet ", "$$", "for $\\mathcal{F}^\\bullet, \\mathcal{G}^\\bullet \\in \\Ob(D(\\mathcal{O}'))$." ], "refs": [], "proofs": [ { "contents": [ "By our construction of derived pullback in", "Lemma \\ref{lemma-derived-base-change}.", "and the existence of resolutions in", "Lemma \\ref{lemma-K-flat-resolution}", "we may replace $\\mathcal{F}^\\bullet$ and $\\mathcal{G}^\\bullet$", "by complexes of $\\mathcal{O}'$-modules which are K-flat and have", "flat terms. In this case", "$\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}'}^{\\mathbf{L}} \\mathcal{G}^\\bullet$", "is just the total complex associated to the double complex", "$\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}'} \\mathcal{G}^\\bullet$.", "The complex", "$\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}'} \\mathcal{G}^\\bullet)$", "is K-flat with flat terms by Lemma \\ref{lemma-tensor-product-K-flat} and", "Modules on Sites, Lemma \\ref{sites-modules-lemma-tensor-flats}.", "Hence the isomorphism of the lemma comes from the isomorphism", "$$", "\\text{Tot}(f^*\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}}", "f^*\\mathcal{G}^\\bullet)", "\\longrightarrow", "f^*\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}'} \\mathcal{G}^\\bullet)", "$$", "whose constituents are the isomorphisms", "$f^*\\mathcal{F}^p \\otimes_{\\mathcal{O}} f^*\\mathcal{G}^q \\to", "f^*(\\mathcal{F}^p \\otimes_{\\mathcal{O}'} \\mathcal{G}^q)$ of", "Modules on Sites, Lemma \\ref{sites-modules-lemma-tensor-product-pullback}." ], "refs": [ "sites-cohomology-lemma-derived-base-change", "sites-cohomology-lemma-K-flat-resolution", "sites-cohomology-lemma-tensor-product-K-flat", "sites-modules-lemma-tensor-flats", "sites-modules-lemma-tensor-product-pullback" ], "ref_ids": [ 4242, 4236, 4230, 14207, 14189 ] } ], "ref_ids": [] }, { "id": 4245, "type": "theorem", "label": "sites-cohomology-lemma-variant-derived-pullback", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-variant-derived-pullback", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$", "be a morphism of ringed topoi. There is a canonical bifunctorial", "isomorphism", "$$", "\\mathcal{F}^\\bullet", "\\otimes_\\mathcal{O}^{\\mathbf{L}}", "Lf^*\\mathcal{G}^\\bullet", "=", "\\mathcal{F}^\\bullet ", "\\otimes_{f^{-1}\\mathcal{O}_Y}^{\\mathbf{L}}", "f^{-1}\\mathcal{G}^\\bullet ", "$$", "for $\\mathcal{F}^\\bullet$ in $D(\\mathcal{O})$ and", "$\\mathcal{G}^\\bullet$ in $D(\\mathcal{O}')$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be an $\\mathcal{O}$-module and let $\\mathcal{G}$", "be an $\\mathcal{O}'$-module. Then", "$\\mathcal{F} \\otimes_{\\mathcal{O}} f^*\\mathcal{G} =", "\\mathcal{F} \\otimes_{f^{-1}\\mathcal{O}'} f^{-1}\\mathcal{G}$", "because", "$f^*\\mathcal{G} =", "\\mathcal{O} \\otimes_{f^{-1}\\mathcal{O}'} f^{-1}\\mathcal{G}$.", "The lemma follows from this and the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4246, "type": "theorem", "label": "sites-cohomology-lemma-check-K-flat-stalks", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-check-K-flat-stalks", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{K}^\\bullet$ be a complex of $\\mathcal{O}$-modules.", "\\begin{enumerate}", "\\item If $\\mathcal{K}^\\bullet$ is K-flat, then for every point $p$", "of the site $\\mathcal{C}$ the complex of $\\mathcal{O}_p$-modules", "$\\mathcal{K}_p^\\bullet$ is K-flat in the sense of", "More on Algebra, Definition \\ref{more-algebra-definition-K-flat}", "\\item If $\\mathcal{C}$ has enough points, then the converse is true.", "\\end{enumerate}" ], "refs": [ "more-algebra-definition-K-flat" ], "proofs": [ { "contents": [ "Proof of (2). If $\\mathcal{C}$ has enough points and", "$\\mathcal{K}_p^\\bullet$ is K-flat for all points $p$ of $\\mathcal{C}$", "then we see that $\\mathcal{K}^\\bullet$ is K-flat because $\\otimes$ and", "direct sums commute with taking stalks and because we can check exactness", "at stalks, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-check-exactness-stalks}.", "\\medskip\\noindent", "Proof of (1). Assume $\\mathcal{K}^\\bullet$ is K-flat.", "Choose a quasi-isomorphism $a : \\mathcal{L}^\\bullet \\to \\mathcal{K}^\\bullet$", "such that $\\mathcal{L}^\\bullet$ is K-flat with flat terms, see", "Lemma \\ref{lemma-K-flat-resolution}. Any pullback", "of $\\mathcal{L}^\\bullet$ is K-flat, see", "Lemma \\ref{lemma-pullback-K-flat}. In particular the stalk", "$\\mathcal{L}_p^\\bullet$ is a K-flat complex of $\\mathcal{O}_p$-modules.", "Thus the cone $C(a)$ on $a$ is a K-flat", "(Lemma \\ref{lemma-K-flat-two-out-of-three})", "acyclic complex of $\\mathcal{O}$-modules and it suffuces", "to show the stalk of $C(a)$ is K-flat", "(by More on Algebra, Lemma \\ref{more-algebra-lemma-K-flat-two-out-of-three}).", "Thus we may assume that $\\mathcal{K}^\\bullet$ is K-flat and acyclic.", "\\medskip\\noindent", "Assume $\\mathcal{K}^\\bullet$ is acyclic and K-flat. Before continuing", "we replace the site $\\mathcal{C}$ by another one as in", "Sites, Lemma \\ref{sites-lemma-topos-good-site}", "to insure that $\\mathcal{C}$ has all finite", "limits. This implies the category of neighbourhoods of $p$ is filtered", "(Sites, Lemma \\ref{sites-lemma-neighbourhoods-directed})", "and the colimit defining the stalk of a sheaf is filtered.", "Let $M$ be a finitely presented $\\mathcal{O}_p$-module.", "It suffices to show that $\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_p} M$", "is acyclic, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-universally-acyclic-K-flat}.", "Since $\\mathcal{O}_p$ is the filtered colimit of $\\mathcal{O}(U)$", "where $U$ runs over the neighbourhoods of $p$, we", "can find a neighbourhood $(U, x)$ of $p$ and a finitely", "presented $\\mathcal{O}(U)$-module $M'$ whose base change", "to $\\mathcal{O}_p$ is $M$, see", "Algebra, Lemma \\ref{algebra-lemma-colimit-category-fp-modules}.", "By Lemma \\ref{lemma-restriction-K-flat}", "we may replace $\\mathcal{C}, \\mathcal{O}, \\mathcal{K}^\\bullet$", "by $\\mathcal{C}/U, \\mathcal{O}_U, \\mathcal{K}^\\bullet|_U$.", "We conclude that we may assume there exists an $\\mathcal{O}$-module", "$\\mathcal{F}$ such that $M \\cong \\mathcal{F}_p$.", "Since $\\mathcal{K}^\\bullet$ is K-flat and acyclic,", "we see that $\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{F}$", "is acyclic (as it computes the derived tensor product by", "definition). Taking stalks is an exact functor, hence we get that", "$\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_p} M$", "is acyclic as desired." ], "refs": [ "sites-modules-lemma-check-exactness-stalks", "sites-cohomology-lemma-K-flat-resolution", "sites-cohomology-lemma-pullback-K-flat", "sites-cohomology-lemma-K-flat-two-out-of-three", "more-algebra-lemma-K-flat-two-out-of-three", "sites-lemma-topos-good-site", "sites-lemma-neighbourhoods-directed", "more-algebra-lemma-universally-acyclic-K-flat", "algebra-lemma-colimit-category-fp-modules", "sites-cohomology-lemma-restriction-K-flat" ], "ref_ids": [ 14160, 4236, 4241, 4231, 10126, 8581, 8601, 10130, 1095, 4229 ] } ], "ref_ids": [ 10620 ] }, { "id": 4247, "type": "theorem", "label": "sites-cohomology-lemma-pullback-K-flat-points", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-pullback-K-flat-points", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$", "be a morphism of ringed topoi. If $\\mathcal{C}$ has enough points, then", "the pullback of a K-flat complex of", "$\\mathcal{O}'$-modules is a K-flat complex of $\\mathcal{O}$-modules." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-check-K-flat-stalks},", "Modules on Sites, Lemma \\ref{sites-modules-lemma-pullback-stalk},", "and", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-K-flat}." ], "refs": [ "sites-cohomology-lemma-check-K-flat-stalks", "sites-modules-lemma-pullback-stalk", "more-algebra-lemma-base-change-K-flat" ], "ref_ids": [ 4246, 14245, 10124 ] } ], "ref_ids": [] }, { "id": 4248, "type": "theorem", "label": "sites-cohomology-lemma-tensor-pull-compatibility", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-tensor-pull-compatibility", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.", "Let $\\mathcal{K}^\\bullet$ and $\\mathcal{M}^\\bullet$", "be complexes of $\\mathcal{O}_\\mathcal{D}$-modules.", "The diagram", "$$", "\\xymatrix{", "Lf^*(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "\\mathcal{M}^\\bullet) \\ar[r] \\ar[d] &", "Lf^*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}", "\\mathcal{M}^\\bullet) \\ar[d] \\\\", "Lf^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L}", "Lf^*\\mathcal{M}^\\bullet \\ar[d] &", "f^*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}", "\\mathcal{M}^\\bullet) \\ar[d] \\\\", "f^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L}", "f^*\\mathcal{M}^\\bullet \\ar[r] &", "\\text{Tot}(f^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}", "f^*\\mathcal{M}^\\bullet)", "}", "$$", "commutes." ], "refs": [], "proofs": [ { "contents": [ "We will use the existence of K-flat resolutions with flat terms", "(Lemma \\ref{lemma-K-flat-resolution}), we will use that derived pullback", "is computed by such complexes (Lemma \\ref{lemma-derived-base-change}),", "and that pullbacks preserve these properties", "(Lemma \\ref{lemma-pullback-K-flat}). If we choose such", "resolutions $\\mathcal{P}^\\bullet \\to \\mathcal{K}^\\bullet$", "and $\\mathcal{Q}^\\bullet \\to \\mathcal{M}^\\bullet$, then", "we see that", "$$", "\\xymatrix{", "Lf^*\\text{Tot}(\\mathcal{P}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}", "\\mathcal{Q}^\\bullet) \\ar[r] \\ar[d] &", "Lf^*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}", "\\mathcal{M}^\\bullet) \\ar[d] \\\\", "f^*\\text{Tot}(\\mathcal{P}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}", "\\mathcal{Q}^\\bullet) \\ar[d] \\ar[r] &", "f^*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}", "\\mathcal{M}^\\bullet) \\ar[d] \\\\", "\\text{Tot}(f^*\\mathcal{P}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}", "f^*\\mathcal{Q}^\\bullet) \\ar[r] &", "\\text{Tot}(f^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}", "f^*\\mathcal{M}^\\bullet)", "}", "$$", "commutes. However, now the left hand side of the diagram", "is the left hand side of the diagram by our choice of", "$\\mathcal{P}^\\bullet$ and $\\mathcal{Q}^\\bullet$ and", "Lemma \\ref{lemma-tensor-product-K-flat}." ], "refs": [ "sites-cohomology-lemma-K-flat-resolution", "sites-cohomology-lemma-derived-base-change", "sites-cohomology-lemma-pullback-K-flat", "sites-cohomology-lemma-tensor-product-K-flat" ], "ref_ids": [ 4236, 4242, 4241, 4230 ] } ], "ref_ids": [] }, { "id": 4249, "type": "theorem", "label": "sites-cohomology-lemma-adjoint", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-adjoint", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$", "be a morphism of ringed topoi.", "The functor $Rf_*$ defined above and", "the functor $Lf^*$ defined in", "Lemma \\ref{lemma-derived-base-change} are adjoint:", "$$", "\\Hom_{D(\\mathcal{O})}(Lf^*\\mathcal{G}^\\bullet, \\mathcal{F}^\\bullet)", "=", "\\Hom_{D(\\mathcal{O}')}(\\mathcal{G}^\\bullet, Rf_*\\mathcal{F}^\\bullet)", "$$", "bifunctorially in $\\mathcal{F}^\\bullet \\in \\Ob(D(\\mathcal{O}))$ and", "$\\mathcal{G}^\\bullet \\in \\Ob(D(\\mathcal{O}'))$." ], "refs": [ "sites-cohomology-lemma-derived-base-change" ], "proofs": [ { "contents": [ "This follows formally from the fact that $Rf_*$ and $Lf^*$ exist, see", "Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors}." ], "refs": [ "derived-lemma-derived-adjoint-functors" ], "ref_ids": [ 1907 ] } ], "ref_ids": [ 4242 ] }, { "id": 4250, "type": "theorem", "label": "sites-cohomology-lemma-derived-pushforward-composition", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-derived-pushforward-composition", "contents": [ "Let", "$f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "and", "$g : (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\to", "(\\Sh(\\mathcal{E}), \\mathcal{O}_\\mathcal{E})$", "be morphisms of ringed topoi.", "Then $Rg_* \\circ Rf_* = R(g \\circ f)_*$ as functors", "$D(\\mathcal{O}_\\mathcal{C}) \\to D(\\mathcal{O}_\\mathcal{E})$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-adjoint} we see that $Rg_* \\circ Rf_*$", "is adjoint to $Lf^* \\circ Lg^*$. We have", "$Lf^* \\circ Lg^* = L(g \\circ f)^*$ by", "Lemma \\ref{lemma-derived-pullback-composition}", "and hence by", "uniqueness of adjoint functors we have $Rg_* \\circ Rf_* = R(g \\circ f)_*$." ], "refs": [ "sites-cohomology-lemma-adjoint", "sites-cohomology-lemma-derived-pullback-composition" ], "ref_ids": [ 4249, 4243 ] } ], "ref_ids": [] }, { "id": 4251, "type": "theorem", "label": "sites-cohomology-lemma-adjoints-push-pull-compatibility", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-adjoints-push-pull-compatibility", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.", "Let $\\mathcal{K}^\\bullet$", "be a complex of $\\mathcal{O}_\\mathcal{C}$-modules.", "The diagram", "$$", "\\xymatrix{", "Lf^*f_*\\mathcal{K}^\\bullet \\ar[r] \\ar[d] &", "f^*f_*\\mathcal{K}^\\bullet \\ar[d] \\\\", "Lf^*Rf_*\\mathcal{K}^\\bullet \\ar[r] &", "\\mathcal{K}^\\bullet", "}", "$$", "coming from $Lf^* \\to f^*$ on complexes, $f_* \\to Rf_*$ on complexes,", "and adjunction $Lf^* \\circ Rf_* \\to \\text{id}$", "commutes in $D(\\mathcal{O}_\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "We will use the existence of K-flat resolutions and", "K-injective resolutions, see Lemmas", "\\ref{lemma-K-flat-resolution}, \\ref{lemma-derived-base-change}, and", "\\ref{lemma-pullback-K-flat} and the discussion above.", "Choose a quasi-isomorphism", "$\\mathcal{K}^\\bullet \\to \\mathcal{I}^\\bullet$ where $\\mathcal{I}^\\bullet$", "is K-injective as a complex of $\\mathcal{O}_\\mathcal{C}$-modules.", "Choose a quasi-isomorphism $\\mathcal{Q}^\\bullet \\to f_*\\mathcal{I}^\\bullet$", "where $\\mathcal{Q}^\\bullet$ is a K-flat complex of", "$\\mathcal{O}_\\mathcal{D}$-modules with flat terms.", "We can choose a K-flat complex of", "$\\mathcal{O}_\\mathcal{D}$-modules $\\mathcal{P}^\\bullet$ with flat terms", "and a diagram of morphisms of complexes", "$$", "\\xymatrix{", "\\mathcal{P}^\\bullet \\ar[r] \\ar[d] &", "f_*\\mathcal{K}^\\bullet \\ar[d] \\\\", "\\mathcal{Q}^\\bullet \\ar[r] & f_*\\mathcal{I}^\\bullet", "}", "$$", "commutative up to homotopy where the top horizontal arrow", "is a quasi-isomorphism. Namely, we can first choose such a", "diagram for some complex $\\mathcal{P}^\\bullet$ because", "the quasi-isomorphisms form a multiplicative system in", "the homotopy category of complexes and then we can choose", "a resolution of $\\mathcal{P}^\\bullet$ by a K-flat complex with flat terms.", "Taking pullbacks we obtain a diagram of morphisms of complexes", "$$", "\\xymatrix{", "f^*\\mathcal{P}^\\bullet \\ar[r] \\ar[d] &", "f^*f_*\\mathcal{K}^\\bullet \\ar[d] \\ar[r] &", "\\mathcal{K}^\\bullet \\ar[d] \\\\", "f^*\\mathcal{Q}^\\bullet \\ar[r] &", "f^*f_*\\mathcal{I}^\\bullet \\ar[r] &", "\\mathcal{I}^\\bullet", "}", "$$", "commutative up to homotopy. The outer rectangle witnesses the", "truth of the statement in the lemma." ], "refs": [ "sites-cohomology-lemma-K-flat-resolution", "sites-cohomology-lemma-derived-base-change", "sites-cohomology-lemma-pullback-K-flat" ], "ref_ids": [ 4236, 4242, 4241 ] } ], "ref_ids": [] }, { "id": 4252, "type": "theorem", "label": "sites-cohomology-lemma-torsion", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-torsion", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\subset \\textit{Ab}(\\mathcal{C})$", "denote the Serre subcategory consisting of torsion abelian sheaves.", "Then the functor $D(\\mathcal{A}) \\to D_\\mathcal{A}(\\mathcal{C})$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "A key observation is that an injective abelian sheaf $\\mathcal{I}$", "is divisible. Namely, if $s \\in \\mathcal{I}(U)$ is a local section,", "then we interpret $s$ as a map", "$s : j_{U!}\\mathbf{Z} \\to \\mathcal{I}$ and we apply the", "defining property of an injective object to", "the injective map of sheaves $n : j_{U!}\\mathbf{Z} \\to j_{U!}\\mathbf{Z}$", "to see that there exists an $s' \\in \\mathcal{I}(U)$ with $ns' = s$.", "\\medskip\\noindent", "For a sheaf $\\mathcal{F}$ denote $\\mathcal{F}_{tor}$ its torsion subsheaf.", "We claim that if $\\mathcal{I}^\\bullet$ is a complex of injective abelian", "sheaves whose cohomology sheaves are torsion, then", "$$", "\\mathcal{I}^\\bullet_{tor} \\to \\mathcal{I}^\\bullet", "$$", "is a quasi-isomorphism. Namely, by flatness of $\\mathbf{Q}$ over $\\mathbf{Z}$", "we have", "$$", "H^p(\\mathcal{I}^\\bullet) \\otimes_\\mathbf{Z} \\mathbf{Q} =", "H^p(\\mathcal{I}^\\bullet \\otimes_\\mathbf{Z} \\mathbf{Q})", "$$", "which is zero because the cohomology sheaves are torsion.", "By divisibility (shown above) we see that", "$\\mathcal{I}^\\bullet \\to \\mathcal{I}^\\bullet \\otimes_\\mathbf{Z} \\mathbf{Q}$", "is surjective with kernel $\\mathcal{I}^\\bullet_{tor}$.", "The claim follows from the long exact sequence of cohomology sheaves", "associated to the short exact sequence you get.", "\\medskip\\noindent", "To prove the lemma we will construct right adjoint", "$T : D(\\mathcal{C}) \\to D(\\mathcal{A})$. Namely, given $K$", "in $D(\\mathcal{C})$ we can represent $K$ by a K-injective complex", "$\\mathcal{I}^\\bullet$ whose cohomology sheaves are injective, see", "Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "Then we set $T(K) = \\mathcal{I}^\\bullet_{tor}$, in other words,", "$T$ is the right derived functor of taking torsion.", "The functor $T$ is a right adjoint to", "$i : D(\\mathcal{A}) \\to D_\\mathcal{A}(\\mathcal{C})$.", "This readily follows from the observation that", "if $\\mathcal{F}^\\bullet$ is a complex of torsion sheaves, then", "$$", "\\Hom_{K(\\mathcal{A})}(\\mathcal{F}^\\bullet, I^\\bullet_{tor}) =", "\\Hom_{K(\\textit{Ab}(\\mathcal{C}))}(\\mathcal{F}^\\bullet, I^\\bullet)", "$$", "in particular $\\mathcal{I}^\\bullet_{tor}$ is a K-injective complex", "of $\\mathcal{A}$. Some details omitted; in case of doubt, it also", "follows from the more general", "Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors}.", "Our claim above gives that $L = T(i(L))$ for $L$ in $D(\\mathcal{A})$", "and $i(T(K)) = K$ if $K$ is in $D_\\mathcal{A}(\\mathcal{C})$.", "Using Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}", "the result follows." ], "refs": [ "injectives-theorem-K-injective-embedding-grothendieck", "derived-lemma-derived-adjoint-functors", "categories-lemma-adjoint-fully-faithful" ], "ref_ids": [ 7768, 1907, 12248 ] } ], "ref_ids": [] }, { "id": 4253, "type": "theorem", "label": "sites-cohomology-lemma-restrict-K-injective-to-open", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-restrict-K-injective-to-open", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of", "$\\mathcal{C}$. The restriction of a K-injective complex of", "$\\mathcal{O}$-modules to $\\mathcal{C}/U$ is a K-injective complex of", "$\\mathcal{O}_U$-modules." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}", "and the fact that the restriction functor has the", "exact left adjoint $j_!$. See discussion above." ], "refs": [ "derived-lemma-adjoint-preserve-K-injectives" ], "ref_ids": [ 1915 ] } ], "ref_ids": [] }, { "id": 4254, "type": "theorem", "label": "sites-cohomology-lemma-unbounded-cohomology-of-open", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-unbounded-cohomology-of-open", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U \\in \\Ob(\\mathcal{C})$.", "For $K$ in $D(\\mathcal{O})$ we have", "$H^p(U, K) = H^p(\\mathcal{C}/U, K|_{\\mathcal{C}/U})$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I}^\\bullet$ be a K-injective complex of $\\mathcal{O}$-modules", "representing $K$. Then", "$$", "H^q(U, K) = H^q(\\Gamma(U, \\mathcal{I}^\\bullet)) =", "H^q(\\Gamma(\\mathcal{C}/U, \\mathcal{I}^\\bullet|_{\\mathcal{C}/U}))", "$$", "by construction of cohomology. By", "Lemma \\ref{lemma-restrict-K-injective-to-open}", "the complex $\\mathcal{I}^\\bullet|_{\\mathcal{C}/U}$ is a K-injective complex", "representing $K|_{\\mathcal{C}/U}$ and the lemma follows." ], "refs": [ "sites-cohomology-lemma-restrict-K-injective-to-open" ], "ref_ids": [ 4253 ] } ], "ref_ids": [] }, { "id": 4255, "type": "theorem", "label": "sites-cohomology-lemma-sheafification-cohomology", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-sheafification-cohomology", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K$ be an object of", "$D(\\mathcal{O})$. The sheafification of", "$$", "U \\mapsto H^q(U, K) = H^q(\\mathcal{C}/U, K|_{\\mathcal{C}/U})", "$$", "is the $q$th cohomology sheaf $H^q(K)$ of $K$." ], "refs": [], "proofs": [ { "contents": [ "The equality $H^q(U, K) = H^q(\\mathcal{C}/U, K|_{\\mathcal{C}/U})$", "holds by Lemma \\ref{lemma-unbounded-cohomology-of-open}.", "Choose a K-injective complex $\\mathcal{I}^\\bullet$ representing $K$.", "Then", "$$", "H^q(U, K) =", "\\frac{\\Ker(\\mathcal{I}^q(U) \\to \\mathcal{I}^{q + 1}(U))}", "{\\Im(\\mathcal{I}^{q - 1}(U) \\to \\mathcal{I}^q(U))}.", "$$", "by our construction of cohomology. Since", "$H^q(K) = \\Ker(\\mathcal{I}^q \\to \\mathcal{I}^{q + 1})/", "\\Im(\\mathcal{I}^{q - 1} \\to \\mathcal{I}^q)$ the result is clear." ], "refs": [ "sites-cohomology-lemma-unbounded-cohomology-of-open" ], "ref_ids": [ 4254 ] } ], "ref_ids": [] }, { "id": 4256, "type": "theorem", "label": "sites-cohomology-lemma-restrict-direct-image-open", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-restrict-direct-image-open", "contents": [ "Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to", "(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites", "corresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.", "Given $V \\in \\mathcal{D}$, set $U = u(V)$ and denote", "$g : (\\mathcal{C}/U, \\mathcal{O}_U) \\to (\\mathcal{D}/V, \\mathcal{O}_V)$", "the induced morphism of ringed sites", "(Modules on Sites, Lemma", "\\ref{sites-modules-lemma-localize-morphism-ringed-sites}).", "Then $(Rf_*E)|_{\\mathcal{D}/V} = Rg_*(E|_{\\mathcal{C}/U})$", "for $E$ in $D(\\mathcal{O}_\\mathcal{C})$." ], "refs": [ "sites-modules-lemma-localize-morphism-ringed-sites" ], "proofs": [ { "contents": [ "Represent $E$ by a K-injective complex $\\mathcal{I}^\\bullet$ of", "$\\mathcal{O}_\\mathcal{C}$-modules. Then", "$Rf_*(E) = f_*\\mathcal{I}^\\bullet$", "and $Rg_*(E|_{\\mathcal{C}/U}) = g_*(\\mathcal{I}^\\bullet|_{\\mathcal{C}/U})$ by", "Lemma \\ref{lemma-restrict-K-injective-to-open}.", "Since it is clear that", "$(f_*\\mathcal{F})|_{\\mathcal{D}/V} = g_*(\\mathcal{F}|_{\\mathcal{C}/U})$", "for any sheaf $\\mathcal{F}$ on $\\mathcal{C}$", "(see Modules on Sites, Lemma", "\\ref{sites-modules-lemma-localize-morphism-ringed-sites} or the more basic", "Sites, Lemma \\ref{sites-lemma-localize-morphism})", "the result follows." ], "refs": [ "sites-cohomology-lemma-restrict-K-injective-to-open", "sites-modules-lemma-localize-morphism-ringed-sites", "sites-lemma-localize-morphism" ], "ref_ids": [ 4253, 14174, 8571 ] } ], "ref_ids": [ 14174 ] }, { "id": 4257, "type": "theorem", "label": "sites-cohomology-lemma-Leray-unbounded", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-Leray-unbounded", "contents": [ "Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to", "(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites", "corresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.", "Then $R\\Gamma(\\mathcal{D}, -) \\circ Rf_* = R\\Gamma(\\mathcal{C}, -)$ as", "functors $D(\\mathcal{O}_\\mathcal{C}) \\to D(\\Gamma(\\mathcal{O}_\\mathcal{D}))$.", "More generally, for $V \\in \\mathcal{D}$ with $U = u(V)$", "we have $R\\Gamma(U, -) = R\\Gamma(V, -) \\circ Rf_*$." ], "refs": [], "proofs": [ { "contents": [ "Consider the punctual topos $pt$ endowed with $\\mathcal{O}_{pt}$", "given by the ring $\\Gamma(\\mathcal{O}_\\mathcal{D})$.", "There is a canonical morphism", "$(\\mathcal{D}, \\mathcal{O}_\\mathcal{D}) \\to (pt, \\mathcal{O}_{pt})$", "of ringed topoi inducing the identification on global sections of", "structure sheaves. Then", "$D(\\mathcal{O}_{pt}) = D(\\Gamma(\\mathcal{O}_\\mathcal{D}))$.", "The assertion", "$R\\Gamma(\\mathcal{D}, -) \\circ Rf_* = R\\Gamma(\\mathcal{C}, -)$", "follows from Lemma \\ref{lemma-derived-pushforward-composition}", "applied to", "$$", "(\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to", "(\\mathcal{D}, \\mathcal{O}_\\mathcal{D}) \\to (pt, \\mathcal{O}_{pt})", "$$", "The second (more general) statement follows from the first statement", "after applying Lemma \\ref{lemma-restrict-direct-image-open}." ], "refs": [ "sites-cohomology-lemma-derived-pushforward-composition", "sites-cohomology-lemma-restrict-direct-image-open" ], "ref_ids": [ 4250, 4256 ] } ], "ref_ids": [] }, { "id": 4258, "type": "theorem", "label": "sites-cohomology-lemma-unbounded-describe-higher-direct-images", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-unbounded-describe-higher-direct-images", "contents": [ "Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to", "(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites", "corresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.", "Let $K$ be in $D(\\mathcal{O}_\\mathcal{C})$. Then $H^i(Rf_*K)$ is the sheaf", "associated to the presheaf", "$$", "V \\mapsto H^i(u(V), K) = H^i(V, Rf_*K)", "$$" ], "refs": [], "proofs": [ { "contents": [ "The equality $H^i(u(V), K) = H^i(V, Rf_*K)$ follows upon taking", "cohomology from the second statement in", "Lemma \\ref{lemma-Leray-unbounded}. Then the statement on sheafification", "follows from Lemma \\ref{lemma-sheafification-cohomology}." ], "refs": [ "sites-cohomology-lemma-Leray-unbounded", "sites-cohomology-lemma-sheafification-cohomology" ], "ref_ids": [ 4257, 4255 ] } ], "ref_ids": [] }, { "id": 4259, "type": "theorem", "label": "sites-cohomology-lemma-modules-abelian-unbounded", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-modules-abelian-unbounded", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O}_\\mathcal{C})$ be a ringed site.", "Let $K$ be an object of $D(\\mathcal{O}_\\mathcal{C})$", "and denote $K_{ab}$ its image in $D(\\underline{\\mathbf{Z}}_\\mathcal{C})$.", "\\begin{enumerate}", "\\item There is a canonical map", "$R\\Gamma(\\mathcal{C}, K) \\to R\\Gamma(\\mathcal{C}, K_{ab})$", "which is an isomorphism in $D(\\textit{Ab})$.", "\\item For any $U \\in \\mathcal{C}$ there is a canonical map", "$R\\Gamma(U, K) \\to R\\Gamma(U, K_{ab})$", "which is an isomorphism in $D(\\textit{Ab})$.", "\\item Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to", "(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites.", "There is a canonical map $Rf_*K \\to Rf_*(K_{ab})$ which", "is an isomorphism in $D(\\underline{\\mathbf{Z}}_\\mathcal{D})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The map is constructed as follows. Choose a K-injective complex", "$\\mathcal{I}^\\bullet$ representing $K$. Choose a quasi-isomorpism", "$\\mathcal{I}^\\bullet \\to \\mathcal{J}^\\bullet$ where $\\mathcal{J}^\\bullet$", "is a K-injective complex of abelian groups. Then the map in", "(1) is given by", "$\\Gamma(\\mathcal{C}, \\mathcal{I}^\\bullet) \\to", "\\Gamma(\\mathcal{C}, \\mathcal{J}^\\bullet)$", "(2) is given by", "$\\Gamma(U, \\mathcal{I}^\\bullet) \\to \\Gamma(U, \\mathcal{J}^\\bullet)$", "and the map in (3) is given by", "$f_*\\mathcal{I}^\\bullet \\to f_*\\mathcal{J}^\\bullet$.", "To show that these maps are isomorphisms, it suffices to prove", "they induce isomorphisms on cohomology groups and cohomology sheaves.", "By Lemmas \\ref{lemma-unbounded-cohomology-of-open} and", "\\ref{lemma-unbounded-describe-higher-direct-images}", "it suffices to show that the map", "$$", "H^0(\\mathcal{C}, K) \\longrightarrow H^0(\\mathcal{C}, K_{ab})", "$$", "is an isomorphism. Observe that", "$$", "H^0(\\mathcal{C}, K) =", "\\Hom_{D(\\mathcal{O}_\\mathcal{C})}(\\mathcal{O}_\\mathcal{C}, K)", "$$", "and similarly for the other group. Choose any complex $\\mathcal{K}^\\bullet$", "of $\\mathcal{O}_\\mathcal{C}$-modules representing $K$. By construction of the", "derived category as a localization we have", "$$", "\\Hom_{D(\\mathcal{O}_\\mathcal{C})}(\\mathcal{O}_\\mathcal{C}, K) =", "\\colim_{s : \\mathcal{F}^\\bullet \\to \\mathcal{O}_\\mathcal{C}}", "\\Hom_{K(\\mathcal{O}_\\mathcal{C})}(\\mathcal{F}^\\bullet, \\mathcal{K}^\\bullet)", "$$", "where the colimit is over quasi-isomorphisms $s$ of complexes of", "$\\mathcal{O}_\\mathcal{C}$-modules. Similarly, we have", "$$", "\\Hom_{D(\\underline{\\mathbf{Z}}_\\mathcal{C})}", "(\\underline{\\mathbf{Z}}_\\mathcal{C}, K) =", "\\colim_{s : \\mathcal{G}^\\bullet \\to \\underline{\\mathbf{Z}}_\\mathcal{C}}", "\\Hom_{K(\\underline{\\mathbf{Z}}_\\mathcal{C})}", "(\\mathcal{G}^\\bullet, \\mathcal{K}^\\bullet)", "$$", "Next, we observe that the quasi-isomorphisms", "$s : \\mathcal{G}^\\bullet \\to \\underline{\\mathbf{Z}}_\\mathcal{C}$", "with $\\mathcal{G}^\\bullet$ bounded above complex of flat", "$\\underline{\\mathbf{Z}}_\\mathcal{C}$-modules is cofinal in the system.", "(This follows from Modules on Sites, Lemma", "\\ref{sites-modules-lemma-module-quotient-flat} and", "Derived Categories, Lemma \\ref{derived-lemma-subcategory-left-resolution};", "see discussion in Section \\ref{section-flat}.)", "Hence we can construct an inverse to the map", "$H^0(\\mathcal{C}, K) \\longrightarrow H^0(\\mathcal{C}, K_{ab})$", "by representing an element $\\xi \\in H^0(\\mathcal{C}, K_{ab})$ by a pair", "$$", "(s : \\mathcal{G}^\\bullet \\to \\underline{\\mathbf{Z}}_\\mathcal{C},", "a : \\mathcal{G}^\\bullet \\to \\mathcal{K}^\\bullet)", "$$", "with $\\mathcal{G}^\\bullet$ a bounded above complex of flat", "$\\underline{\\mathbf{Z}}_\\mathcal{C}$-modules and sending this to", "$$", "(\\mathcal{G}^\\bullet \\otimes_{\\underline{\\mathbf{Z}}_\\mathcal{C}}", "\\mathcal{O}_\\mathcal{C}", "\\to \\mathcal{O}_\\mathcal{C},", "\\mathcal{G}^\\bullet \\otimes_{\\underline{\\mathbf{Z}}_\\mathcal{C}}", "\\mathcal{O}_\\mathcal{C}", "\\to \\mathcal{K}^\\bullet)", "$$", "The only thing to note here is that the first arrow", "is a quasi-isomorphism by", "Lemmas \\ref{lemma-derived-tor-quasi-isomorphism-other-side} and", "\\ref{lemma-bounded-flat-K-flat}.", "We omit the detailed verification that this construction", "is indeed an inverse." ], "refs": [ "sites-cohomology-lemma-unbounded-cohomology-of-open", "sites-cohomology-lemma-unbounded-describe-higher-direct-images", "sites-modules-lemma-module-quotient-flat", "derived-lemma-subcategory-left-resolution", "sites-cohomology-lemma-derived-tor-quasi-isomorphism-other-side", "sites-cohomology-lemma-bounded-flat-K-flat" ], "ref_ids": [ 4254, 4258, 14203, 1835, 4237, 4233 ] } ], "ref_ids": [] }, { "id": 4260, "type": "theorem", "label": "sites-cohomology-lemma-adjoint-lower-shriek-restrict", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-adjoint-lower-shriek-restrict", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an", "object of $\\mathcal{C}$. Denote", "$j : (\\Sh(\\mathcal{C}/U), \\mathcal{O}_U) \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$", "the corresponding localization morphism. The restriction functor", "$D(\\mathcal{O}) \\to D(\\mathcal{O}_U)$ is a right adjoint to", "extension by zero $j_! : D(\\mathcal{O}_U) \\to D(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "We have to show that", "$$", "\\Hom_{D(\\mathcal{O})}(j_!E, F) = \\Hom_{D(\\mathcal{O}_U)}(E, F|_U)", "$$", "Choose a complex $\\mathcal{E}^\\bullet$ of $\\mathcal{O}_U$-modules", "representing $E$ and choose", "a K-injective complex $\\mathcal{I}^\\bullet$ representing $F$.", "By Lemma \\ref{lemma-restrict-K-injective-to-open} the complex", "$\\mathcal{I}^\\bullet|_U$ is K-injective as well. Hence we see that", "the formula above becomes", "$$", "\\Hom_{D(\\mathcal{O})}(j_!\\mathcal{E}^\\bullet, \\mathcal{I}^\\bullet) =", "\\Hom_{D(\\mathcal{O}_U)}(\\mathcal{E}^\\bullet, \\mathcal{I}^\\bullet|_U)", "$$", "which holds as $|_U$ and $j_!$ are adjoint functors", "(Modules on Sites, Lemma \\ref{sites-modules-lemma-extension-by-zero})", "and", "Derived Categories, Lemma \\ref{derived-lemma-K-injective}." ], "refs": [ "sites-cohomology-lemma-restrict-K-injective-to-open", "sites-modules-lemma-extension-by-zero", "derived-lemma-K-injective" ], "ref_ids": [ 4253, 14169, 1908 ] } ], "ref_ids": [] }, { "id": 4261, "type": "theorem", "label": "sites-cohomology-lemma-K-injective-flat", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-K-injective-flat", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a flat morphism", "of ringed topoi. If $\\mathcal{I}^\\bullet$ is a K-injective", "complex of $\\mathcal{O}_\\mathcal{C}$-modules, then", "$f_*\\mathcal{I}^\\bullet$ is K-injective", "as a complex of $\\mathcal{O}_\\mathcal{D}$-modules." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$$", "\\Hom_{K(\\mathcal{O}_\\mathcal{D})}(\\mathcal{F}^\\bullet, f_*\\mathcal{I}^\\bullet)", "=", "\\Hom_{K(\\mathcal{O}_\\mathcal{C})}(f^*\\mathcal{F}^\\bullet, \\mathcal{I}^\\bullet)", "$$", "by", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-adjoint-pullback-pushforward-modules}", "and the fact that $f^*$ is exact as $f$ is assumed to be flat." ], "refs": [ "sites-modules-lemma-adjoint-pullback-pushforward-modules" ], "ref_ids": [ 14155 ] } ], "ref_ids": [] }, { "id": 4262, "type": "theorem", "label": "sites-cohomology-lemma-hom-K-injective", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-hom-K-injective", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}'$ be a map", "of sheaves of rings. If $\\mathcal{I}^\\bullet$ is a K-injective complex of", "$\\mathcal{O}$-modules, then", "$\\SheafHom_\\mathcal{O}(\\mathcal{O}', \\mathcal{I}^\\bullet)$", "is a K-injective complex of $\\mathcal{O}'$-modules." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$\\Hom_{K(\\mathcal{O}')}(\\mathcal{G}^\\bullet,", "\\Hom_\\mathcal{O}(\\mathcal{O}', \\mathcal{I}^\\bullet)) =", "\\Hom_{K(\\mathcal{O})}(\\mathcal{G}^\\bullet, \\mathcal{I}^\\bullet)$", "by Modules on Sites, Lemma \\ref{sites-modules-lemma-adjoint-hom-restrict}." ], "refs": [ "sites-modules-lemma-adjoint-hom-restrict" ], "ref_ids": [ 14196 ] } ], "ref_ids": [] }, { "id": 4263, "type": "theorem", "label": "sites-cohomology-lemma-localize-cartesian-square", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-localize-cartesian-square", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$$", "\\xymatrix{", "X' \\ar[d] \\ar[r] & X \\ar[d] \\\\", "Y' \\ar[r] & Y", "}", "$$", "be a cartesian diagram of $\\mathcal{C}$. Then we have", "$j_{Y'/Y}^{-1} \\circ Rj_{X/Y, *} = Rj_{X'/Y', *} \\circ j_{X'/X}^{-1}$", "as functors $D(\\mathcal{C}/X) \\to D(\\mathcal{C}/Y')$." ], "refs": [], "proofs": [ { "contents": [ "Let $E \\in D(\\mathcal{C}/X)$. Choose a K-injective complex", "$\\mathcal{I}^\\bullet$ of abelian sheaves on $\\mathcal{C}/X$", "representing $E$. By Lemma \\ref{lemma-restrict-K-injective-to-open}", "we see that $j_{X'/X}^{-1}\\mathcal{I}^\\bullet$ is K-injective too.", "Hence we may compute $Rj_{X'/Y'}(j_{X'/X}^{-1}E)$ by", "$j_{X'/Y', *}j_{X'/X}^{-1}\\mathcal{I}^\\bullet$.", "Thus we see that the equality holds by", "Sites, Lemma \\ref{sites-lemma-localize-cartesian-square}." ], "refs": [ "sites-cohomology-lemma-restrict-K-injective-to-open", "sites-lemma-localize-cartesian-square" ], "ref_ids": [ 4253, 8570 ] } ], "ref_ids": [] }, { "id": 4264, "type": "theorem", "label": "sites-cohomology-lemma-localize-cartesian-square-modules", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-localize-cartesian-square-modules", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$$", "\\xymatrix{", "X' \\ar[d] \\ar[r] & X \\ar[d] \\\\", "Y' \\ar[r] & Y", "}", "$$", "be a cartesian diagram of $\\mathcal{C}$. Then we have", "$j_{Y'/Y}^* \\circ Rj_{X/Y, *} = Rj_{X'/Y', *} \\circ j_{X'/X}^*$", "as functors", "$D(\\mathcal{O}_X) \\to D(\\mathcal{O}_{Y'})$." ], "refs": [], "proofs": [ { "contents": [ "Since $j_{Y'/Y}^{-1}\\mathcal{O}_Y = \\mathcal{O}_{Y'}$ we have", "$j_{Y'/Y}^* = Lj_{Y'/Y}^* = j_{Y'/Y}^{-1}$. Similarly we have", "$j_{X'/X}^* = Lj_{X'/X}^* = j_{X'/X}^{-1}$. Thus by", "Lemma \\ref{lemma-modules-abelian-unbounded} it suffices", "to prove the result on derived categories of abelian sheaves", "which we did in", "Lemma \\ref{lemma-localize-cartesian-square}." ], "refs": [ "sites-cohomology-lemma-modules-abelian-unbounded", "sites-cohomology-lemma-localize-cartesian-square" ], "ref_ids": [ 4259, 4263 ] } ], "ref_ids": [] }, { "id": 4265, "type": "theorem", "label": "sites-cohomology-lemma-derived-limit-is-ok", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-derived-limit-is-ok", "contents": [ "Let $\\mathcal{C}$ be a site. Let $K$ be an object of", "$D(\\mathcal{C} \\times \\mathbf{N})$. Set $K_n = i_n^{-1}K$ as above.", "Then", "$$", "R\\lim K \\cong R\\lim K_n", "$$", "in $D(\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "To calculate $R\\lim$ on an object $K$ of $D(\\mathcal{C} \\times \\mathbf{N})$", "we choose a K-injective representative $\\mathcal{I}^\\bullet$ whose terms are", "injective objects of $\\textit{Ab}(\\mathcal{C} \\times \\mathbf{N})$, see", "Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "We may and do think of $\\mathcal{I}^\\bullet$ as an inverse system of", "complexes $(\\mathcal{I}_n^\\bullet)$ and then we see that", "$$", "R\\lim K = \\lim \\mathcal{I}_n^\\bullet", "$$", "where the right hand side is the termwise inverse limit.", "\\medskip\\noindent", "Let $\\mathcal{J} = (\\mathcal{J}_n)$ be an injective object of", "$\\textit{Ab}(\\mathcal{C} \\times \\mathbf{N})$. The morphisms", "$(U, n) \\to (U, n + 1)$ are monomorphisms of", "$\\mathcal{C} \\times \\mathbf{N}$, hence", "$\\mathcal{J}(U, n + 1) \\to \\mathcal{J}(U, n)$ is surjective", "(Lemma \\ref{lemma-restriction-along-monomorphism-surjective}).", "It follows that $\\mathcal{J}_{n + 1} \\to \\mathcal{J}_n$ is", "surjective as a map of presheaves.", "\\medskip\\noindent", "Note that the functor $i_n^{-1}$ has an exact left adjoint $i_{n, !}$.", "Namely, $i_{n, !}\\mathcal{F}$ is the inverse system", "$\\ldots 0 \\to 0 \\to \\mathcal{F} \\to \\ldots \\to \\mathcal{F}$.", "Thus the complexes $i_n^{-1}\\mathcal{I}^\\bullet = \\mathcal{I}_n^\\bullet$", "are K-injective by", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}.", "\\medskip\\noindent", "Because we chose our K-injective complex to have injective terms", "we conclude that", "$$", "0 \\to \\lim \\mathcal{I}_n^\\bullet \\to \\prod \\mathcal{I}_n^\\bullet", "\\to \\prod \\mathcal{I}_n^\\bullet \\to 0", "$$", "is a short exact sequence of complexes of abelian sheaves as it", "is a short exact sequence of complexes of abelian presheaves.", "Moreover, the products in the middle and the right represent", "the products in $D(\\mathcal{C})$, see", "Injectives, Lemma \\ref{injectives-lemma-derived-products} and its", "proof (this is where we use that $\\mathcal{I}_n^\\bullet$ is K-injective).", "Thus $R\\lim K$ is a homotopy limit of the inverse system $(K_n)$", "by definition of homotopy limits in triangulated categories." ], "refs": [ "injectives-theorem-K-injective-embedding-grothendieck", "sites-cohomology-lemma-restriction-along-monomorphism-surjective", "derived-lemma-adjoint-preserve-K-injectives", "injectives-lemma-derived-products" ], "ref_ids": [ 7768, 4212, 1915, 7795 ] } ], "ref_ids": [] }, { "id": 4266, "type": "theorem", "label": "sites-cohomology-lemma-RGamma-commutes-with-Rlim", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-RGamma-commutes-with-Rlim", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. The functors", "$R\\Gamma(\\mathcal{C}, -)$ and $R\\Gamma(U, -)$ for $U \\in \\Ob(\\mathcal{C})$", "commute with $R\\lim$. Moreover, there are", "short exact sequences", "$$", "0 \\to", "R^1\\lim H^{m - 1}(U, K_n) \\to H^m(U, R\\lim K_n) \\to", "\\lim H^m(U, K_n) \\to 0", "$$", "for any inverse system $(K_n)$ in $D(\\mathcal{O})$ and $m \\in \\mathbf{Z}$.", "Similar for $H^m(\\mathcal{C}, R\\lim K_n)$." ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from", "Injectives, Lemma \\ref{injectives-lemma-RF-commutes-with-Rlim}.", "Then we may apply ", "More on Algebra, Remark \\ref{more-algebra-remark-compare-derived-limit}", "to $R\\lim R\\Gamma(U, K_n) = R\\Gamma(U, R\\lim K_n)$ to get the short", "exact sequences." ], "refs": [ "injectives-lemma-RF-commutes-with-Rlim", "more-algebra-remark-compare-derived-limit" ], "ref_ids": [ 7796, 10658 ] } ], "ref_ids": [] }, { "id": 4267, "type": "theorem", "label": "sites-cohomology-lemma-Rf-commutes-with-Rlim", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-Rf-commutes-with-Rlim", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$", "be a morphism of ringed topoi. Then $Rf_*$ commutes with $R\\lim$, i.e.,", "$Rf_*$ commutes with derived limits." ], "refs": [], "proofs": [ { "contents": [ "Let $(K_n)$ be an inverse system of objects of $D(\\mathcal{O})$.", "By induction on $n$ we may choose actual complexes $\\mathcal{K}_n^\\bullet$", "of $\\mathcal{O}$-modules and maps of complexes", "$\\mathcal{K}_{n + 1}^\\bullet \\to \\mathcal{K}_n^\\bullet$ representing the", "maps $K_{n + 1} \\to K_n$ in $D(\\mathcal{O})$. In other words, there exists", "an object $K$ in $D(\\mathcal{C} \\times \\mathbf{N})$ whose associated inverse", "system is the given one. Next, consider the commutative diagram", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C} \\times \\mathbf{N}) \\ar[r]_g \\ar[d]_{f \\times 1} &", "\\Sh(\\mathcal{C}) \\ar[d]_f \\\\", "\\Sh(\\mathcal{C}' \\times \\mathbf{N}) \\ar[r]^{g'} &", "\\Sh(\\mathcal{C}')", "}", "$$", "of morphisms of topoi. It follows that", "$R\\lim R(f \\times 1)_*K = Rf_* R\\lim K$. Working through the definitions", "and using Lemma \\ref{lemma-derived-limit-is-ok}", "we obtain that $R\\lim (Rf_*K_n) = Rf_*(R\\lim K_n)$.", "\\medskip\\noindent", "Alternate proof in case $\\mathcal{C}$ has enough points. Consider the defining", "distinguished triangle", "$$", "R\\lim K_n \\to \\prod K_n \\to \\prod K_n", "$$", "in $D(\\mathcal{O})$. Applying the exact functor $Rf_*$ we obtain", "the distinguished triangle", "$$", "Rf_*(R\\lim K_n) \\to Rf_*\\left(\\prod K_n\\right) \\to Rf_*\\left(\\prod K_n\\right)", "$$", "in $D(\\mathcal{O}')$. Thus we see that it suffices to prove that", "$Rf_*$ commutes with products in the derived category (which are not just", "given by products of complexes, see", "Injectives, Lemma \\ref{injectives-lemma-derived-products}).", "However, since $Rf_*$ is a right adjoint by Lemma \\ref{lemma-adjoint}", "this follows formally (see", "Categories, Lemma \\ref{categories-lemma-adjoint-exact}).", "Caution: Note that we cannot apply", "Categories, Lemma \\ref{categories-lemma-adjoint-exact}", "directly as $R\\lim K_n$ is not a limit in $D(\\mathcal{O})$." ], "refs": [ "sites-cohomology-lemma-derived-limit-is-ok", "injectives-lemma-derived-products", "sites-cohomology-lemma-adjoint", "categories-lemma-adjoint-exact", "categories-lemma-adjoint-exact" ], "ref_ids": [ 4265, 7795, 4249, 12249, 12249 ] } ], "ref_ids": [] }, { "id": 4268, "type": "theorem", "label": "sites-cohomology-lemma-inverse-limit-is-derived-limit", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-inverse-limit-is-derived-limit", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{F}_n)$ be an", "inverse system of $\\mathcal{O}$-modules. Let", "$\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume", "\\begin{enumerate}", "\\item every object of $\\mathcal{C}$ has a covering whose members are elements", "of $\\mathcal{B}$,", "\\item $H^p(U, \\mathcal{F}_n) = 0$ for $p > 0$ and $U \\in \\mathcal{B}$,", "\\item the inverse system $\\mathcal{F}_n(U)$ has vanishing $R^1\\lim$", "for $U \\in \\mathcal{B}$.", "\\end{enumerate}", "Then $R\\lim \\mathcal{F}_n = \\lim \\mathcal{F}_n$ and we have", "$H^p(U, \\lim \\mathcal{F}_n) = 0$ for $p > 0$ and $U \\in \\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "Set $K_n = \\mathcal{F}_n$ and $K = R\\lim \\mathcal{F}_n$. Using the notation", "of Remark \\ref{remark-discuss-derived-limit} and assumption (2) we see that for", "$U \\in \\mathcal{B}$ we have $\\underline{\\mathcal{H}}_n^m(U) = 0$", "when $m \\not = 0$ and $\\underline{\\mathcal{H}}_n^0(U) = \\mathcal{F}_n(U)$.", "From Equation (\\ref{equation-ses-Rlim-over-U}) and assumption (3)", "we see that $\\underline{\\mathcal{H}}^m(U) = 0$", "when $m \\not = 0$ and equal to $\\lim \\mathcal{F}_n(U)$", "when $m = 0$. Sheafifying using (1) we find that", "$\\mathcal{H}^m = 0$ when $m \\not = 0$ and equal to", "$\\lim \\mathcal{F}_n$ when $m = 0$.", "Hence $K = \\lim \\mathcal{F}_n$.", "Since $H^m(U, K) = \\underline{\\mathcal{H}}^m(U) = 0$ for $m > 0$", "(see above) we see that the second assertion holds." ], "refs": [ "sites-cohomology-remark-discuss-derived-limit" ], "ref_ids": [ 4428 ] } ], "ref_ids": [] }, { "id": 4269, "type": "theorem", "label": "sites-cohomology-lemma-cohomology-derived-limit-injective", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cohomology-derived-limit-injective", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(K_n)$ be an", "inverse system in $D(\\mathcal{O})$. Let $V \\in \\Ob(\\mathcal{C})$", "and $m \\in \\mathbf{Z}$. Assume there exist an integer $n(V)$", "and a cofinal system $\\text{Cov}_V$ of coverings of $V$ such that", "for $\\{V_i \\to V\\} \\in \\text{Cov}_V$", "\\begin{enumerate}", "\\item $R^1\\lim H^{m - 1}(V_i, K_n) = 0$, and", "\\item $H^m(V_i, K_n) \\to H^m(V_i, K_{n(V)})$ is injective", "for $n \\geq n(V)$.", "\\end{enumerate}", "Then the map on sections $H^m(R\\lim K_n)(V) \\to H^m(K_{n(V)})(V)$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\gamma \\in H^m(R\\lim K_n)(V)$ map to zero in", "$H^m(K_{n(V)})(V)$. Since $H^m(R\\lim K_n)$ is the sheafification of", "$U \\mapsto H^m(U, R\\lim K_n)$ (by Lemma \\ref{lemma-sheafification-cohomology})", "we can choose $\\{V_i \\to V\\} \\in \\text{Cov}_V$", "and elements $\\tilde\\gamma_i \\in H^m(V_i, R\\lim K_n)$ mapping to", "$\\gamma|_{V_i}$.", "Then $\\tilde\\gamma_i$ maps to $\\tilde\\gamma_{i, n(V)} \\in H^m(V_i, K_{n(V)})$.", "Using that $H^m(K_{n(V)})$ is the sheafification of", "$U \\mapsto H^m(U, K_{n(V)})$", "(by Lemma \\ref{lemma-sheafification-cohomology} again)", "we see that after replacing $\\{V_i \\to V\\}$ by a refinement", "we may assume that $\\tilde\\gamma_{i, n(V)} = 0$ for all $i$.", "For this covering we consider the short exact sequences", "$$", "0 \\to", "R^1\\lim H^{m - 1}(V_i, K_n) \\to H^m(V_i, R\\lim K_n) \\to", "\\lim H^m(V_i, K_n) \\to 0", "$$", "of Lemma \\ref{lemma-RGamma-commutes-with-Rlim}.", "By assumption (1) the group on the left is zero and by", "assumption (2) the group on the right maps injectively", "into $H^m(V_i, K_{n(V)})$. We conclude $\\tilde\\gamma_i = 0$", "and hence $\\gamma = 0$ as desired." ], "refs": [ "sites-cohomology-lemma-sheafification-cohomology", "sites-cohomology-lemma-sheafification-cohomology", "sites-cohomology-lemma-RGamma-commutes-with-Rlim" ], "ref_ids": [ 4255, 4255, 4266 ] } ], "ref_ids": [] }, { "id": 4270, "type": "theorem", "label": "sites-cohomology-lemma-is-limit-per-object", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-is-limit-per-object", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E \\in D(\\mathcal{O})$.", "Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume", "\\begin{enumerate}", "\\item every object of $\\mathcal{C}$ has a covering whose members", "are elements of $\\mathcal{B}$, and", "\\item for every $V \\in \\mathcal{B}$ there exist a function", "$p(V, -) : \\mathbf{Z} \\to \\mathbf{Z}$ and a cofinal system $\\text{Cov}_V$", "of coverings of $V$ such that", "$$", "H^p(V_i, H^{m - p}(E)) = 0", "$$", "for all $\\{V_i \\to V\\} \\in \\text{Cov}_V$ and all integers $p, m$", "satisfying $p > p(V, m)$.", "\\end{enumerate}", "Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$", "is an isomorphism in $D(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "Set $K_n = \\tau_{\\geq -n}E$ and $K = R\\lim K_n$.", "The canonical map $E \\to K$", "comes from the canonical maps $E \\to K_n = \\tau_{\\geq -n}E$.", "We have to show that $E \\to K$ induces an isomorphism", "$H^m(E) \\to H^m(K)$ of cohomology sheaves. In the rest of the", "proof we fix $m$. If $n \\geq -m$, then", "the map $E \\to \\tau_{\\geq -n}E = K_n$ induces an isomorphism", "$H^m(E) \\to H^m(K_n)$.", "To finish the proof it suffices to show that for every $V \\in \\mathcal{B}$", "there exists an integer $n(V) \\geq -m$ such that the map", "$H^m(K)(V) \\to H^m(K_{n(V)})(V)$ is injective. Namely, then", "the composition", "$$", "H^m(E)(V) \\to H^m(K)(V) \\to H^m(K_{n(V)})(V)", "$$", "is a bijection and the second arrow is injective, hence the", "first arrow is bijective. By property (1) this will imply", "$H^m(E) \\to H^m(K)$ is an isomorphism. Set", "$$", "n(V) = 1 + \\max\\{-m, p(V, m - 1) - m, -1 + p(V, m) - m, -2 + p(V, m + 1) - m\\}.", "$$", "so that in any case $n(V) \\geq -m$. Claim: the maps", "$$", "H^{m - 1}(V_i, K_{n + 1}) \\to H^{m - 1}(V_i, K_n)", "\\quad\\text{and}\\quad", "H^m(V_i, K_{n + 1}) \\to H^m(V_i, K_n)", "$$", "are isomorphisms for $n \\geq n(V)$ and $\\{V_i \\to V\\} \\in \\text{Cov}_V$.", "The claim implies conditions", "(1) and (2) of Lemma \\ref{lemma-cohomology-derived-limit-injective}", "are satisfied and hence implies the desired injectivity.", "Recall (Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle})", "that we have distinguished triangles", "$$", "H^{-n - 1}(E)[n + 1] \\to", "K_{n + 1} \\to K_n \\to H^{-n - 1}(E)[n + 2]", "$$", "Looking at the asssociated long exact cohomology sequence the claim follows if", "$$", "H^{m + n}(V_i, H^{-n - 1}(E)),\\quad", "H^{m + n + 1}(V_i, H^{-n - 1}(E)),\\quad", "H^{m + n + 2}(V_i, H^{-n - 1}(E))", "$$", "are zero for $n \\geq n(V)$ and $\\{V_i \\to V\\} \\in \\text{Cov}_V$.", "This follows from our choice of $n(V)$", "and the assumption in the lemma." ], "refs": [ "sites-cohomology-lemma-cohomology-derived-limit-injective", "derived-remark-truncation-distinguished-triangle" ], "ref_ids": [ 4269, 2016 ] } ], "ref_ids": [] }, { "id": 4271, "type": "theorem", "label": "sites-cohomology-lemma-is-limit-spaltenstein", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-is-limit-spaltenstein", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E \\in D(\\mathcal{O})$.", "Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume", "\\begin{enumerate}", "\\item every object of $\\mathcal{C}$ has a covering whose members are", "elements of $\\mathcal{B}$, and", "\\item for every $V \\in \\mathcal{B}$ there exist an integer $d_V \\geq 0$ and", "a cofinal system $\\text{Cov}_V$ of coverings of $V$ such that", "$$", "H^p(V_i, H^q(E)) = 0 \\text{ for }", "\\{V_i \\to V\\} \\in \\text{Cov}_V,\\ p > d_V, \\text{ and }q < 0", "$$", "\\end{enumerate}", "Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$", "is an isomorphism in $D(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-is-limit-per-object}", "with $p(V, m) = d_V + \\max(0, m)$." ], "refs": [ "sites-cohomology-lemma-is-limit-per-object" ], "ref_ids": [ 4270 ] } ], "ref_ids": [] }, { "id": 4272, "type": "theorem", "label": "sites-cohomology-lemma-is-limit", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-is-limit", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E \\in D(\\mathcal{O})$.", "Assume there exists a function $p(-) : \\mathbf{Z} \\to \\mathbf{Z}$", "and a subset $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ such that", "\\begin{enumerate}", "\\item every object of $\\mathcal{C}$ has a covering whose members are", "elements of $\\mathcal{B}$,", "\\item $H^p(V, H^{m - p}(E)) = 0$ for $p > p(m)$ and $V \\in \\mathcal{B}$.", "\\end{enumerate}", "Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$", "is an isomorphism in $D(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-is-limit-per-object}", "with $p(V, m) = p(m)$ and $\\text{Cov}_V$", "equal to the set of coverings $\\{V_i \\to V\\}$ with", "$V_i \\in \\mathcal{B}$ for all $i$." ], "refs": [ "sites-cohomology-lemma-is-limit-per-object" ], "ref_ids": [ 4270 ] } ], "ref_ids": [] }, { "id": 4273, "type": "theorem", "label": "sites-cohomology-lemma-is-limit-dimension", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-is-limit-dimension", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E \\in D(\\mathcal{O})$.", "Assume there exists an integer $d \\geq 0$", "and a subset $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ such that", "\\begin{enumerate}", "\\item every object of $\\mathcal{C}$ has a covering whose members are", "elements of $\\mathcal{B}$,", "\\item $H^p(V, H^q(E)) = 0$ for $p > d$, $q < 0$, and $V \\in \\mathcal{B}$.", "\\end{enumerate}", "Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$", "is an isomorphism in $D(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-is-limit-spaltenstein}", "with $d_V = d$ and $\\text{Cov}_V$", "equal to the set of coverings $\\{V_i \\to V\\}$ with", "$V_i \\in \\mathcal{B}$ for all $i$." ], "refs": [ "sites-cohomology-lemma-is-limit-spaltenstein" ], "ref_ids": [ 4271 ] } ], "ref_ids": [] }, { "id": 4274, "type": "theorem", "label": "sites-cohomology-lemma-cohomology-over-U-trivial", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cohomology-over-U-trivial", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K$", "be an object of $D(\\mathcal{O})$.", "Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume", "\\begin{enumerate}", "\\item every object of $\\mathcal{C}$ has a covering whose members are", "elements of $\\mathcal{B}$,", "\\item $H^p(U, H^q(K)) = 0$ for all $p > 0$, $q \\in \\mathbf{Z}$, and", "$U \\in \\mathcal{B}$.", "\\end{enumerate}", "Then $H^q(U, K) = H^0(U, H^q(K))$ for $q \\in \\mathbf{Z}$", "and $U \\in \\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $K = R\\lim \\tau_{\\geq -n} K$ by", "Lemma \\ref{lemma-is-limit-dimension} with $d = 0$.", "Let $U \\in \\mathcal{B}$. By Equation (\\ref{equation-ses-Rlim-over-U})", "we get a short exact sequence", "$$", "0 \\to R^1\\lim H^{q - 1}(U, \\tau_{\\geq -n}K) \\to", "H^q(U, K) \\to \\lim H^q(U, \\tau_{\\geq -n}K) \\to 0", "$$", "Condition (2) implies", "$H^q(U, \\tau_{\\geq -n} K) = H^0(U, H^q(\\tau_{\\geq -n} K))$", "for all $q$ by using the spectral sequence of", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "The spectral sequence converges because $\\tau_{\\geq -n}K$ is bounded", "below. If $n > -q$ then we have $H^q(\\tau_{\\geq -n}K) = H^q(K)$.", "Thus the systems on the left and the right of the displayed", "short exact sequence are eventually constant with values", "$H^0(U, H^{q - 1}(K))$ and $H^0(U, H^q(K))$ and the lemma follows." ], "refs": [ "sites-cohomology-lemma-is-limit-dimension", "derived-lemma-two-ss-complex-functor" ], "ref_ids": [ 4273, 1871 ] } ], "ref_ids": [] }, { "id": 4275, "type": "theorem", "label": "sites-cohomology-lemma-derived-limit-suitable-system", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-derived-limit-suitable-system", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(K_n)$", "be an inverse system of objects of $D(\\mathcal{O})$.", "Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume", "\\begin{enumerate}", "\\item every object of $\\mathcal{C}$ has a covering whose members are", "elements of $\\mathcal{B}$,", "\\item for all $U \\in \\mathcal{B}$ and all $q \\in \\mathbf{Z}$ we have", "\\begin{enumerate}", "\\item $H^p(U, H^q(K_n)) = 0$ for $p > 0$,", "\\item the inverse system $H^0(U, H^q(K_n))$ has vanishing $R^1\\lim$.", "\\end{enumerate}", "\\end{enumerate}", "Then $H^q(R\\lim K_n) = \\lim H^q(K_n)$ for $q \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Set $K = R\\lim K_n$. We will use notation as in", "Remark \\ref{remark-discuss-derived-limit}. Let $U \\in \\mathcal{B}$.", "By Lemma \\ref{lemma-cohomology-over-U-trivial} and (2)(a)", "we have $H^q(U, K_n) = H^0(U, H^q(K_n))$.", "Using that the functor $R\\Gamma(U, -)$ commutes with", "derived limits we have", "$$", "H^q(U, K) = H^q(R\\lim R\\Gamma(U, K_n)) = \\lim H^0(U, H^q(K_n))", "$$", "where the final equality follows from", "More on Algebra, Remark \\ref{more-algebra-remark-compare-derived-limit}", "and assumption (2)(b). Thus $H^q(U, K)$ is the inverse limit", "the sections of the sheaves $H^q(K_n)$ over $U$. Since", "$\\lim H^q(K_n)$ is a sheaf we find using assumption (1) that $H^q(K)$,", "which is the sheafification of the presheaf $U \\mapsto H^q(U, K)$,", "is equal to $\\lim H^q(K_n)$. This proves the lemma." ], "refs": [ "sites-cohomology-remark-discuss-derived-limit", "sites-cohomology-lemma-cohomology-over-U-trivial", "more-algebra-remark-compare-derived-limit" ], "ref_ids": [ 4428, 4274, 10658 ] } ], "ref_ids": [] }, { "id": 4276, "type": "theorem", "label": "sites-cohomology-lemma-K-injective", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-K-injective", "contents": [ "In the situation described above. Denote", "$\\mathcal{H}^m = H^m(\\mathcal{F}^\\bullet)$ the $m$th cohomology sheaf.", "Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset.", "Let $d \\in \\mathbf{N}$.", "Assume", "\\begin{enumerate}", "\\item every object of $\\mathcal{C}$ has a covering whose members are", "elements of $\\mathcal{B}$,", "\\item for every $U \\in \\mathcal{B}$ we have $H^p(U, \\mathcal{H}^q) = 0$", "for $p > d$ and $q < 0$\\footnote{It suffices if", "$\\forall m$, $\\exists p(m)$, $H^p(U. \\mathcal{H}^{m - p}) = 0$ for", "$p > p(m)$, see Lemma \\ref{lemma-is-limit}.}.", "\\end{enumerate}", "Then (\\ref{equation-into-candidate-K-injective}) is a quasi-isomorphism." ], "refs": [ "sites-cohomology-lemma-is-limit" ], "proofs": [ { "contents": [ "By Derived Categories, Lemma \\ref{derived-lemma-difficulty-K-injectives}", "it suffices to show that the canonical map", "$\\mathcal{F}^\\bullet \\to R\\lim \\tau_{\\geq -n} \\mathcal{F}^\\bullet$", "is an isomorphism. This follows from Lemma \\ref{lemma-is-limit-dimension}." ], "refs": [ "derived-lemma-difficulty-K-injectives", "sites-cohomology-lemma-is-limit-dimension" ], "ref_ids": [ 1927, 4273 ] } ], "ref_ids": [ 4272 ] }, { "id": 4277, "type": "theorem", "label": "sites-cohomology-lemma-inverse-limit-complexes", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-inverse-limit-complexes", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$(\\mathcal{F}_n^\\bullet)$ be an inverse system of complexes of", "$\\mathcal{O}$-modules. Let $m \\in \\mathbf{Z}$. Suppose given", "$\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ and an integer", "$n_0$ such that", "\\begin{enumerate}", "\\item every object of $\\mathcal{C}$ has a covering whose members are", "elements of $\\mathcal{B}$,", "\\item for every $U \\in \\mathcal{B}$", "\\begin{enumerate}", "\\item the systems of abelian groups", "$\\mathcal{F}_n^{m - 2}(U)$ and $\\mathcal{F}_n^{m - 1}(U)$", "have vanishing $R^1\\lim$ (for example these have the Mittag-Leffler property),", "\\item the system of abelian groups $H^{m - 1}(\\mathcal{F}_n^\\bullet(U))$", "has vanishing $R^1\\lim$ (for example it has the Mittag-Leffler property), and", "\\item we have", "$H^m(\\mathcal{F}_n^\\bullet(U)) = H^m(\\mathcal{F}_{n_0}^\\bullet(U))$", "for all $n \\geq n_0$.", "\\end{enumerate}", "\\end{enumerate}", "Then the maps $H^m(\\mathcal{F}^\\bullet) \\to \\lim H^m(\\mathcal{F}_n^\\bullet)", "\\to H^m(\\mathcal{F}_{n_0}^\\bullet)$ are isomorphisms of sheaves where", "$\\mathcal{F}^\\bullet = \\lim \\mathcal{F}_n^\\bullet$ is the termwise", "inverse limit." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\in \\mathcal{B}$. Note that", "$H^m(\\mathcal{F}^\\bullet(U))$ is the cohomology of", "$$", "\\lim_n \\mathcal{F}_n^{m - 2}(U) \\to", "\\lim_n \\mathcal{F}_n^{m - 1}(U) \\to", "\\lim_n \\mathcal{F}_n^m(U) \\to", "\\lim_n \\mathcal{F}_n^{m + 1}(U)", "$$", "in the third spot from the left. By assumptions (2)(a) and (2)(b)", "we may apply", "More on Algebra, Lemma \\ref{more-algebra-lemma-apply-Mittag-Leffler-again}", "to conclude that", "$$", "H^m(\\mathcal{F}^\\bullet(U)) = \\lim H^m(\\mathcal{F}_n^\\bullet(U))", "$$", "By assumption (2)(c) we conclude", "$$", "H^m(\\mathcal{F}^\\bullet(U)) = H^m(\\mathcal{F}_n^\\bullet(U))", "$$", "for all $n \\geq n_0$. By assumption (1) we conclude that the sheafification of", "$U \\mapsto H^m(\\mathcal{F}^\\bullet(U))$ is equal to the sheafification", "of $U \\mapsto H^m(\\mathcal{F}_n^\\bullet(U))$ for all $n \\geq n_0$.", "Thus the inverse system of sheaves $H^m(\\mathcal{F}_n^\\bullet)$ is", "constant for $n \\geq n_0$ with value $H^m(\\mathcal{F}^\\bullet)$ which", "proves the lemma." ], "refs": [ "more-algebra-lemma-apply-Mittag-Leffler-again" ], "ref_ids": [ 10314 ] } ], "ref_ids": [] }, { "id": 4278, "type": "theorem", "label": "sites-cohomology-lemma-olsson-laszlo", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-olsson-laszlo", "contents": [ "\\begin{reference}", "This is \\cite[Proposition 2.1.4]{six-I} with slightly changed", "hypotheses; it is the analogue of \\cite[Proposition 3.13]{Spaltenstein}", "for sites.", "\\end{reference}", "In Situation \\ref{situation-olsson-laszlo} for any", "$E \\in D_\\mathcal{A}(\\mathcal{O})$ the canonical map", "$E \\to R\\lim \\tau_{\\geq -n} E$", "is an isomorphism in $D(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-is-limit-spaltenstein}." ], "refs": [ "sites-cohomology-lemma-is-limit-spaltenstein" ], "ref_ids": [ 4271 ] } ], "ref_ids": [] }, { "id": 4279, "type": "theorem", "label": "sites-cohomology-lemma-olsson-laszlo-modified", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-olsson-laszlo-modified", "contents": [ "In Situation \\ref{situation-olsson-laszlo} let", "$(K_n)$ be an inverse system in $D_\\mathcal{A}^+(\\mathcal{O})$.", "Assume that for every $j$ the inverse system $(H^j(K_n))$", "in $\\mathcal{A}$ is eventually constant with value $\\mathcal{H}^j$. Then", "$H^j(R\\lim K_n) = \\mathcal{H}^j$ for all $j$." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\in \\mathcal{B}$. Let $\\{V_i \\to V\\}$ be in the set", "$\\text{Cov}_V$ of Situation \\ref{situation-olsson-laszlo}.", "Because $K_n$ is bounded below there is a spectral sequence", "$$", "E_2^{p, q} = H^p(V_i, H^q(K_n))", "$$", "converging to $H^{p + q}(V_i, K_n)$. See", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "Observe that $E_2^{p, q} = 0$", "for $p > d_V$ by assumption. Pick $n_0$ such that", "$$", "\\begin{matrix}", "\\mathcal{H}^{j + 1} & = & H^{j + 1}(K_n), \\\\", "\\mathcal{H}^j & = & H^j(K_n), \\\\", "\\ldots, \\\\", "\\mathcal{H}^{j - d_V - 2} & = & H^{j - d_V - 2}(K_n)", "\\end{matrix}", "$$", "for all $n \\geq n_0$. Comparing the spectral sequences above", "for $K_n$ and $K_{n_0}$, we see that for $n \\geq n_0$ the", "cohomology groups $H^{j - 1}(V_i, K_n)$ and $H^j(V_i, K_n)$", "are independent of $n$. It follows that the map on sections", "$H^j(R\\lim K_n)(V) \\to H^j(K_n)(V)$ is injective for $n$ large", "enough (depending on $V$), see", "Lemma \\ref{lemma-cohomology-derived-limit-injective}.", "Since every object of $\\mathcal{C}$ can be covered by elements", "of $\\mathcal{B}$, we conclude that the map", "$H^j(R\\lim K_n) \\to \\mathcal{H}^j$ is injective.", "\\medskip\\noindent", "Surjectivity is shown in a similar manner. Namely, pick", "$U \\in \\Ob(\\mathcal{C})$ and $\\gamma \\in \\mathcal{H}^j(U)$.", "We want to lift $\\gamma$ to a section of $H^j(R\\lim K_n)$", "after replacing $U$ by the members of a covering. Hence we may", "assume $U = V \\in \\mathcal{B}$ by property (1) of", "Situation \\ref{situation-olsson-laszlo}.", "Pick $n_0$ such that", "$$", "\\begin{matrix}", "\\mathcal{H}^{j + 1} & = & H^{j + 1}(K_n), \\\\", "\\mathcal{H}^j & = & H^j(K_n), \\\\", "\\ldots, \\\\", "\\mathcal{H}^{j - d_V - 2} & = & H^{j - d_V - 2}(K_n)", "\\end{matrix}", "$$", "for all $n \\geq n_0$. Choose an element $\\{V_i \\to V\\}$ of", "$\\text{Cov}_V$ such that", "$\\gamma|_{V_i} \\in \\mathcal{H}^j(V_i) = H^j(K_{n_0})(V_i)$", "lifts to an element $\\gamma_{n_0, i} \\in H^j(V_i, K_{n_0})$.", "This is possible because $H^j(K_{n_0})$ is the sheafification", "of $U \\mapsto H^j(U, K_{n_0})$ by Lemma \\ref{lemma-sheafification-cohomology}.", "By the discussion in the first paragraph of the proof we have that", "$H^{j - 1}(V_i, K_n)$ and $H^j(V_i, K_n)$", "are independent of $n \\geq n_0$. Hence", "$\\gamma_{n_0, i}$ lifts to an element", "$\\gamma_i \\in H^j(V_i, R\\lim K_n)$ by", "Lemma \\ref{lemma-RGamma-commutes-with-Rlim}.", "This finishes the proof." ], "refs": [ "derived-lemma-two-ss-complex-functor", "sites-cohomology-lemma-cohomology-derived-limit-injective", "sites-cohomology-lemma-sheafification-cohomology", "sites-cohomology-lemma-RGamma-commutes-with-Rlim" ], "ref_ids": [ 1871, 4269, 4255, 4266 ] } ], "ref_ids": [] }, { "id": 4280, "type": "theorem", "label": "sites-cohomology-lemma-olsson-laszlo-map-version-one", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-olsson-laszlo-map-version-one", "contents": [ "\\begin{reference}", "This is a version of \\cite[Lemma 2.1.10]{six-I} with slightly changed", "hypotheses.", "\\end{reference}", "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$", "be a morphism of ringed topoi.", "Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$", "and $\\mathcal{A}' \\subset \\textit{Mod}(\\mathcal{O}')$", "be weak Serre subcategories. Assume there is an integer $N$ such that", "\\begin{enumerate}", "\\item $\\mathcal{C}, \\mathcal{O}, \\mathcal{A}$ satisfy the", "assumption of Situation \\ref{situation-olsson-laszlo},", "\\item $\\mathcal{C}', \\mathcal{O}', \\mathcal{A}'$ satisfy the", "assumption of Situation \\ref{situation-olsson-laszlo},", "\\item $R^pf_*\\mathcal{F} \\in \\Ob(\\mathcal{A}')$ for", "$p \\geq 0$ and $\\mathcal{F} \\in \\Ob(\\mathcal{A})$,", "\\item $R^pf_*\\mathcal{F} = 0$ for", "$p > N$ and $\\mathcal{F} \\in \\Ob(\\mathcal{A})$,", "\\end{enumerate}", "Then for $K$ in $D_\\mathcal{A}(\\mathcal{O})$ we have", "\\begin{enumerate}", "\\item[(a)] $Rf_*K$ is in $D_{\\mathcal{A}'}(\\mathcal{O}')$,", "\\item[(b)] the map", "$H^j(Rf_*K) \\to H^j(Rf_*(\\tau_{\\geq -n}K))$ is an isomorphism", "for $j \\geq N - n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-olsson-laszlo} we have $K = R\\lim \\tau_{\\geq -n}K$.", "By Lemma \\ref{lemma-Rf-commutes-with-Rlim}", "we have $Rf_*K = R\\lim Rf_*\\tau_{\\geq -n}K$.", "The complexes $Rf_*\\tau_{\\geq -n}K$ are bounded below.", "The spectral sequence", "$$", "E_2^{p, q} = R^pf_*H^q(\\tau_{\\geq -n}K)", "$$", "converging to $H^{p + q}(Rf_*\\tau_{\\geq -n}K)$", "(Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor})", "and assumption (3)", "show that $Rf_*\\tau_{\\geq -n}K$ lies in $D^+_{\\mathcal{A}'}(\\mathcal{O}')$,", "see Homology, Lemma \\ref{homology-lemma-biregular-ss-converges}.", "Observe that for $m \\geq n$ the map", "$$", "Rf_*(\\tau_{\\geq -m}K) \\longrightarrow Rf_*(\\tau_{\\geq -n}K)", "$$", "induces an isomorphism on cohomology sheaves in degrees $j \\geq -n + N$", "by the spectral sequences above. Hence we may apply", "Lemma \\ref{lemma-olsson-laszlo-modified} to conclude." ], "refs": [ "sites-cohomology-lemma-olsson-laszlo", "sites-cohomology-lemma-Rf-commutes-with-Rlim", "derived-lemma-two-ss-complex-functor", "homology-lemma-biregular-ss-converges", "sites-cohomology-lemma-olsson-laszlo-modified" ], "ref_ids": [ 4278, 4267, 1871, 12101, 4279 ] } ], "ref_ids": [] }, { "id": 4281, "type": "theorem", "label": "sites-cohomology-lemma-olsson-laszlo-map-version-two", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-olsson-laszlo-map-version-two", "contents": [ "\\begin{reference}", "This is a version of \\cite[Lemma 2.1.10]{six-I} with slightly changed", "hypotheses.", "\\end{reference}", "Let $f : (\\mathcal{C}, \\mathcal{O}) \\to (\\mathcal{C}', \\mathcal{O}')$", "be a morphism of ringed sites.", "assume moreover there is an integer $N$ such that", "\\begin{enumerate}", "\\item $\\mathcal{C}, \\mathcal{O}, \\mathcal{A}$ satisfy the", "assumption of Situation \\ref{situation-olsson-laszlo},", "\\item $f : (\\mathcal{C}, \\mathcal{O}) \\to (\\mathcal{C}', \\mathcal{O}')$", "and $\\mathcal{A}$ satisfy the assumption of", "Situation \\ref{situation-olsson-laszlo-prime},", "\\item $R^pf_*\\mathcal{F} = 0$ for", "$p > N$ and $\\mathcal{F} \\in \\Ob(\\mathcal{A})$,", "\\end{enumerate}", "Then for $K$ in $D_\\mathcal{A}(\\mathcal{O})$ the map", "$H^j(Rf_*K) \\to H^j(Rf_*(\\tau_{\\geq -n}K))$ is an isomorphism", "for $j \\geq N - n$." ], "refs": [], "proofs": [ { "contents": [ "Let $K$ be in $D_\\mathcal{A}(\\mathcal{O})$.", "By Lemma \\ref{lemma-olsson-laszlo} we have $K = R\\lim \\tau_{\\geq -n}K$.", "By Lemma \\ref{lemma-Rf-commutes-with-Rlim}", "we have $Rf_*K = R\\lim Rf_*(\\tau_{\\geq -n}K)$.", "Let $V' \\in \\mathcal{B}'$ and let $\\{V'_i \\to V'\\}$ be an element", "of $\\text{Cov}_{V'}$. Then we consider", "$$", "H^j(V'_i, Rf_*K) = H^j(u(V'_i), K)", "\\quad\\text{and}\\quad", "H^j(V'_i, Rf_*(\\tau_{\\geq -n}K)) = H^j(u(V'_i), \\tau_{\\geq -n}K)", "$$", "The assumption in Situation \\ref{situation-olsson-laszlo-prime}", "implies that the last group is independent of $n$ for $n$ large enough", "depending on $j$ and $d_{V'}$. Some details omitted.", "We apply this for $j$ and $j - 1$ and via", "Lemma \\ref{lemma-RGamma-commutes-with-Rlim} this gives that", "$$", "H^j(V'_i, Rf_*K) = \\lim H^j(V'_i, Rf_*(\\tau_{\\geq -n} K))", "$$", "and the system on the right is constant for $n$ larger than", "a constant depending only on $d_{V'}$ and $j$.", "Thus Lemma \\ref{lemma-cohomology-derived-limit-injective}", "implies that", "$$", "H^j(Rf_*K)(V') \\longrightarrow", "\\left(\\lim H^j(Rf_*(\\tau_{\\geq -n}K))\\right)(V')", "$$", "is injective. Since the elements $V' \\in \\mathcal{B}'$ cover", "every object of $\\mathcal{C}'$ we conclude that the map", "$H^j(Rf_*K) \\to \\lim H^j(Rf_*(\\tau_{\\geq -n}K))$ is injective.", "The spectral sequence", "$$", "E_2^{p, q} = R^pf_*H^q(\\tau_{\\geq -n}K)", "$$", "converging to $H^{p + q}(Rf_*(\\tau_{\\geq -n}K))$", "(Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor})", "and assumption (3) show that", "$H^j(Rf_*(\\tau_{\\geq -n}K))$ is constant for $n \\geq N - j$.", "Hence $H^j(Rf_*K) \\to H^j(Rf_*(\\tau_{\\geq -n}K))$ is injective", "for $j \\geq N - n$.", "\\medskip\\noindent", "Thus we proved the lemma with ``isomorphism'' in the last line of", "the lemma replaced by ``injective''. However, now choose $j$ and $n$", "with $j \\geq N - n$. Then consider the distinguished triangle", "$$", "\\tau_{\\leq -n - 1}K \\to K \\to \\tau_{\\geq -n}K \\to (\\tau_{\\leq -n - 1}K)[1]", "$$", "See Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}.", "Since $\\tau_{\\geq -n}\\tau_{\\leq -n -1}K = 0$, the", "injectivity already proven for $\\tau_{-n - 1}K$ implies", "$$", "0 = H^j(Rf_*(\\tau_{\\leq -n - 1}K)) =", "H^{j + 1}(Rf_*(\\tau_{\\leq -n - 1}K)) =", "H^{j + 2}(Rf_*(\\tau_{\\leq -n - 1}K)) = \\ldots", "$$", "By the long exact cohomology sequence associated to the distinguished", "triangle", "$$", "Rf_*(\\tau_{\\leq -n - 1}K) \\to Rf_*K \\to Rf_*(\\tau_{\\geq -n}K) \\to", "Rf_*(\\tau_{\\leq -n - 1}K)[1]", "$$", "this implies that $H^j(Rf_*K) \\to H^j(Rf_*(\\tau_{\\geq -n}K))$", "is an isomorphism." ], "refs": [ "sites-cohomology-lemma-olsson-laszlo", "sites-cohomology-lemma-Rf-commutes-with-Rlim", "sites-cohomology-lemma-RGamma-commutes-with-Rlim", "sites-cohomology-lemma-cohomology-derived-limit-injective", "derived-lemma-two-ss-complex-functor", "derived-remark-truncation-distinguished-triangle" ], "ref_ids": [ 4278, 4267, 4266, 4269, 1871, 2016 ] } ], "ref_ids": [] }, { "id": 4282, "type": "theorem", "label": "sites-cohomology-lemma-c-square", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-c-square", "contents": [ "In the situation above, choose a K-injective complex $\\mathcal{I}^\\bullet$", "of $\\mathcal{O}$-modules representing $K$. Using $-1$ times the canonical map", "for one of the four arrows we get maps of complexes", "$$", "\\mathcal{I}^\\bullet(X) \\xrightarrow{\\alpha}", "\\mathcal{I}^\\bullet(Z) \\oplus", "\\mathcal{I}^\\bullet(Y) \\xrightarrow{\\beta}", "\\mathcal{I}^\\bullet(E)", "$$", "with $\\beta \\circ \\alpha = 0$. Thus a canonical map", "$$", "c^K_{X, Z, Y, E} :", "\\mathcal{I}^\\bullet(X)", "\\longrightarrow", "C(\\beta)^\\bullet[-1]", "$$", "This map is canonical in the sense that a different choice", "of K-injective complex representing $K$ determines an isomorphic", "arrow in the derived category of abelian groups. If $c^K_{X, Z, Y, E}$", "is an isomorphism, then using its inverse we obtain a canonical", "distinguished triangle", "$$", "R\\Gamma(X, K) \\to", "R\\Gamma(Z, K) \\oplus", "R\\Gamma(Y, K) \\to", "R\\Gamma(E, K) \\to", "R\\Gamma(X, K)[1]", "$$", "All of these constructions are functorial in $K$." ], "refs": [], "proofs": [ { "contents": [ "This lemma proves itself. For example, if $\\mathcal{J}^\\bullet$", "is a second K-injective complex representing $K$, then", "we can choose a quasi-isomorphism", "$\\mathcal{I}^\\bullet \\to \\mathcal{J}^\\bullet$", "which determines quasi-isomorphisms between all the complexes", "in sight. Details omitted. For the construction of cones", "and the relationship with distinguished triangles see", "Derived Categories, Sections \\ref{derived-section-cones} and", "\\ref{derived-section-homotopy-triangulated}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4283, "type": "theorem", "label": "sites-cohomology-lemma-two-out-of-three-blow-up-square", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-two-out-of-three-blow-up-square", "contents": [ "In the situation above, let $K_1 \\to K_2 \\to K_3 \\to K_1[1]$ be a distinguished", "triangle in $D(\\mathcal{O})$.", "If $c^{K_i}_{X, Z, Y, E}$ is a quasi-isomorphism for", "two $i$ out of $\\{1, 2, 3\\}$, then it is a quasi-isomorphism", "for the third $i$." ], "refs": [], "proofs": [ { "contents": [ "By rotating the triangle we may assume $c^{K_1}_{X, Z, Y, E}$ and", "$c^{K_2}_{X, Z, Y, E}$ are quasi-isomorphisms. Choose a map", "$f : \\mathcal{I}^\\bullet_1 \\to \\mathcal{I}^\\bullet_2$ of", "K-injective complexes of $\\mathcal{O}$-modules representing $K_1 \\to K_2$.", "Then $K_3$ is represented by the K-injective complex", "$C(f)^\\bullet$, see", "Derived Categories, Lemma \\ref{derived-lemma-triangle-K-injective}.", "Then the morphism $c^{K_3}_{X, Z, Y, E}$ is an isomorphism", "as it is the third leg in a map of distinguished triangles", "in $K(\\textit{Ab})$ whose other two legs are quasi-isomorphisms.", "Some details omitted; use", "Derived Categories, Lemma \\ref{derived-lemma-third-isomorphism-triangle}." ], "refs": [ "derived-lemma-triangle-K-injective", "derived-lemma-third-isomorphism-triangle" ], "ref_ids": [ 1909, 1759 ] } ], "ref_ids": [] }, { "id": 4284, "type": "theorem", "label": "sites-cohomology-lemma-square-triangle", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-square-triangle", "contents": [ "In the situation above assume", "\\begin{enumerate}", "\\item $h_X^\\# = h_Y^\\# \\amalg_{h_E^\\#} h_Z^\\#$, and", "\\item $h_E^\\# \\to h_Y^\\#$ is injective.", "\\end{enumerate}", "Then the construction of Lemma \\ref{lemma-c-square}", "produces a distinguished triangle", "$$", "R\\Gamma(X, K) \\to", "R\\Gamma(Z, K) \\oplus", "R\\Gamma(Y, K) \\to", "R\\Gamma(E, K) \\to R\\Gamma(X, K)[1]", "$$", "functorial for $K$ in $D(\\mathcal{C})$." ], "refs": [ "sites-cohomology-lemma-c-square" ], "proofs": [ { "contents": [ "We can represent $K$ by a K-injective complex whose terms are injective", "abelian sheaves, see Section \\ref{section-unbounded}.", "Thus it suffices to show: if $\\mathcal{I}$ is an injective abelian", "sheaf, then", "$$", "0 \\to \\mathcal{I}(X) \\to", "\\mathcal{I}(Z) \\oplus \\mathcal{I}(Y) \\to", "\\mathcal{I}(E) \\to 0", "$$", "is a short exact sequence. The first arrow is injective", "because by condition (1) the map $h_Y \\amalg h_Z \\to h_X$", "becomes surjective after sheafification, which means that", "$\\{Y \\to X, Z \\to X\\}$ can be refined by a covering of $X$.", "The last arrow is surjective because $\\mathcal{I}(Y) \\to \\mathcal{I}(E)$", "is surjective. Namely, we have", "$\\mathcal{I}(E) = \\Hom(\\mathbf{Z}_E^\\#, \\mathcal{I})$,", "$\\mathcal{I}(Y) = \\Hom(\\mathbf{Z}_Y^\\#, \\mathcal{I})$,", "the map $\\mathbf{Z}_E^\\# \\to \\mathbf{Z}_Y^\\#$ is injective", "by (2), and $\\mathcal{I}$ is an injective abelian sheaf.", "Please compare with Modules on Sites, Section", "\\ref{sites-modules-section-free-abelian-sheaf}.", "Finally, suppose we have $s \\in \\mathcal{I}(Y)$ and", "$t \\in \\mathcal{F}(Z)$ mapping to the same element of", "$\\mathcal{I}(E)$. Then $s$ and $t$ define a map", "$$", "s \\amalg t : h_Y^\\# \\amalg h_Z^\\# \\longrightarrow \\mathcal{I}", "$$", "which by assumption factors through $h_Y^\\# \\amalg_{h_E^\\#} h_Z^\\#$.", "Thus by assumption (1) we obtain a unique map", "$h_X^\\# \\to \\mathcal{I}$ which corresponds to an element", "of $\\mathcal{I}(X)$ restricting to $s$ on $Y$ and $t$ on $Z$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 4282 ] }, { "id": 4285, "type": "theorem", "label": "sites-cohomology-lemma-square-triangle-general", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-square-triangle-general", "contents": [ "Let $\\mathcal{C}$ be a site. Consider a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{D} \\ar[r] \\ar[d] & \\mathcal{F} \\ar[d] \\\\", "\\mathcal{E} \\ar[r] & \\mathcal{G}", "}", "$$", "of presheaves of sets on $\\mathcal{C}$ and assume that", "\\begin{enumerate}", "\\item $\\mathcal{G}^\\# =", "\\mathcal{E}^\\# \\amalg_{\\mathcal{D}^\\#} \\mathcal{F}^\\#$, and", "\\item $\\mathcal{D}^\\# \\to \\mathcal{F}^\\#$ is injective.", "\\end{enumerate}", "Then there is a canonical distinguished triangle", "$$", "R\\Gamma(\\mathcal{G}, K) \\to", "R\\Gamma(\\mathcal{E}, K) \\oplus", "R\\Gamma(\\mathcal{F}, K) \\to", "R\\Gamma(\\mathcal{D}, K) \\to", "R\\Gamma(\\mathcal{G}, K)[1]", "$$", "functorial in $K \\in D(\\mathcal{C})$ where $R\\Gamma(\\mathcal{G}, -)$", "is the cohomology discussed in Section \\ref{section-limp}." ], "refs": [], "proofs": [ { "contents": [ "Since sheafification is exact and since", "$R\\Gamma(\\mathcal{G}, -) = R\\Gamma(\\mathcal{G}^\\#, -)$", "we may assume $\\mathcal{D}, \\mathcal{E}, \\mathcal{F}, \\mathcal{G}$", "are sheaves of sets. Moreover, the cohomology", "$R\\Gamma(\\mathcal{G}, -)$ only depends on the topos,", "not on the underlying site. Hence by", "Sites, Lemma \\ref{sites-lemma-topos-good-site}", "we may replace $\\mathcal{C}$ by a ``larger'' site", "with a subcanonical topology such that $\\mathcal{G} = h_X$,", "$\\mathcal{F} = h_Y$,", "$\\mathcal{E} = h_Z$, and", "$\\mathcal{D} = h_E$", "for some objects $X, Y, Z, E$ of $\\mathcal{C}$.", "In this case the result follows from", "Lemma \\ref{lemma-square-triangle}." ], "refs": [ "sites-lemma-topos-good-site", "sites-cohomology-lemma-square-triangle" ], "ref_ids": [ 8581, 4284 ] } ], "ref_ids": [] }, { "id": 4286, "type": "theorem", "label": "sites-cohomology-lemma-downstairs", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-downstairs", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.", "Consider the full subcategory $D' \\subset D(\\mathcal{O}_\\mathcal{D})$", "consisting of objects $K$ such that", "$$", "K \\longrightarrow Rf_*Lf^*K", "$$", "is an isomorphism. Then $D'$ is a saturated triangulated strictly full", "subcategory of $D(\\mathcal{O}_\\mathcal{D})$ and the functor", "$Lf^* : D' \\to D(\\mathcal{O}_\\mathcal{C})$ is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "See Derived Categories, Definition \\ref{derived-definition-saturated}", "for the definition of saturated in this setting. See", "Derived Categories, Lemma \\ref{derived-lemma-triangulated-subcategory}", "for a discussion of triangulated subcategories.", "The canonical map of the lemma is the unit of the adjoint", "pair of functors $(Lf^*, Rf_*)$, see Lemma \\ref{lemma-adjoint}.", "Having said this the proof that $D'$ is a saturated triangulated subcategory", "is omitted; it follows formally from the fact that", "$Lf^*$ and $Rf_*$ are exact functors of triangulated categories.", "The final part follows formally from", "fact that $Lf^*$ and $Rf_*$ are adjoint; compare with", "Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}." ], "refs": [ "derived-definition-saturated", "derived-lemma-triangulated-subcategory", "sites-cohomology-lemma-adjoint", "categories-lemma-adjoint-fully-faithful" ], "ref_ids": [ 1974, 1771, 4249, 12248 ] } ], "ref_ids": [] }, { "id": 4287, "type": "theorem", "label": "sites-cohomology-lemma-upstairs", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-upstairs", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.", "Consider the full subcategory $D' \\subset D(\\mathcal{O}_\\mathcal{C})$", "consisting of objects $K$ such that", "$$", "Lf^*Rf_*K \\longrightarrow K", "$$", "is an isomorphism. Then $D'$ is a saturated triangulated strictly full", "subcategory of $D(\\mathcal{O}_\\mathcal{C})$ and the functor", "$Rf_* : D' \\to D(\\mathcal{O}_\\mathcal{D})$ is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "See Derived Categories, Definition \\ref{derived-definition-saturated}", "for the definition of saturated in this setting. See", "Derived Categories, Lemma \\ref{derived-lemma-triangulated-subcategory}", "for a discussion of triangulated subcategories.", "The canonical map of the lemma is the counit of the adjoint", "pair of functors $(Lf^*, Rf_*)$, see Lemma \\ref{lemma-adjoint}.", "Having said this the proof that $D'$ is a saturated triangulated subcategory", "is omitted; it follows formally from the fact that", "$Lf^*$ and $Rf_*$ are exact functors of triangulated categories.", "The final part follows formally from", "fact that $Lf^*$ and $Rf_*$ are adjoint; compare with", "Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}." ], "refs": [ "derived-definition-saturated", "derived-lemma-triangulated-subcategory", "sites-cohomology-lemma-adjoint", "categories-lemma-adjoint-fully-faithful" ], "ref_ids": [ 1974, 1771, 4249, 12248 ] } ], "ref_ids": [] }, { "id": 4288, "type": "theorem", "label": "sites-cohomology-lemma-bounded-in-image-upstairs", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-bounded-in-image-upstairs", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.", "Let $K$ be an object of $D(\\mathcal{O}_\\mathcal{C})$. Assume", "\\begin{enumerate}", "\\item $f$ is flat,", "\\item $K$ is bounded below,", "\\item $f^*Rf_*H^q(K) \\to H^q(K)$ is an isomorphism.", "\\end{enumerate}", "Then $f^*Rf_*K \\to K$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Observe that $f^*Rf_*K \\to K$ is an isomorphism if and only", "if it is an isomorphism on cohomology sheaves $H^j$. Observe that", "$H^j(f^*Rf_*K) = f^*H^j(Rf_*K) = f^*H^j(Rf_*\\tau_{\\leq j}K) =", "H^j(f^*Rf_*\\tau_{\\leq j}K)$.", "Hence we may assume that $K$ is bounded. Then property (3)", "tells us the cohomology sheaves are in the triangulated", "subcategory $D' \\subset D(\\mathcal{O}_\\mathcal{C})$ of", "Lemma \\ref{lemma-upstairs}. Hence $K$ is in it too." ], "refs": [ "sites-cohomology-lemma-upstairs" ], "ref_ids": [ 4287 ] } ], "ref_ids": [] }, { "id": 4289, "type": "theorem", "label": "sites-cohomology-lemma-bounded-in-image-downstairs", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-bounded-in-image-downstairs", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.", "Let $K$ be an object of $D(\\mathcal{O}_\\mathcal{D})$. Assume", "\\begin{enumerate}", "\\item $f$ is flat,", "\\item $K$ is bounded below,", "\\item $H^q(K) \\to Rf_*f^*H^q(K)$ is an isomorphism.", "\\end{enumerate}", "Then $K \\to Rf_*f^*K$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Observe that $K \\to Rf_*f^*K$ is an isomorphism if and only", "if it is an isomorphism on cohomology sheaves $H^j$. Observe that", "$H^j(Rf_*f^*K) = H^j(Rf_*\\tau_{\\leq j}f^*K) = H^j(Rf_*f^*\\tau_{\\leq j}K)$.", "Hence we may assume that $K$ is bounded. Then property (3)", "tells us the cohomology sheaves are in the triangulated", "subcategory $D' \\subset D(\\mathcal{O}_\\mathcal{D})$ of", "Lemma \\ref{lemma-downstairs}. Hence $K$ is in it too." ], "refs": [ "sites-cohomology-lemma-downstairs" ], "ref_ids": [ 4286 ] } ], "ref_ids": [] }, { "id": 4290, "type": "theorem", "label": "sites-cohomology-lemma-equivalence-bounded", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-equivalence-bounded", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$", "be a morphism of ringed topoi.", "Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$", "and $\\mathcal{A}' \\subset \\textit{Mod}(\\mathcal{O}')$", "be weak Serre subcategories. Assume", "\\begin{enumerate}", "\\item $f$ is flat,", "\\item $f^*$ induces an equivalence of categories", "$\\mathcal{A}' \\to \\mathcal{A}$,", "\\item $\\mathcal{F}' \\to Rf_*f^*\\mathcal{F}'$ is an isomorphism", "for $\\mathcal{F}' \\in \\Ob(\\mathcal{A}')$.", "\\end{enumerate}", "Then", "$f^* : D_{\\mathcal{A}'}^+(\\mathcal{O}') \\to D_\\mathcal{A}^+(\\mathcal{O})$", "is an equivalence of categories with quasi-inverse given by", "$Rf_* : D_\\mathcal{A}^+(\\mathcal{O}) \\to D_{\\mathcal{A}'}^+(\\mathcal{O}')$." ], "refs": [], "proofs": [ { "contents": [ "By assumptions (2) and (3) and", "Lemmas \\ref{lemma-bounded-in-image-upstairs} and \\ref{lemma-downstairs}", "we see that", "$f^* : D_{\\mathcal{A}'}^+(\\mathcal{O}') \\to D_\\mathcal{A}^+(\\mathcal{O})$", "is fully faithful.", "Let $\\mathcal{F} \\in \\Ob(\\mathcal{A})$. Then we can write", "$\\mathcal{F} = f^*\\mathcal{F}'$. Then", "$Rf_*\\mathcal{F} = Rf_* f^*\\mathcal{F}' = \\mathcal{F}'$.", "In particular, we have $R^pf_*\\mathcal{F} = 0$ for $p > 0$", "and $f_*\\mathcal{F} \\in \\Ob(\\mathcal{A}')$.", "Thus for any $K \\in D^+_\\mathcal{A}(\\mathcal{O})$ we see,", "using the spectral sequence $E_2^{p, q} = R^pf_*H^q(K)$", "converging to $R^{p + q}f_*K$,", "that $Rf_*K$ is in $D^+_{\\mathcal{A}'}(\\mathcal{O}')$.", "Of course, it also follows from", "Lemmas \\ref{lemma-bounded-in-image-downstairs} and \\ref{lemma-upstairs}", "that $Rf_* : D_\\mathcal{A}^+(\\mathcal{O}) \\to D_{\\mathcal{A}'}^+(\\mathcal{O}')$", "is fully faithful. Since $f^*$ and $Rf_*$ are adjoint", "we then get the result of the lemma, for example by", "Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}." ], "refs": [ "sites-cohomology-lemma-bounded-in-image-upstairs", "sites-cohomology-lemma-downstairs", "sites-cohomology-lemma-bounded-in-image-downstairs", "sites-cohomology-lemma-upstairs", "categories-lemma-adjoint-fully-faithful" ], "ref_ids": [ 4288, 4286, 4289, 4287, 12248 ] } ], "ref_ids": [] }, { "id": 4291, "type": "theorem", "label": "sites-cohomology-lemma-equivalence-unbounded-one", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-equivalence-unbounded-one", "contents": [ "\\begin{reference}", "This is analogous to \\cite[Theorem 2.2.3]{six-I}.", "\\end{reference}", "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$", "be a morphism of ringed topoi.", "Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$", "and $\\mathcal{A}' \\subset \\textit{Mod}(\\mathcal{O}')$", "be weak Serre subcategories. Assume", "\\begin{enumerate}", "\\item $f$ is flat,", "\\item $f^*$ induces an equivalence of categories", "$\\mathcal{A}' \\to \\mathcal{A}$,", "\\item $\\mathcal{F}' \\to Rf_*f^*\\mathcal{F}'$ is an isomorphism", "for $\\mathcal{F}' \\in \\Ob(\\mathcal{A}')$,", "\\item $\\mathcal{C}, \\mathcal{O}, \\mathcal{A}$ satisfy the", "assumption of Situation \\ref{situation-olsson-laszlo},", "\\item $\\mathcal{C}', \\mathcal{O}', \\mathcal{A}'$ satisfy the", "assumption of Situation \\ref{situation-olsson-laszlo}.", "\\end{enumerate}", "Then $f^* : D_{\\mathcal{A}'}(\\mathcal{O}') \\to D_\\mathcal{A}(\\mathcal{O})$", "is an equivalence of categories with quasi-inverse given by", "$Rf_* : D_\\mathcal{A}(\\mathcal{O}) \\to D_{\\mathcal{A}'}(\\mathcal{O}')$." ], "refs": [], "proofs": [ { "contents": [ "Since $f^*$ is exact, it is clear that $f^*$ defines a functor", "$f^* : D_{\\mathcal{A}'}(\\mathcal{O}') \\to D_\\mathcal{A}(\\mathcal{O})$", "as in the statement of the lemma and that moreover this", "functor commutes with the truncation functors $\\tau_{\\geq -n}$.", "We already know that $f^*$ and $Rf_*$ are quasi-inverse", "equivalence on the corresponding bounded below categories,", "see Lemma \\ref{lemma-equivalence-bounded}.", "By Lemma \\ref{lemma-olsson-laszlo-map-version-one}", "with $N = 0$ we see that $Rf_*$ indeed defines a functor", "$Rf_* : D_\\mathcal{A}(\\mathcal{O}) \\to D_{\\mathcal{A}'}(\\mathcal{O}')$", "and that moreover this functor commutes with", "the truncation functors $\\tau_{\\geq -n}$.", "Thus for $K$ in $D_\\mathcal{A}(\\mathcal{O})$ the map", "$f^*Rf_*K \\to K$ is an isomorphism as this is true", "on trunctions.", "Similarly, for $K'$ in $D_{\\mathcal{A}'}(\\mathcal{O}')$ the map", "$K' \\to Rf_*f^*K'$ is an isomorphism as this is true", "on trunctions.", "This finishes the proof." ], "refs": [ "sites-cohomology-lemma-equivalence-bounded", "sites-cohomology-lemma-olsson-laszlo-map-version-one" ], "ref_ids": [ 4290, 4280 ] } ], "ref_ids": [] }, { "id": 4292, "type": "theorem", "label": "sites-cohomology-lemma-equivalence-unbounded-two", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-equivalence-unbounded-two", "contents": [ "\\begin{reference}", "This is analogous to \\cite[Theorem 2.2.3]{six-I}.", "\\end{reference}", "Let $f : (\\mathcal{C}, \\mathcal{O}) \\to (\\mathcal{C}', \\mathcal{O}')$", "be a morphism of ringed sites.", "Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$", "and $\\mathcal{A}' \\subset \\textit{Mod}(\\mathcal{O}')$", "be weak Serre subcategories. Assume", "\\begin{enumerate}", "\\item $f$ is flat,", "\\item $f^*$ induces an equivalence of categories", "$\\mathcal{A}' \\to \\mathcal{A}$,", "\\item $\\mathcal{F}' \\to Rf_*f^*\\mathcal{F}'$ is an isomorphism", "for $\\mathcal{F}' \\in \\Ob(\\mathcal{A}')$,", "\\item $\\mathcal{C}, \\mathcal{O}, \\mathcal{A}$ satisfy the", "assumption of Situation \\ref{situation-olsson-laszlo},", "\\item $f : (\\mathcal{C}, \\mathcal{O}) \\to (\\mathcal{C}', \\mathcal{O}')$", "and $\\mathcal{A}$ satisfy the assumption of", "Situation \\ref{situation-olsson-laszlo-prime}.", "\\end{enumerate}", "Then $f^* : D_{\\mathcal{A}'}(\\mathcal{O}') \\to D_\\mathcal{A}(\\mathcal{O})$", "is an equivalence of categories with quasi-inverse given by", "$Rf_* : D_\\mathcal{A}(\\mathcal{O}) \\to D_{\\mathcal{A}'}(\\mathcal{O}')$." ], "refs": [], "proofs": [ { "contents": [ "The proof of this lemma is exactly the same as the proof", "of Lemma \\ref{lemma-equivalence-unbounded-one}", "except the reference to Lemma \\ref{lemma-olsson-laszlo-map-version-one}", "is replaced by a reference to", "Lemma \\ref{lemma-olsson-laszlo-map-version-two}." ], "refs": [ "sites-cohomology-lemma-equivalence-unbounded-one", "sites-cohomology-lemma-olsson-laszlo-map-version-one", "sites-cohomology-lemma-olsson-laszlo-map-version-two" ], "ref_ids": [ 4291, 4280, 4281 ] } ], "ref_ids": [] }, { "id": 4293, "type": "theorem", "label": "sites-cohomology-lemma-compare-topologies-derived-adequate-modules", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-compare-topologies-derived-adequate-modules", "contents": [ "With $\\epsilon : (\\mathcal{C}_\\tau, \\mathcal{O}_\\tau) \\to", "(\\mathcal{C}_{\\tau'}, \\mathcal{O}_{\\tau'})$ as above.", "Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset.", "Let $\\mathcal{A} \\subset \\textit{PMod}(\\mathcal{O})$", "be a full subcategory. Assume", "\\begin{enumerate}", "\\item every object of $\\mathcal{A}$ is a sheaf for the $\\tau$-topology,", "\\item $\\mathcal{A}$ is a weak Serre subcategory of", "$\\textit{Mod}(\\mathcal{O}_\\tau)$,", "\\item every object of $\\mathcal{C}$ has a $\\tau'$-covering whose", "members are elements of $\\mathcal{B}$, and", "\\item for every $U \\in \\mathcal{B}$ we have $H^p_\\tau(U, \\mathcal{F}) = 0$,", "$p > 0$ for all $\\mathcal{F} \\in \\mathcal{A}$.", "\\end{enumerate}", "Then $\\mathcal{A}$ is a weak Serre subcategory of", "$\\textit{Mod}(\\mathcal{O}_{\\tau'})$ and there is an equivalence", "of triangulated categories", "$D_\\mathcal{A}(\\mathcal{O}_\\tau) = D_\\mathcal{A}(\\mathcal{O}_{\\tau'})$", "given by $\\epsilon^*$ and $R\\epsilon_*$." ], "refs": [], "proofs": [ { "contents": [ "Since $\\epsilon^{-1}\\mathcal{O}_{\\tau'} = \\mathcal{O}_\\tau$", "we see that $\\epsilon$ is a flat morphism of ringed sites", "and that in fact $\\epsilon^{-1} = \\epsilon^*$ on sheaves", "of modules. By property (1) we can think of every object of", "$\\mathcal{A}$ as a sheaf of $\\mathcal{O}_\\tau$-modules", "and as a sheaf of $\\mathcal{O}_{\\tau'}$-modules.", "In other words, we have fully faithful inclusion functors", "$$", "\\mathcal{A} \\to \\textit{Mod}(\\mathcal{O}_\\tau) \\to", "\\textit{Mod}(\\mathcal{O}_{\\tau'})", "$$", "To avoid confusion we will denote", "$\\mathcal{A}' \\subset \\textit{Mod}(\\mathcal{O}_{\\tau'})$", "the image of $\\mathcal{A}$. Then it is clear that", "$\\epsilon_* : \\mathcal{A} \\to \\mathcal{A}'$ and", "$\\epsilon^* : \\mathcal{A}' \\to \\mathcal{A}$ are", "quasi-inverse equivalences (see discussion preceding", "the lemma and use that objects of $\\mathcal{A}'$ are", "sheaves in the $\\tau$ topology).", "\\medskip\\noindent", "Conditions (3) and (4) imply that $R^p\\epsilon_*\\mathcal{F} = 0$", "for $p > 0$ and $\\mathcal{F} \\in \\Ob(\\mathcal{A})$.", "This is true because $R^p\\epsilon_*$ is the sheaf associated", "to the presheave $U \\mapsto H^p_\\tau(U, \\mathcal{F})$, see", "Lemma \\ref{lemma-higher-direct-images}.", "Thus any exact complex in $\\mathcal{A}$ (which is the same thing", "as an exact complex in $\\textit{Mod}(\\mathcal{O}_\\tau)$", "whose terms are in $\\mathcal{A}$, see", "Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory})", "remains exact upon applying the functor $\\epsilon_*$.", "\\medskip\\noindent", "Consider an exact sequence", "$$", "\\mathcal{F}'_0 \\to \\mathcal{F}'_1 \\to", "\\mathcal{F}'_2 \\to \\mathcal{F}'_3 \\to", "\\mathcal{F}'_4", "$$", "in $\\textit{Mod}(\\mathcal{O}_{\\tau'})$ with", "$\\mathcal{F}'_0, \\mathcal{F}'_1, \\mathcal{F}'_3, \\mathcal{F}'_4$ in", "$\\mathcal{A}'$. Apply the exact functor $\\epsilon^*$ to get", "an exact sequence", "$$", "\\epsilon^*\\mathcal{F}'_0 \\to \\epsilon^*\\mathcal{F}'_1 \\to", "\\epsilon^*\\mathcal{F}'_2 \\to \\epsilon^*\\mathcal{F}'_3 \\to", "\\epsilon^*\\mathcal{F}'_4", "$$", "in $\\textit{Mod}(\\mathcal{O}_\\tau)$. Since $\\mathcal{A}$ is", "a weak Serre subcategory and since", "$\\epsilon^*\\mathcal{F}'_0, \\epsilon^*\\mathcal{F}'_1,", "\\epsilon^*\\mathcal{F}'_3, \\epsilon^*\\mathcal{F}'_4$ are in", "$\\mathcal{A}$, we conclude that $\\epsilon^*\\mathcal{F}_2$", "is in $\\mathcal{A}$ by", "Homology, Definition \\ref{homology-definition-serre-subcategory}.", "Consider the map of sequences", "$$", "\\xymatrix{", "\\mathcal{F}'_0 \\ar[r] \\ar[d] &", "\\mathcal{F}'_1 \\ar[r] \\ar[d] &", "\\mathcal{F}'_2 \\ar[r] \\ar[d] &", "\\mathcal{F}'_3 \\ar[r] \\ar[d] &", "\\mathcal{F}'_4 \\ar[d] \\\\", "\\epsilon_*\\epsilon^*\\mathcal{F}'_0 \\ar[r] &", "\\epsilon_*\\epsilon^*\\mathcal{F}'_1 \\ar[r] &", "\\epsilon_*\\epsilon^*\\mathcal{F}'_2 \\ar[r] &", "\\epsilon_*\\epsilon^*\\mathcal{F}'_3 \\ar[r] &", "\\epsilon_*\\epsilon^*\\mathcal{F}'_4", "}", "$$", "The lower row is exact by the discussion in the preceding", "paragraph. The vertical arrows with index $0$, $1$, $3$, $4$", "are isomorphisms by the discussion in the first paragraph.", "By the $5$ lemma (Homology, Lemma \\ref{homology-lemma-five-lemma})", "we find that $\\mathcal{F}'_2 \\cong \\epsilon_*\\epsilon^*\\mathcal{F}'_2$", "and hence $\\mathcal{F}'_2$ is in $\\mathcal{A}'$.", "In this way we see that $\\mathcal{A}'$ is a weak Serre subcategory", "of $\\textit{Mod}(\\mathcal{O}_{\\tau'})$, see", "Homology, Definition \\ref{homology-definition-serre-subcategory}.", "\\medskip\\noindent", "At this point it makes sense to talk about the", "derived categories $D_\\mathcal{A}(\\mathcal{O}_\\tau)$ and", "$D_{\\mathcal{A}'}(\\mathcal{O}_{\\tau'})$, see", "Derived Categories, Section \\ref{derived-section-triangulated-sub}.", "To finish the proof we show that conditions", "(1) -- (5) of Lemma \\ref{lemma-equivalence-unbounded-two} apply.", "We have already seen (1), (2), (3) above.", "Note that since every object has a $\\tau'$-covering", "by objects of $\\mathcal{B}$, a fortiori every object has", "a $\\tau$-covering by objects of $\\mathcal{B}$. Hence", "condition (4) of Lemma \\ref{lemma-equivalence-unbounded-two} is satisfied.", "Similarly, condition (5) is satisfied as well." ], "refs": [ "sites-cohomology-lemma-higher-direct-images", "homology-lemma-characterize-weak-serre-subcategory", "homology-definition-serre-subcategory", "homology-lemma-five-lemma", "homology-definition-serre-subcategory", "sites-cohomology-lemma-equivalence-unbounded-two", "sites-cohomology-lemma-equivalence-unbounded-two" ], "ref_ids": [ 4189, 12046, 12146, 12030, 12146, 4292, 4292 ] } ], "ref_ids": [] }, { "id": 4294, "type": "theorem", "label": "sites-cohomology-lemma-descent-squares", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-descent-squares", "contents": [ "With $\\epsilon : (\\mathcal{C}_\\tau, \\mathcal{O}_\\tau) \\to", "(\\mathcal{C}_{\\tau'}, \\mathcal{O}_{\\tau'})$ as above.", "Let $A$ be a set and for $\\alpha \\in A$ let", "$$", "\\xymatrix{", "E_\\alpha \\ar[d] \\ar[r] & Y_\\alpha \\ar[d] \\\\", "Z_\\alpha \\ar[r] & X_\\alpha", "}", "$$", "be a commutative diagram in the category $\\mathcal{C}$. Assume that", "\\begin{enumerate}", "\\item a $\\tau'$-sheaf $\\mathcal{F}'$ is a $\\tau$-sheaf if", "$\\mathcal{F}'(X_\\alpha) =", "\\mathcal{F}'(Z_\\alpha) \\times_{\\mathcal{F}'(E_\\alpha)} ", "\\mathcal{F}'(Y_\\alpha)$ for all $\\alpha$,", "\\item for $K'$ in $D(\\mathcal{O}_{\\tau'})$ in the essential image", "of $R\\epsilon_*$ the maps $c^{K'}_{X_\\alpha, Z_\\alpha, Y_\\alpha, E_\\alpha}$", "of Lemma \\ref{lemma-c-square}", "are isomorphisms for all $\\alpha$.", "\\end{enumerate}", "Then $K' \\in D^+(\\mathcal{O}_{\\tau'})$ is in", "the essential image of $R\\epsilon_*$ if and only if", "the maps $c^{K'}_{X_\\alpha, Z_\\alpha, Y_\\alpha, E_\\alpha}$", "are isomorphisms for all $\\alpha$." ], "refs": [ "sites-cohomology-lemma-c-square" ], "proofs": [ { "contents": [ "The ``only if'' direction is implied by assumption (2).", "On the other hand, if $K'$ has a unique nonzero cohomology sheaf,", "then the ``if'' direction follows from assumption (1).", "In general we will use an induction argument to prove the", "``if'' direction. Let us say an object $K'$ of $D^+(\\mathcal{O}_{\\tau'})$", "satisfies (P) if the maps $c^{K'}_{X_\\alpha, Z_\\alpha, Y_\\alpha, E_\\alpha}$", "are isomorphisms for all $\\alpha \\in A$.", "\\medskip\\noindent", "Namely, let $K'$ be an object of $D^+(\\mathcal{O}_{\\tau'})$", "satisfying (P). Choose a bounded below complex", "${\\mathcal{K}'}^\\bullet$ of sheaves of $\\mathcal{O}_{\\tau'}$-modules", "representing $K'$. We will show by induction on $n$ that we may assume", "for $p \\leq n$ we have $(\\mathcal{K}')^p = \\epsilon_*\\mathcal{J}^p$ for some", "injective sheaf $\\mathcal{J}^p$ of $\\mathcal{O}_{\\tau}$-modules.", "The assertion is true for $n \\ll 0$ because $(\\mathcal{K}')^\\bullet$", "is bounded below.", "\\medskip\\noindent", "Induction step. Assume we have $(\\mathcal{K}')^p = \\epsilon_*\\mathcal{J}^p$", "for some injective sheaves $\\mathcal{J}^p$ of $\\mathcal{O}_\\tau$-modules", "for $p \\leq n$. Denote $\\mathcal{J}^\\bullet$ the bounded complex", "of injective $\\mathcal{O}_\\tau$-modules made from these sheaves", "and the maps between them. Consider the short exact sequence of complexes", "$$", "0 \\to \\sigma_{\\geq n + 1}(\\mathcal{K}')^\\bullet \\to", "(\\mathcal{K}')^\\bullet \\to \\epsilon_*\\mathcal{J}^\\bullet \\to 0", "$$", "where $\\sigma_{\\geq n + 1}$ denotes the ``stupid'' truncation.", "By assumption (2) the object $\\epsilon_*\\mathcal{J}^\\bullet$", "of $D(\\mathcal{O}_{\\tau'})$ satisfies (P).", "By Lemma \\ref{lemma-two-out-of-three-blow-up-square}", "we conclude that $\\sigma_{\\geq n + 1}(\\mathcal{K}')^\\bullet$", "satisfies (P).", "We conclude that for $\\alpha \\in A$", "the sequence", "$$", "\\begin{matrix}", "0 \\\\", "\\downarrow \\\\", "H^{n + 1}_{\\tau'}(X_\\alpha, \\sigma_{\\geq n + 1}(\\mathcal{K}')^\\bullet) \\\\", "\\downarrow \\\\", "H^{n + 1}_{\\tau'}(Z_\\alpha, \\sigma_{\\geq n + 1}(\\mathcal{K}')^\\bullet) \\oplus", "H^{n + 1}_{\\tau'}(Y_\\alpha, \\sigma_{\\geq n + 1}(\\mathcal{K}')^\\bullet) \\\\", "\\downarrow \\\\", "H^{n + 1}_{\\tau'}(E_\\alpha, \\sigma_{\\geq n + 1}(\\mathcal{K}')^\\bullet)", "\\end{matrix}", "$$", "is exact by the distinguished triangle of Lemma \\ref{lemma-c-square}", "and the fact that $\\sigma_{\\geq n + 1}(\\mathcal{K}')^\\bullet$", "has vanishing cohomology over $E_\\alpha$ in degrees $< n + 1$.", "We conclude that", "$$", "\\mathcal{F}' = \\Ker((\\mathcal{K}')^{n + 1} \\to (\\mathcal{K}')^{n + 2})", "$$", "is a $\\tau$-sheaf by assumption (1) because the cohomology groups", "above evaluate to", "$\\mathcal{F}'(X_\\alpha)$,", "$\\mathcal{F}'(Z_\\alpha) \\oplus \\mathcal{F}'(Y_\\alpha)$, and", "$\\mathcal{F}'(E_\\alpha)$.", "Thus we may choose an injective $\\mathcal{O}_\\tau$-module", "$\\mathcal{J}^{n + 1}$ and an injection", "$\\mathcal{F}' \\to \\epsilon_*\\mathcal{J}^{n + 1}$.", "Since $\\epsilon_*\\mathcal{J}^{n + 1}$ is also an injective", "$\\mathcal{O}_{\\tau'}$-module (Lemma \\ref{lemma-pushforward-injective-flat})", "we can extend $\\mathcal{F}' \\to \\epsilon_*\\mathcal{J}^{n + 1}$", "to a map", "$(\\mathcal{K}')^{n + 1} \\to \\epsilon_*\\mathcal{J}^{n + 1}$.", "Then the complex $(\\mathcal{K}')^\\bullet$ is quasi-isomorphic to the complex", "$$", "\\ldots \\to", "\\epsilon_*\\mathcal{J}^n \\to", "\\epsilon_*\\mathcal{J}^{n + 1} \\to", "\\frac{\\epsilon_*\\mathcal{J}^{n + 1} \\oplus (\\mathcal{K}')^{n + 2}}{(\\mathcal{K}')^{n + 1}}", "\\to", "(\\mathcal{K}')^{n + 3} \\to \\ldots", "$$", "This finishes the induction step.", "\\medskip\\noindent", "The induction procedure described above actually produces a sequence of", "quasi-isomorphisms of complexes", "$$", "(\\mathcal{K}')^\\bullet \\to", "(\\mathcal{K}'_{n_0})^\\bullet \\to", "(\\mathcal{K}'_{n_0 + 1})^\\bullet \\to", "(\\mathcal{K}'_{n_0 + 2})^\\bullet \\to \\ldots", "$$", "where $(\\mathcal{K}'_n)^\\bullet \\to (\\mathcal{K}'_{n + 1})^\\bullet$", "is an isomorphism in degrees $\\leq n$ and such that", "$(\\mathcal{K}'_n)^p = \\epsilon_*\\mathcal{J}^p$ for $p \\leq n$.", "Taking the ``limit'' of these maps therefore gives", "a quasi-isomorphism $(\\mathcal{K}')^\\bullet \\to \\epsilon_*\\mathcal{J}^\\bullet$", "which proves the lemma." ], "refs": [ "sites-cohomology-lemma-two-out-of-three-blow-up-square", "sites-cohomology-lemma-c-square", "sites-cohomology-lemma-pushforward-injective-flat" ], "ref_ids": [ 4283, 4282, 4218 ] } ], "ref_ids": [ 4282 ] }, { "id": 4295, "type": "theorem", "label": "sites-cohomology-lemma-descent-squares-helper", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-descent-squares-helper", "contents": [ "With $\\epsilon : (\\mathcal{C}_\\tau, \\mathcal{O}_\\tau) \\to", "(\\mathcal{C}_{\\tau'}, \\mathcal{O}_{\\tau'})$ as above. Let", "$$", "\\xymatrix{", "E \\ar[d] \\ar[r] & Y \\ar[d] \\\\", "Z \\ar[r] & X", "}", "$$", "be a commutative diagram in the category $\\mathcal{C}$ such that", "\\begin{enumerate}", "\\item $h_X^\\# = h_Y^\\# \\amalg_{h_E^\\#} h_Z^\\#$, and", "\\item $h_E^\\# \\to h_Y^\\#$ is injective", "\\end{enumerate}", "where ${}^\\#$ denotes $\\tau$-sheafification. Then for", "$K' \\in D(\\mathcal{O}_{\\tau'})$ in the essential image of", "$R\\epsilon_*$ the map $c^{K'}_{X, Z, Y, E}$ of Lemma \\ref{lemma-c-square}", "(using the $\\tau'$-topology) is an isomorphism." ], "refs": [ "sites-cohomology-lemma-c-square" ], "proofs": [ { "contents": [ "This helper lemma is an almost immediate consequence of", "Lemma \\ref{lemma-square-triangle}", "and we strongly urge the reader skip the proof.", "Say $K' = R\\epsilon_*K$. Choose a K-injective complex of", "$\\mathcal{O}_\\tau$-modules $\\mathcal{J}^\\bullet$ representing $K$.", "Then $\\epsilon_*\\mathcal{J}^\\bullet$ is a K-injective complex of", "$\\mathcal{O}_{\\tau'}$-modules representing $K'$, see", "Lemma \\ref{lemma-K-injective-flat}. Next,", "$$", "0 \\to", "\\mathcal{J}^\\bullet(X) \\xrightarrow{\\alpha}", "\\mathcal{J}^\\bullet(Z) \\oplus", "\\mathcal{J}^\\bullet(Y) \\xrightarrow{\\beta}", "\\mathcal{J}^\\bullet(E)", "\\to 0", "$$", "is a short exact sequence of complexes of abelian groups, see", "Lemma \\ref{lemma-square-triangle} and its proof.", "Since this is the same as the sequence of complexes of abelian", "groups which is used to define $c^{K'}_{X, Z, Y, E}$, we conclude." ], "refs": [ "sites-cohomology-lemma-square-triangle", "sites-cohomology-lemma-K-injective-flat", "sites-cohomology-lemma-square-triangle" ], "ref_ids": [ 4284, 4261, 4284 ] } ], "ref_ids": [ 4282 ] }, { "id": 4296, "type": "theorem", "label": "sites-cohomology-lemma-A", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-A", "contents": [ "In Situation \\ref{situation-compare} for $X$ in $\\mathcal{C}$", "denote $\\mathcal{A}_X$", "the objects of $\\textit{Ab}(\\mathcal{C}_\\tau/X)$ of the form", "$\\epsilon_X^{-1}\\mathcal{F}'$ with $\\mathcal{F}'$ in $\\mathcal{A}'_X$.", "Then", "\\begin{enumerate}", "\\item for $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C}_\\tau/X)$", "we have $\\mathcal{F} \\in \\mathcal{A}_X \\Leftrightarrow", "\\epsilon_{X, *}\\mathcal{F} \\in \\mathcal{A}'_X$, and", "\\item $f_\\tau^{-1}$ sends $\\mathcal{A}_Y$ into $\\mathcal{A}_X$", "for any morphism $f : X \\to Y$ of $\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from (\\ref{item-A-sheaf}) and part (2)", "follows from (\\ref{item-restriction-A}) and", "the commutativity of (\\ref{equation-commutative-epsilon}) which gives", "$\\epsilon_X^{-1} \\circ f_{\\tau'}^{-1} = f_\\tau^{-1} \\circ \\epsilon_Y^{-1}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4297, "type": "theorem", "label": "sites-cohomology-lemma-V-implies-C-general", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-V-implies-C-general", "contents": [ "In Situation \\ref{situation-compare} assume $(V_n)$ holds.", "For $f : X \\to Y$ in $\\mathcal{P}$ and $\\mathcal{F}$ in $\\mathcal{A}_X$", "we have $R^if_{\\tau', *}\\epsilon_{X, *}\\mathcal{F} =", "\\epsilon_{Y, *}R^if_{\\tau, *}\\mathcal{F}$ for $i \\leq n$." ], "refs": [], "proofs": [ { "contents": [ "We will use the commutative diagram (\\ref{equation-commutative-epsilon})", "without further mention. In particular have", "$$", "Rf_{\\tau', *}R\\epsilon_{X, *}\\mathcal{F} =", "R\\epsilon_{Y, *}Rf_{\\tau, *}\\mathcal{F}", "$$", "Assumption $(V_n)$ tells us that", "$\\epsilon_{X, *}\\mathcal{F} \\to R\\epsilon_{X, *}\\mathcal{F}$", "is an isomorphism in degrees $\\leq n$. Hence", "$Rf_{\\tau', *}\\epsilon_{X, *}\\mathcal{F} \\to", "Rf_{\\tau', *}R\\epsilon_{X, *}\\mathcal{F}$ is an isomorphism", "in degrees $\\leq n$. We conclude that", "$$", "R^if_{\\tau', *}\\epsilon_{X, *}\\mathcal{F} \\to", "H^i(R\\epsilon_{Y, *}Rf_{\\tau, *}\\mathcal{F})", "$$", "is an isomorphism for $i \\leq n$. We will prove the lemma by looking", "at the second page of the spectral sequence of", "Lemma \\ref{lemma-relative-Leray}", "for $R\\epsilon_{Y, *}Rf_{\\tau, *}\\mathcal{F}$. Here is a picture:", "$$", "\\begin{matrix}", "\\ldots &", "\\ldots &", "\\ldots &", "\\ldots \\\\", "\\epsilon_{Y, *}R^2f_{\\tau, *}\\mathcal{F} &", "R^1\\epsilon_{Y, *}R^2f_{\\tau, *}\\mathcal{F} &", "R^2\\epsilon_{Y, *}R^2f_{\\tau, *}\\mathcal{F} &", "\\ldots \\\\", "\\epsilon_{Y, *}R^1f_{\\tau, *}\\mathcal{F} &", "R^1\\epsilon_{Y, *}R^1f_{\\tau, *}\\mathcal{F} &", "R^2\\epsilon_{Y, *}R^1f_{\\tau, *}\\mathcal{F} &", "\\ldots \\\\", "\\epsilon_{Y, *}f_{\\tau, *}\\mathcal{F} &", "R^1\\epsilon_{Y, *}f_{\\tau, *}\\mathcal{F} &", "R^2\\epsilon_{Y, *}f_{\\tau, *}\\mathcal{F} &", "\\ldots", "\\end{matrix}", "$$", "Let $(C_m)$ be the hypothesis: $R^if_{\\tau', *}\\epsilon_{X, *}\\mathcal{F} =", "\\epsilon_{Y, *}R^if_{\\tau, *}\\mathcal{F}$ for $i \\leq m$. Observe that", "$(C_0)$ holds. We will show that $(C_{m - 1}) \\Rightarrow (C_m)$ for $m < n$.", "Namely, if $(C_{m - 1})$ holds, then for", "$n \\geq p > 0$ and $q \\leq m - 1$ we have", "\\begin{align*}", "R^p\\epsilon_{Y, *}R^qf_{\\tau, *}\\mathcal{F}", "& =", "R^p\\epsilon_{Y, *}", "\\epsilon_Y^{-1} \\epsilon_{Y, *} R^qf_{\\tau, *}\\mathcal{F} \\\\", "& =", "R^p\\epsilon_{Y, *}", "\\epsilon_Y^{-1}R^qf_{\\tau', *}\\epsilon_{X, *}\\mathcal{F} = 0", "\\end{align*}", "First equality as $\\epsilon_Y^{-1}\\epsilon_{Y, *} = \\text{id}$,", "the second by $(C_{m - 1})$, and the final by", "by $(V_n)$ because $\\epsilon_Y^{-1}R^qf_{\\tau', *}\\epsilon_{X, *}\\mathcal{F}$", "is in $\\mathcal{A}_Y$ by (\\ref{item-A-and-P}).", "Looking at the spectral sequence we see that", "$E_2^{0, m} = \\epsilon_{Y, *}R^mf_{\\tau, *}\\mathcal{F}$", "is the only nonzero term $E_2^{p, q}$ with $p + q = m$.", "Recall that $\\text{d}_r^{p, q} : E_r^{p, q} \\to E_r^{p + r, q - r + 1}$.", "Hence there are no nonzero differentials $\\text{d}_r^{p, q}$, $r \\geq 2$", "either emanating or entering this spot. We conclude that", "$H^m(R\\epsilon_{Y, *}Rf_{\\tau, *}\\mathcal{F}) =", "\\epsilon_{Y, *}R^mf_{\\tau, *}\\mathcal{F}$ which implies", "$(C_m)$ by the discussion above.", "\\medskip\\noindent", "Finally, assume $(C_{n - 1})$. The same analysis shows that", "$E_2^{0, n} = \\epsilon_{Y, *}R^nf_{\\tau, *}\\mathcal{F}$", "is the only nonzero term $E_2^{p, q}$ with $p + q = n$.", "We do still have no nonzero differentials entering", "this spot, but there can be a nonzero differential", "emanating it. Namely, the map", "$d_{n + 1}^{0, n} : \\epsilon_{Y, *}R^nf_{\\tau, *}\\mathcal{F} \\to", "R^{n + 1}\\epsilon_{Y, *}f_{\\tau, *}\\mathcal{F}$.", "We conclude that there is an exact sequence", "$$", "0 \\to R^nf_{\\tau', *}\\epsilon_{X, *}\\mathcal{F} \\to ", "\\epsilon_{Y, *}R^nf_{\\tau, *}\\mathcal{F} \\to", "R^{n + 1}\\epsilon_{Y, *}f_{\\tau, *}\\mathcal{F}", "$$", "By (\\ref{item-A-and-P}) and (\\ref{item-A-sheaf}) the sheaf", "$R^nf_{\\tau', *}\\epsilon_{X, *}\\mathcal{F}$", "satisfies the sheaf property for $\\tau$-coverings", "as does $\\epsilon_{Y, *}R^nf_{\\tau, *}\\mathcal{F}$", "(use the description of $\\epsilon_*$ in Section \\ref{section-compare}).", "However, the $\\tau$-sheafification of the $\\tau'$-sheaf", "$R^{n + 1}\\epsilon_{Y, *}f_{\\tau, *}\\mathcal{F}$", "is zero (by locality of cohomology; use", "Lemmas \\ref{lemma-kill-cohomology-class-on-covering} and", "\\ref{lemma-higher-direct-images}).", "Thus $R^nf_{\\tau', *}\\epsilon_{X, *}\\mathcal{F} \\to ", "\\epsilon_{Y, *}R^nf_{\\tau, *}\\mathcal{F}$", "has to be an isomorphism and the proof is complete." ], "refs": [ "sites-cohomology-lemma-relative-Leray", "sites-cohomology-lemma-kill-cohomology-class-on-covering", "sites-cohomology-lemma-higher-direct-images" ], "ref_ids": [ 4222, 4188, 4189 ] } ], "ref_ids": [] }, { "id": 4298, "type": "theorem", "label": "sites-cohomology-lemma-V-implies-cohomology-general", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-V-implies-cohomology-general", "contents": [ "In Situation \\ref{situation-compare} if $(V_n)$ holds, then", "for $X$ in $\\mathcal{C}$ and $L \\in D(\\mathcal{C}_{\\tau'}/X)$", "with $H^i(L) = 0$ for $i < 0$ and $H^i(L)$ in $\\mathcal{A}'_X$", "for $0 \\leq i \\leq n$ we have", "$H^n_{\\tau'}(X, L) = H^n_\\tau(X, \\epsilon_X^{-1}L)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-Leray-unbounded} we have", "$H^n_\\tau(X, \\epsilon_X^{-1}L) =", "H^n_{\\tau'}(X, R\\epsilon_{X, *}\\epsilon_X^{-1}L)$.", "There is a spectral sequence", "$$", "E_2^{p, q} = R^p\\epsilon_{X, *}\\epsilon_X^{-1}H^q(L)", "$$", "converging to $H^{p + q}(R\\epsilon_{X, *}\\epsilon_X^{-1}L)$.", "By $(V_n)$ we have the vanishing of $E_2^{p, q}$ for", "$0 < p \\leq n$ and $0 \\leq q \\leq n$. Thus", "$E_2^{0, q} = \\epsilon_{X, *}\\epsilon_X^{-1}H^q(L) = H^q(L)$", "are the only nonzero terms $E_2^{p, q}$ with $p + q \\leq n$.", "It follows that the map", "$$", "L \\longrightarrow R\\epsilon_{X, *}\\epsilon_X^{-1}L", "$$", "is an isomorphism in degrees $\\leq n$ (small detail omitted).", "Hence we find that", "$H^i_{\\tau'}(X, L) = H^i_{\\tau'}(X, R\\epsilon_{X, *}\\epsilon_X^{-1}L)$", "for $i \\leq n$. Thus the lemma is proved." ], "refs": [ "sites-cohomology-lemma-Leray-unbounded" ], "ref_ids": [ 4257 ] } ], "ref_ids": [] }, { "id": 4299, "type": "theorem", "label": "sites-cohomology-lemma-V-implies-cohomology-extra-general", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-V-implies-cohomology-extra-general", "contents": [ "In Situation \\ref{situation-compare} if $(V_n)$ holds, then for", "$X$ in $\\mathcal{C}$ and $\\mathcal{F}$ in $\\mathcal{A}_X$ the map", "$H^{n + 1}_{\\tau'}(X, \\epsilon_{X, *}\\mathcal{F}) \\to", "H^{n + 1}_\\tau(X, \\mathcal{F})$", "is injective with image those classes which become trivial on", "a $\\tau'$-covering of $X$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\epsilon_X^{-1}\\epsilon_{X, *}\\mathcal{F} = \\mathcal{F}$", "hence the map is given by pulling back cohomology classes", "by $\\epsilon_X$. The Leray spectral sequence (Lemma \\ref{lemma-Leray})", "$$", "E_2^{p, q} = H^p_{\\tau'}(X, R^q\\epsilon_{X, *}\\mathcal{F})", "\\Rightarrow", "H^{p + q}_\\tau(X, \\mathcal{F})", "$$", "combined with the assumed vanishing gives an exact sequence", "$$", "0 \\to", "H^{n + 1}_{\\tau'}(X, \\epsilon_{X, *}\\mathcal{F}) \\to", "H^{n + 1}_\\tau(X, \\mathcal{F}) \\to", "H^0_{\\tau'}(X, R^{n + 1}\\epsilon_{X, *}\\mathcal{F})", "$$", "This is a restatement of the lemma." ], "refs": [ "sites-cohomology-lemma-Leray" ], "ref_ids": [ 4220 ] } ], "ref_ids": [] }, { "id": 4300, "type": "theorem", "label": "sites-cohomology-lemma-make-class-zero-general", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-make-class-zero-general", "contents": [ "In Situation \\ref{situation-compare} let $f : X \\to Y$", "be in $\\mathcal{P}$ such that $\\{X \\to Y\\}$ is a $\\tau$-covering.", "Let $\\mathcal{F}'$ be in $\\mathcal{A}'_Y$. If $n \\geq 0$ and", "$$", "\\theta \\in", "\\text{Equalizer}\\left(", "\\xymatrix{", "H^{n + 1}_{\\tau'}(X, \\mathcal{F}')", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "H^{n + 1}_{\\tau'}(X \\times_Y X, \\mathcal{F}')", "}", "\\right)", "$$", "then there exists a $\\tau'$-covering $\\{Y_i \\to Y\\}$", "such that $\\theta$ restricts to zero in", "$H^{n + 1}_{\\tau'}(Y_i \\times_Y X, \\mathcal{F}')$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $X \\times_Y X$ exists by (\\ref{item-base-change-P}).", "For $Z$ in $\\mathcal{C}/Y$ denote $\\mathcal{F}'|_Z$", "the restriction of $\\mathcal{F}'$ to $\\mathcal{C}_{\\tau'}/Z$.", "Recall that $H^{n + 1}_{\\tau'}(X, \\mathcal{F}') =", "H^{n + 1}(\\mathcal{C}_{\\tau'}/X, \\mathcal{F}'|_X)$, see", "Lemma \\ref{lemma-cohomology-of-open}.", "The lemma asserts that the", "image $\\overline{\\theta} \\in H^0(Y, R^{n + 1}f_{\\tau', *}\\mathcal{F}'|_X)$", "of $\\theta$ is zero. Consider the cartesian diagram", "$$", "\\xymatrix{", "X \\times_Y X \\ar[d]_{\\text{pr}_1} \\ar[r]_{\\text{pr}_2} &", "X \\ar[d]^f \\\\", "X \\ar[r]^f & Y", "}", "$$", "By trivial base change (Lemma \\ref{lemma-localize-cartesian-square})", "we have", "$$", "f_{\\tau'}^{-1}R^{n + 1}f_{\\tau', *}(\\mathcal{F}'|_X) =", "R^{n + 1}\\text{pr}_{1, \\tau', *}(\\mathcal{F}'|_{X \\times_Y X})", "$$", "If $\\text{pr}_1^{-1}\\theta = \\text{pr}_2^{-1}\\theta$,", "then the section $f_{\\tau'}^{-1}\\overline{\\theta}$ of", "$f_{\\tau'}^{-1}R^{n + 1}f_{\\tau', *}(\\mathcal{F}'|_X)$ is zero,", "because it is clear that $\\text{pr}_1^{-1}\\theta$ maps to the zero element in", "$H^0(X, R^{n + 1}\\text{pr}_{1, \\tau', *}(\\mathcal{F}'|_{X \\times_Y X}))$.", "By (\\ref{item-restriction-A}) we have $\\mathcal{F}'|_X$ in $\\mathcal{A}'_X$.", "Thus $\\mathcal{G}' = R^{n + 1}f_{\\tau', *}(\\mathcal{F}'|_X)$", "is an object of $\\mathcal{A}'_Y$ by (\\ref{item-A-and-P}).", "Thus $\\mathcal{G}'$ satisfies the sheaf property for", "$\\tau$-coverings by (\\ref{item-A-sheaf}).", "Since $\\{X \\to Y\\}$ is a $\\tau$-covering we conclude", "that restriction $\\mathcal{G}'(Y) \\to \\mathcal{G}'(X)$ is injective.", "It follows that $\\overline{\\theta}$ is zero." ], "refs": [ "sites-cohomology-lemma-cohomology-of-open", "sites-cohomology-lemma-localize-cartesian-square" ], "ref_ids": [ 4186, 4263 ] } ], "ref_ids": [] }, { "id": 4301, "type": "theorem", "label": "sites-cohomology-lemma-induction-step-V-C-general", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-induction-step-V-C-general", "contents": [ "In Situation \\ref{situation-compare} we have $(V_n) \\Rightarrow (V_{n + 1})$." ], "refs": [], "proofs": [ { "contents": [ "Let $X$ in $\\mathcal{C}$ and $\\mathcal{F}$ in $\\mathcal{A}_X$.", "Let $\\xi \\in H^{n + 1}_\\tau(U, \\mathcal{F})$ for some $U/X$.", "We have to show that $\\xi$ restricts to zero on the members of", "a $\\tau'$-covering of $U$. See", "Lemma \\ref{lemma-higher-direct-images}.", "It follows from this that we may", "replace $U$ by the members of a $\\tau'$-covering of $U$.", "\\medskip\\noindent", "By locality of cohomology", "(Lemma \\ref{lemma-kill-cohomology-class-on-covering})", "we can choose a $\\tau$-covering $\\{U_i \\to U\\}$", "such that $\\xi$ restricts to zero on $U_i$.", "Choose $\\{V_j \\to V\\}$, $\\{f_j : W_j \\to V_j\\}$, and", "$\\{W_{jk} \\to W_j\\}$ as in (\\ref{item-refine-tau-by-P}).", "After replacing both $U$ by $V_j$", "and $\\mathcal{F}$ by its restriction to", "$\\mathcal{C}_\\tau/V_j$, which is allowed by", "(\\ref{item-base-change-P}), we reduce to the", "case discussed in the next paragraph.", "\\medskip\\noindent", "Here $f : X \\to Y$ is an element of $\\mathcal{P}$", "such that $\\{X \\to Y\\}$ is a $\\tau$-covering,", "$\\mathcal{F}$ is an object of $\\mathcal{A}_Y$, and", "$\\xi \\in H^{n + 1}_\\tau(Y, \\mathcal{F})$ is such that", "there exists a $\\tau'$-covering $\\{X_i \\to X\\}_{i \\in I}$", "such that $\\xi$ restricts to zero on $X_i$ for all $i \\in I$.", "Problem: show that $\\xi$ restricts to zero on a $\\tau'$-covering of $Y$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-V-implies-cohomology-extra-general} there exists a", "unique $\\tau'$-cohomology class", "$\\theta \\in H^{n + 1}_{\\tau'}(X, \\epsilon_{X, *}\\mathcal{F})$", "whose image is $\\xi|_X$.", "Since $\\xi|_X$ pulls back to the same class on $X \\times_Y X$", "via the two projections, we find that the same is true for $\\theta$", "(by uniqueness).", "By Lemma \\ref{lemma-make-class-zero-general}", "we see that after replacing $Y$ by the members of a $\\tau'$-covering,", "we may assume that $\\theta = 0$.", "Consequently, we may assume that $\\xi|_X$ is zero.", "\\medskip\\noindent", "Let $f : X \\to Y$ be an element of $\\mathcal{P}$", "such that $\\{X \\to Y\\}$ is a $\\tau$-covering,", "$\\mathcal{F}$ is an object of $\\mathcal{A}_Y$, and", "$\\xi \\in H^{n + 1}_\\tau(Y, \\mathcal{F})$", "maps to zero in $H^{n + 1}_\\tau(X, \\mathcal{F})$.", "Problem: show that $\\xi$ restricts to zero on a $\\tau'$-covering of $Y$.", "\\medskip\\noindent", "The assumptions tell us $\\xi$ maps to zero under the map", "$$", "\\mathcal{F} \\longrightarrow Rf_{\\tau, *}f_\\tau^{-1}\\mathcal{F}", "$$", "Use Lemma \\ref{lemma-Leray-unbounded}.", "A simple argument using the distinguished triangle of truncations", "(Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}) shows that", "$\\xi$ maps to zero under the map", "$$", "\\mathcal{F} \\longrightarrow \\tau_{\\leq n}Rf_{\\tau, *}f_\\tau^{-1}\\mathcal{F}", "$$", "We will compare this with the map $\\epsilon_{Y, *}\\mathcal{F} \\to K$", "where", "$$", "K = \\tau_{\\leq n}Rf_{\\tau', *}f_{\\tau'}^{-1}\\epsilon_{Y, *}\\mathcal{F} =", "\\tau_{\\leq n}Rf_{\\tau', *}\\epsilon_{X, *}f_{\\tau}^{-1}\\mathcal{F}", "$$", "The equality", "$\\epsilon_{X, *} f_\\tau^{-1} = f_{\\tau'}^{-1} \\epsilon_{Y, *}$", "is a property of (\\ref{equation-commutative-epsilon}). Consider the map", "$$", "Rf_{\\tau', *}\\epsilon_{X, *}f_{\\tau}^{-1}\\mathcal{F} \\longrightarrow", "Rf_{\\tau', *}R\\epsilon_{X, *}f_{\\tau}^{-1}\\mathcal{F} =", "R\\epsilon_{Y, *}Rf_{\\tau, *}f_\\tau^{-1}\\mathcal{F}", "$$", "used in the proof of Lemma \\ref{lemma-V-implies-C-general}", "which induces by adjunction a map", "$$", "\\epsilon_Y^{-1} Rf_{\\tau', *}\\epsilon_{X, *}f_{\\tau}^{-1}\\mathcal{F} \\to", "Rf_{\\tau, *}f_\\tau^{-1}\\mathcal{F}", "$$", "Taking trunctions we find a map", "$$", "\\epsilon_Y^{-1}K", "\\longrightarrow", "\\tau_{\\leq n}Rf_{\\tau, *}f_\\tau^{-1}\\mathcal{F}", "$$", "which is an isomorphism by Lemma \\ref{lemma-V-implies-C-general};", "the lemma applies because $f_\\tau^{-1}\\mathcal{F}$ is in $\\mathcal{A}_X$", "by Lemma \\ref{lemma-A}. Choose a distinguished triangle", "$$", "\\epsilon_{Y, *}\\mathcal{F} \\to K \\to L \\to \\epsilon_{Y, *}\\mathcal{F}[1]", "$$", "The map $\\mathcal{F} \\to f_{\\tau, *}f_\\tau^{-1}\\mathcal{F}$", "is injective as $\\{X \\to Y\\}$ is a $\\tau$-covering. Thus", "$\\epsilon_{Y, *}\\mathcal{F} \\to", "\\epsilon_{Y, *}f_{\\tau, *}f_\\tau^{-1}\\mathcal{F} =", "f_{\\tau', *}f_{\\tau'}^{-1}\\epsilon_{Y, *}\\mathcal{F}$", "is injective too. ", "Hence $L$ only has nonzero cohomology sheaves in degrees $0, \\ldots, n$.", "As $f_{\\tau', *}f_{\\tau'}^{-1}\\epsilon_{Y, *}\\mathcal{F}$", "is in $\\mathcal{A}'_Y$ by (\\ref{item-restriction-A}) and", "(\\ref{item-A-and-P}) we conclude that", "$$", "H^0(L) = \\Coker(\\epsilon_{Y, *}\\mathcal{F} \\to", "f_{\\tau', *}f_{\\tau'}^{-1}\\epsilon_{Y, *}\\mathcal{F})", "$$", "is in the weak Serre subcategory $\\mathcal{A}'_Y$. For $1 \\leq i \\leq n$", "we see that $H^i(L) = R^if_{\\tau', *}f_{\\tau'}^{-1}\\epsilon_{Y, *}\\mathcal{F}$", "is in $\\mathcal{A}'_Y$ by (\\ref{item-restriction-A}) and", "(\\ref{item-A-and-P}). Pulling back the distinguished triangle", "above by $\\epsilon_Y$ we get the distinguished triangle", "$$", "\\mathcal{F} \\to \\tau_{\\leq n}Rf_{\\tau, *}f_\\tau^{-1}\\mathcal{F}", "\\to \\epsilon_Y^{-1}L \\to \\mathcal{F}[1]", "$$", "Since $\\xi$ maps to zero in the middle term we find", "that $\\xi$ is the image of an element", "$\\xi' \\in H^n_\\tau(Y, \\epsilon_Y^{-1}L)$.", "By Lemma \\ref{lemma-V-implies-cohomology-general} we have", "$$", "H^n_{\\tau'}(Y, L) = H^n_\\tau(Y, \\epsilon_Y^{-1}L),", "$$", "Thus we may lift $\\xi'$ to an element of $H^n_{\\tau'}(Y, L)$", "and take the boundary into", "$H^{n + 1}_{\\tau'}(Y, \\epsilon_{Y, *}\\mathcal{F})$", "to see that $\\xi$ is in the image of the canonical map", "$H^{n + 1}_{\\tau'}(Y, \\epsilon_{Y, *}\\mathcal{F}) \\to", "H^{n + 1}_\\tau(Y, \\mathcal{F})$.", "By locality of cohomology for", "$H^{n + 1}_{\\tau'}(Y,\\epsilon_{Y, *}\\mathcal{F})$, see", "Lemma \\ref{lemma-kill-cohomology-class-on-covering},", "we conclude." ], "refs": [ "sites-cohomology-lemma-higher-direct-images", "sites-cohomology-lemma-kill-cohomology-class-on-covering", "sites-cohomology-lemma-V-implies-cohomology-extra-general", "sites-cohomology-lemma-make-class-zero-general", "sites-cohomology-lemma-Leray-unbounded", "derived-remark-truncation-distinguished-triangle", "sites-cohomology-lemma-V-implies-C-general", "sites-cohomology-lemma-V-implies-C-general", "sites-cohomology-lemma-A", "sites-cohomology-lemma-V-implies-cohomology-general", "sites-cohomology-lemma-kill-cohomology-class-on-covering" ], "ref_ids": [ 4189, 4188, 4299, 4300, 4257, 2016, 4297, 4297, 4296, 4298, 4188 ] } ], "ref_ids": [] }, { "id": 4302, "type": "theorem", "label": "sites-cohomology-lemma-V-C-all-n-general", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-V-C-all-n-general", "contents": [ "In Situation \\ref{situation-compare} we have that", "$(V_n)$ is true for all $n$. Moreover:", "\\begin{enumerate}", "\\item For $X$ in $\\mathcal{C}$ and", "$K' \\in D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/X)$ the map", "$K' \\to R\\epsilon_{X, *}(\\epsilon_X^{-1}K')$ is an isomorphism.", "\\item For $f : X \\to Y$ in $\\mathcal{P}$ and", "$K' \\in D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/X)$ we have", "$Rf_{\\tau', *}K' \\in D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/Y)$ and", "$\\epsilon_Y^{-1}(Rf_{\\tau', *}K') = Rf_{\\tau, *}(\\epsilon_X^{-1}K')$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $(V_0)$ holds as it is the empty condition.", "Then we get $(V_n)$ for all $n$ by", "Lemma \\ref{lemma-induction-step-V-C-general}.", "\\medskip\\noindent", "Proof of (1). The object $K = \\epsilon_X^{-1}K'$ has cohomology", "sheaves $H^i(K) = \\epsilon_X^{-1}H^i(K')$ in $\\mathcal{A}_X$.", "Hence the spectral sequence", "$$", "E_2^{p, q} = R^p\\epsilon_{X, *} H^q(K) \\Rightarrow", "H^{p + q}(R\\epsilon_{X, *}K)", "$$", "degenerates by $(V_n)$ for all $n$ and we find", "$$", "H^n(R\\epsilon_{X, *}K) = \\epsilon_{X, *}H^n(K) =", "\\epsilon_{X, *}\\epsilon_X^{-1}H^i(K') = H^i(K').", "$$", "again because $H^i(K')$ is in $\\mathcal{A}'_X$.", "Thus the canonical map $K' \\to R\\epsilon_{X, *}(\\epsilon_X^{-1}K')$", "is an isomorphism.", "\\medskip\\noindent", "Proof of (2). Using the spectral sequence", "$$", "E_2^{p, q} = R^pf_{\\tau', *}H^q(K') \\Rightarrow R^{p + q}f_{\\tau', *}K'", "$$", "the fact that $R^pf_{\\tau', *}H^q(K')$ is in", "$\\mathcal{A}'_Y$ by (\\ref{item-A-and-P}),", "the fact that $\\mathcal{A}'_Y$ is a weak Serre subcategory of", "$\\textit{Ab}(\\mathcal{C}_{\\tau'}/Y)$,", "and Homology, Lemma \\ref{homology-lemma-biregular-ss-converges}", "we conclude that", "$Rf_{\\tau', *}K' \\in D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/X)$.", "To finish the proof we have to show the base change map", "$$", "\\epsilon_Y^{-1}(Rf_{\\tau', *}K')", "\\longrightarrow", "Rf_{\\tau, *}(\\epsilon_X^{-1}K')", "$$", "is an isomorphism. Comparing the spectral sequence above to the", "spectral sequence", "$$", "E_2^{p, q} = R^pf_{\\tau, *}H^q(\\epsilon_X^{-1}K')", "\\Rightarrow R^{p + q}f_{\\tau, *}\\epsilon_X^{-1}K'", "$$", "we reduce this to the case where $K'$ has a single nonzero", "cohomology sheaf $\\mathcal{F}'$ in $\\mathcal{A}'_X$; details omitted.", "Then Lemma \\ref{lemma-V-implies-C-general} gives", "$\\epsilon_Y^{-1}R^if_{\\tau', *}\\mathcal{F}' =", "R^if_{\\tau, *}\\epsilon_X^{-1}\\mathcal{F}'$ for all $i$", "and the proof is complete." ], "refs": [ "sites-cohomology-lemma-induction-step-V-C-general", "homology-lemma-biregular-ss-converges", "sites-cohomology-lemma-V-implies-C-general" ], "ref_ids": [ 4301, 12101, 4297 ] } ], "ref_ids": [] }, { "id": 4303, "type": "theorem", "label": "sites-cohomology-lemma-cohomological-descent-general", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cohomological-descent-general", "contents": [ "In Situation \\ref{situation-compare}. For any $X$ in", "$\\mathcal{C}$ the category", "$\\mathcal{A}_X \\subset \\textit{Ab}(\\mathcal{C}_\\tau/X)$", "is a weak Serre subcategory and the functor", "$$", "R\\epsilon_{X, *} :", "D^+_{\\mathcal{A}_X}(\\mathcal{C}_\\tau/X)", "\\longrightarrow", "D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/X)", "$$", "is an equivalence with quasi-inverse given by $\\epsilon_X^{-1}$." ], "refs": [], "proofs": [ { "contents": [ "We need to check the conditions listed in", "Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}", "for $\\mathcal{A}_X$. If $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a map", "in $\\mathcal{A}_X$, then $\\epsilon_{X, *}\\varphi :", "\\epsilon_{X, *}\\mathcal{F} \\to \\epsilon_{X, *}\\mathcal{G}$", "is a map in $\\mathcal{A}'_X$. Hence $\\Ker(\\epsilon_{X, *}\\varphi)$ and", "$\\Coker(\\epsilon_{X, *}\\varphi)$ are objects of $\\mathcal{A}'_X$", "as this is a weak Serre subcategory of $\\textit{Ab}(\\mathcal{C}_{\\tau'}/X)$.", "Applying $\\epsilon_X^{-1}$ we obtain an exact sequence", "$$", "0 \\to", "\\epsilon_X^{-1}\\Ker(\\epsilon_{X, *}\\varphi) \\to", "\\mathcal{F} \\to \\mathcal{G} \\to", "\\epsilon_X^{-1}\\Coker(\\epsilon_{X, *}\\varphi) \\to 0", "$$", "and we see that $\\Ker(\\varphi)$ and $\\Coker(\\varphi)$ are in", "$\\mathcal{A}_X$. Finally, suppose that", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0", "$$", "is a short exact sequence in $\\textit{Ab}(\\mathcal{C}_\\tau/X)$", "with $\\mathcal{F}_1$ and $\\mathcal{F}_3$ in $\\mathcal{A}_X$.", "Then applying $\\epsilon_{X, *}$ we obtain an exact sequence", "$$", "0 \\to ", "\\epsilon_{X, *}\\mathcal{F}_1 \\to", "\\epsilon_{X, *}\\mathcal{F}_2 \\to", "\\epsilon_{X, *}\\mathcal{F}_3 \\to", "R^1\\epsilon_{X, *}\\mathcal{F}_1 = 0", "$$", "Vanishing by Lemma \\ref{lemma-V-C-all-n-general}.", "Hence $\\epsilon_{X, *}\\mathcal{F}_2$ is in $\\mathcal{A}'_X$", "as this is a weak Serre subcategory of $\\textit{Ab}(\\mathcal{C}_{\\tau'}/X)$.", "Pulling back by $\\epsilon_X$ we conclude that", "$\\mathcal{F}_2$ is in $\\mathcal{A}_X$.", "\\medskip\\noindent", "Thus $\\mathcal{A}_X$ is a weak Serre subcategory of", "$\\textit{Ab}(\\mathcal{C}_\\tau/X)$ and it makes sense", "to consider the category $D^+_{\\mathcal{A}_X}(\\mathcal{C}_\\tau/X)$.", "Observe that $\\epsilon_X^{-1} : \\mathcal{A}'_X \\to \\mathcal{A}_X$", "is an equivalence and that", "$\\mathcal{F}' \\to R\\epsilon_{X, *}\\epsilon_X^{-1}\\mathcal{F}'$", "is an isomorphism for $\\mathcal{F}'$ in $\\mathcal{A}'_X$ since we", "have $(V_n)$ for all $n$ by Lemma \\ref{lemma-V-C-all-n-general}.", "Thus we conclude by Lemma \\ref{lemma-equivalence-bounded}." ], "refs": [ "homology-lemma-characterize-weak-serre-subcategory", "sites-cohomology-lemma-V-C-all-n-general", "sites-cohomology-lemma-V-C-all-n-general", "sites-cohomology-lemma-equivalence-bounded" ], "ref_ids": [ 12046, 4302, 4302, 4290 ] } ], "ref_ids": [] }, { "id": 4304, "type": "theorem", "label": "sites-cohomology-lemma-compare-cohomology-general", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-compare-cohomology-general", "contents": [ "In Situation \\ref{situation-compare}. Let $X$ be in $\\mathcal{C}$.", "\\begin{enumerate}", "\\item for $\\mathcal{F}'$ in $\\mathcal{A}'_X$ we have", "$H^n_{\\tau'}(X, \\mathcal{F}') = H^n_\\tau(X, \\epsilon_X^{-1}\\mathcal{F}')$,", "\\item for $K' \\in D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/X)$", "we have $H^n_{\\tau'}(X, K') = H^n_\\tau(X, \\epsilon_X^{-1}K')$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-V-C-all-n-general}", "by Remark \\ref{remark-before-Leray}." ], "refs": [ "sites-cohomology-lemma-V-C-all-n-general", "sites-cohomology-remark-before-Leray" ], "ref_ids": [ 4302, 4423 ] } ], "ref_ids": [] }, { "id": 4305, "type": "theorem", "label": "sites-cohomology-lemma-LC-basic", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-LC-basic", "contents": [ "The category $\\textit{LC}$ has fibre products and a final object and hence", "has arbitrary finite limits. Given morphisms $X \\to Z$ and $Y \\to Z$", "in $\\textit{LC}$ with", "$X$ and $Y$ quasi-compact, then $X \\times_Z Y$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "The final object is the singleton space. Given morphisms $X \\to Z$ and", "$Y \\to Z$ of $\\textit{LC}$ the fibre product $X \\times_Z Y$ is", "a subspace of $X \\times Y$. Hence $X \\times_Z Y$ is Hausdorff as", "$X \\times Y$ is Hausdorff by", "Topology, Section \\ref{topology-section-Hausdorff}.", "\\medskip\\noindent", "If $X$ and $Y$ are quasi-compact, then $X \\times Y$ is quasi-compact by ", "Topology, Theorem \\ref{topology-theorem-tychonov}.", "Since $X \\times_Z Y$ is a closed subset of $X \\times Y$", "(Topology, Lemma \\ref{topology-lemma-fibre-product-closed})", "we find that $X \\times_Z Y$ is quasi-compact by", "Topology, Lemma \\ref{topology-lemma-closed-in-quasi-compact}.", "\\medskip\\noindent", "Finally, returning to the general case, if $x \\in X$ and $y \\in Y$", "we can pick quasi-compact neighbourhoods $x \\in E \\subset X$ and", "$y \\in F \\subset Y$ and we find that $E \\times_Z F$ is a quasi-compact", "neighbourhood of $(x, y)$ by the result above. Thus $X \\times_Z Y$", "is an object of $\\textit{LC}$ by", "Topology, Lemma \\ref{topology-lemma-locally-quasi-compact-Hausdorff}." ], "refs": [ "topology-theorem-tychonov", "topology-lemma-fibre-product-closed", "topology-lemma-closed-in-quasi-compact", "topology-lemma-locally-quasi-compact-Hausdorff" ], "ref_ids": [ 8188, 8193, 8229, 8242 ] } ], "ref_ids": [] }, { "id": 4306, "type": "theorem", "label": "sites-cohomology-lemma-qc", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-qc", "contents": [ "Let $X$ be a Hausdorff and locally quasi-compact space, in other words,", "an object of $\\textit{LC}$.", "\\begin{enumerate}", "\\item If $X' \\to X$ is an isomorphism in $\\textit{LC}$ then", "$\\{X' \\to X\\}$ is a qc covering.", "\\item If $\\{f_i : X_i \\to X\\}_{i\\in I}$ is a qc covering and for each", "$i$ we have a qc covering $\\{g_{ij} : X_{ij} \\to X_i\\}_{j\\in J_i}$, then", "$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a qc covering.", "\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a qc covering", "and $X' \\to X$ is a morphism of $\\textit{LC}$ then", "$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a qc covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) holds by the remark above that open coverings are qc coverings.", "\\medskip\\noindent", "Proof of (2). Let $x \\in X$. Choose $i_1, \\ldots, i_n \\in I$ and", "$E_a \\subset X_{i_a}$ quasi-compact such that $\\bigcup f_{i_a}(E_a)$", "is a neighbourhood of $x$. For every $e \\in E_a$ we can find", "a finite subset $J_e \\subset J_{i_a}$ and quasi-compact", "$F_{e, j} \\subset X_{ij}$, $j \\in J_e$ such that $\\bigcup g_{ij}(F_{e, j})$", "is a neighbourhood of $e$. Since $E_a$ is quasi-compact we find", "a finite collection $e_1, \\ldots, e_{m_a}$ such that", "$$", "E_a \\subset", "\\bigcup\\nolimits_{k = 1, \\ldots, m_a}", "\\bigcup\\nolimits_{j \\in J_{e_k}} g_{ij}(F_{e_k, j})", "$$", "Then we find that", "$$", "\\bigcup\\nolimits_{a = 1, \\ldots, n}", "\\bigcup\\nolimits_{k = 1, \\ldots, m_a}", "\\bigcup\\nolimits_{j \\in J_{e_k}} f_i(g_{ij}(F_{e_k, j}))", "$$", "is a neighbourhood of $x$.", "\\medskip\\noindent", "Proof of (3). Let $x' \\in X'$ be a point. Let $x \\in X$ be its image.", "Choose $i_1, \\ldots, i_n \\in I$ and quasi-compact subsets", "$E_j \\subset X_{i_j}$ such that $\\bigcup f_{i_j}(E_j)$ is a", "neighbourhood of $x$. Choose a quasi-compact neighbourhood $F \\subset X'$", "of $x'$ which maps into the quasi-compact neighbourhood", "$\\bigcup f_{i_j}(E_j)$ of $x$. Then", "$F \\times_X E_j \\subset X' \\times_X X_{i_j}$ is a", "quasi-compact subset and $F$ is the image of the map", "$\\coprod F \\times_X E_j \\to F$. Hence the base change is a", "qc covering and the proof is finished." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4307, "type": "theorem", "label": "sites-cohomology-lemma-proper-surjective-is-qc-covering", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-proper-surjective-is-qc-covering", "contents": [ "Let $f : X \\to Y$ be a morphism of $\\textit{LC}$.", "If $f$ is proper and surjective, then $\\{f : X \\to Y\\}$", "is a qc covering." ], "refs": [], "proofs": [ { "contents": [ "Let $y \\in Y$ be a point. For each $x \\in X_y$ choose a quasi-compact", "neighbourhood $E_x \\subset X$. Choose $x \\in U_x \\subset E_x$ open.", "Since $f$ is proper the fibre $X_y$ is quasi-compact and we find", "$x_1, \\ldots, x_n \\in X_y$ such that", "$X_y \\subset U_{x_1} \\cup \\ldots \\cup U_{x_n}$.", "We claim that $f(E_{x_1}) \\cup \\ldots \\cup f(E_{x_n})$ is a neighbourhood of", "$y$. Namely, as $f$ is closed", "(Topology, Theorem \\ref{topology-theorem-characterize-proper})", "we see that $Z = f(X \\setminus U_{x_1} \\cup \\ldots \\cup U_{x_n})$", "is a closed subset of $Y$ not containing $y$. As $f$ is surjective", "we see that $Y \\setminus Z$ is contained in", "$f(E_{x_1}) \\cup \\ldots \\cup f(E_{x_n})$ as desired." ], "refs": [ "topology-theorem-characterize-proper" ], "ref_ids": [ 8189 ] } ], "ref_ids": [] }, { "id": 4308, "type": "theorem", "label": "sites-cohomology-lemma-describe-pullback-pi", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-describe-pullback-pi", "contents": [ "Let $X$ be an object of $\\textit{LC}_{qc}$. Let $\\mathcal{F}$ be a", "sheaf on $X$. The rule", "$$", "\\textit{LC}_{qc}/X \\longrightarrow \\textit{Sets},\\quad", "(f : Y \\to X) \\longmapsto \\Gamma(Y, f^{-1}\\mathcal{F})", "$$", "is a sheaf and a fortiori also a sheaf on $\\textit{LC}_{Zar}/X$.", "This sheaf is equal to", "$\\pi_X^{-1}\\mathcal{F}$ on $\\textit{LC}_{Zar}/X$ and", "$\\epsilon_X^{-1}\\pi_X^{-1}\\mathcal{F}$ on $\\textit{LC}_{qc}/X$." ], "refs": [], "proofs": [ { "contents": [ "Denote $\\mathcal{G}$ the presheaf given by the formula in the lemma.", "Of course the pullback $f^{-1}$ in the formula denotes usual", "pullback of sheaves on topological spaces. It is immediate", "from the definitions that $\\mathcal{G}$ is a sheaf for the Zar", "topology.", "\\medskip\\noindent", "Let $Y \\to X$ be a morphism in $\\textit{LC}_{qc}$. Let", "$\\mathcal{V} = \\{g_i : Y_i \\to Y\\}_{i \\in I}$ be a qc covering.", "To prove $\\mathcal{G}$ is a sheaf for the qc topology it", "suffices to show that", "$\\mathcal{G}(Y) \\to H^0(\\mathcal{V}, \\mathcal{G})$", "is an isomorphism, see Sites, Section \\ref{sites-section-sheafification}.", "We first point out that the map is injective as a qc covering", "is surjective and we can detect equality of sections at stalks", "(use Sheaves, Lemmas \\ref{sheaves-lemma-sheaf-subset-stalks} and", "\\ref{sheaves-lemma-stalk-pullback-presheaf}). Thus", "$\\mathcal{G}$ is a separated presheaf on $\\textit{LC}_{qc}$", "hence it suffices to show that any element", "$(s_i) \\in H^0(\\mathcal{V}, \\mathcal{G})$", "maps to an element in the image of $\\mathcal{G}(Y)$", "after replacing $\\mathcal{V}$ by a refinement", "(Sites, Theorem \\ref{sites-theorem-plus}).", "\\medskip\\noindent", "Identifying sheaves on $Y_{i, Zar}$ and sheaves on $Y_i$ we find that", "$\\mathcal{G}|_{Y_{i, Zar}}$ is the pullback of $f^{-1}\\mathcal{F}$ under", "the continuous map $g_i : Y_i \\to Y$. Thus we can choose an open covering", "$Y_i = \\bigcup V_{ij}$ such that for each $j$ there is an open", "$W_{ij} \\subset Y$ and a section $t_{ij} \\in \\mathcal{G}(W_{ij})$", "such that $V_{ij}$ maps into $W_{ij}$ and such that", "$s|_{V_{ij}}$ is the pullback of $t_{ij}$. In other words,", "after refining the covering $\\{Y_i \\to Y\\}$ we may assume there", "are opens $W_i \\subset Y$ such that $Y_i \\to Y$ factors through $W_i$", "and sections $t_i$ of $\\mathcal{G}$ over $W_i$ which restrict", "to the given sections $s_i$. Moreover, if $y \\in Y$ is in the image", "of both $Y_i \\to Y$ and $Y_j \\to Y$, then the images $t_{i, y}$", "and $t_{j, y}$ in the stalk $f^{-1}\\mathcal{F}_y$ agree", "(because $s_i$ and $s_j$ agree over $Y_i \\times_Y Y_j$).", "Thus for $y \\in Y$ there is a well defined element $t_y$ of", "$f^{-1}\\mathcal{F}_y$ agreeing with $t_{i, y}$ whenever $y$", "is in the image of $Y_i \\to Y$.", "We will show that the element $(t_y)$ comes from a global section", "of $f^{-1}\\mathcal{F}$ over $Y$ which will finish the proof of the lemma.", "\\medskip\\noindent", "It suffices to show that this is true locally on $Y$, see", "Sheaves, Section \\ref{sheaves-section-sheafification}. Let $y_0 \\in Y$.", "Pick $i_1, \\ldots, i_n \\in I$ and", "quasi-compact subsets $E_j \\subset Y_{i_j}$ such that", "$\\bigcup g_{i_j}(E_j)$ is a neighbourhood of $y_0$.", "Let $V \\subset Y$ be an open neighbourhood of $y_0$ contained", "in $\\bigcup g_{i_j}(E_j)$ and contained in $W_{i_1} \\cap \\ldots \\cap W_{i_n}$.", "Since $t_{i_1, y_0} = \\ldots = t_{i_n, y_0}$, after shrinking $V$", "we may assume the sections $t_{i_j}|_V$, $j = 1, \\ldots, n$ of", "$f^{-1}\\mathcal{F}$ agree. As $V \\subset \\bigcup g_{i_j}(E_j)$", "we see that $(t_y)_{y \\in V}$ comes from this section.", "\\medskip\\noindent", "We still have to show that $\\mathcal{G}$ is equal to", "$\\epsilon_X^{-1}\\pi_X^{-1}\\mathcal{F}$ on $\\textit{LC}_{qc}$,", "resp.\\ $\\pi_X^{-1}\\mathcal{F}$ on $\\textit{LC}_{Zar}$.", "In both cases the pullback is defined by taking the presheaf", "$$", "(f : Y \\to X)", "\\longmapsto", "\\colim_{f(Y) \\subset U \\subset X} \\mathcal{F}(U)", "$$", "and then sheafifying. Sheafifying in the Zar topology", "exactly produces our sheaf $\\mathcal{G}$ and the fact", "that $\\mathcal{G}$ is a qc sheaf, shows that it works as well", "in the qc topology." ], "refs": [ "sheaves-lemma-sheaf-subset-stalks", "sheaves-lemma-stalk-pullback-presheaf", "sites-theorem-plus" ], "ref_ids": [ 14482, 14506, 8492 ] } ], "ref_ids": [] }, { "id": 4309, "type": "theorem", "label": "sites-cohomology-lemma-collect-true-things-Zar", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-collect-true-things-Zar", "contents": [ "Let $X$ be an object of $\\textit{LC}_{Zar}$. Then", "\\begin{enumerate}", "\\item for $\\mathcal{F} \\in \\textit{Ab}(X)$ we have", "$H^n_{Zar}(X, \\pi_X^{-1}\\mathcal{F}) = H^n(X, \\mathcal{F})$,", "\\item $\\pi_{X, *} : \\textit{Ab}(\\textit{LC}_{Zar}/X) \\to \\textit{Ab}(X)$", "is exact,", "\\item the unit $\\text{id} \\to \\pi_{X, *} \\circ \\pi_X^{-1}$", "of the adjunction is an isomorphism, and", "\\item for $K \\in D(X)$ the canonical map", "$K \\to R\\pi_{X, *} \\pi_X^{-1}K$ is an isomorphism.", "\\end{enumerate}", "Let $f : X \\to Y$ be a morphism of $\\textit{LC}_{Zar}$. Then", "\\begin{enumerate}", "\\item[(5)] there is a commutative diagram", "$$", "\\xymatrix{", "\\Sh(\\textit{LC}_{Zar}/X) \\ar[r]_{f_{Zar}} \\ar[d]_{\\pi_X} &", "\\Sh(\\textit{LC}_{Zar}/Y) \\ar[d]^{\\pi_Y} \\\\", "\\Sh(X_{Zar}) \\ar[r]^f &", "\\Sh(Y_{Zar})", "}", "$$", "of topoi,", "\\item[(6)] for $L \\in D^+(Y)$ we have", "$H^n_{Zar}(X, \\pi_Y^{-1}L) = H^n(X, f^{-1}L)$,", "\\item[(7)] if $f$ is proper, then we have", "\\begin{enumerate}", "\\item $\\pi_Y^{-1} \\circ f_* = f_{Zar, *} \\circ \\pi_X^{-1}$ as functors", "$\\Sh(X) \\to \\Sh(\\textit{LC}_{Zar}/Y)$,", "\\item $\\pi_Y^{-1} \\circ Rf_* = Rf_{Zar, *} \\circ \\pi_X^{-1}$ as", "functors $D^+(X) \\to D^+(\\textit{LC}_{Zar}/Y)$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1).", "The equality $H^n_{Zar}(X, \\pi_X^{-1}\\mathcal{F}) = H^n(X, \\mathcal{F})$", "is a general fact coming from the trivial observation that", "coverings of $X$ in $\\textit{LC}_{Zar}$ are the same thing as open", "coverings of $X$. The reader who wishes to see a detailed proof", "should apply Lemma \\ref{lemma-cohomology-bigger-site} to the functor", "$X_{Zar} \\to \\textit{LC}_{Zar}$.", "\\medskip\\noindent", "Proof of (2). This is true because $\\pi_{X, *} = \\tau_X^{-1}$", "for some morphism of topoi $\\tau_X : \\Sh(X_{Zar}) \\to \\Sh(\\textit{LC}_{Zar})$", "as follows from Sites, Lemma \\ref{sites-lemma-bigger-site}", "applied to the functor", "$X_{Zar} \\to \\textit{LC}_{Zar}/X$ used to define $\\pi_X$.", "\\medskip\\noindent", "Proof of (3). This is true because $\\tau_X^{-1} \\circ \\pi_X^{-1}$", "is the identity functor by Sites, Lemma \\ref{sites-lemma-bigger-site}.", "Or you can deduce it from the explicit description of", "$\\pi_X^{-1}$ in Lemma \\ref{lemma-describe-pullback-pi}.", "\\medskip\\noindent", "Proof of (4). Apply (3) to an complex of abelian sheaves representing $K$.", "\\medskip\\noindent", "Proof of (5). The morphism of topoi $f_{Zar}$ comes from an application of", "Sites, Lemma \\ref{sites-lemma-relocalize}", "and in our case comes from the continuous functor", "$Z/Y \\mapsto Z \\times_Y X/X$ by", "Sites, Lemma \\ref{sites-lemma-relocalize-given-fibre-products}.", "The diagram commutes simply because the corresponding", "continuous functors compose correctly", "(see Sites, Lemma \\ref{sites-lemma-composition-morphisms-sites}).", "\\medskip\\noindent", "Proof of (6). We have", "$H^n_{Zar}(X, \\pi_Y^{-1}\\mathcal{G}) =", "H^n_{Zar}(X, f_{Zar}^{-1}\\pi_Y^{-1}\\mathcal{G})$", "for $\\mathcal{G}$ in $\\textit{Ab}(Y)$, see", "Lemma \\ref{lemma-cohomology-of-open}.", "This is equal to $H^n_{Zar}(X, \\pi_X^{-1}f^{-1}\\mathcal{G})$", "by the commutativity of the diagram in (5).", "Hence we conclude by (1) in the case $L$ consists of a single", "sheaf in degree $0$. The general case follows by representing", "$L$ by a bounded below complex of abelian sheaves.", "\\medskip\\noindent", "Proof of (7a). Let $\\mathcal{F}$ be a sheaf on $X$.", "Let $g : Z \\to Y$ be an object of $\\textit{LC}_{Zar}/Y$. Consider the", "fibre product", "$$", "\\xymatrix{", "Z' \\ar[r]_{f'} \\ar[d]_{g'} & Z \\ar[d]^g \\\\", "X \\ar[r]^f & Y", "}", "$$", "Then we have", "$$", "(f_{Zar, *}\\pi_X^{-1}\\mathcal{F})(Z/Y) =", "(\\pi_X^{-1}\\mathcal{F})(Z'/X) =", "\\Gamma(Z', (g')^{-1}\\mathcal{F}) =", "\\Gamma(Z, f'_*(g')^{-1}\\mathcal{F})", "$$", "the second equality by Lemma \\ref{lemma-describe-pullback-pi}.", "On the other hand", "$$", "(\\pi_Y^{-1}f_*\\mathcal{F})(Z/Y) = \\Gamma(Z, g^{-1}f_*\\mathcal{F})", "$$", "again by Lemma \\ref{lemma-describe-pullback-pi}.", "Hence by proper base change for sheaves of sets", "(Cohomology, Lemma \\ref{cohomology-lemma-proper-base-change-sheaves-of-sets})", "we conclude the two sets are canonically isomorphic.", "The isomorphism is compatible with restriction mappings", "and defines an isomorphism", "$\\pi_Y^{-1}f_*\\mathcal{F} = f_{Zar, *}\\pi_X^{-1}\\mathcal{F}$.", "Thus an isomorphism of functors", "$\\pi_Y^{-1} \\circ f_* = f_{Zar, *} \\circ \\pi_X^{-1}$.", "\\medskip\\noindent", "Proof of (7b). Let $K \\in D^+(X)$. By", "Lemma \\ref{lemma-unbounded-describe-higher-direct-images}", "the $n$th cohomology sheaf of", "$Rf_{Zar, *}\\pi_X^{-1}K$ is the sheaf associated to the presheaf", "$$", "(g : Z \\to Y) \\longmapsto H^n_{Zar}(Z', \\pi_X^{-1}K)", "$$", "with notation as above. Observe that", "\\begin{align*}", "H^n_{Zar}(Z', \\pi_X^{-1}K)", "& =", "H^n(Z', (g')^{-1}K) \\\\", "& =", "H^n(Z, Rf'_*(g')^{-1}K) \\\\", "& =", "H^n(Z, g^{-1}Rf_*K) \\\\", "& =", "H^n_{Zar}(Z, \\pi_Y^{-1}Rf_*K)", "\\end{align*}", "The first equality is (6) applied to $K$ and $g' : Z' \\to X$.", "The second equality is Leray for $f' : Z' \\to Z$", "(Cohomology, Lemma \\ref{cohomology-lemma-before-Leray}).", "The third equality is the proper base change theorem", "(Cohomology, Theorem \\ref{cohomology-theorem-proper-base-change}).", "The fourth equality is (6) applied to $g : Z \\to Y$ and $Rf_*K$.", "Thus $Rf_{Zar, *}\\pi_X^{-1}K$ and $\\pi_Y^{-1}Rf_*K$ have the same", "cohomology sheaves. We omit the verification that the", "canonical base change map $\\pi_Y^{-1}Rf_*K \\to Rf_{Zar, *}\\pi_X^{-1}K$", "induces this isomorphism." ], "refs": [ "sites-cohomology-lemma-cohomology-bigger-site", "sites-lemma-bigger-site", "sites-lemma-bigger-site", "sites-cohomology-lemma-describe-pullback-pi", "sites-lemma-relocalize", "sites-lemma-relocalize-given-fibre-products", "sites-lemma-composition-morphisms-sites", "sites-cohomology-lemma-cohomology-of-open", "sites-cohomology-lemma-describe-pullback-pi", "sites-cohomology-lemma-describe-pullback-pi", "cohomology-lemma-proper-base-change-sheaves-of-sets", "sites-cohomology-lemma-unbounded-describe-higher-direct-images", "cohomology-lemma-before-Leray", "cohomology-theorem-proper-base-change" ], "ref_ids": [ 4187, 8548, 8548, 4308, 8559, 8568, 8525, 4186, 4308, 4308, 2081, 4258, 2068, 2031 ] } ], "ref_ids": [] }, { "id": 4310, "type": "theorem", "label": "sites-cohomology-lemma-push-pull-LC", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-push-pull-LC", "contents": [ "Let $f : X \\to Y$ be a morphism of $\\textit{LC}_{qc}$.", "Then there are commutative diagrams of topoi", "$$", "\\vcenter{", "\\xymatrix{", "\\Sh(\\textit{LC}_{qc}/X) \\ar[r]_{f_{qc}} \\ar[d]_{\\epsilon_X} &", "\\Sh(\\textit{LC}_{qc}/Y) \\ar[d]^{\\epsilon_Y} \\\\", "\\Sh(\\textit{LC}_{Zar}/X) \\ar[r]^{f_{Zar}} &", "\\Sh(\\textit{LC}_{Zar}/Y)", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "\\Sh(\\textit{LC}_{qc}/X) \\ar[r]_{f_{qc}} \\ar[d]_{a_X} &", "\\Sh(\\textit{LC}_{qc}/Y) \\ar[d]^{a_Y} \\\\", "\\Sh(X) \\ar[r]^f &", "\\Sh(Y)", "}", "}", "$$", "with $a_X = \\pi_X \\circ \\epsilon_X$, $a_Y = \\pi_X \\circ \\epsilon_X$.", "If $f$ is proper, then $a_Y^{-1} \\circ f_* = f_{qc, *} \\circ a_X^{-1}$." ], "refs": [], "proofs": [ { "contents": [ "The morphism of topoi $f_{qc}$ is the one from", "Sites, Lemma \\ref{sites-lemma-relocalize}", "which in our case comes from the continuous functor", "$Z/Y \\mapsto Z \\times_Y X/X$, see", "Sites, Lemma \\ref{sites-lemma-relocalize-given-fibre-products}.", "The diagram on the left commutes because the corresponding", "continuous functors compose correctly", "(see Sites, Lemma \\ref{sites-lemma-composition-morphisms-sites}).", "The diagram on the right commutes because the one on the left does", "and because of part (5) of Lemma \\ref{lemma-collect-true-things-Zar}.", "\\medskip\\noindent", "Proof of the final assertion. The reader may repeat the proof of part (7a) of", "Lemma \\ref{lemma-collect-true-things-Zar}; we will instead deduce this from it.", "As $\\epsilon_{Y, *}$ is the identity functor on underlying presheaves,", "it reflects isomorphisms. The description", "in Lemma \\ref{lemma-describe-pullback-pi}", "shows that $\\epsilon_{Y, *} \\circ a_Y^{-1} = \\pi_Y^{-1}$", "and similarly for $X$. To show that the canonical map", "$a_Y^{-1}f_*\\mathcal{F} \\to f_{qc, *}a_X^{-1}\\mathcal{F}$", "is an isomorphism, it suffices to show that", "$$", "\\pi_Y^{-1}f_*\\mathcal{F} =", "\\epsilon_{Y, *}a_Y^{-1}f_*\\mathcal{F} \\to", "\\epsilon_{Y, *}f_{qc, *}a_X^{-1}\\mathcal{F} =", "f_{Zar, *}\\epsilon_{X, *}a_X^{-1}\\mathcal{F} =", "f_{Zar, *}\\pi_X^{-1}\\mathcal{F}", "$$", "is an isomorphism. This is part", "(7a) of Lemma \\ref{lemma-collect-true-things-Zar}." ], "refs": [ "sites-lemma-relocalize", "sites-lemma-relocalize-given-fibre-products", "sites-lemma-composition-morphisms-sites", "sites-cohomology-lemma-collect-true-things-Zar", "sites-cohomology-lemma-collect-true-things-Zar", "sites-cohomology-lemma-describe-pullback-pi", "sites-cohomology-lemma-collect-true-things-Zar" ], "ref_ids": [ 8559, 8568, 8525, 4309, 4309, 4308, 4309 ] } ], "ref_ids": [] }, { "id": 4311, "type": "theorem", "label": "sites-cohomology-lemma-compare-qc-zar", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-compare-qc-zar", "contents": [ "Consider the comparison morphism", "$\\epsilon : \\textit{LC}_{qc} \\to \\textit{LC}_{Zar}$.", "Let $\\mathcal{P}$ denote the class of proper maps of topological spaces.", "For $X$ in $\\textit{LC}_{Zar}$ denote", "$\\mathcal{A}'_X \\subset \\textit{Ab}(\\textit{LC}_{Zar}/X)$", "the full subcategory consisting of sheaves of the form", "$\\pi_X^{-1}\\mathcal{F}$ with $\\mathcal{F}$ in $\\textit{Ab}(X)$.", "Then", "(\\ref{item-base-change-P}),", "(\\ref{item-restriction-A}),", "(\\ref{item-A-sheaf}),", "(\\ref{item-A-and-P}), and", "(\\ref{item-refine-tau-by-P})", "of Situation \\ref{situation-compare} hold." ], "refs": [], "proofs": [ { "contents": [ "We first show that $\\mathcal{A}'_X \\subset \\textit{Ab}(\\textit{LC}_{Zar}/X)$", "is a weak Serre subcategory by checking conditions (1), (2), (3), and (4)", "of Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}.", "Parts (1), (2), (3) are immediate as $\\pi_X^{-1}$ is exact and", "fully faithful by Lemma \\ref{lemma-collect-true-things-Zar} part (3). If", "$0 \\to \\pi_X^{-1}\\mathcal{F} \\to \\mathcal{G} \\to \\pi_X^{-1}\\mathcal{F}' \\to 0$", "is a short exact sequence in $\\textit{Ab}(\\textit{LC}_{Zar}/X)$", "then $0 \\to \\mathcal{F} \\to \\pi_{X, *}\\mathcal{G} \\to \\mathcal{F}' \\to 0$", "is exact by Lemma \\ref{lemma-collect-true-things-Zar} part (2).", "Hence $\\mathcal{G} = \\pi_X^{-1}\\pi_{X, *}\\mathcal{G}$ is in", "$\\mathcal{A}'_X$ which checks the final condition.", "\\medskip\\noindent", "Property (\\ref{item-base-change-P}) holds by Lemma \\ref{lemma-LC-basic}", "and the fact that the base change of a proper map is a proper map, see", "Topology, Theorem \\ref{topology-theorem-characterize-proper}.", "\\medskip\\noindent", "Property (\\ref{item-restriction-A}) follows from the commutative", "diagram (5) in Lemma \\ref{lemma-collect-true-things-Zar}.", "\\medskip\\noindent", "Property (\\ref{item-A-sheaf}) is Lemma \\ref{lemma-describe-pullback-pi}.", "\\medskip\\noindent", "Property (\\ref{item-A-and-P}) is Lemma \\ref{lemma-collect-true-things-Zar}", "part (7)(b).", "\\medskip\\noindent", "Proof of (\\ref{item-refine-tau-by-P}). Suppose given a qc covering", "$\\{U_i \\to U\\}$. For $u \\in U$ pick $i_1, \\ldots, i_m \\in I$ and", "quasi-compact subsets $E_j \\subset U_{i_j}$ such that", "$\\bigcup f_{i_j}(E_j)$ is a neighbourhood of $u$.", "Observe that $Y = \\coprod_{j = 1, \\ldots, m} E_j \\to U$", "is proper as a continuous map from a quasi-compact space", "to a Hausdorff one (Topology, Lemma \\ref{topology-lemma-closed-map}).", "Choose an open neighbourhood $u \\in V$ contained in $\\bigcup f_{i_j}(E_j)$.", "Then $Y \\times_U V \\to V$ is a surjective proper morphism and", "hence a $qc$ covering by Lemma \\ref{lemma-proper-surjective-is-qc-covering}.", "Since we can do this for every $u \\in U$ we see that", "(\\ref{item-refine-tau-by-P}) holds." ], "refs": [ "homology-lemma-characterize-weak-serre-subcategory", "sites-cohomology-lemma-collect-true-things-Zar", "sites-cohomology-lemma-collect-true-things-Zar", "sites-cohomology-lemma-LC-basic", "topology-theorem-characterize-proper", "sites-cohomology-lemma-collect-true-things-Zar", "sites-cohomology-lemma-describe-pullback-pi", "sites-cohomology-lemma-collect-true-things-Zar", "topology-lemma-closed-map", "sites-cohomology-lemma-proper-surjective-is-qc-covering" ], "ref_ids": [ 12046, 4309, 4309, 4305, 8189, 4309, 4308, 4309, 8274, 4307 ] } ], "ref_ids": [] }, { "id": 4312, "type": "theorem", "label": "sites-cohomology-lemma-V-C-all-n", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-V-C-all-n", "contents": [ "With notation as above.", "\\begin{enumerate}", "\\item For $X \\in \\Ob(\\textit{LC}_{qc})$ and an abelian sheaf $\\mathcal{F}$", "on $X$ we have $\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$", "and $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$.", "\\item For a proper morphism $f : X \\to Y$ in $\\textit{LC}_{qc}$", "and abelian sheaf $\\mathcal{F}$ on $X$ we have", "$a_Y^{-1}(R^if_*\\mathcal{F}) = R^if_{qc, *}(a_X^{-1}\\mathcal{F})$", "for all $i$.", "\\item For $X \\in \\Ob(\\textit{LC}_{qc})$ and $K$ in $D^+(X)$ the map", "$\\pi_X^{-1}K \\to R\\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism.", "\\item For a proper morphism $f : X \\to Y$ in $\\textit{LC}_{qc}$", "and $K$ in $D^+(X)$ we have $a_Y^{-1}(Rf_*K) = Rf_{qc, *}(a_X^{-1}K)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-compare-qc-zar} the lemmas in", "Section \\ref{section-compare-general} all apply to our current setting.", "To translate the results", "observe that the category $\\mathcal{A}_X$ of Lemma \\ref{lemma-A}", "is the essential image of", "$a_X^{-1} : \\textit{Ab}(X) \\to \\textit{Ab}(\\textit{LC}_{qc}/X)$.", "\\medskip\\noindent", "Part (1) is equivalent to $(V_n)$ for all $n$ which holds by", "Lemma \\ref{lemma-V-C-all-n-general}.", "\\medskip\\noindent", "Part (2) follows by applying $\\epsilon_Y^{-1}$ to the conclusion of", "Lemma \\ref{lemma-V-implies-C-general}.", "\\medskip\\noindent", "Part (3) follows from Lemma \\ref{lemma-V-C-all-n-general} part (1)", "because $\\pi_X^{-1}K$ is in $D^+_{\\mathcal{A}'_X}(\\textit{LC}_{Zar}/X)$", "and $a_X^{-1} = \\epsilon_X^{-1} \\circ a_X^{-1}$.", "\\medskip\\noindent", "Part (4) follows from Lemma \\ref{lemma-V-C-all-n-general} part (2)", "for the same reason." ], "refs": [ "sites-cohomology-lemma-compare-qc-zar", "sites-cohomology-lemma-A", "sites-cohomology-lemma-V-C-all-n-general", "sites-cohomology-lemma-V-implies-C-general", "sites-cohomology-lemma-V-C-all-n-general", "sites-cohomology-lemma-V-C-all-n-general" ], "ref_ids": [ 4311, 4296, 4302, 4297, 4302, 4302 ] } ], "ref_ids": [] }, { "id": 4313, "type": "theorem", "label": "sites-cohomology-lemma-cohomological-descent-LC", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cohomological-descent-LC", "contents": [ "Let $X$ be an object of $\\textit{LC}_{qc}$. For $K \\in D^+(X)$ the map", "$$", "K \\longrightarrow Ra_{X, *}a_X^{-1}K", "$$", "is an isomorphism with $a_X : \\Sh(\\textit{LC}_{qc}/X) \\to \\Sh(X)$ as above." ], "refs": [], "proofs": [ { "contents": [ "We first reduce the statement to the case where", "$K$ is given by a single abelian sheaf. Namely, represent $K$", "by a bounded below complex $\\mathcal{F}^\\bullet$. By the case of a", "sheaf we see that", "$\\mathcal{F}^n = a_{X, *} a_X^{-1} \\mathcal{F}^n$", "and that the sheaves $R^qa_{X, *}a_X^{-1}\\mathcal{F}^n$", "are zero for $q > 0$. By Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "applied to $a_X^{-1}\\mathcal{F}^\\bullet$", "and the functor $a_{X, *}$ we conclude. From now on assume $K = \\mathcal{F}$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-describe-pullback-pi} we have", "$a_{X, *}a_X^{-1}\\mathcal{F} = \\mathcal{F}$. Thus it suffices to show that", "$R^qa_{X, *}a_X^{-1}\\mathcal{F} = 0$ for $q > 0$.", "For this we can use $a_X = \\epsilon_X \\circ \\pi_X$ and", "the Leray spectral sequence Lemma \\ref{lemma-relative-Leray}.", "By Lemma \\ref{lemma-V-C-all-n}", "we have $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$", "and $\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$.", "By Lemma \\ref{lemma-collect-true-things-Zar} we have", "$R^j\\pi_{X, *}(\\pi_X^{-1}\\mathcal{F}) = 0$ for $j > 0$.", "This concludes the proof." ], "refs": [ "derived-lemma-leray-acyclicity", "sites-cohomology-lemma-describe-pullback-pi", "sites-cohomology-lemma-relative-Leray", "sites-cohomology-lemma-V-C-all-n", "sites-cohomology-lemma-collect-true-things-Zar" ], "ref_ids": [ 1844, 4308, 4222, 4312, 4309 ] } ], "ref_ids": [] }, { "id": 4314, "type": "theorem", "label": "sites-cohomology-lemma-compare-cohomology-LC", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-compare-cohomology-LC", "contents": [ "With $X \\in \\Ob(\\textit{LC}_{qc})$ and", "$a_X : \\Sh(\\textit{LC}_{qc}/X) \\to \\Sh(X)$ as above:", "\\begin{enumerate}", "\\item for an abelian sheaf $\\mathcal{F}$ on $X$ we have", "$H^n(X, \\mathcal{F}) = H^n_{qc}(X, a_X^{-1}\\mathcal{F})$,", "\\item for $K \\in D^+(X)$ we have $H^n(X, K) = H^n_{qc}(X, a_X^{-1}K)$.", "\\end{enumerate}", "For example, if $A$ is an abelian group, then we have", "$H^n(X, \\underline{A}) = H^n_{qc}(X, \\underline{A})$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-cohomological-descent-LC}", "by Remark \\ref{remark-before-Leray}." ], "refs": [ "sites-cohomology-lemma-cohomological-descent-LC", "sites-cohomology-remark-before-Leray" ], "ref_ids": [ 4313, 4423 ] } ], "ref_ids": [] }, { "id": 4315, "type": "theorem", "label": "sites-cohomology-lemma-cup-compatible-with-naive", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cup-compatible-with-naive", "contents": [ "In the situation above the following diagram commutes", "$$", "\\xymatrix{", "f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "f_*\\mathcal{M}^\\bullet \\ar[r] \\ar[d]", "&", "Rf_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "Rf_*\\mathcal{M}^\\bullet \\ar[d]^{\\text{Remark \\ref{remark-cup-product}}} \\\\", "\\text{Tot}(", "f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}", "f_*\\mathcal{M}^\\bullet) \\ar[d]_{\\text{naive cup product}} &", "Rf_*(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L}", "\\mathcal{M}^\\bullet) \\ar[d] \\\\", "f_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{C}}", "\\mathcal{M}^\\bullet) \\ar[r] &", "Rf_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{C}}", "\\mathcal{M}^\\bullet)", "}", "$$" ], "refs": [ "sites-cohomology-remark-cup-product" ], "proofs": [ { "contents": [ "By the construction in Remark \\ref{remark-cup-product} we see that", "going around the diagram clockwise the map", "$$", "f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "f_*\\mathcal{M}^\\bullet ", "\\longrightarrow", "Rf_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{C}}", "\\mathcal{M}^\\bullet)", "$$", "is adjoint to the map", "\\begin{align*}", "Lf^*(f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "f_*\\mathcal{M}^\\bullet)", "& =", "Lf^*f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "Lf^*f_*\\mathcal{M}^\\bullet \\\\", "& \\to", "Lf^*Rf_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "Lf^*Rf_*\\mathcal{M}^\\bullet \\\\", "& \\to", "\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "\\mathcal{M}^\\bullet \\\\", "& \\to", "\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{C}}", "\\mathcal{M}^\\bullet)", "\\end{align*}", "By Lemma \\ref{lemma-adjoints-push-pull-compatibility} this is also equal to", "\\begin{align*}", "Lf^*(f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "f_*\\mathcal{M}^\\bullet)", "& =", "Lf^*f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "Lf^*f_*\\mathcal{M}^\\bullet \\\\", "& \\to", "f^*f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "f^*f_*\\mathcal{M}^\\bullet \\\\", "& \\to", "\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "\\mathcal{M}^\\bullet \\\\", "& \\to", "\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{C}}", "\\mathcal{M}^\\bullet)", "\\end{align*}", "Going around anti-clockwise we obtain the map adjoint to the map", "\\begin{align*}", "Lf^*(f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "f_*\\mathcal{M}^\\bullet)", "& \\to", "Lf^*\\text{Tot}(", "f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}", "f_*\\mathcal{M}^\\bullet) \\\\", "& \\to", "Lf^*f_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{C}}", "\\mathcal{M}^\\bullet) \\\\", "& \\to", "Lf^*Rf_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{C}}", "\\mathcal{M}^\\bullet) \\\\", "& \\to", "\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{C}}", "\\mathcal{M}^\\bullet)", "\\end{align*}", "By Lemma \\ref{lemma-adjoints-push-pull-compatibility} this is also equal to", "\\begin{align*}", "Lf^*(f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "f_*\\mathcal{M}^\\bullet)", "& \\to", "Lf^*\\text{Tot}(", "f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}", "f_*\\mathcal{M}^\\bullet) \\\\", "& \\to", "Lf^*f_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{C}}", "\\mathcal{M}^\\bullet) \\\\", "& \\to", "f^*f_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{C}}", "\\mathcal{M}^\\bullet) \\\\", "& \\to", "\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{C}}", "\\mathcal{M}^\\bullet)", "\\end{align*}", "Now the proof is finished by a contemplation of the diagram", "$$", "\\xymatrix{", "Lf^*(f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}", "f_*\\mathcal{M}^\\bullet) \\ar[d] \\ar[rr] & &", "Lf^*f_*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L}", "Lf^*f_*\\mathcal{M}^\\bullet \\ar[d] \\\\", "Lf^*\\text{Tot}(", "f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}", "f_*\\mathcal{M}^\\bullet) \\ar[d]_{naive} \\ar[r] &", "f^*\\text{Tot}(", "f_*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{D}}", "f_*\\mathcal{M}^\\bullet) \\ar[ldd]^{naive} \\ar[dd] &", "f^*f_*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L}", "f^*f_*\\mathcal{M}^\\bullet \\ar[dd] \\ar[ldd] \\\\", "Lf^*f_*\\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{C}}", "\\mathcal{M}^\\bullet) \\ar[d] \\\\", "f^*f_*\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}", "\\mathcal{M}^\\bullet) \\ar[rd] &", "\\text{Tot}(f^*f_*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}", "f^*f_*\\mathcal{M}^\\bullet) \\ar[d] &", "\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L}", "\\mathcal{M}^\\bullet \\ar[ld] \\\\", "& \\text{Tot}(\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_\\mathcal{C}}", "\\mathcal{M}^\\bullet)", "}", "$$", "All of the polygons in this diagram commute. The top one commutes", "by Lemma \\ref{lemma-tensor-pull-compatibility}.", "The square with the two naive cup products commutes because", "$Lf^* \\to f^*$ is functorial in the complex of modules.", "Similarly with the square involving the two maps", "$\\mathcal{A}^\\bullet \\otimes^\\mathbf{L} \\mathcal{B}^\\bullet \\to", "\\text{Tot}(\\mathcal{A}^\\bullet \\otimes \\mathcal{B}^\\bullet)$.", "Finally, the commutativity of the remaining square", "is true on the level of complexes and may be viewed as the", "definiton of the naive cup product (by the adjointness", "of $f^*$ and $f_*$). The proof is finished because", "going around the diagram on the outside are the two maps", "given above." ], "refs": [ "sites-cohomology-remark-cup-product", "sites-cohomology-lemma-adjoints-push-pull-compatibility", "sites-cohomology-lemma-adjoints-push-pull-compatibility", "sites-cohomology-lemma-tensor-pull-compatibility" ], "ref_ids": [ 4427, 4251, 4251, 4248 ] } ], "ref_ids": [ 4427 ] }, { "id": 4316, "type": "theorem", "label": "sites-cohomology-lemma-cup-product-associative", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cup-product-associative", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$", "be a morphism of ringed topoi. The relative cup product of", "Remark \\ref{remark-cup-product} is associative in the sense that", "the diagram", "$$", "\\xymatrix{", "Rf_*K \\otimes_{\\mathcal{O}'}^\\mathbf{L}", "Rf_*L \\otimes_{\\mathcal{O}'}^\\mathbf{L}", "Rf_*M \\ar[r] \\ar[d] &", "Rf_*(K \\otimes_\\mathcal{O}^\\mathbf{L} L)", "\\otimes_{\\mathcal{O}'}^\\mathbf{L} Rf_*M \\ar[d] \\\\", "Rf_*K \\otimes_{\\mathcal{O}'}^\\mathbf{L}", "Rf_*(L \\otimes_\\mathcal{O}^\\mathbf{L} M) \\ar[r] &", "Rf_*(K \\otimes_\\mathcal{O}^\\mathbf{L} ", "L \\otimes_\\mathcal{O}^\\mathbf{L} M)", "}", "$$", "is commutative in $D(\\mathcal{O}')$ for all $K, L, M$ in $D(\\mathcal{O})$." ], "refs": [ "sites-cohomology-remark-cup-product" ], "proofs": [ { "contents": [ "Going around either side we obtain the map adjoint to the obvious map", "\\begin{align*}", "Lf^*(Rf_*K \\otimes_{\\mathcal{O}'}^\\mathbf{L}", "Rf_*L \\otimes_{\\mathcal{O}'}^\\mathbf{L}", "Rf_*M) & =", "Lf^*(Rf_*K) \\otimes_\\mathcal{O}^\\mathbf{L}", "Lf^*(Rf_*L) \\otimes_\\mathcal{O}^\\mathbf{L}", "Lf^*(Rf_*M) \\\\", "& \\to", "K \\otimes_\\mathcal{O}^\\mathbf{L} ", "L \\otimes_\\mathcal{O}^\\mathbf{L} M", "\\end{align*}", "in $D(\\mathcal{O})$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 4427 ] }, { "id": 4317, "type": "theorem", "label": "sites-cohomology-lemma-cup-product-commutative", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cup-product-commutative", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$", "be a morphism of ringed topoi. The relative cup product of", "Remark \\ref{remark-cup-product} is commutative in the sense that", "the diagram", "$$", "\\xymatrix{", "Rf_*K \\otimes_{\\mathcal{O}'}^\\mathbf{L} Rf_*L \\ar[r] \\ar[d]_\\psi &", "Rf_*(K \\otimes_\\mathcal{O}^\\mathbf{L} L) \\ar[d]^{Rf_*\\psi} \\\\", "Rf_*L \\otimes_{\\mathcal{O}'}^\\mathbf{L} Rf_*K \\ar[r] &", "Rf_*(L \\otimes_\\mathcal{O}^\\mathbf{L} K)", "}", "$$", "is commutative in $D(\\mathcal{O}')$ for all $K, L$ in $D(\\mathcal{O})$.", "Here $\\psi$ is the commutativity constraint on the derived category", "(Lemma \\ref{lemma-symmetric-monoidal-derived})." ], "refs": [ "sites-cohomology-remark-cup-product", "sites-cohomology-lemma-symmetric-monoidal-derived" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 4427, 4391 ] }, { "id": 4318, "type": "theorem", "label": "sites-cohomology-lemma-compose-cup-product", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-compose-cup-product", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$", "and $f' : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to", "(\\Sh(\\mathcal{C}''), \\mathcal{O}'')$", "be morphisms of ringed topoi. The relative cup product of", "Remark \\ref{remark-cup-product} is compatible with compositions", "in the sense that the diagram", "$$", "\\xymatrix{", "R(f' \\circ f)_*K \\otimes_{\\mathcal{O}''}^\\mathbf{L} R(f' \\circ f)_*L", "\\ar@{=}[rr] \\ar[d] & &", "Rf'_*Rf_*K \\otimes_{\\mathcal{O}''}^\\mathbf{L} Rf'_*Rf_*L \\ar[d] \\\\", "R(f' \\circ f)_*(K \\otimes_\\mathcal{O}^\\mathbf{L} L) \\ar@{=}[r] &", "Rf'_*Rf_*(K \\otimes_\\mathcal{O}^\\mathbf{L} L) &", "Rf'_*(Rf_*K \\otimes_{\\mathcal{O}'}^\\mathbf{L} Rf_*L) \\ar[l]", "}", "$$", "is commutative in $D(\\mathcal{O}'')$ for all $K, L$ in $D(\\mathcal{O})$." ], "refs": [ "sites-cohomology-remark-cup-product" ], "proofs": [ { "contents": [ "This is true because going around the diagram either way we obtain the map", "adjoint to the map", "\\begin{align*}", "& L(f' \\circ f)^*\\left(R(f' \\circ f)_*K", "\\otimes_{\\mathcal{O}''}^\\mathbf{L}", "R(f' \\circ f)_*L\\right) \\\\", "& =", "L(f' \\circ f)^*R(f' \\circ f)_*K", "\\otimes_\\mathcal{O}^\\mathbf{L}", "L(f' \\circ f)^*R(f' \\circ f)_*L) \\\\", "& \\to", "K \\otimes_\\mathcal{O}^\\mathbf{L} L", "\\end{align*}", "in $D(\\mathcal{O})$. To see this one uses that the composition", "of the counits like so", "$$", "L(f' \\circ f)^*R(f' \\circ f)_* =", "Lf^* L(f')^* Rf'_* Rf_* \\to", "Lf^* Rf_* \\to \\text{id}", "$$", "is the counit for $L(f' \\circ f)^*$ and $R(f' \\circ f)_*$. See", "Categories, Lemma \\ref{categories-lemma-compose-counits}." ], "refs": [ "categories-lemma-compose-counits" ], "ref_ids": [ 12252 ] } ], "ref_ids": [ 4427 ] }, { "id": 4319, "type": "theorem", "label": "sites-cohomology-lemma-compose", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-compose", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Given complexes $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$", "of $\\mathcal{O}$-modules there is an isomorphism", "$$", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet,", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet))", "=", "\\SheafHom^\\bullet(\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O}", "\\mathcal{L}^\\bullet), \\mathcal{M}^\\bullet)", "$$", "of complexes of $\\mathcal{O}$-modules functorial in", "$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is proved in exactly the same way as", "More on Algebra, Lemma \\ref{more-algebra-lemma-compose}." ], "refs": [ "more-algebra-lemma-compose" ], "ref_ids": [ 10198 ] } ], "ref_ids": [] }, { "id": 4320, "type": "theorem", "label": "sites-cohomology-lemma-composition", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-composition", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given complexes", "$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$", "of $\\mathcal{O}$-modules there is a canonical morphism", "$$", "\\text{Tot}\\left(", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet)", "\\otimes_\\mathcal{O}", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet)", "\\right)", "\\longrightarrow", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{M}^\\bullet)", "$$", "of complexes of $\\mathcal{O}$-modules." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is proved in exactly the same way as", "More on Algebra, Lemma \\ref{more-algebra-lemma-composition}." ], "refs": [ "more-algebra-lemma-composition" ], "ref_ids": [ 10199 ] } ], "ref_ids": [] }, { "id": 4321, "type": "theorem", "label": "sites-cohomology-lemma-diagonal-better", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-diagonal-better", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given complexes", "$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$", "of $\\mathcal{O}$-modules there is a canonical morphism", "$$", "\\text{Tot}\\left(", "\\mathcal{K}^\\bullet \\otimes_\\mathcal{O}", "\\SheafHom^\\bullet(\\mathcal{M}^\\bullet, \\mathcal{L}^\\bullet)", "\\right)", "\\longrightarrow", "\\SheafHom^\\bullet(\\mathcal{M}^\\bullet,", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet))", "$$", "of complexes of $\\mathcal{O}$-modules functorial in all three complexes." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is proved in exactly the same way as", "More on Algebra, Lemma \\ref{more-algebra-lemma-diagonal-better}." ], "refs": [ "more-algebra-lemma-diagonal-better" ], "ref_ids": [ 10200 ] } ], "ref_ids": [] }, { "id": 4322, "type": "theorem", "label": "sites-cohomology-lemma-diagonal", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-diagonal", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given complexes", "$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$", "of $\\mathcal{O}$-modules there is a canonical morphism", "$$", "\\mathcal{K}^\\bullet", "\\longrightarrow", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet,", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet))", "$$", "of complexes of $\\mathcal{O}$-modules functorial in both complexes." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is proved in exactly the same way as", "More on Algebra, Lemma \\ref{more-algebra-lemma-diagonal}." ], "refs": [ "more-algebra-lemma-diagonal" ], "ref_ids": [ 10201 ] } ], "ref_ids": [] }, { "id": 4323, "type": "theorem", "label": "sites-cohomology-lemma-evaluate-and-more", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-evaluate-and-more", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given complexes", "$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$", "of $\\mathcal{O}$-modules there is a canonical morphism", "$$", "\\text{Tot}(\\SheafHom^\\bullet(\\mathcal{L}^\\bullet,", "\\mathcal{M}^\\bullet) \\otimes_\\mathcal{O} \\mathcal{K}^\\bullet)", "\\longrightarrow", "\\SheafHom^\\bullet(\\SheafHom^\\bullet(\\mathcal{K}^\\bullet,", "\\mathcal{L}^\\bullet), \\mathcal{M}^\\bullet)", "$$", "of complexes of $\\mathcal{O}$-modules functorial in all three complexes." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is proved in exactly the same way as", "More on Algebra, Lemma \\ref{more-algebra-lemma-evaluate-and-more}." ], "refs": [ "more-algebra-lemma-evaluate-and-more" ], "ref_ids": [ 10202 ] } ], "ref_ids": [] }, { "id": 4324, "type": "theorem", "label": "sites-cohomology-lemma-RHom-into-K-injective", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-RHom-into-K-injective", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{I}^\\bullet$", "be a K-injective complex of $\\mathcal{O}$-modules. Let", "$\\mathcal{L}^\\bullet$ be a complex of $\\mathcal{O}$-modules.", "Then", "$$", "H^0(\\Gamma(U, \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet))) =", "\\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U)", "$$", "for all $U \\in \\Ob(\\mathcal{C})$. Similarly,", "$H^0(\\Gamma(\\mathcal{C},", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet))) =", "\\Hom_{D(\\mathcal{O}_U)}(L, M)$." ], "refs": [], "proofs": [ { "contents": [ "We have", "\\begin{align*}", "H^0(\\Gamma(U, \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet)))", "& =", "\\Hom_{K(\\mathcal{O}_U)}(L|_U, M|_U) \\\\", "& =", "\\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U)", "\\end{align*}", "The first equality is (\\ref{equation-cohomology-hom-complex}).", "The second equality is true because $\\mathcal{I}^\\bullet|_U$", "is K-injective by Lemma \\ref{lemma-restrict-K-injective-to-open}.", "The proof of the last equation is similar except that it uses", "(\\ref{equation-global-cohomology-hom-complex})." ], "refs": [ "sites-cohomology-lemma-restrict-K-injective-to-open" ], "ref_ids": [ 4253 ] } ], "ref_ids": [] }, { "id": 4325, "type": "theorem", "label": "sites-cohomology-lemma-RHom-well-defined", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-RHom-well-defined", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$(\\mathcal{I}')^\\bullet \\to \\mathcal{I}^\\bullet$", "be a quasi-isomorphism of K-injective complexes of $\\mathcal{O}$-modules.", "Let $(\\mathcal{L}')^\\bullet \\to \\mathcal{L}^\\bullet$", "be a quasi-isomorphism of complexes of $\\mathcal{O}$-modules.", "Then", "$$", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, (\\mathcal{I}')^\\bullet)", "\\longrightarrow", "\\SheafHom^\\bullet((\\mathcal{L}')^\\bullet, \\mathcal{I}^\\bullet)", "$$", "is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be the object of $D(\\mathcal{O})$ represented by", "$\\mathcal{I}^\\bullet$ and $(\\mathcal{I}')^\\bullet$.", "Let $L$ be the object of $D(\\mathcal{O})$ represented by", "$\\mathcal{L}^\\bullet$ and $(\\mathcal{L}')^\\bullet$.", "By Lemma \\ref{lemma-RHom-into-K-injective}", "we see that the sheaves", "$$", "H^0(\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, (\\mathcal{I}')^\\bullet))", "\\quad\\text{and}\\quad", "H^0(\\SheafHom^\\bullet((\\mathcal{L}')^\\bullet, \\mathcal{I}^\\bullet))", "$$", "are both equal to the sheaf associated to the presheaf", "$$", "U \\longmapsto \\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U)", "$$", "Thus the map is a quasi-isomorphism." ], "refs": [ "sites-cohomology-lemma-RHom-into-K-injective" ], "ref_ids": [ 4324 ] } ], "ref_ids": [] }, { "id": 4326, "type": "theorem", "label": "sites-cohomology-lemma-RHom-from-K-flat-into-K-injective", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-RHom-from-K-flat-into-K-injective", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{I}^\\bullet$", "be a K-injective complex of $\\mathcal{O}$-modules. Let", "$\\mathcal{L}^\\bullet$ be a K-flat complex of $\\mathcal{O}$-modules.", "Then $\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet)$", "is a K-injective complex of $\\mathcal{O}$-modules." ], "refs": [], "proofs": [ { "contents": [ "Namely, if $\\mathcal{K}^\\bullet$ is an acyclic complex of", "$\\mathcal{O}$-modules, then", "\\begin{align*}", "\\Hom_{K(\\mathcal{O})}(\\mathcal{K}^\\bullet,", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet))", "& =", "H^0(\\Gamma(\\mathcal{C},", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet,", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet)))) \\\\", "& =", "H^0(\\Gamma(\\mathcal{C}, \\SheafHom^\\bullet(\\text{Tot}(", "\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet),", "\\mathcal{I}^\\bullet))) \\\\", "& =", "\\Hom_{K(\\mathcal{O})}(", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet),", "\\mathcal{I}^\\bullet) \\\\", "& =", "0", "\\end{align*}", "The first equality by (\\ref{equation-global-cohomology-hom-complex}).", "The second equality by Lemma \\ref{lemma-compose}.", "The third equality by (\\ref{equation-global-cohomology-hom-complex}).", "The final equality because", "$\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet)$", "is acyclic because $\\mathcal{L}^\\bullet$ is K-flat", "(Definition \\ref{definition-K-flat}) and because $\\mathcal{I}^\\bullet$", "is K-injective." ], "refs": [ "sites-cohomology-lemma-compose", "sites-cohomology-definition-K-flat" ], "ref_ids": [ 4319, 4414 ] } ], "ref_ids": [] }, { "id": 4327, "type": "theorem", "label": "sites-cohomology-lemma-section-RHom-over-U", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-section-RHom-over-U", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L$ be objects", "of $D(\\mathcal{O})$. For every object $U$ of $\\mathcal{C}$ we have", "$$", "H^0(U, R\\SheafHom(L, M)) =", "\\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U)", "$$", "and we have $H^0(\\mathcal{C}, R\\SheafHom(L, M)) =", "\\Hom_{D(\\mathcal{O})}(L, M)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $\\mathcal{I}^\\bullet$ of", "$\\mathcal{O}$-modules representing $M$ and a K-flat complex", "$\\mathcal{L}^\\bullet$ representing $L$. Then", "$\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet)$", "is K-injective by Lemma \\ref{lemma-RHom-from-K-flat-into-K-injective}.", "Hence we can compute cohomology over $U$ by simply taking sections over $U$", "and the result follows from Lemma \\ref{lemma-RHom-into-K-injective}." ], "refs": [ "sites-cohomology-lemma-RHom-from-K-flat-into-K-injective", "sites-cohomology-lemma-RHom-into-K-injective" ], "ref_ids": [ 4326, 4324 ] } ], "ref_ids": [] }, { "id": 4328, "type": "theorem", "label": "sites-cohomology-lemma-internal-hom", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-internal-hom", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L, M$ be objects", "of $D(\\mathcal{O})$. With the construction as described above", "there is a canonical isomorphism", "$$", "R\\SheafHom(K, R\\SheafHom(L, M)) =", "R\\SheafHom(K \\otimes_\\mathcal{O}^\\mathbf{L} L, M)", "$$", "in $D(\\mathcal{O})$ functorial in $K, L, M$", "which recovers (\\ref{equation-internal-hom}) on taking $H^0(\\mathcal{C}, -)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $\\mathcal{I}^\\bullet$ representing", "$M$ and a K-flat complex of $\\mathcal{O}$-modules $\\mathcal{L}^\\bullet$", "representing $L$.", "For any complex of $\\mathcal{O}$-modules $\\mathcal{K}^\\bullet$", "we have", "$$", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet,", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet))", "=", "\\SheafHom^\\bullet(", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet),", "\\mathcal{I}^\\bullet)", "$$", "by Lemma \\ref{lemma-compose}.", "Note that the left hand side represents", "$R\\SheafHom(K, R\\SheafHom(L, M))$ (use", "Lemma \\ref{lemma-RHom-from-K-flat-into-K-injective})", "and that the right hand side represents", "$R\\SheafHom(K \\otimes_\\mathcal{O}^\\mathbf{L} L, M)$.", "This proves the displayed formula of the lemma.", "Taking global sections and using Lemma \\ref{lemma-section-RHom-over-U}", "we obtain (\\ref{equation-internal-hom})." ], "refs": [ "sites-cohomology-lemma-compose", "sites-cohomology-lemma-RHom-from-K-flat-into-K-injective" ], "ref_ids": [ 4319, 4326 ] } ], "ref_ids": [] }, { "id": 4329, "type": "theorem", "label": "sites-cohomology-lemma-restriction-RHom-to-U", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-restriction-RHom-to-U", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L$ be objects", "of $D(\\mathcal{O})$. The construction of $R\\SheafHom(K, L)$", "commutes with restrictions, i.e.,", "for every object $U$ of $\\mathcal{C}$ we have", "$R\\SheafHom(K|_U, L|_U) = R\\SheafHom(K, L)|_U$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the construction and", "Lemma \\ref{lemma-restrict-K-injective-to-open}." ], "refs": [ "sites-cohomology-lemma-restrict-K-injective-to-open" ], "ref_ids": [ 4253 ] } ], "ref_ids": [] }, { "id": 4330, "type": "theorem", "label": "sites-cohomology-lemma-RHom-triangulated", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-RHom-triangulated", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. The bifunctor", "$R\\SheafHom(- , -)$ transforms distinguished triangles into", "distinguished triangles in both variables." ], "refs": [], "proofs": [ { "contents": [ "This follows from the observation that the assignment", "$$", "(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet) \\longmapsto", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet)", "$$", "transforms a termwise split short exact sequences of complexes in either", "variable into a termwise split short exact sequence. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4331, "type": "theorem", "label": "sites-cohomology-lemma-internal-hom-evaluate", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-internal-hom-evaluate", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L, M$ be objects of", "$D(\\mathcal{O})$. There is a canonical morphism", "$$", "R\\SheafHom(L, M) \\otimes_\\mathcal{O}^\\mathbf{L} K", "\\longrightarrow", "R\\SheafHom(R\\SheafHom(K, L), M)", "$$", "in $D(\\mathcal{O})$ functorial in $K, L, M$." ], "refs": [], "proofs": [ { "contents": [ "Choose", "a K-injective complex $\\mathcal{I}^\\bullet$ representing $M$,", "a K-injective complex $\\mathcal{J}^\\bullet$ representing $L$, and", "a K-flat complex $\\mathcal{K}^\\bullet$ representing $K$.", "The map is defined using the map", "$$", "\\text{Tot}(\\SheafHom^\\bullet(\\mathcal{J}^\\bullet,", "\\mathcal{I}^\\bullet) \\otimes_\\mathcal{O} \\mathcal{K}^\\bullet)", "\\longrightarrow", "\\SheafHom^\\bullet(\\SheafHom^\\bullet(\\mathcal{K}^\\bullet,", "\\mathcal{J}^\\bullet), \\mathcal{I}^\\bullet)", "$$", "of Lemma \\ref{lemma-evaluate-and-more}. By our particular", "choice of complexes the left hand side represents", "$R\\SheafHom(L, M) \\otimes_\\mathcal{O}^\\mathbf{L} K$", "and the right hand side represents", "$R\\SheafHom(R\\SheafHom(K, L), M)$. We omit the proof that", "this is functorial in all three objects of $D(\\mathcal{O})$." ], "refs": [ "sites-cohomology-lemma-evaluate-and-more" ], "ref_ids": [ 4323 ] } ], "ref_ids": [] }, { "id": 4332, "type": "theorem", "label": "sites-cohomology-lemma-internal-hom-composition", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-internal-hom-composition", "contents": [ "\\begin{slogan}", "Composition on RSheafHom.", "\\end{slogan}", "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given $K, L, M$ in", "$D(\\mathcal{O})$ there is a canonical morphism", "$$", "R\\SheafHom(L, M) \\otimes_\\mathcal{O}^\\mathbf{L} R\\SheafHom(K, L)", "\\longrightarrow R\\SheafHom(K, M)", "$$", "in $D(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $\\mathcal{I}^\\bullet$ representing $M$,", "a K-injective complex $\\mathcal{J}^\\bullet$ representing $L$, and", "any complex of $\\mathcal{O}$-modules $\\mathcal{K}^\\bullet$ representing $K$.", "By Lemma \\ref{lemma-composition} there is a map of complexes", "$$", "\\text{Tot}\\left(", "\\SheafHom^\\bullet(\\mathcal{J}^\\bullet, \\mathcal{I}^\\bullet)", "\\otimes_\\mathcal{O}", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{J}^\\bullet)", "\\right)", "\\longrightarrow", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{I}^\\bullet)", "$$", "The complexes of $\\mathcal{O}$-modules", "$\\SheafHom^\\bullet(\\mathcal{J}^\\bullet, \\mathcal{I}^\\bullet)$,", "$\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{J}^\\bullet)$, and", "$\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{I}^\\bullet)$", "represent $R\\SheafHom(L, M)$, $R\\SheafHom(K, L)$, and $R\\SheafHom(K, M)$.", "If we choose a K-flat complex $\\mathcal{H}^\\bullet$ and a quasi-isomorphism", "$\\mathcal{H}^\\bullet \\to", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{J}^\\bullet)$,", "then there is a map", "$$", "\\text{Tot}\\left(", "\\SheafHom^\\bullet(\\mathcal{J}^\\bullet, \\mathcal{I}^\\bullet)", "\\otimes_\\mathcal{O} \\mathcal{H}^\\bullet", "\\right)", "\\longrightarrow", "\\text{Tot}\\left(", "\\SheafHom^\\bullet(\\mathcal{J}^\\bullet, \\mathcal{I}^\\bullet)", "\\otimes_\\mathcal{O}", "\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{J}^\\bullet)", "\\right)", "$$", "whose source represents", "$R\\SheafHom(L, M) \\otimes_\\mathcal{O}^\\mathbf{L} R\\SheafHom(K, L)$.", "Composing the two displayed arrows gives the desired map. We omit the", "proof that the construction is functorial." ], "refs": [ "sites-cohomology-lemma-composition" ], "ref_ids": [ 4320 ] } ], "ref_ids": [] }, { "id": 4333, "type": "theorem", "label": "sites-cohomology-lemma-internal-hom-diagonal-better", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-internal-hom-diagonal-better", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given $K, L, M$", "in $D(\\mathcal{O})$ there is a canonical morphism", "$$", "K \\otimes_\\mathcal{O}^\\mathbf{L} R\\SheafHom(M, L)", "\\longrightarrow", "R\\SheafHom(M, K \\otimes_\\mathcal{O}^\\mathbf{L} L)", "$$", "in $D(\\mathcal{O})$ functorial in $K, L, M$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-flat complex $\\mathcal{K}^\\bullet$ representing $K$,", "and a K-injective complex $\\mathcal{I}^\\bullet$ representing $L$, and", "choose any complex of $\\mathcal{O}$-modules $\\mathcal{M}^\\bullet$", "representing $M$. Choose a quasi-isomorphism", "$\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{I}^\\bullet)", "\\to \\mathcal{J}^\\bullet$", "where $\\mathcal{J}^\\bullet$ is K-injective. Then we use the map", "$$", "\\text{Tot}\\left(", "\\mathcal{K}^\\bullet \\otimes_\\mathcal{O}", "\\SheafHom^\\bullet(\\mathcal{M}^\\bullet, \\mathcal{I}^\\bullet)", "\\right)", "\\to", "\\SheafHom^\\bullet(\\mathcal{M}^\\bullet,", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{I}^\\bullet))", "\\to", "\\SheafHom^\\bullet(\\mathcal{M}^\\bullet, \\mathcal{J}^\\bullet)", "$$", "where the first map is the map from Lemma \\ref{lemma-diagonal-better}." ], "refs": [ "sites-cohomology-lemma-diagonal-better" ], "ref_ids": [ 4321 ] } ], "ref_ids": [] }, { "id": 4334, "type": "theorem", "label": "sites-cohomology-lemma-internal-hom-diagonal", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-internal-hom-diagonal", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Given $K, L$ in $D(\\mathcal{O})$ there is a canonical morphism", "$$", "K \\longrightarrow R\\SheafHom(L, K \\otimes_\\mathcal{O}^\\mathbf{L} L)", "$$", "in $D(\\mathcal{O})$ functorial in both $K$ and $L$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-flat complex $\\mathcal{K}^\\bullet$ representing $K$", "and any complex of $\\mathcal{O}$-modules $\\mathcal{L}^\\bullet$", "representing $L$. Choose a K-injective complex $\\mathcal{J}^\\bullet$", "and a quasi-isomorphism", "$\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet)", "\\to \\mathcal{J}^\\bullet$. Then we use", "$$", "\\mathcal{K}^\\bullet \\to", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet,", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet))", "\\to", "\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{J}^\\bullet)", "$$", "where the first map comes from Lemma \\ref{lemma-diagonal}." ], "refs": [ "sites-cohomology-lemma-diagonal" ], "ref_ids": [ 4322 ] } ], "ref_ids": [] }, { "id": 4335, "type": "theorem", "label": "sites-cohomology-lemma-dual", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-dual", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $L$ be an", "object of $D(\\mathcal{O})$. Set $L^\\vee = R\\SheafHom(L, \\mathcal{O})$.", "For $M$ in $D(\\mathcal{O})$ there is a canonical map", "\\begin{equation}", "\\label{equation-eval}", "M \\otimes^\\mathbf{L}_\\mathcal{O} L^\\vee \\longrightarrow R\\SheafHom(L, M)", "\\end{equation}", "which induces a canonical map", "$$", "H^0(\\mathcal{C}, M \\otimes_\\mathcal{O}^\\mathbf{L} L^\\vee)", "\\longrightarrow", "\\Hom_{D(\\mathcal{O})}(L, M)", "$$", "functorial in $M$ in $D(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "The map (\\ref{equation-eval}) is a special case of", "Lemma \\ref{lemma-internal-hom-composition}", "using the identification $M = R\\SheafHom(\\mathcal{O}, M)$." ], "refs": [ "sites-cohomology-lemma-internal-hom-composition" ], "ref_ids": [ 4332 ] } ], "ref_ids": [] }, { "id": 4336, "type": "theorem", "label": "sites-cohomology-lemma-pullback-injective-pre-limp", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-pullback-injective-pre-limp", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous", "functor of sites. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "be the corresponding morphism of topoi. Let $\\mathcal{O}_\\mathcal{D}$", "be a sheaf of rings and let $\\mathcal{I}$ be an injective", "$\\mathcal{O}_\\mathcal{D}$-module. Then", "$H^p(U, g^{-1}\\mathcal{I}) = 0$ for all $p > 0$ and $U \\in \\Ob(\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "The vanishing of the lemma follows from", "Lemma \\ref{lemma-cech-vanish-collection}", "if we can prove vanishing of all higher", "{\\v C}ech cohomology groups", "$\\check H^p(\\mathcal{U}, g^{-1}\\mathcal{I})$", "for any covering $\\mathcal{U} = \\{U_i \\to U\\}$ of $\\mathcal{C}$.", "Since $u$ is continuous, $u(\\mathcal{U}) = \\{u(U_i) \\to u(U)\\}$", "is a covering of $\\mathcal{D}$, and", "$u(U_{i_0} \\times_U \\ldots \\times_U U_{i_n}) =", "u(U_{i_0}) \\times_{u(U)} \\ldots \\times_{u(U)} u(U_{i_n})$.", "Thus we have", "$$", "\\check H^p(\\mathcal{U}, g^{-1}\\mathcal{I}) =", "\\check H^p(u(\\mathcal{U}), \\mathcal{I})", "$$", "because $g^{-1} = u^p$ by Sites, Lemma \\ref{sites-lemma-when-shriek}.", "Since $\\mathcal{I}$ is an injective", "$\\mathcal{O}_\\mathcal{D}$-module these {\\v C}ech cohomology groups vanish, see", "Lemma \\ref{lemma-injective-module-trivial-cech}." ], "refs": [ "sites-cohomology-lemma-cech-vanish-collection", "sites-lemma-when-shriek", "sites-cohomology-lemma-injective-module-trivial-cech" ], "ref_ids": [ 4205, 8545, 4209 ] } ], "ref_ids": [] }, { "id": 4337, "type": "theorem", "label": "sites-cohomology-lemma-existence-derived-lower-shriek", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-existence-derived-lower-shriek", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous", "functor of sites. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be the", "corresponding morphism of topoi. Let $\\mathcal{O}_\\mathcal{D}$", "be a sheaf of rings and set", "$\\mathcal{O}_\\mathcal{C} = g^{-1}\\mathcal{O}_\\mathcal{D}$.", "The functor $g_! : \\textit{Mod}(\\mathcal{O}_\\mathcal{C}) \\to", "\\textit{Mod}(\\mathcal{O}_\\mathcal{D})$", "(see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-lower-shriek-modules})", "has a left derived functor", "$$", "Lg_! : D(\\mathcal{O}_\\mathcal{C}) \\longrightarrow D(\\mathcal{O}_\\mathcal{D})", "$$", "which is left adjoint to $g^*$. Moreover, for $U \\in \\Ob(\\mathcal{C})$ we", "have", "$$", "Lg_!(j_{U!}\\mathcal{O}_U) =", "g_!j_{U!}\\mathcal{O}_U =", "j_{u(U)!} \\mathcal{O}_{u(U)}.", "$$", "where $j_{U!}$ and $j_{u(U)!}$ are extension by zero associated to the", "localization morphism", "$j_U : \\mathcal{C}/U \\to \\mathcal{C}$ and", "$j_{u(U)} : \\mathcal{D}/u(U) \\to \\mathcal{D}$." ], "refs": [ "sites-modules-lemma-lower-shriek-modules" ], "proofs": [ { "contents": [ "We are going to use", "Derived Categories, Proposition \\ref{derived-proposition-left-derived-exists}", "to construct $Lg_!$. To do this we have to verify assumptions", "(1), (2), (3), (4), and (5) of that proposition.", "First, since $g_!$ is a left adjoint", "we see that it is right exact and commutes with all colimits, so", "(5) holds. Conditions (3) and (4) hold because the category of modules", "on a ringed site is a Grothendieck abelian category.", "Let $\\mathcal{P} \\subset \\Ob(\\textit{Mod}(\\mathcal{O}_\\mathcal{C}))$", "be the collection of $\\mathcal{O}_\\mathcal{C}$-modules which are direct", "sums of modules of the form $j_{U!}\\mathcal{O}_U$. Note that", "$g_!j_{U!}\\mathcal{O}_U = j_{u(U)!} \\mathcal{O}_{u(U)}$, see proof of", "Modules on Sites, Lemma \\ref{sites-modules-lemma-lower-shriek-modules}.", "Every $\\mathcal{O}_\\mathcal{C}$-module is a quotient of an object of", "$\\mathcal{P}$, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-module-quotient-flat}.", "Thus (1) holds. Finally, we have to prove (2).", "Let $\\mathcal{K}^\\bullet$ be a bounded above acyclic complex of", "$\\mathcal{O}_\\mathcal{C}$-modules with $\\mathcal{K}^n \\in \\mathcal{P}$", "for all $n$. We have to show that $g_!\\mathcal{K}^\\bullet$ is", "exact. To do this it suffices to show, for every injective", "$\\mathcal{O}_\\mathcal{D}$-module $\\mathcal{I}$ that", "$$", "\\Hom_{D(\\mathcal{O}_\\mathcal{D})}(", "g_!\\mathcal{K}^\\bullet, \\mathcal{I}[n]) = 0", "$$", "for all $n \\in \\mathbf{Z}$. Since $\\mathcal{I}$ is injective we have", "\\begin{align*}", "\\Hom_{D(\\mathcal{O}_\\mathcal{D})}(", "g_!\\mathcal{K}^\\bullet, \\mathcal{I}[n])", "& =", "\\Hom_{K(\\mathcal{O}_\\mathcal{D})}(", "g_!\\mathcal{K}^\\bullet, \\mathcal{I}[n]) \\\\", "& =", "H^n(\\Hom_{\\mathcal{O}_\\mathcal{D}}(", "g_!\\mathcal{K}^\\bullet, \\mathcal{I})) \\\\", "& =", "H^n(\\Hom_{\\mathcal{O}_\\mathcal{C}}(", "\\mathcal{K}^\\bullet, g^{-1}\\mathcal{I}))", "\\end{align*}", "the last equality by the adjointness of $g_!$ and $g^{-1}$.", "\\medskip\\noindent", "The vanishing of this group would be clear if $g^{-1}\\mathcal{I}$", "were an injective $\\mathcal{O}_\\mathcal{C}$-module. But", "$g^{-1}\\mathcal{I}$ isn't necessarily an injective", "$\\mathcal{O}_\\mathcal{C}$-module as $g_!$ isn't exact in", "general. We do know that", "$$", "\\Ext^p_{\\mathcal{O}_\\mathcal{C}}(", "j_{U!}\\mathcal{O}_U, g^{-1}\\mathcal{I}) =", "H^p(U, g^{-1}\\mathcal{I}) = 0 \\text{ for }p \\geq 1", "$$", "Here the first equality follows from", "$\\Hom_{\\mathcal{O}_\\mathcal{C}}(j_{U!}\\mathcal{O}_U, \\mathcal{H}) =", "\\mathcal{H}(U)$ and taking derived functors and the vanishing of", "$H^p(U, g^{-1}\\mathcal{I})$ for $p > 0$ and $U \\in \\Ob(\\mathcal{C})$", "follows from Lemma \\ref{lemma-pullback-injective-pre-limp}.", "Since each $\\mathcal{K}^{-q}$ is a direct sum of modules of the form", "$j_{U!}\\mathcal{O}_U$ we see that", "$$", "\\Ext^p_{\\mathcal{O}_\\mathcal{C}}(\\mathcal{K}^{-q}, g^{-1}\\mathcal{I}) = 0", "\\text{ for }p \\geq 1\\text{ and all }q", "$$", "Let us use the spectral sequence (see", "Example", "\\ref{example-hom-complex-into-sheaf})", "$$", "E_1^{p, q} = \\Ext^p_{\\mathcal{O}_\\mathcal{C}}(", "\\mathcal{K}^{-q}, g^{-1}\\mathcal{I})", "\\Rightarrow", "\\Ext^{p + q}_{\\mathcal{O}_\\mathcal{C}}(", "\\mathcal{K}^\\bullet, g^{-1}\\mathcal{I}) = 0.", "$$", "Note that the spectral sequence abuts to zero as $\\mathcal{K}^\\bullet$", "is acyclic (hence vanishes in the derived category, hence produces", "vanishing ext groups). By the vanishing of higher exts proved above", "the only nonzero terms on the $E_1$ page are the terms", "$E_1^{0, q} = \\Hom_{\\mathcal{O}_\\mathcal{C}}(", "\\mathcal{K}^{-q}, g^{-1}\\mathcal{I})$.", "We conclude that the complex", "$\\Hom_{\\mathcal{O}_\\mathcal{C}}(", "\\mathcal{K}^\\bullet, g^{-1}\\mathcal{I})$", "is acyclic as desired.", "\\medskip\\noindent", "Thus the left derived functor $Lg_!$ exists.", "It is left adjoint to $g^{-1} = g^* = Rg^* = Lg^*$, i.e., we have", "\\begin{equation}", "\\label{equation-to-prove}", "\\Hom_{D(\\mathcal{O}_\\mathcal{C})}(K, g^*L) =", "\\Hom_{D(\\mathcal{O}_\\mathcal{D})}(Lg_!K, L)", "\\end{equation}", "by Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors}.", "This finishes the proof." ], "refs": [ "derived-proposition-left-derived-exists", "sites-modules-lemma-lower-shriek-modules", "sites-modules-lemma-module-quotient-flat", "sites-cohomology-lemma-pullback-injective-pre-limp", "derived-lemma-derived-adjoint-functors" ], "ref_ids": [ 1964, 14262, 14203, 4336, 1907 ] } ], "ref_ids": [ 14262 ] }, { "id": 4338, "type": "theorem", "label": "sites-cohomology-lemma-pullback-injective-limp", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-pullback-injective-limp", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous", "functor of sites. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "be the corresponding morphism of topoi. Let $\\mathcal{O}_\\mathcal{D}$", "be a sheaf of rings and let $\\mathcal{I}$ be an injective", "$\\mathcal{O}_\\mathcal{D}$-module. If", "$g_!^{Sh} : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "commutes with fibre products\\footnote{Holds if $\\mathcal{C}$", "has finite connected limits and $u$ commutes with them, see", "Sites, Lemma \\ref{sites-lemma-preserve-equalizers}.}, then", "$g^{-1}\\mathcal{I}$ is totally acyclic." ], "refs": [ "sites-lemma-preserve-equalizers" ], "proofs": [ { "contents": [ "We will use the criterion of Lemma \\ref{lemma-characterize-limp}.", "Condition (1) holds by Lemma \\ref{lemma-pullback-injective-pre-limp}.", "Let $K' \\to K$ be a surjective map of sheaves of sets on $\\mathcal{C}$.", "Since $g_!^{Sh}$ is a left adjoint,", "we see that $g_!^{Sh}K' \\to g_!^{Sh}K$ is surjective.", "Observe that", "\\begin{align*}", "H^0(K' \\times_K \\ldots \\times_K K', g^{-1}\\mathcal{I}) ", "& =", "H^0(g_!^{Sh}(K' \\times_K \\ldots \\times_K K'), \\mathcal{I}) \\\\", "& =", "H^0(g_!^{Sh}K' \\times_{g_!^{Sh}K} \\ldots \\times_{g_!^{Sh}K} g_!^{Sh}K',", "\\mathcal{I})", "\\end{align*}", "by our assumption on $g_!^{Sh}$. Since $\\mathcal{I}$ is an injective module", "it is totally acyclic by Lemma \\ref{lemma-direct-image-injective-sheaf}", "(applied to the identity). Hence we can use the converse of", "Lemma \\ref{lemma-characterize-limp} to see that the complex", "$$", "0 \\to H^0(K, g^{-1}\\mathcal{I}) \\to H^0(K', g^{-1}\\mathcal{I}) \\to", "H^0(K' \\times_K K', g^{-1}\\mathcal{I}) \\to \\ldots", "$$", "is exact as desired." ], "refs": [ "sites-cohomology-lemma-characterize-limp", "sites-cohomology-lemma-pullback-injective-pre-limp", "sites-cohomology-lemma-direct-image-injective-sheaf", "sites-cohomology-lemma-characterize-limp" ], "ref_ids": [ 4216, 4336, 4217, 4216 ] } ], "ref_ids": [ 8546 ] }, { "id": 4339, "type": "theorem", "label": "sites-cohomology-lemma-pullback-same-cohomology", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-pullback-same-cohomology", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous", "functor of sites. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "be the corresponding morphism of topoi. Let $U \\in \\Ob(\\mathcal{C})$.", "\\begin{enumerate}", "\\item For $M$ in $D(\\mathcal{D})$ we have", "$R\\Gamma(U, g^{-1}M) = R\\Gamma(u(U), M)$.", "\\item If $\\mathcal{O}_\\mathcal{D}$ is a sheaf of rings and", "$\\mathcal{O}_\\mathcal{C} = g^{-1}\\mathcal{O}_\\mathcal{D}$, then", "for $M$ in $D(\\mathcal{O}_\\mathcal{D})$ we have", "$R\\Gamma(U, g^*M) = R\\Gamma(u(U), M)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In the bounded below case (1) and (2) can be seen by representing", "$K$ by a bounded below complex of injectives and using", "Lemma \\ref{lemma-pullback-injective-pre-limp} as well as", "Leray's acyclicity lemma.", "In the unbounded case, first note that", "(1) is a special case of (2). For (2) we can use", "$$", "R\\Gamma(U, g^*M) =", "R\\Hom_{\\mathcal{O}_\\mathcal{C}}(j_{U!}\\mathcal{O}_U, g^*M) =", "R\\Hom_{\\mathcal{O}_\\mathcal{D}}(j_{u(U)!}\\mathcal{O}_{u(U)}, M) =", "R\\Gamma(u(U), M)", "$$", "where the middle equality is a consequence of", "Lemma \\ref{lemma-existence-derived-lower-shriek}." ], "refs": [ "sites-cohomology-lemma-pullback-injective-pre-limp", "sites-cohomology-lemma-existence-derived-lower-shriek" ], "ref_ids": [ 4336, 4337 ] } ], "ref_ids": [] }, { "id": 4340, "type": "theorem", "label": "sites-cohomology-lemma-special-square-cocontinuous", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-special-square-cocontinuous", "contents": [ "Assume given a commutative diagram", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})", "\\ar[r]_{(g', (g')^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} &", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^{(f, f^\\sharp)} \\\\", "(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^{(g, g^\\sharp)} &", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})", "}", "$$", "of ringed topoi. Assume", "\\begin{enumerate}", "\\item $f$, $f'$, $g$, and $g'$ correspond to cocontinuous functors", "$u$, $u'$, $v$, and $v'$ as in", "Sites, Lemma \\ref{sites-lemma-cocontinuous-morphism-topoi},", "\\item $v \\circ u' = u \\circ v'$,", "\\item $v$ and $v'$ are continuous as well as cocontinuous,", "\\item for any object $V'$ of $\\mathcal{D}'$ the functor", "${}^{u'}_{V'}\\mathcal{I} \\to {}^{\\ \\ \\ u}_{v(V')}\\mathcal{I}$", "given by $v$ is cofinal,", "\\item $g^{-1}\\mathcal{O}_{\\mathcal{D}} = \\mathcal{O}_{\\mathcal{D}'}$", "and $(g')^{-1}\\mathcal{O}_{\\mathcal{C}} = \\mathcal{O}_{\\mathcal{C}'}$, and", "\\item $g'_! : \\textit{Ab}(\\mathcal{C}') \\to \\textit{Ab}(\\mathcal{C})$", "is exact\\footnote{Holds if fibre products and equalizers exist in", "$\\mathcal{C}'$ and $v'$ commutes with them, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-exactness-lower-shriek}.}.", "\\end{enumerate}", "Then we have $Rf'_* \\circ (g')^* = g^* \\circ Rf_*$ as functors", "$D(\\mathcal{O}_\\mathcal{C}) \\to D(\\mathcal{O}_{\\mathcal{D}'})$." ], "refs": [ "sites-lemma-cocontinuous-morphism-topoi", "sites-modules-lemma-exactness-lower-shriek" ], "proofs": [ { "contents": [ "We have $g^* = Lg^* = g^{-1}$ and $(g')^* = L(g')^* = (g')^{-1}$", "by condition (5).", "By Lemma \\ref{lemma-modules-abelian-unbounded} it suffices", "to prove the result on the derived category $D(\\mathcal{C})$", "of abelian sheaves. Choose an object $K \\in D(\\mathcal{C})$.", "Let $\\mathcal{I}^\\bullet$ be a K-injective complex of abelian", "sheaves on $\\mathcal{C}$ representing $K$. By", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}", "and assumption (6) we find that $(g')^{-1}\\mathcal{I}^\\bullet$", "is a K-injective complex of abelian sheaves on $\\mathcal{C}'$.", "By Modules on Sites, Lemma", "\\ref{sites-modules-lemma-special-square-cocontinuous}", "we find that $f'_*(g')^{-1}\\mathcal{I}^\\bullet = g^{-1}f_*\\mathcal{I}^\\bullet$.", "Since $f_*\\mathcal{I}^\\bullet$ represents $Rf_*K$ and since", "$f'_*(g')^{-1}\\mathcal{I}^\\bullet$ represents $Rf'_*(g')^{-1}K$", "we conclude." ], "refs": [ "sites-cohomology-lemma-modules-abelian-unbounded", "derived-lemma-adjoint-preserve-K-injectives", "sites-modules-lemma-special-square-cocontinuous" ], "ref_ids": [ 4259, 1915, 14263 ] } ], "ref_ids": [ 8543, 14165 ] }, { "id": 4341, "type": "theorem", "label": "sites-cohomology-lemma-special-square-continuous", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-special-square-continuous", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'}", "\\ar[r]_{(g', (g')^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} &", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^{(f, f^\\sharp)} \\\\", "(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^{(g, g^\\sharp)} &", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})", "}", "$$", "of ringed topoi and suppose we have functors", "$$", "\\xymatrix{", "\\mathcal{C}' \\ar[r]_{v'} &", "\\mathcal{C} \\\\", "\\mathcal{D}' \\ar[r]^v \\ar[u]^{u'} &", "\\mathcal{D} \\ar[u]_u", "}", "$$", "such that (with notation as in", "Sites, Sections \\ref{sites-section-morphism-sites} and", "\\ref{sites-section-cocontinuous-morphism-topoi}) we have", "\\begin{enumerate}", "\\item $u$ and $u'$ are continuous and give rise to the morphisms", "$f$ and $f'$,", "\\item $v$ and $v'$ are cocontinuous giving rise to the morphisms $g$ and $g'$,", "\\item $u \\circ v = v' \\circ u'$,", "\\item $v$ and $v'$ are continuous as well as cocontinuous, and", "\\item $g^{-1}\\mathcal{O}_{\\mathcal{D}} = \\mathcal{O}_{\\mathcal{D}'}$", "and $(g')^{-1}\\mathcal{O}_{\\mathcal{C}} = \\mathcal{O}_{\\mathcal{C}'}$.", "\\end{enumerate}", "Then $Rf'_* \\circ (g')^* = g^* \\circ Rf_*$ as functors", "$D^+(\\mathcal{O}_\\mathcal{C}) \\to D^+(\\mathcal{O}_{\\mathcal{D}'})$.", "If in addition", "\\begin{enumerate}", "\\item[(6)] $g'_! : \\textit{Ab}(\\mathcal{C}') \\to \\textit{Ab}(\\mathcal{C})$", "is exact\\footnote{Holds if fibre products and equalizers exist in", "$\\mathcal{C}'$ and $v'$ commutes with them, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-exactness-lower-shriek}.},", "\\end{enumerate}", "then $Rf'_* \\circ (g')^* = g^* \\circ Rf_*$ as functors", "$D(\\mathcal{O}_\\mathcal{C}) \\to D(\\mathcal{O}_{\\mathcal{D}'})$." ], "refs": [ "sites-modules-lemma-exactness-lower-shriek" ], "proofs": [ { "contents": [ "We have $g^* = Lg^* = g^{-1}$ and $(g')^* = L(g')^* = (g')^{-1}$", "by condition (5).", "By Lemma \\ref{lemma-modules-abelian-unbounded} it suffices", "to prove the result on the derived category $D^+(\\mathcal{C})$ or", "$D(\\mathcal{C})$ of abelian sheaves.", "\\medskip\\noindent", "Choose an object $K \\in D^+(\\mathcal{C})$.", "Let $\\mathcal{I}^\\bullet$ be a bounded below complex of injective abelian", "sheaves on $\\mathcal{C}$ representing $K$. By", "Lemma \\ref{lemma-pullback-injective-pre-limp}", "we see that $H^p(U', (g')^{-1}\\mathcal{I}^q) = 0$ for", "all $p > 0$ and any $q$ and any $U' \\in \\Ob(\\mathcal{C}')$.", "Recall that $R^pf'_*(g')^{-1}\\mathcal{I}^q$ is the sheaf", "associated to the presheaf $V' \\mapsto H^p(u'(V'), (g')^{-1}\\mathcal{I}^q)$,", "see Lemma \\ref{lemma-higher-direct-images}.", "Thus we see that $(g')^{-1}\\mathcal{I}^q$ is right acyclic", "for the functor $f'_*$. By Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "we find that $f'_*(g')^*\\mathcal{I}^\\bullet$", "represents $Rf'_*(g')^{-1}K$.", "By Modules on Sites, Lemma", "\\ref{sites-modules-lemma-special-square-continuous}", "we find that $f'_*(g')^{-1}\\mathcal{I}^\\bullet = g^{-1}f_*\\mathcal{I}^\\bullet$.", "Since $g^{-1}f_*\\mathcal{I}^\\bullet$ represents $g^{-1}Rf_*K$", "we conclude.", "\\medskip\\noindent", "Choose an object $K \\in D(\\mathcal{C})$.", "Let $\\mathcal{I}^\\bullet$ be a K-injective complex of abelian", "sheaves on $\\mathcal{C}$ representing $K$. By", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}", "and assumption (6) we find that $(g')^{-1}\\mathcal{I}^\\bullet$", "is a K-injective complex of abelian sheaves on $\\mathcal{C}'$.", "By Modules on Sites, Lemma", "\\ref{sites-modules-lemma-special-square-continuous}", "we find that $f'_*(g')^{-1}\\mathcal{I}^\\bullet = g^{-1}f_*\\mathcal{I}^\\bullet$.", "Since $f_*\\mathcal{I}^\\bullet$ represents $Rf_*K$ and since", "$f'_*(g')^{-1}\\mathcal{I}^\\bullet$ represents $Rf'_*(g')^{-1}K$", "we conclude." ], "refs": [ "sites-cohomology-lemma-modules-abelian-unbounded", "sites-cohomology-lemma-pullback-injective-pre-limp", "sites-cohomology-lemma-higher-direct-images", "derived-lemma-leray-acyclicity", "sites-modules-lemma-special-square-continuous", "derived-lemma-adjoint-preserve-K-injectives", "sites-modules-lemma-special-square-continuous" ], "ref_ids": [ 4259, 4336, 4189, 1844, 14264, 1915, 14264 ] } ], "ref_ids": [ 14165 ] }, { "id": 4342, "type": "theorem", "label": "sites-cohomology-lemma-fibred-category-with-object", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-fibred-category-with-object", "contents": [ "Assumptions and notation as in Situation \\ref{situation-fibred-category}.", "For $U \\in \\Ob(\\mathcal{C})$ consider the induced morphism", "of topoi", "$$", "\\pi_U : \\Sh(\\mathcal{C}/U) \\longrightarrow \\Sh(\\mathcal{D}/p(U))", "$$", "Then there exists a morphism of topoi", "$$", "\\sigma : \\Sh(\\mathcal{D}/p(U)) \\to \\Sh(\\mathcal{C}/U)", "$$", "such that $\\pi_U \\circ \\sigma = \\text{id}$ and $\\sigma^{-1} = \\pi_{U, *}$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\pi_U$ is the restriction of $\\pi$ to the localizations, see", "Sites, Lemma \\ref{sites-lemma-localize-cocontinuous}.", "For an object $V \\to p(U)$ of $\\mathcal{D}/p(U)$ denote", "$V \\times_{p(U)} U \\to U$ the strongly cartesian morphism of $\\mathcal{C}$", "over $\\mathcal{D}$ which exists as $p$ is a fibred category.", "The functor", "$$", "v : \\mathcal{D}/p(U) \\to \\mathcal{C}/U,\\quad", "V/p(U) \\mapsto V \\times_{p(U)} U/U", "$$", "is continuous by the definition of the topology on $\\mathcal{C}$.", "Moreover, it is a right adjoint to $p$ by the definition of strongly", "cartesian morphisms. Hence we are in the situation discussed in", "Sites, Section \\ref{sites-section-cocontinuous-adjoint}", "and we see that the sheaf $\\pi_{U, *}\\mathcal{F}$", "is equal to $V \\mapsto \\mathcal{F}(V \\times_{p(U)} U)$", "(see especially Sites, Lemma", "\\ref{sites-lemma-have-functor-other-way-morphism}).", "\\medskip\\noindent", "But here we have more. Namely, the functor $v$", "is also cocontinuous (as all morphisms in coverings of $\\mathcal{C}$ ", "are strongly cartesian). Hence $v$ defines a morphism $\\sigma$ as", "indicated in the lemma. The equality $\\sigma^{-1} = \\pi_{U, *}$", "is immediate from the definition. Since $\\pi_U^{-1}\\mathcal{G}$", "is given by the rule $U'/U \\mapsto \\mathcal{G}(p(U')/p(U))$", "it follows that $\\sigma^{-1} \\circ \\pi_U^{-1} = \\text{id}$", "which proves the equality", "$\\pi_U \\circ \\sigma = \\text{id}$." ], "refs": [ "sites-lemma-localize-cocontinuous", "sites-lemma-have-functor-other-way-morphism" ], "ref_ids": [ 8574, 8550 ] } ], "ref_ids": [] }, { "id": 4343, "type": "theorem", "label": "sites-cohomology-lemma-morphism-fibred-categories-with-object", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-morphism-fibred-categories-with-object", "contents": [ "Assumptions and notation as in", "Situation \\ref{situation-morphism-fibred-categories}.", "For $U' \\in \\Ob(\\mathcal{C}')$ set $U = u(U')$ and $V = p'(U')$ and", "consider the induced morphisms of ringed topoi", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}'/U'), \\mathcal{O}_{U'}) \\ar[rd]_{\\pi'_{U'}} \\ar[rr]_{g'} & &", "(\\Sh(\\mathcal{C}), \\mathcal{O}_U) \\ar[ld]^{\\pi_U} \\\\", "& (\\Sh(\\mathcal{D}/V), \\mathcal{O}_V)", "}", "$$", "Then there exists a morphism of topoi", "$$", "\\sigma' : \\Sh(\\mathcal{D}/V) \\to \\Sh(\\mathcal{C}'/U'),", "$$", "such that setting $\\sigma = g' \\circ \\sigma'$ we have", "$\\pi'_{U'} \\circ \\sigma' = \\text{id}$, $\\pi_U \\circ \\sigma = \\text{id}$,", "$(\\sigma')^{-1} = \\pi'_{U', *}$, and $\\sigma^{-1} = \\pi_{U, *}$." ], "refs": [], "proofs": [ { "contents": [ "Let $v' : \\mathcal{D}/V \\to \\mathcal{C}'/U'$ be the functor constructed", "in the proof of Lemma \\ref{lemma-fibred-category-with-object} starting", "with $p' : \\mathcal{C}' \\to \\mathcal{D}'$ and the object $U'$.", "Since $u$ is a $1$-morphism of fibred categories over $\\mathcal{D}$", "it transforms strongly cartesian morphisms into strongly cartesian morphisms,", "hence the functor $v = u \\circ v'$ is the functor of", "the proof of Lemma \\ref{lemma-fibred-category-with-object}", "relative to $p : \\mathcal{C} \\to \\mathcal{D}$ and $U$. Thus our lemma", "follows from that lemma." ], "refs": [ "sites-cohomology-lemma-fibred-category-with-object", "sites-cohomology-lemma-fibred-category-with-object" ], "ref_ids": [ 4342, 4342 ] } ], "ref_ids": [] }, { "id": 4344, "type": "theorem", "label": "sites-cohomology-lemma-properties-lower-shriek-fibred-category", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-properties-lower-shriek-fibred-category", "contents": [ "Assumption and notation as in", "Situation \\ref{situation-morphism-fibred-categories}.", "\\begin{enumerate}", "\\item There are left adjoints", "$g_! : \\textit{Mod}(\\mathcal{O}_{\\mathcal{C}'}) \\to", "\\textit{Mod}(\\mathcal{O}_\\mathcal{C})$ and", "$g_!^{\\textit{Ab}} : \\textit{Ab}(\\mathcal{C}') \\to \\textit{Ab}(\\mathcal{C})$", "to $g^* = g^{-1}$ on modules and on abelian sheaves.", "\\item The diagram", "$$", "\\xymatrix{", "\\textit{Mod}(\\mathcal{O}_{\\mathcal{C}'}) \\ar[d] \\ar[r]_{g_!} &", "\\textit{Mod}(\\mathcal{O}_\\mathcal{C}) \\ar[d] \\\\", "\\textit{Ab}(\\mathcal{C}') \\ar[r]^{g_!^{\\textit{Ab}}} &", "\\textit{Ab}(\\mathcal{C})", "}", "$$", "commutes.", "\\item There are left adjoints", "$Lg_! : D(\\mathcal{O}_{\\mathcal{C}'}) \\to D(\\mathcal{O}_\\mathcal{C})$", "and", "$Lg_!^{\\textit{Ab}} : D(\\mathcal{C}') \\to D(\\mathcal{C})$", "to $g^* = g^{-1}$ on derived categories of modules and abelian sheaves.", "\\item The diagram", "$$", "\\xymatrix{", "D(\\mathcal{O}_{\\mathcal{C}'}) \\ar[d] \\ar[r]_{Lg_!} &", "D(\\mathcal{O}_\\mathcal{C}) \\ar[d] \\\\", "D(\\mathcal{C}') \\ar[r]^{Lg_!^{\\textit{Ab}}} &", "D(\\mathcal{C})", "}", "$$", "commutes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The functor $u$ is continuous and cocontinuous", "Stacks, Lemma \\ref{stacks-lemma-topology-inherited-functorial}.", "Hence the existence of the functors $g_!$, $g_!^{\\textit{Ab}}$,", "$Lg_!$, and $Lg_!^{\\textit{Ab}}$ can be found in", "Modules on Sites, Sections", "\\ref{sites-modules-section-exactness-lower-shriek} and", "\\ref{sites-modules-section-lower-shriek-modules}", "and", "Section \\ref{section-derived-lower-shriek}.", "\\medskip\\noindent", "To prove (2) it suffices to show that the canonical map", "$$", "g_!^{\\textit{Ab}}j_{U'!}\\mathcal{O}_{U'} \\to j_{u(U')!}\\mathcal{O}_{u(U')}", "$$", "is an isomorphism for all objects $U'$ of $\\mathcal{C}'$, see", "Modules on Sites, Remark \\ref{sites-modules-remark-when-shriek-equal}.", "Similarly, to prove (4) it suffices to show that the canonical map", "$$", "Lg_!^{\\textit{Ab}}j_{U'!}\\mathcal{O}_{U'} \\to j_{u(U')!}\\mathcal{O}_{u(U')}", "$$", "is an isomorphism in $D(\\mathcal{C})$ for all objects $U'$ of", "$\\mathcal{C}'$, see Remark \\ref{remark-when-derived-shriek-equal}.", "This will also imply the previous formula hence this is what we will show.", "\\medskip\\noindent", "We will use that for a localization morphism $j$ the", "functors $j_!$ and $j_!^{\\textit{Ab}}$ agree (see", "Modules on Sites, Remark \\ref{sites-modules-remark-localize-shriek-equal})", "and that $j_!$ is exact", "(Modules on Sites, Lemma \\ref{sites-modules-lemma-extension-by-zero-exact}).", "Let us adopt the notation of", "Lemma \\ref{lemma-morphism-fibred-categories-with-object}.", "Since $Lg_!^{\\textit{Ab}} \\circ j_{U'!} = j_{U!} \\circ L(g')^{\\textit{Ab}}_!$", "(by commutativity of Sites, Lemma \\ref{sites-lemma-localize-cocontinuous}", "and uniqueness of adjoint functors) it suffices to prove that", "$L(g')^{\\textit{Ab}}_!\\mathcal{O}_{U'} = \\mathcal{O}_U$. Using the", "results of", "Lemma \\ref{lemma-morphism-fibred-categories-with-object}", "we have for any object $E$ of $D(\\mathcal{C}/u(U'))$ the following", "sequence of equalities", "\\begin{align*}", "\\Hom_{D(\\mathcal{C}/U)}(L(g')_!^{\\textit{Ab}}\\mathcal{O}_{U'}, E)", "& =", "\\Hom_{D(\\mathcal{C}'/U')}(\\mathcal{O}_{U'}, (g')^{-1}E) \\\\", "& =", "\\Hom_{D(\\mathcal{C}'/U')}((\\pi'_{U'})^{-1}\\mathcal{O}_V, (g')^{-1}E) \\\\", "& =", "\\Hom_{D(\\mathcal{D}/V)}(\\mathcal{O}_V, R\\pi'_{U', *}(g')^{-1}E) \\\\", "& =", "\\Hom_{D(\\mathcal{D}/V)}(\\mathcal{O}_V, (\\sigma')^{-1}(g')^{-1}E) \\\\", "& =", "\\Hom_{D(\\mathcal{D}/V)}(\\mathcal{O}_V, \\sigma^{-1}E) \\\\", "& =", "\\Hom_{D(\\mathcal{D}/V)}(\\mathcal{O}_V, \\pi_{U, *}E) \\\\", "& =", "\\Hom_{D(\\mathcal{C}/U)}(\\pi_U^{-1}\\mathcal{O}_V, E) \\\\", "& =", "\\Hom_{D(\\mathcal{C}/U)}(\\mathcal{O}_U, E)", "\\end{align*}", "By Yoneda's lemma we conclude." ], "refs": [ "stacks-lemma-topology-inherited-functorial", "sites-modules-remark-when-shriek-equal", "sites-cohomology-remark-when-derived-shriek-equal", "sites-modules-remark-localize-shriek-equal", "sites-modules-lemma-extension-by-zero-exact", "sites-cohomology-lemma-morphism-fibred-categories-with-object", "sites-lemma-localize-cocontinuous", "sites-cohomology-lemma-morphism-fibred-categories-with-object" ], "ref_ids": [ 8970, 14312, 4433, 14307, 14170, 4343, 8574, 4343 ] } ], "ref_ids": [] }, { "id": 4345, "type": "theorem", "label": "sites-cohomology-lemma-compute-pi-shriek", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-compute-pi-shriek", "contents": [ "Assumptions and notation as in", "Situation \\ref{situation-fibred-category}.", "For $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C})$", "the sheaf $\\pi_!\\mathcal{F}$ is the", "sheaf associated to the presheaf", "$$", "V \\longmapsto \\colim_{\\mathcal{C}_V^{opp}} \\mathcal{F}|_{\\mathcal{C}_V}", "$$", "with restriction maps as indicated in the proof." ], "refs": [], "proofs": [ { "contents": [ "Denote $\\mathcal{H}$ be the rule of the lemma.", "For a morphism $h : V' \\to V$ of $\\mathcal{D}$ there is a", "pullback functor $h^* : \\mathcal{C}_V \\to \\mathcal{C}_{V'}$ of fibre", "categories (Categories, Definition", "\\ref{categories-definition-pullback-functor-fibred-category}).", "Moreover for $U \\in \\Ob(\\mathcal{C}_V)$ there is a", "strongly cartesian morphism $h^*U \\to U$ covering $h$.", "Restriction along these strongly cartesian morphisms defines a", "transformation of functors", "$$", "\\mathcal{F}|_{\\mathcal{C}_V}", "\\longrightarrow", "\\mathcal{F}|_{\\mathcal{C}_{V'}} \\circ h^*.", "$$", "Hence a map $\\mathcal{H}(V) \\to \\mathcal{H}(V')$ between colimits, see", "Categories, Lemma \\ref{categories-lemma-functorial-colimit}.", "\\medskip\\noindent", "To prove the lemma we show that", "$$", "\\Mor_{\\textit{PSh}(\\mathcal{D})}(\\mathcal{H}, \\mathcal{G}) =", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F}, \\pi^{-1}\\mathcal{G})", "$$", "for every sheaf $\\mathcal{G}$ on $\\mathcal{C}$. An element of the", "left hand side is a compatible system of maps", "$\\mathcal{F}(U) \\to \\mathcal{G}(p(U))$ for all $U$ in $\\mathcal{C}$.", "Since $\\pi^{-1}\\mathcal{G}(U) = \\mathcal{G}(p(U))$ by our choice", "of topology on $\\mathcal{C}$ we see the same thing is true for the", "right hand side and we win." ], "refs": [ "categories-definition-pullback-functor-fibred-category", "categories-lemma-functorial-colimit" ], "ref_ids": [ 12389, 12210 ] } ], "ref_ids": [] }, { "id": 4346, "type": "theorem", "label": "sites-cohomology-lemma-initial-final", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-initial-final", "contents": [ "Notation and assumptions as in Example \\ref{example-category-to-point}.", "If $\\mathcal{C}$ has either an initial or a final object, then", "$L\\pi_! \\circ \\pi^{-1} = \\text{id}$ on $D(\\textit{Ab})$, resp.\\ $D(B)$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{C}$ has an initial object, then $\\pi_!$ is computed by", "evaluating on this object and the statement is clear. If $\\mathcal{C}$", "has a final object, then $R\\pi_*$ is computed by evaluating on this", "object, hence $R\\pi_* \\circ \\pi^{-1} \\cong \\text{id}$ on", "$D(\\textit{Ab})$, resp.\\ $D(B)$. This implies that", "$\\pi^{-1} : D(\\textit{Ab}) \\to D(\\mathcal{C})$,", "resp.\\ $\\pi^{-1} : D(B) \\to D(\\underline{B})$ is fully faithful, see", "Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}.", "Then the same lemma implies that $L\\pi_! \\circ \\pi^{-1} = \\text{id}$", "as desired." ], "refs": [ "categories-lemma-adjoint-fully-faithful" ], "ref_ids": [ 12248 ] } ], "ref_ids": [] }, { "id": 4347, "type": "theorem", "label": "sites-cohomology-lemma-change-of-rings", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-change-of-rings", "contents": [ "Notation and assumptions as in Example \\ref{example-category-to-point}.", "Let $B \\to B'$ be a ring map. Consider the commutative diagram", "of ringed topoi", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}), \\underline{B}) \\ar[d]_\\pi &", "(\\Sh(\\mathcal{C}), \\underline{B'}) \\ar[d]^{\\pi'} \\ar[l]^h \\\\", "(*, B) & (*, B') \\ar[l]_f", "}", "$$", "Then $L\\pi_! \\circ Lh^* = Lf^* \\circ L\\pi'_!$." ], "refs": [], "proofs": [ { "contents": [ "Both functors are right adjoint to the obvious functor", "$D(B') \\to D(\\underline{B})$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4348, "type": "theorem", "label": "sites-cohomology-lemma-compute-by-cosimplicial-resolution", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-compute-by-cosimplicial-resolution", "contents": [ "Notation and assumptions as in Example \\ref{example-category-to-point}.", "Let $U_\\bullet$ be a cosimplicial object in $\\mathcal{C}$ such that", "for every $U \\in \\Ob(\\mathcal{C})$ the simplicial set", "$\\Mor_\\mathcal{C}(U_\\bullet, U)$", "is homotopy equivalent to the constant simplicial set on a singleton. Then", "$$", "L\\pi_!(\\mathcal{F}) = \\mathcal{F}(U_\\bullet)", "$$", "in $D(\\textit{Ab})$, resp.\\ $D(B)$ functorially in $\\mathcal{F}$ in", "$\\textit{Ab}(\\mathcal{C})$, resp.\\ $\\textit{Mod}(\\underline{B})$." ], "refs": [], "proofs": [ { "contents": [ "As $L\\pi_!$ agrees for modules and abelian sheaves by", "Lemma \\ref{lemma-properties-lower-shriek-fibred-category}", "it suffices to prove this when $\\mathcal{F}$ is an abelian sheaf.", "For $U \\in \\Ob(\\mathcal{C})$ the abelian sheaf $j_{U!}\\mathbf{Z}_U$", "is a projective object of $\\textit{Ab}(\\mathcal{C})$ since", "$\\Hom(j_{U!}\\mathbf{Z}_U, \\mathcal{F}) = \\mathcal{F}(U)$", "and taking sections is an exact functor as the topology is chaotic.", "Every abelian sheaf is a quotient of a direct sum of $j_{U!}\\mathbf{Z}_U$", "by Modules on Sites, Lemma \\ref{sites-modules-lemma-module-quotient-flat}.", "Thus we can compute $L\\pi_!(\\mathcal{F})$ by choosing a resolution", "$$", "\\ldots \\to \\mathcal{G}^{-1} \\to \\mathcal{G}^0 \\to \\mathcal{F} \\to 0", "$$", "whose terms are direct sums of sheaves of the form above and taking", "$L\\pi_!(\\mathcal{F}) = \\pi_!(\\mathcal{G}^\\bullet)$. Consider the", "double complex", "$A^{\\bullet, \\bullet} = \\mathcal{G}^\\bullet(U_\\bullet)$.", "The map $\\mathcal{G}^0 \\to \\mathcal{F}$ gives a map of complexes", "$A^{0, \\bullet} \\to \\mathcal{F}(U_\\bullet)$.", "Since $\\pi_!$ is computed by taking the colimit over", "$\\mathcal{C}^{opp}$ (Lemma \\ref{lemma-compute-pi-shriek})", "we see that the two compositions", "$\\mathcal{G}^m(U_1) \\to \\mathcal{G}^m(U_0) \\to \\pi_!\\mathcal{G}^m$", "are equal. Thus we obtain a canonical map of complexes", "$$", "\\text{Tot}(A^{\\bullet, \\bullet})", "\\longrightarrow", "\\pi_!(\\mathcal{G}^\\bullet) = L\\pi_!(\\mathcal{F})", "$$", "To prove the lemma it suffices to show that the complexes", "$$", "\\ldots \\to \\mathcal{G}^m(U_1) \\to \\mathcal{G}^m(U_0) \\to", "\\pi_!\\mathcal{G}^m \\to 0", "$$", "are exact, see Homology, Lemma", "\\ref{homology-lemma-double-complex-gives-resolution}.", "Since the sheaves $\\mathcal{G}^m$ are direct sums of the sheaves", "$j_{U!}\\mathbf{Z}_U$ we reduce to $\\mathcal{G} = j_{U!}\\mathbf{Z}_U$.", "The complex $j_{U!}\\mathbf{Z}_U(U_\\bullet)$", "is the complex of abelian groups associated to the free", "$\\mathbf{Z}$-module on the simplicial set", "$\\Mor_\\mathcal{C}(U_\\bullet, U)$ which we assumed to be homotopy", "equivalent to a singleton. We conclude that", "$$", "j_{U!}\\mathbf{Z}_U(U_\\bullet) \\to \\mathbf{Z}", "$$", "is a homotopy equivalence of abelian groups hence a quasi-isomorphism", "(Simplicial, Remark \\ref{simplicial-remark-homotopy-better} and", "Lemma \\ref{simplicial-lemma-homotopy-s-N}). This finishes the proof", "since $\\pi_!j_{U!}\\mathbf{Z}_U = \\mathbf{Z}$", "as was shown in the proof of", "Lemma \\ref{lemma-properties-lower-shriek-fibred-category}." ], "refs": [ "sites-cohomology-lemma-properties-lower-shriek-fibred-category", "sites-modules-lemma-module-quotient-flat", "sites-cohomology-lemma-compute-pi-shriek", "homology-lemma-double-complex-gives-resolution", "simplicial-remark-homotopy-better", "simplicial-lemma-homotopy-s-N", "sites-cohomology-lemma-properties-lower-shriek-fibred-category" ], "ref_ids": [ 4344, 14203, 4345, 12106, 14941, 14876, 4344 ] } ], "ref_ids": [] }, { "id": 4349, "type": "theorem", "label": "sites-cohomology-lemma-get-it-now", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-get-it-now", "contents": [ "Notation and assumptions as in Example \\ref{example-morphism-categories}.", "If there exists a cosimplicial object $U'_\\bullet$ of $\\mathcal{C}'$", "such that Lemma \\ref{lemma-compute-by-cosimplicial-resolution}", "applies to both $U'_\\bullet$ in $\\mathcal{C}'$", "and $u(U'_\\bullet)$ in $\\mathcal{C}$, then we have", "$L\\pi'_! \\circ g^{-1} = L\\pi_!$ as functors", "$D(\\mathcal{C}) \\to D(\\textit{Ab})$,", "resp.\\ $D(\\mathcal{C}, \\underline{B}) \\to D(B)$." ], "refs": [ "sites-cohomology-lemma-compute-by-cosimplicial-resolution" ], "proofs": [ { "contents": [ "Follows immediately from", "Lemma \\ref{lemma-compute-by-cosimplicial-resolution}", "and the fact that $g^{-1}$ is given by precomposing with $u$." ], "refs": [ "sites-cohomology-lemma-compute-by-cosimplicial-resolution" ], "ref_ids": [ 4348 ] } ], "ref_ids": [ 4348 ] }, { "id": 4350, "type": "theorem", "label": "sites-cohomology-lemma-product-categories", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-product-categories", "contents": [ "Let $\\mathcal{C}_i$, $i = 1, 2$ be categories. Let", "$u_i : \\mathcal{C}_1 \\times \\mathcal{C}_2 \\to \\mathcal{C}_i$ be the", "projection functors. Let $B$ be a ring. Let", "$g_i : (\\Sh(\\mathcal{C}_1 \\times \\mathcal{C}_2), \\underline{B}) \\to", "(\\Sh(\\mathcal{C}_i), \\underline{B})$ be the corresponding morphisms", "of ringed topoi, see Example \\ref{example-morphism-categories}. For", "$K_i \\in D(\\mathcal{C}_i, B)$ we have", "$$", "L(\\pi_1 \\times \\pi_2)_!(", "g_1^{-1}K_1 \\otimes_{\\underline{B}}^\\mathbf{L} g_2^{-1}K_2)", "=", "L\\pi_{1, !}(K_1) \\otimes_B^\\mathbf{L} L\\pi_{2, !}(K_2)", "$$", "in $D(B)$ with obvious notation." ], "refs": [], "proofs": [ { "contents": [ "As both sides commute with colimits, it suffices to prove this for", "$K_1 = j_{U!}\\underline{B}_U$ and $K_2 = j_{V!}\\underline{B}_V$", "for $U \\in \\Ob(\\mathcal{C}_1)$ and $V \\in \\Ob(\\mathcal{C}_2)$.", "See construction of $L\\pi_!$ in", "Lemma \\ref{lemma-existence-derived-lower-shriek}.", "In this case", "$$", "g_1^{-1}K_1 \\otimes_{\\underline{B}}^\\mathbf{L} g_2^{-1}K_2 =", "g_1^{-1}K_1 \\otimes_{\\underline{B}} g_2^{-1}K_2 =", "j_{(U, V)!}\\underline{B}_{(U, V)}", "$$", "Verification omitted. Hence the result follows as both the left and", "the right hand side of the formula of the lemma evaluate to $B$, see", "construction of $L\\pi_!$ in Lemma \\ref{lemma-existence-derived-lower-shriek}." ], "refs": [ "sites-cohomology-lemma-existence-derived-lower-shriek", "sites-cohomology-lemma-existence-derived-lower-shriek" ], "ref_ids": [ 4337, 4337 ] } ], "ref_ids": [] }, { "id": 4351, "type": "theorem", "label": "sites-cohomology-lemma-eilenberg-zilber", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-eilenberg-zilber", "contents": [ "Notation and assumptions as in Example \\ref{example-category-to-point}.", "If there exists a cosimplicial object $U_\\bullet$ of $\\mathcal{C}$", "such that Lemma \\ref{lemma-compute-by-cosimplicial-resolution}", "applies, then", "$$", "L\\pi_!(K_1 \\otimes^\\mathbf{L}_{\\underline{B}} K_2) =", "L\\pi_!(K_1) \\otimes^\\mathbf{L}_B L\\pi_!(K_2)", "$$", "for all $K_i \\in D(\\underline{B})$." ], "refs": [ "sites-cohomology-lemma-compute-by-cosimplicial-resolution" ], "proofs": [ { "contents": [ "Consider the diagram of categories and functors", "$$", "\\xymatrix{", "& & \\mathcal{C} \\\\", "\\mathcal{C} \\ar[r]^-u &", "\\mathcal{C} \\times \\mathcal{C} \\ar[rd]^{u_2} \\ar[ru]_{u_1} \\\\", "& & \\mathcal{C}", "}", "$$", "where $u$ is the diagonal functor and $u_i$ are the projection functors.", "This gives morphisms of ringed topoi $g$, $g_1$, $g_2$.", "For any object $(U_1, U_2)$ of $\\mathcal{C}$ we have", "$$", "\\Mor_{\\mathcal{C} \\times \\mathcal{C}}(u(U_\\bullet), (U_1, U_2)) =", "\\Mor_\\mathcal{C}(U_\\bullet, U_1) \\times \\Mor_\\mathcal{C}(U_\\bullet, U_2)", "$$", "which is homotopy equivalent to a point by", "Simplicial, Lemma \\ref{simplicial-lemma-products-homotopy}.", "Thus Lemma \\ref{lemma-get-it-now} gives", "$L\\pi_!(g^{-1}K) = L(\\pi \\times \\pi)_!(K)$ for any $K$ in", "$D(\\mathcal{C} \\times \\mathcal{C}, B)$.", "Take $K = g_1^{-1}K_1 \\otimes_B^\\mathbf{L} g_2^{-1}K_2$.", "Then $g^{-1}K = K_1 \\otimes^\\mathbf{L}_{\\underline{B}} K_2$", "because $g^{-1} = g^* = Lg^*$ commutes with derived tensor product", "(Lemma \\ref{lemma-pullback-tensor-product}).", "To finish we apply Lemma \\ref{lemma-product-categories}." ], "refs": [ "simplicial-lemma-products-homotopy", "sites-cohomology-lemma-get-it-now", "sites-cohomology-lemma-pullback-tensor-product", "sites-cohomology-lemma-product-categories" ], "ref_ids": [ 14875, 4349, 4244, 4350 ] } ], "ref_ids": [ 4348 ] }, { "id": 4352, "type": "theorem", "label": "sites-cohomology-lemma-O-homology-qis", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-O-homology-qis", "contents": [ "Let $\\mathcal{C}$ be a category (endowed with chaotic topology).", "Let $\\mathcal{O} \\to \\mathcal{O}'$ be a map of sheaves of rings on", "$\\mathcal{C}$. Assume", "\\begin{enumerate}", "\\item there exists a cosimplicial object $U_\\bullet$ in $\\mathcal{C}$", "as in Lemma \\ref{lemma-compute-by-cosimplicial-resolution}, and", "\\item $L\\pi_!\\mathcal{O} \\to L\\pi_!\\mathcal{O}'$ is an isomorphism.", "\\end{enumerate}", "For $K$ in $D(\\mathcal{O})$ we have", "$$", "L\\pi_!(K) = L\\pi_!(K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}')", "$$", "in $D(\\textit{Ab})$." ], "refs": [ "sites-cohomology-lemma-compute-by-cosimplicial-resolution" ], "proofs": [ { "contents": [ "Note: in this proof $L\\pi_!$ denotes the left derived functor", "of $\\pi_!$ on abelian sheaves.", "Since $L\\pi_!$ commutes with colimits, it suffices", "to prove this for bounded above complexes of $\\mathcal{O}$-modules", "(compare with argument of", "Derived Categories, Proposition \\ref{derived-proposition-left-derived-exists}", "or just stick to bounded above complexes).", "Every such complex is quasi-isomorphic to a bounded above complex", "whose terms are direct sums of $j_{U!}\\mathcal{O}_U$ with", "$U \\in \\Ob(\\mathcal{C})$, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-module-quotient-flat}.", "Thus it suffices to prove the lemma", "for $j_{U!}\\mathcal{O}_U$. By assumption", "$$", "S_\\bullet = \\Mor_\\mathcal{C}(U_\\bullet, U)", "$$", "is a simplicial set homotopy equivalent to the constant simplicial", "set on a singleton. Set $P_n = \\mathcal{O}(U_n)$ and", "$P'_n = \\mathcal{O}'(U_n)$. Observe that the complex associated to the", "simplicial abelian group", "$$", "X_\\bullet : n \\longmapsto \\bigoplus\\nolimits_{s \\in S_n} P_n", "$$", "computes $L\\pi_!(j_{U!}\\mathcal{O}_U)$ by", "Lemma \\ref{lemma-compute-by-cosimplicial-resolution}.", "Since $j_{U!}\\mathcal{O}_U$ is a flat $\\mathcal{O}$-module we have", "$j_{U!}\\mathcal{O}_U \\otimes^\\mathbf{L}_\\mathcal{O} \\mathcal{O}' =", "j_{U!}\\mathcal{O}'_U$ and $L\\pi_!$ of this is computed by the complex", "associated to the simplicial abelian group", "$$", "X'_\\bullet : n \\longmapsto \\bigoplus\\nolimits_{s \\in S_n} P'_n", "$$", "As the rule which to a simplicial set $T_\\bullet$ associated the simplicial", "abelian group with terms $\\bigoplus_{t \\in T_n} P_n$ is a functor, we see", "that $X_\\bullet \\to P_\\bullet$ is a homotopy equivalence of simplicial", "abelian groups. Similarly, the rule which to a simplicial set", "$T_\\bullet$ associates the simplicial abelian group with terms", "$\\bigoplus_{t \\in T_n} P'_n$ is a functor. Hence $X'_\\bullet \\to P'_\\bullet$", "is a homotopy equivalence of simplicial abelian groups.", "By assumption $P_\\bullet \\to P'_\\bullet$ is a quasi-isomorphism", "(since $P_\\bullet$, resp.\\ $P'_\\bullet$ computes $L\\pi_!\\mathcal{O}$,", "resp. $L\\pi_!\\mathcal{O}'$ by", "Lemma \\ref{lemma-compute-by-cosimplicial-resolution}).", "We conclude that $X_\\bullet$ and $X'_\\bullet$ are quasi-isomorphic as desired." ], "refs": [ "derived-proposition-left-derived-exists", "sites-modules-lemma-module-quotient-flat", "sites-cohomology-lemma-compute-by-cosimplicial-resolution", "sites-cohomology-lemma-compute-by-cosimplicial-resolution" ], "ref_ids": [ 1964, 14203, 4348, 4348 ] } ], "ref_ids": [ 4348 ] }, { "id": 4353, "type": "theorem", "label": "sites-cohomology-lemma-compute-left-derived-pi-shriek-pre", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-compute-left-derived-pi-shriek-pre", "contents": [ "Assumptions and notation as in Situation \\ref{situation-fibred-category}.", "For $\\mathcal{F}$ in $\\textit{PAb}(\\mathcal{C})$ and $n \\geq 0$", "consider the abelian sheaf $L_n(\\mathcal{F})$ on $\\mathcal{D}$", "which is the sheaf associated to the presheaf", "$$", "V \\longmapsto H_n(\\mathcal{C}_V, \\mathcal{F}|_{\\mathcal{C}_V})", "$$", "with restriction maps as indicated in the proof. Then", "$L_n(\\mathcal{F}) = L_n(\\mathcal{F}^\\#)$." ], "refs": [], "proofs": [ { "contents": [ "For a morphism $h : V' \\to V$ of $\\mathcal{D}$ there is a", "pullback functor $h^* : \\mathcal{C}_V \\to \\mathcal{C}_{V'}$ of fibre", "categories (Categories, Definition", "\\ref{categories-definition-pullback-functor-fibred-category}).", "Moreover for $U \\in \\Ob(\\mathcal{C}_V)$ there is a", "strongly cartesian morphism $h^*U \\to U$ covering $h$.", "Restriction along these strongly cartesian morphisms defines a", "transformation of functors", "$$", "\\mathcal{F}|_{\\mathcal{C}_V}", "\\longrightarrow", "\\mathcal{F}|_{\\mathcal{C}_{V'}} \\circ h^*.", "$$", "By Example \\ref{example-morphism-categories}", "we obtain the desired restriction map", "$$", "H_n(\\mathcal{C}_V, \\mathcal{F}|_{\\mathcal{C}_V})", "\\longrightarrow", "H_n(\\mathcal{C}_{V'}, \\mathcal{F}|_{\\mathcal{C}_{V'}})", "$$", "Let us denote $L_{n, p}(\\mathcal{F})$ this presheaf, so that", "$L_n(\\mathcal{F}) = L_{n, p}(\\mathcal{F})^\\#$.", "The canonical map $\\gamma : \\mathcal{F} \\to \\mathcal{F}^+$", "(Sites, Theorem \\ref{sites-theorem-plus})", "defines a canonical", "map $L_{n, p}(\\mathcal{F}) \\to L_{n, p}(\\mathcal{F}^+)$.", "We have to prove this map becomes an isomorphism after sheafification.", "\\medskip\\noindent", "Let us use the computation of homology given in", "Example \\ref{example-left-derived-colimits}. Denote", "$K_\\bullet(\\mathcal{F}|_{\\mathcal{C}_V})$ the complex associated to", "the restriction of $\\mathcal{F}$ to the fibre category $\\mathcal{C}_V$.", "By the remarks above we obtain a presheaf $K_\\bullet(\\mathcal{F})$", "of complexes", "$$", "V \\longmapsto K_\\bullet(\\mathcal{F}|_{\\mathcal{C}_V})", "$$", "whose cohomology presheaves are the presheaves $L_{n, p}(\\mathcal{F})$.", "Thus it suffices to show that", "$$", "K_\\bullet(\\mathcal{F}) \\longrightarrow K_\\bullet(\\mathcal{F}^+)", "$$", "becomes an isomorphism on sheafification.", "\\medskip\\noindent", "Injectivity. Let $V$ be an object of $\\mathcal{D}$ and let", "$\\xi \\in K_n(\\mathcal{F})(V)$ be an element which maps", "to zero in $K_n(\\mathcal{F}^+)(V)$. We have to show there exists a", "covering $\\{V_j \\to V\\}$ such that $\\xi|_{V_j}$ is zero in", "$K_n(\\mathcal{F})(V_j)$. We write", "$$", "\\xi = \\sum (U_{i, n + 1} \\to \\ldots \\to U_{i, 0}, \\sigma_i)", "$$", "with $\\sigma_i \\in \\mathcal{F}(U_{i, 0})$. We arrange it so that", "each sequence of morphisms $U_n \\to \\ldots \\to U_0$ of $\\mathcal{C}_V$", "occurs are most once. Since the sums in the definition", "of the complex $K_\\bullet$ are direct sums, the only way this can map", "to zero in $K_\\bullet(\\mathcal{F}^+)(V)$ is if all $\\sigma_i$ map", "to zero in $\\mathcal{F}^+(U_{i, 0})$. By construction of", "$\\mathcal{F}^+$ there exist coverings $\\{U_{i, 0, j} \\to U_{i, 0}\\}$", "such that $\\sigma_i|_{U_{i, 0, j}}$ is zero. By our construction of", "the topology on $\\mathcal{C}$ we can write $U_{i, 0, j} \\to U_{i, 0}$", "as the pullback (Categories, Definition", "\\ref{categories-definition-pullback-functor-fibred-category})", "of some morphisms $V_{i, j} \\to V$ and moreover each", "$\\{V_{i, j} \\to V\\}$ is a covering. Choose a covering", "$\\{V_j \\to V\\}$ dominating each of the coverings $\\{V_{i, j} \\to V\\}$.", "Then it is clear that $\\xi|_{V_j} = 0$.", "\\medskip\\noindent", "Surjectivity. Proof omitted. Hint: Argue as in the proof of", "injectivity." ], "refs": [ "categories-definition-pullback-functor-fibred-category", "sites-theorem-plus", "categories-definition-pullback-functor-fibred-category" ], "ref_ids": [ 12389, 8492, 12389 ] } ], "ref_ids": [] }, { "id": 4354, "type": "theorem", "label": "sites-cohomology-lemma-compute-left-derived-pi-shriek", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-compute-left-derived-pi-shriek", "contents": [ "Assumptions and notation as in Situation \\ref{situation-fibred-category}.", "For $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C})$ and $n \\geq 0$", "the sheaf $L_n\\pi_!(\\mathcal{F})$ is equal to the sheaf", "$L_n(\\mathcal{F})$ constructed in", "Lemma \\ref{lemma-compute-left-derived-pi-shriek-pre}." ], "refs": [ "sites-cohomology-lemma-compute-left-derived-pi-shriek-pre" ], "proofs": [ { "contents": [ "Consider the sequence of functors $\\mathcal{F} \\mapsto L_n(\\mathcal{F})$", "from $\\textit{PAb}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{C})$.", "Since for each $V \\in \\Ob(\\mathcal{D})$ the sequence of functors", "$H_n(\\mathcal{C}_V, - )$ forms a $\\delta$-functor", "so do the functors $\\mathcal{F} \\mapsto L_n(\\mathcal{F})$.", "Our goal is to show these form a universal $\\delta$-functor.", "In order to do this we construct some abelian presheaves", "on which these functors vanish.", "\\medskip\\noindent", "For $U' \\in \\Ob(\\mathcal{C})$ consider the abelian presheaf", "$\\mathcal{F}_{U'} = j_{U'!}^{\\textit{PAb}}\\mathbf{Z}_{U'}$", "(Modules on Sites, Remark \\ref{sites-modules-remark-localize-presheaves}).", "Recall that", "$$", "\\mathcal{F}_{U'}(U) =", "\\bigoplus\\nolimits_{\\Mor_\\mathcal{C}(U, U')} \\mathbf{Z}", "$$", "If $U$ lies over $V = p(U)$ in $\\mathcal{D})$ and $U'$ lies over $V' = p(U')$", "then any morphism $a : U \\to U'$ factors uniquely as $U \\to h^*U' \\to U'$", "where $h = p(a) : V \\to V'$ (see", "Categories, Definition", "\\ref{categories-definition-pullback-functor-fibred-category}).", "Hence we see that", "$$", "\\mathcal{F}_{U'}|_{\\mathcal{C}_V}", "=", "\\bigoplus\\nolimits_{h \\in \\Mor_\\mathcal{D}(V, V')}", "j_{h^*U'!}\\mathbf{Z}_{h^*U'}", "$$", "where $j_{h^*U'} : \\Sh(\\mathcal{C}_V/h^*U') \\to \\Sh(\\mathcal{C}_V)$", "is the localization morphism. The sheaves $j_{h^*U'!}\\mathbf{Z}_{h^*U'}$", "have vanishing higher homology groups (see", "Example \\ref{example-left-derived-colimits}).", "We conclude that $L_n(\\mathcal{F}_{U'}) = 0$ for all $n > 0$ and all $U'$.", "It follows that any abelian presheaf $\\mathcal{F}$ is a quotient", "of an abelian presheaf $\\mathcal{G}$ with $L_n(\\mathcal{G}) = 0$ for", "all $n > 0$ (Modules on Sites, Lemma", "\\ref{sites-modules-lemma-module-quotient-flat}).", "Since $L_n(\\mathcal{F}) = L_n(\\mathcal{F}^\\#)$ we see", "that the same thing is true for abelian sheaves. Thus", "the sequence of functors $L_n(-)$ is a universal delta functor", "on $\\textit{Ab}(\\mathcal{C})$", "(Homology, Lemma \\ref{homology-lemma-efface-implies-universal}).", "Since we have agreement with", "$H^{-n}(L\\pi_!(-))$ for $n = 0$ by", "Lemma \\ref{lemma-compute-pi-shriek}", "we conclude by uniqueness of universal $\\delta$-functors", "(Homology, Lemma \\ref{homology-lemma-uniqueness-universal-delta-functor})", "and", "Derived Categories, Lemma \\ref{derived-lemma-right-derived-delta-functor}." ], "refs": [ "sites-modules-remark-localize-presheaves", "categories-definition-pullback-functor-fibred-category", "sites-modules-lemma-module-quotient-flat", "homology-lemma-efface-implies-universal", "sites-cohomology-lemma-compute-pi-shriek", "homology-lemma-uniqueness-universal-delta-functor", "derived-lemma-right-derived-delta-functor" ], "ref_ids": [ 14308, 12389, 14203, 12052, 4345, 12053, 1843 ] } ], "ref_ids": [ 4353 ] }, { "id": 4355, "type": "theorem", "label": "sites-cohomology-lemma-compute-left-derived-g-shriek", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-compute-left-derived-g-shriek", "contents": [ "Assumptions and notation as in", "Situation \\ref{situation-morphism-fibred-categories}.", "For an abelian sheaf $\\mathcal{F}'$ on $\\mathcal{C}'$ the sheaf", "$L_ng_!(\\mathcal{F}')$ is the sheaf associated to the presheaf", "$$", "U \\longmapsto H_n(\\mathcal{I}_U, \\mathcal{F}'_U)", "$$", "For notation and restriction maps see proof." ], "refs": [], "proofs": [ { "contents": [ "Say $p(U) = V$. The category $\\mathcal{I}_U$ is the category of pairs", "$(U', \\varphi)$ where $\\varphi : U \\to u(U')$ is a morphism of $\\mathcal{C}$", "with $p(\\varphi) = \\text{id}_V$, i.e., $\\varphi$ is a morphism of the", "fibre category $\\mathcal{C}_V$. Morphisms", "$(U'_1, \\varphi_1) \\to (U'_2, \\varphi_2)$ are given by morphisms", "$a : U'_1 \\to U'_2$ of the fibre category $\\mathcal{C}'_V$ such that", "$\\varphi_2 = u(a) \\circ \\varphi_1$. The presheaf $\\mathcal{F}'_U$ sends", "$(U', \\varphi)$ to $\\mathcal{F}'(U')$.", "We will construct the restriction mappings below.", "\\medskip\\noindent", "Choose a factorization", "$$", "\\xymatrix{", "\\mathcal{C}' \\ar@<1ex>[r]^{u'} &", "\\mathcal{C}'' \\ar[r]^{u''} \\ar@<1ex>[l]^w & \\mathcal{C}", "}", "$$", "of $u$ as in", "Categories, Lemma \\ref{categories-lemma-ameliorate-morphism-fibred-categories}.", "Then $g_! = g''_! \\circ g'_!$ and similarly for derived functors.", "On the other hand, the functor $g'_!$ is exact, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-have-left-adjoint}.", "Thus we get $Lg_!(\\mathcal{F}') = Lg''_!(\\mathcal{F}'')$ where", "$\\mathcal{F}'' = g'_!\\mathcal{F}'$. Note that", "$\\mathcal{F}'' = h^{-1}\\mathcal{F}'$ where", "$h : \\Sh(\\mathcal{C}'') \\to \\Sh(\\mathcal{C}')$ is the morphism of topoi", "associated to $w$, see", "Sites, Lemma \\ref{sites-lemma-have-left-adjoint}.", "The functor $u''$ turns $\\mathcal{C}''$ into a fibred category", "over $\\mathcal{C}$, hence", "Lemma \\ref{lemma-compute-left-derived-pi-shriek}", "applies to the computation of $L_ng''_!$. The result follows as the", "construction of $\\mathcal{C}''$ in the proof of", "Categories, Lemma \\ref{categories-lemma-ameliorate-morphism-fibred-categories}", "shows that the fibre category $\\mathcal{C}''_U$ is equal to", "$\\mathcal{I}_U$. Moreover, $h^{-1}\\mathcal{F}'|_{\\mathcal{C}''_U}$", "is given by the rule described above", "(as $w$ is continuous and cocontinuous by", "Stacks, Lemma \\ref{stacks-lemma-topology-inherited-functorial}", "so we may apply", "Sites, Lemma \\ref{sites-lemma-when-shriek})." ], "refs": [ "categories-lemma-ameliorate-morphism-fibred-categories", "sites-modules-lemma-have-left-adjoint", "sites-lemma-have-left-adjoint", "sites-cohomology-lemma-compute-left-derived-pi-shriek", "categories-lemma-ameliorate-morphism-fibred-categories", "stacks-lemma-topology-inherited-functorial", "sites-lemma-when-shriek" ], "ref_ids": [ 12291, 14167, 8551, 4354, 12291, 8970, 8545 ] } ], "ref_ids": [] }, { "id": 4356, "type": "theorem", "label": "sites-cohomology-lemma-base-change-by-qis", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-base-change-by-qis", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A}_\\bullet \\to \\mathcal{B}_\\bullet$", "be a homomorphism of simplicial sheaves of rings on $\\mathcal{C}$.", "If $L\\pi_!\\mathcal{A}_\\bullet \\to L\\pi_!\\mathcal{B}_\\bullet$ is an", "isomorphism in $D(\\mathcal{C})$, then we have", "$$", "L\\pi_!(K) =", "L\\pi_!(K \\otimes^\\mathbf{L}_{\\mathcal{A}_\\bullet} \\mathcal{B}_\\bullet)", "$$", "for all $K$ in $D(\\mathcal{A}_\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "Let $([n], U)$ be an object of $\\Delta \\times \\mathcal{C}$. Since $L\\pi_!$", "commutes with colimits, it suffices to prove this for bounded above complexes", "of $\\mathcal{O}$-modules (compare with argument of", "Derived Categories, Proposition \\ref{derived-proposition-left-derived-exists}", "or just stick to bounded above complexes). Every such complex is", "quasi-isomorphic to a bounded above complex whose terms are flat modules, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-module-quotient-flat}.", "Thus it suffices to prove the lemma for a flat $\\mathcal{A}_\\bullet$-module", "$\\mathcal{F}$. In this case the derived tensor product is the usual", "tensor product and is a sheaf also. Hence by", "Lemma \\ref{lemma-compute-left-derived-pi-shriek}", "we can compute the cohomology", "sheaves of both sides of the equation by the procedure of", "Lemma \\ref{lemma-compute-left-derived-pi-shriek-pre}.", "Thus it suffices to prove the result for the restriction of", "$\\mathcal{F}$ to the fibre categories (i.e., to $\\Delta \\times U$).", "In this case the result follows from Lemma \\ref{lemma-O-homology-qis}." ], "refs": [ "derived-proposition-left-derived-exists", "sites-modules-lemma-module-quotient-flat", "sites-cohomology-lemma-compute-left-derived-pi-shriek", "sites-cohomology-lemma-compute-left-derived-pi-shriek-pre", "sites-cohomology-lemma-O-homology-qis" ], "ref_ids": [ 1964, 14203, 4354, 4353, 4352 ] } ], "ref_ids": [] }, { "id": 4357, "type": "theorem", "label": "sites-cohomology-lemma-cone", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cone", "contents": [ "The cone on a morphism of strictly perfect complexes is", "strictly perfect." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4358, "type": "theorem", "label": "sites-cohomology-lemma-tensor", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-tensor", "contents": [ "The total complex associated to the tensor product of two", "strictly perfect complexes is strictly perfect." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4359, "type": "theorem", "label": "sites-cohomology-lemma-strictly-perfect-pullback", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-strictly-perfect-pullback", "contents": [ "Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to", "(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi. If $\\mathcal{F}^\\bullet$ is a strictly", "perfect complex of $\\mathcal{O}_\\mathcal{D}$-modules, then", "$f^*\\mathcal{F}^\\bullet$ is a strictly perfect complex of", "$\\mathcal{O}_\\mathcal{C}$-modules." ], "refs": [], "proofs": [ { "contents": [ "We have seen in", "Modules on Sites, Lemma \\ref{sites-modules-lemma-global-pullback}", "that the pullback of a finite free module is finite free. The functor", "$f^*$ is additive functor hence preserves direct summands. The lemma follows." ], "refs": [ "sites-modules-lemma-global-pullback" ], "ref_ids": [ 14168 ] } ], "ref_ids": [] }, { "id": 4360, "type": "theorem", "label": "sites-cohomology-lemma-local-lift-map", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-local-lift-map", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of", "$\\mathcal{C}$. Given a solid diagram of $\\mathcal{O}_U$-modules", "$$", "\\xymatrix{", "\\mathcal{E} \\ar@{..>}[dr] \\ar[r] & \\mathcal{F} \\\\", "& \\mathcal{G} \\ar[u]_p", "}", "$$", "with $\\mathcal{E}$ a direct summand of a finite free", "$\\mathcal{O}_U$-module and $p$ surjective, then there exists a", "covering $\\{U_i \\to U\\}$ such that a dotted arrow", "making the diagram commute exists over each $U_i$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $\\mathcal{E} = \\mathcal{O}_U^{\\oplus n}$ for some $n$.", "In this case finding the dotted arrow is equivalent to lifting the", "images of the basis elements in $\\Gamma(U, \\mathcal{F})$. This is", "locally possible by the characterization of surjective maps of", "sheaves (Sites, Section \\ref{sites-section-sheaves-injective})." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4361, "type": "theorem", "label": "sites-cohomology-lemma-local-homotopy", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-local-homotopy", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object", "of $\\mathcal{C}$.", "\\begin{enumerate}", "\\item Let $\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$", "be a morphism of complexes of $\\mathcal{O}_U$-modules", "with $\\mathcal{E}^\\bullet$ strictly perfect and $\\mathcal{F}^\\bullet$", "acyclic. Then there exists a covering $\\{U_i \\to U\\}$ such that each", "$\\alpha|_{U_i}$ is homotopic to zero.", "\\item Let $\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$", "be a morphism of complexes of $\\mathcal{O}_U$-modules", "with $\\mathcal{E}^\\bullet$ strictly perfect, $\\mathcal{E}^i = 0$", "for $i < a$, and $H^i(\\mathcal{F}^\\bullet) = 0$ for $i \\geq a$.", "Then there exists a covering $\\{U_i \\to U\\}$ such that each", "$\\alpha|_{U_i}$ is homotopic to zero.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from the second, hence we only prove (2).", "We will prove this by induction on the length of the complex", "$\\mathcal{E}^\\bullet$. If $\\mathcal{E}^\\bullet \\cong \\mathcal{E}[-n]$", "for some direct summand $\\mathcal{E}$ of a finite free", "$\\mathcal{O}$-module and integer $n \\geq a$, then the result follows from", "Lemma \\ref{lemma-local-lift-map} and the fact that", "$\\mathcal{F}^{n - 1} \\to \\Ker(\\mathcal{F}^n \\to \\mathcal{F}^{n + 1})$", "is surjective by the assumed vanishing of $H^n(\\mathcal{F}^\\bullet)$.", "If $\\mathcal{E}^i$ is zero except for $i \\in [a, b]$, then we have a", "split exact sequence of complexes", "$$", "0 \\to \\mathcal{E}^b[-b] \\to \\mathcal{E}^\\bullet \\to", "\\sigma_{\\leq b - 1}\\mathcal{E}^\\bullet \\to 0", "$$", "which determines a distinguished triangle in", "$K(\\mathcal{O}_U)$. Hence an exact sequence", "$$", "\\Hom_{K(\\mathcal{O}_U)}(", "\\sigma_{\\leq b - 1}\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)", "\\to", "\\Hom_{K(\\mathcal{O}_U)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)", "\\to", "\\Hom_{K(\\mathcal{O}_U)}(\\mathcal{E}^b[-b], \\mathcal{F}^\\bullet)", "$$", "by the axioms of triangulated categories. The composition", "$\\mathcal{E}^b[-b] \\to \\mathcal{F}^\\bullet$ is homotopic to zero", "on the members of a covering of $U$ by the above,", "whence we may assume our map comes from an element in the", "left hand side of the displayed exact sequence above. This element", "is zero on the members of a covering of $U$ by induction hypothesis." ], "refs": [ "sites-cohomology-lemma-local-lift-map" ], "ref_ids": [ 4360 ] } ], "ref_ids": [] }, { "id": 4362, "type": "theorem", "label": "sites-cohomology-lemma-lift-through-quasi-isomorphism", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-lift-through-quasi-isomorphism", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of", "$\\mathcal{C}$. Given a solid diagram of complexes of $\\mathcal{O}_U$-modules", "$$", "\\xymatrix{", "\\mathcal{E}^\\bullet \\ar@{..>}[dr] \\ar[r]_\\alpha & \\mathcal{F}^\\bullet \\\\", "& \\mathcal{G}^\\bullet \\ar[u]_f", "}", "$$", "with $\\mathcal{E}^\\bullet$ strictly perfect, $\\mathcal{E}^j = 0$ for", "$j < a$ and $H^j(f)$ an isomorphism for $j > a$ and surjective for", "$j = a$, then there exists a covering $\\{U_i \\to U\\}$ and for each $i$", "a dotted arrow over $U_i$ making the diagram commute up to homotopy." ], "refs": [], "proofs": [ { "contents": [ "Our assumptions on $f$ imply the cone $C(f)^\\bullet$ has vanishing", "cohomology sheaves in degrees $\\geq a$.", "Hence Lemma \\ref{lemma-local-homotopy} guarantees there is a", "covering $\\{U_i \\to U\\}$ such that the composition", "$\\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet \\to C(f)^\\bullet$", "is homotopic to zero over $U_i$. Since", "$$", "\\mathcal{G}^\\bullet \\to \\mathcal{F}^\\bullet \\to C(f)^\\bullet \\to", "\\mathcal{G}^\\bullet[1]", "$$", "restricts to a distinguished triangle in $K(\\mathcal{O}_{U_i})$", "we see that we can lift $\\alpha|_{U_i}$ up to homotopy to a map", "$\\alpha_i : \\mathcal{E}^\\bullet|_{U_i} \\to \\mathcal{G}^\\bullet|_{U_i}$", "as desired." ], "refs": [ "sites-cohomology-lemma-local-homotopy" ], "ref_ids": [ 4361 ] } ], "ref_ids": [] }, { "id": 4363, "type": "theorem", "label": "sites-cohomology-lemma-local-actual", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-local-actual", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object", "of $\\mathcal{C}$. Let $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be", "complexes of $\\mathcal{O}_U$-modules with $\\mathcal{E}^\\bullet$ strictly", "perfect.", "\\begin{enumerate}", "\\item For any element", "$\\alpha \\in \\Hom_{D(\\mathcal{O}_U)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$", "there exists a covering $\\{U_i \\to U\\}$ such that", "$\\alpha|_{U_i}$ is given by a morphism of complexes", "$\\alpha_i : \\mathcal{E}^\\bullet|_{U_i} \\to \\mathcal{F}^\\bullet|_{U_i}$.", "\\item Given a morphism of complexes", "$\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$", "whose image in the group", "$\\Hom_{D(\\mathcal{O}_U)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$", "is zero, there exists a covering $\\{U_i \\to U\\}$ such that", "$\\alpha|_{U_i}$ is homotopic to zero.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1).", "By the construction of the derived category we can find a quasi-isomorphism", "$f : \\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$ and a map of complexes", "$\\beta : \\mathcal{E}^\\bullet \\to \\mathcal{G}^\\bullet$ such that", "$\\alpha = f^{-1}\\beta$. Thus the result follows from", "Lemma \\ref{lemma-lift-through-quasi-isomorphism}.", "We omit the proof of (2)." ], "refs": [ "sites-cohomology-lemma-lift-through-quasi-isomorphism" ], "ref_ids": [ 4362 ] } ], "ref_ids": [] }, { "id": 4364, "type": "theorem", "label": "sites-cohomology-lemma-Rhom-strictly-perfect", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-Rhom-strictly-perfect", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes", "of $\\mathcal{O}$-modules with $\\mathcal{E}^\\bullet$ strictly perfect.", "Then the internal hom $R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$", "is represented by the complex $\\mathcal{H}^\\bullet$ with terms", "$$", "\\mathcal{H}^n =", "\\bigoplus\\nolimits_{n = p + q}", "\\SheafHom_\\mathcal{O}(\\mathcal{E}^{-q}, \\mathcal{F}^p)", "$$", "and differential as described in Section \\ref{section-internal-hom}." ], "refs": [], "proofs": [ { "contents": [ "Choose a quasi-isomorphism $\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$", "into a K-injective complex. Let $(\\mathcal{H}')^\\bullet$ be the", "complex with terms", "$$", "(\\mathcal{H}')^n =", "\\prod\\nolimits_{n = p + q}", "\\SheafHom_\\mathcal{O}(\\mathcal{L}^{-q}, \\mathcal{I}^p)", "$$", "which represents $R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$", "by the construction in Section \\ref{section-internal-hom}. It suffices", "to show that the map", "$$", "\\mathcal{H}^\\bullet \\longrightarrow (\\mathcal{H}')^\\bullet", "$$", "is a quasi-isomorphism. Given an object $U$ of $\\mathcal{C}$ we have", "by inspection", "$$", "H^0(\\mathcal{H}^\\bullet(U)) =", "\\Hom_{K(\\mathcal{O}_U)}(\\mathcal{E}^\\bullet|_U, \\mathcal{K}^\\bullet|_U)", "\\to", "H^0((\\mathcal{H}')^\\bullet(U)) =", "\\Hom_{D(\\mathcal{O}_U)}(\\mathcal{E}^\\bullet|_U, \\mathcal{K}^\\bullet|_U)", "$$", "By Lemma \\ref{lemma-local-actual} the sheafification of", "$U \\mapsto H^0(\\mathcal{H}^\\bullet(U))$", "is equal to the sheafification of", "$U \\mapsto H^0((\\mathcal{H}')^\\bullet(U))$. A similar argument can be", "given for the other cohomology sheaves. Thus $\\mathcal{H}^\\bullet$", "is quasi-isomorphic to $(\\mathcal{H}')^\\bullet$ which proves the lemma." ], "refs": [ "sites-cohomology-lemma-local-actual" ], "ref_ids": [ 4363 ] } ], "ref_ids": [] }, { "id": 4365, "type": "theorem", "label": "sites-cohomology-lemma-Rhom-complex-of-direct-summands-finite-free", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-Rhom-complex-of-direct-summands-finite-free", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes", "of $\\mathcal{O}$-modules with", "\\begin{enumerate}", "\\item $\\mathcal{F}^n = 0$ for $n \\ll 0$,", "\\item $\\mathcal{E}^n = 0$ for $n \\gg 0$, and", "\\item $\\mathcal{E}^n$ isomorphic to a direct summand of a finite", "free $\\mathcal{O}$-module.", "\\end{enumerate}", "Then the internal hom $R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$", "is represented by the complex $\\mathcal{H}^\\bullet$ with terms", "$$", "\\mathcal{H}^n =", "\\bigoplus\\nolimits_{n = p + q}", "\\SheafHom_\\mathcal{O}(\\mathcal{E}^{-q}, \\mathcal{F}^p)", "$$", "and differential as described in Section \\ref{section-internal-hom}." ], "refs": [], "proofs": [ { "contents": [ "Choose a quasi-isomorphism $\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$", "where $\\mathcal{I}^\\bullet$ is a bounded below complex of injectives.", "Note that $\\mathcal{I}^\\bullet$ is K-injective", "(Derived Categories, Lemma", "\\ref{derived-lemma-bounded-below-injectives-K-injective}).", "Hence the construction in Section \\ref{section-internal-hom}", "shows that", "$R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$ is ", "represented by the complex $(\\mathcal{H}')^\\bullet$ with terms", "$$", "(\\mathcal{H}')^n =", "\\prod\\nolimits_{n = p + q}", "\\SheafHom_\\mathcal{O}(\\mathcal{E}^{-q}, \\mathcal{I}^p) =", "\\bigoplus\\nolimits_{n = p + q}", "\\SheafHom_\\mathcal{O}(\\mathcal{E}^{-q}, \\mathcal{I}^p)", "$$", "(equality because there are only finitely many nonzero terms).", "Note that $\\mathcal{H}^\\bullet$ is the total complex associated to", "the double complex with terms", "$\\SheafHom_\\mathcal{O}(\\mathcal{E}^{-q}, \\mathcal{F}^p)$", "and similarly for $(\\mathcal{H}')^\\bullet$.", "The natural map $(\\mathcal{H}')^\\bullet \\to \\mathcal{H}^\\bullet$", "comes from a map of double complexes.", "Thus to show this map is a quasi-isomorphism, we may use the spectral", "sequence of a double complex", "(Homology, Lemma \\ref{homology-lemma-first-quadrant-ss})", "$$", "{}'E_1^{p, q} =", "H^p(\\SheafHom_\\mathcal{O}(\\mathcal{E}^{-q}, \\mathcal{F}^\\bullet))", "$$", "converging to $H^{p + q}(\\mathcal{H}^\\bullet)$ and similarly for", "$(\\mathcal{H}')^\\bullet$. To finish the proof of the lemma it", "suffices to show that $\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$", "induces an isomorphism", "$$", "H^p(\\SheafHom_\\mathcal{O}(\\mathcal{E}, \\mathcal{F}^\\bullet))", "\\longrightarrow", "H^p(\\SheafHom_\\mathcal{O}(\\mathcal{E}, \\mathcal{I}^\\bullet))", "$$", "on cohomology sheaves whenever $\\mathcal{E}$ is a direct summand of a", "finite free $\\mathcal{O}$-module. Since this is clear when $\\mathcal{E}$", "is finite free the result follows." ], "refs": [ "derived-lemma-bounded-below-injectives-K-injective", "homology-lemma-first-quadrant-ss" ], "ref_ids": [ 1910, 12105 ] } ], "ref_ids": [] }, { "id": 4366, "type": "theorem", "label": "sites-cohomology-lemma-pseudo-coherent-independent-representative", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-pseudo-coherent-independent-representative", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $E$ be an object of $D(\\mathcal{O})$.", "\\begin{enumerate}", "\\item If $\\mathcal{C}$ has a final object $X$ and if there exist a covering", "$\\{U_i \\to X\\}$, strictly perfect complexes $\\mathcal{E}_i^\\bullet$ of", "$\\mathcal{O}_{U_i}$-modules, and", "maps $\\alpha_i : \\mathcal{E}_i^\\bullet \\to E|_{U_i}$ in", "$D(\\mathcal{O}_{U_i})$ with $H^j(\\alpha_i)$ an isomorphism for $j > m$", "and $H^m(\\alpha_i)$ surjective, then $E$ is $m$-pseudo-coherent.", "\\item If $E$ is $m$-pseudo-coherent, then any complex of $\\mathcal{O}$-modules", "representing $E$ is $m$-pseudo-coherent.", "\\item If for every object $U$ of $\\mathcal{C}$ there exists a covering", "$\\{U_i \\to U\\}$ such that $E|_{U_i}$ is $m$-pseudo-coherent, then", "$E$ is $m$-pseudo-coherent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}^\\bullet$ be any complex representing $E$", "and let $X$, $\\{U_i \\to X\\}$, and $\\alpha_i : \\mathcal{E}_i \\to E|_{U_i}$", "be as in (1). We will show that $\\mathcal{F}^\\bullet$ is $m$-pseudo-coherent", "as a complex, which will prove (1) and (2) in case $\\mathcal{C}$ has a", "final object. By Lemma \\ref{lemma-local-actual}", "we can after refining the covering $\\{U_i \\to X\\}$", "represent the maps $\\alpha_i$ by maps of complexes", "$\\alpha_i : \\mathcal{E}_i^\\bullet \\to \\mathcal{F}^\\bullet|_{U_i}$.", "By assumption", "$H^j(\\alpha_i)$ are isomorphisms for $j > m$, and $H^m(\\alpha_i)$", "is surjective whence $\\mathcal{F}^\\bullet$ is", "$m$-pseudo-coherent.", "\\medskip\\noindent", "Proof of (2). By the above we see that $\\mathcal{F}^\\bullet|_U$ is", "$m$-pseudo-coherent as a complex of $\\mathcal{O}_U$-modules for all", "objects $U$ of $\\mathcal{C}$. It is a formal consequence of the definitions", "that $\\mathcal{F}^\\bullet$ is $m$-pseudo-coherent.", "\\medskip\\noindent", "Proof of (3). Follows from the definitions and", "Sites, Definition \\ref{sites-definition-site} part (2)." ], "refs": [ "sites-cohomology-lemma-local-actual", "sites-definition-site" ], "ref_ids": [ 4363, 8652 ] } ], "ref_ids": [] }, { "id": 4367, "type": "theorem", "label": "sites-cohomology-lemma-pseudo-coherent-pullback", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-pseudo-coherent-pullback", "contents": [ "Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to", "(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed sites. Let $E$ be an object of", "$D(\\mathcal{O}_\\mathcal{C})$. If $E$ is $m$-pseudo-coherent,", "then $Lf^*E$ is $m$-pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "Say $f$ is given by the functor $u : \\mathcal{D} \\to \\mathcal{C}$.", "Let $U$ be an object of $\\mathcal{C}$. By", "Sites, Lemma \\ref{sites-lemma-morphism-of-sites-covering}", "we can find a covering $\\{U_i \\to U\\}$ and for each $i$ a morphism", "$U_i \\to u(V_i)$ for some object $V_i$ of $\\mathcal{D}$.", "By Lemma \\ref{lemma-pseudo-coherent-independent-representative}", "it suffices to show that $Lf^*E|_{U_i}$ is $m$-pseudo-coherent.", "To do this it is enough to show that $Lf^*E|_{u(V_i)}$ is", "$m$-pseudo-coherent, since $Lf^*E|_{U_i}$ is the restriction", "of $Lf^*E|_{u(V_i)}$ to $\\mathcal{C}/U_i$ (via", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-relocalize}).", "By the commutative diagram of", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-localize-morphism-ringed-sites}", "it suffices to prove the lemma for the morphism of ringed", "sites $(\\mathcal{C}/u(V_i), \\mathcal{O}_{u(V_i)}) \\to", "(\\mathcal{D}/V_i, \\mathcal{O}_{V_i})$.", "Thus we may assume $\\mathcal{D}$ has a final object $Y$ such that", "$X = u(Y)$ is a final object of $\\mathcal{C}$.", "\\medskip\\noindent", "Let $\\{V_i \\to Y\\}$ be a covering such that for each $i$ there exists", "a strictly perfect complex $\\mathcal{F}_i^\\bullet$ of", "$\\mathcal{O}_{V_i}$-modules and a morphism", "$\\alpha_i : \\mathcal{F}_i^\\bullet \\to E|_{V_i}$ of $D(\\mathcal{O}_{V_i})$", "such that $H^j(\\alpha_i)$ is an isomorphism", "for $j > m$ and $H^m(\\alpha_i)$ is surjective.", "Arguing as above it suffices to prove the result for", "$(\\mathcal{C}/u(V_i), \\mathcal{O}_{u(V_i)}) \\to", "(\\mathcal{D}/V_i, \\mathcal{O}_{V_i})$. Hence we may assume that", "there exists a strictly perfect complex $\\mathcal{F}^\\bullet$ of", "$\\mathcal{O}_\\mathcal{D}$-modules and a morphism", "$\\alpha : \\mathcal{F}^\\bullet \\to E$ of $D(\\mathcal{O}_\\mathcal{D})$", "such that $H^j(\\alpha)$ is an isomorphism", "for $j > m$ and $H^m(\\alpha)$ is surjective. In this case, choose", "a distinguished triangle", "$$", "\\mathcal{F}^\\bullet \\to E \\to C \\to \\mathcal{F}^\\bullet[1]", "$$", "The assumption on $\\alpha$ means exactly that the cohomology sheaves", "$H^j(C)$ are zero for all $j \\geq m$. Applying $Lf^*$ we obtain", "the distinguished triangle", "$$", "Lf^*\\mathcal{F}^\\bullet \\to Lf^*E \\to Lf^*C \\to Lf^*\\mathcal{F}^\\bullet[1]", "$$", "By the construction of $Lf^*$ as a left derived functor we see that", "$H^j(Lf^*C) = 0$ for $j \\geq m$ (by the dual of Derived Categories, Lemma", "\\ref{derived-lemma-negative-vanishing}). Hence $H^j(Lf^*\\alpha)$ is an", "isomorphism for $j > m$ and $H^m(Lf^*\\alpha)$ is surjective.", "On the other hand, since", "$\\mathcal{F}^\\bullet$ is a bounded above complex of flat", "$\\mathcal{O}_\\mathcal{D}$-modules we see that", "$Lf^*\\mathcal{F}^\\bullet = f^*\\mathcal{F}^\\bullet$.", "Applying Lemma \\ref{lemma-strictly-perfect-pullback} we conclude." ], "refs": [ "sites-lemma-morphism-of-sites-covering", "sites-cohomology-lemma-pseudo-coherent-independent-representative", "sites-modules-lemma-relocalize", "sites-modules-lemma-localize-morphism-ringed-sites", "derived-lemma-negative-vanishing", "sites-cohomology-lemma-strictly-perfect-pullback" ], "ref_ids": [ 8527, 4366, 14172, 14174, 1839, 4359 ] } ], "ref_ids": [] }, { "id": 4368, "type": "theorem", "label": "sites-cohomology-lemma-cone-pseudo-coherent", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cone-pseudo-coherent", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site and $m \\in \\mathbf{Z}$.", "Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\\mathcal{O})$.", "\\begin{enumerate}", "\\item If $K$ is $(m + 1)$-pseudo-coherent and $L$ is $m$-pseudo-coherent", "then $M$ is $m$-pseudo-coherent.", "\\item If $K$ and $M$ are $m$-pseudo-coherent, then $L$ is $m$-pseudo-coherent.", "\\item If $L$ is $(m + 1)$-pseudo-coherent and $M$", "is $m$-pseudo-coherent, then $K$ is $(m + 1)$-pseudo-coherent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $U$ be an object of $\\mathcal{C}$. Choose a covering", "$\\{U_i \\to U\\}$ and maps $\\alpha_i : \\mathcal{K}_i^\\bullet \\to K|_{U_i}$", "in $D(\\mathcal{O}_{U_i})$ with $\\mathcal{K}_i^\\bullet$ strictly perfect and", "$H^j(\\alpha_i)$ isomorphisms for $j > m + 1$ and surjective for $j = m + 1$.", "We may replace $\\mathcal{K}_i^\\bullet$ by", "$\\sigma_{\\geq m + 1}\\mathcal{K}_i^\\bullet$", "and hence we may assume that $\\mathcal{K}_i^j = 0$", "for $j < m + 1$. After refining the covering we may choose", "maps $\\beta_i : \\mathcal{L}_i^\\bullet \\to L|_{U_i}$ in $D(\\mathcal{O}_{U_i})$", "with $\\mathcal{L}_i^\\bullet$ strictly perfect such that", "$H^j(\\beta)$ is an isomorphism for $j > m$ and", "surjective for $j = m$. By", "Lemma \\ref{lemma-lift-through-quasi-isomorphism}", "we can, after refining the covering, find maps of complexes", "$\\gamma_i : \\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet$", "such that the diagrams", "$$", "\\xymatrix{", "K|_{U_i} \\ar[r] & L|_{U_i} \\\\", "\\mathcal{K}_i^\\bullet \\ar[u]^{\\alpha_i} \\ar[r]^{\\gamma_i} &", "\\mathcal{L}_i^\\bullet \\ar[u]_{\\beta_i}", "}", "$$", "are commutative in $D(\\mathcal{O}_{U_i})$ (this requires representing the", "maps $\\alpha_i$, $\\beta_i$ and $K|_{U_i} \\to L|_{U_i}$", "by actual maps of complexes; some details omitted).", "The cone $C(\\gamma_i)^\\bullet$ is strictly perfect (Lemma \\ref{lemma-cone}).", "The commutativity of the diagram implies that there exists a morphism", "of distinguished triangles", "$$", "(\\mathcal{K}_i^\\bullet, \\mathcal{L}_i^\\bullet, C(\\gamma_i)^\\bullet)", "\\longrightarrow", "(K|_{U_i}, L|_{U_i}, M|_{U_i}).", "$$", "It follows from the induced map on long exact cohomology sequences and", "Homology, Lemmas \\ref{homology-lemma-four-lemma} and", "\\ref{homology-lemma-five-lemma}", "that $C(\\gamma_i)^\\bullet \\to M|_{U_i}$ induces an isomorphism", "on cohomology in degrees $> m$ and a surjection in degree $m$.", "Hence $M$ is $m$-pseudo-coherent by", "Lemma \\ref{lemma-pseudo-coherent-independent-representative}.", "\\medskip\\noindent", "Assertions (2) and (3) follow from (1) by rotating the distinguished", "triangle." ], "refs": [ "sites-cohomology-lemma-lift-through-quasi-isomorphism", "sites-cohomology-lemma-cone", "homology-lemma-four-lemma", "homology-lemma-five-lemma", "sites-cohomology-lemma-pseudo-coherent-independent-representative" ], "ref_ids": [ 4362, 4357, 12029, 12030, 4366 ] } ], "ref_ids": [] }, { "id": 4369, "type": "theorem", "label": "sites-cohomology-lemma-tensor-pseudo-coherent", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-tensor-pseudo-coherent", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L$ be objects", "of $D(\\mathcal{O})$.", "\\begin{enumerate}", "\\item If $K$ is $n$-pseudo-coherent and $H^i(K) = 0$ for $i > a$", "and $L$ is $m$-pseudo-coherent and $H^j(L) = 0$ for $j > b$, then", "$K \\otimes_\\mathcal{O}^\\mathbf{L} L$ is $t$-pseudo-coherent", "with $t = \\max(m + a, n + b)$.", "\\item If $K$ and $L$ are pseudo-coherent, then", "$K \\otimes_\\mathcal{O}^\\mathbf{L} L$ is pseudo-coherent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $U$ be an object of $\\mathcal{C}$.", "By replacing $U$ by the members of a covering", "and replacing $\\mathcal{C}$ by the localization $\\mathcal{C}/U$", "we may assume there exist strictly perfect complexes $\\mathcal{K}^\\bullet$", "and $\\mathcal{L}^\\bullet$ and maps", "$\\alpha : \\mathcal{K}^\\bullet \\to K$ and", "$\\beta : \\mathcal{L}^\\bullet \\to L$ with $H^i(\\alpha)$ and isomorphism", "for $i > n$ and surjective for $i = n$ and with", "$H^i(\\beta)$ and isomorphism for $i > m$ and surjective for $i = m$.", "Then the map", "$$", "\\alpha \\otimes^\\mathbf{L} \\beta :", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet)", "\\to K \\otimes_\\mathcal{O}^\\mathbf{L} L", "$$", "induces isomorphisms on cohomology sheaves in degree $i$ for", "$i > t$ and a surjection for $i = t$. This follows from the", "spectral sequence of tors (details omitted).", "\\medskip\\noindent", "Proof of (2). Let $U$ be an object of $\\mathcal{C}$.", "We may first replace $U$ by the members of a covering", "and $\\mathcal{C}$ by the localization $\\mathcal{C}/U$", "to reduce to the case that $K$ and $L$ are bounded above.", "Then the statement follows immediately from case (1)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4370, "type": "theorem", "label": "sites-cohomology-lemma-summands-pseudo-coherent", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-summands-pseudo-coherent", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $m \\in \\mathbf{Z}$.", "If $K \\oplus L$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)", "in $D(\\mathcal{O})$ so are $K$ and $L$." ], "refs": [], "proofs": [ { "contents": [ "Assume that $K \\oplus L$ is $m$-pseudo-coherent. Let $U$ be an object of", "$\\mathcal{C}$. After replacing $U$ by the members of a covering we may", "assume $K \\oplus L \\in D^-(\\mathcal{O}_U)$, hence $L \\in D^-(\\mathcal{O}_U)$.", "Note that there is a distinguished triangle", "$$", "(K \\oplus L, K \\oplus L, L \\oplus L[1]) =", "(K, K, 0) \\oplus (L, L, L \\oplus L[1])", "$$", "see", "Derived Categories, Lemma \\ref{derived-lemma-direct-sum-triangles}.", "By", "Lemma \\ref{lemma-cone-pseudo-coherent}", "we see that $L \\oplus L[1]$ is $m$-pseudo-coherent.", "Hence also $L[1] \\oplus L[2]$ is $m$-pseudo-coherent.", "By induction $L[n] \\oplus L[n + 1]$ is $m$-pseudo-coherent.", "Since $L$ is bounded above we see that $L[n]$ is $m$-pseudo-coherent", "for large $n$. Hence working backwards, using the distinguished triangles", "$$", "(L[n], L[n] \\oplus L[n - 1], L[n - 1])", "$$", "we conclude that $L[n - 1], L[n - 2], \\ldots, L$ are $m$-pseudo-coherent", "as desired." ], "refs": [ "derived-lemma-direct-sum-triangles", "sites-cohomology-lemma-cone-pseudo-coherent" ], "ref_ids": [ 1765, 4368 ] } ], "ref_ids": [] }, { "id": 4371, "type": "theorem", "label": "sites-cohomology-lemma-finite-cohomology", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-finite-cohomology", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K$ be an object of", "$D(\\mathcal{O})$. Let $m \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$", "for $i > m$, then $H^m(K)$ is a finite type $\\mathcal{O}$-module.", "\\item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$", "for $i > m + 1$, then $H^{m + 1}(K)$ is a finitely presented", "$\\mathcal{O}$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $U$ be an object of $\\mathcal{C}$. We have to show that", "$H^m(K)$ is can be generated by finitely many sections over the members of", "a covering of $U$ (see", "Modules on Sites, Definition \\ref{sites-modules-definition-site-local}).", "Thus during the proof we may (finitely often) choose a covering", "$\\{U_i \\to U\\}$ and replace $\\mathcal{C}$ by $\\mathcal{C}/U_i$ and", "$U$ by $U_i$. In particular, by our definitions we may assume there exists", "a strictly perfect complex $\\mathcal{E}^\\bullet$ and a map", "$\\alpha : \\mathcal{E}^\\bullet \\to K$ which induces", "an isomorphism on cohomology in degrees $> m$ and a surjection in degree $m$.", "It suffices to prove the result for $\\mathcal{E}^\\bullet$.", "Let $n$ be the largest integer such that $\\mathcal{E}^n \\not = 0$.", "If $n = m$, then $H^m(\\mathcal{E}^\\bullet)$ is a quotient of", "$\\mathcal{E}^n$ and the result is clear.", "If $n > m$, then $\\mathcal{E}^{n - 1} \\to \\mathcal{E}^n$ is surjective as", "$H^n(E^\\bullet) = 0$. By Lemma \\ref{lemma-local-lift-map}", "we can (after replacing $U$ by the members of a covering)", "find a section of this surjection and write", "$\\mathcal{E}^{n - 1} = \\mathcal{E}' \\oplus \\mathcal{E}^n$.", "Hence it suffices to prove the result for the complex", "$(\\mathcal{E}')^\\bullet$ which is the same as $\\mathcal{E}^\\bullet$", "except has $\\mathcal{E}'$ in degree $n - 1$ and $0$ in degree $n$.", "We win by induction on $n$.", "\\medskip\\noindent", "Proof of (2). Pick an object $U$ of $\\mathcal{C}$.", "As in the proof of (1) we may work locally on $U$.", "Hence we may assume there exists a strictly perfect complex", "$\\mathcal{E}^\\bullet$ and a map", "$\\alpha : \\mathcal{E}^\\bullet \\to K$ which induces", "an isomorphism on cohomology in degrees $> m$ and a surjection in degree $m$.", "As in the proof of (1) we can reduce to the case that $\\mathcal{E}^i = 0$", "for $i > m + 1$. Then we see that", "$H^{m + 1}(K) \\cong H^{m + 1}(\\mathcal{E}^\\bullet) =", "\\Coker(\\mathcal{E}^m \\to \\mathcal{E}^{m + 1})$", "which is of finite presentation." ], "refs": [ "sites-modules-definition-site-local", "sites-cohomology-lemma-local-lift-map" ], "ref_ids": [ 14289, 4360 ] } ], "ref_ids": [] }, { "id": 4372, "type": "theorem", "label": "sites-cohomology-lemma-last-one-flat", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-last-one-flat", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{E}^\\bullet$ be a bounded above complex of flat", "$\\mathcal{O}$-modules with tor-amplitude in $[a, b]$.", "Then $\\Coker(d_{\\mathcal{E}^\\bullet}^{a - 1})$ is a flat", "$\\mathcal{O}$-module." ], "refs": [], "proofs": [ { "contents": [ "As $\\mathcal{E}^\\bullet$ is a bounded above complex of flat modules we see that", "$\\mathcal{E}^\\bullet \\otimes_\\mathcal{O} \\mathcal{F} =", "\\mathcal{E}^\\bullet \\otimes_\\mathcal{O}^{\\mathbf{L}} \\mathcal{F}$", "for any $\\mathcal{O}$-module $\\mathcal{F}$.", "Hence for every $\\mathcal{O}$-module $\\mathcal{F}$ the sequence", "$$", "\\mathcal{E}^{a - 2} \\otimes_\\mathcal{O} \\mathcal{F} \\to", "\\mathcal{E}^{a - 1} \\otimes_\\mathcal{O} \\mathcal{F} \\to", "\\mathcal{E}^a \\otimes_\\mathcal{O} \\mathcal{F}", "$$", "is exact in the middle. Since", "$\\mathcal{E}^{a - 2} \\to \\mathcal{E}^{a - 1} \\to \\mathcal{E}^a \\to", "\\Coker(d^{a - 1}) \\to 0$", "is a flat resolution this implies that", "$\\text{Tor}_1^\\mathcal{O}(\\Coker(d^{a - 1}), \\mathcal{F}) = 0$", "for all $\\mathcal{O}$-modules $\\mathcal{F}$. This means that", "$\\Coker(d^{a - 1})$ is flat, see Lemma \\ref{lemma-flat-tor-zero}." ], "refs": [ "sites-cohomology-lemma-flat-tor-zero" ], "ref_ids": [ 4238 ] } ], "ref_ids": [] }, { "id": 4373, "type": "theorem", "label": "sites-cohomology-lemma-tor-amplitude", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-tor-amplitude", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E$ be an object of", "$D(\\mathcal{O})$. Let $a, b \\in \\mathbf{Z}$ with $a \\leq b$. The following", "are equivalent", "\\begin{enumerate}", "\\item $E$ has tor-amplitude in $[a, b]$.", "\\item $E$ is represented by a complex", "$\\mathcal{E}^\\bullet$ of flat $\\mathcal{O}$-modules with", "$\\mathcal{E}^i = 0$ for $i \\not \\in [a, b]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (2) holds, then we may compute", "$E \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{F} =", "\\mathcal{E}^\\bullet \\otimes_\\mathcal{O} \\mathcal{F}$", "and it is clear that (1) holds.", "\\medskip\\noindent", "Assume that (1) holds. We may represent $E$ by a bounded above complex", "of flat $\\mathcal{O}$-modules $\\mathcal{K}^\\bullet$, see", "Section \\ref{section-flat}.", "Let $n$ be the largest integer such that $\\mathcal{K}^n \\not = 0$.", "If $n > b$, then $\\mathcal{K}^{n - 1} \\to \\mathcal{K}^n$ is surjective as", "$H^n(\\mathcal{K}^\\bullet) = 0$. As $\\mathcal{K}^n$ is flat we see that", "$\\Ker(\\mathcal{K}^{n - 1} \\to \\mathcal{K}^n)$ is flat", "(Modules on Sites, Lemma \\ref{sites-modules-lemma-flat-ses}).", "Hence we may replace $\\mathcal{K}^\\bullet$ by", "$\\tau_{\\leq n - 1}\\mathcal{K}^\\bullet$. Thus, by induction on $n$, we", "reduce to the case that $K^\\bullet$ is a complex of flat", "$\\mathcal{O}$-modules with $\\mathcal{K}^i = 0$ for $i > b$.", "\\medskip\\noindent", "Set $\\mathcal{E}^\\bullet = \\tau_{\\geq a}\\mathcal{K}^\\bullet$.", "Everything is clear except that $\\mathcal{E}^a$ is flat", "which follows immediately from Lemma \\ref{lemma-last-one-flat}", "and the definitions." ], "refs": [ "sites-modules-lemma-flat-ses", "sites-cohomology-lemma-last-one-flat" ], "ref_ids": [ 14205, 4372 ] } ], "ref_ids": [] }, { "id": 4374, "type": "theorem", "label": "sites-cohomology-lemma-bounded-below-tor-amplitude", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-bounded-below-tor-amplitude", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E$ be an object of", "$D(\\mathcal{O})$. Let $a \\in \\mathbf{Z}$. The following", "are equivalent", "\\begin{enumerate}", "\\item $E$ has tor-amplitude in $[a, \\infty]$.", "\\item $E$ can be represented by a K-flat complex $\\mathcal{E}^\\bullet$", "of flat $\\mathcal{O}$-modules with $\\mathcal{E}^i = 0$ for", "$i \\not \\in [a, \\infty]$.", "\\end{enumerate}", "Moreover, we can choose $\\mathcal{E}^\\bullet$ such that any pullback", "by a morphism of ringed sites is a K-flat complex with flat terms." ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is immediate. Assume (1) holds.", "First we choose a K-flat complex $\\mathcal{K}^\\bullet$", "with flat terms representing $E$, see Lemma \\ref{lemma-K-flat-resolution}.", "For any $\\mathcal{O}$-module $\\mathcal{M}$ the cohomology of", "$$", "\\mathcal{K}^{n - 1} \\otimes_\\mathcal{O} \\mathcal{M} \\to", "\\mathcal{K}^n \\otimes_\\mathcal{O} \\mathcal{M} \\to", "\\mathcal{K}^{n + 1} \\otimes_\\mathcal{O} \\mathcal{M}", "$$", "computes $H^n(E \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{M})$.", "This is always zero for $n < a$. Hence if we apply", "Lemma \\ref{lemma-last-one-flat} to the complex", "$\\ldots \\to \\mathcal{K}^{a - 1} \\to \\mathcal{K}^a \\to \\mathcal{K}^{a + 1}$", "we conclude that $\\mathcal{N} = \\Coker(\\mathcal{K}^{a - 1} \\to \\mathcal{K}^a)$", "is a flat $\\mathcal{O}$-module. We set", "$$", "\\mathcal{E}^\\bullet = \\tau_{\\geq a}\\mathcal{K}^\\bullet =", "(\\ldots \\to 0 \\to \\mathcal{N} \\to \\mathcal{K}^{a + 1} \\to \\ldots )", "$$", "The kernel $\\mathcal{L}^\\bullet$ of", "$\\mathcal{K}^\\bullet \\to \\mathcal{E}^\\bullet$ is the complex", "$$", "\\mathcal{L}^\\bullet = (\\ldots \\to \\mathcal{K}^{a - 1} \\to", "\\mathcal{I} \\to 0 \\to \\ldots)", "$$", "where $\\mathcal{I} \\subset \\mathcal{K}^a$ is the image of", "$\\mathcal{K}^{a - 1} \\to \\mathcal{K}^a$.", "Since we have the short exact sequence", "$0 \\to \\mathcal{I} \\to \\mathcal{K}^a \\to \\mathcal{N} \\to 0$", "we see that $\\mathcal{I}$ is a flat $\\mathcal{O}$-module.", "Thus $\\mathcal{L}^\\bullet$ is a bounded", "above complex of flat modules, hence K-flat by", "Lemma \\ref{lemma-bounded-flat-K-flat}.", "It follows that $\\mathcal{E}^\\bullet$ is K-flat by", "Lemma \\ref{lemma-K-flat-two-out-of-three-ses}.", "\\medskip\\noindent", "Proof of the final assertion. Let", "$f : (\\mathcal{C}', \\mathcal{O}') \\to (\\mathcal{C}, \\mathcal{O})$", "be a morphism of ringed sites. By Lemma \\ref{lemma-pullback-K-flat}", "the complex $f^*\\mathcal{K}^\\bullet$ is K-flat with flat terms.", "The complex $f^*\\mathcal{L}^\\bullet$ is K-flat as it is a bounded", "above complex of flat $\\mathcal{O}'$-modules. We have a short exact", "sequence of complexes of $\\mathcal{O}'$-modules", "$$", "0 \\to f^*\\mathcal{L}^\\bullet \\to f^*\\mathcal{K}^\\bullet \\to", "f^*\\mathcal{E}^\\bullet \\to 0", "$$", "because the short exact sequence", "$0 \\to \\mathcal{I} \\to \\mathcal{K}^a \\to \\mathcal{N} \\to 0$", "of flat modules pulls back to a short exact sequence.", "By Lemma \\ref{lemma-K-flat-two-out-of-three-ses}.", "the complex $f^*\\mathcal{E}^\\bullet$ is K-flat and the proof is complete." ], "refs": [ "sites-cohomology-lemma-K-flat-resolution", "sites-cohomology-lemma-last-one-flat", "sites-cohomology-lemma-bounded-flat-K-flat", "sites-cohomology-lemma-K-flat-two-out-of-three-ses", "sites-cohomology-lemma-pullback-K-flat", "sites-cohomology-lemma-K-flat-two-out-of-three-ses" ], "ref_ids": [ 4236, 4372, 4233, 4232, 4241, 4232 ] } ], "ref_ids": [] }, { "id": 4375, "type": "theorem", "label": "sites-cohomology-lemma-tor-amplitude-pullback", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-tor-amplitude-pullback", "contents": [ "Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to", "(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed sites.", "Let $E$ be an object of $D(\\mathcal{O}_\\mathcal{D})$.", "If $E$ has tor amplitude in $[a, b]$,", "then $Lf^*E$ has tor amplitude in $[a, b]$." ], "refs": [], "proofs": [ { "contents": [ "Assume $E$ has tor amplitude in $[a, b]$. By", "Lemma \\ref{lemma-tor-amplitude}", "we can represent $E$ by a complex of", "$\\mathcal{E}^\\bullet$ of flat $\\mathcal{O}$-modules with", "$\\mathcal{E}^i = 0$ for $i \\not \\in [a, b]$. Then", "$Lf^*E$ is represented by $f^*\\mathcal{E}^\\bullet$.", "By Modules on Sites, Lemma \\ref{sites-modules-lemma-pullback-flat}", "the module $f^*\\mathcal{E}^i$ are flat.", "Thus by Lemma \\ref{lemma-tor-amplitude}", "we conclude that $Lf^*E$ has tor amplitude in $[a, b]$." ], "refs": [ "sites-cohomology-lemma-tor-amplitude", "sites-modules-lemma-pullback-flat", "sites-cohomology-lemma-tor-amplitude" ], "ref_ids": [ 4373, 14249, 4373 ] } ], "ref_ids": [] }, { "id": 4376, "type": "theorem", "label": "sites-cohomology-lemma-cone-tor-amplitude", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-cone-tor-amplitude", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(K, L, M, f, g, h)$ be a distinguished", "triangle in $D(\\mathcal{O})$. Let $a, b \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item If $K$ has tor-amplitude in $[a + 1, b + 1]$ and", "$L$ has tor-amplitude in $[a, b]$ then $M$ has", "tor-amplitude in $[a, b]$.", "\\item If $K$ and $M$ have tor-amplitude in $[a, b]$, then", "$L$ has tor-amplitude in $[a, b]$.", "\\item If $L$ has tor-amplitude in $[a + 1, b + 1]$", "and $M$ has tor-amplitude in $[a, b]$, then", "$K$ has tor-amplitude in $[a + 1, b + 1]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This just follows from the long exact cohomology sequence", "associated to a distinguished triangle and the fact that", "$- \\otimes_\\mathcal{O}^{\\mathbf{L}} \\mathcal{F}$", "preserves distinguished triangles.", "The easiest one to prove is (2) and the others follow from it by", "translation." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4377, "type": "theorem", "label": "sites-cohomology-lemma-tensor-tor-amplitude", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-tensor-tor-amplitude", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L$ be objects of", "$D(\\mathcal{O})$. If $K$ has tor-amplitude in $[a, b]$ and", "$L$ has tor-amplitude in $[c, d]$ then $K \\otimes_\\mathcal{O}^\\mathbf{L} L$", "has tor amplitude in $[a + c, b + d]$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: use the spectral sequence for tors." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4378, "type": "theorem", "label": "sites-cohomology-lemma-summands-tor-amplitude", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-summands-tor-amplitude", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $a, b \\in \\mathbf{Z}$.", "For $K$, $L$ objects of $D(\\mathcal{O})$ if $K \\oplus L$ has tor", "amplitude in $[a, b]$ so do $K$ and $L$." ], "refs": [], "proofs": [ { "contents": [ "Clear from the fact that the Tor functors are additive." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4379, "type": "theorem", "label": "sites-cohomology-lemma-bounded", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-bounded", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{I} \\subset \\mathcal{O}$ be a sheaf of ideals.", "Let $K$ be an object of $D(\\mathcal{O})$.", "\\begin{enumerate}", "\\item If $K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}$", "is bounded above, then", "$K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n$", "is uniformly bounded above for all $n$.", "\\item If $K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}$", "as an object of $D(\\mathcal{O}/\\mathcal{I})$ has tor amplitude in $[a, b]$,", "then $K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n$", "as an object of $D(\\mathcal{O}/\\mathcal{I}^n)$", "has tor amplitude in $[a, b]$ for all $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Assume that", "$K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}$", "is bounded above, say", "$H^i(K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}) = 0$", "for $i > b$. Note that we have distinguished triangles", "$$", "K \\otimes_\\mathcal{O}^\\mathbf{L}", "\\mathcal{I}^n/\\mathcal{I}^{n + 1} \\to", "K \\otimes_\\mathcal{O}^\\mathbf{L}", "\\mathcal{O}/\\mathcal{I}^{n + 1} \\to", "K \\otimes_\\mathcal{O}^\\mathbf{L}", "\\mathcal{O}/\\mathcal{I}^n \\to", "K \\otimes_\\mathcal{O}^\\mathbf{L}", "\\mathcal{I}^n/\\mathcal{I}^{n + 1}[1]", "$$", "and that", "$$", "K \\otimes_\\mathcal{O}^\\mathbf{L}", "\\mathcal{I}^n/\\mathcal{I}^{n + 1} =", "\\left(", "K \\otimes_\\mathcal{O}^\\mathbf{L}", "\\mathcal{O}/\\mathcal{I}\\right)", "\\otimes_{\\mathcal{O}/\\mathcal{I}}^\\mathbf{L}", "\\mathcal{I}^n/\\mathcal{I}^{n + 1}", "$$", "By induction we conclude that", "$H^i(K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n) = 0$", "for $i > b$ for all $n$.", "\\medskip\\noindent", "Proof of (2). Assume $K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}$", "as an object of $D(\\mathcal{O}/\\mathcal{I})$ has tor amplitude in $[a, b]$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}/\\mathcal{I}^n$-modules.", "Then we have a finite filtration", "$$", "0 \\subset \\mathcal{I}^{n - 1}\\mathcal{F} \\subset \\ldots", "\\subset \\mathcal{I}\\mathcal{F} \\subset \\mathcal{F}", "$$", "whose successive quotients are sheaves of $\\mathcal{O}/\\mathcal{I}$-modules.", "Thus to prove that $K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n$", "has tor amplitude in $[a, b]$ it suffices to show", "$H^i(K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n", "\\otimes_{\\mathcal{O}/\\mathcal{I}^n}^\\mathbf{L} \\mathcal{G})$", "is zero for $i \\not \\in [a, b]$ for all $\\mathcal{O}/\\mathcal{I}$-modules", "$\\mathcal{G}$. Since", "$$", "\\left(K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n\\right)", "\\otimes_{\\mathcal{O}/\\mathcal{I}^n}^\\mathbf{L} \\mathcal{G}", "=", "\\left(K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}\\right)", "\\otimes_{\\mathcal{O}/\\mathcal{I}}^\\mathbf{L} \\mathcal{G}", "$$", "for every sheaf of $\\mathcal{O}/\\mathcal{I}$-modules $\\mathcal{G}$", "the result follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4380, "type": "theorem", "label": "sites-cohomology-lemma-tor-amplitude-stalk", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-tor-amplitude-stalk", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $E$ be an object of $D(\\mathcal{O})$.", "Let $a, b \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item If $E$ has tor amplitude in $[a, b]$, then for every point $p$", "of the site $\\mathcal{C}$ the object $E_p$ of $D(\\mathcal{O}_p)$", "has tor amplitude in $[a, b]$.", "\\item If $\\mathcal{C}$ has enough points, then the converse is true.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). This follows because taking stalks at $p$ is", "the same as pulling back by the morphism of ringed sites", "$(p, \\mathcal{O}_p) \\to (\\mathcal{C}, \\mathcal{O})$ and hence", "we can apply Lemma \\ref{lemma-tor-amplitude-pullback}.", "\\medskip\\noindent", "Proof of (2). If $\\mathcal{C}$ has enough points, then we can check", "vanishing of", "$H^i(E \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{F})$", "at stalks, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-check-exactness-stalks}.", "Since $H^i(E \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{F})_p =", "H^i(E_p \\otimes_{\\mathcal{O}_p}^\\mathbf{L} \\mathcal{F}_p)$ we conclude." ], "refs": [ "sites-cohomology-lemma-tor-amplitude-pullback", "sites-modules-lemma-check-exactness-stalks" ], "ref_ids": [ 4375, 14160 ] } ], "ref_ids": [] }, { "id": 4381, "type": "theorem", "label": "sites-cohomology-lemma-perfect-independent-representative", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-perfect-independent-representative", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $E$ be an object of $D(\\mathcal{O})$.", "\\begin{enumerate}", "\\item If $\\mathcal{C}$ has a final object $X$ and there exist a", "covering $\\{U_i \\to X\\}$, strictly perfect complexes $\\mathcal{E}_i^\\bullet$", "of $\\mathcal{O}_{U_i}$-modules, and isomorphisms", " $\\alpha_i : \\mathcal{E}_i^\\bullet \\to E|_{U_i}$ in", "$D(\\mathcal{O}_{U_i})$, then $E$ is perfect.", "\\item If $E$ is perfect, then any complex representing $E$ is perfect.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Identical to the proof of", "Lemma \\ref{lemma-pseudo-coherent-independent-representative}." ], "refs": [ "sites-cohomology-lemma-pseudo-coherent-independent-representative" ], "ref_ids": [ 4366 ] } ], "ref_ids": [] }, { "id": 4382, "type": "theorem", "label": "sites-cohomology-lemma-perfect-precise", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-perfect-precise", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $E$ be an object of $D(\\mathcal{O})$.", "Let $a \\leq b$ be integers. If $E$ has tor amplitude in $[a, b]$", "and is $(a - 1)$-pseudo-coherent, then $E$ is perfect." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be an object of $\\mathcal{C}$. After replacing $U$ by the members", "of a covering and $\\mathcal{C}$ by the localization $\\mathcal{C}/U$", "we may assume there exists a strictly perfect complex $\\mathcal{E}^\\bullet$", "and a map $\\alpha : \\mathcal{E}^\\bullet \\to E$ such that $H^i(\\alpha)$ is", "an isomorphism for $i \\geq a$. We may and do replace", "$\\mathcal{E}^\\bullet$ by $\\sigma_{\\geq a - 1}\\mathcal{E}^\\bullet$. Choose a", "distinguished triangle", "$$", "\\mathcal{E}^\\bullet \\to E \\to C \\to \\mathcal{E}^\\bullet[1]", "$$", "From the vanishing of cohomology sheaves of $E$ and $\\mathcal{E}^\\bullet$", "and the assumption on $\\alpha$ we obtain $C \\cong \\mathcal{K}[a - 2]$ with", "$\\mathcal{K} = \\Ker(\\mathcal{E}^{a - 1} \\to \\mathcal{E}^a)$.", "Let $\\mathcal{F}$ be an $\\mathcal{O}$-module.", "Applying $- \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{F}$", "the assumption that $E$ has tor amplitude in $[a, b]$", "implies $\\mathcal{K} \\otimes_\\mathcal{O} \\mathcal{F} \\to", "\\mathcal{E}^{a - 1} \\otimes_\\mathcal{O} \\mathcal{F}$ has image", "$\\Ker(\\mathcal{E}^{a - 1} \\otimes_\\mathcal{O} \\mathcal{F}", "\\to \\mathcal{E}^a \\otimes_\\mathcal{O} \\mathcal{F})$.", "It follows that $\\text{Tor}_1^\\mathcal{O}(\\mathcal{E}', \\mathcal{F}) = 0$", "where $\\mathcal{E}' = \\Coker(\\mathcal{E}^{a - 1} \\to \\mathcal{E}^a)$.", "Hence $\\mathcal{E}'$ is flat (Lemma \\ref{lemma-flat-tor-zero}).", "Thus there exists a covering $\\{U_i \\to U\\}$ such that", "$\\mathcal{E}'|_{U_i}$ is a direct summand of a finite free module by", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-flat-locally-finite-presentation}.", "Thus the complex", "$$", "\\mathcal{E}'|_{U_i} \\to \\mathcal{E}^{a - 1}|_{U_i} \\to \\ldots \\to", "\\mathcal{E}^b|_{U_i}", "$$", "is quasi-isomorphic to $E|_{U_i}$ and $E$ is perfect." ], "refs": [ "sites-cohomology-lemma-flat-tor-zero", "sites-modules-lemma-flat-locally-finite-presentation" ], "ref_ids": [ 4238, 14212 ] } ], "ref_ids": [] }, { "id": 4383, "type": "theorem", "label": "sites-cohomology-lemma-perfect", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-perfect", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $E$ be an object of $D(\\mathcal{O})$.", "The following are equivalent", "\\begin{enumerate}", "\\item $E$ is perfect, and", "\\item $E$ is pseudo-coherent and locally has finite tor dimension.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Let $U$ be an object of $\\mathcal{C}$.", "By definition there exists a covering $\\{U_i \\to U\\}$ such that", "$E|_{U_i}$ is represented by a strictly perfect complex.", "Thus $E$ is pseudo-coherent (i.e., $m$-pseudo-coherent for all $m$) by", "Lemma \\ref{lemma-pseudo-coherent-independent-representative}.", "Moreover, a direct summand of a finite free module is flat, hence", "$E|_{U_i}$ has finite Tor dimension by", "Lemma \\ref{lemma-tor-amplitude}. Thus (2) holds.", "\\medskip\\noindent", "Assume (2). Let $U$ be an object of $\\mathcal{C}$.", "After replacing $U$ by the members of a covering", "we may assume there exist integers $a \\leq b$ such that $E|_U$", "has tor amplitude in $[a, b]$. Since $E|_U$ is $m$-pseudo-coherent", "for all $m$ we conclude using Lemma \\ref{lemma-perfect-precise}." ], "refs": [ "sites-cohomology-lemma-pseudo-coherent-independent-representative", "sites-cohomology-lemma-tor-amplitude", "sites-cohomology-lemma-perfect-precise" ], "ref_ids": [ 4366, 4373, 4382 ] } ], "ref_ids": [] }, { "id": 4384, "type": "theorem", "label": "sites-cohomology-lemma-perfect-pullback", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-perfect-pullback", "contents": [ "Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to", "(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed sites.", "Let $E$ be an object of $D(\\mathcal{O}_\\mathcal{D})$.", "If $E$ is perfect in $D(\\mathcal{O}_\\mathcal{D})$,", "then $Lf^*E$ is perfect in $D(\\mathcal{O}_\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-perfect},", "\\ref{lemma-tor-amplitude-pullback}, and", "\\ref{lemma-pseudo-coherent-pullback}." ], "refs": [ "sites-cohomology-lemma-perfect", "sites-cohomology-lemma-tor-amplitude-pullback", "sites-cohomology-lemma-pseudo-coherent-pullback" ], "ref_ids": [ 4383, 4375, 4367 ] } ], "ref_ids": [] }, { "id": 4385, "type": "theorem", "label": "sites-cohomology-lemma-two-out-of-three-perfect", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-two-out-of-three-perfect", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(K, L, M, f, g, h)$", "be a distinguished triangle in $D(\\mathcal{O})$. If two out of three of", "$K, L, M$ are perfect then the third is also perfect." ], "refs": [], "proofs": [ { "contents": [ "First proof: Combine", "Lemmas \\ref{lemma-perfect}, \\ref{lemma-cone-pseudo-coherent}, and", "\\ref{lemma-cone-tor-amplitude}.", "Second proof (sketch): Say $K$ and $L$ are perfect. Let $U$ be an object", "of $\\mathcal{C}$. After replacing", "$U$ by the members of a covering we may assume that $K|_U$ and $L|_U$", "are represented by strictly perfect complexes $\\mathcal{K}^\\bullet$", "and $\\mathcal{L}^\\bullet$. After replacing $U$ by the members", "of a covering we may assume the map $K|_U \\to L|_U$ is given by", "a map of complexes $\\alpha : \\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet$,", "see Lemma \\ref{lemma-local-actual}.", "Then $M|_U$ is isomorphic to the cone of $\\alpha$ which is strictly", "perfect by Lemma \\ref{lemma-cone}." ], "refs": [ "sites-cohomology-lemma-perfect", "sites-cohomology-lemma-cone-pseudo-coherent", "sites-cohomology-lemma-cone-tor-amplitude", "sites-cohomology-lemma-local-actual", "sites-cohomology-lemma-cone" ], "ref_ids": [ 4383, 4368, 4376, 4363, 4357 ] } ], "ref_ids": [] }, { "id": 4386, "type": "theorem", "label": "sites-cohomology-lemma-tensor-perfect", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-tensor-perfect", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "If $K, L$ are perfect objects of $D(\\mathcal{O})$, then", "so is $K \\otimes_\\mathcal{O}^\\mathbf{L} L$." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemmas \\ref{lemma-perfect}, \\ref{lemma-tensor-pseudo-coherent}, and", "\\ref{lemma-tensor-tor-amplitude}." ], "refs": [ "sites-cohomology-lemma-perfect", "sites-cohomology-lemma-tensor-pseudo-coherent", "sites-cohomology-lemma-tensor-tor-amplitude" ], "ref_ids": [ 4383, 4369, 4377 ] } ], "ref_ids": [] }, { "id": 4387, "type": "theorem", "label": "sites-cohomology-lemma-summands-perfect", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-summands-perfect", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "If $K \\oplus L$ is a perfect object of $D(\\mathcal{O})$, then", "so are $K$ and $L$." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemmas \\ref{lemma-perfect}, \\ref{lemma-summands-pseudo-coherent}, and", "\\ref{lemma-summands-tor-amplitude}." ], "refs": [ "sites-cohomology-lemma-perfect", "sites-cohomology-lemma-summands-pseudo-coherent", "sites-cohomology-lemma-summands-tor-amplitude" ], "ref_ids": [ 4383, 4370, 4378 ] } ], "ref_ids": [] }, { "id": 4388, "type": "theorem", "label": "sites-cohomology-lemma-symmetric-monoidal-cat-complexes", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-symmetric-monoidal-cat-complexes", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed space. The category of complexes", "of $\\mathcal{O}$-modules with tensor product defined by", "$\\mathcal{F}^\\bullet \\otimes \\mathcal{G}^\\bullet =", "\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{G}^\\bullet)$", "is a symmetric monoidal category." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints: as unit $\\mathbf{1}$ we take the complex having", "$\\mathcal{O}$ in degree $0$ and zero in other degrees with", "obvious isomorphisms", "$\\text{Tot}(\\mathbf{1} \\otimes_\\mathcal{O} \\mathcal{G}^\\bullet) =", "\\mathcal{G}^\\bullet$ and", "$\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathbf{1}) =", "\\mathcal{F}^\\bullet$.", "to prove the lemma you have to check the commutativity", "of various diagrams, see Categories, Definitions", "\\ref{categories-definition-monoidal-category} and", "\\ref{categories-definition-symmetric-monoidal-category}.", "The verifications are straightforward in each case." ], "refs": [ "categories-definition-monoidal-category", "categories-definition-symmetric-monoidal-category" ], "ref_ids": [ 12404, 12408 ] } ], "ref_ids": [] }, { "id": 4389, "type": "theorem", "label": "sites-cohomology-lemma-left-dual-complex", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-left-dual-complex", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}^\\bullet$", "be a complex of $\\mathcal{O}$-modules. If $\\mathcal{F}^\\bullet$", "has a left dual in the monoidal category of complexes of", "$\\mathcal{O}$-modules", "(Categories, Definition \\ref{categories-definition-dual})", "then for every object $U$ of $\\mathcal{C}$ there exists a", "covering $\\{U_i \\to U\\}$ such that $\\mathcal{F}^\\bullet|_{U_i}$", "is strictly perfect and the left dual is as constructed in", "Example \\ref{example-dual}." ], "refs": [ "categories-definition-dual" ], "proofs": [ { "contents": [ "By uniqueness of left duals", "(Categories, Remark \\ref{categories-remark-left-dual-adjoint})", "we get the final statement provided we show that $\\mathcal{F}^\\bullet$", "is as stated. Let $\\mathcal{G}^\\bullet, \\eta, \\epsilon$ be a left dual.", "Write $\\eta = \\sum \\eta_n$ and $\\epsilon = \\sum \\epsilon_n$", "where $\\eta_n : \\mathcal{O} \\to", "\\mathcal{F}^n \\otimes_\\mathcal{O} \\mathcal{G}^{-n}$", "and", "$\\epsilon_n : \\mathcal{G}^{-n} \\otimes_\\mathcal{O} \\mathcal{F}^n", "\\to \\mathcal{O}$. Since", "$(1 \\otimes \\epsilon) \\circ (\\eta \\otimes 1) = \\text{id}_{\\mathcal{F}^\\bullet}$", "and", "$(\\epsilon \\otimes 1) \\circ (1 \\otimes \\eta) = \\text{id}_{\\mathcal{G}^\\bullet}$", "by Categories, Definition \\ref{categories-definition-dual} we see immediately", "that we have", "$(1 \\otimes \\epsilon_n) \\circ (\\eta_n \\otimes 1) = \\text{id}_{\\mathcal{F}^n}$", "and", "$(\\epsilon_n \\otimes 1) \\circ (1 \\otimes \\eta_n) =", "\\text{id}_{\\mathcal{G}^{-n}}$.", "In other words, we see that $\\mathcal{G}^{-n}$ is a left dual of", "$\\mathcal{F}^n$ and we see that", "Modules on Sites, Lemma \\ref{sites-modules-lemma-left-dual-module}", "applies to each $\\mathcal{F}^n$. Let $U$ be an object of $\\mathcal{C}$.", "There exists a covering $\\{U_i \\to U\\}$ such that for every", "$i$ only a finite number of $\\eta_n|_{U_i}$ are nonzero.", "Thus after replacing $U$ by $U_i$ we may assume only a finite", "number of $\\eta_n|_U$ are nonzero and by the lemma cited", "this implies only a finite number of $\\mathcal{F}^n|_U$ are ", "nonzero. Using the lemma again we can then find a covering", "$\\{U_i \\to U\\}$ such that each", "$\\mathcal{F}^n|_{U_i}$ is a direct summand of a finite", "free $\\mathcal{O}$-module and the proof is complete." ], "refs": [ "categories-remark-left-dual-adjoint", "categories-definition-dual", "sites-modules-lemma-left-dual-module" ], "ref_ids": [ 12430, 12407, 14211 ] } ], "ref_ids": [ 12407 ] }, { "id": 4390, "type": "theorem", "label": "sites-cohomology-lemma-dual-perfect-complex", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-dual-perfect-complex", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $K$ be a perfect object of $D(\\mathcal{O})$.", "Then $K^\\vee = R\\SheafHom(K, \\mathcal{O})$ is a", "perfect object too and $(K^\\vee)^\\vee \\cong K$. There are", "functorial isomorphisms", "$$", "M \\otimes^\\mathbf{L}_\\mathcal{O} K^\\vee = R\\SheafHom_\\mathcal{O}(K, M)", "$$", "and", "$$", "H^0(\\mathcal{C}, M \\otimes^\\mathbf{L}_\\mathcal{O} K^\\vee) =", "\\Hom_{D(\\mathcal{O})}(K, M)", "$$", "for $M$ in $D(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "We will us without further mention that formation of internal hom commutes", "with restriction (Lemma \\ref{lemma-restriction-RHom-to-U}). Let $U$", "be an arbitrary object of $\\mathcal{C}$. To check that", "$K^\\vee$ is perfect, it suffices to show that there exists a covering", "$\\{U_i \\to U\\}$ such that $K^\\vee|_{U_i}$ is perfect for all $i$.", "There is a canonical map", "$$", "K = R\\SheafHom(\\mathcal{O}_X, \\mathcal{O}_X)", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L} K \\longrightarrow", "R\\SheafHom(R\\SheafHom(K, \\mathcal{O}_X), \\mathcal{O}_X) =", "(K^\\vee)^\\vee", "$$", "see Lemma \\ref{lemma-internal-hom-evaluate}. It suffices to prove there", "is a covering $\\{U_i \\to U\\}$ such that the restriction of this map", "to $\\mathcal{C}/U_i$ is an isomorphism for all $i$.", "By Lemma \\ref{lemma-dual} to see the final statement it suffices to check", "that the map (\\ref{equation-eval})", "$$", "M \\otimes^\\mathbf{L}_\\mathcal{O} K^\\vee", "\\longrightarrow", "R\\SheafHom(K, M)", "$$", "is an isomorphism. This is a local question as well (in the sense above).", "Hence it suffices to prove the lemma when $K$ is represented", "by a strictly perfect complex.", "\\medskip\\noindent", "Assume $K$ is represented by the strictly perfect complex", "$\\mathcal{E}^\\bullet$. Then it follows from", "Lemma \\ref{lemma-Rhom-strictly-perfect}", "that $K^\\vee$ is represented by the complex whose terms are", "$(\\mathcal{E}^n)^\\vee =", "\\SheafHom_\\mathcal{O}(\\mathcal{E}^n, \\mathcal{O})$", "in degree $-n$. Since $\\mathcal{E}^n$ is a direct summand of a finite", "free $\\mathcal{O}$-module, so is $(\\mathcal{E}^n)^\\vee$.", "Hence $K^\\vee$ is represented by a strictly perfect complex too ", "and we see that $K^\\vee$ is perfect.", "The map $K \\to (K^\\vee)^\\vee$ is an isomorphism as it is given up", "to sign by the evaluation maps", "$\\mathcal{E}^n \\to ((\\mathcal{E}^n)^\\vee)^\\vee$ which are", "isomorphisms. To see that (\\ref{equation-eval}) is an isomorphism, represent", "$M$ by a K-flat complex $\\mathcal{F}^\\bullet$.", "By Lemma \\ref{lemma-Rhom-strictly-perfect} the complex", "$R\\SheafHom(K, M)$ is represented by the complex with terms", "$$", "\\bigoplus\\nolimits_{n = p + q}", "\\SheafHom_\\mathcal{O}(\\mathcal{E}^{-q}, \\mathcal{F}^p)", "$$", "On the other hand, the object $M \\otimes^\\mathbf{L}_\\mathcal{O} K^\\vee$", "is represented by the complex with terms", "$$", "\\bigoplus\\nolimits_{n = p + q}", "\\mathcal{F}^p \\otimes_\\mathcal{O} (\\mathcal{E}^{-q})^\\vee", "$$", "Thus the assertion that (\\ref{equation-eval}) is an isomorphism", "reduces to the assertion that the canonical map", "$$", "\\mathcal{F}", "\\otimes_\\mathcal{O}", "\\SheafHom_\\mathcal{O}(\\mathcal{E}, \\mathcal{O})", "\\longrightarrow", "\\SheafHom_\\mathcal{O}(\\mathcal{E}, \\mathcal{F})", "$$", "is an isomorphism when $\\mathcal{E}$ is a direct summand of a finite", "free $\\mathcal{O}$-module and $\\mathcal{F}$ is any $\\mathcal{O}$-module.", "This follows immediately from the corresponding statement when", "$\\mathcal{E}$ is finite free." ], "refs": [ "sites-cohomology-lemma-restriction-RHom-to-U", "sites-cohomology-lemma-internal-hom-evaluate", "sites-cohomology-lemma-dual", "sites-cohomology-lemma-Rhom-strictly-perfect", "sites-cohomology-lemma-Rhom-strictly-perfect" ], "ref_ids": [ 4329, 4331, 4335, 4364, 4364 ] } ], "ref_ids": [] }, { "id": 4391, "type": "theorem", "label": "sites-cohomology-lemma-symmetric-monoidal-derived", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-symmetric-monoidal-derived", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. The derived category", "$D(\\mathcal{O})$ is a symmetric monoidal category with tensor product", "given by derived tensor product with usual associativity and", "commutativity constraints (for sign rules, see", "More on Algebra, Section \\ref{more-algebra-section-sign-rules})." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Compare with Lemma \\ref{lemma-symmetric-monoidal-cat-complexes}." ], "refs": [ "sites-cohomology-lemma-symmetric-monoidal-cat-complexes" ], "ref_ids": [ 4388 ] } ], "ref_ids": [] }, { "id": 4392, "type": "theorem", "label": "sites-cohomology-lemma-left-dual-derived", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-left-dual-derived", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $M$ be an object", "of $D(\\mathcal{O})$. If $M$ has a left dual in the monoidal category", "$D(\\mathcal{O})$ (Categories, Definition \\ref{categories-definition-dual})", "then $M$ is perfect and the left dual is as constructed in", "Example \\ref{example-dual-derived}." ], "refs": [ "categories-definition-dual" ], "proofs": [ { "contents": [ "Let $N, \\eta, \\epsilon$ be a left dual. Observe that for any object", "$U$ of $\\mathcal{C}$ the restriction $N|_U, \\eta|_U, \\epsilon|_U$", "is a left dual for $M|_U$.", "\\medskip\\noindent", "Let $U$ be an object of $\\mathcal{C}$. It suffices to find a covering", "$\\{U_i \\to U\\}_{i \\in I}$ fo $\\mathcal{C}$ such that $M|_{U_i}$ is", "a perfect object of $D(\\mathcal{O}_{U_i})$. Hence we may replace", "$\\mathcal{C}, \\mathcal{O}, M, N, \\eta, \\epsilon$ by", "$\\mathcal{C}/U, \\mathcal{O}_U, M|_U, N|_U, \\eta|_U, \\epsilon|_U$", "and assume $\\mathcal{C}$ has a final object $X$. Moreover, during the", "proof we can (finitely often) replace $X$ by the members of a", "covering $\\{U_i \\to X\\}$ of $X$.", "\\medskip\\noindent", "We are going to use the following argument several times. Choose any", "complex $\\mathcal{M}^\\bullet$", "of $\\mathcal{O}$-modules representing $M$. Choose a K-flat complex", "$\\mathcal{N}^\\bullet$ representing $N$ whose terms are flat", "$\\mathcal{O}$-modules, see Lemma \\ref{lemma-K-flat-resolution}.", "Consider the map", "$$", "\\eta : \\mathcal{O} \\to", "\\text{Tot}(\\mathcal{M}^\\bullet \\otimes_\\mathcal{O} \\mathcal{N}^\\bullet)", "$$", "After replacing $X$ by the members of a covering,", "we can find an integer $N$ and for", "$i = 1, \\ldots, N$ integers $n_i \\in \\mathbf{Z}$ and sections", "$f_i$ and $g_i$ of $\\mathcal{M}^{n_i}$ and $\\mathcal{N}^{-n_i}$", "such that", "$$", "\\eta(1) = \\sum\\nolimits_i f_i \\otimes g_i", "$$", "Let $\\mathcal{K}^\\bullet \\subset \\mathcal{M}^\\bullet$ be any subcomplex", "of $\\mathcal{O}$-modules containing the sections $f_i$", "for $i = 1, \\ldots, N$.", "Since", "$\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{N}^\\bullet)", "\\subset", "\\text{Tot}(\\mathcal{M}^\\bullet \\otimes_\\mathcal{O} \\mathcal{N}^\\bullet)$", "by flatness of the modules $\\mathcal{N}^n$, we see that $\\eta$ factors through", "$$", "\\tilde \\eta :", "\\mathcal{O} \\to", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{N}^\\bullet)", "$$", "Denoting $K$ the object of $D(\\mathcal{O})$ represented by", "$\\mathcal{K}^\\bullet$ we find a commutative diagram", "$$", "\\xymatrix{", "M \\ar[rr]_-{\\eta \\otimes 1} \\ar[rrd]_{\\tilde \\eta \\otimes 1} & &", "M \\otimes^\\mathbf{L} N \\otimes^\\mathbf{L} M", "\\ar[r]_-{1 \\otimes \\epsilon} &", "M \\\\", "& &", "K \\otimes^\\mathbf{L} N \\otimes^\\mathbf{L} M", "\\ar[u] \\ar[r]^-{1 \\otimes \\epsilon} &", "K \\ar[u]", "}", "$$", "Since the composition of the upper row is the identity on $M$", "we conclude that $M$ is a direct summand of $K$ in $D(\\mathcal{O})$.", "\\medskip\\noindent", "As a first use of the argument above, we can choose the subcomplex", "$\\mathcal{K}^\\bullet = \\sigma_{\\geq a} \\tau_{\\leq b}\\mathcal{M}^\\bullet$", "with $a < n_i < b$ for $i = 1, \\ldots, N$. Thus $M$ is a direct", "summand in $D(\\mathcal{O})$ of a bounded complex and we conclude", "we may assume $M$ is in $D^b(\\mathcal{O})$. (Recall that the process", "above involves replacing $X$ by the members of a covering.)", "\\medskip\\noindent", "Since $M$ is in $D^b(\\mathcal{O})$ we may choose", "$\\mathcal{M}^\\bullet$ to be a bounded above complex of", "flat modules (by Modules, Lemma \\ref{modules-lemma-module-quotient-flat} and", "Derived Categories, Lemma \\ref{derived-lemma-subcategory-left-resolution}).", "Then we can choose $\\mathcal{K}^\\bullet = \\sigma_{\\geq a}\\mathcal{M}^\\bullet$", "with $a < n_i$ for $i = 1, \\ldots, N$ in the argument above.", "Thus we find that we may assume $M$ is a direct summand in", "$D(\\mathcal{O})$ of a bounded complex of flat modules.", "In particular, we find $M$ has finite tor amplitude.", "\\medskip\\noindent", "Say $M$ has tor amplitude in $[a, b]$. Assuming $M$ is $m$-pseudo-coherent", "we are going to show that (after replacing $X$ by the members of a covering)", "we may assume $M$ is $(m - 1)$-pseudo-coherent. This will finish the proof by", "Lemma \\ref{lemma-perfect-precise} and the fact that", "$M$ is $(b + 1)$-pseudo-coherent in any case. After replacing $X$", "by the members of a covering we may assume there exists a strictly perfect", "complex $\\mathcal{E}^\\bullet$ and a map $\\alpha : \\mathcal{E}^\\bullet \\to M$", "in $D(\\mathcal{O})$ such that $H^i(\\alpha)$ is an isomorphism for", "$i > m$ and surjective for $i = m$. We may and do assume", "that $\\mathcal{E}^i = 0$ for $i < m$. Choose a distinguished triangle", "$$", "\\mathcal{E}^\\bullet \\to M \\to L \\to \\mathcal{E}^\\bullet[1]", "$$", "Observe that $H^i(L) = 0$ for $i \\geq m$. Thus we may represent", "$L$ by a complex $\\mathcal{L}^\\bullet$ with $\\mathcal{L}^i = 0$", "for $i \\geq m$. The map $L \\to \\mathcal{E}^\\bullet[1]$", "is given by a map of complexes", "$\\mathcal{L}^\\bullet \\to \\mathcal{E}^\\bullet[1]$", "which is zero in all degrees except in degree $m - 1$", "where we obtain a map $\\mathcal{L}^{m - 1} \\to \\mathcal{E}^m$, see", "Derived Categories, Lemma \\ref{derived-lemma-negative-exts}.", "Then $M$ is represented by the complex", "$$", "\\mathcal{M}^\\bullet :", "\\ldots \\to", "\\mathcal{L}^{m - 2} \\to", "\\mathcal{L}^{m - 1} \\to", "\\mathcal{E}^m \\to", "\\mathcal{E}^{m + 1} \\to \\ldots", "$$", "Apply the discussion in the second paragraph to this complex to get", "sections $f_i$ of $\\mathcal{M}^{n_i}$ for $i = 1, \\ldots, N$.", "For $n < m$ let $\\mathcal{K}^n \\subset \\mathcal{L}^n$", "be the $\\mathcal{O}$-submodule generated by the sections", "$f_i$ for $n_i = n$ and $d(f_i)$ for $n_i = n - 1$.", "For $n \\geq m$ set $\\mathcal{K}^n = \\mathcal{E}^n$.", "Clearly, we have a morphism of", "distinguished triangles", "$$", "\\xymatrix{", "\\mathcal{E}^\\bullet \\ar[r] &", "\\mathcal{M}^\\bullet \\ar[r] &", "\\mathcal{L}^\\bullet \\ar[r] &", "\\mathcal{E}^\\bullet[1] \\\\", "\\mathcal{E}^\\bullet \\ar[r] \\ar[u] &", "\\mathcal{K}^\\bullet \\ar[r] \\ar[u] &", "\\sigma_{\\leq m - 1}\\mathcal{K}^\\bullet \\ar[r] \\ar[u] &", "\\mathcal{E}^\\bullet[1] \\ar[u]", "}", "$$", "where all the morphisms are as indicated above.", "Denote $K$ the object of $D(\\mathcal{O})$ corresponding to the complex", "$\\mathcal{K}^\\bullet$.", "By the arguments in the second paragraph of the proof we obtain", "a morphism $s : M \\to K$ in $D(\\mathcal{O})$ such that the composition", "$M \\to K \\to M$ is the identity on $M$. We don't know that the", "diagram", "$$", "\\xymatrix{", "\\mathcal{E}^\\bullet \\ar[r] &", "\\mathcal{K}^\\bullet \\ar@{=}[r] &", "K \\\\", "\\mathcal{E}^\\bullet \\ar[u]^{\\text{id}} \\ar[r]^i &", "\\mathcal{M}^\\bullet \\ar@{=}[r] &", "M \\ar[u]_s", "}", "$$", "commutes, but we do know it commutes after composing with the", "map $K \\to M$. By Lemma \\ref{lemma-local-actual} after replacing", "$X$ by the members of a covering, we may", "assume that $s \\circ i$ is given by a map of complexes", "$\\sigma : \\mathcal{E}^\\bullet \\to \\mathcal{K}^\\bullet$.", "By the same lemma we may assume the composition of $\\sigma$", "with the inclusion $\\mathcal{K}^\\bullet \\subset \\mathcal{M}^\\bullet$", "is homotopic to zero by some homotopy", "$\\{h^i : \\mathcal{E}^i \\to \\mathcal{M}^{i - 1}\\}$.", "Thus, after replacing $\\mathcal{K}^{m - 1}$ by", "$\\mathcal{K}^{m - 1} + \\Im(h^m)$ (note that after doing this", "it is still the case that $\\mathcal{K}^{m - 1}$ is generated", "by finitely many global sections), we see that", "$\\sigma$ itself is homotopic to zero!", "This means that we have a commutative solid diagram", "$$", "\\xymatrix{", "\\mathcal{E}^\\bullet \\ar[r] &", "M \\ar[r] &", "\\mathcal{L}^\\bullet \\ar[r] &", "\\mathcal{E}^\\bullet[1] \\\\", "\\mathcal{E}^\\bullet \\ar[r] \\ar[u] &", "K \\ar[r] \\ar[u] &", "\\sigma_{\\leq m - 1}\\mathcal{K}^\\bullet \\ar[r] \\ar[u] &", "\\mathcal{E}^\\bullet[1] \\ar[u] \\\\", "\\mathcal{E}^\\bullet \\ar[r] \\ar[u] &", "M \\ar[r] \\ar[u]^s &", "\\mathcal{L}^\\bullet \\ar[r] \\ar@{..>}[u] &", "\\mathcal{E}^\\bullet[1] \\ar[u]", "}", "$$", "By the axioms of triangulated categories we obtain a dotted", "arrow fitting into the diagram.", "Looking at cohomology sheaves in degree $m - 1$ we see that we obtain", "$$", "\\xymatrix{", "H^{m - 1}(M) \\ar[r] &", "H^{m - 1}(\\mathcal{L}^\\bullet) \\ar[r] &", "H^m(\\mathcal{E}^\\bullet) \\\\", "H^{m - 1}(K) \\ar[r] \\ar[u] &", "H^{m - 1}(\\sigma_{\\leq m - 1}\\mathcal{K}^\\bullet) \\ar[r] \\ar[u] &", "H^m(\\mathcal{E}^\\bullet) \\ar[u] \\\\", "H^{m - 1}(M) \\ar[r] \\ar[u] &", "H^{m - 1}(\\mathcal{L}^\\bullet) \\ar[r] \\ar[u] &", "H^m(\\mathcal{E}^\\bullet) \\ar[u]", "}", "$$", "Since the vertical compositions are the identity in both the", "left and right column, we conclude the vertical composition", "$H^{m - 1}(\\mathcal{L}^\\bullet) \\to", "H^{m - 1}(\\sigma_{\\leq m - 1}\\mathcal{K}^\\bullet) \\to", "H^{m - 1}(\\mathcal{L}^\\bullet)$ in the middle is surjective!", "In particular $H^{m - 1}(\\sigma_{\\leq m - 1}\\mathcal{K}^\\bullet) \\to", "H^{m - 1}(\\mathcal{L}^\\bullet)$ is surjective.", "Using the induced map of long exact sequences of cohomology", "sheaves from the morphism of triangles above, a diagram chase", "shows this implies $H^i(K) \\to H^i(M)$ is an isomorphism", "for $i \\geq m$ and surjective for $i = m - 1$.", "By construction we can choose an $r \\geq 0$ and a surjection", "$\\mathcal{O}^{\\oplus r} \\to \\mathcal{K}^{m - 1}$. Then the", "composition", "$$", "(\\mathcal{O}^{\\oplus r} \\to \\mathcal{E}^m \\to", "\\mathcal{E}^{m + 1} \\to \\ldots ) \\longrightarrow", "K \\longrightarrow M", "$$", "induces an isomorphism on cohomology sheaves in degrees $\\geq m$ and", "a surjection in degree $m - 1$ and the proof is complete." ], "refs": [ "sites-cohomology-lemma-K-flat-resolution", "modules-lemma-module-quotient-flat", "derived-lemma-subcategory-left-resolution", "sites-cohomology-lemma-perfect-precise", "derived-lemma-negative-exts", "sites-cohomology-lemma-local-actual" ], "ref_ids": [ 4236, 13276, 1835, 4382, 1893, 4363 ] } ], "ref_ids": [ 12407 ] }, { "id": 4393, "type": "theorem", "label": "sites-cohomology-lemma-colim-and-lim-of-duals", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-colim-and-lim-of-duals", "contents": [ "\\begin{slogan}", "Trivial duality for systems of perfect objects.", "\\end{slogan}", "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$(K_n)_{n \\in \\mathbf{N}}$ be a system of perfect objects of $D(\\mathcal{O})$.", "Let $K = \\text{hocolim} K_n$ be the derived colimit", "(Derived Categories, Definition \\ref{derived-definition-derived-colimit}).", "Then for any object $E$ of $D(\\mathcal{O})$ we have", "$$", "R\\SheafHom(K, E) = R\\lim E \\otimes^\\mathbf{L}_\\mathcal{O} K_n^\\vee", "$$", "where $(K_n^\\vee)$ is the inverse system of dual perfect complexes." ], "refs": [ "derived-definition-derived-colimit" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-dual-perfect-complex} we have", "$R\\lim E \\otimes^\\mathbf{L}_\\mathcal{O} K_n^\\vee =", "R\\lim R\\SheafHom(K_n, E)$", "which fits into the distinguished triangle", "$$", "R\\lim R\\SheafHom(K_n, E) \\to", "\\prod R\\SheafHom(K_n, E) \\to", "\\prod R\\SheafHom(K_n, E)", "$$", "Because $K$ similarly fits into the distinguished triangle", "$\\bigoplus K_n \\to \\bigoplus K_n \\to K$ it suffices to show that", "$\\prod R\\SheafHom(K_n, E) = R\\SheafHom(\\bigoplus K_n, E)$.", "This is a formal consequence of (\\ref{equation-internal-hom})", "and the fact that derived tensor product commutes with direct sums." ], "refs": [ "sites-cohomology-lemma-dual-perfect-complex" ], "ref_ids": [ 4390 ] } ], "ref_ids": [ 2001 ] }, { "id": 4394, "type": "theorem", "label": "sites-cohomology-lemma-category-summands-finite-free", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-category-summands-finite-free", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed space.", "Set $R = \\Gamma(\\mathcal{C}, \\mathcal{O})$. The category of", "$\\mathcal{O}$-modules which are summands of finite free", "$\\mathcal{O}$-modules is equivalent to the category of", "finite projective $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "Observe that a finite projective $R$-module is the same thing", "as a summand of a finite free $R$-module.", "The equivalence is given by the functor $\\mathcal{E} \\mapsto", "\\Gamma(\\mathcal{C}, \\mathcal{E})$.", "The inverse functor is given by the following construction.", "Consider the morphism of topoi $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\text{pt})$", "with $f_*$ given by taking global sections and", "$f^{-1}$ by sending a set $S$, i.e., an object of", "$\\Sh(\\text{pt})$, to the constant sheaf with value $S$.", "We obtain a morphism", "$(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\text{pt}), R)$", "of ringed topoi by using the identity map $R \\to f_*\\mathcal{O}$.", "Then the inverse functor is given by $f^*$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4395, "type": "theorem", "label": "sites-cohomology-lemma-invertible-derived", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-invertible-derived", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $M$ be an object", "of $D(\\mathcal{O})$. The following are equivalent", "\\begin{enumerate}", "\\item $M$ is invertible in $D(\\mathcal{O})$, see", "Categories, Definition \\ref{categories-definition-invertible}, and", "\\item there is a locally finite\\footnote{This means that for every", "object $U$ of $\\mathcal{C}$ there is a covering $\\{U_i \\to U\\}$", "such that for every $i$ the sheaf $\\mathcal{O}_n|_{U_i}$ is nonzero", "for only a finite number of $n$.} direct product decomposition", "$$", "\\mathcal{O} = \\prod\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_n", "$$", "and for each $n$ there is an invertible $\\mathcal{O}_n$-module", "$\\mathcal{H}^n$", "(Modules on Sites, Definition \\ref{sites-modules-definition-invertible-sheaf})", "and $M = \\bigoplus \\mathcal{H}^n[-n]$ in $D(\\mathcal{O})$.", "\\end{enumerate}", "If (1) and (2) hold, then $M$ is a perfect object of $D(\\mathcal{O})$. If", "$(\\mathcal{C}, \\mathcal{O})$ is a locally ringed site these condition", "are also equivalent to", "\\begin{enumerate}", "\\item[(3)] for every object $U$ of $\\mathcal{C}$ there exists a", "covering $\\{U_i \\to U\\}$ and for each $i$ an integer $n_i$ such that", "$M|_{U_i}$ is represented by an invertible $\\mathcal{O}_{U_i}$-module", "placed in degree $n_i$.", "\\end{enumerate}" ], "refs": [ "categories-definition-invertible", "sites-modules-definition-invertible-sheaf" ], "proofs": [ { "contents": [ "Assume (2). Consider the object $R\\SheafHom(M, \\mathcal{O})$", "and the composition map", "$$", "R\\SheafHom(M, \\mathcal{O}) \\otimes_\\mathcal{O}^\\mathbf{L} M \\to \\mathcal{O}", "$$", "To prove this is an isomorphism, we may work locally. Thus we may", "assume $\\mathcal{O} = \\prod_{a \\leq n \\leq b} \\mathcal{O}_n$", "and $M = \\bigoplus_{a \\leq n \\leq b} \\mathcal{H}^n[-n]$.", "Then it suffices to show that", "$$", "R\\SheafHom(\\mathcal{H}^m, \\mathcal{O})", "\\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{H}^n", "$$", "is zero if $n \\not = m$ and equal to $\\mathcal{O}_n$ if $n = m$.", "The case $n \\not = m$ follows from the fact that $\\mathcal{O}_n$ and", "$\\mathcal{O}_m$ are flat $\\mathcal{O}$-algebras with", "$\\mathcal{O}_n \\otimes_\\mathcal{O} \\mathcal{O}_m = 0$.", "Using the local structure of invertible $\\mathcal{O}$-modules", "(Modules on Sites, Lemma \\ref{sites-modules-lemma-invertible})", "and working locally", "the isomorphism in case $n = m$ follows in a straightforward manner;", "we omit the details. Because $D(\\mathcal{O})$ is symmetric monoidal,", "we conclude that $M$ is invertible.", "\\medskip\\noindent", "Assume (1). The description in (2) shows that we have a candidate", "for $\\mathcal{O}_n$, namely,", "$\\SheafHom_\\mathcal{O}(H^n(M), H^n(M))$.", "If this is a locally finite family of sheaves of rings", "and if $\\mathcal{O} = \\prod \\mathcal{O}_n$, then we immediately", "obtain the direct sum decomposition $M = \\bigoplus H^n(M)[-n]$", "using the idempotents in $\\mathcal{O}$ coming from the product", "decomposition. This shows that in order to prove (2) we may work", "locally in the following sense. Let $U$ be an object of $\\mathcal{C}$.", "We have to show there exists a covering", "$\\{U_i \\to U\\}$ of $U$ such that with $\\mathcal{O}_n$ as above", "we have the statements above and those of (2) after", "restriction to $\\mathcal{C}/U_i$.", "Thus we may assume $\\mathcal{C}$ has a final object $X$", "and during the proof of (2) we may finitely many times", "replace $X$ by the members of a covering of $X$.", "\\medskip\\noindent", "Choose an object $N$ of $D(\\mathcal{O})$ and an isomorphism", "$M \\otimes_\\mathcal{O}^\\mathbf{L} N \\cong \\mathcal{O}$.", "Then $N$ is a left dual for $M$ in the monoidal category", "$D(\\mathcal{O})$ and we conclude that $M$ is perfect by", "Lemma \\ref{lemma-left-dual-derived}. By symmetry we see that", "$N$ is perfect. After replacing $X$ by the members of a covering,", "we may assume $M$ and $N$ are represented by a strictly perfect", "complexes $\\mathcal{E}^\\bullet$ and $\\mathcal{F}^\\bullet$.", "Then $M \\otimes_\\mathcal{O}^\\mathbf{L} N$ is represented by", "$\\text{Tot}(\\mathcal{E}^\\bullet \\otimes_\\mathcal{O} \\mathcal{F}^\\bullet)$.", "After replacing $X$ by the members of a covering of $X$", "we may assume the mutually inverse isomorphisms", "$\\mathcal{O} \\to M \\otimes_\\mathcal{O}^\\mathbf{L} N$ and", "$M \\otimes_\\mathcal{O}^\\mathbf{L} N \\to \\mathcal{O}$", "are given by maps of complexes", "$$", "\\alpha : \\mathcal{O} \\to", "\\text{Tot}(\\mathcal{E}^\\bullet \\otimes_\\mathcal{O} \\mathcal{F}^\\bullet)", "\\quad\\text{and}\\quad", "\\beta :", "\\text{Tot}(\\mathcal{E}^\\bullet \\otimes_\\mathcal{O} \\mathcal{F}^\\bullet)", "\\to \\mathcal{O}", "$$", "See Lemma \\ref{lemma-local-actual}. Then $\\beta \\circ \\alpha = 1$", "as maps of complexes and $\\alpha \\circ \\beta = 1$ as a morphism", "in $D(\\mathcal{O})$. After replacing $X$ by the members of a covering", "of $X$ we may assume the composition $\\alpha \\circ \\beta$ is homotopic to $1$", "by some homotopy $\\theta$ with components", "$$", "\\theta^n :", "\\text{Tot}^n(\\mathcal{E}^\\bullet \\otimes_\\mathcal{O} \\mathcal{F}^\\bullet)", "\\to", "\\text{Tot}^{n - 1}(", "\\mathcal{E}^\\bullet \\otimes_\\mathcal{O} \\mathcal{F}^\\bullet)", "$$", "by the same lemma as before. Set $R = \\Gamma(\\mathcal{C}, \\mathcal{O})$. By", "Lemma \\ref{lemma-category-summands-finite-free}", "we find that we obtain", "\\begin{enumerate}", "\\item $M^\\bullet = \\Gamma(X, \\mathcal{E}^\\bullet)$", "is a bounded complex of finite projective $R$-modules,", "\\item $N^\\bullet = \\Gamma(X, \\mathcal{F}^\\bullet)$", "is a bounded complex of finite projective $R$-modules,", "\\item $\\alpha$ and $\\beta$ correspond to maps of complexes", "$a : R \\to \\text{Tot}(M^\\bullet \\otimes_R N^\\bullet)$ and", "$b : \\text{Tot}(M^\\bullet \\otimes_R N^\\bullet) \\to R$,", "\\item $\\theta^n$ corresponds to a map", "$h^n : \\text{Tot}^n(M^\\bullet \\otimes_R N^\\bullet) \\to", "\\text{Tot}^{n - 1}(M^\\bullet \\otimes_R N^\\bullet)$, and", "\\item $b \\circ a = 1$ and $b \\circ a - 1 = dh + hd$,", "\\end{enumerate}", "It follows that $M^\\bullet$ and $N^\\bullet$ define", "mutually inverse objects of $D(R)$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-invertible-derived}", "we find a product decomposition $R = \\prod_{a \\leq n \\leq b} R_n$", "and invertible $R_n$-modules $H^n$ such", "that $M^\\bullet \\cong \\bigoplus_{a \\leq n \\leq b} H^n[-n]$.", "This isomorphism in $D(R)$ can be lifted to an morphism", "$$", "\\bigoplus H^n[-n] \\longrightarrow M^\\bullet", "$$", "of complexes because each $H^n$ is projective as an $R$-module.", "Correspondingly, using Lemma \\ref{lemma-category-summands-finite-free} again,", "we obtain an morphism", "$$", "\\bigoplus H^n \\otimes_R \\mathcal{O}[-n] \\to \\mathcal{E}^\\bullet", "$$", "which is an isomorphism in $D(\\mathcal{O})$. Here $M \\otimes_R \\mathcal{O}$", "denotes the functor from finite projective $R$-modules to $\\mathcal{O}$-modules", "constructed in the proof of Lemma \\ref{lemma-category-summands-finite-free}.", "Setting $\\mathcal{O}_n = R_n \\otimes_R \\mathcal{O}$ we conclude", "(2) is true.", "\\medskip\\noindent", "If $(\\mathcal{C}, \\mathcal{O})$ is a locally ringed site,", "then given an object $U$ and a finite product decomposition", "$\\mathcal{O}|_U = \\prod_{a \\leq n \\leq b} \\mathcal{O}_n|_U$", "we can find a covering $\\{U_i \\to U\\}$ such that for every", "$i$ there is at most one $n$ with $\\mathcal{O}_n|_{U_i}$ nonzero.", "This follows readily from part (2) of", "Modules on Sites, Lemma \\ref{sites-modules-lemma-locally-ringed}", "and the definition of locally ringed sites as given in", "Modules on Sites, Definition \\ref{sites-modules-definition-locally-ringed}.", "From this the implication (2) $\\Rightarrow$ (3) is easily seen.", "The implication (3) $\\Rightarrow$ (2) holds without any assumptions", "on the ringed site. We omit the details." ], "refs": [ "sites-modules-lemma-invertible", "sites-cohomology-lemma-left-dual-derived", "sites-cohomology-lemma-local-actual", "sites-cohomology-lemma-category-summands-finite-free", "more-algebra-lemma-invertible-derived", "sites-cohomology-lemma-category-summands-finite-free", "sites-cohomology-lemma-category-summands-finite-free", "sites-modules-lemma-locally-ringed", "sites-modules-definition-locally-ringed" ], "ref_ids": [ 14224, 4392, 4363, 4394, 10575, 4394, 4394, 14253, 14302 ] } ], "ref_ids": [ 12406, 14293 ] }, { "id": 4396, "type": "theorem", "label": "sites-cohomology-lemma-projection-formula", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-projection-formula", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.", "Let $E \\in D(\\mathcal{O}_\\mathcal{C})$ and $K \\in D(\\mathcal{O}_\\mathcal{D})$.", "If $K$ is perfect, then", "$$", "Rf_*E \\otimes^\\mathbf{L}_{\\mathcal{O}_\\mathcal{D}} K =", "Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_\\mathcal{C}} Lf^*K)", "$$", "in $D(\\mathcal{O}_\\mathcal{D})$." ], "refs": [], "proofs": [ { "contents": [ "To check (\\ref{equation-projection-formula-map}) is an isomorphism", "we may work locally on $\\mathcal{D}$, i.e.,", "for any object $V$ of $\\mathcal{D}$ we have to find a covering $\\{V_j \\to V\\}$", "such that the map restricts to an isomorphism on $V_j$. By definition", "of perfect objects, this means we may assume $K$ is represented by", "a strictly perfect complex of $\\mathcal{O}_\\mathcal{D}$-modules.", "Note that, completely generally, the statement is true for", "$K = K_1 \\oplus K_2$, if and only if the statement is true for", "$K_1$ and $K_2$. Hence we may assume $K$ is a finite", "complex of finite free $\\mathcal{O}_\\mathcal{D}$-modules.", "In this case a simple argument involving stupid truncations reduces", "the statement to the case where $K$ is represented by a finite", "free $\\mathcal{O}_\\mathcal{D}$-module. Since the statement is invariant", "under finite direct summands in the $K$ variable, we conclude", "it suffices to prove it for $K = \\mathcal{O}_\\mathcal{D}[n]$", "in which case it is trivial." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4397, "type": "theorem", "label": "sites-cohomology-lemma-w-contractible", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-w-contractible", "contents": [ "Let $\\mathcal{C}$ be a site. Let $U$ be a weakly contractible", "object of $\\mathcal{C}$. Then", "\\begin{enumerate}", "\\item the functor $\\mathcal{F} \\mapsto \\mathcal{F}(U)$ is an exact", "functor $\\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}$,", "\\item $H^p(U, \\mathcal{F}) = 0$", "for every abelian sheaf $\\mathcal{F}$ and all $p \\geq 1$, and", "\\item for any sheaf of groups $\\mathcal{G}$ any $\\mathcal{G}$-torsor", "has a section over $U$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first statement follows immediately from the definition", "(see also Homology, Section \\ref{homology-section-functors}).", "The higher derived functors vanish by", "Derived Categories, Lemma \\ref{derived-lemma-right-derived-exact-functor}.", "Let $\\mathcal{F}$ be a $\\mathcal{G}$-torsor. Then $\\mathcal{F} \\to *$", "is a surjective map of sheaves. Hence (3) follows from the", "definition as well." ], "refs": [ "derived-lemma-right-derived-exact-functor" ], "ref_ids": [ 1845 ] } ], "ref_ids": [] }, { "id": 4398, "type": "theorem", "label": "sites-cohomology-lemma-compact-in-terms-of-generators", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-compact-in-terms-of-generators", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let", "$S \\subset \\Ob(\\mathcal{A})$ be a set of objects such that", "\\begin{enumerate}", "\\item any object of $\\mathcal{A}$ is a quotient of a direct sum", "of elements of $S$, and", "\\item for any $E \\in S$ the functor $\\Hom_\\mathcal{A}(E, -)$", "commutes with direct sums.", "\\end{enumerate}", "Then every compact object of $D(\\mathcal{A})$ is a direct summand", "in $D(\\mathcal{A})$ of a finite complex of finite direct sums of", "elements of $S$." ], "refs": [], "proofs": [ { "contents": [ "Assume $K \\in D(\\mathcal{A})$ is a compact object. Represent $K$ by a complex", "$K^\\bullet$ and consider the map", "$$", "K^\\bullet", "\\longrightarrow", "\\bigoplus\\nolimits_{n \\geq 0} \\tau_{\\geq n} K^\\bullet", "$$", "where we have used the canonical truncations, see", "Homology, Section \\ref{homology-section-truncations}.", "This makes sense as in each degree the direct sum on the right is finite.", "By assumption this map factors through a finite direct sum.", "We conclude that $K \\to \\tau_{\\geq n} K$ is zero for at least one $n$,", "i.e., $K$ is in $D^{-}(R)$.", "\\medskip\\noindent", "We may represent $K$ by a bounded above complex $K^\\bullet$ each of whose", "terms is a direct sum of objects from $S$, see", "Derived Categories, Lemma \\ref{derived-lemma-subcategory-left-resolution}.", "Note that we have", "$$", "K^\\bullet = \\bigcup\\nolimits_{n \\leq 0} \\sigma_{\\geq n}K^\\bullet", "$$", "where we have used the stupid truncations, see", "Homology, Section \\ref{homology-section-truncations}.", "Hence by Derived Categories, Lemmas \\ref{derived-lemma-colim-hocolim} and", "\\ref{derived-lemma-commutes-with-countable-sums}", "we see that $1 : K^\\bullet \\to K^\\bullet$ factors through", "$\\sigma_{\\geq n}K^\\bullet \\to K^\\bullet$ in $D(R)$.", "Thus we see that $1 : K^\\bullet \\to K^\\bullet$ factors as", "$$", "K^\\bullet \\xrightarrow{\\varphi} L^\\bullet \\xrightarrow{\\psi} K^\\bullet", "$$", "in $D(\\mathcal{A})$ for some complex $L^\\bullet$ which is bounded and", "whose terms are direct sums of elements of $S$. Say $L^i$ is zero for", "$i \\not \\in [a, b]$. Let $c$ be the largest integer $\\leq b + 1$ such", "that $L^i$ a finite direct sum of elements of $S$ for $i < c$.", "Claim: if $c < b + 1$, then we can modify $L^\\bullet$ to increase $c$.", "By induction this claim will show we have a factorization", "of $1_K$ as", "$$", "K \\xrightarrow{\\varphi} L \\xrightarrow{\\psi} K", "$$", "in $D(\\mathcal{A})$ where $L$ can be represented by a finite", "complex of finite direct sums of elements of $S$. Note that", "$e = \\varphi \\circ \\psi \\in \\text{End}_{D(\\mathcal{A})}(L)$", "is an idempotent. By Derived Categories,", "Lemma \\ref{derived-lemma-projectors-have-images-triangulated}", "we see that $L = \\Ker(e) \\oplus \\Ker(1 - e)$.", "The map $\\varphi : K \\to L$ induces an isomorphism with", "$\\Ker(1 - e)$ in $D(R)$ and we conclude.", "\\medskip\\noindent", "Proof of the claim. Write $L^c = \\bigoplus_{\\lambda \\in \\Lambda} E_\\lambda$.", "Since $L^{c - 1}$ is a finite direct sum of elements of $S$", "we can by assumption (2) find a finite subset", "$\\Lambda' \\subset \\Lambda$ such that $L^{c - 1} \\to L^c$ factors", "through $\\bigoplus_{\\lambda \\in \\Lambda'} E_\\lambda \\subset L^c$.", "Consider the map of complexes", "$$", "\\pi :", "L^\\bullet", "\\longrightarrow", "(\\bigoplus\\nolimits_{\\lambda \\in \\Lambda \\setminus \\Lambda'} E_\\lambda)[-i]", "$$", "given by the projection onto the factors corresponding to", "$\\Lambda \\setminus \\Lambda'$ in degree $i$.", "By our assumption on $K$ we see that, after possibly replacing $\\Lambda'$ by", "a larger finite subset, we may assume that $\\pi \\circ \\varphi = 0$", "in $D(\\mathcal{A})$.", "Let $(L')^\\bullet \\subset L^\\bullet$ be the kernel of $\\pi$.", "Since $\\pi$ is surjective we get a short exact sequence of complexes,", "which gives a distinguished triangle in $D(\\mathcal{A})$ (see", "Derived Categories, Lemma \\ref{derived-lemma-derived-canonical-delta-functor}).", "Since $\\Hom_{D(\\mathcal{A})}(K, -)$ is homological (see", "Derived Categories, Lemma \\ref{derived-lemma-representable-homological})", "and $\\pi \\circ \\varphi = 0$, we can find a morphism", "$\\varphi' : K^\\bullet \\to (L')^\\bullet$ in $D(\\mathcal{A})$ whose", "composition with $(L')^\\bullet \\to L^\\bullet$ gives $\\varphi$.", "Setting $\\psi'$ equal to the composition of $\\psi$ with", "$(L')^\\bullet \\to L^\\bullet$ we obtain a new factorization.", "Since $(L')^\\bullet$ agrees with $L^\\bullet$ except in degree $c$", "and since $(L')^c = \\bigoplus_{\\lambda \\in \\Lambda'} E_\\lambda$ the", "claim is proved." ], "refs": [ "derived-lemma-subcategory-left-resolution", "derived-lemma-colim-hocolim", "derived-lemma-commutes-with-countable-sums", "derived-lemma-projectors-have-images-triangulated", "derived-lemma-derived-canonical-delta-functor", "derived-lemma-representable-homological" ], "ref_ids": [ 1835, 1922, 1924, 1769, 1814, 1758 ] } ], "ref_ids": [] }, { "id": 4399, "type": "theorem", "label": "sites-cohomology-lemma-compact-objects-if-enough-qc", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-compact-objects-if-enough-qc", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Assume every object of", "$\\mathcal{C}$ has a covering by quasi-compact objects. Then every", "compact object of $D(\\mathcal{O})$ is a direct summand in $D(\\mathcal{O})$", "of a finite complex whose terms are finite direct sums of", "$\\mathcal{O}$-modules of the form $j_!\\mathcal{O}_U$", "where $U$ is a quasi-compact object of $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-compact-in-terms-of-generators}", "where $S \\subset \\Ob(\\textit{Mod}(\\mathcal{O}))$ is the set of modules", "of the form $j_!\\mathcal{O}_U$ with $U \\in \\Ob(\\mathcal{C})$", "quasi-compact. Assumption (1) holds by", "Modules on Sites, Lemma \\ref{sites-modules-lemma-module-quotient-flat}", "and the assumption that every $U$ can be covered by quasi-compact", "objects. Assumption (2) follows as", "$$", "\\Hom_\\mathcal{O}(j_!\\mathcal{O}_U, \\mathcal{F}) = \\mathcal{F}(U)", "$$", "which commutes with direct sums by", "Sites, Lemma \\ref{sites-lemma-directed-colimits-sections}." ], "refs": [ "sites-cohomology-lemma-compact-in-terms-of-generators", "sites-modules-lemma-module-quotient-flat", "sites-lemma-directed-colimits-sections" ], "ref_ids": [ 4398, 14203, 8531 ] } ], "ref_ids": [] }, { "id": 4400, "type": "theorem", "label": "sites-cohomology-lemma-when-jshriek-lower-compact", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-when-jshriek-lower-compact", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of", "$\\mathcal{C}$. Assume the functors $\\mathcal{F} \\mapsto H^p(U, \\mathcal{F})$", "commute with direct sums. Then $\\mathcal{O}$-module $j_!\\mathcal{O}_U$ is a", "compact object of $D^+(\\mathcal{O})$ in the following sense:", "if $M = \\bigoplus_{i \\in I} M_i$ in $D(\\mathcal{O})$ is", "bounded below, then $\\Hom(j_{U!}\\mathcal{O}_U, M) =", "\\bigoplus_{i \\in I} \\Hom(j_{U!}\\mathcal{O}_U, M_i)$." ], "refs": [], "proofs": [ { "contents": [ "Since $\\Hom(j_{U!}\\mathcal{O}_U, -)$ is the same as the functor", "$\\mathcal{F} \\mapsto \\mathcal{F}(U)$ by", "Modules on Sites, Equation", "(\\ref{sites-modules-equation-map-lower-shriek-OU-into-module}) it suffices", "to prove that $H^p(U, M) = \\bigoplus H^p(U, M_i)$.", "Let $\\mathcal{I}_i$, $i \\in I$ be a collection of injective", "$\\mathcal{O}$-modules. By assumption we have", "$$", "H^p(U, \\bigoplus\\nolimits_{i \\in I} \\mathcal{I}_i) =", "\\bigoplus\\nolimits_{i \\in I} H^p(U, \\mathcal{I}_i) = 0", "$$", "for all $p$. Since $M = \\bigoplus M_i$ is bounded below, we", "see that there exists an $a \\in \\mathbf{Z}$ such that $H^n(M_i) = 0$", "for $n < a$. Thus we can choose complexes of injective $\\mathcal{O}$-modues", "$\\mathcal{I}_i^\\bullet$ representing $M_i$", "with $\\mathcal{I}_i^n = 0$ for $n < a$, see", "Derived Categories, Lemma \\ref{derived-lemma-injective-resolutions-exist}.", "By Injectives, Lemma \\ref{injectives-lemma-derived-products}", "we see that the direct sum complex $\\bigoplus \\mathcal{I}_i^\\bullet$", "represents $M$. By Leray acyclicity", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "we see that", "$$", "R\\Gamma(U, M) = \\Gamma(U, \\bigoplus \\mathcal{I}_i^\\bullet) =", "\\bigoplus \\Gamma(U, \\bigoplus \\mathcal{I}_i^\\bullet) =", "\\bigoplus R\\Gamma(U, M_i)", "$$", "as desired." ], "refs": [ "derived-lemma-injective-resolutions-exist", "injectives-lemma-derived-products", "derived-lemma-leray-acyclicity" ], "ref_ids": [ 1851, 7795, 1844 ] } ], "ref_ids": [] }, { "id": 4401, "type": "theorem", "label": "sites-cohomology-lemma-when-jshriek-lower-compact-worked-out", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-when-jshriek-lower-compact-worked-out", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site", "with set of coverings $\\text{Cov}_\\mathcal{C}$.", "Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$, and", "$\\text{Cov} \\subset \\text{Cov}_\\mathcal{C}$", "be subsets. Assume that", "\\begin{enumerate}", "\\item For every $\\mathcal{U} \\in \\text{Cov}$ we have", "$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ with $I$ finite,", "$U, U_i \\in \\mathcal{B}$ and every", "$U_{i_0} \\times_U \\ldots \\times_U U_{i_p} \\in \\mathcal{B}$.", "\\item For every $U \\in \\mathcal{B}$ the coverings of $U$", "occurring in $\\text{Cov}$ is a cofinal system of coverings of $U$.", "\\end{enumerate}", "Then for $U \\in \\mathcal{B}$ the object $j_{U!}\\mathcal{O}_U$ is", "a compact object of $D^+(\\mathcal{O})$ in the following sense:", "if $M = \\bigoplus_{i \\in I} M_i$ in $D(\\mathcal{O})$ is", "bounded below, then $\\Hom(j_{U!}\\mathcal{O}_U, M) =", "\\bigoplus_{i \\in I} \\Hom(j_{U!}\\mathcal{O}_U, M_i)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-when-jshriek-lower-compact}", "and Lemma \\ref{lemma-colim-works-over-collection}." ], "refs": [ "sites-cohomology-lemma-when-jshriek-lower-compact", "sites-cohomology-lemma-colim-works-over-collection" ], "ref_ids": [ 4400, 4224 ] } ], "ref_ids": [] }, { "id": 4402, "type": "theorem", "label": "sites-cohomology-lemma-when-jshriek-compact", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-when-jshriek-compact", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of", "$\\mathcal{C}$. The $\\mathcal{O}$-module $j_!\\mathcal{O}_U$ is a", "compact object of $D(\\mathcal{O})$ if there exists an integer $d$ such that", "\\begin{enumerate}", "\\item $H^p(U, \\mathcal{F}) = 0$ for all $p > d$, and", "\\item the functors $\\mathcal{F} \\mapsto H^p(U, \\mathcal{F})$", "commute with direct sums.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and (2). Recall that $\\Hom(j_!\\mathcal{O}_U, K) = R\\Gamma(U, K)$ for", "$K$ in $D(\\mathcal{O})$. Thus we have to show that $R\\Gamma(U, -)$", "commutes with direct sums. The first assumption means that the functor", "$F = H^0(U, -)$ has finite cohomological dimension. Moreover, the second", "assumption implies any direct sum of injective modules is acyclic for $F$.", "Let $K_i$ be a family of objects of $D(\\mathcal{O})$.", "Choose K-injective representatives $I_i^\\bullet$ with injective terms", "representing $K_i$, see Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "Since we may compute $RF$ by applying $F$ to any complex of acyclics", "(Derived Categories, Lemma \\ref{derived-lemma-unbounded-right-derived})", "and since $\\bigoplus K_i$ is represented by $\\bigoplus I_i^\\bullet$", "(Injectives, Lemma \\ref{injectives-lemma-derived-products})", "we conclude that $R\\Gamma(U, \\bigoplus K_i)$ is represented by", "$\\bigoplus H^0(U, I_i^\\bullet)$. Hence $R\\Gamma(U, -)$ commutes", "with direct sums as desired." ], "refs": [ "injectives-theorem-K-injective-embedding-grothendieck", "derived-lemma-unbounded-right-derived", "injectives-lemma-derived-products" ], "ref_ids": [ 7768, 1917, 7795 ] } ], "ref_ids": [] }, { "id": 4403, "type": "theorem", "label": "sites-cohomology-lemma-quasi-compact-weakly-contractible-compact", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-quasi-compact-weakly-contractible-compact", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$", "be an object of $\\mathcal{C}$ which is quasi-compact and", "weakly contractible. Then", "$j_!\\mathcal{O}_U$ is a compact object of $D(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-when-jshriek-compact} and", "\\ref{lemma-w-contractible} with", "Modules on Sites, Lemma \\ref{sites-modules-lemma-sections-over-quasi-compact}." ], "refs": [ "sites-cohomology-lemma-when-jshriek-compact", "sites-cohomology-lemma-w-contractible", "sites-modules-lemma-sections-over-quasi-compact" ], "ref_ids": [ 4402, 4397, 14215 ] } ], "ref_ids": [] }, { "id": 4404, "type": "theorem", "label": "sites-cohomology-lemma-perfect-is-compact", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-perfect-is-compact", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Assume", "$\\mathcal{C}$ has the following properties", "\\begin{enumerate}", "\\item $\\mathcal{C}$ has a quasi-compact final object $X$,", "\\item every quasi-compact object of $\\mathcal{C}$", "has a cofinal system of coverings which are finite", "and consist of quasi-compact objects,", "\\item for a finite covering $\\{U_i \\to U\\}_{i \\in I}$", "with $U$, $U_i$ quasi-compact the fibre products $U_i \\times_U U_j$ are", "quasi-compact.", "\\end{enumerate}", "Let $K$ be a perfect object of $D(\\mathcal{O})$. Then", "\\begin{enumerate}", "\\item[(a)] $K$ is a compact object of $D^+(\\mathcal{O})$", "in the following sense: if $M = \\bigoplus_{i \\in I} M_i$ is", "bounded below, then $\\Hom(K, M) = \\bigoplus_{i \\in I} \\Hom(K, M_i)$.", "\\item[(b)] If $(\\mathcal{C}, \\mathcal{O})$", "has finite cohomological dimension, i.e., if there exists", "a $d$ such that $H^i(X, \\mathcal{F}) = 0$ for $i > d$ for", "any $\\mathcal{O}$-module $\\mathcal{F}$, then", "$K$ is a compact object of $D(\\mathcal{O})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $K^\\vee$ be the dual of $K$, see", "Lemma \\ref{lemma-dual-perfect-complex}. Then we have", "$$", "\\Hom_{D(\\mathcal{O})}(K, M) =", "H^0(X, K^\\vee \\otimes_\\mathcal{O}^\\mathbf{L} M)", "$$", "functorially in $M$ in $D(\\mathcal{O})$.", "Since $K^\\vee \\otimes_\\mathcal{O}^\\mathbf{L} -$ commutes with", "direct sums it suffices", "to show that $R\\Gamma(X, -)$ commutes with the relevant direct sums.", "\\medskip\\noindent", "Proof of (a). After reformulation as above this is a special", "case of Lemma \\ref{lemma-when-jshriek-lower-compact-worked-out} with $U = X$.", "\\medskip\\noindent", "Proof of (b). Since $R\\Gamma(X, K) = R\\Hom(\\mathcal{O}, K)$", "and since $H^p(X, -)$ commutes with direct sums by", "Lemma \\ref{lemma-colim-works-over-collection}", "this is a special case of", "Lemma \\ref{lemma-when-jshriek-compact}." ], "refs": [ "sites-cohomology-lemma-dual-perfect-complex", "sites-cohomology-lemma-when-jshriek-lower-compact-worked-out", "sites-cohomology-lemma-colim-works-over-collection", "sites-cohomology-lemma-when-jshriek-compact" ], "ref_ids": [ 4390, 4401, 4224, 4402 ] } ], "ref_ids": [] }, { "id": 4405, "type": "theorem", "label": "sites-cohomology-lemma-locally-constant", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-locally-constant", "contents": [ "Let $\\mathcal{C}$ be a site with final object $X$.", "Let $\\Lambda$ be a Noetherian ring.", "Let $K \\in D^b(\\mathcal{C}, \\Lambda)$", "with $H^i(K)$ locally constant sheaves of $\\Lambda$-modules", "of finite type. Then there exists a covering $\\{U_i \\to X\\}$", "such that each $K|_{U_i}$ is represented by", "a complex of locally constant sheaves of $\\Lambda$-modules", "of finite type." ], "refs": [], "proofs": [ { "contents": [ "Let $a \\leq b$ be such that $H^i(K) = 0$ for $i \\not \\in [a, b]$.", "By induction on $b - a$ we will prove there exists a covering", "$\\{U_i \\to X\\}$ such that $K|_{U_i}$ can be represented by a complex", "$\\underline{M^\\bullet}_{U_i}$ with $M^p$ a finite type $\\Lambda$-module", "and $M^p = 0$ for $p \\not \\in [a, b]$. If $b = a$, then", "this is clear. In general, we may replace $X$ by the members", "of a covering and assume that $H^b(K)$ is constant, say", "$H^b(K) = \\underline{M}$. By Modules on Sites, Lemma", "\\ref{sites-modules-lemma-locally-constant-finite-type}", "the module $M$ is a finite $\\Lambda$-module. Choose a surjection", "$\\Lambda^{\\oplus r} \\to M$ given by generators $x_1, \\ldots, x_r$", "of $M$.", "\\medskip\\noindent", "By a slight generalization of", "Lemma \\ref{lemma-kill-cohomology-class-on-covering} (details omitted)", "there exists a covering $\\{U_i \\to X\\}$ such that $x_i \\in H^0(X, H^b(K))$", "lifts to an element of $H^b(U_i, K)$. Thus, after replacing $X$ by the", "$U_i$ we reach the situation where there is a map", "$\\underline{\\Lambda^{\\oplus r}}[-b] \\to K$", "inducing a surjection on cohomology sheaves in degree $b$.", "Choose a distinguished triangle", "$$", "\\underline{\\Lambda^{\\oplus r}}[-b] \\to K \\to L \\to", "\\underline{\\Lambda^{\\oplus r}}[-b + 1]", "$$", "Now the cohomology sheaves of $L$ are nonzero only in the interval", "$[a, b - 1]$, agree with the cohomology sheaves of $K$ in the interval", "$[a, b - 2]$ and there is a short exact sequence", "$$", "0 \\to H^{b - 1}(K) \\to H^{b - 1}(L) \\to", "\\underline{\\Ker(\\Lambda^{\\oplus r} \\to M)} \\to 0", "$$", "in degree $b - 1$. By", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-kernel-finite-locally-constant}", "we see that $H^{b - 1}(L)$ is locally constant of finite type.", "By induction hypothesis we obtain an isomorphism", "$\\underline{M^\\bullet} \\to L$ in $D(\\mathcal{C}, \\underline{\\Lambda})$", "with $M^p$ a finite $\\Lambda$-module and $M^p = 0$ for", "$p \\not \\in [a, b - 1]$. The map $L \\to \\Lambda^{\\oplus r}[-b + 1]$", "gives a map $\\underline{M^{b - 1}} \\to \\underline{\\Lambda^{\\oplus r}}$", "which locally is constant", "(Modules on Sites, Lemma", "\\ref{sites-modules-lemma-morphism-locally-constant}).", "Thus we may assume it is given by a map $M^{b - 1} \\to \\Lambda^{\\oplus r}$.", "The distinguished triangle shows that the composition", "$M^{b - 2} \\to M^{b - 1} \\to \\Lambda^{\\oplus r}$ is zero", "and the axioms of triangulated categories produce an isomorphism", "$$", "\\underline{M^a \\to \\ldots \\to M^{b - 1} \\to \\Lambda^{\\oplus r}}", "\\longrightarrow K", "$$", "in $D(\\mathcal{C}, \\Lambda)$." ], "refs": [ "sites-modules-lemma-locally-constant-finite-type", "sites-cohomology-lemma-kill-cohomology-class-on-covering", "sites-modules-lemma-kernel-finite-locally-constant", "sites-modules-lemma-morphism-locally-constant" ], "ref_ids": [ 14269, 4188, 14273, 14271 ] } ], "ref_ids": [] }, { "id": 4406, "type": "theorem", "label": "sites-cohomology-lemma-map-out-of-locally-constant", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-map-out-of-locally-constant", "contents": [ "Let $\\mathcal{C}$ be a site with final object $X$. Let $\\Lambda$ be a ring. Let", "\\begin{enumerate}", "\\item $K$ a perfect object of $D(\\Lambda)$,", "\\item a finite complex $K^\\bullet$ of finite projective $\\Lambda$-modules", "representing $K$,", "\\item $\\mathcal{L}^\\bullet$ a complex of sheaves of $\\Lambda$-modules, and", "\\item $\\varphi : \\underline{K} \\to \\mathcal{L}^\\bullet$ a map in", "$D(\\mathcal{C}, \\Lambda)$.", "\\end{enumerate}", "Then there exists a covering $\\{U_i \\to X\\}$ and maps of complexes", "$\\alpha_i : \\underline{K}^\\bullet|_{U_i} \\to \\mathcal{L}^\\bullet|_{U_i}$", "representing $\\varphi|_{U_i}$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-local-actual}." ], "refs": [ "sites-cohomology-lemma-local-actual" ], "ref_ids": [ 4363 ] } ], "ref_ids": [] }, { "id": 4407, "type": "theorem", "label": "sites-cohomology-lemma-locally-constant-map", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-locally-constant-map", "contents": [ "Let $\\mathcal{C}$ be a site with final object $X$.", "Let $\\Lambda$ be a ring. Let $K, L$ be objects of", "$D(\\Lambda)$ with $K$ perfect. Let $\\varphi : \\underline{K} \\to \\underline{L}$", "be map in $D(\\mathcal{C}, \\Lambda)$. There exists a covering $\\{U_i \\to X\\}$", "such that $\\varphi|_{U_i}$ is equal to $\\underline{\\alpha_i}$", "for some map $\\alpha_i : K \\to L$ in $D(\\Lambda)$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-map-out-of-locally-constant} and", "Modules on Sites, Lemma \\ref{sites-modules-lemma-morphism-locally-constant}." ], "refs": [ "sites-cohomology-lemma-map-out-of-locally-constant", "sites-modules-lemma-morphism-locally-constant" ], "ref_ids": [ 4406, 14271 ] } ], "ref_ids": [] }, { "id": 4408, "type": "theorem", "label": "sites-cohomology-lemma-locally-constant-tensor-product", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-locally-constant-tensor-product", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a Noetherian ring.", "Let $K, L \\in D^-(\\mathcal{C}, \\Lambda)$. If the cohomology sheaves of", "$K$ and $L$ are locally constant sheaves of $\\Lambda$-modules of", "finite type, then the cohomology sheaves of", "$K \\otimes_\\Lambda^\\mathbf{L} L$", "are locally constant sheaves of $\\Lambda$-modules of finite type." ], "refs": [], "proofs": [ { "contents": [ "We'll prove this as an application of Lemma \\ref{lemma-locally-constant}.", "Note that $H^i(K \\otimes_\\Lambda^\\mathbf{L} L)$ is the same as", "$H^i(\\tau_{\\geq i - 1}K \\otimes_\\Lambda^\\mathbf{L} \\tau_{\\geq i - 1}L)$.", "Thus we may assume $K$ and $L$ are bounded. By", "Lemma \\ref{lemma-locally-constant}", "we may assume that $K$ and $L$ are represented by", "complexes of locally constant sheaves of $\\Lambda$-modules", "of finite type. Then we can replace these complexes by", "bounded above complexes of finite free $\\Lambda$-modules.", "In this case the result is clear." ], "refs": [ "sites-cohomology-lemma-locally-constant", "sites-cohomology-lemma-locally-constant" ], "ref_ids": [ 4405, 4405 ] } ], "ref_ids": [] }, { "id": 4409, "type": "theorem", "label": "sites-cohomology-lemma-locally-constant-bounded", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-lemma-locally-constant-bounded", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a Noetherian ring.", "Let $I \\subset \\Lambda$ be an ideal.", "Let $K \\in D^-(\\mathcal{C}, \\Lambda)$. If the cohomology sheaves of", "$K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$ are locally constant", "sheaves of $\\Lambda/I$-modules of finite type, then the cohomology sheaves of", "$K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$", "are locally constant sheaves of $\\Lambda/I^n$-modules of finite type for all", "$n \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "Recall that the locally constant sheaves of $\\Lambda$-modules of finite type", "form a weak Serre subcategory of all $\\underline{\\Lambda}$-modules, see", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-kernel-finite-locally-constant}.", "Thus the subcategory of $D(\\mathcal{C}, \\Lambda)$ consisting of", "complexes whose cohomology sheaves are locally constant sheaves", "of $\\Lambda$-modules of finite type forms a strictly full, saturated", "triangulated subcategory of $D(\\mathcal{C}, \\Lambda)$, see", "Derived Categories, Lemma \\ref{derived-lemma-cohomology-in-serre-subcategory}.", "Next, consider the distinguished triangles", "$$", "K \\otimes_\\Lambda^\\mathbf{L} \\underline{I^n/I^{n + 1}} \\to", "K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^{n + 1}} \\to", "K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n} \\to", "K \\otimes_\\Lambda^\\mathbf{L} \\underline{I^n/I^{n + 1}}[1]", "$$", "and the isomorphisms", "$$", "K \\otimes_\\Lambda^\\mathbf{L} \\underline{I^n/I^{n + 1}}", "=", "\\left(K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}\\right)", "\\otimes_{\\Lambda/I}^\\mathbf{L} \\underline{I^n/I^{n + 1}}", "$$", "Combined with Lemma \\ref{lemma-locally-constant-tensor-product}", "we obtain the result." ], "refs": [ "sites-modules-lemma-kernel-finite-locally-constant", "derived-lemma-cohomology-in-serre-subcategory", "sites-cohomology-lemma-locally-constant-tensor-product" ], "ref_ids": [ 14273, 1846, 4408 ] } ], "ref_ids": [] }, { "id": 4410, "type": "theorem", "label": "sites-cohomology-proposition-enough-weakly-contractibles", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-proposition-enough-weakly-contractibles", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$", "such that every $U \\in \\mathcal{B}$ is weakly contractible and", "every object of $\\mathcal{C}$ has a covering by elements of $\\mathcal{B}$.", "Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$. Then", "\\begin{enumerate}", "\\item A complex $\\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3$", "of $\\mathcal{O}$-modules is exact, if and only if", "$\\mathcal{F}_1(U) \\to \\mathcal{F}_2(U) \\to \\mathcal{F}_3(U)$", "is exact for all $U \\in \\mathcal{B}$.", "\\item Every object $K$ of $D(\\mathcal{O})$ is a derived limit", "of its canonical truncations: $K = R\\lim \\tau_{\\geq -n} K$.", "\\item Given an inverse system", "$\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1$", "with surjective transition maps, the projection", "$\\lim \\mathcal{F}_n \\to \\mathcal{F}_1$ is surjective.", "\\item Products are exact on $\\textit{Mod}(\\mathcal{O})$.", "\\item Products on $D(\\mathcal{O})$ can be computed by taking", "products of any representative complexes.", "\\item If $(\\mathcal{F}_n)$ is an inverse system of $\\mathcal{O}$-modules,", "then $R^p\\lim \\mathcal{F}_n = 0$ for all $p > 1$ and", "$$", "R^1\\lim \\mathcal{F}_n =", "\\Coker(\\prod \\mathcal{F}_n \\to \\prod \\mathcal{F}_n)", "$$", "where the map is $(x_n) \\mapsto (x_n - f(x_{n + 1}))$.", "\\item If $(K_n)$ is an inverse system of objects of $D(\\mathcal{O})$,", "then there are short exact sequences", "$$", "0 \\to R^1\\lim H^{p - 1}(K_n) \\to H^p(R\\lim K_n) \\to", "\\lim H^p(K_n) \\to 0", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). If the sequence is exact, then evaluating at any", "weakly contractible element of $\\mathcal{C}$ gives an exact", "sequence by Lemma \\ref{lemma-w-contractible}. Conversely, assume that", "$\\mathcal{F}_1(U) \\to \\mathcal{F}_2(U) \\to \\mathcal{F}_3(U)$", "is exact for all $U \\in \\mathcal{B}$.", "Let $V$ be an object of $\\mathcal{C}$ and let", "$s \\in \\mathcal{F}_2(V)$ be an element of the kernel of", "$\\mathcal{F}_2 \\to \\mathcal{F}_3$. By assumption there exists", "a covering $\\{U_i \\to V\\}$ with $U_i \\in \\mathcal{B}$.", "Then $s|_{U_i}$ lifts to a section $s_i \\in \\mathcal{F}_1(U_i)$.", "Thus $s$ is a section of the image sheaf", "$\\Im(\\mathcal{F}_1 \\to \\mathcal{F}_2)$.", "In other words, the sequence", "$\\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3$", "is exact.", "\\medskip\\noindent", "Proof of (2). This holds by Lemma \\ref{lemma-is-limit-dimension} with $d = 0$.", "\\medskip\\noindent", "Proof of (3). Let $(\\mathcal{F}_n)$ be a system as in (2) and set", "$\\mathcal{F} = \\lim \\mathcal{F}_n$. If $U \\in \\mathcal{B}$, then", "$\\mathcal{F}(U) = \\lim \\mathcal{F}_n(U)$", "surjects onto $\\mathcal{F}_1(U)$ as all the transition maps", "$\\mathcal{F}_{n + 1}(U) \\to \\mathcal{F}_n(U)$ are surjective.", "Thus $\\mathcal{F} \\to \\mathcal{F}_1$ is surjective by", "Sites, Definition \\ref{sites-definition-sheaves-injective-surjective}", "and the assumption that every object", "has a covering by elements of $\\mathcal{B}$.", "\\medskip\\noindent", "Proof of (4). Let", "$\\mathcal{F}_{i, 1} \\to \\mathcal{F}_{i, 2} \\to \\mathcal{F}_{i, 3}$", "be a family of exact sequences of $\\mathcal{O}$-modules.", "We want to show that", "$\\prod \\mathcal{F}_{i, 1} \\to \\prod \\mathcal{F}_{i, 2} \\to", "\\prod \\mathcal{F}_{i, 3}$ is exact. We use the criterion of (1).", "Let $U \\in \\mathcal{B}$. Then", "$$", "(\\prod \\mathcal{F}_{i, 1})(U) \\to", "(\\prod \\mathcal{F}_{i, 2})(U) \\to", "(\\prod \\mathcal{F}_{i, 3})(U)", "$$", "is the same as", "$$", "\\prod \\mathcal{F}_{i, 1}(U) \\to", "\\prod \\mathcal{F}_{i, 2}(U) \\to", "\\prod \\mathcal{F}_{i, 3}(U)", "$$", "Each of the sequences", "$\\mathcal{F}_{i, 1}(U) \\to \\mathcal{F}_{i, 2}(U) \\to \\mathcal{F}_{i, 3}(U)$", "are exact by (1). Thus the displayed sequences are exact by", "Homology, Lemma \\ref{homology-lemma-product-abelian-groups-exact}.", "We conclude by (1) again.", "\\medskip\\noindent", "Proof of (5). Follows from (4) and (slightly generalized)", "Derived Categories, Lemma \\ref{derived-lemma-products}.", "\\medskip\\noindent", "Proof of (6) and (7). We refer to Section \\ref{section-derived-limits}", "for a discussion of derived and homotopy limits and their relationship.", "By Derived Categories, Definition \\ref{derived-definition-derived-limit}", "we have a distinguished", "triangle", "$$", "R\\lim K_n \\to \\prod K_n \\to \\prod K_n \\to R\\lim K_n[1]", "$$", "Taking the long exact sequence of cohomology sheaves we obtain", "$$", "H^{p - 1}(\\prod K_n) \\to H^{p - 1}(\\prod K_n) \\to", "H^p(R\\lim K_n) \\to H^p(\\prod K_n) \\to H^p(\\prod K_n)", "$$", "Since products are exact by (4) this becomes", "$$", "\\prod H^{p - 1}(K_n) \\to \\prod H^{p - 1}(K_n) \\to", "H^p(R\\lim K_n) \\to \\prod H^p(K_n) \\to \\prod H^p(K_n)", "$$", "Now we first apply this to the case $K_n = \\mathcal{F}_n[0]$", "where $(\\mathcal{F}_n)$ is as in (6). We conclude that (6) holds.", "Next we apply it to $(K_n)$ as in (7) and we conclude (7) holds." ], "refs": [ "sites-cohomology-lemma-w-contractible", "sites-cohomology-lemma-is-limit-dimension", "sites-definition-sheaves-injective-surjective", "homology-lemma-product-abelian-groups-exact", "derived-lemma-products", "derived-definition-derived-limit" ], "ref_ids": [ 4397, 4273, 8660, 12130, 1925, 2002 ] } ], "ref_ids": [] }, { "id": 4441, "type": "theorem", "label": "fields-theorem-existence-algebraic-closure", "categories": [ "fields" ], "title": "fields-theorem-existence-algebraic-closure", "contents": [ "Every field has an algebraic closure." ], "refs": [], "proofs": [ { "contents": [ "Let $F$ be a field. By Lemma \\ref{lemma-size-algebraic-extension} the", "cardinality of an algebraic extension of $F$ is bounded by", "$\\max(\\aleph_0, |F|)$. Choose a set $S$ containing $F$ with", "$|S| > \\max(\\aleph_0, |F|)$. Let's consider triples", "$(E, \\sigma_E, \\mu_E)$ where", "\\begin{enumerate}", "\\item $E$ is a set with $F \\subset E \\subset S$, and", "\\item $\\sigma_E : E \\times E \\to E$ and $\\mu_E : E \\times E \\to E$", "are maps of sets such that $(E, \\sigma_E, \\mu_E)$ defines the structure", "of a field extension of $F$ (in particular $\\sigma_E(a, b) = a +_F b$", "for $a, b \\in F$ and similarly for $\\mu_E$), and", "\\item $F \\subset E$ is an algebraic field extension.", "\\end{enumerate}", "The collection of all triples $(E, \\sigma_E, \\mu_E)$ forms a set $I$.", "For $i \\in I$ we will denote $E_i = (E_i, \\sigma_i, \\mu_i)$ the", "corresponding field extension to $F$. We define a partial ordering on", "$I$ by declaring $i \\leq i'$ if and only if $E_i \\subset E_{i'}$", "(this makes sense as $E_i$ and $E_{i'}$ are subsets of the same set $S$)", "and we have $\\sigma_i = \\sigma_{i'}|_{E_i \\times E_i}$ and", "$\\mu_i = \\mu_{i'}|_{E_i \\times E_i}$, in other words, $E_{i'}$ is a field", "extension of $E_i$.", "\\medskip\\noindent", "Let $T \\subset I$ be a totally ordered subset. Then it is clear that", "$E_T = \\bigcup_{i \\in T} E_i$ with induced maps $\\sigma_T = \\bigcup \\sigma_i$", "and $\\mu_T = \\bigcup \\mu_i$ is another element of $I$. In other words", "every totally order subset of $I$ has a upper bound in $I$. By Zorn's lemma", "there exists a maximal element $(E, \\sigma_E, \\mu_E)$ in $I$. We claim that", "$E$ is an algebraic closure. Since by definition of $I$ the extension", "$E/F$ is algebraic, it suffices to show that $E$ is algebraically closed.", "\\medskip\\noindent", "To see this we argue by contradiction. Namely, suppose that $E$ is not", "algebraically closed. Then there exists an irreducible polynomial", "$P$ over $E$ of degree $> 1$, see Lemma \\ref{lemma-algebraically-closed}.", "By Lemma \\ref{lemma-finite-is-algebraic} we obtain a nontrivial finite", "extension $E' = E[x]/(P)$. Observe that $E'/F$ is algebraic by", "Lemma \\ref{lemma-algebraic-permanence}.", "Thus the cardinality of $E'$ is $\\leq \\max(\\aleph_0, |F|)$.", "By elementary set theory we can extend the given injection", "$E \\subset S$ to an injection $E' \\to S$. In other words, we may", "think of $E'$ as an element of our set $I$ contradicting the", "maximality of $E$. This contradiction completes the proof." ], "refs": [ "fields-lemma-size-algebraic-extension", "fields-lemma-algebraically-closed", "fields-lemma-finite-is-algebraic", "fields-lemma-algebraic-permanence" ], "ref_ids": [ 4456, 4460, 4452, 4455 ] } ], "ref_ids": [] }, { "id": 4442, "type": "theorem", "label": "fields-theorem-galois-theory", "categories": [ "fields" ], "title": "fields-theorem-galois-theory", "contents": [ "Let $L/K$ be a finite Galois extension with Galois group $G$.", "Then we have $K = L^G$ and the map", "$$", "\\{\\text{subgroups of }G\\}", "\\longrightarrow", "\\{\\text{subextensions }K \\subset M \\subset L\\},\\quad", "H \\longmapsto L^H", "$$", "is a bijection whose inverse maps $M$ to $\\text{Gal}(L/M)$.", "The normal subgroups $H$ of $G$ correspond exactly to those", "subextensions $M$ with $M/K$ Galois." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-galois-goes-up} given a subextension $L/M/K$", "the extension $L/M$ is Galois. Of course $L/M$ is also finite", "(Lemma \\ref{lemma-finite-goes-up}). Thus $|\\text{Gal}(L/M)| = [L : M]$", "by Lemma \\ref{lemma-finite-Galois}.", "Conversely, if $H \\subset G$ is a finite subgroup, then", "$[L : L^H] = |H|$ by Lemma \\ref{lemma-galois-over-fixed-field}.", "It follows formally from these two observations that we obtain", "a bijective correspondence as in the theorem.", "\\medskip\\noindent", "If $H \\subset G$ is normal, then $L^H$ is fixed by the action of", "$G$ and we obtain a canonical map $G/H \\to \\text{Aut}(L^H/K)$.", "This map has to be injective as $\\text{Gal}(L/L^H) = H$. Hence", "$|G/H| = [L^H : K]$ and $L^H$ is Galois by", "Lemma \\ref{lemma-finite-Galois}.", "\\medskip\\noindent", "Conversely, assume that $K \\subset M \\subset L$ with $M/K$ Galois.", "By Lemma \\ref{lemma-lift-maps} we see that every", "element $\\tau \\in \\text{Gal}(L/K)$ induces an element", "$\\tau|_M \\in \\text{Gal}(M/K)$. This induces a homomorphism", "of Galois groups $\\text{Gal}(L/K) \\to \\text{Gal}(M/K)$ whose", "kernel is $H$. Thus $H$ is a normal subgroup." ], "refs": [ "fields-lemma-galois-goes-up", "fields-lemma-finite-goes-up", "fields-lemma-finite-Galois", "fields-lemma-galois-over-fixed-field", "fields-lemma-finite-Galois", "fields-lemma-lift-maps" ], "ref_ids": [ 4505, 4448, 4504, 4507, 4504, 4490 ] } ], "ref_ids": [] }, { "id": 4443, "type": "theorem", "label": "fields-theorem-inifinite-galois-theory", "categories": [ "fields" ], "title": "fields-theorem-inifinite-galois-theory", "contents": [ "Let $L/K$ be a Galois extension. Let $G = \\text{Gal}(L/K)$", "be the Galois group viewed as a profinite topological group", "(Lemma \\ref{lemma-galois-profinite}). Then we have $K = L^G$ and the map", "$$", "\\{\\text{closed subgroups of }G\\}", "\\longrightarrow", "\\{\\text{subextensions }K \\subset M \\subset L\\},\\quad", "H \\longmapsto L^H", "$$", "is a bijection whose inverse maps $M$ to $\\text{Gal}(L/M)$.", "The finite subextensions $M$ correspond exactly to the open", "subgroups $H \\subset G$. The normal closed subgroups $H$ of $G$", "correspond exactly to subextensions $M$ Galois over $K$." ], "refs": [ "fields-lemma-galois-profinite" ], "proofs": [ { "contents": [ "We will use the result of finite Galois theory", "(Theorem \\ref{theorem-galois-theory})", "without further mention.", "Let $S \\subset L$ be a finite subset. There exists a tower", "$L/E/K$ such that $K(S) \\subset E$ and such that", "$E/K$ is finite Galois, see Lemma \\ref{lemma-normal-closure-inside-normal}.", "In other words, we see that $L/K$ is the union of its finite", "Galois subextensions.", "For such an $E$, by Lemma \\ref{lemma-galois-infinite}", "the map $\\text{Gal}(L/K) \\to \\text{Gal}(E/K)$ is surjective", "and continuous, i.e., the kernel is open because the topology", "on $\\text{Gal}(E/K)$ is discrete.", "In particular we see that no element of $M \\setminus K$ is fixed by", "$\\text{Gal}(L/K)$ as $E^{\\text{Gal}(E/K)} = K$.", "This proves that $L^G = K$.", "\\medskip\\noindent", "Lemma \\ref{lemma-galois-goes-up} given a subextension $L/M/K$", "the extension $L/M$ is Galois. It is immediate from the definition", "of the topology on $G$ that the subgroup $\\text{Gal}(L/M)$ is closed.", "By the above applied to $L/M$ we see that $L^{\\text{Gal}(L/M)} = M$", "\\medskip\\noindent", "Conversely, let $H \\subset G$ be a closed subgroup. We claim that", "$H = \\text{Gal}(L/L^H)$. The inclusion $H \\subset \\text{Gal}(L/L^H)$", "is clear. Suppose that $g \\in \\text{Gal}(L/L^H)$. Let $S \\subset L$", "be a finite subset. We will show that the open neighbourhood", "$U_S(g) = \\{g' \\in G \\mid g'(s) = g(s)\\}$ of $g$ meets $H$.", "This implies that $g \\in H$ because $H$ is closed.", "Let $L/E/K$ be a finite Galois subextension containing $K(S)$", "as in the first paragraph of the proof and consider the homomorphism", "$c : \\text{Gal}(L/K) \\to \\text{Gal}(E/K)$.", "Then $L^H \\cap E = E^{c(H)}$. Since $g$ fixes $L^H$ it fixes", "$E^{c(H)}$ and hence $c(g) \\in c(H)$ by finite Galois theory.", "Pick $h \\in H$ with $c(h) = c(g)$. Then $h \\in U_S(g)$ as desired.", "\\medskip\\noindent", "At this point we have established the correspondence between closed", "subgroups and subextensions.", "\\medskip\\noindent", "Assume $H \\subset G$ is open. Arguing as above we find that", "$H$ containes $\\text{Gal}(E/K)$ for some large enough finite", "Galois subextension $E$ and we find that $L^H$ is contained", "in $E$ whence finite over $K$. Conversely, if $M$ is a finite", "subextension, then $M$ is generated by a finite subset $S$", "and the corresponding subgroup is the open subset $U_S(e)$", "where $e \\in G$ is the neutral element.", "\\medskip\\noindent", "Assume that $K \\subset M \\subset L$ with $M/K$ Galois.", "By Lemma \\ref{lemma-galois-infinite} there is a surjective", "continuous homomorphism of Galois groups", "$\\text{Gal}(L/K) \\to \\text{Gal}(M/K)$ whose", "kernel is $\\text{Gal}(L/M)$. Thus $\\text{Gal}(L/M)$ is a normal", "closed subgroup.", "\\medskip\\noindent", "Finally, assume $N \\subset G$ is normal and closed. For any", "$L/E/K$ as in the first paragraph of the proof, the image", "$c(N) \\subset \\text{Gal}(E/K)$ is a normal subgroup.", "Hence $L^N = \\bigcup E^{c(N)}$ is a union of Galois extensions", "of $K$ (by finite Galois theory) whence Galois over $K$." ], "refs": [ "fields-theorem-galois-theory", "fields-lemma-normal-closure-inside-normal", "fields-lemma-galois-infinite", "fields-lemma-galois-goes-up", "fields-lemma-galois-infinite" ], "ref_ids": [ 4442, 4495, 4510, 4505, 4510 ] } ], "ref_ids": [ 4509 ] }, { "id": 4444, "type": "theorem", "label": "fields-lemma-vector-space-is-free", "categories": [ "fields" ], "title": "fields-lemma-vector-space-is-free", "contents": [ "If $k$ is a field, then every $k$-module is free." ], "refs": [], "proofs": [ { "contents": [ "Indeed, by linear algebra we know that a $k$-module (i.e. vector space)", "$V$ has a {\\it basis} $\\mathcal{B} \\subset V$, which defines an isomorphism", "from the free vector space on $\\mathcal{B}$ to $V$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4445, "type": "theorem", "label": "fields-lemma-field-semi-simple", "categories": [ "fields" ], "title": "fields-lemma-field-semi-simple", "contents": [ "Every exact sequence of modules over a field splits." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-vector-space-is-free} as every vector", "space is a projective module." ], "refs": [ "fields-lemma-vector-space-is-free" ], "ref_ids": [ 4444 ] } ], "ref_ids": [] }, { "id": 4446, "type": "theorem", "label": "fields-lemma-field-maps-injective", "categories": [ "fields" ], "title": "fields-lemma-field-maps-injective", "contents": [ "If $F$ is a field and $R$ is a nonzero ring, then any ring homomorphism", "$\\varphi : F \\to R$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Indeed, let $a \\in \\Ker(\\varphi)$ be a nonzero element. Then we have", "$\\varphi(1) = \\varphi(a^{-1} a) = \\varphi(a^{-1}) \\varphi(a) = 0$.", "Thus $1 = \\varphi(1) = 0$ and $R$ is the zero ring." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4447, "type": "theorem", "label": "fields-lemma-field-extension-generated-by-one-element", "categories": [ "fields" ], "title": "fields-lemma-field-extension-generated-by-one-element", "contents": [ "If a field extension $F/k$ is generated by one element, then it is", "$k$-isomorphic either to the rational function field $k(t)/k$ or to one", "of the extensions $k[t]/(P)$ for $P \\in k[t]$ irreducible." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha \\in F$ be such that $F = k(\\alpha)$; by assumption, such an", "$\\alpha$ exists. There is a morphism of rings", "$$", "k[t] \\to F", "$$", "sending the indeterminate $t$ to $\\alpha$. The image is a domain, so the", "kernel is a prime ideal. Thus, it is either $(0)$ or $(P)$ for $P \\in k[t]$", "irreducible.", "\\medskip\\noindent", "If the kernel is $(P)$ for $P \\in k[t]$ irreducible, then the map factors", "through $k[t]/(P)$, and induces a morphism of fields $k[t]/(P) \\to F$. Since", "the image contains $\\alpha$, we see easily that the map is surjective, hence", "an isomorphism. In this case, $k[t]/(P) \\simeq F$.", "\\medskip\\noindent", "If the kernel is trivial, then we have an injection $k[t] \\to F$.", "One may thus define a morphism of the quotient field $k(t)$ into $F$; given a", "quotient $R(t)/Q(t)$ with $R(t), Q(t) \\in k[t]$, we map this to", "$R(\\alpha)/Q(\\alpha)$. The hypothesis that $k[t] \\to F$ is injective implies", "that $Q(\\alpha) \\neq 0$ unless $Q$ is the zero polynomial.", "The quotient field of $k[t]$ is the rational function field $k(t)$, so we get", "a morphism $k(t) \\to F$", "whose image contains $\\alpha$. It is thus surjective, hence an isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4448, "type": "theorem", "label": "fields-lemma-finite-goes-up", "categories": [ "fields" ], "title": "fields-lemma-finite-goes-up", "contents": [ "Let $K/E/F$ be a tower of algebraic field extensions.", "If $K$ is finite over $F$, then $K$ is finite over $E$." ], "refs": [], "proofs": [ { "contents": [ "Direct from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4449, "type": "theorem", "label": "fields-lemma-finite-finitely-generated", "categories": [ "fields" ], "title": "fields-lemma-finite-finitely-generated", "contents": [ "A finite extension of fields is a finitely generated field extension.", "The converse is not true." ], "refs": [], "proofs": [ { "contents": [ "Let $F/E$ be a finite extension of fields. Let $\\alpha_1, \\ldots, \\alpha_n$", "be a basis of $F$ as a vector space over $E$. Then", "$F = E(\\alpha_1, \\ldots, \\alpha_n)$ hence $F/E$ is a finitely generated", "field extension. The converse is not true as follows from", "Example \\ref{example-degree-rational-function-field}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4450, "type": "theorem", "label": "fields-lemma-multiplicativity-degrees", "categories": [ "fields" ], "title": "fields-lemma-multiplicativity-degrees", "contents": [ "Suppose given a tower of fields $F/E/k$. Then", "$$", "[F:k] = [F:E][E:k]", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha_1, \\ldots, \\alpha_n \\in F$ be an $E$-basis for $F$. Let", "$\\beta_1, \\ldots, \\beta_m \\in E$ be a $k$-basis for $E$. Then the claim is", "that the set of products", "$\\{\\alpha_i \\beta_j, 1 \\leq i \\leq n, 1 \\leq j \\leq m\\}$", "is a $k$-basis for $F$. Indeed, let us check first that they span $F$ over $k$.", "\\medskip\\noindent", "By assumption, the $\\{\\alpha_i\\}$ span $F$ over $E$. So if", "$f \\in F$, there are $a_i \\in E$ with", "$$", "f = \\sum\\nolimits_i a_i \\alpha_i,", "$$", "and, for each $i$, we can write $a_i = \\sum b_{ij} \\beta_j$ for some", "$b_{ij} \\in k$. Putting these together, we find", "$$", "f = \\sum\\nolimits_{i,j} b_{ij} \\alpha_i \\beta_j,", "$$", "proving that the $\\{\\alpha_i \\beta_j\\}$ span $F$ over $k$.", "\\medskip\\noindent", "Suppose now that there existed a nontrivial relation", "$$", "\\sum\\nolimits_{i,j} c_{ij} \\alpha_i \\beta_j = 0", "$$", "for the $c_{ij} \\in k$. In that case, we would have", "$$", "\\sum\\nolimits_i \\alpha_i \\left( \\sum\\nolimits_j c_{ij} \\beta_j \\right) = 0,", "$$", "and the inner terms lie in $E$ as the $\\beta_j$ do. Now $E$-linear", "independence of the $\\{\\alpha_i\\}$ shows that the inner sums are all zero.", "Then $k$-linear independence of the $\\{\\beta_j\\}$ shows that the", "$c_{ij}$ all vanish." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4451, "type": "theorem", "label": "fields-lemma-algebraic-goes-up", "categories": [ "fields" ], "title": "fields-lemma-algebraic-goes-up", "contents": [ "Let $K/E/F$ be a tower of field extensions.", "\\begin{enumerate}", "\\item If $\\alpha \\in K$ is algebraic over $F$, then $\\alpha$ is algebraic", "over $E$.", "\\item if $K$ is algebraic over $F$, then $K$ is algebraic over $E$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4452, "type": "theorem", "label": "fields-lemma-finite-is-algebraic", "categories": [ "fields" ], "title": "fields-lemma-finite-is-algebraic", "contents": [ "A finite extension is algebraic. In fact, an extension $E/k$ is algebraic", "if and only if every subextension $k(\\alpha)/k$ generated by some", "$\\alpha \\in E$ is finite." ], "refs": [], "proofs": [ { "contents": [ "Let $E/k$ be finite, say of degree $n$. Choose $\\alpha \\in E$. Then the", "elements $\\{1, \\alpha, \\ldots, \\alpha^n\\}$ are linearly", "dependent over $E$, or we would necessarily have $[E : k] > n$. A relation of", "linear dependence now gives the desired polynomial that $\\alpha$ must satisfy.", "\\medskip\\noindent", "For the last assertion, note that a monogenic extension $k(\\alpha)/k$ is", "finite if and only if $\\alpha$ is algebraic over $k$, by", "Examples \\ref{example-degree-rational-function-field} and", "\\ref{example-degree-simple-algebraic-extension}.", "So if $E/k$ is algebraic, then each $k(\\alpha)/k$, $\\alpha \\in E$, is a finite", "extension, and conversely." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4453, "type": "theorem", "label": "fields-lemma-algebraic-finitely-generated", "categories": [ "fields" ], "title": "fields-lemma-algebraic-finitely-generated", "contents": [ "\\begin{slogan}", "A finitely generated algebraic extension is finite.", "\\end{slogan}", "Let $k$ be a field, and let $\\alpha_1, \\alpha_2, \\ldots, \\alpha_n$ be elements", "of some extension field such that each $\\alpha_i$ is algebraic over $k$. Then", "the extension $k(\\alpha_1, \\ldots, \\alpha_n)/k$ is finite.", "That is, a finitely generated algebraic extension is finite." ], "refs": [], "proofs": [ { "contents": [ "Indeed, each extension", "$k(\\alpha_{1}, \\ldots, \\alpha_{i+1})/k(\\alpha_1, \\ldots, \\alpha_{i})$", "is generated by one element and algebraic, hence finite.", "By multiplicativity of degree (Lemma \\ref{lemma-multiplicativity-degrees})", "we obtain the result." ], "refs": [ "fields-lemma-multiplicativity-degrees" ], "ref_ids": [ 4450 ] } ], "ref_ids": [] }, { "id": 4454, "type": "theorem", "label": "fields-lemma-algebraic-elements", "categories": [ "fields" ], "title": "fields-lemma-algebraic-elements", "contents": [ "Let $E/k$ be a field extension. Then the elements of $E$ algebraic over $k$", "form a subextension of $E/k$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha, \\beta \\in E$ be algebraic over $k$. Then $k(\\alpha, \\beta)/k$", "is a finite extension by Lemma \\ref{lemma-algebraic-finitely-generated}.", "It follows that $k(\\alpha + \\beta) \\subset k(\\alpha, \\beta)$ is a finite", "extension, which implies that $\\alpha + \\beta$ is algebraic by", "Lemma \\ref{lemma-finite-is-algebraic}. Similarly for the difference,", "product and quotient of $\\alpha$ and $\\beta$." ], "refs": [ "fields-lemma-algebraic-finitely-generated", "fields-lemma-finite-is-algebraic" ], "ref_ids": [ 4453, 4452 ] } ], "ref_ids": [] }, { "id": 4455, "type": "theorem", "label": "fields-lemma-algebraic-permanence", "categories": [ "fields" ], "title": "fields-lemma-algebraic-permanence", "contents": [ "Let $E/k$ and $F/E$ be algebraic extensions of fields. Then $F/k$ is an", "algebraic extension of fields." ], "refs": [], "proofs": [ { "contents": [ "Choose $\\alpha \\in F$. Then $\\alpha$ is algebraic over $E$.", "The key observation is that $\\alpha$ is algebraic over a", "finitely generated subextension of $k$.", "That is, there is a finite set $S \\subset E$ such that $\\alpha $ is algebraic", "over $k(S)$: this is clear because being algebraic means that a certain", "polynomial in $E[x]$ that $\\alpha$ satisfies exists, and as $S$ we can take the", "coefficients of this polynomial. It follows that $\\alpha$ is algebraic over", "$k(S)$. In particular, the extension $k(S, \\alpha)/ k(S)$ is finite.", "Since $S$ is a finite set, and $k(S)/k$ is algebraic,", "Lemma \\ref{lemma-algebraic-finitely-generated} shows that", "$k(S)/k$ is finite. Using multiplicativity", "(Lemma \\ref{lemma-multiplicativity-degrees})", "we find that $k(S,\\alpha)/k$ is finite, so $\\alpha$ is algebraic over $k$." ], "refs": [ "fields-lemma-algebraic-finitely-generated", "fields-lemma-multiplicativity-degrees" ], "ref_ids": [ 4453, 4450 ] } ], "ref_ids": [] }, { "id": 4456, "type": "theorem", "label": "fields-lemma-size-algebraic-extension", "categories": [ "fields" ], "title": "fields-lemma-size-algebraic-extension", "contents": [ "Let $E/F$ be an algebraic extension of fields. Then the cardinality $|E|$", "of $E$ is at most $\\max(\\aleph_0, |F|)$." ], "refs": [], "proofs": [ { "contents": [ "Let $S$ be the set of nonconstant polynomials with coefficients in $F$.", "For every $P \\in S$ the set of roots", "$r(P, E) = \\{\\alpha \\in E \\mid P(\\alpha) = 0\\}$", "is finite (details omitted). Moreover, the fact that $E$ is algebraic", "over $F$ implies that $E = \\bigcup_{P \\in S} r(P, E)$.", "It is clear that $S$ has cardinality bounded by $\\max(\\aleph_0, |F|)$", "because the cardinality of a countable product of copies of $F$ has", "cardinality at most $\\max(\\aleph_0, |F|)$.", "Thus so does $E$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4457, "type": "theorem", "label": "fields-lemma-subalgebra-algebraic-extension-field", "categories": [ "fields" ], "title": "fields-lemma-subalgebra-algebraic-extension-field", "contents": [ "Let $E/F$ be a finite or more generally an algebraic extension of fields.", "Any subring $F \\subset R \\subset E$ is a field." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha \\in R$ be nonzero. Then $1, \\alpha, \\alpha^2, \\ldots$", "are contained in $R$. By Lemma \\ref{lemma-finite-is-algebraic}", "we find a nontrivial relation", "$a_0 + a_1 \\alpha + \\ldots + a_d \\alpha^d = 0$.", "We may assume $a_0 \\not = 0$ because if not we can divide the relation", "by $\\alpha$ to decrease $d$. Then we see that", "$$", "a_0 = \\alpha (- a_1 - \\ldots - a_d \\alpha^{d - 1})", "$$", "which proves that the inverse of $\\alpha$ is the element", "$a_0^{-1} (- a_1 - \\ldots - a_d \\alpha^{d - 1})$", "of $R$." ], "refs": [ "fields-lemma-finite-is-algebraic" ], "ref_ids": [ 4452 ] } ], "ref_ids": [] }, { "id": 4458, "type": "theorem", "label": "fields-lemma-algebraic-extension-self-map", "categories": [ "fields" ], "title": "fields-lemma-algebraic-extension-self-map", "contents": [ "Let $E/F$ an algebraic extension of fields. Any $F$-algebra map", "$f : E \\to E$ is an automorphism." ], "refs": [], "proofs": [ { "contents": [ "If $E/F$ is finite, then $f : E \\to E$ is an $F$-linear", "injective map (Lemma \\ref{lemma-field-maps-injective})", "of finite dimensional vector spaces, and hence bijective.", "In general we still see that $f$ is injective.", "Let $\\alpha \\in E$ and let $P \\in F[x]$ be a", "polynomial such that $P(\\alpha) = 0$.", "Let $E' \\subset E$ be the subfield of $E$ generated", "by the roots $\\alpha = \\alpha_1, \\ldots, \\alpha_n$ of $P$ in $E$.", "Then $E'$ is finite over $F$ by Lemma \\ref{lemma-algebraic-finitely-generated}.", "Since $f$ preserves the set of roots, we find that", "$f|_{E'} : E' \\to E'$. Hence $f|_{E'}$ is an isomorphism", "by the first part of the proof and we conclude that $\\alpha$", "is in the image of $f$." ], "refs": [ "fields-lemma-field-maps-injective", "fields-lemma-algebraic-finitely-generated" ], "ref_ids": [ 4446, 4453 ] } ], "ref_ids": [] }, { "id": 4459, "type": "theorem", "label": "fields-lemma-degree-minimal-polynomial", "categories": [ "fields" ], "title": "fields-lemma-degree-minimal-polynomial", "contents": [ "The degree of the minimal polynomial is $[k(\\alpha) : k]$." ], "refs": [], "proofs": [ { "contents": [ "This is just a restatement of the argument in", "Lemma \\ref{lemma-field-extension-generated-by-one-element}: the observation", "is that if $P$ is the minimal polynomial of $\\alpha$, then the map", "$$", "k[x]/(P) \\to k(\\alpha), \\quad x \\mapsto \\alpha", "$$", "is an isomorphism as in the aforementioned proof, and we have counted the", "degree of such an extension (see", "Example \\ref{example-degree-simple-algebraic-extension})." ], "refs": [ "fields-lemma-field-extension-generated-by-one-element" ], "ref_ids": [ 4447 ] } ], "ref_ids": [] }, { "id": 4460, "type": "theorem", "label": "fields-lemma-algebraically-closed", "categories": [ "fields" ], "title": "fields-lemma-algebraically-closed", "contents": [ "Let $F$ be a field. The following are equivalent", "\\begin{enumerate}", "\\item $F$ is algebraically closed,", "\\item every irreducible polynomial over $F$ is linear,", "\\item every nonconstant polynomial over $F$ has a root,", "\\item every nonconstant polynomial over $F$ is a product of linear factors.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $F$ is algebraically closed, then every irreducible polynomial is linear.", "Namely, if there exists an irreducible polynomial of degree $> 1$, then", "this generates a nontrivial finite (hence algebraic) field extension, see", "Example \\ref{example-degree-simple-algebraic-extension}.", "Thus (1) implies (2). If every irreducible polynomial", "is linear, then every irreducible polynomial has a root, whence every", "nonconstant polynomial has a root. Thus (2) implies (3).", "\\medskip\\noindent", "Assume every nonconstant polynomial has a root. Let $P \\in F[x]$", "be nonconstant. If $P(\\alpha) = 0$ with $\\alpha \\in F$, then we see", "that $P = (x - \\alpha)Q$ for some $Q \\in F[x]$ (by division with remainder).", "Thus we can argue by induction on the degree that any nonconstant", "polynomial can be written as a product $c \\prod (x - \\alpha_i)$.", "\\medskip\\noindent", "Finally, suppose that every nonconstant polynomial over $F$ is a product of", "linear factors. Let $E/F$ be an algebraic extension. Then all the simple", "subextensions $F(\\alpha)/F$ of $E$ are necessarily trivial (because the", "only irreducible polynomials are linear by assumption). Thus $E = F$.", "We see that (4) implies (1) and we are done." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4461, "type": "theorem", "label": "fields-lemma-map-into-algebraic-closure", "categories": [ "fields" ], "title": "fields-lemma-map-into-algebraic-closure", "contents": [ "Let $F$ be a field. Let $\\overline{F}$ be an algebraic closure of $F$.", "Let $M/F$ be an algebraic extension. Then there is a morphism of", "$F$-extensions $M \\to \\overline{F}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the set $I$ of pairs $(E, \\varphi)$ where $F \\subset E \\subset M$", "is a subextension and $\\varphi : E \\to \\overline{F}$ is a morphism of", "$F$-extensions. We partially order the set $I$ by declaring", "$(E, \\varphi) \\leq (E', \\varphi')$ if and only if $E \\subset E'$ and", "$\\varphi'|_E = \\varphi$. If $T = \\{(E_t, \\varphi_t)\\} \\subset I$", "is a totally ordered subset, then", "$\\bigcup \\varphi_t : \\bigcup E_t \\to \\overline{F}$ is an element of $I$.", "Thus every totally ordered subset of $I$ has an upper bound.", "By Zorn's lemma there exists a maximal element $(E, \\varphi)$ in $I$.", "We claim that $E = M$, which will finish the proof. If not, then", "pick $\\alpha \\in M$, $\\alpha \\not \\in E$. The $\\alpha$ is algebraic", "over $E$, see Lemma \\ref{lemma-algebraic-goes-up}.", "Let $P$ be the minimal polynomial of $\\alpha$ over $E$.", "Let $P^\\varphi$ be the image of $P$ by $\\varphi$ in $\\overline{F}[x]$.", "Since $\\overline{F}$ is algebraically closed there is a root $\\beta$", "of $P^\\varphi$ in $\\overline{F}$. Then we can extend $\\varphi$ to", "$\\varphi' : E(\\alpha) = E[x]/(P) \\to \\overline{F}$ by mapping", "$x$ to $\\beta$. This contradicts the maximality of $(E, \\varphi)$", "as desired." ], "refs": [ "fields-lemma-algebraic-goes-up" ], "ref_ids": [ 4451 ] } ], "ref_ids": [] }, { "id": 4462, "type": "theorem", "label": "fields-lemma-algebraic-closures-isomorphic", "categories": [ "fields" ], "title": "fields-lemma-algebraic-closures-isomorphic", "contents": [ "Any two algebraic closures of a field are isomorphic." ], "refs": [], "proofs": [ { "contents": [ "Let $F$ be a field. If $M$ and $\\overline{F}$ are algebraic closures of", "$F$, then there exists a morphism of $F$-extensions", "$\\varphi : M \\to \\overline{F}$ by", "Lemma \\ref{lemma-map-into-algebraic-closure}.", "Now the image $\\varphi(M)$ is algebraically closed.", "On the other hand, the extension $\\varphi(M) \\subset \\overline{F}$", "is algebraic by Lemma \\ref{lemma-algebraic-goes-up}.", "Thus $\\varphi(M) = \\overline{F}$." ], "refs": [ "fields-lemma-map-into-algebraic-closure", "fields-lemma-algebraic-goes-up" ], "ref_ids": [ 4461, 4451 ] } ], "ref_ids": [] }, { "id": 4463, "type": "theorem", "label": "fields-lemma-relatively-prime-polynomials", "categories": [ "fields" ], "title": "fields-lemma-relatively-prime-polynomials", "contents": [ "Two polynomials in $k[x]$ are relatively prime precisely when they", "have no common roots in an algebraic closure $\\overline{k}$ of $k$." ], "refs": [], "proofs": [ { "contents": [ "The claim is that any two polynomials $P, Q$ generate $(1)$ in $k[x]$ if and", "only if they generate $(1)$ in $\\overline{k}[x]$. This is a piece of", "linear algebra: a system of linear equations with coefficients in $k$ has", "a solution if and only if it has a solution in any extension of $k$.", "Consequently, we can reduce to the case of an algebraically closed field, in", "which case the result is clear from what we have already proved." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4464, "type": "theorem", "label": "fields-lemma-irreducible-polynomials", "categories": [ "fields" ], "title": "fields-lemma-irreducible-polynomials", "contents": [ "Let $F$ be a field. Let $P \\in F[x]$ be an irreducible polynomial over $F$.", "Let $P' = \\text{d}P/\\text{d}x$ be the derivative of $P$ with respect", "to $x$. Then one of the following two cases happens", "\\begin{enumerate}", "\\item $P$ and $P'$ are relatively prime, or", "\\item $P'$ is the zero polynomial.", "\\end{enumerate}", "The second case can only happen if $F$ has characteristic $p > 0$.", "In this case $P(x) = Q(x^q)$ where $q = p^f$ is a power of $p$ and", "$Q \\in F[x]$ is an irreducible polynomial such that $Q$ and $Q'$", "are relatively prime." ], "refs": [], "proofs": [ { "contents": [ "Note that $P'$ has degree $< \\deg(P)$. Hence if $P$ and $P'$ are not relatively", "prime, then $(P, P') = (R)$ where $R$ is a polynomial of degree $< \\deg(P)$", "contradicting the irreducibility of $P$. This proves we have the dichotomy", "between (1) and (2).", "\\medskip\\noindent", "Assume we are in case (2) and $P = a_d x^d + \\ldots + a_0$. Then", "$P' = da_d x^{d - 1} + \\ldots + a_1$. In characteristic $0$ we see", "that this forces $a_d, \\ldots, a_1 = 0$ which would mean $P$ is constant", "a contradiction. Thus we conclude that the characteristic $p$ is positive.", "In this case the condition $P' = 0$ forces $a_i = 0$ whenever $p$ does", "not divide $i$.", "In other words, $P(x) = P_1(x^p)$ for some nonconstant polynomial $P_1$.", "Clearly, $P_1$ is irreducible as well. By induction on the degree we", "see that $P_1(x) = Q(x^q)$ as in the statement of the lemma, hence", "$P(x) = Q(x^{pq})$ and the lemma is proved." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4465, "type": "theorem", "label": "fields-lemma-separable-goes-up", "categories": [ "fields" ], "title": "fields-lemma-separable-goes-up", "contents": [ "Let $K/E/F$ be a tower of algebraic field extensions.", "\\begin{enumerate}", "\\item If $\\alpha \\in K$ is separable over $F$, then $\\alpha$ is separable", "over $E$.", "\\item if $K$ is separable over $F$, then $K$ is separable over $E$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-irreducible-polynomials} without further mention.", "Let $P$ be the minimal polynomial of $\\alpha$ over $F$.", "Let $Q$ be the minimal polynomial of $\\alpha$ over $E$.", "Then $Q$ divides $P$ in the polynomial ring $E[x]$, say $P = QR$.", "Then $P' = Q'R + QR'$. Thus if $Q' = 0$, then $Q$ divides $P$ and $P'$", "hence $P' = 0$ by the lemma. This proves (1). Part (2)", "follows immediately from (1) and the definitions." ], "refs": [ "fields-lemma-irreducible-polynomials" ], "ref_ids": [ 4464 ] } ], "ref_ids": [] }, { "id": 4466, "type": "theorem", "label": "fields-lemma-recognize-separable", "categories": [ "fields" ], "title": "fields-lemma-recognize-separable", "contents": [ "Let $F$ be a field. An irreducible polynomial $P$ over $F$", "is separable if and only if $P$ has pairwise distinct roots in an", "algebraic closure of $F$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $\\alpha \\in \\overline{F}$ is a root of both $P$ and $P'$.", "Then $P = (x - \\alpha)Q$ for some polynomial $Q$. Taking derivatives", "we obtain $P' = Q + (x - \\alpha)Q'$. Thus $\\alpha$ is a root of $Q$.", "Hence we see that if $P$ and $P'$ have a common root, then $P$", "does not have pairwise distinct roots. Conversely, if $P$ has", "a repeated root, i.e., $(x - \\alpha)^2$ divides $P$, then $\\alpha$", "is a root of both $P$ and $P'$. Combined with", "Lemma \\ref{lemma-relatively-prime-polynomials} this proves the lemma." ], "refs": [ "fields-lemma-relatively-prime-polynomials" ], "ref_ids": [ 4463 ] } ], "ref_ids": [] }, { "id": 4467, "type": "theorem", "label": "fields-lemma-nr-roots-unchanged", "categories": [ "fields" ], "title": "fields-lemma-nr-roots-unchanged", "contents": [ "Let $F$ be a field and let $\\overline{F}$ be an algebraic closure of $F$.", "Let $p > 0$ be the characteristic of $F$. Let $P$ be a polynomial", "over $F$. Then the set of roots of $P$ and $P(x^p)$ in $\\overline{F}$", "have the same cardinality (not counting multiplicity)." ], "refs": [], "proofs": [ { "contents": [ "Clearly, $\\alpha$ is a root of $P(x^p)$ if and only if $\\alpha^p$ is a", "root of $P$. In other words, the roots of $P(x^p)$ are the roots of", "$x^p - \\beta$, where $\\beta$ is a root of $P$. Thus it suffices to show", "that the map $\\overline{F} \\to \\overline{F}$, $\\alpha \\mapsto \\alpha^p$", "is bijective. It is surjective, as $\\overline{F}$ is algebraically closed", "which means that every element has a $p$th root. It is injective because", "$\\alpha^p = \\beta^p$ implies $(\\alpha - \\beta)^p = 0$ because", "the characteristic is $p$. And of course in a field $x^p = 0$ implies", "$x = 0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4468, "type": "theorem", "label": "fields-lemma-count-embeddings", "categories": [ "fields" ], "title": "fields-lemma-count-embeddings", "contents": [ "In Situation \\ref{situation-finitely-generated} the correspondence", "$$", "\\Mor_F(K, \\overline{F})", "\\longrightarrow", "\\{(\\beta_1, \\ldots, \\beta_n)\\text{ as below}\\},", "\\quad", "\\varphi \\longmapsto (\\varphi(\\alpha_1), \\ldots, \\varphi(\\alpha_n))", "$$", "is a bijection. Here the right hand side is the set of $n$-tuples", "$(\\beta_1, \\ldots, \\beta_n)$ of elements of $\\overline{F}$", "such that $\\beta_i$ is a root of $P_i^\\varphi$." ], "refs": [], "proofs": [ { "contents": [ "Let $(\\beta_1, \\ldots, \\beta_n)$ be an element of the right hand side.", "We construct a map of fields corresponding to it by induction.", "Namely, we set $\\varphi_0 : K_0 \\to \\overline{F}$ equal to the given", "map $K_0 = F \\subset \\overline{F}$. Having constructed", "$\\varphi_{i - 1} : K_{i - 1} \\to \\overline{F}$ we observe that", "$K_i = K_{i - 1}[x]/(P_i)$. Hence we can set $\\varphi_i$ equal", "to the unique map $K_i \\to \\overline{F}$ inducing $\\varphi_{i - 1}$", "on $K_{i - 1}$ and mapping $x$ to $\\beta_i$. This works precisely", "as $\\beta_i$ is a root of $P_i^\\varphi$. Uniqueness implies that", "the two constructions are mutually inverse." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4469, "type": "theorem", "label": "fields-lemma-count-embeddings-explicitly", "categories": [ "fields" ], "title": "fields-lemma-count-embeddings-explicitly", "contents": [ "In Situation \\ref{situation-finitely-generated} we have", "$|\\Mor_F(K, \\overline{F})| = \\prod_{i = 1}^n \\deg_s(P_i)$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from Lemma \\ref{lemma-count-embeddings}.", "Observe that a key ingredient we are tacitly using here is the", "well-definedness of the separable degree of an irreducible polynomial", "which was observed just prior to", "Definition \\ref{definition-separable-degree}." ], "refs": [ "fields-lemma-count-embeddings", "fields-definition-separable-degree" ], "ref_ids": [ 4468, 4538 ] } ], "ref_ids": [] }, { "id": 4470, "type": "theorem", "label": "fields-lemma-separably-generated-separable", "categories": [ "fields" ], "title": "fields-lemma-separably-generated-separable", "contents": [ "Assumptions and notation as in Situation \\ref{situation-finitely-generated}.", "If each $P_i$ is separable, i.e., each $\\alpha_i$ is separable over", "$K_{i - 1}$, then", "$$", "|\\Mor_F(K, \\overline{F})| = [K : F]", "$$", "and the field extension $K/F$ is separable. If one of the $\\alpha_i$ is", "not separable over $K_{i - 1}$, then", "$|\\Mor_F(K, \\overline{F})| < [K : F]$." ], "refs": [], "proofs": [ { "contents": [ "If $\\alpha_i$ is separable over $K_{i - 1}$ then", "$\\deg_s(P_i) = \\deg(P_i) = [K_i : K_{i - 1}]$", "(last equality by Lemma \\ref{lemma-degree-minimal-polynomial}).", "By multiplicativity (Lemma \\ref{lemma-multiplicativity-degrees}) we have", "$$", "[K : F] = \\prod [K_i : K_{i - 1}] = \\prod \\deg(P_i) =", "\\prod \\deg_s(P_i) = |\\Mor_F(K, \\overline{F})|", "$$", "where the last equality is Lemma \\ref{lemma-count-embeddings-explicitly}.", "By the exact same argument we get the strict inequality", "$|\\Mor_F(K, \\overline{F})| < [K : F]$ if one of the $\\alpha_i$ is", "not separable over $K_{i - 1}$.", "\\medskip\\noindent", "Finally, assume again that each $\\alpha_i$ is separable over $K_{i - 1}$.", "We will show $K/F$ is separable.", "Let $\\gamma = \\gamma_1 \\in K$ be arbitrary. Then we can find additional", "elements $\\gamma_2, \\ldots, \\gamma_m$ such that", "$K = F(\\gamma_1, \\ldots, \\gamma_m)$ (for example we could take", "$\\gamma_2 = \\alpha_1, \\ldots, \\gamma_{n + 1} = \\alpha_n$).", "Then we see by the last part of the lemma (already proven above)", "that if $\\gamma$ is not separable over $F$ we would have the", "strict inequality $|\\Mor_F(K, \\overline{F})| < [K : F]$", "contradicting the very first part of the lemma (already prove above", "as well)." ], "refs": [ "fields-lemma-degree-minimal-polynomial", "fields-lemma-multiplicativity-degrees", "fields-lemma-count-embeddings-explicitly" ], "ref_ids": [ 4459, 4450, 4469 ] } ], "ref_ids": [] }, { "id": 4471, "type": "theorem", "label": "fields-lemma-separable-equality", "categories": [ "fields" ], "title": "fields-lemma-separable-equality", "contents": [ "Let $K/F$ be a finite extension of fields. Let $\\overline{F}$ be an", "algebraic closure of $F$. Then we have", "$$", "|\\Mor_F(K, \\overline{F})| \\leq [K : F]", "$$", "with equality if and only if $K$ is separable over $F$." ], "refs": [], "proofs": [ { "contents": [ "This is a corollary of Lemma \\ref{lemma-separably-generated-separable}.", "Namely, since $K/F$ is finite we can find finitely many elements", "$\\alpha_1, \\ldots, \\alpha_n \\in K$ generating $K$ over $F$ (for example", "we can choose the $\\alpha_i$ to be a basis of $K$ over $F$).", "If $K/F$ is separable, then each $\\alpha_i$ is separable over", "$F(\\alpha_1, \\ldots, \\alpha_{i - 1})$ by Lemma \\ref{lemma-separable-goes-up}", "and we get equality by Lemma \\ref{lemma-separably-generated-separable}.", "On the other hand, if we have equality, then no matter how we choose", "$\\alpha_1, \\ldots, \\alpha_n$ we get that $\\alpha_1$ is separable over", "$F$ by Lemma \\ref{lemma-separably-generated-separable}. Since we", "can start the sequence with an arbitrary element of $K$ it follows", "that $K$ is separable over $F$." ], "refs": [ "fields-lemma-separably-generated-separable", "fields-lemma-separable-goes-up", "fields-lemma-separably-generated-separable", "fields-lemma-separably-generated-separable" ], "ref_ids": [ 4470, 4465, 4470, 4470 ] } ], "ref_ids": [] }, { "id": 4472, "type": "theorem", "label": "fields-lemma-separable-permanence", "categories": [ "fields" ], "title": "fields-lemma-separable-permanence", "contents": [ "Let $E/k$ and $F/E$ be separable algebraic extensions of fields. Then $F/k$", "is a separable extension of fields." ], "refs": [], "proofs": [ { "contents": [ "Choose $\\alpha \\in F$. Then $\\alpha$ is separable algebraic over $E$.", "Let $P = x^d + \\sum_{i < d} a_i x^i$ be the minimal polynomial of", "$\\alpha$ over $E$. Each $a_i$ is separable algebraic over $k$.", "Consider the tower of fields", "$$", "k \\subset k(a_0) \\subset k(a_0, a_1) \\subset \\ldots \\subset", "k(a_0, \\ldots, a_{d - 1}) \\subset k(a_0, \\ldots, a_{d - 1}, \\alpha)", "$$", "Because $a_i$ is separable algebraic over $k$ it is separable algebraic", "over $k(a_0, \\ldots, a_{i - 1})$ by Lemma \\ref{lemma-separable-goes-up}.", "Finally, $\\alpha$ is separable algebraic over $k(a_0, \\ldots, a_{d - 1})$", "because it is a root of $P$ which is irreducible", "(as it is irreducible over the possibly bigger field $E$)", "and separable (as it is separable over $E$).", "Thus $k(a_0, \\ldots, a_{d - 1}, \\alpha)$ is separable over $k$", "by Lemma \\ref{lemma-separably-generated-separable}", "and we conclude that $\\alpha$ is separable over $k$ as desired." ], "refs": [ "fields-lemma-separable-goes-up", "fields-lemma-separably-generated-separable" ], "ref_ids": [ 4465, 4470 ] } ], "ref_ids": [] }, { "id": 4473, "type": "theorem", "label": "fields-lemma-separable-elements", "categories": [ "fields" ], "title": "fields-lemma-separable-elements", "contents": [ "Let $E/k$ be a field extension. Then the elements of $E$ separable", "over $k$ form a subextension of $E/k$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha, \\beta \\in E$ be separable over $k$. Then $\\beta$ is separable", "over $k(\\alpha)$ by Lemma \\ref{lemma-separable-goes-up}.", "Thus we can apply Lemma \\ref{lemma-separable-permanence} to $k(\\alpha, \\beta)$", "to see that $k(\\alpha, \\beta)$ is separable over $k$." ], "refs": [ "fields-lemma-separable-goes-up", "fields-lemma-separable-permanence" ], "ref_ids": [ 4465, 4472 ] } ], "ref_ids": [] }, { "id": 4474, "type": "theorem", "label": "fields-lemma-independence-characters", "categories": [ "fields" ], "title": "fields-lemma-independence-characters", "contents": [ "Let $L$ be a field. Let $G$ be a monoid, for example a group. Let", "$\\chi_1, \\ldots, \\chi_n : G \\to L$ be pairwise distinct", "homomorphisms of monoids where $L$ is regarded as a monoid", "by multiplication. Then $\\chi_1, \\ldots, \\chi_n$", "are $L$-linearly independent: if $\\lambda_1, \\ldots, \\lambda_n \\in L$", "not all zero, then $\\sum \\lambda_i\\chi_i(g) \\not = 0$", "for some $g \\in G$." ], "refs": [], "proofs": [ { "contents": [ "If $n = 1$ this is true because $\\chi_1(e) = 1$ if $e \\in G$ is the", "neutral (identity) element. We prove the result by induction for $n > 1$.", "Suppose that $\\lambda_1, \\ldots, \\lambda_n \\in L$ not all zero.", "If $\\lambda_i = 0$ for some, then we win by induction on $n$.", "Since we want to show that $\\sum \\lambda_i\\chi_i(g) \\not = 0$", "for some $g \\in G$ we may after dividing by $-\\lambda_n$", "assume that $\\lambda_n = -1$. Then the only way we get in trouble", "is if", "$$", "\\chi_n(g) = \\sum\\nolimits_{i = 1, \\ldots, n - 1} \\lambda_i\\chi_i(g)", "$$", "for all $g \\in G$. Fix $h \\in G$. Then we would also get", "\\begin{align*}", "\\chi_n(h)\\chi_n(g) & = \\chi_n(hg) \\\\", "& = \\sum\\nolimits_{i = 1, \\ldots, n - 1} \\lambda_i\\chi_i(hg) \\\\", "& = \\sum\\nolimits_{i = 1, \\ldots, n - 1} \\lambda_i\\chi_i(h) \\chi_i(g)", "\\end{align*}", "Multiplying the previous relation by $\\chi_n(h)$ and substracting we obtain", "$$", "0 = \\sum\\nolimits_{i = 1, \\ldots, n - 1}", "\\lambda_i (\\chi_n(h) - \\chi_i(h)) \\chi_i(g)", "$$", "for all $g \\in G$. Since $\\lambda_i \\not = 0$ we conclude that", "$\\chi_n(h) = \\chi_i(h)$ for all $i$ by induction.", "The choice of $h$ above was arbitrary, so we conclude", "that $\\chi_i = \\chi_n$ for $i \\leq n - 1$ which contradicts", "the assumption that our characters $\\chi_i$ are pairwise distinct." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4475, "type": "theorem", "label": "fields-lemma-sums-of-powers", "categories": [ "fields" ], "title": "fields-lemma-sums-of-powers", "contents": [ "Let $L$ be a field. Let $n \\geq 1$ and $\\alpha_1, \\ldots, \\alpha_n \\in L$", "pairwise distinct elements of $L$. Then there exists an", "$e \\geq 0$ such that $\\sum_{i = 1, \\ldots, n} \\alpha_i^e \\not = 0$." ], "refs": [], "proofs": [ { "contents": [ "Apply linear independence of characters", "(Lemma \\ref{lemma-independence-characters})", "to the monoid homomorphisms $\\mathbf{Z}_{\\geq 0} \\to L$,", "$e \\mapsto \\alpha_i^e$." ], "refs": [ "fields-lemma-independence-characters" ], "ref_ids": [ 4474 ] } ], "ref_ids": [] }, { "id": 4476, "type": "theorem", "label": "fields-lemma-independence-embeddings", "categories": [ "fields" ], "title": "fields-lemma-independence-embeddings", "contents": [ "Let $K/F$ and $L/F$ be field extensions. Let", "$\\sigma_1, \\ldots, \\sigma_n : K \\to L$ be pairwise distinct", "morphisms of $F$-extensions. Then $\\sigma_1, \\ldots, \\sigma_n$", "are $L$-linearly independent: if $\\lambda_1, \\ldots, \\lambda_n \\in L$", "not all zero, then $\\sum \\lambda_i\\sigma_i(\\alpha) \\not = 0$", "for some $\\alpha \\in K$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-independence-characters} to", "the restrictions of $\\sigma_i$ to the groups of units." ], "refs": [ "fields-lemma-independence-characters" ], "ref_ids": [ 4474 ] } ], "ref_ids": [] }, { "id": 4477, "type": "theorem", "label": "fields-lemma-finite-separable-tensor-alg-closed", "categories": [ "fields" ], "title": "fields-lemma-finite-separable-tensor-alg-closed", "contents": [ "Let $K/F$ and $L/F$ be field extensions with", "$K/F$ finite separable and $L$ algebraically closed.", "Then the map", "$$", "K \\otimes_F L", "\\longrightarrow", "\\prod\\nolimits_{\\sigma \\in \\Hom_F(K, L)} L,\\quad", "\\alpha \\otimes \\beta \\mapsto (\\sigma(\\alpha)\\beta)_\\sigma", "$$", "is an isomorphism of $L$-algebras." ], "refs": [], "proofs": [ { "contents": [ "Choose a basis $\\alpha_1, \\ldots, \\alpha_n$ of $K$ as a vector space over $F$.", "By Lemma \\ref{lemma-separable-equality} (and a tiny omitted argument) the set", "$\\Hom_F(K, L)$ has $n$ elements, say $\\sigma_1, \\ldots, \\sigma_n$.", "In particular, the two sides have the same dimension $n$ as vector", "spaces over $L$. Thus if the map is not an isomorphism, then it", "has a kernel. In other words, there would exist", "$\\mu_j \\in L$, $j = 1, \\ldots, n$ not all zero,", "with $\\sum \\alpha_j \\otimes \\mu_j$ in the kernel.", "In other words, $\\sum \\sigma_i(\\alpha_j)\\mu_j = 0$ for all $i$.", "This would mean the $n \\times n$ matrix with entries", "$\\sigma_i(\\alpha_j)$ is not invertible. Thus we can find", "$\\lambda_1, \\ldots, \\lambda_n \\in L$ not all zero,", "such that $\\sum \\lambda_i\\sigma_i(\\alpha_j) = 0$ for all $j$.", "Now any element $\\alpha \\in K$ can be written as", "$\\alpha = \\sum \\beta_j \\alpha_j$ with $\\beta_j \\in F$ and we would get", "$$", "\\sum \\lambda_i\\sigma_i(\\alpha) =", "\\sum \\lambda_i\\sigma_i(\\sum \\beta_j \\alpha_j) =", "\\sum \\beta_j \\sum \\lambda_i\\sigma_i(\\alpha_j) = 0", "$$", "which contradicts Lemma \\ref{lemma-independence-embeddings}." ], "refs": [ "fields-lemma-separable-equality", "fields-lemma-independence-embeddings" ], "ref_ids": [ 4471, 4476 ] } ], "ref_ids": [] }, { "id": 4478, "type": "theorem", "label": "fields-lemma-take-pth-root", "categories": [ "fields" ], "title": "fields-lemma-take-pth-root", "contents": [ "Let $p$ be a prime number. Let $F$ be a field of characteristic $p$.", "Let $t \\in F$ be an element which does not have a $p$th root in $F$.", "Then the polynomial $x^p - t$ is irreducible over $F$." ], "refs": [], "proofs": [ { "contents": [ "To see this, suppose that we have a factorization", "$x^p - t = f g$. Taking derivatives we get $f' g + f g' = 0$.", "Note that neither $f' = 0$ nor $g' = 0$ as the degrees of $f$ and $g$", "are smaller than $p$. Moreover, $\\deg(f') < \\deg(f)$ and $\\deg(g') < \\deg(g)$.", "We conclude that $f$ and $g$ have a factor in common. Thus if $x^p - t$", "is reducible, then it is of the form $x^p - t = c f^n$ for some irreducible", "$f$, $c \\in F^*$, and $n > 1$. Since $p$ is a prime number this", "implies $n = p$ and $f$ linear, which would imply $x^p - t$ has a root", "in $F$. Contradiction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4479, "type": "theorem", "label": "fields-lemma-purely-inseparable-permanence", "categories": [ "fields" ], "title": "fields-lemma-purely-inseparable-permanence", "contents": [ "Let $E/k$ and $F/E$ be purely inseparable extensions of fields. Then $F/k$", "is a purely inseparable extension of fields." ], "refs": [], "proofs": [ { "contents": [ "Say the characteristic of $k$ is $p$. Choose $\\alpha \\in F$. Then", "$\\alpha^q \\in E$ for some $p$-power $q$. Whereupon $(\\alpha^q)^{q'} \\in k$", "for some $p$-power $q'$. Hence $\\alpha^{qq'} \\in k$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4480, "type": "theorem", "label": "fields-lemma-purely-inseparable-elements", "categories": [ "fields" ], "title": "fields-lemma-purely-inseparable-elements", "contents": [ "Let $E/k$ be a field extension. Then the elements of $E$ purely-inseparable", "over $k$ form a subextension of $E/k$." ], "refs": [], "proofs": [ { "contents": [ "Let $p$ be the characteristic of $k$.", "Let $\\alpha, \\beta \\in E$ be purely inseparable over $k$. Say", "$\\alpha^q \\in k$ and $\\beta^{q'} \\in k$ for some $p$-powers $q, q'$.", "If $q''$ is a $p$-power, then", "$(\\alpha + \\beta)^{q''} = \\alpha^{q''} + \\beta^{q''}$.", "Hence if $q'' \\geq q, q'$, then we conclude that $\\alpha + \\beta$", "is purely inseparable over $k$. Similarly for the difference,", "product and quotient of $\\alpha$ and $\\beta$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4481, "type": "theorem", "label": "fields-lemma-finite-purely-inseparable", "categories": [ "fields" ], "title": "fields-lemma-finite-purely-inseparable", "contents": [ "Let $E/F$ be a finite purely inseparable field extension of", "characteristic $p > 0$. Then there exists a sequence of elements", "$\\alpha_1, \\ldots, \\alpha_n \\in E$ such that we obtain a tower", "of fields", "$$", "E = F(\\alpha_1, \\ldots, \\alpha_n) \\supset", "F(\\alpha_1, \\ldots, \\alpha_{n - 1}) \\supset", "\\ldots", "\\supset F(\\alpha_1) \\supset F", "$$", "such that each intermediate extension is of degree $p$ and comes", "from adjoining a $p$th root. Namely,", "$\\alpha_i^p \\in F(\\alpha_1, \\ldots, \\alpha_{i - 1})$", "is an element which does not have a $p$th root in", "$F(\\alpha_1, \\ldots, \\alpha_{i - 1})$ for $i = 1, \\ldots, n$." ], "refs": [], "proofs": [ { "contents": [ "By induction on the degree of $E/F$. If the degree of the extension is $1$", "then the result is clear (with $n = 0$). If not, then choose", "$\\alpha \\in E$, $\\alpha \\not \\in F$. Say $\\alpha^{p^r} \\in F$ for some", "$r > 0$. Pick $r$ minimal and replace $\\alpha$ by $\\alpha^{p^{r - 1}}$.", "Then $\\alpha \\not \\in F$, but $\\alpha^p \\in F$. Then $t = \\alpha^p$ is not", "a $p$th power in $F$ (because that would imply $\\alpha \\in F$, see", "Lemma \\ref{lemma-nr-roots-unchanged} or its proof).", "Thus $F \\subset F(\\alpha)$ is a subextension of degree $p$", "(Lemma \\ref{lemma-take-pth-root}). By induction we find", "$\\alpha_1, \\ldots, \\alpha_n \\in E$ generating $E/F(\\alpha)$", "satisfying the conclusions of the lemma.", "The sequence $\\alpha, \\alpha_1, \\ldots, \\alpha_n$ does the job", "for the extension $E/F$." ], "refs": [ "fields-lemma-nr-roots-unchanged", "fields-lemma-take-pth-root" ], "ref_ids": [ 4467, 4478 ] } ], "ref_ids": [] }, { "id": 4482, "type": "theorem", "label": "fields-lemma-separable-first", "categories": [ "fields" ], "title": "fields-lemma-separable-first", "contents": [ "\\begin{slogan}", "Any algebraic field extension is uniquely a separable field extension", "followed by a purely inseparable one.", "\\end{slogan}", "Let $E/F$ be an algebraic field extension. There exists a unique subextension", "$E/E_{sep}/F$ such that $E_{sep}/F$ is separable and $E/E_{sep}$ is", "purely inseparable." ], "refs": [], "proofs": [ { "contents": [ "If the characteristic is zero we set $E_{sep} = E$. Assume the characteristic", "is $p > 0$. Let $E_{sep}$ be the set of elements of $E$ which are separable", "over $F$. This is a subextension by Lemma \\ref{lemma-separable-elements}", "and of course $E_{sep}$ is separable over $F$. Given an $\\alpha$ in $E$", "there exists a $p$-power $q$ such that $\\alpha^q$ is separable over $F$.", "Namely, $q$ is that power of $p$ such that the minimal polynomial of", "$\\alpha$ is of the form $P(x^q)$ with $P$ separable algebraic, see", "Lemma \\ref{lemma-irreducible-polynomials}. Hence $E/E_{sep}$ is purely", "inseparable. Uniqueness is clear." ], "refs": [ "fields-lemma-separable-elements", "fields-lemma-irreducible-polynomials" ], "ref_ids": [ 4473, 4464 ] } ], "ref_ids": [] }, { "id": 4483, "type": "theorem", "label": "fields-lemma-separable-degree", "categories": [ "fields" ], "title": "fields-lemma-separable-degree", "contents": [ "Let $K/F$ be a finite extension. Let $\\overline{F}$ be an algebraic", "closure of $F$. Then $[K : F]_s = |\\Mor_F(K, \\overline{F})|$." ], "refs": [], "proofs": [ { "contents": [ "We first prove this when $K/F$ is purely inseparable. Namely, we claim that", "in this case there is a unique map $K \\to \\overline{F}$. This can be", "seen by choosing a sequence of elements $\\alpha_1, \\ldots, \\alpha_n \\in K$", "as in Lemma \\ref{lemma-finite-purely-inseparable}. The irreducible polynomial", "of $\\alpha_i$ over $F(\\alpha_1, \\ldots, \\alpha_{i - 1})$ is $x^p - \\alpha_i^p$.", "Applying Lemma \\ref{lemma-count-embeddings-explicitly} we see that", "$|\\Mor_F(K, \\overline{F})| = 1$. On the other hand, $[K : F]_s = 1$", "in this case hence the equality holds.", "\\medskip\\noindent", "Let's return to a general finite extension $K/F$. In this case", "choose $F \\subset K_s \\subset K$ as in Lemma \\ref{lemma-separable-first}.", "By Lemma \\ref{lemma-separable-equality} we have", "$|\\Mor_F(K_s, \\overline{F})| = [K_s : F] = [K : F]_s$.", "On the other hand, every field map $\\sigma' : K_s \\to \\overline{F}$", "extends to a unique field map $\\sigma : K \\to \\overline{F}$ by the", "result of the previous paragraph. In other words", "$|\\Mor_F(K, \\overline{F})| = |\\Mor_F(K_s, \\overline{F})|$", "and the proof is done." ], "refs": [ "fields-lemma-finite-purely-inseparable", "fields-lemma-count-embeddings-explicitly", "fields-lemma-separable-first", "fields-lemma-separable-equality" ], "ref_ids": [ 4481, 4469, 4482, 4471 ] } ], "ref_ids": [] }, { "id": 4484, "type": "theorem", "label": "fields-lemma-multiplicativity-all-degrees", "categories": [ "fields" ], "title": "fields-lemma-multiplicativity-all-degrees", "contents": [ "Suppose given a tower of algebraic field extensions $K/E/F$. Then", "$$", "[K : F]_s = [K : E]_s [E : F]_s", "\\quad\\text{and}\\quad", "[K : F]_i = [K : E]_i [E : F]_i", "$$" ], "refs": [], "proofs": [ { "contents": [ "We first prove this in case $K$ is finite over $F$. Since we have", "multiplicativity for the usual degree (by", "Lemma \\ref{lemma-multiplicativity-degrees}) it suffices to prove", "one of the two formulas. By Lemma \\ref{lemma-separable-degree} we have", "$[K : F]_s = |\\Mor_F(K, \\overline{F})|$. By the same lemma,", "given any $\\sigma \\in \\Mor_F(E, \\overline{F})$ the number of extensions", "of $\\sigma$ to a map $\\tau : K \\to \\overline{F}$ is $[K : E]_s$.", "Namely, via $E \\cong \\sigma(E) \\subset \\overline{F}$ we can view", "$\\overline{F}$ as an algebraic closure of $E$. Combined with the", "fact that there are $[E : F]_s = |\\Mor_F(E, \\overline{F})|$ choices", "for $\\sigma$ we obtain the result.", "\\medskip\\noindent", "If the extensions are infinite one can write $K$ as the union", "of all finite subextension $F \\subset K' \\subset K$. For each", "$K'$ we set $E' = E \\cap K'$. Then we have the formulas of the", "lemma for $K'/E'/F$ by the first paragraph. Since", "$[K : F]_s = \\sup \\{[K' : F]_s\\}$ and similarly for the other", "degrees (some details omitted) we obtain the result in general." ], "refs": [ "fields-lemma-multiplicativity-degrees", "fields-lemma-separable-degree" ], "ref_ids": [ 4450, 4483 ] } ], "ref_ids": [] }, { "id": 4485, "type": "theorem", "label": "fields-lemma-normal-goes-up", "categories": [ "fields" ], "title": "fields-lemma-normal-goes-up", "contents": [ "Let $K/E/F$ be a tower of algebraic field extensions.", "If $K$ is normal over $F$, then $K$ is normal over $E$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha \\in K$. Let $P$ be the minimal polynomial of $\\alpha$ over $F$.", "Let $Q$ be the minimal polynomial of $\\alpha$ over $E$.", "Then $Q$ divides $P$ in the polynomial ring $E[x]$, say $P = QR$.", "Hence, if $P$ splits completely over $K$, then so does $Q$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4486, "type": "theorem", "label": "fields-lemma-intersect-normal", "categories": [ "fields" ], "title": "fields-lemma-intersect-normal", "contents": [ "Let $F$ be a field. Let $M/F$ be an algebraic extension. Let", "$F \\subset E_i \\subset M$, $i \\in I$ be subextensions with", "$E_i/F$ normal. Then $\\bigcap E_i$ is normal over $F$." ], "refs": [], "proofs": [ { "contents": [ "Direct from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4487, "type": "theorem", "label": "fields-lemma-separable-first-normal", "categories": [ "fields" ], "title": "fields-lemma-separable-first-normal", "contents": [ "Let $E/F$ be a normal algebraic field extension. Then the subextension", "$E/E_{sep}/F$ of Lemma \\ref{lemma-separable-first} is normal." ], "refs": [ "fields-lemma-separable-first" ], "proofs": [ { "contents": [ "If the characteristic is zero, then $E_{sep} = E$, and the result", "is clear. If the characteristic is $p > 0$, then $E_{sep}$", "is the set of elements of $E$ which are separable over $F$.", "Then if $\\alpha \\in E_{sep}$ has minimal polynomial $P$", "write $P = c(x - \\alpha)(x - \\alpha_2) \\ldots (x - \\alpha_d)$", "with $\\alpha_2, \\ldots, \\alpha_d \\in E$. Since", "$P$ is a separable polynomial and since $\\alpha_i$", "is a root of $P$, we conclude $\\alpha_i \\in E_{sep}$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 4482 ] }, { "id": 4488, "type": "theorem", "label": "fields-lemma-characterize-normal", "categories": [ "fields" ], "title": "fields-lemma-characterize-normal", "contents": [ "Let $E/F$ be an algebraic extension of fields. Let $\\overline{F}$ be an", "algebraic closure of $F$. The following are equivalent", "\\begin{enumerate}", "\\item $E$ is normal over $F$, and", "\\item for every pair $\\sigma, \\sigma' \\in \\Mor_F(E, \\overline{F})$ we", "have $\\sigma(E) = \\sigma'(E)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{P}$ be the set of all minimal polynomials over $F$ of", "all elements of $E$. Set", "$$", "T =", "\\{\\beta \\in \\overline{F} \\mid P(\\beta) = 0\\text{ for some }P \\in \\mathcal{P}\\}", "$$", "It is clear that if $E$ is normal over $F$, then $\\sigma(E) = T$", "for all $\\sigma \\in \\Mor_F(E, \\overline{F})$. Thus we see that (1)", "implies (2).", "\\medskip\\noindent", "Conversely, assume (2). Pick $\\beta \\in T$.", "We can find a corresponding $\\alpha \\in E$ whose minimal polynomial", "$P \\in \\mathcal{P}$ annihilates $\\beta$. Because $F(\\alpha) = F[x]/(P)$", "we can find an element $\\sigma_0 \\in \\Mor_F(F(\\alpha), \\overline{F})$ mapping", "$\\alpha$ to $\\beta$. By Lemma \\ref{lemma-map-into-algebraic-closure}", "we can extend $\\sigma_0$ to a $\\sigma \\in \\Mor_F(E, \\overline{F})$.", "Whence we see that $\\beta$ is in the common image of all embeddings", "$\\sigma : E \\to \\overline{F}$. It follows that $\\sigma(E) = T$", "for any $\\sigma$. Fix a $\\sigma$. Now let $P \\in \\mathcal{P}$. Then we", "can write", "$$", "P = (x - \\beta_1) \\ldots (x - \\beta_n)", "$$", "for some $n$ and $\\beta_i \\in \\overline{F}$ by", "Lemma \\ref{lemma-algebraically-closed}. Observe that $\\beta_i \\in T$.", "Thus $\\beta_i = \\sigma(\\alpha_i)$ for some $\\alpha_i \\in E$. Thus", "$P = (x - \\alpha_1) \\ldots (x - \\alpha_n)$ splits completely over $E$.", "This finishes the proof." ], "refs": [ "fields-lemma-map-into-algebraic-closure", "fields-lemma-algebraically-closed" ], "ref_ids": [ 4461, 4460 ] } ], "ref_ids": [] }, { "id": 4489, "type": "theorem", "label": "fields-lemma-normally-generated", "categories": [ "fields" ], "title": "fields-lemma-normally-generated", "contents": [ "Let $E/F$ be an algebraic extension of fields.", "If $E$ is generated by $\\alpha_i \\in E$, $i \\in I$", "over $F$ and if for each $i$ the minimal polynomial", "of $\\alpha_i$ over $F$ splits completely in $E$, then", "$E/F$ is normal." ], "refs": [], "proofs": [ { "contents": [ "Let $P_i$ be the minimal polynomial of $\\alpha_i$ over $F$.", "Let $\\alpha_i = \\alpha_{i, 1}, \\alpha_{i, 2}, \\ldots, \\alpha_{i, d_i}$", "be the roots of $P_i$ over $E$. Given two embeddings", "$\\sigma, \\sigma' : E \\to \\overline{F}$ over $F$ we see that", "$$", "\\{\\sigma(\\alpha_{i, 1}), \\ldots, \\sigma(\\alpha_{i, d_i})\\} =", "\\{\\sigma'(\\alpha_{i, 1}), \\ldots, \\sigma'(\\alpha_{i, d_i})\\}", "$$", "because both sides are equal to the set of roots of $P_i$", "in $\\overline{F}$. The elements $\\alpha_{i, j}$", "generate $E$ over $F$ and we find that $\\sigma(E) = \\sigma'(E)$.", "Hence $E/F$ is normal by Lemma \\ref{lemma-characterize-normal}." ], "refs": [ "fields-lemma-characterize-normal" ], "ref_ids": [ 4488 ] } ], "ref_ids": [] }, { "id": 4490, "type": "theorem", "label": "fields-lemma-lift-maps", "categories": [ "fields" ], "title": "fields-lemma-lift-maps", "contents": [ "Let $L/M/K$ be a tower of algebraic extensions.", "\\begin{enumerate}", "\\item If $M/K$ is normal, then any automorphism $\\tau$ of $L/K$", "induces an automorphism $\\tau|_M : M \\to M$.", "\\item If $L/K$ is normal, then any $K$-algebra map $\\sigma : M \\to L$", "extends to an automorphism of $L$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose an algebraic closure $\\overline{L}$ of $L$", "(Theorem \\ref{theorem-existence-algebraic-closure}).", "\\medskip\\noindent", "Let $\\tau$ be as in (1). Then $\\tau(M) = M$ as subfields of $\\overline{L}$", "by Lemma \\ref{lemma-characterize-normal} and hence", "$\\tau|_M : M \\to M$ is an automorphism.", "\\medskip\\noindent", "Let $\\sigma : M \\to L$ be as in (2).", "By Lemma \\ref{lemma-map-into-algebraic-closure}", "we can extend $\\sigma$ to a map", "$\\tau : L \\to \\overline{L}$, i.e., such that", "$$", "\\xymatrix{", "L \\ar[r]_\\tau & \\overline{L} \\\\", "M \\ar[u] \\ar[ru]_\\sigma & K \\ar[l] \\ar[u]", "}", "$$", "is commutative. By Lemma \\ref{lemma-characterize-normal} we see that", "$\\tau(L) = L$. Hence $\\tau : L \\to L$ is an automorphism which", "extends $\\sigma$." ], "refs": [ "fields-theorem-existence-algebraic-closure", "fields-lemma-characterize-normal", "fields-lemma-map-into-algebraic-closure", "fields-lemma-characterize-normal" ], "ref_ids": [ 4441, 4488, 4461, 4488 ] } ], "ref_ids": [] }, { "id": 4491, "type": "theorem", "label": "fields-lemma-normal-and-automorphisms", "categories": [ "fields" ], "title": "fields-lemma-normal-and-automorphisms", "contents": [ "Let $E/F$ be a finite extension. We have", "$$", "|\\text{Aut}(E/F)| \\leq [E : F]_s", "$$", "with equality if and only if $E$ is normal over $F$." ], "refs": [], "proofs": [ { "contents": [ "Choose an algebraic closure $\\overline{F}$ of $F$. Recall that", "$[E : F]_s = |\\Mor_F(E, \\overline{F})|$. Pick an element", "$\\sigma_0 \\in \\Mor_F(E, \\overline{F})$. Then the map", "$$", "\\text{Aut}(E/F) \\longrightarrow \\Mor_F(E, \\overline{F}),\\quad", "\\tau \\longmapsto \\sigma_0 \\circ \\tau", "$$", "is injective. Thus the inequality. If equality holds, then", "every $\\sigma \\in \\Mor_F(E, \\overline{F})$ is gotten by precomposing", "$\\sigma_0$ by an automorphism. Hence $\\sigma(E) = \\sigma_0(E)$.", "Thus $E$ is normal over $F$ by Lemma \\ref{lemma-characterize-normal}.", "\\medskip\\noindent", "Conversely, assume that $E/F$ is normal. Then by", "Lemma \\ref{lemma-characterize-normal} we have $\\sigma(E) = \\sigma_0(E)$", "for all $\\sigma \\in \\Mor_F(E, \\overline{F})$.", "Thus we get an automorphism of $E$ over $F$ by setting", "$\\tau = \\sigma_0^{-1} \\circ \\sigma$. Whence the map displayed above", "is surjective." ], "refs": [ "fields-lemma-characterize-normal", "fields-lemma-characterize-normal" ], "ref_ids": [ 4488, 4488 ] } ], "ref_ids": [] }, { "id": 4492, "type": "theorem", "label": "fields-lemma-normal-embeddings-differ-by-aut", "categories": [ "fields" ], "title": "fields-lemma-normal-embeddings-differ-by-aut", "contents": [ "Let $L/K$ be an algebraic normal extension of fields.", "Let $E/K$ be an extension of fields. Then either", "there is no $K$-embedding from $L$ to $E$ or", "there is one $\\tau : L \\to E$ and every other", "one is of the form $\\tau \\circ \\sigma$ where $\\sigma \\in \\text{Aut}(L/K)$." ], "refs": [], "proofs": [ { "contents": [ "Given $\\tau$ replace $L$ by $\\tau(L) \\subset E$ and apply", "Lemma \\ref{lemma-lift-maps}." ], "refs": [ "fields-lemma-lift-maps" ], "ref_ids": [ 4490 ] } ], "ref_ids": [] }, { "id": 4493, "type": "theorem", "label": "fields-lemma-splitting-field", "categories": [ "fields" ], "title": "fields-lemma-splitting-field", "contents": [ "Let $F$ be a field. Let $P \\in F[x]$ be a nonconstant polynomial.", "There exists a smallest field extension $E/F$ such that $P$", "splits completely over $E$. Moreover, the field extension $E/F$ is normal", "and unique up to (nonunique) isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Choose an algebraic closure $\\overline{F}$. Then we can write", "$P = c (x - \\beta_1) \\ldots (x - \\beta_n)$ in $\\overline{F}[x]$, see", "Lemma \\ref{lemma-algebraically-closed}. Note that $c \\in F^*$. Set", "$E = F(\\beta_1, \\ldots, \\beta_n)$. Then it is clear that $E$ is", "minimal with the requirement that $P$ splits completely over $E$.", "\\medskip\\noindent", "Next, let $E'$ be another minimal field extension of $F$ such that", "$P$ splits completely over $E'$. Write", "$P = c (x - \\alpha_1) \\ldots (x - \\alpha_n)$ with $c \\in F$ and", "$\\alpha_i \\in E'$. Again it follows from minimality that", "$E' = F(\\alpha_1, \\ldots, \\alpha_n)$. Moreover, if we pick", "any $\\sigma : E' \\to \\overline{F}$", "(Lemma \\ref{lemma-map-into-algebraic-closure})", "then we immediately see that $\\sigma(\\alpha_i) = \\beta_{\\tau(i)}$", "for some permutation $\\tau : \\{1, \\ldots, n\\} \\to \\{1, \\ldots, n\\}$.", "Thus $\\sigma(E') = E$. This implies that $E'$ is a normal extension", "of $F$ by Lemma \\ref{lemma-characterize-normal}", "and that $E \\cong E'$ as extensions of $F$ thereby finishing the proof." ], "refs": [ "fields-lemma-algebraically-closed", "fields-lemma-map-into-algebraic-closure", "fields-lemma-characterize-normal" ], "ref_ids": [ 4460, 4461, 4488 ] } ], "ref_ids": [] }, { "id": 4494, "type": "theorem", "label": "fields-lemma-normal-closure", "categories": [ "fields" ], "title": "fields-lemma-normal-closure", "contents": [ "\\begin{slogan}", "Existence of normal closure of finite extensions of fields.", "\\end{slogan}", "Let $E/F$ be a finite extension of fields. There exists a unique", "smallest finite extension $K/E$ such that $K$ is normal over $F$." ], "refs": [], "proofs": [ { "contents": [ "Choose generators $\\alpha_1, \\ldots, \\alpha_n$ of $E$ over $F$.", "Let $P_1, \\ldots, P_n$ be the minimal polynomials of", "$\\alpha_1, \\ldots, \\alpha_n$ over $F$. Set $P = P_1 \\ldots P_n$.", "Observe that $(x - \\alpha_1) \\ldots (x - \\alpha_n)$ divides $P$, since", "each $(x - \\alpha_i)$ divides $P_i$. Say", "$P = (x - \\alpha_1) \\ldots (x - \\alpha_n)Q$.", "Let $K/E$ be the splitting field of $P$ over $E$.", "We claim that $K$ is the splitting field of $P$ over $F$ as well", "(which implies that $K$ is normal over $F$).", "This is clear because $K/E$ is generated by the roots of", "$Q$ over $E$ and $E$ is generated by the roots of", "$(x - \\alpha_1) \\ldots (x - \\alpha_n)$ over $F$, hence", "$K$ is generated by the roots of $P$ over $F$.", "\\medskip\\noindent", "Uniqueness. Suppose that $K'/E$ is a second smallest extension such that", "$K'/F$ is normal. Choose an algebraic closure $\\overline{F}$ and an", "embedding $\\sigma_0 : E \\to \\overline{F}$. By", "Lemma \\ref{lemma-map-into-algebraic-closure}", "we can extend $\\sigma_0$ to $\\sigma : K \\to \\overline{F}$ and", "$\\sigma' : K' \\to \\overline{F}$.", "By Lemma \\ref{lemma-intersect-normal} we see that", "$\\sigma(K) \\cap \\sigma'(K')$ is normal over $F$.", "By minimality we conclude that $\\sigma(K) = \\sigma(K')$.", "Thus $\\sigma \\circ (\\sigma')^{-1} : K' \\to K$ gives an isomorphism", "of extensions of $E$." ], "refs": [ "fields-lemma-map-into-algebraic-closure", "fields-lemma-intersect-normal" ], "ref_ids": [ 4461, 4486 ] } ], "ref_ids": [] }, { "id": 4495, "type": "theorem", "label": "fields-lemma-normal-closure-inside-normal", "categories": [ "fields" ], "title": "fields-lemma-normal-closure-inside-normal", "contents": [ "Let $L/K$ be an algebraic normal extension.", "\\begin{enumerate}", "\\item If $L/M/K$ is a subextension with $M/K$ finite, then there exists", "a tower $L/M'/M/K$ with $M'/K$ finite and normal.", "\\item If $L/M'/M/K$ is a tower with $M/K$ normal and $M'/M$ finite,", "then there exists a tower $L/M''/M'/M/K$ with $M''/M$", "finite and $M''/K$ normal.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $M'$ be the smallest subextension of $L/K$ containing $M$", "which is normal over $K$. By Lemma \\ref{lemma-normal-closure}", "this is the normal closure of $M/K$ and is finite over $K$.", "\\medskip\\noindent", "Proof of (2). Let $\\alpha_1, \\ldots, \\alpha_n \\in M'$ generate $M'$ over $M$.", "Let $P_1, \\ldots, P_n$ be the minimal polynomials of", "$\\alpha_1, \\ldots, \\alpha_n$ over $K$. Let $\\alpha_{i, j}$ be the roots", "of $P_i$ in $L$. Let $M'' = M(\\alpha_{i, j})$. It follows from", "Lemma \\ref{lemma-normally-generated}", "(applied with the set of generators $M \\cup \\{\\alpha_{i, j}\\}$)", "that $M''$ is normal over $K$." ], "refs": [ "fields-lemma-normal-closure", "fields-lemma-normally-generated" ], "ref_ids": [ 4494, 4489 ] } ], "ref_ids": [] }, { "id": 4496, "type": "theorem", "label": "fields-lemma-normal-closure-tensor-product", "categories": [ "fields" ], "title": "fields-lemma-normal-closure-tensor-product", "contents": [ "Let $L/K$ be a finite extension. Let $M/L$ be the normal", "closure of $L$ over $K$. Then there is a surjective map", "$$", "L \\otimes_K L \\otimes_K \\ldots \\otimes_K L \\longrightarrow M", "$$", "of $K$-algebras where the number of tensors can be taken", "$[L : K]_s \\leq [L : K]$." ], "refs": [], "proofs": [ { "contents": [ "Choose an algebraic closure $\\overline{K}$ of $K$.", "Set $n = [L : K]_s = |\\Mor_K(L, \\overline{K})|$ with equality by", "Lemma \\ref{lemma-separable-degree}.", "Say $\\Mor_K(L, \\overline{K}) = \\{\\sigma_1, \\ldots, \\sigma_n\\}$.", "Let $M' \\subset \\overline{K}$ be the $K$-subalgebra", "generated by $\\sigma_i(L)$, $i = 1, \\ldots, n$.", "It follows from Lemma \\ref{lemma-characterize-normal}", "that $M'$ is normal over $K$ and that it is the", "smallest normal subextension of $\\overline{K}$ containing", "$\\sigma_1(L)$. By uniqueness of normal closure we", "have $M \\cong M'$. Finally, there is a surjective map", "$$", "L \\otimes_K L \\otimes_K \\ldots \\otimes_K L \\longrightarrow M',", "\\quad", "\\lambda_1 \\otimes \\ldots \\otimes \\lambda_n \\longmapsto", "\\sigma_1(\\lambda_1) \\ldots \\sigma_n(\\lambda_n)", "$$", "and note that $n \\leq [L : K]$ by definition." ], "refs": [ "fields-lemma-separable-degree", "fields-lemma-characterize-normal" ], "ref_ids": [ 4483, 4488 ] } ], "ref_ids": [] }, { "id": 4497, "type": "theorem", "label": "fields-lemma-cyclic", "categories": [ "fields" ], "title": "fields-lemma-cyclic", "contents": [ "Let $A$ be an abelian group of exponent dividing $n$ such that", "$\\{x \\in A \\mid dx = 0\\}$ has cardinality at most $d$ for all $d | n$.", "Then $A$ is cyclic of order dividing $n$." ], "refs": [], "proofs": [ { "contents": [ "The conditions imply that $|A| \\leq n$, in particular $A$ is finite.", "The structure of finite abelian groups shows that", "$A = \\mathbf{Z}/e_1\\mathbf{Z} \\oplus \\ldots \\oplus \\mathbf{Z}/e_r\\mathbf{Z}$", "for some integers $1 < e_1 | e_2 | \\ldots | e_r$. This would imply", "that $\\{x \\in A \\mid e_1 x = 0\\}$ has cardinality $e_1^r$. Hence", "$r = 1$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4498, "type": "theorem", "label": "fields-lemma-primitive-element", "categories": [ "fields" ], "title": "fields-lemma-primitive-element", "contents": [ "Let $E/F$ be a finite extension of fields. The following are equivalent", "\\begin{enumerate}", "\\item there exists a primitive element for $E$ over $F$, and", "\\item there are finitely many subextensions $E/K/F$.", "\\end{enumerate}", "Moreover, (1) and (2) hold if $E/F$ is separable." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha \\in E$ be a primitive element. Let $P$ be the minimal", "polynomial of $\\alpha$ over $F$. Let $E \\subset M$ be a splitting", "field for $P$ over $E$, so that", "$P(x) = (x - \\alpha)(x - \\alpha_2) \\ldots (x - \\alpha_n)$ over $M$.", "For ease of notation we set $\\alpha_1 = \\alpha$.", "Next, let $E/K/F$ be a subextension. Let $Q$ be the minimal", "polynomial of $\\alpha$ over $K$. Observe that $\\deg(Q) = [E : K]$.", "Writing $Q = x^d + \\sum_{i < d} a_i x^i$ we claim that", "$K$ is equal to $L = F(a_0, \\ldots, a_{d - 1})$. Indeed $\\alpha$ has degree", "$d$ over $L$ and $L \\subset K$. Hence $[E : L] = [E : K]$ and it follows", "that $[K : L] = 1$, i.e., $K = L$.", "Thus it suffices to show there are at most finitely many possibilities", "for the polynomial $Q$. This is clear because we have a factorization", "$P = QR$ in $K[x]$ in particular in $E[x]$. Since we have unique", "factorization in $E[x]$ there are at most finitely many monic", "factors of $P$ in $E[x]$.", "\\medskip\\noindent", "If $F$ is a finite field (equivalently $E$ is a finite field), then", "$E/F$ has a primitive element by the discussion in", "Section \\ref{section-finite}.", "Next, assume $F$ is infinite and there are at most finitely many proper", "subfields $E/K/F$. List them, say $K_1, \\ldots, K_N$. Then", "each $K_i \\subset E$ is a proper sub $F$-vector space. As $F$ is infinite", "we can find a vector $\\alpha \\in E$ with $\\alpha \\not \\in K_i$ for all $i$", "(a vector space can never be equal to a finite union of proper subvector spaces;", "details omitted). Then $\\alpha$ is a primitive element for $E$ over $F$.", "\\medskip\\noindent", "Having established the equivalence of (1) and (2) we now turn to", "the final statement of the lemma. Choose an algebraic closure", "$\\overline{F}$ of $F$. Enumerate the elements", "$\\sigma_1, \\ldots, \\sigma_n \\in \\Mor_F(E, \\overline{F})$.", "Since $E/F$ is separable we have $n = [E : F]$ by", "Lemma \\ref{lemma-separable-equality}.", "Note that if $i \\not = j$, then", "$$", "V_{ij} = \\Ker(\\sigma_i - \\sigma_j : E \\longrightarrow \\overline{F})", "$$", "is not equal to $E$. Hence arguing as in the preceding paragraph", "we can find $\\alpha \\in E$ with $\\alpha \\not \\in V_{ij}$ for all", "$i \\not = j$. It follows that $|\\Mor_F(F(\\alpha), \\overline{F})| \\geq n$.", "On the other hand $[F(\\alpha) : F] \\leq [E : F]$. Hence equality", "by Lemma \\ref{lemma-separable-equality}", "and we conclude that $E = F(\\alpha)$." ], "refs": [ "fields-lemma-separable-equality", "fields-lemma-separable-equality" ], "ref_ids": [ 4471, 4471 ] } ], "ref_ids": [] }, { "id": 4499, "type": "theorem", "label": "fields-lemma-characteristic-vs-minimal-polynomial", "categories": [ "fields" ], "title": "fields-lemma-characteristic-vs-minimal-polynomial", "contents": [ "Let $L/K$ be a finite extension of fields. Let $\\alpha \\in L$ and let $P$", "be the minimal polynomial of $\\alpha$ over $K$. Then the characteristic", "polynomial of the $K$-linear map $\\alpha : L \\to L$ is equal to", "$P^e$ with $e \\deg(P) = [L : K]$." ], "refs": [], "proofs": [ { "contents": [ "Choose a basis $\\beta_1, \\ldots, \\beta_e$ of $L$ over $K(\\alpha)$.", "Then $e$ satisfies $e \\deg(P) = [L : K]$ by", "Lemmas \\ref{lemma-degree-minimal-polynomial} and", "\\ref{lemma-multiplicativity-degrees}.", "Then we see that $L = \\bigoplus K(\\alpha) \\beta_i$ is a", "direct sum decomposition into $\\alpha$-invariant subspaces", "hence the characteristic polynomial of $\\alpha : L \\to L$", "is equal to the characteristic polynomial of", "$\\alpha : K(\\alpha) \\to K(\\alpha)$ to the power $e$.", "\\medskip\\noindent", "To finish the proof we may assume that $L = K(\\alpha)$.", "In this case by Cayley-Hamilton we see that $\\alpha$", "is a root of the characteristic polynomial. And since the", "characteristic polynomial has the same degree as the minimal", "polynomial, we find that equality holds." ], "refs": [ "fields-lemma-degree-minimal-polynomial", "fields-lemma-multiplicativity-degrees" ], "ref_ids": [ 4459, 4450 ] } ], "ref_ids": [] }, { "id": 4500, "type": "theorem", "label": "fields-lemma-trace-and-norm-from-minimal-polynomial", "categories": [ "fields" ], "title": "fields-lemma-trace-and-norm-from-minimal-polynomial", "contents": [ "Let $L/K$ be a finite extension of fields. Let $\\alpha \\in L$ and let", "$P = x^d + a_1 x^{d - 1} + \\ldots + a_d$", "be the minimal polynomial of $\\alpha$ over $K$. Then", "$$", "\\text{Norm}_{L/K}(\\alpha) = (-1)^{[L : K]} a_d^e", "\\quad\\text{and}\\quad", "\\text{Trace}_{L/K}(\\alpha) = - e a_1", "$$", "where $e d = [L : K]$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-characteristic-vs-minimal-polynomial}", "and the definitions." ], "refs": [ "fields-lemma-characteristic-vs-minimal-polynomial" ], "ref_ids": [ 4499 ] } ], "ref_ids": [] }, { "id": 4501, "type": "theorem", "label": "fields-lemma-trace-and-norm-linear", "categories": [ "fields" ], "title": "fields-lemma-trace-and-norm-linear", "contents": [ "Let $L/K$ be a finite extension of fields. Let $V$ be a finite dimensional", "vector space over $L$. Let $\\varphi : V \\to V$ be an $L$-linear map.", "Then", "$$", "\\text{Trace}_K(\\varphi : V \\to V) =", "\\text{Trace}_{L/K}(\\text{Trace}_L(\\varphi : V \\to V))", "$$", "and", "$$", "\\det\\nolimits_K(\\varphi : V \\to V) =", "\\text{Norm}_{L/K}(\\det\\nolimits_L(\\varphi : V \\to V))", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose an isomorphism $V = L^{\\oplus n}$ so that $\\varphi$ corresponds", "to an $n \\times n$ matrix. In the case of traces, both sides of the formula", "are additive in $\\varphi$. Hence we can assume that $\\varphi$", "corresponds to the matrix with exactly one nonzero entry in the $(i, j)$ spot.", "In this case a direct computation shows both sides are equal.", "\\medskip\\noindent", "In the case of norms both sides are zero if $\\varphi$ has a nonzero kernel.", "Hence we may assume $\\varphi$ corresponds to an element of", "$\\text{GL}_n(L)$. Both sides of the formula are multiplicative in $\\varphi$.", "Since every element of $\\text{GL}_n(L)$ is a product of elementary", "matrices we may assume that $\\varphi$ either looks like", "$$", "E_{12}(\\lambda) =", "\\left(", "\\begin{matrix}", "1 & \\lambda & \\ldots \\\\", "0 & 1 & \\ldots \\\\", "\\ldots & \\ldots & \\ldots", "\\end{matrix}", "\\right)", "\\quad\\text{or}\\quad", "E_1(a) =", "\\left(", "\\begin{matrix}", "a & 0 & \\ldots \\\\", "0 & 1 & \\ldots \\\\", "\\ldots & \\ldots & \\ldots", "\\end{matrix}", "\\right)", "$$", "(because we may also permute the basis elements if we like).", "In both cases the formula is easy to verify by direct computation." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4502, "type": "theorem", "label": "fields-lemma-trace-and-norm-tower", "categories": [ "fields" ], "title": "fields-lemma-trace-and-norm-tower", "contents": [ "Let $M/L/K$ be a tower of finite extensions of fields. Then", "$$", "\\text{Trace}_{M/K} = \\text{Trace}_{L/K} \\circ \\text{Trace}_{M/L}", "\\quad\\text{and}\\quad", "\\text{Norm}_{M/K} = \\text{Norm}_{L/K} \\circ \\text{Norm}_{M/L}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Think of $M$ as a vector space over $L$ and apply", "Lemma \\ref{lemma-trace-and-norm-linear}." ], "refs": [ "fields-lemma-trace-and-norm-linear" ], "ref_ids": [ 4501 ] } ], "ref_ids": [] }, { "id": 4503, "type": "theorem", "label": "fields-lemma-separable-trace-pairing", "categories": [ "fields" ], "title": "fields-lemma-separable-trace-pairing", "contents": [ "Let $L/K$ be a finite extension of fields. The following are equivalent:", "\\begin{enumerate}", "\\item $L/K$ is separable,", "\\item $\\text{Trace}_{L/K}$ is not identically zero, and", "\\item the trace pairing $Q_{L/K}$ is nondegenerate.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (3) implies (2). If (2) holds, then pick $\\gamma \\in L$", "with $\\text{Trace}_{L/K}(\\gamma) \\not = 0$. Then if $\\alpha \\in L$", "is nonzero, we see that $Q_{L/K}(\\alpha, \\gamma/\\alpha) \\not = 0$.", "Hence $Q_{L/K}$ is nondegenerate. This proves the equivalence of", "(2) and (3).", "\\medskip\\noindent", "Suppose that $K$ has characteristic $p$ and $L = K(\\alpha)$ with", "$\\alpha \\not \\in K$ and $\\alpha^p \\in K$. Then $\\text{Trace}_{L/K}(1) = p = 0$.", "For $i = 1, \\ldots, p - 1$ we see that $x^p - \\alpha^{pi}$ is the minimal", "polynomial for $\\alpha^i$ over $K$ and we find", "$\\text{Trace}_{L/K}(\\alpha^i) = 0$ by", "Lemma \\ref{lemma-trace-and-norm-from-minimal-polynomial}.", "Hence for this kind of purely inseparable degree $p$ extension", "we see that $\\text{Trace}_{L/K}$ is identically zero.", "\\medskip\\noindent", "Assume that $L/K$ is not separable. Then there exists a subfield", "$L/K'/K$ such that $L/K'$ is a purely inseparable degree $p$ extension", "as in the previous paragraph, see", "Lemmas \\ref{lemma-separable-first} and \\ref{lemma-finite-purely-inseparable}.", "Hence by Lemma \\ref{lemma-trace-and-norm-tower}", "we see that $\\text{Trace}_{L/K}$ is identically zero.", "\\medskip\\noindent", "Assume on the other hand that $L/K$ is separable.", "By induction on the degree we will show that", "$\\text{Trace}_{L/K}$ is not identically zero.", "Thus by Lemma \\ref{lemma-trace-and-norm-tower} we may assume that", "$L/K$ is generated by a single element $\\alpha$ (use that if", "the trace is nonzero then it is surjective).", "We have to show that $\\text{Trace}_{L/K}(\\alpha^e)$", "is nonzero for some $e \\geq 0$.", "Let $P = x^d + a_1 x^{d - 1} + \\ldots + a_d$ be the minimal", "polynomial of $\\alpha$ over $K$.", "Then $P$ is also the characteristic polynomial of the linear", "maps $\\alpha : L \\to L$, see", "Lemma \\ref{lemma-characteristic-vs-minimal-polynomial}.", "Since $L/k$ is separable we see from Lemma \\ref{lemma-recognize-separable}", "that $P$ has $d$ pairwise distinct roots $\\alpha_1, \\ldots, \\alpha_d$", "in an algebraic closure $\\overline{K}$ of $K$. Thus these are the", "eigenvalues of $\\alpha : L \\to L$.", "By linear algebra, the trace of $\\alpha^e$ is", "equal to $\\alpha_1^e + \\ldots + \\alpha_d^e$.", "Thus we conclude by Lemma \\ref{lemma-sums-of-powers}." ], "refs": [ "fields-lemma-trace-and-norm-from-minimal-polynomial", "fields-lemma-separable-first", "fields-lemma-finite-purely-inseparable", "fields-lemma-trace-and-norm-tower", "fields-lemma-trace-and-norm-tower", "fields-lemma-characteristic-vs-minimal-polynomial", "fields-lemma-recognize-separable", "fields-lemma-sums-of-powers" ], "ref_ids": [ 4500, 4482, 4481, 4502, 4502, 4499, 4466, 4475 ] } ], "ref_ids": [] }, { "id": 4504, "type": "theorem", "label": "fields-lemma-finite-Galois", "categories": [ "fields" ], "title": "fields-lemma-finite-Galois", "contents": [ "Let $E/F$ be a finite extension of fields. Then $E$ is Galois over $F$", "if and only if $|\\text{Aut}(E/F)| = [E : F]$." ], "refs": [], "proofs": [ { "contents": [ "Assume $|\\text{Aut}(E/F)| = [E : F]$. By", "Lemma \\ref{lemma-normal-and-automorphisms} this implies", "that $E/F$ is separable and normal, hence Galois.", "Conversely, if $E/F$ is separable then $[E : F] = [E : F]_s$ and", "if $E/F$ is in addition normal, then", "Lemma \\ref{lemma-normal-and-automorphisms} implies that", "$|\\text{Aut}(E/F)| = [E : F]$." ], "refs": [ "fields-lemma-normal-and-automorphisms", "fields-lemma-normal-and-automorphisms" ], "ref_ids": [ 4491, 4491 ] } ], "ref_ids": [] }, { "id": 4505, "type": "theorem", "label": "fields-lemma-galois-goes-up", "categories": [ "fields" ], "title": "fields-lemma-galois-goes-up", "contents": [ "Let $K/E/F$ be a tower of algebraic field extensions.", "If $K$ is Galois over $F$, then $K$ is Galois over $E$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-normal-goes-up} and \\ref{lemma-separable-goes-up}." ], "refs": [ "fields-lemma-normal-goes-up", "fields-lemma-separable-goes-up" ], "ref_ids": [ 4485, 4465 ] } ], "ref_ids": [] }, { "id": 4506, "type": "theorem", "label": "fields-lemma-normal-closure-galois", "categories": [ "fields" ], "title": "fields-lemma-normal-closure-galois", "contents": [ "Let $L/K$ be a finite separable extension of fields.", "Let $M$ be the normal closure of $L$ over $K$", "(Definition \\ref{definition-normal-closure}).", "Then $M/K$ is Galois." ], "refs": [ "fields-definition-normal-closure" ], "proofs": [ { "contents": [ "The subextension $M/M_{sep}/K$ of Lemma \\ref{lemma-separable-first}", "is normal by Lemma \\ref{lemma-separable-first-normal}.", "Since $L/K$ is separable we have $L \\subset M_{sep}$.", "By minimality $M = M_{sep}$ and the proof is done." ], "refs": [ "fields-lemma-separable-first", "fields-lemma-separable-first-normal" ], "ref_ids": [ 4482, 4487 ] } ], "ref_ids": [ 4544 ] }, { "id": 4507, "type": "theorem", "label": "fields-lemma-galois-over-fixed-field", "categories": [ "fields" ], "title": "fields-lemma-galois-over-fixed-field", "contents": [ "Let $K$ be a field. Let $G$ be a finite group acting faithfully on $K$.", "Then the extension $K/K^G$ is Galois, we have $[K : K^G] = |G|$,", "and the Galois group of the extension is $G$." ], "refs": [], "proofs": [ { "contents": [ "Given $\\alpha \\in K$ consider the orbit $G \\cdot \\alpha \\subset K$", "of $\\alpha$ under the group action. Consider the polynomial", "$$", "P = \\prod\\nolimits_{\\beta \\in G \\cdot \\alpha} (x - \\beta) \\in K[x]", "$$", "The key to the whole lemma is that this polynomial is invariant", "under the action of $G$ and hence has coefficients in $K^G$.", "Namely, for $\\tau \\in G$ we have", "$$", "P^\\tau = \\prod\\nolimits_{\\beta \\in G \\cdot \\alpha} (x - \\tau(\\beta)) =", "\\prod\\nolimits_{\\beta \\in G \\cdot \\alpha} (x - \\beta) = P", "$$", "because the map $\\beta \\mapsto \\tau(\\beta)$ is a permutation of", "the orbit $G \\cdot \\alpha$. Thus $P \\in K^G[x]$. Since also", "$P(\\alpha) = 0$ as $\\alpha$ is an element of its orbit", "we conclude that the extension $K/K^G$ is algebraic. Moreover,", "the minimal polynomial $Q$ of $\\alpha$ over $K^G$ divides the", "polynomial $P$ just constructed. Hence $Q$ is separable", "(by Lemma \\ref{lemma-recognize-separable} for example) and", "we conclude that $K/K^G$ is separable. Thus $K/K^G$ is Galois.", "To finish the proof it suffices to show that $[K : K^G] = |G|$", "since then $G$ will be the Galois group by", "Lemma \\ref{lemma-finite-Galois}.", "\\medskip\\noindent", "Pick finitely many elements $\\alpha_i \\in K$, $i = 1, \\ldots, n$", "such that $\\sigma(\\alpha_i) = \\alpha_i$ for $i = 1, \\ldots, n$ implies", "$\\sigma$ is the neutral element of $G$. Set", "$$", "L = K^G(\\{\\sigma(\\alpha_i); 1 \\leq i \\leq n, \\sigma \\in G\\}) \\subset K", "$$", "and observe that the action of $G$ on $K$ induces an action of $G$ on $L$.", "We will show that $L$ has degree $|G|$ over $K^G$. This will finish the", "proof, since if $L \\subset K$ is proper, then we can add an element", "$\\alpha \\in K$, $\\alpha \\not \\in L$ to our list of elements", "$\\alpha_1, \\ldots, \\alpha_n$ without increasing $L$ which is absurd.", "This reduces us to the case that $K/K^G$ is finite which is", "treated in the next paragraph.", "\\medskip\\noindent", "Assume $K/K^G$ is finite. By Lemma \\ref{lemma-primitive-element}", "we can find $\\alpha \\in K$ such that $K = K^G(\\alpha)$.", "By the construction in the first paragraph of this proof we see", "that $\\alpha$ has degree at most $|G|$ over $K$. However, the", "degree cannot be less than $|G|$ as $G$ acts faithfully on", "$K^G(\\alpha) = L$ by construction and the inequality of", "Lemma \\ref{lemma-normal-and-automorphisms}." ], "refs": [ "fields-lemma-recognize-separable", "fields-lemma-finite-Galois", "fields-lemma-primitive-element", "fields-lemma-normal-and-automorphisms" ], "ref_ids": [ 4466, 4504, 4498, 4491 ] } ], "ref_ids": [] }, { "id": 4508, "type": "theorem", "label": "fields-lemma-ses-galois", "categories": [ "fields" ], "title": "fields-lemma-ses-galois", "contents": [ "Let $L/M/K$ be a tower of fields. Assume $L/K$ and $M/K$ are finite Galois.", "Then we obtain a short exact sequence", "$$", "1 \\to \\text{Gal}(L/M) \\to \\text{Gal}(L/K) \\to \\text{Gal}(M/K) \\to 1", "$$", "of finite groups." ], "refs": [], "proofs": [ { "contents": [ "Namely, by Lemma \\ref{lemma-lift-maps}", "we see that every element $\\tau \\in \\text{Gal}(L/K)$ induces an element", "$\\tau|_M \\in \\text{Gal}(M/K)$ which gives us the homomorphism on the", "right. The map on the left identifies the left group with the kernel", "of the right arrow. The sequence is exact because the sizes", "of the groups work out correctly by multiplicativity of degrees", "in towers of finite extensions", "(Lemma \\ref{lemma-multiplicativity-degrees}).", "One can also use Lemma \\ref{lemma-lift-maps}", "directly to see that the map on the right is surjective." ], "refs": [ "fields-lemma-lift-maps", "fields-lemma-multiplicativity-degrees", "fields-lemma-lift-maps" ], "ref_ids": [ 4490, 4450, 4490 ] } ], "ref_ids": [] }, { "id": 4509, "type": "theorem", "label": "fields-lemma-galois-profinite", "categories": [ "fields" ], "title": "fields-lemma-galois-profinite", "contents": [ "Let $E/F$ be a Galois extension. Endow $\\text{Gal}(E/F)$ with the coarsest", "topology such that", "$$", "\\text{Gal}(E/F) \\times E \\longrightarrow E", "$$", "is continuous when $E$ is given the discrete topology. Then", "\\begin{enumerate}", "\\item for any topological space $X$ and map $X \\to \\text{Aut}(E/F)$", "such that the action $X \\times E \\to E$ is continuous the induced map", "$X \\to \\text{Gal}(E/F)$ is continuous,", "\\item this topology turns $\\text{Gal}(E/F)$ into", "a profinite topological group.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Throughout this proof we think of $E$ as a discrete topological space.", "Recall that the compact open topology on the set of self maps", "$\\text{Map}(E, E)$ is the universal topology such that the action", "$\\text{Map}(E, E) \\times E \\to E$ is continuous. See", "Topology, Example \\ref{topology-example-automorphisms-of-a-set}", "for a precise statement. The topology of the lemma on", "$\\text{Gal}(E/F)$ is the induced topology coming from the", "injective map $\\text{Gal}(E/F) \\to \\text{Map}(E, E)$.", "Hence the universal property (1) follows from the corresponding", "universal property of the compact open topology.", "Since the set of invertible self maps $\\text{Aut}(E)$", "endowed with the compact open topology forms a topological group, see", "Topology, Example \\ref{topology-example-automorphisms-of-a-set},", "and since $\\text{Gal}(E/F) = \\text{Aut}(E/F) \\to \\text{Map}(E, E)$", "factors through $\\text{Aut}(E)$ we obtain a topological group.", "In other words, we are using the injection", "$$", "\\text{Gal}(E/F) \\subset \\text{Aut}(E)", "$$", "to endow $\\text{Gal}(E/F)$ with the induced structure of a topological group", "(see Topology, Section \\ref{topology-section-topological-groups})", "and by construction this is the coarsest structure of a topological", "group such that the action $\\text{Gal}(E/F) \\times E \\to E$ is continuous.", "\\medskip\\noindent", "To show that $\\text{Gal}(E/F)$ is profinite we argue as follows", "(our argument is necessarily nonstandard because we have defined", "the topology before showing that the Galois group is an inverse", "limit of finite groups).", "By Topology, Lemma \\ref{topology-lemma-profinite-group}", "it suffices to show that the underlying", "topological space of $\\text{Gal}(E/F)$ is profinite.", "For any subset $S \\subset E$ consider the set", "$$", "G(S) = \\{ f : S \\to E \\mid", "\\begin{matrix}", "f(\\alpha)\\text{ is a root of the minimal polynomial}\\\\", "\\text{of }\\alpha\\text{ over }F\\text{ for all }\\alpha \\in S", "\\end{matrix}", "\\}", "$$", "Since a polynomial has only a finite number of roots we see that", "$G(S)$ is finite for all $S \\subset E$ finite. If $S \\subset S'$", "then restriction gives a map $G(S') \\to G(S)$. Also, observe", "that if $\\alpha \\in S \\cap F$ and $f \\in G(S)$, then $f(\\alpha) = \\alpha$", "because the minimal polynomial is linear in this case.", "Consider the profinite topological space", "$$", "G = \\lim_{S \\subset E\\text{ finite}} G(S)", "$$", "Consider the canonical map", "$$", "c : \\text{Gal}(E/F) \\longrightarrow G,\\quad", "\\sigma \\longmapsto (\\sigma|_S : S \\to E)_S", "$$", "This is injective and unwinding the definitions the", "reader sees the topology on $\\text{Gal}(E/F)$ as defined above", "is the induced topology from $G$. An element $(f_S) \\in G$ is in", "the image of $c$ exactly if", "(A) $f_S(\\alpha) + f_S(\\beta) = f_S(\\alpha + \\beta)$ and", "(M) $f_S(\\alpha)f_S(\\beta) = f_S(\\alpha\\beta)$ whenever", "this makes sense (i.e.,", "$\\alpha, \\beta, \\alpha + \\beta, \\alpha\\beta \\in S$).", "Namely, this means", "$\\lim f_S : E \\to E$ will be an $F$-algebra map", "and hence an automorphism by", "Lemma \\ref{lemma-algebraic-extension-self-map}.", "The conditions (A) and (M) for a given triple", "$(S, \\alpha, \\beta)$ define a closed subset of", "$G$ and hence $\\text{Gal}(E/F)$ is homeomorphic", "to a closed subset of a profinite space and therefore", "profinite itself." ], "refs": [ "topology-lemma-profinite-group", "fields-lemma-algebraic-extension-self-map" ], "ref_ids": [ 8339, 4458 ] } ], "ref_ids": [] }, { "id": 4510, "type": "theorem", "label": "fields-lemma-galois-infinite", "categories": [ "fields" ], "title": "fields-lemma-galois-infinite", "contents": [ "Let $L/M/K$ be a tower of fields. Assume both $L/K$ and", "$M/K$ are Galois. Then there is a canonical surjective continuous", "homomorphism $c : \\text{Gal}(L/K) \\to \\text{Gal}(M/K)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-lift-maps} given $\\tau : L \\to L$ in", "$\\text{Gal}(L/K)$ the restriction $\\tau|_M : M \\to M$", "is an element of $\\text{Gal}(M/K)$. This defines the homomorphism $c$.", "Continuity follows from the universal property of the topology:", "the action", "$$", "\\text{Gal}(L/K) \\times M \\longrightarrow M,\\quad", "(\\tau, x) \\longmapsto \\tau(x) = c(\\tau)(x)", "$$", "is continuous as $M \\subset L$ and the action", "$\\text{Gal}(L/K) \\times L \\to L$ is continuous.", "Hence continuity of $c$ by part (1) of", "Lemma \\ref{lemma-galois-profinite}.", "Lemma \\ref{lemma-lift-maps} also", "shows that the map is surjective." ], "refs": [ "fields-lemma-lift-maps", "fields-lemma-galois-profinite", "fields-lemma-lift-maps" ], "ref_ids": [ 4490, 4509, 4490 ] } ], "ref_ids": [] }, { "id": 4511, "type": "theorem", "label": "fields-lemma-infinite-galois-limit", "categories": [ "fields" ], "title": "fields-lemma-infinite-galois-limit", "contents": [ "Let $L/K$ be a Galois extension with Galois group $G$.", "Let $\\Lambda$ be the set of finite Galois subextensions,", "i.e., $\\lambda \\in \\Lambda$ corresponds to $L/L_\\lambda/K$", "with $L_\\lambda/K$ finite Galois with Galois group $G_\\lambda$.", "Define a partial ordering on $\\Lambda$ by the rule", "$\\lambda \\geq \\lambda'$ if and only if", "$L_\\lambda \\supset L_{\\lambda'}$. Then", "\\begin{enumerate}", "\\item $\\Lambda$ is a directed partially ordered set,", "\\item $L_\\lambda$ is a system of $K$-extensions over $\\Lambda$", "and $L = \\colim L_\\lambda$,", "\\item $G_\\lambda$ is an inverse system of finite groups over $\\Lambda$,", "the transition maps are surjective, and", "$$", "G = \\lim_{\\lambda \\in \\Lambda} G_\\lambda", "$$", "as a profinite group, and", "\\item each of the projections $G \\to G_\\lambda$ is continuous and surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Every subfield of $L$ containing $K$ is separable over $K$", "(follows immediately from the definition). Let $S \\subset L$", "be a finite subset. Then $K(S)/K$ is finite and there exists", "a tower $L/E/K(S)/K$ such that $E/K$ is finite Galois, see", "Lemma \\ref{lemma-normal-closure-inside-normal}.", "Hence $E = L_\\lambda$ for some $\\lambda \\in \\Lambda$.", "This certainly implies the set $\\Lambda$ is not empty.", "Also, given $\\lambda_1, \\lambda_2 \\in \\Lambda$ we can", "write $L_{\\lambda_i} = K(S_i)$ for finite sets", "$S_1, S_2 \\subset L$ (Lemma \\ref{lemma-finite-finitely-generated}).", "Then there exists a $\\lambda \\in \\Lambda$ such that", "$K(S_1 \\cup S_2) \\subset L_\\lambda$. Hence", "$\\lambda \\geq \\lambda_1, \\lambda_2$ and", "$\\Lambda$ is directed (Categories, Definition", "\\ref{categories-definition-directed-system}).", "Finally, since every element in $L$ is contained in", "$L_\\lambda$ for some $\\lambda \\in \\Lambda$, it follows", "from the description of filtered colimits in", "Categories, Section \\ref{categories-section-directed-colimits}", "that $\\colim L_\\lambda = L$.", "\\medskip\\noindent", "If $\\lambda \\geq \\lambda'$ in $\\Lambda$, then we obtain a", "canonical surjective map $G_\\lambda \\to G_{\\lambda'}$,", "$\\sigma \\mapsto \\sigma|_{L_{\\lambda'}}$", "by Lemma \\ref{lemma-ses-galois}. Thus we get an inverse", "system of finite groups with surjective transition maps.", "\\medskip\\noindent", "Recall that $G = \\text{Aut}(L/K)$. By Lemma \\ref{lemma-galois-infinite}", "the restriction $\\sigma|_{L_\\lambda}$ of a $\\sigma \\in G$ to $L_\\lambda$", "is an element of $G_\\lambda$. Moreover, this procedure gives a continuous", "surjection $G \\to G_\\lambda$. Since the transition mappings", "in the inverse system of $G_\\lambda$ are given by restriction", "also, it is clear that we obtain a canonical continuous map", "$$", "G \\longrightarrow \\lim_{\\lambda \\in \\Lambda} G_\\lambda", "$$", "Continuity by definition of limits in the category of topological", "groups; recall that these limits commute with the forgetful functor", "to the categories of sets and topological spaces by", "Topology, Lemma \\ref{topology-lemma-topological-group-limits}.", "On the other hand, since $L = \\colim L_\\lambda$ it is clear", "that any element of the inverse limit (viewed as a set) defines an", "automorphism of $L$. Thus the map is bijective. Since the topology", "on both sides is profinite, and since a bijective continuous map", "of profinite spaces is a homeomorphism", "(Topology, Lemma \\ref{topology-lemma-bijective-map}), the proof is complete." ], "refs": [ "fields-lemma-normal-closure-inside-normal", "fields-lemma-finite-finitely-generated", "categories-definition-directed-system", "fields-lemma-ses-galois", "fields-lemma-galois-infinite", "topology-lemma-topological-group-limits", "topology-lemma-bijective-map" ], "ref_ids": [ 4495, 4449, 12367, 4508, 4510, 8338, 8275 ] } ], "ref_ids": [] }, { "id": 4512, "type": "theorem", "label": "fields-lemma-ses-infinite-galois", "categories": [ "fields" ], "title": "fields-lemma-ses-infinite-galois", "contents": [ "Let $L/M/K$ be a tower of fields. Assume $L/K$ and $M/K$ are Galois.", "Then we obtain a short exact sequence", "$$", "1 \\to \\text{Gal}(L/M) \\to \\text{Gal}(L/K) \\to \\text{Gal}(M/K) \\to 1", "$$", "of profinite topological groups." ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of Lemma \\ref{lemma-galois-infinite}." ], "refs": [ "fields-lemma-galois-infinite" ], "ref_ids": [ 4510 ] } ], "ref_ids": [] }, { "id": 4513, "type": "theorem", "label": "fields-lemma-C-algebraically-closed", "categories": [ "fields" ], "title": "fields-lemma-C-algebraically-closed", "contents": [ "The field $\\mathbf{C}$ is algebraically closed." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4514, "type": "theorem", "label": "fields-lemma-Kummer", "categories": [ "fields" ], "title": "fields-lemma-Kummer", "contents": [ "Let $K \\subset L$ be a Galois extension of fields whose Galois group is", "$\\mathbf{Z}/n\\mathbf{Z}$. Assume moreover that the characteristic of $K$", "is prime to $n$ and that $K$ contains a primitive $n$th root of $1$.", "Then $L = K[z]$ with $z^n \\in K$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\zeta \\in K$ be a primitive $n$th root of $1$.", "Let $\\sigma$ be a generator of $\\text{Gal}(L/K)$.", "Consider $\\sigma : L \\to L$ as a $K$-linear operator.", "Note that $\\sigma^n - 1 = 0$ as a linear operator.", "Applying linear independence of characters", "(Lemma \\ref{lemma-independence-characters}), we see", "that there cannot be a polynomial over $K$ of degree $< n$", "annihilating $\\sigma$. Hence the minimal polynomial of $\\sigma$", "as a linear operator is $x^n - 1$. ", "Since $\\zeta$ is a root of $x^n - 1$ by linear algebra", "there is a $0 \\neq z \\in L$ such that $\\sigma(z) = \\zeta z$.", "This $z$ satisfies $z^n \\in K$ because", "$\\sigma(z^n) = (\\zeta z)^n = z^n$. Moreover, we see that", "$z, \\sigma(z), \\ldots, \\sigma^{n - 1}(z) =", "z, \\zeta z, \\ldots \\zeta^{n - 1} z$ are pairwise distinct", "which guarantees that $z$ generates $L$ over $K$.", "Hence $L = K[z]$ as required." ], "refs": [ "fields-lemma-independence-characters" ], "ref_ids": [ 4474 ] } ], "ref_ids": [] }, { "id": 4515, "type": "theorem", "label": "fields-lemma-adjoint-pth-root-unity", "categories": [ "fields" ], "title": "fields-lemma-adjoint-pth-root-unity", "contents": [ "Let $K$ be a field with algebraic closure $\\overline{K}$.", "Let $p$ be a prime different from the characteristic of $K$.", "Let $\\zeta \\in \\overline{K}$ be a primitive $p$th root", "of $1$. Then $K(\\zeta)/K$ is a Galois extension of degree dividing $p - 1$." ], "refs": [], "proofs": [ { "contents": [ "The polynomial $x^p - 1$ splits completely over", "$K(\\zeta)$ as its roots are $1, \\zeta, \\zeta^2, \\ldots, \\zeta^{p - 1}$.", "Hence $K(\\zeta)/K$ is a splitting field and hence normal.", "The extension is separable as $x^p - 1$ is a separable polynomial.", "Thus the extension is Galois. Any automorphism of $K(\\zeta)$ over $K$", "sends $\\zeta$ to $\\zeta^i$ for some $1 \\leq i \\leq p - 1$.", "Thus the Galois group is a subgroup of $(\\mathbf{Z}/p\\mathbf{Z})^*$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4516, "type": "theorem", "label": "fields-lemma-subfields-kummer", "categories": [ "fields" ], "title": "fields-lemma-subfields-kummer", "contents": [ "Let $K$ be a field. Let $L/K$ be a finite extension of degree $e$", "which is generated by an element $\\alpha$ with $a = \\alpha^e \\in K$.", "Then any sub extension $L/L'/K$ is generated by $\\alpha^d$ for some $d | e$." ], "refs": [], "proofs": [ { "contents": [ "Observe that for $d | e$ the subfield $K(\\alpha^d)$ has", "$[K(\\alpha^d) : K] = d/e$ and $[L : K(\\alpha^d)] = d$", "and that both extensions $K(\\alpha^d)/K$ and $L/K(\\alpha^d)$", "are extensions as in the lemma.", "\\medskip\\noindent", "We will use induction on the pair of integers $([L : L'], [L' : K])$", "ordered lexicographically.", "Let $p$ be a prime number dividing $e$ and set $d = e/p$.", "If $K(\\alpha^d)$ is contained in $L'$, then we win", "by induction, because then it suffices to prove the lemma", "for $L/L'/K(\\alpha^d)$. If not, then $[L'(\\alpha^d) : L'] = p$", "and by induction hypothesis we have $L'(\\alpha^d) = K(\\alpha^i)$", "for some $i | d$. If $i \\not = 1$ we are done by induction.", "Thus we may assume that $[L : L'] = p$.", "\\medskip\\noindent", "If $e$ is not a power of $p$, then we can do this trick", "again with a second prime number and we win. Thus we may", "assume $e$ is a power of $p$.", "\\medskip\\noindent", "If the characteristic of $K$ is $p$ and $e$ is a $p$th power, then", "$L/K$ is purely inseparable. Hence $L/L'$ is purely inseparable", "of degree $p$ and hence $\\alpha^p \\in L'$.", "Thus $L' = K(\\alpha^p)$ and this case is done.", "\\medskip\\noindent", "The final case is where $e$ is a power of $p$,", "the characteristic of $K$ is not $p$,", "$L/L'$ is a degree $p$ extension, and", "$L = L'(\\alpha^{e/p})$. Claim: this can", "only happen if $e = p$ and $L' = K$. The claim finishes the proof.", "\\medskip\\noindent", "First, we prove the claim when $K$ contains a primitive", "$p$th root of unity $\\zeta$. In this case the degree $p$ extension", "$K(\\alpha^{e/p})/K$ is Galois with Galois group generated", "by the automorphism $\\alpha^{e/p} \\mapsto \\zeta \\alpha^{e/p}$.", "On the other hand, since $L$ is generated by", "$\\alpha^{e/p}$ and $L'$ we see that the map", "$$", "K(\\alpha^{e/p}) \\otimes_K L' \\longrightarrow L", "$$", "is an isomorphism of $K$-algebras (look at dimensions).", "Thus $L$ has an automorphism $\\sigma$ of order $p$ over $K$ sending", "$\\alpha^{e/p}$ to $\\zeta \\alpha^{e/p}$. Then", "$\\sigma(\\alpha) = \\zeta' \\alpha$ for some $e$th root of unity $\\zeta'$", "(as $\\alpha^e$ is in $K$).", "Then on the one hand $(\\zeta')^{e/p} = \\zeta$ and on the other", "hand $\\zeta'$ has to be a $p$th root of $1$ as $\\sigma$", "has order $p$. Thus $e/p = 1$ and the claim has been shown.", "\\medskip\\noindent", "Finally, suppose that $K$ does not contain a primitive", "$p$th root of $1$.", "Choose a primitive $p$th root $\\zeta$ in some algebraic closure", "$\\overline{L}$ of $L$. Consider the diagram", "$$", "\\xymatrix{", "K(\\zeta) \\ar[r] & L(\\zeta) \\\\", "K \\ar[u] \\ar[r] & L \\ar[u]", "}", "$$", "By Lemma \\ref{lemma-adjoint-pth-root-unity}", "the vertical extensions have degree prime to $p$.", "Hence $[L(\\zeta) : K(\\zeta)]$ is divisible by $e$.", "On the other hand, $L(\\zeta)$ is generated by $\\alpha$", "over $K(\\zeta)$ and hence $[L(\\zeta) : K(\\zeta)] \\leq e$.", "Thus $[L(\\zeta) : K(\\zeta)] = e$. Similarly we have", "$[K(\\alpha^{e/p}, \\zeta) : K(\\zeta)] = p$ and", "$[L(\\zeta) : L'(\\zeta)] = p$. Thus the fields", "$K(\\zeta), L'(\\zeta), L(\\zeta)$ and the element $\\alpha$", "fall into the case discussed", "in the previous paragraph we conclude $e = p$ as desired." ], "refs": [ "fields-lemma-adjoint-pth-root-unity" ], "ref_ids": [ 4515 ] } ], "ref_ids": [] }, { "id": 4517, "type": "theorem", "label": "fields-lemma-Artin-Schreier", "categories": [ "fields" ], "title": "fields-lemma-Artin-Schreier", "contents": [ "Let $K \\subset L$ be a Galois extension of fields of characteristic $p > 0$", "with Galois group $\\mathbf{Z}/p\\mathbf{Z}$. Then $L = K[z]$ with", "$z^p - z \\in K$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\sigma$ be a generator of $\\text{Gal}(L/K)$.", "Consider $\\sigma : L \\to L$ as a $K$-linear operator.", "Observe that $\\sigma^p - 1 = 0$ as a linear operator.", "Applying linear independence of characters", "(Lemma \\ref{lemma-independence-characters}),", "there cannot be a polynomial of degree $< p$", "annihilating $\\sigma$. We conclude that the minimal", "polynomial of $\\sigma$ is $x^p - 1 = (x - 1)^p$.", "This implies that there exists $w \\in L$ such that", "$(\\sigma - 1)^{p - 1}(w) = y$ is nonzero. Then", "$\\sigma(y) = y$, i.e., $y \\in K$. Thus", "$z = y^{-1}(\\sigma - 1)^{p - 2}(w)$ satisfies", "$\\sigma(z) = z + 1$. Since $z \\not \\in K$ we have $L = K[z]$.", "Moreover since $\\sigma(z^p - z) = (z + 1)^p - (z + 1) = z^p - z$", "we see that $z^p - z \\in K$ and the proof is complete." ], "refs": [ "fields-lemma-independence-characters" ], "ref_ids": [ 4474 ] } ], "ref_ids": [] }, { "id": 4518, "type": "theorem", "label": "fields-lemma-transcendence-degree", "categories": [ "fields" ], "title": "fields-lemma-transcendence-degree", "contents": [ "Let $E/F$ be a field extension. A transcendence basis of $E$ over $F$ exists.", "Any two transcendence bases have the same cardinality." ], "refs": [], "proofs": [ { "contents": [ "Let $A$ be an algebraically independent subset of $E$. Let $G$ be a subset", "of $E$ containing $A$ that generates $E/F$. We claim we can find a", "transcendence basis $B$ such that $A \\subset B \\subset G$.", "To prove this consider the collection of algebraically independent subsets", "$\\mathcal{B}$ whose members are subsets of $G$ that contain $A$.", "Define a partial ordering on $\\mathcal{B}$ using inclusion.", "Then $\\mathcal{B}$ contains at least one element $A$.", "The union of the elements of a totally ordered subset $T$ of $\\mathcal{B}$", "is an algebraically independent subset of $E$ over $F$ since any algebraic", "dependence relation would have occurred in one of the elements of $T$", "(since polynomials only involve finitely many variables). The union also", "contains $A$ and is contained in $G$. By Zorn's lemma, there is a maximal", "element $B \\in \\mathcal{B}$. Now we claim $E$ is algebraic over $F(B)$.", "This is because if it wasn't then there would be an element", "$f \\in G$ transcendental over $F(B)$ since $F(G) = E$. Then", "$B \\cup\\{f\\}$ wold be algebraically independent contradicting the", "maximality of $B$. Thus $B$ is our transcendence basis.", "\\medskip\\noindent", "Let $B$ and $B'$ be two transcendence bases. Without loss of generality, we", "can assume that $|B'| \\leq |B|$. Now we divide the proof into two cases: the", "first case is that $B$ is an infinite set. Then for each $\\alpha \\in B'$,", "there is a finite set $B_{\\alpha}$ such that $\\alpha$ is algebraic over", "$F(B_{\\alpha})$ since any algebraic dependence relation only uses finitely many", "indeterminates. Then we define $B^* = \\bigcup_{\\alpha\\in B'} B_{\\alpha}$.", "By construction, $B^* \\subset B$, but we claim that in fact the two sets are", "equal. To see this, suppose that they are not equal, say there is an element", "$\\beta \\in B \\setminus B^*$. We know $\\beta$ is algebraic over $F(B')$ which", "is algebraic over $F(B^*)$. Therefore $\\beta$ is algebraic over $F(B^*)$, a", "contradiction. So $|B| \\leq |\\bigcup_{\\alpha \\in B'} B_{\\alpha}|$.", "Now if $B'$ is finite, then so is $B$ so we can assume $B'$ is infinite;", "this means", "$$", "|B| \\leq |\\bigcup\\nolimits_{\\alpha \\in B'} B_{\\alpha}| = |B'|", "$$", "because each $B_\\alpha$ is finite and $B'$ is infinite. Therefore in the", "infinite case, $|B| = |B'|$.", "\\medskip\\noindent", "Now we need to look at the case where $B$ is finite.", "In this case, $B'$ is also finite, so suppose", "$B = \\{\\alpha_1, \\ldots, \\alpha_n\\}$ and", "$B' = \\{\\beta_1, \\ldots, \\beta_m\\}$ with $m \\leq n$.", "We perform induction on $m$: if $m = 0$ then $E/F$ is algebraic so", "$B = \\emptyset$ so $n = 0$. If $m > 0$, there is an irreducible polynomial", "$f \\in F[x, y_1, \\ldots, y_n]$ such that", "$f(\\beta_1, \\alpha_1, \\ldots, \\alpha_n) = 0$ and such that $x$ occurs in $f$.", "Since $\\beta_1$ is not algebraic over $F$, $f$ must involve some $y_i$", "so without loss of generality, assume $f$ uses $y_1$.", "Let $B^* = \\{\\beta_1, \\alpha_2, \\ldots, \\alpha_n\\}$.", "We claim that $B^*$ is a basis for $E/F$. To prove this claim, we see that", "we have a tower of algebraic extensions", "$$", "E/ F(B^*, \\alpha_1) / F(B^*)", "$$", "since $\\alpha_1$ is algebraic over $F(B^*)$.", "Now we claim that $B^*$ (counting multiplicity of elements) is", "algebraically independent over $F$ because if it weren't, then there would be an", "irreducible $g\\in F[x, y_2, \\ldots, y_n]$ such that", "$g(\\beta_1, \\alpha_2, \\ldots, \\alpha_n) = 0$", "which must involve $x$ making $\\beta_1$", "algebraic over $F(\\alpha_2, \\ldots, \\alpha_n)$ which would make $\\alpha_1$", "algebraic over $F(\\alpha_2, \\ldots, \\alpha_n)$ which is impossible.", "So this means that $\\{\\alpha_2, \\ldots, \\alpha_n\\}$ and", "$\\{\\beta_2, \\ldots, \\beta_m\\}$ are bases for $E$ over $F(\\beta_1)$", "which means by induction, $m = n$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4519, "type": "theorem", "label": "fields-lemma-transcendence-degree-tower", "categories": [ "fields" ], "title": "fields-lemma-transcendence-degree-tower", "contents": [ "Let $k \\subset K \\subset L$ be field extensions.", "Then", "$$", "\\text{trdeg}_k(L) =", "\\text{trdeg}_K(L) +", "\\text{trdeg}_k(K).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose a transcendence basis $A \\subset K$ of $K$ over $k$.", "Choose a transcendence basis $B \\subset L$ of $L$ over $K$.", "Then it is straightforward to see that $A \\cup B$ is a transcendence", "basis of $L$ over $k$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4520, "type": "theorem", "label": "fields-lemma-purely-transcendental-degree", "categories": [ "fields" ], "title": "fields-lemma-purely-transcendental-degree", "contents": [ "Let $k'/k$ be a finite extension of fields. Let", "$k'(x_1, \\ldots, x_r)/k(x_1, \\ldots, x_r)$ be the", "induced extension of purely transcendental extensions.", "Then $[k'(x_1, \\ldots, x_r) : k(x_1, \\ldots, x_r)] = [k' : k] < \\infty$." ], "refs": [], "proofs": [ { "contents": [ "By multiplicativity of degrees of extensions", "(Lemma \\ref{lemma-multiplicativity-degrees})", "it suffices to prove this when $k'$ is generated by a single element", "$\\alpha \\in k'$ over $k$. Let $f \\in k[T]$ be the minimal polynomial", "of $\\alpha$ over $k$. Then $k'(x_1, \\ldots, x_r)$ is generated", "by $\\alpha, x_1, \\ldots, x_r$ over $k$ and hence $k'(x_1, \\ldots, x_r)$", "is generated by $\\alpha$ over $k(x_1, \\ldots, x_r)$.", "Thus it suffices to show that $f$ is still irreducible as", "an element of $k(x_1, \\ldots, x_r)[T]$. We only sketch the proof.", "It is clear that $f$ is irreducible as an element of", "$k[x_1, \\ldots, x_r, T]$ for example because $f$ is monic", "as a polynomial in $T$ and any putative factorization in", "$k[x_1, \\ldots, x_r, T]$ would lead to a factorization in $k[T]$", "by setting $x_i$ equal to $0$. By Gauss' lemma we conclude." ], "refs": [ "fields-lemma-multiplicativity-degrees" ], "ref_ids": [ 4450 ] } ], "ref_ids": [] }, { "id": 4521, "type": "theorem", "label": "fields-lemma-algebraic-closure-in-finitely-generated", "categories": [ "fields" ], "title": "fields-lemma-algebraic-closure-in-finitely-generated", "contents": [ "Let $k \\subset K$ be a finitely generated field extension.", "The algebraic closure of $k$ in $K$ is finite over $k$." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1, \\ldots, x_r \\in K$ be a transcendence basis for $K$", "over $k$. Then $n = [K : k(x_1, \\ldots, x_r)] < \\infty$.", "Suppose that $k \\subset k' \\subset K$ with $k'/k$ finite.", "In this case", "$[k'(x_1, \\ldots, x_r) : k(x_1, \\ldots, x_r)] = [k' : k] < \\infty$, see", "Lemma \\ref{lemma-purely-transcendental-degree}.", "Hence", "$$", "[k' : k] = [k'(x_1, \\ldots, x_r) : k(x_1, \\ldots, x_r)]", "< [K : k(x_1, \\ldots, x_r)] = n.", "$$", "In other words, the degrees of finite subextensions are bounded", "and the lemma follows." ], "refs": [ "fields-lemma-purely-transcendental-degree" ], "ref_ids": [ 4520 ] } ], "ref_ids": [] }, { "id": 4522, "type": "theorem", "label": "fields-lemma-normal-case", "categories": [ "fields" ], "title": "fields-lemma-normal-case", "contents": [ "Let $E/F$ be a normal algebraic field extension. There exist subextensions", "$E / E_{sep} /F$ and $E / E_{insep} / F$ such that", "\\begin{enumerate}", "\\item $F \\subset E_{sep}$ is Galois and $E_{sep} \\subset E$", "is purely inseparable,", "\\item $F \\subset E_{insep}$ is purely inseparable and $E_{insep} \\subset E$", "is Galois,", "\\item $E = E_{sep} \\otimes_F E_{insep}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We found the subfield $E_{sep}$ in Lemma \\ref{lemma-separable-first}.", "We set $E_{insep} = E^{\\text{Aut}(E/F)}$. Details omitted." ], "refs": [ "fields-lemma-separable-first" ], "ref_ids": [ 4482 ] } ], "ref_ids": [] }, { "id": 4523, "type": "theorem", "label": "fields-lemma-pth-root", "categories": [ "fields" ], "title": "fields-lemma-pth-root", "contents": [ "Let $K$ be a field of characteristic $p > 0$. Let $K \\subset L$ be a separable", "algebraic extension. Let $\\alpha \\in L$.", "\\begin{enumerate}", "\\item If the coefficients of the minimal polynomial of $\\alpha$", "over $K$ are $p$th powers in $K$ then $\\alpha$ is a $p$th", "power in $L$.", "\\item More generally, if $P \\in K[T]$ is a polynomial such that (a) $\\alpha$", "is a root of $P$, (b) $P$ has pairwise distinct roots in an algebraic closure,", "and (c) all coefficients of $P$ are $p$th powers, then $\\alpha$ is a", "$p$th power in $L$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It follows from the definitions that (2) implies (1). Assume $P$ is as in (2).", "Write $P(T) = \\sum\\nolimits_{i = 0}^d a_i T^{d - i}$ and $a_i = b_i^p$.", "The polynomial $Q(T) = \\sum\\nolimits_{i = 0}^d b_i T^{d - i}$ has distinct", "roots in an algebraic closure as well, because the roots of $Q$", "are the $p$th roots of the roots of $P$. If $\\alpha$ is not a $p$th power,", "then $T^p - \\alpha$ is an irreducible polynomial over $L$", "(Lemma \\ref{lemma-take-pth-root}).", "Moreover $Q$ and $T^p - \\alpha$ have a root in common in", "an algebraic closure $\\overline{L}$.", "Thus $Q$ and $T^p - \\alpha$ are not relatively prime, which", "implies $T^p - \\alpha | Q$ in $L[T]$. This contradicts the fact that", "the roots of $Q$ are pairwise distinct." ], "refs": [ "fields-lemma-take-pth-root" ], "ref_ids": [ 4478 ] } ], "ref_ids": [] }, { "id": 4556, "type": "theorem", "label": "spaces-limits-lemma-characterize-relative-limit-preserving", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-characterize-relative-limit-preserving", "contents": [ "Let $S$ be a scheme. Let $a : F \\to G$ be a transformation of functors", "$(\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $a : F \\to G$ is limit preserving, and", "\\item for every affine scheme $T$ over $S$ which is a", "limit $T = \\lim T_i$ of a directed inverse system of affine", "schemes $T_i$ over $S$ the diagram of sets", "$$", "\\xymatrix{", "\\colim_i F(T_i) \\ar[r] \\ar[d]_a & F(T) \\ar[d]^a \\\\", "\\colim_i G(T_i) \\ar[r] & G(T)", "}", "$$", "is a fibre product diagram.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Consider $T = \\lim_{i \\in I} T_i$ as in (2). Let", "$(y, x_T)$ be an element of the fibre product", "$\\colim_i G(T_i) \\times_{G(T)} F(T)$.", "Then $y$ comes from $y_i \\in G(T_i)$ for some $i$.", "Consider the functor $F_{y_i}$ on $(\\Sch/T_i)_{fppf}$ as in", "Definition \\ref{definition-locally-finite-presentation}.", "We see that $x_T \\in F_{y_i}(T)$. Moreover $T = \\lim_{i' \\geq i} T_{i'}$", "is a directed system of affine schemes over $T_i$. Hence (1) implies", "that $x_T$ the image of a unique element $x$ of", "$\\colim_{i' \\geq i} F_{y_i}(T_{i'})$. Thus $x$ is the unique", "element of $\\colim F(T_i)$ which maps to the pair $(y, x_T)$.", "This proves that (2) holds.", "\\medskip\\noindent", "Assume (2). Let $T$ be a scheme and $y_T \\in G(T)$. We have to show that", "$F_{y_T}$ is limit preserving. Let $T' = \\lim_{i \\in I} T'_i$ be an", "affine scheme over $T$ which is the directed limit of affine scheme $T'_i$", "over $T$. Let $x_{T'} \\in F_{y_T}$. Pick $i \\in I$ which is possible as", "$I$ is a directed set. Denote $y_i \\in F(T'_i)$ the", "image of $y_{T'}$. Then we see that $(y_i, x_{T'})$ is an", "element of the fibre product", "$\\colim_i G(T'_i) \\times_{G(T')} F(T')$.", "Hence by (2) we get a unique element $x$ of $\\colim_i F(T'_i)$", "mapping to $(y_i, x_{T'})$. It is clear that $x$ defines an element", "of $\\colim_i F_y(T'_i)$ mapping to $x_{T'}$ and we win." ], "refs": [ "spaces-limits-definition-locally-finite-presentation" ], "ref_ids": [ 4660 ] } ], "ref_ids": [] }, { "id": 4557, "type": "theorem", "label": "spaces-limits-lemma-composition-locally-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-composition-locally-finite-presentation", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$, $b : G \\to H$ be transformations of functors.", "If $a$ and $b$ are limit preserving, then", "$$", "b \\circ a : F \\longrightarrow H", "$$", "is limit preserving." ], "refs": [], "proofs": [ { "contents": [ "Let $T = \\lim_{i \\in I} T_i$ as in characterization (2) of", "Lemma \\ref{lemma-characterize-relative-limit-preserving}.", "Consider the diagram of sets", "$$", "\\xymatrix{", "\\colim_i F(T_i) \\ar[r] \\ar[d]_a & F(T) \\ar[d]^a \\\\", "\\colim_i G(T_i) \\ar[r] \\ar[d]_b & G(T) \\ar[d]^b \\\\", "\\colim_i H(T_i) \\ar[r] & H(T)", "}", "$$", "By assumption the two squares are fibre product squares. Hence the", "outer rectangle is a fibre product diagram too which proves the lemma." ], "refs": [ "spaces-limits-lemma-characterize-relative-limit-preserving" ], "ref_ids": [ 4556 ] } ], "ref_ids": [] }, { "id": 4558, "type": "theorem", "label": "spaces-limits-lemma-locally-finite-presentation-permanence", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-locally-finite-presentation-permanence", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$, $b : G \\to H$ be transformations of functors.", "If $b \\circ a$ and $b$ are limit preserving, then $a$", "is limit preserving." ], "refs": [], "proofs": [ { "contents": [ "Let $T = \\lim_{i \\in I} T_i$ as in characterization (2) of", "Lemma \\ref{lemma-characterize-relative-limit-preserving}.", "Consider the diagram of sets", "$$", "\\xymatrix{", "\\colim_i F(T_i) \\ar[r] \\ar[d]_a & F(T) \\ar[d]^a \\\\", "\\colim_i G(T_i) \\ar[r] \\ar[d]_b & G(T) \\ar[d]^b \\\\", "\\colim_i H(T_i) \\ar[r] & H(T)", "}", "$$", "By assumption the lower square and the outer rectangle", "are fibre products of sets. Hence the upper square", "is a fibre product square too which proves the lemma." ], "refs": [ "spaces-limits-lemma-characterize-relative-limit-preserving" ], "ref_ids": [ 4556 ] } ], "ref_ids": [] }, { "id": 4559, "type": "theorem", "label": "spaces-limits-lemma-base-change-locally-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-base-change-locally-finite-presentation", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$, $b : H \\to G$ be transformations of functors.", "Consider the fibre product diagram", "$$", "\\xymatrix{", "H \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\", "H \\ar[r]^b & G", "}", "$$", "If $a$ is limit preserving, then the base change $a'$ is limit preserving." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4560, "type": "theorem", "label": "spaces-limits-lemma-fibre-product-locally-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-fibre-product-locally-finite-presentation", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $E, F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$, $b : H \\to G$, and $c : G \\to E$", "be transformations of functors. If $c$, $c \\circ a$, and $c \\circ b$", "are limit preserving, then $F \\times_G H \\to E$ is too." ], "refs": [], "proofs": [ { "contents": [ "Let $T = \\lim_{i \\in I} T_i$ as in characterization (2) of", "Lemma \\ref{lemma-characterize-relative-limit-preserving}.", "Then we have", "$$", "\\colim (F \\times_G H)(T_i) =", "\\colim F(T_i) \\times_{\\colim G(T_i)} \\colim H(T_i)", "$$", "as filtered colimits commute with finite products. Our goal is thus to", "show that", "$$", "\\xymatrix{", "\\colim F(T_i) \\times_{\\colim G(T_i)} \\colim H(T_i) \\ar[r] \\ar[d] &", "F(T) \\times_{G(T)} H(T) \\ar[d] \\\\", "\\colim_i E(T_i) \\ar[r] & E(T)", "}", "$$", "is a fibre product diagram. This follows from the observation that", "given maps of sets $E' \\to E$, $F \\to G$, $H \\to G$, and $G \\to E$", "we have", "$$", "E' \\times_E (F \\times_G H) =", "(E' \\times_E F) \\times_{(E' \\times_E G)} (E' \\times_E H)", "$$", "Some details omitted." ], "refs": [ "spaces-limits-lemma-characterize-relative-limit-preserving" ], "ref_ids": [ 4556 ] } ], "ref_ids": [] }, { "id": 4561, "type": "theorem", "label": "spaces-limits-lemma-sheafify-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-sheafify-finite-presentation", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor.", "If $F$ is limit preserving then its sheafification $F^\\#$ is limit preserving." ], "refs": [], "proofs": [ { "contents": [ "Assume $F$ is limit preserving.", "It suffices to show that $F^+$ is limit preserving, since", "$F^\\# = (F^+)^+$, see", "Sites, Theorem \\ref{sites-theorem-plus}.", "Let $T$ be an affine scheme over $S$, and let $T = \\lim T_i$ be written", "as the directed limit of an inverse system of affine $S$ schemes.", "Recall that $F^+(T)$ is the colimit of $\\check H^0(\\mathcal{V}, F)$", "where the limit is over all coverings of $T$ in $(\\Sch/S)_{fppf}$.", "Any fppf covering of an affine scheme can be refined by a standard", "fppf covering, see", "Topologies, Lemma \\ref{topologies-lemma-fppf-affine}.", "Hence we can write", "$$", "F^+(T)", "=", "\\colim_{\\mathcal{V}\\text{ standard covering }T}", "\\check H^0(\\mathcal{V}, F).", "$$", "Any $\\mathcal{V} = \\{T_k \\to T\\}_{k = 1, \\ldots, n}$", "in the colimit may be written as", "$V_i \\times_{T_i} T$ for some $i$ and some standard fppf covering", "$\\mathcal{V}_i = \\{T_{i, k} \\to T_i\\}_{k = 1, \\ldots, n}$ of $T_i$.", "Denote $\\mathcal{V}_{i'} = \\{T_{i', k} \\to T_{i'}\\}_{k = 1, \\ldots, n}$", "the base change for $i' \\geq i$. Then we see that", "\\begin{align*}", "\\colim_{i' \\geq i} \\check H^0(\\mathcal{V}_i, F)", "& =", "\\colim_{i' \\geq i}", "\\text{Equalizer}(", "\\xymatrix{", "\\prod F(T_{i', k})", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\prod F(T_{i', k} \\times_{T_{i'}} T_{i', l})", "}", "\\\\", "& =", "\\text{Equalizer}(", "\\xymatrix{", "\\colim_{i' \\geq i}", "\\prod F(T_{i', k})", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\colim_{k' \\geq k}", "\\prod F(T_{i', k} \\times_{T_{i'}} T_{i', l})", "}", "\\\\", "& =", "\\text{Equalizer}(", "\\xymatrix{", "\\prod F(T_k)", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\prod F(T_k \\times_T T_l)", "}", "\\\\", "& =", "\\check H^0(\\mathcal{V}, F)", "\\end{align*}", "Here the second equality holds because filtered colimits are exact.", "The third equality holds because $F$ is limit preserving and because", "$\\lim_{i' \\geq i} T_{i', k} = T_k$ and", "$\\lim_{i' \\geq i} T_{i', k} \\times_{T_{i'}} T_{i', l} = T_k \\times_T T_l$", "by Limits, Lemma \\ref{limits-lemma-scheme-over-limit}.", "If we use this for all coverings at the same time we obtain", "\\begin{align*}", "F^+(T)", "& =", "\\colim_{\\mathcal{V}\\text{ standard covering }T} \\check H^0(\\mathcal{V}, F) \\\\", "& =", "\\colim_{i \\in I}", "\\colim_{\\mathcal{V}_i\\text{ standard covering }T_i}", "\\check H^0(T \\times_{T_i}\\mathcal{V}_i, F) \\\\", "& =", "\\colim_{i \\in I} F^+(T_i)", "\\end{align*}", "The switch of the order of the colimits is allowed by", "Categories, Lemma \\ref{categories-lemma-colimits-commute}." ], "refs": [ "sites-theorem-plus", "topologies-lemma-fppf-affine", "limits-lemma-scheme-over-limit", "categories-lemma-colimits-commute" ], "ref_ids": [ 8492, 12473, 15028, 12212 ] } ], "ref_ids": [] }, { "id": 4562, "type": "theorem", "label": "spaces-limits-lemma-sheaf-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-sheaf-finite-presentation", "contents": [ "Let $S$ be a scheme.", "Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor.", "Assume that", "\\begin{enumerate}", "\\item $F$ is a sheaf, and", "\\item there exists an fppf covering $\\{U_j \\to S\\}_{j \\in J}$ such that", "$F|_{(\\Sch/U_j)_{fppf}}$ is limit preserving.", "\\end{enumerate}", "Then $F$ is limit preserving." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be an affine scheme over $S$.", "Let $I$ be a directed set, and let", "$T_i$ be an inverse system of affine schemes over $S$ such that", "$T = \\lim T_i$. We have to show that the canonical", "map $\\colim F(T_i) \\to F(T)$ is bijective.", "\\medskip\\noindent", "Choose some $0 \\in I$ and choose a standard fppf covering", "$\\{V_{0, k} \\to T_{0}\\}_{k = 1, \\ldots, m}$ which refines", "the pullback $\\{U_j \\times_S T_0 \\to T_0\\}$ of the given fppf covering of $S$.", "For each $i \\geq 0$ we set $V_{i, k} = T_i \\times_{T_0} V_{0, k}$, and", "we set $V_k = T \\times_{T_0} V_{0, k}$. Note that", "$V_k = \\lim_{i \\geq 0} V_{i, k}$, see", "Limits, Lemma \\ref{limits-lemma-scheme-over-limit}.", "\\medskip\\noindent", "Suppose that $x, x' \\in \\colim F(T_i)$ map to the same", "element of $F(T)$. Say $x, x'$ are given by elements $x_i, x'_i \\in F(T_i)$", "for some $i \\in I$ (we may choose the same $i$ for both as $I$ is directed).", "By assumption (2) and the fact that $x_i, x'_i$ map to the same element", "of $F(T)$ this implies that", "$$", "x_i|_{V_{i', k}} = x'_i|_{V_{i', k}}", "$$", "for some suitably large $i' \\in I$. We can choose the same $i'$ for each", "$k$ as $k \\in \\{1, \\ldots, m\\}$ ranges over a finite set.", "Since $\\{V_{i', k} \\to T_{i'}\\}$", "is an fppf covering and $F$ is a sheaf this implies that", "$x_i|_{T_{i'}} = x'_i|_{T_{i'}}$ as desired. This proves that the map", "$\\colim F(T_i) \\to F(T)$ is injective.", "\\medskip\\noindent", "To show surjectivity we argue in a similar fashion.", "Let $x \\in F(T)$. By assumption (2) for each $k$ we", "can choose a $i$ such that $x|_{V_k}$ comes from an", "element $x_{i, k} \\in F(V_{i, k})$. As before we may choose a", "single $i$ which works for all $k$. By the injectivity", "proved above we see that", "$$", "x_{i, k}|_{V_{i', k} \\times_{T_{i'}} V_{i', l}}", "=", "x_{i, l}|_{V_{i', k} \\times_{T_{i'}} V_{i', l}}", "$$", "for some large enough $i'$. Hence by the sheaf condition of $F$", "the elements $x_{i, k}|_{V_{i', k}}$ glue to an element $x_{i'} \\in F(T_{i'})$", "as desired." ], "refs": [ "limits-lemma-scheme-over-limit" ], "ref_ids": [ 15028 ] } ], "ref_ids": [] }, { "id": 4563, "type": "theorem", "label": "spaces-limits-lemma-sheafify-finite-presentation-map", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-sheafify-finite-presentation-map", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be functors.", "If $a : F \\to G$ is a transformation which is limit preserving,", "then the induced transformation of sheaves", "$F^\\# \\to G^\\#$ is limit preserving." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $T$ is a scheme and $y \\in G^\\#(T)$.", "We have to show the functor", "$F^\\#_y : (\\Sch/T)_{fppf}^{opp} \\to \\textit{Sets}$", "constructed from $F^\\# \\to G^\\#$ and $y$ as in", "Definition \\ref{definition-locally-finite-presentation}", "is limit preserving.", "By Equation (\\ref{equation-fibre-map-functors})", "we see that $F^\\#_y$ is a sheaf. Choose an fppf covering", "$\\{V_j \\to T\\}_{j \\in J}$ such that $y|_{V_j}$ comes from", "an element $y_j \\in F(V_j)$.", "Note that the restriction of $F^\\#$ to $(\\Sch/V_j)_{fppf}$", "is just $F^\\#_{y_j}$. If we can show that $F^\\#_{y_j}$ is", "limit preserving then", "Lemma \\ref{lemma-sheaf-finite-presentation}", "guarantees that $F^\\#_y$ is limit preserving and", "we win. This reduces us to the case $y \\in G(T)$.", "\\medskip\\noindent", "Let $y \\in G(T)$. In this case we claim that $F^\\#_y = (F_y)^\\#$.", "This follows from", "Equation (\\ref{equation-fibre-map-functors}).", "Thus this case follows from", "Lemma \\ref{lemma-sheafify-finite-presentation}." ], "refs": [ "spaces-limits-definition-locally-finite-presentation", "spaces-limits-lemma-sheaf-finite-presentation", "spaces-limits-lemma-sheafify-finite-presentation" ], "ref_ids": [ 4660, 4562, 4561 ] } ], "ref_ids": [] }, { "id": 4564, "type": "theorem", "label": "spaces-limits-lemma-surjection-is-enough", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-surjection-is-enough", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If for every directed limit $T = \\lim_{i \\in I} T_i$", "of affine schemes over $S$ the map", "$$", "\\colim X(T_i) \\longrightarrow X(T) \\times_{Y(T)} \\colim Y(T_i)", "$$", "is surjective, then $f$ is locally of finite presentation.", "In other words, in", "Proposition \\ref{proposition-characterize-locally-finite-presentation}", "part (2) it suffices to check surjectivity in the criterion of", "Lemma \\ref{lemma-characterize-relative-limit-preserving}." ], "refs": [ "spaces-limits-proposition-characterize-locally-finite-presentation", "spaces-limits-lemma-characterize-relative-limit-preserving" ], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective \\'etale morphism $g : V \\to Y$.", "Next, choose a scheme $U$ and a surjective \\'etale morphism", "$h : U \\to V \\times_Y X$. It suffices to show for $T = \\lim T_i$", "as in the lemma that the map", "$$", "\\colim U(T_i) \\longrightarrow U(T) \\times_{V(T)} \\colim V(T_i)", "$$", "is surjective, because then $U \\to V$ will be locally of finite", "presentation by Limits, Lemma \\ref{limits-lemma-surjection-is-enough}", "(modulo a set theoretic remark exactly as in the proof of", "Proposition \\ref{proposition-characterize-locally-finite-presentation}).", "Thus we take $a : T \\to U$ and $b_i : T_i \\to V$ which determine", "the same morphism $T \\to V$. Picture", "$$", "\\xymatrix{", "T \\ar[d]_a \\ar[rr]_{p_i} & & T_i \\ar[d]^{b_i} \\ar@{..>}[ld] \\\\", "U \\ar[r]^-h & X \\times_Y V \\ar[d] \\ar[r] & V \\ar[d]^g \\\\", "& X \\ar[r]^f & Y", "}", "$$", "By the assumption of the lemma after increasing $i$", "we can find a morphism $c_i : T_i \\to X$ such that", "$h \\circ a = (b_i, c_i) \\circ p_i : T_i \\to V \\times_Y X$", "and such that $f \\circ c_i = g \\circ b_i$.", "Since $h$ is an \\'etale morphism of algebraic spaces", "(and hence locally of finite presentation), we have the surjectivity of", "$$", "\\colim U(T_i) \\longrightarrow U(T) \\times_{(X \\times_Y V)(T)}", "\\colim (X \\times_Y V)(T_i)", "$$", "by Proposition \\ref{proposition-characterize-locally-finite-presentation}.", "Hence after increasing $i$ again we can find the desired", "morphism $a_i : T_i \\to U$ with $a = a_i \\circ p_i$ and", "$b_i = (U \\to V) \\circ a_i$." ], "refs": [ "limits-lemma-surjection-is-enough", "spaces-limits-proposition-characterize-locally-finite-presentation", "spaces-limits-proposition-characterize-locally-finite-presentation" ], "ref_ids": [ 15054, 4655, 4655 ] } ], "ref_ids": [ 4655, 4556 ] }, { "id": 4565, "type": "theorem", "label": "spaces-limits-lemma-directed-inverse-system-has-limit", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-directed-inverse-system-has-limit", "contents": [ "Let $S$ be a scheme. Let $I$ be a directed set.", "Let $(X_i, f_{ii'})$ be an inverse system over $I$", "in the category of algebraic spaces over $S$.", "If the morphisms $f_{ii'} : X_i \\to X_{i'}$ are affine, then the", "limit $X = \\lim_i X_i$ (as an fppf sheaf) is an algebraic space.", "Moreover,", "\\begin{enumerate}", "\\item each of the morphisms $f_i : X \\to X_i$ is affine,", "\\item for any $i \\in I$ and any morphism of algebraic spaces", "$T \\to X_i$ we have", "$$", "X \\times_{X_i} T = \\lim_{i' \\geq i} X_{i'} \\times_{X_i} T.", "$$", "as algebraic spaces over $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) is a formal consequence of the existence of the", "limit $X = \\lim X_i$ as an algebraic space over $S$.", "Choose an element $0 \\in I$ (this is possible as a directed set is nonempty).", "Choose a scheme $U_0$ and a surjective", "\\'etale morphism $U_0 \\to X_0$. Set $R_0 = U_0 \\times_{X_0} U_0$", "so that $X_0 = U_0/R_0$. For $i \\geq 0$ set", "$U_i = X_i \\times_{X_0} U_0$ and", "$R_i = X_i \\times_{X_0} R_0 = U_i \\times_{X_i} U_i$.", "By Limits, Lemma \\ref{limits-lemma-directed-inverse-system-has-limit}", "we see that $U = \\lim_{i \\geq 0} U_i$ and $R = \\lim_{i \\geq 0} R_i$", "are schemes. Moreover, the two morphisms $s, t : R \\to U$ are the base", "change of the two projections $R_0 \\to U_0$ by the morphism", "$U \\to U_0$, in particular \\'etale. The morphism $R \\to U \\times_S U$", "defines an equivalence relation as directed a limit of equivalence relations", "is an equivalence relation. Hence the morphism", "$R \\to U \\times_S U$ is an \\'etale equivalence relation. We claim that", "the natural map", "\\begin{equation}", "\\label{equation-isomorphism-sheaves}", "U/R \\longrightarrow \\lim X_i", "\\end{equation}", "is an isomorphism of fppf sheaves on the category of schemes over $S$.", "The claim implies $X = \\lim X_i$ is an algebraic", "space by Spaces, Theorem \\ref{spaces-theorem-presentation}.", "\\medskip\\noindent", "Let $Z$ be a scheme and let $a : Z \\to \\lim X_i$ be a morphism.", "Then $a = (a_i)$ where $a_i : Z \\to X_i$. Set $W_0 = Z \\times_{a_0, X_0} U_0$.", "Note that $W_0 = Z \\times_{a_i, X_i} U_i$ for all $i \\geq 0$ by our", "choice of $U_i \\to X_i$ above. Hence we obtain a morphism", "$W_0 \\to \\lim_{i \\geq 0} U_i = U$. Since $W_0 \\to Z$ is surjective", "and \\'etale, we conclude that (\\ref{equation-isomorphism-sheaves})", "is a surjective map of sheaves. Finally, suppose that", "$Z$ is a scheme and that $a, b : Z \\to U/R$ are two morphisms", "which are equalized by (\\ref{equation-isomorphism-sheaves}).", "We have to show that $a = b$.", "After replacing $Z$ by the members of an fppf covering", "we may assume there exist morphisms $a', b' : Z \\to U$ which", "give rise to $a$ and $b$. The condition that $a, b$ are", "equalized by (\\ref{equation-isomorphism-sheaves}) means that", "for each $i \\geq 0$ the compositions $a_i', b_i' : Z \\to U \\to U_i$", "are equal as morphisms into $U_i/R_i = X_i$. Hence", "$(a_i', b_i') : Z \\to U_i \\times_S U_i$ factors through", "$R_i$, say by some morphism $c_i : Z \\to R_i$. Since", "$R = \\lim_{i \\geq 0} R_i$ we see that $c = \\lim c_i : Z \\to R$", "is a morphism which shows that $a, b$ are equal as morphisms", "of $Z$ into $U/R$.", "\\medskip\\noindent", "Part (1) follows as we have seen above that", "$U_i \\times_{X_i} X = U$ and $U \\to U_i$ is affine by", "construction." ], "refs": [ "limits-lemma-directed-inverse-system-has-limit", "spaces-theorem-presentation" ], "ref_ids": [ 15027, 8124 ] } ], "ref_ids": [] }, { "id": 4566, "type": "theorem", "label": "spaces-limits-lemma-space-over-limit", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-space-over-limit", "contents": [ "Let $S$ be a scheme. Let $I$ be a directed set.", "Let $(X_i, f_{ii'})$ be an inverse system over $I$ of algebraic spaces", "over $S$ with affine transition maps.", "Let $X = \\lim_i X_i$. Let $0 \\in I$. Suppose that $T \\to X_0$ is a", "morphism of algebraic spaces. Then", "$$", "T \\times_{X_0} X = \\lim_{i \\geq 0} T \\times_{X_0} X_i", "$$", "as algebraic spaces over $S$." ], "refs": [], "proofs": [ { "contents": [ "The limit $X$ is an algebraic space by", "Lemma \\ref{lemma-directed-inverse-system-has-limit}.", "The equality is formal, see", "Categories, Lemma \\ref{categories-lemma-colimits-commute}." ], "refs": [ "spaces-limits-lemma-directed-inverse-system-has-limit", "categories-lemma-colimits-commute" ], "ref_ids": [ 4565, 12212 ] } ], "ref_ids": [] }, { "id": 4567, "type": "theorem", "label": "spaces-limits-lemma-directed-inverse-system-closed-immersions", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-directed-inverse-system-closed-immersions", "contents": [ "Let $S$ be a scheme. Let $I$ be a directed set.", "Let $(X_i, f_{i'i}) \\to (Y_i, g_{i'i})$ be a morphism", "of inverse systems over $I$ of algebraic spaces over $S$.", "Assume", "\\begin{enumerate}", "\\item the morphisms $f_{i'i} : X_{i'} \\to X_i$ are affine,", "\\item the morphisms $g_{i'i} : Y_{i'} \\to Y_i$ are affine,", "\\item the morphisms $X_i \\to Y_i$ are closed immersions.", "\\end{enumerate}", "Then $\\lim X_i \\to \\lim Y_i$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\lim X_i$ and $\\lim Y_i$ exist by", "Lemma \\ref{lemma-directed-inverse-system-has-limit}.", "Pick $0 \\in I$ and choose an affine scheme $V_0$ and an \\'etale morphism", "$V_0 \\to Y_0$. Then the morphisms", "$V_i = Y_i \\times_{Y_0} V_0 \\to U_i = X_i \\times_{Y_0} V_0$", "are closed immersions of affine schemes.", "Hence the morphism $V = Y \\times_{Y_0} V_0 \\to U = X \\times_{Y_0} V_0$", "is a closed immersion because $V = \\lim V_i$, $U = \\lim U_i$", "and because a limit of closed immersions of affine schemes is a", "closed immersion: a filtered colimit of surjective ring maps", "is surjective. Since the \\'etale morphisms $V \\to Y$ form an", "\\'etale covering of $Y$ as we vary our choice of $V_0 \\to Y_0$", "we see that the lemma is true." ], "refs": [ "spaces-limits-lemma-directed-inverse-system-has-limit" ], "ref_ids": [ 4565 ] } ], "ref_ids": [] }, { "id": 4568, "type": "theorem", "label": "spaces-limits-lemma-directed-inverse-system-reduced", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-directed-inverse-system-reduced", "contents": [ "Let $S$ be a scheme. Let $I$ be a directed set.", "Let $(X_i, f_{i'i})$ be an inverse systems over $I$", "of algebraic spaces over $S$. If $X_i$ is reduced", "for all $i$, then $X$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\lim X_i$ exists by", "Lemma \\ref{lemma-directed-inverse-system-has-limit}.", "Pick $0 \\in I$ and choose an affine scheme $V_0$ and an \\'etale morphism", "$U_0 \\to X_0$. Then the affine schemes", "$U_i = X_i \\times_{X_0} U_0$ are reduced.", "Hence $U = X \\times_{X_0} U_0$", "is a reduced affine scheme as a limit of reduced affine schemes:", "a filtered colimit of reduced rings is reduced.", "Since the \\'etale morphisms $U \\to X$ form an", "\\'etale covering of $X$ as we vary our choice of $U_0 \\to X_0$", "we see that the lemma is true." ], "refs": [ "spaces-limits-lemma-directed-inverse-system-has-limit" ], "ref_ids": [ 4565 ] } ], "ref_ids": [] }, { "id": 4569, "type": "theorem", "label": "spaces-limits-lemma-better-characterize-relative-limit-preserving", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-better-characterize-relative-limit-preserving", "contents": [ "Let $S$ be a scheme. Let $X \\to Y$ be a morphism of algebraic spaces", "over $S$. The equivalent conditions (1) and (2) of", "Proposition \\ref{proposition-characterize-locally-finite-presentation}", "are also equivalent to", "\\begin{enumerate}", "\\item[(3)] for every directed limit $T = \\lim T_i$ of quasi-compact", "and quasi-separated algebraic spaces $T_i$ over $S$ with affine", "transition morphisms the diagram of sets", "$$", "\\xymatrix{", "\\colim_i \\Mor(T_i, X) \\ar[r] \\ar[d] & \\Mor(T, X) \\ar[d] \\\\", "\\colim_i \\Mor(T_i, Y) \\ar[r] & \\Mor(T, Y)", "}", "$$", "is a fibre product diagram.", "\\end{enumerate}" ], "refs": [ "spaces-limits-proposition-characterize-locally-finite-presentation" ], "proofs": [ { "contents": [ "It is clear that (3) implies (2). We will assume (2) and prove (3).", "The proof is rather formal and we encourage the reader to find their", "own proof.", "\\medskip\\noindent", "Let us first prove that (3) holds", "when $T_i$ is in addition assumed separated for all $i$.", "Choose $i \\in I$ and choose a surjective \\'etale morphism $U_i \\to T_i$", "where $U_i$ is affine. Using Lemma \\ref{lemma-space-over-limit}", "we see that with $U = U_i \\times_{T_i} T$ and", "$U_{i'} = U_i \\times_{T_i} T_{i'}$ we have $U = \\lim_{i' \\geq i} U_{i'}$.", "Of course $U$ and $U_{i'}$ are affine (see", "Lemma \\ref{lemma-directed-inverse-system-has-limit}).", "Since $T_i$ is separated, the fibre product $V_i = U_i \\times_{T_i} U_i$", "is an affine scheme as well and we obtain affine schemes", "$V = V_i \\times_{T_i} T$ and", "$V_{i'} = V_i \\times_{T_i} T_{i'}$ with $V = \\lim_{i' \\geq i} V_{i'}$.", "Observe that $U \\to T$ and $U_i \\to T_i$ are surjective \\'etale and", "that $V = U \\times_T U$ and $V_{i'} = U_{i'} \\times_{T_{i'}} U_{i'}$.", "Note that $\\Mor(T, X)$ is the equalizer of the two maps", "$\\Mor(U, X) \\to \\Mor(V, X)$; this is true for example because", "$X$ as a sheaf on $(\\Sch/S)_{fppf}$ is the coequalizer", "of the two maps $h_V \\to h_u$. Similarly", "$\\Mor(T_{i'}, X)$ is the equalizer of the", "two maps $\\Mor(U_{i'}, X) \\to \\Mor(V_{i'}, X)$.", "And of course the same thing is true with $X$ replaced with $Y$.", "Condition (2) says that the diagrams of in (3) are fibre products", "in the case of $U = \\lim U_i$ and $V = \\lim V_i$.", "It follows formally that the same thing is true for $T = \\lim T_i$.", "\\medskip\\noindent", "In the general case, choose an affine scheme $U$, an $i \\in I$,", "and a surjective \\'etale morphism $U \\to T_i$. Repeating the", "argument of the previous paragraph we still achieve the proof:", "the schemes $V_{i'}$, $V$ are no longer affine, but they are", "still quasi-compact and", "separated and the result of the preceding paragraph applies." ], "refs": [ "spaces-limits-lemma-space-over-limit", "spaces-limits-lemma-directed-inverse-system-has-limit" ], "ref_ids": [ 4566, 4565 ] } ], "ref_ids": [ 4655 ] }, { "id": 4570, "type": "theorem", "label": "spaces-limits-lemma-inverse-limit-sets", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-inverse-limit-sets", "contents": [ "Let $S$ be a scheme. Let $X = \\lim_{i \\in I} X_i$ be the limit of a directed", "inverse system of algebraic spaces over $S$ with affine transition morphisms", "(Lemma \\ref{lemma-directed-inverse-system-has-limit}). If each $X_i$", "is decent (for example quasi-separated or locally separated)", "then $|X| = \\lim_i |X_i|$ as sets." ], "refs": [ "spaces-limits-lemma-directed-inverse-system-has-limit" ], "proofs": [ { "contents": [ "There is a canonical map $|X| \\to \\lim |X_i|$. Choose $0 \\in I$.", "If $W_0 \\subset X_0$ is an open subspace, then we have", "$f_0^{-1}W_0 = \\lim_{i \\geq 0} f_{i0}^{-1}W_0$, see", "Lemma \\ref{lemma-directed-inverse-system-has-limit}.", "Hence, if we can prove the lemma for inverse systems where $X_0$", "is quasi-compact, then the lemma follows in general. Thus we may and do", "assume $X_0$ is quasi-compact.", "\\medskip\\noindent", "Choose an affine scheme $U_0$ and a surjective \\'etale morphism $U_0 \\to X_0$.", "Set $U_i = X_i \\times_{X_0} U_0$ and $U = X \\times_{X_0} U_0$.", "Set $R_i = U_i \\times_{X_i} U_i$ and $R = U \\times_X U$.", "Recall that $U = \\lim U_i$ and $R = \\lim R_i$, see proof of", "Lemma \\ref{lemma-directed-inverse-system-has-limit}.", "Recall that $|X| = |U|/|R|$ and $|X_i| = |U_i|/|R_i|$. By", "Limits, Lemma \\ref{limits-lemma-topology-limit} we have", "$|U| = \\lim |U_i|$ and $|R| = \\lim |R_i|$.", "\\medskip\\noindent", "Surjectivity of $|X| \\to \\lim |X_i|$. Let $(x_i) \\in \\lim |X_i|$. Denote", "$S_i \\subset |U_i|$ the inverse image of $x_i$. This is a finite nonempty", "set by the definition of decent spaces", "(Decent Spaces, Definition \\ref{decent-spaces-definition-very-reasonable}).", "Hence $\\lim S_i$ is nonempty, see", "Categories, Lemma \\ref{categories-lemma-nonempty-limit}.", "Let $(u_i) \\in \\lim S_i \\subset \\lim |U_i|$. By the above this determines", "a point $u \\in |U|$ which maps to an $x \\in |X|$ mapping to the given", "element $(x_i)$ of $\\lim |X_i|$.", "\\medskip\\noindent", "Injectivity of $|X| \\to \\lim |X_i|$. Suppose that $x, x' \\in |X|$", "map to the same point of $\\lim |X_i|$. Choose lifts $u, u' \\in |U|$", "and denote $u_i, u'_i \\in |U_i|$ the images.", "For each $i$ let $T_i \\subset |R_i|$ be the set of points mapping", "to $(u_i, u'_i) \\in |U_i| \\times |U_i|$. This is a finite", "set by the definition of decent spaces", "(Decent Spaces, Definition \\ref{decent-spaces-definition-very-reasonable}).", "Moreover $T_i$ is nonempty as we've assumed that $x$ and $x'$ map to the", "same point of $X_i$. Hence $\\lim T_i$ is nonempty, see", "Categories, Lemma \\ref{categories-lemma-nonempty-limit}.", "As before let $r \\in |R| = \\lim |R_i|$ be a point corresponding to an", "element of $\\lim T_i$. Then $r$ maps to $(u, u')$ in $|U| \\times |U|$", "by construction and we see that $x = x'$ in $|X|$ as desired.", "\\medskip\\noindent", "Parenthetical statement: A quasi-separated algebraic space is decent, see", "Decent Spaces, Section \\ref{decent-spaces-section-reasonable-decent}", "(the key observation to this is Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-finite-fibres-presentation}).", "A locally separated algebraic space is decent by", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-locally-separated-decent}." ], "refs": [ "spaces-limits-lemma-directed-inverse-system-has-limit", "spaces-limits-lemma-directed-inverse-system-has-limit", "limits-lemma-topology-limit", "decent-spaces-definition-very-reasonable", "categories-lemma-nonempty-limit", "decent-spaces-definition-very-reasonable", "categories-lemma-nonempty-limit", "spaces-properties-lemma-finite-fibres-presentation", "decent-spaces-lemma-locally-separated-decent" ], "ref_ids": [ 4565, 4565, 15036, 9562, 12237, 9562, 12237, 11836, 9512 ] } ], "ref_ids": [ 4565 ] }, { "id": 4571, "type": "theorem", "label": "spaces-limits-lemma-topology-limit", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-topology-limit", "contents": [ "With same notation and assumptions as in Lemma \\ref{lemma-inverse-limit-sets}", "we have $|X| = \\lim_i |X_i|$ as topological spaces." ], "refs": [ "spaces-limits-lemma-inverse-limit-sets" ], "proofs": [ { "contents": [ "We will use the criterion of", "Topology, Lemma \\ref{topology-lemma-characterize-limit}.", "We have seen that $|X| = \\lim_i |X_i|$ as sets in", "Lemma \\ref{lemma-inverse-limit-sets}.", "The maps $f_i : X \\to X_i$ are morphisms of algebraic spaces", "hence determine continuous maps $|X| \\to |X_i|$.", "Thus $f_i^{-1}(U_i)$ is open for each", "open $U_i \\subset |X_i|$. Finally, ", "let $x \\in |X|$ and let $x \\in V \\subset |X|$ be an open", "neighbourhood. We have to find an $i$ and an open neighbourhood", "$W_i \\subset |X_i|$ of the image $x$", "with $f_i^{-1}(W_i) \\subset V$.", "Choose $0 \\in I$. Choose a scheme $U_0$ and a surjective", "\\'etale morphism $U_0 \\to X_0$. Set $U = X \\times_{X_0} U_0$", "and $U_i = X_i \\times_{X_0} U_0$ for $i \\geq 0$.", "Then $U = \\lim_{i \\geq 0} U_i$ in the category of schemes by", "Lemma \\ref{lemma-directed-inverse-system-has-limit}.", "Choose $u \\in U$ mapping to $x$. By the result for schemes", "(Limits, Lemma \\ref{limits-lemma-inverse-limit-top})", "we can find an $i \\geq 0$ and an open neighbourhood", "$E_i \\subset U_i$ of the image of $u$", "whose inverse image in $U$ is contained in the", "inverse image of $V$ in $U$. Then we can set $W_i \\subset |X_i|$", "equal to the image of $E_i$. This works because $|U_i| \\to |X_i|$ is open." ], "refs": [ "topology-lemma-characterize-limit", "spaces-limits-lemma-inverse-limit-sets", "spaces-limits-lemma-directed-inverse-system-has-limit", "limits-lemma-inverse-limit-top" ], "ref_ids": [ 8250, 4570, 4565, 15033 ] } ], "ref_ids": [ 4570 ] }, { "id": 4572, "type": "theorem", "label": "spaces-limits-lemma-limit-nonempty", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-limit-nonempty", "contents": [ "Let $S$ be a scheme. Let $X = \\lim_{i \\in I} X_i$ be the limit of a directed", "inverse system of algebraic spaces over $S$ with affine transition morphisms", "(Lemma \\ref{lemma-directed-inverse-system-has-limit}). If each $X_i$", "is quasi-compact and nonempty, then $|X|$ is nonempty." ], "refs": [ "spaces-limits-lemma-directed-inverse-system-has-limit" ], "proofs": [ { "contents": [ "Choose $0 \\in I$.", "Choose an affine scheme $U_0$ and a surjective \\'etale morphism $U_0 \\to X_0$.", "Set $U_i = X_i \\times_{X_0} U_0$ and $U = X \\times_{X_0} U_0$.", "Then each $U_i$ is a nonempty affine scheme. Hence $U = \\lim U_i$", "is nonempty (Limits, Lemma \\ref{limits-lemma-limit-nonempty}) and thus", "$X$ is nonempty." ], "refs": [ "limits-lemma-limit-nonempty" ], "ref_ids": [ 15034 ] } ], "ref_ids": [ 4565 ] }, { "id": 4573, "type": "theorem", "label": "spaces-limits-lemma-inverse-limit-irreducibles", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-inverse-limit-irreducibles", "contents": [ "Let $S$ be a scheme. Let $X = \\lim_{i \\in I} X_i$ be the limit of a directed", "inverse system of algebraic spaces over $S$ with affine transition morphisms", "(Lemma \\ref{lemma-directed-inverse-system-has-limit}).", "Let $x \\in |X|$ with images $x_i \\in |X_i|$. If each $X_i$ is decent,", "then $\\overline{\\{x\\}} = \\lim_i \\overline{\\{x_i\\}}$ as sets", "and as algebraic spaces if endowed with reduced induced scheme structure." ], "refs": [ "spaces-limits-lemma-directed-inverse-system-has-limit" ], "proofs": [ { "contents": [ "Set $Z = \\overline{\\{x\\}} \\subset |X|$ and", "$Z_i = \\overline{\\{x_i\\}} \\subset |X_i|$.", "Since $|X| \\to |X_i|$ is continuous we see that $Z$ maps into $Z_i$", "for each $i$. Hence we obtain an injective map $Z \\to \\lim Z_i$", "because $|X| = \\lim |X_i|$ as sets (Lemma \\ref{lemma-inverse-limit-sets}).", "Suppose that $x' \\in |X|$ is not in $Z$.", "Then there is an open subset $U \\subset |X|$ with $x' \\in U$", "and $x \\not \\in U$. Since", "$|X| = \\lim |X_i|$ as topological spaces (Lemma \\ref{lemma-topology-limit})", "we can write $U = \\bigcup_{j \\in J} f_j^{-1}(U_j)$", "for some subset $J \\subset I$ and opens $U_j \\subset |X_j|$, see", "Topology, Lemma \\ref{topology-lemma-describe-limits}.", "Then we see that for some $j \\in J$ we have $f_j(x') \\in U_j$", "and $f_j(x) \\not \\in U_j$. In other words, we see that $f_j(x') \\not \\in Z_j$.", "Thus $Z = \\lim Z_i$ as sets.", "\\medskip\\noindent", "Next, endow $Z$ and $Z_i$ with their reduced induced scheme structures, see", "Properties of Spaces, Definition", "\\ref{spaces-properties-definition-reduced-induced-space}.", "The transition morphisms $X_{i'} \\to X_i$ induce affine", "morphisms $Z_{i'} \\to Z_i$ and the projections $X \\to X_i$", "induce compatible morphisms $Z \\to Z_i$.", "Hence we obtain morphisms $Z \\to \\lim Z_i \\to X$ of algebraic spaces.", "By Lemma \\ref{lemma-directed-inverse-system-closed-immersions}", "we see that $\\lim Z_i \\to X$ is a", "closed immersion. By Lemma \\ref{lemma-directed-inverse-system-reduced}", "the algebraic space $\\lim Z_i$ is reduced.", "By the above $Z \\to \\lim Z_i$ is bijective on points.", "By uniqueness of the reduced induced closed subscheme structure", "we find that this morphism is an isomorphism of algebraic spaces." ], "refs": [ "spaces-limits-lemma-inverse-limit-sets", "spaces-limits-lemma-topology-limit", "topology-lemma-describe-limits", "spaces-properties-definition-reduced-induced-space", "spaces-limits-lemma-directed-inverse-system-closed-immersions", "spaces-limits-lemma-directed-inverse-system-reduced" ], "ref_ids": [ 4570, 4571, 8249, 11932, 4567, 4568 ] } ], "ref_ids": [ 4565 ] }, { "id": 4574, "type": "theorem", "label": "spaces-limits-lemma-descend-section", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-section", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent}.", "Suppose that $\\mathcal{F}_0$ is a quasi-coherent sheaf on $X_0$.", "Set $\\mathcal{F}_i = f_{0i}^*\\mathcal{F}_0$ for $i \\geq 0$ and set", "$\\mathcal{F} = f_0^*\\mathcal{F}_0$. Then", "$$", "\\Gamma(X, \\mathcal{F}) = \\colim_{i \\geq 0} \\Gamma(X_i, \\mathcal{F}_i)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose a surjective \\'etale morphism $U_0 \\to X_0$ where $U_0$ is an affine", "scheme (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "Set $U_i = X_i \\times_{X_0} U_0$.", "Set $R_0 = U_0 \\times_{X_0} U_0$ and $R_i = R_0 \\times_{X_0} X_i$.", "In the proof of Lemma \\ref{lemma-directed-inverse-system-has-limit} we have", "seen that there exists a presentation $X = U/R$ with", "$U = \\lim U_i$ and $R = \\lim R_i$.", "Note that $U_i$ and $U$ are affine and that $R_i$ and $R$ are", "quasi-compact and separated (as $X_i$ is quasi-separated). Hence", "Limits, Lemma \\ref{limits-lemma-descend-section}", "implies that", "$$", "\\mathcal{F}(U) = \\colim \\mathcal{F}_i(U_i)", "\\quad\\text{and}\\quad", "\\mathcal{F}(R) = \\colim \\mathcal{F}_i(R_i).", "$$", "The lemma follows as", "$\\Gamma(X, \\mathcal{F}) = \\Ker(\\mathcal{F}(U) \\to \\mathcal{F}(R))$", "and similarly", "$\\Gamma(X_i, \\mathcal{F}_i) =", "\\Ker(\\mathcal{F}_i(U_i) \\to \\mathcal{F}_i(R_i))$" ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-limits-lemma-directed-inverse-system-has-limit" ], "ref_ids": [ 11832, 4565 ] } ], "ref_ids": [] }, { "id": 4575, "type": "theorem", "label": "spaces-limits-lemma-descend-opens", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-opens", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent}.", "For any quasi-compact open subspace $U \\subset X$ there exists an $i$", "and a quasi-compact open $U_i \\subset X_i$ whose inverse image in $X$ is $U$." ], "refs": [], "proofs": [ { "contents": [ "Follows formally from the construction of limits in", "Lemma \\ref{lemma-directed-inverse-system-has-limit}", "and the corresponding result for schemes:", "Limits, Lemma \\ref{limits-lemma-descend-opens}." ], "refs": [ "spaces-limits-lemma-directed-inverse-system-has-limit", "limits-lemma-descend-opens" ], "ref_ids": [ 4565, 15041 ] } ], "ref_ids": [] }, { "id": 4576, "type": "theorem", "label": "spaces-limits-lemma-descend-equality", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-equality", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent}.", "Let $f_0 : Y_0 \\to Z_0$ be a morphism of algebraic spaces over $X_0$.", "Assume (a) $Y_0 \\to X_0$ and $Z_0 \\to X_0$ are representable, (b)", "$Y_0$, $Z_0$ quasi-compact and quasi-separated, (c)", "$f_0$ locally of finite presentation, and", "(d) $Y_0 \\times_{X_0} X \\to Z_0 \\times_{X_0} X$ an isomorphism.", "Then there exists an $i \\geq 0$ such that", "$Y_0 \\times_{X_0} X_i \\to Z_0 \\times_{X_0} X_i$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $U_0$ and a surjective \\'etale morphism $U_0 \\to X_0$.", "Set $U_i = U_0 \\times_{X_0} X_i$ and $U = U_0 \\times_{X_0} X$.", "Apply Limits, Lemma \\ref{limits-lemma-descend-isomorphism}", "to see that $Y_0 \\times_{X_0} U_i \\to Z_0 \\times_{X_0} U_i$", "is an isomorphism of schemes for some $i \\geq 0$ (details omitted).", "As $U_i \\to X_i$ is surjective \\'etale, it follows that", "$Y_0 \\times_{X_0} X_i \\to Z_0 \\times_{X_0} X_i$ is an isomorphism", "(details omitted)." ], "refs": [ "limits-lemma-descend-isomorphism" ], "ref_ids": [ 15066 ] } ], "ref_ids": [] }, { "id": 4577, "type": "theorem", "label": "spaces-limits-lemma-descend-separated", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-separated", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent}.", "If $X$ is separated, then $X_i$ is separated for some $i \\in I$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $U_0$ and a surjective \\'etale morphism $U_0 \\to X_0$.", "For $i \\geq 0$ set $U_i = U_0 \\times_{X_0} X_i$ and set", "$U = U_0 \\times_{X_0} X$. Note that $U_i$ and $U$ are affine schemes", "which come equipped with surjective \\'etale morphisms $U_i \\to X_i$", "and $U \\to X$. Set $R_i = U_i \\times_{X_i} U_i$ and $R = U \\times_X U$", "with projections $s_i, t_i : R_i \\to U_i$ and $s, t : R \\to U$.", "Note that $R_i$ and $R$ are quasi-compact separated schemes (as the", "algebraic spaces $X_i$ and $X$ are quasi-separated). The maps", "$s_i : R_i \\to U_i$ and $s : R \\to U$ are of finite type.", "By definition $X_i$ is separated if and only if", "$(t_i, s_i) : R_i \\to U_i \\times U_i$", "is a closed immersion, and since $X$ is separated by assumption,", "the morphism $(t, s) : R \\to U \\times U$ is a closed immersion. Since", "$R \\to U$ is of finite type, there exists an", "$i$ such that the morphism $R \\to U_i \\times U$ is a closed immersion", "(Limits, Lemma \\ref{limits-lemma-finite-type-eventually-closed}).", "Fix such an $i \\in I$. Apply Limits, Lemma", "\\ref{limits-lemma-descend-closed-immersion-finite-presentation}", "to the system of morphisms $R_{i'} \\to U_i \\times U_{i'}$ for $i' \\geq i$", "(this is permissible as indeed", "$R_{i'} = R_i \\times_{U_i \\times U_i} U_i \\times U_{i'}$)", "to see that $R_{i'} \\to U_i \\times U_{i'}$ is a closed immersion", "for $i'$ sufficiently large. This implies immediately", "that $R_{i'} \\to U_{i'} \\times U_{i'}$ is a closed immersion", "finishing the proof of the lemma." ], "refs": [ "limits-lemma-finite-type-eventually-closed", "limits-lemma-descend-closed-immersion-finite-presentation" ], "ref_ids": [ 15046, 15060 ] } ], "ref_ids": [] }, { "id": 4578, "type": "theorem", "label": "spaces-limits-lemma-limit-is-affine", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-limit-is-affine", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent}.", "If $X$ is affine, then there exists an $i$ such that $X_i$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Choose $0 \\in I$. Choose an affine scheme $U_0$ and a surjective", "\\'etale morphism $U_0 \\to X_0$. Set $U = U_0 \\times_{X_0} X$", "and $U_i = U_0 \\times_{X_0} X_i$ for $i \\geq 0$. Since the transition", "morphisms are affine, the algebraic spaces $U_i$ and $U$ are affine.", "Thus $U \\to X$ is an \\'etale morphism of affine schemes. Hence we", "can write $X = \\Spec(A)$, $U = \\Spec(B)$ and", "$$", "B = A[x_1, \\ldots, x_n]/(g_1, \\ldots, g_n)", "$$", "such that $\\Delta = \\det(\\partial g_\\lambda/\\partial x_\\mu)$ is invertible", "in $B$, see Algebra, Lemma \\ref{algebra-lemma-etale-standard-smooth}.", "Set $A_i = \\mathcal{O}_{X_i}(X_i)$. We have $A = \\colim A_i$ by", "Lemma \\ref{lemma-descend-section}. After increasing $0$ we may assume", "we have $g_{1, i}, \\ldots, g_{n, i} \\in A_i[x_1, \\ldots, x_n]$ mapping to", "$g_1, \\ldots, g_n$. Set", "$$", "B_i = A_i[x_1, \\ldots, x_n]/(g_{1, i}, \\ldots, g_{n, i})", "$$", "for all $i \\geq 0$. Increasing $0$ if necessary we may assume that", "$\\Delta_i = \\det(\\partial g_{\\lambda, i}/\\partial x_\\mu)$ is invertible", "in $B_i$ for all $i \\geq 0$. Thus $A_i \\to B_i$ is an \\'etale ring map.", "After increasing $0$ we may assume also that", "$\\Spec(B_i) \\to \\Spec(A_i)$ is surjective, see", "Limits, Lemma \\ref{limits-lemma-descend-surjective}. Increasing", "$0$ yet again we may choose elements", "$h_{1, i}, \\ldots, h_{n, i} \\in \\mathcal{O}_{U_i}(U_i)$ which map to the", "classes of $x_1, \\ldots, x_n$ in $B = \\mathcal{O}_U(U)$ and such that", "$g_{\\lambda, i}(h_{\\nu, i}) = 0$ in $\\mathcal{O}_{U_i}(U_i)$. Thus", "we obtain a commutative diagram", "\\begin{equation}", "\\label{equation-to-show-cartesian}", "\\vcenter{", "\\xymatrix{", "X_i \\ar[d] & U_i \\ar[l] \\ar[d] \\\\", "\\Spec(A_i) & \\Spec(B_i) \\ar[l]", "}", "}", "\\end{equation}", "By construction $B_i = B_0 \\otimes_{A_0} A_i$ and", "$B = B_0 \\otimes_{A_0} A$. Consider the morphism", "$$", "f_0 : U_0 \\longrightarrow X_0 \\times_{\\Spec(A_0)} \\Spec(B_0)", "$$", "This is a morphism of quasi-compact and quasi-separated algebraic spaces", "representable, separated and \\'etale over $X_0$. The base change of $f_0$", "to $X$ is an isomorphism by our choices. Hence", "Lemma \\ref{lemma-descend-equality}", "guarantees that there exists an $i$ such that the base change of $f_0$", "to $X_i$ is an isomorphism, in other words the diagram", "(\\ref{equation-to-show-cartesian}) is cartesian. Thus", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves}", "applied to the fppf covering $\\{\\Spec(B_i) \\to \\Spec(A_i)\\}$", "combined with Descent, Lemma \\ref{descent-lemma-affine}", "give that $X_i \\to \\Spec(A_i)$ is representable by a scheme", "affine over $\\Spec(A_i)$ as desired. (Of course it then also follows", "that $X_i = \\Spec(A_i)$ but we don't need this.)" ], "refs": [ "algebra-lemma-etale-standard-smooth", "limits-lemma-descend-surjective", "spaces-limits-lemma-descend-equality", "descent-lemma-descent-data-sheaves", "descent-lemma-affine" ], "ref_ids": [ 1230, 15069, 4576, 14751, 14748 ] } ], "ref_ids": [] }, { "id": 4579, "type": "theorem", "label": "spaces-limits-lemma-limit-is-scheme", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-limit-is-scheme", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent}.", "If $X$ is a scheme, then there exists an $i$ such that $X_i$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite affine open covering $X = \\bigcup W_j$.", "By Lemma \\ref{lemma-descend-opens}", "we can find an $i \\in I$ and open subspaces $W_{j, i} \\subset X_i$", "whose base change to $X$ is $W_j \\to X$. By", "Lemma \\ref{lemma-limit-is-affine} we may assume that", "each $W_{j, i}$ is an affine scheme. This means that $X_i$", "is a scheme (see for example", "Properties of Spaces, Section \\ref{spaces-properties-section-schematic})." ], "refs": [ "spaces-limits-lemma-descend-opens", "spaces-limits-lemma-limit-is-affine" ], "ref_ids": [ 4575, 4578 ] } ], "ref_ids": [] }, { "id": 4580, "type": "theorem", "label": "spaces-limits-lemma-finite-type-eventually-closed", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-finite-type-eventually-closed", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $X = \\lim X_i$ be a directed limit of", "algebraic spaces over $B$ with affine transition morphisms.", "Let $Y \\to X$ be a morphism of algebraic spaces over $B$.", "\\begin{enumerate}", "\\item If $Y \\to X$ is a closed immersion, $X_i$ quasi-compact, and", "$Y \\to B$ locally of finite type, then $Y \\to X_i$ is a closed immersion", "for $i$ large enough.", "\\item If $Y \\to X$ is an immersion, $X_i$ quasi-separated, $Y \\to B$", "locally of finite type, and $Y$ quasi-compact, then $Y \\to X_i$ is an", "immersion for $i$ large enough.", "\\item If $Y \\to X$ is an isomorphism, $X_i$ quasi-compact,", "$X_i \\to B$ locally of finite type, the transition morphisms", "$X_{i'} \\to X_i$ are closed immersions, and $Y \\to B$ is locally", "of finite presentation, then $Y \\to X_i$ is an isomorphism for $i$", "large enough.", "\\item If $Y \\to X$ is a monomorphism, $X_i$ quasi-separated,", "$Y \\to B$ locally of finite type, and $Y$ quasi-compact, then", "$Y \\to X_i$ is a monomorphism for $i$ large enough.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Choose $0 \\in I$. As $X_0$ is quasi-compact, we can choose", "an affine scheme $W$ and an \\'etale morphism $W \\to B$ such that the image of", "$|X_0| \\to |B|$ is contained in $|W| \\to |B|$. ", "Choose an affine scheme $U_0$ and an \\'etale morphism", "$U_0 \\to X_0 \\times_B W$ such that $U_0 \\to X_0$ is surjective.", "(This is possible by our choice of $W$ and the fact that $X_0$ is", "quasi-compact; details omitted.)", "Let $V \\to Y$, resp.\\ $U \\to X$, resp.\\ $U_i \\to X_i$ be the base change", "of $U_0 \\to X_0$ (for $i \\geq 0$). It suffices to prove that $V \\to U_i$", "is a closed immersion for $i$ sufficiently large. Thus we reduce", "to proving the result for $V \\to U = \\lim U_i$ over $W$. This follows from", "the case of schemes, which is", "Limits, Lemma \\ref{limits-lemma-finite-type-eventually-closed}.", "\\medskip\\noindent", "Proof of (2). Choose $0 \\in I$. Choose a quasi-compact open subspace", "$X'_0 \\subset X_0$ such that $Y \\to X_0$ factors through $X'_0$.", "After replacing $X_i$ by the inverse image of $X'_0$ for $i \\geq 0$", "we may assume all $X_i'$ are quasi-compact and quasi-separated.", "Let $U \\subset X$ be a quasi-compact open such that $Y \\to X$ factors", "through a closed immersion $Y \\to U$ ($U$ exists as $Y$ is quasi-compact). By", "Lemma \\ref{lemma-descend-opens}", "we may assume that $U = \\lim U_i$ with $U_i \\subset X_i$ quasi-compact", "open. By part (1) we see that $Y \\to U_i$ is a closed immersion for some", "$i$. Thus (2) holds.", "\\medskip\\noindent", "Proof of (3). Choose $0 \\in I$. Choose an affine scheme $U_0$", "and a surjective \\'etale morphism $U_0 \\to X_0$.", "Set $U_i = X_i \\times_{X_0} U_0$,", "$U = X \\times_{X_0} U_0 = Y \\times_{X_0} U_0$. Then $U = \\lim U_i$ is a", "limit of affine schemes, the transition maps of the system are closed", "immersions, and $U \\to U_0$ is of finite presentation (because", "$U \\to B$ is locally of finite presentation and $U_0 \\to B$ is locally", "of finite type and", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-permanence}).", "Thus we've reduced to the following algebra fact: If $A = \\lim A_i$", "is a directed colimit of $R$-algebras with surjective transition", "maps and $A$ of finite presentation over $A_0$, then $A = A_i$ for", "some $i$. Namely, write $A = A_0/(f_1, \\ldots, f_n)$. Pick $i$ such", "that $f_1, \\ldots, f_n$ map to zero under the surjective map $A_0 \\to A_i$.", "\\medskip\\noindent", "Proof of (4). Set $Z_i = Y \\times_{X_i} Y$.", "As the transition morphisms $X_{i'} \\to X_i$ are affine hence separated,", "the transition morphisms $Z_{i'} \\to Z_i$ are closed immersions, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-fibre-product-after-map}.", "We have $\\lim Z_i = Y \\times_X Y = Y$ as $Y \\to X$ is a monomorphism.", "Choose $0 \\in I$. Since $Y \\to X_0$ is locally of finite type", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-permanence-finite-type})", "the morphism $Y \\to Z_0$ is locally of finite presentation", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-diagonal-morphism-finite-type}).", "The morphisms $Z_i \\to Z_0$ are locally of finite type", "(they are closed immersions).", "Finally, $Z_i = Y \\times_{X_i} Y$ is quasi-compact as", "$X_i$ is quasi-separated and $Y$ is quasi-compact.", "Thus part (3) applies to $Y = \\lim_{i \\geq 0} Z_i$ over $Z_0$", "and we conclude $Y = Z_i$ for some $i$. This proves (4) and the lemma." ], "refs": [ "limits-lemma-finite-type-eventually-closed", "spaces-limits-lemma-descend-opens", "spaces-morphisms-lemma-finite-presentation-permanence", "spaces-morphisms-lemma-fibre-product-after-map", "spaces-morphisms-lemma-permanence-finite-type", "spaces-morphisms-lemma-diagonal-morphism-finite-type" ], "ref_ids": [ 15046, 4575, 4846, 4715, 4818, 4847 ] } ], "ref_ids": [] }, { "id": 4581, "type": "theorem", "label": "spaces-limits-lemma-eventually-separated", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-eventually-separated", "contents": [ "Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$.", "Let $X = \\lim X_i$ be a directed limit of algebraic spaces over $Y$", "with affine transition morphisms. Assume", "\\begin{enumerate}", "\\item $Y$ is quasi-separated,", "\\item $X_i$ is quasi-compact and quasi-separated,", "\\item the morphism $X \\to Y$ is separated.", "\\end{enumerate}", "Then $X_i \\to Y$ is separated for all $i$ large enough." ], "refs": [], "proofs": [ { "contents": [ "Let $0 \\in I$. Choose an affine scheme $W$ and an \\'etale morphism", "$W \\to Y$ such that the image of $|W| \\to |Y|$ contains the image of", "$|X_0| \\to |Y|$. This is possible as $X_0$ is quasi-compact.", "It suffices to check that $W \\times_Y X_i \\to W$ is separated", "for some $i \\geq 0$ because the diagonal of $W \\times_Y X_i$", "over $W$ is the base change of $X_i \\to X_i \\times_Y X_i$ by", "the surjective \\'etale morphism $(X_i \\times_Y X_i) \\times_Y W \\to", "X_i \\times_Y X_i$. Since $Y$ is quasi-separated the algebraic spaces", "$W \\times_Y X_i$ are quasi-compact (as well as quasi-separated).", "Thus we may base change to $W$ and assume $Y$ is an affine scheme.", "When $Y$ is an affine scheme, we have to show that $X_i$ is a", "separated algebraic space for $i$ large enough and we are given that", "$X$ is a separated algebraic space. Thus this case follows from", "Lemma \\ref{lemma-descend-separated}." ], "refs": [ "spaces-limits-lemma-descend-separated" ], "ref_ids": [ 4577 ] } ], "ref_ids": [] }, { "id": 4582, "type": "theorem", "label": "spaces-limits-lemma-eventually-affine", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-eventually-affine", "contents": [ "Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$.", "Let $X = \\lim X_i$ be a directed limit of algebraic spaces over $Y$", "with affine transition morphisms. Assume", "\\begin{enumerate}", "\\item $Y$ quasi-compact and quasi-separated,", "\\item $X_i$ quasi-compact and quasi-separated,", "\\item $X \\to Y$ affine.", "\\end{enumerate}", "Then $X_i \\to Y$ is affine for $i$ large enough." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $W$ and a surjective \\'etale morphism $W \\to Y$.", "Then $X \\times_Y W$ is affine and it suffices to check that", "$X_i \\times_Y W$ is affine for some $i$ (Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-affine-local}).", "This follows from Lemma \\ref{lemma-limit-is-affine}." ], "refs": [ "spaces-morphisms-lemma-affine-local", "spaces-limits-lemma-limit-is-affine" ], "ref_ids": [ 4798, 4578 ] } ], "ref_ids": [] }, { "id": 4583, "type": "theorem", "label": "spaces-limits-lemma-eventually-finite", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-eventually-finite", "contents": [ "Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$.", "Let $X = \\lim X_i$ be a directed limit of algebraic spaces", "over $Y$ with affine transition morphisms. Assume", "\\begin{enumerate}", "\\item $Y$ quasi-compact and quasi-separated,", "\\item $X_i$ quasi-compact and quasi-separated,", "\\item the transition morphisms $X_{i'} \\to X_i$ are finite,", "\\item $X_i \\to Y$ locally of finite type", "\\item $X \\to Y$ integral.", "\\end{enumerate}", "Then $X_i \\to Y$ is finite for $i$ large enough." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $W$ and a surjective \\'etale morphism $W \\to Y$.", "Then $X \\times_Y W$ is finite over $W$ and it suffices to check that", "$X_i \\times_Y W$ is finite over $W$ for some $i$ (Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-integral-local}). By", "Lemma \\ref{lemma-limit-is-scheme} this reduces us to the case of schemes.", "In the case of schemes it follows from", "Limits, Lemma \\ref{limits-lemma-eventually-finite}." ], "refs": [ "spaces-morphisms-lemma-integral-local", "spaces-limits-lemma-limit-is-scheme", "limits-lemma-eventually-finite" ], "ref_ids": [ 4940, 4579, 15049 ] } ], "ref_ids": [] }, { "id": 4584, "type": "theorem", "label": "spaces-limits-lemma-eventually-closed-immersion", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-eventually-closed-immersion", "contents": [ "Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$.", "Let $X = \\lim X_i$ be a directed limit of algebraic spaces", "over $Y$ with affine transition morphisms. Assume", "\\begin{enumerate}", "\\item $Y$ quasi-compact and quasi-separated,", "\\item $X_i$ quasi-compact and quasi-separated,", "\\item the transition morphisms $X_{i'} \\to X_i$ are closed immersions,", "\\item $X_i \\to Y$ locally of finite type", "\\item $X \\to Y$ is a closed immersion.", "\\end{enumerate}", "Then $X_i \\to Y$ is a closed immersion for $i$ large enough." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $W$ and a surjective \\'etale morphism $W \\to Y$.", "Then $X \\times_Y W$ is a closed subspace of $W$ and it suffices to check", "that $X_i \\times_Y W$ is a closed subspace $W$ for some $i$", "(Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-closed-immersion-local}). By", "Lemma \\ref{lemma-limit-is-scheme} this reduces us to the case of schemes.", "In the case of schemes it follows from", "Limits, Lemma \\ref{limits-lemma-eventually-closed-immersion}." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-local", "spaces-limits-lemma-limit-is-scheme", "limits-lemma-eventually-closed-immersion" ], "ref_ids": [ 4761, 4579, 15050 ] } ], "ref_ids": [] }, { "id": 4585, "type": "theorem", "label": "spaces-limits-lemma-descend-etale", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-etale", "contents": [ "With notation and assumptions as in", "Situation \\ref{situation-descent-property}. If", "\\begin{enumerate}", "\\item $f$ is \\'etale,", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then $f_i$ is \\'etale for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $V_0$ and a surjective \\'etale morphism", "$V_0 \\to Y_0$. Choose an affine scheme $U_0$ and a surjective \\'etale", "morphism $U_0 \\to V_0 \\times_{Y_0} X_0$. Diagram", "$$", "\\xymatrix{", "U_0 \\ar[d] \\ar[r] & V_0 \\ar[d] \\\\", "X_0 \\ar[r] & Y_0", "}", "$$", "The vertical arrows are surjective and \\'etale by construction.", "We can base change this diagram to $B_i$ or $B$ to get", "$$", "\\vcenter{", "\\xymatrix{", "U_i \\ar[d] \\ar[r] & V_i \\ar[d] \\\\", "X_i \\ar[r] & Y_i", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "}", "$$", "Note that $U_i, V_i, U, V$ are affine schemes,", "the vertical morphisms are surjective \\'etale, and the limit of the", "morphisms $U_i \\to V_i$ is $U \\to V$. Recall that $X_i \\to Y_i$ is \\'etale", "if and only if $U_i \\to V_i$ is", "\\'etale and similarly $X \\to Y$ is \\'etale if and only if", "$U \\to V$ is \\'etale", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-etale-local}).", "Since $f_0$ is locally of finite", "presentation, so is the morphism $U_0 \\to V_0$. Hence the lemma follows", "from Limits, Lemma \\ref{limits-lemma-descend-etale}." ], "refs": [ "spaces-morphisms-lemma-etale-local", "limits-lemma-descend-etale" ], "ref_ids": [ 4905, 15065 ] } ], "ref_ids": [] }, { "id": 4586, "type": "theorem", "label": "spaces-limits-lemma-descend-smooth", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-smooth", "contents": [ "With notation and assumptions as in", "Situation \\ref{situation-descent-property}. If", "\\begin{enumerate}", "\\item $f$ is smooth,", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then $f_i$ is smooth for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $V_0$ and a surjective \\'etale morphism", "$V_0 \\to Y_0$. Choose an affine scheme $U_0$ and a surjective \\'etale", "morphism $U_0 \\to V_0 \\times_{Y_0} X_0$. Diagram", "$$", "\\xymatrix{", "U_0 \\ar[d] \\ar[r] & V_0 \\ar[d] \\\\", "X_0 \\ar[r] & Y_0", "}", "$$", "The vertical arrows are surjective and \\'etale by construction.", "We can base change this diagram to $B_i$ or $B$ to get", "$$", "\\vcenter{", "\\xymatrix{", "U_i \\ar[d] \\ar[r] & V_i \\ar[d] \\\\", "X_i \\ar[r] & Y_i", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "}", "$$", "Note that $U_i, V_i, U, V$ are affine schemes,", "the vertical morphisms are surjective \\'etale, and the limit of the", "morphisms $U_i \\to V_i$ is $U \\to V$. Recall that $X_i \\to Y_i$ is smooth", "if and only if $U_i \\to V_i$ is smooth and similarly", "$X \\to Y$ is smooth if and only if $U \\to V$ is smooth", "(Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-smooth}).", "Since $f_0$ is locally of finite", "presentation, so is the morphism $U_0 \\to V_0$. Hence the lemma follows", "from Limits, Lemma \\ref{limits-lemma-descend-smooth}." ], "refs": [ "spaces-morphisms-definition-smooth", "limits-lemma-descend-smooth" ], "ref_ids": [ 5012, 15064 ] } ], "ref_ids": [] }, { "id": 4587, "type": "theorem", "label": "spaces-limits-lemma-descend-surjective", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-surjective", "contents": [ "With notation and assumptions as in", "Situation \\ref{situation-descent-property}. If", "\\begin{enumerate}", "\\item $f$ is surjective,", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then $f_i$ is surjective for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $V_0$ and a surjective \\'etale morphism", "$V_0 \\to Y_0$. Choose an affine scheme $U_0$ and a surjective \\'etale", "morphism $U_0 \\to V_0 \\times_{Y_0} X_0$. Diagram", "$$", "\\xymatrix{", "U_0 \\ar[d] \\ar[r] & V_0 \\ar[d] \\\\", "X_0 \\ar[r] & Y_0", "}", "$$", "The vertical arrows are surjective and \\'etale by construction.", "We can base change this diagram to $B_i$ or $B$ to get", "$$", "\\vcenter{", "\\xymatrix{", "U_i \\ar[d] \\ar[r] & V_i \\ar[d] \\\\", "X_i \\ar[r] & Y_i", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "}", "$$", "Note that $U_i, V_i, U, V$ are affine schemes, the vertical morphisms are", "surjective \\'etale, the limit of the morphisms $U_i \\to V_i$ is $U \\to V$,", "and the morphisms $U_i \\to X_i \\times_{Y_i} V_i$ and", "$U \\to X \\times_Y V$ are surjective (as base changes of", "$U_0 \\to X_0 \\times_{Y_0} V_0$). In particular, we see that", "$X_i \\to Y_i$ is surjective if and only if $U_i \\to V_i$ is surjective", "and similarly $X \\to Y$ is surjective if and only if $U \\to V$ is surjective.", "Since $f_0$ is locally of finite", "presentation, so is the morphism $U_0 \\to V_0$. Hence the lemma follows", "from the case of schemes (Limits, Lemma \\ref{limits-lemma-descend-surjective})." ], "refs": [ "limits-lemma-descend-surjective" ], "ref_ids": [ 15069 ] } ], "ref_ids": [] }, { "id": 4588, "type": "theorem", "label": "spaces-limits-lemma-descend-universally-injective", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-universally-injective", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}. If", "\\begin{enumerate}", "\\item $f$ is universally injective,", "\\item $f_0$ is locally of finite type,", "\\end{enumerate}", "then $f_i$ is universally injective for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Recall that a morphism $X \\to Y$ is universally injective if and", "only if the diagonal $X \\to X \\times_Y X$ is surjective", "(Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-universally-injective} and", "Lemma \\ref{spaces-morphisms-lemma-universally-injective}).", "Observe that $X_0 \\to X_0 \\times_{Y_0} X_0$ is of locally of finite", "presentation (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-diagonal-morphism-finite-type}).", "Hence the lemma follows from Lemma \\ref{lemma-descend-surjective}", "by considering the morphism $X_0 \\to X_0 \\times_{Y_0} X_0$." ], "refs": [ "spaces-morphisms-definition-universally-injective", "spaces-morphisms-lemma-universally-injective", "spaces-morphisms-lemma-diagonal-morphism-finite-type", "spaces-limits-lemma-descend-surjective" ], "ref_ids": [ 4997, 4793, 4847, 4587 ] } ], "ref_ids": [] }, { "id": 4589, "type": "theorem", "label": "spaces-limits-lemma-descend-affine", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-affine", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}. If", "$f$ is affine, then $f_i$ is affine for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $V_0$ and a surjective \\'etale morphism $V_0 \\to Y_0$.", "Set $V_i = V_0 \\times_{Y_0} Y_i$ and $V = V_0 \\times_{Y_0} Y$.", "Since $f$ is affine we see that $V \\times_Y X = \\lim V_i \\times_{Y_i} X_i$", "is affine. By Lemma \\ref{lemma-limit-is-affine} we see that", "$V_i \\times_{Y_i} X_i$ is affine for some $i \\geq 0$. For this $i$ the morphism", "$f_i$ is affine", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-affine-local})." ], "refs": [ "spaces-limits-lemma-limit-is-affine", "spaces-morphisms-lemma-affine-local" ], "ref_ids": [ 4578, 4798 ] } ], "ref_ids": [] }, { "id": 4590, "type": "theorem", "label": "spaces-limits-lemma-descend-finite", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-finite", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}. If", "\\begin{enumerate}", "\\item $f$ is finite,", "\\item $f_0$ is locally of finite type,", "\\end{enumerate}", "then $f_i$ is finite for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $V_0$ and a surjective \\'etale morphism $V_0 \\to Y_0$.", "Set $V_i = V_0 \\times_{Y_0} Y_i$ and $V = V_0 \\times_{Y_0} Y$.", "Since $f$ is finite we see that $V \\times_Y X = \\lim V_i \\times_{Y_i} X_i$", "is a scheme finite over $V$. By Lemma \\ref{lemma-limit-is-affine} we see that", "$V_i \\times_{Y_i} X_i$ is affine for some $i \\geq 0$. Increasing $i$ if", "necessary we find that $V_i \\times_{Y_i} X_i \\to V_i$ is finite by", "Limits, Lemma \\ref{limits-lemma-descend-finite-finite-presentation}.", "For this $i$ the morphism $f_i$ is finite", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-integral-local})." ], "refs": [ "spaces-limits-lemma-limit-is-affine", "limits-lemma-descend-finite-finite-presentation", "spaces-morphisms-lemma-integral-local" ], "ref_ids": [ 4578, 15058, 4940 ] } ], "ref_ids": [] }, { "id": 4591, "type": "theorem", "label": "spaces-limits-lemma-descend-closed-immersion", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-closed-immersion", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}. If", "\\begin{enumerate}", "\\item $f$ is a closed immersion,", "\\item $f_0$ is locally of finite type,", "\\end{enumerate}", "then $f_i$ is a closed immersion for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $V_0$ and a surjective \\'etale morphism $V_0 \\to Y_0$.", "Set $V_i = V_0 \\times_{Y_0} Y_i$ and $V = V_0 \\times_{Y_0} Y$.", "Since $f$ is a closed immersion we see that", "$V \\times_Y X = \\lim V_i \\times_{Y_i} X_i$", "is a closed subscheme of the affine scheme $V$. By", "Lemma \\ref{lemma-limit-is-affine} we see that", "$V_i \\times_{Y_i} X_i$ is affine for some $i \\geq 0$. Increasing $i$ if", "necessary we find that $V_i \\times_{Y_i} X_i \\to V_i$ is a closed immersion by", "Limits, Lemma \\ref{limits-lemma-descend-closed-immersion-finite-presentation}.", "For this $i$ the morphism $f_i$ is a closed immersion", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-integral-local})." ], "refs": [ "spaces-limits-lemma-limit-is-affine", "limits-lemma-descend-closed-immersion-finite-presentation", "spaces-morphisms-lemma-integral-local" ], "ref_ids": [ 4578, 15060, 4940 ] } ], "ref_ids": [] }, { "id": 4592, "type": "theorem", "label": "spaces-limits-lemma-descend-separated-morphism", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-separated-morphism", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If $f$ is separated, then $f_i$ is separated for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-descend-closed-immersion}", "to the diagonal morphism $\\Delta_{X_0/Y_0} : X_0 \\to X_0 \\times_{Y_0} X_0$.", "(Diagonal morphisms are locally of finite type", "and the fibre product $X_0 \\times_{Y_0} X_0$ is quasi-compact and", "quasi-separated. Some details omitted.)" ], "refs": [ "spaces-limits-lemma-descend-closed-immersion" ], "ref_ids": [ 4591 ] } ], "ref_ids": [] }, { "id": 4593, "type": "theorem", "label": "spaces-limits-lemma-descend-isomorphism", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-isomorphism", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}. If", "\\begin{enumerate}", "\\item $f$ is a isomorphism,", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then $f_i$ is a isomorphism for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Being an isomorphism is equivalent to being \\'etale, universally injective,", "and surjective, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-etale-universally-injective-open}.", "Thus the lemma follows from", "Lemmas \\ref{lemma-descend-etale},", "\\ref{lemma-descend-surjective}, and", "\\ref{lemma-descend-universally-injective}." ], "refs": [ "spaces-morphisms-lemma-etale-universally-injective-open", "spaces-limits-lemma-descend-etale", "spaces-limits-lemma-descend-surjective", "spaces-limits-lemma-descend-universally-injective" ], "ref_ids": [ 4973, 4585, 4587, 4588 ] } ], "ref_ids": [] }, { "id": 4594, "type": "theorem", "label": "spaces-limits-lemma-descend-monomorphism", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-monomorphism", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}. If", "\\begin{enumerate}", "\\item $f$ is a monomorphism,", "\\item $f_0$ is locally of finite type,", "\\end{enumerate}", "then $f_i$ is a monomorphism for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Recall that a morphism is a monomorphism if and only if the diagonal is", "an isomorphism. The morphism $X_0 \\to X_0 \\times_{Y_0} X_0$ is locally of", "finite presentation by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-diagonal-morphism-finite-type}.", "Since $X_0 \\times_{Y_0} X_0$ is quasi-compact and quasi-separated", "we conclude from", "Lemma \\ref{lemma-descend-isomorphism}", "that $\\Delta_i : X_i \\to X_i \\times_{Y_i} X_i$ is an isomorphism for", "some $i \\geq 0$. For this $i$ the morphism $f_i$ is a monomorphism." ], "refs": [ "spaces-morphisms-lemma-diagonal-morphism-finite-type", "spaces-limits-lemma-descend-isomorphism" ], "ref_ids": [ 4847, 4593 ] } ], "ref_ids": [] }, { "id": 4595, "type": "theorem", "label": "spaces-limits-lemma-descend-flat", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-flat", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "Let $\\mathcal{F}_0$ be a quasi-coherent $\\mathcal{O}_{X_0}$-module", "and denote $\\mathcal{F}_i$ the pullback to $X_i$ and $\\mathcal{F}$", "the pullback to $X$. If", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is flat over $Y$,", "\\item $\\mathcal{F}_0$ is of finite presentation, and", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then $\\mathcal{F}_i$ is flat over $Y_i$ for some $i \\geq 0$.", "In particular, if $f_0$ is locally of finite presentation and", "$f$ is flat, then $f_i$ is flat for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $V_0$ and a surjective \\'etale morphism", "$V_0 \\to Y_0$. Choose an affine scheme $U_0$ and a surjective \\'etale", "morphism $U_0 \\to V_0 \\times_{Y_0} X_0$. Diagram", "$$", "\\xymatrix{", "U_0 \\ar[d] \\ar[r] & V_0 \\ar[d] \\\\", "X_0 \\ar[r] & Y_0", "}", "$$", "The vertical arrows are surjective and \\'etale by construction.", "We can base change this diagram to $B_i$ or $B$ to get", "$$", "\\vcenter{", "\\xymatrix{", "U_i \\ar[d] \\ar[r] & V_i \\ar[d] \\\\", "X_i \\ar[r] & Y_i", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "}", "$$", "Note that $U_i, V_i, U, V$ are affine schemes, the vertical morphisms are", "surjective \\'etale, and the limit of the morphisms $U_i \\to V_i$ is", "$U \\to V$. Recall that $\\mathcal{F}_i$ is flat over $Y_i$ if and only if", "$\\mathcal{F}_i|_{U_i}$ is flat over $V_i$ and similarly $\\mathcal{F}$ is flat", "over $Y$ if and only if $\\mathcal{F}|_U$ is flat over $V$", "(Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-flat}).", "Since $f_0$ is locally of finite presentation, so is the morphism", "$U_0 \\to V_0$. Hence the lemma follows", "from Limits, Lemma \\ref{limits-lemma-descend-module-flat-finite-presentation}." ], "refs": [ "spaces-morphisms-definition-flat", "limits-lemma-descend-module-flat-finite-presentation" ], "ref_ids": [ 5007, 15080 ] } ], "ref_ids": [] }, { "id": 4596, "type": "theorem", "label": "spaces-limits-lemma-eventually-proper", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-eventually-proper", "contents": [ "Assumptions and notation as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is proper, and", "\\item $f_0$ is locally of finite type,", "\\end{enumerate}", "then there exists an $i$ such that $f_i$ is proper." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $V_0$ and a surjective \\'etale morphism $V_0 \\to Y_0$.", "Set $V_i = Y_i \\times_{Y_0} V_0$ and $V = Y \\times_{Y_0} V_0$.", "It suffices to prove that the base change of $f_i$ to $V_i$ is", "proper, see Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-proper-local}.", "Thus we may assume $Y_0$ is affine.", "\\medskip\\noindent", "By Lemma \\ref{lemma-descend-separated-morphism} we see that", "$f_i$ is separated for some $i \\geq 0$. Replacing", "$0$ by $i$ we may assume that $f_0$ is separated.", "Observe that $f_0$ is quasi-compact. Thus $f_0$ is separated and", "of finite type. By", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-weak-chow}", "we can choose a diagram", "$$", "\\xymatrix{", "X_0 \\ar[rd] & X_0' \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_{Y_0} \\ar[dl] \\\\", "& Y_0 &", "}", "$$", "where $X_0' \\to \\mathbf{P}^n_{Y_0}$ is an immersion, and", "$\\pi : X_0' \\to X_0$ is proper and surjective. Introduce", "$X' = X_0' \\times_{Y_0} Y$ and $X_i' = X_0' \\times_{Y_0} Y_i$.", "By Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-composition-proper} and", "\\ref{spaces-morphisms-lemma-base-change-proper}", "we see that $X' \\to Y$ is proper. Hence $X' \\to \\mathbf{P}^n_Y$ is", "a closed immersion (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-permanence}). By", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-image-proper-is-proper}", "it suffices to prove that $X'_i \\to Y_i$ is proper for some $i$.", "By Lemma \\ref{lemma-descend-closed-immersion}", "we find that $X'_i \\to \\mathbf{P}^n_{Y_i}$ is", "a closed immersion for $i$ large enough. Then $X'_i \\to Y_i$", "is proper and we win." ], "refs": [ "spaces-morphisms-lemma-proper-local", "spaces-limits-lemma-descend-separated-morphism", "spaces-cohomology-lemma-weak-chow", "spaces-morphisms-lemma-composition-proper", "spaces-morphisms-lemma-base-change-proper", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-morphisms-lemma-image-proper-is-proper", "spaces-limits-lemma-descend-closed-immersion" ], "ref_ids": [ 4916, 4592, 11327, 4918, 4917, 4920, 4921, 4591 ] } ], "ref_ids": [] }, { "id": 4597, "type": "theorem", "label": "spaces-limits-lemma-eventually-relative-dimension", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-eventually-relative-dimension", "contents": [ "Assumptions and notation as in Situation \\ref{situation-descent-property}.", "Let $d \\geq 0$. If", "\\begin{enumerate}", "\\item $f$ has relative dimension $\\leq d$", "(Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-relative-dimension}), and", "\\item $f_0$ is locally of finite type,", "\\end{enumerate}", "then there exists an $i$ such that $f_i$ has relative dimension $\\leq d$." ], "refs": [ "spaces-morphisms-definition-relative-dimension" ], "proofs": [ { "contents": [ "Choose an affine scheme $V_0$ and a surjective \\'etale morphism", "$V_0 \\to Y_0$. Choose an affine scheme $U_0$ and a surjective \\'etale", "morphism $U_0 \\to V_0 \\times_{Y_0} X_0$. Diagram", "$$", "\\xymatrix{", "U_0 \\ar[d] \\ar[r] & V_0 \\ar[d] \\\\", "X_0 \\ar[r] & Y_0", "}", "$$", "The vertical arrows are surjective and \\'etale by construction.", "We can base change this diagram to $B_i$ or $B$ to get", "$$", "\\vcenter{", "\\xymatrix{", "U_i \\ar[d] \\ar[r] & V_i \\ar[d] \\\\", "X_i \\ar[r] & Y_i", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "}", "$$", "Note that $U_i, V_i, U, V$ are affine schemes,", "the vertical morphisms are surjective \\'etale, and the limit of the", "morphisms $U_i \\to V_i$ is $U \\to V$.", "In this situation $X_i \\to Y_i$ has relative dimension $\\leq d$", "if and only if $U_i \\to V_i$ has relative dimension $\\leq d$", "(as defined in Morphisms, Definition", "\\ref{morphisms-definition-relative-dimension-d}).", "To see the equivalence, use that the definition for morphisms", "of algebraic spaces involves Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-dimension-fibre}", "which uses \\'etale localization. The same is true for $X \\to Y$ and $U \\to V$.", "Since $f_0$ is locally of finite type, so is the morphism $U_0 \\to V_0$.", "Hence the lemma follows from the more general", "Limits, Lemma \\ref{limits-lemma-limit-dimension}." ], "refs": [ "morphisms-definition-relative-dimension-d", "spaces-morphisms-definition-dimension-fibre", "limits-lemma-limit-dimension" ], "ref_ids": [ 5559, 5009, 15104 ] } ], "ref_ids": [ 5010 ] }, { "id": 4598, "type": "theorem", "label": "spaces-limits-lemma-descend-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-finite-presentation", "contents": [ "Let $S$ be a scheme. Let $I$ be a directed set.", "Let $(X_i, f_{ii'})$ be an inverse system over $I$ of algebraic spaces", "over $S$. Assume", "\\begin{enumerate}", "\\item the morphisms $f_{ii'} : X_i \\to X_{i'}$ are affine,", "\\item the spaces $X_i$ are quasi-compact and quasi-separated.", "\\end{enumerate}", "Let $X = \\lim_i X_i$. Then the category of algebraic spaces", "of finite presentation over $X$ is the colimit over $I$ of the", "categories of algebraic spaces of finite presentation over $X_i$." ], "refs": [], "proofs": [ { "contents": [ "Pick $0 \\in I$. Choose a surjective \\'etale morphism $U_0 \\to X_0$ where", "$U_0$ is an affine scheme (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "Set $U_i = X_i \\times_{X_0} U_0$. Set $R_0 = U_0 \\times_{X_0} U_0$ and", "$R_i = R_0 \\times_{X_0} X_i$. Denote $s_i, t_i : R_i \\to U_i$ and", "$s, t : R \\to U$ the two projections. In the proof of", "Lemma \\ref{lemma-directed-inverse-system-has-limit} we have", "seen that there exists a presentation $X = U/R$ with", "$U = \\lim U_i$ and $R = \\lim R_i$. Note that $U_i$ and $U$ are affine and", "that $R_i$ and $R$ are quasi-compact and separated (as $X_i$ is", "quasi-separated). Let $Y$ be an algebraic space over $S$ and let", "$Y \\to X$ be a morphism of finite presentation. Set $V = U \\times_X Y$.", "This is an algebraic space of finite presentation over $U$.", "Choose an affine scheme $W$ and a surjective \\'etale morphism $W \\to V$.", "Then $W \\to Y$ is surjective \\'etale as well. Set $R' = W \\times_Y W$", "so that $Y = W/R'$ (see Spaces, Section \\ref{spaces-section-presentations}).", "Note that $W$ is a scheme of finite presentation over $U$ and that $R'$", "is a scheme of finite presentation over $R$ (details omitted).", "By Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can find an index $i$ and a morphism of schemes $W_i \\to U_i$ of", "finite presentation whose base change to $U$ gives $W \\to U$. Similarly", "we can find, after possibly increasing $i$, a scheme $R'_i$ of finite", "presentation over $R_i$ whose base change to $R$ is $R'$.", "The projection morphisms $s', t' : R' \\to W$ are morphisms over", "the projection morphisms $s, t : R \\to U$. Hence we can view $s'$,", "resp.\\ $t'$ as a morphism between schemes of finite presentation over", "$U$ (with structure morphism $R' \\to U$ given by $R' \\to R$ followed", "by $s$, resp.\\ $t$). Hence we can apply", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "again to see that, after possibly increasing $i$, there exist", "morphisms $s'_i, t'_i : R'_i \\to W_i$, whose base change to $U$", "is $S', t'$. By Limits, Lemmas \\ref{limits-lemma-descend-etale} and", "\\ref{limits-lemma-descend-monomorphism}", "we may assume that $s'_i, t'_i$ are \\'etale and that", "$j'_i : R'_i \\to W_i \\times_{X_i} W_i$ is a monomorphism (here we", "view $j'_i$ as a morphism of schemes of finite presentation over $U_i$ via", "one of the projections -- it doesn't matter which one). Setting", "$Y_i = W_i/R'_i$ (see Spaces, Theorem \\ref{spaces-theorem-presentation})", "we obtain an algebraic space of finite presentation", "over $X_i$ whose base change to $X$ is isomorphic to $Y$.", "\\medskip\\noindent", "This shows that every algebraic space of finite presentation over $X$ comes", "from an algebraic space of finite presentation over some $X_i$, i.e.,", "it shows that the functor of the lemma is essentially surjective. To", "show that it is fully faithful, consider an index $0 \\in I$ and two", "algebraic spaces $Y_0, Z_0$ of finite presentation over $X_0$.", "Set $Y_i = X_i \\times_{X_0} Y_0$, $Y = X \\times_{X_0} Y_0$,", "$Z_i = X_i \\times_{X_0} Z_0$, and $Z = X \\times_{X_0} Z_0$. Let", "$\\alpha : Y \\to Z$ be a morphism of algebraic spaces over $X$.", "Choose a surjective \\'etale morphism $V_0 \\to Y_0$ where $V_0$ is", "an affine scheme. Set $V_i = V_0 \\times_{Y_0} Y_i$ and", "$V = V_0 \\times_{Y_0} Y$ which are affine schemes endowed with", "surjective \\'etale morphisms to $Y_i$ and $Y$. The composition", "$V \\to Y \\to Z \\to Z_0$ comes from a (essentially unique) morphism", "$V_i \\to Z_0$ for some $i \\geq 0$ by", "Proposition \\ref{proposition-characterize-locally-finite-presentation}", "(applied to $Z_0 \\to X_0$ which is of finite presentation by assumption).", "After increasing $i$ the two compositions", "$$", "V_i \\times_{Y_i} V_i \\to V_i \\to Z_0", "$$", "are equal as this is true in the limit. Hence we obtain a (essentially unique)", "morphism $Y_i \\to Z_0$. Since this is a morphism over $X_0$", "it induces a morphism into $Z_i = Z_0 \\times_{X_0} X_i$ as desired." ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-limits-lemma-directed-inverse-system-has-limit", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-etale", "limits-lemma-descend-monomorphism", "spaces-theorem-presentation", "spaces-limits-proposition-characterize-locally-finite-presentation" ], "ref_ids": [ 11832, 4565, 15077, 15077, 15065, 15068, 8124, 4655 ] } ], "ref_ids": [] }, { "id": 4599, "type": "theorem", "label": "spaces-limits-lemma-descend-modules-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-modules-finite-presentation", "contents": [ "With notation and assumptions as in", "Lemma \\ref{lemma-descend-finite-presentation}.", "The category of $\\mathcal{O}_X$-modules of finite presentation is the", "colimit over $I$ of the categories $\\mathcal{O}_{X_i}$-modules of finite", "presentation." ], "refs": [ "spaces-limits-lemma-descend-finite-presentation" ], "proofs": [ { "contents": [ "Choose $0 \\in I$. Choose an affine scheme $U_0$ and a surjective", "\\'etale morphism $U_0 \\to X_0$. Set $U_i = X_i \\times_{X_0} U_0$.", "Set $R_0 = U_0 \\times_{X_0} U_0$ and $R_i = R_0 \\times_{X_0} X_i$.", "Denote $s_i, t_i : R_i \\to U_i$ and $s, t : R \\to U$ the two", "projections. In the proof of", "Lemma \\ref{lemma-directed-inverse-system-has-limit} we have", "seen that there exists a presentation $X = U/R$ with", "$U = \\lim U_i$ and $R = \\lim R_i$. Note that $U_i$ and $U$ are affine and", "that $R_i$ and $R$ are quasi-compact and separated (as $X_i$ is", "quasi-separated). Moreover, it is also true that", "$R \\times_{s, U, t} R = \\colim R_i \\times_{s_i, U_i, t_i} R_i$.", "Thus we know that", "$\\QCoh(\\mathcal{O}_U) = \\colim \\QCoh(\\mathcal{O}_{U_i})$,", "$\\QCoh(\\mathcal{O}_R) = \\colim \\QCoh(\\mathcal{O}_{R_i})$,", "and", "$\\QCoh(\\mathcal{O}_{R \\times_{s, U, t} R}) =", "\\colim \\QCoh(\\mathcal{O}_{R_i \\times_{s_i, U_i, t_i} R_i})$ by", "Limits, Lemma \\ref{limits-lemma-descend-modules-finite-presentation}.", "We have $\\QCoh(\\mathcal{O}_X) = \\QCoh(U, R, s, t, c)$ and", "$\\QCoh(\\mathcal{O}_{X_i}) = \\QCoh(U_i, R_i, s_i, t_i, c_i)$,", "see Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-quasi-coherent}.", "Thus the result follows formally." ], "refs": [ "spaces-limits-lemma-directed-inverse-system-has-limit", "limits-lemma-descend-modules-finite-presentation", "spaces-properties-proposition-quasi-coherent" ], "ref_ids": [ 4565, 15078, 11920 ] } ], "ref_ids": [ 4598 ] }, { "id": 4600, "type": "theorem", "label": "spaces-limits-lemma-descend-invertible-modules", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-descend-invertible-modules", "contents": [ "With notation and assumptions as in", "Lemma \\ref{lemma-descend-finite-presentation}.", "Then any invertible $\\mathcal{O}_X$-module is the pullback of an invertible", "$\\mathcal{O}_{X_i}$-module for some $i$." ], "refs": [ "spaces-limits-lemma-descend-finite-presentation" ], "proofs": [ { "contents": [ "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Since", "invertible modules are of finite presentation we can find an $i$", "and modules $\\mathcal{L}_i$ and $\\mathcal{N}_i$ of finite presentation", "over $X_i$ such that $f_i^*\\mathcal{L}_i \\cong \\mathcal{L}$ and", "$f_i^*\\mathcal{N}_i \\cong \\mathcal{L}^{\\otimes -1}$, see", "Lemma \\ref{lemma-descend-modules-finite-presentation}.", "Since pullback commutes with tensor product we see that", "$f_i^*(\\mathcal{L}_i \\otimes_{\\mathcal{O}_{X_i}} \\mathcal{N}_i)$", "is isomorphic to $\\mathcal{O}_X$. Since the tensor product of", "finitely presented modules is finitely presented, the same", "lemma implies that", "$f_{i'i}^*\\mathcal{L}_i", "\\otimes_{\\mathcal{O}_{X_{i'}}} f_{i'i}^*\\mathcal{N}_i$", "is isomorphic to $\\mathcal{O}_{X_{i'}}$ for some $i' \\geq i$.", "It follows that $f_{i'i}^*\\mathcal{L}_i$ is invertible", "(Modules on Sites, Lemma \\ref{sites-modules-lemma-invertible})", "and the proof is complete." ], "refs": [ "spaces-limits-lemma-descend-modules-finite-presentation", "sites-modules-lemma-invertible" ], "ref_ids": [ 4599, 14224 ] } ], "ref_ids": [ 4598 ] }, { "id": 4601, "type": "theorem", "label": "spaces-limits-lemma-colimit-finitely-presented", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-colimit-finitely-presented", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic", "space over $S$. Every quasi-coherent $\\mathcal{O}_X$-module is a", "filtered colimit of finitely presented $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "We may view as an algebraic space over $\\Spec(\\mathbf{Z})$, see", "Spaces, Definition \\ref{spaces-definition-base-change} and", "Properties of Spaces, Definition \\ref{spaces-properties-definition-separated}.", "Thus we may apply Proposition \\ref{proposition-approximate}", "and write $X = \\lim X_i$ with $X_i$ of finite presentation over $\\mathbf{Z}$.", "Thus $X_i$ is a Noetherian algebraic space, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-noetherian}.", "The morphism $X \\to X_i$ is affine, see", "Lemma \\ref{lemma-directed-inverse-system-has-limit}.", "Conclusion by", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-direct-colimit-finite-presentation}." ], "refs": [ "spaces-definition-base-change", "spaces-properties-definition-separated", "spaces-limits-proposition-approximate", "spaces-morphisms-lemma-finite-presentation-noetherian", "spaces-limits-lemma-directed-inverse-system-has-limit", "spaces-cohomology-lemma-direct-colimit-finite-presentation" ], "ref_ids": [ 8183, 11922, 4656, 4843, 4565, 11316 ] } ], "ref_ids": [] }, { "id": 4602, "type": "theorem", "label": "spaces-limits-lemma-directed-colimit-finite-type", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-directed-colimit-finite-type", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then $\\mathcal{F}$ is the directed colimit of its finite type", "quasi-coherent submodules." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{G}, \\mathcal{H} \\subset \\mathcal{F}$ are finite type", "quasi-coherent $\\mathcal{O}_X$-submodules then the image", "of $\\mathcal{G} \\oplus \\mathcal{H} \\to \\mathcal{F}$ is another", "finite type quasi-coherent $\\mathcal{O}_X$-submodule which contains", "both of them. In this way we see that the system is directed.", "To show that $\\mathcal{F}$ is the colimit of this system, write", "$\\mathcal{F} = \\colim_i \\mathcal{F}_i$ as a directed", "colimit of finitely presented quasi-coherent sheaves as in", "Lemma \\ref{lemma-colimit-finitely-presented}.", "Then the images $\\mathcal{G}_i = \\Im(\\mathcal{F}_i \\to \\mathcal{F})$ are", "finite type quasi-coherent subsheaves of $\\mathcal{F}$. Since", "$\\mathcal{F}$ is the colimit of these the result follows." ], "refs": [ "spaces-limits-lemma-colimit-finitely-presented" ], "ref_ids": [ 4601 ] } ], "ref_ids": [] }, { "id": 4603, "type": "theorem", "label": "spaces-limits-lemma-finite-directed-colimit-surjective-maps", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-finite-directed-colimit-surjective-maps", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $\\mathcal{F}$ be a finite type", "quasi-coherent $\\mathcal{O}_X$-module. Then we can write", "$\\mathcal{F} = \\lim \\mathcal{F}_i$ where each $\\mathcal{F}_i$ is an", "$\\mathcal{O}_X$-module of finite presentation and all transition maps", "$\\mathcal{F}_i \\to \\mathcal{F}_{i'}$ surjective." ], "refs": [], "proofs": [ { "contents": [ "Write $\\mathcal{F} = \\colim \\mathcal{G}_i$ as a filtered colimit of", "finitely presented $\\mathcal{O}_X$-modules", "(Lemma \\ref{lemma-colimit-finitely-presented}).", "We claim that $\\mathcal{G}_i \\to \\mathcal{F}$ is surjective for some $i$.", "Namely, choose an \\'etale surjection $U \\to X$ where $U$ is an affine scheme.", "Choose finitely many sections $s_k \\in \\mathcal{F}(U)$ generating", "$\\mathcal{F}|_U$. Since $U$ is affine we see that $s_k$ is in the image", "of $\\mathcal{G}_i \\to \\mathcal{F}$ for $i$ large enough. Hence", "$\\mathcal{G}_i \\to \\mathcal{F}$ is surjective for $i$ large enough.", "Choose such an $i$ and let $\\mathcal{K} \\subset \\mathcal{G}_i$ be the", "kernel of the map $\\mathcal{G}_i \\to \\mathcal{F}$. Write", "$\\mathcal{K} = \\colim \\mathcal{K}_a$", "as the filtered colimit of its finite type quasi-coherent submodules", "(Lemma \\ref{lemma-directed-colimit-finite-type}). Then", "$\\mathcal{F} = \\colim \\mathcal{G}_i/\\mathcal{K}_a$ is a solution", "to the problem posed by the lemma." ], "refs": [ "spaces-limits-lemma-colimit-finitely-presented", "spaces-limits-lemma-directed-colimit-finite-type" ], "ref_ids": [ 4601, 4602 ] } ], "ref_ids": [] }, { "id": 4604, "type": "theorem", "label": "spaces-limits-lemma-algebra-directed-colimit-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-algebra-directed-colimit-finite-presentation", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$.", "Let $\\mathcal{A}$ be a quasi-coherent $\\mathcal{O}_X$-algebra.", "Then $\\mathcal{A}$ is a directed colimit of finitely presented", "quasi-coherent $\\mathcal{O}_X$-algebras." ], "refs": [], "proofs": [ { "contents": [ "First we write $\\mathcal{A} = \\colim_i \\mathcal{F}_i$ as a directed", "colimit of finitely presented quasi-coherent sheaves as in", "Lemma \\ref{lemma-colimit-finitely-presented}.", "For each $i$ let $\\mathcal{B}_i = \\text{Sym}(\\mathcal{F}_i)$ be the", "symmetric algebra on $\\mathcal{F}_i$ over $\\mathcal{O}_X$. Write", "$\\mathcal{I}_i = \\Ker(\\mathcal{B}_i \\to \\mathcal{A})$. Write", "$\\mathcal{I}_i = \\colim_j \\mathcal{F}_{i, j}$ where", "$\\mathcal{F}_{i, j}$ is a finite type quasi-coherent submodule of", "$\\mathcal{I}_i$, see", "Lemma \\ref{lemma-directed-colimit-finite-type}.", "Set $\\mathcal{I}_{i, j} \\subset \\mathcal{I}_i$", "equal to the $\\mathcal{B}_i$-ideal generated by $\\mathcal{F}_{i, j}$.", "Set $\\mathcal{A}_{i, j} = \\mathcal{B}_i/\\mathcal{I}_{i, j}$.", "Then $\\mathcal{A}_{i, j}$ is a quasi-coherent finitely presented", "$\\mathcal{O}_X$-algebra. Define $(i, j) \\leq (i', j')$ if", "$i \\leq i'$ and the map $\\mathcal{B}_i \\to \\mathcal{B}_{i'}$", "maps the ideal $\\mathcal{I}_{i, j}$ into the ideal $\\mathcal{I}_{i', j'}$.", "Then it is clear that $\\mathcal{A} = \\colim_{i, j} \\mathcal{A}_{i, j}$." ], "refs": [ "spaces-limits-lemma-colimit-finitely-presented", "spaces-limits-lemma-directed-colimit-finite-type" ], "ref_ids": [ 4601, 4602 ] } ], "ref_ids": [] }, { "id": 4605, "type": "theorem", "label": "spaces-limits-lemma-algebra-directed-colimit-finite-type", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-algebra-directed-colimit-finite-type", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic", "space over $S$. Let $\\mathcal{A}$ be a quasi-coherent $\\mathcal{O}_X$-algebra.", "Then $\\mathcal{A}$ is the directed colimit of its finite type", "quasi-coherent $\\mathcal{O}_X$-subalgebras." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Compare with the proof of", "Lemma \\ref{lemma-directed-colimit-finite-type}." ], "refs": [ "spaces-limits-lemma-directed-colimit-finite-type" ], "ref_ids": [ 4602 ] } ], "ref_ids": [] }, { "id": 4606, "type": "theorem", "label": "spaces-limits-lemma-finite-algebra-directed-colimit-finite-finitely-presented", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-finite-algebra-directed-colimit-finite-finitely-presented", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $\\mathcal{A}$ be a finite quasi-coherent", "$\\mathcal{O}_X$-algebra. Then $\\mathcal{A} = \\colim \\mathcal{A}_i$", "is a directed colimit of finite and finitely presented quasi-coherent", "$\\mathcal{O}_X$-algebras with surjective transition maps." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-directed-colimit-surjective-maps}", "there exists a finitely presented $\\mathcal{O}_X$-module", "$\\mathcal{F}$ and a surjection $\\mathcal{F} \\to \\mathcal{A}$.", "Using the algebra structure we obtain a surjection", "$$", "\\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F}) \\longrightarrow \\mathcal{A}", "$$", "Denote $\\mathcal{J}$ the kernel. Write $\\mathcal{J} = \\colim \\mathcal{E}_i$", "as a filtered colimit of finite type $\\mathcal{O}_X$-submodules", "$\\mathcal{E}_i$ (Lemma \\ref{lemma-directed-colimit-finite-type}). Set", "$$", "\\mathcal{A}_i = \\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F})/(\\mathcal{E}_i)", "$$", "where $(\\mathcal{E}_i)$ indicates the ideal sheaf generated by", "the image of $\\mathcal{E}_i \\to \\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F})$.", "Then each $\\mathcal{A}_i$ is a finitely presented $\\mathcal{O}_X$-algebra,", "the transition maps are surjective, and $\\mathcal{A} = \\colim \\mathcal{A}_i$.", "To finish the proof we still", "have to show that $\\mathcal{A}_i$ is a finite $\\mathcal{O}_X$-algebra", "for $i$ sufficiently large. To do this we choose an \\'etale surjective", "map $U \\to X$ where $U$ is an affine scheme. Take generators", "$f_1, \\ldots, f_m \\in \\Gamma(U, \\mathcal{F})$.", "As $\\mathcal{A}(U)$ is a finite $\\mathcal{O}_X(U)$-algebra we", "see that for each $j$ there exists a monic polynomial", "$P_j \\in \\mathcal{O}(U)[T]$ such that $P_j(f_j)$ is zero in $\\mathcal{A}(U)$.", "Since $\\mathcal{A} = \\colim \\mathcal{A}_i$ by construction, we", "have $P_j(f_j) = 0$ in $\\mathcal{A}_i(U)$ for all sufficiently large $i$.", "For such $i$ the algebras $\\mathcal{A}_i$ are finite." ], "refs": [ "spaces-limits-lemma-finite-directed-colimit-surjective-maps", "spaces-limits-lemma-directed-colimit-finite-type" ], "ref_ids": [ 4603, 4602 ] } ], "ref_ids": [] }, { "id": 4607, "type": "theorem", "label": "spaces-limits-lemma-integral-algebra-directed-colimit-finite", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-integral-algebra-directed-colimit-finite", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $\\mathcal{A}$ be an integral quasi-coherent", "$\\mathcal{O}_X$-algebra. Then", "\\begin{enumerate}", "\\item $\\mathcal{A}$ is the directed colimit of its finite", "quasi-coherent $\\mathcal{O}_X$-subalgebras, and", "\\item $\\mathcal{A}$ is a directed colimit of finite and finitely presented", "$\\mathcal{O}_X$-algebras.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-algebra-directed-colimit-finite-type} we have", "$\\mathcal{A} = \\colim \\mathcal{A}_i$ where", "$\\mathcal{A}_i \\subset \\mathcal{A}$ runs through the", "quasi-coherent $\\mathcal{O}_X$-sub algebras of finite type.", "Any finite type quasi-coherent $\\mathcal{O}_X$-subalgebra", "of $\\mathcal{A}$ is finite (use Algebra, Lemma", "\\ref{algebra-lemma-characterize-finite-in-terms-of-integral}", "on affine schemes \\'etale over $X$). This proves (1).", "\\medskip\\noindent", "To prove (2), write $\\mathcal{A} = \\colim \\mathcal{F}_i$", "as a colimit of finitely presented $\\mathcal{O}_X$-modules using", "Lemma \\ref{lemma-colimit-finitely-presented}.", "For each $i$, let $\\mathcal{J}_i$ be the kernel of the map", "$$", "\\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F}_i) \\longrightarrow \\mathcal{A}", "$$", "For $i' \\geq i$ there is an induced map $\\mathcal{J}_i \\to \\mathcal{J}_{i'}$", "and we have $\\mathcal{A} =", "\\colim \\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F}_i)/\\mathcal{J}_i$.", "Moreover, the quasi-coherent $\\mathcal{O}_X$-algebras", "$\\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F}_i)/\\mathcal{J}_i$", "are finite (see above). Write $\\mathcal{J}_i = \\colim \\mathcal{E}_{ik}$", "as a colimit of finitely presented $\\mathcal{O}_X$-modules.", "Given $i' \\geq i$ and $k$ there exists a $k'$ such that we", "have a map $\\mathcal{E}_{ik} \\to \\mathcal{E}_{i'k'}$", "making", "$$", "\\xymatrix{", "\\mathcal{J}_i \\ar[r] & \\mathcal{J}_{i'} \\\\", "\\mathcal{E}_{ik} \\ar[u] \\ar[r] & \\mathcal{E}_{i'k'} \\ar[u]", "}", "$$", "commute. This follows from", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-finite-presentation-quasi-compact-colimit}.", "This induces a map", "$$", "\\mathcal{A}_{ik} =", "\\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F}_i)/(\\mathcal{E}_{ik})", "\\longrightarrow", "\\text{Sym}^*_{\\mathcal{O}_X}(\\mathcal{F}_{i'})/(\\mathcal{E}_{i'k'}) =", "\\mathcal{A}_{i'k'}", "$$", "where $(\\mathcal{E}_{ik})$ denotes the ideal generated by $\\mathcal{E}_{ik}$.", "The quasi-coherent $\\mathcal{O}_X$-algebras $\\mathcal{A}_{ki}$", "are of finite presentation and finite for $k$ large enough", "(see proof of", "Lemma \\ref{lemma-finite-algebra-directed-colimit-finite-finitely-presented}).", "Finally, we have", "$$", "\\colim \\mathcal{A}_{ik} = \\colim \\mathcal{A}_i = \\mathcal{A}", "$$", "Namely, the first equality was shown in the proof of", "Lemma \\ref{lemma-finite-algebra-directed-colimit-finite-finitely-presented}", "and the second equality because $\\mathcal{A}$ is the colimit of", "the modules $\\mathcal{F}_i$." ], "refs": [ "spaces-limits-lemma-algebra-directed-colimit-finite-type", "algebra-lemma-characterize-finite-in-terms-of-integral", "spaces-limits-lemma-colimit-finitely-presented", "spaces-cohomology-lemma-finite-presentation-quasi-compact-colimit", "spaces-limits-lemma-finite-algebra-directed-colimit-finite-finitely-presented", "spaces-limits-lemma-finite-algebra-directed-colimit-finite-finitely-presented" ], "ref_ids": [ 4605, 484, 4601, 11279, 4606, 4606 ] } ], "ref_ids": [] }, { "id": 4608, "type": "theorem", "label": "spaces-limits-lemma-extend", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-extend", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $U \\subset X$ be a quasi-compact open.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\mathcal{G} \\subset \\mathcal{F}|_U$ be a quasi-coherent", "$\\mathcal{O}_U$-submodule which is of finite type. Then", "there exists a quasi-coherent submodule $\\mathcal{G}' \\subset \\mathcal{F}$", "which is of finite type such that $\\mathcal{G}'|_U = \\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Denote $j : U \\to X$ the inclusion morphism. As $X$ is quasi-separated", "and $U$ quasi-compact, the morphism $j$ is quasi-compact. Hence", "$j_*\\mathcal{G} \\subset j_*\\mathcal{F}|_U$ are quasi-coherent modules", "on $X$ (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-pushforward}).", "Let $\\mathcal{H} =", "\\Ker(j_*\\mathcal{G} \\oplus \\mathcal{F} \\to j_*\\mathcal{F}|_U)$.", "Then $\\mathcal{H}|_U = \\mathcal{G}$. By", "Lemma \\ref{lemma-directed-colimit-finite-type}", "we can find a finite type quasi-coherent submodule", "$\\mathcal{H}' \\subset \\mathcal{H}$ such that", "$\\mathcal{H}'|_U = \\mathcal{H}|_U = \\mathcal{G}$.", "Set $\\mathcal{G}' = \\Im(\\mathcal{H}' \\to \\mathcal{F})$", "to conclude." ], "refs": [ "spaces-morphisms-lemma-pushforward", "spaces-limits-lemma-directed-colimit-finite-type" ], "ref_ids": [ 4760, 4602 ] } ], "ref_ids": [] }, { "id": 4609, "type": "theorem", "label": "spaces-limits-lemma-relative-approximation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-relative-approximation", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume that", "\\begin{enumerate}", "\\item $X$ is quasi-compact and quasi-separated, and", "\\item $Y$ is quasi-separated.", "\\end{enumerate}", "Then $X = \\lim X_i$ is a limit of a directed inverse system of algebraic spaces", "$X_i$ of finite presentation over $Y$ with affine transition morphisms", "over $Y$." ], "refs": [], "proofs": [ { "contents": [ "Since $|f|(|X|)$ is quasi-compact we may replace $Y$ by a quasi-compact", "open subspace whose set of points contains $|f|(|X|)$. Hence we may assume", "$Y$ is quasi-compact as well. Write $X = \\lim X_a$ and $Y = \\lim Y_b$ as in", "Proposition \\ref{proposition-approximate}, i.e., with $X_a$ and $Y_b$", "of finite type over $\\mathbf{Z}$ and with affine transition morphisms.", "By Proposition \\ref{proposition-characterize-locally-finite-presentation}", "we find that for each $b$ there exists an $a$ and a morphism", "$f_{a, b} : X_a \\to Y_b$ making the diagram", "$$", "\\xymatrix{", "X \\ar[d] \\ar[r] & Y \\ar[d] \\\\", "X_a \\ar[r] & Y_b", "}", "$$", "commute. Moreover the same proposition implies that, given a second", "triple $(a', b', f_{a', b'})$, there exists an $a'' \\geq a'$ such that", "the compositions $X_{a''} \\to X_a \\to X_b$ and", "$X_{a''} \\to X_{a'} \\to X_{b'} \\to X_b$ are equal.", "Consider the set of triples $(a, b, f_{a, b})$ endowed with the preordering", "$$", "(a, b, f_{a, b}) \\geq (a', b', f_{a', b'})", "\\Leftrightarrow", "a \\geq a',\\ b' \\geq b,\\text{ and }", "f_{a', b'} \\circ h_{a, a'} = g_{b', b} \\circ f_{a, b}", "$$", "where $h_{a, a'} : X_a \\to X_{a'}$ and $g_{b', b} : Y_{b'} \\to Y_b$", "are the transition morphisms. The remarks above show that this system", "is directed. It follows formally from the equalities", "$X = \\lim X_a$ and $Y = \\lim Y_b$ that", "$$", "X = \\lim_{(a, b, f_{a, b})} X_a \\times_{f_{a, b}, Y_b} Y.", "$$", "where the limit is over our directed system above. The transition morphisms", "$X_a \\times_{Y_b} Y \\to X_{a'} \\times_{Y_{b'}} Y$ are affine as", "the composition", "$$", "X_a \\times_{Y_b} Y \\to X_a \\times_{Y_{b'}} Y \\to X_{a'} \\times_{Y_{b'}} Y", "$$", "where the first morphism is a closed immersion (by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-fibre-product-after-map})", "and the second is a base change of an affine morphism", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-affine})", "and the composition of affine morphisms is affine", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-composition-affine}).", "The morphisms $f_{a, b}$ are of finite presentation", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation} and", "\\ref{spaces-morphisms-lemma-finite-presentation-permanence})", "and hence the base changes $X_a \\times_{f_{a, b}, S_b} S \\to S$", "are of finite presentation", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-finite-presentation})." ], "refs": [ "spaces-limits-proposition-approximate", "spaces-limits-proposition-characterize-locally-finite-presentation", "spaces-morphisms-lemma-fibre-product-after-map", "spaces-morphisms-lemma-base-change-affine", "spaces-morphisms-lemma-composition-affine", "spaces-morphisms-lemma-noetherian-finite-type-finite-presentation", "spaces-morphisms-lemma-finite-presentation-permanence", "spaces-morphisms-lemma-base-change-finite-presentation" ], "ref_ids": [ 4656, 4655, 4715, 4800, 4799, 4844, 4846, 4840 ] } ], "ref_ids": [] }, { "id": 4610, "type": "theorem", "label": "spaces-limits-lemma-affine-morphism-is-limit", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-affine-morphism-is-limit", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic", "spaces over $S$. If $Y$ quasi-compact and", "quasi-separated, then $X$ is a directed limit $X = \\lim X_i$", "with each $X_i$ affine and of finite presentation over $Y$." ], "refs": [], "proofs": [ { "contents": [ "Consider the quasi-coherent $\\mathcal{O}_Y$-module", "$\\mathcal{A} = f_*\\mathcal{O}_X$. By", "Lemma \\ref{lemma-algebra-directed-colimit-finite-presentation}", "we can write $\\mathcal{A} = \\colim \\mathcal{A}_i$ as a directed", "colimit of finitely presented", "$\\mathcal{O}_Y$-algebras $\\mathcal{A}_i$.", "Set $X_i = \\underline{\\Spec}_Y(\\mathcal{A}_i)$, see", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-relative-spec}.", "By construction $X_i \\to Y$ is affine and of finite presentation", "and $X = \\lim X_i$." ], "refs": [ "spaces-limits-lemma-algebra-directed-colimit-finite-presentation", "spaces-morphisms-definition-relative-spec" ], "ref_ids": [ 4604, 4999 ] } ], "ref_ids": [] }, { "id": 4611, "type": "theorem", "label": "spaces-limits-lemma-integral-limit-finite-and-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-integral-limit-finite-and-finite-presentation", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an integral morphism of algebraic", "spaces over $S$. Assume $Y$ quasi-compact and quasi-separated.", "Then $X$ can be written as a directed limit $X = \\lim X_i$", "where $X_i$ are finite and of finite presentation over $Y$." ], "refs": [], "proofs": [ { "contents": [ "Consider the finite quasi-coherent $\\mathcal{O}_Y$-module", "$\\mathcal{A} = f_*\\mathcal{O}_X$. By", "Lemma \\ref{lemma-integral-algebra-directed-colimit-finite}", "we can write $\\mathcal{A} = \\colim \\mathcal{A}_i$ as a directed", "colimit of finite and finitely presented $\\mathcal{O}_Y$-algebras", "$\\mathcal{A}_i$.", "Set $X_i = \\underline{\\Spec}_Y(\\mathcal{A}_i)$, see", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-relative-spec}.", "By construction $X_i \\to Y$ is finite and of finite presentation and", "$X = \\lim X_i$." ], "refs": [ "spaces-limits-lemma-integral-algebra-directed-colimit-finite", "spaces-morphisms-definition-relative-spec" ], "ref_ids": [ 4607, 4999 ] } ], "ref_ids": [] }, { "id": 4612, "type": "theorem", "label": "spaces-limits-lemma-finite-in-finite-and-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-finite-in-finite-and-finite-presentation", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a finite morphism of algebraic", "spaces over $S$. Assume $Y$ quasi-compact and quasi-separated.", "Then $X$ can be written as a directed limit $X = \\lim X_i$", "where the transition maps are closed immersions and the objects", "$X_i$ are finite and of finite presentation over $Y$." ], "refs": [], "proofs": [ { "contents": [ "Consider the finite quasi-coherent $\\mathcal{O}_Y$-module", "$\\mathcal{A} = f_*\\mathcal{O}_X$. By", "Lemma \\ref{lemma-finite-algebra-directed-colimit-finite-finitely-presented}", "we can write $\\mathcal{A} = \\colim \\mathcal{A}_i$ as a directed", "colimit of finite and finitely presented $\\mathcal{O}_Y$-algebras", "$\\mathcal{A}_i$ with surjective transition maps.", "Set $X_i = \\underline{\\Spec}_Y(\\mathcal{A}_i)$, see", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-relative-spec}.", "By construction $X_i \\to Y$ is finite and of finite presentation,", "the transition maps are closed immersions, and $X = \\lim X_i$." ], "refs": [ "spaces-limits-lemma-finite-algebra-directed-colimit-finite-finitely-presented", "spaces-morphisms-definition-relative-spec" ], "ref_ids": [ 4606, 4999 ] } ], "ref_ids": [] }, { "id": 4613, "type": "theorem", "label": "spaces-limits-lemma-closed-is-limit-closed-and-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-closed-is-limit-closed-and-finite-presentation", "contents": [ "\\begin{slogan}", "Closed immersions of qcqs algebraic spaces can be approximated", "by finitely presented closed immersions.", "\\end{slogan}", "Let $S$ be a scheme. Let $f : X \\to Y$ be a closed immersion of algebraic", "spaces over $S$. Assume $Y$ quasi-compact and quasi-separated.", "Then $X$ can be written as a directed limit $X = \\lim X_i$", "where the transition maps are closed immersions and the morphisms", "$X_i \\to Y$ are closed immersions of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\subset \\mathcal{O}_Y$ be the quasi-coherent sheaf", "of ideals defining $X$ as a closed subspace of $Y$. By", "Lemma \\ref{lemma-directed-colimit-finite-type}", "we can write $\\mathcal{I} = \\colim \\mathcal{I}_i$ as the", "filtered colimit of its finite type quasi-coherent submodules.", "Let $X_i$ be the closed subspace of $X$ cut out by $\\mathcal{I}_i$.", "Then $X_i \\to Y$ is a closed immersion of finite presentation,", "and $X = \\lim X_i$. Some details omitted." ], "refs": [ "spaces-limits-lemma-directed-colimit-finite-type" ], "ref_ids": [ 4602 ] } ], "ref_ids": [] }, { "id": 4614, "type": "theorem", "label": "spaces-limits-lemma-quasi-affine-closed-in-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-quasi-affine-closed-in-finite-presentation", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume", "\\begin{enumerate}", "\\item $f$ is locally of finite type and quasi-affine, and", "\\item $Y$ is quasi-compact and quasi-separated.", "\\end{enumerate}", "Then there exists a morphism of finite presentation", "$f' : X' \\to Y$ and a closed immersion $X \\to X'$ over $Y$." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-characterize-quasi-affine}", "we can find a factorization $X \\to Z \\to Y$ where", "$X \\to Z$ is a quasi-compact open immersion and", "$Z \\to Y$ is affine. Write $Z = \\lim Z_i$ with $Z_i$ affine and", "of finite presentation over $Y$ (Lemma \\ref{lemma-affine-morphism-is-limit}).", "For some $0 \\in I$ we can find a quasi-compact open $U_0 \\subset Z_0$", "such that $X$ is isomorphic to the inverse image of $U_0$ in $Z$", "(Lemma \\ref{lemma-descend-opens}). Let $U_i$ be the inverse image of", "$U_0$ in $Z_i$, so $U = \\lim U_i$. By", "Lemma \\ref{lemma-finite-type-eventually-closed}", "we see that $X \\to U_i$ is a closed immersion for some $i$ large enough.", "Setting $X' = U_i$ finishes the proof." ], "refs": [ "spaces-morphisms-lemma-characterize-quasi-affine", "spaces-limits-lemma-affine-morphism-is-limit", "spaces-limits-lemma-descend-opens", "spaces-limits-lemma-finite-type-eventually-closed" ], "ref_ids": [ 4810, 4610, 4575, 4580 ] } ], "ref_ids": [] }, { "id": 4615, "type": "theorem", "label": "spaces-limits-lemma-finite-type-closed-in-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-finite-type-closed-in-finite-presentation", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume:", "\\begin{enumerate}", "\\item $f$ is of locally of finite type.", "\\item $X$ is quasi-compact and quasi-separated, and", "\\item $Y$ is quasi-compact and quasi-separated.", "\\end{enumerate}", "Then there exists a morphism of finite presentation", "$f' : X' \\to Y$ and a closed immersion $X \\to X'$ of", "algebraic spaces over $Y$." ], "refs": [], "proofs": [ { "contents": [ "By Proposition \\ref{proposition-approximate}", "we can write $X = \\lim_i X_i$ with $X_i$ quasi-separated of finite type over", "$\\mathbf{Z}$ and with transition morphisms $f_{ii'} : X_i \\to X_{i'}$ affine.", "Consider the commutative diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[rd] & X_{i, Y} \\ar[r] \\ar[d] & X_i \\ar[d] \\\\", "& Y \\ar[r] & \\Spec(\\mathbf{Z})", "}", "$$", "Note that $X_i$ is of finite presentation over $\\Spec(\\mathbf{Z})$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation}.", "Hence the base change $X_{i, Y} \\to Y$ is of finite presentation by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-finite-presentation}.", "Observe that $\\lim X_{i, Y} = X \\times Y$ and that $X \\to X \\times Y$ is a", "monomorphism. By Lemma \\ref{lemma-finite-type-eventually-closed}", "we see that $X \\to X_{i, Y}$ is a monomorphism for $i$ large enough.", "Fix such an $i$. Note that $X \\to X_{i, Y}$ is locally of finite type", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-permanence-finite-type})", "and a monomorphism, hence separated and locally quasi-finite", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}).", "Hence $X \\to X_{i, Y}$ is representable.", "Hence $X \\to X_{i, Y}$ is quasi-affine because we can use the", "principle Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}", "and the result for morphisms of schemes More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-quasi-finite-separated-quasi-affine}.", "Thus Lemma \\ref{lemma-quasi-affine-closed-in-finite-presentation}", "gives a factorization $X \\to X' \\to X_{i, Y}$", "with $X \\to X'$ a closed immersion and $X' \\to X_{i, Y}$ of finite", "presentation. Finally, $X' \\to Y$ is of finite presentation as a", "composition of morphisms of finite presentation", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-composition-finite-presentation})." ], "refs": [ "spaces-limits-proposition-approximate", "spaces-morphisms-lemma-noetherian-finite-type-finite-presentation", "spaces-morphisms-lemma-base-change-finite-presentation", "spaces-limits-lemma-finite-type-eventually-closed", "spaces-morphisms-lemma-permanence-finite-type", "spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "spaces-lemma-representable-transformations-property-implication", "more-morphisms-lemma-quasi-finite-separated-quasi-affine", "spaces-limits-lemma-quasi-affine-closed-in-finite-presentation", "spaces-morphisms-lemma-composition-finite-presentation" ], "ref_ids": [ 4656, 4844, 4840, 4580, 4818, 4838, 8136, 13900, 4614, 4839 ] } ], "ref_ids": [] }, { "id": 4616, "type": "theorem", "label": "spaces-limits-lemma-proper-limit-of-proper-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-proper-limit-of-proper-finite-presentation", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of algebraic", "spaces over $S$ with $Y$ quasi-compact and quasi-separated. Then", "$X = \\lim X_i$ is a directed limit of algebraic spaces $X_i$", "proper and of finite presentation over $Y$ and with transition", "morphisms and morphisms $X \\to X_i$ closed immersions." ], "refs": [], "proofs": [ { "contents": [ "By Proposition \\ref{proposition-separated-closed-in-finite-presentation}", "we can find a closed immersion $X \\to X'$ with $X'$ separated and of", "finite presentation over $Y$. By", "Lemma \\ref{lemma-closed-is-limit-closed-and-finite-presentation}", "we can write $X = \\lim X_i$ with $X_i \\to X'$ a closed immersion of", "finite presentation. We claim that for all $i$ large enough", "the morphism $X_i \\to Y$ is proper which finishes the proof.", "\\medskip\\noindent", "To prove this we may assume that $Y$ is an affine scheme, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-proper-local}.", "Next, we use the weak version of Chow's lemma, see", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-weak-chow},", "to find a diagram", "$$", "\\xymatrix{", "X' \\ar[rd] & X'' \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_Y \\ar[dl] \\\\", "& Y &", "}", "$$", "where $X'' \\to \\mathbf{P}^n_Y$ is an immersion, and", "$\\pi : X'' \\to X'$ is proper and surjective. Denote", "$X'_i \\subset X''$, resp.\\ $\\pi^{-1}(X)$ the scheme theoretic inverse image of", "$X_i \\subset X'$, resp.\\ $X \\subset X'$.", "Then $\\lim X'_i = \\pi^{-1}(X)$. Since $\\pi^{-1}(X) \\to Y$ is proper", "(Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-composition-proper}),", "we see that $\\pi^{-1}(X) \\to \\mathbf{P}^n_Y$ is a closed immersion", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-universally-closed-permanence} and", "\\ref{spaces-morphisms-lemma-immersion-when-closed}).", "Hence for $i$ large enough", "we find that $X'_i \\to \\mathbf{P}^n_Y$ is a closed immersion by", "Lemma \\ref{lemma-eventually-closed-immersion}.", "Thus $X'_i$ is proper over $Y$.", "For such $i$ the morphism $X_i \\to Y$ is proper by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-image-proper-is-proper}." ], "refs": [ "spaces-limits-proposition-separated-closed-in-finite-presentation", "spaces-limits-lemma-closed-is-limit-closed-and-finite-presentation", "spaces-morphisms-lemma-proper-local", "spaces-cohomology-lemma-weak-chow", "spaces-morphisms-lemma-composition-proper", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-morphisms-lemma-immersion-when-closed", "spaces-limits-lemma-eventually-closed-immersion", "spaces-morphisms-lemma-image-proper-is-proper" ], "ref_ids": [ 4657, 4613, 4916, 11327, 4918, 4920, 4763, 4584, 4921 ] } ], "ref_ids": [] }, { "id": 4617, "type": "theorem", "label": "spaces-limits-lemma-proper-limit-of-proper-finite-presentation-noetherian", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-proper-limit-of-proper-finite-presentation-noetherian", "contents": [ "Let $f : X \\to Y$ be a proper morphism of algebraic spaces over $\\mathbf{Z}$", "with $Y$ quasi-compact and quasi-separated. Then there exists a directed", "set $I$, an inverse system $(f_i : X_i \\to Y_i)$ of morphisms of algebraic", "spaces over $I$, such that the transition morphisms $X_i \\to X_{i'}$", "and $Y_i \\to Y_{i'}$ are affine, such that $f_i$ is proper and of", "finite presentation, such that $Y_i$ is of finite presentation over", "$\\mathbf{Z}$, and such that $(X \\to Y) = \\lim (X_i \\to Y_i)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-proper-limit-of-proper-finite-presentation}", "we can write $X = \\lim_{k \\in K} X_k$ with $X_k \\to Y$ proper and", "of finite presentation. Next, by absolute Noetherian approximation", "(Proposition \\ref{proposition-approximate}) we can", "write $Y = \\lim_{j \\in J} Y_j$ with $Y_j$ of finite presentation", "over $\\mathbf{Z}$.", "For each $k$ there exists a $j$ and a morphism $X_{k, j} \\to Y_j$", "of finite presentation with $X_k \\cong Y \\times_{Y_j} X_{k, j}$", "as algebraic spaces over $Y$, see", "Lemma \\ref{lemma-descend-finite-presentation}.", "After increasing $j$ we may assume $X_{k, j} \\to Y_j$ is proper, see", "Lemma \\ref{lemma-eventually-proper}. The set $I$ will be consist", "of these pairs $(k, j)$ and the corresponding morphism is $X_{k, j} \\to Y_j$.", "For every $k' \\geq k$ we can find a $j' \\geq j$ and a morphism", "$X_{j', k'} \\to X_{j, k}$ over $Y_{j'} \\to Y_j$ whose base change to $Y$", "gives the morphism $X_{k'} \\to X_k$ (follows again from", "Lemma \\ref{lemma-descend-finite-presentation}).", "These morphisms form the transition morphisms of the system. Some details", "omitted." ], "refs": [ "spaces-limits-lemma-proper-limit-of-proper-finite-presentation", "spaces-limits-proposition-approximate", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-eventually-proper", "spaces-limits-lemma-descend-finite-presentation" ], "ref_ids": [ 4616, 4656, 4598, 4596, 4598 ] } ], "ref_ids": [] }, { "id": 4618, "type": "theorem", "label": "spaces-limits-lemma-eventually-proper-support", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-eventually-proper-support", "contents": [ "Assumptions and notation as in Situation \\ref{situation-descent-property}.", "Let $\\mathcal{F}_0$ be a quasi-coherent $\\mathcal{O}_{X_0}$-module.", "Denote $\\mathcal{F}$ and $\\mathcal{F}_i$ the pullbacks of", "$\\mathcal{F}_0$ to $X$ and $X_i$. Assume", "\\begin{enumerate}", "\\item $f_0$ is locally of finite type,", "\\item $\\mathcal{F}_0$ is of finite type,", "\\item the scheme theoretic support of $\\mathcal{F}$ is proper over $Y$.", "\\end{enumerate}", "Then the scheme theoretic support of $\\mathcal{F}_i$ is proper over $Y_i$", "for some $i$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $X_0$ by the scheme theoretic support of $\\mathcal{F}_0$.", "By Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-support-finite-type}", "this guarantees that $X_i$ is the support of $\\mathcal{F}_i$ and $X$ is the", "support of $\\mathcal{F}$. Then, if $Z \\subset X$ denotes the scheme", "theoretic support of $\\mathcal{F}$, we see that $Z \\to X$ is a universal", "homeomorphism. We conclude that $X \\to Y$ is proper as this is true for", "$Z \\to Y$ by assumption, see", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-is-proper}.", "By Lemma \\ref{lemma-eventually-proper} we see that $X_i \\to Y$ is proper", "for some $i$. Then it follows that the scheme theoretic support $Z_i$ of", "$\\mathcal{F}_i$ is proper over $Y$ by", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-closed-immersion-proper} and", "\\ref{spaces-morphisms-lemma-composition-proper}." ], "refs": [ "spaces-morphisms-lemma-support-finite-type", "morphisms-lemma-image-proper-is-proper", "spaces-limits-lemma-eventually-proper", "spaces-morphisms-lemma-closed-immersion-proper", "spaces-morphisms-lemma-composition-proper" ], "ref_ids": [ 4777, 5413, 4596, 4919, 4918 ] } ], "ref_ids": [] }, { "id": 4619, "type": "theorem", "label": "spaces-limits-lemma-embedding-into-affine-over-ls-qs", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-embedding-into-affine-over-ls-qs", "contents": [ "Let $S$ be a scheme. Let $f : U \\to X$ be a morphism of algebraic", "spaces over $S$. Assume $U$ is an affine scheme, $f$ is locally of", "finite type, and $X$ quasi-separated and locally separated.", "Then there exists an immersion $U \\to \\mathbf{A}^n_X$ over $X$." ], "refs": [], "proofs": [ { "contents": [ "Say $U = \\Spec(A)$. Write $A = \\colim A_i$ as a filtered colimit", "of finite type $\\mathbf{Z}$-subalgebras. For each $i$ the morphism", "$U \\to U_i = \\Spec(A_i)$ induces a morphism", "$$", "U \\longrightarrow X \\times U_i", "$$", "over $X$. In the limit the morphism $U \\to X \\times U$ is an immersion", "as $X$ is locally separated, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-semi-diagonal}.", "By Lemma \\ref{lemma-finite-type-eventually-closed}", "we see that $U \\to X \\times U_i$ is an immersion for some $i$.", "Since $U_i$ is isomorphic to a closed subscheme of", "$\\mathbf{A}^n_{\\mathbf{Z}}$ the lemma follows." ], "refs": [ "spaces-morphisms-lemma-semi-diagonal", "spaces-limits-lemma-finite-type-eventually-closed" ], "ref_ids": [ 4716, 4580 ] } ], "ref_ids": [] }, { "id": 4620, "type": "theorem", "label": "spaces-limits-lemma-embedding-into-affine-over-qs", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-embedding-into-affine-over-qs", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic", "spaces over $S$. Assume $X$ Noetherian and $f$ of finite presentation.", "Then there exists a dense open $V \\subset Y$ and an immersion", "$V \\to \\mathbf{A}^n_X$." ], "refs": [], "proofs": [ { "contents": [ "The assumptions imply that $Y$ is Noetherian", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-noetherian}).", "Then $Y$ is quasi-separated, hence has a dense open subscheme", "(Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}).", "Thus we may assume that $Y$ is a Noetherian scheme.", "By removing intersections of irreducible components of $Y$", "(use Topology, Lemma \\ref{topology-lemma-Noetherian} and", "Properties, Lemma \\ref{properties-lemma-Noetherian-topology})", "we may assume that $Y$ is a disjoint union of irreducible", "Noetherian schemes. Since there is an immersion", "$$", "\\mathbf{A}^n_X \\amalg \\mathbf{A}^m_X", "\\longrightarrow", "\\mathbf{A}^{\\max(n, m) + 1}_X", "$$", "(details omitted) we see that it suffices to prove the result in case", "$Y$ is irreducible.", "\\medskip\\noindent", "Assume $Y$ is an irreducible scheme. Let $T \\subset |X|$ be the closure of the", "image of $f : Y \\to X$. Note that since $|Y|$ and $|X|$ are sober topological", "spaces (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-separated-sober})", "$T$ is irreducible with a unique generic point $\\xi$ which is the", "image of the generic point $\\eta$ of $Y$.", "Let $\\mathcal{I} \\subset X$ be a quasi-coherent sheaf of ideals", "cutting out the reduced induced space structure on $T$", "(Properties of Spaces, Definition", "\\ref{spaces-properties-definition-reduced-induced-space}).", "Since $\\mathcal{O}_{Y, \\eta}$ is an Artinian local ring we see", "that for some $n > 0$ we have $f^{-1}\\mathcal{I}^n \\mathcal{O}_{Y, \\eta} = 0$.", "As $f^{-1}\\mathcal{I}\\mathcal{O}_Y$ is a finite type quasi-coherent ideal", "we conclude that $f^{-1}\\mathcal{I}^n\\mathcal{O}_V = 0$ for", "some nonempty open $V \\subset Y$. Let $Z \\subset X$ be the closed subspace", "cut out by $\\mathcal{I}^n$. By construction $V \\to Y \\to X$ factors through", "$Z$. Because $\\mathbf{A}^n_Z \\to \\mathbf{A}^n_X$ is an immersion,", "we may replace $X$ by $Z$ and $Y$ by $V$.", "Hence we reach the situation where $Y$ and $X$ are irreducible and", "$Y \\to X$ maps the generic point of $Y$ onto the generic point of $X$.", "\\medskip\\noindent", "Assume $Y$ and $X$ are irreducible, $Y$ is a scheme,", "and $Y \\to X$ maps the generic point of", "$Y$ onto the generic point of $X$. By Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}", "$X$ has a dense open subscheme $U \\subset X$. Choose a nonempty affine", "open $V \\subset Y$ whose image in $X$ is contained in $U$. By", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-affine-finite-type-over-S}", "we may factor $V \\to U$ as $V \\to \\mathbf{A}^n_U \\to U$. Composing", "with $\\mathbf{A}^n_U \\to \\mathbf{A}^n_X$ we obtain the desired immersion." ], "refs": [ "spaces-morphisms-lemma-finite-presentation-noetherian", "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme", "topology-lemma-Noetherian", "properties-lemma-Noetherian-topology", "spaces-properties-lemma-quasi-separated-sober", "spaces-properties-definition-reduced-induced-space", "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme", "morphisms-lemma-quasi-affine-finite-type-over-S" ], "ref_ids": [ 4843, 11917, 8220, 2954, 11852, 11932, 11917, 5392 ] } ], "ref_ids": [] }, { "id": 4621, "type": "theorem", "label": "spaces-limits-lemma-quasi-coherent-finite-type-ideals", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-quasi-coherent-finite-type-ideals", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space.", "Let $U \\subset X$ be an open subspace. The following are equivalent:", "\\begin{enumerate}", "\\item $U \\to X$ is quasi-compact,", "\\item $U$ is quasi-compact, and", "\\item there exists a finite type quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_X$ such that", "$|X| \\setminus |U| = |V(\\mathcal{I})|$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $W$ be an affine scheme and let $\\varphi : W \\to X$ be a surjective", "\\'etale morphism, see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}.", "If (1) holds, then $\\varphi^{-1}(U) \\to W$ is quasi-compact, hence", "$\\varphi^{-1}(U)$ is quasi-compact, hence $U$ is quasi-compact", "(as $|\\varphi^{-1}(U)| \\to |U|$ is surjective). If (2) holds, then", "$\\varphi^{-1}(U)$ is quasi-compact because $\\varphi$ is quasi-compact", "since $X$ is quasi-separated (Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence}).", "Hence $\\varphi^{-1}(U) \\to W$ is a quasi-compact morphism of schemes by", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-finite-type-ideals}.", "It follows that $U \\to X$ is quasi-compact by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-quasi-compact-local}.", "Thus (1) and (2) are equivalent.", "\\medskip\\noindent", "Assume (1) and (2). By", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-reduced-closed-subspace} there exists", "a unique quasi-coherent sheaf of ideals $\\mathcal{J}$ cutting", "out the reduced induced closed subspace structure on $|X| \\setminus |U|$.", "Note that $\\mathcal{J}|_U = \\mathcal{O}_U$ which is an", "$\\mathcal{O}_U$-modules of finite type.", "As $U$ is quasi-compact it follows from", "Lemma \\ref{lemma-directed-colimit-finite-type}", "that there exists a quasi-coherent subsheaf", "$\\mathcal{I} \\subset \\mathcal{J}$ which is of finite type", "and has the property that $\\mathcal{I}|_U = \\mathcal{J}|_U$.", "Then $|X| \\setminus |U| = |V(\\mathcal{I})|$ and we obtain (3). Conversely,", "if $\\mathcal{I}$ is as in (3), then $\\varphi^{-1}(U) \\subset W$", "is a quasi-compact open by the lemma for schemes", "(Properties, Lemma \\ref{properties-lemma-quasi-coherent-finite-type-ideals})", "applied to $\\varphi^{-1}\\mathcal{I}$ on $W$.", "Thus (2) holds." ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence", "properties-lemma-quasi-coherent-finite-type-ideals", "spaces-morphisms-lemma-quasi-compact-local", "spaces-properties-lemma-reduced-closed-subspace", "spaces-limits-lemma-directed-colimit-finite-type", "properties-lemma-quasi-coherent-finite-type-ideals" ], "ref_ids": [ 11832, 4744, 3033, 4742, 11846, 4602, 3033 ] } ], "ref_ids": [] }, { "id": 4622, "type": "theorem", "label": "spaces-limits-lemma-sections-annihilated-by-ideal", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-sections-annihilated-by-ideal", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Consider the sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}'$", "which associates to every object $U$ of $X_\\etale$ the module", "$$", "\\mathcal{F}'(U)", "=", "\\{s \\in \\mathcal{F}(U) \\mid", "\\mathcal{I}s = 0\\}", "$$", "Assume $\\mathcal{I}$ is of finite type. Then", "\\begin{enumerate}", "\\item $\\mathcal{F}'$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules,", "\\item for affine $U$ in $X_\\etale$ we have", "$\\mathcal{F}'(U) = \\{s \\in \\mathcal{F}(U) \\mid \\mathcal{I}(U)s = 0\\}$, and", "\\item $\\mathcal{F}'_x = \\{s \\in \\mathcal{F}_x \\mid \\mathcal{I}_x s = 0\\}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that the rule defining $\\mathcal{F}'$ gives a subsheaf", "of $\\mathcal{F}$. Hence we may work \\'etale locally on $X$ to verify", "the other statements. Thus the lemma reduces to the case of schemes", "which is", "Properties, Lemma \\ref{properties-lemma-sections-annihilated-by-ideal}." ], "refs": [ "properties-lemma-sections-annihilated-by-ideal" ], "ref_ids": [ 3034 ] } ], "ref_ids": [] }, { "id": 4623, "type": "theorem", "label": "spaces-limits-lemma-push-sections-annihilated-by-ideal", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-push-sections-annihilated-by-ideal", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism", "of algebraic spaces over $S$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_Y$ be a quasi-coherent", "sheaf of ideals of finite type. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. Let $\\mathcal{F}' \\subset \\mathcal{F}$", "be the subsheaf of sections annihilated by $f^{-1}\\mathcal{I}\\mathcal{O}_X$.", "Then $f_*\\mathcal{F}' \\subset f_*\\mathcal{F}$ is the subsheaf", "of sections annihilated by $\\mathcal{I}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: The assumption that $f$ is quasi-compact and", "quasi-separated implies that $f_*\\mathcal{F}$ is quasi-coherent", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-pushforward})", "so that Lemma \\ref{lemma-sections-annihilated-by-ideal} applies", "to $\\mathcal{I}$ and $f_*\\mathcal{F}$." ], "refs": [ "spaces-morphisms-lemma-pushforward", "spaces-limits-lemma-sections-annihilated-by-ideal" ], "ref_ids": [ 4760, 4622 ] } ], "ref_ids": [] }, { "id": 4624, "type": "theorem", "label": "spaces-limits-lemma-sections-supported-on-closed-subset", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-sections-supported-on-closed-subset", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $T \\subset |X|$ be a closed subset and let $U \\subset X$ be", "the open subspace such that $T \\amalg |U| = |X|$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Consider the sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}'$", "which associates to every object $\\varphi : W \\to X$ of", "$X_\\etale$ the module", "$$", "\\mathcal{F}'(W)", "=", "\\{s \\in \\mathcal{F}(W) \\mid", "\\text{the support of }s\\text{ is contained in }|\\varphi|^{-1}(T)\\}", "$$", "If $U \\to X$ is quasi-compact, then", "\\begin{enumerate}", "\\item for $W$ affine there exist a finitely generated", "ideal $I \\subset \\mathcal{O}_X(W)$ such that $|\\varphi|^{-1}(T) = V(I)$,", "\\item for $W$ and $I$ as in (1) we have", "$\\mathcal{F}'(W) = \\{x \\in \\mathcal{F}(W) \\mid", "I^nx = 0 \\text{ for some } n\\}$,", "\\item $\\mathcal{F}'$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that the rule defining $\\mathcal{F}'$ gives a subsheaf", "of $\\mathcal{F}$. Hence we may work \\'etale locally on $X$ to verify", "the other statements. Thus the lemma reduces to the case of schemes", "which is", "Properties, Lemma \\ref{properties-lemma-sections-supported-on-closed-subset}." ], "refs": [ "properties-lemma-sections-supported-on-closed-subset" ], "ref_ids": [ 3036 ] } ], "ref_ids": [] }, { "id": 4625, "type": "theorem", "label": "spaces-limits-lemma-push-sections-supported-on-closed-subset", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-push-sections-supported-on-closed-subset", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism", "of algebraic spaces over $S$. Let $T \\subset |Y|$ be a closed subset.", "Assume $|Y| \\setminus T$ corresponds to an open subspace $V \\subset Y$", "such that $V \\to Y$ is quasi-compact. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. Let $\\mathcal{F}' \\subset \\mathcal{F}$", "be the subsheaf of sections supported on $|f|^{-1}T$.", "Then $f_*\\mathcal{F}' \\subset f_*\\mathcal{F}$ is the subsheaf", "of sections supported on $T$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints: $|X| \\setminus |f|^{-1}T$ is the support of the open subspace", "$U = f^{-1}V \\subset X$. Since $V \\to Y$ is quasi-compact, so is", "$U \\to X$ (by base change). The assumption that $f$ is quasi-compact and", "quasi-separated implies that $f_*\\mathcal{F}$ is quasi-coherent.", "Hence Lemma \\ref{lemma-sections-supported-on-closed-subset}", "applies to $T$ and $f_*\\mathcal{F}$ as well as to", "$|f|^{-1}T$ and $\\mathcal{F}$. The equality of the given quasi-coherent", "modules is immediate from the definitions." ], "refs": [ "spaces-limits-lemma-sections-supported-on-closed-subset" ], "ref_ids": [ 4624 ] } ], "ref_ids": [] }, { "id": 4626, "type": "theorem", "label": "spaces-limits-lemma-affine", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-affine", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume that $f$ is surjective and finite, and assume that $X$", "is affine. Then $Y$ is affine." ], "refs": [], "proofs": [ { "contents": [ "We may and do view $f : X \\to Y$ as a morphism of algebraic space over", "$\\Spec(\\mathbf{Z})$ (see", "Spaces, Definition \\ref{spaces-definition-base-change}).", "Note that a finite morphism is affine and universally closed, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-integral-universally-closed}.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-image-universally-closed-separated}", "we see that $Y$ is a separated algebraic space.", "As $f$ is surjective and $X$ is quasi-compact we see that $Y$ is", "quasi-compact.", "\\medskip\\noindent", "By Lemma \\ref{lemma-finite-in-finite-and-finite-presentation}", "we can write $X = \\lim X_a$ with each $X_a \\to Y$ finite and of", "finite presentation. By", "Lemma \\ref{lemma-limit-is-affine}", "we see that $X_a$ is affine for $a$ large enough.", "Hence we may and do assume that $f : X \\to Y$ is finite, surjective, and", "of finite presentation.", "\\medskip\\noindent", "By Proposition \\ref{proposition-approximate} we may write", "$Y = \\lim Y_i$ as a directed limit of algebraic", "spaces of finite presentation over $\\mathbf{Z}$.", "By Lemma \\ref{lemma-descend-finite-presentation} we can", "find $0 \\in I$ and a morphism $X_0 \\to Y_0$ of finite presentation", "such that $X_i = X_0 \\times_{Y_0} Y_i$ for $i \\geq 0$", "and such that $X = \\lim_i X_i$. By", "Lemma \\ref{lemma-descend-finite}", "we see that $X_i \\to Y_i$ is finite for $i$ large enough.", "By Lemma \\ref{lemma-descend-surjective}", "we see that $X_i \\to Y_i$ is surjective for $i$ large enough.", "By Lemma \\ref{lemma-limit-is-affine} we see that $X_i$ is", "affine for $i$ large enough. Hence for $i$ large enough we can apply", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian}", "to conclude that $Y_i$ is affine. This implies that $Y$ is affine and", "we conclude." ], "refs": [ "spaces-definition-base-change", "spaces-morphisms-lemma-integral-universally-closed", "spaces-morphisms-lemma-image-universally-closed-separated", "spaces-limits-lemma-finite-in-finite-and-finite-presentation", "spaces-limits-lemma-limit-is-affine", "spaces-limits-proposition-approximate", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-finite", "spaces-limits-lemma-descend-surjective", "spaces-limits-lemma-limit-is-affine", "spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian" ], "ref_ids": [ 8183, 4944, 4750, 4612, 4578, 4656, 4598, 4590, 4587, 4578, 11326 ] } ], "ref_ids": [] }, { "id": 4627, "type": "theorem", "label": "spaces-limits-lemma-reduction-scheme", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-reduction-scheme", "contents": [ "\\begin{reference}", "\\cite[3.1.12]{CLO}", "\\end{reference}", "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "If $X_{red}$ is a scheme, then $X$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "Let $U' \\subset X_{red}$ be an open affine subscheme.", "Let $U \\subset X$ be the open subspace corresponding to the open", "$|U'| \\subset |X_{red}| = |X|$. Then $U' \\to U$ is surjective and", "integral. Hence $U$ is affine by", "Proposition \\ref{proposition-affine}.", "Thus every point is contained in an open subscheme of $X$, i.e.,", "$X$ is a scheme." ], "refs": [ "spaces-limits-proposition-affine" ], "ref_ids": [ 4658 ] } ], "ref_ids": [] }, { "id": 4628, "type": "theorem", "label": "spaces-limits-lemma-integral-universally-bijective-scheme", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-integral-universally-bijective-scheme", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is integral and induces a bijection $|X| \\to |Y|$.", "Then $X$ is a scheme if and only if $Y$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "An integral morphism is representable by definition, hence if $Y$", "is a scheme, so is $X$. Conversely, assume that $X$ is a scheme.", "Let $U \\subset X$ be an affine open. An integral morphism is", "closed and $|f|$ is bijective, hence $|f|(|U|) \\subset |Y|$", "is open as the complement of $|f|(|X| \\setminus |U|)$. Let", "$V \\subset Y$ be the open subspace with $|V| = |f|(|U|)$, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-open-subspaces}.", "Then $U \\to V$ is integral and surjective, hence", "$V$ is an affine scheme by Proposition \\ref{proposition-affine}.", "This concludes the proof." ], "refs": [ "spaces-properties-lemma-open-subspaces", "spaces-limits-proposition-affine" ], "ref_ids": [ 11823, 4658 ] } ], "ref_ids": [] }, { "id": 4629, "type": "theorem", "label": "spaces-limits-lemma-check-closed-infinitesimally", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-check-closed-infinitesimally", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to B$ and $B' \\to B$ be morphisms of algebraic spaces over $S$.", "Assume", "\\begin{enumerate}", "\\item $B' \\to B$ is a closed immersion,", "\\item $|B'| \\to |B|$ is bijective,", "\\item $X \\times_B B' \\to B'$ is a closed immersion, and", "\\item $X \\to B$ is of finite type or $B' \\to B$ is of finite presentation.", "\\end{enumerate}", "Then $f : X \\to B$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Assumptions (1) and (2) imply that $B_{red} = B'_{red}$.", "Set $X' = X \\times_B B'$. Then $X' \\to X$ is closed immersion", "and $X'_{red} = X_{red}$. Let $U \\to B$ be an \\'etale morphism", "with $U$ affine. Then $X' \\times_B U \\to X \\times_B U$ is a", "closed immersion of algebraic spaces inducing an isomorphism", "on underlying reduced spaces. Since $X' \\times_B U$ is a scheme", "(as $B' \\to B$ and $X' \\to B'$ are representable) so is", "$X \\times_B U$ by Lemma \\ref{lemma-reduction-scheme}.", "Hence $X \\to B$ is representable too. Thus we reduce to the", "case of schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-check-closed-infinitesimally}." ], "refs": [ "spaces-limits-lemma-reduction-scheme", "morphisms-lemma-check-closed-infinitesimally" ], "ref_ids": [ 4627, 5456 ] } ], "ref_ids": [] }, { "id": 4630, "type": "theorem", "label": "spaces-limits-lemma-minimal-closed-subspace", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-minimal-closed-subspace", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. If $X$ is not a scheme, then there exists", "a closed subspace $Z \\subset X$ such that $Z$ is not a scheme, but", "every proper closed subspace $Z' \\subset Z$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "We prove this by Zorn's lemma. Let $\\mathcal{Z}$ be the set", "of closed subspaces $Z$ which are not schemes ordered by inclusion.", "By assumption $\\mathcal{Z}$ contains $X$, hence is nonempty.", "If $Z_\\alpha$ is a totally ordered subset of $\\mathcal{Z}$, then", "$Z = \\bigcap Z_\\alpha$ is in $\\mathcal{Z}$. Namely,", "$$", "Z = \\lim Z_\\alpha", "$$", "and the transition morphisms are affine.", "Thus we may apply Lemma \\ref{lemma-limit-is-scheme} to see that if $Z$", "were a scheme, then so would one of the $Z_\\alpha$.", "(This works even if $Z = \\emptyset$, but note that by", "Lemma \\ref{lemma-limit-nonempty} this cannot happen.)", "Thus $\\mathcal{Z}$ has minimal elements by Zorn's lemma." ], "refs": [ "spaces-limits-lemma-limit-is-scheme", "spaces-limits-lemma-limit-nonempty" ], "ref_ids": [ 4579, 4572 ] } ], "ref_ids": [] }, { "id": 4631, "type": "theorem", "label": "spaces-limits-lemma-minimal-nonscheme", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-minimal-nonscheme", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Assume that every proper closed subspace", "$Z \\subset X$ is a scheme, but $X$ is not a scheme. Then $X$ is reduced", "and irreducible." ], "refs": [], "proofs": [ { "contents": [ "We see that $X$ is reduced by Lemma \\ref{lemma-reduction-scheme}.", "Choose closed subsets $T_1 \\subset |X|$ and $T_2 \\subset |X|$ such that", "$|X| = T_1 \\cup T_2$. If $T_1$ and $T_2$ are proper closed subsets,", "then the corresponding reduced induced closed subspaces $Z_1, Z_2 \\subset X$", "(Properties of Spaces, Definition", "\\ref{spaces-properties-definition-reduced-induced-space})", "are schemes and so is $Z = Z_1 \\times_X Z_2 = Z_1 \\cap Z_2$ as a closed", "subscheme of either $Z_1$ or $Z_2$. Observe that the coproduct", "$Z_1 \\amalg_Z Z_2$ exists in the category of schemes, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-pushout-along-closed-immersions}.", "One way to proceed, is to show that $Z_1 \\amalg_Z Z_2$ is isomorphic to $X$,", "but we cannot use this here as the material on pushouts of algebraic", "spaces comes later in the theory. Instead we will use", "Lemma \\ref{lemma-affine} to find an affine neighbourhood of every point.", "Namely, let $x \\in |X|$. If $x \\not \\in Z_1$, then $x$ has a neighbourhood", "which is a scheme, namely, $X \\setminus Z_1$. Similarly if $x \\not \\in Z_2$.", "If $x \\in Z = Z_1 \\cap Z_2$, then we choose an affine open", "$U \\subset Z_1 \\amalg_Z Z_2$ containing $z$. Then $U_1 = Z_1 \\cap U$", "and $U_2 = Z_2 \\cap U$ are affine opens whose intersections with", "$Z$ agree. Since $|Z_1| = T_1$ and $|Z_2| = T_2$ are closed subsets of", "$|X|$ which intersect in $|Z|$, we find an open $W \\subset |X|$", "with $W \\cap T_1 = |U_1|$ and $W \\cap T_2 = |U_2|$. Let $W$ denote the", "corresponding open subspace of $X$. Then $x \\in |W|$ and the morphism", "$U_1 \\amalg U_2 \\to W$ is a surjective finite morphism whose source", "is an affine scheme. Thus $W$ is an affine scheme by", "Lemma \\ref{lemma-affine}." ], "refs": [ "spaces-limits-lemma-reduction-scheme", "spaces-properties-definition-reduced-induced-space", "more-morphisms-lemma-pushout-along-closed-immersions", "spaces-limits-lemma-affine", "spaces-limits-lemma-affine" ], "ref_ids": [ 4627, 11932, 14051, 4626, 4626 ] } ], "ref_ids": [] }, { "id": 4632, "type": "theorem", "label": "spaces-limits-lemma-enough-local", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-enough-local", "contents": [ "Let $f: X \\to S$ be a quasi-compact and quasi-separated morphism from an", "algebraic space to a scheme $S$. If for every $x \\in |X|$ with image", "$s = f(x) \\in S$ the algebraic space $X \\times_S \\Spec(\\mathcal{O}_{S,s})$", "is a scheme, then $X$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in |X|$. It suffices to find an open neighbourhood $U$ of", "$s = f(x)$ such that $X \\times_S U$ is a scheme.", "As $X \\times_S \\Spec(\\mathcal{O}_{S, s})$ is a scheme, then, since", "$\\mathcal{O}_{S, s} = \\colim \\mathcal{O}_S(U)$ where the colimit is", "over affine open neighbourhoods of $s$ in $S$ we see that", "$$", "X \\times_S \\Spec(\\mathcal{O}_{S, s}) = \\lim X \\times_S U", "$$", "By Lemma \\ref{lemma-limit-is-scheme} we see that $X \\times_S U$", "is a scheme for some $U$." ], "refs": [ "spaces-limits-lemma-limit-is-scheme" ], "ref_ids": [ 4579 ] } ], "ref_ids": [] }, { "id": 4633, "type": "theorem", "label": "spaces-limits-lemma-maximal-ideal", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-maximal-ideal", "contents": [ "Let $\\varphi : X \\to \\Spec(A)$ be a quasi-compact and quasi-separated", "morphism from an algebraic space to an affine scheme.", "If $X$ is not a scheme, then there exists an ideal $I \\subset A$", "such that the base change $X_{A/I}$ is not a scheme, but", "for every $I \\subset I'$, $I \\not = I'$ the base change", "$X_{A/I'}$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "We prove this by Zorn's lemma. Let $\\mathcal{I}$ be the set", "of ideals $I$ such that $X_{A/I}$ is not a scheme. By", "assumption $\\mathcal{I}$ contains $(0)$. If $I_\\alpha$ is", "a chain of ideals in $\\mathcal{I}$, then", "$I = \\bigcup I_\\alpha$ is in $\\mathcal{I}$. Namely,", "$A/I = \\colim A/I_\\alpha$, hence", "$$", "X_{A/I} = \\lim X_{A/I_\\alpha}", "$$", "Thus we may apply Lemma \\ref{lemma-limit-is-scheme} to see that if $X_{A/I}$", "were a scheme, then so would be one of the $X_{A/I_\\alpha}$.", "Thus $\\mathcal{I}$ has maximal elements by Zorn's lemma." ], "refs": [ "spaces-limits-lemma-limit-is-scheme" ], "ref_ids": [ 4579 ] } ], "ref_ids": [] }, { "id": 4634, "type": "theorem", "label": "spaces-limits-lemma-relative-glueing", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-relative-glueing", "contents": [ "Let $S = U \\cup W$ be an open covering of a scheme. Then the functor", "$$", "FP_S \\longrightarrow FP_U \\times_{FP_{U \\cap W}} FP_W", "$$", "given by base change is an equivalence where $FP_T$", "is the category of algebraic spaces of finite presentation over", "the scheme $T$." ], "refs": [], "proofs": [ { "contents": [ "First, since $S = U \\cup W$ is a Zariski covering, we see that the", "category of sheaves on $(\\Sch/S)_{fppf}$ is equivalent to the category", "of triples $(\\mathcal{F}_U, \\mathcal{F}_W, \\varphi)$ where", "$\\mathcal{F}_U$ is a sheaf on $(\\Sch/U)_{fppf}$,", "$\\mathcal{F}_W$ is a sheaf on $(\\Sch/W)_{fppf}$, and", "$$", "\\varphi :", "\\mathcal{F}_U|_{(\\Sch/U \\cap W)_{fppf}}", "\\longrightarrow", "\\mathcal{F}_W|_{(\\Sch/U \\cap W)_{fppf}}", "$$", "is an isomorphism. See Sites, Lemma \\ref{sites-lemma-mapping-property-glue}", "(note that no other gluing data are necessary because", "$U \\times_S U = U$, $W \\times_S W = W$ and that the cocycle", "condition is automatic for the same reason).", "Now, if the sheaf $\\mathcal{F}$ on $(\\Sch/S)_{fppf}$", "maps to $(\\mathcal{F}_U, \\mathcal{F}_W, \\varphi)$", "via this equivalence, then $\\mathcal{F}$ is an algebraic space", "if and only if $\\mathcal{F}_U$ and $\\mathcal{F}_W$ are algebraic spaces.", "This follows immediately from", "Algebraic Spaces, Lemma \\ref{spaces-lemma-glueing-algebraic-spaces}", "as $\\mathcal{F}_U \\to \\mathcal{F}$ and $\\mathcal{F}_W \\to \\mathcal{F}$", "are representable by open immersions and cover $\\mathcal{F}$.", "Finally, in this case the algebraic space $\\mathcal{F}$ is of finite", "presentation over $S$ if and only if $\\mathcal{F}_U$ is of finite presentation", "over $U$ and $\\mathcal{F}_W$ is of finite presentation over $W$", "by Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-quasi-compact-local},", "\\ref{spaces-morphisms-lemma-separated-local}, and", "\\ref{spaces-morphisms-lemma-finite-presentation-local}." ], "refs": [ "sites-lemma-mapping-property-glue", "spaces-lemma-glueing-algebraic-spaces", "spaces-morphisms-lemma-quasi-compact-local", "spaces-morphisms-lemma-separated-local", "spaces-morphisms-lemma-finite-presentation-local" ], "ref_ids": [ 8565, 8148, 4742, 4722, 4841 ] } ], "ref_ids": [] }, { "id": 4635, "type": "theorem", "label": "spaces-limits-lemma-glueing-near-closed-point", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-glueing-near-closed-point", "contents": [ "Let $S$ be a scheme. Let $s \\in S$ be a closed point such that", "$U = S \\setminus \\{s\\} \\to S$ is quasi-compact. With", "$V = \\Spec(\\mathcal{O}_{S, s}) \\setminus \\{s\\}$ there is", "an equivalence of categories", "$$", "FP_S \\longrightarrow FP_U \\times_{FP_V} FP_{\\Spec(\\mathcal{O}_{S, s})}", "$$", "where $FP_T$ is the category of algebraic spaces of finite presentation", "over $T$." ], "refs": [], "proofs": [ { "contents": [ "Let $W \\subset S$ be an open neighbourhood of $s$. The functor", "$$", "FP_S \\to FP_U \\times_{FP_{W \\setminus \\{s\\}}} FP_W", "$$", "is an equivalence of categories by Lemma \\ref{lemma-relative-glueing}.", "We have $\\mathcal{O}_{S, s} = \\colim \\mathcal{O}_W(W)$ where", "$W$ runs over the affine open neighbourhoods of $s$.", "Hence $\\Spec(\\mathcal{O}_{S, s}) = \\lim W$ where $W$", "runs over the affine open neighbourhoods of $s$.", "Thus the category of algebraic spaces of finite presentation", "over $\\Spec(\\mathcal{O}_{S, s})$ is the limit of the", "category of algebraic spaces of finite presentation over", "$W$ where $W$ runs over the affine open neighbourhoods", "of $s$, see", "Lemma \\ref{lemma-descend-finite-presentation}.", "For every affine open $s \\in W$ we see that $U \\cap W$", "is quasi-compact as $U \\to S$ is quasi-compact.", "Hence $V = \\lim W \\cap U = \\lim W \\setminus \\{s\\}$ is a limit of", "quasi-compact and quasi-separated schemes (see", "Limits, Lemma \\ref{limits-lemma-directed-inverse-system-has-limit}).", "Thus also the category of algebraic spaces of finite presentation", "over $V$ is the limit of the", "categories of algebraic spaces of finite presentation over", "$W \\cap U$ where $W$ runs over the affine open neighbourhoods", "of $s$. The lemma follows formally from a combination", "of these results." ], "refs": [ "spaces-limits-lemma-relative-glueing", "spaces-limits-lemma-descend-finite-presentation", "limits-lemma-directed-inverse-system-has-limit" ], "ref_ids": [ 4634, 4598, 15027 ] } ], "ref_ids": [] }, { "id": 4636, "type": "theorem", "label": "spaces-limits-lemma-glueing-near-point", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-glueing-near-point", "contents": [ "Let $S$ be a scheme. Let $U \\subset S$ be a retrocompact open.", "Let $s \\in S$ be a point in the complement of $U$. With", "$V = \\Spec(\\mathcal{O}_{S, s}) \\cap U$ there is", "an equivalence of categories", "$$", "\\colim_{s \\in U' \\supset U\\text{ open}} FP_{U'}", "\\longrightarrow", "FP_U \\times_{FP_V} FP_{\\Spec(\\mathcal{O}_{S, s})}", "$$", "where $FP_T$ is the category of algebraic spaces of finite presentation", "over $T$." ], "refs": [], "proofs": [ { "contents": [ "Let $W \\subset S$ be an open neighbourhood of $s$. By", "Lemma \\ref{lemma-relative-glueing} the functor", "$$", "FP_{U \\cup W}", "\\longrightarrow", "FP_U \\times_{FP_{U \\cap W}} FP_W", "$$", "is an equivalence of categories. We have", "$\\mathcal{O}_{S, s} = \\colim \\mathcal{O}_W(W)$ where", "$W$ runs over the affine open neighbourhoods of $s$.", "Hence $\\Spec(\\mathcal{O}_{S, s}) = \\lim W$ where $W$", "runs over the affine open neighbourhoods of $s$.", "Thus the category of algebraic spaces of finite presentation", "over $\\Spec(\\mathcal{O}_{S, s})$ is the limit of the", "category of algebraic spaces of finite presentation over", "$W$ where $W$ runs over the affine open neighbourhoods", "of $s$, see", "Lemma \\ref{lemma-descend-finite-presentation}.", "For every affine open $s \\in W$ we see that $U \\cap W$", "is quasi-compact as $U \\to S$ is quasi-compact.", "Hence $V = \\lim W \\cap U$ is a limit of", "quasi-compact and quasi-separated schemes (see", "Limits, Lemma \\ref{limits-lemma-directed-inverse-system-has-limit}).", "Thus also the category of algebraic spaces of finite presentation", "over $V$ is the limit of the", "categories of algebraic spaces of finite presentation over", "$W \\cap U$ where $W$ runs over the affine open neighbourhoods", "of $s$. The lemma follows formally from a combination", "of these results." ], "refs": [ "spaces-limits-lemma-relative-glueing", "spaces-limits-lemma-descend-finite-presentation", "limits-lemma-directed-inverse-system-has-limit" ], "ref_ids": [ 4634, 4598, 15027 ] } ], "ref_ids": [] }, { "id": 4637, "type": "theorem", "label": "spaces-limits-lemma-glueing-near-multiple-closed-points", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-glueing-near-multiple-closed-points", "contents": [ "Let $S$ be a scheme. Let $s_1, \\ldots, s_n \\in S$ be", "pairwise distinct closed points such that", "$U = S \\setminus \\{s_1, \\ldots, s_n\\} \\to S$ is quasi-compact. With", "$S_i = \\Spec(\\mathcal{O}_{S, s_i})$ and $U_i = S_i \\setminus \\{s_i\\}$", "there is an equivalence of categories", "$$", "FP_S \\longrightarrow", "FP_U \\times_{(FP_{U_1} \\times \\ldots \\times FP_{U_n})}", "(FP_{S_1} \\times \\ldots \\times FP_{S_n})", "$$", "where $FP_T$ is the category of algebraic spaces of finite presentation", "over $T$." ], "refs": [], "proofs": [ { "contents": [ "For $n = 1$ this is Lemma \\ref{lemma-glueing-near-closed-point}.", "For $n > 1$ the lemma can be proved in exactly the same way or it", "can be deduced from it. For example, suppose that $f_i : X_i \\to S_i$", "are objects of $FP_{S_i}$ and $f : X \\to U$ is an object", "of $FP_U$ and we're given isomorphisms $X_i \\times_{S_i} U_i = X \\times_U U_i$.", "By Lemma \\ref{lemma-glueing-near-closed-point} we can find", "a morphism $f' : X' \\to U' = S \\setminus \\{s_1, \\ldots, s_{n - 1}\\}$", "which is of finite presentation, which is isomorphic to", "$X_i$ over $S_i$, which is isomorphic to $X$ over $U$, and", "these isomorphisms are compatible with the given isomorphism", "$X_i \\times_{S_n} U_n = X \\times_U U_n$.", "Then we can apply induction to", "$f_i : X_i \\to S_i$, $i \\leq n - 1$,", "$f' : X' \\to U'$, and the induced", "isomorphisms $X_i \\times_{S_i} U_i = X' \\times_{U'} U_i$, $i \\leq n - 1$.", "This shows essential surjectivity. We omit the proof of", "fully faithfulness." ], "refs": [ "spaces-limits-lemma-glueing-near-closed-point", "spaces-limits-lemma-glueing-near-closed-point" ], "ref_ids": [ 4635, 4635 ] } ], "ref_ids": [] }, { "id": 4638, "type": "theorem", "label": "spaces-limits-lemma-excision-modifications", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-excision-modifications", "contents": [ "Let $S$ be a scheme. Consider a separated \\'etale morphism", "$f : V \\to W$ of algebraic spaces over $S$.", "Assume there exists a", "closed subspace $T \\subset W$ such that $f^{-1}T \\to T$ is", "an isomorphism. Then, with $W^0 = W \\setminus T$ and", "$V^0 = f^{-1}W^0$ the base change functor", "$$", "\\left\\{", "\\begin{matrix}", "g : X \\to W\\text{ morphism of algebraic spaces} \\\\", "g^{-1}(W^0) \\to W^0\\text{ is an isomorphism}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "h : Y \\to V\\text{ morphism of algebraic spaces} \\\\", "h^{-1}(V^0) \\to V^0\\text{ is an isomorphism}", "\\end{matrix}", "\\right\\}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "Since $V \\to W$ is separated we see that", "$V \\times_W V = \\Delta(V) \\amalg U$ for some open and closed subspace", "$U$ of $V \\times_W V$. By the assumption that $f^{-1}T \\to T$ is an", "isomorphism we see that $U \\times_W T = \\emptyset$, i.e., the two", "projections $U \\to V$ maps into $V^0$.", "\\medskip\\noindent", "Given $h : Y \\to V$ in the right hand category, consider the", "contravariant functor $X$ on $(\\Sch/S)_{fppf}$ defined by the rule", "$$", "X(T) = \\{(w, y) \\mid", "w : T \\to W,\\ y : T \\times_{w, W} V \\to Y\\text{ morphism over }V\\}", "$$", "Denote $g : X \\to W$ the map sending $(w, y) \\in X(T)$ to $w \\in W(T)$.", "Since $h^{-1}V^0 \\to V^0$ is an isomorphism, we see that if", "$w : T \\to W$ maps into $W^0$, then there is a unique choice for $h$.", "In other words $X \\times_{g, W} W^0 = W^0$. On the other hand, consider", "a $T$-valued point $(w, y, v)$ of $X \\times_{g, W, f} V$.", "Then $w = f \\circ v$ and", "$$", "y : T \\times_{f \\circ v, W} V \\longrightarrow V", "$$", "is a morphism over $V$. Consider the morphism", "$$", "T \\times_{f \\circ v, W} V \\xrightarrow{(v, \\text{id}_V)}", "V \\times_W V = V \\amalg U", "$$", "The inverse image of $V$ is $T$ embedded via", "$(\\text{id}_T, v) : T \\to T \\times_{f \\circ v, W} V$.", "The composition $y' = y \\circ (\\text{id}_T, v) : T \\to Y$", "is a morphism with $v = h \\circ y'$ which determines $y$ because the", "restriction of $y$ to the other part is uniquely determined as", "$U$ maps into $V^0$ by the second projection. It follows that", "$X \\times_{g, W, f} V \\to Y$, $(w, y, v) \\mapsto y'$ is an isomorphism.", "\\medskip\\noindent", "Thus if we can show that $X$ is an algebraic space, then we are done.", "Since $V \\to W$ is separated and \\'etale it is representable by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable}", "(and Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-etale-locally-quasi-finite}).", "Of course $W^0 \\to W$ is representable and \\'etale as it is an", "open immersion. Thus", "$$", "W^0 \\amalg Y = X \\times_{g, W} W^0 \\amalg X \\times_{g, W, f} V", "= X \\times_{g, W} (W^0 \\amalg V) \\longrightarrow X", "$$", "is representable, surjective, and \\'etale by Spaces, Lemmas", "\\ref{spaces-lemma-base-change-representable-transformations} and", "\\ref{spaces-lemma-base-change-representable-transformations-property}.", "Thus $X$ is an algebraic", "space by Spaces, Lemma", "\\ref{spaces-lemma-etale-locally-representable-by-space-gives-space}." ], "refs": [ "spaces-morphisms-lemma-locally-quasi-finite-separated-representable", "spaces-morphisms-lemma-etale-locally-quasi-finite", "spaces-lemma-base-change-representable-transformations", "spaces-lemma-base-change-representable-transformations-property", "spaces-lemma-etale-locally-representable-by-space-gives-space" ], "ref_ids": [ 4972, 4908, 8127, 8133, 8155 ] } ], "ref_ids": [] }, { "id": 4639, "type": "theorem", "label": "spaces-limits-lemma-excision-modifications-properties", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-excision-modifications-properties", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-excision-modifications}.", "Let $g : X \\to W$ correspond to $h : Y \\to V$ via the equivalence.", "Then $g$ is quasi-compact, quasi-separated, separated, locally of finite", "presentation, of finite presentation, locally of finite type, of finite type,", "proper, integral, finite, and add more here if and only if", "$h$ is so." ], "refs": [ "spaces-limits-lemma-excision-modifications" ], "proofs": [ { "contents": [ "If $g$ is quasi-compact, quasi-separated, separated, locally of finite", "presentation, of finite presentation, locally of finite type, of finite type,", "proper, finite, so is $h$ as a base change of $g$ by", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-quasi-compact},", "\\ref{spaces-morphisms-lemma-base-change-separated},", "\\ref{spaces-morphisms-lemma-base-change-finite-presentation},", "\\ref{spaces-morphisms-lemma-base-change-finite-type},", "\\ref{spaces-morphisms-lemma-base-change-proper},", "\\ref{spaces-morphisms-lemma-base-change-integral}.", "Conversely, let $P$ be a property of morphisms of algebraic", "spaces which is \\'etale local on the base and which holds for", "the identity morphism of any algebraic space.", "Since $\\{W^0 \\to W, V \\to W\\}$ is an \\'etale", "covering, to prove that $g$ has $P$ it suffices to show", "that $h$ has $P$. Thus we conclude using", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-quasi-compact-local},", "\\ref{spaces-morphisms-lemma-separated-local},", "\\ref{spaces-morphisms-lemma-finite-presentation-local},", "\\ref{spaces-morphisms-lemma-finite-type-local},", "\\ref{spaces-morphisms-lemma-proper-local},", "\\ref{spaces-morphisms-lemma-integral-local}." ], "refs": [ "spaces-morphisms-lemma-base-change-quasi-compact", "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-base-change-finite-presentation", "spaces-morphisms-lemma-base-change-finite-type", "spaces-morphisms-lemma-base-change-proper", "spaces-morphisms-lemma-base-change-integral", "spaces-morphisms-lemma-quasi-compact-local", "spaces-morphisms-lemma-separated-local", "spaces-morphisms-lemma-finite-presentation-local", "spaces-morphisms-lemma-finite-type-local", "spaces-morphisms-lemma-proper-local", "spaces-morphisms-lemma-integral-local" ], "ref_ids": [ 4738, 4714, 4840, 4815, 4917, 4942, 4742, 4722, 4841, 4816, 4916, 4940 ] } ], "ref_ids": [ 4638 ] }, { "id": 4640, "type": "theorem", "label": "spaces-limits-lemma-modifications", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-modifications", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $x \\in |X|$ be a closed point such that $U = X \\setminus \\{x\\} \\to X$", "is quasi-compact. With", "$V = \\Spec(\\mathcal{O}_{X, x}^h) \\setminus \\{\\mathfrak m_x^h\\}$", "the base change functor", "$$", "\\left\\{", "\\begin{matrix}", "f : Y \\to X\\text{ of finite presentation} \\\\", "f^{-1}(U) \\to U\\text{ is an isomorphism}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "g : Y \\to \\Spec(\\mathcal{O}_{X, x}^h)\\text{ of finite presentation} \\\\", "g^{-1}(V) \\to V\\text{ is an isomorphism}", "\\end{matrix}", "\\right\\}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "Let $a : (W, w) \\to (X, x)$ be an elementary \\'etale neighbourhood of $x$", "with $W$ affine as in", "Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-decent-space-elementary-etale-neighbourhood}.", "Since $x$ is a closed point of $X$ and $w$ is the unique point of $W$", "lying over $x$, we see that $w$ is a closed point of $W$. Since $a$", "is \\'etale and identifies residue fields at $x$ and $w$, it", "follows that $a$ induces an isomorphism $a^{-1}x \\to x$ (as closed", "subspaces of $X$ and $W$). Thus we may apply", "Lemma \\ref{lemma-excision-modifications} and", "\\ref{lemma-excision-modifications-properties}", "to reduce the problem to the case where $X$ is an affine scheme.", "\\medskip\\noindent", "Assume $X$ is an affine scheme. Recall that $\\mathcal{O}_{X, x}^h$", "is the colimit of $\\Gamma(U, \\mathcal{O}_U)$ over affine", "elementary \\'etale neighbourhoods $(U, u) \\to (X, x)$.", "Recall that the category of these neighbourhoods is", "cofiltered, see Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-elementary-etale-neighbourhoods} or", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-elementary-etale-neighbourhoods}.", "Then $\\Spec(\\mathcal{O}_{X, x}^h) = \\lim U$ and", "$V = \\lim U \\setminus \\{u\\}$", "(Lemma \\ref{lemma-directed-inverse-system-has-limit})", "where the limits are taken over the same category. Thus by", "Lemma \\ref{lemma-descend-finite-presentation}", "The category on the right is the colimit of the categories", "for the pairs $(U, u)$. And by the material in the first", "paragraph, each of these categories is equivalent to the", "category for the pair $(X, x)$. This finishes the proof." ], "refs": [ "decent-spaces-lemma-decent-space-elementary-etale-neighbourhood", "spaces-limits-lemma-excision-modifications", "spaces-limits-lemma-excision-modifications-properties", "decent-spaces-lemma-elementary-etale-neighbourhoods", "more-morphisms-lemma-elementary-etale-neighbourhoods", "spaces-limits-lemma-directed-inverse-system-has-limit", "spaces-limits-lemma-descend-finite-presentation" ], "ref_ids": [ 9488, 4638, 4639, 9489, 13868, 4565, 4598 ] } ], "ref_ids": [] }, { "id": 4641, "type": "theorem", "label": "spaces-limits-lemma-separate", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-separate", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$ be", "morphisms of algebraic spaces over $S$. Let $z \\in |Z|$ and let", "$T \\subset |X \\times_Y Z|$ be a closed subset", "with $z \\not \\in \\Im(T \\to |Z|)$.", "If $f$ is quasi-compact, then there exists", "an \\'etale neighbourhood $(V, v) \\to (Z, z)$,", "a commutative diagram", "$$", "\\xymatrix{", "V \\ar[d] \\ar[r]_a & Z' \\ar[d]^b \\\\", "Z \\ar[r]^g & Y,", "}", "$$", "and a closed subset $T' \\subset |X \\times_Y Z'|$ such that", "\\begin{enumerate}", "\\item the morphism $b : Z' \\to Y$ is locally of finite presentation,", "\\item with $z' = a(v)$ we have $z' \\not \\in \\Im(T' \\to |Z'|)$, and", "\\item the inverse image of $T$ in $|X \\times_Y V|$", "maps into $T'$ via $|X \\times_Y V| \\to |X \\times_Y Z'|$.", "\\end{enumerate}", "Moreover, we may assume $V$ and $Z'$ are affine schemes and if $Z$", "is a scheme we may assume $V$ is an affine open neighbourhood of $z$." ], "refs": [], "proofs": [ { "contents": [ "We will deduce this from the corresponding result for morphisms of schemes.", "Let $y \\in |Y|$ be the image of $z$. First we choose an affine \\'etale", "neighbourhood $(U, u) \\to (Y, y)$ and then we choose an affine \\'etale", "neighbourhood $(V, v) \\to (Z, z)$ such that the morphism $V \\to Y$", "factors through $U$. Then we may replace", "\\begin{enumerate}", "\\item $X \\to Y$ by $X \\times_Y U \\to U$,", "\\item $Z \\to Y$ by $V \\to U$,", "\\item $z$ by $v$, and", "\\item $T$ by its inverse image in", "$|(X \\times_Y U) \\times_U V| = |X \\times_Y V|$.", "\\end{enumerate}", "In fact, below we will show that after replacing $V$ by an affine", "open neighbourhood of $v$ there will be a morphism $a : V \\to Z'$ for", "some $Z' \\to U$ of finite presentation and a closed subset $T'$", "of $|(X \\times_Y U) \\times_U Z'| = |X \\times_Y Z'|$ such that", "$T$ maps into $T'$ and $a(v) \\not \\in \\Im(T' \\to |Z'|)$.", "Thus we may and do assume that $Z$ and $Y$ are affine schemes", "with the proviso that we need to find a solution where $V$", "is an open neighbourhood of $z$.", "\\medskip\\noindent", "Since $f$ is quasi-compact and $Y$ is affine, the algebraic space", "$X$ is quasi-compact. Choose an affine scheme $W$ and a surjective", "\\'etale morphism $W \\to X$. Let $T_W \\subset |W \\times_Y Z|$", "be the inverse image of $T$. Then $z$ is not in the image of", "$T_W$. By the schemes case (Limits, Lemma \\ref{limits-lemma-separate})", "we can find an open neighbourhood $V \\subset Z$ of $z$", "a commutative diagram of schemes", "$$", "\\xymatrix{", "V \\ar[d] \\ar[r]_a & Z' \\ar[d]^b \\\\", "Z \\ar[r]^g & Y,", "}", "$$", "and a closed subset $T' \\subset |W \\times_Y Z'|$ such that", "\\begin{enumerate}", "\\item the morphism $b : Z' \\to Y$ is locally of finite presentation,", "\\item with $z' = a(z)$ we have $z' \\not \\in \\Im(T' \\to Z')$, and", "\\item $T_W \\cap |W \\times_Y V|$ maps into $T'$ via", "$|W \\times_Y V| \\to |W \\times_Y Z'|$.", "\\end{enumerate}", "The commutative diagram", "$$", "\\xymatrix{", "W \\times_Y V \\ar[r]_b \\ar[d]_c & W \\times_Y Z' \\ar[d]_q \\\\", "X \\times_Y V \\ar[r]^a & X \\times_Y Z'", "}", "$$", "is cartesian. The vertical maps are surjective \\'etale hence surjective", "and open. Also $T_1 = T_W \\cap |W \\times_Y V|$ is the inverse image of", "$T_2 = T \\cap |X \\times_Y V|$ by $c$. By Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-points-cartesian}", "we get $b(T_1) = q^{-1}(a(T_2))$. By", "Topology, Lemma \\ref{topology-lemma-open-morphism-quotient-topology}", "we get", "$$", "q^{-1}(\\overline{a(T_1)}) = \\overline{q^{-1}(a(T_1))} =", "\\overline{b(T_2)} \\subset T'", "$$", "As $q$ is surjective the image of $\\overline{a(T_1)} \\to |Z'|$", "does not contain $z'$ since the same is true for $T'$. This", "concludes the proof." ], "refs": [ "limits-lemma-separate", "spaces-properties-lemma-points-cartesian", "topology-lemma-open-morphism-quotient-topology" ], "ref_ids": [ 15094, 11819, 8203 ] } ], "ref_ids": [] }, { "id": 4642, "type": "theorem", "label": "spaces-limits-lemma-test-universally-closed", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-test-universally-closed", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a quasi-compact morphism of algebraic spaces over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally closed,", "\\item for every morphism $Z \\to Y$ which is locally of finite presentation", "the map $|X \\times_Y Z| \\to |Z|$ is closed, and", "\\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$", "such that $|\\mathbf{A}^n \\times (X \\times_Y V)| \\to |\\mathbf{A}^n \\times V|$", "is closed for all $n \\geq 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2).", "Suppose that $|X \\times_Y Z| \\to |Z|$ is not closed for some", "morphism of algebraic spaces $Z \\to Y$ over $S$. This means that there", "exists some closed subset $T \\subset |X \\times_Y Z|$", "such that $\\Im(T \\to |Z|)$ is not closed. Pick $z \\in |Z|$", "in the closure of the image of $T$ but not in the image.", "Apply Lemma \\ref{lemma-separate}.", "We find an \\'etale neighbourhood $(V, v) \\to (Z, z)$, a commutative diagram", "$$", "\\xymatrix{", "V \\ar[d] \\ar[r]_a & Z' \\ar[d]^b \\\\", "Z \\ar[r]^g & Y,", "}", "$$", "and a closed subset $T' \\subset |X \\times_Y Z'|$ such that", "\\begin{enumerate}", "\\item the morphism $b : Z' \\to Y$ is locally of finite presentation,", "\\item with $z' = a(v)$ we have $z' \\not \\in \\Im(T' \\to |Z'|)$, and", "\\item the inverse image of $T$ in $|X \\times_Y V|$ maps into $T'$ via", "$|X \\times_Y V| \\to |X \\times_Y Z'|$.", "\\end{enumerate}", "We claim that $z'$ is in the closure of $\\Im(T' \\to |Z'|)$", "which implies that $|X \\times_Y Z'| \\to |Z'|$ is not closed.", "The claim shows that (2) implies (1).", "To see the claim is true we suggest the reader contemplate the", "following commutative diagram", "$$", "\\xymatrix{", "X \\times_Y Z \\ar[d] &", "X \\times_Y V \\ar[l] \\ar[d] \\ar[r] &", "X \\times_Y Z' \\ar[d] \\\\", "Z & V \\ar[l] \\ar[r]^a & Z'", "}", "$$", "Let $T_V \\subset |X \\times_Y V|$ be the inverse image of $T$.", "By Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}", "the image of $T_V$ in $|V|$ is the inverse image of the image", "of $T$ in $|Z|$. Then since $z$ is in the closure of the image of", "$T \\to |Z|$ and since $|V| \\to |Z|$ is open, we see that $v$ is in", "the closure of the image of $T_V \\to |V|$. Since the image of", "$T_V$ in $|X \\times_Y Z'|$ is contained in $|T'|$ it follows", "immediately that $z' = a(v)$ is in the closure of the image of $T'$.", "\\medskip\\noindent", "It is clear that (1) implies (3). Let $V \\to Y$ be as in (3).", "If we can show that $X \\times_Y V \\to V$ is universally closed,", "then $f$ is universally closed by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-local}.", "Thus it suffices to show that $f : X \\to Y$ satisfies (2)", "if $f$ is a quasi-compact morphism of algebraic spaces,", "$Y$ is a scheme, and $|\\mathbf{A}^n \\times X| \\to |\\mathbf{A}^n \\times Y|$", "is closed for all $n$. Let $Z \\to Y$ be locally of finite presentation.", "We have to show the map $|X \\times_Y Z| \\to |Z|$ is closed.", "This question is \\'etale local on $Z$ hence we may assume $Z$", "is affine (some details omitted). Since $Y$ is a scheme, $Z$ is affine,", "and $Z \\to Y$ is locally of finite presentation we can find", "an immersion $Z \\to \\mathbf{A}^n \\times Y$, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-affine-finite-type-over-S}.", "Consider the cartesian diagram", "$$", "\\vcenter{", "\\xymatrix{", "X \\times_Y Z \\ar[d] \\ar[r] & \\mathbf{A}^n \\times X \\ar[d] \\\\", "Z \\ar[r] & \\mathbf{A}^n \\times Y", "}", "}", "\\quad", "\\begin{matrix}", "\\text{inducing the} \\\\", "\\text{cartesian square}", "\\end{matrix}", "\\quad", "\\vcenter{", "\\xymatrix{", "|X \\times_Y Z| \\ar[d] \\ar[r] & |\\mathbf{A}^n \\times X| \\ar[d] \\\\", "|Z| \\ar[r] & |\\mathbf{A}^n \\times Y|", "}", "}", "$$", "of topological spaces whose horizontal arrows are homeomorphisms", "onto locally closed subsets (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-subspace-induced-topology}).", "Thus every closed subset $T$", "of $|X \\times_Y Z|$ is the pullback of a closed subset $T'$ of", "$|\\mathbf{A}^n \\times Y|$. Since the assumption is that the image", "of $T'$ in $|\\mathbf{A}^n \\times X|$ is closed we conclude that", "the image of $T$ in $|Z|$ is closed as desired." ], "refs": [ "spaces-limits-lemma-separate", "spaces-properties-lemma-points-cartesian", "spaces-morphisms-lemma-universally-closed-local", "morphisms-lemma-quasi-affine-finite-type-over-S", "spaces-properties-lemma-subspace-induced-topology" ], "ref_ids": [ 4641, 11819, 4748, 5392, 11844 ] } ], "ref_ids": [] }, { "id": 4643, "type": "theorem", "label": "spaces-limits-lemma-limited-base-change", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-limited-base-change", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a", "morphism of algebraic spaces over $S$.", "Assume $f$ separated and of finite type.", "The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is proper.", "\\item For any morphism $Y \\to Z$ which is locally of finite presentation", "the map $|X \\times_Y Z| \\to |Z|$ is closed, and", "\\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$", "such that $|\\mathbf{A}^n \\times (X \\times_Y V)| \\to |\\mathbf{A}^n \\times V|$", "is closed for all $n \\geq 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In view of the fact that a proper morphism is the same thing as", "a separated, finite type, and universally closed morphism, this", "lemma is a special case of Lemma \\ref{lemma-test-universally-closed}." ], "refs": [ "spaces-limits-lemma-test-universally-closed" ], "ref_ids": [ 4642 ] } ], "ref_ids": [] }, { "id": 4644, "type": "theorem", "label": "spaces-limits-lemma-reach-point-closure-Noetherian", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-reach-point-closure-Noetherian", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ finite type and $Y$ locally Noetherian.", "Let $y \\in |Y|$ be a point in the closure of the image of $|f|$.", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d]^f \\\\", "\\Spec(A) \\ar[r] & Y", "}", "$$", "where $A$ is a discrete valuation ring and $K$ is its field of fractions", "mapping the closed point of $\\Spec(A)$ to $y$. Moreover, we can assume", "that the point $x \\in |X|$ corresponding to $\\Spec(K) \\to X$ is a", "codimension $0$ point\\footnote{See discussion in", "Properties of Spaces, Section \\ref{spaces-properties-section-generic-points}.}", "and that $K$ is the residue field of a point", "on a scheme \\'etale over $X$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $V$, a point $v \\in V$ and an \\'etale morphism", "$V \\to Y$ mapping $v$ to $y$. The map $|V| \\to |Y|$ is open and by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}", "the image of $|X \\times_Y V| \\to |V|$ is the inverse image of the", "image of $|f|$. We conclude that the point $v$ is in the closure of the", "image of $|X \\times_Y V| \\to |V|$. If we prove the lemma for", "$X \\times_Y V \\to V$ and the point $v$, then the lemma follows for", "$f$ and $y$. In this way we reduce to the situation described in the", "next paragraph.", "\\medskip\\noindent", "Assume we have $f : X \\to Y$ and $y \\in |Y|$ as in the lemma where", "$Y$ is an affine scheme. Since $f$ is quasi-compact, we conclude that", "$X$ is quasi-compact. Hence we can choose an affine scheme $W$ and", "a surjective \\'etale morphism $W \\to X$. Then the image of", "$|f|$ is the same as the image of $W \\to Y$. In this way we reduce", "to the case of schemes which is", "Limits, Lemma \\ref{limits-lemma-reach-point-closure-Noetherian}." ], "refs": [ "spaces-properties-lemma-points-cartesian", "limits-lemma-reach-point-closure-Noetherian" ], "ref_ids": [ 11819, 15097 ] } ], "ref_ids": [] }, { "id": 4645, "type": "theorem", "label": "spaces-limits-lemma-refined-valuative-criterion-proper", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-refined-valuative-criterion-proper", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $h : U \\to X$ be", "morphisms of algebraic spaces over $S$. Assume that $Y$ is", "locally Noetherian, that $f$ and $h$ are of finite type,", "that $f$ is separated, and that the image of $|h| : |U| \\to |X|$", "is dense in $|X|$. If given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\", "\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & Y", "}", "$$", "where $A$ is a discrete valuation ring with field of fractions $K$, there", "exists a dotted arrow making the diagram commute, then $f$ is proper." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove that $f$ is universally closed.", "Let $V \\to Y$ be an \\'etale morphism where $V$ is an affine scheme.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-local}", "it suffices to prove that the base change $X \\times_Y V \\to V$ is", "universally closed. By Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-points-cartesian}", "the image $I$ of $|U \\times_Y V| \\to |X \\times_Y V|$", "is the inverse image of the image of $|h|$. Since", "$|X \\times_Y V| \\to |X|$ is open", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-open})", "we conclude that $I$ is dense in $|X \\times_Y V|$.", "Therefore the assumptions of the lemma are", "satisfied for the morphisms $U \\times_Y V \\to X \\times_Y V \\to V$.", "Hence we may assume $Y$ is an affine scheme.", "\\medskip\\noindent", "Assume $Y$ is an affine scheme. Then $U$ is quasi-compact. Choose", "an affine scheme and a surjective \\'etale morphism $W \\to U$.", "Then we may and do replace $U$ by $W$ and assume that $U$ is affine.", "By the weak version of Chow's lemma", "(Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-weak-chow})", "we can choose a surjective proper morphism $X' \\to X$", "where $X'$ is a scheme. Then $U' = X' \\times_X U$ is a scheme", "and $U' \\to X'$ is of finite type. We may replace $X'$ by", "the scheme theoretic image of $h' : U' \\to X'$ and hence $h'(U')$", "is dense in $X'$. We claim that for every diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & U' \\ar[r]^h & X' \\ar[d]^{f'} \\\\", "\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & Y", "}", "$$", "where $A$ is a discrete valuation ring with field of fractions $K$, there", "exists a dotted arrow making the diagram commute. Namely, we first get an", "arrow $\\Spec(A) \\to X$ by the assumption of the lemma and then we lift this", "to an arrow $\\Spec(A) \\to X'$ using the valuative criterion for properness", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-characterize-proper}).", "The morphism $X' \\to Y$ is separated", "as a composition of a proper and a separated morphism.", "Thus by the case of schemes the morphism $X' \\to Y$ is proper", "(Limits, Lemma", "\\ref{limits-lemma-refined-valuative-criterion-proper}).", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-image-proper-is-proper}", "we conclude that $X \\to Y$ is proper." ], "refs": [ "spaces-morphisms-lemma-universally-closed-local", "spaces-properties-lemma-points-cartesian", "spaces-properties-lemma-etale-open", "spaces-cohomology-lemma-weak-chow", "spaces-morphisms-lemma-characterize-proper", "limits-lemma-refined-valuative-criterion-proper", "spaces-morphisms-lemma-image-proper-is-proper" ], "ref_ids": [ 4748, 11819, 11860, 11327, 4938, 15101, 4921 ] } ], "ref_ids": [] }, { "id": 4646, "type": "theorem", "label": "spaces-limits-lemma-refined-valuative-criterion-separated", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-refined-valuative-criterion-separated", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $h : U \\to X$ be", "morphisms of algebraic spaces over $S$. Assume that $Y$ is", "locally Noetherian, that $f$ is locally of finite type and quasi-separated,", "that $h$ is of finite type, and that the image of $|h| : |U| \\to |X|$", "is dense in $|X|$.", "If given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\", "\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & Y", "}", "$$", "where $A$ is a discrete valuation ring with field of fractions $K$, there", "exists at most one dotted arrow making the diagram commute, then $f$ is", "separated." ], "refs": [], "proofs": [ { "contents": [ "We will apply Lemma \\ref{lemma-refined-valuative-criterion-proper}", "to the morphisms $U \\to X$ and $\\Delta : X \\to X \\times_Y X$.", "We check the conditions. Observe that $\\Delta$ is quasi-compact because", "$f$ is quasi-separated. Of course $\\Delta$ is locally of finite type and", "separated (true for any diagonal morphism).", "Finally, suppose given a commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^\\Delta \\\\", "\\Spec(A) \\ar[rr]^{(a, b)} \\ar@{-->}[rru] & & X \\times_Y X", "}", "$$", "where $A$ is a discrete valuation ring with field of fractions $K$.", "Then $a$ and $b$ give two dotted arrows in the diagram of the lemma", "and have to be equal. Hence as dotted arrow we can use $a = b$", "which gives existence. This finishes the proof." ], "refs": [ "spaces-limits-lemma-refined-valuative-criterion-proper" ], "ref_ids": [ 4645 ] } ], "ref_ids": [] }, { "id": 4647, "type": "theorem", "label": "spaces-limits-lemma-refined-valuative-criterion-universally-closed", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-refined-valuative-criterion-universally-closed", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ and $h : U \\to X$ be morphisms of algebraic spaces over $S$.", "Assume that $Y$ is locally Noetherian, that $f$ and $h$ are of finite type, and", "that $h(U)$ is dense in $X$. If given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\", "\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & Y", "}", "$$", "where $A$ is a discrete valuation ring with field of fractions $K$, there", "exists a unique dotted arrow making the diagram commute, then $f$ is proper." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-refined-valuative-criterion-separated} and", "\\ref{lemma-refined-valuative-criterion-proper}." ], "refs": [ "spaces-limits-lemma-refined-valuative-criterion-separated", "spaces-limits-lemma-refined-valuative-criterion-proper" ], "ref_ids": [ 4646, 4645 ] } ], "ref_ids": [] }, { "id": 4648, "type": "theorem", "label": "spaces-limits-lemma-good-diagram", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-good-diagram", "contents": [ "In Situation \\ref{situation-limit-noetherian}.", "Let $X \\to B$ be a quasi-separated and finite type", "morphism of algebraic spaces.", "Then there exists an $i \\in I$ and a diagram", "\\begin{equation}", "\\label{equation-good-diagram}", "\\vcenter{", "\\xymatrix{", "X \\ar[r] \\ar[d] & W \\ar[d] \\\\", "B \\ar[r] & B_i", "}", "}", "\\end{equation}", "such that $W \\to B_i$ is of finite type and such that", "the induced morphism $X \\to B \\times_{B_i} W$ is a closed", "immersion." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-type-closed-in-finite-presentation}", "we can find a closed immersion $X \\to X'$", "over $B$ where $X'$ is an algebraic space of finite presentation over $B$.", "By Lemma \\ref{lemma-descend-finite-presentation}", "we can find an $i$ and a morphism of finite presentation", "$X'_i \\to B_i$ whose pull back is $X'$. Set $W = X'_i$." ], "refs": [ "spaces-limits-lemma-finite-type-closed-in-finite-presentation", "spaces-limits-lemma-descend-finite-presentation" ], "ref_ids": [ 4615, 4598 ] } ], "ref_ids": [] }, { "id": 4649, "type": "theorem", "label": "spaces-limits-lemma-limit-from-good-diagram", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-limit-from-good-diagram", "contents": [ "In Situation \\ref{situation-limit-noetherian}.", "Let $X \\to B$ be a quasi-separated and finite type morphism", "of algebraic spaces. Given $i \\in I$ and a diagram", "$$", "\\vcenter{", "\\xymatrix{", "X \\ar[r] \\ar[d] & W \\ar[d] \\\\", "B \\ar[r] & B_i", "}", "}", "$$", "as in (\\ref{equation-good-diagram}) for $i' \\geq i$ let", "$X_{i'}$ be the scheme theoretic image of $X \\to B_{i'} \\times_{B_i} W$.", "Then $X = \\lim_{i' \\geq i} X_{i'}$." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is quasi-compact and quasi-separated formation of the", "scheme theoretic image of $X \\to B_{i'} \\times_{B_i} W$", "commutes with \\'etale localization", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image}).", "Hence we may and do assume $W$ is affine and maps into an affine", "$U_i$ \\'etale over $B_i$. Then", "$$", "B_{i'} \\times_{B_i} W =", "B_{i'} \\times_{B_i} U_i \\times_{U_i} W =", "U_{i'} \\times_{U_i} W", "$$", "where $U_{i'} = B_{i'} \\times_{B_i} U_i$ is affine as the transition", "morphisms are affine. Thus the lemma follows from the case of schemes", "which is", "Limits, Lemma \\ref{limits-lemma-limit-from-good-diagram}." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image", "limits-lemma-limit-from-good-diagram" ], "ref_ids": [ 4780, 15120 ] } ], "ref_ids": [] }, { "id": 4650, "type": "theorem", "label": "spaces-limits-lemma-morphism-good-diagram", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-morphism-good-diagram", "contents": [ "In Situation \\ref{situation-limit-noetherian}.", "Let $f : X \\to Y$ be a morphism of algebraic spaces quasi-separated", "and of finite type over $B$. Let", "$$", "\\vcenter{", "\\xymatrix{", "X \\ar[r] \\ar[d] & W \\ar[d] \\\\", "B \\ar[r] & B_{i_1}", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "Y \\ar[r] \\ar[d] & V \\ar[d] \\\\", "B \\ar[r] & B_{i_2}", "}", "}", "$$", "be diagrams as in (\\ref{equation-good-diagram}). Let", "$X = \\lim_{i \\geq i_1} X_i$ and", "$Y = \\lim_{i \\geq i_2} Y_i$ be the corresponding", "limit descriptions as in Lemma \\ref{lemma-limit-from-good-diagram}.", "Then there exists an $i_0 \\geq \\max(i_1, i_2)$ and a morphism", "$$", "(f_i)_{i \\geq i_0} : (X_i)_{i \\geq i_0} \\to (Y_i)_{i \\geq i_0}", "$$", "of inverse systems over $(B_i)_{i \\geq i_0}$ such that", "such that $f = \\lim_{i \\geq i_0} f_i$.", "If $(g_i)_{i \\geq i_0} : (X_i)_{i \\geq i_0} \\to (Y_i)_{i \\geq i_0}$", "is a second morphism of inverse systems over $(B_i)_{i \\geq i_0}$ such that", "such that $f = \\lim_{i \\geq i_0} g_i$", "then $f_i = g_i$ for all $i \\gg i_0$." ], "refs": [ "spaces-limits-lemma-limit-from-good-diagram" ], "proofs": [ { "contents": [ "Since $V \\to B_{i_2}$ is of finite presentation and", "$X = \\lim_{i \\geq i_1} X_i$ we can appeal to Proposition", "\\ref{proposition-characterize-locally-finite-presentation}", "as improved by Lemma \\ref{lemma-better-characterize-relative-limit-preserving}", "to find an $i_0 \\geq \\max(i_1, i_2)$ and a morphism $h : X_{i_0} \\to V$", "over $B_{i_2}$ such that $X \\to X_{i_0} \\to V$ is equal to $X \\to Y \\to V$.", "For $i \\geq i_0$ we get a commutative solid diagram", "$$", "\\xymatrix{", "X \\ar[d] \\ar[r] &", "X_i \\ar[r] \\ar@{..>}[d] \\ar@/_2pc/[dd] |!{[d];[ld]}\\hole &", "X_{i_0} \\ar[d]^h \\\\", "Y \\ar[r] \\ar[d] & Y_i \\ar[r] \\ar[d] & V \\ar[d] \\\\", "B \\ar[r] & B_i \\ar[r] & B_{i_0}", "}", "$$", "Since $X \\to X_i$ has scheme theoretically dense image", "and since $Y_i$ is the scheme theoretic image of", "$Y \\to B_i \\times_{B_{i_2}} V$", "we find that the morphism $X_i \\to B_i \\times_{B_{i_2}} V$", "induced by the diagram factors through $Y_i$", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-factor-factor}).", "This proves existence.", "\\medskip\\noindent", "Uniqueness. Let $E_i \\to X_i$ be the equalizer of $f_i$ and $g_i$", "for $i \\geq i_0$. We have", "$E_i = Y_i \\times_{\\Delta, Y_i \\times_{B_i} Y_i, (f_i, g_i)} X_i$.", "Hence $E_i \\to X_i$ is a monomorphism of finite presentation as a base", "change of the diagonal of $Y_i$ over $B_i$, see", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-properties-diagonal} and", "\\ref{spaces-morphisms-lemma-diagonal-morphism-finite-type}.", "Since $X_i$ is a closed subspace of $B_i \\times_{B_{i_0}} X_{i_0}$", "and similarly for $Y_i$ we see that", "$$", "E_i =", "X_i \\times_{(B_i \\times_{B_{i_0}} X_{i_0})} (B_i \\times_{B_{i_0}} E_{i_0}) =", "X_i \\times_{X_{i_0}} E_{i_0}", "$$", "Similarly, we have $X = X \\times_{X_{i_0}} E_{i_0}$. Hence we conclude", "that $E_i = X_i$ for $i$ large enough by Lemma \\ref{lemma-descend-isomorphism}." ], "refs": [ "spaces-limits-proposition-characterize-locally-finite-presentation", "spaces-limits-lemma-better-characterize-relative-limit-preserving", "spaces-morphisms-lemma-factor-factor", "spaces-morphisms-lemma-properties-diagonal", "spaces-morphisms-lemma-diagonal-morphism-finite-type", "spaces-limits-lemma-descend-isomorphism" ], "ref_ids": [ 4655, 4569, 4783, 4712, 4847, 4593 ] } ], "ref_ids": [ 4649 ] }, { "id": 4651, "type": "theorem", "label": "spaces-limits-lemma-morphism-good-diagram-flat", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-morphism-good-diagram-flat", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}.", "If $f$ is flat and of finite presentation, then", "there exists an $i_3 > i_0$ such that for $i \\geq i_3$ we have", "$f_i$ is flat, $X_i = Y_i \\times_{Y_{i_3}} X_{i_3}$, and", "$X = Y \\times_{Y_{i_3}} X_{i_3}$." ], "refs": [ "spaces-limits-lemma-morphism-good-diagram" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-descend-finite-presentation}", "we can choose an $i \\geq i_2$ and a morphism", "$U \\to Y_i$ of finite presentation such that $X = Y \\times_{Y_i} U$", "(this is where we use that $f$ is of finite presentation).", "After increasing $i$ we may assume that $U \\to Y_i$ is flat, see", "Lemma \\ref{lemma-descend-flat}.", "As discussed in Remark \\ref{remark-finite-type-gives-well-defined-system}", "we may and do replace the initial diagram used to define the system", "$(X_i)_{i \\geq i_1}$ by the system corresponding to", "$X \\to U \\to B_i$. Thus $X_{i'}$ for $i' \\geq i$ is defined as", "the scheme theoretic image of $X \\to B_{i'} \\times_{B_i} U$.", "\\medskip\\noindent", "Because $U \\to Y_i$ is flat (this is where we use that $f$ is flat),", "because $X = Y \\times_{Y_i} U$, and", "because the scheme theoretic image of $Y \\to Y_i$ is $Y_i$,", "we see that the scheme theoretic image of $X \\to U$ is $U$", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image}).", "Observe that $Y_{i'} \\to B_{i'} \\times_{B_i} Y_i$ is a closed", "immersion for $i' \\geq i$ by construction of the system of $Y_j$.", "Then the same argument as above shows that the scheme theoretic image", "of $X \\to B_{i'} \\times_{B_i} U$", "is equal to the closed subspace $Y_{i'} \\times_{Y_i} U$.", "Thus we see that $X_{i'} = Y_{i'} \\times_{Y_i} U$ for all $i' \\geq i$", "and hence the lemma holds with $i_3 = i$." ], "refs": [ "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-flat", "spaces-limits-remark-finite-type-gives-well-defined-system", "spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image" ], "ref_ids": [ 4598, 4595, 4665, 4861 ] } ], "ref_ids": [ 4650 ] }, { "id": 4652, "type": "theorem", "label": "spaces-limits-lemma-morphism-good-diagram-smooth", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-morphism-good-diagram-smooth", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}.", "If $f$ is smooth, then there exists an $i_3 > i_0$ such that for", "$i \\geq i_3$ we have $f_i$ is smooth." ], "refs": [ "spaces-limits-lemma-morphism-good-diagram" ], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-morphism-good-diagram-flat} and", "\\ref{lemma-descend-smooth}." ], "refs": [ "spaces-limits-lemma-morphism-good-diagram-flat", "spaces-limits-lemma-descend-smooth" ], "ref_ids": [ 4651, 4586 ] } ], "ref_ids": [ 4650 ] }, { "id": 4653, "type": "theorem", "label": "spaces-limits-lemma-morphism-good-diagram-proper", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-morphism-good-diagram-proper", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}.", "If $f$ is proper, then there exists an $i_3 \\geq i_0$ such that for", "$i \\geq i_3$ we have $f_i$ is proper." ], "refs": [ "spaces-limits-lemma-morphism-good-diagram" ], "proofs": [ { "contents": [ "By the discussion in ", "Remark \\ref{remark-finite-type-gives-well-defined-system}", "the choice of $i_1$ and $W$ fitting into a diagram as in", "(\\ref{equation-good-diagram}) is immaterial for the truth of", "the lemma. Thus we choose $W$ as follows.", "First we choose a closed immersion $X \\to X'$", "with $X' \\to Y$ proper and of finite presentation, see", "Lemma \\ref{lemma-proper-limit-of-proper-finite-presentation}.", "Then we choose an $i_3 \\geq i_2$ and a proper morphism $W \\to Y_{i_3}$", "such that $X' = Y \\times_{Y_{i_3}} W$. This is possible because", "$Y = \\lim_{i \\geq i_2} Y_i$ and", "Lemmas \\ref{lemma-relative-approximation} and \\ref{lemma-eventually-proper}.", "With this choice of $W$ it is immediate from the construction that", "for $i \\geq i_3$ the algebraic space $X_i$ is a closed subspace of", "$Y_i \\times_{Y_{i_3}} W \\subset B_i \\times_{B_{i_3}} W$", "and hence proper over $Y_i$." ], "refs": [ "spaces-limits-remark-finite-type-gives-well-defined-system", "spaces-limits-lemma-proper-limit-of-proper-finite-presentation", "spaces-limits-lemma-relative-approximation", "spaces-limits-lemma-eventually-proper" ], "ref_ids": [ 4665, 4616, 4609, 4596 ] } ], "ref_ids": [ 4650 ] }, { "id": 4654, "type": "theorem", "label": "spaces-limits-lemma-good-diagram-fibre-product", "categories": [ "spaces-limits" ], "title": "spaces-limits-lemma-good-diagram-fibre-product", "contents": [ "In Situation \\ref{situation-limit-noetherian} suppose that we have a", "cartesian diagram", "$$", "\\xymatrix{", "X^1 \\ar[r]_p \\ar[d]_q & X^3 \\ar[d]^a \\\\", "X^2 \\ar[r]^b & X^4", "}", "$$", "of algebraic spaces quasi-separated and of finite type over $B$.", "For each $j = 1, 2, 3, 4$ choose $i_j \\in I$ and a diagram", "$$", "\\xymatrix{", "X^j \\ar[r] \\ar[d] & W^j \\ar[d] \\\\", "B \\ar[r] & B_{i_j}", "}", "$$", "as in (\\ref{equation-good-diagram}). Let", "$X^j = \\lim_{i \\geq i_j} X^j_i$ be the corresponding limit descriptions", "as in Lemma \\ref{lemma-morphism-good-diagram}.", "Let $(a_i)_{i \\geq i_5}$, $(b_i)_{i \\geq i_6}$, $(p_i)_{i \\geq i_7}$, and", "$(q_i)_{i \\geq i_8}$ be the corresponding morphisms of inverse systems", "contructed in Lemma \\ref{lemma-morphism-good-diagram}. Then there exists an", "$i_9 \\geq \\max(i_5, i_6, i_7, i_8)$ such that for $i \\geq i_9$ we have", "$a_i \\circ p_i = b_i \\circ q_i$ and such that", "$$", "(q_i, p_i) : X^1_i \\longrightarrow X^2_i \\times_{b_i, X^4_i, a_i} X^3_i", "$$", "is a closed immersion.", "If $a$ and $b$ are flat and of finite presentation, then there exists an", "$i_{10} \\geq \\max(i_5, i_6, i_7, i_8, i_9)$ such that for $i \\geq i_{10}$", "the last displayed morphism is an isomorphism." ], "refs": [ "spaces-limits-lemma-morphism-good-diagram", "spaces-limits-lemma-morphism-good-diagram" ], "proofs": [ { "contents": [ "According to the discussion in", "Remark \\ref{remark-finite-type-gives-well-defined-system}", "the choice of $W^1$ fitting into a diagram as in", "(\\ref{equation-good-diagram}) is immaterial for the truth of", "the lemma. Thus we may choose $W^1 = W^2 \\times_{W^4} W^3$.", "Then it is immediate from the construction of $X^1_i$ that ", "$a_i \\circ p_i = b_i \\circ q_i$ and that", "$$", "(q_i, p_i) : X^1_i \\longrightarrow X^2_i \\times_{b_i, X^4_i, a_i} X^3_i", "$$", "is a closed immersion.", "\\medskip\\noindent", "If $a$ and $b$ are flat and of finite presentation, then so are", "$p$ and $q$ as base changes of $a$ and $b$. Thus we can apply", "Lemma \\ref{lemma-morphism-good-diagram-flat}", "to each of $a$, $b$, $p$, $q$, and $a \\circ p = b \\circ q$.", "It follows that there exists an $i_9 \\in I$ such that", "$$", "(q_i, p_i) : X^1_i \\to X^2_i \\times_{X^4_i} X^3_i", "$$", "is the base change of $(q_{i_9}, p_{i_9})$ by the morphism", "by the morphism $X^4_i \\to X^4_{i_9}$ for all $i \\geq i_9$.", "We conclude that $(q_i, p_i)$ is an isomorphism for all sufficiently", "large $i$ by Lemma \\ref{lemma-descend-isomorphism}." ], "refs": [ "spaces-limits-remark-finite-type-gives-well-defined-system", "spaces-limits-lemma-morphism-good-diagram-flat", "spaces-limits-lemma-descend-isomorphism" ], "ref_ids": [ 4665, 4651, 4593 ] } ], "ref_ids": [ 4650, 4650 ] }, { "id": 4655, "type": "theorem", "label": "spaces-limits-proposition-characterize-locally-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-proposition-characterize-locally-finite-presentation", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is a morphism of algebraic spaces which is", "locally of finite presentation, see", "Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-locally-finite-presentation}.", "\\item The morphism $f : X \\to Y$ is limit preserving as", "a transformation of functors, see", "Definition \\ref{definition-locally-finite-presentation}.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-locally-finite-presentation", "spaces-limits-definition-locally-finite-presentation" ], "proofs": [ { "contents": [ "Assume (1). Let $T$ be a scheme and let $y \\in Y(T)$. We have to show that", "$T \\times_Y X$ is limit preserving over $T$ in the sense of", "Definition \\ref{definition-locally-finite-presentation}.", "Hence we are reduced to proving that if $X$ is an algebraic space which", "is locally of finite presentation over $S$ as an algebraic space, then it", "is limit preserving as a functor", "$X : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "To see this choose a presentation $X = U/R$, see", "Spaces, Definition \\ref{spaces-definition-presentation}.", "It follows from", "Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-locally-finite-presentation}", "that both $U$ and $R$ are schemes which are locally of finite presentation", "over $S$. Hence by", "Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}", "we have", "$$", "U(T) = \\colim U(T_i), \\quad", "R(T) = \\colim R(T_i)", "$$", "whenever $T = \\lim_i T_i$ in $(\\Sch/S)_{fppf}$. It follows", "that the presheaf", "$$", "(\\Sch/S)_{fppf}^{opp} \\longrightarrow \\textit{Sets}, \\quad", "W \\longmapsto U(W)/R(W)", "$$", "is limit preserving. Hence by", "Lemma \\ref{lemma-sheafify-finite-presentation}", "its sheafification $X = U/R$ is limit preserving too.", "\\medskip\\noindent", "Assume (2). Choose a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$. Next, choose a scheme $U$ and a surjective \\'etale morphism", "$U \\to V \\times_Y X$. By", "Lemma \\ref{lemma-base-change-locally-finite-presentation}", "the transformation of functors $V \\times_Y X \\to V$ is limit preserving. By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-etale-locally-finite-presentation}", "the morphism of algebraic spaces $U \\to V \\times_Y X$ is locally", "of finite presentation, hence limit preserving as", "a transformation of functors by the first part of the proof. By", "Lemma \\ref{lemma-composition-locally-finite-presentation}", "the composition $U \\to V \\times_Y X \\to V$ is limit preserving", "as a transformation of functors. Hence", "the morphism of schemes $U \\to V$ is locally of finite presentation by", "Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}", "(modulo a set theoretic remark, see last paragraph of the proof).", "This means, by definition, that (1) holds.", "\\medskip\\noindent", "Set theoretic remark. Let $U \\to V$ be a morphism of", "$(\\Sch/S)_{fppf}$. In the statement of", "Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}", "we characterize $U \\to V$ as being locally of finite presentation", "if for {\\it all} directed inverse systems $(T_i, f_{ii'})$ of affine schemes", "over $V$ we have $U(T) = \\colim V(T_i)$, but in the current setting", "we may only consider affine schemes $T_i$ over $V$ which are (isomorphic to)", "an object of $(\\Sch/S)_{fppf}$. So we have to make sure that there", "are enough affines in $(\\Sch/S)_{fppf}$ to make the proof work.", "Inspecting the proof of (2) $\\Rightarrow$ (1) of", "Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}", "we see that the question reduces to the case that $U$ and $V$ are affine.", "Say $U = \\Spec(A)$ and $V = \\Spec(B)$. By construction", "of $(\\Sch/S)_{fppf}$ the spectrum of any ring of cardinality", "$\\leq |B|$ is isomorphic to an object of $(\\Sch/S)_{fppf}$.", "Hence it suffices to observe that in the \"only if\" part of the proof of", "Algebra, Lemma \\ref{algebra-lemma-characterize-finite-presentation}", "only $A$-algebras of cardinality $\\leq |B|$ are used." ], "refs": [ "spaces-limits-definition-locally-finite-presentation", "spaces-definition-presentation", "spaces-morphisms-definition-locally-finite-presentation", "limits-proposition-characterize-locally-finite-presentation", "spaces-limits-lemma-sheafify-finite-presentation", "spaces-limits-lemma-base-change-locally-finite-presentation", "spaces-morphisms-lemma-etale-locally-finite-presentation", "spaces-limits-lemma-composition-locally-finite-presentation", "limits-proposition-characterize-locally-finite-presentation", "limits-proposition-characterize-locally-finite-presentation", "limits-proposition-characterize-locally-finite-presentation", "algebra-lemma-characterize-finite-presentation" ], "ref_ids": [ 4660, 8177, 5006, 15127, 4561, 4559, 4911, 4557, 15127, 15127, 15127, 1092 ] } ], "ref_ids": [ 5006, 4660 ] }, { "id": 4656, "type": "theorem", "label": "spaces-limits-proposition-approximate", "categories": [ "spaces-limits" ], "title": "spaces-limits-proposition-approximate", "contents": [ "\\begin{reference}", "Our proof follows closely the proof given in \\cite[Theorem 1.2.2]{CLO}.", "\\end{reference}", "Let $X$ be a quasi-compact and quasi-separated algebraic space over", "$\\Spec(\\mathbf{Z})$. There exist a directed set $I$", "and an inverse system of algebraic spaces $(X_i, f_{ii'})$ over $I$", "such that", "\\begin{enumerate}", "\\item the transition morphisms $f_{ii'}$ are affine", "\\item each $X_i$ is quasi-separated and of finite type over", "$\\mathbf{Z}$, and", "\\item $X = \\lim X_i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We apply Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated}", "to get open subspaces $U_p \\subset X$, schemes $V_p$, and morphisms", "$f_p : V_p \\to U_p$ with properties as stated. Note that", "$f_n : V_n \\to U_n$ is an \\'etale morphism of algebraic spaces", "whose restriction to the inverse image of $T_n = (V_n)_{red}$ is an", "isomorphism. Hence $f_n$ is an isomorphism, for example by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-etale-universally-injective-open}.", "In particular $U_n$ is a quasi-compact and separated scheme.", "Thus we can write $U_n = \\lim U_{n, i}$ as a directed limit", "of schemes of finite type over $\\mathbf{Z}$ with affine transition", "morphisms, see Limits, Proposition \\ref{limits-proposition-approximate}.", "Thus, applying descending induction on $p$, we see that we have reduced", "to the problem posed in the following paragraph.", "\\medskip\\noindent", "Here we have $U \\subset X$, $U = \\lim U_i$, $Z \\subset X$, and", "$f : V \\to X$ with the following properties", "\\begin{enumerate}", "\\item $X$ is a quasi-compact and quasi-separated algebraic space,", "\\item $V$ is a quasi-compact and separated scheme,", "\\item $U \\subset X$ is a quasi-compact open subspace,", "\\item $(U_i, g_{ii'})$ is a directed inverse system of", "quasi-separated algebraic spaces", "of finite type over $\\mathbf{Z}$ with affine transition morphisms", "whose limit is $U$,", "\\item $Z \\subset X$ is a closed subspace such that $|X| = |U| \\amalg |Z|$,", "\\item $f : V \\to X$ is a surjective \\'etale morphism such that", "$f^{-1}(Z) \\to Z$ is an isomorphism.", "\\end{enumerate}", "Problem: Show that the conclusion of the proposition holds for $X$.", "\\medskip\\noindent", "Note that $W = f^{-1}(U) \\subset V$ is a quasi-compact open subscheme", "\\'etale over $U$. Hence we may apply", "Lemmas \\ref{lemma-descend-finite-presentation} and \\ref{lemma-descend-etale}", "to find an index $0 \\in I$ and an \\'etale morphism $W_0 \\to U_0$", "of finite presentation whose base change to $U$ produces $W$. Setting", "$W_i = W_0 \\times_{U_0} U_i$ we see that $W = \\lim_{i \\geq 0} W_i$. After", "increasing $0$ we may assume the $W_i$ are schemes, see", "Lemma \\ref{lemma-limit-is-scheme}.", "Moreover, $W_i$ is of finite type over $\\mathbf{Z}$.", "\\medskip\\noindent", "Apply Limits, Lemma \\ref{limits-lemma-approximate} to", "$W = \\lim_{i \\geq 0} W_i$ and the inclusion $W \\subset V$. Replace $I$", "by the directed set $J$ found in that lemma. This allows us", "to write $V$ as a directed limit $V = \\lim V_i$ of finite type schemes over", "$\\mathbf{Z}$ with affine transition maps such that each $V_i$ contains", "$W_i$ as an open subscheme (compatible with transition morphisms).", "For each $i$ we can form the push out", "$$", "\\xymatrix{", "W_i \\ar[r] \\ar[d]_\\Delta & V_i \\ar[d] \\\\", "W_i \\times_{U_i} W_i \\ar[r] & R_i", "}", "$$", "in the category of schemes. Namely, the left vertical and upper horizontal", "arrows are open immersions of schemes. In other words, we can construct", "$R_i$ as the glueing of $V_i$ and $W_i \\times_{U_i} W_i$ along the common open", "$W_i$ (see Schemes, Section \\ref{schemes-section-glueing-schemes}). Note that", "the \\'etale projection maps $W_i \\times_{U_i} W_i \\to W_i$ extend", "to \\'etale morphisms $s_i, t_i : R_i \\to V_i$. It is clear that the", "morphism $j_i = (t_i, s_i) : R_i \\to V_i \\times V_i$ is an \\'etale", "equivalence relation on $V_i$. Note that $W_i \\times_{U_i} W_i$ is", "quasi-compact (as $U_i$ is quasi-separated and $W_i$ quasi-compact)", "and $V_i$ is quasi-compact, hence $R_i$ is quasi-compact. For", "$i \\geq i'$ the diagram", "\\begin{equation}", "\\label{equation-cartesian}", "\\vcenter{", "\\xymatrix{", "R_i \\ar[r] \\ar[d]_{s_i} & R_{i'} \\ar[d]^{s_{i'}} \\\\", "V_i \\ar[r] & V_{i'}", "}", "}", "\\end{equation}", "is cartesian because", "$$", "(W_{i'} \\times_{U_{i'}} W_{i'}) \\times_{U_{i'}} U_i =", "W_{i'} \\times_{U_{i'}} U_i \\times_{U_i} U_i \\times_{U_{i'}} W_{i'} =", "W_i \\times_{U_i} W_i.", "$$", "Consider the algebraic space $X_i = V_i/R_i$ (see", "Spaces, Theorem \\ref{spaces-theorem-presentation}).", "As $V_i$ is of finite type over $\\mathbf{Z}$ and $R_i$ is quasi-compact", "we see that $X_i$ is quasi-separated and of finite type over $\\mathbf{Z}$", "(see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quasi-separated}", "and", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-surjection-from-quasi-compact} and", "\\ref{spaces-morphisms-lemma-finite-type-local}).", "As the construction of $R_i$ above is compatible", "with transition morphisms, we obtain morphisms of algebraic spaces", "$X_i \\to X_{i'}$ for $i \\geq i'$. The commutative diagrams", "$$", "\\xymatrix{", "V_i \\ar[r] \\ar[d] & V_{i'} \\ar[d] \\\\", "X_i \\ar[r] & X_{i'}", "}", "$$", "are cartesian as (\\ref{equation-cartesian}) is cartesian, see", "Groupoids, Lemma \\ref{groupoids-lemma-criterion-fibre-product}.", "Since $V_i \\to V_{i'}$ is affine, this implies that $X_i \\to X_{i'}$", "is affine, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-affine-local}.", "Thus we can form the limit $X' = \\lim X_i$ by", "Lemma \\ref{lemma-directed-inverse-system-has-limit}.", "We claim that $X \\cong X'$ which finishes the proof of the proposition.", "\\medskip\\noindent", "Proof of the claim. Set $R = \\lim R_i$.", "By construction the algebraic space $X'$ comes", "equipped with a surjective \\'etale morphism $V \\to X'$ such that", "$$", "V \\times_{X'} V \\cong R", "$$", "(use Lemma \\ref{lemma-directed-inverse-system-has-limit}).", "By construction $\\lim W_i \\times_{U_i} W_i = W \\times_U W$ and $V = \\lim V_i$", "so that $R$ is the union of $W \\times_U W$ and $V$ glued along $W$.", "Property (6) implies the projections $V \\times_X V \\to V$ are isomorphisms", "over $f^{-1}(Z) \\subset V$. Hence the scheme $V \\times_X V$ is the union", "of the opens $\\Delta_{V/X}(V)$ and $W \\times_U W$ which intersect", "along $\\Delta_{W/X}(W)$. We conclude that there exists a unique isomorphism", "$R \\cong V \\times_X V$ compatible with the projections to $V$.", "Since $V \\to X$ and $V \\to X'$ are surjective \\'etale we see that", "$$", "X = V/ V \\times_X V = V/R = V/V \\times_{X'} V = X'", "$$", "by Spaces, Lemma \\ref{spaces-lemma-space-presentation} and we win." ], "refs": [ "decent-spaces-lemma-filter-quasi-compact-quasi-separated", "spaces-morphisms-lemma-etale-universally-injective-open", "limits-proposition-approximate", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-etale", "spaces-limits-lemma-limit-is-scheme", "limits-lemma-approximate", "spaces-theorem-presentation", "spaces-properties-lemma-quasi-separated", "spaces-morphisms-lemma-surjection-from-quasi-compact", "spaces-morphisms-lemma-finite-type-local", "groupoids-lemma-criterion-fibre-product", "spaces-morphisms-lemma-affine-local", "spaces-limits-lemma-directed-inverse-system-has-limit", "spaces-limits-lemma-directed-inverse-system-has-limit", "spaces-lemma-space-presentation" ], "ref_ids": [ 9480, 4973, 15126, 4598, 4585, 4579, 15053, 8124, 11834, 4740, 4816, 9652, 4798, 4565, 4565, 8149 ] } ], "ref_ids": [] }, { "id": 4657, "type": "theorem", "label": "spaces-limits-proposition-separated-closed-in-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-proposition-separated-closed-in-finite-presentation", "contents": [ "Let $S$ be a scheme. $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume", "\\begin{enumerate}", "\\item $f$ is of finite type and separated, and", "\\item $Y$ is quasi-compact and quasi-separated.", "\\end{enumerate}", "Then there exists a separated morphism of finite presentation", "$f' : X' \\to Y$ and a closed immersion $X \\to X'$ over $Y$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-type-closed-in-finite-presentation}", "there is a closed immersion $X \\to Z$ with $Z/Y$ of", "finite presentation. Let $\\mathcal{I} \\subset \\mathcal{O}_Z$", "be the quasi-coherent sheaf of ideals defining $X$ as a closed", "subscheme of $Y$. By", "Lemma \\ref{lemma-directed-colimit-finite-type}", "we can write $\\mathcal{I}$ as a directed colimit", "$\\mathcal{I} = \\colim_{a \\in A} \\mathcal{I}_a$ of its", "quasi-coherent sheaves of ideals of finite type.", "Let $X_a \\subset Z$ be the closed subspace defined by $\\mathcal{I}_a$.", "These form an inverse system indexed by $A$.", "The transition morphisms $X_a \\to X_{a'}$ are affine because", "they are closed immersions. Each $X_a$ is quasi-compact and quasi-separated", "since it is a closed subspace of $Z$ and $Z$ is quasi-compact and", "quasi-separated by our assumptions.", "We have $X = \\lim_a X_a$ as follows directly from the", "fact that $\\mathcal{I} = \\colim_{a \\in A} \\mathcal{I}_a$.", "Each of the morphisms $X_a \\to Z$ is of finite presentation, see", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-finite-presentation}.", "Hence the morphisms $X_a \\to Y$ are of finite presentation.", "Thus it suffices to show that $X_a \\to Y$ is separated for some", "$a \\in A$. This follows from Lemma \\ref{lemma-eventually-separated}", "as we have assumed that $X \\to Y$ is separated." ], "refs": [ "spaces-limits-lemma-finite-type-closed-in-finite-presentation", "spaces-limits-lemma-directed-colimit-finite-type", "morphisms-lemma-closed-immersion-finite-presentation", "spaces-limits-lemma-eventually-separated" ], "ref_ids": [ 4615, 4602, 5243, 4581 ] } ], "ref_ids": [] }, { "id": 4658, "type": "theorem", "label": "spaces-limits-proposition-affine", "categories": [ "spaces-limits" ], "title": "spaces-limits-proposition-affine", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume that $f$ is surjective and integral, and assume that $X$", "is affine. Then $Y$ is affine." ], "refs": [], "proofs": [ { "contents": [ "We may and do view $f : X \\to Y$ as a morphism of algebraic spaces over", "$\\Spec(\\mathbf{Z})$ (see", "Spaces, Definition \\ref{spaces-definition-base-change}).", "Note that integral morphisms are affine and universally closed, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-integral-universally-closed}.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-image-universally-closed-separated}", "we see that $Y$ is a separated algebraic space.", "As $f$ is surjective and $X$ is quasi-compact we see that $Y$ is", "quasi-compact.", "\\medskip\\noindent", "Consider the sheaf $\\mathcal{A} = f_*\\mathcal{O}_X$.", "This is a quasi-coherent sheaf of $\\mathcal{O}_Y$-algebras, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-pushforward}.", "By Lemma \\ref{lemma-colimit-finitely-presented}", "we can write $\\mathcal{A} = \\colim_i \\mathcal{F}_i$ as a filtered", "colimit of finite type $\\mathcal{O}_Y$-modules. Let", "$\\mathcal{A}_i \\subset \\mathcal{A}$ be the $\\mathcal{O}_Y$-subalgebra", "generated by $\\mathcal{F}_i$. Since the map of algebras", "$\\mathcal{O}_Y \\to \\mathcal{A}$ is integral, we see that each $\\mathcal{A}_i$", "is a finite quasi-coherent $\\mathcal{O}_Y$-algebra. Hence", "$$", "X_i = \\underline{\\Spec}_Y(\\mathcal{A}_i) \\longrightarrow Y", "$$", "is a finite morphism of algebraic spaces. Here ", "$\\underline{\\Spec}$ is the construction of Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-affine-equivalence-algebras}.", "It is clear", "that $X = \\lim_i X_i$. Hence by", "Lemma \\ref{lemma-limit-is-affine}", "we see that for $i$ sufficiently large the scheme $X_i$ is affine.", "Moreover, since $X \\to Y$ factors through each $X_i$ we see that", "$X_i \\to Y$ is surjective. Hence we conclude that $Y$ is affine by", "Lemma \\ref{lemma-affine}." ], "refs": [ "spaces-definition-base-change", "spaces-morphisms-lemma-integral-universally-closed", "spaces-morphisms-lemma-image-universally-closed-separated", "spaces-morphisms-lemma-pushforward", "spaces-limits-lemma-colimit-finitely-presented", "spaces-morphisms-lemma-affine-equivalence-algebras", "spaces-limits-lemma-limit-is-affine", "spaces-limits-lemma-affine" ], "ref_ids": [ 8183, 4944, 4750, 4760, 4601, 4802, 4578, 4626 ] } ], "ref_ids": [] }, { "id": 4659, "type": "theorem", "label": "spaces-limits-proposition-there-is-a-scheme-finite-over", "categories": [ "spaces-limits" ], "title": "spaces-limits-proposition-there-is-a-scheme-finite-over", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$.", "\\begin{enumerate}", "\\item There exists a surjective finite morphism $Y \\to X$", "of finite presentation where $Y$ is a scheme,", "\\item given a surjective \\'etale morphism $U \\to X$ we may choose", "$Y \\to X$ such that for every $y \\in Y$ there is an open neighbourhood", "$V \\subset Y$ such that $V \\to X$ factors through $U$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is the special case of (2) with $U = X$.", "Let $Y \\to X$ be as in", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-there-is-a-scheme-integral-over}.", "Choose a finite affine open covering $Y = \\bigcup V_j$ such that $V_j \\to X$", "factors through $U$. We can write $Y = \\lim Y_i$ with", "$Y_i \\to X$ finite and of finite presentation, see", "Lemma \\ref{lemma-integral-limit-finite-and-finite-presentation}.", "For large enough $i$ the algebraic space $Y_i$ is a scheme, see", "Lemma \\ref{lemma-limit-is-scheme}.", "For large enough $i$ we can find affine opens $V_{i, j} \\subset Y_i$", "whose inverse image in $Y$ recovers $V_j$, see", "Lemma \\ref{lemma-descend-opens}.", "For even larger $i$ the morphisms $V_j \\to U$ over $X$ come", "from morphisms $V_{i, j} \\to U$ over $X$, see Proposition", "\\ref{proposition-characterize-locally-finite-presentation}.", "This finishes the proof." ], "refs": [ "decent-spaces-lemma-there-is-a-scheme-integral-over", "spaces-limits-lemma-integral-limit-finite-and-finite-presentation", "spaces-limits-lemma-limit-is-scheme", "spaces-limits-lemma-descend-opens", "spaces-limits-proposition-characterize-locally-finite-presentation" ], "ref_ids": [ 9483, 4611, 4579, 4575, 4655 ] } ], "ref_ids": [] }, { "id": 4666, "type": "theorem", "label": "stacks-geometry-lemma-deformation-category", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-deformation-category", "contents": [ "In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$", "be a morphism, where $k$ is a finite type field over $S$.", "Then $\\mathcal{F}_{\\mathcal{X}, k, x_0}$", "is a deformation category and $T\\mathcal{F}_{\\mathcal{X}, k, x_0}$", "and $\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0})$", "are finite dimensional $k$-vector spaces." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine open $\\Spec(\\Lambda) \\subset S$ such that", "$\\Spec(k) \\to S$ factors through it.", "By Artin's Axioms, Section \\ref{artin-section-predeformation-categories}", "we obtain a predeformation category", "$\\mathcal{F}_{\\mathcal{X}, k, x_0}$", "over the category $\\mathcal{C}_\\Lambda$.", "(As pointed out in locus citatus this category only depends", "on the morphism $\\Spec(k) \\to S$ and not on the choice of", "$\\Lambda$.) By Artin's Axioms, Lemmas", "\\ref{artin-lemma-deformation-category} and", "\\ref{artin-lemma-algebraic-stack-RS}", "$\\mathcal{F}_{\\mathcal{X}, k, x_0}$ is actually a deformation category.", "By Artin's Axioms, Lemma \\ref{artin-lemma-finite-dimension}", "we find that $T\\mathcal{F}_{\\mathcal{X}, k, x_0}$", "and $\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0})$", "are finite dimensional $k$-vector spaces." ], "refs": [ "artin-lemma-deformation-category", "artin-lemma-algebraic-stack-RS", "artin-lemma-finite-dimension" ], "ref_ids": [ 11357, 11355, 11359 ] } ], "ref_ids": [] }, { "id": 4667, "type": "theorem", "label": "stacks-geometry-lemma-versal-ring", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-versal-ring", "contents": [ "In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$", "be a morphism, where $k$ is a finite type field over $S$.", "Then a versal ring to $\\mathcal{X}$ at $x_0$ exists. Given a pair", "$A$, $A'$ of these, then $A \\cong A'[[t_1, \\ldots, t_r]]$", "or $A' \\cong A[[t_1, \\ldots, t_r]]$ as $S$-algebras", "for some $r$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-deformation-category} and", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-versal-object-existence}", "(note that the assumptions of this lemma hold by", "Formal Deformation Theory, Lemmas", "\\ref{formal-defos-lemma-RS-implies-S1-S2} and", "Definition \\ref{formal-defos-definition-deformation-category}).", "By the uniquness result of", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-minimal-versal}", "there exists a ``minimal'' versal ring $A$ of $\\mathcal{X}$ at $x_0$", "such that any other versal ring of $\\mathcal{X}$ at $x_0$ is", "isomorphic to $A[[t_1, \\ldots, t_r]]$ for some $r$.", "This clearly implies the second statement." ], "refs": [ "stacks-geometry-lemma-deformation-category", "formal-defos-lemma-versal-object-existence", "formal-defos-lemma-RS-implies-S1-S2", "formal-defos-definition-deformation-category", "formal-defos-lemma-minimal-versal" ], "ref_ids": [ 4666, 3458, 3469, 3530, 3462 ] } ], "ref_ids": [] }, { "id": 4668, "type": "theorem", "label": "stacks-geometry-lemma-versal-ring-field-extension", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-versal-ring-field-extension", "contents": [ "In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$", "be a morphism, where $k$ is a finite type field over $S$.", "Let $l/k$ be a finite extension of fields and denote", "$x_{l, 0} : \\Spec(l) \\to \\mathcal{X}$ the induced morphism.", "Given a versal ring $A$ to $\\mathcal{X}$ at $x_0$ there exists", "a versal ring $A'$ to $\\mathcal{X}$ at $x_{l, 0}$ such that", "there is a $S$-algebra map $A \\to A'$ which induces the given", "field extension $l/k$ and is formally smooth in the $\\mathfrak m_{A'}$-adic", "topology." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Artin's Axioms, Lemma \\ref{artin-lemma-change-of-field}", "and", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-change-of-field-versal-ring}.", "(We also use that $\\mathcal{X}$ satisfies (RS) by", "Artin's Axioms, Lemma \\ref{artin-lemma-algebraic-stack-RS}.)" ], "refs": [ "artin-lemma-change-of-field", "formal-defos-lemma-change-of-field-versal-ring", "artin-lemma-algebraic-stack-RS" ], "ref_ids": [ 11358, 3509, 11355 ] } ], "ref_ids": [] }, { "id": 4669, "type": "theorem", "label": "stacks-geometry-lemma-compare-versal-ring-completion", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-compare-versal-ring-completion", "contents": [ "In Situation \\ref{situation-versal} let $x : U \\to \\mathcal{X}$ be a", "morphism where $U$ is a scheme locally of finite type over $S$.", "Let $u_0 \\in U$ be a finite type point.", "Set $k = \\kappa(u_0)$ and denote $x_0 : \\Spec(k) \\to \\mathcal{X}$", "the induced map. The following are equivalent", "\\begin{enumerate}", "\\item $x$ is versal at $u_0$", "(Artin's Axioms, Definition \\ref{artin-definition-versal}),", "\\item $\\hat x : \\mathcal{F}_{U, k, u_0} \\to \\mathcal{F}_{\\mathcal{X}, k, x_0}$", "is smooth,", "\\item the formal object associated to", "$x|_{\\Spec(\\mathcal{O}_{U, u_0}^\\wedge)}$ is versal, and", "\\item there is an open neighbourhood $U' \\subset U$ of $x$ such that", "$x|_{U'} : U' \\to \\mathcal{X}$ is smooth.", "\\end{enumerate}", "Moreover, in this case the completion $\\mathcal{O}_{U, u_0}^\\wedge$", "is a versal ring to $\\mathcal{X}$ at $x_0$." ], "refs": [ "artin-definition-versal" ], "proofs": [ { "contents": [ "Since $U \\to S$ is locally of finite type (as a composition of such morphisms),", "we see that $\\Spec(k) \\to S$ is of finite type (again as a composition).", "Thus the statement makes sense. The equivalence of (1) and (2)", "is the definition of $x$ being versal at $u_0$.", "The equivalence of (1) and (3) is", "Artin's Axioms, Lemma \\ref{artin-lemma-versality-matches}.", "Thus (1), (2), and (3) are equivalent.", "\\medskip\\noindent", "If $x|_{U'}$ is smooth, then the functor", "$\\hat x : \\mathcal{F}_{U, k, u_0} \\to \\mathcal{F}_{\\mathcal{X}, k, x_0}$", "is smooth by Artin's Axioms, Lemma", "\\ref{artin-lemma-formally-smooth-on-deformation-categories}.", "Thus (4) implies (1), (2), and (3).", "For the converse, assume $x$ is versal at $u_0$.", "Choose a surjective smooth morphism $y : V \\to \\mathcal{X}$ where $V$", "is a scheme. Set $Z = V \\times_\\mathcal{X} U$ and pick a finite type", "point $z_0 \\in |Z|$ lying over $u_0$ (this is possible by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-type-points-surjective-morphism}).", "By Artin's Axioms, Lemma \\ref{artin-lemma-base-change-versal}", "the morphism $Z \\to V$ is smooth at $z_0$.", "By definition we can find an open neighbourhood $W \\subset Z$", "of $z_0$ such that $W \\to V$ is smooth. Since $Z \\to U$ is open,", "let $U' \\subset U$ be the image of $W$. Then we see that", "$U' \\to \\mathcal{X}$ is smooth by our definition of smooth morphisms", "of stacks.", "\\medskip\\noindent", "The final statement follows from the definitions as", "$\\mathcal{O}_{U, u_0}^\\wedge$", "prorepresents $\\mathcal{F}_{U, k, u_0}$." ], "refs": [ "artin-lemma-versality-matches", "artin-lemma-formally-smooth-on-deformation-categories", "spaces-morphisms-lemma-finite-type-points-surjective-morphism", "artin-lemma-base-change-versal" ], "ref_ids": [ 11368, 11352, 4825, 11371 ] } ], "ref_ids": [ 11422 ] }, { "id": 4670, "type": "theorem", "label": "stacks-geometry-lemma-characterize-smoothness", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-characterize-smoothness", "contents": [ "In Situation \\ref{situation-versal}. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$", "be a morphism such that $\\Spec(k) \\to S$ is of finite type with image $s$.", "Let $A$ be a versal ring to $\\mathcal{X}$ at $x_0$. The following", "are equivalent", "\\begin{enumerate}", "\\item $x_0$ is in the smooth locus of $\\mathcal{X} \\to S$", "(Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-where-smooth}),", "\\item $\\mathcal{O}_{S, s} \\to A$ is formally smooth in the", "$\\mathfrak m_A$-adic topology, and", "\\item $\\mathcal{F}_{\\mathcal{X}, k, x_0}$ is unobstructed.", "\\end{enumerate}" ], "refs": [ "stacks-morphisms-lemma-where-smooth" ], "proofs": [ { "contents": [ "The equivalence of (2) and (3) follows immediately from", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-smooth-power-series-classical}.", "\\medskip\\noindent", "Note that $\\mathcal{O}_{S, s} \\to A$ is formally smooth in the", "$\\mathfrak m_A$-adic topology if and only if", "$\\mathcal{O}_{S, s} \\to A' = A[[t_1, \\ldots, t_r]]$", "is formally smooth in the $\\mathfrak m_{A'}$-adic topology.", "Hence (2) does not depend on the choice", "of our versal ring by Lemma \\ref{lemma-versal-ring}.", "Next, let $l/k$ be a finite extension and choose", "$A \\to A'$ as in Lemma \\ref{lemma-versal-ring-field-extension}.", "If $\\mathcal{O}_{S, s} \\to A$ is formally smooth in the", "$\\mathfrak m_A$-adic topology, then $\\mathcal{O}_{S, s} \\to A'$", "is formally smooth in the $\\mathfrak m_{A'}$-adic topology, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-compose-formally-smooth}.", "Conversely, if $\\mathcal{O}_{S, s} \\to A'$", "is formally smooth in the $\\mathfrak m_{A'}$-adic topology,", "then $\\mathcal{O}_{S, s}^\\wedge \\to A'$ and $A \\to A'$ are regular", "(More on Algebra, Proposition \\ref{more-algebra-proposition-fs-regular})", "and hence $\\mathcal{O}_{S, s}^\\wedge \\to A$ is regular", "(More on Algebra, Lemma \\ref{more-algebra-lemma-regular-permanence}),", "hence $\\mathcal{O}_{S, s} \\to A$ is formally smooth in the", "$\\mathfrak m_A$-adic topology (same lemma as before).", "Thus the equivalence of (2) and (1) holds for", "$k$ and $x_0$ if and only if it holds for $l$ and $x_{0, l}$.", "\\medskip\\noindent", "Choose a scheme $U$ and a smooth morphism $U \\to \\mathcal{X}$ such", "that $\\Spec(k) \\times_\\mathcal{X} U$ is nonempty. Choose a finite", "extension $l/k$ and a point $w_0 : \\Spec(l) \\to \\Spec(k) \\times_\\mathcal{X} U$.", "Let $u_0 \\in U$ be the image of $w_0$.", "We may apply the above to $l/k$ and to $l/\\kappa(u_0)$", "to see that we can reduce to $u_0$. Thus we may assume", "$A = \\mathcal{O}_{U, u_0}^\\wedge$, see", "Lemma \\ref{lemma-compare-versal-ring-completion}.", "Observe that $x_0$ is in the smooth locus of $\\mathcal{X} \\to S$", "if and only if $u_0$ is in the smooth locus of $U \\to S$, see for example", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-where-smooth}.", "Thus the equivalence of (1) and (2) follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-formally-smooth-finite-type}." ], "refs": [ "formal-defos-lemma-smooth-power-series-classical", "stacks-geometry-lemma-versal-ring", "stacks-geometry-lemma-versal-ring-field-extension", "more-algebra-lemma-compose-formally-smooth", "more-algebra-proposition-fs-regular", "more-algebra-lemma-regular-permanence", "stacks-geometry-lemma-compare-versal-ring-completion", "stacks-morphisms-lemma-where-smooth", "more-algebra-lemma-formally-smooth-finite-type" ], "ref_ids": [ 3437, 4667, 4668, 10018, 10579, 10039, 4669, 7543, 10026 ] } ], "ref_ids": [ 7543 ] }, { "id": 4671, "type": "theorem", "label": "stacks-geometry-lemma-Artin-approximation-by-smooth-morphism", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-Artin-approximation-by-smooth-morphism", "contents": [ "In Situation \\ref{situation-versal}. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$", "be a morphism such that $\\Spec(k) \\to S$ is of finite type with image $s$.", "Let $A$ be a versal ring to $\\mathcal{X}$ at $x_0$.", "If $\\mathcal{O}_{S, s}$ is a G-ring, then we may find a smooth morphism", "$U \\to \\mathcal{X}$ whose source is a scheme and a point", "$u_0 \\in U$ with residue field $k$, such that", "\\begin{enumerate}", "\\item $\\Spec(k) \\to U \\to \\mathcal{X}$ coincides with the given morphism $x_0$,", "\\item there is an isomorphism $\\mathcal{O}_{U, u_0}^\\wedge \\cong A$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $(\\xi_n, f_n)$ be the versal formal object over $A$.", "By Artin's Axioms, Lemma \\ref{artin-lemma-effective}", "we know that $\\xi = (A, \\xi_n, f_n)$ is effective.", "By assumption $\\mathcal{X}$ is locally of finite presentation over $S$", "(use Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-noetherian-finite-type-finite-presentation}),", "and hence limit preserving by Limits of Stacks, Proposition", "\\ref{stacks-limits-proposition-characterize-locally-finite-presentation}.", "Thus Artin approximation as in", "Artin's Axioms, Lemma \\ref{artin-lemma-approximate-versal}", "shows that we may find a morphism $U \\to \\mathcal{X}$ with", "source a finite type $S$-scheme, containing a point $u_0 \\in U$", "of residue field $k$ satisfying (1) and (2) such that $U \\to \\mathcal{X}$", "is versal at $u_0$. By Lemma \\ref{lemma-compare-versal-ring-completion}", "after shrinking $U$ we may assume $U \\to \\mathcal{X}$ is smooth." ], "refs": [ "artin-lemma-effective", "stacks-morphisms-lemma-noetherian-finite-type-finite-presentation", "stacks-limits-proposition-characterize-locally-finite-presentation", "artin-lemma-approximate-versal", "stacks-geometry-lemma-compare-versal-ring-completion" ], "ref_ids": [ 11362, 7503, 15024, 11372, 4669 ] } ], "ref_ids": [] }, { "id": 4672, "type": "theorem", "label": "stacks-geometry-lemma-versal-ring-flat", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-versal-ring-flat", "contents": [ "In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$", "be a morphism, where $k$ is a finite type field over $S$.", "Let $A$ be a versal ring to $\\mathcal{X}$ at $x_0$.", "Then the morphism $\\Spec(A) \\to \\mathcal{X}$ of", "Remark \\ref{remark-upgrade} is flat." ], "refs": [ "stacks-geometry-remark-upgrade" ], "proofs": [ { "contents": [ "If the local ring of $S$ at the image point is a G-ring, then this", "follows immediately from", "Lemma \\ref{lemma-Artin-approximation-by-smooth-morphism}", "and the fact that the map from a Noetherian local ring to its", "completion is flat. In general we prove it as follows.", "\\medskip\\noindent", "Step I. If $A$ and $A'$ are two versal rings to $\\mathcal{X}$ at $x_0$,", "then the result is true for $A$ if and only if it is true for $A'$.", "Namely, after possible swapping $A$ and $A'$, we may assume there is", "a formally smooth map $\\varphi : A \\to A'$ such that the composition", "$$", "\\Spec(A') \\to \\Spec(A) \\to \\mathcal{X}", "$$", "is the morphism $\\Spec(A') \\to \\mathcal{X}$, see", "Lemma \\ref{lemma-versal-ring} and", "Remark \\ref{remark-upgrade}.", "Since $A \\to A'$ is faithfully flat we obtain the equivalence from", "Morphisms of Stacks, Lemmas \\ref{stacks-morphisms-lemma-composition-flat} and", "\\ref{stacks-morphisms-lemma-flat-permanence}.", "\\medskip\\noindent", "Step II. Let $l/k$ be a finite extension of fields. Let", "$x_{l, 0} : \\Spec(l) \\to \\mathcal{X}$ be the induced morphism.", "Let $A$ be a versal ring to $\\mathcal{X}$ at $x_0$ and", "let $A \\to A'$ be as in Lemma \\ref{lemma-versal-ring-field-extension}.", "Then again the composition", "$$", "\\Spec(A') \\to \\Spec(A) \\to \\mathcal{X}", "$$", "is the morphism $\\Spec(A') \\to \\mathcal{X}$, see Remark \\ref{remark-upgrade}.", "Arguing as before and using step I to see choice of versal rings", "is irrelevant, we see that the lemma holds for $x_0$ if and only", "if it holds for $x_{l, 0}$.", "\\medskip\\noindent", "Step III. Choose a scheme $U$ and a surjective smooth morphism", "$U \\to \\mathcal{X}$. Then we can choose a finite type point", "$z_0$ on $Z = U \\times_\\mathcal{X} x_0$ (this is a nonempty", "algebraic space). Let $u_0 \\in U$ be the image of $z_0$ in $U$.", "Choose a scheme and a surjective \\'etale map $W \\to Z$", "such that $z_0$ is the image of a closed point $w_0 \\in W$", "(see Morphisms of Spaces, Section", "\\ref{spaces-morphisms-section-points-finite-type}).", "Since $W \\to \\Spec(k)$ and $W \\to U$ are of finite type,", "we see that $\\kappa(w_0)/k$ and $\\kappa(w_0)/\\kappa(u_0)$", "are finite extensions of fields", "(see Morphisms, Section \\ref{morphisms-section-points-finite-type}).", "Applying Step II twice we may replace $x_0$ by", "$u_0 \\to U \\to \\mathcal{X}$. Then we see our morphism is the composition", "$$", "\\Spec(\\mathcal{O}_{U, u_0}^\\wedge) \\to U \\to \\mathcal{X}", "$$", "The first arrow is flat because completion of Noetherian local rings", "are flat (Algebra, Lemma \\ref{algebra-lemma-completion-flat})", "and the second arrow is flat as", "a smooth morphism is flat. The composition is flat as composition", "preserves flatness." ], "refs": [ "stacks-geometry-lemma-Artin-approximation-by-smooth-morphism", "stacks-geometry-lemma-versal-ring", "stacks-geometry-remark-upgrade", "stacks-morphisms-lemma-composition-flat", "stacks-morphisms-lemma-flat-permanence", "stacks-geometry-lemma-versal-ring-field-extension", "stacks-geometry-remark-upgrade", "algebra-lemma-completion-flat" ], "ref_ids": [ 4671, 4667, 4704, 7494, 7497, 4668, 4704, 870 ] } ], "ref_ids": [ 4704 ] }, { "id": 4673, "type": "theorem", "label": "stacks-geometry-lemma-map-of-components", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-map-of-components", "contents": [ "Let $f : U \\to \\mathcal{X}$ be a smooth morphism from a scheme", "to a locally Noetherian algebraic stack. The closure of the image of any", "irreducible component of $|U|$ is an irreducible component of $|\\mathcal{X}|$.", "If $U \\to \\mathcal{X}$ is surjective, then all irreducible components of", "$|\\mathcal{X}|$ are obtained in this way." ], "refs": [], "proofs": [ { "contents": [ "The map $|U| \\to |\\mathcal{X}|$ is continuous and open by", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-topology-points}.", "Let $T \\subset |U|$ be an irreducible component. Since $U$ is locally", "Noetherian, we can find a nonempty affine open $W \\subset U$ contained in $T$.", "Then $f(T) \\subset |\\mathcal{X}|$ is irreducible and contains the", "nonempty open subset $f(W)$. Thus the closure of $f(T)$ is irreducible and", "contains a nonempty open. It follows that this closure is an irreducible", "component.", "\\medskip\\noindent", "Assume $U \\to \\mathcal{X}$ is surjective and let $Z \\subset |\\mathcal{X}|$", "be an irreducible component. Choose a Noetherian open subset $V$", "of $|\\mathcal{X}|$ meeting $Z$. After removing the other irreducible", "components from $V$ we may assume that $V \\subset Z$.", "Take an irreducible component of the nonempty", "open $f^{-1}(V) \\subset |U|$ and let $T \\subset |U|$ be its closure.", "This is an irreducible component of $|U|$ and the closure of $f(T)$", "must agree with $Z$ by our choice of $T$." ], "refs": [ "stacks-properties-lemma-topology-points" ], "ref_ids": [ 8867 ] } ], "ref_ids": [] }, { "id": 4674, "type": "theorem", "label": "stacks-geometry-lemma-multiplicities", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-multiplicities", "contents": [ "Let $U \\to X$ be a smooth morphism of locally Noetherian schemes.", "Let $T'$ is an irreducible component of $U$. Let $T$ be the", "irreducible component of $X$ obtained as the closure of the", "image of $T'$. Then $m_{T', U} = m_{T, X}$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\xi'$ for the generic point of $T'$, and $\\xi$ for the", "generic point of $T$. Let $A = \\mathcal{O}_{X, \\xi}$ and", "$B = \\mathcal{O}_{U, \\xi'}$. We need to show that", "$\\text{length}_A A = \\text{length}_B B$. Since", "$A \\to B$ is a flat local homomorphism of rings", "(since smooth morphisms are flat), we have", "$$", "\\text{length}_A(A) \\text{length}_B(B/\\mathfrak m_A B) =", "\\text{length}_B(B)", "$$", "by Algebra, Lemma \\ref{algebra-lemma-pullback-module}. Thus it suffices", "to show $\\mathfrak m_A B = \\mathfrak m_B$, or equivalently, that", "$B/\\mathfrak m_A B$ is reduced. Since $U \\to X$ is smooth,", "so is its base change $U_{\\xi} \\to \\Spec \\kappa(\\xi)$. As $U_{\\xi}$ is a", "smooth scheme over a field, it is reduced, and thus so its local ring", "at any point", "(Varieties, Lemma \\ref{varieties-lemma-smooth-geometrically-normal}).", "In particular,", "$$", "B/\\mathfrak m_A B =", "\\mathcal{O}_{U, \\xi'}/\\mathfrak m_{X, \\xi}\\mathcal{O}_{U, \\xi'} =", "\\mathcal{O}_{U_\\xi, \\xi'}", "$$", "is reduced, as required." ], "refs": [ "algebra-lemma-pullback-module", "varieties-lemma-smooth-geometrically-normal" ], "ref_ids": [ 640, 11005 ] } ], "ref_ids": [] }, { "id": 4675, "type": "theorem", "label": "stacks-geometry-lemma-multiplicity", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-multiplicity", "contents": [ "Let $U_1 \\to \\mathcal{X}$ and $U_2 \\to \\mathcal{X}$ be two smooth", "morphisms from schemes to a locally Noetherian algebraic stack $\\mathcal{X}$.", "Let $T_1'$ and $T_2'$ be irreducible components of $|U_1|$", "and $|U_2|$ respectively. Assume the closures of the images of", "$T_1'$ and $T_2'$ are the same irreducible component $T$ of $|\\mathcal{X}|$.", "Then $m_{T_1', U_1} = m_{T_2', U_2}$." ], "refs": [], "proofs": [ { "contents": [ "Let $V_1$ and $V_2$ be dense subsets of $T_1'$ and $T'_2$, respectively,", "that are open in $U_1$ and $U_2$ respectively (see proof of", "Lemma \\ref{lemma-map-of-components}).", "The images of $|V_1|$ and $|V_2|$ in $|\\mathcal{X}|$ are non-empty open", "subsets of the irreducible subset $T$, and therefore have non-empty", "intersection. By", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-points-cartesian},", "the map $|V_1 \\times_\\mathcal{X} V_2| \\to |V_1| \\times_{|\\mathcal{X}|} |V_2|$", "is surjective. Consequently $V_1 \\times_\\mathcal{X} V_2$", "is a non-empty algebraic space; we may therefore choose an", "\\'etale surjection $V \\to V_1 \\times_\\mathcal{X} V_2$", "whose source is a (non-empty) scheme.", "If we let $T'$ be any irreducible component of $V$,", "then Lemma \\ref{lemma-map-of-components} shows that the closure of", "the image of $T'$ in $U_1$ (respectively $U_2$) is equal to $T'_1$", "(respectively $T'_2$).", "\\medskip\\noindent", "Applying Lemma \\ref{lemma-multiplicities} twice we find", "that", "$$", "m_{T_1', U_1} = m_{T', V} = m_{T_2', U_2},", "$$", "as required." ], "refs": [ "stacks-geometry-lemma-map-of-components", "stacks-properties-lemma-points-cartesian", "stacks-geometry-lemma-map-of-components", "stacks-geometry-lemma-multiplicities" ], "ref_ids": [ 4673, 8864, 4673, 4674 ] } ], "ref_ids": [] }, { "id": 4676, "type": "theorem", "label": "stacks-geometry-lemma-branches", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-branches", "contents": [ "In the situation of Definition \\ref{definition-formal-branches}", "there is a canonical surjection from the set of formal branches of", "$\\mathcal{X}$ through $x_0$ to the set of irreducible components of", "$|\\mathcal{X}|$ containing $x_0$ in $|\\mathcal{X}|$." ], "refs": [ "stacks-geometry-definition-formal-branches" ], "proofs": [ { "contents": [ "Let $A$ be as in Definition \\ref{definition-formal-branches} and", "let $\\Spec(A) \\to \\mathcal{X}$ be as in Remark \\ref{remark-upgrade}.", "We claim that the generic point of an irreducible component of", "$\\Spec(A)$ maps to a generic point of an irreducible component", "of $|\\mathcal{X}|$. Choose a scheme $U$ and a surjective", "smooth morphism $U \\to \\mathcal{X}$. Consider the diagram", "$$", "\\xymatrix{", "\\Spec(A) \\times_\\mathcal{X} U \\ar[d]_p \\ar[r]_-q & U \\ar[d]^f \\\\", "\\Spec(A) \\ar[r]^j & \\mathcal{X}", "}", "$$", "By Lemma \\ref{lemma-versal-ring-flat} we see that $j$ is flat.", "Hence $q$ is flat. On the other hand, $f$ is surjective smooth", "hence $p$ is surjective smooth. This implies that any generic point", "$\\eta \\in \\Spec(A)$ of an irreducible component is the image of a", "codimension $0$ point $\\eta'$ of the algebraic space", "$\\Spec(A) \\times_\\mathcal{X} U$ (see", "Properties of Spaces, Section \\ref{spaces-properties-section-generic-points}", "for notation and use going down on \\'etale local rings).", "Since $q$ is flat, $q(\\eta')$ is a codimension $0$ point of $U$", "(same argument).", "Since $U$ is a scheme, $q(\\eta')$ is the generic point", "of an irreducible component of $U$. Thus the closure of the image", "of $q(\\eta')$ in $|\\mathcal{X}|$ is an irreducible component", "by Lemma \\ref{lemma-map-of-components} as claimed.", "\\medskip\\noindent", "Clearly the claim provides a mechanism for defining the desired map.", "To see that it is surjective, we choose $u_0 \\in U$ mapping to", "$x_0$ in $|\\mathcal{X}|$. Choose an affine open $U' \\subset U$", "neighbourhood of $u_0$. After shrinking $U'$ we may assume every", "irreducible component of $U'$ passes through $u_0$. Then we may", "replace $\\mathcal{X}$ by the open substack corresponding to the", "image of $|U'| \\to |\\mathcal{X}|$. Thus we may assume $U$ is affine", "has a point $u_0$ mapping to $x_0 \\in |\\mathcal{X}|$", "and every irreducible component of $U$ passes through $u_0$.", "By Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-points-cartesian}", "there is a point $t \\in |\\Spec(A) \\times_\\mathcal{X} U|$", "mapping to the closed point of $\\Spec(A)$ and to $u_0$.", "Using going down for the flat local ring homomorphisms", "$$", "A \\longrightarrow", "\\mathcal{O}_{\\Spec(A) \\times_\\mathcal{X} U, \\overline{t}}", "\\longleftarrow \\mathcal{O}_{U, u_0}", "$$", "we see that every minimal prime of $\\mathcal{O}_{U, u_0}$", "is the image of a minimal prime of the local ring in the middle", "and such a minimal prime maps to a minimal prime of $A$.", "This proves the surjectivity. Some details omitted." ], "refs": [ "stacks-geometry-definition-formal-branches", "stacks-geometry-remark-upgrade", "stacks-geometry-lemma-versal-ring-flat", "stacks-geometry-lemma-map-of-components", "stacks-properties-lemma-points-cartesian" ], "ref_ids": [ 4698, 4704, 4672, 4673, 8864 ] } ], "ref_ids": [ 4698 ] }, { "id": 4677, "type": "theorem", "label": "stacks-geometry-lemma-branches-multiplicity", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-branches-multiplicity", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack locally of finite type", "over a locally Noetherian scheme $S$. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$", "is a morphism where $k$ is a field of finite type over $S$ with", "image $s \\in S$. If $\\mathcal{O}_{S, s}$ is a G-ring, then", "the map of Lemma \\ref{lemma-branches} preserves multiplicities." ], "refs": [ "stacks-geometry-lemma-branches" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-Artin-approximation-by-smooth-morphism} we", "may assume there is a smooth morphism $U \\to \\mathcal{X}$", "where $U$ is a scheme and a $k$-valued point $u_0$ of $U$", "such that $\\mathcal{O}_{U, u_0}^\\wedge$ is a versal ring to", "$\\mathcal{X}$ at $x_0$. By construction of our map in", "the proof of Lemma \\ref{lemma-branches} (which simplifies", "greatly because $A = \\mathcal{O}_{U, u_0}^\\wedge$) we find", "that it suffices to show: the multiplicity of an irreducible", "component of $U$ passing through $u_0$ is the same as the", "multiplicity of any irreducible component of", "$\\Spec(\\mathcal{O}_{U, u_0}^\\wedge)$ mapping into it.", "\\medskip\\noindent", "Translated into commutative algebra we find the following:", "Let $C = \\mathcal{O}_{U, u_0}$. This is essentially of finite type", "over $\\mathcal{O}_{S, s}$ and hence is a G-ring (More on Algebra,", "Proposition \\ref{more-algebra-proposition-finite-type-over-G-ring}).", "Then $A = C^\\wedge$. Therefore $C \\to A$ is a regular ring map.", "Let $\\mathfrak q \\subset C$ be a minimal prime and let $\\mathfrak p \\subset A$", "be a minimal prime lying over $\\mathfrak q$. Then", "$$", "R = C_\\mathfrak p \\longrightarrow A_\\mathfrak p = R'", "$$", "is a regular ring map of Artinian local rings. For such a ring map", "it is always the case that", "$$", "\\text{length}_R R = \\text{length}_{R'} R'", "$$", "This is what we have to show because the left hand side", "is the multiplicity of our component on $U$ and the right hand", "side is the multiplicity of our component on $\\Spec(A)$.", "To see the equality, first we use that", "$$", "\\text{length}_R(R) \\text{length}_{R'}(R'/\\mathfrak m_R R') =", "\\text{length}_{R'}(R')", "$$", "by Algebra, Lemma \\ref{algebra-lemma-pullback-module}. Thus it suffices", "to show $\\mathfrak m_R R' = \\mathfrak m_{R'}$, which is a consequence", "of being a regular homomorphism of zero dimensional local rings." ], "refs": [ "stacks-geometry-lemma-Artin-approximation-by-smooth-morphism", "stacks-geometry-lemma-branches", "more-algebra-proposition-finite-type-over-G-ring", "algebra-lemma-pullback-module" ], "ref_ids": [ 4671, 4676, 10581, 640 ] } ], "ref_ids": [ 4676 ] }, { "id": 4678, "type": "theorem", "label": "stacks-geometry-lemma-behaviour-of-dimensions-wrt-smooth-morphisms", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-behaviour-of-dimensions-wrt-smooth-morphisms", "contents": [ "If $f: U \\to X$ is a smooth morphism of locally Noetherian algebraic", "spaces, and", "if $u \\in |U|$ with image $x \\in |X|$, then", "$$", "\\dim_u (U) = \\dim_x(X) + \\dim_{u} (U_x)", "$$", "where $\\dim_u (U_x)$ is defined via", "Definition \\ref{definition-relative-dimension}." ], "refs": [ "stacks-geometry-definition-relative-dimension" ], "proofs": [ { "contents": [ "See Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-smoothness-dimension-spaces}", "noting that the definition of $\\dim_u (U_x)$ used here coincides with", "the definition used there, by Remark \\ref{remark-relative-dimension} (2)." ], "refs": [ "spaces-morphisms-lemma-smoothness-dimension-spaces", "stacks-geometry-remark-relative-dimension" ], "ref_ids": [ 4894, 4707 ] } ], "ref_ids": [ 4700 ] }, { "id": 4679, "type": "theorem", "label": "stacks-geometry-lemma-dimension-for-stacks", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-dimension-for-stacks", "contents": [ "If $\\mathcal{X}$ is a locally Noetherian algebraic stack and", "$x \\in |\\mathcal{X}|$. Let $U \\to \\mathcal{X}$ be a smooth morphism", "from an algebraic space to $\\mathcal{X}$, let $u$ be any point of $|U|$", "mapping to $x$. Then we have", "$$", "\\dim_x(\\mathcal{X}) = \\dim_u(U) - \\dim_{u}(U_x)", "$$", "where the relative dimension $\\dim_u(U_x)$ is defined", "by Definition \\ref{definition-relative-dimension} and the", "dimension of $\\mathcal{X}$ at $x$ is as in", "Properties of Stacks, Definition", "\\ref{stacks-properties-definition-dimension-at-point}." ], "refs": [ "stacks-geometry-definition-relative-dimension", "stacks-properties-definition-dimension-at-point" ], "proofs": [ { "contents": [ "Lemma \\ref{lemma-behaviour-of-dimensions-wrt-smooth-morphisms}", "can be used to verify that the right hand side", "$\\dim_u(U) + \\dim_u(U_x)$ is independent of the choice", "of the smooth morphism $U \\to \\mathcal{X}$ and $u \\in |U|$.", "We omit the details. In particular, we may assume $U$ is a scheme.", "In this case we can compute $\\dim_u(U_x)$", "by choosing the representative of $x$", "to be the composite $\\Spec \\kappa(u) \\to U \\to \\mathcal{X}$, where", "the first morphism is the canonical one with image $u \\in U$.", "Then, if we write $R = U \\times_{\\mathcal{X}} U$, and let", "$e : U \\to R$ denote the diagonal morphism, the invariance of", "relative dimension under base-change shows that", "$\\dim_u(U_x) = \\dim_{e(u)}(R_u)$. Thus we see that", "the right hand side is equal to", "$\\dim_u (U) - \\dim_{e(u)}(R_u) = \\dim_x(\\mathcal{X})$ as desired." ], "refs": [ "stacks-geometry-lemma-behaviour-of-dimensions-wrt-smooth-morphisms" ], "ref_ids": [ 4678 ] } ], "ref_ids": [ 4700, 8924 ] }, { "id": 4680, "type": "theorem", "label": "stacks-geometry-lemma-base-change-invariance-of-relative-dimension", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-base-change-invariance-of-relative-dimension", "contents": [ "Suppose given", "a Cartesian square of morphisms of locally Noetherian stacks", "$$", "\\xymatrix{", "\\mathcal{T}' \\ar[d]\\ar[r] & \\mathcal{T} \\ar[d] \\\\", "\\mathcal{X}' \\ar[r] & \\mathcal{X}", "}", "$$", "in which the vertical morphisms are locally of finite type.", "If $t' \\in |\\mathcal{T}'|$,", "with images $t$, $x'$, and $x$ in $|\\mathcal{T}|$, $|\\mathcal{X}'|$, and", "$|\\mathcal{X}|$", "respectively, then $\\dim_{t'}(\\mathcal{T}'_{x'}) = \\dim_{t}(\\mathcal{T}_x).$" ], "refs": [], "proofs": [ { "contents": [ "Both sides can (by definition) be computed as the", "dimension of the same fibre product." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4681, "type": "theorem", "label": "stacks-geometry-lemma-behaviour-of-dimensions-wrt-smooth-morphisms-stacky", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-behaviour-of-dimensions-wrt-smooth-morphisms-stacky", "contents": [ "If $f: \\mathcal{U} \\to \\mathcal{X}$ is a smooth morphism of locally Noetherian", "algebraic stacks, and", "if $u \\in |\\mathcal{U}|$ with image $x \\in |\\mathcal{X}|$,", "then", "$$", "\\dim_u (\\mathcal{U}) = \\dim_x(\\mathcal{X}) + \\dim_{u} (\\mathcal{U}_x).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose a smooth surjective morphism $V \\to \\mathcal{U}$ whose source", "is a scheme, and let $v\\in |V|$ be a point mapping to $u$.", "Then the composite $V \\to \\mathcal{U} \\to \\mathcal{X}$ is also smooth,", "and by Lemma \\ref{lemma-behaviour-of-dimensions-wrt-smooth-morphisms}", "we have $\\dim_x(\\mathcal{X}) = \\dim_v(V) - \\dim_v(V_x),$", "while $\\dim_u(\\mathcal{U}) = \\dim_v(V) - \\dim_v(V_u).$", "Thus", "$$", "\\dim_u(\\mathcal{U}) - \\dim_x(\\mathcal{X}) = \\dim_v (V_x) - \\dim_v (V_u).", "$$", "\\medskip\\noindent", "Choose a representative $\\Spec k \\to \\mathcal{X}$ of $x$", "and choose a point $v' \\in | V \\times_{\\mathcal{X}} \\Spec k|$ lying over", "$v$, with image $u'$ in $|\\mathcal{U}\\times_{\\mathcal{X}} \\Spec k|$;", "then by definition", "$\\dim_u(\\mathcal{U}_x) = \\dim_{u'}(\\mathcal{U}\\times_{\\mathcal{X}} \\Spec k),$", "and", "$\\dim_v(V_x) = \\dim_{v'}(V\\times_{\\mathcal{X}} \\Spec k).$", "\\medskip\\noindent", "Now $V\\times_{\\mathcal{X}} \\Spec k \\to \\mathcal{U}\\times_{\\mathcal{X}}\\Spec k$", "is a smooth surjective morphism (being the base-change", "of such a morphism) whose source is an algebraic space", "(since $V$ and $\\Spec k$ are schemes, and $\\mathcal{X}$", "is an algebraic stack). Thus, again by definition,", "we have", "\\begin{align*}", "\\dim_{u'}(\\mathcal{U}\\times_{\\mathcal{X}} \\Spec k)", "& =", "\\dim_{v'}(V\\times_{\\mathcal{X}} \\Spec k) -", "\\dim_{v'}(V \\times_{\\mathcal{X}} \\Spec k)_{u'}) \\\\", "& = \\dim_v(V_x) -", "\\dim_{v'}( (V\\times_{\\mathcal{X}} \\Spec k)_{u'}).", "\\end{align*}", "Now $V\\times_{\\mathcal{X}} \\Spec k \\cong", "V\\times_{\\mathcal{U}} (\\mathcal{U}\\times_{\\mathcal{X}} \\Spec k),$", "and so", "Lemma \\ref{lemma-base-change-invariance-of-relative-dimension}", "shows that", "$\\dim_{v'}((V\\times_{\\mathcal{X}} \\Spec k)_{u'}) = \\dim_v(V_u).$", "Putting everything together, we find that", "$$", "\\dim_u(\\mathcal{U}) - \\dim_x(\\mathcal{X}) =", "\\dim_u(\\mathcal{U}_x),", "$$", "as required." ], "refs": [ "stacks-geometry-lemma-behaviour-of-dimensions-wrt-smooth-morphisms", "stacks-geometry-lemma-base-change-invariance-of-relative-dimension" ], "ref_ids": [ 4678, 4680 ] } ], "ref_ids": [] }, { "id": 4682, "type": "theorem", "label": "stacks-geometry-lemma-relative-dimension-is-semi-continuous", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-relative-dimension-is-semi-continuous", "contents": [ "Let $f: \\mathcal{T} \\to \\mathcal{X}$ be a locally of finite type morphism of", "algebraic stacks.", "\\begin{enumerate}", "\\item", "The function $t \\mapsto \\dim_t(\\mathcal{T}_{f(t)})$ is upper semi-continuous", "on $|\\mathcal{T}|$.", "\\item If $f$ is smooth, then", "the function $t \\mapsto \\dim_t(\\mathcal{T}_{f(t)})$ is locally constant", "on $|\\mathcal{T}|$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose to begin with that $\\mathcal{T}$ is a scheme $T$,", "let $U \\to \\mathcal{X}$ be a smooth surjective morphism whose source", "is a scheme, and let $T' = T \\times_{\\mathcal{X}} U$.", "Let $f': T' \\to U$ be the pull-back of $f$ over $U$,", "and let $g: T' \\to T$ be the projection.", "\\medskip\\noindent", "Lemma \\ref{lemma-base-change-invariance-of-relative-dimension}", "shows that $\\dim_{t'}(T'_{f'(t')}) = \\dim_{g(t')}(T_{f(g(t'))}),$", "for $t' \\in T'$, while,", "since $g$ is smooth and surjective (being the base-change", "of a smooth surjective morphism) the map induced by $g$ on underlying", "topological spaces is continuous and open", "(by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-topology-points}), and", "surjective. Thus it suffices to note that part (1) for the morphism $f'$", "follows from", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-openness-bounded-dimension-fibres}, and part (2)", "from either of Morphisms, Lemma", "\\ref{morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension}", "or", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-omega-finite-locally-free}", "(each of which gives the result for schemes, from which", "the analogous results for algebraic spaces can", "be deduced exactly as in", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-openness-bounded-dimension-fibres}.", "\\medskip\\noindent", "Now return to the general case,", "and choose a smooth surjective morphism", "$h:V \\to \\mathcal{T}$ whose source is a scheme.", "If $v \\in V$, then, essentially by definition,", "we have", "$$", "\\dim_{h(v)}(\\mathcal{T}_{f(h(v))}) =", "\\dim_{v}(V_{f(h(v))}) - \\dim_{v}(V_{h(v)}).", "$$", "Since $V$ is a scheme, we have proved that the first", "of the terms on the right hand side of this equality", "is upper semi-continuous (and even locally", "constant if $f$ is smooth), while the second term is", "in fact locally constant.", "Thus their difference is upper semi-continuous", "(and locally constant if $f$ is smooth),", "and hence the function", "$\\dim_{h(v)}(\\mathcal{T}_{f(h(v))})$", "is upper semi-continuous on $|V|$ (and locally", "constant if $f$ is smooth).", "Since the morphism $|V| \\to |\\mathcal{T}|$ is open and surjective,", "the lemma follows." ], "refs": [ "stacks-geometry-lemma-base-change-invariance-of-relative-dimension", "spaces-properties-lemma-topology-points", "spaces-morphisms-lemma-openness-bounded-dimension-fibres", "morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension", "morphisms-lemma-smooth-omega-finite-locally-free", "spaces-morphisms-lemma-openness-bounded-dimension-fibres" ], "ref_ids": [ 4680, 11822, 4873, 5286, 5334, 4873 ] } ], "ref_ids": [] }, { "id": 4683, "type": "theorem", "label": "stacks-geometry-lemma-dimension-achieved-by-finite-type-point", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-dimension-achieved-by-finite-type-point", "contents": [ "If $X$ is a finite dimensional scheme,", "then there exists a closed (and hence finite type) point $x \\in X$", "such that $\\dim_x X = \\dim X$." ], "refs": [], "proofs": [ { "contents": [ "Let $d = \\dim X$,", "and choose a maximal strictly decreasing", "chain of irreducible closed subsets of $X$,", "say", "\\begin{equation}", "\\label{equation-maximal-chain}", "Z_0 \\supset Z_1 \\supset \\ldots \\supset Z_d.", "\\end{equation}", "The subset $Z_d$ is a minimal irreducible closed subset of $X$,", "and thus any point of $Z_d$ is a generic point of $Z_d$.", "Since the underlying topological space of the scheme $X$ is sober,", "we conclude that $Z_d$ is a singleton, consisting of a single", "closed point $x \\in X$.", "If $U$ is", "any neighbourhood of $x$, then", "the chain", "$$", "U\\cap Z_0 \\supset U\\cap Z_1 \\supset \\ldots \\supset U\\cap Z_d = Z_d =", "\\{x\\}", "$$", "is then a strictly descending chain of irreducible", "closed subsets of $U$, showing that $\\dim U \\geq d$.", "Thus we find that $\\dim_x X \\geq d$. The other inequality", "being obvious, the lemma is proved." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4684, "type": "theorem", "label": "stacks-geometry-lemma-constancy-of-dimension", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-constancy-of-dimension", "contents": [ "If $X$ is an irreducible, Jacobson, catenary, and locally Noetherian", "scheme of finite dimension,", "then $\\dim U = \\dim X$ for every", "non-empty open subset $U$ of $X$.", "Equivalently, $\\dim_x X$ is a constant function on $X$." ], "refs": [], "proofs": [ { "contents": [ "The equivalence of the two claims follows directly from the", "definitions. Suppose, then, that $U\\subset X$ is a non-empty open", "subset.", "Certainly $\\dim U \\leq \\dim X$, and we have to show", "that $\\dim U \\geq \\dim X.$", "Write $d = \\dim X$, and choose a maximal strictly", "decreasing chain of irreducible closed subsets", "of $X$, say", "$$", "X = Z_0 \\supset Z_1 \\supset \\ldots \\supset Z_d.", "$$", "Since $X$ is Jacobson, the minimal irreducible closed", "subset $Z_d$ is equal to $\\{x\\}$ for some closed", "point $x$.", "\\medskip\\noindent", "If $x \\in U,$ then", "$$", "U = U \\cap Z_0 \\supset U\\cap Z_1 \\supset \\ldots \\supset", "U\\cap Z_d = \\{x\\}", "$$", "is a strictly decreasing chain of irreducible closed", "subsets of $U$, and so we conclude that $\\dim U \\geq d$,", "as required. Thus we may suppose that $x \\not\\in U.$", "\\medskip\\noindent", "Consider the flat morphism $\\Spec \\mathcal{O}_{X,x} \\to X$.", "The non-empty (and hence dense) open subset $U$ of $X$", "pulls back to an open subset $V \\subset \\Spec \\mathcal{O}_{X,x}$.", "Replacing $U$ by a non-empty quasi-compact, and hence", "Noetherian, open subset, we may assume that the inclusion", "$U \\to X$ is a quasi-compact morphism. Since the", "formation of scheme-theoretic images of quasi-compact", "morphisms commutes with flat base-change", "Morphisms, Lemma", "\\ref{morphisms-lemma-flat-base-change-scheme-theoretic-image}", "we see that $V$ is dense in $\\Spec \\mathcal{O}_{X,x}$,", "and so in particular non-empty,", "and of course $x \\not\\in V.$ (Here we use $x$ also to denote", "the closed point of $\\Spec \\mathcal{O}_{X,x}$, since its image", "is equal to the given point $x \\in X$.)", "Now $\\Spec \\mathcal{O}_{X,x} \\setminus \\{x\\}$ is Jacobson", "Properties, Lemma", "\\ref{properties-lemma-complement-closed-point-Jacobson}", "and hence $V$ contains a closed point $z$", "of $\\Spec \\mathcal{O}_{X,x} \\setminus \\{x\\}$. The closure", "in $X$ of the image of $z$ is then an irreducible", "closed subset $Z$ of $X$ containing $x$, whose intersection", "with $U$ is non-empty, and", "for which there is no irreducible closed", "subset properly contained in $Z$", "and properly containing $\\{x\\}$", "(because pull-back to $\\Spec \\mathcal{O}_{X,x}$ induces", "a bijection between irreducible closed subsets of $X$", "containing $x$ and irreducible closed subsets of $\\Spec", "\\mathcal{O}_{X,x}$).", "Since $U \\cap Z$ is a non-empty closed subset of $U$,", "it contains a point $u$ that is closed in $X$ (since", "$X$ is Jacobson), and since $U\\cap Z$", "is a non-empty (and hence dense) open subset of the irreducible set $Z$", "(which contains a point not lying in $U$, namely $x$),", "the inclusion $\\{u\\} \\subset U\\cap Z$ is proper.", "\\medskip\\noindent", "As $X$ is catenary, the chain", "$$", "X = Z_0 \\supset Z \\supset \\{x\\} = Z_d", "$$", "can be refined to a chain of length $d+1$, which must then", "be of the form", "$$", "X = Z_0 \\supset W_1 \\supset \\ldots \\supset W_{d-1} = Z \\supset \\{x\\} = Z_d.", "$$", "Since $U\\cap Z$ is non-empty, we then find that", "$$", "U = U \\cap Z_0 \\supset U \\cap W_1\\supset \\ldots \\supset U\\cap W_{d-1}", "= U\\cap Z \\supset \\{u\\}", "$$", "is a strictly decreasing chain of irreducible closed subsets", "of $U$ of length $d+1$, showing that $\\dim U \\geq d$,", "as required." ], "refs": [ "morphisms-lemma-flat-base-change-scheme-theoretic-image", "properties-lemma-complement-closed-point-Jacobson" ], "ref_ids": [ 5273, 2965 ] } ], "ref_ids": [] }, { "id": 4685, "type": "theorem", "label": "stacks-geometry-lemma-catenary-covers", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-catenary-covers", "contents": [ "If $\\mathcal{X}$ is a pseudo-catenary locally Noetherian algebraic", "stack, and if $\\mathcal{Y} \\to \\mathcal{X}$ is a locally of finite type", "morphism,", "then there exists a smooth surjective morphism $V \\to \\mathcal{Y}$", "whose source is a universally catenary scheme; thus", "$\\mathcal{Y}$ is again pseudo-catenary." ], "refs": [], "proofs": [ { "contents": [ "By assumption we may find a smooth surjective morphism", "$U \\to \\mathcal{X}$ whose source is a universally catenary scheme.", "The base-change $U\\times_{\\mathcal{X}} \\mathcal{Y}$ is then an algebraic", "stack; let $V \\to U\\times_{\\mathcal{X}} \\mathcal{Y}$ be a smooth", "surjective morphism whose source is a scheme.", "The composite $V \\to U\\times_{\\mathcal{X}} \\mathcal{Y} \\to \\mathcal{Y}$ is then", "smooth and surjective (being a composite of smooth and", "surjective morphisms), while the morphism $V \\to U\\times_{\\mathcal{X}}", "\\mathcal{Y} \\to U$ is locally of finite type (being a composite", "of morphisms that are locally finite type). Since $U$", "is universally catenary, we see that $V$ is universally catenary", "(by Morphisms, Lemma", "\\ref{morphisms-lemma-universally-catenary-local}),", "as claimed." ], "refs": [ "morphisms-lemma-universally-catenary-local" ], "ref_ids": [ 5214 ] } ], "ref_ids": [] }, { "id": 4686, "type": "theorem", "label": "stacks-geometry-lemma-irreducible-implies-equidimensional", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-irreducible-implies-equidimensional", "contents": [ "If $\\mathcal{X}$ is", "a Jacobson, pseudo-catenary, and locally Noetherian algebraic stack", "for which $|\\mathcal{X}|$ is irreducible,", "then $\\dim_x(\\mathcal{X})$ is a constant function on $|\\mathcal{X}|$." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that $\\dim_x(\\mathcal{X})$ is locally constant on", "$|\\mathcal{X}|$,", "since it will then necessarily be constant (as $|\\mathcal{X}|$ is connected,", "being irreducible). Since $\\mathcal{X}$ is pseudo-catenary,", "we may find a smooth surjective morphism $U \\to \\mathcal{X}$ with $U$", "being a universally catenary scheme. If $\\{U_i\\}$ is an", "cover of $U$ by quasi-compact open subschemes, we may replace", "$U$ by $\\coprod U_i,$, and", "it suffices to show that", "the function $u \\mapsto \\dim_{f(u)}(\\mathcal{X})$ is locally constant on $U_i$.", "Since we check this for one $U_i$ at a time, we now drop the subscript,", "and write simply $U$ rather than $U_i$.", "Since $U$ is quasi-compact, it", "is the union of a finite number of irreducible components,", "say $T_1 \\cup \\ldots \\cup T_n$. Note that each $T_i$ is Jacobson,", "catenary, and locally Noetherian,", "being a closed subscheme of the Jacobson, catenary, and locally Noetherian", "scheme $U$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-behaviour-of-dimensions-wrt-smooth-morphisms}, we have", "$\\dim_{f(u)}(\\mathcal{X}) = \\dim_{u}(U) - \\dim_{u}(U_{f(u)}).$", "Lemma \\ref{lemma-relative-dimension-is-semi-continuous} (2)", "shows that the second term in the right hand expression is locally", "constant on $U$, as $f$ is smooth,", "and hence we must show that $\\dim_u(U)$", "is locally constant on $U$. Since $\\dim_u(U)$ is the maximum", "of the dimensions $\\dim_u T_i$, as $T_i$ ranges over the components", "of $U$ containing $u$, it suffices to show", "that if a point $u$ lies on two distinct components,", "say $T_i$ and $T_j$ (with $i \\neq j$),", "then $\\dim_u T_i = \\dim_u T_j$,", "and then to note that $t\\mapsto \\dim_t T$ is a constant", "function on an irreducible Jacobson,", "catenary, and locally Noetherian scheme $T$", "(as follows from Lemma \\ref{lemma-constancy-of-dimension}).", "\\medskip\\noindent", "Let $V = T_i \\setminus (\\bigcup_{i' \\neq i} T_{i'})$", "and $W = T_j \\setminus (\\bigcup_{i' \\neq j} T_{i'})$.", "Then each of $V$ and $W$ is a non-empty open subset of $U$,", "and so each has non-empty open image in $|\\mathcal{X}|$. As $|\\mathcal{X}|$ is", "irreducible,", "these two non-empty open subsets of $|\\mathcal{X}|$ have a non-empty", "intersection.", "Let $x$ be a point lying in this intersection, and let $v \\in V$ and", "$w\\in W$ be points mapping to $x$.", "We then find that", "$$", "\\dim T_i = \\dim V = \\dim_v (U) = \\dim_x (\\mathcal{X}) + \\dim_v (U_x)", "$$", "and similarly that", "$$", "\\dim T_j = \\dim W = \\dim_w (U) = \\dim_x (\\mathcal{X}) + \\dim_w (U_x).", "$$", "Since $u \\mapsto \\dim_u (U_{f(u)})$ is locally constant on $U$,", "and since $T_i \\cup T_j$ is connected (being the union of two irreducible,", "hence connected, sets that have non-empty intersection),", "we see that $\\dim_v (U_x) = \\dim_w(U_x)$,", "and hence, comparing the preceding two equations,", "that $\\dim T_i = \\dim T_j$, as required." ], "refs": [ "stacks-geometry-lemma-behaviour-of-dimensions-wrt-smooth-morphisms", "stacks-geometry-lemma-relative-dimension-is-semi-continuous", "stacks-geometry-lemma-constancy-of-dimension" ], "ref_ids": [ 4678, 4682, 4684 ] } ], "ref_ids": [] }, { "id": 4687, "type": "theorem", "label": "stacks-geometry-lemma-closed-immersions", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-closed-immersions", "contents": [ "If $\\mathcal{Z} \\hookrightarrow \\mathcal{X}$ is a closed immersion", "of locally Noetherian schemes,", "and if $z \\in |\\mathcal{Z}|$ has image $x \\in |\\mathcal{X}|$,", "then $\\dim_z (\\mathcal{Z}) \\leq \\dim_x(\\mathcal{X})$." ], "refs": [], "proofs": [ { "contents": [ "Choose a smooth surjective morphism", "$U\\to \\mathcal{X}$ whose source is a scheme;", "the base-changed morphism", "$V = U\\times_{\\mathcal{X}} \\mathcal{Z} \\to \\mathcal{Z}$", "is then also smooth and surjective, and the projection", "$V \\to U$ is a closed immersion.", "If $v \\in |V|$ maps to $z \\in |\\mathcal{Z}|$, and", "if we let $u$ denote the image of $v$ in $|U|$,", "then clearly", "$\\dim_v(V) \\leq \\dim_u(U)$,", "while", "$\\dim_v (V_z) = \\dim_u(U_x)$,", "by Lemma \\ref{lemma-base-change-invariance-of-relative-dimension}.", "Thus", "$$", "\\dim_z(\\mathcal{Z}) = \\dim_v(V) - \\dim_v(V_z)", "\\leq \\dim_u(U) - \\dim_u(U_x) = \\dim_x(\\mathcal{X}),", "$$", "as claimed." ], "refs": [ "stacks-geometry-lemma-base-change-invariance-of-relative-dimension" ], "ref_ids": [ 4680 ] } ], "ref_ids": [] }, { "id": 4688, "type": "theorem", "label": "stacks-geometry-lemma-dimension-via-components", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-dimension-via-components", "contents": [ "If $\\mathcal{X}$ is a locally Noetherian algebraic stack, and if", "$x \\in |\\mathcal{X}|$,", "then $\\dim_x(\\mathcal{X}) = \\sup_{\\mathcal{T}} \\{ \\dim_x(\\mathcal{T}) \\} $,", "where $\\mathcal{T}$ runs over all the irreducible components", "of $|\\mathcal{X}|$ passing through $x$ (endowed with their", "induced reduced structure)." ], "refs": [], "proofs": [ { "contents": [ "Lemma \\ref{lemma-closed-immersions}", "shows that", "$\\dim_x (\\mathcal{T}) \\leq \\dim_x(\\mathcal{X})$ for each", "irreducible component $\\mathcal{T}$ passing through", "the point $x$. Thus to prove the lemma,", "it suffices to show that", "\\begin{equation}", "\\label{equation-desired-inequality}", "\\dim_x(\\mathcal{X}) \\leq", "\\sup_{\\mathcal{T}} \\{\\dim_x(\\mathcal{T})\\}.", "\\end{equation}", "Let $U\\to\\mathcal{X}$ be a smooth cover by a scheme. If $T$ is an irreducible", "component of $U$ then we let $\\mathcal{T}$ denote the closure of its image", "in $\\mathcal{X}$, which is an irreducible component of $\\mathcal{X}$. Let", "$u \\in U$ be", "a point mapping to $x$. Then we have", "$\\dim_x(\\mathcal{X})=\\dim_uU-\\dim_uU_x=\\sup_T\\dim_uT-\\dim_uU_x$, where the", "supremum is over the irreducible components of $U$ passing", "through $u$. Choose a component $T$ for which the supremum", "is achieved, and note that", "$\\dim_x(\\mathcal{T})=\\dim_uT-\\dim_u T_x$.", "The desired inequality (\\ref{equation-desired-inequality})", "now follows from the evident inequality $\\dim_u T_x \\leq \\dim_u U_x.$", "(Note that if $\\Spec k \\to \\mathcal{X}$ is a representative of $x$,", "then $T\\times_{\\mathcal{X}} \\Spec k$ is a closed subspace of", "$U\\times_{\\mathcal{X}}", "\\Spec k$.)" ], "refs": [ "stacks-geometry-lemma-closed-immersions" ], "ref_ids": [ 4687 ] } ], "ref_ids": [] }, { "id": 4689, "type": "theorem", "label": "stacks-geometry-lemma-dimension-at-finite-type-point", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-dimension-at-finite-type-point", "contents": [ "If $\\mathcal{X}$ is a locally Noetherian algebraic stack, and if", "$x \\in |\\mathcal{X}|$, then", "for any open substack $\\mathcal{V}$ of $\\mathcal{X}$ containing $x$,", "there is a finite type point $x_0 \\in |\\mathcal{V}|$ such that", "$\\dim_{x_0}(\\mathcal{X}) = \\dim_x(\\mathcal{V})$." ], "refs": [], "proofs": [ { "contents": [ "Choose a smooth surjective", "morphism $f:U \\to \\mathcal{X}$ whose source is a scheme, and consider the", "function $u \\mapsto \\dim_{f(u)}(\\mathcal{X});$", "since the morphism $|U| \\to |\\mathcal{X}|$ induced by $f$ is open (as $f$", "is smooth) as well as surjective (by assumption),", "and takes finite type points to finite type points (by the very definition", "of the finite type points of $|\\mathcal{X}|$),", "it suffices to show that for any $u \\in U$, and any open neighbourhood of $u$,", "there is a finite type point $u_0$ in this neighbourhood such that", "$\\dim_{f(u_0)}(\\mathcal{X}) = \\dim_{f(u)}(\\mathcal{X}).$", "Since, with this reformulation", "of the problem, the surjectivity of $f$ is no longer required,", "we may replace $U$ by the open neighbourhood of the point $u$ in question,", "and thus reduce to the problem of showing that for each $u \\in U$,", "there is a finite type point $u_0 \\in U$ such that", "$\\dim_{f(u_0)}(\\mathcal{X}) = \\dim_{f(u)}(\\mathcal{X}).$", "By Lemma \\ref{lemma-behaviour-of-dimensions-wrt-smooth-morphisms}", "$\\dim_{f(u)}(\\mathcal{X}) = \\dim_u(U) - \\dim_u(U_{f(u)}),$", "while", "$\\dim_{f(u_0)}(\\mathcal{X}) = \\dim_{u_0}(U) - \\dim_{u_0}(U_{f(u_0)}).$", "Since $f$ is smooth, the expression $\\dim_{u_0}(U_{f(u_0)})$ is locally", "constant as $u_0$ varies over $U$ (by", "Lemma \\ref{lemma-relative-dimension-is-semi-continuous} (2)),", "and so shrinking $U$ further around", "$u$ if necessary, we may assume it is constant. Thus the problem", "becomes to show that we may find a finite type point $u_0 \\in U$", "for which $\\dim_{u_0}(U) = \\dim_u(U)$.", "Since by definition $\\dim_u U$ is the minimum of the dimensions", "$\\dim V$, as $V$ ranges over the open neighbourhoods $V$ of $u$", "in $U$, we may shrink $U$ down further around $u$ so that", "$\\dim_u U = \\dim U$.", "The existence of desired point $u_0$ then follows from", "Lemma \\ref{lemma-dimension-achieved-by-finite-type-point}." ], "refs": [ "stacks-geometry-lemma-behaviour-of-dimensions-wrt-smooth-morphisms", "stacks-geometry-lemma-relative-dimension-is-semi-continuous", "stacks-geometry-lemma-dimension-achieved-by-finite-type-point" ], "ref_ids": [ 4678, 4682, 4683 ] } ], "ref_ids": [] }, { "id": 4690, "type": "theorem", "label": "stacks-geometry-lemma-monomorphing-a-component-in-of-the-right-dimension", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-monomorphing-a-component-in-of-the-right-dimension", "contents": [ "Let $\\mathcal{T} \\hookrightarrow \\mathcal{X}$ be a locally", "of finite type monomorphism of algebraic stacks,", "with $\\mathcal{X}$ (and thus also $\\mathcal{T}$)", "being Jacobson, pseudo-catenary, and locally Noetherian.", "Suppose further that $\\mathcal{T}$ is irreducible", "of some (finite) dimension $d$, and that $\\mathcal{X}$ is reduced", "and of dimension less", "than or equal to $d$.", "Then there is a non-empty open substack $\\mathcal{V}$ of $\\mathcal{T}$ such", "that the induced", "monomorphism $\\mathcal{V} \\hookrightarrow \\mathcal{X}$ is an open immersion", "which identifies", "$\\mathcal{V}$ with an open subset of an irreducible component of $\\mathcal{X}$." ], "refs": [], "proofs": [ { "contents": [ "Choose a smooth surjective morphism $f:U \\to \\mathcal{X}$ with source a scheme,", "necessarily reduced since $\\mathcal{X}$ is,", "and write $U' = \\mathcal{T}\\times_{\\mathcal{X}} U$. The base-changed morphism", "$U' \\to U$ is a monomorphism of algebraic spaces, locally of finite", "type, and thus representable", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable} and", "\\ref{spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite};", "since $U$ is a scheme, so is $U'$.", "The projection $f': U' \\to \\mathcal{T}$ is again a smooth surjection.", "Let $u' \\in U'$, with image $u \\in U$.", "Lemma \\ref{lemma-base-change-invariance-of-relative-dimension}", "shows that $\\dim_{u'}(U'_{f(u')}) = \\dim_u(U_{f(u)}),$", "while $\\dim_{f'(u')}(\\mathcal{T}) =d", "\\geq \\dim_{f(u)}(\\mathcal{X})$ by", "Lemma \\ref{lemma-irreducible-implies-equidimensional}", "and our assumptions on $\\mathcal{T}$ and $\\mathcal{X}$.", "Thus we see that", "\\begin{equation}", "\\label{equation-dim-inequality}", "\\dim_{u'} (U') = \\dim_{u'} (U'_{f(u')}) + \\dim_{f'(u')}(\\mathcal{T})", "\\\\", "\\geq \\dim_u (U_{f(u)}) + \\dim_{f(u)}(\\mathcal{X}) = \\dim_u (U).", "\\end{equation}", "Since $U' \\to U$ is a monomorphism, locally of finite type,", "it is in particular unramified,", "and so by the \\'etale local structure of unramified morphisms", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-finite-unramified-etale-local},", "we may find a commutative diagram", "$$", "\\xymatrix{", "V' \\ar[r]\\ar[d] & V \\ar[d] \\\\", "U' \\ar[r] & U", "}", "$$", "in which the scheme $V'$ is non-empty,", "the vertical arrows are \\'etale,", "and the upper horizontal arrow is a closed immersion.", "Replacing $V$ by a quasi-compact open subset", "whose image has non-empty intersection with the image of $U'$,", "and replacing $V'$ by the preimage of $V$, we may further", "assume that $V$ (and thus $V'$) is quasi-compact.", "Since $V$ is also locally Noetherian,", "it is thus Noetherian, and so is the union of finitely many irreducible", "components.", "\\medskip\\noindent", "Since \\'etale morphisms preserve pointwise dimension", "Descent, Lemma \\ref{descent-lemma-dimension-at-point-local}", "we deduce from (\\ref{equation-dim-inequality})", "that for any point $v' \\in V'$,", "with image $v \\in V$, we have", "$\\dim_{v'}( V') \\geq \\dim_v(V)$.", "In particular, the image of $V'$ can't be contained in the intersection", "of two distinct irreducible components of $V$, and so we may find", "at least one irreducible open subset of $V$ which has non-empty intersection", "with $V'$; replacing $V$ by this subset, we may assume that $V$ is integral", "(being both reduced and irreducible). From the preceding inequality", "on dimensions, we conclude that the closed immersion $V' \\hookrightarrow V$", "is in fact an isomorphism.", "If we let $W$ denote the image of $V'$", "in $U'$, then $W$ is a non-empty", "open subset of $U'$ (as \\'etale morphisms are open),", "and the induced monomorphism $W \\to U$ is \\'etale", "(since it is so \\'etale locally on the source, i.e.\\ after pulling back", "to $V'$), and hence is an open immersion (being an \\'etale monomorphism).", "Thus, if we let $\\mathcal{V}$ denote the image of $W$ in $\\mathcal{T}$,", "then $\\mathcal{V}$ is a dense (equivalently, non-empty) open substack of", "$\\mathcal{T}$,", "whose image is dense in an irreducible component of $\\mathcal{X}$.", "Finally,", "we note that the morphism is $\\mathcal{V} \\to \\mathcal{X}$ is smooth", "(since its composite", "with the smooth morphism $W\\to \\mathcal{V}$ is smooth),", "and also a monomorphism, and thus is an open immersion." ], "refs": [ "spaces-morphisms-lemma-locally-quasi-finite-separated-representable", "spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "stacks-geometry-lemma-base-change-invariance-of-relative-dimension", "stacks-geometry-lemma-irreducible-implies-equidimensional", "etale-lemma-finite-unramified-etale-local", "descent-lemma-dimension-at-point-local" ], "ref_ids": [ 4972, 4838, 4680, 4686, 10711, 14660 ] } ], "ref_ids": [] }, { "id": 4691, "type": "theorem", "label": "stacks-geometry-lemma-dims-of-images", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-dims-of-images", "contents": [ "Let $f: \\mathcal{T} \\to \\mathcal{X}$ be a locally of finite type", "morphism of Jacobson, pseudo-catenary, and locally Noetherian", "algebraic stacks,", "whose source is irreducible and whose target is quasi-separated,", "and let $\\mathcal{Z} \\hookrightarrow \\mathcal{X}$ denote the scheme-theoretic", "image of $\\mathcal{T}$.", "Then for every finite type point $t \\in |T|$,", "we have that", "$\\dim_t( \\mathcal{T}_{f(t)}) \\geq \\dim \\mathcal{T} - \\dim \\mathcal{Z}$,", "and there is a non-empty (equivalently, dense)", "open subset of $|\\mathcal{T}|$ over which equality holds." ], "refs": [], "proofs": [ { "contents": [ "Replacing $\\mathcal{X}$ by $\\mathcal{Z}$, we may and do assume that $f$ is", "scheme theoretically dominant,", "and also that $\\mathcal{X}$ is irreducible.", "By the upper semi-continuity of fibre dimensions", "(Lemma \\ref{lemma-relative-dimension-is-semi-continuous} (1)),", "it suffices to prove that the equality", "$\\dim_t( \\mathcal{T}_{f(t)}) =\\dim \\mathcal{T} - \\dim \\mathcal{Z}$", "holds for $t$ lying in", "some non-empty open substack of $\\mathcal{T}$.", "For this reason, in the argument we are always free", "to replace $\\mathcal{T}$ by a non-empty open substack.", "\\medskip\\noindent", "Let $T' \\to \\mathcal{T}$ be a smooth surjective morphism whose source", "is a scheme, and let $T$ be a non-empty quasi-compact open subset", "of $T'$. Since $\\mathcal{Y}$ is quasi-separated, we find", "that $T \\to \\mathcal{Y}$ is quasi-compact", "(by Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-quasi-compact-permanence}, applied to the morphisms", "$T \\to \\mathcal{Y} \\to \\Spec \\mathbf{Z}$).", "Thus, if we replace $\\mathcal{T}$ by the image of $T$ in $\\mathcal{T}$,", "then we may assume (appealing to", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-surjection-from-quasi-compact}", "that the morphism $f:\\mathcal{T} \\to \\mathcal{X}$ is quasi-compact.", "\\medskip\\noindent", "If we choose a smooth surjection $U \\to \\mathcal{X}$ with $U$ a scheme,", "then Lemma \\ref{lemma-map-of-components} ensures that", "we may find an irreducible open subset $V$ of $U$ such", "that $V \\to \\mathcal{X}$ is smooth and scheme-theoretically dominant.", "Since scheme-theoretic dominance for quasi-compact morphisms", "is preserved by flat base-change,", "the base-change $\\mathcal{T} \\times_{\\mathcal{X}} V \\to V$", "of the scheme-theoretically", "dominant morphism $f$ is again", "scheme-theoretically dominant. We let $Z$ denote a scheme", "admitting a smooth surjection onto this fibre product;", "then $Z \\to \\mathcal{T} \\times_{\\mathcal{X}} V \\to V$", "is again scheme-theoretically dominant.", "Thus we may find an irreducible", "component $C$ of $Z$ which scheme-theoretically", "dominates $V$.", "Since the composite $Z \\to \\mathcal{T}\\times_{\\mathcal{X}} V \\to \\mathcal{T}$", "is smooth,", "and since $\\mathcal{T}$ is irreducible,", "Lemma \\ref{lemma-map-of-components} shows that any irreducible", "component of the source has dense image in $|\\mathcal{T}|$.", "We now replace", "$C$ by a non-empty open subset $W$ which is disjoint from every other", "irreducible component of $Z$, and", "then replace $\\mathcal{T}$ and $\\mathcal{X}$ by the images of $W$", "and $V$", "(and apply Lemma \\ref{lemma-irreducible-implies-equidimensional}", "to see that this", "doesn't change the dimension of either $\\mathcal{T}$ or $\\mathcal{X}$).", "If we let $\\mathcal{W}$ denote the image of the morphism", "$W \\to \\mathcal{T}\\times_{\\mathcal{X}} V$,", "then $\\mathcal{W}$ is open in $\\mathcal{T}\\times_{\\mathcal{X}} V$ (since the", "morphism $W \\to \\mathcal{T}\\times_{\\mathcal{X}} V$ is smooth),", "and is irreducible (being the image of an irreducible", "scheme). Thus we end up with a commutative diagram", "$$", "\\xymatrix{", "W \\ar[dr] \\ar[r] & \\mathcal{W} \\ar[r] \\ar[d] & V \\ar[d] \\\\", "& \\mathcal{T} \\ar[r] & \\mathcal{X}", "}", "$$", "in which $W$ and $V$ are schemes,", "the vertical arrows are smooth and surjective,", "the diagonal arrows and the left-hand", "upper horizontal arroware smooth,", "and the induced morphism $\\mathcal{W} \\to \\mathcal{T}\\times_{\\mathcal{X}} V$ is", "an open immersion. Using this diagram, together with the definitions", "of the various dimensions involved in", "the statement of the lemma, we will reduce our verification", "of the lemma to the case of schemes, where it is known.", "\\medskip\\noindent", "Fix $w \\in |W|$ with image $w' \\in |\\mathcal{W}|$,", "image $t \\in |\\mathcal{T}|$, image $v$ in $|V|$,", "and image $x$ in $|\\mathcal{X}|$.", "Essentially by definition (using the", "fact that $\\mathcal{W}$ is open in $\\mathcal{T}\\times_{\\mathcal{X}} V$, and that", "the fibre of a base-change is the base-change of the fibre),", "we obtain the equalities", "$$", "\\dim_v V_x = \\dim_{w'} \\mathcal{W}_t", "$$", "and", "$$", "\\dim_t \\mathcal{T}_x = \\dim_{w'} \\mathcal{W}_v.", "$$", "By Lemma \\ref{lemma-behaviour-of-dimensions-wrt-smooth-morphisms}", "(the diagonal arrow and right-hand vertical", "arrow in our diagram realise $W$ and $V$ as smooth covers by", "schemes of the stacks $\\mathcal{T}$ and $\\mathcal{X}$), we find that", "$$", "\\dim_t \\mathcal{T} = \\dim_w W - \\dim_w W_t", "$$", "and", "$$", "\\dim_x \\mathcal{X} = \\dim_v V - \\dim_v V_x.", "$$", "Combining the equalities, we find that", "$$", "\\dim_t \\mathcal{T}_x - \\dim_t \\mathcal{T} + \\dim_x \\mathcal{X}", "= \\dim_{w'} \\mathcal{W}_v - \\dim_w W + \\dim_w W_t + \\dim_v V -", "\\dim_{w'} \\mathcal{W}_t", "$$", "Since $W \\to \\mathcal{W}$ is a smooth surjection, the same is true", "if we base-change over the morphism $\\Spec \\kappa(v) \\to V$", "(thinking of $W \\to \\mathcal{W}$ as a morphism over $V$),", "and from this smooth morphism we obtain the first of the following", "two equalities", "$$", "\\dim_w W_v - \\dim_{w'} \\mathcal{W}_v = \\dim_w (W_v)_{w'} = \\dim_w W_{w'};", "$$", "the second equality follows via a direct comparison of the", "two fibres involved.", "Similarly, if we think of $W \\to \\mathcal{W}$ as a morphism of schemes", "over $\\mathcal{T}$, and base-change over some representative of the point", "$t \\in |\\mathcal{T}|$, we obtain the equalities", "$$", "\\dim_w W_t - \\dim_{w'} \\mathcal{W}_t = \\dim_w (W_t)_{w'} = \\dim_w W_{w'}.", "$$", "Putting everything together, we find that", "$$", "\\dim_t \\mathcal{T}_x - \\dim_t \\mathcal{T} + \\dim_x \\mathcal{X}", "= \\dim_w W_v - \\dim_w W + \\dim_v V.", "$$", "Our goal is to show that the left-hand side of this equality", "vanishes for a non-empty open subset", "of $t$. As $w$ varies over a non-empty open subset of $W$,", "its image $t \\in |\\mathcal{T}|$ varies over a non-empty open", "subset of $|\\mathcal{T}|$ (as $W \\to \\mathcal{T}$ is smooth).", "\\medskip\\noindent", "We are therefore reduced to showing that if $W\\to V$ is a", "scheme-theoretically dominant morphism of irreducible locally", "Noetherian schemes that is locally of finite type,", "then there is a non-empty open subset of", "points $w\\in W$ such that $\\dim_w W_v =\\dim_w W - \\dim_v V$", "(where $v$ denotes the image of $w$ in $V$).", "This is a standard fact,", "whose proof we recall for the convenience of the reader.", "\\medskip\\noindent", "We may replace $W$ and $V$ by their underlying reduced subschemes", "without altering the validity (or not) of this equation,", "and thus we may assume that they are in fact integral schemes.", "Since $\\dim_w W_v$ is locally constant on $W,$ replacing $W$", "by a non-empty open subset if necessary, we may assume that $\\dim_w W_v$", "is constant, say equal to $d$. Choosing this open subset to be affine,", "we may also assume that the morphism $W\\to V$ is in fact of finite type.", "Replacing $V$ by a non-empty open subset if necessary", "(and then pulling back $W$ over this open subset; the resulting pull-back", "is non-empty, since the flat base-change of a quasi-compact", "and scheme-theoretically", "dominant morphism remains scheme-theoretically dominant),", "we may furthermore assume that $W$ is flat over $V$.", "The morphism $W\\to V$ is thus of relative dimension $d$", "in the sense of", "Morphisms, Definition", "\\ref{morphisms-definition-relative-dimension-d}", "and it follows from", "Morphisms, Lemma \\ref{morphisms-lemma-rel-dimension-dimension}", "that $\\dim_w(W) = \\dim_v(V) + d,$ as required." ], "refs": [ "stacks-geometry-lemma-relative-dimension-is-semi-continuous", "stacks-morphisms-lemma-quasi-compact-permanence", "stacks-morphisms-lemma-surjection-from-quasi-compact", "stacks-geometry-lemma-map-of-components", "stacks-geometry-lemma-map-of-components", "stacks-geometry-lemma-irreducible-implies-equidimensional", "stacks-geometry-lemma-behaviour-of-dimensions-wrt-smooth-morphisms", "morphisms-definition-relative-dimension-d", "morphisms-lemma-rel-dimension-dimension" ], "ref_ids": [ 4682, 7427, 7426, 4673, 4673, 4686, 4678, 5559, 5288 ] } ], "ref_ids": [] }, { "id": 4692, "type": "theorem", "label": "stacks-geometry-lemma-dims-of-images-two", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-dims-of-images-two", "contents": [ "Let $f: \\mathcal{T} \\to \\mathcal{X}$ be a locally of finite type", "morphism of Jacobson, pseudo-catenary, and locally Noetherian", "algebraic stacks", "which is quasi-DM,", "whose source is irreducible and whose target is quasi-separated,", "and let $\\mathcal{Z} \\hookrightarrow \\mathcal{X}$ denote the scheme-theoretic", "image of $\\mathcal{T}$.", "Then $\\dim \\mathcal{Z} \\leq \\dim \\mathcal{T}$,", "and furthermore, exactly one of the following two conditions holds:", "\\begin{enumerate}", "\\item for every finite type point $t \\in |T|,$", "we have", "$\\dim_t(\\mathcal{T}_{f(t)}) > 0,$ in which", "case $\\dim \\mathcal{Z} < \\dim \\mathcal{T}$; or", "\\item $\\mathcal{T}$ and $\\mathcal{Z}$", "are of the same dimension.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As was observed in the preceding remark,", "the dimension of a quasi-DM stack is always non-negative,", "from which we conclude that $\\dim_t \\mathcal{T}_{f(t)} \\geq 0$", "for all $t \\in |\\mathcal{T}|$, with the equality", "$$", "\\dim_t \\mathcal{T}_{f(t)} = \\dim_t \\mathcal{T} - \\dim_{f(t)} \\mathcal{Z}", "$$", "holding", "for a dense open subset of points $t\\in |\\mathcal{T}|$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4693, "type": "theorem", "label": "stacks-geometry-lemma-dimension-local-ring-pre", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-dimension-local-ring-pre", "contents": [ "Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.", "Let $U \\to \\mathcal{X}$ be a smooth morphism and let $u \\in U$.", "Then", "$$", "\\dim(\\mathcal{O}_{U, \\overline{u}}) -", "\\dim(\\mathcal{O}_{R_u, e(\\overline{u})}) =", "2\\dim(\\mathcal{O}_{U, \\overline{u}}) -", "\\dim(\\mathcal{O}_{R, e(\\overline{u})})", "$$", "Here $R = U \\times_\\mathcal{X} U$ with projections $s, t : R \\to U$ and", "diagonal $e : U \\to R$ and $R_u$ is the fibre of $s : R \\to U$ over $u$." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$s : \\mathcal{O}_{U, \\overline{u}} \\to \\mathcal{O}_{R, e(\\overline{u})}$", "is a flat local homomorphism of Noetherian local rings and hence", "$$", "\\dim(\\mathcal{O}_{R, e(\\overline{u})}) =", "\\dim(\\mathcal{O}_{U, \\overline{u}}) +", "\\dim(\\mathcal{O}_{R_u, e(\\overline{u})})", "$$", "by", "Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}." ], "refs": [ "algebra-lemma-dimension-base-fibre-equals-total" ], "ref_ids": [ 987 ] } ], "ref_ids": [] }, { "id": 4694, "type": "theorem", "label": "stacks-geometry-lemma-dimension-local-ring", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-dimension-local-ring", "contents": [ "Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.", "Let $x \\in |\\mathcal{X}|$ be a finite type point ", "Morphisms of Stacks, Definition", "\\ref{stacks-morphisms-definition-finite-type-point}).", "Let $d \\in \\mathbf{Z}$.", "The following are equivalent", "\\begin{enumerate}", "\\item there exists a scheme $U$, a smooth morphism $U \\to \\mathcal{X}$,", "and a finite type point $u \\in U$ mapping to $x$ such that", "$2\\dim(\\mathcal{O}_{U, \\overline{u}}) -", "\\dim(\\mathcal{O}_{R, e(\\overline{u})}) = d$, and", "\\item for any scheme $U$, a smooth morphism $U \\to \\mathcal{X}$,", "and finite type point $u \\in U$ mapping to $x$ we have", "$2\\dim(\\mathcal{O}_{U, \\overline{u}}) -", "\\dim(\\mathcal{O}_{R, e(\\overline{u})}) = d$.", "\\end{enumerate}", "Here $R = U \\times_\\mathcal{X} U$ with projections $s, t : R \\to U$ and", "diagonal $e : U \\to R$ and $R_u$ is the fibre of $s : R \\to U$ over $u$." ], "refs": [ "stacks-morphisms-definition-finite-type-point" ], "proofs": [ { "contents": [ "Suppose we have two smooth neighbourhoods $(U, u)$ and $(U', u')$", "of $x$ with $u$ and $u'$ finite type points. After shrinking $U$", "and $U'$ we may assume that $u$ and $u'$ are closed points", "(by definition of finite type points). Then we choose", "a surjective \\'etale morphism $W \\to U \\times_\\mathcal{X} U'$.", "Let $W_u$ be the fibre of $W \\to U$ over $u$ and", "let $W_{u'}$ be the fibre of $W \\to U'$ over $u'$.", "Since $u$ and $u'$ map to the same point of $|\\mathcal{X}|$", "we see that $W_u \\cap W_{u'}$ is nonempty.", "Hence we may choose a closed point $w \\in W$ mapping to", "both $u$ and $u'$. This reduces us to the discussion in the", "next paragraph.", "\\medskip\\noindent", "Assume $(U', u') \\to (U, u)$ is a smooth morphism of smooth", "neightbourhoods of $x$ with $u$ and $u'$ closed points.", "Goal: prove the invariant defined for $(U, u)$ is the same", "as the invariant defined for $(U', u')$.", "To see this observe that $\\mathcal{O}_{U, u} \\to \\mathcal{O}_{U', u'}$", "is a flat local homomorphism of Noetherian local rings and hence", "$$", "\\dim(\\mathcal{O}_{U', \\overline{u}'}) =", "\\dim(\\mathcal{O}_{U, \\overline{u}}) +", "\\dim(\\mathcal{O}_{U'_u, \\overline{u}'})", "$$", "by Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}.", "(We omit working through all the steps to relate properties of local", "rings and their strict henselizations, see", "More on Algebra, Section \\ref{more-algebra-section-permanence-henselization}).", "On the other hand we have", "$$", "R' = U' \\times_{U, t} R \\times_{s, U} U'", "$$", "Thus we see that", "$$", "\\dim(\\mathcal{O}_{R', e(\\overline{u}')}) =", "\\dim(\\mathcal{O}_{R, e(\\overline{u})}) +", "\\dim(\\mathcal{O}_{U'_u \\times_u U'_u, (\\overline{u}', \\overline{u}')})", "$$", "To prove the lemma it suffices to show that", "$$", "\\dim(\\mathcal{O}_{U'_u \\times_u U'_u, (\\overline{u}', \\overline{u}')}) =", "2\\dim(\\mathcal{O}_{U'_u, \\overline{u}'})", "$$", "Observe that this isn't always true (example: if $U'_u$ is a curve", "and $u'$ is the generic point of this curve). However, we know", "that $u'$ is a closed point of the algebraic space $U'_u$ locally", "of finite type over $u$. In this case the equality holds because, first", "$\\dim_{(u', u')}(U'_u \\times_u U'_u) = 2\\dim_{u'}(U'_u)$ by", "Varieties, Lemma \\ref{varieties-lemma-dimension-product-locally-algebraic}", "and second the agreement of dimension with dimension of local rings", "in closed points of locally algebraic schemes, see", "Varieties, Lemma \\ref{varieties-lemma-dimension-locally-algebraic}.", "We omit the translation of these results for schemes into the", "language of algebraic spaces." ], "refs": [ "algebra-lemma-dimension-base-fibre-equals-total", "varieties-lemma-dimension-product-locally-algebraic", "varieties-lemma-dimension-locally-algebraic" ], "ref_ids": [ 987, 10991, 10989 ] } ], "ref_ids": [ 7615 ] }, { "id": 4695, "type": "theorem", "label": "stacks-geometry-lemma-dimension-formula", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-lemma-dimension-formula", "contents": [ "Suppose that $\\mathcal{X}$ is an algebraic stack, locally of finite type", "over a locally Noetherian scheme $S$. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$", "be a morphism where $k$ is a field of finite type over $S$. Represent", "$\\mathcal{F}_{\\mathcal{X}, k, x_0}$ as in Remark \\ref{remark-groupoid-defo}", "by a cogroupoid $(A, B, s, t, c)$ of Noetherian complete local $S$-algebras", "with residue field $k$. Then", "$$", "\\text{the dimension of the local ring of }\\mathcal{X}\\text{ at }x_0 =", "2\\dim A - \\dim B", "$$" ], "refs": [ "stacks-geometry-remark-groupoid-defo" ], "proofs": [ { "contents": [ "Let $s \\in S$ be the image of $x_0$. If $\\mathcal{O}_{S, s}$", "is a G-ring (a condition that is almost always satisfied in practice),", "then we can prove the lemma as follows.", "By Lemma \\ref{lemma-Artin-approximation-by-smooth-morphism},", "we may find a smooth morphism $U \\to \\mathcal{X}$, whose source is a scheme,", "containing a point $u_0 \\in U$ of residue field $k$, such that induced", "morphism $\\Spec(k) \\to U \\to \\mathcal{X}$ coincides with $x_0$", "and such that $A = \\mathcal{O}_{U, u_0}^\\wedge$.", "Write $R = U \\times_\\mathcal{X} U$. Then we may identify", "$\\mathcal{O}_{R, e(u_0)}^\\wedge$ with $B$.", "Hence the equality follows from the definitions.", "\\medskip\\noindent", "In the rest of this proof we explain how to prove the lemma in", "general, but we urge the reader to skip this.", "\\medskip\\noindent", "First let us show that the right hand side is independent of the choice", "of $(A, B, s, t, c)$. Namely, suppose that $(A', B', s', t', c')$", "is a second choice. Since $A$ and $A'$ are versal rings to $\\mathcal{X}$", "at $x_0$, we can choose, after possibly switching $A$ and $A'$,", "a formally smooth map $A \\to A'$ compatible with the given versal", "formal objects $\\xi$ and $\\xi'$ over $A$ and $A'$.", "Recall that $\\widehat{\\mathcal{C}}_\\Lambda$ has", "coproducts and that these are given by completed tensor product", "over $\\Lambda$, see Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-CLambdahat-coproducts}.", "Then $B$ prorepresents the functor of isomorphisms between the", "two pushforwards of $\\xi$ to $A \\widehat{\\otimes}_\\Lambda A$.", "Similarly for $B'$. We conclude that", "$$", "B' =", "B \\otimes_{(A \\widehat{\\otimes}_\\Lambda A)}", "(A' \\widehat{\\otimes}_\\Lambda A')", "$$", "It is straightforward to see that", "$$", "A \\widehat{\\otimes}_\\Lambda A \\longrightarrow", "A \\widehat{\\otimes}_\\Lambda A' \\longrightarrow", "A' \\widehat{\\otimes}_\\Lambda A'", "$$", "is formally smooth of relative dimension equal to $2$ times the", "relative dimension of the formally smooth map $A \\to A'$.", "(This follows from general principles, but also because", "in this particular case $A'$ is a power series ring over $A$", "in $r$ variables.) Hence $B \\to B'$ is formally smooth of", "relative dimension $2(\\dim(A') - \\dim(A))$ as desired.", "\\medskip\\noindent", "Next, let $l/k$ be a finite extension. let", "$x_{l, 0} : \\Spec(l) \\to \\mathcal{X}$ be", "the induced point. We claim that the right hand side of the formula", "is the same for $x_0$ as it is for $x_{l, 0}$.", "This can be shown by choosing $A \\to A'$ as in", "Lemma \\ref{lemma-versal-ring-field-extension}", "and arguing exactly as in the preceding paragraph.", "We omit the details.", "\\medskip\\noindent", "Finally, arguing as in the proof of Lemma \\ref{lemma-versal-ring-flat}", "we can use the compatibilities in the previous two paragraphs", "to reduce to the case (discussed in the first paragraph)", "where $A$ is the complete local ring of $U$ at $u_0$ for some", "scheme smooth over $\\mathcal{X}$ and finite type point $u_0$.", "Details omitted." ], "refs": [ "stacks-geometry-lemma-Artin-approximation-by-smooth-morphism", "formal-defos-lemma-CLambdahat-coproducts", "stacks-geometry-lemma-versal-ring-field-extension", "stacks-geometry-lemma-versal-ring-flat" ], "ref_ids": [ 4671, 3423, 4668, 4672 ] } ], "ref_ids": [ 4705 ] }, { "id": 4711, "type": "theorem", "label": "spaces-morphisms-theorem-chevalley", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-theorem-chevalley", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume $f$ is quasi-compact and locally of finite presentation.", "Then the image of every \\'etale locally constructible subset of $|X|$ is", "an \\'etale locally constructible subset of $|Y|$." ], "refs": [], "proofs": [ { "contents": [ "Let $E \\subset |X|$ be \\'etale locally constructible.", "Let $V \\to Y$ be an \\'etale morphism with $V$ affine.", "It suffices to show that the inverse image of $f(E)$ in", "$V$ is constructible, see Properties of Spaces, Definition", "\\ref{spaces-properties-definition-locally-constructible}.", "Since $f$ is quasi-compact $V \\times_Y X$ is a quasi-compact algebraic space.", "Choose an affine scheme $U$ and a surjective \\'etale morphism", "$U \\to V \\times_Y X$ (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "By Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}", "the inverse image of $f(E)$ in $V$ is the image under $U \\to V$", "of the inverse image of $E$ in $U$.", "Thus the result follows from the case of schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-chevalley}." ], "refs": [ "spaces-properties-definition-locally-constructible", "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-properties-lemma-points-cartesian", "morphisms-lemma-chevalley" ], "ref_ids": [ 11928, 11832, 11819, 5250 ] } ], "ref_ids": [] }, { "id": 4712, "type": "theorem", "label": "spaces-morphisms-lemma-properties-diagonal", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-properties-diagonal", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\Delta_{X/Y} : X \\to X \\times_Y X$ be the diagonal morphism.", "Then", "\\begin{enumerate}", "\\item $\\Delta_{X/Y}$ is representable,", "\\item $\\Delta_{X/Y}$ is locally of finite type,", "\\item $\\Delta_{X/Y}$ is a monomorphism,", "\\item $\\Delta_{X/Y}$ is separated, and", "\\item $\\Delta_{X/Y}$ is locally quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We are going to use the fact that $\\Delta_{X/S}$ is", "representable (by definition of an algebraic space) and that", "it satisfies properties (2) -- (5), see", "Spaces, Lemma \\ref{spaces-lemma-properties-diagonal}.", "Note that we have a factorization", "$$", "X", "\\longrightarrow", "X \\times_Y X", "\\longrightarrow", "X \\times_S X", "$$", "of the diagonal $\\Delta_{X/S} : X \\to X \\times_S X$. Since", "$X \\times_Y X \\to X \\times_S X$ is a monomorphism, and since", "$\\Delta_{X/S}$ is representable, it follows formally that", "$\\Delta_{X/Y}$ is representable. In particular, the rest of", "the statements now make sense, see", "Section \\ref{section-representable}.", "\\medskip\\noindent", "Choose a surjective \\'etale morphism $U \\to X$, with $U$ a scheme.", "Consider the diagram", "$$", "\\xymatrix{", "R = U \\times_X U \\ar[r] \\ar[d] &", "U \\times_Y U \\ar[d] \\ar[r] &", "U \\times_S U \\ar[d] \\\\", "X \\ar[r] & X \\times_Y X \\ar[r] & X \\times_S X", "}", "$$", "Both squares are cartesian, hence so is the outer rectangle.", "The top row consists of schemes, and the vertical arrows", "are surjective \\'etale morphisms. By", "Spaces, Lemma \\ref{spaces-lemma-representable-morphisms-spaces-property}", "the properties (2) -- (5) for $\\Delta_{X/Y}$ are equivalent to those of", "$R \\to U \\times_Y U$. In the proof of", "Spaces, Lemma \\ref{spaces-lemma-properties-diagonal}", "we have seen that $R \\to U \\times_S U$ has properties (2) -- (5).", "The morphism $U \\times_Y U \\to U \\times_S U$ is a monomorphism", "of schemes. These facts imply that $R \\to U \\times_Y U$ have", "properties (2) -- (5).", "\\medskip\\noindent", "Namely: For (3), note that $R \\to U \\times_Y U$", "is a monomorphism as the composition", "$R \\to U \\times_S U$ is a monomorphism. For (2), note that", "$R \\to U \\times_Y U$ is locally of finite type, as the", "composition $R \\to U \\times_S U$ is locally of finite type", "(Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type}).", "A monomorphism which is locally of finite type is locally quasi-finite", "because it has finite fibres", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-fibre}), hence (5).", "A monomorphism is separated", "(Schemes, Lemma \\ref{schemes-lemma-monomorphism-separated}), hence (4)." ], "refs": [ "spaces-lemma-properties-diagonal", "spaces-lemma-representable-morphisms-spaces-property", "spaces-lemma-properties-diagonal", "morphisms-lemma-permanence-finite-type", "morphisms-lemma-finite-fibre", "schemes-lemma-monomorphism-separated" ], "ref_ids": [ 8163, 8157, 8163, 5204, 5227, 7722 ] } ], "ref_ids": [] }, { "id": 4713, "type": "theorem", "label": "spaces-morphisms-lemma-trivial-implications", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-trivial-implications", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. If $f$ is separated, then $f$ is locally separated and", "$f$ is quasi-separated." ], "refs": [], "proofs": [ { "contents": [ "This is true, via the general principle", "Spaces,", "Lemma \\ref{spaces-lemma-representable-transformations-property-implication},", "because a closed immersion of schemes is an immersion and is quasi-compact." ], "refs": [ "spaces-lemma-representable-transformations-property-implication" ], "ref_ids": [ 8136 ] } ], "ref_ids": [] }, { "id": 4714, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-separated", "contents": [ "All of the separation axioms listed in Definition \\ref{definition-separated}", "are stable under base change." ], "refs": [ "spaces-morphisms-definition-separated" ], "proofs": [ { "contents": [ "Let $f : X \\to Y$ and $Y' \\to Y$ be morphisms of algebraic spaces.", "Let $f' : X' \\to Y'$ be the base change of $f$ by $Y' \\to Y$. Then", "$\\Delta_{X'/Y'}$ is the base change of $\\Delta_{X/Y}$ by", "the morphism $X' \\times_{Y'} X' \\to X \\times_Y X$. By the results of", "Section \\ref{section-representable}", "each of the properties of the diagonal used in", "Definition \\ref{definition-separated}", "is stable under base change. Hence the lemma is true." ], "refs": [ "spaces-morphisms-definition-separated" ], "ref_ids": [ 4984 ] } ], "ref_ids": [ 4984 ] }, { "id": 4715, "type": "theorem", "label": "spaces-morphisms-lemma-fibre-product-after-map", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-fibre-product-after-map", "contents": [ "\\begin{slogan}", "The top arrow of a ``magic diagram'' of algebraic spaces has nice", "immersion-like properties, and under separatedness hypotheses", "these get stronger.", "\\end{slogan}", "Let $S$ be a scheme. Let $f : X \\to Z$, $g : Y \\to Z$ and $Z \\to T$", "be morphisms of algebraic spaces over $S$. Consider the induced morphism", "$i : X \\times_Z Y \\to X \\times_T Y$. Then", "\\begin{enumerate}", "\\item $i$ is representable, locally of finite type, locally quasi-finite,", "separated and a monomorphism,", "\\item if $Z \\to T$ is locally separated, then $i$ is an immersion,", "\\item if $Z \\to T$ is separated, then $i$ is a closed immersion, and", "\\item if $Z \\to T$ is quasi-separated, then $i$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By general category theory the following diagram", "$$", "\\xymatrix{", "X \\times_Z Y \\ar[r]_i \\ar[d] & X \\times_T Y \\ar[d] \\\\", "Z \\ar[r]^-{\\Delta_{Z/T}} \\ar[r] & Z \\times_T Z", "}", "$$", "is a fibre product diagram. Hence $i$ is the base change of the", "diagonal morphism $\\Delta_{Z/T}$. Thus the lemma follows", "from Lemma \\ref{lemma-properties-diagonal}, and the material in", "Section \\ref{section-representable}." ], "refs": [ "spaces-morphisms-lemma-properties-diagonal" ], "ref_ids": [ 4712 ] } ], "ref_ids": [] }, { "id": 4716, "type": "theorem", "label": "spaces-morphisms-lemma-semi-diagonal", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-semi-diagonal", "contents": [ "\\begin{slogan}", "Properties of the graph of a morphism of algebraic spaces", "as a consequence of separation properties of the target.", "\\end{slogan}", "Let $S$ be a scheme. Let $T$ be an algebraic space over $S$.", "Let $g : X \\to Y$ be a morphism of algebraic spaces over $T$.", "Consider the graph $i : X \\to X \\times_T Y$ of $g$. Then", "\\begin{enumerate}", "\\item $i$ is representable, locally of finite type, locally quasi-finite,", "separated and a monomorphism,", "\\item if $Y \\to T$ is locally separated, then $i$ is an immersion,", "\\item if $Y \\to T$ is separated, then $i$ is a closed immersion, and", "\\item if $Y \\to T$ is quasi-separated, then $i$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-fibre-product-after-map}", "applied to the morphism $X = X \\times_Y Y \\to X \\times_T Y$." ], "refs": [ "spaces-morphisms-lemma-fibre-product-after-map" ], "ref_ids": [ 4715 ] } ], "ref_ids": [] }, { "id": 4717, "type": "theorem", "label": "spaces-morphisms-lemma-section-immersion", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-section-immersion", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to T$ be a morphism of algebraic spaces over $S$.", "Let $s : T \\to X$ be a section of $f$ (in a formula", "$f \\circ s = \\text{id}_T$). Then", "\\begin{enumerate}", "\\item $s$ is representable, locally of finite type, locally quasi-finite,", "separated and a monomorphism,", "\\item if $f$ is locally separated, then $s$ is an immersion,", "\\item if $f$ is separated, then $s$ is a closed immersion, and", "\\item if $f$ is quasi-separated, then $s$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-semi-diagonal} applied to", "$g = s$ so the morphism $i = s : T \\to T \\times_T X$." ], "refs": [ "spaces-morphisms-lemma-semi-diagonal" ], "ref_ids": [ 4716 ] } ], "ref_ids": [] }, { "id": 4718, "type": "theorem", "label": "spaces-morphisms-lemma-composition-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-separated", "contents": [ "All of the separation axioms listed in Definition \\ref{definition-separated}", "are stable under composition of morphisms." ], "refs": [ "spaces-morphisms-definition-separated" ], "proofs": [ { "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces", "to which the axiom in question applies.", "The diagonal $\\Delta_{X/Z}$ is the composition", "$$", "X \\longrightarrow X \\times_Y X \\longrightarrow X \\times_Z X.", "$$", "Our separation axiom is defined by requiring the diagonal", "to have some property $\\mathcal{P}$. By", "Lemma \\ref{lemma-fibre-product-after-map} above we see that", "the second arrow also has this property. Hence the lemma follows", "since the composition of (representable) morphisms with property", "$\\mathcal{P}$ also is a morphism with property $\\mathcal{P}$, see", "Section \\ref{section-representable}." ], "refs": [ "spaces-morphisms-lemma-fibre-product-after-map" ], "ref_ids": [ 4715 ] } ], "ref_ids": [ 4984 ] }, { "id": 4719, "type": "theorem", "label": "spaces-morphisms-lemma-separated-over-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-separated-over-separated", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item If $Y$ is separated and $f$ is separated, then $X$ is separated.", "\\item If $Y$ is quasi-separated and $f$ is quasi-separated, then", "$X$ is quasi-separated.", "\\item If $Y$ is locally separated and $f$ is locally separated, then", "$X$ is locally separated.", "\\item If $Y$ is separated over $S$ and $f$ is separated, then", "$X$ is separated over $S$.", "\\item If $Y$ is quasi-separated over $S$ and $f$ is quasi-separated, then", "$X$ is quasi-separated over $S$.", "\\item If $Y$ is locally separated over $S$ and $f$ is locally separated, then", "$X$ is locally separated over $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (4), (5), and (6) follow immediately from", "Lemma \\ref{lemma-composition-separated}", "and", "Spaces, Definition \\ref{spaces-definition-separated}.", "Parts (1), (2), and (3) reduce to parts (4), (5), and (6) by thinking", "of $X$ and $Y$ as algebraic spaces over $\\Spec(\\mathbf{Z})$, see", "Properties of Spaces, Definition \\ref{spaces-properties-definition-separated}." ], "refs": [ "spaces-morphisms-lemma-composition-separated", "spaces-definition-separated", "spaces-properties-definition-separated" ], "ref_ids": [ 4718, 8181, 11922 ] } ], "ref_ids": [] }, { "id": 4720, "type": "theorem", "label": "spaces-morphisms-lemma-compose-after-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-compose-after-separated", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item If $g \\circ f$ is separated then so is $f$.", "\\item If $g \\circ f$ is locally separated then so is $f$.", "\\item If $g \\circ f$ is quasi-separated then so is $f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the factorization", "$$", "X \\to X \\times_Y X \\to X \\times_Z X", "$$", "of the diagonal morphism of $g \\circ f$. In any case the last morphism", "is a monomorphism. Hence for any scheme $T$ and morphism", "$T \\to X \\times_Y X$ we have the equality", "$$", "X \\times_{(X \\times_Y X)} T = X \\times_{(X \\times_Z X)} T.", "$$", "Hence the result is clear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4721, "type": "theorem", "label": "spaces-morphisms-lemma-separated-implies-morphism-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-separated-implies-morphism-separated", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item If $X$ is separated then $X$ is separated over $S$.", "\\item If $X$ is locally separated then $X$ is locally separated over $S$.", "\\item If $X$ is quasi-separated then $X$ is quasi-separated over $S$.", "\\end{enumerate}", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item[(4)] If $X$ is separated over $S$ then $f$ is separated.", "\\item[(5)] If $X$ is locally separated over $S$ then $f$ is locally separated.", "\\item[(6)] If $X$ is quasi-separated over $S$ then $f$ is quasi-separated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (4), (5), and (6) follow immediately from", "Lemma \\ref{lemma-compose-after-separated}", "and", "Spaces, Definition \\ref{spaces-definition-separated}.", "Parts (1), (2), and (3) follow from parts (4), (5), and (6) by", "thinking of $X$ and $Y$ as algebraic spaces over", "$\\Spec(\\mathbf{Z})$, see", "Properties of Spaces, Definition \\ref{spaces-properties-definition-separated}." ], "refs": [ "spaces-morphisms-lemma-compose-after-separated", "spaces-definition-separated", "spaces-properties-definition-separated" ], "ref_ids": [ 4720, 8181, 11922 ] } ], "ref_ids": [] }, { "id": 4722, "type": "theorem", "label": "spaces-morphisms-lemma-separated-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-separated-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{P}$ be any of the separation", "axioms of Definition \\ref{definition-separated}.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is $\\mathcal{P}$,", "\\item for every scheme $Z$ and morphism $Z \\to Y$ the", "base change $Z \\times_Y X \\to Z$ of $f$ is $\\mathcal{P}$,", "\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the", "base change $Z \\times_Y X \\to Z$ of $f$ is $\\mathcal{P}$,", "\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the", "algebraic space $Z \\times_Y X$ is $\\mathcal{P}$ (see", "Properties of Spaces, Definition \\ref{spaces-properties-definition-separated}),", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that the base change $V \\times_Y X \\to V$ has", "$\\mathcal{P}$, and", "\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each", "of the morphisms $f^{-1}(Y_i) \\to Y_i$ has $\\mathcal{P}$.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-separated", "spaces-properties-definition-separated" ], "proofs": [ { "contents": [ "We will repeatedly use", "Lemma \\ref{lemma-base-change-separated}", "without further mention. In particular, it is clear that", "(1) implies (2) and (2) implies (3).", "\\medskip\\noindent", "Let us prove that (3) and (4) are equivalent. Note that if $Z$ is an affine", "scheme, then the morphism $Z \\to \\Spec(\\mathbf{Z})$ is a separated", "morphism as a morphism of algebraic spaces over $\\Spec(\\mathbf{Z})$.", "If $Z \\times_Y X \\to Z$ is $\\mathcal{P}$, then", "$Z \\times_Y X \\to \\Spec(\\mathbf{Z})$ is $\\mathcal{P}$", "as a composition (see", "Lemma \\ref{lemma-composition-separated}). Hence the algebraic", "space $Z \\times_Y X$ is $\\mathcal{P}$. Conversely, if the algebraic", "space $Z \\times_Y X$ is $\\mathcal{P}$, then", "$Z \\times_Y X \\to \\Spec(\\mathbf{Z})$ is $\\mathcal{P}$, and", "hence by", "Lemma \\ref{lemma-compose-after-separated}", "we see that $Z \\times_Y X \\to Z$ is $\\mathcal{P}$.", "\\medskip\\noindent", "Let us prove that (3) implies (5). Assume (3). Let $V$ be a scheme", "and let $V \\to Y$ be \\'etale surjective. We have to show that", "$V \\times_Y X \\to V$ has property $\\mathcal{P}$. In other words,", "we have to show that the morphism", "$$", "V \\times_Y X \\longrightarrow", "(V \\times_Y X) \\times_V (V \\times_Y X) = V \\times_Y X \\times_Y X", "$$", "has the corresponding property (i.e., is a closed immersion, immersion,", "or quasi-compact). Let $V = \\bigcup V_j$ be an", "affine open covering of $V$. By assumption we know that each of the morphisms", "$$", "V_j \\times_Y X \\longrightarrow V_j \\times_Y X \\times_Y X", "$$", "does have the corresponding property. Since being a closed immersion,", "immersion, quasi-compact immersion, or quasi-compact is Zariski local", "on the target, and since the $V_j$ cover $V$ we get the desired conclusion.", "\\medskip\\noindent", "Let us prove that (5) implies (1). Let $V \\to Y$ be as in (5).", "Then we have the fibre product diagram", "$$", "\\xymatrix{", "V \\times_Y X \\ar[r] \\ar[d] &", "X \\ar[d] \\\\", "V \\times_Y X \\times_Y X \\ar[r] &", "X \\times_Y X", "}", "$$", "By assumption the left vertical arrow is a closed immersion,", "immersion, quasi-compact immersion, or quasi-compact. It follows from", "Spaces, Lemma \\ref{spaces-lemma-descent-representable-transformations-property}", "that also the right vertical arrow is a closed immersion,", "immersion, quasi-compact immersion, or quasi-compact.", "\\medskip\\noindent", "It is clear that (1) implies (6) by taking the covering $Y = Y$.", "Assume $Y = \\bigcup Y_i$ is as in (6). Choose schemes $V_i$ and", "surjective \\'etale morphisms $V_i \\to Y_i$. Note that the morphisms", "$V_i \\times_Y X \\to V_i$ have $\\mathcal{P}$ as they are base changes", "of the morphisms $f^{-1}(Y_i) \\to Y_i$. Set $V = \\coprod V_i$.", "Then $V \\to Y$ is a morphism as in (5) (details omitted). Hence", "(6) implies (5) and we are done." ], "refs": [ "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-composition-separated", "spaces-morphisms-lemma-compose-after-separated", "spaces-lemma-descent-representable-transformations-property" ], "ref_ids": [ 4714, 4718, 4720, 8134 ] } ], "ref_ids": [ 4984, 11922 ] }, { "id": 4723, "type": "theorem", "label": "spaces-morphisms-lemma-match-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-match-separated", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item The morphism $f$ is locally separated.", "\\item The morphism $f$ is (quasi-)separated in the sense of", "Definition \\ref{definition-separated}", "above if and only if $f$ is (quasi-)separated in the sense of", "Section \\ref{section-representable}.", "\\end{enumerate}", "In particular, if $f : X \\to Y$ is a morphism of schemes over $S$, then", "$f$ is (quasi-)separated in the sense of", "Definition \\ref{definition-separated}", "if and only if $f$ is (quasi-)separated as a morphism of schemes." ], "refs": [ "spaces-morphisms-definition-separated", "spaces-morphisms-definition-separated" ], "proofs": [ { "contents": [ "This is the equivalence of (1) and (2) of", "Lemma \\ref{lemma-separated-local}", "combined with the fact that any morphism of schemes is locally separated, see", "Schemes, Lemma \\ref{schemes-lemma-diagonal-immersion}." ], "refs": [ "spaces-morphisms-lemma-separated-local", "schemes-lemma-diagonal-immersion" ], "ref_ids": [ 4722, 7707 ] } ], "ref_ids": [ 4984, 4984 ] }, { "id": 4724, "type": "theorem", "label": "spaces-morphisms-lemma-surjective-representable", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-surjective-representable", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable", "morphism of algebraic spaces over $S$. Then", "$f$ is surjective (in the sense of Section \\ref{section-representable})", "if and only if $|f| : |X| \\to |Y|$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Namely, if $f : X \\to Y$ is representable, then it is surjective if and only if", "for every scheme $T$ and every morphism $T \\to Y$ the base change", "$f_T : T \\times_Y X \\to T$ of $f$ is a surjective morphism of schemes,", "in other words, if and only if $|f_T|$ is surjective. By", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}", "the map $|T \\times_Y X| \\to |T| \\times_{|Y|} |X|$ is always surjective.", "Hence $|f_T| : |T \\times_Y X| \\to |T|$ is surjective if $|f| : |X| \\to |Y|$", "is surjective. Conversely, if $|f_T|$ is surjective", "for every $T \\to Y$ as above, then by taking $T$ to be the spectrum of a", "field we conclude that $|X| \\to |Y|$ is surjective." ], "refs": [ "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 11819 ] } ], "ref_ids": [] }, { "id": 4725, "type": "theorem", "label": "spaces-morphisms-lemma-surjective-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-surjective-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is surjective,", "\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism", "$Z \\times_Y X \\to Z$ is surjective,", "\\item for every affine scheme $Z$ and any morphism", "$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is surjective,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is a surjective morphism,", "\\item there exists a scheme $U$ and a surjective \\'etale morphism", "$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$", "is surjective,", "\\item there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes and the vertical arrows are surjective \\'etale", "such that the top horizontal arrow is surjective, and", "\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that", "each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4726, "type": "theorem", "label": "spaces-morphisms-lemma-composition-surjective", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-surjective", "contents": [ "The composition of surjective morphisms is surjective." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4727, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-surjective", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-surjective", "contents": [ "The base change of a surjective morphism is surjective." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}." ], "refs": [ "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 11819 ] } ], "ref_ids": [] }, { "id": 4728, "type": "theorem", "label": "spaces-morphisms-lemma-characterize-representable-universally-open", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-characterize-representable-universally-open", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of", "algebraic spaces over $S$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally open", "(in the sense of Section \\ref{section-representable}), and", "\\item for every morphism of algebraic spaces $Z \\to Y$ the morphism of", "topological spaces $|Z \\times_Y X| \\to |Z|$ is open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1), and let $Z \\to Y$ be as in (2). Choose a scheme $V$ and", "a surjective \\'etale morphism $V \\to Y$. By assumption the morphism", "of schemes $V \\times_Y X \\to V$ is universally open. By", "Properties of Spaces, Section \\ref{spaces-properties-section-points}", "in the commutative diagram", "$$", "\\xymatrix{", "|V \\times_Y X| \\ar[r] \\ar[d] & |Z \\times_Y X| \\ar[d] \\\\", "|V| \\ar[r] & |Z|", "}", "$$", "the horizontal arrows are open and surjective, and moreover", "$$", "|V \\times_Y X| \\longrightarrow |V| \\times_{|Z|} |Z \\times_Y X|", "$$", "is surjective. Hence as the left", "vertical arrow is open it follows that the right vertical arrow is", "open. This proves (2). The implication (2) $\\Rightarrow$ (1) is", "immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4729, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-universally-open", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-universally-open", "contents": [ "The base change of a universally open morphism of algebraic spaces", "by any morphism of algebraic spaces is universally open." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4730, "type": "theorem", "label": "spaces-morphisms-lemma-composition-open", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-open", "contents": [ "The composition of a pair of (universally) open morphisms of algebraic spaces", "is (universally) open." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4731, "type": "theorem", "label": "spaces-morphisms-lemma-universally-open-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-universally-open-local", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally open,", "\\item for every scheme $Z$ and every morphism $Z \\to Y$", "the projection $|Z \\times_Y X| \\to |Z|$ is open,", "\\item for every affine scheme $Z$ and every morphism $Z \\to Y$", "the projection $|Z \\times_Y X| \\to |Z|$ is open, and", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is a universally open morphism", "of algebraic spaces, and", "\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that", "each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is universally open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We omit the proof that (1) implies (2), and that (2) implies (3).", "\\medskip\\noindent", "Assume (3). Choose a surjective \\'etale morphism $V \\to Y$.", "We are going to show that $V \\times_Y X \\to V$ is a universally", "open morphism of algebraic spaces. Let $Z \\to V$ be a morphism", "from an algebraic space to $V$. Let $W \\to Z$ be a surjective \\'etale", "morphism where $W = \\coprod W_i$ is a disjoint union of affine schemes, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-cover-by-union-affines}.", "Then we have the following commutative diagram", "$$", "\\xymatrix{", "\\coprod_i |W_i \\times_Y X| \\ar@{=}[r] \\ar[d] &", "|W \\times_Y X| \\ar[r] \\ar[d] &", "|Z \\times_Y X| \\ar[d] \\ar@{=}[r] &", "|Z \\times_V (V \\times_Y X)| \\ar[ld] \\\\", "\\coprod |W_i| \\ar@{=}[r] &", "|W| \\ar[r] &", "|Z|", "}", "$$", "We have to show the south-east arrow is open. The middle horizontal", "arrows are surjective and open", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-open}).", "By assumption (3), and the fact that", "$W_i$ is affine we see that the left vertical arrows are open. Hence", "it follows that the right vertical arrow is open.", "\\medskip\\noindent", "Assume $V \\to Y$ is as in (4). We will show that $f$ is universally open.", "Let $Z \\to Y$ be a morphism of algebraic spaces. Consider the", "diagram", "$$", "\\xymatrix{", "|(V \\times_Y Z) \\times_V (V \\times_Y X)| \\ar@{=}[r] \\ar[rd] &", "|V \\times_Y X| \\ar[r] \\ar[d] &", "|Z \\times_Y X| \\ar[d] \\\\", " &", "|V \\times_Y Z| \\ar[r] &", "|Z|", "}", "$$", "The south-west arrow is open by assumption. The horizontal arrows are", "surjective and open because the corresponding morphisms of", "algebraic spaces are \\'etale (see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-open}).", "It follows that the right vertical arrow is open.", "\\medskip\\noindent", "Of course (1) implies (5) by taking the covering $Y = Y$.", "Assume $Y = \\bigcup Y_i$ is as in (5). Then for any $Z \\to Y$", "we get a corresponding Zariski covering $Z = \\bigcup Z_i$ such that", "the base change of $f$ to $Z_i$ is open. By a simple topological", "argument this implies that $Z \\times_Y X \\to Z$ is open. Hence (1) holds." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-properties-lemma-etale-open", "spaces-properties-lemma-etale-open" ], "ref_ids": [ 11830, 11860, 11860 ] } ], "ref_ids": [] }, { "id": 4732, "type": "theorem", "label": "spaces-morphisms-lemma-space-over-field-universally-open", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-space-over-field-universally-open", "contents": [ "Let $S$ be a scheme. Let $p : X \\to \\Spec(k)$ be a morphism of", "algebraic spaces over $S$ where $k$ is a field. Then", "$p : X \\to \\Spec(k)$ is universally open." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "The composition $U \\to \\Spec(k)$ is universally open (as a morphism", "of schemes) by", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open}.", "Let $Z \\to \\Spec(k)$ be a morphism of schemes. Then", "$U \\times_{\\Spec(k)} Z \\to X \\times_{\\Spec(k)} Z$ is surjective,", "see", "Lemma \\ref{lemma-base-change-surjective}.", "Hence the first of the maps", "$$", "|U \\times_{\\Spec(k)} Z| \\to |X \\times_{\\Spec(k)} Z| \\to |Z|", "$$", "is surjective. Since the composition is open by the above we conclude that", "the second map is open as well. Whence $p$ is universally open by", "Lemma \\ref{lemma-universally-open-local}." ], "refs": [ "morphisms-lemma-scheme-over-field-universally-open", "spaces-morphisms-lemma-base-change-surjective", "spaces-morphisms-lemma-universally-open-local" ], "ref_ids": [ 5254, 4727, 4731 ] } ], "ref_ids": [] }, { "id": 4733, "type": "theorem", "label": "spaces-morphisms-lemma-characterize-representable-universally-submersive", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-characterize-representable-universally-submersive", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of", "algebraic spaces over $S$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally submersive", "(in the sense of Section \\ref{section-representable}), and", "\\item for every morphism of algebraic spaces $Z \\to Y$ the morphism of", "topological spaces $|Z \\times_Y X| \\to |Z|$ is submersive.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1), and let $Z \\to Y$ be as in (2). Choose a scheme $V$ and", "a surjective \\'etale morphism $V \\to Y$. By assumption the morphism", "of schemes $V \\times_Y X \\to V$ is universally submersive. By", "Properties of Spaces, Section \\ref{spaces-properties-section-points}", "in the commutative diagram", "$$", "\\xymatrix{", "|V \\times_Y X| \\ar[r] \\ar[d] & |Z \\times_Y X| \\ar[d] \\\\", "|V| \\ar[r] & |Z|", "}", "$$", "the horizontal arrows are open and surjective, and moreover", "$$", "|V \\times_Y X| \\longrightarrow |V| \\times_{|Z|} |Z \\times_Y X|", "$$", "is surjective. Hence as the left vertical arrow is submersive", "it follows that the right vertical arrow is submersive.", "This proves (2). The implication (2) $\\Rightarrow$ (1) is", "immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4734, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-universally-submersive", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-universally-submersive", "contents": [ "The base change of a universally submersive morphism of algebraic spaces", "by any morphism of algebraic spaces is universally submersive." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4735, "type": "theorem", "label": "spaces-morphisms-lemma-composition-universally-submersive", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-universally-submersive", "contents": [ "The composition of a pair of (universally) submersive morphisms of", "algebraic spaces is (universally) submersive." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4736, "type": "theorem", "label": "spaces-morphisms-lemma-characterize-representable-quasi-compact", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-characterize-representable-quasi-compact", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is quasi-compact", "(in the sense of Section \\ref{section-representable}), and", "\\item for every quasi-compact algebraic space $Z$ and any morphism", "$Z \\to Y$ the algebraic space $Z \\times_Y X$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1), and let $Z \\to Y$ be a morphism of algebraic spaces with", "$Z$ quasi-compact. By", "Properties of Spaces,", "Definition \\ref{spaces-properties-definition-quasi-compact}", "there exists a quasi-compact scheme $U$ and a surjective \\'etale", "morphism $U \\to Z$. Since $f$ is representable and quasi-compact", "we see by definition that $U \\times_Y X$ is a scheme, and that", "$U \\times_Y X \\to U$ is quasi-compact. Hence $U \\times_Y X$ is", "a quasi-compact scheme. The morphism $U \\times_Y X \\to Z \\times_Y X$", "is \\'etale and surjective (as the base change of the representable", "\\'etale and surjective morphism $U \\to Z$, see", "Section \\ref{section-representable}).", "Hence by definition $Z \\times_Y X$ is quasi-compact.", "\\medskip\\noindent", "Assume (2). Let $Z \\to Y$ be a morphism, where $Z$ is a scheme.", "We have to show that $p : Z \\times_Y X \\to Z$ is quasi-compact.", "Let $U \\subset Z$ be affine open. Then $p^{-1}(U) = U \\times_Y Z$", "and the scheme $U \\times_Y Z$ is quasi-compact by assumption (2).", "Hence $p$ is quasi-compact, see", "Schemes, Section \\ref{schemes-section-quasi-compact}." ], "refs": [ "spaces-properties-definition-quasi-compact" ], "ref_ids": [ 11925 ] } ], "ref_ids": [] }, { "id": 4737, "type": "theorem", "label": "spaces-morphisms-lemma-quasi-compact-is-quasi-compact", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-quasi-compact-is-quasi-compact", "contents": [ "Let $S$ be a scheme. If $f : X \\to Y$ is a quasi-compact morphism of", "algebraic spaces over $S$, then the underlying map", "$|f| : |X| \\to |Y|$ of topological space is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset |Y|$ be quasi-compact open. By Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-open-subspaces}", "there is an open subspace $Y' \\subset Y$ with $V = |Y'|$.", "Then $Y'$ is a quasi-compact algebraic space by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quasi-compact-space}", "and hence $X' = Y' \\times_Y X$ is a quasi-compact algebraic", "space by Definition \\ref{definition-quasi-compact}.", "On the other hand, $X' \\subset X$ is an open subspace", "(Spaces, Lemma \\ref{spaces-lemma-base-change-immersions})", "and $|X'| = |f|^{-1}(|X'|) = |f|^{-1}(V)$ by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}.", "We conclude using", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quasi-compact-space}", "again that $|X'|$ is a quasi-compact open of $|X|$ as desired." ], "refs": [ "spaces-properties-lemma-open-subspaces", "spaces-properties-lemma-quasi-compact-space", "spaces-morphisms-definition-quasi-compact", "spaces-lemma-base-change-immersions", "spaces-properties-lemma-points-cartesian", "spaces-properties-lemma-quasi-compact-space" ], "ref_ids": [ 11823, 11827, 4988, 8161, 11819, 11827 ] } ], "ref_ids": [] }, { "id": 4738, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-quasi-compact", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-quasi-compact", "contents": [ "The base change of a quasi-compact morphism of algebraic spaces", "by any morphism of algebraic spaces is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Transitivity of fibre products." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4739, "type": "theorem", "label": "spaces-morphisms-lemma-composition-quasi-compact", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-quasi-compact", "contents": [ "The composition of a pair of quasi-compact morphisms of algebraic spaces", "is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Transitivity of fibre products." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4740, "type": "theorem", "label": "spaces-morphisms-lemma-surjection-from-quasi-compact", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-surjection-from-quasi-compact", "contents": [ "\\begin{slogan}", "The image of a quasi-compact algebraic space under a surjective morphism", "is quasi-compact.", "\\end{slogan}", "Let $S$ be a scheme.", "\\begin{enumerate}", "\\item If $X \\to Y$ is a surjective morphism of algebraic spaces over $S$,", "and $X$ is quasi-compact then $Y$ is quasi-compact.", "\\item If", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & &", "Y \\ar[dl]^q \\\\", "& Z", "}", "$$", "is a commutative diagram of morphisms of algebraic spaces over $S$", "and $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $X$ is quasi-compact and $X \\to Y$ is surjective. By", "Definition \\ref{definition-surjective}", "the map $|X| \\to |Y|$ is surjective, hence we see $Y$ is quasi-compact by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quasi-compact-space}", "and the topological fact that the image of a quasi-compact space under a", "continuous map is quasi-compact, see", "Topology, Lemma \\ref{topology-lemma-image-quasi-compact}.", "Let $f, p, q$ be as in (2).", "Let $T \\to Z$ be a morphism whose source is a quasi-compact algebraic space.", "By assumption $T \\times_Z X$ is quasi-compact. By", "Lemma \\ref{lemma-base-change-surjective}", "the morphism $T \\times_Z X \\to T \\times_Z Y$ is surjective.", "Hence by part (1) we see $T \\times_Z Y$ is quasi-compact too.", "Thus $q$ is quasi-compact." ], "refs": [ "spaces-morphisms-definition-surjective", "spaces-properties-lemma-quasi-compact-space", "topology-lemma-image-quasi-compact", "spaces-morphisms-lemma-base-change-surjective" ], "ref_ids": [ 4985, 11827, 8233, 4727 ] } ], "ref_ids": [] }, { "id": 4741, "type": "theorem", "label": "spaces-morphisms-lemma-descent-quasi-compact", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-descent-quasi-compact", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $g : Y' \\to Y$ be a universally open and surjective morphism of", "algebraic spaces such that the base change $f' : X' \\to Y'$ is quasi-compact.", "Then $f$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\to Y$ be a morphism of algebraic spaces with $Z$ quasi-compact.", "As $g$ is universally open and surjective, we see that", "$Y' \\times_Y Z \\to Z$ is open and surjective. As every point of", "$|Y' \\times_Y Z|$ has a fundamental system of quasi-compact open", "neighbourhoods (see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-space-locally-quasi-compact})", "we can find a quasi-compact open $W \\subset |Y' \\times_Y Z|$", "which surjects onto $Z$. Denote", "$f'' : W \\times_Y X \\to W$ the base change of $f'$ by $W \\to Y'$.", "By assumption $W \\times_Y X$ is quasi-compact. As $W \\to Z$ is surjective", "we see that $W \\times_Y X \\to Z \\times_Y X$ is surjective.", "Hence $Z \\times_Y X$ is quasi-compact by", "Lemma \\ref{lemma-surjection-from-quasi-compact}.", "Thus $f$ is quasi-compact." ], "refs": [ "spaces-properties-lemma-space-locally-quasi-compact", "spaces-morphisms-lemma-surjection-from-quasi-compact" ], "ref_ids": [ 11829, 4740 ] } ], "ref_ids": [] }, { "id": 4742, "type": "theorem", "label": "spaces-morphisms-lemma-quasi-compact-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-quasi-compact-local", "contents": [ "\\begin{slogan}", "Quasi-compact morphisms of algebraic spaces are preserved under pullback", "and local on the target.", "\\end{slogan}", "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is quasi-compact,", "\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism of", "algebraic spaces $Z \\times_Y X \\to Z$ is quasi-compact,", "\\item for every affine scheme $Z$ and any morphism", "$Z \\to Y$ the algebraic space $Z \\times_Y X$ is quasi-compact,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is a quasi-compact morphism", "of algebraic spaces, and", "\\item there exists a surjective \\'etale morphism", "$Y' \\to Y$ of algebraic spaces such that $Y' \\times_Y X \\to Y'$", "is a quasi-compact morphism of algebraic spaces, and", "\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that", "each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-base-change-quasi-compact}", "without further mention.", "It is clear that (1) implies (2) and that (2) implies (3).", "Assume (3). Let $Z$ be a quasi-compact algebraic space over $S$,", "and let $Z \\to Y$ be a morphism. By", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}", "there exists an affine scheme $U$ and a surjective \\'etale morphism", "$U \\to Z$. Then $U \\times_Y X \\to Z \\times_Y X$ is a surjective", "morphism of algebraic spaces, see", "Lemma \\ref{lemma-base-change-surjective}.", "By assumption $|U \\times_Y X|$ is quasi-compact. It surjects", "onto $|Z \\times_Y X|$, hence we conclude that $|Z \\times_Y X|$", "is quasi-compact, see", "Topology, Lemma \\ref{topology-lemma-image-quasi-compact}.", "This proves that (3) implies (1).", "\\medskip\\noindent", "The implications (1) $\\Rightarrow$ (4), (4) $\\Rightarrow$ (5) are clear.", "The implication (5) $\\Rightarrow$ (1) follows from", "Lemma \\ref{lemma-descent-quasi-compact}", "and the fact that an \\'etale morphism of algebraic spaces is universally open", "(see discussion following", "Definition \\ref{definition-open}).", "\\medskip\\noindent", "Of course (1) implies (6) by taking the covering $Y = Y$.", "Assume $Y = \\bigcup Y_i$ is as in (6). Let $Z$ be affine and let", "$Z \\to Y$ be a morphism. Then there exists a finite standard affine", "covering $Z = Z_1 \\cup \\ldots \\cup Z_n$ such that each $Z_j \\to Y$", "factors through $Y_{i_j}$ for some $i_j$. Hence the algebraic space", "$$", "Z_j \\times_Y X = Z_j \\times_{Y_{i_j}} f^{-1}(Y_{i_j})", "$$", "is quasi-compact. Since", "$Z \\times_Y X = \\bigcup_{j = 1, \\ldots, n} Z_j \\times_Y X$", "is a Zariski covering we see that", "$|Z \\times_Y X| = \\bigcup_{j = 1, \\ldots, n} |Z_j \\times_Y X|$", "(see Properties of Spaces, Lemma \\ref{spaces-properties-lemma-open-subspaces})", "is a finite union of quasi-compact spaces, hence quasi-compact.", "Thus we see that (6) implies (3)." ], "refs": [ "spaces-morphisms-lemma-base-change-quasi-compact", "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-morphisms-lemma-base-change-surjective", "topology-lemma-image-quasi-compact", "spaces-morphisms-lemma-descent-quasi-compact", "spaces-morphisms-definition-open", "spaces-properties-lemma-open-subspaces" ], "ref_ids": [ 4738, 11832, 4727, 8233, 4741, 4986, 11823 ] } ], "ref_ids": [] }, { "id": 4743, "type": "theorem", "label": "spaces-morphisms-lemma-quasi-compact-permanence", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-quasi-compact-permanence", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces over $S$.", "If $g \\circ f$ is quasi-compact and $g$ is quasi-separated", "then $f$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "This is true because $f$ equals the composition", "$(1, f) : X \\to X \\times_Z Y \\to Y$. The first map", "is quasi-compact by Lemma \\ref{lemma-section-immersion}", "because it is a section of the quasi-separated morphism $X \\times_Z Y \\to X$", "(a base change of $g$, see Lemma \\ref{lemma-base-change-separated}).", "The second map is quasi-compact as it", "is the base change of $f$, see", "Lemma \\ref{lemma-base-change-quasi-compact}.", "And compositions of quasi-compact", "morphisms are quasi-compact, see Lemma \\ref{lemma-composition-quasi-compact}." ], "refs": [ "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-base-change-quasi-compact", "spaces-morphisms-lemma-composition-quasi-compact" ], "ref_ids": [ 4714, 4738, 4739 ] } ], "ref_ids": [] }, { "id": 4744, "type": "theorem", "label": "spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence", "contents": [ "Let $f : X \\to Y$ be a morphism of algebraic spaces", "over a scheme $S$.", "\\begin{enumerate}", "\\item If $X$ is quasi-compact and $Y$ is quasi-separated, then $f$ is", "quasi-compact.", "\\item If $X$ is quasi-compact and quasi-separated and $Y$ is quasi-separated,", "then $f$ is quasi-compact and quasi-separated.", "\\item A fibre product of quasi-compact and quasi-separated algebraic spaces", "is quasi-compact and quasi-separated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from", "Lemma \\ref{lemma-quasi-compact-permanence}", "with $Z = S = \\Spec(\\mathbf{Z})$.", "Part (2) follows from (1) and", "Lemma \\ref{lemma-compose-after-separated}.", "For (3) let $X \\to Y$ and $Z \\to Y$ be morphisms", "of quasi-compact and quasi-separated algebraic spaces. Then", "$X \\times_Y Z \\to Z$ is quasi-compact and quasi-separated as a", "base change of $X \\to Y$ using (2) and", "Lemmas \\ref{lemma-base-change-quasi-compact} and", "\\ref{lemma-base-change-separated}.", "Hence $X \\times_Y Z$ is quasi-compact and quasi-separated as", "an algebraic space quasi-compact and quasi-separated over", "$Z$, see", "Lemmas \\ref{lemma-separated-over-separated} and", "\\ref{lemma-composition-quasi-compact}." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-permanence", "spaces-morphisms-lemma-compose-after-separated", "spaces-morphisms-lemma-base-change-quasi-compact", "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-separated-over-separated", "spaces-morphisms-lemma-composition-quasi-compact" ], "ref_ids": [ 4743, 4720, 4738, 4714, 4719, 4739 ] } ], "ref_ids": [] }, { "id": 4745, "type": "theorem", "label": "spaces-morphisms-lemma-characterize-representable-universally-closed", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-characterize-representable-universally-closed", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of", "algebraic spaces over $S$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally closed", "(in the sense of Section \\ref{section-representable}), and", "\\item for every morphism of algebraic spaces $Z \\to Y$ the morphism of", "topological spaces $|Z \\times_Y X| \\to |Z|$ is closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1), and let $Z \\to Y$ be as in (2). Choose a scheme $V$ and", "a surjective \\'etale morphism $V \\to Y$. By assumption the morphism", "of schemes $V \\times_Y X \\to V$ is universally closed. By", "Properties of Spaces, Section \\ref{spaces-properties-section-points}", "in the commutative diagram", "$$", "\\xymatrix{", "|V \\times_Y X| \\ar[r] \\ar[d] & |Z \\times_Y X| \\ar[d] \\\\", "|V| \\ar[r] & |Z|", "}", "$$", "the horizontal arrows are open and surjective, and moreover", "$$", "|V \\times_Y X| \\longrightarrow |V| \\times_{|Z|} |Z \\times_Y X|", "$$", "is surjective. Hence as the left", "vertical arrow is closed it follows that the right vertical arrow is", "closed. This proves (2). The implication (2) $\\Rightarrow$ (1) is", "immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4746, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-universally-closed", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-universally-closed", "contents": [ "The base change of a universally closed morphism of algebraic spaces", "by any morphism of algebraic spaces is universally closed." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4747, "type": "theorem", "label": "spaces-morphisms-lemma-composition-universally-closed", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-universally-closed", "contents": [ "The composition of a pair of (universally) closed morphisms of algebraic spaces", "is (universally) closed." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4748, "type": "theorem", "label": "spaces-morphisms-lemma-universally-closed-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-universally-closed-local", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally closed,", "\\item for every scheme $Z$ and every morphism $Z \\to Y$", "the projection $|Z \\times_Y X| \\to |Z|$ is closed,", "\\item for every affine scheme $Z$ and every morphism $Z \\to Y$", "the projection $|Z \\times_Y X| \\to |Z|$ is closed,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is a universally closed morphism", "of algebraic spaces, and", "\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that", "each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is universally closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We omit the proof that (1) implies (2), and that (2) implies (3).", "\\medskip\\noindent", "Assume (3). Choose a surjective \\'etale morphism $V \\to Y$.", "We are going to show that $V \\times_Y X \\to V$ is a universally", "closed morphism of algebraic spaces. Let $Z \\to V$ be a morphism", "from an algebraic space to $V$. Let $W \\to Z$ be a surjective \\'etale", "morphism where $W = \\coprod W_i$ is a disjoint union of affine schemes, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-cover-by-union-affines}.", "Then we have the following commutative diagram", "$$", "\\xymatrix{", "\\coprod_i |W_i \\times_Y X| \\ar@{=}[r] \\ar[d] &", "|W \\times_Y X| \\ar[r] \\ar[d] &", "|Z \\times_Y X| \\ar[d] \\ar@{=}[r] &", "|Z \\times_V (V \\times_Y X)| \\ar[ld] \\\\", "\\coprod |W_i| \\ar@{=}[r] &", "|W| \\ar[r] &", "|Z|", "}", "$$", "We have to show the south-east arrow is closed. The middle horizontal", "arrows are surjective and open", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-open}).", "By assumption (3), and the fact that", "$W_i$ is affine we see that the left vertical arrows are closed. Hence", "it follows that the right vertical arrow is closed.", "\\medskip\\noindent", "Assume (4). We will show that $f$ is universally closed.", "Let $Z \\to Y$ be a morphism of algebraic spaces. Consider the", "diagram", "$$", "\\xymatrix{", "|(V \\times_Y Z) \\times_V (V \\times_Y X)| \\ar@{=}[r] \\ar[rd] &", "|V \\times_Y X| \\ar[r] \\ar[d] &", "|Z \\times_Y X| \\ar[d] \\\\", " &", "|V \\times_Y Z| \\ar[r] &", "|Z|", "}", "$$", "The south-west arrow is closed by assumption. The horizontal arrows are", "surjective and open because the corresponding morphisms of", "algebraic spaces are \\'etale (see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-open}).", "It follows that the right vertical arrow is closed.", "\\medskip\\noindent", "Of course (1) implies (5) by taking the covering $Y = Y$.", "Assume $Y = \\bigcup Y_i$ is as in (5). Then for any $Z \\to Y$", "we get a corresponding Zariski covering $Z = \\bigcup Z_i$ such that", "the base change of $f$ to $Z_i$ is closed. By a simple topological", "argument this implies that $Z \\times_Y X \\to Z$ is closed. Hence (1) holds." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-properties-lemma-etale-open", "spaces-properties-lemma-etale-open" ], "ref_ids": [ 11830, 11860, 11860 ] } ], "ref_ids": [] }, { "id": 4749, "type": "theorem", "label": "spaces-morphisms-lemma-universally-closed-quasi-compact", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-universally-closed-quasi-compact", "contents": [ "Let $S$ be a scheme.", "A universally closed morphism of algebraic spaces over $S$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "This proof is a repeat of the proof in the case of schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-universally-closed-quasi-compact}.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume that $f$ is not quasi-compact.", "Our goal is to show that $f$ is not universally closed. By", "Lemma \\ref{lemma-quasi-compact-local}", "there exists an affine scheme $Z$ and a morphism $Z \\to Y$", "such that $Z \\times_Y X \\to Z$ is not quasi-compact. To achieve our goal", "it suffices to show that $Z \\times_Y X \\to Z$ is not universally closed,", "hence we may assume that $Y = \\Spec(B)$ for some ring $B$.", "\\medskip\\noindent", "Write $X = \\bigcup_{i \\in I} X_i$ where the $X_i$ are quasi-compact", "open subspaces of $X$. For example, choose a surjective \\'etale morphism", "$U \\to X$ where $U$ is a scheme, choose an affine open covering", "$U = \\bigcup U_i$ and let $X_i \\subset X$ be the image of $U_i$.", "We will use later that the morphisms $X_i \\to Y$ are quasi-compact, see", "Lemma \\ref{lemma-quasi-compact-permanence}.", "Let $T = \\Spec(B[a_i ; i \\in I])$. Let $T_i = D(a_i) \\subset T$.", "Let $Z \\subset T \\times_Y X$ be the reduced closed subspace whose underlying", "closed set of points is", "$|T \\times_Y Z| \\setminus \\bigcup_{i \\in I} |T_i \\times_Y X_i|$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-reduced-closed-subspace}.", "(Note that $T_i \\times_Y X_i$ is an open subspace of $T \\times_Y X$ as", "$T_i \\to T$ and $X_i \\to X$ are open immersions, see", "Spaces, Lemmas \\ref{spaces-lemma-base-change-immersions} and", "\\ref{spaces-lemma-composition-immersions}.) Here is a diagram", "$$", "\\xymatrix{", "Z \\ar[r] \\ar[rd] &", "T \\times_Y X \\ar[d]^{f_T} \\ar[r]_q &", "X \\ar[d]^f \\\\", "& T \\ar[r]^p & Y", "}", "$$", "It suffices to prove that the image $f_T(|Z|)$ is not closed in $|T|$.", "\\medskip\\noindent", "We claim there exists a point $y \\in Y$ such that there is no", "affine open neighborhood $V$ of $y$ in $Y$ such that $X_V$ is quasi-compact.", "If not then we can cover $Y$ with finitely many such $V$ and for", "each $V$ the morphism $Y_V \\to V$ is quasi-compact by", "Lemma \\ref{lemma-quasi-compact-permanence}", "and then", "Lemma \\ref{lemma-quasi-compact-local}", "implies $f$ quasi-compact, a contradiction. Fix a $y \\in Y$ as in the claim.", "\\medskip\\noindent", "Let $t \\in T$ be the point lying over $y$ with $\\kappa(t) = \\kappa(y)$", "such that $a_i = 1$ in $\\kappa(t)$ for all $i$. Suppose $z \\in |Z|$ with", "$f_T(z) = t$. Then $q(t) \\in X_i$ for some $i$. Hence $f_T(z) \\not \\in T_i$", "by construction of $Z$, which contradicts the fact that $t \\in T_i$ by", "construction. Hence we see that $t \\in |T| \\setminus f_T(|Z|)$.", "\\medskip\\noindent", "Assume $f_T(|Z|)$ is closed in $|T|$. Then there exists an element", "$g \\in B[a_i; i \\in I]$ with $f_T(|Z|) \\subset V(g)$ but $t \\not \\in V(g)$.", "Hence the image of $g$ in $\\kappa(t)$ is nonzero. In particular some", "coefficient of $g$ has nonzero image in $\\kappa(y)$. Hence this coefficient is", "invertible on some affine open neighborhood $V$ of $y$. Let $J$ be the finite", "set of $j \\in I$ such that the variable $a_j$ appears in $g$.", "Since $X_V$ is not quasi-compact and each $X_{i, V}$ is quasi-compact,", "we may choose a point $x \\in |X_V| \\setminus \\bigcup_{j \\in J} |X_{j, V}|$.", "In other words, $x \\in |X| \\setminus \\bigcup_{j \\in J} |X_j|$ and $x$ lies", "above some $v \\in V$. Since $g$ has a coefficient that is invertible on $V$,", "we can find a point $t' \\in T$ lying above $v$ such that $t' \\not \\in V(g)$ and", "$t' \\in V(a_i)$ for all $i \\notin J$. This is true because", "$V(a_i; i \\in I \\setminus J) = \\Spec(B[a_j; j\\in J])$", "and the set of points of this scheme lying over $v$ is bijective", "with $\\Spec(\\kappa(v)[a_j; j \\in J])$ and $g$ restricts to", "a nonzero element of this polynomial ring by construction.", "In other words $t' \\not \\in T_i$ for each $i \\not \\in J$. By", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}", "we can find a point $z$ of $X \\times_Y T$ mapping to $x \\in X$ and to", "$t' \\in T$. Since $x \\not \\in |X_j|$ for $j \\in J$ and $t' \\not \\in T_i$", "for $i \\in I \\setminus J$ we see that $z \\in |Z|$. On the other hand", "$f_T(z) = t' \\not \\in V(g)$ which contradicts $f_T(Z) \\subset V(g)$.", "Thus the assumption ``$f_T(|Z|)$ closed'' is wrong and we conclude indeed", "that $f_T$ is not closed as desired." ], "refs": [ "morphisms-lemma-universally-closed-quasi-compact", "spaces-morphisms-lemma-quasi-compact-local", "spaces-morphisms-lemma-quasi-compact-permanence", "spaces-properties-lemma-reduced-closed-subspace", "spaces-lemma-base-change-immersions", "spaces-lemma-composition-immersions", "spaces-morphisms-lemma-quasi-compact-permanence", "spaces-morphisms-lemma-quasi-compact-local", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 5412, 4742, 4743, 11846, 8161, 8160, 4743, 4742, 11819 ] } ], "ref_ids": [] }, { "id": 4750, "type": "theorem", "label": "spaces-morphisms-lemma-image-universally-closed-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-image-universally-closed-separated", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $f : X \\to Y$ be a surjective universally closed", "morphism of algebraic spaces over $B$.", "\\begin{enumerate}", "\\item If $X$ is quasi-separated, then $Y$ is quasi-separated.", "\\item If $X$ is separated, then $Y$ is separated.", "\\item If $X$ is quasi-separated over $B$, then $Y$ is quasi-separated over $B$.", "\\item If $X$ is separated over $B$, then $Y$ is separated over $B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1) and (2) are a consequence of (3) and (4) for", "$S = B = \\Spec(\\mathbf{Z})$ (see", "Properties of Spaces,", "Definition \\ref{spaces-properties-definition-separated}).", "Consider the commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] \\ar[rr]_{\\Delta_{X/B}} & & X \\times_B X \\ar[d] \\\\", "Y \\ar[rr]^{\\Delta_{Y/B}} & & Y \\times_B Y", "}", "$$", "The left vertical arrow is surjective (i.e., universally surjective).", "The right vertical arrow is universally closed as a composition", "of the universally closed morphisms", "$X \\times_B X \\to X \\times_B Y \\to Y \\times_B Y$. Hence it is also", "quasi-compact, see", "Lemma \\ref{lemma-universally-closed-quasi-compact}.", "\\medskip\\noindent", "Assume $X$ is quasi-separated over $B$, i.e., $\\Delta_{X/B}$ is", "quasi-compact. Then if $Z$ is quasi-compact and $Z \\to Y \\times_B Y$ is", "a morphism, then $Z \\times_{Y \\times_B Y} X \\to Z \\times_{Y \\times_B Y} Y$", "is surjective and $Z \\times_{Y \\times_B Y} X$ is quasi-compact by our remarks", "above. We conclude that $\\Delta_{Y/B}$ is quasi-compact, i.e., $Y$", "is quasi-separated over $B$.", "\\medskip\\noindent", "Assume $X$ is separated over $B$, i.e., $\\Delta_{X/B}$ is a closed", "immersion. Then if $Z$ is affine, and $Z \\to Y \\times_B Y$ is", "a morphism, then $Z \\times_{Y \\times_B Y} X \\to Z \\times_{Y \\times_B Y} Y$", "is surjective and $Z \\times_{Y \\times_B Y} X \\to Z$ is universally closed", "by our remarks above. We conclude that $\\Delta_{Y/B}$ is universally closed.", "It follows that $\\Delta_{Y/B}$ is representable, locally of finite type, a", "monomorphism (see", "Lemma \\ref{lemma-properties-diagonal})", "and universally closed, hence a closed immersion, see", "\\'Etale Morphisms,", "Lemma \\ref{etale-lemma-characterize-closed-immersion}", "(and also the abstract principle", "Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}).", "Thus $Y$ is separated over $B$." ], "refs": [ "spaces-properties-definition-separated", "spaces-morphisms-lemma-universally-closed-quasi-compact", "spaces-morphisms-lemma-properties-diagonal", "etale-lemma-characterize-closed-immersion", "spaces-lemma-representable-transformations-property-implication" ], "ref_ids": [ 11922, 4749, 4712, 10702, 8136 ] } ], "ref_ids": [] }, { "id": 4751, "type": "theorem", "label": "spaces-morphisms-lemma-monomorphism", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-monomorphism", "contents": [ "Let $S$ be a scheme.", "Let $j : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $j$ is a monomorphism (as in Definition \\ref{definition-monomorphism}),", "\\item $j$ is a monomorphism in the category of algebraic spaces over $S$, and", "\\item the diagonal morphism $\\Delta_{X/Y} : X \\to X \\times_Y X$ is", "an isomorphism.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-monomorphism" ], "proofs": [ { "contents": [ "Note that $X \\times_Y X$ is both the fibre product in the category of", "sheaves on $(\\Sch/S)_{fppf}$ and the fibre product in the category", "of algebraic spaces over $S$, see", "Spaces, Lemma \\ref{spaces-lemma-fibre-product-spaces}.", "The equivalence of (1) and (3) is a general characterization", "of injective maps of sheaves on any site.", "The equivalence of (2) and (3) is a characterization of monomorphisms", "in any category with fibre products." ], "refs": [ "spaces-lemma-fibre-product-spaces" ], "ref_ids": [ 8143 ] } ], "ref_ids": [ 4990 ] }, { "id": 4752, "type": "theorem", "label": "spaces-morphisms-lemma-monomorphism-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-monomorphism-separated", "contents": [ "A monomorphism of algebraic spaces is separated." ], "refs": [], "proofs": [ { "contents": [ "This is true because an isomorphism is a closed immersion,", "and Lemma \\ref{lemma-monomorphism} above." ], "refs": [ "spaces-morphisms-lemma-monomorphism" ], "ref_ids": [ 4751 ] } ], "ref_ids": [] }, { "id": 4753, "type": "theorem", "label": "spaces-morphisms-lemma-composition-monomorphism", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-monomorphism", "contents": [ "A composition of monomorphisms is a monomorphism." ], "refs": [], "proofs": [ { "contents": [ "True because a composition of injective sheaf maps is injective." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4754, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-monomorphism", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-monomorphism", "contents": [ "The base change of a monomorphism is a monomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is a general fact about fibre products in a category of sheaves." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4755, "type": "theorem", "label": "spaces-morphisms-lemma-monomorphism-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-monomorphism-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is a monomorphism,", "\\item for every scheme $Z$ and morphism $Z \\to Y$ the", "base change $Z \\times_Y X \\to Z$ of $f$ is a monomorphism,", "\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the", "base change $Z \\times_Y X \\to Z$ of $f$ is a monomorphism,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that the base change $V \\times_Y X \\to V$ is a", "monomorphism, and", "\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each", "of the morphisms $f^{-1}(Y_i) \\to Y_i$ is a monomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use without further mention that a base change of a monomorphism", "is a monomorphism, see", "Lemma \\ref{lemma-base-change-monomorphism}.", "In particular it is clear that", "(1) $\\Rightarrow$ (2) $\\Rightarrow$ (3) $\\Rightarrow$ (4)", "(by taking $V$ to be a disjoint union of affine schemes \\'etale over $Y$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-cover-by-union-affines}).", "Let $V$ be a scheme, and let $V \\to Y$ be a surjective \\'etale morphism.", "If $V \\times_Y X \\to V$ is a monomorphism, then it", "follows that $X \\to Y$ is a monomorphism. Namely, given any", "cartesian diagram of sheaves", "$$", "\\vcenter{", "\\xymatrix{", "\\mathcal{F} \\ar[r]_a \\ar[d]_b & \\mathcal{G} \\ar[d]^c \\\\", "\\mathcal{H} \\ar[r]^d & \\mathcal{I}", "}", "}", "\\quad", "\\quad", "\\mathcal{F} = \\mathcal{H} \\times_\\mathcal{I} \\mathcal{G}", "$$", "if $c$ is a surjection of sheaves, and $a$ is injective, then also", "$d$ is injective. Thus (4) implies (1). Proof of the equivalence of", "(5) and (1) is omitted." ], "refs": [ "spaces-morphisms-lemma-base-change-monomorphism", "spaces-properties-lemma-cover-by-union-affines" ], "ref_ids": [ 4754, 11830 ] } ], "ref_ids": [] }, { "id": 4756, "type": "theorem", "label": "spaces-morphisms-lemma-immersions-monomorphisms", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-immersions-monomorphisms", "contents": [ "An immersion of algebraic spaces is a monomorphism.", "In particular, any immersion is separated." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be an immersion of algebraic spaces.", "For any morphism $Z \\to Y$ with $Z$ representable the base", "change $Z \\times_Y X \\to Z$ is an immersion of schemes, hence", "a monomorphism, see", "Schemes, Lemma \\ref{schemes-lemma-immersions-monomorphisms}.", "Hence $f$ is representable, and a monomorphism." ], "refs": [ "schemes-lemma-immersions-monomorphisms" ], "ref_ids": [ 7727 ] } ], "ref_ids": [] }, { "id": 4757, "type": "theorem", "label": "spaces-morphisms-lemma-monomorphism-toward-field", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-monomorphism-toward-field", "contents": [ "Let $S$ be a scheme. Let $k$ be a field and let $Z \\to \\Spec(k)$", "be a monomorphism of algebraic spaces over $S$. Then either", "$Z = \\emptyset$ or $Z = \\Spec(k)$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemmas \\ref{lemma-monomorphism-separated} and", "\\ref{lemma-separated-over-separated}", "we see that $Z$ is a separated algebraic space. Hence there exists an", "open dense subspace $Z' \\subset Z$ which is a scheme, see", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}.", "By", "Schemes, Lemma \\ref{schemes-lemma-mono-towards-spec-field}", "we see that either $Z' = \\emptyset$ or $Z' \\cong \\Spec(k)$.", "In the first case we conclude that $Z = \\emptyset$ and in the", "second case we conclude that $Z' = Z = \\Spec(k)$", "as $Z \\to \\Spec(k)$ is a monomorphism which is an", "isomorphism over $Z'$." ], "refs": [ "spaces-morphisms-lemma-monomorphism-separated", "spaces-morphisms-lemma-separated-over-separated", "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme", "schemes-lemma-mono-towards-spec-field" ], "ref_ids": [ 4752, 4719, 11917, 7729 ] } ], "ref_ids": [] }, { "id": 4758, "type": "theorem", "label": "spaces-morphisms-lemma-monomorphism-injective-points", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-monomorphism-injective-points", "contents": [ "Let $S$ be a scheme. If $X \\to Y$ is a monomorphism of algebraic spaces", "over $S$, then $|X| \\to |Y|$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4759, "type": "theorem", "label": "spaces-morphisms-lemma-compute-pushforward", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-compute-pushforward", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $U \\to X$ be a surjective \\'etale morphism from a scheme to $X$.", "Set $R = U \\times_X U$ and denote $t, s : R \\to U$ the projection", "morphisms as usual. Denote $a : U \\to Y$ and $b : R \\to Y$ the induced", "morphisms. For any object $\\mathcal{F}$ of $\\textit{Mod}(\\mathcal{O}_X)$", "there exists an exact sequence", "$$", "0 \\to f_*\\mathcal{F} \\to a_*(\\mathcal{F}|_U) \\to b_*(\\mathcal{F}|_R)", "$$", "where the second arrow is the difference $t^* - s^*$." ], "refs": [], "proofs": [ { "contents": [ "We denote $\\mathcal{F}$ also its extension to a sheaf of modules on", "$X_{spaces, \\etale}$, see", "Properties of Spaces,", "Remark \\ref{spaces-properties-remark-explain-equivalence}.", "Let $V \\to Y$ be an object of $Y_\\etale$. Then $V \\times_Y X$ is an", "object of $X_{spaces, \\etale}$, and by definition", "$f_*\\mathcal{F}(V) = \\mathcal{F}(V \\times_Y X)$. Since $U \\to X$ is", "surjective \\'etale, we see that $\\{V \\times_Y U \\to V \\times_Y X\\}$", "is a covering. Also, we have", "$(V \\times_Y U) \\times_X (V \\times_Y U) = V \\times_Y R$.", "Hence, by the sheaf condition of $\\mathcal{F}$ on", "$X_{spaces, \\etale}$ we have a short exact sequence", "$$", "0 \\to \\mathcal{F}(V \\times_Y X)", "\\to \\mathcal{F}(V \\times_Y U) \\to \\mathcal{F}(V \\times_Y R)", "$$", "where the second arrow is the difference of restricting via $t$ or $s$.", "This exact sequence is functorial in $V$ and hence we obtain the lemma." ], "refs": [ "spaces-properties-remark-explain-equivalence" ], "ref_ids": [ 11953 ] } ], "ref_ids": [] }, { "id": 4760, "type": "theorem", "label": "spaces-morphisms-lemma-pushforward", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-pushforward", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $f$ is quasi-compact and quasi-separated, then $f_*$ transforms", "quasi-coherent $\\mathcal{O}_X$-modules into", "quasi-coherent $\\mathcal{O}_Y$-modules." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. We have to show that", "$f_*\\mathcal{F}$ is a quasi-coherent sheaf on $Y$. For this it suffices", "to show that for any affine scheme $V$ and \\'etale morphism $V \\to Y$", "the restriction of $f_*\\mathcal{F}$ to $V$ is quasi-coherent, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-characterize-quasi-coherent}.", "Let $f' : V \\times_Y X \\to V$", "be the base change of $f$ by $V \\to Y$. Note that $f'$ is also", "quasi-compact and quasi-separated, see", "Lemmas \\ref{lemma-base-change-quasi-compact} and", "\\ref{lemma-base-change-separated}.", "By (\\ref{equation-representable-pushforward})", "we know that the restriction of $f_*\\mathcal{F}$ to $V$ is $f'_*$ of the", "restriction of $\\mathcal{F}$ to $V \\times_Y X$. Hence", "we may replace $f$ by $f'$, and assume that $Y$ is an affine scheme.", "\\medskip\\noindent", "Assume $Y$ is an affine scheme. Since $f$ is quasi-compact we see that $X$", "is quasi-compact. Thus we may choose an affine scheme $U$ and a surjective", "\\'etale morphism $U \\to X$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-quasi-compact-affine-cover}.", "By Lemma \\ref{lemma-compute-pushforward} we get an exact sequence", "$$", "0 \\to f_*\\mathcal{F} \\to a_*(\\mathcal{F}|_U) \\to b_*(\\mathcal{F}|_R).", "$$", "where $R = U \\times_X U$.", "As $X \\to Y$ is quasi-separated we see that $R \\to U \\times_Y U$", "is a quasi-compact monomorphism. This implies that $R$ is a quasi-compact", "separated scheme (as $U$ and $Y$ are affine at this point).", "Hence $a : U \\to Y$ and $b : R \\to Y$ are quasi-compact and", "quasi-separated morphisms of schemes. Thus by", "Descent,", "Proposition \\ref{descent-proposition-equivalence-quasi-coherent-functorial}", "the sheaves $a_*(\\mathcal{F}|_U)$ and $b_*(\\mathcal{F}|_R)$", "are quasi-coherent (see also the discussion preceding this lemma).", "This implies that $f_*\\mathcal{F}$ is a kernel of", "quasi-coherent modules, and hence itself quasi-coherent, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-properties-quasi-coherent}." ], "refs": [ "spaces-properties-lemma-characterize-quasi-coherent", "spaces-morphisms-lemma-base-change-quasi-compact", "spaces-morphisms-lemma-base-change-separated", "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-morphisms-lemma-compute-pushforward", "descent-proposition-equivalence-quasi-coherent-functorial", "spaces-properties-lemma-properties-quasi-coherent" ], "ref_ids": [ 11911, 4738, 4714, 11832, 4759, 14756, 11912 ] } ], "ref_ids": [] }, { "id": 4761, "type": "theorem", "label": "spaces-morphisms-lemma-closed-immersion-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-closed-immersion-local", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is a closed immersion (resp.\\ open immersion, resp.\\ immersion),", "\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism", "$Z \\times_Y X \\to Z$ is a closed immersion (resp.\\ open immersion,", "resp.\\ immersion),", "\\item for every affine scheme $Z$ and any morphism", "$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is a closed immersion", "(resp.\\ open immersion, resp.\\ immersion),", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is a closed immersion", "(resp.\\ open immersion, resp.\\ immersion), and", "\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that", "each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is a closed immersion", "(resp.\\ open immersion, resp.\\ immersion).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Using that a base change of a", "closed immersion (resp.\\ open immersion, resp.\\ immersion)", "is another one it is clear that (1) implies (2) and (2) implies (3).", "Also (3) implies (4) since we can take $V$ to be a disjoint union of", "affines, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-cover-by-union-affines}.", "\\medskip\\noindent", "Assume $V \\to Y$ is as in (4).", "Let $\\mathcal{P}$ be the property", "closed immersion (resp.\\ open immersion, resp.\\ immersion)", "of morphisms of schemes. Note that property $\\mathcal{P}$", "is preserved under any base change and fppf local on the", "base (see Section \\ref{section-representable}).", "Moreover, morphisms of type $\\mathcal{P}$ are separated and", "locally quasi-finite (in each of the three cases, see", "Schemes, Lemma \\ref{schemes-lemma-immersions-monomorphisms}, and", "Morphisms, Lemma \\ref{morphisms-lemma-immersion-locally-quasi-finite}).", "Hence by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend}", "the morphisms of type $\\mathcal{P}$ satisfy descent for fppf covering. Thus", "Spaces, Lemma \\ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale}", "applies and we see that $X \\to Y$ is representable and has property", "$\\mathcal{P}$, in other words (1) holds.", "\\medskip\\noindent", "The equivalence of (1) and (5) follows from the fact that", "$\\mathcal{P}$ is Zariski local on the target (since we saw", "above that $\\mathcal{P}$ is in fact fppf local on the target)." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "schemes-lemma-immersions-monomorphisms", "morphisms-lemma-immersion-locally-quasi-finite", "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "spaces-lemma-morphism-sheaves-with-P-effective-descent-etale" ], "ref_ids": [ 11830, 7727, 5236, 13949, 8158 ] } ], "ref_ids": [] }, { "id": 4762, "type": "theorem", "label": "spaces-morphisms-lemma-immersion-permanence", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-immersion-permanence", "contents": [ "Let $S$ be a scheme.", "Let $Z \\to Y \\to X$ be morphisms of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item If $Z \\to X$ is representable, locally of finite type, locally", "quasi-finite, separated, and a monomorphism, then $Z \\to Y$ is", "representable, locally of finite type, locally quasi-finite,", "separated, and a monomorphism.", "\\item If $Z \\to X$ is an immersion and $Y \\to X$ is locally separated,", "then $Z \\to Y$ is an immersion.", "\\item If $Z \\to X$ is a closed immersion and $Y \\to X$ is separated,", "then $Z \\to Y$ is a closed immersion.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In each case the proof is to contemplate the commutative diagram", "$$", "\\xymatrix{", "Z \\ar[r] \\ar[rd] & Y \\times_X Z \\ar[r] \\ar[d] & Z \\ar[d] \\\\", "& Y \\ar[r] & X", "}", "$$", "where the composition of the top horizontal arrows is the identity.", "Let us prove (1). The first horizontal arrow is a section of", "$Y \\times_X Z \\to Z$, whence representable, locally of finite type,", "locally quasi-finite, separated, and a monomorphism by", "Lemma \\ref{lemma-section-immersion}.", "The arrow $Y \\times_X Z \\to Y$ is a base change of $Z \\to X$", "hence is representable, locally of finite type,", "locally quasi-finite, separated, and a monomorphism", "(as each of these properties of morphisms of schemes is stable", "under base change, see", "Spaces, Remark \\ref{spaces-remark-list-properties-stable-base-change}).", "Hence the same is true for the composition (as each of these properties of", "morphisms of schemes is stable under composition, see Spaces, Remark", "\\ref{spaces-remark-list-properties-stable-composition}).", "This proves (1). The other results are proved in exactly the same manner." ], "refs": [ "spaces-remark-list-properties-stable-base-change", "spaces-remark-list-properties-stable-composition" ], "ref_ids": [ 8184, 8185 ] } ], "ref_ids": [] }, { "id": 4763, "type": "theorem", "label": "spaces-morphisms-lemma-immersion-when-closed", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-immersion-when-closed", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic", "spaces over $S$. Then $|i| : |Z| \\to |X|$ is a homeomorphism onto a", "locally closed subset, and $i$ is a closed immersion if and only if", "the image $|i|(|Z|) \\subset |X|$ is a closed subset." ], "refs": [], "proofs": [ { "contents": [ "The first statement is Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-subspace-induced-topology}.", "Let $U$ be a scheme and let $U \\to X$ be a surjective \\'etale morphism.", "By assumption $T = U \\times_X Z$ is a scheme and the morphism $j : T \\to U$", "is an immersion of schemes. By Lemma \\ref{lemma-closed-immersion-local}", "the morphism $i$ is a closed immersion if and only if $j$ is a closed", "immersion. By Schemes, Lemma \\ref{schemes-lemma-immersion-when-closed}", "this is true if and only if $j(T)$ is closed in $U$.", "However, the subset $j(T) \\subset U$ is the inverse image of", "$|i|(|Z|) \\subset |X|$, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}.", "This finishes the proof." ], "refs": [ "spaces-properties-lemma-subspace-induced-topology", "spaces-morphisms-lemma-closed-immersion-local", "schemes-lemma-immersion-when-closed", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 11844, 4761, 7671, 11819 ] } ], "ref_ids": [] }, { "id": 4764, "type": "theorem", "label": "spaces-morphisms-lemma-factor-the-other-way", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-factor-the-other-way", "contents": [ "Let $S$ be a scheme. Let $Z \\to X$ be an immersion of algebraic spaces over", "$S$. Assume $Z \\to X$ is quasi-compact.", "There exists a factorization $Z \\to \\overline{Z} \\to X$ where", "$Z \\to \\overline{Z}$ is an open immersion and $\\overline{Z} \\to X$", "is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $U \\to X$ be surjective \\'etale.", "As usual denote $R = U \\times_X U$ with projections", "$s, t : R \\to U$. Set $T = Z \\times_U X$. Let $\\overline{T} \\subset U$", "be the scheme theoretic image of $T \\to U$. Note that", "$s^{-1}\\overline{T} = t^{-1}\\overline{T}$ as taking", "scheme theoretic images of quasi-compact morphisms commute", "with flat base change, see", "Morphisms, Lemma \\ref{morphisms-lemma-flat-base-change-scheme-theoretic-image}.", "Hence we obtain a closed subspace $\\overline{Z} \\subset X$ whose", "pullback to $U$ is $\\overline{T}$, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-subspaces-presentation}.", "By Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-immersion}", "the morphism $T \\to \\overline{T}$", "is an open immersion. It follows that $Z \\to \\overline{Z}$ is", "an open immersion and we win." ], "refs": [ "morphisms-lemma-flat-base-change-scheme-theoretic-image", "spaces-properties-lemma-subspaces-presentation", "morphisms-lemma-quasi-compact-immersion" ], "ref_ids": [ 5273, 11845, 5154 ] } ], "ref_ids": [] }, { "id": 4765, "type": "theorem", "label": "spaces-morphisms-lemma-closed-immersion-ideals", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-closed-immersion-ideals", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "For every closed immersion $i : Z \\to X$ the sheaf", "$i_*\\mathcal{O}_Z$ is a quasi-coherent $\\mathcal{O}_X$-module, the map", "$i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective and its", "kernel is a quasi-coherent sheaf of ideals. The rule", "$Z \\mapsto \\Ker(\\mathcal{O}_X \\to i_*\\mathcal{O}_Z)$", "defines an inclusion reversing bijection", "$$", "\\begin{matrix}", "\\text{closed subspaces}\\\\", "Z \\subset X", "\\end{matrix}", "\\longrightarrow", "\\begin{matrix}", "\\text{quasi-coherent sheaves}\\\\", "\\text{of ideals }\\mathcal{I} \\subset \\mathcal{O}_X", "\\end{matrix}", "$$", "Moreover, given a closed subscheme $Z$ corresponding to the quasi-coherent", "sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$ a morphism of algebraic", "spaces $h : Y \\to X$ factors through $Z$ if and only if the map", "$h^*\\mathcal{I} \\to h^*\\mathcal{O}_X = \\mathcal{O}_Y$ is zero." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be a surjective \\'etale morphism whose source is a scheme.", "Consider the diagram", "$$", "\\xymatrix{", "U \\times_X Z \\ar[r] \\ar[d]_{i'} & Z \\ar[d]^i \\\\", "U \\ar[r] & X", "}", "$$", "By", "Lemma \\ref{lemma-closed-immersion-local}", "we see that $i$ is a closed immersion", "if and only if $i'$ is a closed immersion. By", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-pushforward-etale-base-change-modules}", "we see that $i'_*\\mathcal{O}_{U \\times_X Z}$ is the restriction of", "$i_*\\mathcal{O}_Z$ to $U$. Hence the assertions on", "$\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ are equivalent to the", "corresponding assertions on", "$\\mathcal{O}_U \\to i'_*\\mathcal{O}_{U \\times_X Z}$.", "And since $i'$ is a closed immersion of schemes, these results follow from", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion}.", "\\medskip\\noindent", "Let us prove that given a quasi-coherent", "sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$ the formula", "$$", "Z(T) = \\{h : T \\to X \\mid h^*\\mathcal{I} \\to \\mathcal{O}_T", "\\text{ is zero}\\}", "$$", "defines a closed subspace of $X$. It is clearly a subfunctor of $X$.", "To show that $Z \\to X$ is representable by closed immersions, let", "$\\varphi : U \\to X$ be a morphism from a scheme towards $X$. Then", "$Z \\times_X U$ is represented by the analogous subfunctor of $U$ corresponding", "to the sheaf of ideals $\\Im(\\varphi^*\\mathcal{I} \\to \\mathcal{O}_U)$. By", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-pullback-quasi-coherent}", "the $\\mathcal{O}_U$-module $\\varphi^*\\mathcal{I}$ is quasi-coherent", "on $U$, and hence $\\Im(\\varphi^*\\mathcal{I} \\to \\mathcal{O}_U)$", "is a quasi-coherent sheaf of ideals on $U$. By", "Schemes, Lemma \\ref{schemes-lemma-characterize-closed-subspace}", "we conclude that $Z \\times_X U$ is represented by the closed subscheme", "of $U$ associated to $\\Im(\\varphi^*\\mathcal{I} \\to \\mathcal{O}_U)$.", "Thus $Z$ is a closed subspace of $X$.", "\\medskip\\noindent", "In the formula for $Z$ above the inputs $T$ are schemes since algebraic", "spaces are sheaves on $(\\Sch/S)_{fppf}$. We omit the verification", "that the same formula remains true if $T$ is an algebraic space." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-local", "spaces-properties-lemma-pushforward-etale-base-change-modules", "morphisms-lemma-closed-immersion", "spaces-properties-lemma-pullback-quasi-coherent", "schemes-lemma-characterize-closed-subspace" ], "ref_ids": [ 4761, 11898, 5125, 11907, 7648 ] } ], "ref_ids": [] }, { "id": 4766, "type": "theorem", "label": "spaces-morphisms-lemma-closed-immersion-quasi-compact", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-closed-immersion-quasi-compact", "contents": [ "A closed immersion of algebraic spaces is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Schemes, Lemma \\ref{schemes-lemma-closed-immersion-quasi-compact}", "by general principles, see", "Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}." ], "refs": [ "schemes-lemma-closed-immersion-quasi-compact", "spaces-lemma-representable-transformations-property-implication" ], "ref_ids": [ 7700, 8136 ] } ], "ref_ids": [] }, { "id": 4767, "type": "theorem", "label": "spaces-morphisms-lemma-closed-immersion-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-closed-immersion-separated", "contents": [ "A closed immersion of algebraic spaces is separated." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Schemes, Lemma \\ref{schemes-lemma-immersions-monomorphisms}", "by general principles, see", "Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}." ], "refs": [ "schemes-lemma-immersions-monomorphisms", "spaces-lemma-representable-transformations-property-implication" ], "ref_ids": [ 7727, 8136 ] } ], "ref_ids": [] }, { "id": 4768, "type": "theorem", "label": "spaces-morphisms-lemma-closed-immersion-push-pull", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-closed-immersion-push-pull", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic", "spaces over $S$.", "\\begin{enumerate}", "\\item The functor", "$$", "i_{small, *} :", "\\Sh(Z_\\etale)", "\\longrightarrow", "\\Sh(X_\\etale)", "$$", "is fully faithful and its essential image is those sheaves of sets", "$\\mathcal{F}$ on $X_\\etale$ whose restriction to $X \\setminus Z$ is", "isomorphic to $*$, and", "\\item the functor", "$$", "i_{small, *} :", "\\textit{Ab}(Z_\\etale)", "\\longrightarrow", "\\textit{Ab}(X_\\etale)", "$$", "is fully faithful and its essential image is those abelian sheaves on", "$X_\\etale$ whose support is contained in $|Z|$.", "\\end{enumerate}", "In both cases $i_{small}^{-1}$ is a left inverse to the functor", "$i_{small, *}$." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $U \\to X$ be surjective \\'etale.", "Set $V = Z \\times_X U$. Then $V$ is a scheme and $i' : V \\to U$ is", "a closed immersion of schemes. By", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-pushforward-etale-base-change}", "for any sheaf $\\mathcal{G}$ on $Z$ we have", "$$", "(i_{small}^{-1}i_{small, *}\\mathcal{G})|_V =", "(i')_{small}^{-1}i'_{small, *}(\\mathcal{G}|_V)", "$$", "By", "\\'Etale Cohomology, Proposition", "\\ref{etale-cohomology-proposition-closed-immersion-pushforward}", "the map", "$(i')_{small}^{-1}i'_{small, *}(\\mathcal{G}|_V) \\to \\mathcal{G}|_V$", "is an isomorphism. Since $V \\to Z$ is surjective and \\'etale this implies", "that $i_{small}^{-1}i_{small, *}\\mathcal{G} \\to \\mathcal{G}$ is an", "isomorphism. This clearly implies that $i_{small, *}$ is fully faithful, see", "Sites, Lemma \\ref{sites-lemma-exactness-properties}.", "To prove the statement on the essential image, consider a sheaf of sets", "$\\mathcal{F}$ on $X_\\etale$ whose restriction to $X \\setminus Z$ is", "isomorphic to $*$. As in the proof of", "\\'Etale Cohomology, Proposition", "\\ref{etale-cohomology-proposition-closed-immersion-pushforward}", "we consider the adjunction mapping", "$$", "\\mathcal{F} \\longrightarrow i_{small, *}i_{small}^{-1}\\mathcal{F}.", "$$", "As in the first part we see that the restriction of this map to", "$U$ is an isomorphism by the corresponding result for the case of", "schemes. Since $U$ is an \\'etale covering of $X$ we", "conclude it is an isomorphism." ], "refs": [ "spaces-properties-lemma-pushforward-etale-base-change", "etale-cohomology-proposition-closed-immersion-pushforward", "sites-lemma-exactness-properties", "etale-cohomology-proposition-closed-immersion-pushforward" ], "ref_ids": [ 11867, 6700, 8618, 6700 ] } ], "ref_ids": [] }, { "id": 4769, "type": "theorem", "label": "spaces-morphisms-lemma-stalk-push-closed", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-stalk-push-closed", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic", "spaces over $S$. Let $\\overline{z}$ be a geometric point of $Z$ with", "image $\\overline{x}$ in $X$. Then", "$(i_{small, *}\\mathcal{F})_{\\overline{z}} = \\mathcal{F}_{\\overline{x}}$", "for any sheaf $\\mathcal{F}$ on $Z_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "Choose an \\'etale neighbourhood $(U, \\overline{u})$ of $\\overline{x}$.", "Then the stalk $(i_{small, *}\\mathcal{F})_{\\overline{z}}$", "is the stalk of $i_{small, *}\\mathcal{F}|_U$ at $\\overline{u}$.", "By Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-pushforward-etale-base-change}", "we may replace $X$ by $U$ and $Z$ by $Z \\times_X U$.", "Then $Z \\to X$ is a closed immersion of schemes and the result is", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-stalk-pushforward-closed-immersion}." ], "refs": [ "spaces-properties-lemma-pushforward-etale-base-change", "etale-cohomology-lemma-stalk-pushforward-closed-immersion" ], "ref_ids": [ 11867, 6461 ] } ], "ref_ids": [] }, { "id": 4770, "type": "theorem", "label": "spaces-morphisms-lemma-closed-immersion-rings", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-closed-immersion-rings", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic", "spaces over $S$. Let $\\mathcal{A}$ be a sheaf of rings on $X_\\etale$.", "Let $\\mathcal{B}$ be a sheaf of rings on $Z_\\etale$.", "Let $\\varphi : \\mathcal{A} \\to i_{small, *}\\mathcal{B}$", "be a homomorphism of sheaves of rings so that we obtain a", "morphism of ringed topoi", "$$", "f : (\\Sh(Z_\\etale), \\mathcal{B}) \\longrightarrow (\\Sh(X_\\etale), \\mathcal{A}).", "$$", "For a sheaf of $\\mathcal{A}$-modules $\\mathcal{F}$ and a", "sheaf of $\\mathcal{B}$-modules $\\mathcal{G}$ the canonical map", "$$", "\\mathcal{F} \\otimes_\\mathcal{A} f_*\\mathcal{G}", "\\longrightarrow", "f_*(f^*\\mathcal{F} \\otimes_\\mathcal{B} \\mathcal{G}).", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The map is the map adjoint to the map", "$$", "f^*\\mathcal{F} \\otimes_\\mathcal{B}", "f^* f_*\\mathcal{G} =", "f^*(\\mathcal{F} \\otimes_\\mathcal{A} f_*\\mathcal{G})", "\\longrightarrow", "f^*\\mathcal{F} \\otimes_\\mathcal{B} \\mathcal{G}", "$$", "coming from $\\text{id} : f^*\\mathcal{F} \\to f^*\\mathcal{F}$", "and the adjunction map $f^* f_*\\mathcal{G} \\to \\mathcal{G}$.", "To see this map is an isomorphism, we may check on stalks", "(Properties of Spaces, Theorem", "\\ref{spaces-properties-theorem-exactness-stalks}).", "Let $\\overline{z} : \\Spec(k) \\to Z$ be a geometric point with", "image $\\overline{x} = i \\circ \\overline{z} : \\Spec(k) \\to X$.", "Working out what our maps does on stalks, we see that we", "have to show", "$$", "\\mathcal{F}_{\\overline{x}}", "\\otimes_{\\mathcal{A}_{\\overline{x}}}", "\\mathcal{G}_{\\overline{z}} =", "(\\mathcal{F}_{\\overline{x}}", "\\otimes_{\\mathcal{A}_{\\overline{x}}}", "\\mathcal{B}_{\\overline{z}}) \\otimes_{\\mathcal{B}_{\\overline{z}}}", "\\mathcal{G}_{\\overline{z}}", "$$", "which holds true. Here we have used that", "taking tensor products commutes with taking stalks, the", "behaviour of stalks under pullback", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-stalk-pullback}, and", "the behaviour of stalks under pushforward along a closed immersion", "Lemma \\ref{lemma-stalk-push-closed}." ], "refs": [ "spaces-properties-theorem-exactness-stalks", "spaces-properties-lemma-stalk-pullback", "spaces-morphisms-lemma-stalk-push-closed" ], "ref_ids": [ 11813, 11875, 4769 ] } ], "ref_ids": [] }, { "id": 4771, "type": "theorem", "label": "spaces-morphisms-lemma-i-star-equivalence", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-i-star-equivalence", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic", "spaces over $S$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent", "sheaf of ideals cutting out $Z$.", "\\begin{enumerate}", "\\item For any $\\mathcal{O}_X$-module $\\mathcal{F}$ the adjunction map", "$\\mathcal{F} \\to i_*i^*\\mathcal{F}$ induces an isomorphism", "$\\mathcal{F}/\\mathcal{I}\\mathcal{F} \\cong i_*i^*\\mathcal{F}$.", "\\item The functor $i^*$ is a left inverse to $i_*$, i.e., for any", "$\\mathcal{O}_Z$-module $\\mathcal{G}$ the adjunction map", "$i^*i_*\\mathcal{G} \\to \\mathcal{G}$ is an isomorphism.", "\\item The functor", "$$", "i_* :", "\\QCoh(\\mathcal{O}_Z)", "\\longrightarrow", "\\QCoh(\\mathcal{O}_X)", "$$", "is exact, fully faithful, with essential image those quasi-coherent", "$\\mathcal{O}_X$-modules $\\mathcal{F}$ such that $\\mathcal{I}\\mathcal{F} = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "During this proof we work exclusively with sheaves on", "the small \\'etale sites, and we use $i_*, i^{-1}, \\ldots$", "to denote pushforward and pullback of sheaves of abelian groups", "instead of $i_{small, *}, i_{small}^{-1}$.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. By", "Lemma \\ref{lemma-closed-immersion-rings} applied with", "$\\mathcal{A} = \\mathcal{O}_X$ and", "$\\mathcal{G} = \\mathcal{B} = \\mathcal{O}_Z$ we see that", "$i_*i^*\\mathcal{F} = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_Z$.", "By", "Lemma \\ref{lemma-closed-immersion-ideals}", "we see that we have a short exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_X \\to i_*\\mathcal{O}_Z \\to 0", "$$", "It follows from properties of the tensor product that", "$\\mathcal{F} \\otimes_{\\mathcal{O}_X} i_*\\mathcal{O}_Z", "= \\mathcal{F}/\\mathcal{I}\\mathcal{F}$. This proves (1) (except", "that we omit the verification that the map is induced by the", "adjunction mapping).", "\\medskip\\noindent", "Let $\\mathcal{G}$ be any $\\mathcal{O}_Z$-module. By", "Lemma \\ref{lemma-closed-immersion-push-pull}", "we see that $i^{-1}i_*\\mathcal{G} = \\mathcal{G}$.", "Hence to prove (2) we have to show that the canonical map", "$\\mathcal{G} \\otimes_{i^{-1}\\mathcal{O}_X} \\mathcal{O}_Z \\to \\mathcal{G}$", "is an isomorphism. This follows from general properties of tensor products", "if we can show that $i^{-1}\\mathcal{O}_X \\to \\mathcal{O}_Z$ is surjective. By", "Lemma \\ref{lemma-closed-immersion-push-pull}", "it suffices to prove that", "$i_*i^{-1}\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$", "is surjective. Since the surjective map $\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$", "factors through this map we see that (2) holds.", "\\medskip\\noindent", "Finally we prove the most interesting part of the lemma, namely part (3).", "A closed immersion is quasi-compact and separated, see", "Lemmas \\ref{lemma-closed-immersion-quasi-compact} and", "\\ref{lemma-closed-immersion-separated}. Hence", "Lemma \\ref{lemma-pushforward}", "applies and the pushforward of a quasi-coherent", "sheaf on $Z$ is indeed a quasi-coherent sheaf on $X$.", "Thus we obtain our functor", "$i^{QCoh}_* : \\QCoh(\\mathcal{O}_Z)", "\\to \\QCoh(\\mathcal{O}_X)$.", "It is clear from part (2) that $i^{QCoh}_*$ is fully faithful since", "it has a left inverse, namely $i^*$.", "\\medskip\\noindent", "Now we turn to the description of the essential image of the", "functor $i_*$. It is clear that $\\mathcal{I}(i_*\\mathcal{G}) = 0$", "for any $\\mathcal{O}_Z$-module, since $\\mathcal{I}$ is the kernel", "of the map $\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ which is the map", "we use to put an $\\mathcal{O}_X$-module structure on $i_*\\mathcal{G}$.", "Next, suppose that $\\mathcal{F}$ is any quasi-coherent", "$\\mathcal{O}_X$-module such that $\\mathcal{I}\\mathcal{F} = 0$.", "Then we see that $\\mathcal{F}$ is an $i_*\\mathcal{O}_Z$-module", "because $i_*\\mathcal{O}_Z = \\mathcal{O}_X/\\mathcal{I}$. Hence in", "particular its support is contained in $|Z|$. We apply", "Lemma \\ref{lemma-closed-immersion-push-pull}", "to see that $\\mathcal{F} \\cong i_*\\mathcal{G}$ for some", "$\\mathcal{O}_Z$-module $\\mathcal{G}$. The only small detail left over", "is to see why $\\mathcal{G}$ is quasi-coherent. This is true", "because $\\mathcal{G} \\cong i^*\\mathcal{F}$ by part (2) and", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-pullback-quasi-coherent}." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-rings", "spaces-morphisms-lemma-closed-immersion-ideals", "spaces-morphisms-lemma-closed-immersion-push-pull", "spaces-morphisms-lemma-closed-immersion-push-pull", "spaces-morphisms-lemma-closed-immersion-quasi-compact", "spaces-morphisms-lemma-closed-immersion-separated", "spaces-morphisms-lemma-pushforward", "spaces-morphisms-lemma-closed-immersion-push-pull", "spaces-properties-lemma-pullback-quasi-coherent" ], "ref_ids": [ 4770, 4765, 4768, 4768, 4766, 4767, 4760, 4768, 11907 ] } ], "ref_ids": [] }, { "id": 4772, "type": "theorem", "label": "spaces-morphisms-lemma-largest-quasi-coherent-subsheaf", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-largest-quasi-coherent-subsheaf", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\mathcal{G} \\subset \\mathcal{F}$ be a $\\mathcal{O}_X$-submodule.", "There exists a unique quasi-coherent $\\mathcal{O}_X$-submodule", "$\\mathcal{G}' \\subset \\mathcal{G}$ with the following property:", "For every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{H}$ the map", "$$", "\\Hom_{\\mathcal{O}_X}(\\mathcal{H}, \\mathcal{G}')", "\\longrightarrow", "\\Hom_{\\mathcal{O}_X}(\\mathcal{H}, \\mathcal{G})", "$$", "is bijective. In particular $\\mathcal{G}'$ is the largest quasi-coherent", "$\\mathcal{O}_X$-submodule of $\\mathcal{F}$ contained in $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{G}_a$, $a \\in A$ be the set of quasi-coherent", "$\\mathcal{O}_X$-submodules contained in $\\mathcal{G}$.", "Then the image $\\mathcal{G}'$ of", "$$", "\\bigoplus\\nolimits_{a \\in A} \\mathcal{G}_a \\longrightarrow \\mathcal{F}", "$$", "is quasi-coherent as the image of a map of quasi-coherent sheaves", "on $X$ is quasi-coherent and since a direct sum of quasi-coherent sheaves", "is quasi-coherent, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-properties-quasi-coherent}.", "The module $\\mathcal{G}'$ is contained in $\\mathcal{G}$. Hence this is the", "largest quasi-coherent $\\mathcal{O}_X$-module contained in $\\mathcal{G}$.", "\\medskip\\noindent", "To prove the formula, let $\\mathcal{H}$ be a quasi-coherent", "$\\mathcal{O}_X$-module and let $\\alpha : \\mathcal{H} \\to \\mathcal{G}$", "be an $\\mathcal{O}_X$-module map. The image of the composition", "$\\mathcal{H} \\to \\mathcal{G} \\to \\mathcal{F}$ is quasi-coherent", "as the image of a map of quasi-coherent sheaves. Hence it is contained", "in $\\mathcal{G}'$. Hence $\\alpha$ factors through $\\mathcal{G}'$", "as desired." ], "refs": [ "spaces-properties-lemma-properties-quasi-coherent" ], "ref_ids": [ 11912 ] } ], "ref_ids": [] }, { "id": 4773, "type": "theorem", "label": "spaces-morphisms-lemma-i-upper-shriek", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-i-upper-shriek", "contents": [ "Let $S$ be a scheme.", "Let $i : Z \\to X$ be a closed immersion of algebraic spaces over $S$.", "There is a functor\\footnote{This is likely nonstandard notation.}", "$i^! : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Z)$", "which is a right adjoint to $i_*$. (Compare", "Modules, Lemma \\ref{modules-lemma-i-star-right-adjoint}.)" ], "refs": [ "modules-lemma-i-star-right-adjoint" ], "proofs": [ { "contents": [ "Given quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{G}$ we consider", "the subsheaf $\\mathcal{H}_Z(\\mathcal{G})$ of $\\mathcal{G}$ of local sections", "annihilated by $\\mathcal{I}$. By", "Lemma \\ref{lemma-largest-quasi-coherent-subsheaf}", "there is a canonical largest quasi-coherent $\\mathcal{O}_X$-submodule", "$\\mathcal{H}_Z(\\mathcal{G})'$. By construction we have", "$$", "\\Hom_{\\mathcal{O}_X}(i_*\\mathcal{F}, \\mathcal{H}_Z(\\mathcal{G})')", "=", "\\Hom_{\\mathcal{O}_X}(i_*\\mathcal{F}, \\mathcal{G})", "$$", "for any quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{F}$.", "Hence we can set $i^!\\mathcal{G} = i^*(\\mathcal{H}_Z(\\mathcal{G})')$.", "Details omitted." ], "refs": [ "spaces-morphisms-lemma-largest-quasi-coherent-subsheaf" ], "ref_ids": [ 4772 ] } ], "ref_ids": [ 13233 ] }, { "id": 4774, "type": "theorem", "label": "spaces-morphisms-lemma-scheme-theoretic-intersection", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-scheme-theoretic-intersection", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $Z, Y \\subset X$ be closed subspaces.", "Let $Z \\cap Y$ be the scheme theoretic intersection of $Z$ and $Y$.", "Then $Z \\cap Y \\to Z$ and $Z \\cap Y \\to Y$ are closed immersions", "and", "$$", "\\xymatrix{", "Z \\cap Y \\ar[r] \\ar[d] & Z \\ar[d] \\\\", "Y \\ar[r] & X", "}", "$$", "is a cartesian diagram of algebraic spaces over $S$, i.e.,", "$Z \\cap Y = Z \\times_X Y$." ], "refs": [], "proofs": [ { "contents": [ "The morphisms $Z \\cap Y \\to Z$ and $Z \\cap Y \\to Y$ are closed immersions", "by Lemma \\ref{lemma-closed-immersion-ideals}.", "Since formation of the scheme theoretic intersection commutes", "with \\'etale localization we conclude the diagram is cartesian", "by the case of schemes. See", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-intersection}." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-ideals", "morphisms-lemma-scheme-theoretic-intersection" ], "ref_ids": [ 4765, 5139 ] } ], "ref_ids": [] }, { "id": 4775, "type": "theorem", "label": "spaces-morphisms-lemma-scheme-theoretic-union", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-scheme-theoretic-union", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $Y, Z \\subset X$ be closed subspaces.", "Let $Y \\cup Z$ be the scheme theoretic union of $Y$ and $Z$.", "Let $Y \\cap Z$ be the scheme theoretic intersection of $Y$ and $Z$.", "Then $Y \\to Y \\cup Z$ and $Z \\to Y \\cup Z$ are closed immersions,", "there is a short exact sequence", "$$", "0 \\to \\mathcal{O}_{Y \\cup Z} \\to \\mathcal{O}_Y \\times \\mathcal{O}_Z", "\\to \\mathcal{O}_{Y \\cap Z} \\to 0", "$$", "of $\\mathcal{O}_Z$-modules, and the diagram", "$$", "\\xymatrix{", "Y \\cap Z \\ar[r] \\ar[d] & Y \\ar[d] \\\\", "Z \\ar[r] & Y \\cup Z", "}", "$$", "is cocartesian in the category of algebraic spaces over $S$, i.e.,", "$Y \\cup Z = Y \\amalg_{Y \\cap Z} Z$." ], "refs": [], "proofs": [ { "contents": [ "The morphisms $Y \\to Y \\cup Z$ and $Z \\to Y \\cup Z$ are closed immersions", "by Lemma \\ref{lemma-closed-immersion-ideals}. In the short exact sequence", "we use the equivalence of Lemma \\ref{lemma-i-star-equivalence} to think of", "quasi-coherent modules on closed subspaces of $X$ as quasi-coherent modules", "on $X$. For the first map in the sequence we use the canonical maps", "$\\mathcal{O}_{Y \\cup Z} \\to \\mathcal{O}_Y$ and", "$\\mathcal{O}_{Y \\cup Z} \\to \\mathcal{O}_Z$", "and for the second map we use the canonical map", "$\\mathcal{O}_Y \\to \\mathcal{O}_{Y \\cap Z}$ and", "the negative of the canonical map", "$\\mathcal{O}_Z \\to \\mathcal{O}_{Y \\cap Z}$. Then to check", "exactness we may work \\'etale locally and deduce exactness", "from the case of schemes", "(Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-union}).", "\\medskip\\noindent", "To show the diagram is cocartesian, suppose we are given an algebraic space", "$T$ over $S$ and morphisms $f : Y \\to T$, $g : Z \\to T$ agreeing as morphisms", "$Y \\cap Z \\to T$. Goal: Show there exists a unique morphism", "$h : Y \\cup Z \\to T$ agreeing with $f$ and $g$.", "To construct $h$ we may work \\'etale locally on $Y \\cup Z$", "(as $Y \\cup Z$ is an \\'etale sheaf being an algebraic space).", "Hence we may assume that $X$ is a scheme.", "In this case we know that $Y \\cup Z$ is the pushout", "of $Y$ and $Z$ along $Y \\cap Z$ in the category of schemes", "by Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-union}.", "Choose a scheme $T'$ and a surjective \\'etale morphism $T' \\to T$.", "Set $Y' = T' \\times_{T, f} Y$ and $Z' = T' \\times_{T, g} Z$.", "Then $Y'$ and $Z'$ are schemes and we have a canonical isomorphism", "$\\varphi : Y' \\times_Y (Y \\cap Z) \\to Z' \\times_Z (Y \\cap Z)$", "of schemes. By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-pushout-along-closed-immersions}", "the pushout $W' = Y' \\amalg_{Y' \\times_Y (Y \\cap Z), \\varphi} Z'$", "exists in the category of schemes.", "The morphism $W' \\to Y \\cup Z$ is \\'etale by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-pushout-along-closed-immersions-properties-above}.", "It is surjective as $Y' \\to Y$ and $Z' \\to Z$ are surjective.", "The morphisms $f' : Y' \\to T'$ and $g' : Z' \\to T'$", "glue to a unique morphism of schemes $h' : W' \\to T'$.", "By uniqueness the composition $W' \\to T' \\to T$", "descends to the desired morphism $h : Y \\cup Z \\to T$.", "Some details omitted." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-ideals", "spaces-morphisms-lemma-i-star-equivalence", "morphisms-lemma-scheme-theoretic-union", "morphisms-lemma-scheme-theoretic-union", "more-morphisms-lemma-pushout-along-closed-immersions", "more-morphisms-lemma-pushout-along-closed-immersions-properties-above" ], "ref_ids": [ 4765, 4771, 5140, 5140, 14051, 14052 ] } ], "ref_ids": [] }, { "id": 4776, "type": "theorem", "label": "spaces-morphisms-lemma-support-covering", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-support-covering", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $U$ be a scheme and let $\\varphi : U \\to X$ be an \\'etale morphism.", "Then", "$$", "\\text{Supp}(\\varphi^*\\mathcal{F}) = |\\varphi|^{-1}(\\text{Supp}(\\mathcal{F}))", "$$", "where the left hand side is the support of $\\varphi^*\\mathcal{F}$ as a", "quasi-coherent module on the scheme $U$." ], "refs": [], "proofs": [ { "contents": [ "Let $u\\in U$ be a (usual) point and let $\\overline{x}$ be a", "geometric point lying over $u$. By", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-stalk-quasi-coherent}", "we have", "$(\\varphi^*\\mathcal{F})_u \\otimes_{\\mathcal{O}_{U, u}}", "\\mathcal{O}_{X, \\overline{x}} = \\mathcal{F}_{\\overline{x}}$.", "Since $\\mathcal{O}_{U, u} \\to \\mathcal{O}_{X, \\overline{x}}$", "is the strict henselization by", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring}", "we see that it is faithfully flat (see", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-dumb-properties-henselization}).", "Thus we see that $(\\varphi^*\\mathcal{F})_u = 0$ if and only if", "$\\mathcal{F}_{\\overline{x}} = 0$. This proves the lemma." ], "refs": [ "spaces-properties-lemma-stalk-quasi-coherent", "spaces-properties-lemma-describe-etale-local-ring", "more-algebra-lemma-dumb-properties-henselization" ], "ref_ids": [ 11909, 11884, 10055 ] } ], "ref_ids": [] }, { "id": 4777, "type": "theorem", "label": "spaces-morphisms-lemma-support-finite-type", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-support-finite-type", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Then", "\\begin{enumerate}", "\\item The support of $\\mathcal{F}$ is closed.", "\\item For a geometric point $\\overline{x}$ lying over $x \\in |X|$ we have", "$$", "x \\in \\text{Supp}(\\mathcal{F})", "\\Leftrightarrow", "\\mathcal{F}_{\\overline{x}} \\not = 0", "\\Leftrightarrow", "\\mathcal{F}_{\\overline{x}} \\otimes_{\\mathcal{O}_{X, \\overline{x}}}", "\\kappa(\\overline{x}) \\not = 0.", "$$", "\\item For any morphism of algebraic spaces $f : Y \\to X$ the pullback", "$f^*\\mathcal{F}$ is of finite type as well and we have", "$\\text{Supp}(f^*\\mathcal{F}) = f^{-1}(\\text{Supp}(\\mathcal{F}))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$.", "By Lemma \\ref{lemma-support-covering} the inverse image of the support of", "$\\mathcal{F}$ is the support of $\\varphi^*\\mathcal{F}$ which is closed by", "Morphisms, Lemma \\ref{morphisms-lemma-support-finite-type}.", "Thus (1) follows from the definition of the topology on $|X|$.", "\\medskip\\noindent", "The first equivalence in (2) is the definition of support.", "The second equivalence follows from Nakayama's lemma, see", "Algebra, Lemma \\ref{algebra-lemma-NAK}.", "\\medskip\\noindent", "Let $f : Y \\to X$ be as in (3). Note that $f^*\\mathcal{F}$ is of finite type", "by Properties of Spaces, Section", "\\ref{spaces-properties-section-properties-modules}.", "For the final assertion, let $\\overline{y}$ be a geometric point of $Y$", "mapping to the geometric point $\\overline{x}$ on $X$. Recall that", "$$", "(f^*\\mathcal{F})_{\\overline{y}} =", "\\mathcal{F}_{\\overline{x}} \\otimes_{\\mathcal{O}_{X, \\overline{x}}}", "\\mathcal{O}_{Y, \\overline{y}},", "$$", "see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-stalk-pullback-quasi-coherent}.", "Hence $(f^*\\mathcal{F})_{\\overline{y}} \\otimes \\kappa(\\overline{y})$", "is nonzero if and only if", "$\\mathcal{F}_{\\overline{x}} \\otimes \\kappa(\\overline{x})$ is nonzero.", "By (2) this implies $x \\in \\text{Supp}(\\mathcal{F})$ if and only", "if $y \\in \\text{Supp}(f^*\\mathcal{F})$, which is the content of", "assertion (3)." ], "refs": [ "spaces-morphisms-lemma-support-covering", "morphisms-lemma-support-finite-type", "algebra-lemma-NAK", "spaces-properties-lemma-stalk-pullback-quasi-coherent" ], "ref_ids": [ 4776, 5143, 401, 11910 ] } ], "ref_ids": [] }, { "id": 4778, "type": "theorem", "label": "spaces-morphisms-lemma-scheme-theoretic-support", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-scheme-theoretic-support", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "There exists a smallest closed subspace $i : Z \\to X$ such that there", "exists a quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{G}$ with", "$i_*\\mathcal{G} \\cong \\mathcal{F}$. Moreover:", "\\begin{enumerate}", "\\item If $U$ is a scheme and $\\varphi : U \\to X$ is an \\'etale morphism", "then $Z \\times_X U$ is the scheme theoretic support of $\\varphi^*\\mathcal{F}$.", "\\item The quasi-coherent sheaf $\\mathcal{G}$ is unique up to unique", "isomorphism.", "\\item The quasi-coherent sheaf $\\mathcal{G}$ is of finite type.", "\\item The support of $\\mathcal{G}$ and of $\\mathcal{F}$ is $|Z|$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$.", "Let $R = U \\times_X U$ with projections $s, t : R \\to U$.", "Let $i' : Z' \\to U$ be the scheme theoretic support of $\\varphi^*\\mathcal{F}$", "and let $\\mathcal{G}'$ be the (unique up to unique isomorphism)", "finite type quasi-coherent $\\mathcal{O}_{Z'}$-module", "with $i'_*\\mathcal{G}' = \\varphi^*\\mathcal{F}$, see", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-support}.", "As $s^*\\varphi^*\\mathcal{F} = t^*\\varphi^*\\mathcal{F}$ we see that", "$R' = s^{-1}Z' = t^{-1}Z'$ as closed subschemes of $R$ by", "Morphisms, Lemma \\ref{morphisms-lemma-flat-pullback-support}.", "Thus we may apply Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-subspaces-presentation}", "to find a closed subspace $i : Z \\to X$ whose pullback to $U$ is $Z'$.", "Writing $s', t' : R' \\to Z'$ the projections and", "$j' : R' \\to R$ the given closed immersion, we see that", "$$", "j'_* (s')^*\\mathcal{G}' = s^* i'_*\\mathcal{G}' =", "s^*\\varphi^*\\mathcal{F} = t^*\\varphi^*\\mathcal{F} =", "t^*i'_*\\mathcal{G}' = j'_*(t')^*\\mathcal{G}'", "$$", "(the first and the last equality by Cohomology of Schemes,", "Lemma \\ref{coherent-lemma-flat-base-change-cohomology}).", "Hence the uniqueness of", "Morphisms, Lemma \\ref{morphisms-lemma-flat-pullback-support}", "applied to $R' \\to R$ gives an isomorphism", "$\\alpha : (t')^*\\mathcal{G}' \\to (s')^*\\mathcal{G}'$", "compatible with the canonical isomorphism", "$t^*\\varphi^*\\mathcal{F} = s^*\\varphi^*\\mathcal{F}$", "via $j'_*$. Clearly $\\alpha$ satisfies the cocycle condition, hence", "we may apply", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-quasi-coherent}", "to obtain a quasi-coherent module $\\mathcal{G}$ on $Z$ whose restriction", "to $Z'$ is $\\mathcal{G}'$ compatible with $\\alpha$.", "Again using the equivalence of the proposition mentioned above", "(this time for $X$) we conclude that $i_*\\mathcal{G} \\cong \\mathcal{F}$.", "\\medskip\\noindent", "This proves existence. The other properties of the lemma follow", "by comparing with the result for schemes using", "Lemma \\ref{lemma-support-covering}.", "Detailed proofs omitted." ], "refs": [ "morphisms-lemma-scheme-theoretic-support", "morphisms-lemma-flat-pullback-support", "spaces-properties-lemma-subspaces-presentation", "coherent-lemma-flat-base-change-cohomology", "morphisms-lemma-flat-pullback-support", "spaces-properties-proposition-quasi-coherent", "spaces-morphisms-lemma-support-covering" ], "ref_ids": [ 5144, 5271, 11845, 3298, 5271, 11920, 4776 ] } ], "ref_ids": [] }, { "id": 4779, "type": "theorem", "label": "spaces-morphisms-lemma-scheme-theoretic-image", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-scheme-theoretic-image", "contents": [ "\\begin{slogan}", "The scheme-theoretic image of a morphism of algebraic spaces exists.", "\\end{slogan}", "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. There exists a closed subspace $Z \\subset Y$ such that $f$ factors", "through $Z$ and such that for any other closed subspace $Z' \\subset Y$", "such that $f$ factors through $Z'$ we have $Z \\subset Z'$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} = \\Ker(\\mathcal{O}_Y \\to f_*\\mathcal{O}_X)$.", "If $\\mathcal{I}$ is quasi-coherent then we just take $Z$ to be the", "closed subscheme determined by $\\mathcal{I}$, see", "Lemma \\ref{lemma-closed-immersion-ideals}.", "In general the lemma requires us to show that there exists", "a largest quasi-coherent sheaf of ideals $\\mathcal{I}'$ contained in", "$\\mathcal{I}$.", "This follows from Lemma \\ref{lemma-largest-quasi-coherent-subsheaf}." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-ideals", "spaces-morphisms-lemma-largest-quasi-coherent-subsheaf" ], "ref_ids": [ 4765, 4772 ] } ], "ref_ids": [] }, { "id": 4780, "type": "theorem", "label": "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $Z \\subset Y$ be the scheme theoretic image of $f$.", "If $f$ is quasi-compact then", "\\begin{enumerate}", "\\item the sheaf of ideals", "$\\mathcal{I} = \\Ker(\\mathcal{O}_Y \\to f_*\\mathcal{O}_X)$", "is quasi-coherent,", "\\item the scheme theoretic image $Z$ is the closed subspace", "corresponding to $\\mathcal{I}$,", "\\item for any \\'etale morphism $V \\to Y$ the scheme theoretic image of", "$X \\times_Y V \\to V$ is equal to $Z \\times_Y V$, and", "\\item the image $|f|(|X|) \\subset |Z|$ is a dense subset of $|Z|$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove (3) it suffices to prove (1) and (2) since the", "formation of $\\mathcal{I}$ commutes with \\'etale localization.", "If (1) holds then in the proof of Lemma \\ref{lemma-scheme-theoretic-image}", "we showed (2). Let us prove that $\\mathcal{I}$ is quasi-coherent.", "Since the property of being quasi-coherent is \\'etale local we may", "assume $Y$ is an affine scheme. As $f$ is quasi-compact,", "we can find an affine scheme $U$ and a surjective \\'etale morphism", "$U \\to X$. Denote $f'$ the composition $U \\to X \\to Y$.", "Then $f_*\\mathcal{O}_X$ is a subsheaf of $f'_*\\mathcal{O}_U$,", "and hence $\\mathcal{I} = \\Ker(\\mathcal{O}_Y \\to \\mathcal{O}_{X'})$.", "By Lemma \\ref{lemma-pushforward}", "the sheaf $f'_*\\mathcal{O}_U$ is quasi-coherent on $Y$. Hence $\\mathcal{I}$", "is quasi-coherent as a kernel of a map between coherent modules.", "Finally, part (4) follows from parts (1), (2), and (3) as the ideal", "$\\mathcal{I}$ will be the unit ideal in any point of $|Y|$ which is", "not contained in the closure of $|f|(|X|)$." ], "refs": [ "spaces-morphisms-lemma-scheme-theoretic-image", "spaces-morphisms-lemma-pushforward" ], "ref_ids": [ 4779, 4760 ] } ], "ref_ids": [] }, { "id": 4781, "type": "theorem", "label": "spaces-morphisms-lemma-scheme-theoretic-image-reduced", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-scheme-theoretic-image-reduced", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $X$ is reduced. Then", "\\begin{enumerate}", "\\item the scheme theoretic image $Z$ of $f$ is the reduced induced algebraic", "space structure on $\\overline{|f|(|X|)}$, and", "\\item for any \\'etale morphism $V \\to Y$ the scheme theoretic image of", "$X \\times_Y V \\to V$ is equal to $Z \\times_Y V$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is true because the reduced induced algebraic space structure on", "$\\overline{|f|(|X|)}$ is the smallest closed subspace", "of $Y$ through which $f$ factors, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-map-into-reduction}.", "Part (2) follows from (1), the fact that $|V| \\to |Y|$ is open, and the", "fact that being reduced is preserved under \\'etale localization." ], "refs": [ "spaces-properties-lemma-map-into-reduction" ], "ref_ids": [ 11847 ] } ], "ref_ids": [] }, { "id": 4782, "type": "theorem", "label": "spaces-morphisms-lemma-reach-points-scheme-theoretic-image", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-reach-points-scheme-theoretic-image", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a quasi-compact morphism of algebraic spaces over $S$.", "Let $Z$ be the scheme theoretic image of $f$.", "Let $z \\in |Z|$. There exists a valuation ring $A$ with", "fraction field $K$ and a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[rr] \\ar[d] & & X \\ar[d] \\ar[ld] \\\\", "\\Spec(A) \\ar[r] & Z \\ar[r] & Y", "}", "$$", "such that the closed point of $\\Spec(A)$ maps to $z$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $V$ with a point $z' \\in V$", "and an \\'etale morphism $V \\to Y$ mapping $z'$ to $z$.", "Let $Z' \\subset V$ be the scheme theoretic image of $X \\times_Y V \\to V$.", "By Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image} we have", "$Z' = Z \\times_Y V$. Thus $z' \\in Z'$.", "Since $f$ is quasi-compact and $V$ is affine we see that", "$X \\times_Y V$ is quasi-compact. Hence", "there exists an affine scheme $W$ and a surjective \\'etale", "morphism $W \\to X \\times_Y V$. Then $Z' \\subset V$ is also the", "scheme theoretic image of $W \\to V$.", "By Morphisms, Lemma \\ref{morphisms-lemma-reach-points-scheme-theoretic-image}", "we can choose a diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] &", "W \\ar[r] \\ar[d] &", "X \\times_Y V \\ar[d] \\ar[r] &", "X \\ar[d] \\\\", "\\Spec(A) \\ar[r] &", "Z' \\ar[r] &", "V \\ar[r] &", "Y", "}", "$$", "such that the closed point of $\\Spec(A)$ maps to $z'$.", "Composing with $Z' \\to Z$ and $W \\to X \\times_Y V \\to X$", "we obtain a solution." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image", "morphisms-lemma-reach-points-scheme-theoretic-image" ], "ref_ids": [ 4780, 5147 ] } ], "ref_ids": [] }, { "id": 4783, "type": "theorem", "label": "spaces-morphisms-lemma-factor-factor", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-factor-factor", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X_1 \\ar[d] \\ar[r]_{f_1} & Y_1 \\ar[d] \\\\", "X_2 \\ar[r]^{f_2} & Y_2", "}", "$$", "be a commutative diagram of algebraic spaces over $S$.", "Let $Z_i \\subset Y_i$, $i = 1, 2$ be", "the scheme theoretic image of $f_i$. Then the morphism", "$Y_1 \\to Y_2$ induces a morphism $Z_1 \\to Z_2$ and a", "commutative diagram", "$$", "\\xymatrix{", "X_1 \\ar[r] \\ar[d] & Z_1 \\ar[d] \\ar[r] & Y_1 \\ar[d] \\\\", "X_2 \\ar[r] & Z_2 \\ar[r] & Y_2", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "The scheme theoretic inverse image of $Z_2$ in $Y_1$", "is a closed subspace of $Y_1$ through", "which $f_1$ factors. Hence $Z_1$ is contained in this.", "This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4784, "type": "theorem", "label": "spaces-morphisms-lemma-scheme-theoretic-image-of-partial-section", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-scheme-theoretic-image-of-partial-section", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a separated morphism of algebraic spaces over $S$.", "Let $V \\subset Y$ be an open subspace such that $V \\to Y$ is quasi-compact.", "Let $s : V \\to X$ be a morphism such that $f \\circ s = \\text{id}_V$.", "Let $Y'$ be the scheme theoretic image of $s$.", "Then $Y' \\to Y$ is an isomorphism over $V$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-quasi-compact-permanence}", "the morphism $s : V \\to X$ is quasi-compact.", "Hence the construction of the scheme theoretic image $Y'$", "of $s$ commutes with restriction to opens by", "Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image}.", "In particular, we see that $Y' \\cap f^{-1}(V)$ is the", "scheme theoretic image of a section of the separated", "morphism $f^{-1}(V) \\to V$. Since a section of a separated", "morphism is a closed immersion", "(Lemma \\ref{lemma-section-immersion}),", "we conclude that", "$Y' \\cap f^{-1}(V) \\to V$ is an isomorphism as desired." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-permanence", "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image" ], "ref_ids": [ 4743, 4780 ] } ], "ref_ids": [] }, { "id": 4785, "type": "theorem", "label": "spaces-morphisms-lemma-scheme-theoretically-dense-representable", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-scheme-theoretically-dense-representable", "contents": [ "Let $S$ be a scheme. Let $W \\subset S$ be a scheme theoretically", "dense open subscheme", "(Morphisms, Definition \\ref{morphisms-definition-scheme-theoretically-dense}).", "Let $f : X \\to S$ be a morphism of schemes which is flat, locally of", "finite presentation, and locally quasi-finite.", "Then $f^{-1}(W)$ is scheme theoretically dense in $X$." ], "refs": [ "morphisms-definition-scheme-theoretically-dense" ], "proofs": [ { "contents": [ "We will use the characterization of Morphisms, Lemma", "\\ref{morphisms-lemma-characterize-scheme-theoretically-dense}.", "Assume $V \\subset X$ is an open and $g \\in \\Gamma(V, \\mathcal{O}_V)$", "is a function which restricts to zero on $f^{-1}(W) \\cap V$.", "We have to show that $g = 0$. Assume $g \\not = 0$ to get a", "contradiction. By", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-go-down-with-annihilators}", "we may shrink $V$, find an open $U \\subset S$ fitting into a", "commutative diagram", "$$", "\\xymatrix{", "V \\ar[r] \\ar[d]_\\pi & X \\ar[d]^f \\\\", "U \\ar[r] & S,", "}", "$$", "a quasi-coherent subsheaf $\\mathcal{F} \\subset \\mathcal{O}_U$, an integer", "$r > 0$, and an injective $\\mathcal{O}_U$-module map", "$\\mathcal{F}^{\\oplus r} \\to \\pi_*\\mathcal{O}_V$", "whose image contains $g|_V$. Say", "$(g_1, \\ldots, g_r) \\in \\Gamma(U, \\mathcal{F}^{\\oplus r})$ maps to $g$.", "Then we see that $g_i|_{W \\cap U} = 0$ because $g|_{f^{-1}W \\cap V} = 0$.", "Hence $g_i = 0$ because $\\mathcal{F} \\subset \\mathcal{O}_U$ and", "$W$ is scheme theoretically dense in $S$.", "This implies $g = 0$ which is the desired contradiction." ], "refs": [ "morphisms-lemma-characterize-scheme-theoretically-dense", "more-morphisms-lemma-go-down-with-annihilators" ], "ref_ids": [ 5152, 13911 ] } ], "ref_ids": [ 5540 ] }, { "id": 4786, "type": "theorem", "label": "spaces-morphisms-lemma-scheme-theoretically-dense", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-scheme-theoretically-dense", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $U \\subset X$ be an open subspace.", "The following are equivalent", "\\begin{enumerate}", "\\item for every \\'etale morphism $\\varphi : V \\to X$ (of algebraic spaces)", "the scheme theoretic closure of $\\varphi^{-1}(U)$ in $V$ is equal to $V$,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$\\varphi : V \\to X$ such that the scheme theoretic closure of", "$\\varphi^{-1}(U)$ in $V$ is equal to $V$,", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that if $V \\to V'$ is a morphism of algebraic spaces \\'etale", "over $X$, and $Z \\subset V$, resp.\\ $Z' \\subset V'$ is the scheme theoretic", "closure of $U \\times_X V$, resp.\\ $U \\times_X V'$ in $V$, resp.\\ $V'$,", "then $Z$ maps into $Z'$. Thus if $V \\to V'$ is surjective and \\'etale", "then $Z = V$ implies $Z' = V'$. Next, note that an \\'etale morphism is", "flat, locally of finite presentation, and locally quasi-finite", "(see Morphisms, Section \\ref{morphisms-section-etale}).", "Thus Lemma \\ref{lemma-scheme-theoretically-dense-representable}", "implies that if $V$ and $V'$ are schemes, then $Z' = V'$ implies", "$Z = V$. A formal argument using that every algebraic space has an", "\\'etale covering by a scheme shows that (1) and (2) are equivalent." ], "refs": [ "spaces-morphisms-lemma-scheme-theoretically-dense-representable" ], "ref_ids": [ 4785 ] } ], "ref_ids": [] }, { "id": 4787, "type": "theorem", "label": "spaces-morphisms-lemma-scheme-theoretically-dense-quasi-compact", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-scheme-theoretically-dense-quasi-compact", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U \\subset X$ be an open subspace.", "If $U \\to X$ is quasi-compact, then $U$", "is scheme theoretically dense in $X$ if and only if the scheme theoretic", "closure of $U$ in $X$ is $X$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image} part (3)." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image" ], "ref_ids": [ 4780 ] } ], "ref_ids": [] }, { "id": 4788, "type": "theorem", "label": "spaces-morphisms-lemma-characterize-scheme-theoretically-dense", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-characterize-scheme-theoretically-dense", "contents": [ "Let $S$ be a scheme.", "Let $j : U \\to X$ be an open immersion of algebraic spaces over $S$.", "Then $U$ is scheme theoretically dense in $X$ if and only if", "$\\mathcal{O}_X \\to j_*\\mathcal{O}_U$ is injective." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{O}_X \\to j_*\\mathcal{O}_U$ is injective,", "then the same is true when restricted to any", "algebraic space $V$ \\'etale over $X$.", "Hence the scheme theoretic closure of $U \\times_X V$ in $V$", "is equal to $V$, see proof of", "Lemma \\ref{lemma-scheme-theoretic-image}.", "Conversely, assume the scheme theoretic", "closure of $U \\times_X V$ is equal to $V$ for all $V$ \\'etale over $X$.", "Suppose that $\\mathcal{O}_X \\to j_*\\mathcal{O}_U$ is not injective.", "Then we can find an affine, say $V = \\Spec(A)$, \\'etale over $X$", "and a nonzero element $f \\in A$ such that $f$ maps to zero in", "$\\Gamma(V \\times_X U, \\mathcal{O})$. In this case the scheme theoretic", "closure of $V \\times_X U$ in $V$ is clearly contained in $\\Spec(A/(f))$", "a contradiction." ], "refs": [ "spaces-morphisms-lemma-scheme-theoretic-image" ], "ref_ids": [ 4779 ] } ], "ref_ids": [] }, { "id": 4789, "type": "theorem", "label": "spaces-morphisms-lemma-intersection-scheme-theoretically-dense", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-intersection-scheme-theoretically-dense", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "If $U$, $V$ are scheme theoretically dense", "open subspaces of $X$, then so is $U \\cap V$." ], "refs": [], "proofs": [ { "contents": [ "Let $W \\to X$ be any \\'etale morphism. Consider the map", "$\\mathcal{O}(W) \\to \\mathcal{O}(W \\times_X V)", "\\to \\mathcal{O}(W \\times_X (V \\cap U))$.", "By Lemma \\ref{lemma-characterize-scheme-theoretically-dense}", "both maps are injective. Hence the composite is injective.", "Hence by Lemma \\ref{lemma-characterize-scheme-theoretically-dense}", "$U \\cap V$ is scheme theoretically dense in $X$." ], "refs": [ "spaces-morphisms-lemma-characterize-scheme-theoretically-dense", "spaces-morphisms-lemma-characterize-scheme-theoretically-dense" ], "ref_ids": [ 4788, 4788 ] } ], "ref_ids": [] }, { "id": 4790, "type": "theorem", "label": "spaces-morphisms-lemma-quasi-compact-immersion", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-quasi-compact-immersion", "contents": [ "Let $S$ be a scheme. Let $h : Z \\to X$ be an immersion of algebraic spaces", "over $S$. Assume either $Z \\to X$ is quasi-compact or $Z$ is reduced.", "Let $\\overline{Z} \\subset X$ be the scheme theoretic image of $h$.", "Then the morphism $Z \\to \\overline{Z}$ is an open immersion", "which identifies $Z$ with a scheme theoretically dense open", "subspace of $\\overline{Z}$. Moreover, $Z$ is topologically", "dense in $\\overline{Z}$." ], "refs": [], "proofs": [ { "contents": [ "In both cases the formation of $\\overline{Z}$ commutes with", "\\'etale localization, see", "Lemmas \\ref{lemma-quasi-compact-scheme-theoretic-image} and", "\\ref{lemma-scheme-theoretic-image-reduced}.", "Hence this lemma follows from the case of schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-immersion}." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image", "spaces-morphisms-lemma-scheme-theoretic-image-reduced", "morphisms-lemma-quasi-compact-immersion" ], "ref_ids": [ 4780, 4781, 5154 ] } ], "ref_ids": [] }, { "id": 4791, "type": "theorem", "label": "spaces-morphisms-lemma-equality-of-morphisms", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-equality-of-morphisms", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $f, g : X \\to Y$ be morphisms of algebraic spaces over $B$.", "Let $U \\subset X$ be an open subspace such that", "$f|_U = g|_U$. If the scheme theoretic closure of $U$", "in $X$ is $X$ and $Y \\to B$ is separated, then $f = g$." ], "refs": [], "proofs": [ { "contents": [ "As $Y \\to B$ is separated the fibre product", "$Y \\times_{\\Delta, Y \\times_B Y, (f, g)} X$ is a closed subspace $Z \\subset X$. ", "As $f|_U = g|_U$ we see that $U \\subset Z$. Hence $Z = X$ as $U$ is assumed", "scheme theoretically dense in $X$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4792, "type": "theorem", "label": "spaces-morphisms-lemma-universally-injective-representable", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-universally-injective-representable", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable", "morphism of algebraic spaces over $S$. Then $f$ is universally injective", "(in the sense of Section \\ref{section-representable})", "if and only if for all fields $K$ the map $X(K) \\to Y(K)$ is injective." ], "refs": [], "proofs": [ { "contents": [ "We are going to use", "Morphisms, Lemma \\ref{morphisms-lemma-universally-injective}", "without further mention.", "Suppose that $f$ is universally injective. Then for any field $K$ and any", "morphism $\\Spec(K) \\to Y$ the morphism of schemes", "$\\Spec(K) \\times_Y X \\to \\Spec(K)$ is universally injective.", "Hence there exists at most one section of the morphism", "$\\Spec(K) \\times_Y X \\to \\Spec(K)$. Hence the map", "$X(K) \\to Y(K)$ is injective. Conversely, suppose that for every field $K$", "the map $X(K) \\to Y(K)$ is injective. Let $T \\to Y$ be a morphism from a", "scheme into $Y$, and consider the base change $f_T : T \\times_Y X \\to T$.", "For any field $K$ we have", "$$", "(T \\times_Y X)(K) = T(K) \\times_{Y(K)} X(K)", "$$", "by definition of the fibre product, and hence the injectivity of", "$X(K) \\to Y(K)$ guarantees the injectivity of", "$(T \\times_Y X)(K) \\to T(K)$ which means that $f_T$ is universally injective", "as desired." ], "refs": [ "morphisms-lemma-universally-injective" ], "ref_ids": [ 5167 ] } ], "ref_ids": [] }, { "id": 4793, "type": "theorem", "label": "spaces-morphisms-lemma-universally-injective", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-universally-injective", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item the map $X(K) \\to Y(K)$ is injective for every field $K$ over $S$", "\\item for every morphism $Y' \\to Y$ of algebraic spaces over $S$", "the induced map $|Y' \\times_Y X| \\to |Y'|$ is injective, and", "\\item the diagonal morphism $X \\to X \\times_Y X$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Let $g : Y' \\to Y$ be a morphism of algebraic", "spaces, and denote $f' : Y' \\times_Y X \\to Y'$ the base change of $f$.", "Let $K_i$, $i = 1, 2$ be fields and let", "$\\varphi_i : \\Spec(K_i) \\to Y' \\times_Y X$ be morphisms", "such that $f' \\circ \\varphi_1$ and $f' \\circ \\varphi_2$ define the", "same element of $|Y'|$. By definition this means there exists a", "field $\\Omega$ and embeddings $\\alpha_i : K_i \\subset \\Omega$ such that", "the two morphisms", "$f' \\circ \\varphi_i \\circ \\alpha_i : \\Spec(\\Omega) \\to Y'$ are equal.", "Here is the corresponding commutative diagram", "$$", "\\xymatrix{", "\\Spec(\\Omega) \\ar@/_5ex/[ddrr] \\ar[rd]^{\\alpha_1} \\ar[r]_{\\alpha_2} &", "\\Spec(K_2) \\ar[rd]^{\\varphi_2} \\\\", "& \\Spec(K_1) \\ar[r]^{\\varphi_1} &", "Y' \\times_Y X \\ar[d]^{f'} \\ar[r]^{g'} &", " X \\ar[d]^f \\\\", "& & Y' \\ar[r]^g & Y.", "}", "$$", "In particular the compositions $g \\circ f' \\circ \\varphi_i \\circ \\alpha_i$", "are equal. By assumption (1) this implies that the morphism", "$g' \\circ \\varphi_i \\circ \\alpha_i$ are equal, where $g' : Y' \\times_Y X \\to X$", "is the projection. By the universal property of the fibre product we conclude", "that the morphisms", "$\\varphi_i \\circ \\alpha_i : \\Spec(\\Omega) \\to Y' \\times_Y X$ are", "equal. In other words $\\varphi_1$ and $\\varphi_2$ define the same point", "of $Y' \\times_Y X$. We conclude that (2) holds.", "\\medskip\\noindent", "Assume (2). Let $K$ be a field over $S$, and let $a, b : \\Spec(K) \\to X$", "be two morphisms such that $f \\circ a = f \\circ b$. Denote", "$c : \\Spec(K) \\to Y$ the common value. By assumption", "$|\\Spec(K) \\times_{c, Y} X| \\to |\\Spec(K)|$ is injective.", "This means there exists a field $\\Omega$ and embeddings", "$\\alpha_i : K \\to \\Omega$ such that", "$$", "\\xymatrix{", "\\Spec(\\Omega) \\ar[r]_{\\alpha_1} \\ar[d]_{\\alpha_2} &", "\\Spec(K) \\ar[d]^a \\\\", "\\Spec(K) \\ar[r]^-b &", "\\Spec(K) \\times_{c, Y} X", "}", "$$", "is commutative. Composing with the projection to $\\Spec(K)$", "we see that $\\alpha_1 = \\alpha_2$. Denote the common value $\\alpha$.", "Then we see that $\\{\\alpha : \\Spec(\\Omega) \\to \\Spec(K)\\}$", "is a fpqc covering of $\\Spec(K)$ such that the two morphisms", "$a, b$ become equal on the members of the covering. By", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-sheaf-fpqc}", "we conclude that $a = b$. We conclude that (1) holds.", "\\medskip\\noindent", "Assume (3). Let $x, x' \\in |X|$ be a pair of points such that", "$f(x) = f(x')$ in $|Y|$. By", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}", "we see there exists a $x'' \\in |X \\times_Y X|$ whose projections", "are $x$ and $x'$. By assumption and", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-characterize-surjective}", "there exists a $x''' \\in |X|$ with $\\Delta_{X/Y}(x''') = x''$.", "Thus $x = x'$. In other words $f$ is injective.", "Since condition (3) is stable under base change we see that $f$", "satisfies (2).", "\\medskip\\noindent", "Assume (2). Then in particular $|X \\times_Y X| \\to |X|$ is injective", "which implies immediately that", "$|\\Delta_{X/Y}| : |X| \\to |X \\times_Y X|$ is surjective, which", "implies that $\\Delta_{X/Y}$ is surjective by", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-characterize-surjective}." ], "refs": [ "spaces-properties-proposition-sheaf-fpqc", "spaces-properties-lemma-points-cartesian", "spaces-properties-lemma-characterize-surjective", "spaces-properties-lemma-characterize-surjective" ], "ref_ids": [ 11919, 11819, 11820, 11820 ] } ], "ref_ids": [] }, { "id": 4794, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-universally-injective", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-universally-injective", "contents": [ "The base change of a universally injective morphism is universally injective." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4795, "type": "theorem", "label": "spaces-morphisms-lemma-universally-injective-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-universally-injective-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is universally injective,", "\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism", "$Z \\times_Y X \\to Z$ is universally injective,", "\\item for every affine scheme $Z$ and any morphism", "$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is universally injective,", "\\item there exists a scheme $Z$ and a surjective morphism", "$Z \\to Y$ such that $Z \\times_Y X \\to Z$ is universally injective, and", "\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that", "each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is universally injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use that being universally injective is preserved under base change", "(Lemma \\ref{lemma-base-change-universally-injective})", "without further mention in this proof.", "It is clear that (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3)", "$\\Rightarrow$ (4).", "\\medskip\\noindent", "Assume $g : Z \\to Y$ as in (4). Let $y : \\Spec(K) \\to Y$ be a", "morphism from the spectrum of a field into $Y$. By assumption we", "can find an extension field $\\alpha : K \\subset K'$ and a morphism", "$z : \\Spec(K') \\to Z$ such that $y \\circ \\alpha = g \\circ z$", "(with obvious abuse of notation). By assumption the", "morphism $Z \\times_Y X \\to Z$ is universally injective, hence there", "is at most one", "lift of $g \\circ z : \\Spec(K') \\to Y$ to a morphism into $X$.", "Since $\\{\\alpha : \\Spec(K') \\to \\Spec(K)\\}$ is a", "fpqc covering this implies there is at most one lift of", "$y : \\Spec(K) \\to Y$ to a morphism into $X$, see", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-sheaf-fpqc}.", "Thus we see that (1) holds.", "\\medskip\\noindent", "We omit the verification that (5) is equivalent to (1)." ], "refs": [ "spaces-morphisms-lemma-base-change-universally-injective", "spaces-properties-proposition-sheaf-fpqc" ], "ref_ids": [ 4794, 11919 ] } ], "ref_ids": [] }, { "id": 4796, "type": "theorem", "label": "spaces-morphisms-lemma-composition-universally-injective", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-universally-injective", "contents": [ "A composition of universally injective morphisms is universally injective." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4797, "type": "theorem", "label": "spaces-morphisms-lemma-affine-representable", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-affine-representable", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism", "of algebraic spaces over $S$. Then", "$f$ is affine (in the sense of Section \\ref{section-representable})", "if and only if for all affine schemes $Z$", "and morphisms $Z \\to Y$ the scheme $X \\times_Y Z$ is affine." ], "refs": [], "proofs": [ { "contents": [ "This follows directly from the definition of an affine morphism of schemes", "(Morphisms, Definition \\ref{morphisms-definition-affine})." ], "refs": [ "morphisms-definition-affine" ], "ref_ids": [ 5544 ] } ], "ref_ids": [] }, { "id": 4798, "type": "theorem", "label": "spaces-morphisms-lemma-affine-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-affine-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is representable and affine,", "\\item $f$ is affine,", "\\item for every affine scheme $V$ and \\'etale morphism $V \\to Y$", "the scheme $X \\times_Y V$ is affine,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is affine, and", "\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such", "that each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is affine.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2), that (2) implies (3), and that", "(3) implies (4) by taking $V$ to be a disjoint union of affines", "\\'etale over $Y$, see Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-cover-by-union-affines}.", "Assume $V \\to Y$ is as in (4). Then for every affine open $W$ of $V$ we see", "that $W \\times_Y X$ is an affine open of $V \\times_Y X$. Hence by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme}", "we conclude that $V \\times_Y X$ is a scheme. Moreover the morphism", "$V \\times_Y X \\to V$ is affine. This means we can apply", "Spaces,", "Lemma \\ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale}", "because the class of affine morphisms satisfies all the required", "properties (see", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-affine} and", "Descent, Lemmas \\ref{descent-lemma-descending-property-affine}", "and \\ref{descent-lemma-affine}). The conclusion of applying this lemma", "is that $f$ is representable and affine, i.e., (1) holds.", "\\medskip\\noindent", "The equivalence of (1) and (5) follows from the fact that being", "affine is Zariski local on the target (the reference above shows", "that being affine is in fact fpqc local on the target)." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-properties-lemma-subscheme", "spaces-lemma-morphism-sheaves-with-P-effective-descent-etale", "morphisms-lemma-base-change-affine", "descent-lemma-descending-property-affine", "descent-lemma-affine" ], "ref_ids": [ 11830, 11848, 8158, 5176, 14683, 14748 ] } ], "ref_ids": [] }, { "id": 4799, "type": "theorem", "label": "spaces-morphisms-lemma-composition-affine", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-affine", "contents": [ "The composition of affine morphisms is affine." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Transitivity of fibre products." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4800, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-affine", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-affine", "contents": [ "The base change of an affine morphism is affine." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Transitivity of fibre products." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4801, "type": "theorem", "label": "spaces-morphisms-lemma-closed-immersion-affine", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-closed-immersion-affine", "contents": [ "A closed immersion is affine." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from the corresponding statement for morphisms of", "schemes, see Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-affine}." ], "refs": [ "morphisms-lemma-closed-immersion-affine" ], "ref_ids": [ 5177 ] } ], "ref_ids": [] }, { "id": 4802, "type": "theorem", "label": "spaces-morphisms-lemma-affine-equivalence-algebras", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-affine-equivalence-algebras", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "There is an anti-equivalence of categories", "$$", "\\begin{matrix}", "\\text{algebraic spaces} \\\\", "\\text{affine over }X", "\\end{matrix}", "\\longleftrightarrow", "\\begin{matrix}", "\\text{quasi-coherent sheaves} \\\\", "\\text{of }\\mathcal{O}_X\\text{-algebras}", "\\end{matrix}", "$$", "which associates to $f : Y \\to X$ the sheaf $f_*\\mathcal{O}_Y$.", "Moreover, this equivalence is compatible with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "This lemma is the analogue of Morphisms, Lemma", "\\ref{morphisms-lemma-affine-equivalence-algebras}.", "Let $\\mathcal{A}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras.", "We will construct an affine morphism of algebraic spaces", "$\\pi : Y = \\underline{\\Spec}_X(\\mathcal{A}) \\to X$ with", "$\\pi_*\\mathcal{O}_Y \\cong \\mathcal{A}$. To do this, choose a scheme", "$U$ and a surjective \\'etale morphism $\\varphi : U \\to X$. As usual", "denote $R = U \\times_X U$ with projections $s, t : R \\to U$. Denote", "$\\psi : R \\to X$ the composition $\\psi = \\varphi \\circ s = \\varphi \\circ t$.", "By the aforementioned lemma there exists an affine morphisms of schemes", "$\\pi_0 : V \\to U$ and $\\pi_1 : W \\to R$ with", "$\\pi_{0, *}\\mathcal{O}_V \\cong \\varphi^*\\mathcal{A}$ and", "$\\pi_{1, *}\\mathcal{O}_W \\cong \\psi^*\\mathcal{A}$.", "Since the construction is compatible with base change there exist", "morphisms $s', t' : W \\to V$ such that the diagrams", "$$", "\\vcenter{", "\\xymatrix{", "W \\ar[r]_{s'} \\ar[d] & V \\ar[d] \\\\", "R \\ar[r]^s & U", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "W \\ar[r]_{t'} \\ar[d] & V \\ar[d] \\\\", "R \\ar[r]^t & U", "}", "}", "$$", "are cartesian. It follows that $s', t'$ are \\'etale. It is a formal", "consequence of the above that $(t', s') : W \\to V \\times_S V$ is a", "monomorphism. We omit the verification that $W \\to V \\times_S V$ is an", "equivalence relation (hint: think about the pullback of $\\mathcal{A}$", "to $U \\times_X U \\times_X U = R \\times_{s, U, t} R$).", "The quotient sheaf $Y = V/W$", "is an algebraic space, see", "Spaces, Theorem \\ref{spaces-theorem-presentation}.", "By Groupoids, Lemma \\ref{groupoids-lemma-criterion-fibre-product}", "we see that $Y \\times_X U \\cong V$. Hence $Y \\to X$ is affine by", "Lemma \\ref{lemma-affine-local}. Finally, the isomorphism of", "$$", "(Y \\times_X U \\to U)_*\\mathcal{O}_{Y \\times_X U} =", "\\pi_{0, *}\\mathcal{O}_V \\cong \\varphi^*\\mathcal{A}", "$$", "is compatible with glueing isomorphisms, whence", "$(Y \\to X)_*\\mathcal{O}_Y \\cong \\mathcal{A}$ by", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-quasi-coherent}.", "We omit the verification that this construction is compatible with", "base change." ], "refs": [ "morphisms-lemma-affine-equivalence-algebras", "spaces-theorem-presentation", "groupoids-lemma-criterion-fibre-product", "spaces-morphisms-lemma-affine-local", "spaces-properties-proposition-quasi-coherent" ], "ref_ids": [ 5173, 8124, 9652, 4798, 11920 ] } ], "ref_ids": [] }, { "id": 4803, "type": "theorem", "label": "spaces-morphisms-lemma-affine-equivalence-modules", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-affine-equivalence-modules", "contents": [ "Let $S$ be a scheme.", "Let $f : Y \\to X$ be an affine morphism of algebraic spaces over $S$.", "Let $\\mathcal{A} = f_*\\mathcal{O}_Y$.", "The functor $\\mathcal{F} \\mapsto f_*\\mathcal{F}$ induces", "an equivalence of categories", "$$", "\\left\\{", "\\begin{matrix}", "\\text{category of quasi-coherent}\\\\", "\\mathcal{O}_Y\\text{-modules}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{category of quasi-coherent}\\\\", "\\mathcal{A}\\text{-modules}", "\\end{matrix}", "\\right\\}", "$$", "Moreover, an $\\mathcal{A}$-module is", "quasi-coherent as an $\\mathcal{O}_X$-module if and only if", "it is quasi-coherent as an $\\mathcal{A}$-module." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4804, "type": "theorem", "label": "spaces-morphisms-lemma-affine-permanence", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-affine-permanence", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Suppose $g : X \\to Y$ is a morphism of algebraic spaces over $B$.", "\\begin{enumerate}", "\\item If $X$ is affine over $B$ and $\\Delta : Y \\to Y \\times_B Y$ is affine,", "then $g$ is affine.", "\\item If $X$ is affine over $B$ and $Y$ is separated over $B$,", "then $g$ is affine.", "\\item A morphism from an affine scheme to an algebraic space with affine", "diagonal is affine.", "\\item A morphism from an affine scheme to a separated algebraic space", "is affine.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). The base change $X \\times_B Y \\to Y$ is affine by", "Lemma \\ref{lemma-base-change-affine}.", "The morphism $(1, g) : X \\to X \\times_B Y$ is the base change of", "$Y \\to Y \\times_B Y$ by the morphism $X \\times_B Y \\to Y \\times_B Y$.", "Hence it is affine by", "Lemma \\ref{lemma-base-change-affine}.", "The composition of affine morphisms is affine", "(see Lemma \\ref{lemma-composition-affine}) and (1) follows.", "Part (2) follows from (1) as a closed immersion is affine", "(see Lemma \\ref{lemma-closed-immersion-affine}) and $Y/B$ separated", "means $\\Delta$ is a closed immersion. Parts (3) and (4) are special", "cases of (1) and (2)." ], "refs": [ "spaces-morphisms-lemma-base-change-affine", "spaces-morphisms-lemma-base-change-affine", "spaces-morphisms-lemma-composition-affine", "spaces-morphisms-lemma-closed-immersion-affine" ], "ref_ids": [ 4800, 4800, 4799, 4801 ] } ], "ref_ids": [] }, { "id": 4805, "type": "theorem", "label": "spaces-morphisms-lemma-Artinian-affine", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-Artinian-affine", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$.", "Let $A$ be an Artinian ring. Any morphism $\\Spec(A) \\to X$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be an \\'etale morphism with $U$ affine. To prove the", "lemma we have to show that $\\Spec(A) \\times_X U$ is affine, see", "Lemma \\ref{lemma-affine-local}. Since $X$ is quasi-separated the", "scheme $\\Spec(A) \\times_X U$ is quasi-compact. Moreover, the", "projection morphism $\\Spec(A) \\times_X U \\to \\Spec(A)$ is \\'etale.", "Hence this morphism has finite discrete fibers and moreover", "the topology on $\\Spec(A)$ is discrete. Thus $\\Spec(A) \\times_X U$", "is a scheme whose underlying topological", "space is a finite discrete set. We are done by", "Schemes, Lemma \\ref{schemes-lemma-scheme-finite-discrete-affine}." ], "refs": [ "spaces-morphisms-lemma-affine-local", "schemes-lemma-scheme-finite-discrete-affine" ], "ref_ids": [ 4798, 7678 ] } ], "ref_ids": [] }, { "id": 4806, "type": "theorem", "label": "spaces-morphisms-lemma-quasi-affine-representable", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-quasi-affine-representable", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable", "morphism of algebraic spaces over $S$. Then", "$f$ is quasi-affine (in the sense of Section \\ref{section-representable})", "if and only if for all affine schemes $Z$", "and morphisms $Z \\to Y$ the scheme $X \\times_Y Z$ is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "This follows directly from the definition of a quasi-affine morphism", "of schemes", "(Morphisms, Definition \\ref{morphisms-definition-quasi-affine})." ], "refs": [ "morphisms-definition-quasi-affine" ], "ref_ids": [ 5546 ] } ], "ref_ids": [] }, { "id": 4807, "type": "theorem", "label": "spaces-morphisms-lemma-quasi-affine-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-quasi-affine-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is representable and quasi-affine,", "\\item $f$ is quasi-affine,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is quasi-affine, and", "\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such", "that each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is quasi-affine.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2) and that (2) implies (3) by taking", "$V$ to be a disjoint union of affines \\'etale over $Y$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-cover-by-union-affines}.", "Assume $V \\to Y$ is as in (3). Then for every affine open $W$ of $V$ we see", "that $W \\times_Y X$ is a quasi-affine open of $V \\times_Y X$. Hence by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme}", "we conclude that $V \\times_Y X$ is a scheme. Moreover the morphism", "$V \\times_Y X \\to V$ is quasi-affine. This means we can apply", "Spaces,", "Lemma \\ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale}", "because the class of quasi-affine morphisms satisfies all the required", "properties (see", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-quasi-affine} and", "Descent, Lemmas \\ref{descent-lemma-descending-property-quasi-affine}", "and \\ref{descent-lemma-quasi-affine}). The conclusion of applying this lemma", "is that $f$ is representable and quasi-affine, i.e., (1) holds.", "\\medskip\\noindent", "The equivalence of (1) and (4) follows from the fact that being", "quasi-affine is Zariski local on the target (the reference above shows", "that being quasi-affine is in fact fpqc local on the target)." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-properties-lemma-subscheme", "spaces-lemma-morphism-sheaves-with-P-effective-descent-etale", "morphisms-lemma-base-change-quasi-affine", "descent-lemma-descending-property-quasi-affine", "descent-lemma-quasi-affine" ], "ref_ids": [ 11830, 11848, 8158, 5187, 14685, 14750 ] } ], "ref_ids": [] }, { "id": 4808, "type": "theorem", "label": "spaces-morphisms-lemma-composition-quasi-affine", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-quasi-affine", "contents": [ "The composition of quasi-affine morphisms is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4809, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-quasi-affine", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-quasi-affine", "contents": [ "The base change of a quasi-affine morphism is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4810, "type": "theorem", "label": "spaces-morphisms-lemma-characterize-quasi-affine", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-characterize-quasi-affine", "contents": [ "Let $S$ be a scheme.", "A quasi-compact and quasi-separated morphism of algebraic spaces", "$f : Y \\to X$ is quasi-affine if and only if the canonical factorization", "$Y \\to \\underline{\\Spec}_X(f_*\\mathcal{O}_Y)$", "(Remark \\ref{remark-factorization-quasi-compact-quasi-separated})", "is an open immersion." ], "refs": [ "spaces-morphisms-remark-factorization-quasi-compact-quasi-separated" ], "proofs": [ { "contents": [ "Let $U \\to X$ be a surjective morphism where $U$ is a scheme.", "Since we may check whether $f$ is quasi-affine after base change to", "$U$ (Lemma \\ref{lemma-quasi-affine-local}), since $f_*\\mathcal{O}_Y|_V$", "is equal to $(Y \\times_X U \\to U)_*\\mathcal{O}_{Y \\times_X U}$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-pushforward-etale-base-change-modules}), and", "since formation of relative spectrum commutes with base change", "(Lemma \\ref{lemma-affine-equivalence-algebras}),", "we see that the assertion reduces to the case that $X$ is a scheme.", "If $X$ is a scheme and either $f$ is quasi-affine or", "$Y \\to \\underline{\\Spec}_X(f_*\\mathcal{O}_Y)$ is an open immersion,", "then $Y$ is a scheme as well. Thus we reduce to", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-quasi-affine}." ], "refs": [ "spaces-morphisms-lemma-quasi-affine-local", "spaces-properties-lemma-pushforward-etale-base-change-modules", "spaces-morphisms-lemma-affine-equivalence-algebras", "morphisms-lemma-characterize-quasi-affine" ], "ref_ids": [ 4807, 11898, 4802, 5185 ] } ], "ref_ids": [ 5032 ] }, { "id": 4811, "type": "theorem", "label": "spaces-morphisms-lemma-local-source-target", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-local-source-target", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes", "which is \\'etale local on the source-and-target.", "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Consider commutative diagrams", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$ and $V$ are schemes and the vertical arrows are \\'etale.", "The following are equivalent", "\\begin{enumerate}", "\\item for any diagram as above the morphism $h$ has property", "$\\mathcal{P}$, and", "\\item for some diagram as above with $a : U \\to X$ surjective", "the morphism $h$ has property $\\mathcal{P}$.", "\\end{enumerate}", "If $X$ and $Y$ are representable, then this is also equivalent to", "$f$ (as a morphism of schemes) having property $\\mathcal{P}$.", "If $\\mathcal{P}$ is also preserved under any base change, and", "fppf local on the base, then for representable morphisms $f$ this", "is also equivalent to $f$ having property $\\mathcal{P}$ in the sense", "of Section \\ref{section-representable}." ], "refs": [], "proofs": [ { "contents": [ "Let us prove the equivalence of (1) and (2).", "The implication (1) $\\Rightarrow$ (2) is immediate (taking into account", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}).", "Assume", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_h & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "\\quad\\quad", "\\xymatrix{", "U' \\ar[d] \\ar[r]_{h'} & V' \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "are two diagrams as in the lemma. Assume $U \\to X$ is", "surjective and $h$ has property $\\mathcal{P}$. To show that (2)", "implies (1) we have to prove that $h'$ has $\\mathcal{P}$. To do", "this consider the diagram", "$$", "\\xymatrix{", "U \\ar[d]_h &", "U \\times_X U' \\ar[l] \\ar[d]^{(h, h')} \\ar[r] &", "U' \\ar[d]^{h'} \\\\", "V &", "V \\times_Y V' \\ar[l] \\ar[r] &", "V'", "}", "$$", "By", "Descent, Lemma \\ref{descent-lemma-local-source-target-characterize}", "we see that $h$ has $\\mathcal{P}$ implies $(h, h')$ has $\\mathcal{P}$", "and since $U \\times_X U' \\to U'$ is surjective this implies (by the", "same lemma) that $h'$ has $\\mathcal{P}$.", "\\medskip\\noindent", "If $X$ and $Y$ are representable, then", "Descent, Lemma \\ref{descent-lemma-local-source-target-characterize}", "applies which shows that (1) and (2) are equivalent to $f$ having", "$\\mathcal{P}$.", "\\medskip\\noindent", "Finally, suppose $f$ is representable, and $U, V, a, b, h$ are", "as in part (2) of the lemma, and that $\\mathcal{P}$ is preserved under", "arbitrary base change. We have to show that for any scheme", "$Z$ and morphism $Z \\to X$ the base change $Z \\times_Y X \\to Z$", "has property $\\mathcal{P}$. Consider the diagram", "$$", "\\xymatrix{", "Z \\times_Y U \\ar[d] \\ar[r] &", "Z \\times_Y V \\ar[d] \\\\", "Z \\times_Y X \\ar[r] &", "Z", "}", "$$", "Note that the top horizontal arrow is a base change of $h$ and", "hence has property $\\mathcal{P}$. The left vertical arrow is \\'etale", "and surjective and the right vertical arrow is \\'etale. Thus", "Descent, Lemma \\ref{descent-lemma-local-source-target-characterize}", "once again kicks in and shows that $Z \\times_Y X \\to Z$", "has property $\\mathcal{P}$." ], "refs": [ "spaces-lemma-lift-morphism-presentations", "descent-lemma-local-source-target-characterize", "descent-lemma-local-source-target-characterize", "descent-lemma-local-source-target-characterize" ], "ref_ids": [ 8159, 14720, 14720, 14720 ] } ], "ref_ids": [] }, { "id": 4812, "type": "theorem", "label": "spaces-morphisms-lemma-local-source-target-at-point", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-local-source-target-at-point", "contents": [ "Let $\\mathcal{Q}$ be a property of morphisms of germs which is", "\\'etale local on the source-and-target.", "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $x \\in |X|$ be a point of $X$.", "Consider the diagrams", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "\\quad\\quad", "\\xymatrix{", "u \\ar[d] \\ar[r] & v \\ar[d] \\\\", "x \\ar[r] & y", "}", "$$", "where $U$ and $V$ are schemes, $a, b$ are \\'etale, and $u, v, x, y$", "are points of the corresponding spaces.", "The following are equivalent", "\\begin{enumerate}", "\\item for any diagram as above we have $\\mathcal{Q}((U, u) \\to (V, v))$, and", "\\item for some diagram as above we have $\\mathcal{Q}((U, u) \\to (V, v))$.", "\\end{enumerate}", "If $X$ and $Y$ are representable, then this is also", "equivalent to $\\mathcal{Q}((X, x) \\to (Y, y))$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Very similar to the proof of", "Lemma \\ref{lemma-local-source-target}." ], "refs": [ "spaces-morphisms-lemma-local-source-target" ], "ref_ids": [ 4811 ] } ], "ref_ids": [] }, { "id": 4813, "type": "theorem", "label": "spaces-morphisms-lemma-local-source-target-global-implies-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-local-source-target-global-implies-local", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes", "which is \\'etale local on the source-and-target.", "Consider the property $\\mathcal{Q}$ of morphisms", "of germs associated to $\\mathcal{P}$ in", "Descent, Lemma \\ref{descent-lemma-local-source-target-global-implies-local}.", "Then", "\\begin{enumerate}", "\\item $\\mathcal{Q}$ is \\'etale local on the source-and-target.", "\\item given a morphism of algebraic spaces $f : X \\to Y$ and $x \\in |X|$", "the following are equivalent", "\\begin{enumerate}", "\\item $f$ has $\\mathcal{Q}$ at $x$, and", "\\item there is an open neighbourhood $X' \\subset X$ of $x$", "such that $X' \\to Y$ has $\\mathcal{P}$.", "\\end{enumerate}", "\\item given a morphism of algebraic spaces $f : X \\to Y$", "the following are equivalent:", "\\begin{enumerate}", "\\item $f$ has $\\mathcal{P}$,", "\\item for every $x \\in |X|$ the morphism $f$ has $\\mathcal{Q}$ at $x$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [ "descent-lemma-local-source-target-global-implies-local" ], "proofs": [ { "contents": [ "See", "Descent, Lemma \\ref{descent-lemma-local-source-target-global-implies-local}", "for (1). The implication (1)(a) $\\Rightarrow$ (2)(b) follows on letting", "$X' = a(U) \\subset X$ given a diagram as in", "Lemma \\ref{lemma-local-source-target-at-point}.", "The implication (2)(b) $\\Rightarrow$ (1)(a) is clear.", "The equivalence of (3)(a) and (3)(b) follows from the corresponding", "result for morphisms of schemes, see", "Descent, Lemma \\ref{descent-lemma-local-source-target-local-implies-global}." ], "refs": [ "descent-lemma-local-source-target-global-implies-local", "spaces-morphisms-lemma-local-source-target-at-point", "descent-lemma-local-source-target-local-implies-global" ], "ref_ids": [ 14725, 4812, 14726 ] } ], "ref_ids": [ 14725 ] }, { "id": 4814, "type": "theorem", "label": "spaces-morphisms-lemma-composition-finite-type", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-finite-type", "contents": [ "The composition of finite type morphisms is of finite type.", "The same holds for locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "See Remark \\ref{remark-composition-P} and", "Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-type}." ], "refs": [ "spaces-morphisms-remark-composition-P", "morphisms-lemma-composition-finite-type" ], "ref_ids": [ 5033, 5199 ] } ], "ref_ids": [] }, { "id": 4815, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-finite-type", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-finite-type", "contents": [ "A base change of a finite type morphism is finite type.", "The same holds for locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "See Remark \\ref{remark-base-change-P} and", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-type}." ], "refs": [ "spaces-morphisms-remark-base-change-P", "morphisms-lemma-base-change-finite-type" ], "ref_ids": [ 5034, 5200 ] } ], "ref_ids": [] }, { "id": 4816, "type": "theorem", "label": "spaces-morphisms-lemma-finite-type-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-type-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is locally of finite type,", "\\item for every $x \\in |X|$ the morphism $f$ is of finite type at $x$,", "\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism", "$Z \\times_Y X \\to Z$ is locally of finite type,", "\\item for every affine scheme $Z$ and any morphism", "$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is locally of finite type,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is locally of finite type,", "\\item there exists a scheme $U$ and a surjective \\'etale morphism", "$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$", "is locally of finite type,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes and the vertical arrows are \\'etale", "the top horizontal arrow is locally of finite type,", "\\item there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes, the vertical arrows are \\'etale, $U \\to X$", "is surjective, and the top horizontal arrow is locally of finite type, and", "\\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$,", "and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that", "each morphism $X_{ij} \\to Y_i$ is locally of finite type.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Each of the conditions (2), (3), (4), (5), (6), (7), and (9)", "imply condition (8) in a straightforward manner. For example, if", "(5) holds, then we can choose a scheme $V$ which is a disjoint", "union of affines and a surjective morphism $V \\to Y$", "(see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-cover-by-union-affines}).", "Then $V \\times_Y X \\to V$ is locally of finite type by (5).", "Choose a scheme $U$ and a surjective \\'etale morphism", "$U \\to V \\times_Y X$. Then $U \\to V$ is locally of finite", "type by Lemma \\ref{lemma-composition-finite-type}.", "Hence (8) is true.", "\\medskip\\noindent", "The conditions (1), (7), and (8) are equivalent by definition.", "\\medskip\\noindent", "To finish the proof, we show that (1) implies all of the conditions", "(2), (3), (4), (5), (6), and (9). For (2) this is immediate.", "For (3), (4), (5), and (9) this follows from the fact that being", "locally of finite type is preserved under base change, see", "Lemma \\ref{lemma-base-change-finite-type}.", "For (6) we can take $U = X$ and we're done." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-morphisms-lemma-composition-finite-type", "spaces-morphisms-lemma-base-change-finite-type" ], "ref_ids": [ 11830, 4814, 4815 ] } ], "ref_ids": [] }, { "id": 4817, "type": "theorem", "label": "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $f$ is locally of finite type and $Y$ is locally Noetherian,", "then $X$ is locally Noetherian." ], "refs": [], "proofs": [ { "contents": [ "Let", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "be a commutative diagram where $U$, $V$ are schemes and the vertical arrows", "are surjective \\'etale. If $f$ is locally of finite type, then", "$U \\to V$ is locally of finite type. If $Y$ is locally Noetherian, then", "$V$ is locally Noetherian. By", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}", "we see that $U$ is locally Noetherian, which means that $X$ is locally", "Noetherian." ], "refs": [ "morphisms-lemma-finite-type-noetherian" ], "ref_ids": [ 5202 ] } ], "ref_ids": [] }, { "id": 4818, "type": "theorem", "label": "spaces-morphisms-lemma-permanence-finite-type", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-permanence-finite-type", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$, $g : Y \\to Z$ be morphisms of algebraic spaces over $S$.", "If $g \\circ f : X \\to Z$ is locally of finite type, then $f : X \\to Y$", "is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "We can find a diagram", "$$", "\\xymatrix{", "U \\ar[r] \\ar[d] & V \\ar[r] \\ar[d] & W \\ar[d] \\\\", "X \\ar[r] & Y \\ar[r] & Z", "}", "$$", "where $U$, $V$, $W$ are schemes, the vertical arrows are \\'etale and surjective,", "see", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}.", "At this point we can use", "Lemma \\ref{lemma-finite-type-local}", "and", "Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type}", "to conclude." ], "refs": [ "spaces-lemma-lift-morphism-presentations", "spaces-morphisms-lemma-finite-type-local", "morphisms-lemma-permanence-finite-type" ], "ref_ids": [ 8159, 4816, 5204 ] } ], "ref_ids": [] }, { "id": 4819, "type": "theorem", "label": "spaces-morphisms-lemma-immersion-locally-finite-type", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-immersion-locally-finite-type", "contents": [ "An immersion is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "Follows from the general principle", "Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}", "and", "Morphisms, Lemmas \\ref{morphisms-lemma-immersion-locally-finite-type}." ], "refs": [ "spaces-lemma-representable-transformations-property-implication", "morphisms-lemma-immersion-locally-finite-type" ], "ref_ids": [ 8136, 5201 ] } ], "ref_ids": [] }, { "id": 4820, "type": "theorem", "label": "spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is locally of finite type. The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is surjective, and", "\\item for every algebraically closed field $k$ over $S$ the induced", "map $X(k) \\to Y(k)$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$, $V$ schemes over $S$ and vertical arrows surjective", "and \\'etale, see", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}.", "Since $f$ is locally of finite type we see that $U \\to V$ is locally of", "finite type.", "\\medskip\\noindent", "Assume (1) and let $\\overline{y} \\in Y(k)$. Then $U \\to Y$ is", "surjective and locally of finite type by", "Lemmas \\ref{lemma-composition-surjective} and", "\\ref{lemma-composition-finite-type}.", "Let $Z = U \\times_{Y, \\overline{y}} \\Spec(k)$. This is a scheme.", "The projection $Z \\to \\Spec(k)$", "is surjective and locally of finite type by", "Lemmas \\ref{lemma-base-change-surjective} and", "\\ref{lemma-base-change-finite-type}.", "It follows from", "Varieties, Lemma \\ref{varieties-lemma-locally-finite-type-Jacobson}", "that $Z$ has a $k$ valued point $\\overline{z}$. The image", "$\\overline{x} \\in X(k)$ of $\\overline{z}$ maps to $\\overline{y}$", "as desired.", "\\medskip\\noindent", "Assume (2). By", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-characterize-surjective}", "it suffices to show that $|X| \\to |Y|$ is surjective.", "Let $y \\in |Y|$. Choose a $u \\in U$ mapping to $y$.", "Let $k \\supset \\kappa(u)$ be an algebraic closure. Denote", "$\\overline{u} \\in U(k)$ the corresponding point and", "$\\overline{y} \\in Y(k)$ its image. By assumption there exists", "a $\\overline{x} \\in X(k)$ mapping to $\\overline{y}$.", "Then it is clear that the image $x \\in |X|$ of $\\overline{x}$", "maps to $y$." ], "refs": [ "spaces-lemma-lift-morphism-presentations", "spaces-morphisms-lemma-composition-surjective", "spaces-morphisms-lemma-composition-finite-type", "spaces-morphisms-lemma-base-change-surjective", "spaces-morphisms-lemma-base-change-finite-type", "varieties-lemma-locally-finite-type-Jacobson", "spaces-properties-lemma-characterize-surjective" ], "ref_ids": [ 8159, 4726, 4814, 4727, 4815, 10964, 11820 ] } ], "ref_ids": [] }, { "id": 4821, "type": "theorem", "label": "spaces-morphisms-lemma-large-enough", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-large-enough", "contents": [ "Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$.", "\\begin{enumerate}", "\\item As $k$ ranges over all algebraically closed fields over $S$", "the collection of geometric points $\\overline{y} \\in Y(k)$ cover all of $|Y|$.", "\\item As $k$ ranges over all algebraically closed fields over $S$ with", "$|k| \\geq \\lambda(Y)$ and $|k| > \\lambda(X)$ the geometric points", "$\\overline{y} \\in Y(k)$ cover all of $|Y|$.", "\\item For any geometric point", "$\\overline{s} : \\Spec(k) \\to S$ where", "$k$ has cardinality $> \\lambda(X)$ the map", "$$", "X(k) \\longrightarrow |X_s|", "$$", "is surjective.", "\\item Let $X \\to Y$ be a morphism of algebraic spaces over $S$.", "For any geometric point $\\overline{s} : \\Spec(k) \\to S$ where", "$k$ has cardinality $> \\lambda(X)$ the map", "$$", "X(k) \\longrightarrow |X| \\times_{|Y|} Y(k)", "$$", "is surjective.", "\\item Let $X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item the map $X \\to Y$ is surjective,", "\\item for all algebraically closed fields $k$ over $S$ with", "$|k| > \\lambda(X)$, and $|k| \\geq \\lambda(Y)$ the map $X(k) \\to Y(k)$", "is surjective.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove part (1) choose a surjective \\'etale morphism $V \\to Y$ where", "$V$ is a scheme. For each $v \\in V$ choose an algebraic closure", "$\\kappa(v) \\subset k_v$. Consider the morphisms", "$\\overline{x} : \\Spec(k_v) \\to V \\to Y$.", "By construction of $|Y|$ these cover $|Y|$.", "\\medskip\\noindent", "To prove part (2) we will use the following two facts whose proofs we omit:", "(\\romannumeral1) If $K$ is a field and $\\overline{K}$ is algebraic closure then", "$|\\overline{K}| \\leq \\max\\{\\aleph_0, |K|\\}$.", "(\\romannumeral2) For any algebraically closed field $k$ and any cardinal", "$\\aleph$, $\\aleph \\geq |k|$ there exists an extension of algebraically", "closed fields $k \\subset k'$ with $|k'| = \\aleph$.", "Now we set $\\aleph = \\max\\{\\lambda(X), \\lambda(Y)\\}^+$.", "Here $\\lambda^+ > \\lambda$ indicates the next bigger cardinal, see", "Sets, Section \\ref{sets-section-cardinals}.", "Now (\\romannumeral1)", "implies that the fields $k_u$ constructed in the first paragraph", "of the proof all have cardinality bounded by $\\lambda(X)$. Hence", "by (\\romannumeral2) we can find extensions $k_u \\subset k'_u$ such that", "$|k'_u| = \\aleph$. The morphisms", "$\\overline{x}' : \\Spec(k'_u) \\to X$ cover $|X|$ as desired.", "To really finish the proof of (2) we need to show that the schemes", "$\\Spec(k'_u)$ are (isomorphic to) objects of $\\Sch_{fppf}$", "because our conventions are that all schemes are objects of", "$\\Sch_{fppf}$; the rest of this paragraph should be skipped by", "anyone who is not interested in set theoretical considerations. By", "construction there exists an object $T$ of $\\Sch_{fppf}$ such that", "$\\lambda(X)$ and $\\lambda(Y)$ are bounded by $\\text{size}(T)$.", "By our construction of the category $\\Sch_{fppf}$ in", "Topologies, Definitions \\ref{topologies-definition-big-fppf-site}", "as the category $\\Sch_\\alpha$ constructed in", "Sets, Lemma \\ref{sets-lemma-construct-category}", "we see that any scheme whose size is $\\leq \\text{size}(T)^+$", "is isomorphic to an object of $\\Sch_{fppf}$. See", "the expression for the function $Bound$ in", "Sets, Equation (\\ref{sets-equation-bound}).", "Since $\\aleph \\leq \\text{size}(T)^+$ we conclude.", "\\medskip\\noindent", "The notation $X_s$ in part (3) means", "the fibre product $\\Spec(\\kappa(s)) \\times_S X$, where $s \\in S$", "is the point corresponding to $\\overline{s}$. Hence part (2) follows", "from (4) with $Y = \\Spec(\\kappa(s))$.", "\\medskip\\noindent", "Let us prove (4). Let $X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $k$ be an algebraically closed field over $S$ of cardinality", "$> \\lambda(X)$. Let $\\overline{y} \\in Y(k)$ and $x \\in |X|$ which map to the", "same element $y$ of $|Y|$.", "We have to find $\\overline{x} \\in X(k)$ mapping to $x$ and $\\overline{y}$.", "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$, $V$ schemes over $S$ and vertical arrows surjective", "and \\'etale, see", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}.", "Choose a $u \\in |U|$ which maps to $x$, and denote $v \\in |V|$ the image.", "We will think of $u = \\Spec(\\kappa(u))$ and", "$v = \\Spec(\\kappa(v))$ as schemes.", "Note that $V \\times_Y \\Spec(k)$ is a scheme \\'etale over $k$.", "Hence it is a disjoint union of spectra of finite separable", "extensions of $k$, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}.", "As $v$ maps to $y$ we see that $v \\times_Y \\Spec(k)$ is a", "nonempty scheme. As $v \\to V$ is a monomorphism, we see that", "$v \\times_Y \\Spec(k) \\to V \\times_Y \\Spec(k)$", "is a monomorphism. Hence $v \\times_Y \\Spec(k)$ is a disjoint", "union of spectra of finite separable extensions of $k$, by", "Schemes, Lemma \\ref{schemes-lemma-mono-towards-spec-field}.", "We conclude that the morphism", "$v \\times_Y \\Spec(k) \\to \\Spec(k)$", "has a section, i.e., we can find a morphism", "$\\overline{v} : \\Spec(k) \\to V$ lying over $v$ and over", "$\\overline{y}$. Finally we consider the scheme", "$$", "u \\times_{V, \\overline{v}} \\Spec(k)", "=", "\\Spec(\\kappa(u) \\otimes_{\\kappa(v)} k)", "$$", "where $\\kappa(v) \\to k$ is the field map defining the morphism", "$\\overline{v}$. Since the cardinality of $k$ is larger than the cardinality", "of $\\kappa(u)$ by assumption we may apply", "Algebra, Lemma \\ref{algebra-lemma-base-change-Jacobson}", "to see that any maximal ideal", "$\\mathfrak m \\subset \\kappa(u) \\otimes_{\\kappa(v)} k$", "has a residue field which is algebraic over $k$ and hence equal to $k$.", "Such a maximal ideal will hence produce a morphism", "$\\overline{u} : \\Spec(k) \\to U$ lying over $u$ and mapping to", "$\\overline{v}$. The composition $\\Spec(k) \\to U \\to X$", "will be the desired geometric point $\\overline{x} \\in X(k)$.", "This concludes the proof of part (4).", "\\medskip\\noindent", "Part (5) is a formal consequence of parts (2) and (4) and", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-characterize-surjective}." ], "refs": [ "topologies-definition-big-fppf-site", "sets-lemma-construct-category", "spaces-lemma-lift-morphism-presentations", "morphisms-lemma-etale-over-field", "schemes-lemma-mono-towards-spec-field", "algebra-lemma-base-change-Jacobson", "spaces-properties-lemma-characterize-surjective" ], "ref_ids": [ 12541, 8789, 8159, 5364, 7729, 474, 11820 ] } ], "ref_ids": [] }, { "id": 4822, "type": "theorem", "label": "spaces-morphisms-lemma-point-finite-type", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-point-finite-type", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$. The following are equivalent:", "\\begin{enumerate}", "\\item There exists a morphism $\\Spec(k) \\to X$", "which is locally of finite type and represents $x$.", "\\item There exists a scheme $U$, a closed point $u \\in U$, and an \\'etale", "morphism $\\varphi : U \\to X$ such that $\\varphi(u) = x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $u \\in U$ and $U \\to X$ be as in (2). Then", "$\\Spec(\\kappa(u)) \\to U$ is of finite type, and $U \\to X$ is", "representable and locally of finite type (by the general principle", "Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}", "and", "Morphisms, Lemmas \\ref{morphisms-lemma-etale-locally-finite-presentation} and", "\\ref{morphisms-lemma-finite-presentation-finite-type}). Hence we see", "(1) holds by", "Lemma \\ref{lemma-composition-finite-type}.", "\\medskip\\noindent", "Conversely, assume $\\Spec(k) \\to X$ is locally of finite type and", "represents $x$. Let $U \\to X$ be a surjective \\'etale morphism where $U$", "is a scheme. By assumption $U \\times_X \\Spec(k) \\to U$ is locally", "of finite type. Pick a finite type point $v$ of", "$U \\times_X \\Spec(k)$ (there exists at least one, see", "Morphisms, Lemma \\ref{morphisms-lemma-identify-finite-type-points}).", "By", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-points-morphism}", "the image $u \\in U$ of $v$ is a finite type point of $U$.", "Hence by", "Morphisms, Lemma \\ref{morphisms-lemma-identify-finite-type-points}", "after shrinking $U$ we may assume that $u$ is a closed point of $U$, i.e.,", "(2) holds." ], "refs": [ "spaces-lemma-representable-transformations-property-implication", "morphisms-lemma-etale-locally-finite-presentation", "morphisms-lemma-finite-presentation-finite-type", "spaces-morphisms-lemma-composition-finite-type", "morphisms-lemma-identify-finite-type-points", "morphisms-lemma-finite-type-points-morphism", "morphisms-lemma-identify-finite-type-points" ], "ref_ids": [ 8136, 5368, 5244, 4814, 5207, 5208, 5207 ] } ], "ref_ids": [] }, { "id": 4823, "type": "theorem", "label": "spaces-morphisms-lemma-identify-finite-type-points", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-identify-finite-type-points", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We have", "$$", "X_{\\text{ft-pts}} =", "\\bigcup\\nolimits_{\\varphi : U \\to X\\text{ \\'etale }} |\\varphi|(U_0)", "$$", "where $U_0$ is the set of closed points of $U$.", "Here we may let $U$ range over all schemes \\'etale over $X$ or over all", "affine schemes \\'etale over $X$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from", "Lemma \\ref{lemma-point-finite-type}." ], "refs": [ "spaces-morphisms-lemma-point-finite-type" ], "ref_ids": [ 4822 ] } ], "ref_ids": [] }, { "id": 4824, "type": "theorem", "label": "spaces-morphisms-lemma-finite-type-points-morphism", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-type-points-morphism", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $f$ is locally of finite type, then", "$f(X_{\\text{ft-pts}}) \\subset Y_{\\text{ft-pts}}$." ], "refs": [], "proofs": [ { "contents": [ "Take $x \\in X_{\\text{ft-pts}}$. Represent $x$ by a locally finite type morphism", "$x : \\Spec(k) \\to X$. Then $f \\circ x$ is locally of finite type by", "Lemma \\ref{lemma-composition-finite-type}.", "Hence $f(x) \\in Y_{\\text{ft-pts}}$." ], "refs": [ "spaces-morphisms-lemma-composition-finite-type" ], "ref_ids": [ 4814 ] } ], "ref_ids": [] }, { "id": 4825, "type": "theorem", "label": "spaces-morphisms-lemma-finite-type-points-surjective-morphism", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-type-points-surjective-morphism", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $f$ is locally of finite type and surjective, then", "$f(X_{\\text{ft-pts}}) = Y_{\\text{ft-pts}}$." ], "refs": [], "proofs": [ { "contents": [ "We have $f(X_{\\text{ft-pts}}) \\subset Y_{\\text{ft-pts}}$ by", "Lemma \\ref{lemma-finite-type-points-morphism}.", "Let $y \\in |Y|$ be a finite type point. Represent $y$ by a morphism", "$\\Spec(k) \\to Y$ which is locally of finite type.", "As $f$ is surjective the algebraic space", "$X_k = \\Spec(k) \\times_Y X$ is nonempty, therefore", "has a finite type point $x \\in |X_k|$ by", "Lemma \\ref{lemma-identify-finite-type-points}.", "Now $X_k \\to X$ is a morphism which is locally of finite type as a base change", "of $\\Spec(k) \\to Y$", "(Lemma \\ref{lemma-base-change-finite-type}).", "Hence the image of $x$ in $X$ is a finite type point by", "Lemma \\ref{lemma-finite-type-points-morphism}", "which maps to $y$ by construction." ], "refs": [ "spaces-morphisms-lemma-finite-type-points-morphism", "spaces-morphisms-lemma-identify-finite-type-points", "spaces-morphisms-lemma-base-change-finite-type", "spaces-morphisms-lemma-finite-type-points-morphism" ], "ref_ids": [ 4824, 4823, 4815, 4824 ] } ], "ref_ids": [] }, { "id": 4826, "type": "theorem", "label": "spaces-morphisms-lemma-enough-finite-type-points", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-enough-finite-type-points", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "For any locally closed subset $T \\subset |X|$ we have", "$$", "T \\not = \\emptyset", "\\Rightarrow", "T \\cap X_{\\text{ft-pts}} \\not = \\emptyset.", "$$", "In particular, for any closed subset $T \\subset |X|$ we", "see that $T \\cap X_{\\text{ft-pts}}$ is dense in $T$." ], "refs": [], "proofs": [ { "contents": [ "Let $i : Z \\to X$ be the reduced induce subspace structure on $T$, see", "Remark \\ref{remark-space-structure-locally-closed-subset}.", "Any immersion is locally of finite type, see", "Lemma \\ref{lemma-immersion-locally-finite-type}.", "Hence by", "Lemma \\ref{lemma-finite-type-points-morphism}", "we see $Z_{\\text{ft-pts}} \\subset X_{\\text{ft-pts}} \\cap T$.", "Finally, any nonempty affine scheme $U$ with an \\'etale morphism towards", "$Z$ has at least one closed point. Hence $Z$ has at least one", "finite type point by", "Lemma \\ref{lemma-identify-finite-type-points}.", "The lemma follows." ], "refs": [ "spaces-morphisms-remark-space-structure-locally-closed-subset", "spaces-morphisms-lemma-immersion-locally-finite-type", "spaces-morphisms-lemma-finite-type-points-morphism", "spaces-morphisms-lemma-identify-finite-type-points" ], "ref_ids": [ 5030, 4819, 4824, 4823 ] } ], "ref_ids": [] }, { "id": 4827, "type": "theorem", "label": "spaces-morphisms-lemma-point-finite-type-monomorphism", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-point-finite-type-monomorphism", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$. The following are equivalent:", "\\begin{enumerate}", "\\item $x$ is a finite type point,", "\\item there exists an algebraic space $Z$ whose underlying topological space", "$|Z|$ is a singleton, and a morphism $f : Z \\to X$ which is locally of finite", "type such that $\\{x\\} = |f|(|Z|)$, and", "\\item there exists an algebraic space $Z$ and a morphism $f : Z \\to X$", "with the following properties:", "\\begin{enumerate}", "\\item there is a surjective \\'etale morphism", "$z : \\Spec(k) \\to Z$ where $k$ is a field,", "\\item $f$ is locally of finite type,", "\\item $f$ is a monomorphism, and", "\\item $x = f(z)$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $x$ is a finite type point. Choose an affine scheme $U$,", "a closed point $u \\in U$, and an \\'etale morphism $\\varphi : U \\to X$", "with $\\varphi(u) = x$, see", "Lemma \\ref{lemma-identify-finite-type-points}.", "Set $u = \\Spec(\\kappa(u))$ as usual. The projection morphisms", "$u \\times_X u \\to u$ are the compositions", "$$", "u \\times_X u \\to u \\times_X U \\to u \\times_X X = u", "$$", "where the first arrow is a closed immersion (a base change of", "$u \\to U$) and the second arrow is \\'etale (a base change of the \\'etale", "morphism $U \\to X$). Hence $u \\times_X U$ is a disjoint union of spectra", "of finite separable extensions of $k$ (see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field})", "and therefore the closed subscheme $u \\times_X u$ is a disjoint union of", "finite separable extension of $k$, i.e., $u \\times_X u \\to u$ is \\'etale. By", "Spaces, Theorem \\ref{spaces-theorem-presentation}", "we see that $Z = u/u \\times_X u$ is an algebraic space. By construction", "the diagram", "$$", "\\xymatrix{", "u \\ar[d] \\ar[r] & U \\ar[d] \\\\", "Z \\ar[r] & X", "}", "$$", "is commutative with \\'etale vertical arrows. Hence $Z \\to X$ is locally", "of finite type (see", "Lemma \\ref{lemma-finite-type-local}).", "By construction the morphism $Z \\to X$ is a monomorphism and", "the image of $z$ is $x$. Thus (3) holds.", "\\medskip\\noindent", "It is clear that (3) implies (2).", "If (2) holds then $x$ is a finite type point of $X$ by", "Lemma \\ref{lemma-finite-type-points-morphism}", "(and", "Lemma \\ref{lemma-enough-finite-type-points}", "to see that $Z_{\\text{ft-pts}}$ is nonempty, i.e., the unique point of", "$Z$ is a finite type point of $Z$)." ], "refs": [ "spaces-morphisms-lemma-identify-finite-type-points", "morphisms-lemma-etale-over-field", "spaces-theorem-presentation", "spaces-morphisms-lemma-finite-type-local", "spaces-morphisms-lemma-finite-type-points-morphism", "spaces-morphisms-lemma-enough-finite-type-points" ], "ref_ids": [ 4823, 5364, 8124, 4816, 4824, 4826 ] } ], "ref_ids": [] }, { "id": 4828, "type": "theorem", "label": "spaces-morphisms-lemma-finite-type-nagata", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-type-nagata", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $Y$ is Nagata and $f$ locally of finite type then $X$ is Nagata." ], "refs": [], "proofs": [ { "contents": [ "Let $V$ be a scheme and let $V \\to Y$ be a surjective \\'etale morphism.", "Let $U$ be a scheme and let $U \\to X \\times_Y V$ be a surjective \\'etale", "morphism. If $Y$ is Nagata, then $V$ is a Nagata scheme.", "If $X \\to Y$ is locally of finite type, then $U \\to V$ is locally", "of finite type. Hence $V$ is a Nagata scheme by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-nagata}.", "Then $X$ is Nagata by definition." ], "refs": [ "morphisms-lemma-finite-type-nagata" ], "ref_ids": [ 5218 ] } ], "ref_ids": [] }, { "id": 4829, "type": "theorem", "label": "spaces-morphisms-lemma-ubiquity-nagata", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-ubiquity-nagata", "contents": [ "The following types of algebraic spaces are Nagata.", "\\begin{enumerate}", "\\item Any algebraic space locally of finite type over a Nagata scheme.", "\\item Any algebraic space locally of finite type over a field.", "\\item Any algebraic space locally of finite type over a", "Noetherian complete local ring.", "\\item Any algebraic space locally of finite type over $\\mathbf{Z}$.", "\\item Any algebraic space locally of finite type over a Dedekind ring of", "characteristic zero.", "\\item And so on.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first property holds by Lemma \\ref{lemma-finite-type-nagata}.", "Thus the others hold as well, see", "Morphisms, Lemma \\ref{morphisms-lemma-ubiquity-nagata}." ], "refs": [ "spaces-morphisms-lemma-finite-type-nagata", "morphisms-lemma-ubiquity-nagata" ], "ref_ids": [ 4828, 5219 ] } ], "ref_ids": [] }, { "id": 4830, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-quasi-finite-locus", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-quasi-finite-locus", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y' \\to Y$ be morphisms", "of algebraic spaces over $S$. Denote $f' : X' \\to Y'$ the base change", "of $f$ by $g$. Denote $g' : X' \\to X$ the projection.", "Assume $f$ is locally of finite type.", "Let $W \\subset |X|$, resp.\\ $W' \\subset |X'|$", "be the set of points where $f$, resp.\\ $f'$ is quasi-finite.", "\\begin{enumerate}", "\\item $W \\subset |X|$ and $W' \\subset |X'|$ are open,", "\\item $W' = (g')^{-1}(W)$, i.e., formation of the locus where", "$f$ is quasi-finite commutes with base change,", "\\item the base change of a locally quasi-finite morphism is", "locally quasi-finite, and", "\\item the base change of a quasi-finite morphism is quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to V \\times_Y X$.", "Choose a scheme $V'$ and a surjective \\'etale morphism $V' \\to Y' \\times_Y V$.", "Set $U' = V' \\times_V U$ so that $U' \\to X'$ is a surjective \\'etale", "morphism as well. Picture", "$$", "\\vcenter{", "\\xymatrix{", "U' \\ar[d] \\ar[r] & U \\ar[d] \\\\", "V' \\ar[r] & V", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "X' \\ar[d] \\ar[r] & X \\ar[d] \\\\", "Y' \\ar[r] & Y", "}", "}", "$$", "Choose $u \\in |U|$ with image $x \\in |X|$.", "The property of being \"locally quasi-finite\" is \\'etale local on", "the source-and-target, see", "Descent, Remark \\ref{descent-remark-list-local-source-target}.", "Hence Lemmas \\ref{lemma-local-source-target-at-point} and", "\\ref{lemma-local-source-target-global-implies-local} apply and", "we see that $f : X \\to Y$ is quasi-finite at $x$", "if and only if $U \\to V$ is quasi-finite at $u$.", "Similarly for $f' : X' \\to Y'$ and the morphism $U' \\to V'$.", "Hence parts (1), (2), and (3) reduce to", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-quasi-finite} and", "\\ref{morphisms-lemma-quasi-finite-points-open}.", "Part (4) follows from (3) and Lemma \\ref{lemma-base-change-quasi-compact}." ], "refs": [ "descent-remark-list-local-source-target", "spaces-morphisms-lemma-local-source-target-at-point", "spaces-morphisms-lemma-local-source-target-global-implies-local", "morphisms-lemma-base-change-quasi-finite", "morphisms-lemma-quasi-finite-points-open", "spaces-morphisms-lemma-base-change-quasi-compact" ], "ref_ids": [ 14798, 4812, 4813, 5233, 5521, 4738 ] } ], "ref_ids": [] }, { "id": 4831, "type": "theorem", "label": "spaces-morphisms-lemma-composition-quasi-finite", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-quasi-finite", "contents": [ "The composition of quasi-finite morphisms is quasi-finite.", "The same holds for locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "See Remark \\ref{remark-composition-P} and", "Morphisms, Lemma \\ref{morphisms-lemma-composition-quasi-finite}." ], "refs": [ "spaces-morphisms-remark-composition-P", "morphisms-lemma-composition-quasi-finite" ], "ref_ids": [ 5033, 5232 ] } ], "ref_ids": [] }, { "id": 4832, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-quasi-finite", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-quasi-finite", "contents": [ "A base change of a quasi-finite morphism is quasi-finite.", "The same holds for locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of Lemma \\ref{lemma-base-change-quasi-finite-locus}." ], "refs": [ "spaces-morphisms-lemma-base-change-quasi-finite-locus" ], "ref_ids": [ 4830 ] } ], "ref_ids": [] }, { "id": 4833, "type": "theorem", "label": "spaces-morphisms-lemma-locally-quasi-finite", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-locally-quasi-finite", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces.", "Assume $f$ is locally of finite type. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is locally quasi-finite,", "\\item for every morphism $\\Spec(k) \\to Y$ where $k$ is a field", "the space $|X_k|$ is discrete. Here $X_k = \\Spec(k) \\times_Y X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f$ is locally quasi-finite. Let $\\Spec(k) \\to Y$ be as", "in (2). Choose a surjective \\'etale morphism $U \\to X$ where $U$ is a scheme.", "Then $U_k = \\Spec(k) \\times_Y U \\to X_k$ is an", "\\'etale morphism of algebraic spaces by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-base-change-etale}.", "By", "Lemma \\ref{lemma-base-change-quasi-finite}", "we see that $X_k \\to \\Spec(k)$ is locally quasi-finite. By", "definition this means that $U_k \\to \\Spec(k)$ is locally", "quasi-finite. Hence $|U_k|$ is discrete by", "Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-fibres}.", "Since $|U_k| \\to |X_k|$ is surjective and open we conclude that $|X_k|$", "is discrete.", "\\medskip\\noindent", "Conversely, assume (2). Choose a surjective \\'etale morphism $V \\to Y$", "where $V$ is a scheme. Choose a surjective \\'etale morphism", "$U \\to V \\times_Y X$ where $U$ is a scheme.", "Note that $U \\to V$ is locally of finite type as $f$ is locally of finite type.", "Picture", "$$", "\\xymatrix{", "U \\ar[r] \\ar[rd] & X \\times_Y V \\ar[d] \\ar[r] & V \\ar[d] \\\\", "& X \\ar[r] & Y", "}", "$$", "If $f$ is not locally quasi-finite then $U \\to V$ is not locally quasi-finite.", "Hence there exists a specialization $u \\leadsto u'$ for some $u, u' \\in U$", "lying over the same point $v \\in V$, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}.", "We claim that $u, u'$ do not have the same image in", "$X_v = \\Spec(\\kappa(v)) \\times_Y X$", "which will contradict the assumption that $|X_v|$ is discrete as desired.", "Let $d = \\text{trdeg}_{\\kappa(v)}(\\kappa(u))$ and", "$d' = \\text{trdeg}_{\\kappa(v)}(\\kappa(u'))$.", "Then we see that $d > d'$ by", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-specialization}.", "Note that $U_v$ (the fibre of $U \\to V$ over $v$) is the fibre product of", "$U$ and $X_v$ over $X \\times_Y V$, hence $U_v \\to X_v$ is \\'etale (as a", "base change of the \\'etale morphism $U \\to X \\times_Y V$). If", "$u, u' \\in U_v$ map to the same element of $|X_v|$ then there exists a point", "$r \\in R_v = U_v \\times_{X_v} U_v$ with $t(r) = u$ and $s(r) = u'$, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}.", "Note that $s, t : R_v \\to U_v$ are \\'etale morphisms of schemes", "over $\\kappa(v)$, hence $\\kappa(u) \\subset \\kappa(r) \\supset \\kappa(u')$", "are finite separable extensions of fields over $\\kappa(v)$ (see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}).", "We conclude that the transcendence degrees are equal.", "This contradiction finishes the proof." ], "refs": [ "spaces-properties-lemma-base-change-etale", "spaces-morphisms-lemma-base-change-quasi-finite", "morphisms-lemma-locally-quasi-finite-fibres", "morphisms-lemma-quasi-finite-at-point-characterize", "morphisms-lemma-dimension-fibre-specialization", "spaces-properties-lemma-points-cartesian", "morphisms-lemma-etale-over-field" ], "ref_ids": [ 11858, 4832, 5228, 5226, 5283, 11819, 5364 ] } ], "ref_ids": [] }, { "id": 4834, "type": "theorem", "label": "spaces-morphisms-lemma-quasi-finite-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-quasi-finite-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is locally quasi-finite,", "\\item for every $x \\in |X|$ the morphism $f$ is quasi-finite at $x$,", "\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism", "$Z \\times_Y X \\to Z$ is locally quasi-finite,", "\\item for every affine scheme $Z$ and any morphism", "$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is locally quasi-finite,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is locally quasi-finite,", "\\item there exists a scheme $U$ and a surjective \\'etale morphism", "$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$", "is locally quasi-finite,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes and the vertical arrows are \\'etale", "the top horizontal arrow is locally quasi-finite,", "\\item there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes, the vertical arrows are \\'etale, and", "$U \\to X$ is surjective such that the top horizontal arrow is", "locally quasi-finite, and", "\\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$,", "and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that", "each morphism $X_{ij} \\to Y_i$ is locally quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4835, "type": "theorem", "label": "spaces-morphisms-lemma-immersion-quasi-finite", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-immersion-quasi-finite", "contents": [ "An immersion is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4836, "type": "theorem", "label": "spaces-morphisms-lemma-permanence-quasi-finite", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-permanence-quasi-finite", "contents": [ "Let $S$ be a scheme.", "Let $X \\to Y \\to Z$ be morphisms of algebraic spaces over $S$.", "If $X \\to Z$ is locally quasi-finite, then $X \\to Y$", "is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\ar[r] & W \\ar[d] \\\\", "X \\ar[r] & Y \\ar[r] & Z", "}", "$$", "with vertical arrows \\'etale and surjective. (See", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}.)", "Apply", "Morphisms, Lemma \\ref{morphisms-lemma-permanence-quasi-finite}", "to the top row." ], "refs": [ "spaces-lemma-lift-morphism-presentations", "morphisms-lemma-permanence-quasi-finite" ], "ref_ids": [ 8159, 5237 ] } ], "ref_ids": [] }, { "id": 4837, "type": "theorem", "label": "spaces-morphisms-lemma-quasi-finite-at-a-finite-number-of-points", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-quasi-finite-at-a-finite-number-of-points", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a finite type", "morphism of algebraic spaces over $S$. Let $y \\in |Y|$.", "There are at most finitely many", "points of $|X|$ lying over $y$ at which $f$ is quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Choose a field $k$ and a morphism $\\Spec(k) \\to Y$ in the equivalence", "class determined by $y$. The fibre $X_k = \\Spec(k) \\times_Y X$ is an", "algebraic space of finite type over a field, in particular quasi-compact.", "The map $|X_k| \\to |X|$ surjects onto the fibre of $|X| \\to |Y|$", "over $y$ (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-points-cartesian}).", "Moreover, the set of points where $X_k \\to \\Spec(k)$ is", "quasi-finite maps onto the set of points lying over $y$ where", "$f$ is quasi-finite by Lemma \\ref{lemma-base-change-quasi-finite-locus}.", "Choose an affine scheme $U$ and a surjective \\'etale morphism $U \\to X_k$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "Then $U \\to \\Spec(k)$ is a morphism of finite type and there are at", "most a finite number of points where this morphism is quasi-finite,", "see Morphisms, Lemma", "\\ref{morphisms-lemma-quasi-finite-at-a-finite-number-of-points}.", "Since $X_k \\to \\Spec(k)$ is quasi-finite at a point $x'$ if and only", "if it is the image of a point of $U$ where $U \\to \\Spec(k)$ is", "quasi-finite, we conclude." ], "refs": [ "spaces-properties-lemma-points-cartesian", "spaces-morphisms-lemma-base-change-quasi-finite-locus", "spaces-properties-lemma-quasi-compact-affine-cover", "morphisms-lemma-quasi-finite-at-a-finite-number-of-points" ], "ref_ids": [ 11819, 4830, 11832, 5234 ] } ], "ref_ids": [] }, { "id": 4838, "type": "theorem", "label": "spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $f$ is locally of finite type and a monomorphism, then $f$", "is separated and locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "A monomorphism is separated, see", "Lemma \\ref{lemma-monomorphism-separated}.", "By", "Lemma \\ref{lemma-quasi-finite-local}", "it suffices to prove the lemma after performing a base change", "by $Z \\to Y$ with $Z$ affine. Hence we may assume that $Y$ is an", "affine scheme. Choose an affine scheme $U$ and an \\'etale morphism", "$U \\to X$. Since $X \\to Y$ is locally of finite type the morphism", "of affine schemes $U \\to Y$ is of finite type.", "Since $X \\to Y$ is a monomorphism we have $U \\times_X U = U \\times_Y U$.", "In particular the maps $U \\times_Y U \\to U$ are \\'etale.", "Let $y \\in Y$. Then either $U_y$ is empty, or", "$\\Spec(\\kappa(u)) \\times_{\\Spec(\\kappa(y))} U_y$", "is isomorphic to the fibre of $U \\times_Y U \\to U$ over $u$ for", "some $u \\in U$ lying over $y$. This implies that the fibres of", "$U \\to Y$ are finite discrete sets (as $U \\times_Y U \\to U$", "is an \\'etale morphism of affine schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}).", "Hence $U \\to Y$ is quasi-finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}.", "As $U \\to X$ was an arbitrary \\'etale morphism with $U$ affine", "this implies that $X \\to Y$ is locally quasi-finite." ], "refs": [ "spaces-morphisms-lemma-monomorphism-separated", "spaces-morphisms-lemma-quasi-finite-local", "morphisms-lemma-etale-over-field", "morphisms-lemma-quasi-finite-at-point-characterize" ], "ref_ids": [ 4752, 4834, 5364, 5226 ] } ], "ref_ids": [] }, { "id": 4839, "type": "theorem", "label": "spaces-morphisms-lemma-composition-finite-presentation", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-finite-presentation", "contents": [ "The composition of morphisms of finite presentation is of finite presentation.", "The same holds for locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "See Remark \\ref{remark-composition-P} and", "Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-presentation}.", "Also use the result for quasi-compact and for quasi-separated morphisms", "(Lemmas \\ref{lemma-composition-quasi-compact} and", "\\ref{lemma-composition-separated})." ], "refs": [ "spaces-morphisms-remark-composition-P", "morphisms-lemma-composition-finite-presentation", "spaces-morphisms-lemma-composition-quasi-compact", "spaces-morphisms-lemma-composition-separated" ], "ref_ids": [ 5033, 5239, 4739, 4718 ] } ], "ref_ids": [] }, { "id": 4840, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-finite-presentation", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-finite-presentation", "contents": [ "A base change of a morphism of finite presentation is of finite presentation", "The same holds for locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "See Remark \\ref{remark-base-change-P} and", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-presentation}.", "Also use the result for quasi-compact and for quasi-separated morphisms", "(Lemmas \\ref{lemma-base-change-quasi-compact} and", "\\ref{lemma-base-change-separated})." ], "refs": [ "spaces-morphisms-remark-base-change-P", "morphisms-lemma-base-change-finite-presentation", "spaces-morphisms-lemma-base-change-quasi-compact", "spaces-morphisms-lemma-base-change-separated" ], "ref_ids": [ 5034, 5240, 4738, 4714 ] } ], "ref_ids": [] }, { "id": 4841, "type": "theorem", "label": "spaces-morphisms-lemma-finite-presentation-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-presentation-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item for every $x \\in |X|$ the morphism $f$ is of finite presentation at $x$,", "\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism", "$Z \\times_Y X \\to Z$ is locally of finite presentation,", "\\item for every affine scheme $Z$ and any morphism", "$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is locally of finite presentation,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is", "locally of finite presentation,", "\\item there exists a scheme $U$ and a surjective \\'etale morphism", "$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$", "is locally of finite presentation,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes and the vertical arrows are \\'etale", "the top horizontal arrow is locally of finite presentation,", "\\item there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes, the vertical arrows are \\'etale, and", "$U \\to X$ is surjective such that the top horizontal arrow is", "locally of finite presentation, and", "\\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$,", "and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that", "each morphism $X_{ij} \\to Y_i$ is locally of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4842, "type": "theorem", "label": "spaces-morphisms-lemma-finite-presentation-finite-type", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-presentation-finite-type", "contents": [ "A morphism which is locally of finite presentation is locally of finite type.", "A morphism of finite presentation is of finite type." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a morphism of algebraic spaces which is locally of", "finite presentation. This means there exists a diagram as in", "Lemma \\ref{lemma-local-source-target}", "with $h$ locally of finite presentation and surjective vertical arrow $a$. By", "Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-finite-type}", "$h$ is locally of finite type.", "Hence $X \\to Y$ is locally of finite type by definition.", "If $f$ is of finite presentation then it is quasi-compact and", "it follows that $f$ is of finite type." ], "refs": [ "spaces-morphisms-lemma-local-source-target", "morphisms-lemma-finite-presentation-finite-type" ], "ref_ids": [ 4811, 5244 ] } ], "ref_ids": [] }, { "id": 4843, "type": "theorem", "label": "spaces-morphisms-lemma-finite-presentation-noetherian", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-presentation-noetherian", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $f$ is of finite presentation and $Y$ is Noetherian,", "then $X$ is Noetherian." ], "refs": [], "proofs": [ { "contents": [ "Assume $f$ is of finite presentation and $Y$ Noetherian. By", "Lemmas \\ref{lemma-finite-presentation-finite-type} and", "\\ref{lemma-locally-finite-type-locally-noetherian}", "we see that $X$ is locally Noetherian. As $f$ is quasi-compact", "and $Y$ is quasi-compact we see that $X$ is quasi-compact.", "As $f$ is of finite presentation it is quasi-separated (see", "Definition \\ref{definition-locally-finite-presentation})", "and as $Y$ is Noetherian it is quasi-separated (see", "Properties of Spaces,", "Definition \\ref{spaces-properties-definition-noetherian}).", "Hence $X$ is quasi-separated by", "Lemma \\ref{lemma-separated-over-separated}.", "Hence we have checked all three conditions of", "Properties of Spaces,", "Definition \\ref{spaces-properties-definition-noetherian}", "and we win." ], "refs": [ "spaces-morphisms-lemma-finite-presentation-finite-type", "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "spaces-morphisms-definition-locally-finite-presentation", "spaces-properties-definition-noetherian", "spaces-morphisms-lemma-separated-over-separated", "spaces-properties-definition-noetherian" ], "ref_ids": [ 4842, 4817, 5006, 11946, 4719, 11946 ] } ], "ref_ids": [] }, { "id": 4844, "type": "theorem", "label": "spaces-morphisms-lemma-noetherian-finite-type-finite-presentation", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-noetherian-finite-type-finite-presentation", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item If $Y$ is locally Noetherian and $f$ locally of finite type", "then $f$ is locally of finite presentation.", "\\item If $Y$ is locally Noetherian and $f$ of finite type and quasi-separated", "then $f$ is of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f : X \\to Y$ locally of finite type and $Y$ locally Noetherian.", "This means there exists a diagram as in", "Lemma \\ref{lemma-local-source-target}", "with $h$ locally of finite type and surjective vertical arrow $a$. By", "Morphisms, Lemma", "\\ref{morphisms-lemma-noetherian-finite-type-finite-presentation}", "$h$ is locally of finite presentation.", "Hence $X \\to Y$ is locally of finite presentation by definition.", "This proves (1).", "If $f$ is of finite type and quasi-separated then it is also", "quasi-compact and quasi-separated and (2) follows immediately." ], "refs": [ "spaces-morphisms-lemma-local-source-target", "morphisms-lemma-noetherian-finite-type-finite-presentation" ], "ref_ids": [ 4811, 5245 ] } ], "ref_ids": [] }, { "id": 4845, "type": "theorem", "label": "spaces-morphisms-lemma-finite-presentation-quasi-compact-quasi-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-presentation-quasi-compact-quasi-separated", "contents": [ "Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$ which is", "quasi-compact and quasi-separated. If $X$ is of finite presentation over", "$Y$, then $X$ is quasi-compact and quasi-separated." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4846, "type": "theorem", "label": "spaces-morphisms-lemma-finite-presentation-permanence", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-presentation-permanence", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ and $Y \\to Z$ be morphisms of algebraic spaces over $S$.", "If $X$ is locally of finite presentation over $Z$, and", "$Y$ is locally of finite type over $Z$, then $f$ is locally", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $W$ and a surjective \\'etale morphism $W \\to Z$.", "Then choose a scheme $V$ and a surjective \\'etale morphism $V \\to W \\times_Z Y$.", "Finally choose a scheme $U$ and a surjective \\'etale morphism", "$U \\to V \\times_Y X$. By definition $U$ is locally of finite presentation", "over $W$ and $V$ is locally of finite type over $W$. By", "Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-permanence}", "the morphism $U \\to V$ is locally of finite presentation.", "Hence $f$ is locally of finite presentation." ], "refs": [ "morphisms-lemma-finite-presentation-permanence" ], "ref_ids": [ 5247 ] } ], "ref_ids": [] }, { "id": 4847, "type": "theorem", "label": "spaces-morphisms-lemma-diagonal-morphism-finite-type", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-diagonal-morphism-finite-type", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ with diagonal $\\Delta : X \\to X \\times_Y X$. If $f$ is locally of", "finite type then $\\Delta$ is locally of finite presentation. If $f$ is", "quasi-separated and locally of finite type, then $\\Delta$ is of finite", "presentation." ], "refs": [], "proofs": [ { "contents": [ "Note that $\\Delta$ is a morphism over $X$ (via the second", "projection $X \\times_Y X \\to X$). Assume $f$ is locally of finite type.", "Note that $X$ is of finite presentation over $X$ and $X \\times_Y X$ is", "of finite type over $X$ (by Lemma \\ref{lemma-base-change-finite-type}).", "Thus the first statement holds by", "Lemma \\ref{lemma-finite-presentation-permanence}.", "The second statement follows from the first, the definitions, and", "the fact that a diagonal morphism is separated", "(Lemma \\ref{lemma-properties-diagonal})." ], "refs": [ "spaces-morphisms-lemma-base-change-finite-type", "spaces-morphisms-lemma-finite-presentation-permanence", "spaces-morphisms-lemma-properties-diagonal" ], "ref_ids": [ 4815, 4846, 4712 ] } ], "ref_ids": [] }, { "id": 4848, "type": "theorem", "label": "spaces-morphisms-lemma-open-immersion-locally-finite-presentation", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-open-immersion-locally-finite-presentation", "contents": [ "An open immersion of algebraic spaces is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "An open immersion is by definition representable, hence we can", "use the general principle", "Spaces,", "Lemma \\ref{spaces-lemma-representable-transformations-property-implication}", "and", "Morphisms,", "Lemma \\ref{morphisms-lemma-open-immersion-locally-finite-presentation}." ], "refs": [ "spaces-lemma-representable-transformations-property-implication", "morphisms-lemma-open-immersion-locally-finite-presentation" ], "ref_ids": [ 8136, 5241 ] } ], "ref_ids": [] }, { "id": 4849, "type": "theorem", "label": "spaces-morphisms-lemma-closed-immersion-finite-presentation", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-closed-immersion-finite-presentation", "contents": [ "A closed immersion $i : Z \\to X$ is of finite presentation if and only if", "the associated quasi-coherent sheaf of ideals", "$\\mathcal{I} = \\Ker(\\mathcal{O}_X \\to i_*\\mathcal{O}_Z)$", "is of finite type (as an $\\mathcal{O}_X$-module)." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $U \\to X$ be a surjective \\'etale morphism.", "By Lemma \\ref{lemma-finite-presentation-local}", "we see that $i' : Z \\times_X U \\to U$ is of finite presentation if and", "only if $i$ is. By Properties of Spaces, Section", "\\ref{spaces-properties-section-properties-modules}", "we see that $\\mathcal{I}$ is of finite type if and only if", "$\\mathcal{I}|_U = \\Ker(\\mathcal{O}_U \\to i'_*\\mathcal{O}_{Z \\times_X U})$", "is. Hence the result follows from the case of schemes, see Morphisms,", "Lemma \\ref{morphisms-lemma-closed-immersion-finite-presentation}." ], "refs": [ "spaces-morphisms-lemma-finite-presentation-local", "morphisms-lemma-closed-immersion-finite-presentation" ], "ref_ids": [ 4841, 5243 ] } ], "ref_ids": [] }, { "id": 4850, "type": "theorem", "label": "spaces-morphisms-lemma-inverse-image-constructible", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-inverse-image-constructible", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $E \\subset |Y|$ be a subset.", "If $E$ is \\'etale locally constructible in $Y$, then", "$f^{-1}(E)$ is \\'etale locally constructible in $X$." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective \\'etale morphism $\\varphi : V \\to Y$.", "Choose a scheme $U$ and a surjective \\'etale morphism", "$U \\to V \\times_Y X$. Then $U \\to X$ is surjective \\'etale", "and the inverse image of $f^{-1}(E)$ in $U$ is the inverse", "image of $\\varphi^{-1}(E)$ by $U \\to V$. Thus the lemma follows", "from the case of schemes for $U \\to V$", "(Morphisms, Lemma \\ref{morphisms-lemma-inverse-image-constructible})", "and the definition (Properties of Spaces, Definition", "\\ref{spaces-properties-definition-locally-constructible})." ], "refs": [ "morphisms-lemma-inverse-image-constructible", "spaces-properties-definition-locally-constructible" ], "ref_ids": [ 5249, 11928 ] } ], "ref_ids": [] }, { "id": 4851, "type": "theorem", "label": "spaces-morphisms-lemma-flat-is-flat-at-all-points", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-flat-is-flat-at-all-points", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Then $f$ is flat if and only if $f$ is flat at all points of $|X|$." ], "refs": [], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$ and $V$ are schemes, the vertical arrows are \\'etale, and", "$a$ is surjective. By definition $f$ is flat if and only if $h$ is flat", "(Definition \\ref{definition-P}).", "By definition $f$ is flat at $x \\in |X|$ if and only if $h$ is flat", "at some (equivalently any) $u \\in U$ which maps to $x$", "(Definition \\ref{definition-P-at-point}).", "Thus the lemma follows from the fact that a morphism of schemes", "is flat if and only if it is flat at all points of the source", "(Morphisms, Definition \\ref{morphisms-definition-flat})." ], "refs": [ "spaces-morphisms-definition-P", "spaces-morphisms-definition-P-at-point", "morphisms-definition-flat" ], "ref_ids": [ 5001, 5002, 5557 ] } ], "ref_ids": [] }, { "id": 4852, "type": "theorem", "label": "spaces-morphisms-lemma-composition-flat", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-flat", "contents": [ "The composition of flat morphisms is flat." ], "refs": [], "proofs": [ { "contents": [ "See Remark \\ref{remark-composition-P} and", "Morphisms, Lemma \\ref{morphisms-lemma-composition-flat}." ], "refs": [ "spaces-morphisms-remark-composition-P", "morphisms-lemma-composition-flat" ], "ref_ids": [ 5033, 5263 ] } ], "ref_ids": [] }, { "id": 4853, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-flat", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-flat", "contents": [ "The base change of a flat morphism is flat." ], "refs": [], "proofs": [ { "contents": [ "See Remark \\ref{remark-base-change-P} and", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat}." ], "refs": [ "spaces-morphisms-remark-base-change-P", "morphisms-lemma-base-change-flat" ], "ref_ids": [ 5034, 5265 ] } ], "ref_ids": [] }, { "id": 4854, "type": "theorem", "label": "spaces-morphisms-lemma-flat-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-flat-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is flat,", "\\item for every $x \\in |X|$ the morphism $f$ is flat at $x$,", "\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism", "$Z \\times_Y X \\to Z$ is flat,", "\\item for every affine scheme $Z$ and any morphism", "$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is flat,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is flat,", "\\item there exists a scheme $U$ and a surjective \\'etale morphism", "$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$", "is flat,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes and the vertical arrows are \\'etale", "the top horizontal arrow is flat,", "\\item there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes, the vertical arrows are \\'etale, and", "$U \\to X$ is surjective such that the top horizontal arrow is flat, and", "\\item there exists a Zariski coverings $Y = \\bigcup Y_i$ and", "$f^{-1}(Y_i) = \\bigcup X_{ij}$ such that", "each morphism $X_{ij} \\to Y_i$ is flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4855, "type": "theorem", "label": "spaces-morphisms-lemma-fppf-open", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-fppf-open", "contents": [ "A flat morphism locally of finite presentation is universally open." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a flat morphism locally of finite presentation", "of algebraic spaces over $S$. Choose a diagram", "$$", "\\xymatrix{", "U \\ar[r]_\\alpha \\ar[d] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$ and $V$ are schemes and the vertical arrows are surjective and", "\\'etale, see", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}.", "By", "Lemmas \\ref{lemma-flat-local} and \\ref{lemma-finite-presentation-local}", "the morphism $\\alpha$ is flat and locally of finite presentation.", "Hence by", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}", "we see that $\\alpha$ is universally open.", "Hence $X \\to Y$ is universally open according to", "Lemma \\ref{lemma-universally-open-local}." ], "refs": [ "spaces-lemma-lift-morphism-presentations", "spaces-morphisms-lemma-flat-local", "spaces-morphisms-lemma-finite-presentation-local", "morphisms-lemma-fppf-open", "spaces-morphisms-lemma-universally-open-local" ], "ref_ids": [ 8159, 4854, 4841, 5267, 4731 ] } ], "ref_ids": [] }, { "id": 4856, "type": "theorem", "label": "spaces-morphisms-lemma-fpqc-quotient-topology", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-fpqc-quotient-topology", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a flat, quasi-compact, surjective morphism of", "algebraic spaces over $S$.", "A subset $T \\subset |Y|$ is open (resp.\\ closed) if and only", "$f^{-1}(|T|)$ is open (resp.\\ closed) in $|X|$.", "In other words $f$ is submersive, and in fact universally submersive." ], "refs": [], "proofs": [ { "contents": [ "Choose affine schemes $V_i$ and \\'etale morphisms $V_i \\to Y$ such that", "$V = \\coprod V_i \\to Y$ is surjective, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-cover-by-union-affines}.", "For each $i$ the algebraic space $V_i \\times_Y X$ is quasi-compact.", "Hence we can find an affine scheme $U_i$ and a surjective \\'etale morphism", "$U_i \\to V_i \\times_Y X$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-quasi-compact-affine-cover}.", "Then the composition $U_i \\to V_i \\times_Y X \\to V_i$ is a surjective,", "flat morphism of affines.", "Of course then $U = \\coprod U_i \\to X$ is surjective and \\'etale", "and $U = V \\times_Y X$. Moreover, the morphism $U \\to V$ is the", "disjoint union of the morphisms $U_i \\to V_i$. Hence $U \\to V$ is surjective,", "quasi-compact and flat. Consider the diagram", "$$", "\\xymatrix{", "U \\ar[r] \\ar[d] & X \\ar[d] \\\\", "V \\ar[r] & Y", "}", "$$", "By definition of the topology on $|Y|$ the set $T$ is closed", "(resp.\\ open) if and only if $g^{-1}(T) \\subset |V|$ is closed", "(resp.\\ open). The same holds for", "$f^{-1}(T)$ and its inverse image in $|U|$.", "Since $U \\to V$ is quasi-compact, surjective, and flat we win by", "Morphisms, Lemma \\ref{morphisms-lemma-fpqc-quotient-topology}." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-properties-lemma-quasi-compact-affine-cover", "morphisms-lemma-fpqc-quotient-topology" ], "ref_ids": [ 11830, 11832, 5269 ] } ], "ref_ids": [] }, { "id": 4857, "type": "theorem", "label": "spaces-morphisms-lemma-flat-at-point-etale-local-rings", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-flat-at-point-etale-local-rings", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\overline{x}$ be a geometric point of $X$ lying over the point", "$x \\in |X|$. Let $\\overline{y} = f \\circ \\overline{x}$. The following", "are equivalent", "\\begin{enumerate}", "\\item $f$ is flat at $x$, and", "\\item the map on \\'etale local rings", "$\\mathcal{O}_{Y, \\overline{y}} \\to \\mathcal{O}_{X, \\overline{x}}$", "is flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$ and $V$ are schemes, $a, b$ are \\'etale, and", "$u \\in U$ mapping to $x$. We can find a geometric point", "$\\overline{u} : \\Spec(k) \\to U$ lying over $u$ with", "$\\overline{x} = a \\circ \\overline{u}$, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-geometric-lift-to-usual}.", "Set $\\overline{v} = h \\circ \\overline{u}$ with image $v \\in V$.", "We know that", "$$", "\\mathcal{O}_{X, \\overline{x}} = \\mathcal{O}_{U, u}^{sh}", "\\quad\\text{and}\\quad", "\\mathcal{O}_{Y, \\overline{y}} = \\mathcal{O}_{V, v}^{sh}", "$$", "see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring}.", "We obtain a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{O}_{U, u} \\ar[r] &", "\\mathcal{O}_{X, \\overline{x}} \\\\", "\\mathcal{O}_{V, v} \\ar[u] \\ar[r] &", "\\mathcal{O}_{Y, \\overline{y}} \\ar[u]", "}", "$$", "of local rings with flat horizontal arrows. We have to show that the", "left vertical arrow is flat if and only if the right vertical arrow is.", "Algebra, Lemma \\ref{algebra-lemma-flatness-descends-more-general}", "tells us $\\mathcal{O}_{U, u}$ is flat over $\\mathcal{O}_{V, v}$", "if and only if $\\mathcal{O}_{X, \\overline{x}}$ is flat over", "$\\mathcal{O}_{V, v}$. Hence the result follows from", "More on Flatness, Lemma \\ref{flat-lemma-flat-up-down-henselization}." ], "refs": [ "spaces-properties-lemma-geometric-lift-to-usual", "spaces-properties-lemma-describe-etale-local-ring", "algebra-lemma-flatness-descends-more-general", "flat-lemma-flat-up-down-henselization" ], "ref_ids": [ 11871, 11884, 529, 5982 ] } ], "ref_ids": [] }, { "id": 4858, "type": "theorem", "label": "spaces-morphisms-lemma-flat-morphism-sites", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-flat-morphism-sites", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Then $f$ is flat if and only if the morphism of sites", "$", "(f_{small}, f^\\sharp) :", "(X_\\etale, \\mathcal{O}_X)", "\\to", "(Y_\\etale, \\mathcal{O}_Y)", "$", "associated to $f$ is flat." ], "refs": [], "proofs": [ { "contents": [ "Flatness of $(f_{small}, f^\\sharp)$ is defined in terms of", "flatness of $\\mathcal{O}_X$ as a $f_{small}^{-1}\\mathcal{O}_Y$-module.", "This can be checked at stalks, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-check-flat-stalks}", "and", "Properties of Spaces, Theorem \\ref{spaces-properties-theorem-exactness-stalks}.", "But we've already seen that flatness of $f$ can be checked on stalks, see", "Lemma \\ref{lemma-flat-at-point-etale-local-rings}." ], "refs": [ "sites-modules-lemma-check-flat-stalks", "spaces-properties-theorem-exactness-stalks", "spaces-morphisms-lemma-flat-at-point-etale-local-rings" ], "ref_ids": [ 14251, 11813, 4857 ] } ], "ref_ids": [] }, { "id": 4859, "type": "theorem", "label": "spaces-morphisms-lemma-flat-pullback-support", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-flat-pullback-support", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces", "over $S$. Let $\\mathcal{F}$ be a finite type quasi-coherent", "$\\mathcal{O}_X$-module with scheme theoretic support $Z \\subset X$.", "If $f$ is flat, then $f^{-1}(Z)$ is the scheme theoretic support of", "$f^*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Using the characterization of the scheme theoretic support", "as given in Lemma \\ref{lemma-scheme-theoretic-support}", "and using the characterization of flat morphisms in terms of", "\\'etale coverings in Lemma \\ref{lemma-flat-local}", "we reduce to the case of schemes which is", "Morphisms, Lemma \\ref{morphisms-lemma-flat-pullback-support}." ], "refs": [ "spaces-morphisms-lemma-scheme-theoretic-support", "spaces-morphisms-lemma-flat-local", "morphisms-lemma-flat-pullback-support" ], "ref_ids": [ 4778, 4854, 5271 ] } ], "ref_ids": [] }, { "id": 4860, "type": "theorem", "label": "spaces-morphisms-lemma-flat-morphism-scheme-theoretically-dense-open", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-flat-morphism-scheme-theoretically-dense-open", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a flat morphism of algebraic spaces over $S$.", "Let $V \\to Y$ be a quasi-compact open immersion. If $V$", "is scheme theoretically dense in $Y$, then $f^{-1}V$", "is scheme theoretically dense in $X$." ], "refs": [], "proofs": [ { "contents": [ "Using the characterization of scheme theoretically dense opens", "in Lemma \\ref{lemma-scheme-theoretically-dense}", "and using the characterization of flat morphisms in terms of", "\\'etale coverings in Lemma \\ref{lemma-flat-local}", "we reduce to the case of schemes which is", "Morphisms, Lemma", "\\ref{morphisms-lemma-flat-morphism-scheme-theoretically-dense-open}." ], "refs": [ "spaces-morphisms-lemma-scheme-theoretically-dense", "spaces-morphisms-lemma-flat-local", "morphisms-lemma-flat-morphism-scheme-theoretically-dense-open" ], "ref_ids": [ 4786, 4854, 5272 ] } ], "ref_ids": [] }, { "id": 4861, "type": "theorem", "label": "spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a flat morphism of algebraic spaces", "over $S$. Let $g : V \\to Y$ be a quasi-compact morphism of algebraic spaces.", "Let $Z \\subset Y$ be the scheme theoretic image of $g$ and let $Z' \\subset X$", "be the scheme theoretic image of the base change $V \\times_Y X \\to X$.", "Then $Z' = f^{-1}Z$." ], "refs": [], "proofs": [ { "contents": [ "Let $Y' \\to Y$ be a surjective \\'etale morphism such that $Y'$ is a", "disjoint union of affine schemes (Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-cover-by-union-affines}).", "Let $X' \\to X \\times_Y Y'$ be a surjective \\'etale morphism such", "that $X'$ is a disjoint union of affine schemes.", "By Lemma \\ref{lemma-flat-local} the morphism $X' \\to Y'$ is flat.", "Set $V' = V \\times_Y Y'$.", "By Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image}", "the inverse image of $Z$ in $Y'$ is the scheme theoretic", "image of $V' \\to Y'$ and the inverse image of $Z'$ in $X'$", "is the scheme theoretic image of $V' \\times_{Y'} X' \\to X'$.", "Since $X' \\to X$ is surjective \\'etale, it suffices to prove", "the result in the case of the morphisms $X' \\to Y'$ and $V' \\to Y'$.", "Thus we may assume $X$ and $Y$ are affine schemes.", "In this case $V$ is a quasi-compact algebraic space.", "Choose an affine scheme $W$ and a surjective \\'etale morphism", "$W \\to V$ (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "It is clear that the scheme theoretic image of $V \\to Y$", "agrees with the scheme theoretic image of $W \\to Y$ and", "similarly for $V \\times_Y X \\to Y$ and $W \\times_Y X \\to X$.", "Thus we reduce to the case of schemes which is", "Morphisms, Lemma", "\\ref{morphisms-lemma-flat-base-change-scheme-theoretic-image}." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-morphisms-lemma-flat-local", "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image", "spaces-properties-lemma-quasi-compact-affine-cover", "morphisms-lemma-flat-base-change-scheme-theoretic-image" ], "ref_ids": [ 11830, 4854, 4780, 11832, 5273 ] } ], "ref_ids": [] }, { "id": 4862, "type": "theorem", "label": "spaces-morphisms-lemma-flat-at-point", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-flat-at-point", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $x \\in |X|$. The following are equivalent", "\\begin{enumerate}", "\\item for some commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$ and $V$ are schemes, $a, b$ are \\'etale, and", "$u \\in U$ mapping to $x$ the module $a^*\\mathcal{F}$ is flat at $u$ over $V$,", "\\item the stalk $\\mathcal{F}_{\\overline{x}}$ is flat over", "the \\'etale local ring $\\mathcal{O}_{Y, \\overline{y}}$", "where $\\overline{x}$ is any geometric point lying over", "$x$ and $\\overline{y} = f \\circ \\overline{x}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "During this proof we fix a geometric proof", "$\\overline{x} : \\Spec(k) \\to X$ over $x$ and", "we denote $\\overline{y} = f \\circ \\overline{x}$ its image in $Y$.", "Given a diagram as in (1) we can find a geometric point", "$\\overline{u} : \\Spec(k) \\to U$ lying over $u$ with", "$\\overline{x} = a \\circ \\overline{u}$, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-geometric-lift-to-usual}.", "Set $\\overline{v} = h \\circ \\overline{u}$ with image $v \\in V$.", "We know that", "$$", "\\mathcal{O}_{X, \\overline{x}} = \\mathcal{O}_{U, u}^{sh}", "\\quad\\text{and}\\quad", "\\mathcal{O}_{Y, \\overline{y}} = \\mathcal{O}_{V, v}^{sh}", "$$", "see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring}.", "We obtain a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{O}_{U, u} \\ar[r] &", "\\mathcal{O}_{X, \\overline{x}} \\\\", "\\mathcal{O}_{V, v} \\ar[u] \\ar[r] &", "\\mathcal{O}_{Y, \\overline{y}} \\ar[u]", "}", "$$", "of local rings. Finally, we have", "$$", "\\mathcal{F}_{\\overline{x}} =", "(\\varphi^*\\mathcal{F})_u \\otimes_{\\mathcal{O}_{U, u}}", "\\mathcal{O}_{X, \\overline{x}}", "$$", "by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-stalk-quasi-coherent}.", "Thus", "Algebra, Lemma \\ref{algebra-lemma-flatness-descends-more-general}", "tells us $(\\varphi^*\\mathcal{F})_u$ is flat over $\\mathcal{O}_{V, v}$", "if and only if $\\mathcal{F}_{\\overline{x}}$ is flat over $\\mathcal{O}_{V, v}$.", "Hence the result follows from", "More on Flatness, Lemma \\ref{flat-lemma-flat-up-down-henselization}." ], "refs": [ "spaces-properties-lemma-geometric-lift-to-usual", "spaces-properties-lemma-describe-etale-local-ring", "spaces-properties-lemma-stalk-quasi-coherent", "algebra-lemma-flatness-descends-more-general", "flat-lemma-flat-up-down-henselization" ], "ref_ids": [ 11871, 11884, 11909, 529, 5982 ] } ], "ref_ids": [] }, { "id": 4863, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-module-flat", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-module-flat", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "be a cartesian diagram of algebraic spaces over $S$. Let $x' \\in |X'|$", "with image $x \\in |X|$. Let $\\mathcal{F}$ be a quasi-coherent", "sheaf on $X$ and denote $\\mathcal{F}' = (g')^*\\mathcal{F}$.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is flat at $x$ over $Y$", "then $\\mathcal{F}'$ is flat at $x'$ over $Y'$.", "\\item If $g$ is flat at $f'(x')$ and", "$\\mathcal{F}'$ is flat at $x'$ over $Y'$, then", "$\\mathcal{F}$ is flat at $x$ over $Y$.", "\\end{enumerate}", "In particular, if $\\mathcal{F}$ is flat over $Y$, then", "$\\mathcal{F}'$ is flat over $Y'$." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to V \\times_Y X$.", "Choose a scheme $V'$ and a surjective \\'etale morphism $V' \\to V \\times_Y Y'$.", "Then $U' = V' \\times_V U$ is a scheme endowed with a surjective \\'etale", "morphism $U' = V' \\times_V U \\to Y' \\times_Y X = X'$. Pick $u' \\in U'$", "mapping to $x' \\in |X'|$. Then we can check flatness of", "$\\mathcal{F}'$ at $x'$ over $Y'$ in terms of flatness of", "$\\mathcal{F}'|_{U'}$ at $u'$ over $V'$. Hence the lemma follows from", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-flat-locus-base-change}." ], "refs": [ "more-morphisms-lemma-flat-locus-base-change" ], "ref_ids": [ 13766 ] } ], "ref_ids": [] }, { "id": 4864, "type": "theorem", "label": "spaces-morphisms-lemma-composition-module-flat", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-module-flat", "contents": [ "Let $S$ be a scheme. Let $X \\to Y \\to Z$ be morphisms of algebraic spaces", "over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $x \\in |X|$ with image $y \\in |Y|$.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is flat at $x$ over $Y$ and", "$Y$ is flat at $y$ over $Z$, then $\\mathcal{F}$ is flat at", "$x$ over $Z$.", "\\item Let $x : \\Spec(K) \\to X$ be a representative of $x$. If", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is flat at $x$ over $Y$,", "\\item $x^*\\mathcal{F} \\not = 0$, and", "\\item $\\mathcal{F}$ is flat at $x$ over $Z$,", "\\end{enumerate}", "then $Y$ is flat at $y$ over $Z$.", "\\item Let $\\overline{x}$ be a geometric point of $X$ lying over $x$", "with image $\\overline{y}$ in $Y$. If $\\mathcal{F}_{\\overline{x}}$ is a", "faithfully flat $\\mathcal{O}_{Y, \\overline{y}}$-module and", "$\\mathcal{F}$ is flat at $x$ over $Z$, then", "$Y$ is flat at $y$ over $Z$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Pick $\\overline{x}$ and $\\overline{y}$ as in part (3) and denote", "$\\overline{z}$ the induced geometric point of $Z$. Via the", "characterization of flatness in", "Lemmas \\ref{lemma-flat-at-point} and", "\\ref{lemma-flat-at-point-etale-local-rings}", "the lemma reduces to a purely algebraic question on the local", "ring map $\\mathcal{O}_{Z, \\overline{z}} \\to \\mathcal{O}_{Y, \\overline{y}}$", "and the module $\\mathcal{F}_{\\overline{x}}$.", "Part (1) follows from", "Algebra, Lemma \\ref{algebra-lemma-composition-flat}.", "We remark that condition (2)(b) guarantees that", "$\\mathcal{F}_{\\overline{x}}/", "\\mathfrak m_{\\overline{y}} \\mathcal{F}_{\\overline{x}}$", "is nonzero. Hence (2)(a) $+$ (2)(b) imply that $\\mathcal{F}_{\\overline{x}}$", "is a faithfully flat $\\mathcal{O}_{Y, \\overline{y}}$-module, see", "Algebra, Lemma \\ref{algebra-lemma-ff}.", "Thus (2) is a special case of (3).", "Finally, (3) follows from", "Algebra, Lemma \\ref{algebra-lemma-flat-permanence}." ], "refs": [ "spaces-morphisms-lemma-flat-at-point", "spaces-morphisms-lemma-flat-at-point-etale-local-rings", "algebra-lemma-composition-flat", "algebra-lemma-ff", "algebra-lemma-flat-permanence" ], "ref_ids": [ 4862, 4857, 524, 535, 530 ] } ], "ref_ids": [] }, { "id": 4865, "type": "theorem", "label": "spaces-morphisms-lemma-flat-permanence", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-flat-permanence", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$, $g : Y \\to Z$ be morphisms of", "algebraic spaces over $S$. Let $\\mathcal{G}$ be a quasi-coherent sheaf on $Y$.", "Let $x \\in |X|$ with image $y \\in |Y|$.", "If $f$ is flat at $x$, then", "$$", "\\mathcal{G}\\text{ flat over }Z\\text{ at }y", "\\Leftrightarrow", "f^*\\mathcal{G}\\text{ flat over }Z\\text{ at }x.", "$$", "In particular: If $f$ is surjective and flat, then", "$\\mathcal{G}$ is flat over $Z$, if and only if", "$f^*\\mathcal{G}$ is flat over $Z$." ], "refs": [], "proofs": [ { "contents": [ "Pick a geometric point $\\overline{x}$ of $X$ and denote", "$\\overline{y}$ the image in $Y$ and $\\overline{z}$ the image in $Z$.", "Via the characterization of flatness in", "Lemmas \\ref{lemma-flat-at-point} and", "\\ref{lemma-flat-at-point-etale-local-rings}", "and the description of the stalk of $f^*\\mathcal{G}$ at $\\overline{x}$ of", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-stalk-pullback-quasi-coherent}", "the lemma reduces to a purely algebraic question on the local", "ring maps", "$\\mathcal{O}_{Z, \\overline{z}} \\to \\mathcal{O}_{Y, \\overline{y}}", "\\to \\mathcal{O}_{X, \\overline{x}}$", "and the module $\\mathcal{G}_{\\overline{y}}$.", "This algebraic statement is", "Algebra, Lemma \\ref{algebra-lemma-flatness-descends-more-general}." ], "refs": [ "spaces-morphisms-lemma-flat-at-point", "spaces-morphisms-lemma-flat-at-point-etale-local-rings", "spaces-properties-lemma-stalk-pullback-quasi-coherent", "algebra-lemma-flatness-descends-more-general" ], "ref_ids": [ 4862, 4857, 11910, 529 ] } ], "ref_ids": [] }, { "id": 4866, "type": "theorem", "label": "spaces-morphisms-lemma-pf-flat-module-open", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-pf-flat-module-open", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume $f$ locally finite presentation, $\\mathcal{F}$ of", "finite type, $X = \\text{Supp}(\\mathcal{F})$, and", "$\\mathcal{F}$ flat over $Y$. Then $f$ is universally open." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjective \\'etale morphism $\\varphi : V \\to Y$ where $V$ is a scheme.", "Choose a surjective \\'etale morphism $U \\to V \\times_Y X$ where $U$ is a scheme.", "Then it suffices to prove the lemma for $U \\to V$ and the quasi-coherent", "$\\mathcal{O}_V$-module $\\varphi^*\\mathcal{F}$.", "Hence this lemma follows from the case of schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-pf-flat-module-open}." ], "refs": [ "morphisms-lemma-pf-flat-module-open" ], "ref_ids": [ 5268 ] } ], "ref_ids": [] }, { "id": 4867, "type": "theorem", "label": "spaces-morphisms-lemma-compare-tr-deg", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-compare-tr-deg", "contents": [ "Let $S$ be a scheme. Let $X \\to Y \\to Z$ be morphisms of algebraic", "spaces over $S$. Let $x \\in |X|$ and let $y \\in |Y|$, $z \\in |Z|$", "be the images. Assume $X \\to Y$ is locally quasi-finite and $Y \\to Z$", "locally of finite type. Then the transcendence degree of $x/z$", "is equal to the transcendence degree of $y/z$." ], "refs": [], "proofs": [ { "contents": [ "We can choose commutative diagrams", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\ar[r] & W \\ar[d] \\\\", "X \\ar[r] & Y \\ar[r] & Z", "}", "\\quad\\quad", "\\xymatrix{", "u \\ar[d] \\ar[r] & v \\ar[d] \\ar[r] & w \\ar[d] \\\\", "x \\ar[r] & y \\ar[r] & z", "}", "$$", "where $U, V, W$ are schemes and the vertical arrows are \\'etale.", "By definition the morphism $U \\to V$ is locally quasi-finite", "which implies that $\\kappa(v) \\subset \\kappa(u)$ is finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-residue-field-quasi-finite}.", "Hence the result is clear." ], "refs": [ "morphisms-lemma-residue-field-quasi-finite" ], "ref_ids": [ 5225 ] } ], "ref_ids": [] }, { "id": 4868, "type": "theorem", "label": "spaces-morphisms-lemma-jacobson-finite-type-points", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-jacobson-finite-type-points", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. If $f$ is locally of finite type, $Y$ is Jacobson", "(Properties of Spaces, Remark", "\\ref{spaces-properties-remark-list-properties-local-etale-topology}),", "and $x \\in |X|$ is a finite type point of $X$,", "then the transcendence degree of $x/f(x)$ is $0$." ], "refs": [ "spaces-properties-remark-list-properties-local-etale-topology" ], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X \\times_Y V$.", "By Lemma \\ref{lemma-finite-type-points-surjective-morphism}", "we can find a finite type point $u \\in U$ mapping to $x$.", "After shrinking $U$ we may assume $u \\in U$ is closed", "(Morphisms, Lemma \\ref{morphisms-lemma-identify-finite-type-points}).", "Let $v \\in V$ be the image of $u$. By", "Morphisms, Lemma \\ref{morphisms-lemma-jacobson-finite-type-points}", "the extension $\\kappa(u)/\\kappa(v)$ is finite.", "This finishes the proof." ], "refs": [ "spaces-morphisms-lemma-finite-type-points-surjective-morphism", "morphisms-lemma-identify-finite-type-points", "morphisms-lemma-jacobson-finite-type-points" ], "ref_ids": [ 4825, 5207, 5211 ] } ], "ref_ids": [ 11950 ] }, { "id": 4869, "type": "theorem", "label": "spaces-morphisms-lemma-rel-dimension-dimension", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-rel-dimension-dimension", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian", "algebraic spaces over $S$ which is flat, locally of finite type and of", "relative dimension $d$. For every point $x$ in $|X|$ with image", "$y$ in $|Y|$ we have $\\dim_x(X) = \\dim_y(Y) + d$." ], "refs": [], "proofs": [ { "contents": [ "By definition of the dimension of an algebraic space", "at a point (Properties of Spaces, Definition", "\\ref{spaces-properties-definition-dimension-at-point})", "and by definition of having relative dimension $d$,", "this reduces to the corresponding statement for schemes", "(Morphisms, Lemma \\ref{morphisms-lemma-rel-dimension-dimension})." ], "refs": [ "spaces-properties-definition-dimension-at-point", "morphisms-lemma-rel-dimension-dimension" ], "ref_ids": [ 11929, 5288 ] } ], "ref_ids": [] }, { "id": 4870, "type": "theorem", "label": "spaces-morphisms-lemma-dimension-fibre-at-a-point", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-dimension-fibre-at-a-point", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $x \\in |X|$.", "Assume $f$ is locally of finite type.", "Then we have", "$$", "\\begin{matrix}", "\\text{relative dimension of }f\\text{ at }x \\\\", "= \\\\", "\\text{dimension of local ring of the fibre of }f\\text{ at }x \\\\", "+ \\\\", "\\text{transcendence degree of }x/f(x)", "\\end{matrix}", "$$", "where the notation is as in", "Definition \\ref{definition-dimension-fibre}." ], "refs": [ "spaces-morphisms-definition-dimension-fibre" ], "proofs": [ { "contents": [ "This follows immediately from", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}", "applied to $h : U \\to V$ and $u \\in U$", "as in", "Lemma \\ref{lemma-local-source-target-at-point}." ], "refs": [ "morphisms-lemma-dimension-fibre-at-a-point", "spaces-morphisms-lemma-local-source-target-at-point" ], "ref_ids": [ 5277, 4812 ] } ], "ref_ids": [ 5009 ] }, { "id": 4871, "type": "theorem", "label": "spaces-morphisms-lemma-dimension-fibre-at-a-point-additive", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-dimension-fibre-at-a-point-additive", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces over $S$.", "Let $x \\in |X|$ and set $y = f(x)$.", "Assume $f$ and $g$ locally of finite type.", "Then", "\\begin{enumerate}", "\\item", "$$", "\\begin{matrix}", "\\text{relative dimension of }g \\circ f\\text{ at }x \\\\", "\\leq \\\\", "\\text{relative dimension of }f\\text{ at }x \\\\", "+ \\\\", "\\text{relative dimension of }g\\text{ at }y", "\\end{matrix}", "$$", "\\item equality holds in (1) if for some morphism $\\Spec(k) \\to Z$", "from the spectrum of a field in the class of $g(f(x)) = g(y)$", "the morphism $X_k \\to Y_k$ is flat at $x$, for example if $f$ is flat at $x$,", "\\item", "$$", "\\begin{matrix}", "\\text{transcendence degree of }x/g(f(x)) \\\\", "= \\\\", "\\text{transcendence degree of }x/f(x) \\\\", "+ \\\\", "\\text{transcendence degree of }f(x)/g(f(x))", "\\end{matrix}", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\ar[r] & W \\ar[d] \\\\", "X \\ar[r] & Y \\ar[r] & Z", "}", "$$", "with $U, V, W$ schemes and vertical arrows \\'etale and surjective. (See", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}.)", "Choose $u \\in U$ mapping to $x$. Set $v, w$ equal to the images", "of $u$ in $V, W$.", "Apply", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point-additive}", "to the top row and the points $u, v, w$. Details omitted." ], "refs": [ "spaces-lemma-lift-morphism-presentations", "morphisms-lemma-dimension-fibre-at-a-point-additive" ], "ref_ids": [ 8159, 5278 ] } ], "ref_ids": [] }, { "id": 4872, "type": "theorem", "label": "spaces-morphisms-lemma-dimension-fibre-after-base-change", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-dimension-fibre-after-base-change", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "be a fibre product diagram of algebraic spaces over $S$.", "Let $x' \\in |X'|$. Set $x = g'(x')$. Assume $f$ locally of finite type.", "Then", "\\begin{enumerate}", "\\item", "$$", "\\begin{matrix}", "\\text{relative dimension of }f\\text{ at }x \\\\", "= \\\\", "\\text{relative dimension of }f'\\text{ at }x'", "\\end{matrix}", "$$", "\\item we have", "$$", "\\begin{matrix}", "\\text{dimension of local ring of the fibre of }f'\\text{ at }x' \\\\", "- \\\\", "\\text{dimension of local ring of the fibre of }f\\text{ at }x \\\\", "= \\\\", "\\text{transcendence degree of }x/f(x) \\\\", "- \\\\", "\\text{transcendence degree of }x'/f'(x')", "\\end{matrix}", "$$", "and the common value is $\\geq 0$,", "\\item given $x$ and $y' \\in |Y'|$ mapping to the same $y \\in |Y|$", "there exists a choice of $x'$ such that the integer in (2) is $0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a surjective \\'etale morphism $V \\to Y$ with $V$ a scheme.", "Choose a surjective \\'etale morphism $U \\to V \\times_Y X$ with $U$ a scheme.", "Choose a surjective \\'etale morphism $V' \\to V \\times_Y Y'$ with $V'$ a scheme.", "Set $U' = V' \\times_V U$.", "Then the induced morphism $U' \\to X'$ is also surjective and", "\\'etale (argument omitted). Choose $u' \\in U'$", "mapping to $x'$. At this point parts (1) and (2) follow by applying", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}", "to the diagram of schemes involving $U', U, V', V$ and the point $u'$.", "To prove (3) first choose $v \\in V$ mapping to $y$.", "Then using Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-points-cartesian}", "we can choose $v' \\in V'$ mapping to $y'$ and $v$ and", "$u \\in U$ mapping to $x$ and $v$. Finally, according to", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}", "we can choose $u' \\in U'$ mapping to $v'$ and $u$ such that", "the integer is zero. Then taking $x' \\in |X'|$ the image of $u'$ works." ], "refs": [ "morphisms-lemma-dimension-fibre-after-base-change", "spaces-properties-lemma-points-cartesian", "morphisms-lemma-dimension-fibre-after-base-change" ], "ref_ids": [ 5279, 11819, 5279 ] } ], "ref_ids": [] }, { "id": 4873, "type": "theorem", "label": "spaces-morphisms-lemma-openness-bounded-dimension-fibres", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-openness-bounded-dimension-fibres", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $n \\geq 0$. Assume $f$ is locally of finite type.", "The set", "$$", "W_n = \\{x \\in |X|", "\\text{ such that the relative dimension of }f\\text{ at } x \\leq n\\}", "$$", "is open in $|X|$." ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram", "$$", "\\xymatrix{", "U \\ar[r]_h \\ar[d]_a & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$ and $V$ are schemes and the vertical arrows are surjective and", "\\'etale, see", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}.", "By", "Morphisms, Lemma \\ref{morphisms-lemma-openness-bounded-dimension-fibres}", "the set $U_n$ of points where $h$ has relative dimension", "$\\leq n$ is open in $U$. By our definition of relative dimension", "for morphisms of algebraic spaces at points we see that", "$U_n = a^{-1}(W_n)$.", "The lemma follows by definition of the topology on $|X|$." ], "refs": [ "spaces-lemma-lift-morphism-presentations", "morphisms-lemma-openness-bounded-dimension-fibres" ], "ref_ids": [ 8159, 5280 ] } ], "ref_ids": [] }, { "id": 4874, "type": "theorem", "label": "spaces-morphisms-lemma-openness-bounded-dimension-fibres-finite-presentation", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-openness-bounded-dimension-fibres-finite-presentation", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "Let $n \\geq 0$. Assume $f$ is locally of finite presentation.", "The open", "$$", "W_n = \\{x \\in |X|", "\\text{ such that the relative dimension of }f\\text{ at } x \\leq n\\}", "$$", "of Lemma \\ref{lemma-openness-bounded-dimension-fibres}", "is retrocompact in $|X|$. (See", "Topology, Definition \\ref{topology-definition-quasi-compact}.)" ], "refs": [ "spaces-morphisms-lemma-openness-bounded-dimension-fibres", "topology-definition-quasi-compact" ], "proofs": [ { "contents": [ "Choose a diagram", "$$", "\\xymatrix{", "U \\ar[r]_h \\ar[d]_a & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$ and $V$ are schemes and the vertical arrows are surjective and", "\\'etale, see", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}.", "In the proof of", "Lemma \\ref{lemma-openness-bounded-dimension-fibres}", "we have seen that $a^{-1}(W_n) = U_n$ is the corresponding set", "for the morphism $h$. By", "Morphisms, Lemma", "\\ref{morphisms-lemma-openness-bounded-dimension-fibres-finite-presentation}", "we see that $U_n$ is retrocompact in $U$.", "The lemma follows by definition of the topology on $|X|$, compare with", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-space-locally-quasi-compact}", "and its proof." ], "refs": [ "spaces-lemma-lift-morphism-presentations", "spaces-morphisms-lemma-openness-bounded-dimension-fibres", "morphisms-lemma-openness-bounded-dimension-fibres-finite-presentation", "spaces-properties-lemma-space-locally-quasi-compact" ], "ref_ids": [ 8159, 4873, 5282, 11829 ] } ], "ref_ids": [ 4873, 8360 ] }, { "id": 4875, "type": "theorem", "label": "spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume $f$ is locally of finite type.", "Then $f$ is locally quasi-finite if and only if $f$ has relative", "dimension $0$ at each $x \\in |X|$." ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram", "$$", "\\xymatrix{", "U \\ar[r]_h \\ar[d]_a & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$ and $V$ are schemes and the vertical arrows are surjective and", "\\'etale, see", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}.", "The definitions imply that", "$h$ is locally quasi-finite if and only if $f$ is locally quasi-finite,", "and that $f$ has relative dimension $0$ at all $x \\in |X|$ if and", "only if $h$ has relative dimension $0$ at all $u \\in U$.", "Hence the result follows from the result for $h$ which is", "Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}." ], "refs": [ "spaces-lemma-lift-morphism-presentations", "morphisms-lemma-locally-quasi-finite-rel-dimension-0" ], "ref_ids": [ 8159, 5287 ] } ], "ref_ids": [] }, { "id": 4876, "type": "theorem", "label": "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume $f$ is locally of finite type.", "Then there exists a canonical open subspace $X' \\subset X$ such that", "$f|_{X'} : X' \\to Y$ is locally quasi-finite, and such that the", "relative dimension of $f$ at any $x \\in |X|$, $x \\not \\in |X'|$ is", "$\\geq 1$. Formation of $X'$ commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemmas \\ref{lemma-openness-bounded-dimension-fibres},", "\\ref{lemma-locally-quasi-finite-rel-dimension-0}, and", "\\ref{lemma-dimension-fibre-after-base-change}." ], "refs": [ "spaces-morphisms-lemma-openness-bounded-dimension-fibres", "spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0", "spaces-morphisms-lemma-dimension-fibre-after-base-change" ], "ref_ids": [ 4873, 4875, 4872 ] } ], "ref_ids": [] }, { "id": 4877, "type": "theorem", "label": "spaces-morphisms-lemma-quasi-finite-at-point", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-quasi-finite-at-point", "contents": [ "Let $S$ be a scheme. Consider a cartesian diagram", "$$", "\\xymatrix{", "X \\ar[d] & F \\ar[l]^p \\ar[d] \\\\", "Y & \\Spec(k) \\ar[l]", "}", "$$", "where $X \\to Y$ is a morphism of algebraic spaces over $S$ which is", "locally of finite type and where $k$ is a field over $S$.", "Let $z \\in |F|$ be such that $\\dim_z(F) = 0$. Then, after replacing $X$", "by an open subspace containing $p(z)$, the morphism", "$$", "X \\longrightarrow Y", "$$", "is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Let $X' \\subset X$ be the open subspace over which $f$ is locally quasi-finite", "found in", "Lemma \\ref{lemma-locally-finite-type-quasi-finite-part}.", "Since the formation of $X'$ commutes with arbitrary base change we see", "that $z \\in X' \\times_Y \\Spec(k)$. Hence the lemma is clear." ], "refs": [ "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part" ], "ref_ids": [ 4876 ] } ], "ref_ids": [] }, { "id": 4878, "type": "theorem", "label": "spaces-morphisms-lemma-dimension-formula-general", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-dimension-formula-general", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. Assume $Y$ is locally Noetherian and $f$ locally", "of finite type. Let $x \\in |X|$ with image $y \\in |Y|$.", "Then we have", "\\begin{align*}", "& \\text{the dimension of the local ring of }X\\text{ at }x \\leq \\\\", "& \\text{the dimension of the local ring of }Y\\text{ at }y + E - \\\\", "& \\text{ the transcendence degree of }x/y", "\\end{align*}", "Here $E$ is the maximum of the transcendence degrees of $\\xi/f(\\xi)$", "where $\\xi \\in |X|$ runs over the points specializing to $x$ at", "which the local ring of $X$ has dimension $0$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $V$, an \\'etale morphism $V \\to Y$, and a point", "$v \\in V$ mapping to $y$. Choose an affine scheme $U$ , an \\'etale morphism", "$U \\to X \\times_Y V$ and a point $u \\in U$ mapping to $v$ in $V$ and $x$", "in $X$. Unwinding Definition \\ref{definition-dimension-fibre} and", "Properties of Spaces, Definition", "\\ref{spaces-properties-definition-dimension-local-ring}", "we have to show that", "$$", "\\dim(\\mathcal{O}_{U, u}) \\leq", "\\dim(\\mathcal{O}_{V, v}) + E - \\text{trdeg}_{\\kappa(v)}(\\kappa(u))", "$$", "Let $\\xi_U \\in U$ be a generic point of an irreducible component of", "$U$ which contains $u$. Then $\\xi_U$ maps to a point $\\xi \\in |X|$", "which is in the list used to define the quantity $E$ and in fact", "every $\\xi$ used in the definition of $E$ occurs in this manner", "(small detail omitted). In particular, there are only a finite", "number of these $\\xi$ and we can take the maximum (i.e., it really", "is a maximum and not a supremum).", "The transcendence degree of $\\xi$ over $f(\\xi)$ is", "$\\text{trdeg}_{\\kappa(\\xi_V)}(\\kappa(\\xi_U))$ where $\\xi_V \\in V$", "is the image of $\\xi_U$. Thus the lemma follows from", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-formula-general}." ], "refs": [ "spaces-morphisms-definition-dimension-fibre", "spaces-properties-definition-dimension-local-ring", "morphisms-lemma-dimension-formula-general" ], "ref_ids": [ 5009, 11931, 5494 ] } ], "ref_ids": [] }, { "id": 4879, "type": "theorem", "label": "spaces-morphisms-lemma-alteration-dimension-general", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-alteration-dimension-general", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume", "$Y$ is locally Noetherian and $f$ is locally of finite type.", "Then", "$$", "\\dim(X) \\leq \\dim(Y) + E", "$$", "where $E$ is the supremum of the transcendence degrees of", "$\\xi/f(\\xi)$ where $\\xi$ runs through the points at", "which the local ring of $X$ has dimension $0$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of Lemma \\ref{lemma-dimension-formula-general}", "and Properties of Spaces, Lemma \\ref{spaces-properties-lemma-dimension}." ], "refs": [ "spaces-morphisms-lemma-dimension-formula-general", "spaces-properties-lemma-dimension" ], "ref_ids": [ 4878, 11841 ] } ], "ref_ids": [] }, { "id": 4880, "type": "theorem", "label": "spaces-morphisms-lemma-composition-syntomic", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-syntomic", "contents": [ "The composition of syntomic morphisms is syntomic." ], "refs": [], "proofs": [ { "contents": [ "See Remark \\ref{remark-composition-P} and", "Morphisms, Lemma \\ref{morphisms-lemma-composition-syntomic}." ], "refs": [ "spaces-morphisms-remark-composition-P", "morphisms-lemma-composition-syntomic" ], "ref_ids": [ 5033, 5290 ] } ], "ref_ids": [] }, { "id": 4881, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-syntomic", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-syntomic", "contents": [ "The base change of a syntomic morphism is syntomic." ], "refs": [], "proofs": [ { "contents": [ "See Remark \\ref{remark-base-change-P} and", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-syntomic}." ], "refs": [ "spaces-morphisms-remark-base-change-P", "morphisms-lemma-base-change-syntomic" ], "ref_ids": [ 5034, 5291 ] } ], "ref_ids": [] }, { "id": 4882, "type": "theorem", "label": "spaces-morphisms-lemma-syntomic-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-syntomic-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is syntomic,", "\\item for every $x \\in |X|$ the morphism $f$ is syntomic at $x$,", "\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism", "$Z \\times_Y X \\to Z$ is syntomic,", "\\item for every affine scheme $Z$ and any morphism", "$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is syntomic,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is a syntomic morphism,", "\\item there exists a scheme $U$ and a surjective \\'etale morphism", "$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$", "is syntomic,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes and the vertical arrows are \\'etale", "the top horizontal arrow is syntomic,", "\\item there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes, the vertical arrows are \\'etale, and", "$U \\to X$ is surjective such that the top horizontal arrow is syntomic, and", "\\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$,", "and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that", "each morphism $X_{ij} \\to Y_i$ is syntomic.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4883, "type": "theorem", "label": "spaces-morphisms-lemma-syntomic-locally-finite-presentation", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-syntomic-locally-finite-presentation", "contents": [ "A syntomic morphism is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from the case of schemes", "(Morphisms, Lemma \\ref{morphisms-lemma-syntomic-locally-finite-presentation})." ], "refs": [ "morphisms-lemma-syntomic-locally-finite-presentation" ], "ref_ids": [ 5293 ] } ], "ref_ids": [] }, { "id": 4884, "type": "theorem", "label": "spaces-morphisms-lemma-syntomic-flat", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-syntomic-flat", "contents": [ "A syntomic morphism is flat." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from the case of schemes", "(Morphisms, Lemma \\ref{morphisms-lemma-syntomic-flat})." ], "refs": [ "morphisms-lemma-syntomic-flat" ], "ref_ids": [ 5294 ] } ], "ref_ids": [] }, { "id": 4885, "type": "theorem", "label": "spaces-morphisms-lemma-syntomic-open", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-syntomic-open", "contents": [ "A syntomic morphism is universally open." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemmas \\ref{lemma-syntomic-locally-finite-presentation},", "\\ref{lemma-syntomic-flat}, and", "\\ref{lemma-fppf-open}." ], "refs": [ "spaces-morphisms-lemma-syntomic-locally-finite-presentation", "spaces-morphisms-lemma-syntomic-flat", "spaces-morphisms-lemma-fppf-open" ], "ref_ids": [ 4883, 4884, 4855 ] } ], "ref_ids": [] }, { "id": 4886, "type": "theorem", "label": "spaces-morphisms-lemma-composition-smooth", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-smooth", "contents": [ "The composition of smooth morphisms is smooth." ], "refs": [], "proofs": [ { "contents": [ "See Remark \\ref{remark-composition-P} and", "Morphisms, Lemma \\ref{morphisms-lemma-composition-smooth}." ], "refs": [ "spaces-morphisms-remark-composition-P", "morphisms-lemma-composition-smooth" ], "ref_ids": [ 5033, 5326 ] } ], "ref_ids": [] }, { "id": 4887, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-smooth", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-smooth", "contents": [ "The base change of a smooth morphism is smooth." ], "refs": [], "proofs": [ { "contents": [ "See Remark \\ref{remark-base-change-P} and", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-smooth}." ], "refs": [ "spaces-morphisms-remark-base-change-P", "morphisms-lemma-base-change-smooth" ], "ref_ids": [ 5034, 5327 ] } ], "ref_ids": [] }, { "id": 4888, "type": "theorem", "label": "spaces-morphisms-lemma-smooth-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-smooth-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is smooth,", "\\item for every $x \\in |X|$ the morphism $f$ is smooth at $x$,", "\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism", "$Z \\times_Y X \\to Z$ is smooth,", "\\item for every affine scheme $Z$ and any morphism", "$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is smooth,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is a smooth morphism,", "\\item there exists a scheme $U$ and a surjective \\'etale morphism", "$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$", "is smooth,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes and the vertical arrows are \\'etale", "the top horizontal arrow is smooth,", "\\item there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes, the vertical arrows are \\'etale, and $U \\to X$ is", "surjective such that the top horizontal arrow is smooth, and", "\\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$,", "and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that", "each morphism $X_{ij} \\to Y_i$ is smooth.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4889, "type": "theorem", "label": "spaces-morphisms-lemma-smooth-locally-finite-presentation", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-smooth-locally-finite-presentation", "contents": [ "A smooth morphism of algebraic spaces is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Let $X \\to Y$ be a smooth morphism of algebraic spaces. By", "definition this means there exists a diagram as in", "Lemma \\ref{lemma-local-source-target}", "with $h$ smooth and surjective vertical arrow $a$. By", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-locally-finite-presentation}", "$h$ is locally of finite presentation. Hence $X \\to Y$ is", "locally of finite presentation by definition." ], "refs": [ "spaces-morphisms-lemma-local-source-target", "morphisms-lemma-smooth-locally-finite-presentation" ], "ref_ids": [ 4811, 5330 ] } ], "ref_ids": [] }, { "id": 4890, "type": "theorem", "label": "spaces-morphisms-lemma-smooth-locally-finite-type", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-smooth-locally-finite-type", "contents": [ "A smooth morphism of algebraic spaces is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemmas \\ref{lemma-smooth-locally-finite-presentation} and", "\\ref{lemma-finite-presentation-finite-type}." ], "refs": [ "spaces-morphisms-lemma-smooth-locally-finite-presentation", "spaces-morphisms-lemma-finite-presentation-finite-type" ], "ref_ids": [ 4889, 4842 ] } ], "ref_ids": [] }, { "id": 4891, "type": "theorem", "label": "spaces-morphisms-lemma-smooth-flat", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-smooth-flat", "contents": [ "A smooth morphism of algebraic spaces is flat." ], "refs": [], "proofs": [ { "contents": [ "Let $X \\to Y$ be a smooth morphism of algebraic spaces. By", "definition this means there exists a diagram as in", "Lemma \\ref{lemma-local-source-target}", "with $h$ smooth and surjective vertical arrow $a$. By", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-locally-finite-presentation}", "$h$ is flat. Hence $X \\to Y$ is flat by definition." ], "refs": [ "spaces-morphisms-lemma-local-source-target", "morphisms-lemma-smooth-locally-finite-presentation" ], "ref_ids": [ 4811, 5330 ] } ], "ref_ids": [] }, { "id": 4892, "type": "theorem", "label": "spaces-morphisms-lemma-smooth-syntomic", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-smooth-syntomic", "contents": [ "A smooth morphism of algebraic spaces is syntomic." ], "refs": [], "proofs": [ { "contents": [ "Let $X \\to Y$ be a smooth morphism of algebraic spaces. By", "definition this means there exists a diagram as in", "Lemma \\ref{lemma-local-source-target}", "with $h$ smooth and surjective vertical arrow $a$. By", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-syntomic}", "$h$ is syntomic. Hence $X \\to Y$ is syntomic by definition." ], "refs": [ "spaces-morphisms-lemma-local-source-target", "morphisms-lemma-smooth-syntomic" ], "ref_ids": [ 4811, 5329 ] } ], "ref_ids": [] }, { "id": 4893, "type": "theorem", "label": "spaces-morphisms-lemma-where-smooth", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-where-smooth", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. There is a maximal open subspace $U \\subset X$", "such that $f|_U : U \\to Y$ is smooth. Moreover, formation of", "this open commutes with base change by", "\\begin{enumerate}", "\\item morphisms which are flat and", "locally of finite presentation,", "\\item flat morphisms provided $f$ is locally of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The existence of $U$ follows from the fact that the property", "of being smooth is Zariski (and even \\'etale) local on the source, see", "Lemma \\ref{lemma-smooth-local}. Moreover, this lemma allows", "us to translate properties (1) and (2) into the case", "of morphisms of schemes. The case of schemes is", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-smooth}.", "Some details omitted." ], "refs": [ "spaces-morphisms-lemma-smooth-local", "morphisms-lemma-set-points-where-fibres-smooth" ], "ref_ids": [ 4888, 5336 ] } ], "ref_ids": [] }, { "id": 4894, "type": "theorem", "label": "spaces-morphisms-lemma-smoothness-dimension-spaces", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-smoothness-dimension-spaces", "contents": [ "Let $X$ and $Y$ be locally Noetherian algebraic spaces over a scheme", "$S$, and let $f : X \\to Y$ be a smooth morphism.", "For every point $x \\in |X|$ with image $y \\in |Y|$,", "$$", "\\dim_x(X) = \\dim_y(Y) + \\dim_x(X_y)", "$$", "where $\\dim_x(X_y)$ is the relative dimension of $f$ at $x$ as", "in Definition \\ref{definition-dimension-fibre}." ], "refs": [ "spaces-morphisms-definition-dimension-fibre" ], "proofs": [ { "contents": [ "By definition of the dimension of an algebraic space", "at a point (Properties of Spaces, Definition", "\\ref{spaces-properties-definition-dimension-at-point}),", "this reduces to the corresponding statement for schemes", "(Morphisms, Lemma \\ref{morphisms-lemma-smoothness-dimension})." ], "refs": [ "spaces-properties-definition-dimension-at-point", "morphisms-lemma-smoothness-dimension" ], "ref_ids": [ 11929, 5342 ] } ], "ref_ids": [ 5009 ] }, { "id": 4895, "type": "theorem", "label": "spaces-morphisms-lemma-unramified-G-unramified", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-unramified-G-unramified", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Then $f$ is G-unramified if and only if $f$ is unramified and", "locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Consider any diagram as in", "Lemma \\ref{lemma-local-source-target}.", "Then all we are saying is that the morphism $h$ is", "G-unramified if and only if it is unramified and locally of finite", "presentation. This is clear from", "Morphisms, Definition \\ref{morphisms-definition-unramified}." ], "refs": [ "spaces-morphisms-lemma-local-source-target", "morphisms-definition-unramified" ], "ref_ids": [ 4811, 5566 ] } ], "ref_ids": [] }, { "id": 4896, "type": "theorem", "label": "spaces-morphisms-lemma-composition-unramified", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-unramified", "contents": [ "The composition of unramified morphisms is unramified." ], "refs": [], "proofs": [ { "contents": [ "See Remark \\ref{remark-composition-P} and", "Morphisms, Lemma \\ref{morphisms-lemma-composition-unramified}." ], "refs": [ "spaces-morphisms-remark-composition-P", "morphisms-lemma-composition-unramified" ], "ref_ids": [ 5033, 5345 ] } ], "ref_ids": [] }, { "id": 4897, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-unramified", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-unramified", "contents": [ "The base change of an unramified morphism is unramified." ], "refs": [], "proofs": [ { "contents": [ "See Remark \\ref{remark-base-change-P} and", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-unramified}." ], "refs": [ "spaces-morphisms-remark-base-change-P", "morphisms-lemma-base-change-unramified" ], "ref_ids": [ 5034, 5346 ] } ], "ref_ids": [] }, { "id": 4898, "type": "theorem", "label": "spaces-morphisms-lemma-unramified-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-unramified-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is unramified,", "\\item for every $x \\in |X|$ the morphism $f$ is unramified at $x$,", "\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism", "$Z \\times_Y X \\to Z$ is unramified,", "\\item for every affine scheme $Z$ and any morphism", "$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is unramified,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is an unramified morphism,", "\\item there exists a scheme $U$ and a surjective \\'etale morphism", "$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$", "is unramified,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes and the vertical arrows are \\'etale", "the top horizontal arrow is unramified,", "\\item there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes, the vertical arrows are \\'etale, and", "$U \\to X$ is surjective such that the top horizontal arrow is unramified, and", "\\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$,", "and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that", "each morphism $X_{ij} \\to Y_i$ is unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4899, "type": "theorem", "label": "spaces-morphisms-lemma-unramified-locally-finite-type", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-unramified-locally-finite-type", "contents": [ "An unramified morphism of algebraic spaces is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "Via a diagram as in", "Lemma \\ref{lemma-local-source-target}", "this translates into", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-locally-finite-type}." ], "refs": [ "spaces-morphisms-lemma-local-source-target", "morphisms-lemma-unramified-locally-finite-type" ], "ref_ids": [ 4811, 5350 ] } ], "ref_ids": [] }, { "id": 4900, "type": "theorem", "label": "spaces-morphisms-lemma-unramified-quasi-finite", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-unramified-quasi-finite", "contents": [ "If $f$ is unramified at $x$ then $f$ is quasi-finite at $x$.", "In particular, an unramified morphism is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Via a diagram as in", "Lemma \\ref{lemma-local-source-target}", "this translates into", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-quasi-finite}." ], "refs": [ "spaces-morphisms-lemma-local-source-target", "morphisms-lemma-unramified-quasi-finite" ], "ref_ids": [ 4811, 5351 ] } ], "ref_ids": [] }, { "id": 4901, "type": "theorem", "label": "spaces-morphisms-lemma-immersion-unramified", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-immersion-unramified", "contents": [ "An immersion of algebraic spaces is unramified." ], "refs": [], "proofs": [ { "contents": [ "Let $i : X \\to Y$ be an immersion of algebraic spaces. Choose a scheme", "$V$ and a surjective \\'etale morphism $V \\to Y$. Then $V \\times_Y X \\to V$", "is an immersion of schemes, hence unramified (see", "Morphisms, Lemmas \\ref{morphisms-lemma-open-immersion-unramified} and", "\\ref{morphisms-lemma-closed-immersion-unramified}).", "Thus by definition $i$ is unramified." ], "refs": [ "morphisms-lemma-open-immersion-unramified", "morphisms-lemma-closed-immersion-unramified" ], "ref_ids": [ 5348, 5349 ] } ], "ref_ids": [] }, { "id": 4902, "type": "theorem", "label": "spaces-morphisms-lemma-diagonal-unramified-morphism", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-diagonal-unramified-morphism", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item If $f$ is unramified, then the diagonal morphism", "$\\Delta_{X/Y} : X \\to X \\times_Y X$ is an open immersion.", "\\item If $f$ is locally of finite type", "and $\\Delta_{X/Y}$ is an open immersion, then $f$ is unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We know in any case that $\\Delta_{X/Y}$ is a representable monomorphism, see", "Lemma \\ref{lemma-properties-diagonal}.", "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X \\times_Y V$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[rr]_-{\\Delta_{U/V}} & &", "U \\times_V U \\ar[d] \\ar[r] &", "V \\ar[d]^{\\Delta_{V/Y}} \\\\", "X \\ar[rr]^-{\\Delta_{X/Y}} & &", "X \\times_Y X \\ar[r] &", "V \\times_Y V", "}", "$$", "with cartesian right square. The left vertical arrow is surjective \\'etale.", "The right vertical arrow is \\'etale as a morphism between schemes", "\\'etale over $Y$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-etale-permanence}.", "Hence the middle vertical arrow is \\'etale too (but it need not be", "surjective).", "\\medskip\\noindent", "Assume $f$ is unramified. Then $U \\to V$ is unramified, hence", "$\\Delta_{U/V}$ is an open immersion by", "Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism}.", "Looking at the left square of the diagram above we conclude that", "$\\Delta_{X/Y}$ is an \\'etale morphism, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-etale-local}.", "Hence $\\Delta_{X/Y}$ is a representable \\'etale monomorphism, which", "implies that it is an open immersion by", "\\'Etale Morphisms, Theorem \\ref{etale-theorem-etale-radicial-open}.", "(See also", "Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}", "for the translation from schemes language into the language of functors.)", "\\medskip\\noindent", "Assume that $f$ is locally of finite type and that $\\Delta_{X/Y}$", "is an open immersion. This implies that $U \\to V$ is locally of finite", "type too (by definition of a morphism of algebraic spaces which is", "locally of finite type). Looking at the displayed diagram above", "we conclude that $\\Delta_{U/V}$ is \\'etale as a morphism between", "schemes \\'etale over $X \\times_Y X$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-etale-permanence}.", "But since $\\Delta_{U/V}$ is the diagonal of a morphism between schemes", "we see that it is in any case an immersion, see", "Schemes, Lemma \\ref{schemes-lemma-diagonal-immersion}.", "Hence it is an open immersion, and we conclude", "that $U \\to V$ is unramified by", "Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism}.", "This in turn means that $f$ is unramified by definition." ], "refs": [ "spaces-morphisms-lemma-properties-diagonal", "spaces-properties-lemma-etale-permanence", "morphisms-lemma-diagonal-unramified-morphism", "spaces-properties-lemma-etale-local", "etale-theorem-etale-radicial-open", "spaces-lemma-representable-transformations-property-implication", "spaces-properties-lemma-etale-permanence", "schemes-lemma-diagonal-immersion", "morphisms-lemma-diagonal-unramified-morphism" ], "ref_ids": [ 4712, 11859, 5354, 11856, 10692, 8136, 11859, 7707, 5354 ] } ], "ref_ids": [] }, { "id": 4903, "type": "theorem", "label": "spaces-morphisms-lemma-where-unramified", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-where-unramified", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\", "& Z", "}", "$$", "of algebraic spaces over $S$. Assume that $X \\to Z$ is locally of", "finite type. Then there exists an open subspace $U(f) \\subset X$", "such that $|U(f)| \\subset |X|$ is the set of points where $f$ is unramified.", "Moreover, for any morphism of algebraic spaces $Z' \\to Z$, if", "$f' : X' \\to Y'$ is the base change of $f$ by $Z' \\to Z$, then", "$U(f')$ is the inverse image of $U(f)$ under the projection $X' \\to X$." ], "refs": [], "proofs": [ { "contents": [ "This lemma is the analogue of", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-unramified}", "and in fact we will deduce the lemma from it. By", "Definition \\ref{definition-unramified}", "the set $\\{x \\in |X| : f \\text{ is unramified at }x\\}$ is", "open in $X$. Hence we only need to prove the final statement. By", "Lemma \\ref{lemma-permanence-finite-type}", "the morphism $X \\to Y$ is locally of finite type. By", "Lemma \\ref{lemma-base-change-finite-type}", "the morphism $X' \\to Y'$ is locally of finite type.", "\\medskip\\noindent", "Choose a scheme $W$ and a surjective \\'etale morphism $W \\to Z$.", "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to W \\times_Z Y$.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to V \\times_Y X$.", "Finally, choose a scheme $W'$ and a surjective \\'etale morphism", "$W' \\to W \\times_Z Z'$.", "Set $V' = W' \\times_W V$ and $U' = W' \\times_W U$, so that we obtain", "surjective \\'etale morphisms $V' \\to Y'$ and $U' \\to X'$.", "We will use without further mention an \\'etale morphism of algebraic spaces", "induces an open map of associated topological spaces (see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-etale-open}).", "This combined with", "Lemma \\ref{lemma-unramified-local}", "implies that $U(f)$ is the image in $|X|$ of the set $T$ of points in $U$", "where the morphism $U \\to V$ is unramified. Similarly, $U(f')$ is the image", "in $|X'|$ of the set $T'$ of points in $U'$ where the morphism $U' \\to V'$", "is unramified. Now, by construction the diagram", "$$", "\\xymatrix{", "U' \\ar[r] \\ar[d] & U \\ar[d] \\\\", "V' \\ar[r] & V", "}", "$$", "is cartesian (in the category of schemes). Hence the aforementioned", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-unramified}", "applies to show that $T'$ is the inverse image of $T$. Since", "$|U'| \\to |X'|$ is surjective this implies the lemma." ], "refs": [ "morphisms-lemma-set-points-where-fibres-unramified", "spaces-morphisms-definition-unramified", "spaces-morphisms-lemma-permanence-finite-type", "spaces-morphisms-lemma-base-change-finite-type", "spaces-properties-lemma-etale-open", "spaces-morphisms-lemma-unramified-local", "morphisms-lemma-set-points-where-fibres-unramified" ], "ref_ids": [ 5356, 5013, 4818, 4815, 11860, 4898, 5356 ] } ], "ref_ids": [] }, { "id": 4904, "type": "theorem", "label": "spaces-morphisms-lemma-permanence-unramified", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-permanence-unramified", "contents": [ "Let $S$ be a scheme.", "Let $X \\to Y \\to Z$ be morphisms of algebraic spaces over $S$.", "If $X \\to Z$ is unramified, then $X \\to Y$ is unramified." ], "refs": [], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\ar[r] & W \\ar[d] \\\\", "X \\ar[r] & Y \\ar[r] & Z", "}", "$$", "with vertical arrows \\'etale and surjective. (See", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}.)", "Apply", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-permanence}", "to the top row." ], "refs": [ "spaces-lemma-lift-morphism-presentations", "morphisms-lemma-unramified-permanence" ], "ref_ids": [ 8159, 5357 ] } ], "ref_ids": [] }, { "id": 4905, "type": "theorem", "label": "spaces-morphisms-lemma-etale-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-etale-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is \\'etale,", "\\item for every $x \\in |X|$ the morphism $f$ is \\'etale at $x$,", "\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism", "$Z \\times_Y X \\to Z$ is \\'etale,", "\\item for every affine scheme $Z$ and any morphism", "$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is \\'etale,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is an \\'etale morphism,", "\\item there exists a scheme $U$ and a surjective \\'etale morphism", "$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$", "is \\'etale,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes and the vertical arrows are \\'etale", "the top horizontal arrow is \\'etale,", "\\item there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes, the vertical arrows are \\'etale, and", "$U \\to X$ surjective such that the top horizontal arrow is \\'etale, and", "\\item there exist Zariski coverings $Y = \\bigcup Y_i$ and", "$f^{-1}(Y_i) = \\bigcup X_{ij}$ such that each morphism", "$X_{ij} \\to Y_i$ is \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Combine", "Properties of Spaces, Lemmas", "\\ref{spaces-properties-lemma-etale-local},", "\\ref{spaces-properties-lemma-base-change-etale} and", "\\ref{spaces-properties-lemma-composition-etale}.", "Some details omitted." ], "refs": [ "spaces-properties-lemma-etale-local", "spaces-properties-lemma-base-change-etale", "spaces-properties-lemma-composition-etale" ], "ref_ids": [ 11856, 11858, 11857 ] } ], "ref_ids": [] }, { "id": 4906, "type": "theorem", "label": "spaces-morphisms-lemma-composition-etale", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-etale", "contents": [ "The composition of two \\'etale morphisms of algebraic spaces", "is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "This is a copy of", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-composition-etale}." ], "refs": [ "spaces-properties-lemma-composition-etale" ], "ref_ids": [ 11857 ] } ], "ref_ids": [] }, { "id": 4907, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-etale", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-etale", "contents": [ "The base change of an \\'etale morphism of algebraic spaces", "by any morphism of algebraic spaces is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "This is a copy of", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-base-change-etale}." ], "refs": [ "spaces-properties-lemma-base-change-etale" ], "ref_ids": [ 11858 ] } ], "ref_ids": [] }, { "id": 4908, "type": "theorem", "label": "spaces-morphisms-lemma-etale-locally-quasi-finite", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-etale-locally-quasi-finite", "contents": [ "An \\'etale morphism of algebraic spaces is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Let $X \\to Y$ be an \\'etale morphism of algebraic spaces, see", "Properties of Spaces, Definition \\ref{spaces-properties-definition-etale}.", "By", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-local}", "we see this means there exists a diagram as in", "Lemma \\ref{lemma-local-source-target}", "with $h$ \\'etale and surjective vertical arrow $a$. By", "Morphisms, Lemma \\ref{morphisms-lemma-etale-locally-quasi-finite}", "$h$ is locally quasi-finite. Hence $X \\to Y$ is locally quasi-finite", "by definition." ], "refs": [ "spaces-properties-definition-etale", "spaces-properties-lemma-etale-local", "spaces-morphisms-lemma-local-source-target", "morphisms-lemma-etale-locally-quasi-finite" ], "ref_ids": [ 11933, 11856, 4811, 5363 ] } ], "ref_ids": [] }, { "id": 4909, "type": "theorem", "label": "spaces-morphisms-lemma-etale-smooth", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-etale-smooth", "contents": [ "An \\'etale morphism of algebraic spaces is smooth." ], "refs": [], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Lemma \\ref{lemma-etale-locally-quasi-finite}.", "It uses the fact that an \\'etale morphism of schemes is smooth", "(by definition of an \\'etale morphism of schemes)." ], "refs": [ "spaces-morphisms-lemma-etale-locally-quasi-finite" ], "ref_ids": [ 4908 ] } ], "ref_ids": [] }, { "id": 4910, "type": "theorem", "label": "spaces-morphisms-lemma-etale-flat", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-etale-flat", "contents": [ "An \\'etale morphism of algebraic spaces is flat." ], "refs": [], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Lemma \\ref{lemma-etale-locally-quasi-finite}.", "It uses", "Morphisms, Lemma \\ref{morphisms-lemma-etale-flat}." ], "refs": [ "spaces-morphisms-lemma-etale-locally-quasi-finite", "morphisms-lemma-etale-flat" ], "ref_ids": [ 4908, 5369 ] } ], "ref_ids": [] }, { "id": 4911, "type": "theorem", "label": "spaces-morphisms-lemma-etale-locally-finite-presentation", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-etale-locally-finite-presentation", "contents": [ "\\begin{slogan}", "\\'Etale implies locally of finite presentation.", "\\end{slogan}", "An \\'etale morphism of algebraic spaces is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Lemma \\ref{lemma-etale-locally-quasi-finite}.", "It uses", "Morphisms, Lemma \\ref{morphisms-lemma-etale-locally-finite-presentation}." ], "refs": [ "spaces-morphisms-lemma-etale-locally-quasi-finite", "morphisms-lemma-etale-locally-finite-presentation" ], "ref_ids": [ 4908, 5368 ] } ], "ref_ids": [] }, { "id": 4912, "type": "theorem", "label": "spaces-morphisms-lemma-etale-locally-finite-type", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-etale-locally-finite-type", "contents": [ "An \\'etale morphism of algebraic spaces is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "An \\'etale morphism is locally of finite presentation", "and a morphism locally of finite presentation is locally of finite type,", "see", "Lemmas \\ref{lemma-etale-locally-finite-presentation} and", "\\ref{lemma-finite-presentation-finite-type}." ], "refs": [ "spaces-morphisms-lemma-etale-locally-finite-presentation", "spaces-morphisms-lemma-finite-presentation-finite-type" ], "ref_ids": [ 4911, 4842 ] } ], "ref_ids": [] }, { "id": 4913, "type": "theorem", "label": "spaces-morphisms-lemma-etale-unramified", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-etale-unramified", "contents": [ "An \\'etale morphism of algebraic spaces is unramified." ], "refs": [], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Lemma \\ref{lemma-etale-locally-quasi-finite}.", "It uses", "Morphisms, Lemma \\ref{morphisms-lemma-etale-smooth-unramified}." ], "refs": [ "spaces-morphisms-lemma-etale-locally-quasi-finite", "morphisms-lemma-etale-smooth-unramified" ], "ref_ids": [ 4908, 5362 ] } ], "ref_ids": [] }, { "id": 4914, "type": "theorem", "label": "spaces-morphisms-lemma-etale-permanence", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-etale-permanence", "contents": [ "Let $S$ be a scheme. Let $X, Y$ be algebraic spaces \\'etale over", "an algebraic space $Z$. Any morphism $X \\to Y$ over $Z$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "This is a copy of", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-permanence}." ], "refs": [ "spaces-properties-lemma-etale-permanence" ], "ref_ids": [ 11859 ] } ], "ref_ids": [] }, { "id": 4915, "type": "theorem", "label": "spaces-morphisms-lemma-unramified-flat-lfp-etale", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-unramified-flat-lfp-etale", "contents": [ "A locally finitely presented, flat, unramified morphism of algebraic", "spaces is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Let $X \\to Y$ be a locally finitely presented, flat, unramified morphism", "of algebraic spaces. By", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-local}", "we see this means there exists a diagram as in", "Lemma \\ref{lemma-local-source-target}", "with $h$ locally finitely presented, flat, unramified", "and surjective vertical arrow $a$. By", "Morphisms, Lemma \\ref{morphisms-lemma-flat-unramified-etale}", "$h$ is \\'etale. Hence $X \\to Y$ is \\'etale by definition." ], "refs": [ "spaces-properties-lemma-etale-local", "spaces-morphisms-lemma-local-source-target", "morphisms-lemma-flat-unramified-etale" ], "ref_ids": [ 11856, 4811, 5373 ] } ], "ref_ids": [] }, { "id": 4916, "type": "theorem", "label": "spaces-morphisms-lemma-proper-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-proper-local", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is proper,", "\\item for every scheme $Z$ and every morphism $Z \\to Y$", "the projection $Z \\times_Y X \\to Z$ is proper,", "\\item for every affine scheme $Z$ and every morphism $Z \\to Y$", "the projection $Z \\times_Y X \\to Z$ is proper,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is proper, and", "\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that", "each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is proper.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-separated-local},", "\\ref{lemma-finite-type-local},", "\\ref{lemma-quasi-compact-local}, and", "\\ref{lemma-universally-closed-local}." ], "refs": [ "spaces-morphisms-lemma-separated-local", "spaces-morphisms-lemma-finite-type-local", "spaces-morphisms-lemma-quasi-compact-local", "spaces-morphisms-lemma-universally-closed-local" ], "ref_ids": [ 4722, 4816, 4742, 4748 ] } ], "ref_ids": [] }, { "id": 4917, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-proper", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-proper", "contents": [ "A base change of a proper morphism is proper." ], "refs": [], "proofs": [ { "contents": [ "See", "Lemmas \\ref{lemma-base-change-separated},", "\\ref{lemma-base-change-finite-type}, and", "\\ref{lemma-base-change-universally-closed}." ], "refs": [ "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-base-change-finite-type", "spaces-morphisms-lemma-base-change-universally-closed" ], "ref_ids": [ 4714, 4815, 4746 ] } ], "ref_ids": [] }, { "id": 4918, "type": "theorem", "label": "spaces-morphisms-lemma-composition-proper", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-proper", "contents": [ "A composition of proper morphisms is proper." ], "refs": [], "proofs": [ { "contents": [ "See", "Lemmas \\ref{lemma-composition-separated},", "\\ref{lemma-composition-finite-type}, and", "\\ref{lemma-composition-universally-closed}." ], "refs": [ "spaces-morphisms-lemma-composition-separated", "spaces-morphisms-lemma-composition-finite-type", "spaces-morphisms-lemma-composition-universally-closed" ], "ref_ids": [ 4718, 4814, 4747 ] } ], "ref_ids": [] }, { "id": 4919, "type": "theorem", "label": "spaces-morphisms-lemma-closed-immersion-proper", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-closed-immersion-proper", "contents": [ "A closed immersion of algebraic spaces is a proper morphism of", "algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "As a closed immersion is by definition representable this follows from", "Spaces,", "Lemma \\ref{spaces-lemma-representable-transformations-property-implication}", "and the corresponding result for morphisms of schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-proper}." ], "refs": [ "spaces-lemma-representable-transformations-property-implication", "morphisms-lemma-closed-immersion-proper" ], "ref_ids": [ 8136, 5410 ] } ], "ref_ids": [] }, { "id": 4920, "type": "theorem", "label": "spaces-morphisms-lemma-universally-closed-permanence", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-universally-closed-permanence", "contents": [ "Let $S$ be a scheme.", "Consider a commutative diagram of algebraic spaces", "$$", "\\xymatrix{", "X \\ar[rr] \\ar[rd] & &", "Y \\ar[ld] \\\\", "& B &", "}", "$$", "over $S$.", "\\begin{enumerate}", "\\item If $X \\to B$ is universally closed and $Y \\to B$ is", "separated, then the morphism $X \\to Y$ is universally closed.", "In particular, the image of $|X|$ in $|Y|$ is closed.", "\\item If $X \\to B$ is proper and $Y \\to B$ is separated, then", "the morphism $X \\to Y$ is proper.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $X \\to B$ is universally closed and $Y \\to B$ is separated.", "We factor the morphism as $X \\to X \\times_B Y \\to Y$.", "The first morphism is a closed immersion, see", "Lemma \\ref{lemma-semi-diagonal}", "hence universally closed.", "The projection $X \\times_B Y \\to Y$ is the base change", "of a universally closed morphism and hence", "universally closed, see", "Lemma \\ref{lemma-base-change-universally-closed}.", "Thus $X \\to Y$ is universally closed as the composition", "of universally closed morphisms, see", "Lemma \\ref{lemma-composition-universally-closed}.", "This proves (1). To deduce (2) combine (1) with", "Lemmas \\ref{lemma-compose-after-separated},", "\\ref{lemma-quasi-compact-permanence}, and", "\\ref{lemma-permanence-finite-type}." ], "refs": [ "spaces-morphisms-lemma-semi-diagonal", "spaces-morphisms-lemma-base-change-universally-closed", "spaces-morphisms-lemma-composition-universally-closed", "spaces-morphisms-lemma-compose-after-separated", "spaces-morphisms-lemma-quasi-compact-permanence", "spaces-morphisms-lemma-permanence-finite-type" ], "ref_ids": [ 4716, 4746, 4747, 4720, 4743, 4818 ] } ], "ref_ids": [] }, { "id": 4921, "type": "theorem", "label": "spaces-morphisms-lemma-image-proper-is-proper", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-image-proper-is-proper", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $B$.", "If $X$ is universally closed over $B$ and $f$ is surjective then", "$Y$ is universally closed over $B$. In particular, if also $Y$ is", "separated and of finite type over $B$, then $Y$ is proper over $B$." ], "refs": [], "proofs": [ { "contents": [ "Assume $X$ is universally closed and $f$ surjective.", "Denote $p : X \\to B$, $q : Y \\to B$ the structure morphisms.", "Let $B' \\to B$ be a morphism of algebraic spaces over $S$.", "The base change $f' : X_{B'} \\to Y_{B'}$ is surjective", "(Lemma \\ref{lemma-base-change-surjective}), and the base", "change $p' : X_{B'} \\to B'$ is closed.", "If $T \\subset Y_{B'}$ is closed, then $(f')^{-1}(T) \\subset X_{B'}$", "is closed, hence $p'((f')^{-1}(T)) = q'(T)$ is closed.", "So $q'$ is closed." ], "refs": [ "spaces-morphisms-lemma-base-change-surjective" ], "ref_ids": [ 4727 ] } ], "ref_ids": [] }, { "id": 4922, "type": "theorem", "label": "spaces-morphisms-lemma-scheme-theoretic-image-is-proper", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-scheme-theoretic-image-is-proper", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X \\ar[rr]_h \\ar[rd]_f & & Y \\ar[ld]^g \\\\", "& B", "}", "$$", "be a commutative diagram of morphism of algebraic spaces over $S$.", "Assume", "\\begin{enumerate}", "\\item $X \\to B$ is a proper morphism,", "\\item $Y \\to B$ is separated and locally of finite type,", "\\end{enumerate}", "Then the scheme theoretic image $Z \\subset Y$ of $h$", "is proper over $B$ and $X \\to Z$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "The scheme theoretic image of $h$ is constructed in Section", "\\ref{section-scheme-theoretic-image}.", "Observe that $h$ is quasi-compact", "(Lemma \\ref{lemma-quasi-compact-quasi-separated-permanence})", "hence $|h|(|X|) \\subset |Z|$", "is dense (Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image}).", "On the other hand $|h|(|X|)$ is closed in $|Y|$", "(Lemma \\ref{lemma-universally-closed-permanence})", "hence $X \\to Z$ is surjective.", "Thus $Z \\to B$ is a proper (Lemma \\ref{lemma-image-proper-is-proper})." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence", "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-morphisms-lemma-image-proper-is-proper" ], "ref_ids": [ 4744, 4780, 4920, 4921 ] } ], "ref_ids": [] }, { "id": 4923, "type": "theorem", "label": "spaces-morphisms-lemma-separated-diagonal-proper", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-separated-diagonal-proper", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is separated,", "\\item $\\Delta_{X/Y} : X \\to X \\times_Y X$ is universally closed, and", "\\item $\\Delta_{X/Y} : X \\to X \\times_Y X$ is proper.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (3) follows from", "Lemma \\ref{lemma-closed-immersion-proper}.", "We will use", "Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}", "without further mention in the rest of the proof.", "Recall that $\\Delta_{X/Y}$ is a representable", "monomorphism which is locally of finite type, see", "Lemma \\ref{lemma-properties-diagonal}.", "Since proper $\\Rightarrow$ universally closed for morphisms of schemes", "we conclude that (3) implies (2).", "If $\\Delta_{X/Y}$ is universally closed then", "\\'Etale Morphisms,", "Lemma \\ref{etale-lemma-characterize-closed-immersion}", "implies that it is a closed immersion. Thus (2) $\\Rightarrow$ (1)", "and we win." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-proper", "spaces-lemma-representable-transformations-property-implication", "spaces-morphisms-lemma-properties-diagonal", "etale-lemma-characterize-closed-immersion" ], "ref_ids": [ 4919, 8136, 4712, 10702 ] } ], "ref_ids": [] }, { "id": 4924, "type": "theorem", "label": "spaces-morphisms-lemma-valuative-criterion-representable", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-valuative-criterion-representable", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume $f$ is representable. The following are equivalent", "\\begin{enumerate}", "\\item $f$ satisfies the existence part of the valuative criterion", "as in Definition \\ref{definition-valuative-criterion},", "\\item given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$, there exists", "a dotted arrow, i.e., $f$ satisfies the existence part of the valuative", "criterion as in", "Schemes, Definition \\ref{schemes-definition-valuative-criterion}.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-valuative-criterion", "schemes-definition-valuative-criterion" ], "proofs": [ { "contents": [ "It suffices to show that given a commutative diagram of the form", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\", "\\Spec(A') \\ar[r] \\ar[rru]^\\varphi & \\Spec(A) \\ar[r] & Y", "}", "$$", "as in Definition \\ref{definition-valuative-criterion}, then we can", "find a morphism $\\Spec(A) \\to X$ fitting into the diagram too.", "Set $X_A = \\Spec(A) \\times_Y Y$. As $f$ is representable we see", "that $X_A$ is a scheme. The morphism $\\varphi$ gives a morphism", "$\\varphi' : \\Spec(A') \\to X_A$. Let $x \\in X_A$ be the image of", "the closed point of $\\varphi' : \\Spec(A') \\to X_A$. Then we", "have the following commutative diagram of rings", "$$", "\\xymatrix{", "K' & K \\ar[l] & \\mathcal{O}_{X_A, x} \\ar[l] \\ar[lld] \\\\", "A' \\ar[u] & A \\ar[l] & A \\ar[l] \\ar[u]", "}", "$$", "Since $A$ is a valuation ring, and since $A'$ dominates $A$, we see", "that $K \\cap A' = A$. Hence the ring map $\\mathcal{O}_{X_A, x} \\to K$", "has image contained in $A$. Whence a morphism $\\Spec(A) \\to X_A$ (see", "Schemes, Section \\ref{schemes-section-points})", "as desired." ], "refs": [ "spaces-morphisms-definition-valuative-criterion" ], "ref_ids": [ 5016 ] } ], "ref_ids": [ 5016, 7755 ] }, { "id": 4925, "type": "theorem", "label": "spaces-morphisms-lemma-finite-separable-enough", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-separable-enough", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ satisfies the existence part of the valuative criterion", "as in Definition \\ref{definition-valuative-criterion},", "\\item $f$ satisfies the existence part of the valuative criterion", "as in Definition \\ref{definition-valuative-criterion} modified by", "requiring the extension $K \\subset K'$ to be finite separable.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-valuative-criterion", "spaces-morphisms-definition-valuative-criterion" ], "proofs": [ { "contents": [ "We have to show that (1) implies (2). Suppose given a diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\", "\\Spec(A') \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] & Y", "}", "$$", "as in Definition \\ref{definition-valuative-criterion} with $K \\subset K'$", "arbitrary. Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "Then", "$$", "\\Spec(A') \\times_X U \\longrightarrow \\Spec(A')", "$$", "is surjective \\'etale. Let $p$ be a point of $\\Spec(A') \\times_X U$", "mapping to the closed point of $\\Spec(A')$. Let $p' \\leadsto p$", "be a generalization of $p$ mapping to the generic point of $\\Spec(A')$.", "Such a generalization exists because generalizations lift along flat", "morphisms of schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat}.", "Then $p'$ corresponds to a point of the scheme $\\Spec(K') \\times_X U$.", "Note that", "$$", "\\Spec(K') \\times_X U", "=", "\\Spec(K') \\times_{\\Spec(K)} (\\Spec(K) \\times_X U)", "$$", "Hence $p'$ maps to a point $q' \\in \\Spec(K) \\times_X U$ whose", "residue field is a finite separable extension of $K$. Finally,", "$p' \\leadsto p$ maps to a specialization $u' \\leadsto u$ on the", "scheme $U$. With all this notation we get the following diagram of", "rings", "$$", "\\xymatrix{", "\\kappa(p') & & \\kappa(q') \\ar[ll] & \\kappa(u') \\ar[l] \\\\", "& \\mathcal{O}_{\\Spec(A') \\times_X U, p} \\ar[lu] & &", "\\mathcal{O}_{U, u} \\ar[ll] \\ar[u] \\\\", "K' \\ar[uu] & A' \\ar[l] \\ar[u] & A \\ar[l] \\ar'[u][uu]", "}", "$$", "This means that the ring $B \\subset \\kappa(q')$ generated by", "the images of $A$ and $\\mathcal{O}_{U, u}$ maps to a subring", "of $\\kappa(p')$ contained in the image $B'$ of", "$\\mathcal{O}_{\\Spec(A') \\times_X U, p} \\to \\kappa(p')$.", "Note that $B'$ is a local ring. Let $\\mathfrak m \\subset B$", "be the maximal ideal. By construction $A \\cap \\mathfrak m$,", "(resp.\\ $\\mathcal{O}_{U, u} \\cap \\mathfrak m$, resp.\\ $A' \\cap \\mathfrak m$)", "is the maximal ideal of $A$ (resp.\\ $\\mathcal{O}_{U, u}$, resp.\\ $A'$).", "Set $\\mathfrak q = B \\cap \\mathfrak m$. This is a", "prime ideal such that $A \\cap \\mathfrak q$ is the maximal ideal of $A$.", "Hence $B_{\\mathfrak q} \\subset \\kappa(q')$ is a local ring dominating", "$A$. By", "Algebra, Lemma \\ref{algebra-lemma-dominate}", "we can find a valuation ring $A_1 \\subset \\kappa(q')$", "with field of fractions $\\kappa(q')$", "dominating $B_{\\mathfrak q}$. The (local) ring map", "$\\mathcal{O}_{U, u} \\to A_1$ gives a morphism", "$\\Spec(A_1) \\to U \\to X$", "such that the diagram", "$$", "\\xymatrix{", "\\Spec(\\kappa(q')) \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\", "\\Spec(A_1) \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] & Y", "}", "$$", "is commutative. Since the fraction field of $A_1$ is $\\kappa(q')$", "and since $\\kappa(q')/K$ is finite", "separable by construction the lemma is proved." ], "refs": [ "spaces-morphisms-definition-valuative-criterion", "morphisms-lemma-generalizations-lift-flat", "algebra-lemma-dominate" ], "ref_ids": [ 5016, 5266, 608 ] } ], "ref_ids": [ 5016, 5016 ] }, { "id": 4926, "type": "theorem", "label": "spaces-morphisms-lemma-push-down-solution", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-push-down-solution", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a separated morphism of", "algebraic spaces over $S$. Suppose given a diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\", "\\Spec(A') \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y", "}", "$$", "as in Definition \\ref{definition-valuative-criterion} with $K \\subset K'$", "arbitrary. Then the dotted arrow exists making the diagram commute." ], "refs": [ "spaces-morphisms-definition-valuative-criterion" ], "proofs": [ { "contents": [ "We have to show that we can find a morphism $\\Spec(A) \\to X$ fitting", "into the diagram.", "\\medskip\\noindent", "Consider the base change $X_A = \\Spec(A) \\times_Y X$ of $X$.", "Then $X_A \\to \\Spec(A)$ is a separated morphism of algebraic", "spaces (Lemma \\ref{lemma-base-change-separated}). Base changing", "all the morphisms of the diagram above we obtain", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X_A \\ar[d] \\\\", "\\Spec(A') \\ar[r] \\ar[rru] & \\Spec(A) \\ar@{=}[r] & \\Spec(A)", "}", "$$", "Thus we may replace $X$ by $X_A$, assume that $Y = \\Spec(A)$ and", "that we have a diagram as above. We may and do replace $X$ by", "a quasi-compact open subspace containing the image of $|\\Spec(A')| \\to |X|$.", "\\medskip\\noindent", "The morphism $\\Spec(A') \\to X$ is quasi-compact by", "Lemma \\ref{lemma-quasi-compact-permanence}.", "Let $Z \\subset X$ be the scheme theoretic image of", "$\\Spec(A') \\to X$. Then $Z$ is a reduced", "(Lemma \\ref{lemma-scheme-theoretic-image-reduced}),", "quasi-compact (as a closed subspace of $X$), separated", "(as a closed subspace of $X$) algebraic space over $A$.", "Consider the base change", "$$", "\\Spec(K') = \\Spec(A') \\times_{\\Spec(A)} \\Spec(K) \\to", "X \\times_{\\Spec(A)} \\Spec(K) = X_K", "$$", "of the morphism $\\Spec(A') \\to X$ by the flat morphism of schemes", "$\\Spec(K) \\to \\Spec(A)$. By", "Lemma \\ref{lemma-flat-base-change-scheme-theoretic-image}", "we see that the scheme theoretic image of this morphism", "is the base change $Z_K$ of $Z$. On the other hand, by", "assumption (i.e., the commutative diagram above)", "this morphism factors through a morphism", "$\\Spec(K) \\to Z_K$ which is a section to the structure morphism", "$Z_K \\to \\Spec(K)$. As $Z_K$ is separated, this section is a", "closed immersion (Lemma \\ref{lemma-section-immersion}).", "We conclude that $Z_K = \\Spec(K)$.", "\\medskip\\noindent", "Let $V \\to Z$ be a surjective \\'etale morphism with $V$ an", "affine scheme (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "Say $V = \\Spec(B)$. Then $V \\times_Z \\Spec(A') = \\Spec(C)$", "is affine as $Z$ is separated. Note that $B \\to C$ is injective", "as $V$ is the scheme theoretic image of $V \\times_Z \\Spec(A') \\to V$", "by Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image}.", "On the other hand, $A' \\to C$ is \\'etale as corresponds", "to the base change of $V \\to Z$.", "Since $A'$ is a torsion free $A$-module,", "the flatness of $A' \\to C$ implies $C$ is a torsion free", "$A$-module, hence $B$ is a torsion free $A$-module.", "Note that being torsion free as an $A$-module is", "equivalent to being flat (More on Algebra, Lemma", "\\ref{more-algebra-lemma-valuation-ring-torsion-free-flat}).", "Next, we write", "$$", "V \\times_Z V = \\Spec(B')", "$$", "Note that the two ring maps $B \\to B'$ are \\'etale as $V \\to Z$", "is \\'etale. The canonical surjective map $B \\otimes_A B \\to B'$", "becomes an isomorphism after tensoring with $K$ over $A$", "because $Z_K = \\Spec(K)$. However, $B \\otimes_A B$ is torsion", "free as an $A$-module by our remarks above. Thus $B' = B \\otimes_A B$.", "It follows that the base change of the ring map", "$A \\to B$ by the faithfully flat ring map $A \\to B$", "is \\'etale (note that $\\Spec(B) \\to \\Spec(A)$ is surjective", "as $X \\to \\Spec(A)$ is surjective). Hence $A \\to B$ is", "\\'etale (Descent, Lemma \\ref{descent-lemma-descending-property-etale}),", "in other words, $V \\to X$ is \\'etale.", "Since we have $V \\times_Z V = V \\times_{\\Spec(A)} V$", "we conclude that $Z = \\Spec(A)$ as algebraic spaces", "(for example by Spaces, Lemma \\ref{spaces-lemma-space-presentation})", "and the proof is complete." ], "refs": [ "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-quasi-compact-permanence", "spaces-morphisms-lemma-scheme-theoretic-image-reduced", "spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image", "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image", "more-algebra-lemma-valuation-ring-torsion-free-flat", "descent-lemma-descending-property-etale", "spaces-lemma-space-presentation" ], "ref_ids": [ 4714, 4743, 4781, 4861, 11832, 4780, 9920, 14694, 8149 ] } ], "ref_ids": [ 5016 ] }, { "id": 4927, "type": "theorem", "label": "spaces-morphisms-lemma-usual-enough", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-usual-enough", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a separated morphism of", "algebraic spaces over $S$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ satisfies the existence part of the valuative criterion", "as in Definition \\ref{definition-valuative-criterion},", "\\item given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$, there exists", "a dotted arrow, i.e., $f$ satisfies the existence part of the valuative", "criterion as in", "Schemes, Definition \\ref{schemes-definition-valuative-criterion}.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-valuative-criterion", "schemes-definition-valuative-criterion" ], "proofs": [ { "contents": [ "We have to show that (1) implies (2). Suppose given a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] & Y", "}", "$$", "as in part (2). By (1) there exists a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\", "\\Spec(A') \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] & Y", "}", "$$", "as in Definition \\ref{definition-valuative-criterion} with $K \\subset K'$", "arbitrary. By Lemma \\ref{lemma-push-down-solution} we can find a morphism", "$\\Spec(A) \\to X$ fitting into the diagram, i.e., (2) holds." ], "refs": [ "spaces-morphisms-definition-valuative-criterion", "spaces-morphisms-lemma-push-down-solution" ], "ref_ids": [ 5016, 4926 ] } ], "ref_ids": [ 5016, 7755 ] }, { "id": 4928, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-valuative-criteria", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-valuative-criteria", "contents": [ "The base change of a morphism of algebraic spaces which satisfies the", "existence part of (resp.\\ uniqueness part of) the valuative criterion", "by any morphism of algebraic spaces satisfies the", "existence part of (resp.\\ uniqueness part of) the valuative criterion." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a morphism of algebraic spaces over the scheme $S$.", "Let $Z \\to Y$ be any morphism of algebraic spaces over $S$.", "Consider a solid commutative diagram of the following shape", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & Z \\times_Y X \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] \\ar@{-->}[rru] & Z \\ar[r] & Y", "}", "$$", "Then the set of north-west dotted arrows making the diagram commute", "is in 1-1 correspondence with the set of west-north-west dotted arrows", "making the diagram commute. This proves the lemma in the case of", "``uniqueness''. For the existence part, assume $f$ satisfies the existence", "part of the valuative criterion. If we are given a solid commutative", "diagram as above, then by assumption there exists an extension $K \\subset K'$", "of fields and a valuation ring $A' \\subset K'$ dominating $A$ and", "a morphism $\\Spec(A') \\to X$ fitting into the following commutative", "diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] &", "\\Spec(K) \\ar[r] & Z \\times_Y X \\ar[r] & X \\ar[d] \\\\", "\\Spec(A') \\ar[r] \\ar[rrru] & \\Spec(A) \\ar[r] & Z \\ar[r] & Y", "}", "$$", "And by the remarks above the skew arrow corresponds to an arrow", "$\\Spec(A') \\to Z \\times_Y X$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4929, "type": "theorem", "label": "spaces-morphisms-lemma-composition-valuative-criteria", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-valuative-criteria", "contents": [ "The composition of two morphisms of algebraic spaces which satisfy the", "(existence part of, resp.\\ uniqueness part of) the valuative criterion", "satisfies the (existence part of, resp.\\ uniqueness part of) the valuative", "criterion." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$, $g : Y \\to Z$ be morphisms of algebraic spaces over the", "scheme $S$. Consider a solid commutative diagram of the following shape", "$$", "\\xymatrix{", "\\Spec(K) \\ar[dd] \\ar[r] & X \\ar[d]^f \\\\", "& Y \\ar[d]^g \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] \\ar@{-->}[ruu] & Z", "}", "$$", "If we have the uniqueness part for $g$, then there exists at", "most one north-west dotted arrow making the diagram commute.", "If we also have the uniqueness part for $f$, then we have", "at most one north-north-west dotted arrow making the diagram", "commute. The proof in the existence case comes from contemplating", "the following diagram", "$$", "\\xymatrix{", "\\Spec(K'') \\ar[r] \\ar[dd] &", "\\Spec(K') \\ar[r] &", "\\Spec(K) \\ar[r] &", "X \\ar[d]^f \\\\", "& & & Y \\ar[d]^g \\\\", "\\Spec(A'') \\ar[r] \\ar[rrruu] &", "\\Spec(A') \\ar[r] \\ar[rru] &", "\\Spec(A) \\ar[r] &", "Z", "}", "$$", "Namely, the existence part for $g$ gives us the extension $K'$, the", "valuation ring $A'$ and the arrow $\\Spec(A') \\to Y$, whereupon", "the existence part for $f$ gives us the extension $K''$, the", "valuation ring $A''$ and the arrow $\\Spec(A'') \\to X$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4930, "type": "theorem", "label": "spaces-morphisms-lemma-quasi-compact-existence-universally-closed", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-quasi-compact-existence-universally-closed", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume", "\\begin{enumerate}", "\\item $f$ is quasi-compact, and", "\\item $f$ satisfies the existence part of the valuative criterion.", "\\end{enumerate}", "Then $f$ is universally closed." ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-base-change-quasi-compact}", "and \\ref{lemma-base-change-valuative-criteria}", "properties (1) and (2) are preserved under", "any base change. By Lemma \\ref{lemma-universally-closed-local}", "we only have to show that $|T \\times_Y X| \\to |T|$ is closed,", "whenever $T$ is an affine scheme over $S$ mapping into $Y$. Hence it", "suffices to prove: If $Y$ is an affine scheme, $f : X \\to Y$ is quasi-compact", "and satisfies the existence part of the valuative criterion, then", "$f : |X| \\to |Y|$ is closed. In this situation $X$ is a quasi-compact", "algebraic space. By", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-quasi-compact-affine-cover}", "there exists an affine scheme $U$ and a surjective \\'etale morphism", "$\\varphi : U \\to X$. Let $T \\subset |X|$ closed. The inverse image", "$\\varphi^{-1}(T) \\subset U$ is closed, and hence is the set of points", "of an affine closed subscheme $Z \\subset U$. Thus, by", "Algebra, Lemma \\ref{algebra-lemma-image-stable-specialization-closed}", "we see that $f(T) = f(\\varphi(|Z|)) \\subset |Y|$ is closed if it is", "closed under specialization.", "\\medskip\\noindent", "Let $y' \\leadsto y$ be a specialization in $Y$ with $y' \\in f(T)$.", "Choose a point $x' \\in T \\subset |X|$ mapping to $y'$ under $f$.", "We may represent $x'$ by a morphism $\\Spec(K) \\to X$", "for some field $K$. Thus we have the following diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_-{x'} \\ar[d] & X \\ar[d]^f \\\\", "\\Spec(\\mathcal{O}_{Y, y}) \\ar[r] & Y,", "}", "$$", "see", "Schemes, Section \\ref{schemes-section-points}", "for the existence of the left vertical map.", "Choose a valuation ring $A \\subset K$ dominating the image of", "the ring map $\\mathcal{O}_{Y, y} \\to K$ (this is possible since", "the image is a local ring and not a field as $y' \\not = y$, see", "Algebra, Lemma \\ref{algebra-lemma-dominate}).", "By assumption there exists a field extension $K \\subset K'$ and a", "valuation ring $A' \\subset K'$ dominating $A$, and a morphism", "$\\Spec(A') \\to X$ fitting into the commutative diagram.", "Since $A'$ dominates $A$, and $A$ dominates $\\mathcal{O}_{Y, y}$", "we see that the closed point of $\\Spec(A')$ maps to", "a point $x \\in X$ with $f(x) = y$ which is a specialization of $x'$.", "Hence $x \\in T$ as $T$ is closed, and hence $y \\in f(T)$ as desired." ], "refs": [ "spaces-morphisms-lemma-base-change-quasi-compact", "spaces-morphisms-lemma-base-change-valuative-criteria", "spaces-morphisms-lemma-universally-closed-local", "spaces-properties-lemma-quasi-compact-affine-cover", "algebra-lemma-image-stable-specialization-closed", "algebra-lemma-dominate" ], "ref_ids": [ 4738, 4928, 4748, 11832, 551, 608 ] } ], "ref_ids": [] }, { "id": 4931, "type": "theorem", "label": "spaces-morphisms-lemma-characterize-universally-closed-quasi-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-characterize-universally-closed-quasi-separated", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a", "morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item If $f$ is quasi-separated and universally closed, then", "$f$ satisfies the existence part of the valuative criterion.", "\\item If $f$ is quasi-compact and quasi-separated, then", "$f$ is universally closed if and only if the existence part of the", "valuative criterion holds.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (1) is true then combined with", "Lemma \\ref{lemma-quasi-compact-existence-universally-closed}", "we obtain (2). Assume $f$ is quasi-separated and universally closed.", "Assume given a diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] & Y", "}", "$$", "as in Definition \\ref{definition-valuative-criterion}.", "A formal argument shows that the existence of the desired diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\", "\\Spec(A') \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] & Y", "}", "$$", "follows from existence in the case of the morphism $X_A \\to \\Spec(A)$.", "Since being quasi-separated and universally closed are preserved", "by base change, the lemma follows from the result in the next paragraph.", "\\medskip\\noindent", "Consider a solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_-x \\ar[d] & X \\ar[d]^f \\\\", "\\Spec(A) \\ar@{=}[r] \\ar@{..>}[ru] & \\Spec(A)", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$.", "By Lemma \\ref{lemma-quasi-compact-permanence} and the fact", "that $f$ is quasi-separated we have that", "the morphism $x$ is quasi-compact.", "Since $f$ is universally closed, we have in particular", "that $|f|(\\overline{\\{x\\}})$ is closed in $\\Spec(A)$.", "Since this image contains the generic point of $\\Spec(A)$", "there exists a point $x' \\in |X|$ in the closure", "of $x$ mapping to the closed point of $\\Spec(A)$.", "By Lemma \\ref{lemma-reach-points-scheme-theoretic-image}", "we can find a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[d] \\\\", "\\Spec(A') \\ar[r] & X", "}", "$$", "such that the closed point of $\\Spec(A')$ maps to $x' \\in |X|$.", "It follows that $\\Spec(A') \\to \\Spec(A)$ maps the closed point", "to the closed point, i.e., $A'$ dominates $A$ and this finishes the proof." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-existence-universally-closed", "spaces-morphisms-definition-valuative-criterion", "spaces-morphisms-lemma-quasi-compact-permanence", "spaces-morphisms-lemma-reach-points-scheme-theoretic-image" ], "ref_ids": [ 4930, 5016, 4743, 4782 ] } ], "ref_ids": [] }, { "id": 4932, "type": "theorem", "label": "spaces-morphisms-lemma-characterize-universally-closed-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-characterize-universally-closed-separated", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is quasi-compact and separated. Then the following", "are equivalent", "\\begin{enumerate}", "\\item $f$ is universally closed,", "\\item the existence part of the valuative criterion holds", "as in Definition \\ref{definition-valuative-criterion}, and", "\\item given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$, there exists", "a dotted arrow, i.e., $f$ satisfies the existence part of the valuative", "criterion as in", "Schemes, Definition \\ref{schemes-definition-valuative-criterion}.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-valuative-criterion", "schemes-definition-valuative-criterion" ], "proofs": [ { "contents": [ "Since $f$ is separated parts (2) and (3) are equivalent by", "Lemma \\ref{lemma-usual-enough}.", "The equivalence of (3) and (1) follows from", "Lemma \\ref{lemma-characterize-universally-closed-quasi-separated}." ], "refs": [ "spaces-morphisms-lemma-usual-enough", "spaces-morphisms-lemma-characterize-universally-closed-quasi-separated" ], "ref_ids": [ 4927, 4931 ] } ], "ref_ids": [ 5016, 7755 ] }, { "id": 4933, "type": "theorem", "label": "spaces-morphisms-lemma-lift-valuation-ring-through-flat-morphism", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-lift-valuation-ring-through-flat-morphism", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a flat morphism", "of algebraic spaces over $S$.", "Let $\\Spec(A) \\to Y$ be a morphism where $A$ is a", "valuation ring. If the closed point of $\\Spec(A)$ maps to a", "point of $|Y|$ in the image of $|X| \\to |Y|$, then there exists", "a commutative diagram", "$$", "\\xymatrix{", "\\Spec(A') \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] & Y", "}", "$$", "where $A \\to A'$ is an extension of valuation rings", "(More on Algebra, Definition", "\\ref{more-algebra-definition-extension-valuation-rings})." ], "refs": [ "more-algebra-definition-extension-valuation-rings" ], "proofs": [ { "contents": [ "The base change $X_A \\to \\Spec(A)$ is flat", "(Lemma \\ref{lemma-base-change-flat}) and the closed point of", "$\\Spec(A)$ is in the image of $|X_A| \\to |\\Spec(A)|$", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}).", "Thus we may assume $Y = \\Spec(A)$. Let $U \\to X$ be a surjective", "\\'etale morphism where $U$ is a scheme. Let $u \\in U$ map to", "the closed point of $\\Spec(A)$. Consider the flat local ring map", "$A \\to B = \\mathcal{O}_{U, u}$. By", "Algebra, Lemma \\ref{algebra-lemma-ff-rings}", "there exists a prime ideal $\\mathfrak q \\subset B$ such that", "$\\mathfrak q$ lies over $(0) \\subset A$. By", "Algebra, Lemma \\ref{algebra-lemma-dominate}", "we can find a valuation ring $A' \\subset \\kappa(\\mathfrak q)$", "dominating $B/\\mathfrak q$. The induced morphism", "$\\Spec(A') \\to U \\to X$ is a solution to the problem", "posed by the lemma." ], "refs": [ "spaces-morphisms-lemma-base-change-flat", "spaces-properties-lemma-points-cartesian", "algebra-lemma-ff-rings", "algebra-lemma-dominate" ], "ref_ids": [ 4853, 11819, 536, 608 ] } ], "ref_ids": [ 10647 ] }, { "id": 4934, "type": "theorem", "label": "spaces-morphisms-lemma-refined-valuative-criterion-universally-closed", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-refined-valuative-criterion-universally-closed", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ and $h : U \\to X$ be morphisms of", "algebraic spaces over $S$. If", "\\begin{enumerate}", "\\item $f$ and $h$ are quasi-compact,", "\\item $|h|(|U|)$ is dense in $|X|$, and", "\\end{enumerate}", "given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r] & X \\ar[d] \\\\", "\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & Y", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$", "\\begin{enumerate}", "\\item[(3)] there exists at most one dotted arrow making the diagram", "commute, and", "\\item[(4)] there exists an extension $K \\subset K'$ of fields, a", "valuation ring $A' \\subset K'$ dominating $A$ and a morphism", "$\\Spec(A') \\to X$ such that the following diagram commutes", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & U \\ar[r] & X \\ar[d] \\\\", "\\Spec(A') \\ar[r] \\ar[rrru] & \\Spec(A) \\ar[rr] & & Y", "}", "$$", "\\end{enumerate}", "then $f$ is universally closed. If moreover", "\\begin{enumerate}", "\\item[(5)] $f$ is quasi-separated", "\\end{enumerate}", "then $f$ is separated and universally closed." ], "refs": [], "proofs": [ { "contents": [ "Assume (1), (2), (3), and (4).", "We will verify the existence part of the valuative criterion for $f$", "which will imply $f$ is universally closed by", "Lemma \\ref{lemma-quasi-compact-existence-universally-closed}.", "To do this, consider a commutative diagram", "\\begin{equation}", "\\label{equation-start-with}", "\\vcenter{", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] & Y", "}", "}", "\\end{equation}", "where $A$ is a valuation ring and $K$ is the fraction field of $A$.", "Note that since valuation rings and fields are reduced, we may", "replace $U$, $X$, and $S$ by their respective reductions by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-map-into-reduction}.", "In this case the assumption that $h(U)$ is dense means that", "the scheme theoretic image of $h : U \\to X$ is $X$, see", "Lemma \\ref{lemma-scheme-theoretic-image-reduced}.", "\\medskip\\noindent", "Reduction to the case $Y$ affine. Choose an \\'etale morphism", "$\\Spec(R) \\to Y$ such that the closed point of $\\Spec(A)$ maps", "to an element of $\\Im(|\\Spec(R)| \\to |Y|)$. By", "Lemma \\ref{lemma-lift-valuation-ring-through-flat-morphism}", "we can find a local ring map $A \\to A'$ of valuation rings", "and a morphism $\\Spec(A') \\to \\Spec(R)$ fitting into a commutative", "diagram", "$$", "\\xymatrix{", "\\Spec(A') \\ar[r] \\ar[d] & \\Spec(R) \\ar[d] \\\\", "\\Spec(A) \\ar[r] & Y", "}", "$$", "Since in Definition \\ref{definition-valuative-criterion}", "we allow for extensions of valuation rings", "it is clear that we may replace $A$ by $A'$, $Y$ by $\\Spec(R)$,", "$X$ by $X \\times_Y \\Spec(R)$ and $U$ by $U \\times_Y \\Spec(R)$.", "\\medskip\\noindent", "From now on we assume that $Y = \\Spec(R)$ is an affine scheme.", "Let $\\Spec(B) \\to X$ be an \\'etale morphism from an affine scheme", "such that the morphism $\\Spec(K) \\to X$ is in the image of", "$|\\Spec(B)| \\to |X|$. Since we may replace $K$ by an extension", "$K' \\supset K$ and $A$ by a valuation ring $A' \\subset K'$", "dominating $A$ (which exists by", "Algebra, Lemma \\ref{algebra-lemma-dominate}),", "we may assume the morphism $\\Spec(K) \\to X$ factors through $\\Spec(B)$", "(by definition of $|X|$). In other words, we may think of $K$ as a $B$-algebra.", "Choose a polynomial algebra $P$ over $B$ and a $B$-algebra surjection", "$P \\to K$. Then $\\Spec(P) \\to X$ is flat as a composition", "$\\Spec(P) \\to \\Spec(B) \\to X$. Hence the scheme theoretic image", "of the morphism $U \\times_X \\Spec(P) \\to \\Spec(P)$ is $\\Spec(P)$ by", "Lemma \\ref{lemma-flat-base-change-scheme-theoretic-image}.", "By Lemma \\ref{lemma-reach-points-scheme-theoretic-image}", "we can find a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & U \\times_X \\Spec(P) \\ar[d] \\\\", "\\Spec(A') \\ar[r] & \\Spec(P)", "}", "$$", "where $A'$ is a valuation ring and $K'$ is the fraction field of $A'$", "such that the closed point of $\\Spec(A')$ maps to", "$\\Spec(K) \\subset \\Spec(P)$. In other words, there is a $B$-algebra map", "$\\varphi : K \\to A'/\\mathfrak m_{A'}$. Choose a valuation ring", "$A'' \\subset A'/\\mathfrak m_{A'}$ dominating $\\varphi(A)$ with", "field of fractions $K'' = A'/\\mathfrak m_{A'}$", "(Algebra, Lemma \\ref{algebra-lemma-dominate}). We set", "$$", "C = \\{\\lambda \\in A' \\mid \\lambda \\bmod \\mathfrak m_{A'} \\in A''\\}.", "$$", "which is a valuation ring by", "Algebra, Lemma \\ref{algebra-lemma-stack-valuation-rings}.", "As $C$ is an $R$-algebra with fraction field $K'$, we obtain a solid", "commutative diagram", "$$", "\\xymatrix{", "\\Spec(K'_1) \\ar@{-->}[r] \\ar@{-->}[d] &", "\\Spec(K') \\ar[r] \\ar[d] & U \\ar[r] & X \\ar[d] \\\\", "\\Spec(C_1) \\ar@{-->}[r] \\ar@{-->}[rrru] & \\Spec(C) \\ar[rr] & & Y", "}", "$$", "as in the statement of the lemma. Thus assumption (4) produces", "$C \\to C_1$ and the dotted arrows making the diagram commute.", "Let $A_1' = (C_1)_\\mathfrak p$ be the localization of $C_1$", "at a prime $\\mathfrak p \\subset C_1$ lying over", "$\\mathfrak m_{A'} \\subset C$. Since $C \\to C_1$ is flat by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-valuation-ring-torsion-free-flat}", "such a prime $\\mathfrak p$ exists by", "Algebra, Lemmas \\ref{algebra-lemma-local-flat-ff} and", "\\ref{algebra-lemma-ff-rings}.", "Note that $A'$ is the localization of $C$ at $\\mathfrak m_{A'}$", "and that $A'_1$ is a valuation ring", "(Algebra, Lemma \\ref{algebra-lemma-make-valuation-rings}).", "In other words, $A' \\to A'_1$ is a local ring map of valuation", "rings. Assumption (3) implies", "$$", "\\xymatrix{", "\\Spec(A'_1) \\ar[r] \\ar[d] & \\Spec(C_1) \\ar[r] & X \\\\", "\\Spec(A') \\ar[r] & \\Spec(P) \\ar[r] & \\Spec(B) \\ar[u]", "}", "$$", "commutes. Hence the restriction of the morphism $\\Spec(C_1) \\to X$", "to $\\Spec(C_1/\\mathfrak p)$ restricts to the composition", "$$", "\\Spec(\\kappa(\\mathfrak p)) \\to", "\\Spec(A'/\\mathfrak m_{A'}) = \\Spec(K'') \\to", "\\Spec(K) \\to X", "$$", "on the generic point of $\\Spec(C_1/\\mathfrak p)$. Moreover,", "$C_1/\\mathfrak p$ is a valuation ring", "(Algebra, Lemma \\ref{algebra-lemma-make-valuation-rings})", "dominating $A''$ which dominates $A$.", "Thus the morphism $\\Spec(C_1/\\mathfrak p) \\to X$ witnesses the", "existence part of the valuative criterion for the diagram", "(\\ref{equation-start-with}) as desired.", "\\medskip\\noindent", "Next, suppose that (5) is satisfied as well, i.e., the morphism", "$\\Delta : X \\to X \\times_S X$ is quasi-compact. In this case", "assumptions (1) -- (4) hold for $h$ and $\\Delta$. Hence the first", "part of the proof shows that $\\Delta$ is universally closed.", "By Lemma \\ref{lemma-separated-diagonal-proper} we conclude that", "$f$ is separated." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-existence-universally-closed", "spaces-properties-lemma-map-into-reduction", "spaces-morphisms-lemma-scheme-theoretic-image-reduced", "spaces-morphisms-lemma-lift-valuation-ring-through-flat-morphism", "spaces-morphisms-definition-valuative-criterion", "algebra-lemma-dominate", "spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image", "spaces-morphisms-lemma-reach-points-scheme-theoretic-image", "algebra-lemma-dominate", "algebra-lemma-stack-valuation-rings", "more-algebra-lemma-valuation-ring-torsion-free-flat", "algebra-lemma-local-flat-ff", "algebra-lemma-ff-rings", "algebra-lemma-make-valuation-rings", "algebra-lemma-make-valuation-rings", "spaces-morphisms-lemma-separated-diagonal-proper" ], "ref_ids": [ 4930, 11847, 4781, 4933, 5016, 608, 4861, 4782, 608, 615, 9920, 537, 536, 614, 614, 4923 ] } ], "ref_ids": [] }, { "id": 4935, "type": "theorem", "label": "spaces-morphisms-lemma-separated-implies-valuative", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-separated-implies-valuative", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $f$ is separated, then $f$ satisfies the uniqueness", "part of the valuative criterion." ], "refs": [], "proofs": [ { "contents": [ "Let a diagram as in Definition \\ref{definition-valuative-criterion}", "be given. Suppose there are two distinct morphisms", "$a, b : \\Spec(A) \\to X$ fitting into the diagram.", "Let $Z \\subset \\Spec(A)$ be the equalizer of $a$ and $b$.", "Then $Z = \\Spec(A) \\times_{(a, b), X \\times_Y X, \\Delta} X$.", "If $f$ is separated, then $\\Delta$ is a closed immersion, and", "this is a closed subscheme of $\\Spec(A)$. By assumption it contains", "the generic point of $\\Spec(A)$. Since $A$ is a domain", "this implies $Z = \\Spec(A)$. Hence $a = b$ as desired." ], "refs": [ "spaces-morphisms-definition-valuative-criterion" ], "ref_ids": [ 5016 ] } ], "ref_ids": [] }, { "id": 4936, "type": "theorem", "label": "spaces-morphisms-lemma-valuative-criterion-separatedness", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-valuative-criterion-separatedness", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume", "\\begin{enumerate}", "\\item the morphism $f$ is quasi-separated, and", "\\item the morphism $f$ satisfies the uniqueness", "part of the valuative criterion.", "\\end{enumerate}", "Then $f$ is separated." ], "refs": [], "proofs": [ { "contents": [ "Assumption (1) means $\\Delta_{X/Y}$ is quasi-compact.", "We claim the morphism", "$\\Delta_{X/Y} : X \\to X \\times_Y X$ satisfies the existence", "part of the valuative criterion.", "Let a solid commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & X \\times_Y X", "}", "$$", "be given. The lower right arrow corresponds to a", "pair of morphisms $a, b : \\Spec(A) \\to X$ over $Y$.", "By assumption (2) we see that $a = b$. Hence using $a$ as the dotted", "arrow works. Hence", "Lemma \\ref{lemma-quasi-compact-existence-universally-closed}", "applies, and we see that $\\Delta_{X/Y}$ is universally closed.", "Since always $\\Delta_{X/Y}$ is locally of finite type and", "separated, we conclude from", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-characterize-finite}", "that $\\Delta_{X/Y}$ is a finite morphism (also, use the", "general principle of", "Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}).", "At this point $\\Delta_{X/Y}$ is a representable, finite monomorphism,", "hence a closed immersion by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-monomorphism-closed}." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-existence-universally-closed", "more-morphisms-lemma-characterize-finite", "spaces-lemma-representable-transformations-property-implication", "morphisms-lemma-finite-monomorphism-closed" ], "ref_ids": [ 4930, 13903, 8136, 5449 ] } ], "ref_ids": [] }, { "id": 4937, "type": "theorem", "label": "spaces-morphisms-lemma-characterize-separated-and-universally-closed", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-characterize-separated-and-universally-closed", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is quasi-compact and quasi-separated. Then the", "following are equivalent", "\\begin{enumerate}", "\\item $f$ is separated and universally closed,", "\\item the valuative criterion holds as in Definition", "\\ref{definition-valuative-criterion},", "\\item given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$, there exists", "a unique dotted arrow, i.e., $f$ satisfies the valuative", "criterion as in", "Schemes, Definition \\ref{schemes-definition-valuative-criterion}.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-valuative-criterion", "schemes-definition-valuative-criterion" ], "proofs": [ { "contents": [ "Since $f$ is quasi-separated, the uniqueness", "part of the valutative criterion implies $f$ is separated", "(Lemma \\ref{lemma-valuative-criterion-separatedness}).", "Conversely, if $f$ is separated, then it satisfies the", "uniqueness part of the valuative criterion", "(Lemma \\ref{lemma-separated-implies-valuative}).", "Having said this, we see that in each of the three cases the", "morphism $f$ is separated and satisfies the uniqueness part", "of the valuative criterion. In this case the lemma is a formal", "consequence of", "Lemma \\ref{lemma-characterize-universally-closed-separated}." ], "refs": [ "spaces-morphisms-lemma-valuative-criterion-separatedness", "spaces-morphisms-lemma-separated-implies-valuative", "spaces-morphisms-lemma-characterize-universally-closed-separated" ], "ref_ids": [ 4936, 4935, 4932 ] } ], "ref_ids": [ 5016, 7755 ] }, { "id": 4938, "type": "theorem", "label": "spaces-morphisms-lemma-characterize-proper", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-characterize-proper", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is of finite type and quasi-separated. Then the", "following are equivalent", "\\begin{enumerate}", "\\item $f$ is proper,", "\\item the valuative criterion holds as in Definition", "\\ref{definition-valuative-criterion},", "\\item given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$, there exists", "a unique dotted arrow, i.e., $f$ satisfies the valuative", "criterion as in", "Schemes, Definition \\ref{schemes-definition-valuative-criterion}.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-valuative-criterion", "schemes-definition-valuative-criterion" ], "proofs": [ { "contents": [ "Formal consequence of", "Lemma \\ref{lemma-characterize-separated-and-universally-closed}", "and the definitions." ], "refs": [ "spaces-morphisms-lemma-characterize-separated-and-universally-closed" ], "ref_ids": [ 4937 ] } ], "ref_ids": [ 5016, 7755 ] }, { "id": 4939, "type": "theorem", "label": "spaces-morphisms-lemma-integral-representable", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-integral-representable", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable", "morphism of algebraic spaces over $S$. Then", "$f$ is integral, resp.\\ finite", "(in the sense of Section \\ref{section-representable}),", "if and only if for all affine schemes $Z$", "and morphisms $Z \\to Y$ the scheme $X \\times_Y Z$ is affine and", "integral, resp.\\ finite, over $Z$." ], "refs": [], "proofs": [ { "contents": [ "This follows directly from the definition of an integral (resp.\\ finite)", "morphism of schemes", "(Morphisms, Definition \\ref{morphisms-definition-integral})." ], "refs": [ "morphisms-definition-integral" ], "ref_ids": [ 5573 ] } ], "ref_ids": [] }, { "id": 4940, "type": "theorem", "label": "spaces-morphisms-lemma-integral-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-integral-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is representable and integral (resp.\\ finite),", "\\item $f$ is integral (resp.\\ finite),", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is integral (resp. finite), and", "\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that", "each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is integral (resp.\\ finite).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2) and that (2) implies (3) by taking", "$V$ to be a disjoint union of affines \\'etale over $Y$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-cover-by-union-affines}.", "Assume $V \\to Y$ is as in (3). Then for every affine open $W$ of $V$ we see", "that $W \\times_Y X$ is an affine open of $V \\times_Y X$. Hence by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme}", "we conclude that $V \\times_Y X$ is a scheme. Moreover the morphism", "$V \\times_Y X \\to V$ is affine. This means we can apply", "Spaces,", "Lemma \\ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale}", "because the class of integral (resp.\\ finite) morphisms", "satisfies all the required properties (see", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-finite} and", "Descent, Lemmas", "\\ref{descent-lemma-descending-property-integral},", "\\ref{descent-lemma-descending-property-finite},", "and \\ref{descent-lemma-affine}).", "The conclusion of applying this lemma is that $f$ is representable", "and integral (resp.\\ finite), i.e., (1) holds.", "\\medskip\\noindent", "The equivalence of (1) and (4) follows from the fact that being", "integral (resp.\\ finite) is Zariski local on the target (the", "reference above shows that being integral or finite is in fact", "fpqc local on the target)." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-properties-lemma-subscheme", "spaces-lemma-morphism-sheaves-with-P-effective-descent-etale", "morphisms-lemma-base-change-finite", "descent-lemma-descending-property-integral", "descent-lemma-descending-property-finite", "descent-lemma-affine" ], "ref_ids": [ 11830, 11848, 8158, 5440, 14687, 14688, 14748 ] } ], "ref_ids": [] }, { "id": 4941, "type": "theorem", "label": "spaces-morphisms-lemma-composition-integral", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-integral", "contents": [ "The composition of integral (resp.\\ finite) morphisms is integral", "(resp.\\ finite)." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4942, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-integral", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-integral", "contents": [ "The base change of an integral (resp.\\ finite) morphism is integral", "(resp.\\ finite)." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4943, "type": "theorem", "label": "spaces-morphisms-lemma-finite-integral", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-integral", "contents": [ "A finite morphism of algebraic spaces is integral.", "An integral morphism of algebraic spaces", "which is locally of finite type is finite." ], "refs": [], "proofs": [ { "contents": [ "In both cases the morphism is representable, and you can check the condition", "after a base change by an affine scheme mapping into $Y$, see", "Lemmas \\ref{lemma-integral-local}. Hence this lemma follows from the", "same lemma for the case of schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-integral}." ], "refs": [ "spaces-morphisms-lemma-integral-local", "morphisms-lemma-finite-integral" ], "ref_ids": [ 4940, 5438 ] } ], "ref_ids": [] }, { "id": 4944, "type": "theorem", "label": "spaces-morphisms-lemma-integral-universally-closed", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-integral-universally-closed", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is integral, and", "\\item $f$ is affine and universally closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In both cases the morphism is representable, and you can check the condition", "after a base change by an affine scheme mapping into $Y$, see", "Lemmas \\ref{lemma-integral-local},", "\\ref{lemma-affine-local}, and", "\\ref{lemma-universally-closed-local}.", "Hence the result follows from", "Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed}." ], "refs": [ "spaces-morphisms-lemma-integral-local", "spaces-morphisms-lemma-affine-local", "spaces-morphisms-lemma-universally-closed-local", "morphisms-lemma-integral-universally-closed" ], "ref_ids": [ 4940, 4798, 4748, 5441 ] } ], "ref_ids": [] }, { "id": 4945, "type": "theorem", "label": "spaces-morphisms-lemma-finite-quasi-finite", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-quasi-finite", "contents": [ "A finite morphism of algebraic spaces is quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a morphism of algebraic spaces.", "By", "Definition \\ref{definition-integral}", "and", "Lemmas \\ref{lemma-quasi-compact-local} and", "\\ref{lemma-quasi-finite-local}", "both properties may be checked after base change to an affine over $Y$,", "i.e., we may assume $Y$ affine.", "If $f$ is finite then $X$ is a scheme.", "Hence the result follows from the corresponding result for schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-quasi-finite}." ], "refs": [ "spaces-morphisms-definition-integral", "spaces-morphisms-lemma-quasi-compact-local", "spaces-morphisms-lemma-quasi-finite-local", "morphisms-lemma-finite-quasi-finite" ], "ref_ids": [ 5017, 4742, 4834, 5444 ] } ], "ref_ids": [] }, { "id": 4946, "type": "theorem", "label": "spaces-morphisms-lemma-finite-proper", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-proper", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over", "$S$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is finite, and", "\\item $f$ is affine and proper.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In both cases the morphism is representable, and you can check the condition", "after base change to an affine scheme mapping into $Y$, see", "Lemmas \\ref{lemma-integral-local}, \\ref{lemma-affine-local}, and", "\\ref{lemma-proper-local}. Hence the result follows from", "Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}." ], "refs": [ "spaces-morphisms-lemma-integral-local", "spaces-morphisms-lemma-affine-local", "spaces-morphisms-lemma-proper-local", "morphisms-lemma-finite-proper" ], "ref_ids": [ 4940, 4798, 4916, 5445 ] } ], "ref_ids": [] }, { "id": 4947, "type": "theorem", "label": "spaces-morphisms-lemma-closed-immersion-finite", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-closed-immersion-finite", "contents": [ "A closed immersion is finite (and a fortiori integral)." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4948, "type": "theorem", "label": "spaces-morphisms-lemma-finite-union-finite", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-union-finite", "contents": [ "Let $S$ be a scheme.", "Let $X_i \\to Y$, $i = 1, \\ldots, n$ be finite morphisms of", "algebraic spaces over $S$.", "Then $X_1 \\amalg \\ldots \\amalg X_n \\to Y$ is finite too." ], "refs": [], "proofs": [ { "contents": [ "Follows from the case of schemes", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-union-finite})", "by \\'etale localization." ], "refs": [ "morphisms-lemma-finite-union-finite" ], "ref_ids": [ 5447 ] } ], "ref_ids": [] }, { "id": 4949, "type": "theorem", "label": "spaces-morphisms-lemma-finite-permanence", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-permanence", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item If $g \\circ f$ is finite and $g$ separated then $f$ is finite.", "\\item If $g \\circ f$ is integral and $g$ separated then $f$ is integral.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $g \\circ f$ is finite (resp.\\ integral) and $g$ separated.", "The base change $X \\times_Z Y \\to Y$ is finite (resp.\\ integral) by", "Lemma \\ref{lemma-base-change-integral}.", "The morphism $X \\to X \\times_Z Y$ is", "a closed immersion as $Y \\to Z$ is separated, see", "Lemma \\ref{lemma-section-immersion}.", "A closed immersion is finite (resp.\\ integral),", "see Lemma \\ref{lemma-closed-immersion-finite}.", "The composition of finite (resp.\\ integral) morphisms is finite", "(resp.\\ integral),", "see Lemma \\ref{lemma-composition-integral}. Thus we win." ], "refs": [ "spaces-morphisms-lemma-base-change-integral", "spaces-morphisms-lemma-closed-immersion-finite", "spaces-morphisms-lemma-composition-integral" ], "ref_ids": [ 4942, 4947, 4941 ] } ], "ref_ids": [] }, { "id": 4950, "type": "theorem", "label": "spaces-morphisms-lemma-finite-locally-free-representable", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-locally-free-representable", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism", "of algebraic spaces over $S$. Then $f$ is finite locally free", "(in the sense of Section \\ref{section-representable})", "if and only if $f$ is affine and the sheaf $f_*\\mathcal{O}_X$ is", "a finite locally free $\\mathcal{O}_Y$-module." ], "refs": [], "proofs": [ { "contents": [ "Assume $f$ is finite locally free (as defined in", "Section \\ref{section-representable}). This means that", "for every morphism $V \\to Y$ whose source is a scheme the", "base change $f' : V \\times_Y X \\to V$ is a finite locally free morphism", "of schemes. This in turn means (by the definition of a finite locally", "free morphism of schemes) that", "$f'_*\\mathcal{O}_{V \\times_Y X}$", "is a finite locally free $\\mathcal{O}_V$-module. We may choose $V \\to Y$", "to be surjective and \\'etale. By", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-pushforward-etale-base-change-modules}", "we conclude the restriction of $f_*\\mathcal{O}_X$ to $V$ is", "finite locally free. Hence by", "Modules on Sites, Lemma \\ref{sites-modules-lemma-local-final-object}", "applied to the sheaf $f_*\\mathcal{O}_X$ on $Y_{spaces, \\etale}$", "we conclude that $f_*\\mathcal{O}_X$ is finite locally free.", "\\medskip\\noindent", "Conversely, assume $f$ is affine and that $f_*\\mathcal{O}_X$ is a finite", "locally free $\\mathcal{O}_Y$-module. Let $V$ be a scheme, and let", "$V \\to Y$ be a surjective \\'etale morphism. Again by", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-pushforward-etale-base-change-modules}", "we see that $f'_*\\mathcal{O}_{V \\times_Y X}$ is finite locally free.", "Hence $f' : V \\times_Y X \\to V$ is finite locally free (as it is also affine).", "By", "Spaces,", "Lemma \\ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale}", "we conclude that $f$ is finite locally free (use", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-locally-free}", "Descent, Lemmas \\ref{descent-lemma-descending-property-finite-locally-free}", "and \\ref{descent-lemma-affine}). Thus we win." ], "refs": [ "spaces-properties-lemma-pushforward-etale-base-change-modules", "sites-modules-lemma-local-final-object", "spaces-properties-lemma-pushforward-etale-base-change-modules", "spaces-lemma-morphism-sheaves-with-P-effective-descent-etale", "morphisms-lemma-base-change-finite-locally-free", "descent-lemma-descending-property-finite-locally-free", "descent-lemma-affine" ], "ref_ids": [ 11898, 14185, 11898, 8158, 5473, 14695, 14748 ] } ], "ref_ids": [] }, { "id": 4951, "type": "theorem", "label": "spaces-morphisms-lemma-finite-locally-free-local", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-locally-free-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is representable and finite locally free,", "\\item $f$ is finite locally free,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that $V \\times_Y X \\to V$ is finite locally free, and", "\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that", "each morphism $f^{-1}(Y_i) \\to Y_i$ is finite locally free.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2) and that (2) implies (3) by taking", "$V$ to be a disjoint union of affines \\'etale over $Y$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-cover-by-union-affines}.", "Assume $V \\to Y$ is as in (3). Then for every affine open $W$ of $V$ we see", "that $W \\times_Y X$ is an affine open of $V \\times_Y X$. Hence by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme}", "we conclude that $V \\times_Y X$ is a scheme. Moreover the morphism", "$V \\times_Y X \\to V$ is affine. This means we can apply", "Spaces,", "Lemma \\ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale}", "because the class of finite locally free morphisms", "satisfies all the required properties (see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-locally-free}", "Descent, Lemmas \\ref{descent-lemma-descending-property-finite-locally-free}", "and \\ref{descent-lemma-affine}).", "The conclusion of applying this lemma is that $f$ is representable", "and finite locally free, i.e., (1) holds.", "\\medskip\\noindent", "The equivalence of (1) and (4) follows from the fact that being", "finite locally free is Zariski local on the target (the reference above shows", "that being finite locally free is in fact fpqc local on the target)." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-properties-lemma-subscheme", "spaces-lemma-morphism-sheaves-with-P-effective-descent-etale", "morphisms-lemma-base-change-finite-locally-free", "descent-lemma-descending-property-finite-locally-free", "descent-lemma-affine" ], "ref_ids": [ 11830, 11848, 8158, 5473, 14695, 14748 ] } ], "ref_ids": [] }, { "id": 4952, "type": "theorem", "label": "spaces-morphisms-lemma-composition-finite-locally-free", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-finite-locally-free", "contents": [ "The composition of finite locally free morphisms is finite locally free." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4953, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-finite-locally-free", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-finite-locally-free", "contents": [ "The base change of a finite locally free morphism is finite locally free." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4954, "type": "theorem", "label": "spaces-morphisms-lemma-finite-flat", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-flat", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is finite locally free,", "\\item $f$ is finite, flat, and locally of finite presentation.", "\\end{enumerate}", "If $Y$ is locally Noetherian these are also equivalent to", "\\begin{enumerate}", "\\item[(3)] $f$ is finite and flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In each of the three cases the morphism is representable and you", "can check the property after base change by a surjective \\'etale morphism", "$V \\to Y$, see", "Lemmas \\ref{lemma-integral-local},", "\\ref{lemma-finite-locally-free-local},", "\\ref{lemma-flat-local}, and", "\\ref{lemma-finite-presentation-local}.", "If $Y$ is locally Noetherian, then $V$ is locally Noetherian.", "Hence the result follows from the corresponding result", "in the schemes case, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}." ], "refs": [ "spaces-morphisms-lemma-integral-local", "spaces-morphisms-lemma-finite-locally-free-local", "spaces-morphisms-lemma-flat-local", "spaces-morphisms-lemma-finite-presentation-local", "morphisms-lemma-finite-flat" ], "ref_ids": [ 4940, 4951, 4854, 4841, 5471 ] } ], "ref_ids": [] }, { "id": 4955, "type": "theorem", "label": "spaces-morphisms-lemma-rational-map-from-reduced-to-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-rational-map-from-reduced-to-separated", "contents": [ "Let $S$ be a scheme. Let $X$ and $Y$ be algebraic spaces over $S$.", "Assume $X$ is reduced and $Y$ is separated over $S$. Let", "$\\varphi$ be a rational map from $X$ to $Y$ with domain of definition", "$U \\subset X$. Then there exists a unique morphism $f : U \\to Y$", "of algebraic spaces representing $\\varphi$." ], "refs": [], "proofs": [ { "contents": [ "Let $(V, g)$ and $(V', g')$ be representatives of $\\varphi$. Then", "$g, g'$ agree on a dense open subspace $W \\subset V \\cap V'$.", "On the other hand, the equalizer $E$ of $g|_{V \\cap V'}$ and $g'|_{V \\cap V'}$", "is a closed subspace of $V \\cap V'$ because it is the base change", "of $\\Delta : Y \\to Y \\times_S Y$ by the morphism", "$V \\cap V' \\to Y \\times_S Y$ given by $g|_{V \\cap V'}$ and $g'|_{V \\cap V'}$.", "Now $W \\subset E$ implies that $|E| = |V \\cap V'|$. As $V \\cap V'$", "is reduced we conclude $E = V \\cap V'$ scheme theoretically, i.e.,", "$g|_{V \\cap V'} = g'|_{V \\cap V'}$, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-map-into-reduction}.", "It follows that we can glue the representatives $g : V \\to Y$ of $\\varphi$", "to a morphism $f : U \\to Y$ because $\\coprod V \\to U$ is a surjection of", "fppf sheaves and", "$\\coprod_{V, V'} V \\cap V' = (\\coprod V) \\times_U (\\coprod V)$." ], "refs": [ "spaces-properties-lemma-map-into-reduction" ], "ref_ids": [ 11847 ] } ], "ref_ids": [] }, { "id": 4956, "type": "theorem", "label": "spaces-morphisms-lemma-birational", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-birational", "contents": [ "Let $S$ be a scheme. Let $X$ and $Y$ be algebraic", "space over $S$ with $|X|$ and $|Y|$ irreducible.", "Then $X$ and $Y$ are birational if and only if", "there are nonempty open subspaces $U \\subset X$ and $V \\subset Y$", "which are isomorphic as algebraic spaces over $S$." ], "refs": [], "proofs": [ { "contents": [ "Assume $X$ and $Y$ are birational. Let $f : U \\to Y$ and $g : V \\to X$", "define inverse dominant rational maps from $X$ to $Y$ and from $Y$ to $X$.", "After shrinking $U$ we may assume $f : U \\to Y$ factors through $V$.", "As $g \\circ f$ is the identity as a dominant rational map, we see that", "the composition $U \\to V \\to X$ is the identity on a dense open of $U$.", "Thus after replacing $U$ by a smaller open we may assume that", "$U \\to V \\to X$ is the inclusion of $U$ into $X$.", "By symmetry we find there exists an open subspace $V' \\subset V$", "such that $g|_{V'} : V' \\to X$ factors through $U \\subset X$", "and such that $V' \\to U \\to Y$ is the identity.", "The inverse image of $|V'|$ by $|U| \\to |V|$ is an open of $|U|$", "and hence equal to $|U'|$ for some open subspace $U' \\subset U$, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-open-subspaces}.", "Then $U' \\subset U \\to V$ factors as $U' \\to V'$.", "Similarly $V' \\to U$ factors as $V' \\to U'$.", "The reader finds that $U' \\to V'$", "and $V' \\to U'$ are mutually inverse", "morphisms of algebraic spaces over $S$", "and the proof is complete." ], "refs": [ "spaces-properties-lemma-open-subspaces" ], "ref_ids": [ 11823 ] } ], "ref_ids": [] }, { "id": 4957, "type": "theorem", "label": "spaces-morphisms-lemma-integral-closure", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-integral-closure", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{A}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras.", "There exists a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras", "$\\mathcal{A}' \\subset \\mathcal{A}$ such that", "for any affine object $U$ of $X_\\etale$ the ring", "$\\mathcal{A}'(U) \\subset \\mathcal{A}(U)$ is", "the integral closure of $\\mathcal{O}_X(U)$ in $\\mathcal{A}(U)$." ], "refs": [], "proofs": [ { "contents": [ "By Properties of Spaces, Lemma \\ref{spaces-properties-lemma-alternative}", "it suffices to prove that the rule given above defines a quasi-coherent", "module on $X_{affine, \\etale}$. To see this it suffices", "to show the following: Let $U_1 \\to U_2$ be a morphism of affine objects of", "$X_\\etale$. Say $U_i = \\Spec(R_i)$.", "Say $\\mathcal{A}|_{(U_1)_\\etale}$", "is the quasi-coherent sheaf associated to", "the $R_2$-algebra $A$. Let $A' \\subset A$ be the integral closure", "of $R_2$ in $A$. Then $A' \\otimes_{R_2} R_1$ is the integral closure", "of $R_1$ in $A \\otimes_{R_2} R_1$. This is", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-commutes-etale}." ], "refs": [ "spaces-properties-lemma-alternative", "algebra-lemma-integral-closure-commutes-etale" ], "ref_ids": [ 11863, 1251 ] } ], "ref_ids": [] }, { "id": 4958, "type": "theorem", "label": "spaces-morphisms-lemma-properties-normalization", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-properties-normalization", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated", "morphism of algebraic spaces over $S$. Let $Y \\to X' \\to X$ be the", "normalization of $X$ in $Y$.", "\\begin{enumerate}", "\\item If $W \\to X$ is an \\'etale morphism of algebraic spaces over $S$,", "then $W \\times_X X'$ is the normalization of $W$ in $W \\times_X Y$.", "\\item If $Y$ and $X$ are representable, then $Y'$ is representable", "and is canonically isomorphic to the normalization of the scheme $X$", "in the scheme $Y$ as constructed in", "Morphisms, Section \\ref{morphisms-section-normalization}.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is immediate from the construction that the formation", "of the normalization of $X$ in $Y$ commutes with \\'etale", "base change, i.e., part (1) holds. On the other hand, if", "$X$ and $Y$ are schemes, then for $U \\subset X$ affine open,", "$f_*\\mathcal{O}_Y(U) = \\mathcal{O}_Y(f^{-1}(U))$ and hence", "$\\nu^{-1}(U)$ is the spectrum of exactly the same ring as", "we get in the corresponding construction for schemes." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4959, "type": "theorem", "label": "spaces-morphisms-lemma-characterize-normalization", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-characterize-normalization", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated", "morphism of algebraic spaces over $S$. The factorization $f = \\nu \\circ f'$,", "where $\\nu : X' \\to X$ is the normalization of $X$ in $Y$ is characterized", "by the following two properties:", "\\begin{enumerate}", "\\item the morphism $\\nu$ is integral, and", "\\item for any factorization $f = \\pi \\circ g$, with $\\pi : Z \\to X$", "integral, there exists a commutative diagram", "$$", "\\xymatrix{", "Y \\ar[d]_{f'} \\ar[r]_g & Z \\ar[d]^\\pi \\\\", "X' \\ar[ru]^h \\ar[r]^\\nu & X", "}", "$$", "for a unique morphism $h : X' \\to Z$.", "\\end{enumerate}", "Moreover, in (2) the morphism $h : X' \\to Z$ is the normalization of", "$Z$ in $Y$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{O}' \\subset f_*\\mathcal{O}_Y$ be the integral closure of", "$\\mathcal{O}_X$ as in Definition \\ref{definition-normalization-X-in-Y}.", "The morphism $\\nu$ is integral by construction, which proves (1).", "Assume given a factorization $f = \\pi \\circ g$ with $\\pi : Z \\to X$", "integral as in (2). By Definition \\ref{definition-integral}", "$\\pi$ is affine, and hence $Z$ is the relative", "spectrum of a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras $\\mathcal{B}$.", "The morphism $g : X \\to Z$ corresponds to a map of $\\mathcal{O}_X$-algebras", "$\\chi : \\mathcal{B} \\to f_*\\mathcal{O}_Y$. Since $\\mathcal{B}(U)$ is", "integral over $\\mathcal{O}_X(U)$ for every affine $U$ \\'etale over $X$", "(by Definition \\ref{definition-integral})", "we see from Lemma \\ref{lemma-integral-closure}", "that $\\chi(\\mathcal{B}) \\subset \\mathcal{O}'$.", "By the functoriality of the relative spectrum", "Lemma \\ref{lemma-affine-equivalence-algebras}", "this provides us with a unique morphism", "$h : X' \\to Z$. We omit the verification that the diagram commutes.", "\\medskip\\noindent", "It is clear that (1) and (2) characterize the", "factorization $f = \\nu \\circ f'$ since it characterizes it", "as an initial object in a category. The morphism $h$ in (2)", "is integral by Lemma \\ref{lemma-finite-permanence}.", "Given a factorization $g = \\pi' \\circ g'$ with $\\pi' : Z' \\to Z$", "integral, we get a factorization $f = (\\pi \\circ \\pi') \\circ g'$ and", "we get a morphism $h' : X' \\to Z'$. Uniqueness implies that", "$\\pi' \\circ h' = h$. Hence the characterization (1), (2) applies", "to the morphism $h : X' \\to Z$ which gives the last statement of the lemma." ], "refs": [ "spaces-morphisms-definition-normalization-X-in-Y", "spaces-morphisms-definition-integral", "spaces-morphisms-definition-integral", "spaces-morphisms-lemma-integral-closure", "spaces-morphisms-lemma-affine-equivalence-algebras", "spaces-morphisms-lemma-finite-permanence" ], "ref_ids": [ 5026, 5017, 5017, 4957, 4802, 4949 ] } ], "ref_ids": [] }, { "id": 4960, "type": "theorem", "label": "spaces-morphisms-lemma-normalization-in-reduced", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-normalization-in-reduced", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and", "quasi-separated morphism of algebraic spaces over $S$.", "Let $X' \\to X$ be the normalization of $X$ in $Y$.", "If $Y$ is reduced, so is $X'$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that a subring of a reduced ring is reduced.", "Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4961, "type": "theorem", "label": "spaces-morphisms-lemma-normalization-generic", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-normalization-generic", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated", "morphism of schemes. Let $X' \\to X$ be the normalization of $X$ in $Y$.", "If $x' \\in |X'|$ is a point of codimension $0$", "(Properties of Spaces, Definition", "\\ref{spaces-properties-definition-dimension-local-ring}), then", "$x'$ is the image of some $y \\in |Y|$ of codimension $0$." ], "refs": [ "spaces-properties-definition-dimension-local-ring" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-properties-normalization} and the definitions, we may", "assume that $X = \\Spec(A)$ is affine. Then $X' = \\Spec(A')$ where $A'$ is", "the integral closure of $A$ in $\\Gamma(Y, \\mathcal{O}_Y)$ and $x'$ corresponds", "to a minimal prime of $A'$. Choose a surjective \\'etale", "morphism $V \\to Y$ where $V = \\Spec(B)$ is affine. Then", "$A' \\to B$ is injective, hence every minimal prime of $A'$ is", "the image of a minimal prime of $B$, see", "Algebra, Lemma \\ref{algebra-lemma-injective-minimal-primes-in-image}.", "The lemma follows." ], "refs": [ "spaces-morphisms-lemma-properties-normalization", "algebra-lemma-injective-minimal-primes-in-image" ], "ref_ids": [ 4958, 445 ] } ], "ref_ids": [ 11931 ] }, { "id": 4962, "type": "theorem", "label": "spaces-morphisms-lemma-normalization-in-disjoint-union", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-normalization-in-disjoint-union", "contents": [ "Let $S$ be a scheme.", "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of", "algebraic spaces over $S$.", "Suppose that $Y = Y_1 \\amalg Y_2$ is a disjoint union of two", "algebraic spaces.", "Write $f_i = f|_{Y_i}$. Let $X_i'$ be the normalization of $X$ in $Y_i$.", "Then $X_1' \\amalg X_2'$ is the normalization of $X$ in $Y$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4963, "type": "theorem", "label": "spaces-morphisms-lemma-normalization-in-universally-closed", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-normalization-in-universally-closed", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a quasi-compact, quasi-separated and", "universally closed morphisms of algebraic spaces over $S$.", "Then $f_*\\mathcal{O}_X$ is integral over $\\mathcal{O}_Y$. In other", "words, the normalization of $Y$ in $X$ is equal to the factorization", "$$", "X \\longrightarrow \\underline{\\Spec}_Y(f_*\\mathcal{O}_X)", "\\longrightarrow Y", "$$", "of Remark \\ref{remark-factorization-quasi-compact-quasi-separated}." ], "refs": [ "spaces-morphisms-remark-factorization-quasi-compact-quasi-separated" ], "proofs": [ { "contents": [ "The question is \\'etale local on $Y$, hence we may reduce to the", "case where $Y = \\Spec(R)$ is affine. Let $h \\in \\Gamma(X, \\mathcal{O}_X)$.", "We have to show that $h$ satisfies a monic equation over $R$. Think of $h$", "as a morphism as in the following commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_h \\ar[rd]_f & & \\mathbf{A}^1_Y \\ar[ld] \\\\", "& Y &", "}", "$$", "Let $Z \\subset \\mathbf{A}^1_Y$ be the scheme theoretic image of $h$,", "see Definition \\ref{definition-scheme-theoretic-image}.", "The morphism $h$ is quasi-compact as $f$ is quasi-compact and", "$\\mathbf{A}^1_Y \\to Y$ is separated, see", "Lemma \\ref{lemma-quasi-compact-permanence}.", "By Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image} the", "morphism $X \\to Z$ has dense image on underlying topological spaces. By", "Lemma \\ref{lemma-universally-closed-permanence} the morphism", "$X \\to Z$ is closed. Hence $h(X) = Z$ (set theoretically).", "Thus we can use", "Lemma \\ref{lemma-image-proper-is-proper}", "to conclude that $Z \\to Y$ is universally closed (and even proper).", "Since $Z \\subset \\mathbf{A}^1_Y$, we see that $Z \\to Y$ is affine", "and proper, hence integral by Lemma \\ref{lemma-integral-universally-closed}.", "Writing $\\mathbf{A}^1_Y = \\Spec(R[T])$ we conclude that", "the ideal $I \\subset R[T]$ of $Z$ contains a monic polynomial", "$P(T) \\in R[T]$. Hence $P(h) = 0$ and we win." ], "refs": [ "spaces-morphisms-definition-scheme-theoretic-image", "spaces-morphisms-lemma-quasi-compact-permanence", "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-morphisms-lemma-image-proper-is-proper", "spaces-morphisms-lemma-integral-universally-closed" ], "ref_ids": [ 4994, 4743, 4780, 4920, 4921, 4944 ] } ], "ref_ids": [ 5032 ] }, { "id": 4964, "type": "theorem", "label": "spaces-morphisms-lemma-normalization-in-integral", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-normalization-in-integral", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be an integral morphism", "of algebraic spaces over $S$.", "Then the integral closure of $X$ in $Y$ is equal to $Y$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-integral-universally-closed} this is a special case of", "Lemma \\ref{lemma-normalization-in-universally-closed}." ], "refs": [ "spaces-morphisms-lemma-integral-universally-closed", "spaces-morphisms-lemma-normalization-in-universally-closed" ], "ref_ids": [ 4944, 4963 ] } ], "ref_ids": [] }, { "id": 4965, "type": "theorem", "label": "spaces-morphisms-lemma-nagata-normalization-finite", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-nagata-normalization-finite", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume that", "\\begin{enumerate}", "\\item $Y$ is Nagata,", "\\item $f$ is quasi-separated of finite type,", "\\item $X$ is reduced.", "\\end{enumerate}", "Then the normalization $\\nu : Y' \\to Y$ of $Y$ in $X$ is finite." ], "refs": [], "proofs": [ { "contents": [ "The question is \\'etale local on $Y$, see", "Lemma \\ref{lemma-properties-normalization}.", "Thus we may assume $Y = \\Spec(R)$ is affine.", "Then $R$ is a Noetherian Nagata ring", "and we have to show that the integral closure of", "$R$ in $\\Gamma(X, \\mathcal{O}_X)$ is finite over $R$.", "Since $f$ is quasi-compact we see that $X$ is quasi-compact.", "Choose an affine scheme $U$ and a surjective \\'etale", "morphism $U \\to X$ (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "Then $\\Gamma(X, \\mathcal{O}_X) \\subset \\Gamma(U, \\mathcal{O}_X)$.", "Since $R$ is Noetherian it suffices to show that", "the integral closure of $R$ in $\\Gamma(U, \\mathcal{O}_U)$", "is finite over $R$. As $U \\to Y$ is of finite type", "this follows from", "Morphisms, Lemma \\ref{morphisms-lemma-nagata-normalization-finite}." ], "refs": [ "spaces-morphisms-lemma-properties-normalization", "spaces-properties-lemma-quasi-compact-affine-cover", "morphisms-lemma-nagata-normalization-finite" ], "ref_ids": [ 4958, 11832, 5510 ] } ], "ref_ids": [] }, { "id": 4966, "type": "theorem", "label": "spaces-morphisms-lemma-prepare-normalization", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-prepare-normalization", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item there is a surjective \\'etale morphism $U \\to X$ where $U$", "is as scheme such that every quasi-compact open of $U$ has", "finitely many irreducible components,", "\\item for every scheme $U$ and every \\'etale morphism", "$U \\to X$ every quasi-compact open of $U$ has finitely many", "irreducible components, and", "\\item for every quasi-compact algebraic space $Y$ \\'etale over $X$", "the space $|Y|$ has finitely many irreducible components.", "\\end{enumerate}", "If $X$ is representable this means that every quasi-compact open of $X$", "has finitely many irreducible components." ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) and the final statement follow from", "Descent, Lemma \\ref{descent-lemma-locally-finite-nr-irred-local-fppf} and", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-type-property}.", "It is clear that (3) implies (1) and (2). Conversely, assume (2) and", "let $Y \\to X$ be an \\'etale morphism of algebraic spaces with $Y$", "quasi-compact. Then we can choose an affine scheme $V$ and a surjective", "\\'etale morphism $V \\to Y$ (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "Since $V$ has finitely many irreducible components by (2) and since", "$|V| \\to |Y|$ is surjective and continuous, we conclude that", "$|Y|$ has finitely many irreducible components." ], "refs": [ "descent-lemma-locally-finite-nr-irred-local-fppf", "spaces-properties-lemma-type-property", "spaces-properties-lemma-quasi-compact-affine-cover" ], "ref_ids": [ 14650, 11837, 11832 ] } ], "ref_ids": [] }, { "id": 4967, "type": "theorem", "label": "spaces-morphisms-lemma-normalization", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-normalization", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying", "the equivalent conditions of Lemma \\ref{lemma-prepare-normalization}.", "Then there exists an integral morphism of algebraic spaces", "$$", "X^\\nu \\longrightarrow X", "$$", "such that for every scheme $U$ and \\'etale morphism $U \\to X$", "the fibre product $X^\\nu \\times_X U$ is the normalization of $U$." ], "refs": [ "spaces-morphisms-lemma-prepare-normalization" ], "proofs": [ { "contents": [ "Let $U \\to X$ be a surjective \\'etale morphism where $U$ is a scheme.", "Set $R = U \\times_X U$ with projections $s, t : R \\to U$ and", "$j = (t, s) : R \\to U \\times_S U$ so that $X = U/R$, see", "Spaces, Lemma \\ref{spaces-lemma-space-presentation}.", "The assumption on $X$ means that the normalization $U^\\nu$ of $U$", "is defined, see Morphisms, Definition", "\\ref{morphisms-definition-normalization}.", "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-normalization-and-smooth}", "taking normalization commutes with \\'etale morphisms of schemes.", "Thus we see that the normalization $R^\\nu$ of $R$", "is isomorphic to both $R \\times_{s, U} U^\\nu$ and $U^\\nu \\times_{U, t} R$.", "Thus we obtain two \\'etale morphisms", "$s^\\nu : R^\\nu \\to U^\\nu$ and $t^\\nu : R^\\nu \\to U^\\nu$", "of schemes. The induced morphism $j^\\nu : R^\\nu \\to U^\\nu \\times_S U^\\nu$", "is a monomorphism as $R^\\nu$ is a subscheme of the restriction of", "$R$ to $U^\\nu$. A formal computation with fibre products shows that", "$R^\\nu \\times_{s^\\nu, U^\\nu, t^\\nu} R^\\nu$ is the normalization", "of $R \\times_{s, U, t} R$. Hence the (\\'etale) morphism", "$c : R \\times_{s, U, t} R \\to R$ extends to $c^\\nu$ as well.", "Combined we see that we obtain an \\'etale equivalence relation.", "Setting $X^\\nu = U^\\nu/R^\\nu$", "(Spaces, Theorem \\ref{spaces-theorem-presentation})", "we see that we have $U^\\nu = X^\\nu \\times_X U$ by", "Groupoids, Lemma \\ref{groupoids-lemma-criterion-fibre-product}.", "We omit the verification that this property then holds for every", "\\'etale morphism from a scheme to $X$." ], "refs": [ "spaces-lemma-space-presentation", "morphisms-definition-normalization", "more-morphisms-lemma-normalization-and-smooth", "spaces-theorem-presentation", "groupoids-lemma-criterion-fibre-product" ], "ref_ids": [ 8149, 5592, 13775, 8124, 9652 ] } ], "ref_ids": [ 4966 ] }, { "id": 4968, "type": "theorem", "label": "spaces-morphisms-lemma-normalization-reduced", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-normalization-reduced", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying the", "equivalent conditions of Lemma \\ref{lemma-prepare-normalization}.", "The normalization morphism $\\nu$ factors through the reduction $X_{red}$", "and $X^\\nu \\to X_{red}$ is the normalization of $X_{red}$." ], "refs": [ "spaces-morphisms-lemma-prepare-normalization" ], "proofs": [ { "contents": [ "We may check this \\'etale locally on $X$ and hence reduce to the case", "of schemes which is", "Morphisms, Lemma \\ref{morphisms-lemma-normalization-reduced}.", "Some details omitted." ], "refs": [ "morphisms-lemma-normalization-reduced" ], "ref_ids": [ 5512 ] } ], "ref_ids": [ 4966 ] }, { "id": 4969, "type": "theorem", "label": "spaces-morphisms-lemma-normalization-normal", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-normalization-normal", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying the", "equivalent conditions of Lemma \\ref{lemma-prepare-normalization}.", "\\begin{enumerate}", "\\item The normalization $X^\\nu$ is normal.", "\\item The morphism $\\nu : X^\\nu \\to X$ is integral and surjective.", "\\item The map $|\\nu| : |X^\\nu| \\to |X|$ induces a bijection between", "the sets of points of codimension $0$ (Properties of Spaces,", "Definition \\ref{spaces-properties-definition-dimension-local-ring}).", "\\item Let $Z \\to X$ be a morphism. Assume $Z$ is a normal algebraic space", "and that for $z \\in |Z|$ we have: $z$ has codimension $0$ in", "$Z \\Rightarrow f(z)$ has codimension $0$ in $X$. Then", "there exists a unique factorization $Z \\to X^\\nu \\to X$.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-lemma-prepare-normalization", "spaces-properties-definition-dimension-local-ring" ], "proofs": [ { "contents": [ "Properties (1), (2), and (3) follow from the corresponding results", "for schemes (Morphisms, Lemma \\ref{morphisms-lemma-normalization-normal})", "combined with the fact that a point of a scheme is a generic", "point of an irreducible component if and only if the dimension", "of the local ring is zero", "(Properties, Lemma \\ref{properties-lemma-generic-point}).", "\\medskip\\noindent", "Let $Z \\to X$ be a morphism as in (4). Let $U$ be a scheme and let", "$U \\to X$ be a surjective \\'etale morphism. Choose a scheme $V$ and", "a surjective \\'etale morphism $V \\to U \\times_X Z$. The condition on", "geometric points assures us that $V \\to U$ maps generic points of", "irreducible components of $V$ to generic points of irreducible", "components of $U$. Thus we obtain a unique factorization", "$V \\to U^\\nu \\to U$ by", "Morphisms, Lemma \\ref{morphisms-lemma-normalization-normal}.", "The uniqueness guarantees us that the two maps", "$$", "V \\times_{U \\times_X Z} V \\to V \\to U^\\nu", "$$", "agree because these maps are the unique factorization of the map", "$V \\times_{U \\times_X Z} V \\to V \\to U$.", "Since the algebraic space $U \\times_X Z$ is equal to the quotient", "$V/V \\times_{U \\times_X Z} V$", "(see Spaces, Section \\ref{spaces-section-presentations})", "we find a canonical morphism", "$U \\times_X Z \\to U^\\nu$. Picture", "$$", "\\xymatrix{", "U \\times_X Z \\ar[r] \\ar[d] & U^\\nu \\ar[r] \\ar[d] & U \\ar[d] \\\\", "Z \\ar@/_/[rr] \\ar@{..>}[r] & X^\\nu \\ar[r] & X", "}", "$$", "To obtain the dotted arrow we note that the construction of the", "arrow $U \\times_X Z$ is functorial in the \\'etale morphism $U \\to X$", "(precise formulation and proof omitted).", "Hence if we set $R = U \\times_X U$ with projections $s, t : R \\to U$,", "then we obtain a morphism $R \\times_X Z \\to R^\\nu$ commuting with", "$s, t : R \\to U$ and $s^\\nu, t^\\nu : R^\\nu \\to U^\\nu$.", "Recall that $X^\\nu = U^\\nu/R^\\nu$, see proof of", "Lemma \\ref{lemma-normalization}. Since $X = U/R$ a", "simple sheaf theoretic argument", "shows that $Z = (U \\times_X Z)/(R \\times_X Z)$. Thus the morphisms", "$U \\times_X Z \\to U^\\nu$ and $R \\times_X Z \\to R^\\nu$ define a", "morphism $Z \\to X^\\nu$ as desired." ], "refs": [ "morphisms-lemma-normalization-normal", "properties-lemma-generic-point", "morphisms-lemma-normalization-normal", "spaces-morphisms-lemma-normalization" ], "ref_ids": [ 5515, 2980, 5515, 4967 ] } ], "ref_ids": [ 4966, 11931 ] }, { "id": 4970, "type": "theorem", "label": "spaces-morphisms-lemma-nagata-normalization", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-nagata-normalization", "contents": [ "Let $S$ be a scheme. Let $X$ be a Nagata algebraic space over $S$.", "The normalization $\\nu : X^\\nu \\to X$ is a finite morphism." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ being Nagata is locally Noetherian,", "Definition \\ref{definition-normalization} applies.", "By construction of $X^\\nu$ in Lemma \\ref{lemma-normalization}", "we immediately reduce to the case of schemes which is", "Morphisms, Lemma \\ref{morphisms-lemma-nagata-normalization}." ], "refs": [ "spaces-morphisms-definition-normalization", "spaces-morphisms-lemma-normalization", "morphisms-lemma-nagata-normalization" ], "ref_ids": [ 5027, 4967, 5520 ] } ], "ref_ids": [] }, { "id": 4971, "type": "theorem", "label": "spaces-morphisms-lemma-neighbourhood-scheme", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-neighbourhood-scheme", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "V' \\ar[r] \\ar[rd] & T' \\times_T X \\ar[r] \\ar[d] & X \\ar[d] \\\\", "& T' \\ar[r] & T", "}", "$$", "of algebraic spaces over $S$. Assume", "\\begin{enumerate}", "\\item $T' \\to T$ is an \\'etale morphism of affine schemes,", "\\item $X \\to T$ is a separated, locally quasi-finite morphism,", "\\item $V'$ is an open subspace of $T' \\times_T X$, and", "\\item $V' \\to T'$ is quasi-affine.", "\\end{enumerate}", "In this situation the image $U$ of $V'$ in $X$ is a quasi-compact", "open subspace of $X$ which is representable." ], "refs": [], "proofs": [ { "contents": [ "We first make some trivial observations.", "Note that $V'$ is representable by Lemma \\ref{lemma-quasi-affine-local}.", "It is also quasi-compact (as a quasi-affine scheme over an affine scheme, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-affine-separated}).", "Since $T' \\times_T X \\to X$ is \\'etale", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-base-change-etale})", "the map $|T' \\times_T X| \\to |X|$ is open, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-open}.", "Let $U \\subset X$ be the open subspace corresponding to the image of", "$|V'|$, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-open-subspaces}.", "As $|V'|$ is quasi-compact we see that $|U|$ is quasi-compact, hence", "$U$ is a quasi-compact algebraic space, by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quasi-compact-space}.", "\\medskip\\noindent", "By", "Morphisms,", "Lemma \\ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded}", "the morphism $T' \\to T$ is universally bounded. Hence we can do induction on", "the integer $n$ bounding the degree of the fibres of $T' \\to T$, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-universally-bounded}", "for a description of this integer in the case of an \\'etale morphism.", "If $n = 1$, then $T' \\to T$ is an open immersion (see", "\\'Etale Morphisms, Theorem \\ref{etale-theorem-etale-radicial-open}),", "and the result is clear. Assume $n > 1$.", "\\medskip\\noindent", "Consider the affine scheme $T'' = T' \\times_T T'$.", "As $T' \\to T$ is \\'etale we have a decomposition (into open and closed affine", "subschemes) $T'' = \\Delta(T') \\amalg T^*$. Namely $\\Delta = \\Delta_{T'/T}$", "is open by", "Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism}", "and closed because $T' \\to T$ is separated as a morphism of affines.", "As a base change the degrees of the fibres of the second projection", "$\\text{pr}_1 : T' \\times_T T' \\to T'$ are bounded by $n$, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-universally-bounded}.", "On the other hand, $\\text{pr}_1|_{\\Delta(T')} : \\Delta(T') \\to T'$ is", "an isomorphism and every fibre has exactly one point.", "Thus, on applying", "Morphisms, Lemma \\ref{morphisms-lemma-etale-universally-bounded}", "we conclude the degrees of the fibres of the restriction", "$\\text{pr}_1|_{T^*} : T^* \\to T'$ are bounded by $n - 1$.", "Hence the induction hypothesis applied to the diagram", "$$", "\\xymatrix{", "p_0^{-1}(V') \\cap X^* \\ar[r] \\ar[rd] &", "X^* \\ar[r]_{p_1|_{X^*}} \\ar[d] &", "X' \\ar[d] \\\\", "& T^* \\ar[r]^{\\text{pr}_1|_{T^*}} & T'", "}", "$$", "gives that $p_1(p_0^{-1}(V') \\cap X^*)$", "is a quasi-compact scheme. Here we set", "$X'' = T'' \\times_T X$, $X^* = T^* \\times_T X$, and $X' = T' \\times_T X$,", "and $p_0, p_1 : X'' \\to X'$ are the base changes of $\\text{pr}_0, \\text{pr}_1$.", "Most of the hypotheses of the lemma imply", "by base change the corresponding hypothesis for the diagram above.", "For example $p_0^{-1}(V') = T'' \\times_{T'} V'$", "is a scheme quasi-affine over $T''$ as a base change. Some", "verifications omitted.", "\\medskip\\noindent", "By", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme}", "we conclude that", "$$", "p_1(p_0^{-1}(V')) =", "V' \\cup p_1(p_0^{-1}(V') \\cap X^*)", "$$", "is a quasi-compact scheme. Moreover, it is clear that", "$p_1(p_0^{-1}(V'))$ is the inverse image of the", "quasi-compact open subspace $U \\subset X$ discussed in the", "first paragraph of the proof. In other words, $T' \\times_T U$ is a scheme!", "Note that $T' \\times_T U$ is quasi-compact and", "separated and locally quasi-finite over $T'$, as", "$T' \\times_T X \\to T'$ is locally quasi-finite and separated", "being a base change of the original morphism $X \\to T$ (see", "Lemmas \\ref{lemma-base-change-separated} and", "\\ref{lemma-base-change-quasi-finite}).", "This implies by", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-quasi-finite-separated-quasi-affine}", "that $T' \\times_T U \\to T'$ is quasi-affine.", "\\medskip\\noindent", "By", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves}", "this gives a descent datum on $T' \\times_T U / T'$", "relative to the \\'etale covering $\\{T' \\to W\\}$, where $W \\subset T$", "is the image of the morphism $T' \\to T$.", "Because $U'$ is quasi-affine over $T'$ we see from", "Descent, Lemma \\ref{descent-lemma-quasi-affine}", "that this datum is effective, and by the last part of", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves}", "this implies that $U$ is a scheme as desired.", "Some minor details omitted." ], "refs": [ "spaces-morphisms-lemma-quasi-affine-local", "morphisms-lemma-quasi-affine-separated", "spaces-properties-lemma-base-change-etale", "spaces-properties-lemma-etale-open", "spaces-properties-lemma-open-subspaces", "spaces-properties-lemma-quasi-compact-space", "morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded", "morphisms-lemma-etale-universally-bounded", "etale-theorem-etale-radicial-open", "morphisms-lemma-diagonal-unramified-morphism", "morphisms-lemma-base-change-universally-bounded", "morphisms-lemma-etale-universally-bounded", "spaces-properties-lemma-subscheme", "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-base-change-quasi-finite", "more-morphisms-lemma-quasi-finite-separated-quasi-affine", "descent-lemma-descent-data-sheaves", "descent-lemma-quasi-affine", "descent-lemma-descent-data-sheaves" ], "ref_ids": [ 4807, 5184, 11858, 11860, 11823, 11827, 5531, 5530, 10692, 5354, 5527, 5530, 11848, 4714, 4832, 13900, 14751, 14750, 14751 ] } ], "ref_ids": [] }, { "id": 4972, "type": "theorem", "label": "spaces-morphisms-lemma-locally-quasi-finite-separated-representable", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-locally-quasi-finite-separated-representable", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. If $f$ is locally quasi-finite and separated, then", "$f$ is representable." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from", "Proposition \\ref{proposition-locally-quasi-finite-separated-over-scheme}", "and the fact that being locally quasi-finite and separated is", "preserved under any base change, see", "Lemmas \\ref{lemma-base-change-quasi-finite} and", "\\ref{lemma-base-change-separated}." ], "refs": [ "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "spaces-morphisms-lemma-base-change-quasi-finite", "spaces-morphisms-lemma-base-change-separated" ], "ref_ids": [ 4983, 4832, 4714 ] } ], "ref_ids": [] }, { "id": 4973, "type": "theorem", "label": "spaces-morphisms-lemma-etale-universally-injective-open", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-etale-universally-injective-open", "contents": [ "\\begin{slogan}", "Universally injective \\'etale maps are open immersions.", "\\end{slogan}", "Let $S$ be a scheme. Let $f : X \\to Y$ be an \\'etale and universally", "injective morphism of algebraic spaces over $S$. Then $f$ is an open", "immersion." ], "refs": [], "proofs": [ { "contents": [ "Let $T \\to Y$ be a morphism from a scheme into $Y$.", "If we can show that $X \\times_Y T \\to T$ is an open immersion, then we", "are done. Since being \\'etale and being universally injective are", "properties of morphisms stable under base change (see", "Lemmas \\ref{lemma-base-change-etale} and", "\\ref{lemma-base-change-universally-injective})", "we may assume that $Y$ is a scheme. Note that the", "diagonal $\\Delta_{X/Y} : X \\to X \\times_Y X$ is \\'etale, a monomorphism, and", "surjective by", "Lemma \\ref{lemma-universally-injective}.", "Hence we see that $\\Delta_{X/Y}$ is an isomorphism (see", "Spaces, Lemma", "\\ref{spaces-lemma-surjective-flat-locally-finite-presentation}),", "in particular we see that $X$ is separated over $Y$.", "It follows that $X$ is a scheme too, by", "Proposition \\ref{proposition-locally-quasi-finite-separated-over-scheme}.", "Finally, $X \\to Y$ is an open immersion by the fundamental theorem", "for \\'etale morphisms of schemes, see", "\\'Etale Morphisms, Theorem \\ref{etale-theorem-etale-radicial-open}." ], "refs": [ "spaces-morphisms-lemma-base-change-etale", "spaces-morphisms-lemma-base-change-universally-injective", "spaces-morphisms-lemma-universally-injective", "spaces-lemma-surjective-flat-locally-finite-presentation", "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "etale-theorem-etale-radicial-open" ], "ref_ids": [ 4907, 4794, 4793, 8137, 4983, 10692 ] } ], "ref_ids": [] }, { "id": 4974, "type": "theorem", "label": "spaces-morphisms-lemma-finite-type-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-finite-type-separated", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$ which is representable, of finite type, and separated.", "Let $Y'$ be the normalization of $Y$ in $X$. Picture:", "$$", "\\xymatrix{", "X \\ar[rd]_f \\ar[rr]_{f'} & & Y' \\ar[ld]^\\nu \\\\", "& Y &", "}", "$$", "Then there exists an open subspace $U' \\subset Y'$ such that", "\\begin{enumerate}", "\\item $(f')^{-1}(U') \\to U'$ is an isomorphism, and", "\\item $(f')^{-1}(U') \\subset X$ is the set of points at which", "$f$ is quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $W \\to Y$ be a surjective \\'etale morphism where $W$ is a scheme.", "Then $W \\times_Y X$ is a scheme as well. By", "Lemma \\ref{lemma-properties-normalization}", "the algebraic space $W \\times_Y Y'$ is representable and is", "the normalization of the scheme $W$ in the scheme $W \\times_Y X$. Picture", "$$", "\\xymatrix{", "W \\times_Y X \\ar[rd]_{(1, f)} \\ar[rr]_{(1, f')} & &", "W \\times_Y Y' \\ar[ld]^{(1, \\nu)} \\\\", "& W &", "}", "$$", "By More on Morphisms, Lemma \\ref{more-morphisms-lemma-finite-type-separated}", "the result of the lemma holds over $W$. Let $V' \\subset W \\times_Y Y'$", "be the open subscheme such that", "\\begin{enumerate}", "\\item $(1, f')^{-1}(V') \\to V'$ is an isomorphism, and", "\\item $(1, f')^{-1}(V') \\subset W \\times_Y X$ is the set of points at which", "$(1, f)$ is quasi-finite.", "\\end{enumerate}", "By Lemma \\ref{lemma-locally-finite-type-quasi-finite-part}", "there is a maximal open set of points $U \\subset X$ where $f$", "is quasi-finite and $W \\times_Y U = (1, f')^{-1}(V')$.", "The morphism $f'|_U : U \\to Y'$ is an open immersion", "by Lemma \\ref{lemma-closed-immersion-local}", "as its base change to $W$ is the isomorphism $(1, f')^{-1}(V') \\to V'$", "followed by the open immersion $V' \\to W \\times_Y Y'$.", "Setting $U' = \\Im(U \\to Y')$ finishes the proof", "(omitted: the verification that $(f')^{-1}(U') = U$)." ], "refs": [ "spaces-morphisms-lemma-properties-normalization", "more-morphisms-lemma-finite-type-separated", "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part", "spaces-morphisms-lemma-closed-immersion-local" ], "ref_ids": [ 4958, 13899, 4876, 4761 ] } ], "ref_ids": [] }, { "id": 4975, "type": "theorem", "label": "spaces-morphisms-lemma-quasi-finite-separated-quasi-affine", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-quasi-finite-separated-quasi-affine", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume $f$ is quasi-finite and separated.", "Let $Y'$ be the normalization of $Y$ in $X$.", "Picture:", "$$", "\\xymatrix{", "X \\ar[rd]_f \\ar[rr]_{f'} & & Y' \\ar[ld]^\\nu \\\\", "& Y &", "}", "$$", "Then $f'$ is a quasi-compact open immersion and $\\nu$ is integral.", "In particular $f$ is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-locally-quasi-finite-separated-representable}", "the morphism $f$ is representable. Hence we may apply", "Lemma \\ref{lemma-finite-type-separated}. Thus there exists an open", "subspace $U' \\subset Y'$ such that", "$(f')^{-1}(U') = X$ (!) and $X \\to U'$ is an isomorphism! In other", "words, $f'$ is an open immersion. Note that $f'$ is quasi-compact as", "$f$ is quasi-compact and $\\nu : Y' \\to Y$ is separated", "(Lemma \\ref{lemma-quasi-compact-permanence}).", "Hence for every affine scheme $Z$ and morphism $Z \\to Y$ the", "fibre product $Z \\times_Y X$ is a quasi-compact open subscheme", "of the affine scheme $Z \\times_Y Y'$. Hence $f$ is quasi-affine by", "definition." ], "refs": [ "spaces-morphisms-lemma-locally-quasi-finite-separated-representable", "spaces-morphisms-lemma-finite-type-separated", "spaces-morphisms-lemma-quasi-compact-permanence" ], "ref_ids": [ 4972, 4974, 4743 ] } ], "ref_ids": [] }, { "id": 4976, "type": "theorem", "label": "spaces-morphisms-lemma-universal-homeomorphism-representable", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-universal-homeomorphism-representable", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$.", "Then $f$ is a universal homeomorphism", "(in the sense of Section \\ref{section-representable}) if and only", "if for every morphism of algebraic spaces $Z \\to Y$ the base change", "map $Z \\times_Y X \\to Z$ induces a homeomorphism", "$|Z \\times_Y X| \\to |Z|$." ], "refs": [], "proofs": [ { "contents": [ "If for every morphism of algebraic spaces $Z \\to Y$ the base change", "map $Z \\times_Y X \\to Z$ induces a homeomorphism", "$|Z \\times_Y X| \\to |Z|$, then the same is true whenever $Z$ is a scheme,", "which formally implies that $f$ is a universal homeomorphism in the", "sense of", "Section \\ref{section-representable}.", "Conversely, if $f$ is a universal homeomorphism in the sense of", "Section \\ref{section-representable}", "then $X \\to Y$ is integral, universally injective and surjective", "(by Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}", "and", "Morphisms, Lemma \\ref{morphisms-lemma-universal-homeomorphism}).", "Hence $f$ is universally closed, see", "Lemma \\ref{lemma-integral-universally-closed}", "and universally injective and (universally) surjective, i.e.,", "$f$ is a universal homeomorphism." ], "refs": [ "spaces-lemma-representable-transformations-property-implication", "morphisms-lemma-universal-homeomorphism", "spaces-morphisms-lemma-integral-universally-closed" ], "ref_ids": [ 8136, 5454, 4944 ] } ], "ref_ids": [] }, { "id": 4977, "type": "theorem", "label": "spaces-morphisms-lemma-base-change-universal-homeomorphism", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-base-change-universal-homeomorphism", "contents": [ "The base change of a universal homeomorphism of algebraic spaces", "by any morphism of algebraic spaces is a universal homeomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4978, "type": "theorem", "label": "spaces-morphisms-lemma-composition-universal-homeomorphism", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-composition-universal-homeomorphism", "contents": [ "The composition of a pair of universal homeomorphisms of", "algebraic spaces is a universal homeomorphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 4979, "type": "theorem", "label": "spaces-morphisms-lemma-reduction-universal-homeomorphism", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-reduction-universal-homeomorphism", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The canonical closed immersion $X_{red} \\to X$ (see", "Properties of Spaces, Definition", "\\ref{spaces-properties-definition-reduced-induced-space})", "is a universal homeomorphism." ], "refs": [ "spaces-properties-definition-reduced-induced-space" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 11932 ] }, { "id": 4980, "type": "theorem", "label": "spaces-morphisms-lemma-integral-universally-injective-push-pull", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-lemma-integral-universally-injective-push-pull", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a universally injective,", "integral morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item The functor", "$$", "f_{small, *} : \\Sh(Y_\\etale) \\longrightarrow \\Sh(X_\\etale)", "$$", "is fully faithful and its essential image is those sheaves of sets", "$\\mathcal{F}$ on $X_\\etale$ whose restriction to $|X| \\setminus f(|Y|)$", "is isomorphic to $*$, and", "\\item the functor", "$$", "f_{small, *} : \\textit{Ab}(Y_\\etale) \\longrightarrow \\textit{Ab}(X_\\etale)", "$$", "is fully faithful and its essential image is those abelian sheaves on", "$Y_\\etale$ whose support is contained in $f(|Y|)$.", "\\end{enumerate}", "In both cases $f_{small}^{-1}$ is a left inverse to the functor $f_{small, *}$." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is integral it is universally closed", "(Lemma \\ref{lemma-integral-universally-closed}).", "In particular, $f(|Y|)$ is a closed subset of $|X|$", "and the statements make sense.", "The rest of the proof is identical to the proof of", "Lemma \\ref{lemma-closed-immersion-push-pull}", "except that we use", "\\'Etale Cohomology, Proposition", "\\ref{etale-cohomology-proposition-integral-universally-injective-pushforward}", "instead of", "\\'Etale Cohomology, Proposition", "\\ref{etale-cohomology-proposition-closed-immersion-pushforward}." ], "refs": [ "spaces-morphisms-lemma-integral-universally-closed", "spaces-morphisms-lemma-closed-immersion-push-pull", "etale-cohomology-proposition-integral-universally-injective-pushforward", "etale-cohomology-proposition-closed-immersion-pushforward" ], "ref_ids": [ 4944, 4768, 6701, 6700 ] } ], "ref_ids": [] }, { "id": 4981, "type": "theorem", "label": "spaces-morphisms-proposition-generic-flatness-reduced", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-proposition-generic-flatness-reduced", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.", "Assume", "\\begin{enumerate}", "\\item $Y$ is reduced,", "\\item $f$ is of finite type, and", "\\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module.", "\\end{enumerate}", "Then there exists an open dense subspace $W \\subset Y$ such that", "the base change $X_W \\to W$ of $f$ is flat, locally of finite presentation, and", "quasi-compact and such that $\\mathcal{F}|_{X_W}$ is flat over $W$ and of", "finite presentation over $\\mathcal{O}_{X_W}$." ], "refs": [], "proofs": [ { "contents": [ "Let $V$ be a scheme and let $V \\to Y$ be a surjective \\'etale morphism.", "Let $X_V = V \\times_Y X$ and let $\\mathcal{F}_V$ be the restriction of", "$\\mathcal{F}$ to $X_V$. Suppose that the result holds for the morphism", "$X_V \\to V$ and the sheaf $\\mathcal{F}_V$. Then there exists an open subscheme", "$V' \\subset V$ such that $X_{V'} \\to V'$ is flat and of finite presentation", "and $\\mathcal{F}_{V'}$ is an $\\mathcal{O}_{X_{V'}}$-module of finite", "presentation flat over $V'$. Let $W \\subset Y$ be the image of", "the \\'etale morphism $V' \\to Y$, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-image-open}.", "Then $V' \\to W$ is a surjective \\'etale morphism, hence we see that", "$X_W \\to W$ is flat, locally of finite presentation, and quasi-compact by", "Lemmas \\ref{lemma-finite-presentation-local},", "\\ref{lemma-flat-local}, and", "\\ref{lemma-quasi-compact-local}.", "By the discussion in", "Properties of Spaces, Section", "\\ref{spaces-properties-section-properties-modules}", "we see that $\\mathcal{F}_W$ is of finite presentation as a", "$\\mathcal{O}_{X_W}$-module and by", "Lemma \\ref{lemma-base-change-module-flat}", "we see that $\\mathcal{F}_W$ is flat over $W$. This argument", "reduces the proposition to the case where $Y$ is a scheme.", "\\medskip\\noindent", "Suppose we can prove the proposition when $Y$ is an affine scheme.", "Let $f : X \\to Y$ be a finite type morphism of algebraic spaces", "over $S$ with $Y$ a scheme, and let $\\mathcal{F}$ be a finite type,", "quasi-coherent $\\mathcal{O}_X$-module. Choose an affine open covering", "$Y = \\bigcup V_j$. By assumption we can find dense open $W_j \\subset V_j$", "such that $X_{W_j} \\to W_j$ is flat, locally of finite presentation, and", "quasi-compact and such that $\\mathcal{F}|_{X_{W_j}}$ is flat over $W_j$", "and of finite presentation as an $\\mathcal{O}_{X_{W_j}}$-module. In this", "situation we simply take $W = \\bigcup W_j$ and we win. Hence we reduce", "the proposition to the case where $Y$ is an affine scheme.", "\\medskip\\noindent", "Let $Y$ be an affine scheme over $S$, let", "$f : X \\to Y$ be a finite type morphism of algebraic spaces", "over $S$, and let $\\mathcal{F}$ be a finite type,", "quasi-coherent $\\mathcal{O}_X$-module. Since $f$ is of finite type", "it is quasi-compact, hence $X$ is quasi-compact. Thus we can find", "an affine scheme $U$ and a surjective \\'etale morphism $U \\to X$, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}.", "Note that $U \\to Y$ is of finite type (this is what it means for", "$f$ to be of finite type in this case). Hence we can apply", "Morphisms, Proposition \\ref{morphisms-proposition-generic-flatness-reduced}", "to see that there exists a dense open $W \\subset Y$ such that", "$U_W \\to W$ is flat and of finite presentation and such that", "$\\mathcal{F}|_{U_W}$ is flat over $W$ and of finite presentation", "as an $\\mathcal{O}_{U_W}$-module. According to our definitions this means that", "the base change $X_W \\to W$ of $f$ is flat, locally of finite presentation,", "and quasi-compact and $\\mathcal{F}|_{X_W}$ is flat over $W$ and of", "finite presentation over $\\mathcal{O}_{X_W}$." ], "refs": [ "spaces-properties-lemma-etale-image-open", "spaces-morphisms-lemma-finite-presentation-local", "spaces-morphisms-lemma-flat-local", "spaces-morphisms-lemma-quasi-compact-local", "spaces-morphisms-lemma-base-change-module-flat", "spaces-properties-lemma-quasi-compact-affine-cover", "morphisms-proposition-generic-flatness-reduced" ], "ref_ids": [ 11825, 4841, 4854, 4742, 4863, 11832, 5534 ] } ], "ref_ids": [] }, { "id": 4982, "type": "theorem", "label": "spaces-morphisms-proposition-generic-flatness-reduced-quasi-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-proposition-generic-flatness-reduced-quasi-separated", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.", "Assume", "\\begin{enumerate}", "\\item $Y$ is reduced,", "\\item $f$ is quasi-separated,", "\\item $f$ is of finite type, and", "\\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module.", "\\end{enumerate}", "Then there exists an open dense subspace $W \\subset Y$ such that", "the base change $X_W \\to W$ of $f$ is flat and of finite presentation", "and such that $\\mathcal{F}|_{X_W}$ is flat over $W$ and of", "finite presentation over $\\mathcal{O}_{X_W}$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from", "Proposition \\ref{proposition-generic-flatness-reduced}", "and the fact that ``of finite presentation'' $=$", "``locally of finite presentation'' $+$ ``quasi-compact'' $+$", "``quasi-separated''." ], "refs": [ "spaces-morphisms-proposition-generic-flatness-reduced" ], "ref_ids": [ 4981 ] } ], "ref_ids": [] }, { "id": 4983, "type": "theorem", "label": "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to T$ be a morphism of algebraic spaces over $S$.", "Assume", "\\begin{enumerate}", "\\item $T$ is representable,", "\\item $f$ is locally quasi-finite, and", "\\item $f$ is separated.", "\\end{enumerate}", "Then $X$ is representable." ], "refs": [], "proofs": [ { "contents": [ "Let $T = \\bigcup T_i$ be an affine open covering of the scheme $T$.", "If we can show that the open subspaces $X_i = f^{-1}(T_i)$ are", "representable, then $X$ is representable, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme}.", "Note that $X_i = T_i \\times_T X$ and that locally quasi-finite and", "separated are both stable under base change, see", "Lemmas \\ref{lemma-base-change-separated} and", "\\ref{lemma-base-change-quasi-finite}.", "Hence we may assume $T$ is an affine scheme.", "\\medskip\\noindent", "By", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-union-of-quasi-compact}", "there exists a Zariski covering $X = \\bigcup X_i$", "such that each $X_i$ has a surjective \\'etale covering by", "an affine scheme. By", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme}", "again it suffices to prove the proposition for each $X_i$.", "Hence we may assume there exists an affine scheme $U$ and a", "surjective \\'etale morphism $U \\to X$. This reduces us to the", "situation in the next paragraph.", "\\medskip\\noindent", "Assume we have", "$$", "U \\longrightarrow X \\longrightarrow T", "$$", "where $U$ and $T$ are affine schemes, $U \\to X$ is \\'etale surjective, and", "$X \\to T$ is separated and locally quasi-finite. By", "Lemmas \\ref{lemma-etale-locally-quasi-finite} and", "\\ref{lemma-composition-quasi-finite}", "the morphism $U \\to T$ is locally quasi-finite.", "Since $U$ and $T$ are affine it is quasi-finite.", "Set $R = U \\times_X U$. Then $X = U/R$, see", "Spaces, Lemma \\ref{spaces-lemma-space-presentation}.", "As $X \\to T$ is separated the", "morphism $R \\to U \\times_T U$ is a closed immersion, see", "Lemma \\ref{lemma-fibre-product-after-map}.", "In particular $R$ is an affine scheme also.", "As $U \\to X$ is \\'etale the projection morphisms", "$t, s : R \\to U$ are \\'etale as well. In particular $s$ and $t$ are", "quasi-finite, flat and of finite presentation (see", "Morphisms, Lemmas \\ref{morphisms-lemma-etale-locally-quasi-finite},", "\\ref{morphisms-lemma-etale-flat} and", "\\ref{morphisms-lemma-etale-locally-finite-presentation}).", "\\medskip\\noindent", "Let $(U, R, s, t, c)$ be the groupoid associated to the \\'etale", "equivalence relation $R$ on $U$. Let $u \\in U$ be a point, and", "denote $p \\in T$ its image. We are going to use", "More on Groupoids,", "Lemma \\ref{more-groupoids-lemma-quasi-finite-over-base-j-proper}", "for the groupoid $(U, R, s, t, c)$ over the scheme $T$ with", "points $p$ and $u$ as above.", "By the discussion in the previous paragraph all the", "assumptions (1) -- (7) of that lemma are satisfied.", "Hence we get an \\'etale neighbourhood", "$(T', p') \\to (T, p)$ and disjoint union decompositions", "$$", "U_{T'} = U' \\amalg W, \\quad", "R_{T'} = R' \\amalg W'", "$$", "and $u' \\in U'$ satisfying conclusions", "(a), (b), (c), (d), (e), (f), (g), and (h) of the aforementioned", "More on Groupoids,", "Lemma \\ref{more-groupoids-lemma-quasi-finite-over-base-j-proper}.", "We may and do assume that $T'$ is affine (after possibly shrinking $T'$).", "Conclusion (h) implies that $R' = U' \\times_{X_{T'}} U'$ with projection", "mappings identified with the restrictions of $s'$ and $t'$.", "Thus $(U', R', s'|_{R'}, t'|_{R'}, c'|_{R' \\times_{t', U', s'} R'})$ of", "conclusion (g) is an \\'etale equivalence relation. By", "Spaces, Lemma \\ref{spaces-lemma-finding-opens}", "we conclude that $U'/R'$ is an open subspace of $X_{T'}$. By conclusion (d)", "the schemes $U'$, $R'$ are affine and the morphisms", "$s'|_{R'}, t'|_{R'}$ are finite \\'etale. Hence", "Groupoids, Proposition \\ref{groupoids-proposition-finite-flat-equivalence}", "kicks in and we see that $U'/R'$ is an affine scheme.", "\\medskip\\noindent", "We conclude that for every pair of points $(u, p)$ as above we can", "find an \\'etale neighbourhood $(T', p') \\to (T, p)$ with", "$\\kappa(p) = \\kappa(p')$ and a point $u' \\in U_{T'}$ mapping to $u$", "such that the image $x'$ of $u'$ in $|X_{T'}|$ has an open neighbourhood", "$V'$ in $X_{T'}$ which is an affine scheme. We apply", "Lemma \\ref{lemma-neighbourhood-scheme}", "to obtain an open subspace $W \\subset X$ which is a scheme, and", "which contains $x$ (the image of $u$ in $|X|$).", "Since this works for every $x$ we see that $X$", "is a scheme by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme}.", "This ends the proof." ], "refs": [ "spaces-properties-lemma-subscheme", "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-base-change-quasi-finite", "spaces-properties-lemma-union-of-quasi-compact", "spaces-properties-lemma-subscheme", "spaces-morphisms-lemma-etale-locally-quasi-finite", "spaces-morphisms-lemma-composition-quasi-finite", "spaces-lemma-space-presentation", "spaces-morphisms-lemma-fibre-product-after-map", "morphisms-lemma-etale-locally-quasi-finite", "morphisms-lemma-etale-flat", "morphisms-lemma-etale-locally-finite-presentation", "more-groupoids-lemma-quasi-finite-over-base-j-proper", "more-groupoids-lemma-quasi-finite-over-base-j-proper", "spaces-lemma-finding-opens", "groupoids-proposition-finite-flat-equivalence", "spaces-morphisms-lemma-neighbourhood-scheme", "spaces-properties-lemma-subscheme" ], "ref_ids": [ 11848, 4714, 4832, 11831, 11848, 4908, 4831, 8149, 4715, 5363, 5369, 5368, 2492, 2492, 8151, 9669, 4971, 11848 ] } ], "ref_ids": [] }, { "id": 5036, "type": "theorem", "label": "weil-lemma-composition-correspondences", "categories": [ "weil" ], "title": "weil-lemma-composition-correspondences", "contents": [ "We have the following for correspondences:", "\\begin{enumerate}", "\\item composition of correspondences is $\\mathbf{Q}$-bilinear", "and associative,", "\\item there is a canonical isomorphism", "$$", "\\CH_{-r}(X) \\otimes \\mathbf{Q} = \\text{Corr}^r(X, \\Spec(k))", "$$", "such that pullback by correspondences corresponds to composition,", "\\item there is a canonical isomorphism", "$$", "\\CH^r(X) \\otimes \\mathbf{Q} = \\text{Corr}^r(\\Spec(k), X)", "$$", "such that pushforward by correspondences corresponds to composition,", "\\item composition of correspondences is compatible with pushforward and", "pullback of cycles.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Bilinearity follows immediately from the linearity of pushforward", "and pullback and the bilinearity of the intersection product.", "To prove associativity, say we have", "$X, Y, Z, W$ and $c \\in \\text{Corr}(X, Y)$, $c' \\in \\text{Corr}(Y, Z)$, and", "$c'' \\in \\text{Corr}(Z, W)$. Then we have", "\\begin{align*}", "c'' \\circ (c' \\circ c)", "& =", "\\text{pr}^{134}_{14, *}(", "\\text{pr}^{134, *}_{13}", "\\text{pr}^{123}_{13, *}(\\text{pr}^{123, *}_{12}c \\cdot", "\\text{pr}^{123, *}_{23}c')", "\\cdot \\text{pr}^{134, *}_{34}c'') \\\\", "& =", "\\text{pr}^{134}_{14, *}(", "\\text{pr}^{1234}_{134, *}", "\\text{pr}^{1234, *}_{123}(\\text{pr}^{123, *}_{12}c \\cdot", "\\text{pr}^{123, *}_{23}c')", "\\cdot \\text{pr}^{134, *}_{34}c'') \\\\", "& =", "\\text{pr}^{134}_{14, *}(", "\\text{pr}^{1234}_{134, *}", "(\\text{pr}^{1234, *}_{12}c \\cdot", "\\text{pr}^{1234, *}_{23}c')", "\\cdot \\text{pr}^{134, *}_{34}c'') \\\\", "& =", "\\text{pr}^{134}_{14, *}", "\\text{pr}^{1234}_{134, *}", "((\\text{pr}^{1234, *}_{12}c \\cdot", "\\text{pr}^{1234, *}_{23}c')", "\\cdot \\text{pr}^{1234, *}_{34}c'') \\\\", "& =", "\\text{pr}^{1234}_{14, *}(", "(\\text{pr}^{1234, *}_{12}c \\cdot", "\\text{pr}^{1234, *}_{23}c') \\cdot", "\\text{pr}^{1234, *}_{34}c'')", "\\end{align*}", "Here we use the notation", "$$", "p^{1234}_{134} : X \\times Y \\times Z \\times W \\to X \\times Z \\times W", "\\quad\\text{and}\\quad", "p^{134}_{14} : X \\times Z \\times W \\to X \\times W", "$$", "the projections and similarly for other indices.", "The first equality is the definition of the composition.", "The second equality holds because", "$\\text{pr}^{134, *}_{13} \\text{pr}^{123}_{13, *} =", "\\text{pr}^{1234}_{134, *} \\text{pr}^{1234, *}_{123}$", "by Chow Homology, Lemma \\ref{chow-lemma-flat-pullback-proper-pushforward}.", "The third equality holds because intersection product commutes", "with the gysin map for $p^{1234}_{123}$ (which is given by flat pullback), see", "Chow Homology, Lemma \\ref{chow-lemma-lci-gysin-product}.", "The fourth equality follows from the projection formula for", "$p^{1234}_{134}$, see Chow Homology, Lemma \\ref{chow-lemma-projection-formula}.", "The fourth equality is that proper pushforward is compatible", "with composition, see Chow Homology, Lemma \\ref{chow-lemma-compose-pushforward}.", "Since intersection product is associative by", "Chow Homology, Lemma \\ref{chow-lemma-associative}", "this concludes the proof of associativity of composition of correspondences.", "\\medskip\\noindent", "We omit the proofs of (2) and (3) as these are essentially proved by", "carefully bookkeeping where various cycles live and in what (co)dimension.", "\\medskip\\noindent", "The statement on pushforward and pullback of cycles", "means that $(c' \\circ c)^*(\\alpha) = c^*((c')^*(\\alpha))$ and", "$(c' \\circ c)_*(\\alpha) = (c')_*(c_*(\\alpha))$.", "This follows on combining (1), (2), and (3)." ], "refs": [ "chow-lemma-flat-pullback-proper-pushforward", "chow-lemma-lci-gysin-product", "chow-lemma-projection-formula", "chow-lemma-compose-pushforward", "chow-lemma-associative" ], "ref_ids": [ 5682, 5847, 5848, 5674, 5845 ] } ], "ref_ids": [] }, { "id": 5037, "type": "theorem", "label": "weil-lemma-category-correspondences", "categories": [ "weil" ], "title": "weil-lemma-category-correspondences", "contents": [ "Smooth projective schemes over $k$ with correspondences and composition", "of correspondences as defined above form a graded category over", "$\\mathbf{Q}$", "(Differential Graded Algebra, Definition \\ref{dga-definition-graded-category})." ], "refs": [ "dga-definition-graded-category" ], "proofs": [ { "contents": [ "Everything is clear from the construction and", "Lemma \\ref{lemma-composition-correspondences}", "except for the existence of identity morphisms.", "Given a smooth projective scheme $X$ consider", "the class $[\\Delta]$ of the diagonal $\\Delta \\subset X \\times X$", "in $\\text{Corr}^0(X, X)$. We note that $\\Delta$ is", "equal to the graph of the identity $\\text{id}_X : X \\to X$", "which is a fact we will use below.", "\\medskip\\noindent", "To prove that $[\\Delta]$ can serve as an identity we have to show that", "$[\\Delta] \\circ c = c$ and $c' \\circ [\\Delta] = c'$ for any correspondences", "$c \\in \\text{Corr}^r(Y, X)$ and $c' \\in \\text{Corr}^s(X, Y)$.", "For the second case we have to show that", "$$", "c' = \\text{pr}_{13, *}(\\text{pr}_{12}^*[\\Delta] \\cdot \\text{pr}_{23}^*c')", "$$", "where $\\text{pr}_{12} : X \\times X \\times Y \\to X \\times X$ is the", "projection and simlarly for $\\text{pr}_{13}$ and $\\text{pr}_{23}$.", "We may write $c' = \\sum a_i [Z_i]$ for some integral closed subschemes", "$Z_i \\subset X \\times Y$ and rational numers $a_i$. Thus it clearly", "suffices to show that", "$$", "[Z] = \\text{pr}_{13, *}(\\text{pr}_{12}^*[\\Delta] \\cdot \\text{pr}_{23}^*[Z])", "$$", "in the chow group of $X \\times Y$ for any integral closed subscheme $Z$", "of $X \\times Y$. After replacing $X$ and $Y$ by the", "irreducible component containing the image of $Z$ under the two projections", "we may assume $X$ and $Y$ are integral as well. Then we have to show", "$$", "[Z] = \\text{pr}_{13, *}([\\Delta \\times Y] \\cdot [X \\times Z])", "$$", "Denote $Z' \\subset X \\times X \\times Y$ the image of $Z$ by the morphism", "$(\\Delta, 1) : X \\times Y \\to X \\times X \\times Y$. Then $Z'$", "is a closed subscheme of $X \\times X \\times Y$ isomorphic to $Z$ and", "$Z' = \\Delta \\times Y \\cap X \\times Z$ scheme theoretically.", "By Chow Homology, Lemma \\ref{chow-lemma-intersect-properly}\\footnote{The", "reader verifies that $\\dim(Z') = \\dim(\\Delta \\times Y) + \\dim(X \\times Z) -", "\\dim(X \\times X \\times Y)$ and that $Z'$ has a unique generic point", "mapping to the generic point of $Z$ (where the local ring is CM)", "and to some point of $X$ (where the local ring is CM). Thus all the", "hypothese of the lemma are indeed verified.}", "we conclude that", "$$", "[Z'] = [\\Delta \\times Y] \\cdot [X \\times Z]", "$$", "Since $Z'$ maps isomorphically to $Z$ by $\\text{pr}_{13}$ also", "we conclude. The verification that", "$[\\Delta] \\circ c = c$ is similar and we omit it." ], "refs": [ "weil-lemma-composition-correspondences", "chow-lemma-intersect-properly" ], "ref_ids": [ 5036, 5849 ] } ], "ref_ids": [ 13150 ] }, { "id": 5038, "type": "theorem", "label": "weil-lemma-contravariant-functor", "categories": [ "weil" ], "title": "weil-lemma-contravariant-functor", "contents": [ "There is a contravariant functor from the category of smooth", "projective schemes over $k$ to the category of correspondences", "which is the identity on objects and sends $f : Y \\to X$ to", "the element $[\\Gamma_f] \\in \\text{Corr}^0(X, Y)$." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-category-correspondences}", "we have seen that this construction sends identities to", "identities. To finish the proof we have to show if $g : Z \\to Y$", "is another morphism of smooth projective schemes over $k$, then we have", "$[\\Gamma_g] \\circ [\\Gamma_f] = [\\Gamma_{f \\circ g}]$ in", "$\\text{Corr}^0(X, Z)$. Arguing as in the proof of", "Lemma \\ref{lemma-category-correspondences} we see that it", "suffices to show", "$$", "[\\Gamma_{f \\circ g}] =", "\\text{pr}_{13, *}([\\Gamma_f \\times Z] \\cdot [X \\times \\Gamma_g])", "$$", "in $\\CH^*(X \\times Z)$ when $X$, $Y$, $Z$ are integral.", "Denote $Z' \\subset X \\times Y \\times Z$ the image of the closed immersion", "$(f \\circ g, g, 1) : Z \\to X \\times Y \\times Z$.", "Then $Z' = \\Gamma_f \\times Z \\cap X \\times \\Gamma_g$", "scheme theoretically and we conclude using", "Chow Homology, Lemma \\ref{chow-lemma-intersect-properly}", "that", "$$", "[Z'] = [\\Gamma_f \\times Z] \\cdot [X \\times \\Gamma_g]", "$$", "Since it is clear that $\\text{pr}_{13, *}([Z']) = [\\Gamma_{f \\circ g}]$", "the proof is complete." ], "refs": [ "weil-lemma-category-correspondences", "weil-lemma-category-correspondences", "chow-lemma-intersect-properly" ], "ref_ids": [ 5037, 5037, 5849 ] } ], "ref_ids": [] }, { "id": 5039, "type": "theorem", "label": "weil-lemma-functor-and-cycles", "categories": [ "weil" ], "title": "weil-lemma-functor-and-cycles", "contents": [ "Let $f : Y \\to X$ be a morphism of smooth projective schemes over $k$.", "Let $[\\Gamma_f] \\in \\text{Corr}^0(X, Y)$ be as in", "Example \\ref{example-graph-correspondence}. Then", "\\begin{enumerate}", "\\item pushforward of cycles by the correspondence $[\\Gamma_f]$", "agrees with the gysin map $f^! : \\CH^*(Y) \\to \\CH^*(X)$,", "\\item pullback of cycles by the correspondence $[\\Gamma_f]$", "agrees with the pushforward map $f_* : \\CH_*(Y) \\to \\CH_*(X)$,", "\\item if $X$ and $Y$ are equidimensional of dimensions $d$ and $e$,", "then", "\\begin{enumerate}", "\\item pushforward of cycles by the correspondence", "$[\\Gamma_f^t]$ of Remark \\ref{remark-transpose}", "corresponds to pushforward of cycles by $f$, and", "\\item pullback of cycles by the correspondence", "$[\\Gamma_f^t]$ of Remark \\ref{remark-transpose}", "corresponds to the gysin map $f^!$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [ "weil-remark-transpose", "weil-remark-transpose" ], "proofs": [ { "contents": [ "Proof of (1). Recall that", "$[\\Gamma_f]_*(\\alpha) =", "\\text{pr}_{2, *}([\\Gamma_f] \\cdot \\text{pr}_1^*\\alpha)$.", "We have", "$$", "[\\Gamma_f] \\cdot \\text{pr}_1^*\\alpha =", "(f, 1)_*((f, 1)^! \\text{pr}_1^*\\alpha) =", "(f, 1)_*((f, 1)^! \\text{pr}_1^!\\alpha) =", "(f, 1)_*(f^!\\alpha)", "$$", "The first equality by Chow Homology, Lemma", "\\ref{chow-lemma-intersect-regularly-embedded}.", "The second by ", "Chow Homology, Lemma \\ref{chow-lemma-lci-gysin-flat}.", "The third because $\\text{pr}_1 \\circ (f, 1) = f$ and", "Chow Homology, Lemma \\ref{chow-lemma-lci-gysin-composition}.", "Then we coclude because", "$\\text{pr}_{2, *} \\circ (f, 1)_* = 1_*$ by", "Chow Homology, Lemma \\ref{chow-lemma-compose-pushforward}.", "\\medskip\\noindent", "Proof of (2). Recall that $[\\Gamma_f]_*(\\beta) =", "\\text{pr}_{1, *}([\\Gamma_f] \\cdot \\text{pr}_2^*\\beta)$.", "Arguing exactly as above we have", "$$", "[\\Gamma_f] \\cdot \\text{pr}_2^*\\beta = (f, 1)_*\\beta", "$$", "Thus the result follows as before.", "\\medskip\\noindent", "Proof of (3). Proved in exactly the same manner as above." ], "refs": [ "chow-lemma-intersect-regularly-embedded", "chow-lemma-lci-gysin-flat", "chow-lemma-lci-gysin-composition", "chow-lemma-compose-pushforward" ], "ref_ids": [ 5850, 5832, 5833, 5674 ] } ], "ref_ids": [ 5117, 5117 ] }, { "id": 5040, "type": "theorem", "label": "weil-lemma-tensor-product", "categories": [ "weil" ], "title": "weil-lemma-tensor-product", "contents": [ "The tensor product of correspondences defined above turns the category of", "correspondences into a symmetric monoidal category with unit $\\Spec(k)$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5041, "type": "theorem", "label": "weil-lemma-prep-dual", "categories": [ "weil" ], "title": "weil-lemma-prep-dual", "contents": [ "Let $f : Y \\to X$ be a morphism of smooth projective schemes over $k$.", "Assume $X$ and $Y$ equidimensional of dimensions $d$ and $e$.", "Denote $a = [\\Gamma_f] \\in \\text{Corr}^0(X, Y)$ and", "$a^t = [\\Gamma_f^t] \\in \\text{Corr}^{d - e}(Y, X)$.", "Set", "$\\eta_X = [\\Gamma_{X \\to X \\times X}] \\in \\text{Corr}^0(X \\times X, X)$,", "$\\eta_Y = [\\Gamma_{Y \\to Y \\times Y}] \\in \\text{Corr}^0(Y \\times Y, Y)$,", "$[X] \\in \\text{Corr}^{-d}(X, \\Spec(k))$, and", "$[Y] \\in \\text{Corr}^{-e}(Y, \\Spec(k))$. The diagram", "$$", "\\xymatrix{", "X \\otimes Y \\ar[r]_{a \\otimes \\text{id}} \\ar[d]_{\\text{id} \\otimes a^t} &", "Y \\otimes Y \\ar[r]_{\\eta_Y} &", "Y \\ar[d]^{[Y]} \\\\", "X \\otimes X \\ar[r]^{\\eta_X} &", "X \\ar[r]^{[X]} &", "\\Spec(k)", "}", "$$", "is commutative in the category of correspondences." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\text{Corr}^r(W, \\Spec(k)) = \\CH_{-r}(W)$ for any", "smooth projective scheme $W$ over $k$", "and given $c \\in \\text{Corr}^s(W', W)$ the composition", "with $c$ agrees with pullback by $c$ as a map", "$\\CH_{-r}(W) \\to \\CH_{-r - s}(W')$", "(Lemma \\ref{lemma-composition-correspondences}).", "Finally, we have Lemma \\ref{lemma-functor-and-cycles}", "which tells us how to convert this into usual", "pushforward and pullback of cycles.", "We have", "$$", "(a \\otimes \\text{id})^* \\eta_Y^* [Y] =", "(a \\otimes \\text{id})^* [\\Delta_Y] =", "(f \\times \\text{id})_*\\Delta_Y = [\\Gamma_f]", "$$", "and the other way around we get", "$$", "(\\text{id} \\otimes a^t)^* \\eta_X^* [X] =", "(\\text{id} \\otimes a^t)^* [\\Delta_X] =", "(\\text{id} \\times f)^![\\Delta_X] = [\\Gamma_f]", "$$", "The last equality follows from", "Chow Homology, Lemma \\ref{chow-lemma-lci-gysin-easy}.", "In other words, going either way around the diagram we", "obtain the element of $\\text{Corr}^d(X \\times Y, \\Spec(k))$", "corresponding to the cycle $\\Gamma_f \\subset X \\times Y$." ], "refs": [ "weil-lemma-composition-correspondences", "weil-lemma-functor-and-cycles", "chow-lemma-lci-gysin-easy" ], "ref_ids": [ 5036, 5039, 5835 ] } ], "ref_ids": [] }, { "id": 5042, "type": "theorem", "label": "weil-lemma-motives", "categories": [ "weil" ], "title": "weil-lemma-motives", "contents": [ "The category $M_k$ whose objects are motives over $k$ and morphisms", "are morphisms of motives over $k$ is a $\\mathbf{Q}$-linear category.", "There is a contravariant functor", "$$", "h : \\{\\text{smooth projective schemes over }k\\} \\longrightarrow M_k", "$$", "defined by $h(X) = (X, 1, 0)$ and $h(f) = [\\Gamma_f]$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-contravariant-functor}." ], "refs": [ "weil-lemma-contravariant-functor" ], "ref_ids": [ 5038 ] } ], "ref_ids": [] }, { "id": 5043, "type": "theorem", "label": "weil-lemma-Karoubian", "categories": [ "weil" ], "title": "weil-lemma-Karoubian", "contents": [ "The category $M_k$ is Karoubian." ], "refs": [], "proofs": [ { "contents": [ "Let $M = (X, p, m)$ be a motive and let $a \\in \\Mor(M, M)$", "be a projector. Then $a = a \\circ a$ both in $\\Mor(M, M)$", "as well as in $\\text{Corr}^0(X, X)$. Set $N = (X, a, m)$.", "Since we have $a = p \\circ a \\circ a$ in $\\text{Corr}^0(X, X)$", "we see that $a : N \\to M$ is a morphism of $M_k$.", "Next, suppose that $b : (Y, q, n) \\to M$ is a morphism", "such that $(1 - a) \\circ b = 0$. Then $b = a \\circ b$ as well as", "$b = b \\circ q$. Hence $b$ is a morphism $b : (Y, q, n) \\to N$.", "Thus we see that the projector $1 - a$ has a kernel, namely $N$", "and we find that $M_k$ is Karoubian, see", "Homology, Definition \\ref{homology-definition-karoubian}." ], "refs": [ "homology-definition-karoubian" ], "ref_ids": [ 12136 ] } ], "ref_ids": [] }, { "id": 5044, "type": "theorem", "label": "weil-lemma-motives-monoidal", "categories": [ "weil" ], "title": "weil-lemma-motives-monoidal", "contents": [ "The category $M_k$ with tensor product defined as above", "is symmetric monoidal with the obvious associativity and commutativity", "constraints and with unit $\\mathbf{1} = (\\Spec(k), 1, 0)$." ], "refs": [], "proofs": [ { "contents": [ "Follows readily from Lemma \\ref{lemma-tensor-product}. Details omitted." ], "refs": [ "weil-lemma-tensor-product" ], "ref_ids": [ 5040 ] } ], "ref_ids": [] }, { "id": 5045, "type": "theorem", "label": "weil-lemma-inverse-h2", "categories": [ "weil" ], "title": "weil-lemma-inverse-h2", "contents": [ "With notation as in Example \\ref{example-decompose-P1}", "\\begin{enumerate}", "\\item", "the motive $(X, c_0, 0)$ is isomorphic to the motive", "$\\mathbf{1} = (\\Spec(k), 1, 0)$.", "\\item", "the motive $(X, c_2, 0)$ is isomorphic to the motive", "$\\mathbf{1}(-1) = (\\Spec(k), 1, -1)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-contravariant-functor} without further mention.", "The structure morphism $X \\to \\Spec(k)$ gives a correspondence", "$a \\in \\text{Corr}^0(\\Spec(k), X)$. On the other hand, the rational", "point $x$ is a morphism $\\Spec(k) \\to X$ which gives a correspondence", "$b \\in \\text{Corr}^0(X, \\Spec(k))$. We have $b \\circ a = 1$ as a", "correspondence on $\\Spec(k)$. The composition $a \\circ b$ corresponds", "to the graph of the composition $X \\to x \\to X$ which is", "$c_0 = [x \\times X]$. Thus $a = a \\circ b \\circ a = c_0 \\circ a$", "and $b = a \\circ b \\circ a = b \\circ c_0$.", "Hence, unwinding the definitions, we see that", "$a$ and $b$ are mutually inverse morphisms", "$a : (\\Spec(k), 1, 0) \\to (X, c_0, 0)$ and", "$b : (X, c_0, 0) \\to (\\Spec(k), 1, 0)$.", "\\medskip\\noindent", "We will proceed exactly as above to prove the second statement.", "Denote", "$$", "a' \\in \\text{Corr}^1(\\Spec(k), X) = \\CH^1(X)", "$$", "the class of the point $x$. Denote", "$$", "b' \\in \\text{Corr}^{-1}(X, \\Spec(k)) = \\CH_1(X)", "$$", "the class of $[X]$. We have $b' \\circ a' = 1$ as a correspondence", "on $\\Spec(k)$ because $[x] \\cdot [X] = [x]$ on", "$X = \\Spec(k) \\times X \\times \\Spec(k)$. Computing the", "intersection product $\\text{pr}_{12}^*b' \\cdot \\text{pr}_{23}^*a'$", "on $X \\times \\Spec(k) \\times X$ gives the cycle", "$X \\times \\Spec(k) \\times x$. Hence", "the composition $a' \\circ b'$ is equal to $c_2$ as a", "correspondence on $X$. Thus $a' = a' \\circ b \\circ a' = c_2 \\circ a'$", "and $b' = b' \\circ a' \\circ b' = b' \\circ c_2$. Recall that", "$$", "\\Mor((\\Spec(k), 1, -1), (X, c_2, 0)) =", "c_2 \\circ \\text{Corr}^1(\\Spec(k), X)", "\\subset", "\\text{Corr}^1(\\Spec(k), X)", "$$", "and", "$$", "\\Mor((X, c_2, 0), (\\Spec(k), 1, -1)) =", "\\text{Corr}^{-1}(X, \\Spec(k)) \\circ c_2", "\\subset", "\\text{Corr}^{-1}(X, \\Spec(k))", "$$", "Hence, we see that $a'$ and $b'$ are mutually inverse morphisms", "$a' : (\\Spec(k), 1, -1) \\to (X, c_0, 0)$ and", "$b' : (X, c_0, 0) \\to (\\Spec(k), 1, -1)$." ], "refs": [ "weil-lemma-contravariant-functor" ], "ref_ids": [ 5038 ] } ], "ref_ids": [] }, { "id": 5046, "type": "theorem", "label": "weil-lemma-additive", "categories": [ "weil" ], "title": "weil-lemma-additive", "contents": [ "The category $M_k$ is additive." ], "refs": [], "proofs": [ { "contents": [ "Let $(Y, p, m)$ and $(Z, q, n)$ be motives. If $n = m$, then a", "direct sum is given by $(Y \\amalg Z, p + q, m)$, with obvious notation.", "Details omitted.", "\\medskip\\noindent", "Suppose that $n < m$. Let $X$, $c_2$ be as in", "Example \\ref{example-decompose-P1}. Then we consider", "\\begin{align*}", "(Z, q, n)", "& =", "(Z, q, m) \\otimes (\\Spec(k), 1, -1) \\otimes \\ldots \\otimes", "(\\Spec(k), 1, -1) \\\\", "& \\cong", "(Z, q, m) \\otimes (X, c_2, 0) \\otimes \\ldots \\otimes (X, c_2, 0) \\\\", "& \\cong", "(Z \\times X^{m - n}, q \\otimes c_2 \\otimes \\ldots \\otimes c_2, m)", "\\end{align*}", "where we have used Lemma \\ref{lemma-inverse-h2}.", "This reduces us to the case discussed in the first paragraph." ], "refs": [ "weil-lemma-inverse-h2" ], "ref_ids": [ 5045 ] } ], "ref_ids": [] }, { "id": 5047, "type": "theorem", "label": "weil-lemma-decompose-P1", "categories": [ "weil" ], "title": "weil-lemma-decompose-P1", "contents": [ "In $M_k$ we have $h(\\mathbf{P}^1_k) \\cong \\mathbf{1} \\oplus \\mathbf{1}(-1)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Example \\ref{example-decompose-P1} and", "Lemma \\ref{lemma-inverse-h2}." ], "refs": [ "weil-lemma-inverse-h2" ], "ref_ids": [ 5045 ] } ], "ref_ids": [] }, { "id": 5048, "type": "theorem", "label": "weil-lemma-characterize-motives", "categories": [ "weil" ], "title": "weil-lemma-characterize-motives", "contents": [ "Let $X$, $c_2$ be as in Example \\ref{example-decompose-P1}.", "Let $\\mathcal{C}$ be a $\\mathbf{Q}$-linear Karoubian symmetric", "monoidal category. Any $\\mathbf{Q}$-linear functor", "$$", "F :", "\\left\\{", "\\begin{matrix}", "\\text{smooth projective schemes over }k\\\\", "\\text{morphisms are correspondences of degree }0", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\mathcal{C}", "$$", "of symmetric monoidal categories such that the image of $F(c_2)$ on", "$F(X)$ is an invertible object, factors uniquely through a functor", "$F : M_k \\to \\mathcal{C}$ of symmetric monoidal categories." ], "refs": [], "proofs": [ { "contents": [ "Denote $U$ in $\\mathcal{C}$ the invertible object which is assumed to exist", "in the statement of the lemma. We extend $F$ to motives by setting", "$$", "F(X, p, m) = \\left(\\text{the image of", "the projector }F(p)\\text{ in }F(X)\\right) \\otimes U^{\\otimes -m}", "$$", "which makes sense because $U$ is invertible and because $\\mathcal{C}$", "is Karoubian. An important feature of this choice is that", "$F(X, c_2, 0) = U$. Observe that", "\\begin{align*}", "F((X, p, m) \\otimes (Y, q, n))", "& =", "F(X \\times Y, p \\otimes q, m + n) \\\\", "& =", "\\left(\\text{the image of }F(p \\otimes q)\\text{ in }F(X \\times Y)\\right)", "\\otimes U^{\\otimes -m - n} \\\\", "& =", "F(X, p, m) \\otimes F(Y, q, n)", "\\end{align*}", "Thus we see that our rule is compatible with tensor products on", "the level of objects (details omitted).", "\\medskip\\noindent", "Next, we extend $F$ to morphisms of motives. Suppose that", "$$", "a \\in", "\\Hom((Y, p, m), (Z, q, n)) =", "q \\circ \\text{Corr}^{n - m}(Y, Z) \\circ p \\subset \\text{Corr}^{n - m}(Y, Z)", "$$", "is a morphism. If $n = m$, then $a$ is a correspondence of degree $0$", "and we can use $F(a) : F(Y) \\to F(Z)$ to get the desired map", "$F(Y, p, m) \\to F(Z, q, n)$. If $n < m$ we get canonical identifications", "\\begin{align*}", "s : F((Z, q, n))", "& \\to", "F(Z, q, m) \\otimes U^{m - n} \\\\", "& \\to", "F(Z, q, m) \\otimes F(X, c_2, 0) \\otimes \\ldots \\otimes F(X, c_2, 0) \\\\", "& \\to", "F((Z, q, m) \\otimes (X, c_2, 0) \\otimes \\ldots \\otimes (X, c_2, 0)) \\\\", "& \\to", "F((Z \\times X^{m - n}, q \\otimes c_2 \\otimes \\ldots \\otimes c_2, m))", "\\end{align*}", "Namely, for the first isomorphism we use the definition of $F$ on motives", "above. For the second, we use the choice of $U$. For the third we use", "the compatibility of $F$ on tensor products of motives. The fourth", "is the definition of tensor products on motives. On the other hand, since", "we similarly have an isomorphism", "$$", "\\sigma : (Z, q, n) \\to", "(Z \\times X^{m - n}, q \\otimes c_2 \\otimes \\ldots \\otimes c_2, m)", "$$", "(see proof of Lemma \\ref{lemma-additive}). Composing $a$ with this isomorphism", "gives", "$$", "\\sigma \\circ a \\in", "\\Hom((Y, p, m),", "(Z \\times X^{m - n}, q \\otimes c_2 \\otimes \\ldots \\otimes c_2, m))", "$$", "Putting everything together we obtain", "$$", "s^{-1} \\circ F(\\sigma \\circ a) :", "F(Y, p, m) \\to", "F(Z, q, n)", "$$", "If $n > m$ we similarly define isomorphisms", "$$", "t : F((Y, p, m)) \\to", "F((Y \\times X^{n - m}, p \\otimes c_2 \\otimes \\ldots \\otimes c_2, n))", "$$", "and", "$$", "\\tau : (Y, p, m)) \\to", "(Y \\times X^{n - m}, p \\otimes c_2 \\otimes \\ldots \\otimes c_2, n)", "$$", "and we set $F(a) = F(a \\circ \\tau^{-1}) \\circ t$.", "We omit the verification that this construction defines a functor", "of symmetric monoidal categories." ], "refs": [ "weil-lemma-additive" ], "ref_ids": [ 5046 ] } ], "ref_ids": [] }, { "id": 5049, "type": "theorem", "label": "weil-lemma-dual", "categories": [ "weil" ], "title": "weil-lemma-dual", "contents": [ "Let $X$ be a smooth projective scheme over $k$ which is equidimensional", "of dimension $d$. Then $h(X)(d)$ is a left dual to $h(X)$ in $M_k$." ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-composition-correspondences}", "without further mention. We compute", "$$", "\\Hom(\\mathbf{1}, h(X) \\otimes h(X)(d)) =", "\\text{Corr}^d(\\Spec(k), X \\times X) = \\CH^d(X \\times X)", "$$", "Here we have $\\eta = [\\Delta]$. On the other hand, we have", "$$", "\\Hom(h(X)(d) \\otimes h(X), \\mathbf{1}) =", "\\text{Corr}^{-d}(X \\times X, \\Spec(k)) = \\CH_d(X \\times X)", "$$", "and here we have the class $\\epsilon = [\\Delta]$", "of the diagonal as well. The composition of the correspondence", "$[\\Delta] \\otimes 1$ with $1 \\otimes [\\Delta]$ either way", "is the correspondence $[\\Delta] = 1$ in $\\text{Corr}^0(X, X)$ which proves", "the required diagrams of", "Categories, Definition \\ref{categories-definition-dual} commute.", "Namely, observe that", "$$", "[\\Delta] \\otimes 1 \\in \\text{Corr}^d(X, X \\times X \\times X) =", "\\CH^{2d}(X \\times X \\times X \\times X)", "$$", "is given by the class of the cycle", "$\\text{pr}^{1234, -1}_{23}(\\Delta) \\cap \\text{pr}^{1234, -1}_{14}(\\Delta)$ with", "obvious notation. Similarly, the class", "$$", "1 \\otimes [\\Delta] \\in \\text{Corr}^{-d}(X \\times X \\times X, X) =", "\\CH^{2d}(X \\times X \\times X \\times X)", "$$", "is given by the class of the cycle", "$\\text{pr}^{1234, -1}_{23}(\\Delta) \\cap \\text{pr}^{1234, -1}_{14}(\\Delta)$.", "The composition $(1 \\otimes [\\Delta]) \\circ ([\\Delta] \\otimes 1)$", "is by definition the pushforward $\\text{pr}^{12345}_{15, *}$", "of the intersection product", "$$", "[\\text{pr}^{12345, -1}_{23}(\\Delta) \\cap \\text{pr}^{12345, -1}_{14}(\\Delta)]", "\\cdot", "[\\text{pr}^{12345, -1}_{34}(\\Delta) \\cap \\text{pr}^{12345, -1}_{15}(\\Delta)]", "=", "[\\text{small diagonal in } X^5]", "$$", "which is equal to $\\Delta$ as desired. We omit the proof of the formula", "for the composition in the other order." ], "refs": [ "weil-lemma-composition-correspondences", "categories-definition-dual" ], "ref_ids": [ 5036, 12407 ] } ], "ref_ids": [] }, { "id": 5050, "type": "theorem", "label": "weil-lemma-dual-general", "categories": [ "weil" ], "title": "weil-lemma-dual-general", "contents": [ "Every object of $M_k$ has a left dual." ], "refs": [], "proofs": [ { "contents": [ "Let $M = (X, p, m)$ be an object of $M_k$. Then $M$ is a summand of", "$(X, 0, m) = h(X)(m)$.", "By Homology, Lemma \\ref{homology-lemma-Karoubian-dual}", "it suffices to show that", "$h(X)(m) = h(X) \\otimes \\mathbf{1}(m)$ has a dual.", "By construction $\\mathbf{1}(-m)$ is a left dual of $\\mathbf{1}(m)$.", "Hence it suffices to show that $h(X)$ has a left dual, see", "Categories, Lemma \\ref{categories-lemma-tensor-dual}.", "Let $X = \\coprod X_i$ be the decomposition of $X$ into", "irreducible components. Then $h(X) = \\bigoplus h(X_i)$", "and it suffices to show that $h(X_i)$ has a left dual, see", "Homology, Lemma \\ref{homology-lemma-additive-dual}.", "This follows from Lemma \\ref{lemma-dual}." ], "refs": [ "homology-lemma-Karoubian-dual", "categories-lemma-tensor-dual", "homology-lemma-additive-dual", "weil-lemma-dual" ], "ref_ids": [ 12072, 12326, 12071, 5049 ] } ], "ref_ids": [] }, { "id": 5051, "type": "theorem", "label": "weil-lemma-chow-groups-representable", "categories": [ "weil" ], "title": "weil-lemma-chow-groups-representable", "contents": [ "Let $k$ be a base field. The functor $\\CH^i(-)$ on the category", "of motives $M_k$ is representable by $\\mathbf{1}(-i)$, i.e., we", "have", "$$", "\\CH^i(M) = \\Hom_{M_k}(\\mathbf{1}(-i), M)", "$$", "functorially in $M$ in $M_k$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions and", "Lemma \\ref{lemma-composition-correspondences}." ], "refs": [ "weil-lemma-composition-correspondences" ], "ref_ids": [ 5036 ] } ], "ref_ids": [] }, { "id": 5052, "type": "theorem", "label": "weil-lemma-manin", "categories": [ "weil" ], "title": "weil-lemma-manin", "contents": [ "Let $k$ be a base field. Let $c : M \\to N$ be a morphism of motives.", "If for every smooth projective scheme $X$ over $k$ the map", "$c \\otimes 1 : M \\otimes h(X) \\to N \\otimes h(X)$ induces an isomorphism on", "Chow groups, then $c$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Any object $L$ of $M_k$ is a summand of $h(X)(m)$ for some smooth projective", "scheme $X$ over $k$ and some $m \\in \\mathbf{Z}$. Observe that the Chow groups", "of $M \\otimes h(X)(m)$ are the same as the Chow groups of of $M \\otimes h(X)$", "up to a shift in degrees. Hence our assumption implies", "that $c \\otimes 1 : M \\otimes L \\to N \\otimes L$ induces an isomorphism on", "Chow groups for every object $L$ of $M_k$. By", "Lemma \\ref{lemma-chow-groups-representable}", "we see that", "$$", "\\Hom_{M_k}(\\mathbf{1}, M \\otimes L) \\to", "\\Hom_{M_k}(\\mathbf{1}, N \\otimes L)", "$$", "is an isomorphism for every $L$. Since every object of $M_k$ has a left dual", "(Lemma \\ref{lemma-dual-general}) we conclude that", "$$", "\\Hom_{M_k}(K, M) \\to \\Hom_{M_k}(K, N)", "$$", "is an isomorphism for every object $K$ of $M_k$, see", "Categories, Lemma \\ref{categories-lemma-left-dual}.", "We conclude by the Yoneda lemma", "(Categories, Lemma \\ref{categories-lemma-yoneda})." ], "refs": [ "weil-lemma-chow-groups-representable", "weil-lemma-dual-general", "categories-lemma-left-dual", "categories-lemma-yoneda" ], "ref_ids": [ 5051, 5050, 12325, 12203 ] } ], "ref_ids": [] }, { "id": 5053, "type": "theorem", "label": "weil-lemma-projective-space-bundle-formula", "categories": [ "weil" ], "title": "weil-lemma-projective-space-bundle-formula", "contents": [ "In the situation above, the map", "$$", "\\sum\\nolimits_{i = 0, \\ldots, r - 1} c_i :", "\\bigoplus\\nolimits_{i = 0, \\ldots, r - 1} h(X)(i)", "\\longrightarrow", "h(P)", "$$", "is an isomorphism in the category of motives." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-manin} it suffices to show that", "our map defines an isomorphism on Chow groups of motives", "after taking the product with any smooth projective scheme $Z$.", "Observe that $P \\times Z \\to X \\times Z$ is the projective", "bundle associated to the pullback of $\\mathcal{E}$ to $X \\times Z$.", "Hence the statement on Chow groups is true by the projective space bundle", "formula given in", "Chow Homology, Lemma \\ref{chow-lemma-chow-ring-projective-bundle}.", "Namely, pushforward of cycles along $[\\Gamma_p]$ is given by pullback", "of cycles by $p$ according to Lemma \\ref{lemma-functor-and-cycles} and", "Chow Homology, Lemma \\ref{chow-lemma-lci-gysin-flat}. Hence pushforward", "along $c_i$ sends $\\alpha$ to $c_1(\\mathcal{O}_P(1))^i \\cap p^*\\alpha$.", "Some details omitted." ], "refs": [ "weil-lemma-manin", "chow-lemma-chow-ring-projective-bundle", "weil-lemma-functor-and-cycles", "chow-lemma-lci-gysin-flat" ], "ref_ids": [ 5052, 5743, 5039, 5832 ] } ], "ref_ids": [] }, { "id": 5054, "type": "theorem", "label": "weil-lemma-diagonal-projective-bundle", "categories": [ "weil" ], "title": "weil-lemma-diagonal-projective-bundle", "contents": [ "Let $p : P \\to X$ be as in Lemma \\ref{lemma-projective-space-bundle-formula}.", "The class $[\\Delta_P]$ of the diagonal of $P$ in $\\CH^*(P \\times P)$", "can be written", "as", "$$", "[\\Delta_P] =", "\\left(\\sum\\nolimits_{i = 0, \\ldots, r - 1}", "{r - 1 \\choose i} c_{r - 1 - i}(\\text{pr}_1^*\\mathcal{S}^\\vee) \\cap", "c_1(\\text{pr}_2^*\\mathcal{O}_P(1))^i\\right)", "\\cap", "(p \\times p)^*[\\Delta_X]", "$$", "where $\\mathcal{S}$ is the kernel of the canonical surjection", "$p^*\\mathcal{E} \\to \\mathcal{O}_P(1)$." ], "refs": [ "weil-lemma-projective-space-bundle-formula" ], "proofs": [ { "contents": [ "Observe that $(p \\times p)^*[\\Delta_X] = [P \\times_X P]$.", "Since $\\Delta_P \\subset P \\times_X P \\subset P \\times P$", "and since capping with Chern classes commutes with proper pushforward", "(Chow Homology, Lemma \\ref{chow-lemma-pushforward-cap-cj})", "it suffices to show that the class of", "$\\Delta_P \\subset P \\times_X P$ in $\\CH^*(P \\times_X P)$", "is equal to", "$$", "\\left(\\sum\\nolimits_{i = 0, \\ldots, r - 1}", "{r - 1 \\choose i} c_{r - 1 - i}(q_1^*\\mathcal{S}^\\vee) \\cap", "c_1(q_2^*\\mathcal{O}_P(1))^i\\right)", "\\cap", "[P \\times_X P]", "$$", "where $q_i : P \\times_X P \\to P$, $i = 1, 2$ are the projections.", "Set $q = p \\circ q_1 = p \\circ q_2 : P \\times_X P \\to X$.", "Consider the maps", "$$", "q_1^*\\mathcal{S} \\otimes q_2^*\\mathcal{O}_P(-1) \\to", "q^*\\mathcal{E} \\otimes q^*\\mathcal{E}^\\vee \\to", "\\mathcal{O}_{P \\times_X P}", "$$", "where the final arrow is the pullback by $q$ of the evaluation map", "$\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{E}^\\vee \\to \\mathcal{O}_X$.", "The source of the composition is a module locally free of rank $r - 1$", "and a local calculation shows that this map vanishes exactly along", "$\\Delta_P$. By Chow Homology, Lemma \\ref{chow-lemma-top-chern-class}", "the class $[\\Delta_P]$ is the top Chern class of the dual", "$$", "q_1^*\\mathcal{S}^\\vee \\otimes q_2^*\\mathcal{O}_P(1)", "$$", "The desired result follows from Chow Homology, Lemma", "\\ref{chow-lemma-chern-classes-E-tensor-L}." ], "refs": [ "chow-lemma-pushforward-cap-cj", "chow-lemma-top-chern-class", "chow-lemma-chern-classes-E-tensor-L" ], "ref_ids": [ 5748, 5765, 5753 ] } ], "ref_ids": [ 5053 ] }, { "id": 5055, "type": "theorem", "label": "weil-lemma-pushforward-classical", "categories": [ "weil" ], "title": "weil-lemma-pushforward-classical", "contents": [ "Assume given (D1) and (D3) satisfying (A). For $f : X \\to Y$", "a morphism of smooth projective varieties we have", "$f_*(f^*b \\cup a) = b \\cup f_*a$. If $g : Y \\to Z$ is a second morphism", "of smooth projective varieties, then $g_* \\circ f_* = (g \\circ f)_*$." ], "refs": [], "proofs": [ { "contents": [ "The first equality holds because", "$$", "\\int_Y c \\cup b \\cup f_*a =", "\\int_X f^*c \\cup f^*b \\cup a =", "\\int_Y c \\cup f_*(f^*b \\cup a).", "$$", "The second equality holds because", "$$", "\\int_Z c \\cup (g \\circ f)_*a = \\int_X (g \\circ f)^*c \\cup a =", "\\int_X f^* g^* c \\cup a = \\int_Y g^*c \\cup f_*a = \\int_Z c \\cup g_*f_*a", "$$", "This ends the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5056, "type": "theorem", "label": "weil-lemma-degrees-cycles-classical", "categories": [ "weil" ], "title": "weil-lemma-degrees-cycles-classical", "contents": [ "Let $H^*$ be a classical Weil cohomology theory", "(Definition \\ref{definition-weil-cohomology-theory-classical}).", "Let $X$ be a smooth projective variety of dimension $d$. The diagram", "$$", "\\xymatrix{", "\\CH^d(X) \\ar[r]_-\\gamma \\ar@{=}[d] &", "H^{2d}(X) \\ar[d]^{\\int_X} \\\\", "\\CH_0(X) \\ar[r]^\\deg & F", "}", "$$", "commutes where $\\deg : \\CH_0(X) \\to \\mathbf{Z}$ is the degree of", "zero cycles discussed in Chow Homology, Section", "\\ref{chow-section-degree-zero-cycles}." ], "refs": [ "weil-definition-weil-cohomology-theory-classical" ], "proofs": [ { "contents": [ "The result holds for $\\Spec(k)$ by axiom (C)(d). Let $x : \\Spec(k) \\to X$", "be a closed point of $X$. Then we have $\\gamma([x]) = x_*\\gamma([\\Spec(k)])$", "in $H^{2d}(X)$ by axiom (C)(b). Hence $\\int_X \\gamma([x]) = 1$ by the", "definition of $x_*$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 5115 ] }, { "id": 5057, "type": "theorem", "label": "weil-lemma-trace-product-classical", "categories": [ "weil" ], "title": "weil-lemma-trace-product-classical", "contents": [ "Let $H^*$ be a classical Weil cohomology theory", "(Definition \\ref{definition-weil-cohomology-theory-classical}).", "Let $X$ and $Y$ be smooth projective varieties.", "Then $\\int_{X \\times Y} = \\int_X \\otimes \\int_Y$." ], "refs": [ "weil-definition-weil-cohomology-theory-classical" ], "proofs": [ { "contents": [ "Say $\\dim(X) = d$ and $\\dim(Y) = e$. By axiom (B) we have", "$H^{2d + 2e}(X \\times Y) = H^{2d}(X) \\otimes H^{2e}(Y)$", "and by axiom (A)(d) this is $1$-dimensional.", "By Lemma \\ref{lemma-degrees-cycles-classical}", "this $1$-dimensional vector space generated by the", "class $\\gamma([x \\times y])$ of a closed point $(x, y)$ and", "$\\int_{X \\times Y} \\gamma([x \\times y]) = 1$.", "Since $\\gamma([x \\times y]) = \\gamma([x]) \\otimes \\gamma([y])$", "by axioms (C)(a) and (C)(c) and since $\\int_X \\gamma([x]) = 1$ and", "$\\int_Y \\gamma([y]) = 1$ we conclude." ], "refs": [ "weil-lemma-degrees-cycles-classical" ], "ref_ids": [ 5056 ] } ], "ref_ids": [ 5115 ] }, { "id": 5058, "type": "theorem", "label": "weil-lemma-pr2star-classical", "categories": [ "weil" ], "title": "weil-lemma-pr2star-classical", "contents": [ "Let $H^*$ be a classical Weil cohomology theory", "(Definition \\ref{definition-weil-cohomology-theory-classical}).", "Let $X$ and $Y$ be smooth projective varieties.", "Then $\\text{pr}_{2, *} : H^*(X \\times Y) \\to H^*(Y)$", "sends $a \\otimes b$ to $(\\int_X a) b$." ], "refs": [ "weil-definition-weil-cohomology-theory-classical" ], "proofs": [ { "contents": [ "This is equivalent to the result of Lemma \\ref{lemma-trace-product-classical}." ], "refs": [ "weil-lemma-trace-product-classical" ], "ref_ids": [ 5057 ] } ], "ref_ids": [ 5115 ] }, { "id": 5059, "type": "theorem", "label": "weil-lemma-class-diagonal-classical", "categories": [ "weil" ], "title": "weil-lemma-class-diagonal-classical", "contents": [ "Let $H^*$ be a classical Weil cohomology theory", "(Definition \\ref{definition-weil-cohomology-theory-classical}).", "Let $X$ be a smooth projective variety of dimension $d$.", "Choose a basis $e_{i, j}, j = 1, \\ldots, \\beta_i$ of $H^i(X)$ over $F$.", "Using K\\\"unneth write", "$$", "\\gamma([\\Delta]) =", "\\sum\\nolimits_{i = 0, \\ldots, 2d}", "\\sum\\nolimits_j e_{i, j} \\otimes e'_{2d - i , j}", "\\quad\\text{in}\\quad", "\\bigoplus\\nolimits_i H^i(X) \\otimes_F H^{2d - i}(X)", "$$", "with $e'_{2d - i, j} \\in H^{2d - i}(X)$.", "Then $\\int_X e_{i, j} \\cup e'_{2d - i, j'} = (-1)^i\\delta_{jj'}$." ], "refs": [ "weil-definition-weil-cohomology-theory-classical" ], "proofs": [ { "contents": [ "Recall that $\\Delta^* : H^*(X \\times X) \\to H^*(X)$ is equal to the", "cup product map $H^*(X) \\otimes_F H^*(X) \\to H^*(X)$, see", "Remark \\ref{remark-replace-cup-product-classical}. On the other hand we have", "$\\gamma([\\Delta]) = \\Delta_*\\gamma([X]) = \\Delta_*1$ by", "axiom (C)(b) and the fact that $\\gamma([X]) = 1$. Namely,", "$[X] \\cdot [X] = [X]$ hence by axiom (C)(c) the cohomology class", "$\\gamma([X])$ is $0$ or $1$ in the $1$-dimensional $F$-algebra $H^0(X)$;", "here we have also used axioms (A)(d) and (A)(b).", "But $\\gamma([X])$ cannot be zero as $[X] \\cdot [x] = [x]$", "for a closed point $x$ of $X$ and we have the nonvanishing", "of $\\gamma([x])$ by Lemma \\ref{lemma-degrees-cycles-classical}.", "Hence", "$$", "\\int_{X \\times X} \\gamma([\\Delta]) \\cup a \\otimes b =", "\\int_{X \\times X} \\Delta_*1 \\cup a \\otimes b =", "\\int_X a \\cup b", "$$", "by the definition of $\\Delta_*$. On the other hand, we have", "$$", "\\int_{X \\times X} (\\sum e_{i, j} \\otimes e'_{2d -i , j}) \\cup a \\otimes b =", "\\sum (\\int_X a \\cup e_{i, j})(\\int_X e'_{2d - i, j} \\cup b)", "$$", "by Lemma \\ref{lemma-trace-product-classical}; note that we made", "two switches of order so that the sign is $1$.", "Thus if we choose $a$ such that $\\int_X a \\cup e_{i, j} = 1$", "and all other pairings equal to zero, then we conclude that", "$\\int_X e'_{2d - i, j} \\cup b = \\int_X a \\cup b$ for all $b$, i.e.,", "$e'_{2d - i, j} = a$. This proves the lemma." ], "refs": [ "weil-remark-replace-cup-product-classical", "weil-lemma-degrees-cycles-classical", "weil-lemma-trace-product-classical" ], "ref_ids": [ 5119, 5056, 5057 ] } ], "ref_ids": [ 5115 ] }, { "id": 5060, "type": "theorem", "label": "weil-lemma-square-diagonal-classical", "categories": [ "weil" ], "title": "weil-lemma-square-diagonal-classical", "contents": [ "Let $H^*$ be a classical Weil cohomology theory", "(Definition \\ref{definition-weil-cohomology-theory-classical}).", "Let $X$ be a smooth projective variety. We have", "$$", "\\sum\\nolimits_{i = 0, \\ldots, 2\\dim(X)} (-1)^i\\dim_F H^i(X) =", "\\deg([\\Delta] \\cdot [\\Delta]) = \\deg(c_d(\\mathcal{T}_X) \\cap [X])", "$$" ], "refs": [ "weil-definition-weil-cohomology-theory-classical" ], "proofs": [ { "contents": [ "Equality on the right. We have", "$[\\Delta] \\cdot [\\Delta] = \\Delta_*(\\Delta^![\\Delta])$", "(Chow Homology, Lemma \\ref{chow-lemma-intersect-regularly-embedded}).", "Since $\\Delta_*$ preserves degrees of $0$-cycles it suffices to compute", "the degree of $\\Delta^![\\Delta]$. The class $\\Delta^![\\Delta]$ is given", "by capping $[\\Delta]$ with", "the top Chern class of the normal sheaf of $\\Delta \\subset X \\times X$", "(Chow Homology, Lemma \\ref{chow-lemma-gysin-fundamental}).", "Since the conormal sheaf of $\\Delta$ is $\\Omega_{X/k}$", "(Morphisms, Lemma \\ref{morphisms-lemma-differentials-diagonal})", "we see that the normal sheaf is equal to the tangent sheaf", "$\\mathcal{T}_X = \\SheafHom_{\\mathcal{O}_X}(\\Omega_{X/k}, \\mathcal{O}_X)$", "as desired.", "\\medskip\\noindent", "Equality on the left. By Lemma \\ref{lemma-degrees-cycles-classical} we have", "\\begin{align*}", "\\deg([\\Delta] \\cdot [\\Delta])", "& =", "\\int_{X \\times X} \\gamma([\\Delta]) \\cup \\gamma([\\Delta]) \\\\", "& =", "\\int_{X \\times X} \\Delta_*1 \\cup \\gamma([\\Delta]) \\\\", "& =", "\\int_{X \\times X} \\Delta_*(\\Delta^*\\gamma([\\Delta])) \\\\", "& =", "\\int_X \\Delta^*\\gamma([\\Delta])", "\\end{align*}", "Write $\\gamma([\\Delta]) = \\sum e_{i, j} \\otimes e'_{2d - i , j}$", "as in Lemma \\ref{lemma-class-diagonal-classical}.", "Recalling that $\\Delta^*$ is given by cup product we obtain", "$$", "\\int_X \\sum\\nolimits_{i, j} e_{i, j} \\cup e'_{2d - i, j} =", "\\sum\\nolimits_{i, j} \\int_X e_{i, j} \\cup e'_{2d - i, j} =", "\\sum\\nolimits_{i, j} (-1)^i = \\sum (-1)^i\\beta_i", "$$", "as desired." ], "refs": [ "chow-lemma-intersect-regularly-embedded", "chow-lemma-gysin-fundamental", "morphisms-lemma-differentials-diagonal", "weil-lemma-degrees-cycles-classical", "weil-lemma-class-diagonal-classical" ], "ref_ids": [ 5850, 5814, 5311, 5056, 5059 ] } ], "ref_ids": [ 5115 ] }, { "id": 5061, "type": "theorem", "label": "weil-lemma-from-functor-to-weil-classical", "categories": [ "weil" ], "title": "weil-lemma-from-functor-to-weil-classical", "contents": [ "Let $k$ be an algebraically closed field. Let $F$ be a field of", "characteristic $0$. Consider a $\\mathbf{Q}$-linear functor", "$$", "G : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces}", "$$", "of symmetric monoidal categories such that $G(\\mathbf{1}(1))$", "is nonzero only in degree $-2$. Then we obtain data (D1), (D2), (D3)", "satisfying all of (A), (B), (C) except for possibly (A)(c) and (A)(d)." ], "refs": [], "proofs": [ { "contents": [ "We obtain a contravariant functor from the category of smooth", "projective varieties to the category of graded $F$-vector spaces", "by setting $H^*(X) = G(h(X))$. By assumption we have a canonical", "isomorphism", "$$", "H^*(X \\times Y) = G(h(X \\times Y)) = G(h(X) \\otimes h(Y)) =", "G(h(X)) \\otimes G(h(Y)) = H^*(X) \\otimes H^*(Y)", "$$", "compatible with pullbacks. Using pullback along the diagonal", "$\\Delta : X \\to X \\times X$ we obtain a canonical map", "$$", "H^*(X) \\otimes H^*(X) = H^*(X \\times X) \\to H^*(X)", "$$", "of graded vector spaces compatible with pullbacks.", "This defines a functorial graded $F$-algebra structure on", "$H^*(X)$. Since $\\Delta$ commutes with the commutativity", "constraint $h(X) \\otimes h(X) \\to h(X) \\otimes h(X)$ (switching the factors)", "and since $G$ is a functor of symmetric monoidal categories (so compatible with", "commutativity constraints), and by our convention in", "Homology, Example \\ref{homology-example-graded-vector-spaces}", "we conclude that $H^*(X)$ is a graded", "commutative algebra! Hence we get our datum (D1).", "\\medskip\\noindent", "Since $\\mathbf{1}(1)$ is invertible in the category of motives", "we see that $G(\\mathbf{1}(1))$ is invertible in the category of", "graded $F$-vector spaces. Thus $\\sum_i \\dim_F G^i(\\mathbf{1}(1)) = 1$.", "By assumption we only get something nonzero in degree $-2$ and we may", "choose an isomorphism $F[2] \\to G(\\mathbf{1}(1))$ of graded $F$-vector spaces.", "Here and below $F[n]$ means the graded $F$-vector space which has", "$F$ in degree $-n$ and zero elsewhere. Using compatibility with", "tensor products, we find for all $n \\in \\mathbf{Z}$ an isomorphism", "$F[2n] \\to G(\\mathbf{1}(n))$ compatible with tensor products.", "\\medskip\\noindent", "Let $X$ be a smooth projective variety. By", "Lemma \\ref{lemma-composition-correspondences} we have", "$$", "\\CH^r(X) \\otimes \\mathbf{Q} = \\text{Corr}^r(\\Spec(k), X) =", "\\Hom(\\mathbf{1}(-r), h(X))", "$$", "Applying the functor $G$ we obtain", "$$", "\\gamma :", "\\CH^r(X) \\otimes \\mathbf{Q} \\longrightarrow", "\\Hom(G(\\mathbf{1}(-r)), H^*(X)) = H^{2r}(X)", "$$", "This is the datum (D2).", "\\medskip\\noindent", "Let $X$ be a smooth projective variety of dimension $d$. By", "Lemma \\ref{lemma-composition-correspondences} we have", "$$", "\\Mor(h(X)(d), \\mathbf{1}) = \\Mor((X, 1, d), (\\Spec(k), 1, 0)) =", "\\text{Corr}^{-d}(X, \\Spec(k)) = \\CH_d(X)", "$$", "Thus the class of the cycle $[X]$ in $\\CH_d(X)$ defines a morphism", "$h(X)(d) \\to \\mathbf{1}$. Applying $G$ we obtain", "$$", "H^*(X) \\otimes F[-2d] = G(h(X)(d)) \\longrightarrow G(\\mathbf{1}) = F", "$$", "This map is zero except in degree $0$ where we obtain", "$\\int_X : H^{2d}(X) \\to F$. This is the datum (D3).", "\\medskip\\noindent", "Let $X$ be a smooth projective variety of dimension $d$.", "By Lemma \\ref{lemma-dual}", "we know that $h(X)(d)$ is a left dual to $h(X)$. Hence", "$G(h(X)(d)) = H^*(X) \\otimes F[-2d]$ is a left dual to", "$H^*(X)$ in the category of graded $F$-vector spaces.", "By Homology, Lemma \\ref{homology-lemma-left-dual-graded-vector-spaces}", "we find that $\\sum_i \\dim_F H^i(X) < \\infty$ and that", "$\\epsilon : h(X)(d) \\otimes h(X) \\to \\mathbf{1}$ produces", "nondegenerate pairings $H^{2d - i}(X) \\otimes_F H^i(X) \\to F$.", "In the proof of Lemma \\ref{lemma-dual} we have seen that", "$\\epsilon$ is given by $[\\Delta]$ via the identifications", "$$", "\\Hom(h(X)(d) \\otimes h(X), \\mathbf{1}) =", "\\text{Corr}^{-d}(X \\times X, \\Spec(k)) =", "\\CH_d(X \\times X)", "$$", "Thus $\\epsilon$ is the composition of $[X] : h(X)(d) \\to \\mathbf{1}$", "and $h(\\Delta)(d) : h(X)(d) \\otimes h(X) \\to h(X)(d)$. It follows", "that the pairings above are given by cup product followed by", "$\\int_X$. This proves axiom (A) parts (a) and (b).", "\\medskip\\noindent", "Axiom (B) follows from the assumption that $G$ is compatible", "with tensor structures and our construction of the cup product above.", "\\medskip\\noindent", "Axiom (C). Our construction of $\\gamma$ takes a cycle $\\alpha$ on $X$,", "interprets it as a correspondence $a$ from $\\Spec(k)$ to $X$ of some degree,", "and then applies $G$. If $f : Y \\to X$ is a morphism of smooth projective", "varieties, then $f^!\\alpha$ is the pushforward (!) of $\\alpha$", "by the correspondence $[\\Gamma_f]$ from $X$ to $Y$, see", "Lemma \\ref{lemma-functor-and-cycles}. Hence", "$f^!\\alpha$ viewed as a correspondence from $\\Spec(k)$ to $Y$", "is equal to $a \\circ [\\Gamma_f]$, see", "Lemma \\ref{lemma-composition-correspondences}.", "Since $G$ is a functor, we conclude", "$\\gamma$ is compatible with pullbacks, i.e., axiom (C)(a) holds.", "\\medskip\\noindent", "Let $f : Y \\to X$ be a morphism of smooth projective varieties and", "let $\\beta \\in \\CH^r(Y)$ be a cycle on $Y$. We have to show that", "$$", "\\int_Y \\gamma(\\beta) \\cup f^*c = \\int_X \\gamma(f_*\\beta) \\cup c", "$$", "for all $c \\in H^*(X)$. Let $a, a^t, \\eta_X, \\eta_Y, [X], [Y]$", "be as in Lemma \\ref{lemma-prep-dual}.", "Let $b$ be $\\beta$ viewed as a correspondence from $\\Spec(k)$ to $Y$", "of degree $r$. Then $f_*\\beta$ viewed as a correspondence from", "$\\Spec(k)$ to $X$ is equal to $a^t \\circ b$, see", "Lemmas \\ref{lemma-functor-and-cycles} and", "\\ref{lemma-composition-correspondences}.", "The displayed equality above holds if we can show that", "$$", "h(X) = \\mathbf{1} \\otimes h(X)", "\\xrightarrow{b \\otimes 1}", "h(Y)(r) \\otimes h(X)", "\\xrightarrow{1 \\otimes a}", "h(Y)(r) \\otimes h(Y)", "\\xrightarrow{\\eta_Y}", "h(Y)(r)", "\\xrightarrow{[Y]}", "\\mathbf{1}(r - e)", "$$", "is equal to", "$$", "h(X) = \\mathbf{1} \\otimes h(X)", "\\xrightarrow{a^t \\circ b \\otimes 1}", "h(X)(r + d - e) \\otimes h(X)", "\\xrightarrow{\\eta_X}", "h(X)(r + d - e)", "\\xrightarrow{[X]}", "\\mathbf{1}(r - e)", "$$", "This follows immediately from Lemma \\ref{lemma-prep-dual}.", "Thus we have axiom (C)(b).", "\\medskip\\noindent", "To prove axiom (C)(c) we use the discussion in", "Remark \\ref{remark-replace-cup-product-classical}.", "Hence it suffices to prove that $\\gamma$ is compatible with", "exterior products. Let $X$, $Y$ be smooth projective varieties and", "let $\\alpha$, $\\beta$ be cycles on them. Denote", "$a$, $b$ the corresponding correspondences from $\\Spec(k)$ to", "$X$, $Y$. Then $\\alpha \\times \\beta$ corresponds to the", "correspondence $a \\otimes b$ from $\\Spec(k)$ to $X \\otimes Y = X \\times Y$.", "Hence the requirement follows from the fact that $G$ is", "compatible with the tensor structures on both sides.", "\\medskip\\noindent", "Axiom (C)(d) follows because the cycle $[\\Spec(k)]$", "corresponds to the identity morphism on $h(\\Spec(k))$.", "This finishes the proof of the lemma." ], "refs": [ "weil-lemma-composition-correspondences", "weil-lemma-composition-correspondences", "weil-lemma-dual", "homology-lemma-left-dual-graded-vector-spaces", "weil-lemma-dual", "weil-lemma-functor-and-cycles", "weil-lemma-composition-correspondences", "weil-lemma-prep-dual", "weil-lemma-functor-and-cycles", "weil-lemma-composition-correspondences", "weil-lemma-prep-dual", "weil-remark-replace-cup-product-classical" ], "ref_ids": [ 5036, 5036, 5049, 12073, 5049, 5039, 5036, 5041, 5039, 5036, 5041, 5119 ] } ], "ref_ids": [] }, { "id": 5062, "type": "theorem", "label": "weil-lemma-from-weil-to-functor-classical", "categories": [ "weil" ], "title": "weil-lemma-from-weil-to-functor-classical", "contents": [ "Let $k$ be an algebraically closed field. Let $F$ be a field of", "characteristic $0$. Let $H^*$ be a classical Weil cohomology theory.", "Then we can construct a $\\mathbf{Q}$-linear functor", "$$", "G : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces}", "$$", "of symmetric monoidal categories such that $H^*(X) = G(h(X))$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-characterize-motives} it suffices to construct a functor", "$G$ on the category of smooth projective schemes over $k$", "with morphisms given by correspondences of degree $0$ such that", "the image of $G(c_2)$ on $G(\\mathbf{P}^1)$ is an invertible graded", "$F$-vector space.", "Since every smooth projective scheme is canonically a disjoint", "union of smooth projective varieties, it suffices to construct", "$G$ on the category whose objects are smooth projective varieties", "and whose morphisms are correspondences of degree $0$. (Some details", "omitted.)", "\\medskip\\noindent", "Given a smooth projective variety $X$ we set $G(X) = H^*(X)$.", "\\medskip\\noindent", "Given a correspondence $c \\in \\text{Corr}^0(X, Y)$ between smooth", "projective varieties we consider the map", "$G(c) : G(X) = H^*(X) \\to G(Y) = H^*(Y)$ given by the rule", "$$", "a \\longmapsto", "G(c)(a) = \\text{pr}_{2, *}(\\gamma(c) \\cup \\text{pr}_1^*a)", "$$", "It is clear that $G(c)$ is additive in $c$ and hence $\\mathbf{Q}$-linear.", "Compatibility of $\\gamma$ with pullbacks, pushforwards, and", "intersection products given by axioms (C)(a), (C)(b), and (C)(c)", "shows that we have", "$G(c' \\circ c) = G(c') \\circ G(c)$ if $c' \\in \\text{Corr}^0(Y, Z)$.", "Namely, for $a \\in H^*(X)$ we have", "\\begin{align*}", "(G(c') \\circ G(c))(a)", "& =", "\\text{pr}^{23}_{3, *}(\\gamma(c') \\cup", "\\text{pr}^{23, *}_2(\\text{pr}^{12}_{2, *}(\\gamma(c) \\cup", "\\text{pr}^{12, *}_1a))) \\\\", "& =", "\\text{pr}^{23}_{3, *}(\\gamma(c') \\cup", "\\text{pr}^{123}_{23, *}(\\text{pr}^{123, *}_{12}(\\gamma(c) \\cup", "\\text{pr}^{12, *}_1 a))) \\\\", "& =", "\\text{pr}^{23}_{3, *}", "\\text{pr}^{123}_{23, *}(", "\\text{pr}^{123, *}_{23}\\gamma(c') \\cup", "\\text{pr}^{123, *}_{12}\\gamma(c) \\cup", "\\text{pr}^{123, *}_1 a) \\\\", "& =", "\\text{pr}^{23}_{3, *}", "\\text{pr}^{123}_{23, *}(", "\\gamma(\\text{pr}^{123, *}_{23}c') \\cup", "\\gamma(\\text{pr}^{123, *}_{12}c) \\cup", "\\text{pr}^{123, *}_1 a) \\\\", "& =", "\\text{pr}^{13}_{3, *}", "\\text{pr}^{123}_{13, *}(", "\\gamma(\\text{pr}^{123, *}_{23}c' \\cdot \\text{pr}^{123, *}_{12}c) \\cup", "\\text{pr}^{123, *}_1 a) \\\\", "& =", "\\text{pr}^{13}_{3, *}(", "\\gamma(\\text{pr}^{123}_{13, *}(", "\\text{pr}^{123, *}_{23}c' \\cdot \\text{pr}^{123, *}_{12}c)) \\cup", "\\text{pr}^{13, *}_1 a) \\\\", "& =", "G(c' \\circ c)(a)", "\\end{align*}", "with obvious notation. The first equality follows from the definitions.", "The second equality holds because", "$\\text{pr}^{23, *}_2 \\circ \\text{pr}^{12}_{2, *} =", "\\text{pr}^{123}_{23, *} \\circ \\text{pr}^{123, *}_{12}$", "as follows immediately from the description of pushforward", "along projections given in Lemma \\ref{lemma-pr2star-classical}.", "The third equality holds by Lemma \\ref{lemma-pushforward-classical}", "and the fact that $H^*$ is a functor.", "The fourth equalith holds by axiom (C)(a) and the fact that", "the gysin map agrees with flat pullback for flat morphisms", "(Chow Homology, Lemma \\ref{chow-lemma-lci-gysin-flat}).", "The fifth equality uses axiom (C)(c) as well as", "Lemma \\ref{lemma-pushforward-classical} to see that", "$\\text{pr}^{23}_{3, *} \\circ \\text{pr}^{123}_{23, *} =", "\\text{pr}^{13}_{3, *} \\circ \\text{pr}^{123}_{13, *}$.", "The sixth equality uses the projection formula from", "Lemma \\ref{lemma-pushforward-classical} as well as", "axiom (C)(b) to see that $", "\\text{pr}^{123}_{13, *}", "\\gamma(\\text{pr}^{123, *}_{23}c' \\cdot \\text{pr}^{123, *}_{12}c) =", "\\gamma(\\text{pr}^{123}_{13, *}(", "\\text{pr}^{123, *}_{23}c' \\cdot \\text{pr}^{123, *}_{12}c))$.", "Finally, the last equality is the definition.", "\\medskip\\noindent", "To finish the proof that $G$ is a functor,", "we have to show identities are preserved. In other words, if", "$1 = [\\Delta] \\in \\text{Corr}^0(X, X)$ is the identity", "in the category of correspondences (see", "Lemma \\ref{lemma-category-correspondences} and its proof),", "then we have to show that $G([\\Delta]) = \\text{id}$.", "This follows from the determination of", "$\\gamma([\\Delta])$ in Lemma \\ref{lemma-class-diagonal-classical}", "and Lemma \\ref{lemma-pr2star-classical}.", "This finishes the construction of $G$ as a functor on", "smooth projective varieties and correspondences of degree $0$.", "\\medskip\\noindent", "It follows from axioms (A)(c) and (A)(d) that", "$G(\\Spec(k)) = H^*(\\Spec(k))$ is canonically isomorphic to $F$", "as an $F$-algebra.", "The K\\\"unneth axiom (B) shows our functor is compatible with tensor products.", "Thus our functor is a functor of symmetric monoidal categories.", "\\medskip\\noindent", "We still have to check that the image of $G(c_2)$ on $G(\\mathbf{P}^1)$", "is an invertible graded $F$-vector space (in particular we don't know yet", "that $G$ extends to $M_k$).", "By axiom (A)(d) the map $\\int_{\\mathbf{P}^1} : H^2(\\mathbf{P}^1) \\to F$", "is an isomorphism. By axiom (A)(b) we see that $\\dim_F H^0(\\mathbf{P}^1) = 1$.", "By Lemma \\ref{lemma-square-diagonal-classical} and axiom (A)(c)", "we obtain $2 - \\dim_F H^1(\\mathbf{P}^1) = c_1(T_{\\mathbf{P}^1}) = 2$.", "Hence $H^1(\\mathbf{P}^1) = 0$. Thus", "$$", "G(\\mathbf{P}^1) = H^0(\\mathbf{P}^1) \\oplus H^2(\\mathbf{P}^1)", "$$", "Recall that $1 = c_0 + c_2$ is a decomposition of the identity", "into a sum of orthogonal idempotents in", "$\\text{Corr}^0(\\mathbf{P}^1, \\mathbf{P}^1)$, see", "Example \\ref{example-decompose-P1}. We have $c_0 = a \\circ b$ where", "$a \\in \\text{Corr}^0(\\Spec(k), \\mathbf{P}^1)$ and", "$b \\in \\text{Corr}^0(\\mathbf{P}^1, \\Spec(k))$ and where", "$b \\circ a = 1$ in $\\text{Corr}^0(\\Spec(k), \\Spec(k))$, see proof of", "Lemma \\ref{lemma-inverse-h2}. Since $F = G(\\Spec(k))$, it follows from", "functoriality that $G(c_0)$ is the projector onto the summand", "$H^0(\\mathbf{P}^1) \\subset G(\\mathbf{P}^1)$. Hence", "$G(c_2)$ must necessarily be the projection onto $H^2(\\mathbf{P}^1)$", "and the proof is complete." ], "refs": [ "weil-lemma-characterize-motives", "weil-lemma-pr2star-classical", "weil-lemma-pushforward-classical", "chow-lemma-lci-gysin-flat", "weil-lemma-pushforward-classical", "weil-lemma-pushforward-classical", "weil-lemma-category-correspondences", "weil-lemma-class-diagonal-classical", "weil-lemma-pr2star-classical", "weil-lemma-square-diagonal-classical", "weil-lemma-inverse-h2" ], "ref_ids": [ 5048, 5058, 5055, 5832, 5055, 5055, 5037, 5059, 5058, 5060, 5045 ] } ], "ref_ids": [] }, { "id": 5063, "type": "theorem", "label": "weil-lemma-generated-by-separable", "categories": [ "weil" ], "title": "weil-lemma-generated-by-separable", "contents": [ "Let $k$ be a field. Let $X$ be a smooth projective scheme over $k$.", "Then $\\CH_0(X)$ is generated by classes of closed points whose residue", "fields are separable over $k$." ], "refs": [], "proofs": [ { "contents": [ "The lemma is immediate if $k$ has characteristic $0$ or is perfect.", "Thus we may assume $k$ is an infinite field of characteristic $p > 0$.", "\\medskip\\noindent", "We may assume $X$ is irreducible of dimension $d$.", "Then $k' = H^0(X, \\mathcal{O}_X)$ is a finite separable field", "extension of $k$ and that $X$ is geometrically integral over $k'$.", "See Varieties, Lemmas \\ref{varieties-lemma-smooth-geometrically-normal},", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}, and", "\\ref{varieties-lemma-baby-stein}. We may and do replace $k$ by $k'$", "and assume that $X$ is geometrically integral.", "\\medskip\\noindent", "Let $x \\in X$ be a closed point. To prove the lemma we are going to show that", "$[x] \\in \\CH_0(X)$ is rationally equivalent to an integer linear", "combination of classes of closed points whose residue fields", "are separable over $k$. Choose an ample invertible", "$\\mathcal{O}_X$-module $\\mathcal{L}$. Set", "$$", "V = \\{s \\in H^0(X, \\mathcal{L}) \\mid s(x) = 0 \\}", "$$", "After replacing $\\mathcal{L}$ by a power we may assume", "(a) $\\mathcal{L}$ is very ample, (b) $V$ generates", "$\\mathcal{L}$ over $X \\setminus x$, (c) the morphism", "$X \\setminus x \\to \\mathbf{P}(V)$ is an immersion, (d)", "the map $V \\to \\mathfrak m_x\\mathcal{L}_x/\\mathfrak m_x^2\\mathcal{L}_x$", "is surjective, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-ample-very-ample},", "Varieties, Lemma \\ref{varieties-lemma-generate-over-complement}, and", "Properties, Proposition \\ref{properties-proposition-characterize-ample}.", "Consider the set", "$$", "V^d \\supset U =", "\\{", "(s_1, \\ldots, s_d) \\in V^d \\mid s_1, \\ldots, s_d", "\\text{ generate }", "\\mathfrak m_x\\mathcal{L}_x/\\mathfrak m_x^2\\mathcal{L}_x", "\\text{ over }\\kappa(x)", "\\}", "$$", "Since $\\mathcal{O}_{X, x}$ is a regular local ring of dimension $d$", "we have $\\dim_{\\kappa(x)}(\\mathfrak m_x/\\mathfrak m_x^2) = d$", "and hence we see that $U$ is a nonempty (Zariski) open of $V^d$.", "For $(s_1, \\ldots, s_d) \\in U$ set $H_i = Z(s_i)$. Since", "$s_1, \\ldots, s_d$ generate $\\mathfrak m_x\\mathcal{L}_x$", "we see that", "$$", "H_1 \\cap \\ldots \\cap H_d = x \\amalg Z", "$$", "scheme theoretically for some closed subscheme $Z \\subset X$.", "By Bertini (in the form of Varieties, Lemma \\ref{varieties-lemma-bertini})", "for a general element $s_1 \\in V$ the scheme $H_1 \\cap (X \\setminus x)$", "is smooth over $k$ of dimension $d - 1$.", "Having chosen $s_1$, for a general element", "$s_2 \\in V$ the scheme $H_1 \\cap H_2 \\cap (X \\setminus x)$", "is smooth over $k$ of dimension $d - 2$. And so on.", "We conclude that for sufficiently general", "$(s_1, \\ldots, s_d) \\in U$ the scheme $Z$ is \\'etale over $\\Spec(k)$.", "In particular $H_1 \\cap \\ldots \\cap H_d$ has dimension $0$", "and hence", "$$", "[H_1] \\cdot \\ldots \\cdot [H_d] = [x] + [Z]", "$$", "in $\\CH_0(X)$ by repeated application of", "Chow Homology, Lemma \\ref{chow-lemma-intersect-properly} (details omitted).", "This finishes the proof as it shows that $[x] \\sim_{rat} - [Z] + [Z']$", "where $Z' = H'_1 \\cap \\ldots \\cap H'_d$ is a general complete", "intersection of vanishing loci of sufficiently general sections", "of $\\mathcal{L}$ which will be \\'etale over $k$ by the same argument", "as before." ], "refs": [ "varieties-lemma-smooth-geometrically-normal", "varieties-lemma-proper-geometrically-reduced-global-sections", "varieties-lemma-baby-stein", "morphisms-lemma-finite-type-ample-very-ample", "varieties-lemma-generate-over-complement", "properties-proposition-characterize-ample", "varieties-lemma-bertini", "chow-lemma-intersect-properly" ], "ref_ids": [ 11005, 10948, 10949, 5395, 11131, 3067, 11132, 5849 ] } ], "ref_ids": [] }, { "id": 5064, "type": "theorem", "label": "weil-lemma-chow-limit", "categories": [ "weil" ], "title": "weil-lemma-chow-limit", "contents": [ "Let $K/k$ be an algebraic field extension. Let $X$ be a finite type", "scheme over $k$. Then $\\CH_i(X_K) = \\colim \\CH_i(X_{k'})$ where the", "colimit is over the subextensions $K/k'/k$ with $k'/k$ finite." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Chow Homology, Lemma \\ref{chow-lemma-chow-limit}." ], "refs": [ "chow-lemma-chow-limit" ], "ref_ids": [ 5866 ] } ], "ref_ids": [] }, { "id": 5065, "type": "theorem", "label": "weil-lemma-divide-difference-points", "categories": [ "weil" ], "title": "weil-lemma-divide-difference-points", "contents": [ "Let $k$ be a field. Let $X$ be a geometrically irreducible", "smooth projective scheme over $k$. Let $x, x' \\in X$ be $k$-rational points.", "Let $n$ be an integer invertible in $k$.", "Then there exists a finite separable extension $k'/k$ such that", "the pullback of $[x] - [x']$ to $X_{k'}$", "is divisible by $n$ in $\\CH_0(X_{k'})$." ], "refs": [], "proofs": [ { "contents": [ "Let $k'$ be a separable algebraic closure of $k$. Suppose that we can show", "the the pullback of $[x] - [x']$ to $X_{k'}$ is divisible by $n$ in", "$\\CH_0(X_{k'})$. Then we conclude by Lemma \\ref{lemma-chow-limit}.", "Thus we may and do assume $k$ is separably algebraically closed.", "\\medskip\\noindent", "Suppose $\\dim(X) > 1$. Let $\\mathcal{L}$ be an ample invertible sheaf on $X$.", "Set", "$$", "V = \\{s \\in H^0(X, \\mathcal{L}) \\mid s(x) = 0\\text{ and }s(x') = 0 \\}", "$$", "After replacing $\\mathcal{L}$ by a power we see that for", "a general $v \\in V$ the corresponding divisor $H_v \\subset X$ is smooth", "away from $x$ and $x'$, see", "Varieties, Lemmas \\ref{varieties-lemma-generate-over-complement} and", "\\ref{varieties-lemma-bertini}. To find $v$ we use that $k$ is infinite (being", "separably algebraically closed).", "If we choose $s$ general, then the image of $s$ in", "$\\mathfrak m_x\\mathcal{L}_x/\\mathfrak m_x^2\\mathcal{L}_x$", "will be nonzero, which implies that $H_v$ is smooth at $x$", "(details omitted). Similarly for $x'$. Thus $H_v$ is smooth.", "By Varieties, Lemma \\ref{varieties-lemma-connectedness-ample-divisor}", "(applied to the base change of everything", "to the algebraic closure of $k$)", "we see that $H_v$ is geometrically connected.", "It suffices to prove the result for", "$[x] - [x']$ seen as an element of $\\CH_0(H_v)$.", "In this way we reduce to the case of a curve.", "\\medskip\\noindent", "Assume $X$ is a curve. Then we see that $\\mathcal{O}_X(x - x')$", "defines a $k$-rational point $g$ of $J = \\underline{\\Pic}^0_{X/k}$, see", "Picard Schemes of Curves, Lemma \\ref{pic-lemma-picard-pieces}.", "Recall that $J$ is a proper smooth variety over $k$", "which is also a group scheme over $k$ (same reference).", "Hence $J$ is geometrically integral", "(see Varieties, Lemma \\ref{varieties-lemma-geometrically-connected-criterion}", "and \\ref{varieties-lemma-smooth-geometrically-normal}).", "In other words, $J$ is an abelian variety, see", "Groupoids, Definition \\ref{groupoids-definition-abelian-variety}.", "Thus $[n] : J \\to J$ is finite \\'etale by", "Groupoids, Proposition \\ref{groupoids-proposition-review-abelian-varieties}", "(this is where we use $n$ is invertible in $k$).", "Since $k$ is separably closed we conclude that $g = [n](g')$", "for some $g' \\in J(k)$. If $\\mathcal{L}$ is the degree $0$", "invertible module on $X$ corresponding to $g'$, then we conclude", "that $\\mathcal{O}_X(x - x') \\cong \\mathcal{L}^{\\otimes n}$ as desired." ], "refs": [ "weil-lemma-chow-limit", "varieties-lemma-generate-over-complement", "varieties-lemma-bertini", "varieties-lemma-connectedness-ample-divisor", "pic-lemma-picard-pieces", "varieties-lemma-geometrically-connected-criterion", "varieties-lemma-smooth-geometrically-normal", "groupoids-definition-abelian-variety", "groupoids-proposition-review-abelian-varieties" ], "ref_ids": [ 5064, 11131, 11132, 11135, 12571, 10925, 11005, 9675, 9668 ] } ], "ref_ids": [] }, { "id": 5066, "type": "theorem", "label": "weil-lemma-kernel-to-closure", "categories": [ "weil" ], "title": "weil-lemma-kernel-to-closure", "contents": [ "Let $K/k$ be an algebraic extension of fields.", "Let $X$ be a finite type scheme over $k$.", "The kernel of the map $\\CH_i(X) \\to \\CH_i(X_K)$", "constructed in Lemma \\ref{lemma-chow-limit}", "is torsion." ], "refs": [ "weil-lemma-chow-limit" ], "proofs": [ { "contents": [ "It clearly suffices to show that the kernel", "of flat pullback $\\CH_i(X) \\to \\CH_i(X_{k'})$", "by $\\pi : X_{k'} \\to X$ is torsion", "for any finite extension $k'/k$. This is clear because", "$\\pi_* \\pi^* \\alpha = [k' : k] \\alpha$ by", "Chow Homology, Lemma \\ref{chow-lemma-finite-flat}." ], "refs": [ "chow-lemma-finite-flat" ], "ref_ids": [ 5683 ] } ], "ref_ids": [ 5064 ] }, { "id": 5067, "type": "theorem", "label": "weil-lemma-smash-nilpotence", "categories": [ "weil" ], "title": "weil-lemma-smash-nilpotence", "contents": [ "\\begin{reference}", "\\cite{nilpotence}", "\\end{reference}", "Let $k$ be a field. Let $X$ be a geometrically irreducible", "smooth projective scheme over $k$. Let $x, x' \\in X$ be $k$-rational points.", "For $n$ large enough the class of the zero cycle", "$$", "([x] - [x']) \\times \\ldots \\times ([x] - [x']) \\in", "\\CH_0(X^n)", "$$", "is torsion." ], "refs": [], "proofs": [ { "contents": [ "If we can show this after base change to the algebraic closure of $k$,", "then the result follows over $k$ because the kernel of pullback", "is torsion by Lemma \\ref{lemma-kernel-to-closure}.", "Hence we may and do assume $k$ is algebraically closed.", "\\medskip\\noindent", "Using Bertini we can choose a smooth curve $C \\subset X$ passing through", "$x$ and $x'$. See proof of Lemma \\ref{lemma-divide-difference-points}.", "Hence we may assume $X$ is a curve.", "\\medskip\\noindent", "Assume $X$ is a curve and $k$ is algebraically closed.", "Write $S^n(X) = \\underline{\\Hilbfunctor}^n_{X/k}$ with notation as in", "Picard Schemes of Curves, Sections \\ref{pic-section-hilbert-scheme-points}", "and \\ref{pic-section-divisors}. There is a canonical morphism", "$$", "\\pi : X^n \\longrightarrow S^n(X)", "$$", "which sends the $k$-rational point $(x_1, \\ldots, x_n)$ to the $k$-rational", "point corresponding to the divisor $[x_1] + \\ldots + [x_n]$ on $X$.", "There is a faithful action of the symmetric group $S_n$ on $X^n$.", "The morphism $\\pi$ is $S_n$-invariant and the fibres of $\\pi$ are", "$S_n$-orbits (set theoretically). Finally, $\\pi$ is finite flat of", "degree $n!$, see Picard Schemes of Curves, Lemma", "\\ref{pic-lemma-universal-object}.", "\\medskip\\noindent", "Let $\\alpha_n$ be the zero cycle on $X^n$ given by the formula in the", "statement of the lemma. Let $\\mathcal{L} = \\mathcal{O}_X(x - x')$. Then", "$c_1(\\mathcal{L}) \\cap [X] = [x] - [x']$. Thus", "$$", "\\alpha_n = c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_n) \\cap [X^n]", "$$", "where $\\mathcal{L}_i = \\text{pr}_i^*\\mathcal{L}$ and $\\text{pr}_i : X^n \\to X$", "is the $i$th projection. By either", "Divisors, Lemma \\ref{divisors-lemma-finite-locally-free-has-norm} or", "Divisors, Lemma \\ref{divisors-lemma-norm-in-normal-case}", "there is a norm for $\\pi$. Set", "$\\mathcal{N} = \\text{Norm}_\\pi(\\mathcal{L}_1)$, ", "see Divisors, Lemma \\ref{divisors-lemma-norm-invertible}. We have", "$$", "\\pi^*\\mathcal{N} =", "(\\mathcal{L}_1 \\otimes \\ldots \\otimes \\mathcal{L}_n)^{\\otimes (n - 1)!}", "$$", "in $\\Pic(X^n)$ by a calculation. Deails omitted; hint: this follows from", "the fact that", "$\\text{Norm}_\\pi : \\pi_*\\mathcal{O}_{X^n} \\to \\mathcal{O}_{S^n(X)}$", "composed with the natural map $\\pi_*\\mathcal{O}_{S^n(X)} \\to \\mathcal{O}_{X^n}$", "is equal to the product over all $\\sigma \\in S_n$ of the action of", "$\\sigma$ on $\\pi_*\\mathcal{O}_{X^n}$. Consider", "$$", "\\beta_n = c_1(\\mathcal{N})^n \\cap [S^n(X)]", "$$", "in $\\CH_0(S^n(X))$. Observe that", "$c_1(\\mathcal{L}_i) \\cap c_1(\\mathcal{L}_i) = 0$", "because $\\mathcal{L}_i$ is pulled back from a curve, see", "Chow Homology, Lemma \\ref{chow-lemma-vanish-above-dimension}. Thus we see that", "\\begin{align*}", "\\pi^*\\beta_n", "& =", "((n - 1)!)^n", "(\\sum\\nolimits_{i = 1, \\ldots, n} c_1(\\mathcal{L}_i))^n \\cap [X^n] \\\\", "& =", "((n - 1)!)^n n^n ", "c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_n) \\cap [X^n] \\\\", "& =", "(n!)^n \\alpha_n", "\\end{align*}", "Thus it suffices to show that $\\beta_n$ is torsion.", "\\medskip\\noindent", "There is a canonical morphism", "$$", "f : S^n(X) \\longrightarrow \\underline{\\Picardfunctor}^n_{X/k}", "$$", "See Picard Schemes of Curves, Lemma \\ref{pic-lemma-picard-pieces}.", "For $n \\geq 2g - 1$ this morphism is a projective space bundle", "(details omitted; compare with the", "proof of Picard Schemes of Curves, Lemma \\ref{pic-lemma-picard-pieces}).", "The invertible sheaf $\\mathcal{N}$ is trivial on the", "fibres of $f$, see below. Thus by the projective space bundle formula", "(Chow Homology, Lemma \\ref{chow-lemma-chow-ring-projective-bundle})", "we see that $\\mathcal{N} = f^*\\mathcal{M}$ for some invertible", "module $\\mathcal{M}$ on $\\underline{\\Picardfunctor}^n_{X/k}$.", "Of course, then we see that", "$$", "c_1(\\mathcal{N})^n = f^*(c_1(\\mathcal{M})^n)", "$$", "is zero because $n > g = \\dim(\\underline{\\Picardfunctor}^n_{X/k})$", "and we can use Chow Homology, Lemma \\ref{chow-lemma-vanish-above-dimension}", "as before.", "\\medskip\\noindent", "We still have to show that $\\mathcal{N}$ is trivial on a fibre $F$", "of $f$. Since the fibres of $f$ are projective spaces and since", "$\\Pic(\\mathbf{P}^m_k) = \\mathbf{Z}$", "(Divisors, Lemma \\ref{divisors-lemma-Pic-projective-space-UFD}),", "this can be shown by computing the degree of $\\mathcal{N}$", "on a line contained in the fibre. Instead we will prove it by", "proving that $\\mathcal{N}$ is algebraically", "equivalent to zero. First we claim there is a connected finite type scheme $T$", "over $k$, an invertible module $\\mathcal{L}'$ on $T \\times X$ and", "$k$-rational points $p, q \\in T$ such that", "$\\mathcal{M}_p \\cong \\mathcal{O}_X$ and $\\mathcal{M}_q = \\mathcal{L}$.", "Namely, since $\\mathcal{L} = \\mathcal{O}_X(x - x')$ we can take", "$T = X$, $p = x'$, $q = x$, and", "$\\mathcal{L}' = \\mathcal{O}_{X \\times X}(\\Delta)", "\\otimes \\text{pr}_2^*\\mathcal{O}_X(-x')$.", "Then we let $\\mathcal{L}'_i$ on", "$T \\times X^n$ for $i = 1, \\ldots, n$", "be the pullback of $\\mathcal{L}'$ by", "$\\text{id}_T \\times \\text{pr}_i : T \\times X^n \\to T \\times X$.", "Finally, we let", "$\\mathcal{N}' = \\text{Norm}_{\\text{id}_T \\times \\pi}(\\mathcal{L}'_1)$", "on $T \\times S^n(X)$.", "By construction we have $\\mathcal{N}'_p = \\mathcal{O}_{S^n(X)}$", "and $\\mathcal{N}'_q = \\mathcal{N}$.", "We conclude that", "$$", "\\mathcal{N}'|_{T \\times F}", "$$", "is an invertible module on $T \\times F \\cong T \\times \\mathbf{P}^m_k$", "whose fibre over $p$ is the trivial invertible module and whose fibre", "over $q$ is $\\mathcal{N}|_F$. Since the euler characteristic", "of the trivial bundle is $1$ and since this euler characteristic", "is locally constant in families (Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-chi-locally-constant-geometric})", "we conclude $\\chi(F, \\mathcal{N}^{\\otimes s}|_F) = 1$", "for all $s \\in \\mathbf{Z}$. This can happen only if", "$\\mathcal{N}|_F \\cong \\mathcal{O}_F$ (see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cohomology-projective-space-over-ring})", "and the proof is complete. Some details omitted." ], "refs": [ "weil-lemma-kernel-to-closure", "weil-lemma-divide-difference-points", "pic-lemma-universal-object", "divisors-lemma-finite-locally-free-has-norm", "divisors-lemma-norm-in-normal-case", "divisors-lemma-norm-invertible", "chow-lemma-vanish-above-dimension", "pic-lemma-picard-pieces", "pic-lemma-picard-pieces", "chow-lemma-chow-ring-projective-bundle", "chow-lemma-vanish-above-dimension", "divisors-lemma-Pic-projective-space-UFD", "perfect-lemma-chi-locally-constant-geometric", "coherent-lemma-cohomology-projective-space-over-ring" ], "ref_ids": [ 5066, 5065, 12562, 7968, 7969, 7964, 5737, 12571, 12571, 5743, 5737, 8034, 7063, 3304 ] } ], "ref_ids": [] }, { "id": 5068, "type": "theorem", "label": "weil-lemma-pushforward", "categories": [ "weil" ], "title": "weil-lemma-pushforward", "contents": [ "Assume given (D0), (D1), and (D3) satisfying (A). For $f : X \\to Y$", "a morphism of nonempty equidimensional smooth projective schemes over $k$", "we have $f_*(f^*b \\cup a) = b \\cup f_*a$. If $g : Y \\to Z$ is a second morphism", "with $Z$ nonempty smooth projective and equidimensional, then", "$g_* \\circ f_* = (g \\circ f)_*$." ], "refs": [], "proofs": [ { "contents": [ "The first equality holds because", "$$", "\\int_Y c \\cup b \\cup f_*a =", "\\int_X f^*c \\cup f^*b \\cup a =", "\\int_Y c \\cup f_*(f^*b \\cup a).", "$$", "The second equality holds because", "$$", "\\int_Z c \\cup (g \\circ f)_*a = \\int_X (g \\circ f)^*c \\cup a =", "\\int_X f^* g^* c \\cup a = \\int_Y g^*c \\cup f_*a = \\int_Z c \\cup g_*f_*a", "$$", "This ends the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5069, "type": "theorem", "label": "weil-lemma-pr2star", "categories": [ "weil" ], "title": "weil-lemma-pr2star", "contents": [ "Assume given (D0), (D1), and (D3) satisfying (A) and (B).", "Let $X$ and $Y$ be nonempty smooth projective schemes over $k$", "equidimensional of dimensions $d$ and $e$. Then", "$\\text{pr}_{2, *} : H^*(X \\times Y)(d + e) \\to H^*(Y)(e)$ sends", "$a \\otimes b$ to $(\\int_X a) b$." ], "refs": [], "proofs": [ { "contents": [ "This follows from axioms (B)(a) and (B)(b)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5070, "type": "theorem", "label": "weil-lemma-base", "categories": [ "weil" ], "title": "weil-lemma-base", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).", "Then $H^i(\\Spec(k)) = 0$ for $i \\not = 0$ and there is a", "unique $F$-algebra isomorphism $F = H^0(\\Spec(k))$.", "We have $\\gamma([\\Spec(k)]) = 1$ and $\\int_{\\Spec(k)} 1 = 1$." ], "refs": [], "proofs": [ { "contents": [ "By axiom (C)(d) we see that $H^0(\\Spec(k))$ is nonzero and even", "$\\gamma([\\Spec(k)])$ is nonzero.", "Since $\\Spec(k) \\times \\Spec(k) = \\Spec(k)$ we get", "$$", "H^*(\\Spec(k)) \\otimes_F H^*(\\Spec(k)) = H^*(\\Spec(k))", "$$", "by axiom (B)(a) which implies (look at dimensions) that only", "$H^0$ is nonzero and moreover has dimension $1$. Thus", "$F = H^0(\\Spec(k))$ via the unique $F$-algebra isomorphism", "given by mapping $1 \\in F$ to $1 \\in H^0(\\Spec(k))$.", "Since $[\\Spec(k)] \\cdot [\\Spec(k)] = [\\Spec(k)]$ in the", "Chow ring of $\\Spec(k)$ we conclude that", "$\\gamma([\\Spec(k)) \\cup \\gamma([\\Spec(k)]) = \\gamma([\\Spec(k)])$", "by axiom (C)(c). Since we already know that $\\gamma([\\Spec(k)])$ is nonzero", "we conclude that it has to be equal to $1$.", "Finally, we have $\\int_{\\Spec(k)} 1 = 1$ since", "$\\int_{\\Spec(k)} \\gamma([\\Spec(k)]) = 1$", "by axiom (C)(d)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5071, "type": "theorem", "label": "weil-lemma-unit", "categories": [ "weil" ], "title": "weil-lemma-unit", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).", "Let $X$ be a smooth projective scheme over $k$.", "If $X = \\emptyset$, then $H^*(X) = 0$.", "If $X$ is nonempty, then $\\gamma([X]) = 1$ and $1 \\not = 0$ in $H^0(X)$." ], "refs": [], "proofs": [ { "contents": [ "First assume $X$ is nonempty.", "Observe that $[X]$ is the pullback of $[\\Spec(k)]$ by the structure morphism", "$p : X \\to \\Spec(k)$. Hence we get $\\gamma([X]) = 1$ by axiom (C)(a)", "and Lemma \\ref{lemma-base}. Let $X' \\subset X$ be an irreducible component.", "By functoriality it suffices to show $1 \\not = 0$ in $H^0(X')$.", "Thus we may and do assume $X$ is irreducible, and in particular", "nonempty and equidimensional, say of dimension $d$.", "To see that $1 \\not = 0$ it suffices to show that $H^*(X)$ is nonzero.", "\\medskip\\noindent", "Let $x \\in X$ be a closed point whose residue field $k'$", "is separable over $k$, see", "Varieties, Lemma \\ref{varieties-lemma-smooth-separable-closed-points-dense}.", "Let $i : \\Spec(k') \\to X$ be the inclusion morphism.", "Denote $p : X \\to \\Spec(k)$ is the structure morphism.", "Observe that", "$p_*i_*[\\Spec(k')] = [k' : k][\\Spec(k)]$ in $\\CH_0(\\Spec(k))$.", "Using axiom (C)(b) twice and Lemma \\ref{lemma-base}", "we conclude that", "$$", "p_*i_*\\gamma([\\Spec(k')]) = \\gamma([k' : k][\\Spec(k)]) = [k' : k]", "\\in F = H^0(\\Spec(k))", "$$", "is nonzero. Thus $i_*\\gamma([\\Spec(k)]) \\in H^{2d}(X)(d)$ is nonzero", "(because it maps to something nonzero via $p_*$). This concludes the proof", "in case $X$ is nonempty.", "\\medskip\\noindent", "Finally, we consider the case of the empty scheme. Axiom (B)(a) gives", "$H^*(\\emptyset) \\otimes H^*(\\emptyset) = H^*(\\emptyset)$ and", "we get that $H^*(\\emptyset)$ is either zero or $1$-dimensional", "in degree $0$. Then axiom (B)(a) again shows that", "$H^*(\\emptyset) \\otimes H^*(X) = H^*(\\emptyset)$ for", "all smooth projective schemes $X$ over $k$. Using axiom (A)(b)", "and the nonvanishing of $H^0(X)$ we've seen above", "we find that $H^*(X)$ is nonzero in at least two degrees", "if $\\dim(X) > 0$. This then forces $H^*(\\emptyset)$ to be zero." ], "refs": [ "weil-lemma-base", "varieties-lemma-smooth-separable-closed-points-dense", "weil-lemma-base" ], "ref_ids": [ 5070, 11007, 5070 ] } ], "ref_ids": [] }, { "id": 5072, "type": "theorem", "label": "weil-lemma-push-unit", "categories": [ "weil" ], "title": "weil-lemma-push-unit", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).", "Let $i : X \\to Y$ be a closed immersion of nonempty smooth projective", "equidimensional schemes over $k$. Then", "$\\gamma([X]) = i_*1$ in $H^{2c}(Y)(c)$ where $c = \\dim(Y) - \\dim(X)$." ], "refs": [], "proofs": [ { "contents": [ "This is true because $1 = \\gamma([X])$ in $H^0(X)$ by Lemma \\ref{lemma-unit}", "and then we can apply axiom (C)(b)." ], "refs": [ "weil-lemma-unit" ], "ref_ids": [ 5071 ] } ], "ref_ids": [] }, { "id": 5073, "type": "theorem", "label": "weil-lemma-class-diagonal", "categories": [ "weil" ], "title": "weil-lemma-class-diagonal", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).", "Let $X$ be a nonempty smooth projective scheme over $k$ equidimensional", "of dimension $d$. Choose a basis $e_{i, j}, j = 1, \\ldots, \\beta_i$ of", "$H^i(X)$ over $F$. Using K\\\"unneth write", "$$", "\\gamma([\\Delta]) =", "\\sum\\nolimits_i", "\\sum\\nolimits_j e_{i, j} \\otimes e'_{2d - i , j}", "\\quad\\text{in}\\quad", "\\bigoplus\\nolimits_i H^i(X) \\otimes_F H^{2d - i}(X)(d)", "$$", "with $e'_{2d - i, j} \\in H^{2d - i}(X)(d)$.", "Then $\\int_X e_{i, j} \\cup e'_{2d - i, j'} = (-1)^i\\delta_{jj'}$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\Delta^* : H^*(X \\times X) \\to H^*(X)$ is equal to the", "cup product map $H^*(X) \\otimes_F H^*(X) \\to H^*(X)$, see", "Remark \\ref{remark-replace-cup-product}. On the other", "hand, recall that $\\gamma([\\Delta]) = \\Delta_*1$ (Lemma \\ref{lemma-push-unit})", "and hence", "$$", "\\int_{X \\times X} \\gamma([\\Delta]) \\cup a \\otimes b =", "\\int_{X \\times X} \\Delta_*1 \\cup a \\otimes b =", "\\int_X a \\cup b", "$$", "by Lemma \\ref{lemma-pushforward}.", "On the other hand, we have", "$$", "\\int_{X \\times X} (\\sum e_{i, j} \\otimes e'_{2d -i , j}) \\cup a \\otimes b =", "\\sum (\\int_X a \\cup e_{i, j})(\\int_X e'_{2d - i, j} \\cup b)", "$$", "by axiom (B)(b); note that we made two switches of order so that the sign", "for each term is $1$. Thus if we choose $a$ such that", "$\\int_X a \\cup e_{i, j} = 1$ and all other pairings equal to zero, then", "we conclude that $\\int_X e'_{2d - i, j} \\cup b = \\int_X a \\cup b$", "for all $b$, i.e., $e'_{2d - i, j} = a$. This proves the lemma." ], "refs": [ "weil-remark-replace-cup-product", "weil-lemma-push-unit", "weil-lemma-pushforward" ], "ref_ids": [ 5120, 5072, 5068 ] } ], "ref_ids": [] }, { "id": 5074, "type": "theorem", "label": "weil-lemma-cohomology-P1", "categories": [ "weil" ], "title": "weil-lemma-cohomology-P1", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).", "Then $H^*(\\mathbf{P}^1_k)$ is $1$-dimensional in dimensions $0$ and $2$", "and zero in other degrees." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in \\mathbf{P}^1_k$ be a $k$-rational point. Observe that", "$\\Delta = \\text{pr}_1^*x + \\text{pr}_2^*x$ as divisors on", "$\\mathbf{P}^1_k \\times \\mathbf{P}^1_k$. Using axiom (C)(a)", "and additivity of $\\gamma$ we see that", "$$", "\\gamma([\\Delta]) =", "\\text{pr}_1^*\\gamma([x]) +", "\\text{pr}_2^*\\gamma([x]) =", "\\gamma([x]) \\otimes 1 + 1 \\otimes \\gamma([x])", "$$", "in $H^*(\\mathbf{P}^1_k \\times \\mathbf{P}^1_k) =", "H^*(\\mathbf{P}^1_k) \\otimes_F H^*(\\mathbf{P}^1_k)$.", "However, by Lemma \\ref{lemma-class-diagonal}", "we know that $\\gamma([\\Delta])$ cannot be written", "as a sum of fewer than $\\sum \\beta_i$ pure tensors", "where $\\beta_i = \\dim_F H^i(\\mathbf{P}^1_k)$.", "Thus we see that $\\sum \\beta_i \\leq 2$.", "By Lemma \\ref{lemma-unit} we have $H^0(\\mathbf{P}^1_k) \\not = 0$.", "By Poincar\\'e duality, more precisely axiom (A)(b),", "we have $\\beta_0 = \\beta_2$. Therefore the lemma holds." ], "refs": [ "weil-lemma-class-diagonal", "weil-lemma-unit" ], "ref_ids": [ 5073, 5071 ] } ], "ref_ids": [] }, { "id": 5075, "type": "theorem", "label": "weil-lemma-weil-additive", "categories": [ "weil" ], "title": "weil-lemma-weil-additive", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).", "If $X$ and $Y$ are smooth projective schemes over $k$, then", "$H^*(X \\amalg Y) \\to H^*(X) \\times H^*(Y)$,", "$a \\mapsto (i^*a, j^*a)$ is an isomorphism where $i$, $j$", "are the coprojections." ], "refs": [], "proofs": [ { "contents": [ "If $X$ or $Y$ is empty, then this is true because", "$H^*(\\emptyset) = 0$ by Lemma \\ref{lemma-unit}.", "Thus we may assume both $X$ and $Y$ are nonempty.", "\\medskip\\noindent", "We first show that the map is injective. First, observe that", "we can find morphisms $X' \\to X$ and $Y' \\to Y$", "of smooth projective schemes so that $X'$ and $Y'$ are", "equidimensional of the same dimension and such that", "$X' \\to X$ and $Y' \\to Y$ each have a section. Namely,", "decompose $X = \\coprod X_d$ and $Y = \\coprod Y_e$", "into open and closed subschemes equidimensional of", "dimension $d$ and $e$. Then take", "$X' = \\coprod X_d \\times \\mathbf{P}^{n - d}$", "and $Y' = \\coprod Y_e \\times \\mathbf{P}^{n - e}$ for some", "$n$ sufficiently large. Thus pullback by", "$X' \\amalg Y' \\to X \\amalg Y$ is injective", "(because there is a section) and", "it suffices to show the injectivity for $X', Y'$", "as we do in the next parapgrah.", "\\medskip\\noindent", "Let us show the map is injective when $X$ and $Y$ are equidimensional", "of the same dimension $d$.", "Observe that $[X \\amalg Y] = [X] + [Y]$ in $\\CH^0(X \\amalg Y)$", "and that $[X]$ and $[Y]$ are orthogonal idempotents in $\\CH^0(X \\amalg Y)$.", "Thus", "$$", "1 = \\gamma([X \\amalg Y] = \\gamma([X]) + \\gamma([Y]) = i_*1 + j_*1", "$$", "is a decomposition into orthogonal idempotents. Here we have used", "Lemmas \\ref{lemma-unit} and \\ref{lemma-push-unit} and axiom (C)(c).", "Then we see that", "$$", "a = a \\cup 1 = a \\cup i_*1 + a \\cup j_*1 =", "i_*(i^*a) + j_*(j^*a)", "$$", "by the projection formula (Lemma \\ref{lemma-pushforward}) and hence the map", "is injective.", "\\medskip\\noindent", "We show the map is surjective. Write $e = \\gamma([X])$ and $f = \\gamma([Y])$", "viewed as elements in $H^0(X \\amalg Y)$. We have", "$i^*e = 1$, $i^*f = 0$, $j^*e = 0$, and $j^*f = 1$ by axiom (C)(a).", "Hence if $i^* : H^*(X \\amalg Y) \\to H^*(X)$", "and $j^* : H^*(X \\amalg Y) \\to H^*(Y)$ are surjective, then", "so is $(i^*, j^*)$. Namely, for $a, a' \\in H^*(X \\amalg Y)$", "we have", "$$", "(i^*a, j^*a') = (i^*(a \\cup e + a' \\cup f), j^*(a \\cup e + a' \\cup f))", "$$", "By symmetry it suffices to show $i^* : H^*(X \\amalg Y) \\to H^*(X)$", "is surjective. If there is a morphism $Y \\to X$, then there is a morphism", "$g : X \\amalg Y \\to X$ with $g \\circ i = \\text{id}_X$ and we conclude.", "To finish the proof, observe that in order to prove", "$i^*$ is surjective, it suffices to do so after tensoring", "by a nonzero graded $F$-vector space. Hence by axiom (B)(b)", "and nonvanishing of cohomology (Lemma \\ref{lemma-unit})", "it suffices to prove $i^*$ is surjective after replacing", "$X$ and $Y$ by $X \\times \\Spec(k')$ and $Y \\times \\Spec(k')$", "for some finite separable extension $k'/k$.", "If we choose $k'$ such that there exists a closed point", "$x \\in X$ with $\\kappa(x) = k'$ (and this is possible by", "Varieties, Lemma \\ref{varieties-lemma-smooth-separable-closed-points-dense})", "then there is a morphism $Y \\times \\Spec(k') \\to X \\times \\Spec(k')$", "and we find that the proof is complete." ], "refs": [ "weil-lemma-unit", "weil-lemma-unit", "weil-lemma-push-unit", "weil-lemma-pushforward", "weil-lemma-unit", "varieties-lemma-smooth-separable-closed-points-dense" ], "ref_ids": [ 5071, 5071, 5072, 5068, 5071, 11007 ] } ], "ref_ids": [] }, { "id": 5076, "type": "theorem", "label": "weil-lemma-from-functor-to-weil", "categories": [ "weil" ], "title": "weil-lemma-from-functor-to-weil", "contents": [ "Let $k$ be a field. Let $F$ be a field of characteristic $0$.", "Assume given a $\\mathbf{Q}$-linear functor", "$$", "G : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces}", "$$", "of symmetric monoidal categories such that $G(\\mathbf{1}(1))$", "is nonzero only in degree $-2$. Then we obtain data (D0), (D1), (D2), and (D3)", "satisfying all of (A), (B), and (C) above." ], "refs": [], "proofs": [ { "contents": [ "This proof is the same as the proof of", "Lemma \\ref{lemma-from-functor-to-weil-classical};", "we urge the reader to read the proof of that lemma instead.", "\\medskip\\noindent", "We obtain a contravariant functor from the category of smooth", "projective schemes over $k$ to the category of graded $F$-vector spaces", "by setting $H^*(X) = G(h(X))$. By assumption we have a canonical", "isomorphism", "$$", "H^*(X \\times Y) = G(h(X \\times Y)) = G(h(X) \\otimes h(Y)) =", "G(h(X)) \\otimes G(h(Y)) = H^*(X) \\otimes H^*(Y)", "$$", "compatible with pullbacks. Using pullback along the diagonal", "$\\Delta : X \\to X \\times X$ we obtain a canonical map", "$$", "H^*(X) \\otimes H^*(X) = H^*(X \\times X) \\to H^*(X)", "$$", "of graded vector spaces compatible with pullbacks.", "This defines a functorial graded $F$-algebra structure on", "$H^*(X)$. Since $\\Delta$ commutes with the commutativity", "constraint $h(X) \\otimes h(X) \\to h(X) \\otimes h(X)$ (switching the factors)", "and since $G$ is a functor of symmetric monoidal categories (so compatible with", "commutativity constraints), and by our convention in", "Homology, Example \\ref{homology-example-graded-vector-spaces}", "we conclude that $H^*(X)$ is a graded", "commutative algebra! Hence we get our datum (D1).", "\\medskip\\noindent", "Since $\\mathbf{1}(1)$ is invertible in the category of motives", "we see that $G(\\mathbf{1}(1))$ is invertible in the category of", "graded $F$-vector spaces. Thus $\\sum_i \\dim_F G^i(\\mathbf{1}(1)) = 1$.", "By assumption we only get something nonzero in degree $-2$.", "Our datum (D0) is the vector space $F(1) = G^{-2}(\\mathbf{1}(1))$.", "Since $G$ is a symmetric monoidal functor we see that", "$F(n) = G^{-2n}(\\mathbf{1}(n))$ for all $n \\in \\mathbf{Z}$.", "It follows that", "$$", "H^{2r}(X)(r) = G^{2r}(h(X)) \\otimes G^{-2r}(\\mathbf{1}(r)) =", "G^0(h(X)(r))", "$$", "a formula we will frequently use below.", "\\medskip\\noindent", "Let $X$ be a smooth projective scheme over $k$. By", "Lemma \\ref{lemma-composition-correspondences} we have", "$$", "\\CH^r(X) \\otimes \\mathbf{Q} = \\text{Corr}^r(\\Spec(k), X) =", "\\Hom(\\mathbf{1}(-r), h(X)) = \\Hom(\\mathbf{1}, h(X)(r))", "$$", "Applying the functor $G$ this maps into", "$\\Hom(G(\\mathbf{1}), G(h(X)(r)))$.", "By taking the image of $1$ in $G^0(\\mathbf{1}) = F$", "into $G^0(h(X)(r)) = H^{2r}(X)(r)$ we obtain", "$$", "\\gamma :", "\\CH^r(X) \\otimes \\mathbf{Q} \\longrightarrow H^{2r}(X)(r)", "$$", "This is the datum (D2).", "\\medskip\\noindent", "Let $X$ be a nonempty smooth projective scheme over $k$", "which is equidimensional of dimension $d$. By", "Lemma \\ref{lemma-composition-correspondences} we have", "$$", "\\Mor(h(X)(d), \\mathbf{1}) = \\Mor((X, 1, d), (\\Spec(k), 1, 0)) =", "\\text{Corr}^{-d}(X, \\Spec(k)) = \\CH_d(X)", "$$", "Thus the class of the cycle $[X]$ in $\\CH_d(X)$ defines a morphism", "$h(X)(d) \\to \\mathbf{1}$. Applying $G$ and taking degree $0$", "parts we obtain", "$$", "H^{2d}(X)(d) = G^0(h(X)(d)) \\longrightarrow G^0(\\mathbf{1}) = F", "$$", "This map $\\int_X : H^{2d}(X)(d) \\to F$ is the datum (D3).", "\\medskip\\noindent", "Let $X$ be a smooth projective scheme over $k$ which is", "nonempty and equidimensional of dimension $d$. By Lemma \\ref{lemma-dual}", "we know that $h(X)(d)$ is a left dual to $h(X)$. Hence", "$G(h(X)(d)) = H^*(X) \\otimes_F F(d)[2d]$", "is a left dual to $H^*(X)$ in the category of graded $F$-vector spaces.", "Here $[n]$ is the shift functor on graded vector spaces.", "By Homology, Lemma \\ref{homology-lemma-left-dual-graded-vector-spaces}", "we find that $\\sum_i \\dim_F H^i(X) < \\infty$ and that", "$\\epsilon : h(X)(d) \\otimes h(X) \\to \\mathbf{1}$ produces", "nondegenerate pairings $H^{2d - i}(X)(d) \\otimes_F H^i(X) \\to F$.", "In the proof of Lemma \\ref{lemma-dual} we have seen that", "$\\epsilon$ is given by $[\\Delta]$ via the identifications", "$$", "\\Hom(h(X)(d) \\otimes h(X), \\mathbf{1}) =", "\\text{Corr}^{-d}(X \\times X, \\Spec(k)) =", "\\CH_d(X \\times X)", "$$", "Thus $\\epsilon$ is the composition of $[X] : h(X)(d) \\to \\mathbf{1}$", "and $h(\\Delta)(d) : h(X)(d) \\otimes h(X) \\to h(X)(d)$. It follows", "that the pairings above are given by cup product followed by", "$\\int_X$. This proves axiom (A).", "\\medskip\\noindent", "Axiom (B) follows from the assumption that $G$ is compatible", "with tensor structures and our construction of the cup product above.", "\\medskip\\noindent", "Axiom (C). Our construction of $\\gamma$ takes a cycle $\\alpha$ on $X$,", "interprets it a correspondence $a$ from $\\Spec(k)$ to $X$ of some degree,", "and then applies $G$. If $f : Y \\to X$ is a morphism of nonempty", "equidimensional smooth projective schemes over $k$, then", "$f^!\\alpha$ is the pushforward (!) of $\\alpha$", "by the correspondence $[\\Gamma_f]$ from $X$ to $Y$, see", "Lemma \\ref{lemma-functor-and-cycles}. Hence", "$f^!\\alpha$ viewed as a correspondence from $\\Spec(k)$ to $Y$", "is equal to $a \\circ [\\Gamma_f]$, see", "Lemma \\ref{lemma-composition-correspondences}.", "Since $G$ is a functor, we conclude", "$\\gamma$ is compatible with pullbacks, i.e., axiom (C)(a) holds.", "\\medskip\\noindent", "Let $f : Y \\to X$ be a morphism of nonempty equidimensional", "smooth projective schemes over $k$ and", "let $\\beta \\in \\CH^r(Y)$ be a cycle on $Y$. We have to show that", "$$", "\\int_Y \\gamma(\\beta) \\cup f^*c = \\int_X \\gamma(f_*\\beta) \\cup c", "$$", "for all $c \\in H^*(X)$. Let $a, a^t, \\eta_X, \\eta_Y, [X], [Y]$", "be as in Lemma \\ref{lemma-prep-dual}.", "Let $b$ be $\\beta$ viewed as a correspondence from $\\Spec(k)$ to $Y$", "of degree $r$. Then $f_*\\beta$ viewed as a correspondence from", "$\\Spec(k)$ to $X$ is equal to $a^t \\circ b$, see", "Lemmas \\ref{lemma-functor-and-cycles} and", "\\ref{lemma-composition-correspondences}.", "The displayed equality above holds if we can show that", "$$", "h(X) = \\mathbf{1} \\otimes h(X)", "\\xrightarrow{b \\otimes 1}", "h(Y)(r) \\otimes h(X)", "\\xrightarrow{1 \\otimes a}", "h(Y)(r) \\otimes h(Y)", "\\xrightarrow{\\eta_Y}", "h(Y)(r)", "\\xrightarrow{[Y]}", "\\mathbf{1}(r - e)", "$$", "is equal to", "$$", "h(X) = \\mathbf{1} \\otimes h(X)", "\\xrightarrow{a^t \\circ b \\otimes 1}", "h(X)(r + d - e) \\otimes h(X)", "\\xrightarrow{\\eta_X}", "h(X)(r + d - e)", "\\xrightarrow{[X]}", "\\mathbf{1}(r - e)", "$$", "This follows immediately from Lemma \\ref{lemma-prep-dual}.", "Thus we have axiom (C)(b).", "\\medskip\\noindent", "To prove axiom (C)(c) we use the discussion in", "Remark \\ref{remark-replace-cup-product-classical}.", "Hence it suffices to prove that $\\gamma$ is compatible with", "exterior products. Let $X$, $Y$ be nonempty smooth projective", "schemes over $k$ and let $\\alpha$, $\\beta$ be cycles on them. Denote", "$a$, $b$ the corresponding correspondences from $\\Spec(k)$ to", "$X$, $Y$. Then $\\alpha \\times \\beta$ corresponds to the", "correspondence $a \\otimes b$ from $\\Spec(k)$ to $X \\otimes Y = X \\times Y$.", "Hence the requirement follows from the fact that $G$ is", "compatible with the tensor structures on both sides.", "\\medskip\\noindent", "Axiom (C)(d) follows because the cycle $[\\Spec(k)]$", "corresponds to the identity morphism on $h(\\Spec(k))$.", "This finishes the proof of the lemma." ], "refs": [ "weil-lemma-from-functor-to-weil-classical", "weil-lemma-composition-correspondences", "weil-lemma-composition-correspondences", "weil-lemma-dual", "homology-lemma-left-dual-graded-vector-spaces", "weil-lemma-dual", "weil-lemma-functor-and-cycles", "weil-lemma-composition-correspondences", "weil-lemma-prep-dual", "weil-lemma-functor-and-cycles", "weil-lemma-composition-correspondences", "weil-lemma-prep-dual", "weil-remark-replace-cup-product-classical" ], "ref_ids": [ 5061, 5036, 5036, 5049, 12073, 5049, 5039, 5036, 5041, 5039, 5036, 5041, 5119 ] } ], "ref_ids": [] }, { "id": 5077, "type": "theorem", "label": "weil-lemma-from-weil-to-functor", "categories": [ "weil" ], "title": "weil-lemma-from-weil-to-functor", "contents": [ "Let $k$ be a field. Let $F$ be a field of characteristic $0$. Given", "(D0), (D1), (D2), and (D3) satisfying (A), (B), and (C)", "we can construct a $\\mathbf{Q}$-linear functor", "$$", "G : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces}", "$$", "of symmetric monoidal categories such that $H^*(X) = G(h(X))$." ], "refs": [], "proofs": [ { "contents": [ "The proof of this lemma is the same as the proof of", "Lemma \\ref{lemma-from-weil-to-functor-classical};", "we urge the reader to read the proof of that lemma instead.", "\\medskip\\noindent", "By Lemma \\ref{lemma-characterize-motives} it suffices to construct a functor", "$G$ on the category of smooth projective schemes over $k$", "with morphisms given by correspondences of degree $0$ such that", "the image of $G(c_2)$ on $G(\\mathbf{P}^1_k)$ is an invertible graded", "$F$-vector space.", "\\medskip\\noindent", "Let $X$ be a smooth projective scheme over $k$. There is a canonical", "decomposition", "$$", "X = \\coprod\\nolimits_{0 \\leq d \\le \\dim(X)} X_d", "$$", "into open and closed subschemes such that $X_d$ is equidimensional", "of dimension $d$. By Lemma \\ref{lemma-weil-additive} we have correspondingly", "$$", "H^*(X) \\longrightarrow \\prod\\nolimits_{0 \\leq d \\le \\dim(X)} H^*(X_d)", "$$", "If $Y$ is a second smooth projective scheme over $k$", "and we similarly decompose $Y = \\coprod Y_e$, then", "$$", "\\text{Corr}^0(X, Y) = \\bigoplus \\text{Corr}^0(X_d, Y_e)", "$$", "As well we have $X \\otimes Y = \\coprod X_d \\otimes Y_e$ in the", "category of correspondences. From these observations it follows", "that it suffices to construct $G$ on the category whose objects", "are equidimensional smooth projective schemes over $k$", "and whose morphisms are correspondences of degree $0$. (Some details", "omitted.)", "\\medskip\\noindent", "Given an equdimensional smooth projective scheme", "$X$ over $k$ we set $G(X) = H^*(X)$. Observe that $G(X) = 0$", "if $X = \\emptyset$ (Lemma \\ref{lemma-unit}). Thus maps", "from and to $G(\\emptyset)$ are zero and we may and do", "assume our schemes are nonempty in the discussions below.", "\\medskip\\noindent", "Given a correspondence $c \\in \\text{Corr}^0(X, Y)$ between", "nonempty equidmensional smooth projective schemes over $k$", "we consider the map $G(c) : G(X) = H^*(X) \\to G(Y) = H^*(Y)$", "given by the rule", "$$", "a \\longmapsto", "G(c)(a) = \\text{pr}_{2, *}(\\gamma(c) \\cup \\text{pr}_1^*a)", "$$", "It is clear that $G(c)$ is additive in $c$ and hence $\\mathbf{Q}$-linear.", "Compatibility of $\\gamma$ with pullbacks, pushforwards, and", "intersection products given by axioms (C)(a), (C)(b), and (C)(c)", "shows that we have", "$G(c' \\circ c) = G(c') \\circ G(c)$ if $c' \\in \\text{Corr}^0(Y, Z)$.", "Namely, for $a \\in H^*(X)$ we have", "\\begin{align*}", "(G(c') \\circ G(c))(a)", "& =", "\\text{pr}^{23}_{3, *}(\\gamma(c') \\cup", "\\text{pr}^{23, *}_2(\\text{pr}^{12}_{2, *}(\\gamma(c) \\cup", "\\text{pr}^{12, *}_1a))) \\\\", "& =", "\\text{pr}^{23}_{3, *}(\\gamma(c') \\cup", "\\text{pr}^{123}_{23, *}(\\text{pr}^{123, *}_{12}(\\gamma(c) \\cup", "\\text{pr}^{12, *}_1 a))) \\\\", "& =", "\\text{pr}^{23}_{3, *}", "\\text{pr}^{123}_{23, *}(", "\\text{pr}^{123, *}_{23}\\gamma(c') \\cup", "\\text{pr}^{123, *}_{12}\\gamma(c) \\cup", "\\text{pr}^{123, *}_1 a) \\\\", "& =", "\\text{pr}^{23}_{3, *}", "\\text{pr}^{123}_{23, *}(", "\\gamma(\\text{pr}^{123, *}_{23}c') \\cup", "\\gamma(\\text{pr}^{123, *}_{12}c) \\cup", "\\text{pr}^{123, *}_1 a) \\\\", "& =", "\\text{pr}^{13}_{3, *}", "\\text{pr}^{123}_{13, *}(", "\\gamma(\\text{pr}^{123, *}_{23}c' \\cdot \\text{pr}^{123, *}_{12}c) \\cup", "\\text{pr}^{123, *}_1 a) \\\\", "& =", "\\text{pr}^{13}_{3, *}(", "\\gamma(\\text{pr}^{123}_{13, *}(\\text{pr}^{123, *}_{23}c' \\cdot", "\\text{pr}^{123, *}_{12}c)) \\cup", "\\text{pr}^{13, *}_1 a) \\\\", "& =", "G(c' \\circ c)(a)", "\\end{align*}", "with obvious notation. The first equality follows from the definitions.", "The second equality holds because", "$\\text{pr}^{23, *}_2 \\circ \\text{pr}^{12}_{2, *} =", "\\text{pr}^{123}_{23, *} \\circ \\text{pr}^{123, *}_{12}$", "as follows immediately from the description of pushforward", "along projections given in Lemma \\ref{lemma-pr2star}.", "The third equality holds by Lemma \\ref{lemma-pushforward}", "and the fact that $H^*$ is a functor.", "The fourth equalith holds by axiom (C)(a) and the fact that", "the gysin map agrees with flat pullback for flat morphisms", "(Chow Homology, Lemma \\ref{chow-lemma-lci-gysin-flat}).", "The fifth equality uses axiom (C)(c) as well as", "Lemma \\ref{lemma-pushforward} to see that", "$\\text{pr}^{23}_{3, *} \\circ \\text{pr}^{123}_{23, *} =", "\\text{pr}^{13}_{3, *} \\circ \\text{pr}^{123}_{13, *}$.", "The sixth equality uses the projection formula from", "Lemma \\ref{lemma-pushforward} as well as", "axiom (C)(b) to see that $", "\\text{pr}^{123}_{13, *}", "\\gamma(\\text{pr}^{123, *}_{23}c' \\cdot \\text{pr}^{123, *}_{12}c) =", "\\gamma(\\text{pr}^{123}_{13, *}(", "\\text{pr}^{123, *}_{23}c' \\cdot \\text{pr}^{123, *}_{12}c))$.", "Finally, the last equality is the definition.", "\\medskip\\noindent", "To finish the proof that $G$ is a functor,", "we have to show identities are preserved. In other words, if", "$1 = [\\Delta] \\in \\text{Corr}^0(X, X)$ is the identity in the category", "of correspondences (Lemma \\ref{lemma-category-correspondences}),", "then we have to show that $G([\\Delta]) = \\text{id}$.", "This follows from the determination", "of $\\gamma([\\Delta])$ in Lemma \\ref{lemma-class-diagonal}", "and Lemma \\ref{lemma-pr2star}.", "This finishes the construction of $G$ as a functor on", "smooth projective schemes over $k$ and correspondences of degree $0$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-base} we have that", "$G(\\Spec(k)) = H^*(\\Spec(k))$ is canonically isomorphic to $F$", "as an $F$-algebra. The K\\\"unneth axiom (B)(a)", "shows our functor is compatible with tensor products.", "Thus our functor is a functor of symmetric monoidal categories.", "\\medskip\\noindent", "We still have to check that the image of $G(c_2)$ on", "$G(\\mathbf{P}^1_k) = H^*(\\mathbf{P}^1_k)$", "is an invertible graded $F$-vector space (in particular we don't know yet", "that $G$ extends to $M_k$). By Lemma \\ref{lemma-cohomology-P1}", "we only have nonzero cohomology in degrees $0$ and $2$", "both of dimension $1$. We have $1 = c_0 + c_2$ is a decomposition", "of the identity into a sum of orthogonal idempotents in", "$\\text{Corr}^0(\\mathbf{P}^1_k, \\mathbf{P}^1_k)$, see", "Example \\ref{example-decompose-P1}. Further we have $c_0 = a \\circ b$ where", "$a \\in \\text{Corr}^0(\\Spec(k), \\mathbf{P}^1_k)$ and", "$b \\in \\text{Corr}^0(\\mathbf{P}^1_k, \\Spec(k))$ and where", "$b \\circ a = 1$ in $\\text{Corr}^0(\\Spec(k), \\Spec(k))$, see proof of", "Lemma \\ref{lemma-inverse-h2}. Thus $G(c_0)$ is the projector", "onto the degree $0$ part. It follows that $G(c_2)$ must", "be the projector onto the degree $2$ part and the proof is complete." ], "refs": [ "weil-lemma-from-weil-to-functor-classical", "weil-lemma-characterize-motives", "weil-lemma-weil-additive", "weil-lemma-unit", "weil-lemma-pr2star", "weil-lemma-pushforward", "chow-lemma-lci-gysin-flat", "weil-lemma-pushforward", "weil-lemma-pushforward", "weil-lemma-category-correspondences", "weil-lemma-class-diagonal", "weil-lemma-pr2star", "weil-lemma-base", "weil-lemma-cohomology-P1", "weil-lemma-inverse-h2" ], "ref_ids": [ 5062, 5048, 5075, 5071, 5069, 5068, 5832, 5068, 5068, 5037, 5073, 5069, 5070, 5074, 5045 ] } ], "ref_ids": [] }, { "id": 5078, "type": "theorem", "label": "weil-lemma-trace-disjoint-union", "categories": [ "weil" ], "title": "weil-lemma-trace-disjoint-union", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).", "Let $X, Y$ be nonempty smooth projective schemes both equidimensional", "of dimension $d$ over $k$. Then $\\int_{X \\amalg Y} = \\int_X + \\int_Y$." ], "refs": [], "proofs": [ { "contents": [ "Denote $i : X \\to X \\amalg Y$ and $j : Y \\to X \\amalg Y$ be the coprojections.", "By Lemma \\ref{lemma-weil-additive} the map", "$(i^*, j^*) : H^*(X \\amalg Y) \\to H^*(X) \\times H^*(Y)$ is an isomorphism.", "The statement of the lemma means that under the isomorphism", "$(i^*, j^*) : H^{2d}(X \\amalg Y)(d) \\to H^{2d}(X)(d) \\oplus H^{2d}(Y)(d)$", "the map $\\int_X + \\int_Y$ is tranformed into $\\int_{X \\amalg Y}$.", "This is true because", "$$", "\\int_{X \\amalg Y} a =", "\\int_{X \\amalg Y} i_*(i^*a) + j_*(j^*a) =", "\\int_X i^*a + \\int_Y j^*a", "$$", "where the equality $a = i_*(i^*a) + j_*(j^*a)$ was shown in", "the proof of Lemma \\ref{lemma-weil-additive}." ], "refs": [ "weil-lemma-weil-additive", "weil-lemma-weil-additive" ], "ref_ids": [ 5075, 5075 ] } ], "ref_ids": [] }, { "id": 5079, "type": "theorem", "label": "weil-lemma-dim-0", "categories": [ "weil" ], "title": "weil-lemma-dim-0", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).", "Let $X$ be a smooth projective scheme of dimension zero over $k$.", "Then", "\\begin{enumerate}", "\\item $H^i(X) = 0$ for $i \\not = 0$,", "\\item $H^0(X)$ is a finite separable algebra over $F$,", "\\item $\\dim_F H^0(X) = \\deg(X \\to \\Spec(F))$,", "\\item $\\int_X : H^0(X) \\to F$ is the trace map,", "\\item $\\gamma([X]) = 1$, and", "\\item $\\int_X \\gamma([X]) = \\deg(X \\to \\Spec(k))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We can write $X = \\Spec(k')$ where $k'$ is a finite separable", "algebra over $k$. Observe that $\\deg(X \\to \\Spec(k)) = [k' : k]$.", "Choose a finite Galois extension $k''/k$ containing each of the", "factors of $k'$. (Recall that a finite separable $k$-algebra is", "a product of finite separable field extension of $k$.)", "Set $\\Sigma = \\Hom_k(k', k'')$. Then we get", "$$", "k' \\otimes_k k'' = \\prod\\nolimits_{\\sigma \\in \\Sigma} k''", "$$", "Setting $Y = \\Spec(k'')$ axioms (B)(a) and Lemma \\ref{lemma-weil-additive} give", "$$", "H^*(X) \\otimes_F H^*(Y) =", "\\prod\\nolimits_{\\sigma \\in \\Sigma} H^*(Y)", "$$", "as graded commutative $F$-algebras. By Lemma \\ref{lemma-unit} the", "$F$-algebra $H^*(Y)$ is nonzero. Comparing dimensions on either side", "of the displayed equation we conclude that $H^*(X)$ sits only in degree $0$", "and $\\dim_F H^0(X) = [k' : k]$. Applying this to $Y$ we get", "$H^*(Y) = H^0(Y)$. Since", "$$", "H^0(X) \\otimes_F H^0(Y) = H^0(Y) \\times \\ldots \\times H^0(Y)", "$$", "as $F$-algebras, it follows that $H^0(X)$ is a separable $F$-algebra", "because we may check this after the faithfully flat base change", "$F \\to H^0(Y)$.", "\\medskip\\noindent", "The displayed isomorphism above is given by the map", "$$", "H^0(X) \\otimes_F H^0(Y) \\longrightarrow", "\\prod\\nolimits_{\\sigma \\in \\Sigma} H^0(Y),\\quad", "a \\otimes b \\longmapsto \\prod\\nolimits_\\sigma \\Spec(\\sigma)^*a \\cup b", "$$", "Via this isomorphism we have $\\int_{X \\times Y} = \\sum_\\sigma \\int_Y$ by", "Lemma \\ref{lemma-trace-disjoint-union}. Thus", "$$", "\\int_X a = \\text{pr}_{1, *}(a \\otimes 1) = \\sum \\Spec(\\sigma)^*a", "$$", "in $H^0(Y)$; the first equality by Lemma \\ref{lemma-pr2star}", "and the second by the observation we just made. Choose an", "algebraic closure $\\overline{F}$ and", "a $F$-algebra map $\\tau : H^0(Y) \\to \\overline{F}$.", "The isomorphism above base changes to the isomorphism", "$$", "H^0(X) \\otimes_F \\overline{F} \\longrightarrow", "\\prod\\nolimits_{\\sigma \\in \\Sigma} \\overline{F},\\quad", "a \\otimes b \\longmapsto \\prod\\nolimits_\\sigma \\tau(\\Spec(\\sigma)^*a) b", "$$", "It follows that $a \\mapsto \\tau(\\Spec(\\sigma)^*a)$ is a full set", "of embeddings of $H^0(X)$ into $\\overline{F}$. Applying $\\tau$", "to the formula for $\\int_X a$ obtained above we conclude", "that $\\int_X$ is the trace map.", "By Lemma \\ref{lemma-unit} we have $\\gamma([X]) = 1$.", "Finally, we have $\\int_X \\gamma([X]) = \\deg(X \\to \\Spec(k))$", "because $\\gamma([X]) = 1$ and the trace of $1$ is equal to $[k' : k]$" ], "refs": [ "weil-lemma-weil-additive", "weil-lemma-unit", "weil-lemma-trace-disjoint-union", "weil-lemma-pr2star", "weil-lemma-unit" ], "ref_ids": [ 5075, 5071, 5078, 5069, 5071 ] } ], "ref_ids": [] }, { "id": 5080, "type": "theorem", "label": "weil-lemma-degrees-cycles", "categories": [ "weil" ], "title": "weil-lemma-degrees-cycles", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).", "Let $X$ be a nonempty smooth projective scheme", "equidimensional of dimension $d$ over $k$. The diagram", "$$", "\\xymatrix{", "\\CH^d(X) \\ar[r]_-\\gamma \\ar@{=}[d] &", "H^{2d}(X)(d) \\ar[d]^{\\int_X} \\\\", "\\CH_0(X) \\ar[r]^\\deg & F", "}", "$$", "commutes where $\\deg : \\CH_0(X) \\to \\mathbf{Z}$ is the degree of", "zero cycles discussed in Chow Homology, Section", "\\ref{chow-section-degree-zero-cycles}." ], "refs": [], "proofs": [ { "contents": [ "Let $x$ be a closed point of $X$ whose residue field is separable", "over $k$. View $x$ as a scheme and denote $i : x \\to X$ the inclusion morphism.", "To avoid confusion denote $\\gamma' : \\CH_0(x) \\to H^0(x)$ the cycle class map", "for $x$. Then we have", "$$", "\\int_X \\gamma([x]) = \\int_X \\gamma(i_*[x]) =", "\\int_X i_*\\gamma'([x]) = \\int_x \\gamma'([x]) = \\deg(x \\to \\Spec(k))", "$$", "The second equality is axiom (C)(b) and the third equality is", "the definition of $i_*$ on cohomology. The final equality is", "Lemma \\ref{lemma-dim-0}. This proves the lemma", "because $\\CH_0(X)$ is generated by the classes of points $x$ as above", "by Lemma \\ref{lemma-generated-by-separable}." ], "refs": [ "weil-lemma-dim-0", "weil-lemma-generated-by-separable" ], "ref_ids": [ 5079, 5063 ] } ], "ref_ids": [] }, { "id": 5081, "type": "theorem", "label": "weil-lemma-square-diagonal", "categories": [ "weil" ], "title": "weil-lemma-square-diagonal", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).", "Let $X$ be a nonempty smooth projective scheme over $k$ which is", "equidimensional of dimension $d$. We have", "$$", "\\sum\\nolimits_i (-1)^i\\dim_F H^i(X) =", "\\deg(\\Delta \\cdot \\Delta) = \\deg(c_d(\\mathcal{T}_{X/k}))", "$$" ], "refs": [], "proofs": [ { "contents": [ "Equality on the right. We have", "$[\\Delta] \\cdot [\\Delta] = \\Delta_*(\\Delta^![\\Delta])$", "(Chow Homology, Lemma \\ref{chow-lemma-intersect-regularly-embedded}).", "Since $\\Delta_*$ preserves degrees of $0$-cycles it suffices to compute", "the degree of $\\Delta^![\\Delta]$. The class $\\Delta^![\\Delta]$ is given", "by capping $[\\Delta]$ with", "the top Chern class of the normal sheaf of $\\Delta \\subset X \\times X$", "(Chow Homology, Lemma \\ref{chow-lemma-gysin-fundamental}).", "Since the conormal sheaf of $\\Delta$ is $\\Omega_{X/k}$", "(Morphisms, Lemma \\ref{morphisms-lemma-differentials-diagonal})", "we see that the normal sheaf is equal to the tangent sheaf", "$\\mathcal{T}_{X/k} = \\SheafHom_{\\mathcal{O}_X}(\\Omega_{X/k}, \\mathcal{O}_X)$", "as desired.", "\\medskip\\noindent", "Equality on the left. By Lemma \\ref{lemma-degrees-cycles} we have", "\\begin{align*}", "\\deg([\\Delta] \\cdot [\\Delta])", "& =", "\\int_{X \\times X} \\gamma([\\Delta]) \\cup \\gamma([\\Delta]) \\\\", "& =", "\\int_{X \\times X} \\Delta_*1 \\cup \\gamma([\\Delta]) \\\\", "& =", "\\int_{X \\times X} \\Delta_*(\\Delta^*\\gamma([\\Delta])) \\\\", "& =", "\\int_X \\Delta^*\\gamma([\\Delta])", "\\end{align*}", "We have used Lemmas \\ref{lemma-push-unit} and", "\\ref{lemma-pushforward}.", "Write $\\gamma([\\Delta]) = \\sum e_{i, j} \\otimes e'_{2d - i , j}$", "as in Lemma \\ref{lemma-class-diagonal}.", "Recalling that $\\Delta^*$ is given by cup product", "(Remark \\ref{remark-replace-cup-product}) we obtain", "$$", "\\int_X \\sum\\nolimits_{i, j} e_{i, j} \\cup e'_{2d - i, j} =", "\\sum\\nolimits_{i, j} \\int_X e_{i, j} \\cup e'_{2d - i, j} =", "\\sum\\nolimits_{i, j} (-1)^i = \\sum (-1)^i\\beta_i", "$$", "as desired." ], "refs": [ "chow-lemma-intersect-regularly-embedded", "chow-lemma-gysin-fundamental", "morphisms-lemma-differentials-diagonal", "weil-lemma-degrees-cycles", "weil-lemma-push-unit", "weil-lemma-pushforward", "weil-lemma-class-diagonal", "weil-remark-replace-cup-product" ], "ref_ids": [ 5850, 5814, 5311, 5080, 5072, 5068, 5073, 5120 ] } ], "ref_ids": [] }, { "id": 5082, "type": "theorem", "label": "weil-lemma-algebra-relations", "categories": [ "weil" ], "title": "weil-lemma-algebra-relations", "contents": [ "Let $F$ be a field of characteristic $0$.", "Let $F'$ and $F_i$, $i = 1, \\ldots, r$", "be finite separable $F$-algebras. Let $A$ be a finite $F$-algebra.", "Let $\\sigma, \\sigma' : A \\to F'$ and $\\sigma_i : A \\to F_i$", "be $F$-algebra maps. Assume $\\sigma$ and $\\sigma'$ surjective.", "If there is a relation", "$$", "\\text{Tr}_{F'/F} \\circ \\sigma - \\text{Tr}_{F'/F} \\circ \\sigma' =", "n(\\sum m_i \\text{Tr}_{F_i/F} \\circ \\sigma_i)", "$$", "where $n > 1$ and $m_i$ are integers, then $\\sigma = \\sigma'$." ], "refs": [], "proofs": [ { "contents": [ "We may write $A = \\prod A_j$ as a finite product of", "local Artinian $F$-algebras $(A_j, \\mathfrak m_j, \\kappa_j)$, see", "Algebra, Lemma \\ref{algebra-lemma-finite-dimensional-algebra} and", "Proposition \\ref{algebra-proposition-dimension-zero-ring}.", "Denote $A' = \\prod \\kappa_j$ where the product is over those $j$", "such that $\\kappa_j/k$ is separable. Then each of the maps", "$\\sigma, \\sigma', \\sigma_i$ factors over the map", "$A \\to A'$. After replacing $A$ by $A'$ we may assume", "$A$ is a finite separable $F$-algebra.", "\\medskip\\noindent", "Choose an algebraic closure $\\overline{F}$. Set", "$\\overline{A} = A \\otimes_F \\overline{F}$,", "$\\overline{F}' = F' \\otimes_F \\overline{F}$, and", "$\\overline{F}_i = F_i \\otimes_F \\overline{F}$.", "We can base change $\\sigma$, $\\sigma'$, $\\sigma_i$", "to get $\\overline{F}$ algebra maps $\\overline{A} \\to \\overline{F}'$", "and $\\overline{A} \\to \\overline{F}_i$. Moreover", "$\\text{Tr}_{\\overline{F}'/\\overline{F}}$ is the base", "change of $\\text{Tr}_{F'/F}$ and similarly for", "$\\text{Tr}_{F_i/F}$. Thus we may replace", "$F$ by $\\overline{F}$ and we reduce to the case discussed in", "the next paragraph.", "\\medskip\\noindent", "Assume $F$ is algebraically closed and $A$ a finite separable $F$-algebra.", "Then each of $A$, $F'$, $F_i$ is a product of copies of $F$.", "Let us say an element $e$ of a product", "$F \\times \\ldots \\times F$ of copies of $F$ is a minimal idempotent", "if it generates one of the factors, i.e., if", "$e = (0, \\ldots, 0, 1, 0, \\ldots, 0)$. Let $e \\in A$ be a minimal idempotent.", "Since $\\sigma$ and $\\sigma'$ ", "are surjective, we see that $\\sigma(e)$ and $\\sigma'(e)$ are minimal", "idempotents or zero. If $\\sigma \\not = \\sigma'$, then we can choose", "a minimal idempotent $e \\in A$ such that $\\sigma(e) = 0$ and", "$\\sigma'(e) \\not = 0$ or vice versa. Then", "$\\text{Tr}_{F'/F}(\\sigma(e)) = 0$ and", "$\\text{Tr}_{F'/F}(\\sigma'(e)) = 1$ or vice versa.", "On the other hand, $\\sigma_i(e)$ is an idempotent", "and hence $\\text{Tr}_{F_i/F}(\\sigma_i(e)) = r_i$ is an integer.", "We conclude that", "$$", "-1 = \\sum n m_i r_i = n (\\sum m_i r_i)", "\\quad\\text{or}\\quad", "1 = \\sum n m_i r_i = n (\\sum m_i r_i)", "$$", "which is impossible." ], "refs": [ "algebra-lemma-finite-dimensional-algebra", "algebra-proposition-dimension-zero-ring" ], "ref_ids": [ 642, 1410 ] } ], "ref_ids": [] }, { "id": 5083, "type": "theorem", "label": "weil-lemma-relations-classes-points", "categories": [ "weil" ], "title": "weil-lemma-relations-classes-points", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).", "Let $k'/k$ be a finite separable extension.", "Let $X$ be a smooth projective scheme over $k'$.", "Let $x, x' \\in X$ be $k'$-rational points.", "If $\\gamma(x) \\not = \\gamma(x')$, then", "$[x] - [x']$ is not divisible by any integer $n > 1$ in $\\CH_0(X)$." ], "refs": [], "proofs": [ { "contents": [ "If $x$ and $x'$ lie on distinct irreducible components of $X$, then", "the result is obvious. Thus we may $X$ irreducible of dimension $d$.", "Say $[x] - [x']$ is divisible by $n > 1$ in $\\CH_0(X)$.", "We may write $[x] - [x'] = n(\\sum m_i [x_i])$ in $\\CH_0(X)$", "for some $x_i \\in X$ closed points", "whose residue fields are separable over $k$ by", "Lemma \\ref{lemma-generated-by-separable}.", "Then", "$$", "\\gamma([x]) - \\gamma([x']) = n (\\sum m_i \\gamma([x_i]))", "$$", "in $H^{2d}(X)(d)$. Denote $i^*, (i')^*, i_i^*$ the pullback maps", "$H^0(X) \\to H^0(x)$, $H^0(X) \\to H^0(x')$, $H^0(X) \\to H^0(x_i)$.", "Recall that $H^0(x)$ is a finite separable $F$-algebra", "and that $\\int_x : H^0(x) \\to F$ is the trace map", "(Lemma \\ref{lemma-dim-0}) which we will denote $\\text{Tr}_x$.", "Similarly for $x'$ and $x_i$. Then by Poincar\\'e duality in the form of", "axiom (A)(b) the equation above is dual to", "$$", "\\text{Tr}_x \\circ i^* - \\text{Tr}_{x'} \\circ (i')^* =", "n(\\sum m_i \\text{Tr}_{x_i} \\circ i_i^*)", "$$", "which takes place in $\\Hom_F(H^0(X), F)$. Finally, observe that", "$i^*$ and $(i')^*$ are surjective as $x$ and $x'$ are $k'$-rational", "points and hence the compositions $H^0(\\Spec(k')) \\to H^0(X) \\to H^0(x)$", "and $H^0(\\Spec(k')) \\to H^0(X) \\to H^0(x')$ are isomorphisms.", "By Lemma \\ref{lemma-algebra-relations} we conclude that $i^* = (i')^*$", "which contradicts the assumption that $\\gamma([x]) \\not = \\gamma([x'])$." ], "refs": [ "weil-lemma-generated-by-separable", "weil-lemma-dim-0", "weil-lemma-algebra-relations" ], "ref_ids": [ 5063, 5079, 5082 ] } ], "ref_ids": [] }, { "id": 5084, "type": "theorem", "label": "weil-lemma-classes-points", "categories": [ "weil" ], "title": "weil-lemma-classes-points", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).", "Let $k'/k$ be a finite separable extension. Let $X$ be a geometrically", "irreducible smooth projective scheme over $k'$ of dimension $d$.", "Then $\\gamma : \\CH_0(X) \\to H^{2d}(X)(d)$ factors through", "$\\deg : \\CH_0(X) \\to \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-generated-by-separable} it suffices to show: given", "closed points $x, x' \\in X$ whose residue fields are separable over $k$", "we have $\\deg(x') \\gamma([x]) = \\deg(x) \\gamma([x'])$.", "\\medskip\\noindent", "We first reduce to the case of $k'$-rational points. Let $k''/k'$ be a", "Galois extension such that $\\kappa(x)$ and $\\kappa(x')$ embed into $k''$", "over $k$. Set $Y = X \\times_{\\Spec(k')} \\Spec(k'')$ and denote $p : Y \\to X$", "the projection. By our choice of $k''/k'$ there exists a", "$k''$-rational point $y$, resp.\\ $y'$ on $Y$ mapping to $x$, resp.\\ $x'$.", "Then $p_*[y] = [k'' : \\kappa(x)][x]$ and", "$p_*[y'] = [k'' : \\kappa(x')][x']$ in $\\CH_0(X)$.", "By compatibility with pushforwards given in axiom (C)(b)", "it suffices to prove $\\gamma([y]) = \\gamma([y'])$ in $\\CH^{2d}(Y)(d)$.", "This reduces us to the discussion in the next paragraph.", "\\medskip\\noindent", "Assume $x$ and $x'$ are $k'$-rational points. By", "Lemma \\ref{lemma-divide-difference-points} there", "exists a finite separable extension $k''/k'$ of fields", "such that the pullback $[y] - [y']$", "of the difference $[x] - [x']$ becomes divisible", "by an integer $n > 1$ on $Y = X \\times_{\\Spec(k')} \\Spec(k'')$.", "(Note that $y, y' \\in Y$ are $k''$-rational points.)", "By Lemma \\ref{lemma-relations-classes-points} we have", "$\\gamma([y]) = \\gamma([y'])$ in $H^{2d}(Y)(d)$.", "By compatibility with pushforward in axiom (C)(b)", "we conclude the same for $x$ and $x'$." ], "refs": [ "weil-lemma-generated-by-separable", "weil-lemma-divide-difference-points", "weil-lemma-relations-classes-points" ], "ref_ids": [ 5063, 5065, 5083 ] } ], "ref_ids": [] }, { "id": 5085, "type": "theorem", "label": "weil-lemma-injective", "categories": [ "weil" ], "title": "weil-lemma-injective", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let", "$f : X \\to Y$ be a dominant morphism of irreducible smooth projective schemes", "over $k$. Then $H^*(Y) \\to H^*(X)$ is injective." ], "refs": [], "proofs": [ { "contents": [ "There exists an integral closed subscheme $Z \\subset X$ of the same", "dimension as $Y$ mapping onto $Y$. Thus $f_*[Z] = m[Y]$ for some $m > 0$.", "Then $f_* \\gamma([Z]) = m \\gamma([Y]) = m$ in $H^*(Y)$ because of", "Lemma \\ref{lemma-unit}. Hence by the projection formula", "(Lemma \\ref{lemma-pushforward})", "we have $f_*(f^*a \\cup \\gamma([Z])) = m a$ and we conclude." ], "refs": [ "weil-lemma-unit", "weil-lemma-pushforward" ], "ref_ids": [ 5071, 5068 ] } ], "ref_ids": [] }, { "id": 5086, "type": "theorem", "label": "weil-lemma-otimes", "categories": [ "weil" ], "title": "weil-lemma-otimes", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let", "$k''/k'/k$ be finite separable algebras and let $X$ be a", "smooth projective scheme over $k'$. Then", "$$", "H^*(X) \\otimes_{H^0(\\Spec(k'))} H^0(\\Spec(k'')) =", "H^*(X \\times_{\\Spec(k')} \\Spec(k''))", "$$" ], "refs": [], "proofs": [ { "contents": [ "We will use the results of Lemma \\ref{lemma-dim-0} without further mention.", "Write", "$$", "k' \\otimes_k k'' = k'' \\times l", "$$", "for some finite separable $k'$-algebra $l$. Write", "$F' = H^0(\\Spec(k'))$, $F'' = H^0(\\Spec(k''))$, and $G = H^0(\\Spec(l))$.", "Since $\\Spec(k') \\times \\Spec(k'') = \\Spec(k'') \\amalg \\Spec(l)$ we", "deduce from axiom (B)(a) and Lemma \\ref{lemma-weil-additive}", "that we have", "$$", "F' \\otimes_F F'' = F'' \\times G", "$$", "The map from left to right identifies $F''$ with $F' \\otimes_{F'} F''$.", "By the same token we have", "$$", "H^*(X) \\otimes_F F'' = H^*(X \\times_{\\Spec(k')} \\Spec(k''))", "\\times H^*(X \\times_{\\Spec(k')} \\Spec(l))", "$$", "as modules over $F' \\otimes_F F'' = F'' \\times G$. This proves the lemma." ], "refs": [ "weil-lemma-dim-0", "weil-lemma-weil-additive" ], "ref_ids": [ 5079, 5075 ] } ], "ref_ids": [] }, { "id": 5087, "type": "theorem", "label": "weil-lemma-H-0-separable", "categories": [ "weil" ], "title": "weil-lemma-H-0-separable", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).", "Let $X$ be a smooth projective scheme over $k$.", "Set $k' = \\Gamma(X, \\mathcal{O}_X)$. The following are equivalent", "\\begin{enumerate}", "\\item there exist finitely many closed points $x_1, \\ldots, x_r \\in X$", "whose residue fields are separable over $k$ such that", "$H^0(X) \\to H^0(x_1) \\oplus \\ldots \\oplus H^0(x_r)$ is injective,", "\\item the map $H^0(\\Spec(k')) \\to H^0(X)$ is an isomorphism.", "\\end{enumerate}", "If $X$ is equidimensional of dimension $d$, these are also equivalent to", "\\begin{enumerate}", "\\item[(3)] the classes of closed points generate $H^{2d}(X)(d)$", "as a module over $H^0(X)$.", "\\end{enumerate}", "If this is true, then $H^0(X)$ is a finite separable algebra over $F$." ], "refs": [], "proofs": [ { "contents": [ "We observe that the statement makes sense because $k'$ is a finite separable", "algebra over $k$ (Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections})", "and hence $\\Spec(k')$ is smooth and projective over $k$.", "The compatibility of $H^*$ with direct sums", "(Lemmas \\ref{lemma-weil-additive} and \\ref{lemma-trace-disjoint-union})", "shows that it suffices to prove the lemma when $X$ is connected.", "Hence we may assume $X$ is irreducible and we have to show the", "equivalence of (1), (2), and (3). Set $d = \\dim(X)$.", "This implies that $k'$ is a field finite separable", "over $k$ and that $X$ is geometrically irreducible over $k'$, see", "Varieties, Lemmas", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections} and", "\\ref{varieties-lemma-baby-stein}.", "\\medskip\\noindent", "By Lemma \\ref{lemma-generated-by-separable} we see that the closed", "points in (3) may be assumed to have separable residue fields over $k$.", "By axioms (A)(a) and (A)(b) we see that conditions (1) and (3) are equivalent.", "\\medskip\\noindent", "If (2) holds, then pick any closed point $x \\in X$ whose residue field", "is finite separable over $k'$. Then", "$H^0(\\Spec(k')) = H^0(X) \\to H^0(x)$ is injective for example by", "Lemma \\ref{lemma-injective}.", "\\medskip\\noindent", "Assume the equivalent conditions (1) and (3) hold. Choose", "$x_1, \\ldots, x_r \\in X$ as in (1). Choose a finite separable", "extension $k''/k'$. By Lemma \\ref{lemma-otimes} we have", "$$", "H^0(X) \\otimes_{H^0(\\Spec(k'))} H^0(\\Spec(k'')) =", "H^0(X \\times_{\\Spec(k')} \\Spec(k''))", "$$", "Thus in order to show that", "$H^0(\\Spec(k')) \\to H^0(X)$ is an isomorphism", "we may replace $k'$ by $k''$. Thus we may assume $x_1, \\ldots, x_r$", "are $k'$-rational points (this replaces each $x_i$ with multiple", "points, so $r$ is increased in this step). By Lemma \\ref{lemma-classes-points}", "$\\gamma(x_1) = \\gamma(x_2) = \\ldots = \\gamma(x_r)$.", "By axiom (A)(b) all the maps $H^0(X) \\to H^0(x_i)$", "are the same. This means (2) holds.", "\\medskip\\noindent", "Finally, Lemma \\ref{lemma-dim-0} implies", "$H^0(X)$ is a separable $F$-algebra if (1) holds." ], "refs": [ "varieties-lemma-proper-geometrically-reduced-global-sections", "weil-lemma-weil-additive", "weil-lemma-trace-disjoint-union", "varieties-lemma-proper-geometrically-reduced-global-sections", "varieties-lemma-baby-stein", "weil-lemma-generated-by-separable", "weil-lemma-injective", "weil-lemma-otimes", "weil-lemma-classes-points", "weil-lemma-dim-0" ], "ref_ids": [ 10948, 5075, 5078, 10948, 10949, 5063, 5085, 5086, 5084, 5079 ] } ], "ref_ids": [] }, { "id": 5088, "type": "theorem", "label": "weil-lemma-negative-cohomology", "categories": [ "weil" ], "title": "weil-lemma-negative-cohomology", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).", "If there exists a smooth projective scheme $Y$ over $k$ such that", "$H^i(Y)$ is nonzero for some $i < 0$, then there exists an equidimensional", "smooth projective scheme $X$ over $k$ such that the equivalent conditions", "of Lemma \\ref{lemma-H-0-separable} fail for $X$." ], "refs": [ "weil-lemma-H-0-separable" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-weil-additive} we may assume $Y$ is irreducible", "and a fortiori equidimensional. If $i$ is odd, then after replacing", "$Y$ by $Y \\times Y$ we find an example where $Y$ is equidimensional", "and $i = -2l$ for some $l > 0$. Set $X = Y \\times (\\mathbf{P}^1_k)^l$.", "Using axiom (B)(a) we obtain", "$$", "H^0(X) \\supset H^0(Y) \\oplus", "H^i(Y) \\otimes_F H^2(\\mathbf{P}^1_k)^{\\otimes_F l}", "$$", "with both summands nonzero. Thus it is clear that $H^0(X)$ cannot be", "isomorphic to $H^0$ of the spectrum of", "$\\Gamma(X, \\mathcal{O}_X) = \\Gamma(Y, \\mathcal{O}_Y)$", "as this falls into the first summand." ], "refs": [ "weil-lemma-weil-additive" ], "ref_ids": [ 5075 ] } ], "ref_ids": [ 5087 ] }, { "id": 5089, "type": "theorem", "label": "weil-lemma-splitting-principle", "categories": [ "weil" ], "title": "weil-lemma-splitting-principle", "contents": [ "In the situation above. Let $X \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{E}_i$", "be a finite collection of locally free $\\mathcal{O}_X$-modules of rank $r_i$.", "There exists a morphism $p : P \\to X$ in $\\mathcal{C}$ such that", "\\begin{enumerate}", "\\item $p^* : A(X) \\to A(P)$ is injective,", "\\item each $p^*\\mathcal{E}_i$ has a filtration whose successive quotients", "$\\mathcal{L}_{i, 1}, \\ldots, \\mathcal{L}_{i, r_i}$", "are invertible $\\mathcal{O}_P$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may assume $r_i \\geq 1$ for all $i$. We will prove the lemma by induction", "on $\\sum (r_i - 1)$. If this integer is $0$, then $\\mathcal{E}_i$", "is invertible for all $i$ and we conclude by taking $\\pi = \\text{id}_X$.", "If not, then we can pick an $i$ such that $r_i > 1$ and consider the", "projective bundle $p : P \\to X$ associated to $\\mathcal{E}_i$.", "We have a short exact sequence", "$$", "0 \\to \\mathcal{F} \\to p^*\\mathcal{E}_i \\to \\mathcal{O}_P(1) \\to 0", "$$", "of finite locally free $\\mathcal{O}_P$-modules of ranks $r_i - 1$,", "$r_i$, and $1$. Observe that $p^* : A(X) \\to A(P)$ is injective", "by assumption. By the induction hypothesis applied to the finite locally free", "$\\mathcal{O}_P$-modules $\\mathcal{F}$ and $p^*\\mathcal{E}_{i'}$", "for $i' \\not = i$, we find a morphism $p' : P' \\to P$ with", "properties stated as in the lemma. Then the composition", "$p \\circ p' : P' \\to X$ does the job." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5090, "type": "theorem", "label": "weil-lemma-chern-classes-E-tensor-L", "categories": [ "weil" ], "title": "weil-lemma-chern-classes-E-tensor-L", "contents": [ "Let $X \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{E}$ be a finite locally free", "$\\mathcal{O}_X$-module. Let $\\mathcal{L}$ be an invertible", "$\\mathcal{O}_X$-module. Then", "$$", "c^A_i({\\mathcal E} \\otimes {\\mathcal L})", "=", "\\sum\\nolimits_{j = 0}^i", "\\binom{r - i + j}{j} c^A_{i - j}({\\mathcal E}) \\cup c^A_1({\\mathcal L})^j", "$$" ], "refs": [], "proofs": [ { "contents": [ "By the construction of $c^A_i$ we may assume $\\mathcal{E}$ has", "constant rank $r$. Let $p : P \\to X$ and $p' : P' \\to X$ be the", "projective bundle associated to $\\mathcal{E}$ and", "$\\mathcal{E} \\otimes \\mathcal{L}$.", "Then there is an isomorphism $g : P \\to P'$ such that", "$g^*\\mathcal{O}_{P'}(1) = \\mathcal{O}_P(1) \\otimes p^*\\mathcal{L}$.", "See Constructions, Lemma \\ref{constructions-lemma-twisting-and-proj}.", "Thus", "$$", "g^*c_1^A(\\mathcal{O}_{P'}(1)) =", "c_1^A(\\mathcal{O}_P(1)) + p^*c_1^A(\\mathcal{L})", "$$", "The desired equality follows formally from this and the definition", "of Chern classes using equation (\\ref{equation-chern-classes})." ], "refs": [ "constructions-lemma-twisting-and-proj" ], "ref_ids": [ 12650 ] } ], "ref_ids": [] }, { "id": 5091, "type": "theorem", "label": "weil-lemma-adams-and-chern", "categories": [ "weil" ], "title": "weil-lemma-adams-and-chern", "contents": [ "In the situation above let $X \\in \\Ob(\\mathcal{C})$.", "If $\\psi^2$ is as in", "Chow Homology, Lemma \\ref{chow-lemma-second-adams-operator}", "and $c^A$ and $ch^A$ are as in", "Propositions \\ref{proposition-chern-class} and", "\\ref{proposition-chern-character}", "then we have $c^A_i(\\psi^2(\\alpha)) = 2^i c^A_i(\\alpha)$ and", "$ch^A_i(\\psi^2(\\alpha)) = 2^i ch^A_i(\\alpha)$", "for all $\\alpha \\in K_0(\\textit{Vect}(X))$." ], "refs": [ "chow-lemma-second-adams-operator", "weil-proposition-chern-class", "weil-proposition-chern-character" ], "proofs": [ { "contents": [ "Observe that the map $\\prod_{i \\geq 0} A^i(X) \\to \\prod_{i \\geq 0} A^i(X)$", "multiplying by $2^i$ on $A^i(X)$ is a ring map. Hence, since $\\psi^2$", "is also a ring map, it suffices to prove the formulas for additive generators", "of $K_0(\\textit{Vect}(X))$. Thus we may assume $\\alpha = [\\mathcal{E}]$", "for some finite locally free $\\mathcal{O}_X$-module $\\mathcal{E}$.", "By construction of the Chern classes of $\\mathcal{E}$ we immediately", "reduce to the case where $\\mathcal{E}$ has constant rank $r$.", "In this case, we can choose a projective smooth morphism $p : P \\to X$", "such that restriction $A^*(X) \\to A^*(P)$ is injective", "and such that $p^*\\mathcal{E}$ has a finite filtration whose", "graded parts are invertible $\\mathcal{O}_P$-modules $\\mathcal{L}_j$, see", "Lemma \\ref{lemma-splitting-principle}. Then", "$[p^*\\mathcal{E}] = \\sum [\\mathcal{L}_j]$ and hence", "$\\psi^2([p^*\\mathcal{E}]) = \\sum [\\mathcal{L}_j^{\\otimes 2}]$", "by definition of $\\psi^2$. Setting $x_j = c^A_1(\\mathcal{L}_j)$", "we have", "$$", "c^A(\\alpha) = \\prod (1 + x_j)", "\\quad\\text{and}\\quad", "c^A(\\psi^2(\\alpha)) = \\prod (1 + 2 x_j)", "$$", "in $\\prod A^i(P)$ and we have", "$$", "ch^A(\\alpha) = \\sum \\exp(x_j)", "\\quad\\text{and}\\quad", "ch^A(\\psi^2(\\alpha)) = \\sum \\exp(2 x_j)", "$$", "in $\\prod A^i(P)$. From these formulas the desired result follows." ], "refs": [ "weil-lemma-splitting-principle" ], "ref_ids": [ 5089 ] } ], "ref_ids": [ 5824, 5111, 5112 ] }, { "id": 5092, "type": "theorem", "label": "weil-lemma-lambda-operations", "categories": [ "weil" ], "title": "weil-lemma-lambda-operations", "contents": [ "Let $X$ be a scheme. There are maps", "$$", "\\lambda^r : K_0(\\textit{Vect}(X)) \\longrightarrow K_0(\\textit{Vect}(X))", "$$", "which sends $[\\mathcal{E}]$ to $[\\wedge^r(\\mathcal{E})]$", "when $\\mathcal{E}$ is a finite locally free $\\mathcal{O}_X$-module", "and which are compatible with pullbacks." ], "refs": [], "proofs": [ { "contents": [ "Consider the ring $R = K_0(\\textit{Vect}(X))[[t]]$ where $t$ is a", "variable. For a finite locally free $\\mathcal{O}_X$-module", "$\\mathcal{E}$ we set", "$$", "c(\\mathcal{E}) = \\sum\\nolimits_{i = 0}^\\infty [\\wedge^i(\\mathcal{E})] t^i", "$$", "in $R$. We claim that given a short exact sequence", "$$", "0 \\to \\mathcal{E}' \\to \\mathcal{E} \\to \\mathcal{E}'' \\to 0", "$$", "of finite locally free $\\mathcal{O}_X$-modules", "we have $c(\\mathcal{E}) = c(\\mathcal{E}') c(\\mathcal{E}'')$.", "The claim implies that $c$ extends to a map", "$$", "c : K_0(\\textit{Vect}(X)) \\longrightarrow R", "$$", "which converts addition in $K_0(\\textit{Vect}(X))$ to multiplication in $R$.", "Writing $c(\\alpha) = \\sum \\lambda^i(\\alpha) t^i$ we obtain the desired", "operators $\\lambda^i$.", "\\medskip\\noindent", "To see the claim, we consider the short exact sequence as a", "filtration on $\\mathcal{E}$ with $2$ steps. We obtain an induced", "filtration on $\\wedge^r(\\mathcal{E})$ with $r + 1$ steps and", "subquotients", "$$", "\\wedge^r(\\mathcal{E}'),", "\\wedge^{r - 1}(\\mathcal{E}') \\otimes \\mathcal{E}'',", "\\wedge^{r - 2}(\\mathcal{E}') \\otimes \\wedge^2(\\mathcal{E}''), \\ldots", "\\wedge^r(\\mathcal{E}'')", "$$", "Thus we see that $[\\wedge^r(\\mathcal{E})]$ is equal to", "$$", "\\sum\\nolimits_{i = 0}^r", "[\\wedge^{r - i}(\\mathcal{E}')] [\\wedge^i(\\mathcal{E}'')]", "$$", "and the result follows easily from this and elementary algebra." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5093, "type": "theorem", "label": "weil-lemma-chern-classes", "categories": [ "weil" ], "title": "weil-lemma-chern-classes", "contents": [ "Assume given (D0), (D1), and (D2') satisfying axioms (A1), (A2), (A3), and (A4).", "There is a unique rule which assigns to every smooth projective $X$ over $k$", "a ring homomorphism", "$$", "ch^H :", "K_0(\\textit{Vec}(X))", "\\longrightarrow", "\\prod\\nolimits_{i \\geq 0} H^{2i}(X)(i)", "$$", "compatible with pullbacks such that", "$ch^H(\\mathcal{L}) = \\exp(c_1^H(\\mathcal{L}))$", "for any invertible $\\mathcal{O}_X$-module $\\mathcal{L}$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Proposition \\ref{proposition-chern-character}", "applied to the category of smooth projective schemes over $k$,", "the functor $A : X \\mapsto \\bigoplus_{i \\geq 0} H^{2i}(X)(i)$,", "and the map $c_1^H$." ], "refs": [ "weil-proposition-chern-character" ], "ref_ids": [ 5112 ] } ], "ref_ids": [] }, { "id": 5094, "type": "theorem", "label": "weil-lemma-cycle-classes", "categories": [ "weil" ], "title": "weil-lemma-cycle-classes", "contents": [ "Assume given (D0), (D1), and (D2') satisfying axioms (A1), (A2), (A3), and (A4).", "There is a unique rule which assigns to every smooth projective $X$ over $k$", "a graded ring homomorphism", "$$", "\\gamma : \\CH^*(X) \\longrightarrow \\bigoplus\\nolimits_{i \\geq 0} H^{2i}(X)(i)", "$$", "compatible with pullbacks such that $ch^H(\\alpha) = \\gamma(ch(\\alpha))$", "for $\\alpha$ in $K_0(\\textit{Vect}(X))$." ], "refs": [], "proofs": [ { "contents": [ "Recall that we have an isomorphism", "$$", "K_0(\\textit{Vect}(X)) \\otimes \\mathbf{Q}", "\\longrightarrow \\CH^*(X) \\otimes \\mathbf{Q},\\quad", "\\alpha \\longmapsto ch(\\alpha) \\cap [X]", "$$", "see Chow Homology, Lemma \\ref{chow-lemma-K-tensor-Q}. It is an isomorphism", "of rings by Chow Homology, Remark \\ref{chow-remark-chern-character-K}.", "We define $\\gamma$ by the formula $\\gamma(\\alpha) = ch^H(\\alpha')$", "where $ch^H$ is as in Lemma \\ref{lemma-chern-classes} and", "$\\alpha' \\in K_0(\\textit{Vect}(X))$ is such that", "$ch(\\alpha') \\cap [X] = \\alpha$ in $\\CH^*(X) \\otimes \\mathbf{Q}$.", "\\medskip\\noindent", "The construction $\\alpha \\mapsto \\gamma(\\alpha)$ is compatible", "with pullbacks because both $ch^H$ and taking Chern classes", "is compatible with pullbacks, see", "Lemma \\ref{lemma-chern-classes} and", "Chow Homology, Remark \\ref{chow-remark-gysin-chern-classes}.", "\\medskip\\noindent", "We still have to see that $\\gamma$ is graded.", "Let $\\psi^2 : K_0(\\textit{Vect}(X)) \\to K_0(\\textit{Vect}(X))$", "be the second Adams operator, see Chow Homology,", "Lemma \\ref{chow-lemma-second-adams-operator}.", "If $\\alpha \\in \\CH^i(X)$ and", "$\\alpha' \\in K_0(\\textit{Vect}(X)) \\otimes \\mathbf{Q}$", "is the unique element with $ch(\\alpha') \\cap [X] = \\alpha$,", "then we have seen in", "Chow Homology, Section \\ref{chow-section-intersection-regular}", "that $\\psi^2(\\alpha') = 2^i \\alpha'$.", "Hence we conclude that $ch^H(\\alpha') \\in H^{2i}(X)(i)$", "by Lemma \\ref{lemma-adams-and-chern} as desired." ], "refs": [ "chow-lemma-K-tensor-Q", "chow-remark-chern-character-K", "weil-lemma-chern-classes", "weil-lemma-chern-classes", "chow-remark-gysin-chern-classes", "chow-lemma-second-adams-operator", "weil-lemma-adams-and-chern" ], "ref_ids": [ 5828, 5957, 5093, 5093, 5963, 5824, 5091 ] } ], "ref_ids": [] }, { "id": 5095, "type": "theorem", "label": "weil-lemma-divide-pullback-good-blowing-up", "categories": [ "weil" ], "title": "weil-lemma-divide-pullback-good-blowing-up", "contents": [ "Let $b : X' \\to X$ be the blowing up of a smooth projective", "scheme over $k$ in a smooth closed subscheme $Z \\subset X$.", "Picture", "$$", "\\xymatrix{", "E \\ar[r]_j \\ar[d]_\\pi & X' \\ar[d]^b \\\\", "Z \\ar[r]^i & X", "}", "$$", "Assume there exists an element of $K_0(X)$ whose restriction to", "$Z$ is equal to the class of $\\mathcal{C}_{Z/X}$ in $K_0(Z)$.", "Assume every irreducible component of $Z$ has codimension $r$ in $X$.", "Then there exists a cycle $\\theta \\in \\CH^{r - 1}(X')$", "such that $b^![Z] = [E] \\cdot \\theta$ in $\\CH^r(X')$ and", "$\\pi_*j^!(\\theta) = [Z]$ in $\\CH^r(Z)$." ], "refs": [], "proofs": [ { "contents": [ "The scheme $X$ is smooth and projective over $k$ and hence we have", "$K_0(X) = K_0(\\textit{Vect}(X))$. See", "Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-resolution-property-ample} and", "\\ref{perfect-lemma-K-is-old-K}.", "Let $\\alpha \\in K_0(\\text{Vect}(X))$ be an element", "whose restriction to $Z$ is $[\\mathcal{F}]$.", "By Chow Homology, Lemma \\ref{chow-lemma-minus-adams-operator}", "there exists an element $\\alpha^\\vee$ which restricts to", "$\\mathcal{C}_{Z/X}^\\vee$. By the blow up formula", "(Chow Homology, Lemma \\ref{chow-lemma-blow-up-formula})", "we have", "$$", "b^![Z] = b^!i_*[Z] = j_* res(b^!)([Z]) =", "j_*(c_{r - 1}(\\mathcal{F}^\\vee) \\cap \\pi^*[Z]) =", "j_*(c_{r - 1}(\\mathcal{F}^\\vee) \\cap [E])", "$$", "where $\\mathcal{F}$ is the kernel of the surjection", "$\\pi^*\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{E/X'}$.", "Observe that $b^*\\alpha^\\vee - [\\mathcal{O}_{X'}(E)]$", "is an element of $K_0(\\text{Vect}(X'))$ which", "restricts to $[\\pi^*\\mathcal{C}_{Z/X}^\\vee] - [\\mathcal{C}_{E/X'}^\\vee] =", "[\\mathcal{F}^\\vee]$ on $E$. Since capping with Chern classes", "commutes with $j_*$ we conclude that the above is equal to", "$$", "c_{r - 1}(b^*\\alpha^\\vee - [\\mathcal{O}_{X'}(E)]) \\cap [E]", "$$", "in the chow group of $X'$. Hence we see that setting", "$$", "\\theta = c_{r - 1}(b^*\\alpha^\\vee - [\\mathcal{O}_{X'}(E)]) \\cap [X']", "$$", "we get the first relation $\\theta \\cdot [E] = b^![Z]$", "for example by Chow Homology, Lemma \\ref{chow-lemma-identify-chow-for-smooth}.", "For the second relation observe that", "$$", "j^!\\theta = j^!(c_{r - 1}(b^*\\alpha^\\vee - [\\mathcal{O}_{X'}(E)]) \\cap [X'])", "= c_{r - 1}(\\mathcal{F}^\\vee) \\cap j^![X'] =", "c_{r - 1}(\\mathcal{F}^\\vee) \\cap [E]", "$$", "in the chow groups of $E$. To prove that $\\pi_*$ of this is equal to $[Z]$ it", "suffices to prove that the degree of the codimension $r - 1$ cycle", "$(-1)^{r - 1}c_{r - 1}(\\mathcal{F}) \\cap [E]$ on the fibres of $\\pi$ is $1$.", "This is a computation we omit." ], "refs": [ "perfect-lemma-resolution-property-ample", "perfect-lemma-K-is-old-K", "chow-lemma-minus-adams-operator", "chow-lemma-blow-up-formula", "chow-lemma-identify-chow-for-smooth" ], "ref_ids": [ 7086, 7100, 5825, 5837, 5846 ] } ], "ref_ids": [] }, { "id": 5096, "type": "theorem", "label": "weil-lemma-A5-A6-imply", "categories": [ "weil" ], "title": "weil-lemma-A5-A6-imply", "contents": [ "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A4)", "and (A7). Let $X$ be a smooth projective scheme over $k$. Let $Z \\subset X$", "be a smooth closed subscheme such that every irreducible component of $Z$", "has codimension $r$ in $X$. Assume the class of", "$\\mathcal{C}_{Z/X}$ in $K_0(Z)$ is the restriction of an element of $K_0(X)$.", "If $a \\in H^*(X)$ and $a|_Z = 0$ in $H^*(Z)$, then", "$\\gamma([Z]) \\cup a = 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $b : X' \\to X$ be the blowing up. By (A7) it suffices to show", "that", "$$", "b^*(\\gamma([Z]) \\cup a) = b^*\\gamma([Z]) \\cup b^*a = 0", "$$", "By Lemma \\ref{lemma-divide-pullback-good-blowing-up} we have", "$$", "b^*\\gamma([Z]) = \\gamma(b^*[Z]) =", "\\gamma([E] \\cdot \\theta) =", "\\gamma([E]) \\cup \\gamma(\\theta)", "$$", "Hence because $b^*a$ restricts to zero on $E$ and since", "$\\gamma([E]) = c^H_1(\\mathcal{O}_{X'}(E))$ we get what we want from (A4)." ], "refs": [ "weil-lemma-divide-pullback-good-blowing-up" ], "ref_ids": [ 5095 ] } ], "ref_ids": [] }, { "id": 5097, "type": "theorem", "label": "weil-lemma-poincare-duality", "categories": [ "weil" ], "title": "weil-lemma-poincare-duality", "contents": [ "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).", "Then axiom (A) of Section \\ref{section-axioms} holds with", "$\\int_X = \\lambda$ as in axiom (A6)." ], "refs": [], "proofs": [ { "contents": [ "Let $X$ be a nonempty smooth projective scheme over $k$ which is", "equidimensional of dimension $d$. We will show that the graded $F$-vector space", "$H^*(X)(d)[2d]$ is a left dual to $H^*(X)$. This will prove what we want by", "Homology, Lemma \\ref{homology-lemma-left-dual-graded-vector-spaces}. We are", "going to use axiom (A5) which in particular says that", "$$", "H^*(X \\times X)(d) =", "\\bigoplus H^i(X) \\otimes H^j(X)(d) =", "\\bigoplus H^i(X)(d) \\otimes H^j(X)", "$$", "Define a map", "$$", "\\eta : F \\longrightarrow H^*(X \\times X)(d)", "$$", "by multiplying by $\\gamma([\\Delta]) \\in H^{2d}(X \\times X)(d)$.", "On the other hand, define a map", "$$", "\\epsilon :", "H^*(X \\times X)(d) \\longrightarrow H^*(X)(d) \\xrightarrow{\\lambda} F", "$$", "by first using pullback $\\Delta^*$ by the diagonal morphism", "$\\Delta : X \\to X \\times X$ and then using the $F$-linear map", "$\\lambda : H^{2d}(X)(d) \\to F$ of axiom (A6) precomposed", "by the projection $H^*(X)(d) \\to H^{2d}(X)(d)$.", "In order to show that $H^*(X)(d)$ is a left dual to $H^*(X)$", "we have to show that the composition of the maps", "$$", "\\eta \\otimes 1 :", "H^*(X) \\longrightarrow H^*(X \\times X \\times X)(d)", "$$", "and", "$$", "1 \\otimes \\epsilon : H^*(X \\times X \\times X)(d) \\longrightarrow H^*(X)", "$$", "is the identity. If $a \\in H^*(X)$ then we see", "that the composition maps $a$ to", "$$", "(1 \\otimes \\lambda)(\\Delta_{23}^*(q_{12}^*\\gamma([\\Delta]) \\cup q_3^*a)) =", "(1 \\otimes \\lambda)(\\gamma([\\Delta]) \\cup p_2^*a)", "$$", "where $q_i : X \\times X \\times X \\to X$ and", "$q_{ij} : X \\times X \\times X \\to X \\times X$ are the projections,", "$\\Delta_{23} : X \\times X \\to X \\times X \\times X$ is the diagonal, and", "$p_i : X \\times X \\to X$ are the projections.", "The equality holds because $\\Delta_{23}^*(q_{12}^*\\gamma([\\Delta]) =", "\\Delta_{23}^*\\gamma([\\Delta \\times X]) = \\gamma([\\Delta])$", "and because $\\Delta_{23}^* q_3^*a = p_2^*a$.", "Since $\\gamma([\\Delta]) \\cup p_1^*a = \\gamma([\\Delta]) \\cup p_2^*a$", "(see below) the above simplifies to", "$$", "(1 \\otimes \\lambda)(\\gamma([\\Delta]) \\cup p_1^*a) = a", "$$", "by our choice of $\\lambda$ as desired. The second condition", "$(\\epsilon \\otimes 1) \\circ (1 \\otimes \\eta) = \\text{id}$", "of Categories, Definition \\ref{categories-definition-dual}", "is proved in exactly the", "same manner.", "\\medskip\\noindent", "Note that $p_1^*a$ and $\\text{pr}_2^*a$ restrict to the same", "cohomology class on $\\Delta \\subset X \\times X$. Moreover we", "have $\\mathcal{C}_{\\Delta/X \\times X} = \\Omega^1_\\Delta$ which", "is the restriction of $p_1^*\\Omega^1_X$. Hence", "Lemma \\ref{lemma-A5-A6-imply} implies", "$\\gamma([\\Delta]) \\cup p_1^*a = \\gamma([\\Delta]) \\cup p_2^*a$", "and the proof is complete." ], "refs": [ "homology-lemma-left-dual-graded-vector-spaces", "categories-definition-dual", "weil-lemma-A5-A6-imply" ], "ref_ids": [ 12073, 12407, 5096 ] } ], "ref_ids": [] }, { "id": 5098, "type": "theorem", "label": "weil-lemma-trace-product", "categories": [ "weil" ], "title": "weil-lemma-trace-product", "contents": [ "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).", "Then axiom (B) of Section \\ref{section-axioms} holds." ], "refs": [], "proofs": [ { "contents": [ "Axiom (B)(a) is immediate from axiom (A5).", "Let $X$ and $Y$ be nonempty smooth projective schemes over $k$", "equidimensional of dimensions $d$ and $e$. To see that axiom (B)(b)", "holds, observe that the diagonal $\\Delta_{X \\times Y}$ of $X \\times Y$", "is the intersection product of the pullbacks of the diagonals", "$\\Delta_X$ of $X$ and $\\Delta_Y$ of $Y$ by the projections", "$p : X \\times Y \\times X \\times Y \\to X \\times X$ and", "$q : X \\times Y \\times X \\times Y \\to Y \\times Y$.", "Compatibility of $\\gamma$ with intersection products then gives", "that", "$$", "\\gamma([\\Delta_{X \\times Y}]) \\in", "H^{2d + 2e}(X \\times Y \\times X \\times Y)(d + e)", "$$", "is the cup product of the pullbacks of $\\gamma([\\Delta_X])$", "and $\\gamma([\\Delta_Y])$ by $p$ and $q$. Write", "$$", "\\gamma([\\Delta_{X \\times Y}]) = \\sum \\eta_{X \\times Y, i}", "\\text{ with }", "\\eta_{X \\times Y, i} \\in", "H^i(X \\times Y) \\otimes H^{2d + 2e - i}(X \\times Y)(d + e)", "$$", "and simiarly $\\gamma([\\Delta_X]) = \\sum \\eta_{X, i}$ and", "$\\gamma([\\Delta_Y]) = \\sum \\eta_{Y, i}$. The observation above", "implies we have", "$$", "\\eta_{X \\times Y, 0} =", "\\sum\\nolimits_{i \\in \\mathbf{Z}} p^*\\eta_{X, i} \\cup q^*\\eta_{Y, -i}", "$$", "(If our cohomology theory vanishes in negative degrees, which will", "be true in almost all cases, then only the term for $i = 0$ contributes", "and $\\eta_{X \\times Y, 0}$ lies in", "$H^0(X) \\otimes H^0(Y) \\otimes H^{2d}(X)(d) \\otimes H^{2e}(Y)(e)$ as expected,", "but we don't need this.) Since $\\lambda_X : H^{2d}(X)(d) \\to F$ and ", "$\\lambda_Y : H^{2e}(Y)(e) \\to F$ send $\\eta_{X, 0}$ and $\\eta_{Y, 0}$", "to $1$ in $H^*(X)$ and $H^*(Y)$, we see that $\\lambda_X \\otimes \\lambda_Y$", "sends $\\eta_{X \\times Y, 0}$ to $1$ in", "$H^*(X) \\otimes H^*(Y) = H^*(X \\times Y)$ and the proof is complete." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5099, "type": "theorem", "label": "weil-lemma-trace-base", "categories": [ "weil" ], "title": "weil-lemma-trace-base", "contents": [ "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).", "Then axiom (C)(d) of Section \\ref{section-axioms} holds." ], "refs": [], "proofs": [ { "contents": [ "We have $\\gamma([\\Spec(k)]) = 1 \\in H^*(\\Spec(k))$ by construction.", "Since", "$$", "H^0(\\Spec(k)) = F,\\quad", "H^0(\\Spec(k) \\times \\Spec(k)) = H^0(\\Spec(k)) \\otimes_F H^0(\\Spec(k))", "$$", "the map $\\int_{\\Spec(k)} = \\lambda$ of axiom (A6) must send $1$ to $1$", "because we have seen that", "$\\int_{\\Spec(k) \\times \\Spec(k)} = \\int_{\\Spec(k)} \\int_{\\Spec(k)}$", "in Lemma \\ref{lemma-trace-product}." ], "refs": [ "weil-lemma-trace-product" ], "ref_ids": [ 5098 ] } ], "ref_ids": [] }, { "id": 5100, "type": "theorem", "label": "weil-lemma-ok-for-projective-bundle", "categories": [ "weil" ], "title": "weil-lemma-ok-for-projective-bundle", "contents": [ "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).", "Let $p : P \\to X$ be as in axiom (A3) with $X$ nonempty equidimensional.", "Then $\\gamma$ commutes with pushforward along $p$." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove this on generators for $\\CH_*(P)$.", "Thus it suffices to prove this for a cycle class of the", "form $\\xi^i \\cdot p^*\\alpha$ where $0 \\leq i \\leq r - 1$", "and $\\alpha \\in \\CH_*(X)$. Note that $p_*(\\xi^i \\cdot p^*\\alpha) = 0$", "if $i < r - 1$ and $p_*(\\xi^{r - 1} \\cdot p^*\\alpha) = \\alpha$.", "On the other hand, we have", "$\\gamma(\\xi^i \\cdot p^*\\alpha) = c^i \\cup p^*\\gamma(\\alpha)$", "and by the projection formula (Lemma \\ref{lemma-pushforward})", "we have", "$$", "p_*\\gamma(\\xi^i \\cdot p^*\\alpha) = p_*(c^i) \\cup \\gamma(\\alpha)", "$$", "Thus it suffices to show that $p_*c^i = 0$ for $i < r - 1$ and", "$p_*c^{r - 1} = 1$. Equivalently, it suffices to prove that", "$\\lambda_P : H^{2d + 2r - 2}(P)(d + r - 1) \\to F$ defined by", "the rules", "$$", "\\lambda_P(c^i \\cup p^*(a)) =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & i < r - 1 \\\\", "\\int_X(a) & \\text{if} & i = r - 1", "\\end{matrix}", "\\right.", "$$", "satisfies the condition of axiom (A5). This follows from the", "computation of the class of the diagonal of $P$ in", "Lemma \\ref{lemma-diagonal-projective-bundle}." ], "refs": [ "weil-lemma-pushforward", "weil-lemma-diagonal-projective-bundle" ], "ref_ids": [ 5068, 5054 ] } ], "ref_ids": [] }, { "id": 5101, "type": "theorem", "label": "weil-lemma-integrate-1", "categories": [ "weil" ], "title": "weil-lemma-integrate-1", "contents": [ "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).", "If $k'/k$ is a Galois extension, then we have", "$\\int_{\\Spec(k')} 1 = [k' : k]$." ], "refs": [], "proofs": [ { "contents": [ "We observe that", "$$", "\\Spec(k') \\times \\Spec(k') =", "\\coprod\\nolimits_{\\sigma \\in \\text{Gal}(k'/k)}", "(\\Spec(\\sigma) \\times \\text{id})^{-1} \\Delta", "$$", "as cycles on $\\Spec(k') \\times \\Spec(k')$.", "Our construction of $\\gamma$ always sends $[X]$ to $1$ in $H^0(X)$. Thus", "$1 \\otimes 1 = 1 = \\sum (\\Spec(\\sigma) \\times \\text{id})^*\\gamma([\\Delta])$.", "Denote $\\lambda : H^0(\\Spec(k')) \\to F$ the map from", "axiom (A6), in other words $(\\text{id} \\otimes \\lambda)(\\gamma(\\Delta)) = 1$", "in $H^0(\\Spec(k'))$. We obtain", "\\begin{align*}", "\\lambda(1) 1", "& =", "(\\text{id} \\otimes \\lambda)(1 \\otimes 1) \\\\", "& =", "(\\text{id} \\otimes \\lambda)(", "\\sum (\\Spec(\\sigma) \\times \\text{id})^*\\gamma([\\Delta])) \\\\", "& =", "\\sum (\\Spec(\\sigma) \\times \\text{id})^*(", "(\\text{id} \\otimes \\lambda)(\\gamma([\\Delta])) \\\\", "& =", "\\sum (\\Spec(\\sigma) \\times \\text{id})^*(1) \\\\", "& =", "[k' : k]", "\\end{align*}", "Since $\\lambda$ is another name for $\\int_{\\Spec(k')}$", "(Remark \\ref{remark-trace}) the proof is complete." ], "refs": [ "weil-remark-trace" ], "ref_ids": [ 5122 ] } ], "ref_ids": [] }, { "id": 5102, "type": "theorem", "label": "weil-lemma-enough", "categories": [ "weil" ], "title": "weil-lemma-enough", "contents": [ "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).", "In order to show that $\\gamma$ commutes with pushforward it suffices", "to show that $i_*(1) = \\gamma([Z])$ if $i : Z \\to X$ is a closed", "immersion of nonempty smooth projective equidimensional schemes over $k$." ], "refs": [], "proofs": [ { "contents": [ "We will use without further mention that we've constructed our", "cycle class map $\\gamma$ in Lemma \\ref{lemma-cycle-classes}", "compatible with intersection products and pullbacks and that", "we've already shown axioms", "(A), (B), (C)(a), (C)(c), and (C)(d) of Section \\ref{section-axioms}, see", "Lemma \\ref{lemma-poincare-duality},", "Remark \\ref{remark-trace}, and", "Lemmas \\ref{lemma-trace-product} and \\ref{lemma-trace-base}.", "In particular, we may use (for example) Lemma \\ref{lemma-pushforward}", "to see that pushforward on $H^*$ is compatible with composition", "and satisfies the projection formula.", "\\medskip\\noindent", "Let $f : X \\to Y$ be a morphism of nonempty", "equidimensional smooth projective schemes over $k$.", "We are trying to show $f_*\\gamma(\\alpha) = \\gamma(f_*\\alpha)$", "for any cycle class $\\alpha$ on $X$.", "We can write $\\alpha$ as a $\\mathbf{Q}$-linear combination of products of", "Chern classes of locally free $\\mathcal{O}_X$-modules", "(Chow Homology, Lemma \\ref{chow-lemma-K-tensor-Q}).", "Thus we may assume $\\alpha$ is a product of Chern classes of", "finite locally free $\\mathcal{O}_X$-modules", "$\\mathcal{E}_1, \\ldots, \\mathcal{E}_r$.", "Pick $p : P \\to X$ as in the splitting principle", "(Chow Homology, Lemma \\ref{chow-lemma-splitting-principle}).", "By Chow Homology, Remark \\ref{chow-remark-the-proof-shows-more}", "we see that $p$ is a composition of projective space bundles and", "that $\\alpha = p_*(\\xi_1 \\cap \\ldots \\cap \\xi_d \\cap \\cdot p^*\\alpha)$", "where $\\xi_i$ are first Chern classes of invertible modules.", "By Lemma \\ref{lemma-ok-for-projective-bundle}", "we know that $p_*$ commutes with cycle classes.", "Thus it suffices to prove the property for the composition", "$f \\circ p$. Since $p^*\\mathcal{E}_1, \\ldots, p^*\\mathcal{E}_r$", "have filtrations whose successive quotients are invertible", "modules, this reduces us to the case where $\\alpha$ is", "of the form $\\xi_1 \\cap \\ldots \\cap \\xi_t \\cap [X]$", "for some first Chern classes $\\xi_i$ of invertible modules $\\mathcal{L}_i$.", "\\medskip\\noindent", "Assume", "$\\alpha = c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_t) \\cap [X]$", "for some invertible modules $\\mathcal{L}_i$ on $X$.", "Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.", "For $n \\gg 0$ the invertible $\\mathcal{O}_X$-modules", "$\\mathcal{L}^{\\otimes n}$ and", "$\\mathcal{L}_1 \\otimes \\mathcal{L}^{\\otimes n}$ are both", "very ample on $X$ over $k$, see", "Morphisms, Lemma \\ref{morphisms-lemma-invertible-add-enough-ample-very-ample}.", "Since $c_1(\\mathcal{L}_1) = c_1(\\mathcal{L}_1 \\otimes \\mathcal{L}^{\\otimes n})", "- c_1(\\mathcal{L}^{\\otimes n})$ this reduces us to the case where", "$\\mathcal{L}_1$ is very ample. Repeating this with $\\mathcal{L}_i$", "for $i = 2, \\ldots, t$ we reduce to the case where $\\mathcal{L}_i$", "is very ample on $X$ over $k$ for all $i = 1, \\ldots, t$.", "\\medskip\\noindent", "Assume $k$ is infinite and $\\alpha = ", "c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_t) \\cap [X]$", "for some very ample invertible modules $\\mathcal{L}_i$ on $X$ over $k$.", "By Bertini in the form of Varieties, Lemma \\ref{varieties-lemma-bertini}", "we can successively find regular sections $s_i$ of $\\mathcal{L}_i$", "such that the schemes $Z(s_1) \\cap \\ldots \\cap Z(s_i)$", "are smooth over $k$ and of codimension $i$ in $X$.", "By the construction of capping with the first class of", "an invertible module (going back to", "Chow Homology, Definition \\ref{chow-definition-divisor-invertible-sheaf}),", "this reduces us to the case where $\\alpha = [Z]$", "for some nonempty smooth closed subscheme $Z \\subset X$ which", "is equidimensional.", "\\medskip\\noindent", "Assume $\\alpha = [Z]$ where $Z \\subset X$ is a smooth closed subscheme.", "Choose a closed embedding $X \\to \\mathbf{P}^n$. We can factor $f$ as", "$$", "X \\to Y \\times \\mathbf{P}^n \\to Y", "$$", "Since we know the result for the second morphism by", "Lemma \\ref{lemma-ok-for-projective-bundle}", "it suffices to prove the result when", "$\\alpha = [Z]$ where $i : Z \\to X$ is a closed immersion ", "and $f$ is a closed immersion.", "Then $j = f \\circ i$ is a closed embedding too.", "Using the hypothesis for $i$ and $j$ we win.", "\\medskip\\noindent", "We still have to prove the lemma in case $k$ is finite. We urge the", "reader to skip the rest of the proof. Everything we said above continues", "to work, except that we do not know we can choose the sections", "$s_i$ cutting out our $Z$ over $k$ as $k$ is finite. However, we do", "know that we can find $s_i$ over the algebraic closure $\\overline{k}$", "of $k$ (by the same lemma). This means that we can", "find a finite extension $k'/k$ such that our sections $s_i$", "are defined over $k'$. Denote $\\pi : X_{k'} \\to X$ the projection.", "The arguments above shows that we get the desired conclusion", "(from the assumption in the lemma)", "for the cycle $\\pi^*\\alpha$ and the morphism", "$f \\circ \\pi : X_{k'} \\to Y$.", "We have $\\pi_*\\pi^*\\alpha = [k' : k] \\alpha$, see", "Chow Homology, Lemma \\ref{chow-lemma-finite-flat}.", "On the other hand, we have", "$$", "\\pi_*\\gamma(\\pi^*\\alpha) = \\pi_*\\pi^*\\gamma(\\alpha) =", "\\gamma(\\alpha) \\pi_*1", "$$", "by the projection formula for our cohomology theory. Observe", "that $\\pi$ is a projection (!) and hence we have", "$\\pi_*(1) = \\int_{\\Spec(k')}(1) 1$ by", "Lemma \\ref{lemma-pr2star}. Thus to finish the proof in the", "finite field case, it suffices to prove that", "$\\int_{\\Spec(k')}(1) = [k' : k]$ which we do in", "Lemma \\ref{lemma-integrate-1}." ], "refs": [ "weil-lemma-cycle-classes", "weil-lemma-poincare-duality", "weil-remark-trace", "weil-lemma-trace-product", "weil-lemma-trace-base", "weil-lemma-pushforward", "chow-lemma-K-tensor-Q", "chow-lemma-splitting-principle", "chow-remark-the-proof-shows-more", "weil-lemma-ok-for-projective-bundle", "morphisms-lemma-invertible-add-enough-ample-very-ample", "varieties-lemma-bertini", "chow-definition-divisor-invertible-sheaf", "weil-lemma-ok-for-projective-bundle", "chow-lemma-finite-flat", "weil-lemma-pr2star", "weil-lemma-integrate-1" ], "ref_ids": [ 5094, 5097, 5122, 5098, 5099, 5068, 5828, 5762, 5949, 5100, 5398, 11132, 5913, 5100, 5683, 5069, 5101 ] } ], "ref_ids": [] }, { "id": 5103, "type": "theorem", "label": "weil-lemma-grassmanian", "categories": [ "weil" ], "title": "weil-lemma-grassmanian", "contents": [ "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).", "Given integers $0 < l < n$ and a nonempty equidimensional", "smooth projective scheme $X$ over $k$ consider the projection morphism", "$p : X \\times \\mathbf{G}(l, n) \\to X$.", "Then $\\gamma$ commutes with pushforward along $p$." ], "refs": [], "proofs": [ { "contents": [ "If $l = 1$ or $l = n - 1$ then $p$ is a projective bundle and", "the result follows from Lemma \\ref{lemma-ok-for-projective-bundle}.", "In general there exists a morphism", "$$", "h : Y \\to X \\times \\mathbf{G}(l, n)", "$$", "such that both $h$ and $p \\circ h$ are compositions of projective", "space bundles. Namely, denote $\\mathbf{G}(1, 2, \\ldots, l; n)$", "the partial flag variety. Then the morphism", "$$", "\\mathbf{G}(1, 2, \\ldots, l; n) \\to \\mathbf{G}(l, n)", "$$", "is a compostion of projective space bundles and similarly the", "structure morphism $\\mathbf{G}(1, 2, \\ldots, l; n) \\to \\Spec(k)$", "is of this form. Thus we may set $Y = X \\times \\mathbf{G}(1, 2, \\ldots, l; n)$.", "Since every cycle on $X \\times \\mathbf{G}(l, n)$ is the pushforward of", "a cycle on $Y$, the result for $Y \\to X$ and the result for", "$Y \\to X \\times \\mathbf{G}(l, n)$ imply the result for $p$." ], "refs": [ "weil-lemma-ok-for-projective-bundle" ], "ref_ids": [ 5100 ] } ], "ref_ids": [] }, { "id": 5104, "type": "theorem", "label": "weil-lemma-enough-better", "categories": [ "weil" ], "title": "weil-lemma-enough-better", "contents": [ "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).", "In order to show that $\\gamma$ commutes with pushforward it suffices", "to show that $i_*(1) = \\gamma([Z])$ if $i : Z \\to X$ is a closed", "immersion of nonempty smooth projective equidimensional schemes over $k$", "such that the class of $\\mathcal{C}_{Z/X}$ in $K_0(Z)$ is the", "pullback of a class in $K_0(X)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-enough} it suffices to show that $i_*(1) = \\gamma([Z])$", "if $i : Z \\to X$ is a closed immersion of nonempty", "smooth projective equidimensional", "schemes over $k$. Say $Z$ has codimension $r$ in $X$.", "Let $\\mathcal{L}$ be a sufficiently ample invertible module on $X$.", "Choose $n > 0$ and a surjection", "$$", "\\mathcal{O}_Z^{\\oplus n} \\to \\mathcal{C}_{Z/X} \\otimes \\mathcal{L}|_Z", "$$", "This gives a morphism $g : Z \\to \\mathbf{G}(n - r, n)$", "to the Grassmanian over $k$, see", "Constructions, Section \\ref{constructions-section-grassmannian}.", "Consider the composition", "$$", "Z \\to X \\times \\mathbf{G}(n - r, n) \\to X", "$$", "Pushforward along the second morphism is compatible with classes", "of cycles by Lemma \\ref{lemma-grassmanian}. The conormal sheaf $\\mathcal{C}$", "of the closed immersion $Z \\to X \\times \\mathbf{G}(n - r, n)$ sits in", "a short exact sequence", "$$", "0 \\to \\mathcal{C}_{Z/X} \\to \\mathcal{C} \\to", "g^*\\Omega_{\\mathbf{G}(n - r, n)} \\to 0", "$$", "See More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-two-unramified-morphisms-formally-smooth}.", "Since $\\mathcal{C}_{Z/X} \\otimes \\mathcal{L}|_Z$ is the pull", "back of a finite locally free sheaf on $\\mathbf{G}(n - r, n)$", "we conclude that the class of $\\mathcal{C}$ in $K_0(Z)$", "is the pullback of a class in $K_0(X \\times \\mathbf{G}(n - r, n))$.", "Hence we have the property for $Z \\to X \\times \\mathbf{G}(n - r, n)$", "and we conclude." ], "refs": [ "weil-lemma-enough", "weil-lemma-grassmanian", "more-morphisms-lemma-two-unramified-morphisms-formally-smooth" ], "ref_ids": [ 5102, 5103, 13740 ] } ], "ref_ids": [] }, { "id": 5105, "type": "theorem", "label": "weil-lemma-injective-H0", "categories": [ "weil" ], "title": "weil-lemma-injective-H0", "contents": [ "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).", "If $k''/k'/k$ are finite separable field extensions, then", "$H^0(\\Spec(k')) \\to H^0(\\Spec(k''))$ is injective." ], "refs": [], "proofs": [ { "contents": [ "We may replace $k''$ by its normal closure over $k$", "which is Galois over $k$, see", "Fields, Lemma \\ref{fields-lemma-normal-closure-galois}.", "Then $k''$ is Galois over $k'$ as well, see", "Fields, Lemma \\ref{fields-lemma-galois-goes-up}.", "We deduce we have an isomorphism", "$$", "k' \\otimes_k k'' \\longrightarrow", "\\prod\\nolimits_{\\sigma \\in \\text{Gal}(k''/k')} k'',\\quad", "\\eta \\otimes \\zeta \\longmapsto (\\eta \\sigma(\\zeta))_\\sigma", "$$", "This produces an isomorphism", "$\\coprod_\\sigma \\Spec(k'') \\to \\Spec(k') \\times \\Spec(k'')$", "which on cohomology produces the isomorphism", "$$", "H^*(\\Spec(k')) \\otimes_F H^*(\\Spec(k''))", "\\to", "\\prod\\nolimits_\\sigma H^*(\\Spec(k'')),\\quad", "a' \\otimes a'' \\longmapsto (\\pi^*a' \\cup \\Spec(\\sigma)^*a'')_\\sigma", "$$", "where $\\pi : \\Spec(k'') \\to \\Spec(k')$ is the morphism", "corresponding to the inclusion of $k'$ in $k''$.", "We conclude the lemma is true by taking $a'' = 1$." ], "refs": [ "fields-lemma-normal-closure-galois", "fields-lemma-galois-goes-up" ], "ref_ids": [ 4506, 4505 ] } ], "ref_ids": [] }, { "id": 5106, "type": "theorem", "label": "weil-lemma-pushforward-blowup", "categories": [ "weil" ], "title": "weil-lemma-pushforward-blowup", "contents": [ "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8).", "Let $b : X' \\to X$ be a blowing up of a smooth projective scheme $X$", "over $k$ which is nonempty equidimensional of dimension $d$", "in a nonwhere dense smooth center $Z$. Then $b_*(1) = 1$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $X$ by a connected component of $X$ (some details", "omitted). Thus we may assume $X$ is connected and hence irreducible.", "Set $k' = \\Gamma(X, \\mathcal{O}_X) = \\Gamma(X', \\mathcal{O}_{X'})$;", "we omit the proof of the equality. Choose a closed point $x' \\in X'$", "which isn't contained in the exceptional divisor and whose residue field", "$k''$ is separable over $k$; this is possible by", "Varieties, Lemma \\ref{varieties-lemma-smooth-separable-closed-points-dense}.", "Denote $x \\in X$ the image (whose residue field is equal to $k''$", "as well of course). Consider the diagram", "$$", "\\xymatrix{", "x' \\times X' \\ar[r] \\ar[d] & X' \\times X' \\ar[d] \\\\", "x \\times X \\ar[r] & X \\times X", "}", "$$", "The class of the diagonal $\\Delta = \\Delta_X$ pulls back to the class of the", "``diagonal point'' $\\delta_x : x \\to x \\times X$ and similarly for the class of", "the diagonal $\\Delta'$. On the other hand, the diagonal point $\\delta_x$", "pulls back to the diagonal point $\\delta_{x'}$ by the left vertical arrow.", "Write $\\gamma([\\Delta]) = \\sum \\eta_i$ with", "$\\eta_i \\in H^i(X) \\otimes H^{2d - i}(X)(d)$ and", "$\\gamma([\\Delta']) = \\sum \\eta'_i$ with", "$\\eta'_i \\in H^i(X') \\otimes H^{2d - i}(X')(d)$.", "The arguments above show that $\\eta_0$ and $\\eta'_0$ map to the same", "class in", "$$", "H^0(x') \\otimes_F H^{2d}(X')(d)", "$$", "We have $H^0(\\Spec(k')) = H^0(X) = H^0(X')$ by axiom (A8).", "By Lemma \\ref{lemma-injective-H0} this common value maps injectively", "into $H^0(x')$. We conclude that $\\eta_0$ maps to $\\eta'_0$ by the map", "$$", "H^0(X) \\otimes_F H^{2d}(X)(d)", "\\longrightarrow", "H^0(X') \\otimes_F H^{2d}(X')(d)", "$$", "This means that $\\int_X$ is equal to $\\int_{X'}$ composed with", "the pullback map. This proves the lemma." ], "refs": [ "varieties-lemma-smooth-separable-closed-points-dense", "weil-lemma-injective-H0" ], "ref_ids": [ 11007, 5105 ] } ], "ref_ids": [] }, { "id": 5107, "type": "theorem", "label": "weil-lemma-done", "categories": [ "weil" ], "title": "weil-lemma-done", "contents": [ "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8).", "Then the cycle class map $\\gamma$ commutes with pushforward." ], "refs": [], "proofs": [ { "contents": [ "Let $i : Z \\to X$ be as in Lemma \\ref{lemma-enough-better}. Consider", "the diagram", "$$", "\\xymatrix{", "E \\ar[r]_j \\ar[d]_\\pi & X' \\ar[d]^b \\\\", "Z \\ar[r]^i & X", "}", "$$", "Let $\\theta \\in \\CH^{r - 1}(X')$ be as in", "Lemma \\ref{lemma-divide-pullback-good-blowing-up}.", "Then $\\pi_*j^!\\theta = [Z]$ in $\\CH_*(Z)$ implies that", "$\\pi_*\\gamma(j^!\\theta) = 1$ by Lemma \\ref{lemma-ok-for-projective-bundle}", "because $\\pi$ is a projective space bundle.", "Hence we see that", "$$", "i_*(1) = i_*(\\pi_*(\\gamma(j^!\\theta))) =", "b_*j_*(j^*\\gamma(\\theta)) =", "b_*(j_*(1) \\cup \\gamma(\\theta))", "$$", "We have $j_*(1) = \\gamma([E])$ by (A9). Thus this is equal to", "$$", "b_*(\\gamma([E]) \\cup \\gamma(\\theta)) =", "b_*(\\gamma([E] \\cdot \\theta)) =", "b_*(\\gamma(b^*[Z])) =", "b_*b^*\\gamma([Z]) = b_*(1) \\cup \\gamma([Z])", "$$", "Since $b_*(1) = 1$ by Lemma \\ref{lemma-pushforward-blowup} the", "proof is complete." ], "refs": [ "weil-lemma-enough-better", "weil-lemma-divide-pullback-good-blowing-up", "weil-lemma-ok-for-projective-bundle", "weil-lemma-pushforward-blowup" ], "ref_ids": [ 5104, 5095, 5100, 5106 ] } ], "ref_ids": [] }, { "id": 5108, "type": "theorem", "label": "weil-lemma-check-over-extension", "categories": [ "weil" ], "title": "weil-lemma-check-over-extension", "contents": [ "Let $k'/k$ be an extension of fields. Let $F'/F$ be an extension", "of fields of characteristic $0$. Assume given", "\\begin{enumerate}", "\\item data (D0), (D1), (D2') for $k$ and $F$ denoted", "$F(1), H^*, c_1^H$,", "\\item data (D0), (D1), (D2') for $k'$ and $F'$ denoted", "$F'(1), (H')^*, c_1^{H'}$, and", "\\item an isomorphism $F(1) \\otimes_F F' \\to F'(1)$, functorial isomorphisms", "$H^*(X) \\otimes_F F' \\to (H')^*(X_{k'})$ on the category of smooth projective", "schemes $X$ over $k$ such that the diagrams", "$$", "\\xymatrix{", "\\Pic(X) \\ar[r]_{c_1^H} \\ar[d] & H^2(X)(1) \\ar[d] \\\\", "\\Pic(X_{k'}) \\ar[r]^{c_1^{H'}} & (H')^2(X_{k'})(1)", "}", "$$", "commute.", "\\end{enumerate}", "In this case, if $F'(1), (H')^*, c_1^{H'}$ satisfy axioms (A1) -- (A9),", "then the same is true for $F(1), H^*, c_1^H$." ], "refs": [], "proofs": [ { "contents": [ "We go by the axioms one by one.", "\\medskip\\noindent", "Axiom (A1). We have to show $H^*(\\emptyset) = 0$ and that", "$(i^*, j^*) : H^*(X \\amalg Y) \\to H^*(X) \\times H^*(Y)$", "is an isomorphism where $i$ and $j$ are the coprojections.", "By the functorial nature of the isomorphisms", "$H^*(X) \\otimes_F F' \\to (H')^*(X_{k'})$ this", "follows from linear algebra: if $\\varphi : V \\to W$ is an $F$-linear map", "of $F$-vector spaces, then $\\varphi$ is an isomorphism if and only if", "$\\varphi_{F'} : V \\otimes_F F' \\to W \\otimes_F F'$ is an isomorphism.", "\\medskip\\noindent", "Axiom (A2). This means that given a morphism $f : X \\to Y$ of smooth projective", "schemes over $k$ and an invertible $\\mathcal{O}_Y$-module $\\mathcal{N}$", "we have $f^*c_1^H(\\mathcal{L}) = c_1^H(f^*\\mathcal{L})$. This is immediately", "clear from the corresponding property for $c_1^{H'}$, the commutative", "diagrams in the lemma, and the fact that the canonical map", "$V \\to V \\otimes_F F'$ is injective for any $F$-vector space $V$.", "\\medskip\\noindent", "Axiom (A3). This follows from the principle stated in the proof of", "axiom (A1) and compatibility of $c_1^H$ and $c_1^{H'}$.", "\\medskip\\noindent", "Axiom (A4). Let $i : Y \\to X$ be the inclusion of an effective", "Cartier divisor over $k$ with both $X$ and $Y$ smooth and projective", "over $k$. For $a \\in H^*(X)$ with", "$i^*a = 0$ we have to show $a \\cup c_1^H(\\mathcal{O}_X(Y)) = 0$.", "Denote $a' \\in (H')^*(X_{k'})$ the image of $a$.", "The assumption implies that $(i')^*a' = 0$ where $i' : Y_{k'} \\to X_{k'}$", "is the base change of $i$. Hence we get", "$a' \\cup c_1^{H'}(\\mathcal{O}_{X_{k'}}(Y_{k'})) = 0$ by the axiom", "for $(H')^*$. Since $a' \\cup c_1^{H'}(\\mathcal{O}_{X_{k'}}(Y_{k'}))$", "is the image of $a \\cup c_1^H(\\mathcal{O}_X(Y))$ we conclude by", "the princple stated in the proof of axiom (A2).", "\\medskip\\noindent", "Axiom (A5). This means that $H^*(\\Spec(k)) = F$ and that for $X$ and $Y$ smooth", "projective over $k$ the map $H^*(X) \\otimes_F H^*(Y) \\to H^*(X \\times Y)$,", "$a \\otimes b \\mapsto p^*(a) \\cup q^*(b)$ is an isomorphism", "where $p$ and $q$ are the projections. This follows from the principle", "stated in the proof of axiom (A1).", "\\medskip\\noindent", "We interrupt the flow of the arguments to show that for every", "smooth projective scheme $X$ over $k$ the diagram", "$$", "\\xymatrix{", "\\CH^*(X) \\ar[r]_-\\gamma \\ar[d]_{g^*} & \\bigoplus H^{2i}(X)(i) \\ar[d] \\\\", "\\CH^*(X_{k'}) \\ar[r]^-{\\gamma'} & \\bigoplus (H')^{2i}(X_{k'})(i)", "}", "$$", "commutes. Observe that we have $\\gamma$ as we know axioms", "(A1) -- (A4) already; see Lemma \\ref{lemma-cycle-classes}.", "Also, the left vertical arrow is the one discussed in", "Chow Homology, Section \\ref{chow-section-change-base}", "for the morphism of base schemes $g : \\Spec(k') \\to \\Spec(k)$.", "More precisely, it is the map given in", "Chow Homology, Lemma \\ref{chow-lemma-pullback-base-change}.", "Pick $\\alpha \\in \\CH^*(X)$. Write $\\alpha = ch(\\beta) \\cap [X]$", "in $\\CH^*(X) \\otimes \\mathbf{Q}$", "for some $\\beta \\in K_0(\\textit{Vect}(X)) \\otimes \\mathbf{Q}$", "so that $\\gamma(\\alpha) = ch^{H}(\\beta)$; this is our construction of $\\gamma$.", "Since the map of Chow Homology, Lemma \\ref{chow-lemma-pullback-base-change}", "is compatible with capping with Chern classes by", "Chow Homology, Lemma \\ref{chow-lemma-pullback-base-change-chern-classes}", "we see that $g^*\\alpha = ch((X_{k'} \\to X)^*\\beta) \\cap [X_{k'}]$.", "Hence $\\gamma'(g^*\\alpha) = ch^{H'}((X_{k'} \\to X)^*\\beta)$.", "Thus commutativity of the diagram will hold if for any locally", "free $\\mathcal{O}_X$-module $\\mathcal{E}$ of rank $r$ and $0 \\leq i \\leq r$", "the element $c_i^H(\\mathcal{E})$ of $H^{2i}(X)(i)$", "maps to the element $c_i^{H'}(\\mathcal{E}_{k'})$ in $(H')^{2i}(X_{k'})(i)$.", "Because we have the projective space bundle formula for both", "$X$ and $X'$ we may replace $X$ by a projective space bundle", "over $X$ finitely many times to show this. Thus we may assume", "$\\mathcal{E}$ has a filtration whose graded pieces are", "invertible $\\mathcal{O}_X$-modules", "$\\mathcal{L}_1, \\ldots, \\mathcal{L}_r$.", "See Chow Homology, Lemma \\ref{chow-lemma-splitting-principle} and", "Remark \\ref{chow-remark-the-proof-shows-more}.", "Then $c^H_i(\\mathcal{E}$ is the $i$th elementary symmetric polynomial", "in $c^H_1(\\mathcal{L}_1), \\ldots, c^H_1(\\mathcal{L}_r)$", "and we conclude by our assumption that we have agreement for", "first Chern classes.", "\\medskip\\noindent", "Axiom (A6). Suppose given $F$-vector spaces", "$V$, $W$, an element $v \\in V$, and a tensor $\\xi \\in V \\otimes_F W$.", "Denote $V' = V \\otimes_F F'$, $W' = W \\otimes_F F'$ and $v'$, $\\xi'$", "the images of $v$, $\\xi$ in $V'$, $V' \\otimes_{F'} W'$. The linear algebra", "principle we will use in the proof of axiom (A6) is the following:", "there exists an $F$-linear map $\\lambda : W \\to F$ such that", "$(1 \\otimes \\lambda)\\xi = v$ if and only if there exists an $F'$-linear", "map $\\lambda' : W \\otimes_F F' \\to F'$ such that", "$(1 \\otimes \\lambda')\\xi' = v'$.", "\\medskip\\noindent", "Let $X$ be a nonempty equidimensional smooth projective scheme", "over $k$ of dimension $d$. Denote $\\gamma = \\gamma([\\Delta])$", "in $H^{2d}(X \\times X)(d)$ (unadorned fibre products will be over $k$).", "Observe/recall that this makes sense as we know axioms (A1) -- (A4) already;", "see Lemma \\ref{lemma-cycle-classes}. We may decompose", "$$", "\\gamma = \\sum \\gamma_i, \\quad", "\\gamma_i \\in H^i(X) \\otimes_F H^{2d - i}(X)(d)", "$$", "in the K\\\"unneth decomposition. Similarly, denote", "$\\gamma' = \\gamma([\\Delta']) = \\sum \\gamma'_i$", "in $(H')^{2d}(X_{k'} \\times_{k'} X_{k'})(d)$.", "By the linear algebra princple mentioned above, it suffices", "to show that $\\gamma_0$ maps to $\\gamma'_0$ in", "$(H')^0(X) \\otimes_{F'} (H')^{2d}(X')(d)$.", "By the compatibility of K\\\"unneth decompositions", "we see that it suffice to show that $\\gamma$ maps to", "$\\gamma'$ in", "$$", "(H')^{2d}(X_{k'} \\times_{k'} X_{k'})(d) = (H')^{2d}((X \\times X)_{k'})(d)", "$$", "Since $\\Delta_{k'} = \\Delta'$ this follows from the discussion above.", "\\medskip\\noindent", "Axiom (A7). This follows from the linear algebra fact: a", "linear map $V \\to W$ of $F$-vector spaces is injective", "if and only if $V \\otimes_F F' \\to W \\otimes_F F'$ is injective.", "\\medskip\\noindent", "Axiom (A8). Follows from the linear algebra fact used in", "the proof of axiom (A1).", "\\medskip\\noindent", "Axiom (A9). Let $X$ be a nonempty smooth projective scheme over $k$", "equidimensional of dimension $d$. Let $i : Y \\to X$ be a nonempty", "effective Cartier divisor smooth over $k$.", "Let $\\lambda_Y$ and $\\lambda_X$ be as in axiom (A6) for $X$ and $Y$.", "We have to show: for $a \\in H^{2d - 2}(X)(d - 1)$", "we have $\\lambda_Y(i^*(a)) = \\lambda_X(a \\cup c_1^H(\\mathcal{O}_X(Y))$.", "By Remark \\ref{remark-trace}", "we know that $\\lambda_X : H^{2d}(X)(d) \\to F$ and", "$\\lambda_Y : H^{2d - 2}(Y)(d - 1)$ are uniquely", "determined by the requirement in axiom (A6).", "Having said this, it follows from our proof of axiom (A6) for $H^*$ above", "that $\\lambda_X \\otimes \\text{id}_{F'}$ corresponds to $\\lambda_{X_{k'}}$", "via the given identification $H^{2d}(X)(d) \\otimes_F F' = H^{2d}(X_{k'})(d)$.", "Thus the fact that we know axiom (A9) for $F'(1), (H')^*, c_1^{H'}$", "implies the axiom for $F(1), H^*, c_1^H$ by a diagram chase.", "This completes the proof of the lemma." ], "refs": [ "weil-lemma-cycle-classes", "chow-lemma-pullback-base-change", "chow-lemma-pullback-base-change", "chow-lemma-pullback-base-change-chern-classes", "chow-lemma-splitting-principle", "chow-remark-the-proof-shows-more", "weil-lemma-cycle-classes", "weil-remark-trace" ], "ref_ids": [ 5094, 5860, 5860, 5864, 5762, 5949, 5094, 5122 ] } ], "ref_ids": [] }, { "id": 5109, "type": "theorem", "label": "weil-proposition-weil-cohomology-theory-classical", "categories": [ "weil" ], "title": "weil-proposition-weil-cohomology-theory-classical", "contents": [ "Let $k$ be an algebraically closed field. Let $F$ be a field of", "characteristic $0$. A classical Weil cohomology theory is the same thing", "as a $\\mathbf{Q}$-linear functor", "$$", "G : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces}", "$$", "of symmetric monoidal categories together with an isomorphism", "$F[2] \\to G(\\mathbf{1}(1))$ of graded $F$-vector spaces such that", "in addition", "\\begin{enumerate}", "\\item $G(h(X))$ lives in nonnegative degrees, and", "\\item $\\dim_F G^0(h(X)) = 1$", "\\end{enumerate}", "for any smooth projective variety $X$." ], "refs": [], "proofs": [ { "contents": [ "Given $G$ and $F[2] \\to G(\\mathbf{1}(1))$ by setting $H^*(X) = G(h(X))$", "we obtain data (D1), (D2), and (D3) satisfying all of (A), (B), and (C)", "except for possibly (A)(c) and (A)(d), see", "Lemma \\ref{lemma-from-functor-to-weil-classical} and its proof.", "Observe that assumptions (1) and (2) imply axioms (A)(c) and (A)(d)", "in the presence of the known axioms (A)(a) and (A)(b).", "\\medskip\\noindent", "Conversely, given $H^*$ we get a functor $G$ by the construction of", "Lemma \\ref{lemma-from-weil-to-functor-classical}.", "Let $X = \\mathbf{P}^1, c_0, c_2$ be as in Example \\ref{example-decompose-P1}.", "We have constructed an isomorphism $1(-1) \\to (X, c_2, 0)$ of motives in", "Lemma \\ref{lemma-inverse-h2}. In the proof of", "Lemma \\ref{lemma-from-weil-to-functor-classical} we have seen that", "$G(1(-1)) = G(X, c_2, 0) = H^2(\\mathbf{P}^1)[-2]$.", "Hence the isomorphism $\\int_{\\mathbf{P}^1} : H^2(\\mathbf{P}^1) \\to F$", "of axiom (A)(d) gives an isomorphism $G(1(-1)) \\to F[-2]$ which", "determines an isomorphism $F[2] \\to G(\\mathbf{1}(1))$.", "Finally, since $G(h(X)) = H^*(X)$ assumptions (1) and (2)", "follow from axiom (A)." ], "refs": [ "weil-lemma-from-functor-to-weil-classical", "weil-lemma-from-weil-to-functor-classical", "weil-lemma-inverse-h2", "weil-lemma-from-weil-to-functor-classical" ], "ref_ids": [ 5061, 5062, 5045, 5062 ] } ], "ref_ids": [] }, { "id": 5110, "type": "theorem", "label": "weil-proposition-weil-cohomology-theory", "categories": [ "weil" ], "title": "weil-proposition-weil-cohomology-theory", "contents": [ "Let $k$ be a field. Let $F$ be a field of characteristic $0$. There is a", "$1$-to-$1$ correspondence between the following", "\\begin{enumerate}", "\\item data (D0), (D1), (D2), and (D3) satisfying (A), (B), and(C), and", "\\item $\\mathbf{Q}$-linear symmetric monoidal functors", "$$", "G : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces}", "$$", "such that $G(\\mathbf{1}(1))$ is nonzero only in degree $-2$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Given $G$ as in (2) by setting $H^*(X) = G(h(X))$ we obtain data", "(D0), (D1), (D2), and (D3) satisfying (A), (B), and (C),", "see Lemma \\ref{lemma-from-functor-to-weil} and its proof.", "\\medskip\\noindent", "Conversely, given data (D0), (D1), (D2), and (D3)", "satisfying (A), (B), and (C) we get a functor $G$ as in (2)", "by the construction of the proof of Lemma \\ref{lemma-from-weil-to-functor}.", "\\medskip\\noindent", "We omit the detailed proof that these constructions are inverse", "to each other." ], "refs": [ "weil-lemma-from-functor-to-weil", "weil-lemma-from-weil-to-functor" ], "ref_ids": [ 5076, 5077 ] } ], "ref_ids": [] }, { "id": 5111, "type": "theorem", "label": "weil-proposition-chern-class", "categories": [ "weil" ], "title": "weil-proposition-chern-class", "contents": [ "In the situation above there is a unique rule which assigns to", "every $X \\in \\Ob(\\mathcal{C})$ a ``total Chern class''", "$$", "c^A : K_0(\\textit{Vect}(X)) \\longrightarrow \\prod\\nolimits_{i \\geq 0} A^i(X)", "$$", "with the following properties", "\\begin{enumerate}", "\\item For $X \\in \\Ob(\\mathcal{C})$ we have", "$c^A(\\alpha + \\beta) = c^A(\\alpha) c^A(\\beta)$", "and $c^A(0) = 1$.", "\\item If $f : X' \\to X$ is a morphism of $\\mathcal{C}$, then", "$f^* \\circ c^A = c^A \\circ f^*$.", "\\item Given $X \\in \\Ob(\\mathcal{C})$ and $\\mathcal{L} \\in \\Pic(X)$", "we have $c^A([\\mathcal{L}]) = 1 + c_1^A(\\mathcal{L})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X \\in \\Ob(\\mathcal{C})$ and let $\\mathcal{E}$ be a finite", "locally free $\\mathcal{O}_X$-module. We first show how to define", "an element $c^A(\\mathcal{E}) \\in A(X)$.", "\\medskip\\noindent", "As a first step, let $X = \\bigcup X_r$ be the decomposition into", "open and closed subschemes such that $\\mathcal{E}|_{X_r}$ has", "constant rank $r$. Since $X$ is quasi-compact, this decomposition", "is finite. Hence $A(X) = \\prod A(X_r)$. Thus it suffices to define", "$c^A(\\mathcal{E})$ when $\\mathcal{E}$ has constant rank $r$. In this", "case let $p : P \\to X$ be the projective bundle of $\\mathcal{E}$.", "We can uniquely define elements $c_i^A(\\mathcal{E}) \\in A^i(X)$", "for $i \\geq 0$ such that $c_0^A(\\mathcal{E}) = 1$ and the equation", "\\begin{equation}", "\\label{equation-chern-classes}", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_P(1))^i \\cup p^*c^A_{r - i}(\\mathcal{E})", "= 0", "\\end{equation}", "is true. As usual we set", "$c^A(\\mathcal{E}) = c_0^A(\\mathcal{E}) + c_1^A(\\mathcal{E}) + \\ldots", "+ c_r^A(\\mathcal{E})$ in $A(X)$.", "\\medskip\\noindent", "If $\\mathcal{E}$ is invertible, then", "$c^A(\\mathcal{E}) = 1 + c_1^A(\\mathcal{L})$.", "This follows immediately from the construction above.", "\\medskip\\noindent", "The elements $c_i^A(\\mathcal{E})$ are in the center of $A(X)$.", "Namely, to prove this we may assume $\\mathcal{E}$ has constant rank $r$.", "Let $p : P \\to X$ be the corresponding projective bundle.", "if $a \\in A(X)$ then $p^*a \\cup (-1)^r c_1(\\mathcal{O}_P(1))^r =", "(-1)^r c_1(\\mathcal{O}_P(1))^r \\cup p^*a$ and hence we must have the same", "for all the other terms in the expression defining $c_i^A(\\mathcal{E})$", "as well and we conclude.", "\\medskip\\noindent", "If $f : X' \\to X$ is a morphism of $\\mathcal{C}$, then", "$f^*c_i^A(\\mathcal{E}) = c_i^A(f^*\\mathcal{E})$.", "Namely, to prove this we may assume $\\mathcal{E}$ has constant rank $r$.", "Let $p : P \\to X$ and $p' : P' \\to X'$ be the projective", "bundles corresponding to $\\mathcal{E}$ and $f^*\\mathcal{E}$.", "The induced morphism $g : P' \\to P$ is a morphism of $\\mathcal{C}$.", "The pullback by $g$ of the equality defining $c_i^A(\\mathcal{E})$", "is the corresponding equation for $f^*\\mathcal{E}$ and we conclude.", "\\medskip\\noindent", "Let $X \\in \\Ob(\\mathcal{C})$. Consider a short exact sequence", "$$", "0 \\to \\mathcal{L} \\to \\mathcal{E} \\to \\mathcal{F} \\to 0", "$$", "of finite locally free $\\mathcal{O}_X$-modules with $\\mathcal{L}$ invertible.", "Then", "$$", "c^A(\\mathcal{E}) = c^A(\\mathcal{L}) c^A(\\mathcal{F})", "$$", "Namely, by the construction of $c^A_i$ we may assume $\\mathcal{E}$ has", "constant rank $r + 1$ and $\\mathcal{F}$ has constant rank $r$.", "The inclusion", "$$", "i : P' = \\mathbf{P}(\\mathcal{F}) \\longrightarrow \\mathbf{P}(\\mathcal{E}) = P", "$$", "is a morphism of $\\mathcal{C}$ and it is the zero scheme of a regular", "section of the invertible module", "$\\mathcal{L}^{\\otimes -1} \\otimes \\mathcal{O}_P(1)$.", "The element", "$$", "\\sum\\nolimits_{i = 0}^r (-1)^i c_1^A(\\mathcal{O}_P(1))^i \\cup", "p^*c^A_i(\\mathcal{F})", "$$", "pulls back to zero on $P'$ by definition. Hence we see that", "$$", "\\left(c_1^A(\\mathcal{O}_P(1)) - c_1^A(\\mathcal{L})\\right) \\cup", "\\left(\\sum\\nolimits_{i = 0}^r (-1)^i c_1^A(\\mathcal{O}_P(1))^i \\cup", "p^*c^A_i(\\mathcal{F})\\right) = 0", "$$", "in $A^*(P)$ by assumption (5) on our cohomology $A$.", "By definition of $c_1^A(\\mathcal{E})$", "this gives the desired equality.", "\\medskip\\noindent", "Let $X \\in \\Ob(\\mathcal{C})$. Consider a short exact sequence", "$$", "0 \\to \\mathcal{E} \\to \\mathcal{F} \\to \\mathcal{G} \\to 0", "$$", "of finite locally free $\\mathcal{O}_X$-modules. Then", "$$", "c^A(\\mathcal{F}) = c^A(\\mathcal{E}) c^A(\\mathcal{G})", "$$", "Namely, by the construction of $c^A_i$ we may assume", "$\\mathcal{E}$, $\\mathcal{F}$, and $\\mathcal{G}$ have", "constant ranks $r$, $s$, and $t$. We prove it by induction on $r$.", "The case $r = 1$ was done above. If $r > 1$, then it suffices to check", "this after pulling back by the morphism $\\mathbf{P}(\\mathcal{E}^\\vee) \\to X$.", "Thus we may assume we have an invertible submodule", "$\\mathcal{L} \\subset \\mathcal{E}$ such that both", "$\\mathcal{E}' = \\mathcal{E}/\\mathcal{L}$ and", "$\\mathcal{F}' = \\mathcal{E}/\\mathcal{L}$ are finite locally free", "(of ranks $s - 1$ and $t - 1$). Then we have", "$$", "c^A(\\mathcal{E}) = c^A(\\mathcal{L}) c^A(\\mathcal{E}')", "\\quad\\text{and}\\quad", "c^A(\\mathcal{F}) = c^A(\\mathcal{L}) c^A(\\mathcal{F}')", "$$", "Since we have the short exact sequence", "$$", "0 \\to \\mathcal{E}' \\to \\mathcal{F}' \\to \\mathcal{G} \\to 0", "$$", "we see by induction hypothesis that", "$$", "c^A(\\mathcal{F}') = c^A(\\mathcal{E}') c^A(\\mathcal{G})", "$$", "Thus the result follows from a formal calculation.", "\\medskip\\noindent", "At this point for $X \\in \\Ob(\\mathcal{C})$", "we can define $c^A : K_0(\\textit{Vect}(X)) \\to A(X)$.", "Namely, we send a generator $[\\mathcal{E}]$ to $c^A(\\mathcal{E})$", "and we extend multiplicatively. Thus for example", "$c^A(-[\\mathcal{E}]) = c^A(\\mathcal{E})^{-1}$ is the formal", "inverse of $a^A([\\mathcal{E}])$.", "The multiplicativity in short exact sequences shown above", "guarantees that this works.", "\\medskip\\noindent", "Uniqueness. Suppose $X \\in \\Ob(\\mathcal{C})$ and $\\mathcal{E}$", "is a finite locally free $\\mathcal{O}_X$-module. We want to show", "that conditions (1), (2), and (3) of the lemma uniquely determine", "$c^A([\\mathcal{E}])$. To prove this we may assume $\\mathcal{E}$", "has constant rank $r$; this already uses (2). Then we may use induction on $r$.", "If $r = 1$, then uniqueness follows from (3).", "If $r > 1$ we pullback using (2) to the projective bundle $p : P \\to X$", "and we see that we may assume we have a short exact sequence", "$0 \\to \\mathcal{E}' \\to \\mathcal{E} \\to \\mathcal{E}'' \\to 0$", "with $\\mathcal{E}'$ and $\\mathcal{E}''$ having lower rank.", "By induction hypothesis $c^A(\\mathcal{E}')$ and $c^A(\\mathcal{E}'')$", "are uniquely determined. Thus uniqueness for $\\mathcal{E}$ by", "the axiom (1)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5112, "type": "theorem", "label": "weil-proposition-chern-character", "categories": [ "weil" ], "title": "weil-proposition-chern-character", "contents": [ "In the situation above assume $A(X)$ is a $\\mathbf{Q}$-algebra for all", "$X \\in \\Ob(\\mathcal{C})$. Then there is a unique rule which assigns to", "every $X \\in \\Ob(\\mathcal{C})$ a ``chern character''", "$$", "ch^A : K_0(\\textit{Vect}(X)) \\longrightarrow", "\\prod\\nolimits_{i \\geq 0} A^i(X)", "$$", "with the following properties", "\\begin{enumerate}", "\\item $ch^A$ is a ring map for all $X \\in \\Ob(\\mathcal{C})$.", "\\item If $f : X' \\to X$ is a morphism of $\\mathcal{C}$, then", "$f^* \\circ ch^A = ch^A \\circ f^*$.", "\\item Given $X \\in \\Ob(\\mathcal{C})$ and $\\mathcal{L} \\in \\Pic(X)$", "we have $ch^A([\\mathcal{L}]) = \\exp(c_1^A(\\mathcal{L}))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X \\in \\Ob(\\mathcal{C})$ and let $\\mathcal{E}$ be a finite", "locally free $\\mathcal{O}_X$-module. We first show how to define", "the rank $r^A(\\mathcal{E}) \\in A^0(X)$. Namely, let $X = \\bigcup X_r$", "be the decomposition into open and closed subschemes such that", "$\\mathcal{E}|_{X_r}$ has constant rank $r$. Since $X$ is quasi-compact, this", "decomposition is finite, say $X = X_0 \\amalg X_1 \\amalg \\ldots \\amalg X_n$.", "Then $A(X) = A(X_0) \\times A(X_1) \\times \\ldots \\times A(X_n)$. Thus we", "can define $r^A(\\mathcal{E}) = (0, 1, \\ldots, n) \\in A^0(X)$.", "\\medskip\\noindent", "Let $P_p(c_1, \\ldots, c_p)$ be the polynomials constructed in", "Chow Homology, Example \\ref{chow-example-power-sum}.", "Then we can define", "$$", "ch^A(\\mathcal{E}) = r^A(\\mathcal{E}) +", "\\sum\\nolimits_{i \\geq 1} (1/i!)", "P_i(c^A_1(\\mathcal{E}), \\ldots, c^A_i(\\mathcal{E}))", "\\in \\prod\\nolimits_{i \\geq 0} A^i(X)", "$$", "where $ci^A$ are the Chern classes of", "Proposition \\ref{proposition-chern-class}.", "It follows immediately that we have property (2) and (3) of the lemma.", "\\medskip\\noindent", "We still have to show the following three statements", "\\begin{enumerate}", "\\item If $0 \\to \\mathcal{E}_1 \\to \\mathcal{E} \\to \\mathcal{E}_2 \\to 0$", "is a short exact sequence of finite locally free $\\mathcal{O}_X$-modules", "on $X \\in \\Ob(\\mathcal{C})$, then", "$ch^A(\\mathcal{E}) = ch^A(\\mathcal{E}_1) + ch^A(\\mathcal{E}_2)$.", "\\item If $\\mathcal{E}_1$ and $\\mathcal{E}_2 \\to 0$ are finite locally free", "$\\mathcal{O}_X$-modules on $X \\in \\Ob(\\mathcal{C})$, then", "$ch^A(\\mathcal{E}_1 \\otimes \\mathcal{E}_2) =", "ch^A(\\mathcal{E}_1) ch^A(\\mathcal{E}_2)$.", "\\end{enumerate}", "Namely, the first will prove that $ch^A$ factors through", "$K_0(\\textit{Vect}(X))$ and the first and the second will combined", "show that $ch^A$ is a ring map.", "\\medskip\\noindent", "To prove these statements we can reduce to the case where $\\mathcal{E}_1$", "and $\\mathcal{E}_2$ have constant ranks $r_1$ and $r_2$. In this case the", "equalities in $A^0(X)$ are immediate. To prove the equalities in higher", "degrees, by Lemma \\ref{lemma-splitting-principle} we may", "assume that $\\mathcal{E}_1$ and $\\mathcal{E}_2$ have filtrations", "whose graded pieces are invertible modules", "$\\mathcal{L}_{1, j}$, $j = 1, \\ldots, r_1$ and", "$\\mathcal{L}_{2, j}$, $j = 1, \\ldots, r_2$.", "Using the multiplicativity of Chern classes we get", "$$", "c_i^A(\\mathcal{E}_1) =", "s_i(c_1^A(\\mathcal{L}_{1, 1}), \\ldots, c_1^A(\\mathcal{L}_{1, r_1}))", "$$", "where $s_i$ is the $i$th elementary symmetric function as in", "Chow Homology, Example \\ref{chow-example-power-sum}.", "Similarly for $c_i^A(\\mathcal{E}_2)$. In case (1) we get", "$$", "c_i^A(\\mathcal{E}) =", "s_i(c_1^A(\\mathcal{L}_{1, 1}), \\ldots, c_1^A(\\mathcal{L}_{1, r_1}),", "c_1^A(\\mathcal{L}_{2, 1}), \\ldots, c_1^A(\\mathcal{L}_{2, r_2}))", "$$", "and for case (2) we get", "$$", "c_i^A(\\mathcal{E}_1 \\otimes \\mathcal{E}_2) =", "s_i(c_1^A(\\mathcal{L}_{1, 1}) + c_1^A(\\mathcal{L}_{2, 1}),", "\\ldots, c_1^A(\\mathcal{L}_{1, r_1}) + c_1^A(\\mathcal{L}_{2, r_2}))", "$$", "By the definition of the polynomials $P_i$ we see that this means", "$$", "P_i(c^A_1(\\mathcal{E}_1), \\ldots, c^A_i(\\mathcal{E}_1)) =", "\\sum\\nolimits_{j = 1, \\ldots, r_1} c_1^A(\\mathcal{L}_{1, j})^i", "$$", "and similarly for $\\mathcal{E}_2$. In case (1) we have also", "$$", "P_i(c^A_1(\\mathcal{E}), \\ldots, c^A_i(\\mathcal{E})) =", "\\sum\\nolimits_{j = 1, \\ldots, r_1} c_1^A(\\mathcal{L}_{1, j})^i +", "\\sum\\nolimits_{j = 1, \\ldots, r_2} c_1^A(\\mathcal{L}_{2, j})^i", "$$", "In case (2) we get accordingly", "$$", "P_i(c^A_1(\\mathcal{E}_1 \\otimes \\mathcal{E}_2), \\ldots,", "c^A_i(\\mathcal{E}_1 \\otimes \\mathcal{E}_2)) =", "\\sum\\nolimits_{j = 1, \\ldots, r_1}", "\\sum\\nolimits_{j' = 1, \\ldots, r_2}", "(c_1^A(\\mathcal{L}_{1, j}) + c_1^A(\\mathcal{L}_{2, j'}))^i", "$$", "Thus the desired equalities are now consequences of elementary", "identities between symmetric polynomials.", "\\medskip\\noindent", "We omit the proof of uniqueness." ], "refs": [ "weil-proposition-chern-class", "weil-lemma-splitting-principle" ], "ref_ids": [ 5111, 5089 ] } ], "ref_ids": [] }, { "id": 5113, "type": "theorem", "label": "weil-proposition-get-weil", "categories": [ "weil" ], "title": "weil-proposition-get-weil", "contents": [ "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8).", "Then we have a Weil cohomology theory." ], "refs": [], "proofs": [ { "contents": [ "We have axioms (A), (B) and (C)(a), (C)(c), and (C)(d) of", "Section \\ref{section-axioms} by", "Lemmas \\ref{lemma-poincare-duality}, \\ref{lemma-trace-product}, and", "\\ref{lemma-trace-base}.", "We have axiom (C)(b) by", "Lemma \\ref{lemma-done}.", "Finally, the additional condition of", "Definition \\ref{definition-weil-cohomology-theory}", "holds because it is the same as our axiom (A8)." ], "refs": [ "weil-lemma-poincare-duality", "weil-lemma-trace-product", "weil-lemma-trace-base", "weil-lemma-done", "weil-definition-weil-cohomology-theory" ], "ref_ids": [ 5097, 5098, 5099, 5107, 5116 ] } ], "ref_ids": [] }, { "id": 5123, "type": "theorem", "label": "morphisms-theorem-chevalley", "categories": [ "morphisms" ], "title": "morphisms-theorem-chevalley", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume $f$ is quasi-compact and locally of finite presentation.", "Then the image of every locally constructible subset is locally constructible." ], "refs": [], "proofs": [ { "contents": [ "Let $E \\subset X$ be locally constructible.", "We have to show that $f(E)$ is locally constructible too.", "We will show that $f(E) \\cap V$ is constructible for any affine", "open $V \\subset Y$. Thus we reduce to the case where $Y$ is affine.", "In this case $X$ is quasi-compact. Hence we can write", "$X = U_1 \\cup \\ldots \\cup U_n$ with each $U_i$ affine open in $X$.", "If $E \\subset X$ is locally constructible, then each $E \\cap U_i$", "is constructible, see", "Properties, Lemma \\ref{properties-lemma-locally-constructible}.", "Hence, since $f(E) = \\bigcup f(E \\cap U_i)$ and since finite unions of", "constructible sets are constructible, this reduces us to the case where $X$", "is affine. In this case the result is", "Algebra, Theorem \\ref{algebra-theorem-chevalley}." ], "refs": [ "properties-lemma-locally-constructible", "algebra-theorem-chevalley" ], "ref_ids": [ 2938, 315 ] } ], "ref_ids": [] }, { "id": 5124, "type": "theorem", "label": "morphisms-theorem-main-theorem", "categories": [ "morphisms" ], "title": "morphisms-theorem-main-theorem", "contents": [ "Let $f : Y \\to X$ be an affine morphism of schemes.", "Assume $f$ is of finite type.", "Let $X'$ be the normalization of $X$ in $Y$. Picture:", "$$", "\\xymatrix{", "Y \\ar[rd]_f \\ar[rr]_{f'} & & X' \\ar[ld]^\\nu \\\\", "& X &", "}", "$$", "Then there exists an open subscheme $U' \\subset X'$ such that", "\\begin{enumerate}", "\\item $(f')^{-1}(U') \\to U'$ is an isomorphism, and", "\\item $(f')^{-1}(U') \\subset Y$ is the set of points at which", "$f$ is quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "There is an immediate reduction to the case where $X$ and hence $Y$", "are affine. Say $X = \\Spec(R)$ and $Y = \\Spec(A)$.", "Then $X' = \\Spec(A')$, where $A'$ is the integral closure of", "$R$ in $A$, see Definitions \\ref{definition-integral-closure}", "and \\ref{definition-normalization-X-in-Y}. By", "Algebra, Theorem \\ref{algebra-theorem-main-theorem}", "for every $y \\in Y$ at which $f$ is quasi-finite, there exists an", "open $U'_y \\subset X'$ such that $(f')^{-1}(U'_y) \\to U'_y$", "is an isomorphism. Set $U' = \\bigcup U'_y$ where $y \\in Y$ ranges", "over all points where $f$ is quasi-finite. It remains to show that", "$f$ is quasi-finite at all points of $(f')^{-1}(U')$.", "If $y \\in (f')^{-1}(U')$ with image $x \\in X$, then we see that", "$Y_x \\to X'_x$ is an isomorphism in a neighbourhood of $y$. Hence", "there is no point of $Y_x$ which specializes to $y$, since this", "is true for $f'(y)$ in $X'_x$, see Lemma \\ref{lemma-integral-fibres}.", "By Lemma \\ref{lemma-quasi-finite-at-point-characterize} part (3)", "this implies $f$ is quasi-finite at $y$." ], "refs": [ "morphisms-definition-integral-closure", "morphisms-definition-normalization-X-in-Y", "algebra-theorem-main-theorem", "morphisms-lemma-integral-fibres", "morphisms-lemma-quasi-finite-at-point-characterize" ], "ref_ids": [ 5590, 5591, 325, 5442, 5226 ] } ], "ref_ids": [] }, { "id": 5125, "type": "theorem", "label": "morphisms-lemma-closed-immersion", "categories": [ "morphisms" ], "title": "morphisms-lemma-closed-immersion", "contents": [ "Let $i : Z \\to X$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $i$ is a closed immersion.", "\\item For every affine open $\\Spec(R) = U \\subset X$,", "there exists an ideal $I \\subset R$ such that", "$i^{-1}(U) = \\Spec(R/I)$ as schemes over $U = \\Spec(R)$.", "\\item There exists an affine open covering $X = \\bigcup_{j \\in J} U_j$,", "$U_j = \\Spec(R_j)$ and for every $j \\in J$ there exists", "an ideal $I_j \\subset R_j$ such that", "$i^{-1}(U_j) = \\Spec(R_j/I_j)$ as schemes over $U_j = \\Spec(R_j)$.", "\\item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset", "of $X$ and $i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective.", "\\item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset", "of $X$, the map $i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective,", "and the kernel $\\Ker(i^\\sharp)\\subset \\mathcal{O}_X$ is a quasi-coherent", "sheaf of ideals.", "\\item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset", "of $X$, the map $i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective,", "and the kernel $\\Ker(i^\\sharp)\\subset \\mathcal{O}_X$ is a", "sheaf of ideals which is locally generated by sections.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Condition (6) is our definition of a closed immersion, see Schemes,", "Definitions \\ref{schemes-definition-closed-immersion-locally-ringed-spaces}", "and \\ref{schemes-definition-immersion}.", "So (6) $\\Leftrightarrow$ (1). We have (1) $\\Rightarrow$ (2) by", "Schemes, Lemma \\ref{schemes-lemma-closed-subspace-scheme}.", "Trivially (2) $\\Rightarrow$ (3).", "\\medskip\\noindent", "Assume (3). Each of the morphisms", "$\\Spec(R_j/I_j) \\to \\Spec(R_j)$ is", "a closed immersion, see", "Schemes, Example \\ref{schemes-example-closed-immersion-affines}.", "Hence $i^{-1}(U_j) \\to U_j$ is a homeomorphism onto its image", "and $i^\\sharp|_{U_j}$ is surjective. Hence $i$ is a homeomorphism", "onto its image and $i^\\sharp$ is surjective since this may be", "checked locally. We conclude that (3) $\\Rightarrow$ (4).", "\\medskip\\noindent", "The implication (4) $\\Rightarrow$ (1) is", "Schemes, Lemma \\ref{schemes-lemma-characterize-closed-immersions}.", "The implication (5) $\\Rightarrow$ (6) is trivial.", "And the implication (6) $\\Rightarrow$ (5) follows", "from Schemes, Lemma \\ref{schemes-lemma-closed-subspace-scheme}." ], "refs": [ "schemes-definition-closed-immersion-locally-ringed-spaces", "schemes-definition-immersion", "schemes-lemma-closed-subspace-scheme", "schemes-lemma-characterize-closed-immersions", "schemes-lemma-closed-subspace-scheme" ], "ref_ids": [ 7737, 7743, 7670, 7731, 7670 ] } ], "ref_ids": [] }, { "id": 5126, "type": "theorem", "label": "morphisms-lemma-closed-immersion-ideals", "categories": [ "morphisms" ], "title": "morphisms-lemma-closed-immersion-ideals", "contents": [ "Let $X$ be a scheme. Let $i : Z \\to X$ and $i' : Z' \\to X$", "be closed immersions and consider the ideal sheaves", "$\\mathcal{I} = \\Ker(i^\\sharp)$ and $\\mathcal{I}' = \\Ker((i')^\\sharp)$", "of $\\mathcal{O}_X$.", "\\begin{enumerate}", "\\item The morphism $i : Z \\to X$ factors as $Z \\to Z' \\to X$", "for some $a : Z \\to Z'$ if and only if $\\mathcal{I}' \\subset \\mathcal{I}$.", "If this happens, then $a$ is a closed immersion.", "\\item We have $Z \\cong Z'$ over $X$ if and only if", "$\\mathcal{I} = \\mathcal{I}'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from our discussion of closed subspaces in", "Schemes, Section \\ref{schemes-section-closed-immersion} especially", "Schemes, Lemmas", "\\ref{schemes-lemma-closed-immersion} and", "\\ref{schemes-lemma-characterize-closed-subspace}.", "It also follows in a straightforward way from characterization", "(3) in Lemma \\ref{lemma-closed-immersion} above." ], "refs": [ "schemes-lemma-closed-immersion", "schemes-lemma-characterize-closed-subspace", "morphisms-lemma-closed-immersion" ], "ref_ids": [ 7647, 7648, 5125 ] } ], "ref_ids": [] }, { "id": 5127, "type": "theorem", "label": "morphisms-lemma-closed-immersion-bijection-ideals", "categories": [ "morphisms" ], "title": "morphisms-lemma-closed-immersion-bijection-ideals", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a sheaf of ideals.", "The following are equivalent:", "\\begin{enumerate}", "\\item $\\mathcal{I}$ is locally generated by", "sections as a sheaf of $\\mathcal{O}_X$-modules,", "\\item $\\mathcal{I}$ is quasi-coherent as", "a sheaf of $\\mathcal{O}_X$-modules, and", "\\item there exists a closed immersion $i : Z \\to X$ of schemes whose", "corresponding sheaf of ideals $\\Ker(i^\\sharp)$ is equal to $\\mathcal{I}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is immediate from", "Schemes, Lemma \\ref{schemes-lemma-closed-subspace-scheme}.", "If (1) holds, then there is a closed subspace $i : Z \\to X$", "with $\\mathcal{I} = \\Ker(i^\\sharp)$ by", "Schemes, Definition \\ref{schemes-definition-closed-subspace}", "and Example \\ref{schemes-example-closed-subspace}.", "By Schemes, Lemma \\ref{schemes-lemma-closed-subspace-scheme}", "this is a closed immersion of schemes and (3) holds.", "Conversely, if (3) holds, then (2) holds by", "Schemes, Lemma \\ref{schemes-lemma-closed-subspace-scheme}", "(which applies because a closed immersion of schemes is a fortiori a", "closed immersion of locally ringed spaces)." ], "refs": [ "schemes-lemma-closed-subspace-scheme", "schemes-definition-closed-subspace", "schemes-lemma-closed-subspace-scheme", "schemes-lemma-closed-subspace-scheme" ], "ref_ids": [ 7670, 7738, 7670, 7670 ] } ], "ref_ids": [] }, { "id": 5128, "type": "theorem", "label": "morphisms-lemma-base-change-closed-immersion", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-closed-immersion", "contents": [ "The base change of a closed immersion is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "See Schemes, Lemma \\ref{schemes-lemma-base-change-immersion}." ], "refs": [ "schemes-lemma-base-change-immersion" ], "ref_ids": [ 7695 ] } ], "ref_ids": [] }, { "id": 5129, "type": "theorem", "label": "morphisms-lemma-composition-closed-immersion", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-closed-immersion", "contents": [ "A composition of closed immersions is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "We have seen this in", "Schemes, Lemma \\ref{schemes-lemma-composition-immersion},", "but here is another", "proof. Namely, it follows from the characterization (3) of closed immersions", "in Lemma \\ref{lemma-closed-immersion}. Since if $I \\subset R$", "is an ideal, and $\\overline{J} \\subset R/I$ is an ideal, then", "$\\overline{J} = J/I$ for some ideal $J \\subset R$ which contains", "$I$ and $(R/I)/\\overline{J} = R/J$." ], "refs": [ "schemes-lemma-composition-immersion", "morphisms-lemma-closed-immersion" ], "ref_ids": [ 7732, 5125 ] } ], "ref_ids": [] }, { "id": 5130, "type": "theorem", "label": "morphisms-lemma-closed-immersion-quasi-compact", "categories": [ "morphisms" ], "title": "morphisms-lemma-closed-immersion-quasi-compact", "contents": [ "A closed immersion is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "This lemma is a duplicate of", "Schemes, Lemma \\ref{schemes-lemma-closed-immersion-quasi-compact}." ], "refs": [ "schemes-lemma-closed-immersion-quasi-compact" ], "ref_ids": [ 7700 ] } ], "ref_ids": [] }, { "id": 5131, "type": "theorem", "label": "morphisms-lemma-closed-immersion-separated", "categories": [ "morphisms" ], "title": "morphisms-lemma-closed-immersion-separated", "contents": [ "A closed immersion is separated." ], "refs": [], "proofs": [ { "contents": [ "This lemma is a special case of", "Schemes, Lemma \\ref{schemes-lemma-immersions-monomorphisms}." ], "refs": [ "schemes-lemma-immersions-monomorphisms" ], "ref_ids": [ 7727 ] } ], "ref_ids": [] }, { "id": 5132, "type": "theorem", "label": "morphisms-lemma-immersion-permanence", "categories": [ "morphisms" ], "title": "morphisms-lemma-immersion-permanence", "contents": [ "Let $Z \\to Y \\to X$ be morphisms of schemes.", "\\begin{enumerate}", "\\item If $Z \\to X$ is an immersion, then $Z \\to Y$ is an immersion.", "\\item If $Z \\to X$ is a quasi-compact immersion and $Y \\to X$ is", "quasi-separated, then $Z \\to Y$ is a quasi-compact immersion.", "\\item If $Z \\to X$ is a closed immersion and $Y \\to X$ is separated,", "then $Z \\to Y$ is a closed immersion.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In each case the proof is to contemplate the commutative diagram", "$$", "\\xymatrix{", "Z \\ar[r] \\ar[rd] & Y \\times_X Z \\ar[r] \\ar[d] & Z \\ar[d] \\\\", "& Y \\ar[r] & X", "}", "$$", "where the composition of the top horizontal arrows is the identity.", "Let us prove (1). The first horizontal arrow is a section of", "$Y \\times_X Z \\to Z$, whence an immersion by", "Schemes, Lemma \\ref{schemes-lemma-section-immersion}.", "The arrow $Y \\times_X Z \\to Y$ is a base change of $Z \\to X$ hence", "an immersion (Schemes, Lemma \\ref{schemes-lemma-base-change-immersion}).", "Finally, a composition of immersions is an immersion", "(Schemes, Lemma \\ref{schemes-lemma-composition-immersion}). This proves (1).", "The other two results are proved in exactly the same manner." ], "refs": [ "schemes-lemma-base-change-immersion", "schemes-lemma-composition-immersion" ], "ref_ids": [ 7695, 7732 ] } ], "ref_ids": [] }, { "id": 5133, "type": "theorem", "label": "morphisms-lemma-factor-quasi-compact-immersion", "categories": [ "morphisms" ], "title": "morphisms-lemma-factor-quasi-compact-immersion", "contents": [ "Let $h : Z \\to X$ be an immersion.", "If $h$ is quasi-compact, then we can factor", "$h = i \\circ j$ with $j : Z \\to \\overline{Z}$ an", "open immersion and $i : \\overline{Z} \\to X$ a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Note that $h$ is quasi-compact and quasi-separated (see", "Schemes, Lemma \\ref{schemes-lemma-immersions-monomorphisms}).", "Hence $h_*\\mathcal{O}_Z$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules", "by Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}.", "This implies that", "$\\mathcal{I} = \\Ker(\\mathcal{O}_X \\to h_*\\mathcal{O}_Z)$", "is a quasi-coherent sheaf of ideals, see", "Schemes, Section \\ref{schemes-section-quasi-coherent}.", "Let $\\overline{Z} \\subset X$ be the closed subscheme corresponding", "to $\\mathcal{I}$, see Lemma \\ref{lemma-closed-immersion-bijection-ideals}.", "By Schemes, Lemma \\ref{schemes-lemma-characterize-closed-subspace}", "the morphism $h$ factors as", "$h = i \\circ j$ where $i : \\overline{Z} \\to X$ is the inclusion morphism.", "To see that $j$ is an open immersion, choose an open subscheme", "$U \\subset X$ such that $h$ induces a closed immersion of $Z$", "into $U$. Then it is clear that $\\mathcal{I}|_U$ is the", "sheaf of ideals corresponding to the closed immersion $Z \\to U$.", "Hence we see that $Z = \\overline{Z} \\cap U$." ], "refs": [ "schemes-lemma-immersions-monomorphisms", "schemes-lemma-push-forward-quasi-coherent", "morphisms-lemma-closed-immersion-bijection-ideals", "schemes-lemma-characterize-closed-subspace" ], "ref_ids": [ 7727, 7730, 5127, 7648 ] } ], "ref_ids": [] }, { "id": 5134, "type": "theorem", "label": "morphisms-lemma-factor-reduced-immersion", "categories": [ "morphisms" ], "title": "morphisms-lemma-factor-reduced-immersion", "contents": [ "Let $h : Z \\to X$ be an immersion.", "If $Z$ is reduced, then we can factor", "$h = i \\circ j$ with $j : Z \\to \\overline{Z}$ an", "open immersion and $i : \\overline{Z} \\to X$ a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Let $\\overline{Z} \\subset X$ be the closure of $h(Z)$ with the reduced", "induced closed subscheme structure, see", "Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme}.", "By Schemes, Lemma \\ref{schemes-lemma-map-into-reduction}", "the morphism $h$ factors as", "$h = i \\circ j$ with $i : \\overline{Z} \\to X$ the inclusion morphism", "and $j : Z \\to \\overline{Z}$. From the definition of an immersion we", "see there exists an open subscheme $U \\subset X$ such that", "$h$ factors through a closed immersion into $U$. Hence", "$\\overline{Z} \\cap U$ and $h(Z)$ are reduced closed subschemes", "of $U$ with the same underlying closed set. Hence by the uniqueness", "in Schemes, Lemma \\ref{schemes-lemma-reduced-closed-subscheme}", "we see that $h(Z) \\cong \\overline{Z} \\cap U$.", "So $j$ induces an isomorphism of $Z$ with $\\overline{Z} \\cap U$.", "In other words $j$ is an open immersion." ], "refs": [ "schemes-definition-reduced-induced-scheme", "schemes-lemma-map-into-reduction", "schemes-lemma-reduced-closed-subscheme" ], "ref_ids": [ 7745, 7682, 7681 ] } ], "ref_ids": [] }, { "id": 5135, "type": "theorem", "label": "morphisms-lemma-check-immersion", "categories": [ "morphisms" ], "title": "morphisms-lemma-check-immersion", "contents": [ "Let $f : Y \\to X$ be a morphism of schemes. If for all $y \\in Y$", "there is an open subscheme $f(y) \\in U \\subset X$ such that", "$f|_{f^{-1}(U)} : f^{-1}(U) \\to U$ is an immersion, then $f$ is", "an immersion." ], "refs": [], "proofs": [ { "contents": [ "This statement follows readily from the discussion of closed subschemes", "at the end of Schemes, Section \\ref{schemes-section-immersions}", "but we will also give a detailed proof.", "Let $Z \\subset X$ be the closure of $f(X)$. Since taking closures", "commutes with restricting to opens, we see from the assumption that", "$f(Y) \\subset Z$ is open. Hence $Z' = Z \\setminus f(Y)$ is closed.", "Hence $X' = X \\setminus Z'$ is an open subscheme of $X$ and", "$f$ factors as $f : Y \\to X'$ followed by the inclusion.", "If $y \\in Y$ and $U \\subset X$ is as in the statement", "of the lemma, then $U' = X' \\cap U$ is an open neighbourhood", "of $f'(y)$ such that $(f')^{-1}(U') \\to U'$ is an immersion", "(Lemma \\ref{lemma-immersion-permanence}) with closed image.", "Hence it is a closed immersion, see", "Schemes, Lemma \\ref{schemes-lemma-immersion-when-closed}.", "Since being a closed immersion is local on the target", "(for example by Lemma \\ref{lemma-closed-immersion})", "we conclude that $f'$ is a closed immersion as desired." ], "refs": [ "morphisms-lemma-immersion-permanence", "schemes-lemma-immersion-when-closed", "morphisms-lemma-closed-immersion" ], "ref_ids": [ 5132, 7671, 5125 ] } ], "ref_ids": [] }, { "id": 5136, "type": "theorem", "label": "morphisms-lemma-i-star-equivalence", "categories": [ "morphisms" ], "title": "morphisms-lemma-i-star-equivalence", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes. Let", "$\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent sheaf of ideals", "cutting out $Z$. The functor", "$$", "i_* :", "\\QCoh(\\mathcal{O}_Z)", "\\longrightarrow", "\\QCoh(\\mathcal{O}_X)", "$$", "is exact, fully faithful, with essential image those quasi-coherent", "$\\mathcal{O}_X$-modules $\\mathcal{G}$ such that $\\mathcal{I}\\mathcal{G} = 0$." ], "refs": [], "proofs": [ { "contents": [ "A closed immersion is quasi-compact and separated, see", "Lemmas \\ref{lemma-closed-immersion-quasi-compact} and", "\\ref{lemma-closed-immersion-separated}. Hence", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "applies and the pushforward of a quasi-coherent", "sheaf on $Z$ is indeed a quasi-coherent sheaf on $X$.", "\\medskip\\noindent", "By Modules, Lemma \\ref{modules-lemma-i-star-equivalence}", "the functor $i_*$ is fully faithful.", "\\medskip\\noindent", "Now we turn to the description of the essential image of the", "functor $i_*$. We have $\\mathcal{I}(i_*\\mathcal{F}) = 0$", "for any quasi-coherent $\\mathcal{O}_Z$-module, for example by", "Modules, Lemma \\ref{modules-lemma-i-star-equivalence}.", "Next, suppose that $\\mathcal{G}$", "is any quasi-coherent $\\mathcal{O}_X$-module such that", "$\\mathcal{I}\\mathcal{G} = 0$. It suffices to show that the canonical map", "$$", "\\mathcal{G} \\longrightarrow i_* i^*\\mathcal{G}", "$$", "is an isomorphism\\footnote{This was proved in a more general situation", "in the proof of Modules, Lemma \\ref{modules-lemma-i-star-equivalence}.}.", "In the case of schemes and quasi-coherent modules, working affine locally", "on $X$ and using Lemma \\ref{lemma-closed-immersion} and", "Schemes, Lemma \\ref{schemes-lemma-widetilde-pullback}", "it suffices to prove the following algebraic statement: Given a ring", "$R$, an ideal $I$ and an $R$-module $N$ such that $IN = 0$ the canonical map", "$$", "N \\longrightarrow N \\otimes_R R/I,\\quad", "n \\longmapsto n \\otimes 1", "$$", "is an isomorphism of $R$-modules. Proof of this easy algebra fact is omitted." ], "refs": [ "morphisms-lemma-closed-immersion-quasi-compact", "morphisms-lemma-closed-immersion-separated", "schemes-lemma-push-forward-quasi-coherent", "modules-lemma-i-star-equivalence", "modules-lemma-i-star-equivalence", "modules-lemma-i-star-equivalence", "morphisms-lemma-closed-immersion", "schemes-lemma-widetilde-pullback" ], "ref_ids": [ 5130, 5131, 7730, 13260, 13260, 13260, 5125, 7662 ] } ], "ref_ids": [] }, { "id": 5137, "type": "theorem", "label": "morphisms-lemma-largest-quasi-coherent-subsheaf", "categories": [ "morphisms" ], "title": "morphisms-lemma-largest-quasi-coherent-subsheaf", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. Let $\\mathcal{G} \\subset \\mathcal{F}$", "be a $\\mathcal{O}_X$-submodule. There exists a unique quasi-coherent", "$\\mathcal{O}_X$-submodule $\\mathcal{G}' \\subset \\mathcal{G}$", "with the following property: For every quasi-coherent $\\mathcal{O}_X$-module", "$\\mathcal{H}$ the map", "$$", "\\Hom_{\\mathcal{O}_X}(\\mathcal{H}, \\mathcal{G}')", "\\longrightarrow", "\\Hom_{\\mathcal{O}_X}(\\mathcal{H}, \\mathcal{G})", "$$", "is bijective. In particular $\\mathcal{G}'$ is the largest quasi-coherent", "$\\mathcal{O}_X$-submodule of $\\mathcal{F}$ contained in $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{G}_a$, $a \\in A$ be the set of quasi-coherent", "$\\mathcal{O}_X$-submodules contained in $\\mathcal{G}$.", "Then the image $\\mathcal{G}'$ of", "$$", "\\bigoplus\\nolimits_{a \\in A} \\mathcal{G}_a \\longrightarrow \\mathcal{F}", "$$", "is quasi-coherent as the image of a map of quasi-coherent sheaves", "on $X$ is quasi-coherent and since a direct sum of quasi-coherent sheaves", "is quasi-coherent, see", "Schemes, Section \\ref{schemes-section-quasi-coherent}.", "The module $\\mathcal{G}'$ is contained in $\\mathcal{G}$. Hence this is the", "largest quasi-coherent $\\mathcal{O}_X$-module contained in $\\mathcal{G}$.", "\\medskip\\noindent", "To prove the formula, let $\\mathcal{H}$ be a quasi-coherent", "$\\mathcal{O}_X$-module and let $\\alpha : \\mathcal{H} \\to \\mathcal{G}$", "be an $\\mathcal{O}_X$-module map. The image of the composition", "$\\mathcal{H} \\to \\mathcal{G} \\to \\mathcal{F}$ is quasi-coherent", "as the image of a map of quasi-coherent sheaves. Hence it is contained", "in $\\mathcal{G}'$. Hence $\\alpha$ factors through $\\mathcal{G}'$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5138, "type": "theorem", "label": "morphisms-lemma-i-upper-shriek", "categories": [ "morphisms" ], "title": "morphisms-lemma-i-upper-shriek", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes.", "There is a functor\\footnote{This is likely nonstandard notation.}", "$i^! : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Z)$", "which is a right adjoint to $i_*$. (Compare", "Modules, Lemma \\ref{modules-lemma-i-star-right-adjoint}.)" ], "refs": [ "modules-lemma-i-star-right-adjoint" ], "proofs": [ { "contents": [ "Given quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{G}$ we consider", "the subsheaf $\\mathcal{H}_Z(\\mathcal{G})$ of $\\mathcal{G}$ of local sections", "annihilated by $\\mathcal{I}$. By", "Lemma \\ref{lemma-largest-quasi-coherent-subsheaf}", "there is a canonical largest quasi-coherent $\\mathcal{O}_X$-submodule", "$\\mathcal{H}_Z(\\mathcal{G})'$. By construction we have", "$$", "\\Hom_{\\mathcal{O}_X}(i_*\\mathcal{F}, \\mathcal{H}_Z(\\mathcal{G})')", "=", "\\Hom_{\\mathcal{O}_X}(i_*\\mathcal{F}, \\mathcal{G})", "$$", "for any quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{F}$.", "Hence we can set $i^!\\mathcal{G} = i^*(\\mathcal{H}_Z(\\mathcal{G})')$.", "Details omitted." ], "refs": [ "morphisms-lemma-largest-quasi-coherent-subsheaf" ], "ref_ids": [ 5137 ] } ], "ref_ids": [ 13233 ] }, { "id": 5139, "type": "theorem", "label": "morphisms-lemma-scheme-theoretic-intersection", "categories": [ "morphisms" ], "title": "morphisms-lemma-scheme-theoretic-intersection", "contents": [ "Let $X$ be a scheme. Let $Z, Y \\subset X$ be closed subschemes.", "Let $Z \\cap Y$ be the scheme theoretic intersection of $Z$ and $Y$.", "Then $Z \\cap Y \\to Z$ and $Z \\cap Y \\to Y$ are closed immersions", "and", "$$", "\\xymatrix{", "Z \\cap Y \\ar[r] \\ar[d] & Z \\ar[d] \\\\", "Y \\ar[r] & X", "}", "$$", "is a cartesian diagram of schemes, i.e., $Z \\cap Y = Z \\times_X Y$." ], "refs": [], "proofs": [ { "contents": [ "The morphisms $Z \\cap Y \\to Z$ and $Z \\cap Y \\to Y$ are closed immersions", "by Lemma \\ref{lemma-closed-immersion-ideals}.", "Let $U = \\Spec(A)$ be an affine open of $X$ and let $Z \\cap U$ and $Y \\cap U$", "correspond to the ideals $I \\subset A$ and $J \\subset A$. Then", "$Z \\cap Y \\cap U$ corresponds to $I + J \\subset A$. Since", "$A/I \\otimes_A A/J = A/(I + J)$ we see that the diagram is", "cartesian by our description of fibre products of schemes", "in Schemes, Section \\ref{schemes-section-fibre-products}." ], "refs": [ "morphisms-lemma-closed-immersion-ideals" ], "ref_ids": [ 5126 ] } ], "ref_ids": [] }, { "id": 5140, "type": "theorem", "label": "morphisms-lemma-scheme-theoretic-union", "categories": [ "morphisms" ], "title": "morphisms-lemma-scheme-theoretic-union", "contents": [ "Let $S$ be a scheme. Let $X, Y \\subset S$ be closed subschemes.", "Let $X \\cup Y$ be the scheme theoretic union of $X$ and $Y$.", "Let $X \\cap Y$ be the scheme theoretic intersection of $X$ and $Y$.", "Then $X \\to X \\cup Y$ and $Y \\to X \\cup Y$ are closed immersions, there is a", "short exact sequence", "$$", "0 \\to \\mathcal{O}_{X \\cup Y} \\to \\mathcal{O}_X \\times \\mathcal{O}_Y", "\\to \\mathcal{O}_{X \\cap Y} \\to 0", "$$", "of $\\mathcal{O}_S$-modules, and the diagram", "$$", "\\xymatrix{", "X \\cap Y \\ar[r] \\ar[d] & X \\ar[d] \\\\", "Y \\ar[r] & X \\cup Y", "}", "$$", "is cocartesian in the category of schemes, i.e.,", "$X \\cup Y = X \\amalg_{X \\cap Y} Y$." ], "refs": [], "proofs": [ { "contents": [ "The morphisms $X \\to X \\cup Y$ and $Y \\to X \\cup Y$ are closed immersions", "by Lemma \\ref{lemma-closed-immersion-ideals}. In the short exact sequence", "we use the equivalence of Lemma \\ref{lemma-i-star-equivalence} to think of", "quasi-coherent modules on closed subschemes of $S$ as quasi-coherent modules", "on $S$. For the first map in the sequence we use the canonical maps", "$\\mathcal{O}_{X \\cup Y} \\to \\mathcal{O}_X$ and", "$\\mathcal{O}_{X \\cup Y} \\to \\mathcal{O}_Y$", "and for the second map we use the canonical map", "$\\mathcal{O}_X \\to \\mathcal{O}_{X \\cap Y}$ and", "the negative of the canonical map", "$\\mathcal{O}_Y \\to \\mathcal{O}_{X \\cap Y}$. Then to check", "exactness we may work affine locally.", "Let $U = \\Spec(A)$ be an affine open of $S$ and let $X \\cap U$ and $Y \\cap U$", "correspond to the ideals $I \\subset A$ and $J \\subset A$. Then", "$(X \\cup Y) \\cap U$ corresponds to $I \\cap J \\subset A$", "and $X \\cap Y \\cap U$ corresponds to $I + J \\subset A$.", "Thus exactness follows from the exactness of", "$$", "0 \\to A/I \\cap J \\to A/I \\times A/J \\to A/(I + J) \\to 0", "$$", "To show the diagram is cocartesian, suppose we are given a scheme $T$", "and morphisms of schemes $f : X \\to T$, $g : Y \\to T$ agreeing", "as morphisms $X \\cap Y \\to T$. Goal: Show there exists a unique", "morphism $h : X \\cup Y \\to T$ agreeing with $f$ and $g$.", "To construct $h$ we may work affine locally on $X \\cup Y$, see", "Schemes, Section \\ref{schemes-section-glueing-schemes}.", "If $s \\in X$, $s \\not \\in Y$, then $X \\to X \\cup Y$ is", "an isomorphism in a neighbourhood of $s$ and it is clear", "how to construct $h$. Similarly for $s \\in Y$, $s \\not \\in X$.", "For $s \\in X \\cap Y$ we can pick an affine open", "$V = \\Spec(B) \\subset T$ containing $f(s) = g(s)$.", "Then we can choose an affine open $U = \\Spec(A) \\subset S$", "containing $s$ such that $f(X \\cap U)$ and $g(Y \\cap U)$", "are contained in $V$. The morphisms $f|_{X \\cap U}$", "and $g|_{Y \\cap V}$ into $V$ correspond to ring maps", "$$", "B \\to A/I", "\\quad\\text{and}\\quad", "B \\to A/J", "$$", "which agree as maps into $A/(I + J)$. By the short exact sequence", "displayed above there is a unique lift of these ring homomorphism", "to a ring map $B \\to A/I \\cap J$ as desired." ], "refs": [ "morphisms-lemma-closed-immersion-ideals", "morphisms-lemma-i-star-equivalence" ], "ref_ids": [ 5126, 5136 ] } ], "ref_ids": [] }, { "id": 5141, "type": "theorem", "label": "morphisms-lemma-support-affine-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-support-affine-open", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $\\Spec(A) = U \\subset X$ be an affine open, and set", "$M = \\Gamma(U, \\mathcal{F})$.", "Let $x \\in U$, and let $\\mathfrak p \\subset A$ be the corresponding prime.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathfrak p$ is in the support of $M$, and", "\\item $x$ is in the support of $\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from the equality $\\mathcal{F}_x = M_{\\mathfrak p}$, see", "Schemes, Lemma \\ref{schemes-lemma-spec-sheaves}", "and the definitions." ], "refs": [ "schemes-lemma-spec-sheaves" ], "ref_ids": [ 7651 ] } ], "ref_ids": [] }, { "id": 5142, "type": "theorem", "label": "morphisms-lemma-support-closed-specialization", "categories": [ "morphisms" ], "title": "morphisms-lemma-support-closed-specialization", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "The support of $\\mathcal{F}$ is closed under specialization." ], "refs": [], "proofs": [ { "contents": [ "If $x' \\leadsto x$ is a specialization and $\\mathcal{F}_x = 0$", "then $\\mathcal{F}_{x'}$ is zero, as $\\mathcal{F}_{x'}$ is a localization", "of the module $\\mathcal{F}_x$. Hence the complement of", "$\\text{Supp}(\\mathcal{F})$ is closed under generalization." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5143, "type": "theorem", "label": "morphisms-lemma-support-finite-type", "categories": [ "morphisms" ], "title": "morphisms-lemma-support-finite-type", "contents": [ "Let $\\mathcal{F}$ be a finite type quasi-coherent module", "on a scheme $X$. Then", "\\begin{enumerate}", "\\item The support of $\\mathcal{F}$ is closed.", "\\item For $x \\in X$ we have", "$$", "x \\in \\text{Supp}(\\mathcal{F})", "\\Leftrightarrow", "\\mathcal{F}_x \\not = 0", "\\Leftrightarrow", "\\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) \\not = 0.", "$$", "\\item For any morphism of schemes $f : Y \\to X$ the pullback", "$f^*\\mathcal{F}$ is of finite type as well and we have", "$\\text{Supp}(f^*\\mathcal{F}) = f^{-1}(\\text{Supp}(\\mathcal{F}))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is a reformulation of", "Modules, Lemma \\ref{modules-lemma-support-finite-type-closed}.", "You can also combine", "Lemma \\ref{lemma-support-affine-open},", "Properties, Lemma \\ref{properties-lemma-finite-type-module},", "and", "Algebra, Lemma \\ref{algebra-lemma-support-closed}", "to see this. The first equivalence in (2) is the definition", "of support, and the second equivalence follows from", "Nakayama's lemma, see", "Algebra, Lemma \\ref{algebra-lemma-NAK}.", "Let $f : Y \\to X$ be a morphism of schemes. Note that", "$f^*\\mathcal{F}$ is of finite type by", "Modules, Lemma \\ref{modules-lemma-pullback-finite-type}.", "For the final assertion, let $y \\in Y$ with image $x \\in X$.", "Recall that", "$$", "(f^*\\mathcal{F})_y =", "\\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\mathcal{O}_{Y, y},", "$$", "see", "Sheaves, Lemma \\ref{sheaves-lemma-stalk-pullback-modules}.", "Hence $(f^*\\mathcal{F})_y \\otimes \\kappa(y)$ is nonzero", "if and only if $\\mathcal{F}_x \\otimes \\kappa(x)$ is nonzero.", "By (2) this implies $x \\in \\text{Supp}(\\mathcal{F})$ if and only", "if $y \\in \\text{Supp}(f^*\\mathcal{F})$, which is the content of", "assertion (3)." ], "refs": [ "modules-lemma-support-finite-type-closed", "morphisms-lemma-support-affine-open", "properties-lemma-finite-type-module", "algebra-lemma-support-closed", "algebra-lemma-NAK", "modules-lemma-pullback-finite-type", "sheaves-lemma-stalk-pullback-modules" ], "ref_ids": [ 13240, 5141, 3002, 543, 401, 13236, 14523 ] } ], "ref_ids": [] }, { "id": 5144, "type": "theorem", "label": "morphisms-lemma-scheme-theoretic-support", "categories": [ "morphisms" ], "title": "morphisms-lemma-scheme-theoretic-support", "contents": [ "Let $\\mathcal{F}$ be a finite type quasi-coherent module", "on a scheme $X$. There exists a smallest closed subscheme", "$i : Z \\to X$ such that there exists a quasi-coherent", "$\\mathcal{O}_Z$-module $\\mathcal{G}$ with", "$i_*\\mathcal{G} \\cong \\mathcal{F}$. Moreover:", "\\begin{enumerate}", "\\item If $\\Spec(A) \\subset X$ is any affine open, and", "$\\mathcal{F}|_{\\Spec(A)} = \\widetilde{M}$ then", "$Z \\cap \\Spec(A) = \\Spec(A/I)$ where $I = \\text{Ann}_A(M)$.", "\\item The quasi-coherent sheaf $\\mathcal{G}$ is unique up to unique", "isomorphism.", "\\item The quasi-coherent sheaf $\\mathcal{G}$ is of finite type.", "\\item The support of $\\mathcal{G}$ and of $\\mathcal{F}$ is $Z$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $i' : Z' \\to X$ is a closed subscheme which satisfies the", "description on open affines from the lemma. Then by", "Lemma \\ref{lemma-i-star-equivalence}", "we see that $\\mathcal{F} \\cong i'_*\\mathcal{G}'$ for some unique", "quasi-coherent sheaf $\\mathcal{G}'$ on $Z'$. Furthermore, it is clear", "that $Z'$ is the smallest closed subscheme with this property (by the", "same lemma). Finally, using", "Properties, Lemma \\ref{properties-lemma-finite-type-module}", "and", "Algebra, Lemma \\ref{algebra-lemma-finite-over-subring}", "it follows that $\\mathcal{G}'$ is of finite type. We have", "$\\text{Supp}(\\mathcal{G}') = Z$ by", "Algebra, Lemma \\ref{algebra-lemma-support-closed}.", "Hence, in order to prove the lemma it suffices to show that", "the characterization in (1) actually does define a closed subscheme.", "And, in order to do this it suffices to prove that the given rule", "produces a quasi-coherent sheaf of ideals, see", "Lemma \\ref{lemma-closed-immersion-bijection-ideals}.", "This comes down to the following algebra fact: If $A$ is a ring, $f \\in A$,", "and $M$ is a finite $A$-module, then", "$\\text{Ann}_A(M)_f = \\text{Ann}_{A_f}(M_f)$.", "We omit the proof." ], "refs": [ "morphisms-lemma-i-star-equivalence", "properties-lemma-finite-type-module", "algebra-lemma-finite-over-subring", "algebra-lemma-support-closed", "morphisms-lemma-closed-immersion-bijection-ideals" ], "ref_ids": [ 5136, 3002, 332, 543, 5127 ] } ], "ref_ids": [] }, { "id": 5145, "type": "theorem", "label": "morphisms-lemma-scheme-theoretic-image", "categories": [ "morphisms" ], "title": "morphisms-lemma-scheme-theoretic-image", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. There exists a closed", "subscheme $Z \\subset Y$ such that $f$ factors through $Z$ and such", "that for any other closed subscheme $Z' \\subset Y$ such that $f$", "factors through $Z'$ we have $Z \\subset Z'$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} = \\Ker(\\mathcal{O}_Y \\to f_*\\mathcal{O}_X)$.", "If $\\mathcal{I}$ is quasi-coherent then we just take $Z$ to be the", "closed subscheme determined by $\\mathcal{I}$, see", "Lemma \\ref{lemma-closed-immersion-bijection-ideals}. This works by", "Schemes, Lemma \\ref{schemes-lemma-characterize-closed-subspace}.", "In general the same lemma requires us to show that there exists", "a largest quasi-coherent sheaf of ideals $\\mathcal{I}'$ contained in", "$\\mathcal{I}$.", "This follows from Lemma \\ref{lemma-largest-quasi-coherent-subsheaf}." ], "refs": [ "morphisms-lemma-closed-immersion-bijection-ideals", "schemes-lemma-characterize-closed-subspace", "morphisms-lemma-largest-quasi-coherent-subsheaf" ], "ref_ids": [ 5127, 7648, 5137 ] } ], "ref_ids": [] }, { "id": 5146, "type": "theorem", "label": "morphisms-lemma-quasi-compact-scheme-theoretic-image", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-compact-scheme-theoretic-image", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $Z \\subset Y$ be the scheme theoretic image of $f$.", "If $f$ is quasi-compact then", "\\begin{enumerate}", "\\item the sheaf of ideals", "$\\mathcal{I} = \\Ker(\\mathcal{O}_Y \\to f_*\\mathcal{O}_X)$", "is quasi-coherent,", "\\item the scheme theoretic image $Z$ is the closed subscheme", "determined by $\\mathcal{I}$,", "\\item for any open $U \\subset Y$ the scheme theoretic image of", "$f|_{f^{-1}(U)} : f^{-1}(U) \\to U$ is equal to $Z \\cap U$, and", "\\item the image $f(X) \\subset Z$ is a dense subset of $Z$, in other", "words the morphism $X \\to Z$ is dominant", "(see Definition \\ref{definition-dominant}).", "\\end{enumerate}" ], "refs": [ "morphisms-definition-dominant" ], "proofs": [ { "contents": [ "Part (4) follows from part (3). To show (3) it suffices", "to prove (1) since the formation of $\\mathcal{I}$ commutes with restriction to", "open subschemes of $Y$. And if (1) holds then in the proof of", "Lemma \\ref{lemma-scheme-theoretic-image}", "we showed (2). Thus it suffices to prove that $\\mathcal{I}$ is quasi-coherent.", "Since the property of being quasi-coherent is", "local we may assume $Y$ is affine. As $f$ is quasi-compact,", "we can find a finite affine open covering", "$X = \\bigcup_{i = 1, \\ldots, n} U_i$. Denote $f'$ the composition", "$$", "X' = \\coprod U_i \\longrightarrow X \\longrightarrow Y.", "$$", "Then $f_*\\mathcal{O}_X$ is a subsheaf of $f'_*\\mathcal{O}_{X'}$,", "and hence $\\mathcal{I} = \\Ker(\\mathcal{O}_Y \\to f'_*\\mathcal{O}_{X'})$.", "By Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "the sheaf $f'_*\\mathcal{O}_{X'}$ is quasi-coherent on $Y$. Hence we win." ], "refs": [ "morphisms-lemma-scheme-theoretic-image", "schemes-lemma-push-forward-quasi-coherent" ], "ref_ids": [ 5145, 7730 ] } ], "ref_ids": [ 5541 ] }, { "id": 5147, "type": "theorem", "label": "morphisms-lemma-reach-points-scheme-theoretic-image", "categories": [ "morphisms" ], "title": "morphisms-lemma-reach-points-scheme-theoretic-image", "contents": [ "Let $f : X \\to Y$ be a quasi-compact morphism.", "Let $Z$ be the scheme theoretic image of $f$.", "Let $z \\in Z$\\footnote{By", "Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image} set-theoretically", "$Z$ agrees with the closure of $f(X)$ in $Y$.}.", "There exists a valuation ring $A$ with", "fraction field $K$ and a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[rr] \\ar[d] & & X \\ar[d] \\ar[ld] \\\\", "\\Spec(A) \\ar[r] & Z \\ar[r] & Y", "}", "$$", "such that the closed point of $\\Spec(A)$ maps to $z$. In particular", "any point of $Z$ is the specialization of a point of $f(X)$." ], "refs": [ "morphisms-lemma-quasi-compact-scheme-theoretic-image" ], "proofs": [ { "contents": [ "Let $z \\in \\Spec(R) = V \\subset Y$ be an affine open", "neighbourhood of $z$. By", "Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image}", "the intersection $Z \\cap V$ is the scheme theoretic image of", "$f^{-1}(V) \\to V$. Hence we may replace $Y$ by $V$", "and assume $Y = \\Spec(R)$ is affine.", "In this case $X$ is quasi-compact as $f$ is quasi-compact.", "Say $X = U_1 \\cup \\ldots \\cup U_n$", "is a finite affine open covering. Write $U_i = \\Spec(A_i)$.", "Let $I = \\Ker(R \\to A_1 \\times \\ldots \\times A_n)$.", "By Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image}", "again we see that $Z$ corresponds to the closed subscheme", "$\\Spec(R/I)$ of $Y$. If $\\mathfrak p \\subset R$ is", "the prime corresponding to $z$, then we see that", "$I_{\\mathfrak p} \\subset R_{\\mathfrak p}$ is not an", "equality. Hence (as localization is exact, see", "Algebra, Proposition \\ref{algebra-proposition-localization-exact})", "we see that", "$R_{\\mathfrak p} \\to", "(A_1)_{\\mathfrak p} \\times \\ldots \\times (A_n)_{\\mathfrak p}$", "is not zero. Hence one of the rings $(A_i)_{\\mathfrak p}$ is not zero.", "Hence there exists an $i$ and a prime $\\mathfrak q_i \\subset A_i$", "lying over a prime $\\mathfrak p_i \\subset \\mathfrak p$.", "By Algebra, Lemma \\ref{algebra-lemma-dominate} we can choose a valuation ring", "$A \\subset K = \\kappa(\\mathfrak q_i)$ dominating", "the local ring", "$R_{\\mathfrak p}/\\mathfrak p_iR_{\\mathfrak p} \\subset \\kappa(\\mathfrak q_i)$.", "This gives the desired diagram. Some details omitted." ], "refs": [ "morphisms-lemma-quasi-compact-scheme-theoretic-image", "morphisms-lemma-quasi-compact-scheme-theoretic-image", "algebra-proposition-localization-exact", "algebra-lemma-dominate" ], "ref_ids": [ 5146, 5146, 1402, 608 ] } ], "ref_ids": [ 5146 ] }, { "id": 5148, "type": "theorem", "label": "morphisms-lemma-factor-factor", "categories": [ "morphisms" ], "title": "morphisms-lemma-factor-factor", "contents": [ "Let", "$$", "\\xymatrix{", "X_1 \\ar[d] \\ar[r]_{f_1} & Y_1 \\ar[d] \\\\", "X_2 \\ar[r]^{f_2} & Y_2", "}", "$$", "be a commutative diagram of schemes. Let $Z_i \\subset Y_i$, $i = 1, 2$ be", "the scheme theoretic image of $f_i$. Then the morphism", "$Y_1 \\to Y_2$ induces a morphism $Z_1 \\to Z_2$ and a", "commutative diagram", "$$", "\\xymatrix{", "X_1 \\ar[r] \\ar[d] & Z_1 \\ar[d] \\ar[r] & Y_1 \\ar[d] \\\\", "X_2 \\ar[r] & Z_2 \\ar[r] & Y_2", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "The scheme theoretic inverse image of $Z_2$ in $Y_1$", "is a closed subscheme of $Y_1$ through", "which $f_1$ factors. Hence $Z_1$ is contained in this.", "This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5149, "type": "theorem", "label": "morphisms-lemma-scheme-theoretic-image-reduced", "categories": [ "morphisms" ], "title": "morphisms-lemma-scheme-theoretic-image-reduced", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "If $X$ is reduced, then the scheme theoretic image of $f$ is", "the reduced induced scheme structure on $\\overline{f(X)}$." ], "refs": [], "proofs": [ { "contents": [ "This is true because the reduced induced scheme structure on $\\overline{f(X)}$", "is clearly the smallest closed subscheme of $Y$ through which $f$ factors,", "see", "Schemes, Lemma \\ref{schemes-lemma-map-into-reduction}." ], "refs": [ "schemes-lemma-map-into-reduction" ], "ref_ids": [ 7682 ] } ], "ref_ids": [] }, { "id": 5150, "type": "theorem", "label": "morphisms-lemma-scheme-theoretic-image-of-partial-section", "categories": [ "morphisms" ], "title": "morphisms-lemma-scheme-theoretic-image-of-partial-section", "contents": [ "Let $f : X \\to Y$ be a separated morphism of schemes.", "Let $V \\subset Y$ be a retrocompact open. Let $s : V \\to X$", "be a morphism such that $f \\circ s = \\text{id}_V$.", "Let $Y'$ be the scheme theoretic image of $s$.", "Then $Y' \\to Y$ is an isomorphism over $V$." ], "refs": [], "proofs": [ { "contents": [ "The assumption that $V$ is retrocompact in $Y$", "(Topology, Definition \\ref{topology-definition-quasi-compact})", "means that $V \\to Y$ is a quasi-compact morphism.", "By Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}", "the morphism $s : V \\to X$ is quasi-compact.", "Hence the construction of the scheme theoretic image $Y'$", "of $s$ commutes with restriction to opens by", "Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image}.", "In particular, we see that $Y' \\cap f^{-1}(V)$ is the", "scheme theoretic image of a section of the separated", "morphism $f^{-1}(V) \\to V$. Since a section of a separated", "morphism is a closed immersion", "(Schemes, Lemma \\ref{schemes-lemma-section-immersion}),", "we conclude that", "$Y' \\cap f^{-1}(V) \\to V$ is an isomorphism as desired." ], "refs": [ "topology-definition-quasi-compact", "schemes-lemma-quasi-compact-permanence", "morphisms-lemma-quasi-compact-scheme-theoretic-image" ], "ref_ids": [ 8360, 7716, 5146 ] } ], "ref_ids": [] }, { "id": 5151, "type": "theorem", "label": "morphisms-lemma-scheme-theoretically-dense-quasi-compact", "categories": [ "morphisms" ], "title": "morphisms-lemma-scheme-theoretically-dense-quasi-compact", "contents": [ "Let $X$ be a scheme.", "Let $U \\subset X$ be an open subscheme.", "If the inclusion morphism $U \\to X$ is quasi-compact, then $U$", "is scheme theoretically dense in $X$ if and only if the scheme theoretic", "closure of $U$ in $X$ is $X$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image} part (3)." ], "refs": [ "morphisms-lemma-quasi-compact-scheme-theoretic-image" ], "ref_ids": [ 5146 ] } ], "ref_ids": [] }, { "id": 5152, "type": "theorem", "label": "morphisms-lemma-characterize-scheme-theoretically-dense", "categories": [ "morphisms" ], "title": "morphisms-lemma-characterize-scheme-theoretically-dense", "contents": [ "Let $j : U \\to X$ be an open immersion of schemes.", "Then $U$ is scheme theoretically dense in $X$ if and only if", "$\\mathcal{O}_X \\to j_*\\mathcal{O}_U$ is injective." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{O}_X \\to j_*\\mathcal{O}_U$ is injective,", "then the same is true when restricted to any open $V$ of $X$.", "Hence the scheme theoretic closure of $U \\cap V$ in $V$", "is equal to $V$, see proof of Lemma \\ref{lemma-scheme-theoretic-image}.", "Conversely, suppose that the scheme theoretic", "closure of $U \\cap V$ is equal to $V$ for all opens $V$.", "Suppose that $\\mathcal{O}_X \\to j_*\\mathcal{O}_U$ is not injective.", "Then we can find an affine open, say $\\Spec(A) = V \\subset X$", "and a nonzero element $f \\in A$ such that $f$ maps to zero in", "$\\Gamma(V \\cap U, \\mathcal{O}_X)$. In this case the scheme theoretic", "closure of $V \\cap U$ in $V$ is clearly contained in $\\Spec(A/(f))$", "a contradiction." ], "refs": [ "morphisms-lemma-scheme-theoretic-image" ], "ref_ids": [ 5145 ] } ], "ref_ids": [] }, { "id": 5153, "type": "theorem", "label": "morphisms-lemma-intersection-scheme-theoretically-dense", "categories": [ "morphisms" ], "title": "morphisms-lemma-intersection-scheme-theoretically-dense", "contents": [ "Let $X$ be a scheme. If $U$, $V$ are scheme theoretically dense", "open subschemes of $X$, then so is $U \\cap V$." ], "refs": [], "proofs": [ { "contents": [ "Let $W \\subset X$ be any open.", "Consider the map", "$\\mathcal{O}_X(W) \\to \\mathcal{O}_X(W \\cap V)", "\\to \\mathcal{O}_X(W \\cap V \\cap U)$.", "By Lemma \\ref{lemma-characterize-scheme-theoretically-dense}", "both maps are injective. Hence the composite is injective.", "Hence by Lemma \\ref{lemma-characterize-scheme-theoretically-dense}", "$U \\cap V$ is scheme theoretically dense in $X$." ], "refs": [ "morphisms-lemma-characterize-scheme-theoretically-dense", "morphisms-lemma-characterize-scheme-theoretically-dense" ], "ref_ids": [ 5152, 5152 ] } ], "ref_ids": [] }, { "id": 5154, "type": "theorem", "label": "morphisms-lemma-quasi-compact-immersion", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-compact-immersion", "contents": [ "Let $h : Z \\to X$ be an immersion. Assume either $h$ is quasi-compact", "or $Z$ is reduced. Let $\\overline{Z} \\subset X$ be the scheme theoretic", "image of $h$. Then the morphism $Z \\to \\overline{Z}$ is an open immersion", "which identifies $Z$ with a scheme theoretically dense open", "subscheme of $\\overline{Z}$. Moreover, $Z$ is topologically", "dense in $\\overline{Z}$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-factor-quasi-compact-immersion} or", "Lemma \\ref{lemma-factor-reduced-immersion} we can factor", "$Z \\to X$ as $Z \\to \\overline{Z}_1 \\to X$ with $Z \\to \\overline{Z}_1$", "open and $\\overline{Z}_1 \\to X$ closed. On the other hand, let", "$Z \\to \\overline{Z} \\subset X$ be the scheme theoretic closure of", "$Z \\to X$. We conclude that $\\overline{Z} \\subset \\overline{Z}_1$.", "Since $Z$ is an open subscheme of $\\overline{Z}_1$ it follows", "that $Z$ is an open subscheme of $\\overline{Z}$ as well.", "In the case that $Z$ is reduced we know that $Z \\subset \\overline{Z}_1$", "is topologically dense by the construction of $\\overline{Z}_1$ in", "the proof of Lemma \\ref{lemma-factor-reduced-immersion}.", "Hence $\\overline{Z}_1$ and $\\overline{Z}$ have the same", "underlying topological spaces. Thus $\\overline{Z} \\subset \\overline{Z}_1$", "is a closed immersion into a reduced scheme which induces a bijection", "on underlying topological spaces, and hence it is an isomorphism.", "In the case that $Z \\to X$ is quasi-compact we argue as follows:", "The assertion that $Z$ is scheme theoretically dense in", "$\\overline{Z}$ follows from", "Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image} part (3).", "The last assertion follows from", "Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image} part (4)." ], "refs": [ "morphisms-lemma-factor-quasi-compact-immersion", "morphisms-lemma-factor-reduced-immersion", "morphisms-lemma-factor-reduced-immersion", "morphisms-lemma-quasi-compact-scheme-theoretic-image", "morphisms-lemma-quasi-compact-scheme-theoretic-image" ], "ref_ids": [ 5133, 5134, 5134, 5146, 5146 ] } ], "ref_ids": [] }, { "id": 5155, "type": "theorem", "label": "morphisms-lemma-reduced-scheme-theoretically-dense", "categories": [ "morphisms" ], "title": "morphisms-lemma-reduced-scheme-theoretically-dense", "contents": [ "Let $X$ be a reduced scheme and let $U \\subset X$ be an open subscheme.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $U$ is topologically dense in $X$,", "\\item the scheme theoretic closure of $U$ in $X$ is $X$, and", "\\item $U$ is scheme theoretically dense in $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-quasi-compact-immersion}", "and the fact that a closed subscheme $Z$ of $X$ whose", "underlying topological space equals $X$ must be equal to $X$", "as a scheme." ], "refs": [ "morphisms-lemma-quasi-compact-immersion" ], "ref_ids": [ 5154 ] } ], "ref_ids": [] }, { "id": 5156, "type": "theorem", "label": "morphisms-lemma-reduced-subscheme-closure", "categories": [ "morphisms" ], "title": "morphisms-lemma-reduced-subscheme-closure", "contents": [ "Let $X$ be a scheme and let $U \\subset X$ be a reduced open subscheme.", "Then the following are equivalent", "\\begin{enumerate}", "\\item the scheme theoretic closure of $U$ in $X$ is $X$, and", "\\item $U$ is scheme theoretically dense in $X$.", "\\end{enumerate}", "If this holds then $X$ is a reduced scheme." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-quasi-compact-immersion}", "and the fact that the scheme theoretic closure of $U$ in $X$ is", "reduced by", "Lemma \\ref{lemma-scheme-theoretic-image-reduced}." ], "refs": [ "morphisms-lemma-quasi-compact-immersion", "morphisms-lemma-scheme-theoretic-image-reduced" ], "ref_ids": [ 5154, 5149 ] } ], "ref_ids": [] }, { "id": 5157, "type": "theorem", "label": "morphisms-lemma-equality-of-morphisms", "categories": [ "morphisms" ], "title": "morphisms-lemma-equality-of-morphisms", "contents": [ "Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$.", "Let $f, g : X \\to Y$ be morphisms of schemes over $S$.", "Let $U \\subset X$ be an open subscheme such that", "$f|_U = g|_U$. If the scheme theoretic closure of $U$", "in $X$ is $X$ and $Y \\to S$ is separated, then $f = g$." ], "refs": [], "proofs": [ { "contents": [ "Follows from the definitions and", "Schemes, Lemma \\ref{schemes-lemma-where-are-they-equal}." ], "refs": [ "schemes-lemma-where-are-they-equal" ], "ref_ids": [ 7708 ] } ], "ref_ids": [] }, { "id": 5158, "type": "theorem", "label": "morphisms-lemma-generic-points-in-image-dominant", "categories": [ "morphisms" ], "title": "morphisms-lemma-generic-points-in-image-dominant", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "If every generic point of every irreducible component of $S$", "is in the image of $f$, then $f$ is dominant." ], "refs": [], "proofs": [ { "contents": [ "This is a topological fact which follows directly from the fact that", "the topological space underlying a scheme is sober, see", "Schemes, Lemma \\ref{schemes-lemma-scheme-sober}, and that", "every point of $S$ is contained in an irreducible component of", "$S$, see Topology, Lemma \\ref{topology-lemma-irreducible}." ], "refs": [ "schemes-lemma-scheme-sober", "topology-lemma-irreducible" ], "ref_ids": [ 7672, 8213 ] } ], "ref_ids": [] }, { "id": 5159, "type": "theorem", "label": "morphisms-lemma-quasi-compact-dominant", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-compact-dominant", "contents": [ "\\begin{slogan}", "Morphisms whose image contains the generic points are dominant", "\\end{slogan}", "Let $f : X \\to S$ be a quasi-compact morphism of schemes.", "Then $f$ is dominant (if and) only if for every irreducible", "component $Z \\subset S$ the generic point of $Z$ is in the", "image of $f$." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset S$ be an affine open.", "Because $f$ is quasi-compact we may choose finitely many affine", "opens $U_i \\subset f^{-1}(V)$, $i = 1, \\ldots, n$ covering", "$f^{-1}(V)$. Consider the morphism of affines", "$$", "f' :", "\\coprod\\nolimits_{i = 1, \\ldots, n} U_i", "\\longrightarrow", "V.", "$$", "A disjoint union of affines is affine, see", "Schemes, Lemma \\ref{schemes-lemma-disjoint-union-affines}.", "Generic points of irreducible components of $V$", "are exactly the generic points of the irreducible components of", "$S$ that meet $V$. Also, $f$ is dominant if and only if $f'$ is dominant", "no matter what choices of $V, n, U_i$ we make above. Thus we", "have reduced the lemma to the case of a morphism of affine schemes.", "The affine case is", "Algebra, Lemma \\ref{algebra-lemma-image-dense-generic-points}." ], "refs": [ "schemes-lemma-disjoint-union-affines", "algebra-lemma-image-dense-generic-points" ], "ref_ids": [ 7659, 446 ] } ], "ref_ids": [] }, { "id": 5160, "type": "theorem", "label": "morphisms-lemma-quasi-compact-generic-point-not-in-image", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-compact-generic-point-not-in-image", "contents": [ "Let $f : X \\to S$ be a quasi-compact morphism of schemes.", "Let $\\eta \\in S$ be a generic point of an irreducible", "component of $S$. If $\\eta \\not \\in f(X)$ then there", "exists an open neighbourhood $V \\subset S$ of $\\eta$", "such that $f^{-1}(V) = \\emptyset$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset S$ be the scheme theoretic image of $f$.", "We have to show that $\\eta \\not \\in Z$.", "This follows from", "Lemma \\ref{lemma-reach-points-scheme-theoretic-image}", "but can also be seen as follows.", "By Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image}", "the morphism $X \\to Z$ is dominant, which by", "Lemma \\ref{lemma-quasi-compact-dominant}", "means all the generic points of all irreducible components of $Z$", "are in the image of $X \\to Z$. By assumption we see that", "$\\eta \\not \\in Z$ since $\\eta$ would be the generic", "point of some irreducible component of $Z$ if it were in $Z$." ], "refs": [ "morphisms-lemma-reach-points-scheme-theoretic-image", "morphisms-lemma-quasi-compact-scheme-theoretic-image", "morphisms-lemma-quasi-compact-dominant" ], "ref_ids": [ 5147, 5146, 5159 ] } ], "ref_ids": [] }, { "id": 5161, "type": "theorem", "label": "morphisms-lemma-dominant-finite-number-irreducible-components", "categories": [ "morphisms" ], "title": "morphisms-lemma-dominant-finite-number-irreducible-components", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Suppose that $X$ has finitely many irreducible components.", "Then $f$ is dominant (if and) only if for every irreducible", "component $Z \\subset S$ the generic point of $Z$ is in the", "image of $f$. If so, then $S$ has finitely many irreducible", "components as well." ], "refs": [], "proofs": [ { "contents": [ "Assume $f$ is dominant.", "Say $X = Z_1 \\cup Z_2 \\cup \\ldots \\cup Z_n$ is the decomposition", "of $X$ into irreducible components. Let $\\xi_i \\in Z_i$ be", "its generic point, so $Z_i = \\overline{\\{\\xi_i\\}}$.", "Note that $f(Z_i)$ is an irreducible subset of $S$.", "Hence", "$$", "S = \\overline{f(X)} = \\bigcup \\overline{f(Z_i)} =", "\\bigcup \\overline{\\{f(\\xi_i)\\}}", "$$", "is a finite union of irreducible subsets whose generic", "points are in the image of $f$. The lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5162, "type": "theorem", "label": "morphisms-lemma-dominant-between-integral", "categories": [ "morphisms" ], "title": "morphisms-lemma-dominant-between-integral", "contents": [ "Let $f : X \\to Y$ be a morphism of integral schemes. The following", "are equivalent", "\\begin{enumerate}", "\\item $f$ is dominant,", "\\item $f$ maps the generic point of $X$ to the generic point of $Y$,", "\\item for some nonempty affine opens $U \\subset X$ and $V \\subset Y$", "with $f(U) \\subset V$ the ring map $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$", "is injective,", "\\item for all nonempty affine opens $U \\subset X$ and $V \\subset Y$", "with $f(U) \\subset V$ the ring map $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$", "is injective,", "\\item for some $x \\in X$ with image $y = f(x) \\in Y$ the local ring", "map $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ is injective, and", "\\item for all $x \\in X$ with image $y = f(x) \\in Y$ the local ring", "map $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ is injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows from", "Lemma \\ref{lemma-dominant-finite-number-irreducible-components}.", "Let $U \\subset X$ and $V \\subset Y$ be nonempty affine opens with", "$f(U) \\subset V$. Recall that the rings $A = \\mathcal{O}_X(U)$", "and $B = \\mathcal{O}_Y(V)$ are integral domains.", "The morphism $f|_U : U \\to V$ corresponds to a ring map", "$\\varphi : B \\to A$. The generic points of $X$ and $Y$ correspond", "to the prime ideals $(0) \\subset A$ and $(0) \\subset B$. Thus", "(2) is equivalent to the condition $(0) = \\varphi^{-1}((0))$,", "i.e., to the condition that $\\varphi$ is injective.", "In this way we see that (2), (3), and (4) are equivalent.", "Similarly, given $x$ and $y$ as in (5) the local rings", "$\\mathcal{O}_{X, x}$ and $\\mathcal{O}_{Y, y}$ are domains", "and the prime ideals $(0) \\subset \\mathcal{O}_{X, x}$", "and $(0) \\subset \\mathcal{O}_{Y, y}$ correspond to the", "generic points of $X$ and $Y$ (via the identification of", "the spectrum of the local ring at $x$", "with the set of points specializing to $x$, see", "Schemes, Lemma \\ref{schemes-lemma-specialize-points}).", "Thus we can argue in the exact same manner as above", "to see that (2), (5), and (6) are equivalent." ], "refs": [ "morphisms-lemma-dominant-finite-number-irreducible-components", "schemes-lemma-specialize-points" ], "ref_ids": [ 5161, 7684 ] } ], "ref_ids": [] }, { "id": 5163, "type": "theorem", "label": "morphisms-lemma-composition-surjective", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-surjective", "contents": [ "The composition of surjective morphisms is surjective." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5164, "type": "theorem", "label": "morphisms-lemma-when-point-maps-to-pair", "categories": [ "morphisms" ], "title": "morphisms-lemma-when-point-maps-to-pair", "contents": [ "Let $X$ and $Y$ be schemes over a base scheme $S$. Given points $x \\in X$ and", "$y \\in Y$, there is a point of $X \\times_S Y$ mapping to $x$ and $y$ under the", "projections if and only if $x$ and $y$ lie above the same point of $S$." ], "refs": [], "proofs": [ { "contents": [ "The condition is obviously necessary, and the converse follows from the proof", "of Schemes, Lemma \\ref{schemes-lemma-points-fibre-product}." ], "refs": [ "schemes-lemma-points-fibre-product" ], "ref_ids": [ 7693 ] } ], "ref_ids": [] }, { "id": 5165, "type": "theorem", "label": "morphisms-lemma-base-change-surjective", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-surjective", "contents": [ "The base change of a surjective morphism is surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $f: X \\to Y$ be a morphism of schemes over a base scheme $S$.", "If $S' \\to S$ is a morphism of schemes, let $p: X_{S'} \\to X$", "and $q: Y_{S'} \\to Y$ be the canonical projections. The commutative", "square", "$$", "\\xymatrix{", "X_{S'} \\ar[d]_{f_{S'}} \\ar[r]_p & X \\ar[d]^{f} \\\\", "Y_{S'} \\ar[r]^{q} & Y.", "}", "$$", "identifies $X_{S'}$ as a fibre product of $X \\to Y$ and", "$Y_{S'} \\to Y$. Let $Z$ be a subset of the underlying topological", "space of $X$. Then $q^{-1}(f(Z)) = f_{S'}(p^{-1}(Z))$, because", "$y' \\in q^{-1}(f(Z))$ if and only if $q(y') = f(x)$ for some $x \\in Z$,", "if and only if, by Lemma \\ref{lemma-when-point-maps-to-pair}, there exists", "$x' \\in X_{S'}$ such that $f_{S'}(x') = y'$ and $p(x') = x$. In particular", "taking $Z = X$ we see that if $f$ is surjective so is the base change", "$f_{S'}: X_{S'} \\to Y_{S'}$." ], "refs": [ "morphisms-lemma-when-point-maps-to-pair" ], "ref_ids": [ 5164 ] } ], "ref_ids": [] }, { "id": 5166, "type": "theorem", "label": "morphisms-lemma-surjection-from-quasi-compact", "categories": [ "morphisms" ], "title": "morphisms-lemma-surjection-from-quasi-compact", "contents": [ "Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & &", "Y \\ar[dl]^q \\\\", "& Z", "}", "$$", "be a commutative diagram of morphisms of schemes.", "If $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Let $W \\subset Z$ be a quasi-compact open. By assumption $p^{-1}(W)$", "is quasi-compact. Hence by", "Topology, Lemma \\ref{topology-lemma-image-quasi-compact}", "the inverse image $q^{-1}(W) = f(p^{-1}(W))$ is quasi-compact too.", "This proves the lemma." ], "refs": [ "topology-lemma-image-quasi-compact" ], "ref_ids": [ 8233 ] } ], "ref_ids": [] }, { "id": 5167, "type": "theorem", "label": "morphisms-lemma-universally-injective", "categories": [ "morphisms" ], "title": "morphisms-lemma-universally-injective", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item For every field $K$ the induced map", "$\\Mor(\\Spec(K), X) \\to \\Mor(\\Spec(K), S)$", "is injective.", "\\item The morphism $f$ is universally injective.", "\\item The morphism $f$ is radicial.", "\\item The diagonal morphism $\\Delta_{X/S} : X \\longrightarrow X \\times_S X$", "is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $K$ be a field, and let $s : \\Spec(K) \\to S$ be a morphism.", "Giving a morphism $x : \\Spec(K) \\to X$ such that $f \\circ x = s$", "is the same as giving a section of the projection", "$X_K = \\Spec(K) \\times_S X \\to \\Spec(K)$, which in turn", "is the same as giving a point $x \\in X_K$ whose residue field is $K$.", "Hence we see that (2) implies (1).", "\\medskip\\noindent", "Conversely, suppose that (1) holds. Assume that $x, x' \\in X_{S'}$", "map to the same point $s' \\in S'$. Choose a commutative diagram", "$$", "\\xymatrix{", "K & \\kappa(x) \\ar[l] \\\\", "\\kappa(x') \\ar[u] & \\kappa(s') \\ar[l] \\ar[u]", "}", "$$", "of fields. By Schemes, Lemma \\ref{schemes-lemma-characterize-points}", "we get two morphisms $a, a' : \\Spec(K) \\to X_{S'}$. One corresponding", "to the point $x$ and the embedding $\\kappa(x) \\subset K$ and", "the other corresponding to the point $x'$ and the embedding", "$\\kappa(x') \\subset K$. Also we have $f' \\circ a = f' \\circ a'$.", "Condition (1) now implies that the compositions of $a$ and $a'$ with", "$X_{S'} \\to X$ are equal. Since $X_{S'}$ is the fibre product", "of $S'$ and $X$ over $S$ we see that $a = a'$. Hence $x = x'$.", "Thus (1) implies (2).", "\\medskip\\noindent", "If there are two different points $x, x' \\in X$ mapping to the same point of $s$", "then (2) is violated.", "If for some $s = f(x)$, $x \\in X$ the field extension", "$\\kappa(s) \\subset \\kappa(x)$ is not purely inseparable, then", "we may find a field extension $\\kappa(s) \\subset K$ such that", "$\\kappa(x)$ has two $\\kappa(s)$-homomorphisms into $K$. By", "Schemes, Lemma \\ref{schemes-lemma-characterize-points} this", "implies that the map", "$\\Mor(\\Spec(K), X) \\to \\Mor(\\Spec(K), S)$", "is not injective, and hence (1) is violated.", "Thus we see that the equivalent conditions (1) and (2) imply", "$f$ is radicial, i.e., they imply (3).", "\\medskip\\noindent", "Assume (3). By", "Schemes, Lemma \\ref{schemes-lemma-characterize-points}", "a morphism $\\Spec(K) \\to X$ is given by a pair $(x, \\kappa(x) \\to K)$.", "Property (3) says exactly that associating to the pair", "$(x, \\kappa(x) \\to K)$ the pair $(s, \\kappa(s) \\to \\kappa(x) \\to K)$", "is injective. In other words (1) holds. At this point we know that", "(1), (2) and (3) are all equivalent.", "\\medskip\\noindent", "Finally, we prove the equivalence of (4) with (1), (2) and (3).", "A point of $X \\times_S X$ is given by a quadruple", "$(x_1, x_2, s, \\mathfrak p)$, where $x_1, x_2 \\in X$,", "$f(x_1) = f(x_2) = s$ and", "$\\mathfrak p \\subset \\kappa(x_1) \\otimes_{\\kappa(s)} \\kappa(x_2)$", "is a prime ideal, see", "Schemes, Lemma \\ref{schemes-lemma-points-fibre-product}.", "If $f$ is universally injective, then", "by taking $S'=X$ in the definition of universally injective,", "$\\Delta_{X/S}$ must be surjective since it is a section of", "the injective morphism", "$X \\times_S X \\longrightarrow X$.", "Conversely, if", "$\\Delta_{X/S}$ is surjective, then always $x_1 = x_2 = x$ and there", "is exactly one such prime ideal $\\mathfrak p$, which means that", "$\\kappa(s) \\subset \\kappa(x)$ is purely inseparable.", "Hence $f$ is radicial.", "Alternatively, if $\\Delta_{X/S}$ is surjective,", "then for any $S' \\to S$ the base", "change $\\Delta_{X_{S'}/S'}$ is surjective which implies that $f$", "is universally injective. This finishes the proof of the lemma." ], "refs": [ "schemes-lemma-characterize-points", "schemes-lemma-characterize-points", "schemes-lemma-characterize-points", "schemes-lemma-points-fibre-product" ], "ref_ids": [ 7685, 7685, 7685, 7693 ] } ], "ref_ids": [] }, { "id": 5168, "type": "theorem", "label": "morphisms-lemma-universally-injective-separated", "categories": [ "morphisms" ], "title": "morphisms-lemma-universally-injective-separated", "contents": [ "A universally injective morphism is separated." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemma \\ref{lemma-universally-injective}", "with the remark that $X \\to S$ is separated if and only if the image", "of $\\Delta_{X/S}$ is closed in $X \\times_S X$, see", "Schemes, Definition \\ref{schemes-definition-separated}", "and the discussion following it." ], "refs": [ "morphisms-lemma-universally-injective", "schemes-definition-separated" ], "ref_ids": [ 5167, 7756 ] } ], "ref_ids": [] }, { "id": 5169, "type": "theorem", "label": "morphisms-lemma-base-change-universally-injective", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-universally-injective", "contents": [ "A base change of a universally injective morphism is universally injective." ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5170, "type": "theorem", "label": "morphisms-lemma-composition-universally-injective", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-universally-injective", "contents": [ "A composition of radicial morphisms is radicial, and so the same holds", "for the equivalent condition of being universally injective." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5171, "type": "theorem", "label": "morphisms-lemma-affine-separated", "categories": [ "morphisms" ], "title": "morphisms-lemma-affine-separated", "contents": [ "An affine morphism is separated and quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be affine. Quasi-compactness is immediate from", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-affine}.", "We will show $f$ is separated using", "Schemes, Lemma \\ref{schemes-lemma-characterize-separated}. Let", "$x_1, x_2 \\in X$ be points of $X$ which map to the same point $s \\in S$.", "Choose any affine open $W \\subset S$ containing $s$. By assumption", "$f^{-1}(W)$ is affine. Apply the lemma cited with $U = V = f^{-1}(W)$." ], "refs": [ "schemes-lemma-quasi-compact-affine", "schemes-lemma-characterize-separated" ], "ref_ids": [ 7697, 7710 ] } ], "ref_ids": [] }, { "id": 5172, "type": "theorem", "label": "morphisms-lemma-characterize-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-characterize-affine", "contents": [ "\\begin{reference}", "\\cite[II, Corollary 1.3.2]{EGA}", "\\end{reference}", "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is affine.", "\\item There exists an affine open covering $S = \\bigcup W_j$", "such that each $f^{-1}(W_j)$ is affine.", "\\item There exists a quasi-coherent sheaf of $\\mathcal{O}_S$-algebras", "$\\mathcal{A}$ and an isomorphism", "$X \\cong \\underline{\\Spec}_S(\\mathcal{A})$", "of schemes over $S$. See", "Constructions, Section \\ref{constructions-section-spec} for notation.", "\\end{enumerate}", "Moreover, in this case $X = \\underline{\\Spec}_S(f_*\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "It is obvious that (1) implies (2).", "\\medskip\\noindent", "Assume $S = \\bigcup_{j \\in J} W_j$ is an affine open covering such that", "each $f^{-1}(W_j)$ is affine. By", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-affine} we see", "that $f$ is quasi-compact. By", "Schemes, Lemma \\ref{schemes-lemma-characterize-quasi-separated}", "we see the morphism $f$ is quasi-separated. Hence by", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent} the", "sheaf $\\mathcal{A} = f_*\\mathcal{O}_X$ is a quasi-coherent sheaf", "of $\\mathcal{O}_X$-algebras. Thus we have the scheme", "$g : Y = \\underline{\\Spec}_S(\\mathcal{A}) \\to S$ over $S$.", "The identity map", "$\\text{id} : \\mathcal{A} = f_*\\mathcal{O}_X \\to f_*\\mathcal{O}_X$", "provides, via the definition of the relative spectrum,", "a morphism $can : X \\to Y$ over $S$, see", "Constructions, Lemma \\ref{constructions-lemma-canonical-morphism}.", "By assumption and the lemma just cited", "the restriction $can|_{f^{-1}(W_j)} : f^{-1}(W_j) \\to g^{-1}(W_j)$", "is an isomorphism. Thus $can$ is an isomorphism.", "We have shown that (2) implies (3).", "\\medskip\\noindent", "Assume (3). By Constructions, Lemma \\ref{constructions-lemma-spec-properties}", "we see that the inverse image of every affine open is affine, and hence", "the morphism is affine by definition." ], "refs": [ "schemes-lemma-quasi-compact-affine", "schemes-lemma-characterize-quasi-separated", "schemes-lemma-push-forward-quasi-coherent", "constructions-lemma-canonical-morphism", "constructions-lemma-spec-properties" ], "ref_ids": [ 7697, 7709, 7730, 12591, 12590 ] } ], "ref_ids": [] }, { "id": 5173, "type": "theorem", "label": "morphisms-lemma-affine-equivalence-algebras", "categories": [ "morphisms" ], "title": "morphisms-lemma-affine-equivalence-algebras", "contents": [ "Let $S$ be a scheme. There is an anti-equivalence of categories", "$$", "\\begin{matrix}", "\\text{Schemes affine} \\\\", "\\text{over }S", "\\end{matrix}", "\\longleftrightarrow", "\\begin{matrix}", "\\text{quasi-coherent sheaves} \\\\", "\\text{of }\\mathcal{O}_S\\text{-algebras}", "\\end{matrix}", "$$", "which associates to $f : X \\to S$ the sheaf $f_*\\mathcal{O}_X$.", "Moreover, this equivalence is compatible with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "The functor from right to left is given by $\\underline{\\Spec}_S$.", "The two functors are mutually inverse by", "Lemma \\ref{lemma-characterize-affine} and", "Constructions, Lemma \\ref{constructions-lemma-spec-properties} part (3).", "The final statement is", "Constructions, Lemma \\ref{constructions-lemma-spec-properties} part (2)." ], "refs": [ "morphisms-lemma-characterize-affine", "constructions-lemma-spec-properties", "constructions-lemma-spec-properties" ], "ref_ids": [ 5172, 12590, 12590 ] } ], "ref_ids": [] }, { "id": 5174, "type": "theorem", "label": "morphisms-lemma-affine-equivalence-modules", "categories": [ "morphisms" ], "title": "morphisms-lemma-affine-equivalence-modules", "contents": [ "Let $f : X \\to S$ be an affine morphism of schemes.", "Let $\\mathcal{A} = f_*\\mathcal{O}_X$.", "The functor $\\mathcal{F} \\mapsto f_*\\mathcal{F}$ induces", "an equivalence of categories", "$$", "\\left\\{", "\\begin{matrix}", "\\text{category of quasi-coherent}\\\\", "\\mathcal{O}_X\\text{-modules}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{category of quasi-coherent}\\\\", "\\mathcal{A}\\text{-modules}", "\\end{matrix}", "\\right\\}", "$$", "Moreover, an $\\mathcal{A}$-module is", "quasi-coherent as an $\\mathcal{O}_S$-module if and only if", "it is quasi-coherent as an $\\mathcal{A}$-module." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5175, "type": "theorem", "label": "morphisms-lemma-composition-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-affine", "contents": [ "The composition of affine morphisms is affine." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be affine morphisms.", "Let $U \\subset Z$ be affine open. Then $g^{-1}(U)$ is affine", "by assumption on $g$. Whereupon $f^{-1}(g^{-1}(U))$ is affine", "by assumption on $f$. Hence $(g \\circ f)^{-1}(U)$ is affine." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5176, "type": "theorem", "label": "morphisms-lemma-base-change-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-affine", "contents": [ "The base change of an affine morphism is affine." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be an affine morphism. Let $S' \\to S$ be any morphism.", "Denote $f' : X_{S'} = S' \\times_S X \\to S'$ the base change of $f$.", "For every $s' \\in S'$ there exists an open affine neighbourhood", "$s' \\in V \\subset S'$ which maps into some open affine $U \\subset S$.", "By assumption $f^{-1}(U)$ is affine. By the material in", "Schemes, Section \\ref{schemes-section-fibre-products} we see", "that $f^{-1}(U)_V = V \\times_U f^{-1}(U)$ is affine and equal", "to $(f')^{-1}(V)$. This proves that $S'$ has an open covering by", "affines whose inverse image under $f'$ is affine. We conclude", "by Lemma \\ref{lemma-characterize-affine} above." ], "refs": [ "morphisms-lemma-characterize-affine" ], "ref_ids": [ 5172 ] } ], "ref_ids": [] }, { "id": 5177, "type": "theorem", "label": "morphisms-lemma-closed-immersion-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-closed-immersion-affine", "contents": [ "A closed immersion is affine." ], "refs": [], "proofs": [ { "contents": [ "The first indication of this is", "Schemes, Lemma \\ref{schemes-lemma-closed-immersion-affine-case}.", "See Schemes, Lemma \\ref{schemes-lemma-closed-subspace-scheme}", "for a complete statement." ], "refs": [ "schemes-lemma-closed-immersion-affine-case", "schemes-lemma-closed-subspace-scheme" ], "ref_ids": [ 7668, 7670 ] } ], "ref_ids": [] }, { "id": 5178, "type": "theorem", "label": "morphisms-lemma-affine-s-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-affine-s-open", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{L})$.", "The inclusion morphism $j : X_s \\to X$ is affine." ], "refs": [], "proofs": [ { "contents": [ "This follows from Properties, Lemma \\ref{properties-lemma-affine-cap-s-open}", "and the definition." ], "refs": [ "properties-lemma-affine-cap-s-open" ], "ref_ids": [ 3042 ] } ], "ref_ids": [] }, { "id": 5179, "type": "theorem", "label": "morphisms-lemma-affine-permanence", "categories": [ "morphisms" ], "title": "morphisms-lemma-affine-permanence", "contents": [ "Suppose $g : X \\to Y$ is a morphism of schemes over $S$.", "\\begin{enumerate}", "\\item If $X$ is affine over $S$ and $\\Delta : Y \\to Y \\times_S Y$ is affine,", "then $g$ is affine.", "\\item If $X$ is affine over $S$ and $Y$ is separated over $S$,", "then $g$ is affine.", "\\item A morphism from an affine scheme to a scheme with affine", "diagonal is affine.", "\\item A morphism from an affine scheme to a separated scheme is affine.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). The base change $X \\times_S Y \\to Y$ is affine by", "Lemma \\ref{lemma-base-change-affine}.", "The morphism $(1, g) : X \\to X \\times_S Y$ is the base change of", "$Y \\to Y \\times_S Y$ by the morphism $X \\times_S Y \\to Y \\times_S Y$.", "Hence it is affine by", "Lemma \\ref{lemma-base-change-affine}.", "The composition of affine morphisms is affine", "(see Lemma \\ref{lemma-composition-affine}) and (1) follows.", "Part (2) follows from (1) as a closed immersion is affine", "(see Lemma \\ref{lemma-closed-immersion-affine}) and $Y/S$ separated", "means $\\Delta$ is a closed immersion. Parts (3) and (4) are special", "cases of (1) and (2)." ], "refs": [ "morphisms-lemma-base-change-affine", "morphisms-lemma-base-change-affine", "morphisms-lemma-composition-affine", "morphisms-lemma-closed-immersion-affine" ], "ref_ids": [ 5176, 5176, 5175, 5177 ] } ], "ref_ids": [] }, { "id": 5180, "type": "theorem", "label": "morphisms-lemma-morphism-affines-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-morphism-affines-affine", "contents": [ "A morphism between affine schemes is affine." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-affine-permanence} with", "$S = \\Spec(\\mathbf{Z})$. It also follows directly from the", "equivalence of (1) and (2) in Lemma \\ref{lemma-characterize-affine}." ], "refs": [ "morphisms-lemma-affine-permanence", "morphisms-lemma-characterize-affine" ], "ref_ids": [ 5179, 5172 ] } ], "ref_ids": [] }, { "id": 5181, "type": "theorem", "label": "morphisms-lemma-Artinian-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-Artinian-affine", "contents": [ "Let $S$ be a scheme.", "Let $A$ be an Artinian ring.", "Any morphism $\\Spec(A) \\to S$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5182, "type": "theorem", "label": "morphisms-lemma-get-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-get-affine", "contents": [ "Let $j : Y \\to X$ be an immersion of schemes.", "Assume there exists an open $U \\subset X$ with complement", "$Z = X \\setminus U$ such that", "\\begin{enumerate}", "\\item $U \\to X$ is affine,", "\\item $j^{-1}(U) \\to U$ is affine, and", "\\item $j(Y) \\cap Z$ is closed.", "\\end{enumerate}", "Then $j$ is affine. In particular, if $X$ is affine, so is $Y$." ], "refs": [], "proofs": [ { "contents": [ "By Schemes, Definition \\ref{schemes-definition-immersion} there exists an", "open subscheme $W \\subset X$ such that $j$ factors as a closed immersion", "$i : Y \\to W$ followed by the inclusion morphism $W \\to X$.", "Since a closed immersion is affine", "(Lemma \\ref{lemma-closed-immersion-affine}),", "we see that for every $x \\in W$ there is an affine open", "neighbourhood of $x$ in $X$ whose inverse image under $j$ is affine.", "If $x \\in U$, then the same thing is true by assumption (2).", "Finally, assume $x \\in Z$ and $x \\not \\in W$. Then $x \\not \\in j(Y) \\cap Z$.", "By assumption (3) we can find an affine open neighbourhood", "$V \\subset X$ of $x$ which does not meet $j(Y) \\cap Z$.", "Then $j^{-1}(V) = j^{-1}(V \\cap U)$ which is affine", "by assumptions (1) and (2). It follows that $j$ is affine by", "Lemma \\ref{lemma-characterize-affine}." ], "refs": [ "schemes-definition-immersion", "morphisms-lemma-closed-immersion-affine", "morphisms-lemma-characterize-affine" ], "ref_ids": [ 7743, 5177, 5172 ] } ], "ref_ids": [] }, { "id": 5183, "type": "theorem", "label": "morphisms-lemma-affine-diagonal", "categories": [ "morphisms" ], "title": "morphisms-lemma-affine-diagonal", "contents": [ "Let $X$ be a scheme such that for every point $x \\in X$ there exists", "an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ and a global", "section $s \\in \\Gamma(X, \\mathcal{L})$ such that $x \\in X_s$ and", "$X_s$ is affine. Then the diagonal of $X$ is an affine morphism." ], "refs": [], "proofs": [ { "contents": [ "Given invertible $\\mathcal{O}_X$-modules $\\mathcal{L}$, $\\mathcal{M}$", "and global sections $s \\in \\Gamma(X, \\mathcal{L})$,", "$t \\in \\Gamma(X, \\mathcal{M})$ such that $X_s$ and $X_t$ are affine", "we have to prove $X_s \\cap X_t$ is affine. Namely, then", "Lemma \\ref{lemma-characterize-affine}", "applied to $\\Delta : X \\to X \\times X$ and the fact that", "$\\Delta^{-1}(X_s \\times X_t) = X_s \\cap X_t$ shows that $\\Delta$", "is affine. The fact that $X_s \\cap X_t$ is affine follows from", "Properties, Lemma \\ref{properties-lemma-affine-cap-s-open}." ], "refs": [ "morphisms-lemma-characterize-affine", "properties-lemma-affine-cap-s-open" ], "ref_ids": [ 5172, 3042 ] } ], "ref_ids": [] }, { "id": 5184, "type": "theorem", "label": "morphisms-lemma-quasi-affine-separated", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-affine-separated", "contents": [ "A quasi-affine morphism is separated and quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be quasi-affine.", "Quasi-compactness is immediate from", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-affine}.", "Let $U \\subset S$ be an affine open. If we can show that", "$f^{-1}(U)$ is a separated scheme, then $f$ is separated", "(Schemes, Lemma \\ref{schemes-lemma-characterize-separated}", "shows that being separated is local on the base).", "By assumption $f^{-1}(U)$ is isomorphic to an open subscheme", "of an affine scheme. An affine scheme is separated and hence", "every open subscheme of an affine scheme is separated as desired." ], "refs": [ "schemes-lemma-quasi-compact-affine", "schemes-lemma-characterize-separated" ], "ref_ids": [ 7697, 7710 ] } ], "ref_ids": [] }, { "id": 5185, "type": "theorem", "label": "morphisms-lemma-characterize-quasi-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-characterize-quasi-affine", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is quasi-affine.", "\\item There exists an affine open covering $S = \\bigcup W_j$", "such that each $f^{-1}(W_j)$ is quasi-affine.", "\\item There exists a quasi-coherent sheaf of $\\mathcal{O}_S$-algebras", "$\\mathcal{A}$ and a quasi-compact open immersion", "$$", "\\xymatrix{", "X \\ar[rr] \\ar[rd] & & \\underline{\\Spec}_S(\\mathcal{A}) \\ar[dl] \\\\", "& S &", "}", "$$", "over $S$.", "\\item Same as in (3) but with $\\mathcal{A} = f_*\\mathcal{O}_X$", "and the horizontal arrow the canonical morphism of", "Constructions, Lemma \\ref{constructions-lemma-canonical-morphism}.", "\\end{enumerate}" ], "refs": [ "constructions-lemma-canonical-morphism" ], "proofs": [ { "contents": [ "It is obvious that (1) implies (2) and that (4) implies (3).", "\\medskip\\noindent", "Assume $S = \\bigcup_{j \\in J} W_j$ is an affine open covering such that", "each $f^{-1}(W_j)$ is quasi-affine. By", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-affine} we see", "that $f$ is quasi-compact. By", "Schemes, Lemma \\ref{schemes-lemma-characterize-quasi-separated}", "we see the morphism $f$ is quasi-separated. Hence by", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent} the", "sheaf $\\mathcal{A} = f_*\\mathcal{O}_X$ is a quasi-coherent sheaf", "of $\\mathcal{O}_X$-algebras. Thus we have the scheme", "$g : Y = \\underline{\\Spec}_S(\\mathcal{A}) \\to S$ over $S$.", "The identity map", "$\\text{id} : \\mathcal{A} = f_*\\mathcal{O}_X \\to f_*\\mathcal{O}_X$", "provides, via the definition of the relative spectrum,", "a morphism $can : X \\to Y$ over $S$, see", "Constructions, Lemma \\ref{constructions-lemma-canonical-morphism}.", "By assumption, the lemma just cited, and", "Properties, Lemma \\ref{properties-lemma-quasi-affine}", "the restriction $can|_{f^{-1}(W_j)} : f^{-1}(W_j) \\to g^{-1}(W_j)$", "is a quasi-compact open immersion. Thus $can$ is a quasi-compact", "open immersion. We have shown that (2) implies (4).", "\\medskip\\noindent", "Assume (3). Choose any affine open $U \\subset S$.", "By Constructions, Lemma \\ref{constructions-lemma-spec-properties}", "we see that the inverse image of $U$ in the relative spectrum", "is affine. Hence we conclude that $f^{-1}(U)$ is quasi-affine", "(note that quasi-compactness is encoded in (3) as well).", "Thus (3) implies (1)." ], "refs": [ "schemes-lemma-quasi-compact-affine", "schemes-lemma-characterize-quasi-separated", "schemes-lemma-push-forward-quasi-coherent", "constructions-lemma-canonical-morphism", "properties-lemma-quasi-affine", "constructions-lemma-spec-properties" ], "ref_ids": [ 7697, 7709, 7730, 12591, 3009, 12590 ] } ], "ref_ids": [ 12591 ] }, { "id": 5186, "type": "theorem", "label": "morphisms-lemma-composition-quasi-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-quasi-affine", "contents": [ "The composition of quasi-affine morphisms is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be quasi-affine morphisms.", "Let $U \\subset Z$ be affine open. Then $g^{-1}(U)$ is quasi-affine", "by assumption on $g$. Let $j : g^{-1}(U) \\to V$ be a quasi-compact", "open immersion into an affine scheme $V$.", "By Lemma \\ref{lemma-characterize-quasi-affine} above", "we see that $f^{-1}(g^{-1}(U))$", "is a quasi-compact open subscheme of the relative spectrum", "$\\underline{\\Spec}_{g^{-1}(U)}(\\mathcal{A})$ for", "some quasi-coherent sheaf of $\\mathcal{O}_{g^{-1}(U)}$-algebras", "$\\mathcal{A}$. By", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "the sheaf $\\mathcal{A}' = j_*\\mathcal{A}$", "is a quasi-coherent sheaf of $\\mathcal{O}_V$-algebras", "with the property that $j^*\\mathcal{A}' = \\mathcal{A}$.", "Hence we get a commutative diagram", "$$", "\\xymatrix{", "f^{-1}(g^{-1}(U)) \\ar[r] &", "\\underline{\\Spec}_{g^{-1}(U)}(\\mathcal{A})", "\\ar[r] \\ar[d] &", "\\underline{\\Spec}_V(\\mathcal{A}') \\ar[d] \\\\", "& g^{-1}(U) \\ar[r]^j & V", "}", "$$", "with the square being a fibre square,", "see Constructions, Lemma \\ref{constructions-lemma-spec-properties}.", "Note that the upper right corner is an affine scheme.", "Hence $(g \\circ f)^{-1}(U)$ is quasi-affine." ], "refs": [ "morphisms-lemma-characterize-quasi-affine", "schemes-lemma-push-forward-quasi-coherent", "constructions-lemma-spec-properties" ], "ref_ids": [ 5185, 7730, 12590 ] } ], "ref_ids": [] }, { "id": 5187, "type": "theorem", "label": "morphisms-lemma-base-change-quasi-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-quasi-affine", "contents": [ "The base change of a quasi-affine morphism is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be a quasi-affine morphism.", "By Lemma \\ref{lemma-characterize-quasi-affine} above", "we can find a quasi-coherent sheaf", "of $\\mathcal{O}_S$-algebras $\\mathcal{A}$ and a quasi-compact", "open immersion $X \\to \\underline{\\Spec}_S(\\mathcal{A})$", "over $S$.", "Let $g : S' \\to S$ be any morphism.", "Denote $f' : X_{S'} = S' \\times_S X \\to S'$ the base change of $f$.", "Since the base change of a quasi-compact open immersion is", "a quasi-compact open immersion we see that", "$X_{S'} \\to \\underline{\\Spec}_{S'}(g^*\\mathcal{A})$", "is a quasi-compact open immersion", "(we have used Schemes, Lemmas", "\\ref{schemes-lemma-quasi-compact-preserved-base-change} and", "\\ref{schemes-lemma-base-change-immersion} and", "Constructions, Lemma \\ref{constructions-lemma-spec-properties}).", "By Lemma \\ref{lemma-characterize-quasi-affine} again", "we conclude that $X_{S'} \\to S'$ is quasi-affine." ], "refs": [ "morphisms-lemma-characterize-quasi-affine", "schemes-lemma-quasi-compact-preserved-base-change", "schemes-lemma-base-change-immersion", "constructions-lemma-spec-properties", "morphisms-lemma-characterize-quasi-affine" ], "ref_ids": [ 5185, 7698, 7695, 12590, 5185 ] } ], "ref_ids": [] }, { "id": 5188, "type": "theorem", "label": "morphisms-lemma-quasi-compact-immersion-quasi-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-compact-immersion-quasi-affine", "contents": [ "A quasi-compact immersion is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "Let $X \\to S$ be a quasi-compact immersion. We have to show the", "inverse image of every affine open is quasi-affine. Hence,", "assuming $S$ is an affine scheme, we have to show", "$X$ is quasi-affine. By Lemma \\ref{lemma-quasi-compact-immersion}", "the morphism $X \\to S$ factors as $X \\to Z \\to S$ where $Z$ is a closed", "subscheme of $S$ and $X \\subset Z$ is a quasi-compact open.", "Since $S$ is affine Lemma \\ref{lemma-closed-immersion} implies", "$Z$ is affine. Hence we win." ], "refs": [ "morphisms-lemma-quasi-compact-immersion", "morphisms-lemma-closed-immersion" ], "ref_ids": [ 5154, 5125 ] } ], "ref_ids": [] }, { "id": 5189, "type": "theorem", "label": "morphisms-lemma-affine-quasi-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-affine-quasi-affine", "contents": [ "Let $S$ be a scheme. Let $X$ be an affine scheme.", "A morphism $f : X \\to S$ is quasi-affine if and only if it is quasi-compact.", "In particular any morphism from an affine scheme to a quasi-separated", "scheme is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset S$ be an affine open. Then $f^{-1}(V)$ is an open subscheme", "of the affine scheme $X$, hence quasi-affine if and only if it is", "quasi-compact. This proves the first assertion. The quasi-compactness of any", "$f : X \\to S$ where $X$ is affine and $S$ quasi-separated follows from", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}", "applied to $X \\to S \\to \\Spec(\\mathbf{Z})$." ], "refs": [ "schemes-lemma-quasi-compact-permanence" ], "ref_ids": [ 7716 ] } ], "ref_ids": [] }, { "id": 5190, "type": "theorem", "label": "morphisms-lemma-quasi-affine-permanence", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-affine-permanence", "contents": [ "Suppose $g : X \\to Y$ is a morphism of schemes over $S$.", "If $X$ is quasi-affine over $S$ and $Y$ is quasi-separated over $S$,", "then $g$ is quasi-affine. In particular, any morphism from a", "quasi-affine scheme to a quasi-separated scheme is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "The base change $X \\times_S Y \\to Y$ is quasi-affine by", "Lemma \\ref{lemma-base-change-quasi-affine}.", "The morphism $X \\to X \\times_S Y$ is", "a quasi-compact immersion as $Y \\to S$ is quasi-separated, see", "Schemes, Lemma \\ref{schemes-lemma-section-immersion}.", "A quasi-compact immersion is quasi-affine by", "Lemma \\ref{lemma-quasi-compact-immersion-quasi-affine}", "and the composition of quasi-affine morphisms is quasi-affine", "(see Lemma \\ref{lemma-composition-quasi-affine}). Thus we win." ], "refs": [ "morphisms-lemma-base-change-quasi-affine", "morphisms-lemma-quasi-compact-immersion-quasi-affine", "morphisms-lemma-composition-quasi-affine" ], "ref_ids": [ 5187, 5188, 5186 ] } ], "ref_ids": [] }, { "id": 5191, "type": "theorem", "label": "morphisms-lemma-locally-P", "categories": [ "morphisms" ], "title": "morphisms-lemma-locally-P", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $P$ be a property of ring maps.", "Let $U$ be an affine open of $X$,", "and $V$ an affine open of $S$ such that", "$f(U) \\subset V$.", "If $f$ is locally of type $P$ and $P$ is local,", "then $P(\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U))$ holds." ], "refs": [], "proofs": [ { "contents": [ "As $f$ is locally of type $P$ for every $u \\in U$ there exists an", "affine open $U_u \\subset X$ mapping into an affine open $V_u \\subset S$", "such that $P(\\mathcal{O}_S(V_u) \\to \\mathcal{O}_X(U_u))$ holds.", "Choose an open neighbourhood $U'_u \\subset U \\cap U_u$ of $u$", "which is standard affine open in both $U$ and $U_u$, see", "Schemes, Lemma \\ref{schemes-lemma-standard-open-two-affines}.", "By Definition \\ref{definition-property-local} (1)(b)", "we see that $P(\\mathcal{O}_S(V_u) \\to \\mathcal{O}_X(U'_u))$ holds.", "Hence we may assume that $U_u \\subset U$ is a standard affine open.", "Choose an open neighbourhood $V'_u \\subset V \\cap V_u$", "of $f(u)$ which is standard affine open in both $V$ and $V_u$, see", "Schemes, Lemma \\ref{schemes-lemma-standard-open-two-affines}.", "Then $U'_u = f^{-1}(V'_u) \\cap U_u$ is a standard affine open", "of $U_u$ (hence of $U$) and we have", "$P(\\mathcal{O}_S(V'_u) \\to \\mathcal{O}_X(U'_u))$ by", "Definition \\ref{definition-property-local} (1)(a).", "Hence we may assume both $U_u \\subset U$ and $V_u \\subset V$", "are standard affine open. Applying", "Definition \\ref{definition-property-local} (1)(b)", "one more time we conclude that $P(\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U_u))$", "holds. Because $U$ is quasi-compact we may choose a finite number", "of points $u_1, \\ldots, u_n \\in U$ such that", "$$", "U = U_{u_1} \\cup \\ldots \\cup U_{u_n}.", "$$", "By Definition \\ref{definition-property-local} (1)(c)", "we conclude that $P(\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U))$ holds." ], "refs": [ "schemes-lemma-standard-open-two-affines", "morphisms-definition-property-local", "schemes-lemma-standard-open-two-affines", "morphisms-definition-property-local", "morphisms-definition-property-local", "morphisms-definition-property-local" ], "ref_ids": [ 7675, 5547, 7675, 5547, 5547, 5547 ] } ], "ref_ids": [] }, { "id": 5192, "type": "theorem", "label": "morphisms-lemma-locally-P-characterize", "categories": [ "morphisms" ], "title": "morphisms-lemma-locally-P-characterize", "contents": [ "Let $P$ be a local property of ring maps.", "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is locally of type $P$.", "\\item For every affine opens $U \\subset X$, $V \\subset S$", "with $f(U) \\subset V$ we have $P(\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U))$.", "\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$", "and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$", "is locally of type $P$.", "\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$", "and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that $P(\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i))$ holds, for all", "$j\\in J, i\\in I_j$.", "\\end{enumerate}", "Moreover, if $f$ is locally of type $P$ then for", "any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$", "the restriction $f|_U : U \\to V$ is locally of type $P$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-locally-P} above." ], "refs": [ "morphisms-lemma-locally-P" ], "ref_ids": [ 5191 ] } ], "ref_ids": [] }, { "id": 5193, "type": "theorem", "label": "morphisms-lemma-composition-type-P", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-type-P", "contents": [ "Let $P$ be a property of ring maps.", "Assume $P$ is local and stable under composition.", "The composition of morphisms locally of type $P$ is", "locally of type $P$." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms locally of type $P$.", "Let $x \\in X$. Choose an affine open neighbourhood $W \\subset Z$ of", "$g(f(x))$. Choose an affine open neighbourhood $V \\subset g^{-1}(W)$", "of $f(x)$. Choose an affine open neighbourhood $U \\subset f^{-1}(V)$", "of $x$. By Lemma \\ref{lemma-locally-P-characterize} the ring maps", "$\\mathcal{O}_Z(W) \\to \\mathcal{O}_Y(V)$ and", "$\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$ satisfy $P$.", "Hence $\\mathcal{O}_Z(W) \\to \\mathcal{O}_X(U)$ satisfies $P$", "as $P$ is assumed stable under composition." ], "refs": [ "morphisms-lemma-locally-P-characterize" ], "ref_ids": [ 5192 ] } ], "ref_ids": [] }, { "id": 5194, "type": "theorem", "label": "morphisms-lemma-base-change-type-P", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-type-P", "contents": [ "Let $P$ be a property of ring maps.", "Assume $P$ is local and stable under base change.", "The base change of a morphism locally of type $P$", "is locally of type $P$." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be a morphism locally of type $P$.", "Let $S' \\to S$ be any morphism. Denote", "$f' : X_{S'} = S' \\times_S X \\to S'$ the base change of $f$.", "For every $s' \\in S'$ there exists an open affine neighbourhood", "$s' \\in V' \\subset S'$ which maps into some open affine $V \\subset S$.", "By Lemma \\ref{lemma-locally-P-characterize} the open $f^{-1}(V)$ is a", "union of affines $U_i$ such that the ring maps", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U_i)$ all satisfy $P$.", "By the material in Schemes, Section \\ref{schemes-section-fibre-products}", "we see that $f^{-1}(U)_{V'} = V' \\times_V f^{-1}(V)$ is", "the union of the affine opens $V' \\times_V U_i$.", "Since $\\mathcal{O}_{X_{S'}}(V' \\times_V U_i) =", "\\mathcal{O}_{S'}(V') \\otimes_{\\mathcal{O}_S(V)} \\mathcal{O}_X(U_i)$", "we see that the ring maps", "$\\mathcal{O}_{S'}(V') \\to \\mathcal{O}_{X_{S'}}(V' \\times_V U_i)$", "satisfy $P$ as $P$ is assumed stable under base change." ], "refs": [ "morphisms-lemma-locally-P-characterize" ], "ref_ids": [ 5192 ] } ], "ref_ids": [] }, { "id": 5195, "type": "theorem", "label": "morphisms-lemma-properties-local", "categories": [ "morphisms" ], "title": "morphisms-lemma-properties-local", "contents": [ "The following properties of a ring map $R \\to A$ are local.", "\\begin{enumerate}", "\\item (Isomorphism on local rings.)", "For every prime $\\mathfrak q$ of $A$ lying over $\\mathfrak p \\subset R$", "the ring map $R \\to A$ induces an isomorphism", "$R_{\\mathfrak p} \\to A_{\\mathfrak q}$.", "\\item (Open immersion.)", "For every prime $\\mathfrak q$ of $A$ there exists an $f \\in R$,", "$\\varphi(f) \\not \\in \\mathfrak q$ such that the ring map $\\varphi : R \\to A$", "induces an isomorphism $R_f \\to A_f$.", "\\item (Reduced fibres.)", "For every prime $\\mathfrak p$ of $R$ the fibre ring", "$A \\otimes_R \\kappa(\\mathfrak p)$ is reduced.", "\\item (Fibres of dimension at most $n$.)", "For every prime $\\mathfrak p$ of $R$ the fibre ring", "$A \\otimes_R \\kappa(\\mathfrak p)$ has Krull dimension at most $n$.", "\\item (Locally Noetherian on the target.)", "The ring map $R \\to A$ has the property that $A$ is Noetherian.", "\\item Add more here as needed\\footnote{But only those properties", "that are not already dealt with separately elsewhere.}.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5196, "type": "theorem", "label": "morphisms-lemma-properties-base-change", "categories": [ "morphisms" ], "title": "morphisms-lemma-properties-base-change", "contents": [ "The following properties of ring maps are stable under base change.", "\\begin{enumerate}", "\\item (Isomorphism on local rings.)", "For every prime $\\mathfrak q$ of $A$ lying over $\\mathfrak p \\subset R$", "the ring map $R \\to A$ induces an isomorphism", "$R_{\\mathfrak p} \\to A_{\\mathfrak q}$.", "\\item (Open immersion.)", "For every prime $\\mathfrak q$ of $A$ there exists an $f \\in R$,", "$\\varphi(f) \\not \\in \\mathfrak q$ such that the ring map $\\varphi : R \\to A$", "induces an isomorphism $R_f \\to A_f$.", "\\item Add more here as needed\\footnote{But only those properties", "that are not already dealt with separately elsewhere.}.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5197, "type": "theorem", "label": "morphisms-lemma-properties-composition", "categories": [ "morphisms" ], "title": "morphisms-lemma-properties-composition", "contents": [ "The following properties of ring maps are stable under composition.", "\\begin{enumerate}", "\\item (Isomorphism on local rings.)", "For every prime $\\mathfrak q$ of $A$ lying over $\\mathfrak p \\subset R$", "the ring map $R \\to A$ induces an isomorphism", "$R_{\\mathfrak p} \\to A_{\\mathfrak q}$.", "\\item (Open immersion.)", "For every prime $\\mathfrak q$ of $A$ there exists an $f \\in R$,", "$\\varphi(f) \\not \\in \\mathfrak q$ such that the ring map $\\varphi : R \\to A$", "induces an isomorphism $R_f \\to A_f$.", "\\item (Locally Noetherian on the target.)", "The ring map $R \\to A$ has the property that $A$ is Noetherian.", "\\item Add more here as needed\\footnote{But only those properties", "that are not already dealt with separately elsewhere.}.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5198, "type": "theorem", "label": "morphisms-lemma-locally-finite-type-characterize", "categories": [ "morphisms" ], "title": "morphisms-lemma-locally-finite-type-characterize", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is locally of finite type.", "\\item For all affine opens $U \\subset X$, $V \\subset S$", "with $f(U) \\subset V$ the ring map", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is of finite type.", "\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$", "and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$", "is locally of finite type.", "\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$", "and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is", "of finite type, for all $j\\in J, i\\in I_j$.", "\\end{enumerate}", "Moreover, if $f$ is locally of finite type then for", "any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$", "the restriction $f|_U : U \\to V$ is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-locally-P} if we show that", "the property ``$R \\to A$ is of finite type'' is local.", "We check conditions (a), (b) and (c) of Definition", "\\ref{definition-property-local}.", "By Algebra, Lemma \\ref{algebra-lemma-base-change-finiteness}", "being of finite type is stable under base change and hence", "we conclude (a) holds. By the same lemma being of finite type", "is stable under composition and trivially for any ring", "$R$ the ring map $R \\to R_f$ is of finite type.", "We conclude (b) holds. Finally, property (c) is true", "according to Algebra, Lemma \\ref{algebra-lemma-cover-upstairs}." ], "refs": [ "morphisms-lemma-locally-P", "morphisms-definition-property-local", "algebra-lemma-base-change-finiteness", "algebra-lemma-cover-upstairs" ], "ref_ids": [ 5191, 5547, 373, 412 ] } ], "ref_ids": [] }, { "id": 5199, "type": "theorem", "label": "morphisms-lemma-composition-finite-type", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-finite-type", "contents": [ "The composition of two morphisms which are locally of finite type is", "locally of finite type. The same is true for morphisms of finite type." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-locally-finite-type-characterize}", "we saw that being of finite type is a local property of ring maps.", "Hence the first statement of the lemma follows from", "Lemma \\ref{lemma-composition-type-P} combined", "with the fact that being of finite type is a property of ring maps that is", "stable under composition, see", "Algebra, Lemma \\ref{algebra-lemma-compose-finite-type}.", "By the above and the fact that compositions of", "quasi-compact morphisms are quasi-compact, see", "Schemes, Lemma \\ref{schemes-lemma-composition-quasi-compact}", "we see that the composition of morphisms of finite type is", "of finite type." ], "refs": [ "morphisms-lemma-locally-finite-type-characterize", "morphisms-lemma-composition-type-P", "algebra-lemma-compose-finite-type", "schemes-lemma-composition-quasi-compact" ], "ref_ids": [ 5198, 5193, 333, 7699 ] } ], "ref_ids": [] }, { "id": 5200, "type": "theorem", "label": "morphisms-lemma-base-change-finite-type", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-finite-type", "contents": [ "The base change of a morphism which is locally of finite type", "is locally of finite type. The same is true for morphisms of", "finite type." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-locally-finite-type-characterize}", "we saw that being of finite type is a local property of ring maps.", "Hence the first statement of the lemma follows from", "Lemma \\ref{lemma-base-change-type-P} combined", "with the fact that being of finite type is a property of ring maps that is", "stable under base change, see", "Algebra, Lemma \\ref{algebra-lemma-base-change-finiteness}.", "By the above and the fact that a base change of a", "quasi-compact morphism is quasi-compact, see", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-preserved-base-change}", "we see that the base change of a morphism of finite type is", "a morphism of finite type." ], "refs": [ "morphisms-lemma-locally-finite-type-characterize", "morphisms-lemma-base-change-type-P", "algebra-lemma-base-change-finiteness", "schemes-lemma-quasi-compact-preserved-base-change" ], "ref_ids": [ 5198, 5194, 373, 7698 ] } ], "ref_ids": [] }, { "id": 5201, "type": "theorem", "label": "morphisms-lemma-immersion-locally-finite-type", "categories": [ "morphisms" ], "title": "morphisms-lemma-immersion-locally-finite-type", "contents": [ "A closed immersion is of finite type.", "An immersion is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "This is true because an open immersion is a local isomorphism,", "and a closed immersion is obviously of finite type." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5202, "type": "theorem", "label": "morphisms-lemma-finite-type-noetherian", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-type-noetherian", "contents": [ "Let $f : X \\to S$ be a morphism.", "If $S$ is (locally) Noetherian and $f$ (locally) of finite type", "then $X$ is (locally) Noetherian." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the fact that a ring", "of finite type over a Noetherian ring is Noetherian,", "see Algebra, Lemma \\ref{algebra-lemma-Noetherian-permanence}.", "(Also: use the fact that the source of a quasi-compact morphism", "with quasi-compact target is quasi-compact.)" ], "refs": [ "algebra-lemma-Noetherian-permanence" ], "ref_ids": [ 448 ] } ], "ref_ids": [] }, { "id": 5203, "type": "theorem", "label": "morphisms-lemma-finite-type-Noetherian-quasi-separated", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-type-Noetherian-quasi-separated", "contents": [ "Let $f : X \\to S$ be locally of finite type with $S$ locally Noetherian.", "Then $f$ is quasi-separated." ], "refs": [], "proofs": [ { "contents": [ "In fact, it is true that $X$ is quasi-separated, see", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian-quasi-separated}", "and Lemma \\ref{lemma-finite-type-noetherian} above.", "Then apply Schemes, Lemma \\ref{schemes-lemma-compose-after-separated}", "to conclude that $f$ is quasi-separated." ], "refs": [ "properties-lemma-locally-Noetherian-quasi-separated", "morphisms-lemma-finite-type-noetherian", "schemes-lemma-compose-after-separated" ], "ref_ids": [ 2953, 5202, 7715 ] } ], "ref_ids": [] }, { "id": 5204, "type": "theorem", "label": "morphisms-lemma-permanence-finite-type", "categories": [ "morphisms" ], "title": "morphisms-lemma-permanence-finite-type", "contents": [ "Let $X \\to Y$ be a morphism of schemes over a base scheme $S$.", "If $X$ is locally of finite type over $S$, then $X \\to Y$", "is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "Via Lemma \\ref{lemma-locally-finite-type-characterize} this translates", "into the following algebra", "fact: Given ring maps $A \\to B \\to C$ such that $A \\to C$ is", "of finite type, then $B \\to C$ is of finite type.", "(See", "Algebra, Lemma \\ref{algebra-lemma-compose-finite-type})." ], "refs": [ "morphisms-lemma-locally-finite-type-characterize", "algebra-lemma-compose-finite-type" ], "ref_ids": [ 5198, 333 ] } ], "ref_ids": [] }, { "id": 5205, "type": "theorem", "label": "morphisms-lemma-point-finite-type", "categories": [ "morphisms" ], "title": "morphisms-lemma-point-finite-type", "contents": [ "Let $S$ be a scheme. Let $k$ be a field.", "Let $f : \\Spec(k) \\to S$ be a morphism.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is of finite type.", "\\item The morphism $f$ is locally of finite type.", "\\item There exists an affine open $U = \\Spec(R)$ of $S$", "such that $f$ corresponds to a finite ring map $R \\to k$.", "\\item There exists an affine open $U = \\Spec(R)$ of $S$", "such that the image of $f$ consists of a closed point $u$ in $U$", "and the field extension $\\kappa(u) \\subset k$ is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is obvious as $\\Spec(k)$", "is a singleton and hence any morphism from it is quasi-compact.", "\\medskip\\noindent", "Suppose $f$ is locally of finite type. Choose any affine open", "$\\Spec(R) = U \\subset S$ such that the image of $f$", "is contained in $U$, and the ring map $R \\to k$", "is of finite type. Let $\\mathfrak p \\subset R$ be the kernel.", "Then $R/\\mathfrak p \\subset k$ is of finite type. By", "Algebra, Lemma \\ref{algebra-lemma-field-finite-type-over-domain}", "there exist a $\\overline{f} \\in R/\\mathfrak p$ such that", "$(R/\\mathfrak p)_{\\overline{f}}$ is a field and", "$(R/\\mathfrak p)_{\\overline{f}} \\to k$ is a finite field", "extension. If $f \\in R$ is a lift of $\\overline{f}$, then", "we see that $k$ is a finite $R_f$-module. Thus (2) $\\Rightarrow$ (3).", "\\medskip\\noindent", "Suppose that $\\Spec(R) = U \\subset S$ is an affine open", "such that $f$ corresponds to a finite ring map $R \\to k$.", "Then $f$ is locally of finite type", "by Lemma \\ref{lemma-locally-finite-type-characterize}.", "Thus (3) $\\Rightarrow$ (2).", "\\medskip\\noindent", "Suppose $R \\to k$ is finite. The image of $R \\to k$ is a field", "over which $k$ is finite by", "Algebra, Lemma \\ref{algebra-lemma-integral-under-field}.", "Hence the kernel of $R \\to k$ is a maximal ideal.", "Thus (3) $\\Rightarrow$ (4).", "\\medskip\\noindent", "The implication (4) $\\Rightarrow$ (3) is immediate." ], "refs": [ "algebra-lemma-field-finite-type-over-domain", "morphisms-lemma-locally-finite-type-characterize", "algebra-lemma-integral-under-field" ], "ref_ids": [ 466, 5198, 496 ] } ], "ref_ids": [] }, { "id": 5206, "type": "theorem", "label": "morphisms-lemma-artinian-finite-type", "categories": [ "morphisms" ], "title": "morphisms-lemma-artinian-finite-type", "contents": [ "Let $S$ be a scheme.", "Let $A$ be an Artinian local ring with residue field $\\kappa$.", "Let $f : \\Spec(A) \\to S$ be a morphism of schemes.", "Then $f$ is of finite type if and only if the composition", "$\\Spec(\\kappa) \\to \\Spec(A) \\to S$ is of finite type." ], "refs": [], "proofs": [ { "contents": [ "Since the morphism $\\Spec(\\kappa) \\to \\Spec(A)$ is of finite", "type it is clear that if $f$ is of finite type so is the composition", "$\\Spec(\\kappa) \\to S$ (see Lemma \\ref{lemma-composition-finite-type}).", "For the converse, note that $\\Spec(A) \\to S$ maps into some affine open", "$U = \\Spec(B)$ of $S$ as $\\Spec(A)$ has only one point. To finish", "apply Algebra, Lemma", "\\ref{algebra-lemma-essentially-of-finite-type-into-artinian-local}", "to $B \\to A$." ], "refs": [ "morphisms-lemma-composition-finite-type", "algebra-lemma-essentially-of-finite-type-into-artinian-local" ], "ref_ids": [ 5199, 649 ] } ], "ref_ids": [] }, { "id": 5207, "type": "theorem", "label": "morphisms-lemma-identify-finite-type-points", "categories": [ "morphisms" ], "title": "morphisms-lemma-identify-finite-type-points", "contents": [ "Let $S$ be a scheme. We have", "$$", "S_{\\text{ft-pts}} = \\bigcup\\nolimits_{U \\subset S\\text{ open}} U_0", "$$", "where $U_0$ is the set of closed points of $U$.", "Here we may let $U$ range over all opens or over all affine opens of $S$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-point-finite-type}." ], "refs": [ "morphisms-lemma-point-finite-type" ], "ref_ids": [ 5205 ] } ], "ref_ids": [] }, { "id": 5208, "type": "theorem", "label": "morphisms-lemma-finite-type-points-morphism", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-type-points-morphism", "contents": [ "Let $f : T \\to S$ be a morphism of schemes.", "If $f$ is locally of finite type, then", "$f(T_{\\text{ft-pts}}) \\subset S_{\\text{ft-pts}}$." ], "refs": [], "proofs": [ { "contents": [ "If $T$ is the spectrum of a field this is Lemma \\ref{lemma-point-finite-type}.", "In general it follows since the composition of morphisms locally of finite", "type is locally of finite type (Lemma \\ref{lemma-composition-finite-type})." ], "refs": [ "morphisms-lemma-point-finite-type", "morphisms-lemma-composition-finite-type" ], "ref_ids": [ 5205, 5199 ] } ], "ref_ids": [] }, { "id": 5209, "type": "theorem", "label": "morphisms-lemma-finite-type-points-surjective-morphism", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-type-points-surjective-morphism", "contents": [ "Let $f : T \\to S$ be a morphism of schemes.", "If $f$ is locally of finite type and surjective, then", "$f(T_{\\text{ft-pts}}) = S_{\\text{ft-pts}}$." ], "refs": [], "proofs": [ { "contents": [ "We have $f(T_{\\text{ft-pts}}) \\subset S_{\\text{ft-pts}}$ by", "Lemma \\ref{lemma-finite-type-points-morphism}.", "Let $s \\in S$ be a finite type point. As $f$ is surjective the scheme", "$T_s = \\Spec(\\kappa(s)) \\times_S T$ is nonempty, therefore", "has a finite type point $t \\in T_s$ by", "Lemma \\ref{lemma-identify-finite-type-points}.", "Now $T_s \\to T$ is a morphism of finite type as a base change", "of $s \\to S$", "(Lemma \\ref{lemma-base-change-finite-type}).", "Hence the image of $t$ in $T$ is a finite type point by", "Lemma \\ref{lemma-finite-type-points-morphism}", "which maps to $s$ by construction." ], "refs": [ "morphisms-lemma-finite-type-points-morphism", "morphisms-lemma-identify-finite-type-points", "morphisms-lemma-base-change-finite-type", "morphisms-lemma-finite-type-points-morphism" ], "ref_ids": [ 5208, 5207, 5200, 5208 ] } ], "ref_ids": [] }, { "id": 5210, "type": "theorem", "label": "morphisms-lemma-enough-finite-type-points", "categories": [ "morphisms" ], "title": "morphisms-lemma-enough-finite-type-points", "contents": [ "Let $S$ be a scheme.", "For any locally closed subset $T \\subset S$ we have", "$$", "T \\not = \\emptyset", "\\Rightarrow", "T \\cap S_{\\text{ft-pts}} \\not = \\emptyset.", "$$", "In particular, for any closed subset $T \\subset S$ we", "see that $T \\cap S_{\\text{ft-pts}}$ is dense in $T$." ], "refs": [], "proofs": [ { "contents": [ "Note that $T$ carries a scheme structure (see", "Schemes, Lemma \\ref{schemes-lemma-reduced-closed-subscheme})", "such that $T \\to S$ is a locally closed immersion.", "Any locally closed immersion is locally of finite type,", "see Lemma \\ref{lemma-immersion-locally-finite-type}.", "Hence by Lemma \\ref{lemma-finite-type-points-morphism}", "we see $T_{\\text{ft-pts}} \\subset S_{\\text{ft-pts}}$.", "Finally, any nonempty affine open of $T$ has at least one closed point", "which is a finite type point of $T$ by", "Lemma \\ref{lemma-identify-finite-type-points}." ], "refs": [ "schemes-lemma-reduced-closed-subscheme", "morphisms-lemma-immersion-locally-finite-type", "morphisms-lemma-finite-type-points-morphism", "morphisms-lemma-identify-finite-type-points" ], "ref_ids": [ 7681, 5201, 5208, 5207 ] } ], "ref_ids": [] }, { "id": 5211, "type": "theorem", "label": "morphisms-lemma-jacobson-finite-type-points", "categories": [ "morphisms" ], "title": "morphisms-lemma-jacobson-finite-type-points", "contents": [ "Let $S$ be a scheme. The following are equivalent:", "\\begin{enumerate}", "\\item the scheme $S$ is Jacobson,", "\\item $S_{\\text{ft-pts}}$ is the set of closed points of $S$,", "\\item for all $T \\to S$ locally of finite type", "closed points map to closed points, and", "\\item for all $T \\to S$ locally of finite type", "closed points $t \\in T$ map to closed points $s \\in S$ with", "$\\kappa(s) \\subset \\kappa(t)$ finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have trivially (4) $\\Rightarrow$ (3) $\\Rightarrow$ (2).", "Lemma \\ref{lemma-enough-finite-type-points} shows that (2) implies (1).", "Hence it suffices to show that (1) implies (4).", "Suppose that $T \\to S$ is locally of finite type.", "Choose $t \\in T$ closed and let $s \\in S$ be the image.", "Choose affine open neighbourhoods $\\Spec(R) = U \\subset S$ of $s$ and", "$\\Spec(A) = V \\subset T$ of $t$ with $V$ mapping into $U$.", "The induced ring map $R \\to A$ is of finite type", "(see Lemma \\ref{lemma-locally-finite-type-characterize}) and $R$ is Jacobson", "by Properties, Lemma \\ref{properties-lemma-locally-jacobson}.", "Thus the result follows from", "Algebra, Proposition \\ref{algebra-proposition-Jacobson-permanence}." ], "refs": [ "morphisms-lemma-enough-finite-type-points", "morphisms-lemma-locally-finite-type-characterize", "properties-lemma-locally-jacobson", "algebra-proposition-Jacobson-permanence" ], "ref_ids": [ 5210, 5198, 2964, 1405 ] } ], "ref_ids": [] }, { "id": 5212, "type": "theorem", "label": "morphisms-lemma-Jacobson-universally-Jacobson", "categories": [ "morphisms" ], "title": "morphisms-lemma-Jacobson-universally-Jacobson", "contents": [ "Let $S$ be a Jacobson scheme.", "Any scheme locally of finite type over $S$ is Jacobson." ], "refs": [], "proofs": [ { "contents": [ "This is clear from", "Algebra, Proposition \\ref{algebra-proposition-Jacobson-permanence}", "(and Properties, Lemma \\ref{properties-lemma-locally-jacobson} and", "Lemma \\ref{lemma-locally-finite-type-characterize})." ], "refs": [ "algebra-proposition-Jacobson-permanence", "properties-lemma-locally-jacobson", "morphisms-lemma-locally-finite-type-characterize" ], "ref_ids": [ 1405, 2964, 5198 ] } ], "ref_ids": [] }, { "id": 5213, "type": "theorem", "label": "morphisms-lemma-ubiquity-Jacobson-schemes", "categories": [ "morphisms" ], "title": "morphisms-lemma-ubiquity-Jacobson-schemes", "contents": [ "The following types of schemes are Jacobson.", "\\begin{enumerate}", "\\item Any scheme locally of finite type over a field.", "\\item Any scheme locally of finite type over $\\mathbf{Z}$.", "\\item Any scheme locally of finite type over a $1$-dimensional", "Noetherian domain with infinitely many primes.", "\\item A scheme of the form $\\Spec(R) \\setminus \\{\\mathfrak m\\}$", "where $(R, \\mathfrak m)$ is a Noetherian local ring.", "Also any scheme locally of finite type over it.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-Jacobson-universally-Jacobson} without mention.", "The spectrum of a field is clearly Jacobson.", "The spectrum of $\\mathbf{Z}$ is Jacobson, see", "Algebra, Lemma \\ref{algebra-lemma-pid-jacobson}.", "For (3) see", "Algebra, Lemma \\ref{algebra-lemma-noetherian-dim-1-Jacobson}.", "For (4) see", "Properties, Lemma \\ref{properties-lemma-complement-closed-point-Jacobson}." ], "refs": [ "morphisms-lemma-Jacobson-universally-Jacobson", "algebra-lemma-pid-jacobson", "algebra-lemma-noetherian-dim-1-Jacobson", "properties-lemma-complement-closed-point-Jacobson" ], "ref_ids": [ 5212, 471, 690, 2965 ] } ], "ref_ids": [] }, { "id": 5214, "type": "theorem", "label": "morphisms-lemma-universally-catenary-local", "categories": [ "morphisms" ], "title": "morphisms-lemma-universally-catenary-local", "contents": [ "Let $S$ be a locally Noetherian scheme. The following are equivalent", "\\begin{enumerate}", "\\item $S$ is universally catenary,", "\\item there exists an open covering of $S$ all of whose members are", "universally catenary schemes,", "\\item for every affine open $\\Spec(R) = U \\subset S$ the ring", "$R$ is universally catenary, and", "\\item there exists an affine open covering $S = \\bigcup U_i$ such", "that each $U_i$ is the spectrum of a universally catenary ring.", "\\end{enumerate}", "Moreover, in this case any scheme locally of finite type over $S$", "is universally catenary as well." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-immersion-locally-finite-type} an open immersion", "is locally of finite type. A composition of morphisms locally of", "finite type is locally of finite type", "(Lemma \\ref{lemma-composition-finite-type}). Thus it is clear that if $S$ is", "universally catenary then any open and any scheme locally of finite", "type over $S$ is universally catenary as well. This proves the final", "statement of the lemma and that (1) implies (2).", "\\medskip\\noindent", "If $\\Spec(R)$ is a universally catenary scheme, then every", "scheme $\\Spec(A)$ with $A$ a finite type $R$-algebra is", "catenary. Hence all these rings $A$ are catenary by", "Algebra, Lemma \\ref{algebra-lemma-catenary}.", "Thus $R$ is universally catenary. Combined with the remarks above we", "conclude that (1) implies (3), and (2) implies (4). Of course", "(3) implies (4) trivially.", "\\medskip\\noindent", "To finish the proof we show that (4) implies (1).", "Assume (4) and let $X \\to S$ be a morphism locally of finite type.", "We can find an affine open covering $X = \\bigcup V_j$ such that", "each $V_j \\to S$ maps into one of the $U_i$. By", "Lemma \\ref{lemma-locally-finite-type-characterize}", "the induced ring map $\\mathcal{O}(U_i) \\to \\mathcal{O}(V_j)$ is", "of finite type. Hence $\\mathcal{O}(V_j)$ is catenary. Hence", "$X$ is catenary by Properties, Lemma \\ref{properties-lemma-catenary-local}." ], "refs": [ "morphisms-lemma-immersion-locally-finite-type", "morphisms-lemma-composition-finite-type", "algebra-lemma-catenary", "morphisms-lemma-locally-finite-type-characterize", "properties-lemma-catenary-local" ], "ref_ids": [ 5201, 5199, 931, 5198, 2983 ] } ], "ref_ids": [] }, { "id": 5215, "type": "theorem", "label": "morphisms-lemma-universally-catenary-local-rings-universally-catenary", "categories": [ "morphisms" ], "title": "morphisms-lemma-universally-catenary-local-rings-universally-catenary", "contents": [ "Let $S$ be a locally Noetherian scheme.", "The following are equivalent:", "\\begin{enumerate}", "\\item $S$ is universally catenary, and", "\\item all local rings $\\mathcal{O}_{S, s}$ of $S$ are universally catenary.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume that all local rings of $S$ are universally catenary.", "Let $f : X \\to S$ be locally of finite type.", "We know that $X$ is catenary if and only if $\\mathcal{O}_{X, x}$ is", "catenary for all $x \\in X$. If $f(x) = s$, then $\\mathcal{O}_{X, x}$", "is essentially of finite type over $\\mathcal{O}_{S, s}$. Hence", "$\\mathcal{O}_{X, x}$ is catenary by the assumption that", "$\\mathcal{O}_{S, s}$ is universally catenary.", "\\medskip\\noindent", "Conversely, assume that $S$ is universally catenary. Let $s \\in S$.", "We may replace $S$ by an affine open neighbourhood of $s$ by", "Lemma \\ref{lemma-universally-catenary-local}. Say $S = \\Spec(R)$", "and $s$ corresponds to the prime ideal $\\mathfrak p$. Any finite", "type $R_{\\mathfrak p}$-algebra $A'$ is of the form", "$A_{\\mathfrak p}$ for some finite type $R$-algebra $A$.", "By assumption (and Lemma \\ref{lemma-universally-catenary-local} if you like)", "the ring $A$ is catenary, and hence $A'$ (a localization of $A$) is", "catenary. Thus $R_{\\mathfrak p}$ is universally catenary." ], "refs": [ "morphisms-lemma-universally-catenary-local", "morphisms-lemma-universally-catenary-local" ], "ref_ids": [ 5214, 5214 ] } ], "ref_ids": [] }, { "id": 5216, "type": "theorem", "label": "morphisms-lemma-catenary-check-irreducible", "categories": [ "morphisms" ], "title": "morphisms-lemma-catenary-check-irreducible", "contents": [ "Let $S$ be a locally Noetherian scheme. Then $S$ is universally catenary", "if and only if the irreducible components of $S$ are universally catenary." ], "refs": [], "proofs": [ { "contents": [ "Omitted. For the affine case, please see", "Algebra, Lemma \\ref{algebra-lemma-catenary-check-irreducible}." ], "refs": [ "algebra-lemma-catenary-check-irreducible" ], "ref_ids": [ 936 ] } ], "ref_ids": [] }, { "id": 5217, "type": "theorem", "label": "morphisms-lemma-ubiquity-uc", "categories": [ "morphisms" ], "title": "morphisms-lemma-ubiquity-uc", "contents": [ "The following types of schemes are universally catenary.", "\\begin{enumerate}", "\\item Any scheme locally of finite type over a field.", "\\item Any scheme locally of finite type over a Cohen-Macaulay scheme.", "\\item Any scheme locally of finite type over $\\mathbf{Z}$.", "\\item Any scheme locally of finite type over a $1$-dimensional", "Noetherian domain.", "\\item And so on.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "All of these follow from the fact that a", "Cohen-Macaulay ring is universally catenary, see", "Algebra, Lemma \\ref{algebra-lemma-CM-ring-catenary}.", "Also, use the last assertion of", "Lemma \\ref{lemma-universally-catenary-local}.", "Some details omitted." ], "refs": [ "algebra-lemma-CM-ring-catenary", "morphisms-lemma-universally-catenary-local" ], "ref_ids": [ 937, 5214 ] } ], "ref_ids": [] }, { "id": 5218, "type": "theorem", "label": "morphisms-lemma-finite-type-nagata", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-type-nagata", "contents": [ "Let $f : X \\to S$ be a morphism.", "If $S$ is Nagata and $f$ locally of finite type then $X$ is Nagata.", "If $S$ is universally Japanese", "and $f$ locally of finite type then $X$ is universally Japanese." ], "refs": [], "proofs": [ { "contents": [ "For ``universally Japanese'' this follows from", "Algebra, Lemma \\ref{algebra-lemma-universally-japanese}.", "For ``Nagata'' this follows from", "Algebra, Proposition \\ref{algebra-proposition-nagata-universally-japanese}." ], "refs": [ "algebra-lemma-universally-japanese", "algebra-proposition-nagata-universally-japanese" ], "ref_ids": [ 1349, 1430 ] } ], "ref_ids": [] }, { "id": 5219, "type": "theorem", "label": "morphisms-lemma-ubiquity-nagata", "categories": [ "morphisms" ], "title": "morphisms-lemma-ubiquity-nagata", "contents": [ "The following types of schemes are Nagata.", "\\begin{enumerate}", "\\item Any scheme locally of finite type over a field.", "\\item Any scheme locally of finite type over a Noetherian complete local ring.", "\\item Any scheme locally of finite type over $\\mathbf{Z}$.", "\\item Any scheme locally of finite type over a Dedekind ring of", "characteristic zero.", "\\item And so on.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-type-nagata} we only need to show that", "the rings mentioned above are Nagata rings. For this see", "Algebra, Proposition \\ref{algebra-proposition-ubiquity-nagata}." ], "refs": [ "morphisms-lemma-finite-type-nagata", "algebra-proposition-ubiquity-nagata" ], "ref_ids": [ 5218, 1431 ] } ], "ref_ids": [] }, { "id": 5220, "type": "theorem", "label": "morphisms-lemma-J", "categories": [ "morphisms" ], "title": "morphisms-lemma-J", "contents": [ "Let $X$ be a locally Noetherian scheme. The following are equivalent", "\\begin{enumerate}", "\\item $X$ is J-2,", "\\item there exists an open covering of $X$ all of whose members are", "J-2 schemes,", "\\item for every affine open $\\Spec(R) = U \\subset X$ the ring", "$R$ is J-2, and", "\\item there exists an affine open covering $S = \\bigcup U_i$ such", "that each $\\mathcal{O}(U_i)$ is J-2 for all $i$.", "\\end{enumerate}", "Moreover, in this case any scheme locally of finite type over $X$", "is J-2 as well." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-immersion-locally-finite-type} an open immersion", "is locally of finite type. A composition of morphisms locally of", "finite type is locally of finite type", "(Lemma \\ref{lemma-composition-finite-type}). Thus it is clear that if $X$ is", "J-2 then any open and any scheme locally of finite", "type over $X$ is J-2 as well. This proves the final", "statement of the lemma.", "\\medskip\\noindent", "If $\\Spec(R)$ is J-2, then for every finite type $R$-algebra $A$", "the regular locus of the scheme $\\Spec(A)$ is open. Hence $R$ is", "J-2, by definition (see", "More on Algebra, Definition \\ref{more-algebra-definition-J}).", "Combined with the remarks above we conclude that (1) implies (3), and", "(2) implies (4). Of course (1) $\\Rightarrow$ (2) and", "(3) $\\Rightarrow$ (4) trivially.", "\\medskip\\noindent", "To finish the proof we show that (4) implies (1).", "Assume (4) and let $Y \\to X$ be a morphism locally of finite type.", "We can find an affine open covering $Y = \\bigcup V_j$ such that", "each $V_j \\to X$ maps into one of the $U_i$. By", "Lemma \\ref{lemma-locally-finite-type-characterize}", "the induced ring map $\\mathcal{O}(U_i) \\to \\mathcal{O}(V_j)$ is", "of finite type. Hence the regular locus of", "$V_j = \\Spec(\\mathcal{O}(V_j))$ is open. Since", "$\\text{Reg}(Y) \\cap V_j = \\text{Reg}(V_j)$ we conclude that", "$\\text{Reg}(Y)$ is open as desired." ], "refs": [ "morphisms-lemma-immersion-locally-finite-type", "morphisms-lemma-composition-finite-type", "more-algebra-definition-J", "morphisms-lemma-locally-finite-type-characterize" ], "ref_ids": [ 5201, 5199, 10615, 5198 ] } ], "ref_ids": [] }, { "id": 5221, "type": "theorem", "label": "morphisms-lemma-ubiquity-J-2", "categories": [ "morphisms" ], "title": "morphisms-lemma-ubiquity-J-2", "contents": [ "The following types of schemes are J-2.", "\\begin{enumerate}", "\\item Any scheme locally of finite type over a field.", "\\item Any scheme locally of finite type over a Noetherian complete local ring.", "\\item Any scheme locally of finite type over $\\mathbf{Z}$.", "\\item Any scheme locally of finite type over a Noetherian local ring", "of dimension $1$.", "\\item Any scheme locally of finite type over a Nagata ring of dimension $1$.", "\\item Any scheme locally of finite type over a Dedekind ring of", "characteristic zero.", "\\item And so on.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-J} we only need to show that", "the rings mentioned above are J-2. For this see", "More on Algebra, Proposition \\ref{more-algebra-proposition-ubiquity-J-2}." ], "refs": [ "morphisms-lemma-J", "more-algebra-proposition-ubiquity-J-2" ], "ref_ids": [ 5220, 10578 ] } ], "ref_ids": [] }, { "id": 5222, "type": "theorem", "label": "morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre", "categories": [ "morphisms" ], "title": "morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point. Set $s = f(x)$.", "If $\\kappa(x)/\\kappa(s)$", "is an algebraic field extension, then", "\\begin{enumerate}", "\\item $x$ is a closed point of its fibre, and", "\\item if in addition $s$ is a closed point of $S$, then", "$x$ is a closed point of $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The second statement follows from the first by elementary topology.", "According to Schemes, Lemma \\ref{schemes-lemma-fibre-topological}", "to prove the first statement", "we may replace $X$ by $X_s$ and $S$ by $\\Spec(\\kappa(s))$.", "Thus we may assume that $S = \\Spec(k)$ is the spectrum of a field.", "In this case, let $\\Spec(A) = U \\subset X$ be any affine open", "containing $x$. The point $x$ corresponds to a prime ideal", "$\\mathfrak q \\subset A$ such that $k \\subset \\kappa(\\mathfrak q)$", "is an algebraic field extension. By", "Algebra, Lemma \\ref{algebra-lemma-finite-residue-extension-closed}", "we see that $\\mathfrak q$ is a maximal ideal, i.e., $x \\in U$ is a", "closed point. Since the affine opens form", "a basis of the topology of $X$ we conclude that $\\{x\\}$ is closed." ], "refs": [ "schemes-lemma-fibre-topological", "algebra-lemma-finite-residue-extension-closed" ], "ref_ids": [ 7696, 472 ] } ], "ref_ids": [] }, { "id": 5223, "type": "theorem", "label": "morphisms-lemma-closed-point-fibre-locally-finite-type", "categories": [ "morphisms" ], "title": "morphisms-lemma-closed-point-fibre-locally-finite-type", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point. Set $s = f(x)$.", "Assume $f$ is locally of finite type.", "Then $x$ is a closed point of its fibre", "if and only if $\\kappa(s) \\subset \\kappa(x)$ is", "a finite field extension." ], "refs": [], "proofs": [ { "contents": [ "If the extension is finite, then $x$ is a closed point of", "the fibre by", "Lemma \\ref{lemma-algebraic-residue-field-extension-closed-point-fibre}", "above. For the converse, assume that $x$ is a closed point", "of its fibre. Choose affine opens $\\Spec(A) = U \\subset X$", "and $\\Spec(R) = V \\subset S$ such that $f(U) \\subset V$.", "By Lemma \\ref{lemma-locally-finite-type-characterize} the ring map", "$R \\to A$ is of finite type. Let $\\mathfrak q \\subset A$,", "resp.\\ $\\mathfrak p \\subset R$ be the prime ideal corresponding", "to $x$, resp.\\ $s$. Consider the fibre ring", "$\\overline{A} = A \\otimes_R \\kappa(\\mathfrak p)$.", "Let $\\overline{\\mathfrak q}$ be the prime of $\\overline{A}$", "corresponding to $\\mathfrak q$. The assumption that $x$", "is a closed point of its fibre implies that $\\overline{\\mathfrak q}$", "is a maximal ideal of $\\overline{A}$. Since $\\overline{A}$", "is an algebra of finite type over the field $\\kappa(\\mathfrak p)$", "we see by the Hilbert Nullstellensatz, see", "Algebra, Theorem \\ref{algebra-theorem-nullstellensatz},", "that $\\kappa(\\overline{\\mathfrak q})$ is a finite extension", "of $\\kappa(\\mathfrak p)$.", "Since $\\kappa(s) = \\kappa(\\mathfrak p)$ and", "$\\kappa(x) = \\kappa(\\mathfrak q) = \\kappa(\\overline{\\mathfrak q})$", "we win." ], "refs": [ "morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre", "morphisms-lemma-locally-finite-type-characterize", "algebra-theorem-nullstellensatz" ], "ref_ids": [ 5222, 5198, 316 ] } ], "ref_ids": [] }, { "id": 5224, "type": "theorem", "label": "morphisms-lemma-base-change-closed-point-fibre-locally-finite-type", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-closed-point-fibre-locally-finite-type", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $g : S' \\to S$ be any morphism. Denote $f' : X' \\to S'$ the base change.", "If $x' \\in X'$ maps to a point $x \\in X$ which is closed in $X_{f(x)}$", "then $x'$ is closed in $X'_{f'(x')}$." ], "refs": [], "proofs": [ { "contents": [ "The residue field $\\kappa(x')$ is a quotient of", "$\\kappa(f'(x')) \\otimes_{\\kappa(f(x))} \\kappa(x)$, see", "Schemes, Lemma \\ref{schemes-lemma-points-fibre-product}.", "Hence it is a finite extension of $\\kappa(f'(x'))$ as", "$\\kappa(x)$ is a finite extension of $\\kappa(f(x))$ by", "Lemma \\ref{lemma-closed-point-fibre-locally-finite-type}.", "Thus we see that $x'$ is closed in its fibre by applying that lemma", "one more time." ], "refs": [ "schemes-lemma-points-fibre-product", "morphisms-lemma-closed-point-fibre-locally-finite-type" ], "ref_ids": [ 7693, 5223 ] } ], "ref_ids": [] }, { "id": 5225, "type": "theorem", "label": "morphisms-lemma-residue-field-quasi-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-residue-field-quasi-finite", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point. Set $s = f(x)$.", "If $f$ is quasi-finite at $x$, then the residue field", "extension $\\kappa(s) \\subset \\kappa(x)$ is finite." ], "refs": [], "proofs": [ { "contents": [ "This is clear from Algebra, Definition \\ref{algebra-definition-quasi-finite}." ], "refs": [ "algebra-definition-quasi-finite" ], "ref_ids": [ 1522 ] } ], "ref_ids": [] }, { "id": 5226, "type": "theorem", "label": "morphisms-lemma-quasi-finite-at-point-characterize", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-finite-at-point-characterize", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point. Set $s = f(x)$.", "Let $X_s$ be the fibre of $f$ at $s$.", "Assume $f$ is locally of finite type.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is quasi-finite at $x$.", "\\item The point $x$ is isolated in $X_s$.", "\\item The point $x$ is closed in $X_s$", "and there is no point $x' \\in X_s$, $x' \\not = x$", "which specializes to $x$.", "\\item For any pair of affine opens", "$\\Spec(A) = U \\subset X$, $\\Spec(R) = V \\subset S$ with", "$f(U) \\subset V$ and $x \\in U$ corresponding to $\\mathfrak q \\subset A$", "the ring map $R \\to A$ is quasi-finite at $\\mathfrak q$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f$ is quasi-finite at $x$. By assumption there exist opens", "$U \\subset X$, $V \\subset S$ such that $f(U) \\subset V$, $x \\in U$", "and $x$ an isolated point of $U_s$. Hence $\\{x\\} \\subset U_s$ is an open", "subset. Since $U_s = U \\cap X_s \\subset X_s$ is also open we conclude", "that $\\{x\\} \\subset X_s$ is an open subset also. Thus we conclude", "that $x$ is an isolated point of $X_s$.", "\\medskip\\noindent", "Note that $X_s$ is a Jacobson scheme by", "Lemma \\ref{lemma-ubiquity-Jacobson-schemes}", "(and", "Lemma \\ref{lemma-base-change-finite-type}).", "If $x$ is isolated in $X_s$, i.e., $\\{x\\} \\subset X_s$ is open,", "then $\\{x\\}$ contains a closed point (by the Jacobson property), hence", "$x$ is closed in $X_s$. It is clear that there is no point $x' \\in X_s$,", "distinct from $x$, specializing to $x$.", "\\medskip\\noindent", "Assume that $x$ is closed in $X_s$ and that there is no point $x' \\in X_s$,", "distinct from $x$, specializing to $x$. Consider a pair of affine opens", "$\\Spec(A) = U \\subset X$, $\\Spec(R) = V \\subset S$ with", "$f(U) \\subset V$ and $x \\in U$. Let $\\mathfrak q \\subset A$ correspond to", "$x$ and $\\mathfrak p \\subset R$ correspond to $s$.", "By Lemma \\ref{lemma-locally-finite-type-characterize} the ring map", "$R \\to A$ is of finite type. Consider the fibre ring", "$\\overline{A} = A \\otimes_R \\kappa(\\mathfrak p)$.", "Let $\\overline{\\mathfrak q}$ be the prime of $\\overline{A}$ corresponding", "to $\\mathfrak q$. Since $\\Spec(\\overline{A})$ is an open subscheme of", "the fibre $X_s$ we see that $\\overline{q}$ is a maximal ideal", "of $\\overline{A}$ and that there is no point of $\\Spec(\\overline{A})$", "specializing to $\\overline{\\mathfrak q}$.", "This implies that $\\dim(\\overline{A}_{\\overline{q}}) = 0$.", "Hence by", "Algebra, Definition \\ref{algebra-definition-quasi-finite}", "we see that $R \\to A$ is quasi-finite at $\\mathfrak q$, i.e.,", "$X \\to S$ is quasi-finite at $x$ by definition.", "\\medskip\\noindent", "At this point we have shown conditions (1) -- (3) are all equivalent.", "It is clear that (4) implies (1). And it is also clear that", "(2) implies (4) since if $x$ is an isolated point of $X_s$", "then it is also an isolated point of $U_s$ for any open $U$", "which contains it." ], "refs": [ "morphisms-lemma-ubiquity-Jacobson-schemes", "morphisms-lemma-base-change-finite-type", "morphisms-lemma-locally-finite-type-characterize", "algebra-definition-quasi-finite" ], "ref_ids": [ 5213, 5200, 5198, 1522 ] } ], "ref_ids": [] }, { "id": 5227, "type": "theorem", "label": "morphisms-lemma-finite-fibre", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-fibre", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $s \\in S$. Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite type, and", "\\item $f^{-1}(\\{s\\})$ is a finite set.", "\\end{enumerate}", "Then $X_s$ is a finite discrete topological space, and", "$f$ is quasi-finite at each point of $X$ lying over $s$." ], "refs": [], "proofs": [ { "contents": [ "Suppose $T$ is a scheme which (a) is locally of finite type", "over a field $k$, and (b) has finitely many points. Then", "Lemma \\ref{lemma-ubiquity-Jacobson-schemes} shows $T$ is a", "Jacobson scheme. A finite Jacobson space is discrete, see", "Topology, Lemma \\ref{topology-lemma-finite-jacobson}.", "Apply this remark to the fibre $X_s$ which is locally of finite type over", "$\\Spec(\\kappa(s))$ to see the first statement. Finally, apply", "Lemma \\ref{lemma-quasi-finite-at-point-characterize} to see the second." ], "refs": [ "morphisms-lemma-ubiquity-Jacobson-schemes", "topology-lemma-finite-jacobson", "morphisms-lemma-quasi-finite-at-point-characterize" ], "ref_ids": [ 5213, 8280, 5226 ] } ], "ref_ids": [] }, { "id": 5228, "type": "theorem", "label": "morphisms-lemma-locally-quasi-finite-fibres", "categories": [ "morphisms" ], "title": "morphisms-lemma-locally-quasi-finite-fibres", "contents": [ "\\begin{slogan}", "Finite type morphisms with discrete fibers are quasi-finite.", "\\end{slogan}", "Let $f : X \\to S$ be a morphism of schemes.", "Assume $f$ is locally of finite type.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $f$ is locally quasi-finite,", "\\item for every $s \\in S$ the fibre $X_s$ is a discrete topological space, and", "\\item for every morphism $\\Spec(k) \\to S$ where $k$ is a field", "the base change $X_k$ has an underlying discrete topological space.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is immediate that (3) implies (2).", "Lemma \\ref{lemma-quasi-finite-at-point-characterize}", "shows that (2) is equivalent to (1).", "Assume (2) and let $\\Spec(k) \\to S$ be as in (3).", "Denote $s \\in S$ the image of $\\Spec(k) \\to S$.", "Then $X_k$ is the base change of $X_s$ via", "$\\Spec(k) \\to \\Spec(\\kappa(s))$. Hence every", "point of $X_k$ is closed by", "Lemma \\ref{lemma-base-change-closed-point-fibre-locally-finite-type}.", "As $X_k \\to \\Spec(k)$ is locally of finite type (by", "Lemma \\ref{lemma-base-change-finite-type}),", "we may apply", "Lemma \\ref{lemma-quasi-finite-at-point-characterize}", "to conclude that every point of $X_k$ is isolated, i.e., $X_k$ has", "a discrete underlying topological space." ], "refs": [ "morphisms-lemma-quasi-finite-at-point-characterize", "morphisms-lemma-base-change-closed-point-fibre-locally-finite-type", "morphisms-lemma-base-change-finite-type", "morphisms-lemma-quasi-finite-at-point-characterize" ], "ref_ids": [ 5226, 5224, 5200, 5226 ] } ], "ref_ids": [] }, { "id": 5229, "type": "theorem", "label": "morphisms-lemma-quasi-finite-locally-quasi-compact", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-finite-locally-quasi-compact", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Then $f$ is quasi-finite if and only if $f$ is", "locally quasi-finite and quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Assume $f$ is quasi-finite. It is quasi-compact by Definition", "\\ref{definition-finite-type}. Let $x \\in X$.", "We see that $f$ is quasi-finite at $x$ by", "Lemma \\ref{lemma-quasi-finite-at-point-characterize}.", "Hence $f$ is quasi-compact and locally quasi-finite.", "\\medskip\\noindent", "Assume $f$ is quasi-compact and locally quasi-finite.", "Then $f$ is of finite type. Let $x \\in X$ be a point.", "By Lemma \\ref{lemma-quasi-finite-at-point-characterize}", "we see that $x$ is an isolated point of its fibre.", "The lemma is proved." ], "refs": [ "morphisms-definition-finite-type", "morphisms-lemma-quasi-finite-at-point-characterize", "morphisms-lemma-quasi-finite-at-point-characterize" ], "ref_ids": [ 5549, 5226, 5226 ] } ], "ref_ids": [] }, { "id": 5230, "type": "theorem", "label": "morphisms-lemma-quasi-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-finite", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is quasi-finite, and", "\\item $f$ is locally of finite type, quasi-compact, and has finite fibres.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f$ is quasi-finite. In particular $f$ is locally of finite type", "and quasi-compact (since it is of finite type). Let $s \\in S$. Since", "every $x \\in X_s$ is isolated in $X_s$ we see that", "$X_s = \\bigcup_{x \\in X_s} \\{x\\}$ is an open covering. As $f$", "is quasi-compact, the fibre $X_s$ is quasi-compact. Hence we see", "that $X_s$ is finite.", "\\medskip\\noindent", "Conversely, assume $f$ is locally of finite type, quasi-compact", "and has finite fibres. Then it is locally quasi-finite by", "Lemma \\ref{lemma-finite-fibre}. Hence it is quasi-finite by", "Lemma \\ref{lemma-quasi-finite-locally-quasi-compact}." ], "refs": [ "morphisms-lemma-finite-fibre", "morphisms-lemma-quasi-finite-locally-quasi-compact" ], "ref_ids": [ 5227, 5229 ] } ], "ref_ids": [] }, { "id": 5231, "type": "theorem", "label": "morphisms-lemma-locally-quasi-finite-characterize", "categories": [ "morphisms" ], "title": "morphisms-lemma-locally-quasi-finite-characterize", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is locally quasi-finite.", "\\item For every pair of affine opens $U \\subset X$, $V \\subset S$", "with $f(U) \\subset V$ the ring map", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is quasi-finite.", "\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$", "and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$", "is locally quasi-finite.", "\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$", "and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is", "quasi-finite, for all $j\\in J, i\\in I_j$.", "\\end{enumerate}", "Moreover, if $f$ is locally quasi-finite then for", "any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$", "the restriction $f|_U : U \\to V$ is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "For a ring map $R \\to A$ let us define", "$P(R \\to A)$ to mean ``$R \\to A$ is quasi-finite''", "(see remark above lemma).", "We claim that $P$ is a local property of ring maps.", "We check conditions (a), (b) and (c) of Definition", "\\ref{definition-property-local}. In the proof of", "Lemma \\ref{lemma-locally-finite-type-characterize}", "we have seen that (a), (b) and (c) hold for the property", "of being ``of finite type''. Note that, for a finite type ring map", "$R \\to A$, the property $R \\to A$ is quasi-finite at $\\mathfrak q$", "depends only on the local ring $A_{\\mathfrak q}$ as an", "algebra over $R_{\\mathfrak p}$ where $\\mathfrak p = R \\cap \\mathfrak q$", "(usual abuse of notation). Using these remarks (a), (b) and (c) of", "Definition \\ref{definition-property-local} follow immediately.", "For example, suppose $R \\to A$ is a ring map", "such that all of the ring maps $R \\to A_{a_i}$ are quasi-finite", "for $a_1, \\ldots, a_n \\in A$ generating the unit ideal.", "We conclude that $R \\to A$ is of finite type. Also, for any", "prime $\\mathfrak q \\subset A$ the local ring $A_{\\mathfrak q}$", "is isomorphic as an $R$-algebra to the local ring", "$(A_{a_i})_{\\mathfrak q_i}$ for some $i$ and some", "$\\mathfrak q_i \\subset A_{a_i}$. Hence we conclude that", "$R \\to A$ is quasi-finite at $\\mathfrak q$.", "\\medskip\\noindent", "We conclude that Lemma \\ref{lemma-locally-P} applies with $P$", "as in the previous paragraph.", "Hence it suffices to prove that $f$ is locally quasi-finite is", "equivalent to $f$ is locally of type $P$. Since $P(R \\to A)$", "is ``$R \\to A$ is quasi-finite'' which means $R \\to A$ is", "quasi-finite at every prime of $A$, this follows from", "Lemma \\ref{lemma-quasi-finite-at-point-characterize}." ], "refs": [ "morphisms-definition-property-local", "morphisms-lemma-locally-finite-type-characterize", "morphisms-definition-property-local", "morphisms-lemma-locally-P", "morphisms-lemma-quasi-finite-at-point-characterize" ], "ref_ids": [ 5547, 5198, 5547, 5191, 5226 ] } ], "ref_ids": [] }, { "id": 5232, "type": "theorem", "label": "morphisms-lemma-composition-quasi-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-quasi-finite", "contents": [ "The composition of two morphisms which are locally quasi-finite is", "locally quasi-finite. The same is true for quasi-finite morphisms." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-locally-quasi-finite-characterize}", "we saw that $P = $``quasi-finite'' is a local property of ring maps,", "and that a morphism of schemes is locally quasi-finite if and only if", "it is locally of type $P$ as in Definition \\ref{definition-locally-P}.", "Hence the first statement of the lemma follows from", "Lemma \\ref{lemma-composition-type-P} combined", "with the fact that being quasi-finite is a property of ring maps that is", "stable under composition, see", "Algebra, Lemma \\ref{algebra-lemma-quasi-finite-composition}.", "By the above, Lemma \\ref{lemma-quasi-finite-locally-quasi-compact}", "and the fact that compositions of", "quasi-compact morphisms are quasi-compact, see", "Schemes, Lemma \\ref{schemes-lemma-composition-quasi-compact}", "we see that the composition of quasi-finite morphisms is", "quasi-finite." ], "refs": [ "morphisms-lemma-locally-quasi-finite-characterize", "morphisms-definition-locally-P", "morphisms-lemma-composition-type-P", "algebra-lemma-quasi-finite-composition", "morphisms-lemma-quasi-finite-locally-quasi-compact", "schemes-lemma-composition-quasi-compact" ], "ref_ids": [ 5231, 5548, 5193, 1053, 5229, 7699 ] } ], "ref_ids": [] }, { "id": 5233, "type": "theorem", "label": "morphisms-lemma-base-change-quasi-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-quasi-finite", "contents": [ "\\begin{slogan}", "(Locally) quasi-finite morphisms are stable under base change.", "\\end{slogan}", "Let $f : X \\to S$ be a morphism of schemes.", "Let $g : S' \\to S$ be a morphism of schemes.", "Denote $f' : X' \\to S'$ the base change of $f$ by $g$", "and denote $g' : X' \\to X$ the projection.", "Assume $X$ is locally of finite type over $S$.", "\\begin{enumerate}", "\\item Let $U \\subset X$ (resp.\\ $U' \\subset X'$)", "be the set of points where $f$ (resp.\\ $f'$) is quasi-finite.", "Then $U' = U \\times_S S' = (g')^{-1}(U)$.", "\\item The base change of a locally quasi-finite morphism is", "locally quasi-finite.", "\\item The base change of a quasi-finite morphism is", "quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first and second assertion follow from the corresponding", "algebra result, see", "Algebra, Lemma \\ref{algebra-lemma-quasi-finite-base-change}", "(combined with the fact that $f'$ is also locally of finite type by", "Lemma \\ref{lemma-base-change-finite-type}).", "By the above, Lemma \\ref{lemma-quasi-finite-locally-quasi-compact}", "and the fact that a base change of a", "quasi-compact morphism is quasi-compact, see", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-preserved-base-change}", "we see that the base change of a quasi-finite morphism", "is quasi-finite." ], "refs": [ "algebra-lemma-quasi-finite-base-change", "morphisms-lemma-base-change-finite-type", "morphisms-lemma-quasi-finite-locally-quasi-compact", "schemes-lemma-quasi-compact-preserved-base-change" ], "ref_ids": [ 1054, 5200, 5229, 7698 ] } ], "ref_ids": [] }, { "id": 5234, "type": "theorem", "label": "morphisms-lemma-quasi-finite-at-a-finite-number-of-points", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-finite-at-a-finite-number-of-points", "contents": [ "Let $f : X \\to S$ be a morphism of schemes of finite type.", "Let $s \\in S$. There are at most finitely many points", "of $X$ lying over $s$ at which $f$ is quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "The fibre $X_s$ is a scheme of finite type over a field,", "hence Noetherian (Lemma \\ref{lemma-finite-type-noetherian}).", "Hence the topology on $X_s$ is Noetherian (Properties, Lemma", "\\ref{properties-lemma-Noetherian-topology})", "and can have at most a finite number of isolated points (by elementary", "topology). Thus our lemma follows from", "Lemma \\ref{lemma-quasi-finite-at-point-characterize}." ], "refs": [ "morphisms-lemma-finite-type-noetherian", "properties-lemma-Noetherian-topology", "morphisms-lemma-quasi-finite-at-point-characterize" ], "ref_ids": [ 5202, 2954, 5226 ] } ], "ref_ids": [] }, { "id": 5235, "type": "theorem", "label": "morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "If $f$ is locally of finite type and a monomorphism, then $f$", "is separated and locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "A monomorphism is separated by Schemes, Lemma", "\\ref{schemes-lemma-monomorphism-separated}.", "A monomorphism is injective, hence we get $f$", "is quasi-finite at every $x \\in X$ for example by", "Lemma \\ref{lemma-quasi-finite-at-point-characterize}." ], "refs": [ "schemes-lemma-monomorphism-separated", "morphisms-lemma-quasi-finite-at-point-characterize" ], "ref_ids": [ 7722, 5226 ] } ], "ref_ids": [] }, { "id": 5236, "type": "theorem", "label": "morphisms-lemma-immersion-locally-quasi-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-immersion-locally-quasi-finite", "contents": [ "Any immersion is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "This is true because an open immersion is a local isomorphism", "and a closed immersion is clearly quasi-finite." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5237, "type": "theorem", "label": "morphisms-lemma-permanence-quasi-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-permanence-quasi-finite", "contents": [ "Let $X \\to Y$ be a morphism of schemes over a base scheme $S$.", "Let $x \\in X$. If $X \\to S$ is quasi-finite at $x$, then", "$X \\to Y$ is quasi-finite at $x$.", "If $X$ is locally quasi-finite over $S$, then $X \\to Y$", "is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Via Lemma \\ref{lemma-locally-quasi-finite-characterize} this translates", "into the following algebra", "fact: Given ring maps $A \\to B \\to C$ such that $A \\to C$ is", "quasi-finite, then $B \\to C$ is quasi-finite.", "This follows from", "Algebra, Lemma \\ref{algebra-lemma-four-rings}", "with $R = A$, $S = S' = C$ and $R' = B$." ], "refs": [ "morphisms-lemma-locally-quasi-finite-characterize", "algebra-lemma-four-rings" ], "ref_ids": [ 5231, 1052 ] } ], "ref_ids": [] }, { "id": 5238, "type": "theorem", "label": "morphisms-lemma-locally-finite-presentation-characterize", "categories": [ "morphisms" ], "title": "morphisms-lemma-locally-finite-presentation-characterize", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is locally of finite presentation.", "\\item For every affine opens $U \\subset X$, $V \\subset S$", "with $f(U) \\subset V$ the ring map", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is of finite presentation.", "\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$", "and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$", "is locally of finite presentation.", "\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$", "and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is", "of finite presentation, for all $j\\in J, i\\in I_j$.", "\\end{enumerate}", "Moreover, if $f$ is locally of finite presentation then for", "any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$", "the restriction $f|_U : U \\to V$ is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-locally-P} if we show that", "the property ``$R \\to A$ is of finite presentation'' is local.", "We check conditions (a), (b) and (c) of Definition", "\\ref{definition-property-local}.", "By Algebra, Lemma \\ref{algebra-lemma-base-change-finiteness}", "being of finite presentation is stable under base change and hence", "we conclude (a) holds. By the same lemma being of finite presentation", "is stable under composition and trivially for any ring", "$R$ the ring map $R \\to R_f$ is of finite presentation.", "We conclude (b) holds. Finally, property (c) is true", "according to Algebra, Lemma \\ref{algebra-lemma-cover-upstairs}." ], "refs": [ "morphisms-lemma-locally-P", "morphisms-definition-property-local", "algebra-lemma-base-change-finiteness", "algebra-lemma-cover-upstairs" ], "ref_ids": [ 5191, 5547, 373, 412 ] } ], "ref_ids": [] }, { "id": 5239, "type": "theorem", "label": "morphisms-lemma-composition-finite-presentation", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-finite-presentation", "contents": [ "The composition of two morphisms which are locally of finite presentation is", "locally of finite presentation.", "The same is true for morphisms of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-locally-finite-presentation-characterize}", "we saw that being of finite presentation is a local property of ring maps.", "Hence the first statement of the lemma follows from", "Lemma \\ref{lemma-composition-type-P} combined", "with the fact that being of finite presentation", "is a property of ring maps that is", "stable under composition, see", "Algebra, Lemma \\ref{algebra-lemma-compose-finite-type}.", "By the above and the fact that compositions of", "quasi-compact, quasi-separated morphisms are", "quasi-compact and quasi-separated, see", "Schemes, Lemmas \\ref{schemes-lemma-composition-quasi-compact}", "and \\ref{schemes-lemma-separated-permanence}", "we see that the composition of morphisms of finite presentation is", "of finite presentation." ], "refs": [ "morphisms-lemma-locally-finite-presentation-characterize", "morphisms-lemma-composition-type-P", "algebra-lemma-compose-finite-type", "schemes-lemma-composition-quasi-compact", "schemes-lemma-separated-permanence" ], "ref_ids": [ 5238, 5193, 333, 7699, 7714 ] } ], "ref_ids": [] }, { "id": 5240, "type": "theorem", "label": "morphisms-lemma-base-change-finite-presentation", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-finite-presentation", "contents": [ "The base change of a morphism which is locally of finite presentation", "is locally of finite presentation. The same is true for morphisms of", "finite presentation." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-locally-finite-presentation-characterize}", "we saw that being of finite presentation is a local property of ring maps.", "Hence the first statement of the lemma follows from", "Lemma \\ref{lemma-composition-type-P} combined", "with the fact that being of finite presentation", "is a property of ring maps that is", "stable under base change, see", "Algebra, Lemma \\ref{algebra-lemma-base-change-finiteness}.", "By the above and the fact that a base change of a", "quasi-compact, quasi-separated morphism is quasi-compact", "and quasi-separated, see", "Schemes, Lemmas \\ref{schemes-lemma-quasi-compact-preserved-base-change}", "and \\ref{schemes-lemma-separated-permanence}", "we see that the base change of a morphism of finite presentation is", "a morphism of finite presentation." ], "refs": [ "morphisms-lemma-locally-finite-presentation-characterize", "morphisms-lemma-composition-type-P", "algebra-lemma-base-change-finiteness", "schemes-lemma-quasi-compact-preserved-base-change", "schemes-lemma-separated-permanence" ], "ref_ids": [ 5238, 5193, 373, 7698, 7714 ] } ], "ref_ids": [] }, { "id": 5241, "type": "theorem", "label": "morphisms-lemma-open-immersion-locally-finite-presentation", "categories": [ "morphisms" ], "title": "morphisms-lemma-open-immersion-locally-finite-presentation", "contents": [ "Any open immersion is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "This is true because an open immersion is a local isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5242, "type": "theorem", "label": "morphisms-lemma-quasi-compact-open-immersion-finite-presentation", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-compact-open-immersion-finite-presentation", "contents": [ "Any open immersion is of finite presentation if and only if", "it is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "We have seen (Lemma \\ref{lemma-open-immersion-locally-finite-presentation})", "that an open immersion is locally of finite presentation.", "We have seen (Schemes, Lemma \\ref{schemes-lemma-immersions-monomorphisms})", "that an immersion is separated and hence quasi-separated. From this", "and Definition \\ref{definition-finite-presentation} the lemma follows." ], "refs": [ "morphisms-lemma-open-immersion-locally-finite-presentation", "schemes-lemma-immersions-monomorphisms", "morphisms-definition-finite-presentation" ], "ref_ids": [ 5241, 7727, 5554 ] } ], "ref_ids": [] }, { "id": 5243, "type": "theorem", "label": "morphisms-lemma-closed-immersion-finite-presentation", "categories": [ "morphisms" ], "title": "morphisms-lemma-closed-immersion-finite-presentation", "contents": [ "\\begin{slogan}", "Closed immersions of finite presentation correspond", "to quasi-coherent sheaves of ideals of finite type.", "\\end{slogan}", "A closed immersion $i : Z \\to X$ is of finite presentation if and only if", "the associated quasi-coherent sheaf of ideals", "$\\mathcal{I} = \\Ker(\\mathcal{O}_X \\to i_*\\mathcal{O}_Z)$", "is of finite type (as an $\\mathcal{O}_X$-module)." ], "refs": [], "proofs": [ { "contents": [ "On any affine open $\\Spec(R) \\subset X$ we have", "$i^{-1}(\\Spec(R)) = \\Spec(R/I)$ and", "$\\mathcal{I} = \\widetilde{I}$. Moreover, $\\mathcal{I}$", "is of finite type if and only if $I$ is a finite $R$-module", "for every such affine open (see", "Properties, Lemma \\ref{properties-lemma-finite-type-module}).", "And $R/I$ is of finite presentation", "over $R$ if and only if $I$ is a finite $R$-module. Hence we win." ], "refs": [ "properties-lemma-finite-type-module" ], "ref_ids": [ 3002 ] } ], "ref_ids": [] }, { "id": 5244, "type": "theorem", "label": "morphisms-lemma-finite-presentation-finite-type", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-presentation-finite-type", "contents": [ "A morphism which is locally of finite presentation is locally of finite type.", "A morphism of finite presentation is of finite type." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5245, "type": "theorem", "label": "morphisms-lemma-noetherian-finite-type-finite-presentation", "categories": [ "morphisms" ], "title": "morphisms-lemma-noetherian-finite-type-finite-presentation", "contents": [ "\\begin{slogan}", "Over a locally Noetherian base, finite type is finite presentation.", "\\end{slogan}", "Let $f : X \\to S$ be a morphism.", "\\begin{enumerate}", "\\item If $S$ is locally Noetherian and $f$ locally of finite type", "then $f$ is locally of finite presentation.", "\\item If $S$ is locally Noetherian and $f$ of finite type", "then $f$ is of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from the fact that a ring", "of finite type over a Noetherian ring is of finite presentation, see Algebra,", "Lemma \\ref{algebra-lemma-Noetherian-finite-type-is-finite-presentation}.", "Suppose that $f$ is of finite type and $S$ is locally Noetherian.", "Then $f$ is quasi-compact and locally of finite presentation by (1).", "Hence it suffices to prove that $f$ is quasi-separated.", "This follows from Lemma \\ref{lemma-finite-type-Noetherian-quasi-separated}", "(and Lemma \\ref{lemma-finite-presentation-finite-type})." ], "refs": [ "algebra-lemma-Noetherian-finite-type-is-finite-presentation", "morphisms-lemma-finite-type-Noetherian-quasi-separated", "morphisms-lemma-finite-presentation-finite-type" ], "ref_ids": [ 451, 5203, 5244 ] } ], "ref_ids": [] }, { "id": 5246, "type": "theorem", "label": "morphisms-lemma-finite-presentation-quasi-compact-quasi-separated", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-presentation-quasi-compact-quasi-separated", "contents": [ "Let $S$ be a scheme which is quasi-compact and quasi-separated.", "If $X$ is of finite presentation over $S$, then $X$ is quasi-compact", "and quasi-separated." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5247, "type": "theorem", "label": "morphisms-lemma-finite-presentation-permanence", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-presentation-permanence", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes over $S$.", "\\begin{enumerate}", "\\item If $X$ is locally of finite presentation over $S$ and", "$Y$ is locally of finite type over $S$, then $f$ is locally", "of finite presentation.", "\\item If $X$ is of finite presentation over $S$ and $Y$ is quasi-separated", "and locally of finite type over $S$, then $f$ is of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Via Lemma \\ref{lemma-locally-finite-presentation-characterize}", "this translates into the following algebra", "fact: Given ring maps $A \\to B \\to C$ such that $A \\to C$ is", "of finite presentation and $A \\to B$ is of finite type,", "then $B \\to C$ is of finite presentation. See", "Algebra, Lemma \\ref{algebra-lemma-compose-finite-type}.", "\\medskip\\noindent", "Part (2) follows from (1) and", "Schemes, Lemmas \\ref{schemes-lemma-compose-after-separated} and", "\\ref{schemes-lemma-quasi-compact-permanence}." ], "refs": [ "morphisms-lemma-locally-finite-presentation-characterize", "algebra-lemma-compose-finite-type", "schemes-lemma-compose-after-separated", "schemes-lemma-quasi-compact-permanence" ], "ref_ids": [ 5238, 333, 7715, 7716 ] } ], "ref_ids": [] }, { "id": 5248, "type": "theorem", "label": "morphisms-lemma-diagonal-morphism-finite-type", "categories": [ "morphisms" ], "title": "morphisms-lemma-diagonal-morphism-finite-type", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes with diagonal", "$\\Delta : X \\to X \\times_Y X$. If $f$ is locally of finite type", "then $\\Delta$ is locally of finite presentation. If $f$ is quasi-separated", "and locally of finite type, then $\\Delta$ is of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Note that $\\Delta$ is a morphism of schemes over $X$ (via the second", "projection $X \\times_Y X \\to X$). Assume $f$ is locally of finite type.", "Note that $X$ is of finite presentation over $X$ and $X \\times_Y X$ is", "locally of finite type over $X$ (by Lemma \\ref{lemma-base-change-finite-type}).", "Thus the first statement holds by", "Lemma \\ref{lemma-finite-presentation-permanence}.", "The second statement follows from the first, the definitions, and", "the fact that a diagonal morphism is a monomorphism, hence separated", "(Schemes, Lemma \\ref{schemes-lemma-monomorphism-separated})." ], "refs": [ "morphisms-lemma-base-change-finite-type", "morphisms-lemma-finite-presentation-permanence", "schemes-lemma-monomorphism-separated" ], "ref_ids": [ 5200, 5247, 7722 ] } ], "ref_ids": [] }, { "id": 5249, "type": "theorem", "label": "morphisms-lemma-inverse-image-constructible", "categories": [ "morphisms" ], "title": "morphisms-lemma-inverse-image-constructible", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $E \\subset Y$ be a subset.", "If $E$ is (locally) constructible in $Y$, then $f^{-1}(E)$ is (locally)", "constructible in $X$." ], "refs": [], "proofs": [ { "contents": [ "To show that the inverse image of every constructible subset is constructible", "it suffices to show that the inverse image of every retrocompact open $V$", "of $Y$ is retrocompact in $X$, see", "Topology, Lemma \\ref{topology-lemma-inverse-images-constructibles}.", "The significance of $V$ being retrocompact", "in $Y$ is just that the open immersion $V \\to Y$ is quasi-compact.", "Hence the base change $f^{-1}(V) = X \\times_Y V \\to X$ is quasi-compact", "too, see", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-preserved-base-change}.", "Hence we see $f^{-1}(V)$ is retrocompact in $X$.", "Suppose $E$ is locally constructible in $Y$.", "Choose $x \\in X$. Choose an affine neighbourhood $V$ of $f(x)$ and", "an affine neighbourhood $U \\subset X$ of $x$ such that $f(U) \\subset V$.", "Thus we think of $f|_U : U \\to V$ as a morphism into $V$. By", "Properties, Lemma \\ref{properties-lemma-locally-constructible}", "we see that $E \\cap V$ is constructible in $V$. By the constructible case", "we see that $(f|_U)^{-1}(E \\cap V)$ is constructible in $U$.", "Since $(f|_U)^{-1}(E \\cap V) = f^{-1}(E) \\cap U$ we win." ], "refs": [ "topology-lemma-inverse-images-constructibles", "schemes-lemma-quasi-compact-preserved-base-change", "properties-lemma-locally-constructible" ], "ref_ids": [ 8254, 7698, 2938 ] } ], "ref_ids": [] }, { "id": 5250, "type": "theorem", "label": "morphisms-lemma-chevalley", "categories": [ "morphisms" ], "title": "morphisms-lemma-chevalley", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume", "\\begin{enumerate}", "\\item $f$ is quasi-compact and locally of finite presentation, and", "\\item $Y$ is quasi-compact and quasi-separated.", "\\end{enumerate}", "Then the image of every constructible subset of $X$ is constructible in $Y$." ], "refs": [], "proofs": [ { "contents": [ "By", "Properties,", "Lemma \\ref{properties-lemma-constructible-quasi-compact-quasi-separated}", "it suffices to prove this lemma in case $Y$ is affine.", "In this case $X$ is quasi-compact. Hence we can write", "$X = U_1 \\cup \\ldots \\cup U_n$ with each $U_i$ affine open in $X$.", "If $E \\subset X$ is constructible, then each $E \\cap U_i$ is constructible", "too, see", "Topology,", "Lemma \\ref{topology-lemma-open-immersion-constructible-inverse-image}.", "Hence, since $f(E) = \\bigcup f(E \\cap U_i)$ and since finite unions of", "constructible sets are constructible, this reduces us to the case where", "$X$ is affine. In this case the result is", "Algebra, Theorem \\ref{algebra-theorem-chevalley}." ], "refs": [ "properties-lemma-constructible-quasi-compact-quasi-separated", "topology-lemma-open-immersion-constructible-inverse-image", "algebra-theorem-chevalley" ], "ref_ids": [ 2942, 8255, 315 ] } ], "ref_ids": [] }, { "id": 5251, "type": "theorem", "label": "morphisms-lemma-constructible-containing-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-constructible-containing-open", "contents": [ "Let $X$ be a scheme. Let $x \\in X$. Let $E \\subset X$ be a locally", "constructible subset. If $\\{x' \\mid x' \\leadsto x\\} \\subset E$,", "then $E$ contains an open neighbourhood of $x$." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\{x' \\mid x' \\leadsto x\\} \\subset E$.", "We may assume $X$ is affine.", "In this case $E$ is constructible, see", "Properties, Lemma \\ref{properties-lemma-locally-constructible}.", "In particular, also the complement $E^c$ is constructible. By", "Algebra, Lemma \\ref{algebra-lemma-constructible-is-image}", "we can find a morphism of affine schemes $f : Y \\to X$ such that", "$E^c = f(Y)$. Let $Z \\subset X$ be the scheme theoretic image of $f$. By", "Lemma \\ref{lemma-reach-points-scheme-theoretic-image}", "and the assumption $\\{x' \\mid x' \\leadsto x\\} \\subset E$", "we see that $x \\not \\in Z$. Hence $X \\setminus Z \\subset E$ is an", "open neighbourhood of $x$ contained in $E$." ], "refs": [ "properties-lemma-locally-constructible", "algebra-lemma-constructible-is-image", "morphisms-lemma-reach-points-scheme-theoretic-image" ], "ref_ids": [ 2938, 435, 5147 ] } ], "ref_ids": [] }, { "id": 5252, "type": "theorem", "label": "morphisms-lemma-locally-finite-presentation-universally-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-locally-finite-presentation-universally-open", "contents": [ "Let $f : X \\to S$ be a morphism.", "\\begin{enumerate}", "\\item If $f$ is locally of finite presentation and generalizations lift", "along $f$, then $f$ is open.", "\\item If $f$ is locally of finite presentation and generalizations lift", "along every base change of $f$, then $f$ is universally open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove the first assertion.", "This reduces to the case where both $X$ and $S$ are affine.", "In this case the result follows from", "Algebra, Lemma \\ref{algebra-lemma-going-up-down-specialization}", "and Proposition \\ref{algebra-proposition-fppf-open}." ], "refs": [ "algebra-lemma-going-up-down-specialization", "algebra-proposition-fppf-open" ], "ref_ids": [ 549, 1407 ] } ], "ref_ids": [] }, { "id": 5253, "type": "theorem", "label": "morphisms-lemma-composition-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-open", "contents": [ "A composition of (universally) open morphisms is (universally) open." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5254, "type": "theorem", "label": "morphisms-lemma-scheme-over-field-universally-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-scheme-over-field-universally-open", "contents": [ "Let $k$ be a field. Let $X$ be a scheme over $k$.", "The structure morphism $X \\to \\Spec(k)$ is universally open." ], "refs": [], "proofs": [ { "contents": [ "Let $S \\to \\Spec(k)$ be a morphism.", "We have to show that the base change $X_S \\to S$ is open.", "The question is local on $S$ and $X$, hence we may assume that", "$S$ and $X$ are affine. In this case the result is", "Algebra, Lemma \\ref{algebra-lemma-map-into-tensor-algebra-open}." ], "refs": [ "algebra-lemma-map-into-tensor-algebra-open" ], "ref_ids": [ 555 ] } ], "ref_ids": [] }, { "id": 5255, "type": "theorem", "label": "morphisms-lemma-open-generizing", "categories": [ "morphisms" ], "title": "morphisms-lemma-open-generizing", "contents": [ "\\begin{reference}", "Follows from the implication (a) $\\Rightarrow$ (b) in", "\\cite[IV, Corollary 1.10.4]{EGA}", "\\end{reference}", "Let $\\varphi : X \\to Y$ be a morphism of schemes.", "If $\\varphi$ is open, then $\\varphi$ is generizing", "(i.e., generalizations lift along $\\varphi$).", "If $\\varphi$ is universally open, then $\\varphi$ is", "universally generizing." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\varphi$ is open.", "Let $y' \\leadsto y$ be a specialization of points of $Y$.", "Let $x \\in X$ with $\\varphi(x) = y$.", "Choose affine opens $U \\subset X$ and $V \\subset Y$ such that", "$\\varphi(U) \\subset V$ and $x \\in U$. Then also $y' \\in V$. Hence we", "may replace $X$ by $U$ and $Y$ by $V$ and assume $X$, $Y$ affine.", "The affine case is", "Algebra, Lemma \\ref{algebra-lemma-open-going-down}", "(combined with", "Algebra, Lemma \\ref{algebra-lemma-going-up-down-specialization})." ], "refs": [ "algebra-lemma-open-going-down", "algebra-lemma-going-up-down-specialization" ], "ref_ids": [ 548, 549 ] } ], "ref_ids": [] }, { "id": 5256, "type": "theorem", "label": "morphisms-lemma-descent-quasi-compact", "categories": [ "morphisms" ], "title": "morphisms-lemma-descent-quasi-compact", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $g : Y' \\to Y$ be open and surjective such that the base change", "$f' : X' \\to Y'$ is quasi-compact. Then $f$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset Y$ be a quasi-compact open. As $g$ is open and surjective", "we can find a quasi-compact open $W' \\subset W$ such that $g(W') = V$.", "By assumption $(f')^{-1}(W')$ is quasi-compact. The image of", "$(f')^{-1}(W')$ in $X$ is equal to $f^{-1}(V)$, see", "Lemma \\ref{lemma-when-point-maps-to-pair}.", "Hence $f^{-1}(V)$ is quasi-compact as the image of a quasi-compact space, see", "Topology, Lemma \\ref{topology-lemma-image-quasi-compact}.", "Thus $f$ is quasi-compact." ], "refs": [ "morphisms-lemma-when-point-maps-to-pair", "topology-lemma-image-quasi-compact" ], "ref_ids": [ 5164, 8233 ] } ], "ref_ids": [] }, { "id": 5257, "type": "theorem", "label": "morphisms-lemma-base-change-universally-submersive", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-universally-submersive", "contents": [ "The base change of a universally submersive morphism of schemes", "by any morphism of schemes is universally submersive." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5258, "type": "theorem", "label": "morphisms-lemma-composition-universally-submersive", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-universally-submersive", "contents": [ "The composition of a pair of (universally) submersive morphisms of", "schemes is (universally) submersive." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5259, "type": "theorem", "label": "morphisms-lemma-flat-module-characterize", "categories": [ "morphisms" ], "title": "morphisms-lemma-flat-module-characterize", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.", "The following are equivalent", "\\begin{enumerate}", "\\item The sheaf $\\mathcal{F}$ is flat over $S$.", "\\item For every affine opens $U \\subset X$, $V \\subset S$", "with $f(U) \\subset V$ the $\\mathcal{O}_S(V)$-module $\\mathcal{F}(U)$ is flat.", "\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$", "and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that each of the modules $\\mathcal{F}|_{U_i}$ is", "flat over $V_j$, for all $j\\in J, i\\in I_j$.", "\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$", "and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that $\\mathcal{F}(U_i)$ is a flat $\\mathcal{O}_S(V_j)$-module, for all", "$j\\in J, i\\in I_j$.", "\\end{enumerate}", "Moreover, if $\\mathcal{F}$ is flat over $S$ then for", "any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$", "the restriction $\\mathcal{F}|_U$ is flat over $V$." ], "refs": [], "proofs": [ { "contents": [ "Let $R \\to A$ be a ring map. Let $M$ be an $A$-module.", "If $M$ is $R$-flat, then for all primes", "$\\mathfrak q$ the module $M_{\\mathfrak q}$ is flat over $R_{\\mathfrak p}$", "with $\\mathfrak p$ the prime of $R$ lying under $\\mathfrak q$. Conversely, if", "$M_{\\mathfrak q}$ is flat over $R_{\\mathfrak p}$ for all primes $\\mathfrak q$", "of $A$, then $M$ is flat over $R$. See", "Algebra, Lemma \\ref{algebra-lemma-flat-localization}.", "This equivalence easily implies the statements of the lemma." ], "refs": [ "algebra-lemma-flat-localization" ], "ref_ids": [ 538 ] } ], "ref_ids": [] }, { "id": 5260, "type": "theorem", "label": "morphisms-lemma-flat-characterize", "categories": [ "morphisms" ], "title": "morphisms-lemma-flat-characterize", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is flat.", "\\item For every affine opens $U \\subset X$, $V \\subset S$", "with $f(U) \\subset V$ the ring map", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is flat.", "\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$", "and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$", "is flat.", "\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$", "and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is flat, for all", "$j\\in J, i\\in I_j$.", "\\end{enumerate}", "Moreover, if $f$ is flat then for", "any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$", "the restriction $f|_U : U \\to V$ is flat." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-flat-module-characterize}", "above." ], "refs": [ "morphisms-lemma-flat-module-characterize" ], "ref_ids": [ 5259 ] } ], "ref_ids": [] }, { "id": 5261, "type": "theorem", "label": "morphisms-lemma-pushforward-flat-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-pushforward-flat-affine", "contents": [ "Let $f : X \\to Y$ be an affine morphism of schemes over a base scheme $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then $\\mathcal{F}$ is flat over $S$ if and only if", "$f_*\\mathcal{F}$ is flat over $S$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-flat-module-characterize} and the fact that", "$f$ is an affine morphism, this reduces us to the affine case.", "Say $X \\to Y \\to S$ corresponds to the ring maps $C \\leftarrow B \\leftarrow A$.", "Let $N$ be the $C$-module corresponding to $\\mathcal{F}$.", "Recall that $f_*\\mathcal{F}$ corresponds to $N$ viewed as a $B$-module, see", "Schemes, Lemma \\ref{schemes-lemma-widetilde-pullback}.", "Thus the result is clear." ], "refs": [ "morphisms-lemma-flat-module-characterize", "schemes-lemma-widetilde-pullback" ], "ref_ids": [ 5259, 7662 ] } ], "ref_ids": [] }, { "id": 5262, "type": "theorem", "label": "morphisms-lemma-composition-module-flat", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-module-flat", "contents": [ "Let $X \\to Y \\to Z$ be morphisms of schemes. Let $\\mathcal{F}$ be a", "quasi-coherent $\\mathcal{O}_X$-module. Let $x \\in X$ with image $y$ in $Y$.", "If $\\mathcal{F}$ is flat over $Y$ at $x$, and $Y$ is flat over $Z$ at", "$y$, then $\\mathcal{F}$ is flat over $Z$ at $x$." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-composition-flat}." ], "refs": [ "algebra-lemma-composition-flat" ], "ref_ids": [ 524 ] } ], "ref_ids": [] }, { "id": 5263, "type": "theorem", "label": "morphisms-lemma-composition-flat", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-flat", "contents": [ "The composition of flat morphisms is flat." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-composition-module-flat}." ], "refs": [ "morphisms-lemma-composition-module-flat" ], "ref_ids": [ 5262 ] } ], "ref_ids": [] }, { "id": 5264, "type": "theorem", "label": "morphisms-lemma-base-change-module-flat", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-module-flat", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.", "Let $g : S' \\to S$ be a morphism of schemes.", "Denote $g' : X' = X_{S'} \\to X$ the projection.", "Let $x' \\in X'$ be a point with image $x = g'(x') \\in X$.", "If $\\mathcal{F}$ is flat over $S$ at $x$, then", "$(g')^*\\mathcal{F}$ is flat over $S'$ at $x'$.", "In particular, if $\\mathcal{F}$ is flat over $S$, then", "$(g')^*\\mathcal{F}$ is flat over $S'$." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-flat-base-change}." ], "refs": [ "algebra-lemma-flat-base-change" ], "ref_ids": [ 527 ] } ], "ref_ids": [] }, { "id": 5265, "type": "theorem", "label": "morphisms-lemma-base-change-flat", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-flat", "contents": [ "The base change of a flat morphism is flat." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-base-change-module-flat}." ], "refs": [ "morphisms-lemma-base-change-module-flat" ], "ref_ids": [ 5264 ] } ], "ref_ids": [] }, { "id": 5266, "type": "theorem", "label": "morphisms-lemma-generalizations-lift-flat", "categories": [ "morphisms" ], "title": "morphisms-lemma-generalizations-lift-flat", "contents": [ "Let $f : X \\to S$ be a flat morphism of schemes.", "Then generalizations lift along $f$, see", "Topology, Definition \\ref{topology-definition-lift-specializations}." ], "refs": [ "topology-definition-lift-specializations" ], "proofs": [ { "contents": [ "See Algebra, Section \\ref{algebra-section-going-up}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8366 ] }, { "id": 5267, "type": "theorem", "label": "morphisms-lemma-fppf-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-fppf-open", "contents": [ "A flat morphism locally of finite presentation is universally open." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-generalizations-lift-flat} and", "Lemma \\ref{lemma-locally-finite-presentation-universally-open} above.", "We can also argue directly as follows.", "\\medskip\\noindent", "Let $f : X \\to S$ be flat locally of finite presentation.", "To show $f$ is open it suffices to show that we may cover", "$X$ by open affines $X = \\bigcup U_i$ such that $U_i \\to S$", "is open. By definition we may cover $X$ by", "affine opens $U_i \\subset X$ such that each $U_i$ maps", "into an affine open $V_i \\subset S$ and such that", "the induced ring map $\\mathcal{O}_S(V_i) \\to \\mathcal{O}_X(U_i)$ is", "of finite presentation. Thus $U_i \\to V_i$ is open by", "Algebra, Proposition \\ref{algebra-proposition-fppf-open}.", "The lemma follows." ], "refs": [ "morphisms-lemma-generalizations-lift-flat", "morphisms-lemma-locally-finite-presentation-universally-open", "algebra-proposition-fppf-open" ], "ref_ids": [ 5266, 5252, 1407 ] } ], "ref_ids": [] }, { "id": 5268, "type": "theorem", "label": "morphisms-lemma-pf-flat-module-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-pf-flat-module-open", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume $f$ locally finite presentation, $\\mathcal{F}$ of", "finite type, $X = \\text{Supp}(\\mathcal{F})$, and", "$\\mathcal{F}$ flat over $Y$. Then $f$ is universally open." ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-base-change-module-flat},", "\\ref{lemma-base-change-finite-presentation}, and", "\\ref{lemma-support-finite-type}", "the assumptions are preserved under base change.", "By Lemma \\ref{lemma-locally-finite-presentation-universally-open}", "it suffices to show that generalizations lift along $f$.", "This follows from Algebra, Lemma \\ref{algebra-lemma-going-down-flat-module}." ], "refs": [ "morphisms-lemma-base-change-module-flat", "morphisms-lemma-base-change-finite-presentation", "morphisms-lemma-support-finite-type", "morphisms-lemma-locally-finite-presentation-universally-open", "algebra-lemma-going-down-flat-module" ], "ref_ids": [ 5264, 5240, 5143, 5252, 557 ] } ], "ref_ids": [] }, { "id": 5269, "type": "theorem", "label": "morphisms-lemma-fpqc-quotient-topology", "categories": [ "morphisms" ], "title": "morphisms-lemma-fpqc-quotient-topology", "contents": [ "\\begin{reference}", "\\cite[Expose VIII, Corollaire 4.3]{SGA1} and", "\\cite[IV, Corollaire 2.3.12]{EGA}", "\\end{reference}", "Let $f : X \\to Y$ be a quasi-compact, surjective, flat morphism.", "A subset $T \\subset Y$ is open (resp.\\ closed) if and only", "$f^{-1}(T)$ is open (resp.\\ closed). In other words, $f$ is", "a submersive morphism." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $Y$, hence we may assume that $Y$ is affine.", "In this case $X$ is quasi-compact as $f$ is quasi-compact.", "Write $X = X_1 \\cup \\ldots \\cup X_n$ as a finite union of affine opens.", "Then $f' : X' = X_1 \\amalg \\ldots \\amalg X_n \\to Y$ is a surjective", "flat morphism of affine schemes. Note that for $T \\subset Y$ we have", "$(f')^{-1}(T) = f^{-1}(T) \\cap X_1 \\amalg \\ldots \\amalg f^{-1}(T) \\cap X_n$.", "Hence, $f^{-1}(T)$ is open if and only if $(f')^{-1}(T)$ is open.", "Thus we may assume both $X$ and $Y$ are affine.", "\\medskip\\noindent", "Let $f : \\Spec(B) \\to \\Spec(A)$ be a surjective morphism of affine schemes", "corresponding to a flat ring map $A \\to B$. Suppose that $f^{-1}(T)$ is", "closed, say $f^{-1}(T) = V(J)$ for $J \\subset B$ an ideal. Then", "$T = f(f^{-1}(T)) = f(V(J))$ is the image of $\\Spec(B/J) \\to \\Spec(A)$", "(here we use that $f$ is surjective). On the other hand, generalizations", "lift along $f$ (Lemma \\ref{lemma-generalizations-lift-flat}).", "Hence by Topology, Lemma \\ref{topology-lemma-lift-specializations-images}", "we see that $Y \\setminus T = f(X \\setminus f^{-1}(T))$ is stable under", "generalization. Hence $T$ is stable under specialization", "(Topology, Lemma \\ref{topology-lemma-open-closed-specialization}).", "Thus $T$ is closed by", "Algebra, Lemma \\ref{algebra-lemma-image-stable-specialization-closed}." ], "refs": [ "morphisms-lemma-generalizations-lift-flat", "topology-lemma-lift-specializations-images", "topology-lemma-open-closed-specialization", "algebra-lemma-image-stable-specialization-closed" ], "ref_ids": [ 5266, 8286, 8283, 551 ] } ], "ref_ids": [] }, { "id": 5270, "type": "theorem", "label": "morphisms-lemma-flat-permanence", "categories": [ "morphisms" ], "title": "morphisms-lemma-flat-permanence", "contents": [ "Let $h : X \\to Y$ be a morphism of schemes over $S$.", "Let $\\mathcal{G}$ be a quasi-coherent sheaf on $Y$.", "Let $x \\in X$ with $y = h(x) \\in Y$. If $h$ is flat at $x$, then", "$$", "\\mathcal{G}\\text{ flat over }S\\text{ at }y", "\\Leftrightarrow", "h^*\\mathcal{G}\\text{ flat over }S\\text{ at }x.", "$$", "In particular: If $h$ is surjective and flat, then", "$\\mathcal{G}$ is flat over $S$, if and only if", "$h^*\\mathcal{G}$ is flat over $S$. If $h$ is surjective and", "flat, and $X$ is flat over $S$, then $Y$ is flat over $S$." ], "refs": [], "proofs": [ { "contents": [ "You can prove this by applying", "Algebra, Lemma \\ref{algebra-lemma-flatness-descends-more-general}.", "Here is a direct proof. Let $s \\in S$ be the image of $y$.", "Consider the local ring maps", "$\\mathcal{O}_{S, s} \\to \\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$.", "By assumption the ring map $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$", "is faithfully flat, see", "Algebra, Lemma \\ref{algebra-lemma-local-flat-ff}.", "Let $N = \\mathcal{G}_y$. Note that", "$h^*\\mathcal{G}_x = N \\otimes_{\\mathcal{O}_{Y, y}} \\mathcal{O}_{X, x}$, see", "Sheaves, Lemma \\ref{sheaves-lemma-stalk-pullback-modules}.", "Let $M' \\to M$ be an injection of $\\mathcal{O}_{S, s}$-modules.", "By the faithful flatness mentioned above we have", "\\begin{align*}", "\\Ker(", "M' \\otimes_{\\mathcal{O}_{S, s}} N \\to M \\otimes_{\\mathcal{O}_{S, s}} N)", "\\otimes_{\\mathcal{O}_{Y, y}} \\mathcal{O}_{X, x} \\\\", "=", "\\Ker(", "M' \\otimes_{\\mathcal{O}_{S, s}} N", "\\otimes_{\\mathcal{O}_{Y, y}} \\mathcal{O}_{X, x}", "\\to", "M \\otimes_{\\mathcal{O}_{S, s}} N", "\\otimes_{\\mathcal{O}_{Y, y}} \\mathcal{O}_{X, x})", "\\end{align*}", "Hence the equivalence of the lemma follows from the second characterization", "of flatness in", "Algebra, Lemma \\ref{algebra-lemma-flat}." ], "refs": [ "algebra-lemma-flatness-descends-more-general", "algebra-lemma-local-flat-ff", "sheaves-lemma-stalk-pullback-modules", "algebra-lemma-flat" ], "ref_ids": [ 529, 537, 14523, 525 ] } ], "ref_ids": [] }, { "id": 5271, "type": "theorem", "label": "morphisms-lemma-flat-pullback-support", "categories": [ "morphisms" ], "title": "morphisms-lemma-flat-pullback-support", "contents": [ "Let $f : Y \\to X$ be a morphism of schemes. Let $\\mathcal{F}$ be", "a finite type quasi-coherent $\\mathcal{O}_X$-module with scheme", "theoretic support $Z \\subset X$. If $f$ is flat,", "then $f^{-1}(Z)$ is the scheme theoretic support of $f^*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Using the characterization of scheme theoretic support on affines", "as given in Lemma \\ref{lemma-scheme-theoretic-support} we reduce to", "Algebra, Lemma \\ref{algebra-lemma-annihilator-flat-base-change}." ], "refs": [ "morphisms-lemma-scheme-theoretic-support", "algebra-lemma-annihilator-flat-base-change" ], "ref_ids": [ 5144, 542 ] } ], "ref_ids": [] }, { "id": 5272, "type": "theorem", "label": "morphisms-lemma-flat-morphism-scheme-theoretically-dense-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-flat-morphism-scheme-theoretically-dense-open", "contents": [ "Let $f : X \\to Y$ be a flat morphism of schemes. Let $V \\subset Y$ be", "a retrocompact open which is scheme theoretically dense. Then $f^{-1}V$", "is scheme theoretically dense in $X$." ], "refs": [], "proofs": [ { "contents": [ "We will use the characterization of", "Lemma \\ref{lemma-characterize-scheme-theoretically-dense}.", "We have to show that for any open $U \\subset X$ the map", "$\\mathcal{O}_X(U) \\to \\mathcal{O}_X(U \\cap f^{-1}V)$ is injective.", "It suffices to prove this when $U$ is an affine open which maps into", "an affine open $W \\subset Y$. Say $W = \\Spec(A)$ and $U = \\Spec(B)$.", "Then $V \\cap W = D(f_1) \\cup \\ldots \\cup D(f_n)$ for some", "$f_i \\in A$, see", "Algebra, Lemma \\ref{algebra-lemma-qc-open}.", "Thus we have to show that", "$B \\to B_{f_1} \\times \\ldots \\times B_{f_n}$ is injective.", "We are given that $A \\to A_{f_1} \\times \\ldots \\times A_{f_n}$ is injective", "and that $A \\to B$ is flat. Since $B_{f_i} = A_{f_i} \\otimes_A B$ we win." ], "refs": [ "morphisms-lemma-characterize-scheme-theoretically-dense", "algebra-lemma-qc-open" ], "ref_ids": [ 5152, 432 ] } ], "ref_ids": [] }, { "id": 5273, "type": "theorem", "label": "morphisms-lemma-flat-base-change-scheme-theoretic-image", "categories": [ "morphisms" ], "title": "morphisms-lemma-flat-base-change-scheme-theoretic-image", "contents": [ "\\begin{slogan}", "Taking scheme theoretic images commutes with flat base change", "in the quasi-compact case", "\\end{slogan}", "Let $f : X \\to Y$ be a flat morphism of schemes. Let $g : V \\to Y$ be a", "quasi-compact morphism of schemes. Let $Z \\subset Y$ be the scheme theoretic", "image of $g$ and let $Z' \\subset X$ be the scheme theoretic image of the", "base change $V \\times_Y X \\to X$. Then $Z' = f^{-1}Z$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $Z$ is cut out by", "$\\mathcal{I} = \\Ker(\\mathcal{O}_Y \\to g_*\\mathcal{O}_V)$", "and $Z'$ is cut out by", "$\\mathcal{I}' = \\Ker(\\mathcal{O}_X \\to", "(V \\times_Y X \\to X)_*\\mathcal{O}_{V \\times_Y X})$, see", "Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image}.", "Hence the question is local on $X$ and $Y$ and we may assume $X$ and $Y$", "affine. Note that we may replace $V$ by $\\coprod V_i$ where", "$V = V_1 \\cup \\ldots \\cup V_n$ is a finite affine open covering.", "Hence we may assume $g$ is affine. In this case", "$(V \\times_Y X \\to X)_*\\mathcal{O}_{V \\times_Y X}$ is the pullback", "of $g_*\\mathcal{O}_V$ by $f$. Since $f$ is flat we conclude that", "$f^*\\mathcal{I} = \\mathcal{I}'$ and the lemma holds." ], "refs": [ "morphisms-lemma-quasi-compact-scheme-theoretic-image" ], "ref_ids": [ 5146 ] } ], "ref_ids": [] }, { "id": 5274, "type": "theorem", "label": "morphisms-lemma-characterize-flat-closed-immersions", "categories": [ "morphisms" ], "title": "morphisms-lemma-characterize-flat-closed-immersions", "contents": [ "Let $X$ be a scheme. The rule which associates to a closed subscheme", "of $X$ its underlying closed subset defines a bijection", "$$", "\\left\\{", "\\begin{matrix}", "\\text{closed subschemes }Z \\subset X \\\\", "\\text{such that }Z \\to X\\text{ is flat}", "\\end{matrix}", "\\right\\}", "\\leftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{closed subsets }Z \\subset X \\\\", "\\text{closed under generalizations}", "\\end{matrix}", "\\right\\}", "$$", "If $Z \\subset X$ is such a closed subscheme, every morphism of schemes", "$g : Y \\to X$ with $g(Y) \\subset Z$ set theoretically factors (scheme", "theoretically) through $Z$." ], "refs": [], "proofs": [ { "contents": [ "The affine case of the bijection is", "Algebra, Lemma \\ref{algebra-lemma-pure-open-closed-specializations}.", "For general schemes $X$ the bijection follows by covering $X$ by affines", "and glueing. Details omitted. For the final assertion, observe that the", "projection $Z \\times_{X, g} Y \\to Y$ is a flat", "(Lemma \\ref{lemma-base-change-flat}) closed immersion which", "is bijective on underlying topological spaces and hence must be", "an isomorphism by the bijection esthablished in the first part of the proof." ], "refs": [ "algebra-lemma-pure-open-closed-specializations", "morphisms-lemma-base-change-flat" ], "ref_ids": [ 962, 5265 ] } ], "ref_ids": [] }, { "id": 5275, "type": "theorem", "label": "morphisms-lemma-flat-closed-immersions-finite-presentation", "categories": [ "morphisms" ], "title": "morphisms-lemma-flat-closed-immersions-finite-presentation", "contents": [ "A flat closed immersion of finite presentation", "is the open immersion of an open and closed subscheme." ], "refs": [], "proofs": [ { "contents": [ "The affine case is", "Algebra, Lemma \\ref{algebra-lemma-finitely-generated-pure-ideal}.", "In general the lemma follows by covering $X$ by affines.", "Details omitted." ], "refs": [ "algebra-lemma-finitely-generated-pure-ideal" ], "ref_ids": [ 963 ] } ], "ref_ids": [] }, { "id": 5276, "type": "theorem", "label": "morphisms-lemma-finite-flat-is-finite-locally-free", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-flat-is-finite-locally-free", "contents": [ "Let $X$ be a scheme. The following are equivalent", "\\begin{enumerate}", "\\item every finite flat quasi-coherent $\\mathcal{O}_X$-module is", "finite locally free, and", "\\item every closed subset $Z \\subset X$ which is closed under generalizations", "is open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In the affine case this is", "Algebra, Lemma \\ref{algebra-lemma-finite-flat-module-finitely-presented}.", "The scheme case does not follow directly from the affine case, so we", "simply repeat the arguments.", "\\medskip\\noindent", "Assume (1). Consider a closed immersion $i : Z \\to X$ such that $i$ is flat.", "Then $i_*\\mathcal{O}_Z$ is quasi-coherent and flat, hence finite locally", "free by (1). Thus $Z = \\text{Supp}(i_*\\mathcal{O}_Z)$ is also open and we see", "that (2) holds. Hence the implication (1) $\\Rightarrow$ (2) follows from", "the characterization of flat closed immersions in", "Lemma \\ref{lemma-characterize-flat-closed-immersions}.", "\\medskip\\noindent", "For the converse assume that $X$ satisfies (2).", "Let $\\mathcal{F}$ be a finite flat quasi-coherent $\\mathcal{O}_X$-module.", "The support $Z = \\text{Supp}(\\mathcal{F})$ of $\\mathcal{F}$ is closed, see", "Modules, Lemma \\ref{modules-lemma-support-finite-type-closed}.", "On the other hand, if $x \\leadsto x'$ is a specialization, then by", "Algebra, Lemma \\ref{algebra-lemma-finite-flat-local}", "the module $\\mathcal{F}_{x'}$ is free over $\\mathcal{O}_{X, x'}$, and", "$$", "\\mathcal{F}_x =", "\\mathcal{F}_{x'} \\otimes_{\\mathcal{O}_{X, x'}} \\mathcal{O}_{X, x}.", "$$", "Hence", "$x' \\in \\text{Supp}(\\mathcal{F}) \\Rightarrow x \\in \\text{Supp}(\\mathcal{F})$,", "in other words, the support is closed under generalization.", "As $X$ satisfies (2) we see that the support of $\\mathcal{F}$", "is open and closed. The modules $\\wedge^i(\\mathcal{F})$, $i = 1, 2, 3, \\ldots$", "are finite flat quasi-coherent $\\mathcal{O}_X$-modules", "also, see", "Modules, Section \\ref{modules-section-symmetric-exterior}.", "Note that", "$\\text{Supp}(\\wedge^{i + 1}(\\mathcal{F})) \\subset", "\\text{Supp}(\\wedge^i(\\mathcal{F}))$.", "Thus we see that there exists a decomposition", "$$", "X = U_0 \\amalg U_1 \\amalg U_2 \\amalg \\ldots", "$$", "by open and closed subsets such that the support of", "$\\wedge^i(\\mathcal{F})$ is $U_i \\cup U_{i + 1} \\cup \\ldots$ for all $i$.", "Let $x$ be a point of $X$, and say $x \\in U_r$.", "Note that", "$\\wedge^i(\\mathcal{F})_x \\otimes \\kappa(x) =", "\\wedge^i(\\mathcal{F}_x \\otimes \\kappa(x))$.", "Hence, $x \\in U_r$ implies that $\\mathcal{F}_x \\otimes \\kappa(x)$", "is a vector space of dimension $r$. By Nakayama's lemma, see", "Algebra, Lemma \\ref{algebra-lemma-NAK}", "we can choose an affine open neighbourhood $U \\subset U_r \\subset X$", "of $x$ and sections $s_1, \\ldots, s_r \\in \\mathcal{F}(U)$ such that", "the induced map", "$$", "\\mathcal{O}_U^{\\oplus r} \\longrightarrow \\mathcal{F}|_U, \\quad", "(f_1, \\ldots, f_r) \\longmapsto \\sum f_i s_i", "$$", "is surjective. This means that", "$\\wedge^r(\\mathcal{F}|_U)$ is a finite flat quasi-coherent", "$\\mathcal{O}_U$-module whose support is all of $U$.", "By the above it is generated by a single element, namely", "$s_1 \\wedge \\ldots \\wedge s_r$. Hence", "$\\wedge^r(\\mathcal{F}|_U) \\cong \\mathcal{O}_U/\\mathcal{I}$", "for some quasi-coherent sheaf of ideals $\\mathcal{I}$", "such that $\\mathcal{O}_U/\\mathcal{I}$ is flat over $\\mathcal{O}_U$ and", "such that $V(\\mathcal{I}) = U$.", "It follows that $\\mathcal{I} = 0$ by applying", "Lemma \\ref{lemma-characterize-flat-closed-immersions}.", "Thus $s_1 \\wedge \\ldots \\wedge s_r$ is a basis for", "$\\wedge^r(\\mathcal{F}|_U)$ and it follows that the displayed map is injective", "as well as surjective. This proves that $\\mathcal{F}$ is finite locally free", "as desired." ], "refs": [ "algebra-lemma-finite-flat-module-finitely-presented", "morphisms-lemma-characterize-flat-closed-immersions", "modules-lemma-support-finite-type-closed", "algebra-lemma-finite-flat-local", "algebra-lemma-NAK", "morphisms-lemma-characterize-flat-closed-immersions" ], "ref_ids": [ 964, 5274, 13240, 797, 401, 5274 ] } ], "ref_ids": [] }, { "id": 5277, "type": "theorem", "label": "morphisms-lemma-dimension-fibre-at-a-point", "categories": [ "morphisms" ], "title": "morphisms-lemma-dimension-fibre-at-a-point", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ and set $s = f(x)$.", "Assume $f$ is locally of finite type.", "Then", "$$", "\\dim_x(X_s) =", "\\dim(\\mathcal{O}_{X_s, x}) + \\text{trdeg}_{\\kappa(s)}(\\kappa(x)).", "$$" ], "refs": [], "proofs": [ { "contents": [ "This immediately reduces to the case $S = s$, and $X$ affine.", "In this case the result follows from", "Algebra, Lemma \\ref{algebra-lemma-dimension-at-a-point-finite-type-field}." ], "refs": [ "algebra-lemma-dimension-at-a-point-finite-type-field" ], "ref_ids": [ 1007 ] } ], "ref_ids": [] }, { "id": 5278, "type": "theorem", "label": "morphisms-lemma-dimension-fibre-at-a-point-additive", "categories": [ "morphisms" ], "title": "morphisms-lemma-dimension-fibre-at-a-point-additive", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to S$ be morphisms of schemes.", "Let $x \\in X$ and set $y = f(x)$, $s = g(y)$.", "Assume $f$ and $g$ locally of finite type.", "Then", "$$", "\\dim_x(X_s) \\leq \\dim_x(X_y) + \\dim_y(Y_s).", "$$", "Moreover, equality holds if $\\mathcal{O}_{X_s, x}$ is flat", "over $\\mathcal{O}_{Y_s, y}$, which holds for example if $\\mathcal{O}_{X, x}$", "is flat over $\\mathcal{O}_{Y, y}$." ], "refs": [], "proofs": [ { "contents": [ "Note that $\\text{trdeg}_{\\kappa(s)}(\\kappa(x)) =", "\\text{trdeg}_{\\kappa(y)}(\\kappa(x)) + \\text{trdeg}_{\\kappa(s)}(\\kappa(y))$.", "Thus by Lemma \\ref{lemma-dimension-fibre-at-a-point} the statement", "is equivalent to", "$$", "\\dim(\\mathcal{O}_{X_s, x})", "\\leq", "\\dim(\\mathcal{O}_{X_y, x}) + \\dim(\\mathcal{O}_{Y_s, y}).", "$$", "For this see Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-total}.", "For the flat case see", "Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}." ], "refs": [ "morphisms-lemma-dimension-fibre-at-a-point", "algebra-lemma-dimension-base-fibre-total", "algebra-lemma-dimension-base-fibre-equals-total" ], "ref_ids": [ 5277, 986, 987 ] } ], "ref_ids": [] }, { "id": 5279, "type": "theorem", "label": "morphisms-lemma-dimension-fibre-after-base-change", "categories": [ "morphisms" ], "title": "morphisms-lemma-dimension-fibre-after-base-change", "contents": [ "Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "be a fibre product diagram of schemes. Assume $f$ locally of finite type.", "Suppose that $x' \\in X'$, $x = g'(x')$, $s' = f'(x')$ and", "$s = g(s') = f(x)$. Then", "\\begin{enumerate}", "\\item $\\dim_x(X_s) = \\dim_{x'}(X'_{s'})$,", "\\item if $F$ is the fibre of the morphism $X'_{s'} \\to X_s$", "over $x$, then", "$$", "\\dim(\\mathcal{O}_{F, x'}) =", "\\dim(\\mathcal{O}_{X'_{s'}, x'}) - \\dim(\\mathcal{O}_{X_s, x}) =", "\\text{trdeg}_{\\kappa(s)}(\\kappa(x)) -", "\\text{trdeg}_{\\kappa(s')}(\\kappa(x'))", "$$", "In particular $\\dim(\\mathcal{O}_{X'_{s'}, x'}) \\geq \\dim(\\mathcal{O}_{X_s, x})$", "and $\\text{trdeg}_{\\kappa(s)}(\\kappa(x)) \\geq", "\\text{trdeg}_{\\kappa(s')}(\\kappa(x'))$.", "\\item given $s', s, x$ there exists a choice of $x'$ such that", "$\\dim(\\mathcal{O}_{X'_{s'}, x'}) = \\dim(\\mathcal{O}_{X_s, x})$ and", "$\\text{trdeg}_{\\kappa(s)}(\\kappa(x)) = \\text{trdeg}_{\\kappa(s')}(\\kappa(x'))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows immediately from", "Algebra,", "Lemma \\ref{algebra-lemma-dimension-at-a-point-preserved-field-extension}.", "Parts (2) and (3) from", "Algebra, Lemma \\ref{algebra-lemma-inequalities-under-field-extension}." ], "refs": [ "algebra-lemma-dimension-at-a-point-preserved-field-extension", "algebra-lemma-inequalities-under-field-extension" ], "ref_ids": [ 1010, 1011 ] } ], "ref_ids": [] }, { "id": 5280, "type": "theorem", "label": "morphisms-lemma-openness-bounded-dimension-fibres", "categories": [ "morphisms" ], "title": "morphisms-lemma-openness-bounded-dimension-fibres", "contents": [ "\\begin{reference}", "\\cite[IV Theorem 13.1.3]{EGA}", "\\end{reference}", "Let $f : X \\to S$ be a morphism of schemes.", "Let $n \\geq 0$. Assume $f$ is locally of finite type.", "The set", "$$", "U_n = \\{x \\in X \\mid \\dim_x X_{f(x)} \\leq n\\}", "$$", "is open in $X$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from", "Algebra,", "Lemma \\ref{algebra-lemma-dimension-fibres-bounded-open-upstairs}" ], "refs": [ "algebra-lemma-dimension-fibres-bounded-open-upstairs" ], "ref_ids": [ 1075 ] } ], "ref_ids": [] }, { "id": 5281, "type": "theorem", "label": "morphisms-lemma-morphism-finite-type-bounded-dimension", "categories": [ "morphisms" ], "title": "morphisms-lemma-morphism-finite-type-bounded-dimension", "contents": [ "Let $f : X \\to Y$ be a morphism of finite type with $Y$ quasi-compact.", "Then the dimension of the fibres of $f$ is bounded." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-openness-bounded-dimension-fibres}", "the set $U_n \\subset X$ of points where the dimension of the fibre", "is $\\leq n$ is open. Since $f$ is of finite type, every point is", "contained in some $U_n$ (because the dimension of a finite", "type algebra over a field is finite). Since $Y$ is quasi-compact and $f$ is of", "finite type, we see that $X$ is quasi-compact. Hence $X = U_n$ for", "some $n$." ], "refs": [ "morphisms-lemma-openness-bounded-dimension-fibres" ], "ref_ids": [ 5280 ] } ], "ref_ids": [] }, { "id": 5282, "type": "theorem", "label": "morphisms-lemma-openness-bounded-dimension-fibres-finite-presentation", "categories": [ "morphisms" ], "title": "morphisms-lemma-openness-bounded-dimension-fibres-finite-presentation", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $n \\geq 0$. Assume $f$ is locally of finite presentation.", "The open", "$$", "U_n = \\{x \\in X \\mid \\dim_x X_{f(x)} \\leq n\\}", "$$", "of Lemma \\ref{lemma-openness-bounded-dimension-fibres} is retrocompact", "in $X$. (See Topology, Definition \\ref{topology-definition-quasi-compact}.)" ], "refs": [ "morphisms-lemma-openness-bounded-dimension-fibres", "topology-definition-quasi-compact" ], "proofs": [ { "contents": [ "The topological space $X$ has a basis for its topology consisting of", "affine opens $U \\subset X$ such that the induced morphism", "$f|_U : U \\to S$ factors through an affine open $V \\subset S$. Hence", "it is enough to show that $U \\cap U_n$ is quasi-compact for such a $U$.", "Note that $U_n \\cap U$ is the same as the open", "$\\{x \\in U \\mid \\dim_x U_{f(x)} \\leq n\\}$. This reduces us to the case", "where $X$ and $S$ are affine. In this case the lemma follows from", "Algebra,", "Lemma \\ref{algebra-lemma-dimension-fibres-bounded-quasi-compact-open-upstairs}", "(and Lemma \\ref{lemma-locally-finite-presentation-characterize})." ], "refs": [ "algebra-lemma-dimension-fibres-bounded-quasi-compact-open-upstairs", "morphisms-lemma-locally-finite-presentation-characterize" ], "ref_ids": [ 1077, 5238 ] } ], "ref_ids": [ 5280, 8360 ] }, { "id": 5283, "type": "theorem", "label": "morphisms-lemma-dimension-fibre-specialization", "categories": [ "morphisms" ], "title": "morphisms-lemma-dimension-fibre-specialization", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\leadsto x'$ be a nontrivial specialization of points in $X$", "lying over the same point $s \\in S$. Assume $f$ is locally of finite type.", "Then", "\\begin{enumerate}", "\\item $\\dim_x(X_s) \\leq \\dim_{x'}(X_s)$,", "\\item $\\dim(\\mathcal{O}_{X_s, x}) < \\dim(\\mathcal{O}_{X_s, x'})$, and", "\\item $\\text{trdeg}_{\\kappa(s)}(\\kappa(x)) >", "\\text{trdeg}_{\\kappa(s)}(\\kappa(x'))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from the fact that any open of $X_s$ containing $x'$", "also contains $x$. Part (2) follows since $\\mathcal{O}_{X_s, x}$ is a", "localization of $\\mathcal{O}_{X_s, x'}$ at a prime ideal, hence any chain", "of prime ideals in $\\mathcal{O}_{X_s, x}$ is part of a strictly longer", "chain of primes in $\\mathcal{O}_{X_s, x'}$. The last inequality follows from", "Algebra, Lemma \\ref{algebra-lemma-tr-deg-specialization}." ], "refs": [ "algebra-lemma-tr-deg-specialization" ], "ref_ids": [ 1006 ] } ], "ref_ids": [] }, { "id": 5284, "type": "theorem", "label": "morphisms-lemma-base-change-relative-dimension-d", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-relative-dimension-d", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "If $f$ has relative dimension $d$, then so does any base change of $f$.", "Same for relative dimension $\\leq d$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from", "Lemma \\ref{lemma-dimension-fibre-after-base-change}." ], "refs": [ "morphisms-lemma-dimension-fibre-after-base-change" ], "ref_ids": [ 5279 ] } ], "ref_ids": [] }, { "id": 5285, "type": "theorem", "label": "morphisms-lemma-composition-relative-dimension-d", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-relative-dimension-d", "contents": [ "Let $f : X \\to Y$, $g : Y \\to Z$ be locally of finite type.", "If $f$ has relative dimension $\\leq d$ and $g$ has relative dimension $\\leq e$", "then $g \\circ f$ has relative dimension $\\leq d + e$.", "If", "\\begin{enumerate}", "\\item $f$ has relative dimension $d$,", "\\item $g$ has relative dimension $e$, and", "\\item $f$ is flat,", "\\end{enumerate}", "then $g \\circ f$ has relative dimension $d + e$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from Lemma \\ref{lemma-dimension-fibre-at-a-point-additive}." ], "refs": [ "morphisms-lemma-dimension-fibre-at-a-point-additive" ], "ref_ids": [ 5278 ] } ], "ref_ids": [] }, { "id": 5286, "type": "theorem", "label": "morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension", "categories": [ "morphisms" ], "title": "morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension", "contents": [ "\\begin{slogan}", "Cohen-Macaulay morphisms decompose into clopens of pure relative dimension", "\\end{slogan}", "Let $f : X \\to S$ be a morphism of schemes.", "Assume that", "\\begin{enumerate}", "\\item $f$ is flat,", "\\item $f$ is locally of finite presentation, and", "\\item for all $s \\in S$ the fibre $X_s$ is Cohen-Macaulay", "(Properties, Definition \\ref{properties-definition-Cohen-Macaulay})", "\\end{enumerate}", "Then there exist open and closed subschemes $X_d \\subset X$", "such that $X = \\coprod_{d \\geq 0} X_d$ and $f|_{X_d} : X_d \\to S$", "has relative dimension $d$." ], "refs": [ "properties-definition-Cohen-Macaulay" ], "proofs": [ { "contents": [ "This is immediate from", "Algebra, Lemma", "\\ref{algebra-lemma-relative-dimension-CM}." ], "refs": [ "algebra-lemma-relative-dimension-CM" ], "ref_ids": [ 1128 ] } ], "ref_ids": [ 3074 ] }, { "id": 5287, "type": "theorem", "label": "morphisms-lemma-locally-quasi-finite-rel-dimension-0", "categories": [ "morphisms" ], "title": "morphisms-lemma-locally-quasi-finite-rel-dimension-0", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume $f$ is locally of finite type.", "Let $x \\in X$ with $s = f(x)$.", "Then $f$ is quasi-finite at $x$ if and only if $\\dim_x(X_s) = 0$.", "In particular, $f$ is locally quasi-finite if and only if $f$ has relative", "dimension $0$." ], "refs": [], "proofs": [ { "contents": [ "If $f$ is quasi-finite at $x$ then $\\kappa(x)$ is a finite extension of", "$\\kappa(s)$ (by", "Lemma \\ref{lemma-residue-field-quasi-finite})", "and $x$ is isolated in $X_s$ (by", "Lemma \\ref{lemma-quasi-finite-at-point-characterize}),", "hence $\\dim_x(X_s) = 0$ by", "Lemma \\ref{lemma-dimension-fibre-at-a-point}.", "Conversely, if $\\dim_x(X_s) = 0$ then by", "Lemma \\ref{lemma-dimension-fibre-at-a-point}", "we see $\\kappa(s) \\subset \\kappa(x)$ is algebraic and", "there are no other points of $X_s$ specializing to $x$.", "Hence $x$ is closed in its fibre by", "Lemma \\ref{lemma-algebraic-residue-field-extension-closed-point-fibre}", "and by", "Lemma \\ref{lemma-quasi-finite-at-point-characterize} (3)", "we conclude that $f$ is quasi-finite at $x$." ], "refs": [ "morphisms-lemma-residue-field-quasi-finite", "morphisms-lemma-quasi-finite-at-point-characterize", "morphisms-lemma-dimension-fibre-at-a-point", "morphisms-lemma-dimension-fibre-at-a-point", "morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre", "morphisms-lemma-quasi-finite-at-point-characterize" ], "ref_ids": [ 5225, 5226, 5277, 5277, 5222, 5226 ] } ], "ref_ids": [] }, { "id": 5288, "type": "theorem", "label": "morphisms-lemma-rel-dimension-dimension", "categories": [ "morphisms" ], "title": "morphisms-lemma-rel-dimension-dimension", "contents": [ "Let $f : X \\to Y$ be a morphism of locally Noetherian schemes", "which is flat, locally of finite type and of relative dimension $d$.", "For every point $x$ in $X$ with image", "$y$ in $Y$ we have $\\dim_x(X) = \\dim_y(Y) + d$." ], "refs": [], "proofs": [ { "contents": [ "After shrinking $X$ and $Y$ to open neighborhoods of $x$ and $y$,", "we can assume that $\\dim(X) = \\dim_x(X)$ and $\\dim(Y) = \\dim_y(Y)$,", "by definition of the dimension of a scheme at a point", "(Properties, Definition \\ref{properties-definition-dimension}).", "The morphism $f$ is open by Lemmas", "\\ref{lemma-noetherian-finite-type-finite-presentation} and", "\\ref{lemma-fppf-open}.", "Hence we can shrink $Y$ to arrange that $f$ is surjective.", "It remains to show that $\\dim(X) = \\dim(Y) + d$.", "\\medskip\\noindent", "Let $a$ be a point in $X$ with image $b$ in $Y$. By", "Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total},", "$$", "\\dim(\\mathcal{O}_{X,a}) = \\dim(\\mathcal{O}_{Y,b}) + \\dim(\\mathcal{O}_{X_b, a}).", "$$", "Taking the supremum over all points $a$ in $X$, it follows", "that $\\dim(X) = \\dim(Y) + d$, as we want, see", "Properties, Lemma \\ref{properties-lemma-dimension}." ], "refs": [ "properties-definition-dimension", "morphisms-lemma-noetherian-finite-type-finite-presentation", "morphisms-lemma-fppf-open", "algebra-lemma-dimension-base-fibre-equals-total", "properties-lemma-dimension" ], "ref_ids": [ 3076, 5245, 5267, 987, 2978 ] } ], "ref_ids": [] }, { "id": 5289, "type": "theorem", "label": "morphisms-lemma-syntomic-characterize", "categories": [ "morphisms" ], "title": "morphisms-lemma-syntomic-characterize", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is syntomic.", "\\item For every affine opens $U \\subset X$, $V \\subset S$", "with $f(U) \\subset V$ the ring map", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is syntomic.", "\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$", "and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$", "is syntomic.", "\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$", "and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is", "syntomic, for all $j\\in J, i\\in I_j$.", "\\end{enumerate}", "Moreover, if $f$ is syntomic then for", "any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$", "the restriction $f|_U : U \\to V$ is syntomic." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-locally-P} if we show that", "the property ``$R \\to A$ is syntomic'' is local.", "We check conditions (a), (b) and (c) of Definition", "\\ref{definition-property-local}.", "By Algebra, Lemma \\ref{algebra-lemma-base-change-syntomic}", "being syntomic is stable under base change and hence", "we conclude (a) holds. By", "Algebra, Lemma \\ref{algebra-lemma-composition-syntomic}", "being syntomic is stable under composition and trivially for any ring", "$R$ the ring map $R \\to R_f$ is syntomic.", "We conclude (b) holds. Finally, property (c) is true", "according to Algebra, Lemma \\ref{algebra-lemma-local-syntomic}." ], "refs": [ "morphisms-lemma-locally-P", "morphisms-definition-property-local", "algebra-lemma-base-change-syntomic", "algebra-lemma-composition-syntomic", "algebra-lemma-local-syntomic" ], "ref_ids": [ 5191, 5547, 1176, 1187, 1177 ] } ], "ref_ids": [] }, { "id": 5290, "type": "theorem", "label": "morphisms-lemma-composition-syntomic", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-syntomic", "contents": [ "The composition of two morphisms which are syntomic is syntomic." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-syntomic-characterize}", "we saw that being syntomic is a local property of ring maps.", "Hence the first statement of the lemma follows from", "Lemma \\ref{lemma-composition-type-P} combined", "with the fact that being syntomic is a property of ring maps that is", "stable under composition, see", "Algebra, Lemma \\ref{algebra-lemma-composition-syntomic}." ], "refs": [ "morphisms-lemma-syntomic-characterize", "morphisms-lemma-composition-type-P", "algebra-lemma-composition-syntomic" ], "ref_ids": [ 5289, 5193, 1187 ] } ], "ref_ids": [] }, { "id": 5291, "type": "theorem", "label": "morphisms-lemma-base-change-syntomic", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-syntomic", "contents": [ "The base change of a morphism which is syntomic is syntomic." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-syntomic-characterize}", "we saw that being syntomic is a local property of ring maps.", "Hence the lemma follows from", "Lemma \\ref{lemma-composition-type-P} combined", "with the fact that being syntomic is a property of ring maps that is", "stable under base change, see", "Algebra, Lemma \\ref{algebra-lemma-base-change-syntomic}." ], "refs": [ "morphisms-lemma-syntomic-characterize", "morphisms-lemma-composition-type-P", "algebra-lemma-base-change-syntomic" ], "ref_ids": [ 5289, 5193, 1176 ] } ], "ref_ids": [] }, { "id": 5292, "type": "theorem", "label": "morphisms-lemma-open-immersion-syntomic", "categories": [ "morphisms" ], "title": "morphisms-lemma-open-immersion-syntomic", "contents": [ "Any open immersion is syntomic." ], "refs": [], "proofs": [ { "contents": [ "This is true because an open immersion is a local isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5293, "type": "theorem", "label": "morphisms-lemma-syntomic-locally-finite-presentation", "categories": [ "morphisms" ], "title": "morphisms-lemma-syntomic-locally-finite-presentation", "contents": [ "A syntomic morphism is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "True because a syntomic ring map is of finite presentation by", "definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5294, "type": "theorem", "label": "morphisms-lemma-syntomic-flat", "categories": [ "morphisms" ], "title": "morphisms-lemma-syntomic-flat", "contents": [ "A syntomic morphism is flat." ], "refs": [], "proofs": [ { "contents": [ "True because a syntomic ring map is flat by definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5295, "type": "theorem", "label": "morphisms-lemma-syntomic-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-syntomic-open", "contents": [ "A syntomic morphism is universally open." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemmas \\ref{lemma-syntomic-locally-finite-presentation},", "\\ref{lemma-syntomic-flat}, and", "\\ref{lemma-fppf-open}." ], "refs": [ "morphisms-lemma-syntomic-locally-finite-presentation", "morphisms-lemma-syntomic-flat", "morphisms-lemma-fppf-open" ], "ref_ids": [ 5293, 5294, 5267 ] } ], "ref_ids": [] }, { "id": 5296, "type": "theorem", "label": "morphisms-lemma-local-complete-intersection", "categories": [ "morphisms" ], "title": "morphisms-lemma-local-complete-intersection", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme locally of finite type over $k$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $X$ is a local complete intersection over $k$,", "\\item for every $x \\in X$ there exists an affine open", "$U = \\Spec(R) \\subset X$ neighbourhood of $x$", "such that $R \\cong k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "is a global complete intersection over $k$, and", "\\item for every $x \\in X$ the local ring $\\mathcal{O}_{X, x}$", "is a complete intersection over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The corresponding algebra results can be found in", "Algebra, Lemmas \\ref{algebra-lemma-lci-at-prime} and", "\\ref{algebra-lemma-lci-global}." ], "refs": [ "algebra-lemma-lci-at-prime", "algebra-lemma-lci-global" ], "ref_ids": [ 1170, 1171 ] } ], "ref_ids": [] }, { "id": 5297, "type": "theorem", "label": "morphisms-lemma-syntomic-locally-standard-syntomic", "categories": [ "morphisms" ], "title": "morphisms-lemma-syntomic-locally-standard-syntomic", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point", "with image $s = f(x)$. Let $V \\subset S$ be an affine open neighbourhood", "of $s$. The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is syntomic at $x$.", "\\item There exist an affine open $U \\subset X$ with $x \\in U$ and", "$f(U) \\subset V$ such that $f|_U : U \\to V$ is standard syntomic.", "\\item The morphism $f$ is of finite presentation at $x$, the local ring map", "$\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$", "is flat and $\\mathcal{O}_{X, x}/\\mathfrak m_s \\mathcal{O}_{X, x}$", "is a complete intersection over $\\kappa(s)$ (see", "Algebra, Definition \\ref{algebra-definition-lci-local-ring}).", "\\end{enumerate}" ], "refs": [ "algebra-definition-lci-local-ring" ], "proofs": [ { "contents": [ "Follows from the definitions and", "Algebra, Lemma \\ref{algebra-lemma-syntomic}." ], "refs": [ "algebra-lemma-syntomic" ], "ref_ids": [ 1185 ] } ], "ref_ids": [ 1531 ] }, { "id": 5298, "type": "theorem", "label": "morphisms-lemma-syntomic-flat-fibres", "categories": [ "morphisms" ], "title": "morphisms-lemma-syntomic-flat-fibres", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "If $f$ is flat, locally of finite presentation, and all", "fibres $X_s$ are local complete intersections, then $f$", "is syntomic." ], "refs": [], "proofs": [ { "contents": [ "Clear from Lemmas", "\\ref{lemma-local-complete-intersection} and", "\\ref{lemma-syntomic-locally-standard-syntomic}", "and the isomorphisms of local rings", "$", "\\mathcal{O}_{X, x}/\\mathfrak m_s \\mathcal{O}_{X, x}", "\\cong", "\\mathcal{O}_{X_s, x}", "$." ], "refs": [ "morphisms-lemma-local-complete-intersection", "morphisms-lemma-syntomic-locally-standard-syntomic" ], "ref_ids": [ 5296, 5297 ] } ], "ref_ids": [] }, { "id": 5299, "type": "theorem", "label": "morphisms-lemma-set-points-where-fibres-lci", "categories": [ "morphisms" ], "title": "morphisms-lemma-set-points-where-fibres-lci", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume $f$ locally of finite type. Formation of the set", "$$", "T = \\{x \\in X \\mid \\mathcal{O}_{X_{f(x)}, x}", "\\text{ is a complete intersection over }\\kappa(f(x))\\}", "$$", "commutes with arbitrary base change:", "For any morphism $g : S' \\to S$, consider", "the base change $f' : X' \\to S'$ of $f$ and the", "projection $g' : X' \\to X$. Then the corresponding", "set $T'$ for the morphism $f'$ is equal to $T' = (g')^{-1}(T)$.", "In particular, if $f$ is assumed flat, and locally of finite", "presentation then the same holds for the open set of points", "where $f$ is syntomic." ], "refs": [], "proofs": [ { "contents": [ "Let $s' \\in S'$ be a point, and let $s = g(s')$. Then we have", "$$", "X'_{s'} =", "\\Spec(\\kappa(s')) \\times_{\\Spec(\\kappa(s))} X_s", "$$", "In other words the fibres of the base change are the base changes", "of the fibres. Hence the first part is equivalent to", "Algebra, Lemma \\ref{algebra-lemma-lci-field-change-local}.", "The second part follows from the first because in that case", "$T$ is the set of points where $f$ is syntomic according to", "Lemma \\ref{lemma-syntomic-locally-standard-syntomic}." ], "refs": [ "algebra-lemma-lci-field-change-local", "morphisms-lemma-syntomic-locally-standard-syntomic" ], "ref_ids": [ 1172, 5297 ] } ], "ref_ids": [] }, { "id": 5300, "type": "theorem", "label": "morphisms-lemma-standard-syntomic-relative-dimension", "categories": [ "morphisms" ], "title": "morphisms-lemma-standard-syntomic-relative-dimension", "contents": [ "Let $R$ be a ring.", "Let $R \\to A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ be a relative", "global complete intersection. Set $S = \\Spec(R)$ and", "$X = \\Spec(A)$. Consider the morphism", "$f : X \\to S$ associated to the ring map $R \\to A$.", "The function $x \\mapsto \\dim_x(X_{f(x)})$ is constant with value $n - c$." ], "refs": [], "proofs": [ { "contents": [ "By Algebra,", "Definition \\ref{algebra-definition-relative-global-complete-intersection}", "$R \\to A$ being a relative global complete intersection means", "all nonzero fibre rings have dimension $n - c$.", "Thus for a prime $\\mathfrak p$ of $R$ the fibre ring", "$\\kappa(\\mathfrak p)[x_1, \\ldots, x_n]/(\\overline{f}_1, \\ldots, \\overline{f}_c)$", "is either zero or a global complete intersection ring of dimension $n - c$.", "By the discussion following", "Algebra, Definition \\ref{algebra-definition-lci-field}", "this implies it is equidimensional of dimension $n - c$.", "Whence the lemma." ], "refs": [ "algebra-definition-relative-global-complete-intersection", "algebra-definition-lci-field" ], "ref_ids": [ 1533, 1530 ] } ], "ref_ids": [] }, { "id": 5301, "type": "theorem", "label": "morphisms-lemma-syntomic-relative-dimension", "categories": [ "morphisms" ], "title": "morphisms-lemma-syntomic-relative-dimension", "contents": [ "Let $f : X \\to S$ be a syntomic morphism. The function", "$x \\mapsto \\dim_x(X_{f(x)})$ is locally constant on $X$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-syntomic-locally-standard-syntomic}", "the morphism $f$ locally looks like a standard", "syntomic morphism of affines. Hence the result follows", "from Lemma \\ref{lemma-standard-syntomic-relative-dimension}." ], "refs": [ "morphisms-lemma-syntomic-locally-standard-syntomic", "morphisms-lemma-standard-syntomic-relative-dimension" ], "ref_ids": [ 5297, 5300 ] } ], "ref_ids": [] }, { "id": 5302, "type": "theorem", "label": "morphisms-lemma-syntomic-permanence", "categories": [ "morphisms" ], "title": "morphisms-lemma-syntomic-permanence", "contents": [ "Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & &", "Y \\ar[dl]^q \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes. Assume that", "\\begin{enumerate}", "\\item $f$ is surjective and syntomic,", "\\item $p$ is syntomic, and", "\\item $q$ is locally of finite presentation\\footnote{In fact, if", "$f$ is surjective, flat, and of finite presentation and $p$ is syntomic,", "then both $q$ and $f$ are syntomic, see", "Descent, Lemma \\ref{descent-lemma-syntomic-permanence}.}.", "\\end{enumerate}", "Then $q$ is syntomic." ], "refs": [ "descent-lemma-syntomic-permanence" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-flat-permanence} we see that $q$ is flat.", "Hence it suffices to show that the fibres of $Y \\to S$ are", "local complete intersections, see Lemma \\ref{lemma-syntomic-flat-fibres}.", "Let $s \\in S$. Consider the morphism $X_s \\to Y_s$.", "This is a base change of the morphism $X \\to Y$ and hence", "surjective, and syntomic (Lemma \\ref{lemma-base-change-syntomic}).", "For the same reason $X_s$ is syntomic over $\\kappa(s)$.", "Moreover, $Y_s$ is locally of finite type over $\\kappa(s)$", "(Lemma \\ref{lemma-base-change-finite-type}). In this way", "we reduce to the case where $S$ is the spectrum of a field.", "\\medskip\\noindent", "Assume $S = \\Spec(k)$. Let $y \\in Y$. Choose an affine", "open $\\Spec(A) \\subset Y$ neighbourhood of $y$. Let", "$\\Spec(B) \\subset X$ be an affine open such that", "$f(\\Spec(B)) \\subset \\Spec(A)$, containing", "a point $x \\in X$ such that $f(x) = y$. Choose a surjection", "$k[x_1, \\ldots, x_n] \\to A$ with kernel $I$.", "Choose a surjection $A[y_1, \\ldots, y_m] \\to B$, which gives", "rise in turn to a surjection $k[x_i, y_j] \\to B$ with kernel $J$.", "Let $\\mathfrak q \\subset k[x_i, y_j]$ be the prime corresponding", "to $y \\in \\Spec(B)$ and let $\\mathfrak p \\subset k[x_i]$ the prime", "corresponding to $x \\in \\Spec(A)$.", "Since $x$ maps to $y$ we have $\\mathfrak p = \\mathfrak q \\cap k[x_i]$.", "Consider the following commutative diagram of local rings:", "$$", "\\xymatrix{", "\\mathcal{O}_{X, x} \\ar@{=}[r] &", "B_{\\mathfrak q} &", "k[x_1, \\ldots, x_n, y_1, \\ldots, y_m]_{\\mathfrak q} \\ar[l] \\\\", "\\mathcal{O}_{Y, y} \\ar@{=}[r] \\ar[u] & A_{\\mathfrak p} \\ar[u] &", "k[x_1, \\ldots, x_n]_{\\mathfrak p} \\ar[l] \\ar[u]", "}", "$$", "We claim that the hypotheses of", "Algebra, Lemma \\ref{algebra-lemma-lci-permanence-initial} are satisfied.", "Conditions (1) and (2) are trivial. Condition (4) follows as", "$X \\to Y$ is flat. Condition (3) follows as the rings", "$\\mathcal{O}_{Y, y}$ and", "$\\mathcal{O}_{X_y, x} = \\mathcal{O}_{X, x}/\\mathfrak m_y\\mathcal{O}_{X, x}$", "are complete intersection rings by our assumptions that", "$f$ and $p$ are syntomic, see", "Lemma \\ref{lemma-syntomic-locally-standard-syntomic}.", "The output of Algebra, Lemma \\ref{algebra-lemma-lci-permanence-initial}", "is exactly that $\\mathcal{O}_{Y, y}$ is a complete intersection", "ring! Hence by Lemma \\ref{lemma-syntomic-locally-standard-syntomic}", "again we see that $Y$ is syntomic over $k$ at $y$ as desired." ], "refs": [ "morphisms-lemma-flat-permanence", "morphisms-lemma-syntomic-flat-fibres", "morphisms-lemma-base-change-syntomic", "morphisms-lemma-base-change-finite-type", "algebra-lemma-lci-permanence-initial", "morphisms-lemma-syntomic-locally-standard-syntomic", "algebra-lemma-lci-permanence-initial", "morphisms-lemma-syntomic-locally-standard-syntomic" ], "ref_ids": [ 5270, 5298, 5291, 5200, 1174, 5297, 1174, 5297 ] } ], "ref_ids": [ 14645 ] }, { "id": 5303, "type": "theorem", "label": "morphisms-lemma-affine-conormal", "categories": [ "morphisms" ], "title": "morphisms-lemma-affine-conormal", "contents": [ "Let $i : Z \\to X$ be an immersion. The conormal sheaf", "of $i$ has the following properties:", "\\begin{enumerate}", "\\item Let $U \\subset X$ be any open subscheme such that $i$", "factors as $Z \\xrightarrow{i'} U \\to X$ where $i'$ is a closed", "immersion. Let $\\mathcal{I} = \\Ker((i')^\\sharp) \\subset \\mathcal{O}_U$.", "Then", "$$", "\\mathcal{C}_{Z/X} = (i')^*\\mathcal{I}\\quad\\text{and}\\quad", "i'_*\\mathcal{C}_{Z/X} = \\mathcal{I}/\\mathcal{I}^2", "$$", "\\item", "For any affine open $\\Spec(R) = U \\subset X$", "such that $Z \\cap U = \\Spec(R/I)$ there is a", "canonical isomorphism", "$\\Gamma(Z \\cap U, \\mathcal{C}_{Z/X}) = I/I^2$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Mostly clear from the definitions. Note that given a ring $R$ and", "an ideal $I$ of $R$ we have $I/I^2 = I \\otimes_R R/I$. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5304, "type": "theorem", "label": "morphisms-lemma-conormal-functorial", "categories": [ "morphisms" ], "title": "morphisms-lemma-conormal-functorial", "contents": [ "Let", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\", "Z' \\ar[r]^{i'} & X'", "}", "$$", "be a commutative diagram in the category of schemes.", "Assume $i$, $i'$ immersions. There is a canonical map", "of $\\mathcal{O}_Z$-modules", "$$", "f^*\\mathcal{C}_{Z'/X'}", "\\longrightarrow", "\\mathcal{C}_{Z/X}", "$$", "characterized by the following property: For every pair of affine opens", "$(\\Spec(R) = U \\subset X, \\Spec(R') = U' \\subset X')$ with", "$f(U) \\subset U'$ such that", "$Z \\cap U = \\Spec(R/I)$ and $Z' \\cap U' = \\Spec(R'/I')$", "the induced map", "$$", "\\Gamma(Z' \\cap U', \\mathcal{C}_{Z'/X'}) = I'/I'^2", "\\longrightarrow", "I/I^2 = \\Gamma(Z \\cap U, \\mathcal{C}_{Z/X})", "$$", "is the one induced by the ring map $f^\\sharp : R' \\to R$ which", "has the property $f^\\sharp(I') \\subset I$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\partial Z' = \\overline{Z'} \\setminus Z'$ and", "$\\partial Z = \\overline{Z} \\setminus Z$. These are closed subsets of $X'$ and", "of $X$. Replacing $X'$ by $X' \\setminus \\partial Z'$ and $X$ by", "$X \\setminus \\big(g^{-1}(\\partial Z') \\cup \\partial Z\\big)$ we", "see that we may assume that $i$ and $i'$ are closed immersions.", "\\medskip\\noindent", "The fact that $g \\circ i$ factors through $i'$ implies that", "$g^*\\mathcal{I}'$ maps into $\\mathcal{I}$ under the canonical", "map $g^*\\mathcal{I}' \\to \\mathcal{O}_X$, see", "Schemes, Lemmas", "\\ref{schemes-lemma-characterize-closed-subspace} and", "\\ref{schemes-lemma-restrict-map-to-closed}.", "Hence we get an induced map of quasi-coherent sheaves", "$g^*(\\mathcal{I}'/(\\mathcal{I}')^2) \\to \\mathcal{I}/\\mathcal{I}^2$.", "Pulling back by $i$ gives", "$i^*g^*(\\mathcal{I}'/(\\mathcal{I}')^2) \\to i^*(\\mathcal{I}/\\mathcal{I}^2)$.", "Note that $i^*(\\mathcal{I}/\\mathcal{I}^2) = \\mathcal{C}_{Z/X}$.", "On the other hand,", "$i^*g^*(\\mathcal{I}'/(\\mathcal{I}')^2) =", "f^*(i')^*(\\mathcal{I}'/(\\mathcal{I}')^2) = f^*\\mathcal{C}_{Z'/X'}$.", "This gives the desired map.", "\\medskip\\noindent", "Checking that the map is locally described as the given map", "$I'/(I')^2 \\to I/I^2$ is a matter of unwinding the definitions", "and is omitted. Another observation is that given any", "$x \\in i(Z)$ there do exist affine open neighbourhoods $U$, $U'$", "with $f(U) \\subset U'$ and $Z \\cap U$ as well as $U' \\cap Z'$", "closed such that $x \\in U$. Proof omitted. Hence the requirement", "of the lemma indeed characterizes the map (and could have been used", "to define it)." ], "refs": [ "schemes-lemma-characterize-closed-subspace", "schemes-lemma-restrict-map-to-closed" ], "ref_ids": [ 7648, 7649 ] } ], "ref_ids": [] }, { "id": 5305, "type": "theorem", "label": "morphisms-lemma-conormal-functorial-flat", "categories": [ "morphisms" ], "title": "morphisms-lemma-conormal-functorial-flat", "contents": [ "Let", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\", "Z' \\ar[r]^{i'} & X'", "}", "$$", "be a fibre product diagram in the category of schemes with", "$i$, $i'$ immersions. Then the canonical map", "$f^*\\mathcal{C}_{Z'/X'} \\to \\mathcal{C}_{Z/X}$ of", "Lemma \\ref{lemma-conormal-functorial}", "is surjective. If $g$ is flat, then it is an isomorphism." ], "refs": [ "morphisms-lemma-conormal-functorial" ], "proofs": [ { "contents": [ "Let $R' \\to R$ be a ring map, and $I' \\subset R'$ an ideal.", "Set $I = I'R$. Then $I'/(I')^2 \\otimes_{R'} R \\to I/I^2$ is", "surjective. If $R' \\to R$ is flat, then $I = I' \\otimes_{R'} R$", "and $I^2 = (I')^2 \\otimes_{R'} R$ and we see the map is an", "isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 5304 ] }, { "id": 5306, "type": "theorem", "label": "morphisms-lemma-transitivity-conormal", "categories": [ "morphisms" ], "title": "morphisms-lemma-transitivity-conormal", "contents": [ "Let $Z \\to Y \\to X$ be immersions of schemes. Then there is a canonical", "exact sequence", "$$", "i^*\\mathcal{C}_{Y/X} \\to", "\\mathcal{C}_{Z/X} \\to", "\\mathcal{C}_{Z/Y} \\to 0", "$$", "where the maps come from", "Lemma \\ref{lemma-conormal-functorial}", "and $i : Z \\to Y$ is the first morphism." ], "refs": [ "morphisms-lemma-conormal-functorial" ], "proofs": [ { "contents": [ "Via", "Lemma \\ref{lemma-conormal-functorial}", "this translates into the following algebra fact. Suppose that", "$C \\to B \\to A$ are surjective ring maps. Let $I = \\Ker(B \\to A)$,", "$J = \\Ker(C \\to A)$ and $K = \\Ker(C \\to B)$. Then", "there is an exact sequence", "$$", "K/K^2 \\otimes_B A \\to J/J^2 \\to I/I^2 \\to 0.", "$$", "This follows immediately from the observation that $I = J/K$." ], "refs": [ "morphisms-lemma-conormal-functorial" ], "ref_ids": [ 5304 ] } ], "ref_ids": [ 5304 ] }, { "id": 5307, "type": "theorem", "label": "morphisms-lemma-universal-derivation-universal", "categories": [ "morphisms" ], "title": "morphisms-lemma-universal-derivation-universal", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. The map", "$$", "\\Hom_{\\mathcal{O}_X}(\\Omega_{X/S}, \\mathcal{F})", "\\longrightarrow", "\\text{Der}_S(\\mathcal{O}_X, \\mathcal{F}),\\quad", "\\alpha \\longmapsto \\alpha \\circ \\text{d}_{X/S}", "$$", "is an isomorphism of functors $\\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Sets}$." ], "refs": [], "proofs": [ { "contents": [ "This is just a restatement of the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5308, "type": "theorem", "label": "morphisms-lemma-differentials-restrict-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-differentials-restrict-open", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $U \\subset X$, $V \\subset S$ be open subschemes such", "that $f(U) \\subset V$. Then there is a unique isomorphism", "$\\Omega_{X/S}|_U = \\Omega_{U/V}$ of $\\mathcal{O}_U$-modules such that", "$\\text{d}_{X/S}|_U = \\text{d}_{U/V}$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Modules, Lemma \\ref{modules-lemma-localize-differentials}", "if we use the canonical identification", "$f^{-1}\\mathcal{O}_S|_U = (f|_U)^{-1}\\mathcal{O}_V$." ], "refs": [ "modules-lemma-localize-differentials" ], "ref_ids": [ 13312 ] } ], "ref_ids": [] }, { "id": 5309, "type": "theorem", "label": "morphisms-lemma-affine-case-derivation", "categories": [ "morphisms" ], "title": "morphisms-lemma-affine-case-derivation", "contents": [ "Let $R \\to A$ be a ring map. Let $\\mathcal{F}$", "be a sheaf of $\\mathcal{O}_X$-modules", "on $X = \\Spec(A)$. Set $S = \\Spec(R)$.", "The rule which associates to an $S$-derivation on $\\mathcal{F}$", "its action on global sections defines a bijection between", "the set of $S$-derivations of $\\mathcal{F}$ and the set of", "$R$-derivations on $M = \\Gamma(X, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Let $D : A \\to M$ be an $R$-derivation. We have to show there exists", "a unique $S$-derivation on $\\mathcal{F}$ which gives rise to", "$D$ on global sections. Let $U = D(f) \\subset X$ be a standard affine open.", "Any element of $\\Gamma(U, \\mathcal{O}_X)$ is of the form", "$a/f^n$ for some $a \\in A$ and $n \\geq 0$. By the Leibniz rule", "we have", "$$", "D(a)|_U = a/f^n D(f^n)|_U + f^n D(a/f^n)", "$$", "in $\\Gamma(U, \\mathcal{F})$. Since $f$ acts invertibly", "on $\\Gamma(U, \\mathcal{F})$ this completely determines", "the value of $D(a/f^n) \\in \\Gamma(U, \\mathcal{F})$.", "This proves uniqueness. Existence follows by simply defining", "$$", "D(a/f^n) := (1/f^n) D(a)|_U - a/f^{2n} D(f^n)|_U", "$$", "and proving this has all the desired properties (on the basis", "of standard opens of $X$). Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5310, "type": "theorem", "label": "morphisms-lemma-differentials-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-differentials-affine", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. For any pair of affine opens", "$\\Spec(A) = U \\subset X$, $\\Spec(R) = V \\subset S$ with $f(U) \\subset V$", "there is a unique isomorphism", "$$", "\\Gamma(U, \\Omega_{X/S}) = \\Omega_{A/R}.", "$$", "compatible with $\\text{d}_{X/S}$ and $\\text{d} : A \\to \\Omega_{A/R}$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-differentials-restrict-open} we may replace", "$X$ and $S$ by $U$ and $V$. Thus we may assume $X = \\Spec(A)$", "and $S = \\Spec(R)$ and we have to show the lemma with $U = X$", "and $V = S$. Consider the $A$-module $M = \\Gamma(X, \\Omega_{X/S})$", "together with the $R$-derivation $\\text{d}_{X/S} : A \\to M$.", "Let $N$ be another $A$-module and denote $\\widetilde{N}$ the quasi-coherent", "$\\mathcal{O}_X$-module associated to $N$, see", "Schemes, Section \\ref{schemes-section-quasi-coherent-affine}.", "Precomposing by $\\text{d}_{X/S} : A \\to M$ we get an arrow", "$$", "\\alpha : \\Hom_A(M, N) \\longrightarrow \\text{Der}_R(A, N)", "$$", "Using Lemmas \\ref{lemma-universal-derivation-universal} and", "\\ref{lemma-affine-case-derivation} we get identifications", "$$", "\\Hom_{\\mathcal{O}_X}(\\Omega_{X/S}, \\widetilde{N}) =", "\\text{Der}_S(\\mathcal{O}_X, \\widetilde{N}) =", "\\text{Der}_R(A, N)", "$$", "Taking global sections determines an arrow", "$\\Hom_{\\mathcal{O}_X}(\\Omega_{X/S}, \\widetilde{N}) \\to \\Hom_R(M, N)$.", "Combining this arrow and the identifications above we get an arrow", "$$", "\\beta : \\text{Der}_R(A, N) \\longrightarrow \\Hom_R(M, N)", "$$", "Checking what happens on global sections, we find that $\\alpha$", "and $\\beta$ are each others inverse. Hence we see that", "$\\text{d}_{X/S} : A \\to M$ satisfies", "the same universal property as $\\text{d} : A \\to \\Omega_{A/R}$,", "see Algebra, Lemma \\ref{algebra-lemma-universal-omega}. Thus the Yoneda lemma", "(Categories, Lemma \\ref{categories-lemma-yoneda})", "implies there is a unique isomorphism of $A$-modules", "$M \\cong \\Omega_{A/R}$ compatible with derivations." ], "refs": [ "morphisms-lemma-differentials-restrict-open", "morphisms-lemma-universal-derivation-universal", "morphisms-lemma-affine-case-derivation", "algebra-lemma-universal-omega", "categories-lemma-yoneda" ], "ref_ids": [ 5308, 5307, 5309, 1129, 12203 ] } ], "ref_ids": [] }, { "id": 5311, "type": "theorem", "label": "morphisms-lemma-differentials-diagonal", "categories": [ "morphisms" ], "title": "morphisms-lemma-differentials-diagonal", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. There is a canonical", "isomorphism between $\\Omega_{X/S}$ and the conormal sheaf of", "the diagonal morphism $\\Delta_{X/S} : X \\longrightarrow X \\times_S X$." ], "refs": [], "proofs": [ { "contents": [ "We first establish the existence of a couple of ``global'' sheaves", "and global maps of sheaves, and further down we describe", "the constructions over some affine opens.", "\\medskip\\noindent", "Recall that $\\Delta = \\Delta_{X/S} : X \\to X \\times_S X$", "is an immersion, see Schemes, Lemma \\ref{schemes-lemma-diagonal-immersion}.", "Let $\\mathcal{J}$ be the ideal sheaf of the immersion", "which lives over some open subscheme $W$ of $X \\times_S X$", "such that $\\Delta(X) \\subset W$ is closed. Let us take the one that", "was found in the proof of", "Schemes, Lemma \\ref{schemes-lemma-diagonal-immersion}.", "Note that the sheaf of rings $\\mathcal{O}_W/\\mathcal{J}^2$", "is supported on $\\Delta(X)$. Moreover it sits in a", "short exact sequence of sheaves", "$$", "0 \\to \\mathcal{J}/\\mathcal{J}^2", "\\to \\mathcal{O}_W/\\mathcal{J}^2", "\\to \\Delta_*\\mathcal{O}_X", "\\to 0.", "$$", "Using $\\Delta^{-1}$ we can think of this as a surjection of", "sheaves of $f^{-1}\\mathcal{O}_S$-algebras with kernel the", "conormal sheaf of $\\Delta$ (see Definition \\ref{definition-conormal-sheaf}", "and Lemma \\ref{lemma-affine-conormal}).", "$$", "0 \\to \\mathcal{C}_{X/X \\times_S X}", "\\to \\Delta^{-1}(\\mathcal{O}_W/\\mathcal{J}^2)", "\\to \\mathcal{O}_X", "\\to 0", "$$", "This places us in the situation of", "Modules, Lemma \\ref{modules-lemma-double-structure-gives-derivation}.", "The projection morphisms $p_i : X \\times_S X \\to X$, $i = 1, 2$ induce", "maps of sheaves of rings", "$(p_i)^\\sharp : (p_i)^{-1}\\mathcal{O}_X \\to \\mathcal{O}_{X \\times_S X}$.", "We may restrict to $W$ and quotient by $\\mathcal{J}^2$ to get", "$(p_i)^{-1}\\mathcal{O}_X \\to \\mathcal{O}_W/\\mathcal{J}^2$.", "Since $\\Delta^{-1}p_i^{-1}\\mathcal{O}_X = \\mathcal{O}_X$", "we get maps", "$$", "s_i : \\mathcal{O}_X \\to \\Delta^{-1}(\\mathcal{O}_W/\\mathcal{J}^2).", "$$", "Both $s_1$ and $s_2$ are sections to the map", "$\\Delta^{-1}(\\mathcal{O}_W/\\mathcal{J}^2) \\to \\mathcal{O}_X$,", "as in Modules, Lemma \\ref{modules-lemma-double-structure-gives-derivation}.", "Thus we get an $S$-derivation", "$\\text{d} = s_2 - s_1 : \\mathcal{O}_X \\to \\mathcal{C}_{X/X \\times_S X}$.", "By the universal property of the module of differentials we find a", "unique $\\mathcal{O}_X$-linear map", "$$", "\\Omega_{X/S} \\longrightarrow \\mathcal{C}_{X/X \\times_S X},\\quad", "f\\text{d}g \\longmapsto fs_2(g) - fs_1(g)", "$$", "To see the map is an isomorphism, let us work this out over suitable affine", "opens. We can cover $X$ by affine opens $\\Spec(A) = U \\subset X$", "whose image is contained in an affine open $\\Spec(R) = V \\subset S$.", "According to the proof of Schemes, Lemma \\ref{schemes-lemma-diagonal-immersion}", "$U \\times_V U \\subset X \\times_S X$ is an affine open", "contained in the open $W$ mentioned above. Also", "$U \\times_V U = \\Spec(A \\otimes_R A)$.", "The sheaf $\\mathcal{J}$ corresponds to the ideal", "$J = \\Ker(A \\otimes_R A \\to A)$.", "The short exact sequence to the short exact sequence", "of $A \\otimes_R A$-modules", "$$", "0 \\to J/J^2 \\to (A \\otimes_R A)/J^2 \\to A \\to 0", "$$", "The sections $s_i$ correspond to the ring maps", "$$", "A \\longrightarrow (A \\otimes_R A)/J^2,\\quad", "s_1 : a \\mapsto a \\otimes 1,\\quad", "s_2 : a \\mapsto 1 \\otimes a.", "$$", "By Lemma \\ref{lemma-affine-conormal} we have", "$\\Gamma(U, \\mathcal{C}_{X/X \\times_S X}) = J/J^2$ and", "by Lemma \\ref{lemma-differentials-affine}", "we have $\\Gamma(U, \\Omega_{X/S}) = \\Omega_{A/R}$.", "The map above is the map $a \\text{d}b \\mapsto a \\otimes b - ab \\otimes 1$", "which is shown to be an isomorphism in", "Algebra, Lemma \\ref{algebra-lemma-differentials-diagonal}." ], "refs": [ "schemes-lemma-diagonal-immersion", "schemes-lemma-diagonal-immersion", "morphisms-definition-conormal-sheaf", "morphisms-lemma-affine-conormal", "modules-lemma-double-structure-gives-derivation", "modules-lemma-double-structure-gives-derivation", "schemes-lemma-diagonal-immersion", "morphisms-lemma-affine-conormal", "morphisms-lemma-differentials-affine", "algebra-lemma-differentials-diagonal" ], "ref_ids": [ 7707, 7707, 5562, 5303, 13317, 13317, 7707, 5303, 5310, 1139 ] } ], "ref_ids": [] }, { "id": 5312, "type": "theorem", "label": "morphisms-lemma-functoriality-differentials", "categories": [ "morphisms" ], "title": "morphisms-lemma-functoriality-differentials", "contents": [ "Let", "$$", "\\xymatrix{", "X' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\", "S' \\ar[r] & S", "}", "$$", "be a commutative diagram of schemes. The canonical map", "$\\mathcal{O}_X \\to f_*\\mathcal{O}_{X'}$ composed with the map", "$f_*\\text{d}_{X'/S'} : f_*\\mathcal{O}_{X'} \\to f_*\\Omega_{X'/S'}$ is a", "$S$-derivation. Hence we obtain a canonical map of $\\mathcal{O}_X$-modules", "$\\Omega_{X/S} \\to f_*\\Omega_{X'/S'}$, and by", "adjointness of $f_*$ and $f^*$ a", "canonical $\\mathcal{O}_{X'}$-module homomorphism", "$$", "c_f : f^*\\Omega_{X/S} \\longrightarrow \\Omega_{X'/S'}.", "$$", "It is uniquely characterized by the property that", "$f^*\\text{d}_{X/S}(h)$ maps to $\\text{d}_{X'/S'}(f^* h)$", "for any local section $h$ of $\\mathcal{O}_X$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Modules, Lemma \\ref{modules-lemma-functoriality-differentials-ringed-spaces}.", "In the case of schemes we can also use the functoriality of", "the conormal sheaves (see Lemma \\ref{lemma-conormal-functorial}) and", "Lemma \\ref{lemma-differentials-diagonal} to define $c_f$.", "Or we can use the characterization in the last line of the lemma to", "glue maps defined on affine patches", "(see Algebra, Equation (\\ref{algebra-equation-functorial-omega}))." ], "refs": [ "modules-lemma-functoriality-differentials-ringed-spaces", "morphisms-lemma-conormal-functorial", "morphisms-lemma-differentials-diagonal" ], "ref_ids": [ 13318, 5304, 5311 ] } ], "ref_ids": [] }, { "id": 5313, "type": "theorem", "label": "morphisms-lemma-triangle-differentials", "categories": [ "morphisms" ], "title": "morphisms-lemma-triangle-differentials", "contents": [ "Let $f : X \\to Y$, $g : Y \\to S$ be morphisms of schemes.", "Then there is a canonical exact sequence", "$$", "f^*\\Omega_{Y/S} \\to \\Omega_{X/S} \\to \\Omega_{X/Y} \\to 0", "$$", "where the maps come from applications of", "Lemma \\ref{lemma-functoriality-differentials}." ], "refs": [ "morphisms-lemma-functoriality-differentials" ], "proofs": [ { "contents": [ "This is the sheafified version of", "Algebra, Lemma \\ref{algebra-lemma-exact-sequence-differentials}." ], "refs": [ "algebra-lemma-exact-sequence-differentials" ], "ref_ids": [ 1133 ] } ], "ref_ids": [ 5312 ] }, { "id": 5314, "type": "theorem", "label": "morphisms-lemma-base-change-differentials", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-differentials", "contents": [ "Let $X \\to S$ be a morphism of schemes.", "Let $g : S' \\to S$ be a morphism of schemes.", "Let $X' = X_{S'}$ be the base change of $X$.", "Denote $g' : X' \\to X$ the projection.", "Then the map", "$$", "(g')^*\\Omega_{X/S} \\to \\Omega_{X'/S'}", "$$", "of Lemma \\ref{lemma-functoriality-differentials} is an isomorphism." ], "refs": [ "morphisms-lemma-functoriality-differentials" ], "proofs": [ { "contents": [ "This is the sheafified version of", "Algebra, Lemma \\ref{algebra-lemma-differentials-base-change}." ], "refs": [ "algebra-lemma-differentials-base-change" ], "ref_ids": [ 1138 ] } ], "ref_ids": [ 5312 ] }, { "id": 5315, "type": "theorem", "label": "morphisms-lemma-differential-product", "categories": [ "morphisms" ], "title": "morphisms-lemma-differential-product", "contents": [ "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes with the same", "target. Let $p : X \\times_S Y \\to X$ and $q : X \\times_S Y \\to Y$ be the", "projection morphisms. The maps from", "Lemma \\ref{lemma-functoriality-differentials}", "$$", "p^*\\Omega_{X/S} \\oplus q^*\\Omega_{Y/S}", "\\longrightarrow", "\\Omega_{X \\times_S Y/S}", "$$", "give an isomorphism." ], "refs": [ "morphisms-lemma-functoriality-differentials" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-base-change-differentials} the composition", "$p^*\\Omega_{X/S} \\to \\Omega_{X \\times_S Y/S} \\to \\Omega_{X \\times_S Y/Y}$", "is an isomorphism, and similarly for $q$. Moreover, the cokernel", "of $p^*\\Omega_{X/S} \\to \\Omega_{X \\times_S Y/S}$ is", "$\\Omega_{X \\times_S Y/X}$ by", "Lemma \\ref{lemma-triangle-differentials}. The result follows." ], "refs": [ "morphisms-lemma-base-change-differentials", "morphisms-lemma-triangle-differentials" ], "ref_ids": [ 5314, 5313 ] } ], "ref_ids": [ 5312 ] }, { "id": 5316, "type": "theorem", "label": "morphisms-lemma-finite-type-differentials", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-type-differentials", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "If $f$ is locally of finite type, then $\\Omega_{X/S}$ is", "a finite type $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "Immediate from", "Algebra, Lemma \\ref{algebra-lemma-differentials-finitely-generated},", "Lemma \\ref{lemma-differentials-affine},", "Lemma \\ref{lemma-locally-finite-type-characterize}, and", "Properties, Lemma \\ref{properties-lemma-finite-type-module}." ], "refs": [ "algebra-lemma-differentials-finitely-generated", "morphisms-lemma-differentials-affine", "morphisms-lemma-locally-finite-type-characterize", "properties-lemma-finite-type-module" ], "ref_ids": [ 1142, 5310, 5198, 3002 ] } ], "ref_ids": [] }, { "id": 5317, "type": "theorem", "label": "morphisms-lemma-finite-presentation-differentials", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-presentation-differentials", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "If $f$ is locally of finite presentation, then $\\Omega_{X/S}$ is", "an $\\mathcal{O}_X$-module of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Immediate from", "Algebra, Lemma \\ref{algebra-lemma-differentials-finitely-presented},", "Lemma \\ref{lemma-differentials-affine},", "Lemma \\ref{lemma-locally-finite-presentation-characterize}, and", "Properties, Lemma \\ref{properties-lemma-finite-presentation-module}." ], "refs": [ "algebra-lemma-differentials-finitely-presented", "morphisms-lemma-differentials-affine", "morphisms-lemma-locally-finite-presentation-characterize", "properties-lemma-finite-presentation-module" ], "ref_ids": [ 1141, 5310, 5238, 3003 ] } ], "ref_ids": [] }, { "id": 5318, "type": "theorem", "label": "morphisms-lemma-immersion-differentials", "categories": [ "morphisms" ], "title": "morphisms-lemma-immersion-differentials", "contents": [ "If $X \\to S$ is an immersion, or more generally a monomorphism, then", "$\\Omega_{X/S}$ is zero." ], "refs": [], "proofs": [ { "contents": [ "This is true because $\\Delta_{X/S}$ is an isomorphism in this case", "and hence has trivial conormal sheaf. Hence $\\Omega_{X/S} = 0$", "by Lemma \\ref{lemma-differentials-diagonal}. The algebraic version is", "Algebra, Lemma \\ref{algebra-lemma-trivial-differential-surjective}." ], "refs": [ "morphisms-lemma-differentials-diagonal", "algebra-lemma-trivial-differential-surjective" ], "ref_ids": [ 5311, 1131 ] } ], "ref_ids": [] }, { "id": 5319, "type": "theorem", "label": "morphisms-lemma-differentials-relative-immersion", "categories": [ "morphisms" ], "title": "morphisms-lemma-differentials-relative-immersion", "contents": [ "Let $i : Z \\to X$ be an immersion of schemes over $S$.", "There is a canonical exact sequence", "$$", "\\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0", "$$", "where the first arrow is induced by $\\text{d}_{X/S}$", "and the second arrow comes from Lemma \\ref{lemma-functoriality-differentials}." ], "refs": [ "morphisms-lemma-functoriality-differentials" ], "proofs": [ { "contents": [ "This is the sheafified version of", "Algebra, Lemma \\ref{algebra-lemma-differential-seq}. However", "we should make sure we can define the first arrow globally.", "Hence we explain the meaning of ``induced by $\\text{d}_{X/S}$'' here.", "Namely, we may assume that $i$ is a closed immersion by", "shrinking $X$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be the sheaf of ideals corresponding to $Z \\subset X$.", "Then $\\text{d}_{X/S} : \\mathcal{I} \\to \\Omega_{X/S}$", "maps the subsheaf $\\mathcal{I}^2 \\subset \\mathcal{I}$ to", "$\\mathcal{I}\\Omega_{X/S}$. Hence it induces a map", "$\\mathcal{I}/\\mathcal{I}^2 \\to \\Omega_{X/S}/\\mathcal{I}\\Omega_{X/S}$", "which is $\\mathcal{O}_X/\\mathcal{I}$-linear.", "By Lemma \\ref{lemma-i-star-equivalence} this corresponds to a map", "$\\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/S}$ as desired." ], "refs": [ "algebra-lemma-differential-seq", "morphisms-lemma-i-star-equivalence" ], "ref_ids": [ 1135, 5136 ] } ], "ref_ids": [ 5312 ] }, { "id": 5320, "type": "theorem", "label": "morphisms-lemma-differentials-relative-immersion-section", "categories": [ "morphisms" ], "title": "morphisms-lemma-differentials-relative-immersion-section", "contents": [ "Let $i : Z \\to X$ be an immersion of schemes over $S$, and", "assume $i$ (locally) has a left inverse. Then the canonical", "sequence", "$$", "0 \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0", "$$", "of", "Lemma \\ref{lemma-differentials-relative-immersion}", "is (locally) split exact. In particular, if $s : S \\to X$ is a section", "of the structure morphism $X \\to S$ then the map", "$\\mathcal{C}_{S/X} \\to s^*\\Omega_{X/S}$ induced by", "$\\text{d}_{X/S}$ is an isomorphism." ], "refs": [ "morphisms-lemma-differentials-relative-immersion" ], "proofs": [ { "contents": [ "Follows from", "Algebra, Lemma \\ref{algebra-lemma-differential-seq-split}.", "Clarification: if $g : X \\to Z$ is a left inverse of $i$, then", "$i^*c_g$ is a right inverse of the map $i^*\\Omega_{X/S} \\to \\Omega_{Z/S}$.", "Also, if $s$ is a section, then it is an immersion $s : Z = S \\to X$", "over $S$ (see", "Schemes, Lemma \\ref{schemes-lemma-section-immersion})", "and in that case $\\Omega_{Z/S} = 0$." ], "refs": [ "algebra-lemma-differential-seq-split" ], "ref_ids": [ 1136 ] } ], "ref_ids": [ 5319 ] }, { "id": 5321, "type": "theorem", "label": "morphisms-lemma-two-immersions", "categories": [ "morphisms" ], "title": "morphisms-lemma-two-immersions", "contents": [ "Let", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[rd]_j & X \\ar[d] \\\\", "& Y", "}", "$$", "be a commutative diagram of schemes where $i$ and $j$ are immersions.", "Then there is a canonical exact sequence", "$$", "\\mathcal{C}_{Z/Y} \\to", "\\mathcal{C}_{Z/X} \\to", "i^*\\Omega_{X/Y} \\to 0", "$$", "where the first arrow comes from", "Lemma \\ref{lemma-conormal-functorial}", "and the second from", "Lemma \\ref{lemma-differentials-relative-immersion}." ], "refs": [ "morphisms-lemma-conormal-functorial", "morphisms-lemma-differentials-relative-immersion" ], "proofs": [ { "contents": [ "The algebraic version of this is", "Algebra, Lemma \\ref{algebra-lemma-application-NL}." ], "refs": [ "algebra-lemma-application-NL" ], "ref_ids": [ 1155 ] } ], "ref_ids": [ 5304, 5319 ] }, { "id": 5322, "type": "theorem", "label": "morphisms-lemma-affine-case-differential-operators", "categories": [ "morphisms" ], "title": "morphisms-lemma-affine-case-differential-operators", "contents": [ "Let $R \\to A$ be a ring map. Denote $f : X \\to S$ the corresponding", "morphism of affine schemes. Let $\\mathcal{F}$ and $\\mathcal{G}$", "be $\\mathcal{O}_X$-modules. If $\\mathcal{F}$ is quasi-coherent then", "the map", "$$", "\\text{Diff}^k_{X/S}(\\mathcal{F}, \\mathcal{G}) \\to", "\\text{Diff}^k_{A/R}(\\Gamma(X, \\mathcal{F}), \\Gamma(X, \\mathcal{G}))", "$$", "sending a differential operator to its action on global sections", "is bijective." ], "refs": [], "proofs": [ { "contents": [ "Write $\\mathcal{F} = \\widetilde{M}$ for some $A$-module $M$. Set", "$N = \\Gamma(X, \\mathcal{G})$. Let $D : M \\to N$ be a differential", "operator of order $k$. We have to show there exists", "a unique differential operator $\\mathcal{F} \\to \\mathcal{G}$ of order $k$", "which gives rise to $D$ on global sections. Let $U = D(f) \\subset X$", "be a standard affine open. Then $\\mathcal{F}(U) = M_f$ is the localization.", "By Algebra, Lemma \\ref{algebra-lemma-invert-system-differential-operators}", "the differential operator $D$ extends to a unique differential operator", "$$", "D_f : \\mathcal{F}(U) = \\widetilde{M}(U) = M_f \\to N_f = \\widetilde{N}(U)", "$$", "The uniqueness shows that these maps $D_f$ glue to give a map of sheaves", "$\\widetilde{M} \\to \\widetilde{N}$ on the basis of all standard opens", "of $X$. Hence we get a unique map of sheaves", "$\\widetilde{D} : \\widetilde{M} \\to \\widetilde{N}$ agreeing with these maps", "by the material in Sheaves, Section \\ref{sheaves-section-bases}.", "Since $\\widetilde{D}$ is given by differential operators of order $k$ on", "the standard opens, we find that $\\widetilde{D}$ is a differential operator", "of order $k$ (small detail omitted).", "Finally, we can post-compose with the canonical $\\mathcal{O}_X$-module map", "$c : \\widetilde{N} \\to \\mathcal{G}$", "(Schemes, Lemma \\ref{schemes-lemma-compare-constructions})", "to get $c \\circ \\widetilde{D} : \\mathcal{F} \\to \\mathcal{G}$", "which is a differential operator of order $k$ by", "Modules, Lemma \\ref{modules-lemma-composition-differential-operators}.", "This proves existence. We omit the proof of uniqueness." ], "refs": [ "algebra-lemma-invert-system-differential-operators", "schemes-lemma-compare-constructions", "modules-lemma-composition-differential-operators" ], "ref_ids": [ 1149, 7660, 13320 ] } ], "ref_ids": [] }, { "id": 5323, "type": "theorem", "label": "morphisms-lemma-base-change-differential-operators", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-differential-operators", "contents": [ "Let $a : X \\to S$ and $b : Y \\to S$ be morphisms of schemes.", "Let $\\mathcal{F}$ and $\\mathcal{F}'$ be quasi-coherent $\\mathcal{O}_X$-modules.", "Let $D : \\mathcal{F} \\to \\mathcal{F}'$ be a differential operator", "of order $k$ on $X/S$. Let $\\mathcal{G}$ be a quasi-coherent", "$\\mathcal{O}_Y$-module. Then there is a unique differential", "operator", "$$", "D' :", "\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_S Y}}", "\\text{pr}_2^*\\mathcal{G}", "\\longrightarrow", "\\text{pr}_1^*\\mathcal{F}' \\otimes_{\\mathcal{O}_{X \\times_S Y}}", "\\text{pr}_2^*\\mathcal{G}", "$$", "of order $k$ on $X \\times_S Y / Y$ such that", "$", "D'(s \\otimes t) = D(s) \\otimes t", "$", "for local sections $s$ of $\\mathcal{F}$ and $t$ of $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "In case $X$, $Y$, and $S$ are affine, this follows, via", "Lemma \\ref{lemma-affine-case-differential-operators},", "from the corresponding algebra result, see", "Algebra, Lemma \\ref{algebra-lemma-base-change-differential-operators}.", "In general, one uses coverings by affines", "(for example as in Schemes, Lemma", "\\ref{schemes-lemma-affine-covering-fibre-product})", "to construct $D'$ globally. Details omitted." ], "refs": [ "morphisms-lemma-affine-case-differential-operators", "algebra-lemma-base-change-differential-operators", "schemes-lemma-affine-covering-fibre-product" ], "ref_ids": [ 5322, 1150, 7692 ] } ], "ref_ids": [] }, { "id": 5324, "type": "theorem", "label": "morphisms-lemma-smooth-characterize", "categories": [ "morphisms" ], "title": "morphisms-lemma-smooth-characterize", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is smooth.", "\\item For every affine opens $U \\subset X$, $V \\subset S$", "with $f(U) \\subset V$ the ring map", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is smooth.", "\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$", "and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$", "is smooth.", "\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$", "and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is", "smooth, for all $j\\in J, i\\in I_j$.", "\\end{enumerate}", "Moreover, if $f$ is smooth then for", "any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$", "the restriction $f|_U : U \\to V$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-locally-P} if we show that", "the property ``$R \\to A$ is smooth'' is local.", "We check conditions (a), (b) and (c) of Definition", "\\ref{definition-property-local}.", "By Algebra, Lemma \\ref{algebra-lemma-base-change-smooth}", "being smooth is stable under base change and hence", "we conclude (a) holds. By", "Algebra, Lemma \\ref{algebra-lemma-compose-smooth}", "being smooth is stable under composition and for any ring", "$R$ the ring map $R \\to R_f$ is (standard) smooth.", "We conclude (b) holds. Finally, property (c) is true", "according to Algebra, Lemma \\ref{algebra-lemma-locally-smooth}." ], "refs": [ "morphisms-lemma-locally-P", "morphisms-definition-property-local", "algebra-lemma-base-change-smooth", "algebra-lemma-compose-smooth", "algebra-lemma-locally-smooth" ], "ref_ids": [ 5191, 5547, 1191, 1198, 1197 ] } ], "ref_ids": [] }, { "id": 5325, "type": "theorem", "label": "morphisms-lemma-smooth-flat-smooth-fibres", "categories": [ "morphisms" ], "title": "morphisms-lemma-smooth-flat-smooth-fibres", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "If $f$ is flat, locally of finite presentation, and all", "fibres $X_s$ are smooth, then $f$", "is smooth." ], "refs": [], "proofs": [ { "contents": [ "Follows from Algebra, Lemma \\ref{algebra-lemma-flat-fibre-smooth}." ], "refs": [ "algebra-lemma-flat-fibre-smooth" ], "ref_ids": [ 1200 ] } ], "ref_ids": [] }, { "id": 5326, "type": "theorem", "label": "morphisms-lemma-composition-smooth", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-smooth", "contents": [ "The composition of two morphisms which are smooth is smooth." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-smooth-characterize}", "we saw that being smooth is a local property of ring maps.", "Hence the first statement of the lemma follows from", "Lemma \\ref{lemma-composition-type-P} combined", "with the fact that being smooth is a property of ring maps that is", "stable under composition, see", "Algebra, Lemma \\ref{algebra-lemma-compose-smooth}." ], "refs": [ "morphisms-lemma-smooth-characterize", "morphisms-lemma-composition-type-P", "algebra-lemma-compose-smooth" ], "ref_ids": [ 5324, 5193, 1198 ] } ], "ref_ids": [] }, { "id": 5327, "type": "theorem", "label": "morphisms-lemma-base-change-smooth", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-smooth", "contents": [ "The base change of a morphism which is smooth is smooth." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-smooth-characterize}", "we saw that being smooth is a local property of ring maps.", "Hence the lemma follows from", "Lemma \\ref{lemma-composition-type-P} combined", "with the fact that being smooth is a property of ring maps that is", "stable under base change, see", "Algebra, Lemma \\ref{algebra-lemma-base-change-smooth}." ], "refs": [ "morphisms-lemma-smooth-characterize", "morphisms-lemma-composition-type-P", "algebra-lemma-base-change-smooth" ], "ref_ids": [ 5324, 5193, 1191 ] } ], "ref_ids": [] }, { "id": 5328, "type": "theorem", "label": "morphisms-lemma-open-immersion-smooth", "categories": [ "morphisms" ], "title": "morphisms-lemma-open-immersion-smooth", "contents": [ "Any open immersion is smooth." ], "refs": [], "proofs": [ { "contents": [ "This is true because an open immersion is a local isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5329, "type": "theorem", "label": "morphisms-lemma-smooth-syntomic", "categories": [ "morphisms" ], "title": "morphisms-lemma-smooth-syntomic", "contents": [ "A smooth morphism is syntomic." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-smooth-syntomic}." ], "refs": [ "algebra-lemma-smooth-syntomic" ], "ref_ids": [ 1195 ] } ], "ref_ids": [] }, { "id": 5330, "type": "theorem", "label": "morphisms-lemma-smooth-locally-finite-presentation", "categories": [ "morphisms" ], "title": "morphisms-lemma-smooth-locally-finite-presentation", "contents": [ "A smooth morphism is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "True because a smooth ring map is of finite presentation by", "definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5331, "type": "theorem", "label": "morphisms-lemma-smooth-flat", "categories": [ "morphisms" ], "title": "morphisms-lemma-smooth-flat", "contents": [ "A smooth morphism is flat." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-syntomic-flat} and \\ref{lemma-smooth-syntomic}." ], "refs": [ "morphisms-lemma-syntomic-flat", "morphisms-lemma-smooth-syntomic" ], "ref_ids": [ 5294, 5329 ] } ], "ref_ids": [] }, { "id": 5332, "type": "theorem", "label": "morphisms-lemma-smooth-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-smooth-open", "contents": [ "A smooth morphism is universally open." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemmas \\ref{lemma-smooth-flat},", "\\ref{lemma-smooth-locally-finite-presentation}, and", "\\ref{lemma-fppf-open}.", "Or alternatively, combine", "Lemmas \\ref{lemma-smooth-syntomic},", "\\ref{lemma-syntomic-open}." ], "refs": [ "morphisms-lemma-smooth-flat", "morphisms-lemma-smooth-locally-finite-presentation", "morphisms-lemma-fppf-open", "morphisms-lemma-smooth-syntomic", "morphisms-lemma-syntomic-open" ], "ref_ids": [ 5331, 5330, 5267, 5329, 5295 ] } ], "ref_ids": [] }, { "id": 5333, "type": "theorem", "label": "morphisms-lemma-smooth-locally-standard-smooth", "categories": [ "morphisms" ], "title": "morphisms-lemma-smooth-locally-standard-smooth", "contents": [ "\\begin{slogan}", "Smooth morphisms are locally standard smooth.", "\\end{slogan}", "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point.", "Let $V \\subset S$ be an affine open neighbourhood of $f(x)$.", "The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is smooth at $x$.", "\\item There exist an affine open $U \\subset X$,", "with $x \\in U$ and $f(U) \\subset V$ such that the", "induced morphism $f|_U : U \\to V$ is standard smooth.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows from the definitions and", "Algebra, Lemmas \\ref{algebra-lemma-standard-smooth}", "and \\ref{algebra-lemma-smooth-syntomic}." ], "refs": [ "algebra-lemma-standard-smooth", "algebra-lemma-smooth-syntomic" ], "ref_ids": [ 1193, 1195 ] } ], "ref_ids": [] }, { "id": 5334, "type": "theorem", "label": "morphisms-lemma-smooth-omega-finite-locally-free", "categories": [ "morphisms" ], "title": "morphisms-lemma-smooth-omega-finite-locally-free", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume $f$ is smooth.", "Then the module of differentials $\\Omega_{X/S}$ of $X$ over $S$", "is finite locally free and", "$$", "\\text{rank}_x(\\Omega_{X/S}) = \\dim_x(X_{f(x)})", "$$", "for every $x \\in X$." ], "refs": [], "proofs": [ { "contents": [ "The statement is local on $X$ and $S$.", "By Lemma \\ref{lemma-smooth-locally-standard-smooth}", "above we may assume that $f$ is a standard smooth morphism of affines.", "In this case the result follows from", "Algebra, Lemma \\ref{algebra-lemma-standard-smooth}", "(and the definition of a relative global complete intersection, see", "Algebra,", "Definition \\ref{algebra-definition-relative-global-complete-intersection})." ], "refs": [ "morphisms-lemma-smooth-locally-standard-smooth", "algebra-lemma-standard-smooth", "algebra-definition-relative-global-complete-intersection" ], "ref_ids": [ 5333, 1193, 1533 ] } ], "ref_ids": [] }, { "id": 5335, "type": "theorem", "label": "morphisms-lemma-smooth-at-point", "categories": [ "morphisms" ], "title": "morphisms-lemma-smooth-at-point", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$.", "Set $s = f(x)$.", "Assume $f$ is locally of finite presentation.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is smooth at $x$.", "\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$", "is flat and $X_s \\to \\Spec(\\kappa(s))$ is smooth at $x$.", "\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$", "is flat and the $\\mathcal{O}_{X, x}$-module $\\Omega_{X/S, x}$", "can be generated by at most $\\dim_x(X_{f(x)})$ elements.", "\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$", "is flat and the $\\kappa(x)$-vector space", "$$", "\\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) =", "\\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)", "$$", "can be generated by at most $\\dim_x(X_{f(x)})$ elements.", "\\item There exist affine opens $U \\subset X$,", "and $V \\subset S$ such that $x \\in U$, $f(U) \\subset V$ and the", "induced morphism $f|_U : U \\to V$ is standard smooth.", "\\item There exist affine opens $\\Spec(A) = U \\subset X$", "and $\\Spec(R) = V \\subset S$ with $x \\in U$ corresponding", "to $\\mathfrak q \\subset A$, and $f(U) \\subset V$", "such that there exists a presentation", "$$", "A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)", "$$", "with", "$$", "g =", "\\det", "\\left(", "\\begin{matrix}", "\\partial f_1/\\partial x_1 &", "\\partial f_2/\\partial x_1 &", "\\ldots &", "\\partial f_c/\\partial x_1 \\\\", "\\partial f_1/\\partial x_2 &", "\\partial f_2/\\partial x_2 &", "\\ldots &", "\\partial f_c/\\partial x_2 \\\\", "\\ldots & \\ldots & \\ldots & \\ldots \\\\", "\\partial f_1/\\partial x_c &", "\\partial f_2/\\partial x_c &", "\\ldots &", "\\partial f_c/\\partial x_c", "\\end{matrix}", "\\right)", "$$", "mapping to an element of $A$ not in $\\mathfrak q$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that if $f$ is smooth at $x$, then we see from Lemma", "\\ref{lemma-smooth-locally-standard-smooth} that (5) holds, and (6) is a slightly", "weakened version of (5). Moreover, $f$ smooth implies that the ring", "map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$ is flat (see", "Lemma \\ref{lemma-smooth-flat}) and that $\\Omega_{X/S}$ is", "finite locally free of rank equal to", "$\\dim_x(X_s)$ (see Lemma \\ref{lemma-smooth-omega-finite-locally-free}).", "Thus (1) implies (3) and (4). By Lemma \\ref{lemma-base-change-smooth}", "we also see that (1) implies (2).", "\\medskip\\noindent", "By Lemma \\ref{lemma-base-change-differentials}", "the module of differentials $\\Omega_{X_s/s}$ of the fibre $X_s$", "over $\\kappa(s)$ is the pullback of the module of differentials", "$\\Omega_{X/S}$ of $X$ over $S$. Hence the displayed equality in", "part (4) of the lemma. By Lemma \\ref{lemma-finite-type-differentials}", "these modules are of finite type. Hence the minimal number of", "generators of the modules", "$\\Omega_{X/S, x}$ and $\\Omega_{X_s/s, x}$ is the same and equal to the", "dimension of this $\\kappa(x)$-vector space by Nakayama's Lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK}). This in particular shows that", "(3) and (4) are equivalent.", "\\medskip\\noindent", "Algebra, Lemma \\ref{algebra-lemma-flat-fibre-smooth} shows that", "(2) implies (1).", "Algebra, Lemma \\ref{algebra-lemma-characterize-smooth-over-field}", "shows that (3) and (4) imply (2). Finally, (6) implies (5)", "see for example Algebra, Example \\ref{algebra-example-make-standard-smooth}", "and (5) implies (1) by Algebra, Lemma \\ref{algebra-lemma-standard-smooth}." ], "refs": [ "morphisms-lemma-smooth-locally-standard-smooth", "morphisms-lemma-smooth-flat", "morphisms-lemma-smooth-omega-finite-locally-free", "morphisms-lemma-base-change-smooth", "morphisms-lemma-base-change-differentials", "morphisms-lemma-finite-type-differentials", "algebra-lemma-NAK", "algebra-lemma-flat-fibre-smooth", "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-standard-smooth" ], "ref_ids": [ 5333, 5331, 5334, 5327, 5314, 5316, 401, 1200, 1223, 1193 ] } ], "ref_ids": [] }, { "id": 5336, "type": "theorem", "label": "morphisms-lemma-set-points-where-fibres-smooth", "categories": [ "morphisms" ], "title": "morphisms-lemma-set-points-where-fibres-smooth", "contents": [ "Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "be a cartesian diagram of schemes. Let $W \\subset X$, resp.\\ $W' \\subset X'$", "be the open subscheme of points where $f$, resp.\\ $f'$ is smooth.", "Then $W' = (g')^{-1}(W)$ if", "\\begin{enumerate}", "\\item $f$ is flat and locally of finite presentation, or", "\\item $f$ is locally of finite presentation and $g$ is flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume first that $f$ locally of finite type. Consider the set", "$$", "T = \\{x \\in X \\mid X_{f(x)}\\text{ is smooth over }\\kappa(f(x))\\text{ at }x\\}", "$$", "and the corresponding set $T' \\subset X'$ for $f'$. Then we claim", "$T' = (g')^{-1}(T)$. Namely, let $s' \\in S'$ be a point, and let", "$s = g(s')$. Then we have", "$$", "X'_{s'} =", "\\Spec(\\kappa(s')) \\times_{\\Spec(\\kappa(s))} X_s", "$$", "In other words the fibres of the base change are the base changes", "of the fibres. Hence the claim is equivalent to", "Algebra, Lemma \\ref{algebra-lemma-smooth-field-change-local}.", "\\medskip\\noindent", "Thus case (1) follows because in case (1) $T$ is the (open) set of points", "where $f$ is smooth by Lemma \\ref{lemma-smooth-at-point}.", "\\medskip\\noindent", "In case (2) let $x' \\in W'$. Then $g'$ is flat at $x'$", "(Lemma \\ref{lemma-base-change-module-flat}) and", "$g \\circ f$ is flat at $x'$ (Lemma \\ref{lemma-composition-module-flat}).", "It follows that $f$ is flat at $x = g'(x')$", "by Lemma \\ref{lemma-flat-permanence}. On the other hand, since", "$x' \\in T'$ (Lemma \\ref{lemma-base-change-smooth})", "we see that $x \\in T$. Hence $f$ is smooth at $x$ by", "Lemma \\ref{lemma-smooth-at-point}." ], "refs": [ "algebra-lemma-smooth-field-change-local", "morphisms-lemma-smooth-at-point", "morphisms-lemma-base-change-module-flat", "morphisms-lemma-composition-module-flat", "morphisms-lemma-flat-permanence", "morphisms-lemma-base-change-smooth", "morphisms-lemma-smooth-at-point" ], "ref_ids": [ 1202, 5335, 5264, 5262, 5270, 5327, 5335 ] } ], "ref_ids": [] }, { "id": 5337, "type": "theorem", "label": "morphisms-lemma-triangle-differentials-smooth", "categories": [ "morphisms" ], "title": "morphisms-lemma-triangle-differentials-smooth", "contents": [ "Let $f : X \\to Y$, $g : Y \\to S$ be morphisms of schemes.", "Assume $f$ is smooth. Then", "$$", "0 \\to f^*\\Omega_{Y/S} \\to \\Omega_{X/S} \\to \\Omega_{X/Y} \\to 0", "$$", "(see Lemma \\ref{lemma-triangle-differentials}) is short exact." ], "refs": [ "morphisms-lemma-triangle-differentials" ], "proofs": [ { "contents": [ "The algebraic version of this lemma is the following:", "Given ring maps $A \\to B \\to C$ with $B \\to C$ smooth, then the sequence", "$$", "0 \\to C \\otimes_B \\Omega_{B/A} \\to \\Omega_{C/A} \\to \\Omega_{C/B} \\to 0", "$$", "of", "Algebra, Lemma \\ref{algebra-lemma-exact-sequence-differentials}", "is exact. This is", "Algebra, Lemma \\ref{algebra-lemma-triangle-differentials-smooth}." ], "refs": [ "algebra-lemma-exact-sequence-differentials", "algebra-lemma-triangle-differentials-smooth" ], "ref_ids": [ 1133, 1217 ] } ], "ref_ids": [ 5313 ] }, { "id": 5338, "type": "theorem", "label": "morphisms-lemma-differentials-relative-immersion-smooth", "categories": [ "morphisms" ], "title": "morphisms-lemma-differentials-relative-immersion-smooth", "contents": [ "Let $i : Z \\to X$ be an immersion of schemes over $S$.", "Assume that $Z$ is smooth over $S$. Then the", "canonical exact sequence", "$$", "0 \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0", "$$", "of", "Lemma \\ref{lemma-differentials-relative-immersion}", "is short exact." ], "refs": [ "morphisms-lemma-differentials-relative-immersion" ], "proofs": [ { "contents": [ "The algebraic version of this lemma is the following:", "Given ring maps $A \\to B \\to C$ with $A \\to C$ smooth and $B \\to C$", "surjective with kernel $J$, then the sequence", "$$", "0 \\to J/J^2 \\to C \\otimes_B \\Omega_{B/A} \\to \\Omega_{C/A} \\to 0", "$$", "of", "Algebra, Lemma \\ref{algebra-lemma-differential-seq}", "is exact. This is", "Algebra, Lemma \\ref{algebra-lemma-differential-seq-smooth}." ], "refs": [ "algebra-lemma-differential-seq", "algebra-lemma-differential-seq-smooth" ], "ref_ids": [ 1135, 1218 ] } ], "ref_ids": [ 5319 ] }, { "id": 5339, "type": "theorem", "label": "morphisms-lemma-two-immersions-smooth", "categories": [ "morphisms" ], "title": "morphisms-lemma-two-immersions-smooth", "contents": [ "Let", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[rd]_j & X \\ar[d] \\\\", "& Y", "}", "$$", "be a commutative diagram of schemes where $i$ and $j$ are immersions", "and $X \\to Y$ is smooth.", "Then the canonical exact sequence", "$$", "0 \\to \\mathcal{C}_{Z/Y} \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/Y} \\to 0", "$$", "of", "Lemma \\ref{lemma-two-immersions}", "is exact." ], "refs": [ "morphisms-lemma-two-immersions" ], "proofs": [ { "contents": [ "The algebraic version of this lemma is the following:", "Given ring maps $A \\to B \\to C$ with $A \\to C$ surjective and $A \\to B$", "smooth, then the sequence", "$$", "0 \\to I/I^2 \\to J/J^2 \\to C \\otimes_B \\Omega_{B/A} \\to 0", "$$", "of", "Algebra, Lemma \\ref{algebra-lemma-application-NL}", "is exact. This is", "Algebra, Lemma \\ref{algebra-lemma-application-NL-smooth}." ], "refs": [ "algebra-lemma-application-NL", "algebra-lemma-application-NL-smooth" ], "ref_ids": [ 1155, 1219 ] } ], "ref_ids": [ 5321 ] }, { "id": 5340, "type": "theorem", "label": "morphisms-lemma-smooth-permanence", "categories": [ "morphisms" ], "title": "morphisms-lemma-smooth-permanence", "contents": [ "Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & &", "Y \\ar[dl]^q \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes. Assume that", "\\begin{enumerate}", "\\item $f$ is surjective, and smooth,", "\\item $p$ is smooth, and", "\\item $q$ is locally of finite presentation\\footnote{In fact this", "is implied by (1) and (2), see", "Descent, Lemma \\ref{descent-lemma-flat-finitely-presented-permanence}.", "Moreover, it suffices to assume $f$ is surjective, flat and locally", "of finite presentation, see", "Descent, Lemma \\ref{descent-lemma-smooth-permanence}.}.", "\\end{enumerate}", "Then $q$ is smooth." ], "refs": [ "descent-lemma-flat-finitely-presented-permanence", "descent-lemma-smooth-permanence" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-flat-permanence} we see that $q$ is flat.", "Pick a point $y \\in Y$. Pick a point $x \\in X$ mapping to $y$.", "Suppose $f$ has relative dimension $a$ at $x$ and $p$ has relative", "dimension $b$ at $x$. By Lemma \\ref{lemma-smooth-omega-finite-locally-free}", "this means that $\\Omega_{X/S, x}$ is free of rank $b$ and $\\Omega_{X/Y, x}$", "is free of rank $a$. By the short exact sequence", "of Lemma \\ref{lemma-triangle-differentials-smooth}", "this means that $(f^*\\Omega_{Y/S})_x$ is free", "of rank $b - a$. By Nakayama's Lemma this implies that", "$\\Omega_{Y/S, y}$ can be generated by $b - a$ elements.", "Also, by Lemma \\ref{lemma-dimension-fibre-at-a-point-additive} we see that", "$\\dim_y(Y_s) = b - a$. Hence we conclude that", "$Y \\to S$ is smooth at $y$ by Lemma \\ref{lemma-smooth-at-point} part (2)." ], "refs": [ "morphisms-lemma-flat-permanence", "morphisms-lemma-smooth-omega-finite-locally-free", "morphisms-lemma-triangle-differentials-smooth", "morphisms-lemma-dimension-fibre-at-a-point-additive", "morphisms-lemma-smooth-at-point" ], "ref_ids": [ 5270, 5334, 5337, 5278, 5335 ] } ], "ref_ids": [ 14642, 14644 ] }, { "id": 5341, "type": "theorem", "label": "morphisms-lemma-section-smooth-morphism", "categories": [ "morphisms" ], "title": "morphisms-lemma-section-smooth-morphism", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\sigma : S \\to X$ be a section of $f$.", "Let $s \\in S$ be a point such that $f$ is smooth at $x = \\sigma(s)$.", "Then there exist affine open neighbourhoods", "$\\Spec(A) = U \\subset S$ of $s$ and $\\Spec(B) = V \\subset X$", "of $x$ such that", "\\begin{enumerate}", "\\item $f(V) \\subset U$ and $\\sigma(U) \\subset V$,", "\\item with $I = \\Ker(\\sigma^\\# : B \\to A)$ the module $I/I^2$", "is a free $A$-module, and", "\\item $B^\\wedge \\cong A[[x_1, \\ldots, x_d]]$ as $A$-algebras where", "$B^\\wedge$ denotes the completion of $B$ with respect to $I$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Pick an affine open $U \\subset S$ containing $s$", "Pick an affine open $V \\subset f^{-1}(U)$ containing $x$.", "Pick an affine open $U' \\subset \\sigma^{-1}(V)$ containing $s$.", "Note that $V' = f^{-1}(U') \\cap V$ is affine as it is equal to the", "fibre product $V' = U' \\times_U V$. Then $U'$ and $V'$ satisfy (1).", "Write $U' = \\Spec(A')$ and $V' = \\Spec(B')$. By", "Algebra, Lemma \\ref{algebra-lemma-section-smooth}", "the module $I'/(I')^2$ is finite locally free as a $A'$-module.", "Hence after replacing $U'$ by a smaller affine open $U'' \\subset U'$", "and $V'$ by $V'' = V' \\cap f^{-1}(U'')$ we obtain the situation where", "$I''/(I'')^2$ is free, i.e., (2) holds. In this case (3) holds also by", "Algebra, Lemma \\ref{algebra-lemma-section-smooth}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5342, "type": "theorem", "label": "morphisms-lemma-smoothness-dimension", "categories": [ "morphisms" ], "title": "morphisms-lemma-smoothness-dimension", "contents": [ "Let $f : X \\to Y$ be a smooth morphism of locally Noetherian schemes.", "For every point $x$ in $X$ with image $y$ in $Y$,", "$$", "\\dim_x(X) = \\dim_y(Y) + \\dim_x(X_y),", "$$", "where $X_y$ denotes the fiber over $y$." ], "refs": [], "proofs": [ { "contents": [ "After replacing $X$ by an open neighborhood of $x$,", "there is a natural number $d$ such that all fibers", "of $X \\to Y$ have dimension $d$ at every point, see", "Lemma \\ref{lemma-smooth-omega-finite-locally-free}.", "Then $f$ is flat (Lemma \\ref{lemma-smooth-flat}),", "locally of finite type (Lemma \\ref{lemma-smooth-locally-finite-presentation}),", "and of relative dimension $d$. Hence the result follows from", "Lemma \\ref{lemma-rel-dimension-dimension}." ], "refs": [ "morphisms-lemma-smooth-omega-finite-locally-free", "morphisms-lemma-smooth-flat", "morphisms-lemma-smooth-locally-finite-presentation", "morphisms-lemma-rel-dimension-dimension" ], "ref_ids": [ 5334, 5331, 5330, 5288 ] } ], "ref_ids": [] }, { "id": 5343, "type": "theorem", "label": "morphisms-lemma-unramified-omega-zero", "categories": [ "morphisms" ], "title": "morphisms-lemma-unramified-omega-zero", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Then", "\\begin{enumerate}", "\\item $f$ is unramified if and only if $f$ is locally of finite type", "and $\\Omega_{X/S} = 0$, and", "\\item $f$ is G-unramified if and only if $f$ is locally of finite presentation", "and $\\Omega_{X/S} = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By definition a ring map $R \\to A$ is unramified (resp.\\ G-unramified)", "if and only if it is of finite type (resp.\\ finite presentation)", "and $\\Omega_{A/R} = 0$. Hence the lemma follows", "directly from the definitions and Lemma \\ref{lemma-differentials-affine}." ], "refs": [ "morphisms-lemma-differentials-affine" ], "ref_ids": [ 5310 ] } ], "ref_ids": [] }, { "id": 5344, "type": "theorem", "label": "morphisms-lemma-unramified-characterize", "categories": [ "morphisms" ], "title": "morphisms-lemma-unramified-characterize", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is unramified (resp.\\ G-unramified).", "\\item For every affine open $U \\subset X$, $V \\subset S$", "with $f(U) \\subset V$ the ring map", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is unramified (resp.\\ G-unramified).", "\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$", "and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$", "is unramified (resp.\\ G-unramified).", "\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$", "and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is", "unramified (resp.\\ G-unramified), for all $j\\in J, i\\in I_j$.", "\\end{enumerate}", "Moreover, if $f$ is unramified (resp.\\ G-unramified) then for", "any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$", "the restriction $f|_U : U \\to V$ is unramified (resp.\\ G-unramified)." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-locally-P} if we show that", "the property ``$R \\to A$ is unramified'' is local.", "We check conditions (a), (b) and (c) of Definition", "\\ref{definition-property-local}.", "These properties are proved in", "Algebra, Lemma \\ref{algebra-lemma-unramified}." ], "refs": [ "morphisms-lemma-locally-P", "morphisms-definition-property-local", "algebra-lemma-unramified" ], "ref_ids": [ 5191, 5547, 1266 ] } ], "ref_ids": [] }, { "id": 5345, "type": "theorem", "label": "morphisms-lemma-composition-unramified", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-unramified", "contents": [ "The composition of two morphisms which are unramified is unramified.", "The same holds for G-unramified morphisms." ], "refs": [], "proofs": [ { "contents": [ "The proof of Lemma \\ref{lemma-unramified-characterize}", "shows that being unramified (resp.\\ G-unramified)", "is a local property of ring maps.", "Hence the first statement of the lemma follows from", "Lemma \\ref{lemma-composition-type-P} combined", "with the fact that being unramified (resp.\\ G-unramified)", "is a property of ring maps that is stable under composition, see", "Algebra, Lemma \\ref{algebra-lemma-unramified}." ], "refs": [ "morphisms-lemma-unramified-characterize", "morphisms-lemma-composition-type-P", "algebra-lemma-unramified" ], "ref_ids": [ 5344, 5193, 1266 ] } ], "ref_ids": [] }, { "id": 5346, "type": "theorem", "label": "morphisms-lemma-base-change-unramified", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-unramified", "contents": [ "The base change of a morphism which is unramified is unramified.", "The same holds for G-unramified morphisms." ], "refs": [], "proofs": [ { "contents": [ "The proof of Lemma \\ref{lemma-unramified-characterize}", "shows that being unramified (resp.\\ G-unramified)", "is a local property of ring maps. Hence the lemma follows from", "Lemma \\ref{lemma-base-change-type-P} combined", "with the fact that being unramified (resp.\\ G-unramified)", "is a property of ring maps that is stable under base change, see", "Algebra, Lemma \\ref{algebra-lemma-unramified}." ], "refs": [ "morphisms-lemma-unramified-characterize", "morphisms-lemma-base-change-type-P", "algebra-lemma-unramified" ], "ref_ids": [ 5344, 5194, 1266 ] } ], "ref_ids": [] }, { "id": 5347, "type": "theorem", "label": "morphisms-lemma-noetherian-unramified", "categories": [ "morphisms" ], "title": "morphisms-lemma-noetherian-unramified", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Assume $S$ is locally Noetherian.", "Then $f$ is unramified if and only if $f$ is G-unramified." ], "refs": [], "proofs": [ { "contents": [ "Follows from the definitions and", "Lemma \\ref{lemma-noetherian-finite-type-finite-presentation}." ], "refs": [ "morphisms-lemma-noetherian-finite-type-finite-presentation" ], "ref_ids": [ 5245 ] } ], "ref_ids": [] }, { "id": 5348, "type": "theorem", "label": "morphisms-lemma-open-immersion-unramified", "categories": [ "morphisms" ], "title": "morphisms-lemma-open-immersion-unramified", "contents": [ "Any open immersion is G-unramified." ], "refs": [], "proofs": [ { "contents": [ "This is true because an open immersion is a local isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5349, "type": "theorem", "label": "morphisms-lemma-closed-immersion-unramified", "categories": [ "morphisms" ], "title": "morphisms-lemma-closed-immersion-unramified", "contents": [ "A closed immersion $i : Z \\to X$ is unramified.", "It is G-unramified if and only if the associated quasi-coherent sheaf of", "ideals $\\mathcal{I} = \\Ker(\\mathcal{O}_X \\to i_*\\mathcal{O}_Z)$", "is of finite type (as an $\\mathcal{O}_X$-module)." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-closed-immersion-finite-presentation} and", "Algebra, Lemma \\ref{algebra-lemma-unramified}." ], "refs": [ "morphisms-lemma-closed-immersion-finite-presentation", "algebra-lemma-unramified" ], "ref_ids": [ 5243, 1266 ] } ], "ref_ids": [] }, { "id": 5350, "type": "theorem", "label": "morphisms-lemma-unramified-locally-finite-type", "categories": [ "morphisms" ], "title": "morphisms-lemma-unramified-locally-finite-type", "contents": [ "An unramified morphism is locally of finite type.", "A G-unramified morphism is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "An unramified ring map is of finite type by definition.", "A G-unramified ring map is of finite presentation by definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5351, "type": "theorem", "label": "morphisms-lemma-unramified-quasi-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-unramified-quasi-finite", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "If $f$ is unramified at $x$ then $f$ is quasi-finite at $x$.", "In particular, an unramified morphism is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-unramified-quasi-finite}." ], "refs": [ "algebra-lemma-unramified-quasi-finite" ], "ref_ids": [ 1269 ] } ], "ref_ids": [] }, { "id": 5352, "type": "theorem", "label": "morphisms-lemma-unramified-over-field", "categories": [ "morphisms" ], "title": "morphisms-lemma-unramified-over-field", "contents": [ "Fibres of unramified morphisms.", "\\begin{enumerate}", "\\item Let $X$ be a scheme over a field $k$.", "The structure morphism $X \\to \\Spec(k)$ is unramified if", "and only if $X$ is a disjoint union of spectra of finite separable", "field extensions of $k$.", "\\item If $f : X \\to S$ is an unramified morphism then for every $s \\in S$", "the fibre $X_s$ is a disjoint union of spectra of finite separable field", "extensions of $\\kappa(s)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) follows from part (1) and", "Lemma \\ref{lemma-base-change-unramified}.", "Let us prove part (1).", "We first use Algebra, Lemma \\ref{algebra-lemma-characterize-unramified}.", "This lemma implies that if $X$ is a disjoint union of spectra of finite", "separable field extensions of $k$ then $X \\to \\Spec(k)$ is unramified.", "Conversely, suppose that $X \\to \\Spec(k)$ is unramified.", "By Algebra, Lemma \\ref{algebra-lemma-unramified-at-prime} for every $x \\in X$", "the residue field extension $k \\subset \\kappa(x)$ is", "finite separable. Since $X \\to \\Spec(k)$ is locally", "quasi-finite (Lemma \\ref{lemma-unramified-quasi-finite})", "we see that all points of $X$ are isolated closed points, see", "Lemma \\ref{lemma-quasi-finite-at-point-characterize}.", "Thus $X$ is a discrete space, in particular the disjoint union", "of the spectra of its local rings. By", "Algebra, Lemma \\ref{algebra-lemma-unramified-at-prime} again these", "local rings are fields, and we win." ], "refs": [ "morphisms-lemma-base-change-unramified", "algebra-lemma-characterize-unramified", "algebra-lemma-unramified-at-prime", "morphisms-lemma-unramified-quasi-finite", "morphisms-lemma-quasi-finite-at-point-characterize", "algebra-lemma-unramified-at-prime" ], "ref_ids": [ 5346, 1270, 1268, 5351, 5226, 1268 ] } ], "ref_ids": [] }, { "id": 5353, "type": "theorem", "label": "morphisms-lemma-unramified-etale-fibres", "categories": [ "morphisms" ], "title": "morphisms-lemma-unramified-etale-fibres", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item If $f$ is unramified then for any $x \\in X$ the field extension", "$\\kappa(f(x)) \\subset \\kappa(x)$ is finite separable.", "\\item If $f$ is locally of finite type, and for every", "$s \\in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable", "field extensions of $\\kappa(s)$ then $f$ is unramified.", "\\item If $f$ is locally of finite presentation, and for every", "$s \\in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable", "field extensions of $\\kappa(s)$ then $f$ is G-unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows from Algebra, Lemmas", "\\ref{algebra-lemma-unramified-at-prime} and", "\\ref{algebra-lemma-characterize-unramified}." ], "refs": [ "algebra-lemma-unramified-at-prime", "algebra-lemma-characterize-unramified" ], "ref_ids": [ 1268, 1270 ] } ], "ref_ids": [] }, { "id": 5354, "type": "theorem", "label": "morphisms-lemma-diagonal-unramified-morphism", "categories": [ "morphisms" ], "title": "morphisms-lemma-diagonal-unramified-morphism", "contents": [ "Let $f : X \\to S$ be a morphism.", "\\begin{enumerate}", "\\item If $f$ is unramified, then the diagonal morphism", "$\\Delta : X \\to X \\times_S X$ is an open immersion.", "\\item If $f$ is locally of finite type", "and $\\Delta$ is an open immersion, then $f$ is unramified.", "\\item If $f$ is locally of finite presentation and $\\Delta$ is an open", "immersion, then $f$ is G-unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from", "Algebra, Lemma \\ref{algebra-lemma-diagonal-unramified}.", "The second statement from the fact that $\\Omega_{X/S}$", "is the conormal sheaf of the diagonal morphism", "(Lemma \\ref{lemma-differentials-diagonal})", "and hence clearly zero if $\\Delta$ is an open immersion." ], "refs": [ "algebra-lemma-diagonal-unramified", "morphisms-lemma-differentials-diagonal" ], "ref_ids": [ 1267, 5311 ] } ], "ref_ids": [] }, { "id": 5355, "type": "theorem", "label": "morphisms-lemma-unramified-at-point", "categories": [ "morphisms" ], "title": "morphisms-lemma-unramified-at-point", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$.", "Set $s = f(x)$.", "Assume $f$ is locally of finite type (resp.\\ locally of finite presentation).", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is unramified (resp.\\ G-unramified) at $x$.", "\\item The fibre $X_s$ is unramified over $\\kappa(s)$ at $x$.", "\\item The $\\mathcal{O}_{X, x}$-module $\\Omega_{X/S, x}$ is zero.", "\\item The $\\mathcal{O}_{X_s, x}$-module $\\Omega_{X_s/s, x}$ is zero.", "\\item The $\\kappa(x)$-vector space", "$$", "\\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) =", "\\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)", "$$", "is zero.", "\\item We have $\\mathfrak m_s\\mathcal{O}_{X, x} = \\mathfrak m_x$", "and the field extension $\\kappa(s) \\subset \\kappa(x)$ is finite", "separable.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that if $f$ is unramified at $x$, then", "we see that $\\Omega_{X/S} = 0$ in a neighbourhood of $x$", "by the definitions and the results on modules of differentials", "in Section \\ref{section-sheaf-differentials}. Hence (1) implies", "(3) and the vanishing of the right hand vector space in (5).", "It also implies (2) because by", "Lemma \\ref{lemma-base-change-differentials}", "the module of differentials $\\Omega_{X_s/s}$ of the fibre $X_s$", "over $\\kappa(s)$ is the pullback of the module of differentials", "$\\Omega_{X/S}$ of $X$ over $S$. This fact on modules of differentials", "also implies the displayed equality of vector spaces in part (4). By", "Lemma \\ref{lemma-finite-type-differentials}", "the modules $\\Omega_{X/S, x}$ and $\\Omega_{X_s/s, x}$ are of finite type.", "Hence the modules $\\Omega_{X/S, x}$ and $\\Omega_{X_s/s, x}$ are zero if and only", "if the corresponding $\\kappa(x)$-vector space in (4) is zero by", "Nakayama's Lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK}).", "This in particular shows that (3), (4) and (5) are equivalent.", "The support of $\\Omega_{X/S}$ is closed in $X$, see", "Modules, Lemma \\ref{modules-lemma-support-finite-type-closed}.", "Assumption (3) implies that $x$ is not in the support.", "Hence $\\Omega_{X/S}$ is zero in a neighbourhood of $x$, which", "implies (1). The equivalence of (1) and (3) applied to $X_s \\to s$", "implies the equivalence of (2) and (4).", "At this point we have seen that (1) -- (5) are equivalent.", "\\medskip\\noindent", "Alternatively you can use Algebra, Lemma \\ref{algebra-lemma-unramified}", "to see the equivalence of (1) -- (5) more directly.", "\\medskip\\noindent", "The equivalence of (1) and (6) follows from Lemma", "\\ref{lemma-unramified-etale-fibres}.", "It also follows more directly from", "Algebra, Lemmas \\ref{algebra-lemma-unramified-at-prime} and", "\\ref{algebra-lemma-characterize-unramified}." ], "refs": [ "morphisms-lemma-base-change-differentials", "morphisms-lemma-finite-type-differentials", "algebra-lemma-NAK", "modules-lemma-support-finite-type-closed", "algebra-lemma-unramified", "morphisms-lemma-unramified-etale-fibres", "algebra-lemma-unramified-at-prime", "algebra-lemma-characterize-unramified" ], "ref_ids": [ 5314, 5316, 401, 13240, 1266, 5353, 1268, 1270 ] } ], "ref_ids": [] }, { "id": 5356, "type": "theorem", "label": "morphisms-lemma-set-points-where-fibres-unramified", "categories": [ "morphisms" ], "title": "morphisms-lemma-set-points-where-fibres-unramified", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume $f$ locally of finite type. Formation of the open set", "\\begin{align*}", "T", "& =", "\\{x \\in X \\mid X_{f(x)}\\text{ is unramified over }\\kappa(f(x))\\text{ at }x\\} \\\\", "& =", "\\{x \\in X \\mid X\\text{ is unramified over }S\\text{ at }x\\}", "\\end{align*}", "commutes with arbitrary base change:", "For any morphism $g : S' \\to S$, consider", "the base change $f' : X' \\to S'$ of $f$ and the", "projection $g' : X' \\to X$. Then the corresponding", "set $T'$ for the morphism $f'$ is equal to $T' = (g')^{-1}(T)$.", "If $f$ is assumed locally of finite presentation then the same holds", "for the open set of points where $f$ is G-unramified." ], "refs": [], "proofs": [ { "contents": [ "Let $s' \\in S'$ be a point, and let $s = g(s')$. Then we have", "$$", "X'_{s'} =", "\\Spec(\\kappa(s')) \\times_{\\Spec(\\kappa(s))} X_s", "$$", "In other words the fibres of the base change are the base changes", "of the fibres. In particular", "$$", "\\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x')", "=", "\\Omega_{X'_{s'}/s', x'} \\otimes_{\\mathcal{O}_{X'_{s'}, x'}} \\kappa(x')", "$$", "see", "Lemma \\ref{lemma-base-change-differentials}.", "Whence $x' \\in T'$ if and only if $x \\in T$ by", "Lemma \\ref{lemma-unramified-at-point}.", "The second part follows from the first because in that case", "$T$ is the (open) set of points where $f$ is G-unramified according to", "Lemma \\ref{lemma-unramified-at-point}." ], "refs": [ "morphisms-lemma-base-change-differentials", "morphisms-lemma-unramified-at-point", "morphisms-lemma-unramified-at-point" ], "ref_ids": [ 5314, 5355, 5355 ] } ], "ref_ids": [] }, { "id": 5357, "type": "theorem", "label": "morphisms-lemma-unramified-permanence", "categories": [ "morphisms" ], "title": "morphisms-lemma-unramified-permanence", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes over $S$.", "\\begin{enumerate}", "\\item If $X$ is unramified over $S$, then $f$ is unramified.", "\\item If $X$ is G-unramified over $S$ and $Y$ of finite type over $S$, then", "$f$ is G-unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume that $X$ is unramified over $S$.", "By Lemma \\ref{lemma-permanence-finite-type} we see that $f$", "is locally of finite type.", "By assumption we have $\\Omega_{X/S} = 0$. Hence", "$\\Omega_{X/Y} = 0$ by Lemma \\ref{lemma-triangle-differentials}. Thus", "$f$ is unramified. If $X$ is G-unramified over $S$ and $Y$ of finite type", "over $S$, then by", "Lemma \\ref{lemma-finite-presentation-permanence}", "we see that $f$ is locally of finite presentation and we conclude", "that $f$ is G-unramified." ], "refs": [ "morphisms-lemma-permanence-finite-type", "morphisms-lemma-triangle-differentials", "morphisms-lemma-finite-presentation-permanence" ], "ref_ids": [ 5204, 5313, 5247 ] } ], "ref_ids": [] }, { "id": 5358, "type": "theorem", "label": "morphisms-lemma-value-at-one-point", "categories": [ "morphisms" ], "title": "morphisms-lemma-value-at-one-point", "contents": [ "Let $S$ be a scheme.", "Let $X$, $Y$ be schemes over $S$.", "Let $f, g : X \\to Y$ be morphisms over $S$. Let $x \\in X$.", "Assume that", "\\begin{enumerate}", "\\item the structure morphism $Y \\to S$ is unramified,", "\\item $f(x) = g(x)$ in $Y$, say $y = f(x) = g(x)$, and", "\\item the induced maps $f^\\sharp, g^\\sharp : \\kappa(y) \\to \\kappa(x)$", "are equal.", "\\end{enumerate}", "Then there exists an open neighbourhood of $x$ in $X$ on which", "$f$ and $g$ are equal." ], "refs": [], "proofs": [ { "contents": [ "Consider the morphism $(f, g) : X \\to Y \\times_S Y$. By assumption (1) and", "Lemma \\ref{lemma-diagonal-unramified-morphism}", "the inverse image of $\\Delta_{Y/S}(Y)$ is open in $X$.", "And assumptions (2) and (3) imply that $x$ is in this open subset." ], "refs": [ "morphisms-lemma-diagonal-unramified-morphism" ], "ref_ids": [ 5354 ] } ], "ref_ids": [] }, { "id": 5359, "type": "theorem", "label": "morphisms-lemma-etale-characterize", "categories": [ "morphisms" ], "title": "morphisms-lemma-etale-characterize", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is \\'etale.", "\\item For every affine opens $U \\subset X$, $V \\subset S$", "with $f(U) \\subset V$ the ring map", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is \\'etale.", "\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$", "and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$", "is \\'etale.", "\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$", "and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is", "\\'etale, for all $j\\in J, i\\in I_j$.", "\\end{enumerate}", "Moreover, if $f$ is \\'etale then for", "any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$", "the restriction $f|_U : U \\to V$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-locally-P} if we show that", "the property ``$R \\to A$ is \\'etale'' is local.", "We check conditions (a), (b) and (c) of Definition", "\\ref{definition-property-local}.", "These all follow from Algebra, Lemma \\ref{algebra-lemma-etale}." ], "refs": [ "morphisms-lemma-locally-P", "morphisms-definition-property-local", "algebra-lemma-etale" ], "ref_ids": [ 5191, 5547, 1231 ] } ], "ref_ids": [] }, { "id": 5360, "type": "theorem", "label": "morphisms-lemma-composition-etale", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-etale", "contents": [ "The composition of two morphisms which are \\'etale is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-etale-characterize}", "we saw that being \\'etale is a local property of ring maps.", "Hence the first statement of the lemma follows from", "Lemma \\ref{lemma-composition-type-P} combined", "with the fact that being \\'etale is a property of ring maps that is", "stable under composition, see", "Algebra, Lemma \\ref{algebra-lemma-etale}." ], "refs": [ "morphisms-lemma-etale-characterize", "morphisms-lemma-composition-type-P", "algebra-lemma-etale" ], "ref_ids": [ 5359, 5193, 1231 ] } ], "ref_ids": [] }, { "id": 5361, "type": "theorem", "label": "morphisms-lemma-base-change-etale", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-etale", "contents": [ "The base change of a morphism which is \\'etale is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-etale-characterize}", "we saw that being \\'etale is a local property of ring maps.", "Hence the lemma follows from", "Lemma \\ref{lemma-composition-type-P} combined", "with the fact that being \\'etale is a property of ring maps that is", "stable under base change, see", "Algebra, Lemma \\ref{algebra-lemma-etale}." ], "refs": [ "morphisms-lemma-etale-characterize", "morphisms-lemma-composition-type-P", "algebra-lemma-etale" ], "ref_ids": [ 5359, 5193, 1231 ] } ], "ref_ids": [] }, { "id": 5362, "type": "theorem", "label": "morphisms-lemma-etale-smooth-unramified", "categories": [ "morphisms" ], "title": "morphisms-lemma-etale-smooth-unramified", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$. Then $f$ is \\'etale at $x$ if and only if $f$ is", "smooth and unramified at $x$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5363, "type": "theorem", "label": "morphisms-lemma-etale-locally-quasi-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-etale-locally-quasi-finite", "contents": [ "An \\'etale morphism is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-etale-smooth-unramified}", "an \\'etale morphism is unramified. By", "Lemma \\ref{lemma-unramified-quasi-finite}", "an unramified morphism is locally quasi-finite." ], "refs": [ "morphisms-lemma-etale-smooth-unramified", "morphisms-lemma-unramified-quasi-finite" ], "ref_ids": [ 5362, 5351 ] } ], "ref_ids": [] }, { "id": 5364, "type": "theorem", "label": "morphisms-lemma-etale-over-field", "categories": [ "morphisms" ], "title": "morphisms-lemma-etale-over-field", "contents": [ "\\begin{slogan}", "Description of the \\'etale schemes over fields and fibres", "of \\'etale morphisms.", "\\end{slogan}", "Fibres of \\'etale morphisms.", "\\begin{enumerate}", "\\item Let $X$ be a scheme over a field $k$.", "The structure morphism $X \\to \\Spec(k)$ is \\'etale if", "and only if $X$ is a disjoint union of spectra of finite separable", "field extensions of $k$.", "\\item If $f : X \\to S$ is an \\'etale morphism, then for every $s \\in S$ the", "fibre $X_s$ is a disjoint union of spectra of finite separable field", "extensions of $\\kappa(s)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "You can deduce this from Lemma \\ref{lemma-unramified-over-field}", "via Lemma \\ref{lemma-etale-smooth-unramified} above.", "Here is a direct proof.", "\\medskip\\noindent", "We will use Algebra, Lemma \\ref{algebra-lemma-etale-over-field}.", "Hence it is clear that if $X$ is a disjoint union of spectra of finite", "separable field extensions of $k$ then $X \\to \\Spec(k)$ is \\'etale.", "Conversely, suppose that $X \\to \\Spec(k)$ is \\'etale. Then for any affine", "open $U \\subset X$ we see that $U$ is a finite disjoint union of spectra", "of finite separable field extensions of $k$. Hence all points of $X$", "are closed points (see", "Lemma \\ref{lemma-algebraic-residue-field-extension-closed-point-fibre}", "for example). Thus $X$ is a discrete space and we win." ], "refs": [ "morphisms-lemma-unramified-over-field", "morphisms-lemma-etale-smooth-unramified", "algebra-lemma-etale-over-field", "morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre" ], "ref_ids": [ 5352, 5362, 1232, 5222 ] } ], "ref_ids": [] }, { "id": 5365, "type": "theorem", "label": "morphisms-lemma-etale-flat-etale-fibres", "categories": [ "morphisms" ], "title": "morphisms-lemma-etale-flat-etale-fibres", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "If $f$ is flat, locally of finite presentation, and for every $s \\in S$", "the fibre $X_s$ is a disjoint union of spectra of finite separable", "field extensions of $\\kappa(s)$, then $f$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "You can deduce this from", "Algebra, Lemma \\ref{algebra-lemma-characterize-etale}.", "Here is another proof.", "\\medskip\\noindent", "By Lemma \\ref{lemma-etale-over-field} a fibre $X_s$ is \\'etale", "and hence smooth over $s$. By Lemma \\ref{lemma-smooth-flat-smooth-fibres}", "we see that $X \\to S$ is smooth.", "By Lemma \\ref{lemma-unramified-etale-fibres}", "we see that $f$ is unramified. We conclude by", "Lemma \\ref{lemma-etale-smooth-unramified}." ], "refs": [ "algebra-lemma-characterize-etale", "morphisms-lemma-etale-over-field", "morphisms-lemma-smooth-flat-smooth-fibres", "morphisms-lemma-unramified-etale-fibres", "morphisms-lemma-etale-smooth-unramified" ], "ref_ids": [ 1235, 5364, 5325, 5353, 5362 ] } ], "ref_ids": [] }, { "id": 5366, "type": "theorem", "label": "morphisms-lemma-open-immersion-etale", "categories": [ "morphisms" ], "title": "morphisms-lemma-open-immersion-etale", "contents": [ "Any open immersion is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "This is true because an open immersion is a local isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5367, "type": "theorem", "label": "morphisms-lemma-etale-syntomic", "categories": [ "morphisms" ], "title": "morphisms-lemma-etale-syntomic", "contents": [ "An \\'etale morphism is syntomic." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-smooth-syntomic} and use that an", "\\'etale morphism is the same as a smooth morphism of relative dimension $0$." ], "refs": [ "algebra-lemma-smooth-syntomic" ], "ref_ids": [ 1195 ] } ], "ref_ids": [] }, { "id": 5368, "type": "theorem", "label": "morphisms-lemma-etale-locally-finite-presentation", "categories": [ "morphisms" ], "title": "morphisms-lemma-etale-locally-finite-presentation", "contents": [ "An \\'etale morphism is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "True because an \\'etale ring map is of finite presentation by", "definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5369, "type": "theorem", "label": "morphisms-lemma-etale-flat", "categories": [ "morphisms" ], "title": "morphisms-lemma-etale-flat", "contents": [ "An \\'etale morphism is flat." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-syntomic-flat} and \\ref{lemma-etale-syntomic}." ], "refs": [ "morphisms-lemma-syntomic-flat", "morphisms-lemma-etale-syntomic" ], "ref_ids": [ 5294, 5367 ] } ], "ref_ids": [] }, { "id": 5370, "type": "theorem", "label": "morphisms-lemma-etale-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-etale-open", "contents": [ "An \\'etale morphism is open." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-etale-flat},", "\\ref{lemma-etale-locally-finite-presentation}, and", "\\ref{lemma-fppf-open}." ], "refs": [ "morphisms-lemma-etale-flat", "morphisms-lemma-etale-locally-finite-presentation", "morphisms-lemma-fppf-open" ], "ref_ids": [ 5369, 5368, 5267 ] } ], "ref_ids": [] }, { "id": 5371, "type": "theorem", "label": "morphisms-lemma-etale-locally-standard-etale", "categories": [ "morphisms" ], "title": "morphisms-lemma-etale-locally-standard-etale", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point.", "Let $V \\subset S$ be an affine open neighbourhood of $f(x)$.", "The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is \\'etale at $x$.", "\\item There exist an affine open $U \\subset X$ with", "$x \\in U$ and $f(U) \\subset V$ such that the", "induced morphism $f|_U : U \\to V$ is standard \\'etale", "(see Definition \\ref{definition-etale}).", "\\end{enumerate}" ], "refs": [ "morphisms-definition-etale" ], "proofs": [ { "contents": [ "Follows from the definitions and", "Algebra, Proposition \\ref{algebra-proposition-etale-locally-standard}." ], "refs": [ "algebra-proposition-etale-locally-standard" ], "ref_ids": [ 1427 ] } ], "ref_ids": [ 5567 ] }, { "id": 5372, "type": "theorem", "label": "morphisms-lemma-etale-at-point", "categories": [ "morphisms" ], "title": "morphisms-lemma-etale-at-point", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$.", "Set $s = f(x)$.", "Assume $f$ is locally of finite presentation.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is \\'etale at $x$.", "\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$", "is flat and $X_s \\to \\Spec(\\kappa(s))$ is \\'etale at $x$.", "\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$", "is flat and $X_s \\to \\Spec(\\kappa(s))$ is unramified at $x$.", "\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$", "is flat and the $\\mathcal{O}_{X, x}$-module $\\Omega_{X/S, x}$", "is zero.", "\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$", "is flat and the $\\kappa(x)$-vector space", "$$", "\\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) =", "\\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)", "$$", "is zero.", "\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$", "is flat, we have $\\mathfrak m_s\\mathcal{O}_{X, x} = \\mathfrak m_x$ and", "the field extension $\\kappa(s) \\subset \\kappa(x)$ is finite", "separable.", "\\item There exist affine opens $U \\subset X$,", "and $V \\subset S$ such that $x \\in U$, $f(U) \\subset V$ and the", "induced morphism $f|_U : U \\to V$ is standard smooth", "of relative dimension $0$.", "\\item There exist affine opens $\\Spec(A) = U \\subset X$", "and $\\Spec(R) = V \\subset S$ with $x \\in U$ corresponding", "to $\\mathfrak q \\subset A$, and $f(U) \\subset V$", "such that there exists a presentation", "$$", "A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)", "$$", "with", "$$", "g =", "\\det", "\\left(", "\\begin{matrix}", "\\partial f_1/\\partial x_1 &", "\\partial f_2/\\partial x_1 &", "\\ldots &", "\\partial f_n/\\partial x_1 \\\\", "\\partial f_1/\\partial x_2 &", "\\partial f_2/\\partial x_2 &", "\\ldots &", "\\partial f_n/\\partial x_2 \\\\", "\\ldots & \\ldots & \\ldots & \\ldots \\\\", "\\partial f_1/\\partial x_n &", "\\partial f_2/\\partial x_n &", "\\ldots &", "\\partial f_n/\\partial x_n", "\\end{matrix}", "\\right)", "$$", "mapping to an element of $A$ not in $\\mathfrak q$.", "\\item There exist affine opens $U \\subset X$,", "and $V \\subset S$ such that $x \\in U$, $f(U) \\subset V$ and the", "induced morphism $f|_U : U \\to V$ is standard \\'etale.", "\\item There exist affine opens $\\Spec(A) = U \\subset X$", "and $\\Spec(R) = V \\subset S$ with $x \\in U$ corresponding", "to $\\mathfrak q \\subset A$, and $f(U) \\subset V$", "such that there exists a presentation", "$$", "A = R[x]_Q/(P) = R[x, 1/Q]/(P)", "$$", "with $P, Q \\in R[x]$, $P$ monic and $P' = \\text{d}P/\\text{d}x$ mapping to", "an element of $A$ not in $\\mathfrak q$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Use Lemma \\ref{lemma-etale-locally-standard-etale}", "and the definitions to see that (1) implies all", "of the other conditions. For each of the conditions", "(2) -- (10) combine Lemmas \\ref{lemma-smooth-at-point}", "and \\ref{lemma-unramified-at-point} to see", "that (1) holds by showing $f$ is both smooth and unramified", "at $x$ and applying Lemma \\ref{lemma-etale-smooth-unramified}.", "Some details omitted." ], "refs": [ "morphisms-lemma-etale-locally-standard-etale", "morphisms-lemma-smooth-at-point", "morphisms-lemma-unramified-at-point", "morphisms-lemma-etale-smooth-unramified" ], "ref_ids": [ 5371, 5335, 5355, 5362 ] } ], "ref_ids": [] }, { "id": 5373, "type": "theorem", "label": "morphisms-lemma-flat-unramified-etale", "categories": [ "morphisms" ], "title": "morphisms-lemma-flat-unramified-etale", "contents": [ "A morphism is \\'etale at a point if and only if it is flat and G-unramified", "at that point.", "A morphism is \\'etale if and only if it is flat and G-unramified." ], "refs": [], "proofs": [ { "contents": [ "This is clear from Lemmas \\ref{lemma-etale-at-point}", "and \\ref{lemma-unramified-at-point}." ], "refs": [ "morphisms-lemma-etale-at-point", "morphisms-lemma-unramified-at-point" ], "ref_ids": [ 5372, 5355 ] } ], "ref_ids": [] }, { "id": 5374, "type": "theorem", "label": "morphisms-lemma-set-points-where-fibres-etale", "categories": [ "morphisms" ], "title": "morphisms-lemma-set-points-where-fibres-etale", "contents": [ "Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "be a cartesian diagram of schemes. Let $W \\subset X$, resp.\\ $W' \\subset X'$", "be the open subscheme of points where $f$, resp.\\ $f'$ is \\'etale.", "Then $W' = (g')^{-1}(W)$ if", "\\begin{enumerate}", "\\item $f$ is flat and locally of finite presentation, or", "\\item $f$ is locally of finite presentation and $g$ is flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume first that $f$ locally of finite type. Consider the set", "$$", "T = \\{x \\in X \\mid f\\text{ is unramified at }x\\}", "$$", "and the corresponding set $T' \\subset X'$ for $f'$. Then", "$T' = (g')^{-1}(T)$ by", "Lemma \\ref{lemma-set-points-where-fibres-unramified}.", "\\medskip\\noindent", "Thus case (1) follows because in case (1) $T$ is the (open) set of points", "where $f$ is \\'etale by Lemma \\ref{lemma-flat-unramified-etale}.", "\\medskip\\noindent", "In case (2) let $x' \\in W'$. Then $g'$ is flat at $x'$", "(Lemma \\ref{lemma-base-change-module-flat}) and", "$g \\circ f'$ is flat at $x'$ (Lemma \\ref{lemma-composition-module-flat}).", "It follows that $f$ is flat at $x = g'(x')$", "by Lemma \\ref{lemma-flat-permanence}. On the other hand, since", "$x' \\in T'$ (Lemma \\ref{lemma-base-change-smooth})", "we see that $x \\in T$. Hence $f$ is \\'etale at $x$ by", "Lemma \\ref{lemma-etale-at-point}." ], "refs": [ "morphisms-lemma-set-points-where-fibres-unramified", "morphisms-lemma-flat-unramified-etale", "morphisms-lemma-base-change-module-flat", "morphisms-lemma-composition-module-flat", "morphisms-lemma-flat-permanence", "morphisms-lemma-base-change-smooth", "morphisms-lemma-etale-at-point" ], "ref_ids": [ 5356, 5373, 5264, 5262, 5270, 5327, 5372 ] } ], "ref_ids": [] }, { "id": 5375, "type": "theorem", "label": "morphisms-lemma-etale-permanence", "categories": [ "morphisms" ], "title": "morphisms-lemma-etale-permanence", "contents": [ "\\begin{slogan}", "Cancellation law for \\'etale morphisms", "\\end{slogan}", "Let $f : X \\to Y$ be a morphism of schemes over $S$.", "If $X$ and $Y$ are \\'etale over $S$, then", "$f$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-map-between-etale}." ], "refs": [ "algebra-lemma-map-between-etale" ], "ref_ids": [ 1236 ] } ], "ref_ids": [] }, { "id": 5376, "type": "theorem", "label": "morphisms-lemma-etale-permanence-two", "categories": [ "morphisms" ], "title": "morphisms-lemma-etale-permanence-two", "contents": [ "Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & &", "Y \\ar[dl]^q \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes. Assume that", "\\begin{enumerate}", "\\item $f$ is surjective, and \\'etale,", "\\item $p$ is \\'etale, and", "\\item $q$ is locally of finite presentation\\footnote{In fact this", "is implied by (1) and (2), see", "Descent, Lemma \\ref{descent-lemma-flat-finitely-presented-permanence}.", "Moreover, it suffices to assume that $f$ is surjective, flat and", "locally of finite presentation, see", "Descent, Lemma \\ref{descent-lemma-smooth-permanence}.}.", "\\end{enumerate}", "Then $q$ is \\'etale." ], "refs": [ "descent-lemma-flat-finitely-presented-permanence", "descent-lemma-smooth-permanence" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-smooth-permanence} we see that $q$ is smooth.", "Thus we only need to see that $q$ has relative dimension $0$.", "This follows from Lemma \\ref{lemma-dimension-fibre-at-a-point-additive}", "and the fact that $f$ and $p$ have relative dimension $0$." ], "refs": [ "morphisms-lemma-smooth-permanence", "morphisms-lemma-dimension-fibre-at-a-point-additive" ], "ref_ids": [ 5340, 5278 ] } ], "ref_ids": [ 14642, 14644 ] }, { "id": 5377, "type": "theorem", "label": "morphisms-lemma-smooth-etale-over-affine-space", "categories": [ "morphisms" ], "title": "morphisms-lemma-smooth-etale-over-affine-space", "contents": [ "\\begin{slogan}", "Smooth schemes are \\'etale-locally like affine spaces.", "\\end{slogan}", "Let $\\varphi : X \\to Y$ be a morphism of schemes. Let $x \\in X$.", "Let $V \\subset Y$ be an affine open neighbourhood of $\\varphi(x)$.", "If $\\varphi$ is smooth at $x$, then there exists an integer $d \\geq 0$", "and an affine open $U \\subset X$ with $x \\in U$ and", "$\\varphi(U) \\subset V$ such that there exists a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] & U \\ar[l] \\ar[d] \\ar[r]_-\\pi & \\mathbf{A}^d_V \\ar[ld] \\\\", "Y & V \\ar[l]", "}", "$$", "where $\\pi$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-smooth-locally-standard-smooth}", "we can find an affine open $U$ as in the lemma such that", "$\\varphi|_U : U \\to V$ is standard smooth. Write", "$U = \\Spec(A)$ and $V = \\Spec(R)$ so that we can write", "$$", "A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)", "$$", "with", "$$", "g =", "\\det", "\\left(", "\\begin{matrix}", "\\partial f_1/\\partial x_1 &", "\\partial f_2/\\partial x_1 &", "\\ldots &", "\\partial f_c/\\partial x_1 \\\\", "\\partial f_1/\\partial x_2 &", "\\partial f_2/\\partial x_2 &", "\\ldots &", "\\partial f_c/\\partial x_2 \\\\", "\\ldots & \\ldots & \\ldots & \\ldots \\\\", "\\partial f_1/\\partial x_c &", "\\partial f_2/\\partial x_c &", "\\ldots &", "\\partial f_c/\\partial x_c", "\\end{matrix}", "\\right)", "$$", "mapping to an invertible element of $A$. Then it is clear that", "$R[x_{c + 1}, \\ldots, x_n] \\to A$ is standard smooth of relative", "dimension $0$. Hence it is smooth of relative dimension $0$.", "In other words the ring map $R[x_{c + 1}, \\ldots, x_n] \\to A$", "is \\'etale. As $\\mathbf{A}^{n - c}_V = \\Spec(R[x_{c + 1}, \\ldots, x_n])$", "the lemma with $d = n - c$." ], "refs": [ "morphisms-lemma-smooth-locally-standard-smooth" ], "ref_ids": [ 5333 ] } ], "ref_ids": [] }, { "id": 5378, "type": "theorem", "label": "morphisms-lemma-ample-power-ample", "categories": [ "morphisms" ], "title": "morphisms-lemma-ample-power-ample", "contents": [ "Let $X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $n \\geq 1$. Then $\\mathcal{L}$ is $f$-ample if and only if", "$\\mathcal{L}^{\\otimes n}$ is $f$-ample." ], "refs": [], "proofs": [ { "contents": [ "This follows from Properties, Lemma \\ref{properties-lemma-ample-power-ample}." ], "refs": [ "properties-lemma-ample-power-ample" ], "ref_ids": [ 3040 ] } ], "ref_ids": [] }, { "id": 5379, "type": "theorem", "label": "morphisms-lemma-relatively-ample-separated", "categories": [ "morphisms" ], "title": "morphisms-lemma-relatively-ample-separated", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "If there exists an $f$-ample invertible sheaf, then", "$f$ is separated." ], "refs": [], "proofs": [ { "contents": [ "Being separated is local on the base (see", "Schemes, Lemma \\ref{schemes-lemma-characterize-separated} for example;", "it also follows easily from the definition).", "Hence we may assume $S$ is affine and $X$", "has an ample invertible sheaf. In this case the", "result follows from", "Properties, Lemma \\ref{properties-lemma-ample-separated}." ], "refs": [ "schemes-lemma-characterize-separated", "properties-lemma-ample-separated" ], "ref_ids": [ 7710, 3046 ] } ], "ref_ids": [] }, { "id": 5380, "type": "theorem", "label": "morphisms-lemma-characterize-relatively-ample", "categories": [ "morphisms" ], "title": "morphisms-lemma-characterize-relatively-ample", "contents": [ "\\begin{reference}", "\\cite[II, Proposition 4.6.3]{EGA}", "\\end{reference}", "Let $f : X \\to S$ be a quasi-compact morphism of schemes.", "Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "The following are equivalent:", "\\begin{enumerate}", "\\item The invertible sheaf $\\mathcal{L}$ is $f$-ample.", "\\item There exists an open covering $S = \\bigcup V_i$", "such that each $\\mathcal{L}|_{f^{-1}(V_i)}$ is ample", "relative to $f^{-1}(V_i) \\to V_i$.", "\\item There exists an affine open covering $S = \\bigcup V_i$", "such that each $\\mathcal{L}|_{f^{-1}(V_i)}$ is ample.", "\\item There exists a quasi-coherent graded $\\mathcal{O}_S$-algebra", "$\\mathcal{A}$ and a map of graded $\\mathcal{O}_X$-algebras", "$\\psi : f^*\\mathcal{A} \\to \\bigoplus_{d \\geq 0} \\mathcal{L}^{\\otimes d}$", "such that $U(\\psi) = X$ and", "$$", "r_{\\mathcal{L}, \\psi} :", "X", "\\longrightarrow", "\\underline{\\text{Proj}}_S(\\mathcal{A})", "$$", "is an open immersion (see Constructions, Lemma", "\\ref{constructions-lemma-invertible-map-into-relative-proj} for notation).", "\\item The morphism $f$ is quasi-separated and", "part (4) above holds with", "$\\mathcal{A} = f_*(\\bigoplus_{d \\geq 0} \\mathcal{L}^{\\otimes d})$", "and $\\psi$ the adjunction mapping.", "\\item Same as (4) but just requiring $r_{\\mathcal{L}, \\psi}$", "to be an immersion.", "\\end{enumerate}" ], "refs": [ "constructions-lemma-invertible-map-into-relative-proj" ], "proofs": [ { "contents": [ "It is immediate from the definition that (1) implies (2) and", "(2) implies (3). It is clear that (5) implies (4).", "\\medskip\\noindent", "Assume (3) holds for the affine open covering $S = \\bigcup V_i$.", "We are going to show (5) holds.", "Since each $f^{-1}(V_i)$ has an ample invertible sheaf we see", "that $f^{-1}(V_i)$ is separated", "(Properties, Lemma \\ref{properties-lemma-ample-separated}).", "Hence $f$ is separated. By", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "we see that $\\mathcal{A} = f_*(\\bigoplus_{d \\geq 0} \\mathcal{L}^{\\otimes d})$", "is a quasi-coherent graded $\\mathcal{O}_S$-algebra.", "Denote $\\psi : f^*\\mathcal{A} \\to \\bigoplus_{d \\geq 0} \\mathcal{L}^{\\otimes d}$", "the adjunction mapping.", "The description of the open $U(\\psi)$ in", "Constructions, Section", "\\ref{constructions-section-invertible-relative-proj}", "and the definition of ampleness", "of $\\mathcal{L}|_{f^{-1}(V_i)}$ show that $U(\\psi) = X$.", "Moreover, Constructions,", "Lemma \\ref{constructions-lemma-invertible-map-into-relative-proj} part (3)", "shows that the restriction of $r_{\\mathcal{L}, \\psi}$ to", "$f^{-1}(V_i)$ is the same as the morphism from", "Properties, Lemma \\ref{properties-lemma-map-into-proj}", "which is an open immersion according to", "Properties, Lemma \\ref{properties-lemma-ample-immersion-into-proj}.", "Hence (5) holds.", "\\medskip\\noindent", "Let us show that (4) implies (1). Assume (4).", "Denote $\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$", "the structure morphism. Choose $V \\subset S$ affine open. By", "Constructions, Definition \\ref{constructions-definition-relative-proj}", "we see that $\\pi^{-1}(V) \\subset \\underline{\\text{Proj}}_S(\\mathcal{A})$", "is equal to $\\text{Proj}(A)$ where $A = \\mathcal{A}(V)$", "as a graded ring. Hence $r_{\\mathcal{L}, \\psi}$ maps", "$f^{-1}(V)$ isomorphically onto", "a quasi-compact open of $\\text{Proj}(A)$.", "Moreover, $\\mathcal{L}^{\\otimes d}$ is isomorphic to the pullback of", "$\\mathcal{O}_{\\text{Proj}(A)}(d)$ for some $d \\geq 1$.", "(See part (3) of Constructions,", "Lemma \\ref{constructions-lemma-invertible-map-into-relative-proj}", "and the final statement of Constructions,", "Lemma \\ref{constructions-lemma-invertible-map-into-proj}.)", "This implies that $\\mathcal{L}|_{f^{-1}(V)}$ is ample by", "Properties, Lemmas \\ref{properties-lemma-open-in-proj-ample}", "and \\ref{properties-lemma-ample-power-ample}.", "\\medskip\\noindent", "Assume (6). By the equivalence of (1) - (5) above we see that the", "property of being relatively ample on $X/S$ is local on $S$. Hence", "we may assume that $S$ is affine, and we have to show that", "$\\mathcal{L}$ is ample on $X$. In this case the morphism", "$r_{\\mathcal{L}, \\psi}$ is identified with the morphism, also denoted", "$r_{\\mathcal{L}, \\psi} : X \\to \\text{Proj}(A)$ associated to the map", "$\\psi : A = \\mathcal{A}(V) \\to \\Gamma_*(X, \\mathcal{L})$.", "(See references above.) As above we also see that", "$\\mathcal{L}^{\\otimes d}$ is the pullback of the sheaf", "$\\mathcal{O}_{\\text{Proj}(A)}(d)$ for some $d \\geq 1$.", "Moreover, since $X$ is quasi-compact we", "see that $X$ gets identified with a closed subscheme of a", "quasi-compact open subscheme $Y \\subset \\text{Proj}(A)$.", "By", "Constructions, Lemma", "\\ref{constructions-lemma-ample-on-proj}", "(see also", "Properties, Lemma", "\\ref{properties-lemma-open-in-proj-ample})", "we see that $\\mathcal{O}_Y(d')$ is an ample invertible sheaf on", "$Y$ for some $d' \\geq 1$. Since the restriction of an ample", "sheaf to a closed subscheme is ample, see", "Properties, Lemma", "\\ref{properties-lemma-ample-on-closed}", "we conclude that the pullback of", "$\\mathcal{O}_Y(d')$ is ample. Combining these results with", "Properties, Lemma", "\\ref{properties-lemma-ample-power-ample}", "we conclude that $\\mathcal{L}$ is ample as desired." ], "refs": [ "properties-lemma-ample-separated", "schemes-lemma-push-forward-quasi-coherent", "constructions-lemma-invertible-map-into-relative-proj", "properties-lemma-map-into-proj", "properties-lemma-ample-immersion-into-proj", "constructions-definition-relative-proj", "constructions-lemma-invertible-map-into-relative-proj", "constructions-lemma-invertible-map-into-proj", "properties-lemma-open-in-proj-ample", "properties-lemma-ample-power-ample", "constructions-lemma-ample-on-proj", "properties-lemma-open-in-proj-ample", "properties-lemma-ample-on-closed", "properties-lemma-ample-power-ample" ], "ref_ids": [ 3046, 7730, 12649, 3047, 3049, 12665, 12649, 12628, 3050, 3040, 12606, 3050, 3041, 3040 ] } ], "ref_ids": [ 12649 ] }, { "id": 5381, "type": "theorem", "label": "morphisms-lemma-ample-over-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-ample-over-affine", "contents": [ "\\begin{reference}", "\\cite[II Corollary 4.6.6]{EGA}", "\\end{reference}", "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Assume $S$ affine.", "Then $\\mathcal{L}$ is $f$-relatively ample if and only", "if $\\mathcal{L}$ is ample on $X$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-characterize-relatively-ample}", "and the definitions." ], "refs": [ "morphisms-lemma-characterize-relatively-ample" ], "ref_ids": [ 5380 ] } ], "ref_ids": [] }, { "id": 5382, "type": "theorem", "label": "morphisms-lemma-quasi-affine-O-ample", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-affine-O-ample", "contents": [ "\\begin{reference}", "\\cite[II Proposition 5.1.6]{EGA}", "\\end{reference}", "Let $f : X \\to S$ be a morphism of schemes. Then $f$ is quasi-affine", "if and only if $\\mathcal{O}_X$ is $f$-relatively ample." ], "refs": [], "proofs": [ { "contents": [ "Follows from Properties, Lemma \\ref{properties-lemma-quasi-affine-O-ample}", "and the definitions." ], "refs": [ "properties-lemma-quasi-affine-O-ample" ], "ref_ids": [ 3053 ] } ], "ref_ids": [] }, { "id": 5383, "type": "theorem", "label": "morphisms-lemma-pullback-ample-tensor-relatively-ample", "categories": [ "morphisms" ], "title": "morphisms-lemma-pullback-ample-tensor-relatively-ample", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes, $\\mathcal{M}$", "an invertible $\\mathcal{O}_Y$-module, and $\\mathcal{L}$ an", "invertible $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item If $\\mathcal{L}$ is $f$-ample and $\\mathcal{M}$", "is ample, then $\\mathcal{L} \\otimes f^*\\mathcal{M}^{\\otimes a}$ is ample", "for $a \\gg 0$.", "\\item If $\\mathcal{M}$ is ample", "and $f$ quasi-affine, then $f^*\\mathcal{M}$ is ample.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{L}$ is $f$-ample and $\\mathcal{M}$ ample.", "By assumption $Y$ and $f$ are quasi-compact (see", "Definition \\ref{definition-relatively-ample} and", "Properties, Definition \\ref{properties-definition-ample}).", "Hence $X$ is quasi-compact. Pick $x \\in X$. We can choose $m \\geq 1$", "and $t \\in \\Gamma(Y, \\mathcal{M}^{\\otimes m})$ such that $Y_t$", "is affine and $f(x) \\in Y_t$. Since $\\mathcal{L}$ restricts to an", "ample invertible sheaf on $f^{-1}(Y_t) = X_{f^*t}$", "we can choose $n \\geq 1$ and $s \\in \\Gamma(X_{f^*t}, \\mathcal{L}^{\\otimes n})$", "with $x \\in (X_{f^*t})_s$ with $(X_{f^*t})_s$ affine.", "By Properties, Lemma \\ref{properties-lemma-invert-s-sections}", "there exists an integer $e \\geq 1$ and a section", "$s' \\in \\Gamma(X, \\mathcal{L}^{\\otimes n} \\otimes f^*\\mathcal{M}^{\\otimes em})$", "which restricts to $s(f^*t)^e$ on $X_{f^*t}$. For any $b > 0$", "consider the section $s'' = s'(f^*t)^b$ of", "$\\mathcal{L}^{\\otimes n} \\otimes f^*\\mathcal{M}^{\\otimes (e + b)m}$.", "Then $X_{s''} = (X_{f^*t})_s$ is an affine open of $X$ containing $x$.", "Picking $b$ such that $n$ divides $e + b$ we see", "$\\mathcal{L}^{\\otimes n} \\otimes f^*\\mathcal{M}^{\\otimes (e + b)m}$", "is the $n$th power of $\\mathcal{L} \\otimes f^*\\mathcal{M}^{\\otimes a}$", "for some $a$ and we can get any $a$ divisible by $m$ and big enough.", "Since $X$ is quasi-compact a finite number of these affine opens", "cover $X$. We conclude that for some $a$ sufficiently divisible and", "large enough the invertible sheaf", "$\\mathcal{L} \\otimes f^*\\mathcal{M}^{\\otimes a}$ is ample on $X$.", "On the other hand, we know that $\\mathcal{M}^{\\otimes c}$", "(and hence its pullback to $X$) is globally generated for all $c \\gg 0$", "by Properties, Proposition \\ref{properties-proposition-characterize-ample}.", "Thus $\\mathcal{L} \\otimes f^*\\mathcal{M}^{\\otimes a + c}$ is ample", "(Properties, Lemma \\ref{properties-lemma-ample-tensor-globally-generated})", "for $c \\gg 0$ and (1) is proved.", "\\medskip\\noindent", "Part (2) follows from Lemma \\ref{lemma-quasi-affine-O-ample},", "Properties, Lemma \\ref{properties-lemma-ample-power-ample}, and", "part (1)." ], "refs": [ "morphisms-definition-relatively-ample", "properties-definition-ample", "properties-lemma-invert-s-sections", "properties-proposition-characterize-ample", "properties-lemma-ample-tensor-globally-generated", "morphisms-lemma-quasi-affine-O-ample", "properties-lemma-ample-power-ample" ], "ref_ids": [ 5568, 3088, 3005, 3067, 3043, 5382, 3040 ] } ], "ref_ids": [] }, { "id": 5384, "type": "theorem", "label": "morphisms-lemma-ample-composition", "categories": [ "morphisms" ], "title": "morphisms-lemma-ample-composition", "contents": [ "Let $g : Y \\to S$ and $f : X \\to Y$ be morphisms of schemes.", "Let $\\mathcal{M}$ be an invertible $\\mathcal{O}_Y$-module.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "If $S$ is quasi-compact, $\\mathcal{M}$ is $g$-ample, and", "$\\mathcal{L}$ is $f$-ample, then", "$\\mathcal{L} \\otimes f^*\\mathcal{M}^{\\otimes a}$", "is $g \\circ f$-ample for $a \\gg 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $S = \\bigcup_{i = 1, \\ldots, n} V_i$ be a finite affine open covering.", "By Lemma \\ref{lemma-characterize-relatively-ample}", "it suffices to prove that ", "$\\mathcal{L} \\otimes f^*\\mathcal{M}^{\\otimes a}$", "is ample on $(g \\circ f)^{-1}(V_i)$ for $i = 1, \\ldots, n$.", "Thus the lemma follows from", "Lemma \\ref{lemma-pullback-ample-tensor-relatively-ample}." ], "refs": [ "morphisms-lemma-characterize-relatively-ample", "morphisms-lemma-pullback-ample-tensor-relatively-ample" ], "ref_ids": [ 5380, 5383 ] } ], "ref_ids": [] }, { "id": 5385, "type": "theorem", "label": "morphisms-lemma-ample-base-change", "categories": [ "morphisms" ], "title": "morphisms-lemma-ample-base-change", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $S' \\to S$ be a morphism of schemes.", "Let $f' : X' \\to S'$ be the base change of $f$ and denote", "$\\mathcal{L}'$ the pullback of $\\mathcal{L}$ to $X'$.", "If $\\mathcal{L}$ is $f$-ample, then $\\mathcal{L}'$ is $f'$-ample." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-characterize-relatively-ample} it suffices", "to find an affine open covering $S' = \\bigcup U'_i$", "such that $\\mathcal{L}'$ restricts to an ample invertible", "sheaf on $(f')^{-1}(U_i')$ for all $i$. We may choose $U'_i$", "mapping into an affine open $U_i \\subset S$. In this case the", "morphism $(f')^{-1}(U'_i) \\to f^{-1}(U_i)$ is affine as a base", "change of the affine morphism $U'_i \\to U_i$", "(Lemma \\ref{lemma-base-change-affine}). Thus", "$\\mathcal{L}'|_{(f')^{-1}(U'_i)}$ is ample by", "Lemma \\ref{lemma-pullback-ample-tensor-relatively-ample}." ], "refs": [ "morphisms-lemma-characterize-relatively-ample", "morphisms-lemma-base-change-affine", "morphisms-lemma-pullback-ample-tensor-relatively-ample" ], "ref_ids": [ 5380, 5176, 5383 ] } ], "ref_ids": [] }, { "id": 5386, "type": "theorem", "label": "morphisms-lemma-ample-permanence", "categories": [ "morphisms" ], "title": "morphisms-lemma-ample-permanence", "contents": [ "Let $g : Y \\to S$ and $f : X \\to Y$ be morphisms of schemes.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "If $\\mathcal{L}$ is $g \\circ f$-ample and $f$ is", "quasi-compact\\footnote{This follows if $g$ is quasi-separated by", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}.}", "then $\\mathcal{L}$ is $f$-ample." ], "refs": [ "schemes-lemma-quasi-compact-permanence" ], "proofs": [ { "contents": [ "Assume $f$ is quasi-compact and $\\mathcal{L}$ is $g \\circ f$-ample.", "Let $U \\subset S$ be an affine open and let $V \\subset Y$ be", "an affine open with $g(V) \\subset U$.", "Then $\\mathcal{L}|_{(g \\circ f)^{-1}(U)}$ is ample on", "$(g \\circ f)^{-1}(U)$ by assumption.", "Since $f^{-1}(V) \\subset (g \\circ f)^{-1}(U)$ we see that", "$\\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)$ by", "Properties, Lemma \\ref{properties-lemma-ample-on-locally-closed}.", "Namely, $f^{-1}(V) \\to (g \\circ f)^{-1}(U)$ is a quasi-compact", "open immersion by", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}", "as $(g \\circ f)^{-1}(U)$ is separated", "(Properties, Lemma \\ref{properties-lemma-ample-separated})", "and $f^{-1}(V)$ is quasi-compact (as $f$ is quasi-compact).", "Thus we conclude that $\\mathcal{L}$ is $f$-ample by", "Lemma \\ref{lemma-characterize-relatively-ample}." ], "refs": [ "properties-lemma-ample-on-locally-closed", "schemes-lemma-quasi-compact-permanence", "properties-lemma-ample-separated", "morphisms-lemma-characterize-relatively-ample" ], "ref_ids": [ 3051, 7716, 3046, 5380 ] } ], "ref_ids": [ 7716 ] }, { "id": 5387, "type": "theorem", "label": "morphisms-lemma-ample-very-ample", "categories": [ "morphisms" ], "title": "morphisms-lemma-ample-very-ample", "contents": [ "\\begin{reference}", "\\cite[II, Proposition 4.6.2]{EGA}", "\\end{reference}", "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "If $f$ is quasi-compact and $\\mathcal{L}$ is a relatively", "very ample invertible sheaf, then $\\mathcal{L}$ is a relatively", "ample invertible sheaf." ], "refs": [], "proofs": [ { "contents": [ "By definition there exists quasi-coherent $\\mathcal{O}_S$-module", "$\\mathcal{E}$ and an immersion $i : X \\to \\mathbf{P}(\\mathcal{E})$", "over $S$ such that", "$\\mathcal{L} \\cong i^*\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(1)$.", "Set $\\mathcal{A} = \\text{Sym}(\\mathcal{E})$, so", "$\\mathbf{P}(\\mathcal{E}) = \\underline{\\text{Proj}}_S(\\mathcal{A})$", "by definition. The graded $\\mathcal{O}_S$-algebra $\\mathcal{A}$", "comes equipped with a map", "$$", "\\psi :", "\\mathcal{A} \\to", "\\bigoplus\\nolimits_{n \\geq 0}", "\\pi_*\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(n) \\to", "\\bigoplus\\nolimits_{n \\geq 0}", "f_*\\mathcal{L}^{\\otimes n}", "$$", "where the second arrow uses the identification", "$\\mathcal{L} \\cong i^*\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(1)$.", "By adjointness of $f_*$ and $f^*$ we get a morphism", "$\\psi : f^*\\mathcal{A} \\to \\bigoplus_{n \\geq 0}\\mathcal{L}^{\\otimes n}$.", "We omit the verification that the morphism $r_{\\mathcal{L}, \\psi}$", "associated to this map is exactly the immersion $i$.", "Hence the result follows from", "part (6) of Lemma \\ref{lemma-characterize-relatively-ample}." ], "refs": [ "morphisms-lemma-characterize-relatively-ample" ], "ref_ids": [ 5380 ] } ], "ref_ids": [] }, { "id": 5388, "type": "theorem", "label": "morphisms-lemma-relatively-very-ample-separated", "categories": [ "morphisms" ], "title": "morphisms-lemma-relatively-very-ample-separated", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "If $\\mathcal{L}$ is relatively very ample on $X/S$ then", "$f$ is separated." ], "refs": [], "proofs": [ { "contents": [ "Being separated is local on the base (see", "Schemes, Section \\ref{schemes-section-separation-axioms}).", "An immersion is separated", "(see Schemes, Lemma \\ref{schemes-lemma-immersions-monomorphisms}).", "Hence the lemma follows since locally $X$ has an immersion into", "the homogeneous spectrum of a graded ring which is separated, see", "Constructions, Lemma \\ref{constructions-lemma-proj-separated}." ], "refs": [ "schemes-lemma-immersions-monomorphisms", "constructions-lemma-proj-separated" ], "ref_ids": [ 7727, 12597 ] } ], "ref_ids": [] }, { "id": 5389, "type": "theorem", "label": "morphisms-lemma-relatively-very-ample", "categories": [ "morphisms" ], "title": "morphisms-lemma-relatively-very-ample", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "Assume $f$ is quasi-compact. The following are", "equivalent", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is relatively very ample on $X/S$,", "\\item there exists an open covering $S = \\bigcup V_j$ such", "that $\\mathcal{L}|_{f^{-1}(V_j)}$ is relatively very ample", "on $f^{-1}(V_j)/V_j$ for all $j$,", "\\item there exists a quasi-coherent sheaf of graded", "$\\mathcal{O}_S$-algebras $\\mathcal{A}$ generated in degree", "$1$ over $\\mathcal{O}_S$ and a map of graded $\\mathcal{O}_X$-algebras", "$\\psi : f^*\\mathcal{A} \\to \\bigoplus_{n \\geq 0} \\mathcal{L}^{\\otimes n}$", "such that $f^*\\mathcal{A}_1 \\to \\mathcal{L}$ is surjective and the", "associated morphism", "$r_{\\mathcal{L}, \\psi} : X \\to \\underline{\\text{Proj}}_S(\\mathcal{A})$", "is an immersion, and", "\\item $f$ is quasi-separated, the canonical map", "$\\psi : f^*f_*\\mathcal{L} \\to \\mathcal{L}$ is surjective, and", "the associated map $r_{\\mathcal{L}, \\psi} : X \\to \\mathbf{P}(f_*\\mathcal{L})$", "is an immersion.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2). It is also clear that", "(4) implies (1); the hypothesis of quasi-separation", "in (4) is used to guarantee that $f_*\\mathcal{L}$ is quasi-coherent via", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}.", "\\medskip\\noindent", "Assume (2). We will prove (4).", "Let $S = \\bigcup V_j$ be an open covering as in (2).", "Set $X_j = f^{-1}(V_j)$ and $f_j : X_j \\to V_j$ the", "restriction of $f$. We see that $f$ is separated by", "Lemma \\ref{lemma-relatively-very-ample-separated} (as being", "separated is local on the base). By assumption there exists a", "quasi-coherent $\\mathcal{O}_{V_j}$-module $\\mathcal{E}_j$ and an immersion", "$i_j : X_j \\to \\mathbf{P}(\\mathcal{E}_j)$ with", "$\\mathcal{L}|_{X_j} \\cong i_j^*\\mathcal{O}_{\\mathbf{P}(\\mathcal{E}_j)}(1)$.", "The morphism $i_j$ corresponds to a surjection", "$f_j^*\\mathcal{E}_j \\to \\mathcal{L}|_{X_j}$, see", "Constructions, Section \\ref{constructions-section-projective-bundle}.", "This map is adjoint to a map $\\mathcal{E}_j \\to f_*\\mathcal{L}|_{V_j}$", "such that the composition", "$$", "f_j^*\\mathcal{E}_j \\to (f^*f_*\\mathcal{L})|_{X_j} \\to \\mathcal{L}|_{X_j}", "$$", "is surjective. We conclude that", "$\\psi : f^*f_*\\mathcal{L} \\to \\mathcal{L}$ is surjective. Let", "$r_{\\mathcal{L}, \\psi} : X \\to \\mathbf{P}(f_*\\mathcal{L})$ be the", "associated morphism. We still have to show that $r_{\\mathcal{L}, \\psi}$", "is an immersion; we urge the reader to prove this for themselves.", "The $\\mathcal{O}_{V_j}$-module map $\\mathcal{E}_j \\to f_*\\mathcal{L}|_{V_j}$", "determines a homomorphism on symmetric algebras, which in turn defines a", "morphism", "$$", "\\mathbf{P}(f_*\\mathcal{L}|_{V_j}) \\supset U_j \\longrightarrow", "\\mathbf{P}(\\mathcal{E}_j)", "$$", "where $U_j$ is the open subscheme of", "Constructions, Lemma \\ref{constructions-lemma-morphism-relative-proj}.", "The compatibility of $\\psi$ with $\\mathcal{E}_j \\to f_*\\mathcal{L}|_{V_j}$", "shows that $r_{\\mathcal{L}, \\psi}(X_j) \\subset U_j$ and that there is a", "factorization", "$$", "\\xymatrix{", "X_j \\ar[r]^-{r_{\\mathcal{L}, \\psi}} & U_j \\ar[r] &", "\\mathbf{P}(\\mathcal{E}_j)", "}", "$$", "We omit the verification.", "This shows that $r_{\\mathcal{L}, \\psi}$ is an immersion.", "\\medskip\\noindent", "At this point we see that (1), (2) and (4) are equivalent.", "Clearly (4) implies (3). Assume (3). We will prove (1).", "Let $\\mathcal{A}$ be a quasi-coherent sheaf of graded $\\mathcal{O}_S$-algebras", "generated in degree $1$ over $\\mathcal{O}_S$. Consider the map of", "graded $\\mathcal{O}_S$-algebras $\\text{Sym}(\\mathcal{A}_1) \\to \\mathcal{A}$.", "This is surjective by hypothesis and hence induces a closed immersion", "$$", "\\underline{\\text{Proj}}_S(\\mathcal{A})", "\\longrightarrow", "\\mathbf{P}(\\mathcal{A}_1)", "$$", "which pulls back $\\mathcal{O}(1)$ to $\\mathcal{O}(1)$,", "see Constructions, Lemma", "\\ref{constructions-lemma-surjective-generated-degree-1-map-relative-proj}.", "Hence it is clear that (3) implies (1)." ], "refs": [ "schemes-lemma-push-forward-quasi-coherent", "morphisms-lemma-relatively-very-ample-separated", "constructions-lemma-morphism-relative-proj", "constructions-lemma-surjective-generated-degree-1-map-relative-proj" ], "ref_ids": [ 7730, 5388, 12644, 12648 ] } ], "ref_ids": [] }, { "id": 5390, "type": "theorem", "label": "morphisms-lemma-very-ample-base-change", "categories": [ "morphisms" ], "title": "morphisms-lemma-very-ample-base-change", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $S' \\to S$ be a morphism of schemes.", "Let $f' : X' \\to S'$ be the base change of $f$ and denote", "$\\mathcal{L}'$ the pullback of $\\mathcal{L}$ to $X'$.", "If $\\mathcal{L}$ is $f$-very ample, then $\\mathcal{L}'$ is $f'$-very ample." ], "refs": [], "proofs": [ { "contents": [ "By Definition \\ref{definition-very-ample} there exists there exist a", "quasi-coherent $\\mathcal{O}_S$-module $\\mathcal{E}$ and an immersion", "$i : X \\to \\mathbf{P}(\\mathcal{E})$ over $S$ such that", "$\\mathcal{L} \\cong i^*\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(1)$.", "The base change of $\\mathbf{P}(\\mathcal{E})$ to $S'$ is", "the projective bundle associated to the pullback $\\mathcal{E}'$", "of $\\mathcal{E}$ and the pullback of", "$\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(1)$", "is", "$\\mathcal{O}_{\\mathbf{P}(\\mathcal{E}')}(1)$, see", "Constructions, Lemma \\ref{constructions-lemma-relative-proj-base-change}.", "Finally, the base change", "of an immersion is an immersion", "(Schemes, Lemma \\ref{schemes-lemma-base-change-immersion})." ], "refs": [ "morphisms-definition-very-ample", "constructions-lemma-relative-proj-base-change", "schemes-lemma-base-change-immersion" ], "ref_ids": [ 5569, 12641, 7695 ] } ], "ref_ids": [] }, { "id": 5391, "type": "theorem", "label": "morphisms-lemma-very-ample-finite-type-over-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-very-ample-finite-type-over-affine", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "Assume that", "\\begin{enumerate}", "\\item the invertible sheaf $\\mathcal{L}$ is very ample on $X/S$,", "\\item the morphism $X \\to S$ is of finite type, and", "\\item $S$ is affine.", "\\end{enumerate}", "Then there exists an $n \\geq 0$ and an immersion", "$i : X \\to \\mathbf{P}^n_S$ over $S$ such that", "$\\mathcal{L} \\cong i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1)$." ], "refs": [], "proofs": [ { "contents": [ "Assume (1), (2) and (3).", "Condition (3) means $S = \\Spec(R)$ for some ring $R$.", "Condition (1) means by definition", "there exists a quasi-coherent $\\mathcal{O}_S$-module", "$\\mathcal{E}$ and an immersion $\\alpha : X \\to \\mathbf{P}(\\mathcal{E})$", "such that $\\mathcal{L} = \\alpha^*\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(1)$.", "Write $\\mathcal{E} = \\widetilde{M}$ for some $R$-module $M$.", "Thus we have", "$$", "\\mathbf{P}(\\mathcal{E}) = \\text{Proj}(\\text{Sym}_R(M)).", "$$", "Since $\\alpha$ is an immersion, and since the topology of", "$\\text{Proj}(\\text{Sym}_R(M))$ is generated by the standard", "opens $D_{+}(f)$, $f \\in \\text{Sym}_R^d(M)$, $d \\geq 1$,", "we can find for each $x \\in X$ an", "$f \\in \\text{Sym}_R^d(M)$, $d \\geq 1$, with $\\alpha(x) \\in D_{+}(f)$", "such that", "$$", "\\alpha|_{\\alpha^{-1}(D_{+}(f))} : \\alpha^{-1}(D_{+}(f)) \\to D_{+}(f)", "$$", "is a closed immersion.", "Condition (2) implies $X$ is quasi-compact. Hence we can find", "a finite collection of elements", "$f_j \\in \\text{Sym}_R^{d_j}(M)$, $d_j \\geq 1$", "such that for each $f = f_j$ the displayed map above", "is a closed immersion and such that $\\alpha(X) \\subset \\bigcup D_{+}(f_j)$.", "Write $U_j = \\alpha^{-1}(D_{+}(f_j))$. Note that $U_j$ is affine", "as a closed subscheme of the affine scheme $D_{+}(f_j)$.", "Write $U_j = \\Spec(A_j)$. Condition (2) also implies that", "$A_j$ is of finite type over $R$, see", "Lemma \\ref{lemma-locally-finite-type-characterize}.", "Choose finitely many $x_{j, k} \\in A_j$ which", "generate $A_j$ as a $R$-algebra. Since $\\alpha|_{U_j}$ is a closed", "immersion we see that $x_{j, k}$ is the image of an element", "$$", "f_{j, k}/f_j^{e_{j, k}} \\in \\text{Sym}_R(M)_{(f_j)}", "=", "\\Gamma(D_{+}(f_j), \\mathcal{O}_{\\text{Proj}(\\text{Sym}_R(M))}).", "$$", "Finally, choose $n \\geq 1$ and elements $y_0, \\ldots, y_n \\in M$ such that each", "of the polynomials $f_j, f_{j, k} \\in \\text{Sym}_R(M)$ is a polynomial", "in the elements $y_t$ with coefficients in $R$.", "Consider the graded ring map", "$$", "\\psi : R[Y_0, \\ldots, Y_n] \\longrightarrow \\text{Sym}_R(M),", "\\quad Y_i \\longmapsto y_i.", "$$", "Denote $F_j$, $F_{j, k}$ the elements of $R[Y_0, \\ldots, Y_n]$ such", "that $\\psi(F_j) = f_j$ and $\\psi(F_{j, k}) = f_{j, k}$.", "By Constructions, Lemma \\ref{constructions-lemma-morphism-proj}", "we obtain an open subscheme", "$$", "U(\\psi) \\subset \\text{Proj}(\\text{Sym}_R(M))", "$$", "and a morphism", "$r_\\psi : U(\\psi) \\to \\mathbf{P}^n_R$. This morphism", "satisfies $r_\\psi^{-1}(D_{+}(F_j)) = D_{+}(f_j)$, and hence we see", "that $\\alpha(X) \\subset U(\\psi)$. Moreover, it is clear", "that", "$$", "i = r_\\psi \\circ \\alpha : X \\longrightarrow \\mathbf{P}^n_R", "$$", "is still an immersion since", "$i^\\sharp(F_{j, k}/F_j^{e_{j, k}}) = x_{j, k} \\in", "A_j = \\Gamma(U_j, \\mathcal{O}_X)$", "by construction. Moreover, the morphism $r_\\psi$ comes", "equipped with a map", "$\\theta : r_\\psi^*\\mathcal{O}_{\\mathbf{P}^n_R}(1)", "\\to \\mathcal{O}_{\\text{Proj}(\\text{Sym}_R(M))}(1)|_{U(\\psi)}$", "which is an isomorphism in this case (for construction $\\theta$", "see lemma cited above; some details omitted).", "Since the original map $\\alpha$ was assumed to have the", "property that", "$\\mathcal{L} = \\alpha^*\\mathcal{O}_{\\text{Proj}(\\text{Sym}_R(M))}(1)$", "we win." ], "refs": [ "morphisms-lemma-locally-finite-type-characterize", "constructions-lemma-morphism-proj" ], "ref_ids": [ 5198, 12608 ] } ], "ref_ids": [] }, { "id": 5392, "type": "theorem", "label": "morphisms-lemma-quasi-affine-finite-type-over-S", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-affine-finite-type-over-S", "contents": [ "Let $\\pi : X \\to S$ be a morphism of schemes.", "Assume that $X$ is quasi-affine and that $\\pi$ is locally of finite type.", "Then there exist $n \\geq 0$ and an immersion $i : X \\to \\mathbf{A}^n_S$", "over $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $A = \\Gamma(X, \\mathcal{O}_X)$. By assumption $X$ is quasi-compact", "and is identified with an open subscheme of $\\Spec(A)$, see", "Properties, Lemma \\ref{properties-lemma-quasi-affine}.", "Moreover, the set of opens $X_f$, for those $f \\in A$ such that $X_f$ is", "affine, forms a basis for the topology of $X$, see the proof of", "Properties, Lemma \\ref{properties-lemma-quasi-affine}.", "Hence we can find a finite number of $f_j \\in A$, $j = 1, \\ldots, m$ such that", "$X = \\bigcup X_{f_j}$, and such that $\\pi(X_{f_j}) \\subset V_j$ for", "some affine open $V_j \\subset S$. By", "Lemma \\ref{lemma-locally-finite-type-characterize}", "the ring maps $\\mathcal{O}(V_j) \\to \\mathcal{O}(X_{f_j}) = A_{f_j}$", "are of finite type. Thus we may choose $a_1, \\ldots, a_N \\in A$ such that", "the elements $a_1, \\ldots, a_N, 1/f_j$ generate", "$A_{f_j}$ over $\\mathcal{O}(V_j)$ for each $j$. Take $n = m + N$ and", "let", "$$", "i : X \\longrightarrow \\mathbf{A}^n_S", "$$", "be the morphism given by the global sections", "$f_1, \\ldots, f_m, a_1, \\ldots, a_N$ of the structure sheaf of $X$.", "Let $D(x_j) \\subset \\mathbf{A}^n_S$ be the open subscheme where the", "$j$th coordinate function is nonzero.", "Then for $1 \\leq j \\leq m$ we have $i^{-1}(D(x_j)) = X_{f_j}$ and", "the induced morphism $X_{f_j} \\to D(x_j)$ factors through the affine", "open $\\Spec(\\mathcal{O}(V_j)[x_1, \\ldots, x_n, 1/x_j])$", "of $D(x_j)$. Since the ring map", "$\\mathcal{O}(V_j)[x_1, \\ldots, x_n, 1/x_j] \\to A_{f_j}$ is", "surjective by construction we conclude that $i^{-1}(D(x_j)) \\to D(x_j)$", "is an immersion as desired." ], "refs": [ "properties-lemma-quasi-affine", "properties-lemma-quasi-affine", "morphisms-lemma-locally-finite-type-characterize" ], "ref_ids": [ 3009, 3009, 5198 ] } ], "ref_ids": [] }, { "id": 5393, "type": "theorem", "label": "morphisms-lemma-quasi-projective-finite-type-over-S", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-projective-finite-type-over-S", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "Assume that", "\\begin{enumerate}", "\\item the invertible sheaf $\\mathcal{L}$ is ample on $X$, and", "\\item the morphism $X \\to S$ is locally of finite type.", "\\end{enumerate}", "Then there exists a $d_0 \\geq 1$ such that for every $d \\geq d_0$", "there exists an $n \\geq 0$ and an immersion", "$i : X \\to \\mathbf{P}^n_S$ over $S$ such that", "$\\mathcal{L}^{\\otimes d} \\cong i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1)$." ], "refs": [], "proofs": [ { "contents": [ "Let", "$A = \\Gamma_*(X, \\mathcal{L}) =", "\\bigoplus_{d \\geq 0} \\Gamma(X, \\mathcal{L}^{\\otimes d})$.", "By Properties, Proposition \\ref{properties-proposition-characterize-ample}", "the set of affine opens $X_a$ with $a \\in A_{+}$ homogeneous forms", "a basis for the topology of $X$. Hence we can find finitely", "many such elements $a_0, \\ldots, a_n \\in A_{+}$ such that", "\\begin{enumerate}", "\\item we have $X = \\bigcup_{i = 0, \\ldots, n} X_{a_i}$,", "\\item each $X_{a_i}$ is affine, and", "\\item each $X_{a_i}$ maps into an affine open $V_i \\subset S$.", "\\end{enumerate}", "By Lemma \\ref{lemma-locally-finite-type-characterize}", "we see that the ring maps", "$\\mathcal{O}_S(V_i) \\to \\mathcal{O}_X(X_{a_i})$ are", "of finite type. Hence we can find finitely many", "elements $f_{ij} \\in \\mathcal{O}_X(X_{a_i})$, $j = 1, \\ldots, n_i$", "which generate $\\mathcal{O}_X(X_{a_i})$ as an $\\mathcal{O}_S(V_i)$-algebra.", "By Properties, Lemma \\ref{properties-lemma-invert-s-sections}", "we may write each", "$f_{ij}$ as $a_{ij}/a_i^{e_{ij}}$ for some", "$a_{ij} \\in A_{+}$ homogeneous. Let $N$ be a positive integer which", "is a common multiple of all the degrees of the elements", "$a_i$, $a_{ij}$. Consider the elements", "$$", "a_i^{N/\\deg(a_i)}, \\ a_{ij}a_i^{(N/\\deg(a_i)) - e_{ij}} \\in A_N.", "$$", "By construction these generate the invertible sheaf", "$\\mathcal{L}^{\\otimes N}$ over $X$. Hence they give rise", "to a morphism", "$$", "j : X \\longrightarrow", "\\mathbf{P}_S^{m}", "\\quad", "\\text{with } m = n + \\sum n_i", "$$", "over $S$, see Constructions, Lemma \\ref{constructions-lemma-projective-space}", "and Definition \\ref{constructions-definition-projective-space}.", "Moreover, $j^*\\mathcal{O}_{\\mathbf{P}_S}(1) = \\mathcal{L}^{\\otimes N}$.", "We name the homogeneous coordinates $T_0, \\ldots, T_n, T_{ij}$", "instead of $T_0, \\ldots, T_m$.", "For $i = 0, \\ldots, n$ we have $i^{-1}(D_{+}(T_i)) = X_{a_i}$.", "Moreover, pulling back the element $T_{ij}/T_i$ via $j^\\sharp$ we", "get the element $f_{ij} \\in \\mathcal{O}_X(X_{a_i})$.", "Hence the morphism $j$ restricted to $X_{a_i}$", "gives a closed immersion of $X_{a_i}$ into the affine open", "$D_{+}(T_i) \\cap \\mathbf{P}^m_{V_i}$ of $\\mathbf{P}^N_S$.", "Hence we conclude that the morphism $j$ is an immersion.", "This implies the lemma holds for some $d$ and $n$ which is enough", "in virtually all applications.", "\\medskip\\noindent", "This proves that for one $d_2 \\geq 1$", "(namely $d_2 = N$ above), some $m \\geq 0$ there exists some", "immersion $j : X \\to \\mathbf{P}^m_S$ given by global sections", "$s'_0, \\ldots, s'_m \\in \\Gamma(X, \\mathcal{L}^{\\otimes d_2})$.", "By Properties, Proposition \\ref{properties-proposition-characterize-ample}", "we know there exists an integer", "$d_1$ such that $\\mathcal{L}^{\\otimes d}$ is globally generated", "for all $d \\geq d_1$. Set $d_0 = d_1 + d_2$. We claim that", "the lemma holds with this value of $d_0$. Namely, given", "an integer $d \\geq d_0$ we may choose $s''_1, \\ldots, s''_t", "\\in \\Gamma(X, \\mathcal{L}^{\\otimes d - d_2})$ which generate", "$\\mathcal{L}^{\\otimes d - d_2}$ over $X$. Set $k = (m + 1)t$ and", "denote $s_0, \\ldots, s_k$ the collection of sections", "$s'_\\alpha s''_\\beta$, $\\alpha = 0, \\ldots, m$,", "$\\beta = 1, \\ldots, t$. These generate $\\mathcal{L}^{\\otimes d}$", "over $X$ and therefore define a morphism", "$$", "i : X \\longrightarrow \\mathbf{P}^{k - 1}_S", "$$", "such that $i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1) \\cong \\mathcal{L}^{\\otimes d}$.", "To see that $i$ is an immersion, observe that $i$ is the composition", "$$", "X \\longrightarrow \\mathbf{P}^m_S \\times_S \\mathbf{P}^{t - 1}_S", "\\longrightarrow \\mathbf{P}^{k - 1}_S", "$$", "where the first morphism is $(j, j')$ with $j'$ given by", "$s''_1, \\ldots, s''_t$ and the", "second morphism is the Segre embedding", "(Constructions, Lemma \\ref{constructions-lemma-segre-embedding}).", "Since $j$ is an immersion, so is $(j, j')$", "(apply Lemma \\ref{lemma-immersion-permanence}", "to $X \\to \\mathbf{P}^m_S \\times_S \\mathbf{P}^{t - 1}_S", "\\to \\mathbf{P}^m_S$). Thus $i$ is a composition of", "immersions and hence an immersion", "(Schemes, Lemma \\ref{schemes-lemma-composition-immersion})." ], "refs": [ "properties-proposition-characterize-ample", "morphisms-lemma-locally-finite-type-characterize", "properties-lemma-invert-s-sections", "constructions-lemma-projective-space", "constructions-definition-projective-space", "properties-proposition-characterize-ample", "constructions-lemma-segre-embedding", "morphisms-lemma-immersion-permanence", "schemes-lemma-composition-immersion" ], "ref_ids": [ 3067, 5198, 3005, 12621, 12664, 3067, 12624, 5132, 7732 ] } ], "ref_ids": [] }, { "id": 5394, "type": "theorem", "label": "morphisms-lemma-finite-type-over-affine-ample-very-ample", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-type-over-affine-ample-very-ample", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Assume $S$ affine and $f$ of finite type.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is ample on $X$,", "\\item $\\mathcal{L}$ is $f$-ample,", "\\item $\\mathcal{L}^{\\otimes d}$ is $f$-very ample for some $d \\geq 1$,", "\\item $\\mathcal{L}^{\\otimes d}$ is $f$-very ample for all $d \\gg 1$,", "\\item for some $d \\geq 1$ there exist $n \\geq 1$ and an immersion", "$i : X \\to \\mathbf{P}^n_S$ such that", "$\\mathcal{L}^{\\otimes d} \\cong i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1)$, and", "\\item for all $d \\gg 1$ there exist $n \\geq 1$ and an immersion", "$i : X \\to \\mathbf{P}^n_S$ such that", "$\\mathcal{L}^{\\otimes d} \\cong i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is Lemma \\ref{lemma-ample-over-affine}.", "The implication (2) $\\Rightarrow$ (6) is", "Lemma \\ref{lemma-quasi-projective-finite-type-over-S}.", "Trivially (6) implies (5).", "As $\\mathbf{P}^n_S$ is a projective bundle over $S$ (see", "Constructions, Lemma \\ref{constructions-lemma-projective-space-bundle})", "we see that", "(5) implies (3) and (6) implies (4) from the definition of a", "relatively very ample sheaf.", "Trivially (4) implies (3). To finish we have to show that", "(3) implies (2) which follows from Lemma \\ref{lemma-ample-very-ample}", "and Lemma \\ref{lemma-ample-power-ample}." ], "refs": [ "morphisms-lemma-ample-over-affine", "morphisms-lemma-quasi-projective-finite-type-over-S", "constructions-lemma-projective-space-bundle", "morphisms-lemma-ample-very-ample", "morphisms-lemma-ample-power-ample" ], "ref_ids": [ 5381, 5393, 12652, 5387, 5378 ] } ], "ref_ids": [] }, { "id": 5395, "type": "theorem", "label": "morphisms-lemma-finite-type-ample-very-ample", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-type-ample-very-ample", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Assume $S$ quasi-compact and $f$ of finite type.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is $f$-ample,", "\\item $\\mathcal{L}^{\\otimes d}$ is $f$-very ample for some $d \\geq 1$,", "\\item $\\mathcal{L}^{\\otimes d}$ is $f$-very ample for all $d \\gg 1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Trivially (3) implies (2). Lemma \\ref{lemma-ample-very-ample} guarantees that", "(2) implies (1) since a morphism of finite type is quasi-compact", "by definition. Assume that $\\mathcal{L}$ is $f$-ample. Choose a finite affine", "open covering $S = V_1 \\cup \\ldots \\cup V_m$. Write $X_i = f^{-1}(V_i)$.", "By Lemma \\ref{lemma-finite-type-over-affine-ample-very-ample} above we see", "there exists a $d_0$ such that $\\mathcal{L}^{\\otimes d}$ is", "relatively very ample on $X_i/V_i$ for all $d \\geq d_0$. Hence we conclude", "(1) implies (3) by Lemma \\ref{lemma-relatively-very-ample}." ], "refs": [ "morphisms-lemma-ample-very-ample", "morphisms-lemma-finite-type-over-affine-ample-very-ample", "morphisms-lemma-relatively-very-ample" ], "ref_ids": [ 5387, 5394, 5389 ] } ], "ref_ids": [] }, { "id": 5396, "type": "theorem", "label": "morphisms-lemma-characterize-very-ample-on-finite-type", "categories": [ "morphisms" ], "title": "morphisms-lemma-characterize-very-ample-on-finite-type", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "Assume $f$ is of finite type.", "The following are equivalent:", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is $f$-relatively very ample, and", "\\item there exist an open covering $S = \\bigcup V_j$,", "for each $j$ an integer $n_j$, and immersions", "$$", "i_j :", "X_j = f^{-1}(V_j) = V_j \\times_S X", "\\longrightarrow", "\\mathbf{P}^{n_j}_{V_j}", "$$", "over $V_j$ such that", "$\\mathcal{L}|_{X_j} \\cong i_j^*\\mathcal{O}_{\\mathbf{P}^{n_j}_{V_j}}(1)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We see that (1) implies (2) by taking an affine open covering of $S$", "and applying Lemma \\ref{lemma-very-ample-finite-type-over-affine} to", "each of the restrictions of $f$ and", "$\\mathcal{L}$. We see that (2) implies (1) by", "Lemma \\ref{lemma-relatively-very-ample}." ], "refs": [ "morphisms-lemma-very-ample-finite-type-over-affine", "morphisms-lemma-relatively-very-ample" ], "ref_ids": [ 5391, 5389 ] } ], "ref_ids": [] }, { "id": 5397, "type": "theorem", "label": "morphisms-lemma-characterize-ample-on-finite-type", "categories": [ "morphisms" ], "title": "morphisms-lemma-characterize-ample-on-finite-type", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "Assume $f$ is of finite type.", "The following are equivalent:", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is $f$-relatively ample, and", "\\item there exist an open covering $S = \\bigcup V_j$,", "for each $j$ an integers $d_j \\geq 1$,", "$n_j \\geq 0$, and immersions", "$$", "i_j :", "X_j = f^{-1}(V_j) = V_j \\times_S X", "\\longrightarrow", "\\mathbf{P}^{n_j}_{V_j}", "$$", "over $V_j$ such that", "$\\mathcal{L}^{\\otimes d_j}|_{X_j} \\cong", "i_j^*\\mathcal{O}_{\\mathbf{P}^{n_j}_{V_j}}(1)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We see that (1) implies (2) by taking an affine open covering of $S$", "and applying Lemma \\ref{lemma-finite-type-over-affine-ample-very-ample} to", "each of the restrictions of $f$ and", "$\\mathcal{L}$. We see that (2) implies (1) by", "Lemma \\ref{lemma-characterize-relatively-ample}." ], "refs": [ "morphisms-lemma-finite-type-over-affine-ample-very-ample", "morphisms-lemma-characterize-relatively-ample" ], "ref_ids": [ 5394, 5380 ] } ], "ref_ids": [] }, { "id": 5398, "type": "theorem", "label": "morphisms-lemma-invertible-add-enough-ample-very-ample", "categories": [ "morphisms" ], "title": "morphisms-lemma-invertible-add-enough-ample-very-ample", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{N}$, $\\mathcal{L}$ be invertible $\\mathcal{O}_X$-modules.", "Assume $S$ is quasi-compact, $f$ is of finite type, and $\\mathcal{L}$", "is $f$-ample. Then", "$\\mathcal{N} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}$", "is $f$-very ample for all $d \\gg 1$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-characterize-very-ample-on-finite-type}", "we reduce to the case $S$ is affine. Combining", "Lemma \\ref{lemma-finite-type-over-affine-ample-very-ample} and", "Properties, Proposition \\ref{properties-proposition-characterize-ample}", "we can find an integer $d_0$ such that", "$\\mathcal{N} \\otimes \\mathcal{L}^{\\otimes d_0}$", "is globally generated. Choose global sections", "$s_0, \\ldots, s_n$ of $\\mathcal{N} \\otimes \\mathcal{L}^{\\otimes d_0}$", "which generate it. This determines a morphism", "$j : X \\to \\mathbf{P}^n_S$ over $S$. By", "Lemma \\ref{lemma-finite-type-over-affine-ample-very-ample}", "we can also pick an integer $d_1$ such that for all $d \\geq d_1$", "there exist sections $t_{d, 0}, \\ldots, t_{d, n(d)}$", "of $\\mathcal{L}^{\\otimes d}$ which generate it and define", "an immersion", "$$", "j_d = \\varphi_{\\mathcal{L}^{\\otimes d}, t_{d, 0}, \\ldots, t_{d, n(d)}} :", "X", "\\longrightarrow", "\\mathbf{P}^{n(d)}_S", "$$", "over $S$. Then for $d \\geq d_0 + d_1$ we can consider the", "morphism", "$$", "\\varphi_{\\mathcal{N} \\otimes \\mathcal{L}^{\\otimes d},", "s_j \\otimes t_{d - d_0, i}} :", "X", "\\longrightarrow", "\\mathbf{P}^{(n + 1)(n(d - d_0) + 1) - 1}_S", "$$", "This morphism is an immersion as it is the composition", "$$", "S \\to \\mathbf{P}^n_S \\times_S \\mathbf{P}^{n(d - d_0)}_S", "\\to \\mathbf{P}^{(n + 1)(n(d - d_0) + 1) - 1}_S", "$$", "where the first morphism is $(j, j_{d - d_0})$ and", "the second is the Segre embedding", "(Constructions, Lemma \\ref{constructions-lemma-segre-embedding}).", "Since $j$ is an immersion, so is $(j, j_{d - d_0})$", "(apply Lemma \\ref{lemma-immersion-permanence}). We have a composition of", "immersions and hence an immersion", "(Schemes, Lemma \\ref{schemes-lemma-composition-immersion})." ], "refs": [ "morphisms-lemma-characterize-very-ample-on-finite-type", "morphisms-lemma-finite-type-over-affine-ample-very-ample", "properties-proposition-characterize-ample", "morphisms-lemma-finite-type-over-affine-ample-very-ample", "constructions-lemma-segre-embedding", "morphisms-lemma-immersion-permanence", "schemes-lemma-composition-immersion" ], "ref_ids": [ 5396, 5394, 3067, 5394, 12624, 5132, 7732 ] } ], "ref_ids": [] }, { "id": 5399, "type": "theorem", "label": "morphisms-lemma-base-change-quasi-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-quasi-projective", "contents": [ "A base change of a quasi-projective morphism is quasi-projective." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemmas \\ref{lemma-base-change-finite-type} and", "\\ref{lemma-ample-base-change}." ], "refs": [ "morphisms-lemma-base-change-finite-type", "morphisms-lemma-ample-base-change" ], "ref_ids": [ 5200, 5385 ] } ], "ref_ids": [] }, { "id": 5400, "type": "theorem", "label": "morphisms-lemma-composition-quasi-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-quasi-projective", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to S$ be morphisms of schemes.", "If $S$ is quasi-compact and $f$ and $g$ are quasi-projective,", "then $g \\circ f$ is quasi-projective." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemmas \\ref{lemma-composition-finite-type} and", "\\ref{lemma-ample-composition}." ], "refs": [ "morphisms-lemma-composition-finite-type", "morphisms-lemma-ample-composition" ], "ref_ids": [ 5199, 5384 ] } ], "ref_ids": [] }, { "id": 5401, "type": "theorem", "label": "morphisms-lemma-quasi-projective-properties", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-projective-properties", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. If $f$ is quasi-projective,", "or H-quasi-projective or locally quasi-projective, then $f$ is", "separated of finite type." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5402, "type": "theorem", "label": "morphisms-lemma-H-quasi-projective-quasi-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-H-quasi-projective-quasi-projective", "contents": [ "A H-quasi-projective morphism is quasi-projective." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5403, "type": "theorem", "label": "morphisms-lemma-characterize-locally-quasi-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-characterize-locally-quasi-projective", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is locally quasi-projective.", "\\item There exists an open covering $S = \\bigcup V_j$ such", "that each $f^{-1}(V_j) \\to V_j$ is H-quasi-projective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-H-quasi-projective-quasi-projective}", "we see that (2) implies (1). Assume (1).", "The question is local on $S$ and hence we may assume $S$ is affine,", "$X$ of finite type over $S$ and", "$\\mathcal{L}$ is a relatively ample invertible sheaf on $X/S$.", "By Lemma \\ref{lemma-finite-type-over-affine-ample-very-ample}", "we may assume $\\mathcal{L}$ is ample on $X$.", "By Lemma \\ref{lemma-quasi-projective-finite-type-over-S} we see that there", "exists an immersion of $X$ into", "a projective space over $S$, i.e., $X$ is H-quasi-projective over $S$", "as desired." ], "refs": [ "morphisms-lemma-H-quasi-projective-quasi-projective", "morphisms-lemma-finite-type-over-affine-ample-very-ample", "morphisms-lemma-quasi-projective-finite-type-over-S" ], "ref_ids": [ 5402, 5394, 5393 ] } ], "ref_ids": [] }, { "id": 5404, "type": "theorem", "label": "morphisms-lemma-quasi-affine-finite-type-quasi-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-affine-finite-type-quasi-projective", "contents": [ "\\begin{reference}", "\\cite[II, Proposition 5.3.4 (i)]{EGA}", "\\end{reference}", "A quasi-affine morphism of finite type is quasi-projective." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-quasi-affine-O-ample}." ], "refs": [ "morphisms-lemma-quasi-affine-O-ample" ], "ref_ids": [ 5382 ] } ], "ref_ids": [] }, { "id": 5405, "type": "theorem", "label": "morphisms-lemma-quasi-projective-permanence", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-projective-permanence", "contents": [ "Let $g : Y \\to S$ and $f : X \\to Y$ be morphisms of schemes.", "If $g \\circ f$ is quasi-projective and $f$ is", "quasi-compact\\footnote{This follows if $g$ is quasi-separated by", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}.},", "then $f$ is quasi-projective." ], "refs": [ "schemes-lemma-quasi-compact-permanence" ], "proofs": [ { "contents": [ "Observe that $f$ is of finite type by", "Lemma \\ref{lemma-permanence-finite-type}.", "Thus the lemma follows from Lemma \\ref{lemma-ample-permanence}", "and the definitions." ], "refs": [ "morphisms-lemma-permanence-finite-type", "morphisms-lemma-ample-permanence" ], "ref_ids": [ 5204, 5386 ] } ], "ref_ids": [ 7716 ] }, { "id": 5406, "type": "theorem", "label": "morphisms-lemma-universally-closed-local-on-the-base", "categories": [ "morphisms" ], "title": "morphisms-lemma-universally-closed-local-on-the-base", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is universally closed.", "\\item There exists an open covering $S = \\bigcup V_j$ such", "that $f^{-1}(V_j) \\to V_j$ is universally closed for all indices $j$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5407, "type": "theorem", "label": "morphisms-lemma-proper-local-on-the-base", "categories": [ "morphisms" ], "title": "morphisms-lemma-proper-local-on-the-base", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is proper.", "\\item There exists an open covering $S = \\bigcup V_j$ such", "that $f^{-1}(V_j) \\to V_j$ is proper for all indices $j$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5408, "type": "theorem", "label": "morphisms-lemma-composition-proper", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-proper", "contents": [ "The composition of proper morphisms is proper.", "The same is true for universally closed morphisms." ], "refs": [], "proofs": [ { "contents": [ "A composition of closed morphisms is closed.", "If $X \\to Y \\to Z$ are universally closed morphisms", "and $Z' \\to Z$ is any morphism, then we see that", "$Z' \\times_Z X = (Z' \\times_Z Y) \\times_Y X \\to Z' \\times_Z Y$", "is closed and $Z' \\times_Z Y \\to Z'$ is closed.", "Hence the result for universally closed morphisms.", "We have seen that ``separated'' and ``finite type''", "are preserved under compositions", "(Schemes, Lemma \\ref{schemes-lemma-separated-permanence} and", "Lemma \\ref{lemma-composition-finite-type}). Hence the", "result for proper morphisms." ], "refs": [ "schemes-lemma-separated-permanence", "morphisms-lemma-composition-finite-type" ], "ref_ids": [ 7714, 5199 ] } ], "ref_ids": [] }, { "id": 5409, "type": "theorem", "label": "morphisms-lemma-base-change-proper", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-proper", "contents": [ "The base change of a proper morphism is proper.", "The same is true for universally closed morphisms." ], "refs": [], "proofs": [ { "contents": [ "This is true by definition for universally closed morphisms.", "It is true for separated morphisms", "(Schemes, Lemma \\ref{schemes-lemma-separated-permanence}).", "It is true for morphisms of finite type", "(Lemma \\ref{lemma-base-change-finite-type}).", "Hence it is true for proper morphisms." ], "refs": [ "schemes-lemma-separated-permanence", "morphisms-lemma-base-change-finite-type" ], "ref_ids": [ 7714, 5200 ] } ], "ref_ids": [] }, { "id": 5410, "type": "theorem", "label": "morphisms-lemma-closed-immersion-proper", "categories": [ "morphisms" ], "title": "morphisms-lemma-closed-immersion-proper", "contents": [ "A closed immersion is proper, hence a fortiori universally closed." ], "refs": [], "proofs": [ { "contents": [ "The base change of a closed immersion is a closed immersion", "(Schemes, Lemma \\ref{schemes-lemma-base-change-immersion}).", "Hence it is universally closed.", "A closed immersion is separated", "(Schemes, Lemma \\ref{schemes-lemma-immersions-monomorphisms}).", "A closed immersion is of finite type", "(Lemma \\ref{lemma-immersion-locally-finite-type}).", "Hence a closed immersion is proper." ], "refs": [ "schemes-lemma-base-change-immersion", "schemes-lemma-immersions-monomorphisms", "morphisms-lemma-immersion-locally-finite-type" ], "ref_ids": [ 7695, 7727, 5201 ] } ], "ref_ids": [] }, { "id": 5411, "type": "theorem", "label": "morphisms-lemma-image-proper-scheme-closed", "categories": [ "morphisms" ], "title": "morphisms-lemma-image-proper-scheme-closed", "contents": [ "Suppose given a commutative diagram of schemes", "$$", "\\xymatrix{", "X \\ar[rr] \\ar[rd] & &", "Y \\ar[ld] \\\\", "& S &", "}", "$$", "with $Y$ separated over $S$.", "\\begin{enumerate}", "\\item If $X \\to S$ is universally closed, then the morphism", "$X \\to Y$ is universally closed.", "\\item If $X$ is proper over $S$, then the morphism $X \\to Y$ is proper.", "\\end{enumerate}", "In particular, in both cases the image of $X$ in $Y$ is closed." ], "refs": [], "proofs": [ { "contents": [ "Assume that $X \\to S$ is universally closed (resp.\\ proper).", "We factor the morphism as $X \\to X \\times_S Y \\to Y$.", "The first morphism is a closed immersion, see", "Schemes, Lemma \\ref{schemes-lemma-semi-diagonal}.", "Hence the first morphism is proper (Lemma \\ref{lemma-closed-immersion-proper}).", "The projection $X \\times_S Y \\to Y$ is the base change", "of a universally closed (resp.\\ proper) morphism and hence", "universally closed (resp.\\ proper), see Lemma \\ref{lemma-base-change-proper}.", "Thus $X \\to Y$ is universally closed (resp.\\ proper) as the composition", "of universally closed (resp.\\ proper) morphisms", "(Lemma \\ref{lemma-composition-proper})." ], "refs": [ "schemes-lemma-semi-diagonal", "morphisms-lemma-closed-immersion-proper", "morphisms-lemma-base-change-proper", "morphisms-lemma-composition-proper" ], "ref_ids": [ 7712, 5410, 5409, 5408 ] } ], "ref_ids": [] }, { "id": 5412, "type": "theorem", "label": "morphisms-lemma-universally-closed-quasi-compact", "categories": [ "morphisms" ], "title": "morphisms-lemma-universally-closed-quasi-compact", "contents": [ "\\begin{reference}", "Due to Bjorn Poonen.", "\\end{reference}", "A universally closed morphism of schemes is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be a morphism. Assume that $f$ is not quasi-compact.", "Our goal is to show that $f$ is not universally closed. By", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-affine}", "there exists an affine open $V \\subset S$ such that $f^{-1}(V)$ is", "not quasi-compact. To achieve our goal it suffices to show that", "$f^{-1}(V) \\to V$ is not universally closed, hence we may assume that", "$S = \\Spec(A)$ for some ring $A$.", "\\medskip\\noindent", "Write $X = \\bigcup_{i \\in I} X_i$ where the $X_i$ are affine open subschemes", "of $X$. Let $T = \\Spec(A[y_i ; i \\in I])$.", "Let $T_i = D(y_i) \\subset T$. Let $Z$ be the closed set", "$(X \\times_S T) - \\bigcup_{i \\in I} (X_i \\times_S T_i)$. It suffices to", "prove that the image $f_T(Z)$ of $Z$ under $f_T : X \\times_S T \\to T$", "is not closed.", "\\medskip\\noindent", "There exists a point $s \\in S$ such that there is no", "neighborhood $U$ of $s$ in $S$ such that $X_U$ is quasi-compact.", "Otherwise we could cover $S$ with finitely many such $U$ and", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-affine}", "would imply $f$ quasi-compact. Fix such an $s \\in S$.", "\\medskip\\noindent", "First we check that $f_T(Z_s) \\ne T_s$. Let $t \\in T$ be the point", "lying over $s$ with $\\kappa(t) = \\kappa(s)$ such that $y_i = 1$ in", "$\\kappa(t)$ for all $i$.", "Then $t \\in T_i$ for all $i$, and the fiber of $Z_s \\to T_s$ above", "$t$ is isomorphic to $(X - \\bigcup_{i \\in I} X_i)_s$, which is empty.", "Thus $t \\in T_s - f_T(Z_s)$.", "\\medskip\\noindent", "Assume $f_T(Z)$ is closed in $T$. Then there exists an element", "$g \\in A[y_i; i \\in I]$ with $f_T(Z) \\subset V(g)$ but $t \\not \\in V(g)$.", "Hence the image of $g$ in $\\kappa(t)$ is nonzero. In particular some", "coefficient of $g$ has nonzero image in $\\kappa(s)$. Hence this coefficient is", "invertible on some neighborhood $U$ of $s$. Let $J$ be the finite set of", "$j \\in I$ such that $y_j$ appears in $g$. Since $X_U$ is not quasi-compact,", "we may choose a point $x \\in X - \\bigcup_{j \\in J} X_j$ lying above some", "$u \\in U$. Since $g$ has a coefficient that is invertible on $U$, we can", "find a point $t' \\in T$ lying above $u$ such that $t' \\not \\in V(g)$ and", "$t' \\in V(y_i)$ for all $i \\notin J$. This is true because", "$V(y_i; i \\in I, i \\not\\in J) = \\Spec(A[t_j; j\\in J])$", "and the set of points of this scheme lying over $u$ is bijective", "with $\\Spec(\\kappa(u)[t_j; j \\in J])$. In other words $t' \\notin T_i$", "for each $i \\notin J$. By", "Schemes, Lemma \\ref{schemes-lemma-points-fibre-product}", "we can find a point $z$ of $X \\times_S T$ mapping to $x \\in X$ and to", "$t' \\in T$. Since $x \\not \\in X_j$ for $j \\in J$ and $t' \\not \\in T_i$", "for $i \\in I \\setminus J$ we see that $z \\in Z$. On the other hand", "$f_T(z) = t' \\not \\in V(g)$ which contradicts $f_T(Z) \\subset V(g)$.", "Thus the assumption ``$f_T(Z)$ closed'' is wrong and we conclude indeed", "that $f_T$ is not closed, as desired." ], "refs": [ "schemes-lemma-quasi-compact-affine", "schemes-lemma-quasi-compact-affine", "schemes-lemma-points-fibre-product" ], "ref_ids": [ 7697, 7697, 7693 ] } ], "ref_ids": [] }, { "id": 5413, "type": "theorem", "label": "morphisms-lemma-image-proper-is-proper", "categories": [ "morphisms" ], "title": "morphisms-lemma-image-proper-is-proper", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$.", "If $X$ is universally closed over $S$ and $f$ is surjective then", "$Y$ is universally closed over $S$. In particular, if also $Y$ is", "separated and locally of finite type over $S$, then $Y$ is proper over $S$." ], "refs": [], "proofs": [ { "contents": [ "Assume $X$ is universally closed and $f$ surjective.", "Denote $p : X \\to S$, $q : Y \\to S$ the structure morphisms.", "Let $S' \\to S$ be a morphism of schemes. The base change", "$f' : X_{S'} \\to Y_{S'}$ is surjective", "(Lemma \\ref{lemma-base-change-surjective}), and the base", "change $p' : X_{S'} \\to S'$ is closed.", "If $T \\subset Y_{S'}$ is closed, then $(f')^{-1}(T) \\subset X_{S'}$", "is closed, hence $p'((f')^{-1}(T)) = q'(T)$ is closed.", "So $q'$ is closed. This proves the first statement.", "Thus $Y \\to S$ is quasi-compact by", "Lemma \\ref{lemma-universally-closed-quasi-compact}", "and hence $Y \\to S$ is proper by definition", "if in addition $Y \\to S$ is locally of finite type and separated." ], "refs": [ "morphisms-lemma-base-change-surjective", "morphisms-lemma-universally-closed-quasi-compact" ], "ref_ids": [ 5165, 5412 ] } ], "ref_ids": [] }, { "id": 5414, "type": "theorem", "label": "morphisms-lemma-scheme-theoretic-image-is-proper", "categories": [ "morphisms" ], "title": "morphisms-lemma-scheme-theoretic-image-is-proper", "contents": [ "Suppose given a commutative diagram of schemes", "$$", "\\xymatrix{", "X \\ar[rr]_h \\ar[rd]_f & & Y \\ar[ld]^g \\\\", "& S", "}", "$$", "Assume", "\\begin{enumerate}", "\\item $X \\to S$ is a universally closed (for example proper) morphism, and", "\\item $Y \\to S$ is separated and locally of finite type.", "\\end{enumerate}", "Then the scheme theoretic image $Z \\subset Y$ of $h$", "is proper over $S$ and $X \\to Z$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "The scheme theoretic image of $h$ is constructed in Section", "\\ref{section-scheme-theoretic-image}. Since $f$ is quasi-compact", "(Lemma \\ref{lemma-universally-closed-quasi-compact})", "we find that $h$ is quasi-compact", "(Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}).", "Hence $h(X) \\subset Z$", "is dense (Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image}).", "On the other hand $h(X)$ is closed in $Y$", "(Lemma \\ref{lemma-image-proper-scheme-closed})", "hence $X \\to Z$ is surjective.", "Thus $Z \\to S$ is a proper (Lemma \\ref{lemma-image-proper-is-proper})." ], "refs": [ "morphisms-lemma-universally-closed-quasi-compact", "schemes-lemma-quasi-compact-permanence", "morphisms-lemma-quasi-compact-scheme-theoretic-image", "morphisms-lemma-image-proper-scheme-closed", "morphisms-lemma-image-proper-is-proper" ], "ref_ids": [ 5412, 7716, 5146, 5411, 5413 ] } ], "ref_ids": [] }, { "id": 5415, "type": "theorem", "label": "morphisms-lemma-image-universally-closed-separated", "categories": [ "morphisms" ], "title": "morphisms-lemma-image-universally-closed-separated", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a surjective universally closed", "morphism of schemes over $S$.", "\\begin{enumerate}", "\\item If $X$ is quasi-separated, then $Y$ is quasi-separated.", "\\item If $X$ is separated, then $Y$ is separated.", "\\item If $X$ is quasi-separated over $S$, then $Y$ is quasi-separated over $S$.", "\\item If $X$ is separated over $S$, then $Y$ is separated over $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1) and (2) are a consequence of (3) and (4) for", "$S = \\Spec(\\mathbf{Z})$ (see", "Schemes, Definition \\ref{schemes-definition-separated}).", "Consider the commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] \\ar[rr]_{\\Delta_{X/S}} & & X \\times_S X \\ar[d] \\\\", "Y \\ar[rr]^{\\Delta_{Y/S}} & & Y \\times_S Y", "}", "$$", "The left vertical arrow is surjective (i.e., universally surjective).", "The right vertical arrow is universally closed as a composition", "of the universally closed morphisms", "$X \\times_S X \\to X \\times_S Y \\to Y \\times_S Y$. Hence it is also", "quasi-compact, see", "Lemma \\ref{lemma-universally-closed-quasi-compact}.", "\\medskip\\noindent", "Assume $X$ is quasi-separated over $S$, i.e., $\\Delta_{X/S}$ is", "quasi-compact. If $V \\subset Y \\times_S Y$ is a quasi-compact", "open, then $V \\times_{Y \\times_S Y} X \\to \\Delta_{Y/S}^{-1}(V)$", "is surjective and $V \\times_{Y \\times_S Y} X$ is quasi-compact by our remarks", "above. We conclude that $\\Delta_{Y/S}$ is quasi-compact, i.e., $Y$", "is quasi-separated over $S$.", "\\medskip\\noindent", "Assume $X$ is separated over $S$, i.e., $\\Delta_{X/S}$ is a closed", "immersion. Then $X \\to Y \\times_S Y$ is closed as a", "composition of closed morphisms. Since $X \\to Y$ is", "surjective, it follows that $\\Delta_{Y/S}(Y)$ is closed in $Y \\times_S Y$.", "Hence $Y$ is separated over $S$ by the discussion following", "Schemes, Definition \\ref{schemes-definition-separated}." ], "refs": [ "schemes-definition-separated", "morphisms-lemma-universally-closed-quasi-compact", "schemes-definition-separated" ], "ref_ids": [ 7756, 5412, 7756 ] } ], "ref_ids": [] }, { "id": 5416, "type": "theorem", "label": "morphisms-lemma-characterize-proper", "categories": [ "morphisms" ], "title": "morphisms-lemma-characterize-proper", "contents": [ "\\begin{reference}", "\\cite[II Theorem 7.3.8]{EGA}", "\\end{reference}", "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes", "over $S$. Assume $f$ is of finite type and quasi-separated.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $f$ is proper,", "\\item $f$ satisfies the valuative criterion", "(Schemes, Definition \\ref{schemes-definition-valuative-criterion}),", "\\item given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$, there exists", "a unique dotted arrow making the diagram commute.", "\\end{enumerate}" ], "refs": [ "schemes-definition-valuative-criterion" ], "proofs": [ { "contents": [ "Part (3) is a reformulation of (2). Thus the lemma is a formal", "consequence of", "Schemes, Proposition \\ref{schemes-proposition-characterize-universally-closed}", "and Lemma \\ref{schemes-lemma-valuative-criterion-separatedness}", "and the definitions." ], "refs": [ "schemes-proposition-characterize-universally-closed", "schemes-lemma-valuative-criterion-separatedness" ], "ref_ids": [ 7733, 7720 ] } ], "ref_ids": [ 7755 ] }, { "id": 5417, "type": "theorem", "label": "morphisms-lemma-refined-valuative-criterion-universally-closed", "categories": [ "morphisms" ], "title": "morphisms-lemma-refined-valuative-criterion-universally-closed", "contents": [ "Let $f : X \\to S$ and $h : U \\to X$ be morphisms of schemes.", "Assume that $f$ and $h$ are quasi-compact and that $h(U)$ is dense in $X$.", "If given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\", "\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & S", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$, there", "exists a unique dotted arrow making the diagram commute, then $f$", "is universally closed. If moreover $f$ is quasi-separated, then", "$f$ is separated." ], "refs": [], "proofs": [ { "contents": [ "To prove $f$ is universally closed we will verify the existence part of the", "valuative criterion for $f$ which suffices by", "Schemes, Proposition \\ref{schemes-proposition-characterize-universally-closed}.", "To do this, consider a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] & S", "}", "$$", "where $A$ is a valuation ring and $K$ is the fraction field of $A$.", "Note that since valuation rings and fields are reduced, we may", "replace $U$, $X$, and $S$ by their respective reductions", "by Schemes, Lemma \\ref{schemes-lemma-map-into-reduction}.", "In this case the assumption that $h(U)$ is dense means that", "the scheme theoretic image of $h : U \\to X$ is $X$, see", "Lemma \\ref{lemma-scheme-theoretic-image-reduced}.", "We may also replace $S$ by an affine open through which", "the morphism $\\Spec(A) \\to S$ factors. Thus we may", "assume that $S = \\Spec(R)$.", "\\medskip\\noindent", "Let $\\Spec(B) \\subset X$ be an affine open through which", "the morphism $\\Spec(K) \\to X$ factors. Choose a polynomial", "algebra $P$ over $B$ and a $B$-algebra surjection $P \\to K$.", "Then $\\Spec(P) \\to X$ is flat. Hence the scheme theoretic image", "of the morphism $U \\times_X \\Spec(P) \\to \\Spec(P)$ is $\\Spec(P)$ by", "Lemma \\ref{lemma-flat-base-change-scheme-theoretic-image}.", "By Lemma \\ref{lemma-reach-points-scheme-theoretic-image}", "we can find a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & U \\times_X \\Spec(P) \\ar[d] \\\\", "\\Spec(A') \\ar[r] & \\Spec(P)", "}", "$$", "where $A'$ is a valuation ring and $K'$ is the fraction field of $A'$", "such that the closed point of $\\Spec(A')$ maps to $\\Spec(K) \\subset \\Spec(P)$.", "In other words, there is a $B$-algebra map", "$\\varphi : K \\to A'/\\mathfrak m_{A'}$. Choose a valuation ring", "$A'' \\subset A'/\\mathfrak m_{A'}$ dominating $\\varphi(A)$ with", "field of fractions $K'' = A'/\\mathfrak m_{A'}$", "(Algebra, Lemma \\ref{algebra-lemma-dominate}). We set", "$$", "C = \\{\\lambda \\in A' \\mid \\lambda \\bmod \\mathfrak m_{A'} \\in A''\\}.", "$$", "which is a valuation ring by", "Algebra, Lemma \\ref{algebra-lemma-stack-valuation-rings}.", "As $C$ is an $R$-algebra with fraction field $K'$, we obtain a", "commutative diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & U \\ar[r] & X \\ar[d] \\\\", "\\Spec(C) \\ar[rr] \\ar@{-->}[rru] & & S", "}", "$$", "as in the statement of the lemma. Thus a dotted arrow fitting into", "the diagram as indicated. By the uniqueness assumption of the lemma", "the composition $\\Spec(A') \\to \\Spec(C) \\to X$ agrees with the", "given morphism $\\Spec(A') \\to \\Spec(P) \\to \\Spec(B) \\subset X$.", "Hence the restriction of the morphism to the spectrum of", "$C/\\mathfrak m_{A'} = A''$ induces the given morphism", "$\\Spec(K'') = \\Spec(A'/\\mathfrak m_{A'}) \\to \\Spec(K) \\to X$.", "Let $x \\in X$ be the image of the closed point of $\\Spec(A'') \\to X$.", "The image of the induced ring map $\\mathcal{O}_{X, x} \\to A''$", "is a local subring which is contained in $K \\subset K''$.", "Since $A$ is maximal for the relation of domination in $K$", "and since $A \\subset A''$, we have $A = K \\cap A''$. We conclude", "that $\\mathcal{O}_{X, x} \\to A''$ factors through $A \\subset A''$.", "In this way we obtain our desired arrow $\\Spec(A) \\to X$.", "\\medskip\\noindent", "Finally, assume $f$ is quasi-separated. Then $\\Delta : X \\to X \\times_S X$", "is quasi-compact. Given a solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^\\Delta \\\\", "\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & X \\times_S X", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$, there", "exists a unique dotted arrow making the diagram commute. Namely,", "the lower horizontal arrow is the same thing as a pair of morphisms", "$\\Spec(A) \\to X$ which can serve as the dotted arrow in the diagram", "of the lemma. Thus the required uniqueness shows that the lower", "horizontal arrow factors through $\\Delta$.", "Hence we can apply the result we just proved to", "$\\Delta : X \\to X \\times_S X$ and $h : U \\to X$ and conclude that", "$\\Delta$ is universally closed. Clearly this means that $f$", "is separated." ], "refs": [ "schemes-proposition-characterize-universally-closed", "schemes-lemma-map-into-reduction", "morphisms-lemma-scheme-theoretic-image-reduced", "morphisms-lemma-flat-base-change-scheme-theoretic-image", "morphisms-lemma-reach-points-scheme-theoretic-image", "algebra-lemma-dominate", "algebra-lemma-stack-valuation-rings" ], "ref_ids": [ 7733, 7682, 5149, 5273, 5147, 608, 615 ] } ], "ref_ids": [] }, { "id": 5418, "type": "theorem", "label": "morphisms-lemma-morphism-defined-local-ring", "categories": [ "morphisms" ], "title": "morphisms-lemma-morphism-defined-local-ring", "contents": [ "Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$.", "Let $s \\in S$ and $x \\in X$, $y \\in Y$ points over $s$.", "\\begin{enumerate}", "\\item Let $f, g : X \\to Y$ be morphisms over $S$ such that", "$f(x) = g(x) = y$ and", "$f^\\sharp_x = g^\\sharp_x : \\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$.", "Then there is an open neighbourhood $U \\subset X$ with", "$f|_U = g|_U$ in the following cases", "\\begin{enumerate}", "\\item $Y$ is locally of finite type over $S$,", "\\item $X$ is integral,", "\\item $X$ is locally Noetherian, or", "\\item $X$ is reduced with finitely many irreducible components.", "\\end{enumerate}", "\\item Let $\\varphi : \\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$", "be a local $\\mathcal{O}_{S, s}$-algebra map. Then there exists", "an open neighbourhood $U \\subset X$ of $x$ and a morphism $f : U \\to Y$", "mapping $x$ to $y$ with $f^\\sharp_x = \\varphi$ in the following cases", "\\begin{enumerate}", "\\item $Y$ is locally of finite presentation over $S$,", "\\item $Y$ is locally of finite type and $X$ is integral,", "\\item $Y$ is locally of finite type and $X$ is locally Noetherian, or", "\\item $Y$ is locally of finite type and $X$ is reduced with finitely", "many irreducible components.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). We may replace $X$, $Y$, $S$ by suitable affine open", "neighbourhoods of $x$, $y$, $s$ and reduce to the following algebra", "problem: given a ring $R$, two $R$-algebra maps $\\varphi, \\psi : B \\to A$", "such that", "\\begin{enumerate}", "\\item $R \\to B$ is of finite type, or $A$ is a domain, or $A$", "is Noetherian, or $A$ is reduced and has finitely many minimal primes,", "\\item the two maps $B \\to A_\\mathfrak p$ are the same for", "some prime $\\mathfrak p \\subset A$,", "\\end{enumerate}", "show that $\\varphi, \\psi$ define the same map $B \\to A_g$ for", "a suitable $g \\in A$, $g \\not \\in \\mathfrak p$. If $R \\to B$ is of", "finite type, let $t_1, \\ldots, t_m \\in B$ be generators of $B$", "as an $R$-algebra. For each $j$ we can find", "$g_j \\in A$, $g_j \\not \\in \\mathfrak p$", "such that $\\varphi(t_j)$ and $\\psi(t_j)$ have the same image in", "$A_{g_j}$. Then we set $g = \\prod g_j$.", "In the other cases (if $A$ is a domain, Noetherian, or reduced", "with finitely many minimal primes), we can find a $g \\in A$,", "$g \\not \\in \\mathfrak p$ such that $A_g \\subset A_\\mathfrak p$.", "See Algebra, Lemma \\ref{algebra-lemma-subring-of-local-ring}.", "Thus the maps $B \\to A_g$ are equal as desired.", "\\medskip\\noindent", "Proof of (2). To do this we may replace $X$, $Y$, and $S$ by suitable affine", "opens. Say $X = \\Spec(A)$, $Y = \\Spec(B)$, and $S = \\Spec(R)$.", "Let $\\mathfrak p \\subset A$ be the prime ideal corresponding to $x$.", "Let $\\mathfrak q \\subset B$ be the prime corresponding to $y$.", "Then $\\varphi$ is a local $R$-algebra map", "$\\varphi : B_\\mathfrak q \\to A_\\mathfrak p$.", "If $R \\to B$ is a ring map of finite presentation, then there exists a", "$g \\in A \\setminus \\mathfrak p$ and an $R$-algebra map $B \\to A_g$ such that", "$$", "\\xymatrix{", "B_\\mathfrak q \\ar[r]_\\varphi & A_\\mathfrak p \\\\", "B \\ar[u] \\ar[r] & A_g \\ar[u]", "}", "$$", "commutes, see", "Algebra, Lemmas \\ref{algebra-lemma-characterize-finite-presentation} and", "\\ref{algebra-lemma-localization-colimit}.", "The induced morphism $\\Spec(A_g) \\to \\Spec(B)$ works.", "If $B$ is of finite type over $R$, let $t_1, \\ldots, t_m \\in B$ be", "generators of $B$ as an $R$-algebra. Then we can choose", "$g_j \\in A$, $g_j \\not \\in \\mathfrak p$", "such that $\\varphi(t_j) \\in \\Im(A_{g_j} \\to A_\\mathfrak p)$.", "Thus after replacing $A$ by $A[1/\\prod g_j]$ we may assume", "that $B$ maps into the image of $A \\to A_\\mathfrak p$.", "If we can find a $g \\in A$, $g \\not \\in \\mathfrak p$", "such that $A_g \\to A_\\mathfrak p$ is injective, then", "we'll get the desired $R$-algebra map $B \\to A_g$.", "Thus the proof is finished by another application of", "See Algebra, Lemma \\ref{algebra-lemma-subring-of-local-ring}." ], "refs": [ "algebra-lemma-subring-of-local-ring", "algebra-lemma-characterize-finite-presentation", "algebra-lemma-localization-colimit", "algebra-lemma-subring-of-local-ring" ], "ref_ids": [ 456, 1092, 348, 456 ] } ], "ref_ids": [] }, { "id": 5419, "type": "theorem", "label": "morphisms-lemma-extend-across", "categories": [ "morphisms" ], "title": "morphisms-lemma-extend-across", "contents": [ "Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Let $x \\in X$.", "Let $U \\subset X$ be an open and let $f : U \\to Y$ be a morphism over $S$.", "Assume", "\\begin{enumerate}", "\\item $x$ is in the closure of $U$,", "\\item $X$ is reduced with finitely many irreducible components or", "$X$ is Noetherian,", "\\item $\\mathcal{O}_{X, x}$ is a valuation ring,", "\\item $Y \\to S$ is proper", "\\end{enumerate}", "Then there exists an open $U \\subset U' \\subset X$ containing", "$x$ and an $S$-morphism $f' : U' \\to Y$ extending $f$." ], "refs": [], "proofs": [ { "contents": [ "It is harmless to replace $X$ by an open neighbourhood of $x$ in $X$", "(small detail omitted). By Properties, Lemma", "\\ref{properties-lemma-ring-affine-open-injective-local-ring}", "we may assume $X$ is affine with", "$\\Gamma(X, \\mathcal{O}_X) \\subset \\mathcal{O}_{X, x}$.", "In particular $X$ is integral with a unique generic point $\\xi$", "whose residue field is the fraction field $K$ of the", "valuation ring $\\mathcal{O}_{X, x}$.", "Since $x$ is in the closure of $U$ we see that $U$ is not", "empty, hence $U$ contains $\\xi$. Thus by the valuative criterion", "of properness (Lemma \\ref{lemma-characterize-proper})", "there is a morphism $t : \\Spec(\\mathcal{O}_{X, x}) \\to Y$", "fitting into a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[d]_\\xi \\ar[r] & \\Spec(\\mathcal{O}_{X, x}) \\ar[d]_t \\\\", "U \\ar[r]^f & Y", "}", "$$", "of morphisms of schemes over $S$. Applying", "Lemma \\ref{lemma-morphism-defined-local-ring}", "with $y = t(x)$ and $\\varphi = t^\\sharp_x$ we obtain an open", "neighbourhood $V \\subset X$ of $x$ and a morphism $g : V \\to Y$", "over $S$ which sends $x$ to $y$ and such that $g^\\sharp_x = t^\\sharp_x$.", "As $Y \\to S$ is separated, the equalizer $E$ of $f|_{U \\cap V}$", "and $g|_{U \\cap V}$ is a closed subscheme of $U \\cap V$, see", "Schemes, Lemma \\ref{schemes-lemma-where-are-they-equal}.", "Since $f$ and $g$ determine the same morphism $\\Spec(K) \\to Y$", "by construction we see that $E$ contains the generic point", "of the integral scheme $U \\cap V$. Hence $E = U \\cap V$ and", "we conclude that $f$ and $g$ glue to a morphism $U' = U \\cup V \\to Y$", "as desired." ], "refs": [ "properties-lemma-ring-affine-open-injective-local-ring", "morphisms-lemma-characterize-proper", "morphisms-lemma-morphism-defined-local-ring", "schemes-lemma-where-are-they-equal" ], "ref_ids": [ 3065, 5416, 5418, 7708 ] } ], "ref_ids": [] }, { "id": 5420, "type": "theorem", "label": "morphisms-lemma-H-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-H-projective", "contents": [ "An H-projective morphism is H-quasi-projective.", "An H-projective morphism is projective." ], "refs": [], "proofs": [ { "contents": [ "The first statement is immediate from the definitions.", "The second holds as $\\mathbf{P}^n_S$ is a projective bundle over $S$, see", "Constructions, Lemma \\ref{constructions-lemma-projective-space-bundle}." ], "refs": [ "constructions-lemma-projective-space-bundle" ], "ref_ids": [ 12652 ] } ], "ref_ids": [] }, { "id": 5421, "type": "theorem", "label": "morphisms-lemma-characterize-locally-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-characterize-locally-projective", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is locally projective.", "\\item There exists an open covering $S = \\bigcup U_i$ such", "that each $f^{-1}(U_i) \\to U_i$ is H-projective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-H-projective} we see that (2) implies (1). Assume (1).", "For every point $s \\in S$ we can find $\\Spec(R) = U \\subset S$", "an affine open neighbourhood of $s$ such that $X_U$ is isomorphic to a", "closed subscheme of $\\mathbf{P}(\\mathcal{E})$ for some finite type,", "quasi-coherent sheaf of $\\mathcal{O}_U$-modules $\\mathcal{E}$.", "Write $\\mathcal{E} = \\widetilde{M}$ for some finite type", "$R$-module $M$ (see", "Properties, Lemma \\ref{properties-lemma-finite-type-module}).", "Choose generators $x_0, \\ldots, x_n \\in M$ of $M$ as an $R$-module.", "Consider the surjective graded $R$-algebra map", "$$", "R[X_0, \\ldots, X_n] \\longrightarrow \\text{Sym}_R(M).", "$$", "According to", "Constructions, Lemma \\ref{constructions-lemma-surjective-graded-rings-map-proj}", "the corresponding morphism", "$$", "\\mathbf{P}(\\mathcal{E}) \\to \\mathbf{P}^n_R", "$$", "is a closed immersion. Hence we conclude that $f^{-1}(U)$ is isomorphic", "to a closed subscheme of $\\mathbf{P}^n_U$ (as a scheme over $U$).", "In other words: (2) holds." ], "refs": [ "morphisms-lemma-H-projective", "properties-lemma-finite-type-module", "constructions-lemma-surjective-graded-rings-map-proj" ], "ref_ids": [ 5420, 3002, 12610 ] } ], "ref_ids": [] }, { "id": 5422, "type": "theorem", "label": "morphisms-lemma-locally-projective-proper", "categories": [ "morphisms" ], "title": "morphisms-lemma-locally-projective-proper", "contents": [ "A locally projective morphism is proper." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be locally projective.", "In order to show that $f$ is proper we may work locally on the", "base, see Lemma \\ref{lemma-proper-local-on-the-base}.", "Hence, by Lemma \\ref{lemma-characterize-locally-projective}", "above we may assume there exists a closed immersion $X \\to \\mathbf{P}^n_S$.", "By Lemmas \\ref{lemma-composition-proper}", "and \\ref{lemma-closed-immersion-proper} it suffices to prove that", "$\\mathbf{P}^n_S \\to S$ is proper. Since", "$\\mathbf{P}^n_S \\to S$ is the base change of", "$\\mathbf{P}^n_{\\mathbf{Z}} \\to \\Spec(\\mathbf{Z})$ it suffices", "to show that $\\mathbf{P}^n_{\\mathbf{Z}} \\to \\Spec(\\mathbf{Z})$", "is proper, see Lemma \\ref{lemma-base-change-proper}.", "By Constructions, Lemma \\ref{constructions-lemma-proj-separated} the scheme", "$\\mathbf{P}^n_{\\mathbf{Z}}$ is separated.", "By Constructions, Lemma \\ref{constructions-lemma-proj-quasi-compact} the scheme", "$\\mathbf{P}^n_{\\mathbf{Z}}$ is quasi-compact.", "It is clear that $\\mathbf{P}^n_{\\mathbf{Z}} \\to \\Spec(\\mathbf{Z})$", "is locally of finite type since $\\mathbf{P}^n_{\\mathbf{Z}}$ is", "covered by the affine opens $D_{+}(X_i)$ each of which is the", "spectrum of the finite type $\\mathbf{Z}$-algebra", "$$", "\\mathbf{Z}[X_0/X_i, \\ldots, X_n/X_i].", "$$", "Finally, we have to show that", "$\\mathbf{P}^n_{\\mathbf{Z}} \\to \\Spec(\\mathbf{Z})$", "is universally closed. This follows from", "Constructions, Lemma \\ref{constructions-lemma-proj-valuative-criterion}", "and the valuative criterion (see Schemes,", "Proposition \\ref{schemes-proposition-characterize-universally-closed})." ], "refs": [ "morphisms-lemma-proper-local-on-the-base", "morphisms-lemma-characterize-locally-projective", "morphisms-lemma-composition-proper", "morphisms-lemma-closed-immersion-proper", "morphisms-lemma-base-change-proper", "constructions-lemma-proj-separated", "constructions-lemma-proj-quasi-compact", "constructions-lemma-proj-valuative-criterion", "schemes-proposition-characterize-universally-closed" ], "ref_ids": [ 5407, 5421, 5408, 5410, 5409, 12597, 12598, 12600, 7733 ] } ], "ref_ids": [] }, { "id": 5423, "type": "theorem", "label": "morphisms-lemma-proper-ample-locally-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-proper-ample-locally-projective", "contents": [ "Let $f : X \\to S$ be a proper morphism of schemes. If there exists", "an $f$-ample invertible sheaf on $X$, then $f$ is locally projective." ], "refs": [], "proofs": [ { "contents": [ "If there exists an $f$-ample invertible sheaf, then we can locally", "on $S$ find an immersion $i : X \\to \\mathbf{P}^n_S$, see", "Lemma \\ref{lemma-finite-type-over-affine-ample-very-ample}. Since $X \\to S$", "is proper the morphism $i$ is a closed immersion, see", "Lemma \\ref{lemma-image-proper-scheme-closed}." ], "refs": [ "morphisms-lemma-finite-type-over-affine-ample-very-ample", "morphisms-lemma-image-proper-scheme-closed" ], "ref_ids": [ 5394, 5411 ] } ], "ref_ids": [] }, { "id": 5424, "type": "theorem", "label": "morphisms-lemma-H-projective-composition", "categories": [ "morphisms" ], "title": "morphisms-lemma-H-projective-composition", "contents": [ "A composition of H-projective morphisms is H-projective." ], "refs": [], "proofs": [ { "contents": [ "Suppose $X \\to Y$ and $Y \\to Z$ are H-projective.", "Then there exist closed immersions $X \\to \\mathbf{P}^n_Y$", "over $Y$, and $Y \\to \\mathbf{P}^m_Z$ over $Z$.", "Consider the following diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d] &", "\\mathbf{P}^n_Y \\ar[r] \\ar[dl] &", "\\mathbf{P}^n_{\\mathbf{P}^m_Z} \\ar[dl] \\ar@{=}[r] &", "\\mathbf{P}^n_Z \\times_Z \\mathbf{P}^m_Z \\ar[r] &", "\\mathbf{P}^{nm + n + m}_Z \\ar[ddllll] \\\\", "Y \\ar[r] \\ar[d] & \\mathbf{P}^m_Z \\ar[dl] & \\\\", "Z & &", "}", "$$", "Here the rightmost top horizontal arrow is the Segre embedding,", "see Constructions, Lemma \\ref{constructions-lemma-segre-embedding}.", "The diagram identifies", "$X$ as a closed subscheme of $\\mathbf{P}^{nm + n + m}_Z$ as desired." ], "refs": [ "constructions-lemma-segre-embedding" ], "ref_ids": [ 12624 ] } ], "ref_ids": [] }, { "id": 5425, "type": "theorem", "label": "morphisms-lemma-H-projective-base-change", "categories": [ "morphisms" ], "title": "morphisms-lemma-H-projective-base-change", "contents": [ "A base change of a H-projective morphism is H-projective." ], "refs": [], "proofs": [ { "contents": [ "This is true because the base change of projective space", "over a scheme is projective space, and the fact that the base", "change of a closed immersion is a closed immersion, see", "Schemes, Lemma \\ref{schemes-lemma-base-change-immersion}." ], "refs": [ "schemes-lemma-base-change-immersion" ], "ref_ids": [ 7695 ] } ], "ref_ids": [] }, { "id": 5426, "type": "theorem", "label": "morphisms-lemma-base-change-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-projective", "contents": [ "A base change of a (locally) projective morphism is (locally) projective." ], "refs": [], "proofs": [ { "contents": [ "This is true because the base change of a projective bundle", "over a scheme is a projective bundle, the pullback of", "a finite type $\\mathcal{O}$-module is of finite type", "(Modules, Lemma \\ref{modules-lemma-pullback-finite-type})", "and the fact that the base", "change of a closed immersion is a closed immersion, see", "Schemes, Lemma \\ref{schemes-lemma-base-change-immersion}.", "Some details omitted." ], "refs": [ "modules-lemma-pullback-finite-type", "schemes-lemma-base-change-immersion" ], "ref_ids": [ 13236, 7695 ] } ], "ref_ids": [] }, { "id": 5427, "type": "theorem", "label": "morphisms-lemma-projective-quasi-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-projective-quasi-projective", "contents": [ "A projective morphism is quasi-projective." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be a projective morphism. Choose a closed immersion", "$i : X \\to \\mathbf{P}(\\mathcal{E})$ where $\\mathcal{E}$ is a quasi-coherent,", "finite type $\\mathcal{O}_S$-module. Then", "$\\mathcal{L} = i^*\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(1)$ is $f$-very ample.", "Since $f$ is proper (Lemma \\ref{lemma-locally-projective-proper})", "it is quasi-compact. Hence Lemma \\ref{lemma-ample-very-ample} implies", "that $\\mathcal{L}$ is $f$-ample. Since $f$ is proper it is of finite type.", "Thus we've checked all the defining properties of quasi-projective", "holds and we win." ], "refs": [ "morphisms-lemma-locally-projective-proper", "morphisms-lemma-ample-very-ample" ], "ref_ids": [ 5422, 5387 ] } ], "ref_ids": [] }, { "id": 5428, "type": "theorem", "label": "morphisms-lemma-H-quasi-projective-open-H-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-H-quasi-projective-open-H-projective", "contents": [ "Let $f : X \\to S$ be a H-quasi-projective morphism.", "Then $f$ factors as $X \\to X' \\to S$ where $X \\to X'$ is an", "open immersion and $X' \\to S$ is H-projective." ], "refs": [], "proofs": [ { "contents": [ "By definition we can factor $f$ as a quasi-compact immersion", "$i : X \\to \\mathbf{P}^n_S$ followed by the projection $\\mathbf{P}^n_S \\to S$.", "By Lemma \\ref{lemma-quasi-compact-immersion} there exists a closed", "subscheme $X' \\subset \\mathbf{P}^n_S$ such that $i$ factors through", "an open immersion $X \\to X'$. The lemma follows." ], "refs": [ "morphisms-lemma-quasi-compact-immersion" ], "ref_ids": [ 5154 ] } ], "ref_ids": [] }, { "id": 5429, "type": "theorem", "label": "morphisms-lemma-quasi-projective-open-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-projective-open-projective", "contents": [ "Let $f : X \\to S$ be a quasi-projective morphism with $S$ quasi-compact", "and quasi-separated. Then $f$ factors as $X \\to X' \\to S$ where $X \\to X'$", "is an open immersion and $X' \\to S$ is projective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{L}$ be $f$-ample. Since $f$ is of finite type and $S$ is", "quasi-compact $\\mathcal{L}^{\\otimes n}$ is $f$-very ample for some $n > 0$, see", "Lemma \\ref{lemma-finite-type-ample-very-ample}.", "Replace $\\mathcal{L}$ by $\\mathcal{L}^{\\otimes n}$.", "Write $\\mathcal{F} = f_*\\mathcal{L}$. This is a quasi-coherent", "$\\mathcal{O}_S$-module by ", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "(quasi-projective morphisms are quasi-compact and", "separated, see Lemma \\ref{lemma-quasi-projective-properties}). By", "Properties, Lemma \\ref{properties-lemma-directed-colimit-finite-presentation}", "we can find a directed set $I$ and a system of", "finite type quasi-coherent $\\mathcal{O}_S$-modules $\\mathcal{E}_i$", "over $I$ such that $\\mathcal{F} = \\colim \\mathcal{E}_i$.", "Consider the compositions", "$\\psi_i : f^*\\mathcal{E}_i \\to f^*\\mathcal{F} \\to \\mathcal{L}$.", "Choose a finite affine open covering $S = \\bigcup_{j = 1, \\ldots, m} V_j$.", "For each $j$ we can choose sections", "$$", "s_{j, 0}, \\ldots, s_{j, n_j} \\in", "\\Gamma(f^{-1}(V_j), \\mathcal{L}) = f_*\\mathcal{L}(V_j) = \\mathcal{F}(V_j)", "$$", "which generate $\\mathcal{L}$ over $f^{-1}V_j$ and define an immersion", "$$", "f^{-1}V_j \\longrightarrow \\mathbf{P}^{n_j}_{V_j},", "$$", "see Lemma \\ref{lemma-very-ample-finite-type-over-affine}.", "Choose $i$ such that there exist sections $e_{j, t} \\in \\mathcal{E}_i(V_j)$", "mapping to $s_{j, t}$ in $\\mathcal{F}$ for all $j = 1, \\ldots, m$ and", "$t = 1, \\ldots, n_j$. Then the map $\\psi_i$ is surjective", "as the sections $f^*e_{j, t}$ have the same image as the sections $s_{j, t}$", "which generate $\\mathcal{L}|_{f^{-1}V_j}$. Whence we obtain a morphism", "$$", "r_{\\mathcal{L}, \\psi_i} : X \\longrightarrow \\mathbf{P}(\\mathcal{E}_i)", "$$", "over $S$ such that over $V_j$ we have a factorization", "$$", "f^{-1}V_j \\to \\mathbf{P}(\\mathcal{E}_i)|_{V_j} \\to \\mathbf{P}^{n_j}_{V_j}", "$$", "of the immersion given above. It follows that $r_{\\mathcal{L}, \\psi_i}|_{V_j}$", "is an immersion, see Lemma \\ref{lemma-immersion-permanence}.", "Since $S = \\bigcup V_j$ we conclude that $r_{\\mathcal{L}, \\psi_i}$", "is an immersion.", "Note that $r_{\\mathcal{L}, \\psi_i}$ is quasi-compact as", "$X \\to S$ is quasi-compact and $\\mathbf{P}(\\mathcal{E}_i) \\to S$ is separated", "(see Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}).", "By Lemma \\ref{lemma-quasi-compact-immersion} there exists a closed", "subscheme $X' \\subset \\mathbf{P}(\\mathcal{E}_i)$ such that $i$ factors", "through an open immersion $X \\to X'$. Then $X' \\to S$ is projective", "by definition and we win." ], "refs": [ "morphisms-lemma-finite-type-ample-very-ample", "schemes-lemma-push-forward-quasi-coherent", "morphisms-lemma-quasi-projective-properties", "properties-lemma-directed-colimit-finite-presentation", "morphisms-lemma-very-ample-finite-type-over-affine", "morphisms-lemma-immersion-permanence", "schemes-lemma-quasi-compact-permanence", "morphisms-lemma-quasi-compact-immersion" ], "ref_ids": [ 5395, 7730, 5401, 3024, 5391, 5132, 7716, 5154 ] } ], "ref_ids": [] }, { "id": 5430, "type": "theorem", "label": "morphisms-lemma-projective-is-quasi-projective-proper", "categories": [ "morphisms" ], "title": "morphisms-lemma-projective-is-quasi-projective-proper", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $f : X \\to S$ be a morphism of schemes. Then", "\\begin{enumerate}", "\\item $f$ is projective if and only if $f$ is quasi-projective and proper, and", "\\item $f$ is H-projective if and only if $f$ is H-quasi-projective and proper.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $f$ is projective, then $f$ is quasi-projective by", "Lemma \\ref{lemma-projective-quasi-projective} and proper by", "Lemma \\ref{lemma-locally-projective-proper}. Conversely, if", "$X \\to S$ is quasi-projective and proper, then we can choose", "an open immersion $X \\to X'$ with $X' \\to S$ projective by", "Lemma \\ref{lemma-quasi-projective-open-projective}.", "Since $X \\to S$ is proper, we see that $X$ is closed in $X'$", "(Lemma \\ref{lemma-image-proper-scheme-closed}), i.e.,", "$X \\to X'$ is a (open and) closed immersion.", "Since $X'$ is isomorphic to a closed subscheme of a projective", "bundle over $S$ (Definition \\ref{definition-projective})", "we see that the same thing is true for $X$, i.e.,", "$X \\to S$ is a projective morphism. This proves (1).", "The proof of (2) is the same, except it uses", "Lemmas \\ref{lemma-H-projective} and", "\\ref{lemma-H-quasi-projective-open-H-projective}." ], "refs": [ "morphisms-lemma-projective-quasi-projective", "morphisms-lemma-locally-projective-proper", "morphisms-lemma-quasi-projective-open-projective", "morphisms-lemma-image-proper-scheme-closed", "morphisms-definition-projective", "morphisms-lemma-H-projective", "morphisms-lemma-H-quasi-projective-open-H-projective" ], "ref_ids": [ 5427, 5422, 5429, 5411, 5572, 5420, 5428 ] } ], "ref_ids": [] }, { "id": 5431, "type": "theorem", "label": "morphisms-lemma-composition-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-projective", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to S$ be morphisms of schemes.", "If $S$ is quasi-compact and quasi-separated and $f$ and $g$ are projective,", "then $g \\circ f$ is projective." ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-projective-quasi-projective} and", "\\ref{lemma-locally-projective-proper}", "we see that $f$ and $g$ are quasi-projective and proper.", "By Lemmas \\ref{lemma-composition-proper} and", "\\ref{lemma-composition-quasi-projective}", "we see that $g \\circ f$ is proper and quasi-projective.", "Thus $g \\circ f$ is projective by", "Lemma \\ref{lemma-projective-is-quasi-projective-proper}." ], "refs": [ "morphisms-lemma-projective-quasi-projective", "morphisms-lemma-locally-projective-proper", "morphisms-lemma-composition-proper", "morphisms-lemma-composition-quasi-projective", "morphisms-lemma-projective-is-quasi-projective-proper" ], "ref_ids": [ 5427, 5422, 5408, 5400, 5430 ] } ], "ref_ids": [] }, { "id": 5432, "type": "theorem", "label": "morphisms-lemma-projective-permanence", "categories": [ "morphisms" ], "title": "morphisms-lemma-projective-permanence", "contents": [ "Let $g : Y \\to S$ and $f : X \\to Y$ be morphisms of schemes.", "If $g \\circ f$ is projective and $g$ is separated,", "then $f$ is projective." ], "refs": [], "proofs": [ { "contents": [ "Choose a closed immersion $X \\to \\mathbf{P}(\\mathcal{E})$ where $\\mathcal{E}$", "is a quasi-coherent, finite type $\\mathcal{O}_S$-module. Then", "we get a morphism $X \\to \\mathbf{P}(\\mathcal{E}) \\times_S Y$.", "This morphism is a closed immersion because it is the composition", "$$", "X \\to X \\times_S Y \\to \\mathbf{P}(\\mathcal{E}) \\times_S Y", "$$", "where the first morphism is a closed immersion by", "Schemes, Lemma \\ref{schemes-lemma-semi-diagonal}", "(and the fact that $g$ is separated) and", "the second as the base change of a closed immersion.", "Finally, the fibre product $\\mathbf{P}(\\mathcal{E}) \\times_S Y$", "is isomorphic to $\\mathbf{P}(g^*\\mathcal{E})$ and pullback", "preserves quasi-coherent, finite type modules." ], "refs": [ "schemes-lemma-semi-diagonal" ], "ref_ids": [ 7712 ] } ], "ref_ids": [] }, { "id": 5433, "type": "theorem", "label": "morphisms-lemma-projective-over-quasi-projective-is-H-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-projective-over-quasi-projective-is-H-projective", "contents": [ "Let $S$ be a scheme which admits an ample invertible sheaf. Then", "\\begin{enumerate}", "\\item any projective morphism $X \\to S$ is H-projective, and", "\\item any quasi-projective morphism $X \\to S$ is H-quasi-projective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The assumptions on $S$ imply that $S$ is quasi-compact and separated, see", "Properties, Definition \\ref{properties-definition-ample} and", "Lemma \\ref{properties-lemma-ample-immersion-into-proj}", "and Constructions, Lemma \\ref{constructions-lemma-proj-separated}.", "Hence Lemma \\ref{lemma-quasi-projective-open-projective}", "applies and we see that (1) implies (2).", "Let $\\mathcal{E}$ be a finite type quasi-coherent $\\mathcal{O}_S$-module.", "By our definition of projective morphisms it suffices to show that", "$\\mathbf{P}(\\mathcal{E}) \\to S$ is H-projective.", "If $\\mathcal{E}$ is generated by finitely many global sections,", "then the corresponding surjection $\\mathcal{O}_S^{\\oplus n} \\to \\mathcal{E}$", "induces a closed immersion", "$$", "\\mathbf{P}(\\mathcal{E}) \\longrightarrow", "\\mathbf{P}(\\mathcal{O}_S^{\\oplus n}) = \\mathbf{P}^n_S", "$$", "as desired. In general, let $\\mathcal{L}$ be an invertible sheaf on $S$.", "By Properties, Proposition \\ref{properties-proposition-characterize-ample}", "there exists an integer $n$ such that", "$\\mathcal{E} \\otimes_{\\mathcal{O}_S} \\mathcal{L}^{\\otimes n}$", "is globally generated by finitely many sections. Since", "$\\mathbf{P}(\\mathcal{E}) =", "\\mathbf{P}(\\mathcal{E} \\otimes_{\\mathcal{O}_S} \\mathcal{L}^{\\otimes n})$ by", "Constructions, Lemma \\ref{constructions-lemma-twisting-and-proj}", "this finishes the proof." ], "refs": [ "properties-definition-ample", "properties-lemma-ample-immersion-into-proj", "constructions-lemma-proj-separated", "morphisms-lemma-quasi-projective-open-projective", "properties-proposition-characterize-ample", "constructions-lemma-twisting-and-proj" ], "ref_ids": [ 3088, 3049, 12597, 5429, 3067, 12650 ] } ], "ref_ids": [] }, { "id": 5434, "type": "theorem", "label": "morphisms-lemma-proper-ample-is-proj", "categories": [ "morphisms" ], "title": "morphisms-lemma-proper-ample-is-proj", "contents": [ "Let $f : X \\to S$ be a universally closed morphism.", "Let $\\mathcal{L}$ be an $f$-ample invertible $\\mathcal{O}_X$-module.", "Then the canonical morphism", "$$", "r : X", "\\longrightarrow", "\\underline{\\text{Proj}}_S", "\\left(", "\\bigoplus\\nolimits_{d \\geq 0} f_*\\mathcal{L}^{\\otimes d}", "\\right)", "$$", "of Lemma \\ref{lemma-characterize-relatively-ample} is an isomorphism." ], "refs": [ "morphisms-lemma-characterize-relatively-ample" ], "proofs": [ { "contents": [ "Observe that $f$ is quasi-compact because the existence", "of an $f$-ample invertible module forces $f$ to be quasi-compact.", "By the lemma cited the morphism $r$ is an open immersion.", "On the other hand, the image of $r$ is closed by", "Lemma \\ref{lemma-image-proper-scheme-closed}", "(the target of $r$ is separated over $S$ by Constructions,", "Lemma \\ref{constructions-lemma-relative-proj-separated}).", "Finally, the image of $r$ is dense by", "Properties, Lemma \\ref{properties-lemma-ample-immersion-into-proj}", "(here we also use that it was shown in the proof of", "Lemma \\ref{lemma-characterize-relatively-ample}", "that the morphism $r$ over affine opens of $S$", "is given by the canonical morphism of", "Properties, Lemma \\ref{properties-lemma-map-into-proj}).", "Thus we conclude that $r$ is a surjective open immersion, i.e.,", "an isomorphism." ], "refs": [ "morphisms-lemma-image-proper-scheme-closed", "constructions-lemma-relative-proj-separated", "properties-lemma-ample-immersion-into-proj", "morphisms-lemma-characterize-relatively-ample", "properties-lemma-map-into-proj" ], "ref_ids": [ 5411, 12640, 3049, 5380, 3047 ] } ], "ref_ids": [ 5380 ] }, { "id": 5435, "type": "theorem", "label": "morphisms-lemma-proper-ample-delete-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-proper-ample-delete-affine", "contents": [ "Let $f : X \\to S$ be a universally closed morphism. Let $\\mathcal{L}$", "be an $f$-ample invertible $\\mathcal{O}_X$-module. Let", "$s \\in \\Gamma(X, \\mathcal{L})$. Then $X_s \\to S$ is an affine morphism." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $S$ (Lemma \\ref{lemma-characterize-affine})", "hence we may assume $S$ is affine. By Lemma \\ref{lemma-proper-ample-is-proj}", "we can write $X = \\text{Proj}(A)$ where $A$ is a graded", "ring and $s$ corresponds to $f \\in A_1$ and", "$X_s = D_+(f)$ (Properties, Lemma \\ref{properties-lemma-map-into-proj})", "which proves the lemma by construction of $\\text{Proj}(A)$, see", "Constructions, Section \\ref{constructions-section-proj}." ], "refs": [ "morphisms-lemma-characterize-affine", "morphisms-lemma-proper-ample-is-proj", "properties-lemma-map-into-proj" ], "ref_ids": [ 5172, 5434, 3047 ] } ], "ref_ids": [] }, { "id": 5436, "type": "theorem", "label": "morphisms-lemma-integral-local", "categories": [ "morphisms" ], "title": "morphisms-lemma-integral-local", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is integral.", "\\item There exists an affine open covering $S = \\bigcup U_i$ such that", "each $f^{-1}(U_i)$ is affine and", "$\\mathcal{O}_S(U_i) \\to \\mathcal{O}_X(f^{-1}(U_i))$ is integral.", "\\item There exists an open covering $S = \\bigcup U_i$", "such that each $f^{-1}(U_i) \\to U_i$ is integral.", "\\end{enumerate}", "Moreover, if $f$ is integral then for every open subscheme", "$U \\subset S$ the morphism $f : f^{-1}(U) \\to U$ is integral." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-integral-local}.", "Some details omitted." ], "refs": [ "algebra-lemma-integral-local" ], "ref_ids": [ 492 ] } ], "ref_ids": [] }, { "id": 5437, "type": "theorem", "label": "morphisms-lemma-finite-local", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-local", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is finite.", "\\item There exists an affine open covering $S = \\bigcup U_i$ such that", "each $f^{-1}(U_i)$ is affine and", "$\\mathcal{O}_S(U_i) \\to \\mathcal{O}_X(f^{-1}(U_i))$ is finite.", "\\item There exists an open covering $S = \\bigcup U_i$", "such that each $f^{-1}(U_i) \\to U_i$ is finite.", "\\end{enumerate}", "Moreover, if $f$ is finite then for every open subscheme", "$U \\subset S$ the morphism $f : f^{-1}(U) \\to U$ is finite." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-integral-local}.", "Some details omitted." ], "refs": [ "algebra-lemma-integral-local" ], "ref_ids": [ 492 ] } ], "ref_ids": [] }, { "id": 5438, "type": "theorem", "label": "morphisms-lemma-finite-integral", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-integral", "contents": [ "A finite morphism is integral.", "An integral morphism which is locally of finite type is finite." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-finite-is-integral}", "and Lemma \\ref{algebra-lemma-characterize-finite-in-terms-of-integral}." ], "refs": [ "algebra-lemma-finite-is-integral", "algebra-lemma-characterize-finite-in-terms-of-integral" ], "ref_ids": [ 482, 484 ] } ], "ref_ids": [] }, { "id": 5439, "type": "theorem", "label": "morphisms-lemma-composition-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-finite", "contents": [ "A composition of finite morphisms is finite.", "Same is true for integral morphisms." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemmas \\ref{algebra-lemma-finite-transitive}", "and \\ref{algebra-lemma-integral-transitive}." ], "refs": [ "algebra-lemma-finite-transitive", "algebra-lemma-integral-transitive" ], "ref_ids": [ 337, 485 ] } ], "ref_ids": [] }, { "id": 5440, "type": "theorem", "label": "morphisms-lemma-base-change-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-finite", "contents": [ "A base change of a finite morphism is finite.", "Same is true for integral morphisms." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-base-change-integral}." ], "refs": [ "algebra-lemma-base-change-integral" ], "ref_ids": [ 491 ] } ], "ref_ids": [] }, { "id": 5441, "type": "theorem", "label": "morphisms-lemma-integral-universally-closed", "categories": [ "morphisms" ], "title": "morphisms-lemma-integral-universally-closed", "contents": [ "\\begin{slogan}", "integral $=$ affine $+$ universally closed", "\\end{slogan}", "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is integral, and", "\\item $f$ is affine and universally closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). An integral morphism is affine by definition.", "A base change of an integral morphism is integral so in order to prove (2)", "it suffices to show that an integral morphism is closed.", "This follows from Algebra, Lemmas \\ref{algebra-lemma-integral-going-up}", "and \\ref{algebra-lemma-going-up-closed}.", "\\medskip\\noindent", "Assume (2). We may assume $f$ is the morphism", "$f : \\Spec(A) \\to \\Spec(R)$ coming from a ring map", "$R \\to A$. Let $a$ be an element of $A$. We have to show", "that $a$ is integral over $R$, i.e. that in the kernel", "$I$ of the map $R[x] \\to A$ sending $x$ to $a$ there is", "a monic polynomial. Consider the ring", "$B = A[x]/(ax -1)$ and let $J$ be the kernel of the", "composition $R[x]\\to A[x] \\to B$. If $f\\in J$ there exists", "$q\\in A[x]$ such that $f = (ax-1)q$ in $A[x]$ so if", "$f = \\sum_i f_ix^i$ and $q = \\sum_iq_ix^i$, for all $i \\geq 0$", "we have $f_i = aq_{i-1} - q_i$. For $n \\geq \\deg q + 1$ the polynomial", "$$", "\\sum\\nolimits_{i \\geq 0} f_i x^{n - i} =", "\\sum\\nolimits_{i \\geq 0} (a q_{i - 1} - q_i) x^{n - i} =", "(a - x) \\sum\\nolimits_{i \\geq 0} q_i x^{n - i - 1}", "$$", "is clearly in $I$; if $f_0 = 1$ this polynomial is also monic, so we", "are reduced to prove that $J$ contains a polynomial with constant term $1$.", "We do it by proving $\\Spec(R[x]/(J + (x))$ is empty.", "\\medskip\\noindent", "Since $f$ is universally closed the base change $\\Spec(A[x]) \\to \\Spec(R[x])$", "is closed. Hence the image of the closed subset $\\Spec(B) \\subset \\Spec(A[x])$", "is the closed subset $\\Spec(R[x]/J) \\subset \\Spec(R[x])$, see", "Example \\ref{example-scheme-theoretic-image} and", "Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image}.", "In particular $\\Spec(B) \\to \\Spec(R[x]/J)$ is surjective. Consider the", "following diagram where every square is a pullback:", "$$", "\\xymatrix{", "\\Spec(B) \\ar@{->>}[r]^g &", "\\Spec(R[x]/J) \\ar[r] &", "\\Spec(R[x])\\\\", "\\emptyset \\ar[u] \\ar[r] &", "\\Spec(R[x]/(J + (x)))\\ar[u] \\ar[r] &", "\\Spec(R) \\ar[u]^0", "}", "$$", "The bottom left corner is empty because it is the spectrum of", "$R\\otimes_{R[x]} B$ where the map $R[x]\\to B$ sends $x$ to an", "invertible element and $R[x]\\to R$ sends $x$ to $0$. Since $g$", "is surjective this implies $\\Spec(R[x]/(J + (x)))$ is empty, as", "we wanted to show." ], "refs": [ "algebra-lemma-integral-going-up", "algebra-lemma-going-up-closed", "morphisms-lemma-quasi-compact-scheme-theoretic-image" ], "ref_ids": [ 500, 552, 5146 ] } ], "ref_ids": [] }, { "id": 5442, "type": "theorem", "label": "morphisms-lemma-integral-fibres", "categories": [ "morphisms" ], "title": "morphisms-lemma-integral-fibres", "contents": [ "Let $f : X \\to S$ be an integral morphism.", "Then every point of $X$ is closed in its fibre." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-integral-no-inclusion}." ], "refs": [ "algebra-lemma-integral-no-inclusion" ], "ref_ids": [ 498 ] } ], "ref_ids": [] }, { "id": 5443, "type": "theorem", "label": "morphisms-lemma-integral-dimension", "categories": [ "morphisms" ], "title": "morphisms-lemma-integral-dimension", "contents": [ "Let $f : X \\to Y$ be an integral morphism. Then $\\dim(X) \\leq \\dim(Y)$.", "If $f$ is surjective then $\\dim(X) = \\dim(Y)$." ], "refs": [], "proofs": [ { "contents": [ "Since the dimension of $X$ and $Y$ is the supremum of the dimensions", "of the members of an affine open covering, we may assume $Y$ and $X$", "are affine. The inequality follows from", "Algebra, Lemma \\ref{algebra-lemma-integral-dim-up}.", "The equality then follows from", "Algebra, Lemmas \\ref{algebra-lemma-dimension-going-up}", "and \\ref{algebra-lemma-integral-going-up}." ], "refs": [ "algebra-lemma-integral-dim-up", "algebra-lemma-dimension-going-up", "algebra-lemma-integral-going-up" ], "ref_ids": [ 984, 982, 500 ] } ], "ref_ids": [] }, { "id": 5444, "type": "theorem", "label": "morphisms-lemma-finite-quasi-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-quasi-finite", "contents": [ "A finite morphism is quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "This is implied by Algebra, Lemma \\ref{algebra-lemma-quasi-finite}", "and Lemma \\ref{lemma-quasi-finite-locally-quasi-compact}.", "Alternatively, all points in fibres are closed points by", "Lemma \\ref{lemma-integral-fibres} (and the fact that a finite", "morphism is integral) and use", "Lemma \\ref{lemma-quasi-finite-at-point-characterize} (3) to", "see that $f$ is quasi-finite at $x$ for all $x \\in X$." ], "refs": [ "algebra-lemma-quasi-finite", "morphisms-lemma-quasi-finite-locally-quasi-compact", "morphisms-lemma-integral-fibres", "morphisms-lemma-quasi-finite-at-point-characterize" ], "ref_ids": [ 1050, 5229, 5442, 5226 ] } ], "ref_ids": [] }, { "id": 5445, "type": "theorem", "label": "morphisms-lemma-finite-proper", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-proper", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is finite, and", "\\item $f$ is affine and proper.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows formally from", "Lemma \\ref{lemma-integral-universally-closed},", "the fact that a finite morphism is integral and separated,", "the fact that a proper morphism is the same thing as", "a finite type, separated, universally closed morphism,", "and the fact that an integral morphism of finite type is", "finite (Lemma \\ref{lemma-finite-integral})." ], "refs": [ "morphisms-lemma-integral-universally-closed", "morphisms-lemma-finite-integral" ], "ref_ids": [ 5441, 5438 ] } ], "ref_ids": [] }, { "id": 5446, "type": "theorem", "label": "morphisms-lemma-closed-immersion-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-closed-immersion-finite", "contents": [ "A closed immersion is finite (and a fortiori integral)." ], "refs": [], "proofs": [ { "contents": [ "True because a closed immersion is affine", "(Lemma \\ref{lemma-closed-immersion-affine})", "and a surjective ring map is finite and integral." ], "refs": [ "morphisms-lemma-closed-immersion-affine" ], "ref_ids": [ 5177 ] } ], "ref_ids": [] }, { "id": 5447, "type": "theorem", "label": "morphisms-lemma-finite-union-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-union-finite", "contents": [ "Let $X_i \\to Y$, $i = 1, \\ldots, n$ be finite morphisms of schemes.", "Then $X_1 \\amalg \\ldots \\amalg X_n \\to Y$ is finite too." ], "refs": [], "proofs": [ { "contents": [ "Follows from the algebra fact that if $R \\to A_i$, $i = 1, \\ldots, n$", "are finite ring maps, then $R \\to A_1 \\times \\ldots \\times A_n$ is finite too." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5448, "type": "theorem", "label": "morphisms-lemma-finite-permanence", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-permanence", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms.", "\\begin{enumerate}", "\\item If $g \\circ f$ is finite and $g$ separated then $f$ is finite.", "\\item If $g \\circ f$ is integral and $g$ separated then $f$ is integral.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $g \\circ f$ is finite (resp.\\ integral) and $g$ separated.", "The base change $X \\times_Z Y \\to Y$ is finite (resp.\\ integral) by", "Lemma \\ref{lemma-base-change-finite}.", "The morphism $X \\to X \\times_Z Y$ is", "a closed immersion as $Y \\to Z$ is separated, see", "Schemes, Lemma \\ref{schemes-lemma-section-immersion}.", "A closed immersion is finite (resp.\\ integral),", "see Lemma \\ref{lemma-closed-immersion-finite}.", "The composition of finite (resp.\\ integral) morphisms is finite", "(resp.\\ integral),", "see Lemma \\ref{lemma-composition-finite}. Thus we win." ], "refs": [ "morphisms-lemma-base-change-finite", "morphisms-lemma-closed-immersion-finite", "morphisms-lemma-composition-finite" ], "ref_ids": [ 5440, 5446, 5439 ] } ], "ref_ids": [] }, { "id": 5449, "type": "theorem", "label": "morphisms-lemma-finite-monomorphism-closed", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-monomorphism-closed", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "If $f$ is finite and a monomorphism, then $f$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "This reduces to", "Algebra, Lemma \\ref{algebra-lemma-finite-epimorphism-surjective}." ], "refs": [ "algebra-lemma-finite-epimorphism-surjective" ], "ref_ids": [ 952 ] } ], "ref_ids": [] }, { "id": 5450, "type": "theorem", "label": "morphisms-lemma-finite-projective", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-projective", "contents": [ "A finite morphism is projective." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be a finite morphism. Then $f_*\\mathcal{O}_X$ is", "a quasi-coherent $\\mathcal{O}_S$-module", "(Lemma \\ref{lemma-affine-equivalence-algebras})", "of finite type", "(by our definition of finite morphisms and", "Properties, Lemma \\ref{properties-lemma-finite-type-module}).", "We claim there is a closed immersion", "$$", "\\sigma :", "X", "\\longrightarrow", "\\mathbf{P}(f_*\\mathcal{O}_X) =", "\\underline{\\text{Proj}}_S(\\text{Sym}^*_{\\mathcal{O}_S}(f_*\\mathcal{O}_X))", "$$", "over $S$, which finishes", "the proof. Namely, we let $\\sigma$ be the morphism which corresponds", "(via Constructions, Lemma \\ref{constructions-lemma-apply-relative})", "to the surjection", "$$", "f^*f_*\\mathcal{O}_X \\longrightarrow \\mathcal{O}_X", "$$", "coming from the adjunction map $f^*f_* \\to \\text{id}$. Then $\\sigma$", "is a closed immersion by", "Schemes, Lemma \\ref{schemes-lemma-semi-diagonal} and", "Constructions, Lemma \\ref{constructions-lemma-projective-bundle-separated}." ], "refs": [ "morphisms-lemma-affine-equivalence-algebras", "properties-lemma-finite-type-module", "constructions-lemma-apply-relative", "schemes-lemma-semi-diagonal", "constructions-lemma-projective-bundle-separated" ], "ref_ids": [ 5173, 3002, 12642, 7712, 12651 ] } ], "ref_ids": [] }, { "id": 5451, "type": "theorem", "label": "morphisms-lemma-base-change-universal-homeomorphism", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-universal-homeomorphism", "contents": [ "The base change of a universal homeomorphism of schemes", "by any morphism of schemes is a universal homeomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5452, "type": "theorem", "label": "morphisms-lemma-composition-universal-homeomorphism", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-universal-homeomorphism", "contents": [ "The composition of a pair of universal homeomorphisms of", "schemes is a universal homeomorphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5453, "type": "theorem", "label": "morphisms-lemma-homeomorphism-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-homeomorphism-affine", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. If $f$ is a homeomorphism", "onto a closed subset of $Y$ then $f$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Let $y \\in Y$ be a point. If $y \\not \\in f(X)$, then there exists", "an affine neighbourhood of $y$ which is disjoint from $f(X)$.", "If $y \\in f(X)$, let $x \\in X$ be the unique point of $X$ mapping to $y$.", "Let $y \\in V$ be an affine open neighbourhood.", "Let $U \\subset X$ be an affine open neighbourhood of $x$ which maps into $V$.", "Since $f(U) \\subset V \\cap f(X)$ is open in the induced topology by our", "assumption on $f$ we may choose a", "$h \\in \\Gamma(V, \\mathcal{O}_Y)$ such that $y \\in D(h)$", "and $D(h) \\cap f(X) \\subset f(U)$. Denote $h' \\in \\Gamma(U, \\mathcal{O}_X)$", "the restriction of $f^\\sharp(h)$ to $U$. Then we see that", "$D(h') \\subset U$ is equal to $f^{-1}(D(h))$. In other words, every point", "of $Y$ has an open neighbourhood whose inverse image is affine.", "Thus $f$ is affine, see", "Lemma \\ref{lemma-characterize-affine}." ], "refs": [ "morphisms-lemma-characterize-affine" ], "ref_ids": [ 5172 ] } ], "ref_ids": [] }, { "id": 5454, "type": "theorem", "label": "morphisms-lemma-universal-homeomorphism", "categories": [ "morphisms" ], "title": "morphisms-lemma-universal-homeomorphism", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. The following are", "equivalent:", "\\begin{enumerate}", "\\item $f$ is a universal homeomorphism, and", "\\item $f$ is integral, universally injective and surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f$ is a universal homeomorphism. By", "Lemma \\ref{lemma-homeomorphism-affine}", "we see that $f$ is affine. Since $f$ is clearly universally closed we", "see that $f$ is integral by", "Lemma \\ref{lemma-integral-universally-closed}.", "It is also clear that $f$ is universally injective and surjective.", "\\medskip\\noindent", "Assume $f$ is integral, universally injective and surjective. By", "Lemma \\ref{lemma-integral-universally-closed}", "$f$ is universally closed. Since it is also universally bijective (see", "Lemma \\ref{lemma-base-change-surjective})", "we see that it is a universal homeomorphism." ], "refs": [ "morphisms-lemma-homeomorphism-affine", "morphisms-lemma-integral-universally-closed", "morphisms-lemma-integral-universally-closed", "morphisms-lemma-base-change-surjective" ], "ref_ids": [ 5453, 5441, 5441, 5165 ] } ], "ref_ids": [] }, { "id": 5455, "type": "theorem", "label": "morphisms-lemma-reduction-universal-homeomorphism", "categories": [ "morphisms" ], "title": "morphisms-lemma-reduction-universal-homeomorphism", "contents": [ "Let $X$ be a scheme. The canonical closed immersion $X_{red} \\to X$ (see", "Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme})", "is a universal homeomorphism." ], "refs": [ "schemes-definition-reduced-induced-scheme" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 7745 ] }, { "id": 5456, "type": "theorem", "label": "morphisms-lemma-check-closed-infinitesimally", "categories": [ "morphisms" ], "title": "morphisms-lemma-check-closed-infinitesimally", "contents": [ "Let $f : X \\to S$ and $S' \\to S$ be morphisms of schemes.", "Assume", "\\begin{enumerate}", "\\item $S' \\to S$ is a closed immersion,", "\\item $S' \\to S$ is bijective on points,", "\\item $X \\times_S S' \\to S'$ is a closed immersion, and", "\\item $X \\to S$ is of finite type or $S' \\to S$ is of finite presentation.", "\\end{enumerate}", "Then $f : X \\to S$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Assumptions (1) and (2) imply that $S' \\to S$ is a universal homeomorphism", "(for example because $S_{red} = S'_{red}$ and using", "Lemma \\ref{lemma-reduction-universal-homeomorphism}).", "Hence (3) implies that $X \\to S$ is homeomorphism onto a", "closed subset of $S$. Then $X \\to S$ is affine by", "Lemma \\ref{lemma-homeomorphism-affine}.", "Let $U \\subset S$ be an affine open, say $U = \\Spec(A)$. Then $S' = \\Spec(A/I)$", "by (1) for a locally nilpotent ideal $I$ by (2). As $f$ is affine we see that", "$f^{-1}(U) = \\Spec(B)$.", "Assumption (4) tells us $B$ is a finite type $A$-algebra", "(Lemma \\ref{lemma-locally-finite-type-characterize}) or", "that $I$ is finitely generated", "(Lemma \\ref{lemma-closed-immersion-finite-presentation}).", "Assumption (3) is that $A/I \\to B/IB$ is surjective. From", "Algebra, Lemma \\ref{algebra-lemma-surjective-mod-locally-nilpotent}", "if $A \\to B$ is of finite type", "or Algebra, Lemma \\ref{algebra-lemma-NAK} if $I$ is finitely generated", "and hence nilpotent we deduce that $A \\to B$ is surjective.", "This means that $f$ is a closed immersion, see", "Lemma \\ref{lemma-closed-immersion}." ], "refs": [ "morphisms-lemma-reduction-universal-homeomorphism", "morphisms-lemma-homeomorphism-affine", "morphisms-lemma-locally-finite-type-characterize", "morphisms-lemma-closed-immersion-finite-presentation", "algebra-lemma-surjective-mod-locally-nilpotent", "algebra-lemma-NAK", "morphisms-lemma-closed-immersion" ], "ref_ids": [ 5455, 5453, 5198, 5243, 1087, 401, 5125 ] } ], "ref_ids": [] }, { "id": 5457, "type": "theorem", "label": "morphisms-lemma-subalgebra-inherits", "categories": [ "morphisms" ], "title": "morphisms-lemma-subalgebra-inherits", "contents": [ "Let $A \\to B$ be a ring map such that the induced morphism of", "schemes $f : \\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism,", "resp.\\ a universal homeomorphism inducing isomorphisms on residue fields,", "resp.\\ universally closed,", "resp.\\ universally closed and universally injective.", "Then for any $A$-subalgebra $B' \\subset B$", "the same thing is true for $f' : \\Spec(B') \\to \\Spec(A)$." ], "refs": [], "proofs": [ { "contents": [ "If $f$ is universally closed, then", "$B$ is integral over $A$ by Lemma \\ref{lemma-integral-universally-closed}.", "Hence $B'$ is integral over $A$ and $f'$", "is universally closed (by the same lemma).", "This proves the case where $f$ is universally closed.", "\\medskip\\noindent", "Continuing, we see that $B$ is integral over $B'$ ", "(Algebra, Lemma \\ref{algebra-lemma-integral-permanence})", "which implies $\\Spec(B) \\to \\Spec(B')$ is surjective", "(Algebra, Lemma \\ref{algebra-lemma-integral-overring-surjective}).", "Thus if $A \\to B$ induces purely inseparable extensions of residue fields,", "then the same is true for $A \\to B'$. This proves the case", "where $f$ is universally closed and universally injective, see", "Lemma \\ref{lemma-universally-injective}.", "\\medskip\\noindent", "The case where $f$ is a universal homeomorphism follows from", "the remarks above, Lemma \\ref{lemma-universal-homeomorphism},", "and the obvious observation that if $f$ is surjective, then so is $f'$.", "\\medskip\\noindent", "If $A \\to B$ induces isomorphisms on residue fields, then", "so does $A \\to B'$ (see argument in second paragraph).", "In this way we see that the lemma holds in the remaining case." ], "refs": [ "morphisms-lemma-integral-universally-closed", "algebra-lemma-integral-permanence", "algebra-lemma-integral-overring-surjective", "morphisms-lemma-universally-injective", "morphisms-lemma-universal-homeomorphism" ], "ref_ids": [ 5441, 493, 495, 5167, 5454 ] } ], "ref_ids": [] }, { "id": 5458, "type": "theorem", "label": "morphisms-lemma-colimit-inherits", "categories": [ "morphisms" ], "title": "morphisms-lemma-colimit-inherits", "contents": [ "Let $A$ be a ring. Let $B = \\colim B_\\lambda$ be a filtered colimit", "of $A$-algebras. If each $f_\\lambda : \\Spec(B_\\lambda) \\to \\Spec(A)$", "is a universal homeomorphism,", "resp.\\ a universal homeomorphism inducing isomorphisms on residue fields,", "resp.\\ universally closed,", "resp.\\ universally closed and universally injective,", "then the same thing is true for $f : \\Spec(B) \\to \\Spec(A)$." ], "refs": [], "proofs": [ { "contents": [ "If $f_\\lambda$ is universally closed, then", "$B_\\lambda$ is integral over $A$ by", "Lemma \\ref{lemma-integral-universally-closed}.", "Hence $B$ is integral over $A$ and $f$", "is universally closed (by the same lemma).", "This proves the case where each $f_\\lambda$ is universally closed.", "\\medskip\\noindent", "For a prime $\\mathfrak q \\subset B$ lying over $\\mathfrak p \\subset A$", "denote $\\mathfrak q_\\lambda \\subset B_\\lambda$ the inverse image.", "Then $\\kappa(\\mathfrak q) = \\colim \\kappa(\\mathfrak q_\\lambda)$.", "Thus if $A \\to B_\\lambda$ induces purely inseparable extensions", "of residue fields, then the same is true for $A \\to B$. This proves the case", "where $f_\\lambda$ is universally closed and universally injective, see", "Lemma \\ref{lemma-universally-injective}.", "\\medskip\\noindent", "The case where $f$ is a universal homeomorphism follows from", "the remarks above and Lemma \\ref{lemma-universal-homeomorphism}", "combined with the fact that prime ideals in $B$ are the same thing", "as compatible sequences of prime ideals in all of the $B_\\lambda$.", "\\medskip\\noindent", "If $A \\to B_\\lambda$ induces isomorphisms on residue fields, then", "so does $A \\to B$ (see argument in second paragraph).", "In this way we see that the lemma holds in the remaining case." ], "refs": [ "morphisms-lemma-integral-universally-closed", "morphisms-lemma-universally-injective", "morphisms-lemma-universal-homeomorphism" ], "ref_ids": [ 5441, 5167, 5454 ] } ], "ref_ids": [] }, { "id": 5459, "type": "theorem", "label": "morphisms-lemma-special-elements-and-localization", "categories": [ "morphisms" ], "title": "morphisms-lemma-special-elements-and-localization", "contents": [ "Let $A \\subset B$ be a ring extension. Let $S \\subset A$ be a", "multiplicative subset. Let $n \\geq 1$ and", "$b_i \\in B$ for $1 \\leq i \\leq n$. If the set", "$$", "\\{x \\in S^{-1}B \\mid", "x \\not \\in S^{-1}A\\text{ and } b_i x^i \\in S^{-1}A\\text{ for }i = 1, \\ldots, n\\}", "$$", "is nonempty, then so is", "$$", "\\{x \\in B \\mid", "x \\not \\in A\\text{ and } b_i x^i \\in A\\text{ for }i = 1, \\ldots, n\\}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: clear denominators." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5460, "type": "theorem", "label": "morphisms-lemma-nth-and-nplusone-implies-square-and-cube", "categories": [ "morphisms" ], "title": "morphisms-lemma-nth-and-nplusone-implies-square-and-cube", "contents": [ "Let $A \\subset B$ be a ring extension. If there exists", "$b \\in B$, $b \\not \\in A$ and an integer $n \\geq 2$ with", "$b^n \\in A$ and $b^{n + 1} \\in A$, then there exists a", "$b' \\in B$, $b' \\not \\in A$", "with $(b')^2 \\in A$ and $(b')^3 \\in A$." ], "refs": [], "proofs": [ { "contents": [ "Let $b$ and $n$ be as in the lemma.", "Then all sufficiently large powers of $b$ are in $A$.", "Namely, $(b^n)^k(b^{n + 1})^i = b^{(k + i)n + i}$", "which implies any power $b^m$ with $m \\geq n^2$ is in $A$.", "Hence if $i \\geq 1$ is the largest integer such that", "$b^i \\not \\in A$, then $(b^i)^2 \\in A$ and $(b^i)^3 \\in A$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5461, "type": "theorem", "label": "morphisms-lemma-square-and-cube", "categories": [ "morphisms" ], "title": "morphisms-lemma-square-and-cube", "contents": [ "Let $A \\subset B$ be a ring extension such that $\\Spec(B) \\to \\Spec(A)$", "is a universal homeomorphism inducing isomorphisms on residue fields.", "If $A \\not = B$, then there exists a $b \\in B$, $b \\not \\in A$ with", "$b^2 \\in A$ and $b^3 \\in A$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $A \\subset B$ is integral", "(Lemma \\ref{lemma-integral-universally-closed}).", "By Lemma \\ref{lemma-subalgebra-inherits}", "we may assume that $B$ is generated by a single element", "over $A$. Hence $B$ is finite over $A$", "(Algebra, Lemma \\ref{algebra-lemma-characterize-finite-in-terms-of-integral}).", "Hence the support of $B/A$ as an $A$-module is", "closed and not empty (Algebra, Lemmas", "\\ref{algebra-lemma-support-closed} and \\ref{algebra-lemma-support-zero}).", "Let $\\mathfrak p \\subset A$ be a minimal prime", "of the support. After replacing $A \\subset B$ by", "$A_\\mathfrak p \\subset B_\\mathfrak p$ (permissible by", "Lemma \\ref{lemma-special-elements-and-localization})", "we may assume that $(A, \\mathfrak m)$ is a local ring,", "that $B$ is finite over $A$, and that $B/A$ has support $\\{\\mathfrak m\\}$", "as an $A$-module. Since $B/A$ is a finite module,", "we see that $I = \\text{Ann}_A(B/A)$ satisfies $\\mathfrak m = \\sqrt{I}$", "(Algebra, Lemma \\ref{algebra-lemma-support-closed}).", "Let $\\mathfrak m' \\subset B$ be the unique prime ideal lying over", "$\\mathfrak m$. Because $\\Spec(B) \\to \\Spec(A)$ is a homeomorphism,", "we find that $\\mathfrak m' = \\sqrt{IB}$.", "For $f \\in \\mathfrak m'$ pick $n \\geq 1$ such that", "$f^n \\in IB$. Then also $f^{n + 1} \\in IB$.", "Since $IB \\subset A$ by our choice of $I$ we conclude that", "$f^n, f^{n + 1} \\in A$. Using", "Lemma \\ref{lemma-nth-and-nplusone-implies-square-and-cube}", "we conclude our lemma is true if $\\mathfrak m' \\not \\subset A$.", "However, if $\\mathfrak m' \\subset A$, then $\\mathfrak m' = \\mathfrak m$", "and we conclude that $A = B$ as the residue fields", "are isomorphic as well by assumption. This contradiction", "finishes the proof." ], "refs": [ "morphisms-lemma-integral-universally-closed", "morphisms-lemma-subalgebra-inherits", "algebra-lemma-characterize-finite-in-terms-of-integral", "algebra-lemma-support-closed", "algebra-lemma-support-zero", "morphisms-lemma-special-elements-and-localization", "algebra-lemma-support-closed", "morphisms-lemma-nth-and-nplusone-implies-square-and-cube" ], "ref_ids": [ 5441, 5457, 484, 543, 541, 5459, 543, 5460 ] } ], "ref_ids": [] }, { "id": 5462, "type": "theorem", "label": "morphisms-lemma-pth-power-and-multiple", "categories": [ "morphisms" ], "title": "morphisms-lemma-pth-power-and-multiple", "contents": [ "Let $A \\subset B$ be a ring extension such that $\\Spec(B) \\to \\Spec(A)$", "is a universal homeomorphism.", "If $A \\not = B$, then either there exists a $b \\in B$, $b \\not \\in A$ with", "$b^2 \\in A$ and $b^3 \\in A$ or there exists a prime number $p$", "and a $b \\in B$, $b \\not \\in A$ with $pb \\in A$ and $b^p \\in A$." ], "refs": [], "proofs": [ { "contents": [ "The argument is almost exactly the same as in the proof of", "Lemma \\ref{lemma-square-and-cube} but we write everything", "out to make sure it works.", "\\medskip\\noindent", "Recall that $A \\subset B$ is integral", "(Lemma \\ref{lemma-integral-universally-closed}).", "By Lemma \\ref{lemma-subalgebra-inherits}", "we may assume that $B$ is generated by a single element", "over $A$. Hence $B$ is finite over $A$", "(Algebra, Lemma \\ref{algebra-lemma-characterize-finite-in-terms-of-integral}).", "Hence the support of $B/A$ as an $A$-module is", "closed and not empty (Algebra, Lemmas", "\\ref{algebra-lemma-support-closed} and \\ref{algebra-lemma-support-zero}).", "Let $\\mathfrak p \\subset A$ be a minimal prime", "of the support. After replacing $A \\subset B$ by", "$A_\\mathfrak p \\subset B_\\mathfrak p$ (permissible by", "Lemma \\ref{lemma-special-elements-and-localization})", "we may assume that $(A, \\mathfrak m)$ is a local ring,", "that $B$ is finite over $A$, and that $B/A$ has support $\\{\\mathfrak m\\}$", "as an $A$-module. Since $B/A$ is a finite module,", "we see that $I = \\text{Ann}_A(B/A)$ satisfies $\\mathfrak m = \\sqrt{I}$", "(Algebra, Lemma \\ref{algebra-lemma-support-closed}).", "Let $\\mathfrak m' \\subset B$ be the unique prime ideal lying over", "$\\mathfrak m$. Because $\\Spec(B) \\to \\Spec(A)$ is a homeomorphism,", "we find that $\\mathfrak m' = \\sqrt{IB}$.", "For $f \\in \\mathfrak m'$ pick $n \\geq 1$ such that", "$f^n \\in IB$. Then also $f^{n + 1} \\in IB$.", "Since $IB \\subset A$ by our choice of $I$ we conclude that", "$f^n, f^{n + 1} \\in A$. Using", "Lemma \\ref{lemma-nth-and-nplusone-implies-square-and-cube}", "we conclude our lemma is true if $\\mathfrak m' \\not \\subset A$.", "If $\\mathfrak m' \\subset A$, then $\\mathfrak m' = \\mathfrak m$.", "Since $A \\not = B$ we conclude the map", "$\\kappa = A/\\mathfrak m \\to B/\\mathfrak m' = \\kappa'$", "of residue fields cannot be an isomorphism. By", "Lemma \\ref{lemma-universally-injective} we conclude", "that the characteristic of $\\kappa$ is a prime number $p$", "and that the extension $\\kappa'/\\kappa$ is purely inseparable.", "Pick $b \\in B$ whose image in $\\kappa'$ is an element", "not contained in $\\kappa$ but whose $p$th power is in $\\kappa$.", "Then $b \\not \\in A$, $b^p \\in A$, and $pb \\in A$", "(because $pb \\in \\mathfrak m' = \\mathfrak m \\subset A$)", "as desired." ], "refs": [ "morphisms-lemma-square-and-cube", "morphisms-lemma-integral-universally-closed", "morphisms-lemma-subalgebra-inherits", "algebra-lemma-characterize-finite-in-terms-of-integral", "algebra-lemma-support-closed", "algebra-lemma-support-zero", "morphisms-lemma-special-elements-and-localization", "algebra-lemma-support-closed", "morphisms-lemma-nth-and-nplusone-implies-square-and-cube", "morphisms-lemma-universally-injective" ], "ref_ids": [ 5461, 5441, 5457, 484, 543, 541, 5459, 543, 5460, 5167 ] } ], "ref_ids": [] }, { "id": 5463, "type": "theorem", "label": "morphisms-lemma-universal-homeo-iso-if-invert-p", "categories": [ "morphisms" ], "title": "morphisms-lemma-universal-homeo-iso-if-invert-p", "contents": [ "Let $p$ be a prime number. Let $A \\to B$ be a ring map", "which induces an isomorphism $A[1/p] \\to B[1/p]$", "(for example if $p$ is nilpotent in $A$).", "The following are equivalent", "\\begin{enumerate}", "\\item $\\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism, and", "\\item the kernel of $A \\to B$ is a locally nilpotent ideal and", "for every $b \\in B$ there exists a $p$-power $q$ with $qb$ and $b^q$", "in the image of $A \\to B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (2) holds, then (1) holds by Algebra, Lemma \\ref{algebra-lemma-p-ring-map}.", "Assume (1). Then the kernel of $A \\to B$ consists of nilpotent", "elements by Algebra, Lemma \\ref{algebra-lemma-image-dense-generic-points}.", "Thus we may replace $A$ by the image of $A \\to B$ and assume that", "$A \\subset B$. By Algebra, Lemma \\ref{algebra-lemma-help-with-powers}", "the set", "$$", "B' = \\{b \\in B \\mid p^nb, b^{p^n} \\in A\\text{ for some }n \\geq 0\\}", "$$", "is an $A$-subalgebra of $B$ (being closed under products is trivial).", "We have to show $B' = B$. If not, then", "according to Lemma \\ref{lemma-pth-power-and-multiple}", "there exists a $b \\in B$, $b \\not \\in B'$ with either", "$b^2, b^3 \\in B'$ or there exists a prime number $\\ell$", "with $\\ell b, b^\\ell \\in B'$.", "We will show both cases lead to a contradiction, thereby proving the lemma.", "\\medskip\\noindent", "Since $A[1/p] = B[1/p]$ we can choose a $p$-power $q$ such that $qb \\in A$.", "\\medskip\\noindent", "If $b^2, b^3 \\in B'$ then also $b^q \\in B'$. By definition of $B'$", "we find that $(b^q)^{q'} \\in A$ for some $p$-power $q'$.", "Then $qq'b, b^{qq'} \\in A$ whence $b \\in B'$ which is a contradiction.", "\\medskip\\noindent", "Assume now there exists a prime number $\\ell$ with $\\ell b, b^\\ell \\in B'$.", "If $\\ell \\not = p$ then $\\ell b \\in B'$ and $qb \\in A \\subset B'$ imply", "$b \\in B'$ a contradiction. Thus $\\ell = p$ and $b^p \\in B'$", "and we get a contradiction exactly as before." ], "refs": [ "algebra-lemma-p-ring-map", "algebra-lemma-image-dense-generic-points", "algebra-lemma-help-with-powers", "morphisms-lemma-pth-power-and-multiple" ], "ref_ids": [ 582, 446, 580, 5462 ] } ], "ref_ids": [] }, { "id": 5464, "type": "theorem", "label": "morphisms-lemma-make-universal-homeo", "categories": [ "morphisms" ], "title": "morphisms-lemma-make-universal-homeo", "contents": [ "Let $A$ be a ring. Let $x, y \\in A$.", "\\begin{enumerate}", "\\item If $x^3 = y^2$ in $A$, then $A \\to B = A[t]/(t^2 - x, t^3 - y)$", "induces bijections on residue fields and a", "universal homeomorphism on spectra.", "\\item If there is a prime number $p$ such that $p^px = y^p$ in $A$, then", "$A \\to B = A[t]/(t^p - x, pt - y)$ induces a universal", "homeomorphism on spectra.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion of Lemma \\ref{lemma-universal-homeomorphism}", "to check this. In both cases the ring map is integral. Thus it suffices to show", "that given a field $k$ and a ring map $\\varphi : A \\to k$", "the $k$-algebra $B \\otimes_A k$ has a unique prime ideal", "whose residue field is equal to $k$ in case (1) and", "purely inseparable over $k$ in case (2). See", "Lemma \\ref{lemma-universally-injective}.", "\\medskip\\noindent", "In case (1) set $\\lambda = 0$ if $\\varphi(x) = 0$ and set", "$\\lambda = \\varphi(y)/\\varphi(x)$ if not. Then", "$B = k[t]/(t^2 - \\lambda^2, t^3 - \\lambda^2)$.", "Thus the result is clear.", "\\medskip\\noindent", "In case (2) if the characteristic of $k$ is $p$, then", "we obtain $\\varphi(y) = 0$ and $B = k[t]/(t^p - \\varphi(x))$", "which is a local Artinian $k$-algebra whose residue field", "is either $k$ or a degree $p$ purely inseparable extension of $k$.", "If the characteristic of $k$ is not $p$, then setting", "$\\lambda = \\varphi(y)/p$ we see $B = k[t]/(t - \\lambda) = k$", "and we conclude as well." ], "refs": [ "morphisms-lemma-universal-homeomorphism", "morphisms-lemma-universally-injective" ], "ref_ids": [ 5454, 5167 ] } ], "ref_ids": [] }, { "id": 5465, "type": "theorem", "label": "morphisms-lemma-universal-homeo-limit", "categories": [ "morphisms" ], "title": "morphisms-lemma-universal-homeo-limit", "contents": [ "Let $A \\to B$ be a ring map.", "\\begin{enumerate}", "\\item If $A \\to B$ induces a universal homeomorphism on spectra,", "then $B = \\colim B_i$ is a filtered colimit of finitely presented $A$-algebras", "$B_i$ such that $A \\to B_i$ induces a universal homeomorphism on spectra.", "\\item If $A \\to B$ induces isomorphisms on residue fields and", "a universal homeomorphism on spectra, then $B = \\colim B_i$ is a", "filtered colimit of finitely presented $A$-algebras $B_i$ such that", "$A \\to B_i$ induces isomorphisms on residue fields and a", "universal homeomorphism on spectra.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). We will use the criterion of", "Algebra, Lemma \\ref{algebra-lemma-when-colimit}.", "Let $A \\to C$ be of finite presentation and let $\\varphi : C \\to B$", "be an $A$-algebra map. Let $B' = \\varphi(C) \\subset B$ be the image.", "Then $A \\to B'$ induces a universal homeomorphism on spectra by", "Lemma \\ref{lemma-subalgebra-inherits}.", "By Algebra, Lemma \\ref{algebra-lemma-ring-colimit-fp}", "we can write $B' = \\colim_{i \\in I} B_i$", "with $A \\to B_i$ of finite presentation and surjective", "transition maps.", "By Algebra, Lemma \\ref{algebra-lemma-characterize-finite-presentation}", "we can choose an index $0 \\in I$", "and a factorization $C \\to B_0 \\to B'$ of the map $C \\to B'$.", "We claim that $\\Spec(B_i) \\to \\Spec(A)$ is a universal homeomorphism", "for $i$ sufficiently large. The claim finishes the proof of (1).", "\\medskip\\noindent", "Proof of the claim. By Lemma \\ref{lemma-reduction-universal-homeomorphism}", "the ring map $A_{red} \\to B'_{red}$ induces a universal homeomorphism", "on spectra. Thus $A_{red} \\subset B'_{red}$ by", "Algebra, Lemma \\ref{algebra-lemma-image-dense-generic-points}.", "Setting $A' = \\Im(A \\to B')$ we have surjections $A \\to A' \\to A_{red}$", "inducing bijections $\\Spec(A_{red}) = \\Spec(A') = \\Spec(A)$.", "Thus $A' \\subset B'$ induces a universal homeomorphism on spectra.", "By Proposition \\ref{proposition-universal-homeomorphism}", "and the fact that $B'$ is finite type over $A'$ we can", "find $n$ and $b'_1, \\ldots, b'_n \\in B'$ such that", "$B' = A'[b'_1, \\ldots, b'_n]$", "and such that for $j = 1, \\ldots, n$ we have", "\\begin{enumerate}", "\\item $(b'_j)^2, (b'_j)^3 \\in A'[b'_1, \\ldots, b'_{j - 1}]$, or", "\\item there exists a prime number $p$ with", "$pb'_j, (b'_j)^p \\in A'[b'_1, \\ldots, b'_{j - 1}]$.", "\\end{enumerate}", "Choose $b_1, \\ldots, b_n \\in B_0$ lifting $b'_1, \\ldots, b'_n$.", "For $i \\geq 0$ denote $b_{j, i}$ the image of $b_j$ in $B_i$.", "For large enough $i$ we will have for $j = 1, \\ldots, n$", "\\begin{enumerate}", "\\item $b_{j, i}^2, b_{j, i}^3 \\in A_i[b_{1, i}, \\ldots, b_{j - 1, i}]$, or", "\\item there exists a prime number $p$ with", "$pb_{j, i}, b_{j, i}^p \\in A_i[b_{1, i}, \\ldots, b_{j - 1, i}]$.", "\\end{enumerate}", "Here $A_i \\subset B_i$ is the image of $A \\to B_i$. Observe that $A \\to A_i$", "is a surjective ring map whose kernel is a locally nilpotent ideal.", "After increasing $i$ more if necessary, we may assume $B_i$", "is generated by $b_1, \\ldots, b_n$ over $A_i$, in other words", "$B_i = A_i[b_1, \\ldots, b_n]$.", "By Algebra, Lemmas \\ref{algebra-lemma-p-ring-map} and", "\\ref{algebra-lemma-2-3-ring-map} we conclude that", "$A \\to A_i \\to A_i[b_1] \\to \\ldots \\to A_i[b_1, \\ldots, b_n] = B_i$", "induce universal homeomorphisms on spectra. This finishes the proof", "of the claim.", "\\medskip\\noindent", "The proof of (2) is exactly the same." ], "refs": [ "algebra-lemma-when-colimit", "morphisms-lemma-subalgebra-inherits", "algebra-lemma-ring-colimit-fp", "algebra-lemma-characterize-finite-presentation", "morphisms-lemma-reduction-universal-homeomorphism", "algebra-lemma-image-dense-generic-points", "morphisms-proposition-universal-homeomorphism", "algebra-lemma-p-ring-map", "algebra-lemma-2-3-ring-map" ], "ref_ids": [ 1093, 5457, 1091, 1092, 5455, 446, 5536, 582, 579 ] } ], "ref_ids": [] }, { "id": 5466, "type": "theorem", "label": "morphisms-lemma-seminormal-local-property", "categories": [ "morphisms" ], "title": "morphisms-lemma-seminormal-local-property", "contents": [ "Being seminormal or being absolutely weakly normal", "is a local property of rings, see", "Properties, Definition \\ref{properties-definition-property-local}." ], "refs": [ "properties-definition-property-local" ], "proofs": [ { "contents": [ "Suppose that $A$ is seminormal and $f \\in A$. Let $x', y' \\in A_f$", "with $(x')^3 = (y')^2$. Write $x' = x/f^{2n}$ and $y' = y/f^{3n}$", "for some $n \\geq 0$ and $x, y \\in A$. After replacing $x, y$", "by $f^{2m}x, f^{3m}y$ and $n$ by $n + m$, we see that", "$x^3 = y^2$ in $A$. Then we find a unique $a \\in A$ with", "$x = a^2$ and $y = a^3$. Setting $a' = a/f^n$ we get", "$x' = (a')^2$ and $y' = (a')^3$ as desired. Uniqueness", "of $a'$ follows from uniqueness of $a$.", "In exactly the same manner the reader shows that if $A$ is", "absolutely weakly normal,", "then $A_f$ is absolutely weakly normal.", "\\medskip\\noindent", "Assume $A$ is a ring and $f_1, \\ldots, f_n \\in A$ generate the unit", "ideal. Assume $A_{f_i}$ is seminormal for each $i$.", "Let $x, y \\in A$ with $x^3 = y^2$. For each $i$ we find a unique", "$a_i \\in A_{f_i}$ with $x = a_i^2$ and $y = a_i^3$ in $A_{f_i}$.", "By the uniqueness and the result of the first paragraph (which", "tells us that $A_{f_if_j}$ is seminormal) we see that", "$a_i$ and $a_j$ map to the same element of $A_{f_if_j}$.", "By Algebra, Lemma \\ref{algebra-lemma-standard-covering}", "we find a unique $a \\in A$", "mapping to $a_i$ in $A_{f_i}$ for all $i$.", "Then $x = a^2$ and $y = a^3$ by the same token. Clearly this $a$ is unique.", "Thus $A$ is seminormal. If we assume $A_{f_i}$ is absolutely weakly normal,", "then the exact same argument shows that $A$ is absolutely weakly normal." ], "refs": [ "algebra-lemma-standard-covering" ], "ref_ids": [ 414 ] } ], "ref_ids": [ 3069 ] }, { "id": 5467, "type": "theorem", "label": "morphisms-lemma-locally-seminormal", "categories": [ "morphisms" ], "title": "morphisms-lemma-locally-seminormal", "contents": [ "Let $X$ be a scheme. The following are equivalent:", "\\begin{enumerate}", "\\item The scheme $X$ is seminormal.", "\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$", "is seminormal.", "\\item There exists an affine open covering $X = \\bigcup U_i$ such that", "each $\\mathcal{O}_X(U_i)$ is seminormal.", "\\item There exists an open covering $X = \\bigcup X_j$", "such that each open subscheme $X_j$ is seminormal.", "\\end{enumerate}", "Moreover, if $X$ is seminormal", "then every open subscheme is seminormal.", "The same statements are true with ``seminormal'' replaced by", "``absolutely weakly normal''." ], "refs": [], "proofs": [ { "contents": [ "Combine Properties, Lemma \\ref{properties-lemma-locally-P} and", "Lemma \\ref{lemma-seminormal-local-property}." ], "refs": [ "properties-lemma-locally-P", "morphisms-lemma-seminormal-local-property" ], "ref_ids": [ 2948, 5466 ] } ], "ref_ids": [] }, { "id": 5468, "type": "theorem", "label": "morphisms-lemma-seminormal-reduced", "categories": [ "morphisms" ], "title": "morphisms-lemma-seminormal-reduced", "contents": [ "A seminormal scheme or ring is reduced. A fortiori the same", "is true for absolutely weakly normal schemes or rings." ], "refs": [], "proofs": [ { "contents": [ "Let $A$ be a ring.", "If $a \\in A$ is nonzero but $a^2 = 0$, then $a^2 = 0^2$ and", "$a^3 = 0^3$ and hence $A$ is not seminormal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5469, "type": "theorem", "label": "morphisms-lemma-seminormalization-ring", "categories": [ "morphisms" ], "title": "morphisms-lemma-seminormalization-ring", "contents": [ "Let $A$ be a ring.", "\\begin{enumerate}", "\\item The category of ring maps $A \\to B$ inducing a", "universal homeomorphism on spectra has a final object $A \\to A^{awn}$.", "\\item Given $A \\to B$ in the category of (1) the resulting map", "$B \\to A^{awn}$ is an isomorphism if and only if $B$ is", "absolutely weakly normal.", "\\item The category of ring maps $A \\to B$ inducing isomorphisms on", "residue fields and a universal homeomorphism on spectra has a final", "object $A \\to A^{sn}$.", "\\item Given $A \\to B$ in the category of (3) the resulting map", "$B \\to A^{sn}$ is an isomorphism if and only if $B$ is seminormal.", "\\end{enumerate}", "For any ring map $\\varphi : A \\to A'$ there are unique maps", "$\\varphi^{awn} : A^{awn} \\to (A')^{awn}$ and", "$\\varphi^{sn} : A^{sn} \\to (A')^{sn}$ compatible with $\\varphi$." ], "refs": [], "proofs": [ { "contents": [ "We prove (1) and (2) and we omit the proof of (3) and (4) and", "the final statement. Consider the category of $A$-algebras of the form", "$$", "B = A[x_1, \\ldots, x_n]/J", "$$", "where $J$ is a finitely generated ideal such that $A \\to B$ defines a", "universal homeomorphism on spectra. We claim this category is directed", "(Categories, Definition \\ref{categories-definition-directed}). Namely, given", "$$", "B = A[x_1, \\ldots, x_n]/J", "\\quad\\text{and}\\quad", "B' = A[x_1, \\ldots, x_{n'}]/J'", "$$", "then we can consider", "$$", "B'' = A[x_1, \\ldots, x_{n + n'}]/J''", "$$", "where $J''$ is generated by the elements of $J$ and the elements", "$f(x_{n + 1}, \\ldots, x_{n + n'})$ where $f \\in J'$.", "Then we have $A$-algebra homomorphisms $B \\to B''$ and $B' \\to B''$", "which induce an isomorphism $B \\otimes_A B' \\to B''$.", "It follows from", "Lemmas \\ref{lemma-base-change-universal-homeomorphism} and", "\\ref{lemma-composition-universal-homeomorphism}", "that $\\Spec(B'') \\to \\Spec(A)$ is a universal homeomorphism and", "hence $A \\to B''$ is in our category.", "Finally, given $\\varphi, \\varphi' : B \\to B'$", "in our category with $B$ as displayed above,", "then we consider the quotient $B''$ of $B'$", "by the ideal generated by $\\varphi(x_i) - \\varphi'(x_i)$, $i = 1, \\ldots, n$.", "Since $\\Spec(B') = \\Spec(B)$ we see that $\\Spec(B'') \\to \\Spec(B')$", "is a bijective closed immersion hence a universal homeomorphism.", "Thus $B''$ is in our category and $\\varphi, \\varphi'$ are equalized", "by $B' \\to B''$. This completes the proof of our claim. We set", "$$", "A^{awn} = \\colim B", "$$", "where the colimit is over the category just described. Observe that", "$A \\to A^{awn}$ induces a universal homeomorphism on spectra by", "Lemma \\ref{lemma-colimit-inherits} (this is where we use the category", "is directed).", "\\medskip\\noindent", "Given a ring map $A \\to B$ of finite presentation inducing a universal", "homeomorphism on spectra, we get a canonical map $B \\to A^{awn}$ by the", "very construction of $A^{awn}$. Since every $A \\to B$ as in (1)", "is a filtered colimit of $A \\to B$ as in (1) of finite presentation", "(Lemma \\ref{lemma-universal-homeo-limit}), we see that", "$A \\to A^{awn}$ is final in the category (1).", "\\medskip\\noindent", "Let $x, y \\in A^{awn}$ be elements such that $x^3 = y^2$.", "Then $A^{awn} \\to A^{awn}[t]/(t^2 - x, t^3 - y)$ induces", "a universal homeomorphism on spectra by Lemma \\ref{lemma-make-universal-homeo}.", "Thus $A \\to A^{awn}[t]/(t^2 - x, t^3 - y)$ is in the category (1)", "and we obtain a unique $A$-algebra map", "$A^{awn}[t]/(t^2 - x, t^3 - y) \\to A^{awn}$.", "The image $a \\in A^{awn}$ of $t$ is therefore the unique", "element such that $a^2 = x$ and $a^3 = y$ in $A^{awn}$.", "In exactly the same manner, given a prime $p$ and $x, y \\in A^{awn}$", "with $p^px = y^p$ we find a unique $a \\in A^{awn}$ with", "$a^p = x$ and $pq = y$. Thus $A^{awn}$ is absolutely weakly normal", "by definition.", "\\medskip\\noindent", "Finally, let $A \\to B$ be in the category (1) with $B$ absolutely weakly normal.", "Since $A^{awn} \\to B^{awn}$ is a universal homeomorphism and ", "since $A^{awn}$ is reduced (Lemma \\ref{lemma-seminormal-reduced}) we find", "$A^{awn} \\subset B^{awn}$", "(see Algebra, Lemma \\ref{algebra-lemma-image-dense-generic-points}).", "If this inclusion is not an equality, then", "Lemma \\ref{lemma-pth-power-and-multiple}", "implies there is an element $b \\in B^{awn}$, $b \\not \\in A^{awn}$", "such that either $b^2, b^3 \\in A^{awn}$ or $pb, b^p \\in A^{awn}$", "for some prime number $p$. However, by the existence and uniqueness in", "Definition \\ref{definition-seminormal-ring} this forces $b \\in A^{awn}$", "and hence we obtain the contradiction that finishes the proof." ], "refs": [ "categories-definition-directed", "morphisms-lemma-base-change-universal-homeomorphism", "morphisms-lemma-composition-universal-homeomorphism", "morphisms-lemma-colimit-inherits", "morphisms-lemma-universal-homeo-limit", "morphisms-lemma-make-universal-homeo", "morphisms-lemma-seminormal-reduced", "algebra-lemma-image-dense-generic-points", "morphisms-lemma-pth-power-and-multiple", "morphisms-definition-seminormal-ring" ], "ref_ids": [ 12363, 5451, 5452, 5458, 5465, 5464, 5468, 446, 5462, 5575 ] } ], "ref_ids": [] }, { "id": 5470, "type": "theorem", "label": "morphisms-lemma-seminormalization", "categories": [ "morphisms" ], "title": "morphisms-lemma-seminormalization", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item The category of universal homeomorphisms $Y \\to X$ has", "an initial object $X^{awn} \\to X$.", "\\item Given $Y \\to X$ in the category of (1) the resulting morphism", "$X^{awn} \\to Y$ is an isomorphism if and only if $Y$ is", "absolutely weakly normal.", "\\item The category of universal homeomorphisms $Y \\to X$ which", "induce ismomorphisms on residue fields has an initial object", "$X^{sn} \\to X$.", "\\item Given $Y \\to X$ in the category of (3) the resulting morphism", "$X^{sn} \\to Y$ is an isomorphism if and only if $Y$ is seminormal.", "\\end{enumerate}", "For any morphism $h : X' \\to X$ of schemes there are unique morphisms", "$h^{awn} : (X')^{awn} \\to X^{awn}$ and $h^{sn} : (X')^{sn} \\to X^{sn}$", "compatible with $h$." ], "refs": [], "proofs": [ { "contents": [ "We will prove (3) and (4) and omit the proof of (1) and (2).", "Let $h : X' \\to X$ be a morphism of schemes. If (3) holds for $X$", "and $X'$, then $X' \\times_X X^{sn} \\to X'$ is a universal homeomorphism", "and hence we get a unique morphism $(X')^{sn} \\to X' \\times_X X^{sn}$", "over $X'$ by the universal property of $(X')^{sn} \\to X'$. Composed with the", "projection $X' \\times_X X^{sn} \\to X^{sn}$ we obtain $h^{sn}$.", "If in addition (4) holds for $X$ and $X'$", "and $h$ is an open immersion, then $X' \\times_X X^{sn}$ is seminormal", "(Lemma \\ref{lemma-locally-seminormal}) and we deduce that", "$(X')^{sn} \\to X' \\times_X X^{sn}$ is an isomorphism.", "\\medskip\\noindent", "Recall that any universal homeomorphism is affine, see", "Lemma \\ref{lemma-homeomorphism-affine}. Thus if $X$ is affine", "then (3) and (4) follow immediately from", "Lemma \\ref{lemma-seminormalization-ring}.", "Let $X$ be a scheme and let $\\mathcal{B}$ be the set of affine opens", "of $X$. For each $U \\in \\mathcal{B}$ we obtain $U^{sn} \\to U$", "and for $V \\subset U$, $V, U \\in \\mathcal{B}$ we obtain a canonical", "isomorphism $\\rho_{V, U} : V^{sn} \\to V \\times_U U^{sn}$ by the discussion", "in the previous paragraph. Thus by relative glueing", "(Constructions, Lemma \\ref{constructions-lemma-relative-glueing})", "we obtain a morphism $X^{sn} \\to X$ which restricts", "to $U^{sn}$ over $U$ compatibly with the $\\rho_{V, U}$.", "Next, let $Y \\to X$ be a universal homeomorphism.", "Then $U \\times_X Y \\to U$ is a universal homeomorphism for $U \\in \\mathcal{B}$", "and we obtain a unique morphism $g_U : U^{sn} \\to U \\times_X Y$ over $U$.", "These $g_U$ are compatible with the morphisms $\\rho_{V, U}$; details", "omitted. Hence there is a unique morphism $g : X^{sn} \\to Y$", "over $X$ agreeing with $g_U$ over $U$, see", "Constructions, Remark \\ref{constructions-remark-relative-glueing-functorial}.", "This proves (3) for $X$. The proof of (4) is similar; details omitted." ], "refs": [ "morphisms-lemma-locally-seminormal", "morphisms-lemma-homeomorphism-affine", "morphisms-lemma-seminormalization-ring", "constructions-lemma-relative-glueing", "constructions-remark-relative-glueing-functorial" ], "ref_ids": [ 5467, 5453, 5469, 12581, 12668 ] } ], "ref_ids": [] }, { "id": 5471, "type": "theorem", "label": "morphisms-lemma-finite-flat", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-flat", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is finite locally free,", "\\item $f$ is finite, flat, and locally of finite presentation.", "\\end{enumerate}", "If $S$ is locally Noetherian these are also equivalent to", "\\begin{enumerate}", "\\item[(3)] $f$ is finite and flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset S$ be affine open. In all three cases", "the morphism is affine hence $f^{-1}(V)$ is affine.", "Thus we may write $V = \\Spec(R)$ and $f^{-1}(V) = \\Spec(A)$", "for some $R$-algebra $A$.", "Assume (1). This means we can cover $S$ by affine opens", "$V = \\Spec(R)$ such that $A$ is finite free as an $R$-module.", "Then $R \\to A$ is of finite presentation by", "Algebra, Lemma \\ref{algebra-lemma-finite-finite-type}.", "Thus (2) holds. Conversely, assume (2).", "For every affine open $V = \\Spec(R)$ of $S$ the ring map", "$R \\to A$ is finite and of finite presentation and $A$ is flat", "as an $R$-module. By", "Algebra, Lemma \\ref{algebra-lemma-finite-finitely-presented-extension}", "we see that $A$ is finitely presented as an $R$-module.", "Thus Algebra, Lemma \\ref{algebra-lemma-finite-projective}", "implies $A$ is finite locally free. Thus (1) holds.", "The Noetherian case follows as a finite module over a Noetherian ring", "is a finitely presented module, see Algebra,", "Lemma \\ref{algebra-lemma-Noetherian-finite-type-is-finite-presentation}." ], "refs": [ "algebra-lemma-finite-finite-type", "algebra-lemma-finite-finitely-presented-extension", "algebra-lemma-finite-projective", "algebra-lemma-Noetherian-finite-type-is-finite-presentation" ], "ref_ids": [ 338, 501, 795, 451 ] } ], "ref_ids": [] }, { "id": 5472, "type": "theorem", "label": "morphisms-lemma-composition-finite-locally-free", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-finite-locally-free", "contents": [ "A composition of finite locally free morphisms is finite locally free." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5473, "type": "theorem", "label": "morphisms-lemma-base-change-finite-locally-free", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-finite-locally-free", "contents": [ "A base change of a finite locally free morphism is finite locally free." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5474, "type": "theorem", "label": "morphisms-lemma-finite-locally-free", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-locally-free", "contents": [ "Let $f : X \\to S$ be a finite locally free morphism of schemes.", "There exists a disjoint union decomposition", "$S = \\coprod_{d \\geq 0} S_d$ by open and closed subschemes", "such that setting $X_d = f^{-1}(S_d)$ the restrictions", "$f|_{X_d}$ are finite locally free morphisms $X_d \\to S_d$", "of degree $d$." ], "refs": [], "proofs": [ { "contents": [ "This is true because a finite locally free sheaf locally has", "a well defined rank. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5475, "type": "theorem", "label": "morphisms-lemma-massage-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-massage-finite", "contents": [ "Let $f : Y \\to X$ be a finite morphism with $X$ affine.", "There exists a diagram", "$$", "\\xymatrix{", "Z' \\ar[rd] &", "Y' \\ar[l]^i \\ar[d] \\ar[r] &", "Y \\ar[d] \\\\", " & X' \\ar[r] & X", "}", "$$", "where", "\\begin{enumerate}", "\\item $Y' \\to Y$ and $X' \\to X$ are surjective finite locally free,", "\\item $Y' = X' \\times_X Y$,", "\\item $i : Y' \\to Z'$ is a closed immersion,", "\\item $Z' \\to X'$ is finite locally free, and", "\\item $Z' = \\bigcup_{j = 1, \\ldots, m} Z'_j$ is a (set theoretic)", "finite union of closed subschemes, each of which maps isomorphically", "to $X'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $X = \\Spec(A)$ and $Y = \\Spec(B)$. See also", "More on Algebra, Section \\ref{more-algebra-section-descent-flatness-integral}.", "Let $x_1, \\ldots, x_n \\in B$ be generators of $B$ over $A$.", "For each $i$ we can choose a monic polynomial $P_i(T) \\in A[T]$", "such that $P(x_i) = 0$ in $B$. By", "Algebra, Lemma \\ref{algebra-lemma-adjoin-roots}", "(applied $n$ times) there exists a finite locally free ring", "extension $A \\subset A'$ such that each $P_i$ splits completely:", "$$", "P_i(T) = \\prod\\nolimits_{k = 1, \\ldots, d_i} (T - \\alpha_{ik})", "$$", "for certain $\\alpha_{ik} \\in A'$. Set", "$$", "C = A'[T_1, \\ldots, T_n]/(P_1(T_1), \\ldots, P_n(T_n))", "$$", "and $B' = A' \\otimes_A B$. The map $C \\to B'$, $T_i \\mapsto 1 \\otimes x_i$", "is an $A'$-algebra surjection. Setting $X' = \\Spec(A')$,", "$Y' = \\Spec(B')$ and $Z' = \\Spec(C)$ we see that", "(1) -- (4) hold. Part (5) holds because set theoretically", "$\\Spec(C)$ is the union of the closed subschemes", "cut out by the ideals", "$$", "(T_1 - \\alpha_{1k_1}, T_2 - \\alpha_{2k_2}, \\ldots, T_n - \\alpha_{nk_n})", "$$", "for any $1 \\leq k_i \\leq d_i$." ], "refs": [ "algebra-lemma-adjoin-roots" ], "ref_ids": [ 1179 ] } ], "ref_ids": [] }, { "id": 5476, "type": "theorem", "label": "morphisms-lemma-image-nowhere-dense-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-image-nowhere-dense-finite", "contents": [ "Let $f : Y \\to X$ be a finite morphism of schemes.", "Let $T \\subset Y$ be a closed nowhere dense subset of $Y$.", "Then $f(T) \\subset X$ is a closed nowhere dense subset of $X$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-proper} we know that $f(T) \\subset X$ is closed.", "Let $X = \\bigcup X_i$ be an affine covering.", "Since $T$ is nowhere dense in $Y$, we see that also $T \\cap f^{-1}(X_i)$", "is nowhere dense in $f^{-1}(X_i)$. Hence if we can prove the theorem in the", "affine case, then we see that $f(T) \\cap X_i$ is nowhere dense.", "This then implies that $T$ is nowhere dense in $X$ by", "Topology, Lemma \\ref{topology-lemma-nowhere-dense-local}.", "\\medskip\\noindent", "Assume $X$ is affine. Choose a diagram", "$$", "\\xymatrix{", "Z' \\ar[rd] &", "Y' \\ar[l]^i \\ar[d]^{f'} \\ar[r]_a &", "Y \\ar[d]^f \\\\", " & X' \\ar[r]^b & X", "}", "$$", "as in Lemma \\ref{lemma-massage-finite}. The morphisms $a$, $b$ are", "open since they are finite locally free", "(Lemmas \\ref{lemma-finite-flat} and \\ref{lemma-fppf-open}).", "Hence $T' = a^{-1}(T)$ is nowhere dense, see", "Topology, Lemma \\ref{topology-lemma-open-inverse-image-closed-nowhere-dense}.", "The morphism $b$ is surjective and open.", "Hence, if we can prove $f'(T') = b^{-1}(f(T))$ is", "nowhere dense, then $f(T)$ is nowhere dense, see", "Topology, Lemma \\ref{topology-lemma-open-inverse-image-closed-nowhere-dense}.", "As $i$ is a closed immersion, by", "Topology, Lemma \\ref{topology-lemma-closed-image-nowhere-dense}", "we see that $i(T') \\subset Z'$ is closed and nowhere dense.", "Thus we have reduced the problem to the case discussed", "in the following paragraph.", "\\medskip\\noindent", "Assume that $Y = \\bigcup_{i = 1, \\ldots, n} Y_i$ is a finite union of", "closed subsets, each mapping isomorphically to $X$. Consider", "$T_i = Y_i \\cap T$. If each of the $T_i$ is nowhere dense in $Y_i$,", "then each $f(T_i)$ is nowhere dense in $X$ as $Y_i \\to X$ is an isomorphism.", "Hence $f(T) = f(T_i)$ is a finite union of nowhere dense closed", "subsets of $X$ and we win, see", "Topology, Lemma \\ref{topology-lemma-nowhere-dense}.", "Suppose not, say $T_1$ contains a nonempty open $V \\subset Y_1$.", "We are going to show this leads to a contradiction.", "Consider $Y_2 \\cap V \\subset V$. This is either", "a proper closed subset, or equal to $V$. In the first case we replace", "$V$ by $V \\setminus V \\cap Y_2$, so $V \\subset T_1$ is open in $Y_1$ and", "does not meet $Y_2$. In the second case we have", "$V \\subset Y_1 \\cap Y_2$ is open in both $Y_1$ and $Y_2$.", "Repeat sequentially with $i = 3, \\ldots, n$. The result is a disjoint", "union decomposition", "$$", "\\{1, \\ldots, n\\} = I_1 \\amalg I_2, \\quad 1 \\in I_1", "$$", "and an open $V$ of $Y_1$ contained in $T_1$ such that $V \\subset Y_i$", "for $i \\in I_1$ and $V \\cap Y_i = \\emptyset$ for $i \\in I_2$. Set", "$U = f(V)$. This is an open of $X$ since $f|_{Y_1} : Y_1 \\to X$ is", "an isomorphism. Then", "$$", "f^{-1}(U) = V\\ \\amalg\\ \\bigcup\\nolimits_{i \\in I_2} (Y_i \\cap f^{-1}(U))", "$$", "As $\\bigcup_{i \\in I_2} Y_i$ is closed, this implies that", "$V \\subset f^{-1}(U)$ is open, hence $V \\subset Y$ is open.", "This contradicts the assumption that $T$ is nowhere dense in $Y$, as desired." ], "refs": [ "morphisms-lemma-finite-proper", "topology-lemma-nowhere-dense-local", "morphisms-lemma-massage-finite", "morphisms-lemma-finite-flat", "morphisms-lemma-fppf-open", "topology-lemma-open-inverse-image-closed-nowhere-dense", "topology-lemma-open-inverse-image-closed-nowhere-dense", "topology-lemma-closed-image-nowhere-dense", "topology-lemma-nowhere-dense" ], "ref_ids": [ 5445, 8296, 5475, 5471, 5267, 8298, 8298, 8297, 8294 ] } ], "ref_ids": [] }, { "id": 5477, "type": "theorem", "label": "morphisms-lemma-rational-map-finite-presentation", "categories": [ "morphisms" ], "title": "morphisms-lemma-rational-map-finite-presentation", "contents": [ "Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Assume $X$ has", "finitely many irreducible components with generic points", "$x_1, \\ldots, x_n$. Let $s_i \\in S$ be the image of $x_i$.", "Consider the map", "$$", "\\left\\{", "\\begin{matrix}", "S\\text{-rational maps} \\\\", "\\text{from }X\\text{ to }Y", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "(y_1, \\varphi_1, \\ldots, y_n, \\varphi_n)\\text{ where }", "y_i \\in Y\\text{ lies over }s_i\\text{ and}\\\\", "\\varphi_i : \\mathcal{O}_{Y, y_i} \\to \\mathcal{O}_{X, x_i}", "\\text{ is a local }\\mathcal{O}_{S, s_i}\\text{-algebra map}", "\\end{matrix}", "\\right\\}", "$$", "which sends $f : U \\to Y$ to the $2n$-tuple with", "$y_i = f(x_i)$ and $\\varphi_i = f^\\sharp_{x_i}$. Then", "\\begin{enumerate}", "\\item If $Y \\to S$ is locally of finite type, then the map is injective.", "\\item If $Y \\to S$ is locally of finite presentation, then the map is bijective.", "\\item If $Y \\to S$ is locally of finite type and $X$ reduced,", "then the map is bijective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that any dense open of $X$ contains the points $x_i$", "so the construction makes sense. To prove (1) or (2)", "we may replace $X$ by any dense open. Thus if $Z_1, \\ldots, Z_n$", "are the irreducible components of $X$, then we may replace", "$X$ by $X \\setminus \\bigcup_{i \\not = j} Z_i \\cap Z_j$.", "After doing this $X$ is the disjoint union of its irreducible", "components (viewed as open and closed subschemes). Then both the", "right hand side and the left hand side of the arrow are products", "over the irreducible components and we reduce to the case where", "$X$ is irreducible.", "\\medskip\\noindent", "Assume $X$ is irreducible with generic point $x$ lying over $s \\in S$.", "Part (1) follows from part (1) of", "Lemma \\ref{lemma-morphism-defined-local-ring}.", "Parts (2) and (3) follow from part (2) of the same lemma." ], "refs": [ "morphisms-lemma-morphism-defined-local-ring" ], "ref_ids": [ 5418 ] } ], "ref_ids": [] }, { "id": 5478, "type": "theorem", "label": "morphisms-lemma-integral-scheme-rational-functions", "categories": [ "morphisms" ], "title": "morphisms-lemma-integral-scheme-rational-functions", "contents": [ "Let $X$ be a scheme with finitely many irreducible components", "$X_1, \\ldots, X_n$. If $\\eta_i \\in X_i$ is the generic point, then", "$$", "R(X) = \\mathcal{O}_{X, \\eta_1} \\times \\ldots \\times \\mathcal{O}_{X, \\eta_n}", "$$", "If $X$ is reduced this is equal to $\\prod \\kappa(\\eta_i)$.", "If $X$ is integral then $R(X) = \\mathcal{O}_{X, \\eta} = \\kappa(\\eta)$", "is a field." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be an open dense subset. Then", "$U_i = (U \\cap X_i) \\setminus (\\bigcup_{j \\not = i} X_j)$", "is nonempty open as it contained $\\eta_i$, contained in $X_i$,", "and $\\bigcup U_i \\subset U \\subset X$ is dense.", "Thus the identification in the lemma comes from the string of equalities", "\\begin{align*}", "R(X)", "& =", "\\colim_{U \\subset X\\text{ open dense}} \\Mor(U, \\mathbf{A}^1_\\mathbf{Z}) \\\\", "& =", "\\colim_{U \\subset X\\text{ open dense}} \\mathcal{O}_X(U) \\\\", "& =", "\\colim_{\\eta_i \\in U_i \\subset X\\text{ open}} \\prod \\mathcal{O}_X(U_i) \\\\", "& =", "\\prod \\colim_{\\eta_i \\in U_i \\subset X\\text{ open}} \\mathcal{O}_X(U_i) \\\\", "& =", "\\prod \\mathcal{O}_{X, \\eta_i}", "\\end{align*}", "where the second equality is", "Schemes, Example \\ref{schemes-example-global-sections}.", "The final statement follows from", "Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring}." ], "refs": [ "algebra-lemma-minimal-prime-reduced-ring" ], "ref_ids": [ 418 ] } ], "ref_ids": [] }, { "id": 5479, "type": "theorem", "label": "morphisms-lemma-distinct-local-rings", "categories": [ "morphisms" ], "title": "morphisms-lemma-distinct-local-rings", "contents": [ "Let $X$ be an integral separated scheme.", "Let $Z_1$, $Z_2$ be distinct irreducible closed subsets of $X$.", "Let $\\eta_i$ be the generic point of $Z_i$.", "If $Z_1 \\not\\subset Z_2$, then", "$\\mathcal{O}_{X, \\eta_1} \\not \\subset \\mathcal{O}_{X, \\eta_2}$", "as subrings of $R(X)$.", "In particular, if $Z_1 = \\{x\\}$ consists of one closed point $x$,", "there exists a function regular in a neighborhood of $x$", "which is not in $\\mathcal{O}_{X, \\eta_{2}}$." ], "refs": [], "proofs": [ { "contents": [ "First observe that under the assumption of $X$ being separated,", "there is a unique map of schemes", "$\\Spec(\\mathcal{O}_{X, \\eta_2}) \\to X$ over $X$", "such that the composition", "$$", "\\Spec(R(X)) \\longrightarrow", "\\Spec(\\mathcal{O}_{X, \\eta_2}) \\longrightarrow X", "$$", "is the canonical map $\\Spec(R(X)) \\to X$.", "Namely, there is the canonical map", "$can : \\Spec(\\mathcal{O}_{X, \\eta_2}) \\to X$, see", "Schemes, Equation (\\ref{schemes-equation-canonical-morphism}).", "Given a second morphism $a$ to $X$, we have that $a$ agrees with $can$", "on the generic point of", "$\\Spec(\\mathcal{O}_{X, \\eta_2})$ by assumption.", "Now $X$ being separated guarantees that the subset in", "$\\Spec(\\mathcal{O}_{X, \\eta_2})$ where", "these two maps agree is closed, see", "Schemes, Lemma \\ref{schemes-lemma-where-are-they-equal}.", "Hence $a = can$ on all of $\\Spec(\\mathcal{O}_{X, \\eta_2})$.", "\\medskip\\noindent", "Assume $Z_1 \\not \\subset Z_2$ and assume on the contrary that", "$\\mathcal{O}_{X, \\eta_{1}} \\subset \\mathcal{O}_{X, \\eta_{2}}$", "as subrings of $R(X)$. Then we would obtain a second morphism", "$$", "\\Spec(\\mathcal{O}_{X, \\eta_{2}}) \\longrightarrow", "\\Spec(\\mathcal{O}_{X, \\eta_{1}}) \\longrightarrow", "X.", "$$", "By the above this composition would have to be equal to $can$.", "This implies that $\\eta_2$ specializes to $\\eta_1$ (see", "Schemes, Lemma \\ref{schemes-lemma-specialize-points}).", "But this contradicts our assumption $Z_1 \\not \\subset Z_2$." ], "refs": [ "schemes-lemma-where-are-they-equal", "schemes-lemma-specialize-points" ], "ref_ids": [ 7708, 7684 ] } ], "ref_ids": [] }, { "id": 5480, "type": "theorem", "label": "morphisms-lemma-rational-map-from-reduced-to-separated", "categories": [ "morphisms" ], "title": "morphisms-lemma-rational-map-from-reduced-to-separated", "contents": [ "Let $X$ and $Y$ be schemes. Assume $X$ reduced and $Y$ separated. Let", "$\\varphi$ be a rational map from $X$ to $Y$ with domain of definition", "$U \\subset X$. Then there exists a unique morphism $f : U \\to Y$", "representing $\\varphi$. If $X$ and $Y$ are schemes over a separated", "scheme $S$ and if $\\varphi$ is an $S$-rational map, then $f$ is a", "morphism over $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $(V, g)$ and $(V', g')$ be representatives of $\\varphi$. Then", "$g, g'$ agree on a dense open subscheme $W \\subset V \\cap V'$.", "On the other hand, the equalizer $E$ of $g|_{V \\cap V'}$ and $g'|_{V \\cap V'}$", "is a closed subscheme of $V \\cap V'$ (Schemes, Lemma", "\\ref{schemes-lemma-where-are-they-equal}). Now $W \\subset E$", "implies that $E = V \\cap V'$ set theoretically. As $V \\cap V'$", "is reduced we conclude $E = V \\cap V'$ scheme theoretically, i.e.,", "$g|_{V \\cap V'} = g'|_{V \\cap V'}$. It follows that we can glue the", "representatives $g : V \\to Y$ of $\\varphi$ to a morphism $f : U \\to Y$, see", "Schemes, Lemma \\ref{schemes-lemma-glue}.", "We omit the proof of the final statement." ], "refs": [ "schemes-lemma-where-are-they-equal", "schemes-lemma-glue" ], "ref_ids": [ 7708, 7686 ] } ], "ref_ids": [] }, { "id": 5481, "type": "theorem", "label": "morphisms-lemma-birational-integral", "categories": [ "morphisms" ], "title": "morphisms-lemma-birational-integral", "contents": [ "Let $X$ and $Y$ be irreducible schemes.", "\\begin{enumerate}", "\\item The schemes $X$ and $Y$ are birational if and only if they have", "isomorphic nonempty opens.", "\\item Assume $X$ and $Y$ are schemes over a base scheme $S$. Then", "$X$ and $Y$ are $S$-birational if and only if there are nonempty", "opens $U \\subset X$ and $V \\subset Y$ which are $S$-isomorphic.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $X$ and $Y$ are birational. Let $f : U \\to Y$ and $g : V \\to X$", "define inverse dominant rational maps from $X$ to $Y$ and from $Y$ to $X$.", "We may assume $V$ affine. We may replace $U$ by an affine open of $f^{-1}(V)$.", "As $g \\circ f$ is the identity as a dominant rational map, we see that", "the composition $U \\to V \\to X$ is the identity on a dense open of $U$.", "Thus after replacing $U$ by a smaller affine open we may assume that", "$U \\to V \\to X$ is the inclusion of $U$ into $X$. It follows that", "$U \\to V$ is an immersion", "(apply Schemes, Lemma \\ref{schemes-lemma-section-immersion}", "to $U \\to g^{-1}(U) \\to U$).", "However, switching the roles of $U$ and $V$ and redoing the argument", "above, we see that there exists a nonempty affine open $V' \\subset V$", "such that the inclusion factors as $V' \\to U \\to V$. Then $V' \\to U$ is", "necessarily an open immersion. Namely, $V' \\to f^{-1}(V') \\to V'$ are", "monomorphisms", "(Schemes, Lemma \\ref{schemes-lemma-immersions-monomorphisms})", "composing to the identity, hence isomorphisms.", "Thus $V'$ is isomorphic to an open of both $X$ and $Y$.", "In the $S$-rational maps case, the exact same argument works." ], "refs": [ "schemes-lemma-immersions-monomorphisms" ], "ref_ids": [ 7727 ] } ], "ref_ids": [] }, { "id": 5482, "type": "theorem", "label": "morphisms-lemma-birational-dominant", "categories": [ "morphisms" ], "title": "morphisms-lemma-birational-dominant", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes having finitely", "many irreducible components. If $f$ is birational then", "$f$ is dominant." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-generic-points-in-image-dominant}", "and the definition." ], "refs": [ "morphisms-lemma-generic-points-in-image-dominant" ], "ref_ids": [ 5158 ] } ], "ref_ids": [] }, { "id": 5483, "type": "theorem", "label": "morphisms-lemma-birational-generic-fibres", "categories": [ "morphisms" ], "title": "morphisms-lemma-birational-generic-fibres", "contents": [ "Let $f : X \\to Y$ be a birational morphism of schemes having finitely", "many irreducible components. If $y \\in Y$ is the generic point of", "an irreducible component, then the base change", "$X \\times_Y \\Spec(\\mathcal{O}_{Y, y}) \\to \\Spec(\\mathcal{O}_{Y, y})$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "We may assume $Y = \\Spec(B)$ is affine and irreducible.", "Then $X$ is irreducible too. If we prove the result for any nonempty", "affine open $U \\subset X$, then the result holds for $X$", "(small argument omitted). Hence we may assume $X$", "is affine too, say $X = \\Spec(A)$.", "Let $y \\in Y$ correspond to the minimal prime $\\mathfrak q \\subset B$.", "By assumption $A$ has a unique minimal prime $\\mathfrak p$ lying", "over $\\mathfrak q$ and $B_\\mathfrak q \\to A_\\mathfrak p$ is an isomorphism.", "It follows that $A_\\mathfrak q \\to \\kappa(\\mathfrak p)$ is surjective,", "hence $\\mathfrak p A_\\mathfrak q$ is a maximal ideal. On the other hand", "$\\mathfrak p A_\\mathfrak q$ is the unique minimal prime of $A_\\mathfrak q$.", "We conclude that $\\mathfrak p A_\\mathfrak q$ is the unique prime of", "$A_\\mathfrak q$ and that $A_\\mathfrak q = A_\\mathfrak p$. Since", "$A_\\mathfrak q = A \\otimes_B B_\\mathfrak q$ the lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5484, "type": "theorem", "label": "morphisms-lemma-birational-birational", "categories": [ "morphisms" ], "title": "morphisms-lemma-birational-birational", "contents": [ "Let $f : X \\to Y$ be a birational morphism of schemes having finitely", "many irreducible components over a base scheme $S$. Assume one of the", "following conditions is satisfied", "\\begin{enumerate}", "\\item $f$ is locally of finite type and $Y$ reduced,", "\\item $f$ is locally of finite presentation.", "\\end{enumerate}", "Then there exist dense opens $U \\subset X$ and $V \\subset Y$", "such that $f(U) \\subset V$ and $f|_U : U \\to V$ is an isomorphism.", "In particular if $X$ and $Y$ are irreducible, then", "$X$ and $Y$ are $S$-birational." ], "refs": [], "proofs": [ { "contents": [ "There is an immediate reduction to the case where $X$ and $Y$ are irreducible", "which we omit. Moreover, after shrinking further and we may assume $X$ and", "$Y$ are affine, say $X = \\Spec(A)$ and $Y = \\Spec(B)$.", "By assumption $A$, resp.\\ $B$ has a unique minimal prime $\\mathfrak p$,", "resp.\\ $\\mathfrak q$, the prime $\\mathfrak p$ lies over $\\mathfrak q$, and", "$B_\\mathfrak q = A_\\mathfrak p$.", "By Lemma \\ref{lemma-birational-generic-fibres} we have", "$B_\\mathfrak q = A_\\mathfrak q = A_\\mathfrak p$.", "\\medskip\\noindent", "Suppose $B \\to A$ is of finite type, say $A = B[x_1, \\ldots, x_n]$.", "There exist a $b_i \\in B$ and $g_i \\in B \\setminus \\mathfrak q$", "such that $b_i/g_i$ maps to the image of $x_i$ in $A_\\mathfrak q$.", "Hence $b_i - g_ix_i$ maps to zero in $A_{g_i'}$ for some", "$g_i' \\in B \\setminus \\mathfrak q$. Setting $g = \\prod g_i g'_i$", "we see that $B_g \\to A_g$ is surjective. If moreover $Y$ is reduced,", "then the map $B_g \\to B_\\mathfrak q$ is injective and hence", "$B_g \\to A_g$ is injective as well. This proves case (1).", "\\medskip\\noindent", "Proof of (2). By the argument given in the previous paragraph we may", "assume that $B \\to A$ is surjective. As $f$ is locally of finite presentation", "the kernel $J \\subset B$ is a finitely generated ideal. Say", "$J = (b_1, \\ldots, b_r)$. Since $B_\\mathfrak q = A_\\mathfrak q$", "there exist $g_i \\in B \\setminus \\mathfrak q$ such that", "$g_i b_i = 0$. Setting $g = \\prod g_i$ we see that $B_g \\to A_g$", "is an isomorphism." ], "refs": [ "morphisms-lemma-birational-generic-fibres" ], "ref_ids": [ 5483 ] } ], "ref_ids": [] }, { "id": 5485, "type": "theorem", "label": "morphisms-lemma-criterion-birational-finite-presentation", "categories": [ "morphisms" ], "title": "morphisms-lemma-criterion-birational-finite-presentation", "contents": [ "Let $S$ be a scheme. Let $X$ and $Y$ be irreducible schemes", "locally of finite presentation over $S$. Let $x \\in X$ and $y \\in Y$", "be the generic points. The following are equivalent", "\\begin{enumerate}", "\\item $X$ and $Y$ are $S$-birational,", "\\item there exist nonempty opens of $X$ and $Y$", "which are $S$-isomorphic, and", "\\item $x$ and $y$ map to the same point $s$ of $S$ and", "$\\mathcal{O}_{X, x}$ and $\\mathcal{O}_{Y, y}$ are isomorphic as", "$\\mathcal{O}_{S, s}$-algebras.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have seen the equivalence of (1) and (2) in", "Lemma \\ref{lemma-birational-integral}.", "It is immediate that (2) implies (3).", "To finish we assume (3) holds and we prove (1).", "By Lemma \\ref{lemma-rational-map-finite-presentation}", "there is a rational map $f : U \\to Y$", "which sends $x \\in U$ to $y$ and induces", "the given isomorphism $\\mathcal{O}_{Y, y} \\cong \\mathcal{O}_{X, x}$.", "Thus $f$ is a birational morphism and hence induces", "an isomorphism on nonempty opens", "by Lemma \\ref{lemma-birational-birational}.", "This finishes the proof." ], "refs": [ "morphisms-lemma-birational-integral", "morphisms-lemma-rational-map-finite-presentation", "morphisms-lemma-birational-birational" ], "ref_ids": [ 5481, 5477, 5484 ] } ], "ref_ids": [] }, { "id": 5486, "type": "theorem", "label": "morphisms-lemma-common-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-common-open", "contents": [ "Let $S$ be a scheme. Let $X$ and $Y$ be integral schemes locally", "of finite type over $S$. Let $x \\in X$ and $y \\in Y$ be the generic points.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ and $Y$ are $S$-birational,", "\\item there exist nonempty opens of $X$ and $Y$ which are $S$-isomorphic, and", "\\item $x$ and $y$ map to the same point $s \\in S$ and", "$\\kappa(x) \\cong \\kappa(y)$ as $\\kappa(s)$-extensions.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have seen the equivalence of (1) and (2) in", "Lemma \\ref{lemma-birational-integral}.", "It is immediate that (2) implies (3).", "To finish we assume (3) holds and we prove (1).", "Observe that $\\mathcal{O}_{X, x} = \\kappa(x)$ and", "$\\mathcal{O}_{Y, y} = \\kappa(y)$ by", "Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring}.", "By Lemma \\ref{lemma-rational-map-finite-presentation}", "there is a rational map $f : U \\to Y$", "which sends $x \\in U$ to $y$ and induces", "the given isomorphism $\\mathcal{O}_{Y, y} \\cong \\mathcal{O}_{X, x}$.", "Thus $f$ is a birational morphism and hence induces", "an isomorphism on nonempty opens", "by Lemma \\ref{lemma-birational-birational}.", "This finishes the proof." ], "refs": [ "morphisms-lemma-birational-integral", "algebra-lemma-minimal-prime-reduced-ring", "morphisms-lemma-rational-map-finite-presentation", "morphisms-lemma-birational-birational" ], "ref_ids": [ 5481, 418, 5477, 5484 ] } ], "ref_ids": [] }, { "id": 5487, "type": "theorem", "label": "morphisms-lemma-generically-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-generically-finite", "contents": [ "Let $X$, $Y$ be schemes.", "Let $f : X \\to Y$ be locally of finite type.", "Let $\\eta \\in Y$ be a generic point of an irreducible component", "of $Y$. The following are equivalent:", "\\begin{enumerate}", "\\item the set $f^{-1}(\\{\\eta\\})$ is finite,", "\\item there exist affine opens $U_i \\subset X$, $i = 1, \\ldots, n$", "and $V \\subset Y$ with $f(U_i) \\subset V$,", "$\\eta \\in V$ and $f^{-1}(\\{\\eta\\}) \\subset \\bigcup U_i$", "such that each $f|_{U_i} : U_i \\to V$ is finite.", "\\end{enumerate}", "If $f$ is quasi-separated, then these are also equivalent to", "\\begin{enumerate}", "\\item[(3)] there exist affine opens $V \\subset Y$,", "and $U \\subset X$ with $f(U) \\subset V$,", "$\\eta \\in V$ and $f^{-1}(\\{\\eta\\}) \\subset U$", "such that $f|_U : U \\to V$ is finite.", "\\end{enumerate}", "If $f$ is quasi-compact and quasi-separated,", "then these are also equivalent to", "\\begin{enumerate}", "\\item[(4)] there exists an affine open $V \\subset Y$, $\\eta \\in V$", "such that $f^{-1}(V) \\to V$ is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The question is local on the base. Hence we may replace $Y$ by an", "affine neighbourhood of $\\eta$, and we may and do assume throughout", "the proof below that $Y$ is affine, say $Y = \\Spec(R)$.", "\\medskip\\noindent", "It is clear that (2) implies (1).", "Assume that $f^{-1}(\\{\\eta\\}) = \\{\\xi_1, \\ldots, \\xi_n\\}$ is finite.", "Choose affine opens $U_i \\subset X$ with $\\xi_i \\in U_i$.", "By Algebra, Lemma \\ref{algebra-lemma-generically-finite} we see", "that after replacing $Y$ by a standard open in", "$Y$ each of the morphisms $U_i \\to Y$ is finite.", "In other words (2) holds.", "\\medskip\\noindent", "It is clear that (3) implies (1). Assume $f$ is quasi-separated and (1).", "Write $f^{-1}(\\{\\eta\\}) = \\{\\xi_1, \\ldots, \\xi_n\\}$. There are no", "specializations among the $\\xi_i$ by Lemma \\ref{lemma-finite-fibre}.", "Since each $\\xi_i$ maps to the generic", "point $\\eta$ of an irreducible component of $Y$, there cannot be a", "nontrivial specialization $\\xi \\leadsto \\xi_i$ in $X$", "(since $\\xi$ would map to $\\eta$ as well).", "We conclude each $\\xi_i$ is a generic point of an", "irreducible component of $X$.", "Since $Y$ is affine and $f$ quasi-separated we see $X$ is quasi-separated.", "By Properties, Lemma \\ref{properties-lemma-maximal-points-affine}", "we can find an affine open $U \\subset X$ containing each $\\xi_i$.", "By Algebra, Lemma \\ref{algebra-lemma-generically-finite} we see", "that after replacing $Y$ by a standard open in", "$Y$ the morphisms $U \\to Y$ is finite.", "In other words (3) holds.", "\\medskip\\noindent", "It is clear that (4) implies all of (1) -- (3) with no further assumptions", "on $f$. Suppose that $f$ is quasi-compact and quasi-separated. We have to", "show that the equivalent conditions (1) -- (3) imply (4).", "Let $U$, $V$ be as in (3). Replace $Y$ by $V$. Since $f$ is quasi-compact", "and $Y$ is quasi-compact (being affine) we see that $X$ is quasi-compact.", "Hence $Z = X \\setminus U$ is quasi-compact, hence the morphism", "$f|_Z : Z \\to Y$ is quasi-compact. By construction of $Z$ we see that", "$\\eta \\not \\in f(Z)$. Hence by", "Lemma \\ref{lemma-quasi-compact-generic-point-not-in-image}", "we see that there exists an affine open", "neighbourhood $V'$ of $\\eta$ in $Y$ such that $f^{-1}(V') \\cap Z = \\emptyset$.", "Then we have $f^{-1}(V') \\subset U$ and this means", "that $f^{-1}(V') \\to V'$ is finite." ], "refs": [ "algebra-lemma-generically-finite", "morphisms-lemma-finite-fibre", "properties-lemma-maximal-points-affine", "algebra-lemma-generically-finite", "morphisms-lemma-quasi-compact-generic-point-not-in-image" ], "ref_ids": [ 1056, 5227, 3059, 1056, 5160 ] } ], "ref_ids": [] }, { "id": 5488, "type": "theorem", "label": "morphisms-lemma-quasi-finiteness-over-generic-point", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-finiteness-over-generic-point", "contents": [ "Let $X$, $Y$ be schemes. Let $f : X \\to Y$ be locally of finite type.", "Let $X^0$, resp.\\ $Y^0$ denote the set of generic points of irreducible", "components of $X$, resp.\\ $Y$. Let $\\eta \\in Y^0$. The following are", "equivalent", "\\begin{enumerate}", "\\item $f^{-1}(\\{\\eta\\}) \\subset X^0$,", "\\item $f$ is quasi-finite at all points lying over $\\eta$,", "\\item $f$ is quasi-finite at all $\\xi \\in X^0$ lying over $\\eta$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Condition (1) implies there are no specializations among the points", "of the fibre $X_\\eta$. Hence (2) holds by", "Lemma \\ref{lemma-quasi-finite-at-point-characterize}.", "The implication (2) $\\Rightarrow$ (3) is immediate.", "Since $\\eta$ is a generic point of $Y$, the generic points of $X_\\eta$", "are generic points of $X$. Hence (3) and ", "Lemma \\ref{lemma-quasi-finite-at-point-characterize}", "imply the generic points of $X_\\eta$ are also closed.", "Thus all points of $X_\\eta$ are generic and we see that (1) holds." ], "refs": [ "morphisms-lemma-quasi-finite-at-point-characterize", "morphisms-lemma-quasi-finite-at-point-characterize" ], "ref_ids": [ 5226, 5226 ] } ], "ref_ids": [] }, { "id": 5489, "type": "theorem", "label": "morphisms-lemma-finite-over-dense-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-over-dense-open", "contents": [ "Let $X$, $Y$ be schemes. Let $f : X \\to Y$ be locally of finite type.", "Let $X^0$, resp.\\ $Y^0$ denote the set of generic points of irreducible", "components of $X$, resp.\\ $Y$. Assume", "\\begin{enumerate}", "\\item $X^0$ and $Y^0$ are finite and $f^{-1}(Y^0) = X^0$,", "\\item either $f$ is quasi-compact or $f$ is separated.", "\\end{enumerate}", "Then there exists a dense open $V \\subset Y$", "such that $f^{-1}(V) \\to V$ is finite." ], "refs": [], "proofs": [ { "contents": [ "Since $Y$ has finitely many irreducible components, we can find a dense", "open which is a disjoint union of its irreducible components. Thus we may", "assume $Y$ is irreducible affine with generic point $\\eta$. Then the fibre", "over $\\eta$ is finite as $X^0$ is finite.", "\\medskip\\noindent", "Assume $f$ is separated and $Y$ irreducible affine. Choose $V \\subset Y$", "and $U \\subset X$ as in Lemma \\ref{lemma-generically-finite} part (3).", "Since $f|_U : U \\to V$ is finite, we see that $U \\subset f^{-1}(V)$", "is closed as well as open (Lemmas \\ref{lemma-image-proper-scheme-closed} and", "\\ref{lemma-finite-proper}). Thus $f^{-1}(V) = U \\amalg W$ for some", "open subscheme $W$ of $X$. However, since $U$ contains all the generic", "points of $X$ we conclude that $W = \\emptyset$ as desired.", "\\medskip\\noindent", "Assume $f$ is quasi-compact and $Y$ irreducible affine. Then $X$ is", "quasi-compact, hence there exists a dense open subscheme $U \\subset X$", "which is separated", "(Properties, Lemma \\ref{properties-lemma-quasi-compact-dense-open-separated}).", "Since the set of generic points $X^0$ is finite, we see that $X^0 \\subset U$.", "Thus $\\eta \\not \\in f(X \\setminus U)$. Since $X \\setminus U \\to Y$ is", "quasi-compact, we conclude that there is a nonempty open $V \\subset Y$", "such that $f^{-1}(V) \\subset U$, see Lemma \\ref{lemma-quasi-compact-dominant}.", "After replacing $X$ by $f^{-1}(V)$ and $Y$ by $V$ we reduce to the", "separated case which we dealt with in the preceding paragraph." ], "refs": [ "morphisms-lemma-generically-finite", "morphisms-lemma-image-proper-scheme-closed", "morphisms-lemma-finite-proper", "properties-lemma-quasi-compact-dense-open-separated", "morphisms-lemma-quasi-compact-dominant" ], "ref_ids": [ 5487, 5411, 5445, 3060, 5159 ] } ], "ref_ids": [] }, { "id": 5490, "type": "theorem", "label": "morphisms-lemma-birational-isomorphism-over-dense-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-birational-isomorphism-over-dense-open", "contents": [ "Let $X$, $Y$ be schemes. Let $f : X \\to Y$ be a birational morphism", "between schemes which have finitely many irreducible components.", "Assume", "\\begin{enumerate}", "\\item either $f$ is quasi-compact or $f$ is separated, and", "\\item either $f$ is locally of finite type and $Y$ is reduced or", "$f$ is locally of finite presentation.", "\\end{enumerate}", "Then there exists a dense open $V \\subset Y$", "such that $f^{-1}(V) \\to V$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-over-dense-open} we may assume that $f$ is finite.", "Since $Y$ has finitely many irreducible components, we can find a dense", "open which is a disjoint union of its irreducible components. Thus we may", "assume $Y$ is irreducible. By Lemma \\ref{lemma-birational-birational} we find", "a nonempty open $U \\subset X$ such that $f|_U : U \\to Y$ is an open immersion.", "After removing the closed (as $f$ finite)", "subset $f(X \\setminus U)$ from $Y$ we see that $f$ is an isomorphism." ], "refs": [ "morphisms-lemma-finite-over-dense-open", "morphisms-lemma-birational-birational" ], "ref_ids": [ 5489, 5484 ] } ], "ref_ids": [] }, { "id": 5491, "type": "theorem", "label": "morphisms-lemma-finite-degree", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-degree", "contents": [ "Let $X$, $Y$ be integral schemes.", "Let $f : X \\to Y$ be locally of finite type.", "Assume $f$ is dominant.", "The following are equivalent:", "\\begin{enumerate}", "\\item the extension $R(Y) \\subset R(X)$ has", "transcendence degree $0$,", "\\item the extension $R(Y) \\subset R(X)$ is finite,", "\\item there exist nonempty affine opens $U \\subset X$", "and $V \\subset Y$ such that $f(U) \\subset V$", "and $f|_U : U \\to V$ is finite, and", "\\item the generic point of $X$ is the only point of $X$ mapping to", "the generic point of $Y$.", "\\end{enumerate}", "If $f$ is separated or if $f$ is quasi-compact, then these are", "also equivalent to", "\\begin{enumerate}", "\\item[(5)] there exists a nonempty affine open $V \\subset Y$ such", "that $f^{-1}(V) \\to V$ is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose any affine opens $\\Spec(A) = U \\subset X$", "and $\\Spec(R) = V \\subset Y$ such that $f(U) \\subset V$.", "Then $R$ and $A$ are domains by definition. The ring map", "$R \\to A$ is of finite type", "(Lemma \\ref{lemma-locally-finite-type-characterize}).", "By Lemma \\ref{lemma-dominant-finite-number-irreducible-components}", "the generic point of $X$ maps to the generic point of $Y$", "hence $R \\to A$ is injective. Let $K = R(Y)$ be the fraction field", "of $R$ and $L = R(X)$ the fraction field of $A$. Then $K \\subset L$", "is a finitely generated field extension. Hence we see that", "(1) is equivalent to (2).", "\\medskip\\noindent", "Suppose (2) holds. Let $x_1, \\ldots, x_n \\in A$ be generators", "of $A$ over $R$. By assumption there exist nonzero polynomials", "$P_i(X) \\in R[X]$ such that $P_i(x_i) = 0$. Let $f_i \\in R$ be the", "leading coefficient of $P_i$. Then we conclude that", "$R_{f_1 \\ldots f_n} \\to A_{f_1 \\ldots f_n}$ is finite, i.e., (3) holds.", "Note that (3) implies (2). So now we see that (1), (2) and (3) are all", "equivalent.", "\\medskip\\noindent", "Let $\\eta$ be the generic point of $X$, and let $\\eta' \\in Y$ be the", "generic point of $Y$. Assume (4). Then", "$\\dim_\\eta(X_{\\eta'}) = 0$ and we see that $R(X) = \\kappa(\\eta)$ has", "transcendence degree $0$ over $R(Y) = \\kappa(\\eta')$ by", "Lemma \\ref{lemma-dimension-fibre-at-a-point}.", "In other words (1) holds. Assume the equivalent conditions (1), (2) and", "(3). Suppose that $x \\in X$ is a point mapping to $\\eta'$.", "As $x$ is a specialization of $\\eta$,", "this gives inclusions $R(Y) \\subset \\mathcal{O}_{X, x} \\subset R(X)$,", "which implies $\\mathcal{O}_{X, x}$ is a field, see", "Algebra, Lemma \\ref{algebra-lemma-integral-over-field}.", "Hence $x = \\eta$. Thus we see that (1) -- (4)", "are all equivalent.", "\\medskip\\noindent", "It is clear that (5) implies (3) with no additional assumptions on", "$f$. What remains is to prove that if $f$ is either separated or", "quasi-compact, then the equivalent conditions (1) -- (4) imply (5).", "This follows from Lemma \\ref{lemma-finite-over-dense-open}." ], "refs": [ "morphisms-lemma-locally-finite-type-characterize", "morphisms-lemma-dominant-finite-number-irreducible-components", "morphisms-lemma-dimension-fibre-at-a-point", "algebra-lemma-integral-over-field", "morphisms-lemma-finite-over-dense-open" ], "ref_ids": [ 5198, 5161, 5277, 497, 5489 ] } ], "ref_ids": [] }, { "id": 5492, "type": "theorem", "label": "morphisms-lemma-degree-composition", "categories": [ "morphisms" ], "title": "morphisms-lemma-degree-composition", "contents": [ "Let $X$, $Y$, $Z$ be integral schemes.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be dominant morphisms locally", "of finite type. Assume that $[R(X) : R(Y)] < \\infty$ and", "$[R(Y) : R(Z)] < \\infty$. Then", "$$", "\\deg(X/Z) = \\deg(X/Y) \\deg(Y/Z).", "$$" ], "refs": [], "proofs": [ { "contents": [ "This comes from the multiplicativity of degrees in towers", "of finite extensions of fields, see", "Fields, Lemma \\ref{fields-lemma-multiplicativity-degrees}." ], "refs": [ "fields-lemma-multiplicativity-degrees" ], "ref_ids": [ 4450 ] } ], "ref_ids": [] }, { "id": 5493, "type": "theorem", "label": "morphisms-lemma-dimension-formula", "categories": [ "morphisms" ], "title": "morphisms-lemma-dimension-formula", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$, and set $s = f(x)$.", "Assume", "\\begin{enumerate}", "\\item $S$ is locally Noetherian,", "\\item $f$ is locally of finite type,", "\\item $X$ and $S$ integral, and", "\\item $f$ dominant.", "\\end{enumerate}", "We have", "\\begin{equation}", "\\label{equation-dimension-formula}", "\\dim(\\mathcal{O}_{X, x})", "\\leq", "\\dim(\\mathcal{O}_{S, s}) + \\text{trdeg}_{R(S)}R(X)", "- \\text{trdeg}_{\\kappa(s)} \\kappa(x).", "\\end{equation}", "Moreover, equality holds if $S$ is universally catenary." ], "refs": [], "proofs": [ { "contents": [ "The corresponding algebra statement is", "Algebra, Lemma \\ref{algebra-lemma-dimension-formula}." ], "refs": [ "algebra-lemma-dimension-formula" ], "ref_ids": [ 990 ] } ], "ref_ids": [] }, { "id": 5494, "type": "theorem", "label": "morphisms-lemma-dimension-formula-general", "categories": [ "morphisms" ], "title": "morphisms-lemma-dimension-formula-general", "contents": [ "Let $S$ be a scheme. Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$, and set $s = f(x)$. Assume $S$ is locally Noetherian", "and $f$ is locally of finite type,", "We have", "\\begin{equation}", "\\label{equation-dimension-formula-general}", "\\dim(\\mathcal{O}_{X, x})", "\\leq", "\\dim(\\mathcal{O}_{S, s}) + E - \\text{trdeg}_{\\kappa(s)} \\kappa(x).", "\\end{equation}", "where $E$ is the maximum of $\\text{trdeg}_{\\kappa(f(\\xi))}(\\kappa(\\xi))$", "where $\\xi$ runs over the generic points of irreducible components", "of $X$ containing $x$." ], "refs": [], "proofs": [ { "contents": [ "Let $X_1, \\ldots, X_n$ be the irreducible components of $X$ containing $x$", "endowed with their reduced induced scheme structure.", "These correspond to the minimal primes $\\mathfrak q_i$ of", "$\\mathcal{O}_{X, x}$ and hence there are finitely many of them", "(Schemes, Lemma \\ref{schemes-lemma-specialize-points} and", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-irreducible-components}).", "Then", "$\\dim(\\mathcal{O}_{X, x}) = \\max \\dim(\\mathcal{O}_{X, x}/\\mathfrak q_i)", "= \\max \\dim(\\mathcal{O}_{X_i, x})$.", "The $\\xi$'s occurring in the definition of $E$ are exactly the", "generic points $\\xi_i \\in X_i$. Let $Z_i = \\overline{\\{f(\\xi_i)\\}} \\subset S$", "endowed with the reduced induced scheme structure.", "The composition $X_i \\to X \\to S$ factors through $Z_i$", "(Schemes, Lemma \\ref{schemes-lemma-map-into-reduction}). Thus we may apply", "the dimension formula (Lemma \\ref{lemma-dimension-formula})", "to see that", "$\\dim(\\mathcal{O}_{X_i, x}) \\leq \\dim(\\mathcal{O}_{Z_i, x}) +", "\\text{trdeg}_{\\kappa(f(\\xi))}(\\kappa(\\xi)) -", "\\text{trdeg}_{\\kappa(s)} \\kappa(x)$. Putting everything together", "we obtain the lemma." ], "refs": [ "schemes-lemma-specialize-points", "algebra-lemma-Noetherian-irreducible-components", "schemes-lemma-map-into-reduction", "morphisms-lemma-dimension-formula" ], "ref_ids": [ 7684, 453, 7682, 5493 ] } ], "ref_ids": [] }, { "id": 5495, "type": "theorem", "label": "morphisms-lemma-dimension-function-propagates", "categories": [ "morphisms" ], "title": "morphisms-lemma-dimension-function-propagates", "contents": [ "Let $S$ be a locally Noetherian and universally catenary scheme.", "Let $\\delta : S \\to \\mathbf{Z}$ be a dimension function.", "Let $f : X \\to S$ be a morphism of schemes.", "Assume $f$ locally of finite type.", "Then the map", "\\begin{align*}", "\\delta = \\delta_{X/S} : X & \\longrightarrow \\mathbf{Z} \\\\", "x & \\longmapsto \\delta(f(x)) + \\text{trdeg}_{\\kappa(f(x))} \\kappa(x)", "\\end{align*}", "is a dimension function on $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be locally of finite type.", "Let $x \\leadsto y$, $x \\not = y$ be a specialization in $X$.", "We have to show that $\\delta_{X/S}(x) > \\delta_{X/S}(y)$ and", "that $\\delta_{X/S}(x) = \\delta_{X/S}(y) + 1$ if $y$ is an", "immediate specialization of $x$.", "\\medskip\\noindent", "Choose an affine open $V \\subset S$ containing the image of", "$y$ and choose an affine open $U \\subset X$ mapping into $V$", "and containing $y$. We may clearly replace $X$ by $U$ and", "$S$ by $V$. Thus we may assume that $X = \\Spec(A)$", "and $S = \\Spec(R)$ and that $f$ is given by a ring", "map $R \\to A$. The ring $R$ is universally catenary", "(Lemma \\ref{lemma-universally-catenary-local})", "and the map $R \\to A$ is of finite type", "(Lemma \\ref{lemma-locally-finite-type-characterize}).", "\\medskip\\noindent", "Let $\\mathfrak q \\subset A$ be the prime ideal corresponding", "to the point $x$ and let $\\mathfrak p \\subset R$ be the prime", "ideal corresponding to $f(x)$. The restriction $\\delta'$ of $\\delta$", "to $S' = \\Spec(R/\\mathfrak p) \\subset S$ is a dimension", "function. The ring $R/\\mathfrak p$ is universally catenary.", "The restriction of $\\delta_{X/S}$ to $X' = \\Spec(A/\\mathfrak q)$", "is clearly equal to the function $\\delta_{X'/S'}$ constructed", "using the dimension function $\\delta'$. Hence we may assume", "in addition to the above that $R \\subset A$ are domains, in", "other words that $X$ and $S$ are integral schemes, and that", "$x$ is the generic point of $X$ and $f(x)$ is the generic point of $S$.", "\\medskip\\noindent", "Note that $\\mathcal{O}_{X, x} = R(X)$ and that since $x \\leadsto y$,", "$x \\not = y$, the spectrum of $\\mathcal{O}_{X, y}$ has at least", "two points", "(Schemes, Lemma \\ref{schemes-lemma-specialize-points})", "hence $\\dim(\\mathcal{O}_{X, y}) > 0$ .", "If $y$ is an immediate specialization of $x$, then", "$\\Spec(\\mathcal{O}_{X, y}) = \\{x, y\\}$ and", "$\\dim(\\mathcal{O}_{X, y}) = 1$.", "\\medskip\\noindent", "Write $s = f(x)$ and $t = f(y)$. We compute", "\\begin{align*}", "\\delta_{X/S}(x) - \\delta_{X/S}(y)", "& =", "\\delta(s) + \\text{trdeg}_{\\kappa(s)} \\kappa(x)", "- \\delta(t) - \\text{trdeg}_{\\kappa(t)} \\kappa(y) \\\\", "& =", "\\delta(s) - \\delta(t) +", "\\text{trdeg}_{R(S)} R(X) - \\text{trdeg}_{\\kappa(t)} \\kappa(y) \\\\", "& =", "\\delta(s) - \\delta(t) + \\dim(\\mathcal{O}_{X, y})", "- \\dim(\\mathcal{O}_{S, t})", "\\end{align*}", "where we use equality in (\\ref{equation-dimension-formula}) in", "the last step. Since $\\delta$ is a dimension function on the scheme $S$", "and $s \\in S$ is the generic point, the difference", "$\\delta(s) - \\delta(t)$ is equal to $\\text{codim}(\\overline{\\{t\\}}, S)$ by", "Topology, Lemma \\ref{topology-lemma-dimension-function-catenary}.", "This is equal to $\\dim(\\mathcal{O}_{S, t})$ by", "Properties, Lemma", "\\ref{properties-lemma-codimension-local-ring}.", "Hence we conclude that", "$$", "\\delta_{X/S}(x) - \\delta_{X/S}(y) = \\dim(\\mathcal{O}_{X, y})", "$$", "and the lemma follows from what we said above about $\\dim(\\mathcal{O}_{X, y})$." ], "refs": [ "morphisms-lemma-universally-catenary-local", "morphisms-lemma-locally-finite-type-characterize", "schemes-lemma-specialize-points", "topology-lemma-dimension-function-catenary", "properties-lemma-codimension-local-ring" ], "ref_ids": [ 5214, 5198, 7684, 8291, 2979 ] } ], "ref_ids": [] }, { "id": 5496, "type": "theorem", "label": "morphisms-lemma-alteration-dimension", "categories": [ "morphisms" ], "title": "morphisms-lemma-alteration-dimension", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume that", "\\begin{enumerate}", "\\item $Y$ is locally Noetherian,", "\\item $X$ and $Y$ are integral schemes,", "\\item $f$ is dominant, and", "\\item $f$ is locally of finite type.", "\\end{enumerate}", "Then we have", "$$", "\\dim(X) \\leq \\dim(Y) + \\text{trdeg}_{R(Y)} R(X).", "$$", "If $f$ is closed\\footnote{For example if $f$ is proper, see", "Definition \\ref{definition-proper}.} then equality holds." ], "refs": [ "morphisms-definition-proper" ], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be as in the lemma.", "Let $\\xi_0 \\leadsto \\xi_1 \\leadsto \\ldots \\leadsto \\xi_e$ be", "a sequence of specializations in $X$. Set $x = \\xi_e$", "and $y = f(x)$. Observe that $e \\leq \\dim(\\mathcal{O}_{X, x})$", "as the given specializations occur in the spectrum of", "$\\mathcal{O}_{X, x}$, see Schemes, Lemma \\ref{schemes-lemma-specialize-points}.", "By the dimension formula, Lemma \\ref{lemma-dimension-formula},", "we see that", "\\begin{align*}", "e & \\leq \\dim(\\mathcal{O}_{X, x}) \\\\", "& \\leq \\dim(\\mathcal{O}_{Y, y}) +", "\\text{trdeg}_{R(Y)} R(X) - \\text{trdeg}_{\\kappa(y)} \\kappa(x) \\\\", "& \\leq \\dim(\\mathcal{O}_{Y, y}) + \\text{trdeg}_{R(Y)} R(X)", "\\end{align*}", "Hence we conclude that $e \\leq \\dim(Y) + \\text{trdeg}_{R(Y)} R(X)$ as desired.", "\\medskip\\noindent", "Next, assume $f$ is also closed.", "Say $\\overline{\\xi}_0 \\leadsto \\overline{\\xi}_1 \\leadsto \\ldots", "\\leadsto \\overline{\\xi}_d$ is a sequence of specializations in $Y$.", "We want to show that $\\dim(X) \\geq d + r$.", "We may assume that $\\overline{\\xi}_0 = \\eta$ is the generic point of $Y$.", "The generic fibre $X_\\eta$ is a scheme locally of finite type over", "$\\kappa(\\eta) = R(Y)$. It is nonempty as $f$ is dominant. Hence by", "Lemma \\ref{lemma-ubiquity-Jacobson-schemes} it is a Jacobson scheme.", "Thus by Lemma \\ref{lemma-jacobson-finite-type-points}", "we can find a closed point $\\xi_0 \\in X_\\eta$ and the extension", "$\\kappa(\\eta) \\subset \\kappa(\\xi_0)$ is", "a finite extension. Note that", "$\\mathcal{O}_{X, \\xi_0} = \\mathcal{O}_{X_\\eta, \\xi_0}$ because", "$\\eta$ is the generic point of $Y$. Hence we see that", "$\\dim(\\mathcal{O}_{X, \\xi_0}) = r$ by Lemma \\ref{lemma-dimension-formula}", "applied to the scheme $X_\\eta$ over the universally catenary", "scheme $\\Spec(\\kappa(\\eta))$ (see Lemma \\ref{lemma-ubiquity-uc})", "and the point $\\xi_0$. This means that we can find", "$\\xi_{-r} \\leadsto \\ldots \\leadsto \\xi_{-1} \\leadsto \\xi_0$", "in $X$. On the other hand, as $f$ is closed specializations", "lift along $f$, see", "Topology, Lemma \\ref{topology-lemma-closed-open-map-specialization}.", "Thus, as $\\xi_0$ lies", "over $\\eta = \\overline{\\xi}_0$ we can", "find specializations $\\xi_0 \\leadsto \\xi_1 \\leadsto \\ldots \\leadsto \\xi_d$", "lying over $\\overline{\\xi}_0 \\leadsto \\overline{\\xi}_1 \\leadsto \\ldots", "\\leadsto \\overline{\\xi}_d$. In other words we have", "$$", "\\xi_{-r} \\leadsto \\ldots \\leadsto \\xi_{-1} \\leadsto \\xi_0", "\\leadsto \\xi_1 \\leadsto \\ldots \\leadsto \\xi_d", "$$", "which means that $\\dim(X) \\geq d + r$ as desired." ], "refs": [ "schemes-lemma-specialize-points", "morphisms-lemma-dimension-formula", "morphisms-lemma-ubiquity-Jacobson-schemes", "morphisms-lemma-jacobson-finite-type-points", "morphisms-lemma-dimension-formula", "morphisms-lemma-ubiquity-uc", "topology-lemma-closed-open-map-specialization" ], "ref_ids": [ 7684, 5493, 5213, 5211, 5493, 5217, 8287 ] } ], "ref_ids": [ 5571 ] }, { "id": 5497, "type": "theorem", "label": "morphisms-lemma-alteration-dimension-general", "categories": [ "morphisms" ], "title": "morphisms-lemma-alteration-dimension-general", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume that", "$Y$ is locally Noetherian and $f$ is locally of finite type.", "Then", "$$", "\\dim(X) \\leq \\dim(Y) + E", "$$", "where $E$ is the supremum of $\\text{trdeg}_{\\kappa(f(\\xi))}(\\kappa(\\xi))$", "where $\\xi$ runs through the generic points of the irreducible components", "of $X$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of Lemma \\ref{lemma-dimension-formula-general}", "and Properties, Lemma \\ref{properties-lemma-dimension}." ], "refs": [ "morphisms-lemma-dimension-formula-general", "properties-lemma-dimension" ], "ref_ids": [ 5494, 2978 ] } ], "ref_ids": [] }, { "id": 5498, "type": "theorem", "label": "morphisms-lemma-integral-closure", "categories": [ "morphisms" ], "title": "morphisms-lemma-integral-closure", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent sheaf", "of $\\mathcal{O}_X$-algebras. The subsheaf $\\mathcal{A}' \\subset \\mathcal{A}$", "defined by the rule", "$$", "U \\longmapsto \\{f \\in \\mathcal{A}(U) \\mid", "f_x \\in \\mathcal{A}_x \\text{ integral over } \\mathcal{O}_{X, x}", "\\text{ for all }x \\in U\\}", "$$", "is a quasi-coherent $\\mathcal{O}_X$-algebra, the stalk $\\mathcal{A}'_x$", "is the integral closure of $\\mathcal{O}_{X, x}$ in $\\mathcal{A}_x$, and", "for any affine open $U \\subset X$ the ring", "$\\mathcal{A}'(U) \\subset \\mathcal{A}(U)$ is", "the integral closure of $\\mathcal{O}_X(U)$ in $\\mathcal{A}(U)$." ], "refs": [], "proofs": [ { "contents": [ "This is a subsheaf by the local nature of the conditions.", "It is an $\\mathcal{O}_X$-algebra by", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-is-ring}.", "Let $U \\subset X$ be an affine open. Say $U = \\Spec(R)$", "and say $\\mathcal{A}$ is the quasi-coherent sheaf associated to", "the $R$-algebra $A$. Then according to", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-stalks}", "the value of $\\mathcal{A}'$ over $U$ is given by the integral", "closure $A'$ of $R$ in $A$. This proves the last assertion of", "the lemma. To prove that $\\mathcal{A}'$ is quasi-coherent, it", "suffices to show that $\\mathcal{A}'(D(f)) = A'_f$. This follows", "from the fact that integral closure and localization commute, see", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-localize}.", "The same fact shows that the stalks are as advertised." ], "refs": [ "algebra-lemma-integral-closure-is-ring", "algebra-lemma-integral-closure-stalks", "algebra-lemma-integral-closure-localize" ], "ref_ids": [ 486, 490, 489 ] } ], "ref_ids": [] }, { "id": 5499, "type": "theorem", "label": "morphisms-lemma-characterize-normalization", "categories": [ "morphisms" ], "title": "morphisms-lemma-characterize-normalization", "contents": [ "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes.", "The factorization $f = \\nu \\circ f'$, where $\\nu : X' \\to X$ is the", "normalization of $X$ in $Y$ is characterized by the following", "two properties:", "\\begin{enumerate}", "\\item the morphism $\\nu$ is integral, and", "\\item for any factorization $f = \\pi \\circ g$, with $\\pi : Z \\to X$", "integral, there exists a commutative diagram", "$$", "\\xymatrix{", "Y \\ar[d]_{f'} \\ar[r]_g & Z \\ar[d]^\\pi \\\\", "X' \\ar[ru]^h \\ar[r]^\\nu & X", "}", "$$", "for some unique morphism $h : X' \\to Z$.", "\\end{enumerate}", "Moreover, the morphism $f' : Y \\to X'$ is dominant and in (2) the", "morphism $h : X' \\to Z$ is the normalization of $Z$ in $Y$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{O}' \\subset f_*\\mathcal{O}_Y$ be the integral closure of", "$\\mathcal{O}_X$ as in Definition \\ref{definition-normalization-X-in-Y}.", "The morphism $\\nu$ is integral by construction, which proves (1).", "Assume given a factorization $f = \\pi \\circ g$ with $\\pi : Z \\to X$", "integral as in (2). By Definition \\ref{definition-integral}", "$\\pi$ is affine, and hence $Z$ is the relative", "spectrum of a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras $\\mathcal{B}$.", "The morphism $g : Y \\to Z$ corresponds to a map of $\\mathcal{O}_X$-algebras", "$\\chi : \\mathcal{B} \\to f_*\\mathcal{O}_Y$. Since $\\mathcal{B}(U)$ is", "integral over $\\mathcal{O}_X(U)$ for every affine open $U \\subset X$", "(by Definition \\ref{definition-integral})", "we see from Lemma \\ref{lemma-integral-closure}", "that $\\chi(\\mathcal{B}) \\subset \\mathcal{O}'$.", "By the functoriality of the relative spectrum", "Lemma \\ref{lemma-affine-equivalence-algebras}", "this provides us with a unique morphism", "$h : X' \\to Z$. We omit the verification that the diagram commutes.", "\\medskip\\noindent", "It is clear that (1) and (2) characterize the factorization $f = \\nu \\circ f'$", "since it characterizes it as an initial object in a category.", "\\medskip\\noindent", "From the universal property in (2) we see that $f'$ does not factor through", "a proper closed subscheme of $X'$. Hence the scheme theoretic image of $f'$", "is $X'$. Since $f'$ is quasi-compact (by Schemes, Lemma", "\\ref{schemes-lemma-quasi-compact-permanence} and the fact that $\\nu$ is", "separated as an affine morphism) we see that $f'(Y)$ is dense in $X'$.", "Hence $f'$ is dominant.", "\\medskip\\noindent", "The morphism $h$ in (2) is integral by Lemma \\ref{lemma-finite-permanence}.", "Given a factorization $g = \\pi' \\circ g'$ with $\\pi' : Z' \\to Z$", "integral, we get a factorization $f = (\\pi \\circ \\pi') \\circ g'$ and", "we get a morphism $h' : X' \\to Z'$. Uniqueness implies that", "$\\pi' \\circ h' = h$. Hence the characterization (1), (2) applies", "to the morphism $h : X' \\to Z$ which gives the last statement of the lemma." ], "refs": [ "morphisms-definition-normalization-X-in-Y", "morphisms-definition-integral", "morphisms-definition-integral", "morphisms-lemma-integral-closure", "morphisms-lemma-affine-equivalence-algebras", "schemes-lemma-quasi-compact-permanence", "morphisms-lemma-finite-permanence" ], "ref_ids": [ 5591, 5573, 5573, 5498, 5173, 7716, 5448 ] } ], "ref_ids": [] }, { "id": 5500, "type": "theorem", "label": "morphisms-lemma-functoriality-normalization", "categories": [ "morphisms" ], "title": "morphisms-lemma-functoriality-normalization", "contents": [ "Let", "$$", "\\xymatrix{", "Y_2 \\ar[d]_{f_2} \\ar[r] & Y_1 \\ar[d]^{f_1} \\\\", "X_2 \\ar[r] & X_1", "}", "$$", "be a commutative diagram of morphisms of schemes.", "Assume $f_1$, $f_2$ quasi-compact and quasi-separated.", "Let $f_i = \\nu_i \\circ f_i'$, $i = 1, 2$", "be the canonical factorizations, where $\\nu_i : X_i' \\to X_i$ is", "the normalization of $X_i$ in $Y_i$. Then there exists a unique", "arrow $X'_2 \\to X'_1$ fitting into a", "commutative diagram", "$$", "\\xymatrix{", "Y_2 \\ar[d]_{f_2'} \\ar[r] & Y_1 \\ar[d]^{f_1'} \\\\", "X_2' \\ar[d]_{\\nu_2} \\ar[r] & X_1' \\ar[d]^{\\nu_1} \\\\", "X_2 \\ar[r] & X_1", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-characterize-normalization} (1)", "and \\ref{lemma-base-change-finite}", "the base change $X_2 \\times_{X_1} X'_1 \\to X_2$", "is integral. Note that $f_2$ factors through this morphism.", "Hence we get a unique morphism", "$X'_2 \\to X_2 \\times_{X_1} X'_1$ from", "Lemma \\ref{lemma-characterize-normalization} (2).", "This gives the arrow $X'_2 \\to X'_1$ fitting into", "the commutative diagram and uniqueness follows as well." ], "refs": [ "morphisms-lemma-characterize-normalization", "morphisms-lemma-base-change-finite", "morphisms-lemma-characterize-normalization" ], "ref_ids": [ 5499, 5440, 5499 ] } ], "ref_ids": [] }, { "id": 5501, "type": "theorem", "label": "morphisms-lemma-normalization-localization", "categories": [ "morphisms" ], "title": "morphisms-lemma-normalization-localization", "contents": [ "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes.", "Let $U \\subset X$ be an open subscheme and set $V = f^{-1}(U)$.", "Then the normalization of $U$ in $V$ is the inverse image of $U$", "in the normalization of $X$ in $Y$." ], "refs": [], "proofs": [ { "contents": [ "Clear from the construction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5502, "type": "theorem", "label": "morphisms-lemma-normalization-is-normalization", "categories": [ "morphisms" ], "title": "morphisms-lemma-normalization-is-normalization", "contents": [ "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes.", "Let $X'$ be the normalization of $X$ in $Y$. Then the normalization of", "$X'$ in $Y$ is $X'$." ], "refs": [], "proofs": [ { "contents": [ "If $Y \\to X'' \\to X'$ is the normalization of $X'$ in $Y$, then", "we can apply Lemma \\ref{lemma-characterize-normalization}", "to the composition $X'' \\to X$ to get a canonical morphism", "$h : X' \\to X''$ over $X$. We omit the verification that the", "morphisms $h$ and $X'' \\to X'$ are mutually inverse (using uniqueness", "of the factorization in the lemma)." ], "refs": [ "morphisms-lemma-characterize-normalization" ], "ref_ids": [ 5499 ] } ], "ref_ids": [] }, { "id": 5503, "type": "theorem", "label": "morphisms-lemma-normalization-in-reduced", "categories": [ "morphisms" ], "title": "morphisms-lemma-normalization-in-reduced", "contents": [ "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes.", "Let $X' \\to X$ be the normalization of $X$ in $Y$. If $Y$ is reduced, so", "is $X'$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that a subring of a reduced ring is reduced.", "Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5504, "type": "theorem", "label": "morphisms-lemma-normalization-generic", "categories": [ "morphisms" ], "title": "morphisms-lemma-normalization-generic", "contents": [ "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes.", "Let $X' \\to X$ be the normalization of $X$ in $Y$. Every generic point of", "an irreducible component of $X'$ is the image of a generic point of", "an irreducible component of $Y$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-normalization-localization} we may assume $X = \\Spec(A)$", "is affine. Choose a finite affine open covering $Y = \\bigcup \\Spec(B_i)$.", "Then $X' = \\Spec(A')$ and the morphisms $\\Spec(B_i) \\to Y \\to X'$", "jointly define an injective $A$-algebra map $A' \\to \\prod B_i$.", "Thus the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-injective-minimal-primes-in-image}." ], "refs": [ "morphisms-lemma-normalization-localization", "algebra-lemma-injective-minimal-primes-in-image" ], "ref_ids": [ 5501, 445 ] } ], "ref_ids": [] }, { "id": 5505, "type": "theorem", "label": "morphisms-lemma-normalization-in-disjoint-union", "categories": [ "morphisms" ], "title": "morphisms-lemma-normalization-in-disjoint-union", "contents": [ "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes.", "Suppose that $Y = Y_1 \\amalg Y_2$ is a disjoint union of two schemes.", "Write $f_i = f|_{Y_i}$. Let $X_i'$ be the normalization of $X$ in $Y_i$.", "Then $X_1' \\amalg X_2'$ is the normalization of $X$ in $Y$." ], "refs": [], "proofs": [ { "contents": [ "In terms of integral closures this corresponds to the following fact:", "Let $A \\to B$ be a ring map. Suppose that $B = B_1 \\times B_2$.", "Let $A_i'$ be the integral closure of $A$ in $B_i$. Then", "$A_1' \\times A_2'$ is the integral closure of $A$ in $B$.", "The reason this works is that the elements $(1, 0)$ and $(0, 1)$ of $B$", "are idempotents and hence integral over $A$. Thus the integral closure", "$A'$ of $A$ in $B$ is a product and it is not hard to see that the factors", "are the integral closures $A'_i$ as described above (some details", "omitted)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5506, "type": "theorem", "label": "morphisms-lemma-normalization-in-universally-closed", "categories": [ "morphisms" ], "title": "morphisms-lemma-normalization-in-universally-closed", "contents": [ "Let $f : X \\to S$ be a quasi-compact, quasi-separated and", "universally closed morphisms of schemes.", "Then $f_*\\mathcal{O}_X$ is integral over $\\mathcal{O}_S$. In other", "words, the normalization of $S$ in $X$ is equal to the factorization", "$$", "X \\longrightarrow \\underline{\\Spec}_S(f_*\\mathcal{O}_X)", "\\longrightarrow S", "$$", "of Constructions, Lemma \\ref{constructions-lemma-canonical-morphism}." ], "refs": [ "constructions-lemma-canonical-morphism" ], "proofs": [ { "contents": [ "The question is local on $S$, hence we may assume $S = \\Spec(R)$", "is affine. Let $h \\in \\Gamma(X, \\mathcal{O}_X)$. We have to show that", "$h$ satisfies a monic equation over $R$. Think of $h$ as a morphism", "as in the following commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_h \\ar[rd]_f & & \\mathbf{A}^1_S \\ar[ld] \\\\", "& S &", "}", "$$", "Let $Z \\subset \\mathbf{A}^1_S$ be the scheme theoretic image of $h$,", "see Definition \\ref{definition-scheme-theoretic-image}.", "The morphism $h$ is quasi-compact as $f$ is quasi-compact and", "$\\mathbf{A}^1_S \\to S$ is separated, see", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}.", "By Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image} the", "morphism $X \\to Z$ is dominant. By", "Lemma \\ref{lemma-image-proper-scheme-closed} the morphism", "$X \\to Z$ is closed. Hence $h(X) = Z$ (set theoretically).", "Thus we can use", "Lemma \\ref{lemma-image-proper-is-proper}", "to conclude that $Z \\to S$ is universally closed (and even proper).", "Since $Z \\subset \\mathbf{A}^1_S$, we see that $Z \\to S$ is affine", "and proper, hence integral by Lemma \\ref{lemma-integral-universally-closed}.", "Writing $\\mathbf{A}^1_S = \\Spec(R[T])$ we conclude that", "the ideal $I \\subset R[T]$ of $Z$ contains a monic polynomial", "$P(T) \\in R[T]$. Hence $P(h) = 0$ and we win." ], "refs": [ "morphisms-definition-scheme-theoretic-image", "schemes-lemma-quasi-compact-permanence", "morphisms-lemma-quasi-compact-scheme-theoretic-image", "morphisms-lemma-image-proper-scheme-closed", "morphisms-lemma-image-proper-is-proper", "morphisms-lemma-integral-universally-closed" ], "ref_ids": [ 5539, 7716, 5146, 5411, 5413, 5441 ] } ], "ref_ids": [ 12591 ] }, { "id": 5507, "type": "theorem", "label": "morphisms-lemma-normalization-in-integral", "categories": [ "morphisms" ], "title": "morphisms-lemma-normalization-in-integral", "contents": [ "Let $f : Y \\to X$ be an integral morphism.", "Then the normalization of $X$ in $Y$ is equal to $Y$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-integral-universally-closed} this is a special case of", "Lemma \\ref{lemma-normalization-in-universally-closed}." ], "refs": [ "morphisms-lemma-integral-universally-closed", "morphisms-lemma-normalization-in-universally-closed" ], "ref_ids": [ 5441, 5506 ] } ], "ref_ids": [] }, { "id": 5508, "type": "theorem", "label": "morphisms-lemma-normal-normalization", "categories": [ "morphisms" ], "title": "morphisms-lemma-normal-normalization", "contents": [ "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism", "of schemes. Let $X'$ be the normalization of $X$ in $Y$. Assume", "\\begin{enumerate}", "\\item $Y$ is a normal scheme,", "\\item quasi-compact opens of $Y$ have finitely many irreducible components.", "\\end{enumerate}", "Then $X'$ is a disjoint union of integral normal schemes. Moreover,", "the morphism $Y \\to X'$ is dominant and induces a bijection of", "irreducible components." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be an affine open. Consider the", "inverse image $U'$ of $U$ in $X'$.", "Set $V = f^{-1}(U)$. By Lemma \\ref{lemma-normalization-localization}", "we $V \\to U' \\to U$ is the normalization of $U$ in $V$. Say", "$U = \\Spec(A)$. Then $V$ is quasi-compact, and hence has a finite number of", "irreducible components by assumption. Hence", "$V = \\coprod_{i = 1, \\ldots n} V_i$ is a finite disjoint union of", "normal integral schemes by", "Properties, Lemma \\ref{properties-lemma-normal-locally-finite-nr-irreducibles}.", "By Lemma \\ref{lemma-normalization-in-disjoint-union}", "we see that $U' = \\coprod_{i = 1, \\ldots, n} U_i'$,", "where $U'_i$ is the normalization of $U$ in $V_i$.", "By Properties, Lemma \\ref{properties-lemma-normal-integral-sections}", "we see that $B_i = \\Gamma(V_i, \\mathcal{O}_{V_i})$ is a normal domain.", "Note that $U_i' = \\Spec(A_i')$, where $A_i' \\subset B_i$", "is the integral closure of $A$ in $B_i$, see", "Lemma \\ref{lemma-integral-closure}. By", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-in-normal}", "we see that $A_i' \\subset B_i$ is a normal domain.", "Hence $U' = \\coprod U_i'$ is a finite union of normal integral schemes", "and hence is normal.", "\\medskip\\noindent", "As $X'$ has an open covering by the schemes $U'$ we conclude from", "Properties, Lemma \\ref{properties-lemma-locally-normal} that $X'$ is normal.", "On the other hand, each $U'$ is a finite disjoint union of irreducible", "schemes, hence every quasi-compact open of $X'$ has finitely many irreducible", "components (by a topological argument which we omit). Thus $X'$", "is a disjoint union of normal integral schemes by", "Properties, Lemma \\ref{properties-lemma-normal-locally-finite-nr-irreducibles}.", "It is clear from the description of $X'$ above that $Y \\to X'$", "is dominant and induces a bijection on irreducible components", "$V \\to U'$ for every affine open $U \\subset X$. The bijection of", "irreducible components for the morphism $Y \\to X'$", "follows from this by a topological argument (omitted)." ], "refs": [ "morphisms-lemma-normalization-localization", "properties-lemma-normal-locally-finite-nr-irreducibles", "morphisms-lemma-normalization-in-disjoint-union", "properties-lemma-normal-integral-sections", "morphisms-lemma-integral-closure", "algebra-lemma-integral-closure-in-normal", "properties-lemma-locally-normal", "properties-lemma-normal-locally-finite-nr-irreducibles" ], "ref_ids": [ 5501, 2969, 5505, 2972, 5498, 503, 2966, 2969 ] } ], "ref_ids": [] }, { "id": 5509, "type": "theorem", "label": "morphisms-lemma-nagata-normalization-finite-general", "categories": [ "morphisms" ], "title": "morphisms-lemma-nagata-normalization-finite-general", "contents": [ "Let $f : X \\to S$ be a morphism. Assume that", "\\begin{enumerate}", "\\item $S$ is a Nagata scheme,", "\\item $f$ is quasi-compact and quasi-separated,", "\\item quasi-compact opens of $X$ have finitely many irreducible components,", "\\item if $x \\in X$ is a generic point of an irreducible component,", "then the field extension $\\kappa(f(x)) \\subset \\kappa(x)$ is finitely", "generated, and", "\\item $X$ is reduced.", "\\end{enumerate}", "Then the normalization $\\nu : S' \\to S$ of $S$ in $X$ is finite." ], "refs": [], "proofs": [ { "contents": [ "There is an immediate reduction to the case $S = \\Spec(R)$", "where $R$ is a Nagata ring by assumption (1). We have to show that", "the integral closure $A$ of $R$ in $\\Gamma(X, \\mathcal{O}_X)$ is", "finite over $R$. Since $f$ is quasi-compact by assumption (2) we can write", "$X = \\bigcup_{i = 1, \\ldots, n} U_i$ with each $U_i$ affine.", "Say $U_i = \\Spec(B_i)$. Each $B_i$ is reduced by assumption (5)", "and has finitely many minimal primes", "$\\mathfrak q_{i1}, \\ldots, \\mathfrak q_{im_i}$", "by assumption (3) and", "Algebra, Lemma \\ref{algebra-lemma-irreducible}.", "We have", "$$", "\\Gamma(X, \\mathcal{O}_X) \\subset B_1 \\times \\ldots \\times B_n", "\\subset", "\\prod\\nolimits_{i = 1, \\ldots, n}", "\\prod\\nolimits_{j = 1, \\ldots, m_i} (B_i)_{\\mathfrak q_{ij}}", "$$", "the second inclusion by", "Algebra, Lemma \\ref{algebra-lemma-reduced-ring-sub-product-fields}.", "We have $\\kappa(\\mathfrak q_{ij}) = (B_i)_{\\mathfrak q_{ij}}$ by", "Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring}.", "Hence the integral closure $A$ of $R$ in $\\Gamma(X, \\mathcal{O}_X)$", "is contained in the product of the integral closures $A_{ij}$ of", "$R$ in $\\kappa(\\mathfrak q_{ij})$. Since $R$ is Noetherian", "it suffices to show that $A_{ij}$ is a finite $R$-module for each $i, j$.", "Let $\\mathfrak p_{ij} \\subset R$ be the image of $\\mathfrak q_{ij}$.", "As $\\kappa(\\mathfrak p_{ij}) \\subset \\kappa(\\mathfrak q_{ij})$", "is a finitely generated field extension by assumption (4),", "we see that $R \\to \\kappa(\\mathfrak q_{ij})$ is essentially of finite type.", "Thus $R \\to A_{ij}$ is finite by Algebra, Lemma", "\\ref{algebra-lemma-nagata-in-reduced-finite-type-finite-integral-closure}." ], "refs": [ "algebra-lemma-irreducible", "algebra-lemma-reduced-ring-sub-product-fields", "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-nagata-in-reduced-finite-type-finite-integral-closure" ], "ref_ids": [ 422, 419, 418, 1347 ] } ], "ref_ids": [] }, { "id": 5510, "type": "theorem", "label": "morphisms-lemma-nagata-normalization-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-nagata-normalization-finite", "contents": [ "Let $f : X \\to S$ be a morphism. Assume that", "\\begin{enumerate}", "\\item $S$ is a Nagata scheme,", "\\item $f$ is of finite type,", "\\item $X$ is reduced.", "\\end{enumerate}", "Then the normalization $\\nu : S' \\to S$ of $S$ in $X$ is finite." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-nagata-normalization-finite-general}.", "Namely, (2) holds as the finite type morphism $f$ is quasi-compact", "by definition and quasi-separated by", "Lemma \\ref{lemma-finite-type-Noetherian-quasi-separated}.", "Condition (3) holds because $X$ is locally Noetherian by", "Lemma \\ref{lemma-finite-type-noetherian}. Finally, condition (4)", "holds because a finite type morphism induces finitely generated", "residue field extensions." ], "refs": [ "morphisms-lemma-nagata-normalization-finite-general", "morphisms-lemma-finite-type-Noetherian-quasi-separated", "morphisms-lemma-finite-type-noetherian" ], "ref_ids": [ 5509, 5203, 5202 ] } ], "ref_ids": [] }, { "id": 5511, "type": "theorem", "label": "morphisms-lemma-relative-normalization-normal-codim-1", "categories": [ "morphisms" ], "title": "morphisms-lemma-relative-normalization-normal-codim-1", "contents": [ "Let $f : Y \\to X$ be a finite type morphism of schemes with $Y$ reduced", "and $X$ Nagata. Let $X'$ be the normalization of $X$ in $Y$.", "Let $x' \\in X'$ be a point such that", "\\begin{enumerate}", "\\item $\\dim(\\mathcal{O}_{X', x'}) = 1$, and", "\\item the fibre of $Y \\to X'$ over $x'$ is empty.", "\\end{enumerate}", "Then $\\mathcal{O}_{X', x'}$ is a discrete valuation ring." ], "refs": [], "proofs": [ { "contents": [ "We can replace $X$ by an affine neighbourhood of the image of $x'$.", "Hence we may assume $X = \\Spec(A)$ with $A$ Nagata.", "By Lemma \\ref{lemma-nagata-normalization-finite}", "the morphism $X' \\to X$ is finite. Hence we can write", "$X' = \\Spec(A')$ for a finite $A$-algebra $A'$.", "By Lemma \\ref{lemma-normalization-is-normalization}", "after replacing $X$ by $X'$", "we reduce to the case described in the next paragraph.", "\\medskip\\noindent", "The case $X = X' = \\Spec(A)$ with $A$ Noetherian.", "Let $\\mathfrak p \\subset A$ be the prime ideal", "corresponding to our point $x'$. Choose $g \\in \\mathfrak p$", "not contained in any minimal prime of $A$ (use prime avoidance", "and the fact that $A$ has finitely many minimal primes, see", "Algebra, Lemmas \\ref{algebra-lemma-silly} and", "\\ref{algebra-lemma-Noetherian-irreducible-components}).", "Set $Z = f^{-1}V(g) \\subset Y$; it is a closed subscheme of $Y$.", "Then $f(Z)$ does not contain any generic point by choice of $g$", "and does not contain $x'$ because $x'$ is not in the image of $f$.", "The closure of $f(Z)$ is the set of specializations of points", "of $f(Z)$ by Lemma \\ref{lemma-reach-points-scheme-theoretic-image}.", "Thus the closure of $f(Z)$ does not contain $x'$ because", "the condition $\\dim(\\mathcal{O}_{X', x'}) = 1$ implies only", "the generic points of $X = X'$ specialize to $x'$.", "In other words, after replacing $X$ by an affine open", "neighbourhood of $x'$ we may assume that $f^{-1}V(g) = \\emptyset$.", "Thus $g$ maps to an invertible global function on $Y$ and", "we obtain a factorization", "$$", "A \\to A_g \\to \\Gamma(Y, \\mathcal{O}_Y)", "$$", "Since $X = X'$ this implies that $A$ is equal to the integral", "closure of $A$ in $A_g$. By", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-localize}", "we conclude that $A_\\mathfrak p$ is the integral closure", "of $A_\\mathfrak p$ in $A_\\mathfrak p[1/g]$.", "By our choice of $g$, since $\\dim(A_\\mathfrak p) = 1$", "and since $A$ is reduced we see that $A_\\mathfrak p[1/g]$", "is a finite product of fields (the product of the residue fields", "of the minimal primes contained in $\\mathfrak p$). Hence $A_\\mathfrak p$ is", "normal (Algebra, Lemma \\ref{algebra-lemma-characterize-reduced-ring-normal})", "and the proof is complete. Some details omitted." ], "refs": [ "morphisms-lemma-nagata-normalization-finite", "morphisms-lemma-normalization-is-normalization", "algebra-lemma-silly", "algebra-lemma-Noetherian-irreducible-components", "morphisms-lemma-reach-points-scheme-theoretic-image", "algebra-lemma-integral-closure-localize", "algebra-lemma-characterize-reduced-ring-normal" ], "ref_ids": [ 5510, 5502, 378, 453, 5147, 489, 515 ] } ], "ref_ids": [] }, { "id": 5512, "type": "theorem", "label": "morphisms-lemma-normalization-reduced", "categories": [ "morphisms" ], "title": "morphisms-lemma-normalization-reduced", "contents": [ "Let $X$ be a scheme such that every quasi-compact open has", "finitely many irreducible components. The normalization morphism", "$\\nu$ factors through the reduction $X_{red}$ and $X^\\nu \\to X_{red}$", "is the normalization of $X_{red}$." ], "refs": [], "proofs": [ { "contents": [ "Let $f : Y \\to X$ be the morphism (\\ref{equation-generic-points}).", "We get a factorization $Y \\to X_{red} \\to X$ of $f$ from", "Schemes, Lemma \\ref{schemes-lemma-map-into-reduction}.", "By Lemma \\ref{lemma-characterize-normalization} we obtain a canonical", "morphism $X^\\nu \\to X_{red}$", "and that $X^\\nu$ is the normalization of $X_{red}$ in $Y$.", "The lemma follows as $Y \\to X_{red}$ is identical to the morphism", "(\\ref{equation-generic-points}) constructed for $X_{red}$." ], "refs": [ "schemes-lemma-map-into-reduction", "morphisms-lemma-characterize-normalization" ], "ref_ids": [ 7682, 5499 ] } ], "ref_ids": [] }, { "id": 5513, "type": "theorem", "label": "morphisms-lemma-description-normalization", "categories": [ "morphisms" ], "title": "morphisms-lemma-description-normalization", "contents": [ "Let $X$ be a reduced scheme such that every quasi-compact open has", "finitely many irreducible components. Let $\\Spec(A) = U \\subset X$", "be an affine open. Then", "\\begin{enumerate}", "\\item $A$ has finitely many minimal primes", "$\\mathfrak q_1, \\ldots, \\mathfrak q_t$,", "\\item the total ring of fractions $Q(A)$ of $A$ is", "$Q(A/\\mathfrak q_1) \\times \\ldots \\times Q(A/\\mathfrak q_t)$,", "\\item the integral closure $A'$ of $A$ in $Q(A)$ is the product of", "the integral closures of the domains $A/\\mathfrak q_i$", "in the fields $Q(A/\\mathfrak q_i)$, and", "\\item $\\nu^{-1}(U)$ is identified with the spectrum of $A'$ where", "$\\nu : X^\\nu \\to X$ is the normalization morphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Minimal primes correspond to irreducible components", "(Algebra, Lemma \\ref{algebra-lemma-irreducible}),", "hence we have (1) by assumption. Then", "$(0) = \\mathfrak q_1 \\cap \\ldots \\cap \\mathfrak q_t$ because $A$ is reduced", "(Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}).", "Then we have", "$Q(A) = \\prod A_{\\mathfrak q_i} = \\prod \\kappa(\\mathfrak q_i)$", "by Algebra, Lemmas \\ref{algebra-lemma-total-ring-fractions-no-embedded-points}", "and \\ref{algebra-lemma-minimal-prime-reduced-ring}.", "This proves (2). Part (3) follows from", "Algebra, Lemma \\ref{algebra-lemma-characterize-reduced-ring-normal},", "or Lemma \\ref{lemma-normalization-in-disjoint-union}.", "Part (4) holds because it is clear that $f^{-1}(U) \\to U$ is the morphism", "$$", "\\Spec\\left(\\prod \\kappa(\\mathfrak q_i)\\right)", "\\longrightarrow", "\\Spec(A)", "$$", "where $f : Y \\to X$ is the morphism (\\ref{equation-generic-points})." ], "refs": [ "algebra-lemma-irreducible", "algebra-lemma-Zariski-topology", "algebra-lemma-total-ring-fractions-no-embedded-points", "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-characterize-reduced-ring-normal", "morphisms-lemma-normalization-in-disjoint-union" ], "ref_ids": [ 422, 389, 421, 418, 515, 5505 ] } ], "ref_ids": [] }, { "id": 5514, "type": "theorem", "label": "morphisms-lemma-stalk-normalization", "categories": [ "morphisms" ], "title": "morphisms-lemma-stalk-normalization", "contents": [ "Let $X$ be a scheme such that every quasi-compact open has a finite", "number of irreducible components. Let $\\nu : X^\\nu \\to X$", "be the normalization of $X$. Let $x \\in X$. Then the following", "are canonically isomorphic as $\\mathcal{O}_{X, x}$-algebras", "\\begin{enumerate}", "\\item the stalk $(\\nu_*\\mathcal{O}_{X^\\nu})_x$,", "\\item the integral closure of $\\mathcal{O}_{X, x}$ in", "the total ring of fractions of $(\\mathcal{O}_{X, x})_{red}$,", "\\item the integral closure of $\\mathcal{O}_{X, x}$ in the product", "of the residue fields of the minimal primes of $\\mathcal{O}_{X, x}$", "(and there are finitely many of these).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "After replacing $X$ by an affine open neighbourhood", "of $x$ we may assume that $X$ has finitely many irreducible", "components and that $x$ is contained in each of them.", "Then the stalk $(\\nu_*\\mathcal{O}_{X^\\nu})_x$ is the", "integral closure of $A = \\mathcal{O}_{X, x}$ in the product $L$", "of the residue fields of the minimal primes of $A$.", "This follows from the construction of the normalization and", "Lemma \\ref{lemma-integral-closure}.", "Alternatively, you can use Lemma", "\\ref{lemma-description-normalization}", "and the fact that normalization commutes with localization", "(Algebra, Lemma \\ref{algebra-lemma-integral-closure-localize}).", "Since $A_{red}$ has finitely many minimal primes", "(because these correspond exactly to the generic points", "of the irreducible components of $X$ passing through $x$)", "we see that $L$ is the total ring of fractions of $A_{red}$", "(Algebra, Lemma \\ref{algebra-lemma-total-ring-fractions-no-embedded-points}).", "Thus our ring is also the integral closure of $A$ in the total", "ring of fractions of $A_{red}$." ], "refs": [ "morphisms-lemma-integral-closure", "morphisms-lemma-description-normalization", "algebra-lemma-integral-closure-localize", "algebra-lemma-total-ring-fractions-no-embedded-points" ], "ref_ids": [ 5498, 5513, 489, 421 ] } ], "ref_ids": [] }, { "id": 5515, "type": "theorem", "label": "morphisms-lemma-normalization-normal", "categories": [ "morphisms" ], "title": "morphisms-lemma-normalization-normal", "contents": [ "Let $X$ be a scheme such that every quasi-compact open has", "finitely many irreducible components.", "\\begin{enumerate}", "\\item The normalization $X^\\nu$ is a disjoint union of integral normal schemes.", "\\item The morphism $\\nu : X^\\nu \\to X$ is integral, surjective, and", "induces a bijection on irreducible components.", "\\item For any integral morphism $\\alpha : X' \\to X$ such that for", "$U \\subset X$ quasi-compact open the inverse image $\\alpha^{-1}(U)$ has", "finitely many irreducible components and", "$\\alpha|_{\\alpha^{-1}(U)} : \\alpha^{-1}(U) \\to U$ is birational\\footnote{This", "awkward formulation is necessary as we've only defined what", "it means for a morphism to be birational if the source and target", "have finitely many irreducible components. It suffices if", "$X'_{red} \\to X_{red}$ satisfies the condition.}", "there exists a factorization", "$X^\\nu \\to X' \\to X$ and $X^\\nu \\to X'$ is the normalization of $X'$.", "\\item For any morphism $Z \\to X$ with $Z$ a normal scheme", "such that each irreducible component of $Z$ dominates an irreducible", "component of $X$ there exists a unique factorization $Z \\to X^\\nu \\to X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $f : Y \\to X$ be as in (\\ref{equation-generic-points}).", "The scheme $X^\\nu$ is a disjoint union of normal integral schemes", "because $Y$ is normal and every affine open of $Y$ has finitely", "many irreducible components, see", "Lemma \\ref{lemma-normal-normalization}. This proves (1).", "Alternatively one can deduce (1) from", "Lemmas \\ref{lemma-normalization-reduced} and", "\\ref{lemma-description-normalization}.", "\\medskip\\noindent", "The morphism $\\nu$ is integral by Lemma \\ref{lemma-characterize-normalization}.", "By Lemma \\ref{lemma-normal-normalization} the", "morphism $Y \\to X^\\nu$ induces a bijection on irreducible components,", "and by construction of $Y$ this implies that $X^\\nu \\to X$ induces", "a bijection on irreducible components. By construction $f : Y \\to X$", "is dominant, hence also $\\nu$ is dominant. Since an integral morphism is", "closed (Lemma \\ref{lemma-integral-universally-closed}) this implies that", "$\\nu$ is surjective. This proves (2).", "\\medskip\\noindent", "Suppose that $\\alpha : X' \\to X$ is as in (3). It is clear that", "$X'$ satisfies the assumptions under which the normalization", "is defined. Let $f' : Y' \\to X'$ be the morphism", "(\\ref{equation-generic-points}) constructed starting with $X'$.", "As $\\alpha$ is birational it is clear that $Y' = Y$ and $f = \\alpha \\circ f'$.", "Hence the factorization $X^\\nu \\to X' \\to X$ exists", "and $X^\\nu \\to X'$ is the normalization of $X'$ by", "Lemma \\ref{lemma-characterize-normalization}. This proves (3).", "\\medskip\\noindent", "Let $g : Z \\to X$ be a morphism whose domain is a normal scheme", "and such that every irreducible component dominates an irreducible", "component of $X$. By Lemma \\ref{lemma-normalization-reduced}", "we have $X^\\nu = X_{red}^\\nu$ and by", "Schemes, Lemma \\ref{schemes-lemma-map-into-reduction}", "$Z \\to X$ factors through $X_{red}$. Hence we may replace $X$ by", "$X_{red}$ and assume $X$ is reduced. Moreover, as the factorization", "is unique it suffices to construct it locally on $Z$.", "Let $W \\subset Z$ and $U \\subset X$ be affine opens", "such that $g(W) \\subset U$. Write $U = \\Spec(A)$ and", "$W = \\Spec(B)$, with $g|_W$ given by $\\varphi : A \\to B$.", "We will use the results of Lemma \\ref{lemma-description-normalization} freely.", "Let $\\mathfrak p_1, \\ldots, \\mathfrak p_t$ be the minimal primes of $A$.", "As $Z$ is normal, we see that $B$ is a normal", "ring, in particular reduced. Moreover, by assumption any minimal", "prime $\\mathfrak q \\subset B$ we have that $\\varphi^{-1}(\\mathfrak q)$", "is a minimal prime of $A$. Hence if $x \\in A$ is a nonzerodivisor, i.e.,", "$x \\not \\in \\bigcup \\mathfrak p_i$, then $\\varphi(x)$ is a nonzerodivisor", "in $B$. Thus we obtain a canonical ring map $Q(A) \\to Q(B)$. As $B$ is", "normal it is equal to its integral closure in $Q(B)$ (see", "Algebra, Lemma \\ref{algebra-lemma-normal-ring-integrally-closed}).", "Hence we see that the integral closure $A' \\subset Q(A)$ of $A$", "maps into $B$ via the canonical map $Q(A) \\to Q(B)$.", "Since $\\nu^{-1}(U) = \\Spec(A')$ this gives the canonical", "factorization $W \\to \\nu^{-1}(U) \\to U$ of $\\nu|_W$.", "We omit the verification that it is unique." ], "refs": [ "morphisms-lemma-normal-normalization", "morphisms-lemma-normalization-reduced", "morphisms-lemma-description-normalization", "morphisms-lemma-characterize-normalization", "morphisms-lemma-normal-normalization", "morphisms-lemma-integral-universally-closed", "morphisms-lemma-characterize-normalization", "morphisms-lemma-normalization-reduced", "schemes-lemma-map-into-reduction", "morphisms-lemma-description-normalization", "algebra-lemma-normal-ring-integrally-closed" ], "ref_ids": [ 5508, 5512, 5513, 5499, 5508, 5441, 5499, 5512, 7682, 5513, 511 ] } ], "ref_ids": [] }, { "id": 5516, "type": "theorem", "label": "morphisms-lemma-normalization-in-terms-of-components", "categories": [ "morphisms" ], "title": "morphisms-lemma-normalization-in-terms-of-components", "contents": [ "Let $X$ be a scheme such that every quasi-compact open has", "finitely many irreducible components. Let $Z_i \\subset X$, $i \\in I$", "be the irreducible components of $X$ endowed with the reduced", "induced structure. Let $Z_i^\\nu \\to Z_i$ be the normalization.", "Then $\\coprod_{i \\in I} Z_i^\\nu \\to X$ is the normalization of $X$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $X$ is reduced, see Lemma \\ref{lemma-normalization-reduced}.", "Then the lemma follows either from the local description in", "Lemma \\ref{lemma-description-normalization}", "or from Lemma \\ref{lemma-normalization-normal} part (3) because", "$\\coprod Z_i \\to X$ is integral and birational (as $X$ is reduced", "and has locally finitely many irreducible components)." ], "refs": [ "morphisms-lemma-normalization-reduced", "morphisms-lemma-description-normalization", "morphisms-lemma-normalization-normal" ], "ref_ids": [ 5512, 5513, 5515 ] } ], "ref_ids": [] }, { "id": 5517, "type": "theorem", "label": "morphisms-lemma-normalization-birational", "categories": [ "morphisms" ], "title": "morphisms-lemma-normalization-birational", "contents": [ "Let $X$ be a reduced scheme with finitely many irreducible components.", "Then the normalization morphism $X^\\nu \\to X$ is birational." ], "refs": [], "proofs": [ { "contents": [ "The normalization induces a bijection of irreducible components by", "Lemma \\ref{lemma-normalization-normal}. Let $\\eta \\in X$ be a generic", "point of an irreducible component of $X$ and let $\\eta^\\nu \\in X^\\nu$", "be the generic point of the corresponding irreducible component of $X^\\nu$.", "Then $\\eta^\\nu \\mapsto \\eta$ and to finish the proof we have to show that", "$\\mathcal{O}_{X, \\eta} \\to \\mathcal{O}_{X^\\nu, \\eta^\\nu}$", "is an isomorphism, see Definition \\ref{definition-birational}.", "Because $X$ and $X^\\nu$ are reduced, we see that both local rings", "are equal to their residue fields", "(Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring}).", "On the other hand, by the construction of the normalization", "as the normalization of $X$ in $Y = \\coprod \\Spec(\\kappa(\\eta))$", "we see that we have", "$\\kappa(\\eta) \\subset \\kappa(\\eta^\\nu) \\subset \\kappa(\\eta)$", "and the proof is complete." ], "refs": [ "morphisms-lemma-normalization-normal", "morphisms-definition-birational", "algebra-lemma-minimal-prime-reduced-ring" ], "ref_ids": [ 5515, 5586, 418 ] } ], "ref_ids": [] }, { "id": 5518, "type": "theorem", "label": "morphisms-lemma-finite-birational-over-normal", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-birational-over-normal", "contents": [ "A finite (or even integral) birational morphism $f : X \\to Y$", "of integral schemes with $Y$ normal is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset Y$ be an affine open", "with inverse image $U \\subset X$ which is an affine open too.", "Since $f$ is a birational morphism of integral schemes, the homomorphism", "$\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$ is an injective map of domains", "which induces an isomorphism of fraction fields. As $Y$ is normal,", "the ring $\\mathcal{O}_Y(V)$ is integrally closed in the fraction field.", "Since $f$ is finite (or integral) every element of $\\mathcal{O}_X(U)$", "is integral over $\\mathcal{O}_Y(V)$. We conclude that", "$\\mathcal{O}_Y(V) = \\mathcal{O}_X(U)$. This proves that $f$ is an", "isomorphism as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5519, "type": "theorem", "label": "morphisms-lemma-Japanese-normalization", "categories": [ "morphisms" ], "title": "morphisms-lemma-Japanese-normalization", "contents": [ "Let $X$ be an integral, Japanese scheme.", "The normalization $\\nu : X^\\nu \\to X$ is a finite morphism." ], "refs": [], "proofs": [ { "contents": [ "Follows from the definition", "(Properties, Definition \\ref{properties-definition-nagata}) and", "Lemma \\ref{lemma-description-normalization}. Namely, in this case", "the lemma says that $\\nu^{-1}(\\Spec(A))$ is the spectrum", "of the integral closure of $A$ in its field of fractions." ], "refs": [ "properties-definition-nagata", "morphisms-lemma-description-normalization" ], "ref_ids": [ 3079, 5513 ] } ], "ref_ids": [] }, { "id": 5520, "type": "theorem", "label": "morphisms-lemma-nagata-normalization", "categories": [ "morphisms" ], "title": "morphisms-lemma-nagata-normalization", "contents": [ "Let $X$ be a Nagata scheme.", "The normalization $\\nu : X^\\nu \\to X$ is a finite morphism." ], "refs": [], "proofs": [ { "contents": [ "Note that a Nagata scheme is locally Noetherian, thus", "Definition \\ref{definition-normalization}", "does apply. The lemma is now a special case of", "Lemma \\ref{lemma-nagata-normalization-finite-general}", "but we can also prove it directly as follows.", "Write $X^\\nu \\to X$ as the composition", "$X^\\nu \\to X_{red} \\to X$. As $X_{red} \\to X$ is a closed immersion", "it is finite. Hence it suffices to prove the lemma for a reduced", "Nagata scheme (by Lemma \\ref{lemma-composition-finite}).", "Let $\\Spec(A) = U \\subset X$ be an affine open.", "By Lemma \\ref{lemma-description-normalization} we have", "$\\nu^{-1}(U) = \\Spec(\\prod A_i')$ where $A_i'$ is the integral", "closure of $A/\\mathfrak q_i$ in its fraction field. As $A$ is a Nagata", "ring (see Properties, Lemma \\ref{properties-lemma-locally-nagata})", "each of the ring extensions", "$A/\\mathfrak q_i \\subset A'_i$ are finite. Hence $A \\to \\prod A'_i$", "is a finite ring map and we win." ], "refs": [ "morphisms-definition-normalization", "morphisms-lemma-nagata-normalization-finite-general", "morphisms-lemma-composition-finite", "morphisms-lemma-description-normalization", "properties-lemma-locally-nagata" ], "ref_ids": [ 5592, 5509, 5439, 5513, 2995 ] } ], "ref_ids": [] }, { "id": 5521, "type": "theorem", "label": "morphisms-lemma-quasi-finite-points-open", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-finite-points-open", "contents": [ "\\begin{slogan}", "The locally quasi-finite locus of a morphism is open", "\\end{slogan}", "Let $f : X \\to S$ be a morphism of schemes.", "The set of points of $X$ where $f$ is quasi-finite is an open", "$U \\subset X$. The induced morphism $U \\to S$ is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Suppose $f$ is quasi-finite at $x$.", "Let $x \\in U = \\Spec(A) \\subset X$, $V = \\Spec(R) \\subset S$", "be affine opens as in Definition \\ref{definition-quasi-finite}.", "By either Theorem \\ref{theorem-main-theorem} above or", "Algebra, Lemma \\ref{algebra-lemma-quasi-finite-open},", "the set of primes $\\mathfrak q$ at which $R \\to A$ is quasi-finite", "is open in $\\Spec(A)$. Since these all correspond to points", "of $X$ where $f$ is quasi-finite we get the first statement.", "The second statement is obvious." ], "refs": [ "morphisms-definition-quasi-finite", "morphisms-theorem-main-theorem", "algebra-lemma-quasi-finite-open" ], "ref_ids": [ 5553, 5124, 1066 ] } ], "ref_ids": [] }, { "id": 5522, "type": "theorem", "label": "morphisms-lemma-quasi-finite-affine", "categories": [ "morphisms" ], "title": "morphisms-lemma-quasi-finite-affine", "contents": [ "Let $f : Y \\to X$ be a morphism of schemes.", "Assume", "\\begin{enumerate}", "\\item $X$ and $Y$ are affine, and", "\\item $f$ is quasi-finite.", "\\end{enumerate}", "Then there exists a diagram", "$$", "\\xymatrix{", "Y \\ar[rd]_f \\ar[rr]_j & & Z \\ar[ld]^\\pi \\\\", "& X &", "}", "$$", "with $Z$ affine, $\\pi$ finite and $j$ an open immersion." ], "refs": [], "proofs": [ { "contents": [ "This is", "Algebra, Lemma \\ref{algebra-lemma-quasi-finite-open-integral-closure}", "reformulated in the language of schemes." ], "refs": [ "algebra-lemma-quasi-finite-open-integral-closure" ], "ref_ids": [ 1067 ] } ], "ref_ids": [] }, { "id": 5523, "type": "theorem", "label": "morphisms-lemma-image-nowhere-dense-quasi-finite", "categories": [ "morphisms" ], "title": "morphisms-lemma-image-nowhere-dense-quasi-finite", "contents": [ "Let $f : Y \\to X$ be a quasi-finite morphism of schemes.", "Let $T \\subset Y$ be a closed nowhere dense subset of $Y$.", "Then $f(T) \\subset X$ is a nowhere dense subset of $X$." ], "refs": [], "proofs": [ { "contents": [ "As in the proof of Lemma \\ref{lemma-image-nowhere-dense-finite} this", "reduces immediately to the case where the base $X$ is affine.", "In this case $Y = \\bigcup_{i = 1, \\ldots, n} Y_i$ is a finite union", "of affine opens (as $f$ is quasi-compact). Since each $T \\cap Y_i$", "is nowhere dense, and since a finite union of nowhere dense sets is", "nowhere dense (see", "Topology, Lemma \\ref{topology-lemma-nowhere-dense}),", "it suffices to prove that the image $f(T \\cap Y_i)$ is nowhere dense in $X$.", "This reduces us to the case where both $X$ and $Y$ are affine. At this point", "we apply Lemma \\ref{lemma-quasi-finite-affine} above to get a diagram", "$$", "\\xymatrix{", "Y \\ar[rd]_f \\ar[rr]_j & & Z \\ar[ld]^\\pi \\\\", "& X &", "}", "$$", "with $Z$ affine, $\\pi$ finite and $j$ an open immersion.", "Set $\\overline{T} = \\overline{j(T)} \\subset Z$. By", "Topology, Lemma \\ref{topology-lemma-image-nowhere-dense-open}", "we see $\\overline{T}$ is nowhere dense in $Z$.", "Since $f(T) \\subset \\pi(\\overline{T})$", "the lemma follows from the corresponding result in the finite case, see", "Lemma \\ref{lemma-image-nowhere-dense-finite}." ], "refs": [ "morphisms-lemma-image-nowhere-dense-finite", "topology-lemma-nowhere-dense", "morphisms-lemma-quasi-finite-affine", "topology-lemma-image-nowhere-dense-open", "morphisms-lemma-image-nowhere-dense-finite" ], "ref_ids": [ 5476, 8294, 5522, 8295, 5476 ] } ], "ref_ids": [] }, { "id": 5524, "type": "theorem", "label": "morphisms-lemma-characterize-universally-bounded", "categories": [ "morphisms" ], "title": "morphisms-lemma-characterize-universally-bounded", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $n \\geq 0$.", "The following are equivalent:", "\\begin{enumerate}", "\\item the integer $n$ bounds the degrees of the fibres of $f$, and", "\\item for every morphism $\\Spec(k) \\to Y$, where $k$ is a field,", "the fibre product $X_k = \\Spec(k) \\times_Y X$ is finite over $k$", "of degree $\\leq n$.", "\\end{enumerate}", "In this case the fibres of $f$ are universally bounded and the schemes", "$X_k$ have at most $n$ points. More precisely, if", "$X_k = \\{x_1, \\ldots, x_t\\}$, then we have", "$$", "n \\geq \\sum\\nolimits_{i = 1, \\ldots, t} [\\kappa(x_i) : k]", "$$" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is trivial. The other implication", "holds because if the image of $\\Spec(k) \\to Y$ is $y$, then", "$X_k = \\Spec(k) \\times_{\\Spec(\\kappa(y))} X_y$. By definition the", "fibres of $f$ being universally bounded means that some $n$ exists.", "Finally, suppose that $X_k = \\Spec(A)$. Then $\\dim_k A = n$.", "Hence $A$ is Artinian, all prime ideals are maximal ideals $\\mathfrak m_i$,", "and $A$ is the product of the localizations at these maximal ideals.", "See Algebra, Lemmas \\ref{algebra-lemma-finite-dimensional-algebra}", "and \\ref{algebra-lemma-artinian-finite-length}. Then $\\mathfrak m_i$", "corresponds to $x_i$, we have", "$A_{\\mathfrak m_i} = \\mathcal{O}_{X_k, x_i}$", "and hence there is a surjection", "$A \\to \\bigoplus \\kappa(\\mathfrak m_i) = \\bigoplus \\kappa(x_i)$", "which implies the inequality in the statement of the lemma", "by linear algebra." ], "refs": [ "algebra-lemma-finite-dimensional-algebra", "algebra-lemma-artinian-finite-length" ], "ref_ids": [ 642, 646 ] } ], "ref_ids": [] }, { "id": 5525, "type": "theorem", "label": "morphisms-lemma-finite-locally-free-universally-bounded", "categories": [ "morphisms" ], "title": "morphisms-lemma-finite-locally-free-universally-bounded", "contents": [ "If $f$ is a finite locally free morphism of degree $d$, then", "$d$ bounds the degree of the fibres of $f$." ], "refs": [], "proofs": [ { "contents": [ "This is true because any base change of $f$ is", "finite locally free of degree $d$", "(Lemma \\ref{lemma-base-change-finite-locally-free})", "and hence the fibres of $f$ all have degree $d$." ], "refs": [ "morphisms-lemma-base-change-finite-locally-free" ], "ref_ids": [ 5473 ] } ], "ref_ids": [] }, { "id": 5526, "type": "theorem", "label": "morphisms-lemma-composition-universally-bounded", "categories": [ "morphisms" ], "title": "morphisms-lemma-composition-universally-bounded", "contents": [ "A composition of morphisms with universally bounded fibres", "is a morphism with universally bounded fibres. More precisely,", "assume that $n$ bounds the degrees of the fibres of $f : X \\to Y$ and", "$m$ bounds the degrees of $g : Y \\to Z$.", "Then $nm$ bounds the degrees of the fibres of $g \\circ f : X \\to Z$." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ have universally bounded fibres.", "Say that $\\deg(X_y/\\kappa(y)) \\leq n$ for all $y \\in Y$, and that", "$\\deg(Y_z/\\kappa(z)) \\leq m$ for all $z \\in Z$.", "Let $z \\in Z$ be a point. By assumption the scheme", "$Y_z$ is finite over $\\Spec(\\kappa(z))$.", "In particular, the underlying topological space of $Y_z$", "is a finite discrete set. The fibres of the morphism", "$f_z : X_z \\to Y_z$ are the fibres of $f$ at the corresponding", "points of $Y$, which are finite discrete sets by the reasoning above.", "Hence we conclude that the underlying topological space", "of $X_z$ is a finite discrete set as well. Thus $X_z$ is an affine", "scheme (this is a nice exercise; it also follows for example from", "Properties, Lemma \\ref{properties-lemma-maximal-points-affine}", "applied to the set of all points of $X_z$). Write $X_z = \\Spec(A)$,", "$Y_z = \\Spec(B)$, and $k = \\kappa(z)$. Then $k \\to B \\to A$", "and we know that (a) $\\dim_k(B) \\leq m$, and (b) for every maximal", "ideal $\\mathfrak m \\subset B$ we have", "$\\dim_{\\kappa(\\mathfrak m)}(A/\\mathfrak mA) \\leq n$.", "We claim this implies that $\\dim_k(A) \\leq nm$.", "Note that $B$ is the product of its localizations $B_{\\mathfrak m}$, for", "example because $Y_z$ is a disjoint union of $1$-point schemes, or by", "Algebra, Lemmas \\ref{algebra-lemma-finite-dimensional-algebra} and", "\\ref{algebra-lemma-artinian-finite-length}.", "So we see that", "$\\dim_k(B) = \\sum_{\\mathfrak m} \\dim_k(B_{\\mathfrak m})$ and", "$\\dim_k(A) = \\sum_{\\mathfrak m} \\dim_k(A_{\\mathfrak m})$ where", "in both cases $\\mathfrak m$ runs over the maximal ideals of", "$B$ (not of $A$). By the above, and Nakayama's Lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK})", "we see that each $A_{\\mathfrak m}$ is a quotient of", "$B_{\\mathfrak m}^{\\oplus n}$ as a $B_{\\mathfrak m}$-module. Hence", "$\\dim_k(A_{\\mathfrak m}) \\leq n \\dim_k(B_{\\mathfrak m})$. Putting", "everything together we see that", "$$", "\\dim_k(A) = \\sum\\nolimits_{\\mathfrak m} \\dim_ta\t(A_{\\mathfrak m})", "\\leq \\sum\\nolimits_{\\mathfrak m} n \\dim_k(B_{\\mathfrak m})", "= n \\dim_k(B) \\leq nm", "$$", "as desired." ], "refs": [ "properties-lemma-maximal-points-affine", "algebra-lemma-finite-dimensional-algebra", "algebra-lemma-artinian-finite-length", "algebra-lemma-NAK" ], "ref_ids": [ 3059, 642, 646, 401 ] } ], "ref_ids": [] }, { "id": 5527, "type": "theorem", "label": "morphisms-lemma-base-change-universally-bounded", "categories": [ "morphisms" ], "title": "morphisms-lemma-base-change-universally-bounded", "contents": [ "A base change of a morphism with universally bounded fibres is", "a morphism with universally bounded fibres. More precisely, if", "$n$ bounds the degrees of the fibres of $f : X \\to Y$ and $Y' \\to Y$", "is any morphism, then the degrees of the fibres of the base change", "$f' : Y' \\times_Y X \\to Y'$ is also bounded by $n$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the result of", "Lemma \\ref{lemma-characterize-universally-bounded}." ], "refs": [ "morphisms-lemma-characterize-universally-bounded" ], "ref_ids": [ 5524 ] } ], "ref_ids": [] }, { "id": 5528, "type": "theorem", "label": "morphisms-lemma-descent-universally-bounded", "categories": [ "morphisms" ], "title": "morphisms-lemma-descent-universally-bounded", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $Y' \\to Y$ be a morphism of schemes, and let", "$f' : X' = X_{Y'} \\to Y'$ be the base change of $f$.", "If $Y' \\to Y$ is surjective and $f'$ has universally bounded fibres,", "then $f$ has universally bounded fibres. More precisely, if $n$ bounds", "the degree of the fibres of $f'$, then also $n$ bounds the degrees", "of the fibres of $f$." ], "refs": [], "proofs": [ { "contents": [ "Let $n \\geq 0$ be an integer bounding the degrees of the fibres of $f'$.", "We claim that $n$ works for $f$ also. Namely, if $y \\in Y$ is a point,", "then choose a point $y' \\in Y'$ lying over $y$ and observe that", "$$", "X'_{y'} = \\Spec(\\kappa(y')) \\times_{\\Spec(\\kappa(y))} X_y.", "$$", "Since $X'_{y'}$ is assumed finite of degree $\\leq n$ over $\\kappa(y')$", "it follows that also $X_y$ is finite of degree $\\leq n$ over $\\kappa(y)$.", "(Some details omitted.)" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5529, "type": "theorem", "label": "morphisms-lemma-immersion-universally-bounded", "categories": [ "morphisms" ], "title": "morphisms-lemma-immersion-universally-bounded", "contents": [ "An immersion has universally bounded fibres." ], "refs": [], "proofs": [ { "contents": [ "The integer $n = 1$ works in the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5530, "type": "theorem", "label": "morphisms-lemma-etale-universally-bounded", "categories": [ "morphisms" ], "title": "morphisms-lemma-etale-universally-bounded", "contents": [ "Let $f : X \\to Y$ be an \\'etale morphism of schemes.", "Let $n \\geq 0$. The following are equivalent", "\\begin{enumerate}", "\\item the integer $n$ bounds the degrees of the fibres,", "\\item for every field $k$ and morphism $\\Spec(k) \\to Y$ the", "base change $X_k = \\Spec(k) \\times_Y X$ has at most $n$ points, and", "\\item for every $y \\in Y$ and every separable algebraic closure", "$\\kappa(y) \\subset \\kappa(y)^{sep}$ the scheme", "$X_{\\kappa(y)^{sep}}$ has at most $n$ points.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-characterize-universally-bounded}", "and the fact that the fibres $X_y$ are disjoint unions of spectra of finite", "separable field extensions of $\\kappa(y)$, see", "Lemma \\ref{lemma-etale-over-field}." ], "refs": [ "morphisms-lemma-characterize-universally-bounded", "morphisms-lemma-etale-over-field" ], "ref_ids": [ 5524, 5364 ] } ], "ref_ids": [] }, { "id": 5531, "type": "theorem", "label": "morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded", "categories": [ "morphisms" ], "title": "morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume that", "\\begin{enumerate}", "\\item $f$ is locally quasi-finite, and", "\\item $X$ is quasi-compact.", "\\end{enumerate}", "Then $f$ has universally bounded fibres." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is quasi-compact, there exists a finite affine open covering", "$X = \\bigcup_{i = 1, \\ldots, n} U_i$ and affine opens $V_i \\subset Y$,", "$i = 1, \\ldots, n$ such that $f(U_i) \\subset V_i$.", "Because of the local nature of ``local quasi-finiteness''", "(see Lemma \\ref{lemma-quasi-finite-at-point-characterize} part (4))", "we see that the morphisms $f|_{U_i} : U_i \\to V_i$ are locally", "quasi-finite morphisms of affines, hence quasi-finite, see", "Lemma \\ref{lemma-quasi-finite-locally-quasi-compact}.", "For $y \\in Y$ it is clear that $X_y = \\bigcup_{y \\in V_i} (U_i)_y$", "is an open covering. Hence it suffices to prove the lemma", "for a quasi-finite morphism of affines (namely, if $n_i$ works", "for the morphism $f|_{U_i} : U_i \\to V_i$, then $\\sum n_i$", "works for $f$).", "\\medskip\\noindent", "Assume $f : X \\to Y$ is a quasi-finite morphism of affines.", "By Lemma \\ref{lemma-quasi-finite-affine}", "we can find a diagram", "$$", "\\xymatrix{", "X \\ar[rd]_f \\ar[rr]_j & & Z \\ar[ld]^\\pi \\\\", "& Y &", "}", "$$", "with $Z$ affine, $\\pi$ finite and $j$ an open immersion. Since", "$j$ has universally bounded fibres", "(Lemma \\ref{lemma-immersion-universally-bounded})", "this reduces us to showing that $\\pi$ has universally bounded", "fibres (Lemma \\ref{lemma-composition-universally-bounded}).", "\\medskip\\noindent", "This reduces us to a morphism of the form", "$\\Spec(B) \\to \\Spec(A)$ where", "$A \\to B$ is finite. Say $B$ is generated by $x_1, \\ldots, x_n$", "over $A$ and say $P_i(T) \\in A[T]$ is a monic polynomial of degree", "$d_i$ such that $P_i(x_i) = 0$ in $B$ (a finite ring extension", "is integral, see", "Algebra, Lemma \\ref{algebra-lemma-finite-is-integral}).", "With these notations it is clear that", "$$", "\\bigoplus\\nolimits_{0 \\leq e_i < d_i, i = 1, \\ldots n} A", "\\longrightarrow", "B, \\quad", "(a_{(e_1, \\ldots, e_n)}) \\longmapsto", "\\sum a_{(e_1, \\ldots, e_n)} x_1^{e_1} \\ldots x_n^{e_n}", "$$", "is a surjective $A$-module map. Thus for any prime $\\mathfrak p \\subset A$", "this induces a surjective map $\\kappa(\\mathfrak p)$-vector spaces", "$$", "\\kappa(\\mathfrak p)^{\\oplus d_1 \\ldots d_n} \\longrightarrow", "B \\otimes_A \\kappa(\\mathfrak p)", "$$", "In other words, the integer $d_1 \\ldots d_n$ works in the definition", "of a morphism with universally bounded fibres." ], "refs": [ "morphisms-lemma-quasi-finite-at-point-characterize", "morphisms-lemma-quasi-finite-locally-quasi-compact", "morphisms-lemma-quasi-finite-affine", "morphisms-lemma-immersion-universally-bounded", "morphisms-lemma-composition-universally-bounded", "algebra-lemma-finite-is-integral" ], "ref_ids": [ 5226, 5229, 5522, 5529, 5526, 482 ] } ], "ref_ids": [] }, { "id": 5532, "type": "theorem", "label": "morphisms-lemma-universally-bounded-permanence", "categories": [ "morphisms" ], "title": "morphisms-lemma-universally-bounded-permanence", "contents": [ "Consider a commutative diagram of morphisms of schemes", "$$", "\\xymatrix{", "X \\ar[rd]_g \\ar[rr]_f & & Y \\ar[ld]^h \\\\", "& Z &", "}", "$$", "If $g$ has universally bounded fibres, and $f$ is surjective and flat,", "then also $h$ has universally bounded fibres. More precisely, if $n$", "bounds the degree of the fibres of $g$, then also $n$ bounds the", "degree of the fibres of $h$." ], "refs": [], "proofs": [ { "contents": [ "Assume $g$ has universally bounded fibres, and $f$ is surjective and flat.", "Say the degree of the fibres of $g$ is bounded by $n \\in \\mathbf{N}$.", "We claim $n$ also works for $h$.", "Let $z \\in Z$. Consider the morphism of schemes $X_z \\to Y_z$.", "It is flat and surjective. By assumption $X_z$ is a finite scheme", "over $\\kappa(z)$, in particular it is the spectrum of an", "Artinian ring (by", "Algebra, Lemma \\ref{algebra-lemma-finite-dimensional-algebra}).", "By Lemma \\ref{lemma-Artinian-affine} the morphism $X_z \\to Y_z$ is affine", "in particular quasi-compact. It follows from", "Lemma \\ref{lemma-fpqc-quotient-topology}", "that $Y_z$ is a finite discrete as this holds for $X_z$.", "Hence $Y_z$ is an affine scheme (this is a nice exercise; it also follows", "for example from", "Properties, Lemma \\ref{properties-lemma-maximal-points-affine}", "applied to the set of all points of $Y_z$).", "Write $Y_z = \\Spec(B)$ and $X_z = \\Spec(A)$.", "Then $A$ is faithfully flat over $B$, so $B \\subset A$.", "Hence $\\dim_k(B) \\leq \\dim_k(A) \\leq n$ as desired." ], "refs": [ "algebra-lemma-finite-dimensional-algebra", "morphisms-lemma-Artinian-affine", "morphisms-lemma-fpqc-quotient-topology", "properties-lemma-maximal-points-affine" ], "ref_ids": [ 642, 5181, 5269, 3059 ] } ], "ref_ids": [] }, { "id": 5533, "type": "theorem", "label": "morphisms-proposition-generic-flatness", "categories": [ "morphisms" ], "title": "morphisms-proposition-generic-flatness", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.", "Assume", "\\begin{enumerate}", "\\item $S$ is integral,", "\\item $f$ is of finite type, and", "\\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module.", "\\end{enumerate}", "Then there exists an open dense subscheme $U \\subset S$ such that", "$X_U \\to U$ is flat and of finite presentation and such that", "$\\mathcal{F}|_{X_U}$ is flat over $U$ and of finite presentation", "over $\\mathcal{O}_{X_U}$." ], "refs": [], "proofs": [ { "contents": [ "As $S$ is integral it is irreducible (see", "Properties, Lemma \\ref{properties-lemma-characterize-integral})", "and any nonempty open is dense. Hence we may replace", "$S$ by an affine open of $S$ and assume that $S = \\Spec(A)$ is", "affine. As $S$ is integral we see that $A$ is a domain.", "As $f$ is of finite type, it is quasi-compact, so $X$ is quasi-compact.", "Hence we can find a finite affine open cover", "$X = \\bigcup_{i = 1, \\ldots, n} X_i$. Write $X_i = \\Spec(B_i)$.", "Then $B_i$ is a finite type $A$-algebra, see", "Lemma \\ref{lemma-locally-finite-type-characterize}.", "Moreover there are finite type", "$B_i$-modules $M_i$ such that $\\mathcal{F}|_{X_i}$ is the", "quasi-coherent sheaf associated to the $B_i$-module $M_i$, see", "Properties, Lemma \\ref{properties-lemma-finite-type-module}.", "Next, for each pair of indices $i, j$ choose an ideal $I_{ij} \\subset B_i$", "such that $X_i \\setminus X_i \\cap X_j = V(I_{ij})$ inside", "$X_i = \\Spec(B_i)$. Set $M_{ij} = B_i/I_{ij}$ and think", "of it as a $B_i$-module. Then $V(I_{ij}) = \\text{Supp}(M_{ij})$", "and $M_{ij}$ is a finite $B_i$-module.", "\\medskip\\noindent", "At this point we apply", "Algebra, Lemma \\ref{algebra-lemma-generic-flatness}", "the pairs $(A \\to B_i, M_{ij})$ and to the", "pairs $(A \\to B_i, M_i)$. Thus we obtain", "nonzero $f_{ij}, f_i \\in A$ such that (a) $A_{f_{ij}} \\to B_{i, f_{ij}}$", "is flat and of finite presentation and $M_{ij, f_{ij}}$ is flat", "over $A_{f_{ij}}$ and of finite presentation over $B_{i, f_{ij}}$, and", "(b) $B_{i, f_i}$ is flat and of finite presentation over $A_f$ and", "$M_{i, f_i}$ is flat and of finite presentation over $B_{i, f_i}$. Set", "$f = (\\prod f_i) (\\prod f_{ij})$.", "We claim that taking $U = D(f)$ works.", "\\medskip\\noindent", "To prove our claim we may replace $A$ by $A_f$, i.e.,", "perform the base change by $U = \\Spec(A_f) \\to S$.", "After this base change we see that each of $A \\to B_i$ is", "flat and of finite presentation and that $M_i$, $M_{ij}$ are flat over $A$", "and of finite presentation over $B_i$.", "This already proves that $X \\to S$ is quasi-compact,", "locally of finite presentation, flat, and that $\\mathcal{F}$", "is flat over $S$ and of finite presentation over $\\mathcal{O}_X$, see", "Lemma \\ref{lemma-locally-finite-presentation-characterize}", "and", "Properties, Lemma \\ref{properties-lemma-finite-presentation-module}.", "Since $M_{ij}$ is of finite presentation over $B_i$ we see that", "$X_i \\cap X_j = X_i \\setminus \\text{Supp}(M_{ij})$ is a quasi-compact", "open of $X_i$, see", "Algebra, Lemma \\ref{algebra-lemma-support-finite-presentation-constructible}.", "Hence we see that $X \\to S$ is quasi-separated by", "Schemes, Lemma \\ref{schemes-lemma-characterize-quasi-separated}.", "This proves the proposition." ], "refs": [ "properties-lemma-characterize-integral", "morphisms-lemma-locally-finite-type-characterize", "properties-lemma-finite-type-module", "algebra-lemma-generic-flatness", "morphisms-lemma-locally-finite-presentation-characterize", "properties-lemma-finite-presentation-module", "algebra-lemma-support-finite-presentation-constructible", "schemes-lemma-characterize-quasi-separated" ], "ref_ids": [ 2947, 5198, 3002, 1015, 5238, 3003, 546, 7709 ] } ], "ref_ids": [] }, { "id": 5534, "type": "theorem", "label": "morphisms-proposition-generic-flatness-reduced", "categories": [ "morphisms" ], "title": "morphisms-proposition-generic-flatness-reduced", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.", "Assume", "\\begin{enumerate}", "\\item $S$ is reduced,", "\\item $f$ is of finite type, and", "\\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module.", "\\end{enumerate}", "Then there exists an open dense subscheme $U \\subset S$ such that", "$X_U \\to U$ is flat and of finite presentation and such that", "$\\mathcal{F}|_{X_U}$ is flat over $U$ and of finite presentation", "over $\\mathcal{O}_{X_U}$." ], "refs": [], "proofs": [ { "contents": [ "For the impatient reader: This proof is a repeat of the proof of", "Proposition \\ref{proposition-generic-flatness}", "using", "Algebra, Lemma \\ref{algebra-lemma-generic-flatness-reduced}", "instead of", "Algebra, Lemma \\ref{algebra-lemma-generic-flatness}.", "\\medskip\\noindent", "Since being flat and being of finite presentation is local on the", "base, see", "Lemmas \\ref{lemma-flat-module-characterize} and", "\\ref{lemma-locally-finite-presentation-characterize},", "we may work affine locally on $S$. Thus we may assume that", "$S = \\Spec(A)$, where $A$ is a reduced ring (see", "Properties, Lemma \\ref{properties-lemma-characterize-reduced}).", "As $f$ is of finite type, it is quasi-compact, so $X$ is quasi-compact.", "Hence we can find a finite affine open cover", "$X = \\bigcup_{i = 1, \\ldots, n} X_i$. Write $X_i = \\Spec(B_i)$.", "Then $B_i$ is a finite type $A$-algebra, see", "Lemma \\ref{lemma-locally-finite-type-characterize}.", "Moreover there are finite type", "$B_i$-modules $M_i$ such that $\\mathcal{F}|_{X_i}$ is the", "quasi-coherent sheaf associated to the $B_i$-module $M_i$, see", "Properties, Lemma \\ref{properties-lemma-finite-type-module}.", "Next, for each pair of indices $i, j$ choose an ideal $I_{ij} \\subset B_i$", "such that $X_i \\setminus X_i \\cap X_j = V(I_{ij})$ inside", "$X_i = \\Spec(B_i)$. Set $M_{ij} = B_i/I_{ij}$ and think", "of it as a $B_i$-module. Then $V(I_{ij}) = \\text{Supp}(M_{ij})$", "and $M_{ij}$ is a finite $B_i$-module.", "\\medskip\\noindent", "At this point we apply", "Algebra, Lemma \\ref{algebra-lemma-generic-flatness-reduced}", "the pairs $(A \\to B_i, M_{ij})$ and to the pairs $(A \\to B_i, M_i)$.", "Thus we obtain dense opens", "$U(A \\to B_i, M_{ij}) \\subset S$ and dense opens", "$U(A \\to B_i, M_i) \\subset S$ with notation as in", "Algebra, Equation (\\ref{algebra-equation-good-locus}).", "Since a finite intersection of dense opens is dense open, we see that", "$$", "U =", "\\bigcap\\nolimits_{i, j} U(A \\to B_i, M_{ij})", "\\quad\\cap\\quad", "\\bigcap\\nolimits_i U(A \\to B_i, M_i)", "$$", "is open and dense in $S$. We claim that $U$ is the desired open.", "\\medskip\\noindent", "Pick $u \\in U$. By definition of the loci $U(A \\to B_i, M_{ij})$", "and $U(A \\to B, M_i)$ there exist $f_{ij}, f_i \\in A$ such that", "(a) $u \\in D(f_i)$ and $u \\in D(f_{ij})$,", "(b) $A_{f_{ij}} \\to B_{i, f_{ij}}$ is flat and of finite presentation", "and $M_{ij, f_{ij}}$ is flat over $A_{f_{ij}}$ and of finite presentation", "over $B_{i, f_{ij}}$, and", "(c) $B_{i, f_i}$ is flat and of finite presentation over $A_f$ and", "$M_{i, f_i}$ is flat and of finite presentation over $B_{i, f_i}$. Set", "$f = (\\prod f_i) (\\prod f_{ij})$.", "Now it suffices to prove that $X \\to S$ is flat and of finite presentation", "over $D(f)$ and that $\\mathcal{F}$ restricted to $X_{D(f)}$ is", "flat over $D(f)$ and of finite presentation over the structure sheaf", "of $X_{D(f)}$.", "\\medskip\\noindent", "Hence we may replace $A$ by $A_f$, i.e.,", "perform the base change by $\\Spec(A_f) \\to S$.", "After this base change we see that each of $A \\to B_i$ is", "flat and of finite presentation and that $M_i$, $M_{ij}$ are flat over $A$", "and of finite presentation over $B_i$.", "This already proves that $X \\to S$ is quasi-compact,", "locally of finite presentation, flat, and that $\\mathcal{F}$", "is flat over $S$ and of finite presentation over $\\mathcal{O}_X$, see", "Lemma \\ref{lemma-locally-finite-presentation-characterize}", "and", "Properties, Lemma \\ref{properties-lemma-finite-presentation-module}.", "Since $M_{ij}$ is of finite presentation over $B_i$ we see that", "$X_i \\cap X_j = X_i \\setminus \\text{Supp}(M_{ij})$ is a quasi-compact", "open of $X_i$, see", "Algebra, Lemma \\ref{algebra-lemma-support-finite-presentation-constructible}.", "Hence we see that $X \\to S$ is quasi-separated by", "Schemes, Lemma \\ref{schemes-lemma-characterize-quasi-separated}.", "This proves the proposition." ], "refs": [ "morphisms-proposition-generic-flatness", "algebra-lemma-generic-flatness-reduced", "algebra-lemma-generic-flatness", "morphisms-lemma-flat-module-characterize", "morphisms-lemma-locally-finite-presentation-characterize", "properties-lemma-characterize-reduced", "morphisms-lemma-locally-finite-type-characterize", "properties-lemma-finite-type-module", "algebra-lemma-generic-flatness-reduced", "morphisms-lemma-locally-finite-presentation-characterize", "properties-lemma-finite-presentation-module", "algebra-lemma-support-finite-presentation-constructible", "schemes-lemma-characterize-quasi-separated" ], "ref_ids": [ 5533, 1019, 1015, 5259, 5238, 2945, 5198, 3002, 1019, 5238, 3003, 546, 7709 ] } ], "ref_ids": [] }, { "id": 5535, "type": "theorem", "label": "morphisms-proposition-universal-homeomorphism-equal-residue-fields", "categories": [ "morphisms" ], "title": "morphisms-proposition-universal-homeomorphism-equal-residue-fields", "contents": [ "Let $A \\subset B$ be a ring extension. The following are equivalent", "\\begin{enumerate}", "\\item $\\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism inducing", "isomorphisms on residue fields, and", "\\item every finite subset $E \\subset B$ is contained in an extension", "$$", "A[b_1, \\ldots, b_n] \\subset B", "$$", "such that $b_i^2, b_i^3 \\in A[b_1, \\ldots, b_{i - 1}]$ for $i = 1, \\ldots, n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). By transfinite induction we construct for each ordinal $\\alpha$", "an $A$-subalgebra $B_\\alpha \\subset B$ as follows. Set $B_0 = A$. If $\\alpha$", "is a limit ordinal, then we set $B_\\alpha = \\colim_{\\beta < \\alpha} B_\\beta$.", "If $\\alpha = \\beta + 1$, then either", "$B_\\beta = B$ in which case we set $B_\\alpha = B_\\beta$ or", "$B_\\beta \\not = B$, in which case we apply Lemma \\ref{lemma-square-and-cube}", "to choose a $b_\\alpha \\in B$, $b_\\alpha \\not \\in B_\\beta$ with", "$b_\\alpha^2, b_\\alpha^3 \\in B_\\beta$", "and we set $B_\\alpha = B_\\beta[b_\\alpha] \\subset B$.", "Clearly, $B = \\colim B_\\alpha$ (in fact $B = B_\\alpha$ for some", "ordinal $\\alpha$ as one sees by looking at cardinalities).", "We will prove, by transfinite induction, that (2) holds for", "$A \\to B_\\alpha$ for every ordinal $\\alpha$. It is clear for", "$\\alpha = 0$. Assume the statement holds for every $\\beta < \\alpha$", "and let $E \\subset B_\\alpha$ be a finite subset.", "If $\\alpha$ is a limit ordinal, then", "$B_\\alpha = \\bigcup_{\\beta < \\alpha} B_\\beta$ and we see", "that $E \\subset B_\\beta$ for some $\\beta < \\alpha$ which", "proves the result in this case. If $\\alpha = \\beta + 1$,", "then $B_\\alpha = B_\\beta[b_\\alpha]$. Thus any $e \\in E$", "can be written as a polynomial $e = \\sum d_{e, i}b_\\alpha^i$", "with $d_{e, i} \\in B_\\beta$. Let $D \\subset B_\\beta$", "be the set $D = \\{d_{e, i}\\} \\cup \\{b_\\alpha^2, b_\\alpha^3\\}$.", "By induction assumption", "there exists an $A$-subalgebra $A[b_1, \\ldots, b_n] \\subset B_\\beta$", "as in the statement of the lemma containing $D$.", "Then $A[b_1, \\ldots, b_n, b_\\alpha] \\subset B_\\alpha$", "is an $A$-subalgebra of $B_\\alpha$ as in the statement", "of the lemma containing $E$.", "\\medskip\\noindent", "Assume (2). Write $B = \\colim B_\\lambda$ as the colimit of its", "finite $A$-subalgebras.", "By Lemma \\ref{lemma-colimit-inherits} it suffices to show", "that $\\Spec(B_\\lambda) \\to \\Spec(A)$ is a", "universal homeomorphism inducing isomorphisms on residue fields.", "Compositions of universally closed morphisms are universally closed", "and the same thing for morphisms which induce isomorphisms on residue fields.", "Thus it suffices to show that if $A \\subset B$ and $B$ is generated", "by a single element $b$ with $b^2, b^3 \\in A$, then (1) holds.", "Such an extension is integral and hence $\\Spec(B) \\to \\Spec(A)$ is", "universally closed and surjective", "(Lemma \\ref{lemma-integral-universally-closed} and", "Algebra, Lemma \\ref{algebra-lemma-integral-overring-surjective}).", "Note that $(b^2)^3 = (b^3)^2$ in $A$.", "For any ring map $\\varphi : A \\to K$ to a field $K$ we see that there", "exists a $\\lambda \\in K$ with $\\varphi(b^2) = \\lambda^2$ and", "$\\varphi(b^3) = \\lambda^3$. Namely, $\\lambda = 0$ if $\\varphi(b^2) = 0$", "and $\\lambda = \\varphi(b^3)/\\varphi(b^2)$ if not. Thus", "$B \\otimes_A K$ is a quotient of $K[x]/(x^2 - \\lambda^2, x^3 - \\lambda^3)$.", "This ring has exactly one prime with residue field $K$.", "This implies that $\\Spec(B) \\to \\Spec(A)$ is bijective and induces", "isomorphisms on residue fields. Combined with universal closedness", "this shows (1) is true, see Lemmas \\ref{lemma-universal-homeomorphism}", "and \\ref{lemma-universally-injective}." ], "refs": [ "morphisms-lemma-square-and-cube", "morphisms-lemma-colimit-inherits", "morphisms-lemma-integral-universally-closed", "algebra-lemma-integral-overring-surjective", "morphisms-lemma-universal-homeomorphism", "morphisms-lemma-universally-injective" ], "ref_ids": [ 5461, 5458, 5441, 495, 5454, 5167 ] } ], "ref_ids": [] }, { "id": 5536, "type": "theorem", "label": "morphisms-proposition-universal-homeomorphism", "categories": [ "morphisms" ], "title": "morphisms-proposition-universal-homeomorphism", "contents": [ "Let $A \\subset B$ be a ring extension. The following are equivalent", "\\begin{enumerate}", "\\item $\\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism, and", "\\item every finite subset $E \\subset B$ is contained in an extension", "$$", "A[b_1, \\ldots, b_n] \\subset B", "$$", "such that for $i = 1, \\ldots, n$ we have", "\\begin{enumerate}", "\\item $b_i^2, b_i^3 \\in A[b_1, \\ldots, b_{i - 1}]$, or", "\\item there exists a prime number $p$ with", "$pb_i, b_i^p \\in A[b_1, \\ldots, b_{i - 1}]$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Proposition \\ref{proposition-universal-homeomorphism-equal-residue-fields}", "except for the following changes:", "\\begin{enumerate}", "\\item Use Lemma \\ref{lemma-pth-power-and-multiple} instead of", "Lemma \\ref{lemma-square-and-cube} which means that for each successor", "ordinal $\\alpha = \\beta + 1$ we either have", "$b_\\alpha^2, b_\\alpha^3 \\in B_\\beta$ or we have a prime $p$ and", "$pb_\\alpha, b_\\alpha^p \\in B_\\beta$.", "\\item If $\\alpha$ is a successor ordinal, then take", "$D = \\{d_{e, i}\\} \\cup \\{b_\\alpha^2, b_\\alpha^3\\}$ or", "take $D = \\{d_{e, i}\\} \\cup \\{pb_\\alpha, b_\\alpha^p\\}$ depending", "on which case $\\alpha$ falls into.", "\\item In the proof of (2) $\\Rightarrow$ (1) we also need to consider", "the case where $B$ is generated over $A$ by a single element $b$", "with $pb, b^p \\in B$ for some prime number $p$. Here $A \\subset B$", "defines a universal homeomorphism for example by", "Algebra, Lemma \\ref{algebra-lemma-p-ring-map}.", "\\end{enumerate}", "This finishes the proof." ], "refs": [ "morphisms-proposition-universal-homeomorphism-equal-residue-fields", "morphisms-lemma-pth-power-and-multiple", "morphisms-lemma-square-and-cube", "algebra-lemma-p-ring-map" ], "ref_ids": [ 5535, 5462, 5461, 582 ] } ], "ref_ids": [] }, { "id": 5605, "type": "theorem", "label": "smoothing-theorem-popescu", "categories": [ "smoothing" ], "title": "smoothing-theorem-popescu", "contents": [ "Any regular homomorphism of Noetherian rings is a filtered colimit", "of smooth ring maps." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-reduce-to-field}", "it suffices to prove this for $k \\to \\Lambda$", "where $\\Lambda$ is Noetherian and geometrically regular over $k$.", "Let $k \\to A \\to \\Lambda$ be a factorization with $A$ a finite type", "$k$-algebra. It suffices to construct a factorization", "$A \\to B \\to \\Lambda$ with $B$ of finite type such that", "$\\mathfrak h_B = \\Lambda$, see Lemma \\ref{lemma-final-solve}.", "Hence we may perform Noetherian induction on the ideal $\\mathfrak h_A$.", "Pick a prime $\\mathfrak q \\supset \\mathfrak h_A$ such that", "$\\mathfrak q$ is minimal over $\\mathfrak h_A$.", "It now suffices to resolve $k \\to A \\to \\Lambda \\supset \\mathfrak q$", "(as defined in the text following Situation \\ref{situation-local}).", "If the characteristic of $k$ is zero, this follows from", "Lemma \\ref{lemma-resolve-special}.", "If the characteristic of $k$ is $p > 0$, this follows from", "Lemma \\ref{lemma-resolve-general}." ], "refs": [ "smoothing-lemma-reduce-to-field", "smoothing-lemma-final-solve", "smoothing-lemma-resolve-special", "smoothing-lemma-resolve-general" ], "ref_ids": [ 5632, 5612, 5638, 5642 ] } ], "ref_ids": [] }, { "id": 5606, "type": "theorem", "label": "smoothing-theorem-approximation-property", "categories": [ "smoothing" ], "title": "smoothing-theorem-approximation-property", "contents": [ "Let $R$ be a Noetherian local ring. Let", "$f_1, \\ldots, f_m \\in R[x_1, \\ldots, x_n]$.", "Suppose that $(a_1, \\ldots, a_n) \\in (R^\\wedge)^n$ is a solution", "in $R^\\wedge$. If $R$ is a henselian G-ring, then for every integer", "$N$ there exists a solution $(b_1, \\ldots, b_n) \\in R^n$ in $R$ such that", "$a_i - b_i \\in \\mathfrak m^NR^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "Let $c_i \\in R$ be an element such that $a_i - c_i \\in \\mathfrak m^N$.", "Choose generators $\\mathfrak m^N = (d_1, \\ldots, d_M)$.", "Write $a_i = c_i + \\sum a_{i, l} d_l$.", "Consider the polynomial ring $R[x_{i, l}]$ and the elements", "$$", "g_j = f_j(c_1 + \\sum x_{1, l} d_l , \\ldots, c_n + \\sum x_{n, l} d_{n, l})", "\\in R[x_{i, l}]", "$$", "The system of equations $g_j = 0$ has the solution $(a_{i, l})$.", "Suppose that we can show that $g_j$ as a solution $(b_{i, l})$ in $R$.", "Then it follows that $b_i = c_i + \\sum b_{i, l}d_l$ is a solution", "of $f_j = 0$ which is congruent to $a_i$ modulo $\\mathfrak m^N$.", "Thus it suffices to show that solvability over $R^\\wedge$ implies", "solvability over $R$.", "\\medskip\\noindent", "Let $A \\subset R^\\wedge$ be the $R$-subalgebra generated by", "$a_1, \\ldots, a_n$. Since we've assumed $R$ is a G-ring, i.e.,", "that $R \\to R^\\wedge$ is regular, we see that", "there exists a factorization", "$$", "A \\to B \\to R^\\wedge", "$$", "with $B$ smooth over $R$, see Theorem \\ref{theorem-popescu}.", "Denote $\\kappa = R/\\mathfrak m$ the residue field. It is also", "the residue field of $R^\\wedge$, so we get a commutative diagram", "$$", "\\xymatrix{", "B \\ar[rd] \\ar@{..>}[r] & R' \\ar@{..>}[d] \\\\", "R \\ar[r] \\ar[u] & \\kappa", "}", "$$", "Since the vertical arrow is smooth,", "More on Algebra, Lemma \\ref{more-algebra-lemma-lift-section-smooth-morphism}", "implies that there exists an \\'etale ring map $R \\to R'$", "which induces an isomorphism $R/\\mathfrak m \\to R'/\\mathfrak mR'$", "and an $R$-algebra map $B \\to R'$ making the diagram above commute.", "Since $R$ is henselian we see that $R \\to R'$ has a section, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-henselian}.", "Let $b_i \\in R$ be the image of $a_i$ under the ring maps", "$A \\to B \\to R' \\to R$. Since all of these maps are $R$-algebra", "maps, we see that $(b_1, \\ldots, b_n)$ is a solution in $R$." ], "refs": [ "smoothing-theorem-popescu", "algebra-lemma-characterize-henselian" ], "ref_ids": [ 5605, 1276 ] } ], "ref_ids": [] }, { "id": 5607, "type": "theorem", "label": "smoothing-theorem-approximation-property-variant", "categories": [ "smoothing" ], "title": "smoothing-theorem-approximation-property-variant", "contents": [ "Let $R$ be a Noetherian local ring. Let", "$f_1, \\ldots, f_m \\in R[x_1, \\ldots, x_n]$.", "Suppose that $(a_1, \\ldots, a_n) \\in (R^\\wedge)^n$ is a solution.", "If $R$ is a G-ring, then for every integer $N$ there exist", "\\begin{enumerate}", "\\item an \\'etale ring map $R \\to R'$,", "\\item a maximal ideal $\\mathfrak m' \\subset R'$ lying over $\\mathfrak m$", "\\item a solution $(b_1, \\ldots, b_n) \\in (R')^n$ in $R'$", "\\end{enumerate}", "such that $\\kappa(\\mathfrak m) = \\kappa(\\mathfrak m')$ and", "$a_i - b_i \\in (\\mathfrak m')^NR^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "We could deduce this theorem from Theorem \\ref{theorem-approximation-property}", "using that the henselization $R^h$ is a G-ring by", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-G-ring}", "and writing $R^h$ as a directed colimit of \\'etale extension $R'$.", "Instead we prove this by redoing the proof of the previous theorem", "in this case.", "\\medskip\\noindent", "Let $c_i \\in R$ be an element such that $a_i - c_i \\in \\mathfrak m^N$.", "Choose generators $\\mathfrak m^N = (d_1, \\ldots, d_M)$.", "Write $a_i = c_i + \\sum a_{i, l} d_l$.", "Consider the polynomial ring $R[x_{i, l}]$ and the elements", "$$", "g_j = f_j(c_1 + \\sum x_{1, l} d_l , \\ldots, c_n + \\sum x_{n, l} d_{n, l})", "\\in R[x_{i, l}]", "$$", "The system of equations $g_j = 0$ has the solution $(a_{i, l})$.", "Suppose that we can show that $g_j$ as a solution $(b_{i, l})$ in $R'$", "for some \\'etale ring map $R \\to R'$ endowed with a maximal ideal", "$\\mathfrak m'$ such that $\\kappa(\\mathfrak m) = \\kappa(\\mathfrak m')$.", "Then it follows that $b_i = c_i + \\sum b_{i, l}d_l$ is a solution", "of $f_j = 0$ which is congruent to $a_i$ modulo $(\\mathfrak m')^N$.", "Thus it suffices to show that solvability over $R^\\wedge$ implies", "solvability over some \\'etale ring extension which induces a trivial", "residue field extension at some prime over $\\mathfrak m$.", "\\medskip\\noindent", "Let $A \\subset R^\\wedge$ be the $R$-subalgebra generated by", "$a_1, \\ldots, a_n$. Since we've assumed $R$ is a G-ring, i.e.,", "that $R \\to R^\\wedge$ is regular, we see that", "there exists a factorization", "$$", "A \\to B \\to R^\\wedge", "$$", "with $B$ smooth over $R$, see Theorem \\ref{theorem-popescu}.", "Denote $\\kappa = R/\\mathfrak m$ the residue field. It is also", "the residue field of $R^\\wedge$, so we get a commutative diagram", "$$", "\\xymatrix{", "B \\ar[rd] \\ar@{..>}[r] & R' \\ar@{..>}[d] \\\\", "R \\ar[r] \\ar[u] & \\kappa", "}", "$$", "Since the vertical arrow is smooth,", "More on Algebra, Lemma \\ref{more-algebra-lemma-lift-section-smooth-morphism}", "implies that there exists an \\'etale ring map $R \\to R'$", "which induces an isomorphism $R/\\mathfrak m \\to R'/\\mathfrak mR'$", "and an $R$-algebra map $B \\to R'$ making the diagram above commute.", "Let $b_i \\in R'$ be the image of $a_i$ under the ring maps", "$A \\to B \\to R'$. Since all of these maps are $R$-algebra", "maps, we see that $(b_1, \\ldots, b_n)$ is a solution in $R'$." ], "refs": [ "smoothing-theorem-approximation-property", "more-algebra-lemma-henselization-G-ring", "smoothing-theorem-popescu" ], "ref_ids": [ 5606, 10089, 5605 ] } ], "ref_ids": [] }, { "id": 5608, "type": "theorem", "label": "smoothing-lemma-find-strictly-standard", "categories": [ "smoothing" ], "title": "smoothing-lemma-find-strictly-standard", "contents": [ "Let $R$ be a ring. Let $A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$.", "Let $\\mathfrak q \\subset A$ be a prime ideal. Assume $R \\to A$ is smooth", "at $\\mathfrak q$. Then there exists an $a \\in A$, $a \\not \\in \\mathfrak q$,", "an integer $c$, $0 \\leq c \\leq \\min(n, m)$, subsets", "$U \\subset \\{1, \\ldots, n\\}$, $V \\subset \\{1, \\ldots, m\\}$", "of cardinality $c$ such that", "$$", "a = a' \\det(\\partial f_j/\\partial x_i)_{j \\in V, i \\in U}", "$$", "for some $a' \\in A$ and", "$$", "a f_\\ell \\in (f_j, j \\in V) + (f_1, \\ldots, f_m)^2", "$$", "for all $\\ell \\in \\{1, \\ldots, m\\}$." ], "refs": [], "proofs": [ { "contents": [ "Set $I = (f_1, \\ldots, f_m)$ so that the naive cotangent", "complex of $A$ over $R$ is homotopy equivalent to", "$I/I^2 \\to \\bigoplus A\\text{d}x_i$, see", "Algebra, Lemma \\ref{algebra-lemma-NL-homotopy}.", "We will use the formation of the naive cotangent complex commutes with", "localization, see Algebra, Section \\ref{algebra-section-netherlander},", "especially Algebra, Lemma \\ref{algebra-lemma-localize-NL}.", "By Algebra, Definitions \\ref{algebra-definition-smooth} and", "\\ref{algebra-definition-smooth-at-prime}", "we see that $(I/I^2)_a \\to \\bigoplus A_a\\text{d}x_i$", "is a split injection for some $a \\in A$, $a \\not \\in \\mathfrak q$.", "After renumbering $x_1, \\ldots, x_n$ and $f_1, \\ldots, f_m$ we may", "assume that $f_1, \\ldots, f_c$ form a basis for", "the vector space $I/I^2 \\otimes_A \\kappa(\\mathfrak q)$ and that", "$\\text{d}x_{c + 1}, \\ldots, \\text{d}x_n$ map to a basis of", "$\\Omega_{A/R} \\otimes_A \\kappa(\\mathfrak q)$. Hence after replacing $a$", "by $aa'$ for some $a' \\in A$, $a' \\not \\in \\mathfrak q$ we may assume", "$f_1, \\ldots, f_c$ form a basis for $(I/I^2)_a$ and that", "$\\text{d}x_{c + 1}, \\ldots, \\text{d}x_n$ map to a basis of", "$(\\Omega_{A/R})_a$. In this situation $a^N$ for some large integer", "$N$ satisfies the conditions of the lemma (with $U = V = \\{1, \\ldots, c\\}$)." ], "refs": [ "algebra-lemma-NL-homotopy", "algebra-lemma-localize-NL", "algebra-definition-smooth", "algebra-definition-smooth-at-prime" ], "ref_ids": [ 1151, 1161, 1534, 1536 ] } ], "ref_ids": [] }, { "id": 5609, "type": "theorem", "label": "smoothing-lemma-parse-equation-strictly-standard-one", "categories": [ "smoothing" ], "title": "smoothing-lemma-parse-equation-strictly-standard-one", "contents": [ "Let $R$ be a ring. Let $A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$", "and write $I = (f_1, \\ldots, f_m)$. Let $a \\in A$. Then", "(\\ref{equation-strictly-standard-one}) implies", "there exists an $A$-linear map", "$\\psi : \\bigoplus\\nolimits_{i = 1, \\ldots, n} A \\text{d}x_i \\to A^{\\oplus c}$", "such that the composition", "$$", "A^{\\oplus c} \\xrightarrow{(f_1, \\ldots, f_c)}", "I/I^2 \\xrightarrow{f \\mapsto \\text{d}f}", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} A \\text{d}x_i", "\\xrightarrow{\\psi}", "A^{\\oplus c}", "$$", "is multiplication by $a$. Conversely, if such a $\\psi$ exists, then", "$a^c$ satisfies (\\ref{equation-strictly-standard-one})." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Algebra, Lemma \\ref{algebra-lemma-matrix-left-inverse}." ], "refs": [ "algebra-lemma-matrix-left-inverse" ], "ref_ids": [ 381 ] } ], "ref_ids": [] }, { "id": 5610, "type": "theorem", "label": "smoothing-lemma-elkik", "categories": [ "smoothing" ], "title": "smoothing-lemma-elkik", "contents": [ "Let $R \\to A$ be a ring map of finite presentation.", "The singular ideal $H_{A/R}$ is the radical of the ideal", "generated by strictly standard elements in $A$ over $R$", "and also the radical of the ideal generated by elementary", "standard elements in $A$ over $R$." ], "refs": [], "proofs": [ { "contents": [ "Assume $a$ is strictly standard in $A$ over $R$. We claim that", "$A_a$ is smooth over $R$, which proves that $a \\in H_{A/R}$. Namely,", "let $A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$, $c$, and $a' \\in A$", "be as in Definition \\ref{definition-strictly-standard}.", "Write $I = (f_1, \\ldots, f_m)$ so that the naive cotangent", "complex of $A$ over $R$ is given by $I/I^2 \\to \\bigoplus A\\text{d}x_i$.", "Assumption (\\ref{equation-strictly-standard-two})", "implies that $(I/I^2)_a$ is generated by the classes of $f_1, \\ldots, f_c$.", "Assumption (\\ref{equation-strictly-standard-one}) implies", "that the differential $(I/I^2)_a \\to \\bigoplus A_a\\text{d}x_i$", "has a left inverse, see", "Lemma \\ref{lemma-parse-equation-strictly-standard-one}.", "Hence $R \\to A_a$ is smooth by definition and", "Algebra, Lemma \\ref{algebra-lemma-localize-NL}.", "\\medskip\\noindent", "Let $H_e, H_s \\subset A$ be the radical of the ideal generated by", "elementary, resp.\\ strictly standard elements of $A$ over $R$.", "By definition and what we just proved we have", "$H_e \\subset H_s \\subset H_{A/R}$. The inclusion $H_{A/R} \\subset H_e$", "follows from Lemma \\ref{lemma-find-strictly-standard}." ], "refs": [ "smoothing-definition-strictly-standard", "algebra-lemma-localize-NL", "smoothing-lemma-find-strictly-standard" ], "ref_ids": [ 5648, 1161, 5608 ] } ], "ref_ids": [] }, { "id": 5611, "type": "theorem", "label": "smoothing-lemma-strictly-standard-base-change", "categories": [ "smoothing" ], "title": "smoothing-lemma-strictly-standard-base-change", "contents": [ "Let $R \\to A$ be a ring map of finite presentation.", "Let $R \\to R'$ be a ring map. If $a \\in A$ is elementary,", "resp.\\ strictly standard in $A$ over $R$, then $a \\otimes 1$", "is elementary, resp.\\ strictly standard in $A \\otimes_R R'$ over $R'$." ], "refs": [], "proofs": [ { "contents": [ "If $A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$ is a presentation", "of $A$ over $R$, then", "$A \\otimes_R R' = R'[x_1, \\ldots, x_n]/(f'_1, \\ldots, f'_m)$", "is a presentation of $A \\otimes_R R'$ over $R'$. Here $f'_j$ is", "the image of $f_j$ in $R'[x_1, \\ldots, x_n]$.", "Hence the result follows from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5612, "type": "theorem", "label": "smoothing-lemma-final-solve", "categories": [ "smoothing" ], "title": "smoothing-lemma-final-solve", "contents": [ "Let $R \\to A \\to \\Lambda$ be ring maps with $A$ of finite presentation", "over $R$. Assume that $H_{A/R} \\Lambda = \\Lambda$. Then there exists", "a factorization $A \\to B \\to \\Lambda$ with $B$ smooth over $R$." ], "refs": [], "proofs": [ { "contents": [ "Choose $f_1, \\ldots, f_r \\in H_{A/R}$ and", "$\\lambda_1, \\ldots, \\lambda_r \\in \\Lambda$ such that", "$\\sum f_i\\lambda_i = 1$ in $\\Lambda$. Set", "$B = A[x_1, \\ldots, x_r]/(f_1x_1 + \\ldots + f_rx_r - 1)$", "and define $B \\to \\Lambda$ by mapping $x_i$ to $\\lambda_i$.", "Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5613, "type": "theorem", "label": "smoothing-lemma-improve-presentation", "categories": [ "smoothing" ], "title": "smoothing-lemma-improve-presentation", "contents": [ "Let $R$ be a ring and let $A$ be a finitely presented $R$-algebra.", "There exists finite type $R$-algebra map $A \\to C$ which has a", "retraction with the following two properties", "\\begin{enumerate}", "\\item for each $a \\in A$ such that $R \\to A_a$ is a local complete", "intersection (More on Algebra, Definition", "\\ref{more-algebra-definition-local-complete-intersection})", "the ring $C_a$ is smooth over $A_a$ and has a presentation", "$C_a = R[y_1, \\ldots, y_m]/J$ such that $J/J^2$ is free over $C_a$, and", "\\item for each $a \\in A$ such that $A_a$ is smooth over $R$ the", "module $\\Omega_{C_a/R}$ is free over $C_a$.", "\\end{enumerate}" ], "refs": [ "more-algebra-definition-local-complete-intersection" ], "proofs": [ { "contents": [ "Choose a presentation $A = R[x_1, \\ldots, x_n]/I$ and write", "$I = (f_1, \\ldots, f_m)$. Define the $A$-module $K$ by the short exact sequence", "$$", "0 \\to K \\to A^{\\oplus m} \\to I/I^2 \\to 0", "$$", "where the $j$th basis vector $e_j$ in the middle is mapped to the class of", "$f_j$ on the right. Set", "$$", "C = \\text{Sym}^*_A(I/I^2).", "$$", "The retraction is just the projection onto the degree $0$ part of $C$.", "We have a surjection $R[x_1, \\ldots, x_n, y_1, \\ldots, y_m] \\to C$", "which maps $y_j$ to the class of $f_j$ in $I/I^2$. The kernel $J$ of this", "map is generated by the elements $f_1, \\ldots, f_m$ and by elements", "$\\sum h_j y_j$ with $h_j \\in R[x_1, \\ldots, x_n]$ such that", "$\\sum h_j e_j$ defines an element of $K$. By", "Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL}", "applied to $R \\to A \\to C$ and the presentations above and", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-cotangent-complex-symmetric-algebra}", "there is a short exact sequence", "\\begin{equation}", "\\label{equation-sequence}", "I/I^2 \\otimes_A C \\to J/J^2 \\to K \\otimes_A C \\to 0", "\\end{equation}", "of $C$-modules. Let $h \\in R[x_1, \\ldots, x_n]$ be an element", "with image $a \\in A$. We will use as presentations for the localized rings", "$$", "A_a = R[x_0, x_1, \\ldots, x_n]/I'", "\\quad\\text{and}\\quad", "C_a = R[x_0, x_1, \\ldots, x_n, y_1, \\ldots, y_m]/J'", "$$", "where $I' = (hx_0 - 1, I)$ and $J' = (hx_0 - 1, J)$. Hence", "$I'/(I')^2 = A_a \\oplus (I/I^2)_a$ as $A_a$-modules and", "$J'/(J')^2 = C_a \\oplus (J/J^2)_a$ as $C_a$-modules.", "Thus we obtain", "\\begin{equation}", "\\label{equation-sequence-localized}", "C_a \\oplus I/I^2 \\otimes_A C_a \\to", "C_a \\oplus (J/J^2)_a \\to", "K \\otimes_A C_a \\to 0", "\\end{equation}", "as the sequence of", "Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL}", "corresponding to $R \\to A_a \\to C_a$ and the presentations above.", "\\medskip\\noindent", "Next, assume that $a \\in A$ is such that $A_a$ is a local complete", "intersection over $R$. Then $(I/I^2)_a$ is finite projective over $A_a$, see", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-quasi-regular-ideal-finite-projective}.", "Hence we see $K_a \\oplus (I/I^2)_a \\cong A_a^{\\oplus m}$ is free.", "In particular $K_a$ is finite projective too.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-transitive-lci-at-end}", "the sequence (\\ref{equation-sequence-localized}) is exact on the left.", "Hence", "$$", "J'/(J')^2 \\cong", "C_a \\oplus I/I^2 \\otimes_A C_a \\oplus K \\otimes_A C_a \\cong", "C_a^{\\oplus m + 1}", "$$", "This proves (1). Finally, suppose that in addition $A_a$ is smooth over", "$R$. Then the same presentation shows that $\\Omega_{C_a/R}$", "is the cokernel of the map", "$$", "J'/(J')^2 \\longrightarrow", "\\bigoplus\\nolimits_i C_a\\text{d}x_i \\oplus \\bigoplus\\nolimits_j C_a\\text{d}y_j", "$$", "The summand $C_a$ of $J'/(J')^2$ in the decomposition above", "corresponds to $hx_0 - 1$ and hence maps", "isomorphically to the summand $C_a\\text{d}x_0$. The summand", "$I/I^2 \\otimes_A C_a$ of $J'/(J')^2$ maps injectively to", "$\\bigoplus_{i = 1, \\ldots, n} C_a\\text{d}x_i$", "with quotient $\\Omega_{A_a/R} \\otimes_{A_a} C_a$. The summand", "$K \\otimes_A C_a$ maps injectively to", "$\\bigoplus_{j \\geq 1} C_a\\text{d}y_j$ with quotient isomorphic to", "$I/I^2 \\otimes_A C_a$. Thus the cokernel of the last displayed", "map is the module", "$I/I^2 \\otimes_A C_a \\oplus \\Omega_{A_a/R} \\otimes_{A_a} C_a$.", "Since $(I/I^2)_a \\oplus \\Omega_{A_a/R}$ is", "free (from the definition of smooth ring maps) we see that (2) holds." ], "refs": [ "algebra-lemma-exact-sequence-NL", "more-algebra-lemma-cotangent-complex-symmetric-algebra", "algebra-lemma-exact-sequence-NL", "more-algebra-lemma-quasi-regular-ideal-finite-projective", "more-algebra-lemma-transitive-lci-at-end" ], "ref_ids": [ 1153, 9850, 1153, 9996, 10003 ] } ], "ref_ids": [ 10609 ] }, { "id": 5614, "type": "theorem", "label": "smoothing-lemma-syntomic-complete-intersection", "categories": [ "smoothing" ], "title": "smoothing-lemma-syntomic-complete-intersection", "contents": [ "Let $R \\to A$ be a syntomic ring map. Then there exists a smooth $R$-algebra", "map $A \\to C$ with a retraction such that $C$ is a global relative complete", "intersection over $R$, i.e.,", "$$", "C \\cong R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)", "$$", "flat over $R$ and all fibres of dimension $n - c$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-improve-presentation} to get $A \\to C$.", "By Algebra, Lemma \\ref{algebra-lemma-huber}", "we can write $C = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "with $f_i$ mapping to a basis of $J/J^2$.", "The ring map $R \\to C$ is syntomic (hence flat)", "as it is a composition of a syntomic and a smooth ring map.", "The dimension of the fibres is $n - c$ by", "Algebra, Lemma \\ref{algebra-lemma-lci}", "(the fibres are local complete intersections, so the lemma applies)." ], "refs": [ "smoothing-lemma-improve-presentation", "algebra-lemma-huber", "algebra-lemma-lci" ], "ref_ids": [ 5613, 1178, 1167 ] } ], "ref_ids": [] }, { "id": 5615, "type": "theorem", "label": "smoothing-lemma-smooth-standard-smooth", "categories": [ "smoothing" ], "title": "smoothing-lemma-smooth-standard-smooth", "contents": [ "Let $R \\to A$ be a smooth ring map. Then there exists a smooth $R$-algebra", "map $A \\to B$ with a retraction such that $B$ is standard smooth over", "$R$, i.e.,", "$$", "B \\cong R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)", "$$", "and $\\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}$", "is invertible in $B$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-syntomic-complete-intersection}", "to get a smooth $R$-algebra map $A \\to C$ with a retraction such that", "$C = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "is a relative global complete intersection over $R$. As $C$ is smooth", "over $R$ we have a short exact sequence", "$$", "0 \\to", "\\bigoplus\\nolimits_{j = 1, \\ldots, c} C f_j \\to", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} C\\text{d}x_i \\to", "\\Omega_{C/R} \\to 0", "$$", "Since $\\Omega_{C/R}$ is a projective $C$-module this sequence is split.", "Choose a left inverse $t$ to the first map. Say", "$t(\\text{d}x_i) = \\sum c_{ij} f_j$", "so that $\\sum_i \\frac{\\partial f_j}{\\partial x_i} c_{i\\ell} = \\delta_{j\\ell}$", "(Kronecker delta). Let", "$$", "B' = C[y_1, \\ldots, y_c] =", "R[x_1, \\ldots, x_n, y_1, \\ldots, y_c]/(f_1, \\ldots, f_c)", "$$", "The $R$-algebra map $C \\to B'$ has a retraction given by mapping $y_j$ to zero.", "We claim that the map", "$$", "R[z_1, \\ldots, z_n] \\longrightarrow B',\\quad", "z_i \\longmapsto x_i - \\sum\\nolimits_j c_{ij} y_j", "$$", "is \\'etale at every point in the image of $\\Spec(C) \\to \\Spec(B')$.", "In $\\Omega_{B'/R[z_1, \\ldots, z_n]}$ we have", "$$", "0 =", "\\text{d}f_j - \\sum\\nolimits_i \\frac{\\partial f_j}{\\partial x_i} \\text{d}z_i", "\\equiv", "\\sum\\nolimits_{i, \\ell}", "\\frac{\\partial f_j}{\\partial x_i} c_{i\\ell} \\text{d}y_\\ell", "\\equiv", "\\text{d}y_j \\bmod (y_1, \\ldots, y_c)\\Omega_{B'/R[z_1, \\ldots, z_n]}", "$$", "Since $0 = \\text{d}z_i = \\text{d}x_i$ modulo", "$\\sum B'\\text{d}y_j + (y_1, \\ldots, y_c)\\Omega_{B'/R[z_1, \\ldots, z_n]}$", "we conclude that", "$$", "\\Omega_{B'/R[z_1, \\ldots, z_n]}/", "(y_1, \\ldots, y_c)\\Omega_{B'/R[z_1, \\ldots, z_n]} = 0.", "$$", "As $\\Omega_{B'/R[z_1, \\ldots, z_n]}$ is a finite $B'$-module", "by Nakayama's lemma there exists a $g \\in 1 + (y_1, \\ldots, y_c)$", "that $(\\Omega_{B'/R[z_1, \\ldots, z_n]})_g = 0$. This proves that", "$R[z_1, \\ldots, z_n] \\to B'_g$ is unramified, see", "Algebra, Definition \\ref{algebra-definition-unramified}.", "For any ring map $R \\to k$ where $k$ is a field we obtain an", "unramified ring map $k[z_1, \\ldots, z_n] \\to (B'_g) \\otimes_R k$", "between smooth $k$-algebras of dimension $n$. It follows that", "$k[z_1, \\ldots, z_n] \\to (B'_g) \\otimes_R k$ is flat by", "Algebra, Lemmas \\ref{algebra-lemma-CM-over-regular-flat} and", "\\ref{algebra-lemma-characterize-smooth-kbar}. By the crit\\`ere", "de platitude par fibre", "(Algebra, Lemma \\ref{algebra-lemma-criterion-flatness-fibre})", "we conclude that $R[z_1, \\ldots, z_n] \\to B'_g$ is flat.", "Finally, Algebra, Lemma \\ref{algebra-lemma-characterize-etale}", "implies that $R[z_1, \\ldots, z_n] \\to B'_g$ is \\'etale.", "Set $B = B'_g$. Note that $C \\to B$ is smooth and has a retraction,", "so also $A \\to B$ is smooth and has a retraction.", "Moreover, $R[z_1, \\ldots, z_n] \\to B$ is \\'etale.", "By Algebra, Lemma \\ref{algebra-lemma-etale-standard-smooth}", "we can write", "$$", "B = R[z_1, \\ldots, z_n, w_1, \\ldots, w_c]/(g_1, \\ldots, g_c)", "$$", "with $\\det(\\partial g_j/\\partial w_i)$ invertible in $B$.", "This proves the lemma." ], "refs": [ "smoothing-lemma-syntomic-complete-intersection", "algebra-definition-unramified", "algebra-lemma-CM-over-regular-flat", "algebra-lemma-characterize-smooth-kbar", "algebra-lemma-criterion-flatness-fibre", "algebra-lemma-characterize-etale", "algebra-lemma-etale-standard-smooth" ], "ref_ids": [ 5614, 1544, 1107, 1222, 1114, 1235, 1230 ] } ], "ref_ids": [] }, { "id": 5616, "type": "theorem", "label": "smoothing-lemma-colimit-standard-smooth", "categories": [ "smoothing" ], "title": "smoothing-lemma-colimit-standard-smooth", "contents": [ "Let $R \\to \\Lambda$ be a ring map. If $\\Lambda$ is a filtered colimit of", "smooth $R$-algebras, then $\\Lambda$ is a filtered colimit of standard", "smooth $R$-algebras." ], "refs": [], "proofs": [ { "contents": [ "Let $A \\to \\Lambda$ be an $R$-algebra map with $A$", "of finite presentation over $R$. According to", "Algebra, Lemma \\ref{algebra-lemma-when-colimit}", "we have to factor this map through a standard smooth algebra, and", "we know we can factor it as $A \\to B \\to \\Lambda$ with $B$ smooth", "over $R$. Choose an $R$-algebra map $B \\to C$ with a retraction", "$C \\to B$ such that $C$ is standard smooth over $R$, see", "Lemma \\ref{lemma-smooth-standard-smooth}.", "Then the desired factorization is $A \\to B \\to C \\to B \\to \\Lambda$." ], "refs": [ "algebra-lemma-when-colimit", "smoothing-lemma-smooth-standard-smooth" ], "ref_ids": [ 1093, 5615 ] } ], "ref_ids": [] }, { "id": 5617, "type": "theorem", "label": "smoothing-lemma-standard-smooth-include-generators", "categories": [ "smoothing" ], "title": "smoothing-lemma-standard-smooth-include-generators", "contents": [ "Let $R \\to A$ be a standard smooth ring map.", "Let $E \\subset A$ be a finite subset of order $|E| = n$.", "Then there exists a presentation", "$A = R[x_1, \\ldots, x_{n + m}]/(f_1, \\ldots, f_c)$ with $c \\geq n$,", "with $\\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}$", "invertible in $A$, and such that $E$ is the set of congruence classes of", "$x_1, \\ldots, x_n$." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $A = R[y_1, \\ldots, y_m]/(g_1, \\ldots, g_d)$", "such that the image of", "$\\det(\\partial g_j/\\partial y_i)_{i, j = 1, \\ldots, d}$", "is invertible in $A$. Choose an enumerations $E = \\{a_1, \\ldots, a_n\\}$", "and choose $h_i \\in R[y_1, \\ldots, y_m]$ whose image in $A$ is $a_i$.", "Consider the presentation", "$$", "A = R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]/", "(x_1 - h_1, \\ldots, x_n - h_n, g_1, \\ldots, g_d)", "$$", "and set $c = n + d$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5618, "type": "theorem", "label": "smoothing-lemma-compare-standard", "categories": [ "smoothing" ], "title": "smoothing-lemma-compare-standard", "contents": [ "Let $R \\to A$ be a ring map of finite presentation.", "Let $a \\in A$. Consider the following conditions on $a$:", "\\begin{enumerate}", "\\item $A_a$ is smooth over $R$,", "\\item $A_a$ is smooth over $R$ and $\\Omega_{A_a/R}$ is stably free,", "\\item $A_a$ is smooth over $R$ and $\\Omega_{A_a/R}$ is free,", "\\item $A_a$ is standard smooth over $R$,", "\\item $a$ is strictly standard in $A$ over $R$,", "\\item $a$ is elementary standard in $A$ over $R$.", "\\end{enumerate}", "Then we have", "\\begin{enumerate}", "\\item[(a)] (4) $\\Rightarrow$ (3) $\\Rightarrow$ (2) $\\Rightarrow$ (1),", "\\item[(b)] (6) $\\Rightarrow$ (5),", "\\item[(c)] (6) $\\Rightarrow$ (4),", "\\item[(d)] (5) $\\Rightarrow$ (2),", "\\item[(e)] (2) $\\Rightarrow$ the elements $a^e$, $e \\geq e_0$ are", "strictly standard in $A$ over $R$,", "\\item[(f)] (4) $\\Rightarrow$ the elements $a^e$, $e \\geq e_0$ are", "elementary standard in $A$ over $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (a) is clear from the definitions and", "Algebra, Lemma \\ref{algebra-lemma-standard-smooth}.", "Part (b) is clear from Definition \\ref{definition-strictly-standard}.", "\\medskip\\noindent", "Proof of (c). Choose a presentation", "$A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$ such that", "(\\ref{equation-elementary-standard-one}) and", "(\\ref{equation-elementary-standard-two}) hold.", "Choose $h \\in R[x_1, \\ldots, x_n]$ mapping to $a$. Then", "$$", "A_a = R[x_0, x_1, \\ldots, x_n]/(x_0h - 1, f_1, \\ldots, f_m).", "$$", "Write $J = (x_0h - 1, f_1, \\ldots, f_m)$.", "By (\\ref{equation-elementary-standard-two}) we see that the $A_a$-module", "$J/J^2$ is generated by $x_0h - 1, f_1, \\ldots, f_c$", "over $A_a$. Hence, as in the proof of Algebra, Lemma \\ref{algebra-lemma-huber},", "we can choose a $g \\in 1 + J$ such that", "$$", "A_a = R[x_0, \\ldots, x_n, x_{n + 1}]/", "(x_0h - 1, f_1, \\ldots, f_m, gx_{n + 1} - 1).", "$$", "At this point (\\ref{equation-elementary-standard-one})", "implies that $R \\to A_a$ is standard smooth (use the coordinates", "$x_0, x_1, \\ldots, x_c, x_{n + 1}$ to take derivatives).", "\\medskip\\noindent", "Proof of (d). Choose a presentation", "$A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$ such that", "(\\ref{equation-strictly-standard-one}) and", "(\\ref{equation-strictly-standard-two}) hold.", "Write $I = (f_1, \\ldots, f_m)$.", "We already know that $A_a$ is smooth over $R$, see", "Lemma \\ref{lemma-elkik}. By", "Lemma \\ref{lemma-parse-equation-strictly-standard-one}", "we see that $(I/I^2)_a$ is free on $f_1, \\ldots, f_c$", "and maps isomorphically to a direct summand of", "$\\bigoplus A_a \\text{d}x_i$. Since", "$\\Omega_{A_a/R} = (\\Omega_{A/R})_a$", "is the cokernel of the map", "$(I/I^2)_a \\to \\bigoplus A_a \\text{d}x_i$", "we conclude that it is stably free.", "\\medskip\\noindent", "Proof of (e). Choose a presentation", "$A = R[x_1, \\ldots, x_n]/I$ with $I$ finitely generated.", "By assumption we have a short exact sequence", "$$", "0 \\to (I/I^2)_a \\to \\bigoplus\\nolimits_{i = 1, \\ldots, n} A_a\\text{d}x_i \\to", "\\Omega_{A_a/R} \\to 0", "$$", "which is split exact. Hence we see that", "$(I/I^2)_a \\oplus \\Omega_{A_a/R}$ is a free $A_a$-module.", "Since $\\Omega_{A_a/R}$ is stably free we see that $(I/I^2)_a$", "is stably free as well. Thus replacing the presentation chosen", "above by $A = R[x_1, \\ldots, x_n, x_{n + 1}, \\ldots, x_{n + r}]/J$ with", "$J = (I, x_{n + 1}, \\ldots, x_{n + r})$ for some $r$ we get that $(J/J^2)_a$", "is (finite) free. Choose $f_1, \\ldots, f_c \\in J$ which map to a basis of", "$(J/J^2)_a$. Extend this to a list of generators", "$f_1, \\ldots, f_m \\in J$. Consider the presentation", "$A = R[x_1, \\ldots, x_{n + r}]/(f_1, \\ldots, f_m)$.", "Then (\\ref{equation-strictly-standard-two}) holds for $a^e$", "for all sufficiently large $e$ by construction. Moreover, since", "$(J/J^2)_a \\to \\bigoplus\\nolimits_{i = 1, \\ldots, n + r} A_a\\text{d}x_i$", "is a split injection we can find an $A_a$-linear left inverse.", "Writing this left inverse in terms of the basis $f_1, \\ldots, f_c$", "and clearing denominators we find a linear map", "$\\psi_0 : A^{\\oplus n + r} \\to A^{\\oplus c}$ such that", "$$", "A^{\\oplus c} \\xrightarrow{(f_1, \\ldots, f_c)}", "J/J^2 \\xrightarrow{f \\mapsto \\text{d}f}", "\\bigoplus\\nolimits_{i = 1, \\ldots, n + r} A \\text{d}x_i", "\\xrightarrow{\\psi_0}", "A^{\\oplus c}", "$$", "is multiplication by $a^{e_0}$ for some $e_0 \\geq 1$. By", "Lemma \\ref{lemma-parse-equation-strictly-standard-one}", "we see (\\ref{equation-strictly-standard-one})", "holds for all $a^{ce_0}$ and hence for $a^e$ for all $e$ with $e \\geq ce_0$.", "\\medskip\\noindent", "Proof of (f). Choose a presentation", "$A_a = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ such that", "$\\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}$", "is invertible in $A_a$. We may assume that for some", "$m < n$ the classes of the elements $x_1, \\ldots, x_m$", "correspond $a_i/1$ where $a_1, \\ldots, a_m \\in A$ are generators of $A$", "over $R$, see Lemma \\ref{lemma-standard-smooth-include-generators}.", "After replacing $x_i$ by $a^Nx_i$ for $m < i \\leq n$", "we may assume the class of $x_i$ is $a_i/1 \\in A_a$ for some", "$a_i \\in A$. Consider the ring map", "$$", "\\Psi : R[x_1, \\ldots, x_n] \\longrightarrow A,\\quad", "x_i \\longmapsto a_i.", "$$", "This is a surjective ring map. By replacing $f_j$ by $a^Nf_j$ we may", "assume that $f_j \\in R[x_1, \\ldots, x_n]$ and that", "$\\Psi(f_j) = 0$ (since after all $f_j(a_1/1, \\ldots, a_n/1) = 0$", "in $A_a$). Let $J = \\Ker(\\Psi)$. Then $A = R[x_1, \\ldots, x_n]/J$", "is a presentation and $f_1, \\ldots, f_c \\in J$ are elements such that", "$(J/J^2)_a$ is freely generated by $f_1, \\ldots, f_c$ and such", "that $\\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}$", "maps to an invertible element of $A_a$. It follows that", "(\\ref{equation-elementary-standard-one}) and", "(\\ref{equation-elementary-standard-two})", "hold for $a^e$ and all large enough $e$ as desired." ], "refs": [ "algebra-lemma-standard-smooth", "smoothing-definition-strictly-standard", "algebra-lemma-huber", "smoothing-lemma-elkik", "smoothing-lemma-standard-smooth-include-generators" ], "ref_ids": [ 1193, 5648, 1178, 5610, 5617 ] } ], "ref_ids": [] }, { "id": 5619, "type": "theorem", "label": "smoothing-lemma-neron-functorial", "categories": [ "smoothing" ], "title": "smoothing-lemma-neron-functorial", "contents": [ "In Situation \\ref{situation-neron} N\\'eron's blowup is functorial", "in the following sense", "\\begin{enumerate}", "\\item if $a \\in A$, $a \\not \\in \\mathfrak p$, then N\\'eron's blowup", "of $A_a$ is $A'_a$, and", "\\item if $B \\to A$ is a surjection of flat finite type $R$-algebras", "with kernel $I$, then $A'$ is the quotient of $B'/IB'$ by its", "$\\pi$-power torsion.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Both (1) and (2) are special cases of", "Algebra, Lemma \\ref{algebra-lemma-blowup-base-change}.", "In fact, whenever we have $A_1 \\to A_2 \\to \\Lambda$ such that", "$\\mathfrak p_1 A_2 = \\mathfrak p_2$, we have that $A_2'$ is", "the quotient of $A_1' \\otimes_{A_1} A_2$ by its $\\pi$-power torsion." ], "refs": [ "algebra-lemma-blowup-base-change" ], "ref_ids": [ 753 ] } ], "ref_ids": [] }, { "id": 5620, "type": "theorem", "label": "smoothing-lemma-neron-blowup-smooth", "categories": [ "smoothing" ], "title": "smoothing-lemma-neron-blowup-smooth", "contents": [ "In Situation \\ref{situation-neron} assume that $R \\to A$ is smooth", "at $\\mathfrak p$ and that $R/\\pi R \\subset \\Lambda/\\pi \\Lambda$", "is a separable field extension. Then $R \\to A'$ is smooth at", "$\\mathfrak p'$ and there is a short exact sequence", "$$", "0 \\to", "\\Omega_{A/R} \\otimes_A A'_{\\mathfrak p'} \\to", "\\Omega_{A'/R, \\mathfrak p'} \\to", "(A'/\\pi A')_{\\mathfrak p'}^{\\oplus c} \\to 0", "$$", "where $c = \\dim((A/\\pi A)_\\mathfrak p)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-neron-functorial} we may replace $A$ by a localization", "at an element not in $\\mathfrak p$; we will use this without further mention.", "Write $\\kappa = R/\\pi R$. Since smoothness is stable under base change", "(Algebra, Lemma \\ref{algebra-lemma-base-change-smooth})", "we see that $A/\\pi A$ is smooth over $\\kappa$ at $\\mathfrak p$.", "Hence $(A/\\pi A)_\\mathfrak p$ is a regular local ring", "(Algebra, Lemma \\ref{algebra-lemma-characterize-smooth-over-field}).", "Choose $g_1, \\ldots, g_c \\in \\mathfrak p$ which map to", "a regular system of parameters in $(A/\\pi A)_\\mathfrak p$.", "Then we see that $\\mathfrak p = (\\pi, g_1, \\ldots, g_c)$", "after possibly replacing $A$ by a localization.", "Note that $\\pi, g_1, \\ldots, g_c$ is a regular sequence", "in $A_\\mathfrak p$ (first $\\pi$ is a nonzerodivisor and", "then Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}", "for the rest of the sequence).", "After replacing $A$ by a localization we may assume that", "$\\pi, g_1, \\ldots, g_c$ is a regular sequence in $A$", "(Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-neighbourhood}).", "It follows that", "$$", "A' = A[y_1, \\ldots, y_c]/(\\pi y_1 - g_1, \\ldots, \\pi y_c - g_c) =", "A[y_1, \\ldots, y_c]/I", "$$", "by More on Algebra, Lemma \\ref{more-algebra-lemma-blowup-regular-sequence}.", "In the following we will use the definition of smoothness using the", "naive cotangent complex (Algebra, Definition \\ref{algebra-definition-smooth})", "and the criterion of Algebra, Lemma \\ref{algebra-lemma-smooth-at-point}", "without further mention.", "The exact sequence of Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL}", "for $R \\to A[y_1, \\ldots, y_c] \\to A'$ looks like this", "$$", "0 \\to H_1(\\NL_{A'/R}) \\to I/I^2 \\to", "\\Omega_{A/R} \\otimes_A A' \\oplus", "\\bigoplus\\nolimits_{i = 1, \\ldots, c} A' \\text{d}y_i \\to", "\\Omega_{A'/R} \\to 0", "$$", "where the class of $\\pi y_i - g_i$ in $I/I^2$ is mapped", "to $- \\text{d}g_i + \\pi \\text{d}y_i$ in the next term.", "Here we have used Algebra, Lemma \\ref{algebra-lemma-NL-surjection}", "to compute $\\NL_{A'/A[y_1, \\ldots, y_c]}$ and we have used that", "$R \\to A[y_1, \\ldots, y_c]$ is smooth, so", "$H_1(\\NL_{A[y_1, \\ldots, y_c]/R}) = 0$ and", "$\\Omega_{A[y_1, \\ldots, y_c]/R}$ is a finite projective (a fortiori flat)", "$A[y_1, \\ldots, y_c]$-module which is in fact the direct sum", "of $\\Omega_{A/R} \\otimes_A A[y_1, \\ldots, y_c]$ and a free", "module with basis $\\text{d}y_i$. To finish the proof it", "suffices to show that $\\text{d}g_1, \\ldots, \\text{d}g_c$", "forms part of a basis for the finite free module $\\Omega_{A/R, \\mathfrak p}$.", "Namely, this will show $(I/I^2)_\\mathfrak p$ is free on $\\pi y_i - g_i$,", "the localization at $\\mathfrak p$ of the middle map in the sequence is", "injective, so $H_1(\\NL_{A'/R})_\\mathfrak p = 0$, and", "that the cokernel $\\Omega_{A'/R, \\mathfrak p}$ is finite free.", "To do this it suffices to show", "that the images of $\\text{d}g_i$ are $\\kappa(\\mathfrak p)$-linearly", "independent in", "$\\Omega_{A/R, \\mathfrak p}/\\pi = \\Omega_{(A/\\pi A)/\\kappa, \\mathfrak p}$", "(equality by Algebra, Lemma \\ref{algebra-lemma-differentials-base-change}).", "Since $\\kappa \\subset \\kappa(\\mathfrak p) \\subset \\Lambda/\\pi \\Lambda$", "we see that $\\kappa(\\mathfrak p)$ is separable over $\\kappa$", "(Algebra, Definition \\ref{algebra-definition-separable-field-extension}).", "The desired linear independence now follows from", "Algebra, Lemma \\ref{algebra-lemma-computation-differential}." ], "refs": [ "smoothing-lemma-neron-functorial", "algebra-lemma-base-change-smooth", "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-regular-ring-CM", "algebra-lemma-regular-sequence-in-neighbourhood", "more-algebra-lemma-blowup-regular-sequence", "algebra-definition-smooth", "algebra-lemma-smooth-at-point", "algebra-lemma-exact-sequence-NL", "algebra-lemma-NL-surjection", "algebra-lemma-differentials-base-change", "algebra-definition-separable-field-extension", "algebra-lemma-computation-differential" ], "ref_ids": [ 5619, 1191, 1223, 941, 741, 9990, 1534, 1196, 1153, 1154, 1138, 1460, 1224 ] } ], "ref_ids": [] }, { "id": 5621, "type": "theorem", "label": "smoothing-lemma-neron-when-smooth", "categories": [ "smoothing" ], "title": "smoothing-lemma-neron-when-smooth", "contents": [ "In Situation \\ref{situation-neron} assume that $R \\to A$ is smooth", "at $\\mathfrak q$ and that we have a surjection of $R$-algebras", "$B \\to A$ with kernel $I$. Assume $R \\to B$ smooth at", "$\\mathfrak p_B = (B \\to A)^{-1}\\mathfrak p$. If the cokernel of", "$$", "I/I^2 \\otimes_A \\Lambda \\to \\Omega_{B/R} \\otimes_B \\Lambda", "$$", "is a free $\\Lambda$-module, then $R \\to A$ is smooth at $\\mathfrak p$." ], "refs": [], "proofs": [ { "contents": [ "The cokernel of the map $I/I^2 \\to \\Omega_{B/R} \\otimes_B A$", "is $\\Omega_{A/R}$, see Algebra, Lemma \\ref{algebra-lemma-differential-seq}.", "Let $d = \\dim_\\mathfrak q(A/R)$ be the relative dimension of $R \\to A$", "at $\\mathfrak q$, i.e., the dimension of $\\Spec(A[1/\\pi])$ at $\\mathfrak q$.", "See Algebra, Definition \\ref{algebra-definition-relative-dimension}.", "Then $\\Omega_{A/R, \\mathfrak q}$ is free over $A_\\mathfrak q$ of rank $d$", "(Algebra, Lemma \\ref{algebra-lemma-characterize-smooth-over-field}).", "Thus if the hypothesis of the lemma holds,", "then $\\Omega_{A/R} \\otimes_A \\Lambda$ is free of rank $d$.", "It follows that $\\Omega_{A/R} \\otimes_A \\kappa(\\mathfrak p)$", "has dimension $d$ (as it is true upon tensoring with $\\Lambda/\\pi \\Lambda$).", "Since $R \\to A$ is flat and since $\\mathfrak p$", "is a specialization of $\\mathfrak q$, we see that", "$\\dim_\\mathfrak p(A/R) \\geq d$ by Algebra, Lemma", "\\ref{algebra-lemma-dimension-fibres-bounded-open-upstairs}.", "Then it follows that $R \\to A$ is smooth at $\\mathfrak p$ by", "Algebra, Lemmas \\ref{algebra-lemma-flat-fibre-smooth} and", "\\ref{algebra-lemma-characterize-smooth-over-field}." ], "refs": [ "algebra-lemma-differential-seq", "algebra-definition-relative-dimension", "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-dimension-fibres-bounded-open-upstairs", "algebra-lemma-flat-fibre-smooth", "algebra-lemma-characterize-smooth-over-field" ], "ref_ids": [ 1135, 1524, 1223, 1075, 1200, 1223 ] } ], "ref_ids": [] }, { "id": 5622, "type": "theorem", "label": "smoothing-lemma-neron-desingularization", "categories": [ "smoothing" ], "title": "smoothing-lemma-neron-desingularization", "contents": [ "In Situation \\ref{situation-neron}", "assume that $R \\to A$ is smooth at $\\mathfrak q$", "and that $R/\\pi R \\subset \\Lambda/\\pi \\Lambda$ is a separable", "extension of fields. Then after a finite number of affine N\\'eron", "blowups the algebra $A$ becomes smooth over $R$ at $\\mathfrak p$." ], "refs": [], "proofs": [ { "contents": [ "We choose an $R$-algebra $B$ and a surjection $B \\to A$. Set", "$\\mathfrak p_B = (B \\to A)^{-1}(\\mathfrak p)$ and denote $r$", "the relative dimension of $R \\to B$ at $\\mathfrak p_B$. We choose $B$", "such that $R \\to B$ is smooth at $\\mathfrak p_B$.", "For example we can take $B$ to be a polynomial algebra in $r$ variables", "over $R$. Consider the complex", "$$", "I/I^2 \\otimes_A \\Lambda \\longrightarrow \\Omega_{B/R} \\otimes_B \\Lambda", "$$", "of Lemma \\ref{lemma-neron-when-smooth}. By the structure of finite modules", "over $\\Lambda$ (More on Algebra, Lemma \\ref{more-algebra-lemma-modules-PID})", "we see that the cokernel looks like", "$$", "\\Lambda^{\\oplus d} \\oplus", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} \\Lambda/\\pi^{e_i} \\Lambda", "$$", "for some $d \\geq 0$, $n \\geq 0$, and $e_i \\geq 1$. Observe that $d$", "is the relative dimension of $A/R$ at $\\mathfrak q$", "(Algebra, Lemma \\ref{algebra-lemma-characterize-smooth-over-field}).", "If the defect $e = \\sum_{i = 1, \\ldots, n} e_i$ is zero, then we are done by", "Lemma \\ref{lemma-neron-when-smooth}.", "\\medskip\\noindent", "Next, we consider what happens when we perform the N\\'eron blowup.", "Recall that $A'$ is the quotient of $B'/IB'$ by its $\\pi$-power", "torsion (Lemma \\ref{lemma-neron-functorial}) and that $R \\to B'$ is smooth at", "$\\mathfrak p_{B'}$ (Lemma \\ref{lemma-neron-blowup-smooth}).", "Thus after blowup we have exactly the same setup. Picture", "$$", "\\xymatrix{", "0 \\ar[r] & I' \\ar[r] & B' \\ar[r] & A' \\ar[r] & 0 \\\\", "0 \\ar[r] & I \\ar[u] \\ar[r] & B \\ar[u] \\ar[r] & A \\ar[r] \\ar[u] & 0", "}", "$$", "Since $I \\subset \\mathfrak p_B$, we see that $I \\to I'$", "factors through $\\pi I'$. Looking at the induced map of", "complexes we get", "$$", "\\xymatrix{", "I'/(I')^2 \\otimes_{A'} \\Lambda \\ar[r] &", "\\Omega_{B'/R} \\otimes_{B'} \\Lambda \\ar@{=}[r] & M' \\\\", "I/I^2 \\otimes_A \\Lambda \\ar[r] \\ar[u] &", "\\Omega_{B/R} \\otimes_B \\Lambda \\ar[u] \\ar@{=}[r] & M", "}", "$$", "Then $M \\subset M'$ are finite free $\\Lambda$-modules with quotient", "$M'/M$ annihilated by $\\pi$, see Lemma \\ref{lemma-neron-blowup-smooth}.", "Let $N \\subset M$ and $N' \\subset M'$ be the images of the horizontal", "maps and denote $Q = M/N$ and $Q' = M'/N'$.", "We obtain a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "N' \\ar[r] &", "M' \\ar[r] &", "Q' \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "N \\ar[r] \\ar[u] &", "M \\ar[r] \\ar[u] &", "Q \\ar[r] \\ar[u] &", "0", "}", "$$", "Then $N \\subset N'$ are free $\\Lambda$-modules of rank $r - d$.", "Since $I$ maps into $\\pi I'$ we see that $N \\subset \\pi N'$.", "\\medskip\\noindent", "Let $K = \\Lambda_\\pi$ be the fraction field of $\\Lambda$.", "We have a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "N' \\ar[r] &", "N'_K \\cap M' \\ar[r] &", "Q'_{tor} \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "N \\ar[r] \\ar[u] &", "N_K \\cap M \\ar[r] \\ar[u] &", "Q_{tor} \\ar[r] \\ar[u] &", "0", "}", "$$", "whose rows are short exact sequences. This shows that the change in defect", "is given by", "$$", "e - e' =", "\\text{length}(Q_{tor}) - \\text{length}(Q'_{tor})", "=", "\\text{length}(N'/N) - \\text{length}(N'_K \\cap M' / N_K \\cap M)", "$$", "Since $M'/M$ is annihilated by $\\pi$, so is $N'_K \\cap M' / N_K \\cap M$,", "and its length is at most $\\text{dim}_K(N_K)$.", "Since $N \\subset \\pi N'$ we get $\\text{length}(N'/N) \\ge \\text{dim}_K(N_K)$,", "with equality if and only if $N = \\pi N'$.", "\\medskip\\noindent", "To finish the proof we have to show that $N$ is strictly smaller", "than $\\pi N'$ when $A$ is not smooth at $\\mathfrak p$;", "this is the key computation one has to do in N\\'eron's argument.", "To do this, we consider the exact sequence", "$$", "I/I^2 \\otimes_B \\kappa(\\mathfrak p_B)", "\\to \\Omega_{B/R} \\otimes_B \\kappa(\\mathfrak p_B)", "\\to \\Omega_{A/R} \\otimes_A \\kappa(\\mathfrak p) \\to 0", "$$", "(follows from Algebra, Lemma \\ref{algebra-lemma-differential-seq}).", "Since $R \\to A$ is not smooth at $\\mathfrak p$ we see that the dimension $s$ of", "$\\Omega_{A/R} \\otimes_A \\kappa(\\mathfrak p)$", "is bigger than $d$. On the other hand", "the first arrow factors through the injective map", "$$", "\\mathfrak p B_\\mathfrak p/\\mathfrak p^2 B_\\mathfrak p", "\\to \\Omega_{B/R} \\otimes_B \\kappa(\\mathfrak p_B)", "$$", "of Algebra, Lemma \\ref{algebra-lemma-computation-differential};", "note that $\\kappa(\\mathfrak p)$ is separable over $k$", "by our assumption on $R/\\pi R \\subset \\Lambda/\\pi \\Lambda$.", "Hence we conclude that we can find generators", "$g_1, \\ldots, g_t \\in I$ such that $g_j \\in \\mathfrak p^2$", "for $j > r - s$. Then the images of $g_j$ in $A'$ are in $\\pi^2 I'$", "for $j > r - s$. Since $r - s < r - d$", "we find that at least one of the minimal generators", "of $N$ becomes divisible by $\\pi^2$ in $N'$.", "Thus we see that $e$ decreases by at least $1$ and we win." ], "refs": [ "smoothing-lemma-neron-when-smooth", "more-algebra-lemma-modules-PID", "algebra-lemma-characterize-smooth-over-field", "smoothing-lemma-neron-when-smooth", "smoothing-lemma-neron-functorial", "smoothing-lemma-neron-blowup-smooth", "smoothing-lemma-neron-blowup-smooth", "algebra-lemma-differential-seq", "algebra-lemma-computation-differential" ], "ref_ids": [ 5621, 10561, 1223, 5621, 5619, 5620, 5620, 1135, 1224 ] } ], "ref_ids": [] }, { "id": 5623, "type": "theorem", "label": "smoothing-lemma-neron-colimit", "categories": [ "smoothing" ], "title": "smoothing-lemma-neron-colimit", "contents": [ "\\begin{slogan}", "Unramified extensions of DVRs are ind-smooth AKA N\\'eron desingularization", "\\end{slogan}", "Let $R \\subset \\Lambda$ be an extension of discrete valuation", "rings which has ramification index $1$ and induces a separable", "extension of residue fields and of fraction fields.", "Then $\\Lambda$ is a filtered colimit of smooth $R$-algebras." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-when-colimit} it suffices to show", "that any $R \\to A \\to \\Lambda$ as in Situation \\ref{situation-neron}", "can be factored as $A \\to B \\to \\Lambda$ with $B$ a", "smooth $R$-algebra. After replacing $A$ by its image in $\\Lambda$", "we may assume that $A$ is a domain whose fraction field $K$", "is a subfield of the fraction field of $\\Lambda$.", "In particular, $A$ is separable over the fraction field of $R$", "by our assumptions. Then $R \\to A$ is smooth at $\\mathfrak q = (0)$ by", "Algebra, Lemma \\ref{algebra-lemma-smooth-at-generic-point}.", "After a finite number of N\\'eron blowups, we may assume $R \\to A$", "is smooth at $\\mathfrak p$, see Lemma \\ref{lemma-neron-desingularization}.", "Then, after replacing $A$ by a localization", "at an element $a \\in A$, $a \\not \\in \\mathfrak p$ it becomes", "smooth over $R$ and the lemma is proved." ], "refs": [ "algebra-lemma-when-colimit", "algebra-lemma-smooth-at-generic-point", "smoothing-lemma-neron-desingularization" ], "ref_ids": [ 1093, 1228, 5622 ] } ], "ref_ids": [] }, { "id": 5624, "type": "theorem", "label": "smoothing-lemma-lift-once", "categories": [ "smoothing" ], "title": "smoothing-lemma-lift-once", "contents": [ "Let $R \\to \\Lambda$ be a ring map. Let $I \\subset R$ be an ideal.", "Assume that", "\\begin{enumerate}", "\\item $I^2 = 0$, and", "\\item $\\Lambda/I\\Lambda$ is a filtered colimit of smooth $R/I$-algebras.", "\\end{enumerate}", "Let $\\varphi : A \\to \\Lambda$ be an $R$-algebra map with $A$ of finite", "presentation over $R$. Then there exists a factorization", "$$", "A \\to B/J \\to \\Lambda", "$$", "where $B$ is a smooth $R$-algebra and $J \\subset IB$ is a finitely generated", "ideal." ], "refs": [], "proofs": [ { "contents": [ "Choose a factorization", "$$", "A/IA \\to \\bar B \\to \\Lambda/I\\Lambda", "$$", "with $\\bar B$ standard smooth over $R/I$; this is possible by", "assumption and Lemma \\ref{lemma-colimit-standard-smooth}. Write", "$$", "\\bar B = A/IA[t_1, \\ldots, t_r]/(\\bar g_1, \\ldots, \\bar g_s)", "$$", "and say $\\bar B \\to \\Lambda/I\\Lambda$ maps $t_i$ to the class", "of $\\lambda_i$ modulo $I\\Lambda$. Choose", "$g_1, \\ldots, g_s \\in A[t_1, \\ldots, t_r]$ lifting", "$\\bar g_1, \\ldots, \\bar g_s$. Write", "$\\varphi(g_i)(\\lambda_1, \\ldots, \\lambda_r) =", "\\sum \\epsilon_{ij} \\mu_{ij}$", "for some $\\epsilon_{ij} \\in I$ and $\\mu_{ij} \\in \\Lambda$. Define", "$$", "A' = A[t_1, \\ldots, t_r, \\delta_{i, j}]/", "(g_i - \\sum \\epsilon_{ij} \\delta_{ij})", "$$", "and consider the map", "$$", "A' \\longrightarrow \\Lambda,\\quad", "a \\longmapsto \\varphi(a),\\quad", "t_i \\longmapsto \\lambda_i,\\quad", "\\delta_{ij} \\longmapsto \\mu_{ij}", "$$", "We have", "$$", "A'/IA' = A/IA[t_1, \\ldots, t_r]/(\\bar g_1, \\ldots, \\bar g_s)[\\delta_{ij}]", "\\cong \\bar B[\\delta_{ij}]", "$$", "This is a standard smooth algebra over $R/I$ as $\\bar B$ is standard", "smooth. Choose a presentation", "$A'/IA' = R/I[x_1, \\ldots, x_n]/(\\bar f_1, \\ldots, \\bar f_c)$ with", "$\\det(\\partial \\bar f_j/\\partial x_i)_{i, j = 1, \\ldots, c}$ invertible in", "$A'/IA'$. Choose lifts $f_1, \\ldots, f_c \\in R[x_1, \\ldots, x_n]$ of", "$\\bar f_1, \\ldots, \\bar f_c$. Then", "$$", "B = R[x_1, \\ldots, x_n, x_{n + 1}]/", "(f_1, \\ldots, f_c,", "x_{n + 1}\\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c} - 1)", "$$", "is smooth over $R$. Since smooth ring maps are formally smooth", "(Algebra, Proposition \\ref{algebra-proposition-smooth-formally-smooth})", "there exists an $R$-algebra map $B \\to A'$ which is an isomorphism", "modulo $I$. Then $B \\to A'$ is surjective by Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK}).", "Thus $A' = B/J$ with $J \\subset IB$ finitely generated (see", "Algebra, Lemma \\ref{algebra-lemma-finite-presentation-independent})." ], "refs": [ "smoothing-lemma-colimit-standard-smooth", "algebra-proposition-smooth-formally-smooth", "algebra-lemma-NAK", "algebra-lemma-finite-presentation-independent" ], "ref_ids": [ 5616, 1426, 401, 334 ] } ], "ref_ids": [] }, { "id": 5625, "type": "theorem", "label": "smoothing-lemma-lift-twice", "categories": [ "smoothing" ], "title": "smoothing-lemma-lift-twice", "contents": [ "Let $R \\to \\Lambda$ be a ring map. Let $I \\subset R$ be an ideal.", "Assume that", "\\begin{enumerate}", "\\item $I^2 = 0$,", "\\item $\\Lambda/I\\Lambda$ is a filtered colimit of smooth $R/I$-algebras, and", "\\item $R \\to \\Lambda$ is flat.", "\\end{enumerate}", "Let $\\varphi : B \\to \\Lambda$ be an $R$-algebra map with $B$", "smooth over $R$. Let $J \\subset IB$ be a finitely generated ideal", "such that $\\varphi(J) = 0$.", "Then there exists $R$-algebra maps", "$$", "B \\xrightarrow{\\alpha} B' \\xrightarrow{\\beta} \\Lambda", "$$", "such that $B'$ is smooth over $R$, such that $\\alpha(J) = 0$ and", "such that $\\beta \\circ \\alpha = \\varphi \\bmod I\\Lambda$." ], "refs": [], "proofs": [ { "contents": [ "If we can prove the lemma in case $J = (h)$, then we can prove the", "lemma by induction on the number of generators of $J$. Namely, suppose", "that $J$ can be generated by $n$ elements $h_1, \\ldots, h_n$ and the", "lemma holds for all cases where $J$ is generated by $n - 1$ elements.", "Then we apply the case $n = 1$ to produce $B \\to B' \\to \\Lambda$", "where the first map kills of $h_n$. Then we let $J'$ be the", "ideal of $B'$ generated by the images of $h_1, \\ldots, h_{n - 1}$", "and we apply the case for $n - 1$ to produce $B' \\to B'' \\to \\Lambda$.", "It is easy to verify that $B \\to B'' \\to \\Lambda$ does the job.", "\\medskip\\noindent", "Assume $J = (h)$ and write $h = \\sum \\epsilon_i b_i$", "for some $\\epsilon_i \\in I$ and $b_i \\in B$. Note that", "$0 = \\varphi(h) = \\sum \\epsilon_i \\varphi(b_i)$.", "As $\\Lambda$ is flat over $R$, the equational criterion for", "flatness (Algebra, Lemma \\ref{algebra-lemma-flat-eq})", "implies that we can find $\\lambda_j \\in \\Lambda$,", "$j = 1, \\ldots, m$ and $a_{ij} \\in R$ such that", "$\\varphi(b_i) = \\sum_j a_{ij} \\lambda_j$ and $\\sum_i \\epsilon_i a_{ij} = 0$.", "Set", "$$", "C = B[x_1, \\ldots, x_m]/(b_i - \\sum a_{ij} x_j)", "$$", "with $C \\to \\Lambda$ given by $\\varphi$ and $x_j \\mapsto \\lambda_j$.", "Choose a factorization", "$$", "C \\to B'/J' \\to \\Lambda", "$$", "as in Lemma \\ref{lemma-lift-once}. Since $B$ is smooth over $R$ we can", "lift the map $B \\to C \\to B'/J'$ to a map $\\psi : B \\to B'$. We claim that", "$\\psi(h) = 0$. Namely, the fact that $\\psi$ agrees with", "$B \\to C \\to B'/J'$ mod $I$ implies that", "$$", "\\psi(b_i) = \\sum a_{ij} \\xi_j + \\theta_i", "$$", "for some $\\xi_i \\in B'$ and $\\theta_i \\in IB'$. Hence we see that", "$$", "\\psi(h) = \\psi(\\sum \\epsilon_i b_i) =", "\\sum \\epsilon_i a_{ij} \\xi_j + \\sum \\epsilon_i \\theta_i = 0", "$$", "because of the relations above and the fact that $I^2 = 0$." ], "refs": [ "algebra-lemma-flat-eq", "smoothing-lemma-lift-once" ], "ref_ids": [ 531, 5624 ] } ], "ref_ids": [] }, { "id": 5626, "type": "theorem", "label": "smoothing-lemma-lifting", "categories": [ "smoothing" ], "title": "smoothing-lemma-lifting", "contents": [ "Let $R$ be a Noetherian ring. Let $\\Lambda$ be an $R$-algebra.", "Let $\\pi \\in R$ and assume that $\\text{Ann}_R(\\pi) = \\text{Ann}_R(\\pi^2)$ and", "$\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$.", "Suppose we have $R$-algebra maps", "$R/\\pi^2R \\to \\bar C \\to \\Lambda/\\pi^2\\Lambda$", "with $\\bar C$ of finite presentation.", "Then there exists an $R$-algebra homomorphism", "$D \\to \\Lambda$ and a commutative diagram", "$$", "\\xymatrix{", "R/\\pi^2R \\ar[r] \\ar[d] &", "\\bar C \\ar[r] \\ar[d] &", "\\Lambda/\\pi^2\\Lambda \\ar[d] \\\\", "R/\\pi R \\ar[r] &", "D/\\pi D \\ar[r] &", "\\Lambda/\\pi \\Lambda", "}", "$$", "with the following properties", "\\begin{enumerate}", "\\item[(a)] $D$ is of finite presentation,", "\\item[(b)] $R \\to D$ is smooth at any prime $\\mathfrak q$ with", "$\\pi \\not \\in \\mathfrak q$,", "\\item[(c)] $R \\to D$ is smooth at any prime $\\mathfrak q$ with", "$\\pi \\in \\mathfrak q$ lying over a prime of $\\bar C$ where", "$R/\\pi^2 R \\to \\bar C$ is smooth, and", "\\item[(d)] $\\bar C/\\pi \\bar C \\to D/\\pi D$ is smooth at any prime", "lying over a prime of $\\bar C$ where $R/\\pi^2R \\to \\bar C$ is smooth.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We choose a presentation", "$$", "\\bar C = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)", "$$", "We also denote $I = (f_1, \\ldots, f_m)$ and $\\bar I$ the image of", "$I$ in $R/\\pi^2R[x_1, \\ldots, x_n]$. Since $R$ is Noetherian, so is", "$\\bar C$. Hence the smooth locus of $R/\\pi^2 R \\to \\bar C$", "is quasi-compact, see", "Topology, Lemma \\ref{topology-lemma-Noetherian}.", "Applying", "Lemma \\ref{lemma-find-strictly-standard}", "we may choose a finite list of elements", "$a_1, \\ldots, a_r \\in R[x_1, \\ldots, x_n]$ such that", "\\begin{enumerate}", "\\item the union of the open subspaces", "$\\Spec(\\bar C_{a_k}) \\subset \\Spec(\\bar C)$", "cover the smooth locus of $R/\\pi^2 R \\to \\bar C$, and", "\\item for each $k = 1, \\ldots, r$ there exists a finite subset", "$E_k \\subset \\{1, \\ldots, m\\}$ such that", "$(\\bar I/\\bar I^2)_{a_k}$ is freely generated by the classes of", "$f_j$, $j \\in E_k$.", "\\end{enumerate}", "Set $I_k = (f_j, j \\in E_k) \\subset I$ and denote $\\bar I_k$ the", "image of $I_k$ in $R/\\pi^2R[x_1, \\ldots, x_n]$.", "By (2) and Nakayama's lemma we see that $(\\bar I/\\bar I_k)_{a_k}$", "is annihilated by $1 + b'_k$ for some $b'_k \\in \\bar I_{a_k}$.", "Suppose $b'_k$ is the image of $b_k/(a_k)^N$ for some $b_k \\in I$", "and some integer $N$. After replacing $a_k$ by $a_kb_k$ we get", "\\begin{enumerate}", "\\item[(3)] $(\\bar I_k)_{a_k} = (\\bar I)_{a_k}$.", "\\end{enumerate}", "Thus, after possibly replacing $a_k$ by a high power, we may write", "\\begin{enumerate}", "\\item[(4)]", "$a_k f_\\ell = \\sum\\nolimits_{j \\in E_k} h_{k, \\ell}^jf_j + \\pi^2 g_{k, \\ell}$", "\\end{enumerate}", "for any $\\ell \\in \\{1, \\ldots, m\\}$ and some", "$h_{i, \\ell}^j, g_{i, \\ell} \\in R[x_1, \\ldots, x_n]$.", "If $\\ell \\in E_k$ we choose $h_{k, \\ell}^j = a_k\\delta_{\\ell, j}$", "(Kronecker delta) and $g_{k, \\ell} = 0$. Set", "$$", "D = R[x_1, \\ldots, x_n, z_1, \\ldots, z_m]/", "(f_j - \\pi z_j, p_{k, \\ell}).", "$$", "Here $j \\in \\{1, \\ldots, m\\}$, $k \\in \\{1, \\ldots, r\\}$,", "$\\ell \\in \\{1, \\ldots, m\\}$, and", "$$", "p_{k, \\ell} = a_k z_\\ell - \\sum\\nolimits_{j \\in E_k} h_{k, \\ell}^j z_j", "- \\pi g_{k, \\ell}.", "$$", "Note that for $\\ell \\in E_k$ we have $p_{k, \\ell} = 0$ by our choices above.", "\\medskip\\noindent", "The map $R \\to D$ is the given one.", "Say $\\bar C \\to \\Lambda/\\pi^2\\Lambda$ maps $x_i$", "to the class of $\\lambda_i$ modulo $\\pi^2$. For an element", "$f \\in R[x_1, \\ldots, x_n]$ we denote $f(\\lambda) \\in \\Lambda$", "the result of substituting $\\lambda_i$ for $x_i$. Then we know that", "$f_j(\\lambda) = \\pi^2 \\mu_j$ for some $\\mu_j \\in \\Lambda$.", "Define $D \\to \\Lambda$ by the rules $x_i \\mapsto \\lambda_i$ and", "$z_j \\mapsto \\pi\\mu_j$. This is well defined because", "\\begin{align*}", "p_{k, \\ell} & \\mapsto", "a_k(\\lambda) \\pi \\mu_\\ell -", "\\sum\\nolimits_{j \\in E_k} h_{k, \\ell}^j(\\lambda) \\pi \\mu_j", "- \\pi g_{k, \\ell}(\\lambda) \\\\", "& =", "\\pi\\left(a_k(\\lambda) \\mu_\\ell -", "\\sum\\nolimits_{j \\in E_k} h_{k, \\ell}^j(\\lambda) \\mu_j", "- g_{k, \\ell}(\\lambda)\\right)", "\\end{align*}", "Substituting $x_i = \\lambda_i$ in (4) above we see that the expression", "inside the brackets is annihilated by $\\pi^2$, hence it is annihilated", "by $\\pi$ as we have assumed", "$\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$.", "The map $\\bar C \\to D/\\pi D$ is determined by $x_i \\mapsto x_i$", "(clearly well defined). Thus we are done if we can prove (b), (c), and (d).", "\\medskip\\noindent", "Using (4) we obtain the following key equality", "\\begin{align*}", "\\pi p_{k, \\ell} & =", "\\pi a_k z_\\ell - \\sum\\nolimits_{j \\in E_k} \\pi h_{k, \\ell}^jz_j", "- \\pi^2 g_{k, \\ell} \\\\", "& =", "- a_k (f_\\ell - \\pi z_\\ell) + a_k f_\\ell +", "\\sum\\nolimits_{j \\in E_k} h_{k, \\ell}^j (f_j - \\pi z_j) -", "\\sum\\nolimits_{j \\in E_k} h_{k, \\ell}^j f_j - \\pi^2 g_{k, \\ell} \\\\", "& =", "-a_k(f_\\ell - \\pi z_\\ell) +", "\\sum\\nolimits_{j \\in E_k} h_{k, \\ell}^j(f_j - \\pi z_j)", "\\end{align*}", "The end result is an element of the ideal generated by $f_j - \\pi z_j$.", "In particular, we see that $D[1/\\pi]$ is isomorphic to", "$R[1/\\pi][x_1, \\ldots, x_n, z_1, \\ldots, z_m]/(f_j - \\pi z_j)$", "which is isomorphic to $R[1/\\pi][x_1, \\ldots, x_n]$ hence smooth", "over $R$. This proves (b).", "\\medskip\\noindent", "For fixed $k \\in \\{1, \\ldots, r\\}$ consider the ring", "$$", "D_k = R[x_1, \\ldots, x_n, z_1, \\ldots, z_m]/", "(f_j - \\pi z_j, j \\in E_k, p_{k, \\ell})", "$$", "The number of equations is $m = |E_k| + (m - |E_k|)$ as $p_{k, \\ell}$", "is zero if $\\ell \\in E_k$. Also, note that", "\\begin{align*}", "(D_k/\\pi D_k)_{a_k}", "& =", "R/\\pi R[x_1, \\ldots, x_n, 1/a_k, z_1, \\ldots, z_m]/", "(f_j, j \\in E_k, p_{k, \\ell}) \\\\", "& =", "(\\bar C/\\pi \\bar C)_{a_k}[z_1, \\ldots, z_m]/", "(a_kz_\\ell - \\sum\\nolimits_{j \\in E_k} h_{k, \\ell}^j z_j) \\\\", "& \\cong", "(\\bar C/\\pi \\bar C)_{a_k}[z_j, j \\in E_k]", "\\end{align*}", "In particular $(D_k/\\pi D_k)_{a_k}$ is smooth over $(\\bar C/\\pi \\bar C)_{a_k}$.", "By our choice of $a_k$ we have that $(\\bar C/\\pi \\bar C)_{a_k}$ is smooth", "over $R/\\pi R$ of relative dimension $n - |E_k|$, see (2). Hence for a prime", "$\\mathfrak q_k \\subset D_k$ containing $\\pi$ and lying over", "$\\Spec(\\bar C_{a_k})$ the fibre ring of $R \\to D_k$", "is smooth at $\\mathfrak q_k$ of dimension $n$. Thus $R \\to D_k$ is syntomic", "at $\\mathfrak q_k$ by our count of the number of equations above, see", "Algebra, Lemma \\ref{algebra-lemma-localize-relative-complete-intersection}.", "Hence $R \\to D_k$ is smooth at $\\mathfrak q_k$, see", "Algebra, Lemma \\ref{algebra-lemma-flat-fibre-smooth}.", "\\medskip\\noindent", "To finish the proof, let $\\mathfrak q \\subset D$ be a prime", "containing $\\pi$ lying over a prime where $R/\\pi^2 R \\to \\bar C$", "is smooth. Then $a_k \\not \\in \\mathfrak q$ for some $k$ by (1).", "We will show that the surjection $D_k \\to D$ induces", "an isomorphism on local rings at $\\mathfrak q$. Since we know that", "the ring maps $\\bar C/\\pi \\bar C \\to D_k/\\pi D_k$ and", "$R \\to D_k$ are smooth at the corresponding prime $\\mathfrak q_k$", "by the preceding paragraph this will prove (c) and (d) and thus", "finish the proof.", "\\medskip\\noindent", "First, note that for any $\\ell$ the equation", "$\\pi p_{k, \\ell} = -a_k(f_\\ell - \\pi z_\\ell) +", "\\sum_{j \\in E_k} h_{k, \\ell}^j (f_j - \\pi z_j)$ proved above shows that", "$f_\\ell - \\pi z_\\ell$ maps to zero in $(D_k)_{a_k}$ and in particular", "in $(D_k)_{\\mathfrak q_k}$.", "The relations (4) imply that $a_k f_\\ell =", "\\sum_{j \\in E_k} h_{k, \\ell}^j f_j$ in $I/I^2$.", "Since $(\\bar I_k/\\bar I_k^2)_{a_k}$ is free on $f_j$, $j \\in E_k$", "we see that", "$$", "a_{k'} h_{k, \\ell}^j -", "\\sum\\nolimits_{j' \\in E_{k'}} h_{k', \\ell}^{j'} h_{k, j'}^j", "$$", "is zero in $\\bar C_{a_k}$ for every $k, k', \\ell$ and $j \\in E_k$.", "Hence we can find a large integer $N$ such that", "$$", "a_k^N\\left(", "a_{k'} h_{k, \\ell}^j -", "\\sum\\nolimits_{j' \\in E_{k'}} h_{k', \\ell}^{j'} h_{k, j'}^j", "\\right)", "$$", "is in $I_k + \\pi^2R[x_1, \\ldots, x_n]$. Computing modulo $\\pi$ we have", "\\begin{align*}", "&", "a_kp_{k', \\ell} - a_{k'}p_{k, \\ell} + \\sum h_{k', \\ell}^{j'} p_{k, j'}", "\\\\", "&", "=", "- a_k \\sum h_{k', \\ell}^{j'} z_{j'}", "+ a_{k'} \\sum h_{k, \\ell}^j z_j", "+ \\sum h_{k', \\ell}^{j'} a_k z_{j'}", "- \\sum \\sum h_{k', \\ell}^{j'} h_{k, j'}^j z_j \\\\", "&", "=", "\\sum \\left(", "a_{k'} h_{k, \\ell}^j", "- \\sum h_{k', \\ell}^{j'} h_{k, j'}^j", "\\right) z_j", "\\end{align*}", "with Einstein summation convention. Combining with the above we see", "$a_k^{N + 1} p_{k', \\ell}$ is contained in the ideal generated", "by $I_k$ and $\\pi$ in $R[x_1, \\ldots, x_n, z_1, \\ldots, z_m]$.", "Thus $p_{k', \\ell}$ maps into $\\pi (D_k)_{a_k}$. On the other hand,", "the equation", "$$", "\\pi p_{k', \\ell} =", "-a_{k'} (f_\\ell - \\pi z_\\ell) +", "\\sum\\nolimits_{j' \\in E_{k'}} h_{k', \\ell}^{j'}(f_{j'} - \\pi z_{j'})", "$$", "shows that $\\pi p_{k', \\ell}$ is zero in $(D_k)_{a_k}$.", "Since we have assumed that $\\text{Ann}_R(\\pi) = \\text{Ann}_R(\\pi^2)$", "and since $(D_k)_{\\mathfrak q_k}$ is smooth hence flat over $R$", "we see that", "$\\text{Ann}_{(D_k)_{\\mathfrak q_k}}(\\pi) =", "\\text{Ann}_{(D_k)_{\\mathfrak q_k}}(\\pi^2)$.", "We conclude that $p_{k', \\ell}$ maps to zero as well, hence", "$D_{\\mathfrak q} = (D_k)_{\\mathfrak q_k}$ and we win." ], "refs": [ "topology-lemma-Noetherian", "smoothing-lemma-find-strictly-standard", "algebra-lemma-localize-relative-complete-intersection", "algebra-lemma-flat-fibre-smooth" ], "ref_ids": [ 8220, 5608, 1181, 1200 ] } ], "ref_ids": [] }, { "id": 5627, "type": "theorem", "label": "smoothing-lemma-desingularize", "categories": [ "smoothing" ], "title": "smoothing-lemma-desingularize", "contents": [ "Let $R$ be a Noetherian ring.", "Let $\\Lambda$ be an $R$-algebra. Let $\\pi \\in R$ and", "assume that $\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$. Let", "$A \\to \\Lambda$ be an $R$-algebra map with $A$ of finite", "presentation. Assume", "\\begin{enumerate}", "\\item the image of $\\pi$ is strictly standard in $A$ over $R$, and", "\\item there exists a section $\\rho : A/\\pi^4 A \\to R/\\pi^4 R$", "which is compatible with the map to $\\Lambda/\\pi^4 \\Lambda$.", "\\end{enumerate}", "Then we can find $R$-algebra maps $A \\to B \\to \\Lambda$ with $B$", "of finite presentation such that $\\mathfrak a B \\subset H_{B/R}$ where", "$\\mathfrak a = \\text{Ann}_R(\\text{Ann}_R(\\pi^2)/\\text{Ann}_R(\\pi))$." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation", "$$", "A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)", "$$", "and $0 \\leq c \\leq \\min(n, m)$ such that", "(\\ref{equation-strictly-standard-one}) holds for $\\pi$ and such that", "\\begin{equation}", "\\label{equation-star}", "\\pi f_{c + j} \\in (f_1, \\ldots, f_c) + (f_1, \\ldots, f_m)^2", "\\end{equation}", "for $j = 1, \\ldots, m - c$. Say $\\rho$ maps $x_i$ to the class of", "$r_i \\in R$. Then we can replace $x_i$ by $x_i - r_i$. Hence we may", "assume $\\rho(x_i) = 0$ in $R/\\pi^4 R$. This implies that", "$f_j(0) \\in \\pi^4R$ and that $A \\to \\Lambda$ maps $x_i$", "to $\\pi^4\\lambda_i$ for some $\\lambda_i \\in \\Lambda$. Write", "$$", "f_j = f_j(0) + \\sum\\nolimits_{i = 1, \\ldots, n} r_{ji} x_i + \\text{h.o.t.}", "$$", "This implies that the constant term of $\\partial f_j/\\partial x_i$ is", "$r_{ji}$. Apply $\\rho$ to (\\ref{equation-strictly-standard-one})", "for $\\pi$ and we see that", "$$", "\\pi = \\sum\\nolimits_{I \\subset \\{1, \\ldots, n\\},\\ |I| = c}", "r_I \\det(r_{ji})_{j = 1, \\ldots, c,\\ i \\in I} \\bmod \\pi^4R", "$$", "for some $r_I \\in R$. Thus we have", "$$", "u\\pi = \\sum\\nolimits_{I \\subset \\{1, \\ldots, n\\},\\ |I| = c}", "r_I \\det(r_{ji})_{j = 1, \\ldots, c,\\ i \\in I}", "$$", "for some $u \\in 1 + \\pi^3R$. By", "Algebra, Lemma \\ref{algebra-lemma-matrix-left-inverse}", "this implies there exists a $n \\times c$ matrix $(s_{ik})$ such that", "$$", "u\\pi \\delta_{jk} = \\sum\\nolimits_{i = 1, \\ldots, n} r_{ji}s_{ik}\\quad", "\\text{for all } j, k = 1, \\ldots, c", "$$", "(Kronecker delta). We introduce auxiliary variables", "$v_1, \\ldots, v_c, w_1, \\ldots, w_n$ and we set", "$$", "h_i = x_i - \\pi^2 \\sum\\nolimits_{j = 1, \\ldots c} s_{ij} v_j - \\pi^3 w_i", "$$", "In the following we will use that", "$$", "R[x_1, \\ldots, x_n, v_1, \\ldots, v_c, w_1, \\ldots, w_n]/", "(h_1, \\ldots, h_n) = R[v_1, \\ldots, v_c, w_1, \\ldots, w_n]", "$$", "without further mention. In", "$R[x_1, \\ldots, x_n, v_1, \\ldots, v_c, w_1, \\ldots, w_n]/", "(h_1, \\ldots, h_n)$ we have", "\\begin{align*}", "f_j & = f_j(x_1 - h_1, \\ldots, x_n - h_n) \\\\", "& =", "\\pi^2 \\sum\\nolimits_{k = 1}^c", "\\left(\\sum\\nolimits_{i = 1}^n r_{ji} s_{ik}\\right) v_k", "+", "\\pi^3 \\sum\\nolimits_{i = 1}^n r_{ji}w_i \\bmod \\pi^4 \\\\", "& =", "\\pi^3 v_j + \\pi^3 \\sum\\nolimits_{i = 1}^n r_{ji}w_i \\bmod \\pi^4", "\\end{align*}", "for $1 \\leq j \\leq c$. Hence we can choose elements", "$g_j \\in R[v_1, \\ldots, v_c, w_1, \\ldots, w_n]$", "such that $g_j = v_j + \\sum r_{ji}w_i \\bmod \\pi$", "and such that $f_j = \\pi^3 g_j$ in the $R$-algebra", "$R[x_1, \\ldots, x_n, v_1, \\ldots, v_c, w_1, \\ldots, w_n]/", "(h_1, \\ldots, h_n)$. We set", "$$", "B = R[x_1, \\ldots, x_n, v_1, \\ldots, v_c, w_1, \\ldots, w_n]/", "(f_1, \\ldots, f_m, h_1, \\ldots, h_n, g_1, \\ldots, g_c).", "$$", "The map $A \\to B$ is clear. We define $B \\to \\Lambda$ by mapping", "$x_i \\to \\pi^4\\lambda_i$, $v_i \\mapsto 0$, and $w_i \\mapsto \\pi \\lambda_i$.", "Then it is clear that the elements $f_j$ and $h_i$ are mapped to zero", "in $\\Lambda$. Moreover, it is clear that $g_i$ is mapped to an element", "$t$ of $\\pi\\Lambda$ such that $\\pi^3t = 0$ (as $f_i = \\pi^3 g_i$ modulo", "the ideal generated by the $h$'s). Hence our assumption that", "$\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$ implies that $t = 0$.", "Thus we are done if we can prove the statement about smoothness.", "\\medskip\\noindent", "Note that $B_\\pi \\cong A_\\pi[v_1, \\ldots, v_c]$ because the equations", "$g_i = 0$ are implied by $f_i = 0$. Hence $B_\\pi$ is smooth over $R$", "as $A_\\pi$ is smooth over $R$ by the assumption that $\\pi$ is strictly", "standard in $A$ over $R$, see", "Lemma \\ref{lemma-elkik}.", "\\medskip\\noindent", "Set $B' = R[v_1, \\ldots, v_c, w_1, \\ldots, w_n]/(g_1, \\ldots, g_c)$.", "As $g_i = v_i + \\sum r_{ji}w_i \\bmod \\pi$ we see that", "$B'/\\pi B' = R/\\pi R[w_1, \\ldots, w_n]$. Hence", "$R \\to B'$ is smooth of relative dimension $n$ at every", "point of $V(\\pi)$ by", "Algebra, Lemmas", "\\ref{algebra-lemma-localize-relative-complete-intersection} and", "\\ref{algebra-lemma-flat-fibre-smooth}", "(the first lemma shows it is syntomic at those primes, in particular", "flat, whereupon the second lemma shows it is smooth).", "\\medskip\\noindent", "Let $\\mathfrak q \\subset B$ be a prime with $\\pi \\in \\mathfrak q$ and", "for some $r \\in \\mathfrak a$, $r \\not \\in \\mathfrak q$.", "Denote $\\mathfrak q' = B' \\cap \\mathfrak q$.", "We claim the surjection $B' \\to B$ induces an isomorphism of local", "rings $(B')_{\\mathfrak q'} \\to B_\\mathfrak q$. This will", "conclude the proof of the lemma. Note that $B_\\mathfrak q$ is the", "quotient of $(B')_{\\mathfrak q'}$ by the ideal generated by", "$f_{c + j}$, $j = 1, \\ldots, m - c$. We observe two things:", "first the image of $f_{c + j}$ in $(B')_{\\mathfrak q'}$ is", "divisible by $\\pi^2$ and", "second the image of $\\pi f_{c + j}$ in $(B')_{\\mathfrak q'}$", "can be written as $\\sum b_{j_1 j_2} f_{c + j_1}f_{c + j_2}$ by", "(\\ref{equation-star}). Thus we see that the image of each $\\pi f_{c + j}$", "is contained in the ideal generated by the elements $\\pi^2 f_{c + j'}$.", "Hence $\\pi f_{c + j} = 0$ in $(B')_{\\mathfrak q'}$ as this is a", "Noetherian local ring, see", "Algebra, Lemma \\ref{algebra-lemma-intersect-powers-ideal-module-zero}.", "As $R \\to (B')_{\\mathfrak q'}$ is flat we see that", "$$", "\\left(\\text{Ann}_R(\\pi^2)/\\text{Ann}_R(\\pi)\\right)", "\\otimes_R (B')_{\\mathfrak q'}", "=", "\\text{Ann}_{(B')_{\\mathfrak q'}}(\\pi^2)/\\text{Ann}_{(B')_{\\mathfrak q'}}(\\pi)", "$$", "Because $r \\in \\mathfrak a$ is invertible in", "$(B')_{\\mathfrak q'}$ we see that this module is zero.", "Hence we see that the image of $f_{c + j}$ is zero in", "$(B')_{\\mathfrak q'}$ as desired." ], "refs": [ "algebra-lemma-matrix-left-inverse", "smoothing-lemma-elkik", "algebra-lemma-localize-relative-complete-intersection", "algebra-lemma-flat-fibre-smooth", "algebra-lemma-intersect-powers-ideal-module-zero" ], "ref_ids": [ 381, 5610, 1181, 1200, 627 ] } ], "ref_ids": [] }, { "id": 5628, "type": "theorem", "label": "smoothing-lemma-desingularize-strictly-standard", "categories": [ "smoothing" ], "title": "smoothing-lemma-desingularize-strictly-standard", "contents": [ "Let $R$ be a Noetherian ring. Let $\\Lambda$ be an $R$-algebra.", "Let $\\pi \\in R$ and assume that $\\text{Ann}_R(\\pi) = \\text{Ann}_R(\\pi^2)$ and", "$\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$.", "Let $A \\to \\Lambda$ and $D \\to \\Lambda$ be $R$-algebra maps with", "$A$ and $D$ of finite presentation. Assume", "\\begin{enumerate}", "\\item $\\pi$ is strictly standard in $A$ over $R$, and", "\\item there exists an $R$-algebra map $A/\\pi^4 A \\to D/\\pi^4 D$ compatible", "with the maps to $\\Lambda/\\pi^4 \\Lambda$.", "\\end{enumerate}", "Then we can find an $R$-algebra map $B \\to \\Lambda$ with $B$ of finite", "presentation and $R$-algebra maps $A \\to B$ and $D \\to B$", "compatible with the maps to $\\Lambda$ such that $H_{D/R}B \\subset H_{B/D}$", "and $H_{D/R}B \\subset H_{B/R}$." ], "refs": [], "proofs": [ { "contents": [ "We apply Lemma \\ref{lemma-desingularize} to", "$$", "D \\longrightarrow A \\otimes_R D \\longrightarrow \\Lambda", "$$", "and the image of $\\pi$ in $D$. By", "Lemma \\ref{lemma-strictly-standard-base-change}", "we see that $\\pi$ is strictly standard in $A \\otimes_R D$ over $D$.", "As our section $\\rho : (A \\otimes_R D)/\\pi^4 (A \\otimes_R D) \\to D/\\pi^4 D$", "we take the map induced by the map in (2). Thus", "Lemma \\ref{lemma-desingularize} applies and we obtain a factorization", "$A \\otimes_R D \\to B \\to \\Lambda$ with $B$ of finite presentation", "and $\\mathfrak a B \\subset H_{B/D}$ where", "$$", "\\mathfrak a = \\text{Ann}_D(\\text{Ann}_D(\\pi^2)/\\text{Ann}_D(\\pi)).", "$$", "For any prime $\\mathfrak q$ of $D$ such that $D_\\mathfrak q$ is flat over $R$", "we have", "$\\text{Ann}_{D_\\mathfrak q}(\\pi^2)/\\text{Ann}_{D_\\mathfrak q}(\\pi) = 0$", "because annihilators of elements commutes with flat base change and", "we assumed $\\text{Ann}_R(\\pi) = \\text{Ann}_R(\\pi^2)$. Because $D$ is", "Noetherian we see that $\\text{Ann}_D(\\pi^2)/\\text{Ann}_D(\\pi)$ is a finite", "$D$-module, hence formation of its annihilator commutes with localization.", "Thus we see that $\\mathfrak a \\not \\subset \\mathfrak q$. Hence we see", "that $D \\to B$ is smooth at any prime of $B$ lying over $\\mathfrak q$.", "Since any prime of $D$ where $R \\to D$ is smooth is one where", "$D_\\mathfrak q$ is flat over $R$ we conclude that $H_{D/R}B \\subset H_{B/D}$.", "The final inclusion $H_{D/R}B \\subset H_{B/R}$ follows because compositions", "of smooth ring maps are smooth", "(Algebra, Lemma \\ref{algebra-lemma-compose-smooth})." ], "refs": [ "smoothing-lemma-desingularize", "smoothing-lemma-strictly-standard-base-change", "smoothing-lemma-desingularize", "algebra-lemma-compose-smooth" ], "ref_ids": [ 5627, 5611, 5627, 1198 ] } ], "ref_ids": [] }, { "id": 5629, "type": "theorem", "label": "smoothing-lemma-desingularize-lifting-apply", "categories": [ "smoothing" ], "title": "smoothing-lemma-desingularize-lifting-apply", "contents": [ "Let $R$ be a Noetherian ring. Let $\\Lambda$ be an $R$-algebra.", "Let $\\pi \\in R$ and assume that $\\text{Ann}_R(\\pi) = \\text{Ann}_R(\\pi^2)$ and", "$\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$.", "Let $A \\to \\Lambda$ be an $R$-algebra map with", "$A$ of finite presentation and assume $\\pi$ is strictly standard", "in $A$ over $R$. Let", "$$", "A/\\pi^8A \\to \\bar C \\to \\Lambda/\\pi^8\\Lambda", "$$", "be a factorization with $\\bar C$ of finite presentation.", "Then we can find a factorization $A \\to B \\to \\Lambda$ with $B$ of finite", "presentation such that $R_\\pi \\to B_\\pi$ is smooth and such that", "$$", "H_{\\bar C/(R/\\pi^8 R)} \\cdot \\Lambda/\\pi^8\\Lambda", "\\subset", "\\sqrt{H_{B/R} \\Lambda} \\bmod \\pi^8\\Lambda.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-lifting} to get $R \\to D \\to \\Lambda$", "with a factorization", "$\\bar C/\\pi^4\\bar C \\to D/\\pi^4 D \\to \\Lambda/\\pi^4\\Lambda$", "such that $R \\to D$ is smooth at any prime not containing $\\pi$", "and at any prime lying over a prime of $\\bar C/\\pi^4\\bar C$", "where $R/\\pi^8 R \\to \\bar C$ is smooth.", "By Lemma \\ref{lemma-desingularize-strictly-standard}", "we can find a finitely presented $R$-algebra $B$ and", "factorizations $A \\to B \\to \\Lambda$ and $D \\to B \\to \\Lambda$", "such that $H_{D/R}B \\subset H_{B/R}$. We omit the verification that", "this is a solution to the problem posed by the lemma." ], "refs": [ "smoothing-lemma-lifting", "smoothing-lemma-desingularize-strictly-standard" ], "ref_ids": [ 5626, 5628 ] } ], "ref_ids": [] }, { "id": 5630, "type": "theorem", "label": "smoothing-lemma-product", "categories": [ "smoothing" ], "title": "smoothing-lemma-product", "contents": [ "Let $R_i \\to \\Lambda_i$, $i = 1, 2$ be as in Situation \\ref{situation-global}.", "If PT holds for $R_i \\to \\Lambda_i$, $i = 1, 2$, then PT holds for", "$R_1 \\times R_2 \\to \\Lambda_1 \\times \\Lambda_2$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: A product of colimits is a colimit." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5631, "type": "theorem", "label": "smoothing-lemma-delocalize-base", "categories": [ "smoothing" ], "title": "smoothing-lemma-delocalize-base", "contents": [ "Let $R \\to A \\to \\Lambda$ be ring maps with $A$ of finite presentation", "over $R$. Let $S \\subset R$ be a multiplicative", "set. Let $S^{-1}A \\to B' \\to S^{-1}\\Lambda$ be a factorization with", "$B'$ smooth over $S^{-1}R$. Then we can find a factorization", "$A \\to B \\to \\Lambda$ such that some $s \\in S$ maps to an", "elementary standard element (Definition \\ref{definition-strictly-standard})", "in $B$ over $R$." ], "refs": [ "smoothing-definition-strictly-standard" ], "proofs": [ { "contents": [ "We first apply Lemma \\ref{lemma-smooth-standard-smooth} to $S^{-1}R \\to B'$.", "Thus we may assume $B'$ is standard smooth over $S^{-1}R$.", "Write $A = R[x_1, \\ldots, x_n]/(g_1, \\ldots, g_t)$ and say", "$x_i \\mapsto \\lambda_i$ in $\\Lambda$. We may write", "$B' = S^{-1}R[x_1, \\ldots, x_{n + m}]/(f_1, \\ldots, f_c)$", "for some $c \\geq n$ where", "$\\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}$", "is invertible in $B'$ and such that $A \\to B'$ is given by $x_i \\mapsto x_i$,", "see Lemma \\ref{lemma-standard-smooth-include-generators}.", "After multiplying $x_i$, $i > n$ by an element of $S$ and correspondingly", "modifying the equations $f_j$ we may assume $B' \\to S^{-1}\\Lambda$ maps", "$x_i$ to $\\lambda_i/1$ for some $\\lambda_i \\in \\Lambda$ for $i > n$.", "Choose a relation", "$$", "1 =", "a_0 \\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}", "+", "\\sum\\nolimits_{j = 1, \\ldots, c} a_jf_j", "$$", "for some $a_j \\in S^{-1}R[x_1, \\ldots, x_{n + m}]$. Since each element of $S$", "is invertible in $B'$ we may (by clearing denominators) assume that", "$f_j, a_j \\in R[x_1, \\ldots, x_{n + m}]$ and that", "$$", "s_0 = a_0 \\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}", "+", "\\sum\\nolimits_{j = 1, \\ldots, c} a_jf_j", "$$", "for some $s_0 \\in S$. Since $g_j$ maps to zero in", "$S^{-1}R[x_1, \\ldots, x_{n + m}]/(f_1, \\ldots, x_c)$", "we can find elements $s_j \\in S$ such that $s_j g_j = 0$ in", "$R[x_1, \\ldots, x_{n + m}]/(f_1, \\ldots, f_c)$.", "Since $f_j$ maps to zero in $S^{-1}\\Lambda$ we can find $s'_j \\in S$", "such that $s'_j f_j(\\lambda_1, \\ldots, \\lambda_{n + m}) = 0$ in", "$\\Lambda$. Consider the ring", "$$", "B = R[x_1, \\ldots, x_{n + m}]/", "(s'_1f_1, \\ldots, s'_cf_c, g_1, \\ldots, g_t)", "$$", "and the factorization $A \\to B \\to \\Lambda$ with $B \\to \\Lambda$ given by", "$x_i \\mapsto \\lambda_i$. We claim that $s = s_0s_1 \\ldots s_ts'_1 \\ldots s'_c$", "is elementary standard in $B$ over $R$ which finishes the proof.", "Namely, $s_j g_j \\in (f_1, \\ldots, f_c)$ and hence", "$sg_j \\in (s'_1f_1, \\ldots, s'_cf_c)$. Finally, we have", "$$", "a_0\\det(\\partial s'_jf_j/\\partial x_i)_{i, j = 1, \\ldots, c}", "+", "\\sum\\nolimits_{j = 1, \\ldots, c}", "(s'_1 \\ldots \\hat{s'_j} \\ldots s'_c) a_j s'_jf_j", "=", "s_0s'_1\\ldots s'_c", "$$", "which divides $s$ as desired." ], "refs": [ "smoothing-lemma-smooth-standard-smooth", "smoothing-lemma-standard-smooth-include-generators" ], "ref_ids": [ 5615, 5617 ] } ], "ref_ids": [ 5648 ] }, { "id": 5632, "type": "theorem", "label": "smoothing-lemma-reduce-to-field", "categories": [ "smoothing" ], "title": "smoothing-lemma-reduce-to-field", "contents": [ "\\begin{slogan}", "Proving Popescu approximation reduces to algebras over a field", "\\end{slogan}", "If for every Situation \\ref{situation-global} where $R$", "is a field PT holds, then PT holds in general." ], "refs": [], "proofs": [ { "contents": [ "Assume PT holds for any Situation \\ref{situation-global} where $R$ is a field.", "Let $R \\to \\Lambda$ be as in Situation \\ref{situation-global} arbitrary.", "Note that $R/I \\to \\Lambda/I\\Lambda$ is another regular ring map", "of Noetherian rings, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-regular-base-change}.", "Consider the set of ideals", "$$", "\\mathcal{I} = \\{I \\subset R \\mid R/I \\to \\Lambda/I\\Lambda", "\\text{ does not have PT}\\}", "$$", "We have to show that $\\mathcal{I}$ is empty. If this set is nonempty,", "then it contains a maximal element because $R$ is Noetherian.", "Replacing $R$ by $R/I$ and $\\Lambda$ by $\\Lambda/I$ we obtain a", "situation where PT holds for $R/I \\to \\Lambda/I\\Lambda$ for any", "nonzero ideal of $R$. In particular, we see by applying", "Proposition \\ref{proposition-lift}", "that $R$ is a reduced ring.", "\\medskip\\noindent", "Let $A \\to \\Lambda$ be an $R$-algebra homomorphism with $A$ of", "finite presentation. We have to find a factorization $A \\to B \\to \\Lambda$", "with $B$ smooth over $R$, see Algebra, Lemma \\ref{algebra-lemma-when-colimit}.", "\\medskip\\noindent", "Let $S \\subset R$ be the set of nonzerodivisors and", "consider the total ring of fractions $Q = S^{-1}R$ of $R$. We know that", "$Q = K_1 \\times \\ldots \\times K_n$ is a product of fields, see", "Algebra, Lemmas \\ref{algebra-lemma-total-ring-fractions-no-embedded-points} and", "\\ref{algebra-lemma-Noetherian-irreducible-components}.", "By Lemma \\ref{lemma-product} and our assumption", "PT holds for the ring map $S^{-1}R \\to S^{-1}\\Lambda$.", "Hence we can find a factorization $S^{-1}A \\to B' \\to S^{-1}\\Lambda$", "with $B'$ smooth over $S^{-1}R$.", "\\medskip\\noindent", "We apply Lemma \\ref{lemma-delocalize-base}", "and find a factorization $A \\to B \\to \\Lambda$ such that", "some $\\pi \\in S$ is elementary standard in $B$ over $R$.", "After replacing $A$ by $B$ we may assume that $\\pi$ is", "elementary standard, hence strictly standard in $A$. We know that", "$R/\\pi^8R \\to \\Lambda/\\pi^8\\Lambda$ satisfies PT.", "Hence we can find a factorization", "$R/\\pi^8 R \\to A/\\pi^8A \\to \\bar C \\to \\Lambda/\\pi^8\\Lambda$", "with $R/\\pi^8 R \\to \\bar C$ smooth. By", "Lemma \\ref{lemma-lifting}", "we can find an $R$-algebra map $D \\to \\Lambda$ with $D$ smooth over $R$", "and a factorization", "$R/\\pi^4 R \\to A/\\pi^4A \\to D/\\pi^4D \\to \\Lambda/\\pi^4\\Lambda$.", "By Lemma \\ref{lemma-desingularize-strictly-standard}", "we can find $A \\to B \\to \\Lambda$ with $B$ smooth over $R$", "which finishes the proof." ], "refs": [ "more-algebra-lemma-regular-base-change", "smoothing-proposition-lift", "algebra-lemma-when-colimit", "algebra-lemma-total-ring-fractions-no-embedded-points", "algebra-lemma-Noetherian-irreducible-components", "smoothing-lemma-product", "smoothing-lemma-delocalize-base", "smoothing-lemma-lifting", "smoothing-lemma-desingularize-strictly-standard" ], "ref_ids": [ 10035, 5646, 1093, 421, 453, 5630, 5631, 5626, 5628 ] } ], "ref_ids": [] }, { "id": 5633, "type": "theorem", "label": "smoothing-lemma-lift-solution", "categories": [ "smoothing" ], "title": "smoothing-lemma-lift-solution", "contents": [ "Let $R \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in", "Situation \\ref{situation-local}. Let $r \\geq 1$ and", "$\\pi_1, \\ldots, \\pi_r \\in R$ map to elements of $\\mathfrak q$. Assume", "\\begin{enumerate}", "\\item for $i = 1, \\ldots, r$ we have", "$$", "\\text{Ann}_{R/(\\pi_1^8, \\ldots, \\pi_{i - 1}^8)R}(\\pi_i)", "=", "\\text{Ann}_{R/(\\pi_1^8, \\ldots, \\pi_{i - 1}^8)R}(\\pi_i^2)", "$$", "and", "$$", "\\text{Ann}_{\\Lambda/(\\pi_1^8, \\ldots, \\pi_{i - 1}^8)\\Lambda}(\\pi_i)", "=", "\\text{Ann}_{\\Lambda/(\\pi_1^8, \\ldots, \\pi_{i - 1}^8)\\Lambda}(\\pi_i^2)", "$$", "\\item for $i = 1, \\ldots, r$ the element $\\pi_i$ maps to a strictly", "standard element in $A$ over $R$.", "\\end{enumerate}", "Then, if", "$$", "R/(\\pi_1^8, \\ldots, \\pi_r^8)R \\to A/(\\pi_1^8, \\ldots, \\pi_r^8)A", "\\to \\Lambda/(\\pi_1^8, \\ldots, \\pi_r^8)\\Lambda \\supset", "\\mathfrak q/(\\pi_1^8, \\ldots, \\pi_r^8)\\Lambda", "$$", "can be resolved, so can $R \\to A \\to \\Lambda \\supset \\mathfrak q$." ], "refs": [], "proofs": [ { "contents": [ "We are going to prove this by induction on $r$.", "\\medskip\\noindent", "The case $r = 1$. Here the assumption is that there exists a", "factorization $A/\\pi_1^8 \\to \\bar C \\to \\Lambda/\\pi_1^8$", "which resolves the situation modulo $\\pi_1^8$. Conditions (1) and", "(2) are the assumptions needed to apply", "Lemma \\ref{lemma-desingularize-lifting-apply}.", "Thus we can ``lift'' the resolution $\\bar C$", "to a resolution of $R \\to A \\to \\Lambda \\supset \\mathfrak q$.", "\\medskip\\noindent", "The case $r > 1$. In this case we apply the induction hypothesis for $r - 1$", "to the situation", "$R/\\pi_1^8 \\to A/\\pi_1^8 \\to \\Lambda/\\pi_1^8", "\\supset \\mathfrak q/\\pi_1^8\\Lambda$.", "Note that property (2) is preserved by", "Lemma \\ref{lemma-strictly-standard-base-change}." ], "refs": [ "smoothing-lemma-desingularize-lifting-apply", "smoothing-lemma-strictly-standard-base-change" ], "ref_ids": [ 5629, 5611 ] } ], "ref_ids": [] }, { "id": 5634, "type": "theorem", "label": "smoothing-lemma-delocalize-weak", "categories": [ "smoothing" ], "title": "smoothing-lemma-delocalize-weak", "contents": [ "\\begin{reference}", "\\cite[Lemma 12.2]{swan} or", "\\cite[Lemma 2]{popescu-GND}", "\\end{reference}", "Let $R \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in", "Situation \\ref{situation-local}. Let $\\mathfrak p = R \\cap \\mathfrak q$.", "Assume that $\\mathfrak q$ is minimal over $\\mathfrak h_A$ and that", "$R_\\mathfrak p \\to A_\\mathfrak p \\to \\Lambda_\\mathfrak q", "\\supset \\mathfrak q\\Lambda_\\mathfrak q$ can be resolved.", "Then there exists a factorization $A \\to C \\to \\Lambda$ with $C$ of", "finite presentation such that $H_{C/R} \\Lambda \\not \\subset \\mathfrak q$." ], "refs": [], "proofs": [ { "contents": [ "Let $A_\\mathfrak p \\to C \\to \\Lambda_\\mathfrak q$ be a resolution of", "$R_\\mathfrak p \\to A_\\mathfrak p \\to \\Lambda_\\mathfrak q", "\\supset \\mathfrak q\\Lambda_\\mathfrak q$. By our assumption", "that $\\mathfrak q$ is minimal over $\\mathfrak h_A$ this", "means that $H_{C/R_\\mathfrak p} \\Lambda_\\mathfrak q = \\Lambda_\\mathfrak q$.", "By Lemma \\ref{lemma-final-solve}", "we may assume that $C$ is smooth over $R_\\mathfrak p$.", "By Lemma \\ref{lemma-smooth-standard-smooth} we may assume that", "$C$ is standard smooth over $R_\\mathfrak p$.", "Write $A = R[x_1, \\ldots, x_n]/(g_1, \\ldots, g_t)$ and say", "$A \\to \\Lambda$ is given by $x_i \\mapsto \\lambda_i$.", "Write $C = R_\\mathfrak p[x_1, \\ldots, x_{n + m}]/(f_1, \\ldots, f_c)$", "for some $c \\geq n$ such that $A \\to C$ maps $x_i$ to $x_i$ and such that", "$\\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}$", "is invertible in $C$, see", "Lemma \\ref{lemma-standard-smooth-include-generators}.", "After clearing denominators we may assume", "$f_1, \\ldots, f_c$ are elements of $R[x_1, \\ldots, x_{n + m}]$.", "Of course", "$\\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}$", "is not invertible in $R[x_1, \\ldots, x_{n + m}]/(f_1, \\ldots, f_c)$", "but it becomes invertible after inverting some element $s_0 \\in R$,", "$s_0 \\not \\in \\mathfrak p$.", "As $g_j$ maps to zero under $R[x_1, \\ldots, x_n] \\to A \\to C$", "we can find $s_j \\in R$, $s_j \\not \\in \\mathfrak p$ such that", "$s_j g_j$ is zero in $R[x_1, \\ldots, x_{n + m}]/(f_1, \\ldots, f_c)$.", "Write $f_j = F_j(x_1, \\ldots, x_{n + m}, 1)$", "for some polynomial", "$F_j \\in R[x_1, \\ldots, x_n, X_{n + 1}, \\ldots, X_{n + m + 1}]$", "homogeneous in $X_{n + 1}, \\ldots, X_{n + m + 1}$.", "Pick $\\lambda_{n + i} \\in \\Lambda$, $i = 1, \\ldots, m + 1$ with", "$\\lambda_{n + m + 1} \\not \\in \\mathfrak q$ such that $x_{n + i}$ maps to", "$\\lambda_{n + i}/\\lambda_{n + m + 1}$ in $\\Lambda_\\mathfrak q$.", "Then", "\\begin{align*}", "F_j(\\lambda_1, \\ldots, \\lambda_{n + m + 1})", "& =", "(\\lambda_{n + m + 1})^{\\deg(F_j)} F_j(\\lambda_1, \\ldots, \\lambda_n,", "\\frac{\\lambda_{n + 1}}{\\lambda_{n + m + 1}}, \\ldots,", "\\frac{\\lambda_{n + m}}{\\lambda_{n + m + 1}}, 1) \\\\", "& =", "(\\lambda_{n + m + 1})^{\\deg(F_j)} f_j(\\lambda_1, \\ldots, \\lambda_n,", "\\frac{\\lambda_{n + 1}}{\\lambda_{n + m + 1}}, \\ldots,", "\\frac{\\lambda_{n + m}}{\\lambda_{n + m + 1}}) \\\\", "& = 0", "\\end{align*}", "in $\\Lambda_\\mathfrak q$. Thus we can find", "$\\lambda_0 \\in \\Lambda$, $\\lambda_0 \\not \\in \\mathfrak q$ such that", "$\\lambda_0 F_j(\\lambda_1, \\ldots, \\lambda_{n + m + 1}) = 0$", "in $\\Lambda$. Now we set $B$ equal to", "$$", "R[x_0, \\ldots, x_{n + m + 1}]/", "(g_1, \\ldots, g_t, x_0F_1(x_1, \\ldots, x_{n + m + 1}), \\ldots,", "x_0F_c(x_1, \\ldots, x_{n + m + 1}))", "$$", "which we map to $\\Lambda$ by mapping $x_i$ to $\\lambda_i$.", "Let $b$ be the image of $x_0 x_{n + m + 1} s_0 s_1 \\ldots s_t$ in $B$.", "Then $B_b$ is isomorphic to", "$$", "R_{s_0s_1 \\ldots s_t}[x_0, x_1, \\ldots, x_{n + m + 1}, 1/x_0x_{n + m + 1}]/", "(f_1, \\ldots, f_c)", "$$", "which is smooth over $R$ by construction.", "Since $b$ does not map to an element of $\\mathfrak q$, we win." ], "refs": [ "smoothing-lemma-final-solve", "smoothing-lemma-smooth-standard-smooth", "smoothing-lemma-standard-smooth-include-generators" ], "ref_ids": [ 5612, 5615, 5617 ] } ], "ref_ids": [] }, { "id": 5635, "type": "theorem", "label": "smoothing-lemma-delocalize-height-zero", "categories": [ "smoothing" ], "title": "smoothing-lemma-delocalize-height-zero", "contents": [ "Let $R \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in", "Situation \\ref{situation-local}. Let $\\mathfrak p = R \\cap \\mathfrak q$.", "Assume", "\\begin{enumerate}", "\\item $\\mathfrak q$ is minimal over $\\mathfrak h_A$,", "\\item $R_\\mathfrak p \\to A_\\mathfrak p \\to \\Lambda_\\mathfrak q", "\\supset \\mathfrak q\\Lambda_\\mathfrak q$ can be resolved, and", "\\item $\\dim(\\Lambda_\\mathfrak q) = 0$.", "\\end{enumerate}", "Then $R \\to A \\to \\Lambda \\supset \\mathfrak q$ can be resolved." ], "refs": [], "proofs": [ { "contents": [ "By (3) the ring $\\Lambda_\\mathfrak q$ is Artinian local hence", "$\\mathfrak q\\Lambda_\\mathfrak q$ is nilpotent. Thus", "$(\\mathfrak h_A)^N \\Lambda_\\mathfrak q = 0$ for some $N > 0$.", "Thus there exists a $\\lambda \\in \\Lambda$, $\\lambda \\not \\in \\mathfrak q$", "such that $\\lambda (\\mathfrak h_A)^N = 0$ in $\\Lambda$.", "Say $H_{A/R} = (a_1, \\ldots, a_r)$ so that $\\lambda a_i^N = 0$", "in $\\Lambda$. By Lemma \\ref{lemma-delocalize-weak} we can find a factorization", "$A \\to C \\to \\Lambda$ with $C$ of finite presentation such that", "$\\mathfrak h_C \\not \\subset \\mathfrak q$.", "Write $C = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$.", "Set", "$$", "B = A[x_1, \\ldots, x_n, y_1, \\ldots, y_r, z, t_{ij}]/", "(f_j - \\sum y_i t_{ij}, zy_i)", "$$", "where $t_{ij}$ is a set of $rm$ variables.", "Note that there is a map $B \\to C[y_i, z]/(y_iz)$ given by setting $t_{ij}$", "equal to zero. The map $B \\to \\Lambda$ is the composition", "$B \\to C[y_i, z]/(y_iz) \\to \\Lambda$ where $C[y_i, z]/(y_iz) \\to \\Lambda$", "is the given map $C \\to \\Lambda$, maps $z$ to $\\lambda$, and maps", "$y_i$ to the image of $a_i^N$ in $\\Lambda$.", "\\medskip\\noindent", "We claim that $B$ is a solution for $R \\to A \\to \\Lambda \\supset \\mathfrak q$.", "First note that $B_z$ is isomorphic to $C[y_1, \\ldots, y_r, z, z^{-1}]$", "and hence is smooth. On the other hand,", "$B_{y_\\ell} \\cong A[x_i, y_i, y_\\ell^{-1}, t_{ij}, i \\not = \\ell]$", "which is smooth over $A$. Thus we see that $z$ and $a_\\ell y_\\ell$", "(compositions of smooth maps are smooth) are all", "elements of $H_{B/R}$. This proves the lemma." ], "refs": [ "smoothing-lemma-delocalize-weak" ], "ref_ids": [ 5634 ] } ], "ref_ids": [] }, { "id": 5636, "type": "theorem", "label": "smoothing-lemma-ogoma", "categories": [ "smoothing" ], "title": "smoothing-lemma-ogoma", "contents": [ "Let $A$ be a Noetherian ring and let $M$ be a finite $A$-module.", "Let $S \\subset A$ be a multiplicative set. If $\\pi \\in A$ and", "$\\Ker(\\pi : S^{-1}M \\to S^{-1}M) =", "\\Ker(\\pi^2 : S^{-1}M \\to S^{-1}M)$", "then there exists an $s \\in S$ such that for any $n > 0$ we have", "$\\Ker(s^n\\pi : M \\to M) = \\Ker((s^n\\pi)^2 : M \\to M)$." ], "refs": [], "proofs": [ { "contents": [ "Let $K = \\Ker(\\pi : M \\to M)$ and", "$K' = \\{m \\in M \\mid \\pi^2 m = 0\\text{ in }S^{-1}M\\}$ and", "$Q = K'/K$. Note that $S^{-1}Q = 0$ by assumption. Since $A$", "is Noetherian we see that $Q$ is a finite $A$-module.", "Hence we can find an $s \\in S$ such that $s$ annihilates $Q$.", "Then $s$ works." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5637, "type": "theorem", "label": "smoothing-lemma-find-sequence", "categories": [ "smoothing" ], "title": "smoothing-lemma-find-sequence", "contents": [ "Let $\\Lambda$ be a Noetherian ring. Let $I \\subset \\Lambda$ be an ideal.", "Let $I \\subset \\mathfrak q$ be a prime. Let $n, e$ be positive integers", "Assume that $\\mathfrak q^n\\Lambda_\\mathfrak q \\subset I\\Lambda_\\mathfrak q$", "and that $\\Lambda_\\mathfrak q$ is a regular local ring of dimension $d$.", "Then there exists an $n > 0$ and", "$\\pi_1, \\ldots, \\pi_d \\in \\Lambda$ such that", "\\begin{enumerate}", "\\item $(\\pi_1, \\ldots, \\pi_d)\\Lambda_\\mathfrak q =", "\\mathfrak q\\Lambda_\\mathfrak q$,", "\\item $\\pi_1^n, \\ldots, \\pi_d^n \\in I$, and", "\\item for $i = 1, \\ldots, d$ we have", "$$", "\\text{Ann}_{\\Lambda/(\\pi_1^e, \\ldots, \\pi_{i - 1}^e)\\Lambda}(\\pi_i) =", "\\text{Ann}_{\\Lambda/(\\pi_1^e, \\ldots, \\pi_{i - 1}^e)\\Lambda}(\\pi_i^2).", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Set $S = \\Lambda \\setminus \\mathfrak q$ so that", "$\\Lambda_\\mathfrak q = S^{-1}\\Lambda$.", "First pick $\\pi_1, \\ldots, \\pi_d$ with (1) which is possible", "as $\\Lambda_\\mathfrak q$ is regular. By assumption", "$\\pi_i^n \\in I\\Lambda_\\mathfrak q$. Thus we can find", "$s_1, \\ldots, s_d \\in S$ such that $s_i\\pi_i^n \\in I$.", "Replacing $\\pi_i$ by $s_i\\pi_i$ we get (2).", "Note that (1) and (2) are preserved by further multiplying by elements of $S$.", "Suppose that (3) holds for $i = 1, \\ldots, t$ for some", "$t \\in \\{0, \\ldots, d\\}$. Note that", "$\\pi_1, \\ldots, \\pi_d$ is a regular sequence in $S^{-1}\\Lambda$, see", "Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}.", "In particular $\\pi_1^e, \\ldots, \\pi_t^e, \\pi_{t + 1}$ is a", "regular sequence in $S^{-1}\\Lambda = \\Lambda_\\mathfrak q$ by", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-powers}.", "Hence we see that", "$$", "\\text{Ann}_{S^{-1}\\Lambda/(\\pi_1^e, \\ldots, \\pi_{i - 1}^e)}(\\pi_i) =", "\\text{Ann}_{S^{-1}\\Lambda/(\\pi_1^e, \\ldots, \\pi_{i - 1}^e)}(\\pi_i^2).", "$$", "Thus we get (3) for $i = t + 1$ after replacing $\\pi_{t + 1}$ by $s\\pi_{t + 1}$", "for some $s \\in S$ by Lemma \\ref{lemma-ogoma}. By induction on $t$ this", "produces a sequence satisfying (1), (2), and (3)." ], "refs": [ "algebra-lemma-regular-ring-CM", "algebra-lemma-regular-sequence-powers", "smoothing-lemma-ogoma" ], "ref_ids": [ 941, 744, 5636 ] } ], "ref_ids": [] }, { "id": 5638, "type": "theorem", "label": "smoothing-lemma-resolve-special", "categories": [ "smoothing" ], "title": "smoothing-lemma-resolve-special", "contents": [ "Let $k \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in", "Situation \\ref{situation-local} where", "\\begin{enumerate}", "\\item $k$ is a field,", "\\item $\\Lambda$ is Noetherian,", "\\item $\\mathfrak q$ is minimal over $\\mathfrak h_A$,", "\\item $\\Lambda_\\mathfrak q$ is a regular local ring, and", "\\item the field extension $k \\subset \\kappa(\\mathfrak q)$ is separable.", "\\end{enumerate}", "Then $k \\to A \\to \\Lambda \\supset \\mathfrak q$ can be resolved." ], "refs": [], "proofs": [ { "contents": [ "Set $d = \\dim \\Lambda_\\mathfrak q$. Set $R = k[x_1, \\ldots, x_d]$.", "Choose $n > 0$ such that", "$\\mathfrak q^n\\Lambda_\\mathfrak q \\subset \\mathfrak h_A\\Lambda_\\mathfrak q$", "which is possible as $\\mathfrak q$ is minimal over $\\mathfrak h_A$.", "Choose generators $a_1, \\ldots, a_r$ of $H_{A/R}$. Set", "$$", "B = A[x_1, \\ldots, x_d, z_{ij}]/(x_i^n - \\sum z_{ij}a_j)", "$$", "Each $B_{a_j}$ is smooth over $R$ it is a polynomial", "algebra over $A_{a_j}[x_1, \\ldots, x_d]$ and $A_{a_j}$ is smooth over $k$.", "Hence $B_{x_i}$ is smooth over $R$. Let $B \\to C$ be the $R$-algebra", "map constructed in Lemma \\ref{lemma-improve-presentation}", "which comes with a $R$-algebra retraction $C \\to B$. In particular", "a map $C \\to \\Lambda$ fitting into the diagram above.", "By construction $C_{x_i}$ is a smooth $R$-algebra with", "$\\Omega_{C_{x_i}/R}$ free. Hence we can find $c > 0$", "such that $x_i^c$ is strictly standard in $C/R$, see", "Lemma \\ref{lemma-compare-standard}.", "Now choose $\\pi_1, \\ldots, \\pi_d \\in \\Lambda$ as in", "Lemma \\ref{lemma-find-sequence}", "where $n = n$, $e = 8c$, $\\mathfrak q = \\mathfrak q$ and $I = \\mathfrak h_A$.", "Write $\\pi_i^n = \\sum \\lambda_{ij} a_j$ for some $\\pi_{ij} \\in \\Lambda$.", "There is a map $B \\to \\Lambda$ given by $x_i \\mapsto \\pi_i$", "and $z_{ij} \\mapsto \\lambda_{ij}$. Set $R = k[x_1, \\ldots, x_d]$.", "Diagram", "$$", "\\xymatrix{", "R \\ar[r] & B \\ar[rd] \\\\", "k \\ar[u] \\ar[r] & A \\ar[u] \\ar[r] & \\Lambda", "}", "$$", "Now we apply", "Lemma \\ref{lemma-lift-solution}", "to $R \\to C \\to \\Lambda \\supset \\mathfrak q$", "and the sequence of elements $x_1^c, \\ldots, x_d^c$ of $R$.", "Assumption (2) is clear. Assumption (1) holds for $R$", "by inspection and for $\\Lambda$ by our choice of", "$\\pi_1, \\ldots, \\pi_d$. (Note that if", "$\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$, then we have", "$\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^c)$ for all $c > 0$.)", "Thus it suffices to resolve", "$$", "R/(x_1^e, \\ldots, x_d^e) \\to C/(x_1^e, \\ldots, x_d^e) \\to", "\\Lambda/(\\pi_1^e, \\ldots, \\pi_d^e) \\supset", "\\mathfrak q/(\\pi_1^e, \\ldots, \\pi_d^e)", "$$", "for $e = 8c$. By", "Lemma \\ref{lemma-delocalize-height-zero}", "it suffices to resolve this after localizing at $\\mathfrak q$.", "But since $x_1, \\ldots, x_d$ map to a regular sequence", "in $\\Lambda_\\mathfrak q$ we see that $R_\\mathfrak p \\to \\Lambda_\\mathfrak q$", "is flat, see Algebra, Lemma \\ref{algebra-lemma-flat-over-regular}. Hence", "$$", "R_\\mathfrak p/(x_1^e, \\ldots, x_d^e) \\to", "\\Lambda_\\mathfrak q/(\\pi_1^e, \\ldots, \\pi_d^e)", "$$", "is a flat ring map of Artinian local rings.", "Moreover, this map induces a separable field extension", "on residue fields by assumption. Thus this map is a filtered colimit", "of smooth algebras by", "Algebra, Lemma \\ref{algebra-lemma-colimit-syntomic}", "and Proposition \\ref{proposition-lift}.", "Existence of the desired solution follows from", "Algebra, Lemma \\ref{algebra-lemma-when-colimit}." ], "refs": [ "smoothing-lemma-improve-presentation", "smoothing-lemma-compare-standard", "smoothing-lemma-find-sequence", "smoothing-lemma-lift-solution", "smoothing-lemma-delocalize-height-zero", "algebra-lemma-flat-over-regular", "algebra-lemma-colimit-syntomic", "smoothing-proposition-lift", "algebra-lemma-when-colimit" ], "ref_ids": [ 5613, 5618, 5637, 5633, 5635, 1108, 1323, 5646, 1093 ] } ], "ref_ids": [] }, { "id": 5639, "type": "theorem", "label": "smoothing-lemma-helper", "categories": [ "smoothing" ], "title": "smoothing-lemma-helper", "contents": [ "Let $k$ be a field of characteristic $p > 0$.", "Let $(\\Lambda, \\mathfrak m, K)$ be an Artinian local $k$-algebra.", "Assume that $\\dim H_1(L_{K/k}) < \\infty$.", "Then $\\Lambda$ is a filtered colimit of Artinian", "local $k$-algebras $A$ with each map $A \\to \\Lambda$ flat, with", "$\\mathfrak m_A \\Lambda = \\mathfrak m$, and with", "$A$ essentially of finite type over $k$." ], "refs": [], "proofs": [ { "contents": [ "Note that the flatness of $A \\to \\Lambda$ implies that $A \\to \\Lambda$", "is injective, so the lemma really tells us that $\\Lambda$ is a", "directed union of these types of subrings $A \\subset \\Lambda$.", "Let $n$ be the minimal integer such that $\\mathfrak m^n = 0$.", "We will prove this lemma by induction on $n$. The case $n = 1$ is clear", "as a field extension is a union of finitely generated field extensions.", "\\medskip\\noindent", "Pick $\\lambda_1, \\ldots, \\lambda_d \\in \\mathfrak m$ which generate", "$\\mathfrak m$. As $K$ is formally smooth over $\\mathbf{F}_p$ (see", "Algebra, Lemma \\ref{algebra-lemma-formally-smooth-extensions-easy}) we can", "find a ring map $\\sigma : K \\to \\Lambda$ which is a section of the", "quotient map $\\Lambda \\to K$. In general $\\sigma$ is {\\bf not}", "a $k$-algebra map. Given $\\sigma$ we define", "$$", "\\Psi_\\sigma : K[x_1, \\ldots, x_d] \\longrightarrow \\Lambda", "$$", "using $\\sigma$ on elements of $K$ and mapping $x_i$ to $\\lambda_i$.", "Claim: there exists a $\\sigma : K \\to \\Lambda$", "and a subfield $k \\subset F \\subset K$ finitely generated over $k$", "such that the image of $k$ in $\\Lambda$ is contained in", "$\\Psi_\\sigma(F[x_1, \\ldots, x_d])$.", "\\medskip\\noindent", "We will prove the claim by induction on the least integer $n$ such that", "$\\mathfrak m^n = 0$. It is clear for $n = 1$. If $n > 1$ set", "$I = \\mathfrak m^{n - 1}$ and $\\Lambda' = \\Lambda/I$.", "By induction we may assume", "given $\\sigma' : K \\to \\Lambda'$ and $k \\subset F' \\subset K$ finitely", "generated such that the image of $k \\to \\Lambda \\to \\Lambda'$", "is contained in $A' = \\Psi_{\\sigma'}(F'[x_1, \\ldots, x_d])$.", "Denote $\\tau' : k \\to A'$ the induced map.", "Choose a lift $\\sigma : K \\to \\Lambda$ of $\\sigma'$ (this is possible", "by the formal smoothness of $K/\\mathbf{F}_p$ we mentioned above).", "For later reference we note that we can change $\\sigma$ to", "$\\sigma + D$ for some derivation $D : K \\to I$.", "Set $A = F[x_1, \\ldots, x_d]/(x_1, \\ldots, x_d)^n$.", "Then $\\Psi_\\sigma$ induces a ring map", "$\\Psi_\\sigma : A \\to \\Lambda$. The composition with the", "quotient map $\\Lambda \\to \\Lambda'$ induces a surjective", "map $A \\to A'$ with nilpotent kernel.", "Choose a lift $\\tau : k \\to A$ of $\\tau'$ (possible as $k/\\mathbf{F}_p$", "is formally smooth). Thus we obtain two maps $k \\to \\Lambda$, namely", "$\\Psi_\\sigma \\circ \\tau : k \\to \\Lambda$ and the given map $i : k \\to \\Lambda$.", "These maps agree modulo $I$, whence the difference is a", "derivation $\\theta = i - \\Psi_\\sigma \\circ \\tau : k \\to I$.", "Note that if we change $\\sigma$ into $\\sigma + D$ then we change", "$\\theta$ into $\\theta - D|_k$.", "\\medskip\\noindent", "Choose a set of elements $\\{y_j\\}_{j \\in J}$ of $k$ whose differentials", "$\\text{d}y_j$ form a basis of $\\Omega_{k/\\mathbf{F}_p}$. The Jacobi-Zariski", "sequence for $\\mathbf{F}_p \\subset k \\subset K$ is", "$$", "0 \\to H_1(L_{K/k}) \\to \\Omega_{k/\\mathbf{F}_p} \\otimes K \\to", "\\Omega_{K/\\mathbf{F}_p} \\to \\Omega_{K/k} \\to 0", "$$", "As $\\dim H_1(L_{K/k}) < \\infty$ we can find a finite subset $J_0 \\subset J$", "such that the image of the first map is contained in", "$\\bigoplus_{j \\in J_0} K\\text{d}y_j$. Hence the elements", "$\\text{d}y_j$, $j \\in J \\setminus J_0$ map to $K$-linearly independent", "elements of $\\Omega_{K/\\mathbf{F}_p}$. Therefore we can choose", "a $D : K \\to I$ such that $\\theta - D|_k = \\xi \\circ \\text{d}$", "where $\\xi$ is a composition", "$$", "\\Omega_{k/\\mathbf{F}_p} = \\bigoplus\\nolimits_{j \\in J} k \\text{d}y_j", "\\longrightarrow \\bigoplus\\nolimits_{j \\in J_0} k \\text{d}y_j", "\\longrightarrow I", "$$", "Let $f_j = \\xi(\\text{d}y_j) \\in I$ for $j \\in J_0$.", "Change $\\sigma$ into $\\sigma + D$ as above. Then we see that", "$\\theta(a) = \\sum_{j \\in J_0} a_j f_j$ for $a \\in k$ where", "$\\text{d}a = \\sum a_j \\text{d}y_j$ in $\\Omega_{k/\\mathbf{F}_p}$.", "Note that $I$ is generated by the monomials", "$\\lambda^E = \\lambda_1^{e_1} \\ldots \\lambda_d^{e_d}$ of", "total degree $|E| = \\sum e_i = n - 1$ in $\\lambda_1, \\ldots, \\lambda_d$.", "Write $f_j = \\sum_E c_{j, E} \\lambda^E$ with $c_{j, E} \\in K$.", "Replace $F'$ by $F = F'(c_{j, E})$. Then the claim holds.", "\\medskip\\noindent", "Choose $\\sigma$ and $F$ as in the claim. The kernel of $\\Psi_\\sigma$ is", "generated by finitely many polynomials", "$g_1, \\ldots, g_t \\in K[x_1, \\ldots, x_d]$ and we may assume their", "coefficients are in $F$ after enlarging $F$ by adjoining finitely many", "elements. In this case it is clear that the map", "$A = F[x_1, \\ldots, x_d]/(g_1, \\ldots, g_t) \\to", "K[x_1, \\ldots, x_d]/(g_1, \\ldots, g_t) = \\Lambda$ is flat.", "By the claim $A$ is a $k$-subalgebra of $\\Lambda$.", "It is clear that $\\Lambda$ is the filtered colimit of these", "algebras, as $K$ is the filtered union of the subfields $F$.", "Finally, these algebras are essentially of finite type over $k$ by", "Algebra, Lemma", "\\ref{algebra-lemma-essentially-of-finite-type-into-artinian-local}." ], "refs": [ "algebra-lemma-formally-smooth-extensions-easy", "algebra-lemma-essentially-of-finite-type-into-artinian-local" ], "ref_ids": [ 1320, 649 ] } ], "ref_ids": [] }, { "id": 5640, "type": "theorem", "label": "smoothing-lemma-solution-modulo", "categories": [ "smoothing" ], "title": "smoothing-lemma-solution-modulo", "contents": [ "Let $k$ be a field of characteristic $p > 0$.", "Let $\\Lambda$ be a Noetherian geometrically regular $k$-algebra.", "Let $\\mathfrak q \\subset \\Lambda$ be a prime ideal.", "Let $n \\geq 1$ be an integer and let", "$E \\subset \\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q$", "be a finite subset.", "Then we can find $m \\geq 0$ and", "$\\varphi : k[y_1, \\ldots, y_m] \\to \\Lambda$ with the following properties", "\\begin{enumerate}", "\\item setting $\\mathfrak p = \\varphi^{-1}(\\mathfrak q)$ we have", "$\\mathfrak q\\Lambda_\\mathfrak q = \\mathfrak p \\Lambda_\\mathfrak q$", "and $k[y_1, \\ldots, y_m]_\\mathfrak p \\to \\Lambda_\\mathfrak q$ is flat,", "\\item there is a factorization by homomorphisms of local Artinian rings", "$$", "k[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n k[y_1, \\ldots, y_m]_\\mathfrak p", "\\to D \\to", "\\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q", "$$", "where the first arrow is essentially smooth and the second is flat,", "\\item $E$ is contained in $D$ modulo $\\mathfrak q^n\\Lambda_\\mathfrak q$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Set $\\bar \\Lambda = \\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q$.", "Note that $\\dim H_1(L_{\\kappa(\\mathfrak q)/k}) < \\infty$ by", "More on Algebra, Proposition", "\\ref{more-algebra-proposition-characterization-geometrically-regular}.", "Pick $A \\subset \\bar \\Lambda$ containing $E$ such that $A$ is local", "Artinian, essentially of finite type over $k$, the map", "$A \\to \\bar \\Lambda$ is flat, and $\\mathfrak m_A$ generates the maximal", "ideal of $\\bar \\Lambda$, see Lemma \\ref{lemma-helper}.", "Denote $F = A/\\mathfrak m_A$ the residue field so that $k \\subset F \\subset K$.", "Pick $\\lambda_1, \\ldots, \\lambda_t \\in \\Lambda$ which map", "to elements of $A$ in $\\bar \\Lambda$ such that moreover the images", "of $\\text{d}\\lambda_1, \\ldots, \\text{d}\\lambda_t$ form a basis", "of $\\Omega_{F/k}$. Consider the map", "$\\varphi' : k[y_1, \\ldots, y_t] \\to \\Lambda$ sending $y_j$ to $\\lambda_j$.", "Set $\\mathfrak p' = (\\varphi')^{-1}(\\mathfrak q)$. By", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-geometrically-regular-over-field}", "the ring map $k[y_1, \\ldots, y_t]_{\\mathfrak p'} \\to \\Lambda_\\mathfrak q$", "is flat and $\\Lambda_\\mathfrak q/\\mathfrak p' \\Lambda_\\mathfrak q$ is", "regular. Thus we can choose further elements", "$\\lambda_{t + 1}, \\ldots, \\lambda_m \\in \\Lambda$", "which map into $A \\subset \\bar \\Lambda$ and which", "map to a regular system of parameters of", "$\\Lambda_\\mathfrak q/\\mathfrak p' \\Lambda_\\mathfrak q$.", "We obtain $\\varphi : k[y_1, \\ldots, y_m] \\to \\Lambda$ having", "property (1) such that", "$k[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n k[y_1, \\ldots, y_m]_\\mathfrak p", "\\to \\bar\\Lambda$", "factors through $A$. Thus", "$k[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n k[y_1, \\ldots, y_m]_\\mathfrak p", "\\to A$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-flatness-descends-more-general}.", "By construction the residue field extension $\\kappa(\\mathfrak p) \\subset F$", "is finitely generated and $\\Omega_{F/\\kappa(\\mathfrak p)} = 0$. Hence it is", "finite separable by", "More on Algebra, Lemma \\ref{more-algebra-lemma-cartier-equality}.", "Thus", "$k[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n k[y_1, \\ldots, y_m]_\\mathfrak p", "\\to A$", "is finite by Algebra, Lemma", "\\ref{algebra-lemma-essentially-of-finite-type-into-artinian-local}.", "Finally, we conclude that it is \\'etale by", "Algebra, Lemma \\ref{algebra-lemma-characterize-etale}.", "Since an \\'etale ring map is certainly essentially smooth we win." ], "refs": [ "more-algebra-proposition-characterization-geometrically-regular", "smoothing-lemma-helper", "more-algebra-lemma-geometrically-regular-over-field", "algebra-lemma-flatness-descends-more-general", "more-algebra-lemma-cartier-equality", "algebra-lemma-essentially-of-finite-type-into-artinian-local", "algebra-lemma-characterize-etale" ], "ref_ids": [ 10576, 5639, 10009, 529, 10006, 649, 1235 ] } ], "ref_ids": [] }, { "id": 5641, "type": "theorem", "label": "smoothing-lemma-enlarge-solution-modulo", "categories": [ "smoothing" ], "title": "smoothing-lemma-enlarge-solution-modulo", "contents": [ "Let $\\varphi : k[y_1, \\ldots, y_m] \\to \\Lambda$, $n$, $\\mathfrak q$,", "$\\mathfrak p$ and", "$$", "k[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n \\to", "D \\to \\Lambda_\\mathfrak q/\\mathfrak q^n \\Lambda_\\mathfrak q", "$$", "be as in Lemma \\ref{lemma-solution-modulo}. Then for any", "$\\lambda \\in \\Lambda \\setminus \\mathfrak q$", "there exists an integer $q > 0$ and a factorization", "$$", "k[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n \\to", "D \\to D' \\to \\Lambda_\\mathfrak q/\\mathfrak q^n \\Lambda_\\mathfrak q", "$$", "such that $D \\to D'$ is an essentially smooth map of local Artinian rings,", "the last arrow is flat, and $\\lambda^q$ is in $D'$." ], "refs": [ "smoothing-lemma-solution-modulo" ], "proofs": [ { "contents": [ "Set $\\bar \\Lambda = \\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q$.", "Let $\\bar \\lambda$ be the image of $\\lambda$ in $\\bar \\Lambda$.", "Let $\\alpha \\in \\kappa(\\mathfrak q)$ be the image of $\\lambda$ in the", "residue field.", "Let $k \\subset F \\subset \\kappa(\\mathfrak q)$ be the residue field of $D$.", "If $\\alpha$ is in $F$ then we can find an", "$x \\in D$ such that $x \\bar\\lambda = 1 \\bmod \\mathfrak q$. Hence", "$(x \\bar \\lambda)^q = 1 \\bmod (\\mathfrak q)^q$ if $q$ is divisible by $p$.", "Hence $\\bar\\lambda^q$ is in $D$. If $\\alpha$ is", "transcendental over $F$, then we can take $D' = (D[\\bar \\lambda])_\\mathfrak m$", "equal to the subring generated by $D$ and $\\bar \\lambda$ localized", "at $\\mathfrak m = D[\\bar \\lambda] \\cap \\mathfrak q \\bar \\Lambda$.", "This works because $D[\\bar \\lambda]$ is in fact a polynomial algebra", "over $D$ in this case. Finally, if $\\lambda \\bmod \\mathfrak q$ is", "algebraic over $F$, then we can find a $p$-power $q$ such that", "$\\alpha^q$ is separable algebraic over $F$, see", "Fields, Section \\ref{fields-section-algebraic}.", "Note that $D$ and $\\bar\\Lambda$ are henselian local rings, see", "Algebra, Lemma \\ref{algebra-lemma-local-dimension-zero-henselian}.", "Let $D \\to D'$ be a finite \\'etale extension", "whose residue field extension is $F \\subset F(\\alpha^q)$, see", "Algebra, Lemma \\ref{algebra-lemma-henselian-cat-finite-etale}.", "Since $\\bar\\Lambda$ is henselian and $F(\\alpha^q)$ is contained", "in its residue field we can find a factorization", "$D' \\to \\bar \\Lambda$. By the first part of the argument", "we see that $\\bar\\lambda^{qq'} \\in D'$ for some $q' > 0$." ], "refs": [ "algebra-lemma-local-dimension-zero-henselian", "algebra-lemma-henselian-cat-finite-etale" ], "ref_ids": [ 1283, 1280 ] } ], "ref_ids": [ 5640 ] }, { "id": 5642, "type": "theorem", "label": "smoothing-lemma-resolve-general", "categories": [ "smoothing" ], "title": "smoothing-lemma-resolve-general", "contents": [ "Let $k \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in", "Situation \\ref{situation-local} where", "\\begin{enumerate}", "\\item $k$ is a field of characteristic $p > 0$,", "\\item $\\Lambda$ is Noetherian and geometrically regular over $k$,", "\\item $\\mathfrak q$ is minimal over $\\mathfrak h_A$.", "\\end{enumerate}", "Then $k \\to A \\to \\Lambda \\supset \\mathfrak q$ can be resolved." ], "refs": [], "proofs": [ { "contents": [ "The lemma is proven by the following steps in the given order.", "We will justify each of these steps below.", "\\begin{enumerate}", "\\item", "\\label{item-power}", "Pick an integer $N > 0$ such that", "$\\mathfrak q^N\\Lambda_\\mathfrak q \\subset H_{A/k}\\Lambda_\\mathfrak q$.", "\\item", "\\label{item-generators}", "Pick generators $a_1, \\ldots, a_t \\in A$ of the ideal $H_{A/R}$.", "\\item", "\\label{item-dimension}", "Set $d = \\dim(\\Lambda_\\mathfrak q)$.", "\\item", "\\label{item-standardizer}", "Set $B = A[x_1, \\ldots, x_d, z_{ij}]/(x_i^{2N} - \\sum z_{ij}a_j)$.", "\\item", "\\label{item-elkik}", "Consider $B$ as a $k[x_1, \\ldots, x_d]$-algebra and let", "$B \\to C$ be as in", "Lemma \\ref{lemma-improve-presentation}.", "We also obtain a section $C \\to B$.", "\\item", "\\label{item-strictly-standard}", "Choose $c > 0$ such that each $x_i^c$", "is strictly standard in $C$ over $k[x_1, \\ldots, x_d]$.", "\\item", "\\label{item-set-n}", "Set $n = N + dc$ and $e = 8c$.", "\\item", "\\label{item-set-E}", "Let $E \\subset \\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q$", "be the images of generators of $A$ as a $k$-algebra.", "\\item", "\\label{item-NP}", "Choose an integer $m$ and a $k$-algebra map", "$\\varphi : k[y_1, \\ldots, y_m] \\to \\Lambda$", "and a factorization by local Artinian rings", "$$", "k[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n k[y_1, \\ldots, y_m]_\\mathfrak p", "\\to D \\to", "\\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q", "$$", "such that the first arrow is essentially smooth, the second is flat,", "$E$ is contained in $D$, with $\\mathfrak p = \\varphi^{-1}(\\mathfrak q)$", "the map $k[y_1, \\ldots, y_m]_\\mathfrak p \\to \\Lambda_\\mathfrak q$ is", "flat, and $\\mathfrak p \\Lambda_\\mathfrak q = \\mathfrak q \\Lambda_\\mathfrak q$.", "\\item", "\\label{item-choose-pii}", "Choose $\\pi_1, \\ldots, \\pi_d \\in \\mathfrak p$ which map to a", "regular system of parameters of $k[y_1, \\ldots, y_m]_\\mathfrak p$.", "\\item", "\\label{item-set-R}", "Let $R = k[y_1, \\ldots, y_m, t_1, \\ldots, t_m]$ and $\\gamma_i = \\pi_i t_i$.", "\\item", "\\label{item-modify-pii}", "If necessary modify the choice of $\\pi_i$ such that", "for $i = 1, \\ldots, d$ we have ", "$$", "\\text{Ann}_{R/(\\gamma_1^e, \\ldots, \\gamma_{i - 1}^e)R}(\\gamma_i)", "=", "\\text{Ann}_{R/(\\gamma_1^e, \\ldots, \\gamma_{i - 1}^e)R}(\\gamma_i^2)", "$$", "\\item", "\\label{item-choose-deltai}", "There exist $\\delta_1, \\ldots, \\delta_d \\in \\Lambda$,", "$\\delta_i \\not \\in \\mathfrak q$ and a factorization", "$D \\to D' \\to \\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q$", "with $D'$ local Artinian, $D \\to D'$ essentially smooth, the map", "$D' \\to \\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q$", "flat such that, with $\\pi_i' = \\delta_i \\pi_i$, we have for", "$i = 1, \\ldots, d$", "\\begin{enumerate}", "\\item $(\\pi_i')^{2N} = \\sum a_j\\lambda_{ij}$ in $\\Lambda$ where", "$\\lambda_{ij} \\bmod \\mathfrak q^n\\Lambda_\\mathfrak q$ is an element of $D'$,", "\\item $\\text{Ann}_{\\Lambda/({\\pi'}_1^e, \\ldots, {\\pi'}_{i - 1}^e)}({\\pi'}_i) =", "\\text{Ann}_{\\Lambda/({\\pi'}_1^e, \\ldots, {\\pi'}_{i - 1}^e)}({\\pi'}_i^2)$,", "\\item $\\delta_i \\bmod \\mathfrak q^n\\Lambda_\\mathfrak q$ is an element of $D'$.", "\\end{enumerate}", "\\item", "\\label{item-map-B-Lambda}", "Define $B \\to \\Lambda$ by sending $x_i$ to $\\pi'_i$ and", "$z_{ij}$ to $\\lambda_{ij}$ found above. Define $C \\to \\Lambda$", "by composing the map $B \\to \\Lambda$ with the retraction $C \\to B$.", "\\item", "\\label{item-set-map-R}", "Map $R \\to \\Lambda$ by $\\varphi$ on $k[y_1, \\ldots, y_m]$", "and by sending $t_i$ to $\\delta_i$. Further introduce a map", "$$", "k[x_1, \\ldots, x_d]", "\\longrightarrow", "R = k[y_1, \\ldots, y_m, t_1, \\ldots, t_d]", "$$", "by sending $x_i$ to $\\gamma_i = \\pi_i t_i$.", "\\item", "\\label{item-first-solve}", "It suffices to resolve", "$$", "R", "\\to", "C \\otimes_{k[x_1, \\ldots, x_d]} R", "\\to", "\\Lambda \\supset \\mathfrak q", "$$", "\\item", "\\label{item-set-I}", "Set $I = (\\gamma_1^e, \\ldots, \\gamma_d^e) \\subset R$.", "\\item", "\\label{item-second-resolve}", "It suffices to resolve", "$$", "R/I", "\\to", "C \\otimes_{k[x_1, \\ldots, x_d]} R/I", "\\to", "\\Lambda/I\\Lambda \\supset \\mathfrak q/I\\Lambda", "$$", "\\item", "\\label{item-set-r}", "We denote $\\mathfrak r \\subset R = k[y_1, \\ldots, y_m, t_1, \\ldots, t_d]$", "the inverse image of $\\mathfrak q$.", "\\item", "\\label{item-third-resolve}", "It suffices to resolve", "$$", "(R/I)_\\mathfrak r \\to", "C \\otimes_{k[x_1, \\ldots, x_d]} (R/I)_\\mathfrak r \\to", "\\Lambda_\\mathfrak q/I\\Lambda_\\mathfrak q", "\\supset", "\\mathfrak q\\Lambda_\\mathfrak q/I\\Lambda_\\mathfrak q", "$$", "\\item", "\\label{item-fourth-resolve}", "Set $J = (\\pi_1^e, \\ldots, \\pi_d^e)$ in $k[y_1, \\ldots, y_m]$.", "\\item", "\\label{item-fifth-resolve}", "It suffices to resolve", "$$", "(R/JR)_\\mathfrak p \\to", "C \\otimes_{k[x_1, \\ldots, x_d]} (R/JR)_\\mathfrak p \\to", "\\Lambda_\\mathfrak q/J\\Lambda_\\mathfrak q", "\\supset", "\\mathfrak q\\Lambda_\\mathfrak q/J\\Lambda_\\mathfrak q", "$$", "\\item", "\\label{item-sixth-resolve}", "It suffices to resolve", "$$", "(R/\\mathfrak p^nR)_\\mathfrak p \\to", "C \\otimes_{k[x_1, \\ldots, x_d]} (R/\\mathfrak p^nR)_\\mathfrak p \\to", "\\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q", "\\supset", "\\mathfrak q\\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q", "$$", "\\item", "\\label{item-seventh-resolve}", "It suffices to resolve", "$$", "(R/\\mathfrak p^nR)_\\mathfrak p \\to", "B \\otimes_{k[x_1, \\ldots, x_d]} (R/\\mathfrak p^nR)_\\mathfrak p \\to", "\\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q", "\\supset", "\\mathfrak q\\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q", "$$", "\\item", "\\label{item-eighth-resolve}", "The ring $D'[t_1, \\ldots, t_d]$ is given the structure of an", "$R_\\mathfrak p/\\mathfrak p^nR_\\mathfrak p$-algebra by the given map", "$k[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n k[y_1, \\ldots, y_m]_\\mathfrak p", "\\to D'$ and by sending $t_i$ to $t_i$. It suffices to find a factorization", "$$", "B \\otimes_{k[x_1, \\ldots, x_d]} (R/\\mathfrak p^nR)_\\mathfrak p", "\\to D'[t_1, \\ldots, t_d] \\to", "\\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q", "$$", "where the second arrow sends $t_i$ to $\\delta_i$ and induces the given", "homomorphism $D' \\to \\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q$.", "\\item", "\\label{item-done}", "Such a factorization exists by our choice of $D'$ above.", "\\end{enumerate}", "We now give the justification for each of the steps, except that we", "skip justifying the steps which just introduce notation.", "\\medskip\\noindent", "Ad (\\ref{item-power}). This is possible as $\\mathfrak q$ is minimal", "over $\\mathfrak h_A = \\sqrt{H_{A/k}\\Lambda}$.", "\\medskip\\noindent", "Ad (\\ref{item-strictly-standard}). Note that $A_{a_i}$ is smooth", "over $k$. Hence $B_{a_j}$, which is isomorphic to a polynomial", "algebra over $A_{a_j}[x_1, \\ldots, x_d]$, is smooth over", "$k[x_1, \\ldots, x_d]$. Thus $B_{x_i}$ is smooth over $k[x_1, \\ldots, x_d]$.", "By Lemma \\ref{lemma-improve-presentation}", "we see that $C_{x_i}$ is smooth over $k[x_1, \\ldots, x_d]$", "with finite free module of differentials. Hence some power of", "$x_i$ is strictly standard in $C$ over $k[x_1, \\ldots, x_n]$", "by Lemma \\ref{lemma-compare-standard}.", "\\medskip\\noindent", "Ad (\\ref{item-NP}). This follows by applying Lemma \\ref{lemma-solution-modulo}.", "\\medskip\\noindent", "Ad (\\ref{item-choose-pii}). Since", "$k[y_1, \\ldots, y_m]_\\mathfrak p \\to \\Lambda_\\mathfrak q$ is", "flat and $\\mathfrak p \\Lambda_\\mathfrak q = \\mathfrak q \\Lambda_\\mathfrak q$", "by construction", "we see that $\\dim(k[y_1, \\ldots, y_m]_\\mathfrak p) = d$ by", "Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}.", "Thus we can find $\\pi_1, \\ldots, \\pi_d \\in \\Lambda$ which map to", "a regular system of parameters in $\\Lambda_\\mathfrak q$.", "\\medskip\\noindent", "Ad (\\ref{item-modify-pii}). By", "Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}", "any permutation of the sequence $\\pi_1, \\ldots, \\pi_d$ is a", "regular sequence in $k[y_1, \\ldots, y_m]_\\mathfrak p$. Hence", "$\\gamma_1 = \\pi_1 t_1, \\ldots, \\gamma_d = \\pi_d t_d$ is a regular", "sequence in", "$R_\\mathfrak p = k[y_1, \\ldots, y_m]_\\mathfrak p[t_1, \\ldots, t_d]$, see", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-polynomial-ring}.", "Let $S = k[y_1, \\ldots, y_m] \\setminus \\mathfrak p$ so that", "$R_\\mathfrak p = S^{-1}R$. Note that $\\pi_1, \\ldots, \\pi_d$", "and $\\gamma_1, \\ldots, \\gamma_d$", "remain regular sequences if we multiply our $\\pi_i$ by elements of $S$.", "Suppose that", "$$", "\\text{Ann}_{R/(\\gamma_1^e, \\ldots, \\gamma_{i - 1}^e)R}(\\gamma_i)", "=", "\\text{Ann}_{R/(\\gamma_1^e, \\ldots, \\gamma_{i - 1}^e)R}(\\gamma_i^2)", "$$", "holds for $i = 1, \\ldots, t$ for some $t \\in \\{0, \\ldots, d\\}$. Note that", "$\\gamma_1^e, \\ldots, \\gamma_t^e, \\gamma_{t + 1}$ is a", "regular sequence in $S^{-1}R$ by", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-powers}.", "Hence we see that", "$$", "\\text{Ann}_{S^{-1}R/(\\gamma_1^e, \\ldots, \\gamma_{i - 1}^e)}(\\gamma_i) =", "\\text{Ann}_{S^{-1}R/(\\gamma_1^e, \\ldots, \\gamma_{i - 1}^e)}(\\gamma_i^2).", "$$", "Thus we get", "$$", "\\text{Ann}_{R/(\\gamma_1^e, \\ldots, \\gamma_t^e)R}(\\gamma_{t + 1})", "=", "\\text{Ann}_{R/(\\gamma_1^e, \\ldots, \\gamma_t^e)R}(\\gamma_{t + 1}^2)", "$$", "after replacing $\\pi_{t + 1}$ by $s\\pi_{t + 1}$ for some $s \\in S$ by", "Lemma \\ref{lemma-ogoma}. By induction on $t$ this produces the desired", "sequence.", "\\medskip\\noindent", "Ad (\\ref{item-choose-deltai}). Let $S = \\Lambda \\setminus \\mathfrak q$", "so that $\\Lambda_\\mathfrak q = S^{-1}\\Lambda$. Set", "$\\bar \\Lambda = \\Lambda_\\mathfrak q/\\mathfrak q^n \\Lambda_\\mathfrak q$.", "Suppose that we have a $t \\in \\{0, \\ldots, d\\}$ and", "$\\delta_1, \\ldots, \\delta_t \\in S$ and a factorization", "$D \\to D' \\to \\bar \\Lambda$ as in (\\ref{item-choose-deltai})", "such that (a), (b), (c) hold for $i = 1, \\ldots, t$. We have", "$\\pi_{t + 1}^N \\in H_{A/k}\\Lambda_\\mathfrak q$", "as $\\mathfrak q^N \\Lambda_\\mathfrak q \\subset H_{A/k}\\Lambda_\\mathfrak q$", "by (\\ref{item-power}). Hence", "$\\pi_{t + 1}^N \\in H_{A/k} \\bar\\Lambda$. Hence", "$\\pi_{t + 1}^N \\in H_{A/k}D'$ as $D' \\to \\bar \\Lambda$", "is faithfully flat, see", "Algebra, Lemma \\ref{algebra-lemma-faithfully-flat-universally-injective}.", "Recall that $H_{A/k} = (a_1, \\ldots, a_t)$.", "Say $\\pi_{t + 1}^N = \\sum a_j d_j$ in $D'$ and choose", "$c_j \\in \\Lambda_\\mathfrak q$ lifting $d_j \\in D'$. Then", "$\\pi_{t + 1}^N = \\sum c_j a_j + \\epsilon$ with", "$\\epsilon \\in \\mathfrak q^n\\Lambda_\\mathfrak q \\subset", "\\mathfrak q^{n - N}H_{A/k}\\Lambda_\\mathfrak q$.", "Write $\\epsilon = \\sum a_j c'_j$ for some", "$c'_j \\in \\mathfrak q^{n - N}\\Lambda_\\mathfrak q$.", "Hence $\\pi_{t + 1}^{2N} = \\sum (\\pi_{t + 1}^N c_j + \\pi_{t + 1}^N c'_j) a_j$.", "Note that $\\pi_{t + 1}^Nc'_j$ maps to zero in $\\bar \\Lambda$; this trivial", "but key observation will ensure later that (a) holds.", "Now we choose $s \\in S$ such that there exist", "$\\mu_{t + 1j} \\in \\Lambda$ such that on the one hand", "$\\pi_{t + 1}^N c_j + \\pi_{t + 1}^N c'_j = \\mu_{t + 1j}/s^{2N}$", "in $S^{-1}\\Lambda$ and on the other", "$(s \\pi_{t + 1})^{2N} = \\sum \\mu_{t + 1j}a_j$", "in $\\Lambda$ (minor detail omitted). We may further replace $s$ by", "a power and enlarge $D'$ such that $s$ maps to an element of $D'$.", "With these choices $\\mu_{t + 1j}$ maps to $s^{2N}d_j$ which is", "an element of $D'$. Note that $\\pi_1, \\ldots, \\pi_d$ are a regular", "sequence of parameters in $S^{-1}\\Lambda$ by our", "choice of $\\varphi$. Hence $\\pi_1, \\ldots, \\pi_d$ forms a regular sequence", "in $\\Lambda_\\mathfrak q$ by", "Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}.", "It follows that ${\\pi'}_1^e, \\ldots, {\\pi'}_t^e, s\\pi_{t + 1}$ is a", "regular sequence in $S^{-1}\\Lambda$ by", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-powers}.", "Thus we get", "$$", "\\text{Ann}_{S^{-1}\\Lambda/({\\pi'}_1^e, \\ldots, {\\pi'}_t^e)}(s\\pi_{t + 1}) =", "\\text{Ann}_{S^{-1}\\Lambda/({\\pi'}_1^e, \\ldots, {\\pi'}_t^e)}((s\\pi_{t + 1})^2).", "$$", "Hence we may apply Lemma \\ref{lemma-ogoma} to find an $s' \\in S$", "such that", "$$", "\\text{Ann}_{\\Lambda/({\\pi'}_1^e, \\ldots, {\\pi'}_t^e)}((s')^qs\\pi_{t + 1})", "=", "\\text{Ann}_{\\Lambda/({\\pi'}_1^e, \\ldots, {\\pi'}_t^e)}(((s')^qs\\pi_{t + 1})^2).", "$$", "for any $q > 0$. By Lemma \\ref{lemma-enlarge-solution-modulo}", "we can choose $q$ and enlarge $D'$ such that $(s')^q$ maps to an element", "of $D'$. Setting $\\delta_{t + 1} = (s')^qs$ and we conclude that", "(a), (b), (c) hold for $i = 1, \\ldots, t + 1$. For (a) note that", "$\\lambda_{t + 1j} = (s')^{2Nq}\\mu_{t + 1j}$ works.", "By induction on $t$ we win.", "\\medskip\\noindent", "Ad (\\ref{item-first-solve}). By construction the radical of", "$H_{(C \\otimes_{k[x_1, \\ldots, x_d]} R)/R} \\Lambda$ contains", "$\\mathfrak h_A$. Namely, the elements $a_j \\in H_{A/k}$", "map to elements of $H_{B/k[x_1, \\ldots, x_n]}$, hence map to elements", "of $H_{C/k[x_1, \\ldots, x_n]}$, hence $a_j \\otimes 1$ map to elements of", "$H_{C \\otimes_{k[x_1, \\ldots, x_d]} R/R}$. Moreover, if we have a solution", "$C \\otimes_{k[x_1, \\ldots, x_n]} R \\to T \\to \\Lambda$ of", "$$", "R", "\\to", "C \\otimes_{k[x_1, \\ldots, x_d]} R", "\\to", "\\Lambda \\supset \\mathfrak q", "$$", "then $H_{T/R} \\subset H_{T/k}$ as $R$ is smooth over $k$.", "Hence $T$ will also be a solution for", "the original situation $k \\to A \\to \\Lambda \\supset \\mathfrak q$.", "\\medskip\\noindent", "Ad (\\ref{item-second-resolve}). Follows on applying", "Lemma \\ref{lemma-lift-solution} to", "$R \\to C \\otimes_{k[x_1, \\ldots, x_d]} R", "\\to \\Lambda \\supset \\mathfrak q$ and the sequence of", "elements $\\gamma_1^c, \\ldots, \\gamma_d^c$. We note that since $x_i^c$", "are strictly standard in $C$ over $k[x_1, \\ldots, x_d]$ the elements", "$\\gamma_i^c$ are strictly standard in $C \\otimes_{k[x_1, \\ldots, x_d]} R$", "over $R$ by Lemma \\ref{lemma-strictly-standard-base-change}.", "The other assumption of Lemma \\ref{lemma-lift-solution} holds by steps", "(\\ref{item-modify-pii}) and (\\ref{item-choose-deltai}).", "\\medskip\\noindent", "Ad (\\ref{item-third-resolve}). Apply Lemma \\ref{lemma-delocalize-height-zero}", "to the situation in (\\ref{item-second-resolve}). In the rest of the", "arguments the target ring is local Artinian, hence we are looking for", "a factorization by a smooth algebra $T$ over the source ring.", "\\medskip\\noindent", "Ad (\\ref{item-fifth-resolve}).", "Suppose that $C \\otimes_{k[x_1, \\ldots, x_d]} (R/JR)_\\mathfrak p \\to", "T \\to \\Lambda_\\mathfrak q/J\\Lambda_\\mathfrak q$ is a solution to", "$$", "(R/JR)_\\mathfrak p \\to", "C \\otimes_{k[x_1, \\ldots, x_d]} (R/JR)_\\mathfrak p \\to", "\\Lambda_\\mathfrak q/J\\Lambda_\\mathfrak q", "\\supset", "\\mathfrak q\\Lambda_\\mathfrak q/J\\Lambda_\\mathfrak q", "$$", "Then $C \\otimes_{k[x_1, \\ldots, x_d]} (R/I)_\\mathfrak r \\to T_\\mathfrak r \\to", "\\Lambda_\\mathfrak q/I\\Lambda_\\mathfrak q$", "is a solution to the situation in (\\ref{item-third-resolve}).", "\\medskip\\noindent", "Ad (\\ref{item-sixth-resolve}). Our $n = N + dc$ is large enough so that", "$\\mathfrak p^nk[y_1, \\ldots, y_m]_\\mathfrak p \\subset J_\\mathfrak p$", "and $\\mathfrak q^n \\Lambda_\\mathfrak q \\subset J\\Lambda_\\mathfrak q$.", "Hence if we have a solution", "$C \\otimes_{k[x_1, \\ldots, x_d]} (R/\\mathfrak p^nR)_\\mathfrak p \\to", "T \\to \\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q$", "of (\\ref{item-fifth-resolve}", "then we can take $T/JT$ as the solution for", "(\\ref{item-sixth-resolve}).", "\\medskip\\noindent", "Ad (\\ref{item-seventh-resolve}). This is true because we have a", "section $C \\to B$ in the category of $R$-algebras.", "\\medskip\\noindent", "Ad (\\ref{item-eighth-resolve}). This is true because $D'$ is", "essentially smooth over the local Artinian ring", "$k[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n k[y_1, \\ldots, y_m]_\\mathfrak p$", "and", "$$", "R_\\mathfrak p/\\mathfrak p^nR_\\mathfrak p =", "k[y_1, \\ldots, y_m]_\\mathfrak p/", "\\mathfrak p^n k[y_1, \\ldots, y_m]_\\mathfrak p[t_1, \\ldots, t_d].", "$$", "Hence $D'[t_1, \\ldots, t_d]$ is a filtered colimit of smooth", "$R_\\mathfrak p/\\mathfrak p^nR_\\mathfrak p$-algebras and", "$B \\otimes_{k[x_1, \\ldots, x_d]} (R_\\mathfrak p/\\mathfrak p^nR_\\mathfrak p)$", "factors through one of these.", "\\medskip\\noindent", "Ad (\\ref{item-done}). The final twist of the proof is that we cannot", "just use the map $B \\to D'$ which maps $x_i$ to the image of $\\pi_i'$", "in $D'$ and $z_{ij}$ to the image of $\\lambda_{ij}$ in $D'$", "because we need the diagram", "$$", "\\xymatrix{", "B \\ar[r] & D'[t_1, \\ldots, t_d] \\\\", "k[x_1, \\ldots, x_d] \\ar[r] \\ar[u] &", "R_\\mathfrak p/\\mathfrak p^nR_\\mathfrak p \\ar[u]", "}", "$$", "to commute and we need the composition", "$B \\to D'[t_1, \\ldots, t_d] \\to", "\\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q$", "to be the map of (\\ref{item-map-B-Lambda}).", "This requires us to map $x_i$ to the image of", "$\\pi_i t_i$ in $D'[t_1, \\ldots, t_d]$.", "Hence we map $z_{ij}$ to the image of", "$\\lambda_{ij} t_i^{2N} / \\delta_i^{2N}$ in $D'[t_1, \\ldots, t_d]$", "and everything is clear." ], "refs": [ "smoothing-lemma-improve-presentation", "smoothing-lemma-improve-presentation", "smoothing-lemma-compare-standard", "smoothing-lemma-solution-modulo", "algebra-lemma-dimension-base-fibre-equals-total", "algebra-lemma-regular-ring-CM", "algebra-lemma-regular-sequence-in-polynomial-ring", "algebra-lemma-regular-sequence-powers", "smoothing-lemma-ogoma", "algebra-lemma-faithfully-flat-universally-injective", "algebra-lemma-regular-ring-CM", "algebra-lemma-regular-sequence-powers", "smoothing-lemma-ogoma", "smoothing-lemma-enlarge-solution-modulo", "smoothing-lemma-lift-solution", "smoothing-lemma-strictly-standard-base-change", "smoothing-lemma-lift-solution", "smoothing-lemma-delocalize-height-zero" ], "ref_ids": [ 5613, 5613, 5618, 5640, 987, 941, 745, 744, 5636, 814, 941, 744, 5636, 5641, 5633, 5611, 5633, 5635 ] } ], "ref_ids": [] }, { "id": 5643, "type": "theorem", "label": "smoothing-lemma-approximation-property-variant", "categories": [ "smoothing" ], "title": "smoothing-lemma-approximation-property-variant", "contents": [ "Let $R$ be a Noetherian ring. Let $\\mathfrak p \\subset R$ be a prime ideal. Let", "$f_1, \\ldots, f_m \\in R[x_1, \\ldots, x_n]$.", "Suppose that $(a_1, \\ldots, a_n) \\in ((R_\\mathfrak p)^\\wedge)^n$ is a solution.", "If $R_\\mathfrak p$ is a G-ring, then for every integer $N$ there exist", "\\begin{enumerate}", "\\item an \\'etale ring map $R \\to R'$,", "\\item a prime ideal $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$", "\\item a solution $(b_1, \\ldots, b_n) \\in (R')^n$ in $R'$", "\\end{enumerate}", "such that $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$ and", "$a_i - b_i \\in (\\mathfrak p')^N(R'_{\\mathfrak p'})^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "By Theorem \\ref{theorem-approximation-property-variant}", "we can find a solution $(b'_1, \\ldots, b'_n)$ in some ring", "$R''$ \\'etale over $R_\\mathfrak p$ which comes with a", "prime ideal $\\mathfrak p''$ lying over $\\mathfrak p$ such", "that $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p'')$ and", "$a_i - b'_i \\in (\\mathfrak p'')^N(R''_{\\mathfrak p''})^\\wedge$.", "We can write", "$R'' = R' \\otimes_R R_\\mathfrak p$ for some \\'etale $R$-algebra $R'$", "(see Algebra, Lemma \\ref{algebra-lemma-etale}). After replacing", "$R'$ by a principal localization if necessary we may assume", "$(b'_1, \\ldots, b'_n)$ come from a solution $(b_1, \\ldots, b_n)$", "in $R'$. Setting $\\mathfrak p' = R' \\cap \\mathfrak p''$ we", "see that $R''_{\\mathfrak p''} = R'_{\\mathfrak p'}$ which finishes the proof." ], "refs": [ "smoothing-theorem-approximation-property-variant", "algebra-lemma-etale" ], "ref_ids": [ 5607, 1231 ] } ], "ref_ids": [] }, { "id": 5644, "type": "theorem", "label": "smoothing-lemma-henselian-pair", "categories": [ "smoothing" ], "title": "smoothing-lemma-henselian-pair", "contents": [ "\\begin{slogan}", "Approximation for henselian pairs.", "\\end{slogan}", "Let $(A, I)$ be a henselian pair with $A$ Noetherian.", "Let $A^\\wedge$ be the $I$-adic completion", "of $A$. Assume at least one of the following", "conditions holds", "\\begin{enumerate}", "\\item $A \\to A^\\wedge$ is a regular ring map,", "\\item $A$ is a Noetherian G-ring, or", "\\item $(A, I)$ is the henselization", "(More on Algebra, Lemma \\ref{more-algebra-lemma-henselization})", "of a pair $(B, J)$ where $B$ is a Noetherian G-ring.", "\\end{enumerate}", "Given $f_1, \\ldots, f_m \\in A[x_1, \\ldots, x_n]$", "and $\\hat{a}_1, \\ldots, \\hat{a}_n \\in A^\\wedge$ such that", "$f_j(\\hat{a}_1, \\ldots, \\hat{a}_n) = 0$", "for $j = 1, \\ldots, m$, for every $N \\geq 1$ there exist", "$a_1, \\ldots, a_n \\in A$ such that", "$\\hat{a}_i - a_i \\in I^N$ and such that $f_j(a_1, \\ldots, a_n) = 0$", "for $j = 1, \\ldots, m$." ], "refs": [ "more-algebra-lemma-henselization" ], "proofs": [ { "contents": [ "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-henselization-pair-G-ring}", "we see that (3) implies (2). By More on Algebra, Lemma", "\\ref{more-algebra-lemma-map-G-ring-to-completion-regular}", "we see that (2) implies (1).", "Thus it suffices to prove the lemma in case $A \\to A^\\wedge$ is", "a regular ring map.", "\\medskip\\noindent", "Let $\\hat{a}_1, \\ldots, \\hat{a}_n$ be as in the statement of the lemma.", "By Theorem \\ref{theorem-popescu} we can find a factorization", "$A \\to B \\to A^\\wedge$ with $A \\to P$ smooth and", "$b_1, \\ldots, b_n \\in B$ with $f_j(b_1, \\ldots, b_n) = 0$ in $B$.", "Denote $\\sigma : B \\to A^\\wedge \\to A/I^N$ the composition.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-lift-section-smooth-morphism}", "we can find an \\'etale ring", "map $A \\to A'$ which induces an isomorphism $A/I^N \\to A'/I^NA'$", "and an $A$-algebra map $\\tilde \\sigma : B \\to A'$ lifting $\\sigma$.", "Since $(A, I)$ is henselian, there is an $A$-algebra map $\\chi : A' \\to A$,", "see More on Algebra, Lemma \\ref{more-algebra-lemma-characterize-henselian-pair}.", "Then setting $a_i = \\chi(\\tilde \\sigma(b_i))$ gives a solution." ], "refs": [ "more-algebra-lemma-henselization-pair-G-ring", "more-algebra-lemma-map-G-ring-to-completion-regular", "smoothing-theorem-popescu", "more-algebra-lemma-characterize-henselian-pair" ], "ref_ids": [ 10093, 10092, 5605, 9861 ] } ], "ref_ids": [ 9871 ] }, { "id": 5645, "type": "theorem", "label": "smoothing-proposition-lift-smooth", "categories": [ "smoothing" ], "title": "smoothing-proposition-lift-smooth", "contents": [ "\\begin{slogan}", "Smooth and syntomic algebras lift along surjections", "\\end{slogan}", "Let $R \\to R_0$ be a surjective ring map with kernel $I$.", "\\begin{enumerate}", "\\item If $R_0 \\to A_0$ is a syntomic ring map, then there exists a syntomic", "ring map $R \\to A$ such that $A/IA \\cong A_0$.", "\\item If $R_0 \\to A_0$ is a smooth ring map, then there exists a smooth", "ring map $R \\to A$ such that $A/IA \\cong A_0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $R_0 \\to A_0$ syntomic, in particular a local complete intersection", "(More on Algebra, Lemma \\ref{more-algebra-lemma-syntomic-lci}).", "Choose a presentation $A_0 = R_0[x_1, \\ldots, x_n]/J_0$. Set", "$C_0 = \\text{Sym}^*_{A_0}(J_0/J_0^2)$. Note that $J_0/J_0^2$ is a finite", "projective $A_0$-module (Algebra, Lemma", "\\ref{algebra-lemma-syntomic-presentation-ideal-mod-squares}).", "By Lemma \\ref{lemma-improve-presentation} the ring map", "$A_0 \\to C_0$ is smooth and we can find a presentation", "$C_0 = R_0[y_1, \\ldots, y_m]/K_0$ with $K_0/K_0^2$ free over $C_0$.", "By Algebra, Lemma \\ref{algebra-lemma-huber} we can assume", "$C_0 = R_0[y_1, \\ldots, y_m]/(\\overline{f}_1, \\ldots, \\overline{f}_c)$", "where $\\overline{f}_1, \\ldots, \\overline{f}_c$ maps to a basis of", "$K_0/K_0^2$ over $C_0$. Choose", "$f_1, \\ldots, f_c \\in R[y_1, \\ldots, y_c]$ lifting", "$\\overline{f}_1, \\ldots, \\overline{f}_c$ and set", "$$", "C = R[y_1, \\ldots, y_m]/(f_1, \\ldots, f_c)", "$$", "By construction $C_0 = C/IC$. By Algebra, Lemma", "\\ref{algebra-lemma-localize-relative-complete-intersection}", "we can after replacing $C$ by $C_g$ assume that $C$ is a relative", "global complete intersection over $R$.", "We conclude that there exists a finite projective $A_0$-module", "$P_0$ such that $C_0 = \\text{Sym}^*_{A_0}(P_0)$", "is isomorphic to $C/IC$ for some syntomic $R$-algebra $C$.", "\\medskip\\noindent", "Choose an integer $n$ and a direct sum decomposition", "$A_0^{\\oplus n} = P_0 \\oplus Q_0$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-lift-projective-module}", "we can find an \\'etale ring map $C \\to C'$ which induces", "an isomorphism $C/IC \\to C'/IC'$ and a finite projective", "$C'$-module $Q$ such that $Q/IQ$ is isomorphic to", "$Q_0 \\otimes_{A_0} C/IC$.", "Then $D = \\text{Sym}_{C'}^*(Q)$ is a smooth $C'$-algebra (see", "More on Algebra, Lemma \\ref{more-algebra-lemma-symmetric-algebra-smooth}).", "Picture", "$$", "\\xymatrix{", "R \\ar[d] \\ar[rr] & &", "C \\ar[r] \\ar[d] &", "C' \\ar[r] \\ar[d] &", "D \\ar[d] \\\\", "R/I \\ar[r] &", "A_0 \\ar[r] &", "C/IC \\ar[r]^{\\cong} &", "C'/IC' \\ar[r] &", "D/ID", "}", "$$", "Observe that our choice of $Q$ gives", "\\begin{align*}", "D/ID & =", "\\text{Sym}_{C/IC}^*(Q_0 \\otimes_{A_0} C/IC) \\\\", "& =", "\\text{Sym}_{A_0}^*(Q_0) \\otimes_{A_0} C/IC \\\\", "& =", "\\text{Sym}_{A_0}^*(Q_0) \\otimes_{A_0}", "\\text{Sym}_{A_0}^*(P_0) \\\\", "& =", "\\text{Sym}_{A_0}^*(Q_0 \\oplus P_0) \\\\", "& =", "\\text{Sym}_{A_0}^*(A_0^{\\oplus n}) \\\\", "& =", "A_0[x_1, \\ldots, x_n]", "\\end{align*}", "Choose $f_1, \\ldots, f_n \\in D$ which map to $x_1, \\ldots, x_n$", "in $D/ID = A_0[x_1, \\ldots, x_n]$. Set $A = D/(f_1, \\ldots, f_n)$.", "Note that $A_0 = A/IA$. We claim that $R \\to A$ is syntomic", "in a neighbourhood of $V(IA)$. If the claim is true, then we can", "find a $f \\in A$ mapping to $1 \\in A_0$ such that $A_f$ is syntomic", "over $R$ and the proof of (1) is finished.", "\\medskip\\noindent", "Proof of the claim. Observe that $R \\to D$ is syntomic as a composition", "of the syntomic ring map $R \\to C$, the \\'etale ring map $C \\to C'$ and the", "smooth ring map $C' \\to D$ (Algebra, Lemmas", "\\ref{algebra-lemma-composition-syntomic} and", "\\ref{algebra-lemma-smooth-syntomic}).", "The question is local on $\\Spec(D)$, hence we", "may assume that $D$ is a relative global complete intersection", "(Algebra, Lemma \\ref{algebra-lemma-syntomic}).", "Say $D = R[y_1, \\ldots, y_m]/(g_1, \\ldots, g_s)$.", "Let $f'_1, \\ldots, f'_n \\in R[y_1, \\ldots, y_m]$ be lifts of", "$f_1, \\ldots, f_n$. Then we can apply", "Algebra, Lemma \\ref{algebra-lemma-localize-relative-complete-intersection}", "to get the claim.", "\\medskip\\noindent", "Proof of (2). Since a smooth ring map is syntomic, we can find", "a syntomic ring map $R \\to A$ such that $A_0 = A/IA$.", "By assumption the fibres of $R \\to A$ are smooth over primes in $V(I)$", "hence $R \\to A$ is smooth in an open neighbourhood of $V(IA)$", "(Algebra, Lemma \\ref{algebra-lemma-flat-fibre-smooth}).", "Thus we can replace $A$ by a localization to obtain the result we want." ], "refs": [ "more-algebra-lemma-syntomic-lci", "algebra-lemma-syntomic-presentation-ideal-mod-squares", "smoothing-lemma-improve-presentation", "algebra-lemma-huber", "algebra-lemma-localize-relative-complete-intersection", "more-algebra-lemma-lift-projective-module", "more-algebra-lemma-symmetric-algebra-smooth", "algebra-lemma-composition-syntomic", "algebra-lemma-smooth-syntomic", "algebra-lemma-syntomic", "algebra-lemma-localize-relative-complete-intersection", "algebra-lemma-flat-fibre-smooth" ], "ref_ids": [ 10002, 1186, 5613, 1178, 1181, 9849, 9851, 1187, 1195, 1185, 1181, 1200 ] } ], "ref_ids": [] }, { "id": 5646, "type": "theorem", "label": "smoothing-proposition-lift", "categories": [ "smoothing" ], "title": "smoothing-proposition-lift", "contents": [ "\\begin{slogan}", "Ind-smoothness of an algebra is stable under infinitesimal deformations", "\\end{slogan}", "Let $R \\to \\Lambda$ be a ring map. Let $I \\subset R$ be an ideal.", "Assume that", "\\begin{enumerate}", "\\item $I$ is nilpotent,", "\\item $\\Lambda/I\\Lambda$ is a filtered colimit of smooth $R/I$-algebras, and", "\\item $R \\to \\Lambda$ is flat.", "\\end{enumerate}", "Then $\\Lambda$ is a filtered colimit of smooth $R$-algebras." ], "refs": [], "proofs": [ { "contents": [ "Since $I^n = 0$ for some $n$, it follows by induction on $n$ that", "it suffices to consider the case where $I^2 = 0$. Let", "$\\varphi : A \\to \\Lambda$ be an $R$-algebra map with $A$ of finite", "presentation over $R$. We have to find a factorization $A \\to B \\to \\Lambda$", "with $B$ smooth over $R$, see Algebra, Lemma \\ref{algebra-lemma-when-colimit}.", "By Lemma \\ref{lemma-lift-once} we may assume that", "$A = B/J$ with $B$ smooth over $R$ and $J \\subset IB$", "a finitely generated ideal. By", "Lemma \\ref{lemma-lift-twice}", "we can find a (possibly noncommutative) diagram", "$$", "\\xymatrix{", "B \\ar[rr]_\\alpha \\ar[rd]_\\varphi & & B' \\ar[ld]^\\beta \\\\", "& \\Lambda", "}", "$$", "of $R$-algebras which commutes modulo $I$ and such that $\\alpha(J) = 0$.", "The map", "$$", "D : B \\longrightarrow I\\Lambda,\\quad", "b \\longmapsto \\varphi(b) - \\beta(\\alpha(b))", "$$", "is a derivation over $R$ hence we can write it as", "$D = \\xi \\circ \\text{d}_{B/R}$ for some $B$-linear map", "$\\xi : \\Omega_{B/R} \\to I\\Lambda$. Since $\\Omega_{B/R}$ is a", "finite projective $B$-module we can write", "$\\xi = \\sum_{i = 1, \\ldots, n} \\epsilon_i \\Xi_i$", "for some $\\epsilon_i \\in I$ and $B$-linear maps", "$\\Xi_i : \\Omega_{B/R} \\to \\Lambda$.", "(Details omitted. Hint: write $\\Omega_{B/R}$ as a direct sum of", "a finite free module to reduce to the finite free case.)", "We define", "$$", "B'' = \\text{Sym}^*_{B'}\\left(\\bigoplus\\nolimits_{i = 1, \\ldots, n}", "\\Omega_{B/R} \\otimes_{B, \\alpha} B'\\right)", "$$", "and we define $\\beta' : B'' \\to \\Lambda$ by", "$\\beta$ on $B'$ and by", "$$", "\\beta'|_{i\\text{th summand }\\Omega_{B/R} \\otimes_{B, \\alpha} B'} =", "\\Xi_i \\otimes \\beta", "$$", "and $\\alpha' : B \\to B''$ by", "$$", "\\alpha'(b) =", "\\alpha(b) \\oplus \\sum \\epsilon_i \\text{d}_{B/R}(b) \\otimes 1", "\\oplus 0 \\oplus \\ldots", "$$", "At this point the diagram", "$$", "\\xymatrix{", "B \\ar[rr]_{\\alpha'} \\ar[rd]_\\varphi & & B'' \\ar[ld]^{\\beta'} \\\\", "& \\Lambda", "}", "$$", "does commute. Moreover, it is direct from the definitions that", "$\\alpha'(J) = 0$ as $I^2 = 0$. Hence the desired factorization." ], "refs": [ "algebra-lemma-when-colimit", "smoothing-lemma-lift-once", "smoothing-lemma-lift-twice" ], "ref_ids": [ 1093, 5624, 5625 ] } ], "ref_ids": [] }, { "id": 5649, "type": "theorem", "label": "chow-lemma-additivity-periodic-length", "categories": [ "chow" ], "title": "chow-lemma-additivity-periodic-length", "contents": [ "Let $R$ be a ring. Suppose that we have a short exact sequence of", "$2$-periodic complexes", "$$", "0 \\to (M_1, N_1, \\varphi_1, \\psi_1)", "\\to (M_2, N_2, \\varphi_2, \\psi_2)", "\\to (M_3, N_3, \\varphi_3, \\psi_3)", "\\to 0", "$$", "If two out of three have cohomology modules of finite length so does", "the third and we have", "$$", "e_R(M_2, N_2, \\varphi_2, \\psi_2) =", "e_R(M_1, N_1, \\varphi_1, \\psi_1) +", "e_R(M_3, N_3, \\varphi_3, \\psi_3).", "$$" ], "refs": [], "proofs": [ { "contents": [ "We abbreviate $A = (M_1, N_1, \\varphi_1, \\psi_1)$,", "$B = (M_2, N_2, \\varphi_2, \\psi_2)$ and $C = (M_3, N_3, \\varphi_3, \\psi_3)$.", "We have a long exact cohomology sequence", "$$", "\\ldots", "\\to H^1(C)", "\\to H^0(A)", "\\to H^0(B)", "\\to H^0(C)", "\\to H^1(A)", "\\to H^1(B)", "\\to H^1(C)", "\\to \\ldots", "$$", "This gives a finite exact sequence", "$$", "0 \\to I", "\\to H^0(A)", "\\to H^0(B)", "\\to H^0(C)", "\\to H^1(A)", "\\to H^1(B)", "\\to K \\to 0", "$$", "with $0 \\to K \\to H^1(C) \\to I \\to 0$ a filtration. By additivity of", "the length function (Algebra, Lemma \\ref{algebra-lemma-length-additive})", "we see the result." ], "refs": [ "algebra-lemma-length-additive" ], "ref_ids": [ 631 ] } ], "ref_ids": [] }, { "id": 5650, "type": "theorem", "label": "chow-lemma-finite-periodic-length", "categories": [ "chow" ], "title": "chow-lemma-finite-periodic-length", "contents": [ "Let $R$ be a ring. If $(M, N, \\varphi, \\psi)$ is a $2$-periodic complex", "such that $M$, $N$ have finite length, then", "$e_R(M, N, \\varphi, \\psi) = \\text{length}_R(M) - \\text{length}_R(N)$.", "In particular, if $(M, \\varphi, \\psi)$ is a $(2, 1)$-periodic complex", "such that $M$ has finite length, then", "$e_R(M, \\varphi, \\psi) = 0$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the additity of", "Lemma \\ref{lemma-additivity-periodic-length}", "and the short exact sequence", "$0 \\to (M, 0, 0, 0) \\to (M, N, \\varphi, \\psi) \\to (0, N, 0, 0) \\to 0$." ], "refs": [ "chow-lemma-additivity-periodic-length" ], "ref_ids": [ 5649 ] } ], "ref_ids": [] }, { "id": 5651, "type": "theorem", "label": "chow-lemma-compare-periodic-lengths", "categories": [ "chow" ], "title": "chow-lemma-compare-periodic-lengths", "contents": [ "Let $R$ be a ring. Let $f : (M, \\varphi, \\psi) \\to (M', \\varphi', \\psi')$", "be a map of $(2, 1)$-periodic complexes whose cohomology modules", "have finite length. If $\\Ker(f)$ and $\\Coker(f)$ have finite length,", "then $e_R(M, \\varphi, \\psi) = e_R(M', \\varphi', \\psi')$." ], "refs": [], "proofs": [ { "contents": [ "Apply the additivity of Lemma \\ref{lemma-additivity-periodic-length}", "and observe that $(\\Ker(f), \\varphi, \\psi)$ and", "$(\\Coker(f), \\varphi', \\psi')$ have vanishing multiplicity by", "Lemma \\ref{lemma-finite-periodic-length}." ], "refs": [ "chow-lemma-additivity-periodic-length", "chow-lemma-finite-periodic-length" ], "ref_ids": [ 5649, 5650 ] } ], "ref_ids": [] }, { "id": 5652, "type": "theorem", "label": "chow-lemma-length-multiplication", "categories": [ "chow" ], "title": "chow-lemma-length-multiplication", "contents": [ "Let $R$ be a Noetherian local ring.", "Let $M$ be a finite $R$-module. Let $x \\in R$. Assume that", "\\begin{enumerate}", "\\item $\\dim(\\text{Supp}(M)) \\leq 1$, and", "\\item $\\dim(\\text{Supp}(M/xM)) \\leq 0$.", "\\end{enumerate}", "Write", "$\\text{Supp}(M) = \\{\\mathfrak m, \\mathfrak q_1, \\ldots, \\mathfrak q_t\\}$.", "Then", "$$", "e_R(M, 0, x) =", "\\sum\\nolimits_{i = 1, \\ldots, t}", "\\text{ord}_{R/\\mathfrak q_i}(x)", "\\text{length}_{R_{\\mathfrak q_i}}(M_{\\mathfrak q_i}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "We first make some preparatory remarks.", "The result of the lemma holds if $M$ has finite length, i.e., if $t = 0$,", "because both the left hand side and the right hand side are zero", "in this case, see Lemma \\ref{lemma-finite-periodic-length}.", "Also, if we have a short exact sequence $0 \\to M \\to M' \\to M'' \\to 0$", "of modules satisfying (1) and (2), then lemma for 2 out of 3", "of these implies the lemma for the third by the", "additivity of length (Algebra, Lemma \\ref{algebra-lemma-length-additive}) and", "additivty of multiplicities (Lemma \\ref{lemma-additivity-periodic-length}).", "\\medskip\\noindent", "Denote $M_i$ the image of $M$ in $M_{\\mathfrak q_i}$, so", "$\\text{Supp}(M_i) = \\{\\mathfrak m, \\mathfrak q_i\\}$.", "The kernel and cokernel of the map $M \\to \\bigoplus M_i$", "have support $\\{\\mathfrak m\\}$ and hence have finite length.", "By our preparatory remarks, it follows that it suffices to", "prove the lemma for each $M_i$. Thus we may assume that", "$\\text{Supp}(M) = \\{\\mathfrak m, \\mathfrak q\\}$.", "In this case we have a finite filtration", "$M \\supset \\mathfrak qM \\supset \\mathfrak q^2M \\supset \\ldots \\supset", "\\mathfrak q^nM = 0$ by Algebra, Lemma", "\\ref{algebra-lemma-Noetherian-power-ideal-kills-module}.", "Again additivity shows that it suffices to prove the lemma", "in the case $M$ is annihilated by $\\mathfrak q$.", "In this case we can view $M$ as a $R/\\mathfrak q$-module,", "i.e., we may assume that $R$ is a Noetherian local domain", "of dimension $1$ with fraction field $K$.", "Dividing by the torsion submodule, i.e., by the", "kernel of $M \\to M \\otimes_R K = V$ (the torsion has", "finite length hence is handled by our preliminary remarks)", "we may assume that $M \\subset V$ is a lattice", "(Algebra, Definition \\ref{algebra-definition-lattice}).", "Then $x : M \\to M$ is injective and", "$\\text{length}_R(M/xM) = d(M, xM)$", "(Algebra, Definition \\ref{algebra-definition-distance}). Since", "$\\text{length}_K(V) = \\dim_K(V)$", "we see that $\\det(x : V \\to V) = x^{\\dim_K(V)}$ and", "$\\text{ord}_R(\\det(x : V \\to V)) = \\dim_K(V) \\text{ord}_R(x)$.", "Thus the desired equality follows from", "Algebra, Lemma \\ref{algebra-lemma-order-vanishing-determinant}", "in this case." ], "refs": [ "chow-lemma-finite-periodic-length", "algebra-lemma-length-additive", "chow-lemma-additivity-periodic-length", "algebra-lemma-Noetherian-power-ideal-kills-module", "algebra-definition-lattice", "algebra-definition-distance", "algebra-lemma-order-vanishing-determinant" ], "ref_ids": [ 5650, 631, 5649, 694, 1520, 1521, 1046 ] } ], "ref_ids": [] }, { "id": 5653, "type": "theorem", "label": "chow-lemma-additivity-divisors-restricted", "categories": [ "chow" ], "title": "chow-lemma-additivity-divisors-restricted", "contents": [ "Let $R$ be a Noetherian local ring.", "Let $x \\in R$. If $M$ is a finite Cohen-Macaulay module over $R$", "with $\\dim(\\text{Supp}(M)) = 1$ and $\\dim(\\text{Supp}(M/xM)) = 0$, then", "$$", "\\text{length}_R(M/xM)", "=", "\\sum\\nolimits_i \\text{length}_R(R/(x, \\mathfrak q_i))", "\\text{length}_{R_{\\mathfrak q_i}}(M_{\\mathfrak q_i}).", "$$", "where $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ are the", "minimal primes of the support of $M$. If $I \\subset R$ is an ideal", "such that $x$ is a nonzerodivisor on $R/I$ and $\\dim(R/I) = 1$, then", "$$", "\\text{length}_R(R/(x, I))", "=", "\\sum\\nolimits_i \\text{length}_R(R/(x, \\mathfrak q_i))", "\\text{length}_{R_{\\mathfrak q_i}}((R/I)_{\\mathfrak q_i})", "$$", "where $\\mathfrak q_1, \\ldots, \\mathfrak q_n$ are the minimal", "primes over $I$." ], "refs": [], "proofs": [ { "contents": [ "These are special cases of Lemma \\ref{lemma-length-multiplication}." ], "refs": [ "chow-lemma-length-multiplication" ], "ref_ids": [ 5652 ] } ], "ref_ids": [] }, { "id": 5654, "type": "theorem", "label": "chow-lemma-powers-period-length-zero", "categories": [ "chow" ], "title": "chow-lemma-powers-period-length-zero", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Let $\\varphi : M \\to M$ be an endomorphism and $n > 0$", "such that $\\varphi^n = 0$ and such that $\\Ker(\\varphi)/\\Im(\\varphi^{n - 1})$", "has finite length as an $R$-module.", "Then", "$$", "e_R(M, \\varphi^i, \\varphi^{n - i}) = 0", "$$", "for $i = 0, \\ldots, n$." ], "refs": [], "proofs": [ { "contents": [ "The cases $i = 0, n$ are trivial as $\\varphi^0 = \\text{id}_M$ by convention.", "Let us think of $M$ as an $R[t]$-module where multiplication by $t$", "is given by $\\varphi$. Let us write", "$K_i = \\Ker(t^i : M \\to M)$ and", "$$", "a_i = \\text{length}_R(K_i/t^{n - i}M),\\quad", "b_i = \\text{length}_R(K_i/tK_{i + 1}),\\quad", "c_i = \\text{length}_R(K_1/t^iK_{i + 1})", "$$", "Boundary values are $a_0 = a_n = b_0 = c_0 = 0$.", "The $c_i$ are integers for $i < n$ as $K_1/t^iK_{i + 1}$", "is a quotient of $K_1/t^{n - 1}M$ which is assumed to have finite length.", "We will use frequently that $K_i \\cap t^jM = t^jK_{i + j}$.", "For $0 < i < n - 1$ we have an exact sequence", "$$", "0 \\to", "K_1/t^{n - i - 1}K_{n - i} \\to", "K_{i + 1}/t^{n - i - 1}M \\xrightarrow{t} K_i/t^{n - i}M", "\\to K_i/tK_{i + 1} \\to 0", "$$", "By induction on $i$ we conclude that $a_i$ and $b_i$ are", "integers for $i < n$ and that", "$$", "c_{n - i - 1} - a_{i + 1} + a_i - b_i = 0", "$$", "For $0 < i < n - 1$ there is a short exact sequence", "$$", "0 \\to", "K_i/tK_{i + 1} \\to", "K_{i + 1}/tK_{i + 2} \\xrightarrow{t^i}", "K_1/t^{i + 1}K_{i + 2} \\to", "K_1/t^iK_{i + 1} \\to 0", "$$", "which gives", "$$", "b_i - b_{i + 1} + c_{i + 1} - c_i = 0", "$$", "Since $b_0 = c_0$ we conclude that $b_i = c_i$ for $i < n$.", "Then we see that", "$$", "a_2 = a_1 + b_{n - 2} - b_1,\\quad", "a_3 = a_2 + b_{n - 3} - b_2,\\quad \\ldots", "$$", "It is straighforward to see that this implies $a_i = a_{n - i}$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5655, "type": "theorem", "label": "chow-lemma-multiply-period-length", "categories": [ "chow" ], "title": "chow-lemma-multiply-period-length", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring. Let", "$(M, \\varphi, \\psi)$ be a $(2, 1)$-periodic complex over $R$", "with $M$ finite and with cohomology groups of finite length over $R$.", "Let $x \\in R$ be such that $\\dim(\\text{Supp}(M/xM)) \\leq 0$. Then", "$$", "e_R(M, x\\varphi, \\psi) = e_R(M, \\varphi, \\psi) - e_R(\\Im(\\varphi), 0, x)", "$$", "and", "$$", "e_R(M, \\varphi, x\\psi) = e_R(M, \\varphi, \\psi) + e_R(\\Im(\\psi), 0, x)", "$$" ], "refs": [], "proofs": [ { "contents": [ "We will only prove the first formula as the second is proved", "in exactly the same manner.", "Let $M' = M[x^\\infty]$ be the $x$-power torsion submodule of $M$.", "Consider the short exact sequence $0 \\to M' \\to M \\to M'' \\to 0$.", "Then $M''$ is $x$-power torsion free (More on Algebra, Lemma", "\\ref{more-algebra-lemma-divide-by-torsion}).", "Since $\\varphi$, $\\psi$ map $M'$ into $M'$", "we obtain a short exact sequence", "$$", "0 \\to (M', \\varphi', \\psi') \\to (M, \\varphi, \\psi) \\to", "(M'', \\varphi'', \\psi'') \\to 0", "$$", "of $(2, 1)$-periodic complexes. Also, we get a short exact sequence", "$0 \\to M' \\cap \\Im(\\varphi) \\to \\Im(\\varphi) \\to \\Im(\\varphi'') \\to 0$.", "We have", "$e_R(M', \\varphi, \\psi) = e_R(M', x\\varphi, \\psi) =", "e_R(M' \\cap \\Im(\\varphi), 0, x) = 0$", "by Lemma \\ref{lemma-compare-periodic-lengths}.", "By additivity (Lemma \\ref{lemma-additivity-periodic-length})", "we see that it suffices to prove the lemma for $(M'', \\varphi'', \\psi'')$.", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $x : M \\to M$ is injective.", "In this case $\\Ker(x\\varphi) = \\Ker(\\varphi)$.", "On the other hand we have a short exact sequence", "$$", "0 \\to \\Im(\\varphi)/x\\Im(\\varphi) \\to", "\\Ker(\\psi)/\\Im(x\\varphi) \\to \\Ker(\\psi)/\\Im(\\varphi) \\to 0", "$$", "This together with (\\ref{equation-multiplicity-coker-ker}) proves the formula." ], "refs": [ "more-algebra-lemma-divide-by-torsion", "chow-lemma-compare-periodic-lengths", "chow-lemma-additivity-periodic-length" ], "ref_ids": [ 10335, 5651, 5649 ] } ], "ref_ids": [] }, { "id": 5656, "type": "theorem", "label": "chow-lemma-glue-at-max", "categories": [ "chow" ], "title": "chow-lemma-glue-at-max", "contents": [ "Let $A$ be a Noetherian ring. Let $\\mathfrak m_1, \\ldots, \\mathfrak m_r$", "be pairwise distinct maximal ideals of $A$. For $i = 1, \\ldots, r$ let", "$\\varphi_i : A_{\\mathfrak m_i} \\to B_i$ be a ring map whose", "kernel and cokernel are annihilated by a power", "of $\\mathfrak m_i$. Then there exists a ring map $\\varphi : A \\to B$ such", "that", "\\begin{enumerate}", "\\item the localization of $\\varphi$ at $\\mathfrak m_i$ is", "isomorphic to $\\varphi_i$, and", "\\item $\\Ker(\\varphi)$ and $\\Coker(\\varphi)$ are annihilated", "by a power of $\\mathfrak m_1 \\cap \\ldots \\cap \\mathfrak m_r$.", "\\end{enumerate}", "Moreover, if each $\\varphi_i$ is finite, injective, or", "surjective then so is $\\varphi$." ], "refs": [], "proofs": [ { "contents": [ "Set $I = \\mathfrak m_1 \\cap \\ldots \\cap \\mathfrak m_r$. Set", "$A_i = A_{\\mathfrak m_i}$ and $A' = \\prod A_i$.", "Then $IA' = \\prod \\mathfrak m_i A_i$ and $A \\to A'$", "is a flat ring map such that $A/I \\cong A'/IA'$.", "Thus we may use More on Algebra, Lemma", "\\ref{more-algebra-lemma-application-formal-glueing}", "to see that there exists an $A$-module map $\\varphi : A \\to B$", "with $\\varphi_i$ isomorphic to the localization of $\\varphi$", "at $\\mathfrak m_i$. Then we can use the discussion in", "More on Algebra, Remark \\ref{more-algebra-remark-formal-glueing-algebras}", "to endow $B$ with an $A$-algebra structure", "matching the given $A$-algebra structure on $B_i$.", "The final statement of the lemma follows easily from", "the fact that $\\Ker(\\varphi)_{\\mathfrak m_i} \\cong \\Ker(\\varphi_i)$", "and $\\Coker(\\varphi)_{\\mathfrak m_i} \\cong \\Coker(\\varphi_i)$." ], "refs": [ "more-algebra-lemma-application-formal-glueing", "more-algebra-remark-formal-glueing-algebras" ], "ref_ids": [ 10352, 10663 ] } ], "ref_ids": [] }, { "id": 5657, "type": "theorem", "label": "chow-lemma-Noetherian-domain-dim-1-two-elements", "categories": [ "chow" ], "title": "chow-lemma-Noetherian-domain-dim-1-two-elements", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $1$.", "Let $a, b \\in R$ be nonzerodivisors.", "There exists a finite ring extension $R \\subset R'$", "with $R'/R$ annihilated by a power of $\\mathfrak m$", "and nonzerodivisors $t, a', b' \\in R'$ such that", "$a = ta'$ and $b = tb'$ and $R' = a'R' + b'R'$." ], "refs": [], "proofs": [ { "contents": [ "If $a$ or $b$ is a unit, then the lemma is true with $R = R'$.", "Thus we may assume $a, b \\in \\mathfrak m$.", "Set $I = (a, b)$. The idea is to blow up $R$ in $I$.", "Instead of doing the algebraic argument we work geometrically.", "Let $X = \\text{Proj}(\\bigoplus_{d \\geq 0} I^d)$.", "By Divisors, Lemma", "\\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}", "the morphism $X \\to \\Spec(R)$ is an isomorphism over", "the punctured spectrum $U = \\Spec(R) \\setminus \\{\\mathfrak m\\}$.", "Thus we may and do view $U$ as an open subscheme of $X$.", "The morphism $X \\to \\Spec(R)$ is projective by", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-projective}.", "Also, every generic point of $X$ lies in $U$, for example", "by Divisors, Lemma \\ref{divisors-lemma-blow-up-and-irreducible-components}.", "It follows from Varieties, Lemma \\ref{varieties-lemma-finite-in-codim-1}", "that $X \\to \\Spec(R)$ is finite. Thus $X = \\Spec(R')$ is", "affine and $R \\to R'$ is finite. We have $R_a \\cong R'_a$ as $U = D(a)$.", "Hence a power of $a$ annihilates the finite $R$-module $R'/R$.", "As $\\mathfrak m = \\sqrt{(a)}$ we see that $R'/R$ is annihilated", "by a power of $\\mathfrak m$. By", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}", "we see that $IR'$ is a locally principal ideal.", "Since $R'$ is semi-local we see that $IR'$ is principal,", "see Algebra, Lemma \\ref{algebra-lemma-locally-free-semi-local-free},", "say $IR' = (t)$. Then we have $a = a't$ and $b = b't$ and everything is", "clear." ], "refs": [ "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "divisors-lemma-blowing-up-projective", "divisors-lemma-blow-up-and-irreducible-components", "varieties-lemma-finite-in-codim-1", "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "algebra-lemma-locally-free-semi-local-free" ], "ref_ids": [ 8054, 8063, 8060, 10978, 8054, 799 ] } ], "ref_ids": [] }, { "id": 5658, "type": "theorem", "label": "chow-lemma-not-infinitely-divisible", "categories": [ "chow" ], "title": "chow-lemma-not-infinitely-divisible", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $1$.", "Let $a, b \\in R$ be nonzerodivisors with $a \\in \\mathfrak m$.", "There exists an integer $n = n(R, a, b)$ such that for a finite ring", "extension $R \\subset R'$ if $b = a^m c$ for some $c \\in R'$, then $m \\leq n$." ], "refs": [], "proofs": [ { "contents": [ "Choose a minimal prime $\\mathfrak q \\subset R$. Observe that", "$\\dim(R/\\mathfrak q) = 1$, in particular $R/\\mathfrak q$ is not a field.", "We can choose a discrete valuation ring $A$ dominating $R/\\mathfrak q$", "with the same fraction field, see", "Algebra, Lemma \\ref{algebra-lemma-dominate-by-dimension-1}. Observe that", "$a$ and $b$ map to nonzero elements of $A$ as nonzerodivisors in $R$", "are not contained in $\\mathfrak q$. Let $v$ be the discrete valuation on $A$.", "Then $v(a) > 0$ as $a \\in \\mathfrak m$.", "We claim $n = v(b)/v(a)$ works.", "\\medskip\\noindent", "Let $R \\subset R'$ be given. Set $A' = A \\otimes_R R'$.", "Since $\\Spec(R') \\to \\Spec(R)$ is surjective", "(Algebra, Lemma \\ref{algebra-lemma-integral-overring-surjective})", "also $\\Spec(A') \\to \\Spec(A)$ is surjective", "(Algebra, Lemma \\ref{algebra-lemma-surjective-spec-radical-ideal}).", "Pick a prime $\\mathfrak q' \\subset A'$ lying over $(0) \\subset A$.", "Then $A \\subset A'' = A'/\\mathfrak q'$ is a finite extension of rings", "(again inducing a surjection on spectra).", "Pick a maximal ideal $\\mathfrak m'' \\subset A''$", "lying over the maximal ideal of $A$ and a discrete valuation ring", "$A'''$ dominating $A''_{\\mathfrak m''}$ (see lemma cited above).", "Then $A \\to A'''$ is an extension of discrete valuation rings", "and we have $b = a^m c$ in $A'''$. Thus $v'''(b) \\geq mv'''(a)$.", "Since $v''' = ev$ where $e$ is the ramification index", "of $A'''/A$, we find that $m \\leq n$ as desired." ], "refs": [ "algebra-lemma-dominate-by-dimension-1", "algebra-lemma-integral-overring-surjective", "algebra-lemma-surjective-spec-radical-ideal" ], "ref_ids": [ 1020, 495, 443 ] } ], "ref_ids": [] }, { "id": 5659, "type": "theorem", "label": "chow-lemma-prepare-tame-symbol", "categories": [ "chow" ], "title": "chow-lemma-prepare-tame-symbol", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring of dimension $1$.", "Let $r \\geq 2$ and let $a_1, \\ldots, a_r \\in A$ be nonzerodivisors", "not all units.", "Then there exist", "\\begin{enumerate}", "\\item a finite ring extension $A \\subset B$ with", "$B/A$ annihilated by a power of $\\mathfrak m$,", "\\item for each of maximal ideal $\\mathfrak m_j \\subset B$", "a nonzerodivisor $\\pi_j \\in B_j = B_{\\mathfrak m_j}$, and", "\\item factorizations $a_i = u_{i, j} \\pi_j^{e_{i, j}}$ in $B_j$", "with $u_{i, j} \\in B_j$ units and $e_{i, j} \\geq 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since at least one $a_i$ is not a unit and we find that $\\mathfrak m$", "is not an associated prime of $A$. Moreover, for any $A \\subset B$", "as in the statement $\\mathfrak m$ is not an associated prime of $B$", "and $\\mathfrak m_j$ is not an associate prime of $B_j$.", "Keeping this in mind will help check the arguments below.", "\\medskip\\noindent", "First, we claim that it suffices to prove the lemma for $r = 2$.", "We will argue this by induction on $r$; we suggest the reader", "skip the proof. Suppose we are given $A \\subset B$ and $\\pi_j$ in", "$B_j = B_{\\mathfrak m_j}$ and factorizations", "$a_i = u_{i, j} \\pi_j^{e_{i, j}}$ for $i = 1, \\ldots, r - 1$ in $B_j$", "with $u_{i, j} \\in B_j$ units and $e_{i, j} \\geq 0$.", "Then by the case $r = 2$ for $\\pi_j$ and $a_r$ in $B_j$", "we can find extensions $B_j \\subset C_j$ and for every maximal ideal", "$\\mathfrak m_{j, k}$ of $C_j$ a nonzerodivisor", "$\\pi_{j, k} \\in C_{j, k} = (C_j)_{\\mathfrak m_{j, k}}$", "and factorizations", "$$", "\\pi_j = v_{j, k} \\pi_{j, k}^{f_{j, k}}", "\\quad\\text{and}\\quad", "a_r = w_{j, k} \\pi_{j, k}^{g_{j, k}}", "$$", "as in the lemma. There exists a unique finite extension $B \\subset C$", "with $C/B$ annihilated by a power of $\\mathfrak m$ such", "that $C_j \\cong C_{\\mathfrak m_j}$ for all $j$, see", "Lemma \\ref{lemma-glue-at-max}.", "The maximal ideals of $C$ correspond $1$-to-$1$", "to the maximal ideals $\\mathfrak m_{j, k}$ in the localizations", "and in these localizations we have", "$$", "a_i = u_{i, j} \\pi_j^{e_{i, j}} =", "u_{i, j} v_{j, k}^{e_{i, j}} \\pi_{j, k}^{e_{i, j}f_{j, k}}", "$$", "for $i \\leq r - 1$. Since $a_r$ factors correctly too the", "proof of the induction step is complete.", "\\medskip\\noindent", "Proof of the case $r = 2$. We will use induction on", "$$", "\\ell = \\min(\\text{length}_A(A/a_1A),\\ \\text{length}_A(A/a_2A)).", "$$", "If $\\ell = 0$, then either $a_1$ or $a_2$ is a unit and", "the lemma holds with $A = B$. Thus we may and do assume $\\ell > 0$.", "\\medskip\\noindent", "Suppose we have a finite extension of rings $A \\subset A'$ such that", "$A'/A$ is annihilated by a power of $\\mathfrak m$ and such that", "$\\mathfrak m$ is not an associated prime of $A'$.", "Let $\\mathfrak m_1, \\ldots, \\mathfrak m_r \\subset A'$", "be the maximal ideals and set $A'_i = A'_{\\mathfrak m_i}$.", "If we can solve the problem for $a_1, a_2$ in each $A'_i$,", "then we can apply Lemma \\ref{lemma-glue-at-max}", "to produce a solution for $a_1, a_2$ in $A$.", "Choose $x \\in \\{a_1, a_2\\}$ such that $\\ell = \\text{length}_A(A/xA)$.", "By Lemma \\ref{lemma-compare-periodic-lengths} ", "and (\\ref{equation-multiplicity-coker-ker})", "we have $\\text{length}_A(A/xA) = \\text{length}_A(A'/xA')$.", "On the other hand, we have", "$$", "\\text{length}_A(A'/xA') =", "\\sum [\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m)]", "\\text{length}_{A'_i}(A'_i/xA'_i)", "$$", "by Algebra, Lemma \\ref{algebra-lemma-pushdown-module}.", "Since $x \\in \\mathfrak m$ we see that each term on the right hand side", "is positive. We conclude that the induction hypothesis applies", "to $a_1, a_2$ in each $A'_i$ if $r > 1$ or if $r = 1$ and", "$[\\kappa(\\mathfrak m_1) : \\kappa(\\mathfrak m)] > 1$.", "We conclude that we may assume each $A'$ as above is local with", "the same residue field as $A$.", "\\medskip\\noindent", "Applying the discussion of the previous paragraph,", "we may replace $A$ by the ring constructed in", "Lemma \\ref{lemma-Noetherian-domain-dim-1-two-elements}", "for $a_1, a_2 \\in A$. Then since $A$ is local we find,", "after possibly switching $a_1$ and $a_2$, that $a_2 \\in (a_1)$.", "Write $a_2 = a_1^m c$ with $m > 0$ maximal. In fact, by", "Lemma \\ref{lemma-not-infinitely-divisible}", "we may assume $m$ is maximal even after replacing $A$", "by any finite extension $A \\subset A'$ as in the previous paragraph.", "If $c$ is a unit, then we are done. If not, then we replace", "$A$ by the ring constructed in", "Lemma \\ref{lemma-Noetherian-domain-dim-1-two-elements}", "for $a_1, c \\in A$. Then either (1) $c = a_1 c'$ or", "(2) $a_1 = c a'_1$. The first case cannot happen since", "it would give $a_2 = a_1^{m + 1} c'$ contradicting the", "maximality of $m$. In the second case we get", "$a_1 = c a'_1$ and $a_2 = c^{m + 1} (a'_1)^m$.", "Then it suffices to prove the lemma for $A$ and $c, a'_1$.", "If $a'_1$ is a unit we're done and if not, then", "$\\text{length}_A(A/cA) < \\ell$ because $cA$ is a strictly", "bigger ideal than $a_1A$. Thus we win by induction hypothesis." ], "refs": [ "chow-lemma-glue-at-max", "chow-lemma-glue-at-max", "chow-lemma-compare-periodic-lengths", "algebra-lemma-pushdown-module", "chow-lemma-Noetherian-domain-dim-1-two-elements", "chow-lemma-not-infinitely-divisible", "chow-lemma-Noetherian-domain-dim-1-two-elements" ], "ref_ids": [ 5656, 5656, 5651, 639, 5657, 5658, 5657 ] } ], "ref_ids": [] }, { "id": 5660, "type": "theorem", "label": "chow-lemma-well-defined-tame-symbol", "categories": [ "chow" ], "title": "chow-lemma-well-defined-tame-symbol", "contents": [ "The formula (\\ref{equation-tame-symbol}) determines a", "well defined element of $\\kappa(\\mathfrak m)^*$. In other words, the", "right hand side does not depend on the choice of the", "local factorizations or the choice of $B$." ], "refs": [], "proofs": [ { "contents": [ "Independence of choice of factorizations. Suppose we have", "a Noetherian $1$-dimensional local ring $B$, elements $a_1, a_2 \\in B$,", "and nonzerodivisors $\\pi, \\theta$ such that we can write", "$$", "a_1 = u_1 \\pi^{e_1} = v_1 \\theta^{f_1},\\quad", "a_2 = u_2 \\pi^{e_2} = v_2 \\theta^{f_2}", "$$", "with $e_i, f_i \\geq 0$ integers and $u_i, v_i$ units in $B$.", "Observe that this implies", "$$", "a_1^{e_2} = u_1^{e_2}u_2^{-e_1}a_2^{e_1},\\quad", "a_1^{f_2} = v_1^{f_2}v_2^{-f_1}a_2^{f_1}", "$$", "On the other hand, setting", "$m = \\text{length}_B(B/\\pi B)$ and $k = \\text{length}_B(B/\\theta B)$", "we find $e_2 m = \\text{length}_B(B/a_2 B) = f_2 k$.", "Expanding $a_1^{e_2m} = a_1^{f_2 k}$ using the above we find", "$$", "(u_1^{e_2}u_2^{-e_1})^m = (v_1^{f_2}v_2^{-f_1})^k", "$$", "This proves the desired equality up to signs. To see the signs", "work out we have to show $me_1e_2$ is even if and only if", "$kf_1f_2$ is even. This follows as both $me_2 = kf_2$ and", "$me_1 = kf_1$ (same argument as above).", "\\medskip\\noindent", "Independence of choice of $B$. Suppose given two extensions", "$A \\subset B$ and $A \\subset B'$ as in Lemma \\ref{lemma-prepare-tame-symbol}.", "Then", "$$", "C = (B \\otimes_A B')/(\\mathfrak m\\text{-power torsion})", "$$", "will be a third one. Thus we may assume we have", "$A \\subset B \\subset C$ and factorizations over the", "local rings of $B$ and we have to show that using", "the same factorizations over the local rings of $C$", "gives the same element of $\\kappa(\\mathfrak m)$.", "By transitivity of norms", "(Fields, Lemma \\ref{fields-lemma-trace-and-norm-tower})", "this comes down to the following problem:", "if $B$ is a Noetherian local ring of dimension $1$", "and $\\pi \\in B$ is a nonzerodivisor, then", "$$", "\\lambda^m = \\prod \\text{Norm}_{\\kappa_k/\\kappa}(\\lambda)^{m_k}", "$$", "Here we have used the following notation:", "(1) $\\kappa$ is the residue field of $B$,", "(2) $\\lambda$ is an element of $\\kappa$,", "(3) $\\mathfrak m_k \\subset C$ are the maximal ideals of $C$,", "(4) $\\kappa_k = \\kappa(\\mathfrak m_k)$ is the residue field of", "$C_k = C_{\\mathfrak m_k}$,", "(5) $m = \\text{length}_B(B/\\pi B)$, and", "(6) $m_k = \\text{length}_{C_k}(C_k/\\pi C_k)$.", "The displayed equality holds because", "$\\text{Norm}_{\\kappa_k/\\kappa}(\\lambda) = \\lambda^{[\\kappa_k : \\kappa]}$", "as $\\lambda \\in \\kappa$ and because $m = \\sum m_k[\\kappa_k:\\kappa]$.", "First, we have $m = \\text{length}_B(B/xB) = \\text{length}_B(C/\\pi C)$", "by Lemma \\ref{lemma-compare-periodic-lengths} ", "and (\\ref{equation-multiplicity-coker-ker}).", "Finally, we have $\\text{length}_B(C/\\pi C) = \\sum m_k[\\kappa_k:\\kappa]$", "by Algebra, Lemma \\ref{algebra-lemma-pushdown-module}." ], "refs": [ "chow-lemma-prepare-tame-symbol", "fields-lemma-trace-and-norm-tower", "chow-lemma-compare-periodic-lengths", "algebra-lemma-pushdown-module" ], "ref_ids": [ 5659, 4502, 5651, 639 ] } ], "ref_ids": [] }, { "id": 5661, "type": "theorem", "label": "chow-lemma-tame-symbol", "categories": [ "chow" ], "title": "chow-lemma-tame-symbol", "contents": [ "The tame symbol (\\ref{equation-tame-symbol}) satisfies", "(\\ref{item-bilinear-better}), (\\ref{item-skew-better}),", "(\\ref{item-normalization}), (\\ref{item-1-x-better}) and hence", "gives a map $\\partial_A : Q(A)^* \\times Q(A)^* \\to \\kappa(\\mathfrak m)^*$", "satisfying (\\ref{item-bilinear}), (\\ref{item-skew}), (\\ref{item-1-x})." ], "refs": [], "proofs": [ { "contents": [ "Let us prove (\\ref{item-bilinear-better}).", "Let $a_1, a_2, a_3 \\in A$ be nonzerodivisors.", "Choose $A \\subset B$ as in Lemma \\ref{lemma-prepare-tame-symbol}", "for $a_1, a_2, a_3$. Then the equality", "$$", "\\partial_A(a_1a_2, a_3) = \\partial_A(a_1, a_3) \\partial_A(a_2, a_3)", "$$", "follows from the equality", "$$", "(-1)^{(e_{1, j} + e_{2, j})e_{3, j}}", "(u_{1, j}u_{2, j})^{e_{3, j}}u_{3, j}^{-e_{1, j} - e_{2, j}} =", "(-1)^{e_{1, j}e_{3, j}}", "u_{1, j}^{e_{3, j}}u_{3, j}^{-e_{1, j}}", "(-1)^{e_{2, j}e_{3, j}}", "u_{2, j}^{e_{3, j}}u_{3, j}^{-e_{2, j}}", "$$", "in $B_j$. Properties (\\ref{item-skew-better}) and", "(\\ref{item-normalization}) are equally immediate.", "\\medskip\\noindent", "Let us prove (\\ref{item-1-x-better}). Let $a_1, a_2, a_1 - a_2 \\in A$", "be nonzerodivisors and set $a_3 = a_1 - a_2$.", "Choose $A \\subset B$ as in Lemma \\ref{lemma-prepare-tame-symbol}", "for $a_1, a_2, a_3$. Then it suffices to show", "$$", "(-1)^{e_{1, j}e_{2, j} + e_{1, j}e_{3, j} + e_{2, j}e_{3, j} + e_{2, j}}", "u_{1, j}^{e_{2, j} - e_{3, j}}", "u_{2, j}^{e_{3, j} - e_{1, j}}", "u_{3, j}^{e_{1, j} - e_{2, j}} \\bmod \\mathfrak m_j = 1", "$$", "This is clear if $e_{1, j} = e_{2, j} = e_{3, j}$.", "Say $e_{1, j} > e_{2, j}$. Then we see that $e_{3, j} = e_{2, j}$", "because $a_3 = a_1 - a_2$ and we see that $u_{3, j}$", "has the same residue class as $-u_{2, j}$. Hence", "the formula is true -- the signs work out as well", "and this verification is the reason for the choice of signs", "in (\\ref{equation-tame-symbol}).", "The other cases are handled in exactly the same manner." ], "refs": [ "chow-lemma-prepare-tame-symbol", "chow-lemma-prepare-tame-symbol" ], "ref_ids": [ 5659, 5659 ] } ], "ref_ids": [] }, { "id": 5662, "type": "theorem", "label": "chow-lemma-norm-down-tame-symbol", "categories": [ "chow" ], "title": "chow-lemma-norm-down-tame-symbol", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring of dimension $1$.", "Let $A \\subset B$ be a finite ring extension with $B/A$", "annihilated by a power of $\\mathfrak m$ and $\\mathfrak m$ not", "an associated prime of $B$.", "For $a, b \\in A$ nonzerodivisors we have", "$$", "\\partial_A(a, b) = \\prod", "\\text{Norm}_{\\kappa(\\mathfrak m_j)/\\kappa(\\mathfrak m)}(\\partial_{B_j}(a, b))", "$$", "where the product is over the maximal ideals $\\mathfrak m_j$ of $B$", "and $B_j = B_{\\mathfrak m_j}$." ], "refs": [], "proofs": [ { "contents": [ "Choose $B_j \\subset C_j$ as in", "Lemma \\ref{lemma-prepare-tame-symbol} for $a, b$.", "By Lemma \\ref{lemma-glue-at-max} we can choose a finite ring", "extension $B \\subset C$ with $C_j \\cong C_{\\mathfrak m_j}$ for all $j$.", "Let $\\mathfrak m_{j, k} \\subset C$ be the maximal ideals of $C$", "lying over $\\mathfrak m_j$. Let", "$$", "a = u_{j, k}\\pi_{j, k}^{f_{j, k}},\\quad", "b = v_{j, k}\\pi_{j, k}^{g_{j, k}}", "$$", "be the local factorizations which exist by our choice of", "$C_j \\cong C_{\\mathfrak m_j}$. By definition we have", "$$", "\\partial_A(a, b) = ", "\\prod\\nolimits_{j, k}", "\\text{Norm}_{\\kappa(\\mathfrak m_{j, k})/\\kappa(\\mathfrak m)}", "((-1)^{f_{j, k}g_{j, k}}u_{j, k}^{g_{j, k}}v_{j, k}^{-f_{j, k}}", "\\bmod \\mathfrak m_{j, k})^{m_{j, k}}", "$$", "and", "$$", "\\partial_{B_j}(a, b) = ", "\\prod\\nolimits_k", "\\text{Norm}_{\\kappa(\\mathfrak m_{j, k})/\\kappa(\\mathfrak m_j)}", "((-1)^{f_{j, k}g_{j, k}}u_{j, k}^{g_{j, k}}v_{j, k}^{-f_{j, k}}", "\\bmod \\mathfrak m_{j, k})^{m_{j, k}}", "$$", "The result follows by transitivity of norms", "for $\\kappa(\\mathfrak m_{j, k})/\\kappa(\\mathfrak m_j)/\\kappa(\\mathfrak m)$, see", "Fields, Lemma \\ref{fields-lemma-trace-and-norm-tower}." ], "refs": [ "chow-lemma-prepare-tame-symbol", "chow-lemma-glue-at-max", "fields-lemma-trace-and-norm-tower" ], "ref_ids": [ 5659, 5656, 4502 ] } ], "ref_ids": [] }, { "id": 5663, "type": "theorem", "label": "chow-lemma-tame-symbol-formally-smooth", "categories": [ "chow" ], "title": "chow-lemma-tame-symbol-formally-smooth", "contents": [ "Let $(A, \\mathfrak m, \\kappa) \\to (A', \\mathfrak m', \\kappa')$", "be a local homomorphism of Noetherian local rings of dimension $1$.", "If $A \\to A'$ is flat, $\\mathfrak m' = \\mathfrak m A'$, and $\\kappa'/\\kappa$", "is separable, then for $a_1, a_2 \\in A$ nonzerodivisors the tame symbol", "$\\partial_A(a_1, a_2)$ maps to $\\partial_{A'}(a_1, a_2)$." ], "refs": [], "proofs": [ { "contents": [ "If $a_1, a_2$ are both units, then $\\partial_A(a_1, a_2) = 1$", "and $\\partial_{A'}(a_1, a_2) = 1$ and the result is true.", "If not, then we can choose a ring extension $A \\subset B$ and", "local factorizations as in Lemma \\ref{lemma-prepare-tame-symbol}.", "Set $B' = A' \\otimes_A B$. Since $A'$ is flat over $A$ we see", "that $A' \\subset B'$ is a ring extension with $B'/A'$ annihilated", "by a power of $\\mathfrak m'$. Let $\\mathfrak m_1, \\ldots, \\mathfrak m_m$", "be the maximal ideals of $B$. For each $j \\in \\{1, \\ldots, m\\}$", "denote $\\kappa_j = \\kappa(\\mathfrak m_j)$ the residue field.", "Then", "$$", "\\kappa_j \\otimes_\\kappa \\kappa' =", "\\prod\\nolimits_{l = 1, \\ldots, n_j} \\kappa'_{j, l}", "$$", "is a product of fields each finite over $\\kappa'$ because $\\kappa'/\\kappa$", "is a separable field extension", "(Algebra, Lemma \\ref{algebra-lemma-separable-extension-preserves-reducedness}).", "It follows that $B'$ has corresponding maximal ideals $\\mathfrak m'_{j, l}$", "lying over $\\mathfrak m_j$. As factorizations in", "$B'_{j, l} = B'_{\\mathfrak m'_{j, l}}$ we use the image of the factorizations", "$a_i = u_{i, j} \\pi_j^{e_{i, j}}$ given to us in $B_j$.", "Thus we obtain", "$$", "\\partial_A(a_1, a_2) = \\prod\\nolimits_j", "\\text{Norm}_{\\kappa_j/\\kappa}", "((-1)^{e_{1, j}e_{2, j}}u_{1, j}^{e_{2, j}}u_{2, j}^{-e_{1, j}}", "\\bmod \\mathfrak m_j)^{m_j}", "$$", "by definition and similarly", "$$", "\\partial_{A'}(a_1, a_2) = \\prod\\nolimits_{j, l}", "\\text{Norm}_{\\kappa'_{j, l}/\\kappa'}", "((-1)^{e_{1, j}e_{2, j}}u_{1, j}^{e_{2, j}}u_{2, j}^{-e_{1, j}}", "\\bmod \\mathfrak m'_{j, l})^{m_j}", "$$", "To finish the proof we observe that if $u \\in \\kappa_j$", "has image $u_l \\in \\kappa'_{j, l}$, then", "$\\text{Norm}_{\\kappa_j/\\kappa}(u)$ in $\\kappa$ maps to", "$\\prod_l \\text{Norm}_{\\kappa'_{j, l}/\\kappa'}(u_l)$ in $\\kappa'$.", "This follows from the fact that taking determinants of linear", "maps commutes with ground field extension." ], "refs": [ "chow-lemma-prepare-tame-symbol", "algebra-lemma-separable-extension-preserves-reducedness" ], "ref_ids": [ 5659, 565 ] } ], "ref_ids": [] }, { "id": 5664, "type": "theorem", "label": "chow-lemma-perpare-key", "categories": [ "chow" ], "title": "chow-lemma-perpare-key", "contents": [ "Let $(A, \\mathfrak m)$ be a $2$-dimensional Noetherian local ring.", "Let $t \\in \\mathfrak m$ be a nonzerodivisor. Say", "$V(t) = \\{\\mathfrak m, \\mathfrak q_1, \\ldots, \\mathfrak q_r\\}$.", "Let $A_{\\mathfrak q_i} \\subset B_i$ be a finite ring", "extension with $B_i/A_{\\mathfrak q_i}$ annihilated by a power of", "$t$. Then there exists a finite extension $A \\subset B$ of", "local rings identifying residue fields", "with $B_i \\cong B_{\\mathfrak q_i}$ and $B/A$ annihilated", "by a power of $t$." ], "refs": [], "proofs": [ { "contents": [ "Choose $n > 0$ such that $B_i \\subset t^{-n}A_{\\mathfrak q_i}$.", "Let $M \\subset t^{-n}A$, resp.\\ $M' \\subset t^{-2n}A$ be the", "$A$-submodule consisting of elements mapping to $B_i$ in", "$t^{-n}A_{\\mathfrak q_i}$, resp.\\ $t^{-2n}A_{\\mathfrak q_i}$.", "Then $M \\subset M'$ are finite $A$-modules as $A$ is Noetherian", "and $M_{\\mathfrak q_i} = M'_{\\mathfrak q_i} = B_i$ as localization", "is exact. Thus $M'/M$ is annihilated by $\\mathfrak m^c$ for some", "$c > 0$. Observe that $M \\cdot M \\subset M'$ under the multiplication", "$t^{-n}A \\times t^{-n}A \\to t^{-2n}A$. Hence", "$B = A + \\mathfrak m^{c + 1}M$ is a finite $A$-algebra with the correct", "localizations. We omit the verification that $B$ is local with", "maximal ideal $\\mathfrak m + \\mathfrak m^{c + 1}M$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5665, "type": "theorem", "label": "chow-lemma-key-nonzerodivisors", "categories": [ "chow" ], "title": "chow-lemma-key-nonzerodivisors", "contents": [ "Let $(A, \\mathfrak m)$ be a $2$-dimensional Noetherian local ring.", "Let $a, b \\in A$ be nonzerodivisors.", "Then we have", "$$", "\\sum", "\\text{ord}_{A/\\mathfrak q}(\\partial_{A_{\\mathfrak q}}(a, b))", "=", "0", "$$", "where the sum is over the height $1$ primes $\\mathfrak q$ of $A$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathfrak q$ is a height $1$ prime of $A$ such that $a, b$", "map to a unit of $A_\\mathfrak q$, then $\\partial_{A_\\mathfrak q}(a, b) = 1$.", "Thus the sum is finite. In fact, if", "$V(ab) = \\{\\mathfrak m, \\mathfrak q_1, \\ldots, \\mathfrak q_r\\}$", "then the sum is over $i = 1, \\ldots, r$.", "For each $i$ we pick an extension $A_{\\mathfrak q_i} \\subset B_i$", "as in Lemma \\ref{lemma-prepare-tame-symbol} for $a, b$.", "By Lemma \\ref{lemma-perpare-key} with $t = ab$ and the given list of primes", "we may assume we have a finite local extension $A \\subset B$", "with $B/A$ annihilated by a power of $ab$ and such that", "for each $i$ the $B_{\\mathfrak q_i} \\cong B_i$.", "Observe that if $\\mathfrak q_{i, j}$ are the primes of $B$", "lying over $\\mathfrak q_i$ then we have", "$$", "\\text{ord}_{A/\\mathfrak q_i}(\\partial_{A_{\\mathfrak q_i}}(a, b))", "=", "\\sum\\nolimits_j", "\\text{ord}_{B/\\mathfrak q_{i, j}}(\\partial_{B_{\\mathfrak q_{i, j}}}(a, b))", "$$", "by Lemma \\ref{lemma-norm-down-tame-symbol} and", "Algebra, Lemma \\ref{algebra-lemma-finite-extension-dim-1}.", "Thus we may replace $A$ by $B$ and", "reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume for each $i$ there is a nonzerodivisor", "$\\pi_i \\in A_{\\mathfrak q_i}$ and units $u_i, v_i \\in A_{\\mathfrak q_i}$", "such that for some integers $e_i, f_i \\geq 0$ we have", "$$", "a = u_i \\pi_i^{e_i},\\quad b = v_i \\pi_i^{f_i}", "$$", "in $A_{\\mathfrak q_i}$. Setting", "$m_i = \\text{length}_{A_{\\mathfrak q_i}}(A_{\\mathfrak q_i}/\\pi_i)$", "we have", "$\\partial_{A_{\\mathfrak q_i}}(a, b) =", "((-1)^{e_if_i}u_i^{f_i}v_i^{-e_i})^{m_i}$ by definition.", "Since $a, b$ are nonzerodivisors the", "$(2, 1)$-periodic complex $(A/(ab), a, b)$ has vanishing cohomology.", "Denote $M_i$ the image of $A/(ab)$ in $A_{\\mathfrak q_i}/(ab)$.", "Then we have a map", "$$", "A/(ab) \\longrightarrow \\bigoplus M_i", "$$", "whose kernel and cokernel are supported in $\\{\\mathfrak m\\}$", "and hence have finite length. Thus we see that", "$$", "\\sum e_A(M_i, a, b) = 0", "$$", "by Lemma \\ref{lemma-compare-periodic-lengths}. Hence it suffices to show", "$e_A(M_i, a, b) =", "- \\text{ord}_{A/\\mathfrak q_i}(\\partial_{A_{\\mathfrak q_i}}(a, b))$.", "\\medskip\\noindent", "Let us prove this first, in case $\\pi_i, u_i, v_i$ are the images", "of elements $\\pi_i, u_i, v_i \\in A$ (using the same symbols", "should not cause any confusion). In this case we get", "\\begin{align*}", "e_A(M_i, a, b) & =", "e_A(M_i, u_i\\pi_i^{e_i}, v_i\\pi_i^{f_i}) \\\\", "& =", "e_A(M_i, \\pi_i^{e_i}, \\pi_i^{f_i}) -", "e_A(\\pi_i^{e_i}M_i, 0, u_i) +", "e_A(\\pi_i^{f_i}M_i, 0, v_i) \\\\", "& =", "0 -", "f_im_i\\text{ord}_{A/\\mathfrak q_i}(u_i) +", "e_im_i\\text{ord}_{A/\\mathfrak q_i}(v_i) \\\\", "& =", "-m_i\\text{ord}_{A/\\mathfrak q_i}(u_i^{f_i}v_i^{-e_i}) =", "-\\text{ord}_{A/\\mathfrak q_i}(\\partial_{A_{\\mathfrak q_i}}(a, b))", "\\end{align*}", "The second equality holds by Lemma \\ref{lemma-multiply-period-length}.", "Observe that", "$M_i \\subset (M_i)_{\\mathfrak q_i} = A_{\\mathfrak q_i}/(\\pi_i^{e_i + f_i})$", "and", "$(\\pi_i^{e_i}M_i)_{\\mathfrak q_i} \\cong A_{\\mathfrak q_i}/\\pi_i^{f_i}$ and", "$(\\pi_i^{f_i}M_i)_{\\mathfrak q_i} \\cong A_{\\mathfrak q_i}/\\pi_i^{e_i}$.", "The $0$ in the third equality comes from", "Lemma \\ref{lemma-powers-period-length-zero}", "and the other two terms come from", "Lemma \\ref{lemma-length-multiplication}.", "The last two equalities follow from multiplicativity of", "the order function and from the definition of our tame symbol.", "\\medskip\\noindent", "In general, we may first choose $c \\in A$, $c \\not \\in \\mathfrak q_i$", "such that $c\\pi_i \\in A$. After replacing $\\pi_i$ by $c\\pi_i$", "and $u_i$ by $c^{-e_i}u_i$ and $v_i$ by $c^{-f_i}v_i$", "we may and do assume $\\pi_i$ is in $A$.", "Next, choose an $c \\in A$, $c \\not \\in \\mathfrak q_i$", "with $cu_i, cv_i \\in A$. Then we observe that", "$$", "e_A(M_i, ca, cb) = e_A(M_i, a, b) - e_A(aM_i, 0, c) + e_A(bM_i, 0, c)", "$$", "by Lemma \\ref{lemma-length-multiplication}.", "On the other hand, we have", "$$", "\\partial_{A_{\\mathfrak q_i}}(ca, cb) =", "c^{m_i(f_i - e_i)}\\partial_{A_{\\mathfrak q_i}}(a, b)", "$$", "in $\\kappa(\\mathfrak q_i)^*$ because $c$ is a unit in $A_{\\mathfrak q_i}$.", "The arguments in the previous paragraph show that", "$e_A(M_i, ca, cb) = -", "\\text{ord}_{A/\\mathfrak q_i}(\\partial_{A_{\\mathfrak q_i}}(ca, cb))$.", "Thus it suffices to prove", "$$", "e_A(aM_i, 0, c) = \\text{ord}_{A/\\mathfrak q_i}(c^{m_if_i})", "\\quad\\text{and}\\quad", "e_A(bM_i, 0, c) = \\text{ord}_{A/\\mathfrak q_i}(c^{m_ie_i})", "$$", "and this follows from Lemma \\ref{lemma-length-multiplication}", "by the description (see above)", "of what happens when we localize at $\\mathfrak q_i$." ], "refs": [ "chow-lemma-prepare-tame-symbol", "chow-lemma-perpare-key", "chow-lemma-norm-down-tame-symbol", "algebra-lemma-finite-extension-dim-1", "chow-lemma-compare-periodic-lengths", "chow-lemma-multiply-period-length", "chow-lemma-powers-period-length-zero", "chow-lemma-length-multiplication", "chow-lemma-length-multiplication", "chow-lemma-length-multiplication" ], "ref_ids": [ 5659, 5664, 5662, 1047, 5651, 5655, 5654, 5652, 5652, 5652 ] } ], "ref_ids": [] }, { "id": 5666, "type": "theorem", "label": "chow-lemma-milnor-gersten-low-degree", "categories": [ "chow" ], "title": "chow-lemma-milnor-gersten-low-degree", "contents": [ "\\begin{reference}", "When $A$ is an excellent ring this is \\cite[Proposition 1]{Kato-Milnor-K}.", "\\end{reference}", "Let $A$ be a $2$-dimensional Noetherian local domain with fraction field $K$.", "Let $f, g \\in K^*$.", "Let $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ be the height", "$1$ primes $\\mathfrak q$ of $A$ such that either $f$ or $g$ is not an", "element of $A^*_{\\mathfrak q}$.", "Then we have", "$$", "\\sum\\nolimits_{i = 1, \\ldots, t}", "\\text{ord}_{A/\\mathfrak q_i}(\\partial_{A_{\\mathfrak q_i}}(f, g))", "=", "0", "$$", "We can also write this as", "$$", "\\sum\\nolimits_{\\text{height}(\\mathfrak q) = 1}", "\\text{ord}_{A/\\mathfrak q}(\\partial_{A_{\\mathfrak q}}(f, g))", "=", "0", "$$", "since at any height $1$ prime $\\mathfrak q$", "of $A$ where $f, g \\in A^*_{\\mathfrak q}$", "we have $\\partial_{A_{\\mathfrak q}}(f, g) = 1$." ], "refs": [], "proofs": [ { "contents": [ "Since the tame symbols $\\partial_{A_{\\mathfrak q}}(f, g)$ are", "bilinear and the order functions $\\text{ord}_{A/\\mathfrak q}$", "are additive it suffices to prove the formula when", "$f$ and $g$ are elements of $A$. This case is proven in", "Lemma \\ref{lemma-key-nonzerodivisors}." ], "refs": [ "chow-lemma-key-nonzerodivisors" ], "ref_ids": [ 5665 ] } ], "ref_ids": [] }, { "id": 5667, "type": "theorem", "label": "chow-lemma-delta-is-dimension", "categories": [ "chow" ], "title": "chow-lemma-delta-is-dimension", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Assume in addition $S$ is a Jacobson scheme, and $\\delta(s) = 0$ for every", "closed point $s$ of $S$. Let $X$ be locally of finite type over $S$.", "Let $Z \\subset X$ be an integral closed subscheme and let", "$\\xi \\in Z$ be its generic point. The following integers are the same:", "\\begin{enumerate}", "\\item $\\delta_{X/S}(\\xi)$,", "\\item $\\dim(Z)$, and", "\\item $\\dim(\\mathcal{O}_{Z, z})$ where $z$ is a closed point of $Z$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X \\to S$, $\\xi \\in Z \\subset X$ be as in the lemma.", "Since $X$ is locally of finite type over $S$ we see that", "$X$ is Jacobson, see", "Morphisms, Lemma \\ref{morphisms-lemma-Jacobson-universally-Jacobson}.", "Hence closed points of $X$ are dense in every closed subset of $Z$", "and map to closed points of $S$. Hence given any chain", "of irreducible closed subsets of $Z$ we can end it with a closed point of $Z$.", "It follows that $\\dim(Z) = \\sup_z(\\dim(\\mathcal{O}_{Z, z})$", "(see Properties, Lemma \\ref{properties-lemma-codimension-local-ring})", "where $z \\in Z$ runs over the closed points of $Z$.", "Note that $\\dim(\\mathcal{O}_{Z, z}) = \\delta(\\xi) - \\delta(z)$", "by the properties of a dimension function.", "For each closed $z \\in Z$ the field extension", "$\\kappa(z) \\supset \\kappa(f(z))$ is finite, see Morphisms,", "Lemma \\ref{morphisms-lemma-jacobson-finite-type-points}.", "Hence $\\delta_{X/S}(z) = \\delta(f(z)) = 0$ for $z \\in Z$ closed.", "It follows that all three integers are equal." ], "refs": [ "morphisms-lemma-Jacobson-universally-Jacobson", "properties-lemma-codimension-local-ring", "morphisms-lemma-jacobson-finite-type-points" ], "ref_ids": [ 5212, 2979, 5211 ] } ], "ref_ids": [] }, { "id": 5668, "type": "theorem", "label": "chow-lemma-multiplicity-finite", "categories": [ "chow" ], "title": "chow-lemma-multiplicity-finite", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $Z \\subset X$ be a closed subscheme.", "\\begin{enumerate}", "\\item Let $Z' \\subset Z$ be an irreducible component and", "let $\\xi \\in Z'$ be its generic point.", "Then", "$$", "\\text{length}_{\\mathcal{O}_{X, \\xi}} \\mathcal{O}_{Z, \\xi} < \\infty", "$$", "\\item If $\\dim_\\delta(Z) \\leq k$ and $\\xi \\in Z$ with", "$\\delta(\\xi) = k$, then $\\xi$ is a generic point of an", "irreducible component of $Z$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $Z' \\subset Z$, $\\xi \\in Z'$ be as in (1).", "Then $\\dim(\\mathcal{O}_{Z, \\xi}) = 0$ (for example by", "Properties, Lemma \\ref{properties-lemma-codimension-local-ring}).", "Hence $\\mathcal{O}_{Z, \\xi}$ is Noetherian", "local ring of dimension zero, and hence has finite length over", "itself (see", "Algebra, Proposition \\ref{algebra-proposition-dimension-zero-ring}).", "Hence, it also has finite length over $\\mathcal{O}_{X, \\xi}$, see", "Algebra, Lemma \\ref{algebra-lemma-pushdown-module}.", "\\medskip\\noindent", "Assume $\\xi \\in Z$ and $\\delta(\\xi) = k$.", "Consider the closure $Z' = \\overline{\\{\\xi\\}}$. It is an irreducible", "closed subscheme with $\\dim_\\delta(Z') = k$ by definition.", "Since $\\dim_\\delta(Z) = k$ it must be an irreducible component", "of $Z$. Hence we see (2) holds." ], "refs": [ "properties-lemma-codimension-local-ring", "algebra-proposition-dimension-zero-ring", "algebra-lemma-pushdown-module" ], "ref_ids": [ 2979, 1410, 639 ] } ], "ref_ids": [] }, { "id": 5669, "type": "theorem", "label": "chow-lemma-length-finite", "categories": [ "chow" ], "title": "chow-lemma-length-finite", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item The collection of irreducible components of the support of", "$\\mathcal{F}$ is locally finite.", "\\item Let $Z' \\subset \\text{Supp}(\\mathcal{F})$", "be an irreducible component and", "let $\\xi \\in Z'$ be its generic point.", "Then", "$$", "\\text{length}_{\\mathcal{O}_{X, \\xi}} \\mathcal{F}_\\xi < \\infty", "$$", "\\item If $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$", "and $\\xi \\in Z$ with $\\delta(\\xi) = k$, then $\\xi$ is a", "generic point of an irreducible component of $\\text{Supp}(\\mathcal{F})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-support-closed}", "the support $Z$ of $\\mathcal{F}$ is a closed subset of $X$.", "We may think of $Z$ as a reduced closed subscheme of $X$", "(Schemes, Lemma \\ref{schemes-lemma-reduced-closed-subscheme}).", "Hence (1) follows from", "Divisors, Lemma \\ref{divisors-lemma-components-locally-finite} applied to $Z$", "and (3) follows from", "Lemma \\ref{lemma-multiplicity-finite} applied to $Z$.", "\\medskip\\noindent", "Let $\\xi \\in Z'$ be as in (2). In this case for any specialization", "$\\xi' \\leadsto \\xi$ in $X$ we have $\\mathcal{F}_{\\xi'} = 0$.", "Recall that the non-maximal primes of $\\mathcal{O}_{X, \\xi}$ correspond", "to the points of $X$ specializing to $\\xi$", "(Schemes, Lemma \\ref{schemes-lemma-specialize-points}).", "Hence $\\mathcal{F}_\\xi$ is a finite $\\mathcal{O}_{X, \\xi}$-module", "whose support is $\\{\\mathfrak m_\\xi\\}$. Hence it has finite length", "by Algebra, Lemma \\ref{algebra-lemma-support-point}." ], "refs": [ "coherent-lemma-coherent-support-closed", "schemes-lemma-reduced-closed-subscheme", "divisors-lemma-components-locally-finite", "chow-lemma-multiplicity-finite", "schemes-lemma-specialize-points", "algebra-lemma-support-point" ], "ref_ids": [ 3314, 7681, 8022, 5668, 7684, 693 ] } ], "ref_ids": [] }, { "id": 5670, "type": "theorem", "label": "chow-lemma-cycle-closed-coherent", "categories": [ "chow" ], "title": "chow-lemma-cycle-closed-coherent", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $Z \\subset X$ be a closed subscheme.", "If $\\dim_\\delta(Z) \\leq k$, then $[Z]_k = [{\\mathcal O}_Z]_k$." ], "refs": [], "proofs": [ { "contents": [ "This is because in this case the multiplicities $m_{Z', Z}$ and", "$m_{Z', \\mathcal{O}_Z}$ agree by definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5671, "type": "theorem", "label": "chow-lemma-additivity-sheaf-cycle", "categories": [ "chow" ], "title": "chow-lemma-additivity-sheaf-cycle", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0$", "be a short exact sequence of coherent sheaves on $X$.", "Assume that the $\\delta$-dimension of the supports", "of $\\mathcal{F}$, $\\mathcal{G}$, and $\\mathcal{H}$ is $\\leq k$.", "Then $[\\mathcal{G}]_k = [\\mathcal{F}]_k + [\\mathcal{H}]_k$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from additivity of lengths, see", "Algebra, Lemma \\ref{algebra-lemma-length-additive}." ], "refs": [ "algebra-lemma-length-additive" ], "ref_ids": [ 631 ] } ], "ref_ids": [] }, { "id": 5672, "type": "theorem", "label": "chow-lemma-equal-dimension", "categories": [ "chow" ], "title": "chow-lemma-equal-dimension", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $f : X \\to Y$ be a morphism.", "Assume $X$, $Y$ integral and $\\dim_\\delta(X) = \\dim_\\delta(Y)$.", "Then either $f(X)$ is contained in a proper closed subscheme", "of $Y$, or $f$ is dominant and the extension of function fields", "$R(Y) \\subset R(X)$ is finite." ], "refs": [], "proofs": [ { "contents": [ "The closure $\\overline{f(X)} \\subset Y$ is irreducible as $X$", "is irreducible (Topology, Lemmas", "\\ref{topology-lemma-image-irreducible-space} and", "\\ref{topology-lemma-irreducible}).", "If $\\overline{f(X)} \\not = Y$, then we are done.", "If $\\overline{f(X)} = Y$, then $f$ is dominant and by", "Morphisms,", "Lemma \\ref{morphisms-lemma-dominant-finite-number-irreducible-components}", "we see that the generic point $\\eta_Y$ of $Y$ is in the image of $f$.", "Of course this implies that $f(\\eta_X) = \\eta_Y$, where $\\eta_X \\in X$", "is the generic point of $X$. Since $\\delta(\\eta_X) = \\delta(\\eta_Y)$", "we see that $R(Y) = \\kappa(\\eta_Y) \\subset \\kappa(\\eta_X) = R(X)$", "is an extension of transcendence degree $0$.", "Hence $R(Y) \\subset R(X)$ is a finite extension by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-degree}", "(which applies by", "Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type})." ], "refs": [ "topology-lemma-image-irreducible-space", "topology-lemma-irreducible", "morphisms-lemma-dominant-finite-number-irreducible-components", "morphisms-lemma-finite-degree", "morphisms-lemma-permanence-finite-type" ], "ref_ids": [ 8212, 8213, 5161, 5491, 5204 ] } ], "ref_ids": [] }, { "id": 5673, "type": "theorem", "label": "chow-lemma-quasi-compact-locally-finite", "categories": [ "chow" ], "title": "chow-lemma-quasi-compact-locally-finite", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $f : X \\to Y$ be a morphism.", "Assume $f$ is quasi-compact, and $\\{Z_i\\}_{i \\in I}$ is a locally", "finite collection of closed subsets of $X$.", "Then $\\{\\overline{f(Z_i)}\\}_{i \\in I}$ is a locally finite", "collection of closed subsets of $Y$." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset Y$ be a quasi-compact open subset.", "Since $f$ is quasi-compact the open $f^{-1}(V)$ is", "quasi-compact. Hence the set", "$\\{i \\in I \\mid Z_i \\cap f^{-1}(V) \\not = \\emptyset \\}$", "is finite by a simple topological argument which we omit.", "Since this is the same as the set", "$$", "\\{i \\in I \\mid f(Z_i) \\cap V \\not = \\emptyset \\} =", "\\{i \\in I \\mid \\overline{f(Z_i)} \\cap V \\not = \\emptyset \\}", "$$", "the lemma is proved." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5674, "type": "theorem", "label": "chow-lemma-compose-pushforward", "categories": [ "chow" ], "title": "chow-lemma-compose-pushforward", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$, and $Z$ be locally of finite type over $S$.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms.", "Then $g_* \\circ f_* = (g \\circ f)_*$ as maps $Z_k(X) \\to Z_k(Z)$." ], "refs": [], "proofs": [ { "contents": [ "Let $W \\subset X$ be an integral closed subscheme of dimension $k$.", "Consider $W' = f(W) \\subset Y$ and $W'' = g(f(W)) \\subset Z$.", "Since $f$, $g$ are proper we see that $W'$ (resp.\\ $W''$) is", "an integral closed subscheme of $Y$ (resp.\\ $Z$).", "We have to show that $g_*(f_*[W]) = (g \\circ f)_*[W]$.", "If $\\dim_\\delta(W'') < k$, then both sides are zero.", "If $\\dim_\\delta(W'') = k$, then we see the induced morphisms", "$$", "W \\longrightarrow", "W' \\longrightarrow", "W''", "$$", "both satisfy the hypotheses of Lemma \\ref{lemma-equal-dimension}. Hence", "$$", "g_*(f_*[W]) = \\deg(W/W')\\deg(W'/W'')[W''],", "\\quad", "(g \\circ f)_*[W] = \\deg(W/W'')[W''].", "$$", "Then we can apply", "Morphisms, Lemma \\ref{morphisms-lemma-degree-composition}", "to conclude." ], "refs": [ "chow-lemma-equal-dimension", "morphisms-lemma-degree-composition" ], "ref_ids": [ 5672, 5492 ] } ], "ref_ids": [] }, { "id": 5675, "type": "theorem", "label": "chow-lemma-exact-sequence-closed", "categories": [ "chow" ], "title": "chow-lemma-exact-sequence-closed", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. Let $X_1, X_2 \\subset X$", "be closed subschemes such that $X = X_1 \\cup X_2$ set theoretically.", "For every $k \\in \\mathbf{Z}$ the sequence of abelian groups", "$$", "\\xymatrix{", "Z_k(X_1 \\cap X_2) \\ar[r] &", "Z_k(X_1) \\oplus Z_k(X_2) \\ar[r] &", "Z_k(X) \\ar[r] &", "0", "}", "$$", "is exact. Here $X_1 \\cap X_2$ is the scheme theoretic intersection and", "the maps are the pushforward maps with one multiplied by $-1$." ], "refs": [], "proofs": [ { "contents": [ "First assume $X$ is quasi-compact. Then $Z_k(X)$ is a free $\\mathbf{Z}$-module", "with basis given by the elements $[Z]$ where $Z \\subset X$ is integral", "closed of $\\delta$-dimension $k$. The groups", "$Z_k(X_1)$, $Z_k(X_2)$, $Z_k(X_1 \\cap X_2)$ are free on the subset of these", "$Z$ such that $Z \\subset X_1$, $Z \\subset X_2$, $Z \\subset X_1 \\cap X_2$.", "This immediately proves the lemma in this case. The general case is similar", "and the proof is omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5676, "type": "theorem", "label": "chow-lemma-cycle-push-sheaf", "categories": [ "chow" ], "title": "chow-lemma-cycle-push-sheaf", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ be a proper morphism of schemes which are", "locally of finite type over $S$.", "\\begin{enumerate}", "\\item Let $Z \\subset X$ be a closed subscheme with $\\dim_\\delta(Z) \\leq k$.", "Then", "$$", "f_*[Z]_k = [f_*{\\mathcal O}_Z]_k.", "$$", "\\item Let $\\mathcal{F}$ be a coherent sheaf on $X$ such that", "$\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$. Then", "$$", "f_*[\\mathcal{F}]_k = [f_*{\\mathcal F}]_k.", "$$", "\\end{enumerate}", "Note that the statement makes sense since $f_*\\mathcal{F}$ and", "$f_*\\mathcal{O}_Z$ are coherent $\\mathcal{O}_Y$-modules by", "Cohomology of Schemes, Proposition", "\\ref{coherent-proposition-proper-pushforward-coherent}." ], "refs": [ "coherent-proposition-proper-pushforward-coherent" ], "proofs": [ { "contents": [ "Part (1) follows from (2) and Lemma \\ref{lemma-cycle-closed-coherent}.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Assume that $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$.", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-support-closed}", "there exists a closed subscheme $i : Z \\to X$ and a coherent", "$\\mathcal{O}_Z$-module $\\mathcal{G}$ such that", "$i_*\\mathcal{G} \\cong \\mathcal{F}$ and such that the support", "of $\\mathcal{F}$ is $Z$. Let $Z' \\subset Y$ be the scheme theoretic image", "of $f|_Z : Z \\to Y$. Consider the commutative diagram of schemes", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[d]_{f|_Z} &", "X \\ar[d]^f \\\\", "Z' \\ar[r]^{i'} & Y", "}", "$$", "We have $f_*\\mathcal{F} = f_*i_*\\mathcal{G} = i'_*(f|_Z)_*\\mathcal{G}$", "by going around the diagram in two ways. Suppose we know the result holds", "for closed immersions and for $f|_Z$. Then we see that", "$$", "f_*[\\mathcal{F}]_k = f_*i_*[\\mathcal{G}]_k", "= (i')_*(f|_Z)_*[\\mathcal{G}]_k =", "(i')_*[(f|_Z)_*\\mathcal{G}]_k =", "[(i')_*(f|_Z)_*\\mathcal{G}]_k = [f_*\\mathcal{F}]_k", "$$", "as desired. The case of a closed immersion is straightforward (omitted).", "Note that $f|_Z : Z \\to Z'$ is a dominant morphism (see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}).", "Thus we have reduced to the case where", "$\\dim_\\delta(X) \\leq k$ and $f : X \\to Y$ is proper and dominant.", "\\medskip\\noindent", "Assume $\\dim_\\delta(X) \\leq k$ and $f : X \\to Y$ is proper and dominant.", "Since $f$ is dominant, for every irreducible component $Z \\subset Y$", "with generic point $\\eta$ there exists a point $\\xi \\in X$ such", "that $f(\\xi) = \\eta$. Hence $\\delta(\\eta) \\leq \\delta(\\xi) \\leq k$.", "Thus we see that in the expressions", "$$", "f_*[\\mathcal{F}]_k = \\sum n_Z[Z],", "\\quad", "\\text{and}", "\\quad", "[f_*\\mathcal{F}]_k = \\sum m_Z[Z].", "$$", "whenever $n_Z \\not = 0$, or $m_Z \\not = 0$ the integral closed", "subscheme $Z$ is actually an irreducible component of $Y$ of", "$\\delta$-dimension $k$. Pick such an integral closed subscheme", "$Z \\subset Y$ and denote $\\eta$ its generic point. Note that for", "any $\\xi \\in X$ with $f(\\xi) = \\eta$ we have $\\delta(\\xi) \\geq k$", "and hence $\\xi$ is a generic point of an irreducible component", "of $X$ of $\\delta$-dimension $k$ as well", "(see Lemma \\ref{lemma-multiplicity-finite}). Since $f$ is quasi-compact", "and $X$ is locally Noetherian, there can be only finitely many of", "these and hence $f^{-1}(\\{\\eta\\})$ is finite.", "By Morphisms, Lemma \\ref{morphisms-lemma-generically-finite} there exists", "an open neighbourhood $\\eta \\in V \\subset Y$ such that $f^{-1}(V) \\to V$", "is finite. Replacing $Y$ by $V$ and $X$ by $f^{-1}(V)$ we reduce to the", "case where $Y$ is affine, and $f$ is finite.", "\\medskip\\noindent", "Write $Y = \\Spec(R)$ and $X = \\Spec(A)$ (possible as", "a finite morphism is affine).", "Then $R$ and $A$ are Noetherian rings and $A$ is finite over $R$.", "Moreover $\\mathcal{F} = \\widetilde{M}$ for some finite $A$-module", "$M$. Note that $f_*\\mathcal{F}$ corresponds to $M$ viewed as an $R$-module.", "Let $\\mathfrak p \\subset R$ be the minimal prime corresponding", "to $\\eta \\in Y$. The coefficient of $Z$ in $[f_*\\mathcal{F}]_k$", "is clearly $\\text{length}_{R_{\\mathfrak p}}(M_{\\mathfrak p})$.", "Let $\\mathfrak q_i$, $i = 1, \\ldots, t$ be the primes of $A$", "lying over $\\mathfrak p$. Then $A_{\\mathfrak p} = \\prod A_{\\mathfrak q_i}$", "since $A_{\\mathfrak p}$ is an Artinian ring being finite over the", "dimension zero local Noetherian ring $R_{\\mathfrak p}$.", "Clearly the coefficient of $Z$ in $f_*[\\mathcal{F}]_k$ is", "$$", "\\sum\\nolimits_{i = 1, \\ldots, t}", "[\\kappa(\\mathfrak q_i) : \\kappa(\\mathfrak p)]", "\\text{length}_{A_{\\mathfrak q_i}}(M_{\\mathfrak q_i})", "$$", "Hence the desired equality follows from", "Algebra, Lemma \\ref{algebra-lemma-pushdown-module}." ], "refs": [ "chow-lemma-cycle-closed-coherent", "coherent-lemma-coherent-support-closed", "morphisms-lemma-quasi-compact-scheme-theoretic-image", "chow-lemma-multiplicity-finite", "morphisms-lemma-generically-finite", "algebra-lemma-pushdown-module" ], "ref_ids": [ 5670, 3314, 5146, 5668, 5487, 639 ] } ], "ref_ids": [ 3401 ] }, { "id": 5677, "type": "theorem", "label": "chow-lemma-flat-inverse-image-dimension", "categories": [ "chow" ], "title": "chow-lemma-flat-inverse-image-dimension", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $f : X \\to Y$ be a morphism.", "Assume $f$ is flat of relative dimension $r$.", "For any closed subset $Z \\subset Y$ we have", "$$", "\\dim_\\delta(f^{-1}(Z)) = \\dim_\\delta(Z) + r.", "$$", "provided $f^{-1}(Z)$ is nonempty.", "If $Z$ is irreducible and $Z' \\subset f^{-1}(Z)$ is an irreducible", "component, then $Z'$ dominates $Z$ and", "$\\dim_\\delta(Z') = \\dim_\\delta(Z) + r$." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove the final statement.", "We may replace $Y$ by the integral closed subscheme $Z$ and", "$X$ by the scheme theoretic inverse image $f^{-1}(Z) = Z \\times_Y X$.", "Hence we may assume $Z = Y$ is integral and $f$ is a flat morphism", "of relative dimension $r$. Since $Y$ is locally Noetherian the", "morphism $f$ which is locally of finite type,", "is actually locally of finite presentation. Hence", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}", "applies and we see that $f$ is open.", "Let $\\xi \\in X$ be a generic point of an irreducible component", "of $X$. By the openness of $f$ we see that $f(\\xi)$ is the", "generic point $\\eta$ of $Z = Y$. Note that $\\dim_\\xi(X_\\eta) = r$", "by assumption that $f$ has relative dimension $r$. On the other", "hand, since $\\xi$ is a generic point of $X$ we see that", "$\\mathcal{O}_{X, \\xi} = \\mathcal{O}_{X_\\eta, \\xi}$ has only one", "prime ideal and hence has dimension $0$. Thus by", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}", "we conclude that the transcendence", "degree of $\\kappa(\\xi)$ over $\\kappa(\\eta)$ is $r$.", "In other words, $\\delta(\\xi) = \\delta(\\eta) + r$ as desired." ], "refs": [ "morphisms-lemma-fppf-open", "morphisms-lemma-dimension-fibre-at-a-point" ], "ref_ids": [ 5267, 5277 ] } ], "ref_ids": [] }, { "id": 5678, "type": "theorem", "label": "chow-lemma-inverse-image-locally-finite", "categories": [ "chow" ], "title": "chow-lemma-inverse-image-locally-finite", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $f : X \\to Y$ be a morphism.", "Assume $\\{Z_i\\}_{i \\in I}$ is a locally", "finite collection of closed subsets of $Y$.", "Then $\\{f^{-1}(Z_i)\\}_{i \\in I}$ is a locally finite", "collection of closed subsets of $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be a quasi-compact open subset.", "Since the image $f(U) \\subset Y$ is a quasi-compact subset", "there exists a quasi-compact open $V \\subset Y$ such that", "$f(U) \\subset V$. Note that", "$$", "\\{i \\in I \\mid f^{-1}(Z_i) \\cap U \\not = \\emptyset \\}", "\\subset", "\\{i \\in I \\mid Z_i \\cap V \\not = \\emptyset \\}.", "$$", "Since the right hand side is finite by assumption we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5679, "type": "theorem", "label": "chow-lemma-exact-sequence-open", "categories": [ "chow" ], "title": "chow-lemma-exact-sequence-open", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $U \\subset X$ be an open subscheme, and denote", "$i : Y = X \\setminus U \\to X$ as a reduced closed subscheme of $X$.", "For every $k \\in \\mathbf{Z}$ the sequence", "$$", "\\xymatrix{", "Z_k(Y) \\ar[r]^{i_*} & Z_k(X) \\ar[r]^{j^*} & Z_k(U) \\ar[r] & 0", "}", "$$", "is an exact complex of abelian groups." ], "refs": [], "proofs": [ { "contents": [ "First assume $X$ is quasi-compact. Then $Z_k(X)$ is a free $\\mathbf{Z}$-module", "with basis given by the elements $[Z]$ where $Z \\subset X$ is integral", "closed of $\\delta$-dimension $k$. Such a basis element maps", "either to the basis element $[Z \\cap U]$ or to zero if $Z \\subset Y$.", "Hence the lemma is clear in this case. The general case is similar", "and the proof is omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5680, "type": "theorem", "label": "chow-lemma-compose-flat-pullback", "categories": [ "chow" ], "title": "chow-lemma-compose-flat-pullback", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X, Y, Z$ be locally of finite type over $S$.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be flat morphisms of relative dimensions", "$r$ and $s$. Then $g \\circ f$ is flat of relative dimension", "$r + s$ and", "$$", "f^* \\circ g^* = (g \\circ f)^*", "$$", "as maps $Z_k(Z) \\to Z_{k + r + s}(X)$." ], "refs": [], "proofs": [ { "contents": [ "The composition is flat of relative dimension $r + s$ by", "Morphisms, Lemma \\ref{morphisms-lemma-composition-relative-dimension-d}.", "Suppose that", "\\begin{enumerate}", "\\item $W \\subset Z$ is a closed integral subscheme of $\\delta$-dimension $k$,", "\\item $W' \\subset Y$ is a closed integral subscheme of $\\delta$-dimension", "$k + s$ with $W' \\subset g^{-1}(W)$, and", "\\item $W'' \\subset Y$ is a closed integral subscheme of $\\delta$-dimension", "$k + s + r$ with $W'' \\subset f^{-1}(W')$.", "\\end{enumerate}", "We have to show that the coefficient $n$ of $[W'']$ in", "$(g \\circ f)^*[W]$ agrees with the coefficient $m$ of", "$[W'']$ in $f^*(g^*[W])$. That it suffices to check the lemma in these", "cases follows from Lemma \\ref{lemma-flat-inverse-image-dimension}.", "Let $\\xi'' \\in W''$, $\\xi' \\in W'$", "and $\\xi \\in W$ be the generic points. Consider the local rings", "$A = \\mathcal{O}_{Z, \\xi}$, $B = \\mathcal{O}_{Y, \\xi'}$", "and $C = \\mathcal{O}_{X, \\xi''}$. Then we have local flat ring maps", "$A \\to B$, $B \\to C$ and moreover", "$$", "n = \\text{length}_C(C/\\mathfrak m_AC),", "\\quad", "\\text{and}", "\\quad", "m = \\text{length}_C(C/\\mathfrak m_BC) \\text{length}_B(B/\\mathfrak m_AB)", "$$", "Hence the equality follows from", "Algebra, Lemma \\ref{algebra-lemma-pullback-transitive}." ], "refs": [ "morphisms-lemma-composition-relative-dimension-d", "chow-lemma-flat-inverse-image-dimension", "algebra-lemma-pullback-transitive" ], "ref_ids": [ 5285, 5677, 641 ] } ], "ref_ids": [] }, { "id": 5681, "type": "theorem", "label": "chow-lemma-pullback-coherent", "categories": [ "chow" ], "title": "chow-lemma-pullback-coherent", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X, Y$ be locally of finite type over $S$.", "Let $f : X \\to Y$ be a flat morphism of relative dimension $r$.", "\\begin{enumerate}", "\\item Let $Z \\subset Y$ be a closed subscheme with", "$\\dim_\\delta(Z) \\leq k$. Then we have", "$\\dim_\\delta(f^{-1}(Z)) \\leq k + r$", "and $[f^{-1}(Z)]_{k + r} = f^*[Z]_k$ in $Z_{k + r}(X)$.", "\\item Let $\\mathcal{F}$ be a coherent sheaf on $Y$ with", "$\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$.", "Then we have $\\dim_\\delta(\\text{Supp}(f^*\\mathcal{F})) \\leq k + r$", "and", "$$", "f^*[{\\mathcal F}]_k = [f^*{\\mathcal F}]_{k+r}", "$$", "in $Z_{k + r}(X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The statements on dimensions follow immediately from", "Lemma \\ref{lemma-flat-inverse-image-dimension}.", "Part (1) follows from part (2) by Lemma \\ref{lemma-cycle-closed-coherent}", "and the fact that $f^*\\mathcal{O}_Z = \\mathcal{O}_{f^{-1}(Z)}$.", "\\medskip\\noindent", "Proof of (2).", "As $X$, $Y$ are locally Noetherian we may apply", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-Noetherian} to see", "that $\\mathcal{F}$ is of finite type, hence $f^*\\mathcal{F}$ is", "of finite type (Modules, Lemma \\ref{modules-lemma-pullback-finite-type}),", "hence $f^*\\mathcal{F}$ is coherent", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-Noetherian} again).", "Thus the lemma makes sense. Let $W \\subset Y$ be an integral closed", "subscheme of $\\delta$-dimension $k$, and let $W' \\subset X$ be", "an integral closed subscheme of dimension $k + r$ mapping into $W$", "under $f$. We have to show that the coefficient $n$ of", "$[W']$ in $f^*[{\\mathcal F}]_k$ agrees with the coefficient", "$m$ of $[W']$ in $[f^*{\\mathcal F}]_{k+r}$. Let $\\xi \\in W$ and", "$\\xi' \\in W'$ be the generic points. Let", "$A = \\mathcal{O}_{Y, \\xi}$, $B = \\mathcal{O}_{X, \\xi'}$", "and set $M = \\mathcal{F}_\\xi$ as an $A$-module. (Note that", "$M$ has finite length by our dimension assumptions, but we", "actually do not need to verify this. See", "Lemma \\ref{lemma-length-finite}.)", "We have $f^*\\mathcal{F}_{\\xi'} = B \\otimes_A M$.", "Thus we see that", "$$", "n = \\text{length}_B(B \\otimes_A M)", "\\quad", "\\text{and}", "\\quad", "m = \\text{length}_A(M) \\text{length}_B(B/\\mathfrak m_AB)", "$$", "Thus the equality follows from", "Algebra, Lemma \\ref{algebra-lemma-pullback-module}." ], "refs": [ "chow-lemma-flat-inverse-image-dimension", "chow-lemma-cycle-closed-coherent", "coherent-lemma-coherent-Noetherian", "modules-lemma-pullback-finite-type", "coherent-lemma-coherent-Noetherian", "chow-lemma-length-finite", "algebra-lemma-pullback-module" ], "ref_ids": [ 5677, 5670, 3308, 13236, 3308, 5669, 640 ] } ], "ref_ids": [] }, { "id": 5682, "type": "theorem", "label": "chow-lemma-flat-pullback-proper-pushforward", "categories": [ "chow" ], "title": "chow-lemma-flat-pullback-proper-pushforward", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "be a fibre product diagram of schemes locally of finite type over $S$.", "Assume $f : X \\to Y$ proper and $g : Y' \\to Y$ flat of relative dimension $r$.", "Then also $f'$ is proper and $g'$ is flat of relative dimension $r$.", "For any $k$-cycle $\\alpha$ on $X$ we have", "$$", "g^*f_*\\alpha = f'_*(g')^*\\alpha", "$$", "in $Z_{k + r}(Y')$." ], "refs": [], "proofs": [ { "contents": [ "The assertion that $f'$ is proper follows from", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-proper}.", "The assertion that $g'$ is flat of relative dimension $r$ follows from", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-relative-dimension-d}", "and \\ref{morphisms-lemma-base-change-flat}.", "It suffices to prove the equality of cycles when $\\alpha = [W]$", "for some integral closed subscheme $W \\subset X$ of $\\delta$-dimension $k$.", "Note that in this case we have $\\alpha = [\\mathcal{O}_W]_k$, see", "Lemma \\ref{lemma-cycle-closed-coherent}.", "By Lemmas \\ref{lemma-cycle-push-sheaf} and", "\\ref{lemma-pullback-coherent} it therefore suffices", "to show that $f'_*(g')^*\\mathcal{O}_W$ is isomorphic to", "$g^*f_*\\mathcal{O}_W$. This follows from cohomology and", "base change, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}." ], "refs": [ "morphisms-lemma-base-change-proper", "morphisms-lemma-base-change-relative-dimension-d", "morphisms-lemma-base-change-flat", "chow-lemma-cycle-closed-coherent", "chow-lemma-cycle-push-sheaf", "chow-lemma-pullback-coherent", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 5409, 5284, 5265, 5670, 5676, 5681, 3298 ] } ], "ref_ids": [] }, { "id": 5683, "type": "theorem", "label": "chow-lemma-finite-flat", "categories": [ "chow" ], "title": "chow-lemma-finite-flat", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $f : X \\to Y$ be a finite locally free morphism", "of degree $d$ (see", "Morphisms, Definition \\ref{morphisms-definition-finite-locally-free}).", "Then $f$ is both proper and flat of relative dimension $0$, and", "$$", "f_*f^*\\alpha = d\\alpha", "$$", "for every $\\alpha \\in Z_k(Y)$." ], "refs": [ "morphisms-definition-finite-locally-free" ], "proofs": [ { "contents": [ "A finite locally free morphism is flat and finite by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-flat},", "and a finite morphism is proper", "by Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}.", "We omit showing that a finite", "morphism has relative dimension $0$. Thus the formula makes sense.", "To prove it, let $Z \\subset Y$ be an integral closed subscheme", "of $\\delta$-dimension $k$. It suffices to prove the formula", "for $\\alpha = [Z]$. Since the base change of a finite locally free", "morphism is finite locally free", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-locally-free})", "we see that $f_*f^*\\mathcal{O}_Z$ is a finite locally free sheaf of", "rank $d$ on $Z$. Hence", "$$", "f_*f^*[Z] = f_*f^*[\\mathcal{O}_Z]_k =", "[f_*f^*\\mathcal{O}_Z]_k = d[Z]", "$$", "where we have used Lemmas \\ref{lemma-pullback-coherent} and", "\\ref{lemma-cycle-push-sheaf}." ], "refs": [ "morphisms-lemma-finite-flat", "morphisms-lemma-finite-proper", "morphisms-lemma-base-change-finite-locally-free", "chow-lemma-pullback-coherent", "chow-lemma-cycle-push-sheaf" ], "ref_ids": [ 5471, 5445, 5473, 5681, 5676 ] } ], "ref_ids": [ 5578 ] }, { "id": 5684, "type": "theorem", "label": "chow-lemma-divisor-delta-dimension", "categories": [ "chow" ], "title": "chow-lemma-divisor-delta-dimension", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. Assume $X$ is", "integral.", "\\begin{enumerate}", "\\item If $Z \\subset X$ is an integral closed subscheme, then", "the following are equivalent:", "\\begin{enumerate}", "\\item $Z$ is a prime divisor,", "\\item $Z$ has codimension $1$ in $X$, and", "\\item $\\dim_\\delta(Z) = \\dim_\\delta(X) - 1$.", "\\end{enumerate}", "\\item If $Z$ is an irreducible component of an effective Cartier", "divisor on $X$, then $\\dim_\\delta(Z) = \\dim_\\delta(X) - 1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from the definition of a prime divisor", "(Divisors, Definition \\ref{divisors-definition-Weil-divisor})", "and the definition of a dimension function", "(Topology, Definition \\ref{topology-definition-dimension-function}).", "Let $\\xi \\in Z$ be the generic point of an irreducible component $Z$ of", "an effective Cartier divisor $D \\subset X$.", "Then $\\dim(\\mathcal{O}_{D, \\xi}) = 0$ and", "$\\mathcal{O}_{D, \\xi} = \\mathcal{O}_{X, \\xi}/(f)$ for some", "nonzerodivisor $f \\in \\mathcal{O}_{X, \\xi}$ (Divisors,", "Lemma \\ref{divisors-lemma-effective-Cartier-in-points}).", "Then $\\dim(\\mathcal{O}_{X, \\xi}) = 1$ by", "Algebra, Lemma \\ref{algebra-lemma-one-equation}. Hence $Z$ is as in (1) by", "Properties, Lemma \\ref{properties-lemma-codimension-local-ring}", "and the proof is complete." ], "refs": [ "divisors-definition-Weil-divisor", "topology-definition-dimension-function", "divisors-lemma-effective-Cartier-in-points", "properties-lemma-codimension-local-ring" ], "ref_ids": [ 8106, 8367, 7946, 2979 ] } ], "ref_ids": [] }, { "id": 5685, "type": "theorem", "label": "chow-lemma-finite-in-codimension-one", "categories": [ "chow" ], "title": "chow-lemma-finite-in-codimension-one", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\xi \\in Y$ be a point.", "Assume that", "\\begin{enumerate}", "\\item $X$, $Y$ are integral,", "\\item $Y$ is locally Noetherian", "\\item $f$ is proper, dominant and $R(Y) \\subset R(X)$ is finite, and", "\\item $\\dim(\\mathcal{O}_{Y, \\xi}) = 1$.", "\\end{enumerate}", "Then there exists an open neighbourhood $V \\subset Y$ of $\\xi$", "such that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite." ], "refs": [], "proofs": [ { "contents": [ "This lemma is a special case of", "Varieties, Lemma \\ref{varieties-lemma-finite-in-codim-1}.", "Here is a direct argument in this case.", "By Cohomology of Schemes,", "Lemma \\ref{coherent-lemma-proper-finite-fibre-finite-in-neighbourhood}", "it suffices to prove that $f^{-1}(\\{\\xi\\})$ is finite.", "We replace $Y$ by an affine open, say $Y = \\Spec(R)$.", "Note that $R$ is Noetherian, as $Y$ is assumed locally Noetherian.", "Since $f$ is proper it is quasi-compact. Hence we can find a finite", "affine open covering $X = U_1 \\cup \\ldots \\cup U_n$ with", "each $U_i = \\Spec(A_i)$. Note that $R \\to A_i$ is a", "finite type injective homomorphism of domains such that", "the induced extension of fraction fields is finite.", "Thus the lemma follows", "from Algebra, Lemma \\ref{algebra-lemma-finite-in-codim-1}." ], "refs": [ "varieties-lemma-finite-in-codim-1", "coherent-lemma-proper-finite-fibre-finite-in-neighbourhood", "algebra-lemma-finite-in-codim-1" ], "ref_ids": [ 10978, 3366, 991 ] } ], "ref_ids": [] }, { "id": 5686, "type": "theorem", "label": "chow-lemma-flat-pullback-principal-divisor", "categories": [ "chow" ], "title": "chow-lemma-flat-pullback-principal-divisor", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$. Assume $X$, $Y$", "are integral and $n = \\dim_\\delta(Y)$.", "Let $f : X \\to Y$ be a flat morphism of relative dimension $r$.", "Let $g \\in R(Y)^*$. Then", "$$", "f^*(\\text{div}_Y(g)) = \\text{div}_X(g)", "$$", "in $Z_{n + r - 1}(X)$." ], "refs": [], "proofs": [ { "contents": [ "Note that since $f$ is flat it is dominant so that", "$f$ induces an embedding $R(Y) \\subset R(X)$, and hence", "we may think of $g$ as an element of $R(X)^*$.", "Let $Z \\subset X$ be an integral closed subscheme of", "$\\delta$-dimension $n + r - 1$. Let $\\xi \\in Z$", "be its generic point. If $\\dim_\\delta(f(Z)) > n - 1$,", "then we see that the coefficient of $[Z]$ in the left and", "right hand side of the equation is zero.", "Hence we may assume that $Z' = \\overline{f(Z)}$ is an", "integral closed subscheme of $Y$ of $\\delta$-dimension $n - 1$.", "Let $\\xi' = f(\\xi)$. It is the generic point of $Z'$.", "Set $A = \\mathcal{O}_{Y, \\xi'}$, $B = \\mathcal{O}_{X, \\xi}$.", "The ring map $A \\to B$ is a flat local homomorphism of", "Noetherian local domains of dimension $1$.", "We have $g$ in the fraction field of $A$. What we have to show is that", "$$", "\\text{ord}_A(g) \\text{length}_B(B/\\mathfrak m_AB)", "=", "\\text{ord}_B(g).", "$$", "This follows from Algebra, Lemma \\ref{algebra-lemma-pullback-module}", "(details omitted)." ], "refs": [ "algebra-lemma-pullback-module" ], "ref_ids": [ 640 ] } ], "ref_ids": [] }, { "id": 5687, "type": "theorem", "label": "chow-lemma-proper-pushforward-alteration", "categories": [ "chow" ], "title": "chow-lemma-proper-pushforward-alteration", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$. Assume $X$, $Y$", "are integral and $n = \\dim_\\delta(X) = \\dim_\\delta(Y)$.", "Let $p : X \\to Y$ be a dominant proper morphism.", "Let $f \\in R(X)^*$. Set", "$$", "g = \\text{Nm}_{R(X)/R(Y)}(f).", "$$", "Then we have", "$p_*\\text{div}(f) = \\text{div}(g)$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset Y$ be an integral closed subscheme of $\\delta$-dimension", "$n - 1$. We want to show that the coefficient of $[Z]$ in", "$p_*\\text{div}(f)$ and $\\text{div}(g)$ are equal. We may apply", "Lemma \\ref{lemma-finite-in-codimension-one}", "to the morphism $p : X \\to Y$ and the generic point $\\xi \\in Z$.", "Hence we may replace $Y$ by an", "affine open neighbourhood of $\\xi$ and assume that $p : X \\to Y$ is finite.", "Write $Y = \\Spec(R)$ and $X = \\Spec(A)$ with $p$ induced", "by a finite homomorphism $R \\to A$ of Noetherian domains which induces", "an finite field extension $L/K$ of fraction fields.", "Now we have $f \\in L$, $g = \\text{Nm}(f) \\in K$,", "and a prime $\\mathfrak p \\subset R$ with $\\dim(R_{\\mathfrak p}) = 1$.", "The coefficient of $[Z]$ in $\\text{div}_Y(g)$ is", "$\\text{ord}_{R_\\mathfrak p}(g)$.", "The coefficient of $[Z]$ in $p_*\\text{div}_X(f)$ is", "$$", "\\sum\\nolimits_{\\mathfrak q\\text{ lying over }\\mathfrak p}", "[\\kappa(\\mathfrak q) : \\kappa(\\mathfrak p)]", "\\text{ord}_{A_{\\mathfrak q}}(f)", "$$", "The desired equality therefore follows from", "Algebra, Lemma \\ref{algebra-lemma-finite-extension-dim-1}." ], "refs": [ "chow-lemma-finite-in-codimension-one", "algebra-lemma-finite-extension-dim-1" ], "ref_ids": [ 5685, 1047 ] } ], "ref_ids": [] }, { "id": 5688, "type": "theorem", "label": "chow-lemma-rational-function", "categories": [ "chow" ], "title": "chow-lemma-rational-function", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. Assume $X$ is", "integral and $n = \\dim_\\delta(X)$. Let $f \\in R(X)^*$.", "Let $U \\subset X$ be a nonempty open such that $f$", "corresponds to a section $f \\in \\Gamma(U, \\mathcal{O}_X^*)$.", "Let $Y \\subset X \\times_S \\mathbf{P}^1_S$ be the", "closure of the graph of $f : U \\to \\mathbf{P}^1_S$.", "Then", "\\begin{enumerate}", "\\item the projection morphism $p : Y \\to X$ is proper,", "\\item $p|_{p^{-1}(U)} : p^{-1}(U) \\to U$ is an isomorphism,", "\\item the pullbacks $Y_0 = q^{-1}D_0$ and $Y_\\infty = q^{-1}D_\\infty$", "via the morphism $q : Y \\to \\mathbf{P}^1_S$ are defined", "(Divisors, Definition", "\\ref{divisors-definition-pullback-effective-Cartier-divisor}),", "\\item we have", "$$", "\\text{div}_Y(f) = [Y_0]_{n - 1} - [Y_\\infty]_{n - 1}", "$$", "\\item we have", "$$", "\\text{div}_X(f) = p_*\\text{div}_Y(f)", "$$", "\\item if we view $Y_0$ and $Y_\\infty$ as closed subschemes of $X$", "via the morphism $p$ then we have", "$$", "\\text{div}_X(f) = [Y_0]_{n - 1} - [Y_\\infty]_{n - 1}", "$$", "\\end{enumerate}" ], "refs": [ "divisors-definition-pullback-effective-Cartier-divisor" ], "proofs": [ { "contents": [ "Since $X$ is integral, we see that $U$ is integral.", "Hence $Y$ is integral, and $(1, f)(U) \\subset Y$ is an open dense subscheme.", "Also, note that the closed subscheme $Y \\subset X \\times_S \\mathbf{P}^1_S$", "does not depend on the choice of the open $U$, since after all it is", "the closure of the one point set $\\{\\eta'\\} = \\{(1, f)(\\eta)\\}$", "where $\\eta \\in X$ is the generic point. Having said this let us", "prove the assertions of the lemma.", "\\medskip\\noindent", "For (1) note that $p$ is the composition of the closed immersion", "$Y \\to X \\times_S \\mathbf{P}^1_S = \\mathbf{P}^1_X$ with the proper", "morphism $\\mathbf{P}^1_X \\to X$. As a composition of proper morphisms", "is proper (Morphisms, Lemma \\ref{morphisms-lemma-composition-proper})", "we conclude.", "\\medskip\\noindent", "It is clear that $Y \\cap U \\times_S \\mathbf{P}^1_S = (1, f)(U)$.", "Thus (2) follows. It also follows that $\\dim_\\delta(Y) = n$.", "\\medskip\\noindent", "Note that $q(\\eta') = f(\\eta)$ is not contained in $D_0$ or $D_\\infty$", "since $f \\in R(X)^*$. Hence (3) by", "Divisors, Lemma \\ref{divisors-lemma-pullback-effective-Cartier-defined}.", "We obtain $\\dim_\\delta(Y_0) = n - 1$", "and $\\dim_\\delta(Y_\\infty) = n - 1$ from", "Lemma \\ref{lemma-divisor-delta-dimension}.", "\\medskip\\noindent", "Consider the effective Cartier divisor $Y_0$.", "At every point $\\xi \\in Y_0$ we have $f \\in \\mathcal{O}_{Y, \\xi}$ and", "the local equation for $Y_0$ is given by $f$.", "In particular, if $\\delta(\\xi) = n - 1$ so $\\xi$ is the generic point", "of a integral closed subscheme $Z$ of $\\delta$-dimension $n - 1$,", "then we see that the coefficient of $[Z]$ in $\\text{div}_Y(f)$ is", "$$", "\\text{ord}_Z(f) =", "\\text{length}_{\\mathcal{O}_{Y, \\xi}}", "(\\mathcal{O}_{Y, \\xi}/f\\mathcal{O}_{Y, \\xi}) =", "\\text{length}_{\\mathcal{O}_{Y, \\xi}}", "(\\mathcal{O}_{Y_0, \\xi})", "$$", "which is the coefficient of $[Z]$ in $[Y_0]_{n - 1}$. A similar", "argument using the rational function $1/f$ shows that", "$-[Y_\\infty]$ agrees with the terms with negative coefficients in", "the expression for $\\text{div}_Y(f)$. Hence (4) follows.", "\\medskip\\noindent", "Note that $D_0 \\to S$ is an isomorphism. Hence we see that", "$X \\times_S D_0 \\to X$ is an isomorphism as well. Clearly", "we have $Y_0 = Y \\cap X \\times_S D_0$ (scheme theoretic intersection)", "inside $X \\times_S \\mathbf{P}^1_S$. Hence it is really the case that", "$Y_0 \\to X$ is a closed immersion. It follows that", "$$", "p_*\\mathcal{O}_{Y_0} = \\mathcal{O}_{Y'_0}", "$$", "where $Y'_0 \\subset X$ is the image of $Y_0 \\to X$.", "By Lemma \\ref{lemma-cycle-push-sheaf} we", "have $p_*[Y_0]_{n - 1} = [Y'_0]_{n - 1}$. The same", "is true for $D_\\infty$ and $Y_\\infty$. Hence (6) is a consequence of (5).", "Finally, (5) follows immediately from", "Lemma \\ref{lemma-proper-pushforward-alteration}." ], "refs": [ "morphisms-lemma-composition-proper", "divisors-lemma-pullback-effective-Cartier-defined", "chow-lemma-divisor-delta-dimension", "chow-lemma-cycle-push-sheaf", "chow-lemma-proper-pushforward-alteration" ], "ref_ids": [ 5408, 7936, 5684, 5676, 5687 ] } ], "ref_ids": [ 8091 ] }, { "id": 5689, "type": "theorem", "label": "chow-lemma-curve-principal-divisor", "categories": [ "chow" ], "title": "chow-lemma-curve-principal-divisor", "contents": [ "Let $K$ be any field. Let $X$ be a $1$-dimensional integral scheme", "endowed with a proper morphism $c : X \\to \\Spec(K)$.", "Let $f \\in K(X)^*$ be an invertible rational function.", "Then", "$$", "\\sum\\nolimits_{x \\in X \\text{ closed}}", "[\\kappa(x) : K] \\text{ord}_{\\mathcal{O}_{X, x}}(f)", "=", "0", "$$", "where $\\text{ord}$ is as in", "Algebra, Definition \\ref{algebra-definition-ord}.", "In other words, $c_*\\text{div}(f) = 0$." ], "refs": [ "algebra-definition-ord" ], "proofs": [ { "contents": [ "Consider the diagram", "$$", "\\xymatrix{", "Y \\ar[r]_p \\ar[d]_q & X \\ar[d]^c \\\\", "\\mathbf{P}^1_K \\ar[r]^-{c'} & \\Spec(K)", "}", "$$", "that we constructed in Lemma \\ref{lemma-rational-function}", "starting with $X$ and the rational function $f$ over $S = \\Spec(K)$.", "We will use all the results of this lemma without further mention.", "We have to show that $c_*\\text{div}_X(f) = c_*p_*\\text{div}_Y(f) = 0$.", "This is the same as proving that $c'_*q_*\\text{div}_Y(f) = 0$.", "If $q(Y)$ is a closed point of $\\mathbf{P}^1_K$ then we", "see that $\\text{div}_X(f) = 0$ and the lemma holds.", "Thus we may assume that $q$ is dominant.", "Suppose we can show that $q : Y \\to \\mathbf{P}^1_K$ is finite", "locally free of degree $d$ (see", "Morphisms, Definition \\ref{morphisms-definition-finite-locally-free}).", "Since $\\text{div}_Y(f) = [q^{-1}D_0]_0 - [q^{-1}D_\\infty]_0$", "we see (by definition of flat pullback) that", "$\\text{div}_Y(f) = q^*([D_0]_0 - [D_\\infty]_0)$.", "Then by Lemma \\ref{lemma-finite-flat} we get", "$q_*\\text{div}_Y(f) = d([D_0]_0 - [D_\\infty]_0)$.", "Since clearly $c'_*[D_0]_0 = c'_*[D_\\infty]_0$ we win.", "\\medskip\\noindent", "It remains to show that $q$ is finite locally free.", "(It will automatically have some given degree as $\\mathbf{P}^1_K$", "is connected.)", "Since $\\dim(\\mathbf{P}^1_K) = 1$ we see that $q$ is finite for example", "by Lemma \\ref{lemma-finite-in-codimension-one}.", "All local rings of $\\mathbf{P}^1_K$ at", "closed points are regular local rings of dimension $1$", "(in other words discrete valuation rings), since they are", "localizations of $K[T]$ (see", "Algebra, Lemma \\ref{algebra-lemma-dim-affine-space}).", "Hence for $y\\in Y$ closed the local ring $\\mathcal{O}_{Y, y}$", "will be flat over $\\mathcal{O}_{\\mathbf{P}^1_K, q(y)}$ as soon as", "it is torsion free (More on Algebra, Lemma", "\\ref{more-algebra-lemma-dedekind-torsion-free-flat}).", "This is obviously the case as", "$\\mathcal{O}_{Y, y}$ is a domain and $q$ is dominant.", "Thus $q$ is flat. Hence $q$ is finite locally free by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}." ], "refs": [ "chow-lemma-rational-function", "morphisms-definition-finite-locally-free", "chow-lemma-finite-flat", "chow-lemma-finite-in-codimension-one", "algebra-lemma-dim-affine-space", "more-algebra-lemma-dedekind-torsion-free-flat", "morphisms-lemma-finite-flat" ], "ref_ids": [ 5688, 5578, 5683, 5685, 992, 9921, 5471 ] } ], "ref_ids": [ 1519 ] }, { "id": 5690, "type": "theorem", "label": "chow-lemma-restrict-to-open", "categories": [ "chow" ], "title": "chow-lemma-restrict-to-open", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be a scheme locally of finite type over $S$.", "Let $U \\subset X$ be an open subscheme, and denote", "$i : Y = X \\setminus U \\to X$ as a reduced closed subscheme of $X$.", "Let $k \\in \\mathbf{Z}$.", "Suppose $\\alpha, \\beta \\in Z_k(X)$.", "If $\\alpha|_U \\sim_{rat} \\beta|_U$ then there exist a cycle", "$\\gamma \\in Z_k(Y)$ such that", "$$", "\\alpha \\sim_{rat} \\beta + i_*\\gamma.", "$$", "In other words, the sequence", "$$", "\\xymatrix{", "\\CH_k(Y) \\ar[r]^{i_*} & \\CH_k(X) \\ar[r]^{j^*} & \\CH_k(U) \\ar[r] & 0", "}", "$$", "is an exact complex of abelian groups." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{W_j\\}_{j \\in J}$ be a locally finite collection of integral closed", "subschemes of $U$ of $\\delta$-dimension $k + 1$, and let $f_j \\in R(W_j)^*$", "be elements such that $(\\alpha - \\beta)|_U = \\sum (i_j)_*\\text{div}(f_j)$", "as in the definition. Set $W_j' \\subset X$ equal", "to the closure of $W_j$. Suppose that $V \\subset X$ is a quasi-compact", "open. Then also $V \\cap U$ is quasi-compact open in $U$ as", "$V$ is Noetherian. Hence the set", "$\\{j \\in J \\mid W_j \\cap V \\not = \\emptyset\\}", "= \\{j \\in J \\mid W'_j \\cap V \\not = \\emptyset\\}$", "is finite since $\\{W_j\\}$ is locally finite. In other words we see that", "$\\{W'_j\\}$ is also locally finite. Since $R(W_j) = R(W'_j)$ we see", "that", "$$", "\\alpha - \\beta - \\sum (i'_j)_*\\text{div}(f_j)", "$$", "is a cycle supported on $Y$ and the lemma follows (see", "Lemma \\ref{lemma-exact-sequence-open})." ], "refs": [ "chow-lemma-exact-sequence-open" ], "ref_ids": [ 5679 ] } ], "ref_ids": [] }, { "id": 5691, "type": "theorem", "label": "chow-lemma-exact-sequence-closed-chow", "categories": [ "chow" ], "title": "chow-lemma-exact-sequence-closed-chow", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. Let $X_1, X_2 \\subset X$", "be closed subschemes such that $X = X_1 \\cup X_2$ set theoretically.", "For every $k \\in \\mathbf{Z}$ the sequence of abelian groups", "$$", "\\xymatrix{", "\\CH_k(X_1 \\cap X_2) \\ar[r] &", "\\CH_k(X_1) \\oplus \\CH_k(X_2) \\ar[r] &", "\\CH_k(X) \\ar[r] &", "0", "}", "$$", "is exact. Here $X_1 \\cap X_2$ is the scheme theoretic intersection and the", "maps are the pushforward maps with one multiplied by $-1$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-exact-sequence-closed} the arrow", "$\\CH_k(X_1) \\oplus \\CH_k(X_2) \\to \\CH_k(X)$ is surjective.", "Suppose that $(\\alpha_1, \\alpha_2)$ maps to zero under this map.", "Write $\\alpha_1 = \\sum n_{1, i}[W_{1, i}]$ and", "$\\alpha_2 = \\sum n_{2, i}[W_{2, i}]$. Then we obtain a locally", "finite collection $\\{W_j\\}_{j \\in J}$ of integral closed", "subschemes of $X$ of $\\delta$-dimension $k + 1$ and $f_j \\in R(W_j)^*$", "such that", "$$", "\\sum n_{1, i}[W_{1, i}] + \\sum n_{2, i}[W_{2, i}] = \\sum (i_j)_*\\text{div}(f_j)", "$$", "as cycles on $X$ where $i_j : W_j \\to X$ is the inclusion morphism.", "Choose a disjoint union decomposition $J = J_1 \\amalg J_2$ such that", "$W_j \\subset X_1$ if $j \\in J_1$ and $W_j \\subset X_2$ if $j \\in J_2$.", "(This is possible because the $W_j$ are integral.) Then we can write", "the equation above as", "$$", "\\sum n_{1, i}[W_{1, i}] - \\sum\\nolimits_{j \\in J_1} (i_j)_*\\text{div}(f_j) =", "- \\sum n_{2, i}[W_{2, i}] + \\sum\\nolimits_{j \\in J_2} (i_j)_*\\text{div}(f_j)", "$$", "Hence this expression is a cycle (!) on $X_1 \\cap X_2$. In other words", "the element $(\\alpha_1, \\alpha_2)$ is in the image of the first arrow", "and the proof is complete." ], "refs": [ "chow-lemma-exact-sequence-closed" ], "ref_ids": [ 5675 ] } ], "ref_ids": [] }, { "id": 5692, "type": "theorem", "label": "chow-lemma-prepare-flat-pullback-rational-equivalence", "categories": [ "chow" ], "title": "chow-lemma-prepare-flat-pullback-rational-equivalence", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be schemes locally of finite type over $S$.", "Assume $Y$ integral with $\\dim_\\delta(Y) = k$.", "Let $f : X \\to Y$ be a flat morphism of", "relative dimension $r$. Then for $g \\in R(Y)^*$ we have", "$$", "f^*\\text{div}_Y(g) =", "\\sum n_j i_{j, *}\\text{div}_{X_j}(g \\circ f|_{X_j})", "$$", "as $(k + r - 1)$-cycles on $X$ where the sum is over the irreducible", "components $X_j$ of $X$ and $n_j$ is the multiplicity of $X_j$ in $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset X$ be an integral closed subscheme of $\\delta$-dimension", "$k + r - 1$. We have to show that the coefficient $n$ of $[Z]$ in", "$f^*\\text{div}(g)$ is equal to the coefficient", "$m$ of $[Z]$ in $\\sum i_{j, *} \\text{div}(g \\circ f|_{X_j})$.", "Let $Z'$ be the closure of $f(Z)$ which is an integral closed", "subscheme of $Y$. By Lemma \\ref{lemma-flat-inverse-image-dimension}", "we have $\\dim_\\delta(Z') \\geq k - 1$. Thus either $Z' = Y$", "or $Z'$ is a prime divisor on $Y$. If $Z' = Y$, then the coefficients", "$n$ and $m$ are both zero: this is clear for $n$ by definition", "of $f^*$ and follows for $m$ because $g \\circ f|_{X_j}$ is", "a unit in any point of $X_j$ mapping to the generic point of $Y$.", "Hence we may assume that $Z' \\subset Y$ is a prime divisor.", "\\medskip\\noindent", "We are going to translate the equality of $n$ and $m$ into algebra.", "Namely, let $\\xi' \\in Z'$ and $\\xi \\in Z$ be the generic points.", "Set $A = \\mathcal{O}_{Y, \\xi'}$ and $B = \\mathcal{O}_{X, \\xi}$.", "Note that $A$, $B$ are Noetherian, $A \\to B$ is flat, local,", "$A$ is a domain, and $\\mathfrak m_AB$ is an ideal of definition", "of the local ring $B$. The rational function $g$ is an element", "of the fraction field $Q(A)$ of $A$.", "By construction, the closed subschemes $X_j$", "which meet $\\xi$ correspond $1$-to-$1$ with minimal primes", "$$", "\\mathfrak q_1, \\ldots, \\mathfrak q_s \\subset B", "$$", "The integers $n_j$ are the corresponding lengths", "$$", "n_i = \\text{length}_{B_{\\mathfrak q_i}}(B_{\\mathfrak q_i})", "$$", "The rational functions $g \\circ f|_{X_j}$ correspond to the image", "$g_i \\in \\kappa(\\mathfrak q_i)^*$ of $g \\in Q(A)$.", "Putting everything together we see that", "$$", "n = \\text{ord}_A(g) \\text{length}_B(B/\\mathfrak m_AB)", "$$", "and that", "$$", "m = \\sum \\text{ord}_{B/\\mathfrak q_i}(g_i)", "\\text{length}_{B_{\\mathfrak q_i}}(B_{\\mathfrak q_i})", "$$", "Writing $g = x/y$ for some nonzero $x, y \\in A$ we see that it suffices", "to prove", "$$", "\\text{length}_A(A/(x)) \\text{length}_B(B/\\mathfrak m_AB) =", "\\text{length}_B(B/xB)", "$$", "(equality uses Algebra, Lemma \\ref{algebra-lemma-pullback-module})", "equals", "$$", "\\sum\\nolimits_{i = 1, \\ldots, s}", "\\text{length}_{B/\\mathfrak q_i}(B/(x, \\mathfrak q_i))", "\\text{length}_{B_{\\mathfrak q_i}}(B_{\\mathfrak q_i})", "$$", "and similarly for $y$. As $A \\to B$ is flat it follows that $x$", "is a nonzerodivisor in $B$. Hence the desired equality follows from", "Lemma \\ref{lemma-additivity-divisors-restricted}." ], "refs": [ "chow-lemma-flat-inverse-image-dimension", "algebra-lemma-pullback-module", "chow-lemma-additivity-divisors-restricted" ], "ref_ids": [ 5677, 640, 5653 ] } ], "ref_ids": [] }, { "id": 5693, "type": "theorem", "label": "chow-lemma-flat-pullback-rational-equivalence", "categories": [ "chow" ], "title": "chow-lemma-flat-pullback-rational-equivalence", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be schemes locally of finite type over $S$.", "Let $f : X \\to Y$ be a flat morphism of relative dimension $r$.", "Let $\\alpha \\sim_{rat} \\beta$ be rationally equivalent $k$-cycles on $Y$.", "Then $f^*\\alpha \\sim_{rat} f^*\\beta$ as $(k + r)$-cycles on $X$." ], "refs": [], "proofs": [ { "contents": [ "What do we have to show? Well, suppose we are given a collection", "$$", "i_j : W_j \\longrightarrow Y", "$$", "of closed immersions, with each $W_j$ integral of $\\delta$-dimension $k + 1$", "and rational functions $g_j \\in R(W_j)^*$. Moreover, assume that", "the collection $\\{i_j(W_j)\\}_{j \\in J}$ is locally finite on $Y$.", "Then we have to show that", "$$", "f^*(\\sum i_{j, *}\\text{div}(g_j)) = \\sum f^*i_{j, *}\\text{div}(g_j)", "$$", "is rationally equivalent to zero on $X$. The sum on the right", "makes sense as $\\{W_j\\}$ is locally finite in $X$ by", "Lemma \\ref{lemma-inverse-image-locally-finite}.", "\\medskip\\noindent", "Consider the fibre products", "$$", "i'_j : W'_j = W_j \\times_Y X \\longrightarrow X.", "$$", "and denote $f_j : W'_j \\to W_j$ the first projection.", "By Lemma \\ref{lemma-flat-pullback-proper-pushforward}", "we can write the sum above as", "$$", "\\sum i'_{j, *}(f_j^*\\text{div}(g_j))", "$$", "By Lemma \\ref{lemma-prepare-flat-pullback-rational-equivalence}", "we see that each $f_j^*\\text{div}(g_j)$ is rationally equivalent", "to zero on $W'_j$. Hence each $i'_{j, *}(f_j^*\\text{div}(g_j))$", "is rationally equivalent to zero. Then the same is true for", "the displayed sum by the discussion in", "Remark \\ref{remark-infinite-sums-rational-equivalences}." ], "refs": [ "chow-lemma-inverse-image-locally-finite", "chow-lemma-flat-pullback-proper-pushforward", "chow-lemma-prepare-flat-pullback-rational-equivalence", "chow-remark-infinite-sums-rational-equivalences" ], "ref_ids": [ 5678, 5682, 5692, 5931 ] } ], "ref_ids": [] }, { "id": 5694, "type": "theorem", "label": "chow-lemma-proper-pushforward-rational-equivalence", "categories": [ "chow" ], "title": "chow-lemma-proper-pushforward-rational-equivalence", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be schemes locally of finite type over $S$.", "Let $p : X \\to Y$ be a proper morphism.", "Suppose $\\alpha, \\beta \\in Z_k(X)$ are rationally equivalent.", "Then $p_*\\alpha$ is rationally equivalent to $p_*\\beta$." ], "refs": [], "proofs": [ { "contents": [ "What do we have to show? Well, suppose we are given a collection", "$$", "i_j : W_j \\longrightarrow X", "$$", "of closed immersions, with each $W_j$ integral of $\\delta$-dimension $k + 1$", "and rational functions $f_j \\in R(W_j)^*$.", "Moreover, assume that", "the collection $\\{i_j(W_j)\\}_{j \\in J}$ is locally finite on $X$.", "Then we have to show that", "$$", "p_*\\left(\\sum i_{j, *}\\text{div}(f_j)\\right)", "$$", "is rationally equivalent to zero on $X$.", "\\medskip\\noindent", "Note that the sum is equal to", "$$", "\\sum p_*i_{j, *}\\text{div}(f_j).", "$$", "Let $W'_j \\subset Y$ be the integral closed subscheme which is the", "image of $p \\circ i_j$. The collection $\\{W'_j\\}$ is locally finite", "in $Y$ by Lemma \\ref{lemma-quasi-compact-locally-finite}.", "Hence it suffices to show, for a given $j$, that either", "$p_*i_{j, *}\\text{div}(f_j) = 0$ or that it", "is equal to $i'_{j, *}\\text{div}(g_j)$ for some $g_j \\in R(W'_j)^*$.", "\\medskip\\noindent", "The arguments above therefore reduce us to the case of a since", "integral closed subscheme $W \\subset X$ of $\\delta$-dimension $k + 1$.", "Let $f \\in R(W)^*$. Let $W' = p(W)$ as above.", "We get a commutative diagram of morphisms", "$$", "\\xymatrix{", "W \\ar[r]_i \\ar[d]_{p'} & X \\ar[d]^p \\\\", "W' \\ar[r]^{i'} & Y", "}", "$$", "Note that $p_*i_*\\text{div}(f) = i'_*(p')_*\\text{div}(f)$ by", "Lemma \\ref{lemma-compose-pushforward}. As explained above", "we have to show that $(p')_*\\text{div}(f)$", "is the divisor of a rational function on $W'$ or zero.", "There are three cases to distinguish.", "\\medskip\\noindent", "The case $\\dim_\\delta(W') < k$. In this case automatically", "$(p')_*\\text{div}(f) = 0$ and there is nothing to prove.", "\\medskip\\noindent", "The case $\\dim_\\delta(W') = k$. Let us show that $(p')_*\\text{div}(f) = 0$", "in this case. Let $\\eta \\in W'$ be the generic point.", "Note that $c : W_\\eta \\to \\Spec(K)$", "is a proper integral curve over $K = \\kappa(\\eta)$", "whose function field $K(W_\\eta)$ is identified with $R(W)$.", "Here is a diagram", "$$", "\\xymatrix{", "W_\\eta \\ar[r] \\ar[d]_c & W \\ar[d]^{p'} \\\\", "\\Spec(K) \\ar[r] & W'", "}", "$$", "Let us denote $f_\\eta \\in K(W_\\eta)^*$ the rational function", "corresponding to $f \\in R(W)^*$.", "Moreover, the closed points $\\xi$ of $W_\\eta$ correspond $1 - 1$ to the", "closed integral subschemes $Z = Z_\\xi \\subset W$ of $\\delta$-dimension $k$", "with $p'(Z) = W'$. Note that the multiplicity", "of $Z_\\xi$ in $\\text{div}(f)$ is equal to", "$\\text{ord}_{\\mathcal{O}_{W_\\eta, \\xi}}(f_\\eta)$ simply because the", "local rings $\\mathcal{O}_{W_\\eta, \\xi}$ and $\\mathcal{O}_{W, \\xi}$", "are identified (as subrings of their fraction fields).", "Hence we see that the multiplicity of $[W']$ in", "$(p')_*\\text{div}(f)$ is equal to the multiplicity of", "$[\\Spec(K)]$ in $c_*\\text{div}(f_\\eta)$.", "By Lemma \\ref{lemma-curve-principal-divisor} this is zero.", "\\medskip\\noindent", "The case $\\dim_\\delta(W') = k + 1$. In this case", "Lemma \\ref{lemma-proper-pushforward-alteration} applies,", "and we see that indeed $p'_*\\text{div}(f) = \\text{div}(g)$", "for some $g \\in R(W')^*$ as desired." ], "refs": [ "chow-lemma-quasi-compact-locally-finite", "chow-lemma-compose-pushforward", "chow-lemma-curve-principal-divisor", "chow-lemma-proper-pushforward-alteration" ], "ref_ids": [ 5673, 5674, 5689, 5687 ] } ], "ref_ids": [] }, { "id": 5695, "type": "theorem", "label": "chow-lemma-rational-equivalence-family", "categories": [ "chow" ], "title": "chow-lemma-rational-equivalence-family", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be a scheme locally of finite type over $S$.", "Let $W \\subset X \\times_S \\mathbf{P}^1_S$ be an integral", "closed subscheme of $\\delta$-dimension $k + 1$.", "Assume $W \\not = W_0$, and $W \\not = W_\\infty$. Then", "\\begin{enumerate}", "\\item $W_0$, $W_\\infty$ are effective Cartier divisors of $W$,", "\\item $W_0$, $W_\\infty$ can be viewed as closed subschemes", "of $X$ and", "$$", "[W_0]_k \\sim_{rat} [W_\\infty]_k,", "$$", "\\item for any locally finite family of", "integral closed subschemes", "$W_i \\subset X \\times_S \\mathbf{P}^1_S$", "of $\\delta$-dimension $k + 1$ with $W_i \\not = (W_i)_0$ and", "$W_i \\not = (W_i)_\\infty$ we have", "$\\sum ([(W_i)_0]_k - [(W_i)_\\infty]_k) \\sim_{rat} 0$", "on $X$, and", "\\item for any $\\alpha \\in Z_k(X)$ with $\\alpha \\sim_{rat} 0$", "there exists a locally finite family of", "integral closed subschemes $W_i \\subset X \\times_S \\mathbf{P}^1_S$", "as above such that $\\alpha = \\sum ([(W_i)_0]_k - [(W_i)_\\infty]_k)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from", "Divisors, Lemma \\ref{divisors-lemma-pullback-effective-Cartier-defined}", "since the generic point", "of $W$ is not mapped into $D_0$ or $D_\\infty$ under the projection", "$X \\times_S \\mathbf{P}^1_S \\to \\mathbf{P}^1_S$ by assumption.", "\\medskip\\noindent", "Since $X \\times_S D_0 \\to X$ is a closed immersion, we see that $W_0$", "is isomorphic to a closed subscheme of $X$. Similarly for $W_\\infty$.", "The morphism $p : W \\to X$ is proper as a composition of", "the closed immersion $W \\to X \\times_S \\mathbf{P}^1_S$ and the", "proper morphism $X \\times_S \\mathbf{P}^1_S \\to X$. By", "Lemma \\ref{lemma-rational-function} we have", "$[W_0]_k \\sim_{rat} [W_\\infty]_k$ as cycles on $W$. Hence part (2) follows from", "Lemma \\ref{lemma-proper-pushforward-rational-equivalence} as clearly", "$p_*[W_0]_k = [W_0]_k$ and similarly for $W_\\infty$.", "\\medskip\\noindent", "The only content of statement (3) is, given parts (1) and (2), that", "the collection $\\{(W_i)_0, (W_i)_\\infty\\}$ is a locally finite collection", "of closed subschemes of $X$. This is clear.", "\\medskip\\noindent", "Suppose that $\\alpha \\sim_{rat} 0$.", "By definition this means there exist integral closed subschemes", "$V_i \\subset X$ of $\\delta$-dimension $k + 1$ and rational", "functions $f_i \\in R(V_i)^*$ such that the family", "$\\{V_i\\}_{i \\in I}$ is locally finite in $X$ and such that", "$\\alpha = \\sum (V_i \\to X)_*\\text{div}(f_i)$.", "Let", "$$", "W_i \\subset V_i \\times_S \\mathbf{P}^1_S \\subset X \\times_S \\mathbf{P}^1_S", "$$", "be the closure of the graph of the rational map $f_i$ as in", "Lemma \\ref{lemma-rational-function}.", "Then we have that $(V_i \\to X)_*\\text{div}(f_i)$", "is equal to $[(W_i)_0]_k - [(W_i)_\\infty]_k$ by that same lemma.", "Hence the result is clear." ], "refs": [ "divisors-lemma-pullback-effective-Cartier-defined", "chow-lemma-rational-function", "chow-lemma-proper-pushforward-rational-equivalence", "chow-lemma-rational-function" ], "ref_ids": [ 7936, 5688, 5694, 5688 ] } ], "ref_ids": [] }, { "id": 5696, "type": "theorem", "label": "chow-lemma-closed-subscheme-cross-p1", "categories": [ "chow" ], "title": "chow-lemma-closed-subscheme-cross-p1", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be a scheme locally of finite type over $S$.", "Let $Z$ be a closed subscheme of $X \\times \\mathbf{P}^1$.", "Assume", "\\begin{enumerate}", "\\item $\\dim_\\delta(Z) \\leq k + 1$,", "\\item $\\dim_\\delta(Z_0) \\leq k$, $\\dim_\\delta(Z_\\infty) \\leq k$, and", "\\item for any embedded point $\\xi$ (Divisors, Definition", "\\ref{divisors-definition-embedded}) of $Z$ either", "$\\xi \\not \\in Z_0 \\cup Z_\\infty$ or $\\delta(\\xi) < k$.", "\\end{enumerate}", "Then $[Z_0]_k \\sim_{rat} [Z_\\infty]_k$ as $k$-cycles on $X$." ], "refs": [ "divisors-definition-embedded" ], "proofs": [ { "contents": [ "Let $\\{W_i\\}_{i \\in I}$ be the collection of irreducible", "components of $Z$ which have $\\delta$-dimension $k + 1$.", "Write", "$$", "[Z]_{k + 1} = \\sum n_i[W_i]", "$$", "with $n_i > 0$ as per definition. Note that $\\{W_i\\}$", "is a locally finite collection of closed subsets of", "$X \\times_S \\mathbf{P}^1_S$ by", "Divisors, Lemma \\ref{divisors-lemma-components-locally-finite}.", "We claim that", "$$", "[Z_0]_k = \\sum n_i[(W_i)_0]_k", "$$", "and similarly for $[Z_\\infty]_k$. If we prove this then the lemma", "follows from Lemma \\ref{lemma-rational-equivalence-family}.", "\\medskip\\noindent", "Let $Z' \\subset X$ be an integral closed subscheme of $\\delta$-dimension $k$.", "To prove the equality above it suffices to show that the coefficient $n$", "of $[Z']$ in $[Z_0]_k$ is the same as the coefficient $m$ of", "$[Z']$ in $\\sum n_i[(W_i)_0]_k$. Let $\\xi' \\in Z'$ be the generic point.", "Set $\\xi = (\\xi', 0) \\in X \\times_S \\mathbf{P}^1_S$.", "Consider the local ring $A = \\mathcal{O}_{X \\times_S \\mathbf{P}^1_S, \\xi}$.", "Let $I \\subset A$ be the ideal cutting out $Z$, in other words so that", "$A/I = \\mathcal{O}_{Z, \\xi}$. Let $t \\in A$ be the element cutting", "out $X \\times_S D_0$ (i.e., the coordinate of $\\mathbf{P}^1$ at zero", "pulled back). By our choice of $\\xi' \\in Z'$ we have $\\delta(\\xi) = k$", "and hence $\\dim(A/I) = 1$. Since $\\xi$ is not an embedded point by", "assumption (3) we see that $A/I$ is Cohen-Macaulay. Since $\\dim_\\delta(Z_0)", "= k$ we see that $\\dim(A/(t, I)) = 0$ which implies that $t$", "is a nonzerodivisor on $A/I$. Finally, the irreducible closed subschemes", "$W_i$ passing through $\\xi$ correspond to the minimal primes", "$I \\subset \\mathfrak q_i$ over $I$. The multiplicities $n_i$ correspond", "to the lengths $\\text{length}_{A_{\\mathfrak q_i}}(A/I)_{\\mathfrak q_i}$.", "Hence we see that", "$$", "n = \\text{length}_A(A/(t, I))", "$$", "and", "$$", "m = \\sum", "\\text{length}_A(A/(t, \\mathfrak q_i))", "\\text{length}_{A_{\\mathfrak q_i}}(A/I)_{\\mathfrak q_i}", "$$", "Thus the result follows from", "Lemma \\ref{lemma-additivity-divisors-restricted}." ], "refs": [ "divisors-lemma-components-locally-finite", "chow-lemma-rational-equivalence-family", "chow-lemma-additivity-divisors-restricted" ], "ref_ids": [ 8022, 5695, 5653 ] } ], "ref_ids": [ 8083 ] }, { "id": 5697, "type": "theorem", "label": "chow-lemma-coherent-sheaf-cross-p1", "categories": [ "chow" ], "title": "chow-lemma-coherent-sheaf-cross-p1", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be a scheme locally of finite type over $S$.", "Let $\\mathcal{F}$ be a coherent sheaf on $X \\times \\mathbf{P}^1$.", "Let $i_0, i_\\infty : X \\to X \\times \\mathbf{P}^1$ be the closed immersion", "such that $i_t(x) = (x, t)$. Denote $\\mathcal{F}_0 = i_0^*\\mathcal{F}$ and", "$\\mathcal{F}_\\infty = i_\\infty^*\\mathcal{F}$.", "Assume", "\\begin{enumerate}", "\\item $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k + 1$,", "\\item $\\dim_\\delta(\\text{Supp}(\\mathcal{F}_0)) \\leq k$,", "$\\dim_\\delta(\\text{Supp}(\\mathcal{F}_\\infty)) \\leq k$, and", "\\item for any embedded associated point $\\xi$ of $\\mathcal{F}$ either", "$\\xi \\not \\in (X \\times \\mathbf{P}^1)_0 \\cup (X \\times \\mathbf{P}^1)_\\infty$", "or $\\delta(\\xi) < k$.", "\\end{enumerate}", "Then $[\\mathcal{F}_0]_k \\sim_{rat} [\\mathcal{F}_\\infty]_k$ as $k$-cycles on $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{W_i\\}_{i \\in I}$ be the collection of irreducible", "components of $\\text{Supp}(\\mathcal{F})$", "which have $\\delta$-dimension $k + 1$.", "Write", "$$", "[\\mathcal{F}]_{k + 1} = \\sum n_i[W_i]", "$$", "with $n_i > 0$ as per definition. Note that $\\{W_i\\}$", "is a locally finite collection of closed subsets of", "$X \\times_S \\mathbf{P}^1_S$ by Lemma \\ref{lemma-length-finite}.", "We claim that", "$$", "[\\mathcal{F}_0]_k = \\sum n_i[(W_i)_0]_k", "$$", "and similarly for $[\\mathcal{F}_\\infty]_k$. If we prove this then the lemma", "follows from Lemma \\ref{lemma-rational-equivalence-family}.", "\\medskip\\noindent", "Let $Z' \\subset X$ be an integral closed subscheme of $\\delta$-dimension $k$.", "To prove the equality above it suffices to show that the coefficient $n$", "of $[Z']$ in $[\\mathcal{F}_0]_k$ is the same as the coefficient $m$ of", "$[Z']$ in $\\sum n_i[(W_i)_0]_k$. Let $\\xi' \\in Z'$ be the generic point.", "Set $\\xi = (\\xi', 0) \\in X \\times_S \\mathbf{P}^1_S$.", "Consider the local ring $A = \\mathcal{O}_{X \\times_S \\mathbf{P}^1_S, \\xi}$.", "Let $M = \\mathcal{F}_\\xi$ as an $A$-module.", "Let $t \\in A$ be the element cutting out $X \\times_S D_0$", "(i.e., the coordinate of $\\mathbf{P}^1$ at zero pulled back).", "By our choice of $\\xi' \\in Z'$ we have $\\delta(\\xi) = k$", "and hence $\\dim(\\text{Supp}(M)) = 1$. Since $\\xi$ is not an associated point", "of $\\mathcal{F}$ by assumption (3) we see that $M$ is a Cohen-Macaulay module.", "Since $\\dim_\\delta(\\text{Supp}(\\mathcal{F}_0)) = k$", "we see that $\\dim(\\text{Supp}(M/tM)) = 0$ which implies that $t$", "is a nonzerodivisor on $M$. Finally, the irreducible closed subschemes", "$W_i$ passing through $\\xi$ correspond to the minimal primes", "$\\mathfrak q_i$ of $\\text{Ass}(M)$. The multiplicities $n_i$ correspond", "to the lengths $\\text{length}_{A_{\\mathfrak q_i}}M_{\\mathfrak q_i}$.", "Hence we see that", "$$", "n = \\text{length}_A(M/tM)", "$$", "and", "$$", "m = \\sum", "\\text{length}_A(A/(t, \\mathfrak q_i)A)", "\\text{length}_{A_{\\mathfrak q_i}}M_{\\mathfrak q_i}", "$$", "Thus the result follows from", "Lemma \\ref{lemma-additivity-divisors-restricted}." ], "refs": [ "chow-lemma-length-finite", "chow-lemma-rational-equivalence-family", "chow-lemma-additivity-divisors-restricted" ], "ref_ids": [ 5669, 5695, 5653 ] } ], "ref_ids": [] }, { "id": 5698, "type": "theorem", "label": "chow-lemma-Serre-subcategories", "categories": [ "chow" ], "title": "chow-lemma-Serre-subcategories", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be a scheme locally of finite type over $S$.", "The categories $\\textit{Coh}_{\\leq k}(X)$ are Serre subcategories", "of the abelian category $\\textit{Coh}(X)$." ], "refs": [], "proofs": [ { "contents": [ "The definition of a Serre subcategory is", "Homology, Definition \\ref{homology-definition-serre-subcategory}.", "The proof of the lemma is straightforward and omitted." ], "refs": [ "homology-definition-serre-subcategory" ], "ref_ids": [ 12146 ] } ], "ref_ids": [] }, { "id": 5699, "type": "theorem", "label": "chow-lemma-cycles-k-group", "categories": [ "chow" ], "title": "chow-lemma-cycles-k-group", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be a scheme locally of finite type over $S$.", "The maps", "$$", "Z_k(X)", "\\longrightarrow", "K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X)),", "\\quad", "\\sum n_Z[Z] \\mapsto", "\\left[\\bigoplus\\nolimits_{n_Z > 0} \\mathcal{O}_Z^{\\oplus n_Z}\\right]", "-", "\\left[\\bigoplus\\nolimits_{n_Z < 0} \\mathcal{O}_Z^{\\oplus -n_Z}\\right]", "$$", "and", "$$", "K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))", "\\longrightarrow", "Z_k(X),\\quad", "\\mathcal{F} \\longmapsto [\\mathcal{F}]_k", "$$", "are mutually inverse isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "Note that if $\\sum n_Z[Z]$ is in $Z_k(X)$, then", "the direct sums", "$\\bigoplus\\nolimits_{n_Z > 0} \\mathcal{O}_Z^{\\oplus n_Z}$ and", "$\\bigoplus\\nolimits_{n_Z < 0} \\mathcal{O}_Z^{\\oplus -n_Z}$", "are coherent sheaves on $X$ since the family $\\{Z \\mid n_Z > 0\\}$", "is locally finite on $X$.", "The map $\\mathcal{F} \\to [\\mathcal{F}]_k$ is additive", "on $\\textit{Coh}_{\\leq k}(X)$, see", "Lemma \\ref{lemma-additivity-sheaf-cycle}. And $[\\mathcal{F}]_k = 0$", "if $\\mathcal{F} \\in \\textit{Coh}_{\\leq k - 1}(X)$. By part (1)", "of Homology, Lemma \\ref{homology-lemma-serre-subcategory-K-groups}", "this implies that the second map is well defined too.", "It is clear that the composition of the first map with the second", "map is the identity.", "\\medskip\\noindent", "Conversely, say we start with a coherent sheaf $\\mathcal{F}$", "on $X$. Write $[\\mathcal{F}]_k = \\sum_{i \\in I} n_i[Z_i]$", "with $n_i > 0$ and $Z_i \\subset X$, $i \\in I$", "pairwise distinct integral closed subschemes of $\\delta$-dimension $k$.", "We have to show that", "$$", "[\\mathcal{F}] = [\\bigoplus\\nolimits_{i \\in I} \\mathcal{O}_{Z_i}^{\\oplus n_i}]", "$$", "in $K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))$.", "Denote $\\xi_i \\in Z_i$ the generic point.", "If we set", "$$", "\\mathcal{F}' = \\Ker(\\mathcal{F} \\to \\bigoplus \\xi_{i, *}\\mathcal{F}_{\\xi_i})", "$$", "then $\\mathcal{F}'$ is the maximal coherent submodule of $\\mathcal{F}$", "whose support has dimension $\\leq k - 1$. In particular $\\mathcal{F}$", "and $\\mathcal{F}/\\mathcal{F}'$ have the same class in", "$K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))$.", "Thus after replacing $\\mathcal{F}$ by $\\mathcal{F}/\\mathcal{F}'$", "we may and do assume that the kernel $\\mathcal{F}'$ displayed", "above is zero.", "\\medskip\\noindent", "For each $i \\in I$ we choose a filtration", "$$", "\\mathcal{F}_{\\xi_i} = \\mathcal{F}_i^0 \\supset \\mathcal{F}_i^1 \\supset", "\\ldots \\supset \\mathcal{F}_i^{n_i} = 0", "$$", "such that the successive quotients are of dimension $1$ over the residue", "field at $\\xi_i$. This is possible as the length of $\\mathcal{F}_{\\xi_i}$", "over $\\mathcal{O}_{X, \\xi_i}$ is $n_i$.", "For $p > n_i$ set $\\mathcal{F}_i^p = 0$. For $p \\geq 0$ we denote", "$$", "\\mathcal{F}^p =", "\\Ker\\left(\\mathcal{F} \\longrightarrow \\bigoplus", "\\xi_{i, *}(\\mathcal{F}_{\\xi_i}/\\mathcal{F}_i^p)\\right)", "$$", "Then $\\mathcal{F}^p$ is coherent, $\\mathcal{F}^0 = \\mathcal{F}$, and", "$\\mathcal{F}^p/\\mathcal{F}^{p + 1}$ is isomorphic to a free", "$\\mathcal{O}_{Z_i}$-module of rank $1$ (if $n_i > p$) or $0$", "(if $n_i \\leq p$) in an open neighbourhood of $\\xi_i$. Moreover,", "$\\mathcal{F}' = \\bigcap \\mathcal{F}^p = 0$. Since every quasi-compact", "open $U \\subset X$ contains only a finite number of $\\xi_i$", "we conclude that $\\mathcal{F}^p|_U$ is zero for $p \\gg 0$.", "Hence $\\bigoplus_{p \\geq 0} \\mathcal{F}^p$ is a coherent", "$\\mathcal{O}_X$-module. Consider the short exact sequences", "$$", "0 \\to", "\\bigoplus\\nolimits_{p > 0} \\mathcal{F}^p \\to", "\\bigoplus\\nolimits_{p \\geq 0} \\mathcal{F}^p \\to", "\\bigoplus\\nolimits_{p > 0} \\mathcal{F}^p/\\mathcal{F}^{p + 1} \\to 0", "$$", "and", "$$", "0 \\to", "\\bigoplus\\nolimits_{p > 0} \\mathcal{F}^p \\to", "\\bigoplus\\nolimits_{p \\geq 0} \\mathcal{F}^p \\to", "\\mathcal{F} \\to 0", "$$", "of coherent $\\mathcal{O}_X$-modules. This already shows that", "$$", "[\\mathcal{F}] = [\\bigoplus \\mathcal{F}^p/\\mathcal{F}^{p + 1}]", "$$", "in $K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))$.", "Next, for every $p \\geq 0$ and $i \\in I$ such that $n_i > p$", "we choose a nonzero ideal sheaf $\\mathcal{I}_{i, p} \\subset \\mathcal{O}_{Z_i}$", "and a map $\\mathcal{I}_{i, p} \\to \\mathcal{F}^p/\\mathcal{F}^{p + 1}$ on $X$", "which is an isomorphism over the open neighbourhood of $\\xi_i$", "mentioned above. This is possible by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-extend-coherent}.", "Then we consider the short exact sequence", "$$", "0 \\to", "\\bigoplus\\nolimits_{p \\geq 0, i \\in I, n_i > p} \\mathcal{I}_{i, p}", "\\to", "\\bigoplus \\mathcal{F}^p/\\mathcal{F}^{p + 1} \\to", "\\mathcal{Q} \\to 0", "$$", "and the short exact sequence", "$$", "0 \\to", "\\bigoplus\\nolimits_{p \\geq 0, i \\in I, n_i > p} \\mathcal{I}_{i, p}", "\\to", "\\bigoplus\\nolimits_{p \\geq 0, i \\in I, n_i > p} \\mathcal{O}_{Z_i}", "\\to", "\\mathcal{Q}' \\to 0", "$$", "Observe that both $\\mathcal{Q}$ and $\\mathcal{Q}'$ are zero in a neighbourhood", "of the points $\\xi_i$ and that they are supported on $\\bigcup Z_i$.", "Hence $\\mathcal{Q}$ and $\\mathcal{Q}'$ are in", "$\\textit{Coh}_{\\leq k - 1}(X)$.", "Since", "$$", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{O}_{Z_i}^{\\oplus n_i} \\cong", "\\bigoplus\\nolimits_{p \\geq 0, i \\in I, n_i > p} \\mathcal{O}_{Z_i}", "$$", "this concludes the proof." ], "refs": [ "chow-lemma-additivity-sheaf-cycle", "homology-lemma-serre-subcategory-K-groups", "coherent-lemma-extend-coherent" ], "ref_ids": [ 5671, 12051, 3323 ] } ], "ref_ids": [] }, { "id": 5700, "type": "theorem", "label": "chow-lemma-finite-cycles-k-group", "categories": [ "chow" ], "title": "chow-lemma-finite-cycles-k-group", "contents": [ "Let $\\pi : X \\to Y$ be a finite morphism of schemes locally of finite type", "over $(S, \\delta)$ as in Situation \\ref{situation-setup}. Then", "$\\pi_* : \\textit{Coh}(X) \\to \\textit{Coh}(Y)$ is an exact functor", "which sends $\\textit{Coh}_{\\leq k}(X)$ into $\\textit{Coh}_{\\leq k}(Y)$", "and induces homomorphisms on $K_0$ of these categories and", "their quotients. The maps of Lemma \\ref{lemma-cycles-k-group}", "fit into a commutative diagram", "$$", "\\xymatrix{", "Z_k(X) \\ar[d]^{\\pi_*} \\ar[r] &", "K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))", "\\ar[d]^{\\pi_*} \\ar[r] &", "Z_k(X) \\ar[d]^{\\pi_*} \\\\", "Z_k(Y) \\ar[r] &", "K_0(\\textit{Coh}_{\\leq k}(Y)/\\textit{Coh}_{\\leq k - 1}(Y)) \\ar[r] &", "Z_k(Y)", "}", "$$" ], "refs": [ "chow-lemma-cycles-k-group" ], "proofs": [ { "contents": [ "A finite morphism is affine, hence pushforward of quasi-coherent", "modules along $\\pi$ is an exact functor by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-vanishing}.", "A finite morphism is proper, hence $\\pi_*$ sends coherent sheaves", "to coherent sheaves, see Cohomology of Schemes, Proposition", "\\ref{coherent-proposition-proper-pushforward-coherent}.", "The statement on dimensions of supports is clear.", "Commutativity on the right follows immediately from", "Lemma \\ref{lemma-cycle-push-sheaf}.", "Since the horizontal arrows are bijections, we find that", "we have commutativity on the left as well." ], "refs": [ "coherent-lemma-relative-affine-vanishing", "coherent-proposition-proper-pushforward-coherent", "chow-lemma-cycle-push-sheaf" ], "ref_ids": [ 3283, 3401, 5676 ] } ], "ref_ids": [ 5699 ] }, { "id": 5701, "type": "theorem", "label": "chow-lemma-from-chow-to-K", "categories": [ "chow" ], "title": "chow-lemma-from-chow-to-K", "contents": [ "Let $X$ be a scheme locally of finite type over $(S, \\delta)$", "as in Situation \\ref{situation-setup}. There is a canonical map", "$$", "\\CH_k(X)", "\\longrightarrow", "K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))", "$$", "induced by the map", "$Z_k(X) \\to K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))$", "from Lemma \\ref{lemma-cycles-k-group}." ], "refs": [ "chow-lemma-cycles-k-group" ], "proofs": [ { "contents": [ "We have to show that an element $\\alpha$ of $Z_k(X)$ which is rationally", "equivalent to zero, is mapped to zero in", "$K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))$.", "Write $\\alpha = \\sum (i_j)_*\\text{div}(f_j)$ as in", "Definition \\ref{definition-rational-equivalence}.", "Observe that", "$$", "\\pi = \\coprod i_j : W = \\coprod W_j \\longrightarrow X", "$$", "is a finite morphism as each $i_j : W_j \\to X$ is a closed immersion", "and the family of $W_j$ is locally finite in $X$. Hence we may use", "Lemma \\ref{lemma-finite-cycles-k-group} to reduce to the case of $W$.", "Since $W$ is a disjoint union of intregral scheme, we reduce", "to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $X$ is integral of $\\delta$-dimension $k + 1$.", "Let $f$ be a nonzero rational function on $X$.", "Let $\\alpha = \\text{div}(f)$. We have to show that", "$\\alpha$ is mapped to zero in", "$K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the ideal of denominators", "of $f$, see Divisors, Definition", "\\ref{divisors-definition-regular-meromorphic-ideal-denominators}.", "Then we have short exact sequences", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_X \\to \\mathcal{O}_X/\\mathcal{I} \\to 0", "$$", "and", "$$", "0 \\to \\mathcal{I} \\xrightarrow{f} \\mathcal{O}_X \\to", "\\mathcal{O}_X/f\\mathcal{I} \\to 0", "$$", "See Divisors, Lemma", "\\ref{divisors-lemma-regular-meromorphic-ideal-denominators}.", "We claim that", "$$", "[\\mathcal{O}_X/\\mathcal{I}]_k - [\\mathcal{O}_X/f\\mathcal{I}]_k =", "\\text{div}(f)", "$$", "The claim implies the element $\\alpha = \\text{div}(f)$ is represented by", "$[\\mathcal{O}_X/\\mathcal{I}] - [\\mathcal{O}_X/f\\mathcal{I}]$", "in $K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))$.", "Then the short exact sequences show that this element maps to", "zero in $K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))$.", "\\medskip\\noindent", "To prove the claim, let $Z \\subset X$ be an integral closed subscheme", "of $\\delta$-dimension $k$ and let $\\xi \\in Z$ be its generic point.", "Then $I = \\mathcal{I}_\\xi \\subset A = \\mathcal{O}_{X, \\xi}$", "is an ideal such that $fI \\subset A$. Now the coefficient of", "$[Z]$ in $\\text{div}(f)$ is $\\text{ord}_A(f)$. (Of course as usual", "we identify the function field of $X$ with the fraction field of $A$.)", "On the other hand, the coefficient of $[Z]$ in", "$[\\mathcal{O}_X/\\mathcal{I}] - [\\mathcal{O}_X/f\\mathcal{I}]$", "is", "$$", "\\text{length}_A(A/I) - \\text{length}_A(A/fI)", "$$", "Using the distance fuction of", "Algebra, Definition \\ref{algebra-definition-distance}", "we can rewrite this as", "$$", "d(A, I) - d(A, fI) = d(I, fI) = \\text{ord}_A(f)", "$$", "The equalities hold by Algebra, Lemmas", "\\ref{algebra-lemma-properties-distance-function} and", "\\ref{algebra-lemma-order-vanishing-determinant}.", "(Using these lemmas isn't necessary, but convenient.)" ], "refs": [ "chow-definition-rational-equivalence", "chow-lemma-finite-cycles-k-group", "divisors-definition-regular-meromorphic-ideal-denominators", "divisors-lemma-regular-meromorphic-ideal-denominators", "algebra-definition-distance", "algebra-lemma-properties-distance-function", "algebra-lemma-order-vanishing-determinant" ], "ref_ids": [ 5912, 5700, 8105, 8012, 1521, 1045, 1046 ] } ], "ref_ids": [ 5699 ] }, { "id": 5702, "type": "theorem", "label": "chow-lemma-K-coherent-supported-on-closed", "categories": [ "chow" ], "title": "chow-lemma-K-coherent-supported-on-closed", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $Z \\subset X$ be a closed", "subscheme. Denote $\\textit{Coh}_Z(X) \\subset \\textit{Coh}(X)$", "the Serre subcategory of coherent $\\mathcal{O}_X$-modules whose", "set theoretic support is contained in $Z$. Then the exact inclusion", "functor $\\textit{Coh}(Z) \\to \\textit{Coh}_Z(X)$ induces", "an isomorphism", "$$", "K'_0(Z) = K_0(\\textit{Coh}(Z)) \\longrightarrow K_0(\\textit{Coh}_Z(X))", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be an object of $\\textit{Coh}_Z(X)$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent", "ideal sheaf of $Z$. Consider the descending filtration", "$$", "\\ldots \\subset", "\\mathcal{F}^p = \\mathcal{I}^p \\mathcal{F} \\subset", "\\mathcal{F}^{p - 1} \\subset \\ldots \\subset \\mathcal{F}^0 = \\mathcal{F}", "$$", "Exactly as in the proof of Lemma \\ref{lemma-from-chow-to-K} this filtration", "is locally finite and hence", "$\\bigoplus_{p \\geq 0} \\mathcal{F}^p$,", "$\\bigoplus_{p \\geq 1} \\mathcal{F}^p$, and", "$\\bigoplus_{p \\geq 0} \\mathcal{F}^p/\\mathcal{F}^{p + 1}$", "are coherent $\\mathcal{O}_X$-modules supported on $Z$.", "Hence we get", "$$", "[\\mathcal{F}] =", "[\\bigoplus\\nolimits_{p \\geq 0} \\mathcal{F}^p/\\mathcal{F}^{p + 1}]", "$$", "in $K_0(\\textit{Coh}_Z(X))$ exactly as in the proof of", "Lemma \\ref{lemma-from-chow-to-K}. Since the coherent module", "$\\bigoplus_{p \\geq 0} \\mathcal{F}^p/\\mathcal{F}^{p + 1}$", "is annihilated by $\\mathcal{I}$ we conclude that", "$[\\mathcal{F}]$ is in the image. Actually, we claim that the map", "$$", "\\mathcal{F} \\longmapsto ", "c(\\mathcal{F}) =", "[\\bigoplus\\nolimits_{p \\geq 0} \\mathcal{F}^p/\\mathcal{F}^{p + 1}]", "$$", "factors through $K_0(\\textit{Coh}_Z(X))$ and is an inverse to", "the map in the statement of the lemma. To see this all we have", "to show is that if", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0", "$$", "is a short exact sequence in $\\textit{Coh}_Z(X)$, then we", "get $c(\\mathcal{G}) = c(\\mathcal{F}) + c(\\mathcal{H})$.", "Observe that for all $q \\geq 0$ we have a short exact sequence", "$$", "0 \\to", "(\\mathcal{F} \\cap \\mathcal{I}^q\\mathcal{G})/", "(\\mathcal{F} \\cap \\mathcal{I}^{q + 1}\\mathcal{G}) \\to", "\\mathcal{G}^q/\\mathcal{G}^{q + 1} \\to", "\\mathcal{H}^q/\\mathcal{H}^{q + 1} \\to 0", "$$", "For $p, q \\geq 0$ consider the coherent submodule", "$$", "\\mathcal{F}^{p, q} = \\mathcal{I}^p\\mathcal{F} \\cap \\mathcal{I}^q\\mathcal{G}", "$$", "Arguing exactly as above and using that the filtrations", "$\\mathcal{F}^p = \\mathcal{I}^p\\mathcal{F}$ and", "$\\mathcal{F} \\cap \\mathcal{I}^q\\mathcal{G}$ are locally finite,", "we find that", "$$", "[\\bigoplus\\nolimits_{p \\geq 0} \\mathcal{F}^p/\\mathcal{F}^{p + 1}] =", "[\\bigoplus\\nolimits_{p, q \\geq 0}", "\\mathcal{F}^{p, q}/(\\mathcal{F}^{p + 1, q} + \\mathcal{F}^{p, q + 1})] =", "[\\bigoplus\\nolimits_{q \\geq 0}", "(\\mathcal{F} \\cap \\mathcal{I}^q\\mathcal{G})/", "(\\mathcal{F} \\cap \\mathcal{I}^{q + 1}\\mathcal{G})]", "$$", "in $K_0(\\textit{Coh}(Z))$. Combined with the exact sequences above we obtain", "the desired result. Some details omitted." ], "refs": [ "chow-lemma-from-chow-to-K", "chow-lemma-from-chow-to-K" ], "ref_ids": [ 5701, 5701 ] } ], "ref_ids": [] }, { "id": 5703, "type": "theorem", "label": "chow-lemma-compute-c1", "categories": [ "chow" ], "title": "chow-lemma-compute-c1", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. Assume $X$ is", "integral and $n = \\dim_\\delta(X)$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{L})$ be a nonzero global section.", "Then", "$$", "\\text{div}_\\mathcal{L}(s) = [Z(s)]_{n - 1}", "$$", "in $Z_{n - 1}(X)$ and", "$$", "c_1(\\mathcal{L}) \\cap [X] = [Z(s)]_{n - 1}", "$$", "in $\\CH_{n - 1}(X)$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset X$ be an integral closed subscheme of", "$\\delta$-dimension $n - 1$. Let $\\xi \\in Z$ be its generic", "point. Choose a generator $s_\\xi \\in \\mathcal{L}_\\xi$.", "Write $s = fs_\\xi$ for some $f \\in \\mathcal{O}_{X, \\xi}$.", "By definition of $Z(s)$, see", "Divisors, Definition \\ref{divisors-definition-zero-scheme-s}", "we see that $Z(s)$ is cut out by a quasi-coherent", "sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$ such", "that $\\mathcal{I}_\\xi = (f)$. Hence", "$\\text{length}_{\\mathcal{O}_{X, x}}(\\mathcal{O}_{Z(s), \\xi})", "=", "\\text{length}_{\\mathcal{O}_{X, x}}(\\mathcal{O}_{X, \\xi}/(f))", "=", "\\text{ord}_{\\mathcal{O}_{X, x}}(f)$ as desired." ], "refs": [ "divisors-definition-zero-scheme-s" ], "ref_ids": [ 8094 ] } ], "ref_ids": [] }, { "id": 5704, "type": "theorem", "label": "chow-lemma-flat-pullback-divisor-invertible-sheaf", "categories": [ "chow" ], "title": "chow-lemma-flat-pullback-divisor-invertible-sheaf", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$. Assume $X$, $Y$", "are integral and $n = \\dim_\\delta(Y)$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_Y$-module.", "Let $f : X \\to Y$ be a flat morphism of relative dimension $r$.", "Let $\\mathcal{L}$ be an invertible sheaf on $Y$. Then", "$$", "f^*(c_1(\\mathcal{L}) \\cap [Y]) = c_1(f^*\\mathcal{L}) \\cap [X]", "$$", "in $\\CH_{n + r - 1}(X)$." ], "refs": [], "proofs": [ { "contents": [ "Let $s$ be a nonzero meromorphic section of $\\mathcal{L}$.", "We will show that actually", "$f^*\\text{div}_\\mathcal{L}(s) = \\text{div}_{f^*\\mathcal{L}}(f^*s)$", "and hence the lemma holds.", "To see this let $\\xi \\in Y$ be a point and let $s_\\xi \\in \\mathcal{L}_\\xi$", "be a generator. Write $s = gs_\\xi$ with $g \\in R(X)^*$.", "Then there is an open neighbourhood $V \\subset Y$ of $\\xi$", "such that $s_\\xi \\in \\mathcal{L}(V)$ and such that $s_\\xi$ generates", "$\\mathcal{L}|_V$. Hence we see that", "$$", "\\text{div}_\\mathcal{L}(s)|_V = \\text{div}(g)|_V.", "$$", "In exactly the same way, since $f^*s_\\xi$ generates $\\mathcal{L}$", "over $f^{-1}(V)$ and since $f^*s = g f^*s_\\xi$ we also", "have", "$$", "\\text{div}_\\mathcal{L}(f^*s)|_{f^{-1}(V)}", "=", "\\text{div}(g)|_{f^{-1}(V)}.", "$$", "Thus the desired equality of cycles over $f^{-1}(V)$ follows from the", "corresponding result for pullbacks of principal divisors, see", "Lemma \\ref{lemma-flat-pullback-principal-divisor}." ], "refs": [ "chow-lemma-flat-pullback-principal-divisor" ], "ref_ids": [ 5686 ] } ], "ref_ids": [] }, { "id": 5705, "type": "theorem", "label": "chow-lemma-c1-cap-additive", "categories": [ "chow" ], "title": "chow-lemma-c1-cap-additive", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{L}$, $\\mathcal{N}$ be an invertible sheaves on $X$.", "Then", "$$", "c_1(\\mathcal{L}) \\cap \\alpha + c_1(\\mathcal{N}) \\cap \\alpha =", "c_1(\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}) \\cap \\alpha", "$$", "in $\\CH_k(X)$ for every $\\alpha \\in Z_{k + 1}(X)$. Moreover,", "$c_1(\\mathcal{O}_X) \\cap \\alpha = 0$ for all $\\alpha$." ], "refs": [], "proofs": [ { "contents": [ "The additivity follows directly from", "Divisors, Lemma \\ref{divisors-lemma-c1-additive}", "and the definitions. To see that $c_1(\\mathcal{O}_X) \\cap \\alpha = 0$", "consider the section $1 \\in \\Gamma(X, \\mathcal{O}_X)$. This restricts", "to an everywhere nonzero section on any integral closed subscheme", "$W \\subset X$. Hence $c_1(\\mathcal{O}_X) \\cap [W] = 0$ as desired." ], "refs": [ "divisors-lemma-c1-additive" ], "ref_ids": [ 8027 ] } ], "ref_ids": [] }, { "id": 5706, "type": "theorem", "label": "chow-lemma-prepare-geometric-cap", "categories": [ "chow" ], "title": "chow-lemma-prepare-geometric-cap", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $Y$ be locally of finite type over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_Y$-module.", "Let $s \\in \\Gamma(Y, \\mathcal{L})$.", "Assume", "\\begin{enumerate}", "\\item $\\dim_\\delta(Y) \\leq k + 1$,", "\\item $\\dim_\\delta(Z(s)) \\leq k$, and", "\\item for every generic point $\\xi$ of an irreducible component of", "$Z(s)$ of $\\delta$-dimension $k$ the multiplication by $s$", "induces an injection $\\mathcal{O}_{Y, \\xi} \\to \\mathcal{L}_\\xi$.", "\\end{enumerate}", "Write $[Y]_{k + 1} = \\sum n_i[Y_i]$ where $Y_i \\subset Y$ are the", "irreducible components of $Y$ of $\\delta$-dimension $k + 1$.", "Set $s_i = s|_{Y_i} \\in \\Gamma(Y_i, \\mathcal{L}|_{Y_i})$. Then", "\\begin{equation}", "\\label{equation-equal-as-cycles}", "[Z(s)]_k = \\sum n_i[Z(s_i)]_k", "\\end{equation}", "as $k$-cycles on $Y$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset Y$ be an integral closed subscheme of", "$\\delta$-dimension $k$. Let $\\xi \\in Z$ be its generic point.", "We want to compare the coefficient $n$ of $[Z]$ in the expression", "$\\sum n_i[Z(s_i)]_k$ with the coefficient $m$ of $[Z]$ in the", "expression $[Z(s)]_k$. Choose a generator $s_\\xi \\in \\mathcal{L}_\\xi$.", "Write $A = \\mathcal{O}_{Y, \\xi}$, $L = \\mathcal{L}_\\xi$.", "Then $L = As_\\xi$. Write $s = f s_\\xi$ for some (unique) $f \\in A$.", "Hypothesis (3) means that $f : A \\to A$ is injective.", "Since $\\dim_\\delta(Y) \\leq k + 1$ and $\\dim_\\delta(Z) = k$", "we have $\\dim(A) = 0$ or $1$. We have", "$$", "m = \\text{length}_A(A/(f))", "$$", "which is finite in either case.", "\\medskip\\noindent", "If $\\dim(A) = 0$, then $f : A \\to A$ being injective", "implies that $f \\in A^*$. Hence in this case $m$ is zero.", "Moreover, the condition $\\dim(A) = 0$ means that $\\xi$", "does not lie on any irreducible component of $\\delta$-dimension", "$k + 1$, i.e., $n = 0$ as well.", "\\medskip\\noindent", "Now, let $\\dim(A) = 1$.", "Since $A$ is a Noetherian local ring it has finitely", "many minimal primes $\\mathfrak q_1, \\ldots, \\mathfrak q_t$.", "These correspond 1-1 with the $Y_i$ passing through $\\xi'$.", "Moreover $n_i = \\text{length}_{A_{\\mathfrak q_i}}(A_{\\mathfrak q_i})$.", "Also, the multiplicity of $[Z]$ in $[Z(s_i)]_k$ is", "$\\text{length}_A(A/(f, \\mathfrak q_i))$.", "Hence the equation to prove in this case is", "$$", "\\text{length}_A(A/(f))", "=", "\\sum \\text{length}_{A_{\\mathfrak q_i}}(A_{\\mathfrak q_i})", "\\text{length}_A(A/(f, \\mathfrak q_i))", "$$", "which follows from", "Lemma \\ref{lemma-additivity-divisors-restricted}." ], "refs": [ "chow-lemma-additivity-divisors-restricted" ], "ref_ids": [ 5653 ] } ], "ref_ids": [] }, { "id": 5707, "type": "theorem", "label": "chow-lemma-geometric-cap", "categories": [ "chow" ], "title": "chow-lemma-geometric-cap", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $Y \\subset X$ be a closed subscheme.", "Let $s \\in \\Gamma(Y, \\mathcal{L}|_Y)$.", "Assume", "\\begin{enumerate}", "\\item $\\dim_\\delta(Y) \\leq k + 1$,", "\\item $\\dim_\\delta(Z(s)) \\leq k$, and", "\\item for every generic point $\\xi$ of an irreducible component of", "$Z(s)$ of $\\delta$-dimension $k$ the multiplication by $s$", "induces an injection", "$\\mathcal{O}_{Y, \\xi} \\to (\\mathcal{L}|_Y)_\\xi$\\footnote{For example,", "this holds if $s$ is a regular section of $\\mathcal{L}|_Y$.}.", "\\end{enumerate}", "Then", "$$", "c_1(\\mathcal{L}) \\cap [Y]_{k + 1} = [Z(s)]_k", "$$", "in $\\CH_k(X)$." ], "refs": [], "proofs": [ { "contents": [ "Write", "$$", "[Y]_{k + 1} = \\sum n_i[Y_i]", "$$", "where $Y_i \\subset Y$ are the irreducible components of", "$Y$ of $\\delta$-dimension $k + 1$ and $n_i > 0$.", "By assumption the restriction", "$s_i = s|_{Y_i} \\in \\Gamma(Y_i, \\mathcal{L}|_{Y_i})$ is not", "zero, and hence is a regular section. By Lemma \\ref{lemma-compute-c1}", "we see that $[Z(s_i)]_k$ represents $c_1(\\mathcal{L}|_{Y_i})$.", "Hence by definition", "$$", "c_1(\\mathcal{L}) \\cap [Y]_{k + 1} = \\sum n_i[Z(s_i)]_k", "$$", "Thus the result follows from Lemma \\ref{lemma-prepare-geometric-cap}." ], "refs": [ "chow-lemma-compute-c1", "chow-lemma-prepare-geometric-cap" ], "ref_ids": [ 5703, 5706 ] } ], "ref_ids": [] }, { "id": 5708, "type": "theorem", "label": "chow-lemma-prepare-flat-pullback-cap-c1", "categories": [ "chow" ], "title": "chow-lemma-prepare-flat-pullback-cap-c1", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $f : X \\to Y$ be a flat morphism of relative dimension $r$.", "Let $\\mathcal{L}$ be an invertible sheaf on $Y$.", "Assume $Y$ is integral and $n = \\dim_\\delta(Y)$.", "Let $s$ be a nonzero meromorphic section of $\\mathcal{L}$.", "Then we have", "$$", "f^*\\text{div}_\\mathcal{L}(s) = \\sum n_i\\text{div}_{f^*\\mathcal{L}|_{X_i}}(s_i)", "$$", "in $Z_{n + r - 1}(X)$. Here the sum is over the irreducible", "components $X_i \\subset X$ of $\\delta$-dimension $n + r$,", "the section $s_i = f|_{X_i}^*(s)$ is the pullback of $s$, and", "$n_i = m_{X_i, X}$ is the multiplicity of $X_i$ in $X$." ], "refs": [], "proofs": [ { "contents": [ "To prove this equality of cycles, we may work locally on $Y$.", "Hence we may assume $Y$ is affine and $s = p/q$ for some nonzero", "sections $p \\in \\Gamma(Y, \\mathcal{L})$ and $q \\in \\Gamma(Y, \\mathcal{O})$.", "If we can show both", "$$", "f^*\\text{div}_\\mathcal{L}(p) =", "\\sum n_i\\text{div}_{f^*\\mathcal{L}|_{X_i}}(p_i)", "\\quad\\text{and}\\quad", "f^*\\text{div}_\\mathcal{O}(q) =", "\\sum n_i\\text{div}_{\\mathcal{O}_{X_i}}(q_i)", "$$", "(with obvious notations) then we win by the", "additivity, see Divisors, Lemma \\ref{divisors-lemma-c1-additive}.", "Thus we may assume that $s \\in \\Gamma(Y, \\mathcal{L})$.", "In this case we may apply the equality", "(\\ref{equation-equal-as-cycles}) to see that", "$$", "[Z(f^*(s))]_{k + r - 1} =", "\\sum n_i\\text{div}_{f^*\\mathcal{L}|_{X_i}}(s_i)", "$$", "where $f^*(s) \\in f^*\\mathcal{L}$ denotes the pullback of $s$ to $X$.", "On the other hand we have", "$$", "f^*\\text{div}_\\mathcal{L}(s) = f^*[Z(s)]_{k - 1}", "= [f^{-1}(Z(s))]_{k + r - 1},", "$$", "by Lemmas \\ref{lemma-compute-c1} and \\ref{lemma-pullback-coherent}.", "Since $Z(f^*(s)) = f^{-1}(Z(s))$ we win." ], "refs": [ "divisors-lemma-c1-additive", "chow-lemma-compute-c1", "chow-lemma-pullback-coherent" ], "ref_ids": [ 8027, 5703, 5681 ] } ], "ref_ids": [] }, { "id": 5709, "type": "theorem", "label": "chow-lemma-flat-pullback-cap-c1", "categories": [ "chow" ], "title": "chow-lemma-flat-pullback-cap-c1", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $f : X \\to Y$ be a flat morphism of relative dimension $r$.", "Let $\\mathcal{L}$ be an invertible sheaf on $Y$.", "Let $\\alpha$ be a $k$-cycle on $Y$.", "Then", "$$", "f^*(c_1(\\mathcal{L}) \\cap \\alpha) = c_1(f^*\\mathcal{L}) \\cap f^*\\alpha", "$$", "in $\\CH_{k + r - 1}(X)$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\alpha = \\sum n_i[W_i]$. We will show that", "$$", "f^*(c_1(\\mathcal{L}) \\cap [W_i]) = c_1(f^*\\mathcal{L}) \\cap f^*[W_i]", "$$", "in $\\CH_{k + r - 1}(X)$ by producing a rational equivalence", "on the closed subscheme $f^{-1}(W_i)$ of $X$.", "By the discussion in", "Remark \\ref{remark-infinite-sums-rational-equivalences}", "this will prove the equality of the lemma is true.", "\\medskip\\noindent", "Let $W \\subset Y$ be an integral closed subscheme of $\\delta$-dimension $k$.", "Consider the closed subscheme $W' = f^{-1}(W) = W \\times_Y X$", "so that we have the fibre product diagram", "$$", "\\xymatrix{", "W' \\ar[r] \\ar[d]_h & X \\ar[d]^f \\\\", "W \\ar[r] & Y", "}", "$$", "We have to show that", "$f^*(c_1(\\mathcal{L}) \\cap [W]) = c_1(f^*\\mathcal{L}) \\cap f^*[W]$.", "Choose a nonzero meromorphic section $s$ of $\\mathcal{L}|_W$.", "Let $W'_i \\subset W'$ be the irreducible components of", "$\\delta$-dimension $k + r$. Write $[W']_{k + r} = \\sum n_i[W'_i]$", "with $n_i$ the multiplicity of $W'_i$ in $W'$ as per definition.", "So $f^*[W] = \\sum n_i[W'_i]$ in $Z_{k + r}(X)$.", "Since each $W'_i \\to W$ is dominant we", "see that $s_i = s|_{W'_i}$ is a nonzero meromorphic section for", "each $i$. By Lemma \\ref{lemma-prepare-flat-pullback-cap-c1}", "we have the following equality of cycles", "$$", "h^*\\text{div}_{\\mathcal{L}|_W}(s) =", "\\sum n_i\\text{div}_{f^*\\mathcal{L}|_{W'_i}}(s_i)", "$$", "in $Z_{k + r - 1}(W')$. This finishes the proof since", "the left hand side is a cycle on $W'$ which pushes to", "$f^*(c_1(\\mathcal{L}) \\cap [W])$ in $\\CH_{k + r - 1}(X)$", "and the right hand side is a cycle on $W'$ which pushes to", "$c_1(f^*\\mathcal{L}) \\cap f^*[W]$ in $\\CH_{k + r - 1}(X)$." ], "refs": [ "chow-remark-infinite-sums-rational-equivalences", "chow-lemma-prepare-flat-pullback-cap-c1" ], "ref_ids": [ 5931, 5708 ] } ], "ref_ids": [] }, { "id": 5710, "type": "theorem", "label": "chow-lemma-equal-c1-as-cycles", "categories": [ "chow" ], "title": "chow-lemma-equal-c1-as-cycles", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $f : X \\to Y$ be a proper morphism.", "Let $\\mathcal{L}$ be an invertible sheaf on $Y$.", "Let $s$ be a nonzero meromorphic section $s$ of $\\mathcal{L}$ on $Y$.", "Assume $X$, $Y$ integral, $f$ dominant, and $\\dim_\\delta(X) = \\dim_\\delta(Y)$.", "Then", "$$", "f_*\\left(\\text{div}_{f^*\\mathcal{L}}(f^*s)\\right) =", "[R(X) : R(Y)]\\text{div}_\\mathcal{L}(s).", "$$", "as cycles on $Y$. In particular", "$$", "f_*(c_1(f^*\\mathcal{L}) \\cap [X]) = c_1(\\mathcal{L}) \\cap [Y].", "$$" ], "refs": [], "proofs": [ { "contents": [ "The last equation follows from the first since $f_*[X] = [R(X) : R(Y)][Y]$", "by definition. It turns out that we can re-use", "Lemma \\ref{lemma-proper-pushforward-alteration}", "to prove this. Namely, since we are trying to prove an equality", "of cycles, we may work locally on $Y$. Hence we may assume", "that $\\mathcal{L} = \\mathcal{O}_Y$. In this case $s$", "corresponds to a rational function $g \\in R(Y)$, and", "we are simply trying to prove", "$$", "f_*\\left(\\text{div}_X(g)\\right) =", "[R(X) : R(Y)]\\text{div}_Y(g).", "$$", "Comparing with the result of the aforementioned", "Lemma \\ref{lemma-proper-pushforward-alteration}", "we see this true since", "$\\text{Nm}_{R(X)/R(Y)}(g) = g^{[R(X) : R(Y)]}$", "as $g \\in R(Y)^*$." ], "refs": [ "chow-lemma-proper-pushforward-alteration", "chow-lemma-proper-pushforward-alteration" ], "ref_ids": [ 5687, 5687 ] } ], "ref_ids": [] }, { "id": 5711, "type": "theorem", "label": "chow-lemma-pushforward-cap-c1", "categories": [ "chow" ], "title": "chow-lemma-pushforward-cap-c1", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $p : X \\to Y$ be a proper morphism.", "Let $\\alpha \\in Z_{k + 1}(X)$.", "Let $\\mathcal{L}$ be an invertible sheaf on $Y$.", "Then", "$$", "p_*(c_1(p^*\\mathcal{L}) \\cap \\alpha) = c_1(\\mathcal{L}) \\cap p_*\\alpha", "$$", "in $\\CH_k(Y)$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $p$ has the property that for every integral", "closed subscheme $W \\subset X$ the map $p|_W : W \\to Y$", "is a closed immersion. Then, by definition of capping", "with $c_1(\\mathcal{L})$ the lemma holds.", "\\medskip\\noindent", "We will use this remark to reduce to a special case. Namely,", "write $\\alpha = \\sum n_i[W_i]$ with $n_i \\not = 0$ and $W_i$ pairwise", "distinct. Let $W'_i \\subset Y$ be the image of $W_i$ (as an integral", "closed subscheme). Consider the diagram", "$$", "\\xymatrix{", "X' = \\coprod W_i \\ar[r]_-q \\ar[d]_{p'} & X \\ar[d]^p \\\\", "Y' = \\coprod W'_i \\ar[r]^-{q'} & Y.", "}", "$$", "Since $\\{W_i\\}$ is locally finite on $X$, and $p$ is proper", "we see that $\\{W'_i\\}$ is locally finite on $Y$ and that", "$q, q', p'$ are also proper morphisms.", "We may think of $\\sum n_i[W_i]$ also as a $k$-cycle", "$\\alpha' \\in Z_k(X')$. Clearly $q_*\\alpha' = \\alpha$.", "We have", "$q_*(c_1(q^*p^*\\mathcal{L}) \\cap \\alpha')", "= c_1(p^*\\mathcal{L}) \\cap q_*\\alpha'$", "and", "$(q')_*(c_1((q')^*\\mathcal{L}) \\cap p'_*\\alpha') =", "c_1(\\mathcal{L}) \\cap q'_*p'_*\\alpha'$ by the initial", "remark of the proof. Hence it suffices to prove the lemma", "for the morphism $p'$ and the cycle $\\sum n_i[W_i]$.", "Clearly, this means we may assume $X$, $Y$ integral,", "$f : X \\to Y$ dominant and $\\alpha = [X]$.", "In this case the result follows from", "Lemma \\ref{lemma-equal-c1-as-cycles}." ], "refs": [ "chow-lemma-equal-c1-as-cycles" ], "ref_ids": [ 5710 ] } ], "ref_ids": [] }, { "id": 5712, "type": "theorem", "label": "chow-lemma-key-formula", "categories": [ "chow" ], "title": "chow-lemma-key-formula", "contents": [ "In the situation above the cycle", "$$", "\\sum", "(Z_i \\to X)_*\\left(", "\\text{ord}_{B_i}(f_i) \\text{div}_{\\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) -", "\\text{ord}_{B_i}(g_i) \\text{div}_{\\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) \\right)", "$$", "is equal to the cycle", "$$", "\\sum (Z_i \\to X)_*\\text{div}(\\partial_{B_i}(f_i, g_i))", "$$" ], "refs": [], "proofs": [ { "contents": [ "First, let us examine what happens if we replace $s_i$ by $us_i$", "for some unit $u$ in $B_i$. Then $f_i$ gets replaced by $u^{-1} f_i$.", "Thus the first part of the first expression of the lemma is unchanged", "and in the second part we add", "$$", "-\\text{ord}_{B_i}(g_i)\\text{div}(u|_{Z_i})", "$$", "(where $u|_{Z_i}$ is the image of $u$ in the residue field) by", "Divisors, Lemma \\ref{divisors-lemma-divisor-meromorphic-well-defined}", "and in the second expression we add", "$$", "\\text{div}(\\partial_{B_i}(u^{-1}, g_i))", "$$", "by bi-linearity of the tame symbol. These terms agree by property", "(\\ref{item-normalization}) of the tame symbol.", "\\medskip\\noindent", "Let $Z \\subset X$ be an irreducible closed with $\\dim_\\delta(Z) = n - 2$.", "To show that the coefficients of $Z$ of the two cycles of the lemma", "is the same, we may do a replacement $s_i \\mapsto us_i$ as in the previous", "paragraph. In exactly the same way one shows that we may do a replacement", "$t_i \\mapsto vt_i$ for some unit $v$ of $B_i$.", "\\medskip\\noindent", "Since we are proving the equality of cycles we may argue one coefficient", "at a time. Thus we choose an irreducible closed $Z \\subset X$", "with $\\dim_\\delta(Z) = n - 2$ and compare coefficients. Let $\\xi \\in Z$", "be the generic point and set $A = \\mathcal{O}_{X, \\xi}$. This is a Noetherian", "local domain of dimension $2$. Choose generators $\\sigma$ and $\\tau$", "for $\\mathcal{L}_\\xi$ and $\\mathcal{N}_\\xi$. After shrinking $X$, we may", "and do assume $\\sigma$ and $\\tau$ define trivializations", "of the invertible sheaves $\\mathcal{L}$ and $\\mathcal{N}$ over all of $X$.", "Because $Z_i$ is locally", "finite after shrinking $X$ we may assume $Z \\subset Z_i$ for all $i \\in I$", "and that $I$ is finite. Then $\\xi_i$ corresponds to a prime", "$\\mathfrak q_i \\subset A$ of height $1$.", "We may write $s_i = a_i \\sigma$ and $t_i = b_i \\tau$", "for some $a_i$ and $b_i$ units in $A_{\\mathfrak q_i}$.", "By the remarks above, it suffices to prove the lemma when", "$a_i = b_i = 1$ for all $i$.", "\\medskip\\noindent", "Assume $a_i = b_i = 1$ for all $i$. Then the first expression of the", "lemma is zero, because we choose $\\sigma$ and $\\tau$ to be trivializing", "sections. Write $s = f\\sigma$ and $t = g \\tau$ with $f$ and $g$ in the", "fraction field of $A$. By the previous paragraph we have reduced to the case", "$f_i = f$ and $g_i = g$ for all $i$. Moreover, for a height $1$ prime", "$\\mathfrak q$ of $A$ which is not in $\\{\\mathfrak q_i\\}$ we have", "that both $f$ and $g$ are units in $A_\\mathfrak q$ (by our choice of", "the family $\\{Z_i\\}$ in the discussion preceding the lemma). Thus", "the coefficient of $Z$ in the second expression of the lemma is", "$$", "\\sum\\nolimits_i \\text{ord}_{A/\\mathfrak q_i}(\\partial_{B_i}(f, g))", "$$", "which is zero by the key Lemma \\ref{lemma-milnor-gersten-low-degree}." ], "refs": [ "divisors-lemma-divisor-meromorphic-well-defined", "chow-lemma-milnor-gersten-low-degree" ], "ref_ids": [ 8026, 5666 ] } ], "ref_ids": [] }, { "id": 5713, "type": "theorem", "label": "chow-lemma-commutativity-on-integral", "categories": [ "chow" ], "title": "chow-lemma-commutativity-on-integral", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Assume $X$ integral and $\\dim_\\delta(X) = n$.", "Let $\\mathcal{L}$, $\\mathcal{N}$ be invertible on $X$.", "Choose a nonzero meromorphic section $s$ of $\\mathcal{L}$", "and a nonzero meromorphic section $t$ of $\\mathcal{N}$.", "Set $\\alpha = \\text{div}_\\mathcal{L}(s)$ and", "$\\beta = \\text{div}_\\mathcal{N}(t)$.", "Then", "$$", "c_1(\\mathcal{N}) \\cap \\alpha", "=", "c_1(\\mathcal{L}) \\cap \\beta", "$$", "in $\\CH_{n - 2}(X)$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the key Lemma \\ref{lemma-key-formula}", "and the discussion preceding it." ], "refs": [ "chow-lemma-key-formula" ], "ref_ids": [ 5712 ] } ], "ref_ids": [] }, { "id": 5714, "type": "theorem", "label": "chow-lemma-factors", "categories": [ "chow" ], "title": "chow-lemma-factors", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{L}$ be invertible on $X$.", "The operation $\\alpha \\mapsto c_1(\\mathcal{L}) \\cap \\alpha$", "factors through rational equivalence to give an operation", "$$", "c_1(\\mathcal{L}) \\cap - : \\CH_{k + 1}(X) \\to \\CH_k(X)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha \\in Z_{k + 1}(X)$, and $\\alpha \\sim_{rat} 0$.", "We have to show that $c_1(\\mathcal{L}) \\cap \\alpha$", "as defined in Definition \\ref{definition-cap-c1} is zero.", "By Definition \\ref{definition-rational-equivalence} there", "exists a locally finite family $\\{W_j\\}$ of integral closed", "subschemes with $\\dim_\\delta(W_j) = k + 2$ and rational functions", "$f_j \\in R(W_j)^*$ such that", "$$", "\\alpha = \\sum (i_j)_*\\text{div}_{W_j}(f_j)", "$$", "Note that $p : \\coprod W_j \\to X$ is a proper morphism,", "and hence $\\alpha = p_*\\alpha'$ where $\\alpha' \\in Z_{k + 1}(\\coprod W_j)$", "is the sum of the principal divisors $\\text{div}_{W_j}(f_j)$.", "By Lemma \\ref{lemma-pushforward-cap-c1} we have", "$c_1(\\mathcal{L}) \\cap \\alpha = p_*(c_1(p^*\\mathcal{L}) \\cap \\alpha')$.", "Hence it suffices to show that each", "$c_1(\\mathcal{L}|_{W_j}) \\cap \\text{div}_{W_j}(f_j)$ is zero.", "In other words we may assume that $X$ is integral and", "$\\alpha = \\text{div}_X(f)$ for some $f \\in R(X)^*$.", "\\medskip\\noindent", "Assume $X$ is integral and $\\alpha = \\text{div}_X(f)$ for some $f \\in R(X)^*$.", "We can think of $f$ as a regular meromorphic section of the invertible", "sheaf $\\mathcal{N} = \\mathcal{O}_X$. Choose a meromorphic section", "$s$ of $\\mathcal{L}$ and denote $\\beta = \\text{div}_\\mathcal{L}(s)$.", "By Lemma \\ref{lemma-commutativity-on-integral}", "we conclude that", "$$", "c_1(\\mathcal{L}) \\cap \\alpha = c_1(\\mathcal{O}_X) \\cap \\beta.", "$$", "However, by Lemma \\ref{lemma-c1-cap-additive} we see that the right hand side", "is zero in $\\CH_k(X)$ as desired." ], "refs": [ "chow-definition-cap-c1", "chow-definition-rational-equivalence", "chow-lemma-pushforward-cap-c1", "chow-lemma-commutativity-on-integral", "chow-lemma-c1-cap-additive" ], "ref_ids": [ 5914, 5912, 5711, 5713, 5705 ] } ], "ref_ids": [] }, { "id": 5715, "type": "theorem", "label": "chow-lemma-cap-commutative", "categories": [ "chow" ], "title": "chow-lemma-cap-commutative", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{L}$, $\\mathcal{N}$ be invertible on $X$.", "For any $\\alpha \\in \\CH_{k + 2}(X)$ we have", "$$", "c_1(\\mathcal{L}) \\cap c_1(\\mathcal{N}) \\cap \\alpha", "=", "c_1(\\mathcal{N}) \\cap c_1(\\mathcal{L}) \\cap \\alpha", "$$", "as elements of $\\CH_k(X)$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\alpha = \\sum m_j[Z_j]$ for some locally finite", "collection of integral closed subschemes $Z_j \\subset X$", "with $\\dim_\\delta(Z_j) = k + 2$.", "Consider the proper morphism $p : \\coprod Z_j \\to X$.", "Set $\\alpha' = \\sum m_j[Z_j]$ as a $(k + 2)$-cycle on", "$\\coprod Z_j$. By several applications of", "Lemma \\ref{lemma-pushforward-cap-c1} we see that", "$c_1(\\mathcal{L}) \\cap c_1(\\mathcal{N}) \\cap \\alpha", "= p_*(c_1(p^*\\mathcal{L}) \\cap c_1(p^*\\mathcal{N}) \\cap \\alpha')$", "and", "$c_1(\\mathcal{N}) \\cap c_1(\\mathcal{L}) \\cap \\alpha", "= p_*(c_1(p^*\\mathcal{N}) \\cap c_1(p^*\\mathcal{L}) \\cap \\alpha')$.", "Hence it suffices to prove the formula in case $X$ is integral", "and $\\alpha = [X]$. In this case the result follows", "from Lemma \\ref{lemma-commutativity-on-integral} and the definitions." ], "refs": [ "chow-lemma-pushforward-cap-c1", "chow-lemma-commutativity-on-integral" ], "ref_ids": [ 5711, 5713 ] } ], "ref_ids": [] }, { "id": 5716, "type": "theorem", "label": "chow-lemma-support-cap-effective-Cartier", "categories": [ "chow" ], "title": "chow-lemma-support-cap-effective-Cartier", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally", "of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in", "Definition \\ref{definition-gysin-homomorphism}. Let $\\alpha$ be a", "$(k + 1)$-cycle on $X$. Then $i_*i^*\\alpha = c_1(\\mathcal{L}) \\cap \\alpha$", "in $\\CH_k(X)$. In particular, if $D$ is an effective Cartier divisor, then", "$D \\cdot \\alpha = c_1(\\mathcal{O}_X(D)) \\cap \\alpha$." ], "refs": [ "chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "Write $\\alpha = \\sum n_j[W_j]$ where $i_j : W_j \\to X$ are integral closed", "subschemes with $\\dim_\\delta(W_j) = k$.", "Since $D$ is the zero scheme of $s$ we see that $D \\cap W_j$ is the zero scheme", "of the restriction $s|_{W_j}$. Hence for each $j$ such that", "$W_j \\not \\subset D$ we have", "$c_1(\\mathcal{L}) \\cap [W_j] = [D \\cap W_j]_k$", "by Lemma \\ref{lemma-geometric-cap}. So we have", "$$", "c_1(\\mathcal{L}) \\cap \\alpha", "=", "\\sum\\nolimits_{W_j \\not \\subset D} n_j[D \\cap W_j]_k", "+", "\\sum\\nolimits_{W_j \\subset D}", "n_j i_{j, *}(c_1(\\mathcal{L})|_{W_j}) \\cap [W_j])", "$$", "in $\\CH_k(X)$ by Definition \\ref{definition-cap-c1}.", "The right hand side matches (termwise) the pushforward of the class", "$i^*\\alpha$ on $D$ from Definition \\ref{definition-gysin-homomorphism}.", "Hence we win." ], "refs": [ "chow-lemma-geometric-cap", "chow-definition-cap-c1", "chow-definition-gysin-homomorphism" ], "ref_ids": [ 5707, 5914, 5915 ] } ], "ref_ids": [ 5915 ] }, { "id": 5717, "type": "theorem", "label": "chow-lemma-easy-gysin", "categories": [ "chow" ], "title": "chow-lemma-easy-gysin", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $(\\mathcal{L}, s, i : D \\to X)$ be as in", "Definition \\ref{definition-gysin-homomorphism}.", "\\begin{enumerate}", "\\item Let $Z \\subset X$ be a closed subscheme such", "that $\\dim_\\delta(Z) \\leq k + 1$ and such that", "$D \\cap Z$ is an effective Cartier divisor on $Z$. Then", "$i^*[Z]_{k + 1} = [D \\cap Z]_k$.", "\\item Let $\\mathcal{F}$ be a coherent sheaf on $X$", "such that $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k + 1$ and", "$s : \\mathcal{F} \\to \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}$", "is injective. Then", "$$", "i^*[\\mathcal{F}]_{k + 1} = [i^*\\mathcal{F}]_k", "$$", "in $\\CH_k(D)$.", "\\end{enumerate}" ], "refs": [ "chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "Assume $Z \\subset X$ as in (1). Then set $\\mathcal{F} = \\mathcal{O}_Z$.", "The assumption that $D \\cap Z$ is an effective Cartier divisor is", "equivalent to the assumption that", "$s : \\mathcal{F} \\to \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}$", "is injective. Moreover $[Z]_{k + 1} = [\\mathcal{F}]_{k + 1}]$", "and $[D \\cap Z]_k = [\\mathcal{O}_{D \\cap Z}]_k = [i^*\\mathcal{F}]_k$.", "See Lemma \\ref{lemma-cycle-closed-coherent}.", "Hence part (1) follows from part (2).", "\\medskip\\noindent", "Write $[\\mathcal{F}]_{k + 1} = \\sum m_j[W_j]$ with $m_j > 0$", "and pairwise distinct integral closed subschemes $W_j \\subset X$", "of $\\delta$-dimension $k + 1$. The assumption that", "$s : \\mathcal{F} \\to \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}$", "is injective implies that $W_j \\not \\subset D$ for all $j$.", "By definition we see that", "$$", "i^*[\\mathcal{F}]_{k + 1} = \\sum [D \\cap W_j]_k.", "$$", "We claim that", "$$", "\\sum [D \\cap W_j]_k = [i^*\\mathcal{F}]_k", "$$", "as cycles.", "Let $Z \\subset D$ be an integral closed subscheme of $\\delta$-dimension", "$k$. Let $\\xi \\in Z$ be its generic point. Let $A = \\mathcal{O}_{X, \\xi}$.", "Let $M = \\mathcal{F}_\\xi$. Let $f \\in A$ be an element generating the", "ideal of $D$, i.e., such that $\\mathcal{O}_{D, \\xi} = A/fA$.", "By assumption $\\dim(\\text{Supp}(M)) = 1$,", "the map $f : M \\to M$ is injective, and", "$\\text{length}_A(M/fM) < \\infty$. Moreover, $\\text{length}_A(M/fM)$", "is the coefficient of $[Z]$ in $[i^*\\mathcal{F}]_k$. On the", "other hand, let $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ be the minimal", "primes in the support of $M$. Then", "$$", "\\sum", "\\text{length}_{A_{\\mathfrak q_i}}(M_{\\mathfrak q_i})", "\\text{ord}_{A/\\mathfrak q_i}(f)", "$$", "is the coefficient of $[Z]$ in $\\sum [D \\cap W_j]_k$.", "Hence we see the equality by", "Lemma \\ref{lemma-additivity-divisors-restricted}." ], "refs": [ "chow-lemma-cycle-closed-coherent", "chow-lemma-additivity-divisors-restricted" ], "ref_ids": [ 5670, 5653 ] } ], "ref_ids": [ 5915 ] }, { "id": 5718, "type": "theorem", "label": "chow-lemma-closed-in-X-gysin", "categories": [ "chow" ], "title": "chow-lemma-closed-in-X-gysin", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X' \\to X$ be a proper morphism of schemes", "locally of finite type over $S$.", "Let $(\\mathcal{L}, s, i : D \\to X)$ be as in", "Definition \\ref{definition-gysin-homomorphism}.", "Form the diagram", "$$", "\\xymatrix{", "D' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\", "D \\ar[r]^i & X", "}", "$$", "as in Remark \\ref{remark-pullback-pairs}.", "For any $(k + 1)$-cycle $\\alpha'$ on $X'$ we have", "$i^*f_*\\alpha' = g_*(i')^*\\alpha'$ in $\\CH_k(D)$", "(this makes sense as $f_*$ is defined on the level of cycles)." ], "refs": [ "chow-definition-gysin-homomorphism", "chow-remark-pullback-pairs" ], "proofs": [ { "contents": [ "Suppose $\\alpha = [W']$ for some integral closed subscheme", "$W' \\subset X'$. Let $W = f(W') \\subset X$. In case $W' \\not \\subset D'$,", "then $W \\not \\subset D$ and we see that", "$$", "[W' \\cap D']_k = \\text{div}_{\\mathcal{L}'|_{W'}}({s'|_{W'}})", "\\quad\\text{and}\\quad", "[W \\cap D]_k = \\text{div}_{\\mathcal{L}|_W}(s|_W)", "$$", "and hence $f_*$ of the first cycle equals the second cycle by", "Lemma \\ref{lemma-equal-c1-as-cycles}. Hence the", "equality holds as cycles. In case $W' \\subset D'$, then", "$W \\subset D$ and $f_*(c_1(\\mathcal{L}|_{W'}) \\cap [W'])$", "is equal to $c_1(\\mathcal{L}|_W) \\cap [W]$ in $\\CH_k(W)$ by the second", "assertion of Lemma \\ref{lemma-equal-c1-as-cycles}.", "By Remark \\ref{remark-infinite-sums-rational-equivalences}", "the result follows for general $\\alpha'$." ], "refs": [ "chow-lemma-equal-c1-as-cycles", "chow-lemma-equal-c1-as-cycles", "chow-remark-infinite-sums-rational-equivalences" ], "ref_ids": [ 5710, 5710, 5931 ] } ], "ref_ids": [ 5915, 5936 ] }, { "id": 5719, "type": "theorem", "label": "chow-lemma-gysin-flat-pullback", "categories": [ "chow" ], "title": "chow-lemma-gysin-flat-pullback", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X' \\to X$", "be a flat morphism of relative dimension $r$ of schemes locally of finite type", "over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in", "Definition \\ref{definition-gysin-homomorphism}. Form the diagram", "$$", "\\xymatrix{", "D' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\", "D \\ar[r]^i & X", "}", "$$", "as in Remark \\ref{remark-pullback-pairs}.", "For any $(k + 1)$-cycle $\\alpha$ on $X$ we have", "$(i')^*f^*\\alpha = g^*i^*\\alpha'$ in $\\CH_{k + r}(D)$", "(this makes sense as $f^*$ is defined on the level of cycles)." ], "refs": [ "chow-definition-gysin-homomorphism", "chow-remark-pullback-pairs" ], "proofs": [ { "contents": [ "Suppose $\\alpha = [W]$ for some integral closed subscheme", "$W \\subset X$. Let $W' = f^{-1}(W) \\subset X'$. In case $W \\not \\subset D$,", "then $W' \\not \\subset D'$ and we see that", "$$", "W' \\cap D' = g^{-1}(W \\cap D)", "$$", "as closed subschemes of $D'$. Hence the", "equality holds as cycles, see Lemma \\ref{lemma-pullback-coherent}.", "In case $W \\subset D$, then $W' \\subset D'$ and $W' = g^{-1}(W)$", "with $[W']_{k + 1 + r} = g^*[W]$ and equality holds in", "$\\CH_{k + r}(D')$ by Lemma \\ref{lemma-flat-pullback-cap-c1}.", "By Remark \\ref{remark-infinite-sums-rational-equivalences}", "the result follows for general $\\alpha'$." ], "refs": [ "chow-lemma-pullback-coherent", "chow-lemma-flat-pullback-cap-c1", "chow-remark-infinite-sums-rational-equivalences" ], "ref_ids": [ 5681, 5709, 5931 ] } ], "ref_ids": [ 5915, 5936 ] }, { "id": 5720, "type": "theorem", "label": "chow-lemma-gysin-factors-general", "categories": [ "chow" ], "title": "chow-lemma-gysin-factors-general", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $X$ be integral and $n = \\dim_\\delta(X)$.", "Let $i : D \\to X$ be an effective Cartier divisor.", "Let $\\mathcal{N}$ be an invertible $\\mathcal{O}_X$-module", "and let $t$ be a nonzero meromorphic section of $\\mathcal{N}$.", "Then $i^*\\text{div}_\\mathcal{N}(t) = c_1(\\mathcal{N}) \\cap [D]_{n - 1}$", "in $\\CH_{n - 2}(D)$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\text{div}_\\mathcal{N}(t) = \\sum \\text{ord}_{Z_i, \\mathcal{N}}(t)[Z_i]$", "for some integral closed subschemes $Z_i \\subset X$ of $\\delta$-dimension", "$n - 1$. We may assume that the family $\\{Z_i\\}$ is locally", "finite, that $t \\in \\Gamma(U, \\mathcal{N}|_U)$ is a generator", "where $U = X \\setminus \\bigcup Z_i$, and that every irreducible component", "of $D$ is one of the $Z_i$, see", "Divisors, Lemmas \\ref{divisors-lemma-components-locally-finite},", "\\ref{divisors-lemma-divisor-locally-finite}, and", "\\ref{divisors-lemma-divisor-meromorphic-locally-finite}.", "\\medskip\\noindent", "Set $\\mathcal{L} = \\mathcal{O}_X(D)$. Denote", "$s \\in \\Gamma(X, \\mathcal{O}_X(D)) = \\Gamma(X, \\mathcal{L})$", "the canonical section. We will apply the discussion of", "Section \\ref{section-key} to our current situation.", "For each $i$ let $\\xi_i \\in Z_i$ be its generic point. Let", "$B_i = \\mathcal{O}_{X, \\xi_i}$. For each $i$ we pick generators", "$s_i \\in \\mathcal{L}_{\\xi_i}$ and $t_i \\in \\mathcal{N}_{\\xi_i}$", "over $B_i$ but we insist that we pick $s_i = s$ if $Z_i \\not \\subset D$.", "Write $s = f_i s_i$ and $t = g_i t_i$ with $f_i, g_i \\in B_i$.", "Then $\\text{ord}_{Z_i, \\mathcal{N}}(t) = \\text{ord}_{B_i}(g_i)$.", "On the other hand, we have $f_i \\in B_i$ and", "$$", "[D]_{n - 1} = \\sum \\text{ord}_{B_i}(f_i)[Z_i]", "$$", "because of our choices of $s_i$. We claim that", "$$", "i^*\\text{div}_\\mathcal{N}(t) =", "\\sum \\text{ord}_{B_i}(g_i) \\text{div}_{\\mathcal{L}|_{Z_i}}(s_i|_{Z_i})", "$$", "as cycles. More precisely, the right hand side is a cycle", "representing the left hand side. Namely, this is clear by our", "formula for $\\text{div}_\\mathcal{N}(t)$ and the fact that", "$\\text{div}_{\\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) = [Z(s_i|_{Z_i})]_{n - 2} =", "[Z_i \\cap D]_{n - 2}$ when $Z_i \\not \\subset D$ because in", "that case $s_i|_{Z_i} = s|_{Z_i}$ is a regular section, see", "Lemma \\ref{lemma-compute-c1}. Similarly,", "$$", "c_1(\\mathcal{N}) \\cap [D]_{n - 1} =", "\\sum \\text{ord}_{B_i}(f_i) \\text{div}_{\\mathcal{N}|_{Z_i}}(t_i|_{Z_i})", "$$", "The key formula (Lemma \\ref{lemma-key-formula}) gives the equality", "$$", "\\sum \\left(", "\\text{ord}_{B_i}(f_i) \\text{div}_{\\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) -", "\\text{ord}_{B_i}(g_i) \\text{div}_{\\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) \\right) =", "\\sum \\text{div}_{Z_i}(\\partial_{B_i}(f_i, g_i))", "$$", "of cycles. If $Z_i \\not \\subset D$, then $f_i = 1$ and hence", "$\\text{div}_{Z_i}(\\partial_{B_i}(f_i, g_i)) = 0$. Thus we get a rational", "equivalence between our specific cycles representing", "$i^*\\text{div}_\\mathcal{N}(t)$ and $c_1(\\mathcal{N}) \\cap [D]_{n - 1}$", "on $D$. This finishes the proof." ], "refs": [ "divisors-lemma-components-locally-finite", "divisors-lemma-divisor-locally-finite", "divisors-lemma-divisor-meromorphic-locally-finite", "chow-lemma-compute-c1", "chow-lemma-key-formula" ], "ref_ids": [ 8022, 8023, 8025, 5703, 5712 ] } ], "ref_ids": [] }, { "id": 5721, "type": "theorem", "label": "chow-lemma-gysin-factors", "categories": [ "chow" ], "title": "chow-lemma-gysin-factors", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $(\\mathcal{L}, s, i : D \\to X)$ be as in", "Definition \\ref{definition-gysin-homomorphism}.", "The Gysin homomorphism factors through rational equivalence to", "give a map $i^* : \\CH_{k + 1}(X) \\to \\CH_k(D)$." ], "refs": [ "chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "Let $\\alpha \\in Z_{k + 1}(X)$ and assume that $\\alpha \\sim_{rat} 0$.", "This means there exists a locally finite collection of integral", "closed subschemes $W_j \\subset X$ of $\\delta$-dimension $k + 2$", "and $f_j \\in R(W_j)^*$ such that", "$\\alpha = \\sum i_{j, *}\\text{div}_{W_j}(f_j)$.", "Set $X' = \\coprod W_i$ and consider the diagram", "$$", "\\xymatrix{", "D' \\ar[d]_q \\ar[r]_{i'} & X' \\ar[d]^p \\\\", "D \\ar[r]^i & X", "}", "$$", "of Remark \\ref{remark-pullback-pairs}. Since $X' \\to X$ is proper", "we see that $i^*p_* = q_*(i')^*$ by Lemma \\ref{lemma-closed-in-X-gysin}.", "As we know that $q_*$ factors through rational equivalence", "(Lemma \\ref{lemma-proper-pushforward-rational-equivalence}), it suffices", "to prove the result for $\\alpha' = \\sum \\text{div}_{W_j}(f_j)$", "on $X'$. Clearly this reduces us to the case where $X$ is integral", "and $\\alpha = \\text{div}(f)$ for some $f \\in R(X)^*$.", "\\medskip\\noindent", "Assume $X$ is integral and $\\alpha = \\text{div}(f)$ for some $f \\in R(X)^*$.", "If $X = D$, then we see that $i^*\\alpha$ is equal", "to $c_1(\\mathcal{L}) \\cap \\alpha$.", "This is rationally equivalent to zero by Lemma \\ref{lemma-factors}.", "If $D \\not = X$, then we see that $i^*\\text{div}_X(f)$ is equal to", "$c_1(\\mathcal{O}_D) \\cap [D]_{n - 1}$ in $\\CH_{n - 2}(D)$ by", "Lemma \\ref{lemma-gysin-factors-general}. Of course", "capping with $c_1(\\mathcal{O}_D)$ is the zero map", "(Lemma \\ref{lemma-c1-cap-additive})." ], "refs": [ "chow-remark-pullback-pairs", "chow-lemma-closed-in-X-gysin", "chow-lemma-proper-pushforward-rational-equivalence", "chow-lemma-factors", "chow-lemma-gysin-factors-general", "chow-lemma-c1-cap-additive" ], "ref_ids": [ 5936, 5718, 5694, 5714, 5720, 5705 ] } ], "ref_ids": [ 5915 ] }, { "id": 5722, "type": "theorem", "label": "chow-lemma-gysin-back", "categories": [ "chow" ], "title": "chow-lemma-gysin-back", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally", "of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in", "Definition \\ref{definition-gysin-homomorphism}. Then", "$i^*i_* : \\CH_k(D) \\to \\CH_{k - 1}(D)$ sends $\\alpha$ to", "$c_1(\\mathcal{L}|_D) \\cap \\alpha$." ], "refs": [ "chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "This is immediate from the definition of $i_*$ on cycles", "and the definition of $i^*$ given in", "Definition \\ref{definition-gysin-homomorphism}." ], "refs": [ "chow-definition-gysin-homomorphism" ], "ref_ids": [ 5915 ] } ], "ref_ids": [ 5915 ] }, { "id": 5723, "type": "theorem", "label": "chow-lemma-gysin-commutes-cap-c1", "categories": [ "chow" ], "title": "chow-lemma-gysin-commutes-cap-c1", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be", "locally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$", "be a triple as in Definition \\ref{definition-gysin-homomorphism}.", "Let $\\mathcal{N}$ be an invertible $\\mathcal{O}_X$-module.", "Then $i^*(c_1(\\mathcal{N}) \\cap \\alpha) = c_1(i^*\\mathcal{N}) \\cap i^*\\alpha$", "in $\\CH_{k - 2}(D)$ for all $\\alpha \\in \\CH_k(X)$." ], "refs": [ "chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "With exactly the same proof as in Lemma \\ref{lemma-gysin-factors}", "this follows from Lemmas", "\\ref{lemma-pushforward-cap-c1},", "\\ref{lemma-cap-commutative}, and", "\\ref{lemma-gysin-factors-general}." ], "refs": [ "chow-lemma-gysin-factors", "chow-lemma-pushforward-cap-c1", "chow-lemma-cap-commutative", "chow-lemma-gysin-factors-general" ], "ref_ids": [ 5721, 5711, 5715, 5720 ] } ], "ref_ids": [ 5915 ] }, { "id": 5724, "type": "theorem", "label": "chow-lemma-gysin-commutes-gysin", "categories": [ "chow" ], "title": "chow-lemma-gysin-commutes-gysin", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally", "of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ and", "$(\\mathcal{L}', s', i' : D' \\to X)$ be two triples as in", "Definition \\ref{definition-gysin-homomorphism}. Then the diagram", "$$", "\\xymatrix{", "\\CH_k(X) \\ar[r]_{i^*} \\ar[d]_{(i')^*} & \\CH_{k - 1}(D) \\ar[d]^{j^*} \\\\", "\\CH_{k - 1}(D') \\ar[r]^{(j')^*} & \\CH_{k - 2}(D \\cap D')", "}", "$$", "commutes where each of the maps is a gysin map." ], "refs": [ "chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "Denote $j : D \\cap D' \\to D$ and $j' : D \\cap D' \\to D'$ the closed", "immersions corresponding to $(\\mathcal{L}|_{D'}, s|_{D'}$ and", "$(\\mathcal{L}'_D, s|_D)$. We have to show that", "$(j')^*i^*\\alpha = j^* (i')^*\\alpha$ for all $\\alpha \\in \\CH_k(X)$.", "Let $W \\subset X$ be an integral closed subscheme of dimension $k$.", "Let us prove the equality in case $\\alpha = [W]$. We will deduce", "it from the key formula.", "\\medskip\\noindent", "We let $\\sigma$ be a nonzero meromorphic section of $\\mathcal{L}|_W$", "which we require to be equal to $s|_W$ if $W \\not \\subset D$.", "We let $\\sigma'$ be a nonzero meromorphic section of $\\mathcal{L}'|_W$", "which we require to be equal to $s'|_W$ if $W \\not \\subset D'$.", "Write", "$$", "\\text{div}_{\\mathcal{L}|_W}(\\sigma) =", "\\sum \\text{ord}_{Z_i, \\mathcal{L}|_W}(\\sigma)[Z_i] = \\sum n_i[Z_i]", "$$", "and similarly", "$$", "\\text{div}_{\\mathcal{L}'|_W}(\\sigma') =", "\\sum \\text{ord}_{Z_i, \\mathcal{L}'|_W}(\\sigma')[Z_i] = \\sum n'_i[Z_i]", "$$", "as in the discussion in Section \\ref{section-key}.", "Then we see that $Z_i \\subset D$ if $n_i \\not = 0$ and", "$Z'_i \\subset D'$ if $n'_i \\not = 0$. For each $i$, let $\\xi_i \\in Z_i$", "be the generic point. As in Section \\ref{section-key} we choose", "for each $i$ an element", "$\\sigma_i \\in \\mathcal{L}_{\\xi_i}$, resp.\\ $\\sigma'_i \\in \\mathcal{L}'_{\\xi_i}$", "which generates over $B_i = \\mathcal{O}_{W, \\xi_i}$", "and which is equal to the image of", "$s$, resp.\\ $s'$ if $Z_i \\not \\subset D$, resp.\\ $Z_i \\not \\subset D'$.", "Write $\\sigma = f_i \\sigma_i$ and $\\sigma' = f'_i\\sigma'_i$ so that", "$n_i = \\text{ord}_{B_i}(f_i)$ and", "$n'_i = \\text{ord}_{B_i}(f'_i)$.", "From our definitions it follows that", "$$", "(j')^*i^*[W] =", "\\sum \\text{ord}_{B_i}(f_i) \\text{div}_{\\mathcal{L}'|_{Z_i}}(\\sigma'_i|_{Z_i})", "$$", "as cycles and", "$$", "j^*(i')^*[W] =", "\\sum \\text{ord}_{B_i}(f'_i) \\text{div}_{\\mathcal{L}|_{Z_i}}(\\sigma_i|_{Z_i})", "$$", "The key formula (Lemma \\ref{lemma-key-formula}) now gives the equality", "$$", "\\sum \\left(", "\\text{ord}_{B_i}(f_i) \\text{div}_{\\mathcal{L}'|_{Z_i}}(\\sigma'_i|_{Z_i}) -", "\\text{ord}_{B_i}(f'_i) \\text{div}_{\\mathcal{L}|_{Z_i}}(\\sigma_i|_{Z_i})", "\\right) =", "\\sum \\text{div}_{Z_i}(\\partial_{B_i}(f_i, f'_i))", "$$", "of cycles. Note that $\\text{div}_{Z_i}(\\partial_{B_i}(f_i, f'_i)) = 0$ if", "$Z_i \\not \\subset D \\cap D'$ because in this case either $f_i = 1$", "or $f'_i = 1$. Thus we get a rational equivalence between our specific", "cycles representing $(j')^*i^*[W]$ and $j^*(i')^*[W]$ on $D \\cap D' \\cap W$.", "By Remark \\ref{remark-infinite-sums-rational-equivalences}", "the result follows for general $\\alpha$." ], "refs": [ "chow-lemma-key-formula", "chow-remark-infinite-sums-rational-equivalences" ], "ref_ids": [ 5712, 5931 ] } ], "ref_ids": [ 5915 ] }, { "id": 5725, "type": "theorem", "label": "chow-lemma-relative-effective-cartier", "categories": [ "chow" ], "title": "chow-lemma-relative-effective-cartier", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $p : X \\to Y$ be a flat morphism of relative dimension $r$.", "Let $i : D \\to X$ be a relative effective Cartier divisor", "(Divisors, Definition", "\\ref{divisors-definition-relative-effective-Cartier-divisor}).", "Let $\\mathcal{L} = \\mathcal{O}_X(D)$.", "For any $\\alpha \\in \\CH_{k + 1}(Y)$ we have", "$$", "i^*p^*\\alpha = (p|_D)^*\\alpha", "$$", "in $\\CH_{k + r}(D)$ and", "$$", "c_1(\\mathcal{L}) \\cap p^*\\alpha = i_* ((p|_D)^*\\alpha)", "$$", "in $\\CH_{k + r}(X)$." ], "refs": [ "divisors-definition-relative-effective-Cartier-divisor" ], "proofs": [ { "contents": [ "Let $W \\subset Y$ be an integral closed subscheme of $\\delta$-dimension", "$k + 1$. By Divisors, Lemma \\ref{divisors-lemma-relative-Cartier}", "we see that $D \\cap p^{-1}W$ is an effective", "Cartier divisor on $p^{-1}W$. By Lemma \\ref{lemma-easy-gysin}", "we get the first equality in", "$$", "i^*[p^{-1}W]_{k + r + 1} =", "[D \\cap p^{-1}W]_{k + r} =", "[(p|_D)^{-1}(W)]_{k + r}.", "$$", "and the second because $D \\cap p^{-1}(W) = (p|_D)^{-1}(W)$ as schemes.", "Since by definition $p^*[W] = [p^{-1}W]_{k + r + 1}$ we see that", "$i^*p^*[W] = (p|_D)^*[W]$ as cycles. If $\\alpha = \\sum m_j[W_j]$ is a", "general $k + 1$ cycle, then we get", "$i^*\\alpha = \\sum m_j i^*p^*[W_j] = \\sum m_j(p|_D)^*[W_j]$ as cycles.", "This proves then first equality. To deduce the second from the", "first apply Lemma \\ref{lemma-support-cap-effective-Cartier}." ], "refs": [ "divisors-lemma-relative-Cartier", "chow-lemma-easy-gysin", "chow-lemma-support-cap-effective-Cartier" ], "ref_ids": [ 7972, 5717, 5716 ] } ], "ref_ids": [ 8095 ] }, { "id": 5726, "type": "theorem", "label": "chow-lemma-pullback-affine-fibres-surjective", "categories": [ "chow" ], "title": "chow-lemma-pullback-affine-fibres-surjective", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $f : X \\to Y$ be a flat morphism of relative dimension $r$.", "Assume that for every $y \\in Y$, there exists an open neighbourhood", "$U \\subset Y$ such that $f|_{f^{-1}(U)} : f^{-1}(U) \\to U$", "is identified with the morphism $U \\times \\mathbf{A}^r \\to U$.", "Then $f^* : \\CH_k(Y) \\to \\CH_{k + r}(X)$ is surjective for all", "$k \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha \\in \\CH_{k + r}(X)$.", "Write $\\alpha = \\sum m_j[W_j]$ with $m_j \\not = 0$ and", "$W_j$ pairwise distinct integral closed subschemes of", "$\\delta$-dimension $k + r$. Then the family $\\{W_j\\}$", "is locally finite in $X$. For any quasi-compact open", "$V \\subset Y$ we see that $f^{-1}(V) \\cap W_j$", "is nonempty only for finitely many $j$. Hence the", "collection $Z_j = \\overline{f(W_j)}$ of closures", "of images is a locally finite collection of integral", "closed subschemes of $Y$.", "\\medskip\\noindent", "Consider the fibre product diagrams", "$$", "\\xymatrix{", "f^{-1}(Z_j) \\ar[r] \\ar[d]_{f_j} & X \\ar[d]^f \\\\", "Z_j \\ar[r] & Y", "}", "$$", "Suppose that $[W_j] \\in Z_{k + r}(f^{-1}(Z_j))$", "is rationally equivalent to $f_j^*\\beta_j$ for some", "$k$-cycle $\\beta_j \\in \\CH_k(Z_j)$. Then", "$\\beta = \\sum m_j \\beta_j$ will be a $k$-cycle on $Y$", "and $f^*\\beta = \\sum m_j f_j^*\\beta_j$ will be rationally", "equivalent to $\\alpha$ (see", "Remark \\ref{remark-infinite-sums-rational-equivalences}).", "This reduces us to the case $Y$ integral, and", "$\\alpha = [W]$ for some integral closed subscheme", "of $X$ dominating $Y$. In particular we may", "assume that $d = \\dim_\\delta(Y) < \\infty$.", "\\medskip\\noindent", "Hence we can use induction on $d = \\dim_\\delta(Y)$.", "If $d < k$, then $\\CH_{k + r}(X) = 0$ and the lemma holds.", "By assumption there exists a dense open $V \\subset Y$ such", "that $f^{-1}(V) \\cong V \\times \\mathbf{A}^r$ as schemes over $V$.", "Suppose that we can show that $\\alpha|_{f^{-1}(V)} = f^*\\beta$", "for some $\\beta \\in Z_k(V)$. By Lemma \\ref{lemma-exact-sequence-open}", "we see that", "$\\beta = \\beta'|_V$ for some $\\beta' \\in Z_k(Y)$.", "By the exact sequence", "$\\CH_k(f^{-1}(Y \\setminus V)) \\to \\CH_k(X) \\to \\CH_k(f^{-1}(V))$", "of Lemma \\ref{lemma-restrict-to-open}", "we see that $\\alpha - f^*\\beta'$ comes from", "a cycle $\\alpha' \\in \\CH_{k + r}(f^{-1}(Y \\setminus V))$.", "Since $\\dim_\\delta(Y \\setminus V) < d$ we win by", "induction on $d$.", "\\medskip\\noindent", "Thus we may assume that $X = Y \\times \\mathbf{A}^r$.", "In this case we can factor $f$ as", "$$", "X = Y \\times \\mathbf{A}^r \\to", "Y \\times \\mathbf{A}^{r - 1} \\to \\ldots \\to", "Y \\times \\mathbf{A}^1 \\to Y.", "$$", "Hence it suffices to do the case $r = 1$. By the argument in the", "second paragraph of the proof we are reduced to the case", "$\\alpha = [W]$, $Y$ integral, and $W \\to Y$ dominant.", "Again we can do induction on $d = \\dim_\\delta(Y)$.", "If $W = Y \\times \\mathbf{A}^1$, then $[W] = f^*[Y]$.", "Lastly, $W \\subset Y \\times \\mathbf{A}^1$ is a proper inclusion,", "then $W \\to Y$ induces a finite field extension $R(Y) \\subset R(W)$.", "Let $P(T) \\in R(Y)[T]$ be the monic irreducible polynomial such", "that the generic fibre of $W \\to Y$ is cut out by $P$ in", "$\\mathbf{A}^1_{R(Y)}$. Let $V \\subset Y$ be a nonempty open such", "that $P \\in \\Gamma(V, \\mathcal{O}_Y)[T]$, and such that", "$W \\cap f^{-1}(V)$ is still cut out by $P$. Then we see that", "$\\alpha|_{f^{-1}(V)} \\sim_{rat} 0$ and hence $\\alpha \\sim_{rat} \\alpha'$", "for some cycle $\\alpha'$ on $(Y \\setminus V) \\times \\mathbf{A}^1$.", "By induction on the dimension we win." ], "refs": [ "chow-remark-infinite-sums-rational-equivalences", "chow-lemma-exact-sequence-open", "chow-lemma-restrict-to-open" ], "ref_ids": [ 5931, 5679, 5690 ] } ], "ref_ids": [] }, { "id": 5727, "type": "theorem", "label": "chow-lemma-linebundle", "categories": [ "chow" ], "title": "chow-lemma-linebundle", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let", "$$", "p :", "L = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{L}))", "\\longrightarrow", "X", "$$", "be the associated vector bundle over $X$.", "Then $p^* : \\CH_k(X) \\to \\CH_{k + 1}(L)$ is an isomorphism for all $k$." ], "refs": [], "proofs": [ { "contents": [ "For surjectivity see Lemma \\ref{lemma-pullback-affine-fibres-surjective}.", "Let $o : X \\to L$ be the zero section of $L \\to X$, i.e., the morphism", "corresponding to the surjection $\\text{Sym}^*(\\mathcal{L}) \\to \\mathcal{O}_X$", "which maps $\\mathcal{L}^{\\otimes n}$ to zero for all $n > 0$.", "Then $p \\circ o = \\text{id}_X$ and $o(X)$ is an effective", "Cartier divisor on $L$. Hence by Lemma \\ref{lemma-relative-effective-cartier}", "we see that $o^* \\circ p^* = \\text{id}$ and we conclude that $p^*$ is", "injective too." ], "refs": [ "chow-lemma-pullback-affine-fibres-surjective", "chow-lemma-relative-effective-cartier" ], "ref_ids": [ 5726, 5725 ] } ], "ref_ids": [] }, { "id": 5728, "type": "theorem", "label": "chow-lemma-linebundle-formulae", "categories": [ "chow" ], "title": "chow-lemma-linebundle-formulae", "contents": [ "In the situation of Lemma \\ref{lemma-linebundle} denote $o : X \\to L$", "the zero section (see proof of the lemma). Then we have", "\\begin{enumerate}", "\\item $o(X)$ is the zero scheme of a regular global section of", "$p^*\\mathcal{L}^{\\otimes -1}$,", "\\item $o_* : \\CH_k(X) \\to \\CH_k(L)$ as $o$ is a closed immersion,", "\\item $o^* : \\CH_{k + 1}(L) \\to \\CH_k(X)$ as $o(X)$", "is an effective Cartier divisor,", "\\item $o^* p^* : \\CH_k(X) \\to \\CH_k(X)$ is the identity map,", "\\item $o_*\\alpha = - p^*(c_1(\\mathcal{L}) \\cap \\alpha)$ for any", "$\\alpha \\in \\CH_k(X)$, and", "\\item $o^* o_* : \\CH_k(X) \\to \\CH_{k - 1}(X)$ is equal to the map", "$\\alpha \\mapsto - c_1(\\mathcal{L}) \\cap \\alpha$.", "\\end{enumerate}" ], "refs": [ "chow-lemma-linebundle" ], "proofs": [ { "contents": [ "Since $p_*\\mathcal{O}_L = \\text{Sym}^*(\\mathcal{L})$ we have", "$p_*(p^*\\mathcal{L}^{\\otimes -1}) =", "\\text{Sym}^*(\\mathcal{L}) \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes -1}$", "by the projection formula", "(Cohomology, Lemma \\ref{cohomology-lemma-projection-formula})", "and the section mentioned in (1) is", "the canonical trivialization $\\mathcal{O}_X \\to", "\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes -1}$.", "We omit the proof that the vanishing locus of", "this section is precisely $o(X)$. This proves (1).", "\\medskip\\noindent", "Parts (2), (3), and (4) we've seen in the course of the proof of", "Lemma \\ref{lemma-linebundle}. Of course (4) is the first", "formula in Lemma \\ref{lemma-relative-effective-cartier}.", "\\medskip\\noindent", "Part (5) follows from the second formula in", "Lemma \\ref{lemma-relative-effective-cartier},", "additivity of capping with $c_1$ (Lemma \\ref{lemma-c1-cap-additive}),", "and the fact that capping with $c_1$ commutes with flat pullback", "(Lemma \\ref{lemma-flat-pullback-cap-c1}).", "\\medskip\\noindent", "Part (6) follows from Lemma \\ref{lemma-gysin-back}", "and the fact that $o^*p^*\\mathcal{L} = \\mathcal{L}$." ], "refs": [ "cohomology-lemma-projection-formula", "chow-lemma-linebundle", "chow-lemma-relative-effective-cartier", "chow-lemma-relative-effective-cartier", "chow-lemma-c1-cap-additive", "chow-lemma-flat-pullback-cap-c1", "chow-lemma-gysin-back" ], "ref_ids": [ 2243, 5727, 5725, 5725, 5705, 5709, 5722 ] } ], "ref_ids": [ 5727 ] }, { "id": 5729, "type": "theorem", "label": "chow-lemma-decompose-section", "categories": [ "chow" ], "title": "chow-lemma-decompose-section", "contents": [ "Let $Y$ be a scheme. Let $\\mathcal{L}_i$, $i = 1, 2$ be invertible", "$\\mathcal{O}_Y$-modules. Let $s$ be a global section of", "$\\mathcal{L}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{L}_2$.", "Denote $i : D \\to X$ the zero scheme of $s$.", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "D_1 \\ar[r]_{i_1} \\ar[d]_{p_1} &", "L \\ar[d]^p &", "D_2 \\ar[l]^{i_2} \\ar[d]^{p_2} \\\\", "D \\ar[r]^i &", "Y &", "D \\ar[l]_i", "}", "$$", "and sections $s_i$ of $p^*\\mathcal{L}_i$ such that", "the following hold:", "\\begin{enumerate}", "\\item $p^*s = s_1 \\otimes s_2$,", "\\item $p$ is of finite type and flat of relative dimension $1$,", "\\item $D_i$ is the zero scheme of $s_i$,", "\\item $D_i \\cong", "\\underline{\\Spec}(\\text{Sym}^*(\\mathcal{L}_{1 - i}^{\\otimes -1})|_D))$", "over $D$ for $i = 1, 2$,", "\\item $p^{-1}D = D_1 \\cup D_2$ (scheme theoretic union),", "\\item $D_1 \\cap D_2$ (scheme theoretic intersection) maps", "isomorphically to $D$, and", "\\item $D_1 \\cap D_2 \\to D_i$", "is the zero section of the line bundle $D_i \\to D$ for $i = 1, 2$.", "\\end{enumerate}", "Moreover, the formation of this diagram and the sections $s_i$", "commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "Let $p : X \\to Y$ be the relative spectrum of the quasi-coherent", "sheaf of $\\mathcal{O}_Y$-algebras", "$$", "\\mathcal{A} =", "\\left(\\bigoplus\\nolimits_{a_1, a_2 \\geq 0}", "\\mathcal{L}_1^{\\otimes -a_1} \\otimes_{\\mathcal{O}_Y}", "\\mathcal{L}_2^{\\otimes -a_2}\\right)/\\mathcal{J}", "$$", "where $\\mathcal{J}$ is the ideal generated by local sections of", "the form $st - t$ for $t$ a local section of any summand", "$\\mathcal{L}_1^{\\otimes -a_1} \\otimes \\mathcal{L}_2^{\\otimes -a_2}$", "with $a_1, a_2 > 0$. The sections $s_i$ viewed as maps", "$p^*\\mathcal{L}_i^{\\otimes -1} \\to \\mathcal{O}_X$ are defined as the adjoints", "of the maps $\\mathcal{L}_i^{\\otimes -1} \\to \\mathcal{A} = p_*\\mathcal{O}_X$.", "For any $y \\in Y$ we can choose an affine", "open $V \\subset Y$, say $V = \\Spec(B)$, containing $y$ and", "trivializations $z_i : \\mathcal{O}_V \\to \\mathcal{L}_i^{\\otimes -1}|_V$.", "Observe that $f = s(z_1z_2) \\in A$ cuts out the closed subscheme $D$.", "Then clearly", "$$", "p^{-1}(V) = \\Spec(B[z_1, z_2]/(z_1 z_2 - f))", "$$", "Since $D_i$ is cut out by $z_i$ everything is clear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5730, "type": "theorem", "label": "chow-lemma-decompose-section-formulae", "categories": [ "chow" ], "title": "chow-lemma-decompose-section-formulae", "contents": [ "In the situation of Lemma \\ref{lemma-decompose-section}", "assume $Y$ is locally of finite type over $(S, \\delta)$ as in", "Situation \\ref{situation-setup}. Then we have", "$i_1^*p^*\\alpha = p_1^*i^*\\alpha$", "in $\\CH_k(D_1)$ for all $\\alpha \\in \\CH_k(Y)$." ], "refs": [], "proofs": [ { "contents": [ "Let $W \\subset Y$ be an integral closed subscheme of $\\delta$-dimension $k$.", "We distinguish two cases.", "\\medskip\\noindent", "Assume $W \\subset D$. Then", "$i^*[W] = c_1(\\mathcal{L}_1) \\cap [W] + c_1(\\mathcal{L}_2) \\cap [W]$", "in $\\CH_{k - 1}(D)$ by our definition of gysin homomorphisms and the", "additivity of Lemma \\ref{lemma-c1-cap-additive}.", "Hence $p_1^*i^*[W] =", "p_1^*(c_1(\\mathcal{L}_1) \\cap [W]) + p_1^*(c_1(\\mathcal{L}_2) \\cap [W])$.", "On the other hand, we have", "$p^*[W] = [p^{-1}(W)]_{k + 1}$ by construction of flat pullback.", "And $p^{-1}(W) = W_1 \\cup W_2$ (scheme theoretically)", "where $W_i = p_i^{-1}(W)$ is a line bundle over $W$", "by the lemma (since formation of the diagram commutes with base change).", "Then $[p^{-1}(W)]_{k + 1} = [W_1] + [W_2]$ as $W_i$ are integral closed", "subschemes of $L$ of $\\delta$-dimension $k + 1$. Hence", "\\begin{align*}", "i_1^*p^*[W]", "& =", "i_1^*[p^{-1}(W)]_{k + 1} \\\\", "& =", "i_1^*([W_1] + [W_2]) \\\\", "& =", "c_1(p_1^*\\mathcal{L}_1) \\cap [W_1] + [W_1 \\cap W_2]_k \\\\", "& =", "c_1(p_1^*\\mathcal{L}_1) \\cap p_1^*[W] + [W_1 \\cap W_2]_k \\\\", "& =", "p_1^*(c_1(\\mathcal{L}_1) \\cap [W]) + [W_1 \\cap W_2]_k", "\\end{align*}", "by construction of gysin homomorphisms, the definition of flat pullback", "(for the second equality), and compatibility of $c_1 \\cap -$", "with flat pullback (Lemma \\ref{lemma-flat-pullback-cap-c1}).", "Since $W_1 \\cap W_2$ is the zero section of the line bundle", "$W_1 \\to W$ we see from Lemma \\ref{lemma-linebundle-formulae}", "that $[W_1 \\cap W_2]_k = p_1^*(c_1(\\mathcal{L}_2) \\cap [W])$.", "Note that here we use the fact that $D_1$ is the line bundle", "which is the relative spectrum of the inverse of $\\mathcal{L}_2$.", "Thus we get the same thing as before.", "\\medskip\\noindent", "Assume $W \\not \\subset D$. In this case, both $i_1^*p^*[W]$", "and $p_1^*i^*[W]$ are represented by the $k - 1$ cycle associated", "to the scheme theoretic inverse image of $W$ in $D_1$." ], "refs": [ "chow-lemma-c1-cap-additive", "chow-lemma-flat-pullback-cap-c1", "chow-lemma-linebundle-formulae" ], "ref_ids": [ 5705, 5709, 5728 ] } ], "ref_ids": [] }, { "id": 5731, "type": "theorem", "label": "chow-lemma-normal-cone-effective-Cartier", "categories": [ "chow" ], "title": "chow-lemma-normal-cone-effective-Cartier", "contents": [ "In Situation \\ref{situation-setup} let $X$ be a scheme locally", "of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$", "be a triple as in Definition \\ref{definition-gysin-homomorphism}.", "There exists a commutative diagram", "$$", "\\xymatrix{", "D' \\ar[r]_{i'} \\ar[d]_p & X' \\ar[d]^g \\\\", "D \\ar[r]^i & X", "}", "$$", "such that", "\\begin{enumerate}", "\\item $p$ and $g$ are of finite type and flat of relative dimension $1$,", "\\item $p^* : \\CH_k(D) \\to \\CH_{k + 1}(D')$ is injective for all $k$,", "\\item $D' \\subset X'$ is the zero scheme of a global section", "$s' \\in \\Gamma(X', \\mathcal{O}_{X'})$,", "\\item $p^*i^* = (i')^*g^*$ as maps $\\CH_k(X) \\to \\CH_k(D')$.", "\\end{enumerate}", "Moreover, these properties remain true after arbitrary base change", "by morphisms $Y \\to X$ which are locally of finite type." ], "refs": [ "chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "Observe that $(i')^*$ is defined because we have the triple", "$(\\mathcal{O}_{X'}, s', i' : D' \\to X')$ as in", "Definition \\ref{definition-gysin-homomorphism}. Thus the statement makes sense.", "\\medskip\\noindent", "Set $\\mathcal{L}_1 = \\mathcal{O}_X$, $\\mathcal{L}_2 = \\mathcal{L}$", "and apply Lemma \\ref{lemma-decompose-section} with the section $s$ of", "$\\mathcal{L} = \\mathcal{L}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{L}_2$.", "Take $D' = D_1$. The results now follow from the lemma, from", "Lemma \\ref{lemma-decompose-section-formulae}", "and injectivity by", "Lemma \\ref{lemma-linebundle}." ], "refs": [ "chow-definition-gysin-homomorphism", "chow-lemma-linebundle" ], "ref_ids": [ 5915, 5727 ] } ], "ref_ids": [ 5915 ] }, { "id": 5732, "type": "theorem", "label": "chow-lemma-flat-pullback-bivariant", "categories": [ "chow" ], "title": "chow-lemma-flat-pullback-bivariant", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ be a flat morphism of relative dimension $r$", "between schemes locally of finite type over $S$.", "Then the rule that to $Y' \\to Y$ assigns", "$(f')^* : \\CH_k(Y') \\to \\CH_{k + r}(X')$ where $X' = X \\times_Y Y'$", "is a bivariant class of degree $-r$." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemmas \\ref{lemma-flat-pullback-rational-equivalence},", "\\ref{lemma-compose-flat-pullback},", "\\ref{lemma-flat-pullback-proper-pushforward}, and", "\\ref{lemma-gysin-flat-pullback}." ], "refs": [ "chow-lemma-flat-pullback-rational-equivalence", "chow-lemma-compose-flat-pullback", "chow-lemma-flat-pullback-proper-pushforward", "chow-lemma-gysin-flat-pullback" ], "ref_ids": [ 5693, 5680, 5682, 5719 ] } ], "ref_ids": [] }, { "id": 5733, "type": "theorem", "label": "chow-lemma-gysin-bivariant", "categories": [ "chow" ], "title": "chow-lemma-gysin-bivariant", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $(\\mathcal{L}, s, i : D \\to X)$ be a triple as in", "Definition \\ref{definition-gysin-homomorphism}.", "Then the rule that to $f : X' \\to X$ assigns", "$(i')^* : \\CH_k(X') \\to \\CH_{k - 1}(D')$ where $D' = D \\times_X X'$", "is a bivariant class of degree $1$." ], "refs": [ "chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-gysin-factors},", "\\ref{lemma-closed-in-X-gysin},", "\\ref{lemma-gysin-flat-pullback}, and", "\\ref{lemma-gysin-commutes-gysin}." ], "refs": [ "chow-lemma-gysin-factors", "chow-lemma-closed-in-X-gysin", "chow-lemma-gysin-flat-pullback", "chow-lemma-gysin-commutes-gysin" ], "ref_ids": [ 5721, 5718, 5719, 5724 ] } ], "ref_ids": [ 5915 ] }, { "id": 5734, "type": "theorem", "label": "chow-lemma-push-proper-bivariant", "categories": [ "chow" ], "title": "chow-lemma-push-proper-bivariant", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of", "schemes locally of finite type over $S$.", "Let $c \\in A^p(X \\to Z)$ and assume $f$ is proper.", "Then the rule that to $Z' \\to Z$ assigns", "$\\alpha \\longmapsto f'_*(c \\cap \\alpha)$", "is a bivariant class denoted $f_* \\circ c \\in A^p(Y \\to Z)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-compose-pushforward},", "\\ref{lemma-flat-pullback-proper-pushforward}, and", "\\ref{lemma-closed-in-X-gysin}." ], "refs": [ "chow-lemma-compose-pushforward", "chow-lemma-flat-pullback-proper-pushforward", "chow-lemma-closed-in-X-gysin" ], "ref_ids": [ 5674, 5682, 5718 ] } ], "ref_ids": [] }, { "id": 5735, "type": "theorem", "label": "chow-lemma-cap-c1-bivariant", "categories": [ "chow" ], "title": "chow-lemma-cap-c1-bivariant", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Then the rule that to $f : X' \\to X$ assigns", "$c_1(f^*\\mathcal{L}) \\cap - : \\CH_k(X') \\to \\CH_{k - 1}(X')$", "is a bivariant class of degree $1$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-factors},", "\\ref{lemma-pushforward-cap-c1},", "\\ref{lemma-flat-pullback-cap-c1}, and", "\\ref{lemma-gysin-commutes-cap-c1}." ], "refs": [ "chow-lemma-factors", "chow-lemma-pushforward-cap-c1", "chow-lemma-flat-pullback-cap-c1", "chow-lemma-gysin-commutes-cap-c1" ], "ref_ids": [ 5714, 5711, 5709, 5723 ] } ], "ref_ids": [] }, { "id": 5736, "type": "theorem", "label": "chow-lemma-c1-center", "categories": [ "chow" ], "title": "chow-lemma-c1-center", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Then", "\\begin{enumerate}", "\\item $c_1(\\mathcal{L}) \\in A^1(X)$ is in the center of $A^*(X)$ and", "\\item if $f : X' \\to X$ is locally of finite type and $c \\in A^*(X' \\to X)$,", "then $c \\circ c_1(\\mathcal{L}) = c_1(f^*\\mathcal{L}) \\circ c$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Of course (2) implies (1).", "Let $p : L \\to X$ be as in Lemma \\ref{lemma-linebundle} and let $o : X \\to L$", "be the zero section. Denote $p' : L' \\to X'$ and $o' : X' \\to L'$", "their base changes. By Lemma \\ref{lemma-linebundle-formulae} we have", "$$", "p^*(c_1(\\mathcal{L}) \\cap \\alpha) = - o_* \\alpha", "\\quad\\text{and}\\quad", "(p')^*(c_1(f^*\\mathcal{L}) \\cap \\alpha') = - o'_* \\alpha'", "$$", "Since $c$ is a bivariant class we have", "\\begin{align*}", "(p')^*(c \\cap c_1(\\mathcal{L}) \\cap \\alpha)", "& =", "c \\cap p^*(c_1(\\mathcal{L}) \\cap \\alpha) \\\\", "& =", "- c \\cap o_* \\alpha \\\\", "& =", "- o'_*(c \\cap \\alpha) \\\\", "& =", "(p')^*(c_1(f^*\\mathcal{L}) \\cap c \\cap \\alpha)", "\\end{align*}", "Since $(p')^*$ is injective by one of the lemmas cited above we obtain", "$c \\cap c_1(\\mathcal{L}) \\cap \\alpha =", "c_1(f^*\\mathcal{L}) \\cap c \\cap \\alpha$.", "The same is true after any base change by $Y \\to X$ locally of finite type", "and hence we have the equality of bivariant classes stated in (2)." ], "refs": [ "chow-lemma-linebundle", "chow-lemma-linebundle-formulae" ], "ref_ids": [ 5727, 5728 ] } ], "ref_ids": [] }, { "id": 5737, "type": "theorem", "label": "chow-lemma-vanish-above-dimension", "categories": [ "chow" ], "title": "chow-lemma-vanish-above-dimension", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a", "finite type scheme over $S$ which has an ample invertible sheaf.", "Assume $d = \\dim(X) < \\infty$ (here we really mean dimension and", "not $\\delta$-dimension).", "Then for any invertible sheaves $\\mathcal{L}_1, \\ldots, \\mathcal{L}_{d + 1}$", "on $X$ we have", "$c_1(\\mathcal{L}_1) \\circ \\ldots \\circ c_1(\\mathcal{L}_{d + 1}) = 0$", "in $A^{d + 1}(X)$." ], "refs": [], "proofs": [ { "contents": [ "We prove this by induction on $d$. The base case $d = 0$ is true because", "in this case $X$ is a finite set of closed points and hence every invertible", "module is trivial. Assume $d > 0$. By Divisors, Lemma", "\\ref{divisors-lemma-quasi-projective-Noetherian-pic-effective-Cartier}", "we can write $\\mathcal{L}_{d + 1} \\cong \\mathcal{O}_X(D) \\otimes", "\\mathcal{O}_X(D')^{\\otimes -1}$ for some effective Cartier divisors", "$D, D' \\subset X$. Then $c_1(\\mathcal{L}_{d + 1})$ is the difference", "of $c_1(\\mathcal{O}_X(D))$ and $c_1(\\mathcal{O}_X(D'))$ and hence", "we may assume $\\mathcal{L}_{d + 1} = \\mathcal{O}_X(D)$ for some", "effective Cartier divisor.", "\\medskip\\noindent", "Denote $i : D \\to X$ the inclusion morphism and denote", "$i^* \\in A^1(D \\to X)$ the bivariant class given by the", "gysin hommomorphism as in Lemma \\ref{lemma-gysin-bivariant}.", "We have $i_* \\circ i^* = c_1(\\mathcal{L}_{d + 1})$", "in $A^1(X)$ by Lemma \\ref{lemma-support-cap-effective-Cartier}", "(and Lemma \\ref{lemma-push-proper-bivariant}", "to make sense of the left hand side).", "Since $c_1(\\mathcal{L}_i)$ commutes with", "both $i_*$ and $i^*$ (by definition of bivariant classes)", "we conclude that", "$$", "c_1(\\mathcal{L}_1) \\circ \\ldots \\circ c_1(\\mathcal{L}_{d + 1}) =", "i_* \\circ c_1(\\mathcal{L}_1) \\circ \\ldots \\circ c_1(\\mathcal{L}_d) \\circ i^* =", "i_* \\circ c_1(\\mathcal{L}_1|_D) \\circ \\ldots \\circ c_1(\\mathcal{L}_d|_D)", "\\circ i^*", "$$", "Thus we conclude by induction on $d$. Namely, we have $\\dim(D) < d$", "as none of the generic points of $X$ are in $D$." ], "refs": [ "divisors-lemma-quasi-projective-Noetherian-pic-effective-Cartier", "chow-lemma-gysin-bivariant", "chow-lemma-support-cap-effective-Cartier", "chow-lemma-push-proper-bivariant" ], "ref_ids": [ 7956, 5733, 5716, 5734 ] } ], "ref_ids": [] }, { "id": 5738, "type": "theorem", "label": "chow-lemma-factors-through-rational-equivalence", "categories": [ "chow" ], "title": "chow-lemma-factors-through-rational-equivalence", "contents": [ "\\begin{reference}", "Very weak form of \\cite[Theorem 17.1]{F}", "\\end{reference}", "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$.", "Let $p \\in \\mathbf{Z}$. Suppose given a rule", "which assigns to every locally of finite type morphism $Y' \\to Y$", "and every $k$ a map", "$$", "c \\cap - : Z_k(Y') \\longrightarrow \\CH_{k - p}(X')", "$$", "where $Y' = X' \\times_X Y$, satisfying condition (3) of", "Definition \\ref{definition-bivariant-class}", "whenever $\\mathcal{L}'|_{D'} \\cong \\mathcal{O}_{D'}$. Then", "$c \\cap -$ factors through rational equivalence." ], "refs": [ "chow-definition-bivariant-class" ], "proofs": [ { "contents": [ "The statement makes sense because given a triple", "$(\\mathcal{L}, s, i : D \\to X)$ as in", "Definition \\ref{definition-gysin-homomorphism}", "such that $\\mathcal{L}|_D \\cong \\mathcal{O}_D$, then", "the operation $i^*$ is defined on the level of cycles, see", "Remark \\ref{remark-gysin-on-cycles}.", "Let $\\alpha \\in Z_k(X')$ be a cycle which is rationally equivalent to zero.", "We have to show that $c \\cap \\alpha = 0$. By", "Lemma \\ref{lemma-rational-equivalence-family}", "there exists a cycle $\\beta \\in Z_{k + 1}(X' \\times \\mathbf{P}^1)$", "such that $\\alpha = i_0^*\\beta - i_\\infty^*\\beta$", "where $i_0, i_\\infty : X' \\to X' \\times \\mathbf{P}^1$ are the", "closed immersions of $X'$ over $0, \\infty$. Since these are", "examples of effective Cartier divisors with trivial normal", "bundles, we see that $c \\cap i_0^*\\beta = j_0^*(c \\cap \\beta)$", "and $c \\cap i_\\infty^*\\beta = j_\\infty^*(c \\cap \\beta)$", "where $j_0, j_\\infty : Y' \\to Y' \\times \\mathbf{P}^1$ are", "closed immersions as before. Since", "$j_0^*(c \\cap \\beta) \\sim_{rat} j_\\infty^*(c \\cap \\beta)$", "(follows from Lemma \\ref{lemma-rational-equivalence-family}) we conclude." ], "refs": [ "chow-definition-gysin-homomorphism", "chow-remark-gysin-on-cycles", "chow-lemma-rational-equivalence-family", "chow-lemma-rational-equivalence-family" ], "ref_ids": [ 5915, 5935, 5695, 5695 ] } ], "ref_ids": [ 5916 ] }, { "id": 5739, "type": "theorem", "label": "chow-lemma-bivariant-weaker", "categories": [ "chow" ], "title": "chow-lemma-bivariant-weaker", "contents": [ "\\begin{reference}", "Weak form of \\cite[Theorem 17.1]{F}", "\\end{reference}", "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$.", "Let $p \\in \\mathbf{Z}$. Suppose given a rule", "which assigns to every locally of finite type morphism $Y' \\to Y$", "and every $k$ a map", "$$", "c \\cap - : \\CH_k(Y') \\longrightarrow \\CH_{k - p}(X')", "$$", "where $Y' = X' \\times_X Y$, satisfying conditions (1), (2) of", "Definition \\ref{definition-bivariant-class}", "and condition (3) whenever $\\mathcal{L}'|_{D'} \\cong \\mathcal{O}_{D'}$. Then", "$c \\cap -$ is a bivariant class." ], "refs": [ "chow-definition-bivariant-class" ], "proofs": [ { "contents": [ "Let $Y' \\to Y$ be a morphism of schemes which is locally of finite type.", "Let $(\\mathcal{L}', s', i' : D' \\to Y')$ be as in", "Definition \\ref{definition-gysin-homomorphism}", "with pullback $(\\mathcal{N}', t', j' : E' \\to X')$ to $X'$.", "We have to show that $c \\cap (i')^*\\alpha' = (j')^*(c \\cap \\alpha')$", "for all $\\alpha' \\in \\CH_k(Y')$.", "\\medskip\\noindent", "Denote $g : Y'' \\to Y'$ the smooth morphism of relative", "dimension $1$ with $i'' : D'' \\to Y''$ and $p : D'' \\to D'$", "constructed in Lemma \\ref{lemma-normal-cone-effective-Cartier}.", "(Warning: $D''$ isn't the full inverse image of $D'$.)", "Denote $f : X'' \\to X'$ and $E'' \\subset X''$", "their base changes by $X' \\to Y'$. Picture", "$$", "\\xymatrix{", "& X'' \\ar[rr] \\ar'[d][dd]_h & & Y'' \\ar[dd]^g \\\\", "E'' \\ar[rr] \\ar[dd]_q \\ar[ru]^{j''} & & D'' \\ar[dd]^p \\ar[ru]^{i''} & \\\\", "& X' \\ar'[r][rr] & & Y' \\\\", "E' \\ar[rr] \\ar[ru]^{j'} & & D' \\ar[ru]^{i'}", "}", "$$", "By the properties given in the lemma we know that $\\beta' = (i')^*\\alpha'$", "is the unique element of $\\CH_{k - 1}(D')$ such that", "$p^*\\beta' = (i'')^*g^*\\alpha'$. Similarly, we know that", "$\\gamma' = (j')^*(c \\cap \\alpha')$ is the unique element of", "$\\CH_{k - 1 - p}(E')$ such that $q^*\\gamma' = (j'')^*h^*(c \\cap \\alpha')$.", "Since we know that", "$$", "(j'')^*h^*(c \\cap \\alpha') =", "(j'')^*(c \\cap g^*\\alpha') =", "c \\cap (i'')^*g^*\\alpha'", "$$", "by our assuptions on $c$; note that the modified version of (3)", "assumed in the statement of the lemma applies to $i''$", "and its base change $j''$. We similarly know that", "$$", "q^*(c \\cap \\beta') = c \\cap p^*\\beta'", "$$", "We conclude that $\\gamma' = c \\cap \\beta'$ by the uniqueness pointed", "out above." ], "refs": [ "chow-definition-gysin-homomorphism", "chow-lemma-normal-cone-effective-Cartier" ], "ref_ids": [ 5915, 5731 ] } ], "ref_ids": [ 5916 ] }, { "id": 5740, "type": "theorem", "label": "chow-lemma-bivariant-zero", "categories": [ "chow" ], "title": "chow-lemma-bivariant-zero", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$.", "Let $c \\in A^p(X \\to Y)$. For $Y'' \\to Y' \\to Y$ set", "$X'' = Y'' \\times_Y X$ and $X' = Y' \\times_Y X$.", "The following are equivalent", "\\begin{enumerate}", "\\item $c$ is zero,", "\\item $c \\cap [Y'] = 0$ in $\\CH_*(X')$ for every integral scheme $Y'$", "locally of finite type over $Y$, and", "\\item for every integral scheme $Y'$ locally of finite type over $Y$,", "there exists a proper birational morphism $Y'' \\to Y'$ such that", "$c \\cap [Y''] = 0$ in $\\CH_*(X'')$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implications (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3) are clear.", "Assumption (3) implies (2) because $(Y'' \\to Y')_*[Y''] = [Y']$", "and hence $c \\cap [Y'] = (X'' \\to X')_*(c \\cap [Y''])$ as $c$", "is a bivariant class. Assume (2).", "Let $Y' \\to Y$ be locally of finite type. Let $\\alpha \\in \\CH_k(Y')$.", "Write $\\alpha = \\sum n_i [Y'_i]$ with $Y'_i \\subset Y'$ a locally finite", "collection of integral closed subschemes of $\\delta$-dimension $k$.", "Then we see that $\\alpha$ is pushforward of the cycle", "$\\alpha' = \\sum n_i[Y'_i]$ on $Y'' = \\coprod Y'_i$ under the", "proper morphism $Y'' \\to Y'$. By the properties of bivariant", "classes it suffices to prove that $c \\cap \\alpha' = 0$ in $\\CH_{k - p}(X'')$.", "We have $\\CH_{k - p}(X'') = \\prod \\CH_{k - p}(X'_i)$ where", "$X'_i = Y'_i \\times_Y X$. This follows immediately", "from the definitions. The projection maps", "$\\CH_{k - p}(X'') \\to \\CH_{k - p}(X'_i)$ are given by flat pullback.", "Since capping with $c$ commutes with", "flat pullback, we see that it suffices to show that $c \\cap [Y'_i]$", "is zero in $\\CH_{k - p}(X'_i)$ which is true by assumption." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5741, "type": "theorem", "label": "chow-lemma-disjoint-decomposition-bivariant", "categories": [ "chow" ], "title": "chow-lemma-disjoint-decomposition-bivariant", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$.", "Assume we have disjoint union decompositions", "$X = \\coprod_{i \\in I} X_i$ and $Y = \\coprod_{j \\in J} Y_j$", "by open and closed subschemes", "and a map $a : I \\to J$ of sets such that $f(X_i) \\subset Y_{a(i)}$.", "Then", "$$", "A^p(X \\to Y) = \\prod\\nolimits_{i \\in I} A^p(X_i \\to Y_{a(i)})", "$$" ], "refs": [], "proofs": [ { "contents": [ "Suppose given an element $(c_i) \\in \\prod_i A^p(X_i \\to Y_{a(i)})$.", "Then given $\\beta \\in \\CH_k(Y)$ we can map this to the element of", "$\\CH_{k - p}(X)$ whose restriction to $X_i$ is $c_i \\cap \\beta|_{Y_{a(i)}}$.", "This works because $\\CH_{k - p}(X) = \\prod_i \\CH_{k - p}(X_i)$.", "The same construction works after base change by any $Y' \\to Y$", "locally of finite type and we get $c \\in A^p(X \\to Y)$.", "Thus we obtain a map $\\Psi$ from the right hand side of the formula", "to the left hand side of the formula.", "Conversely, given $c \\in A^p(X \\to Y)$ and an element", "$\\beta_i \\in \\CH_k(Y_{a(i)})$ we can consider the element", "$(c \\cap (Y_{a(i)} \\to Y)_*\\beta_i)|_{X_i}$ in $\\CH_{k - p}(X_i)$.", "The same thing works after base change by any $Y' \\to Y$", "locally of finite type and we get $c_i \\in A^p(X_i \\to Y_{a(i)})$.", "Thus we obtain a map $\\Phi$ from the left hand", "side of the formula to the right hand side of the formula.", "It is immediate that $\\Phi \\circ \\Psi = \\text{id}$.", "For the converse, suppose that $c \\in A^p(X \\to Y)$ and", "$\\beta \\in \\CH_k(Y)$. Say $\\Phi(c) = (c_i)$. Let $j \\in J$.", "Because $c$ commutes with flat pullback we get", "$$", "(c \\cap \\beta)|_{\\coprod_{a(i) = j} X_i} =", "c \\cap \\beta|_{Y_j}", "$$", "Because $c$ commutes with proper pushforward we get", "$$", "(\\coprod\\nolimits_{a(i) = j} X_i \\to X)_*", "((c \\cap \\beta)|_{\\coprod_{a(i) = j} X_i})", "=", "c \\cap (Y_j \\to Y)_*\\beta|_{Y_j}", "$$", "The left hand side is the cycle on $X$ restricting to $(c \\cap \\beta)|_{X_i}$", "on $X_i$ for $i \\in I$ with $a(i) = j$ and $0$ else.", "The right hand side is a cycle on $X$ whose restriction to $X_i$", "is $c_i \\cap \\beta|_{Y_j}$ for $i \\in I$ with $a(i) = j$.", "Thus $c \\cap \\beta = \\Psi((c_i))$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5742, "type": "theorem", "label": "chow-lemma-cap-projective-bundle", "categories": [ "chow" ], "title": "chow-lemma-cap-projective-bundle", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module", "$\\mathcal{E}$ of rank $r$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$", "be the projective bundle associated to $\\mathcal{E}$.", "For any $\\alpha \\in \\CH_k(X)$ the element", "$$", "\\pi_*\\left(", "c_1(\\mathcal{O}_P(1))^s \\cap \\pi^*\\alpha", "\\right)", "\\in", "\\CH_{k + r - 1 - s}(X)", "$$", "is $0$ if $s < r - 1$ and is equal to $\\alpha$ when $s = r - 1$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset X$ be an integral closed subscheme of $\\delta$-dimension $k$.", "Note that $\\pi^*[Z] = [\\pi^{-1}(Z)]$ as $\\pi^{-1}(Z)$ is integral of", "$\\delta$-dimension $r - 1$.", "If $s < r - 1$, then by construction", "$c_1(\\mathcal{O}_P(1))^s \\cap \\pi^*[Z]$", "is represented by a $(k + r - 1 - s)$-cycle supported on", "$\\pi^{-1}(Z)$. Hence the pushforward of this cycle", "is zero for dimension reasons.", "\\medskip\\noindent", "Let $s = r - 1$. By the argument given above we see that", "$\\pi_*(c_1(\\mathcal{O}_P(1))^s \\cap \\pi^*\\alpha) = n [Z]$", "for some $n \\in \\mathbf{Z}$. We want to show that $n = 1$.", "For the same dimension reasons", "as above it suffices to prove this result after replacing $X$ by", "$X \\setminus T$ where $T \\subset Z$ is a proper closed subset.", "Let $\\xi$ be the generic point of $Z$.", "We can choose elements $e_1, \\ldots, e_{r - 1} \\in \\mathcal{E}_\\xi$", "which form part of a basis of $\\mathcal{E}_\\xi$.", "These give rational sections $s_1, \\ldots, s_{r - 1}$", "of $\\mathcal{O}_P(1)|_{\\pi^{-1}(Z)}$ whose common zero set", "is the closure of the image a rational section of", "$\\mathbf{P}(\\mathcal{E}|_Z) \\to Z$ union a closed subset whose", "support maps to a proper closed subset $T$ of $Z$.", "After removing $T$ from $X$ (and correspondingly $\\pi^{-1}(T)$", "from $P$), we see that $s_1, \\ldots, s_n$ form a sequence", "of global sections", "$s_i \\in \\Gamma(\\pi^{-1}(Z), \\mathcal{O}_{\\pi^{-1}(Z)}(1))$", "whose common zero set is the image of a section $Z \\to \\pi^{-1}(Z)$.", "Hence we see successively that", "\\begin{eqnarray*}", "\\pi^*[Z] & = & [\\pi^{-1}(Z)] \\\\", "c_1(\\mathcal{O}_P(1)) \\cap \\pi^*[Z] & = & [Z(s_1)] \\\\", "c_1(\\mathcal{O}_P(1))^2 \\cap \\pi^*[Z] & = & [Z(s_1) \\cap Z(s_2)] \\\\", "\\ldots & = & \\ldots \\\\", "c_1(\\mathcal{O}_P(1))^{r - 1} \\cap \\pi^*[Z] & = &", "[Z(s_1) \\cap \\ldots \\cap Z(s_{r - 1})]", "\\end{eqnarray*}", "by repeated applications of Lemma \\ref{lemma-geometric-cap}.", "Since the pushforward by $\\pi$ of the image of a", "section of $\\pi$ over $Z$ is clearly $[Z]$ we see the result", "when $\\alpha = [Z]$. We omit the verification that these", "arguments imply the result for a general cycle $\\alpha = \\sum n_j [Z_j]$." ], "refs": [ "chow-lemma-geometric-cap" ], "ref_ids": [ 5707 ] } ], "ref_ids": [] }, { "id": 5743, "type": "theorem", "label": "chow-lemma-chow-ring-projective-bundle", "categories": [ "chow" ], "title": "chow-lemma-chow-ring-projective-bundle", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module", "$\\mathcal{E}$ of rank $r$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$", "be the projective bundle associated to $\\mathcal{E}$.", "The map", "$$", "\\bigoplus\\nolimits_{i = 0}^{r - 1}", "\\CH_{k + i}(X)", "\\longrightarrow", "\\CH_{k + r - 1}(P),", "$$", "$$", "(\\alpha_0, \\ldots, \\alpha_{r-1})", "\\longmapsto", "\\pi^*\\alpha_0 +", "c_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\alpha_1", "+ \\ldots +", "c_1(\\mathcal{O}_P(1))^{r - 1} \\cap \\pi^*\\alpha_{r-1}", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Fix $k \\in \\mathbf{Z}$. We first show the map is injective.", "Suppose that $(\\alpha_0, \\ldots, \\alpha_{r - 1})$ is an element", "of the left hand side that maps to zero.", "By Lemma \\ref{lemma-cap-projective-bundle} we see that", "$$", "0 = \\pi_*(\\pi^*\\alpha_0 +", "c_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\alpha_1", "+ \\ldots +", "c_1(\\mathcal{O}_P(1))^{r - 1} \\cap \\pi^*\\alpha_{r-1})", "= \\alpha_{r - 1}", "$$", "Next, we see that", "$$", "0 = \\pi_*(c_1(\\mathcal{O}_P(1)) \\cap (\\pi^*\\alpha_0 +", "c_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\alpha_1", "+ \\ldots +", "c_1(\\mathcal{O}_P(1))^{r - 2} \\cap \\pi^*\\alpha_{r - 2}))", "= \\alpha_{r - 2}", "$$", "and so on. Hence the map is injective.", "\\medskip\\noindent", "It remains to show the map is surjective.", "Let $X_i$, $i \\in I$ be the irreducible components of $X$.", "Then $P_i = \\mathbf{P}(\\mathcal{E}|_{X_i})$, $i \\in I$", "are the irreducible components of $P$. Consider the commutative", "diagram", "$$", "\\xymatrix{", "\\coprod P_i \\ar[d]_{\\coprod \\pi_i} \\ar[r]_p & P \\ar[d]^\\pi \\\\", "\\coprod X_i \\ar[r]^q & X", "}", "$$", "Observe that $p_*$ is surjective. If $\\beta \\in \\CH_k(\\coprod X_i)$", "then $\\pi^* q_* \\beta = p_*(\\coprod \\pi_i)^* \\beta$, see", "Lemma \\ref{lemma-flat-pullback-proper-pushforward}. Similarly for", "capping with $c_1(\\mathcal{O}(1))$ by", "Lemma \\ref{lemma-pushforward-cap-c1}.", "Hence, if the map of the lemma is surjective for each", "of the morphisms $\\pi_i : P_i \\to X_i$, then the map is", "surjective for $\\pi : P \\to X$. Hence we may assume $X$ is irreducible.", "Thus $\\dim_\\delta(X) < \\infty$ and in particular we may use", "induction on $\\dim_\\delta(X)$.", "\\medskip\\noindent", "The result is clear if $\\dim_\\delta(X) < k$.", "Let $\\alpha \\in \\CH_{k + r - 1}(P)$.", "For any locally closed subscheme $T \\subset X$ denote", "$\\gamma_T : \\bigoplus \\CH_{k + i}(T) \\to \\CH_{k + r - 1}(\\pi^{-1}(T))$", "the map", "$$", "\\gamma_T(\\alpha_0, \\ldots, \\alpha_{r - 1})", "= \\pi^*\\alpha_0 + \\ldots +", "c_1(\\mathcal{O}_{\\pi^{-1}(T)}(1))^{r - 1} \\cap \\pi^*\\alpha_{r - 1}.", "$$", "Suppose for some nonempty open $U \\subset X$ we have", "$\\alpha|_{\\pi^{-1}(U)} = \\gamma_U(\\alpha_0, \\ldots, \\alpha_{r - 1})$.", "Then we may choose lifts $\\alpha'_i \\in \\CH_{k + i}(X)$ and we", "see that $\\alpha - \\gamma_X(\\alpha'_0, \\ldots, \\alpha'_{r - 1})$", "is by Lemma \\ref{lemma-restrict-to-open}", "rationally equivalent to a $k$-cycle on $P_Y = \\mathbf{P}(\\mathcal{E}|_Y)$", "where $Y = X \\setminus U$ as a reduced closed subscheme.", "Note that $\\dim_\\delta(Y) < \\dim_\\delta(X)$.", "By induction the result holds", "for $P_Y \\to Y$ and hence the result holds for $\\alpha$.", "Hence we may replace $X$ by any nonempty open of $X$.", "\\medskip\\noindent", "In particular we may assume that $\\mathcal{E} \\cong \\mathcal{O}_X^{\\oplus r}$.", "In this case $\\mathbf{P}(\\mathcal{E}) = X \\times \\mathbf{P}^{r - 1}$.", "Let us use the stratification", "$$", "\\mathbf{P}^{r - 1} = \\mathbf{A}^{r - 1}", "\\amalg \\mathbf{A}^{r - 2}", "\\amalg \\ldots", "\\amalg \\mathbf{A}^0", "$$", "The closure of each stratum is a $\\mathbf{P}^{r - 1 - i}$ which is a", "representative of $c_1(\\mathcal{O}(1))^i \\cap [\\mathbf{P}^{r - 1}]$.", "Hence $P$ has a similar stratification", "$$", "P = U^{r - 1} \\amalg U^{r - 2} \\amalg \\ldots \\amalg U^0", "$$", "Let $P^i$ be the closure of $U^i$. Let $\\pi^i : P^i \\to X$", "be the restriction of $\\pi$ to $P^i$.", "Let $\\alpha \\in \\CH_{k + r - 1}(P)$. By", "Lemma \\ref{lemma-pullback-affine-fibres-surjective}", "we can write $\\alpha|_{U^{r - 1}} = \\pi^*\\alpha_0|_{U^{r - 1}}$", "for some $\\alpha_0 \\in \\CH_k(X)$. Hence the difference", "$\\alpha - \\pi^*\\alpha_0$ is the image of some", "$\\alpha' \\in \\CH_{k + r - 1}(P^{r - 2})$.", "By Lemma \\ref{lemma-pullback-affine-fibres-surjective}", "again we can write", "$\\alpha'|_{U^{r - 2}} = (\\pi^{r - 2})^*\\alpha_1|_{U^{r - 2}}$", "for some $\\alpha_1 \\in \\CH_{k + 1}(X)$.", "By Lemma \\ref{lemma-relative-effective-cartier}", "we see that the image of $(\\pi^{r - 2})^*\\alpha_1$", "represents $c_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\alpha_1$.", "We also see that", "$\\alpha - \\pi^*\\alpha_0 - c_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\alpha_1$", "is the image of some $\\alpha'' \\in \\CH_{k + r - 1}(P^{r - 3})$.", "And so on." ], "refs": [ "chow-lemma-cap-projective-bundle", "chow-lemma-flat-pullback-proper-pushforward", "chow-lemma-pushforward-cap-c1", "chow-lemma-restrict-to-open", "chow-lemma-pullback-affine-fibres-surjective", "chow-lemma-pullback-affine-fibres-surjective", "chow-lemma-relative-effective-cartier" ], "ref_ids": [ 5742, 5682, 5711, 5690, 5726, 5726, 5725 ] } ], "ref_ids": [] }, { "id": 5744, "type": "theorem", "label": "chow-lemma-vectorbundle", "categories": [ "chow" ], "title": "chow-lemma-vectorbundle", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.", "Let", "$$", "p :", "E = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{E}))", "\\longrightarrow", "X", "$$", "be the associated vector bundle over $X$.", "Then $p^* : \\CH_k(X) \\to \\CH_{k + r}(E)$ is an isomorphism for all $k$." ], "refs": [], "proofs": [ { "contents": [ "(For the case of linebundles, see Lemma \\ref{lemma-linebundle}.)", "For surjectivity see Lemma \\ref{lemma-pullback-affine-fibres-surjective}.", "Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$", "be the projective space bundle associated", "to the finite locally free sheaf $\\mathcal{E} \\oplus \\mathcal{O}_X$.", "Let $s \\in \\Gamma(P, \\mathcal{O}_P(1))$ correspond to the global", "section $(0, 1) \\in \\Gamma(X, \\mathcal{E} \\oplus \\mathcal{O}_X)$.", "Let $D = Z(s) \\subset P$. Note that", "$(\\pi|_D : D \\to X , \\mathcal{O}_P(1)|_D)$", "is the projective space bundle associated", "to $\\mathcal{E}$. We denote $\\pi_D = \\pi|_D$ and", "$\\mathcal{O}_D(1) = \\mathcal{O}_P(1)|_D$.", "Moreover, $D$ is an effective", "Cartier divisor on $P$. Hence $\\mathcal{O}_P(D) = \\mathcal{O}_P(1)$", "(see Divisors, Lemma \\ref{divisors-lemma-characterize-OD}).", "Also there is an isomorphism", "$E \\cong P \\setminus D$. Denote $j : E \\to P$ the", "corresponding open immersion.", "For injectivity we use that the kernel of", "$$", "j^* :", "\\CH_{k + r}(P)", "\\longrightarrow", "\\CH_{k + r}(E)", "$$", "are the cycles supported in the effective Cartier divisor $D$,", "see Lemma \\ref{lemma-restrict-to-open}. So if $p^*\\alpha = 0$, then", "$\\pi^*\\alpha = i_*\\beta$ for some $\\beta \\in \\CH_{k + r}(D)$.", "By Lemma \\ref{lemma-chow-ring-projective-bundle} we may write", "$$", "\\beta = \\pi_D^*\\beta_0 +", "\\ldots + c_1(\\mathcal{O}_D(1))^{r - 1} \\cap \\pi_D^* \\beta_{r - 1}.", "$$", "for some $\\beta_i \\in \\CH_{k + i}(X)$.", "By Lemmas \\ref{lemma-relative-effective-cartier}", "and \\ref{lemma-pushforward-cap-c1}", "this implies", "$$", "\\pi^*\\alpha = i_*\\beta =", "c_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\beta_0 +", "\\ldots +", "c_1(\\mathcal{O}_D(1))^r \\cap \\pi^*\\beta_{r - 1}.", "$$", "Since the rank of $\\mathcal{E} \\oplus \\mathcal{O}_X$ is $r + 1$", "this contradicts Lemma \\ref{lemma-pushforward-cap-c1} unless all", "$\\alpha$ and all $\\beta_i$ are zero." ], "refs": [ "chow-lemma-linebundle", "chow-lemma-pullback-affine-fibres-surjective", "divisors-lemma-characterize-OD", "chow-lemma-restrict-to-open", "chow-lemma-chow-ring-projective-bundle", "chow-lemma-relative-effective-cartier", "chow-lemma-pushforward-cap-c1", "chow-lemma-pushforward-cap-c1" ], "ref_ids": [ 5727, 5726, 7944, 5690, 5743, 5725, 5711, 5711 ] } ], "ref_ids": [] }, { "id": 5745, "type": "theorem", "label": "chow-lemma-first-chern-class", "categories": [ "chow" ], "title": "chow-lemma-first-chern-class", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Assume $X$ is integral and $n = \\dim_\\delta(X)$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "The first Chern class of $\\mathcal{L}$ on $X$ of", "Definition \\ref{definition-chern-classes}", "is equal to the Weil divisor associated to $\\mathcal{L}$", "by Definition \\ref{definition-divisor-invertible-sheaf}." ], "refs": [ "chow-definition-chern-classes", "chow-definition-divisor-invertible-sheaf" ], "proofs": [ { "contents": [ "In this proof we use $c_1(\\mathcal{L}) \\cap [X]$ to denote the", "construction of Definition \\ref{definition-divisor-invertible-sheaf}.", "Since $\\mathcal{L}$ has rank $1$ we have", "$\\mathbf{P}(\\mathcal{L}) = X$ and", "$\\mathcal{O}_{\\mathbf{P}(\\mathcal{L})}(1) = \\mathcal{L}$", "by our normalizations. Hence (\\ref{equation-chern-classes})", "reads", "$$", "(-1)^1 c_1(\\mathcal{L}) \\cap c_0 + (-1)^0 c_1 = 0", "$$", "Since $c_0 = [X]$, we conclude $c_1 = c_1(\\mathcal{L}) \\cap [X]$", "as desired." ], "refs": [ "chow-definition-divisor-invertible-sheaf" ], "ref_ids": [ 5913 ] } ], "ref_ids": [ 5919, 5913 ] }, { "id": 5746, "type": "theorem", "label": "chow-lemma-determine-intersections", "categories": [ "chow" ], "title": "chow-lemma-determine-intersections", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.", "Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$ be the projective bundle", "associated to $\\mathcal{E}$.", "For $\\alpha \\in Z_k(X)$ the elements", "$c_j(\\mathcal{E}) \\cap \\alpha$ are the unique elements", "$\\alpha_j$ of $\\CH_{k - j}(X)$", "such that $\\alpha_0 = \\alpha$ and", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_P(1))^i \\cap", "\\pi^*(\\alpha_{r - i}) = 0", "$$", "holds in the Chow group of $P$." ], "refs": [], "proofs": [ { "contents": [ "The uniqueness of $\\alpha_0, \\ldots, \\alpha_r$ such that", "$\\alpha_0 = \\alpha$ and such that", "the displayed equation holds follows from", "the projective space bundle formula", "Lemma \\ref{lemma-chow-ring-projective-bundle}.", "The identity holds by definition for $\\alpha = [W]$ where $W$", "is an integral closed subscheme of $X$.", "For a general $k$-cycle $\\alpha$ on $X$ write", "$\\alpha = \\sum n_a[W_a]$ with $n_a \\not = 0$, and", "$i_a : W_a \\to X$ pairwise distinct integral closed subschemes.", "Then the family $\\{W_a\\}$ is locally finite on $X$.", "Set $P_a = \\pi^{-1}(W_a) = \\mathbf{P}(\\mathcal{E}|_{W_a})$.", "Denote $i'_a : P_a \\to P$ the corresponding closed immersions.", "Consider the fibre product diagram", "$$", "\\xymatrix{", "P' \\ar@{=}[r] \\ar[d]_{\\pi'} &", "\\coprod P_a \\ar[d]_{\\coprod \\pi_a} \\ar[r]_{\\coprod i'_a} &", "P \\ar[d]^\\pi \\\\", "X' \\ar@{=}[r] &", "\\coprod W_a \\ar[r]^{\\coprod i_a} &", "X", "}", "$$", "The morphism $p : X' \\to X$ is proper. Moreover", "$\\pi' : P' \\to X'$ together with the invertible sheaf", "$\\mathcal{O}_{P'}(1) = \\coprod \\mathcal{O}_{P_a}(1)$", "which is also the pullback of $\\mathcal{O}_P(1)$", "is the projective bundle associated to", "$\\mathcal{E}' = p^*\\mathcal{E}$. By definition", "$$", "c_j(\\mathcal{E}) \\cap [\\alpha]", "=", "\\sum i_{a, *}(c_j(\\mathcal{E}|_{W_a}) \\cap [W_a]).", "$$", "Write $\\beta_{a, j} = c_j(\\mathcal{E}|_{W_a}) \\cap [W_a]$", "which is an element of $\\CH_{k - j}(W_a)$. We have", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_{P_a}(1))^i \\cap \\pi_a^*(\\beta_{a, r - i}) = 0", "$$", "for each $a$ by definition. Thus clearly we have", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_{P'}(1))^i \\cap (\\pi')^*(\\beta_{r - i}) = 0", "$$", "with $\\beta_j = \\sum n_a\\beta_{a, j} \\in \\CH_{k - j}(X')$. Denote", "$p' : P' \\to P$ the morphism $\\coprod i'_a$.", "We have $\\pi^*p_*\\beta_j = p'_*(\\pi')^*\\beta_j$", "by Lemma \\ref{lemma-flat-pullback-proper-pushforward}.", "By the projection formula of Lemma \\ref{lemma-pushforward-cap-c1}", "we conclude that", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_P(1))^i \\cap \\pi^*(p_*\\beta_j) = 0", "$$", "Since $p_*\\beta_j$ is a representative of $c_j(\\mathcal{E}) \\cap \\alpha$", "we win." ], "refs": [ "chow-lemma-chow-ring-projective-bundle", "chow-lemma-flat-pullback-proper-pushforward", "chow-lemma-pushforward-cap-c1" ], "ref_ids": [ 5743, 5682, 5711 ] } ], "ref_ids": [] }, { "id": 5747, "type": "theorem", "label": "chow-lemma-cap-chern-class-factors-rational-equivalence", "categories": [ "chow" ], "title": "chow-lemma-cap-chern-class-factors-rational-equivalence", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.", "If $\\alpha \\sim_{rat} \\beta$ are rationally equivalent $k$-cycles", "on $X$ then $c_j(\\mathcal{E}) \\cap \\alpha = c_j(\\mathcal{E}) \\cap \\beta$", "in $\\CH_{k - j}(X)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-determine-intersections} the elements", "$\\alpha_j = c_j(\\mathcal{E}) \\cap \\alpha$, $j \\geq 1$ and", "$\\beta_j = c_j(\\mathcal{E}) \\cap \\beta$, $j \\geq 1$ are uniquely determined", "by the {\\it same} equation in the chow group of the projective", "bundle associated to $\\mathcal{E}$. (This of course relies on the fact that", "flat pullback is compatible with rational equivalence, see", "Lemma \\ref{lemma-flat-pullback-rational-equivalence}.) Hence they are equal." ], "refs": [ "chow-lemma-determine-intersections", "chow-lemma-flat-pullback-rational-equivalence" ], "ref_ids": [ 5746, 5693 ] } ], "ref_ids": [] }, { "id": 5748, "type": "theorem", "label": "chow-lemma-pushforward-cap-cj", "categories": [ "chow" ], "title": "chow-lemma-pushforward-cap-cj", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.", "Let $p : X \\to Y$ be a proper morphism.", "Let $\\alpha$ be a $k$-cycle on $X$.", "Let $\\mathcal{E}$ be a finite locally free sheaf on $Y$.", "Then", "$$", "p_*(c_j(p^*\\mathcal{E}) \\cap \\alpha) = c_j(\\mathcal{E}) \\cap p_*\\alpha", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $(\\pi : P \\to Y, \\mathcal{O}_P(1))$ be the projective bundle associated", "to $\\mathcal{E}$. Then $P_X = X \\times_Y P$ is the projective bundle associated", "to $p^*\\mathcal{E}$ and $\\mathcal{O}_{P_X}(1)$ is the pullback of", "$\\mathcal{O}_P(1)$. Write $\\alpha_j = c_j(p^*\\mathcal{E}) \\cap \\alpha$, so", "$\\alpha_0 = \\alpha$. By Lemma \\ref{lemma-determine-intersections} we have", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_P(1))^i \\cap", "\\pi_X^*(\\alpha_{r - i}) = 0", "$$", "in the chow group of $P_X$. Consider the fibre product diagram", "$$", "\\xymatrix{", "P_X \\ar[r]_-{p'} \\ar[d]_{\\pi_X} & P \\ar[d]^\\pi \\\\", "X \\ar[r]^p & Y", "}", "$$", "Apply proper pushforward $p'_*$", "(Lemma \\ref{lemma-proper-pushforward-rational-equivalence})", "to the displayed equality above. Using", "Lemmas \\ref{lemma-pushforward-cap-c1} and", "\\ref{lemma-flat-pullback-proper-pushforward} we obtain", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_P(1))^i \\cap", "\\pi^*(p_*\\alpha_{r - i}) = 0", "$$", "in the chow group of $P$. By the characterization of", "Lemma \\ref{lemma-determine-intersections} we conclude." ], "refs": [ "chow-lemma-determine-intersections", "chow-lemma-proper-pushforward-rational-equivalence", "chow-lemma-pushforward-cap-c1", "chow-lemma-flat-pullback-proper-pushforward", "chow-lemma-determine-intersections" ], "ref_ids": [ 5746, 5694, 5711, 5682, 5746 ] } ], "ref_ids": [] }, { "id": 5749, "type": "theorem", "label": "chow-lemma-flat-pullback-cap-cj", "categories": [ "chow" ], "title": "chow-lemma-flat-pullback-cap-cj", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $Y$.", "Let $f : X \\to Y$ be a flat morphism of relative dimension $r$.", "Let $\\alpha$ be a $k$-cycle on $Y$.", "Then", "$$", "f^*(c_j(\\mathcal{E}) \\cap \\alpha) = c_j(f^*\\mathcal{E}) \\cap f^*\\alpha", "$$" ], "refs": [], "proofs": [ { "contents": [ "Write $\\alpha_j = c_j(\\mathcal{E}) \\cap \\alpha$, so $\\alpha_0 = \\alpha$.", "By Lemma \\ref{lemma-determine-intersections} we have", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_P(1))^i \\cap", "\\pi^*(\\alpha_{r - i}) = 0", "$$", "in the chow group of the projective bundle", "$(\\pi : P \\to Y, \\mathcal{O}_P(1))$", "associated to $\\mathcal{E}$. Consider the fibre product diagram", "$$", "\\xymatrix{", "P_X = \\mathbf{P}(f^*\\mathcal{E}) \\ar[r]_-{f'} \\ar[d]_{\\pi_X} &", "P \\ar[d]^\\pi \\\\", "X \\ar[r]^f & Y", "}", "$$", "Note that $\\mathcal{O}_{P_X}(1)$ is the pullback of $\\mathcal{O}_P(1)$.", "Apply flat pullback $(f')^*$", "(Lemma \\ref{lemma-flat-pullback-rational-equivalence}) to the displayed", "equation above. By Lemmas \\ref{lemma-flat-pullback-cap-c1} and", "\\ref{lemma-compose-flat-pullback} we see that", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_{P_X}(1))^i \\cap", "\\pi_X^*(f^*\\alpha_{r - i}) = 0", "$$", "holds in the chow group of $P_X$. By the characterization of", "Lemma \\ref{lemma-determine-intersections} we conclude." ], "refs": [ "chow-lemma-determine-intersections", "chow-lemma-flat-pullback-rational-equivalence", "chow-lemma-flat-pullback-cap-c1", "chow-lemma-compose-flat-pullback", "chow-lemma-determine-intersections" ], "ref_ids": [ 5746, 5693, 5709, 5680, 5746 ] } ], "ref_ids": [] }, { "id": 5750, "type": "theorem", "label": "chow-lemma-cap-chern-class-commutes-with-gysin", "categories": [ "chow" ], "title": "chow-lemma-cap-chern-class-commutes-with-gysin", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.", "Let $(\\mathcal{L}, s, i : D \\to X)$ be as in", "Definition \\ref{definition-gysin-homomorphism}.", "Then $c_j(\\mathcal{E}|_D) \\cap i^*\\alpha = i^*(c_j(\\mathcal{E}) \\cap \\alpha)$", "for all $\\alpha \\in \\CH_k(X)$." ], "refs": [ "chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "Write $\\alpha_j = c_j(\\mathcal{E}) \\cap \\alpha$, so $\\alpha_0 = \\alpha$.", "By Lemma \\ref{lemma-determine-intersections} we have", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_P(1))^i \\cap", "\\pi^*(\\alpha_{r - i}) = 0", "$$", "in the chow group of the projective bundle", "$(\\pi : P \\to X, \\mathcal{O}_P(1))$", "associated to $\\mathcal{E}$. Consider the fibre product diagram", "$$", "\\xymatrix{", "P_D = \\mathbf{P}(\\mathcal{E}|_D) \\ar[r]_-{i'} \\ar[d]_{\\pi_D} &", "P \\ar[d]^\\pi \\\\", "D \\ar[r]^i & X", "}", "$$", "Note that $\\mathcal{O}_{P_D}(1)$ is the pullback of $\\mathcal{O}_P(1)$.", "Apply the gysin map $(i')^*$ (Lemma \\ref{lemma-gysin-factors}) to the", "displayed equation above.", "Applying Lemmas \\ref{lemma-gysin-commutes-cap-c1} and", "\\ref{lemma-gysin-flat-pullback} we obtain", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_{P_D}(1))^i \\cap", "\\pi_D^*(i^*\\alpha_{r - i}) = 0", "$$", "in the chow group of $P_D$.", "By the characterization of Lemma \\ref{lemma-determine-intersections}", "we conclude." ], "refs": [ "chow-lemma-determine-intersections", "chow-lemma-gysin-factors", "chow-lemma-gysin-commutes-cap-c1", "chow-lemma-gysin-flat-pullback", "chow-lemma-determine-intersections" ], "ref_ids": [ 5746, 5721, 5723, 5719, 5746 ] } ], "ref_ids": [ 5915 ] }, { "id": 5751, "type": "theorem", "label": "chow-lemma-cap-cp-bivariant", "categories": [ "chow" ], "title": "chow-lemma-cap-cp-bivariant", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module", "of rank $r$. Let $0 \\leq p \\leq r$.", "Then the rule that to $f : X' \\to X$ assigns", "$c_p(f^*\\mathcal{E}) \\cap - : \\CH_k(X') \\to \\CH_{k - p}(X')$", "is a bivariant class of degree $p$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemmas", "\\ref{lemma-cap-chern-class-factors-rational-equivalence},", "\\ref{lemma-pushforward-cap-cj},", "\\ref{lemma-flat-pullback-cap-cj}, and", "\\ref{lemma-cap-chern-class-commutes-with-gysin}", "and Definition \\ref{definition-bivariant-class}." ], "refs": [ "chow-lemma-cap-chern-class-factors-rational-equivalence", "chow-lemma-pushforward-cap-cj", "chow-lemma-flat-pullback-cap-cj", "chow-lemma-cap-chern-class-commutes-with-gysin", "chow-definition-bivariant-class" ], "ref_ids": [ 5747, 5748, 5749, 5750, 5916 ] } ], "ref_ids": [] }, { "id": 5752, "type": "theorem", "label": "chow-lemma-cap-commutative-chern", "categories": [ "chow" ], "title": "chow-lemma-cap-commutative-chern", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $r$.", "Then", "\\begin{enumerate}", "\\item $c_j(\\mathcal{E}) \\in A^j(X)$ is in the center of $A^*(X)$ and", "\\item if $f : X' \\to X$ is locally of finite type and $c \\in A^*(X' \\to X)$,", "then $c \\circ c_j(\\mathcal{E}) = c_j(f^*\\mathcal{E}) \\circ c$.", "\\end{enumerate}", "In particular, if $\\mathcal{F}$ is a second locally free", "$\\mathcal{O}_X$-module on $X$ of rank $s$, then", "$$", "c_i(\\mathcal{E}) \\cap c_j(\\mathcal{F}) \\cap \\alpha", "=", "c_j(\\mathcal{F}) \\cap c_i(\\mathcal{E}) \\cap \\alpha", "$$", "as elements of $\\CH_{k - i - j}(X)$ for all $\\alpha \\in \\CH_k(X)$." ], "refs": [], "proofs": [ { "contents": [ "It is immediate that (2) implies (1).", "Let $\\alpha \\in \\CH_k(X)$. Write $\\alpha_j = c_j(\\mathcal{E}) \\cap \\alpha$, so", "$\\alpha_0 = \\alpha$. By Lemma \\ref{lemma-determine-intersections} we have", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_P(1))^i \\cap", "\\pi^*(\\alpha_{r - i}) = 0", "$$", "in the chow group of the projective bundle", "$(\\pi : P \\to Y, \\mathcal{O}_P(1))$", "associated to $\\mathcal{E}$. Denote $\\pi' : P' \\to X'$ the base change", "of $\\pi$ by $f$. Using Lemma \\ref{lemma-c1-center} and", "the properties of bivariant classes we obtain", "\\begin{align*}", "0 & = c \\cap \\left(\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_P(1))^i \\cap", "\\pi^*(\\alpha_{r - i})\\right) \\\\", "& =", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_{P'}(1))^i \\cap", "(\\pi')^*(c \\cap \\alpha_{r - i})", "\\end{align*}", "in the Chow group of $P'$ (calculation omitted).", "Hence we see that $c \\cap \\alpha_j$ is", "equal to $c_j(f^*\\mathcal{E}) \\cap (c \\cap \\alpha)$ by the characterization", "of Lemma \\ref{lemma-determine-intersections}.", "This proves the lemma." ], "refs": [ "chow-lemma-determine-intersections", "chow-lemma-c1-center", "chow-lemma-determine-intersections" ], "ref_ids": [ 5746, 5736, 5746 ] } ], "ref_ids": [] }, { "id": 5753, "type": "theorem", "label": "chow-lemma-chern-classes-E-tensor-L", "categories": [ "chow" ], "title": "chow-lemma-chern-classes-E-tensor-L", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a finite locally free sheaf of", "rank $r$ on $X$. Let $\\mathcal{L}$ be an invertible", "sheaf on $X$. Then we have", "\\begin{equation}", "\\label{equation-twist}", "c_i({\\mathcal E} \\otimes {\\mathcal L})", "=", "\\sum\\nolimits_{j = 0}^i", "\\binom{r - i + j}{j} c_{i - j}({\\mathcal E}) c_1({\\mathcal L})^j", "\\end{equation}", "in $A^*(X)$." ], "refs": [], "proofs": [ { "contents": [ "This should hold for any triple $(X, \\mathcal{E}, \\mathcal{L})$.", "In particular it should hold when $X$ is integral and by", "Lemma \\ref{lemma-bivariant-zero}", "it is enough to prove", "it holds when capping with $[X]$ for such $X$. Thus assume", "that $X$ is integral. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$,", "resp.\\ $(\\pi' : P' \\to X, \\mathcal{O}_{P'}(1))$ be the", "projective space bundle associated to $\\mathcal{E}$,", "resp.\\ $\\mathcal{E} \\otimes \\mathcal{L}$. Consider the canonical morphism", "$$", "\\xymatrix{", "P \\ar[rd]_\\pi \\ar[rr]_g & & P' \\ar[ld]^{\\pi'} \\\\", "& X &", "}", "$$", "see Constructions, Lemma \\ref{constructions-lemma-twisting-and-proj}.", "It has the property that", "$g^*\\mathcal{O}_{P'}(1)", "= \\mathcal{O}_P(1) \\otimes \\pi^* {\\mathcal L}$.", "This means that we have", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i", "(\\xi + x)^i \\cap \\pi^*(c_{r - i}(\\mathcal{E} \\otimes \\mathcal{L}) \\cap [X])", "=", "0", "$$", "in $\\CH_*(P)$, where $\\xi$ represents", "$c_1(\\mathcal{O}_P(1))$ and $x$", "represents $c_1(\\pi^*\\mathcal{L})$. By simple algebra this", "is equivalent to", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i \\xi^i \\left(", "\\sum\\nolimits_{j = i}^r", "(-1)^{j - i}", "\\binom{j}{i}", "x^{j - i} \\cap", "\\pi^*(c_{r - j}(\\mathcal{E} \\otimes \\mathcal{L}) \\cap [X])", "\\right)", "=", "0", "$$", "Comparing with", "Equation (\\ref{equation-chern-classes}) it follows from this that", "$$", "c_{r - i}(\\mathcal{E}) \\cap [X] =", "\\sum\\nolimits_{j = i}^r", "\\binom{j}{i}", "(-c_1(\\mathcal{L}))^{j - i} \\cap", "c_{r - j}(\\mathcal{E} \\otimes \\mathcal{L}) \\cap [X]", "$$", "Reworking this (getting rid of minus signs, and renumbering) we get", "the desired relation." ], "refs": [ "chow-lemma-bivariant-zero", "constructions-lemma-twisting-and-proj" ], "ref_ids": [ 5740, 12650 ] } ], "ref_ids": [] }, { "id": 5754, "type": "theorem", "label": "chow-lemma-get-rid-of-trivial-subbundle", "categories": [ "chow" ], "title": "chow-lemma-get-rid-of-trivial-subbundle", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$, $\\mathcal{F}$ be finite locally free sheaves", "on $X$ of ranks $r$, $r - 1$ which fit into a short", "exact sequence", "$$", "0 \\to \\mathcal{O}_X \\to \\mathcal{E} \\to \\mathcal{F} \\to 0", "$$", "Then we have", "$$", "c_r(\\mathcal{E}) = 0, \\quad", "c_j(\\mathcal{E}) = c_j(\\mathcal{F}), \\quad j = 0, \\ldots, r - 1", "$$", "in $A^*(X)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-bivariant-zero}", "it suffices to show that if $X$ is integral", "then $c_j(\\mathcal{E}) \\cap [X] = c_j(\\mathcal{F}) \\cap [X]$.", "Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$,", "resp.\\ $(\\pi' : P' \\to X, \\mathcal{O}_{P'}(1))$ denote the", "projective space bundle associated to $\\mathcal{E}$, resp.\\ $\\mathcal{F}$.", "The surjection $\\mathcal{E} \\to \\mathcal{F}$ gives rise", "to a closed immersion", "$$", "i : P' \\longrightarrow P", "$$", "over $X$. Moreover, the element", "$1 \\in \\Gamma(X, \\mathcal{O}_X) \\subset \\Gamma(X, \\mathcal{E})$", "gives rise to a global section $s \\in \\Gamma(P, \\mathcal{O}_P(1))$", "whose zero set is exactly $P'$. Hence $P'$ is an effective Cartier", "divisor on $P$ such that $\\mathcal{O}_P(P') \\cong \\mathcal{O}_P(1)$.", "Hence we see that", "$$", "c_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\alpha = i_*((\\pi')^*\\alpha)", "$$", "for any cycle class $\\alpha$ on $X$ by", "Lemma \\ref{lemma-relative-effective-cartier}.", "By Lemma \\ref{lemma-determine-intersections} we see that", "$\\alpha_j = c_j(\\mathcal{F}) \\cap [X]$, $j = 0, \\ldots, r - 1$", "satisfy", "$$", "\\sum\\nolimits_{j = 0}^{r - 1} (-1)^jc_1(\\mathcal{O}_{P'}(1))^j", "\\cap (\\pi')^*\\alpha_j = 0", "$$", "Pushing this to $P$ and using the remark above as well as", "Lemma \\ref{lemma-pushforward-cap-c1} we get", "$$", "\\sum\\nolimits_{j = 0}^{r - 1}", "(-1)^j c_1(\\mathcal{O}_P(1))^{j + 1}", "\\cap \\pi^*\\alpha_j = 0", "$$", "By the uniqueness of Lemma \\ref{lemma-determine-intersections}", "we conclude that", "$c_r(\\mathcal{E}) \\cap [X] = 0$ and", "$c_j(\\mathcal{E}) \\cap [X] = \\alpha_j = c_j(\\mathcal{F}) \\cap [X]$", "for $j = 0, \\ldots, r - 1$. Hence the lemma holds." ], "refs": [ "chow-lemma-bivariant-zero", "chow-lemma-relative-effective-cartier", "chow-lemma-determine-intersections", "chow-lemma-pushforward-cap-c1", "chow-lemma-determine-intersections" ], "ref_ids": [ 5740, 5725, 5746, 5711, 5746 ] } ], "ref_ids": [] }, { "id": 5755, "type": "theorem", "label": "chow-lemma-additivity-invertible-subsheaf", "categories": [ "chow" ], "title": "chow-lemma-additivity-invertible-subsheaf", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$, $\\mathcal{F}$ be finite locally free sheaves", "on $X$ of ranks $r$, $r - 1$ which fit into a short", "exact sequence", "$$", "0 \\to \\mathcal{L} \\to \\mathcal{E} \\to \\mathcal{F} \\to 0", "$$", "where $\\mathcal{L}$ is an invertible sheaf.", "Then", "$$", "c(\\mathcal{E}) = c(\\mathcal{L}) c(\\mathcal{F})", "$$", "in $A^*(X)$." ], "refs": [], "proofs": [ { "contents": [ "This relation really just says that", "$c_i(\\mathcal{E}) = c_i(\\mathcal{F}) + c_1(\\mathcal{L})c_{i - 1}(\\mathcal{F})$.", "By Lemma \\ref{lemma-get-rid-of-trivial-subbundle}", "we have $c_j(\\mathcal{E} \\otimes \\mathcal{L}^{\\otimes -1})", "= c_j(\\mathcal{F} \\otimes \\mathcal{L}^{\\otimes -1})$ for", "$j = 0, \\ldots, r$ were we set", "$c_r(\\mathcal{F} \\otimes \\mathcal{L}^{-1}) = 0$ by convention.", "Applying Lemma \\ref{lemma-chern-classes-E-tensor-L} we deduce", "$$", "\\sum_{j = 0}^i", "\\binom{r - i + j}{j} (-1)^j c_{i - j}({\\mathcal E}) c_1({\\mathcal L})^j", "=", "\\sum_{j = 0}^i", "\\binom{r - 1 - i + j}{j} (-1)^j c_{i - j}({\\mathcal F}) c_1({\\mathcal L})^j", "$$", "Setting", "$c_i(\\mathcal{E}) = c_i(\\mathcal{F}) + c_1(\\mathcal{L})c_{i - 1}(\\mathcal{F})$", "gives a ``solution'' of this equation. The lemma follows if we show", "that this is the only possible solution. We omit the verification." ], "refs": [ "chow-lemma-get-rid-of-trivial-subbundle", "chow-lemma-chern-classes-E-tensor-L" ], "ref_ids": [ 5754, 5753 ] } ], "ref_ids": [] }, { "id": 5756, "type": "theorem", "label": "chow-lemma-additivity-chern-classes", "categories": [ "chow" ], "title": "chow-lemma-additivity-chern-classes", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be a scheme locally of finite type over $S$.", "Suppose that ${\\mathcal E}$ sits in an", "exact sequence", "$$", "0", "\\to", "{\\mathcal E}_1", "\\to", "{\\mathcal E}", "\\to", "{\\mathcal E}_2", "\\to", "0", "$$", "of finite locally free sheaves $\\mathcal{E}_i$ of rank $r_i$.", "The total Chern classes satisfy", "$$", "c({\\mathcal E}) = c({\\mathcal E}_1) c({\\mathcal E}_2)", "$$", "in $A^*(X)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-bivariant-zero} we may assume that $X$ is integral", "and we have to show the identity when capping against $[X]$.", "By induction on $r_1$. The case $r_1 = 1$ is", "Lemma \\ref{lemma-additivity-invertible-subsheaf}.", "Assume $r_1 > 1$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$", "denote the projective space bundle associated to $\\mathcal{E}_1$. Note that", "\\begin{enumerate}", "\\item $\\pi^* : \\CH_*(X) \\to \\CH_*(P)$ is injective, and", "\\item $\\pi^*\\mathcal{E}_1$ sits in a short exact sequence", "$0 \\to \\mathcal{F} \\to \\pi^*\\mathcal{E}_1 \\to \\mathcal{L} \\to 0$", "where $\\mathcal{L}$ is invertible.", "\\end{enumerate}", "The first assertion follows from the projective space bundle formula", "and the second follows from the definition of a projective space bundle.", "(In fact $\\mathcal{L} = \\mathcal{O}_P(1)$.)", "Let $Q = \\pi^*\\mathcal{E}/\\mathcal{F}$, which sits in an", "exact sequence $0 \\to \\mathcal{L} \\to Q \\to \\pi^*\\mathcal{E}_2 \\to 0$.", "By induction we have", "\\begin{eqnarray*}", "c(\\pi^*\\mathcal{E}) \\cap [P]", "& = &", "c(\\mathcal{F}) \\cap c(\\pi^*\\mathcal{E}/\\mathcal{F}) \\cap [P] \\\\", "& = &", "c(\\mathcal{F}) \\cap c(\\mathcal{L}) \\cap c(\\pi^*\\mathcal{E}_2) \\cap [P] \\\\", "& = &", "c(\\pi^*\\mathcal{E}_1) \\cap c(\\pi^*\\mathcal{E}_2) \\cap [P]", "\\end{eqnarray*}", "Since $[P] = \\pi^*[X]$ we", "win by Lemma \\ref{lemma-flat-pullback-cap-cj}." ], "refs": [ "chow-lemma-bivariant-zero", "chow-lemma-additivity-invertible-subsheaf", "chow-lemma-flat-pullback-cap-cj" ], "ref_ids": [ 5740, 5755, 5749 ] } ], "ref_ids": [] }, { "id": 5757, "type": "theorem", "label": "chow-lemma-chern-filter-by-linebundles", "categories": [ "chow" ], "title": "chow-lemma-chern-filter-by-linebundles", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let ${\\mathcal L}_i$, $i = 1, \\ldots, r$ be invertible", "$\\mathcal{O}_X$-modules on $X$.", "Let $\\mathcal{E}$ be a locally free rank", "$\\mathcal{O}_X$-module endowed with a filtration", "$$", "0 = \\mathcal{E}_0 \\subset \\mathcal{E}_1 \\subset \\mathcal{E}_2", "\\subset \\ldots \\subset \\mathcal{E}_r = \\mathcal{E}", "$$", "such that $\\mathcal{E}_i/\\mathcal{E}_{i - 1} \\cong \\mathcal{L}_i$.", "Set $c_1({\\mathcal L}_i) = x_i$. Then", "$$", "c(\\mathcal{E})", "=", "\\prod\\nolimits_{i = 1}^r (1 + x_i)", "$$", "in $A^*(X)$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-additivity-invertible-subsheaf} and induction." ], "refs": [ "chow-lemma-additivity-invertible-subsheaf" ], "ref_ids": [ 5755 ] } ], "ref_ids": [] }, { "id": 5758, "type": "theorem", "label": "chow-lemma-spell-out-degree-zero-cycle", "categories": [ "chow" ], "title": "chow-lemma-spell-out-degree-zero-cycle", "contents": [ "Let $k$ be a field. Let $X$ be proper over $k$. Let $\\alpha = \\sum n_i[Z_i]$", "be in $Z_0(X)$. Then", "$$", "\\deg(\\alpha) = \\sum n_i\\deg(Z_i)", "$$", "where $\\deg(Z_i)$ is the degree of $Z_i \\to \\Spec(k)$, i.e.,", "$\\deg(Z_i) = \\dim_k \\Gamma(Z_i, \\mathcal{O}_{Z_i})$." ], "refs": [], "proofs": [ { "contents": [ "This is the definition of proper pushforward", "(Definition \\ref{definition-proper-pushforward})." ], "refs": [ "chow-definition-proper-pushforward" ], "ref_ids": [ 5909 ] } ], "ref_ids": [] }, { "id": 5759, "type": "theorem", "label": "chow-lemma-degree-vector-bundle", "categories": [ "chow" ], "title": "chow-lemma-degree-vector-bundle", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $\\leq 1$.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module of constant", "rank. Then", "$$", "\\deg(\\mathcal{E}) = \\deg(c_1(\\mathcal{E}) \\cap [X]_1)", "$$", "where the left hand side is defined in", "Varieties, Definition \\ref{varieties-definition-degree-invertible-sheaf}." ], "refs": [ "varieties-definition-degree-invertible-sheaf" ], "proofs": [ { "contents": [ "Let $C_i \\subset X$, $i = 1, \\ldots, t$ be the irreducible components", "of dimension $1$ with reduced induced scheme structure and let $m_i$ be the", "multiplicity of $C_i$ in $X$. Then $[X]_1 = \\sum m_i[C_i]$ and", "$c_1(\\mathcal{E}) \\cap [X]_1$ is the sum of the pushforwards of the cycles", "$m_i c_1(\\mathcal{E}|_{C_i}) \\cap [C_i]$. Since we have a similar decomposition", "of the degree of $\\mathcal{E}$ by", "Varieties, Lemma \\ref{varieties-lemma-degree-in-terms-of-components}", "it suffices to prove the lemma in case $X$ is a proper curve over $k$.", "\\medskip\\noindent", "Assume $X$ is a proper curve over $k$.", "By Divisors, Lemma \\ref{divisors-lemma-filter-after-modification}", "there exists a modification $f : X' \\to X$ such that $f^*\\mathcal{E}$", "has a filtration whose successive quotients are invertible", "$\\mathcal{O}_{X'}$-modules. Since $f_*[X']_1 = [X]_1$ we conclude", "from Lemma \\ref{lemma-pushforward-cap-cj} that", "$$", "\\deg(c_1(\\mathcal{E}) \\cap [X]_1) = \\deg(c_1(f^*\\mathcal{E}) \\cap [X']_1)", "$$", "Since we have a similar relationship for the degree by", "Varieties, Lemma \\ref{varieties-lemma-degree-birational-pullback}", "we reduce to the case where $\\mathcal{E}$ has a filtration whose", "successive quotients are invertible $\\mathcal{O}_X$-modules.", "In this case, we may use additivity of the degree", "(Varieties, Lemma \\ref{varieties-lemma-degree-additive})", "and of first Chern classes (Lemma \\ref{lemma-additivity-chern-classes})", "to reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $X$ is a proper curve over $k$ and $\\mathcal{E}$ is an", "invertible $\\mathcal{O}_X$-module. By", "Divisors, Lemma", "\\ref{divisors-lemma-quasi-projective-Noetherian-pic-effective-Cartier}", "we see that $\\mathcal{E}$ is isomorphic to", "$\\mathcal{O}_X(D) \\otimes \\mathcal{O}_X(D')^{\\otimes -1}$", "for some effective Cartier divisors $D, D'$ on $X$ (this also uses", "that $X$ is projective, see", "Varieties, Lemma \\ref{varieties-lemma-dim-1-proper-projective} for example).", "By additivity of degree under tensor product of invertible sheaves", "(Varieties, Lemma \\ref{varieties-lemma-degree-tensor-product})", "and additivity of $c_1$ under tensor product of invertible sheaves", "(Lemma \\ref{lemma-c1-cap-additive} or \\ref{lemma-chern-classes-E-tensor-L})", "we reduce to the case $\\mathcal{E} = \\mathcal{O}_X(D)$.", "In this case the left hand side gives $\\deg(D)$", "(Varieties, Lemma \\ref{varieties-lemma-degree-effective-Cartier-divisor})", "and the right hand side gives $\\deg([D]_0)$ by", "Lemma \\ref{lemma-geometric-cap}.", "Since", "$$", "[D]_0 = \\sum\\nolimits_{x \\in D}", "\\text{length}_{\\mathcal{O}_{X, x}}(\\mathcal{O}_{D, x}) [x] =", "\\sum\\nolimits_{x \\in D}", "\\text{length}_{\\mathcal{O}_{D, x}}(\\mathcal{O}_{D, x}) [x]", "$$", "by definition, we see", "$$", "\\deg([D]_0) = \\sum\\nolimits_{x \\in D}", "\\text{length}_{\\mathcal{O}_{D, x}}(\\mathcal{O}_{D, x}) [\\kappa(x) : k] =", "\\dim_k \\Gamma(D, \\mathcal{O}_D) = \\deg(D)", "$$", "The penultimate equality by", "Algebra, Lemma \\ref{algebra-lemma-pushdown-module}", "using that $D$ is affine." ], "refs": [ "varieties-lemma-degree-in-terms-of-components", "divisors-lemma-filter-after-modification", "chow-lemma-pushforward-cap-cj", "varieties-lemma-degree-birational-pullback", "varieties-lemma-degree-additive", "chow-lemma-additivity-chern-classes", "divisors-lemma-quasi-projective-Noetherian-pic-effective-Cartier", "varieties-lemma-dim-1-proper-projective", "varieties-lemma-degree-tensor-product", "chow-lemma-c1-cap-additive", "chow-lemma-chern-classes-E-tensor-L", "varieties-lemma-degree-effective-Cartier-divisor", "chow-lemma-geometric-cap", "algebra-lemma-pushdown-module" ], "ref_ids": [ 11108, 8079, 5748, 11106, 11105, 5756, 7956, 11099, 11109, 5705, 5753, 11111, 5707, 639 ] } ], "ref_ids": [ 11161 ] }, { "id": 5760, "type": "theorem", "label": "chow-lemma-degrees-and-numerical-intersections", "categories": [ "chow" ], "title": "chow-lemma-degrees-and-numerical-intersections", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$.", "Let $Z \\subset X$ be a closed subscheme of dimension $d$.", "Let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible", "$\\mathcal{O}_X$-modules. Then", "$$", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) =", "\\deg(", "c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_d) \\cap [Z]_d)", "$$", "where the left hand side is defined in", "Varieties, Definition \\ref{varieties-definition-intersection-number}.", "In particular,", "$$", "\\deg_\\mathcal{L}(Z) = \\deg(c_1(\\mathcal{L})^d \\cap [Z]_d)", "$$", "if $\\mathcal{L}$ is an ample invertible $\\mathcal{O}_X$-module." ], "refs": [ "varieties-definition-intersection-number" ], "proofs": [ { "contents": [ "We will prove this by induction on $d$. If $d = 0$, then the result is", "true by Varieties, Lemma \\ref{varieties-lemma-chi-tensor-finite}.", "Assume $d > 0$.", "\\medskip\\noindent", "Let $Z_i \\subset Z$, $i = 1, \\ldots, t$ be the irreducible components", "of dimension $d$ with reduced induced scheme structure and let $m_i$ be the", "multiplicity of $Z_i$ in $Z$. Then $[Z]_d = \\sum m_i[Z_i]$ and", "$c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_d) \\cap [Z]_d$", "is the sum of the cycles", "$m_i c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_d) \\cap [Z_i]$.", "Since we have a similar decomposition for", "$(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$ by", "Varieties, Lemma \\ref{varieties-lemma-numerical-polynomial-leading-term}", "it suffices to prove the lemma in case $Z = X$", "is a proper variety of dimension $d$ over $k$.", "\\medskip\\noindent", "By Chow's lemma there exists a birational proper morphism $f : Y \\to X$", "with $Y$ H-projective over $k$. See Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-chow-Noetherian} and Remark", "\\ref{coherent-remark-chow-Noetherian}. Then", "$$", "(f^*\\mathcal{L}_1 \\cdots f^*\\mathcal{L}_d \\cdot Y) =", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot X)", "$$", "by Varieties, Lemma \\ref{varieties-lemma-intersection-number-and-pullback}", "and we have", "$$", "f_*(c_1(f^*\\mathcal{L}_1) \\cap \\ldots \\cap c_1(f^*\\mathcal{L}_d) \\cap [Y]) =", "c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_d) \\cap [X]", "$$", "by Lemma \\ref{lemma-pushforward-cap-c1}. Thus we may replace $X$ by $Y$", "and assume that $X$ is projective over $k$.", "\\medskip\\noindent", "If $X$ is a proper $d$-dimensional projective variety, then we can", "write $\\mathcal{L}_1 = \\mathcal{O}_X(D) \\otimes \\mathcal{O}_X(D')^{\\otimes -1}$", "for some effective Cartier divisors $D, D' \\subset X$", "by Divisors, Lemma", "\\ref{divisors-lemma-quasi-projective-Noetherian-pic-effective-Cartier}.", "By additivity for both sides of the equation", "(Varieties, Lemma \\ref{varieties-lemma-intersection-number-additive} and", "Lemma \\ref{lemma-c1-cap-additive})", "we reduce to the case $\\mathcal{L}_1 = \\mathcal{O}_X(D)$ for some", "effective Cartier divisor $D$.", "By Varieties, Lemma", "\\ref{varieties-lemma-numerical-intersection-effective-Cartier-divisor}", "we have", "$$", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot X) =", "(\\mathcal{L}_2 \\cdots \\mathcal{L}_d \\cdot D)", "$$", "and by Lemma \\ref{lemma-geometric-cap} we have", "$$", "c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_d) \\cap [X] =", "c_1(\\mathcal{L}_2) \\cap \\ldots \\cap c_1(\\mathcal{L}_d) \\cap [D]_{d - 1}", "$$", "Thus we obtain the result from our induction hypothesis." ], "refs": [ "varieties-lemma-chi-tensor-finite", "varieties-lemma-numerical-polynomial-leading-term", "coherent-lemma-chow-Noetherian", "coherent-remark-chow-Noetherian", "varieties-lemma-intersection-number-and-pullback", "chow-lemma-pushforward-cap-c1", "divisors-lemma-quasi-projective-Noetherian-pic-effective-Cartier", "varieties-lemma-intersection-number-additive", "chow-lemma-c1-cap-additive", "varieties-lemma-numerical-intersection-effective-Cartier-divisor", "chow-lemma-geometric-cap" ], "ref_ids": [ 11030, 11122, 3354, 3406, 11126, 5711, 7956, 11124, 5705, 11127, 5707 ] } ], "ref_ids": [ 11162 ] }, { "id": 5761, "type": "theorem", "label": "chow-lemma-locally-equidimensional", "categories": [ "chow" ], "title": "chow-lemma-locally-equidimensional", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. Write", "$\\delta = \\delta_{X/S}$ as in Section \\ref{section-setup}.", "The following are equivalent", "\\begin{enumerate}", "\\item There exists a decomposition $X = \\coprod_{n \\in \\mathbf{Z}} X_n$", "into open and closed subschemes such that $\\delta(\\xi) = n$ whenever", "$\\xi \\in X_n$ is a generic point of an irreducible component of $X_n$.", "\\item For all $x \\in X$ there exists an open neighbourhood $U \\subset X$", "of $x$ and an integer $n$ such that $\\delta(\\xi) = n$ whenever", "$\\xi \\in U$ is a generic point of an irreducible component of $U$.", "\\item For all $x \\in X$ there exists an integer $n_x$ such that", "$\\delta(\\xi) = n_x$ for any generic point $\\xi$ of an irreducible", "component of $X$ containing $x$.", "\\end{enumerate}", "The conditions are satisfied if $X$ is either", "normal or Cohen-Macaulay\\footnote{In fact, it suffices if", "$X$ is $(S_2)$. Compare with Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-catenary-S2-equidimensional}.}." ], "refs": [ "local-cohomology-lemma-catenary-S2-equidimensional" ], "proofs": [ { "contents": [ "It is clear that (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3).", "Conversely, if (3) holds, then we set $X_n = \\{x \\in X \\mid n_x = n\\}$", "and we get a decomposition as in (1). Namely, $X_n$ is open because", "given $x$ the union of the irreducible components of $X$ passing through $x$", "minus the union of the irreducible components of $X$ not passing through $x$", "is an open neighbourhood of $x$. If $X$ is normal, then $X$ is a", "disjoint union of integral schemes", "(Properties, Lemma \\ref{properties-lemma-normal-locally-Noetherian})", "and hence the properties hold.", "If $X$ is Cohen-Macaulay, then", "$\\delta' : X \\to \\mathbf{Z}$, $x \\mapsto -\\dim(\\mathcal{O}_{X, x})$", "is a dimension function on $X$ (see Example \\ref{example-CM-irreducible}).", "Since $\\delta - \\delta'$ is locally constant", "(Topology, Lemma \\ref{topology-lemma-dimension-function-unique})", "and since $\\delta'(\\xi) = 0$ for every generic point $\\xi$ of $X$", "we see that (2) holds." ], "refs": [ "properties-lemma-normal-locally-Noetherian", "topology-lemma-dimension-function-unique" ], "ref_ids": [ 2971, 8292 ] } ], "ref_ids": [ 9702 ] }, { "id": 5762, "type": "theorem", "label": "chow-lemma-splitting-principle", "categories": [ "chow" ], "title": "chow-lemma-splitting-principle", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally", "of finite type over $S$. Let $\\mathcal{E}_i$ be a finite collection of", "locally free $\\mathcal{O}_X$-modules of rank $r_i$. There exists a projective", "flat morphism $\\pi : P \\to X$ of relative dimension $d$ such that", "\\begin{enumerate}", "\\item for any morphism $f : Y \\to X$ the map", "$\\pi_Y^* : \\CH_*(Y) \\to \\CH_{* + d}(Y \\times_X P)$ is injective, and", "\\item each $\\pi^*\\mathcal{E}_i$ has a filtration", "whose successive quotients $\\mathcal{L}_{i, 1}, \\ldots, \\mathcal{L}_{i, r_i}$", "are invertible ${\\mathcal O}_P$-modules.", "\\end{enumerate}", "Moreover, when (1) holds the restriction map $A^*(X) \\to A^*(P)$", "(Remark \\ref{remark-pullback-cohomology}) is injective." ], "refs": [ "chow-remark-pullback-cohomology" ], "proofs": [ { "contents": [ "We may assume $r_i \\geq 1$ for all $i$. We will prove the lemma by induction", "on $\\sum (r_i - 1)$. If this integer is $0$, then $\\mathcal{E}_i$", "is invertible for all $i$ and we conclude by taking $\\pi = \\text{id}_X$.", "If not, then we can pick an $i$ such that $r_i > 1$ and consider the", "morphism $\\pi_i : P_i = \\mathbf{P}(\\mathcal{E}_i) \\to X$.", "We have a short exact sequence", "$$", "0 \\to \\mathcal{F} \\to \\pi_i^*\\mathcal{E}_i \\to \\mathcal{O}_{P_i}(1) \\to 0", "$$", "of finite locally free $\\mathcal{O}_{P_i}$-modules of ranks $r_i - 1$,", "$r_i$, and $1$. Observe that $\\pi_i^*$ is injective on chow groups", "after any base change by the projective bundle formula", "(Lemma \\ref{lemma-chow-ring-projective-bundle}).", "By the induction hypothesis applied to the finite locally free", "$\\mathcal{O}_{P_i}$-modules $\\mathcal{F}$ and $\\pi_{i'}^*\\mathcal{E}_{i'}$", "for $i' \\not = i$, we find a morphism $\\pi : P \\to P_i$ with", "properties stated as in the lemma. Then the composition", "$\\pi_i \\circ \\pi : P \\to X$ does the job. Some details omitted." ], "refs": [ "chow-lemma-chow-ring-projective-bundle" ], "ref_ids": [ 5743 ] } ], "ref_ids": [ 5941 ] }, { "id": 5763, "type": "theorem", "label": "chow-lemma-chern-classes-dual", "categories": [ "chow" ], "title": "chow-lemma-chern-classes-dual", "contents": [ "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module", "with dual $\\mathcal{E}^\\vee$. Then", "$$", "c_i(\\mathcal{E}^\\vee) = (-1)^i c_i(\\mathcal{E})", "$$", "in $A^i(X)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a morphism $\\pi : P \\to X$ as in", "Lemma \\ref{lemma-splitting-principle}.", "By the injectivity of $\\pi^*$ (after any base change)", "it suffices to prove the relation between", "the Chern classes of $\\mathcal{E}$ and $\\mathcal{E}^\\vee$", "after pulling back to $P$. Thus we may assume there", "exist invertible $\\mathcal{O}_X$-modules", "${\\mathcal L}_i$, $i = 1, \\ldots, r$", "and a filtration", "$$", "0 = \\mathcal{E}_0 \\subset \\mathcal{E}_1 \\subset \\mathcal{E}_2", "\\subset \\ldots \\subset \\mathcal{E}_r = \\mathcal{E}", "$$", "such that $\\mathcal{E}_i/\\mathcal{E}_{i - 1} \\cong \\mathcal{L}_i$.", "Then we obtain the dual filtration", "$$", "0 = \\mathcal{E}_r^\\perp \\subset \\mathcal{E}_1^\\perp \\subset \\mathcal{E}_2^\\perp", "\\subset \\ldots \\subset \\mathcal{E}_0^\\perp = \\mathcal{E}^\\vee", "$$", "such that $\\mathcal{E}_{i - 1}^\\perp/\\mathcal{E}_i^\\perp \\cong", "\\mathcal{L}_i^{\\otimes -1}$.", "Set $x_i = c_1(\\mathcal{L}_i)$.", "Then $c_1(\\mathcal{L}_i^{\\otimes -1}) = - x_i$", "by Lemma \\ref{lemma-c1-cap-additive}.", "By Lemma \\ref{lemma-chern-filter-by-linebundles}", "we have", "$$", "c(\\mathcal{E}) = \\prod\\nolimits_{i = 1}^r (1 + x_i)", "\\quad\\text{and}\\quad", "c(\\mathcal{E}^\\vee) = \\prod\\nolimits_{i = 1}^r (1 - x_i)", "$$", "in $A^*(X)$. The result follows from a formal computation", "which we omit." ], "refs": [ "chow-lemma-splitting-principle", "chow-lemma-c1-cap-additive", "chow-lemma-chern-filter-by-linebundles" ], "ref_ids": [ 5762, 5705, 5757 ] } ], "ref_ids": [] }, { "id": 5764, "type": "theorem", "label": "chow-lemma-chern-classes-tensor-product", "categories": [ "chow" ], "title": "chow-lemma-chern-classes-tensor-product", "contents": [ "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ and $\\mathcal{F}$ be a finite locally free", "$\\mathcal{O}_X$-modules of ranks $r$ and $s$. Then we have", "$$", "c_1(\\mathcal{E} \\otimes \\mathcal{F})", "=", "r c_1(\\mathcal{F}) + s c_1(\\mathcal{E})", "$$", "$$", "c_2(\\mathcal{E} \\otimes \\mathcal{F})", "=", "r c_2(\\mathcal{F}) + s c_2(\\mathcal{E}) +", "{r \\choose 2} c_1(\\mathcal{F})^2 +", "(rs - 1) c_1(\\mathcal{F})c_1(\\mathcal{E}) +", "{s \\choose 2} c_1(\\mathcal{E})^2", "$$", "and so on in $A^*(X)$." ], "refs": [], "proofs": [ { "contents": [ "Arguing exactly as in the proof of Lemma \\ref{lemma-chern-classes-dual}", "we may assume we have", "invertible $\\mathcal{O}_X$-modules", "${\\mathcal L}_i$, $i = 1, \\ldots, r$", "${\\mathcal N}_i$, $i = 1, \\ldots, s$", "filtrations", "$$", "0 = \\mathcal{E}_0 \\subset \\mathcal{E}_1 \\subset \\mathcal{E}_2", "\\subset \\ldots \\subset \\mathcal{E}_r = \\mathcal{E}", "\\quad\\text{and}\\quad", "0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset \\mathcal{F}_2", "\\subset \\ldots \\subset \\mathcal{F}_s = \\mathcal{F}", "$$", "such that $\\mathcal{E}_i/\\mathcal{E}_{i - 1} \\cong \\mathcal{L}_i$", "and such that $\\mathcal{F}_j/\\mathcal{F}_{j - 1} \\cong \\mathcal{N}_j$.", "Ordering pairs $(i, j)$ lexicographically", "we obtain a filtration", "$$", "0 \\subset \\ldots \\subset", "\\mathcal{E}_i \\otimes \\mathcal{F}_j", "+", "\\mathcal{E}_{i - 1} \\otimes \\mathcal{F}", "\\subset \\ldots \\subset \\mathcal{E} \\otimes \\mathcal{F}", "$$", "with successive quotients", "$$", "\\mathcal{L}_1 \\otimes \\mathcal{N}_1,", "\\mathcal{L}_1 \\otimes \\mathcal{N}_2,", "\\ldots,", "\\mathcal{L}_1 \\otimes \\mathcal{N}_s,", "\\mathcal{L}_2 \\otimes \\mathcal{N}_1,", "\\ldots,", "\\mathcal{L}_r \\otimes \\mathcal{N}_s", "$$", "By Lemma \\ref{lemma-chern-filter-by-linebundles}", "we have", "$$", "c(\\mathcal{E}) = \\prod (1 + x_i),", "\\quad", "c(\\mathcal{F}) = \\prod (1 + y_j),", "\\quad\\text{and}\\quad", "c(\\mathcal{F}) = \\prod (1 + x_i + y_j),", "$$", "in $A^*(X)$. The result follows from a formal computation", "which we omit." ], "refs": [ "chow-lemma-chern-classes-dual", "chow-lemma-chern-filter-by-linebundles" ], "ref_ids": [ 5763, 5757 ] } ], "ref_ids": [] }, { "id": 5765, "type": "theorem", "label": "chow-lemma-top-chern-class", "categories": [ "chow" ], "title": "chow-lemma-top-chern-class", "contents": [ "In the situation described just above assume $\\dim_\\delta(X') = n$,", "that $f^*\\mathcal{E}$ has constant rank $r$, that", "$\\dim_\\delta(Z(s)) \\leq n - r$, and that for every generic point", "$\\xi \\in Z(s)$ with $\\delta(\\xi) = n - r$ the ideal of $Z(s)$", "in $\\mathcal{O}_{X', \\xi}$ is generated by a regular sequence", "of length $r$. Then", "$$", "c_r(\\mathcal{E}) \\cap [X']_n = [Z(s)]_{n - r}", "$$", "in $\\CH_*(X')$." ], "refs": [], "proofs": [ { "contents": [ "Since $c_r(\\mathcal{E})$ is a bivariant class", "(Lemma \\ref{lemma-cap-cp-bivariant})", "we may assume $X = X'$ and we have to show that", "$c_r(\\mathcal{E}) \\cap [X]_n = [Z(s)]_{n - r}$ in $\\CH_{n - r}(X)$.", "We will prove the lemma by induction on $r \\geq 0$. (The case", "$r = 0$ is trivial.) The case $r = 1$", "is handled by Lemma \\ref{lemma-geometric-cap}. Assume $r > 1$.", "\\medskip\\noindent", "Let $\\pi : P \\to X$ be the projective space bundle associated to", "$\\mathcal{E}$ and consider the short exact sequence", "$$", "0 \\to \\mathcal{E}' \\to \\pi^*\\mathcal{E} \\to \\mathcal{O}_P(1) \\to 0", "$$", "By the projective space bundle formula", "(Lemma \\ref{lemma-chow-ring-projective-bundle})", "it suffices to prove the equality after pulling back by $\\pi$.", "Observe that $\\pi^{-1}Z(s) = Z(\\pi^*s)$ has $\\delta$-dimension", "$\\leq n - 1$ and that the assumption on regular sequences at", "generic points of $\\delta$-dimension $n - 1$ holds by", "flat pullback, see", "Algebra, Lemma \\ref{algebra-lemma-flat-increases-depth}.", "Let $t \\in \\Gamma(P, \\mathcal{O}_P(1))$ be the image of $\\pi^*s$.", "We claim", "$$", "[Z(t)]_{n + r - 2} = c_1(\\mathcal{O}_P(1)) \\cap [P]_{n + r - 1}", "$$", "Assuming the claim we finish the proof as follows.", "The restriction $\\pi^*s|_{Z(t)}$ maps to zero in", "$\\mathcal{O}_P(1)|_{Z(t)}$ hence comes from a unique", "element $s' \\in \\Gamma(Z(t), \\mathcal{E}'|_{Z(t)})$.", "Note that $Z(s') = Z(\\pi^*s)$ as closed subschemes of $P$.", "If $\\xi \\in Z(s')$ is a generic point with $\\delta(\\xi) = n - 1$,", "then the ideal of $Z(s')$ in $\\mathcal{O}_{Z(t), \\xi}$", "can be generated by a regular sequence of length $r - 1$: it is generated by", "$r - 1$ elements which are the images of $r - 1$ elements in", "$\\mathcal{O}_{P, \\xi}$ which together with a generator of the", "ideal of $Z(t)$ in $\\mathcal{O}_{P, \\xi}$ form a regular sequence", "of length $r$ in $\\mathcal{O}_{P, \\xi}$. Hence we can apply the", "induction hypothesis to $s'$ on $Z(t)$ to get", "$c_{r - 1}(\\mathcal{E}') \\cap [Z(t)]_{n + r - 2} = [Z(s')]_{n - 1}$.", "Combining all of the above we obtain", "\\begin{align*}", "c_r(\\pi^*\\mathcal{E}) \\cap [P]_{n + r - 1}", "& =", "c_{r - 1}(\\mathcal{E}') \\cap c_1(\\mathcal{O}_P(1)) \\cap [P]_{n + r - 1} \\\\", "& =", "c_{r - 1}(\\mathcal{E}') \\cap [Z(t)]_{n + r - 2} \\\\", "& =", "[Z(s')]_{n - 1} \\\\", "& = [Z(\\pi^*s)]_{n - 1}", "\\end{align*}", "which is what we had to show.", "\\medskip\\noindent", "Proof of the claim. This will follow from an application of", "the already used Lemma \\ref{lemma-geometric-cap}.", "We have $\\pi^{-1}(Z(s)) = Z(\\pi^*s) \\subset Z(t)$.", "On the other hand, for $x \\in X$ if $P_x \\subset Z(t)$, then", "$t|_{P_x} = 0$ which implies that $s$ is zero in the fibre", "$\\mathcal{E} \\otimes \\kappa(x)$, which implies $x \\in Z(s)$.", "It follows that $\\dim_\\delta(Z(t)) \\leq n + (r - 1) - 1$.", "Finally, let $\\xi \\in Z(t)$ be a generic point with", "$\\delta(\\xi) = n + r - 2$. If $\\xi$ is not the generic point", "of the fibre of $P \\to X$ it is immediate that", "a local equation of $Z(t)$ is a nonzerodivisor in $\\mathcal{O}_{P, \\xi}$", "(because we can check this on the fibre by", "Algebra, Lemma \\ref{algebra-lemma-grothendieck}).", "If $\\xi$ is the generic point of a fibre, then $x = \\pi(\\xi) \\in Z(s)$", "and $\\delta(x) = n + r - 2 - (r - 1) = n - 1$. This is a contradiction", "with $\\dim_\\delta(Z(s)) \\leq n - r$ because $r > 1$", "so this case doesn't happen." ], "refs": [ "chow-lemma-cap-cp-bivariant", "chow-lemma-geometric-cap", "chow-lemma-chow-ring-projective-bundle", "algebra-lemma-flat-increases-depth", "chow-lemma-geometric-cap", "algebra-lemma-grothendieck" ], "ref_ids": [ 5751, 5707, 5743, 740, 5707, 884 ] } ], "ref_ids": [] }, { "id": 5766, "type": "theorem", "label": "chow-lemma-easy-virtual-class", "categories": [ "chow" ], "title": "chow-lemma-easy-virtual-class", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$", "be a scheme locally of finite type over $S$. Let", "$$", "0 \\to \\mathcal{N}' \\to \\mathcal{N} \\to \\mathcal{E} \\to 0", "$$", "be a short exact sequence of finite locally free $\\mathcal{O}_X$-modules.", "Consider the closed embedding", "$$", "i :", "N' = \\underline{\\text{Spec}}_X(\\text{Sym}((\\mathcal{N}')^\\vee))", "\\longrightarrow", "N = \\underline{\\text{Spec}}_X(\\text{Sym}(\\mathcal{N}^\\vee))", "$$", "For $\\alpha \\in \\CH_k(X)$ we have", "$$", "i_*(p')^*\\alpha = p^*(c_{top}(\\mathcal{E}) \\cap \\alpha)", "$$", "where $p' : N' \\to X$ and $p : N \\to X$ are the structure morphisms." ], "refs": [], "proofs": [ { "contents": [ "Here $c_{top}(\\mathcal{E})$ is the bivariant class defined in", "Remark \\ref{remark-top-chern-class}. By its very definition, in", "order to verify the formula, we may assume that $\\mathcal{E}$", "has constant rank. We may similarly assume $\\mathcal{N}'$ and", "$\\mathcal{N}$ have constant ranks, say $r'$ and $r$, so", "$\\mathcal{E}$ has rank $r - r'$ and", "$c_{top}(\\mathcal{E}) = c_{r - r'}(\\mathcal{E})$.", "Observe that $p^*\\mathcal{E}$ has a canonical section", "$$", "s \\in \\Gamma(N, p^*\\mathcal{E}) = \\Gamma(X, p_*p^*\\mathcal{E}) =", "\\Gamma(X, \\mathcal{E} \\otimes_{\\mathcal{O}_X} \\text{Sym}(\\mathcal{N}^\\vee)", "\\supset \\Gamma(X, \\SheafHom(\\mathcal{N}, \\mathcal{E}))", "$$", "corresponding to the surjection $\\mathcal{N} \\to \\mathcal{E}$ given", "in the statement of the lemma. The vanishing scheme of this section", "is exactly $N' \\subset N$. Let $Y \\subset X$ be an integral closed", "subscheme of $\\delta$-dimension $n$. Then we have", "\\begin{enumerate}", "\\item $p^*[Y] = [p^{-1}(Y)]$ since $p^{-1}(Y)$ is integral of", "$\\delta$-dimension $n + r$,", "\\item $(p')^*[Y] = [(p')^{-1}(Y)]$ since $(p')^{-1}(Y)$ is integral of", "$\\delta$-dimension $n + r'$,", "\\item the restriction of $s$ to $p^{-1}Y$ has vanishing scheme", "$(p')^{-1}Y$ and the closed immersion $(p')^{-1}Y \\to p^{-1}Y$", "is a regular immersion (locally cut out by a regular sequence).", "\\end{enumerate}", "We conclude that", "$$", "(p')^*[Y] = c_{r - r'}(p^*\\mathcal{E}) \\cap p^*[Y]", "\\quad\\text{in}\\quad \\CH_*(N)", "$$", "by Lemma \\ref{lemma-top-chern-class}. This proves the lemma." ], "refs": [ "chow-remark-top-chern-class", "chow-lemma-top-chern-class" ], "ref_ids": [ 5947, 5765 ] } ], "ref_ids": [] }, { "id": 5767, "type": "theorem", "label": "chow-lemma-chern-character-additive", "categories": [ "chow" ], "title": "chow-lemma-chern-character-additive", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally", "of finite type over $S$. Let", "$", "0 \\to \\mathcal{E}_1 \\to \\mathcal{E} \\to \\mathcal{E}_2 \\to 0", "$", "be a short exact sequence of finite locally free $\\mathcal{O}_X$-modules.", "Then we have the equality", "$$", "ch(\\mathcal{E}) = ch(\\mathcal{E}_1) + ch(\\mathcal{E}_2)", "$$", "in $A^*(X) \\otimes \\mathbf{Q}$. More precisely, we have", "$P_p(\\mathcal{E}) = P_p(\\mathcal{E}_1) + P_p(\\mathcal{E}_2)$", "in $A^p(X)$ where $P_p$ is as in Example \\ref{example-power-sum}." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove the more precise statement. By", "Section \\ref{section-splitting-principle}", "this follows because if $x_{1, i}$, $i = 1, \\ldots, r_1$", "and $x_{2, i}$, $i = 1, \\ldots, r_2$ are the", "Chern roots of $\\mathcal{E}_1$ and $\\mathcal{E}_2$, then", "$x_{1, 1}, \\ldots, x_{1, r_1}, x_{2, 1}, \\ldots, x_{2, r_2}$", "are the Chern roots of $\\mathcal{E}$. Hence we get the result", "from our choice of $P_p$ in Example \\ref{example-power-sum}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5768, "type": "theorem", "label": "chow-lemma-chern-character-multiplicative", "categories": [ "chow" ], "title": "chow-lemma-chern-character-multiplicative", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally", "of finite type over $S$. Let $\\mathcal{E}_1$ and $\\mathcal{E}_2$", "be finite locally free $\\mathcal{O}_X$-modules.", "Then we have the equality", "$$", "ch(\\mathcal{E}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{E}_2) =", "ch(\\mathcal{E}_1) ch(\\mathcal{E}_2)", "$$", "in $A^*(X) \\otimes \\mathbf{Q}$. More precisely, we have", "$$", "P_p(\\mathcal{E}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{E}_2) =", "\\sum\\nolimits_{p_1 + p_2 = p}", "{p \\choose p_1} P_{p_1}(\\mathcal{E}_1) P_{p_2}(\\mathcal{E}_2)", "$$", "in $A^p(X)$ where $P_p$ is as in Example \\ref{example-power-sum}." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove the more precise statement. By", "Section \\ref{section-splitting-principle}", "this follows because if $x_{1, i}$, $i = 1, \\ldots, r_1$", "and $x_{2, i}$, $i = 1, \\ldots, r_2$ are the", "Chern roots of $\\mathcal{E}_1$ and $\\mathcal{E}_2$, then", "$x_{1, i} + x_{2, j}$, $1 \\leq i \\leq r_1$, $1 \\leq j \\leq r_2$", "are the Chern roots of $\\mathcal{E}_1 \\otimes \\mathcal{E}_2$.", "Hence we get the result from the binomial formula for", "$(x_{1, i} + x_{2, j})^p$ and the", "shape of our polynomials $P_p$ in Example \\ref{example-power-sum}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5769, "type": "theorem", "label": "chow-lemma-chern-character-dual", "categories": [ "chow" ], "title": "chow-lemma-chern-character-dual", "contents": [ "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module", "with dual $\\mathcal{E}^\\vee$. Then", "$ch_i(\\mathcal{E}^\\vee) = (-1)^i ch_i(\\mathcal{E})$ in $A^i(X)$." ], "refs": [], "proofs": [ { "contents": [ "Follows from the corresponding result for Chern classes", "(Lemma \\ref{lemma-chern-classes-dual})." ], "refs": [ "chow-lemma-chern-classes-dual" ], "ref_ids": [ 5763 ] } ], "ref_ids": [] }, { "id": 5770, "type": "theorem", "label": "chow-lemma-pre-derived-chern-class", "categories": [ "chow" ], "title": "chow-lemma-pre-derived-chern-class", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. Let $E \\in D(\\mathcal{O}_X)$", "be an object such that there exists a finite complex $\\mathcal{E}^\\bullet$", "of finite locally free $\\mathcal{O}_X$-modules representing $E$.", "Then $c(\\mathcal{E}^\\bullet) \\in A^*(X)$,", "$ch(\\mathcal{E}^\\bullet) \\in A^*(X) \\otimes \\mathbf{Q}$, and", "$P_p(\\mathcal{E}^\\bullet) \\in A^p(X)$", "are independent of the choice of the complex." ], "refs": [], "proofs": [ { "contents": [ "We prove this for the total Chern class; the other two cases follow", "by the same arguments using", "Lemma \\ref{lemma-chern-character-additive}", "instead of", "Lemma \\ref{lemma-additivity-chern-classes}.", "\\medskip\\noindent", "Suppose we have a second finite complex", "$\\mathcal{F}^\\bullet$ of finite locally free $\\mathcal{O}_X$-modules", "representing $E$.", "Let $g : Y \\to X$ be a morphism locally of finite type with $Y$ integral.", "By Lemma \\ref{lemma-bivariant-zero} it suffices to show that", "with $c(g^*\\mathcal{E}^\\bullet) \\cap [Y]$ is the same as", "$c(g^*\\mathcal{F}^\\bullet) \\cap [Y]$ and it even suffices to prove", "this after replacing $Y$ by an integral scheme proper and birational", "over $Y$. By", "More on Flatness, Lemma \\ref{flat-lemma-blowup-complex-integral}", "we may assume that $H^i(Lg^*E)$ is perfect of tor dimension $\\leq 1$", "for all $i \\in \\mathbf{Z}$.", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $X$ is integral and $H^i(E)$ is a perfect $\\mathcal{O}_X$-module", "of tor dimension $\\leq 1$ for all $i \\in \\mathbf{Z}$. We have to", "show that $c(\\mathcal{E}^\\bullet) \\cap [X]$ is the same as", "$c(\\mathcal{F}^\\bullet) \\cap [X]$. Denote", "$d_\\mathcal{E}^i : \\mathcal{E}^i \\to \\mathcal{E}^{i + 1}$ and", "$d_\\mathcal{F}^i : \\mathcal{F}^i \\to \\mathcal{F}^{i + 1}$", "the differentials of our complexes. By", "More on Flatness, Remark \\ref{flat-remark-when-you-have-a-complex}", "we know that $\\Im(d_\\mathcal{E}^i)$, $\\Ker(d_\\mathcal{E}^i)$,", "$\\Im(d_\\mathcal{F}^i)$, and $\\Ker(d_\\mathcal{F}^i)$", "are finite locally free $\\mathcal{O}_X$-modules for all $i$.", "By additivity (Lemma \\ref{lemma-additivity-chern-classes}) we see that", "$$", "c(\\mathcal{E}^\\bullet) = \\prod\\nolimits_i", "c(\\Ker(d_\\mathcal{E}^i))^{(-1)^i} c(\\Im(d_\\mathcal{E}^i))^{(-1)^i}", "$$", "and similarly for $\\mathcal{F}^\\bullet$. Since we have the", "short exact sequences", "$$", "0 \\to \\Im(d_\\mathcal{E}^i) \\to \\Ker(d_\\mathcal{E}^i) \\to H^i(E) \\to 0", "\\quad\\text{and}\\quad", "0 \\to \\Im(d_\\mathcal{F}^i) \\to \\Ker(d_\\mathcal{F}^i) \\to H^i(E) \\to 0", "$$", "we reduce to the problem stated and solved in the next paragraph.", "\\medskip\\noindent", "Assume $X$ is integral and we have two short exact sequences", "$$", "0 \\to \\mathcal{E}' \\to \\mathcal{E} \\to \\mathcal{Q} \\to 0", "\\quad\\text{and}\\quad", "0 \\to \\mathcal{F}' \\to \\mathcal{F} \\to \\mathcal{Q} \\to 0", "$$", "with $\\mathcal{E}$, $\\mathcal{E}'$, $\\mathcal{F}$, $\\mathcal{F}'$", "finite locally free. Problem: show that", "$c(\\mathcal{E})c(\\mathcal{E}')^{-1} \\cap [X] =", "c(\\mathcal{F})c(\\mathcal{F}')^{-1} \\cap [X]$.", "To do this, consider the short exact sequence", "$$", "0 \\to \\mathcal{G} \\to \\mathcal{E} \\oplus \\mathcal{F} \\to \\mathcal{Q} \\to 0", "$$", "defining $\\mathcal{G}$. Since $\\mathcal{Q}$ has tor dimension $\\leq 1$", "we see that $\\mathcal{G}$ is finite locally free. A diagram chase", "shows that the kernel of the surjection $\\mathcal{G} \\to \\mathcal{F}$", "maps isomorphically to $\\mathcal{E}'$ in $\\mathcal{E}$ and", "the kernel of the surjection $\\mathcal{G} \\to \\mathcal{E}$ maps", "isomorphically to $\\mathcal{F}'$ in $\\mathcal{F}$. (Working affine", "locally this follows from or is equivalent to Schanuel's lemma, see", "Algebra, Lemma \\ref{algebra-lemma-Schanuel}.)", "We conclude that", "$$", "c(\\mathcal{E})c(\\mathcal{F}') = c(\\mathcal{G}) =", "c(\\mathcal{F})c(\\mathcal{E}')", "$$", "as desired." ], "refs": [ "chow-lemma-chern-character-additive", "chow-lemma-additivity-chern-classes", "chow-lemma-bivariant-zero", "flat-lemma-blowup-complex-integral", "flat-remark-when-you-have-a-complex", "chow-lemma-additivity-chern-classes", "algebra-lemma-Schanuel" ], "ref_ids": [ 5767, 5756, 5740, 6181, 6235, 5756, 965 ] } ], "ref_ids": [] }, { "id": 5771, "type": "theorem", "label": "chow-lemma-commutative-chern-perfect", "categories": [ "chow" ], "title": "chow-lemma-commutative-chern-perfect", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $E \\in D(\\mathcal{O}_X)$ be perfect. If the Chern classes", "of $E$ are defined then", "\\begin{enumerate}", "\\item $c_p(E)$ is in the center of the algebra $A^*(X)$ and", "\\item if $f : X' \\to X$ is locally of finite type and $c \\in A^*(X' \\to X)$,", "then $c \\circ c_j(E) = c_j(Lf^*E) \\circ c$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-cap-commutative-chern} and the", "construction." ], "refs": [ "chow-lemma-cap-commutative-chern" ], "ref_ids": [ 5752 ] } ], "ref_ids": [] }, { "id": 5772, "type": "theorem", "label": "chow-lemma-additivity-on-perfect", "categories": [ "chow" ], "title": "chow-lemma-additivity-on-perfect", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. Let", "$$", "E_1 \\to E_2 \\to E_3 \\to E_1[1]", "$$", "be a distinguished triangle of perfect objects in $D(\\mathcal{O}_X)$.", "If $E_1 \\to E_2$ can be represented be a map of bounded complexes", "of finite locally free $\\mathcal{O}_X$-modules, then we have", "$c(E_2) = c(E_1) c(E_3)$, $ch(E_2) = ch(E_1) + ch(E_3)$, and", "$P_p(E_2) = P_p(E_1) + P_p(E_3)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha^\\bullet : \\mathcal{E}_1^\\bullet \\to \\mathcal{E}_2^\\bullet$", "be a map of bounded complexes of finite locally free $\\mathcal{O}_X$-modules", "representing $E_1 \\to E_2$. Then the cone $C(\\alpha)^\\bullet$", "represents $E_3$. Since", "$C(\\alpha)^n = \\mathcal{E}_2^n \\oplus \\mathcal{E}_1^{n + 1}$", "the formulas follow from the additivity and multiplicativity", "of Lemmas \\ref{lemma-chern-character-additive} and", "\\ref{lemma-additivity-chern-classes}." ], "refs": [ "chow-lemma-chern-character-additive", "chow-lemma-additivity-chern-classes" ], "ref_ids": [ 5767, 5756 ] } ], "ref_ids": [] }, { "id": 5773, "type": "theorem", "label": "chow-lemma-chern-classes-perfect-dual", "categories": [ "chow" ], "title": "chow-lemma-chern-classes-perfect-dual", "contents": [ "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.", "Let $E \\in D(\\mathcal{O}_X)$ be a perfect object whose Chern classes are", "defined. Then $c_i(E^\\vee) = (-1)^i c_i(E)$, $P_i(E^\\vee) = (-1)^iP_i(E)$,", "and $ch_i(E^\\vee) = (-1)^ich_i(E)$ in $A^i(X)$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definition and Lemma \\ref{lemma-chern-classes-dual}." ], "refs": [ "chow-lemma-chern-classes-dual" ], "ref_ids": [ 5763 ] } ], "ref_ids": [] }, { "id": 5774, "type": "theorem", "label": "chow-lemma-chern-class-perfect-tensor-invertible", "categories": [ "chow" ], "title": "chow-lemma-chern-class-perfect-tensor-invertible", "contents": [ "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.", "Let $E$ be a perfect object of $D(\\mathcal{O}_X)$ whose Chern classes", "are defined.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then", "$$", "c_i(E \\otimes \\mathcal{L}) =", "\\sum\\nolimits_{j = 0}^i", "\\binom{r - i + j}{j} c_{i - j}(E) c_1(\\mathcal{L})^j", "$$", "provided $E$ has constant rank $r \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "In the case where $E$ is locally free of rank $r$ this is", "Lemma \\ref{lemma-chern-classes-E-tensor-L}. The reader can deduce", "the lemma from this special case by a formal computation.", "An alternative is to use the splitting principle of", "Remark \\ref{remark-splitting-principle-perfect}.", "In this case one ends up having to prove the following", "algebra fact: if we write formally", "$$", "\\frac{\\prod_{a = 1, \\ldots, n} (1 + x_a)}{\\prod_{n = 1, \\ldots, m} (1 + y_b)}", "= 1 + c_1 + c_2 + c_3 + \\ldots", "$$", "with $c_i$ homogeneous of degree $i$", "in $\\mathbf{Z}[x_i, y_j]$ then we have", "$$", "\\frac{\\prod_{a = 1, \\ldots, n} (1 + x_a + t)}{\\prod_{b = 1, \\ldots, m} (1 + y_b + t)}", "= \\sum\\nolimits_{i \\geq 0} \\sum\\nolimits_{j = 0}^i", "\\binom{r - i + j}{j} c_{i - j} t^j", "$$", "where $r = n - m$. We omit the details." ], "refs": [ "chow-lemma-chern-classes-E-tensor-L", "chow-remark-splitting-principle-perfect" ], "ref_ids": [ 5753, 5952 ] } ], "ref_ids": [] }, { "id": 5775, "type": "theorem", "label": "chow-lemma-chern-classes-perfect-tensor-product", "categories": [ "chow" ], "title": "chow-lemma-chern-classes-perfect-tensor-product", "contents": [ "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.", "Let $E$ and $F$ be perfect objects of $D(\\mathcal{O}_X)$ whose Chern classes", "are defined. Then we have", "$$", "c_1(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F) =", "r(E) c_1(\\mathcal{F}) + r(F) c_1(\\mathcal{E})", "$$", "and for $c_2(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F)$ we have the expression", "$$", "r(E) c_2(F) + r(F) c_2(E) + {r(E) \\choose 2} c_1(F)^2 +", "(r(E)r(F) - 1) c_1(F)c_1(E) + {r(F) \\choose 2} c_1(E)^2", "$$", "and so on for higher Chern classes in $A^*(X)$. Similarly, we have", "$ch(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F) = ch(E) ch(F)$", "in $A^*(X) \\otimes \\mathbf{Q}$. More precisely, we have", "$$", "P_p(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F) = \\sum\\nolimits_{p_1 + p_2 = p}", "{p \\choose p_1} P_{p_1}(E) P_{p_2}(F)", "$$", "in $A^p(X)$." ], "refs": [], "proofs": [ { "contents": [ "After representing $E$ and $F$ by bounded complexes of finite locally", "free $\\mathcal{O}_X$-modules this follows by a compuation from the", "corresponding result for vector bundles in", "Lemmas \\ref{lemma-chern-classes-tensor-product} and", "\\ref{lemma-chern-character-multiplicative} by a calculation.", "A better proof is probably to use the splitting principle as in", "Remark \\ref{remark-splitting-principle-perfect}", "and reduce the lemma to computations in polynomial rings ", "which we describe in the next paragraph.", "\\medskip\\noindent", "Let $A$ be a commutative ring (for us this will be the subring of the", "bivariant chow ring of $X$ generated by Chern classes).", "Let $S$ be a finite set together with maps $\\epsilon : S \\to \\{\\pm 1\\}$", "and $f : S \\to A$. Define", "$$", "P_p(S, f , \\epsilon) = \\sum\\nolimits_{s \\in S} \\epsilon(s) f(s)^p", "$$", "in $A$. Given a second triple $(S', \\epsilon', f')$", "the equality that has to be shown for $P_p$ is the equality", "$$", "P_p(S \\times S', f + f' , \\epsilon \\epsilon') = ", "\\sum\\nolimits_{p_1 + p_2 = p}", "{p \\choose p_1} P_{p_1}(S, f, \\epsilon) P_{p_2}(S', f', \\epsilon')", "$$", "To see this is true, one reduces to the polynomial ring on variables", "$S \\amalg S'$ and one shows that each term $f(s)^if'(s')^j$ occurs", "on the left and right hand side with the same coefficient.", "To verify the formulas for $c_1(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F)$", "and $c_2(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F)$ we use the splitting", "principle to reduce to checking these formulae in a torsion free ring.", "Then we use the relationship between $P_j(E)$ and $c_i(E)$ proved", "in Remark \\ref{remark-splitting-principle-perfect}. For example", "$$", "c_1(E \\otimes F) = P_1(E \\otimes F) = r(F)P_1(E) + r(E)P_1(F) =", "r(F)c_1(E) + r(E)c_1(F)", "$$", "the middle equation because $r(E) = P_0(E)$ by definition. Similarly, we have", "\\begin{align*}", "& 2c_2(E \\otimes F) \\\\", "& = c_1(E \\otimes F)^2 - P_2(E \\otimes F) \\\\", "& =", "(r(F)c_1(E) + r(E)c_1(F))^2 -", "r(F)P_2(E) - P_1(E)P_1(F) - r(E)P_2(F) \\\\", "& =", "(r(F)c_1(E) + r(E)c_1(F))^2 -", "r(F)(c_1(E)^2 - 2c_2(E)) - c_1(E)c_1(F) - \\\\", "& \\quad r(E)(c_1(F)^2 - 2c_2(F))", "\\end{align*}", "which the reader can verify agrees with the formula in the statement", "of the lemma up to a factor of $2$." ], "refs": [ "chow-lemma-chern-classes-tensor-product", "chow-lemma-chern-character-multiplicative", "chow-remark-splitting-principle-perfect", "chow-remark-splitting-principle-perfect" ], "ref_ids": [ 5764, 5768, 5952, 5952 ] } ], "ref_ids": [] }, { "id": 5776, "type": "theorem", "label": "chow-lemma-silly", "categories": [ "chow" ], "title": "chow-lemma-silly", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be", "locally of finite type over $S$. Let $i_j : X_j \\to X$, $j = 1, 2$", "be closed immersions such that $X = X_1 \\cup X_2$ set theoretically. Let", "$E_2 \\in D(\\mathcal{O}_{X_2})$ be a perfect object. Assume", "\\begin{enumerate}", "\\item Chern classes of $E_2$ are defined,", "\\item the restriction $E_2|_{X_1 \\cap X_2}$ is zero,", "resp.\\ isomorphic to a finite locally free $\\mathcal{O}_{X_1 \\cap X_2}$-module", "of rank $< p$ sitting in cohomological degree $0$.", "\\end{enumerate}", "Then there is a canonical bivariant class", "$$", "P'_p(E_2),\\text{ resp. }c'_p(E_2) \\in A^p(X_2 \\to X)", "$$", "characterized by the property", "$$", "P'_p(E_2) \\cap i_{2, *} \\alpha_2 = P_p(E_2) \\cap \\alpha_2", "\\quad\\text{and}\\quad", "P'_p(E_2) \\cap i_{1, *} \\alpha_1 = 0,", "$$", "respectively", "$$", "c'_p(E_2) \\cap i_{2, *} \\alpha_2 = c_p(E_2) \\cap \\alpha_2", "\\quad\\text{and}\\quad", "c'_p(E_2) \\cap i_{1, *} \\alpha_1 = 0", "$$", "for $\\alpha_i \\in \\CH_k(X_i)$ and similarly after any base change", "$X' \\to X$ locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "We are going to use the material of Section \\ref{section-pre-derived}", "without further mention.", "\\medskip\\noindent", "Assume $E_2|_{X_1 \\cap X_2}$ is zero.", "Consider a morphism of schemes $X' \\to X$", "which is locally of finite type and denote $i'_j : X'_j \\to X'$ the", "base change of $i_j$. By Lemma \\ref{lemma-exact-sequence-closed-chow}", "we can write any element $\\alpha' \\in \\CH_k(X')$ as", "$i'_{1, *}\\alpha'_1 + i'_{2, *}\\alpha'_2$ where", "$\\alpha'_2 \\in \\CH_k(X'_2)$", "is well defined up to an element in the image of pushforward", "by $X'_1 \\cap X'_2 \\to X'_2$. Then we can set", "$P'_p(E_2) \\cap \\alpha' = P_p(E_2) \\cap \\alpha'_2 \\in \\CH_{k - p}(X'_2)$. This", "is well defined by our assumption that $E_2$ restricts", "to zero on $X_1 \\cap X_2$.", "\\medskip\\noindent", "If $E_2|_{X_1 \\cap X_2}$ is isomorphic to a finite locally free", "$\\mathcal{O}_{X_1 \\cap X_2}$-module of rank $< p$ sitting in", "cohomological degree $0$, then $c_p(E_2|_{X_1 \\cap X_2}) = 0$", "by rank considerations and we can argue in exactly the same manner." ], "refs": [ "chow-lemma-exact-sequence-closed-chow" ], "ref_ids": [ 5691 ] } ], "ref_ids": [] }, { "id": 5777, "type": "theorem", "label": "chow-lemma-silly-independent", "categories": [ "chow" ], "title": "chow-lemma-silly-independent", "contents": [ "In Lemma \\ref{lemma-silly} the bivariant class", "$P'_p(E_2)$, resp.\\ $c'_p(E_2)$ in $A^p(X_2 \\to X)$", "does not depend on the choice of $X_1$." ], "refs": [ "chow-lemma-silly" ], "proofs": [ { "contents": [ "Suppose that $X_1' \\subset X$ is another closed subscheme such that", "$X = X'_1 \\cup X_2$ set theoretically and the restriction", "$E_2|_{X'_1 \\cap X_2}$ is zero, resp.\\ isomorphic to a", "finite locally free $\\mathcal{O}_{X'_1 \\cap X_2}$-module", "of rank $< p$ sitting in cohomological degree $0$.", "Then $X = (X_1 \\cap X'_1) \\cup X_2$. Hence we can write", "any element $\\alpha \\in \\CH_k(X)$ as $i_*\\beta + i_{2, *}\\alpha_2$ with", "$\\alpha_2 \\in \\CH_k(X'_2)$ and $\\beta \\in \\CH_k(X_1 \\cap X'_1)$.", "Thus it is clear that", "$P'_p(E_2) \\cap \\alpha = P_p(E_2) \\cap \\alpha_2 \\in \\CH_{k - p}(X_2)$,", "resp.\\ $c'_p(E_2) \\cap \\alpha = c_p(E_2) \\cap \\alpha_2 \\in \\CH_{k - p}(X_2)$,", "is independent of whether we use $X_1$ or $X'_1$. Similarly", "after any base change." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 5776 ] }, { "id": 5778, "type": "theorem", "label": "chow-lemma-silly-silly", "categories": [ "chow" ], "title": "chow-lemma-silly-silly", "contents": [ "In Lemma \\ref{lemma-silly} say $E_2$ is the restriction of a", "perfect $E \\in D(\\mathcal{O}_X)$ such that $E|_{X_1}$ is zero,", "resp.\\ isomorphic to a finite locally free $\\mathcal{O}_{X_1}$-module", "of rank $< p$ sitting in cohomological degree $0$.", "If Chern classes of $E$ are defined, then", "$i_{2, *} \\circ P'_p(E_2) = P_p(E)$,", "resp.\\ $i_{2, *} \\circ c'_p(E_2) = c_p(E)$", "(with $\\circ$ as in Lemma \\ref{lemma-push-proper-bivariant})." ], "refs": [ "chow-lemma-silly", "chow-lemma-push-proper-bivariant" ], "proofs": [ { "contents": [ "First, assume $E|_{X_1}$ is zero.", "With notations as in the proof of Lemma \\ref{lemma-silly}", "the lemma in this case follows from", "\\begin{align*}", "P_p(E) \\cap \\alpha'", "& =", "i'_{1, *}(P_p(E) \\cap \\alpha'_1) +", "i'_{2, *}(P_p(E) \\cap \\alpha'_2) \\\\", "& =", "i'_{1, *}(P_p(E|_{X_1}) \\cap \\alpha'_1) +", "i'_{2, *}(P'_p(E_2) \\cap \\alpha') \\\\", "& =", "i'_{2, *}(P'_p(E_2) \\cap \\alpha')", "\\end{align*}", "The case where $E|_{X_1}$ is isomorphic to a finite locally free", "$\\mathcal{O}_{X_1}$-module of rank $< p$ sitting in cohomological degree $0$", "is similar." ], "refs": [ "chow-lemma-silly" ], "ref_ids": [ 5776 ] } ], "ref_ids": [ 5776, 5734 ] }, { "id": 5779, "type": "theorem", "label": "chow-lemma-silly-shrink", "categories": [ "chow" ], "title": "chow-lemma-silly-shrink", "contents": [ "In Lemma \\ref{lemma-silly} suppose we have closed subschemes", "$X'_2 \\subset X_2$ and $X_1 \\subset X'_1 \\subset X$ such that", "$X = X'_1 \\cup X'_2$ set theoretically. Assume $E_2|_{X'_1 \\cap X_2}$", "is zero, resp.\\ isomorphic to a finite locally free module", "of rank $< p$ placed in degree $0$. Then we have", "$(X'_2 \\to X_2)_* \\circ P'_p(E_2|_{X'_2}) = P'_p(E_2)$,", "resp.\\ $(X'_2 \\to X_2)_* \\circ c'_p(E_2|_{X'_2}) = c_p(E_2)$", "(with $\\circ$ as in Lemma \\ref{lemma-push-proper-bivariant})." ], "refs": [ "chow-lemma-silly", "chow-lemma-push-proper-bivariant" ], "proofs": [ { "contents": [ "This follows immediately from the characterization of these classes", "in Lemma \\ref{lemma-silly}." ], "refs": [ "chow-lemma-silly" ], "ref_ids": [ 5776 ] } ], "ref_ids": [ 5776, 5734 ] }, { "id": 5780, "type": "theorem", "label": "chow-lemma-silly-commutes", "categories": [ "chow" ], "title": "chow-lemma-silly-commutes", "contents": [ "In Lemma \\ref{lemma-silly} let $f : Y \\to X$ be locally of finite type", "and say $c \\in A^*(Y \\to X)$. Then", "$$", "c \\circ P'_p(E_2) = P'_p(Lf_2^*E_2) \\circ c", "\\quad\\text{resp.}\\quad", "c \\circ c'_p(E_2) = c'_p(Lf_2^*E_2) \\circ c", "$$", "in $A^*(Y_2 \\to Y)$ where $f_2 : Y_2 \\to X_2$ is the base change of $f$." ], "refs": [ "chow-lemma-silly" ], "proofs": [ { "contents": [ "Let $\\alpha \\in \\CH_k(X)$. We may write", "$$", "\\alpha = \\alpha_1 + \\alpha_2", "$$", "with $\\alpha_i \\in \\CH_k(X_i)$; we are omitting the pushforwards", "by the closed immersions $X_i \\to X$. The reader then checks that", "$c'_p(E_2) \\cap \\alpha = c_p(E_2) \\cap \\alpha_2$,", "$c \\cap c'_p(E_2) \\cap \\alpha = c \\cap c_p(E_2) \\cap \\alpha_2$,", "$c \\cap \\alpha = c \\cap \\alpha_1 + c \\cap \\alpha_2$, and", "$c'_p(Lf_2^*E_2) \\cap c \\cap \\alpha = c_p(Lf_2^*E_2) \\cap c \\cap \\alpha_2$.", "We conclude by Lemma \\ref{lemma-commutative-chern-perfect}." ], "refs": [ "chow-lemma-commutative-chern-perfect" ], "ref_ids": [ 5771 ] } ], "ref_ids": [ 5776 ] }, { "id": 5781, "type": "theorem", "label": "chow-lemma-silly-compose", "categories": [ "chow" ], "title": "chow-lemma-silly-compose", "contents": [ "In Lemma \\ref{lemma-silly} assume $E_2|_{X_1 \\cap X_2}$ is zero. Then", "\\begin{align*}", "P'_1(E_2) & = c'_1(E_2), \\\\", "P'_2(E_2) & = c'_1(E_2)^2 - 2c'_2(E_2), \\\\", "P'_3(E_2) & = c'_1(E_2)^3 - 3c'_1(E_2)c'_2(E_2) + 3c'_3(E_2), \\\\", "P'_4(E_2) & = c'_1(E_2)^4 - 4c'_1(E_2)^2c'_2(E_2) +", "4c'_1(E_2)c'_3(E_2) + 2c'_2(E_2)^2 - 4c'_4(E_2),", "\\end{align*}", "and so on with multiplication as in Remark \\ref{remark-ring-loc-classes}." ], "refs": [ "chow-lemma-silly", "chow-remark-ring-loc-classes" ], "proofs": [ { "contents": [ "The statement makes sense because the zero sheaf has rank $< 1$ and", "hence the classes $c'_p(E_2)$ are defined for all $p \\geq 1$. The equalities", "follow immediately from the characterization of the classes produced", "by Lemma \\ref{lemma-silly} and the corresponding result for", "capping with the Chern classes of $E_2$ given in", "Remark \\ref{remark-splitting-principle-perfect}." ], "refs": [ "chow-lemma-silly", "chow-remark-splitting-principle-perfect" ], "ref_ids": [ 5776, 5952 ] } ], "ref_ids": [ 5776, 5942 ] }, { "id": 5782, "type": "theorem", "label": "chow-lemma-silly-sum-c", "categories": [ "chow" ], "title": "chow-lemma-silly-sum-c", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be", "locally of finite type over $S$. Let $i_j : X_j \\to X$, $j = 1, 2$", "be closed immersions such that $X = X_1 \\cup X_2$ set theoretically. Let", "$E, F \\in D(\\mathcal{O}_X)$ be perfect objects. Assume", "\\begin{enumerate}", "\\item Chern classes of $E$ and $F$ are defined,", "\\item the restrictions $E|_{X_1 \\cap X_2}$ and $F|_{X_1 \\cap X_2}$", "are isomorphic to a finite locally free $\\mathcal{O}_{X_1}$-modules", "of rank $< p$ and $< q$ sitting in cohomological degree $0$.", "\\end{enumerate}", "With notation as in Remark \\ref{remark-ring-loc-classes} set", "$$", "c^{(p)}(E) = 1 + c_1(E) + \\ldots + c_{p - 1}(E) +", "c'_p(E|_{X_2}) + c'_{p + 1}(E|_{X_2}) + \\ldots \\in A^{(p)}(X_2 \\to X)", "$$", "with $c'_p(E|_{X_2})$ as in Lemma \\ref{lemma-silly}. Similarly", "for $c^{(q)}(F)$ and $c^{(p + q)}(E \\oplus F)$.", "Then $c^{(p + q)}(E \\oplus F) = c^{(p)}(E)c^{(q)}(F)$", "in $A^{(p + q)}(X_2 \\to X)$." ], "refs": [ "chow-remark-ring-loc-classes", "chow-lemma-silly" ], "proofs": [ { "contents": [ "Immediate from the characterization of the classes in", "Lemma \\ref{lemma-silly} and the additivity in", "Lemma \\ref{lemma-additivity-on-perfect}." ], "refs": [ "chow-lemma-silly", "chow-lemma-additivity-on-perfect" ], "ref_ids": [ 5776, 5772 ] } ], "ref_ids": [ 5942, 5776 ] }, { "id": 5783, "type": "theorem", "label": "chow-lemma-silly-sum-P", "categories": [ "chow" ], "title": "chow-lemma-silly-sum-P", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be", "locally of finite type over $S$. Let $i_j : X_j \\to X$, $j = 1, 2$", "be closed immersions such that $X = X_1 \\cup X_2$ set theoretically. Let", "$E, F \\in D(\\mathcal{O}_{X_2})$ be perfect objects. Assume", "\\begin{enumerate}", "\\item Chern classes of $E$ and $F$ are defined,", "\\item the restrictions $E|_{X_1 \\cap X_2}$ and $F|_{X_1 \\cap X_2}$ are zero,", "\\end{enumerate}", "Denote $P'_p(E), P'_p(F), P'_p(E \\oplus F) \\in A^p(X_2 \\to X)$ for $p \\geq 0$", "the classes constructed in Lemma \\ref{lemma-silly}. Then", "$P'_p(E \\oplus F) = P'_p(E) + P'_p(F)$." ], "refs": [ "chow-lemma-silly" ], "proofs": [ { "contents": [ "Immediate from the characterization of the classes in", "Lemma \\ref{lemma-silly} and the additivity in", "Lemma \\ref{lemma-additivity-on-perfect}." ], "refs": [ "chow-lemma-silly", "chow-lemma-additivity-on-perfect" ], "ref_ids": [ 5776, 5772 ] } ], "ref_ids": [ 5776 ] }, { "id": 5784, "type": "theorem", "label": "chow-lemma-silly-tensor-invertible", "categories": [ "chow" ], "title": "chow-lemma-silly-tensor-invertible", "contents": [ "In Lemma \\ref{lemma-silly} assume $E_2$ has constant rank $0$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then", "$$", "c'_i(E_2 \\otimes \\mathcal{L}) =", "\\sum\\nolimits_{j = 0}^i", "\\binom{- i + j}{j} c'_{i - j}(E_2) c_1(\\mathcal{L})^j", "$$" ], "refs": [ "chow-lemma-silly" ], "proofs": [ { "contents": [ "The assumption on rank implies that $E_2|_{X_1 \\cap X_2}$ is zero.", "Hence $c'_i(E_2)$ is defined for all $i \\geq 1$ and the statement", "makes sense. The actual equality follows", "immediately from Lemma \\ref{lemma-chern-class-perfect-tensor-invertible}", "and the characterization of $c'_i$ in Lemma \\ref{lemma-silly}." ], "refs": [ "chow-lemma-chern-class-perfect-tensor-invertible", "chow-lemma-silly" ], "ref_ids": [ 5774, 5776 ] } ], "ref_ids": [ 5776 ] }, { "id": 5785, "type": "theorem", "label": "chow-lemma-silly-tensor-product", "categories": [ "chow" ], "title": "chow-lemma-silly-tensor-product", "contents": [ "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.", "Let", "$$", "X = X_1 \\cup X_2 = X'_1 \\cup X'_2", "$$", "be two ways of writing $X$ as a set theoretic union of closed subschemes.", "Let $E$, $E'$ be perfect objects of $D(\\mathcal{O}_X)$", "whose Chern classes are defined.", "Assume that $E|_{X_1}$ and $E'|_{X'_1}$ are zero\\footnote{Presumably there", "is a variant of this lemma where we only assume these restrictions are", "isomorphic to a finite locally free modules", "of rank $< p$ and $< p'$.} for $i = 1, 2$. Denote", "\\begin{enumerate}", "\\item $r = P'_0(E) \\in A^0(X_2 \\to X)$ and", "$r' = P'_0(E') \\in A^0(X'_2 \\to X)$,", "\\item $\\gamma_p = c'_p(E|_{X_2}) \\in A^p(X_2 \\to X)$ and", "$\\gamma'_p = c'_p(E'|_{X'_2}) \\in A^p(X'_2 \\to X)$,", "\\item $\\chi_p = P'_p(E|_{X_2}) \\in A^p(X_2 \\to X)$ and", "$\\chi'_p = P'_p(E'|_{X'_2}) \\in A^p(X'_2 \\to X)$", "\\end{enumerate}", "the classes constructed in Lemma \\ref{lemma-silly}. Then we have", "$$", "c'_1((E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E')|_{X_2 \\cap X'_2}) =", "r \\gamma'_1 + r' \\gamma_1", "$$", "in $A^1(X_2 \\cap X'_2 \\to X)$ and", "$$", "c'_2((E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E')|_{X_2 \\cap X'_2}) =", "r \\gamma'_2 + r' \\gamma_2 + {r \\choose 2} (\\gamma'_1)^2 +", "(rr' - 1) \\gamma'_1\\gamma_1 + {r' \\choose 2} \\gamma_1^2", "$$", "in $A^2(X_2 \\cap X'_2 \\to X)$ and so on for higher Chern classes.", "Similarly, we have", "$$", "P'_p((E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E')|_{X_2 \\cap X'_2}) =", "\\sum\\nolimits_{p_1 + p_2 = p}", "{p \\choose p_1} \\chi_{p_1} \\chi'_{p_2}", "$$", "in $A^p(X_2 \\cap X'_2 \\to X)$." ], "refs": [ "chow-lemma-silly" ], "proofs": [ { "contents": [ "First we observe that the statement makes sense. Namely, we have", "$X = (X_2 \\cap X'_2) \\cup Y$ where", "$Y = (X_1 \\cap X'_1) \\cup (X_1 \\cap X'_2) \\cup (X_2 \\cap X'_1)$", "and the object $E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E'$", "restricts to zero on $Y$.", "The actual equalities follow from the characterization", "of our classes in Lemma \\ref{lemma-silly}", "and the equalities of Lemma \\ref{lemma-chern-classes-perfect-tensor-product}.", "We omit the details." ], "refs": [ "chow-lemma-silly", "chow-lemma-chern-classes-perfect-tensor-product" ], "ref_ids": [ 5776, 5775 ] } ], "ref_ids": [ 5776 ] }, { "id": 5786, "type": "theorem", "label": "chow-lemma-gysin-at-infty", "categories": [ "chow" ], "title": "chow-lemma-gysin-at-infty", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. Let", "$b : W \\to \\mathbf{P}^1_X$ be a proper morphism of schemes", "which is an isomorphism over $\\mathbf{A}^1_X$.", "Denote $i_\\infty : W_\\infty \\to W$ the inverse image of the divisor", "$D_\\infty \\subset \\mathbf{P}^1_X$ with complement $\\mathbf{A}^1_X$.", "Then there is a canonical bivariant class", "$$", "C \\in A^0(W_\\infty \\to X)", "$$", "with the property that", "$i_{\\infty, *}(C \\cap \\alpha) = i_{0, *}\\alpha$", "for $\\alpha \\in \\CH_k(X)$ and similarly after any base change by", "$X' \\to X$ locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "Given $\\alpha \\in \\CH_k(X)$ there exists a $\\beta \\in \\CH_{k + 1}(W)$", "restricting to the flat pullback of $\\alpha$ on $b^{-1}(\\mathbf{A}^1_X)$, see", "Lemma \\ref{lemma-exact-sequence-open}.", "A second choice of $\\beta$ differs from $\\beta$ by a cycle", "supported on $W_\\infty$, see", "Lemma \\ref{lemma-restrict-to-open}. Since the normal bundle of the effective", "Cartier divisor $D_\\infty \\subset \\mathbf{P}^1_X$ of", "(\\ref{equation-zero-infty}) is trivial,", "the gysin homomorphism $i_\\infty^*$ kills cycle classes", "supported on $W_\\infty$, see Remark \\ref{remark-gysin-on-cycles}.", "Hence setting $C \\cap \\alpha = i_\\infty^*\\beta$ is well defined.", "\\medskip\\noindent", "Since $W_\\infty$ and $W_0 = X \\times \\{0\\}$", "are the pullbacks of the rationally equivalent effective Cartier divisors", "$D_0, D_\\infty$ in $\\mathbf{P}^1_X$, we see that $i_\\infty^*\\beta$ and", "$i_0^*\\beta$ map to the same cycle class on $W$; namely, both", "represent the class $c_1(\\mathcal{O}_{\\mathbf{P}^1_X}(1)) \\cap \\beta$ by", "Lemma \\ref{lemma-support-cap-effective-Cartier}. By our choice of", "$\\beta$ we have $i_0^*\\beta = \\alpha$ as cycles on", "$W_0 = X \\times \\{0\\}$, see for example", "Lemma \\ref{lemma-relative-effective-cartier}.", "Thus we see that $i_{\\infty, *}(C \\cap \\alpha) = i_{0, *}\\alpha$", "as stated in the lemma.", "\\medskip\\noindent", "Observe that the assumptions on $b$ are preserved by any base change", "by $X' \\to X$ locally of finite type. Hence we get an operation", "$C \\cap - : \\CH_k(X') \\to \\CH_k(W'_\\infty)$ by the same construction as above.", "To see that this family of operations defines a bivariant class,", "we consider the diagram", "$$", "\\xymatrix{", "& & & \\CH_*(X) \\ar[d]^{\\text{flat pullback}} \\\\", "\\CH_{* + 1}(W_\\infty) \\ar[r] \\ar[rd]^0 &", "\\CH_{* + 1}(W) \\ar[d]^{i_\\infty^*} \\ar[rr]^{\\text{flat pullback}} & &", "\\CH_{* + 1}(\\mathbf{A}^1_X) \\ar[r] \\ar@{..>}[lld]^{C \\cap -} &", "0 \\\\", "& \\CH_*(W_\\infty)", "}", "$$", "for $X$ as indicated and the base change of this diagram for any $X' \\to X$.", "We know that flat pullback and $i_\\infty^*$ are bivariant operations, see", "Lemmas \\ref{lemma-flat-pullback-bivariant} and \\ref{lemma-gysin-bivariant}.", "Then a formal argument (involving huge diagrams of schemes and their", "chow groups) shows that the dotted arrow is a bivariant operation." ], "refs": [ "chow-lemma-exact-sequence-open", "chow-lemma-restrict-to-open", "chow-remark-gysin-on-cycles", "chow-lemma-support-cap-effective-Cartier", "chow-lemma-relative-effective-cartier", "chow-lemma-flat-pullback-bivariant", "chow-lemma-gysin-bivariant" ], "ref_ids": [ 5679, 5690, 5935, 5716, 5725, 5732, 5733 ] } ], "ref_ids": [] }, { "id": 5787, "type": "theorem", "label": "chow-lemma-gysin-at-infty-independent", "categories": [ "chow" ], "title": "chow-lemma-gysin-at-infty-independent", "contents": [ "In Lemma \\ref{lemma-gysin-at-infty} let $g : W' \\to W$ be a proper morphism", "which is an isomorphism over $\\mathbf{A}^1_X$. Let", "$C' \\in A^0(W'_\\infty \\to X)$ and $C \\in A^0(W_\\infty \\to X)$", "be the classes constructed in Lemma \\ref{lemma-gysin-at-infty}.", "Then $g_{\\infty, *} \\circ C' = C$ in $A^0(W_\\infty \\to X)$." ], "refs": [ "chow-lemma-gysin-at-infty", "chow-lemma-gysin-at-infty" ], "proofs": [ { "contents": [ "Set $b' = b \\circ g : W' \\to \\mathbf{P}^1_X$. Denote", "$i'_\\infty : W'_\\infty \\to W'$ the inclusion morphism.", "Denote $g_\\infty : W'_\\infty \\to W_\\infty$ the restriction of $g$.", "Given $\\alpha \\in \\CH_k(X)$ choose $\\beta' \\in \\CH_{k + 1}(W')$", "restricting to the flat pullback of $\\alpha$ on $(b')^{-1}\\mathbf{A}^1_X$.", "Then $\\beta = g_*\\beta' \\in \\CH_{k + 1}(W)$ restricts to the", "flat pullback of $\\alpha$ on $b^{-1}\\mathbf{A}^1_X$.", "Then $i_\\infty^*\\beta = g_{\\infty, *}(i'_\\infty)^*\\beta'$", "by Lemma \\ref{lemma-closed-in-X-gysin}.", "This and the corresponding fact after base change by", "morphisms $X' \\to X$ locally of finite type, corresponds", "to the assertion made in the lemma." ], "refs": [ "chow-lemma-closed-in-X-gysin" ], "ref_ids": [ 5718 ] } ], "ref_ids": [ 5786, 5786 ] }, { "id": 5788, "type": "theorem", "label": "chow-lemma-homomorphism-pre", "categories": [ "chow" ], "title": "chow-lemma-homomorphism-pre", "contents": [ "In Lemma \\ref{lemma-gysin-at-infty} we have", "$C \\circ (W_\\infty \\to X)_* \\circ i_\\infty^* = i_\\infty^*$." ], "refs": [ "chow-lemma-gysin-at-infty" ], "proofs": [ { "contents": [ "Let $\\beta \\in \\CH_{k + 1}(W)$. Denote $i_0 : X = X \\times \\{0\\} \\to W$", "the closed immersion of the fibre over $0$ in $\\mathbf{P}^1$. Then", "$(W_\\infty \\to X)_* i_\\infty^* \\beta = i_0^*\\beta$ in $\\CH_k(X)$ because", "$i_{\\infty, *}i_\\infty^*\\beta$ and $i_{0, *}i_0^*\\beta$", "represent the same class on $W$ (for example by", "Lemma \\ref{lemma-support-cap-effective-Cartier})", "and hence pushforward to the same class on $X$.", "The restriction of $\\beta$ to $b^{-1}(\\mathbf{A}^1_X)$", "restricts to the flat pullback of", "$i_0^*\\beta = (W_\\infty \\to X)_* i_\\infty^* \\beta$ because we can check", "this after pullback by $i_0$, see", "Lemmas \\ref{lemma-linebundle} and \\ref{lemma-linebundle-formulae}.", "Hence we may use $\\beta$ when computing the image of", "$(W_\\infty \\to X)_*i_\\infty^*\\beta$ under $C$", "and we get the desired result." ], "refs": [ "chow-lemma-support-cap-effective-Cartier", "chow-lemma-linebundle", "chow-lemma-linebundle-formulae" ], "ref_ids": [ 5716, 5727, 5728 ] } ], "ref_ids": [ 5786 ] }, { "id": 5789, "type": "theorem", "label": "chow-lemma-gysin-at-infty-commutes", "categories": [ "chow" ], "title": "chow-lemma-gysin-at-infty-commutes", "contents": [ "In Lemma \\ref{lemma-gysin-at-infty} let $f : Y \\to X$ be a morphism", "locally of finite type and $c \\in A^*(Y \\to X)$. Then $C \\circ c = c \\circ C$", "in $A^*(W_\\infty \\times_X Y)$." ], "refs": [ "chow-lemma-gysin-at-infty" ], "proofs": [ { "contents": [ "Consider the commutative diagram", "$$", "\\xymatrix{", "W_\\infty \\times_X Y \\ar@{=}[r] &", "W_{Y, \\infty} \\ar[r]_{i_{Y, \\infty}} \\ar[d] &", "W_Y \\ar[r]_{b_Y} \\ar[d] &", "\\mathbf{P}^1_Y \\ar[r]_{p_Y} \\ar[d] &", "Y \\ar[d]^f \\\\", "& W_\\infty \\ar[r]^{i_\\infty} &", "W \\ar[r]^b &", "\\mathbf{P}^1_X \\ar[r]^p &", "X", "}", "$$", "with cartesian squares. For an elemnent $\\alpha \\in \\CH_k(X)$", "choose $\\beta \\in \\CH_{k + 1}(W)$ whose restriction to $b^{-1}(\\mathbf{A}^1_X)$", "is the flat pullback of $\\alpha$. Then $c \\cap \\beta$ is a class", "in $\\CH_*(W_Y)$ whose restriction to $b_Y^{-1}(\\mathbf{A}^1_Y)$", "is the flat pullback of $c \\cap \\alpha$. Next, we have", "$$", "i_{Y, \\infty}^*(c \\cap \\beta) = c \\cap i_\\infty^*\\beta", "$$", "because $c$ is a bivariant class. This exactly says that", "$C \\cap c \\cap \\alpha = c \\cap C \\cap \\alpha$. The same argument", "works after any base change by $X' \\to X$ locally of finite type.", "This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 5786 ] }, { "id": 5790, "type": "theorem", "label": "chow-lemma-localized-chern-pre", "categories": [ "chow" ], "title": "chow-lemma-localized-chern-pre", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be", "locally of finite type over $S$. Let $Z \\subset X$ be a closed subscheme.", "Let", "$$", "b : W \\longrightarrow \\mathbf{P}^1_X", "$$", "be a proper morphism of schemes. Let $Q \\in D(\\mathcal{O}_W)$ be a", "perfect object. Denote $W_\\infty \\subset W$ the inverse image of the divisor", "$D_\\infty \\subset \\mathbf{P}^1_X$ with complement $\\mathbf{A}^1_X$.", "We assume", "\\begin{enumerate}", "\\item[(A0)] Chern classes of $Q$ are defined", "(Section \\ref{section-pre-derived}),", "\\item[(A1)] $b$ is an isomorphism over $\\mathbf{A}^1_X$,", "\\item[(A2)] there exists a closed subscheme $T \\subset W_\\infty$", "containing all points of $W_\\infty$ lying over $X \\setminus Z$ such that", "$Q|_T$ is zero, resp.\\ isomorphic to a finite locally free", "$\\mathcal{O}_T$-module of rank $< p$ sitting in cohomological degree $0$.", "\\end{enumerate}", "Then there exists a canonical bivariant class", "$$", "P'_p(Q),\\text{ resp. }c'_p(Q) \\in A^p(Z \\to X)", "$$", "with", "$(Z \\to X)_* \\circ P'_p(Q) = P_p(Q|_{X \\times \\{0\\}})$,", "resp.\\ $(Z \\to X)_* \\circ c'_p(Q) = c_p(Q|_{X \\times \\{0\\}})$." ], "refs": [], "proofs": [ { "contents": [ "Denote $E \\subset W_\\infty$ the inverse image of $Z$. Then", "$W_\\infty = T \\cup E$ and $b$ induces a proper morphism $E \\to Z$.", "Denote $C \\in A^0(W_\\infty \\to X)$ the bivariant class constructed", "in Lemma \\ref{lemma-gysin-at-infty}. Denote $P'_p(Q|_E)$, resp.\\ $c'_p(Q|_E)$", "in $A^p(E \\to W_\\infty)$ the bivariant class constructed", "in Lemma \\ref{lemma-silly}. This makes sense because", "$(Q|_E)|_{T \\cap E}$ is zero, resp.\\ isomorphic to a finite locally free", "$\\mathcal{O}_{E \\cap T}$-module of rank $< p$ sitting in", "cohomological degree $0$ by assumption (A2). Then we define", "$$", "P'_p(Q) = (E \\to Z)_* \\circ P'_p(Q|_E) \\circ C,\\text{ resp. }", "c'_p(Q) = (E \\to Z)_* \\circ c'_p(Q|_E) \\circ C", "$$", "This is a bivariant class, see Lemma \\ref{lemma-push-proper-bivariant}.", "Since $E \\to Z \\to X$ is equal to $E \\to W_\\infty \\to W \\to X$ we see that", "\\begin{align*}", "(Z \\to X)_* \\circ c'_p(Q)", "& =", "(W \\to X)_* \\circ i_{\\infty, *} \\circ (E \\to W_\\infty)_*", "\\circ c'_p(Q|_E) \\circ C \\\\", "& =", "(W \\to X)_* \\circ i_{\\infty, *} \\circ c_p(Q|_{W_\\infty}) \\circ C \\\\", "& =", "(W \\to X)_* \\circ c_p(Q) \\circ i_{\\infty, *} \\circ C \\\\", "& =", "(W \\to X)_*\\circ c_p(Q) \\circ i_{0, *} \\\\", "& =", "(W \\to X)_* \\circ i_{0, *} \\circ c_p(Q|_{X \\times \\{0\\}}) \\\\", "& =", "c_p(Q|_{X \\times \\{0\\}})", "\\end{align*}", "The second equality holds by Lemma \\ref{lemma-silly-silly}.", "The third equality because $c_p(Q)$ is a bivariant class.", "The fourth equality by Lemma \\ref{lemma-gysin-at-infty}.", "The fifth equality because $c_p(Q)$ is a bivariant class.", "The final equality because $(W_0 \\to W) \\circ (W \\to X)$", "is the identity on $X$ if we identify $W_0$ with $X$ as we've", "done above. The exact same sequence of equations works to", "prove the property for $P'_p(Q)$." ], "refs": [ "chow-lemma-gysin-at-infty", "chow-lemma-silly", "chow-lemma-push-proper-bivariant", "chow-lemma-silly-silly", "chow-lemma-gysin-at-infty" ], "ref_ids": [ 5786, 5776, 5734, 5778, 5786 ] } ], "ref_ids": [] }, { "id": 5791, "type": "theorem", "label": "chow-lemma-localized-chern-pre-independent", "categories": [ "chow" ], "title": "chow-lemma-localized-chern-pre-independent", "contents": [ "In Lemma \\ref{lemma-localized-chern-pre} the bivariant class", "$P'_p(Q)$, resp.\\ $c'_p(Q)$", "is independent of the choice of the closed subscheme $T$.", "Moreover, given a proper morphism $g : W' \\to W$ which is an", "isomorphism over $\\mathbf{A}^1_X$, then setting $Q' = g^*Q$", "we have $P'_p(Q) = P'_p(Q')$, resp.\\ $c'_p(Q) = c'_p(Q')$." ], "refs": [ "chow-lemma-localized-chern-pre" ], "proofs": [ { "contents": [ "The independence of $T$ follows immediately from", "Lemma \\ref{lemma-silly-independent}.", "\\medskip\\noindent", "Let $g : W' \\to W$ be a proper morphism which is an isomorphism over", "$\\mathbf{A}^1_X$. Observe that taking $T' = g^{-1}(T) \\subset W'_\\infty$", "is a closed subscheme satisfying (A2) hence the operator", "$P'_p(Q')$, resp.\\ $c'_p(Q')$ in $A^p(Z \\to X)$", "corresponding to $b' = b \\circ g : W' \\to \\mathbf{P}^1_X$", "and $Q'$ is defined. Denote $E' \\subset W'_\\infty$", "the inverse image of $Z$ in $W'_\\infty$. Recall that", "$$", "c'_p(Q') = (E' \\to Z)_* \\circ c'_p(Q'|_{E'}) \\circ C'", "$$", "with $C' \\in A^0(W'_\\infty \\to X)$ and", "$c'_p(Q'|_{E'}) \\in A^p(E' \\to W'_\\infty)$.", "By Lemma \\ref{lemma-gysin-at-infty-independent} we have", "$g_{\\infty, *} \\circ C' = C$. Observe that $E'$ is also", "the inverse image of $E$ in $W'_\\infty$ by $g_\\infty$.", "Since moreover $Q' = g^*Q$ we find that $c'_p(Q'|_{E'})$ is simply the", "restriction of $c'_p(Q|_E)$ to schemes lying over $W'_\\infty$, see", "Remark \\ref{remark-restriction-bivariant}. Thus we obtain", "\\begin{align*}", "c'_p(Q')", "& = ", "(E' \\to Z)_* \\circ c'_p(Q'|_{E'}) \\circ C' \\\\", "& =", "(E \\to Z)_* \\circ (E' \\to E)_* \\circ c'_p(Q|_E) \\circ C' \\\\", "& =", "(E \\to Z)_* \\circ c'_p(Q|_E) \\circ g_{\\infty, *} \\circ C' \\\\", "& =", "(E \\to Z)_* \\circ c'_p(Q|_E) \\circ C \\\\", "& =", "c'_p(Q)", "\\end{align*}", "In the third equality we used that $c'_p(Q|_E)$", "commutes with proper pushforward as it is a", "bivariant class. The equality $P'_p(Q) = P'_p(Q')$", "is proved in exactly the same way." ], "refs": [ "chow-lemma-silly-independent", "chow-lemma-gysin-at-infty-independent", "chow-remark-restriction-bivariant" ], "ref_ids": [ 5777, 5787, 5938 ] } ], "ref_ids": [ 5790 ] }, { "id": 5792, "type": "theorem", "label": "chow-lemma-homomorphism", "categories": [ "chow" ], "title": "chow-lemma-homomorphism", "contents": [ "In Lemma \\ref{lemma-localized-chern-pre} assume $Q|_T$ is isomorphic", "to a finite locally free $\\mathcal{O}_T$-module of rank $< p$.", "Denote $C \\in A^0(W_\\infty \\to X)$ the class of", "Lemma \\ref{lemma-gysin-at-infty}. Then", "$$", "C \\circ c_p(Q|_{X \\times \\{0\\}}) =", "C \\circ (Z \\to X)_* \\circ c'_p(Q) = c_p(Q|_{W_\\infty}) \\circ C", "$$" ], "refs": [ "chow-lemma-localized-chern-pre", "chow-lemma-gysin-at-infty" ], "proofs": [ { "contents": [ "The first equality holds because $c_p(Q|_{X \\times \\{0\\}}) =", "(Z \\to X)_* \\circ c'_p(Q)$ by Lemma \\ref{lemma-localized-chern-pre}.", "We may prove the second equality one cycle class at a time", "(see Lemma \\ref{lemma-bivariant-zero}). Since the construction of", "the bivariant classes in the lemma is compatible with base change,", "we may assume we have some $\\alpha \\in \\CH_k(X)$ and we have to show that", "$C \\cap (Z \\to X)_*(c'_p(Q) \\cap \\alpha) =", "c_p(Q|_{W_\\infty}) \\cap C \\cap \\alpha$. Observe that", "\\begin{align*}", "C \\cap (Z \\to X)_*(c'_p(Q) \\cap \\alpha)", "& =", "C \\cap (Z \\to X)_* (E \\to Z)_*(c'_p(Q|_E) \\cap C \\cap \\alpha) \\\\", "& =", "C \\cap (W_\\infty \\to X)_*(E \\to W_\\infty)_*(c'_p(Q|_E) \\cap C \\cap \\alpha) \\\\", "& =", "C \\cap (W_\\infty \\to X)_*(E \\to W_\\infty)_*(c'_p(Q|_E) \\cap i_\\infty^*\\beta) \\\\", "& =", "C \\cap (W_\\infty \\to X)_*(c_p(Q|_{W_\\infty}) \\cap i_\\infty^*\\beta) \\\\", "& =", "C \\cap (W_\\infty \\to X)_*i_\\infty^*(c_p(Q) \\cap \\beta) \\\\", "& =", "i_\\infty^*(c_p(Q) \\cap \\beta) \\\\", "& =", "c_p(Q|_{W_\\infty}) \\cap i_\\infty^*\\beta \\\\", "& =", "c_p(Q|_{W_\\infty}) \\cap C \\cap \\alpha", "\\end{align*}", "as desired. For the first equality we used that", "$c'_p(Q) = (E \\to Z)_* \\circ c'_p(Q|_E) \\circ C$ where $E \\subset W_\\infty$", "is the inverse image of $Z$ and $c'_p(Q|_E)$ is the class constructed", "in Lemma \\ref{lemma-silly}. The second equality is just the statement", "that $E \\to Z \\to X$ is equal to $E \\to W_\\infty \\to X$.", "For the third equality we choose $\\beta \\in \\CH_{k + 1}(W)$ whose restriction to", "$b^{-1}(\\mathbf{A}^1_X)$ is the flat pullback of $\\alpha$ so that", "$C \\cap \\alpha = i_\\infty^*\\beta$ by construction. The fourth equality is", "Lemma \\ref{lemma-silly-silly}. The fifth equality is the fact that", "$c_p(Q)$ is a bivariant class and hence commutes with $i_\\infty^*$.", "The sixth equality is Lemma \\ref{lemma-homomorphism-pre}.", "The seventh uses again that $c_p(Q)$ is a bivariant class.", "The final holds as $C \\cap \\alpha = i_\\infty^*\\beta$." ], "refs": [ "chow-lemma-localized-chern-pre", "chow-lemma-bivariant-zero", "chow-lemma-silly", "chow-lemma-silly-silly", "chow-lemma-homomorphism-pre" ], "ref_ids": [ 5790, 5740, 5776, 5778, 5788 ] } ], "ref_ids": [ 5790, 5786 ] }, { "id": 5793, "type": "theorem", "label": "chow-lemma-homomorphism-commute", "categories": [ "chow" ], "title": "chow-lemma-homomorphism-commute", "contents": [ "In Lemma \\ref{lemma-localized-chern-pre} let $Y \\to X$ be a morphism", "locally of finite type and let $c \\in A^*(Y \\to X)$ be a bivariant class.", "Then", "$$", "P'_p(Q) \\circ c = c \\circ P'_p(Q)", "\\quad\\text{resp.}\\quad", "c'_p(Q) \\circ c = c \\circ c'_p(Q)", "$$", "in $A^*(Y \\times_X Z \\to X)$." ], "refs": [ "chow-lemma-localized-chern-pre" ], "proofs": [ { "contents": [ "Let $E \\subset W_\\infty$ be the inverse image of $Z$.", "Recall that $P'_p(Q) = (E \\to Z)_* \\circ P'_p(Q|_E) \\circ C$,", "resp.\\ $c'_p(Q) = (E \\to Z)_* \\circ c'_p(Q|_E) \\circ C$", "where $C$ is as in Lemma \\ref{lemma-gysin-at-infty} and", "$P'_p(Q|_E)$, resp.\\ $c'_p(Q|_E)$ are as in", "Lemma \\ref{lemma-silly}.", "By Lemma \\ref{lemma-gysin-at-infty-commutes}", "we see that $C$ commutes with $c$", "and by Lemma \\ref{lemma-silly-commutes} we see that", "$P'_p(Q|_E)$, resp.\\ $c'_p(Q|_E)$ commutes with $c$.", "Since $c$ is a bivariant class it commutes with proper", "pushforward by $E \\to Z$ by definition. This finishes the proof." ], "refs": [ "chow-lemma-gysin-at-infty", "chow-lemma-silly", "chow-lemma-gysin-at-infty-commutes", "chow-lemma-silly-commutes" ], "ref_ids": [ 5786, 5776, 5789, 5780 ] } ], "ref_ids": [ 5790 ] }, { "id": 5794, "type": "theorem", "label": "chow-lemma-localized-chern-pre-compose", "categories": [ "chow" ], "title": "chow-lemma-localized-chern-pre-compose", "contents": [ "In Lemma \\ref{lemma-localized-chern-pre} assume $Q|_T$ is zero. In", "$A^*(Z \\to X)$ we have", "\\begin{align*}", "P'_1(Q) & = c'_1(Q), \\\\", "P'_2(Q) & = c'_1(Q)^2 - 2c'_2(Q), \\\\", "P'_3(Q) & = c'_1(Q)^3 - 3c'_1(Q)c'_2(Q) + 3c'_3(Q), \\\\", "P'_4(Q) & = c'_1(Q)^4 - 4c'_1(Q)^2c'_2(Q) +", "4c'_1(Q)c'_3(Q) + 2c'_2(Q)^2 - 4c'_4(Q),", "\\end{align*}", "and so on with multiplication as in Remark \\ref{remark-ring-loc-classes}." ], "refs": [ "chow-lemma-localized-chern-pre", "chow-remark-ring-loc-classes" ], "proofs": [ { "contents": [ "The statement makes sense because the zero sheaf has rank $< 1$ and", "hence the classes $c'_p(Q)$ are defined for all $p \\geq 1$.", "In the proof of Lemma \\ref{lemma-localized-chern-pre} we have constructed", "the classes $P'_p(Q)$ and $c'_p(Q)$ using the bivariant class", "$C \\in A^0(W_\\infty \\to X)$ of Lemma \\ref{lemma-gysin-at-infty}", "and the bivariant classes", "$P'_p(Q|_E)$ and $c'_p(Q|_E)$ of Lemma \\ref{lemma-silly} for the restriction", "$Q|_E$ of $Q$ to the inverse image $E$ of $Z$ in $W_\\infty$.", "Observe that by Lemma \\ref{lemma-silly-compose} we have the desired", "relationship between $P'_p(Q|_E)$ and $c'_p(Q|_E)$. Recall that", "$$", "P'_p(Q) = (E \\to Z)_* \\circ P'_p(Q|_E) \\circ C", "\\quad\\text{and}\\quad", "c'_p(Q) = (E \\to Z)_* \\circ c'_p(Q|_E) \\circ C", "$$", "To finish the proof it suffices to show the multiplications defined", "in Remark \\ref{remark-ring-loc-classes} on the classes $a_p = c'_p(Q)$", "and on the classes $b_p = c'_p(Q|_E)$ agree:", "$$", "a_{p_1}a_{p_2} \\ldots a_{p_r} =", "(E \\to Z)_* \\circ b_{p_1}b_{p_2} \\ldots b_{p_r} \\circ C", "$$", "Some details omitted. If $r = 1$, then this is true.", "For $r > 1$ note that by Remark \\ref{remark-res-push} the multiplication in", "Remark \\ref{remark-ring-loc-classes} proceeds", "by inserting $(Z \\to X)_*$, resp.\\ $(E \\to W_\\infty)_*$ in between", "the factors of the product", "$a_{p_1}a_{p_2} \\ldots a_{p_r}$, resp.\\ $b_{p_1}b_{p_2} \\ldots b_{p_r}$", "and taking compositions as bivariant classes.", "Now by Lemma \\ref{lemma-silly} we have", "$$", "(E \\to W_\\infty)_* \\circ b_{p_i} = c_{p_i}(Q|_{W_\\infty})", "$$", "and by Lemma \\ref{lemma-homomorphism} we have", "$$", "C \\circ (Z \\to X)_* \\circ a_{p_i} = c_{p_i}(Q|_{W_\\infty}) \\circ C", "$$", "for $i = 2, \\ldots, r$. A calculation", "shows that the left and right hand side of the desired", "equality both simplify to", "$$", "(E \\to Z)_* \\circ c'_{p_1}(Q|_E) \\circ", "c_{p_2}(Q|_{W_\\infty}) \\circ \\ldots \\circ", "c_{p_r}(Q|_{W_\\infty}) \\circ C", "$$", "and the proof is complete." ], "refs": [ "chow-lemma-localized-chern-pre", "chow-lemma-gysin-at-infty", "chow-lemma-silly", "chow-lemma-silly-compose", "chow-remark-ring-loc-classes", "chow-remark-res-push", "chow-remark-ring-loc-classes", "chow-lemma-silly", "chow-lemma-homomorphism" ], "ref_ids": [ 5790, 5786, 5776, 5781, 5942, 5943, 5942, 5776, 5792 ] } ], "ref_ids": [ 5790, 5942 ] }, { "id": 5795, "type": "theorem", "label": "chow-lemma-localized-chern-pre-sum-c", "categories": [ "chow" ], "title": "chow-lemma-localized-chern-pre-sum-c", "contents": [ "In Lemma \\ref{lemma-localized-chern-pre} assume $Q|_T$ is isomorphic", "to a finite locally free $\\mathcal{O}_T$-module of rank $< p$.", "Assume we have another perfect object $Q' \\in D(\\mathcal{O}_W)$", "whose Chern classes are defined with $Q'|_T$ isomorphic to a", "finite locally free $\\mathcal{O}_T$-module of rank $< p'$ placed", "in cohomological degree $0$. With notation as in", "Remark \\ref{remark-ring-loc-classes} set", "$$", "c^{(p)}(Q) = 1 + c_1(Q|_{X \\times \\{0\\}}) + \\ldots +", "c_{p - 1}(Q|_{X \\times \\{0\\}}) +", "c'_{p}(Q) + c'_{p + 1}(Q) + \\ldots", "$$", "in $A^{(p)}(Z \\to X)$ with $c'_i(Q)$ for $i \\geq p$ as in", "Lemma \\ref{lemma-localized-chern-pre}. Similarly for $c^{(p')}(Q')$ and", "$c^{(p + p')}(Q \\oplus Q')$.", "Then $c^{(p + p')}(Q \\oplus Q') = c^{(p)}(Q)c^{(p')}(Q')$", "in $A^{(p + p')}(Z \\to X)$." ], "refs": [ "chow-lemma-localized-chern-pre", "chow-remark-ring-loc-classes", "chow-lemma-localized-chern-pre" ], "proofs": [ { "contents": [ "Recall that the image of $c'_i(Q)$ in $A^p(X)$ is equal to", "$c_i(Q|_{X \\times \\{0\\}})$ for $i \\geq p$ and similarly for", "$Q'$ and $Q \\oplus Q'$, see Lemma \\ref{lemma-localized-chern-pre}.", "Hence the equality in degrees $< p + p'$ follows from the", "additivity of Lemma \\ref{lemma-additivity-on-perfect}.", "\\medskip\\noindent", "Let's take $n \\geq p + p'$.", "As in the proof of Lemma \\ref{lemma-localized-chern-pre}", "let $E \\subset W_\\infty$ denote the inverse image of $Z$.", "Observe that we have the equality", "$$", "c^{(p + p')}(Q|_E \\oplus Q'|_E) =", "c^{(p)}(Q|_E)c^{(p')}(Q'|_E)", "$$", "in $A^{(p + p')}(E \\to W_\\infty)$ by Lemma \\ref{lemma-silly-sum-c}.", "Since by construction", "$$", "c'_p(Q \\oplus Q') = (E \\to Z)_* \\circ c'_p(Q|_E \\oplus Q'|_E) \\circ C", "$$", "we conclude that suffices to show for all $i + j = n$ we have", "$$", "(E \\to Z)_* \\circ c^{(p)}_i(Q|_E)c^{(p')}_j(Q'|_E) \\circ C", "=", "c^{(p)}_i(Q)c^{(p')}_j(Q')", "$$", "in $A^n(Z \\to X)$ where the multiplication is the one from", "Remark \\ref{remark-ring-loc-classes} on both sides. There are", "three cases, depending on whether $i \\geq p$, $j \\geq p'$, or both.", "\\medskip\\noindent", "Assume $i \\geq p$ and $j \\geq p'$. In this case the products are", "defined by inserting $(E \\to W_\\infty)_*$, resp.\\ $(Z \\to X)_*$ in between", "the two factors and taking compositions as bivariant classes, see", "Remark \\ref{remark-res-push}.", "In other words, we have to show", "$$", "(E \\to Z)_* \\circ c'_i(Q|_E) \\circ", "(E \\to W_\\infty)_* \\circ c'_j(Q'|_E) \\circ C =", "c'_i(Q) \\circ (Z \\to X)_* \\circ c'_j(Q')", "$$", "By Lemma \\ref{lemma-silly} the left hand side is equal to", "$$", "(E \\to Z)_* \\circ c'_i(Q|_E) \\circ c_j(Q'|_{W_\\infty}) \\circ C", "$$", "Since $c'_i(Q) = (E \\to Z)_* \\circ c'_i(Q|_E) \\circ C$", "the right hand side is equal to", "$$", "(E \\to Z)_* \\circ c'_i(Q|_E) \\circ C \\circ (Z \\to X)_* \\circ c'_j(Q')", "$$", "which is immediately seen to be equal to the above", "by Lemma \\ref{lemma-homomorphism}.", "\\medskip\\noindent", "Assume $i \\geq p$ and $j < p$. Unwinding the products", "in this case we have to show", "$$", "(E \\to Z)_* \\circ c'_i(Q|_E) \\circ c_j(Q'|_{W_\\infty}) \\circ C =", "c'_i(Q) \\circ c_j(Q'|_{X \\times \\{0\\}})", "$$", "Again using that $c'_i(Q) = (E \\to Z)_* \\circ c'_i(Q|_E) \\circ C$", "we see that it suffices to show $c_j(Q'|_{W_\\infty}) \\circ C =", "C \\circ c_j(Q'|_{X \\times \\{0\\}})$ which is part of", "Lemma \\ref{lemma-homomorphism}.", "\\medskip\\noindent", "Assume $i < p$ and $j \\geq p'$. Unwinding the products", "in this case we have to show", "$$", "(E \\to Z)_* \\circ c_i(Q|_E) \\circ c'_j(Q'|_E) \\circ C =", "c_i(Q|_{Z \\times \\{0\\}}) \\circ c'_j(Q')", "$$", "However, since $c'_j(Q|_E)$ and $c'_j(Q')$ are", "bivariant classes, they commute with capping with Chern classes", "(Lemma \\ref{lemma-cap-commutative-chern}). Hence it suffices to prove", "$$", "(E \\to Z)_* \\circ c'_j(Q'|_E) \\circ c_i(Q|_{W_\\infty}) \\circ C =", "c'_j(Q') \\circ c_i(Q|_{X \\times \\{0\\}})", "$$", "which we reduces us to the case discussed in the preceding paragraph." ], "refs": [ "chow-lemma-localized-chern-pre", "chow-lemma-additivity-on-perfect", "chow-lemma-localized-chern-pre", "chow-lemma-silly-sum-c", "chow-remark-ring-loc-classes", "chow-remark-res-push", "chow-lemma-silly", "chow-lemma-homomorphism", "chow-lemma-homomorphism", "chow-lemma-cap-commutative-chern" ], "ref_ids": [ 5790, 5772, 5790, 5782, 5942, 5943, 5776, 5792, 5792, 5752 ] } ], "ref_ids": [ 5790, 5942, 5790 ] }, { "id": 5796, "type": "theorem", "label": "chow-lemma-localized-chern-pre-sum-P", "categories": [ "chow" ], "title": "chow-lemma-localized-chern-pre-sum-P", "contents": [ "In Lemma \\ref{lemma-localized-chern-pre} assume $Q|_T$ is zero.", "Assume we have another perfect object $Q' \\in D(\\mathcal{O}_W)$", "whose Chern classes are defined such that the restriction $Q'|_T$ is zero.", "In this case the classes", "$P'_p(Q), P'_p(Q'), P'_p(Q \\oplus Q') \\in A^p(Z \\to X)$", "constructed in Lemma \\ref{lemma-localized-chern-pre}", "satisfy $P'_p(Q \\oplus Q') = P'_p(Q) + P'_p(Q')$." ], "refs": [ "chow-lemma-localized-chern-pre", "chow-lemma-localized-chern-pre" ], "proofs": [ { "contents": [ "This follows immediately from the construction of these", "classes and Lemma \\ref{lemma-silly-sum-P}." ], "refs": [ "chow-lemma-silly-sum-P" ], "ref_ids": [ 5783 ] } ], "ref_ids": [ 5790, 5790 ] }, { "id": 5797, "type": "theorem", "label": "chow-lemma-independent-loc-chern", "categories": [ "chow" ], "title": "chow-lemma-independent-loc-chern", "contents": [ "The localized class constructed above is independent of choices." ], "refs": [], "proofs": [ { "contents": [ "Here are the choices we made above: the bounded complex $\\mathcal{E}^\\bullet$ ", "of finite locally free $\\mathcal{O}_X$-modules representing $E$,", "the blowup $b : W \\to \\mathbf{P}^1_X$, the choice of $\\mathcal{Q}^\\bullet$,", "and the closed subscheme $T'$. In", "Lemma \\ref{lemma-localized-chern-pre-independent}", "we have seen that the class is independent of the choice of $T'$. In", "More on Flatness, Lemma \\ref{flat-lemma-complex-and-divisor-derived}", "we have seen that the blowing up $b : W \\to \\mathbf{P}^1_X$", "and the isomorphism class $Q$ of $\\mathcal{Q}^\\bullet$ in $D(\\mathcal{O}_W)$", "only depend on the isomorphism class of $L\\text{pr}^*E$ in", "$D(\\mathcal{O}_{\\mathbf{P}^1_X})$ where $\\text{pr} : \\mathbf{P}^1_X \\to X$", "is the projection morphism. Since the construction of", "Lemma \\ref{lemma-localized-chern-pre} depends only on the", "isomorphism class $Q$, we conclude." ], "refs": [ "chow-lemma-localized-chern-pre-independent", "flat-lemma-complex-and-divisor-derived", "chow-lemma-localized-chern-pre" ], "ref_ids": [ 5791, 6196, 5790 ] } ], "ref_ids": [] }, { "id": 5798, "type": "theorem", "label": "chow-lemma-base-change-loc-chern", "categories": [ "chow" ], "title": "chow-lemma-base-change-loc-chern", "contents": [ "In the situation above let $f : X' \\to X$ be a morphism of schemes", "which is locally of finite type. Denote $E' = Lf^*E$ and $Z' = f^{-1}(Z)$.", "Then the bivariant class", "$$", "P_p(Z' \\to X', E') \\in A^p(Z' \\to X'),", "\\quad\\text{resp.}\\quad", "c_p(Z' \\to X', E') \\in A^p(Z' \\to X')", "$$", "constructed above using $X', Z', E'$ is the restriction", "(Remark \\ref{remark-restriction-bivariant}) of the", "bivariant class $P_p(Z \\to X, E) \\in A^p(Z \\to X)$,", "resp.\\ $c_p(Z \\to X, E) \\in A^p(Z \\to X)$." ], "refs": [ "chow-remark-restriction-bivariant" ], "proofs": [ { "contents": [ "Choose a bounded complex $\\mathcal{E}^\\bullet$ of finite locally free", "$\\mathcal{O}_X$-modules representing $E$. Denote", "$(\\mathcal{E}')^\\bullet = f^*\\mathcal{E}^\\bullet$.", "Observe that $\\mathbf{P}^1_{X'} \\to \\mathbf{P}^1_X$ is a morphism of", "schemes such that the pullback of the effective Cartier divisor", "$(\\mathbf{P}^1_X)_\\infty$ is the effective Cartier divisor", "$(\\mathbf{P}^1_{X'})_\\infty$. By More on Flatness, Lemma", "\\ref{flat-lemma-complex-and-divisor-blowup-base-change}", "we obtain a commutative diagram", "$$", "\\xymatrix{", "W' \\ar[rd]_{b'} \\ar[r]_-g &", "\\mathbf{P}^1_{X'} \\times_{\\mathbf{P}^1_X} W \\ar[d]_r \\ar[r]_-q &", "W \\ar[d]^b \\\\", "&", "\\mathbf{P}^1_{X'} \\ar[r] &", "\\mathbf{P}^1_X", "}", "$$", "such that $W'$ is the strict transform of $\\mathbf{P}^1_{X'}$", "with respect to $b$ and such that", "$(\\mathcal{Q}')^\\bullet = g^*q^*\\mathcal{Q}^\\bullet$.", "The restriction of the bivariant class $P_p(Z \\to X, E)$,", "resp.\\ $c_p(Z \\to X, E)$ corresponds to the class constructed in", "Lemma \\ref{lemma-localized-chern-pre} using the", "proper morphism $r$ and the complex $q^*\\mathcal{Q}^\\bullet$.", "On the other hand, the bivariant class $P_p(Z' \\to X', E')$,", "resp.\\ $c_p(Z' \\to X', E')$ corresponds to the", "proper morphism $b'$ and the complex $(\\mathcal{Q}')^\\bullet$.", "Thus we conclude by Lemma \\ref{lemma-localized-chern-pre-independent}." ], "refs": [ "flat-lemma-complex-and-divisor-blowup-base-change", "chow-lemma-localized-chern-pre", "chow-lemma-localized-chern-pre-independent" ], "ref_ids": [ 6192, 5790, 5791 ] } ], "ref_ids": [ 5938 ] }, { "id": 5799, "type": "theorem", "label": "chow-lemma-loc-chern-after-pushforward", "categories": [ "chow" ], "title": "chow-lemma-loc-chern-after-pushforward", "contents": [ "In the situation above we have", "$$", "P_p(Z \\to X, E) \\cap i_*\\alpha = P_p(E|_Z) \\cap \\alpha,", "\\quad\\text{resp.}\\quad", "c_p(Z \\to X, E) \\cap i_*\\alpha = c_p(E|_Z) \\cap \\alpha", "$$", "in $\\CH_*(Z)$ for any $\\alpha \\in \\CH_*(Z)$." ], "refs": [], "proofs": [ { "contents": [ "We only prove the second equality and we omit the proof of the first.", "Since $c_p(Z \\to X, E)$ is a bivariant class and since the base", "change of $Z \\to X$ by $Z \\to X$ is $\\text{id} : Z \\to Z$ we have", "$c_p(Z \\to X, E) \\cap i_*\\alpha = c_p(Z \\to X, E) \\cap \\alpha$.", "By Lemma \\ref{lemma-base-change-loc-chern} the restriction of", "$c_p(Z \\to X, E)$ to $Z$ (!) is the localized Chern class for", "$\\text{id} : Z \\to Z$ and $E|_Z$. Thus the result follows from", "(\\ref{equation-defining-property-localized-classes}) with $X = Z$." ], "refs": [ "chow-lemma-base-change-loc-chern" ], "ref_ids": [ 5798 ] } ], "ref_ids": [] }, { "id": 5800, "type": "theorem", "label": "chow-lemma-loc-chern-disjoint", "categories": [ "chow" ], "title": "chow-lemma-loc-chern-disjoint", "contents": [ "In the situation of Definition \\ref{definition-localized-chern}", "if $\\alpha \\in \\CH_k(X)$ has support disjoint from $Z$, then", "$P_p(Z \\to X, E) \\cap \\alpha = 0$, resp.\\ $c_p(Z \\to X, E) \\cap \\alpha = 0$." ], "refs": [ "chow-definition-localized-chern" ], "proofs": [ { "contents": [ "This is immediate from the construction of the localized Chern classes.", "It also follows from the fact that we can compute", "$c_p(Z \\to X, E) \\cap \\alpha$ by first restricting $c_p(Z \\to X, E)$ to", "the support of $\\alpha$, and then using Lemma \\ref{lemma-base-change-loc-chern}", "to see that this restriction is zero." ], "refs": [ "chow-lemma-base-change-loc-chern" ], "ref_ids": [ 5798 ] } ], "ref_ids": [ 5924 ] }, { "id": 5801, "type": "theorem", "label": "chow-lemma-loc-chern-shrink-Z", "categories": [ "chow" ], "title": "chow-lemma-loc-chern-shrink-Z", "contents": [ "In the situation of Definition \\ref{definition-localized-chern}", "assume $Z \\subset Z' \\subset X$ where $Z'$ is a closed subscheme of $X$.", "Then", "$P_p(Z' \\to X, E) = (Z \\to Z')_* \\circ P_p(Z \\to X, E)$,", "resp.\\ $c_p(Z' \\to X, E) = (Z \\to Z')_* \\circ c_p(Z \\to X, E)$", "(with $\\circ$ as in Lemma \\ref{lemma-push-proper-bivariant})." ], "refs": [ "chow-definition-localized-chern", "chow-lemma-push-proper-bivariant" ], "proofs": [ { "contents": [ "This is true because the construction of", "$P_p(Z' \\to X, E)$, resp.\\ $c_p(Z' \\to X, E)$", "uses the exact same morphism $b : W \\to \\mathbf{P}^1_X$", "and $\\mathcal{Q}^\\bullet$. Then we can use", "Lemma \\ref{lemma-silly-shrink} to conclude.", "Some details omitted." ], "refs": [ "chow-lemma-silly-shrink" ], "ref_ids": [ 5779 ] } ], "ref_ids": [ 5924, 5734 ] }, { "id": 5802, "type": "theorem", "label": "chow-lemma-loc-chern-agree", "categories": [ "chow" ], "title": "chow-lemma-loc-chern-agree", "contents": [ "In Lemma \\ref{lemma-silly} say $E_2$ is the restriction of a perfect", "$E \\in D(\\mathcal{O}_X)$ whose Chern classes are defined and", "whose restriction to $X_1$ is zero, resp.\\ isomorphic to a", "finite locally free $\\mathcal{O}_{X_1}$-module", "of rank $< p$ sitting in cohomological degree $0$. Then the", "class $P'_p(E_2)$, resp.\\ $c'_p(E_2)$ of Lemma \\ref{lemma-silly} agrees with", "$P_p(X_2 \\to X, E)$, resp.\\ $c_p(X_2 \\to X, E)$ of", "Definition \\ref{definition-localized-chern}." ], "refs": [ "chow-lemma-silly", "chow-lemma-silly", "chow-definition-localized-chern" ], "proofs": [ { "contents": [ "The assumptions on $E$ imply that there is an open $U \\subset X$", "containing $X_1$ such that $E|_U$ is zero, resp.\\ isomorphic to a finite locally", "free $\\mathcal{O}_U$-module of rank $< p$. See More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-perfect-from-residue-field}.", "Let $Z \\subset X$ be the complement of $U$ in $X$ endowed with", "the reduced induced closed subscheme structure. Then", "$P_p(X_2 \\to X, E) = (Z \\to X_2)_* \\circ P_p(Z \\to X, E)$,", "resp.\\ $c_p(X_2 \\to X, E) = (Z \\to X_2)_* \\circ c_p(Z \\to X, E)$", "by Lemma \\ref{lemma-loc-chern-shrink-Z}.", "Now we can prove that $P_p(X_2 \\to X, E)$, resp.\\ $c_p(X_2 \\to X, E)$", "satisfies the characterization of $P'_p(E_2)$, resp.\\ $c'_p(E_2)$", "given in Lemma \\ref{lemma-silly}. Namely, by the relation", "$P_p(X_2 \\to X, E) = (Z \\to X_2)_* \\circ P_p(Z \\to X, E)$,", "resp.\\ $c_p(X_2 \\to X, E) = (Z \\to X_2)_* \\circ c_p(Z \\to X, E)$", "just proven and the fact that $X_1 \\cap Z = \\emptyset$,", "the composition $P_p(X_2 \\to X, E) \\circ i_{1, *}$,", "resp.\\ $c_p(X_2 \\to X, E) \\circ i_{1, *}$ is zero", "by Lemma \\ref{lemma-loc-chern-disjoint}.", "On the other hand,", "$P_p(X_2 \\to X, E) \\circ i_{2, *} = P_p(E_2)$,", "resp.\\ $c_p(X_2 \\to X, E) \\circ i_{2, *} = c_p(E_2)$", "by Lemma \\ref{lemma-loc-chern-after-pushforward}." ], "refs": [ "more-algebra-lemma-lift-perfect-from-residue-field", "chow-lemma-loc-chern-shrink-Z", "chow-lemma-silly", "chow-lemma-loc-chern-disjoint", "chow-lemma-loc-chern-after-pushforward" ], "ref_ids": [ 10232, 5801, 5776, 5800, 5799 ] } ], "ref_ids": [ 5776, 5776, 5924 ] }, { "id": 5803, "type": "theorem", "label": "chow-lemma-homomorphism-final", "categories": [ "chow" ], "title": "chow-lemma-homomorphism-final", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be", "locally of finite type over $S$. Let $b : W \\longrightarrow \\mathbf{P}^1_X$", "be a proper morphism of schemes. Let $n \\geq 1$. For $i = 1, \\ldots, n$", "let $Z_i \\subset X$ be a closed subscheme, let $Q_i \\in D(\\mathcal{O}_W)$,", "be a perfect object, let $p_i \\geq 0$ be an integer, and let", "$T_i \\subset W_\\infty$, $i = 1, \\ldots, n$ be closed.", "Denote $W_i = b^{-1}(\\mathbf{P}^1_{Z_i})$. Assume", "\\begin{enumerate}", "\\item for $i = 1, \\ldots, n$ the assumption of", "Lemma \\ref{lemma-localized-chern-pre} hold for", "$b, Z_i, Q_i, T_i, p_i$,", "\\item $Q_i|_{W \\setminus W_i}$ is zero, resp.\\ isomorphic to a finite", "locally free module of rank $< p_i$ placed in cohomological degree $0$.", "\\end{enumerate}", "Then $P'_{p_n}(Q_n) \\circ \\ldots \\circ P'_{p_1}(Q_1)$ is equal to", "$$", "(W_{n, \\infty} \\cap \\ldots \\cap W_{1, \\infty} \\to", "Z_n \\cap \\ldots \\cap Z_1)_* \\circ", "P'_{p_n}(Q_n|_{W_{n, \\infty}}) \\circ \\ldots \\circ P'_{p_1}(Q_1|_{W_{1, \\infty}})", "\\circ C", "$$", "in $A^{p_n + \\ldots + p_1}(Z_n \\cap \\ldots \\cap Z_1 \\to X)$,", "resp.\\ $c'_{p_n}(Q_n) \\circ \\ldots \\circ c'_{p_1}(Q_1)$ is equal to", "$$", "(W_{n, \\infty} \\cap \\ldots \\cap W_{1, \\infty} \\to", "Z_n \\cap \\ldots \\cap Z_1)_* \\circ", "c'_{p_n}(Q_n|_{W_{n, \\infty}}) \\circ \\ldots \\circ c'_{p_1}(Q_1|_{W_{1, \\infty}})", "\\circ C", "$$", "in $A^{p_n + \\ldots + p_1}(Z_n \\cap \\ldots \\cap Z_1 \\to X)$." ], "refs": [ "chow-lemma-localized-chern-pre" ], "proofs": [ { "contents": [ "Let us prove the statement on Chern classes by induction on $n$;", "the statement on $P_p(-)$ is proved in the exact same manner.", "The case $n = 1$ is the construction of $c'_{p_1}(Q_1)$.", "For $n > 1$ we have by induction that", "$c'_{p_n}(Q_n) \\circ \\ldots \\circ c'_{p_1}(Q_1)$ is equal to", "$$", "c'_{p_n}(Q_n) \\circ", "(W_{n - 1, \\infty} \\cap \\ldots \\cap W_{1, \\infty} \\to", "Z_{n - 1} \\cap \\ldots \\cap Z_1)_* \\circ", "c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \\infty}) \\circ \\ldots \\circ", "c'_{p_1}(Q_1|_{W_{1, \\infty}})", "\\circ C", "$$", "Observe that the restriction of $c'_{p_n}(Q_n)$ to", "$Z_{n - 1} \\cap \\ldots \\cap Z_1$ is computed by", "$b' : W_{n - 1} \\cap \\ldots \\cap W_1 \\to", "\\mathbf{P}^1_{Z_{n - 1} \\cap \\ldots \\cap Z_1}$", "and the restriction of $Q_n$ to $W_{n - 1} \\cap \\ldots \\cap W_1$.", "Denote $C_{n - 1} \\in A^0(W_{n - 1, \\infty} \\cap \\ldots \\cap W_{1, \\infty} \\to", "Z_{n - 1} \\cap \\ldots \\cap Z_1)$ the class of Lemma \\ref{lemma-gysin-at-infty}.", "Hence the above becomes", "\\begin{align*}", "(W_{n, \\infty} \\cap \\ldots \\cap W_{1, \\infty} \\to", "Z_n \\cap \\ldots \\cap Z_1)_* \\circ", "c'_{p_n}(Q_n|_{W_n, \\infty}) \\circ \\\\", "C_{n - 1} \\circ", "(W_{n - 1, \\infty} \\cap \\ldots \\cap W_{1, \\infty} \\to", "Z_{n - 1} \\cap \\ldots \\cap Z_1)_* \\\\", "\\circ", "c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \\infty}) \\circ \\ldots \\circ", "c'_{p_1}(Q_1|_{W_{1, \\infty}})", "\\circ C", "\\end{align*}", "By Lemma \\ref{lemma-homomorphism-pre}", "we know that the composition", "$C_{n - 1} \\circ (W_{n - 1, \\infty} \\cap \\ldots \\cap W_{1, \\infty} \\to", "Z_{n - 1} \\cap \\ldots \\cap Z_1)_*$", "is the identity on elements in the image of the gysin map", "$$", "(W_{n - 1, \\infty} \\cap \\ldots \\cap W_{1, \\infty} \\to", "W_{n - 1} \\cap \\ldots \\cap W_1)^*", "$$", "Thus it suffices to show that any element in the image of", "$c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \\infty}) \\circ \\ldots \\circ", "c'_{p_1}(Q_1|_{W_{1, \\infty}}) \\circ C$", "is in the image of the gysin map. We may write", "$$", "c'_{p_i}(Q_i|_{W_{i, \\infty}}) = \\text{restriction of } c_{p_i}(W_i \\to W, Q_i)", "\\text{ to } W_{i, \\infty}", "$$", "by Lemma \\ref{lemma-loc-chern-agree} and assumption (2) on $Q_i$", "in the statement of the lemma. Thus, if $\\beta \\in \\CH_{k + 1}(W)$", "restricts to the flat pullback of $\\alpha$ on $b^{-1}(\\mathbf{A}^1_X)$,", "then", "\\begin{align*}", "& c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \\infty}) \\cap \\ldots \\cap", "c'_{p_1}(Q_1|_{W_{1, \\infty}})", "\\cap C \\cap \\alpha \\\\", "& =", "c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \\infty}) \\cap \\ldots \\cap", "c'_{p_1}(Q_1|_{W_{1, \\infty}})", "\\cap i_\\infty^* \\beta \\\\", "& =", "c_{p_{n - 1}}(W_{n - 1} \\to W, Q_{n - 1}) \\cap \\ldots \\cap", "c_{p_{n - 1}}(W_1 \\to W, Q_1) \\cap i_\\infty^* \\beta \\\\", "& =", "(W_{n - 1, \\infty} \\cap \\ldots \\cap W_{1, \\infty} \\to", "W_{n - 1} \\cap \\ldots \\cap W_1)^*", "\\left(c_{p_{n - 1}}(W_{n - 1} \\to W, Q_{n - 1}) \\cap \\ldots \\cap", "c_{p_1}(W_1 \\to W, Q_1) \\cap \\beta\\right)", "\\end{align*}", "as desired." ], "refs": [ "chow-lemma-gysin-at-infty", "chow-lemma-homomorphism-pre", "chow-lemma-loc-chern-agree" ], "ref_ids": [ 5786, 5788, 5802 ] } ], "ref_ids": [ 5790 ] }, { "id": 5804, "type": "theorem", "label": "chow-lemma-independent-loc-chern-bQ", "categories": [ "chow" ], "title": "chow-lemma-independent-loc-chern-bQ", "contents": [ "Assume $(S, \\delta), X, Z, b : W \\to \\mathbf{P}^1_X, Q, T, p$", "satisfy all the assumptions of Lemma \\ref{lemma-localized-chern-pre}.", "Finally, let $F \\in D(\\mathcal{O}_X)$ be a perfect object whose", "Chern classes are defined such that", "\\begin{enumerate}", "\\item the restriction of $Q$ to $b^{-1}(\\mathbf{A}^1_X)$ is", "isomorphic to the pullback of $F$, and", "\\item $F|_{X \\setminus Z}$ is zero, resp.\\ isomorphic to a finite", "locally free $\\mathcal{O}_{X \\setminus Z}$-module of rank $< p$", "sitting in cohomological degree $0$.", "\\end{enumerate}", "Then the class $P'_p(Q)$, resp.\\ $c'_p(Q)$ in $A^p(Z \\to X)$ constructed", "in Lemma \\ref{lemma-localized-chern-pre}", "is equal to $P_p(Z \\to X, F)$, resp.\\ $c_p(Z \\to X, F)$." ], "refs": [ "chow-lemma-localized-chern-pre", "chow-lemma-localized-chern-pre" ], "proofs": [ { "contents": [ "The assumptions are preserved by base change with a morphism", "$X' \\to X$ locally of finite type. Hence it suffices to show that", "$P_p(Z \\to X, F) \\cap \\alpha = P'_p(Q) \\cap \\alpha$,", "resp.\\ $c_p(Z \\to X, F) \\cap \\alpha = c'_p(Q) \\cap \\alpha$", "for any $\\alpha \\in \\CH_k(X)$. Choose $\\beta \\in \\CH_{k + 1}(W)$", "whose restriction to $b^{-1}(\\mathbf{A}^1_X)$ is equal to", "the flat pullback of $\\alpha$ as in the construction of", "$C$ in Lemma \\ref{lemma-gysin-at-infty}.", "Denote $E \\subset W_\\infty$ the inverse image of $Z$.", "\\medskip\\noindent", "Let $U \\subset X$ be the maximal open subscheme such that $F|_U$ is", "zero, resp.\\ isomorphic to a finite locally free $\\mathcal{O}_U$-module", "of rank $< p$ sitting in cohomological degree $0$. Let $V \\subset W$ be", "the maximal open subscheme such that $Q|_V$ is zero, resp.\\ isomorphic", "to a finite locally free $\\mathcal{O}_V$-module of rank $< p$ sitting in", "cohomological degree $0$. By our assumptions on $Q$ and $F$ we have", "$$", "V \\cap b^{-1}(\\mathbf{A}^1_X) = \\mathbf{A}^1_U,\\quad", "V \\cap W_\\infty \\supset T,", "\\quad\\text{and}\\quad", "X \\setminus U \\subset Z", "$$", "Let $Z' = X \\setminus U$ and let $W' \\subset W$ be the scheme", "theoretic closure of $b^{-1}(\\mathbf{A}^1_{Z'})$. The inclusions", "above imply that we have $Z' \\subset Z$, $b(W') \\subset \\mathbf{P}^1_{Z'}$,", "$b' : W' \\to \\mathbf{P}^1_{Z'}$ is an isomorphism over $\\mathbf{A}^1_{Z'}$,", "and that $Q|_{W \\setminus W'}$ is zero, resp.\\ isomorphic to a finite", "locally free $\\mathcal{O}_{W \\setminus W'}$-module of rank $< p$", "sitting in cohomological degree $0$. The lemma follows from", "the following sequence of equalities (the case of $P_p$ is similar)", "\\begin{align*}", "c'_p(Q) \\cap \\alpha", "& =", "(E \\to Z)_*(c'_p(Q|_E) \\cap i_\\infty^*\\beta) \\\\", "& =", "(E \\to Z)_*(c_p(E \\to W_\\infty, Q|_{W_\\infty}) \\cap i_\\infty^*\\beta) \\\\", "& =", "(E \\to Z)_*(W'_\\infty \\to E)_*(c_p(W' \\to W, Q) \\cap i_\\infty^*\\beta) \\\\", "& =", "(W'_\\infty \\to Z)_*((i'_\\infty)^*(c_p(W' \\to W, Q) \\cap \\beta)) \\\\", "& =", "(W'_\\infty \\to Z)_*(C' \\cap c_p(Z' \\to X, F) \\cap \\alpha) \\\\", "& =", "c_p(Z \\to X, F) \\cap \\alpha", "\\end{align*}", "The first equality is the construction of $c'_p(Q)$. ", "The second is Lemma \\ref{lemma-loc-chern-agree}.", "The third expresses the fact that the restriction of", "$c_p(W' \\to W, Q)$ to $W_\\infty$ is equal to", "$c_p(W'_\\infty \\to W_\\infty, Q|_{W_\\infty})$, see for example", "Lemma \\ref{lemma-base-change-loc-chern}, and that", "$c_p(W'_\\infty \\to W_\\infty, Q|_{W_\\infty})$", "pushes forward to $c_p(E \\to W_\\infty, Q|_{W_\\infty})$", "by Lemma \\ref{lemma-loc-chern-shrink-Z}.", "The fourth is commutation of a bivariant class with", "a gysin homomorphism. For the fith, please observe that", "$c_p(W' \\to W, Q)$ and $c_p(Z' \\to X, E)$ restrict to the", "same bivariant class on $b^{-1}(\\mathbf{A}^1_X)$ by", "assumption (1) of the lemma. Hence $c_p(W' \\to W, Q) \\cap \\beta$", "is a class in $\\CH_{k + 1 - p}(W')$ whose restriction to", "$(b')^{-1}(\\mathbf{A}^1_{Z'})$ agrees with the flat pullback", "of $c_p(Z' \\to X, F) \\cap \\alpha$. Thus", "$(i'_\\infty)^*(c_p(W' \\to W, Q) \\cap \\beta)$ is equal to", "$C' \\cap c_p(Z' \\to X, F) \\cap \\alpha$ where", "$C' \\in A^0(W'_\\infty \\to Z')$ is constructed in", "Lemma \\ref{lemma-gysin-at-infty}.", "The final equality holds because $c_p(Z' \\to X, F)$ pushes forward", "to $c_p(Z \\to X, F)$ and because $(W'_\\infty \\to Z)_* \\circ C' = 1$ in $A^0(Z)$.", "In fact we have $(W'_\\infty \\to Z') \\circ C' = 1$ in $A^0(Z')$ as", "follows from the final statement in Lemma \\ref{lemma-gysin-at-infty}." ], "refs": [ "chow-lemma-gysin-at-infty", "chow-lemma-loc-chern-agree", "chow-lemma-base-change-loc-chern", "chow-lemma-loc-chern-shrink-Z", "chow-lemma-gysin-at-infty", "chow-lemma-gysin-at-infty" ], "ref_ids": [ 5786, 5802, 5798, 5801, 5786, 5786 ] } ], "ref_ids": [ 5790, 5790 ] }, { "id": 5805, "type": "theorem", "label": "chow-lemma-loc-chern-character", "categories": [ "chow" ], "title": "chow-lemma-loc-chern-character", "contents": [ "In the situation of Definition \\ref{definition-localized-chern}", "assume $E|_{X \\setminus Z}$ is zero. Then", "\\begin{align*}", "P_1(Z \\to X, E) & = c_1(Z \\to X, E), \\\\", "P_2(Z \\to X, E) & = c_1(Z \\to X, E)^2 - 2c_2(Z \\to X, E), \\\\", "P_3(Z \\to X, E) & = c_1(Z \\to X, E)^3 - 3c_1(Z \\to X, E)c_2(Z \\to X, E)", "+ 3c_3(Z \\to X, E),", "\\end{align*}", "and so on where the products are taken in the algebra $A^{(1)}(Z \\to X)$", "of Remark \\ref{remark-ring-loc-classes}." ], "refs": [ "chow-definition-localized-chern", "chow-remark-ring-loc-classes" ], "proofs": [ { "contents": [ "The statement makes sense because the zero sheaf has rank $< 1$ and", "hence the classes $c_p(Z \\to X, E)$ are defined for all $p \\geq 1$.", "The result itself follows immediately from the more general", "Lemma \\ref{lemma-localized-chern-pre-compose} as the localized Chern", "classes where defined using the procedure of", "Lemma \\ref{lemma-localized-chern-pre}", "in Section \\ref{section-localized-chern}." ], "refs": [ "chow-lemma-localized-chern-pre-compose", "chow-lemma-localized-chern-pre" ], "ref_ids": [ 5794, 5790 ] } ], "ref_ids": [ 5924, 5942 ] }, { "id": 5806, "type": "theorem", "label": "chow-lemma-loc-chern-classes-commute", "categories": [ "chow" ], "title": "chow-lemma-loc-chern-classes-commute", "contents": [ "In the situation of Definition \\ref{definition-localized-chern}", "assume $P_p(Z \\to X, E)$, resp.\\ $c_p(Z \\to X, E)$ is defined.", "Let $Y \\to X$ be locally of finite type and $c \\in A^*(Y \\to X)$.", "Then", "$$", "P_p(Z \\to X, E) \\circ c = c \\circ P_p(Z \\to X, E),", "$$", "respectively", "$$", "c_p(Z \\to X, E) \\circ c = c \\circ c_p(Z \\to X, E)", "$$", "in $A^*(Y \\times_X Z \\to X)$." ], "refs": [ "chow-definition-localized-chern" ], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-homomorphism-commute}.", "Namely, our assumptions say $E$", "is represented to a bounded complex $\\mathcal{E}^\\bullet$", "of finite locally free $\\mathcal{O}_X$-modules. Let", "$$", "b : W \\to \\mathbf{P}^1_X", "\\quad\\text{and}\\quad", "\\mathcal{Q}^\\bullet", "$$", "be the blowing up and complex of $\\mathcal{O}_W$-modules constructed in", "More on Flatness, Section \\ref{flat-section-blowup-complexes-III}.", "Let $T \\subset W_\\infty$ be the closed subscheme whose existence is", "averted in More on Flatness, Lemma \\ref{flat-lemma-graph-construction}.", "Let $T' \\subset T$ be the open and closed subscheme such that", "$\\mathcal{Q}^\\bullet|_{T'}$ is zero, resp.\\ isomorphic to a", "finite locally free sheaf of rank $< p$ placed in degree $0$. By definition", "$$", "c_p(Z \\to X, E) = c'_p(\\mathcal{Q}^\\bullet)", "$$", "as bivariant operations (and not just on cycles over $X$)", "where the right hand side is the bivariant class constructed in", "Lemma \\ref{lemma-localized-chern-pre} using $W, b, \\mathcal{Q}^\\bullet, T'$.", "By Lemma \\ref{lemma-homomorphism-commute} we have", "$$", "P'_p(\\mathcal{Q}^\\bullet) \\circ c = c \\circ P'_p(\\mathcal{Q}^\\bullet)", "\\quad\\text{resp.}\\quad", "c'_p(\\mathcal{Q}^\\bullet) \\circ c = c \\circ c'_p(\\mathcal{Q}^\\bullet)", "$$", "in $A^*(Y \\times_X Z \\to X)$ and we conclude." ], "refs": [ "chow-lemma-homomorphism-commute", "flat-lemma-graph-construction", "chow-lemma-localized-chern-pre", "chow-lemma-homomorphism-commute" ], "ref_ids": [ 5793, 6197, 5790, 5793 ] } ], "ref_ids": [ 5924 ] }, { "id": 5807, "type": "theorem", "label": "chow-lemma-additivity-loc-chern-c", "categories": [ "chow" ], "title": "chow-lemma-additivity-loc-chern-c", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. Let $Z \\to X$ be", "a closed immersion. Let", "$$", "E_1 \\to E_2 \\to E_3 \\to E_1[1]", "$$", "be a distinguished triangle of perfect objects in $D(\\mathcal{O}_X)$.", "Assume", "\\begin{enumerate}", "\\item $E_3 \\to E_1[1]$ can be represented be a map of bounded complexes", "of finite locally free $\\mathcal{O}_X$-modules, and", "\\item the restrictions $E_1|_{X \\setminus Z}$ and $E_3|_{X \\setminus Z}$", "are isomorphic to finite locally free $\\mathcal{O}_{X \\setminus Z}$-modules", "of rank $< p_1$ and $< p_3$ placed in degree $0$.", "\\end{enumerate}", "With notation as in Remark \\ref{remark-loc-chern-classes} we have", "$$", "c^{(p_1 + p_3)}(Z \\to X, E_2) = c^{(p_1)}(Z \\to X, E_1)c^{(p_3)}(Z \\to X, E_3)", "$$", "in $A^{(p_1 + p_3)}(Z \\to X)$." ], "refs": [ "chow-remark-loc-chern-classes" ], "proofs": [ { "contents": [ "Observe that the assumptions imply that $E_2|_{X \\setminus Z}$ is zero,", "resp.\\ isomorphic to a finite locally free $\\mathcal{O}_{X \\setminus Z}$-module", "of rank $< p_1 + p_3$. Thus the statement makes sense. The proof of this", "statement is tricky because the operator $\\eta_\\mathcal{I}$ from", "More on Flatness, Section \\ref{flat-section-eta} used in the construction", "of localized Chern classes doesn't transform distinguished triangles into", "distinguished triangles.", "\\medskip\\noindent", "Let $\\varphi^\\bullet : \\mathcal{E}_3^\\bullet[-1] \\to \\mathcal{E}_1^\\bullet$", "be a map of bounded complexes of finite locally free $\\mathcal{O}_X$-modules", "representing $E_3[-1] \\to E_1$ which exists by assumption. Consider the scheme", "$X' = \\mathbf{A}^1 \\times X$ with projection", "$g : X' \\to X$. Let $Z' = g^{-1}(Z) = \\mathbf{A}^1 \\times Z$.", "Denote $t$ the coordinate on $\\mathbf{A}^1$. Consider the cone", "$\\mathcal{C}^\\bullet$ of the map of complexes", "$$", "t g^*\\varphi^\\bullet :", "g^*\\mathcal{E}_3^\\bullet[-1]", "\\longrightarrow", "g^*\\mathcal{E}_1^\\bullet", "$$", "over $X'$. We obtain a distinguished triangle", "$$", "g^*\\mathcal{E}_1^\\bullet \\to \\mathcal{C}^\\bullet \\to", "g^*\\mathcal{E}_3^\\bullet \\to g^*\\mathcal{E}_1^\\bullet[1]", "$$", "where the first three terms form a termwise split short exact", "sequence of complexes. Clearly $\\mathcal{C}^\\bullet$ is a", "bounded complex of finite locally free $\\mathcal{O}_{X'}$-modules", "whose restriction to $X' \\setminus Z'$ is isomorphic to a", "finite locally free", "$\\mathcal{O}_{X' \\setminus Z'}$-module of rank $< p_1 + p_3$", "placed in degree $0$. Thus we have the localized Chern classes", "$$", "c_p(Z' \\to X', \\mathcal{C}^\\bullet) \\in A^p(Z' \\to X')", "$$", "for $p \\geq p_1 + p_3$. For any $\\alpha \\in \\CH_k(X)$ consider", "$$", "c_p(Z' \\to X', \\mathcal{C}^\\bullet) \\cap g^*\\alpha", "\\in \\CH_{k + 1 - p}(\\mathbf{A}^1 \\times X)", "$$", "If we restrict to $t = 0$, then the map $t g^*\\varphi^\\bullet$", "restricts to zero and $\\mathcal{C}^\\bullet|_{t = 0}$", "is the direct sum of $\\mathcal{E}_1^\\bullet$ and $\\mathcal{E}_3^\\bullet$.", "By compatibility of localized Chern classes with base change", "(Lemma \\ref{lemma-base-change-loc-chern}) we conclude that", "$$", "i_0^* \\circ c^{(p_1 + p_3)}(Z' \\to X', \\mathcal{C}^\\bullet) \\circ g^* =", "c^{(p_1 + p_2)}(Z \\to X, E_1 \\oplus E_3)", "$$", "in $A^{(p_1 + p_3)}(Z \\to X)$. On the other hand, if we restrict to $t = 1$,", "then the map $t g^*\\varphi^\\bullet$", "restricts to $\\varphi$ and $\\mathcal{C}^\\bullet|_{t = 1}$", "is a bounded complex of finite locally free modules representing $E_2$.", "We conclude that", "$$", "i_1^* \\circ c^{(p_1 + p_3)}(Z' \\to X', \\mathcal{C}^\\bullet) \\circ g^* =", "c^{(p_1 + p_2)}(Z \\to X, E_2)", "$$", "in $A^{(p_1 + p_3)}(Z \\to X)$. Since $i_0^* = i_1^*$ by definition of", "rational equivalence (more precisely this follows from the formulae in", "Lemma \\ref{lemma-linebundle-formulae}) we conclude that", "$$", "c^{(p_1 + p_2)}(Z \\to X, E_2) = c^{(p_1 + p_2)}(Z \\to X, E_1 \\oplus E_3)", "$$", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $E_2 = E_1 \\oplus E_3$ and $E_i$ corresponds to $\\mathcal{E}_i^\\bullet$", "for $i = 1, 3$ as above. Set", "$\\mathcal{E}_2^\\bullet = \\mathcal{E}_1^\\bullet \\oplus \\mathcal{E}_3^\\bullet$.", "For $i = 1, 2, 3$ let", "$$", "b_i : W_i \\to \\mathbf{P}^1_X", "\\quad\\text{and}\\quad", "\\mathcal{Q}_i^\\bullet", "$$", "be the blowing up and complex of $\\mathcal{O}_{W_i}$-modules constructed in", "More on Flatness, Section \\ref{flat-section-blowup-complexes-III}.", "Let $T_i \\subset W_{i, \\infty}$ be the closed subscheme whose existence is", "averted in More on Flatness, Lemma \\ref{flat-lemma-graph-construction}.", "Let $T'_i \\subset T_i$ be the open and closed subscheme such that", "$\\mathcal{Q}_i^\\bullet|_{T'_i}$ is isomorphic to a finite locally free", "sheaf of rank $< p_i$ in degree $0$. By definition", "$$", "c_p(Z \\to X, E_i) = c'_p(\\mathcal{Q}_i^\\bullet)", "$$", "where the right hand side is the bivariant class constructed in", "Lemma \\ref{lemma-localized-chern-pre} using", "$W_i, b_i, \\mathcal{Q}_i^\\bullet, T'_i$.", "By Divisors, Lemma \\ref{divisors-lemma-blowing-up-two-ideals} we", "can choose a commutative diagram", "$$", "\\xymatrix{", "W \\ar[dd]^{g_1} \\ar[rd]^{g_2} \\ar[rr]_{g_3} & & W_3 \\ar[dd]^{b_3} \\\\", "& W_2 \\ar[rd]^{b_2} \\\\", "W_1 \\ar[rr]^{b_1} & & \\mathbf{P}^1_X", "}", "$$", "where all morphisms are blowing ups which are isomorphisms over", "$\\mathbf{A}^1_X$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-localized-chern-pre-independent} we may use", "$W$, $b = b_i \\circ g_i$, $g_i^*\\mathcal{Q}_i^\\bullet$, and", "$g_i^{-1}(T'_i)$ to construct $c_p(Z \\to X, E_i)$. The same lemma also", "tells us that we may replace $g_i^{-1}(T'_i)$ with", "$T = g_1^{-1}(T'_1) \\cap g_2^{-1}(T'_2) \\cap g_3^{-1}(T'_3)$", "because this closed subscheme still contains all points of $W_\\infty$", "lying over $X \\setminus Z$. Hence we may use $T$ for each of", "the three constructions. By More on Flatness, Lemma", "\\ref{flat-lemma-complex-and-divisor-eta-pull}", "applied to the morphisms $g_i : W \\to W_i$ we find that", "$$", "g_i^*\\mathcal{Q}_i^\\bullet =", "\\eta_\\mathcal{I}b^*p^*\\mathcal{E}_i^\\bullet", "$$", "where $\\mathcal{I}$ is the invertible ideal sheaf of the effective", "Cartier divisor $W_\\infty$ and", "$p :\\mathbf{P}^1_X \\to X$ is the projection morphism. Since the functor", "$\\eta_\\mathcal{I}$ visibly commutes with direct sums, we see that", "$\\mathcal{Q}_2^\\bullet = \\mathcal{Q}_1^\\bullet \\oplus \\mathcal{Q}_3^\\bullet$.", "Thus the desired equality follows from", "Lemma \\ref{lemma-localized-chern-pre-sum-c}." ], "refs": [ "chow-lemma-base-change-loc-chern", "chow-lemma-linebundle-formulae", "flat-lemma-graph-construction", "chow-lemma-localized-chern-pre", "divisors-lemma-blowing-up-two-ideals", "chow-lemma-localized-chern-pre-independent", "flat-lemma-complex-and-divisor-eta-pull", "chow-lemma-localized-chern-pre-sum-c" ], "ref_ids": [ 5798, 5728, 6197, 5790, 8062, 5791, 6191, 5795 ] } ], "ref_ids": [ 5953 ] }, { "id": 5808, "type": "theorem", "label": "chow-lemma-additivity-loc-chern-P", "categories": [ "chow" ], "title": "chow-lemma-additivity-loc-chern-P", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. Let $Z \\to X$ be", "a closed immersion. Let", "$$", "E_1 \\to E_2 \\to E_3 \\to E_1[1]", "$$", "be a distinguished triangle of perfect objects in $D(\\mathcal{O}_X)$.", "Assume", "\\begin{enumerate}", "\\item $E_3 \\to E_1[1]$ can be represented be a map of bounded complexes", "of finite locally free $\\mathcal{O}_X$-modules, and", "\\item the restrictions $E_1|_{X \\setminus Z}$ and $E_3|_{X \\setminus Z}$", "are zero.", "\\end{enumerate}", "Then we have", "$$", "P_p(Z \\to X, E_2) = P_p(Z \\to X, E_1) + P_p(Z \\to X, E_3)", "$$", "for all $p \\in \\mathbf{Z}$ and consequently", "$ch(Z \\to X, E_2) = ch(Z \\to X, E_1) + ch(Z \\to X, E_3)$." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Lemma \\ref{lemma-additivity-loc-chern-c}", "except it uses", "Lemma \\ref{lemma-localized-chern-pre-sum-P}", "at the very end. For $p > 0$ we can deduce this lemma", "from Lemma \\ref{lemma-additivity-loc-chern-c} with $p_1 = p_3 = 1$", "and the relationship between $P_p(Z \\to X, E)$ and $c_p(Z \\to X, E)$ given in", "Lemma \\ref{lemma-loc-chern-character}. The case $p = 0$ can be shown", "directly (it is only interesting if $X$ has a connected component", "entirely contained in $Z$)." ], "refs": [ "chow-lemma-additivity-loc-chern-c", "chow-lemma-localized-chern-pre-sum-P", "chow-lemma-additivity-loc-chern-c", "chow-lemma-loc-chern-character" ], "ref_ids": [ 5807, 5796, 5807, 5805 ] } ], "ref_ids": [] }, { "id": 5809, "type": "theorem", "label": "chow-lemma-loc-chern-tensor-product", "categories": [ "chow" ], "title": "chow-lemma-loc-chern-tensor-product", "contents": [ "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.", "Let $Z_i \\subset X$, $i = 1, 2$ be closed subschemes. Let $F_i$, $i = 1, 2$", "be perfect objects of $D(\\mathcal{O}_X)$ whose Chern classes are defined.", "Assume that $F_i|_{X \\setminus Z_i}$ is zero\\footnote{Presumably there", "is a variant of this lemma where we only assume $F_i|_{X \\setminus Z_i}$", "is isomorphic to a finite locally free $\\mathcal{O}_{X \\setminus Z_i}$-module", "of rank $< p_i$.} for $i = 1, 2$. Denote", "$r_i = P_0(Z_i \\to X, F_i) \\in A^0(Z_i \\to X)$.", "Then we have", "$$", "c_1(Z_1 \\cap Z_2 \\to X, F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2) =", "r_1 c_1(Z_2 \\to X, F_2) + r_2 c_1(Z_1 \\to X, F_1)", "$$", "in $A^1(Z_1 \\cap Z_2 \\to X)$ and", "\\begin{align*}", "c_2(Z_1 \\cap Z_2 \\to X, F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2)", "& =", "r_1 c_2(Z_2 \\to X, F_2) +", "r_2 c_2(Z_1 \\to X, F_1) + \\\\", "& {r_1 \\choose 2} c_1(Z_2 \\to X, F_2)^2 + \\\\", "& (r_1r_2 - 1) c_1(Z_2 \\to X, F_2)c_1(Z_1 \\to X, F_1) + \\\\", "& {r_2 \\choose 2} c_1(Z_1 \\to X, F_1)^2", "\\end{align*}", "in $A^2(Z_1 \\cap Z_2 \\to X)$ and so on for higher Chern classes.", "Similarly, we have", "$$", "ch(Z_1 \\cap Z_2 \\to X, F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2) =", "ch(Z_1 \\to X, F_1) ch(Z_2 \\to X, F_2)", "$$", "in $A^*(Z_1 \\cap Z_2 \\to X) \\otimes \\mathbf{Q}$. More precisely, we have", "$$", "P_p(Z_1 \\cap Z_2 \\to X, F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2) =", "\\sum\\nolimits_{p_1 + p_2 = p}", "{p \\choose p_1} P_{p_1}(Z_1 \\to X, F_1) P_{p_2}(Z_2 \\to X, F_2)", "$$", "in $A^p(Z_1 \\cap Z_2 \\to X)$." ], "refs": [], "proofs": [ { "contents": [ "Choose proper morphisms $b_i : W_i \\to \\mathbf{P}^1_X$ and", "$Q_i \\in D(\\mathcal{O}_{W_i})$ as well as closed subschemes", "$T_i \\subset W_{i, \\infty}$ as in the construction of", "the localized Chern classes for $F_i$ or more generally as in", "Lemma \\ref{lemma-independent-loc-chern-bQ}. Choose a commutative", "diagram", "$$", "\\xymatrix{", "W \\ar[d]^{g_1} \\ar[rd]^b \\ar[r]_{g_2} & W_2 \\ar[d]^{b_2} \\\\", "W_1 \\ar[r]^{b_1} & \\mathbf{P}^1_X", "}", "$$", "where all morphisms are proper and isomorphisms over", "$\\mathbf{A}^1_X$. For example, we can take $W$ to be the closure", "of the graph of the isomorphism between", "$b_1^{-1}(\\mathbf{A}^1_X)$ and $b_2^{-1}(\\mathbf{A}^1_X)$.", "By Lemma \\ref{lemma-independent-loc-chern-bQ} we may work with", "$W$, $b = b_i \\circ g_i$, $Lg_i^*Q_i$, and", "$g_i^{-1}(T_i)$ to construct the localized Chern classes", "$c_p(Z_i \\to X, F_i)$. Thus we reduce to the situation described", "in the next paragraph.", "\\medskip\\noindent", "Assume we have", "\\begin{enumerate}", "\\item a proper morphism $b : W \\to \\mathbf{P}^1_X$ which is an isomorphism", "over $\\mathbf{A}^1_X$,", "\\item $E_i \\subset W_\\infty$ is the inverse image of $Z_i$,", "\\item perfect objects $Q_i \\in D(\\mathcal{O}_W)$ whose Chern classes", "are defined, such that", "\\begin{enumerate}", "\\item the restriction of $Q_i$ to $b^{-1}(\\mathbf{A}^1_X)$ is", "the pullback of $F_i$, and", "\\item there exists a closed subscheme $T_i \\subset W_\\infty$ containing", "all points of $W_\\infty$ lying over $X \\setminus Z_i$ such that", "$Q_i|_{T_i}$ is zero.", "\\end{enumerate}", "\\end{enumerate}", "By Lemma \\ref{lemma-independent-loc-chern-bQ} we have", "$$", "c_p(Z_i \\to X, F_i) = c'_p(Q_i) =", "(E_i \\to Z_i)_* \\circ c'_p(Q_i|_{E_i}) \\circ C", "$$", "and", "$$", "P_p(Z_i \\to X, F_i) = P'_p(Q_i) =", "(E_i \\to Z_i)_* \\circ P'_p(Q_i|_{E_i}) \\circ C", "$$", "for $i = 1, 2$. Next, we observe that", "$Q = Q_1 \\otimes_{\\mathcal{O}_W}^\\mathbf{L} Q_2$", "satisfies (3)(a) and (3)(b) for $F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2$", "and $T_1 \\cup T_2$. Hence we see that", "$$", "c_p(Z_1 \\cap Z_2 \\to X, F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2) =", "(E_1 \\cap E_2 \\to Z_1 \\cap Z_2)_* \\circ", "c'_p(Q|_{E_1 \\cap E_2}) \\circ C", "$$", "and", "$$", "P_p(Z_1 \\cap Z_2 \\to X, F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2) =", "(E_1 \\cap E_2 \\to Z_1 \\cap Z_2)_* \\circ", "P'_p(Q|_{E_1 \\cap E_2}) \\circ C", "$$", "by the same lemma. By Lemma \\ref{lemma-silly-tensor-product}", "the classes $c'_p(Q|_{E_1 \\cap E_2})$ and $P'_p(Q|_{E_1 \\cap E_2})$", "can be expanded in the correct manner in terms of the classes", "$c'_p(Q_i|_{E_i})$ and $P'_p(Q_i|_{E_i})$. Then finally", "Lemma \\ref{lemma-homomorphism-final}", "tells us that polynomials in $c'_p(Q_i|_{E_i})$ and $P'_p(Q_i|_{E_i})$", "agree with the corresponding polynomials in", "$c'_p(Q_i)$ and $P'_p(Q_i)$ as desired." ], "refs": [ "chow-lemma-independent-loc-chern-bQ", "chow-lemma-independent-loc-chern-bQ", "chow-lemma-independent-loc-chern-bQ", "chow-lemma-silly-tensor-product", "chow-lemma-homomorphism-final" ], "ref_ids": [ 5804, 5804, 5804, 5785, 5803 ] } ], "ref_ids": [] }, { "id": 5810, "type": "theorem", "label": "chow-lemma-pullback-virtual-normal-sheaf", "categories": [ "chow" ], "title": "chow-lemma-pullback-virtual-normal-sheaf", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let", "$$", "\\xymatrix{", "Z' \\ar[r] \\ar[d]_g & X' \\ar[d]^f \\\\", "Z \\ar[r] & X", "}", "$$", "be a cartesian diagram of schemes locally of finite type over $S$", "whose horizontal arrows are closed immersions.", "If $\\mathcal{N}$ is a virtual normal sheaf for $Z$ in $X$, then", "$\\mathcal{N}' = g^*\\mathcal{N}$ is a virtual normal sheaf for", "$Z'$ in $X'$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the surjectivity of the map", "$g^*\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z'/X'}$ proved in", "Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial-flat}." ], "refs": [ "morphisms-lemma-conormal-functorial-flat" ], "ref_ids": [ 5305 ] } ], "ref_ids": [] }, { "id": 5811, "type": "theorem", "label": "chow-lemma-construction-gysin", "categories": [ "chow" ], "title": "chow-lemma-construction-gysin", "contents": [ "The construction above defines a bivariant class\\footnote{The", "notation $A^*(Z \\to X)^\\wedge$ is discussed in", "Remark \\ref{remark-completion-bivariant}.", "If $X$ is quasi-compact, then $A^*(Z \\to X)^\\wedge = A^*(Z \\to X)$.}", "$$", "c(Z \\to X, \\mathcal{N}) \\in A^*(Z \\to X)^\\wedge", "$$", "and moreover the construction is compatible with base change", "as in Lemma \\ref{lemma-pullback-virtual-normal-sheaf}.", "If $\\mathcal{N}$ has constant rank $r$, then", "$c(Z \\to X, \\mathcal{N}) \\in A^r(Z \\to X)$." ], "refs": [ "chow-remark-completion-bivariant", "chow-lemma-pullback-virtual-normal-sheaf" ], "proofs": [ { "contents": [ "Since both $i_* \\circ j^* \\circ C$ and $p^*$ are bivariant classes", "(see Lemmas \\ref{lemma-flat-pullback-bivariant} and", "\\ref{lemma-push-proper-bivariant}) we can use the equation", "$$", "i_* \\circ j^* \\circ C = p^* \\circ c(Z \\to X, \\mathcal{N})", "$$", "(suitably interpreted) to define $c(Z \\to X, \\mathcal{N})$", "as a bivariant class. This works because $p^*$ is always", "bijective on chow groups by Lemma \\ref{lemma-vectorbundle}.", "\\medskip\\noindent", "Let $X' \\to X$, $Z' \\to X'$, and $\\mathcal{N}'$ be as in", "Lemma \\ref{lemma-pullback-virtual-normal-sheaf}. Write", "$c = c(Z \\to X, \\mathcal{N})$ and $c' = c(Z' \\to X', \\mathcal{N}')$.", "The second statement of the lemma means that $c'$ is the restriction of $c$", "as in Remark \\ref{remark-restriction-bivariant}. Since we claim this", "is true for all $X'/X$ locally of finite type, a formal argument", "shows that it suffices to check that $c' \\cap \\alpha' = c \\cap \\alpha'$", "for $\\alpha' \\in \\CH_k(X')$.", "To see this, note that we have a commutative diagram", "$$", "\\xymatrix{", "C_{Z'}X' \\ar[d] \\ar[r] &", "W'_\\infty \\ar[d] \\ar[r] &", "W' \\ar[d] \\ar[r] &", "\\mathbf{P}^1_{X'} \\ar[d] \\\\", "C_ZX \\ar[r] &", "W_\\infty \\ar[r] &", "W \\ar[r] &", "\\mathbf{P}^1_X", "}", "$$", "which induces closed immersions:", "$$", "W' \\to W \\times_{\\mathbf{P}^1_X} \\mathbf{P}^1_{X'},\\quad", "W'_\\infty \\to W_\\infty \\times_X X',\\quad", "C_{Z'}X' \\to C_ZX \\times_Z Z'", "$$", "To get $c \\cap \\alpha'$ we use the class $C \\cap \\alpha'$", "defined using the morphism", "$W \\times_{\\mathbf{P}^1_X} \\mathbf{P}^1_{X'} \\to \\mathbf{P}^1_{X'}$", "in Lemma \\ref{lemma-gysin-at-infty}.", "To get $c' \\cap \\alpha'$ on the other hand, we use the class", "$C' \\cap \\alpha'$ defined using the morphism $W' \\to \\mathbf{P}^1_{X'}$.", "By Lemma \\ref{lemma-gysin-at-infty-independent} the pushforward of", "$C' \\cap \\alpha'$ by the closed immersion", "$W'_\\infty \\to (W \\times_{\\mathbf{P}^1_X} \\mathbf{P}^1_{X'})_\\infty$,", "is equal to $C \\cap \\alpha'$. Hence the same is true for the pullbacks", "to the opens", "$$", "C_{Z'}X' \\subset W'_\\infty,\\quad", "C_ZX \\times_Z Z' \\subset (W \\times_{\\mathbf{P}^1_X} \\mathbf{P}^1_{X'})_\\infty", "$$", "by Lemma \\ref{lemma-flat-pullback-proper-pushforward}.", "Since we have a commutative diagram", "$$", "\\xymatrix{", "C_{Z'} X' \\ar[d] \\ar[r] & N' \\ar@{=}[d] \\\\", "C_ZX \\times_Z Z' \\ar[r] & N \\times_Z Z'", "}", "$$", "these classes pushforward to the same class on $N'$ which", "proves that we obtain the same element $c \\cap \\alpha' = c' \\cap \\alpha'$", "in $\\CH_*(Z')$." ], "refs": [ "chow-lemma-flat-pullback-bivariant", "chow-lemma-push-proper-bivariant", "chow-lemma-vectorbundle", "chow-lemma-pullback-virtual-normal-sheaf", "chow-remark-restriction-bivariant", "chow-lemma-gysin-at-infty", "chow-lemma-gysin-at-infty-independent", "chow-lemma-flat-pullback-proper-pushforward" ], "ref_ids": [ 5732, 5734, 5744, 5810, 5938, 5786, 5787, 5682 ] } ], "ref_ids": [ 5944, 5810 ] }, { "id": 5812, "type": "theorem", "label": "chow-lemma-gysin-decompose", "categories": [ "chow" ], "title": "chow-lemma-gysin-decompose", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme", "locally of finite type over $S$. Let $\\mathcal{N}$ be a virtual normal", "sheaf for a closed subscheme $Z$ of $X$. Suppose that we have a short", "exact sequence $0 \\to \\mathcal{N}' \\to \\mathcal{N} \\to \\mathcal{E} \\to 0$", "of finite locally free $\\mathcal{O}_Z$-modules such that the given surjection", "$\\sigma : \\mathcal{N}^\\vee \\to \\mathcal{C}_{Z/X}$ factors through a map", "$\\sigma' : (\\mathcal{N}')^\\vee \\to \\mathcal{C}_{Z/X}$.", "Then", "$$", "c(Z \\to X, \\mathcal{N}) = c_{top}(\\mathcal{E}) \\circ c(Z \\to X, \\mathcal{N}')", "$$", "as bivariant classes." ], "refs": [], "proofs": [ { "contents": [ "Denote $N' \\to N$ the closed immersion of vector bundles corresponding", "to the surjection $\\mathcal{N}^\\vee \\to (\\mathcal{N}')^\\vee$. Then we", "have closed immersions", "$$", "C_ZX \\to N' \\to N", "$$", "Thus the desired relationship between the bivariant classes follows", "immediately from Lemma \\ref{lemma-easy-virtual-class}." ], "refs": [ "chow-lemma-easy-virtual-class" ], "ref_ids": [ 5766 ] } ], "ref_ids": [] }, { "id": 5813, "type": "theorem", "label": "chow-lemma-gysin-excess", "categories": [ "chow" ], "title": "chow-lemma-gysin-excess", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Consider", "a cartesian diagram", "$$", "\\xymatrix{", "Z' \\ar[r] \\ar[d]_g & X' \\ar[d]^f \\\\", "Z \\ar[r] & X", "}", "$$", "of schemes locally of finite type over $S$ whose horizontal arrows", "are closed immersions. Let $\\mathcal{N}$, resp.\\ $\\mathcal{N}'$", "be a virtual normal sheaf for $Z \\subset X$, resp.\\ $Z' \\to X'$.", "Assume given a short exact sequence", "$0 \\to \\mathcal{N}' \\to g^*\\mathcal{N} \\to \\mathcal{E} \\to 0$", "of finite locally free modules on $Z'$ such that the diagram", "$$", "\\xymatrix{", "g^*\\mathcal{N}^\\vee \\ar[r] \\ar[d] &", "(\\mathcal{N}')^\\vee \\ar[d] \\\\", "g^*\\mathcal{C}_{Z/X} \\ar[r] &", "\\mathcal{C}_{Z'/X'}", "}", "$$", "commutes. Then we have", "$$", "res(c(Z \\to X, \\mathcal{N})) =", "c_{top}(\\mathcal{E}) \\circ c(Z' \\to X', \\mathcal{N}')", "$$", "in $A^*(Z' \\to X')^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-construction-gysin} we have", "$res(c(Z \\to X, \\mathcal{N})) = c(Z' \\to X', g^*\\mathcal{N})$", "and the equality follows from Lemma \\ref{lemma-gysin-decompose}." ], "refs": [ "chow-lemma-construction-gysin", "chow-lemma-gysin-decompose" ], "ref_ids": [ 5811, 5812 ] } ], "ref_ids": [] }, { "id": 5814, "type": "theorem", "label": "chow-lemma-gysin-fundamental", "categories": [ "chow" ], "title": "chow-lemma-gysin-fundamental", "contents": [ "In the situation described just above assume $\\dim_\\delta(Y) = n$", "and that $\\mathcal{C}_{Y \\times_X Z/Z}$ has constant rank $r$.", "Then", "$$", "c(Z \\to X, \\mathcal{N}) \\cap [Y]_n =", "c_{top}(\\mathcal{E}) \\cap [Z \\times_X Y]_{n - r}", "$$", "in $\\CH_*(Z \\times_X Y)$." ], "refs": [], "proofs": [ { "contents": [ "The bivariant class $c_{top}(\\mathcal{E}) \\in A^*(Z \\times_X Y)$ was", "defined in Remark \\ref{remark-top-chern-class}.", "By Lemma \\ref{lemma-construction-gysin} we may replace $X$ by $Y$.", "Thus we may assume $Z \\to X$ is a regular closed immersion", "of codimension $r$, we have $\\dim_\\delta(X) = n$, and we have", "to show that $c(Z \\to X, \\mathcal{N}) \\cap [X]_n =", "c_{top}(\\mathcal{E}) \\cap [Z]_{n - r}$ in $\\CH_*(Z)$.", "By Lemma \\ref{lemma-gysin-decompose} we may even assume", "$\\mathcal{N}^\\vee \\to \\mathcal{C}_{Z/X}$ is an isomorphism.", "In other words, we have to show", "$c(Z \\to X, \\mathcal{C}_{Z/X}^\\vee) \\cap [X]_n = [Z]_{n - r}$ in $\\CH_*(Z)$.", "\\medskip\\noindent", "Let us trace through the steps in the definition of", "$c(Z \\to X, \\mathcal{C}_{Z/X}^\\vee) \\cap [X]_n$. Let", "$b : W \\to \\mathbf{P}^1_X$", "be the blowing up of $\\infty(Z)$. We first have to compute", "$C \\cap [X]_n$ where $C \\in A^0(W_\\infty \\to X)$ is", "the class of Lemma \\ref{lemma-gysin-at-infty}.", "To do this, note that $[W]_{n + 1}$", "is a cycle on $W$ whose restriction to $\\mathbf{A}^1_X$ is", "equal to the flat pullback of $[X]_n$. Hence $C \\cap [X]_n$", "is equal to $i_\\infty^*[W]_{n + 1}$. Since $W_\\infty$ is an", "effective Cartier divisor on $W$ we have", "$i_\\infty^*[W]_{n + 1} = [W_\\infty]_n$, see Lemma \\ref{lemma-easy-gysin}.", "The restriction of this class to the open $C_ZX \\subset W_\\infty$", "is of course just $[C_ZX]_n$. Because $Z \\subset X$ is regularly", "embedded we have", "$$", "\\mathcal{C}_{Z/X, *} = \\text{Sym}(\\mathcal{C}_{Z/X})", "$$", "as graded $\\mathcal{O}_Z$-algebras, see", "Divisors, Lemma \\ref{divisors-lemma-quasi-regular-immersion}.", "Hence $p : N = C_ZX \\to Z$ is the structure morphism of the", "vector bundle associated to the finite locally free module", "$\\mathcal{C}_{Z/X}$ of rank $r$. Then it is clear that", "$p^*[Z]_{n - r} = [C_ZX]_n$ and the proof is complete." ], "refs": [ "chow-remark-top-chern-class", "chow-lemma-construction-gysin", "chow-lemma-gysin-decompose", "chow-lemma-gysin-at-infty", "chow-lemma-easy-gysin", "divisors-lemma-quasi-regular-immersion" ], "ref_ids": [ 5947, 5811, 5812, 5786, 5717, 7992 ] } ], "ref_ids": [] }, { "id": 5815, "type": "theorem", "label": "chow-lemma-gysin-easy", "categories": [ "chow" ], "title": "chow-lemma-gysin-easy", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme", "locally of finite type over $S$. Let $\\mathcal{N}$ be a virtual normal", "sheaf for a closed subscheme $Z$ of $X$. Let $Y \\to X$ be a morphism", "which is locally of finite type. Given integers $r$, $n$ assume", "\\begin{enumerate}", "\\item $\\mathcal{N}$ is locally free of rank $r$,", "\\item every irreducible component of $Y$ has $\\delta$-dimension $n$,", "\\item $\\dim_\\delta(Z \\times_X Y) \\leq n - r$, and", "\\item for $\\xi \\in Z \\times_X Y$ with $\\delta(\\xi) = n - r$", "the local ring $\\mathcal{O}_{Y, \\xi}$ is Cohen-Macaulay.", "\\end{enumerate}", "Then $c(Z \\to X, \\mathcal{N}) \\cap [Y]_n = [Z \\times_X Y]_{n - r}$", "in $\\CH_{n - r}(Z \\times_X Y)$." ], "refs": [], "proofs": [ { "contents": [ "The statement makes sense as $Z \\times_X Y$ is a closed subscheme of $Y$.", "Because $\\mathcal{N}$ has rank $r$ we know that", "$c(Z \\to X, \\mathcal{N}) \\cap [Y]_n$ is in $\\CH_{n - r}(Z \\times_X Y)$.", "Since $\\dim_\\delta(Z \\cap Y) \\leq n - r$ the chow group", "$\\CH_{n - r}(Z \\times_X Y)$ is freely generated by the", "cycle classes of the irreducible components $W \\subset Z \\times_X Y$", "of $\\delta$-dimension $n - r$. Let $\\xi \\in W$ be the generic point.", "By assumption (2) we see that $\\dim(\\mathcal{O}_{Y, \\xi}) = r$.", "On the other hand, since $\\mathcal{N}$ has rank $r$ and since", "$\\mathcal{N}^\\vee \\to \\mathcal{C}_{Z/X}$ is surjective, we see that", "the ideal sheaf of $Z$ is locally cut out by $r$ equations.", "Hence the quasi-coherent ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_Y$", "of $Z \\times_X Y$ in $Y$ is locally generated by $r$ elements.", "Since $\\mathcal{O}_{Y, \\xi}$ is Cohen-Macaulay of dimension $r$", "and since $\\mathcal{I}_\\xi$ is an ideal of definition (as $\\xi$ is", "a generic point of $Z \\times_X Y$) it follows that $\\mathcal{I}_\\xi$", "is generated by a regular sequence", "(Algebra, Lemma \\ref{algebra-lemma-reformulate-CM}).", "By Divisors, Lemma \\ref{divisors-lemma-Noetherian-scheme-regular-ideal}", "we see that $\\mathcal{I}$ is generated by a regular sequence over", "an open neighbourhood $V \\subset Y$ of $\\xi$. By our description of", "$\\CH_{n - r}(Z \\times_X Y)$ it suffices to show that", "$c(Z \\to X, \\mathcal{N}) \\cap [V]_n = [Z \\times_X V]_{n - r}$", "in $\\CH_{n - r}(Z \\times_X V)$. This follows from", "Lemma \\ref{lemma-gysin-fundamental}", "because the excess normal sheaf is $0$ over $V$." ], "refs": [ "algebra-lemma-reformulate-CM", "divisors-lemma-Noetherian-scheme-regular-ideal", "chow-lemma-gysin-fundamental" ], "ref_ids": [ 923, 7988, 5814 ] } ], "ref_ids": [] }, { "id": 5816, "type": "theorem", "label": "chow-lemma-gysin-agrees", "categories": [ "chow" ], "title": "chow-lemma-gysin-agrees", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme", "locally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$", "be a triple as in Definition \\ref{definition-gysin-homomorphism}.", "The gysin homomorphism $i^*$ viewed as an element of $A^1(D \\to X)$", "(see Lemma \\ref{lemma-gysin-bivariant}) is the same as the bivariant class", "$c(D \\to X, \\mathcal{N}) \\in A^1(D \\to X)$", "constructed using $\\mathcal{N} = i^*\\mathcal{L}$", "viewed as a virtual normal sheaf for $D$ in $X$." ], "refs": [ "chow-definition-gysin-homomorphism", "chow-lemma-gysin-bivariant" ], "proofs": [ { "contents": [ "We will use the criterion of Lemma \\ref{lemma-bivariant-zero}.", "Thus we may assume that $X$ is an integral scheme and", "we have to show that $i^*[X]$ is equal to $c \\cap [X]$.", "Let $n = \\dim_\\delta(X)$. As usual, there are two cases.", "\\medskip\\noindent", "If $X = D$, then we see that both classes are represented by", "$c_1(\\mathcal{N}) \\cap [X]_n$. See Lemma \\ref{lemma-gysin-fundamental}", "and Definition \\ref{definition-gysin-homomorphism}.", "\\medskip\\noindent", "If $D \\not = X$, then $D \\to X$ is an effective Cartier divisor", "and in particular a regular closed immersion of codimension $1$.", "Again by Lemma \\ref{lemma-gysin-fundamental} we conclude", "$c(D \\to X, \\mathcal{N}) \\cap [X]_n = [D]_{n - 1}$. The same", "is true by definition for the gysin homomorphism and we conclude", "once again." ], "refs": [ "chow-lemma-bivariant-zero", "chow-lemma-gysin-fundamental", "chow-definition-gysin-homomorphism", "chow-lemma-gysin-fundamental" ], "ref_ids": [ 5740, 5814, 5915, 5814 ] } ], "ref_ids": [ 5915, 5733 ] }, { "id": 5817, "type": "theorem", "label": "chow-lemma-gysin-commutes", "categories": [ "chow" ], "title": "chow-lemma-gysin-commutes", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme", "locally of finite type over $S$. Let $Z \\subset X$ be a closed subscheme", "with virtual normal sheaf $\\mathcal{N}$. Let $Y \\to X$ be locally of", "finite type and $c \\in A^*(Y \\to X)$. Then $c$ and $c(Z \\to X, \\mathcal{N})$", "commute (Remark \\ref{remark-bivariant-commute})." ], "refs": [ "chow-remark-bivariant-commute" ], "proofs": [ { "contents": [ "To check this we may use Lemma \\ref{lemma-bivariant-zero}.", "Thus we may assume $X$ is an integral scheme and we have to show", "$c \\cap c(Z \\to X, \\mathcal{N}) \\cap [X] =", "c(Z \\to X, \\mathcal{N}) \\cap c \\cap [X]$ in $\\CH_*(Z \\times_X Y)$.", "\\medskip\\noindent", "If $Z = X$, then $c(Z \\to X, \\mathcal{N}) = c_{top}(\\mathcal{N})$ by", "Lemma \\ref{lemma-gysin-fundamental} which commutes", "with the bivariant class $c$, see Lemma \\ref{lemma-cap-commutative-chern}.", "\\medskip\\noindent", "Assume that $Z$ is not equal to $X$. By Lemma \\ref{lemma-bivariant-zero}", "it even suffices to prove the result after blowing up $X$ (in a nonzero ideal).", "Let us blowup $X$ in the ideal sheaf of $Z$. This reduces us to the case", "where $Z$ is an effective Cartier divisor, see", "Divisors, Lemma", "\\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor},", "\\medskip\\noindent", "If $Z$ is an effective Cartier divisor, then we have", "$$", "c(Z \\to X, \\mathcal{N}) =", "c_{top}(\\mathcal{E}) \\circ i^*", "$$", "where $i^* \\in A^1(Z \\to X)$ is the gysin homomorphism", "associated to $i : Z \\to X$ (Lemma \\ref{lemma-gysin-bivariant})", "and $\\mathcal{E}$ is the dual of the kernel of", "$\\mathcal{N}^\\vee \\to \\mathcal{C}_{Z/X}$, see", "Lemmas \\ref{lemma-gysin-decompose} and \\ref{lemma-gysin-agrees}.", "Then we conclude because Chern classes are in the center of the", "bivariant ring (in the strong sense formulated in", "Lemma \\ref{lemma-cap-commutative-chern}) and $c$ commutes", "with the gysin homomorphism $i^*$ by definition of bivariant classes." ], "refs": [ "chow-lemma-bivariant-zero", "chow-lemma-gysin-fundamental", "chow-lemma-cap-commutative-chern", "chow-lemma-bivariant-zero", "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "chow-lemma-gysin-bivariant", "chow-lemma-gysin-decompose", "chow-lemma-gysin-agrees", "chow-lemma-cap-commutative-chern" ], "ref_ids": [ 5740, 5814, 5752, 5740, 8054, 5733, 5812, 5816, 5752 ] } ], "ref_ids": [ 5939 ] }, { "id": 5818, "type": "theorem", "label": "chow-lemma-relation-normal-cones", "categories": [ "chow" ], "title": "chow-lemma-relation-normal-cones", "contents": [ "With notation as above we have", "$$", "o^*[C_ZX]_n = [C_Z Y]_{n - 1}", "$$", "in $\\CH_{n - 1}(Y \\times_{o, C_Y X} C_ZX)$." ], "refs": [], "proofs": [ { "contents": [ "Denote $W \\to \\mathbf{P}^1_X$ the blowing up of $\\infty(Z)$ as in", "Section \\ref{section-blowup-Z-first}.", "Similarly, denote $W' \\to \\mathbf{P}^1_X$ the blowing up of $\\infty(Y)$.", "Since $\\infty(Z) \\subset \\infty(Y)$ we get an opposite inclusion", "of ideal sheaves and hence a map of the graded algebras", "defining these blowups. This produces a rational morphism from $W$", "to $W'$ which in fact has a canonical representative", "$$", "W \\supset U \\longrightarrow W'", "$$", "See Constructions, Lemma \\ref{constructions-lemma-morphism-relative-proj}.", "A local calculation (omitted) shows that $U$ contains at least all points", "of $W$ not lying over $\\infty$ and the open subscheme $C_Z X$ of the special", "fibre. After shrinking $U$ we may assume $U_\\infty = C_Z X$ and", "$\\mathbf{A}^1_X \\subset U$. Another local calculation (omitted)", "shows that the morphism $U_\\infty \\to W'_\\infty$", "induces the canonical morphism $C_Z X \\to C_Y X \\subset W'_\\infty$", "of normal cones induced by the inclusion of ideals sheaves", "coming from $Z \\subset Y$. Denote $W'' \\subset W$ the strict transform of", "$\\mathbf{P}^1_Y \\subset \\mathbf{P}^1_X$ in $W$. Then $W''$ is the blowing", "up of $\\mathbf{P}^1_Y$ in $\\infty(Z)$ by", "Divisors, Lemma \\ref{divisors-lemma-strict-transform}", "and hence $(W'' \\cap U)_\\infty = C_ZY$.", "\\medskip\\noindent", "Consider the effective Cartier divisor $i : \\mathbf{P}^1_Y \\to W'$", "from (\\ref{item-find-Z-in-blowup}) and its associated bivariant class", "$i^* \\in A^1(\\mathbf{P}^1_Y \\to W')$ from Lemma \\ref{lemma-gysin-bivariant}.", "We similarly denote $(i'_\\infty)^* \\in A^1(W'_\\infty \\to W')$ the", "gysin map at infinity. Observe that the restriction of $i'_\\infty$", "(Remark \\ref{remark-restriction-bivariant}) to $U$ is the restriction of", "$i_\\infty^* \\in A^1(W_\\infty \\to W)$ to $U$. On the one hand we have", "$$", "(i'_\\infty)^* i^* [U]_{n + 1} =", "i_\\infty^* i^* [U]_{n + 1} =", "i_\\infty^* [(W'' \\cap U)_\\infty]_{n + 1} =", "[C_ZY]_n", "$$", "because $i_\\infty^*$ kills all classes supported over $\\infty$, because", "$i^*[U]$ and $[W'']$ agree as cycles over $\\mathbf{A}^1$, and because", "$C_ZY$ is the fibre of $W'' \\cap U$ over $\\infty$.", "On the other hand, we have", "$$", "(i'_\\infty)^* i^* [U]_{n + 1} =", "i^* i_\\infty^*[U]_{n + 1} =", "i^* [U_\\infty] =", "o^*[C_YX]_n", "$$", "because $(i'_\\infty)^*$ and $i^*$ commute", "(Lemma \\ref{lemma-gysin-commutes-gysin})", "and because the fibre of $i : \\mathbf{P}^1_Y \\to W'$ over $\\infty$", "factors as $o : Y \\to C_YX$ and the open immersion $C_YX \\to W'_\\infty$.", "The lemma follows." ], "refs": [ "constructions-lemma-morphism-relative-proj", "divisors-lemma-strict-transform", "chow-lemma-gysin-bivariant", "chow-remark-restriction-bivariant", "chow-lemma-gysin-commutes-gysin" ], "ref_ids": [ 12644, 8065, 5733, 5938, 5724 ] } ], "ref_ids": [] }, { "id": 5819, "type": "theorem", "label": "chow-lemma-gysin-composition", "categories": [ "chow" ], "title": "chow-lemma-gysin-composition", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $Z \\subset Y \\subset X$ be closed subschemes of a scheme locally", "of finite type over $S$.", "Let $\\mathcal{N}$ be a virtual normal sheaf for $Z \\subset X$.", "Let $\\mathcal{N}'$ be a virtual normal sheaf for $Z \\subset Y$.", "Let $\\mathcal{N}''$ be a virtual normal sheaf for $Y \\subset X$.", "Assume there is a commutative diagram", "$$", "\\xymatrix{", "(\\mathcal{N}'')^\\vee|_Z \\ar[r] \\ar[d] &", "\\mathcal{N}^\\vee \\ar[r] \\ar[d] &", "(\\mathcal{N}')^\\vee \\ar[d] \\\\", "\\mathcal{C}_{Y/X}|_Z \\ar[r] &", "\\mathcal{C}_{Z/X} \\ar[r] &", "\\mathcal{C}_{Z/Y}", "}", "$$", "where the sequence at the bottom is from More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-transitivity-conormal} and the top", "sequence is a short exact sequence. Then", "$$", "c(Z \\to X, \\mathcal{N}) =", "c(Z \\to Y, \\mathcal{N}') \\circ c(Y \\to X, \\mathcal{N}'')", "$$", "in $A^*(Z \\to X)^\\wedge$." ], "refs": [ "more-morphisms-lemma-transitivity-conormal" ], "proofs": [ { "contents": [ "Observe that the assumptions remain satisfied after any base change", "by a morphism $X' \\to X$ which is locally of finite type (the short", "exact sequence of virtual normal sheaves is locally split hence", "remains exact after any base change). Thus to check the", "equality of bivariant classes we may use Lemma \\ref{lemma-bivariant-zero}.", "Thus we may assume $X$ is an integral scheme and we have to show", "$c(Z \\to X, \\mathcal{N}) \\cap [X] =", "c(Z \\to Y, \\mathcal{N}') \\cap c(Y \\to X, \\mathcal{N}'') \\cap [X]$.", "\\medskip\\noindent", "If $Y = X$, then we have", "\\begin{align*}", "c(Z \\to Y, \\mathcal{N}') \\cap c(Y \\to X, \\mathcal{N}'') \\cap [X]", "& =", "c(Z \\to Y, \\mathcal{N}') \\cap c_{top}(\\mathcal{N}'') \\cap [Y] \\\\", "& =", "c_{top}(\\mathcal{N}''|_Z) \\cap c(Z \\to Y, \\mathcal{N}') \\cap [Y] \\\\", "& =", "c(Z \\to X, \\mathcal{N}) \\cap [X] ", "\\end{align*}", "The first equality by Lemma \\ref{lemma-gysin-decompose}.", "The second because Chern classes commute with bivariant classes", "(Lemma \\ref{lemma-cap-commutative-chern}).", "The third equality by Lemma \\ref{lemma-gysin-decompose}.", "\\medskip\\noindent", "Assume $Y \\not = X$. By Lemma \\ref{lemma-bivariant-zero}", "it even suffices to prove the result after blowing up $X$ in a nonzero ideal.", "Let us blowup $X$ in the product of the ideal sheaf of $Y$ and the ideal", "sheaf of $Z$. This reduces us to the case where both $Y$ and $Z$ are", "effective Cartier divisors on $X$, see", "Divisors, Lemmas", "\\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor} and", "\\ref{divisors-lemma-blowing-up-two-ideals}.", "\\medskip\\noindent", "Denote $\\mathcal{N}'' \\to \\mathcal{E}$ the surjection of finite locally", "free $\\mathcal{O}_Z$-modules such that", "$0 \\to \\mathcal{E}^\\vee \\to (\\mathcal{N}'')^\\vee \\to \\mathcal{C}_{Y/X} \\to 0$", "is a short exact sequence. Then $\\mathcal{N} \\to \\mathcal{E}|_Z$", "is a surjection as well. Denote $\\mathcal{N}_1$ the finite locally free kernel", "of this map and observe that $\\mathcal{N}^\\vee \\to \\mathcal{C}_{Z/X}$", "factors through $\\mathcal{N}_1$.", "By Lemma \\ref{lemma-gysin-decompose} we have", "$$", "c(Y \\to X, \\mathcal{N}'') = c_{top}(\\mathcal{E}) \\circ", "c(Y \\to X, \\mathcal{C}_{Y/X}^\\vee)", "$$", "and", "$$", "c(Z \\to X, \\mathcal{N}) = c_{top}(\\mathcal{E}|_Z) \\circ", "c(Z \\to X, \\mathcal{N}_1)", "$$", "Since Chern classes of bundles commute with bivariant classes", "(Lemma \\ref{lemma-cap-commutative-chern})", "it suffices to prove", "$$", "c(Z \\to X, \\mathcal{N}_1) =", "c(Z \\to Y, \\mathcal{N}') \\circ c(Y \\to X, \\mathcal{C}_{Y/X}^\\vee)", "$$", "in $A^*(Z \\to X)$. This we may assume that $\\mathcal{N}'' = \\mathcal{C}_{Y/X}$.", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "In this paragraph $Z$ and $Y$ are effective Cartier divisors on $X$", "integral of dimension $n$, we have $\\mathcal{N}'' = \\mathcal{C}_{Y/X}$.", "In this case $c(Y \\to X, \\mathcal{C}_{Y/X}^\\vee) \\cap [X] = [Y]_{n - 1}$ by", "Lemma \\ref{lemma-gysin-fundamental}. Thus we have to prove that", "$c(Z \\to X, \\mathcal{N}) \\cap [X] = c(Z \\to Y, \\mathcal{N}') \\cap [Y]_{n - 1}$.", "Denote $N$ and $N'$ the vector bundles over $Z$ associated to", "$\\mathcal{N}$ and $\\mathcal{N}'$. Consider the commutative diagram", "$$", "\\xymatrix{", "N' \\ar[r]_i &", "N \\ar[r] &", "(C_Y X) \\times_Y Z \\\\", "C_Z Y \\ar[r] \\ar[u] &", "C_Z X \\ar[u]", "}", "$$", "of cones and vector bundles over $Z$. Observe that $N'$ is a relative", "effective Cartier divisor in $N$ over $Z$ and that", "$$", "\\xymatrix{", "N' \\ar[d] \\ar[r]_i & N \\ar[d] \\\\", "Z \\ar[r]^-o & (C_Y X) \\times_Y Z", "}", "$$", "is cartesian where $o$ is the zero section of the line bundle", "$C_Y X$ over $Y$. By", "Lemma \\ref{lemma-relation-normal-cones} we have $o^*[C_ZX]_n = [C_Z Y]_{n - 1}$", "in", "$$", "\\CH_{n - 1}(Y \\times_{o, C_Y X} C_ZX) =", "\\CH_{n - 1}(Z \\times_{o, (C_Y X) \\times_Y Z} C_ZX)", "$$", "By the cartesian property of", "the square above this implies that", "$$", "i^*[C_ZX]_n = [C_Z Y]_{n - 1}", "$$", "in $\\CH_{n - 1}(N')$. Now observe that", "$\\gamma = c(Z \\to X, \\mathcal{N}) \\cap [X]$ and", "$\\gamma' = c(Z \\to Y, \\mathcal{N}') \\cap [Y]_{n - 1}$", "are characterized by $p^*\\gamma = [C_Z X]_n$ in $\\CH_n(N)$", "and by $(p')^*\\gamma' = [C_Z Y]_{n - 1}$ in $\\CH_{n - 1}(N')$.", "Hence the proof is finished as $i^* \\circ p^* = (p')^*$ by", "Lemma \\ref{lemma-relative-effective-cartier}." ], "refs": [ "chow-lemma-bivariant-zero", "chow-lemma-gysin-decompose", "chow-lemma-cap-commutative-chern", "chow-lemma-gysin-decompose", "chow-lemma-bivariant-zero", "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "divisors-lemma-blowing-up-two-ideals", "chow-lemma-gysin-decompose", "chow-lemma-cap-commutative-chern", "chow-lemma-gysin-fundamental", "chow-lemma-relation-normal-cones", "chow-lemma-relative-effective-cartier" ], "ref_ids": [ 5740, 5812, 5752, 5812, 5740, 8054, 8062, 5812, 5752, 5814, 5818, 5725 ] } ], "ref_ids": [ 13707 ] }, { "id": 5820, "type": "theorem", "label": "chow-lemma-compute-koszul", "categories": [ "chow" ], "title": "chow-lemma-compute-koszul", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$", "be a scheme locally of finite type over $S$. Let $\\mathcal{E}$", "be a locally free $\\mathcal{O}_X$-module of rank $r$.", "Then", "$$", "\\prod\\nolimits_{n = 0, \\ldots, r} c(\\wedge^n \\mathcal{E})^{(-1)^n} =", "1 - (r - 1)! c_r(\\mathcal{E}) + \\ldots", "$$" ], "refs": [], "proofs": [ { "contents": [ "By the splitting principle we can turn this into a calculation in the", "polynomial ring on the Chern roots $x_1, \\ldots, x_r$ of $\\mathcal{E}$. See", "Section \\ref{section-splitting-principle}. Observe that", "$$", "c(\\wedge^n \\mathcal{E}) =", "\\prod\\nolimits_{1 \\leq i_1 < \\ldots < i_n \\leq r}", "(1 + x_{i_1} + \\ldots + x_{i_n})", "$$", "Thus the logarithm of the left hand side of the equation in the lemma is", "$$", "-", "\\sum\\nolimits_{p \\geq 1}", "\\sum\\nolimits_{n = 0}^r", "\\sum\\nolimits_{1 \\leq i_1 < \\ldots < i_n \\leq r}", "\\frac{(-1)^{p + n}}{p}(x_{i_1} + \\ldots + x_{i_n})^p", "$$", "Please notice the minus sign in front. However, we have", "$$", "\\sum\\nolimits_{p \\geq 0}", "\\sum\\nolimits_{n = 0}^r", "\\sum\\nolimits_{1 \\leq i_1 < \\ldots < i_n \\leq r}", "\\frac{(-1)^{p + n}}{p!}(x_{i_1} + \\ldots + x_{i_n})^p", "=", "\\prod (1 - e^{-x_i})", "$$", "Hence we see that the first nonzero term in our Chern class", "is in degree $r$ and equal to the predicted value." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5821, "type": "theorem", "label": "chow-lemma-compute-section", "categories": [ "chow" ], "title": "chow-lemma-compute-section", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$", "be a scheme locally of finite type over $S$. Let $\\mathcal{C}$", "be a locally free $\\mathcal{O}_X$-module of rank $r$. Consider the", "morphisms", "$$", "X = \\underline{\\text{Proj}}_X(\\mathcal{O}_X[T])", "\\xrightarrow{i}", "E = \\underline{\\text{Proj}}_X(\\text{Sym}^*(\\mathcal{C})[T])", "\\xrightarrow{\\pi}", "X", "$$", "Then $c_t(i_*\\mathcal{O}_X) = 0$ for $t = 1, \\ldots, r - 1$ and in", "$A^0(C \\to E)$ we have", "$$", "p^* \\circ \\pi_* \\circ c_r(i_*\\mathcal{O}_X) = (-1)^{r - 1}(r - 1)! j^*", "$$", "where", "$j : C \\to E$ and $p : C \\to X$ are the inclusion and structure", "morphism of the vector bundle", "$C = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{C}))$." ], "refs": [], "proofs": [ { "contents": [ "The canonical map $\\pi^*\\mathcal{C} \\to \\mathcal{O}_E(1)$ vanishes", "exactly along $i(X)$. Hence the Koszul complex on the map", "$$", "\\pi^*\\mathcal{C} \\otimes \\mathcal{O}_E(-1) \\to \\mathcal{O}_E", "$$", "is a resolution of $i_*\\mathcal{O}_X$. In particular we see that", "$i_*\\mathcal{O}_X$ is a perfect object of $D(\\mathcal{O}_E)$", "whose Chern classes are defined. The vanishing of $c_t(i_*\\mathcal{O}_X)$", "for $t = 1, \\ldots, t - 1$ follows from Lemma \\ref{lemma-compute-koszul}.", "This lemma also gives", "$$", "c_r(i_*\\mathcal{O}_X) = - (r - 1)!", "c_r(\\pi^*\\mathcal{C} \\otimes \\mathcal{O}_E(-1))", "$$", "On the other hand, by Lemma \\ref{lemma-chern-classes-dual} we have", "$$", "c_r(\\pi^*\\mathcal{C} \\otimes \\mathcal{O}_E(-1)) =", "(-1)^r c_r(\\pi^*\\mathcal{C}^\\vee \\otimes \\mathcal{O}_E(1))", "$$", "and $\\pi^*\\mathcal{C}^\\vee \\otimes \\mathcal{O}_E(1)$ has a section $s$", "vanishing exactly along $i(X)$.", "\\medskip\\noindent", "After replacing $X$ by a scheme locally of finite type over $X$,", "it suffices to prove that both sides of the equality have the", "same effect on an element $\\alpha \\in \\CH_*(E)$. Since $C \\to X$", "is a vector bundle, every cycle class on $C$ is of the form $p^*\\beta$", "for some $\\beta \\in \\CH_*(X)$ (Lemma \\ref{lemma-vectorbundle}).", "Hence by Lemma \\ref{lemma-restrict-to-open}", "we can write $\\alpha = \\pi^*\\beta + \\gamma$ where $\\gamma$", "is supported on $E \\setminus C$. Using the equalities above", "it suffices to show that", "$$", "p^*(\\pi_*(c_r(\\pi^*\\mathcal{C}^\\vee \\otimes \\mathcal{O}_E(1)) \\cap [W])) =", "j^*[W]", "$$", "when $W \\subset E$ is an integral closed subscheme which", "is either (a) disjoint from $C$ or (b) is of the form $W = \\pi^{-1}Y$", "for some integral closed subscheme $Y \\subset X$.", "Using the section $s$ and Lemma \\ref{lemma-top-chern-class} we find", "in case (a) $c_r(\\pi^*\\mathcal{C}^\\vee \\otimes \\mathcal{O}_E(1)) \\cap [W] = 0$", "and in case (b)", "$c_r(\\pi^*\\mathcal{C}^\\vee \\otimes \\mathcal{O}_E(1)) \\cap [W] = [i(Y)]$.", "The result follows easily from this; details omitted." ], "refs": [ "chow-lemma-compute-koszul", "chow-lemma-chern-classes-dual", "chow-lemma-vectorbundle", "chow-lemma-restrict-to-open", "chow-lemma-top-chern-class" ], "ref_ids": [ 5820, 5763, 5744, 5690, 5765 ] } ], "ref_ids": [] }, { "id": 5822, "type": "theorem", "label": "chow-lemma-agreement-with-loc-chern", "categories": [ "chow" ], "title": "chow-lemma-agreement-with-loc-chern", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $i : Z \\to X$", "be a regular closed immersion of codimension $r$", "between schemes locally of finite type over $S$.", "Let $\\mathcal{N} = \\mathcal{C}_{Z/X}^\\vee$ be the normal sheaf. If $X$", "is quasi-compact and has the resolution property, then", "$c_t(Z \\to X, i_*\\mathcal{O}_Z) = 0$ for $t = 1, \\ldots, r - 1$ and", "$$", "c_r(Z \\to X, i_*\\mathcal{O}_Z) = (-1)^{r - 1} (r - 1)! c(Z \\to X, \\mathcal{N})", "\\quad\\text{in}\\quad", "A^r(Z \\to X)", "$$", "where $c_t(Z \\to X, i_*\\mathcal{O}_Z)$", "is the localized Chern class", "of Definition \\ref{definition-localized-chern}." ], "refs": [ "chow-definition-localized-chern" ], "proofs": [ { "contents": [ "For any $x \\in Z$ we can choose an affine open neighbourhood", "$\\Spec(A) \\subset X$ such that $Z \\cap \\Spec(A) = V(f_1, \\ldots, f_r)$", "where $f_1, \\ldots, f_r \\in A$ is a regular sequence.", "See Divisors, Definition \\ref{divisors-definition-regular-immersion} and", "Lemma \\ref{divisors-lemma-Noetherian-scheme-regular-ideal}.", "Then we see that the Koszul complex on $f_1, \\ldots, f_r$ is", "a resolution of $A/(f_1, \\ldots, f_r)$ for example by", "More on Algebra, Lemma \\ref{more-algebra-lemma-regular-koszul-regular}.", "Hence $A/(f_1, \\ldots, f_r)$ is perfect as an $A$-module.", "It follows that $F = i_*\\mathcal{O}_Z$ is a perfect object of", "$D(\\mathcal{O}_X)$ whose restriction to $X \\setminus Z$ is zero.", "Since $X$ is quasi-compact and quasi-separated", "(Properties, Lemma \\ref{properties-lemma-locally-Noetherian-quasi-separated})", "and has the resolution property we see that $F = i_*\\mathcal{O}_Z$ can be", "represented by a bounded complex of finite locally free modules", "(Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-construct-strictly-perfect}) and hence", "the Chern classes of $F = i_*\\mathcal{O}_Z$ are defined", "(Definition \\ref{definition-defined-on-perfect}). All in all", "we conclude that the statement makes sense.", "\\medskip\\noindent", "Denote $b : W \\to \\mathbf{P}^1_X$ the blowing up in $\\infty(Z)$", "as in Section \\ref{section-blowup-Z-first}. By (\\ref{item-find-Z-in-blowup})", "we have a closed immersion", "$$", "i' : \\mathbf{P}^1_Z \\longrightarrow W", "$$", "We claim that $Q = i'_*\\mathcal{O}_{\\mathbf{P}^1_Z}$", "is a perfect object of", "$D(\\mathcal{O}_W)$ and that $F$ and $Q$ satisfy the assumptions of", "Lemma \\ref{lemma-independent-loc-chern-bQ}.", "\\medskip\\noindent", "Assume the claim. The output of Lemma \\ref{lemma-independent-loc-chern-bQ}", "is that we have", "$$", "c_p(Z \\to X, F) = c'_p(Q) = (E \\to Z)_* \\circ c'_p(Q|_E) \\circ C", "$$", "for all $p \\geq 1$. Observe that $Q|_E$ is equal to the pushforward of", "the structure sheaf of $Z$ via the morphism $Z \\to E$ which is the", "base change of $i'$ by $\\infty$.", "Thus the vanishing of $c_t(Z \\to X, F)$ for $1 \\leq t \\leq r - 1$", "by Lemma \\ref{lemma-compute-section} applied to $E \\to Z$.", "Because $\\mathcal{C}_{Z/X} = \\mathcal{N}^\\vee$", "is locally free the bivariant class $c(Z \\to X, \\mathcal{N})$", "is characterized by the relation", "$$", "j^* \\circ C = p^* \\circ c(Z \\to X, \\mathcal{N})", "$$", "where $j : C_ZX \\to W_\\infty$ and $p : C_ZX \\to Z$ are the given maps.", "(Recall $C \\in A^0(W_\\infty \\to X)$ is the class of", "Lemma \\ref{lemma-gysin-at-infty}.)", "Thus the displayed equation in the statement of the lemma", "follows from the corresponding equation in Lemma \\ref{lemma-compute-section}.", "\\medskip\\noindent", "Proof of the claim. Let $A$ and $f_1, \\ldots, f_r$ be as above.", "Consider the affine open $\\Spec(A[s]) \\subset \\mathbf{P}^1_X$", "as in Section \\ref{section-blowup-Z-first}. Recall that $s = 0$", "defines $(\\mathbf{P}^1_X)_\\infty$ over this open. Hence over", "$\\Spec(A[s])$ we are blowing up in the ideal generated by", "the regular sequence $s, f_1, \\ldots, f_r$. By More on Algebra, Lemma", "\\ref{more-algebra-lemma-blowup-regular-sequence} the $r + 1$", "affine charts are global complete intersections over $A[s]$.", "The chart corresponding to the affine blowup algebra", "$$", "A[s][f_1/s, \\ldots, f_r/s] = A[s, y_1, \\ldots, y_r]/(sy_i - f_i)", "$$", "contains $i'(Z \\cap \\Spec(A))$ as the closed subscheme cut out by", "$y_1, \\ldots, y_r$. Since $y_1, \\ldots, y_r, sy_1 - f_1, \\ldots, sy_r - f_r$", "is a regular sequence in the polynomial ring $A[s, y_1, \\ldots, y_r]$", "we find that $i'$ is a regular immersion. Some details omitted.", "As above we conclude that $Q = i'_*\\mathcal{O}_{\\mathbf{P}^1_Z}$", "is a perfect object of $D(\\mathcal{O}_W)$. Since $W$ also has", "the resolution property (Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-resolution-property-ample-relative})", "we find that the Chern classes of $Q$ are defined. All the", "other assumptions on $F$ and $Q$ in Lemma \\ref{lemma-independent-loc-chern-bQ}", "(and Lemma \\ref{lemma-localized-chern-pre}) are immediately verified." ], "refs": [ "divisors-definition-regular-immersion", "divisors-lemma-Noetherian-scheme-regular-ideal", "more-algebra-lemma-regular-koszul-regular", "properties-lemma-locally-Noetherian-quasi-separated", "perfect-lemma-construct-strictly-perfect", "chow-definition-defined-on-perfect", "chow-lemma-independent-loc-chern-bQ", "chow-lemma-independent-loc-chern-bQ", "chow-lemma-gysin-at-infty", "more-algebra-lemma-blowup-regular-sequence", "perfect-lemma-resolution-property-ample-relative", "chow-lemma-independent-loc-chern-bQ", "chow-lemma-localized-chern-pre" ], "ref_ids": [ 8099, 7988, 9973, 2953, 7093, 5923, 5804, 5804, 5786, 9990, 7087, 5804, 5790 ] } ], "ref_ids": [ 5924 ] }, { "id": 5823, "type": "theorem", "label": "chow-lemma-actual-computation", "categories": [ "chow" ], "title": "chow-lemma-actual-computation", "contents": [ "In the situation of Lemma \\ref{lemma-agreement-with-loc-chern}", "say $\\dim_\\delta(X) = n$. Then we have", "\\begin{enumerate}", "\\item $c_t(Z \\to X, i_*\\mathcal{O}_Z) \\cap [X]_n = 0$ for", "$t = 1, \\ldots, r - 1$,", "\\item $c_r(Z \\to X, i_*\\mathcal{O}_Z) \\cap [X]_n =", "(-1)^{r - 1}(r - 1)![Z]_{n - r}$,", "\\item $ch_t(Z \\to X, i_*\\mathcal{O}_Z) \\cap [X]_n = 0$ for", "$t = 0, \\ldots, r - 1$, and", "\\item $ch_r(Z \\to X, i_*\\mathcal{O}_Z) \\cap [X]_n = [Z]_{n - r}$.", "\\end{enumerate}" ], "refs": [ "chow-lemma-agreement-with-loc-chern" ], "proofs": [ { "contents": [ "Parts (1) and (2) follow immediately from", "Lemma \\ref{lemma-agreement-with-loc-chern}", "combined with Lemma \\ref{lemma-gysin-fundamental}.", "Then we deduce parts (3) and (4) using the relationship", "between $ch_p = (1/p!)P_p$ and $c_p$ given in", "Lemma \\ref{lemma-loc-chern-character}. (Namely,", "$(-1)^{r - 1}(r - 1)!ch_r = c_r$ provided", "$c_1 = c_2 = \\ldots = c_{r - 1} = 0$.)" ], "refs": [ "chow-lemma-agreement-with-loc-chern", "chow-lemma-gysin-fundamental", "chow-lemma-loc-chern-character" ], "ref_ids": [ 5822, 5814, 5805 ] } ], "ref_ids": [ 5822 ] }, { "id": 5824, "type": "theorem", "label": "chow-lemma-second-adams-operator", "categories": [ "chow" ], "title": "chow-lemma-second-adams-operator", "contents": [ "Let $X$ be a scheme. There is a ring map", "$$", "\\psi^2 :", "K_0(\\textit{Vect}(X))", "\\longrightarrow", "K_0(\\textit{Vect}(X))", "$$", "which sends $[\\mathcal{L}]$ to $[\\mathcal{L}^{\\otimes 2}]$", "when $\\mathcal{L}$ is invertible and is compatible with pullbacks." ], "refs": [], "proofs": [ { "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module.", "We will consider the element", "$$", "\\psi^2(\\mathcal{E}) = [\\text{Sym}^2(\\mathcal{E})] - [\\wedge^2(\\mathcal{E})]", "$$", "of $K_0(\\textit{Vect}(X))$.", "\\medskip\\noindent", "Let $X$ be a scheme and consider a short exact sequence", "$$", "0 \\to \\mathcal{E} \\to \\mathcal{F} \\to \\mathcal{G} \\to 0", "$$", "of finite locally free $\\mathcal{O}_X$-modules. Let us think of", "this as a filtration on $\\mathcal{F}$ with $2$ steps. The induced", "filtration on $\\text{Sym}^2(\\mathcal{F})$ has $3$ steps with", "graded pieces $\\text{Sym}^2(\\mathcal{E})$, $\\mathcal{E} \\otimes \\mathcal{F}$,", "and $\\text{Sym}^2(\\mathcal{G})$. Hence", "$$", "[\\text{Sym}^2(\\mathcal{F})] =", "[\\text{Sym}^2(\\mathcal{E})] +", "[\\mathcal{E} \\otimes \\mathcal{F}] +", "[\\text{Sym}^2(\\mathcal{G})]", "$$", "In exactly the same manner one shows that", "$$", "[\\wedge^2(\\mathcal{F})] =", "[\\wedge^2(\\mathcal{E})] +", "[\\mathcal{E} \\otimes \\mathcal{F}] +", "[\\wedge^2(\\mathcal{G})]", "$$", "Thus we see that", "$\\psi^2(\\mathcal{F}) = \\psi^2(\\mathcal{E}) + \\psi^2(\\mathcal{G})$.", "We conclude that we obtain a well defined additive map", "$\\psi^2 : K_0(\\textit{Vect}(X)) \\to K_0(\\textit{Vect}(X))$.", "\\medskip\\noindent", "It is clear that this map commutes with pullbacks.", "\\medskip\\noindent", "We still have to show that $\\psi^2$ is a ring map.", "Let $X$ be a scheme and let $\\mathcal{E}$ and $\\mathcal{F}$", "be finite locally free $\\mathcal{O}_X$-modules.", "Observe that there is a short exact sequence", "$$", "0 \\to \\wedge^2(\\mathcal{E}) \\otimes \\wedge^2(\\mathcal{F}) \\to", "\\text{Sym}^2(\\mathcal{E} \\otimes \\mathcal{F}) \\to", "\\text{Sym}^2(\\mathcal{E}) \\otimes \\text{Sym}^2(\\mathcal{F}) \\to 0", "$$", "where the first map sends $(e \\wedge e') \\otimes (f \\wedge f')$ to", "$(e \\otimes f)(e' \\otimes f') - (e' \\otimes f)(e \\otimes f')$ and", "the second map sends $(e \\otimes f) (e' \\otimes f')$ to $ee' \\otimes ff'$.", "Similarly, there is a short exact sequence", "$$", "0 \\to \\text{Sym}^2(\\mathcal{E}) \\otimes \\wedge^2(\\mathcal{F}) \\to", "\\wedge^2(\\mathcal{E} \\otimes \\mathcal{F}) \\to", "\\wedge^2(\\mathcal{E}) \\otimes \\text{Sym}^2(\\mathcal{F}) \\to 0", "$$", "where the first map sends $e e' \\otimes f \\wedge f'$ to", "$(e \\otimes f) \\wedge (e' \\otimes f') + (e' \\otimes f) \\wedge (e \\otimes f')$", "and the second map sends", "$(e \\otimes f) \\wedge (e' \\otimes f')$ to", "$(e \\wedge e') \\otimes (f f')$.", "As above this proves the map $\\psi^2$ is multiplicative.", "Since it is clear that $\\psi^2(1) = 1$ this concludes the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5825, "type": "theorem", "label": "chow-lemma-minus-adams-operator", "categories": [ "chow" ], "title": "chow-lemma-minus-adams-operator", "contents": [ "Let $X$ be a scheme. There is a ring map", "$\\psi^{-1} : K_0(\\textit{Vect}(X)) \\to K_0(\\textit{Vect}(X))$", "which sends $[\\mathcal{E}]$ to $[\\mathcal{E}^\\vee]$", "when $\\mathcal{E}$ is finite locally free", "and is compatible with pullbacks." ], "refs": [], "proofs": [ { "contents": [ "The only thing to check is that taking duals is compatible with", "short exact sequences and with pullbacks. This is clear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5826, "type": "theorem", "label": "chow-lemma-adams-and-chern", "categories": [ "chow" ], "title": "chow-lemma-adams-and-chern", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. If $\\psi^2$ is", "as in Lemma \\ref{lemma-second-adams-operator} and $c$ and $ch$ are as in", "Remarks \\ref{remark-chern-classes-K} and \\ref{remark-chern-character-K}", "then we have $c_i(\\psi^2(\\alpha)) = 2^i c_i(\\alpha)$ and", "$ch_i(\\psi^2(\\alpha)) = 2^i ch_i(\\alpha)$", "for all $\\alpha \\in K_0(\\textit{Vect}(X))$." ], "refs": [ "chow-lemma-second-adams-operator", "chow-remark-chern-classes-K", "chow-remark-chern-character-K" ], "proofs": [ { "contents": [ "Observe that the map $\\prod_{i \\geq 0} A^i(X) \\to \\prod_{i \\geq 0} A^i(X)$", "multiplying by $2^i$ on $A^i(X)$ is a ring map. Hence, since $\\psi^2$", "is also a ring map, it suffices to prove the formulas for additive generators", "of $K_0(\\textit{Vect}(X))$. Thus we may assume $\\alpha = [\\mathcal{E}]$", "for some finite locally free $\\mathcal{O}_X$-module $\\mathcal{E}$.", "By construction of the Chern classes of $\\mathcal{E}$ we immediately", "reduce to the case where $\\mathcal{E}$ has constant rank $r$, see", "Remark \\ref{remark-extend-to-finite-locally-free}.", "In this case, we can choose a projective smooth morphism $p : P \\to X$", "such that restriction $A^*(X) \\to A^*(P)$ is injective", "and such that $p^*\\mathcal{E}$ has a finite filtration whose", "graded parts are invertible $\\mathcal{O}_P$-modules $\\mathcal{L}_j$, see", "Lemma \\ref{lemma-splitting-principle}. Then", "$[p^*\\mathcal{E}] = \\sum [\\mathcal{L}_j]$ and hence", "$\\psi^2([p^\\mathcal{E}]) = \\sum [\\mathcal{L}_j^{\\otimes 2}]$", "by definition of $\\psi^2$. Setting $x_j = c_1(\\mathcal{L}_j)$", "we have", "$$", "c(\\alpha) = \\prod (1 + x_j)", "\\quad\\text{and}\\quad", "c(\\psi^2(\\alpha)) = \\prod (1 + 2 x_j)", "$$", "in $\\prod A^i(P)$ and we have", "$$", "ch(\\alpha) = \\sum \\exp(x_j)", "\\quad\\text{and}\\quad", "ch(\\psi^2(\\alpha)) = \\sum \\exp(2 x_j)", "$$", "in $\\prod A^i(P)$. From these formulas the desired result follows." ], "refs": [ "chow-remark-extend-to-finite-locally-free", "chow-lemma-splitting-principle" ], "ref_ids": [ 5946, 5762 ] } ], "ref_ids": [ 5824, 5956, 5957 ] }, { "id": 5827, "type": "theorem", "label": "chow-lemma-perf-Z-regular", "categories": [ "chow" ], "title": "chow-lemma-perf-Z-regular", "contents": [ "Let $X$ be a Noetherian regular scheme of finite dimension.", "Let $Z \\subset X$ be a closed subschemes. The maps constructed", "in Remarks \\ref{remark-perf-Z-cohomology-K} and", "\\ref{remark-perf-Z-regular} are mutually inverse and we get", "$K'_0(Z) = K_0(D_{Z, perf}(\\mathcal{O}_X))$." ], "refs": [ "chow-remark-perf-Z-cohomology-K", "chow-remark-perf-Z-regular" ], "proofs": [ { "contents": [ "Clearly the composition", "$$", "K_0(\\textit{Coh}(Z)) \\longrightarrow", "K_0(D_{Z, perf}(\\mathcal{O}_X)) \\longrightarrow", "K_0(\\textit{Coh}(Z))", "$$", "is the identity map. Thus it suffices to show the first arrow is", "surjective. Let $E$ be an object of $D_{Z, perf}(\\mathcal{O}_X)$.", "We are going to use without further mention that $E$ is bounded", "with coherent cohomology and that any such complex is a perfect complex.", "Using the distinguished triangles of canonical truncations the", "reader sees that", "$$", "[E] = \\sum (-1)^i[H^i(E)[0]]", "$$", "in $K_0(D_{Z, perf}(\\mathcal{O}_X))$. Then it suffices to", "show that $[\\mathcal{F}[0]]$ is in the image of the map", "for any coherent $\\mathcal{O}_X$-module set theoretically", "supported on $Z$. Since we can find a finite filtration on", "$\\mathcal{F}$ whose subquotients are $\\mathcal{O}_Z$-modules,", "the proof is complete." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 5958, 5959 ] }, { "id": 5828, "type": "theorem", "label": "chow-lemma-K-tensor-Q", "categories": [ "chow" ], "title": "chow-lemma-K-tensor-Q", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be a quasi-compact regular scheme of finite type over $S$ with", "affine diagonal and $\\delta_{X/S} : X \\to \\mathbf{Z}$ bounded.", "Then the composition", "$$", "K_0(\\textit{Vect}(X)) \\otimes \\mathbf{Q}", "\\longrightarrow", "A^*(X) \\otimes \\mathbf{Q}", "\\longrightarrow", "\\CH_*(X) \\otimes \\mathbf{Q}", "$$", "of the map $ch$ from Remark \\ref{remark-chern-character-K} and", "the map $c \\mapsto c \\cap [X]$ is an isomorphism." ], "refs": [ "chow-remark-chern-character-K" ], "proofs": [ { "contents": [ "We have $K'_0(X) = K_0(X) = K_0(\\textit{Vect}(X))$ by", "Derived Categories of Schemes, Lemmas \\ref{perfect-lemma-Kprime-K} and", "\\ref{perfect-lemma-K-is-old-K}.", "By Remark \\ref{remark-chern-classes-agree}", "the composition given agrees with the map of", "Proposition \\ref{proposition-K-tensor-Q} for $X = Y$.", "Thus the result follows from the proposition." ], "refs": [ "perfect-lemma-Kprime-K", "perfect-lemma-K-is-old-K", "chow-remark-chern-classes-agree", "chow-proposition-K-tensor-Q" ], "ref_ids": [ 7099, 7100, 5962, 5900 ] } ], "ref_ids": [ 5957 ] }, { "id": 5829, "type": "theorem", "label": "chow-lemma-composition-regular-immersion", "categories": [ "chow" ], "title": "chow-lemma-composition-regular-immersion", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $i : X \\to Y$ and $j : Y \\to Z$ be regular immersions", "of schemes locally of finite type over $S$. Then", "$j \\circ i$ is a regular immersion and", "$(j \\circ i)^! = i^! \\circ j^!$." ], "refs": [], "proofs": [ { "contents": [ "The first statement is", "Divisors, Lemma \\ref{divisors-lemma-composition-regular-immersion}.", "By Divisors, Lemma \\ref{divisors-lemma-transitivity-conormal-quasi-regular}", "there is a short exact sequence", "$$", "0 \\to", "i^*(\\mathcal{C}_{Y/Z}) \\to", "\\mathcal{C}_{X/Z} \\to", "\\mathcal{C}_{X/Y} \\to 0", "$$", "Thus the result by the more general Lemma \\ref{lemma-gysin-composition}." ], "refs": [ "divisors-lemma-composition-regular-immersion", "divisors-lemma-transitivity-conormal-quasi-regular", "chow-lemma-gysin-composition" ], "ref_ids": [ 7994, 7993, 5819 ] } ], "ref_ids": [] }, { "id": 5830, "type": "theorem", "label": "chow-lemma-section-smooth", "categories": [ "chow" ], "title": "chow-lemma-section-smooth", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $p : P \\to X$ be a smooth morphism of schemes locally of finite type", "over $S$ and let $s : X \\to P$ be a section. Then $s$ is a", "regular immersion and $1 = s^! \\circ p^*$ in $A^*(X)^\\wedge$", "where $p^* \\in A^*(P \\to X)^\\wedge$ is the bivariant class", "of Lemma \\ref{lemma-flat-pullback-bivariant}." ], "refs": [ "chow-lemma-flat-pullback-bivariant" ], "proofs": [ { "contents": [ "The first statement is Divisors, Lemma", "\\ref{divisors-lemma-section-smooth-regular-immersion}.", "It suffices to show that $s^! \\cap p^*[Z] = [Z]$ in", "$\\CH_*(X)$ for any integral closed subscheme $Z \\subset X$", "as the assumptions are preserved by base change by $X' \\to X$", "locally of finite type. After replacing $P$ by an open neighbourhood", "of $s(Z)$ we may assume $P \\to X$ is smooth of fixed relative dimension $r$.", "Say $\\dim_\\delta(Z) = n$. Then every irreducible component of $p^{-1}(Z)$", "has dimension $r + n$ and $p^*[Z]$ is given by $[p^{-1}(Z)]_{n + r}$.", "Observe that $s(X) \\cap p^{-1}(Z) = s(Z)$ scheme theoretically. Hence by the", "same reference as used above $s(X) \\cap p^{-1}(Z)$ is a closed subscheme", "regularly embedded in $\\overline{p}^{-1}(Z)$ of codimension $r$.", "We conclude by Lemma \\ref{lemma-gysin-fundamental}." ], "refs": [ "chow-lemma-gysin-fundamental" ], "ref_ids": [ 5814 ] } ], "ref_ids": [ 5732 ] }, { "id": 5831, "type": "theorem", "label": "chow-lemma-lci-gysin-well-defined", "categories": [ "chow" ], "title": "chow-lemma-lci-gysin-well-defined", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ be a local complete intersection morphism", "of schemes locally of finite type over $S$.", "The bivariant class $f^!$ is independent of the choice of", "the factorization $f = g \\circ i$ with $g$ smooth (provided", "one exists)." ], "refs": [], "proofs": [ { "contents": [ "Given a second such factorization $f = g' \\circ i'$ we can", "consider the smooth morphism $g'' : P \\times_Y P' \\to Y$, the", "immersion $i'' : X \\to P \\times_Y P'$ and the factorization", "$f = g'' \\circ i''$. Thus we may assume that we have a diagram", "$$", "\\xymatrix{", "& P' \\ar[d]^p \\ar[rd]^{g'} \\\\", "X \\ar[r]^i \\ar[ru]^{i'} & P \\ar[r]^g & Y", "}", "$$", "where $p$ is a smooth morphism. Then $(g')^* = p^* \\circ g^*$", "(Lemma \\ref{lemma-compose-flat-pullback}) and hence it suffices", "to show that $i^! = (i')^! \\circ p^*$", "in $A^*(X \\to P)$. Consider the commutative diagram", "$$", "\\xymatrix{", "& X \\times_P P' \\ar[d]^{\\overline{p}} \\ar[r]_j & P' \\ar[d]^p \\\\", "X \\ar[ru]^s \\ar[r]^1 & X \\ar[r]^i & P", "}", "$$", "where $s =(1, i')$. Then $s$ and $j$ are regular immersions", "(by Divisors, Lemma \\ref{divisors-lemma-section-smooth-regular-immersion}", "and Divisors, Lemma \\ref{divisors-lemma-flat-base-change-regular-immersion})", "and $i' = j \\circ s$. By Lemma \\ref{lemma-composition-regular-immersion}", "we have $(i')^! = s^! \\circ j^!$.", "Since the square is cartesian, the bivariant class $j^!$", "is the restriction (Remark \\ref{remark-restriction-bivariant})", "of $i^!$ to $P'$, see Lemma \\ref{lemma-construction-gysin}.", "Since bivariant classes commute with flat pullbacks", "we find $j^! \\circ p^* = \\overline{p}^* \\circ i^!$.", "Thus it suffices to show that $s^! \\circ \\overline{p}^* = \\text{id}$", "which is done in Lemma \\ref{lemma-section-smooth}." ], "refs": [ "chow-lemma-compose-flat-pullback", "divisors-lemma-flat-base-change-regular-immersion", "chow-lemma-composition-regular-immersion", "chow-remark-restriction-bivariant", "chow-lemma-construction-gysin" ], "ref_ids": [ 5680, 7991, 5829, 5938, 5811 ] } ], "ref_ids": [] }, { "id": 5832, "type": "theorem", "label": "chow-lemma-lci-gysin-flat", "categories": [ "chow" ], "title": "chow-lemma-lci-gysin-flat", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ be a local complete intersection morphism", "of schemes locally of finite type over $S$. If the gysin map", "exists for $f$ and $f$ is flat, then $f^!$ is equal to the", "bivariant class of Lemma \\ref{lemma-flat-pullback-bivariant}." ], "refs": [ "chow-lemma-flat-pullback-bivariant" ], "proofs": [ { "contents": [ "Choose a factorization $f = g \\circ i$ with $i : X \\to P$", "an immersion and $g : P \\to Y$ smooth. Observe that for", "any morphism $Y' \\to Y$ which is locally of finite type,", "the base changes of $f'$, $g'$, $i'$ satisfy the same", "assumptions (see Morphisms, Lemmas \\ref{morphisms-lemma-base-change-smooth}", "and \\ref{morphisms-lemma-base-change-syntomic} and", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-flat-lci}).", "Thus we reduce to proving that $f^*[Y] = i^!(g^*[Y])$ in case $Y$", "is integral, see Lemma \\ref{lemma-bivariant-zero}. Set $n = \\dim_\\delta(Y)$.", "After decomposing $X$ and $P$ into connected components we", "may assume $f$ is flat of relative dimension $r$ and", "$g$ is smooth of relative dimension $t$.", "Then $f^*[Y] = [X]_{n + s}$ and $g^*[Y] = [P]_{n + t}$.", "On the other hand $i$ is a regular immersion of codimension $t - s$.", "Thus $i^![P]_{n + t} = [X]_{n + s}$ (Lemma \\ref{lemma-gysin-fundamental})", "and the proof is complete." ], "refs": [ "morphisms-lemma-base-change-smooth", "morphisms-lemma-base-change-syntomic", "more-morphisms-lemma-flat-lci", "chow-lemma-bivariant-zero", "chow-lemma-gysin-fundamental" ], "ref_ids": [ 5327, 5291, 14006, 5740, 5814 ] } ], "ref_ids": [ 5732 ] }, { "id": 5833, "type": "theorem", "label": "chow-lemma-lci-gysin-composition", "categories": [ "chow" ], "title": "chow-lemma-lci-gysin-composition", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be local complete intersection morphisms", "of schemes locally of finite type over $S$. Assume the gysin", "map exists for $g \\circ f$ and $g$. Then the gysin map exists for $f$", "and $(g \\circ f)^! = f^! \\circ g^!$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $g \\circ f$ is a local complete intersection morphism", "by More on Morphisms, Lemma \\ref{more-morphisms-lemma-composition-lci}", "and hence the statement of the lemma makes sense.", "If $X \\to P$ is an immersion of $X$ into a scheme $P$ smooth over $Z$", "then $X \\to P \\times_Z Y$ is an immersion of $X$ into a scheme smooth", "over $Y$. This prove the first assertion of the lemma.", "Let $Y \\to P'$ be an immersion of $Y$ into a scheme $P'$ smooth over $Z$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d] &", "P \\times_Z Y \\ar[r]_a \\ar[ld]^p &", "P \\times_Z P' \\ar[ld]^q \\\\", "Y \\ar[r]_b \\ar[d] &", "P' \\ar[ld] \\\\", "Z", "}", "$$", "Here the horizontal arrows are regular immersions, the south-west arrows", "are smooth, and the square is cartesian. Whence", "$a^! \\circ q^* = p^* \\circ b^!$ as bivariant classes commute", "with flat pullback. Combining this fact with", "Lemmas \\ref{lemma-composition-regular-immersion} and", "\\ref{lemma-compose-flat-pullback}", "the reader finds the statement of the lemma holds true.", "Small detail omitted." ], "refs": [ "more-morphisms-lemma-composition-lci", "chow-lemma-composition-regular-immersion", "chow-lemma-compose-flat-pullback" ], "ref_ids": [ 14005, 5829, 5680 ] } ], "ref_ids": [] }, { "id": 5834, "type": "theorem", "label": "chow-lemma-lci-gysin-commutes", "categories": [ "chow" ], "title": "chow-lemma-lci-gysin-commutes", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Consider a commutative diagram", "$$", "\\xymatrix{", "X'' \\ar[d] \\ar[r] &", "X' \\ar[d] \\ar[r] &", "X \\ar[d]^f \\\\", "Y'' \\ar[r] &", "Y' \\ar[r] &", "Y", "}", "$$", "of schemes locally of finite type over $S$ with both square cartesian.", "Assume $f : X \\to Y$ is a local complete intersection morphism", "such that the gysin map exists for $f$. Let $c \\in A^*(Y'' \\to Y')$. Denote", "$res(f^!) \\in A^*(X' \\to Y')$ the restriction of $f^!$ to $Y'$", "(Remark \\ref{remark-restriction-bivariant}). Then $c$ and $res(f^!)$ commute", "(Remark \\ref{remark-bivariant-commute})." ], "refs": [ "chow-remark-restriction-bivariant", "chow-remark-bivariant-commute" ], "proofs": [ { "contents": [ "Choose a factorization $f = g \\circ i$ with $g$ smooth and $i$ an immersion.", "Since $f^! = i^! \\circ g^!$ it suffices to prove the lemma for $g^!$", "(which is given by flat pullback) and for $i^!$. The result for flat pullback", "is part of the definition of a bivariant class. The case of $i^!$ follows", "immediately from Lemma \\ref{lemma-gysin-commutes}." ], "refs": [ "chow-lemma-gysin-commutes" ], "ref_ids": [ 5817 ] } ], "ref_ids": [ 5938, 5939 ] }, { "id": 5835, "type": "theorem", "label": "chow-lemma-lci-gysin-easy", "categories": [ "chow" ], "title": "chow-lemma-lci-gysin-easy", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Consider a cartesian diagram", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r] &", "X \\ar[d]^f \\\\", "Y' \\ar[r] &", "Y", "}", "$$", "of schemes locally of finite type over $S$. Assume", "\\begin{enumerate}", "\\item $f$ is a local complete intersection morphism and", "the gysin map exists for $f$,", "\\item $X$, $X'$, $Y$, $Y'$ satisfy the equivalent conditions of", "Lemma \\ref{lemma-locally-equidimensional},", "\\item for $x' \\in X'$ with images $x$, $y'$, and $y$", "in $X$, $Y'$, and $Y$ we have $n_{x'} - n_{y'} = n_x - n_y$", "where $n_{x'}$, $n_x$, $n_{y'}$, and $n_y$ are as in the lemma, and", "\\item for every generic point $\\xi \\in X'$ the local ring", "$\\mathcal{O}_{Y', f'(\\xi)}$ is Cohen-Macaulay.", "\\end{enumerate}", "Then $f^![Y'] = [X']$ where $[Y']$ and $[X']$ are as in", "Remark \\ref{remark-fundamental-class}." ], "refs": [ "chow-lemma-locally-equidimensional", "chow-remark-fundamental-class" ], "proofs": [ { "contents": [ "Recall that $n_{x'}$ is the common value of $\\delta(\\xi)$", "where $\\xi$ is the generic point of an irreducible component", "passing through $x'$. Moreover, the functions", "$x' \\mapsto n_{x'}$, $x \\mapsto n_x$, $y' \\mapsto n_{y'}$, and", "$y \\mapsto n_y$ are locally constant. Let $X'_n$, $X_n$, $Y'_n$,", "and $Y_n$ be the open and closed subscheme of $X'$, $X$, $Y'$, and", "$Y$ where the function has value $n$. Recall that", "$[X'] = \\sum [X'_n]_n$ and $[Y'] = \\sum [Y'_n]_n$.", "Having said this, it is clear that to prove the lemma we", "may replace $X'$ by one of its connected components", "and $X$, $Y'$, $Y$ by the connected component that", "it maps into. Then we know that $X'$, $X$, $Y'$, and", "$Y$ are $\\delta$-equidimensional in the sense that", "each irreducible component has the same $\\delta$-dimension.", "Say $n'$, $n$, $m'$, and $m$ is this common value", "for $X'$, $X$, $Y'$, and $Y$. The last assumption", "means that $n' - m' = n - m$.", "\\medskip\\noindent", "Choose a factorization $f = g \\circ i$ where $i : X \\to P$", "is an immersion and $g : P \\to Y$ is smooth. As $X$ is connected,", "we see that the relative dimension of $P \\to Y$ at points of $i(X)$", "is constant. Hence after replacing $P$ by an open neighbourhood", "of $i(X)$, we may assume that $P \\to Y$ has constant relative dimension", "and $i : X \\to P$ is a closed immersion.", "Denote $g' : Y' \\times_Y P \\to Y'$ the base change of $g$ and denote", "$i' : X' \\to Y' \\times_Y P$ the base change of $i$.", "It is clear that $g^*[Y] = [P]$ and $(g')^*[Y'] = [Y' \\times_Y P]$.", "Finally, if $\\xi' \\in X'$ is a generic point, then", "$\\mathcal{O}_{Y' \\times_Y P, i'(\\xi)}$ is Cohen-Macaulay.", "Namely, the local ring map", "$\\mathcal{O}_{Y', f'(\\xi)} \\to \\mathcal{O}_{Y' \\times_Y P, i'(\\xi)}$", "is flat with regular fibre", "(see Algebra, Section \\ref{algebra-section-smooth-overview}),", "a regular local ring is Cohen-Macaulay", "(Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}),", "$\\mathcal{O}_{Y', f'(\\xi)}$ is Cohen-Macaulay by assumption", "(4) and we get what we want from", "Algebra, Lemma \\ref{algebra-lemma-CM-goes-up}.", "Thus we reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $f$ is a regular closed immersion and $X'$, $X$, $Y'$, and", "$Y$ are $\\delta$-equidimensional of $\\delta$-dimensions", "$n'$, $n$, $m'$, and $m$ and $m' - n' = m - n$.", "In this case we obtain the result immediately from", "Lemma \\ref{lemma-gysin-easy}." ], "refs": [ "algebra-lemma-regular-ring-CM", "algebra-lemma-CM-goes-up", "chow-lemma-gysin-easy" ], "ref_ids": [ 941, 1362, 5815 ] } ], "ref_ids": [ 5761, 5948 ] }, { "id": 5836, "type": "theorem", "label": "chow-lemma-compare-gysin-base-change", "categories": [ "chow" ], "title": "chow-lemma-compare-gysin-base-change", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Consider a cartesian square", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} &", "X \\ar[d]^f \\\\", "Y' \\ar[r]^g &", "Y", "}", "$$", "of schemes locally of finite type over $S$. Assume", "\\begin{enumerate}", "\\item both $f$ and $f'$ are local complete intersection morphisms, and", "\\item the gysin map exists for $f$", "\\end{enumerate}", "Then $\\mathcal{C} = \\Ker(H^{-1}((g')^*\\NL_{X/Y}) \\to H^{-1}(\\NL_{X'/Y'}))$", "is a finite locally free $\\mathcal{O}_{X'}$-module, the gysin map", "exists for $f'$, and we have", "$$", "res(f^!) = c_{top}(\\mathcal{C}^\\vee) \\circ (f')^!", "$$", "in $A^*(X' \\to Y')$." ], "refs": [], "proofs": [ { "contents": [ "The fact that $\\mathcal{C}$ is finite locally free follows immediately", "from More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-lci-bis}.", "Choose a factorization $f = g \\circ i$ with $g : P \\to Y$ smooth and $i$", "an immersion. Then we can factor $f' = g' \\circ i'$ where $g' : P' \\to Y'$", "and $i' : X' \\to P'$ the base changes. Picture", "$$", "\\xymatrix{", "X' \\ar[r] \\ar[d] &", "P' \\ar[r] \\ar[d] &", "Y' \\ar[d] \\\\", "X \\ar[r] &", "P \\ar[r] &", "Y", "}", "$$", "In particular, we see that the gysin map exists for $f'$. By", "More on Morphisms, Lemmas \\ref{more-morphisms-lemma-get-NL}", "we have", "$$", "\\NL_{X/Y} = \\left( \\mathcal{C}_{X/P} \\to i^*\\Omega_{P/Y} \\right)", "$$", "where $\\mathcal{C}_{X/P}$ is the conormal sheaf of the embedding $i$.", "Similarly for the primed version. We have", "$(g')^*i^*\\Omega_{P/Y} = (i')^*\\Omega_{P'/Y'}$ because", "$\\Omega_{P/Y}$ pulls back to $\\Omega_{P'/Y'}$ by", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-differentials}.", "Also, recall that $(g')^*\\mathcal{C}_{X/P} \\to \\mathcal{C}_{X'/P'}$", "is surjective, see", "Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial-flat}.", "We deduce that the sheaf $\\mathcal{C}$ is canonicallly", "isomorphic to the kernel of the map", "$(g')^*\\mathcal{C}_{X/P} \\to \\mathcal{C}_{X'/P'}$", "of finite locally free modules. Recall that $i^!$ is defined", "using $\\mathcal{N} = \\mathcal{C}_{Z/X}^\\vee$ and similarly", "for $(i')^!$. Thus we have", "$$", "res(i^!) = c_{top}(\\mathcal{C}^\\vee) \\circ (i')^!", "$$", "in $A^*(X' \\to P')$ by an application of Lemma \\ref{lemma-gysin-excess}.", "Since finally we have $f^! = i^! \\circ g^*$,", "$(f')^! = (i')^! \\circ (g')^*$, and $(g')^* = res(g^*)$ we conclude." ], "refs": [ "more-algebra-lemma-base-change-lci-bis", "more-morphisms-lemma-get-NL", "morphisms-lemma-base-change-differentials", "morphisms-lemma-conormal-functorial-flat", "chow-lemma-gysin-excess" ], "ref_ids": [ 10312, 13756, 5314, 5305, 5813 ] } ], "ref_ids": [] }, { "id": 5837, "type": "theorem", "label": "chow-lemma-blow-up-formula", "categories": [ "chow" ], "title": "chow-lemma-blow-up-formula", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $i : Z \\to X$ be a regular closed immersion of schemes", "locally of finite type over $S$. Let $b : X' \\to X$ be the", "blowing up with center $Z$. Picture", "$$", "\\xymatrix{", "E \\ar[r]_j \\ar[d]_\\pi & X' \\ar[d]^b \\\\", "Z \\ar[r]^i & X", "}", "$$", "Assume that the gysin map exists for $b$. Then we have", "$$", "res(b^!) = c_{top}(\\mathcal{F}^\\vee) \\circ \\pi^*", "$$", "in $A^*(E \\to Z)$ where $\\mathcal{F}$ is the kernel of the canonical map", "$\\pi^*\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{E/X'}$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the morphism $b$ is a local complete intersection morphism", "by More on Algebra, Lemma \\ref{more-algebra-lemma-blowup-regular-sequence}", "and hence the statement makes sense. Since $Z \\to X$ is a regular", "immersion (and hence a fortiori quasi-regular) we see that $\\mathcal{C}_{Z/X}$", "is finite locally free and the map", "$\\text{Sym}^*(\\mathcal{C}_{Z/X}) \\to \\mathcal{C}_{Z/X, *}$", "is an isomorphism, see", "Divisors, Lemma \\ref{divisors-lemma-quasi-regular-immersion}.", "Since $E = \\text{Proj}(\\mathcal{C}_{Z/X, *})$ we conclude", "that $E = \\mathbf{P}(\\mathcal{C}_{Z/X})$", "is a projective space bundle over $Z$.", "Thus $E \\to Z$ is smooth and certainly a local complete intersection", "morphism. Thus Lemma \\ref{lemma-compare-gysin-base-change}", "applies and we see that", "$$", "res(b^!) = c_{top}(\\mathcal{C}^\\vee) \\circ \\pi^!", "$$", "with $\\mathcal{C}$ as in the statement there.", "Of course $\\pi^* = \\pi^!$ by Lemma \\ref{lemma-lci-gysin-flat}.", "It remains to show that $\\mathcal{F}$ is equal to", "the kernel $\\mathcal{C}$ of the map", "$H^{-1}(j^*\\NL_{X'/X}) \\to H^{-1}(\\NL_{E/Z})$.", "\\medskip\\noindent", "Since $E \\to Z$ is smooth we have $H^{-1}(\\NL_{E/Z}) = 0$, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-NL-smooth}.", "Hence it suffices to show that $\\mathcal{F}$ can be identified", "with $H^{-1}(j^*\\NL_{X'/X})$. By More on Morphisms, Lemmas", "\\ref{more-morphisms-lemma-get-triangle-NL} and", "\\ref{more-morphisms-lemma-NL-immersion} we have an exact sequence", "$$", "0 \\to H^{-1}(j^*\\NL_{X'/X}) \\to H^{-1}(\\NL_{E/X}) \\to", "\\mathcal{C}_{E/X'} \\to \\ldots", "$$", "By the same lemmas applied to $E \\to Z \\to X$ we obtain an isomorphism", "$\\pi^*\\mathcal{C}_{Z/X} = H^{-1}(\\pi^*\\NL_{Z/X}) \\to H^{-1}(\\NL_{E/X})$.", "Thus we conclude." ], "refs": [ "more-algebra-lemma-blowup-regular-sequence", "divisors-lemma-quasi-regular-immersion", "chow-lemma-compare-gysin-base-change", "chow-lemma-lci-gysin-flat", "more-morphisms-lemma-NL-smooth", "more-morphisms-lemma-get-triangle-NL", "more-morphisms-lemma-NL-immersion" ], "ref_ids": [ 9990, 7992, 5836, 5832, 13750, 13754, 13752 ] } ], "ref_ids": [] }, { "id": 5838, "type": "theorem", "label": "chow-lemma-lci-gysin-product-regular", "categories": [ "chow" ], "title": "chow-lemma-lci-gysin-product-regular", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ be a morphism of schemes locally of finite", "type over $S$ such that both $X$ and $Y$ are quasi-compact,", "regular, have affine diagonal, and finite dimension.", "Then $f$ is a local complete intersection morphism.", "Assume moreover the gysin map is defined for $f$. Then", "$$", "f^!(\\alpha \\cdot \\beta) = f^!\\alpha \\cdot f^!\\beta", "$$", "in $\\CH^*(X) \\otimes \\mathbf{Q}$ where the intersection product", "is as in Section \\ref{section-intersection-regular}." ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-morphism-regular-schemes-is-lci}.", "Observe that $f^![Y] = [X]$, see Lemma \\ref{lemma-lci-gysin-easy}.", "Write $\\alpha = ch(\\alpha') \\cap [Y]$ and $\\beta = ch(\\beta') \\cap [Y]$", "where $\\alpha', \\beta' \\in K_0(\\textit{Vect}(X)) \\otimes \\mathbf{Q}$", "as in Section \\ref{section-intersection-regular}.", "Setting $c = ch(\\alpha')$ and $c' = ch(\\beta')$ we find", "$\\alpha \\cdot \\beta = c \\cap c' \\cap [Y]$ by construction.", "By Lemma \\ref{lemma-lci-gysin-commutes} we know that $f^!$", "commutes with both $c$ and $c'$. Hence", "\\begin{align*}", "f^!(\\alpha \\cdot \\beta)", "& =", "f^!(c \\cap c' \\cap [Y]) \\\\", "& =", "c \\cap c' \\cap f^![Y] \\\\", "& =", "c \\cap c' \\cap [X] \\\\", "& =", "(c \\cap [X]) \\cdot (c' \\cap [X]) \\\\", "& =", "(c \\cap f^![Y]) \\cdot (c' \\cap f^![Y]) \\\\", "& =", "f^!(\\alpha) \\cdot f^!(\\beta)", "\\end{align*}", "as desired." ], "refs": [ "more-morphisms-lemma-morphism-regular-schemes-is-lci", "chow-lemma-lci-gysin-easy", "chow-lemma-lci-gysin-commutes" ], "ref_ids": [ 14009, 5835, 5834 ] } ], "ref_ids": [] }, { "id": 5839, "type": "theorem", "label": "chow-lemma-projection-formula-regular", "categories": [ "chow" ], "title": "chow-lemma-projection-formula-regular", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ be a morphism of schemes locally of finite", "type over $S$ such that both $X$ and $Y$ are quasi-compact,", "regular, have affine diagonal, and finite dimension.", "Then $f$ is a local complete intersection morphism.", "Assume moreover the gysin map is defined for $f$", "and that $f$ is proper. Then", "$$", "f_*(\\alpha \\cdot f^!\\beta) = f_*\\alpha \\cdot \\beta", "$$", "in $\\CH^*(Y) \\otimes \\mathbf{Q}$ where the intersection product", "is as in Section \\ref{section-intersection-regular}." ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-morphism-regular-schemes-is-lci}.", "Observe that $f^![Y] = [X]$, see Lemma \\ref{lemma-lci-gysin-easy}.", "Write $\\alpha = ch(\\alpha') \\cap [X]$ and $\\beta = ch(\\beta') \\cap [Y]$", "$\\alpha' \\in K_0(\\textit{Vect}(X)) \\otimes \\mathbf{Q}$ and", "$\\beta' \\in K_0(\\textit{Vect}(Y)) \\otimes \\mathbf{Q}$", "as in Section \\ref{section-intersection-regular}.", "Set $c = ch(\\alpha')$ and $c' = ch(\\beta')$. We have", "\\begin{align*}", "f_*(\\alpha \\cdot f^!\\beta)", "& =", "f_*(c \\cap f^!(c' \\cap [Y]_e)) \\\\", "& =", "f_*(c \\cap c' \\cap f^![Y]_e) \\\\", "& =", "f_*(c \\cap c' \\cap [X]_d) \\\\", "& =", "f_*(c' \\cap c \\cap [X]_d) \\\\", "& =", "c' \\cap f_*(c \\cap [X]_d) \\\\", "& =", "\\beta \\cdot f_*(\\alpha)", "\\end{align*}", "The first equality by the construction of the intersection product.", "By Lemma \\ref{lemma-lci-gysin-commutes} we know that $f^!$", "commutes with $c'$. The fact that Chern classes are in the center", "of the bivariant ring justifies switching the order of capping", "$[X]$ with $c$ and $c'$. Commuting $c'$ with $f_*$ is allowed as $c'$", "is a bivariant class. The final equality is again the construction", "of the intersection product." ], "refs": [ "more-morphisms-lemma-morphism-regular-schemes-is-lci", "chow-lemma-lci-gysin-easy", "chow-lemma-lci-gysin-commutes" ], "ref_ids": [ 14009, 5835, 5834 ] } ], "ref_ids": [] }, { "id": 5840, "type": "theorem", "label": "chow-lemma-diagonal-identity", "categories": [ "chow" ], "title": "chow-lemma-diagonal-identity", "contents": [ "In the situation above we have $\\Delta^! \\circ \\text{pr}_i^! = 1$ in $A^0(X)$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the projections $\\text{pr}_i : X \\times_Y X \\to X$ are", "smooth and hence we have gysin maps for these projections as well.", "Thus the lemma makes sense and is a special case of", "Lemma \\ref{lemma-lci-gysin-composition}." ], "refs": [ "chow-lemma-lci-gysin-composition" ], "ref_ids": [ 5833 ] } ], "ref_ids": [] }, { "id": 5841, "type": "theorem", "label": "chow-lemma-exterior-product-well-defined", "categories": [ "chow" ], "title": "chow-lemma-exterior-product-well-defined", "contents": [ "The map", "$\\times : \\CH_n(X) \\otimes_{\\mathbf{Z}} \\CH_m(Y) \\to \\CH_{n + m}(X \\times_k Y)$", "is well defined." ], "refs": [], "proofs": [ { "contents": [ "A first remark is that if $\\alpha = \\sum n_i[X_i]$", "and $\\beta = \\sum m_j[Y_j]$ with $X_i \\subset X$ and $Y_j \\subset Y$", "locally finite families of integral closed subschemes of", "dimensions $n$ and $m$, then", "$X_i \\times_k Y_j$ is a locally finite", "collection of closed subschemes of $X \\times_k Y$ of", "dimensions $n + m$ and we can indeed consider", "$$", "\\alpha \\times \\beta = \\sum n_i m_j [X_i \\times_k Y_j]_{n + m}", "$$", "as a $(n + m)$-cycle on $X \\times_k Y$. In this way we obtain an", "additive map", "$\\times : Z_n(X) \\otimes_{\\mathbf{Z}} Z_m(Y) \\to Z_{n + m}(X \\times_k Y)$.", "The problem is to show that", "this procedure is compatible with rational equivalence.", "\\medskip\\noindent", "Let $i : X' \\to X$ be the inclusion morphism of", "an integral closed subscheme of dimension $n$.", "Then flat pullback along the morphism $p' : X' \\to \\Spec(k)$ is an element", "$(p')^* \\in A^{-n}(X' \\to \\Spec(k))$ by", "Lemma \\ref{lemma-flat-pullback-bivariant}", "and hence $c' = i_* \\circ (p')^* \\in A^{-n}(X \\to \\Spec(k))$ by", "Lemma \\ref{lemma-push-proper-bivariant}.", "This produces maps", "$$", "c' \\cap - : \\CH_m(Y) \\longrightarrow \\CH_{m + n}(X \\times_k Y)", "$$", "which the reader easily sends $[Y']$ to $[X' \\times_k Y']_{n + m}$", "for any integral closed subscheme $Y' \\subset Y$ of dimension", "$m$. Hence the construction", "$([X'], [Y']) \\mapsto [X' \\times_k Y']_{n + m}$", "factors through rational equivalence in the second variable, i.e.,", "gives a well defined map", "$Z_n(X) \\otimes_{\\mathbf{Z}} \\CH_m(Y) \\to \\CH_{n + m}(X \\times_k Y)$.", "By symmetry the same is true for the other variable and we conclude." ], "refs": [ "chow-lemma-flat-pullback-bivariant", "chow-lemma-push-proper-bivariant" ], "ref_ids": [ 5732, 5734 ] } ], "ref_ids": [] }, { "id": 5842, "type": "theorem", "label": "chow-lemma-chow-cohomology-towards-point", "categories": [ "chow" ], "title": "chow-lemma-chow-cohomology-towards-point", "contents": [ "Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$.", "Then we have a canonical identification", "$$", "A^p(X \\to \\Spec(k)) = \\CH_{-p}(X)", "$$", "for all $p \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the element $[\\Spec(k)] \\in \\CH_0(\\Spec(k))$. We get a map", "$A^p(X \\to \\Spec(k)) \\to \\CH_{-p}(X)$ by sending $c$ to $c \\cap [\\Spec(k)]$.", "\\medskip\\noindent", "Conversely, suppose we have $\\alpha \\in \\CH_{-p}(X)$.", "Then we can define $c_\\alpha \\in A^p(X \\to \\Spec(k))$ as", "follows: given $X' \\to \\Spec(k)$ and $\\alpha' \\in \\CH_n(X')$", "we let", "$$", "c_\\alpha \\cap \\alpha' = \\alpha \\times \\alpha'", "$$", "in $\\CH_{n - p}(X \\times_k X')$. To show that this is a bivariant", "class we write $\\alpha = \\sum n_i[X_i]$ as in", "Definition \\ref{definition-cycles}. Consider the composition", "$$", "\\coprod X_i \\xrightarrow{g} X \\to \\Spec(k)", "$$", "and denote $f : \\coprod X_i \\to \\Spec(k)$ the composition.", "Then $g$ is proper and $f$ is flat of relative dimension $-p$.", "Pullback along $f$ is a bivariant class", "$f^* \\in A^p(\\coprod X_i \\to \\Spec(k))$ by", "Lemma \\ref{lemma-flat-pullback-bivariant}.", "Denote $\\nu \\in A^0(\\coprod X_i)$ the bivariant class", "which multiplies a cycle by $n_i$ on the $i$th component.", "Thus $\\nu \\circ f^* \\in A^p(\\coprod X_i \\to X)$.", "Finally, we have a bivariant class", "$$", "g_* \\circ \\nu \\circ f^*", "$$", "by Lemma \\ref{lemma-push-proper-bivariant}. The reader easily", "verifies that $c_\\alpha$ is equal to this class and hence", "is itself a bivariant class.", "\\medskip\\noindent", "To finish the proof we have to show that the two constructions", "are mutually inverse. Since $c_\\alpha \\cap [\\Spec(k)] = \\alpha$", "this is clear for one of the two directions. For the other, let", "$c \\in A^p(X \\to \\Spec(k))$ and set $\\alpha = c \\cap [\\Spec(k)]$.", "It suffices to prove that", "$$", "c \\cap [X'] = c_\\alpha \\cap [X']", "$$", "when $X'$ is an integral scheme locally of finite type over $\\Spec(k)$,", "see Lemma \\ref{lemma-bivariant-zero}. However, then $p' : X' \\to \\Spec(k)$", "is flat of relative dimension $\\dim(X')$ and hence", "$[X'] = (p')^*[\\Spec(k)]$. Thus the fact that the bivariant classes", "$c$ and $c_\\alpha$ agree on $[\\Spec(k)]$ implies they", "agree when capped against $[X']$ and the proof is complete." ], "refs": [ "chow-definition-cycles", "chow-lemma-flat-pullback-bivariant", "chow-lemma-push-proper-bivariant", "chow-lemma-bivariant-zero" ], "ref_ids": [ 5906, 5732, 5734, 5740 ] } ], "ref_ids": [] }, { "id": 5843, "type": "theorem", "label": "chow-lemma-chow-cohomology-towards-point-commutes", "categories": [ "chow" ], "title": "chow-lemma-chow-cohomology-towards-point-commutes", "contents": [ "Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$.", "Let $c \\in A^p(X \\to \\Spec(k))$. Let $Y \\to Z$ be a morphism of schemes", "locally of finite type over $k$. Let $c' \\in A^q(Y \\to Z)$. Then", "$c \\circ c' = c' \\circ c$ in $A^{p + q}(X \\times_k Y \\to X \\times_k Z)$." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-chow-cohomology-towards-point}", "we have seen that $c$ is given by a combination of", "proper pushforward, multiplying by integers over connected", "components, and flat pullback. Since $c'$ commutes with each of", "these operations by definition of bivariant classes, we conclude.", "Some details omitted." ], "refs": [ "chow-lemma-chow-cohomology-towards-point" ], "ref_ids": [ 5842 ] } ], "ref_ids": [] }, { "id": 5844, "type": "theorem", "label": "chow-lemma-exterior-product-associative", "categories": [ "chow" ], "title": "chow-lemma-exterior-product-associative", "contents": [ "Exterior product is associative. More precisely, let $k$ be a field,", "let $X, Y, Z$ be schemes locally of finite type over $k$, let", "$\\alpha \\in \\CH_*(X)$, $\\beta \\in \\CH_*(Y)$, $\\gamma \\in \\CH_*(Z)$.", "Then $(\\alpha \\times \\beta) \\times \\gamma =", "\\alpha \\times (\\beta \\times \\gamma)$ in $\\CH_*(X \\times_k Y \\times_k Z)$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: associativity of fibre product of schemes." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5845, "type": "theorem", "label": "chow-lemma-associative", "categories": [ "chow" ], "title": "chow-lemma-associative", "contents": [ "The product defined above is associative. More precisely, let $k$ be a field,", "let $X$ be smooth over $k$,", "let $Y, Z, W$ be schemes locally of finite type over $X$, let", "$\\alpha \\in \\CH_*(Y)$, $\\beta \\in \\CH_*(Z)$, $\\gamma \\in \\CH_*(W)$.", "Then $(\\alpha \\cdot \\beta) \\cdot \\gamma =", "\\alpha \\cdot (\\beta \\cdot \\gamma)$ in $\\CH_*(Y \\times_X Z \\times_X W)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-exterior-product-associative} we have", "$(\\alpha \\times \\beta) \\times \\gamma =", "\\alpha \\times (\\beta \\times \\gamma)$ in $\\CH_*(Y \\times_k Z \\times_k W)$.", "Consider the closed immersions", "$$", "\\Delta_{12} : X \\times_k X \\longrightarrow X \\times_k X \\times_k X,", "\\quad (x, x') \\mapsto (x, x, x')", "$$", "and", "$$", "\\Delta_{23} : X \\times_k X \\longrightarrow X \\times_k X \\times_k X,", "\\quad (x, x') \\mapsto (x, x', x')", "$$", "Denote $\\Delta_{12}^!$ and $\\Delta_{23}^!$ the corresponding bivariant", "classes; observe that $\\Delta_{12}^!$ is the restriction", "(Remark \\ref{remark-restriction-bivariant}) of $\\Delta^!$", "to $X \\times_k X \\times_k X$ by the map $\\text{pr}_{12}$ and that", "$\\Delta_{23}^!$ is the restriction of $\\Delta^!$", "to $X \\times_k X \\times_k X$ by the map $\\text{pr}_{23}$.", "Thus clearly the restriction of $\\Delta_{12}^!$ by $\\Delta_{23}$", "is $\\Delta^!$ and the restriction of $\\Delta_{23}^!$ by $\\Delta_{12}$ is", "$\\Delta^!$ too. Thus by Lemma \\ref{lemma-gysin-commutes} we have", "$$", "\\Delta^! \\circ \\Delta_{12}^! =", "\\Delta^! \\circ \\Delta_{23}^! ", "$$", "Now we can prove the lemma by the following sequence of equalities:", "\\begin{align*}", "(\\alpha \\cdot \\beta) \\cdot \\gamma", "& =", "\\Delta^!(\\Delta^!(\\alpha \\times \\beta) \\times \\gamma) \\\\", "& =", "\\Delta^!(\\Delta_{12}^!((\\alpha \\times \\beta) \\times \\gamma)) \\\\", "& =", "\\Delta^!(\\Delta_{23}^!((\\alpha \\times \\beta) \\times \\gamma)) \\\\", "& =", "\\Delta^!(\\Delta_{23}^!(\\alpha \\times (\\beta \\times \\gamma)) \\\\", "& =", "\\Delta^!(\\alpha \\times \\Delta^!(\\beta \\times \\gamma)) \\\\", "& =", "\\alpha \\cdot (\\beta \\cdot \\gamma)", "\\end{align*}", "All equalities are clear from the above except perhaps", "for the second and penultimate one. The equation", "$\\Delta_{23}^!(\\alpha \\times (\\beta \\times \\gamma)) =", "\\alpha \\times \\Delta^!(\\beta \\times \\gamma)$ holds by", "Remark \\ref{remark-commuting-exterior}. Similarly for the second", "equation." ], "refs": [ "chow-lemma-exterior-product-associative", "chow-remark-restriction-bivariant", "chow-lemma-gysin-commutes", "chow-remark-commuting-exterior" ], "ref_ids": [ 5844, 5938, 5817, 5964 ] } ], "ref_ids": [] }, { "id": 5846, "type": "theorem", "label": "chow-lemma-identify-chow-for-smooth", "categories": [ "chow" ], "title": "chow-lemma-identify-chow-for-smooth", "contents": [ "Let $k$ be a field. Let $X$ be a smooth scheme over $k$, equidimensional", "of dimension $d$. The map", "$$", "A^p(X) \\longrightarrow \\CH_{d - p}(X),\\quad", "c \\longmapsto c \\cap [X]_d", "$$", "is an isomorphism. Via this isomorphism composition of bivariant", "classes turns into the intersection product defined above." ], "refs": [], "proofs": [ { "contents": [ "Denote $g : X \\to \\Spec(k)$ the structure morphism.", "The map is the composition of the isomorphisms", "$$", "A^p(X) \\to A^{p - d}(X \\to \\Spec(k)) \\to \\CH_{d - p}(X)", "$$", "The first is the isomorphism $c \\mapsto c \\circ g^*$ of", "Proposition \\ref{proposition-compute-bivariant}", "and the second is the isomorphism $c \\mapsto c \\cap [\\Spec(k)]$ of", "Lemma \\ref{lemma-chow-cohomology-towards-point}.", "From the proof of Lemma \\ref{lemma-chow-cohomology-towards-point}", "we see that the inverse to the second arrow sends $\\alpha \\in \\CH_{d - p}(X)$", "to the bivariant class $c_\\alpha$ which sends $\\beta \\in \\CH_*(Y)$", "for $Y$ locally of finite type over $k$", "to $\\alpha \\times \\beta$ in $\\CH_*(X \\times_k Y)$. From the proof of", "Proposition \\ref{proposition-compute-bivariant} we see the inverse", "to the first arrow in turn sends $c_\\alpha$ to the bivariant class", "which sends $\\beta \\in \\CH_*(Y)$ for $Y \\to X$ locally of finite type", "to $\\Delta^!(\\alpha \\times \\beta) = \\alpha \\cdot \\beta$.", "From this the final result of the lemma follows." ], "refs": [ "chow-proposition-compute-bivariant", "chow-lemma-chow-cohomology-towards-point", "chow-lemma-chow-cohomology-towards-point", "chow-proposition-compute-bivariant" ], "ref_ids": [ 5901, 5842, 5842, 5901 ] } ], "ref_ids": [] }, { "id": 5847, "type": "theorem", "label": "chow-lemma-lci-gysin-product", "categories": [ "chow" ], "title": "chow-lemma-lci-gysin-product", "contents": [ "Let $k$ be a field. Let $f : X \\to Y$ be a morphism of schemes smooth", "over $k$. Then the gysin map is defined for $f$ and", "$f^!(\\alpha \\cdot \\beta) = f^!\\alpha \\cdot f^!\\beta$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $X \\to X \\times_k Y$ is an immersion of $X$ into a scheme", "smooth over $Y$. Hence the gysin map is defined for $f$", "(Definition \\ref{definition-lci-gysin}).", "To prove the formula we may decompose $X$ and $Y$ into their", "connected components, hence we may assume $X$ is smooth over $k$", "and equidimensional of dimension $d$ and $Y$ is smooth over $k$", "and equidimensional of dimension $e$. Observe that", "$f^![Y]_e = [X]_d$ (see for example Lemma \\ref{lemma-lci-gysin-easy}).", "Write $\\alpha = c \\cap [Y]_e$ and $\\beta = c' \\cap [Y]_e$", "and hence $\\alpha \\cdot \\beta = c \\cap c' \\cap [Y]_e$,", "see Lemma \\ref{lemma-identify-chow-for-smooth}.", "By Lemma \\ref{lemma-lci-gysin-commutes} we know that $f^!$", "commutes with both $c$ and $c'$. Hence", "\\begin{align*}", "f^!(\\alpha \\cdot \\beta)", "& =", "f^!(c \\cap c' \\cap [Y]_e) \\\\", "& =", "c \\cap c' \\cap f^![Y]_e \\\\", "& =", "c \\cap c' \\cap [X]_d \\\\", "& =", "(c \\cap [X]_d) \\cdot (c' \\cap [X]_d) \\\\", "& =", "(c \\cap f^![Y]_e) \\cdot (c' \\cap f^![Y]_e) \\\\", "& =", "f^!(\\alpha) \\cdot f^!(\\beta)", "\\end{align*}", "as desired where we have used Lemma \\ref{lemma-identify-chow-for-smooth}", "for $X$ as well.", "\\medskip\\noindent", "An alternative proof can be given by proving that", "$(f \\times f)^!(\\alpha \\times \\beta) = f^!\\alpha \\times f^!\\beta$", "and using Lemma \\ref{lemma-lci-gysin-composition}." ], "refs": [ "chow-definition-lci-gysin", "chow-lemma-lci-gysin-easy", "chow-lemma-identify-chow-for-smooth", "chow-lemma-lci-gysin-commutes", "chow-lemma-identify-chow-for-smooth", "chow-lemma-lci-gysin-composition" ], "ref_ids": [ 5925, 5835, 5846, 5834, 5846, 5833 ] } ], "ref_ids": [] }, { "id": 5848, "type": "theorem", "label": "chow-lemma-projection-formula", "categories": [ "chow" ], "title": "chow-lemma-projection-formula", "contents": [ "Let $k$ be a field. Let $f : X \\to Y$ be a proper morphism of schemes smooth", "over $k$. Then the gysin map is defined for $f$ and", "$f_*(\\alpha \\cdot f^!\\beta) = f_*\\alpha \\cdot \\beta$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $X \\to X \\times_k Y$ is an immersion of $X$ into a scheme", "smooth over $Y$. Hence the gysin map is defined for $f$", "(Definition \\ref{definition-lci-gysin}).", "To prove the formula we may decompose $X$ and $Y$ into their", "connected components, hence we may assume $X$ is smooth over $k$", "and equidimensional of dimension $d$ and $Y$ is smooth over $k$", "and equidimensional of dimension $e$. Observe that", "$f^![Y]_e = [X]_d$ (see for example Lemma \\ref{lemma-lci-gysin-easy}).", "Write $\\alpha = c \\cap [X]_d$ and $\\beta = c' \\cap [Y]_e$,", "see Lemma \\ref{lemma-identify-chow-for-smooth}. We have", "\\begin{align*}", "f_*(\\alpha \\cdot f^!\\beta)", "& =", "f_*(c \\cap f^!(c' \\cap [Y]_e)) \\\\", "& =", "f_*(c \\cap c' \\cap f^![Y]_e) \\\\", "& =", "f_*(c \\cap c' \\cap [X]_d) \\\\", "& =", "f_*(c' \\cap c \\cap [X]_d) \\\\", "& =", "c' \\cap f_*(c \\cap [X]_d) \\\\", "& =", "\\beta \\cdot f_*(\\alpha)", "\\end{align*}", "The first equality by the result of Lemma \\ref{lemma-identify-chow-for-smooth}", "for $X$. By Lemma \\ref{lemma-lci-gysin-commutes} we know that $f^!$", "commutes with $c'$. The commutativity of the intersection", "product justifies switching the order of capping $[X]_d$ with $c$ and $c'$", "(via the lemma). Commuting $c'$ with $f_*$ is allowed as $c'$", "is a bivariant class. The final equality is again the lemma." ], "refs": [ "chow-definition-lci-gysin", "chow-lemma-lci-gysin-easy", "chow-lemma-identify-chow-for-smooth", "chow-lemma-identify-chow-for-smooth", "chow-lemma-lci-gysin-commutes" ], "ref_ids": [ 5925, 5835, 5846, 5846, 5834 ] } ], "ref_ids": [] }, { "id": 5849, "type": "theorem", "label": "chow-lemma-intersect-properly", "categories": [ "chow" ], "title": "chow-lemma-intersect-properly", "contents": [ "Let $k$ be a field. Let $X$ be an integral scheme smooth over $k$.", "Let $Y, Z \\subset X$ be integral closed subschemes. Set", "$d = \\dim(Y) + \\dim(Z) - \\dim(X)$. Assume", "\\begin{enumerate}", "\\item $\\dim(Y \\cap Z) \\leq d$, and", "\\item $\\mathcal{O}_{Y, \\xi}$ and $\\mathcal{O}_{Z, \\xi}$", "are Cohen-Macaulay for every $\\xi \\in Y \\cap Z$ with", "$\\delta(\\xi) = d$.", "\\end{enumerate}", "Then $[Y] \\cdot [Z] = [Y \\cap Z]_d$ in $\\CH_d(X)$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $[Y] \\cdot [Z] = \\Delta^!([Y \\times Z])$ where", "$\\Delta^! = c(\\Delta : X \\to X \\times X, \\mathcal{T}_{X/k})$", "is a higher codimension gysin map", "(Section \\ref{section-gysin-higher-codimension}) with", "$\\mathcal{T}_{X/k} = \\SheafHom(\\Omega_{X/k}, \\mathcal{O}_X)$", "locally free of rank $\\dim(X)$. We have the equality of schemes", "$$", "Y \\cap Z = X \\times_{\\Delta, (X \\times X)} (Y \\times Z)", "$$", "and $\\dim(Y \\times Z) = \\dim(Y) + \\dim(Z)$ and hence conditions ", "(1), (2), and (3) of Lemma \\ref{lemma-gysin-easy} hold.", "Finally, if $\\xi \\in Y \\cap Z$, then we have a flat local", "homomorphism", "$$", "\\mathcal{O}_{Y, \\xi} \\longrightarrow", "\\mathcal{O}_{Y \\times Z, \\xi}", "$$", "whose ``fibre'' is $\\mathcal{O}_{Z, \\xi}$. It follows that if both", "$\\mathcal{O}_{Y, \\xi}$ and $\\mathcal{O}_{Z, \\xi}$", "are Cohen-Macaulay, then so is $\\mathcal{O}_{Y \\times Z, \\xi}$, see", "Algebra, Lemma \\ref{algebra-lemma-CM-goes-up}.", "In this way we see that all the hypotheses of", "Lemma \\ref{lemma-gysin-easy} are satisfied and we conclude." ], "refs": [ "chow-lemma-gysin-easy", "algebra-lemma-CM-goes-up", "chow-lemma-gysin-easy" ], "ref_ids": [ 5815, 1362, 5815 ] } ], "ref_ids": [] }, { "id": 5850, "type": "theorem", "label": "chow-lemma-intersect-regularly-embedded", "categories": [ "chow" ], "title": "chow-lemma-intersect-regularly-embedded", "contents": [ "Let $k$ be a field. Let $X$ be a scheme smooth over $k$. Let $i : Y \\to X$ be", "a regular closed immersion. Let $\\alpha \\in \\CH_*(X)$. If $Y$ is", "equidimensional of dimension $e$, then", "$\\alpha \\cdot [Y]_e = i_*(i^!(\\alpha))$ in $\\CH_*(X)$." ], "refs": [], "proofs": [ { "contents": [ "After decomposing $X$ into connected components we may and do assume $X$", "is equidimensional of dimension $d$. Write $\\alpha = c \\cap [X]_n$", "with $x \\in A^*(X)$, see Lemma \\ref{lemma-identify-chow-for-smooth}. Then", "$$", "i_*(i^!(\\alpha)) = i_*(i^!(c \\cap [X]_n)) =", "i_*(c \\cap i^![X]_n) = i_*(c \\cap [Y]_e) =", "c \\cap i_*[Y]_e = \\alpha \\cdot [Y]_e", "$$", "The first equality by choice of $c$. Then second equality by", "Lemma \\ref{lemma-lci-gysin-commutes}. The third because", "$i^![X]_d = [Y]_e$ in $\\CH_*(Y)$ (Lemma \\ref{lemma-lci-gysin-easy}).", "The fourth because bivariant classes commute with proper pushforward.", "The last equality by Lemma \\ref{lemma-identify-chow-for-smooth}." ], "refs": [ "chow-lemma-identify-chow-for-smooth", "chow-lemma-lci-gysin-commutes", "chow-lemma-lci-gysin-easy", "chow-lemma-identify-chow-for-smooth" ], "ref_ids": [ 5846, 5834, 5835, 5846 ] } ], "ref_ids": [] }, { "id": 5851, "type": "theorem", "label": "chow-lemma-intersection-regular-smooth", "categories": [ "chow" ], "title": "chow-lemma-intersection-regular-smooth", "contents": [ "Let $k$ be a field. Let $X$ be a smooth scheme over $k$ which is", "quasi-compact and has affine diagonal. Then the intersection", "product on $\\CH^*(X)$ constructed in this section agrees", "after tensoring with $\\mathbf{Q}$ with the intersection product", "constructed in Section \\ref{section-intersection-regular}." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha \\in \\CH^i(X)$ and $\\beta \\in \\CH^j(X)$. Write", "$\\alpha = ch(\\alpha') \\cap [X]$ and $\\beta = ch(\\beta') \\cap [X]$", "$\\alpha', \\beta' \\in K_0(\\textit{Vect}(X)) \\otimes \\mathbf{Q}$", "as in Section \\ref{section-intersection-regular}.", "Set $c = ch(\\alpha')$ and $c' = ch(\\beta')$.", "Then the intersection product in Section \\ref{section-intersection-regular}", "produces $c \\cap c' \\cap [X]$. This is the same as $\\alpha \\cdot \\beta$", "by Lemma \\ref{lemma-identify-chow-for-smooth} (or rather the", "generalization that $A^i(X) \\to \\CH^i(X)$, $c \\mapsto c \\cap [X]$", "is an isomorphism for any smooth scheme $X$ over $k$)." ], "refs": [ "chow-lemma-identify-chow-for-smooth" ], "ref_ids": [ 5846 ] } ], "ref_ids": [] }, { "id": 5852, "type": "theorem", "label": "chow-lemma-exterior-product-well-defined-dim-1", "categories": [ "chow" ], "title": "chow-lemma-exterior-product-well-defined-dim-1", "contents": [ "The map $\\times : \\CH_n(X) \\otimes_{\\mathbf{Z}} \\CH_m(Y) \\to", "\\CH_{n + m - 1}(X \\times_S Y)$ is well defined." ], "refs": [], "proofs": [ { "contents": [ "Consider $n$ and $m$ cycles $\\alpha = \\sum_{i \\in I} n_i[X_i]$", "and $\\beta = \\sum_{j \\in J} m_j[Y_j]$ with $X_i \\subset X$ and $Y_j \\subset Y$", "locally finite families of integral closed subschemes of", "$\\delta$-dimensions $n$ and $m$. Let $K \\subset I \\times J$ be the set", "of pairs $(i, j) \\in I \\times J$ such that $X_i$ or $Y_j$ dominates", "an irreducible component of $S$.", "Then $\\{X_i \\times_S Y_j\\}_{(i, j) \\in K}$ is a locally finite", "collection of closed subschemes of $X \\times_S Y$ of", "$\\delta$-dimension $n + m - 1$. This means we can indeed consider", "$$", "\\alpha \\times \\beta =", "\\sum\\nolimits_{(i, j) \\in K} n_i m_j [X_i \\times_S Y_j]_{n + m - 1}", "$$", "as a $(n + m - 1)$-cycle on $X \\times_S Y$. In this way we obtain an", "additive map", "$\\times : Z_n(X) \\otimes_{\\mathbf{Z}} Z_m(Y) \\to Z_{n + m}(X \\times_S Y)$.", "The problem is to show that", "this procedure is compatible with rational equivalence.", "\\medskip\\noindent", "Let $i : X' \\to X$ be the inclusion morphism of an integral closed subscheme", "of $\\delta$-dimension $n$ which dominates an irreducible component", "of $S$. Then $p' : X' \\to S$ is flat of relative dimension $n - 1$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-dedekind-torsion-free-flat}.", "Hence flat pullback along $p'$ is an element", "$(p')^* \\in A^{-n + 1}(X' \\to S)$ by", "Lemma \\ref{lemma-flat-pullback-bivariant}", "and hence $c' = i_* \\circ (p')^* \\in A^{-n + 1}(X \\to S)$ by", "Lemma \\ref{lemma-push-proper-bivariant}.", "This produces maps", "$$", "c' \\cap - : \\CH_m(Y) \\longrightarrow \\CH_{m + n - 1}(X \\times_S Y)", "$$", "which sends $[Y']$ to $[X' \\times_S Y']_{n + m - 1}$ for any", "integral closed subscheme $Y' \\subset Y$ of $\\delta$-dimension $m$.", "\\medskip\\noindent", "Let $i : X' \\to X$ be the inclusion morphism of an integral closed subscheme", "of $\\delta$-dimension $n$ such that the composition $X' \\to X \\to S$ ", "factors through a closed point $s \\in S$. Since $s$ is a closed point", "of the spectrum of a Dedekind domain, we see that $s$ is an effective", "Cartier divisor on $S$ whose normal bundle is trivial. Denote", "$c \\in A^1(s \\to S)$ the gysin homomorphism, see", "Lemma \\ref{lemma-gysin-bivariant}. The morphism $p' : X' \\to s$", "is flat of relative dimension $n$. Hence flat pullback along $p'$", "is an element $(p')^* \\in A^{-n}(X' \\to S)$ by", "Lemma \\ref{lemma-flat-pullback-bivariant}.", "Thus", "$$", "c' = i_* \\circ (p')^* \\circ c \\in A^{-n}(X \\to S)", "$$", "by Lemma \\ref{lemma-push-proper-bivariant}. This produces maps", "$$", "c' \\cap - : \\CH_m(Y) \\longrightarrow \\CH_{m + n - 1}(X \\times_S Y)", "$$", "which for any integral closed subscheme $Y' \\subset Y$", "of $\\delta$-dimension $m$", "sends $[Y']$ to either $[X' \\times_S Y']_{n + m - 1}$ if $Y'$ dominates", "an irreducible component of $S$ or to $0$ if not.", "\\medskip\\noindent", "From the previous two paragraphs we conclude", "the construction $([X'], [Y']) \\mapsto [X' \\times_S Y']_{n + m - 1}$", "factors through rational equivalence in the second variable, i.e.,", "gives a well defined map", "$Z_n(X) \\otimes_{\\mathbf{Z}} \\CH_m(Y) \\to \\CH_{n + m - 1}(X \\times_S Y)$.", "By symmetry the same is true for the other variable and we conclude." ], "refs": [ "more-algebra-lemma-dedekind-torsion-free-flat", "chow-lemma-flat-pullback-bivariant", "chow-lemma-push-proper-bivariant", "chow-lemma-gysin-bivariant", "chow-lemma-flat-pullback-bivariant", "chow-lemma-push-proper-bivariant" ], "ref_ids": [ 9921, 5732, 5734, 5733, 5732, 5734 ] } ], "ref_ids": [] }, { "id": 5853, "type": "theorem", "label": "chow-lemma-chow-cohomology-towards-base-dim-1", "categories": [ "chow" ], "title": "chow-lemma-chow-cohomology-towards-base-dim-1", "contents": [ "Let $(S, \\delta)$ be as above. Let $X$ be a scheme locally of finite type", "over $S$. Then we have a canonical identification", "$$", "A^p(X \\to S) = \\CH_{1 - p}(X)", "$$", "for all $p \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the element $[S]_1 \\in \\CH_1(S)$. We get a map", "$A^p(X \\to S) \\to \\CH_{1 - p}(X)$ by sending $c$ to $c \\cap [S]_1$.", "\\medskip\\noindent", "Conversely, suppose we have $\\alpha \\in \\CH_{1 - p}(X)$.", "Then we can define $c_\\alpha \\in A^p(X \\to S)$ as", "follows: given $X' \\to S$ and $\\alpha' \\in \\CH_n(X')$", "we let", "$$", "c_\\alpha \\cap \\alpha' = \\alpha \\times \\alpha'", "$$", "in $\\CH_{n - p}(X \\times_S X')$. To show that this is a bivariant", "class we write $\\alpha = \\sum_{i \\in I} n_i[X_i]$ as in", "Definition \\ref{definition-cycles}. In particular the morphism", "$$", "g : \\coprod\\nolimits_{i \\in I} X_i \\longrightarrow X", "$$", "is proper. Pick $i \\in I$. If $X_i$ dominates an irreducible component", "of $S$, then the structure morphism $p_i : X_i \\to S$ is flat and we have", "$\\xi_i = p_i^* \\in A^p(X_i \\to S)$. On the other hand, if $p_i$ factors", "as $p'_i : X_i \\to s_i$ followed by the inclusion $s_i \\to S$", "of a closed point, then we have", "$\\xi_i = (p'_i)^* \\circ c_i \\in A^p(X_i \\to S)$", "where $c_i \\in A^1(s_i \\to S)$ is the gysin homomorphism and", "$(p'_i)^*$ is flat pullback. Observe that", "$$", "A^p(\\coprod\\nolimits_{i \\in I} X_i \\to S) =", "\\prod\\nolimits_{i \\in I} A^p(X_i \\to S)", "$$", "Thus we have", "$$", "\\xi = \\sum n_i \\xi_i \\in A^p(\\coprod\\nolimits_{i \\in I} X_i \\to S)", "$$", "Finally, since $g$ is proper we have a bivariant class", "$$", "g_* \\circ \\xi \\in A^p(X \\to S)", "$$", "by Lemma \\ref{lemma-push-proper-bivariant}. The reader easily", "verifies that $c_\\alpha$ is equal to this class", "(please compare with the proof of", "Lemma \\ref{lemma-exterior-product-well-defined-dim-1})", "and hence is itself a bivariant class.", "\\medskip\\noindent", "To finish the proof we have to show that the two constructions", "are mutually inverse. Since $c_\\alpha \\cap [S]_1 = \\alpha$", "this is clear for one of the two directions. For the other, let", "$c \\in A^p(X \\to S)$ and set $\\alpha = c \\cap [S]_1$.", "It suffices to prove that", "$$", "c \\cap [X'] = c_\\alpha \\cap [X']", "$$", "when $X'$ is an integral scheme locally of finite type over $S$,", "see Lemma \\ref{lemma-bivariant-zero}. However, either $p' : X' \\to S$", "is flat of relative dimension $\\dim_\\delta(X') - 1$ and hence", "$[X'] = (p')^*[S]_1$ or $X' \\to S$ factors as $X' \\to s \\to S$", "and hence $[X'] = (p')^*(s \\to S)^*[S]_1$. Thus the fact that the", "bivariant classes $c$ and $c_\\alpha$ agree on $[S]_1$", "implies they agree when capped against $[X']$ (since bivariant classes", "commute with flat pullback and gysin maps) and the proof is complete." ], "refs": [ "chow-definition-cycles", "chow-lemma-push-proper-bivariant", "chow-lemma-exterior-product-well-defined-dim-1", "chow-lemma-bivariant-zero" ], "ref_ids": [ 5906, 5734, 5852, 5740 ] } ], "ref_ids": [] }, { "id": 5854, "type": "theorem", "label": "chow-lemma-chow-cohomology-towards-base-dim-1-commutes", "categories": [ "chow" ], "title": "chow-lemma-chow-cohomology-towards-base-dim-1-commutes", "contents": [ "Let $(S, \\delta)$ be as above. Let $X$ be a scheme locally of finite type", "over $S$. Let $c \\in A^p(X \\to S)$. Let $Y \\to Z$ be a morphism of schemes", "locally of finite type over $S$. Let $c' \\in A^q(Y \\to Z)$. Then", "$c \\circ c' = c' \\circ c$ in $A^{p + q}(X \\times_S Y \\to X \\times_S Z)$." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-chow-cohomology-towards-base-dim-1}", "we have seen that $c$ is given by a combination of", "proper pushforward, multiplying by integers over connected", "components, flat pullback, and gysin maps. Since $c'$ commutes with each of", "these operations by definition of bivariant classes, we conclude.", "Some details omitted." ], "refs": [ "chow-lemma-chow-cohomology-towards-base-dim-1" ], "ref_ids": [ 5853 ] } ], "ref_ids": [] }, { "id": 5855, "type": "theorem", "label": "chow-lemma-exterior-product-associative-dim-1", "categories": [ "chow" ], "title": "chow-lemma-exterior-product-associative-dim-1", "contents": [ "Exterior product is associative. More precisely, let $(S, \\delta)$ be", "as above, let $X, Y, Z$ be schemes locally of finite type over $S$, let", "$\\alpha \\in \\CH_*(X)$, $\\beta \\in \\CH_*(Y)$, $\\gamma \\in \\CH_*(Z)$.", "Then $(\\alpha \\times \\beta) \\times \\gamma =", "\\alpha \\times (\\beta \\times \\gamma)$ in $\\CH_*(X \\times_S Y \\times_S Z)$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: associativity of fibre product of schemes." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5856, "type": "theorem", "label": "chow-lemma-associative-dim-1", "categories": [ "chow" ], "title": "chow-lemma-associative-dim-1", "contents": [ "The product defined above is associative. More precisely, with", "$(S, \\delta)$ as above, let $X$ be smooth over $S$,", "let $Y, Z, W$ be schemes locally of finite type over $X$, let", "$\\alpha \\in \\CH_*(Y)$, $\\beta \\in \\CH_*(Z)$, $\\gamma \\in \\CH_*(W)$.", "Then $(\\alpha \\cdot \\beta) \\cdot \\gamma =", "\\alpha \\cdot (\\beta \\cdot \\gamma)$ in $\\CH_*(Y \\times_X Z \\times_X W)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-exterior-product-associative-dim-1} we have", "$(\\alpha \\times \\beta) \\times \\gamma =", "\\alpha \\times (\\beta \\times \\gamma)$ in $\\CH_*(Y \\times_S Z \\times_S W)$.", "Consider the closed immersions", "$$", "\\Delta_{12} : X \\times_S X \\longrightarrow X \\times_S X \\times_S X,", "\\quad (x, x') \\mapsto (x, x, x')", "$$", "and", "$$", "\\Delta_{23} : X \\times_S X \\longrightarrow X \\times_S X \\times_S X,", "\\quad (x, x') \\mapsto (x, x', x')", "$$", "Denote $\\Delta_{12}^!$ and $\\Delta_{23}^!$ the corresponding bivariant", "classes; observe that $\\Delta_{12}^!$ is the restriction", "(Remark \\ref{remark-restriction-bivariant}) of $\\Delta^!$", "to $X \\times_S X \\times_S X$ by the map $\\text{pr}_{12}$ and that", "$\\Delta_{23}^!$ is the restriction of $\\Delta^!$", "to $X \\times_S X \\times_S X$ by the map $\\text{pr}_{23}$.", "Thus clearly the restriction of $\\Delta_{12}^!$ by $\\Delta_{23}$", "is $\\Delta^!$ and the restriction of $\\Delta_{23}^!$ by $\\Delta_{12}$ is", "$\\Delta^!$ too. Thus by Lemma \\ref{lemma-gysin-commutes} we have", "$$", "\\Delta^! \\circ \\Delta_{12}^! =", "\\Delta^! \\circ \\Delta_{23}^! ", "$$", "Now we can prove the lemma by the following sequence of equalities:", "\\begin{align*}", "(\\alpha \\cdot \\beta) \\cdot \\gamma", "& =", "\\Delta^!(\\Delta^!(\\alpha \\times \\beta) \\times \\gamma) \\\\", "& =", "\\Delta^!(\\Delta_{12}^!((\\alpha \\times \\beta) \\times \\gamma)) \\\\", "& =", "\\Delta^!(\\Delta_{23}^!((\\alpha \\times \\beta) \\times \\gamma)) \\\\", "& =", "\\Delta^!(\\Delta_{23}^!(\\alpha \\times (\\beta \\times \\gamma)) \\\\", "& =", "\\Delta^!(\\alpha \\times \\Delta^!(\\beta \\times \\gamma)) \\\\", "& =", "\\alpha \\cdot (\\beta \\cdot \\gamma)", "\\end{align*}", "All equalities are clear from the above except perhaps", "for the second and penultimate one. The equation", "$\\Delta_{23}^!(\\alpha \\times (\\beta \\times \\gamma)) =", "\\alpha \\times \\Delta^!(\\beta \\times \\gamma)$ holds by", "Remark \\ref{remark-commuting-exterior}. Similarly for the second", "equation." ], "refs": [ "chow-lemma-exterior-product-associative-dim-1", "chow-remark-restriction-bivariant", "chow-lemma-gysin-commutes", "chow-remark-commuting-exterior" ], "ref_ids": [ 5855, 5938, 5817, 5964 ] } ], "ref_ids": [] }, { "id": 5857, "type": "theorem", "label": "chow-lemma-identify-chow-for-smooth-dim-1", "categories": [ "chow" ], "title": "chow-lemma-identify-chow-for-smooth-dim-1", "contents": [ "Let $(S, \\delta)$ be as above. Let $X$ be a smooth scheme over $S$,", "equidimensional of dimension $d$. The map", "$$", "A^p(X) \\longrightarrow \\CH_{d - p}(X),\\quad", "c \\longmapsto c \\cap [X]_d", "$$", "is an isomorphism. Via this isomorphism composition of bivariant", "classes turns into the intersection product defined above." ], "refs": [], "proofs": [ { "contents": [ "Denote $g : X \\to S$ the structure morphism.", "The map is the composition of the isomorphisms", "$$", "A^p(X) \\to A^{p - d + 1}(X \\to S) \\to \\CH_{d - p}(X)", "$$", "The first is the isomorphism $c \\mapsto c \\circ g^*$ of", "Proposition \\ref{proposition-compute-bivariant}", "and the second is the isomorphism $c \\mapsto c \\cap [S]_1$ of", "Lemma \\ref{lemma-chow-cohomology-towards-base-dim-1}.", "From the proof of Lemma \\ref{lemma-chow-cohomology-towards-base-dim-1}", "we see that the inverse to the second arrow sends $\\alpha \\in \\CH_{d - p}(X)$", "to the bivariant class $c_\\alpha$ which sends $\\beta \\in \\CH_*(Y)$", "for $Y$ locally of finite type over $k$", "to $\\alpha \\times \\beta$ in $\\CH_*(X \\times_k Y)$. From the proof of", "Proposition \\ref{proposition-compute-bivariant} we see the inverse", "to the first arrow in turn sends $c_\\alpha$ to the bivariant class", "which sends $\\beta \\in \\CH_*(Y)$ for $Y \\to X$ locally of finite type", "to $\\Delta^!(\\alpha \\times \\beta) = \\alpha \\cdot \\beta$.", "From this the final result of the lemma follows." ], "refs": [ "chow-proposition-compute-bivariant", "chow-lemma-chow-cohomology-towards-base-dim-1", "chow-lemma-chow-cohomology-towards-base-dim-1", "chow-proposition-compute-bivariant" ], "ref_ids": [ 5901, 5853, 5853, 5901 ] } ], "ref_ids": [] }, { "id": 5858, "type": "theorem", "label": "chow-lemma-dimension-base-change", "categories": [ "chow" ], "title": "chow-lemma-dimension-base-change", "contents": [ "In Situation \\ref{situation-setup-base-change} let $X \\to S$ be locally", "of finite type. Denote $X' \\to S'$ the base change by $S' \\to S$.", "If $X$ is integral with $\\dim_\\delta(X) = k$, then", "every irreducible component $Z'$ of $X'$ has $\\dim_{\\delta'}(Z') = k + c$," ], "refs": [], "proofs": [ { "contents": [ "The projection $X' \\to X$ is flat as a base change of the flat morphism", "$S' \\to S$ (Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat}).", "Hence every generic point $x'$ of an irreducible", "component of $X'$ maps to the generic point $x \\in X$ (because generalizations", "lift along $X' \\to X$ by", "Morphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat}).", "Let $s \\in S$ be the image of $x$.", "Recall that the scheme $S'_s = S' \\times_S s$", "has the same underlying topological space as $g^{-1}(\\{s\\})$", "(Schemes, Lemma \\ref{schemes-lemma-fibre-topological}).", "We may view $x'$ as a point of the scheme $S'_s \\times_s x$ which", "comes equipped with a monomorphism $S'_s \\times_s x \\to S' \\times_S X$.", "Of course, $x'$ is a generic point of an irreducible component", "of $S'_s \\times_s x$ as well.", "Using the flatness of $\\Spec(\\kappa(x)) \\to \\Spec(\\kappa(s)) = s$", "and arguing as above, we see that $x'$ maps to a generic point $s'$", "of an irreducible component of $g^{-1}(\\{s\\})$. Hence", "$\\delta'(s') = \\delta(s) + c$ by assumption.", "We have $\\dim_x(X_s) = \\dim_{x'}(X_{s'})$ by", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}.", "Since $x$ is a generic point of an irreducible component $X_s$", "(this is an irreducible scheme but we don't need this) and", "$x'$ is a generic point of an irreducible component of $X'_{s'}$ we conclude", "that $\\text{trdeg}_{\\kappa(s)}(\\kappa(x)) = ", "\\text{trdeg}_{\\kappa(s')}(\\kappa(x'))$", "by Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}.", "Then", "$$", "\\delta_{X'/S'}(x') = \\delta(s') + \\text{trdeg}_{\\kappa(s')}(\\kappa(x')) =", "\\delta(s) + c + \\text{trdeg}_{\\kappa(s)}(\\kappa(x)) = \\delta_{X/S}(x) + c", "$$", "This proves what we want by Definition \\ref{definition-delta-dimension}." ], "refs": [ "morphisms-lemma-base-change-flat", "morphisms-lemma-generalizations-lift-flat", "schemes-lemma-fibre-topological", "morphisms-lemma-dimension-fibre-after-base-change", "morphisms-lemma-dimension-fibre-at-a-point", "chow-definition-delta-dimension" ], "ref_ids": [ 5265, 5266, 7696, 5279, 5277, 5905 ] } ], "ref_ids": [] }, { "id": 5859, "type": "theorem", "label": "chow-lemma-pullback-coherent-base-change", "categories": [ "chow" ], "title": "chow-lemma-pullback-coherent-base-change", "contents": [ "In Situation \\ref{situation-setup-base-change} let $X \\to S$", "locally of finite type and let $X' \\to S$ be the base change by $S' \\to S$.", "\\begin{enumerate}", "\\item Let $Z \\subset X$ be a closed subscheme with", "$\\dim_\\delta(Z) \\leq k$ and base change $Z' \\subset X'$. Then we have", "$\\dim_{\\delta'}(Z')) \\leq k + c$", "and $[Z']_{k + c} = g^*[Z]_k$ in $Z_{k + c}(X')$.", "\\item Let $\\mathcal{F}$ be a coherent sheaf on $X$ with", "$\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$ and base", "change $\\mathcal{F}'$ on $X'$.", "Then we have $\\dim_\\delta(\\text{Supp}(\\mathcal{F}')) \\leq k + c$", "and $g^*[\\mathcal{F}]_k = [\\mathcal{F}']_{k + c}$", "in $Z_{k + c}(X')$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same is the proof of", "Lemma \\ref{lemma-pullback-coherent}", "and we suggest the reader skip it.", "\\medskip\\noindent", "The statements on dimensions follow from", "Lemma \\ref{lemma-dimension-base-change}.", "Part (1) follows from part (2) by Lemma \\ref{lemma-cycle-closed-coherent}", "and the fact that the base change of the coherent module $\\mathcal{O}_Z$", "is $\\mathcal{O}_{Z'}$.", "\\medskip\\noindent", "Proof of (2). As $X$, $X'$ are locally Noetherian we may apply", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-Noetherian} to see", "that $\\mathcal{F}$ is of finite type, hence $\\mathcal{F}'$ is", "of finite type (Modules, Lemma \\ref{modules-lemma-pullback-finite-type}),", "hence $\\mathcal{F}'$ is coherent", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-Noetherian} again).", "Thus the lemma makes sense. Let $W \\subset X$ be an integral closed", "subscheme of $\\delta$-dimension $k$, and let $W' \\subset X'$ be", "an integral closed subscheme of $\\delta'$-dimension $k + c$ mapping into $W$", "under $X' \\to X$. We have to show that the coefficient $n$ of", "$[W']$ in $g^*[\\mathcal{F}]_k$ agrees with the coefficient", "$m$ of $[W']$ in $[\\mathcal{F}']_{k + c}$. Let $\\xi \\in W$ and", "$\\xi' \\in W'$ be the generic points. Let", "$A = \\mathcal{O}_{X, \\xi}$, $B = \\mathcal{O}_{X', \\xi'}$", "and set $M = \\mathcal{F}_\\xi$ as an $A$-module. (Note that", "$M$ has finite length by our dimension assumptions, but we", "actually do not need to verify this. See", "Lemma \\ref{lemma-length-finite}.)", "We have $\\mathcal{F}'_{\\xi'} = B \\otimes_A M$.", "Thus we see that", "$$", "n = \\text{length}_B(B \\otimes_A M)", "\\quad", "\\text{and}", "\\quad", "m = \\text{length}_A(M) \\text{length}_B(B/\\mathfrak m_AB)", "$$", "Thus the equality follows from", "Algebra, Lemma \\ref{algebra-lemma-pullback-module}." ], "refs": [ "chow-lemma-pullback-coherent", "chow-lemma-dimension-base-change", "chow-lemma-cycle-closed-coherent", "coherent-lemma-coherent-Noetherian", "modules-lemma-pullback-finite-type", "coherent-lemma-coherent-Noetherian", "chow-lemma-length-finite", "algebra-lemma-pullback-module" ], "ref_ids": [ 5681, 5858, 5670, 3308, 13236, 3308, 5669, 640 ] } ], "ref_ids": [] }, { "id": 5860, "type": "theorem", "label": "chow-lemma-pullback-base-change", "categories": [ "chow" ], "title": "chow-lemma-pullback-base-change", "contents": [ "In Situation \\ref{situation-setup-base-change} let $X \\to S$ be locally", "of finite type and let $X' \\to S'$ be the base change by $S' \\to S$.", "The map $g^* : Z_k(X) \\to Z_{k + c}(X')$ above factors through rational", "equivalence to give a map", "$$", "g^* : \\CH_k(X) \\longrightarrow \\CH_{k + c}(X')", "$$", "of chow groups." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $\\alpha \\in Z_k(X)$ is a $k$-cycle which is rationally equivalent", "to zero. By Lemma \\ref{lemma-rational-equivalence-family}", "there exists a locally finite family of integral closed subschemes", "$W_i \\subset X \\times \\mathbf{P}^1$ of $\\delta$-dimension $k$", "not contained in the divisors", "$(X \\times \\mathbf{P}^1)_0$ or $(X \\times \\mathbf{P}^1)_\\infty$", "of $X \\times \\mathbf{P}^1$ such that", "$\\alpha = \\sum ([(W_i)_0]_k - [(W_i)_\\infty]_k)$.", "Thus it suffices to prove for $W \\subset X \\times \\mathbf{P}^1$", "integral closed of $\\delta$-dimension $k$ not contained in the divisors", "$(X \\times \\mathbf{P}^1)_0$ or $(X \\times \\mathbf{P}^1)_\\infty$", "of $X \\times \\mathbf{P}^1$ we have", "\\begin{enumerate}", "\\item the base change $W' \\subset X' \\times \\mathbf{P}^1$ satisfies the", "assumptions of Lemma \\ref{lemma-closed-subscheme-cross-p1} with", "$k$ replaced by $k + c$, and", "\\item $g^*[W_0]_k = [(W')_0]_{k + c}$ and", "$g^*[W_\\infty]_k = [(W')_\\infty]_{k + c}$.", "\\end{enumerate}", "Part (2) follows immediately from", "Lemma \\ref{lemma-pullback-coherent-base-change} and the fact that", "$(W')_0$ is the base change of $W_0$ (by associativity of fibre products).", "For part (1), first the statement on dimensions follows", "from Lemma \\ref{lemma-dimension-base-change}.", "Then let $w' \\in (W')_0$ with image $w \\in W_0$", "and $z \\in \\mathbf{P}^1_S$. Denote $t \\in \\mathcal{O}_{\\mathbf{P}^1_S, z}$", "the usual equation for $0 : S \\to \\mathbf{P}^1_S$.", "Since $\\mathcal{O}_{W, w} \\to \\mathcal{O}_{W', w'}$ is flat", "and since $t$ is a nonzerodivisor on $\\mathcal{O}_{W, w}$", "(as $W$ is integral and $W \\not = W_0$) we see that also", "$t$ is a nonzerodivisor in $\\mathcal{O}_{W', w'}$. Hence", "$W'$ has no associated points lying on $W'_0$." ], "refs": [ "chow-lemma-rational-equivalence-family", "chow-lemma-closed-subscheme-cross-p1", "chow-lemma-pullback-coherent-base-change", "chow-lemma-dimension-base-change" ], "ref_ids": [ 5695, 5696, 5859, 5858 ] } ], "ref_ids": [] }, { "id": 5861, "type": "theorem", "label": "chow-lemma-pullback-base-change-pullback", "categories": [ "chow" ], "title": "chow-lemma-pullback-base-change-pullback", "contents": [ "In Situation \\ref{situation-setup-base-change} let $Y \\to X \\to S$ be locally", "of finite type and let $Y' \\to X' \\to S'$ be the base change by $S' \\to S$.", "Assume $f : Y \\to X$ is flat of relative dimension $r$. Then $f' : Y' \\to X'$", "is flat of relative dimension $r$ and the diagram", "$$", "\\xymatrix{", "\\CH_{k + r}(Y) \\ar[r]_{g^*} & \\CH_{k + c + r}(Y') \\\\", "\\CH_k(X) \\ar[r]^{g^*} \\ar[u]^{(f')^*} & \\CH_{k + c}(X') \\ar[u]_{f^*}", "}", "$$", "of chow groups commutes." ], "refs": [], "proofs": [ { "contents": [ "In fact, we claim it commutes on the level of cycles. Namely, let", "$Z \\subset X$ be an integral closed subscheme of $\\delta$-dimension $k$", "and denote $Z' \\subset X'$ its base change. By construction", "we have $g^*[Z] = [Z']_{k + c}$. By Lemma \\ref{lemma-pullback-coherent}", "we have $(f')^*g^*[Z] = [Z' \\times_{X'} Y']_{k + c + r}$.", "Conversely, we have $f^*[Z] = [Z \\times_X Y]_{k + r}$ by", "Definition \\ref{definition-flat-pullback}. By", "Lemma \\ref{lemma-pullback-coherent-base-change}", "we have $g^*f^*[Z] = [(Z \\times_X Y)']_{k + r + c}$.", "Since $(Z \\times_X Y)' = Z' \\times_{X'} Y'$ by", "associativity of fibre product we conclude." ], "refs": [ "chow-lemma-pullback-coherent", "chow-definition-flat-pullback", "chow-lemma-pullback-coherent-base-change" ], "ref_ids": [ 5681, 5910, 5859 ] } ], "ref_ids": [] }, { "id": 5862, "type": "theorem", "label": "chow-lemma-pullback-base-change-pushforward", "categories": [ "chow" ], "title": "chow-lemma-pullback-base-change-pushforward", "contents": [ "In Situation \\ref{situation-setup-base-change} let $Y \\to X \\to S$ be locally", "of finite type and let $Y' \\to X' \\to S'$ be the base change by $S' \\to S$.", "Assume $f : Y \\to X$ is proper. Then $f' : Y' \\to X'$ is proper and the diagram", "$$", "\\xymatrix{", "\\CH_k(Y) \\ar[r]_{g^*} \\ar[d]_{f_*} & \\CH_{k + c}(Y') \\ar[d]^{f'_*} \\\\", "\\CH_k(X) \\ar[r]^{g^*} & \\CH_{k + c}(X')", "}", "$$", "of chow groups commutes." ], "refs": [], "proofs": [ { "contents": [ "In fact, we claim it commutes on the level of cycles. Namely, let", "$Z \\subset Y$ be an integral closed subscheme of $\\delta$-dimension $k$", "and denote $Z' \\subset X'$ its base change. By construction", "we have $g^*[Z] = [Z']_{k + c}$. By Lemma \\ref{lemma-cycle-push-sheaf}", "we have $(f')_*g^*[Z] = [f'_*\\mathcal{O}_{Z'}]_{k + c}$.", "By the same lemma we have $f_*[Z] = [f_*\\mathcal{O}_Z]_k$. By", "Lemma \\ref{lemma-pullback-coherent-base-change}", "we have $g^*f_*[Z] = [(X' \\to X)^*f_*\\mathcal{O}_Z]_{k + r}$.", "Thus it suffices to show that", "$$", "(X' \\to X)^*f_*\\mathcal{O}_Z \\cong f'_*\\mathcal{O}_{Z'}", "$$", "as coherent modules on $X'$. As $X' \\to X$ is flat and as", "$\\mathcal{O}_{Z'} = (Y' \\to Y)^*\\mathcal{O}_Z$, this", "follows from flat base change, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}." ], "refs": [ "chow-lemma-cycle-push-sheaf", "chow-lemma-pullback-coherent-base-change", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 5676, 5859, 3298 ] } ], "ref_ids": [] }, { "id": 5863, "type": "theorem", "label": "chow-lemma-pullback-base-change-c1", "categories": [ "chow" ], "title": "chow-lemma-pullback-base-change-c1", "contents": [ "In Situation \\ref{situation-setup-base-change} let $X \\to S$ be locally", "of finite type and let $X' \\to S'$ be the base change by $S' \\to S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module with", "base change $\\mathcal{L}'$ on $X'$. Then the", "diagram", "$$", "\\xymatrix{", "\\CH_k(X) \\ar[r]_{g^*} \\ar[d]_{c_1(\\mathcal{L}) \\cap -} &", "\\CH_{k + c}(X') \\ar[d]^{c_1(\\mathcal{L}') \\cap -} \\\\", "\\CH_{k - 1}(X) \\ar[r]^{g^*} & \\CH_{k + c - 1}(X')", "}", "$$", "of chow groups commutes." ], "refs": [], "proofs": [ { "contents": [ "Let $p : L \\to X$ be the line bundle associated to $\\mathcal{L}$", "with zero section $o : X \\to L$. For $\\alpha \\in CH_k(X)$ we", "know that $\\beta = c_1(\\mathcal{L}) \\cap \\alpha$", "is the unique element of $\\CH_{k - 1}(X)$ such that", "$o_*\\alpha = - p^*\\beta$, see Lemmas \\ref{lemma-linebundle} and", "\\ref{lemma-linebundle-formulae}.", "The same characterization holds after pullback. Hence the lemma follows from", "Lemmas \\ref{lemma-pullback-base-change-pullback} and", "\\ref{lemma-pullback-base-change-pushforward}." ], "refs": [ "chow-lemma-linebundle", "chow-lemma-linebundle-formulae", "chow-lemma-pullback-base-change-pullback", "chow-lemma-pullback-base-change-pushforward" ], "ref_ids": [ 5727, 5728, 5861, 5862 ] } ], "ref_ids": [] }, { "id": 5864, "type": "theorem", "label": "chow-lemma-pullback-base-change-chern-classes", "categories": [ "chow" ], "title": "chow-lemma-pullback-base-change-chern-classes", "contents": [ "In Situation \\ref{situation-setup-base-change} let $X \\to S$ be locally", "of finite type and let $X' \\to S'$ be the base change by $S' \\to S$.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module of", "rank $r$ with base change $\\mathcal{E}'$ on $X'$. Then the", "diagram", "$$", "\\xymatrix{", "\\CH_k(X) \\ar[r]_{g^*} \\ar[d]_{c_i(\\mathcal{E}) \\cap -} &", "\\CH_{k + c}(X') \\ar[d]^{c_i(\\mathcal{E}') \\cap -} \\\\", "\\CH_{k - i}(X) \\ar[r]^{g^*} & \\CH_{k + c - i}(X')", "}", "$$", "of chow groups commutes for all $i$." ], "refs": [], "proofs": [ { "contents": [ "Set $P = \\mathbf{P}(\\mathcal{E})$. The base change $P'$ of $P$", "is equal to $\\mathbf{P}(\\mathcal{E}')$. Since we already know that", "flat pullback and cupping with $c_1$ of an invertible module", "commute with base change (Lemmas \\ref{lemma-pullback-base-change-pullback} and", "\\ref{lemma-pullback-base-change-c1})", "the lemma follows from the characterization of capping", "with $c_i(\\mathcal{E})$ given in Lemma \\ref{lemma-determine-intersections}." ], "refs": [ "chow-lemma-pullback-base-change-pullback", "chow-lemma-pullback-base-change-c1", "chow-lemma-determine-intersections" ], "ref_ids": [ 5861, 5863, 5746 ] } ], "ref_ids": [] }, { "id": 5865, "type": "theorem", "label": "chow-lemma-compose-base-change", "categories": [ "chow" ], "title": "chow-lemma-compose-base-change", "contents": [ "Let $(S, \\delta)$, $(S', \\delta')$, $(S'', \\delta'')$ be as in", "Situation \\ref{situation-setup}. Let $g : S' \\to S$ and $g' : S'' \\to S'$", "be flat morphisms of schemes and let $c, c' \\in \\mathbf{Z}$", "be integers such that $S, \\delta, S', \\delta', g, c$ and", "$S', \\delta', S'', g', c'$ are as in", "Situation \\ref{situation-setup-base-change}.", "Let $X \\to S$ be locally of finite type and denote $X' \\to S'$", "and $X'' \\to S''$ the base changes by $S' \\to S$ and $S'' \\to S$.", "Then $S, \\delta, S'', \\delta'', g \\circ g', c + c'$ is as in", "Situation \\ref{situation-setup-base-change} and", "the maps $g^* : \\CH_k(X) \\to \\CH_{k + c}(X')$ and", "$(g')^* : \\CH_{k + c}(X') \\to \\CH_{k + c + c'}(X'')$ of", "Lemma \\ref{lemma-pullback-base-change}", "compose to give the map $(g \\circ g')^* : \\CH_k(X) \\to \\CH_{k + c + c'}(X'')$", "of Lemma \\ref{lemma-pullback-base-change}." ], "refs": [ "chow-lemma-pullback-base-change", "chow-lemma-pullback-base-change" ], "proofs": [ { "contents": [ "Let $s \\in S$ and let $s'' \\in S''$ be a generic point of an irreducible", "component of $(g \\circ g')^{-1}(\\{s\\})$. Set $s' = g'(s'')$.", "Clearly, $s''$ is a generic point of an irreducible component of", "$(g')^{-1}(\\{s'\\})$. Moreover, since $g'$ is flat and hence generalizations", "lift along $g'$ (Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat})", "we see that also $s'$ is a generic point of an irreducible component", "of $g^{-1}(\\{s\\})$. Thus by assumption $\\delta'(s') = \\delta(s) + c$", "and $\\delta''(s'') = \\delta'(s') + c'$. We conclude", "$\\delta''(s'') = \\delta(s) + c + c'$ and the first part of the", "statement is true.", "\\medskip\\noindent", "For the second part, let $Z \\subset X$ be an integral closed subscheme", "of $\\delta$-dimension $k$. Denote $Z' \\subset X'$ and $Z'' \\subset X''$", "the base changes. By definition we have $g^*[Z] = [Z']_{k + c}$.", "By Lemma \\ref{lemma-pullback-coherent-base-change} we have", "$(g')^*[Z']_{k + c} = [Z'']_{k + c + c'}$. This proves the final statement." ], "refs": [ "morphisms-lemma-base-change-flat", "chow-lemma-pullback-coherent-base-change" ], "ref_ids": [ 5265, 5859 ] } ], "ref_ids": [ 5860, 5860 ] }, { "id": 5866, "type": "theorem", "label": "chow-lemma-chow-limit", "categories": [ "chow" ], "title": "chow-lemma-chow-limit", "contents": [ "In Situation \\ref{situation-setup-base-change} assume $c = 0$", "and assume that $S' = \\lim_{i \\in I} S_i$ is a filtered limit", "of schemes $S_i$ affine over $S$ such that", "\\begin{enumerate}", "\\item with $\\delta_i$ equal to $S_i \\to S \\xrightarrow{\\delta} \\mathbf{Z}$", "the pair $(S_i, \\delta_i)$ is as in Situation \\ref{situation-setup},", "\\item $S_i, \\delta_i, S, \\delta, S \\to S_i, c = 0$ is as in", "Situation \\ref{situation-setup-base-change},", "\\item $S_i, \\delta_i, S_{i'}, \\delta_{i'}, S_i \\to S_{i'}, c = 0$ ", "for $i \\geq i'$ is as in Situation \\ref{situation-setup-base-change}.", "\\end{enumerate}", "Then for a quasi-compact scheme $X$ of finite type over $S$", "with base change $X'$ and $X_i$ by $S' \\to S$ and $S_i \\to S$ we have", "$\\CH_k(X') = \\colim \\CH_k(X_i)$." ], "refs": [], "proofs": [ { "contents": [ "By the result of Lemma \\ref{lemma-compose-base-change} we obtain", "an inverse system of chow groups $\\CH_k(X_i)$ and a map", "$\\colim \\CH_i(X_i) \\to \\CH_k(X)$.", "We may replace $S$ by a quasi-compact open through which $X \\to S$", "factors, hence we may and do assume all the schemes occuring in", "this proof are Noetherian (and hence quasi-compact and quasi-separated).", "\\medskip\\noindent", "Let us show that this map is surjective. Namely, let $Z' \\subset X'$", "be an integral closed subscheme of $\\delta'$-dimension $k$. By", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can find an $i$ and a morphism $Z_i \\to X_i$ of finite presentation", "whose base change is $Z'$. Afer increasing $i$ we may assume $Z_i$", "is a closed subscheme of $X_i$, see", "Limits, Lemma \\ref{limits-lemma-descend-closed-immersion-finite-presentation}.", "Then $Z' \\to X_i$ factors through $Z_i$ and we may replace $Z_i$", "by the scheme theoretic image of $Z' \\to X_i$. In this way we see", "that we may assume $Z_i$ is an integral closed subscheme of $X_i$.", "By Lemma \\ref{lemma-dimension-base-change} we conclude that", "$\\dim_{\\delta_i}(Z_i) = \\dim_{\\delta'}(Z') = k$.", "Thus $\\CH_k(X_i) \\to \\CH_k(X')$ maps $[Z_i]$ to $[Z']$ and", "we conclude surjectivity holds.", "\\medskip\\noindent", "Let us show that our map is injective. Let $\\alpha_i \\in \\CH_k(X_i)$", "be a cycle whose image $\\alpha' \\in \\CH_k(X')$ is zero.", "Then there exist integral closed subschemes", "$W_l' \\subset X'$, $l = 1, \\ldots, r$ of $\\delta\"$-dimension $k + 1$", "and nonzero rational functions $f'_l$ on $W'_l$", "such that $\\alpha' = \\sum_{l = 1, \\ldots, r} \\text{div}_{W'_l}(f'_l)$.", "Arguing as above we can find an $i$ and integral closed subschemes", "$W_{i, l} \\subset X_i$ of $\\delta_i$-dimension $k + 1$", "whose base change is $W'_l$.", "After increasin $i$ we may assume we have rational functions", "$f_{i, l}$ on $W_{i, l}$. Namely, we may think of $f'_l$ as a", "section of the structure sheaf over a nonempty open $U'_l \\subset W'_l$,", "we can descend these opens by Limits, Lemma \\ref{limits-lemma-descend-opens}", "and after increasing $i$ we may descend $f'_l$ by", "Limits, Lemma \\ref{limits-lemma-descend-section}.", "We claim that", "$$", "\\alpha_i = \\sum\\nolimits_{l = 1, \\ldots, r} \\text{div}_{W_{i, l}}(f_{i, l})", "$$", "after possibly increasing $i$.", "\\medskip\\noindent", "To prove the claim, let $Z'_{l, j} \\subset W'_l$ be a finite", "collection of integral closed subschemes of $\\delta'$-dimension $k$", "such that $f'_l$ is an invertible regular function outside", "$\\bigcup_j Y'_{l, j}$. After increasing $i$ (by the arguments above)", "we may assume there exist integral closed subschemes $Z_{i, l, j} \\subset W_i$", "of $\\delta_i$-dimension $k$ such that $f_{i, l}$ is an", "invertible regular function outside $\\bigcup_j Z_{i, l, j}$.", "Then we may write", "$$", "\\text{div}_{W'_l}(f'_l) = \\sum n_{l, j} [Z'_{l, j}]", "$$", "and", "$$", "\\text{div}_{W_{i, l}}(f_{i, l}) = \\sum n_{i, l, j} [Z_{i, l, j}]", "$$", "To prove the claim it suffices to show that $n_{l, i} = n_{i, l, j}$.", "Namely, this will imply that $\\beta_i =", "\\alpha_i - \\sum\\nolimits_{l = 1, \\ldots, r} \\text{div}_{W_{i, l}}(f_{i, l})$", "is a cycle on $X_i$ whose pullback to $X'$ is zero as a cycle!", "It follows that $\\beta_i$ pulls back to zero as a cycle on $X_{i'}$", "for some $i' \\geq i$ by an easy argument we omit.", "\\medskip\\noindent", "To prove the equality $n_{l, i} = n_{i, l, j}$ we choose a", "generic point $\\xi' \\in Z'_{l, j}$ and we denote", "$\\xi \\in Z_{i, l, j}$ the image which is a generic point also.", "Then the local ring map", "$$", "\\mathcal{O}_{W_{i, l}, \\xi}", "\\longrightarrow", "\\mathcal{O}_{W'_l, \\xi'}", "$$", "is flat as $W'_l \\to W_{i, l}$ is the base change of the flat", "morphism $S' \\to S_i$. We also have", "$\\mathfrak m_\\xi \\mathcal{O}_{W'_l, \\xi'} = \\mathfrak m_{\\xi'}$", "because $Z_{i, l, j}$ pulls back to $Z'_{l, j}$! Thus the equality of", "$$", "n_{l, j} = \\text{ord}_{Z'_{l, j}}(f'_l) =", "\\text{ord}_{\\mathcal{O}_{W'_l, \\xi'}}(f'_l)", "\\quad\\text{and}\\quad", "n_{i, l, j} = \\text{ord}_{Z_{i, l, j}}(f_{i, l}) =", "\\text{ord}_{\\mathcal{O}_{W_{i, l}, \\xi}}(f_{i, l})", "$$", "follows from Algebra, Lemma \\ref{algebra-lemma-pullback-module}", "and the construction of $\\text{ord}$ in", "Algebra, Section \\ref{algebra-section-orders-of-vanishing}." ], "refs": [ "chow-lemma-compose-base-change", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-closed-immersion-finite-presentation", "chow-lemma-dimension-base-change", "limits-lemma-descend-opens", "algebra-lemma-pullback-module" ], "ref_ids": [ 5865, 15077, 15060, 5858, 15041, 640 ] } ], "ref_ids": [] }, { "id": 5867, "type": "theorem", "label": "chow-lemma-dimension-at-most-one", "categories": [ "chow" ], "title": "chow-lemma-dimension-at-most-one", "contents": [ "With notations as above we have $\\dim_\\kappa(\\det_\\kappa(M)) \\leq 1$." ], "refs": [], "proofs": [ { "contents": [ "Fix an admissible sequence $(f_1, \\ldots, f_l)$ of $M$ such that", "$$", "\\text{length}_R(\\langle f_1, \\ldots, f_i\\rangle) = i", "$$", "for $i = 1, \\ldots, l$. Such an admissible sequence exists exactly because", "$M$ has length $l$. We will show that any element of", "$\\det_\\kappa(M)$ is a $\\kappa$-multiple of the symbol", "$[f_1, \\ldots, f_l]$. This will prove the lemma.", "\\medskip\\noindent", "Let $(e_1, \\ldots, e_l)$ be an admissible sequence of $M$.", "It suffices to show that $[e_1, \\ldots, e_l]$ is a multiple", "of $[f_1, \\ldots, f_l]$. First assume that", "$\\langle e_1, \\ldots, e_l\\rangle \\not = M$. Then there exists", "an $i \\in [1, \\ldots, l]$ such that", "$e_i \\in \\langle e_1, \\ldots, e_{i - 1}\\rangle$. It immediately", "follows from the first admissible relation that", "$[e_1, \\ldots, e_n] = 0$ in $\\det_\\kappa(M)$.", "Hence we may assume that $\\langle e_1, \\ldots, e_l\\rangle = M$.", "In particular there exists a smallest index $i \\in \\{1, \\ldots, l\\}$", "such that $f_1 \\in \\langle e_1, \\ldots, e_i\\rangle$. This means", "that $e_i = \\lambda f_1 + x$ with", "$x \\in \\langle e_1, \\ldots, e_{i - 1}\\rangle$ and $\\lambda \\in R^*$.", "By the second admissible relation this means that", "$[e_1, \\ldots, e_l] =", "\\overline{\\lambda}[e_1, \\ldots, e_{i - 1}, f_1, e_{i + 1}, \\ldots, e_l]$.", "Note that $\\mathfrak m f_1 = 0$. Hence by applying the third", "admissible relation $i - 1$ times we see that", "$$", "[e_1, \\ldots, e_l] =", "(-1)^{i - 1}\\overline{\\lambda}", "[f_1, e_1, \\ldots, e_{i - 1}, e_{i + 1}, \\ldots, e_l].", "$$", "Note that it is also the case that", "$ \\langle f_1, e_1, \\ldots, e_{i - 1}, e_{i + 1}, \\ldots, e_l\\rangle = M$.", "By induction suppose we have proven that our original", "symbol is equal to a scalar times", "$$", "[f_1, \\ldots, f_j, e_{j + 1}, \\ldots, e_l]", "$$", "for some admissible sequence $(f_1, \\ldots, f_j, e_{j + 1}, \\ldots, e_l)$", "whose elements generate $M$, i.e., \\ with", "$\\langle f_1, \\ldots, f_j, e_{j + 1}, \\ldots, e_l\\rangle = M$.", "Then we find the smallest $i$ such that", "$f_{j + 1} \\in \\langle f_1, \\ldots, f_j, e_{j + 1}, \\ldots, e_i\\rangle$", "and we go through the same process as above to see that", "$$", "[f_1, \\ldots, f_j, e_{j + 1}, \\ldots, e_l]", "=", "(\\text{scalar}) [f_1, \\ldots, f_j, f_{j + 1}, e_{j + 1},", "\\ldots, \\hat{e_i}, \\ldots, e_l]", "$$", "Continuing in this vein we obtain the desired result." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5868, "type": "theorem", "label": "chow-lemma-compare-det", "categories": [ "chow" ], "title": "chow-lemma-compare-det", "contents": [ "Let $R$ be a local ring with maximal ideal $\\mathfrak m$ and", "residue field $\\kappa$. Let $M$ be a finite length $R$-module", "which is annihilated by $\\mathfrak m$. Let $l = \\dim_\\kappa(M)$.", "Then the map", "$$", "\\det\\nolimits_\\kappa(M) \\longrightarrow \\wedge^l_\\kappa(M),", "\\quad", "[e_1, \\ldots, e_l] \\longmapsto e_1 \\wedge \\ldots \\wedge e_l", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "It is clear that the rule described in the lemma gives a $\\kappa$-linear", "map since all of the admissible relations are satisfied by the usual", "symbols $e_1 \\wedge \\ldots \\wedge e_l$. It is also clearly a surjective", "map. Since by Lemma \\ref{lemma-dimension-at-most-one} the left hand side", "has dimension at most one", "we see that the map is an isomorphism." ], "refs": [ "chow-lemma-dimension-at-most-one" ], "ref_ids": [ 5867 ] } ], "ref_ids": [] }, { "id": 5869, "type": "theorem", "label": "chow-lemma-determinant-dimension-one", "categories": [ "chow" ], "title": "chow-lemma-determinant-dimension-one", "contents": [ "Let $R$ be a local ring with maximal ideal $\\mathfrak m$ and", "residue field $\\kappa$. Let $M$ be a finite length $R$-module.", "The determinant $\\det_\\kappa(M)$ defined above is a $\\kappa$-vector", "space of dimension $1$. It is generated by the symbol", "$[f_1, \\ldots, f_l]$ for any admissible sequence such", "that $\\langle f_1, \\ldots f_l \\rangle = M$." ], "refs": [], "proofs": [ { "contents": [ "We know $\\det_\\kappa(M)$ has dimension at most $1$, and in fact that it", "is generated by $[f_1, \\ldots, f_l]$, by", "Lemma \\ref{lemma-dimension-at-most-one} and its proof.", "We will show by induction on $l = \\text{length}(M)$", "that it is nonzero. For $l = 1$ it follows from Lemma \\ref{lemma-compare-det}.", "Choose a nonzero element $f \\in M$", "with $\\mathfrak m f = 0$. Set $\\overline{M} = M /\\langle f \\rangle$,", "and denote the quotient map $x \\mapsto \\overline{x}$.", "We will define a surjective map", "$$", "\\psi : \\det\\nolimits_k(M) \\to \\det\\nolimits_\\kappa(\\overline{M})", "$$", "which will prove the lemma since by induction the determinant of", "$\\overline{M}$ is nonzero.", "\\medskip\\noindent", "We define $\\psi$ on symbols as follows.", "Let $(e_1, \\ldots, e_l)$ be an admissible sequence.", "If $f \\not \\in \\langle e_1, \\ldots, e_l \\rangle$ then", "we simply set $\\psi([e_1, \\ldots, e_l]) = 0$.", "If $f \\in \\langle e_1, \\ldots, e_l \\rangle$ then we choose", "an $i$ minimal such that $f \\in \\langle e_1, \\ldots, e_i \\rangle$.", "We may write $e_i = \\lambda f + x$ for some unit $\\lambda \\in R$", "and $x \\in \\langle e_1, \\ldots, e_{i - 1} \\rangle$.", "In this case we set", "$$", "\\psi([e_1, \\ldots, e_l]) =", "(-1)^i", "\\overline{\\lambda}[\\overline{e}_1, \\ldots,", "\\overline{e}_{i - 1},", "\\overline{e}_{i + 1}, \\ldots, \\overline{e}_l].", "$$", "Note that it is indeed the case that", "$(\\overline{e}_1, \\ldots,", "\\overline{e}_{i - 1},", "\\overline{e}_{i + 1}, \\ldots, \\overline{e}_l)$", "is an admissible sequence in $\\overline{M}$, so this makes sense.", "Let us show that extending this rule $\\kappa$-linearly to", "linear combinations of symbols does indeed lead to a map on", "determinants. To do this we have to show that the admissible", "relations are mapped to zero.", "\\medskip\\noindent", "Type (a) relations. Suppose we have $(e_1, \\ldots, e_l)$ an", "admissible sequence and for some $1 \\leq a \\leq l$ we have", "$e_a \\in \\langle e_1, \\ldots, e_{a - 1}\\rangle$.", "Suppose that $f \\in \\langle e_1, \\ldots, e_i\\rangle$ with $i$ minimal.", "Then $i \\not = a$ and", "$\\overline{e}_a \\in \\langle \\overline{e}_1, \\ldots,", "\\hat{\\overline{e}_i}, \\ldots, \\overline{e}_{a - 1}\\rangle$ if $i < a$", "or", "$\\overline{e}_a \\in \\langle \\overline{e}_1, \\ldots,", "\\overline{e}_{a - 1}\\rangle$ if $i > a$.", "Thus the same admissible relation for $\\det_\\kappa(\\overline{M})$ forces", "the symbol $[\\overline{e}_1, \\ldots,", "\\overline{e}_{i - 1},", "\\overline{e}_{i + 1}, \\ldots, \\overline{e}_l]$", "to be zero as desired.", "\\medskip\\noindent", "Type (b) relations. Suppose we have $(e_1, \\ldots, e_l)$ an", "admissible sequence and for some $1 \\leq a \\leq l$ we have", "$e_a = \\lambda e'_a + x$ with $\\lambda \\in R^*$, and", "$x \\in \\langle e_1, \\ldots, e_{a - 1}\\rangle$.", "Suppose that $f \\in \\langle e_1, \\ldots, e_i\\rangle$ with $i$ minimal.", "Say $e_i = \\mu f + y$ with $y \\in \\langle e_1, \\ldots, e_{i - 1}\\rangle$.", "If $i < a$ then the desired equality is", "$$", "(-1)^i", "\\overline{\\lambda}", "[\\overline{e}_1,", "\\ldots,", "\\overline{e}_{i - 1},", "\\overline{e}_{i + 1},", "\\ldots,", "\\overline{e}_l]", "=", "(-1)^i", "\\overline{\\lambda}", "[\\overline{e}_1,", "\\ldots,", "\\overline{e}_{i - 1},", "\\overline{e}_{i + 1},", "\\ldots,", "\\overline{e}_{a - 1},", "\\overline{e}'_a,", "\\overline{e}_{a + 1},", "\\ldots,", "\\overline{e}_l]", "$$", "which follows from $\\overline{e}_a = \\lambda \\overline{e}'_a + \\overline{x}$", "and the corresponding admissible relation for $\\det_\\kappa(\\overline{M})$.", "If $i > a$ then the desired equality is", "$$", "(-1)^i", "\\overline{\\lambda}", "[\\overline{e}_1,", "\\ldots,", "\\overline{e}_{i - 1},", "\\overline{e}_{i + 1},", "\\ldots,", "\\overline{e}_l]", "=", "(-1)^i", "\\overline{\\lambda}", "[\\overline{e}_1,", "\\ldots,", "\\overline{e}_{a - 1},", "\\overline{e}'_a,", "\\overline{e}_{a + 1},", "\\ldots,", "\\overline{e}_{i - 1},", "\\overline{e}_{i + 1},", "\\ldots,", "\\overline{e}_l]", "$$", "which follows from $\\overline{e}_a = \\lambda \\overline{e}'_a + \\overline{x}$", "and the corresponding admissible relation for $\\det_\\kappa(\\overline{M})$.", "The interesting case is when $i = a$. In this case we have", "$e_a = \\lambda e'_a + x = \\mu f + y$. Hence also", "$e'_a = \\lambda^{-1}(\\mu f + y - x)$. Thus we see that", "$$", "\\psi([e_1, \\ldots, e_l])", "= (-1)^i \\overline{\\mu}", "[\\overline{e}_1, \\ldots,", "\\overline{e}_{i - 1},", "\\overline{e}_{i + 1}, \\ldots, \\overline{e}_l]", "=", "\\psi(", "\\overline{\\lambda}", "[e_1, \\ldots, e_{a - 1}, e'_a, e_{a + 1}, \\ldots, e_l]", ")", "$$", "as desired.", "\\medskip\\noindent", "Type (c) relations. Suppose that $(e_1, \\ldots, e_l)$", "is an admissible sequence and", "$\\mathfrak m e_a \\subset \\langle e_1, \\ldots, e_{a - 2}\\rangle$.", "Suppose that $f \\in \\langle e_1, \\ldots, e_i\\rangle$ with $i$ minimal.", "Say $e_i = \\lambda f + x$ with $x \\in \\langle e_1, \\ldots, e_{i - 1}\\rangle$.", "We distinguish $4$ cases:", "\\medskip\\noindent", "Case 1: $i < a - 1$. The desired equality is", "\\begin{align*}", "& (-1)^i", "\\overline{\\lambda}", "[\\overline{e}_1,", "\\ldots,", "\\overline{e}_{i - 1},", "\\overline{e}_{i + 1},", "\\ldots,", "\\overline{e}_l] \\\\", "& =", "(-1)^{i + 1}", "\\overline{\\lambda}", "[\\overline{e}_1,", "\\ldots,", "\\overline{e}_{i - 1},", "\\overline{e}_{i + 1},", "\\ldots,", "\\overline{e}_{a - 2},", "\\overline{e}_a,", "\\overline{e}_{a - 1},", "\\overline{e}_{a + 1},", "\\ldots,", "\\overline{e}_l]", "\\end{align*}", "which follows from the type (c) admissible relation for", "$\\det_\\kappa(\\overline{M})$.", "\\medskip\\noindent", "Case 2: $i > a$. The desired equality is", "\\begin{align*}", "& (-1)^i", "\\overline{\\lambda}", "[\\overline{e}_1,", "\\ldots,", "\\overline{e}_{i - 1},", "\\overline{e}_{i + 1},", "\\ldots,", "\\overline{e}_l] \\\\", "& =", "(-1)^{i + 1}", "\\overline{\\lambda}", "[\\overline{e}_1,", "\\ldots,", "\\overline{e}_{a - 2},", "\\overline{e}_a,", "\\overline{e}_{a - 1},", "\\overline{e}_{a + 1},", "\\ldots,", "\\overline{e}_{i - 1},", "\\overline{e}_{i + 1},", "\\ldots,", "\\overline{e}_l]", "\\end{align*}", "which follows from the type (c) admissible relation for", "$\\det_\\kappa(\\overline{M})$.", "\\medskip\\noindent", "Case 3: $i = a$. We write $e_a = \\lambda f + \\mu e_{a - 1} + y$", "with $y \\in \\langle e_1, \\ldots, e_{a - 2}\\rangle$. Then", "$$", "\\psi([e_1, \\ldots, e_l]) =", "(-1)^a", "\\overline{\\lambda}", "[\\overline{e}_1,", "\\ldots,", "\\overline{e}_{a - 1},", "\\overline{e}_{a + 1},", "\\ldots,", "\\overline{e}_l]", "$$", "by definition. If $\\overline{\\mu}$ is nonzero, then we have", "$e_{a - 1} = - \\mu^{-1} \\lambda f + \\mu^{-1}e_a - \\mu^{-1} y$", "and we obtain", "$$", "\\psi(-[e_1, \\ldots, e_{a - 2}, e_a, e_{a - 1}, e_{a + 1}, \\ldots, e_l]) =", "(-1)^a", "\\overline{\\mu^{-1}\\lambda}", "[\\overline{e}_1,", "\\ldots,", "\\overline{e}_{a - 2},", "\\overline{e}_a,", "\\overline{e}_{a + 1},", "\\ldots,", "\\overline{e}_l]", "$$", "by definition. Since in $\\overline{M}$ we have", "$\\overline{e}_a = \\mu \\overline{e}_{a - 1} + \\overline{y}$ we see", "the two outcomes are equal by relation (a) for $\\det_\\kappa(\\overline{M})$.", "If on the other hand $\\overline{\\mu}$ is zero, then we can write", "$e_a = \\lambda f + y$ with $y \\in \\langle e_1, \\ldots, e_{a - 2}\\rangle$", "and we have", "$$", "\\psi(-[e_1, \\ldots, e_{a - 2}, e_a, e_{a - 1}, e_{a + 1}, \\ldots, e_l]) =", "(-1)^a", "\\overline{\\lambda}", "[\\overline{e}_1,", "\\ldots,", "\\overline{e}_{a - 1},", "\\overline{e}_{a + 1},", "\\ldots,", "\\overline{e}_l]", "$$", "which is equal to $\\psi([e_1, \\ldots, e_l])$.", "\\medskip\\noindent", "Case 4: $i = a - 1$. Here we have", "$$", "\\psi([e_1, \\ldots, e_l]) =", "(-1)^{a - 1}", "\\overline{\\lambda}", "[\\overline{e}_1,", "\\ldots,", "\\overline{e}_{a - 2},", "\\overline{e}_a,", "\\ldots,", "\\overline{e}_l]", "$$", "by definition. If $f \\not \\in \\langle e_1, \\ldots, e_{a - 2}, e_a \\rangle$", "then", "$$", "\\psi(-[e_1, \\ldots, e_{a - 2}, e_a, e_{a - 1}, e_{a + 1}, \\ldots, e_l]) =", "(-1)^{a + 1}\\overline{\\lambda}", "[\\overline{e}_1,", "\\ldots,", "\\overline{e}_{a - 2},", "\\overline{e}_a,", "\\ldots,", "\\overline{e}_l]", "$$", "Since $(-1)^{a - 1} = (-1)^{a + 1}$ the two expressions are the same.", "Finally, assume $f \\in \\langle e_1, \\ldots, e_{a - 2}, e_a \\rangle$.", "In this case we see that $e_{a - 1} = \\lambda f + x$ with", "$x \\in \\langle e_1, \\ldots, e_{a - 2}\\rangle$ and", "$e_a = \\mu f + y$ with $y \\in \\langle e_1, \\ldots, e_{a - 2}\\rangle$", "for units $\\lambda, \\mu \\in R$.", "We conclude that both", "$e_a \\in \\langle e_1, \\ldots, e_{a - 1} \\rangle$ and", "$e_{a - 1} \\in \\langle e_1, \\ldots, e_{a - 2}, e_a\\rangle$.", "In this case a relation of type (a) applies to both", "$[e_1, \\ldots, e_l]$ and", "$[e_1, \\ldots, e_{a - 2}, e_a, e_{a - 1}, e_{a + 1}, \\ldots, e_l]$", "and the compatibility of $\\psi$ with these shown above to see that both", "$$", "\\psi([e_1, \\ldots, e_l])", "\\quad\\text{and}\\quad", "\\psi([e_1, \\ldots, e_{a - 2}, e_a, e_{a - 1}, e_{a + 1}, \\ldots, e_l])", "$$", "are zero, as desired.", "\\medskip\\noindent", "At this point we have shown that $\\psi$ is well defined, and all that remains", "is to show that it is surjective. To see this let", "$(\\overline{f}_2, \\ldots, \\overline{f}_l)$ be an admissible sequence", "in $\\overline{M}$. We can choose lifts $f_2, \\ldots, f_l \\in M$, and", "then $(f, f_2, \\ldots, f_l)$ is an admissible sequence in $M$.", "Since $\\psi([f, f_2, \\ldots, f_l]) = [f_2, \\ldots, f_l]$ we win." ], "refs": [ "chow-lemma-dimension-at-most-one", "chow-lemma-compare-det" ], "ref_ids": [ 5867, 5868 ] } ], "ref_ids": [] }, { "id": 5870, "type": "theorem", "label": "chow-lemma-det-exact-sequences", "categories": [ "chow" ], "title": "chow-lemma-det-exact-sequences", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring.", "For every short exact sequence", "$$", "0 \\to K \\to L \\to M \\to 0", "$$", "of finite length $R$-modules there exists a canonical isomorphism", "$$", "\\gamma_{K \\to L \\to M} :", "\\det\\nolimits_\\kappa(K) \\otimes_\\kappa \\det\\nolimits_\\kappa(M)", "\\longrightarrow", "\\det\\nolimits_\\kappa(L)", "$$", "defined by the rule on nonzero symbols", "$$", "[e_1, \\ldots, e_k]", "\\otimes", "[\\overline{f}_1, \\ldots, \\overline{f}_m]", "\\longrightarrow", "[e_1, \\ldots, e_k, f_1, \\ldots, f_m]", "$$", "with the following properties:", "\\begin{enumerate}", "\\item For every isomorphism of short exact sequences, i.e., for", "every commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "K \\ar[r] \\ar[d]^u &", "L \\ar[r] \\ar[d]^v &", "M \\ar[r] \\ar[d]^w &", "0 \\\\", "0 \\ar[r] &", "K' \\ar[r] &", "L' \\ar[r] &", "M' \\ar[r] &", "0", "}", "$$", "with short exact rows and isomorphisms $u, v, w$ we have", "$$", "\\gamma_{K' \\to L' \\to M'} \\circ", "(\\det\\nolimits_\\kappa(u) \\otimes \\det\\nolimits_\\kappa(w))", "=", "\\det\\nolimits_\\kappa(v) \\circ", "\\gamma_{K \\to L \\to M},", "$$", "\\item for every commutative square of finite length $R$-modules", "with exact rows and columns", "$$", "\\xymatrix{", "& 0 \\ar[d] & 0 \\ar[d] & 0 \\ar[d] & \\\\", "0 \\ar[r] & A \\ar[r] \\ar[d] & B \\ar[r] \\ar[d] & C \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] & D \\ar[r] \\ar[d] & E \\ar[r] \\ar[d] & F \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] & G \\ar[r] \\ar[d] & H \\ar[r] \\ar[d] & I \\ar[r] \\ar[d] & 0 \\\\", "& 0 & 0 & 0 &", "}", "$$", "the following diagram is commutative", "$$", "\\xymatrix{", "\\det\\nolimits_\\kappa(A) \\otimes", "\\det\\nolimits_\\kappa(C) \\otimes", "\\det\\nolimits_\\kappa(G) \\otimes", "\\det\\nolimits_\\kappa(I)", "\\ar[dd]_{\\epsilon}", "\\ar[rrr]_-{\\gamma_{A \\to B \\to C} \\otimes \\gamma_{G \\to H \\to I}}", "& & &", "\\det\\nolimits_\\kappa(B) \\otimes", "\\det\\nolimits_\\kappa(H)", "\\ar[d]^{\\gamma_{B \\to E \\to H}}", "\\\\", "& & & \\det\\nolimits_\\kappa(E)", "\\\\", "\\det\\nolimits_\\kappa(A) \\otimes", "\\det\\nolimits_\\kappa(G) \\otimes", "\\det\\nolimits_\\kappa(C) \\otimes", "\\det\\nolimits_\\kappa(I)", "\\ar[rrr]^-{\\gamma_{A \\to D \\to G} \\otimes \\gamma_{C \\to F \\to I}}", "& & &", "\\det\\nolimits_\\kappa(D) \\otimes", "\\det\\nolimits_\\kappa(F)", "\\ar[u]_{\\gamma_{D \\to E \\to F}}", "}", "$$", "where $\\epsilon$ is the switch of the factors in the tensor product", "times $(-1)^{cg}$ with $c = \\text{length}_R(C)$ and $g = \\text{length}_R(G)$,", "and", "\\item the map $\\gamma_{K \\to L \\to M}$ agrees with the usual isomorphism", "if $0 \\to K \\to L \\to M \\to 0$ is actually a short exact sequence", "of $\\kappa$-vector spaces.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The significance of taking nonzero symbols in the explicit description", "of the map $\\gamma_{K \\to L \\to M}$ is simply that if $(e_1, \\ldots, e_l)$", "is an admissible sequence in $K$, and", "$(\\overline{f}_1, \\ldots, \\overline{f}_m)$ is an admissible sequence in", "$M$, then it is not guaranteed that $(e_1, \\ldots, e_l, f_1, \\ldots, f_m)$", "is an admissible sequence in $L$ (where of course $f_i \\in L$ signifies", "a lift of $\\overline{f}_i$). However, if the symbol", "$[e_1, \\ldots, e_l]$ is nonzero in $\\det_\\kappa(K)$, then", "necessarily $K = \\langle e_1, \\ldots, e_k\\rangle$ (see", "proof of Lemma \\ref{lemma-dimension-at-most-one}), and", "in this case it is true that $(e_1, \\ldots, e_k, f_1, \\ldots, f_m)$", "is an admissible sequence.", "Moreover, by the admissible relations of type (b) for $\\det_\\kappa(L)$", "we see that the value of $[e_1, \\ldots, e_k, f_1, \\ldots, f_m]$ in", "$\\det_\\kappa(L)$ is independent of the choice of the lifts", "$f_i$ in this case also. Given this remark, it is clear", "that an admissible relation for $e_1, \\ldots, e_k$ in $K$", "translates into an admissible relation among", "$e_1, \\ldots, e_k, f_1, \\ldots, f_m$ in $L$, and", "similarly for an admissible relation among the", "$\\overline{f}_1, \\ldots, \\overline{f}_m$.", "Thus $\\gamma$ defines a linear map of vector spaces as claimed in the lemma.", "\\medskip\\noindent", "By Lemma \\ref{lemma-determinant-dimension-one} we know", "$\\det_\\kappa(L)$ is generated by any single", "symbol $[x_1, \\ldots, x_{k + m}]$ such that", "$(x_1, \\ldots, x_{k + m})$ is an admissible sequence", "with $L = \\langle x_1, \\ldots, x_{k + m}\\rangle$. Hence it is", "clear that the map $\\gamma_{K \\to L \\to M}$ is surjective and", "hence an isomorphism.", "\\medskip\\noindent", "Property (1) holds because", "\\begin{eqnarray*}", "& & \\det\\nolimits_\\kappa(v)([e_1, \\ldots, e_k, f_1, \\ldots, f_m]) \\\\", "& = &", "[v(e_1), \\ldots, v(e_k), v(f_1), \\ldots, v(f_m)] \\\\", "& = &", "\\gamma_{K' \\to L' \\to M'}([u(e_1), \\ldots, u(e_k)]", "\\otimes [w(f_1), \\ldots, w(f_m)]).", "\\end{eqnarray*}", "Property (2) means that given a symbol", "$[\\alpha_1, \\ldots, \\alpha_a]$ generating $\\det_\\kappa(A)$,", "a symbol $[\\gamma_1, \\ldots, \\gamma_c]$ generating $\\det_\\kappa(C)$,", "a symbol $[\\zeta_1, \\ldots, \\zeta_g]$ generating $\\det_\\kappa(G)$, and", "a symbol $[\\iota_1, \\ldots, \\iota_i]$ generating $\\det_\\kappa(I)$", "we have", "\\begin{eqnarray*}", "& & [\\alpha_1, \\ldots, \\alpha_a, \\tilde\\gamma_1, \\ldots, \\tilde\\gamma_c,", "\\tilde\\zeta_1, \\ldots, \\tilde\\zeta_g, \\tilde\\iota_1, \\ldots, \\tilde\\iota_i] \\\\", "& = &", "(-1)^{cg} [\\alpha_1, \\ldots, \\alpha_a, \\tilde\\zeta_1, \\ldots, \\tilde\\zeta_g,", "\\tilde\\gamma_1, \\ldots, \\tilde\\gamma_c, \\tilde\\iota_1, \\ldots, \\tilde\\iota_i]", "\\end{eqnarray*}", "(for suitable lifts $\\tilde{x}$ in $E$) in $\\det_\\kappa(E)$.", "This holds because we may use the admissible relations of type (c)", "$cg$ times in the following order: move the", "$\\tilde\\zeta_1$ past the elements", "$\\tilde\\gamma_c, \\ldots, \\tilde\\gamma_1$", "(allowed since $\\mathfrak m\\tilde\\zeta_1 \\subset A$),", "then move $\\tilde\\zeta_2$ past the elements", "$\\tilde\\gamma_c, \\ldots, \\tilde\\gamma_1$", "(allowed since $\\mathfrak m\\tilde\\zeta_2 \\subset A + R\\tilde\\zeta_1$),", "and so on.", "\\medskip\\noindent", "Part (3) of the lemma is obvious.", "This finishes the proof." ], "refs": [ "chow-lemma-dimension-at-most-one", "chow-lemma-determinant-dimension-one" ], "ref_ids": [ 5867, 5869 ] } ], "ref_ids": [] }, { "id": 5871, "type": "theorem", "label": "chow-lemma-uniqueness-det", "categories": [ "chow" ], "title": "chow-lemma-uniqueness-det", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be any local ring.", "The functor", "$$", "\\det\\nolimits_\\kappa :", "\\left\\{", "\\begin{matrix}", "\\text{finite length }R\\text{-modules} \\\\", "\\text{with isomorphisms}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "1\\text{-dimensional }\\kappa\\text{-vector spaces} \\\\", "\\text{with isomorphisms}", "\\end{matrix}", "\\right\\}", "$$", "endowed with the maps $\\gamma_{K \\to L \\to M}$ is characterized by", "the following properties", "\\begin{enumerate}", "\\item its restriction to the subcategory of modules annihilated", "by $\\mathfrak m$ is isomorphic to the usual determinant functor", "(see Lemma \\ref{lemma-compare-det}), and", "\\item (1), (2) and (3) of Lemma \\ref{lemma-det-exact-sequences}", "hold.", "\\end{enumerate}" ], "refs": [ "chow-lemma-compare-det", "chow-lemma-det-exact-sequences" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 5868, 5870 ] }, { "id": 5872, "type": "theorem", "label": "chow-lemma-determinant-quotient-ring", "categories": [ "chow" ], "title": "chow-lemma-determinant-quotient-ring", "contents": [ "Let $(R', \\mathfrak m') \\to (R, \\mathfrak m)$ be a local ring", "homomorphism which induces an isomorphism on residue fields $\\kappa$.", "Then for every finite length $R$-module the restriction $M_{R'}$", "is a finite length $R'$-module and there is a canonical isomorphism", "$$", "\\det\\nolimits_{R, \\kappa}(M)", "\\longrightarrow", "\\det\\nolimits_{R', \\kappa}(M_{R'})", "$$", "This isomorphism is functorial in $M$ and compatible with the", "isomorphisms $\\gamma_{K \\to L \\to M}$ of Lemma \\ref{lemma-det-exact-sequences}", "defined for $\\det_{R, \\kappa}$ and $\\det_{R', \\kappa}$." ], "refs": [ "chow-lemma-det-exact-sequences" ], "proofs": [ { "contents": [ "If the length of $M$ as an $R$-module is $l$, then the length", "of $M$ as an $R'$-module (i.e., $M_{R'}$) is $l$ as well, see", "Algebra, Lemma \\ref{algebra-lemma-pushdown-module}.", "Note that an admissible sequence $x_1, \\ldots, x_l$ of $M$", "over $R$ is an admissible sequence of $M$ over $R'$ as $\\mathfrak m'$", "maps into $\\mathfrak m$.", "The isomorphism is obtained by mapping the symbol", "$[x_1, \\ldots, x_l] \\in \\det\\nolimits_{R, \\kappa}(M)$", "to the corresponding symbol", "$[x_1, \\ldots, x_l] \\in \\det\\nolimits_{R', \\kappa}(M)$.", "It is immediate to verify that this is functorial for", "isomorphisms and compatible with the isomorphisms", "$\\gamma$ of Lemma \\ref{lemma-det-exact-sequences}." ], "refs": [ "algebra-lemma-pushdown-module", "chow-lemma-det-exact-sequences" ], "ref_ids": [ 639, 5870 ] } ], "ref_ids": [ 5870 ] }, { "id": 5873, "type": "theorem", "label": "chow-lemma-times-u-determinant", "categories": [ "chow" ], "title": "chow-lemma-times-u-determinant", "contents": [ "Let $R$ be a local ring with residue field $\\kappa$.", "Let $u \\in R^*$ be a unit.", "Let $M$ be a module of finite length over $R$.", "Denote $u_M : M \\to M$ the map multiplication by $u$.", "Then", "$$", "\\det\\nolimits_\\kappa(u_M) :", "\\det\\nolimits_\\kappa(M)", "\\longrightarrow", "\\det\\nolimits_\\kappa(M)", "$$", "is multiplication by $\\overline{u}^l$ where $l = \\text{length}_R(M)$", "and $\\overline{u} \\in \\kappa^*$ is the image of $u$." ], "refs": [], "proofs": [ { "contents": [ "Denote $f_M \\in \\kappa^*$ the element such that", "$\\det\\nolimits_\\kappa(u_M) = f_M \\text{id}_{\\det\\nolimits_\\kappa(M)}$.", "Suppose that $0 \\to K \\to L \\to M \\to 0$ is a short", "exact sequence of finite $R$-modules. Then we see that", "$u_k$, $u_L$, $u_M$ give an isomorphism of short exact sequences.", "Hence by Lemma \\ref{lemma-det-exact-sequences} (1) we conclude that", "$f_K f_M = f_L$.", "This means that by induction on length it suffices to prove the", "lemma in the case of length $1$ where it is trivial." ], "refs": [ "chow-lemma-det-exact-sequences" ], "ref_ids": [ 5870 ] } ], "ref_ids": [] }, { "id": 5874, "type": "theorem", "label": "chow-lemma-periodic-determinant-shift", "categories": [ "chow" ], "title": "chow-lemma-periodic-determinant-shift", "contents": [ "Let $R$ be a local ring with residue field $\\kappa$.", "Let $(M, \\varphi, \\psi)$ be a $(2, 1)$-periodic complex over $R$.", "Assume that $M$ has finite length and that $(M, \\varphi, \\psi)$ is", "exact. Then", "$$", "\\det\\nolimits_\\kappa(M, \\varphi, \\psi)", "\\det\\nolimits_\\kappa(M, \\psi, \\varphi)", "= 1.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5875, "type": "theorem", "label": "chow-lemma-periodic-determinant-sign", "categories": [ "chow" ], "title": "chow-lemma-periodic-determinant-sign", "contents": [ "Let $R$ be a local ring with residue field $\\kappa$.", "Let $(M, \\varphi, \\varphi)$ be a $(2, 1)$-periodic complex over $R$.", "Assume that $M$ has finite length and that $(M, \\varphi, \\varphi)$ is", "exact. Then $\\text{length}_R(M) = 2 \\text{length}_R(\\Im(\\varphi))$", "and", "$$", "\\det\\nolimits_\\kappa(M, \\varphi, \\varphi)", "=", "(-1)^{\\text{length}_R(\\Im(\\varphi))}", "=", "(-1)^{\\frac{1}{2}\\text{length}_R(M)}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Follows directly from the sign rule in the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 5876, "type": "theorem", "label": "chow-lemma-periodic-determinant-easy-case", "categories": [ "chow" ], "title": "chow-lemma-periodic-determinant-easy-case", "contents": [ "Let $R$ be a local ring with residue field $\\kappa$.", "Let $M$ be a finite length $R$-module.", "\\begin{enumerate}", "\\item if $\\varphi : M \\to M$ is an isomorphism then", "$\\det_\\kappa(M, \\varphi, 0) = \\det_\\kappa(\\varphi)$.", "\\item if $\\psi : M \\to M$ is an isomorphism then", "$\\det_\\kappa(M, 0, \\psi) = \\det_\\kappa(\\psi)^{-1}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us prove (1). Set $\\psi = 0$. Then we may, with notation", "as above Definition \\ref{definition-periodic-determinant}, identify", "$K_\\varphi = I_\\psi = 0$, $I_\\varphi = K_\\psi = M$.", "With these identifications, the map", "$$", "\\gamma_\\varphi :", "\\kappa \\otimes \\det\\nolimits_\\kappa(M)", "=", "\\det\\nolimits_\\kappa(K_\\varphi)", "\\otimes", "\\det\\nolimits_\\kappa(I_\\varphi)", "\\longrightarrow", "\\det\\nolimits_\\kappa(M)", "$$", "is identified with $\\det_\\kappa(\\varphi^{-1})$. On the other hand the", "map $\\gamma_\\psi$ is identified with the identity map. Hence", "$\\gamma_\\psi \\circ \\sigma \\circ \\gamma_\\varphi^{-1}$ is equal", "to $\\det_\\kappa(\\varphi)$ in this case. Whence the result.", "We omit the proof of (2)." ], "refs": [ "chow-definition-periodic-determinant" ], "ref_ids": [ 5927 ] } ], "ref_ids": [] }, { "id": 5877, "type": "theorem", "label": "chow-lemma-periodic-determinant", "categories": [ "chow" ], "title": "chow-lemma-periodic-determinant", "contents": [ "Let $R$ be a local ring with residue field $\\kappa$.", "Suppose that we have a short exact sequence of", "$(2, 1)$-periodic complexes", "$$", "0 \\to (M_1, \\varphi_1, \\psi_1)", "\\to (M_2, \\varphi_2, \\psi_2)", "\\to (M_3, \\varphi_3, \\psi_3)", "\\to 0", "$$", "with all $M_i$ of finite length, and each $(M_1, \\varphi_1, \\psi_1)$ exact.", "Then", "$$", "\\det\\nolimits_\\kappa(M_2, \\varphi_2, \\psi_2) =", "\\det\\nolimits_\\kappa(M_1, \\varphi_1, \\psi_1)", "\\det\\nolimits_\\kappa(M_3, \\varphi_3, \\psi_3).", "$$", "in $\\kappa^*$." ], "refs": [], "proofs": [ { "contents": [ "Let us abbreviate", "$I_{\\varphi, i} = \\Im(\\varphi_i)$,", "$K_{\\varphi, i} = \\Ker(\\varphi_i)$,", "$I_{\\psi, i} = \\Im(\\psi_i)$, and", "$K_{\\psi, i} = \\Ker(\\psi_i)$.", "Observe that we have a commutative square", "$$", "\\xymatrix{", "& 0 \\ar[d] & 0 \\ar[d] & 0 \\ar[d] & \\\\", "0 \\ar[r] &", "K_{\\varphi, 1} \\ar[r] \\ar[d] &", "K_{\\varphi, 2} \\ar[r] \\ar[d] &", "K_{\\varphi, 3} \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "M_1 \\ar[r] \\ar[d] &", "M_2 \\ar[r] \\ar[d] &", "M_3 \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "I_{\\varphi, 1} \\ar[r] \\ar[d] &", "I_{\\varphi, 2} \\ar[r] \\ar[d] &", "I_{\\varphi, 3} \\ar[r] \\ar[d] &", "0 \\\\", "& 0 & 0 & 0 &", "}", "$$", "of finite length $R$-modules with exact rows and columns.", "The top row is exact since it can be identified with the", "sequence $I_{\\psi, 1} \\to I_{\\psi, 2} \\to I_{\\psi, 3} \\to 0$", "of images, and similarly for the bottom row. There is a similar diagram", "involving the modules $I_{\\psi, i}$ and $K_{\\psi, i}$.", "By definition $\\det_\\kappa(M_2, \\varphi_2, \\psi_2)$", "corresponds, up to a sign, to the composition of the left vertical maps", "in the following diagram", "$$", "\\xymatrix{", "\\det_\\kappa(M_1) \\otimes", "\\det_\\kappa(M_3) \\ar[r]^\\gamma", "\\ar[d]^{\\gamma^{-1} \\otimes \\gamma^{-1}} &", "\\det_\\kappa(M_2)", "\\ar[d]^{\\gamma^{-1}} \\\\", "\\det\\nolimits_\\kappa(K_{\\varphi, 1})", "\\otimes", "\\det\\nolimits_\\kappa(I_{\\varphi, 1})", "\\otimes", "\\det\\nolimits_\\kappa(K_{\\varphi, 3})", "\\otimes", "\\det\\nolimits_\\kappa(I_{\\varphi, 3})", "\\ar[d]^{\\sigma \\otimes \\sigma}", "\\ar[r]^-{\\gamma \\otimes \\gamma} &", "\\det\\nolimits_\\kappa(K_{\\varphi, 2})", "\\otimes", "\\det\\nolimits_\\kappa(I_{\\varphi, 2})", "\\ar[d]^\\sigma", "\\\\", "\\det\\nolimits_\\kappa(K_{\\psi, 1})", "\\otimes", "\\det\\nolimits_\\kappa(I_{\\psi, 1})", "\\otimes", "\\det\\nolimits_\\kappa(K_{\\psi, 3})", "\\otimes", "\\det\\nolimits_\\kappa(I_{\\psi, 3})", "\\ar[d]^{\\gamma \\otimes \\gamma}", "\\ar[r]^-{\\gamma \\otimes \\gamma}", "&", "\\det\\nolimits_\\kappa(K_{\\psi, 2})", "\\otimes", "\\det\\nolimits_\\kappa(I_{\\psi, 2})", "\\ar[d]^\\gamma \\\\", "\\det_\\kappa(M_1)", "\\otimes", "\\det_\\kappa(M_3) \\ar[r]^\\gamma", "&", "\\det_\\kappa(M_2)", "}", "$$", "The top and bottom squares are commutative up to sign", "by applying Lemma \\ref{lemma-det-exact-sequences} (2).", "The middle square is trivially", "commutative (we are just switching factors). Hence we see", "that", "$\\det\\nolimits_\\kappa(M_2, \\varphi_2, \\psi_2) =", "\\epsilon \\det\\nolimits_\\kappa(M_1, \\varphi_1, \\psi_1)", "\\det\\nolimits_\\kappa(M_3, \\varphi_3, \\psi_3)", "$", "for some sign $\\epsilon$. And the sign can be worked out, namely", "the outer rectangle in the diagram above commutes up to", "\\begin{eqnarray*}", "\\epsilon & = &", "(-1)^{\\text{length}(I_{\\varphi, 1})\\text{length}(K_{\\varphi, 3})", "+ \\text{length}(I_{\\psi, 1})\\text{length}(K_{\\psi, 3})} \\\\", "& = &", "(-1)^{\\text{length}(I_{\\varphi, 1})\\text{length}(I_{\\psi, 3})", "+ \\text{length}(I_{\\psi, 1})\\text{length}(I_{\\varphi, 3})}", "\\end{eqnarray*}", "(proof omitted). It follows easily from this that the signs", "work out as well." ], "refs": [ "chow-lemma-det-exact-sequences" ], "ref_ids": [ 5870 ] } ], "ref_ids": [] }, { "id": 5878, "type": "theorem", "label": "chow-lemma-multiplicativity-determinant", "categories": [ "chow" ], "title": "chow-lemma-multiplicativity-determinant", "contents": [ "Let $R$ be a local ring with residue field $\\kappa$.", "Let $M$ be a finite length $R$-module.", "Let $\\alpha, \\beta, \\gamma$ be endomorphisms of $M$.", "Assume that", "\\begin{enumerate}", "\\item $I_\\alpha = K_{\\beta\\gamma}$, and similarly for any permutation", "of $\\alpha, \\beta, \\gamma$,", "\\item $K_\\alpha = I_{\\beta\\gamma}$, and similarly for any permutation", "of $\\alpha, \\beta, \\gamma$.", "\\end{enumerate}", "Then", "\\begin{enumerate}", "\\item The triple $(M, \\alpha, \\beta\\gamma)$", "is an exact $(2, 1)$-periodic complex.", "\\item The triple $(I_\\gamma, \\alpha, \\beta)$", "is an exact $(2, 1)$-periodic complex.", "\\item The triple $(M/K_\\beta, \\alpha, \\gamma)$", "is an exact $(2, 1)$-periodic complex.", "\\item We have", "$$", "\\det\\nolimits_\\kappa(M, \\alpha, \\beta\\gamma)", "=", "\\det\\nolimits_\\kappa(I_\\gamma, \\alpha, \\beta)", "\\det\\nolimits_\\kappa(M/K_\\beta, \\alpha, \\gamma).", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that the assumptions imply part (1) of the lemma.", "\\medskip\\noindent", "To see part (1) note that the assumptions imply that", "$I_{\\gamma\\alpha} = I_{\\alpha\\gamma}$, and similarly for kernels", "and any other pair of morphisms.", "Moreover, we see that", "$I_{\\gamma\\beta} =I_{\\beta\\gamma} = K_\\alpha \\subset I_\\gamma$ and", "similarly for any other pair. In particular we get a short exact sequence", "$$", "0 \\to I_{\\beta\\gamma} \\to I_\\gamma \\xrightarrow{\\alpha} I_{\\alpha\\gamma} \\to 0", "$$", "and similarly we get a short exact sequence", "$$", "0 \\to I_{\\alpha\\gamma} \\to I_\\gamma \\xrightarrow{\\beta} I_{\\beta\\gamma} \\to 0.", "$$", "This proves $(I_\\gamma, \\alpha, \\beta)$ is an exact $(2, 1)$-periodic", "complex. Hence part (2) of the lemma holds.", "\\medskip\\noindent", "To see that $\\alpha$, $\\gamma$ give well defined endomorphisms", "of $M/K_\\beta$ we have to check that $\\alpha(K_\\beta) \\subset K_\\beta$", "and $\\gamma(K_\\beta) \\subset K_\\beta$. This is true because", "$\\alpha(K_\\beta) = \\alpha(I_{\\gamma\\alpha}) = I_{\\alpha\\gamma\\alpha}", "\\subset I_{\\alpha\\gamma} = K_\\beta$, and similarly in the other case.", "The kernel of the map $\\alpha : M/K_\\beta \\to M/K_\\beta$ is", "$K_{\\beta\\alpha}/K_\\beta = I_\\gamma/K_\\beta$. Similarly,", "the kernel of $\\gamma : M/K_\\beta \\to M/K_\\beta$ is equal to", "$I_\\alpha/K_\\beta$. Hence we conclude that (3) holds.", "\\medskip\\noindent", "We introduce $r = \\text{length}_R(K_\\alpha)$,", "$s = \\text{length}_R(K_\\beta)$ and $t = \\text{length}_R(K_\\gamma)$.", "By the exact sequences above and our hypotheses we have", "$\\text{length}_R(I_\\alpha) = s + t$, $\\text{length}_R(I_\\beta) = r + t$,", "$\\text{length}_R(I_\\gamma) = r + s$, and", "$\\text{length}(M) = r + s + t$.", "Choose", "\\begin{enumerate}", "\\item an admissible sequence $x_1, \\ldots, x_r \\in K_\\alpha$", "generating $K_\\alpha$", "\\item an admissible sequence $y_1, \\ldots, y_s \\in K_\\beta$", "generating $K_\\beta$,", "\\item an admissible sequence $z_1, \\ldots, z_t \\in K_\\gamma$", "generating $K_\\gamma$,", "\\item elements $\\tilde x_i \\in M$ such that $\\beta\\gamma\\tilde x_i = x_i$,", "\\item elements $\\tilde y_i \\in M$ such that $\\alpha\\gamma\\tilde y_i = y_i$,", "\\item elements $\\tilde z_i \\in M$ such that $\\beta\\alpha\\tilde z_i = z_i$.", "\\end{enumerate}", "With these choices the sequence", "$y_1, \\ldots, y_s, \\alpha\\tilde z_1, \\ldots, \\alpha\\tilde z_t$", "is an admissible sequence in $I_\\alpha$ generating it.", "Hence, by Remark \\ref{remark-more-elementary} the determinant", "$D = \\det_\\kappa(M, \\alpha, \\beta\\gamma)$ is the", "unique element of $\\kappa^*$ such that", "\\begin{align*}", "[y_1, \\ldots, y_s,", "\\alpha\\tilde z_1, \\ldots, \\alpha\\tilde z_s,", "\\tilde x_1, \\ldots, \\tilde x_r] \\\\", "= (-1)^{r(s + t)} D", "[x_1, \\ldots, x_r,", "\\gamma\\tilde y_1, \\ldots, \\gamma\\tilde y_s,", "\\tilde z_1, \\ldots, \\tilde z_t]", "\\end{align*}", "By the same remark, we see that", "$D_1 = \\det_\\kappa(M/K_\\beta, \\alpha, \\gamma)$", "is characterized by", "$$", "[y_1, \\ldots, y_s,", "\\alpha\\tilde z_1, \\ldots, \\alpha\\tilde z_t,", "\\tilde x_1, \\ldots, \\tilde x_r]", "=", "(-1)^{rt} D_1", "[y_1, \\ldots, y_s,", "\\gamma\\tilde x_1, \\ldots, \\gamma\\tilde x_r,", "\\tilde z_1, \\ldots, \\tilde z_t]", "$$", "By the same remark, we see that", "$D_2 = \\det_\\kappa(I_\\gamma, \\alpha, \\beta)$ is characterized by", "$$", "[y_1, \\ldots, y_s,", "\\gamma\\tilde x_1, \\ldots, \\gamma\\tilde x_r,", "\\tilde z_1, \\ldots, \\tilde z_t]", "=", "(-1)^{rs} D_2", "[x_1, \\ldots, x_r,", "\\gamma\\tilde y_1, \\ldots, \\gamma\\tilde y_s,", "\\tilde z_1, \\ldots, \\tilde z_t]", "$$", "Combining the formulas above we see that $D = D_1 D_2$", "as desired." ], "refs": [ "chow-remark-more-elementary" ], "ref_ids": [ 5967 ] } ], "ref_ids": [] }, { "id": 5879, "type": "theorem", "label": "chow-lemma-tricky", "categories": [ "chow" ], "title": "chow-lemma-tricky", "contents": [ "Let $R$ be a local ring with residue field $\\kappa$.", "Let $\\alpha : (M, \\varphi, \\psi) \\to (M', \\varphi', \\psi')$", "be a morphism of $(2, 1)$-periodic complexes over $R$.", "Assume", "\\begin{enumerate}", "\\item $M$, $M'$ have finite length,", "\\item $(M, \\varphi, \\psi)$, $(M', \\varphi', \\psi')$ are exact,", "\\item the maps $\\varphi$, $\\psi$ induce the zero map on", "$K = \\Ker(\\alpha)$, and", "\\item the maps $\\varphi$, $\\psi$ induce the zero map on", "$Q = \\Coker(\\alpha)$.", "\\end{enumerate}", "Denote $N = \\alpha(M) \\subset M'$. We obtain two short exact sequences", "of $(2, 1)$-periodic complexes", "$$", "\\begin{matrix}", "0 \\to (N, \\varphi', \\psi') \\to (M', \\varphi', \\psi') \\to (Q, 0, 0) \\to 0 \\\\", "0 \\to (K, 0, 0) \\to (M, \\varphi, \\psi) \\to (N, \\varphi', \\psi') \\to 0", "\\end{matrix}", "$$", "which induce two isomorphisms $\\alpha_i : Q \\to K$, $i = 0, 1$. Then", "$$", "\\det\\nolimits_\\kappa(M, \\varphi, \\psi)", "=", "\\det\\nolimits_\\kappa(\\alpha_0^{-1} \\circ \\alpha_1)", "\\det\\nolimits_\\kappa(M', \\varphi', \\psi')", "$$", "In particular, if $\\alpha_0 = \\alpha_1$, then", "$\\det\\nolimits_\\kappa(M, \\varphi, \\psi) =", "\\det\\nolimits_\\kappa(M', \\varphi', \\psi')$." ], "refs": [], "proofs": [ { "contents": [ "There are (at least) two ways to prove this lemma. One is to produce an", "enormous commutative diagram using the properties of the determinants.", "The other is to use the characterization of the determinants in terms", "of admissible sequences of elements. It is the second approach that we", "will use.", "\\medskip\\noindent", "First let us explain precisely what the maps $\\alpha_i$ are.", "Namely, $\\alpha_0$ is the composition", "$$", "\\alpha_0 : Q = H^0(Q, 0, 0) \\to H^1(N, \\varphi', \\psi') \\to H^2(K, 0, 0) = K", "$$", "and $\\alpha_1$ is the composition", "$$", "\\alpha_1 : Q = H^1(Q, 0, 0) \\to H^2(N, \\varphi', \\psi') \\to H^3(K, 0, 0) = K", "$$", "coming from the boundary maps of the short exact sequences of complexes", "displayed in the lemma. The fact that the", "complexes $(M, \\varphi, \\psi)$, $(M', \\varphi', \\psi')$ are exact", "implies these maps are isomorphisms.", "\\medskip\\noindent", "We will use the notation $I_\\varphi = \\Im(\\varphi)$,", "$K_\\varphi = \\Ker(\\varphi)$ and similarly for the other maps.", "Exactness for $M$ and $M'$", "means that $K_\\varphi = I_\\psi$ and three similar equalities.", "We introduce $k = \\text{length}_R(K)$, $a = \\text{length}_R(I_\\varphi)$,", "$b = \\text{length}_R(I_\\psi)$. Then we see that $\\text{length}_R(M) = a + b$,", "and $\\text{length}_R(N) = a + b - k$, $\\text{length}_R(Q) = k$", "and $\\text{length}_R(M') = a + b$. The exact sequences below will show", "that also $\\text{length}_R(I_{\\varphi'}) = a$ and", "$\\text{length}_R(I_{\\psi'}) = b$.", "\\medskip\\noindent", "The assumption that $K \\subset K_\\varphi = I_\\psi$ means that", "$\\varphi$ factors through $N$ to give an exact sequence", "$$", "0 \\to \\alpha(I_\\psi) \\to N \\xrightarrow{\\varphi\\alpha^{-1}} I_\\psi \\to 0.", "$$", "Here $\\varphi\\alpha^{-1}(x') = y$ means $x' = \\alpha(x)$ and $y = \\varphi(x)$.", "Similarly, we have", "$$", "0 \\to \\alpha(I_\\varphi) \\to N \\xrightarrow{\\psi\\alpha^{-1}} I_\\varphi \\to 0.", "$$", "The assumption that $\\psi'$ induces the zero map on", "$Q$ means that $I_{\\psi'} = K_{\\varphi'} \\subset N$.", "This means the quotient $\\varphi'(N) \\subset I_{\\varphi'}$", "is identified with $Q$. Note that $\\varphi'(N) = \\alpha(I_\\varphi)$.", "Hence we conclude there is an isomorphism", "$$", "\\varphi' : Q \\to I_{\\varphi'}/\\alpha(I_\\varphi)", "$$", "simply described by", "$\\varphi'(x' \\bmod N) = \\varphi'(x') \\bmod \\alpha(I_\\varphi)$.", "In exactly the same way we get", "$$", "\\psi' : Q \\to I_{\\psi'}/\\alpha(I_\\psi)", "$$", "Finally, note that $\\alpha_0$ is the composition", "$$", "\\xymatrix{", "Q \\ar[r]^-{\\varphi'} &", "I_{\\varphi'}/\\alpha(I_\\varphi)", "\\ar[rrr]^-{\\psi\\alpha^{-1}|_{I_{\\varphi'}/\\alpha(I_\\varphi)}} & & &", "K", "}", "$$", "and similarly", "$\\alpha_1 = \\varphi\\alpha^{-1}|_{I_{\\psi'}/\\alpha(I_\\psi)} \\circ \\psi'$.", "\\medskip\\noindent", "To shorten the formulas below we are going to write $\\alpha x$ instead", "of $\\alpha(x)$ in the following. No confusion should result since", "all maps are indicated by Greek letters and elements by Roman letters.", "We are going to choose", "\\begin{enumerate}", "\\item an admissible sequence $z_1, \\ldots, z_k \\in K$", "generating $K$,", "\\item elements $z'_i \\in M$ such that $\\varphi z'_i = z_i$,", "\\item elements $z''_i \\in M$ such that $\\psi z''_i = z_i$,", "\\item elements $x_{k + 1}, \\ldots, x_a \\in I_\\varphi$ such", "that $z_1, \\ldots, z_k, x_{k + 1}, \\ldots, x_a$ is an admissible", "sequence generating $I_\\varphi$,", "\\item elements $\\tilde x_i \\in M$ such that $\\varphi \\tilde x_i = x_i$,", "\\item elements $y_{k + 1}, \\ldots, y_b \\in I_\\psi$ such that", "$z_1, \\ldots, z_k, y_{k + 1}, \\ldots, y_b$ is an admissible", "sequence generating $I_\\psi$,", "\\item elements $\\tilde y_i \\in M$ such that $\\psi \\tilde y_i = y_i$, and", "\\item elements $w_1, \\ldots, w_k \\in M'$ such that", "$w_1 \\bmod N, \\ldots, w_k \\bmod N$ are an admissible sequence", "in $Q$ generating $Q$.", "\\end{enumerate}", "By Remark \\ref{remark-more-elementary} the element", "$D = \\det_\\kappa(M, \\varphi, \\psi) \\in \\kappa^*$ is", "characterized by", "\\begin{eqnarray*}", "& &", "[z_1, \\ldots, z_k,", "x_{k + 1}, \\ldots, x_a,", "z''_1, \\ldots, z''_k,", "\\tilde y_{k + 1}, \\ldots, \\tilde y_b] \\\\", "& = &", "(-1)^{ab} D", "[z_1, \\ldots, z_k,", "y_{k + 1}, \\ldots, y_b,", "z'_1, \\ldots, z'_k,", "\\tilde x_{k + 1}, \\ldots, \\tilde x_a]", "\\end{eqnarray*}", "Note that by the discussion above", "$\\alpha x_{k + 1}, \\ldots, \\alpha x_a, \\varphi w_1, \\ldots, \\varphi w_k$", "is an admissible sequence generating $I_{\\varphi'}$ and", "$\\alpha y_{k + 1}, \\ldots, \\alpha y_b, \\psi w_1, \\ldots, \\psi w_k$", "is an admissible sequence generating $I_{\\psi'}$.", "Hence by Remark \\ref{remark-more-elementary} the element", "$D' = \\det_\\kappa(M', \\varphi', \\psi') \\in \\kappa^*$ is", "characterized by", "\\begin{eqnarray*}", "& &", "[\\alpha x_{k + 1}, \\ldots, \\alpha x_a,", "\\varphi' w_1, \\ldots, \\varphi' w_k,", "\\alpha \\tilde y_{k + 1}, \\ldots, \\alpha \\tilde y_b,", "w_1, \\ldots, w_k]", "\\\\", "& = &", "(-1)^{ab} D'", "[\\alpha y_{k + 1}, \\ldots, \\alpha y_b,", "\\psi' w_1, \\ldots, \\psi' w_k,", "\\alpha \\tilde x_{k + 1}, \\ldots, \\alpha \\tilde x_a,", "w_1, \\ldots, w_k]", "\\end{eqnarray*}", "Note how in the first, resp.\\ second displayed formula", "the first, resp.\\ last $k$ entries of the symbols on both sides", "are the same. Hence these formulas are really equivalent to the", "equalities", "\\begin{eqnarray*}", "& &", "[\\alpha x_{k + 1}, \\ldots, \\alpha x_a,", "\\alpha z''_1, \\ldots, \\alpha z''_k,", "\\alpha \\tilde y_{k + 1}, \\ldots, \\alpha \\tilde y_b] \\\\", "& = &", "(-1)^{ab} D", "[\\alpha y_{k + 1}, \\ldots, \\alpha y_b,", "\\alpha z'_1, \\ldots, \\alpha z'_k,", "\\alpha \\tilde x_{k + 1}, \\ldots, \\alpha \\tilde x_a]", "\\end{eqnarray*}", "and", "\\begin{eqnarray*}", "& &", "[\\alpha x_{k + 1}, \\ldots, \\alpha x_a,", "\\varphi' w_1, \\ldots, \\varphi' w_k,", "\\alpha \\tilde y_{k + 1}, \\ldots, \\alpha \\tilde y_b]", "\\\\", "& = &", "(-1)^{ab} D'", "[\\alpha y_{k + 1}, \\ldots, \\alpha y_b,", "\\psi' w_1, \\ldots, \\psi' w_k,", "\\alpha \\tilde x_{k + 1}, \\ldots, \\alpha \\tilde x_a]", "\\end{eqnarray*}", "in $\\det_\\kappa(N)$. Note that", "$\\varphi' w_1, \\ldots, \\varphi' w_k$", "and", "$\\alpha z''_1, \\ldots, z''_k$", "are admissible sequences generating the module", "$I_{\\varphi'}/\\alpha(I_\\varphi)$. Write", "$$", "[\\varphi' w_1, \\ldots, \\varphi' w_k]", "= \\lambda_0 [\\alpha z''_1, \\ldots, \\alpha z''_k]", "$$", "in $\\det_\\kappa(I_{\\varphi'}/\\alpha(I_\\varphi))$", "for some $\\lambda_0 \\in \\kappa^*$. Similarly,", "write", "$$", "[\\psi' w_1, \\ldots, \\psi' w_k]", "= \\lambda_1 [\\alpha z'_1, \\ldots, \\alpha z'_k]", "$$", "in $\\det_\\kappa(I_{\\psi'}/\\alpha(I_\\psi))$", "for some $\\lambda_1 \\in \\kappa^*$. On the one hand", "it is clear that", "$$", "\\alpha_i([w_1, \\ldots, w_k]) = \\lambda_i[z_1, \\ldots, z_k]", "$$", "for $i = 0, 1$ by our description of $\\alpha_i$ above,", "which means that", "$$", "\\det\\nolimits_\\kappa(\\alpha_0^{-1} \\circ \\alpha_1)", "=", "\\lambda_1/\\lambda_0", "$$", "and", "on the other hand it is clear that", "\\begin{eqnarray*}", "& &", "\\lambda_0 [\\alpha x_{k + 1}, \\ldots, \\alpha x_a,", "\\alpha z''_1, \\ldots, \\alpha z''_k,", "\\alpha \\tilde y_{k + 1}, \\ldots, \\alpha \\tilde y_b] \\\\", "& = &", "[\\alpha x_{k + 1}, \\ldots, \\alpha x_a,", "\\varphi' w_1, \\ldots, \\varphi' w_k,", "\\alpha \\tilde y_{k + 1}, \\ldots, \\alpha \\tilde y_b]", "\\end{eqnarray*}", "and", "\\begin{eqnarray*}", "& &", "\\lambda_1[\\alpha y_{k + 1}, \\ldots, \\alpha y_b,", "\\alpha z'_1, \\ldots, \\alpha z'_k,", "\\alpha \\tilde x_{k + 1}, \\ldots, \\alpha \\tilde x_a] \\\\", "& = &", "[\\alpha y_{k + 1}, \\ldots, \\alpha y_b,", "\\psi' w_1, \\ldots, \\psi' w_k,", "\\alpha \\tilde x_{k + 1}, \\ldots, \\alpha \\tilde x_a]", "\\end{eqnarray*}", "which imply $\\lambda_0 D = \\lambda_1 D'$. The lemma follows." ], "refs": [ "chow-remark-more-elementary", "chow-remark-more-elementary" ], "ref_ids": [ 5967, 5967 ] } ], "ref_ids": [] }, { "id": 5880, "type": "theorem", "label": "chow-lemma-pre-symbol", "categories": [ "chow" ], "title": "chow-lemma-pre-symbol", "contents": [ "Let $A$ be a Noetherian local ring.", "Let $M$ be a finite $A$-module of dimension $1$.", "Assume $\\varphi, \\psi : M \\to M$ are two injective", "$A$-module maps, and assume $\\varphi(\\psi(M)) = \\psi(\\varphi(M))$,", "for example if $\\varphi$ and $\\psi$ commute.", "Then $\\text{length}_R(M/\\varphi\\psi M) < \\infty$", "and $(M/\\varphi\\psi M, \\varphi, \\psi)$ is an exact", "$(2, 1)$-periodic complex." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q$ be a minimal prime of the support of $M$.", "Then $M_{\\mathfrak q}$ is a finite length $A_{\\mathfrak q}$-module,", "see Algebra, Lemma \\ref{algebra-lemma-support-point}.", "Hence both $\\varphi$ and $\\psi$", "induce isomorphisms $M_{\\mathfrak q} \\to M_{\\mathfrak q}$.", "Thus the support of $M/\\varphi\\psi M$ is $\\{\\mathfrak m_A\\}$", "and hence it has finite length (see lemma cited above).", "Finally, the kernel of $\\varphi$ on $M/\\varphi\\psi M$", "is clearly $\\psi M/\\varphi\\psi M$, and hence the kernel", "of $\\varphi$ is the image of $\\psi$ on $M/\\varphi\\psi M$.", "Similarly the other way since $M/\\varphi\\psi M = M/\\psi\\varphi M$", "by assumption." ], "refs": [ "algebra-lemma-support-point" ], "ref_ids": [ 693 ] } ], "ref_ids": [] }, { "id": 5881, "type": "theorem", "label": "chow-lemma-symbol-defined", "categories": [ "chow" ], "title": "chow-lemma-symbol-defined", "contents": [ "Let $A$ be a Noetherian local ring. Let $a, b \\in A$.", "\\begin{enumerate}", "\\item If $M$ is a finite $A$-module of dimension $1$", "such that $a, b$ are nonzerodivisors on $M$, then", "$\\text{length}_A(M/abM) < \\infty$ and", "$(M/abM, a, b)$ is a $(2, 1)$-periodic exact complex.", "\\item If $a, b$ are nonzerodivisors and $\\dim(A) = 1$", "then $\\text{length}_A(A/(ab)) < \\infty$ and", "$(A/(ab), a, b)$ is a $(2, 1)$-periodic exact complex.", "\\end{enumerate}", "In particular, in these cases", "$\\det_\\kappa(M/abM, a, b) \\in \\kappa^*$,", "resp.\\ $\\det_\\kappa(A/(ab), a, b) \\in \\kappa^*$", "are defined." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-pre-symbol}." ], "refs": [ "chow-lemma-pre-symbol" ], "ref_ids": [ 5880 ] } ], "ref_ids": [] }, { "id": 5882, "type": "theorem", "label": "chow-lemma-multiplicativity-symbol", "categories": [ "chow" ], "title": "chow-lemma-multiplicativity-symbol", "contents": [ "Let $A$ be a Noetherian local ring.", "Let $a, b, c \\in A$. Let $M$ be a finite $A$-module", "with $\\dim(\\text{Supp}(M)) = 1$. Assume $a, b, c$ are nonzerodivisors on $M$.", "Then", "$$", "d_M(a, bc) = d_M(a, b) d_M(a, c)", "$$", "and $d_M(a, b)d_M(b, a) = 1$." ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from Lemma \\ref{lemma-multiplicativity-determinant}", "applied to $M/abcM$ and endomorphisms $\\alpha, \\beta, \\gamma$ given by", "multiplication by $a, b, c$.", "The second comes from Lemma \\ref{lemma-periodic-determinant-shift}." ], "refs": [ "chow-lemma-multiplicativity-determinant", "chow-lemma-periodic-determinant-shift" ], "ref_ids": [ 5878, 5874 ] } ], "ref_ids": [] }, { "id": 5883, "type": "theorem", "label": "chow-lemma-symbol-when-equal", "categories": [ "chow" ], "title": "chow-lemma-symbol-when-equal", "contents": [ "Let $A$ be a Noetherian local ring and $M$ a finite $A$-module of", "dimension $1$. Let $a \\in A$ be a nonzerodivisor on $M$.", "Then $d_M(a, a) = (-1)^{\\text{length}_A(M/aM)}$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-periodic-determinant-sign}." ], "refs": [ "chow-lemma-periodic-determinant-sign" ], "ref_ids": [ 5875 ] } ], "ref_ids": [] }, { "id": 5884, "type": "theorem", "label": "chow-lemma-symbol-when-one-is-a-unit", "categories": [ "chow" ], "title": "chow-lemma-symbol-when-one-is-a-unit", "contents": [ "Let $A$ be a Noetherian local ring.", "Let $M$ be a finite $A$-module of dimension $1$.", "Let $b \\in A$ be a nonzerodivisor on $M$, and let $u \\in A^*$.", "Then", "$$", "d_M(u, b) = u^{\\text{length}_A(M/bM)} \\bmod \\mathfrak m_A.", "$$", "In particular, if $M = A$, then", "$d_A(u, b) = u^{\\text{ord}_A(b)} \\bmod \\mathfrak m_A$." ], "refs": [], "proofs": [ { "contents": [ "Note that in this case $M/ubM = M/bM$ on which multiplication", "by $b$ is zero. Hence $d_M(u, b) = \\det_\\kappa(u|_{M/bM})$", "by Lemma \\ref{lemma-periodic-determinant-easy-case}. The lemma", "then follows from Lemma \\ref{lemma-times-u-determinant}." ], "refs": [ "chow-lemma-periodic-determinant-easy-case", "chow-lemma-times-u-determinant" ], "ref_ids": [ 5876, 5873 ] } ], "ref_ids": [] }, { "id": 5885, "type": "theorem", "label": "chow-lemma-symbol-short-exact-sequence", "categories": [ "chow" ], "title": "chow-lemma-symbol-short-exact-sequence", "contents": [ "Let $A$ be a Noetherian local ring.", "Let $a, b \\in A$.", "Let", "$$", "0 \\to M \\to M' \\to M'' \\to 0", "$$", "be a short exact sequence of $A$-modules of dimension $1$", "such that $a, b$ are nonzerodivisors on", "all three $A$-modules.", "Then", "$$", "d_{M'}(a, b) = d_M(a, b) d_{M''}(a, b)", "$$", "in $\\kappa^*$." ], "refs": [], "proofs": [ { "contents": [ "It is easy to see that this leads to a short exact sequence", "of exact $(2, 1)$-periodic complexes", "$$", "0 \\to", "(M/abM, a, b) \\to", "(M'/abM', a, b) \\to", "(M''/abM'', a, b) \\to 0", "$$", "Hence the lemma follows from Lemma \\ref{lemma-periodic-determinant}." ], "refs": [ "chow-lemma-periodic-determinant" ], "ref_ids": [ 5877 ] } ], "ref_ids": [] }, { "id": 5886, "type": "theorem", "label": "chow-lemma-symbol-compare-modules", "categories": [ "chow" ], "title": "chow-lemma-symbol-compare-modules", "contents": [ "Let $A$ be a Noetherian local ring.", "Let $\\alpha : M \\to M'$ be a homomorphism of", "finite $A$-modules of dimension $1$.", "Let $a, b \\in A$. Assume", "\\begin{enumerate}", "\\item $a$, $b$ are nonzerodivisors on both $M$ and $M'$, and", "\\item $\\dim(\\Ker(\\alpha)), \\dim(\\Coker(\\alpha)) \\leq 0$.", "\\end{enumerate}", "Then $d_M(a, b) = d_{M'}(a, b)$." ], "refs": [], "proofs": [ { "contents": [ "If $a \\in A^*$, then the equality follows from the", "equality $\\text{length}(M/bM) = \\text{length}(M'/bM')$", "and Lemma \\ref{lemma-symbol-when-one-is-a-unit}.", "Similarly if $b$ is a unit the lemma holds as well", "(by the symmetry of Lemma \\ref{lemma-multiplicativity-symbol}).", "Hence we may assume that $a, b \\in \\mathfrak m_A$.", "This in particular implies that $\\mathfrak m$ is not", "an associated prime of $M$, and hence $\\alpha : M \\to M'$", "is injective. This permits us to think of $M$ as a submodule of $M'$.", "By assumption $M'/M$ is a finite $A$-module with support", "$\\{\\mathfrak m_A\\}$ and hence has finite length.", "Note that for any third module $M''$ with $M \\subset M'' \\subset M'$", "the maps $M \\to M''$ and $M'' \\to M'$ satisfy the assumptions of the lemma", "as well. This reduces us, by induction on the length of $M'/M$,", "to the case where $\\text{length}_A(M'/M) = 1$.", "Finally, in this case consider the map", "$$", "\\overline{\\alpha} : M/abM \\longrightarrow M'/abM'.", "$$", "By construction the cokernel $Q$ of $\\overline{\\alpha}$ has", "length $1$. Since $a, b \\in \\mathfrak m_A$, they act trivially on", "$Q$. It also follows that the kernel $K$ of $\\overline{\\alpha}$ has", "length $1$ and hence also $a$, $b$ act trivially on $K$.", "Hence we may apply Lemma \\ref{lemma-tricky}. Thus it suffices to see", "that the two maps $\\alpha_i : Q \\to K$ are the same.", "In fact, both maps are equal to the map", "$q = x' \\bmod \\Im(\\overline{\\alpha}) \\mapsto abx' \\in K$.", "We omit the verification." ], "refs": [ "chow-lemma-symbol-when-one-is-a-unit", "chow-lemma-multiplicativity-symbol", "chow-lemma-tricky" ], "ref_ids": [ 5884, 5882, 5879 ] } ], "ref_ids": [] }, { "id": 5887, "type": "theorem", "label": "chow-lemma-compute-symbol-M", "categories": [ "chow" ], "title": "chow-lemma-compute-symbol-M", "contents": [ "Let $A$ be a Noetherian local ring.", "Let $M$ be a finite $A$-module with $\\dim(\\text{Supp}(M)) = 1$.", "Let $a, b \\in A$ nonzerodivisors on $M$.", "Let $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ be the minimal", "primes in the support of $M$. Then", "$$", "d_M(a, b)", "=", "\\prod\\nolimits_{i = 1, \\ldots, t}", "d_{A/\\mathfrak q_i}(a, b)^{", "\\text{length}_{A_{\\mathfrak q_i}}(M_{\\mathfrak q_i})}", "$$", "as elements of $\\kappa^*$." ], "refs": [], "proofs": [ { "contents": [ "Choose a filtration by $A$-submodules", "$$", "0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M", "$$", "such that each quotient $M_j/M_{j - 1}$ is isomorphic", "to $A/\\mathfrak p_j$ for some prime ideal $\\mathfrak p_j$", "of $A$. See Algebra, Lemma \\ref{algebra-lemma-filter-Noetherian-module}.", "For each $j$ we have either $\\mathfrak p_j = \\mathfrak q_i$", "for some $i$, or $\\mathfrak p_j = \\mathfrak m_A$. Moreover,", "for a fixed $i$, the number of $j$ such that", "$\\mathfrak p_j = \\mathfrak q_i$ is equal to", "$\\text{length}_{A_{\\mathfrak q_i}}(M_{\\mathfrak q_i})$ by", "Algebra, Lemma \\ref{algebra-lemma-filter-minimal-primes-in-support}.", "Hence $d_{M_j}(a, b)$ is defined for each $j$ and", "$$", "d_{M_j}(a, b)", "=", "\\left\\{", "\\begin{matrix}", "d_{M_{j - 1}}(a, b) d_{A/\\mathfrak q_i}(a, b) &", "\\text{if} & \\mathfrak p_j = \\mathfrak q_i \\\\", "d_{M_{j - 1}}(a, b) & \\text{if} & \\mathfrak p_j = \\mathfrak m_A", "\\end{matrix}", "\\right.", "$$", "by Lemma \\ref{lemma-symbol-short-exact-sequence} in the first instance", "and Lemma \\ref{lemma-symbol-compare-modules} in the second. Hence the lemma." ], "refs": [ "algebra-lemma-filter-Noetherian-module", "algebra-lemma-filter-minimal-primes-in-support", "chow-lemma-symbol-short-exact-sequence", "chow-lemma-symbol-compare-modules" ], "ref_ids": [ 691, 695, 5885, 5886 ] } ], "ref_ids": [] }, { "id": 5888, "type": "theorem", "label": "chow-lemma-symbol-is-usual-tame-symbol", "categories": [ "chow" ], "title": "chow-lemma-symbol-is-usual-tame-symbol", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "For nonzero $x, y \\in K$ we have", "$$", "d_A(x, y)", "=", "(-1)^{\\text{ord}_A(x)\\text{ord}_A(y)}", "\\frac{x^{\\text{ord}_A(y)}}{y^{\\text{ord}_A(x)}} \\bmod \\mathfrak m_A,", "$$", "in other words the symbol is equal to the usual tame symbol." ], "refs": [], "proofs": [ { "contents": [ "By multiplicativity it suffices to prove this when $x, y \\in A$.", "Let $t \\in A$ be a uniformizer.", "Write $x = t^bu$ and $y = t^bv$ for some $a, b \\geq 0$", "and $u, v \\in A^*$. Set $l = a + b$. Then", "$t^{l - 1}, \\ldots, t^b$ is an admissible sequence in", "$(x)/(xy)$ and $t^{l - 1}, \\ldots, t^a$ is an admissible", "sequence in $(y)/(xy)$. Hence by Remark \\ref{remark-more-elementary}", "we see that $d_A(x, y)$ is characterized by the equation", "$$", "[t^{l - 1}, \\ldots, t^b, v^{-1}t^{b - 1}, \\ldots, v^{-1}]", "=", "(-1)^{ab} d_A(x, y)", "[t^{l - 1}, \\ldots, t^a, u^{-1}t^{a - 1}, \\ldots, u^{-1}].", "$$", "Hence by the admissible relations for the", "symbols $[x_1, \\ldots, x_l]$ we see that", "$$", "d_A(x, y) = (-1)^{ab} u^a/v^b \\bmod \\mathfrak m_A", "$$", "as desired." ], "refs": [ "chow-remark-more-elementary" ], "ref_ids": [ 5967 ] } ], "ref_ids": [] }, { "id": 5889, "type": "theorem", "label": "chow-lemma-symbol-is-steinberg-prepare", "categories": [ "chow" ], "title": "chow-lemma-symbol-is-steinberg-prepare", "contents": [ "Let $A$ be a Noetherian local ring.", "Let $a, b \\in A$.", "Let $M$ be a finite $A$-module of dimension $1$ on", "which each of $a$, $b$, $b - a$ are nonzerodivisors.", "Then", "$$", "d_M(a, b - a)d_M(b, b) = d_M(b, b - a)d_M(a, b)", "$$", "in $\\kappa^*$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-compute-symbol-M} it suffices to show the relation when", "$M = A/\\mathfrak q$ for some prime $\\mathfrak q \\subset A$ with", "$\\dim(A/\\mathfrak q) = 1$.", "\\medskip\\noindent", "In case $M = A/\\mathfrak q$ we may replace $A$ by $A/\\mathfrak q$ and", "$a, b$ by their images in $A/\\mathfrak q$. Hence we may assume $A = M$", "and $A$ a local Noetherian domain of dimension $1$. The reason is", "that the residue field $\\kappa$ of $A$ and $A/\\mathfrak q$ are", "the same and that for any $A/\\mathfrak q$-module $M$ the determinant", "taken over $A$ or over $A/\\mathfrak q$ are canonically identified.", "See Lemma \\ref{lemma-determinant-quotient-ring}.", "\\medskip\\noindent", "It suffices to show the relation when both", "$a, b$ are in the maximal ideal. Namely, the case where one", "or both are units follows from Lemmas \\ref{lemma-symbol-when-one-is-a-unit}", "and \\ref{lemma-symbol-when-equal}.", "\\medskip\\noindent", "Choose an extension $A \\subset A'$ and factorizations", "$a = ta'$, $b = tb'$ as in", "Lemma \\ref{lemma-Noetherian-domain-dim-1-two-elements}.", "Note that also $b - a = t(b' - a')$ and that", "$A' = (a', b') = (a', b' - a') = (b' - a', b')$.", "Here and in the following we think of $A'$ as an $A$-module and", "$a, b, a', b', t$ as $A$-module endomorphisms of $A'$.", "We will use the notation $d^A_{A'}(a', b')$ and so on to indicate", "$$", "d^A_{A'}(a', b')", "=", "\\det\\nolimits_\\kappa(A'/a'b'A', a', b')", "$$", "which is defined by Lemma \\ref{lemma-pre-symbol}. The upper index ${}^A$", "is used to distinguish this from the already defined symbol", "$d_{A'}(a', b')$ which is different (for example because it has values", "in the residue field of $A'$ which may be different from $\\kappa$).", "By Lemma \\ref{lemma-symbol-compare-modules} we see that", "$d_A(a, b) = d^A_{A'}(a, b)$,", "and similarly for the other combinations.", "Using this and multiplicativity we see that it suffices to prove", "$$", "d^A_{A'}(a', b' - a') d^A_{A'}(b', b')", "=", "d^A_{A'}(b', b' - a') d^A_{A'}(a', b')", "$$", "Now, since $(a', b') = A'$ and so on we have", "$$", "\\begin{matrix}", "A'/(a'(b' - a')) & \\cong & A'/(a') \\oplus A'/(b' - a') \\\\", "A'/(b'(b' - a')) & \\cong & A'/(b') \\oplus A'/(b' - a') \\\\", "A'/(a'b') & \\cong & A'/(a') \\oplus A'/(b')", "\\end{matrix}", "$$", "Moreover, note that multiplication by $b' - a'$ on", "$A/(a')$ is equal to multiplication by $b'$, and that", "multiplication by $b' - a'$ on $A/(b')$ is equal to multiplication by $-a'$.", "Using Lemmas", "\\ref{lemma-periodic-determinant-easy-case} and", "\\ref{lemma-periodic-determinant}", "we conclude", "$$", "\\begin{matrix}", "d^A_{A'}(a', b' - a') & = &", "\\det\\nolimits_\\kappa(b'|_{A'/(a')})^{-1}", "\\det\\nolimits_\\kappa(a'|_{A'/(b' - a')}) \\\\", "d^A_{A'}(b', b' - a') & = &", "\\det\\nolimits_\\kappa(-a'|_{A'/(b')})^{-1}", "\\det\\nolimits_\\kappa(b'|_{A'/(b' - a')}) \\\\", "d^A_{A'}(a', b') & = &", "\\det\\nolimits_\\kappa(b'|_{A'/(a')})^{-1}", "\\det\\nolimits_\\kappa(a'|_{A'/(b')})", "\\end{matrix}", "$$", "Hence we conclude that", "$$", "(-1)^{\\text{length}_A(A'/(b'))}", "d^A_{A'}(a', b' - a')", "=", "d^A_{A'}(b', b' - a') d^A_{A'}(a', b')", "$$", "the sign coming from the $-a'$ in the second equality above.", "On the other hand, by Lemma \\ref{lemma-periodic-determinant-sign} we have", "$d^A_{A'}(b', b') = (-1)^{\\text{length}_A(A'/(b'))}$ and the lemma", "is proved." ], "refs": [ "chow-lemma-compute-symbol-M", "chow-lemma-determinant-quotient-ring", "chow-lemma-symbol-when-one-is-a-unit", "chow-lemma-symbol-when-equal", "chow-lemma-Noetherian-domain-dim-1-two-elements", "chow-lemma-pre-symbol", "chow-lemma-symbol-compare-modules", "chow-lemma-periodic-determinant-easy-case", "chow-lemma-periodic-determinant", "chow-lemma-periodic-determinant-sign" ], "ref_ids": [ 5887, 5872, 5884, 5883, 5657, 5880, 5886, 5876, 5877, 5875 ] } ], "ref_ids": [] }, { "id": 5890, "type": "theorem", "label": "chow-lemma-symbol-is-steinberg", "categories": [ "chow" ], "title": "chow-lemma-symbol-is-steinberg", "contents": [ "Let $A$ be a Noetherian local domain of dimension $1$", "with fraction field $K$. For $x \\in K \\setminus \\{0, 1\\}$", "we have", "$$", "d_A(x, 1 -x) = 1", "$$" ], "refs": [], "proofs": [ { "contents": [ "Write $x = a/b$ with $a, b \\in A$.", "The hypothesis implies, since $1 - x = (b - a)/b$,", "that also $b - a \\not = 0$. Hence we compute", "$$", "d_A(x, 1 - x)", "=", "d_A(a, b - a)d_A(a, b)^{-1}d_A(b, b - a)^{-1}d_A(b, b)", "$$", "Thus we have to show that", "$d_A(a, b - a) d_A(b, b) = d_A(b, b - a) d_A(a, b)$.", "This is Lemma \\ref{lemma-symbol-is-steinberg-prepare}." ], "refs": [ "chow-lemma-symbol-is-steinberg-prepare" ], "ref_ids": [ 5889 ] } ], "ref_ids": [] }, { "id": 5891, "type": "theorem", "label": "chow-lemma-key-lemma", "categories": [ "chow" ], "title": "chow-lemma-key-lemma", "contents": [ "Let $R$ be a Noetherian local ring.", "Let $\\mathfrak q \\subset R$ be a prime with $\\dim(R/\\mathfrak q) = 1$.", "Let $\\varphi : M \\to N$ be a homomorphism of finite $R$-modules.", "Assume there exist $x_1, \\ldots, x_l \\in M$ and $y_1, \\ldots, y_l \\in M$", "with the following properties", "\\begin{enumerate}", "\\item $M = \\langle x_1, \\ldots, x_l\\rangle$,", "\\item $\\langle x_1, \\ldots, x_i\\rangle / \\langle x_1, \\ldots, x_{i - 1}\\rangle", "\\cong R/\\mathfrak q$ for $i = 1, \\ldots, l$,", "\\item $N = \\langle y_1, \\ldots, y_l\\rangle$, and", "\\item $\\langle y_1, \\ldots, y_i\\rangle / \\langle y_1, \\ldots, y_{i - 1}\\rangle", "\\cong R/\\mathfrak q$ for $i = 1, \\ldots, l$.", "\\end{enumerate}", "Then $\\varphi$ is injective if and only if $\\varphi_{\\mathfrak q}$ is an", "isomorphism, and in this case we have", "$$", "\\text{length}_R(\\Coker(\\varphi)) = \\text{ord}_{R/\\mathfrak q}(f)", "$$", "where $f \\in \\kappa(\\mathfrak q)$ is the element such that", "$$", "[\\varphi(x_1), \\ldots, \\varphi(x_l)] = f [y_1, \\ldots, y_l]", "$$", "in $\\det_{\\kappa(\\mathfrak q)}(N_{\\mathfrak q})$." ], "refs": [], "proofs": [ { "contents": [ "First, note that the lemma holds in case $l = 1$.", "Namely, in this case $x_1$ is a basis of $M$ over $R/\\mathfrak q$", "and $y_1$ is a basis of $N$ over $R/\\mathfrak q$ and we have", "$\\varphi(x_1) = fy_1$ for some $f \\in R$. Thus $\\varphi$ is injective", "if and only if $f \\not \\in \\mathfrak q$. Moreover,", "$\\Coker(\\varphi) = R/(f, \\mathfrak q)$ and hence the lemma", "holds by definition of $\\text{ord}_{R/q}(f)$", "(see Algebra, Definition \\ref{algebra-definition-ord}).", "\\medskip\\noindent", "In fact, suppose more generally that $\\varphi(x_i) = f_iy_i$ for some", "$f_i \\in R$, $f_i \\not \\in \\mathfrak q$. Then the induced maps", "$$", "\\langle x_1, \\ldots, x_i\\rangle / \\langle x_1, \\ldots, x_{i - 1}\\rangle", "\\longrightarrow", "\\langle y_1, \\ldots, y_i\\rangle / \\langle y_1, \\ldots, y_{i - 1}\\rangle", "$$", "are all injective and have cokernels isomorphic to", "$R/(f_i, \\mathfrak q)$. Hence we see that", "$$", "\\text{length}_R(\\Coker(\\varphi)) = \\sum \\text{ord}_{R/\\mathfrak q}(f_i).", "$$", "On the other hand it is clear that", "$$", "[\\varphi(x_1), \\ldots, \\varphi(x_l)] = f_1 \\ldots f_l [y_1, \\ldots, y_l]", "$$", "in this case from the admissible relation (b) for symbols.", "Hence we see the result holds in this case also.", "\\medskip\\noindent", "We prove the general case by induction on $l$. Assume $l > 1$.", "Let $i \\in \\{1, \\ldots, l\\}$ be minimal such that", "$\\varphi(x_1) \\in \\langle y_1, \\ldots, y_i\\rangle$.", "We will argue by induction on $i$.", "If $i = 1$, then we get a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\langle x_1 \\rangle \\ar[r] \\ar[d] &", "\\langle x_1, \\ldots, x_l \\rangle \\ar[r] \\ar[d] &", "\\langle x_1, \\ldots, x_l \\rangle / \\langle x_1 \\rangle \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\langle y_1 \\rangle \\ar[r] &", "\\langle y_1, \\ldots, y_l \\rangle \\ar[r] &", "\\langle y_1, \\ldots, y_l \\rangle / \\langle y_1 \\rangle \\ar[r] &", "0", "}", "$$", "and the lemma follows from the snake lemma and induction on $l$.", "Assume now that $i > 1$.", "Write $\\varphi(x_1) = a_1 y_1 + \\ldots + a_{i - 1} y_{i - 1} + a y_i$", "with $a_j, a \\in R$ and $a \\not \\in \\mathfrak q$ (since otherwise", "$i$ was not minimal). Set", "$$", "x'_j =", "\\left\\{", "\\begin{matrix}", "x_j & \\text{if} & j = 1 \\\\", "ax_j & \\text{if} & j \\geq 2", "\\end{matrix}", "\\right.", "\\quad\\text{and}\\quad", "y'_j =", "\\left\\{", "\\begin{matrix}", "y_j & \\text{if} & j < i \\\\", "ay_j & \\text{if} & j \\geq i", "\\end{matrix}", "\\right.", "$$", "Let $M' = \\langle x'_1, \\ldots, x'_l \\rangle$ and", "$N' = \\langle y'_1, \\ldots, y'_l \\rangle$.", "Since $\\varphi(x'_1) = a_1 y'_1 + \\ldots + a_{i - 1} y'_{i - 1} + y'_i$", "by construction and since for $j > 1$ we have", "$\\varphi(x'_j) = a\\varphi(x_i) \\in \\langle y'_1, \\ldots, y'_l\\rangle$", "we get a commutative diagram of $R$-modules and maps", "$$", "\\xymatrix{", "M' \\ar[d] \\ar[r]_{\\varphi'} & N' \\ar[d] \\\\", "M \\ar[r]^\\varphi & N", "}", "$$", "By the result of the second paragraph of the proof we know", "that $\\text{length}_R(M/M') = (l - 1)\\text{ord}_{R/\\mathfrak q}(a)$", "and similarly", "$\\text{length}_R(M/M') = (l - i + 1)\\text{ord}_{R/\\mathfrak q}(a)$.", "By a diagram chase this implies that", "$$", "\\text{length}_R(\\Coker(\\varphi')) =", "\\text{length}_R(\\Coker(\\varphi)) + i\\ \\text{ord}_{R/\\mathfrak q}(a).", "$$", "On the other hand, it is clear that writing", "$$", "[\\varphi(x_1), \\ldots, \\varphi(x_l)] = f [y_1, \\ldots, y_l],", "\\quad", "[\\varphi'(x'_1), \\ldots, \\varphi(x'_l)] = f' [y'_1, \\ldots, y'_l]", "$$", "we have $f' = a^if$. Hence it suffices to prove the lemma for the", "case that $\\varphi(x_1) = a_1y_1 + \\ldots a_{i - 1}y_{i - 1} + y_i$,", "i.e., in the case that $a = 1$. Next, recall that", "$$", "[y_1, \\ldots, y_l] = [y_1, \\ldots, y_{i - 1},", "a_1y_1 + \\ldots a_{i - 1}y_{i - 1} + y_i, y_{i + 1}, \\ldots, y_l]", "$$", "by the admissible relations for symbols. The sequence", "$y_1, \\ldots, y_{i - 1},", "a_1y_1 + \\ldots + a_{i - 1}y_{i - 1} + y_i, y_{i + 1}, \\ldots, y_l$", "satisfies the conditions (3), (4) of the lemma also.", "Hence, we may actually", "assume that $\\varphi(x_1) = y_i$. In this case, note that we have", "$\\mathfrak q x_1 = 0$ which implies also $\\mathfrak q y_i = 0$.", "We have", "$$", "[y_1, \\ldots, y_l] =", "- [y_1, \\ldots, y_{i - 2}, y_i, y_{i - 1}, y_{i + 1}, \\ldots, y_l]", "$$", "by the third of the admissible relations defining", "$\\det_{\\kappa(\\mathfrak q)}(N_{\\mathfrak q})$. Hence we may", "replace $y_1, \\ldots, y_l$ by", "the sequence", "$y'_1, \\ldots, y'_l =", "y_1, \\ldots, y_{i - 2}, y_i, y_{i - 1}, y_{i + 1}, \\ldots, y_l$", "(which also satisfies conditions (3) and (4) of the lemma).", "Clearly this decreases the invariant $i$ by $1$ and we win by induction", "on $i$." ], "refs": [ "algebra-definition-ord" ], "ref_ids": [ 1519 ] } ], "ref_ids": [] }, { "id": 5892, "type": "theorem", "label": "chow-lemma-good-sequence-exists", "categories": [ "chow" ], "title": "chow-lemma-good-sequence-exists", "contents": [ "Let $R$ be a local Noetherian ring.", "Let $\\mathfrak q \\subset R$ be a prime ideal.", "Let $M$ be a finite $R$-module such that", "$\\mathfrak q$ is one of the minimal primes of the support of $M$.", "Then there exist $x_1, \\ldots, x_l \\in M$ such that", "\\begin{enumerate}", "\\item the support of $M / \\langle x_1, \\ldots, x_l\\rangle$ does not contain", "$\\mathfrak q$, and", "\\item $\\langle x_1, \\ldots, x_i\\rangle / \\langle x_1, \\ldots, x_{i - 1}\\rangle", "\\cong R/\\mathfrak q$ for $i = 1, \\ldots, l$.", "\\end{enumerate}", "Moreover, in this case $l = \\text{length}_{R_\\mathfrak q}(M_\\mathfrak q)$." ], "refs": [], "proofs": [ { "contents": [ "The condition that $\\mathfrak q$ is a minimal prime in the support", "of $M$ implies that $l = \\text{length}_{R_\\mathfrak q}(M_\\mathfrak q)$", "is finite (see Algebra, Lemma \\ref{algebra-lemma-support-point}).", "Hence we can find $y_1, \\ldots, y_l \\in M_{\\mathfrak q}$", "such that", "$\\langle y_1, \\ldots, y_i\\rangle / \\langle y_1, \\ldots, y_{i - 1}\\rangle", "\\cong \\kappa(\\mathfrak q)$ for $i = 1, \\ldots, l$.", "We can find $f_i \\in R$, $f_i \\not \\in \\mathfrak q$ such that", "$f_i y_i$ is the image of some element $z_i \\in M$.", "Moreover, as $R$ is Noetherian we can write", "$\\mathfrak q = (g_1, \\ldots, g_t)$ for some $g_j \\in R$.", "By assumption $g_j y_i \\in \\langle y_1, \\ldots, y_{i - 1} \\rangle$", "inside the module $M_{\\mathfrak q}$.", "By our choice of $z_i$ we can find some further elements", "$f_{ji} \\in R$, $f_{ij} \\not \\in \\mathfrak q$ such that", "$f_{ij} g_j z_i \\in \\langle z_1, \\ldots, z_{i - 1} \\rangle$", "(equality in the module $M$).", "The lemma follows by taking", "$$", "x_1 = f_{11}f_{12}\\ldots f_{1t}z_1,", "\\quad", "x_2 = f_{11}f_{12}\\ldots f_{1t}f_{21}f_{22}\\ldots f_{2t}z_2,", "$$", "and so on. Namely, since all the elements $f_i, f_{ij}$ are invertible", "in $R_{\\mathfrak q}$ we still have that", "$R_{\\mathfrak q}x_1 + \\ldots + R_{\\mathfrak q}x_i /", "R_{\\mathfrak q}x_1 + \\ldots + R_{\\mathfrak q}x_{i - 1}", "\\cong \\kappa(\\mathfrak q)$ for $i = 1, \\ldots, l$.", "By construction, $\\mathfrak q x_i \\in \\langle x_1, \\ldots, x_{i - 1}\\rangle$.", "Thus $\\langle x_1, \\ldots, x_i\\rangle / \\langle x_1, \\ldots, x_{i - 1}\\rangle$", "is an $R$-module generated by one element, annihilated $\\mathfrak q$", "such that localizing at $\\mathfrak q$ gives a $q$-dimensional", "vector space over $\\kappa(\\mathfrak q)$.", "Hence it is isomorphic to $R/\\mathfrak q$." ], "refs": [ "algebra-lemma-support-point" ], "ref_ids": [ 693 ] } ], "ref_ids": [] }, { "id": 5893, "type": "theorem", "label": "chow-lemma-application-herbrand-quotient", "categories": [ "chow" ], "title": "chow-lemma-application-herbrand-quotient", "contents": [ "Let $R$ be a Noetherian local ring with maximal ideal $\\mathfrak m$.", "Let $M$ be a finite $R$-module, and let $\\psi : M \\to M$ be an", "$R$-module map. Assume that", "\\begin{enumerate}", "\\item $\\Ker(\\psi)$ and $\\Coker(\\psi)$ have finite length, and", "\\item $\\dim(\\text{Supp}(M)) \\leq 1$.", "\\end{enumerate}", "Write", "$\\text{Supp}(M) = \\{\\mathfrak m, \\mathfrak q_1, \\ldots, \\mathfrak q_t\\}$", "and denote $f_i \\in \\kappa(\\mathfrak q_i)^*$ the element such that", "$\\det_{\\kappa(\\mathfrak q_i)}(\\psi_{\\mathfrak q_i}) :", "\\det_{\\kappa(\\mathfrak q_i)}(M_{\\mathfrak q_i})", "\\to \\det_{\\kappa(\\mathfrak q_i)}(M_{\\mathfrak q_i})$", "is multiplication by $f_i$. Then", "we have", "$$", "\\text{length}_R(\\Coker(\\psi))", "-", "\\text{length}_R(\\Ker(\\psi))", "=", "\\sum\\nolimits_{i = 1, \\ldots, t}", "\\text{ord}_{R/\\mathfrak q_i}(f_i).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Recall that $H^0(M, 0, \\psi) = \\Coker(\\psi)$ and", "$H^1(M, 0, \\psi) = \\Ker(\\psi)$, see remarks above", "Definition \\ref{definition-periodic-length}.", "The lemma follows by combining", "Proposition \\ref{proposition-length-determinant-periodic-complex} with", "Lemma \\ref{lemma-periodic-determinant-easy-case}.", "\\medskip\\noindent", "Alternative proof. Reduce to the case", "$\\text{Supp}(M) = \\{\\mathfrak m, \\mathfrak q\\}$", "as in the proof of", "Proposition \\ref{proposition-length-determinant-periodic-complex}.", "Then directly combine", "Lemmas \\ref{lemma-key-lemma} and", "\\ref{lemma-good-sequence-exists}", "to prove this specific case of", "Proposition \\ref{proposition-length-determinant-periodic-complex}.", "There is much less bookkeeping in this case, and the reader is", "encouraged to work this out. Details omitted." ], "refs": [ "chow-definition-periodic-length", "chow-proposition-length-determinant-periodic-complex", "chow-lemma-periodic-determinant-easy-case", "chow-proposition-length-determinant-periodic-complex", "chow-lemma-key-lemma", "chow-lemma-good-sequence-exists", "chow-proposition-length-determinant-periodic-complex" ], "ref_ids": [ 5904, 5902, 5876, 5902, 5891, 5892, 5902 ] } ], "ref_ids": [] }, { "id": 5894, "type": "theorem", "label": "chow-lemma-secondary-ramification", "categories": [ "chow" ], "title": "chow-lemma-secondary-ramification", "contents": [ "\\begin{reference}", "When $A$ is an excellent ring this is \\cite[Proposition 1]{Kato-Milnor-K}.", "\\end{reference}", "Let $A$ be a $2$-dimensional Noetherian local domain with fraction field $K$.", "Let $f, g \\in K^*$.", "Let $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ be the height", "$1$ primes $\\mathfrak q$ of $A$ such that either $f$ or $g$ is not an", "element of $A^*_{\\mathfrak q}$.", "Then we have", "$$", "\\sum\\nolimits_{i = 1, \\ldots, t}", "\\text{ord}_{A/\\mathfrak q_i}(d_{A_{\\mathfrak q_i}}(f, g))", "=", "0", "$$", "We can also write this as", "$$", "\\sum\\nolimits_{\\text{height}(\\mathfrak q) = 1}", "\\text{ord}_{A/\\mathfrak q}(d_{A_{\\mathfrak q}}(f, g))", "=", "0", "$$", "since at any height one prime $\\mathfrak q$", "of $A$ where $f, g \\in A^*_{\\mathfrak q}$", "we have $d_{A_{\\mathfrak q}}(f, g) = 1$ by", "Lemma \\ref{lemma-symbol-when-one-is-a-unit}." ], "refs": [ "chow-lemma-symbol-when-one-is-a-unit" ], "proofs": [ { "contents": [ "Since the tame symbols $d_{A_{\\mathfrak q}}(f, g)$ are additive", "(Lemma \\ref{lemma-multiplicativity-symbol}) and the order", "functions $\\text{ord}_{A/\\mathfrak q}$", "are additive (Algebra, Lemma \\ref{algebra-lemma-ord-additive})", "it suffices to prove the formula when $f = a \\in A$ and", "$g = b \\in A$. In this case we see that we have to show", "$$", "\\sum\\nolimits_{\\text{height}(\\mathfrak q) = 1}", "\\text{ord}_{A/\\mathfrak q}(\\det\\nolimits_\\kappa(A_{\\mathfrak q}/(ab), a, b))", "= 0", "$$", "By Proposition \\ref{proposition-length-determinant-periodic-complex}", "this is equivalent to showing that", "$$", "e_A(A/(ab), a, b) = 0.", "$$", "Since the complex", "$A/(ab) \\xrightarrow{a} A/(ab) \\xrightarrow{b} A/(ab) \\xrightarrow{a} A/(ab)$", "is exact we win." ], "refs": [ "chow-lemma-multiplicativity-symbol", "algebra-lemma-ord-additive", "chow-proposition-length-determinant-periodic-complex" ], "ref_ids": [ 5882, 1043, 5902 ] } ], "ref_ids": [ 5884 ] }, { "id": 5895, "type": "theorem", "label": "chow-lemma-maps-between-coherent-sheaves", "categories": [ "chow" ], "title": "chow-lemma-maps-between-coherent-sheaves", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be a scheme locally of finite type over $S$.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Let", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "\\mathcal{F} \\ar[r]^\\varphi &", "\\mathcal{F} \\ar[r]^\\psi &", "\\mathcal{F} \\ar[r]^\\varphi &", "\\mathcal{F} \\ar[r] & \\ldots", "}", "$$", "be a complex as in Homology, Equation (\\ref{homology-equation-cyclic-complex}).", "Assume that", "\\begin{enumerate}", "\\item $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k + 1$.", "\\item $\\dim_\\delta(\\text{Supp}(H^i(\\mathcal{F}, \\varphi, \\psi))) \\leq k$", "for $i = 0, 1$.", "\\end{enumerate}", "Then we have", "$$", "[H^0(\\mathcal{F}, \\varphi, \\psi)]_k", "\\sim_{rat}", "[H^1(\\mathcal{F}, \\varphi, \\psi)]_k", "$$", "as $k$-cycles on $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{W_j\\}_{j \\in J}$ be the collection of irreducible", "components of $\\text{Supp}(\\mathcal{F})$", "which have $\\delta$-dimension $k + 1$. Note that $\\{W_j\\}$", "is a locally finite collection of closed subsets of", "$X$ by Lemma \\ref{lemma-length-finite}.", "For every $j$, let $\\xi_j \\in W_j$ be the generic point.", "Set", "$$", "f_j = \\det\\nolimits_{\\kappa(\\xi_j)}", "(\\mathcal{F}_{\\xi_j}, \\varphi_{\\xi_j}, \\psi_{\\xi_j})", "\\in", "R(W_j)^*.", "$$", "See Definition \\ref{definition-periodic-determinant} for notation.", "We claim that", "$$", "- [H^0(\\mathcal{F}, \\varphi, \\psi)]_k + [H^1(\\mathcal{F}, \\varphi, \\psi)]_k", "=", "\\sum (W_j \\to X)_*\\text{div}(f_j)", "$$", "If we prove this then the lemma follows.", "\\medskip\\noindent", "Let $Z \\subset X$ be an integral closed subscheme of $\\delta$-dimension $k$.", "To prove the equality above it suffices to show that the coefficient $n$", "of $[Z]$ in", "$", "[H^0(\\mathcal{F}, \\varphi, \\psi)]_k - [H^1(\\mathcal{F}, \\varphi, \\psi)]_k", "$", "is the same as the coefficient $m$ of $[Z]$ in", "$", "\\sum (W_j \\to X)_*\\text{div}(f_j)", "$.", "Let $\\xi \\in Z$ be the generic point.", "Consider the local ring $A = \\mathcal{O}_{X, \\xi}$.", "Let $M = \\mathcal{F}_\\xi$ as an $A$-module.", "Denote $\\varphi, \\psi : M \\to M$ the action of $\\varphi, \\psi$ on", "the stalk.", "By our choice of $\\xi \\in Z$ we have $\\delta(\\xi) = k$", "and hence $\\dim(\\text{Supp}(M)) = 1$.", "Finally, the integral closed subschemes", "$W_j$ passing through $\\xi$ correspond to the minimal primes", "$\\mathfrak q_i$ of $\\text{Supp}(M)$.", "In each case the element $f_j \\in R(W_j)^*$ corresponds to", "the element $\\det_{\\kappa(\\mathfrak q_i)}(M_{\\mathfrak q_i}, \\varphi, \\psi)$", "in $\\kappa(\\mathfrak q_i)^*$. Hence we see that", "$$", "n = - e_A(M, \\varphi, \\psi)", "$$", "and", "$$", "m =", "\\sum", "\\text{ord}_{A/\\mathfrak q_i}", "(\\det\\nolimits_{\\kappa(\\mathfrak q_i)}(M_{\\mathfrak q_i}, \\varphi, \\psi))", "$$", "Thus the result follows from", "Proposition \\ref{proposition-length-determinant-periodic-complex}." ], "refs": [ "chow-lemma-length-finite", "chow-definition-periodic-determinant", "chow-proposition-length-determinant-periodic-complex" ], "ref_ids": [ 5669, 5927, 5902 ] } ], "ref_ids": [] }, { "id": 5896, "type": "theorem", "label": "chow-lemma-cycles-rational-equivalence-K-group", "categories": [ "chow" ], "title": "chow-lemma-cycles-rational-equivalence-K-group", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be a scheme locally of finite type over $S$.", "The map", "$$", "\\CH_k(X) \\longrightarrow", "K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))", "$$", "from Lemma \\ref{lemma-from-chow-to-K} induces a bijection from", "$\\CH_k(X)$ onto the image $B_k(X)$ of the map", "$$", "K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))", "\\longrightarrow", "K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X)).", "$$" ], "refs": [ "chow-lemma-from-chow-to-K" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-cycles-k-group} we have", "$Z_k(X) = K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))$", "compatible with the map of Lemma \\ref{lemma-from-chow-to-K}.", "Thus, suppose we have an element $[A] - [B]$ of", "$K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))$", "which maps to zero in $B_k(X)$, i.e., maps to zero in", "$K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))$.", "We have to show that $[A] - [B]$ corresponds to a cycle", "rationally equivalent to zero on $X$.", "Suppose $[A] = [\\mathcal{A}]$ and $[B] = [\\mathcal{B}]$", "for some coherent sheaves $\\mathcal{A}, \\mathcal{B}$ on", "$X$ supported in $\\delta$-dimension $\\leq k$.", "The assumption that $[A] - [B]$ maps to zero in the group", "$K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))$", "means that there exists coherent sheaves", "$\\mathcal{A}', \\mathcal{B}'$ on $X$ supported in", "$\\delta$-dimension $\\leq k - 1$ such that", "$[\\mathcal{A} \\oplus \\mathcal{A}'] - [\\mathcal{B} \\oplus \\mathcal{B}']$", "is zero in $K_0(\\textit{Coh}_{k + 1}(X))$ (use part (1) of", "Homology, Lemma \\ref{homology-lemma-serre-subcategory-K-groups}).", "By part (2) of", "Homology, Lemma \\ref{homology-lemma-serre-subcategory-K-groups}", "this means there exists a $(2, 1)$-periodic complex", "$(\\mathcal{F}, \\varphi, \\psi)$ in the category $\\textit{Coh}_{\\leq k + 1}(X)$", "such that", "$\\mathcal{A} \\oplus \\mathcal{A}' = H^0(\\mathcal{F}, \\varphi, \\psi)$", "and $\\mathcal{B} \\oplus \\mathcal{B}' = H^1(\\mathcal{F}, \\varphi, \\psi)$.", "By Lemma \\ref{lemma-maps-between-coherent-sheaves}", "this implies that", "$$", "[\\mathcal{A} \\oplus \\mathcal{A}']_k", "\\sim_{rat}", "[\\mathcal{B} \\oplus \\mathcal{B}']_k", "$$", "This proves that $[A] - [B]$ maps to a cycle rationally", "equivalent to zero by the map", "$$", "K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))", "\\longrightarrow Z_k(X)", "$$", "of Lemma \\ref{lemma-cycles-k-group}. This is what we", "had to prove and the proof is complete." ], "refs": [ "chow-lemma-cycles-k-group", "chow-lemma-from-chow-to-K", "homology-lemma-serre-subcategory-K-groups", "homology-lemma-serre-subcategory-K-groups", "chow-lemma-maps-between-coherent-sheaves", "chow-lemma-cycles-k-group" ], "ref_ids": [ 5699, 5701, 12051, 12051, 5895, 5699 ] } ], "ref_ids": [ 5701 ] }, { "id": 5897, "type": "theorem", "label": "chow-lemma-no-embedded-points-modules", "categories": [ "chow" ], "title": "chow-lemma-no-embedded-points-modules", "contents": [ "Let $A$ be a Noetherian local ring.", "Let $M$ be a finite $A$-module.", "Let $a, b \\in A$.", "Assume", "\\begin{enumerate}", "\\item $\\dim(A) = 1$,", "\\item both $a$ and $b$ are nonzerodivisors in $A$,", "\\item $A$ has no embedded primes,", "\\item $M$ has no embedded associated primes,", "\\item $\\text{Supp}(M) = \\Spec(A)$.", "\\end{enumerate}", "Let $I = \\{x \\in A \\mid x(a/b) \\in A\\}$.", "Let $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ be the minimal", "primes of $A$. Then $(a/b)IM \\subset M$ and", "$$", "\\text{length}_A(M/(a/b)IM)", "-", "\\text{length}_A(M/IM)", "=", "\\sum\\nolimits_i", "\\text{length}_{A_{\\mathfrak q_i}}(M_{\\mathfrak q_i})", "\\text{ord}_{A/\\mathfrak q_i}(a/b)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since $M$ has no embedded associated primes, and since", "the support of $M$ is $\\Spec(A)$ we see that", "$\\text{Ass}(M) = \\{\\mathfrak q_1, \\ldots, \\mathfrak q_t\\}$.", "Hence $a$, $b$ are nonzerodivisors on $M$. Note that", "\\begin{align*}", "& \\text{length}_A(M/(a/b)IM) \\\\", "& = \\text{length}_A(bM/aIM) \\\\", "& = \\text{length}_A(M/aIM)", "-", "\\text{length}_A(M/bM) \\\\", "& = \\text{length}_A(M/aM) + \\text{length}_A(aM/aIM) - \\text{length}_A(M/bM) \\\\", "& = \\text{length}_A(M/aM) + \\text{length}_A(M/IM) - \\text{length}_A(M/bM)", "\\end{align*}", "as the injective map $b : M \\to bM$ maps $(a/b)IM$ to $aIM$", "and the injective map $a : M \\to aM$ maps $IM$ to $aIM$.", "Hence the left hand side of the equation of the lemma is", "equal to", "$$", "\\text{length}_A(M/aM) - \\text{length}_A(M/bM).", "$$", "Applying the second formula of", "Lemma \\ref{lemma-additivity-divisors-restricted}", "with $x = a, b$ respectively", "and using Algebra, Definition \\ref{algebra-definition-ord}", "of the $\\text{ord}$-functions we get the result." ], "refs": [ "chow-lemma-additivity-divisors-restricted", "algebra-definition-ord" ], "ref_ids": [ 5653, 1519 ] } ], "ref_ids": [] }, { "id": 5898, "type": "theorem", "label": "chow-lemma-no-embedded-points", "categories": [ "chow" ], "title": "chow-lemma-no-embedded-points", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{K}_X(\\mathcal{L}))$ be a", "meromorphic section of $\\mathcal{L}$.", "Assume", "\\begin{enumerate}", "\\item $\\dim_\\delta(X) \\leq k + 1$,", "\\item $X$ has no embedded points,", "\\item $\\mathcal{F}$ has no embedded associated points,", "\\item the support of $\\mathcal{F}$ is $X$, and", "\\item the section $s$ is regular meromorphic.", "\\end{enumerate}", "In this situation let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be the ideal of denominators of $s$, see", "Divisors,", "Definition \\ref{divisors-definition-regular-meromorphic-ideal-denominators}.", "Then we have the following:", "\\begin{enumerate}", "\\item there are short exact sequences", "$$", "\\begin{matrix}", "0 &", "\\to &", "\\mathcal{I}\\mathcal{F} &", "\\xrightarrow{1} &", "\\mathcal{F} &", "\\to &", "\\mathcal{Q}_1 &", "\\to &", "0 \\\\", "0 &", "\\to &", "\\mathcal{I}\\mathcal{F} &", "\\xrightarrow{s} &", "\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L} &", "\\to &", "\\mathcal{Q}_2 &", "\\to &", "0", "\\end{matrix}", "$$", "\\item the coherent sheaves $\\mathcal{Q}_1$, $\\mathcal{Q}_2$", "are supported in $\\delta$-dimension $\\leq k$,", "\\item the section $s$ restricts to a regular meromorphic", "section $s_i$ on every irreducible component $X_i$ of", "$X$ of $\\delta$-dimension $k + 1$, and", "\\item writing $[\\mathcal{F}]_{k + 1} = \\sum m_i[X_i]$ we have", "$$", "[\\mathcal{Q}_2]_k - [\\mathcal{Q}_1]_k", "=", "\\sum m_i(X_i \\to X)_*\\text{div}_{\\mathcal{L}|_{X_i}}(s_i)", "$$", "in $Z_k(X)$, in particular", "$$", "[\\mathcal{Q}_2]_k - [\\mathcal{Q}_1]_k", "=", "c_1(\\mathcal{L}) \\cap [\\mathcal{F}]_{k + 1}", "$$", "in $\\CH_k(X)$.", "\\end{enumerate}" ], "refs": [ "divisors-definition-regular-meromorphic-ideal-denominators" ], "proofs": [ { "contents": [ "Recall from Divisors, Lemma \\ref{divisors-lemma-make-maps-regular-section}", "the existence of injective maps", "$1 : \\mathcal{I}\\mathcal{F} \\to \\mathcal{F}$ and", "$s : \\mathcal{I}\\mathcal{F} \\to \\mathcal{F} \\otimes_{\\mathcal{O}_X}\\mathcal{L}$", "whose cokernels are supported on a closed nowhere dense subsets $T$.", "Denote $\\mathcal{Q}_i$ there cokernels as in the lemma.", "We conclude that $\\dim_\\delta(\\text{Supp}(\\mathcal{Q}_i)) \\leq k$.", "By Divisors, Lemmas \\ref{divisors-lemma-pullback-meromorphic-sections-defined}", "and \\ref{divisors-lemma-meromorphic-sections-pullback} the pullbacks $s_i$", "are defined and are regular meromorphic sections for $\\mathcal{L}|_{X_i}$.", "The equality of cycles in (4) implies the equality of cycle classes", "in (4). Hence the only remaining thing to show is that", "$$", "[\\mathcal{Q}_2]_k - [\\mathcal{Q}_1]_k", "=", "\\sum m_i(X_i \\to X)_*\\text{div}_{\\mathcal{L}|_{X_i}}(s_i)", "$$", "holds in $Z_k(X)$. To see this, let $Z \\subset X$ be an integral closed", "subscheme of $\\delta$-dimension $k$. Let $\\xi \\in Z$ be the generic point.", "Let $A = \\mathcal{O}_{X, \\xi}$ and $M = \\mathcal{F}_\\xi$.", "Moreover, choose a generator $s_\\xi \\in \\mathcal{L}_\\xi$.", "Then we can write $s = (a/b) s_\\xi$ where $a, b \\in A$ are", "nonzerodivisors. In this case", "$I = \\mathcal{I}_\\xi = \\{x \\in A \\mid x(a/b) \\in A\\}$.", "In this case the coefficient of $[Z]$ in the left hand side is", "$$", "\\text{length}_A(M/(a/b)IM) - \\text{length}_A(M/IM)", "$$", "and the coefficient of $[Z]$ in the right hand side", "is", "$$", "\\sum", "\\text{length}_{A_{\\mathfrak q_i}}(M_{\\mathfrak q_i})", "\\text{ord}_{A/\\mathfrak q_i}(a/b)", "$$", "where $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ are the minimal", "primes of the $1$-dimensional local ring $A$. Hence the result", "follows from Lemma \\ref{lemma-no-embedded-points-modules}." ], "refs": [ "divisors-lemma-pullback-meromorphic-sections-defined", "divisors-lemma-meromorphic-sections-pullback", "chow-lemma-no-embedded-points-modules" ], "ref_ids": [ 8009, 8011, 5897 ] } ], "ref_ids": [ 8105 ] }, { "id": 5899, "type": "theorem", "label": "chow-lemma-coherent-sheaf-cap-c1", "categories": [ "chow" ], "title": "chow-lemma-coherent-sheaf-cap-c1", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Assume $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k + 1$.", "Then the element", "$$", "[\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}]", "-", "[\\mathcal{F}]", "\\in", "K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))", "$$", "lies in the subgroup $B_k(X)$ of", "Lemma \\ref{lemma-cycles-rational-equivalence-K-group} and maps to", "the element $c_1(\\mathcal{L}) \\cap [\\mathcal{F}]_{k + 1}$ via", "the map $B_k(X) \\to \\CH_k(X)$." ], "refs": [ "chow-lemma-cycles-rational-equivalence-K-group" ], "proofs": [ { "contents": [ "Let", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{F} \\to \\mathcal{F}' \\to 0", "$$", "be the short exact sequence constructed in", "Divisors, Lemma \\ref{divisors-lemma-remove-embedded-points}.", "This in particular means that $\\mathcal{F}'$ has no embedded", "associated points.", "Since the support of $\\mathcal{K}$ is nowhere dense in the", "support of $\\mathcal{F}$ we see that", "$\\dim_\\delta(\\text{Supp}(\\mathcal{K})) \\leq k$. We may", "re-apply", "Divisors, Lemma \\ref{divisors-lemma-remove-embedded-points}", "starting with $\\mathcal{K}$ to get a short exact sequence", "$$", "0 \\to \\mathcal{K}'' \\to \\mathcal{K} \\to \\mathcal{K}' \\to 0", "$$", "where now $\\dim_\\delta(\\text{Supp}(\\mathcal{K}'')) < k$", "and $\\mathcal{K}'$ has no embedded associated points.", "Suppose we can prove the lemma for the coherent sheaves", "$\\mathcal{F}'$ and $\\mathcal{K}'$. Then we see", "from the equations", "$$", "[\\mathcal{F}]_{k + 1}", "=", "[\\mathcal{F}']_{k + 1}", "+ [\\mathcal{K}']_{k + 1}", "+ [\\mathcal{K}'']_{k + 1}", "$$", "(use Lemma \\ref{lemma-additivity-sheaf-cycle}),", "$$", "[\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}]", "-", "[\\mathcal{F}]", "=", "[\\mathcal{F}' \\otimes_{\\mathcal{O}_X} \\mathcal{L}]", "-", "[\\mathcal{F}']", "+", "[\\mathcal{K}' \\otimes_{\\mathcal{O}_X} \\mathcal{L}]", "-", "[\\mathcal{K}']", "+", "[\\mathcal{K}'' \\otimes_{\\mathcal{O}_X} \\mathcal{L}]", "-", "[\\mathcal{K}'']", "$$", "(use the $\\otimes \\mathcal{L}$ is exact)", "and the trivial vanishing of $[\\mathcal{K}'']_{k + 1}$ and", "$[\\mathcal{K}'' \\otimes_{\\mathcal{O}_X} \\mathcal{L}]", "- [\\mathcal{K}'']$ in", "$K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))$", "that the result holds", "for $\\mathcal{F}$. What this means is that we may assume that", "the sheaf $\\mathcal{F}$ has no embedded associated points.", "\\medskip\\noindent", "Assume $X$, $\\mathcal{F}$ as in the lemma, and assume in addition", "that $\\mathcal{F}$ has no embedded associated points. Consider the", "sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$, the corresponding", "closed subscheme $i : Z \\to X$ and the coherent $\\mathcal{O}_Z$-module", "$\\mathcal{G}$ constructed in", "Divisors, Lemma \\ref{divisors-lemma-no-embedded-points-endos}.", "Recall that $Z$ is a locally Noetherian scheme without embedded points,", "$\\mathcal{G}$ is a coherent sheaf without embedded", "associated points, with $\\text{Supp}(\\mathcal{G}) = Z$", "and such that $i_*\\mathcal{G} = \\mathcal{F}$.", "Moreover, set $\\mathcal{N} = \\mathcal{L}|_Z$.", "\\medskip\\noindent", "By Divisors, Lemma \\ref{divisors-lemma-regular-meromorphic-section-exists}", "the invertible sheaf $\\mathcal{N}$ has a regular meromorphic section $s$", "over $Z$. Let us denote $\\mathcal{J} \\subset \\mathcal{O}_Z$ the sheaf", "of denominators of $s$. By Lemma \\ref{lemma-no-embedded-points}", "there exist short exact sequences", "$$", "\\begin{matrix}", "0 &", "\\to &", "\\mathcal{J}\\mathcal{G} &", "\\xrightarrow{1} &", "\\mathcal{G} &", "\\to &", "\\mathcal{Q}_1 &", "\\to &", "0 \\\\", "0 &", "\\to &", "\\mathcal{J}\\mathcal{G} &", "\\xrightarrow{s} &", "\\mathcal{G} \\otimes_{\\mathcal{O}_Z} \\mathcal{N} &", "\\to &", "\\mathcal{Q}_2 &", "\\to &", "0", "\\end{matrix}", "$$", "such that $\\dim_\\delta(\\text{Supp}(\\mathcal{Q}_i)) \\leq k$ and", "such that the cycle", "$", "[\\mathcal{Q}_2]_k - [\\mathcal{Q}_1]_k", "$", "is a representative of $c_1(\\mathcal{N}) \\cap [\\mathcal{G}]_{k + 1}$.", "We see (using the fact that", "$i_*(\\mathcal{G} \\otimes \\mathcal{N}) = \\mathcal{F} \\otimes \\mathcal{L}$", "by the projection formula, see", "Cohomology, Lemma \\ref{cohomology-lemma-projection-formula})", "that", "$$", "[\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}]", "-", "[\\mathcal{F}]", "=", "[i_*\\mathcal{Q}_2] - [i_*\\mathcal{Q}_1]", "$$", "in $K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))$.", "This already shows that", "$[\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}] - [\\mathcal{F}]$", "is an element of $B_k(X)$. Moreover we have", "\\begin{eqnarray*}", "[i_*\\mathcal{Q}_2]_k - [i_*\\mathcal{Q}_1]_k", "& = &", "i_*\\left( [\\mathcal{Q}_2]_k - [\\mathcal{Q}_1]_k \\right) \\\\", "& = &", "i_*\\left(c_1(\\mathcal{N}) \\cap [\\mathcal{G}]_{k + 1} \\right) \\\\", "& = &", "c_1(\\mathcal{L}) \\cap i_*[\\mathcal{G}]_{k + 1} \\\\", "& = &", "c_1(\\mathcal{L}) \\cap [\\mathcal{F}]_{k + 1}", "\\end{eqnarray*}", "by the above and Lemmas \\ref{lemma-pushforward-cap-c1}", "and \\ref{lemma-cycle-push-sheaf}. And this agree with the", "image of the element under $B_k(X) \\to \\CH_k(X)$ by definition.", "Hence the lemma is proved." ], "refs": [ "divisors-lemma-remove-embedded-points", "divisors-lemma-remove-embedded-points", "chow-lemma-additivity-sheaf-cycle", "divisors-lemma-no-embedded-points-endos", "chow-lemma-no-embedded-points", "cohomology-lemma-projection-formula", "chow-lemma-pushforward-cap-c1", "chow-lemma-cycle-push-sheaf" ], "ref_ids": [ 7870, 7870, 5671, 7871, 5898, 2243, 5711, 5676 ] } ], "ref_ids": [ 5896 ] }, { "id": 5900, "type": "theorem", "label": "chow-proposition-K-tensor-Q", "categories": [ "chow" ], "title": "chow-proposition-K-tensor-Q", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Assume given a", "closed immersion $X \\to Y$ of schemes locally of finite type over $S$", "with $Y$ regular, quasi-compact, affine diagonal, and", "$\\delta_{Y/S} : Y \\to \\mathbf{Z}$ bounded. Then the composition", "$$", "K'_0(X) \\to", "K_0(D_{X, perf}(\\mathcal{O}_Y)) \\to", "A^*(X \\to Y) \\to", "\\CH_*(X)", "$$", "of the map $\\mathcal{F} \\mapsto \\mathcal{F}[0]$ from", "Remark \\ref{remark-perf-Z-regular}, the map $ch(X \\to Y, -)$ from", "Remark \\ref{remark-localized-chern-character-K}, and", "the map $c \\mapsto c \\cap [Y]$ induces an isomorphism", "$$", "K'_0(X) \\otimes \\mathbf{Q}", "\\longrightarrow", "\\CH_*(X) \\otimes \\mathbf{Q}", "$$", "which depends on the choice of $Y$. Moreover, the canonical map", "$$", "\\CH_k(X) \\otimes \\mathbf{Q}", "\\longrightarrow", "\\text{gr}_k K'_0(X) \\otimes \\mathbf{Q}", "$$", "(see above) is an isomorphism of $\\mathbf{Q}$-vector spaces for all", "$k \\in \\mathbf{Z}$." ], "refs": [ "chow-remark-perf-Z-regular", "chow-remark-localized-chern-character-K" ], "proofs": [ { "contents": [ "Since $Y$ is regular of finite dimension, the construction in", "Remark \\ref{remark-perf-Z-regular} applies.", "We have the resolution property for $Y$ by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-regular-resolution-property}", "and the construction in Remark \\ref{remark-localized-chern-character-K}", "applies. We have that $Y$ is locally equidimensional", "(Lemma \\ref{lemma-locally-equidimensional}) and", "thus the ``fundamental cycle'' $[Y]$ is defined", "as an element of $\\CH_*(Y)$, see Remark \\ref{remark-fundamental-class}.", "Combining this with the map $\\CH_k(X) \\to \\text{gr}_kK'_0(X)$", "constructed above we see that it suffices to prove", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module", "whose support has $\\delta$-dimension $\\leq k$, then", "the composition above sends $[\\mathcal{F}]$ into", "$\\bigoplus_{k' \\leq k} \\CH_{k'}(X) \\otimes \\mathbf{Q}$.", "\\item If $Z \\subset X$ is an integral closed subscheme", "of $\\delta$-dimension $k$, then the composition above", "sends $[\\mathcal{O}_Z]$ to an element whose degree $k$", "part is the class of $[Z]$ in $\\CH_k(X) \\otimes \\mathbf{Q}$.", "\\end{enumerate}", "Namely, if this holds, then our maps induce maps", "$\\text{gr}_kK'_0(X) \\otimes \\mathbf{Q} \\to CH_k(X) \\otimes \\mathbf{Q}$", "which are inverse to the canonical maps ", "$\\CH_k(X) \\otimes \\mathbf{Q} \\to \\text{gr}_k K'_0(X) \\otimes \\mathbf{Q}$", "given above the proposition.", "\\medskip\\noindent", "Given a coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "the composition above sends $[\\mathcal{F}]$ to", "$$", "ch(X \\to Y, \\mathcal{F}[0]) \\cap [Y] \\in \\CH_*(X) \\otimes \\mathbf{Q}", "$$", "If $\\mathcal{F}$ is (set theoretically) supported on a closed subscheme", "$Z \\subset X$, then we have", "$$", "ch(X \\to Y, \\mathcal{F}[0]) = (Z \\to X)_* \\circ ch(Z \\to Y, \\mathcal{F}[0])", "$$", "by Lemma \\ref{lemma-loc-chern-shrink-Z}. We conclude that in this", "case we end up in the image of $\\CH_*(Z) \\to \\CH_*(X)$. Hence", "we get condition (1).", "\\medskip\\noindent", "Let $Z \\subset X$ be an integral closed subscheme of $\\delta$-dimension $k$.", "The composition above sends $[\\mathcal{O}_Z]$ to the element", "$$", "ch(X \\to Y, \\mathcal{O}_Z[0]) \\cap [Y] =", "(Z \\to X)_* ch(Z \\to Y, \\mathcal{O}_Z[0]) \\cap [Y]", "$$", "by the same argument as above.", "Thus it suffices to prove that the degree $k$ part of", "$ch(Z \\to Y, \\mathcal{O}_Z[0]) \\cap [Y] \\in", "\\CH_*(Z) \\otimes \\mathbf{Q}$ is $[Z]$.", "Since $\\CH_k(Z) = \\mathbf{Z}$, in order", "to prove this we may replace $Y$ by an open neighbourhood of the", "generic point $\\xi$ of $Z$. Since the maximal ideal of the regular", "local ring $\\mathcal{O}_{X, \\xi}$ is generated by a", "regular sequence (Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM})", "we may assume the ideal of $Z$ is generated by a regular sequence, see", "Divisors, Lemma \\ref{divisors-lemma-Noetherian-scheme-regular-ideal}.", "Thus we deduce the result from Lemma \\ref{lemma-actual-computation}." ], "refs": [ "chow-remark-perf-Z-regular", "perfect-lemma-regular-resolution-property", "chow-remark-localized-chern-character-K", "chow-lemma-locally-equidimensional", "chow-remark-fundamental-class", "chow-lemma-loc-chern-shrink-Z", "algebra-lemma-regular-ring-CM", "divisors-lemma-Noetherian-scheme-regular-ideal", "chow-lemma-actual-computation" ], "ref_ids": [ 5959, 7090, 5961, 5761, 5948, 5801, 941, 7988, 5823 ] } ], "ref_ids": [ 5959, 5961 ] }, { "id": 5901, "type": "theorem", "label": "chow-proposition-compute-bivariant", "categories": [ "chow" ], "title": "chow-proposition-compute-bivariant", "contents": [ "\\begin{reference}", "\\cite[Proposition 17.4.2]{F}", "\\end{reference}", "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes locally", "of finite type over $S$. If $g$ is smooth of relative dimension $d$, then", "$A^p(X \\to Y) = A^{p - d}(X \\to Z)$." ], "refs": [], "proofs": [ { "contents": [ "We will use that smooth morphisms are local complete intersection", "morphisms whose gysin maps are defined (see Section \\ref{section-koszul}).", "In particular we have $g^! \\in A^{-d}(Y \\to Z)$. Then we can send", "$c \\in A^p(X \\to Y)$ to $c \\circ g^! \\in A^{p - d}(X \\to Z)$.", "\\medskip\\noindent", "Conversely, let $c' \\in A^{p - d}(X \\to Z)$. Denote $res(c')$ the restriction", "(Remark \\ref{remark-restriction-bivariant}) of $c'$ by the morphism $Y \\to Z$.", "Since the diagram", "$$", "\\xymatrix{", "X \\times_Z Y \\ar[r]_{\\text{pr}_2} \\ar[d]_{\\text{pr}_1} & Y \\ar[d]^g \\\\", "X \\ar[r]^f & Z", "}", "$$", "is cartesian we find $res(c') \\in A^{p - d}(X \\times_Z Y \\to Y)$.", "Let $\\Delta : Y \\to Y \\times_Z Y$ be the diagonal and denote", "$res(\\Delta^!)$ the restriction of $\\Delta^!$", "to $X \\times_Z Y$ by the morphism $X \\times_Z Y \\to Y \\times_Z Y$.", "Since the diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d] & X \\times_Z Y \\ar[d] \\\\", "Y \\ar[r]^-\\Delta & Y \\times_Z Y", "}", "$$", "is cartesian we see that $res(\\Delta^!) \\in A^d(X \\to X \\times_Z Y)$.", "Combining these two restrictions we obtain", "$$", "res(\\Delta^!) \\circ res(c') \\in A^p(X \\to Y)", "$$", "Thus we have produced maps $A^p(X \\to Y) \\to A^{p - d}(X \\to Z)$", "and $A^{p - d}(X \\to Z) \\to A^p(X \\to Y)$. To finish the proof we", "will show these maps are mutually inverse.", "\\medskip\\noindent", "Let us start with $c \\in A^p(X \\to Y)$. Consider the diagram", "$$", "\\xymatrix{", "X \\ar[d] \\ar[r] & Y \\ar[d] \\\\", "X \\times_Z Y \\ar[r] \\ar[d]^{\\text{pr}_1} &", "Y \\times_Z Y \\ar[r]_{p_2} \\ar[d]^{p_1} &", "Y \\ar[d]^g \\\\", "X \\ar[r]^f &", "Y \\ar[r]^g &", "Z", "}", "$$", "whose squares are carteisan. The lower two square of this diagram", "show that $res(c \\circ g^!) = res(c) \\cap p_2^!$ where in this formula", "$res(c)$ means the restriction of $c$ via $p_1$. Looking at the upper", "square of the diagram and using Lemma \\ref{lemma-lci-gysin-commutes}", "we get $c \\circ \\Delta^! = res(\\Delta^!) \\circ res(c)$.", "We compute", "\\begin{align*}", "res(\\Delta^!) \\circ res(c \\circ g^!)", "& =", "res(\\Delta^!) \\circ res(c) \\circ p_2^! \\\\", "& =", "c \\circ \\Delta^! \\circ p_2^! \\\\", "& =", "c", "\\end{align*}", "The final equality by Lemma \\ref{lemma-diagonal-identity}.", "\\medskip\\noindent", "Conversely, let us start with $c' \\in A^{p - d}(X \\to Z)$. Looking", "at the lower rectangle of the diagram above we find", "$res(c') \\circ g^! = \\text{pr}_1^! \\circ c'$.", "We compute", "\\begin{align*}", "res(\\Delta^!) \\circ res(c') \\circ g^!", "& =", "res(\\Delta^!) \\circ \\text{pr}_1^! \\circ c' \\\\", "& =", "c'", "\\end{align*}", "The final equality holds because the left two squares of", "the diagram show that", "$\\text{id} = res(\\Delta^! \\circ p_1^!) = res(\\Delta^!) \\circ \\text{pr}_1^!$.", "This finishes the proof." ], "refs": [ "chow-remark-restriction-bivariant", "chow-lemma-lci-gysin-commutes", "chow-lemma-diagonal-identity" ], "ref_ids": [ 5938, 5834, 5840 ] } ], "ref_ids": [] }, { "id": 5902, "type": "theorem", "label": "chow-proposition-length-determinant-periodic-complex", "categories": [ "chow" ], "title": "chow-proposition-length-determinant-periodic-complex", "contents": [ "Let $R$ be a local Noetherian ring with residue field $\\kappa$.", "Suppose that $(M, \\varphi, \\psi)$ is a $(2, 1)$-periodic", "complex over $R$. Assume", "\\begin{enumerate}", "\\item $M$ is a finite $R$-module,", "\\item the cohomology modules of $(M, \\varphi, \\psi)$ are of finite length, and", "\\item $\\dim(\\text{Supp}(M)) = 1$.", "\\end{enumerate}", "Let $\\mathfrak q_i$, $i = 1, \\ldots, t$ be the minimal", "primes of the support of $M$. Then we have\\footnote{", "Obviously we could get rid of the minus sign by redefining", "$\\det_\\kappa(M, \\varphi, \\psi)$ as the inverse of its", "current value, see Definition \\ref{definition-periodic-determinant}.}", "$$", "- e_R(M, \\varphi, \\psi) =", "\\sum\\nolimits_{i = 1, \\ldots, t}", "\\text{ord}_{R/\\mathfrak q_i}\\left(", "\\det\\nolimits_{\\kappa(\\mathfrak q_i)}", "(M_{\\mathfrak q_i}, \\varphi_{\\mathfrak q_i}, \\psi_{\\mathfrak q_i})", "\\right)", "$$" ], "refs": [ "chow-definition-periodic-determinant" ], "proofs": [ { "contents": [ "We first reduce to the case $t = 1$ in the following way.", "Note that", "$\\text{Supp}(M) = \\{\\mathfrak m, \\mathfrak q_1, \\ldots, \\mathfrak q_t\\}$,", "where $\\mathfrak m \\subset R$ is the maximal ideal.", "Let $M_i$ denote the image of $M \\to M_{\\mathfrak q_i}$,", "so $\\text{Supp}(M_i) = \\{\\mathfrak m, \\mathfrak q_i\\}$.", "The map $\\varphi$ (resp.\\ $\\psi$) induces an $R$-module map", "$\\varphi_i : M_i \\to M_i$ (resp.\\ $\\psi_i : M_i \\to M_i$).", "Thus we get a morphism of $(2, 1)$-periodic complexes", "$$", "(M, \\varphi, \\psi) \\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, t} (M_i, \\varphi_i, \\psi_i).", "$$", "The kernel and cokernel of this map have support contained in", "$\\{\\mathfrak m\\}$. Hence by Lemma \\ref{lemma-compare-periodic-lengths}", "we have", "$$", "e_R(M, \\varphi, \\psi) =", "\\sum\\nolimits_{i = 1, \\ldots, t}", "e_R(M_i, \\varphi_i, \\psi_i)", "$$", "On the other hand we clearly have $M_{\\mathfrak q_i} = M_{i, \\mathfrak q_i}$,", "and hence the terms of the right hand side of the formula of the", "lemma are equal to the expressions", "$$", "\\text{ord}_{R/\\mathfrak q_i}\\left(", "\\det\\nolimits_{\\kappa(\\mathfrak q_i)}", "(M_{i, \\mathfrak q_i}, \\varphi_{i, \\mathfrak q_i}, \\psi_{i, \\mathfrak q_i})", "\\right)", "$$", "In other words, if we can prove the lemma for each of the modules", "$M_i$, then the lemma holds. This reduces us to the case $t = 1$.", "\\medskip\\noindent", "Assume we have a $(2, 1)$-periodic complex $(M, \\varphi, \\psi)$", "over a Noetherian local ring with $M$ a finite $R$-module,", "$\\text{Supp}(M) = \\{\\mathfrak m, \\mathfrak q\\}$, and", "finite length cohomology modules. The proof in this case", "follows from Lemma \\ref{lemma-key-lemma} and careful bookkeeping.", "Denote", "$K_\\varphi = \\Ker(\\varphi)$,", "$I_\\varphi = \\Im(\\varphi)$,", "$K_\\psi = \\Ker(\\psi)$, and", "$I_\\psi = \\Im(\\psi)$.", "Since $R$ is Noetherian these are all finite $R$-modules.", "Set", "$$", "a = \\text{length}_{R_{\\mathfrak q}}(I_{\\varphi, \\mathfrak q})", "= \\text{length}_{R_{\\mathfrak q}}(K_{\\psi, \\mathfrak q}),", "\\quad", "b = \\text{length}_{R_{\\mathfrak q}}(I_{\\psi, \\mathfrak q})", "= \\text{length}_{R_{\\mathfrak q}}(K_{\\varphi, \\mathfrak q}).", "$$", "Equalities because the complex becomes exact after localizing at", "$\\mathfrak q$. Note that $l = \\text{length}_{R_{\\mathfrak q}}(M_{\\mathfrak q})$", "is equal to $l = a + b$.", "\\medskip\\noindent", "We are going to use Lemma \\ref{lemma-good-sequence-exists}", "to choose sequences of elements in finite $R$-modules", "$N$ with support contained in $\\{\\mathfrak m, \\mathfrak q\\}$.", "In this case $N_{\\mathfrak q}$ has finite length, say $n \\in \\mathbf{N}$.", "Let us call a sequence $w_1, \\ldots, w_n \\in N$", "with properties (1) and (2) of Lemma \\ref{lemma-good-sequence-exists}", "a ``good sequence''. Note that the quotient", "$N/\\langle w_1, \\ldots, w_n \\rangle$ of $N$ by the submodule generated by", "a good sequence has support (contained in) $\\{\\mathfrak m\\}$", "and hence has finite length (Algebra, Lemma \\ref{algebra-lemma-support-point}).", "Moreover, the symbol", "$[w_1, \\ldots, w_n] \\in \\det_{\\kappa(\\mathfrak q)}(N_{\\mathfrak q})$", "is a generator, see Lemma \\ref{lemma-determinant-dimension-one}.", "\\medskip\\noindent", "Having said this we choose good sequences", "$$", "\\begin{matrix}", "x_1, \\ldots, x_b & \\text{in} & K_\\varphi, &", "t_1, \\ldots, t_a & \\text{in} & K_\\psi, \\\\", "y_1, \\ldots, y_a & \\text{in} & I_\\varphi \\cap \\langle t_1, \\ldots t_a\\rangle, &", "s_1, \\ldots, s_b & \\text{in} & I_\\psi \\cap \\langle x_1, \\ldots, x_b\\rangle.", "\\end{matrix}", "$$", "We will adjust our choices a little bit as follows.", "Choose lifts $\\tilde y_i \\in M$ of $y_i \\in I_\\varphi$", "and $\\tilde s_i \\in M$ of $s_i \\in I_\\psi$. It may not be the case", "that $\\mathfrak q \\tilde y_1 \\subset \\langle x_1, \\ldots, x_b\\rangle$", "and it may not be the case that", "$\\mathfrak q \\tilde s_1 \\subset \\langle t_1, \\ldots, t_a\\rangle$.", "However, using that $\\mathfrak q$ is finitely generated (as in the proof", "of Lemma \\ref{lemma-good-sequence-exists}) we can find a", "$d \\in R$, $d \\not \\in \\mathfrak q$ such that", "$\\mathfrak q d\\tilde y_1 \\subset \\langle x_1, \\ldots, x_b\\rangle$", "and", "$\\mathfrak q d\\tilde s_1 \\subset \\langle t_1, \\ldots, t_a\\rangle$.", "Thus after replacing $y_i$ by $dy_i$,", "$\\tilde y_i$ by $d\\tilde y_i$, $s_i$ by $ds_i$ and $\\tilde s_i$", "by $d\\tilde s_i$ we see that we may assume also that", "$x_1, \\ldots, x_b, \\tilde y_1, \\ldots, \\tilde y_b$", "and $t_1, \\ldots, t_a, \\tilde s_1, \\ldots, \\tilde s_b$", "are good sequences in $M$.", "\\medskip\\noindent", "Finally, we choose a good sequence", "$z_1, \\ldots, z_l$ in the finite $R$-module", "$$", "\\langle", "x_1, \\ldots, x_b, \\tilde y_1, \\ldots, \\tilde y_a", "\\rangle", "\\cap", "\\langle", "t_1, \\ldots, t_a, \\tilde s_1, \\ldots, \\tilde s_b", "\\rangle.", "$$", "Note that this is also a good sequence in $M$.", "\\medskip\\noindent", "Since $I_{\\varphi, \\mathfrak q} = K_{\\psi, \\mathfrak q}$", "there is a unique element $h \\in \\kappa(\\mathfrak q)$ such that", "$[y_1, \\ldots, y_a] = h [t_1, \\ldots, t_a]$", "inside $\\det_{\\kappa(\\mathfrak q)}(K_{\\psi, \\mathfrak q})$.", "Similarly, as $I_{\\psi, \\mathfrak q} = K_{\\varphi, \\mathfrak q}$", "there is a unique element $h \\in \\kappa(\\mathfrak q)$ such that", "$[s_1, \\ldots, s_b] = g [x_1, \\ldots, x_b]$", "inside $\\det_{\\kappa(\\mathfrak q)}(K_{\\varphi, \\mathfrak q})$.", "We can also do this with the three good sequences we have", "in $M$. All in all we get the following identities", "\\begin{align*}", "[y_1, \\ldots, y_a]", "& =", "h [t_1, \\ldots, t_a] \\\\", "[s_1, \\ldots, s_b]", "& =", "g [x_1, \\ldots, x_b] \\\\", "[z_1, \\ldots, z_l]", "& =", "f_\\varphi [x_1, \\ldots, x_b, \\tilde y_1, \\ldots, \\tilde y_a] \\\\", "[z_1, \\ldots, z_l]", "& =", "f_\\psi [t_1, \\ldots, t_a, \\tilde s_1, \\ldots, \\tilde s_b]", "\\end{align*}", "for some $g, h, f_\\varphi, f_\\psi \\in \\kappa(\\mathfrak q)$.", "\\medskip\\noindent", "Having set up all this", "notation let us compute $\\det_{\\kappa(\\mathfrak q)}(M, \\varphi, \\psi)$.", "Namely, consider the element $[z_1, \\ldots, z_l]$.", "Under the map $\\gamma_\\psi \\circ \\sigma \\circ \\gamma_\\varphi^{-1}$", "of Definition \\ref{definition-periodic-determinant} we have", "\\begin{eqnarray*}", "[z_1, \\ldots, z_l] & = &", "f_\\varphi [x_1, \\ldots, x_b, \\tilde y_1, \\ldots, \\tilde y_a] \\\\", "& \\mapsto & f_\\varphi [x_1, \\ldots, x_b] \\otimes [y_1, \\ldots, y_a] \\\\", "& \\mapsto &", "f_\\varphi h/g [t_1, \\ldots, t_a] \\otimes [s_1, \\ldots, s_b] \\\\", "& \\mapsto &", "f_\\varphi h/g [t_1, \\ldots, t_a, \\tilde s_1, \\ldots, \\tilde s_b] \\\\", "& = &", "f_\\varphi h/f_\\psi g [z_1, \\ldots, z_l]", "\\end{eqnarray*}", "This means that", "$\\det_{\\kappa(\\mathfrak q)}", "(M_{\\mathfrak q}, \\varphi_{\\mathfrak q}, \\psi_{\\mathfrak q})$", "is equal to $f_\\varphi h/f_\\psi g$ up to a sign.", "\\medskip\\noindent", "We abbreviate the following quantities", "\\begin{eqnarray*}", "k_\\varphi & = & \\text{length}_R(K_\\varphi/\\langle x_1, \\ldots, x_b\\rangle) \\\\", "k_\\psi & = & \\text{length}_R(K_\\psi/\\langle t_1, \\ldots, t_a\\rangle) \\\\", "i_\\varphi & = & \\text{length}_R(I_\\varphi/\\langle y_1, \\ldots, y_a\\rangle) \\\\", "i_\\psi & = & \\text{length}_R(I_\\psi/\\langle s_1, \\ldots, s_a\\rangle) \\\\", "m_\\varphi & = & \\text{length}_R(M/", "\\langle x_1, \\ldots, x_b, \\tilde y_1, \\ldots, \\tilde y_a\\rangle) \\\\", "m_\\psi & = & \\text{length}_R(M/", "\\langle t_1, \\ldots, t_a, \\tilde s_1, \\ldots, \\tilde s_b\\rangle) \\\\", "\\delta_\\varphi & = & \\text{length}_R(", "\\langle x_1, \\ldots, x_b, \\tilde y_1, \\ldots, \\tilde y_a\\rangle", "\\langle z_1, \\ldots, z_l\\rangle) \\\\", "\\delta_\\psi & = & \\text{length}_R(", "\\langle t_1, \\ldots, t_a, \\tilde s_1, \\ldots, \\tilde s_b\\rangle", "\\langle z_1, \\ldots, z_l\\rangle)", "\\end{eqnarray*}", "Using the exact sequences $0 \\to K_\\varphi \\to M \\to I_\\varphi \\to 0$", "we get $m_\\varphi = k_\\varphi + i_\\varphi$. Similarly we have", "$m_\\psi = k_\\psi + i_\\psi$. We have", "$\\delta_\\varphi + m_\\varphi = \\delta_\\psi + m_\\psi$ since this", "is equal to the colength of $\\langle z_1, \\ldots, z_l \\rangle$", "in $M$. Finally, we have", "$$", "\\delta_\\varphi = \\text{ord}_{R/\\mathfrak q}(f_\\varphi),", "\\quad", "\\delta_\\psi = \\text{ord}_{R/\\mathfrak q}(f_\\psi)", "$$", "by our first application of the key Lemma \\ref{lemma-key-lemma}.", "\\medskip\\noindent", "Next, let us compute the multiplicity of the periodic complex", "\\begin{eqnarray*}", "e_R(M, \\varphi, \\psi) & = &", "\\text{length}_R(K_\\varphi/I_\\psi) - \\text{length}_R(K_\\psi/I_\\varphi) \\\\", "& = &", "\\text{length}_R(", "\\langle x_1, \\ldots, x_b\\rangle/", "\\langle s_1, \\ldots, s_b\\rangle)", "+ k_\\varphi - i_\\psi \\\\", "& & -", "\\text{length}_R(", "\\langle t_1, \\ldots, t_a\\rangle/", "\\langle y_1, \\ldots, y_a\\rangle)", "- k_\\psi + i_\\varphi \\\\", "& = &", "\\text{ord}_{R/\\mathfrak q}(g/h) + k_\\varphi - i_\\psi - k_\\psi + i_\\varphi \\\\", "& = &", "\\text{ord}_{R/\\mathfrak q}(g/h) + m_\\varphi - m_\\psi \\\\", "& = &", "\\text{ord}_{R/\\mathfrak q}(g/h) + \\delta_\\psi - \\delta_\\varphi \\\\", "& = &", "\\text{ord}_{R/\\mathfrak q}(f_\\psi g/f_\\varphi h)", "\\end{eqnarray*}", "where we used the key Lemma \\ref{lemma-key-lemma} twice in the third equality.", "By our computation of $\\det_{\\kappa(\\mathfrak q)}", "(M_{\\mathfrak q}, \\varphi_{\\mathfrak q}, \\psi_{\\mathfrak q})$", "this proves the proposition." ], "refs": [ "chow-lemma-compare-periodic-lengths", "chow-lemma-key-lemma", "chow-lemma-good-sequence-exists", "chow-lemma-good-sequence-exists", "algebra-lemma-support-point", "chow-lemma-determinant-dimension-one", "chow-lemma-good-sequence-exists", "chow-definition-periodic-determinant", "chow-lemma-key-lemma", "chow-lemma-key-lemma" ], "ref_ids": [ 5651, 5891, 5892, 5892, 693, 5869, 5892, 5927, 5891, 5891 ] } ], "ref_ids": [ 5927 ] }, { "id": 5968, "type": "theorem", "label": "flat-theorem-finite-type-flat", "categories": [ "flat" ], "title": "flat-theorem-finite-type-flat", "contents": [ "\\begin{slogan}", "The flat locus is open (non-Noetherian version).", "\\end{slogan}", "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume", "\\begin{enumerate}", "\\item $X \\to S$ is locally of finite presentation,", "\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type, and", "\\item the set of weakly associated points of $S$ is locally finite in $S$.", "\\end{enumerate}", "Then $U = \\{x \\in X \\mid \\mathcal{F}\\text{ flat at }x\\text{ over }S\\}$", "is open in $X$ and $\\mathcal{F}|_U$ is an $\\mathcal{O}_U$-module", "of finite presentation and flat over $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$ be such that $\\mathcal{F}$ is flat at $x$ over $S$.", "We have to find an open neighbourhood of $x$ such that $\\mathcal{F}$ restricts", "to a $S$-flat finitely presented module on this neighbourhood.", "The problem is local on $X$ and $S$, hence we may assume that $X$ and $S$", "are affine. As $\\mathcal{F}_x$ is a finitely presented", "$\\mathcal{O}_{X, x}$-module by", "Lemma \\ref{lemma-finite-type-flat-at-point-local}", "we conclude from", "Algebra, Lemma \\ref{algebra-lemma-construct-fp-module-from-stalk}", "there exists a finitely presented $\\mathcal{O}_X$-module $\\mathcal{F}'$", "and a map $\\varphi : \\mathcal{F}' \\to \\mathcal{F}$ which induces", "an isomorphism $\\varphi_x : \\mathcal{F}'_x \\to \\mathcal{F}_x$. In particular", "we see that $\\mathcal{F}'$ is flat over $S$ at $x$, hence by openness", "of flatness", "More on Morphisms, Theorem \\ref{more-morphisms-theorem-openness-flatness}", "we see that after shrinking $X$ we may assume that", "$\\mathcal{F}'$ is flat over $S$. As $\\mathcal{F}$ is of finite type", "after shrinking $X$ we may assume that $\\varphi$ is surjective, see", "Modules, Lemma \\ref{modules-lemma-finite-type-surjective-on-stalk}", "or alternatively use Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK}).", "By", "Lemma \\ref{lemma-bourbaki-finite-type-general-base}", "we have", "$$", "\\text{WeakAss}_X(\\mathcal{F}') \\subset", "\\bigcup\\nolimits_{s \\in \\text{WeakAss}(S)} \\text{Ass}_{X_s}(\\mathcal{F}'_s)", "$$", "As $\\text{WeakAss}(S)$ is finite by assumption and since", "$\\text{Ass}_{X_s}(\\mathcal{F}'_s)$ is finite by", "Divisors, Lemma \\ref{divisors-lemma-finite-ass}", "we conclude that $\\text{WeakAss}_X(\\mathcal{F}')$ is finite. Using", "Algebra, Lemma \\ref{algebra-lemma-silly}", "we may, after shrinking $X$ once more, assume that", "$\\text{WeakAss}_X(\\mathcal{F}')$ is contained in the generalization", "of $x$. Now consider $\\mathcal{K} = \\Ker(\\varphi)$. We have", "$\\text{WeakAss}_X(\\mathcal{K}) \\subset \\text{WeakAss}_X(\\mathcal{F}')$", "(by", "Divisors, Lemma \\ref{divisors-lemma-ses-weakly-ass})", "but on the other hand, $\\varphi_x$ is an isomorphism, also $\\varphi_{x'}$", "is an isomorphism for all $x' \\leadsto x$. We conclude that", "$\\text{WeakAss}_X(\\mathcal{K}) = \\emptyset$ whence", "$\\mathcal{K} = 0$ by", "Divisors, Lemma \\ref{divisors-lemma-weakly-ass-zero}." ], "refs": [ "flat-lemma-finite-type-flat-at-point-local", "algebra-lemma-construct-fp-module-from-stalk", "more-morphisms-theorem-openness-flatness", "modules-lemma-finite-type-surjective-on-stalk", "algebra-lemma-NAK", "flat-lemma-bourbaki-finite-type-general-base", "divisors-lemma-finite-ass", "algebra-lemma-silly", "divisors-lemma-ses-weakly-ass", "divisors-lemma-weakly-ass-zero" ], "ref_ids": [ 6022, 1083, 13670, 13238, 401, 6042, 7859, 378, 7874, 7875 ] } ], "ref_ids": [] }, { "id": 5969, "type": "theorem", "label": "flat-theorem-flattening-map", "categories": [ "flat" ], "title": "flat-theorem-flattening-map", "contents": [ "In", "Situation \\ref{situation-iso}", "assume", "\\begin{enumerate}", "\\item $f$ is of finite presentation,", "\\item $\\mathcal{F}$ is of finite presentation, flat over $S$, and", "pure relative to $S$, and", "\\item $u$ is surjective.", "\\end{enumerate}", "Then $F_{iso}$ is representable by a closed immersion $Z \\to S$.", "Moreover $Z \\to S$ is of finite presentation if $\\mathcal{G}$ is", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "We will use without further mention that $\\mathcal{F}$ is universally pure", "over $S$, see", "Lemma \\ref{lemma-finite-type-flat-pure-along-fibre-is-universal}.", "By", "Lemma \\ref{lemma-iso-sheaf}", "and", "Descent, Lemmas \\ref{descent-lemma-closed-immersion} and", "\\ref{descent-lemma-descent-data-sheaves}", "the question is local for the \\'etale topology on $S$.", "Hence it suffices to prove, given $s \\in S$, that there exists", "an \\'etale neighbourhood of $(S, s)$ so that the theorem holds.", "\\medskip\\noindent", "Using", "Lemma \\ref{lemma-finite-presentation-flat-along-fibre}", "and after replacing $S$ by an elementary \\'etale neighbourhood of $s$", "we may assume there exists a commutative diagram", "$$", "\\xymatrix{", "X \\ar[dr] & & X' \\ar[ll]^g \\ar[ld] \\\\", "& S &", "}", "$$", "of schemes of finite presentation over $S$,", "where $g$ is \\'etale, $X_s \\subset g(X')$, the schemes $X'$ and $S$ are affine,", "$\\Gamma(X', g^*\\mathcal{F})$ a projective $\\Gamma(S, \\mathcal{O}_S)$-module.", "Note that $g^*\\mathcal{F}$ is universally pure over $S$, see", "Lemma \\ref{lemma-affine-locally-projective-pure}.", "Hence by", "Lemma \\ref{lemma-criterion}", "we see that the open $g(X')$ contains the points of", "$\\text{Ass}_{X/S}(\\mathcal{F})$ lying over $\\Spec(\\mathcal{O}_{S, s})$.", "Set", "$$", "E = \\{t \\in S \\mid \\text{Ass}_{X_t}(\\mathcal{F}_t) \\subset g(X') \\}.", "$$", "By", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-relative-assassin-constructible}", "$E$ is a constructible subset of $S$. We have seen that", "$\\Spec(\\mathcal{O}_{S, s}) \\subset E$. By", "Morphisms, Lemma \\ref{morphisms-lemma-constructible-containing-open}", "we see that $E$ contains an open neighbourhood of $s$. Hence after", "replacing $S$ by a smaller affine neighbourhood of $s$ we may assume that", "$\\text{Ass}_{X/S}(\\mathcal{F}) \\subset g(X')$.", "\\medskip\\noindent", "Since we have assumed that $u$ is surjective we have $F_{iso} = F_{inj}$. From", "Lemma \\ref{lemma-universally-separating}", "it follows that $u : \\mathcal{F} \\to \\mathcal{G}$ is injective if and only if", "$g^*u : g^*\\mathcal{F} \\to g^*\\mathcal{G}$ is injective, and the same remains", "true after any base change. Hence we have reduced to the case where,", "in addition to the assumptions in the theorem, $X \\to S$ is a morphism of", "affine schemes and $\\Gamma(X, \\mathcal{F})$ is a projective", "$\\Gamma(S, \\mathcal{O}_S)$-module. This case follows immediately from", "Lemma \\ref{lemma-flattening-module-map}.", "\\medskip\\noindent", "To see that $Z$ is of finite presentation if $\\mathcal{G}$ is of finite", "presentation, combine", "Lemma \\ref{lemma-iso-sheaf} part (4)", "with", "Limits, Remark \\ref{limits-remark-limit-preserving}." ], "refs": [ "flat-lemma-finite-type-flat-pure-along-fibre-is-universal", "flat-lemma-iso-sheaf", "descent-lemma-closed-immersion", "descent-lemma-descent-data-sheaves", "flat-lemma-finite-presentation-flat-along-fibre", "flat-lemma-affine-locally-projective-pure", "flat-lemma-criterion", "more-morphisms-lemma-relative-assassin-constructible", "morphisms-lemma-constructible-containing-open", "flat-lemma-universally-separating", "flat-lemma-flattening-module-map", "flat-lemma-iso-sheaf", "limits-remark-limit-preserving" ], "ref_ids": [ 6067, 6076, 14749, 14751, 6031, 6064, 6066, 13812, 5251, 6086, 6087, 6076, 15130 ] } ], "ref_ids": [] }, { "id": 5970, "type": "theorem", "label": "flat-theorem-flattening-local", "categories": [ "flat" ], "title": "flat-theorem-flattening-local", "contents": [ "In", "Situation \\ref{situation-flat-at-point}", "assume $A$ is henselian, $B$ is essentially of finite type over $A$, and", "$M$ is a finite $B$-module. Then there exists an ideal", "$I \\subset A$ such that $A/I$ corepresents the functor $F_{lf}$ on the category", "$\\mathcal{C}$. In other words given a local homomorphism of local rings", "$\\varphi : A \\to A'$ with $B' = B \\otimes_A A'$ and $M' = M \\otimes_A A'$", "the following are equivalent:", "\\begin{enumerate}", "\\item $\\forall \\mathfrak q \\in V(\\mathfrak m_{A'}B' + \\mathfrak m_B B')", "\\subset \\Spec(B') :", "M'_{\\mathfrak q}\\text{ is flat over }A'$, and", "\\item $\\varphi(I) = 0$.", "\\end{enumerate}", "If $B$ is essentially of finite presentation over $A$ and $M$", "of finite presentation over $B$, then $I$ is a finitely generated ideal." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite type ring map $A \\to C$ and a finite $C$-module $N$", "and a prime $\\mathfrak q$ of $C$ such that $B = C_{\\mathfrak q}$", "and $M = N_{\\mathfrak q}$. In the following, when we say", "``the theorem holds for $(N/C/A, \\mathfrak q)$ we mean that", "it holds for $(A \\to B, M)$ where $B = C_{\\mathfrak q}$ and", "$M = N_{\\mathfrak q}$. By", "Lemma \\ref{lemma-flat-at-point-go-up}", "the functor $F_{lf}$ is unchanged if we replace $B$ by a local ring", "flat over $B$. Hence, since $A$ is henselian, we may apply", "Lemma \\ref{lemma-existence-algebra}", "and assume that there exists a complete d\\'evissage of", "$N/C/A$ at $\\mathfrak q$.", "\\medskip\\noindent", "Let $(A_i, B_i, M_i, \\alpha_i, \\mathfrak q_i)_{i = 1, \\ldots, n}$", "be such a complete d\\'evissage of $N/C/A$ at $\\mathfrak q$. Let", "$\\mathfrak q'_i \\subset A_i$ be the unique prime lying over", "$\\mathfrak q_i \\subset B_i$ as in", "Definition \\ref{definition-complete-devissage-at-x-algebra}.", "Since $C \\to A_1$ is surjective and $N \\cong M_1$ as $C$-modules,", "we see by", "Lemma \\ref{lemma-flat-at-point-finite}", "it suffices to prove the theorem holds for $(M_1/A_1/A, \\mathfrak q'_1)$.", "Since $B_1 \\to A_1$ is finite and $\\mathfrak q_1$ is the only prime", "of $B_1$ over $\\mathfrak q'_1$ we see that", "$(A_1)_{\\mathfrak q'_1} \\to (B_1)_{\\mathfrak q_1}$ is finite (see", "Algebra, Lemma \\ref{algebra-lemma-unique-prime-over-localize-below} or", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-finite-morphism-single-point-in-fibre}).", "Hence by", "Lemma \\ref{lemma-flat-at-point-finite}", "it suffices to prove the theorem holds for $(M_1/B_1/A, \\mathfrak q_1)$.", "\\medskip\\noindent", "At this point we may assume, by induction on the length $n$ of the", "d\\'evissage, that the theorem holds for $(M_2/B_2/A, \\mathfrak q_2)$.", "(If $n = 1$, then $M_2 = 0$ which is flat over $A$.)", "Reversing the last couple of steps of the previous paragraph, using", "that $M_2 \\cong \\Coker(\\alpha_2)$ as $B_1$-modules, we see", "that the theorem holds for $(\\Coker(\\alpha_1)/B_1/A, \\mathfrak q_1)$.", "\\medskip\\noindent", "Let $A'$ be an object of $\\mathcal{C}$. At this point we use", "Lemma \\ref{lemma-induction-step}", "to see that if $(M_1 \\otimes_A A')_{\\mathfrak q'}$ is flat", "over $A'$ for a prime $\\mathfrak q'$ of $B_1 \\otimes_A A'$", "lying over $\\mathfrak m_{A'}$, then", "$(\\Coker(\\alpha_1) \\otimes_A A')_{\\mathfrak q'}$ is flat over $A'$.", "Hence we conclude that $F_{lf}$ is a subfunctor of the", "functor $F'_{lf}$ associated to the module", "$\\Coker(\\alpha_1)_{\\mathfrak q_1}$ over $(B_1)_{\\mathfrak q_1}$.", "By the previous paragraph we know $F'_{lf}$ is corepresented by", "$A/J$ for some ideal $J \\subset A$. Hence we may replace $A$ by", "$A/J$ and assume that $\\Coker(\\alpha_1)_{\\mathfrak q_1}$ is", "flat over $A$.", "\\medskip\\noindent", "Since $\\Coker(\\alpha_1)$ is a $B_1$-module for which", "there exist a complete d\\'evissage of $N_1/B_1/A$ at $\\mathfrak q_1$", "and since $\\Coker(\\alpha_1)_{\\mathfrak q_1}$ is", "flat over $A$ by", "Lemma \\ref{lemma-complete-devissage-flat-finite-type-module}", "we see that $\\Coker(\\alpha_1)$ is free as an $A$-module, in particular", "flat as an $A$-module. Hence", "Lemma \\ref{lemma-induction-step}", "implies $F_{lf}(A')$ is nonempty if and only if $\\alpha \\otimes 1_{A'}$", "is injective. Let $N_1 = \\Im(\\alpha_1) \\subset M_1$ so that", "we have exact sequences", "$$", "0 \\to N_1 \\to M_1 \\to \\Coker(\\alpha_1) \\to 0", "\\quad\\text{and}\\quad", "B_1^{\\oplus r_1} \\to N_1 \\to 0", "$$", "The flatness of $\\Coker(\\alpha_1)$ implies the first sequence", "is universally exact (see", "Algebra, Lemma \\ref{algebra-lemma-flat-universally-injective}).", "Hence $\\alpha \\otimes 1_{A'}$ is injective if and only if", "$B_1^{\\oplus r_1} \\otimes_A A' \\to N_1 \\otimes_A A'$", "is an isomorphism. Finally,", "Theorem \\ref{theorem-flattening-map}", "applies to show this functor is corepresentable by $A/I$ for some ideal $I$", "and we conclude $F_{lf}$ is corepresentable by $A/I$ also.", "\\medskip\\noindent", "To prove the final statement, suppose that $A \\to B$ is essentially of finite", "presentation and $M$ of finite presentation over $B$. Let $I \\subset A$", "be the ideal such that $F_{lf}$ is corepresented by $A/I$.", "Write $I = \\bigcup I_\\lambda$ where $I_\\lambda$ ranges over the finitely", "generated ideals contained in $I$. Then, since $F_{lf}(A/I) = \\{*\\}$", "we see that $F_{lf}(A/I_\\lambda) = \\{*\\}$ for some $\\lambda$, see", "Lemma \\ref{lemma-flat-at-point} part (2).", "Clearly this implies that $I = I_\\lambda$." ], "refs": [ "flat-lemma-flat-at-point-go-up", "flat-lemma-existence-algebra", "flat-definition-complete-devissage-at-x-algebra", "flat-lemma-flat-at-point-finite", "algebra-lemma-unique-prime-over-localize-below", "more-morphisms-lemma-finite-morphism-single-point-in-fibre", "flat-lemma-flat-at-point-finite", "flat-lemma-induction-step", "flat-lemma-complete-devissage-flat-finite-type-module", "flat-lemma-induction-step", "algebra-lemma-flat-universally-injective", "flat-theorem-flattening-map", "flat-lemma-flat-at-point" ], "ref_ids": [ 6079, 6001, 6214, 6078, 556, 13920, 6078, 6015, 6016, 6015, 809, 5969, 6077 ] } ], "ref_ids": [] }, { "id": 5971, "type": "theorem", "label": "flat-theorem-check-flatness-at-associated-points", "categories": [ "flat" ], "title": "flat-theorem-check-flatness-at-associated-points", "contents": [ "Let $f : X \\to S$ be locally of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.", "Let $x \\in X$ with image $s \\in S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is flat at $x$ over $S$, and", "\\item for every $x' \\in \\text{Ass}_{X_s}(\\mathcal{F}_s)$ which", "specializes to $x$ we have that $\\mathcal{F}$ is flat at $x'$ over $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2) as $\\mathcal{F}_{x'}$ is a localization", "of $\\mathcal{F}_x$ for every point which specializes to $x$.", "Set $A = \\mathcal{O}_{S, s}$, $B = \\mathcal{O}_{X, x}$ and", "$N = \\mathcal{F}_x$. Let $\\Sigma \\subset B$ be the multiplicative", "subset of $B$ of elements which act as nonzerodivisors on $N/\\mathfrak m_AN$.", "Assumption (2) implies that $\\Sigma^{-1}N$ is $A$-flat by the description", "of $\\Spec(\\Sigma^{-1}N)$ in", "Lemma \\ref{lemma-homothety-spectrum}.", "On the other hand, the map $N \\to \\Sigma^{-1}N$ is injective modulo", "$\\mathfrak m_A$ by construction. Hence applying", "Lemma \\ref{lemma-upstairs-finite-type-injective-into-flat-mod-m}", "we win." ], "refs": [ "flat-lemma-homothety-spectrum", "flat-lemma-upstairs-finite-type-injective-into-flat-mod-m" ], "ref_ids": [ 6002, 6093 ] } ], "ref_ids": [] }, { "id": 5972, "type": "theorem", "label": "flat-theorem-flat-dimension-n-representable", "categories": [ "flat" ], "title": "flat-theorem-flat-dimension-n-representable", "contents": [ "In Situation \\ref{situation-flat-dimension-n}.", "Assume moreover that $f$ is of finite presentation, that", "$\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation,", "and that $\\mathcal{F}$ is pure relative to $S$.", "Then $F_n$ is representable by a monomorphism", "$Z_n \\to S$ of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "The functor $F_n$ is a sheaf for the fppf topology by", "Lemma \\ref{lemma-flat-dimension-n}.", "Observe that a monomorphism of finite presentation is", "separated and quasi-finite (Morphisms, Lemma", "\\ref{morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}).", "Hence combining", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves},", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend}", ", and", "Descent, Lemmas \\ref{descent-lemma-descending-property-monomorphism} and", "\\ref{descent-lemma-descending-property-finite-presentation}", "we see that the question is local for the \\'etale topology on $S$.", "\\medskip\\noindent", "In particular the situation is local for the Zariski topology on $S$", "and we may assume that $S$ is affine. In this case the dimension of the", "fibres of $f$ is bounded above, hence we see that $F_n$ is representable", "for $n$ large enough. Thus we may use descending induction on $n$.", "Suppose that we know $F_{n + 1}$ is representable by a monomorphism", "$Z_{n + 1} \\to S$ of finite presentation. Consider the base change", "$X_{n + 1} = Z_{n + 1} \\times_S X$ and the pullback $\\mathcal{F}_{n + 1}$", "of $\\mathcal{F}$ to $X_{n + 1}$. The morphism $Z_{n + 1} \\to S$ is", "quasi-finite as it is a monomorphism of finite presentation, hence", "Lemma \\ref{lemma-quasi-finite-base-change}", "implies that $\\mathcal{F}_{n + 1}$ is pure relative to $Z_{n + 1}$.", "Since $F_n$ is a subfunctor of $F_{n + 1}$ we conclude that in order", "to prove the result for $F_n$ it suffices to prove the result for the", "corresponding functor for the situation", "$\\mathcal{F}_{n + 1}/X_{n + 1}/Z_{n + 1}$.", "In this way we reduce to proving the result for $F_n$ in case", "$S_{n + 1} = S$, i.e., we may assume that $\\mathcal{F}$ is flat", "in dimensions $\\geq n + 1$ over $S$.", "\\medskip\\noindent", "Fix $n$ and assume $\\mathcal{F}$ is flat in dimensions $\\geq n + 1$", "over $S$. To finish the proof we have to show that $F_n$ is representable", "by a monomorphism $Z_n \\to S$ of finite presentation.", "Since the question is local in the \\'etale topology on $S$ it suffices to", "show that for every $s \\in S$ there exists an elementary \\'etale neighbourhood", "$(S', s') \\to (S, s)$ such that the result holds after base change to $S'$.", "Thus by", "Lemma \\ref{lemma-existence-complete}", "we may assume there exist \\'etale morphisms $h_j : Y_j \\to X$,", "$j = 1, \\ldots, m$ such that for each $j$ there exists a complete", "d\\'evissage of $\\mathcal{F}_j/Y_j/S$ over $s$,", "where $\\mathcal{F}_j$ is the pullback of $\\mathcal{F}$ to $Y_j$", "and such that $X_s \\subset \\bigcup h_j(Y_j)$. Note that by", "Lemma \\ref{lemma-localize-flat-dimension-n}", "the sheaves $\\mathcal{F}_j$ are still flat over in", "dimensions $\\geq n + 1$ over $S$.", "Set $W = \\bigcup h_j(Y_j)$, which is a quasi-compact open of $X$.", "As $\\mathcal{F}$ is pure along $X_s$ we see that", "$$", "E = \\{t \\in S \\mid \\text{Ass}_{X_t}(\\mathcal{F}_t) \\subset W \\}.", "$$", "contains all generalizations of $s$. By", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-relative-assassin-constructible}", "$E$ is a constructible subset of $S$. We have seen that", "$\\Spec(\\mathcal{O}_{S, s}) \\subset E$. By", "Morphisms, Lemma \\ref{morphisms-lemma-constructible-containing-open}", "we see that $E$ contains an open neighbourhood of $s$.", "Hence after shrinking $S$ we may assume that $E = S$.", "It follows from", "Lemma \\ref{lemma-localize-flat-dimension-n}", "that it suffices to prove the lemma for the functor $F_n$ associated to", "$X = \\coprod Y_j$ and $\\mathcal{F} = \\coprod \\mathcal{F}_j$.", "If $F_{j, n}$ denotes the functor for $Y_j \\to S$ and the sheaf", "$\\mathcal{F}_i$ we see that $F_n = \\prod F_{j, n}$. Hence it suffices", "to prove each $F_{j, n}$ is representable by some monomorphism", "$Z_{j, n} \\to S$ of finite presentation, since then", "$$", "Z_n = Z_{1, n} \\times_S \\ldots \\times_S Z_{m, n}", "$$", "Thus we have reduced the theorem to the special case handled in", "Lemma \\ref{lemma-flat-dimension-n-representable}." ], "refs": [ "flat-lemma-flat-dimension-n", "morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "descent-lemma-descent-data-sheaves", "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "descent-lemma-descending-property-monomorphism", "descent-lemma-descending-property-finite-presentation", "flat-lemma-quasi-finite-base-change", "flat-lemma-existence-complete", "flat-lemma-localize-flat-dimension-n", "more-morphisms-lemma-relative-assassin-constructible", "morphisms-lemma-constructible-containing-open", "flat-lemma-localize-flat-dimension-n", "flat-lemma-flat-dimension-n-representable" ], "ref_ids": [ 6082, 5235, 14751, 13949, 14696, 14678, 6057, 6000, 6103, 13812, 5251, 6103, 6105 ] } ], "ref_ids": [] }, { "id": 5973, "type": "theorem", "label": "flat-theorem-existence", "categories": [ "flat" ], "title": "flat-theorem-existence", "contents": [ "In Situation \\ref{situation-existence}", "there exists a finitely presented $\\mathcal{O}_X$-module", "$\\mathcal{F}$, flat over $A$, with support proper over $A$,", "such that", "$\\mathcal{F}_n = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X_n}$", "for all $n$ compatibly with the maps $\\varphi_n$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemmas \\ref{lemma-compute-what-it-should-be},", "\\ref{lemma-compute-against-perfect},", "\\ref{lemma-relative-pseudo-coherence},", "\\ref{lemma-compute-over-affine},", "\\ref{lemma-finitely-presented}, and", "\\ref{lemma-proper-support}", "to get an open subscheme $W \\subset X$ containing all points", "lying over $\\Spec(A_n)$", "and a finitely presented $\\mathcal{O}_W$-module $\\mathcal{F}$", "whose support is proper over $A$ with", "$\\mathcal{F}_n = \\mathcal{F} \\otimes_{\\mathcal{O}_W} \\mathcal{O}_{X_n}$", "for all $n \\geq 1$. (This makes sense as $X_n \\subset W$.)", "By Lemma \\ref{lemma-proper-pure} we see that $\\mathcal{F}$", "is universally pure relative to $\\Spec(A)$.", "By Theorem \\ref{theorem-flat-dimension-n-representable}", "(for explanation, see Lemma \\ref{lemma-when-universal-flattening})", "there exists a universal flattening $S' \\to \\Spec(A)$", "of $\\mathcal{F}$ and moreover the morphism $S' \\to \\Spec(A)$", "is a monomorphism of finite presentation.", "Since the base change of $\\mathcal{F}$ to $\\Spec(A_n)$", "is $\\mathcal{F}_n$ we find that $\\Spec(A_n) \\to \\Spec(A)$", "factors (uniquely) through $S'$ for each $n$.", "By Lemma \\ref{lemma-monomorphism-isomorphism}", "we see that $S' = \\Spec(A)$.", "This means that $\\mathcal{F}$ is flat over $A$.", "Finally, since the scheme theoretic support $Z$ of $\\mathcal{F}$", "is proper over $\\Spec(A)$, the morphism $Z \\to X$ is closed.", "Hence the pushforward $(W \\to X)_*\\mathcal{F}$ is supported", "on $W$ and has all the desired properties." ], "refs": [ "flat-lemma-compute-what-it-should-be", "flat-lemma-compute-against-perfect", "flat-lemma-relative-pseudo-coherence", "flat-lemma-compute-over-affine", "flat-lemma-finitely-presented", "flat-lemma-proper-support", "flat-lemma-proper-pure", "flat-theorem-flat-dimension-n-representable", "flat-lemma-when-universal-flattening", "flat-lemma-monomorphism-isomorphism" ], "ref_ids": [ 6107, 6108, 6109, 6110, 6111, 6112, 6061, 5972, 6106, 6113 ] } ], "ref_ids": [] }, { "id": 5974, "type": "theorem", "label": "flat-theorem-existence-derived", "categories": [ "flat" ], "title": "flat-theorem-existence-derived", "contents": [ "In Situation \\ref{situation-existence-derived}", "there exists a pseudo-coherent $K$ in $D(\\mathcal{O}_X)$", "such that $K_n = K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_{X_n}$", "for all $n$ compatibly with the maps $\\varphi_n$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemmas \\ref{lemma-compute-what-it-should-be-derived},", "\\ref{lemma-compute-against-perfect-derived},", "\\ref{lemma-relative-pseudo-coherence-derived}", "to get a pseudo-coherent object $K$ of $D(\\mathcal{O}_X)$.", "Choosing affine opens in Lemma", "\\ref{lemma-compute-over-affine-derived}", "it follows immediately that $K$ restricts to $K_n$ over $X_n$." ], "refs": [ "flat-lemma-compute-what-it-should-be-derived", "flat-lemma-compute-against-perfect-derived", "flat-lemma-relative-pseudo-coherence-derived", "flat-lemma-compute-over-affine-derived" ], "ref_ids": [ 6114, 6115, 6116, 6117 ] } ], "ref_ids": [] }, { "id": 5975, "type": "theorem", "label": "flat-theorem-flatten-module", "categories": [ "flat" ], "title": "flat-theorem-flatten-module", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $X$ be a scheme over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent module on $X$.", "Let $U \\subset S$ be a quasi-compact open. Assume", "\\begin{enumerate}", "\\item $X$ is quasi-compact,", "\\item $X$ is locally of finite presentation over $S$,", "\\item $\\mathcal{F}$ is a module of finite type,", "\\item $\\mathcal{F}_U$ is of finite presentation, and", "\\item $\\mathcal{F}_U$ is flat over $U$.", "\\end{enumerate}", "Then there exists a $U$-admissible blowup $S' \\to S$ such that the", "strict transform $\\mathcal{F}'$ of $\\mathcal{F}$ is an", "$\\mathcal{O}_{X \\times_S S'}$-module of finite presentation and", "flat over $S'$." ], "refs": [], "proofs": [ { "contents": [ "We first prove that we can find a $U$-admissible blowup such that the", "strict transform is flat. The question is \\'etale local on the source", "and the target, see", "Lemma \\ref{lemma-flatten-module-etale-localize} for a precise statement.", "In particular, we may assume that $S = \\Spec(R)$ and $X = \\Spec(A)$", "are affine. For $s \\in S$ write $\\mathcal{F}_s = \\mathcal{F}|_{X_s}$", "(pullback of $\\mathcal{F}$ to the fibre). As $X \\to S$ is of finite type", "$d = \\max_{s \\in S} \\dim(\\text{Supp}(\\mathcal{F}_s))$", "is an integer. We will do induction on $d$.", "\\medskip\\noindent", "Let $x \\in X$ be a point of $X$ lying over $s \\in S$ with", "$\\dim_x(\\text{Supp}(\\mathcal{F}_s)) = d$.", "Apply Lemma \\ref{lemma-elementary-devissage} to get", "$g : X' \\to X$, $e : S' \\to S$, $i : Z' \\to X'$, and $\\pi : Z' \\to Y'$.", "Observe that $Y' \\to S'$ is", "a smooth morphism of affines with geometrically irreducible fibres", "of dimension $d$. Because the problem is \\'etale local it suffices", "to prove the theorem for $g^*\\mathcal{F}/X'/S'$. Because $i : Z' \\to X'$", "is a closed immersion of finite presentation (and since strict transform", "commutes with affine pushforward, see", "Divisors, Lemma \\ref{divisors-lemma-strict-transform-affine})", "it suffices to prove the flattening result for $\\mathcal{G}$.", "Since $\\pi$ is finite (hence also affine) it suffices to prove the", "flattening result for $\\pi_*\\mathcal{G}/Y'/S'$. Thus we may assume", "that $X \\to S$ is a smooth morphism of affines with geometrically", "irreducible fibres of dimension $d$.", "\\medskip\\noindent", "Next, we apply a blowup as in Lemma \\ref{lemma-flatten-module-pre}.", "Doing so we reach the situation where there exists an", "open $V \\subset X$ surjecting onto $S$ such that $\\mathcal{F}|_V$", "is finite locally free. Let $\\xi \\in X$ be the generic point of $X_s$. Let", "$r = \\dim_{\\kappa(\\xi)} \\mathcal{F}_\\xi \\otimes \\kappa(\\xi)$.", "Choose a map $\\alpha : \\mathcal{O}_X^{\\oplus r} \\to \\mathcal{F}$", "which induces an isomorphism", "$\\kappa(\\xi)^{\\oplus r} \\to \\mathcal{F}_\\xi \\otimes \\kappa(\\xi)$.", "Because $\\mathcal{F}$ is locally free over $V$ we find an open neighbourhood", "$W$ of $\\xi$ where $\\alpha$ is an isomorphism. Shrink $S$ to an affine open", "neighbourhood of $s$ such that $W \\to S$ is surjective. Say $\\mathcal{F}$", "is the quasi-coherent module associated to the $A$-module $N$. Since", "$\\mathcal{F}$ is flat over $S$ at all generic points of fibres", "(in fact at all points of $W$), we see that", "$$", "\\alpha_\\mathfrak p : A_\\mathfrak p^{\\oplus r} \\to N_\\mathfrak p", "$$", "is universally injective for all primes $\\mathfrak p$ of $R$, see", "Lemma \\ref{lemma-induction-step}. Hence $\\alpha$ is universally injective,", "see Algebra, Lemma \\ref{algebra-lemma-universally-injective-check-stalks}.", "Set $\\mathcal{H} = \\Coker(\\alpha)$.", "By Divisors, Lemma \\ref{divisors-lemma-strict-transform-universally-injective}", "we see that, given a $U$-admissible blowup $S' \\to S$", "the strict transforms of $\\mathcal{F}'$ and $\\mathcal{H}'$", "fit into an exact sequence", "$$", "0 \\to \\mathcal{O}_{X \\times_S S'}^{\\oplus r} \\to \\mathcal{F}'", "\\to \\mathcal{H}' \\to 0", "$$", "Hence Lemma \\ref{lemma-induction-step} also shows that $\\mathcal{F}'$", "is flat at a point $x'$ if and only if", "$\\mathcal{H}'$ is flat at that point. In particular $\\mathcal{H}_U$ is", "flat over $U$ and $\\mathcal{H}_U$ is a module of finite presentation.", "We may apply the induction hypothesis to $\\mathcal{H}$ to see that", "there exists a $U$-admissible blowup such that the strict transform", "$\\mathcal{H}'$ is flat as desired.", "\\medskip\\noindent", "To finish the proof of the theorem we still have to show that $\\mathcal{F}'$", "is a module of finite presentation (after possibly another", "$U$-admissible blowup). This follows from", "Lemma \\ref{lemma-flat-finite-type-finitely-presented-over-dense-open}", "as we can assume $U \\subset S$ is scheme theoretically dense (see", "third paragraph of Remark \\ref{remark-successive-blowups}).", "This finishes the proof of the theorem." ], "refs": [ "flat-lemma-flatten-module-etale-localize", "flat-lemma-elementary-devissage", "divisors-lemma-strict-transform-affine", "flat-lemma-flatten-module-pre", "flat-lemma-induction-step", "algebra-lemma-universally-injective-check-stalks", "divisors-lemma-strict-transform-universally-injective", "flat-lemma-induction-step", "flat-lemma-flat-finite-type-finitely-presented-over-dense-open", "flat-remark-successive-blowups" ], "ref_ids": [ 6122, 5988, 8067, 6119, 6015, 815, 8070, 6015, 6024, 6234 ] } ], "ref_ids": [] }, { "id": 5976, "type": "theorem", "label": "flat-theorem-nagata", "categories": [ "flat" ], "title": "flat-theorem-nagata", "contents": [ "\\begin{reference}", "See \\cite{Lutkebohmert}, \\cite{Conrad-Nagata}, \\cite{Nagata-1},", "\\cite{Nagata-2}, \\cite{Nagata-3}, and \\cite{Nagata-4}", "\\end{reference}", "Let $S$ be a quasi-compact and quasi-separated scheme. Let", "$X \\to S$ be a separated, finite type morphism.", "Then $X$ has a compactification over $S$." ], "refs": [], "proofs": [ { "contents": [ "We first reduce to the Noetherian case. We strongly urge the reader", "to skip this paragraph. There exists a closed immersion", "$X \\to X'$ with $X' \\to S$ of finite presentation and separated.", "See Limits, Proposition", "\\ref{limits-proposition-separated-closed-in-finite-presentation}.", "If we find a compactification of $X'$ over $S$, then", "taking the scheme theoretic image of $X$ in this will give", "a compactification of $X$ over $S$. Thus we may assume", "$X \\to S$ is separated and of finite presentation.", "We may write $S = \\lim S_i$ as a directed", "limit of a system of Noetherian schemes with affine transition morphisms.", "See Limits, Proposition \\ref{limits-proposition-approximate}.", "We can choose an $i$ and a morphism $X_i \\to S_i$ of finite", "presentation whose base change to $S$ is $X \\to S$, see", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}.", "After increasing $i$ we may assume $X_i \\to S_i$ is separated, see", "Limits, Lemma \\ref{limits-lemma-descend-separated-finite-presentation}.", "If we can find a compactification of $X_i$ over $S_i$, then the", "base change of this to $S$ will be a compactification of $X$ over $S$.", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $S$ is Noetherian. We can choose a finite affine open covering", "$X = \\bigcup_{i = 1, \\ldots, n} U_i$ such that $U_1 \\cap \\ldots \\cap U_n$", "is dense in $X$. This follows from", "Properties, Lemma \\ref{properties-lemma-point-and-maximal-points-affine}", "and the fact that $X$ is quasi-compact with finitely many", "irreducible components. For each $i$ we can choose an $n_i \\geq 0$ and an", "immersion $U_i \\to \\mathbf{A}^{n_i}_S$ by", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-affine-finite-type-over-S}.", "Hence $U_i$ has a compactification over $S$ for $i = 1, \\ldots, n$", "by taking the scheme theoretic image in $\\mathbf{P}^{n_i}_S$.", "Applying Lemma \\ref{lemma-two-compactifications}", "$(n - 1)$ times we conclude that the theorem is true." ], "refs": [ "limits-proposition-separated-closed-in-finite-presentation", "limits-proposition-approximate", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-separated-finite-presentation", "properties-lemma-point-and-maximal-points-affine", "morphisms-lemma-quasi-affine-finite-type-over-S", "flat-lemma-two-compactifications" ], "ref_ids": [ 15128, 15126, 15077, 15061, 3061, 5392, 6138 ] } ], "ref_ids": [] }, { "id": 5977, "type": "theorem", "label": "flat-theorem-pullback-trivial-fibres", "categories": [ "flat" ], "title": "flat-theorem-pullback-trivial-fibres", "contents": [ "Let $p$ be a prime number. Let $Y$ be a quasi-compact and quasi-separated", "scheme over $\\mathbf{F}_p$.", "Let $f : X \\to Y$ be a proper, surjective morphism of finite presentation", "with geometrically connected fibres.", "Then the functor", "$$", "\\colim_F \\textit{Vect}(Y) \\longrightarrow \\colim_F \\textit{Vect}(X)", "$$", "is fully faithful with essential image described as follows.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module.", "Assume for all $y \\in Y$ there exists integers $n_y, r_y \\geq 0$", "such that", "$$", "F^{n_y, *}\\mathcal{E}|_{X_{y, red}}", "\\cong", "\\mathcal{O}_{X_{y, red}}^{\\oplus r_y}", "$$", "Then for some $n \\geq 0$ the $n$th Frobenius power pullback", "$F^{n, *}\\mathcal{E}$ is the pullback of a finite locally free", "$\\mathcal{O}_Y$-module." ], "refs": [], "proofs": [ { "contents": [ "Proof of fully faithfulness. Since vectorbundles on $Y$ are locally", "trivial, this reduces to the statement that", "$$", "\\colim_F \\Gamma(Y, \\mathcal{O}_Y)", "\\longrightarrow", "\\colim_F \\Gamma(X, \\mathcal{O}_X)", "$$", "is bijective. Since $\\{X \\to Y\\}$ is an h covering, this will", "follow from Lemma \\ref{lemma-h-sheaf-colim-F} if we can show that the two maps", "$$", "\\colim_F \\Gamma(X, \\mathcal{O}_X)", "\\longrightarrow", "\\colim_F \\Gamma(X \\times_Y X, \\mathcal{O}_{X \\times_Y X})", "$$", "are equal. Let $g \\in \\Gamma(X, \\mathcal{O}_X)$", "and denote $g_1$ and $g_2$ the two pullbacks of $g$ to $X \\times_Y X$.", "Since $X_{y, red}$ is geometrically connected, we", "see that $H^0(X_{y, red}, \\mathcal{O}_{X_{y, red}})$ is", "a purely inseparable extension of $\\kappa(y)$, see", "Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}.", "Thus $g^q|_{X_{y, red}}$ comes from an element of", "$\\kappa(y)$ for some $p$-power $q$ (which may depend on $y$).", "It follows that $g_1^q$ and $g_2^q$ map to the same", "element of the residue field at any point of", "$(X \\times_Y X)_y = X_y \\times_y X_y$.", "Hence $g_1 - g_2$ restricts to zero on $(X \\times_Y X)_{red}$.", "Hence $(g_1 - g_2)^n = 0$ for some $n$ which we may take", "to be a $p$-power as desired.", "\\medskip\\noindent", "Description of essential image. Let $\\mathcal{E}$ be as in the statement", "of the proposition. We first reduce to the Noetherian case.", "\\medskip\\noindent", "Let $y \\in Y$ be a point and view it as a morphism", "$y \\to Y$ from the spectrum of the residue field into $Y$.", "We can write $y \\to Y$ as a filtered limit of morphisms $Y_i \\to Y$ of", "finite presentation with $Y_i$ affine. (It is best to prove this", "yourself, but it also follows formally from", "Limits, Lemma \\ref{limits-lemma-relative-approximation} and", "\\ref{limits-lemma-limit-affine}.)", "For each $i$ set $Z_i = Y_i \\times_Y X$. Then $X_y = \\lim Z_i$", "and $X_{y, red} = \\lim Z_{i, red}$.", "By Limits, Lemma \\ref{limits-lemma-descend-modules-finite-presentation}", "we can find an $i$ such that", "$F^{n_y, *}\\mathcal{E}|_{Z_{i, red}} \\cong", "\\mathcal{O}_{Z_{i, red}}^{\\oplus r_y}$.", "Fix $i$.", "We have $Z_{i, red} = \\lim Z_{i, j}$ where $Z_{i, j} \\to Z_i$", "is a thickening of finite presentation (Limits, Lemma", "\\ref{limits-lemma-closed-is-limit-closed-and-finite-presentation}).", "Using the same lemma as before we can find a $j$ such that", "$F^{n_y, *}\\mathcal{E}|_{Z_{i, j}} \\cong \\mathcal{O}_{Z_{i, j}}^{\\oplus r_y}$.", "We conclude that for each $y \\in Y$ there exists a morphism", "$Y_y \\to Y$ of finite presentation whose image contains $y$", "and a thickening $Z_y \\to Y_y \\times_Y X$ such that", "$F^{n_y, *}\\mathcal{E}|_{Z_y} \\cong \\mathcal{O}_{Z_y}^{\\oplus r_y}$.", "Observe that the image of $Y_y \\to Y$ is constructible", "(Morphisms, Lemma \\ref{morphisms-lemma-chevalley}).", "Since $Y$ is quasi-compact in the constructible topology", "(Topology, Lemma \\ref{topology-lemma-constructible-hausdorff-quasi-compact} and", "Properties, Lemma \\ref{properties-lemma-quasi-compact-quasi-separated-spectral})", "we conclude that there are a finite number of morphisms", "$$", "Y_1 \\to Y,\\ Y_2 \\to Y,\\ \\ldots,\\ Y_N \\to Y", "$$", "of finite presentation such that $Y = \\bigcup \\Im(Y_a \\to Y)$", "set theoretically and such that for each $a \\in \\{1, \\ldots, N\\}$", "there exist integers $n_a, r_a \\geq 0$ and there is a thickening", "$Z_a \\subset Y_a \\times_Y X$ of finite presentation such that", "$F^{n_a, *}\\mathcal{E}|_{Z_a} \\cong \\mathcal{O}_{Z_a}^{\\oplus r_a}$.", "\\medskip\\noindent", "Formulated in this way, the condition descends to an absolute", "Noetherian approximation. We stronly urge the reader to skip", "this paragraph. First write $Y = \\lim_{i \\in I} Y_i$ as a cofiltered limit", "of schemes of finite type over $\\mathbf{F}_p$ with affine transition", "morphisms (Limits, Lemma \\ref{limits-lemma-relative-approximation}).", "Next, we can assume we have proper morphisms $f_i : X_i \\to Y_i$", "whose base change to $Y$ recovers $f : X \\to Y$, see", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}.", "After increasing $i$ we may assume there exists a finite locally", "free $\\mathcal{O}_{X_i}$-module $\\mathcal{E}_i$ whose pullback", "to $X$ is isomorphic to $\\mathcal{E}$, see", "Limits, Lemma \\ref{limits-lemma-descend-invertible-modules}.", "Pick $0 \\in I$ and denote $E \\subset Y_0$ the constructible subset", "where the geometric fibres of $f_0$ are connected, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-nr-geom-connected-components-constructible}.", "Then $Y \\to Y_0$ maps into $E$, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-base-change-fibres-geometrically-connected}.", "Thus $Y_i \\to Y_0$ maps into $E$ for $i \\gg 0$, see", "Limits, Lemma \\ref{limits-lemma-limit-contained-in-constructible}.", "Hence we see that the fibres of $f_i$ are geometrically connected", "for $i \\gg 0$. By Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "for large enough $i$ we can find morphisms", "$Y_{i, a} \\to Y_i$ of finite type whose base change to $Y$", "recovers $Y_a \\to Y$, $a \\in \\{1, \\ldots, N\\}$.", "After possibly increasing $i$ we can find thickenings", "$Z_{i, a} \\subset Y_{i, a} \\times_{Y_i} X_i$ whose base change", "to $Y_a \\times_Y X$ recovers $Z_a$ (same reference as before", "combined with", "Limits, Lemmas", "\\ref{limits-lemma-descend-closed-immersion-finite-presentation} and", "\\ref{limits-lemma-descend-surjective}).", "Since $Z_a = \\lim Z_{i, a}$ we find that after increasing $i$ we may assume", "$F^{n_a, *}\\mathcal{E}_i|_{Z_{i, a}} \\cong", "\\mathcal{O}_{Z_{i, a}}^{\\oplus r_a}$, see", "Limits, Lemma \\ref{limits-lemma-descend-modules-finite-presentation}.", "Finally, after increasing $i$ one more time we may assume", "$\\coprod Y_{i, a} \\to Y_i$ is surjective by", "Limits, Lemma \\ref{limits-lemma-descend-surjective}.", "At this point all the assumptions hold for $X_i \\to Y_i$", "and $\\mathcal{E}_i$ and we see that it suffices to prove result", "for $X_i \\to Y_i$ and $\\mathcal{E}_i$.", "\\medskip\\noindent", "Assume $Y$ is of finite type over $\\mathbf{F}_p$.", "To prove the result we will use induction on $\\dim(Y)$.", "We are trying to find an object of", "$\\colim_F \\textit{Vect}(Y)$ which pulls back to the", "object of $\\colim_F \\textit{Vect}(X)$ determined by $\\mathcal{E}$.", "By the fully faithfulness already proven and because of", "Proposition \\ref{proposition-h-descent-vector-bundles-p}", "it suffices to construct a descent of $\\mathcal{E}$", "after replacing $Y$ by the members of a h covering", "and $X$ by the corresponding base change. This means", "that we may replace $Y$ by a scheme proper and surjective", "over $Y$ provided this does not increase the dimension of $Y$.", "If $T \\subset T'$ is a thickening of schemes of finite type", "over $\\mathbf{F}_p$ then", "$\\colim_F \\textit{Vect}(T) = \\colim_F \\textit{Vect}(T')$", "as $\\{T \\to T'\\}$ is a h covering such that $T \\times_{T'} T = T$.", "If $T' \\to T$ is a universal homeomorphism of schemes", "of finite type over $\\mathbf{F}_p$, then", "$\\colim_F \\textit{Vect}(T) = \\colim_F \\textit{Vect}(T')$", "as $\\{T \\to T'\\}$ is a h covering such that the diagonal", "$T \\subset T \\times_{T'} T$ is a thickening.", "\\medskip\\noindent", "Using the general remarks made above, we may and do replace", "$X$ by its reduction and we may assume $X$ is reduced.", "Consider the Stein factorization $X \\to Y' \\to Y$, see", "More on Morphisms, Theorem", "\\ref{more-morphisms-theorem-stein-factorization-Noetherian}.", "Then $Y' \\to Y$ is a universal homeomorphism", "of schemes of finite type over $\\mathbf{F}_p$.", "By the above we may replace $Y$ by $Y'$.", "Thus we may assume $f_*\\mathcal{O}_X = \\mathcal{O}_Y$", "and that $Y$ is reduced. This reduces us to the case discussed", "in the next paragraph.", "\\medskip\\noindent", "Assume $Y$ is reduced and $f_*\\mathcal{O}_X = \\mathcal{O}_Y$", "over a dense open subscheme of $Y$.", "Then $X \\to Y$ is flat over a dense open", "subscheme $V \\subset Y$, see", "Morphisms, Proposition \\ref{morphisms-proposition-generic-flatness-reduced}.", "By Lemma \\ref{lemma-flat-after-blowing-up}", "there is a $V$-admissible blowing up $Y' \\to Y$ such that", "the strict transform $X'$ of $X$ is flat over $Y'$.", "Observe that $\\dim(Y') = \\dim(Y)$ as $Y$ and $Y'$ have", "a common dense open subscheme. By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous}", "and the fact that $V \\subset Y'$ is dense", "all fibres of $f' : X' \\to Y'$ are geometrically connected.", "We still have $(f'_*\\mathcal{O}_{X'})|_V = \\mathcal{O}_V$.", "Write", "$$", "Y' \\times_Y X = X' \\cup E \\times_Y X", "$$", "where $E \\subset Y'$ is the exceptional divisor of the blowing up.", "By the general remarks above, it suffices to prove existence", "for $Y' \\times_Y X \\to Y'$ and the restriction of $\\mathcal{E}$", "to $Y' \\times_Y X$.", "Suppose that we find some object $\\xi'$ in $\\colim_F \\textit{Vect}(Y')$", "pulling back to the restriction of $\\mathcal{E}$ to $X'$ (viewed", "as an object of the colimit category).", "By induction on $\\dim(Y)$ we can find an object $\\xi''$ in", "$\\colim_F \\textit{Vect}(E)$ pulling back to the restriction of", "$\\mathcal{E}$ to $E \\times_Y X$. Then the fully faithfullness", "determines a unique isomorphism $\\xi'|_E \\to \\xi''$", "compatible with the given identifications with the restriction", "of $\\mathcal{E}$ to $E \\times_{Y'} X'$. Since", "$$", "\\{E \\times_Y X \\to Y' \\times_Y X, X' \\to Y' \\times_Y X\\}", "$$", "is a h covering given by a pair of closed immersions with", "$$", "(E \\times_Y X) \\times_{(Y' \\times_Y X)} X' = E \\times_{Y'} X'", "$$", "we conclude that $\\xi'$ pulls back to the restriction of", "$\\mathcal{E}$ to $Y' \\times_Y X$. Thus it suffices to find", "$\\xi'$ and we reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $Y$ is reduced, $f$ is flat, and $f_*\\mathcal{O}_X = \\mathcal{O}_Y$", "over a dense open subscheme of $Y$. In this case we consider the", "normalization $Y^\\nu \\to Y$ (Morphisms, Section", "\\ref{morphisms-section-normalization}). This is a finite surjective", "morphism", "(Morphisms, Lemma \\ref{morphisms-lemma-nagata-normalization} and", "\\ref{morphisms-lemma-ubiquity-nagata}) which is an isomorphism", "over a dense open. Hence by our general remarks we may", "replace $Y$ by $Y^\\nu$ and $X$ by $Y^\\nu \\times_Y X$.", "After this replacement we see that $\\mathcal{O}_Y = f_*\\mathcal{O}_X$", "(because the Stein factorization has to be an isomorphism", "in this case; small detail omitted).", "\\medskip\\noindent", "Assume $Y$ is a normal Noetherian scheme, that $f$ is flat, and", "that $f_*\\mathcal{O}_X = \\mathcal{O}_Y$. After replacing $\\mathcal{E}$", "by a suitable Frobenius power pullback, we may assume $\\mathcal{E}$", "is trivial on the scheme theoretic fibres of $f$ at the generic points", "of the irreducible components of $Y$ (because", "$\\colim_F \\textit{Vect}(-)$ is an equivalence on universal", "homeomorphisms, see above). Similarly to the arguments above", "(in the reduction to the Noetherian case) we conclude there is a dense", "open subscheme $V \\subset Y$ such that $\\mathcal{E}|_{f^{-1}(V)}$ is free.", "Let $Z \\subset Y$ be a closed subscheme such that", "$Y = V \\amalg Z$ set theoretically. Let $z_1, \\ldots, z_t \\in Z$", "be the generic points of the irreducible components of $Z$", "of codimension $1$. Then $A_i = \\mathcal{O}_{Y, z_i}$ is", "a discrete valuation ring. Let $n_i$ be the integer found in", "Lemma \\ref{lemma-trivial-fibres-dvr} for the scheme $X_{A_i}$ over $A_i$.", "After replacing $\\mathcal{E}$ by a suitable Frobenius", "power pullback, we may assume $\\mathcal{E}$ is free over", "$X_{A_i/\\mathfrak m_i^{n_i}}$ (again because the colimit category", "is invariant under universal homeomorphisms, see above).", "Then Lemma \\ref{lemma-trivial-fibres-dvr} tells us that", "$\\mathcal{E}$ is free on $X_{A_i}$.", "Thus finally we conclude by applying Lemma \\ref{lemma-trivial-over-dvrs}." ], "refs": [ "flat-lemma-h-sheaf-colim-F", "varieties-lemma-proper-geometrically-reduced-global-sections", "limits-lemma-relative-approximation", "limits-lemma-limit-affine", "limits-lemma-descend-modules-finite-presentation", "limits-lemma-closed-is-limit-closed-and-finite-presentation", "morphisms-lemma-chevalley", "topology-lemma-constructible-hausdorff-quasi-compact", "properties-lemma-quasi-compact-quasi-separated-spectral", "limits-lemma-relative-approximation", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-invertible-modules", "more-morphisms-lemma-nr-geom-connected-components-constructible", "more-morphisms-lemma-base-change-fibres-geometrically-connected", "limits-lemma-limit-contained-in-constructible", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-closed-immersion-finite-presentation", "limits-lemma-descend-surjective", "limits-lemma-descend-modules-finite-presentation", "limits-lemma-descend-surjective", "flat-proposition-h-descent-vector-bundles-p", "more-morphisms-theorem-stein-factorization-Noetherian", "morphisms-proposition-generic-flatness-reduced", "flat-lemma-flat-after-blowing-up", "more-morphisms-lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous", "morphisms-lemma-nagata-normalization", "morphisms-lemma-ubiquity-nagata", "flat-lemma-trivial-fibres-dvr", "flat-lemma-trivial-fibres-dvr", "flat-lemma-trivial-over-dvrs" ], "ref_ids": [ 6168, 10948, 15055, 15043, 15078, 15073, 5250, 8303, 2941, 15055, 15077, 15079, 13832, 13828, 15040, 15077, 15060, 15069, 15078, 15069, 6205, 13674, 5534, 6123, 13946, 5520, 5219, 6177, 6177, 6178 ] } ], "ref_ids": [] }, { "id": 5978, "type": "theorem", "label": "flat-lemma-lift-etale", "categories": [ "flat" ], "title": "flat-lemma-lift-etale", "contents": [ "Let $i : Z \\to X$ be a closed immersion of affine schemes.", "Let $Z' \\to Z$ be an \\'etale morphism with $Z'$ affine.", "Then there exists an \\'etale morphism $X' \\to X$ with $X'$", "affine such that $Z' \\cong Z \\times_X X'$ as schemes over $Z$." ], "refs": [], "proofs": [ { "contents": [ "See", "Algebra, Lemma \\ref{algebra-lemma-lift-etale}." ], "refs": [ "algebra-lemma-lift-etale" ], "ref_ids": [ 1238 ] } ], "ref_ids": [] }, { "id": 5979, "type": "theorem", "label": "flat-lemma-etale-at-point", "categories": [ "flat" ], "title": "flat-lemma-etale-at-point", "contents": [ "Let", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l] \\ar[d] \\\\", "S & S' \\ar[l]", "}", "$$", "be a commutative diagram of schemes with $X' \\to X$ and $S' \\to S$ \\'etale.", "Let $s' \\in S'$ be a point. Then", "$$", "X' \\times_{S'} \\Spec(\\mathcal{O}_{S', s'})", "\\longrightarrow", "X \\times_S \\Spec(\\mathcal{O}_{S', s'})", "$$", "is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "This is true because $X' \\to X_{S'}$ is \\'etale as a morphism of", "schemes \\'etale over $X$, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-permanence}", "and the base change of an \\'etale morphism is \\'etale, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-etale}." ], "refs": [ "morphisms-lemma-etale-permanence", "morphisms-lemma-base-change-etale" ], "ref_ids": [ 5375, 5361 ] } ], "ref_ids": [] }, { "id": 5980, "type": "theorem", "label": "flat-lemma-etale-flat-up-down", "categories": [ "flat" ], "title": "flat-lemma-etale-flat-up-down", "contents": [ "Let $X \\to T \\to S$ be morphisms of schemes with $T \\to S$ \\'etale.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $x \\in X$ be a point. Then", "$$", "\\mathcal{F}\\text{ flat over }S\\text{ at }x", "\\Leftrightarrow", "\\mathcal{F}\\text{ flat over }T\\text{ at }x", "$$", "In particular $\\mathcal{F}$ is flat over $S$ if and only if $\\mathcal{F}$", "is flat over $T$." ], "refs": [], "proofs": [ { "contents": [ "As an \\'etale morphism is a flat morphism (see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-flat})", "the implication ``$\\Leftarrow$'' follows from", "Algebra, Lemma \\ref{algebra-lemma-composition-flat}.", "For the converse assume that $\\mathcal{F}$ is flat at $x$ over $S$.", "Denote $\\tilde x \\in X \\times_S T$ the point lying over $x$ in $X$", "and over the image of $x$ in $T$ in $T$.", "Then $(X \\times_S T \\to X)^*\\mathcal{F}$ is flat at $\\tilde x$ over $T$", "via $\\text{pr}_2 : X \\times_S T \\to T$, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-module-flat}.", "The diagonal $\\Delta_{T/S} : T \\to T \\times_S T$ is an open immersion;", "combine", "Morphisms, Lemmas \\ref{morphisms-lemma-diagonal-unramified-morphism} and", "\\ref{morphisms-lemma-etale-smooth-unramified}.", "So $X$ is identified with open subscheme of $X \\times_S T$,", "the restriction of $\\text{pr}_2$ to this open is the given morphism $X \\to T$,", "the point $\\tilde x$ corresponds to the point $x$ in this open, and", "$(X \\times_S T \\to X)^*\\mathcal{F}$ restricted to this open is $\\mathcal{F}$.", "Whence we see that $\\mathcal{F}$ is flat at $x$ over $T$." ], "refs": [ "morphisms-lemma-etale-flat", "algebra-lemma-composition-flat", "morphisms-lemma-base-change-module-flat", "morphisms-lemma-diagonal-unramified-morphism", "morphisms-lemma-etale-smooth-unramified" ], "ref_ids": [ 5369, 524, 5264, 5354, 5362 ] } ], "ref_ids": [] }, { "id": 5981, "type": "theorem", "label": "flat-lemma-etale-flat-up-down-local-ring", "categories": [ "flat" ], "title": "flat-lemma-etale-flat-up-down-local-ring", "contents": [ "Let $T \\to S$ be an \\'etale morphism. Let $t \\in T$ with image $s \\in S$.", "Let $M$ be a $\\mathcal{O}_{T, t}$-module. Then", "$$", "M\\text{ flat over }\\mathcal{O}_{S, s}", "\\Leftrightarrow", "M\\text{ flat over }\\mathcal{O}_{T, t}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "We may replace $S$ by an affine neighbourhood of $s$ and after that", "$T$ by an affine neighbourhood of $t$.", "Set $\\mathcal{F} = (\\Spec(\\mathcal{O}_{T, t}) \\to T)_*\\widetilde M$.", "This is a quasi-coherent sheaf (see", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "or argue directly)", "on $T$ whose stalk at $t$ is $M$ (details omitted).", "Apply", "Lemma \\ref{lemma-etale-flat-up-down}." ], "refs": [ "schemes-lemma-push-forward-quasi-coherent", "flat-lemma-etale-flat-up-down" ], "ref_ids": [ 7730, 5980 ] } ], "ref_ids": [] }, { "id": 5982, "type": "theorem", "label": "flat-lemma-flat-up-down-henselization", "categories": [ "flat" ], "title": "flat-lemma-flat-up-down-henselization", "contents": [ "Let $S$ be a scheme and $s \\in S$ a point. Denote $\\mathcal{O}_{S, s}^h$", "(resp.\\ $\\mathcal{O}_{S, s}^{sh}$) the henselization (resp.\\ strict", "henselization), see", "Algebra, Definition \\ref{algebra-definition-henselization}.", "Let $M^{sh}$ be a $\\mathcal{O}_{S, s}^{sh}$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $M^{sh}$ is flat over $\\mathcal{O}_{S, s}$,", "\\item $M^{sh}$ is flat over $\\mathcal{O}_{S, s}^h$, and", "\\item $M^{sh}$ is flat over $\\mathcal{O}_{S, s}^{sh}$.", "\\end{enumerate}", "If $M^{sh} = M^h \\otimes_{\\mathcal{O}_{S, s}^h} \\mathcal{O}_{S, s}^{sh}$", "this is also equivalent to", "\\begin{enumerate}", "\\item[(4)] $M^h$ is flat over $\\mathcal{O}_{S, s}$, and", "\\item[(5)] $M^h$ is flat over $\\mathcal{O}_{S, s}^h$.", "\\end{enumerate}", "If $M^h = M \\otimes_{\\mathcal{O}_{S, s}} \\mathcal{O}_{S, s}^h$", "this is also equivalent to", "\\begin{enumerate}", "\\item[(6)] $M$ is flat over $\\mathcal{O}_{S, s}$.", "\\end{enumerate}" ], "refs": [ "algebra-definition-henselization" ], "proofs": [ { "contents": [ "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-dumb-properties-henselization}", "the local ring maps", "$\\mathcal{O}_{S, s} \\to \\mathcal{O}_{S, s}^h \\to \\mathcal{O}_{S, s}^{sh}$", "are faithfully flat.", "Hence (3) $\\Rightarrow$ (2) $\\Rightarrow$ (1) and", "(5) $\\Rightarrow$ (4) follow from", "Algebra, Lemma \\ref{algebra-lemma-composition-flat}.", "By faithful flatness the equivalences (6) $\\Leftrightarrow$ (5) and", "(5) $\\Leftrightarrow$ (3) follow from", "Algebra, Lemma \\ref{algebra-lemma-flatness-descends}.", "Thus it suffices to show that", "(1) $\\Rightarrow$ (2) $\\Rightarrow$ (3) and", "(4) $\\Rightarrow$ (5).", "To prove these we may assume $S$ is an affine scheme.", "\\medskip\\noindent", "Assume (1). By", "Lemma \\ref{lemma-etale-flat-up-down-local-ring}", "we see that $M^{sh}$ is flat over $\\mathcal{O}_{T, t}$ for", "any \\'etale neighbourhood $(T, t) \\to (S, s)$. Since $\\mathcal{O}_{S, s}^h$", "and $\\mathcal{O}_{S, s}^{sh}$ are directed colimits of local rings", "of the form $\\mathcal{O}_{T, t}$ (see", "Algebra, Lemmas \\ref{algebra-lemma-henselization-different}", "and \\ref{algebra-lemma-strict-henselization-different})", "we conclude that $M^{sh}$ is flat over $\\mathcal{O}_{S, s}^h$", "and $\\mathcal{O}_{S, s}^{sh}$ by", "Algebra, Lemma \\ref{algebra-lemma-colimit-rings-flat}.", "Thus (1) implies (2) and (3). Of course this implies also", "(2) $\\Rightarrow$ (3) by replacing $\\mathcal{O}_{S, s}$ by", "$\\mathcal{O}_{S, s}^h$. The same argument applies to prove", "(4) $\\Rightarrow$ (5)." ], "refs": [ "more-algebra-lemma-dumb-properties-henselization", "algebra-lemma-composition-flat", "algebra-lemma-flatness-descends", "flat-lemma-etale-flat-up-down-local-ring", "algebra-lemma-henselization-different", "algebra-lemma-strict-henselization-different", "algebra-lemma-colimit-rings-flat" ], "ref_ids": [ 10055, 524, 528, 5981, 1298, 1304, 526 ] } ], "ref_ids": [ 1546 ] }, { "id": 5983, "type": "theorem", "label": "flat-lemma-tor-amplitude-up-down-henselization", "categories": [ "flat" ], "title": "flat-lemma-tor-amplitude-up-down-henselization", "contents": [ "Let $S$ be a scheme and $s \\in S$ a point. Denote $\\mathcal{O}_{S, s}^h$", "(resp.\\ $\\mathcal{O}_{S, s}^{sh}$) the henselization (resp.\\ strict", "henselization), see", "Algebra, Definition \\ref{algebra-definition-henselization}.", "Let $M^{sh}$ be an object of $D(\\mathcal{O}_{S, s}^{sh})$.", "Let $a, b \\in \\mathbf{Z}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $M^{sh}$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}$,", "\\item $M^{sh}$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}^h$, and", "\\item $M^{sh}$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}^{sh}$.", "\\end{enumerate}", "If $M^{sh} =", "M^h \\otimes_{\\mathcal{O}_{S, s}^h}^\\mathbf{L} \\mathcal{O}_{S, s}^{sh}$", "for $M^h \\in D(\\mathcal{O}_{S, s}^h)$ this is also equivalent to", "\\begin{enumerate}", "\\item[(4)] $M^h$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}$, and", "\\item[(5)] $M^h$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}^h$.", "\\end{enumerate}", "If $M^h = M \\otimes_{\\mathcal{O}_{S, s}}^\\mathbf{L} \\mathcal{O}_{S, s}^h$", "for $M \\in D(\\mathcal{O}_{S, s})$", "this is also equivalent to", "\\begin{enumerate}", "\\item[(6)] $M$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}$.", "\\end{enumerate}" ], "refs": [ "algebra-definition-henselization" ], "proofs": [ { "contents": [ "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-dumb-properties-henselization}", "the local ring maps", "$\\mathcal{O}_{S, s} \\to \\mathcal{O}_{S, s}^h \\to \\mathcal{O}_{S, s}^{sh}$", "are faithfully flat.", "Hence (3) $\\Rightarrow$ (2) $\\Rightarrow$ (1) and", "(5) $\\Rightarrow$ (4) follow from", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-push-tor-amplitude}.", "By faithful flatness the equivalences (6) $\\Leftrightarrow$ (5) and", "(5) $\\Leftrightarrow$ (3) follow from", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-descent-tor-amplitude}.", "Thus it suffices to show that", "(1) $\\Rightarrow$ (3), (2) $\\Rightarrow$ (3), and (4) $\\Rightarrow$ (5).", "\\medskip\\noindent", "Assume (1). In particular $M^{sh}$ has vanishing cohomology", "in degrees $< a$ and $> b$. Hence we can represent $M^{sh}$ by a complex", "$P^\\bullet$ of free $\\mathcal{O}_{X, x}^{sh}$-modules with", "$P^i = 0$ for $i > b$", "(see for example the very general", "Derived Categories, Lemma \\ref{derived-lemma-subcategory-left-resolution}).", "Note that $P^n$ is flat over $\\mathcal{O}_{S, s}$ for all $n$.", "Consider $\\Coker(d_P^{a - 1})$. By More on Algebra, Lemma", "\\ref{more-algebra-lemma-last-one-flat}", "this is a flat $\\mathcal{O}_{S, s}$-module.", "Hence by Lemma \\ref{lemma-flat-up-down-henselization}", "this is a flat $\\mathcal{O}_{S, s}^{sh}$-module.", "Thus $\\tau_{\\geq a}P^\\bullet$ is a complex of", "flat $\\mathcal{O}_{S, s}^{sh}$-modules representing $M^{sh}$", "in $D(\\mathcal{O}_{S, s}^{sh}$ and we find that", "$M^{sh}$ has tor amplitude in $[a, b]$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-tor-amplitude}.", "Thus (1) implies (3). Of course this implies also", "(2) $\\Rightarrow$ (3) by replacing $\\mathcal{O}_{S, s}$ by", "$\\mathcal{O}_{S, s}^h$. The same argument applies to prove", "(4) $\\Rightarrow$ (5)." ], "refs": [ "more-algebra-lemma-dumb-properties-henselization", "more-algebra-lemma-flat-push-tor-amplitude", "more-algebra-lemma-flat-descent-tor-amplitude", "derived-lemma-subcategory-left-resolution", "more-algebra-lemma-last-one-flat", "flat-lemma-flat-up-down-henselization", "more-algebra-lemma-tor-amplitude" ], "ref_ids": [ 10055, 10178, 10184, 1835, 10169, 5982, 10170 ] } ], "ref_ids": [ 1546 ] }, { "id": 5984, "type": "theorem", "label": "flat-lemma-finite-flat-weak-assassin-up-down", "categories": [ "flat" ], "title": "flat-lemma-finite-flat-weak-assassin-up-down", "contents": [ "Let $g : T \\to S$ be a finite flat morphism of schemes.", "Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_S$-module.", "Let $t \\in T$ be a point with image $s \\in S$. Then", "$$", "t \\in \\text{WeakAss}(g^*\\mathcal{G})", "\\Leftrightarrow", "s \\in \\text{WeakAss}(\\mathcal{G})", "$$" ], "refs": [], "proofs": [ { "contents": [ "The implication ``$\\Leftarrow$'' follows immediately from", "Divisors, Lemma \\ref{divisors-lemma-weakly-ass-pullback}.", "Assume $t \\in \\text{WeakAss}(g^*\\mathcal{G})$.", "Let $\\Spec(A) \\subset S$ be an affine open neighbourhood of $s$.", "Let $\\mathcal{G}$ be the quasi-coherent sheaf associated to the $A$-module $M$.", "Let $\\mathfrak p \\subset A$ be the prime ideal corresponding to $s$.", "As $g$ is finite flat we have $g^{-1}(\\Spec(A)) = \\Spec(B)$", "for some finite flat $A$-algebra $B$. Note that", "$g^*\\mathcal{G}$ is the quasi-coherent $\\mathcal{O}_{\\Spec(B)}$-module", "associated to the $B$-module $M \\otimes_A B$ and $g_*g^*\\mathcal{G}$ is the", "quasi-coherent $\\mathcal{O}_{\\Spec(A)}$-module associated to the", "$A$-module $M \\otimes_A B$. By", "Algebra, Lemma \\ref{algebra-lemma-finite-flat-local}", "we have $B_{\\mathfrak p} \\cong A_{\\mathfrak p}^{\\oplus n}$", "for some integer $n \\geq 0$. Note that $n \\geq 1$ as we assumed there", "exists at least one point of $T$ lying over $s$. Hence we see by", "looking at stalks that", "$$", "s \\in \\text{WeakAss}(\\mathcal{G})", "\\Leftrightarrow", "s \\in \\text{WeakAss}(g_*g^*\\mathcal{G})", "$$", "Now the assumption that $t \\in \\text{WeakAss}(g^*\\mathcal{G})$", "implies that $s \\in \\text{WeakAss}(g_*g^*\\mathcal{G})$ by", "Divisors, Lemma \\ref{divisors-lemma-weakly-associated-finite}", "and hence by the above $s \\in \\text{WeakAss}(\\mathcal{G})$." ], "refs": [ "divisors-lemma-weakly-ass-pullback", "algebra-lemma-finite-flat-local", "divisors-lemma-weakly-associated-finite" ], "ref_ids": [ 7886, 797, 7885 ] } ], "ref_ids": [] }, { "id": 5985, "type": "theorem", "label": "flat-lemma-etale-weak-assassin-up-down", "categories": [ "flat" ], "title": "flat-lemma-etale-weak-assassin-up-down", "contents": [ "Let $h : U \\to S$ be an \\'etale morphism of schemes.", "Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_S$-module.", "Let $u \\in U$ be a point with image $s \\in S$. Then", "$$", "u \\in \\text{WeakAss}(h^*\\mathcal{G})", "\\Leftrightarrow", "s \\in \\text{WeakAss}(\\mathcal{G})", "$$" ], "refs": [], "proofs": [ { "contents": [ "After replacing $S$ and $U$ by affine neighbourhoods of $s$ and $u$", "we may assume that $g$ is a standard \\'etale morphism of affines, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-locally-standard-etale}.", "Thus we may assume $S = \\Spec(A)$ and", "$X = \\Spec(A[x, 1/g]/(f))$, where $f$ is monic and $f'$", "is invertible in $A[x, 1/g]$.", "Note that $A[x, 1/g]/(f) = (A[x]/(f))_g$ is also the localization", "of the finite free $A$-algebra $A[x]/(f)$. Hence we may think of", "$U$ as an open subscheme of the scheme $T = \\Spec(A[x]/(f))$", "which is finite locally free over $S$. This reduces us to", "Lemma \\ref{lemma-finite-flat-weak-assassin-up-down}", "above." ], "refs": [ "morphisms-lemma-etale-locally-standard-etale", "flat-lemma-finite-flat-weak-assassin-up-down" ], "ref_ids": [ 5371, 5984 ] } ], "ref_ids": [] }, { "id": 5986, "type": "theorem", "label": "flat-lemma-weakly-associated-henselization", "categories": [ "flat" ], "title": "flat-lemma-weakly-associated-henselization", "contents": [ "Let $S$ be a scheme and $s \\in S$ a point. Denote $\\mathcal{O}_{S, s}^h$", "(resp.\\ $\\mathcal{O}_{S, s}^{sh}$) the henselization (resp.\\ strict", "henselization), see", "Algebra, Definition \\ref{algebra-definition-henselization}.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $s$ is a weakly associated point of $\\mathcal{F}$,", "\\item $\\mathfrak m_s$ is a weakly associated prime of $\\mathcal{F}_s$,", "\\item $\\mathfrak m_s^h$ is a weakly associated prime of", "$\\mathcal{F}_s \\otimes_{\\mathcal{O}_{S, s}} \\mathcal{O}_{S, s}^h$, and", "\\item $\\mathfrak m_s^{sh}$ is a weakly associated prime of", "$\\mathcal{F}_s \\otimes_{\\mathcal{O}_{S, s}} \\mathcal{O}_{S, s}^{sh}$.", "\\end{enumerate}" ], "refs": [ "algebra-definition-henselization" ], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is the definition, see", "Divisors, Definition \\ref{divisors-definition-weakly-associated}.", "The implications (2) $\\Rightarrow$ (3) $\\Rightarrow$ (4)", "follows from Divisors, Lemma \\ref{divisors-lemma-weakly-ass-pullback}", "applied to the flat (More on Algebra, Lemma", "\\ref{more-algebra-lemma-dumb-properties-henselization})", "morphisms", "$$", "\\Spec(\\mathcal{O}_{S, s}^{sh}) \\to", "\\Spec(\\mathcal{O}_{S, s}^h) \\to", "\\Spec(\\mathcal{O}_{S, s})", "$$", "and the closed points. To prove (4) $\\Rightarrow$ (2) we may replace", "$S$ by an affine neighbourhood. Suppose that", "$x \\in \\mathcal{F}_s \\otimes_{\\mathcal{O}_{S, s}} \\mathcal{O}_{S, s}^{sh}$", "is an element whose annihilator has radical equal to $\\mathfrak m_s^{sh}$.", "(See Algebra, Lemma \\ref{algebra-lemma-weakly-ass-local}.)", "Since $\\mathcal{O}_{S, s}^{sh}$ is equal to the limit", "of $\\mathcal{O}_{U, u}$ over \\'etale neighbourhoods", "$f : (U, u) \\to (S, s)$ by Algebra, Lemma", "\\ref{algebra-lemma-strict-henselization-different}", "we may assume that $x$ is the image of some", "$x' \\in \\mathcal{F}_s \\otimes_{\\mathcal{O}_{S, s}} \\mathcal{O}_{U, u}$.", "The local ring map $\\mathcal{O}_{U, u} \\to \\mathcal{O}_{S, s}^{sh}$", "is faithfully flat (as it is the strict henselization), hence", "universally injective", "(Algebra, Lemma \\ref{algebra-lemma-faithfully-flat-universally-injective}).", "It follows that the annihilator of $x'$ is the inverse image of the", "annihilator of $x$. Hence the radical of this annihilator", "is equal to $\\mathfrak m_u$.", "Thus $u$ is a weakly associated point of $f^*\\mathcal{F}$.", "By Lemma \\ref{lemma-etale-weak-assassin-up-down}", "we see that $s$ is a weakly associated point of $\\mathcal{F}$." ], "refs": [ "divisors-definition-weakly-associated", "divisors-lemma-weakly-ass-pullback", "more-algebra-lemma-dumb-properties-henselization", "algebra-lemma-weakly-ass-local", "algebra-lemma-strict-henselization-different", "algebra-lemma-faithfully-flat-universally-injective", "flat-lemma-etale-weak-assassin-up-down" ], "ref_ids": [ 8084, 7886, 10055, 720, 1304, 814, 5985 ] } ], "ref_ids": [ 1546 ] }, { "id": 5987, "type": "theorem", "label": "flat-lemma-sheaf-lives-on-subscheme", "categories": [ "flat" ], "title": "flat-lemma-sheaf-lives-on-subscheme", "contents": [ "Let $f : X \\to S$ be a finite type morphism of affine schemes.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Let $x \\in X$ with image $s = f(x)$ in $S$.", "Set $\\mathcal{F}_s = \\mathcal{F}|_{X_s}$.", "Then there exist a closed immersion $i : Z \\to X$ of finite presentation,", "and a quasi-coherent finite type $\\mathcal{O}_Z$-module $\\mathcal{G}$", "such that $i_*\\mathcal{G} = \\mathcal{F}$ and", "$Z_s = \\text{Supp}(\\mathcal{F}_s)$." ], "refs": [], "proofs": [ { "contents": [ "Say the morphism $f : X \\to S$ is given by the ring map", "$A \\to B$ and that $\\mathcal{F}$ is the quasi-coherent sheaf", "associated to the $B$-module $M$. By", "Morphisms, Lemma \\ref{morphisms-lemma-locally-finite-type-characterize}", "we know that $A \\to B$ is a finite type ring map, and by", "Properties, Lemma \\ref{properties-lemma-finite-type-module}", "we know that $M$ is a finite $B$-module. In particular the", "support of $\\mathcal{F}$ is the closed subscheme of $\\Spec(B)$", "cut out by the annihilator", "$I = \\{x \\in B \\mid xm = 0\\ \\forall m \\in M\\}$ of $M$, see", "Algebra, Lemma \\ref{algebra-lemma-support-closed}.", "Let $\\mathfrak q \\subset B$ be the prime ideal corresponding to $x$", "and let $\\mathfrak p \\subset A$ be the prime ideal corresponding to $s$.", "Note that $X_s = \\Spec(B \\otimes_A \\kappa(\\mathfrak p))$ and", "that $\\mathcal{F}_s$ is the quasi-coherent sheaf associated to the", "$B \\otimes_A \\kappa(\\mathfrak p)$ module $M \\otimes_A \\kappa(\\mathfrak p)$. By", "Morphisms, Lemma \\ref{morphisms-lemma-support-finite-type}", "the support of $\\mathcal{F}_s$ is equal to", "$V(I(B \\otimes_A \\kappa(\\mathfrak p)))$. Since", "$B \\otimes_A \\kappa(\\mathfrak p)$ is of finite type over $\\kappa(\\mathfrak p)$", "there exist finitely many elements $f_1, \\ldots, f_m \\in I$", "such that", "$$", "I(B \\otimes_A \\kappa(\\mathfrak p)) =", "(f_1, \\ldots, f_n)(B \\otimes_A \\kappa(\\mathfrak p)).", "$$", "Denote $i : Z \\to X$ the closed subscheme cut out by", "$(f_1, \\ldots, f_m)$, in a formula $Z = \\Spec(B/(f_1, \\ldots, f_m))$.", "Since $M$ is annihilated by $I$ we can think of $M$ as an", "$B/(f_1, \\ldots, f_m)$-module. In other words, $\\mathcal{F}$ is the", "pushforward of a finite type module on $Z$.", "As $Z_s = \\text{Supp}(\\mathcal{F}_s)$ by construction, this", "proves the lemma." ], "refs": [ "morphisms-lemma-locally-finite-type-characterize", "properties-lemma-finite-type-module", "algebra-lemma-support-closed", "morphisms-lemma-support-finite-type" ], "ref_ids": [ 5198, 3002, 543, 5143 ] } ], "ref_ids": [] }, { "id": 5988, "type": "theorem", "label": "flat-lemma-elementary-devissage", "categories": [ "flat" ], "title": "flat-lemma-elementary-devissage", "contents": [ "Let $f : X \\to S$ be morphism of schemes which is locally of finite type.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Let $x \\in X$ with image $s = f(x)$ in $S$.", "Set $\\mathcal{F}_s = \\mathcal{F}|_{X_s}$ and", "$n = \\dim_x(\\text{Supp}(\\mathcal{F}_s))$.", "Then we can construct", "\\begin{enumerate}", "\\item elementary \\'etale neighbourhoods $g : (X', x') \\to (X, x)$,", "$e : (S', s') \\to (S, s)$,", "\\item a commutative diagram", "$$", "\\xymatrix{", "X \\ar[dd]_f & X' \\ar[dd] \\ar[l]^g & Z' \\ar[l]^i \\ar[d]^\\pi \\\\", "& & Y' \\ar[d]^h \\\\", "S & S' \\ar[l]_e & S' \\ar@{=}[l]", "}", "$$", "\\item a point $z' \\in Z'$ with $i(z') = x'$, $y' = \\pi(z')$, $h(y') = s'$,", "\\item a finite type quasi-coherent $\\mathcal{O}_{Z'}$-module $\\mathcal{G}$,", "\\end{enumerate}", "such that the following properties hold", "\\begin{enumerate}", "\\item $X'$, $Z'$, $Y'$, $S'$ are affine schemes,", "\\item $i$ is a closed immersion of finite presentation,", "\\item $i_*(\\mathcal{G}) \\cong g^*\\mathcal{F}$,", "\\item $\\pi$ is finite and $\\pi^{-1}(\\{y'\\}) = \\{z'\\}$,", "\\item the extension $\\kappa(s') \\subset \\kappa(y')$ is purely transcendental,", "\\item $h$ is smooth of relative dimension $n$", "with geometrically integral fibres.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset S$ be an affine neighbourhood of $s$.", "Let $U \\subset f^{-1}(V)$ be an affine neighbourhood of $x$.", "Then it suffices to prove the lemma for $f|_U : U \\to V$ and", "$\\mathcal{F}|_U$. Hence in the rest of the proof we assume that", "$X$ and $S$ are affine.", "\\medskip\\noindent", "First, suppose that $X_s = \\text{Supp}(\\mathcal{F}_s)$, in particular", "$n = \\dim_x(X_s)$. Apply", "More on Morphisms,", "Lemmas \\ref{more-morphisms-lemma-local-local-structure-finite-type} and", "\\ref{more-morphisms-lemma-local-local-structure-finite-type-affine}.", "This gives us a commutative diagram", "$$", "\\xymatrix{", "X \\ar[dd] & X' \\ar[l]^g \\ar[d]^\\pi \\\\", "& Y' \\ar[d]^h \\\\", "S & S' \\ar[l]_e", "}", "$$", "and point $x' \\in X'$. We set $Z' = X'$, $i = \\text{id}$, and", "$\\mathcal{G} = g^*\\mathcal{F}$ to obtain a solution in this case.", "\\medskip\\noindent", "In general choose a closed immersion $Z \\to X$ and a sheaf", "$\\mathcal{G}$ on $Z$ as in", "Lemma \\ref{lemma-sheaf-lives-on-subscheme}.", "Applying the result of the previous paragraph to $Z \\to S$ and", "$\\mathcal{G}$ we obtain a diagram", "$$", "\\xymatrix{", "X \\ar[dd]_f & Z \\ar[l] \\ar[dd]_{f|_Z} & Z' \\ar[l]^g \\ar[d]^\\pi \\\\", "& & Y' \\ar[d]^h \\\\", "S & S \\ar@{=}[l] & S' \\ar[l]_e", "}", "$$", "and point $z' \\in Z'$ satisfying all the required properties.", "We will use", "Lemma \\ref{lemma-lift-etale}", "to embed $Z'$ into a scheme \\'etale over $X$. We cannot apply the lemma directly", "as we want $X'$ to be a scheme over $S'$. Instead we", "consider the morphisms", "$$", "\\xymatrix{", "Z' \\ar[r] & Z \\times_S S' \\ar[r] & X \\times_S S'", "}", "$$", "The first morphism is \\'etale by", "Morphisms, Lemma \\ref{morphisms-lemma-etale-permanence}.", "The second is a closed immersion as a base change of a closed immersion.", "Finally, as $X$, $S$, $S'$, $Z$, $Z'$ are all affine we may apply", "Lemma \\ref{lemma-lift-etale}", "to get an \\'etale morphism of affine schemes $X' \\to X \\times_S S'$ such that", "$$", "Z' = (Z \\times_S S') \\times_{(X \\times_S S')} X' = Z \\times_X X'.", "$$", "As $Z \\to X$ is a closed immersion of finite presentation, so is $Z' \\to X'$.", "Let $x' \\in X'$ be the point corresponding to $z' \\in Z'$.", "Then the completed diagram", "$$", "\\xymatrix{", "X \\ar[dd] & X' \\ar[dd] \\ar[l] & Z' \\ar[l]^i \\ar[d]^\\pi \\\\", "& & Y' \\ar[d]^h \\\\", "S & S' \\ar[l]_e & S' \\ar@{=}[l]", "}", "$$", "is a solution of the original problem." ], "refs": [ "more-morphisms-lemma-local-local-structure-finite-type", "more-morphisms-lemma-local-local-structure-finite-type-affine", "flat-lemma-sheaf-lives-on-subscheme", "flat-lemma-lift-etale", "morphisms-lemma-etale-permanence", "flat-lemma-lift-etale" ], "ref_ids": [ 13918, 13919, 5987, 5978, 5375, 5978 ] } ], "ref_ids": [] }, { "id": 5989, "type": "theorem", "label": "flat-lemma-devissage-finite-presentation", "categories": [ "flat" ], "title": "flat-lemma-devissage-finite-presentation", "contents": [ "Assumptions and notation as in", "Lemma \\ref{lemma-elementary-devissage}.", "If $f$ is locally of finite presentation", "then $\\pi$ is of finite presentation.", "In this case the following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation", "in a neighbourhood of $x$,", "\\item $\\mathcal{G}$ is an $\\mathcal{O}_{Z'}$-module of finite presentation", "in a neighbourhood of $z'$, and", "\\item $\\pi_*\\mathcal{G}$ is an $\\mathcal{O}_{Y'}$-module of", "finite presentation in a neighbourhood of $y'$.", "\\end{enumerate}", "Still assuming $f$ locally of finite presentation the following are", "equivalent to each other", "\\begin{enumerate}", "\\item[(a)] $\\mathcal{F}_x$ is an $\\mathcal{O}_{X, x}$-module of finite", "presentation,", "\\item[(b)] $\\mathcal{G}_{z'}$ is an $\\mathcal{O}_{Z', z'}$-module of", "finite presentation, and", "\\item[(c)] $(\\pi_*\\mathcal{G})_{y'}$ is an $\\mathcal{O}_{Y', y'}$-module", "of finite presentation.", "\\end{enumerate}" ], "refs": [ "flat-lemma-elementary-devissage" ], "proofs": [ { "contents": [ "Assume $f$ locally of finite presentation. Then $Z' \\to S$ is locally", "of finite presentation as a composition of such, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-presentation}.", "Note that $Y' \\to S$ is also locally of finite presentation as a composition", "of a smooth and an \\'etale morphism. Hence", "Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-permanence}", "implies $\\pi$ is locally of finite presentation.", "Since $\\pi$ is finite we conclude that it is also separated and", "quasi-compact, hence $\\pi$ is actually of finite presentation.", "\\medskip\\noindent", "To prove the equivalence of (1), (2), and (3) we also consider:", "(4) $g^*\\mathcal{F}$ is a $\\mathcal{O}_{X'}$-module of finite presentation", "in a neighbourhood of $x'$. The pullback of a module of finite presentation", "is of finite presentation, see", "Modules, Lemma \\ref{modules-lemma-pullback-finite-presentation}.", "Hence (1) $\\Rightarrow$ (4).", "The \\'etale morphism $g$ is open, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-open}.", "Hence for any open neighbourhood $U' \\subset X'$ of $x'$, the image", "$g(U')$ is an open neighbourhood of $x$ and the map", "$\\{U' \\to g(U')\\}$ is an \\'etale covering. Thus (4) $\\Rightarrow$ (1) by", "Descent, Lemma \\ref{descent-lemma-finite-presentation-descends}.", "Using", "Descent, Lemma \\ref{descent-lemma-finite-finitely-presented-module}", "and some easy topological arguments (see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-finite-morphism-single-point-in-fibre})", "we see that", "(4) $\\Leftrightarrow$ (2) $\\Leftrightarrow$ (3).", "\\medskip\\noindent", "To prove the equivalence of (a), (b), (c) consider the ring maps", "$$", "\\mathcal{O}_{X, x} \\to", "\\mathcal{O}_{X', x'} \\to", "\\mathcal{O}_{Z', z'} \\leftarrow", "\\mathcal{O}_{Y', y'}", "$$", "The first ring map is faithfully flat. Hence", "$\\mathcal{F}_x$ is of finite presentation over $\\mathcal{O}_{X, x}$", "if and only if $g^*\\mathcal{F}_{x'}$ is of finite presentation over", "$\\mathcal{O}_{X', x'}$, see", "Algebra, Lemma \\ref{algebra-lemma-descend-properties-modules}.", "The second ring map is surjective (hence finite) and", "finitely presented by assumption, hence", "$g^*\\mathcal{F}_{x'}$ is of finite presentation over $\\mathcal{O}_{X', x'}$", "if and only if $\\mathcal{G}_{z'}$ is of finite presentation over", "$\\mathcal{O}_{Z', z'}$, see", "Algebra, Lemma \\ref{algebra-lemma-finite-finitely-presented-extension}.", "Because $\\pi$ is finite, of finite presentation, and", "$\\pi^{-1}(\\{y'\\}) = \\{x'\\}$ the ring homomorphism", "$\\mathcal{O}_{Y', y'} \\leftarrow \\mathcal{O}_{Z', z'}$ is finite", "and of finite presentation, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-finite-morphism-single-point-in-fibre}.", "Hence $\\mathcal{G}_{z'}$ is of finite presentation over $\\mathcal{O}_{Z', z'}$", "if and only if $\\pi_*\\mathcal{G}_{y'}$ is of finite presentation over", "$\\mathcal{O}_{Y', y'}$, see", "Algebra, Lemma \\ref{algebra-lemma-finite-finitely-presented-extension}." ], "refs": [ "morphisms-lemma-composition-finite-presentation", "morphisms-lemma-finite-presentation-permanence", "modules-lemma-pullback-finite-presentation", "morphisms-lemma-etale-open", "descent-lemma-finite-presentation-descends", "descent-lemma-finite-finitely-presented-module", "more-morphisms-lemma-finite-morphism-single-point-in-fibre", "algebra-lemma-descend-properties-modules", "algebra-lemma-finite-finitely-presented-extension", "more-morphisms-lemma-finite-morphism-single-point-in-fibre", "algebra-lemma-finite-finitely-presented-extension" ], "ref_ids": [ 5239, 5247, 13250, 5370, 14614, 14620, 13920, 819, 501, 13920, 501 ] } ], "ref_ids": [ 5988 ] }, { "id": 5990, "type": "theorem", "label": "flat-lemma-devissage-flat", "categories": [ "flat" ], "title": "flat-lemma-devissage-flat", "contents": [ "Assumptions and notation as in", "Lemma \\ref{lemma-elementary-devissage}.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is flat over $S$ in a neighbourhood of $x$,", "\\item $\\mathcal{G}$ is flat over $S'$ in a neighbourhood of $z'$, and", "\\item $\\pi_*\\mathcal{G}$ is flat over $S'$ in a neighbourhood of $y'$.", "\\end{enumerate}", "The following are equivalent also", "\\begin{enumerate}", "\\item[(a)] $\\mathcal{F}_x$ is flat over $\\mathcal{O}_{S, s}$,", "\\item[(b)] $\\mathcal{G}_{z'}$ is flat over $\\mathcal{O}_{S', s'}$, and", "\\item[(c)] $(\\pi_*\\mathcal{G})_{y'}$ is flat over $\\mathcal{O}_{S', s'}$.", "\\end{enumerate}" ], "refs": [ "flat-lemma-elementary-devissage" ], "proofs": [ { "contents": [ "To prove the equivalence of (1), (2), and (3) we also consider:", "(4) $g^*\\mathcal{F}$ is flat over $S$ in a neighbourhood of $x'$.", "We will use", "Lemma \\ref{lemma-etale-flat-up-down}", "to equate flatness over $S$ and $S'$ without further mention.", "The \\'etale morphism $g$ is flat and open, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-open}.", "Hence for any open neighbourhood $U' \\subset X'$ of $x'$, the image", "$g(U')$ is an open neighbourhood of $x$ and the map", "$U' \\to g(U')$ is surjective and flat.", "Thus (4) $\\Leftrightarrow$ (1) by", "Morphisms, Lemma \\ref{morphisms-lemma-flat-permanence}.", "Note that", "$$", "\\Gamma(X', g^*\\mathcal{F}) =", "\\Gamma(Z', \\mathcal{G}) =", "\\Gamma(Y', \\pi_*\\mathcal{G})", "$$", "Hence the flatness of $g^*\\mathcal{F}$, $\\mathcal{G}$ and $\\pi_*\\mathcal{G}$", "over $S'$ are all equivalent (this uses that $X'$, $Z'$, $Y'$, and", "$S'$ are all affine). Some omitted topological arguments (compare", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-finite-morphism-single-point-in-fibre})", "regarding affine neighbourhoods now show that", "(4) $\\Leftrightarrow$ (2) $\\Leftrightarrow$ (3).", "\\medskip\\noindent", "To prove the equivalence of (a), (b), (c) consider the commutative diagram", "of local ring maps", "$$", "\\xymatrix{", "\\mathcal{O}_{X', x'} \\ar[r]_\\iota &", "\\mathcal{O}_{Z', z'} &", "\\mathcal{O}_{Y', y'} \\ar[l]^\\alpha &", "\\mathcal{O}_{S', s'} \\ar[l]^\\beta \\\\", "\\mathcal{O}_{X, x} \\ar[u]^\\gamma & & &", "\\mathcal{O}_{S, s} \\ar[lll]_\\varphi \\ar[u]_\\epsilon", "}", "$$", "We will use", "Lemma \\ref{lemma-etale-flat-up-down-local-ring}", "to equate flatness over $\\mathcal{O}_{S, s}$ and $\\mathcal{O}_{S', s'}$", "without further mention.", "The map $\\gamma$ is faithfully flat. Hence", "$\\mathcal{F}_x$ is flat over $\\mathcal{O}_{S, s}$", "if and only if $g^*\\mathcal{F}_{x'}$ is flat over", "$\\mathcal{O}_{S', s'}$, see", "Algebra, Lemma \\ref{algebra-lemma-flatness-descends-more-general}.", "As $\\mathcal{O}_{S', s'}$-modules the modules", "$g^*\\mathcal{F}_{x'}$, $\\mathcal{G}_{z'}$, and", "$\\pi_*\\mathcal{G}_{y'}$ are all isomorphic, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-finite-morphism-single-point-in-fibre}.", "This finishes the proof." ], "refs": [ "flat-lemma-etale-flat-up-down", "morphisms-lemma-etale-open", "morphisms-lemma-flat-permanence", "more-morphisms-lemma-finite-morphism-single-point-in-fibre", "flat-lemma-etale-flat-up-down-local-ring", "algebra-lemma-flatness-descends-more-general", "more-morphisms-lemma-finite-morphism-single-point-in-fibre" ], "ref_ids": [ 5980, 5370, 5270, 13920, 5981, 529, 13920 ] } ], "ref_ids": [ 5988 ] }, { "id": 5991, "type": "theorem", "label": "flat-lemma-elementary-devissage-variant", "categories": [ "flat" ], "title": "flat-lemma-elementary-devissage-variant", "contents": [ "Let $f : X \\to S$ be morphism of schemes which is locally of finite type.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Let $x \\in X$ with image $s = f(x)$ in $S$.", "Then there exists a commutative diagram of pointed schemes", "$$", "\\xymatrix{", "(X, x) \\ar[d]_f & (X', x') \\ar[l]^g \\ar[d] \\\\", "(S, s) & (S', s') \\ar[l] \\\\", "}", "$$", "such that $(S', s') \\to (S, s)$ and $(X', x') \\to (X, x)$", "are elementary \\'etale neighbourhoods, and such that", "$g^*\\mathcal{F}/X'/S'$ has a one step d\\'evissage at $x'$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from", "Definition \\ref{definition-one-step-devissage-at-x}", "and", "Lemma \\ref{lemma-elementary-devissage}." ], "refs": [ "flat-definition-one-step-devissage-at-x", "flat-lemma-elementary-devissage" ], "ref_ids": [ 6207, 5988 ] } ], "ref_ids": [] }, { "id": 5992, "type": "theorem", "label": "flat-lemma-base-change-one-step", "categories": [ "flat" ], "title": "flat-lemma-base-change-one-step", "contents": [ "Let $S$, $X$, $\\mathcal{F}$, $s$ be as in", "Definition \\ref{definition-one-step-devissage}.", "Let $(Z, Y, i, \\pi, \\mathcal{G})$ be a one step d\\'evissage", "of $\\mathcal{F}/X/S$ over $s$.", "Let $(S', s') \\to (S, s)$ be any morphism of pointed schemes.", "Given this data let $X', Z', Y', i', \\pi'$ be the base", "changes of $X, Z, Y, i, \\pi$ via $S' \\to S$.", "Let $\\mathcal{F}'$ be the pullback of $\\mathcal{F}$ to $X'$", "and let $\\mathcal{G}'$ be the pullback of $\\mathcal{G}$ to $Z'$.", "If $S'$ is affine, then $(Z', Y', i', \\pi', \\mathcal{G}')$", "is a one step d\\'evissage of $\\mathcal{F}'/X'/S'$ over $s'$." ], "refs": [ "flat-definition-one-step-devissage" ], "proofs": [ { "contents": [ "Fibre products of affines are affine, see", "Schemes, Lemma \\ref{schemes-lemma-fibre-product-affines}.", "Base change preserves", "closed immersions,", "morphisms of finite presentation,", "finite morphisms,", "smooth morphisms,", "morphisms with geometrically irreducible fibres, and", "morphisms of relative dimension $n$, see", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-closed-immersion},", "\\ref{morphisms-lemma-base-change-finite-presentation},", "\\ref{morphisms-lemma-base-change-finite},", "\\ref{morphisms-lemma-base-change-smooth},", "\\ref{morphisms-lemma-base-change-relative-dimension-d}, and", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-base-change-fibres-geometrically-irreducible}.", "We have $i'_*\\mathcal{G}' \\cong \\mathcal{F}'$ because pushforward", "along the finite morphism $i$ commutes with base change, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-affine-base-change}.", "We have", "$\\dim(\\text{Supp}(\\mathcal{F}_s)) = \\dim(\\text{Supp}(\\mathcal{F}'_{s'}))$", "by", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}", "because", "$$", "\\text{Supp}(\\mathcal{F}_s) \\times_s s' = \\text{Supp}(\\mathcal{F}'_{s'}).", "$$", "This proves the lemma." ], "refs": [ "schemes-lemma-fibre-product-affines", "morphisms-lemma-base-change-closed-immersion", "morphisms-lemma-base-change-finite-presentation", "morphisms-lemma-base-change-finite", "morphisms-lemma-base-change-smooth", "morphisms-lemma-base-change-relative-dimension-d", "more-morphisms-lemma-base-change-fibres-geometrically-irreducible", "coherent-lemma-affine-base-change", "morphisms-lemma-dimension-fibre-after-base-change" ], "ref_ids": [ 7690, 5128, 5240, 5440, 5327, 5284, 13821, 3297, 5279 ] } ], "ref_ids": [ 6206 ] }, { "id": 5993, "type": "theorem", "label": "flat-lemma-base-change-one-step-at-x", "categories": [ "flat" ], "title": "flat-lemma-base-change-one-step-at-x", "contents": [ "Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in", "Definition \\ref{definition-one-step-devissage-at-x}.", "Let $(Z, Y, i, \\pi, \\mathcal{G}, z, y)$ be a one step d\\'evissage", "of $\\mathcal{F}/X/S$ at $x$.", "Let $(S', s') \\to (S, s)$ be a morphism of pointed schemes", "which induces an isomorphism $\\kappa(s) = \\kappa(s')$.", "Let $(Z', Y', i', \\pi', \\mathcal{G}')$ be as constructed in", "Lemma \\ref{lemma-base-change-one-step}", "and let $x' \\in X'$ (resp.\\ $z' \\in Z'$, $y' \\in Y'$) be the", "unique point mapping to both $x \\in X$ (resp.\\ $z \\in Z$, $y \\in Y$)", "and $s' \\in S'$.", "If $S'$ is affine, then $(Z', Y', i', \\pi', \\mathcal{G}', z', y')$", "is a one step d\\'evissage of $\\mathcal{F}'/X'/S'$ at $x'$." ], "refs": [ "flat-definition-one-step-devissage-at-x", "flat-lemma-base-change-one-step" ], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-base-change-one-step}", "$(Z', Y', i', \\pi', \\mathcal{G}')$ is a one step d\\'evissage of", "$\\mathcal{F}'/X'/S'$ over $s'$. Properties (1) -- (4) of", "Definition \\ref{definition-one-step-devissage-at-x}", "hold for $(Z', Y', i', \\pi', \\mathcal{G}', z', y')$", "as the assumption that $\\kappa(s) = \\kappa(s')$ insures that the fibres", "$X'_{s'}$, $Z'_{s'}$, and $Y'_{s'}$ are isomorphic to", "$X_s$, $Z_s$, and $Y_s$." ], "refs": [ "flat-lemma-base-change-one-step", "flat-definition-one-step-devissage-at-x" ], "ref_ids": [ 5992, 6207 ] } ], "ref_ids": [ 6207, 5992 ] }, { "id": 5994, "type": "theorem", "label": "flat-lemma-shrink", "categories": [ "flat" ], "title": "flat-lemma-shrink", "contents": [ "With assumption and notation as in", "Definition \\ref{definition-shrink}", "we have:", "\\begin{enumerate}", "\\item", "\\label{item-shrink-base}", "If $S' \\subset S$ is a standard open neighbourhood of $s$, then", "setting $X' = X_{S'}$, $Z' = Z_{S'}$ and $Y' = Y_{S'}$ we obtain a", "standard shrinking.", "\\item", "\\label{item-shrink-on-Y}", "Let $W \\subset Y$ be a standard open neighbourhood of $y$.", "Then there exists a standard shrinking with $Y' = W \\times_S S'$.", "\\item", "\\label{item-shrink-on-X}", "Let $U \\subset X$ be an open neighbourhood of $x$.", "Then there exists a standard shrinking with $X' \\subset U$.", "\\end{enumerate}" ], "refs": [ "flat-definition-shrink" ], "proofs": [ { "contents": [ "Part (1) is immediate from", "Lemma \\ref{lemma-base-change-one-step-at-x}", "and the fact that the inverse image of a standard open under a morphism", "of affine schemes is a standard open, see", "Algebra, Lemma \\ref{algebra-lemma-spec-functorial}.", "\\medskip\\noindent", "Let $W \\subset Y$ as in (2). Because $Y \\to S$ is smooth it is open, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-open}.", "Hence we can find a standard open neighbourhood $S'$ of $s$", "contained in the image of $W$. Then the fibres of $W_{S'} \\to S'$", "are nonempty open subschemes of the fibres of $Y \\to S$ over $S'$", "and hence geometrically irreducible too. Setting $Y' = W_{S'}$", "and $Z' = \\pi^{-1}(Y')$ we see that $Z' \\subset Z$ is a standard open", "neighbourhood of $z$. Let $\\overline{h} \\in \\Gamma(Z, \\mathcal{O}_Z)$", "be a function such that $Z' = D(\\overline{h})$. As $i : Z \\to X$", "is a closed immersion, we can find a function $h \\in \\Gamma(X, \\mathcal{O}_X)$", "such that $i^\\sharp(h) = \\overline{h}$. Take $X' = D(h) \\subset X$.", "In this way we obtain a standard shrinking as in (2).", "\\medskip\\noindent", "Let $U \\subset X$ be as in (3). We may after shrinking $U$ assume that", "$U$ is a standard open. By", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-finite-morphism-single-point-in-fibre}", "there exists a standard open $W \\subset Y$ neighbourhood of $y$ such", "that $\\pi^{-1}(W) \\subset i^{-1}(U)$. Apply (2) to get a standard", "shrinking $X', S', Z', Y'$ with $Y' = W_{S'}$. Since", "$Z' \\subset \\pi^{-1}(W) \\subset i^{-1}(U)$ we may replace $X'$ by", "$X' \\cap U$ (still a standard open as $U$ is also standard open)", "without violating any of the conditions defining a standard shrinking.", "Hence we win." ], "refs": [ "flat-lemma-base-change-one-step-at-x", "algebra-lemma-spec-functorial", "morphisms-lemma-smooth-open", "more-morphisms-lemma-finite-morphism-single-point-in-fibre" ], "ref_ids": [ 5993, 390, 5332, 13920 ] } ], "ref_ids": [ 6208 ] }, { "id": 5995, "type": "theorem", "label": "flat-lemma-elementary-etale-neighbourhood", "categories": [ "flat" ], "title": "flat-lemma-elementary-etale-neighbourhood", "contents": [ "Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in", "Definition \\ref{definition-one-step-devissage-at-x}.", "Let $(Z, Y, i, \\pi, \\mathcal{G}, z, y)$ be a one step d\\'evissage", "of $\\mathcal{F}/X/S$ at $x$. Let", "$$", "\\xymatrix{", "(Y, y) \\ar[d] & (Y', y') \\ar[l] \\ar[d] \\\\", "(S, s) & (S', s') \\ar[l]", "}", "$$", "be a commutative diagram of pointed schemes such that the horizontal", "arrows are elementary \\'etale neighbourhoods. Then there exists", "a commutative diagram", "$$", "\\xymatrix{", "& & (X'', x'') \\ar[lld] \\ar[d] & (Z'', z'') \\ar[l] \\ar[lld] \\ar[d] \\\\", "(X, x) \\ar[d] & (Z, z) \\ar[l] \\ar[d] &", "(S'', s'') \\ar[lld] & (Y'', y'') \\ar[lld] \\ar[l] \\\\", "(S, s) & (Y, y) \\ar[l]", "}", "$$", "of pointed schemes with the following properties:", "\\begin{enumerate}", "\\item $(S'', s'') \\to (S', s')$ is an elementary \\'etale neighbourhood and", "the morphism $S'' \\to S$ is the composition $S'' \\to S' \\to S$,", "\\item $Y''$ is an open subscheme of $Y' \\times_{S'} S''$,", "\\item $Z'' = Z \\times_Y Y''$,", "\\item $(X'', x'') \\to (X, x)$ is an elementary \\'etale neighbourhood, and", "\\item $(Z'', Y'', i'', \\pi'', \\mathcal{G}'', z'', y'')$ is a one step", "d\\'evissage at $x''$ of the sheaf $\\mathcal{F}''$.", "\\end{enumerate}", "Here $\\mathcal{F}''$ (resp.\\ $\\mathcal{G}''$) is the pullback of", "$\\mathcal{F}$ (resp.\\ $\\mathcal{G}$) via the morphism $X'' \\to X$", "(resp.\\ $Z'' \\to Z$) and $i'' : Z'' \\to X''$ and $\\pi'' : Z'' \\to Y''$", "are as in the diagram." ], "refs": [ "flat-definition-one-step-devissage-at-x" ], "proofs": [ { "contents": [ "Let $(S'', s'') \\to (S', s')$ be any elementary \\'etale neighbourhood", "with $S''$ affine. Let $Y'' \\subset Y' \\times_{S'} S''$ be any affine", "open neighbourhood containing the point $y'' = (y', s'')$. Then we", "obtain an affine $(Z'', z'')$ by (3). Moreover $Z_{S''} \\to X_{S''}$", "is a closed immersion and $Z'' \\to Z_{S''}$ is an \\'etale", "morphism. Hence", "Lemma \\ref{lemma-lift-etale}", "applies and we can find an \\'etale morphism $X'' \\to X_{S'}$ of affines", "such that $Z'' \\cong X'' \\times_{X_{S'}} Z_{S'}$. Denote $i'' : Z'' \\to X''$", "the corresponding closed immersion. Setting $x'' = i''(z'')$ we obtain a", "commutative diagram as in the lemma.", "Properties (1), (2), (3), and (4) hold by construction.", "Thus it suffices to show that (5) holds for a suitable choice of", "$(S'', s'') \\to (S', s')$ and $Y''$.", "\\medskip\\noindent", "We first list those properties which hold for any choice of", "$(S'', s'') \\to (S', s')$ and $Y''$ as in the first paragraph.", "As we have $Z'' = X'' \\times_X Z$ by construction we see that", "$i''_*\\mathcal{G}'' = \\mathcal{F}''$ (with notation as in the", "statement of the lemma), see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-affine-base-change}.", "Set $n = \\dim(\\text{Supp}(\\mathcal{F}_s)) = \\dim_x(\\text{Supp}(\\mathcal{F}_s))$.", "The morphism $Y'' \\to S''$ is smooth of relative dimension $n$", "(because $Y' \\to S'$ is smooth of relative dimension $n$", "as the composition $Y' \\to Y_{S'} \\to S'$ of an \\'etale and", "smooth morphism of relative dimension $n$ and because base change", "preserves smooth morphisms of relative dimension $n$).", "We have $\\kappa(y'') = \\kappa(y)$ and $\\kappa(s) = \\kappa(s'')$", "hence $\\kappa(y'')$ is a purely transcendental extension of $\\kappa(s'')$.", "The morphism of fibres $X''_{s''} \\to X_s$ is an \\'etale morphism of affine", "schemes over $\\kappa(s) = \\kappa(s'')$ mapping the point $x''$ to the", "point $x$ and pulling back $\\mathcal{F}_s$ to $\\mathcal{F}''_{s''}$.", "Hence", "$$", "\\dim(\\text{Supp}(\\mathcal{F}''_{s''})) =", "\\dim(\\text{Supp}(\\mathcal{F}_s)) = n =", "\\dim_x(\\text{Supp}(\\mathcal{F}_s)) =", "\\dim_{x''}(\\text{Supp}(\\mathcal{F}''_{s''}))", "$$", "because dimension is invariant under \\'etale localization, see", "Descent, Lemma \\ref{descent-lemma-dimension-at-point-local}.", "As $\\pi'' : Z'' \\to Y''$ is the base change of $\\pi$ we see that", "$\\pi''$ is finite and as $\\kappa(y) = \\kappa(y'')$ we see that", "$\\pi^{-1}(\\{y''\\}) = \\{z''\\}$.", "\\medskip\\noindent", "At this point we have verified all the conditions of", "Definition \\ref{definition-one-step-devissage}", "except we have not verified that $Y'' \\to S''$ has geometrically", "irreducible fibres. Of course in general this is not going to be", "true, and it is at this point that we will use that", "$\\kappa(s) \\subset \\kappa(y)$ is purely transcendental. Namely,", "let $T \\subset Y'_{s'}$ be the irreducible component of", "$Y'_{s'}$ containing $y' = (y, s')$. Note that $T$ is an open subscheme", "of $Y'_{s'}$ as this is a smooth scheme over $\\kappa(s')$. By", "Varieties,", "Lemma \\ref{varieties-lemma-geometrically-connected-if-connected-and-point}", "we see that $T$ is geometrically connected because $\\kappa(s') = \\kappa(s)$", "is algebraically closed in $\\kappa(y') = \\kappa(y)$.", "As $T$ is smooth we see that $T$ is geometrically irreducible. Hence", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-normal-morphism-irreducible}", "applies and we can find an elementary \\'etale morphism", "$(S'', s'') \\to (S', s')$ and an affine open $Y'' \\subset Y'_{S''}$", "such that all fibres of $Y'' \\to S''$ are geometrically irreducible", "and such that $T = Y''_{s''}$. After shrinking (first $Y''$ and then $S''$)", "we may assume that both $Y''$ and $S''$ are affine.", "This finishes the proof of the lemma." ], "refs": [ "flat-lemma-lift-etale", "coherent-lemma-affine-base-change", "descent-lemma-dimension-at-point-local", "flat-definition-one-step-devissage", "varieties-lemma-geometrically-connected-if-connected-and-point", "more-morphisms-lemma-normal-morphism-irreducible" ], "ref_ids": [ 5978, 3297, 14660, 6206, 10926, 13916 ] } ], "ref_ids": [ 6207 ] }, { "id": 5996, "type": "theorem", "label": "flat-lemma-existence-alpha", "categories": [ "flat" ], "title": "flat-lemma-existence-alpha", "contents": [ "Let $S$, $X$, $\\mathcal{F}$, $s$ be as in", "Definition \\ref{definition-one-step-devissage}.", "Let $(Z, Y, i, \\pi, \\mathcal{G})$ be a one step d\\'evissage", "of $\\mathcal{F}/X/S$ over $s$.", "Let $\\xi \\in Y_s$ be the (unique) generic point.", "Then there exists an integer $r > 0$ and an $\\mathcal{O}_Y$-module map", "$$", "\\alpha : \\mathcal{O}_Y^{\\oplus r} \\longrightarrow \\pi_*\\mathcal{G}", "$$", "such that", "$$", "\\alpha :", "\\kappa(\\xi)^{\\oplus r}", "\\longrightarrow", "(\\pi_*\\mathcal{G})_\\xi \\otimes_{\\mathcal{O}_{Y, \\xi}} \\kappa(\\xi)", "$$", "is an isomorphism. Moreover, in this case we have", "$$", "\\dim(\\text{Supp}(\\Coker(\\alpha)_s)) < \\dim(\\text{Supp}(\\mathcal{F}_s)).", "$$" ], "refs": [ "flat-definition-one-step-devissage" ], "proofs": [ { "contents": [ "By assumption the schemes $S$ and $Y$ are affine.", "Write $S = \\Spec(A)$ and $Y = \\Spec(B)$.", "As $\\pi$ is finite the $\\mathcal{O}_Y$-module $\\pi_*\\mathcal{G}$", "is a finite type quasi-coherent $\\mathcal{O}_Y$-module.", "Hence $\\pi_*\\mathcal{G} = \\widetilde{N}$ for some finite $B$-module $N$.", "Let $\\mathfrak p \\subset B$ be the prime ideal corresponding to $\\xi$.", "To obtain $\\alpha$ set", "$r = \\dim_{\\kappa(\\mathfrak p)} N \\otimes_B \\kappa(\\mathfrak p)$", "and pick $x_1, \\ldots, x_r \\in N$ which form a basis of", "$N \\otimes_B \\kappa(\\mathfrak p)$. Take $\\alpha : B^{\\oplus r} \\to N$", "to be the map given by the formula $\\alpha(b_1, \\ldots, b_r) = \\sum b_ix_i$.", "It is clear that", "$\\alpha : \\kappa(\\mathfrak p)^{\\oplus r} \\to N \\otimes_B \\kappa(\\mathfrak p)$", "is an isomorphism as desired. Finally, suppose $\\alpha$ is any map with this", "property. Then $N' = \\Coker(\\alpha)$ is a finite $B$-module", "such that $N' \\otimes \\kappa(\\mathfrak p) = 0$. By Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK})", "we see that $N'_{\\mathfrak p} = 0$. Since the fibre $Y_s$ is", "geometrically irreducible of dimension $n$ with generic point $\\xi$", "and since we have just seen that $\\xi$ is not in the support of", "$\\Coker(\\alpha)$ the last assertion of the lemma holds." ], "refs": [ "algebra-lemma-NAK" ], "ref_ids": [ 401 ] } ], "ref_ids": [ 6206 ] }, { "id": 5997, "type": "theorem", "label": "flat-lemma-base-change-complete", "categories": [ "flat" ], "title": "flat-lemma-base-change-complete", "contents": [ "Let $S$, $X$, $\\mathcal{F}$, $s$ be as in", "Definition \\ref{definition-complete-devissage}.", "Let $(S', s') \\to (S, s)$ be any morphism of pointed schemes.", "Let $(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k)_{k = 1, \\ldots, n}$", "be a complete d\\'evissage of $\\mathcal{F}/X/S$ over $s$.", "Given this data let $X', Z'_k, Y'_k, i'_k, \\pi'_k$ be the base", "changes of $X, Z_k, Y_k, i_k, \\pi_k$ via $S' \\to S$.", "Let $\\mathcal{F}'$ be the pullback of $\\mathcal{F}$ to $X'$", "and let $\\mathcal{G}'_k$ be the pullback of $\\mathcal{G}_k$ to $Z'_k$.", "Let $\\alpha'_k$ be the pullback of $\\alpha_k$ to $Y'_k$.", "If $S'$ is affine, then", "$(Z'_k, Y'_k, i'_k, \\pi'_k, \\mathcal{G}'_k, \\alpha'_k)_{k = 1, \\ldots, n}$", "is a complete d\\'evissage of $\\mathcal{F}'/X'/S'$ over $s'$." ], "refs": [ "flat-definition-complete-devissage" ], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-base-change-one-step}", "we know that the base change of a one step d\\'evissage is a one step", "d\\'evissage. Hence it suffices to prove that formation of", "$\\Coker(\\alpha_k)$ commutes with base change and that", "condition (2) of", "Definition \\ref{definition-complete-devissage}", "is preserved by base change. The first is true as", "$\\pi'_{k, *}\\mathcal{G}'_k$ is the pullback of", "$\\pi_{k, *}\\mathcal{G}_k$ (by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-affine-base-change})", "and because $\\otimes$ is right exact. The second because", "by the same token we have", "$$", "(\\pi_{k, *}\\mathcal{G}_k)_{\\xi_k}", "\\otimes_{\\mathcal{O}_{Y_k, \\xi_k}} \\kappa(\\xi_k)", "\\otimes_{\\kappa(\\xi_k)} \\kappa(\\xi'_k)", "\\cong", "(\\pi'_{k, *}\\mathcal{G}'_k)_{\\xi'_k}", "\\otimes_{\\mathcal{O}_{Y'_k, \\xi'_k}} \\kappa(\\xi'_k)", "$$", "with obvious notation." ], "refs": [ "flat-lemma-base-change-one-step", "flat-definition-complete-devissage", "coherent-lemma-affine-base-change" ], "ref_ids": [ 5992, 6209, 3297 ] } ], "ref_ids": [ 6209 ] }, { "id": 5998, "type": "theorem", "label": "flat-lemma-base-change-complete-at-x", "categories": [ "flat" ], "title": "flat-lemma-base-change-complete-at-x", "contents": [ "Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in", "Definition \\ref{definition-complete-devissage-at-x}.", "Let $(S', s') \\to (S, s)$ be a morphism of pointed schemes", "which induces an isomorphism $\\kappa(s) = \\kappa(s')$. Let", "$(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k, z_k, y_k)_{k = 1, \\ldots, n}$", "be a complete d\\'evissage of $\\mathcal{F}/X/S$ at $x$.", "Let", "$(Z'_k, Y'_k, i'_k, \\pi'_k, \\mathcal{G}'_k, \\alpha'_k)_{k = 1, \\ldots, n}$", "be as constructed in", "Lemma \\ref{lemma-base-change-complete}", "and let $x' \\in X'$ (resp.\\ $z'_k \\in Z'$, $y'_k \\in Y'$) be the", "unique point mapping to both $x \\in X$ (resp.\\ $z_k \\in Z_k$, $y_k \\in Y_k$)", "and $s' \\in S'$.", "If $S'$ is affine, then", "$(Z'_k, Y'_k, i'_k, \\pi'_k, \\mathcal{G}'_k, \\alpha'_k,", "z'_k, y'_k)_{k = 1, \\ldots, n}$", "is a complete d\\'evissage of $\\mathcal{F}'/X'/S'$ at $x'$." ], "refs": [ "flat-definition-complete-devissage-at-x", "flat-lemma-base-change-complete" ], "proofs": [ { "contents": [ "Combine", "Lemma \\ref{lemma-base-change-complete}", "and", "Lemma \\ref{lemma-base-change-one-step-at-x}." ], "refs": [ "flat-lemma-base-change-complete", "flat-lemma-base-change-one-step-at-x" ], "ref_ids": [ 5997, 5993 ] } ], "ref_ids": [ 6210, 5997 ] }, { "id": 5999, "type": "theorem", "label": "flat-lemma-shrink-complete", "categories": [ "flat" ], "title": "flat-lemma-shrink-complete", "contents": [ "With assumption and notation as in", "Definition \\ref{definition-shrink-complete}", "we have:", "\\begin{enumerate}", "\\item", "\\label{item-shrink-base-complete}", "If $S' \\subset S$ is a standard open neighbourhood of $s$, then", "setting $X' = X_{S'}$, $Z'_k = Z_{S'}$ and $Y'_k = Y_{S'}$ we obtain a", "standard shrinking.", "\\item", "\\label{item-shrink-on-Y-complete}", "Let $W \\subset Y_n$ be a standard open neighbourhood of $y$.", "Then there exists a standard shrinking with $Y'_n = W \\times_S S'$.", "\\item", "\\label{item-shrink-on-X-complete}", "Let $U \\subset X$ be an open neighbourhood of $x$.", "Then there exists a standard shrinking with $X' \\subset U$.", "\\end{enumerate}" ], "refs": [ "flat-definition-shrink-complete" ], "proofs": [ { "contents": [ "Part (1) is immediate from", "Lemmas \\ref{lemma-base-change-complete-at-x} and", "\\ref{lemma-shrink}.", "\\medskip\\noindent", "Proof of (2). For convenience denote $X = Y_0$. We apply", "Lemma \\ref{lemma-shrink} (\\ref{item-shrink-on-Y})", "to find a standard shrinking", "$S', Y'_{n - 1}, Z'_n, Y'_n$", "of the one step d\\'evissage of $\\Coker(\\alpha_{n - 1})/Y_{n - 1}/S$", "at $y_{n - 1}$ with $Y'_n = W \\times_S S'$. We may repeat this procedure", "and find a standard shrinking", "$S'', Y''_{n - 2}, Z''_{n - 1}, Y''_{n - 1}$", "of the one step d\\'evissage of $\\Coker(\\alpha_{n - 2})/Y_{n - 2}/S$", "at $y_{n - 2}$ with $Y''_{n - 1} = Y'_{n - 1} \\times_S S''$.", "We may continue in this manner until we obtain", "$S^{(n)}, Y^{(n)}_0, Z^{(n)}_1, Y^{(n)}_1$.", "At this point it is clear that we obtain our desired standard shrinking", "by taking $S^{(n)}$, $X^{(n)}$, $Z_k^{(n - k)} \\times_S S^{(n)}$, and", "$Y_k^{(n - k)} \\times_S S^{(n)}$ with the desired property.", "\\medskip\\noindent", "Proof of (3). We use induction on the length of the complete", "d\\'evissage. First we apply", "Lemma \\ref{lemma-shrink} (\\ref{item-shrink-on-X})", "to find a standard shrinking", "$S', X', Z'_1, Y'_1$", "of the one step d\\'evissage of $\\mathcal{F}/X/S$ at $x$", "with $X' \\subset U$. If $n = 1$, then we are done.", "If $n > 1$, then by induction we can find a standard shrinking", "$S''$, $Y''_1$, $Z''_k$, and $Y''_k$ of the complete d\\'evissage", "$(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k, z_k, y_k)_{k = 2, \\ldots, n}$", "of $\\Coker(\\alpha_1)/Y_1/S$ at $x$ such that", "$Y''_1 \\subset Y'_1$. Using", "Lemma \\ref{lemma-shrink} (\\ref{item-shrink-on-Y})", "we can find $S''' \\subset S'$, $X''' \\subset X'$, $Z'''_1$ and", "$Y'''_1 = Y''_1 \\times_S S'''$ which is a standard shrinking.", "The solution to our problem is to take", "$$", "S''', X''', Z'''_1, Y'''_1, Z''_2 \\times_S S''',", "Y''_2 \\times_S S''', \\ldots, Z''_n \\times_S S''', Y''_n \\times_S S'''", "$$", "This ends the proof of the lemma." ], "refs": [ "flat-lemma-base-change-complete-at-x", "flat-lemma-shrink", "flat-lemma-shrink", "flat-lemma-shrink", "flat-lemma-shrink" ], "ref_ids": [ 5998, 5994, 5994, 5994, 5994 ] } ], "ref_ids": [ 6211 ] }, { "id": 6000, "type": "theorem", "label": "flat-lemma-existence-complete", "categories": [ "flat" ], "title": "flat-lemma-existence-complete", "contents": [ "Let $X \\to S$ be a finite type morphism of schemes.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Let $s \\in S$ be a point.", "There exists an elementary \\'etale neighbourhood", "$(S', s') \\to (S, s)$ and \\'etale morphisms", "$h_i : Y_i \\to X_{S'}$, $i = 1, \\ldots, n$ such that for each", "$i$ there exists a complete d\\'evissage of $\\mathcal{F}_i/Y_i/S'$ over $s'$,", "where $\\mathcal{F}_i$ is the pullback of $\\mathcal{F}$ to $Y_i$", "and such that $X_s = (X_{S'})_{s'} \\subset \\bigcup h_i(Y_i)$." ], "refs": [], "proofs": [ { "contents": [ "For every point $x \\in X_s$ we can find a diagram", "$$", "\\xymatrix{", "(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\", "(S, s) & (S', s') \\ar[l]", "}", "$$", "of pointed schemes such that the horizontal arrows are elementary", "\\'etale neighbourhoods and such that $g^*\\mathcal{F}/X'/S'$ has a", "complete d\\'evissage at $x'$. As $X \\to S$ is of finite type the", "fibre $X_s$ is quasi-compact, and since each $g : X' \\to X$ as above", "is open we can cover $X_s$ by a finite union of $g(X'_{s'})$.", "Thus we can find a finite family of such diagrams", "$$", "\\vcenter{", "\\xymatrix{", "(X, x) \\ar[d] & (X'_i, x'_i) \\ar[l]^{g_i} \\ar[d] \\\\", "(S, s) & (S'_i, s'_i) \\ar[l]", "}", "}", "\\quad i = 1, \\ldots, n", "$$", "such that $X_s = \\bigcup g_i(X'_i)$. Set", "$S' = S'_1 \\times_S \\ldots \\times_S S'_n$", "and let $Y_i = X_i \\times_{S'_i} S'$ be the base change of $X'_i$ to $S'$. By", "Lemma \\ref{lemma-base-change-complete}", "we see that the pullback of $\\mathcal{F}$ to $Y_i$ has a complete d\\'evissage", "over $s$ and we win." ], "refs": [ "flat-lemma-base-change-complete" ], "ref_ids": [ 5997 ] } ], "ref_ids": [] }, { "id": 6001, "type": "theorem", "label": "flat-lemma-existence-algebra", "categories": [ "flat" ], "title": "flat-lemma-existence-algebra", "contents": [ "Let $R \\to S$ be a finite type ring map.", "Let $M$ be a finite $S$-module.", "Let $\\mathfrak q$ be a prime ideal of $S$.", "There exists an elementary \\'etale localization", "$R' \\to S', \\mathfrak q', \\mathfrak p'$ of", "the ring map $R \\to S$ at $\\mathfrak q$ such that", "there exists a complete d\\'evissage of", "$(M \\otimes_S S')/S'/R'$ at $\\mathfrak q'$." ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of", "Proposition \\ref{proposition-existence-complete-at-x}", "via", "Remark \\ref{remark-same-notion}" ], "refs": [ "flat-proposition-existence-complete-at-x", "flat-remark-same-notion" ], "ref_ids": [ 6198, 6225 ] } ], "ref_ids": [] }, { "id": 6002, "type": "theorem", "label": "flat-lemma-homothety-spectrum", "categories": [ "flat" ], "title": "flat-lemma-homothety-spectrum", "contents": [ "Let $R \\to S$ be a ring map.", "Let $N$ be a $S$-module.", "Assume", "\\begin{enumerate}", "\\item $R$ is a local ring with maximal ideal $\\mathfrak m$,", "\\item $\\overline{S} = S/\\mathfrak m S$ is Noetherian, and", "\\item $\\overline{N} = N/\\mathfrak m_R N$ is a finite $\\overline{S}$-module.", "\\end{enumerate}", "Let $\\Sigma \\subset S$ be the multiplicative subset of elements which are not", "a zerodivisor on $\\overline{N}$. Then $\\Sigma^{-1}S$ is a semi-local ring", "whose spectrum consists of primes $\\mathfrak q \\subset S$ contained in an", "element of $\\text{Ass}_S(\\overline{N})$. Moreover, any maximal", "ideal of $\\Sigma^{-1}S$ corresponds to an associated prime of", "$\\overline{N}$ over $\\overline{S}$." ], "refs": [], "proofs": [ { "contents": [ "Note that", "$\\text{Ass}_S(\\overline{N}) = \\text{Ass}_{\\overline{S}}(\\overline{N})$, see", "Algebra, Lemma \\ref{algebra-lemma-ass-quotient-ring}.", "This is a finite set by", "Algebra, Lemma \\ref{algebra-lemma-finite-ass}.", "Say $\\{\\mathfrak q_1, \\ldots, \\mathfrak q_r\\} = \\text{Ass}_S(\\overline{N})$.", "We have $\\Sigma = S \\setminus (\\bigcup \\mathfrak q_i)$ by", "Algebra, Lemma \\ref{algebra-lemma-ass-zero-divisors}.", "By the description of $\\Spec(\\Sigma^{-1}S)$ in", "Algebra, Lemma \\ref{algebra-lemma-spec-localization}", "and by", "Algebra, Lemma \\ref{algebra-lemma-silly}", "we see that the primes of $\\Sigma^{-1}S$ correspond to the primes of", "$S$ contained in one of the $\\mathfrak q_i$.", "Hence the maximal ideals of $\\Sigma^{-1}S$ correspond one-to-one with the", "maximal (w.r.t.\\ inclusion) elements of the set", "$\\{\\mathfrak q_1, \\ldots, \\mathfrak q_r\\}$. This proves the lemma." ], "refs": [ "algebra-lemma-ass-quotient-ring", "algebra-lemma-finite-ass", "algebra-lemma-ass-zero-divisors", "algebra-lemma-spec-localization", "algebra-lemma-silly" ], "ref_ids": [ 708, 701, 704, 391, 378 ] } ], "ref_ids": [] }, { "id": 6003, "type": "theorem", "label": "flat-lemma-homothety-universally-injective", "categories": [ "flat" ], "title": "flat-lemma-homothety-universally-injective", "contents": [ "Assumption and notation as in", "Lemma \\ref{lemma-homothety-spectrum}.", "Assume moreover that", "\\begin{enumerate}", "\\item $S$ is local and $R \\to S$ is a local homomorphism,", "\\item $S$ is essentially of finite presentation over $R$,", "\\item $N$ is finitely presented over $S$, and", "\\item $N$ is flat over $R$.", "\\end{enumerate}", "Then each $s \\in \\Sigma$ defines a", "universally injective $R$-module map $s : N \\to N$, and the", "map $N \\to \\Sigma^{-1}N$ is $R$-universally injective." ], "refs": [ "flat-lemma-homothety-spectrum" ], "proofs": [ { "contents": [ "By", "Algebra, Lemma \\ref{algebra-lemma-mod-injective-general}", "the sequence $0 \\to N \\to N \\to N/sN \\to 0$ is exact and", "$N/sN$ is flat over $R$. This implies that $s : N \\to N$", "is universally injective, see", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}.", "The map $N \\to \\Sigma^{-1}N$ is universally injective as the directed", "colimit of the maps $s : N \\to N$." ], "refs": [ "algebra-lemma-mod-injective-general", "algebra-lemma-flat-tor-zero" ], "ref_ids": [ 1110, 532 ] } ], "ref_ids": [ 6002 ] }, { "id": 6004, "type": "theorem", "label": "flat-lemma-base-change-universally-flat-local", "categories": [ "flat" ], "title": "flat-lemma-base-change-universally-flat-local", "contents": [ "Let $R \\to S$ be a ring map.", "Let $N$ be an $S$-module.", "Let $S \\to S'$ be a ring map.", "Assume", "\\begin{enumerate}", "\\item $R \\to S$ is a local homomorphism of local rings", "\\item $S$ is essentially of finite presentation over $R$,", "\\item $N$ is of finite presentation over $S$,", "\\item $N$ is flat over $R$,", "\\item $S \\to S'$ is flat, and", "\\item the image of $\\Spec(S') \\to \\Spec(S)$ contains", "all primes $\\mathfrak q$ of $S$ lying over $\\mathfrak m_R$", "such that $\\mathfrak q$ is an associated prime of $N/\\mathfrak m_R N$.", "\\end{enumerate}", "Then $N \\to N \\otimes_S S'$ is $R$-universally injective." ], "refs": [], "proofs": [ { "contents": [ "Set $N' = N \\otimes_R S'$. Consider the commutative diagram", "$$", "\\xymatrix{", "N \\ar[d] \\ar[r] & N' \\ar[d] \\\\", "\\Sigma^{-1}N \\ar[r] & \\Sigma^{-1}N'", "}", "$$", "where $\\Sigma \\subset S$ is the set of elements which are not a", "zerodivisor on $N/\\mathfrak m_R N$. If we can show that the map", "$N \\to \\Sigma^{-1}N'$ is universally injective, then $N \\to N'$", "is too (see", "Algebra, Lemma \\ref{algebra-lemma-universally-injective-permanence}).", "\\medskip\\noindent", "By", "Lemma \\ref{lemma-homothety-spectrum}", "the ring $\\Sigma^{-1}S$ is a semi-local ring whose maximal ideals", "correspond to associated primes of $N/\\mathfrak m_R N$.", "Hence the image of", "$\\Spec(\\Sigma^{-1}S') \\to \\Spec(\\Sigma^{-1}S)$", "contains all these maximal ideals by assumption. By", "Algebra, Lemma \\ref{algebra-lemma-ff-rings}", "the ring map $\\Sigma^{-1}S \\to \\Sigma^{-1}S'$ is faithfully flat.", "Hence $\\Sigma^{-1}N \\to \\Sigma^{-1}N'$, which is the map", "$$", "N \\otimes_S \\Sigma^{-1}S \\longrightarrow N \\otimes_S \\Sigma^{-1}S'", "$$", "is universally injective, see", "Algebra, Lemmas \\ref{algebra-lemma-faithfully-flat-universally-injective} and", "\\ref{algebra-lemma-universally-injective-tensor}.", "Finally, we apply", "Lemma \\ref{lemma-homothety-universally-injective}", "to see that $N \\to \\Sigma^{-1}N$ is universally injective.", "As the composition of universally injective module maps is universally", "injective (see", "Algebra, Lemma \\ref{algebra-lemma-composition-universally-injective})", "we conclude that $N \\to \\Sigma^{-1}N'$ is universally injective and we win." ], "refs": [ "algebra-lemma-universally-injective-permanence", "flat-lemma-homothety-spectrum", "algebra-lemma-ff-rings", "algebra-lemma-faithfully-flat-universally-injective", "algebra-lemma-universally-injective-tensor", "flat-lemma-homothety-universally-injective", "algebra-lemma-composition-universally-injective" ], "ref_ids": [ 813, 6002, 536, 814, 811, 6003, 812 ] } ], "ref_ids": [] }, { "id": 6005, "type": "theorem", "label": "flat-lemma-base-change-universally-flat", "categories": [ "flat" ], "title": "flat-lemma-base-change-universally-flat", "contents": [ "Let $R \\to S$ be a ring map.", "Let $N$ be an $S$-module.", "Let $S \\to S'$ be a ring map.", "Assume", "\\begin{enumerate}", "\\item $R \\to S$ is of finite presentation and $N$ is of finite presentation", "over $S$,", "\\item $N$ is flat over $R$,", "\\item $S \\to S'$ is flat, and", "\\item the image of $\\Spec(S') \\to \\Spec(S)$ contains", "all primes $\\mathfrak q$ such that $\\mathfrak q$ is an associated prime", "of $N \\otimes_R \\kappa(\\mathfrak p)$ where $\\mathfrak p$ is the inverse", "image of $\\mathfrak q$ in $R$.", "\\end{enumerate}", "Then $N \\to N \\otimes_S S'$ is $R$-universally injective." ], "refs": [], "proofs": [ { "contents": [ "By", "Algebra, Lemma \\ref{algebra-lemma-universally-injective-check-stalks}", "it suffices to show that $N_{\\mathfrak q} \\to (N \\otimes_R S')_{\\mathfrak q}$", "is a $R_{\\mathfrak p}$-universally injective for any prime $\\mathfrak q$", "of $S$ lying over $\\mathfrak p$ in $R$. Thus we may apply", "Lemma \\ref{lemma-base-change-universally-flat-local}", "to the ring maps", "$R_{\\mathfrak p} \\to S_{\\mathfrak q} \\to S'_{\\mathfrak q}$", "and the module $N_{\\mathfrak q}$." ], "refs": [ "algebra-lemma-universally-injective-check-stalks", "flat-lemma-base-change-universally-flat-local" ], "ref_ids": [ 815, 6004 ] } ], "ref_ids": [] }, { "id": 6006, "type": "theorem", "label": "flat-lemma-universally-injective-local", "categories": [ "flat" ], "title": "flat-lemma-universally-injective-local", "contents": [ "Let $(R, \\mathfrak m)$ be a local ring. Let $u : M \\to N$ be an $R$-module map.", "If $M$ is a projective $R$-module, $N$ is a flat $R$-module, and", "$\\overline{u} : M/\\mathfrak mM \\to N/\\mathfrak mN$ is injective", "then $u$ is universally injective." ], "refs": [], "proofs": [ { "contents": [ "By", "Algebra, Theorem \\ref{algebra-theorem-projective-free-over-local-ring}", "the module $M$ is free. If we show the result holds for every finitely", "generated direct summand of $M$, then the lemma follows. Hence we may", "assume that $M$ is finite free. Write $N = \\colim_i N_i$ as", "a directed colimit of finite free modules, see", "Algebra, Theorem \\ref{algebra-theorem-lazard}.", "Note that $u : M \\to N$ factors through $N_i$ for some $i$ (as $M$ is finite", "free). Denote $u_i : M \\to N_i$ the corresponding $R$-module map.", "As $\\overline{u}$ is injective we see that", "$\\overline{u_i} : M/\\mathfrak mM \\to N_i/\\mathfrak mN_i$ is", "injective and remains injective on composing with the maps", "$N_i/\\mathfrak mN_i \\to N_{i'}/\\mathfrak mN_{i'}$ for all $i' \\geq i$.", "As $M$ and $N_{i'}$ are finite free over the local ring $R$ this implies", "that $M \\to N_{i'}$ is a split injection for all $i' \\geq i$. Hence", "for any $R$-module $Q$ we see that $M \\otimes_R Q \\to N_{i'} \\otimes_R Q$", "is injective for all $i' \\geq i$. As $- \\otimes_R Q$ commutes with", "colimits we conclude that $M \\otimes_R Q \\to N_{i'} \\otimes_R Q$", "is injective as desired." ], "refs": [ "algebra-theorem-projective-free-over-local-ring", "algebra-theorem-lazard" ], "ref_ids": [ 322, 318 ] } ], "ref_ids": [] }, { "id": 6007, "type": "theorem", "label": "flat-lemma-invert-universally-injective", "categories": [ "flat" ], "title": "flat-lemma-invert-universally-injective", "contents": [ "Assumption and notation as in", "Lemma \\ref{lemma-homothety-spectrum}.", "Assume moreover that $N$ is projective as an $R$-module.", "Then each $s \\in \\Sigma$ defines a", "universally injective $R$-module map $s : N \\to N$, and the", "map $N \\to \\Sigma^{-1}N$ is $R$-universally injective." ], "refs": [ "flat-lemma-homothety-spectrum" ], "proofs": [ { "contents": [ "Pick $s \\in \\Sigma$. By", "Lemma \\ref{lemma-universally-injective-local}", "the map $s : N \\to N$ is universally injective.", "The map $N \\to \\Sigma^{-1}N$ is universally injective as the directed", "colimit of the maps $s : N \\to N$." ], "refs": [ "flat-lemma-universally-injective-local" ], "ref_ids": [ 6006 ] } ], "ref_ids": [ 6002 ] }, { "id": 6008, "type": "theorem", "label": "flat-lemma-completed-direct-sum-ML", "categories": [ "flat" ], "title": "flat-lemma-completed-direct-sum-ML", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $A$ be a set.", "Assume $R$ is Noetherian and complete with respect to $I$. The completion", "$(\\bigoplus\\nolimits_{\\alpha \\in A} R)^\\wedge$", "is flat and Mittag-Leffler." ], "refs": [], "proofs": [ { "contents": [ "By", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-ui-completion-direct-sum-into-product}", "the map $(\\bigoplus\\nolimits_{\\alpha \\in A} R)^\\wedge", "\\to \\prod_{\\alpha \\in A} R$ is universally injective.", "Thus, by", "Algebra, Lemmas \\ref{algebra-lemma-ui-flat-domain} and", "\\ref{algebra-lemma-pure-submodule-ML}", "it suffices to show that $\\prod_{\\alpha \\in A} R$ is flat and Mittag-Leffler.", "By", "Algebra, Proposition \\ref{algebra-proposition-characterize-coherent}", "(and", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-coherent})", "we see that $\\prod_{\\alpha \\in A} R$ is flat.", "Thus we conclude because a product of copies of $R$ is Mittag-Leffler, see", "Algebra, Lemma \\ref{algebra-lemma-product-over-Noetherian-ring}." ], "refs": [ "more-algebra-lemma-ui-completion-direct-sum-into-product", "algebra-lemma-ui-flat-domain", "algebra-lemma-pure-submodule-ML", "algebra-proposition-characterize-coherent", "algebra-lemma-Noetherian-coherent", "algebra-lemma-product-over-Noetherian-ring" ], "ref_ids": [ 9952, 810, 837, 1418, 844, 846 ] } ], "ref_ids": [] }, { "id": 6009, "type": "theorem", "label": "flat-lemma-lift-ML", "categories": [ "flat" ], "title": "flat-lemma-lift-ML", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal.", "Let $M$ be an $R$-module.", "Assume", "\\begin{enumerate}", "\\item $R$ is Noetherian and $I$-adically complete,", "\\item $M$ is flat over $R$, and", "\\item $M/IM$ is a projective $R/I$-module.", "\\end{enumerate}", "Then the $I$-adic completion $M^\\wedge$ is a flat Mittag-Leffler", "$R$-module." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjection $F \\to M$ where $F$ is a free $R$-module. By", "Algebra, Lemma \\ref{algebra-lemma-split-completed-sequence}", "the module $M^\\wedge$ is a direct summand of the module $F^\\wedge$.", "Hence it suffices to prove the lemma for $F$.", "In this case the lemma follows from", "Lemma \\ref{lemma-completed-direct-sum-ML}." ], "refs": [ "algebra-lemma-split-completed-sequence", "flat-lemma-completed-direct-sum-ML" ], "ref_ids": [ 877, 6008 ] } ], "ref_ids": [] }, { "id": 6010, "type": "theorem", "label": "flat-lemma-universally-injective-to-completion", "categories": [ "flat" ], "title": "flat-lemma-universally-injective-to-completion", "contents": [ "Let $R$ be a ring.", "Let $I \\subset R$ be an ideal.", "Let $R \\to S$ be a ring map, and $N$ an $S$-module.", "Assume", "\\begin{enumerate}", "\\item $R$ is a Noetherian ring,", "\\item $S$ is a Noetherian ring,", "\\item $N$ is a finite $S$-module, and", "\\item for any finite $R$-module $Q$, any", "$\\mathfrak q \\in \\text{Ass}_S(Q \\otimes_R N)$", "satisfies $IS + \\mathfrak q \\not = S$.", "\\end{enumerate}", "Then the map $N \\to N^\\wedge$ of $N$ into the $I$-adic completion of $N$", "is universally injective as a map of $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "We have to show that for any finite $R$-module $Q$ the map", "$Q \\otimes_R N \\to Q \\otimes_R N^\\wedge$ is injective, see", "Algebra, Theorem \\ref{algebra-theorem-universally-exact-criteria}.", "As there is a canonical map $Q \\otimes_R N^\\wedge \\to (Q \\otimes_R N)^\\wedge$", "it suffices to prove that the canonical map", "$Q \\otimes_R N \\to (Q \\otimes_R N)^\\wedge$ is injective.", "Hence we may replace $N$ by $Q \\otimes_R N$ and it suffices to prove the", "injectivity for the map $N \\to N^\\wedge$.", "\\medskip\\noindent", "Let $K = \\Ker(N \\to N^\\wedge)$. It suffices to show that", "$K_{\\mathfrak q} = 0$ for $\\mathfrak q \\in \\text{Ass}(N)$ as $N$ is a", "submodule of $\\prod_{\\mathfrak q \\in \\text{Ass}(N)} N_{\\mathfrak q}$, see", "Algebra, Lemma \\ref{algebra-lemma-zero-at-ass-zero}.", "Pick $\\mathfrak q \\in \\text{Ass}(N)$. By the last assumption we see that", "there exists a prime $\\mathfrak q' \\supset IS + \\mathfrak q$.", "Since $K_{\\mathfrak q}$ is a localization of $K_{\\mathfrak q'}$", "it suffices to prove the vanishing of $K_{\\mathfrak q'}$.", "Note that $K = \\bigcap I^nN$, hence", "$K_{\\mathfrak q'} \\subset \\bigcap I^nN_{\\mathfrak q'}$.", "Hence $K_{\\mathfrak q'} = 0$ by", "Algebra, Lemma \\ref{algebra-lemma-intersect-powers-ideal-module-zero}." ], "refs": [ "algebra-theorem-universally-exact-criteria", "algebra-lemma-zero-at-ass-zero", "algebra-lemma-intersect-powers-ideal-module-zero" ], "ref_ids": [ 319, 713, 627 ] } ], "ref_ids": [] }, { "id": 6011, "type": "theorem", "label": "flat-lemma-universally-injective-to-completion-flat", "categories": [ "flat" ], "title": "flat-lemma-universally-injective-to-completion-flat", "contents": [ "Let $R$ be a ring.", "Let $I \\subset R$ be an ideal.", "Let $R \\to S$ be a ring map, and $N$ an $S$-module.", "Assume", "\\begin{enumerate}", "\\item $R$ is a Noetherian ring,", "\\item $S$ is a Noetherian ring,", "\\item $N$ is a finite $S$-module,", "\\item $N$ is flat over $R$, and", "\\item for any prime $\\mathfrak q \\subset S$ which is an associated prime of", "$N \\otimes_R \\kappa(\\mathfrak p)$ where $\\mathfrak p = R \\cap \\mathfrak q$", "we have $IS + \\mathfrak q \\not = S$.", "\\end{enumerate}", "Then the map $N \\to N^\\wedge$ of $N$ into the $I$-adic completion of $N$", "is universally injective as a map of $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-universally-injective-to-completion}", "because", "Algebra, Lemma", "\\ref{algebra-lemma-bourbaki-fibres} and", "Remark \\ref{algebra-remark-bourbaki}", "guarantee that the set of associated primes of tensor products", "$N \\otimes_R Q$ are contained in the set of associated primes of", "the modules $N \\otimes_R \\kappa(\\mathfrak p)$." ], "refs": [ "flat-lemma-universally-injective-to-completion", "algebra-lemma-bourbaki-fibres", "algebra-remark-bourbaki" ], "ref_ids": [ 6010, 719, 1563 ] } ], "ref_ids": [] }, { "id": 6012, "type": "theorem", "label": "flat-lemma-flat-pure-over-complete-projective", "categories": [ "flat" ], "title": "flat-lemma-flat-pure-over-complete-projective", "contents": [ "Let $R$ be a ring.", "Let $I \\subset R$ be an ideal.", "Let $R \\to S$ be a ring map, and $N$ an $S$-module.", "Assume", "\\begin{enumerate}", "\\item $R$ is Noetherian and $I$-adically complete,", "\\item $R \\to S$ is of finite type,", "\\item $N$ is a finite $S$-module,", "\\item $N$ is flat over $R$,", "\\item $N/IN$ is projective as a $R/I$-module, and", "\\item for any prime $\\mathfrak q \\subset S$ which is an associated prime of", "$N \\otimes_R \\kappa(\\mathfrak p)$ where $\\mathfrak p = R \\cap \\mathfrak q$", "we have $IS + \\mathfrak q \\not = S$.", "\\end{enumerate}", "Then $N$ is projective as an $R$-module." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-universally-injective-to-completion-flat}", "the map $N \\to N^\\wedge$ is universally injective.", "By", "Lemma \\ref{lemma-lift-ML}", "the module $N^\\wedge$ is Mittag-Leffler.", "By", "Algebra, Lemma \\ref{algebra-lemma-pure-submodule-ML}", "we conclude that $N$ is Mittag-Leffler.", "Hence $N$ is countably generated, flat and Mittag-Leffler as an $R$-module,", "whence projective by", "Algebra, Lemma \\ref{algebra-lemma-countgen-projective}." ], "refs": [ "flat-lemma-universally-injective-to-completion-flat", "flat-lemma-lift-ML", "algebra-lemma-pure-submodule-ML", "algebra-lemma-countgen-projective" ], "ref_ids": [ 6011, 6009, 837, 850 ] } ], "ref_ids": [] }, { "id": 6013, "type": "theorem", "label": "flat-lemma-fibres-irreducible-flat-projective", "categories": [ "flat" ], "title": "flat-lemma-fibres-irreducible-flat-projective", "contents": [ "Let $R$ be a ring.", "Let $R \\to S$ be a ring map.", "Assume", "\\begin{enumerate}", "\\item $R$ is Noetherian,", "\\item $R \\to S$ is of finite type and flat, and", "\\item every fibre ring $S \\otimes_R \\kappa(\\mathfrak p)$ is", "geometrically integral over $\\kappa(\\mathfrak p)$.", "\\end{enumerate}", "Then $S$ is projective as an $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Consider the set", "$$", "\\{I \\subset R \\mid S/IS\\text{ not projective as }R/I\\text{-module}\\}", "$$", "We have to show this set is empty. To get a contradiction assume it is", "nonempty. Then it contains a maximal element $I$.", "Let $J = \\sqrt{I}$ be its radical. If $I \\not = J$, then", "$S/JS$ is projective as a $R/J$-module, and $S/IS$ is flat over $R/I$", "and $J/I$ is a nilpotent ideal in $R/I$. Applying", "Algebra, Lemma \\ref{algebra-lemma-lift-projective}", "we see that $S/IS$ is a projective $R/I$-module, which is a contradiction.", "Hence we may assume that $I$ is a radical ideal. In other words we", "are reduced to proving the lemma in case $R$ is a reduced ring and", "$S/IS$ is a projective $R/I$-module for every nonzero ideal $I$", "of $R$.", "\\medskip\\noindent", "Assume $R$ is a reduced ring and $S/IS$ is a projective $R/I$-module", "for every nonzero ideal $I$ of $R$. By generic flatness,", "Algebra, Lemma \\ref{algebra-lemma-generic-flatness-Noetherian}", "(applied to a localization $R_g$ which is a domain) or the more general", "Algebra, Lemma \\ref{algebra-lemma-generic-flatness-reduced}", "there exists a nonzero $f \\in R$ such that $S_f$ is free as an", "$R_f$-module. Denote $R^\\wedge = \\lim R/(f^n)$ the $(f)$-adic completion", "of $R$. Note that the ring map", "$$", "R \\longrightarrow R_f \\times R^\\wedge", "$$", "is a faithfully flat ring map, see", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}.", "Hence by faithfully flat descent of projectivity, see", "Algebra, Theorem \\ref{algebra-theorem-ffdescent-projectivity}", "it suffices to prove that $S \\otimes_R R^\\wedge$ is a projective", "$R^\\wedge$-module. To see this we will use the criterion of", "Lemma \\ref{lemma-flat-pure-over-complete-projective}.", "First of all, note that $S/fS = (S \\otimes_R R^\\wedge)/f(S \\otimes_R R^\\wedge)$", "is a projective $R/(f)$-module and that $S \\otimes_R R^\\wedge$ is flat", "and of finite type over $R^\\wedge$ as a base change of such.", "Next, suppose that $\\mathfrak p^\\wedge$ is a prime ideal", "of $R^\\wedge$. Let $\\mathfrak p \\subset R$ be the corresponding prime", "of $R$. As $R \\to S$ has geometrically integral fibre rings, the", "same is true for the fibre rings of any base change. Hence", "$\\mathfrak q^\\wedge = \\mathfrak p^\\wedge(S \\otimes_R R^\\wedge)$,", "is a prime ideals lying over $\\mathfrak p^\\wedge$", "and it is the unique associated prime of", "$S \\otimes_R \\kappa(\\mathfrak p^\\wedge)$. Thus we win if", "$f(S \\otimes_R R^\\wedge) + \\mathfrak q^\\wedge \\not = S \\otimes_R R^\\wedge$.", "This is true because $\\mathfrak p^\\wedge + fR^\\wedge \\not = R^\\wedge$ as", "$f$ lies in the Jacobson radical of the $f$-adically complete ring $R^\\wedge$", "and because $R^\\wedge \\to S \\otimes_R R^\\wedge$ is surjective on spectra", "as its fibres are nonempty (irreducible spaces are nonempty)." ], "refs": [ "algebra-lemma-lift-projective", "algebra-lemma-generic-flatness-Noetherian", "algebra-lemma-generic-flatness-reduced", "algebra-lemma-completion-flat", "algebra-theorem-ffdescent-projectivity", "flat-lemma-flat-pure-over-complete-projective" ], "ref_ids": [ 794, 1013, 1019, 870, 324, 6012 ] } ], "ref_ids": [] }, { "id": 6014, "type": "theorem", "label": "flat-lemma-fibres-irreducible-flat-projective-nonnoetherian", "categories": [ "flat" ], "title": "flat-lemma-fibres-irreducible-flat-projective-nonnoetherian", "contents": [ "Let $R$ be a ring. Let $R \\to S$ be a ring map.", "Assume", "\\begin{enumerate}", "\\item $R \\to S$ is of finite presentation and flat, and", "\\item every fibre ring $S \\otimes_R \\kappa(\\mathfrak p)$ is", "geometrically integral over $\\kappa(\\mathfrak p)$.", "\\end{enumerate}", "Then $S$ is projective as an $R$-module." ], "refs": [], "proofs": [ { "contents": [ "We can find a cocartesian diagram of rings", "$$", "\\xymatrix{", "S_0 \\ar[r] & S \\\\", "R_0 \\ar[u] \\ar[r] & R \\ar[u]", "}", "$$", "such that $R_0$ is of finite type over $\\mathbf{Z}$, the map", "$R_0 \\to S_0$ is of finite type and flat with geometrically integral", "fibres, see", "More on Morphisms,", "Lemmas \\ref{more-morphisms-lemma-Noetherian-approximation-flat},", "\\ref{more-morphisms-lemma-Noetherian-approximation-geometrically-reduced},", "\\ref{more-morphisms-lemma-Noetherian-approximation-geometrically-irreducible},", "and \\ref{more-morphisms-lemma-Noetherian-approximation-combine}.", "By", "Lemma \\ref{lemma-fibres-irreducible-flat-projective}", "we see that $S_0$ is a projective $R_0$-module. Hence $S = S_0 \\otimes_{R_0} R$", "is a projective $R$-module, see", "Algebra, Lemma \\ref{algebra-lemma-ascend-properties-modules}." ], "refs": [ "more-morphisms-lemma-Noetherian-approximation-flat", "more-morphisms-lemma-Noetherian-approximation-geometrically-reduced", "more-morphisms-lemma-Noetherian-approximation-geometrically-irreducible", "more-morphisms-lemma-Noetherian-approximation-combine", "flat-lemma-fibres-irreducible-flat-projective", "algebra-lemma-ascend-properties-modules" ], "ref_ids": [ 13858, 13860, 13861, 13865, 6013, 853 ] } ], "ref_ids": [] }, { "id": 6015, "type": "theorem", "label": "flat-lemma-induction-step", "categories": [ "flat" ], "title": "flat-lemma-induction-step", "contents": [ "Let $(R, \\mathfrak m)$ be a local ring. Let $R \\to S$ be a finitely presented", "flat ring map with geometrically integral fibres. Write", "$\\mathfrak p = \\mathfrak mS$. Let $\\mathfrak q \\subset S$ be a prime ideal", "lying over $\\mathfrak m$. Let $N$ be a finite $S$-module.", "There exist $r \\geq 0$ and an $S$-module map", "$$", "\\alpha : S^{\\oplus r} \\longrightarrow N", "$$", "such that", "$\\alpha : \\kappa(\\mathfrak p)^{\\oplus r} \\to N \\otimes_S \\kappa(\\mathfrak p)$", "is an isomorphism. For any such $\\alpha$ the following are equivalent:", "\\begin{enumerate}", "\\item $N_{\\mathfrak q}$ is $R$-flat,", "\\item $\\alpha$ is $R$-universally injective and", "$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat,", "\\item $\\alpha$ is injective and", "$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat,", "\\item $\\alpha_{\\mathfrak p}$ is an isomorphism and", "$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat, and", "\\item $\\alpha_{\\mathfrak q}$ is injective and", "$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To obtain $\\alpha$ set", "$r = \\dim_{\\kappa(\\mathfrak p)} N \\otimes_S \\kappa(\\mathfrak p)$ and pick", "$x_1, \\ldots, x_r \\in N$ which form a basis of", "$N \\otimes_S \\kappa(\\mathfrak p)$. Define", "$\\alpha(s_1, \\ldots, s_r) = \\sum s_i x_i$. This proves the existence.", "\\medskip\\noindent", "Fix an $\\alpha$. The most interesting implication is", "(1) $\\Rightarrow$ (2) which we prove first. Assume (1).", "Because $S/\\mathfrak mS$ is a domain with fraction field $\\kappa(\\mathfrak p)$", "we see that", "$(S/\\mathfrak mS)^{\\oplus r} \\to", "N_{\\mathfrak p}/\\mathfrak mN_{\\mathfrak p} = N \\otimes_S \\kappa(\\mathfrak p)$", "is injective. Hence by", "Lemmas \\ref{lemma-universally-injective-local} and", "\\ref{lemma-fibres-irreducible-flat-projective-nonnoetherian}.", "the map $S^{\\oplus r} \\to N_{\\mathfrak p}$ is $R$-universally injective.", "It follows that $S^{\\oplus r} \\to N$ is $R$-universally injective, see", "Algebra, Lemma \\ref{algebra-lemma-universally-injective-permanence}.", "Then also the localization $\\alpha_{\\mathfrak q}$ is $R$-universally", "injective, see", "Algebra, Lemma \\ref{algebra-lemma-universally-injective-localize}.", "We conclude that $\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat by", "Algebra, Lemma \\ref{algebra-lemma-ui-flat-domain}.", "\\medskip\\noindent", "The implication (2) $\\Rightarrow$ (3) is immediate. If (3) holds, then", "$\\alpha_{\\mathfrak p}$ is injective as a localization of an injective", "module map. By Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK})", "$\\alpha_{\\mathfrak p}$ is surjective too. Hence (3) $\\Rightarrow$ (4).", "If (4) holds, then $\\alpha_{\\mathfrak p}$ is an isomorphism, so", "$\\alpha$ is injective as $S_{\\mathfrak q} \\to S_{\\mathfrak p}$ is injective.", "Namely, elements of $S \\setminus \\mathfrak p$ are nonzerodivisors on $S$", "by a combination of", "Lemmas \\ref{lemma-invert-universally-injective} and", "\\ref{lemma-fibres-irreducible-flat-projective-nonnoetherian}.", "Hence (4) $\\Rightarrow$ (5). Finally, if (5) holds, then", "$N_{\\mathfrak q}$ is $R$-flat as an extension of flat modules, see", "Algebra, Lemma \\ref{algebra-lemma-flat-ses}.", "Hence (5) $\\Rightarrow$ (1) and the proof is finished." ], "refs": [ "flat-lemma-universally-injective-local", "flat-lemma-fibres-irreducible-flat-projective-nonnoetherian", "algebra-lemma-universally-injective-permanence", "algebra-lemma-universally-injective-localize", "algebra-lemma-ui-flat-domain", "algebra-lemma-NAK", "flat-lemma-invert-universally-injective", "flat-lemma-fibres-irreducible-flat-projective-nonnoetherian", "algebra-lemma-flat-ses" ], "ref_ids": [ 6006, 6014, 813, 816, 810, 401, 6007, 6014, 533 ] } ], "ref_ids": [] }, { "id": 6016, "type": "theorem", "label": "flat-lemma-complete-devissage-flat-finite-type-module", "categories": [ "flat" ], "title": "flat-lemma-complete-devissage-flat-finite-type-module", "contents": [ "Let $(R, \\mathfrak m)$ be a local ring.", "Let $R \\to S$ be a ring map of finite presentation.", "Let $N$ be a finite $S$-module.", "Let $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak m$.", "Assume that $N_{\\mathfrak q}$ is flat over $R$, and", "assume there exists a complete d\\'evissage of $N/S/R$ at $\\mathfrak q$.", "Then $N$ is a finitely presented $S$-module, free as an $R$-module,", "and there exists an isomorphism", "$$", "N \\cong B_1^{\\oplus r_1} \\oplus \\ldots \\oplus B_n^{\\oplus r_n}", "$$", "as $R$-modules where each $B_i$ is a smooth $R$-algebra with geometrically", "irreducible fibres." ], "refs": [], "proofs": [ { "contents": [ "Let $(A_i, B_i, M_i, \\alpha_i, \\mathfrak q_i)_{i = 1, \\ldots, n}$", "be the given complete d\\'evissage. We prove the lemma by induction on $n$.", "Note that $N$ is finitely presented as an $S$-module if and only if", "$M_1$ is finitely presented as an $B_1$-module, see", "Remark \\ref{remark-finite-presentation}.", "Note that $N_{\\mathfrak q} \\cong (M_1)_{\\mathfrak q_1}$ as $R$-modules", "because (a) $N_{\\mathfrak q} \\cong (M_1)_{\\mathfrak q'_1}$ where", "$\\mathfrak q'_1$ is the unique prime in $A_1$ lying over $\\mathfrak q_1$", "and (b) $(A_1)_{\\mathfrak q'_1} = (A_1)_{\\mathfrak q_1}$ by", "Algebra, Lemma \\ref{algebra-lemma-unique-prime-over-localize-below},", "so (c) $(M_1)_{\\mathfrak q'_1} \\cong (M_1)_{\\mathfrak q_1}$.", "Hence $(M_1)_{\\mathfrak q_1}$ is a flat $R$-module. Thus we may replace", "$(S, N)$ by $(B_1, M_1)$ in order to prove the lemma. By", "Lemma \\ref{lemma-induction-step}", "the map $\\alpha_1 : B_1^{\\oplus r_1} \\to M_1$ is $R$-universally injective", "and $\\Coker(\\alpha_1)_{\\mathfrak q}$ is $R$-flat.", "Note that $(A_i, B_i, M_i, \\alpha_i, \\mathfrak q_i)_{i = 2, \\ldots, n}$", "is a complete d\\'evissage of $\\Coker(\\alpha_1)/B_1/R$ at", "$\\mathfrak q_1$. Hence the induction hypothesis", "implies that $\\Coker(\\alpha_1)$ is finitely presented as a", "$B_1$-module, free as an $R$-module, and has a decomposition as in the lemma.", "This implies that $M_1$ is finitely presented as a $B_1$-module, see", "Algebra, Lemma \\ref{algebra-lemma-extension}.", "It further implies that", "$M_1 \\cong B_1^{\\oplus r_1} \\oplus \\Coker(\\alpha_1)$", "as $R$-modules, hence a decomposition as in the lemma.", "Finally, $B_1$ is projective as an $R$-module by", "Lemma \\ref{lemma-fibres-irreducible-flat-projective-nonnoetherian}", "hence free as an $R$-module by", "Algebra, Theorem \\ref{algebra-theorem-projective-free-over-local-ring}.", "This finishes the proof." ], "refs": [ "flat-remark-finite-presentation", "algebra-lemma-unique-prime-over-localize-below", "flat-lemma-induction-step", "algebra-lemma-extension", "flat-lemma-fibres-irreducible-flat-projective-nonnoetherian", "algebra-theorem-projective-free-over-local-ring" ], "ref_ids": [ 6224, 556, 6015, 330, 6014, 322 ] } ], "ref_ids": [] }, { "id": 6017, "type": "theorem", "label": "flat-lemma-open-in-fibre-where-flat", "categories": [ "flat" ], "title": "flat-lemma-open-in-fibre-where-flat", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.", "Let $s \\in S$. Then the set", "$$", "\\{x \\in X_s \\mid \\mathcal{F} \\text{ flat over }S\\text{ at }x\\}", "$$", "is open in the fibre $X_s$." ], "refs": [], "proofs": [ { "contents": [ "Suppose $x \\in U$. Choose an elementary \\'etale neighbourhood", "$(S', s') \\to (S, s)$ and open", "$V \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ as in", "Proposition \\ref{proposition-finite-type-flat-at-point}.", "Note that $X_{s'} = X_s$ as $\\kappa(s) = \\kappa(s')$.", "If $x' \\in V \\cap X_{s'}$, then the pullback of $\\mathcal{F}$ to", "$X \\times_S S'$ is flat over $S'$ at $x'$. Hence $\\mathcal{F}$ is", "flat at $x'$ over $S$, see", "Morphisms, Lemma \\ref{morphisms-lemma-flat-permanence}.", "In other words $X_s \\cap V \\subset U$ is an open neighbourhood", "of $x$ in $U$." ], "refs": [ "flat-proposition-finite-type-flat-at-point", "morphisms-lemma-flat-permanence" ], "ref_ids": [ 6199, 5270 ] } ], "ref_ids": [] }, { "id": 6018, "type": "theorem", "label": "flat-lemma-finite-type-flat-at-point", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-at-point", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $x \\in X$ with image $s \\in S$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite type,", "\\item $\\mathcal{F}$ is of finite type, and", "\\item $\\mathcal{F}$ is flat at $x$ over $S$.", "\\end{enumerate}", "Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$", "and an open subscheme", "$$", "V \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})", "$$", "which contains the unique point of", "$X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ mapping to $x$", "such that the pullback of $\\mathcal{F}$ to $V$ is flat over", "$\\mathcal{O}_{S', s'}$." ], "refs": [], "proofs": [ { "contents": [ "(The only difference between this and", "Proposition \\ref{proposition-finite-type-flat-at-point}", "is that we do not assume $f$ is of finite presentation.)", "The question is local on $X$ and $S$, hence we may assume $X$ and $S$", "are affine. Write $X = \\Spec(B)$, $S = \\Spec(A)$ and write", "$B = A[x_1, \\ldots, x_n]/I$. In other words we obtain a closed immersion", "$i : X \\to \\mathbf{A}^n_S$. Denote $t = i(x) \\in \\mathbf{A}^n_S$.", "We may apply", "Proposition \\ref{proposition-finite-type-flat-at-point}", "to $\\mathbf{A}^n_S \\to S$, the sheaf $i_*\\mathcal{F}$", "and the point $t$. We obtain an elementary", "\\'etale neighbourhood $(S', s') \\to (S, s)$ and an open subscheme", "$$", "W \\subset \\mathbf{A}^n_{\\mathcal{O}_{S', s'}}", "$$", "such that the pullback of $i_*\\mathcal{F}$ to $W$ is flat over", "$\\mathcal{O}_{S', s'}$. This means that", "$V := W \\cap \\big(X \\times_S \\Spec(\\mathcal{O}_{S', s'})\\big)$", "is the desired open subscheme." ], "refs": [ "flat-proposition-finite-type-flat-at-point", "flat-proposition-finite-type-flat-at-point" ], "ref_ids": [ 6199, 6199 ] } ], "ref_ids": [] }, { "id": 6019, "type": "theorem", "label": "flat-lemma-finite-type-flat-along-fibre", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-along-fibre", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $s \\in S$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is of finite presentation,", "\\item $\\mathcal{F}$ is of finite type, and", "\\item $\\mathcal{F}$ is flat over $S$ at every point of the fibre $X_s$.", "\\end{enumerate}", "Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$", "and an open subscheme", "$$", "V \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})", "$$", "which contains the fibre $X_s = X \\times_S s'$ such that the pullback", "of $\\mathcal{F}$ to $V$ is an $\\mathcal{O}_V$-module", "of finite presentation and flat over $\\mathcal{O}_{S', s'}$." ], "refs": [], "proofs": [ { "contents": [ "For every point $x \\in X_s$ we can use", "Proposition \\ref{proposition-finite-type-flat-at-point}", "to find an elementary \\'etale neighbourhood $(S_x, s_x) \\to (S, s)$", "and an open $V_x \\subset X \\times_S \\Spec(\\mathcal{O}_{S_x, s_x})$", "such that $x \\in X_s = X \\times_S s_x$ is contained in $V_x$ and such that", "the pullback of $\\mathcal{F}$ to $V_x$ is an", "$\\mathcal{O}_{V_x}$-module of finite presentation and flat over", "$\\mathcal{O}_{S_x, s_x}$. In particular we may view the fibre", "$(V_x)_{s_x}$ as an open neighbourhood of $x$ in $X_s$.", "Because $X_s$ is quasi-compact we can find a finite number of points", "$x_1, \\ldots, x_n \\in X_s$ such that $X_s$ is the union of", "the $(V_{x_i})_{s_{x_i}}$. Choose an elementary \\'etale neighbourhood", "$(S' , s') \\to (S, s)$ which dominates each of the neighbourhoods", "$(S_{x_i}, s_{x_i})$, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-elementary-etale-neighbourhoods}.", "Set $V = \\bigcup V_i$ where $V_i$ is the inverse images of the open", "$V_{x_i}$ via the morphism", "$$", "X \\times_S \\Spec(\\mathcal{O}_{S', s'})", "\\longrightarrow", "X \\times_S \\Spec(\\mathcal{O}_{S_{x_i}, s_{x_i}})", "$$", "By construction $V$ contains $X_s$ and by construction the pullback", "of $\\mathcal{F}$ to $V$ is an $\\mathcal{O}_V$-module", "of finite presentation and flat over $\\mathcal{O}_{S', s'}$." ], "refs": [ "flat-proposition-finite-type-flat-at-point", "more-morphisms-lemma-elementary-etale-neighbourhoods" ], "ref_ids": [ 6199, 13868 ] } ], "ref_ids": [] }, { "id": 6020, "type": "theorem", "label": "flat-lemma-finite-type-flat-along-fibre-variant", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-along-fibre-variant", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $s \\in S$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is of finite type,", "\\item $\\mathcal{F}$ is of finite type, and", "\\item $\\mathcal{F}$ is flat over $S$ at every point of the fibre $X_s$.", "\\end{enumerate}", "Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$", "and an open subscheme", "$$", "V \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})", "$$", "which contains the fibre $X_s = X \\times_S s'$ such that the pullback", "of $\\mathcal{F}$ to $V$ is flat over $\\mathcal{O}_{S', s'}$." ], "refs": [], "proofs": [ { "contents": [ "(The only difference between this and", "Lemma \\ref{lemma-finite-type-flat-along-fibre}", "is that we do not assume $f$ is of finite presentation.)", "For every point $x \\in X_s$ we can use", "Lemma \\ref{lemma-finite-type-flat-at-point}", "to find an elementary \\'etale neighbourhood $(S_x, s_x) \\to (S, s)$", "and an open $V_x \\subset X \\times_S \\Spec(\\mathcal{O}_{S_x, s_x})$", "such that $x \\in X_s = X \\times_S s_x$ is contained in $V_x$ and such that", "the pullback of $\\mathcal{F}$ to $V_x$ is flat over", "$\\mathcal{O}_{S_x, s_x}$. In particular we may view the fibre", "$(V_x)_{s_x}$ as an open neighbourhood of $x$ in $X_s$.", "Because $X_s$ is quasi-compact we can find a finite number of points", "$x_1, \\ldots, x_n \\in X_s$ such that $X_s$ is the union of", "the $(V_{x_i})_{s_{x_i}}$. Choose an elementary \\'etale neighbourhood", "$(S' , s') \\to (S, s)$ which dominates each of the neighbourhoods", "$(S_{x_i}, s_{x_i})$, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-elementary-etale-neighbourhoods}.", "Set $V = \\bigcup V_i$ where $V_i$ is the inverse images of the open", "$V_{x_i}$ via the morphism", "$$", "X \\times_S \\Spec(\\mathcal{O}_{S', s'})", "\\longrightarrow", "X \\times_S \\Spec(\\mathcal{O}_{S_{x_i}, s_{x_i}})", "$$", "By construction $V$ contains $X_s$ and by construction the pullback", "of $\\mathcal{F}$ to $V$ is flat over $\\mathcal{O}_{S', s'}$." ], "refs": [ "flat-lemma-finite-type-flat-along-fibre", "flat-lemma-finite-type-flat-at-point", "more-morphisms-lemma-elementary-etale-neighbourhoods" ], "ref_ids": [ 6019, 6018, 13868 ] } ], "ref_ids": [] }, { "id": 6021, "type": "theorem", "label": "flat-lemma-finite-type-flat-at-point-X", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-at-point-X", "contents": [ "Let $S$ be a scheme. Let $X$ be locally of finite type over $S$.", "Let $x \\in X$ with image $s \\in S$.", "If $X$ is flat at $x$ over $S$, then there exists an elementary", "\\'etale neighbourhood $(S', s') \\to (S, s)$ and an open subscheme", "$$", "V \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})", "$$", "which contains the unique point of", "$X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ mapping to $x$", "such that $V \\to \\Spec(\\mathcal{O}_{S', s'})$", "is flat and of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$ and $S$, hence we may assume $X$ and $S$", "are affine. Write $X = \\Spec(B)$, $S = \\Spec(A)$ and write", "$B = A[x_1, \\ldots, x_n]/I$. In other words we obtain a closed immersion", "$i : X \\to \\mathbf{A}^n_S$. Denote $t = i(x) \\in \\mathbf{A}^n_S$.", "We may apply", "Proposition \\ref{proposition-finite-type-flat-at-point}", "to $\\mathbf{A}^n_S \\to S$, the sheaf $\\mathcal{F} = i_*\\mathcal{O}_X$", "and the point $t$. We obtain an elementary", "\\'etale neighbourhood $(S', s') \\to (S, s)$ and an open subscheme", "$$", "W \\subset \\mathbf{A}^n_{\\mathcal{O}_{S', s'}}", "$$", "such that the pullback of $i_*\\mathcal{O}_X$ is flat and of finite", "presentation. This means that", "$V := W \\cap \\big(X \\times_S \\Spec(\\mathcal{O}_{S', s'})\\big)$", "is the desired open subscheme." ], "refs": [ "flat-proposition-finite-type-flat-at-point" ], "ref_ids": [ 6199 ] } ], "ref_ids": [] }, { "id": 6022, "type": "theorem", "label": "flat-lemma-finite-type-flat-at-point-local", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-at-point-local", "contents": [ "Let $f : X \\to S$ be a morphism which is locally of finite presentation.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.", "If $x \\in X$ and $\\mathcal{F}$ is flat at $x$ over $S$, then", "$\\mathcal{F}_x$ is an $\\mathcal{O}_{X, x}$-module of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Let $s = f(x)$. By", "Proposition \\ref{proposition-finite-type-flat-at-point}", "there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$", "such that the pullback of $\\mathcal{F}$ to", "$X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ is of", "finite presentation in a neighbourhood of the point $x' \\in X_{s'} = X_s$", "corresponding to $x$. The ring map", "$$", "\\mathcal{O}_{X, x} \\longrightarrow", "\\mathcal{O}_{X \\times_S \\Spec(\\mathcal{O}_{S', s'}), x'}", "=", "\\mathcal{O}_{X \\times_S S', x'}", "$$", "is flat and local as a localization of an \\'etale ring map. Hence", "$\\mathcal{F}_x$ is of finite presentation over $\\mathcal{O}_{X, x}$", "by descent, see", "Algebra, Lemma \\ref{algebra-lemma-descend-properties-modules}", "(and also that a flat local ring map is faithfully flat, see", "Algebra, Lemma \\ref{algebra-lemma-local-flat-ff})." ], "refs": [ "flat-proposition-finite-type-flat-at-point", "algebra-lemma-descend-properties-modules", "algebra-lemma-local-flat-ff" ], "ref_ids": [ 6199, 819, 537 ] } ], "ref_ids": [] }, { "id": 6023, "type": "theorem", "label": "flat-lemma-finite-type-flat-at-point-local-X", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-at-point-local-X", "contents": [ "Let $f : X \\to S$ be a morphism which is locally of finite type.", "Let $x \\in X$ with image $s \\in S$. If $f$ is flat at $x$ over $S$, then", "$\\mathcal{O}_{X, x}$ is essentially of finite presentation over", "$\\mathcal{O}_{S, s}$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $X$ and $S$ affine. Write $X = \\Spec(B)$,", "$S = \\Spec(A)$ and write $B = A[x_1, \\ldots, x_n]/I$.", "In other words we obtain a closed immersion $i : X \\to \\mathbf{A}^n_S$.", "Denote $t = i(x) \\in \\mathbf{A}^n_S$. We may apply", "Lemma \\ref{lemma-finite-type-flat-at-point-local}", "to $\\mathbf{A}^n_S \\to S$, the sheaf $\\mathcal{F} = i_*\\mathcal{O}_X$", "and the point $t$. We conclude that $\\mathcal{O}_{X, x}$ is", "of finite presentation over $\\mathcal{O}_{\\mathbf{A}^n_S, t}$", "which implies what we want." ], "refs": [ "flat-lemma-finite-type-flat-at-point-local" ], "ref_ids": [ 6022 ] } ], "ref_ids": [] }, { "id": 6024, "type": "theorem", "label": "flat-lemma-flat-finite-type-finitely-presented-over-dense-open", "categories": [ "flat" ], "title": "flat-lemma-flat-finite-type-finitely-presented-over-dense-open", "contents": [ "\\begin{slogan}", "$S$-flat and finite type extensions of finitely presented modules", "on a (good) open are also $X$-finitely presented.", "\\end{slogan}", "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a", "quasi-coherent $\\mathcal{O}_X$-module. Let $U \\subset S$ be open.", "Assume", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $\\mathcal{F}$ is of finite type and flat over $S$,", "\\item $U \\subset S$ is retrocompact and scheme theoretically dense,", "\\item $\\mathcal{F}|_{f^{-1}U}$ is of finite presentation.", "\\end{enumerate}", "Then $\\mathcal{F}$ is of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "The problem is local on $X$ and $S$, hence we may assume $X$ and $S$ affine.", "Write $S = \\Spec(A)$ and $X = \\Spec(B)$. Let $N$ be a finite $B$-module such", "that $\\mathcal{F}$ is the quasi-coherent sheaf associated to $N$.", "We have $U = D(f_1) \\cup \\ldots \\cup D(f_n)$ for some $f_i \\in A$, see", "Algebra, Lemma \\ref{algebra-lemma-qc-open}.", "As $U$ is schematically dense the map", "$A \\to A_{f_1} \\times \\ldots \\times A_{f_n}$ is injective.", "Pick a prime $\\mathfrak q \\subset B$ lying over $\\mathfrak p \\subset A$", "corresponding to $x \\in X$ mapping to $s \\in S$.", "By Lemma \\ref{lemma-finite-type-flat-at-point-local}", "the module $N_\\mathfrak q$ is of finite presentation over $B_\\mathfrak q$.", "Choose a surjection $\\varphi : B^{\\oplus m} \\to N$ of $B$-modules.", "Choose $k_1, \\ldots, k_t \\in \\Ker(\\varphi)$ and", "set $N' = B^{\\oplus m}/\\sum Bk_j$. There is a canonical surjection", "$N' \\to N$ and $N$ is the filtered colimit of the $B$-modules $N'$", "constructed in this manner. Thus we see that we can choose", "$k_1, \\ldots, k_t$ such that (a) $N'_{f_i} \\cong N_{f_i}$, $i = 1, \\ldots, n$", "and (b) $N'_\\mathfrak q \\cong N_\\mathfrak q$.", "This in particular implies that $N'_\\mathfrak q$ is flat over $A$.", "By openness of flatness, see", "Algebra, Theorem \\ref{algebra-theorem-openness-flatness}", "we conclude that there exists a $g \\in B$, $g \\not \\in \\mathfrak q$", "such that $N'_g$ is flat over $A$. Consider the commutative diagram", "$$", "\\xymatrix{", "N'_g \\ar[r] \\ar[d] & N_g \\ar[d] \\\\", "\\prod N'_{gf_i} \\ar[r] & \\prod N_{gf_i}", "}", "$$", "The bottom arrow is an isomorphism by choice of $k_1, \\ldots, k_t$.", "The left vertical arrow is an injective map as", "$A \\to \\prod A_{f_i}$ is injective and $N'_g$ is flat over $A$.", "Hence the top horizontal arrow is injective, hence an isomorphism.", "This proves that $N_g$ is of finite presentation over $B_g$.", "We conclude by applying", "Algebra, Lemma \\ref{algebra-lemma-cover}." ], "refs": [ "algebra-lemma-qc-open", "flat-lemma-finite-type-flat-at-point-local", "algebra-theorem-openness-flatness", "algebra-lemma-cover" ], "ref_ids": [ 432, 6022, 326, 411 ] } ], "ref_ids": [] }, { "id": 6025, "type": "theorem", "label": "flat-lemma-flat-finite-type-finitely-presented-over-dense-open-X", "categories": [ "flat" ], "title": "flat-lemma-flat-finite-type-finitely-presented-over-dense-open-X", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $U \\subset S$ be open.", "Assume", "\\begin{enumerate}", "\\item $f$ is locally of finite type and flat,", "\\item $U \\subset S$ is retrocompact and scheme theoretically dense,", "\\item $f|_{f^{-1}U} : f^{-1}U \\to U$ is locally of finite presentation.", "\\end{enumerate}", "Then $f$ is of locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$ and $S$, hence we may assume $X$ and", "$S$ affine. Choose a closed immersion $i : X \\to \\mathbf{A}^n_S$", "and apply", "Lemma \\ref{lemma-flat-finite-type-finitely-presented-over-dense-open}", "to $i_*\\mathcal{O}_X$. Some details omitted." ], "refs": [ "flat-lemma-flat-finite-type-finitely-presented-over-dense-open" ], "ref_ids": [ 6024 ] } ], "ref_ids": [] }, { "id": 6026, "type": "theorem", "label": "flat-lemma-flat-finite-presentation-dimension-over-dense-open", "categories": [ "flat" ], "title": "flat-lemma-flat-finite-presentation-dimension-over-dense-open", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat and locally", "of finite type. Let $U \\subset S$ be a dense open such that", "$X_U \\to U$ has relative dimension $\\leq e$, see", "Morphisms, Definition \\ref{morphisms-definition-relative-dimension-d}.", "If also either", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation, or", "\\item $U \\subset S$ is retrocompact,", "\\end{enumerate}", "then $f$ has relative dimension $\\leq e$." ], "refs": [ "morphisms-definition-relative-dimension-d" ], "proofs": [ { "contents": [ "Proof in case (1). Let $W \\subset X$ be the open subscheme constructed", "and studied in More on Morphisms, Lemmas", "\\ref{more-morphisms-lemma-flat-finite-presentation-CM-open} and", "\\ref{more-morphisms-lemma-flat-finite-presentation-CM-pieces}.", "Note that every generic point of every fibre is contained in $W$,", "hence it suffices to prove the result for $W$. Since", "$W = \\bigcup_{d \\geq 0} U_d$, it suffices to prove that $U_d = \\emptyset$", "for $d > e$. Since $f$ is flat and locally of finite presentation it is open", "hence $f(U_d)$ is open (Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}).", "Thus if $U_d$ is not empty, then $f(U_d) \\cap U \\not = \\emptyset$ as", "desired.", "\\medskip\\noindent", "Proof in case (2). We may replace $S$ by its reduction. Then $U$ is", "scheme theoretically dense. Hence $f$ is locally of finite presentation", "by Lemma \\ref{lemma-flat-finite-type-finitely-presented-over-dense-open-X}.", "In this way we reduce to case (1)." ], "refs": [ "more-morphisms-lemma-flat-finite-presentation-CM-open", "more-morphisms-lemma-flat-finite-presentation-CM-pieces", "morphisms-lemma-fppf-open", "flat-lemma-flat-finite-type-finitely-presented-over-dense-open-X" ], "ref_ids": [ 13789, 13791, 5267, 6025 ] } ], "ref_ids": [ 5559 ] }, { "id": 6027, "type": "theorem", "label": "flat-lemma-proper-flat-finite-over-dense-open", "categories": [ "flat" ], "title": "flat-lemma-proper-flat-finite-over-dense-open", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat and proper.", "Let $U \\subset S$ be a dense open such that $X_U \\to U$ is finite.", "If also either $f$ is locally of finite presentation or", "$U \\subset S$ is retrocompact, then $f$ is finite." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-flat-finite-presentation-dimension-over-dense-open}", "the fibres of $f$ have dimension zero.", "Hence $f$ is quasi-finite", "(Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0})", "whence has finite fibres", "(Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite}).", "Hence $f$ is finite by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-characterize-finite}." ], "refs": [ "flat-lemma-flat-finite-presentation-dimension-over-dense-open", "morphisms-lemma-locally-quasi-finite-rel-dimension-0", "morphisms-lemma-quasi-finite", "more-morphisms-lemma-characterize-finite" ], "ref_ids": [ 6026, 5287, 5230, 13903 ] } ], "ref_ids": [] }, { "id": 6028, "type": "theorem", "label": "flat-lemma-zariski", "categories": [ "flat" ], "title": "flat-lemma-zariski", "contents": [ "Let $f : X \\to S$ be a morphism of schemes and $U \\subset S$ an open. If", "\\begin{enumerate}", "\\item $f$ is separated, locally of finite type, and flat,", "\\item $f^{-1}(U) \\to U$ is an isomorphism, and", "\\item $U \\subset S$ is retrocompact and scheme theoretically dense,", "\\end{enumerate}", "then $f$ is an open immersion." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-flat-finite-type-finitely-presented-over-dense-open-X}", "the morphism $f$ is locally of finite presentation.", "The image $f(X) \\subset S$ is open", "(Morphisms, Lemma \\ref{morphisms-lemma-fppf-open})", "hence we may replace $S$ by $f(X)$. Thus we have to prove that", "$f$ is an isomorphism. We may assume $S$ is affine. We can reduce", "to the case that $X$ is quasi-compact because it suffices to show", "that any quasi-compact open $X' \\subset X$ whose image is $S$", "maps isomorphically to $S$. Thus we may assume $f$ is quasi-compact.", "All the fibers of $f$ have dimension $0$, see", "Lemma \\ref{lemma-flat-finite-presentation-dimension-over-dense-open}.", "Hence $f$ is quasi-finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}.", "Let $s \\in S$. Choose an elementary \\'etale", "neighbourhood $g : (T, t) \\to (S, s)$ such that $X \\times_S T = V \\amalg W$", "with $V \\to T$ finite and $W_t = \\emptyset$, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-splits-off-quasi-finite-part}.", "Denote $\\pi : V \\amalg W \\to T$ the given morphism. Since $\\pi$ is", "flat and locally of finite presentation, we see that $\\pi(V)$ is", "open in $T$ (Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}).", "After shrinking $T$ we may assume that $T = \\pi(V)$.", "Since $f$ is an isomorphism over $U$ we see that $\\pi$ is an", "isomorphism over $g^{-1}U$. Since $\\pi(V) = T$ this implies", "that $\\pi^{-1}g^{-1}U$ is contained in $V$.", "By Morphisms, Lemma", "\\ref{morphisms-lemma-flat-morphism-scheme-theoretically-dense-open}", "we see that $\\pi^{-1}g^{-1}U \\subset V \\amalg W$ is scheme theoretically", "dense. Hence we deduce that $W = \\emptyset$.", "Thus $X \\times_S T = V$ is finite over $T$.", "This implies that $f$ is finite (after replacing $S$ by an", "open neighbourhood of $s$), for example by", "Descent, Lemma \\ref{descent-lemma-descending-property-finite}.", "Then $f$ is finite locally free", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-flat})", "and after shrinking $S$ to a smaller open neighbourhood of $s$", "we see that $f$ is finite locally free of some degree $d$", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-locally-free}).", "But $d = 1$ as is clear from the fact that the degree is $1$ over", "the dense open $U$. Hence $f$ is an isomorphism." ], "refs": [ "flat-lemma-flat-finite-type-finitely-presented-over-dense-open-X", "morphisms-lemma-fppf-open", "flat-lemma-flat-finite-presentation-dimension-over-dense-open", "morphisms-lemma-locally-quasi-finite-rel-dimension-0", "more-morphisms-lemma-etale-splits-off-quasi-finite-part", "morphisms-lemma-fppf-open", "morphisms-lemma-flat-morphism-scheme-theoretically-dense-open", "descent-lemma-descending-property-finite", "morphisms-lemma-finite-flat", "morphisms-lemma-finite-locally-free" ], "ref_ids": [ 6025, 5267, 6026, 5287, 13897, 5267, 5272, 14688, 5471, 5474 ] } ], "ref_ids": [] }, { "id": 6029, "type": "theorem", "label": "flat-lemma-induction-step-fp", "categories": [ "flat" ], "title": "flat-lemma-induction-step-fp", "contents": [ "Let $R$ be a ring. Let $R \\to S$ be a finitely presented", "flat ring map with geometrically integral fibres. Let", "$\\mathfrak q \\subset S$ be a prime ideal lying over the prime", "$\\mathfrak r \\subset R$. Set $\\mathfrak p = \\mathfrak r S$.", "Let $N$ be a finitely presented $S$-module.", "There exists $r \\geq 0$ and an $S$-module map", "$$", "\\alpha : S^{\\oplus r} \\longrightarrow N", "$$", "such that", "$\\alpha : \\kappa(\\mathfrak p)^{\\oplus r} \\to N \\otimes_S \\kappa(\\mathfrak p)$", "is an isomorphism. For any such $\\alpha$ the following are equivalent:", "\\begin{enumerate}", "\\item $N_{\\mathfrak q}$ is $R$-flat,", "\\item there exists an $f \\in R$, $f \\not \\in \\mathfrak r$ such that", "$\\alpha_f : S_f^{\\oplus r} \\to N_f$ is $R_f$-universally injective and", "a $g \\in S$, $g \\not \\in \\mathfrak q$ such that $\\Coker(\\alpha)_g$", "is $R$-flat,", "\\item $\\alpha_{\\mathfrak r}$ is $R_{\\mathfrak r}$-universally injective and", "$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat", "\\item $\\alpha_{\\mathfrak r}$ is injective and", "$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat,", "\\item $\\alpha_{\\mathfrak p}$ is an isomorphism and", "$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat, and", "\\item $\\alpha_{\\mathfrak q}$ is injective and", "$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To obtain $\\alpha$ set", "$r = \\dim_{\\kappa(\\mathfrak p)} N \\otimes_S \\kappa(\\mathfrak p)$ and pick", "$x_1, \\ldots, x_r \\in N$ which form a basis of", "$N \\otimes_S \\kappa(\\mathfrak p)$. Define", "$\\alpha(s_1, \\ldots, s_r) = \\sum s_i x_i$. This proves the existence.", "\\medskip\\noindent", "Fix a choice of $\\alpha$.", "We may apply", "Lemma \\ref{lemma-induction-step}", "to the map", "$\\alpha_{\\mathfrak r} : S_{\\mathfrak r}^{\\oplus r} \\to N_{\\mathfrak r}$.", "Hence we see that (1), (3), (4), (5), and (6) are all equivalent.", "Since it is also clear that (2) implies (3) we see that all we have to", "do is show that (1) implies (2).", "\\medskip\\noindent", "Assume (1). By openness of flatness, see", "Algebra, Theorem \\ref{algebra-theorem-openness-flatness},", "the set", "$$", "U_1 = \\{\\mathfrak q' \\subset S \\mid N_{\\mathfrak q'}\\text{ is flat over }R\\}", "$$", "is open in $\\Spec(S)$. It contains $\\mathfrak q$ by assumption", "and hence $\\mathfrak p$. Because $S^{\\oplus r}$ and $N$ are finitely presented", "$S$-modules the set", "$$", "U_2 = \\{\\mathfrak q' \\subset S \\mid", "\\alpha_{\\mathfrak q'}\\text{ is an isomorphism}\\}", "$$", "is open in $\\Spec(S)$, see", "Algebra, Lemma \\ref{algebra-lemma-map-between-finitely-presented}.", "It contains $\\mathfrak p$ by (5). As $R \\to S$", "is finitely presented and flat the map", "$\\Phi : \\Spec(S) \\to \\Spec(R)$ is open, see", "Algebra, Proposition \\ref{algebra-proposition-fppf-open}.", "For any prime $\\mathfrak r' \\in \\Phi(U_1 \\cap U_2)$ we see that", "there exists a prime $\\mathfrak q'$ lying over $\\mathfrak r'$ such that", "$N_{\\mathfrak q'}$ is flat and such that $\\alpha_{\\mathfrak q'}$ is", "an isomorphism, which implies that $\\alpha \\otimes \\kappa(\\mathfrak p')$", "is an isomorphism where $\\mathfrak p' = \\mathfrak r' S$. Thus", "$\\alpha_{\\mathfrak r'}$ is $R_{\\mathfrak r'}$-universally injective", "by the implication (1) $\\Rightarrow$ (3).", "Hence if we pick $f \\in R$, $f \\not \\in \\mathfrak r$ such that", "$D(f) \\subset \\Phi(U_1 \\cap U_2)$ then we conclude that", "$\\alpha_f$ is $R_f$-universally injective, see", "Algebra, Lemma \\ref{algebra-lemma-universally-injective-check-stalks}.", "The same reasoning also shows that for any", "$\\mathfrak q' \\in U_1 \\cap \\Phi^{-1}(\\Phi(U_1 \\cap U_2))$", "the module $\\Coker(\\alpha)_{\\mathfrak q'}$ is $R$-flat.", "Note that $\\mathfrak q \\in U_1 \\cap \\Phi^{-1}(\\Phi(U_1 \\cap U_2))$.", "Hence we can find a $g \\in S$, $g \\not \\in \\mathfrak q$ such", "that $D(g) \\subset U_1 \\cap \\Phi^{-1}(\\Phi(U_1 \\cap U_2))$", "and we win." ], "refs": [ "flat-lemma-induction-step", "algebra-theorem-openness-flatness", "algebra-lemma-map-between-finitely-presented", "algebra-proposition-fppf-open", "algebra-lemma-universally-injective-check-stalks" ], "ref_ids": [ 6015, 326, 803, 1407, 815 ] } ], "ref_ids": [] }, { "id": 6030, "type": "theorem", "label": "flat-lemma-complete-devissage-flat-finitely-presented-module", "categories": [ "flat" ], "title": "flat-lemma-complete-devissage-flat-finitely-presented-module", "contents": [ "Let $R \\to S$ be a ring map of finite presentation.", "Let $N$ be a finitely presented $S$-module flat over $R$.", "Let $\\mathfrak r \\subset R$ be a prime ideal.", "Assume there exists a complete d\\'evissage of $N/S/R$ over $\\mathfrak r$.", "Then there exists an $f \\in R$, $f \\not \\in \\mathfrak r$", "such that", "$$", "N_f \\cong B_1^{\\oplus r_1} \\oplus \\ldots \\oplus B_n^{\\oplus r_n}", "$$", "as $R$-modules where each $B_i$ is a smooth $R_f$-algebra with geometrically", "irreducible fibres. Moreover, $N_f$ is projective as an $R_f$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $(A_i, B_i, M_i, \\alpha_i)_{i = 1, \\ldots, n}$ be the given", "complete d\\'evissage. We prove the lemma by induction on $n$.", "Note that the assertions of the lemma are entirely about the structure", "of $N$ as an $R$-module. Hence we may replace $N$ by $M_1$, and we", "may think of $M_1$ as a $B_1$-module. See", "Remark \\ref{remark-finite-presentation}", "in order to see why $M_1$ is of finite presentation as a $B_1$-module. By", "Lemma \\ref{lemma-induction-step-fp}", "we may, after replacing $R$ by $R_f$ for some", "$f \\in R$, $f \\not \\in \\mathfrak r$, assume", "the map $\\alpha_1 : B_1^{\\oplus r_1} \\to M_1$ is $R$-universally injective.", "Since $M_1$ and $B_1^{\\oplus r_1}$ are $R$-flat and finitely presented as", "$B_1$-modules we see that $\\Coker(\\alpha_1)$ is $R$-flat", "(Algebra, Lemma \\ref{algebra-lemma-ui-flat-domain})", "and finitely presented as a $B_1$-module. Note that", "$(A_i, B_i, M_i, \\alpha_i)_{i = 2, \\ldots, n}$ is a complete", "d\\'evissage of $\\Coker(\\alpha_1)$. Hence the induction hypothesis", "implies that, after replacing", "$R$ by $R_f$ for some $f \\in R$, $f \\not \\in \\mathfrak r$,", "we may assume that $\\Coker(\\alpha_1)$ has a decomposition", "as in the lemma and is projective. In particular", "$M_1 = B_1^{\\oplus r_1} \\oplus \\Coker(\\alpha_1)$.", "This proves the statement regarding the decomposition.", "The statement on projectivity follows as $B_1$ is projective as", "an $R$-module by", "Lemma \\ref{lemma-fibres-irreducible-flat-projective-nonnoetherian}." ], "refs": [ "flat-remark-finite-presentation", "flat-lemma-induction-step-fp", "algebra-lemma-ui-flat-domain", "flat-lemma-fibres-irreducible-flat-projective-nonnoetherian" ], "ref_ids": [ 6224, 6029, 810, 6014 ] } ], "ref_ids": [] }, { "id": 6031, "type": "theorem", "label": "flat-lemma-finite-presentation-flat-along-fibre", "categories": [ "flat" ], "title": "flat-lemma-finite-presentation-flat-along-fibre", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $s \\in S$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is of finite presentation,", "\\item $\\mathcal{F}$ is of finite presentation, and", "\\item $\\mathcal{F}$ is flat over $S$ at every point of the fibre $X_s$.", "\\end{enumerate}", "Then there exists an elementary \\'etale neighbourhood", "$(S', s') \\to (S, s)$ and a commutative diagram of schemes", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\", "S & S' \\ar[l]", "}", "$$", "such that $g$ is \\'etale, $X_s \\subset g(X')$, the schemes", "$X'$, $S'$ are affine, and such that", "$\\Gamma(X', g^*\\mathcal{F})$ is a projective", "$\\Gamma(S', \\mathcal{O}_{S'})$-module." ], "refs": [], "proofs": [ { "contents": [ "For every point $x \\in X_s$ we can use", "Proposition \\ref{proposition-finite-presentation-flat-at-point}", "to find a commutative diagram", "$$", "\\xymatrix{", "(X, x) \\ar[d] & (Y_x, y_x) \\ar[l]^{g_x} \\ar[d] \\\\", "(S, s) & (S_x, s_x) \\ar[l]", "}", "$$", "whose horizontal arrows are elementary \\'etale neighbourhoods", "such that $Y_x$, $S_x$ are affine and such that", "$\\Gamma(Y_x, g_x^*\\mathcal{F})$ is a projective", "$\\Gamma(S_x, \\mathcal{O}_{S_x})$-module. In particular", "$g_x(Y_x) \\cap X_s$ is an open neighbourhood of $x$ in $X_s$.", "Because $X_s$ is quasi-compact we can find a finite number of points", "$x_1, \\ldots, x_n \\in X_s$ such that $X_s$ is the union of", "the $g_{x_i}(Y_{x_i}) \\cap X_s$. Choose an elementary \\'etale neighbourhood", "$(S' , s') \\to (S, s)$ which dominates each of the neighbourhoods", "$(S_{x_i}, s_{x_i})$, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-elementary-etale-neighbourhoods}.", "We may also assume that $S'$ is affine.", "Set $X' = \\coprod Y_{x_i} \\times_{S_{x_i}} S'$ and endow it with the", "obvious morphism $g : X' \\to X$.", "By construction $g(X')$ contains $X_s$ and", "$$", "\\Gamma(X', g^*\\mathcal{F})", "=", "\\bigoplus \\Gamma(Y_{x_i}, g_{x_i}^*\\mathcal{F})", "\\otimes_{\\Gamma(S_{x_i}, \\mathcal{O}_{S_{x_i}})}", "\\Gamma(S', \\mathcal{O}_{S'}).", "$$", "This is a projective $\\Gamma(S', \\mathcal{O}_{S'})$-module, see", "Algebra, Lemma \\ref{algebra-lemma-ascend-properties-modules}." ], "refs": [ "flat-proposition-finite-presentation-flat-at-point", "more-morphisms-lemma-elementary-etale-neighbourhoods", "algebra-lemma-ascend-properties-modules" ], "ref_ids": [ 6200, 13868, 853 ] } ], "ref_ids": [] }, { "id": 6032, "type": "theorem", "label": "flat-lemma-finite-presentation-flat-at-point-X", "categories": [ "flat" ], "title": "flat-lemma-finite-presentation-flat-at-point-X", "contents": [ "Let $f : X \\to S$ be locally of finite presentation.", "Let $x \\in X$ with image $s \\in S$.", "If $f$ is flat at $x$ over $S$, then there exists a commutative", "diagram of pointed schemes", "$$", "\\xymatrix{", "(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\", "(S, s) & (S', s') \\ar[l]", "}", "$$", "whose horizontal arrows are elementary \\'etale neighbourhoods", "such that $X'$, $S'$ are affine and such that", "$\\Gamma(X', \\mathcal{O}_{X'})$ is a projective", "$\\Gamma(S', \\mathcal{O}_{S'})$-module." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Proposition \\ref{proposition-finite-presentation-flat-at-point}." ], "refs": [ "flat-proposition-finite-presentation-flat-at-point" ], "ref_ids": [ 6200 ] } ], "ref_ids": [] }, { "id": 6033, "type": "theorem", "label": "flat-lemma-finite-presentation-flat-along-fibre-X", "categories": [ "flat" ], "title": "flat-lemma-finite-presentation-flat-along-fibre-X", "contents": [ "Let $f : X \\to S$ be of finite presentation.", "Let $s \\in S$.", "If $X$ is flat over $S$ at all points of $X_s$, then", "there exists an elementary \\'etale neighbourhood", "$(S', s') \\to (S, s)$ and a commutative diagram of schemes", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\", "S & S' \\ar[l]", "}", "$$", "with $g$ \\'etale, $X_s \\subset g(X')$, such that $X'$, $S'$", "are affine, and such that", "$\\Gamma(X', \\mathcal{O}_{X'})$ is a projective", "$\\Gamma(S', \\mathcal{O}_{S'})$-module." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-finite-presentation-flat-along-fibre}." ], "refs": [ "flat-lemma-finite-presentation-flat-along-fibre" ], "ref_ids": [ 6031 ] } ], "ref_ids": [] }, { "id": 6034, "type": "theorem", "label": "flat-lemma-finite-type-flat-at-point-free", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-at-point-free", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $x \\in X$ with image $s \\in S$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $\\mathcal{F}$ is of finite type, and", "\\item $\\mathcal{F}$ is flat at $x$ over $S$.", "\\end{enumerate}", "Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$", "and a commutative diagram of pointed schemes", "$$", "\\xymatrix{", "(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\", "(S, s) & (\\Spec(\\mathcal{O}_{S', s'}), s') \\ar[l]", "}", "$$", "such that $X' \\to X \\times_S \\Spec(\\mathcal{O}_{S', s'})$", "is \\'etale, $\\kappa(x) = \\kappa(x')$, the scheme $X'$ is", "affine of finite presentation over $\\mathcal{O}_{S', s'}$,", "the sheaf $g^*\\mathcal{F}$ is of finite presentation over $\\mathcal{O}_{X'}$,", "and such that $\\Gamma(X', g^*\\mathcal{F})$ is a free", "$\\mathcal{O}_{S', s'}$-module." ], "refs": [], "proofs": [ { "contents": [ "To prove the lemma we may replace $(S, s)$ by any elementary \\'etale", "neighbourhood, and we may also replace $S$ by", "$\\Spec(\\mathcal{O}_{S, s})$. Hence by", "Proposition \\ref{proposition-finite-type-flat-at-point}", "we may assume that $\\mathcal{F}$ is finitely presented and flat over", "$S$ in a neighbourhood of $x$. In this case the result follows from", "Proposition \\ref{proposition-finite-presentation-flat-at-point}", "because", "Algebra, Theorem \\ref{algebra-theorem-projective-free-over-local-ring}", "assures us that projective $=$ free over a local ring." ], "refs": [ "flat-proposition-finite-type-flat-at-point", "flat-proposition-finite-presentation-flat-at-point", "algebra-theorem-projective-free-over-local-ring" ], "ref_ids": [ 6199, 6200, 322 ] } ], "ref_ids": [] }, { "id": 6035, "type": "theorem", "label": "flat-lemma-finite-type-flat-at-point-free-variant", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-at-point-free-variant", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $x \\in X$ with image $s \\in S$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite type,", "\\item $\\mathcal{F}$ is of finite type, and", "\\item $\\mathcal{F}$ is flat at $x$ over $S$.", "\\end{enumerate}", "Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$", "and a commutative diagram of pointed schemes", "$$", "\\xymatrix{", "(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\", "(S, s) & (\\Spec(\\mathcal{O}_{S', s'}), s') \\ar[l]", "}", "$$", "such that $X' \\to X \\times_S \\Spec(\\mathcal{O}_{S', s'})$", "is \\'etale, $\\kappa(x) = \\kappa(x')$, the scheme $X'$ is", "affine, and such that $\\Gamma(X', g^*\\mathcal{F})$ is a free", "$\\mathcal{O}_{S', s'}$-module." ], "refs": [], "proofs": [ { "contents": [ "(The only difference with", "Lemma \\ref{lemma-finite-type-flat-at-point-free}", "is that we do not assume $f$ is of finite presentation.)", "The problem is local on $X$ and $S$. Hence we may assume $X$ and", "$S$ are affine, say $X = \\Spec(B)$ and $S = \\Spec(A)$.", "Since $B$ is a finite type $A$-algebra we can find a surjection", "$A[x_1, \\ldots, x_n] \\to B$. In other words, we can choose a closed", "immersion $i : X \\to \\mathbf{A}^n_S$. Set $t = i(x)$ and", "$\\mathcal{G} = i_*\\mathcal{F}$. Note that $\\mathcal{G}_t \\cong \\mathcal{F}_x$", "are $\\mathcal{O}_{S, s}$-modules. Hence $\\mathcal{G}$ is flat over $S$ at $t$.", "We apply", "Lemma \\ref{lemma-finite-type-flat-at-point-free}", "to the morphism $\\mathbf{A}^n_S \\to S$, the point $t$, and the", "sheaf $\\mathcal{G}$. Thus we can find an", "elementary \\'etale neighbourhood $(S', s') \\to (S, s)$", "and a commutative diagram of pointed schemes", "$$", "\\xymatrix{", "(\\mathbf{A}^n_S, t) \\ar[d] & (Y, y) \\ar[l]^h \\ar[d] \\\\", "(S, s) & (\\Spec(\\mathcal{O}_{S', s'}), s') \\ar[l]", "}", "$$", "such that $Y \\to \\mathbf{A}^n_{\\mathcal{O}_{S', s'}}$", "is \\'etale, $\\kappa(t) = \\kappa(y)$, the scheme $Y$ is", "affine, and such that $\\Gamma(Y, h^*\\mathcal{G})$ is a projective", "$\\mathcal{O}_{S', s'}$-module. Then a solution to the original", "problem is given by the closed subscheme", "$X' = Y \\times_{\\mathbf{A}^n_S} X$ of $Y$." ], "refs": [ "flat-lemma-finite-type-flat-at-point-free", "flat-lemma-finite-type-flat-at-point-free" ], "ref_ids": [ 6034, 6034 ] } ], "ref_ids": [] }, { "id": 6036, "type": "theorem", "label": "flat-lemma-finite-type-flat-along-fibre-free", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-along-fibre-free", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $s \\in S$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is of finite presentation,", "\\item $\\mathcal{F}$ is of finite type, and", "\\item $\\mathcal{F}$ is flat over $S$ at all points of $X_s$.", "\\end{enumerate}", "Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$", "and a commutative diagram of schemes", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\", "S & \\Spec(\\mathcal{O}_{S', s'}) \\ar[l]", "}", "$$", "such that $X' \\to X \\times_S \\Spec(\\mathcal{O}_{S', s'})$", "is \\'etale, $X_s = g((X')_{s'})$, the scheme $X'$ is", "affine of finite presentation over $\\mathcal{O}_{S', s'}$,", "the sheaf $g^*\\mathcal{F}$ is of finite presentation over $\\mathcal{O}_{X'}$,", "and such that $\\Gamma(X', g^*\\mathcal{F})$ is a free", "$\\mathcal{O}_{S', s'}$-module." ], "refs": [], "proofs": [ { "contents": [ "For every point $x \\in X_s$ we can use", "Lemma \\ref{lemma-finite-type-flat-at-point-free}", "to find an elementary \\'etale neighbourhood $(S_x , s_x) \\to (S, s)$", "and a commutative diagram", "$$", "\\xymatrix{", "(X, x) \\ar[d] & (Y_x, y_x) \\ar[l]^{g_x} \\ar[d] \\\\", "(S, s) & (\\Spec(\\mathcal{O}_{S_x, s_x}), s_x) \\ar[l]", "}", "$$", "such that $Y_x \\to X \\times_S \\Spec(\\mathcal{O}_{S_x, s_x})$", "is \\'etale, $\\kappa(x) = \\kappa(y_x)$, the scheme $Y_x$ is affine", "of finite presentation over $\\mathcal{O}_{S_x, s_x}$, the sheaf", "$g_x^*\\mathcal{F}$ is of finite presentation over $\\mathcal{O}_{Y_x}$, and", "such that $\\Gamma(Y_x, g_x^*\\mathcal{F})$ is a free", "$\\mathcal{O}_{S_x, s_x}$-module. In particular", "$g_x((Y_x)_{s_x})$ is an open neighbourhood of $x$ in $X_s$.", "Because $X_s$ is quasi-compact we can find a finite number of points", "$x_1, \\ldots, x_n \\in X_s$ such that $X_s$ is the union of", "the $g_{x_i}((Y_{x_i})_{s_{x_i}})$. Choose an elementary \\'etale neighbourhood", "$(S' , s') \\to (S, s)$ which dominates each of the neighbourhoods", "$(S_{x_i}, s_{x_i})$, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-elementary-etale-neighbourhoods}.", "Set", "$$", "X' = \\coprod Y_{x_i} \\times_{\\Spec(\\mathcal{O}_{S_{x_i}, s_{x_i}})}", "\\Spec(\\mathcal{O}_{S', s'})", "$$", "and endow it with the obvious morphism $g : X' \\to X$.", "By construction $X_s = g(X'_{s'})$ and", "$$", "\\Gamma(X', g^*\\mathcal{F})", "=", "\\bigoplus \\Gamma(Y_{x_i}, g_{x_i}^*\\mathcal{F})", "\\otimes_{\\mathcal{O}_{S_{x_i}, s_{x_i}}}", "\\mathcal{O}_{S', s'}.", "$$", "This is a free $\\mathcal{O}_{S', s'}$-module as a direct sum", "of base changes of free modules. Some minor details omitted." ], "refs": [ "flat-lemma-finite-type-flat-at-point-free", "more-morphisms-lemma-elementary-etale-neighbourhoods" ], "ref_ids": [ 6034, 13868 ] } ], "ref_ids": [] }, { "id": 6037, "type": "theorem", "label": "flat-lemma-finite-type-flat-along-fibre-free-variant", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-along-fibre-free-variant", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $s \\in S$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is of finite type,", "\\item $\\mathcal{F}$ is of finite type, and", "\\item $\\mathcal{F}$ is flat over $S$ at all points of $X_s$.", "\\end{enumerate}", "Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$", "and a commutative diagram of schemes", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\", "S & \\Spec(\\mathcal{O}_{S', s'}) \\ar[l]", "}", "$$", "such that $X' \\to X \\times_S \\Spec(\\mathcal{O}_{S', s'})$", "is \\'etale, $X_s = g((X')_{s'})$, the scheme $X'$ is affine,", "and such that $\\Gamma(X', g^*\\mathcal{F})$ is a free", "$\\mathcal{O}_{S', s'}$-module." ], "refs": [], "proofs": [ { "contents": [ "(The only difference with", "Lemma \\ref{lemma-finite-type-flat-along-fibre-free}", "is that we do not assume $f$ is of finite presentation.)", "For every point $x \\in X_s$ we can use", "Lemma \\ref{lemma-finite-type-flat-at-point-free-variant}", "to find an elementary \\'etale neighbourhood $(S_x , s_x) \\to (S, s)$", "and a commutative diagram", "$$", "\\xymatrix{", "(X, x) \\ar[d] & (Y_x, y_x) \\ar[l]^{g_x} \\ar[d] \\\\", "(S, s) & (\\Spec(\\mathcal{O}_{S_x, s_x}), s_x) \\ar[l]", "}", "$$", "such that $Y_x \\to X \\times_S \\Spec(\\mathcal{O}_{S_x, s_x})$", "is \\'etale, $\\kappa(x) = \\kappa(y_x)$, the scheme $Y_x$ is affine, and", "such that $\\Gamma(Y_x, g_x^*\\mathcal{F})$ is a free", "$\\mathcal{O}_{S_x, s_x}$-module. In particular", "$g_x((Y_x)_{s_x})$ is an open neighbourhood of $x$ in $X_s$.", "Because $X_s$ is quasi-compact we can find a finite number of points", "$x_1, \\ldots, x_n \\in X_s$ such that $X_s$ is the union of", "the $g_{x_i}((Y_{x_i})_{s_{x_i}})$. Choose an elementary \\'etale neighbourhood", "$(S' , s') \\to (S, s)$ which dominates each of the neighbourhoods", "$(S_{x_i}, s_{x_i})$, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-elementary-etale-neighbourhoods}.", "Set", "$$", "X' = \\coprod Y_{x_i} \\times_{\\Spec(\\mathcal{O}_{S_{x_i}, s_{x_i}})}", "\\Spec(\\mathcal{O}_{S', s'})", "$$", "and endow it with the obvious morphism $g : X' \\to X$.", "By construction $X_s = g(X'_{s'})$ and", "$$", "\\Gamma(X', g^*\\mathcal{F})", "=", "\\bigoplus \\Gamma(Y_{x_i}, g_{x_i}^*\\mathcal{F})", "\\otimes_{\\mathcal{O}_{S_{x_i}, s_{x_i}}}", "\\mathcal{O}_{S', s'}.", "$$", "This is a free $\\mathcal{O}_{S', s'}$-module as a direct sum", "of base changes of free modules." ], "refs": [ "flat-lemma-finite-type-flat-along-fibre-free", "flat-lemma-finite-type-flat-at-point-free-variant", "more-morphisms-lemma-elementary-etale-neighbourhoods" ], "ref_ids": [ 6036, 6035, 13868 ] } ], "ref_ids": [] }, { "id": 6038, "type": "theorem", "label": "flat-lemma-weak-bourbaki-pre-pre", "categories": [ "flat" ], "title": "flat-lemma-weak-bourbaki-pre-pre", "contents": [ "Let $R \\to S$ be a ring map of finite presentation. Let $N$ be a", "finitely presented $S$-module. Let $\\mathfrak q \\subset S$ be a prime ideal", "lying over $\\mathfrak p \\subset R$. Set", "$\\overline{S} = S \\otimes_R \\kappa(\\mathfrak p)$,", "$\\overline{\\mathfrak q} = \\mathfrak q \\overline{S}$, and", "$\\overline{N} = N \\otimes_R \\kappa(\\mathfrak p)$. Then", "we can find a $g \\in S$ with", "$g \\not \\in \\mathfrak q$ such that", "$\\overline{g} \\in \\mathfrak r$ for all", "$\\mathfrak r \\in \\text{Ass}_{\\overline{S}}(\\overline{N})$", "such that $\\mathfrak r \\not \\subset \\overline{\\mathfrak q}$." ], "refs": [], "proofs": [ { "contents": [ "Namely, if $\\text{Ass}_{\\overline{S}}(\\overline{N}) =", "\\{\\mathfrak r_1, \\ldots, \\mathfrak r_n\\}$", "(finiteness by Algebra, Lemma \\ref{algebra-lemma-finite-ass}),", "then after renumbering we may assume that", "$$", "\\mathfrak r_1 \\subset \\overline{\\mathfrak q},", "\\ldots,", "\\mathfrak r_r \\subset \\overline{\\mathfrak q}, \\quad", "\\mathfrak r_{r + 1} \\not \\subset \\overline{\\mathfrak q},", "\\ldots,", "\\mathfrak r_n \\not \\subset \\overline{\\mathfrak q}", "$$", "Since $\\overline{\\mathfrak q}$ is a prime ideal we see that the product", "$\\mathfrak r_{r + 1} \\ldots \\mathfrak r_n$ is not contained in", "$\\overline{\\mathfrak q}$ and hence we can pick an element", "$a$ of $\\overline{S}$ contained in", "$\\mathfrak r_{r + 1}, \\ldots, \\mathfrak r_n$ but not in", "$\\overline{\\mathfrak q}$.", "If there exists $g \\in S$ mapping to $a$, then $g$", "works. In general we can find a nonzero element", "$\\lambda \\in \\kappa(\\mathfrak p)$", "such that $\\lambda a$ is the image of a $g \\in S$." ], "refs": [ "algebra-lemma-finite-ass" ], "ref_ids": [ 701 ] } ], "ref_ids": [] }, { "id": 6039, "type": "theorem", "label": "flat-lemma-weak-bourbaki-pre", "categories": [ "flat" ], "title": "flat-lemma-weak-bourbaki-pre", "contents": [ "Let $R \\to S$ be a ring map of finite presentation.", "Let $N$ be a finitely presented $S$-module", "which is flat as an $R$-module. Let $M$ be an $R$-module.", "Let $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak p \\subset R$.", "Then", "$$", "\\mathfrak q \\in \\text{WeakAss}_S(M \\otimes_R N)", "\\Leftrightarrow", "\\Big(", "\\mathfrak p \\in \\text{WeakAss}_R(M)", "\\text{ and }", "\\overline{\\mathfrak q} \\in \\text{Ass}_{\\overline{S}}(\\overline{N})", "\\Big)", "$$", "Here $\\overline{S} = S \\otimes_R \\kappa(\\mathfrak p)$,", "$\\overline{\\mathfrak q} = \\mathfrak q \\overline{S}$, and", "$\\overline{N} = N \\otimes_R \\kappa(\\mathfrak p)$." ], "refs": [], "proofs": [ { "contents": [ "Pick $g \\in S$ as in Lemma \\ref{lemma-weak-bourbaki-pre-pre}.", "Apply Proposition \\ref{proposition-finite-presentation-flat-at-point}", "to the morphism of schemes $\\Spec(S_g) \\to \\Spec(R)$, the quasi-coherent", "module associated to $N_g$, and the points", "corresponding to the primes $\\mathfrak qS_g$ and $\\mathfrak p$. Translating", "into algebra we obtain a commutative diagram of rings", "$$", "\\xymatrix{", "S \\ar[r] & S_g \\ar[r] & S' \\\\", "& R \\ar[lu] \\ar[u] \\ar[r] & R' \\ar[u]", "}", "\\quad\\quad", "\\xymatrix{", "\\mathfrak q \\ar@{-}[r] \\ar@{-}[rd] &", "\\mathfrak qS_g \\ar@{-}[d] \\ar@{-}[r] & \\mathfrak q' \\ar@{-}[d] \\\\", "& \\mathfrak p \\ar@{-}[r] & \\mathfrak p'", "}", "$$", "endowed with primes as shown, the horizontal arrows are \\'etale,", "and $N \\otimes_S S'$ is projective as an $R'$-module. Set", "$N' = N \\otimes_S S'$, $M' = M \\otimes_R R'$,", "$\\overline{S}' = S' \\otimes_{R'} \\kappa(\\mathfrak q')$,", "$\\overline{\\mathfrak q}' = \\mathfrak q' \\overline{S}'$,", "and", "$$", "\\overline{N}' = N' \\otimes_{R'} \\kappa(\\mathfrak p') =", "\\overline{N} \\otimes_{\\overline{S}} \\overline{S}'", "$$", "By Lemma \\ref{lemma-etale-weak-assassin-up-down} we have", "\\begin{align*}", "\\text{WeakAss}_{S'}(M' \\otimes_{R'} N') & =", "(\\Spec(S') \\to \\Spec(S))^{-1}\\text{WeakAss}_S(M \\otimes_R N) \\\\", "\\text{WeakAss}_{R'}(M') & =", "(\\Spec(R') \\to \\Spec(R))^{-1}\\text{WeakAss}_R(M) \\\\", "\\text{Ass}_{\\overline{S}'}(\\overline{N}') & =", "(\\Spec(\\overline{S}') \\to \\Spec(\\overline{S}))^{-1}", "\\text{Ass}_{\\overline{S}}(\\overline{N})", "\\end{align*}", "Use Algebra, Lemma \\ref{algebra-lemma-ass-weakly-ass}", "for $\\overline{N}$ and $\\overline{N}'$. In particular we have", "\\begin{align*}", "\\mathfrak q \\in \\text{WeakAss}_S(M \\otimes_R N)", "& \\Leftrightarrow", "\\mathfrak q' \\in \\text{WeakAss}_{S'}(M' \\otimes_{R'} N') \\\\", "\\mathfrak p \\in \\text{WeakAss}_R(M)", "& \\Leftrightarrow", "\\mathfrak p' \\in \\text{WeakAss}_{R'}(M') \\\\", "\\overline{\\mathfrak q} \\in \\text{Ass}_{\\overline{S}}(\\overline{N})", "& \\Leftrightarrow", "\\overline{\\mathfrak q}' \\in \\text{WeakAss}_{\\overline{S}'}(\\overline{N}')", "\\end{align*}", "Our careful choice of $g$ and the formula for", "$\\text{Ass}_{\\overline{S}'}(\\overline{N}')$ above shows that", "\\begin{equation}", "\\label{equation-key-observation}", "\\text{if }\\mathfrak r' \\in \\text{Ass}_{\\overline{S}'}(\\overline{N}')", "\\text{ lies over }\\mathfrak r \\subset \\overline{S}\\text{ then }", "\\mathfrak r \\subset \\overline{\\mathfrak q}", "\\end{equation}", "This will be a key observation later in the proof. We will use", "the characterization of weakly associated primes given in", "Algebra, Lemma \\ref{algebra-lemma-weakly-ass-local} without further mention.", "\\medskip\\noindent", "Suppose that", "$\\overline{\\mathfrak q} \\not \\in \\text{Ass}_{\\overline{S}}(\\overline{N})$.", "Then", "$\\overline{\\mathfrak q}' \\not \\in \\text{Ass}_{\\overline{S}'}(\\overline{N}')$.", "By", "Algebra, Lemmas \\ref{algebra-lemma-ass-zero-divisors},", "\\ref{algebra-lemma-finite-ass}, and", "\\ref{algebra-lemma-silly}", "there exists an element $\\overline{a}' \\in \\overline{\\mathfrak q}'$", "which is not a zerodivisor on $\\overline{N}'$.", "After replacing $\\overline{a}'$ by $\\lambda \\overline{a}'$ for some nonzero", "$\\lambda \\in \\kappa(\\mathfrak p)$ we can find", "$a' \\in \\mathfrak q'$ mapping to $\\overline{a}'$. By", "Lemma \\ref{lemma-invert-universally-injective}", "the map $a' : N'_{\\mathfrak p'} \\to N'_{\\mathfrak p'}$ is", "$R'_{\\mathfrak p'}$-universally injective. In particular", "we see that $a' : M' \\otimes_{R'} N' \\to M' \\otimes_{R'} N'$ is", "injective after localizing at $\\mathfrak p'$ and hence after", "localizing at $\\mathfrak q'$. Clearly this implies that", "$\\mathfrak q' \\not \\in \\text{WeakAss}_{S'}(M' \\otimes_{R'} N')$.", "We conclude that $\\mathfrak q \\in \\text{WeakAss}_S(M \\otimes_R N)$ implies", "$\\overline{\\mathfrak q} \\in \\text{Ass}_{\\overline{S}}(\\overline{N})$.", "\\medskip\\noindent", "Assume $\\mathfrak q \\in \\text{WeakAss}_S(M \\otimes_R N)$. We want", "to show $\\mathfrak p \\in \\text{WeakAss}_S(M)$.", "Let $z \\in M \\otimes_R N$ be an element such that $\\mathfrak q$", "is minimal over $J = \\text{Ann}_S(z)$.", "Let $f_i \\in \\mathfrak p$, $i \\in I$ be a set of generators of the", "ideal $\\mathfrak p$. Since $\\mathfrak q$ lies over $\\mathfrak p$, for every $i$", "we can choose an $n_i \\geq 1$ and $g_i \\in S$, $g_i \\not \\in \\mathfrak q$", "with $g_i f_i^{n_i} \\in J$, i.e., $g_i f_i^{n_i} z = 0$.", "Let $z' \\in (M' \\otimes_{R'} N')_{\\mathfrak p'}$ be the image of $z$.", "Observe that $z'$ is nonzero because $z$ has nonzero image in", "$(M \\otimes_R N)_\\mathfrak q$ and because $S_\\mathfrak q \\to S'_{\\mathfrak q'}$", "is faithfully flat. We claim that $f_i^{n_i} z' = 0$.", "\\medskip\\noindent", "Proof of the claim: Let $g'_i \\in S'$ be the image of $g_i$.", "By the key observation (\\ref{equation-key-observation})", "we find that the image $\\overline{g}'_i \\in \\overline{S}'$", "is not contained in $\\mathfrak r'$ for any", "$\\mathfrak r' \\in \\text{Ass}_{\\overline{S}'}(\\overline{N})$.", "Hence by Lemma \\ref{lemma-invert-universally-injective}", "we see that $g'_i : N'_{\\mathfrak p'} \\to N'_{\\mathfrak p'}$ is", "$R'_{\\mathfrak p'}$-universally injective. In particular", "we see that $g'_i : M' \\otimes_{R'} N' \\to M' \\otimes_{R'} N'$ is", "injective after localizating at $\\mathfrak p'$. The claim", "follows because $g_i f_i^{n_i} z' = 0$.", "\\medskip\\noindent", "Our claim shows that the annihilator of $z'$ in $R'_{\\mathfrak p'}$", "contains the elements $f_i^{n_i}$. As $R \\to R'$ is \\'etale we have", "$\\mathfrak p'R'_{\\mathfrak p'} = \\mathfrak pR'_{\\mathfrak p'}$", "by Algebra, Lemma \\ref{algebra-lemma-etale-at-prime}.", "Hence the annihilator of $z'$ in $R'_{\\mathfrak p'}$ has radical equal to", "$\\mathfrak p' R_{\\mathfrak p'}$ (here we use $z'$ is not zero).", "On the other hand", "$$", "z' \\in (M' \\otimes_{R'} N')_{\\mathfrak p'} =", "M'_{\\mathfrak p'} \\otimes_{R'_{\\mathfrak p'}} N'_{\\mathfrak p'}", "$$", "The module $N'_{\\mathfrak p'}$ is projective over the local ring", "$R'_{\\mathfrak p'}$ and hence free", "(Algebra, Theorem \\ref{algebra-theorem-projective-free-over-local-ring}).", "Thus we can find a finite free direct summand $F' \\subset N'_{\\mathfrak p'}$", "such that $z' \\in M'_{\\mathfrak p'} \\otimes_{R'_{\\mathfrak p'}} F'$.", "If $F'$ has rank $n$, then we deduce that", "$\\mathfrak p' R'_{\\mathfrak p'} \\in", "\\text{WeakAss}_{R'_{\\mathfrak p'}}({M'_{\\mathfrak p'}}^{\\oplus n})$.", "This implies", "$\\mathfrak p'R'_{\\mathfrak p'} \\in \\text{WeakAss}(M'_{\\mathfrak p'})$", "for example by Algebra, Lemma \\ref{algebra-lemma-weakly-ass}.", "Then $\\mathfrak p' \\in \\text{WeakAss}_{R'}(M')$", "which in turn gives $\\mathfrak p \\in \\text{WeakAss}_R(M)$.", "This finishes the proof of the implication", "``$\\Rightarrow$'' of the equivalence of the lemma.", "\\medskip\\noindent", "Assume that $\\mathfrak p \\in \\text{WeakAss}_R(M)$ and", "$\\overline{\\mathfrak q} \\in \\text{Ass}_{\\overline{S}}(\\overline{N})$.", "We want to show that $\\mathfrak q$ is weakly associated to $M \\otimes_R N$.", "Note that $\\overline{\\mathfrak q}'$ is a maximal element of", "$\\text{Ass}_{\\overline{S}'}(\\overline{N}')$.", "This is a consequence of (\\ref{equation-key-observation})", "and the fact that there are no inclusions among the primes", "of $\\overline{S}'$ lying over $\\overline{\\mathfrak q}$", "(as fibres of \\'etale morphisms are discrete", "Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}).", "Thus, after replacing $R, S, \\mathfrak p, \\mathfrak q, M, N$ by", "$R', S', \\mathfrak p', \\mathfrak q', M', N'$", "we may assume, in addition to the assumptions of the lemma, that", "\\begin{enumerate}", "\\item $\\mathfrak p \\in \\text{WeakAss}_R(M)$,", "\\item $\\overline{\\mathfrak q} \\in \\text{Ass}_{\\overline{S}}(\\overline{N})$,", "\\item $N$ is projective as an $R$-module, and", "\\item $\\overline{\\mathfrak q}$ is maximal in", "$\\text{Ass}_{\\overline{S}}(\\overline{N})$.", "\\end{enumerate}", "There is one more reduction, namely, we may replace", "$R, S, M, N$ by their localizations at $\\mathfrak p$.", "This leads to one more condition, namely,", "\\begin{enumerate}", "\\item[(5)] $R$ is a local ring with maximal ideal $\\mathfrak p$.", "\\end{enumerate}", "We will finish by showing that (1) -- (5) imply", "$\\mathfrak q \\in \\text{WeakAss}(M \\otimes_R N)$.", "\\medskip\\noindent", "Since $R$ is local and $\\mathfrak p \\in \\text{WeakAss}_R(M)$", "we can pick a $y \\in M$ whose annihilator $I$ has radical", "equal to $\\mathfrak p$.", "Write $\\overline{\\mathfrak q} = (\\overline{g}_1, \\ldots, \\overline{g}_n)$", "for some $\\overline{g}_i \\in \\overline{S}$. Choose $g_i \\in S$", "mapping to $\\overline{g}_i$.", "Then $\\mathfrak q = \\mathfrak pS + g_1S + \\ldots + g_nS$.", "Consider the map", "$$", "\\Psi : N/IN \\longrightarrow (N/IN)^{\\oplus n}, \\quad", "z \\longmapsto (g_1z, \\ldots, g_nz).", "$$", "This is a homomorphism of projective $R/I$-modules.", "The local ring $R/I$ is auto-associated", "(More on Algebra, Definition \\ref{more-algebra-definition-auto-ass})", "as $\\mathfrak p/I$ is locally nilpotent.", "The map $\\Psi \\otimes \\kappa(\\mathfrak p)$ is not injective, because", "$\\overline{\\mathfrak q} \\in \\text{Ass}_{\\overline{S}}(\\overline{N})$.", "Hence More on Algebra, Lemma \\ref{more-algebra-lemma-P-fPD-zero}", "implies $\\Psi$ is not injective. Pick $z \\in N/IN$", "nonzero in the kernel of $\\Psi$. The annihilator $J = \\text{Ann}_S(z)$", "contains $IS$ and $g_i$ by construction. Thus", "$\\sqrt{J} \\subset S$ contains $\\mathfrak q$.", "Let $\\mathfrak s \\subset S$ be a prime minimal over $J$.", "Then $\\mathfrak q \\subset \\mathfrak s$,", "$\\mathfrak s$ lies over $\\mathfrak p$, and", "$\\mathfrak s \\in \\text{WeakAss}_S(N/IN)$.", "The last fact by definition of weakly associated primes.", "Apply the ``$\\Rightarrow$'' part of the lemma (which we've already proven)", "to the ring map $R \\to S$ and the modules $R/I$ and $N$", "to conclude that", "$\\overline{\\mathfrak s} \\in \\text{Ass}_{\\overline{S}}(\\overline{N})$.", "Since $\\overline{\\mathfrak q} \\subset \\overline{\\mathfrak s}$", "the maximality of $\\overline{\\mathfrak q}$, see condition (4) above,", "implies that $\\overline{\\mathfrak q} = \\overline{\\mathfrak s}$.", "This shows that $\\mathfrak q = \\mathfrak s$ and we conlude", "what we want." ], "refs": [ "flat-lemma-weak-bourbaki-pre-pre", "flat-proposition-finite-presentation-flat-at-point", "flat-lemma-etale-weak-assassin-up-down", "algebra-lemma-ass-weakly-ass", "algebra-lemma-weakly-ass-local", "algebra-lemma-ass-zero-divisors", "algebra-lemma-finite-ass", "algebra-lemma-silly", "flat-lemma-invert-universally-injective", "flat-lemma-invert-universally-injective", "algebra-lemma-etale-at-prime", "algebra-theorem-projective-free-over-local-ring", "algebra-lemma-weakly-ass", "morphisms-lemma-etale-over-field", "more-algebra-definition-auto-ass", "more-algebra-lemma-P-fPD-zero" ], "ref_ids": [ 6038, 6200, 5985, 727, 720, 704, 701, 378, 6007, 6007, 1233, 322, 722, 5364, 10599, 9890 ] } ], "ref_ids": [] }, { "id": 6040, "type": "theorem", "label": "flat-lemma-bourbaki-finite-type-general-base-at-point", "categories": [ "flat" ], "title": "flat-lemma-bourbaki-finite-type-general-base-at-point", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to S$ be locally of finite type.", "Let $x \\in X$ with image $s \\in S$.", "Let $\\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$.", "Let $\\mathcal{G}$ be a quasi-coherent sheaf on $S$.", "If $\\mathcal{F}$ is flat at $x$ over $S$, then", "$$", "x \\in \\text{WeakAss}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G})", "\\Leftrightarrow", "s \\in \\text{WeakAss}_S(\\mathcal{G})", "\\text{ and }", "x \\in \\text{Ass}_{X_s}(\\mathcal{F}_s).", "$$" ], "refs": [], "proofs": [ { "contents": [ "In this paragraph we reduce to $f$ being of finite presentation.", "The question is local on $X$ and $S$, hence we may assume $X$ and $S$", "are affine. Write $X = \\Spec(B)$, $S = \\Spec(A)$ and write", "$B = A[x_1, \\ldots, x_n]/I$. In other words we obtain a closed immersion", "$i : X \\to \\mathbf{A}^n_S$ over $S$. Denote $t = i(x) \\in \\mathbf{A}^n_S$.", "Note that $i_*\\mathcal{F}$ is a finite type quasi-coherent sheaf on", "$\\mathbf{A}^n_S$ which is flat at $t$ over $S$ and note that", "$$", "i_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}) =", "i_*\\mathcal{F} \\otimes_{\\mathcal{O}_{\\mathbf{A}^n_S}} p^*\\mathcal{G}", "$$", "where $p : \\mathbf{A}^n_S \\to S$ is the projection. Note that", "$t$ is a weakly associated point of", "$i_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G})$", "if and only if $x$ is a weakly associated point of", "$\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}$, see", "Divisors, Lemma \\ref{divisors-lemma-weakly-associated-finite}.", "Similarly $x \\in \\text{Ass}_{X_s}(\\mathcal{F}_s)$ if and only", "if $t \\in \\text{Ass}_{\\mathbf{A}^n_s}((i_*\\mathcal{F})_s)$ (see", "Algebra, Lemma \\ref{algebra-lemma-ass-quotient-ring}).", "Hence it suffices to prove the lemma in case $X = \\mathbf{A}^n_S$.", "Thus we may assume that $X \\to S$ is of finite presentation.", "\\medskip\\noindent", "In this paragraph we reduce to $\\mathcal{F}$ being of finite presentation", "and flat over $S$.", "Choose an elementary \\'etale neighbourhood $e : (S', s') \\to (S, s)$", "and an open $V \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})$", "as in Proposition \\ref{proposition-finite-type-flat-at-point}.", "Let $x' \\in X' = X \\times_S S'$ be the unique point mapping to $x$", "and $s'$. Then it suffices to prove the statement for", "$X' \\to S'$, $x'$, $s'$, $(X' \\to X)^*\\mathcal{F}$, and $e^*\\mathcal{G}$, see", "Lemma \\ref{lemma-etale-weak-assassin-up-down}.", "Let $v \\in V$ the unique point mapping to $x'$", "and let $s' \\in \\Spec(\\mathcal{O}_{S', s'})$ be the closed point.", "Then $\\mathcal{O}_{V, v} = \\mathcal{O}_{X', x'}$", "and $\\mathcal{O}_{\\Spec(\\mathcal{O}_{S', s'}), s'} =", "\\mathcal{O}_{S', s'}$ and similarly for the stalks of pullbacks of", "$\\mathcal{F}$ and $\\mathcal{G}$.", "Also $V_{s'} \\subset X'_{s'}$ is an open subscheme.", "Since the condition of being a weakly associated point", "depend only on the stalk of the sheaf, we may", "replace", "$X' \\to S'$, $x'$, $s'$, $(X' \\to X)^*\\mathcal{F}$, and $e^*\\mathcal{G}$", "by", "$V \\to \\Spec(\\mathcal{O}_{S', s'})$, $v$, $s'$, $(V \\to X)^*\\mathcal{F}$,", "and $(\\Spec(\\mathcal{O}_{S', s'}) \\to S)^*\\mathcal{G}$.", "Thus we may assume that $f$ is of finite presentation and", "$\\mathcal{F}$ of finite presentation and flat over $S$.", "\\medskip\\noindent", "Assume $f$ is of finite presentation and", "$\\mathcal{F}$ of finite presentation and flat over $S$.", "After shrinking $X$ and $S$ to affine neighbourhoods", "of $x$ and $s$, this case is handled by", "Lemma \\ref{lemma-weak-bourbaki-pre}." ], "refs": [ "divisors-lemma-weakly-associated-finite", "algebra-lemma-ass-quotient-ring", "flat-proposition-finite-type-flat-at-point", "flat-lemma-etale-weak-assassin-up-down", "flat-lemma-weak-bourbaki-pre" ], "ref_ids": [ 7885, 708, 6199, 5985, 6039 ] } ], "ref_ids": [] }, { "id": 6041, "type": "theorem", "label": "flat-lemma-weak-bourbaki", "categories": [ "flat" ], "title": "flat-lemma-weak-bourbaki", "contents": [ "Let $R \\to S$ be a ring map which is essentially of finite type.", "Let $N$ be a localization of a finite $S$-module flat over $R$.", "Let $M$ be an $R$-module. Then", "$$", "\\text{WeakAss}_S(M \\otimes_R N)", "=", "\\bigcup\\nolimits_{\\mathfrak p \\in \\text{WeakAss}_R(M)}", "\\text{Ass}_{S \\otimes_R \\kappa(\\mathfrak p)}(N \\otimes_R \\kappa(\\mathfrak p))", "$$" ], "refs": [], "proofs": [ { "contents": [ "This lemma is a translation of", "Lemma \\ref{lemma-bourbaki-finite-type-general-base-at-point}", "into algebra. Details of translation omitted." ], "refs": [ "flat-lemma-bourbaki-finite-type-general-base-at-point" ], "ref_ids": [ 6040 ] } ], "ref_ids": [] }, { "id": 6042, "type": "theorem", "label": "flat-lemma-bourbaki-finite-type-general-base", "categories": [ "flat" ], "title": "flat-lemma-bourbaki-finite-type-general-base", "contents": [ "Let $f : X \\to S$ be a morphism which is locally of finite type.", "Let $\\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$", "which is flat over $S$. Let $\\mathcal{G}$ be a quasi-coherent sheaf on $S$.", "Then we have", "$$", "\\text{WeakAss}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}) =", "\\bigcup\\nolimits_{s \\in \\text{WeakAss}_S(\\mathcal{G})}", "\\text{Ass}_{X_s}(\\mathcal{F}_s)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Lemma \\ref{lemma-bourbaki-finite-type-general-base-at-point}." ], "refs": [ "flat-lemma-bourbaki-finite-type-general-base-at-point" ], "ref_ids": [ 6040 ] } ], "ref_ids": [] }, { "id": 6043, "type": "theorem", "label": "flat-lemma-finite-type-flat-algebra", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-algebra", "contents": [ "Let $R \\to S$ be a ring map of finite presentation.", "Let $M$ be a finite $S$-module. Assume $\\text{WeakAss}_S(S)$ is finite.", "Then", "$$", "U = \\{\\mathfrak q \\subset S \\mid M_{\\mathfrak q}\\text{ flat over }R\\}", "$$", "is open in $\\Spec(S)$ and for every $g \\in S$ such that", "$D(g) \\subset U$ the localization $M_g$ is a finitely presented", "$S_g$-module flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Theorem \\ref{theorem-finite-type-flat}." ], "refs": [ "flat-theorem-finite-type-flat" ], "ref_ids": [ 5968 ] } ], "ref_ids": [] }, { "id": 6044, "type": "theorem", "label": "flat-lemma-finite-type-flat-X", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-X", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite", "type. Assume the set of weakly associated points of $S$ is locally finite", "in $S$. Then the set of points $x \\in X$ where $f$ is flat is an open", "subscheme $U \\subset X$ and $U \\to S$ is flat and locally of finite", "presentation." ], "refs": [], "proofs": [ { "contents": [ "The problem is local on $X$ and $S$, hence we may assume that", "$X$ and $S$ are affine. Then $X \\to S$ corresponds to a finite type", "ring map $A \\to B$. Choose a surjection $A[x_1, \\ldots, x_n] \\to B$", "and consider $B$ as an $A[x_1, \\ldots, x_n]$-module. An application of", "Lemma \\ref{lemma-finite-type-flat-algebra}", "finishes the proof." ], "refs": [ "flat-lemma-finite-type-flat-algebra" ], "ref_ids": [ 6043 ] } ], "ref_ids": [] }, { "id": 6045, "type": "theorem", "label": "flat-lemma-finite-type-flat-over-integral", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-over-integral", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is", "locally of finite type and flat. If $S$ is integral, then $f$", "is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Special case of", "Lemma \\ref{lemma-finite-type-flat-X}." ], "refs": [ "flat-lemma-finite-type-flat-X" ], "ref_ids": [ 6044 ] } ], "ref_ids": [] }, { "id": 6046, "type": "theorem", "label": "flat-lemma-explain-why-pure", "categories": [ "flat" ], "title": "flat-lemma-explain-why-pure", "contents": [ "Let $R$ be a local ring with maximal ideal $\\mathfrak m$.", "Let $R \\to S$ be a ring map. Let $N$ be an $S$-module.", "Assume", "\\begin{enumerate}", "\\item $N$ is projective as an $R$-module, and", "\\item $S/\\mathfrak mS$ is Noetherian and $N/\\mathfrak mN$ is a finite", "$S/\\mathfrak mS$-module.", "\\end{enumerate}", "Then for any prime $\\mathfrak q \\subset S$ which is an associated prime of", "$N \\otimes_R \\kappa(\\mathfrak p)$ where $\\mathfrak p = R \\cap \\mathfrak q$", "we have $\\mathfrak q + \\mathfrak m S \\not = S$." ], "refs": [], "proofs": [ { "contents": [ "Note that the hypotheses of", "Lemmas \\ref{lemma-homothety-spectrum} and", "\\ref{lemma-invert-universally-injective}", "are satisfied. We will use the conclusions of these lemmas without further", "mention. Let $\\Sigma \\subset S$ be the multiplicative set of elements", "which are not zerodivisors on $N/\\mathfrak mN$. The map", "$N \\to \\Sigma^{-1}N$ is $R$-universally injective. Hence we see that", "any $\\mathfrak q \\subset S$ which is an associated prime of", "$N \\otimes_R \\kappa(\\mathfrak p)$ is also an associated prime of", "$\\Sigma^{-1}N \\otimes_R \\kappa(\\mathfrak p)$. Clearly this implies that", "$\\mathfrak q$ corresponds to a prime of $\\Sigma^{-1}S$.", "Thus $\\mathfrak q \\subset \\mathfrak q'$ where $\\mathfrak q'$", "corresponds to an associated prime of $N/\\mathfrak mN$ and we win." ], "refs": [ "flat-lemma-homothety-spectrum", "flat-lemma-invert-universally-injective" ], "ref_ids": [ 6002, 6007 ] } ], "ref_ids": [] }, { "id": 6047, "type": "theorem", "label": "flat-lemma-explain-why-pure-complete", "categories": [ "flat" ], "title": "flat-lemma-explain-why-pure-complete", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal.", "Let $R \\to S$ be a ring map. Let $N$ be an $S$-module.", "If $N$ is $I$-adically complete, then for any $R$-module $M$ and", "for any prime $\\mathfrak q \\subset S$ which is an associated prime of", "$N \\otimes_R M$ we have $\\mathfrak q + I S \\not = S$." ], "refs": [], "proofs": [ { "contents": [ "Let $S^\\wedge$ denote the $I$-adic completion of $S$.", "Note that $N$ is an $S^\\wedge$-module, hence also", "$N \\otimes_R M$ is an $S^\\wedge$-module.", "Let $z \\in N \\otimes_R M$ be an element such that", "$\\mathfrak q = \\text{Ann}_S(z)$. Since $z \\not = 0$ we see", "that $\\text{Ann}_{S^\\wedge}(z) \\not = S^\\wedge$. Hence", "$\\mathfrak q S^\\wedge \\not = S^\\wedge$. Hence there exists a", "maximal ideal $\\mathfrak m \\subset S^\\wedge$ with", "$\\mathfrak q S^\\wedge \\subset \\mathfrak m$. Since", "$IS^\\wedge \\subset \\mathfrak m$ by", "Algebra, Lemma \\ref{algebra-lemma-radical-completion}", "we win." ], "refs": [ "algebra-lemma-radical-completion" ], "ref_ids": [ 862 ] } ], "ref_ids": [] }, { "id": 6048, "type": "theorem", "label": "flat-lemma-explain-why-pure-direct-sum-finite-modules", "categories": [ "flat" ], "title": "flat-lemma-explain-why-pure-direct-sum-finite-modules", "contents": [ "Let $R$ be a local ring with maximal ideal $\\mathfrak m$.", "Let $R \\to S$ be a ring map. Let $N$ be an $S$-module.", "Assume $N$ is isomorphic as an $R$-module to a direct", "sum of finite $R$-modules. Then for any $R$-module $M$ and", "for any prime $\\mathfrak q \\subset S$ which is an associated prime of", "$N \\otimes_R M$ we have $\\mathfrak q + \\mathfrak m S \\not = S$." ], "refs": [], "proofs": [ { "contents": [ "Write $N = \\bigoplus_{i \\in I} M_i$ with each $M_i$ a finite $R$-module.", "Let $M$ be an $R$-module and let $\\mathfrak q \\subset S$ be an associated", "prime of $N \\otimes_R M$ such that $\\mathfrak q + \\mathfrak m S = S$. Let", "$z \\in N \\otimes_R M$ be an element with $\\mathfrak q = \\text{Ann}_S(z)$.", "After modifying the direct sum decomposition a little bit we may assume that", "$z \\in M_1 \\otimes_R M$ for some element $1 \\in I$. Write", "$1 = f + \\sum x_j g_j$ for some $f \\in \\mathfrak q$, $x_j \\in \\mathfrak m$,", "and $g_j \\in S$. For any $g \\in S$ denote $g'$ the $R$-linear map", "$$", "M_1 \\to N \\xrightarrow{g} N \\to M_1", "$$", "where the first arrow is the inclusion map, the second arrow is multiplication", "by $g$ and the third arrow is the projection map. Because each $x_j \\in R$", "we obtain the equality", "$$", "f' + \\sum x_j g'_j = \\text{id}_{M_1} \\in \\text{End}_R(M_1)", "$$", "By Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK})", "we see that $f'$ is surjective, hence by", "Algebra, Lemma \\ref{algebra-lemma-fun}", "we see that $f'$ is an isomorphism. In particular the map", "$$", "M_1 \\otimes_R M \\to N \\otimes_R M \\xrightarrow{f} N \\otimes_R M", "\\to M_1 \\otimes_R M", "$$", "is an isomorphism. This contradicts the assumption that $fz = 0$." ], "refs": [ "algebra-lemma-NAK", "algebra-lemma-fun" ], "ref_ids": [ 401, 388 ] } ], "ref_ids": [] }, { "id": 6049, "type": "theorem", "label": "flat-lemma-explain-why-pure-ML", "categories": [ "flat" ], "title": "flat-lemma-explain-why-pure-ML", "contents": [ "Let $R$ be a henselian local ring with maximal ideal $\\mathfrak m$.", "Let $R \\to S$ be a ring map. Let $N$ be an $S$-module.", "Assume $N$ is countably generated and Mittag-Leffler as an $R$-module.", "Then for any $R$-module $M$ and for any prime $\\mathfrak q \\subset S$", "which is an associated prime of $N \\otimes_R M$ we have", "$\\mathfrak q + \\mathfrak m S \\not = S$." ], "refs": [], "proofs": [ { "contents": [ "This lemma reduces to", "Lemma \\ref{lemma-explain-why-pure-direct-sum-finite-modules}", "by", "Algebra, Lemma \\ref{algebra-lemma-split-ML-henselian}." ], "refs": [ "flat-lemma-explain-why-pure-direct-sum-finite-modules", "algebra-lemma-split-ML-henselian" ], "ref_ids": [ 6048, 1286 ] } ], "ref_ids": [] }, { "id": 6050, "type": "theorem", "label": "flat-lemma-impure-finite-presentation", "categories": [ "flat" ], "title": "flat-lemma-impure-finite-presentation", "contents": [ "In Situation \\ref{situation-pre-pure}.", "If there exists an impurity of $\\mathcal{F}$ above $s$, then", "there exists an impurity $(g : T \\to S, t' \\leadsto t, \\xi)$", "of $\\mathcal{F}$ above $s$ such that $g$ is locally of finite", "presentation and $t$ a closed point of the fibre of $g$ above $s$." ], "refs": [], "proofs": [ { "contents": [ "Let $(g : T \\to S, t' \\leadsto t, \\xi)$ be any impurity of", "$\\mathcal{F}$ above $s$. We apply", "Limits, Lemma \\ref{limits-lemma-separate}", "to $t \\in T$ and $Z = \\overline{\\{\\xi\\}}$ to obtain an open neighbourhood", "$V \\subset T$ of $t$, a commutative diagram", "$$", "\\xymatrix{", "V \\ar[d] \\ar[r]_a & T' \\ar[d]^b \\\\", "T \\ar[r]^g & S,", "}", "$$", "and a closed subscheme $Z' \\subset X_{T'}$ such that", "\\begin{enumerate}", "\\item the morphism $b : T' \\to S$ is locally of finite presentation,", "\\item we have $Z' \\cap X_{a(t)} = \\emptyset$, and", "\\item $Z \\cap X_V$ maps into $Z'$ via the morphism $X_V \\to X_{T'}$.", "\\end{enumerate}", "As $t'$ specializes to $t$ we may replace $T$ by the open neighbourhood", "$V$ of $t$. Thus we have a commutative diagram", "$$", "\\xymatrix{", "X_T \\ar[d] \\ar[r] &", "X_{T'} \\ar[d] \\ar[r] &", "X \\ar[d] \\\\", "T \\ar[r]^a & T' \\ar[r]^b & S", "}", "$$", "where $b \\circ a = g$. Let $\\xi' \\in X_{T'}$ denote the", "image of $\\xi$. By", "Divisors, Lemma \\ref{divisors-lemma-base-change-relative-assassin}", "we see that $\\xi' \\in \\text{Ass}_{X_{T'}/T'}(\\mathcal{F}_{T'})$.", "Moreover, by construction the closure of $\\overline{\\{\\xi'\\}}$", "is contained in the closed subset $Z'$ which avoids the fibre", "$X_{a(t)}$. In this way we see that $(T' \\to S, a(t') \\leadsto a(t), \\xi')$", "is an impurity of $\\mathcal{F}$ above $s$.", "\\medskip\\noindent", "Thus we may assume that $g : T \\to S$ is locally of finite presentation.", "Let $Z = \\overline{\\{\\xi\\}}$. By assumption $Z_t = \\emptyset$. By", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-empty-generic-fibre}", "this means that $Z_{t''} = \\emptyset$ for $t''$ in an open subset", "of $\\overline{\\{t\\}}$. Since the fibre of", "$T \\to S$ over $s$ is a Jacobson scheme, see", "Morphisms, Lemma \\ref{morphisms-lemma-ubiquity-Jacobson-schemes}", "we find that there exist a closed point $t'' \\in \\overline{\\{t\\}}$ such that", "$Z_{t''} = \\emptyset$. Then $(g : T \\to S, t' \\leadsto t'', \\xi)$ is the", "desired impurity." ], "refs": [ "limits-lemma-separate", "divisors-lemma-base-change-relative-assassin", "more-morphisms-lemma-empty-generic-fibre", "morphisms-lemma-ubiquity-Jacobson-schemes" ], "ref_ids": [ 15094, 7890, 13800, 5213 ] } ], "ref_ids": [] }, { "id": 6051, "type": "theorem", "label": "flat-lemma-impure-limit", "categories": [ "flat" ], "title": "flat-lemma-impure-limit", "contents": [ "In Situation \\ref{situation-pre-pure}.", "Let $(g : T \\to S, t' \\leadsto t, \\xi)$ be an impurity of", "$\\mathcal{F}$ above $s$. Assume $T = \\lim_{i \\in I} T_i$", "is a directed limit of affine schemes over $S$. Then for", "some $i$ the triple $(T_i \\to S, t'_i \\leadsto t_i, \\xi_i)$", "is an impurity of $\\mathcal{F}$ above $s$." ], "refs": [], "proofs": [ { "contents": [ "The notation in the statement means this: Let $p_i : T \\to T_i$", "be the projection morphisms, let $t_i = p_i(t)$ and $t'_i = p_i(t')$.", "Finally $\\xi_i \\in X_{T_i}$ is the image of $\\xi$. By", "Divisors, Lemma \\ref{divisors-lemma-base-change-relative-assassin}", "it is true that $\\xi_i$ is a point of the relative", "assassin of $\\mathcal{F}_{T_i}$ over $T_i$. Thus the only point is to", "show that $\\overline{\\{\\xi_i\\}} \\cap X_{t_i} = \\emptyset$ for some $i$.", "\\medskip\\noindent", "First proof. Let $Z_i = \\overline{\\{\\xi_i\\}} \\subset X_{T_i}$", "and $Z = \\overline{\\{\\xi\\}} \\subset X_T$", "endowed with the reduced induced scheme structure.", "Then $Z = \\lim Z_i$ by", "Limits, Lemma \\ref{limits-lemma-inverse-limit-irreducibles}.", "Choose a field $k$ and a morphism $\\Spec(k) \\to T$ whose image is $t$.", "Then", "$$", "\\emptyset =", "Z \\times_T \\Spec(k) = (\\lim Z_i) \\times_{(\\lim T_i)} \\Spec(k)", "= \\lim Z_i \\times_{T_i} \\Spec(k)", "$$", "because limits commute with fibred products (limits commute with limits).", "Each $Z_i \\times_{T_i} \\Spec(k)$ is quasi-compact because $X_{T_i} \\to T_i$", "is of finite type and hence $Z_i \\to T_i$ is of finite type.", "Hence $Z_i \\times_{T_i} \\Spec(k)$ is empty for some $i$ by", "Limits, Lemma \\ref{limits-lemma-limit-nonempty}.", "Since the image of the composition $\\Spec(k) \\to T \\to T_i$ is $t_i$", "we obtain what we want.", "\\medskip\\noindent", "Second proof. Set $Z = \\overline{\\{\\xi\\}}$. Apply", "Limits, Lemma \\ref{limits-lemma-separate}", "to this situation to obtain an open neighbourhood", "$V \\subset T$ of $t$, a commutative diagram", "$$", "\\xymatrix{", "V \\ar[d] \\ar[r]_a & T' \\ar[d]^b \\\\", "T \\ar[r]^g & S,", "}", "$$", "and a closed subscheme $Z' \\subset X_{T'}$ such that", "\\begin{enumerate}", "\\item the morphism $b : T' \\to S$ is locally of finite presentation,", "\\item we have $Z' \\cap X_{a(t)} = \\emptyset$, and", "\\item $Z \\cap X_V$ maps into $Z'$ via the morphism $X_V \\to X_{T'}$.", "\\end{enumerate}", "We may assume $V$ is an affine open of $T$, hence by", "Limits, Lemmas \\ref{limits-lemma-descend-opens} and", "\\ref{limits-lemma-limit-affine}", "we can find an $i$ and an affine open $V_i \\subset T_i$ with", "$V = f_i^{-1}(V_i)$. By", "Limits,", "Proposition \\ref{limits-proposition-characterize-locally-finite-presentation}", "after possibly increasing $i$ a bit we can find a morphism", "$a_i : V_i \\to T'$ such that $a = a_i \\circ f_i|_V$.", "The induced morphism $X_{V_i} \\to X_{T'}$ maps $\\xi_i$ into", "$Z'$. As $Z' \\cap X_{a(t)} = \\emptyset$ we conclude that", "$(T_i \\to S, t'_i \\leadsto t_i, \\xi_i)$ is an impurity of", "$\\mathcal{F}$ above $s$." ], "refs": [ "divisors-lemma-base-change-relative-assassin", "limits-lemma-inverse-limit-irreducibles", "limits-lemma-limit-nonempty", "limits-lemma-separate", "limits-lemma-descend-opens", "limits-lemma-limit-affine", "limits-proposition-characterize-locally-finite-presentation" ], "ref_ids": [ 7890, 15035, 15034, 15094, 15041, 15043, 15127 ] } ], "ref_ids": [] }, { "id": 6052, "type": "theorem", "label": "flat-lemma-quasi-finite-impurity-elementary", "categories": [ "flat" ], "title": "flat-lemma-quasi-finite-impurity-elementary", "contents": [ "In Situation \\ref{situation-pre-pure}.", "If there exists an impurity $(g : T \\to S, t' \\leadsto t, \\xi)$", "of $\\mathcal{F}$ above $s$ with $g$ quasi-finite at $t$, then there", "exists an impurity $(g : T \\to S, t' \\leadsto t, \\xi)$ such that", "$(T, t) \\to (S, s)$ is an elementary \\'etale neighbourhood." ], "refs": [], "proofs": [ { "contents": [ "Let $(g : T \\to S, t' \\leadsto t, \\xi)$ be an impurity of", "$\\mathcal{F}$ above $s$ such that $g$ is quasi-finite at $t$.", "After shrinking $T$ we may assume that $g$ is locally of finite type.", "Apply", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point}", "to $T \\to S$ and $t \\mapsto s$. This gives us a diagram", "$$", "\\xymatrix{", "T \\ar[d] & T \\times_S U \\ar[l] \\ar[d] & V \\ar[l] \\ar[ld] \\\\", "S & U \\ar[l]", "}", "$$", "where $(U, u) \\to (S, s)$ is an elementary \\'etale neighbourhood", "and $V \\subset T \\times_S U$ is an open neighbourhood of $v = (t, u)$", "such that $V \\to U$ is finite and such that $v$ is the unique point of $V$", "lying over $u$. Since the morphism $V \\to T$ is \\'etale", "hence flat we see that there exists a specialization $v' \\leadsto v$ such", "that $v' \\mapsto t'$. Note that $\\kappa(t') \\subset \\kappa(v')$", "is finite separable. Pick any point $\\zeta \\in X_{v'}$ mapping to", "$\\xi \\in X_{t'}$. By", "Divisors, Lemma \\ref{divisors-lemma-base-change-relative-assassin}", "we see that $\\zeta \\in \\text{Ass}_{X_V/V}(\\mathcal{F}_V)$.", "Moreover, the closure $\\overline{\\{\\zeta\\}}$ does not meet", "the fibre $X_v$ as by assumption the closure $\\overline{\\{\\xi\\}}$", "does not meet $X_t$. In other words $(V \\to S, v' \\leadsto v, \\zeta)$", "is an impurity of $\\mathcal{F}$ above $S$.", "\\medskip\\noindent", "Next, let $u' \\in U'$ be the image of $v'$ and let", "$\\theta \\in X_U$ be the image of $\\zeta$.", "Then $\\theta \\mapsto u'$ and $u' \\leadsto u$.", "By", "Divisors, Lemma \\ref{divisors-lemma-base-change-relative-assassin}", "we see that $\\theta \\in \\text{Ass}_{X_U/U}(\\mathcal{F})$.", "Moreover, as $\\pi : X_V \\to X_U$ is finite we see that", "$\\pi\\big(\\overline{\\{\\zeta\\}}\\big) = \\overline{\\{\\pi(\\zeta)\\}}$. Since", "$v$ is the unique point of $V$ lying over $u$ we see that", "$X_u \\cap \\overline{\\{\\pi(\\zeta)\\}} = \\emptyset$ because", "$X_v \\cap \\overline{\\{\\zeta\\}} = \\emptyset$. In this way we conclude that", "$(U \\to S, u' \\leadsto u, \\theta)$ is an impurity of", "$\\mathcal{F}$ above $s$ and we win." ], "refs": [ "more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point", "divisors-lemma-base-change-relative-assassin", "divisors-lemma-base-change-relative-assassin" ], "ref_ids": [ 13892, 7890, 7890 ] } ], "ref_ids": [] }, { "id": 6053, "type": "theorem", "label": "flat-lemma-Noetherian-impurity-quasi-finite", "categories": [ "flat" ], "title": "flat-lemma-Noetherian-impurity-quasi-finite", "contents": [ "In Situation \\ref{situation-pre-pure}.", "Assume that $S$ is locally Noetherian.", "If there exists an impurity of $\\mathcal{F}$ above $s$, then", "there exists an impurity $(g : T \\to S, t' \\leadsto t, \\xi)$", "of $\\mathcal{F}$ above $s$ such that $g$ is quasi-finite at $t$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $S$ by an affine neighbourhood of $s$. By", "Lemma \\ref{lemma-impure-finite-presentation}", "we may assume that we have an impurity $(g : T \\to S, t' \\leadsto t, \\xi)$", "of such that $g$ is locally of finite type and $t$ a closed point of the", "fibre of $g$ above $s$. We may replace $T$ by the reduced induced", "scheme structure on $\\overline{\\{t'\\}}$. Let", "$Z = \\overline{\\{\\xi\\}} \\subset X_T$. By assumption $Z_t = \\emptyset$", "and the image of $Z \\to T$ contains $t'$. By", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-relative-assassin-in-neighbourhood}", "there exists a nonempty open $V \\subset Z$ such that for any", "$w \\in f(V)$ any generic point $\\xi'$ of $V_w$ is in", "$\\text{Ass}_{X_T/T}(\\mathcal{F}_T)$. By", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-nonempty-generic-fibre}", "there exists a nonempty open $W \\subset T$ with $W \\subset f(V)$. By", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-quasi-finite-quasi-section-meeting-nearby-open-X}", "there exists a closed subscheme $T' \\subset T$ such that", "$t \\in T'$, $T' \\to S$ is quasi-finite at $t$, and there exists a point", "$z \\in T' \\cap W$, $z \\leadsto t$ which does not map to $s$.", "Choose any generic point $\\xi'$ of the nonempty scheme $V_z$.", "Then $(T' \\to S, z \\leadsto t, \\xi')$ is the desired impurity." ], "refs": [ "flat-lemma-impure-finite-presentation", "more-morphisms-lemma-relative-assassin-in-neighbourhood", "more-morphisms-lemma-nonempty-generic-fibre" ], "ref_ids": [ 6050, 13808, 13801 ] } ], "ref_ids": [] }, { "id": 6054, "type": "theorem", "label": "flat-lemma-impurity-on-henselization", "categories": [ "flat" ], "title": "flat-lemma-impurity-on-henselization", "contents": [ "In Situation \\ref{situation-pre-pure}.", "If there exists an impurity $(S^h \\to S, s' \\leadsto s, \\xi)$", "of $\\mathcal{F}$ above $s$ then there exists an impurity", "$(T \\to S, t' \\leadsto t, \\xi)$ of $\\mathcal{F}$ above $s$", "where $(T, t) \\to (S, s)$ is an elementary \\'etale neighbourhood." ], "refs": [], "proofs": [ { "contents": [ "We may replace $S$ by an affine neighbourhood of $s$.", "Say $S = \\Spec(A)$ and $s$ corresponds to the prime", "$\\mathfrak p \\subset A$. Then", "$\\mathcal{O}_{S, s}^h = \\colim_{(T, t)} \\Gamma(T, \\mathcal{O}_T)$", "where the limit is over the opposite of the", "cofiltered category of affine elementary \\'etale neighbourhoods", "$(T, t)$ of $(S, s)$, see", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-describe-henselization}", "and its proof. Hence $S^h = \\lim_i T_i$ and we win by", "Lemma \\ref{lemma-impure-limit}." ], "refs": [ "more-morphisms-lemma-describe-henselization", "flat-lemma-impure-limit" ], "ref_ids": [ 13869, 6051 ] } ], "ref_ids": [] }, { "id": 6055, "type": "theorem", "label": "flat-lemma-pure-along-X-s", "categories": [ "flat" ], "title": "flat-lemma-pure-along-X-s", "contents": [ "In Situation \\ref{situation-pre-pure} the following", "are equivalent", "\\begin{enumerate}", "\\item there exists an impurity $(S^h \\to S, s' \\leadsto s, \\xi)$", "of $\\mathcal{F}$ above $s$ where $S^h$ is the henselization of $S$ at $s$,", "\\item there exists an impurity $(T \\to S, t' \\leadsto t, \\xi)$", "of $\\mathcal{F}$ above $s$ such that $(T, t) \\to (S, s)$ is an", "elementary \\'etale neighbourhood, and", "\\item there exists an impurity $(T \\to S, t' \\leadsto t, \\xi)$", "of $\\mathcal{F}$ above $s$ such that $T \\to S$ is quasi-finite at $t$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As an \\'etale morphism is locally quasi-finite it is clear that", "(2) implies (3). We have seen that (3) implies (2) in", "Lemma \\ref{lemma-quasi-finite-impurity-elementary}.", "We have seen that (1) implies (2) in", "Lemma \\ref{lemma-impurity-on-henselization}.", "Finally, if $(T \\to S, t' \\leadsto t, \\xi)$ is an impurity", "of $\\mathcal{F}$ above $s$ such that $(T, t) \\to (S, s)$ is an", "elementary \\'etale neighbourhood, then we can choose a factorization", "$S^h \\to T \\to S$ of the structure morphism $S^h \\to S$.", "Choose any point $s' \\in S^h$ mapping to $t'$ and choose any", "$\\xi' \\in X_{s'}$ mapping to $\\xi \\in X_{t'}$. Then", "$(S^h \\to S, s' \\leadsto s, \\xi')$ is an impurity of", "$\\mathcal{F}$ above $s$. We omit the details." ], "refs": [ "flat-lemma-quasi-finite-impurity-elementary", "flat-lemma-impurity-on-henselization" ], "ref_ids": [ 6052, 6054 ] } ], "ref_ids": [] }, { "id": 6056, "type": "theorem", "label": "flat-lemma-base-change-universally", "categories": [ "flat" ], "title": "flat-lemma-base-change-universally", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is of finite type.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Let $s \\in S$. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is universally pure along $X_s$, and", "\\item for every morphism of pointed schemes $(S', s') \\to (S, s)$", "the pullback $\\mathcal{F}_{S'}$ is pure along $X_{s'}$.", "\\end{enumerate}", "In particular, $\\mathcal{F}$ is universally pure relative to $S$ if and", "only if every base change $\\mathcal{F}_{S'}$ of $\\mathcal{F}$ is", "pure relative to $S'$." ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6057, "type": "theorem", "label": "flat-lemma-quasi-finite-base-change", "categories": [ "flat" ], "title": "flat-lemma-quasi-finite-base-change", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is of finite type.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Let $s \\in S$. Let $(S', s') \\to (S, s)$ be a morphism of pointed schemes.", "If $S' \\to S$ is quasi-finite at $s'$ and $\\mathcal{F}$ is pure along $X_s$,", "then $\\mathcal{F}_{S'}$ is pure along $X_{s'}$." ], "refs": [], "proofs": [ { "contents": [ "It $(T \\to S', t' \\leadsto t, \\xi)$ is an impurity of", "$\\mathcal{F}_{S'}$ above $s'$ with $T \\to S'$ quasi-finite at $t$,", "then $(T \\to S, t' \\to t, \\xi)$ is an impurity of $\\mathcal{F}$", "above $s$ with $T \\to S$ quasi-finite at $t$, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-quasi-finite}.", "Hence the lemma follows immediately from the characterization (2)", "of purity given following", "Definition \\ref{definition-pure}." ], "refs": [ "morphisms-lemma-composition-quasi-finite", "flat-definition-pure" ], "ref_ids": [ 5232, 6216 ] } ], "ref_ids": [] }, { "id": 6058, "type": "theorem", "label": "flat-lemma-Noetherian-base-change", "categories": [ "flat" ], "title": "flat-lemma-Noetherian-base-change", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is of finite type.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Let $s \\in S$. If $\\mathcal{O}_{S, s}$ is Noetherian then", "$\\mathcal{F}$ is pure along $X_s$ if and only if $\\mathcal{F}$", "is universally pure along $X_s$." ], "refs": [], "proofs": [ { "contents": [ "First we may replace $S$ by $\\Spec(\\mathcal{O}_{S, s})$, i.e.,", "we may assume that $S$ is Noetherian. Next, use", "Lemma \\ref{lemma-Noetherian-impurity-quasi-finite}", "and characterization (2) of purity given in discussion following", "Definition \\ref{definition-pure}", "to conclude." ], "refs": [ "flat-lemma-Noetherian-impurity-quasi-finite", "flat-definition-pure" ], "ref_ids": [ 6053, 6216 ] } ], "ref_ids": [] }, { "id": 6059, "type": "theorem", "label": "flat-lemma-flat-descend-pure", "categories": [ "flat" ], "title": "flat-lemma-flat-descend-pure", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is of finite type.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Let $s \\in S$. Let $(S', s') \\to (S, s)$ be a morphism of pointed schemes.", "Assume $S' \\to S$ is flat at $s'$.", "\\begin{enumerate}", "\\item If $\\mathcal{F}_{S'}$ is pure along $X_{s'}$,", "then $\\mathcal{F}$ is pure along $X_s$.", "\\item If $\\mathcal{F}_{S'}$ is universally pure along $X_{s'}$,", "then $\\mathcal{F}$ is universally pure along $X_s$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $(T \\to S, t' \\leadsto t, \\xi)$ be an impurity of", "$\\mathcal{F}$ above $s$. Set $T_1 = T \\times_S S'$, and let $t_1$", "be the unique point of $T_1$ mapping to $t$ and $s'$. Since", "$T_1 \\to T$ is flat at $t_1$, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat},", "there exists a specialization $t'_1 \\leadsto t_1$ lying over", "$t' \\leadsto t$, see", "Algebra, Section \\ref{algebra-section-going-up}.", "Choose a point $\\xi_1 \\in X_{t'_1}$ which corresponds to a generic", "point of $\\Spec(\\kappa(t'_1) \\otimes_{\\kappa(t')} \\kappa(\\xi))$, see", "Schemes, Lemma \\ref{schemes-lemma-points-fibre-product}.", "By", "Divisors, Lemma \\ref{divisors-lemma-base-change-relative-assassin}", "we see that $\\xi_1 \\in \\text{Ass}_{X_{T_1}/T_1}(\\mathcal{F}_{T_1})$.", "As the Zariski closure of $\\{\\xi_1\\}$ in $X_{T_1}$ maps into the", "Zariski closure of $\\{\\xi\\}$ in $X_T$ we conclude that", "this closure is disjoint from $X_{t_1}$. Hence", "$(T_1 \\to S', t'_1 \\leadsto t_1, \\xi_1)$", "is an impurity of $\\mathcal{F}_{S'}$ above $s'$.", "In other words we have proved the contrapositive to part (2) of the", "lemma. Finally, if $(T, t) \\to (S, s)$ is an elementary", "\\'etale neighbourhood, then $(T_1, t_1) \\to (S', s')$ is an", "elementary \\'etale neighbourhood too, and in this way we see that (1) holds." ], "refs": [ "morphisms-lemma-base-change-flat", "schemes-lemma-points-fibre-product", "divisors-lemma-base-change-relative-assassin" ], "ref_ids": [ 5265, 7693, 7890 ] } ], "ref_ids": [] }, { "id": 6060, "type": "theorem", "label": "flat-lemma-supported-on-closed", "categories": [ "flat" ], "title": "flat-lemma-supported-on-closed", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes of finite type over", "a scheme $S$. Let $s \\in S$. Let $\\mathcal{F}$ be a", "finite type, quasi-coherent sheaf on $Z$. Then $\\mathcal{F}$ is", "(universally) pure along $Z_s$ if and only if $i_*\\mathcal{F}$", "is (universally) pure along $X_s$." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Divisors, Lemma \\ref{divisors-lemma-relative-weak-assassin-finite}." ], "refs": [ "divisors-lemma-relative-weak-assassin-finite" ], "ref_ids": [ 7892 ] } ], "ref_ids": [] }, { "id": 6061, "type": "theorem", "label": "flat-lemma-proper-pure", "categories": [ "flat" ], "title": "flat-lemma-proper-pure", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is of finite type.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item If the support of $\\mathcal{F}$ is proper over $S$, then", "$\\mathcal{F}$ is universally pure relative to $S$.", "\\item If $f$ is proper, then", "$\\mathcal{F}$ is universally pure relative to $S$.", "\\item If $f$ is proper, then $X$ is universally pure relative to $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First we reduce (1) to (2). Namely, let $Z \\subset X$ be the", "scheme theoretic support of $\\mathcal{F}$. Let $i : Z \\to X$", "be the corresponding closed immersion and write", "$\\mathcal{F} = i_*\\mathcal{G}$ for some finite type quasi-coherent", "$\\mathcal{O}_Z$-module $\\mathcal{G}$, see", "Morphisms, Section \\ref{morphisms-section-support}.", "In case (1) $Z \\to S$ is proper by assumption.", "Thus by Lemma \\ref{lemma-supported-on-closed} case (1) reduces to case (2).", "\\medskip\\noindent", "Assume $f$ is proper.", "Let $(g : T \\to S, t' \\leadsto t, \\xi)$ be an impurity of $\\mathcal{F}$", "above $s \\in S$. Since $f$ is proper, it is universally closed. Hence", "$f_T : X_T \\to T$ is closed. Since $f_T(\\xi) = t'$ this implies that", "$t \\in f(\\overline{\\{\\xi\\}})$ which is a contradiction." ], "refs": [ "flat-lemma-supported-on-closed" ], "ref_ids": [ 6060 ] } ], "ref_ids": [] }, { "id": 6062, "type": "theorem", "label": "flat-lemma-quasi-finite-pure", "categories": [ "flat" ], "title": "flat-lemma-quasi-finite-pure", "contents": [ "Let $f : X \\to S$ be a separated, finite type morphism of schemes.", "Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module.", "Assume that $\\text{Supp}(\\mathcal{F}_s)$ is finite for every $s \\in S$.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is pure relative to $S$,", "\\item the scheme theoretic support of $\\mathcal{F}$ is finite over $S$, and", "\\item $\\mathcal{F}$ is universally pure relative to $S$.", "\\end{enumerate}", "In particular, given a quasi-finite separated morphism $X \\to S$ we see", "that $X$ is pure relative to $S$ if and only if $X \\to S$ is finite." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset X$ be the scheme theoretic support of $\\mathcal{F}$, see", "Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-support}.", "Then $Z \\to S$ is a separated, finite type morphism of schemes with", "finite fibres. Hence it is separated and quasi-finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite}.", "By", "Lemma \\ref{lemma-supported-on-closed}", "it suffices to prove the lemma for $Z \\to S$ and the sheaf $\\mathcal{F}$", "viewed as a finite type quasi-coherent module on $Z$. Hence we may", "assume that $X \\to S$ is separated and quasi-finite and that", "$\\text{Supp}(\\mathcal{F}) = X$.", "\\medskip\\noindent", "It follows from", "Lemma \\ref{lemma-proper-pure}", "and", "Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}", "that (2) implies (3). Trivially (3) implies (1). Assume (1) holds.", "We will prove that (2) holds. It is clear that we may assume $S$ is affine. By", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite}", "we can find a diagram", "$$", "\\xymatrix{", "X \\ar[rd]_f \\ar[rr]_j & & T \\ar[ld]^\\pi \\\\", "& S &", "}", "$$", "with $\\pi$ finite and $j$ a quasi-compact open immersion.", "If we show that $j$ is closed, then $j$ is a closed immersion", "and we conclude that $f = \\pi \\circ j$ is finite.", "To show that $j$ is closed it suffices to show that specializations", "lift along $j$, see", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-closed}.", "Let $x \\in X$, set $t' = j(x)$ and let $t' \\leadsto t$ be a specialization.", "We have to show $t \\in j(X)$. Set $s' = f(x)$ and $s = \\pi(t)$ so", "$s' \\leadsto s$. By", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical}", "we can find an elementary \\'etale neighbourhood", "$(U, u) \\to (S, s)$ and a decomposition", "$$", "T_U = T \\times_S U = V \\amalg W", "$$", "into open and closed subschemes, such that $V \\to U$ is finite and", "there exists a unique point $v$ of $V$ mapping to $u$, and such that", "$v$ maps to $t$ in $T$. As $V \\to T$ is \\'etale, we can lift", "generalizations, see", "Morphisms, Lemmas \\ref{morphisms-lemma-generalizations-lift-flat} and", "\\ref{morphisms-lemma-etale-flat}.", "Hence there exists a specialization $v' \\leadsto v$ such that $v'$", "maps to $t' \\in T$. In particular we see that $v' \\in X_U \\subset T_U$.", "Denote $u' \\in U$ the image of $t'$. Note that", "$v' \\in \\text{Ass}_{X_U/U}(\\mathcal{F})$ because $X_{u'}$ is a finite", "discrete set and $X_{u'} = \\text{Supp}(\\mathcal{F}_{u'})$.", "As $\\mathcal{F}$ is pure relative to $S$ we see that $v'$ must", "specialize to a point in $X_u$. Since $v$ is the only point of", "$V$ lying over $u$ (and since no point of $W$ can be a specialization", "of $v'$) we see that $v \\in X_u$. Hence $t \\in X$." ], "refs": [ "morphisms-definition-scheme-theoretic-support", "morphisms-lemma-quasi-finite", "flat-lemma-supported-on-closed", "flat-lemma-proper-pure", "morphisms-lemma-finite-proper", "more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "schemes-lemma-quasi-compact-closed", "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical", "morphisms-lemma-generalizations-lift-flat", "morphisms-lemma-etale-flat" ], "ref_ids": [ 5538, 5230, 6060, 6061, 5445, 13901, 7702, 13895, 5266, 5369 ] } ], "ref_ids": [] }, { "id": 6063, "type": "theorem", "label": "flat-lemma-flat-geometrically-integral-fibres-pure", "categories": [ "flat" ], "title": "flat-lemma-flat-geometrically-integral-fibres-pure", "contents": [ "Let $f : X \\to S$ be a finite type, flat morphism of schemes", "with geometrically integral fibres. Then $X$ is universally pure", "over $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\xi \\in X$ with $s' = f(\\xi)$ and $s' \\leadsto s$ a specialization", "of $S$. If $\\xi$ is an associated point of $X_{s'}$, then $\\xi$ is the", "unique generic point because $X_{s'}$ is an integral scheme. Let", "$\\xi_0$ be the unique generic point of $X_s$. As $X \\to S$ is flat", "we can lift $s' \\leadsto s$ to a specialization", "$\\xi' \\leadsto \\xi_0$ in $X$, see", "Morphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat}.", "The $\\xi \\leadsto \\xi'$ because $\\xi$ is the generic point of $X_{s'}$", "hence $\\xi \\leadsto \\xi_0$. This means that $(\\text{id}_S, s' \\to s, \\xi)$", "is not an impurity of $\\mathcal{O}_X$ above $s$. Since the assumption", "that $f$ is finite type, flat with geometrically integral fibres", "is preserved under base change, we see that there doesn't exist an", "impurity after any base change. In this way we see that $X$ is", "universally $S$-pure." ], "refs": [ "morphisms-lemma-generalizations-lift-flat" ], "ref_ids": [ 5266 ] } ], "ref_ids": [] }, { "id": 6064, "type": "theorem", "label": "flat-lemma-affine-locally-projective-pure", "categories": [ "flat" ], "title": "flat-lemma-affine-locally-projective-pure", "contents": [ "Let $f : X \\to S$ be a finite type, affine morphism of schemes.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module", "such that $f_*\\mathcal{F}$ is locally projective on $S$, see", "Properties, Definition \\ref{properties-definition-locally-projective}.", "Then $\\mathcal{F}$ is universally pure over $S$." ], "refs": [ "properties-definition-locally-projective" ], "proofs": [ { "contents": [ "After reducing to the case where $S$ is the spectrum of a henselian", "local ring this follows from", "Lemma \\ref{lemma-explain-why-pure}." ], "refs": [ "flat-lemma-explain-why-pure" ], "ref_ids": [ 6046 ] } ], "ref_ids": [ 3084 ] }, { "id": 6065, "type": "theorem", "label": "flat-lemma-associated-point-specializes", "categories": [ "flat" ], "title": "flat-lemma-associated-point-specializes", "contents": [ "Let $f : X \\to S$ be a morphism of schemes of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.", "Let $s \\in S$.", "Assume that $\\mathcal{F}$ is flat over $S$ at all points of $X_s$.", "Let $x' \\in \\text{Ass}_{X/S}(\\mathcal{F})$ with $f(x') = s'$", "such that $s' \\leadsto s$ is a specialization in $S$. If", "$x'$ specializes to a point of $X_s$, then $x' \\leadsto x$", "with $x \\in \\text{Ass}_{X_s}(\\mathcal{F}_s)$." ], "refs": [], "proofs": [ { "contents": [ "Let $x' \\leadsto t$ be a specialization with $t \\in X_s$.", "We may replace $X$ by an affine neighbourhood of $t$ and $S$ by an", "affine neighbourhood of $s$. Choose a closed immersion", "$i : X \\to \\mathbf{A}^n_S$. Then it suffices to prove the lemma", "for the module $i_*\\mathcal{F}$ on $\\mathbf{A}^n_S$ and the point $i(x')$.", "Hence we may assume $X \\to S$ is of finite presentation.", "\\medskip\\noindent", "Let $x' \\leadsto t$ be a specialization with $t \\in X_s$.", "Set $A = \\mathcal{O}_{S, s}$, $B = \\mathcal{O}_{X, t}$, and", "$N = \\mathcal{F}_t$. Note that $B$ is essentially of finite presentation", "over $A$ and that $N$ is a finite $B$-module flat over $A$.", "Also $N$ is a finitely presented $B$-module by", "Lemma \\ref{lemma-finite-type-flat-at-point-local}.", "Let $\\mathfrak q' \\subset B$ be the prime ideal corresponding to $x'$", "and let $\\mathfrak p' \\subset A$ be the prime ideal corresponding to $s'$.", "The assumption $x' \\in \\text{Ass}_{X/S}(\\mathcal{F})$ means that", "$\\mathfrak q'$ is an associated prime of $N \\otimes_A \\kappa(\\mathfrak p')$.", "Let $\\Sigma \\subset B$ be the multiplicative subset of elements which are", "not zerodivisors on $N/\\mathfrak m_A N$. By", "Lemma \\ref{lemma-homothety-universally-injective}", "the map $N \\to \\Sigma^{-1}N$ is universally injective.", "In particular, we see that", "$N \\otimes_A \\kappa(\\mathfrak p') \\to", "\\Sigma^{-1}N \\otimes_A \\kappa(\\mathfrak p')$", "is injective which implies that $\\mathfrak q'$ is an associated", "prime of $\\Sigma^{-1}N \\otimes_A \\kappa(\\mathfrak p')$ and hence", "$\\mathfrak q'$ is in the image of", "$\\Spec(\\Sigma^{-1}B) \\to \\Spec(B)$. Thus", "Lemma \\ref{lemma-homothety-spectrum}", "implies that $\\mathfrak q' \\subset \\mathfrak q$ for some prime", "$\\mathfrak q \\in \\text{Ass}_B(N/\\mathfrak m_A N)$ (which in particular implies", "that $\\mathfrak m_A = A \\cap \\mathfrak q$). If $x \\in X_s$", "denotes the point corresponding to $\\mathfrak q$, then", "$x \\in \\text{Ass}_{X_s}(\\mathcal{F}_s)$ and $x' \\leadsto x$ as desired." ], "refs": [ "flat-lemma-finite-type-flat-at-point-local", "flat-lemma-homothety-universally-injective", "flat-lemma-homothety-spectrum" ], "ref_ids": [ 6022, 6003, 6002 ] } ], "ref_ids": [] }, { "id": 6066, "type": "theorem", "label": "flat-lemma-criterion", "categories": [ "flat" ], "title": "flat-lemma-criterion", "contents": [ "Let $f : X \\to S$ be a morphism of schemes of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module", "of finite type. Let $s \\in S$. Let $(S', s') \\to (S, s)$ be an", "elementary \\'etale neighbourhood and let", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\", "S & S' \\ar[l]", "}", "$$", "be a commutative diagram of morphisms of schemes. Assume", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is flat over $S$ at all points of $X_s$,", "\\item $X' \\to S'$ is of finite type,", "\\item $g^*\\mathcal{F}$ is pure along $X'_{s'}$,", "\\item $g : X' \\to X$ is \\'etale, and", "\\item $g(X')$ contains $\\text{Ass}_{X_s}(\\mathcal{F}_s)$.", "\\end{enumerate}", "In this situation $\\mathcal{F}$ is pure along $X_s$ if and only", "if the image of $X' \\to X \\times_S S'$ contains the points of", "$\\text{Ass}_{X \\times_S S'/S'}(\\mathcal{F} \\times_S S')$", "lying over points in $S'$ which specialize to $s'$." ], "refs": [], "proofs": [ { "contents": [ "Since the morphism $S' \\to S$ is \\'etale, we see that if $\\mathcal{F}$", "is pure along $X_s$, then $\\mathcal{F} \\times_S S'$ is pure along", "$X_s$, see", "Lemma \\ref{lemma-quasi-finite-base-change}.", "Since purity satisfies flat descent, see", "Lemma \\ref{lemma-flat-descend-pure},", "we see that if $\\mathcal{F} \\times_S S'$ is pure along $X_{s'}$, then", "$\\mathcal{F}$ is pure along $X_s$. Hence we may replace $S$ by $S'$", "and assume that $S = S'$ so that $g : X' \\to X$ is an \\'etale morphism", "between schemes of finite type over $S$. Moreover, we may replace", "$S$ by $\\Spec(\\mathcal{O}_{S, s})$ and assume that $S$ is local.", "\\medskip\\noindent", "First, assume that $\\mathcal{F}$ is pure along $X_s$.", "In this case every point of $\\text{Ass}_{X/S}(\\mathcal{F})$", "specializes to a point of $X_s$ by purity. Hence by", "Lemma \\ref{lemma-associated-point-specializes}", "we see that every point of $\\text{Ass}_{X/S}(\\mathcal{F})$", "specializes to a point of $\\text{Ass}_{X_s}(\\mathcal{F}_s)$.", "Thus every point of $\\text{Ass}_{X/S}(\\mathcal{F})$ is in the", "image of $g$ (as the image is open and contains", "$\\text{Ass}_{X_s}(\\mathcal{F}_s)$).", "\\medskip\\noindent", "Conversely, assume that $g(X')$ contains $\\text{Ass}_{X/S}(\\mathcal{F})$.", "Let $S^h = \\Spec(\\mathcal{O}_{S, s}^h)$ be the henselization", "of $S$ at $s$. Denote $g^h : (X')^h \\to X^h$ the base change of $g$", "by $S^h \\to S$, and denote $\\mathcal{F}^h$ the pullback of $\\mathcal{F}$", "to $X^h$. By", "Divisors, Lemma \\ref{divisors-lemma-base-change-relative-assassin} and", "Remark \\ref{divisors-remark-base-change-relative-assassin}", "the relative assassin $\\text{Ass}_{X^h/S^h}(\\mathcal{F}^h)$", "is the inverse image of $\\text{Ass}_{X/S}(\\mathcal{F})$ via the projection", "$X^h \\to X$. As we have assumed that $g(X')$ contains", "$\\text{Ass}_{X/S}(\\mathcal{F})$ we conclude that the base change", "$g^h((X')^h) = g(X') \\times_S S^h$ contains", "$\\text{Ass}_{X^h/S^h}(\\mathcal{F}^h)$. In this way", "we reduce to the case where $S$ is the spectrum of a henselian local ring.", "Let $x \\in \\text{Ass}_{X/S}(\\mathcal{F})$. To finish the proof of the", "lemma we have to show that $x$ specializes to a point of $X_s$, see", "criterion (4) for purity in discussion following", "Definition \\ref{definition-pure}.", "By assumption there exists a $x' \\in X'$ such that $g(x') = x$.", "As $g : X' \\to X$ is \\'etale, we see that", "$x' \\in \\text{Ass}_{X'/S}(g^*\\mathcal{F})$, see", "Lemma \\ref{lemma-etale-weak-assassin-up-down} (applied to", "the morphism of fibres $X'_w \\to X_w$ where $w \\in S$ is the image of $x'$).", "Since $g^*\\mathcal{F}$ is pure along $X'_s$ we see that $x' \\leadsto y$", "for some $y \\in X'_s$. Hence $x = g(x') \\leadsto g(y)$ and", "$g(y) \\in X_s$ as desired." ], "refs": [ "flat-lemma-quasi-finite-base-change", "flat-lemma-flat-descend-pure", "flat-lemma-associated-point-specializes", "divisors-lemma-base-change-relative-assassin", "divisors-remark-base-change-relative-assassin", "flat-definition-pure", "flat-lemma-etale-weak-assassin-up-down" ], "ref_ids": [ 6057, 6059, 6065, 7890, 8115, 6216, 5985 ] } ], "ref_ids": [] }, { "id": 6067, "type": "theorem", "label": "flat-lemma-finite-type-flat-pure-along-fibre-is-universal", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-pure-along-fibre-is-universal", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $s \\in S$.", "Assume", "\\begin{enumerate}", "\\item $f$ is of finite type,", "\\item $\\mathcal{F}$ is of finite type,", "\\item $\\mathcal{F}$ is flat over $S$ at all points of $X_s$, and", "\\item $\\mathcal{F}$ is pure along $X_s$.", "\\end{enumerate}", "Then $\\mathcal{F}$ is universally pure along $X_s$." ], "refs": [], "proofs": [ { "contents": [ "We first make a preliminary remark. Suppose that $(S', s') \\to (S, s)$", "is an elementary \\'etale neighbourhood. Denote $\\mathcal{F}'$ the", "pullback of $\\mathcal{F}$ to $X' = X \\times_S S'$.", "By the discussion following", "Definition \\ref{definition-pure}", "we see that $\\mathcal{F}'$ is pure along $X'_{s'}$. Moreover, $\\mathcal{F}'$", "is flat over $S'$ along $X'_{s'}$. Then it suffices to prove", "that $\\mathcal{F}'$ is universally pure along $X'_{s'}$. Namely, given", "any morphism $(T, t) \\to (S, s)$ of pointed schemes", "the fibre product $(T', t') = (T \\times_S S', (t, s'))$ is flat over $(T, t)$", "and hence if $\\mathcal{F}_{T'}$ is pure along $X_{t'}$ then", "$\\mathcal{F}_T$ is pure along $X_t$ by", "Lemma \\ref{lemma-flat-descend-pure}.", "Thus during the proof we may always replace $(s, S)$ by an elementary", "\\'etale neighbourhood.", "We may also replace $S$ by $\\Spec(\\mathcal{O}_{S, s})$", "due to the local nature of the problem.", "\\medskip\\noindent", "Choose an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and", "a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\", "S & \\Spec(\\mathcal{O}_{S', s'}) \\ar[l]", "}", "$$", "such that $X' \\to X \\times_S \\Spec(\\mathcal{O}_{S', s'})$", "is \\'etale, $X_s = g((X')_{s'})$, the scheme $X'$ is affine,", "and such that $\\Gamma(X', g^*\\mathcal{F})$ is a free", "$\\mathcal{O}_{S', s'}$-module, see", "Lemma \\ref{lemma-finite-type-flat-along-fibre-free-variant}.", "Note that $X' \\to \\Spec(\\mathcal{O}_{S', s'})$ is of finite type", "(as a quasi-compact morphism which is the composition of an \\'etale morphism", "and the base change of a finite type morphism).", "By our preliminary remarks in the first paragraph of the proof", "we may replace $S$ by $\\Spec(\\mathcal{O}_{S', s'})$. Hence", "we may assume there exists a commutative diagram", "$$", "\\xymatrix{", "X \\ar[dr] & & X' \\ar[ll]^g \\ar[ld] \\\\", "& S &", "}", "$$", "of schemes of finite type over $S$, where $g$ is \\'etale, $X_s \\subset g(X')$,", "with $S$ local with closed point $s$, with $X'$ affine, and with", "$\\Gamma(X', g^*\\mathcal{F})$ a free $\\Gamma(S, \\mathcal{O}_S)$-module.", "Note that in this case $g^*\\mathcal{F}$ is universally pure over $S$, see", "Lemma \\ref{lemma-affine-locally-projective-pure}.", "\\medskip\\noindent", "In this situation we apply", "Lemma \\ref{lemma-criterion}", "to deduce that $\\text{Ass}_{X/S}(\\mathcal{F}) \\subset g(X')$", "from our assumption that $\\mathcal{F}$ is pure along $X_s$", "and flat over $S$ along $X_s$. By", "Divisors, Lemma \\ref{divisors-lemma-base-change-relative-assassin} and", "Remark \\ref{divisors-remark-base-change-relative-assassin}", "we see that for any morphism of pointed schemes", "$(T, t) \\to (S, s)$ we have", "$$", "\\text{Ass}_{X_T/T}(\\mathcal{F}_T) \\subset", "(X_T \\to X)^{-1}(\\text{Ass}_{X/S}(\\mathcal{F})) \\subset", "g(X') \\times_S T = g_T(X'_T).", "$$", "Hence by", "Lemma \\ref{lemma-criterion}", "applied to the base change of our displayed diagram to $(T, t)$", "we conclude that $\\mathcal{F}_T$ is pure along $X_t$ as desired." ], "refs": [ "flat-definition-pure", "flat-lemma-flat-descend-pure", "flat-lemma-finite-type-flat-along-fibre-free-variant", "flat-lemma-affine-locally-projective-pure", "flat-lemma-criterion", "divisors-lemma-base-change-relative-assassin", "divisors-remark-base-change-relative-assassin", "flat-lemma-criterion" ], "ref_ids": [ 6216, 6059, 6037, 6064, 6066, 7890, 8115, 6066 ] } ], "ref_ids": [] }, { "id": 6068, "type": "theorem", "label": "flat-lemma-finite-type-flat-pure-is-universal", "categories": [ "flat" ], "title": "flat-lemma-finite-type-flat-pure-is-universal", "contents": [ "Let $f : X \\to S$ be a finite type morphism of schemes.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Assume $\\mathcal{F}$ is flat over $S$. In this case", "$\\mathcal{F}$ is pure relative to $S$ if and only if $\\mathcal{F}$", "is universally pure relative to $S$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Lemma \\ref{lemma-finite-type-flat-pure-along-fibre-is-universal}", "and the definitions." ], "refs": [ "flat-lemma-finite-type-flat-pure-along-fibre-is-universal" ], "ref_ids": [ 6067 ] } ], "ref_ids": [] }, { "id": 6069, "type": "theorem", "label": "flat-lemma-limit-purity", "categories": [ "flat" ], "title": "flat-lemma-limit-purity", "contents": [ "Let $I$ be a directed set.", "Let $(S_i, g_{ii'})$ be an inverse system of affine schemes over $I$.", "Set $S = \\lim_i S_i$ and $s \\in S$.", "Denote $g_i : S \\to S_i$ the projections and set $s_i = g_i(s)$.", "Suppose that $f : X \\to S$ is a morphism of finite presentation,", "$\\mathcal{F}$ a quasi-coherent $\\mathcal{O}_X$-module of finite presentation", "which is pure along $X_s$ and flat over $S$ at all points of $X_s$.", "Then there exists an $i \\in I$, a morphism of finite presentation", "$X_i \\to S_i$, a quasi-coherent $\\mathcal{O}_{X_i}$-module $\\mathcal{F}_i$", "of finite presentation which is pure along $(X_i)_{s_i}$ and flat over $S_i$", "at all points of $(X_i)_{s_i}$ such that $X \\cong X_i \\times_{S_i} S$", "and such that the pullback of $\\mathcal{F}_i$ to $X$ is isomorphic", "to $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be the set of points where $\\mathcal{F}$ is", "flat over $S$. By", "More on Morphisms, Theorem \\ref{more-morphisms-theorem-openness-flatness}", "this is an open subscheme of $X$. By assumption $X_s \\subset U$.", "As $X_s$ is quasi-compact, we can find a quasi-compact open", "$U' \\subset U$ with $X_s \\subset U'$. By", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can find an $i \\in I$ and a morphism of finite presentation", "$f_i : X_i \\to S_i$ whose base change to $S$ is isomorphic to $f_i$.", "Fix such a choice and set $X_{i'} = X_i \\times_{S_i} S_{i'}$.", "Then $X = \\lim_{i'} X_{i'}$ with affine transition morphisms. By", "Limits, Lemma \\ref{limits-lemma-descend-modules-finite-presentation}", "we can, after possible increasing $i$ assume there exists a", "quasi-coherent $\\mathcal{O}_{X_i}$-module $\\mathcal{F}_i$", "of finite presentation whose base change to $S$ is isomorphic to", "$\\mathcal{F}$. By", "Limits, Lemma \\ref{limits-lemma-descend-opens}", "after possibly increasing $i$ we may assume there exists an", "open $U'_i \\subset X_i$ whose inverse image in $X$ is $U'$.", "Note that in particular $(X_i)_{s_i} \\subset U'_i$. By", "Limits, Lemma \\ref{limits-lemma-descend-module-flat-finite-presentation}", "(after increasing $i$ once more)", "we may assume that $\\mathcal{F}_i$ is flat on $U'_i$.", "In particular we see that $\\mathcal{F}_i$ is flat along $(X_i)_{s_i}$.", "\\medskip\\noindent", "Next, we use", "Lemma \\ref{lemma-finite-presentation-flat-along-fibre}", "to choose an elementary \\'etale neighbourhood", "$(S_i', s_i') \\to (S_i, s_i)$ and a commutative diagram of schemes", "$$", "\\xymatrix{", "X_i \\ar[d] & X_i' \\ar[l]^{g_i} \\ar[d] \\\\", "S_i & S_i' \\ar[l]", "}", "$$", "such that $g_i$ is \\'etale, $(X_i)_{s_i} \\subset g_i(X_i')$, the schemes", "$X_i'$, $S_i'$ are affine, and such that", "$\\Gamma(X_i', g_i^*\\mathcal{F}_i)$ is a projective", "$\\Gamma(S_i', \\mathcal{O}_{S_i'})$-module.", "Note that $g_i^*\\mathcal{F}_i$ is universally pure over $S'_i$, see", "Lemma \\ref{lemma-affine-locally-projective-pure}.", "We may base change the diagram above to a diagram with morphisms", "$(S'_{i'}, s'_{i'}) \\to (S_{i'}, s_{i'})$ and", "$g_{i'} : X'_{i'} \\to X_{i'}$ over $S_{i'}$", "for any $i' \\geq i$ and we may base change the diagram to a diagram", "with morphisms $(S', s') \\to (S, s)$ and $g : X' \\to X$ over $S$.", "\\medskip\\noindent", "At this point we can use our criterion for purity.", "Set $W'_i \\subset X_i \\times_{S_i} S'_i$ equal to the image of the", "\\'etale morphism $X'_i \\to X_i \\times_{S_i} S'_i$. For every $i' \\geq i$", "we have similarly the image $W'_{i'} \\subset X_{i'} \\times_{S_{i'}} S'_{i'}$", "and we have the image $W' \\subset X \\times_S S'$. Taking images commutes", "with base change, hence $W'_{i'} = W'_i \\times_{S'_i} S'_{i'}$ and", "$W' = W_i \\times_{S'_i} S'$. Because", "$\\mathcal{F}$ is pure along $X_s$ the", "Lemma \\ref{lemma-criterion}", "implies that", "\\begin{equation}", "\\label{equation-inclusion}", "f^{-1}(\\Spec(\\mathcal{O}_{S', s'})) \\cap", "\\text{Ass}_{X \\times_S S'/S'}(\\mathcal{F} \\times_S S') \\subset W'", "\\end{equation}", "By", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-relative-assassin-constructible}", "we see that", "$$", "E = \\{t \\in S' \\mid \\text{Ass}_{X_t}(\\mathcal{F}_t) \\subset W' \\}", "\\quad\\text{and}\\quad", "E_{i'} = \\{t \\in S'_{i'} \\mid", "\\text{Ass}_{X_t}(\\mathcal{F}_{i', t}) \\subset W'_{i'} \\}", "$$", "are locally constructible subsets of $S'$ and $S'_{i'}$. By", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-base-change-assassin-in-U}", "we see that $E_{i'}$ is the inverse image of $E_i$ under the morphism", "$S'_{i'} \\to S'_i$ and that $E$ is the inverse image of $E_i$ under", "the morphism $S' \\to S'_i$. Thus", "Equation (\\ref{equation-inclusion})", "is equivalent to the assertion that", "$\\Spec(\\mathcal{O}_{S', s'})$ maps into $E_i$. As", "$\\mathcal{O}_{S', s'} =", "\\colim_{i' \\geq i} \\mathcal{O}_{S'_{i'}, s'_{i'}}$", "we see that $\\Spec(\\mathcal{O}_{S'_{i'}, s'_{i'}})$", "maps into $E_i$ for some $i' \\geq i$, see", "Limits, Lemma \\ref{limits-lemma-limit-contained-in-constructible}.", "Then, applying", "Lemma \\ref{lemma-criterion}", "to the situation over $S_{i'}$,", "we conclude that $\\mathcal{F}_{i'}$ is pure along $(X_{i'})_{s_{i'}}$." ], "refs": [ "more-morphisms-theorem-openness-flatness", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-modules-finite-presentation", "limits-lemma-descend-opens", "limits-lemma-descend-module-flat-finite-presentation", "flat-lemma-finite-presentation-flat-along-fibre", "flat-lemma-affine-locally-projective-pure", "flat-lemma-criterion", "more-morphisms-lemma-relative-assassin-constructible", "more-morphisms-lemma-base-change-assassin-in-U", "limits-lemma-limit-contained-in-constructible", "flat-lemma-criterion" ], "ref_ids": [ 13670, 15077, 15078, 15041, 15080, 6031, 6064, 6066, 13812, 13811, 15040, 6066 ] } ], "ref_ids": [] }, { "id": 6070, "type": "theorem", "label": "flat-lemma-flat-finite-presentation-purity-open", "categories": [ "flat" ], "title": "flat-lemma-flat-finite-presentation-purity-open", "contents": [ "Let $f : X \\to S$ be a morphism of finite presentation.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module", "of finite presentation flat over $S$. Then the set", "$$", "U = \\{s \\in S \\mid \\mathcal{F}\\text{ is pure along }X_s\\}", "$$", "is open in $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $s \\in U$. Using", "Lemma \\ref{lemma-finite-presentation-flat-along-fibre}", "we can find an elementary \\'etale neighbourhood", "$(S', s') \\to (S, s)$ and a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\", "S & S' \\ar[l]", "}", "$$", "such that $g$ is \\'etale, $X_s \\subset g(X')$, the schemes", "$X'$, $S'$ are affine, and such that $\\Gamma(X', g^*\\mathcal{F})$", "is a projective $\\Gamma(S', \\mathcal{O}_{S'})$-module.", "Note that $g^*\\mathcal{F}$ is universally pure over $S'$, see", "Lemma \\ref{lemma-affine-locally-projective-pure}.", "Set $W' \\subset X \\times_S S'$ equal to the image of the \\'etale morphism", "$X' \\to X \\times_S S'$. Note that $W$ is open and quasi-compact over", "$S'$. Set", "$$", "E = \\{t \\in S' \\mid \\text{Ass}_{X_t}(\\mathcal{F}_t) \\subset W' \\}.", "$$", "By", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-relative-assassin-constructible}", "$E$ is a constructible subset of $S'$. By", "Lemma \\ref{lemma-criterion}", "we see that $\\Spec(\\mathcal{O}_{S', s'}) \\subset E$.", "By", "Morphisms, Lemma \\ref{morphisms-lemma-constructible-containing-open}", "we see that $E$ contains an open neighbourhood $V'$ of $s'$.", "Applying", "Lemma \\ref{lemma-criterion}", "once more we see that for any point $s_1$ in the image of $V'$ in $S$", "the sheaf $\\mathcal{F}$ is pure along $X_{s_1}$. Since", "$S' \\to S$ is \\'etale the image of $V'$ in $S$ is open and we win." ], "refs": [ "flat-lemma-finite-presentation-flat-along-fibre", "flat-lemma-affine-locally-projective-pure", "more-morphisms-lemma-relative-assassin-constructible", "flat-lemma-criterion", "morphisms-lemma-constructible-containing-open", "flat-lemma-criterion" ], "ref_ids": [ 6031, 6064, 13812, 6066, 5251, 6066 ] } ], "ref_ids": [] }, { "id": 6071, "type": "theorem", "label": "flat-lemma-injectivity-map-source-flat-pure", "categories": [ "flat" ], "title": "flat-lemma-injectivity-map-source-flat-pure", "contents": [ "Let $f : X \\to S$ be a morphism of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf of finite type on $X$.", "Assume $S$ is local with closed point $s$.", "Assume $\\mathcal{F}$ is pure along $X_s$ and", "that $\\mathcal{F}$ is flat over $S$.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ of quasi-coherent", "$\\mathcal{O}_X$-modules. Then the following are equivalent", "\\begin{enumerate}", "\\item the map on stalks $\\varphi_x$ is injective for all", "$x \\in \\text{Ass}_{X_s}(\\mathcal{F}_s)$, and", "\\item $\\varphi$ is injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{K} = \\Ker(\\varphi)$. Our goal is to prove that", "$\\mathcal{K} = 0$. In order to do this it suffices to prove that", "$\\text{WeakAss}_X(\\mathcal{K}) = \\emptyset$, see", "Divisors, Lemma \\ref{divisors-lemma-weakly-ass-zero}.", "We have", "$\\text{WeakAss}_X(\\mathcal{K}) \\subset \\text{WeakAss}_X(\\mathcal{F})$, see", "Divisors, Lemma \\ref{divisors-lemma-ses-weakly-ass}.", "As $\\mathcal{F}$ is flat we see from", "Lemma \\ref{lemma-bourbaki-finite-type-general-base}", "that $\\text{WeakAss}_X(\\mathcal{F}) \\subset \\text{Ass}_{X/S}(\\mathcal{F})$.", "By purity any point $x'$ of $\\text{Ass}_{X/S}(\\mathcal{F})$", "is a generalization of a point of $X_s$, and hence is the", "specialization of a point $x \\in \\text{Ass}_{X_s}(\\mathcal{F}_s)$, by", "Lemma \\ref{lemma-associated-point-specializes}.", "Hence the injectivity of $\\varphi_x$ implies the injectivity of", "$\\varphi_{x'}$, whence $\\mathcal{K}_{x'} = 0$." ], "refs": [ "divisors-lemma-weakly-ass-zero", "divisors-lemma-ses-weakly-ass", "flat-lemma-bourbaki-finite-type-general-base", "flat-lemma-associated-point-specializes" ], "ref_ids": [ 7875, 7874, 6042, 6065 ] } ], "ref_ids": [] }, { "id": 6072, "type": "theorem", "label": "flat-lemma-flat-finite-presentation-affine-neighbourhood-projective", "categories": [ "flat" ], "title": "flat-lemma-flat-finite-presentation-affine-neighbourhood-projective", "contents": [ "Let $f : X \\to S$ be a morphism which is locally of finite presentation.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module which is", "of finite presentation. Let $x \\in X$ with $s = f(x) \\in S$.", "If $\\mathcal{F}$ is flat at $x$ over $S$ there exists an affine", "elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and", "an affine open $U' \\subset X \\times_S S'$ which contains $x' = (x, s')$", "such that $\\Gamma(U', \\mathcal{F}|_{U'})$ is a projective", "$\\Gamma(S', \\mathcal{O}_{S'})$-module." ], "refs": [], "proofs": [ { "contents": [ "During the proof we may replace $X$ by an open neighbourhood of $x$", "and we may replace $S$ by an elementary \\'etale neighbourhood of $s$.", "Hence, by openness of flatness (see", "More on Morphisms, Theorem \\ref{more-morphisms-theorem-openness-flatness})", "we may assume that $\\mathcal{F}$ is flat over $S$.", "We may assume $S$ and $X$ are affine.", "After shrinking $X$ some more we may assume that any", "point of $\\text{Ass}_{X_s}(\\mathcal{F}_s)$ is a generalization of $x$.", "This property is preserved on replacing $(S, s)$ by an elementary", "\\'etale neighbourhood. Hence we may apply", "Lemma \\ref{lemma-finite-presentation-flat-along-fibre}", "to arrive at the situation where there exists a diagram", "$$", "\\xymatrix{", "X \\ar[dr] & & X' \\ar[ll]^g \\ar[ld] \\\\", "& S &", "}", "$$", "of schemes affine and of finite presentation over $S$,", "where $g$ is \\'etale, $X_s \\subset g(X')$, and with", "$\\Gamma(X', g^*\\mathcal{F})$ a projective $\\Gamma(S, \\mathcal{O}_S)$-module.", "Note that in this case $g^*\\mathcal{F}$ is universally pure over $S$, see", "Lemma \\ref{lemma-affine-locally-projective-pure}.", "\\medskip\\noindent", "Let $U \\subset g(X')$ be an affine open neighbourhood of $x$.", "We claim that $\\mathcal{F}|_U$ is pure along $U_s$. If we prove this, then", "the lemma follows because $\\mathcal{F}|_U$ will be pure relative to $S$", "after shrinking $S$, see", "Lemma \\ref{lemma-flat-finite-presentation-purity-open},", "whereupon the projectivity follows from", "Proposition \\ref{proposition-finite-presentation-flat-pure-is-projective}.", "To prove the claim we have to show, after replacing $(S, s)$", "by an arbitrary elementary \\'etale neighbourhood, that any point $\\xi$ of", "$\\text{Ass}_{U/S}(\\mathcal{F}|_U)$ lying over some", "$s' \\in S$, $s' \\leadsto s$ specializes to a point of $U_s$.", "Since $U \\subset g(X')$ we can find a $\\xi' \\in X'$ with", "$g(\\xi') = \\xi$. Because $g^*\\mathcal{F}$ is pure over $S$, using", "Lemma \\ref{lemma-associated-point-specializes},", "we see there exists a specialization $\\xi' \\leadsto x'$ with", "$x' \\in \\text{Ass}_{X'_s}(g^*\\mathcal{F}_s)$. Then", "$g(x') \\in \\text{Ass}_{X_s}(\\mathcal{F}_s)$ (see for example", "Lemma \\ref{lemma-etale-weak-assassin-up-down}", "applied to the \\'etale morphism $X'_s \\to X_s$ of Noetherian schemes)", "and hence $g(x') \\leadsto x$ by our choice of $X$ above! Since", "$x \\in U$ we conclude that $g(x') \\in U$. Thus", "$\\xi = g(\\xi') \\leadsto g(x') \\in U_s$ as desired." ], "refs": [ "more-morphisms-theorem-openness-flatness", "flat-lemma-finite-presentation-flat-along-fibre", "flat-lemma-affine-locally-projective-pure", "flat-lemma-flat-finite-presentation-purity-open", "flat-proposition-finite-presentation-flat-pure-is-projective", "flat-lemma-associated-point-specializes", "flat-lemma-etale-weak-assassin-up-down" ], "ref_ids": [ 13670, 6031, 6064, 6070, 6202, 6065, 5985 ] } ], "ref_ids": [] }, { "id": 6073, "type": "theorem", "label": "flat-lemma-flat-finite-type-affine-neighbourhood-projective", "categories": [ "flat" ], "title": "flat-lemma-flat-finite-type-affine-neighbourhood-projective", "contents": [ "Let $f : X \\to S$ be a morphism which is locally of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module which is", "of finite type. Let $x \\in X$ with $s = f(x) \\in S$.", "If $\\mathcal{F}$ is flat at $x$ over $S$ there exists an affine", "elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and", "an affine open $U' \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})$", "which contains $x' = (x, s')$ such that", "$\\Gamma(U', \\mathcal{F}|_{U'})$ is a free", "$\\mathcal{O}_{S', s'}$-module." ], "refs": [], "proofs": [ { "contents": [ "The question is Zariski local on $X$ and $S$. Hence we may assume", "that $X$ and $S$ are affine. Then we can find a closed immersion", "$i : X \\to \\mathbf{A}^n_S$ over $S$. It is clear that it suffices to", "prove the lemma for the sheaf $i_*\\mathcal{F}$ on $\\mathbf{A}^n_S$", "and the point $i(x)$. In this way we reduce to the case where $X \\to S$ is", "of finite presentation. After replacing $S$ by", "$\\Spec(\\mathcal{O}_{S', s'})$ and $X$ by an open of", "$X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ we may assume that", "$\\mathcal{F}$ is of finite presentation, see", "Proposition \\ref{proposition-finite-type-flat-at-point}.", "In this case we may appeal to", "Lemma \\ref{lemma-flat-finite-presentation-affine-neighbourhood-projective}", "and", "Algebra, Theorem \\ref{algebra-theorem-projective-free-over-local-ring}", "to conclude." ], "refs": [ "flat-proposition-finite-type-flat-at-point", "flat-lemma-flat-finite-presentation-affine-neighbourhood-projective", "algebra-theorem-projective-free-over-local-ring" ], "ref_ids": [ 6199, 6072, 322 ] } ], "ref_ids": [] }, { "id": 6074, "type": "theorem", "label": "flat-lemma-flat-finite-type-local-colimit-free", "categories": [ "flat" ], "title": "flat-lemma-flat-finite-type-local-colimit-free", "contents": [ "Let $A \\to B$ be a local ring map of local rings which is essentially of", "finite type. Let $N$ be a finite $B$-module which is flat as an $A$-module.", "If $A$ is henselian, then $N$ is a filtered colimit", "$$", "N = \\colim_i F_i", "$$", "of free $A$-modules $F_i$ such that all transition maps", "$u_i : F_i \\to F_{i'}$ of the system induce injective maps", "$\\overline{u}_i : F_i/\\mathfrak m_AF_i \\to F_{i'}/\\mathfrak m_AF_{i'}$.", "Also, $N$ is a Mittag-Leffler $A$-module." ], "refs": [], "proofs": [ { "contents": [ "We can find a morphism of finite type $X \\to S = \\Spec(A)$", "and a point $x \\in X$ lying over the closed point $s$ of $S$ and a finite", "type quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ such that", "$\\mathcal{F}_x \\cong N$ as an $A$-module. After shrinking $X$", "we may assume that each point of $\\text{Ass}_{X_s}(\\mathcal{F}_s)$ specializes", "to $x$. By", "Lemma \\ref{lemma-flat-finite-type-affine-neighbourhood-projective}", "we see that there exists a fundamental system of affine open neighbourhoods", "$U_i \\subset X$ of $x$ such that $\\Gamma(U_i, \\mathcal{F})$ is", "a free $A$-module $F_i$. Note that if $U_{i'} \\subset U_i$, then", "$$", "F_i/\\mathfrak m_AF_i = \\Gamma(U_{i, s}, \\mathcal{F}_s)", "\\longrightarrow", "\\Gamma(U_{i', s}, \\mathcal{F}_s) = F_{i'}/\\mathfrak m_AF_{i'}", "$$", "is injective because a section of the kernel would be supported at", "a closed subset of $X_s$ not meeting $x$ which is a contradiction", "to our choice of $X$ above. Since the maps $F_i \\to F_{i'}$ are", "$A$-universally injective (Lemma \\ref{lemma-universally-injective-local})", "it follows that $N$ is", "Mittag-Leffler by", "Algebra, Lemma \\ref{algebra-lemma-colimit-universally-injective-ML}." ], "refs": [ "flat-lemma-flat-finite-type-affine-neighbourhood-projective", "flat-lemma-universally-injective-local", "algebra-lemma-colimit-universally-injective-ML" ], "ref_ids": [ 6073, 6006, 839 ] } ], "ref_ids": [] }, { "id": 6075, "type": "theorem", "label": "flat-lemma-flat-finite-type-local-valuation-ring-has-content", "categories": [ "flat" ], "title": "flat-lemma-flat-finite-type-local-valuation-ring-has-content", "contents": [ "Let $A \\to B$ be a local ring map of local rings which is essentially of", "finite type. Let $N$ be a finite $B$-module which is flat as an $A$-module.", "If $A$ is a valuation ring, then any element of $N$ has a content ideal", "$I \\subset A$ (More on Algebra, Definition", "\\ref{more-algebra-definition-content-ideal})." ], "refs": [ "more-algebra-definition-content-ideal" ], "proofs": [ { "contents": [ "Let $A \\subset A^h$ be the henselization. Let $B'$ be the localization", "of $B \\otimes_A A^h$ at the maximal ideal", "$\\mathfrak m_B \\otimes A^h + B \\otimes \\mathfrak m_{A^h}$.", "Then $B \\to B'$ is flat, hence faithfully flat.", "Let $N' = N \\otimes_B B'$.", "Let $x \\in N$ and let $x' \\in N'$ be the image.", "We claim that for an ideal $I \\subset A$ we have", "$x \\in IN \\Leftrightarrow x' \\in IN'$.", "Namely, $N/IN \\to N'/IN'$ is the tensor product of $B \\to B'$", "with $N/IN$ and $B \\to B'$ is universally injective by", "Algebra, Lemma \\ref{algebra-lemma-faithfully-flat-universally-injective}.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-valuation-ring}", "and Algebra, Lemma \\ref{algebra-lemma-ideals-valuation-ring}", "the map $A \\to A^h$ defines an inclusion preserving", "bijection $I \\mapsto IA^h$ on sets of ideals. We conclude that", "$x$ has a content ideal in $A$ if and only if $x'$ has a content", "ideal in $A^h$. The assertion for $x' \\in N'$ follows from", "Lemma \\ref{lemma-flat-finite-type-local-colimit-free} and", "Algebra, Lemma \\ref{algebra-lemma-minimal-contains}." ], "refs": [ "algebra-lemma-faithfully-flat-universally-injective", "more-algebra-lemma-henselization-valuation-ring", "algebra-lemma-ideals-valuation-ring", "flat-lemma-flat-finite-type-local-colimit-free", "algebra-lemma-minimal-contains" ], "ref_ids": [ 814, 10553, 622, 6074, 836 ] } ], "ref_ids": [ 10603 ] }, { "id": 6076, "type": "theorem", "label": "flat-lemma-iso-sheaf", "categories": [ "flat" ], "title": "flat-lemma-iso-sheaf", "contents": [ "In Situation \\ref{situation-iso}.", "\\begin{enumerate}", "\\item Each of the functors $F_{iso}$, $F_{inj}$, $F_{surj}$, $F_{zero}$", "satisfies the sheaf property for the fpqc topology.", "\\item If $f$ is quasi-compact and $\\mathcal{G}$ is of finite type,", "then $F_{surj}$ is limit preserving.", "\\item If $f$ is quasi-compact and $\\mathcal{F}$ of finite type, then", "$F_{zero}$ is limit preserving.", "\\item If $f$ is quasi-compact, $\\mathcal{F}$ is of finite type, and", "$\\mathcal{G}$ is of finite presentation, then $F_{iso}$ is limit preserving.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\{T_i \\to T\\}_{i \\in I}$ be an fpqc covering of schemes over $S$.", "Set $X_i = X_{T_i} = X \\times_S T_i$ and $u_i = u_{T_i}$.", "Note that $\\{X_i \\to X_T\\}_{i \\in I}$ is an fpqc covering of $X_T$, see", "Topologies, Lemma \\ref{topologies-lemma-fpqc}.", "In particular, for every $x \\in X_T$ there exists an $i \\in I$", "and an $x_i \\in X_i$ mapping to $x$. Since", "$\\mathcal{O}_{X_T, x} \\to \\mathcal{O}_{X_i, x_i}$ is flat, hence", "faithfully flat (see", "Algebra, Lemma \\ref{algebra-lemma-local-flat-ff})", "we conclude that $(u_i)_{x_i}$ is injective, surjective, bijective, or zero", "if and only if $(u_T)_x$ is injective, surjective, bijective, or zero.", "Whence part (1) of the lemma.", "\\medskip\\noindent", "Proof of (2). Assume $f$ quasi-compact and $\\mathcal{G}$ of finite type.", "Let $T = \\lim_{i \\in I} T_i$ be a directed limit of affine $S$-schemes", "and assume that $u_T$ is surjective.", "Set $X_i = X_{T_i} = X \\times_S T_i$ and", "$u_i = u_{T_i} : \\mathcal{F}_i = \\mathcal{F}_{T_i}", "\\to \\mathcal{G}_i = \\mathcal{G}_{T_i}$.", "To prove part (2) we have to show that $u_i$ is surjective for some $i$.", "Pick $i_0 \\in I$ and replace $I$ by $\\{i \\mid i \\geq i_0\\}$.", "Since $f$ is quasi-compact the scheme $X_{i_0}$ is quasi-compact.", "Hence we may choose affine opens $W_1, \\ldots, W_m \\subset X$", "and an affine open covering", "$X_{i_0} = U_{1, i_0} \\cup \\ldots \\cup U_{m, i_0}$ such that", "$U_{j, i_0}$ maps into $W_j$ under the projection morphism $X_{i_0} \\to X$.", "For any $i \\in I$ let $U_{j, i}$ be the inverse image of $U_{j, i_0}$.", "Setting $U_j = \\lim_i U_{j, i}$ we see that $X_T = U_1 \\cup \\ldots \\cup U_m$", "is an affine open covering of $X_T$. Now it suffices to show, for a given", "$j \\in \\{1, \\ldots, m\\}$ that $u_i|_{U_{j, i}}$ is surjective for some", "$i = i(j) \\in I$. Using", "Properties, Lemma \\ref{properties-lemma-finite-type-module}", "this translates into the following algebra problem:", "Let $A$ be a ring and let $u : M \\to N$ be an $A$-module map.", "Suppose that $R = \\colim_{i \\in I} R_i$ is a directed colimit", "of $A$-algebras. If $N$ is a finite $A$-module and if", "$u \\otimes 1 : M \\otimes_A R \\to N \\otimes_A R$ is surjective, then", "for some $i$ the map", "$u \\otimes 1 : M \\otimes_A R_i \\to N \\otimes_A R_i$ is surjective.", "This is", "Algebra, Lemma \\ref{algebra-lemma-module-map-property-in-colimit} part (2).", "\\medskip\\noindent", "Proof of (3). Exactly the same arguments as given in the proof of (2)", "reduces this to the following algebra problem:", "Let $A$ be a ring and let $u : M \\to N$ be an $A$-module map.", "Suppose that $R = \\colim_{i \\in I} R_i$ is a directed colimit", "of $A$-algebras. If $M$ is a finite $A$-module and if", "$u \\otimes 1 : M \\otimes_A R \\to N \\otimes_A R$ is zero, then", "for some $i$ the map", "$u \\otimes 1 : M \\otimes_A R_i \\to N \\otimes_A R_i$ is zero.", "This is", "Algebra, Lemma \\ref{algebra-lemma-module-map-property-in-colimit} part (1).", "\\medskip\\noindent", "Proof of (4).", "Assume $f$ quasi-compact and $\\mathcal{F}, \\mathcal{G}$ of finite presentation.", "Arguing in exactly the same manner as in the previous paragraph", "(using in addition also", "Properties, Lemma \\ref{properties-lemma-finite-presentation-module})", "part (3) translates into the following algebra statement:", "Let $A$ be a ring and let $u : M \\to N$ be an $A$-module map.", "Suppose that $R = \\colim_{i \\in I} R_i$ is a directed colimit", "of $A$-algebras. Assume $M$ is a finite $A$-module, $N$ is a finitely", "presented $A$-module, and", "$u \\otimes 1 : M \\otimes_A R \\to N \\otimes_A R$ is an isomorphism.", "Then for some $i$ the map", "$u \\otimes 1 : M \\otimes_A R_i \\to N \\otimes_A R_i$ is an isomorphism.", "This is", "Algebra, Lemma \\ref{algebra-lemma-module-map-property-in-colimit} part (3)." ], "refs": [ "topologies-lemma-fpqc", "algebra-lemma-local-flat-ff", "properties-lemma-finite-type-module", "algebra-lemma-module-map-property-in-colimit", "algebra-lemma-module-map-property-in-colimit", "properties-lemma-finite-presentation-module", "algebra-lemma-module-map-property-in-colimit" ], "ref_ids": [ 12498, 537, 3002, 1094, 1094, 3003, 1094 ] } ], "ref_ids": [] }, { "id": 6077, "type": "theorem", "label": "flat-lemma-flat-at-point", "categories": [ "flat" ], "title": "flat-lemma-flat-at-point", "contents": [ "In Situation \\ref{situation-flat-at-point}.", "\\begin{enumerate}", "\\item If $A' \\to A''$ is a flat morphism in $\\mathcal{C}$", "then $F_{lf}(A') = F_{lf}(A'')$.", "\\item If $A \\to B$ is essentially of finite presentation and", "$M$ is a $B$-module of finite presentation, then $F_{lf}$ is limit", "preserving: If $\\{A_i\\}_{i \\in I}$ is a", "directed system of objects of $\\mathcal{C}$, then", "$F_{lf}(\\colim_i A_i) = \\colim_i F_{lf}(A_i)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is a special case of", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-flat-descent-flat-at-primes}.", "Part (2) is a special case of", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-limit-preserving-flat-at-primes}." ], "refs": [ "more-algebra-lemma-flat-descent-flat-at-primes", "more-algebra-lemma-limit-preserving-flat-at-primes" ], "ref_ids": [ 9900, 9901 ] } ], "ref_ids": [] }, { "id": 6078, "type": "theorem", "label": "flat-lemma-flat-at-point-finite", "categories": [ "flat" ], "title": "flat-lemma-flat-at-point-finite", "contents": [ "In Situation \\ref{situation-flat-at-point}. Let $B \\to C$ is a local map of", "local $A$-algebras and $N$ a $C$-module. Denote", "$F'_{lf} : \\mathcal{C} \\to \\textit{Sets}$ the functor associated to the pair", "$(C, N)$. If $M \\cong N$ as $B$-modules and $B \\to C$ is finite, then", "$F_{lf} = F'_{lf}$." ], "refs": [], "proofs": [ { "contents": [ "Let $A'$ be an object of $\\mathcal{C}$. Set $C' = C \\otimes_A A'$", "and $N' = N \\otimes_A A'$ similarly to the definitions of $B'$, $M'$ in", "Situation \\ref{situation-flat-at-point}.", "Note that $M' \\cong N'$ as $B'$-modules.", "The assumption that $B \\to C$ is finite has two consequences:", "(a) $\\mathfrak m_C = \\sqrt{\\mathfrak m_B C}$ and (b)", "$B' \\to C'$ is finite. Consequence (a) implies that", "$$", "V(\\mathfrak m_{A'}C' + \\mathfrak m_C C')", "=", "\\left(", "\\Spec(C') \\to \\Spec(B')", "\\right)^{-1}V(\\mathfrak m_{A'}B' + \\mathfrak m_B B').", "$$", "Suppose $\\mathfrak q \\subset V(\\mathfrak m_{A'}B' + \\mathfrak m_B B')$.", "Then $M'_{\\mathfrak q}$ is flat over $A'$ if and only if", "the $C'_{\\mathfrak q}$-module $N'_{\\mathfrak q}$ is flat over $A'$", "(because these are isomorphic as $A'$-modules) if and only if", "for every maximal ideal $\\mathfrak r$ of $C'_{\\mathfrak q}$", "the module $N'_{\\mathfrak r}$ is flat over $A'$ (see", "Algebra, Lemma \\ref{algebra-lemma-flat-localization}).", "As $B'_{\\mathfrak q} \\to C'_{\\mathfrak q}$ is finite by (b),", "the maximal ideals of $C'_{\\mathfrak q}$ correspond exactly", "to the primes of $C'$ lying over $\\mathfrak q$ (see", "Algebra, Lemma \\ref{algebra-lemma-integral-going-up})", "and these primes are all contained in", "$V(\\mathfrak m_{A'}C' + \\mathfrak m_C C')$ by", "the displayed equation above. Thus the result of the lemma holds." ], "refs": [ "algebra-lemma-flat-localization", "algebra-lemma-integral-going-up" ], "ref_ids": [ 538, 500 ] } ], "ref_ids": [] }, { "id": 6079, "type": "theorem", "label": "flat-lemma-flat-at-point-go-up", "categories": [ "flat" ], "title": "flat-lemma-flat-at-point-go-up", "contents": [ "In", "Situation \\ref{situation-flat-at-point}", "suppose that $B \\to C$ is a flat local homomorphism of local", "rings. Set $N = M \\otimes_B C$. Denote", "$F'_{lf} : \\mathcal{C} \\to \\textit{Sets}$ the functor associated", "to the pair $(C, N)$. Then $F_{lf} = F'_{lf}$." ], "refs": [], "proofs": [ { "contents": [ "Let $A'$ be an object of $\\mathcal{C}$. Set $C' = C \\otimes_A A'$", "and $N' = N \\otimes_A A' = M' \\otimes_{B'} C'$ similarly to the definitions", "of $B'$, $M'$ in", "Situation \\ref{situation-flat-at-point}. Note that", "$$", "V(\\mathfrak m_{A'}B' + \\mathfrak m_B B')", "=", "\\Spec( \\kappa(\\mathfrak m_B) \\otimes_A \\kappa(\\mathfrak m_{A'}) )", "$$", "and similarly for $V(\\mathfrak m_{A'}C' + \\mathfrak m_C C')$.", "The ring map", "$$", "\\kappa(\\mathfrak m_B) \\otimes_A \\kappa(\\mathfrak m_{A'})", "\\longrightarrow", "\\kappa(\\mathfrak m_C) \\otimes_A \\kappa(\\mathfrak m_{A'})", "$$", "is faithfully flat, hence", "$V(\\mathfrak m_{A'}C' + \\mathfrak m_C C') \\to", "V(\\mathfrak m_{A'}B' + \\mathfrak m_B B')$ is surjective.", "Finally, if $\\mathfrak r \\in V(\\mathfrak m_{A'}C' + \\mathfrak m_C C')$", "maps to $\\mathfrak q \\in V(\\mathfrak m_{A'}B' + \\mathfrak m_B B')$, then", "$M'_{\\mathfrak q}$ is flat over $A'$ if and only if", "$N'_{\\mathfrak r}$ is flat over $A'$ because $B' \\to C'$ is flat, see", "Algebra, Lemma \\ref{algebra-lemma-flatness-descends-more-general}.", "The lemma follows formally from these remarks." ], "refs": [ "algebra-lemma-flatness-descends-more-general" ], "ref_ids": [ 529 ] } ], "ref_ids": [] }, { "id": 6080, "type": "theorem", "label": "flat-lemma-free-at-generic-points", "categories": [ "flat" ], "title": "flat-lemma-free-at-generic-points", "contents": [ "In Situation \\ref{situation-free-at-generic-points}.", "\\begin{enumerate}", "\\item The functor $H_p$ satisfies the sheaf property for the fpqc topology.", "\\item If $\\mathcal{F}$ is of finite presentation, then functor $H_p$ is", "limit preserving.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\{T_i \\to T\\}_{i \\in I}$ be an fpqc\\footnote{It is quite easy to", "show that $H_p$", "is a sheaf for the fppf topology using that flat morphisms of finite", "presentation are open. This is all we really need later on. But it is kind", "of fun to prove directly that it also satisfies the sheaf condition", "for the fpqc topology.} covering of schemes over $S$.", "Set $X_i = X_{T_i} = X \\times_S T_i$ and denote $\\mathcal{F}_i$ the", "pullback of $\\mathcal{F}$ to $X_i$. Assume that $\\mathcal{F}_i$ satisfies", "(\\ref{equation-free-at-generic-point-fibre}) for all $i$.", "Pick $t \\in T$ and let $\\xi_t \\in X_T$ denote the generic point of $X_t$.", "We have to show that $\\mathcal{F}$ is free in a neighbourhood of $\\xi_t$.", "For some $i \\in I$ we can find a $t_i \\in T_i$ mapping to $t$.", "Let $\\xi_i \\in X_i$ denote the generic point of $X_{t_i}$, so that", "$\\xi_i$ maps to $\\xi_t$. The fact that $\\mathcal{F}_i$ is free of rank", "$p$ in a neighbourhood of $\\xi_i$ implies that", "$(\\mathcal{F}_i)_{x_i} \\cong \\mathcal{O}_{X_i, x_i}^{\\oplus p}$", "which implies that", "$\\mathcal{F}_{T, \\xi_t} \\cong \\mathcal{O}_{X_T, \\xi_t}^{\\oplus p}$", "as $\\mathcal{O}_{X_T, \\xi_t} \\to \\mathcal{O}_{X_i, x_i}$ is flat, see", "for example", "Algebra, Lemma \\ref{algebra-lemma-finite-projective-descends}.", "Thus there exists an affine neighbourhood $U$ of $\\xi_t$ in $X_T$", "and a surjection", "$\\mathcal{O}_U^{\\oplus p} \\to \\mathcal{F}_U = \\mathcal{F}_T|_U$, see", "Modules, Lemma \\ref{modules-lemma-finite-type-surjective-on-stalk}.", "After shrinking $T$ we may assume that $U \\to T$ is surjective. Hence", "$U \\to T$ is a smooth morphism of affines with geometrically irreducible", "fibres. Moreover, for every $t' \\in T$ we see that the induced map", "$$", "\\alpha :", "\\mathcal{O}_{U, \\xi_{t'}}^{\\oplus p}", "\\longrightarrow", "\\mathcal{F}_{U, \\xi_{t'}}", "$$", "is an isomorphism (since by the same argument as before the module on the right", "is free of rank $p$). It follows from", "Lemma \\ref{lemma-induction-step}", "that", "$$", "\\Gamma(U, \\mathcal{O}_U^{\\oplus p})", "\\otimes_{\\Gamma(T, \\mathcal{O}_T)} \\mathcal{O}_{T, t'}", "\\longrightarrow", "\\Gamma(U, \\mathcal{F}_U)", "\\otimes_{\\Gamma(T, \\mathcal{O}_T)} \\mathcal{O}_{T, t'}", "$$", "is injective for every $t' \\in T$. Hence we see the surjection $\\alpha$ is an", "isomorphism. This finishes the proof of (1).", "\\medskip\\noindent", "Assume that $\\mathcal{F}$ is of finite presentation.", "Let $T = \\lim_{i \\in I} T_i$ be a directed limit of affine $S$-schemes", "and assume that $\\mathcal{F}_T$ satisfies", "(\\ref{equation-free-at-generic-point-fibre}).", "Set $X_i = X_{T_i} = X \\times_S T_i$ and denote $\\mathcal{F}_i$ the", "pullback of $\\mathcal{F}$ to $X_i$.", "Let $U \\subset X_T$ denote the open subscheme of points where", "$\\mathcal{F}_T$ is flat over $T$, see", "More on Morphisms, Theorem \\ref{more-morphisms-theorem-openness-flatness}.", "By assumption every generic point of every fibre is a point of $U$, i.e.,", "$U \\to T$ is a smooth surjective morphism with geometrically irreducible", "fibres. We may shrink $U$ a bit and assume that $U$ is quasi-compact.", "Using", "Limits, Lemma \\ref{limits-lemma-descend-opens}", "we can find an $i \\in I$ and a quasi-compact open $U_i \\subset X_i$", "whose inverse image in $X_T$ is $U$. After increasing $i$ we may", "assume that $\\mathcal{F}_i|_{U_i}$ is flat over $T_i$, see", "Limits, Lemma \\ref{limits-lemma-descend-module-flat-finite-presentation}.", "In particular, $\\mathcal{F}_i|_{U_i}$ is finite locally free", "hence defines a locally constant", "rank function $\\rho : U_i \\to \\{0, 1, 2, \\ldots \\}$.", "Let $(U_i)_p \\subset U_i$ denote the open and closed", "subset where $\\rho$ has value $p$. Let $V_i \\subset T_i$ be the", "image of $(U_i)_p$; note that $V_i$ is open and quasi-compact.", "By assumption the image of $T \\to T_i$ is contained in $V_i$.", "Hence there exists an $i' \\geq i$ such that $T_{i'} \\to T_i$ factors", "through $V_i$ by", "Limits, Lemma \\ref{limits-lemma-descend-opens}.", "Then $\\mathcal{F}_{i'}$ satisfies (\\ref{equation-free-at-generic-point-fibre})", "as desired. Some details omitted." ], "refs": [ "algebra-lemma-finite-projective-descends", "modules-lemma-finite-type-surjective-on-stalk", "flat-lemma-induction-step", "more-morphisms-theorem-openness-flatness", "limits-lemma-descend-opens", "limits-lemma-descend-module-flat-finite-presentation", "limits-lemma-descend-opens" ], "ref_ids": [ 798, 13238, 6015, 13670, 15041, 15080, 15041 ] } ], "ref_ids": [] }, { "id": 6081, "type": "theorem", "label": "flat-lemma-pre-flat-dimension-n", "categories": [ "flat" ], "title": "flat-lemma-pre-flat-dimension-n", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite", "type. Let $n \\geq 0$. The following are equivalent", "\\begin{enumerate}", "\\item for $s \\in S$ the closed subset $Z \\subset X_s$ of points", "where $\\mathcal{F}$ is not flat over $S$ (see", "Lemma \\ref{lemma-open-in-fibre-where-flat})", "satisfies $\\dim(Z) < n$, and", "\\item for $x \\in X$ such that $\\mathcal{F}$ is not flat at $x$", "over $S$ we have $\\text{trdeg}_{\\kappa(f(x))}(\\kappa(x)) < n$.", "\\end{enumerate}", "If this is true, then it remains true after any base change." ], "refs": [ "flat-lemma-open-in-fibre-where-flat" ], "proofs": [ { "contents": [ "Let $x \\in X$ be a point over $s \\in S$.", "Then the dimension of the closure of $\\{x\\}$ in $X_s$", "is $\\text{trdeg}_{\\kappa(s)}(\\kappa(x))$ by", "Varieties, Lemma \\ref{varieties-lemma-dimension-locally-algebraic}.", "Conversely, if $Z \\subset X_s$ is a closed subset", "of dimension $d$, then there exists a point $x \\in Z$", "with $\\text{trdeg}_{\\kappa(s)}(\\kappa(x)) = d$ (same reference).", "Therefore the equivalence of (1) and (2) holds (even fibre by fibre).", "The statement on base change follows from", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-module-flat} and", "\\ref{morphisms-lemma-dimension-fibre-after-base-change}." ], "refs": [ "varieties-lemma-dimension-locally-algebraic", "morphisms-lemma-base-change-module-flat", "morphisms-lemma-dimension-fibre-after-base-change" ], "ref_ids": [ 10989, 5264, 5279 ] } ], "ref_ids": [ 6017 ] }, { "id": 6082, "type": "theorem", "label": "flat-lemma-flat-dimension-n", "categories": [ "flat" ], "title": "flat-lemma-flat-dimension-n", "contents": [ "In Situation \\ref{situation-flat-dimension-n}.", "\\begin{enumerate}", "\\item The functor $F_n$ satisfies the sheaf property for the fpqc topology.", "\\item If $f$ is quasi-compact and locally of finite presentation", "and $\\mathcal{F}$ is of finite presentation, then the functor $F_n$ is", "limit preserving.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\{T_i \\to T\\}_{i \\in I}$ be an fpqc covering of schemes over $S$.", "Set $X_i = X_{T_i} = X \\times_S T_i$ and denote $\\mathcal{F}_i$ the", "pullback of $\\mathcal{F}$ to $X_i$. Assume that $\\mathcal{F}_i$", "is flat over $T_i$ in dimensions $\\geq n$ for all $i$. Let $t \\in T$.", "Choose an index $i$ and a point $t_i \\in T_i$ mapping to $t$.", "Consider the cartesian diagram", "$$", "\\xymatrix{", "X_{\\Spec(\\mathcal{O}_{T, t})} \\ar[d] &", "X_{\\Spec(\\mathcal{O}_{T_i, t_i})} \\ar[d] \\ar[l] \\\\", "\\Spec(\\mathcal{O}_{T, t}) &", "\\Spec(\\mathcal{O}_{T_i, t_i}) \\ar[l]", "}", "$$", "As the lower horizontal morphism is flat we see from", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-flat-locus-base-change}", "that the set $Z_i \\subset X_{t_i}$ where $\\mathcal{F}_i$ is not flat", "over $T_i$ and the set $Z \\subset X_t$ where $\\mathcal{F}_T$ is not flat", "over $T$ are related by the rule $Z_i = Z_{\\kappa(t_i)}$. Hence we see", "that $\\mathcal{F}_T$ is flat over $T$ in dimensions $\\geq n$ by", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}.", "\\medskip\\noindent", "Assume that $f$ is quasi-compact and locally of finite presentation and", "that $\\mathcal{F}$ is of finite presentation.", "In this paragraph we first reduce the proof of (2) to the case where", "$f$ is of finite presentation.", "Let $T = \\lim_{i \\in I} T_i$ be a directed limit of", "affine $S$-schemes and assume that $\\mathcal{F}_T$ is flat in dimensions", "$\\geq n$. Set $X_i = X_{T_i} = X \\times_S T_i$ and denote $\\mathcal{F}_i$", "the pullback of $\\mathcal{F}$ to $X_i$. We have to show that", "$\\mathcal{F}_i$ is flat in dimensions $\\geq n$ for some $i$.", "Pick $i_0 \\in I$ and replace $I$ by $\\{i \\mid i \\geq i_0\\}$.", "Since $T_{i_0}$ is affine (hence quasi-compact) there exist finitely", "many affine opens $W_j \\subset S$, $j = 1, \\ldots, m$ and an affine open", "overing $T_{i_0} = \\bigcup_{j = 1, \\ldots, m} V_{j, i_0}$", "such that $T_{i_0} \\to S$ maps $V_{j, i_0}$ into $W_j$.", "For $i \\geq i_0$ denote $V_{j, i}$ the inverse image of $V_{j, i_0}$", "in $T_i$. If we can show, for each $j$, that there exists an $i$ such that", "$\\mathcal{F}_{V_{j, i_0}}$ is flat in dimensions $\\geq n$, then", "we win. In this way we reduce to the case that $S$ is affine.", "In this case $X$ is quasi-compact and we can choose a finite", "affine open covering $X = W_1 \\cup \\ldots \\cup W_m$. In this case", "the result for $(X \\to S, \\mathcal{F})$ is equivalent to the result for", "$(\\coprod W_j, \\coprod \\mathcal{F}|_{W_j})$. Hence we may assume that", "$f$ is of finite presentation.", "\\medskip\\noindent", "Assume $f$ is of finite presentation and $\\mathcal{F}$ is of finite", "presentation. Let $U \\subset X_T$ denote the open subscheme of points where", "$\\mathcal{F}_T$ is flat over $T$, see", "More on Morphisms, Theorem \\ref{more-morphisms-theorem-openness-flatness}.", "By assumption the dimension of every fibre of $Z = X_T \\setminus U$ over", "$T$ has dimension $< n$. By", "Limits, Lemma \\ref{limits-lemma-approximate-given-relative-dimension}", "we can find a closed subscheme $Z \\subset Z' \\subset X_T$", "such that $\\dim(Z'_t) < n$ for all $t \\in T$ and such that", "$Z' \\to X_T$ is of finite presentation. By", "Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation} and", "\\ref{limits-lemma-descend-closed-immersion-finite-presentation}", "there exists an $i \\in I$ and a closed subscheme $Z'_i \\subset X_i$", "of finite presentation whose base change to $T$ is $Z'$. By", "Limits, Lemma \\ref{limits-lemma-limit-dimension}", "we may assume all fibres of $Z'_i \\to T_i$ have dimension $< n$. By", "Limits, Lemma \\ref{limits-lemma-descend-module-flat-finite-presentation}", "we may assume that $\\mathcal{F}_i|_{X_i \\setminus T'_i}$", "is flat over $T_i$. This implies that $\\mathcal{F}_i$ is", "flat in dimensions $\\geq n$; here we use that $Z' \\to X_T$ is of finite", "presentation, and hence the complement $X_T \\setminus Z'$ is quasi-compact!", "Thus part (2) is proved and the proof of the lemma is complete." ], "refs": [ "more-morphisms-lemma-flat-locus-base-change", "morphisms-lemma-dimension-fibre-after-base-change", "more-morphisms-theorem-openness-flatness", "limits-lemma-approximate-given-relative-dimension", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-closed-immersion-finite-presentation", "limits-lemma-limit-dimension", "limits-lemma-descend-module-flat-finite-presentation" ], "ref_ids": [ 13766, 5279, 13670, 15107, 15077, 15060, 15104, 15080 ] } ], "ref_ids": [] }, { "id": 6083, "type": "theorem", "label": "flat-lemma-flat", "categories": [ "flat" ], "title": "flat-lemma-flat", "contents": [ "In Situation \\ref{situation-flat}.", "\\begin{enumerate}", "\\item The functor $F_{flat}$ satisfies the sheaf property for the fpqc topology.", "\\item If $f$ is quasi-compact and locally of finite presentation", "and $\\mathcal{F}$ is of finite presentation, then the functor", "$F_{flat}$ is limit preserving.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from the following statement: If $T' \\to T$ is a surjective", "flat morphism of schemes over $S$, then $\\mathcal{F}_{T'}$ is flat over $T'$", "if and only if $\\mathcal{F}_T$ is flat over $T$, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-flat-locus-base-change}.", "Part (2) follows from", "Limits, Lemma \\ref{limits-lemma-descend-module-flat-finite-presentation}", "after reducing to the case where $X$ and $S$ are affine (compare with", "the proof of", "Lemma \\ref{lemma-flat-dimension-n})." ], "refs": [ "more-morphisms-lemma-flat-locus-base-change", "limits-lemma-descend-module-flat-finite-presentation", "flat-lemma-flat-dimension-n" ], "ref_ids": [ 13766, 15080, 6082 ] } ], "ref_ids": [] }, { "id": 6084, "type": "theorem", "label": "flat-lemma-generic-flatness-stratification", "categories": [ "flat" ], "title": "flat-lemma-generic-flatness-stratification", "contents": [ "Let $f : X \\to S$ be a morphism of finite presentation between quasi-compact", "and quasi-separated schemes. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module", "of finite presentation. Then there exists a $t \\geq 0$ and closed", "subschemes", "$$", "S \\supset S_0 \\supset S_1 \\supset \\ldots \\supset S_t = \\emptyset", "$$", "such that $S_i \\to S$ is defined by a finite type ideal sheaf,", "$S_0 \\subset S$ is a thickening, and $\\mathcal{F}$ pulled back to", "$X \\times_S (S_i \\setminus S_{i + 1})$ is flat over $S_i \\setminus S_{i + 1}$." ], "refs": [], "proofs": [ { "contents": [ "We can find a cartesian diagram", "$$", "\\xymatrix{", "X \\ar[d] \\ar[r] & X_0 \\ar[d] \\\\", "S \\ar[r] & S_0", "}", "$$", "and a finitely presented $\\mathcal{O}_{X_0}$-module $\\mathcal{F}_0$", "which pulls back to $\\mathcal{F}$ such that $X_0$ and $S_0$ are of", "finite type over $\\mathbf{Z}$. See", "Limits, Proposition \\ref{limits-proposition-approximate} and", "Lemmas \\ref{limits-lemma-descend-finite-presentation} and", "\\ref{limits-lemma-descend-modules-finite-presentation}.", "Thus we may assume $X$ and $S$ are of finite type over $\\mathbf{Z}$", "and $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module.", "\\medskip\\noindent", "Assume $X$ and $S$ are of finite type over $\\mathbf{Z}$", "and $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module.", "In this case every quasi-coherent ideal is of finite type, hence", "we do not have to check the condition that $S_i$ is cut out", "by a finite type ideal. Set $S_0 = S_{red}$ equal to the reduction of $S$.", "By generic flatness as stated in Morphisms, Proposition", "\\ref{morphisms-proposition-generic-flatness-reduced}", "there is a dense open $U_0 \\subset S_0$ such that $\\mathcal{F}$", "pulled back to $X \\times_S U_0$ is flat over $U_0$.", "Let $S_1 \\subset S_0$ be the reduced closed subscheme whose", "underlying closed subset is $S \\setminus U_0$. We continue in this", "way, provided $S_1 \\not = \\emptyset$, to find", "$S_0 \\supset S_1 \\supset \\ldots$. Because $S$", "is Noetherian any descending chain of closed subsets stabilizes", "hence we see that $S_t = \\emptyset$ for some $t \\geq 0$." ], "refs": [ "limits-proposition-approximate", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-modules-finite-presentation", "morphisms-proposition-generic-flatness-reduced" ], "ref_ids": [ 15126, 15077, 15078, 5534 ] } ], "ref_ids": [] }, { "id": 6085, "type": "theorem", "label": "flat-lemma-flattening-stratification-artinian", "categories": [ "flat" ], "title": "flat-lemma-flattening-stratification-artinian", "contents": [ "Let $S$ be the spectrum of an Artinian ring.", "For any scheme $X$ over $S$, and any quasi-coherent $\\mathcal{O}_X$-module", "there exists a universal flattening. In fact the universal flattening", "is given by a closed immersion $S' \\to S$, and hence is a flattening", "stratification for $\\mathcal{F}$ as well." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine open covering $X = \\bigcup U_i$.", "Then $F_{flat}$ is the product of the functors associated to", "each of the pairs $(U_i, \\mathcal{F}|_{U_i})$.", "Hence it suffices to prove the result for each", "$(U_i, \\mathcal{F}|_{U_i})$.", "In the affine case the lemma follows immediately from", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-flattening-artinian-universal-property}." ], "refs": [ "more-algebra-lemma-flattening-artinian-universal-property" ], "ref_ids": [ 9895 ] } ], "ref_ids": [] }, { "id": 6086, "type": "theorem", "label": "flat-lemma-universally-separating", "categories": [ "flat" ], "title": "flat-lemma-universally-separating", "contents": [ "Let $S$ be a scheme.", "Let $g : X' \\to X$ be a flat morphism of schemes over $S$", "with $X$ locally of finite type over $S$.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module", "which is flat over $S$. If $\\text{Ass}_{X/S}(\\mathcal{F}) \\subset g(X')$", "then the canonical map", "$$", "\\mathcal{F} \\longrightarrow g_*g^*\\mathcal{F}", "$$", "is injective, and remains injective after any base change." ], "refs": [], "proofs": [ { "contents": [ "The final assertion means that $\\mathcal{F}_T \\to (g_T)_*g_T^*\\mathcal{F}_T$", "is injective for any morphism $T \\to S$. The assumption", "$\\text{Ass}_{X/S}(\\mathcal{F}) \\subset g(X')$ is preserved by base change, see", "Divisors, Lemma \\ref{divisors-lemma-base-change-relative-assassin} and", "Remark \\ref{divisors-remark-base-change-relative-assassin}.", "The same holds for the assumption of flatness and finite type.", "Hence it suffices to prove the injectivity of the displayed arrow.", "Let $\\mathcal{K} = \\Ker(\\mathcal{F} \\to g_*g^*\\mathcal{F})$.", "Our goal is to prove that $\\mathcal{K} = 0$.", "In order to do this it suffices to prove that", "$\\text{WeakAss}_X(\\mathcal{K}) = \\emptyset$, see", "Divisors, Lemma \\ref{divisors-lemma-weakly-ass-zero}.", "We have", "$\\text{WeakAss}_X(\\mathcal{K}) \\subset \\text{WeakAss}_X(\\mathcal{F})$, see", "Divisors, Lemma \\ref{divisors-lemma-ses-weakly-ass}.", "As $\\mathcal{F}$ is flat we see from", "Lemma \\ref{lemma-bourbaki-finite-type-general-base}", "that $\\text{WeakAss}_X(\\mathcal{F}) \\subset \\text{Ass}_{X/S}(\\mathcal{F})$.", "By assumption any point $x$ of $\\text{Ass}_{X/S}(\\mathcal{F})$", "is the image of some $x' \\in X'$. Since $g$ is flat the", "local ring map $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X', x'}$", "is faithfully flat, hence the map", "$$", "\\mathcal{F}_x", "\\longrightarrow", "g^*\\mathcal{F}_{x'} =", "\\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\mathcal{O}_{X', x'}", "$$", "is injective (see", "Algebra, Lemma \\ref{algebra-lemma-faithfully-flat-universally-injective}).", "This implies that $\\mathcal{K}_x = 0$ as desired." ], "refs": [ "divisors-lemma-base-change-relative-assassin", "divisors-remark-base-change-relative-assassin", "divisors-lemma-weakly-ass-zero", "divisors-lemma-ses-weakly-ass", "flat-lemma-bourbaki-finite-type-general-base", "algebra-lemma-faithfully-flat-universally-injective" ], "ref_ids": [ 7890, 8115, 7875, 7874, 6042, 814 ] } ], "ref_ids": [] }, { "id": 6087, "type": "theorem", "label": "flat-lemma-flattening-module-map", "categories": [ "flat" ], "title": "flat-lemma-flattening-module-map", "contents": [ "Let $A$ be a ring. Let $u : M \\to N$ be a surjective map of $A$-modules.", "If $M$ is projective as an $A$-module, then there exists an ideal", "$I \\subset A$ such that for any ring map $\\varphi : A \\to B$", "the following are equivalent", "\\begin{enumerate}", "\\item $u \\otimes 1 : M \\otimes_A B \\to N \\otimes_A B$ is an", "isomorphism, and", "\\item $\\varphi(I) = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As $M$ is projective we can find a projective $A$-module $C$", "such that $F = M \\oplus C$ is a free $R$-module.", "By replacing $u$ by $u \\oplus 1 : F = M \\oplus C \\to N \\oplus C$", "we see that we may assume $M$ is free. In this case let $I$ be", "the ideal of $A$ generated by coefficients of all the elements of", "$\\Ker(u)$ with respect to some (fixed) basis of $M$.", "The reason this works is that, since $u$ is surjective and", "$\\otimes_A B$ is right exact, $\\Ker(u \\otimes 1)$ is", "the image of $\\Ker(u) \\otimes_A B$ in $M \\otimes_A B$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6088, "type": "theorem", "label": "flat-lemma-Weil-restriction-closed-subschemes", "categories": [ "flat" ], "title": "flat-lemma-Weil-restriction-closed-subschemes", "contents": [ "Let $f:X\\to S$ be a morphism of schemes which is of finite presentation,", "flat, and pure. Let $Y$ be a closed subscheme of $X$. Let $F=f_*Y$ be the", "Weil restriction functor of $Y$ along $f$, defined by", "$$", "F : (\\Sch/S)^{opp} \\to \\textit{Sets}, \\quad", "T \\mapsto", "\\left\\{", "\\begin{matrix}", "\\{*\\} & \\text{if} & Y_T\\to X_T \\text{ is an isomorphism, }\\\\", "\\emptyset & \\text{else.} &", "\\end{matrix}", "\\right.", "$$", "Then $F$ is representable by a closed immersion $Z\\to S$. Moreover", "$Z\\to S$ is of finite presentation if $Y\\to S$ is." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I}$ be the ideal sheaf defining $Y$ in $X$ and let", "$u:\\mathcal{O}_X\\to\\mathcal{O}_X/\\mathcal{I}$ be the surjection.", "Then for an $S$-scheme $T$, the closed immersion $Y_T\\to X_T$", "is an isomorphism if and only if $u_T$ is an isomorphism. Hence", "the result follows from", "Theorem \\ref{theorem-flattening-map}." ], "refs": [ "flat-theorem-flattening-map" ], "ref_ids": [ 5969 ] } ], "ref_ids": [] }, { "id": 6089, "type": "theorem", "label": "flat-lemma-freebie", "categories": [ "flat" ], "title": "flat-lemma-freebie", "contents": [ "Let $S$ be the spectrum of a henselian local ring with closed point $s$.", "Let $X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Let $E \\subset X_s$ be a subset. There exists a closed subscheme", "$Z \\subset S$ with the following property: for any morphism of pointed", "schemes $(T, t) \\to (S, s)$ the following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}_T$ is flat over $T$ at all points of the fibre", "$X_t$ which map to a point of $E \\subset X_s$, and", "\\item $\\Spec(\\mathcal{O}_{T, t}) \\to S$ factors through $Z$.", "\\end{enumerate}", "Moreover, if $X \\to S$ is locally of finite presentation,", "$\\mathcal{F}$ is of finite presentation, and $E \\subset X_s$ is", "closed and quasi-compact, then $Z \\to S$ is of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "For $x \\in X_s$ denote $Z_x \\subset S$ the closed subscheme we found in", "Remark \\ref{remark-flattening-local-scheme-theoretic}.", "Then it is clear that $Z = \\bigcap_{x \\in E} Z_x$ works!", "\\medskip\\noindent", "To prove the final statement assume $X$ locally of finite presentation,", "$\\mathcal{F}$ of finite presentation and $Z$ closed and quasi-compact.", "First, choose finitely many affine opens $W_j \\subset X$ such that", "$E \\subset \\bigcup W_j$. It clearly suffices to prove the", "result for each morphism $W_j \\to S$ with sheaf $\\mathcal{F}|_{X_j}$", "and closed subset $E \\cap W_j$. Hence we may assume $X$ is affine.", "In this case,", "More on Algebra, Lemma \\ref{more-algebra-lemma-limit-preserving-flat-at-primes}", "shows that the functor defined by (1) is ``limit preserving''.", "Hence we can show that $Z \\to S$ is of finite presentation exactly", "as in the last part of the proof of", "Theorem \\ref{theorem-flattening-local}." ], "refs": [ "flat-remark-flattening-local-scheme-theoretic", "more-algebra-lemma-limit-preserving-flat-at-primes", "flat-theorem-flattening-local" ], "ref_ids": [ 6231, 9901, 5970 ] } ], "ref_ids": [] }, { "id": 6090, "type": "theorem", "label": "flat-lemma-Noetherian-finite-type-injective-into-flat-mod-m", "categories": [ "flat" ], "title": "flat-lemma-Noetherian-finite-type-injective-into-flat-mod-m", "contents": [ "If in Situation \\ref{situation-mod-injective} the ring $A$ is Noetherian", "then the lemma holds." ], "refs": [], "proofs": [ { "contents": [ "Applying Algebra, Lemma \\ref{algebra-lemma-mod-injective} we see that", "$u$ is injective and that $N/u(M)$ is flat over $A$. Then $u$ is", "$A$-universally injective", "(Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}) and $N$ is $A$-flat", "(Algebra, Lemma \\ref{algebra-lemma-flat-ses}). Since $B$ is Noetherian", "in this case we see that $N$ is of finite presentation." ], "refs": [ "algebra-lemma-mod-injective", "algebra-lemma-flat-tor-zero", "algebra-lemma-flat-ses" ], "ref_ids": [ 883, 532, 533 ] } ], "ref_ids": [] }, { "id": 6091, "type": "theorem", "label": "flat-lemma-reduce-finite-type-injective-into-flat-mod-m", "categories": [ "flat" ], "title": "flat-lemma-reduce-finite-type-injective-into-flat-mod-m", "contents": [ "Let $A_0$ be a local ring. If the lemma holds for every", "Situation \\ref{situation-mod-injective} with $A = A_0$, with $B$ a", "localization of a polynomial algebra over $A$, and $N$ of finite presentation", "over $B$, then the lemma holds for every", "Situation \\ref{situation-mod-injective} with $A = A_0$." ], "refs": [], "proofs": [ { "contents": [ "Let $A \\to B$, $u : N \\to M$ be as in Situation \\ref{situation-mod-injective}.", "Write $B = C/I$ where $C$ is the localization of a polynomial algebra", "over $A$ at a prime. If we can show that $N$ is finitely presented as", "a $C$-module, then a fortiori this shows that $N$ is finitely presented", "as a $B$-module (see", "Algebra, Lemma \\ref{algebra-lemma-finitely-presented-over-subring}).", "Hence we may assume that $B$ is the localization of a polynomial algebra.", "Next, write $N = B^{\\oplus n}/K$ for some submodule $K \\subset B^{\\oplus n}$.", "Since $B/\\mathfrak m_AB$ is Noetherian (as it is essentially of finite type", "over a field), there exist finitely many elements $k_1, \\ldots, k_s \\in K$", "such that for $K' = \\sum Bk_i$ and $N' = B^{\\oplus n}/K'$ the", "canonical surjection $N' \\to N$ induces an isomorphism", "$N'/\\mathfrak m_AN' \\cong N/\\mathfrak m_AN$.", "Now, if the lemma holds for the composition $u' : N' \\to M$,", "then $u'$ is injective, hence $N' = N$ and $u' = u$. Thus the lemma holds for", "the original situation." ], "refs": [ "algebra-lemma-finitely-presented-over-subring" ], "ref_ids": [ 335 ] } ], "ref_ids": [] }, { "id": 6092, "type": "theorem", "label": "flat-lemma-henselian-finite-type-injective-into-flat-mod-m", "categories": [ "flat" ], "title": "flat-lemma-henselian-finite-type-injective-into-flat-mod-m", "contents": [ "If in Situation \\ref{situation-mod-injective} the ring $A$ is henselian", "then the lemma holds." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove this when $B$ is essentially of finite presentation", "over $A$ and $N$ is of finite presentation over $B$, see", "Lemma \\ref{lemma-reduce-finite-type-injective-into-flat-mod-m}.", "Let us temporarily make the additional assumption that $N$ is flat over $A$.", "Then $N$ is a filtered colimit $N = \\colim_i F_i$", "of free $A$-modules $F_i$ such that the transition maps", "$u_{ii'} : F_i \\to F_{i'}$ are injective modulo $\\mathfrak m_A$, see", "Lemma \\ref{lemma-flat-finite-type-local-colimit-free}.", "Each of the compositions $u_i : F_i \\to M$ is $A$-universally", "injective by", "Lemma \\ref{lemma-universally-injective-local}", "wherefore $u = \\colim u_i$ is $A$-universally injective as desired.", "\\medskip\\noindent", "Assume $A$ is a henselian local ring, $B$ is essentially", "of finite presentation over $A$, $N$ of finite presentation over $B$. By", "Theorem \\ref{theorem-flattening-local}", "there exists a finitely generated ideal $I \\subset A$ such that", "$N/IN$ is flat over $A/I$ and such that $N/I^2N$ is not flat over", "$A/I^2$ unless $I = 0$. The result of the previous paragraph shows that", "the lemma holds for $u \\bmod I : N/IN \\to M/IM$ over $A/I$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "M \\otimes_A I/I^2 \\ar[r] &", "M/I^2M \\ar[r] &", "M/IM \\ar[r] & 0 \\\\", "&", "N \\otimes_A I/I^2 \\ar[r] \\ar[u]^u &", "N/I^2N \\ar[r] \\ar[u]^u &", "N/IN \\ar[r] \\ar[u]^u & 0", "}", "$$", "whose rows are exact by right exactness of $\\otimes$ and the fact that", "$M$ is flat over $A$. Note that the left vertical arrow is the map", "$N/IN \\otimes_{A/I} I/I^2 \\to M/IM \\otimes_{A/I} I/I^2$, hence is", "injective. A diagram chase shows that the lower left arrow is injective,", "i.e., $\\text{Tor}^1_{A/I^2}(I/I^2, M/I^2) = 0$ see", "Algebra, Remark \\ref{algebra-remark-Tor-ring-mod-ideal}.", "Hence $N/I^2N$ is flat over $A/I^2$ by", "Algebra, Lemma \\ref{algebra-lemma-what-does-it-mean}", "a contradiction unless $I = 0$." ], "refs": [ "flat-lemma-reduce-finite-type-injective-into-flat-mod-m", "flat-lemma-flat-finite-type-local-colimit-free", "flat-lemma-universally-injective-local", "flat-theorem-flattening-local", "algebra-remark-Tor-ring-mod-ideal", "algebra-lemma-what-does-it-mean" ], "ref_ids": [ 6091, 6074, 6006, 5970, 1570, 890 ] } ], "ref_ids": [] }, { "id": 6093, "type": "theorem", "label": "flat-lemma-upstairs-finite-type-injective-into-flat-mod-m", "categories": [ "flat" ], "title": "flat-lemma-upstairs-finite-type-injective-into-flat-mod-m", "contents": [ "Let $A \\to B$ be a local ring homomorphism of local rings which is", "essentially of finite type. Let $u : N \\to M$ be a $B$-module map.", "If $N$ is a finite $B$-module, $M$ is flat over $A$, and", "$\\overline{u} : N/\\mathfrak m_A N \\to M/\\mathfrak m_A M$ is injective,", "then $u$ is $A$-universally injective, $N$ is of finite presentation over", "$B$, and $N$ is flat over $A$." ], "refs": [], "proofs": [ { "contents": [ "Let $A \\to A^h$ be the henselization of $A$. Let $B'$ be the localization", "of $B \\otimes_A A^h$ at the maximal ideal", "$\\mathfrak m_B \\otimes A^h + B \\otimes \\mathfrak m_{A^h}$.", "Since $B \\to B'$ is flat (hence faithfully flat, see", "Algebra, Lemma \\ref{algebra-lemma-local-flat-ff}),", "we may replace $A \\to B$ with $A^h \\to B'$,", "the module $M$ by $M \\otimes_B B'$, the module $N$ by $N \\otimes_B B'$,", "and $u$ by $u \\otimes \\text{id}_{B'}$, see", "Algebra, Lemmas \\ref{algebra-lemma-descend-properties-modules} and", "\\ref{algebra-lemma-flatness-descends-more-general}.", "Thus we may assume that $A$ is a henselian local ring.", "In this case our lemma follows from the more general", "Lemma \\ref{lemma-henselian-finite-type-injective-into-flat-mod-m}." ], "refs": [ "algebra-lemma-local-flat-ff", "algebra-lemma-descend-properties-modules", "algebra-lemma-flatness-descends-more-general", "flat-lemma-henselian-finite-type-injective-into-flat-mod-m" ], "ref_ids": [ 537, 819, 529, 6092 ] } ], "ref_ids": [] }, { "id": 6094, "type": "theorem", "label": "flat-lemma-valuation-ring-finite-type-injective-into-flat-mod-m", "categories": [ "flat" ], "title": "flat-lemma-valuation-ring-finite-type-injective-into-flat-mod-m", "contents": [ "If in Situation \\ref{situation-mod-injective} the ring $A$ is a", "valuation ring then the lemma holds." ], "refs": [], "proofs": [ { "contents": [ "Recall that an $A$-module is flat if and only if it is torsion free, see", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-valuation-ring-torsion-free-flat}.", "Let $T \\subset N$ be the $A$-torsion. Then $u(T) = 0$ and", "$N/T$ is $A$-flat. Hence $N/T$ is finitely presented over $B$, see", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-flat-finite-type-valuation-ring-finite-presentation}.", "Thus $T$ is a finite $B$-module, see", "Algebra, Lemma \\ref{algebra-lemma-extension}.", "Since $N/T$ is $A$-flat we see that", "$T/\\mathfrak m_A T \\subset N/\\mathfrak m_A N$, see", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}.", "As $\\overline{u}$ is injective but $u(T) = 0$, we conclude that", "$T/\\mathfrak m_A T = 0$. Hence $T = 0$ by Nakayama's lemma, see", "Algebra, Lemma \\ref{algebra-lemma-NAK}. At this point we have", "proved two out of the three assertions ($N$ is $A$-flat and", "of finite presentation over $B$) and what is left is to show that", "$u$ is universally injective.", "\\medskip\\noindent", "By Algebra, Theorem \\ref{algebra-theorem-universally-exact-criteria}", "it suffices to show that $N \\otimes_A Q \\to M \\otimes_A Q$ is injective", "for every finitely presented $A$-module $Q$. By", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-generalized-valuation-ring-modules}", "we may assume $Q = A/(a)$ with $a \\in \\mathfrak m_A$ nonzero.", "Thus it suffices to show that $N/aN \\to M/aM$ is injective.", "Let $x \\in N$ with $u(x) \\in aM$. By", "Lemma \\ref{lemma-flat-finite-type-local-valuation-ring-has-content}", "we know that $x$ has a content ideal $I \\subset A$. Since", "$I$ is finitely generated", "(More on Algebra, Lemma \\ref{more-algebra-lemma-content-finitely-generated})", "and $A$ is a valuation ring, we have $I = (b)$ for some $b$", "(by Algebra, Lemma \\ref{algebra-lemma-characterize-valuation-ring}).", "By More on Algebra, Lemma \\ref{more-algebra-lemma-equal-content}", "the element $u(x)$ has content ideal $I$ as well.", "Since $u(x) \\in aM$ we see that $(b) \\subset (a)$", "by More on Algebra, Definition \\ref{more-algebra-definition-content-ideal}.", "Since $x \\in bN$ we conclude $x \\in aN$ as desired." ], "refs": [ "more-algebra-lemma-valuation-ring-torsion-free-flat", "more-algebra-lemma-flat-finite-type-valuation-ring-finite-presentation", "algebra-lemma-extension", "algebra-lemma-flat-tor-zero", "algebra-lemma-NAK", "algebra-theorem-universally-exact-criteria", "more-algebra-lemma-generalized-valuation-ring-modules", "flat-lemma-flat-finite-type-local-valuation-ring-has-content", "more-algebra-lemma-content-finitely-generated", "algebra-lemma-characterize-valuation-ring", "more-algebra-lemma-equal-content", "more-algebra-definition-content-ideal" ], "ref_ids": [ 9920, 9947, 330, 532, 401, 319, 10556, 6075, 9940, 620, 9941, 10603 ] } ], "ref_ids": [] }, { "id": 6095, "type": "theorem", "label": "flat-lemma-properties-pure-spreadout", "categories": [ "flat" ], "title": "flat-lemma-properties-pure-spreadout", "contents": [ "In (\\ref{equation-star}) if there exists a pure spreadout, then", "\\begin{enumerate}", "\\item elements of $N$ have content ideals in $A$, and", "\\item if $u : N \\to M$ is a morphism to a flat $A$-module $M$", "such that $N/\\mathfrak m N \\to M/\\mathfrak m M$ is injective", "for all maximal ideals $\\mathfrak m$ of $A$, then $u$ is", "$A$-universally injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose $U$, $N'$ as in the definition of a pure spreadout.", "Any element $x' \\in N'$ has a content ideal in $A$ because", "$N'$ is $A$-projective (this can easily be seen directly, but", "it also follows from More on Algebra, Lemma", "\\ref{more-algebra-lemma-content-exists-flat-Mittag-Leffler} and", "Algebra, Example \\ref{algebra-example-ML}).", "Since $N' \\to N$ is $A$-universally injective, we see that", "the image $x \\in N$ of any $x' \\in N'$ has a content ideal in $A$", "(it is the same as the content ideal of $x'$).", "For a general $x \\in N$ we choose $s \\in S$ such that", "$s x$ is in the image of $N' \\to N$ and we use that $x$ and $sx$", "have the same content ideal.", "\\medskip\\noindent", "Let $u : N \\to M$ be as in (2). To show that $u$ is $A$-universally", "injective, we may replace $A$ by a localization at a maximal ideal", "(small detail omitted). Assume $A$ is local with maximal ideal $\\mathfrak m$.", "Pick $s \\in S$ and consider the composition", "$$", "N' \\to N \\xrightarrow{1/s} N \\xrightarrow{u} M", "$$", "Each of these maps is injective modulo $\\mathfrak m$, hence the composition", "is $A$-universally injective by", "Lemma \\ref{lemma-universally-injective-local}.", "Since $N = \\colim_{s \\in S} (1/s)N'$ we conclude that $u$", "is $A$-inversally injective as a colimit of universally injective maps." ], "refs": [ "more-algebra-lemma-content-exists-flat-Mittag-Leffler", "flat-lemma-universally-injective-local" ], "ref_ids": [ 9942, 6006 ] } ], "ref_ids": [] }, { "id": 6096, "type": "theorem", "label": "flat-lemma-find-pure-spreadout", "categories": [ "flat" ], "title": "flat-lemma-find-pure-spreadout", "contents": [ "In (\\ref{equation-star}) for every $\\mathfrak p \\in \\Spec(A)$", "there is a finitely generated ideal $I \\subset \\mathfrak pA_\\mathfrak p$", "such that over $A_\\mathfrak p/I$ we have a pure spreadout." ], "refs": [], "proofs": [ { "contents": [ "We may replace $A$ by $A_\\mathfrak p$. Thus we may assume $A$ is", "local and $\\mathfrak p$ is the maximal ideal $\\mathfrak m$ of $A$.", "We may write $N = S^{-1}N'$ for some finitely presented $B$-module $N'$", "by clearing denominators in a presentation of $N$ over $S^{-1}B$.", "Since $B/\\mathfrak m B$ is Noetherian, the kernel $K$ of", "$N'/\\mathfrak m N' \\to N/\\mathfrak m N$ is finitely generated.", "Thus we can pick $s \\in S$ such that $K$ is annihilated by $s$.", "After replacing $B$ by $B_s$ which is allowed as it just means passing", "to an affine open subscheme of $\\Spec(B)$, we find that the elements of $S$", "are injective on $N'/\\mathfrak m N'$. At this point we choose", "a local subring $A_0 \\subset A$ essentially of finite type over $\\mathbf{Z}$,", "a finite type ring map $A_0 \\to B_0$ such that $B = A \\otimes_{A_0} B_0$,", "and a finite $B_0$-module $N'_0$ such that", "$N' = B \\otimes_{B_0} N'_0 = A \\otimes_{A_0} N'_0$.", "We claim that $I = \\mathfrak m_{A_0} A$ works.", "Namely, we have", "$$", "N'/IN' = N'_0/\\mathfrak m_{A_0} N'_0 \\otimes_{\\kappa_{A_0}} A/I", "$$", "which is free over $A/I$. Multiplication by the elements of $S$", "is injective after dividing out by the maximal ideal, hence", "$N'/IN' \\to N/IN$ is universally injective for example by", "Lemma \\ref{lemma-invert-universally-injective}." ], "refs": [ "flat-lemma-invert-universally-injective" ], "ref_ids": [ 6007 ] } ], "ref_ids": [] }, { "id": 6097, "type": "theorem", "label": "flat-lemma-universally-injective-if-flat", "categories": [ "flat" ], "title": "flat-lemma-universally-injective-if-flat", "contents": [ "In (\\ref{equation-star}) assume $N$ is $A$-flat, $M$ is a flat $A$-module,", "and $u : N \\to M$ is an $A$-module map such that", "$u \\otimes \\text{id}_{\\kappa(\\mathfrak p)}$ is injective for all", "$\\mathfrak p \\in \\Spec(A)$. Then $u$ is $A$-universally injective." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-check-universally-injective-into-flat}", "it suffices to check that $N/IN \\to M/IM$ is injective for every", "ideal $I \\subset A$. After replacing $A$ by $A/I$ we see that it suffices", "to prove that $u$ is injective.", "\\medskip\\noindent", "Proof that $u$ is injective. Let $x \\in N$ be a nonzero element of the", "kernel of $u$. Then there exists a weakly associated prime $\\mathfrak p$", "of the module $Ax$, see Algebra, Lemma \\ref{algebra-lemma-weakly-ass-zero}.", "Replacing $A$ by $A_\\mathfrak p$ we may assume $A$ is local and", "we find a nonzero element $y \\in Ax$ whose annihilator has radical", "equal to $\\mathfrak m_A$, see", "Algebra, Lemma \\ref{algebra-lemma-weakly-ass-local}.", "Thus $\\text{Supp}(y) \\subset \\Spec(S^{-1}B)$ is nonempty and", "contained in the closed fibre of $\\Spec(S^{-1}B) \\to \\Spec(A)$.", "Let $I \\subset \\mathfrak m_A$ be a", "finitely generated ideal so that we have a pure spreadout over $A/I$, see", "Lemma \\ref{lemma-find-pure-spreadout}. Then $I^n y = 0$ for some $n$. Now", "$y \\in \\text{Ann}_M(I^n) = \\text{Ann}_A(I^n) \\otimes_R N$ by flatness.", "Thus, to get the desired contradiction, it suffices to show that", "$$", "\\text{Ann}_A(I^n) \\otimes_R N", "\\longrightarrow", "\\text{Ann}_A(I^n) \\otimes_R M", "$$", "is injective. Since $N$ and $M$ are flat and since $\\text{Ann}_A(I^n)$", "is annihilated by $I^n$, it suffices to show that", "$Q \\otimes_A N \\to Q \\otimes_A M$ is injective for every $A$-module", "$Q$ annihilated by $I$. This holds by our choice of $I$ and", "Lemma \\ref{lemma-properties-pure-spreadout} part (2)." ], "refs": [ "algebra-lemma-check-universally-injective-into-flat", "algebra-lemma-weakly-ass-zero", "algebra-lemma-weakly-ass-local", "flat-lemma-find-pure-spreadout", "flat-lemma-properties-pure-spreadout" ], "ref_ids": [ 817, 723, 720, 6096, 6095 ] } ], "ref_ids": [] }, { "id": 6098, "type": "theorem", "label": "flat-lemma-big-intersection-is-zero", "categories": [ "flat" ], "title": "flat-lemma-big-intersection-is-zero", "contents": [ "Let $A$ be a local domain which is not a field.", "Let $S$ be a set of finitely generated ideals of $A$.", "Assume that $S$ is closed under products and such that", "$\\bigcup_{I \\in S} V(I)$ is the complement of the generic point of $\\Spec(A)$.", "Then $\\bigcap_{I \\in S} I = (0)$." ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathfrak m_A \\subset A$ is not the generic point of $\\Spec(A)$", "we see that $I \\subset \\mathfrak m_A$ for at least one $I \\in S$.", "Hence $\\bigcap_{I \\in S} I \\subset \\mathfrak m_A$.", "Let $f \\in \\mathfrak m_A$ be nonzero. Then", "$V(f) \\subset \\bigcup_{I \\in S} V(I)$.", "Since the constructible topology on $V(f)$ is quasi-compact", "(Topology, Lemma \\ref{topology-lemma-constructible-hausdorff-quasi-compact}", "and", "Algebra, Lemma \\ref{algebra-lemma-spec-spectral})", "we find that $V(f) \\subset V(I_1) \\cup \\ldots \\cup V(I_n)$", "for some $I_j \\in S$. Because $I_1 \\ldots I_n \\in S$ we see that", "$V(f) \\subset V(I)$ for some $I$. As $I$ is finitely generated", "this implies that $I^m \\subset (f)$ for some $m$ and since", "$S$ is closed under products we see that $I \\subset (f^2)$ for", "some $I \\in S$. Then it is not possible to have $f \\in I$." ], "refs": [ "topology-lemma-constructible-hausdorff-quasi-compact", "algebra-lemma-spec-spectral" ], "ref_ids": [ 8303, 423 ] } ], "ref_ids": [] }, { "id": 6099, "type": "theorem", "label": "flat-lemma-closed-points-complement", "categories": [ "flat" ], "title": "flat-lemma-closed-points-complement", "contents": [ "Let $A$ be a local ring. Let $I, J \\subset A$ be ideals.", "If $J$ is finitely generated and $I \\subset J^n$ for all $n \\geq 1$,", "then $V(I)$ contains the closed points of $\\Spec(A) \\setminus V(J)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p \\subset A$ be a closed point of $\\Spec(A) \\setminus V(J)$.", "We want to show that $I \\subset \\mathfrak p$. If not, then some $f \\in I$", "maps to a nonzero element of $A/\\mathfrak p$. Note that", "$V(J) \\cap \\Spec(A/\\mathfrak p)$ is the set of non-generic points.", "Hence by Lemma \\ref{lemma-big-intersection-is-zero} applied", "to the collection of ideals $J^nA/\\mathfrak p$ we conclude that", "the image of $f$ is zero in $A/\\mathfrak p$." ], "refs": [ "flat-lemma-big-intersection-is-zero" ], "ref_ids": [ 6098 ] } ], "ref_ids": [] }, { "id": 6100, "type": "theorem", "label": "flat-lemma-make-smaller-flatness-ideal", "categories": [ "flat" ], "title": "flat-lemma-make-smaller-flatness-ideal", "contents": [ "Let $A$ be a local ring. Let $I \\subset A$ be an ideal.", "Let $U \\subset \\Spec(A)$ be quasi-compact open.", "Let $M$ be an $A$-module. Assume that", "\\begin{enumerate}", "\\item $M/IM$ is flat over $A/I$,", "\\item $M$ is flat over $U$,", "\\end{enumerate}", "Then $M/I_2M$ is flat over $A/I_2$ where", "$I_2 = \\Ker(I \\to \\Gamma(U, I/I^2))$." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that", "$M \\otimes_A I/I_2 \\to IM/I_2M$ is injective, see", "Algebra, Lemma \\ref{algebra-lemma-what-does-it-mean-again}.", "This is true over $U$ by assumption (2). Thus it suffices to show", "that $M \\otimes_A I/I_2$ injects into its sections over $U$.", "We have $M \\otimes_A I/I_2 = M/IM \\otimes_A I/I_2$ and", "$M/IM$ is a filtered colimit of finite free $A/I$-modules", "(Algebra, Theorem \\ref{algebra-theorem-lazard}).", "Hence it suffices to show that $I/I_2$ injects into its sections", "over $U$, which follows from the construction of $I_2$." ], "refs": [ "algebra-lemma-what-does-it-mean-again", "algebra-theorem-lazard" ], "ref_ids": [ 891, 318 ] } ], "ref_ids": [] }, { "id": 6101, "type": "theorem", "label": "flat-lemma-check-along-closed-fibre", "categories": [ "flat" ], "title": "flat-lemma-check-along-closed-fibre", "contents": [ "Let $S$ be a local scheme with closed point $s$.", "Let $f : X \\to S$ be locally of finite type.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Assume that", "\\begin{enumerate}", "\\item every point of $\\text{Ass}_{X/S}(\\mathcal{F})$ specializes", "to a point of the closed fibre $X_s$\\footnote{For example this holds if", "$f$ is finite type and $\\mathcal{F}$ is pure along $X_s$, or", "if $f$ is proper.},", "\\item $\\mathcal{F}$ is flat over $S$ at every point of $X_s$.", "\\end{enumerate}", "Then $\\mathcal{F}$ is flat over $S$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the fact that it suffices to check for", "flatness at points of the relative assassin of $\\mathcal{F}$", "over $S$ by", "Theorem \\ref{theorem-check-flatness-at-associated-points}." ], "refs": [ "flat-theorem-check-flatness-at-associated-points" ], "ref_ids": [ 5971 ] } ], "ref_ids": [] }, { "id": 6102, "type": "theorem", "label": "flat-lemma-free-at-generic-points-representable", "categories": [ "flat" ], "title": "flat-lemma-free-at-generic-points-representable", "contents": [ "In Situation \\ref{situation-free-at-generic-points}.", "For each $p \\geq 0$ the functor $H_p$", "(\\ref{equation-free-at-generic-points}) is representable", "by a locally closed immersion $S_p \\to S$. If $\\mathcal{F}$", "is of finite presentation, then $S_p \\to S$ is of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "For each $S$ we will prove the statement for all $p \\geq 0$ concurrently.", "The functor $H_p$ is a sheaf for the fppf topology by", "Lemma \\ref{lemma-free-at-generic-points}.", "Hence combining", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves},", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend}", ", and", "Descent, Lemma \\ref{descent-lemma-descending-fppf-property-immersion}", "we see that the question is local for the \\'etale topology on $S$.", "In particular, the question is Zariski local on $S$.", "\\medskip\\noindent", "For $s \\in S$ denote $\\xi_s$ the unique generic point of the fibre $X_s$.", "Note that for every $s \\in S$ the restriction $\\mathcal{F}_s$ of", "$\\mathcal{F}$ is locally free of some rank $p(s) \\geq 0$ in some", "neighbourhood of $\\xi_s$. (As $X_s$ is irreducible", "and smooth this follows from generic flatness for $\\mathcal{F}_s$ over", "$X_s$, see", "Algebra, Lemma \\ref{algebra-lemma-generic-flatness-Noetherian}", "although this is overkill.) For future reference we note that", "$$", "p(s) =", "\\dim_{\\kappa(\\xi_s)}(", "\\mathcal{F}_{\\xi_s} \\otimes_{\\mathcal{O}_{X, \\xi_s}} \\kappa(\\xi_s)", ").", "$$", "In particular $H_{p(s)}(s)$ is nonempty and $H_q(s)$ is empty", "if $q \\not = p(s)$.", "\\medskip\\noindent", "Let $U \\subset X$ be an open subscheme.", "As $f : X \\to S$ is smooth, it is open.", "It is immediate from (\\ref{equation-free-at-generic-points})", "that the functor $H_p$ for the pair $(f|_U : U \\to f(U), \\mathcal{F}|_U)$", "and the functor $H_p$ for the pair", "$(f|_{f^{-1}(f(U))}, \\mathcal{F}|_{f^{-1}(f(U))})$", "are the same. Hence to prove the existence of $S_p$ over $f(U)$ we may", "always replace $X$ by $U$.", "\\medskip\\noindent", "Pick $s \\in S$. There exists an affine open neighbourhood $U$", "of $\\xi_s$ such that $\\mathcal{F}|_U$ can be generated by at most", "$p(s)$ elements. By the arguments above we see that in order to prove", "the statement for $H_{p(s)}$ in an neighbourhood of $s$ we may assume", "that $\\mathcal{F}$ is generated by $p(s)$ elements, i.e., that there exists", "a surjection", "$$", "u : \\mathcal{O}_X^{\\oplus p(s)} \\longrightarrow \\mathcal{F}", "$$", "In this case it is clear that $H_{p(s)}$ is equal to $F_{iso}$", "(\\ref{equation-iso}) for the map $u$ (this follows immediately from", "Lemma \\ref{lemma-injectivity-map-source-flat-pure}", "but also from", "Lemma \\ref{lemma-induction-step-fp}", "after shrinking a bit more so that both $S$ and $X$ are affine.)", "Thus we may apply", "Theorem \\ref{theorem-flattening-map}", "to see that $H_{p(s)}$ is representable by a closed immersion in a", "neighbourhood of $s$.", "\\medskip\\noindent", "The result follows formally from the above.", "Namely, the arguments above show that locally on $S$ the function", "$s \\mapsto p(s)$ is bounded. Hence we may use induction", "on $p = \\max_{s \\in S} p(s)$. The functor $H_p$ is representable", "by a closed immersion $S_p \\to S$ by the above. Replace $S$ by", "$S \\setminus S_p$ which drops the maximum by at least one and", "we win by induction hypothesis.", "\\medskip\\noindent", "Assume $\\mathcal{F}$ is of finite presentation.", "Then $S_p \\to S$ is locally of finite presentation by", "Lemma \\ref{lemma-free-at-generic-points} part (2) combined with", "Limits, Remark \\ref{limits-remark-limit-preserving}.", "Then we redo the induction argument in the paragraph to", "see that each $S_p$ is quasi-compact when $S$ is affine:", "first if $p = \\max_{s \\in S} p(s)$, then $S_p \\subset S$", "is closed (see above) hence quasi-compact. Then $U = S \\setminus S_p$", "is quasi-compact open in $S$ because $S_p \\to S$ is a closed", "immersion of finite presentation (see discussion in Morphisms, Section", "\\ref{morphisms-section-constructible} for example). Then $S_{p - 1} \\to U$", "is a closed immersion of finite presentation, and so $S_{p - 1}$", "is quasi-compact and $U' = S \\setminus (S_p \\cup S_{p - 1})$", "is quasi-compact. And so on." ], "refs": [ "flat-lemma-free-at-generic-points", "descent-lemma-descent-data-sheaves", "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "descent-lemma-descending-fppf-property-immersion", "algebra-lemma-generic-flatness-Noetherian", "flat-lemma-injectivity-map-source-flat-pure", "flat-lemma-induction-step-fp", "flat-theorem-flattening-map", "flat-lemma-free-at-generic-points", "limits-remark-limit-preserving" ], "ref_ids": [ 6080, 14751, 13949, 14698, 1013, 6071, 6029, 5969, 6080, 15130 ] } ], "ref_ids": [] }, { "id": 6103, "type": "theorem", "label": "flat-lemma-localize-flat-dimension-n", "categories": [ "flat" ], "title": "flat-lemma-localize-flat-dimension-n", "contents": [ "In Situation \\ref{situation-flat-dimension-n}.", "Let $h : X' \\to X$ be an \\'etale morphism.", "Set $\\mathcal{F}' = h^*\\mathcal{F}$ and $f' = f \\circ h$.", "Let $F_n'$ be (\\ref{equation-flat-dimension-n})", "associated to $(f' : X' \\to S, \\mathcal{F}')$.", "Then $F_n$ is a subfunctor of $F_n'$ and if", "$h(X') \\supset \\text{Ass}_{X/S}(\\mathcal{F})$, then $F_n = F'_n$." ], "refs": [], "proofs": [ { "contents": [ "Let $T \\to S$ be any morphism. Then $h_T : X'_T \\to X_T$ is \\'etale as a", "base change of the \\'etale morphism $g$. For $t \\in T$ denote", "$Z \\subset X_t$ the set of points where $\\mathcal{F}_T$ is not", "flat over $T$, and similarly denote $Z' \\subset X'_t$ the set of", "points where $\\mathcal{F}'_T$ is not flat over $T$. As", "$\\mathcal{F}'_T = h_T^*\\mathcal{F}_T$ we see that", "$Z' = h_t^{-1}(Z)$, see", "Morphisms, Lemma \\ref{morphisms-lemma-flat-permanence}.", "Hence $Z' \\to Z$ is an \\'etale morphism, so $\\dim(Z') \\leq \\dim(Z)$", "(for example by", "Descent, Lemma \\ref{descent-lemma-dimension-at-point-local}", "or just because an \\'etale morphism is smooth of relative dimension $0$).", "This implies that $F_n \\subset F_n'$.", "\\medskip\\noindent", "Finally, suppose that $h(X') \\supset \\text{Ass}_{X/S}(\\mathcal{F})$", "and that $T \\to S$ is a morphism such that $F_n'(T)$ is nonempty, i.e.,", "such that $\\mathcal{F}'_T$ is flat in dimensions $\\geq n$ over $T$.", "Pick a point $t \\in T$ and let $Z \\subset X_t$ and $Z' \\subset X'_t$", "be as above. To get a contradiction assume that $\\dim(Z) \\geq n$.", "Pick a generic point $\\xi \\in Z$ corresponding to a component", "of dimension $\\geq n$. Let $x \\in \\text{Ass}_{X_t}(\\mathcal{F}_t)$", "be a generalization of $\\xi$. Then $x$ maps to a point of", "$\\text{Ass}_{X/S}(\\mathcal{F})$ by", "Divisors, Lemma \\ref{divisors-lemma-base-change-relative-assassin} and", "Remark \\ref{divisors-remark-base-change-relative-assassin}.", "Thus we see that $x$ is in the image of $h_T$, say", "$x = h_T(x')$ for some $x' \\in X'_T$. But $x' \\not \\in Z'$", "as $x \\leadsto \\xi$ and $\\dim(Z') < n$. Hence $\\mathcal{F}'_T$", "is flat over $T$ at $x'$ which implies that $\\mathcal{F}_T$ is flat", "at $x$ over $T$ (by", "Morphisms, Lemma \\ref{morphisms-lemma-flat-permanence}).", "Since this holds for every such $x$ we conclude", "that $\\mathcal{F}_T$ is flat over $T$ at $\\xi$ by", "Theorem \\ref{theorem-check-flatness-at-associated-points}", "which is the desired contradiction." ], "refs": [ "morphisms-lemma-flat-permanence", "descent-lemma-dimension-at-point-local", "divisors-lemma-base-change-relative-assassin", "divisors-remark-base-change-relative-assassin", "morphisms-lemma-flat-permanence", "flat-theorem-check-flatness-at-associated-points" ], "ref_ids": [ 5270, 14660, 7890, 8115, 5270, 5971 ] } ], "ref_ids": [] }, { "id": 6104, "type": "theorem", "label": "flat-lemma-compare-H-F", "categories": [ "flat" ], "title": "flat-lemma-compare-H-F", "contents": [ "Assume that $X \\to S$ is a smooth morphism of affine schemes", "with geometrically irreducible fibres of dimension $d$ and that", "$\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module of finite", "presentation. Then $F_d = \\coprod_{p = 0, \\ldots, c} H_p$", "for some $c \\geq 0$ with $F_d$ as in", "(\\ref{equation-flat-dimension-n}) and $H_p$ as in", "(\\ref{equation-free-at-generic-points})." ], "refs": [], "proofs": [ { "contents": [ "As $X$ is affine and $\\mathcal{F}$ is quasi-coherent of finite presentation", "we know that $\\mathcal{F}$ can be generated by $c \\geq 0$ elements.", "Then $\\dim_{\\kappa(x)}(\\mathcal{F}_x \\otimes \\kappa(x))$", "in any point $x \\in X$ never exceeds $c$. In particular $H_p = \\emptyset$", "for $p > c$. Moreover, note that there certainly is an inclusion", "$\\coprod H_p \\to F_d$. Having said this the content", "of the lemma is that, if a base change $\\mathcal{F}_T$ is flat in", "dimensions $\\geq d$ over $T$ and if $t \\in T$, then $\\mathcal{F}_T$ is", "free of some rank $r$ in an open neighbourhood $U \\subset X_T$", "of the unique generic point $\\xi$ of $X_t$. Namely, then $H_r$", "contains the image of $U$ which is an open neighbourhood of $t$.", "The existence of $U$ follows from", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-flat-and-free-at-point-fibre}." ], "refs": [ "more-morphisms-lemma-flat-and-free-at-point-fibre" ], "ref_ids": [ 13771 ] } ], "ref_ids": [] }, { "id": 6105, "type": "theorem", "label": "flat-lemma-flat-dimension-n-representable", "categories": [ "flat" ], "title": "flat-lemma-flat-dimension-n-representable", "contents": [ "In Situation \\ref{situation-flat-dimension-n}.", "Let $s \\in S$ let $d \\geq 0$. Assume", "\\begin{enumerate}", "\\item there exists a complete d\\'evissage", "of $\\mathcal{F}/X/S$ over some point $s \\in S$,", "\\item $X$ is of finite presentation over $S$,", "\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation, and", "\\item $\\mathcal{F}$ is flat in dimensions $\\geq d + 1$ over $S$.", "\\end{enumerate}", "Then after possibly replacing $S$ by an open neighbourhood", "of $s$ the functor $F_d$ (\\ref{equation-flat-dimension-n})", "is representable by a monomorphism $Z_d \\to S$ of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "A preliminary remark is that $X$, $S$ are affine schemes and that it", "suffices to prove $F_d$ is representable by a monomorphism of finite", "presentation $Z_d \\to S$ on the category of affine schemes over $S$.", "(Of course we do not require $Z_d$ to be affine.)", "Hence throughout the proof of", "the lemma we work in the category of affine schemes over $S$.", "\\medskip\\noindent", "Let $(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k)_{k = 1, \\ldots, n}$", "be a complete d\\'evissage of $\\mathcal{F}/X/S$ over $s$, see", "Definition \\ref{definition-complete-devissage}.", "We will use induction on the length $n$ of the d\\'evissage.", "Recall that $Y_k \\to S$ is smooth with geometrically irreducible fibres, see", "Definition \\ref{definition-one-step-devissage}.", "Let $d_k$ be the relative dimension of $Y_k$ over $S$.", "Recall that $i_{k, *}\\mathcal{G}_k = \\Coker(\\alpha_k)$ and", "that $i_k$ is a closed immersion.", "By the definitions referenced above we have", "$d_1 = \\dim(\\text{Supp}(\\mathcal{F}_s))$ and", "$$", "d_k = \\dim(\\text{Supp}(\\Coker(\\alpha_{k - 1})_s))", "= \\dim(\\text{Supp}(\\mathcal{G}_{k, s}))", "$$", "for $k = 2, \\ldots, n$. It follows that $d_1 > d_2 > \\ldots > d_n \\geq 0$", "because $\\alpha_k$ is an isomorphism in the generic point of $(Y_k)_s$.", "\\medskip\\noindent", "Note that $i_1$ is a closed immersion and", "$\\mathcal{F} = i_{1, *}\\mathcal{G}_1$.", "Hence for any morphism of schemes $T \\to S$ with $T$ affine,", "we have $\\mathcal{F}_T = i_{1, T, *}\\mathcal{G}_{1, T}$ and", "$i_{1, T}$ is still a closed immersion of schemes over $T$.", "Thus $\\mathcal{F}_T$ is flat in dimensions $\\geq d$ over $T$", "if and only if $\\mathcal{G}_{1, T}$ is flat in dimensions $\\geq d$ over $T$.", "Because $\\pi_1 : Z_1 \\to Y_1$ is finite we see in the same manner that", "$\\mathcal{G}_{1, T}$ is flat in dimensions $\\geq d$ over $T$", "if and only if $\\pi_{1, T, *}\\mathcal{G}_{1, T}$ is flat in dimensions", "$\\geq d$ over $T$. The same arguments work for", "``flat in dimensions $\\geq d + 1$'' and we conclude in particular that", "$\\pi_{1, *}\\mathcal{G}_1$ is flat over $S$ in dimensions $\\geq d + 1$", "by our assumption on $\\mathcal{F}$.", "\\medskip\\noindent", "Suppose that $d_1 > d$. It follows from the discussion above that", "in particular $\\pi_{1, *}\\mathcal{G}_1$ is flat over $S$ at", "the generic point of $(Y_1)_s$. By", "Lemma \\ref{lemma-induction-step-fp}", "we may replace $S$ by an affine neighbourhood of $s$ and assume that", "$\\alpha_1$ is $S$-universally injective. Because $\\alpha_1$ is", "$S$-universally injective, for any morphism $T \\to S$ with $T$ affine,", "we have a short exact sequence", "$$", "0 \\to \\mathcal{O}_{Y_{1, T}}^{\\oplus r_1}", "\\to \\pi_{1, T, *}\\mathcal{G}_{1, T} \\to \\Coker(\\alpha_1)_T \\to 0", "$$", "and still the first arrow is $T$-universally injective. Hence the set", "of points of $(Y_1)_T$ where $\\pi_{1, T, *}\\mathcal{G}_{1, T}$ is flat over", "$T$ is the same as the set of points of $(Y_1)_T$ where", "$\\Coker(\\alpha_1)_T$ is flat over $S$. In this way the question", "reduces to the sheaf $\\Coker(\\alpha_1)$ which has a complete", "d\\'evissage of length $n - 1$ and we win by induction.", "\\medskip\\noindent", "If $d_1 < d$ then $F_d$ is represented by $S$ and we win.", "\\medskip\\noindent", "The last case is the case $d_1 = d$. This case follows from a combination of", "Lemma \\ref{lemma-compare-H-F}", "and", "Lemma \\ref{lemma-free-at-generic-points-representable}." ], "refs": [ "flat-definition-complete-devissage", "flat-definition-one-step-devissage", "flat-lemma-induction-step-fp", "flat-lemma-compare-H-F", "flat-lemma-free-at-generic-points-representable" ], "ref_ids": [ 6209, 6206, 6029, 6104, 6102 ] } ], "ref_ids": [] }, { "id": 6106, "type": "theorem", "label": "flat-lemma-when-universal-flattening", "categories": [ "flat" ], "title": "flat-lemma-when-universal-flattening", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item If $f$ is of finite presentation, $\\mathcal{F}$ is an", "$\\mathcal{O}_X$-module of finite presentation, and $\\mathcal{F}$ is", "pure relative to $S$, then there exists a universal flattening", "$S' \\to S$ of $\\mathcal{F}$. Moreover $S' \\to S$ is a monomorphism", "of finite presentation.", "\\item If $f$ is of finite presentation and $X$ is pure relative to $S$,", "then there exists a universal flattening $S' \\to S$ of $X$.", "Moreover $S' \\to S$ is a monomorphism of finite presentation.", "\\item If $f$ is proper and of finite presentation and $\\mathcal{F}$ is an", "$\\mathcal{O}_X$-module of finite presentation, then there exists a", "universal flattening $S' \\to S$ of $\\mathcal{F}$. Moreover $S' \\to S$ is", "a monomorphism of finite presentation.", "\\item If $f$ is proper and of finite presentation", "then there exists a universal flattening $S' \\to S$ of $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "These statements follow immediately from", "Theorem \\ref{theorem-flat-dimension-n-representable}", "applied to $F_0 = F_{flat}$", "and the fact that if $f$ is proper then $\\mathcal{F}$ is automatically", "pure over the base, see", "Lemma \\ref{lemma-proper-pure}." ], "refs": [ "flat-theorem-flat-dimension-n-representable", "flat-lemma-proper-pure" ], "ref_ids": [ 5972, 6061 ] } ], "ref_ids": [] }, { "id": 6107, "type": "theorem", "label": "flat-lemma-compute-what-it-should-be", "categories": [ "flat" ], "title": "flat-lemma-compute-what-it-should-be", "contents": [ "In Situation \\ref{situation-existence} consider", "$$", "K = R\\lim_{D_\\QCoh(\\mathcal{O}_X)}(\\mathcal{F}_n) =", "DQ_X(R\\lim_{D(\\mathcal{O}_X)}\\mathcal{F}_n)", "$$", "Then $K$ is in $D^b_{\\QCoh}(\\mathcal{O}_X)$ and in fact", "$K$ has nonzero cohomology sheaves only in degrees $\\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Special case of", "Derived Categories of Schemes, Example", "\\ref{perfect-example-inverse-limit-quasi-coherent}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6108, "type": "theorem", "label": "flat-lemma-compute-against-perfect", "categories": [ "flat" ], "title": "flat-lemma-compute-against-perfect", "contents": [ "In Situation \\ref{situation-existence} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be}. For any perfect", "object $E$ of $D(\\mathcal{O}_X)$ we have", "\\begin{enumerate}", "\\item $M = R\\Gamma(X, K \\otimes^\\mathbf{L} E)$ is a perfect object of $D(A)$", "and there is a canonical isomorphism", "$R\\Gamma(X_n, \\mathcal{F}_n \\otimes^\\mathbf{L} E|_{X_n}) =", "M \\otimes_A^\\mathbf{L} A_n$", "in $D(A_n)$,", "\\item $N = R\\Hom_X(E, K)$ is a perfect object of $D(A)$", "and there is a canonical isomorphism", "$R\\Hom_{X_n}(E|_{X_n}, \\mathcal{F}_n) = N \\otimes_A^\\mathbf{L} A_n$", "in $D(A_n)$.", "\\end{enumerate}", "In both statements $E|_{X_n}$ denotes the derived pullback", "of $E$ to $X_n$." ], "refs": [ "flat-lemma-compute-what-it-should-be" ], "proofs": [ { "contents": [ "Proof of (2). Write $E_n = E|_{X_n}$ and", "$N_n = R\\Hom_{X_n}(E_n, \\mathcal{F}_n)$.", "Recall that $R\\Hom_{X_n}(-, -)$ is equal to", "$R\\Gamma(X_n, R\\SheafHom(-, -))$, see", "Cohomology, Section \\ref{cohomology-section-global-RHom}.", "Hence by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-base-change-RHom-perfect}", "we see that $N_n$ is a perfect object of $D(A_n)$", "whose formation commutes with base change. Thus the maps", "$N_n \\otimes_{A_n}^\\mathbf{L} A_{n - 1} \\to N_{n - 1}$", "coming from $\\varphi_n$ are isomorphisms.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-Rlim-perfect-gives-perfect}", "we find that $R\\lim N_n$ is perfect and", "that its base change back to $A_n$ recovers $N_n$.", "On the other hand, the exact functor", "$R\\Hom_X(E, -) : D_\\QCoh(\\mathcal{O}_X) \\to D(A)$", "of triangulated categories commutes with products", "and hence with derived limits, whence", "$$", "R\\Hom_X(E, K) =", "R\\lim R\\Hom_X(E, \\mathcal{F}_n) =", "R\\lim R\\Hom_X(E_n, \\mathcal{F}_n) =", "R\\lim N_n", "$$", "This proves (2). To see that (1) holds, translate it into (2)", "using Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}." ], "refs": [ "perfect-lemma-base-change-RHom-perfect", "more-algebra-lemma-Rlim-perfect-gives-perfect", "cohomology-lemma-dual-perfect-complex" ], "ref_ids": [ 7057, 10409, 2233 ] } ], "ref_ids": [ 6107 ] }, { "id": 6109, "type": "theorem", "label": "flat-lemma-relative-pseudo-coherence", "categories": [ "flat" ], "title": "flat-lemma-relative-pseudo-coherence", "contents": [ "In Situation \\ref{situation-existence} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be}. Then $K$", "is pseudo-coherent relative to $A$." ], "refs": [ "flat-lemma-compute-what-it-should-be" ], "proofs": [ { "contents": [ "Combinging Lemma \\ref{lemma-compute-against-perfect} and", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-perfect-enough}", "we see that $R\\Gamma(X, K \\otimes^\\mathbf{L} E)$", "is pseudo-coherent in $D(A)$ for all pseudo-coherent", "$E$ in $D(\\mathcal{O}_X)$. Thus the lemma follows from", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-characterize-pseudo-coherent}." ], "refs": [ "flat-lemma-compute-against-perfect", "perfect-lemma-perfect-enough", "more-morphisms-lemma-characterize-pseudo-coherent" ], "ref_ids": [ 6108, 7076, 14060 ] } ], "ref_ids": [ 6107 ] }, { "id": 6110, "type": "theorem", "label": "flat-lemma-compute-over-affine", "categories": [ "flat" ], "title": "flat-lemma-compute-over-affine", "contents": [ "In Situation \\ref{situation-existence} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be}. For any quasi-compact", "open $U \\subset X$ we have", "$$", "R\\Gamma(U, K) \\otimes_A^\\mathbf{L} A_n =", "R\\Gamma(U_n, \\mathcal{F}_n)", "$$", "in $D(A_n)$ where $U_n = U \\cap X_n$." ], "refs": [ "flat-lemma-compute-what-it-should-be" ], "proofs": [ { "contents": [ "Fix $n$. By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-computing-sections-as-colim}", "there exists a system of perfect complexes $E_m$", "on $X$ such that", "$R\\Gamma(U, K) = \\text{hocolim} R\\Gamma(X, K \\otimes^\\mathbf{L} E_m)$.", "In fact, this formula holds not just for $K$ but for every object of", "$D_\\QCoh(\\mathcal{O}_X)$.", "Applying this to $\\mathcal{F}_n$", "we obtain", "\\begin{align*}", "R\\Gamma(U_n, \\mathcal{F}_n)", "& =", "R\\Gamma(U, \\mathcal{F}_n) \\\\", "& =", "\\text{hocolim}_m R\\Gamma(X, \\mathcal{F}_n \\otimes^\\mathbf{L} E_m) \\\\", "& =", "\\text{hocolim}_m R\\Gamma(X_n, \\mathcal{F}_n \\otimes^\\mathbf{L} E_m|_{X_n})", "\\end{align*}", "Using Lemma \\ref{lemma-compute-against-perfect}", "and the fact that $- \\otimes_A^\\mathbf{L} A_n$", "commutes with homotopy colimits we obtain the result." ], "refs": [ "perfect-lemma-computing-sections-as-colim", "flat-lemma-compute-against-perfect" ], "ref_ids": [ 7073, 6108 ] } ], "ref_ids": [ 6107 ] }, { "id": 6111, "type": "theorem", "label": "flat-lemma-finitely-presented", "categories": [ "flat" ], "title": "flat-lemma-finitely-presented", "contents": [ "In Situation \\ref{situation-existence} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be}.", "Denote $X_0 \\subset X$ the closed subset", "consisting of points lying over the closed subset", "$\\Spec(A_1) = \\Spec(A_2) = \\ldots$ of $\\Spec(A)$.", "There exists an open $W \\subset X$ containing $X_0$", "such that", "\\begin{enumerate}", "\\item $H^i(K)|_W$ is zero unless $i = 0$,", "\\item $\\mathcal{F} = H^0(K)|_W$ is of finite presentation, and", "\\item $\\mathcal{F}_n = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X_n}$.", "\\end{enumerate}" ], "refs": [ "flat-lemma-compute-what-it-should-be" ], "proofs": [ { "contents": [ "Fix $n \\geq 1$. By construction there is a canonical map", "$K \\to \\mathcal{F}_n$ in $D_\\QCoh(\\mathcal{O}_X)$", "and hence a canonical map $H^0(K) \\to \\mathcal{F}_n$", "of quasi-coherent sheaves. This explains the meaning of part (3).", "\\medskip\\noindent", "Let $x \\in X_0$ be a point.", "We will find an open neighbourhood $W$", "of $x$ such that (1), (2), and (3) are true. Since $X_0$ is quasi-compact", "this will prove the lemma. Let $U \\subset X$ be an affine open", "neighbourhood of $x$. Say $U = \\Spec(B)$.", "Choose a surjection $P \\to B$ with $P$ smooth over $A$.", "By Lemma \\ref{lemma-relative-pseudo-coherence}", "and the definition of relative pseudo-coherence", "there exists a bounded above complex $F^\\bullet$", "of finite free $P$-modules representing", "$Ri_*K$ where $i : U \\to \\Spec(P)$ is the closed", "immersion induced by the presentation.", "Let $M_n$ be the $B$-module corresponding to $\\mathcal{F}_n|_U$.", "By Lemma \\ref{lemma-compute-over-affine}", "$$", "H^i(F^\\bullet \\otimes_A A_n) =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & i \\not = 0 \\\\", "M_n & \\text{if} & i = 0", "\\end{matrix}", "\\right.", "$$", "Let $i$ be the maximal index such that $F^i$ is nonzero.", "If $i \\leq 0$, then (1), (2), and (3) are true.", "If not, then $i > 0$ and we see that the rank of the map", "$$", "F^{i - 1} \\to F^i", "$$", "in the point $x$ is maximal. Hence in an open neighbourhood", "of $x$ inside $\\Spec(P)$ the rank is maximal. Thus after replacing", "$P$ by a principal localization we may assume that the displayed", "map is surjective. Since $F^i$ is finite free we may choose", "a splitting $F^{i - 1} = F' \\oplus F^i$. Then we may", "replace $F^\\bullet$ by the complex", "$$", "\\ldots \\to F^{i - 2} \\to F' \\to 0 \\to \\ldots", "$$", "and we win by induction on $i$." ], "refs": [ "flat-lemma-relative-pseudo-coherence", "flat-lemma-compute-over-affine" ], "ref_ids": [ 6109, 6110 ] } ], "ref_ids": [ 6107 ] }, { "id": 6112, "type": "theorem", "label": "flat-lemma-proper-support", "categories": [ "flat" ], "title": "flat-lemma-proper-support", "contents": [ "In Situation \\ref{situation-existence} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be}. Let $W \\subset X$", "be as in Lemma \\ref{lemma-finitely-presented}.", "Set $\\mathcal{F} = H^0(K)|_W$. Then, after possibly shrinking the open $W$,", "the support of $\\mathcal{F}$ is proper over $A$." ], "refs": [ "flat-lemma-compute-what-it-should-be", "flat-lemma-finitely-presented" ], "proofs": [ { "contents": [ "Fix $n \\geq 1$. Let $I_n = \\Ker(A \\to A_n)$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-limit-henselian}", "the pair $(A, I_n)$ is henselian.", "Let $Z \\subset W$ be the support of $\\mathcal{F}$.", "This is a closed subset as $\\mathcal{F}$ is of finite presentation.", "By part (3) of Lemma \\ref{lemma-finitely-presented}", "we see that $Z \\times_{\\Spec(A)} \\Spec(A_n)$", "is equal to the support of $\\mathcal{F}_n$ and hence", "proper over $\\Spec(A/I)$.", "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-split-off-proper-part-henselian}", "we can write $Z = Z_1 \\amalg Z_2$ with $Z_1, Z_2$ open and", "closed in $Z$, with $Z_1$ proper", "over $A$, and with $Z_1 \\times_{\\Spec(A)} \\Spec(A/I_n)$", "equal to the support of $\\mathcal{F}_n$.", "In other words, $Z_2$ does not meet $X_0$.", "Hence after replacing $W$ by $W \\setminus Z_2$ we obtain the lemma." ], "refs": [ "more-algebra-lemma-limit-henselian", "flat-lemma-finitely-presented", "more-morphisms-lemma-split-off-proper-part-henselian" ], "ref_ids": [ 9858, 6111, 13948 ] } ], "ref_ids": [ 6107, 6111 ] }, { "id": 6113, "type": "theorem", "label": "flat-lemma-monomorphism-isomorphism", "categories": [ "flat" ], "title": "flat-lemma-monomorphism-isomorphism", "contents": [ "Let $A = \\lim A_n$ be a limit of a system of rings", "whose transition maps are surjective and with locally nilpotent", "kernels. Let $S = \\Spec(A)$. Let $T \\to S$ be a monomorphism", "which is locally of finite type. If $\\Spec(A_n) \\to S$", "factors through $T$ for all $n$, then $T = S$." ], "refs": [], "proofs": [ { "contents": [ "Set $S_n = \\Spec(A_n)$. Let $T_0 \\subset T$ be the common", "image of the factorizations $S_n \\to T$. Then $T_0$ is quasi-compact.", "Let $T' \\subset T$ be a quasi-compact open containing $T_0$.", "Then $S_n \\to T$ factors through $T'$.", "If we can show that $T' = S$, then $T' = T = S$.", "Hence we may assume $T$ is quasi-compact.", "\\medskip\\noindent", "Assume $T$ is quasi-compact.", "In this case $T \\to S$ is separated and quasi-finite", "(Morphisms, Lemma", "\\ref{morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}).", "Using Zariski's Main Theorem", "(in the form of More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite})", "we choose a factorization $T \\to W \\to S$ with $W \\to S$ finite", "and $T \\to W$ an open immersion. Write $W = \\Spec(B)$.", "The (unique) factorizations $S_n \\to T$ may be viewed", "as morphisms into $W$ and we obtain", "$$", "A \\longrightarrow B \\longrightarrow \\lim A_n = A", "$$", "Consider the morphism $h : S = \\Spec(A) \\to \\Spec(B) = W$ coming", "from the arrow on the right. Then", "$$", "T \\times_{W, h} S", "$$", "is an open subscheme of $S$ containing the image of $S_n \\to S$ for all $n$.", "To finish the proof it suffices to show that any open $U \\subset S$", "containing the image of $S_n \\to S$ for some $n \\geq 1$ is equal to $S$.", "This is true because $(A, \\Ker(A \\to A_n))$ is a henselian pair", "(More on Algebra, Lemma \\ref{more-algebra-lemma-limit-henselian})", "and hence every closed point of $S$ is contained in the image of $S_n \\to S$." ], "refs": [ "morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "more-algebra-lemma-limit-henselian" ], "ref_ids": [ 5235, 13901, 9858 ] } ], "ref_ids": [] }, { "id": 6114, "type": "theorem", "label": "flat-lemma-compute-what-it-should-be-derived", "categories": [ "flat" ], "title": "flat-lemma-compute-what-it-should-be-derived", "contents": [ "In Situation \\ref{situation-existence-derived} consider", "$$", "K = R\\lim_{D_\\QCoh(\\mathcal{O}_X)}(K_n) =", "DQ_X(R\\lim_{D(\\mathcal{O}_X)} K_n)", "$$", "Then $K$ is in $D^-_{\\QCoh}(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "The functor $DQ_X$ exists because $X$ is quasi-compact and", "quasi-separated, see Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-better-coherator}.", "Since $DQ_X$ is a right adjoint it commutes with products", "and therefore with derived limits. Hence the equality", "in the statement of the lemma.", "\\medskip\\noindent", "By Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-boundedness-better-coherator}", "the functor $DQ_X$ has bounded cohomological dimension.", "Hence it suffices to show that $R\\lim K_n \\in D^-(\\mathcal{O}_X)$.", "To see this, let $U \\subset X$ be an affine open.", "Then there is a canonical exact sequence", "$$", "0 \\to", "R^1\\lim H^{m - 1}(U, K_n) \\to H^m(U, R\\lim K_n) \\to", "\\lim H^m(U, K_n) \\to 0", "$$", "by Cohomology, Lemma \\ref{cohomology-lemma-RGamma-commutes-with-Rlim}.", "Since $U$ is affine and $K_n$ is pseudo-coherent (and hence has", "quasi-coherent cohomology sheaves by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-pseudo-coherent})", "we see that $H^m(U, K_n) = H^m(K_n)(U)$ by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded}.", "Thus we conclude that it suffices to show that $K_n$", "is bounded above independent of $n$.", "\\medskip\\noindent", "Since $K_n$ is pseudo-coherent we have $K_n \\in D^-(\\mathcal{O}_{X_n})$.", "Suppose that $a_n$ is maximal such that $H^{a_n}(K_n)$ is nonzero.", "Of course $a_1 \\leq a_2 \\leq a_3 \\leq \\ldots$.", "Note that $H^{a_n}(K_n)$ is an", "$\\mathcal{O}_{X_n}$-module of finite presentation", "(Cohomology, Lemma \\ref{cohomology-lemma-finite-cohomology}).", "We have $H^{a_n}(K_{n - 1}) =", "H^{a_n}(K_n) \\otimes_{\\mathcal{O}_{X_n}} \\mathcal{O}_{X_{n - 1}}$.", "Since $X_{n - 1} \\to X_n$ is a thickening, it follows from", "Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK}) that if", "$H^{a_n}(K_n) \\otimes_{\\mathcal{O}_{X_n}} \\mathcal{O}_{X_{n - 1}}$", "is zero, then $H^{a_n}(K_n)$ is zero too. Thus $a_n = a_{n - 1}$", "for all $n$ and we conclude." ], "refs": [ "perfect-lemma-better-coherator", "perfect-lemma-boundedness-better-coherator", "cohomology-lemma-RGamma-commutes-with-Rlim", "perfect-lemma-pseudo-coherent", "perfect-lemma-affine-compare-bounded", "cohomology-lemma-finite-cohomology", "algebra-lemma-NAK" ], "ref_ids": [ 7022, 7024, 2160, 6974, 6941, 2212, 401 ] } ], "ref_ids": [] }, { "id": 6115, "type": "theorem", "label": "flat-lemma-compute-against-perfect-derived", "categories": [ "flat" ], "title": "flat-lemma-compute-against-perfect-derived", "contents": [ "In Situation \\ref{situation-existence-derived} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be-derived}. For any perfect", "object $E$ of $D(\\mathcal{O}_X)$ the cohomology", "$$", "M = R\\Gamma(X, K \\otimes^\\mathbf{L} E)", "$$", "is a pseudo-coherent object of $D(A)$ and there is a canonical isomorphism", "$$", "R\\Gamma(X_n, K_n \\otimes^\\mathbf{L} E|_{X_n}) = M \\otimes_A^\\mathbf{L} A_n", "$$", "in $D(A_n)$. Here $E|_{X_n}$ denotes the derived pullback of $E$ to $X_n$." ], "refs": [ "flat-lemma-compute-what-it-should-be-derived" ], "proofs": [ { "contents": [ "Write $E_n = E|_{X_n}$ and", "$M_n = R\\Gamma(X_n, K_n \\otimes^\\mathbf{L} E|_{X_n})$.", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-pseudo-coherent-direct-image-general}", "we see that $M_n$ is a pseudo-coherent object of $D(A_n)$", "whose formation commutes with base change. Thus the maps", "$M_n \\otimes_{A_n}^\\mathbf{L} A_{n - 1} \\to M_{n - 1}$", "coming from $\\varphi_n$ are isomorphisms. By", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-Rlim-pseudo-coherent-gives-pseudo-coherent}", "we find that $R\\lim M_n$ is pseudo-coherent and", "that its base change back to $A_n$ recovers $M_n$.", "On the other hand, the exact functor", "$R\\Gamma(X, -) : D_\\QCoh(\\mathcal{O}_X) \\to D(A)$", "of triangulated categories commutes with products", "and hence with derived limits, whence", "$$", "R\\Gamma(X, E \\otimes^\\mathbf{L} K) =", "R\\lim R\\Gamma(X, E \\otimes^\\mathbf{L} K_n) =", "R\\lim R\\Gamma(X_n, E_n \\otimes^\\mathbf{L} K_n) =", "R\\lim M_n", "$$", "as desired." ], "refs": [ "perfect-lemma-flat-proper-pseudo-coherent-direct-image-general", "more-algebra-lemma-Rlim-pseudo-coherent-gives-pseudo-coherent" ], "ref_ids": [ 7055, 10407 ] } ], "ref_ids": [ 6114 ] }, { "id": 6116, "type": "theorem", "label": "flat-lemma-relative-pseudo-coherence-derived", "categories": [ "flat" ], "title": "flat-lemma-relative-pseudo-coherence-derived", "contents": [ "In Situation \\ref{situation-existence-derived} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be-derived}. Then $K$", "is pseudo-coherent on $X$." ], "refs": [ "flat-lemma-compute-what-it-should-be-derived" ], "proofs": [ { "contents": [ "Combinging Lemma \\ref{lemma-compute-against-perfect-derived} and", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-perfect-enough}", "we see that $R\\Gamma(X, K \\otimes^\\mathbf{L} E)$", "is pseudo-coherent in $D(A)$ for all pseudo-coherent", "$E$ in $D(\\mathcal{O}_X)$. Thus it follows from", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-characterize-pseudo-coherent}", "that $K$ is pseudo-coherent relative to $A$.", "Since $X$ is of flat and of finite presentation", "over $A$, this is the same as being pseudo-coherent on $X$, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-check-relative-pseudo-coherence-on-charts}." ], "refs": [ "flat-lemma-compute-against-perfect-derived", "perfect-lemma-perfect-enough", "more-morphisms-lemma-characterize-pseudo-coherent", "more-morphisms-lemma-check-relative-pseudo-coherence-on-charts" ], "ref_ids": [ 6115, 7076, 14060, 13974 ] } ], "ref_ids": [ 6114 ] }, { "id": 6117, "type": "theorem", "label": "flat-lemma-compute-over-affine-derived", "categories": [ "flat" ], "title": "flat-lemma-compute-over-affine-derived", "contents": [ "In Situation \\ref{situation-existence-derived} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be-derived}. For any quasi-compact", "open $U \\subset X$ we have", "$$", "R\\Gamma(U, K) \\otimes_A^\\mathbf{L} A_n =", "R\\Gamma(U_n, K_n)", "$$", "in $D(A_n)$ where $U_n = U \\cap X_n$." ], "refs": [ "flat-lemma-compute-what-it-should-be-derived" ], "proofs": [ { "contents": [ "Fix $n$. By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-computing-sections-as-colim}", "there exists a system of perfect complexes $E_m$", "on $X$ such that", "$R\\Gamma(U, K) = \\text{hocolim} R\\Gamma(X, K \\otimes^\\mathbf{L} E_m)$.", "In fact, this formula holds not just for $K$ but for every object of", "$D_\\QCoh(\\mathcal{O}_X)$.", "Applying this to $K_n$", "we obtain", "\\begin{align*}", "R\\Gamma(U_n, K_n)", "& =", "R\\Gamma(U, K_n) \\\\", "& =", "\\text{hocolim}_m R\\Gamma(X, K_n \\otimes^\\mathbf{L} E_m) \\\\", "& =", "\\text{hocolim}_m R\\Gamma(X_n, K_n \\otimes^\\mathbf{L} E_m|_{X_n})", "\\end{align*}", "Using Lemma \\ref{lemma-compute-against-perfect-derived}", "and the fact that $- \\otimes_A^\\mathbf{L} A_n$", "commutes with homotopy colimits we obtain the result." ], "refs": [ "perfect-lemma-computing-sections-as-colim", "flat-lemma-compute-against-perfect-derived" ], "ref_ids": [ 7073, 6115 ] } ], "ref_ids": [ 6114 ] }, { "id": 6118, "type": "theorem", "label": "flat-lemma-helper-blowup-affine-space", "categories": [ "flat" ], "title": "flat-lemma-helper-blowup-affine-space", "contents": [ "Let $R$ be a ring and let $f \\in R$. Let $r\\geq 0$ be an integer.", "Let $R \\to S$ be a ring map and let $M$ be an $S$-module. Assume", "\\begin{enumerate}", "\\item $R \\to S$ is of finite presentation and flat,", "\\item every fibre ring $S \\otimes_R \\kappa(\\mathfrak p)$ is", "geometrically integral over $R$,", "\\item $M$ is a finite $S$-module,", "\\item $M_f$ is a finitely presented $S_f$-module,", "\\item for all $\\mathfrak p \\in R$, $f \\not \\in \\mathfrak p$ with", "$\\mathfrak q = \\mathfrak pS$ the module $M_{\\mathfrak q}$ is free", "of rank $r$ over $S_\\mathfrak q$.", "\\end{enumerate}", "Then there exists a finitely generated ideal $I \\subset R$ with", "$V(f) = V(I)$ such that for all $a \\in I$ with $R' = R[\\frac{I}{a}]$", "the quotient", "$$", "M' = (M \\otimes_R R')/a\\text{-power torsion}", "$$", "over $S' = S \\otimes_R R'$ satisfies the following: for every prime", "$\\mathfrak p' \\subset R'$ there exists a $g \\in S'$,", "$g \\not \\in \\mathfrak p'S'$ such that $M'_g$ is a free $S'_g$-module", "of rank $r$." ], "refs": [], "proofs": [ { "contents": [ "This lemma is a generalization of", "More on Algebra, Lemma \\ref{more-algebra-lemma-blowup-module};", "we urge the reader to read that proof first.", "Choose a surjection $S^{\\oplus n} \\to M$, which is possible by (1).", "Choose a finite submodule $K \\subset \\Ker(S^{\\oplus n} \\to M)$", "such that $S^{\\oplus n}/K \\to M$ becomes an isomorphism after inverting $f$.", "This is possible by (4). Set $M_1 = S^{\\oplus n}/K$ and suppose we can", "prove the lemma for $M_1$. Say $I \\subset R$ is the corresponding ideal.", "Then for $a \\in I$ the map", "$$", "M_1' = (M_1 \\otimes_R R')/a\\text{-power torsion}", "\\longrightarrow", "M' = (M \\otimes_R R')/a\\text{-power torsion}", "$$", "is surjective. It is also an isomorphism after inverting $a$ in $R'$", "as $R'_a = R_f$, see Algebra, Lemma \\ref{algebra-lemma-blowup-in-principal}.", "But $a$ is a nonzerodivisor on $M'_1$, whence the displayed map is an", "isomorphism. Thus it suffices to prove the lemma in case $M$ is a finitely", "presented $S$-module.", "\\medskip\\noindent", "Assume $M$ is a finitely presented $S$-module satisfying (3).", "Then $J = \\text{Fit}_r(M) \\subset S$ is a finitely generated ideal.", "By Lemma \\ref{lemma-fibres-irreducible-flat-projective-nonnoetherian}", "we can write $S$ as a direct summand of a free", "$R$-module: $\\bigoplus_{\\alpha \\in A} R = S \\oplus C$.", "For any element $h \\in S$ writing $h = \\sum a_\\alpha$ in the", "decomposition above, we say that the $a_\\alpha$ are the coefficients of $h$.", "Let $I' \\subset R$ be the ideal of coefficients", "of elements of $J$. Multiplication by an element of $S$ defines", "an $R$-linear map $S \\to S$, hence $I'$ is generated by the coefficients", "of the generators of $J$, i.e., $I'$ is a finitely generated ideal.", "We claim that $I = fI'$ works.", "\\medskip\\noindent", "We first check that $V(f) = V(I)$. The inclusion $V(f) \\subset V(I)$ is", "clear. Conversely, if $f \\not \\in \\mathfrak p$, then", "$\\mathfrak q = \\mathfrak p S$ is not an element of $V(J)$ by", "property (5) and More on Algebra, Lemma", "\\ref{more-algebra-lemma-fitting-ideal-generate-locally}.", "Hence there is an", "element of $J$ which does not map to zero in $S \\otimes_R \\kappa(\\mathfrak p)$.", "Thus there exists an element of $I'$ which is not contained in", "$\\mathfrak p$, so $\\mathfrak p \\not \\in V(fI') = V(I)$.", "\\medskip\\noindent", "Let $a \\in I$ and set $R' = R[\\frac{I}{a}]$. We may write $a = fa'$", "for some $a' \\in I'$. By Algebra, Lemmas \\ref{algebra-lemma-affine-blowup} and", "\\ref{algebra-lemma-blowup-add-principal} we see that $I' R' = a'R'$", "and $a'$ is a nonzerodivisor in $R'$. Set $S' = S \\otimes_S R'$.", "Every element $g$ of $JS' = \\text{Fit}_r(M \\otimes_S S')$ can be", "written as $g = \\sum_\\alpha c_\\alpha$ for some $c_\\alpha \\in I'R'$.", "Since $I'R' = a'R'$ we can write $c_\\alpha = a'c'_\\alpha$ for some", "$c'_\\alpha \\in R'$ and $g = (\\sum c'_\\alpha)a' = g' a'$ in $S'$.", "Moreover, there is an $g_0 \\in J$ such that $a' = c_\\alpha$", "for some $\\alpha$. For this element we have $g_0 = g'_0 a'$ in $S'$", "where $g'_0$ is a unit in $S'$.", "Let $\\mathfrak p' \\subset R'$ be", "a prime ideal and $\\mathfrak q' = \\mathfrak p'S'$.", "By the above we see that $JS'_{\\mathfrak q'}$ is the", "principal ideal generated by the nonzerodivisor $a'$.", "It follows from More on Algebra, Lemma", "\\ref{more-algebra-lemma-principal-fitting-ideal}", "that $M'_{\\mathfrak q'}$ can be generated by $r$ elements.", "Since $M'$ is finite, there exist $m_1, \\ldots, m_r \\in M'$ and", "$g \\in S'$, $g \\not \\in \\mathfrak q'$ such that the corresponding map", "$(S')^{\\oplus r} \\to M'$ becomes surjective after inverting $g$.", "\\medskip\\noindent", "Finally, consider the ideal $J' = \\text{Fit}_{k - 1}(M')$.", "Note that $J'S'_g$ is generated by the coefficients of relations between", "$m_1, \\ldots, m_r$ (compatibility of Fitting ideal with base change).", "Thus it suffices to show that $J' = 0$, see", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-fitting-ideal-finite-locally-free}.", "Since $R'_a = R_f$ (Algebra, Lemma \\ref{algebra-lemma-blowup-in-principal})", "and $M'_a = M_f$ we see from (5)", "that $J'_a$ maps to zero in $S_{\\mathfrak q''}$ for any prime", "$\\mathfrak q'' \\subset S'$ of the form $\\mathfrak q'' = \\mathfrak p''S'$", "where $\\mathfrak p'' \\subset R'_a$. Since", "$S'_a \\subset \\prod_{\\mathfrak q''\\text{ as above}} S'_{\\mathfrak q''}$", "(as $(S'_a)_{\\mathfrak p''} \\subset S'_{\\mathfrak q''}$ by", "Lemma \\ref{lemma-base-change-universally-flat})", "we see that $J'R'_a = 0$. Since $a$ is a nonzerodivisor in $R'$ we", "conclude that $J' = 0$ and we win." ], "refs": [ "more-algebra-lemma-blowup-module", "algebra-lemma-blowup-in-principal", "flat-lemma-fibres-irreducible-flat-projective-nonnoetherian", "more-algebra-lemma-fitting-ideal-generate-locally", "algebra-lemma-affine-blowup", "algebra-lemma-blowup-add-principal", "more-algebra-lemma-principal-fitting-ideal", "more-algebra-lemma-fitting-ideal-finite-locally-free", "algebra-lemma-blowup-in-principal", "flat-lemma-base-change-universally-flat" ], "ref_ids": [ 9951, 755, 6014, 9835, 752, 756, 9837, 9836, 755, 6005 ] } ], "ref_ids": [] }, { "id": 6119, "type": "theorem", "label": "flat-lemma-flatten-module-pre", "categories": [ "flat" ], "title": "flat-lemma-flatten-module-pre", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent module on $X$.", "Let $U \\subset S$ be a quasi-compact open. Assume", "\\begin{enumerate}", "\\item $X \\to S$ is affine, of finite presentation, flat,", "geometrically integral fibres,", "\\item $\\mathcal{F}$ is a module of finite type,", "\\item $\\mathcal{F}_U$ is of finite presentation,", "\\item $\\mathcal{F}$ is flat over $S$ at all generic points of", "fibres lying over points of $U$.", "\\end{enumerate}", "Then there exists a $U$-admissible blowup $S' \\to S$", "and an open subscheme $V \\subset X_{S'}$", "such that (a) the strict transform $\\mathcal{F}'$ of $\\mathcal{F}$", "restricts to a finitely locally free $\\mathcal{O}_V$-module and", "(b) $V \\to S'$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Given $\\mathcal{F}/X/S$ and $U \\subset S$ with hypotheses as", "in the lemma, denote $P$ the property ``$\\mathcal{F}$ is flat over $S$ at", "all generic points of fibres''. It is clear that $P$ is preserved under", "strict transform, see", "Divisors, Lemma \\ref{divisors-lemma-strict-transform-flat}", "and Morphisms, Lemma \\ref{morphisms-lemma-base-change-module-flat}.", "It is also clear that $P$ is local on $S$. Hence any and all observations", "of Remark \\ref{remark-successive-blowups} apply to the problem posed by", "the lemma.", "\\medskip\\noindent", "Consider the function $r : U \\to \\mathbf{Z}_{\\geq 0}$ which assigns to", "$u \\in U$ the integer", "$$", "r(u) = \\dim_{\\kappa(\\xi_u)}(\\mathcal{F}_{\\xi_u} \\otimes \\kappa(\\xi_u))", "$$", "where $\\xi_u$ is the generic point of the fibre $X_u$.", "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-flat-and-free-at-point-fibre}", "and the fact that the image of an open in $X_S$ in $S$ is open,", "we see that $r(u)$ is locally constant. Accordingly", "$U = U_0 \\amalg U_1 \\amalg \\ldots \\amalg U_c$ is a finite disjoint", "union of open and closed subschemes where $r$ is constant with value", "$i$ on $U_i$. By", "Divisors, Lemma \\ref{divisors-lemma-separate-disjoint-opens-by-blowing-up}", "we can find a $U$-admissible blowup to decompose $S$ into", "the disjoint union of two schemes, the first containing $U_0$ and", "the second $U_1 \\cup \\ldots \\cup U_c$. Repeating this $c - 1$", "more times we may assume that $S$ is a disjoint union", "$S = S_0 \\amalg S_1 \\amalg \\ldots \\amalg S_c$ with $U_i \\subset S_i$.", "Thus we may assume the function $r$ defined above is constant, say", "with value $r$.", "\\medskip\\noindent", "By Remark \\ref{remark-successive-blowups} we see that we may assume that", "we have an effective Cartier divisor $D \\subset S$ whose support is", "$S \\setminus U$. Another application of Remark \\ref{remark-successive-blowups}", "combined with", "Divisors, Lemma \\ref{divisors-lemma-characterize-effective-Cartier-divisor}", "tells us we may assume that", "$S = \\Spec(R)$ and $D = \\Spec(R/(f))$ for some nonzerodivisor", "$f \\in R$. This case is handled by", "Lemma \\ref{lemma-helper-blowup-affine-space}." ], "refs": [ "divisors-lemma-strict-transform-flat", "morphisms-lemma-base-change-module-flat", "flat-remark-successive-blowups", "more-morphisms-lemma-flat-and-free-at-point-fibre", "divisors-lemma-separate-disjoint-opens-by-blowing-up", "flat-remark-successive-blowups", "flat-remark-successive-blowups", "divisors-lemma-characterize-effective-Cartier-divisor", "flat-lemma-helper-blowup-affine-space" ], "ref_ids": [ 8066, 5264, 6234, 13771, 8074, 6234, 6234, 7927, 6118 ] } ], "ref_ids": [] }, { "id": 6120, "type": "theorem", "label": "flat-lemma-trick-fitting-ideal", "categories": [ "flat" ], "title": "flat-lemma-trick-fitting-ideal", "contents": [ "Let $A \\to C$ be a finite locally free ring map of rank $d$.", "Let $h \\in C$ be an element such that $C_h$ is \\'etale over $A$.", "Let $J \\subset C$ be an ideal. Set $I = \\text{Fit}_0(C/J)$ where we", "think of $C/J$ as a finite $A$-module. Then $IC_h = JJ'$ for some ideal", "$J' \\subset C_h$. If $J$ is finitely generated so are $I$ and $J'$." ], "refs": [], "proofs": [ { "contents": [ "We will use basic properties of Fitting ideals, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-fitting-ideal-basics}.", "Then $IC$ is the Fitting ideal of $C/J \\otimes_A C$.", "Note that $C \\to C \\otimes_A C$, $c \\mapsto 1 \\otimes c$ has a section", "(the multiplication map). By assumption $C \\to C \\otimes_A C$ is \\'etale", "at every prime in the image of $\\Spec(C_h)$ under this section.", "Hence the multiplication map $C \\otimes_A C_h \\to C_h$ is \\'etale", "in particular flat, see Algebra, Lemma \\ref{algebra-lemma-map-between-etale}.", "Hence there exists a $C_h$-algebra such that", "$C \\otimes_A C_h \\cong C_h \\oplus C'$ as $C_h$-algebras, see", "Algebra, Lemma \\ref{algebra-lemma-surjective-flat-finitely-presented}.", "Thus $(C/J) \\otimes_A C_h \\cong (C_h/J_h) \\oplus C'/I'$ as", "$C_h$-modules for some ideal $I' \\subset C'$.", "Hence $IC_h = JJ'$ with $J' = \\text{Fit}_0(C'/I')$ where we view", "$C'/J'$ as a $C_h$-module." ], "refs": [ "more-algebra-lemma-fitting-ideal-basics", "algebra-lemma-map-between-etale", "algebra-lemma-surjective-flat-finitely-presented" ], "ref_ids": [ 9834, 1236, 1237 ] } ], "ref_ids": [] }, { "id": 6121, "type": "theorem", "label": "flat-lemma-push-ideal", "categories": [ "flat" ], "title": "flat-lemma-push-ideal", "contents": [ "Let $A \\to B$ be an \\'etale ring map. Let $a \\in A$ be a nonzerodivisor.", "Let $J \\subset B$ be a finite type ideal with $V(J) \\subset V(aB)$.", "For every $\\mathfrak q \\subset B$ there exists a finite type ideal", "$I \\subset A$ with $V(I) \\subset V(a)$ and", "$g \\in B$, $g \\not \\in \\mathfrak q$ such that", "$IB_g = JJ'$ for some finite type ideal $J' \\subset B_g$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $B$ by a principal localization at", "an element $g \\in B$, $g \\not \\in \\mathfrak q$. Thus we may assume", "that $B$ is standard \\'etale, see", "Algebra, Proposition \\ref{algebra-proposition-etale-locally-standard}.", "Thus we may assume $B$ is a localization", "of $C = A[x]/(f)$ for some monic $f \\in A[x]$ of some degree $d$.", "Say $B = C_h$ for some $h \\in C$. Choose elements $h_1, \\ldots, h_n \\in C$", "which generate $J$", "over $B$. The condition $V(J) \\subset V(aB)$ signifies that", "$a^m = \\sum b_i h_i$ in $B$ for some large $m$. Set $h_{n + 1} = a^m$.", "As in Lemma \\ref{lemma-trick-fitting-ideal} we take", "$I = \\text{Fit}_0(C/(h_1, \\ldots, h_{r + 1}))$.", "Since the module $C/(h_1, \\ldots, h_{r + 1})$", "is annihilated by $a^m$ we see that $a^{dm} \\in I$ which implies", "that $V(I) \\subset V(a)$." ], "refs": [ "algebra-proposition-etale-locally-standard", "flat-lemma-trick-fitting-ideal" ], "ref_ids": [ 1427, 6120 ] } ], "ref_ids": [] }, { "id": 6122, "type": "theorem", "label": "flat-lemma-flatten-module-etale-localize", "categories": [ "flat" ], "title": "flat-lemma-flatten-module-etale-localize", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent module on $X$.", "Let $U \\subset S$ be a quasi-compact open.", "Assume there exist finitely many commutative diagrams", "$$", "\\xymatrix{", "& X_i \\ar[r]_{j_i} \\ar[d] & X \\ar[d] \\\\", "S_i^* \\ar[r] & S_i \\ar[r]^{e_i} & S", "}", "$$", "where", "\\begin{enumerate}", "\\item $e_i : S_i \\to S$ are quasi-compact \\'etale morphisms and", "$S = \\bigcup e_i(S_i)$,", "\\item $j_i : X_i \\to X$ are \\'etale morphisms and", "$X = \\bigcup j_i(X_i)$,", "\\item $S^*_i \\to S_i$ is an $e_i^{-1}(U)$-admissible blowup", "such that the strict transform $\\mathcal{F}_i^*$ of $j_i^*\\mathcal{F}$", "is flat over $S^*_i$.", "\\end{enumerate}", "Then there exists a $U$-admissible blowup $S' \\to S$ such that", "the strict transform of $\\mathcal{F}$ is flat over $S'$." ], "refs": [], "proofs": [ { "contents": [ "We claim that the hypotheses of the lemma are preserved under $U$-admissible", "blowups. Namely, suppose $b : S' \\to S$ is a $U$-admissible blowup in", "the quasi-coherent sheaf of ideals $\\mathcal{I}$. Moreover, let $S^*_i \\to S_i$", "be the blowup in the quasi-coherent sheaf of ideals $\\mathcal{J}_i$.", "Then the collection of morphisms $e'_i : S'_i = S_i \\times_S S' \\to S'$", "and $j'_i : X_i' = X_i \\times_S S' \\to X \\times_S S'$ satisfy conditions", "(1), (2), (3) for the strict transform $\\mathcal{F}'$ of $\\mathcal{F}$", "relative to the blowup $S' \\to S$. First, observe that $S_i'$ is the", "blowup of $S_i$ in the pullback of $\\mathcal{I}$, see", "Divisors, Lemma \\ref{divisors-lemma-flat-base-change-blowing-up}.", "Second, consider the blowup", "$S_i^{\\prime *} \\to S_i'$ of $S_i'$ in the pullback of the ideal", "$\\mathcal{J}_i$.", "By Divisors, Lemma \\ref{divisors-lemma-blowing-up-two-ideals}", "we get a commutative diagram", "$$", "\\xymatrix{", "S_i^{\\prime *} \\ar[r] \\ar[rd] \\ar[d] & S'_i \\ar[d] \\\\", "S_i^* \\ar[r] & S_i", "}", "$$", "and all the morphisms in the diagram above are blowups. Hence by", "Divisors, Lemmas \\ref{divisors-lemma-strict-transform-flat} and", "\\ref{divisors-lemma-strict-transform-composition-blowups}", "we see", "\\begin{align*}", "& \\text{ the strict transform of }(j'_i)^*\\mathcal{F}'\\text{ under }", "S_i^{\\prime *} \\to S_i' \\\\", "= &", "\\text{ the strict transform of }j_i^*\\mathcal{F}\\text{ under }", "S_i^{\\prime *} \\to S_i \\\\", "= &", "\\text{ the strict transform of }\\mathcal{F}_i'\\text{ under }", "S_i^{\\prime *} \\to S_i' \\\\", "= &", "\\text{ the pullback of }\\mathcal{F}_i^*\\text{ via }", "X_i \\times_{S_i} S_i^{\\prime *} \\to X_i", "\\end{align*}", "which is therefore flat over $S_i^{\\prime *}$", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-module-flat}).", "Having said this, we see that all observations of", "Remark \\ref{remark-successive-blowups} apply to the", "problem of finding a $U$-admissible blowup such that the", "strict transform of $\\mathcal{F}$ becomes flat over the base", "under assumptions as in the lemma. In particular, we may assume", "that $S \\setminus U$ is the support of an effective Cartier divisor", "$D \\subset S$. Another application of Remark \\ref{remark-successive-blowups}", "combined with", "Divisors, Lemma \\ref{divisors-lemma-characterize-effective-Cartier-divisor}", "shows we may assume that $S = \\Spec(A)$ and $D = \\Spec(A/(a))$", "for some nonzerodivisor $a \\in A$.", "\\medskip\\noindent", "Pick an $i$ and $s \\in S_i$. Lemma \\ref{lemma-push-ideal}", "implies we can find an open neighbourhood $s \\in W_i \\subset S_i$", "and a finite type quasi-coherent ideal $\\mathcal{I} \\subset \\mathcal{O}_S$", "such that $\\mathcal{I} \\cdot \\mathcal{O}_{W_i} = \\mathcal{J}_i \\mathcal{J}'_i$", "for some finite type quasi-coherent ideal", "$\\mathcal{J}'_i \\subset \\mathcal{O}_{W_i}$", "and such that $V(\\mathcal{I}) \\subset V(a) = S \\setminus U$.", "Since $S_i$ is quasi-compact we can replace $S_i$ by a finite collection", "$W_1, \\ldots, W_n$ of these opens and assume that for each $i$ there exists", "a quasi-coherent sheaf of ideals $\\mathcal{I}_i \\subset \\mathcal{O}_S$ such", "that $\\mathcal{I}_i \\cdot \\mathcal{O}_{S_i} = \\mathcal{J}_i \\mathcal{J}'_i$", "for some finite type quasi-coherent ideal", "$\\mathcal{J}'_i \\subset \\mathcal{O}_{S_i}$.", "As in the discussion of the first paragraph of the proof, consider the", "blowup $S'$ of $S$ in the product $\\mathcal{I}_1 \\ldots \\mathcal{I}_n$", "(this blowup is $U$-admissible by construction). The base change of $S' \\to S$", "to $S_i$ is the blowup in", "$$", "\\mathcal{J}_i \\cdot", "\\mathcal{J}'_i \\mathcal{I}_1 \\ldots \\hat{\\mathcal{I}_i} \\ldots \\mathcal{I}_n", "$$", "which factors through the given blowup $S_i^* \\to S_i$", "(Divisors, Lemma \\ref{divisors-lemma-blowing-up-two-ideals}). In the notation", "of the diagram above this means that $S_i^{\\prime *} = S_i'$. Hence", "after replacing $S$ by $S'$ we arrive in the situation that", "$j_i^*\\mathcal{F}$ is flat over $S_i$. Hence $j_i^*\\mathcal{F}$ is flat", "over $S$, see", "Lemma \\ref{lemma-etale-flat-up-down}.", "By Morphisms, Lemma \\ref{morphisms-lemma-flat-permanence}", "we see that $\\mathcal{F}$ is flat over $S$." ], "refs": [ "divisors-lemma-flat-base-change-blowing-up", "divisors-lemma-blowing-up-two-ideals", "divisors-lemma-strict-transform-flat", "divisors-lemma-strict-transform-composition-blowups", "morphisms-lemma-base-change-module-flat", "flat-remark-successive-blowups", "flat-remark-successive-blowups", "divisors-lemma-characterize-effective-Cartier-divisor", "flat-lemma-push-ideal", "divisors-lemma-blowing-up-two-ideals", "flat-lemma-etale-flat-up-down", "morphisms-lemma-flat-permanence" ], "ref_ids": [ 8053, 8062, 8066, 8069, 5264, 6234, 6234, 7927, 6121, 8062, 5980, 5270 ] } ], "ref_ids": [] }, { "id": 6123, "type": "theorem", "label": "flat-lemma-flat-after-blowing-up", "categories": [ "flat" ], "title": "flat-lemma-flat-after-blowing-up", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $X$ be a scheme over $S$.", "Let $U \\subset S$ be a quasi-compact open.", "Assume", "\\begin{enumerate}", "\\item $X \\to S$ is of finite type and quasi-separated, and", "\\item $X_U \\to U$ is flat and locally of finite presentation.", "\\end{enumerate}", "Then there exists a $U$-admissible blowup $S' \\to S$ such that", "the strict transform of $X$ is flat and of finite presentation", "over $S'$." ], "refs": [], "proofs": [ { "contents": [ "Since $X \\to S$ is quasi-compact and quasi-separated by assumption,", "the strict transform of $X$ with respect to a blowing up $S' \\to S$", "is also quasi-compact and quasi-separated. Hence to prove the lemma", "it suffices to find a $U$-admissible blowup such that the strict", "transform is flat and locally of finite presentation.", "Let $X = W_1 \\cup \\ldots \\cup W_n$ be a finite affine open covering.", "If we can find a $U$-admissible blowup $S_i \\to S$ such that the", "strict transform of $W_i$ is flat and locally of finite presentation,", "then there exists a $U$-admissible blowing up $S' \\to S$ dominating", "all $S_i \\to S$ which does the job (see", "Divisors, Lemma \\ref{divisors-lemma-dominate-admissible-blowups};", "see also Remark \\ref{remark-successive-blowups}).", "Hence we may assume $X$ is affine.", "\\medskip\\noindent", "Assume $X$ is affine. By", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-affine-finite-type-over-S}", "we can choose an immersion $j : X \\to \\mathbf{A}^n_S$ over $S$.", "Let $V \\subset \\mathbf{A}^n_S$ be a quasi-compact open subscheme", "such that $j$ induces a closed immersion $i : X \\to V$ over $S$. Apply", "Theorem \\ref{theorem-flatten-module}", "to $V \\to S$ and the quasi-coherent module $i_*\\mathcal{O}_X$", "to obtain a $U$-admissible blowup $S' \\to S$ such that the strict", "transform of $i_*\\mathcal{O}_X$ is flat over $S'$ and of finite presentation", "over $\\mathcal{O}_{V \\times_S S'}$. Let $X'$ be the strict transform", "of $X$ with respect to $S' \\to S$. Let $i' : X' \\to V \\times_S S'$", "be the induced morphism.", "Since taking strict transform commutes with pushforward along affine", "morphisms (Divisors, Lemma \\ref{divisors-lemma-strict-transform-affine}),", "we see that $i'_*\\mathcal{O}_{X'}$ is flat over $S$ and of", "finite presentation as a $\\mathcal{O}_{V \\times_S S'}$-module.", "This implies the lemma." ], "refs": [ "divisors-lemma-dominate-admissible-blowups", "flat-remark-successive-blowups", "morphisms-lemma-quasi-affine-finite-type-over-S", "flat-theorem-flatten-module", "divisors-lemma-strict-transform-affine" ], "ref_ids": [ 8073, 6234, 5392, 5975, 8067 ] } ], "ref_ids": [] }, { "id": 6124, "type": "theorem", "label": "flat-lemma-finite-after-blowing-up", "categories": [ "flat" ], "title": "flat-lemma-finite-after-blowing-up", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $X$ be a scheme over $S$.", "Let $U \\subset S$ be a quasi-compact open.", "Assume", "\\begin{enumerate}", "\\item $X \\to S$ is proper, and", "\\item $X_U \\to U$ is finite locally free.", "\\end{enumerate}", "Then there exists a $U$-admissible blowup $S' \\to S$ such that", "the strict transform of $X$ is finite locally free over $S'$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-flat-after-blowing-up} we may assume that", "$X \\to S$ is flat and of finite presentation. After replacing", "$S$ by a $U$-admissible blowup if necessary, we may assume", "that $U \\subset S$ is scheme theoretically dense. Then $f$ is", "finite by Lemma \\ref{lemma-proper-flat-finite-over-dense-open}.", "Hence $f$ is finite locally free by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}." ], "refs": [ "flat-lemma-flat-after-blowing-up", "flat-lemma-proper-flat-finite-over-dense-open", "morphisms-lemma-finite-flat" ], "ref_ids": [ 6123, 6027, 5471 ] } ], "ref_ids": [] }, { "id": 6125, "type": "theorem", "label": "flat-lemma-zariski-after-blowup", "categories": [ "flat" ], "title": "flat-lemma-zariski-after-blowup", "contents": [ "Let $\\varphi : X \\to S$ be a separated morphism of finite type with", "$S$ quasi-compact and quasi-separated. Let $U \\subset S$ be a", "quasi-compact open such that $\\varphi^{-1}U \\to U$ is an isomorphism.", "Then there exists a $U$-admissible blowup $S' \\to S$ such that", "the strict transform $X'$ of $X$ is isomorphic to an open subscheme", "of $S'$." ], "refs": [], "proofs": [ { "contents": [ "The discussion in Remark \\ref{remark-successive-blowups} applies.", "Thus we may do a first $U$-admissible blowup and assume the complement", "$S \\setminus U$ is the support of an effective Cartier divisor $D$.", "In particular $U$ is scheme theoretically dense in $S$.", "Next, we do another $U$-admissible blowup to get to the situation where", "$X \\to S$ is flat and of finite presentation, see", "Lemma \\ref{lemma-flat-after-blowing-up}.", "In this case the result follows from Lemma \\ref{lemma-zariski}." ], "refs": [ "flat-remark-successive-blowups", "flat-lemma-flat-after-blowing-up", "flat-lemma-zariski" ], "ref_ids": [ 6234, 6123, 6028 ] } ], "ref_ids": [] }, { "id": 6126, "type": "theorem", "label": "flat-lemma-dominate-modification-by-blowup", "categories": [ "flat" ], "title": "flat-lemma-dominate-modification-by-blowup", "contents": [ "Let $\\varphi : X \\to S$ be a proper morphism with", "$S$ quasi-compact and quasi-separated. Let $U \\subset S$ be a", "quasi-compact open such that $\\varphi^{-1}U \\to U$ is an isomorphism.", "Then there exists a $U$-admissible blowup $S' \\to S$", "which dominates $X$, i.e., such that there exists a factorization", "$S' \\to X \\to S$ of the blowup morphism." ], "refs": [], "proofs": [ { "contents": [ "The discussion in Remark \\ref{remark-successive-blowups} applies.", "Thus we may do a first $U$-admissible blowup and assume the complement", "$S \\setminus U$ is the support of an effective Cartier divisor $D$.", "In particular $U$ is scheme theoretically dense in $S$.", "Choose another $U$-admissible blowup $S' \\to S$ such that the strict", "transform $X'$ of $X$ is an open subscheme of $S'$, see", "Lemma \\ref{lemma-zariski-after-blowup}.", "Since $X' \\to S'$ is proper, and", "$U \\subset S'$ is dense, we see that $X' = S'$. Some details omitted." ], "refs": [ "flat-remark-successive-blowups", "flat-lemma-zariski-after-blowup" ], "ref_ids": [ 6234, 6125 ] } ], "ref_ids": [] }, { "id": 6127, "type": "theorem", "label": "flat-lemma-get-section-after-blowup", "categories": [ "flat" ], "title": "flat-lemma-get-section-after-blowup", "contents": [ "Let $S$ be a scheme. Let $U \\subset W \\subset S$ be open subschemes.", "Let $f : X \\to W$ be a morphism and let $s : U \\to X$ be a", "morphism such that $f \\circ s = \\text{id}_U$. Assume", "\\begin{enumerate}", "\\item $f$ is proper,", "\\item $S$ is quasi-compact and quasi-separated, and", "\\item $U$ and $W$ are quasi-compact.", "\\end{enumerate}", "Then there exists a $U$-admissible blowup $b : S' \\to S$ and a morphism", "$s' : b^{-1}(W) \\to X$ extending $s$ with $f \\circ s' = b|_{b^{-1}(W)}$." ], "refs": [], "proofs": [ { "contents": [ "We may and do replace $X$ by the scheme theoretic image of $s$.", "Then $X \\to W$ is an isomorphism over $U$, see", "Morphisms, Lemma", "\\ref{morphisms-lemma-scheme-theoretic-image-of-partial-section}.", "By Lemma \\ref{lemma-dominate-modification-by-blowup}", "there exists a $U$-admissible blowup $W' \\to W$ and an", "extension $W' \\to X$ of $s$.", "We finish the proof by applying", "Divisors, Lemma \\ref{divisors-lemma-extend-admissible-blowups}", "to extend $W' \\to W$ to a $U$-admissible blowup of $S$." ], "refs": [ "flat-lemma-dominate-modification-by-blowup", "divisors-lemma-extend-admissible-blowups" ], "ref_ids": [ 6126, 8072 ] } ], "ref_ids": [] }, { "id": 6128, "type": "theorem", "label": "flat-lemma-compactifications-cofiltered", "categories": [ "flat" ], "title": "flat-lemma-compactifications-cofiltered", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $X$ be a compactifyable scheme over $S$.", "\\begin{enumerate}", "\\item[(a)] The category of compactifications of $X$ over $S$ is", "cofiltered.", "\\item[(b)] The full subcategory consisting of compactifications", "$j : X \\to \\overline{X}$ such that $j(X)$ is dense and", "scheme theoretically dense in $\\overline{X}$ is initial", "(Categories, Definition \\ref{categories-definition-initial}).", "\\item[(c)] If $f : \\overline{X}' \\to \\overline{X}$ is a morphism", "of compactifications of $X$ such that $j'(X)$ is dense in $\\overline{X}'$,", "then $f^{-1}(j(X)) = j'(X)$.", "\\end{enumerate}" ], "refs": [ "categories-definition-initial" ], "proofs": [ { "contents": [ "To prove part (a) we have to check conditions (1), (2), (3) of", "Categories, Definition \\ref{categories-definition-codirected}.", "Condition (1) holds exactly because we assumed that $X$", "is compactifyable.", "Let $j_i : X \\to \\overline{X}_i$, $i = 1, 2$ be two compactifications.", "Then we can consider the scheme theoretic image $\\overline{X}$", "of $(j_1, j_2) : X \\to \\overline{X}_1 \\times_S \\overline{X}_2$.", "This determines a third compactification $j : X \\to \\overline{X}$", "which dominates both $j_i$:", "$$", "\\xymatrix{", "(X, \\overline{X}_1) & (X, \\overline{X}) \\ar[l] \\ar[r] & (X, \\overline{X}_2)", "}", "$$", "Thus (2) holds. Let $f_1, f_2 : \\overline{X}_1 \\to \\overline{X}_2$", "be two morphisms between compactifications", "$j_i : X \\to \\overline{X}_i$, $i = 1, 2$.", "Let $\\overline{X} \\subset \\overline{X}_1$ be the equalizer of", "$f_1$ and $f_2$. As $\\overline{X}_2 \\to S$ is separated, we see", "that $\\overline{X}$ is a closed subscheme of $\\overline{X}_1$", "and hence proper over $S$. Moreover, we obtain an", "open immersion $X \\to \\overline{X}$ because $f_1|_X = f_2|_X = \\text{id}_X$.", "The morphism $(X \\to \\overline{X}) \\to (j_1 : X \\to \\overline{X}_1)$", "given by the closed immersion $\\overline{X} \\to \\overline{X}_1$", "equalizes $f_1$ and $f_2$ which proves condition (3).", "\\medskip\\noindent", "Proof of (b). Let $j : X \\to \\overline{X}$ be a compactification.", "If $\\overline{X}'$ denotes the scheme theoretic closure of $X$", "in $\\overline{X}$, then $X$ is dense and scheme theoretically dense", "in $\\overline{X}'$ by", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-immersion}.", "This proves the first condition of", "Categories, Definition \\ref{categories-definition-initial}.", "Since we have already shown the category of compactifications", "of $X$ is cofiltered, the second condition of", "Categories, Definition \\ref{categories-definition-initial}", "follows from the first (we omit the solution to this", "categorical exercise).", "\\medskip\\noindent", "Proof of (c). After replacing $\\overline{X}'$ with the scheme theoretic", "closure of $j'(X)$ (which doesn't change the underlying topological space)", "this follows from Morphisms, Lemma", "\\ref{morphisms-lemma-scheme-theoretic-image-of-partial-section}." ], "refs": [ "categories-definition-codirected", "morphisms-lemma-quasi-compact-immersion", "categories-definition-initial", "categories-definition-initial" ], "ref_ids": [ 12364, 5154, 12362, 12362 ] } ], "ref_ids": [ 12362 ] }, { "id": 6129, "type": "theorem", "label": "flat-lemma-compactifyable", "categories": [ "flat" ], "title": "flat-lemma-compactifyable", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to Y$", "be a morphism of schemes over $S$ with $Y$ separated and of finite type", "over $S$ and $X$ compactifyable over $S$. Then $X$ has a compactification", "over $Y$." ], "refs": [], "proofs": [ { "contents": [ "Let $j : X \\to \\overline{X}$ be a compactification of $X$ over $S$.", "Then we let $\\overline{X}'$ be", "the scheme theoretic image of $(j, f) : X \\to \\overline{X} \\times_S Y$.", "The morphism $\\overline{X}' \\to Y$ is proper because", "$\\overline{X} \\times_S Y \\to Y$ is proper as a base change of", "$\\overline{X} \\to S$. On the other hand, since $Y$ is separated", "over $S$, the morphism $(1, f) : X \\to X \\times_S Y$ is a closed", "immersion (Schemes, Lemma \\ref{schemes-lemma-semi-diagonal})", "and hence $X \\to \\overline{X}'$ is an open immersion by Morphisms, Lemma", "\\ref{morphisms-lemma-scheme-theoretic-image-of-partial-section} applied", "to the ``partial section'' $s = (j, f)$ to the projection", "$\\overline{X} \\times_S Y \\to \\overline{X}$." ], "refs": [ "schemes-lemma-semi-diagonal" ], "ref_ids": [ 7712 ] } ], "ref_ids": [] }, { "id": 6130, "type": "theorem", "label": "flat-lemma-right-multiplicative-system", "categories": [ "flat" ], "title": "flat-lemma-right-multiplicative-system", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "The collection of morphisms", "$(u, \\overline{u}) : (X', \\overline{X}') \\to (X, \\overline{X})$", "such that $u$ is an isomorphism forms a right multiplicative system", "(Categories, Definition \\ref{categories-definition-multiplicative-system})", "of arrows in the category of compactifications." ], "refs": [ "categories-definition-multiplicative-system" ], "proofs": [ { "contents": [ "Axiom RMS1 is trivial to verify. Let us check RMS2 holds.", "Suppose given a diagram", "$$", "\\xymatrix{", "& (X', \\overline{X}') \\ar[d]_{(u, \\overline{u})} \\\\", "(Y, \\overline{Y}) \\ar[r]^{(f, \\overline{f})} & (X, \\overline{X})", "}", "$$", "with $u : X' \\to X$ an isomorphism. Then we let $Y' = Y \\times_X X'$", "with the projection map $v : Y' \\to Y$ (an isomorphism). We also", "set $\\overline{Y}' = \\overline{Y} \\times_{\\overline{X}} \\overline{X}'$", "with the projection map $\\overline{v} : \\overline{Y}' \\to \\overline{Y}$", "It is clear that $Y' \\to \\overline{Y}'$ is an open immersion.", "The diagram", "$$", "\\xymatrix{", "(Y', \\overline{Y}') \\ar[r]_{(g, \\overline{g})} \\ar[d]_{(v, \\overline{v})} &", "(X', \\overline{X}') \\ar[d]_{(u, \\overline{u})} \\\\", "(Y, \\overline{Y}) \\ar[r]^{(f, \\overline{f})} & (X, \\overline{X})", "}", "$$", "shows that axiom RMS2 holds.", "\\medskip\\noindent", "Let us check RMS3 holds. Suppose given a pair of morphisms", "$(f, \\overline{f}), (g, \\overline{g}) :", "(X, \\overline{X}) \\to (Y, \\overline{Y})$", "of compactifications and a morphism", "$(v, \\overline{v}) : (Y, \\overline{Y}) \\to (Y', \\overline{Y}')$", "such that $v$ is an isomorphism and such that", "$(v, \\overline{v}) \\circ (f, \\overline{f}) =", "(v, \\overline{v}) \\circ (g, \\overline{g})$. Then $f = g$.", "Hence if we let $\\overline{X}' \\subset \\overline{X}$", "be the equalizer of $\\overline{f}$ and $\\overline{g}$,", "then $(u, \\overline{u}) : (X, \\overline{X}') \\to (X, \\overline{X})$", "will be a morphism of the category of compactifications", "such that $(f, \\overline{f}) \\circ (u, \\overline{u}) =", "(g, \\overline{g}) \\circ (u, \\overline{u})$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12373 ] }, { "id": 6131, "type": "theorem", "label": "flat-lemma-invert-right-multiplicative-system", "categories": [ "flat" ], "title": "flat-lemma-invert-right-multiplicative-system", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "The functor $(X, \\overline{X}) \\mapsto X$ defines an", "equivalence from the category of compactifications localized", "(Categories, Lemma \\ref{categories-lemma-right-localization})", "at the right", "multiplicative system of Lemma \\ref{lemma-right-multiplicative-system}", "to the category of compactifyable schemes over $S$." ], "refs": [ "categories-lemma-right-localization", "flat-lemma-right-multiplicative-system" ], "proofs": [ { "contents": [ "Denote $\\mathcal{C}$ the category of compactifications and", "denote $Q : \\mathcal{C} \\to \\mathcal{C}'$ the localization", "functor of Categories, Lemma", "\\ref{categories-lemma-properties-right-localization}.", "Denote $\\mathcal{D}$ the category of compactifyable schemes", "over $S$. It is clear from the lemma just cited and our", "choice of multiplicative system that we", "obtain a functor $\\mathcal{C}' \\to \\mathcal{D}$.", "This functor is clearly essentially surjective.", "If $f : X \\to Y$ is a morphism of compactifyable", "schemes, then we choose an open immersion $Y \\to \\overline{Y}$", "into a scheme proper over $S$, and then we choose an embedding", "$X \\to \\overline{X}$ into a scheme $\\overline{X}$ proper over", "$\\overline{Y}$ (possible by Lemma \\ref{lemma-compactifyable}", "applied to $X \\to \\overline{Y}$). This gives a morphism", "$(X, \\overline{X}) \\to (Y, \\overline{Y})$ of compactifications", "which produces our given morphism $X \\to Y$.", "Finally, suppose given a pair of morphisms in the", "localized category with the same source and target: say", "$$", "a = ((f, \\overline{f}) : (X', \\overline{X}') \\to (Y, \\overline{Y}),", "(u, \\overline{u}) : (X', \\overline{X}') \\to (X, \\overline{X}))", "$$", "and", "$$", "b = ((g, \\overline{g}) : (X'', \\overline{X}'') \\to (Y, \\overline{Y}),", "(v, \\overline{v}) : (X'', \\overline{X}'') \\to (X, \\overline{X}))", "$$", "which produce the same morphism $X \\to Y$ over $S$, in other words", "$f \\circ u^{-1} = g \\circ v^{-1}$. By", "Categories, Lemma \\ref{categories-lemma-morphisms-right-localization}", "we may assume that $(X', \\overline{X}') = (X'', \\overline{X}'')$", "and $(u, \\overline{u}) = (v, \\overline{v})$. In this case we", "can consider the equalizer $\\overline{X}''' \\subset \\overline{X}'$", "of $\\overline{f}$ and $\\overline{g}$. The morphism", "$(w, \\overline{w}) : (X', \\overline{X}''') \\to (X', \\overline{X}')$ is in", "the multiplicative subset and we see that $a = b$ in the localized", "category by precomposing with $(w, \\overline{w})$." ], "refs": [ "categories-lemma-properties-right-localization", "flat-lemma-compactifyable", "categories-lemma-morphisms-right-localization" ], "ref_ids": [ 12264, 6129, 12262 ] } ], "ref_ids": [ 12261, 6130 ] }, { "id": 6132, "type": "theorem", "label": "flat-lemma-check-separated", "categories": [ "flat" ], "title": "flat-lemma-check-separated", "contents": [ "Let $X \\to S$ be a morphism of schemes. If $X = U \\cup V$", "is an open cover such that $U \\to S$ and $V \\to S$ are separated", "and $U \\cap V \\to U \\times_S V$ is closed, then", "$X \\to S$ is separated." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: check that $\\Delta : X \\to X \\times_S X$ is", "closed by using the open covering of $X \\times_S X$ given by", "$U \\times_S U$, $U \\times_S V$, $V \\times_S U$, and $V \\times_S V$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6133, "type": "theorem", "label": "flat-lemma-separate-disjoint-locally-closed-by-blowing-up", "categories": [ "flat" ], "title": "flat-lemma-separate-disjoint-locally-closed-by-blowing-up", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $U \\subset X$ be a quasi-compact open.", "\\begin{enumerate}", "\\item If $Z_1, Z_2 \\subset X$ are closed subschemes of finite", "presentation such that $Z_1 \\cap Z_2 \\cap U = \\emptyset$, then", "there exists a $U$-admissible blowing up $X' \\to X$", "such that the strict transforms of $Z_1$ and $Z_2$ are disjoint.", "\\item If $T_1, T_2 \\subset U$ are disjoint constructible closed subsets, then", "there is a $U$-admissible blowing up $X' \\to X$ such that the closures of", "$T_1$ and $T_2$ are disjoint.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). The assumption that $Z_i \\to X$ is of finite presentation", "signifies that the quasi-coherent ideal sheaf $\\mathcal{I}_i$ of $Z_i$", "is of finite type, see ", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-finite-presentation}.", "Denote $Z \\subset X$ the closed subscheme", "cut out by the product $\\mathcal{I}_1 \\mathcal{I}_2$.", "Observe that $Z \\cap U$ is the disjoint union", "of $Z_1 \\cap U$ and $Z_2 \\cap U$. By Divisors, Lemma", "\\ref{divisors-lemma-separate-disjoint-opens-by-blowing-up}", "there is a $U \\cap Z$-admissible blowup $Z' \\to Z$ such that", "the strict transforms of $Z_1$ and $Z_2$ are disjoint.", "Denote $Y \\subset Z$ the center of this blowing up.", "Then $Y \\to X$ is a closed immersion of finite presentation as the composition", "of $Y \\to Z$ and $Z \\to X$ (Divisors, Definition", "\\ref{divisors-definition-admissible-blowup}", "and Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-presentation}).", "Thus the blowing up $X' \\to X$ of $Y$ is a $U$-admissible blowing", "up. By general properties of strict transforms, the", "strict transform of $Z_1, Z_2$ with respect to $X' \\to X$", "is the same as the strict transform of $Z_1, Z_2$ with respect", "to $Z' \\to Z$, see", "Divisors, Lemma \\ref{divisors-lemma-strict-transform}.", "Thus (1) is proved.", "\\medskip\\noindent", "Proof of (2). By Properties, Lemma", "\\ref{properties-lemma-quasi-coherent-finite-type-ideals}", "there exists a finite type quasi-coherent sheaf of ideals", "$\\mathcal{J}_i \\subset \\mathcal{O}_U$ such that", "$T_i = V(\\mathcal{J}_i)$ (set theoretically).", "By Properties, Lemma \\ref{properties-lemma-extend}", "there exists a finite type quasi-coherent sheaf", "of ideals $\\mathcal{I}_i \\subset \\mathcal{O}_X$", "whose restriction to $U$ is $\\mathcal{J}_i$.", "Apply the result of part (1) to the closed", "subschemes $Z_i = V(\\mathcal{I}_i)$ to conclude." ], "refs": [ "morphisms-lemma-closed-immersion-finite-presentation", "divisors-lemma-separate-disjoint-opens-by-blowing-up", "divisors-definition-admissible-blowup", "morphisms-lemma-composition-finite-presentation", "divisors-lemma-strict-transform", "properties-lemma-quasi-coherent-finite-type-ideals", "properties-lemma-extend" ], "ref_ids": [ 5243, 8074, 8114, 5239, 8065, 3033, 3019 ] } ], "ref_ids": [] }, { "id": 6134, "type": "theorem", "label": "flat-lemma-blowup-iso-along", "categories": [ "flat" ], "title": "flat-lemma-blowup-iso-along", "contents": [ "Let $f : X \\to Y$ be a proper morphism of quasi-compact and", "quasi-separated schemes. Let $V \\subset Y$ be a quasi-compact open", "and $U = f^{-1}(V)$. Let $T \\subset V$ be a closed subset such that", "$f|_U : U \\to V$ is an isomorphism over an open neighbourhood of $T$", "in $V$. Then there exists a $V$-admissible blowing up $Y' \\to Y$", "such that the strict transform $f' : X' \\to Y'$ of $f$", "is an isomorphism over an open neighbourhood of the closure", "of $T$ in $Y'$." ], "refs": [], "proofs": [ { "contents": [ "Let $T' \\subset V$ be the complement of the maximal open over which", "$f|_U$ is an isomorphism. Then $T', T$ are closed in $V$ and", "$T \\cap T' = \\emptyset$. Since $V$ is a spectral topological", "space, we can find constructible closed subsets $T_c, T'_c$", "with $T \\subset T_c$, $T' \\subset T'_c$ such that", "$T_c \\cap T'_c = \\emptyset$ (choose a quasi-compact", "open $W$ of $V$ containing $T'$ not meeting $T$", "and set $T_c = V \\setminus W$, then choose a quasi-compact", "open $W'$ of $V$ containing $T_c$ not meeting $T'$", "and set $T'_c = V \\setminus W'$).", "By Lemma \\ref{lemma-separate-disjoint-locally-closed-by-blowing-up}", "we may, after replacing $Y$ by a $V$-admissible blowing up,", "assume that $T_c$ and $T'_c$ have disjoint closures in $Y$.", "Set $Y_0 = Y \\setminus \\overline{T}'_c$, $V_0 = V \\setminus T'_c$,", "$U_0 = U \\times_V V_0$, and $X_0 = X \\times_Y Y_0$.", "Since $U_0 \\to V_0$ is an isomorphism, we can find a", "$V_0$-admissible blowing up $Y'_0 \\to Y_0$ such that the", "strict transform $X'_0$ of $X_0$ maps isomorphically to $Y'_0$, see", "Lemma \\ref{lemma-zariski-after-blowup}.", "By Divisors, Lemma \\ref{divisors-lemma-extend-admissible-blowups}", "there exists a $V$-admissible blow up $Y' \\to Y$ whose restriction", "to $Y_0$ is $Y'_0 \\to Y_0$. If $f' : X' \\to Y'$ denotes the", "strict transform of $f$, then we see what we want is true because", "$f'$ restricts to an isomorphism over $Y'_0$." ], "refs": [ "flat-lemma-separate-disjoint-locally-closed-by-blowing-up", "flat-lemma-zariski-after-blowup", "divisors-lemma-extend-admissible-blowups" ], "ref_ids": [ 6133, 6125, 8072 ] } ], "ref_ids": [] }, { "id": 6135, "type": "theorem", "label": "flat-lemma-find-common-blowups", "categories": [ "flat" ], "title": "flat-lemma-find-common-blowups", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $U \\to X_1$ and $U \\to X_2$ be open immersions", "of schemes over $S$ and assume $U$, $X_1$, $X_2$ of finite", "type and separated over $S$. Then there exists a commutative diagram", "$$", "\\xymatrix{", "X_1' \\ar[d] \\ar[r] & X & X_2' \\ar[l] \\ar[d] \\\\", "X_1 & U \\ar[l] \\ar[lu] \\ar[u] \\ar[ru] \\ar[r] & X_2", "}", "$$", "of schemes over $S$ where $X_i' \\to X_i$ is a $U$-admissible", "blowup, $X_i' \\to X$ is an open immersion, and $X$ is separated and finite", "type over $S$." ], "refs": [], "proofs": [ { "contents": [ "Throughout the proof all schemes will be separated of finite type over $S$.", "This in particular implies these schemes are quasi-compact and quasi-separated", "and the morphisms between them are quasi-compact and separated.", "See Schemes, Sections \\ref{schemes-section-quasi-compact} and", "\\ref{schemes-section-separation-axioms}.", "We will use that if $U \\to W$ is an immersion of such schemes over $S$,", "then the scheme theoretic image $Z$ of $U$ in $W$ is a closed subscheme", "of $W$ and $U \\to Z$ is an open immersion, $U \\subset Z$ is scheme", "theoretically dense, and $U \\subset Z$ is dense topologically. See", "Morphisms, Lemma", "\\ref{morphisms-lemma-quasi-compact-immersion}.", "\\medskip\\noindent", "Let $X_{12} \\subset X_1 \\times_S X_2$ be the scheme theoretic image", "of $U \\to X_1 \\times_S X_2$. The projections $p_i : X_{12} \\to X_i$", "induce isomorphisms $p_i^{-1}(U) \\to U$ by", "Morphisms, Lemma", "\\ref{morphisms-lemma-scheme-theoretic-image-of-partial-section}.", "Choose a $U$-admissible blowup $X_i^i \\to X_i$ such that", "the strict transform $X_{12}^i$ of $X_{12}$ is isomorphic to an", "open subscheme of $X_i^i$, see", "Lemma \\ref{lemma-zariski-after-blowup}.", "Let $\\mathcal{I}_i \\subset \\mathcal{O}_{X_i}$ be the corresponding", "finite type quasi-coherent sheaf of ideals.", "Recall that $X_{12}^i \\to X_{12}$ is the blowup in", "$p_i^{-1}\\mathcal{I}_i \\mathcal{O}_{X_{12}}$, see", "Divisors, Lemma \\ref{divisors-lemma-strict-transform}.", "Let $X_{12}'$ be the blowup of $X_{12}$ in", "$p_1^{-1}\\mathcal{I}_1 p_2^{-1}\\mathcal{I}_2 \\mathcal{O}_{X_{12}}$, see", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-two-ideals}", "for what this entails. We obtain in particular a commutative diagram", "$$", "\\xymatrix{", "X_{12}' \\ar[d] \\ar[r] & X_{12}^2 \\ar[d] \\\\", "X_{12}^1 \\ar[r] & X_{12}", "}", "$$", "where all the morphisms are $U$-admissible blowing ups.", "Since $X_{12}^i \\subset X_i^i$ is an open we may choose a $U$-admissible blowup", "$X_i' \\to X_i^i$ restricting to $X_{12}' \\to X_{12}^i$, see", "Divisors, Lemma \\ref{divisors-lemma-extend-admissible-blowups}.", "Then $X_{12}' \\subset X_i'$ is an open subscheme and the diagram", "$$", "\\xymatrix{", "X_{12}' \\ar[d] \\ar[r] & X_i' \\ar[d] \\\\", "X_{12}^i \\ar[r] & X_i^i", "}", "$$", "is commutative with vertical arrows blowing ups and horizontal arrows", "open immersions. Note that $X'_{12} \\to X_1' \\times_S X_2'$ is", "an immersion and proper (use that $X'_{12} \\to X_{12}$ is proper", "and $X_{12} \\to X_1 \\times_S X_2$ is closed and $X_1' \\times_S X_2' \\to", "X_1 \\times_S X_2$ is separated and apply Morphisms, Lemma", "\\ref{morphisms-lemma-image-proper-scheme-closed}).", "Thus $X'_{12} \\to X_1' \\times_S X_2'$ is a closed immersion.", "It follows that if we define $X$ by glueing $X_1'$ and $X_2'$", "along the common open subscheme $X_{12}'$, then $X \\to S$ is of finite type", "and separated (Lemma \\ref{lemma-check-separated}).", "As compositions of $U$-admissible blowups are $U$-admissible blowups", "(Divisors, Lemma \\ref{divisors-lemma-composition-admissible-blowups})", "the lemma is proved." ], "refs": [ "morphisms-lemma-quasi-compact-immersion", "flat-lemma-zariski-after-blowup", "divisors-lemma-strict-transform", "divisors-lemma-blowing-up-two-ideals", "divisors-lemma-extend-admissible-blowups", "morphisms-lemma-image-proper-scheme-closed", "flat-lemma-check-separated", "divisors-lemma-composition-admissible-blowups" ], "ref_ids": [ 5154, 6125, 8065, 8062, 8072, 5411, 6132, 8071 ] } ], "ref_ids": [] }, { "id": 6136, "type": "theorem", "label": "flat-lemma-replaced-by-strict-transform", "categories": [ "flat" ], "title": "flat-lemma-replaced-by-strict-transform", "contents": [ "Let $X \\to S$ and $Y \\to S$ be morphisms of schemes.", "Let $U \\subset X$ be an open subscheme.", "Let $V \\to X \\times_S Y$ be a quasi-compact morphism", "whose composition with the first projection maps into $U$.", "Let $Z \\subset X \\times_S Y$ be the scheme theoretic image of", "$V \\to X \\times_S Y$. Let $X' \\to X$ be a $U$-admissible blowup.", "Then the scheme theoretic image of $V \\to X' \\times_S Y$ is the", "strict transform of $Z$ with respect to the blowing up." ], "refs": [], "proofs": [ { "contents": [ "Denote $Z' \\to Z$ the strict transform. The morphism $Z' \\to X'$", "induces a morphism $Z' \\to X' \\times_S Y$ which is a closed immersion", "(as $Z'$ is a closed subscheme of $X' \\times_X Z$ by definition).", "Thus to finish the proof it suffices to show that the scheme theoretic", "image $Z''$ of $V \\to Z'$ is $Z'$. Observe that $Z'' \\subset Z'$", "is a closed subscheme such that $V \\to Z'$ factors through $Z''$.", "Since both $V \\to X \\times_S Y$ and $V \\to X' \\times_S Y$ are", "quasi-compact (for the latter this follows from Schemes, Lemma", "\\ref{schemes-lemma-quasi-compact-permanence}", "and the fact that $X' \\times_S Y \\to X \\times_S Y$ is separated", "as a base change of a proper morphism), by Morphisms, Lemma", "\\ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}", "we see that $Z \\cap (U \\times_S Y) = Z'' \\cap (U \\times_S Y)$.", "Thus the inclusion morphism $Z'' \\to Z'$ is an isomorphism", "away from the exceptional divisor $E$ of $Z' \\to Z$. However, the", "structure sheaf of $Z'$ does not have any nonzero sections supported", "on $E$ (by definition of strict transforms) and we conclude that", "the surjection $\\mathcal{O}_{Z'} \\to \\mathcal{O}_{Z''}$", "must be an isomorphism." ], "refs": [ "schemes-lemma-quasi-compact-permanence", "morphisms-lemma-quasi-compact-scheme-theoretic-image" ], "ref_ids": [ 7716, 5146 ] } ], "ref_ids": [] }, { "id": 6137, "type": "theorem", "label": "flat-lemma-compactification-dominates", "categories": [ "flat" ], "title": "flat-lemma-compactification-dominates", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme. Let $U$ be a", "scheme of finite type and separated over $S$. Let $V \\subset U$ be a", "quasi-compact open. If $V$ has a compactification $V \\subset Y$", "over $S$, then there exists a $V$-admissible blowing up $Y' \\to Y$ and an", "open $V \\subset V' \\subset Y'$ such that $V \\to U$", "extends to a proper morphism $V' \\to U$." ], "refs": [], "proofs": [ { "contents": [ "Consider the scheme theoretic image $Z \\subset Y \\times_S U$", "of the ``diagonal'' morphism $V \\to Y \\times_S U$. If we replace", "$Y$ by a $V$-admissible blowing up, then $Z$ is replaced by", "the strict transform with respect to this blowing up, see", "Lemma \\ref{lemma-replaced-by-strict-transform}. Hence by", "Lemma \\ref{lemma-zariski-after-blowup} we may assume $Z \\to Y$", "is an open immersion. If $V' \\subset Y$ denotes the image, then we", "see that the induced morphism $V' \\to U$ is proper because the", "projection $Y \\times_S U \\to U$ is proper and $V' \\cong Z$", "is a closed subscheme of $Y \\times_S U$." ], "refs": [ "flat-lemma-replaced-by-strict-transform", "flat-lemma-zariski-after-blowup" ], "ref_ids": [ 6136, 6125 ] } ], "ref_ids": [] }, { "id": 6138, "type": "theorem", "label": "flat-lemma-two-compactifications", "categories": [ "flat" ], "title": "flat-lemma-two-compactifications", "contents": [ "Let $S$ be a Noetherian scheme. Let $U$ be a scheme of finite type", "and separated over $S$. Let $U = U_1 \\cup U_2$ be opens such that", "$U_1$ and $U_2$ have compactifications over $S$ and such that", "$U_1 \\cap U_2$ is dense in $U$. Then $U$ has a compactification over $S$." ], "refs": [], "proofs": [ { "contents": [ "Choose a compactification $U_i \\subset X_i$ for $i = 1, 2$. We may", "assume $U_i$ is scheme theoretically dense in $X_i$. We may assume there", "is an open $V_i \\subset X_i$ and a proper morphism", "$\\psi_i : V_i \\to U$ extending $\\text{id} : U_i \\to U_i$, see", "Lemma \\ref{lemma-compactification-dominates}. Picture", "$$", "\\xymatrix{", "U_i \\ar[r] \\ar[d] & V_i \\ar[r] \\ar[dl]^{\\psi_i} & X_i \\\\", "U", "}", "$$", "If $\\{i, j\\} = \\{1, 2\\}$ denote", "$Z_i = U \\setminus U_j = U_i \\setminus (U_1 \\cap U_2)$", "and", "$Z_j = U \\setminus U_i = U_j \\setminus (U_1 \\cap U_2)$.", "Thus we have", "$$", "U = U_1 \\amalg Z_2 = Z_1 \\amalg U_2 = Z_1 \\amalg (U_1 \\cap U_2) \\amalg Z_2", "$$", "Denote $Z_{i, i} \\subset V_i$ the inverse image of $Z_i$ under $\\psi_i$.", "Observe that $\\psi_i$ is an isomorphism over an open neighbourhood of $Z_i$.", "Denote $Z_{i, j} \\subset V_i$ the inverse image of $Z_j$ under $\\psi_i$.", "Observe that $\\psi_i : Z_{i, j} \\to Z_j$ is a proper morphism.", "Since $Z_i$ and $Z_j$ are disjoint closed subsets of", "$U$, we see that $Z_{i, i}$ and $Z_{i, j}$ are disjoint closed subsets", "of $V_i$.", "\\medskip\\noindent", "Denote $\\overline{Z}_{i, i}$ and $\\overline{Z}_{i, j}$ the closures of", "$Z_{i, i}$ and $Z_{i, j}$ in $X_i$. After replacing $X_i$ by a", "$V_i$-admissible blowup we may assume that", "$\\overline{Z}_{i, i}$ and $\\overline{Z}_{i, j}$ are disjoint, see", "Lemma \\ref{lemma-separate-disjoint-locally-closed-by-blowing-up}.", "We assume this holds for both $X_1$ and $X_2$.", "Observe that this property is preserved if we replace $X_i$", "by a further $V_i$-admissible blowup.", "\\medskip\\noindent", "Set $V_{12} = V_1 \\times_U V_2$. We have an immersion", "$V_{12} \\to X_1 \\times_S X_2$ which is the composition of the closed", "immersion $V_{12} = V_1 \\times_U V_2 \\to V_1 \\times_S V_2$", "(Schemes, Lemma \\ref{schemes-lemma-fibre-product-after-map})", "and the open immersion $V_1 \\times_S V_2 \\to X_1 \\times_S X_2$.", "Let $X_{12} \\subset X_1 \\times_S X_2$ be the scheme theoretic", "image of $V_{12} \\to X_1 \\times_S X_2$. The projection morphisms", "$$", "p_1 : X_{12} \\to X_1", "\\quad\\text{and}\\quad", "p_2 : X_{12} \\to X_2", "$$", "are proper as $X_1$ and $X_2$ are proper over $S$. If we replace $X_1$ by a", "$V_1$-admissible blowing up, then $X_{12}$ is replaced by", "the strict transform with respect to this blowing up, see", "Lemma \\ref{lemma-replaced-by-strict-transform}.", "\\medskip\\noindent", "Denote $\\psi : V_{12} \\to U$ the compositions", "$\\psi = \\psi_1 \\circ p_1|_{V_{12}} = \\psi_2 \\circ p_2|_{V_{12}}$.", "Consider the closed subscheme", "$$", "Z_{12, 2} =", "(p_1|_{V_{12}})^{-1}(Z_{1, 2}) =", "(p_2|_{V_{12}})^{-1}(Z_{2, 2}) =", "\\psi^{-1}(Z_2) \\subset V_{12}", "$$", "The morphism $p_1|_{V_{12}} : V_{12} \\to V_1$ is an isomorphism", "over an open neighbourhood of $Z_{1, 2}$ because $\\psi_2 : V_2 \\to U$", "is an isomorphism over an open neighbourhood of $Z_2$ and", "$V_{12} = V_1 \\times_U V_2$.", "By Lemma \\ref{lemma-blowup-iso-along}", "there exists a $V_1$-admissible blowing up $X_1' \\to X_1$", "such that the strict tranform $p'_1 : X'_{12} \\to X'_1$", "of $p_1$ is an isomorphism over an open neighbourhood of", "the closure of $Z_{1, 2}$ in $X'_1$.", "After replacing $X_1$ by $X'_1$ and $X_{12}$ by $X'_{12}$", "we may assume that $p_1$ is an isomorphism over an open", "neighbourhood of $\\overline{Z}_{1, 2}$.", "\\medskip\\noindent", "The reduction of the previous paragraph tells us that", "$$", "X_{12} \\cap (\\overline{Z}_{1, 2} \\times_S \\overline{Z}_{2, 1}) = \\emptyset", "$$", "where the intersection taken in $X_1 \\times_S X_2$. Namely, the inverse", "image $p_1^{-1}(\\overline{Z}_{1, 2})$ in $X_{12}$ maps isomorphically", "to $\\overline{Z}_{1, 2}$. In particular, we see that $Z_{12, 2}$", "is dense in $p_1^{-1}(\\overline{Z}_{1, 2})$. Thus $p_2$ maps", "$p_1^{-1}(\\overline{Z}_{1, 2})$ into $\\overline{Z}_{2, 2}$.", "Since $\\overline{Z}_{2, 2} \\cap \\overline{Z}_{2, 1} = \\emptyset$", "we conclude.", "\\medskip\\noindent", "Consider the schemes", "$$", "W_i = U \\coprod\\nolimits_{U_i} (X_i \\setminus \\overline{Z}_{i, j}),", "\\quad i = 1, 2", "$$", "obtained by glueing. Let us apply Lemma \\ref{lemma-check-separated}", "to see that $W_i \\to S$ is separated. First,", "$U \\to S$ and $X_i \\to S$ are separated. The immersion", "$U_i \\to U \\times_S (X_i \\setminus \\overline{Z}_{i, j})$", "is closed because any specialization $u_i \\leadsto u$", "with $u_i \\in U_i$ and $u \\in U \\setminus U_i$", "can be lifted uniquely to a specialization", "$u_i \\leadsto v_i$ in $V_i$ along the proper morphism", "$\\psi_i : V_i \\to U$ and then $v_i$ must be in $Z_{i, j}$.", "Thus the image of the immersion is closed, whence the immersion", "is a closed immersion.", "\\medskip\\noindent", "On the other hand, for any valuation ring $A$ over $S$ with fraction field $K$", "and any morphism $\\gamma : \\Spec(K) \\to (U_1 \\cap U_2)$ over $S$, there", "is an $i$ and an extension of $\\gamma$ to a morphism $h_i : \\Spec(A) \\to W_i$.", "Namely, for both $i = 1, 2$ there is a morphism", "$g_i : \\Spec(A) \\to X_i$ extending $\\gamma$ by the", "valuative criterion of properness for $X_i$ over $S$, see", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-proper}.", "Thus we only are in trouble", "if $g_i(\\mathfrak m_A) \\in \\overline{Z}_{i, j}$ for $i = 1, 2$. This is", "impossible by the emptyness of the intersection of $X_{12}$ and", "$\\overline{Z}_{1, 2} \\times_S \\overline{Z}_{2, 1}$ we proved above.", "\\medskip\\noindent", "Consider a diagram", "$$", "\\xymatrix{", "W_1' \\ar[d] \\ar[r] & W & W_2' \\ar[l] \\ar[d] \\\\", "W_1 & U \\ar[l] \\ar[lu] \\ar[u] \\ar[ru] \\ar[r] & W_2", "}", "$$", "as in Lemma \\ref{lemma-find-common-blowups}. By the previous paragraph", "for every solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_\\gamma \\ar[d] & W \\ar[d] \\\\", "\\Spec(A) \\ar@{..>}[ru] \\ar[r] & S", "}", "$$", "where $\\Im(\\gamma) \\subset U_1 \\cap U_2$ there is an $i$ and", "an extension $h_i : \\Spec(A) \\to W_i$ of $\\gamma$.", "Using the valuative criterion of properness for $W'_i \\to W_i$,", "we can then lift $h_i$ to $h'_i : \\Spec(A) \\to W'_i$.", "Hence the dotted arrow in the diagram exists. Since $W$", "is separated over $S$, we see that the arrow is unique as well.", "This implies that $W \\to S$ is universally closed by", "Morphisms, Lemma", "\\ref{morphisms-lemma-refined-valuative-criterion-universally-closed}.", "As $W \\to S$ is already of finite type and separated, we win." ], "refs": [ "flat-lemma-compactification-dominates", "flat-lemma-separate-disjoint-locally-closed-by-blowing-up", "schemes-lemma-fibre-product-after-map", "flat-lemma-replaced-by-strict-transform", "flat-lemma-blowup-iso-along", "flat-lemma-check-separated", "morphisms-lemma-characterize-proper", "flat-lemma-find-common-blowups", "morphisms-lemma-refined-valuative-criterion-universally-closed" ], "ref_ids": [ 6137, 6133, 7711, 6136, 6134, 6132, 5416, 6135, 5417 ] } ], "ref_ids": [] }, { "id": 6139, "type": "theorem", "label": "flat-lemma-equivalence-h-v-locally-finite-presentation", "categories": [ "flat" ], "title": "flat-lemma-equivalence-h-v-locally-finite-presentation", "contents": [ "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms", "of schemes with fixed target with $f_i$ locally of finite", "presentation for all $i$. The following are equivalent", "\\begin{enumerate}", "\\item $\\{X_i \\to X\\}$ is a ph covering, and", "\\item $\\{X_i \\to X\\}$ is a V covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be affine open. Looking at", "Topologies, Definitions \\ref{topologies-definition-ph-covering}", "and \\ref{topologies-definition-V-covering} it suffices to show that", "the base change $\\{X_i \\times_X U \\to U\\}$ can be refined", "by a standard ph covering if and only if it can be refined by", "a standard V covering. Thus we may assume $X$ is affine and we have to show", "$\\{X_i \\to X\\}$ can be refined by a standard ph covering", "if and only if it can be refined by a standard V covering.", "Since a standard ph covering is a standard V covering, see", "Topologies, Lemma \\ref{topologies-lemma-standard-ph-standard-V}", "it suffices to prove the other implication.", "\\medskip\\noindent", "Assume $X$ is affine and assume $\\{f_i : X_i \\to X\\}_{i \\in I}$", "can be refined by a standard V covering", "$\\{g_j : Y_j \\to X\\}_{j = 1, \\ldots, m}$.", "For each $j$ choose an $i_j$ and a morphism", "$h_j : Y_j \\to X_{i_j}$ such that $g_j = f_{i_j} \\circ h_j$.", "Since $Y_j$ is affine hence quasi-compact,", "for each $j$ we can find finitely many affine opens", "$U_{j, k} \\subset X_{i_j}$ such that $\\Im(h_j) \\subset \\bigcup U_{j, k}$.", "Then $\\{U_{j, k} \\to X\\}_{j, k}$ refines $\\{X_i \\to X\\}$", "and is a standard V covering (as it is a finite family of morphisms", "of affines and it inherits the lifting property for valuation rings", "from the corresponding property of $\\{Y_j \\to X\\}$).", "Thus we reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $\\{f_i : X_i \\to X\\}_{i = 1, \\ldots, n}$", "is a standard V covering with $f_i$ of finite presentation.", "We have to show that $\\{X_i \\to X\\}$ can be refined by a standard ph covering.", "Choose a generic flatness stratification", "$$", "X = S \\supset S_0 \\supset S_1 \\supset \\ldots \\supset S_t = \\emptyset", "$$", "as in Lemma \\ref{lemma-generic-flatness-stratification}", "for the finitely presented morphism", "$$", "\\coprod\\nolimits_{i = 1, \\ldots, n} f_i :", "\\coprod\\nolimits_{i = 1, \\ldots, n} X_i", "\\longrightarrow", "X", "$$", "of affines. We are going to use all the properties of the stratification", "without further mention. By construction the base change of each $f_i$ to", "$U_k = S_k \\setminus S_{k + 1}$ is flat.", "Denote $Y_k$ the scheme theoretic closure of $U_k$ in $S_k$. Since", "$U_k \\to S_k$ is a quasi-compact open immersion (see", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-finite-type-ideals}),", "we see that $U_k \\subset Y_k$ is a quasi-compact dense", "(and scheme theoretically dense) open immersion, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}.", "The morphism $\\coprod_{k = 0, \\ldots, t - 1} Y_k \\to X$", "is finite surjective, hence $\\{Y_k \\to X\\}$ is a standard ph covering", "and hence a standard V covering (see above). By the transitivity", "property of standard V coverings", "(Topologies, Lemma \\ref{topologies-lemma-composition-standard-V})", "it suffices to show that the pullback of", "the covering $\\{X_i \\to X\\}$ to each $Y_k$ can be refined by a", "standard V covering. This reduces us to the case described in the", "next paragraph.", "\\medskip\\noindent", "Assume $\\{f_i : X_i \\to X\\}_{i = 1, \\ldots, n}$ is a standard V covering", "with $f_i$ of finite presentation and there is a dense quasi-compact open", "$U \\subset X$ such that $X_i \\times_X U \\to U$ is flat.", "By Theorem \\ref{theorem-flatten-module}", "there is a $U$-admissible blowup $X' \\to X$ such that", "the strict transform $f'_i : X'_i \\to X'$ of $f_i$ is flat.", "Observe that the projective (hence closed) morphism $X' \\to X$", "is surjective as $U \\subset X$ is dense and as $U$ is identified", "with an open of $X'$. After replacing $X'$ by a further", "$U$-admissible blowup if necessary,", "we may also assume $U \\subset X'$ is scheme theoretically dense", "(see Remark \\ref{remark-successive-blowups}).", "Hence for every point $x \\in X'$ there is a valuation ring $V$", "and a morphism $g : \\Spec(V) \\to X'$ such that the generic", "point of $\\Spec(V)$ maps into $U$ and the closed point of", "$\\Spec(V)$ maps to $x$, see Morphisms, Lemma", "\\ref{morphisms-lemma-reach-points-scheme-theoretic-image}.", "Since $\\{X_i \\to X\\}$ is a standard V covering, we can choose", "an extension of valuation rings $V \\subset W$, an index $i$, and a morphism", "$\\Spec(W) \\to X_i$ such that the diagram", "$$", "\\xymatrix{", "\\Spec(W) \\ar[d] \\ar[rr] & & X_i \\ar[d] \\\\", "\\Spec(V) \\ar[r] & X' \\ar[r] & X", "}", "$$", "is commutative. Since $X'_i \\subset X' \\times_X X_i$ is a closed subscheme", "containing the open $U \\times_X X_i$, since $\\Spec(W)$ is an integral scheme,", "and since the induced morphism $h : \\Spec(W) \\to X' \\times_X X_i$ maps", "the generic point of $\\Spec(W)$ into $U \\times_X X_i$, we conclude", "that $h$ factors through the closed subscheme $X'_i \\subset X' \\times_X X_i$.", "We conclude that $\\{f'_i : X'_i \\to X'\\}$ is a V covering.", "In particular, $\\coprod f'_i$ is surjective. In particular", "$\\{X'_i \\to X'\\}$ is an fppf covering. Since an fppf covering is a ph covering", "(More on Morphisms, Lemma \\ref{more-morphisms-lemma-fppf-ph}),", "we can find a standard ph covering $\\{Y_j \\to X'\\}$ refining", "$\\{X'_i \\to X\\}$. Say this covering is given by a proper surjective", "morphism $Y \\to X'$ and a finite affine open covering", "$Y = \\bigcup Y_j$. Then the composition $Y \\to X$ is proper surjective", "and we conclude that $\\{Y_j \\to X\\}$ is a standard ph covering.", "This finishes the proof." ], "refs": [ "topologies-definition-ph-covering", "topologies-definition-V-covering", "topologies-lemma-standard-ph-standard-V", "flat-lemma-generic-flatness-stratification", "properties-lemma-quasi-coherent-finite-type-ideals", "morphisms-lemma-quasi-compact-scheme-theoretic-image", "topologies-lemma-composition-standard-V", "flat-theorem-flatten-module", "flat-remark-successive-blowups", "morphisms-lemma-reach-points-scheme-theoretic-image", "more-morphisms-lemma-fppf-ph" ], "ref_ids": [ 12544, 12551, 12505, 6084, 3033, 5146, 12507, 5975, 6234, 5147, 13927 ] } ], "ref_ids": [] }, { "id": 6140, "type": "theorem", "label": "flat-lemma-Noetherian-h-covering", "categories": [ "flat" ], "title": "flat-lemma-Noetherian-h-covering", "contents": [ "Let $X$ be a Noetherian scheme. Let $\\{X_i \\to X\\}_{i \\in I}$", "be a finite family of finite type morphisms. The following are equivalent", "\\begin{enumerate}", "\\item $\\coprod_{i \\in I} X_i \\to X$ is universally submersive", "(Morphisms, Definition", "\\ref{morphisms-definition-submersive}), and", "\\item $\\{X_i \\to X\\}_{i \\in I}$ is an h covering.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-submersive" ], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) follows from the more general", "Topologies, Lemma \\ref{topologies-lemma-V-covering-universally-submersive}", "and our definition of h covers. Assume $\\coprod X_i \\to X$", "is universally submersive. We will show that $\\{X_i \\to X\\}$", "can be refined by a ph covering; this will suffice by", "Topologies, Lemma \\ref{topologies-lemma-refine-by-ph} and", "our definition of h coverings.", "The argument will be the same as the one used in the proof of", "Lemma \\ref{lemma-equivalence-h-v-locally-finite-presentation}.", "\\medskip\\noindent", "Choose a generic flatness stratification", "$$", "X = S \\supset S_0 \\supset S_1 \\supset \\ldots \\supset S_t = \\emptyset", "$$", "as in Lemma \\ref{lemma-generic-flatness-stratification}", "for the finitely presented morphism", "$$", "\\coprod\\nolimits_{i = 1, \\ldots, n} f_i :", "\\coprod\\nolimits_{i = 1, \\ldots, n} X_i", "\\longrightarrow", "X", "$$", "We are going to use all the properties of the stratification", "without further mention. By construction the base change of each", "$f_i$ to $U_k = S_k \\setminus S_{k + 1}$ is flat.", "Denote $Y_k$ the scheme theoretic closure of $U_k$ in $S_k$. Since", "$U_k \\to S_k$ is a quasi-compact open immersion (all schemes in", "this paragraph are Noetherian),", "we see that $U_k \\subset Y_k$ is a quasi-compact dense", "(and scheme theoretically dense) open immersion, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}.", "The morphism $\\coprod_{k = 0, \\ldots, t - 1} Y_k \\to X$", "is finite surjective, hence $\\{Y_k \\to X\\}$ is a ph covering.", "By the transitivity property of ph coverings", "(Topologies, Lemma \\ref{topologies-lemma-ph})", "it suffices to show that the pullback of", "the covering $\\{X_i \\to X\\}$ to each $Y_k$ can be refined by a", "ph covering. This reduces us to the case described in the", "next paragraph.", "\\medskip\\noindent", "Assume $\\coprod X_i \\to X$ is universally submersive and", "there is a dense open $U \\subset X$ such that", "$X_i \\times_X U \\to U$ is flat for all $i$.", "By Theorem \\ref{theorem-flatten-module}", "there is a $U$-admissible blowup $X' \\to X$ such that", "the strict transform $f'_i : X'_i \\to X'$ of $f_i$ is flat for all $i$.", "Observe that the projective (hence closed) morphism $X' \\to X$", "is surjective as $U \\subset X$ is dense and as $U$ is identified", "with an open of $X'$. After replacing $X'$ by a further", "$U$-admissible blowup if necessary, we may also assume $U \\subset X'$ is dense", "(see Remark \\ref{remark-successive-blowups}).", "Hence for every point $x \\in X'$ there is a discrete valuation ring $A$", "and a morphism $g : \\Spec(A) \\to X'$ such that the generic", "point of $\\Spec(A)$ maps into $U$ and the closed point of", "$\\Spec(A)$ maps to $x$, see Limits, Lemma", "\\ref{limits-lemma-reach-point-closure-Noetherian}.", "Set", "$$", "W = \\Spec(A) \\times_X \\coprod X_i = \\coprod \\Spec(A) \\times_X X_i", "$$", "Since $\\coprod X_i \\to X$ is universally submersive,", "there is a specialization $w' \\leadsto w$ in $W$", "such that $w'$ maps to the generic point of $\\Spec(A)$", "and $w$ maps to the closed point of $\\Spec(A)$.", "(If not, then the closed fibre of $W \\to \\Spec(A)$", "is stable under generalizations, hence open, which", "contradicts the fact that $W \\to \\Spec(A)$ is submersive.)", "Say $w' \\in \\Spec(A) \\times_X X_i$ so of course", "$w \\in \\Spec(A) \\times_X X_i$ as well. Let", "$x'_i \\leadsto x_i$ be the image of $w' \\leadsto w$ in", "$X' \\times_X X_i$. Since $x'_i \\in X'_i$ and since", "$X'_i \\subset X' \\times_X X_i$ is a closed subscheme", "we see that $x_i \\in X'_i$. Since $x_i$ maps to $x \\in X'$", "we conclude that", "$\\coprod X'_i \\to X'$ is surjective! In particular", "$\\{X'_i \\to X'\\}$ is an fppf covering. But an fppf covering is a ph covering", "(More on Morphisms, Lemma \\ref{more-morphisms-lemma-fppf-ph}).", "Since $X' \\to X$ is proper surjective, we conclude", "that $\\{X'_i \\to X\\}$ is a ph covering and the proof is complete." ], "refs": [ "topologies-lemma-V-covering-universally-submersive", "topologies-lemma-refine-by-ph", "flat-lemma-equivalence-h-v-locally-finite-presentation", "flat-lemma-generic-flatness-stratification", "morphisms-lemma-quasi-compact-scheme-theoretic-image", "topologies-lemma-ph", "flat-theorem-flatten-module", "flat-remark-successive-blowups", "limits-lemma-reach-point-closure-Noetherian", "more-morphisms-lemma-fppf-ph" ], "ref_ids": [ 12514, 12484, 6139, 6084, 5146, 12485, 5975, 6234, 15097, 13927 ] } ], "ref_ids": [ 5556 ] }, { "id": 6141, "type": "theorem", "label": "flat-lemma-approximate-h-cover", "categories": [ "flat" ], "title": "flat-lemma-approximate-h-cover", "contents": [ "Let $X$ be an affine scheme. Let $\\{X_i \\to X\\}_{i \\in I}$", "be an h covering. Then there exists a surjective proper morphism", "$$", "Y \\longrightarrow X", "$$", "of finite presentation (!) and a finite affine open covering", "$Y = \\bigcup_{j = 1, \\ldots, m} Y_j$ such that", "$\\{Y_j \\to X\\}_{j = 1, \\ldots, m}$ refines $\\{X_i \\to X\\}_{i \\in I}$." ], "refs": [], "proofs": [ { "contents": [ "By assumption there exists a proper surjective morphism", "$Y \\to X$ and a finite affine open covering", "$Y = \\bigcup_{j = 1, \\ldots, m} Y_j$ such that", "$\\{Y_j \\to X\\}_{j = 1, \\ldots, m}$ refines $\\{X_i \\to X\\}_{i \\in I}$.", "This means that for each $j$ there is an index $i_j \\in I$", "and a morphism $h_j : Y_j \\to X_{i_j}$ over $X$.", "See Definition \\ref{definition-h-covering} and", "Topologies, Definition \\ref{topologies-definition-ph-covering}.", "The problem is that we don't know that $Y \\to X$ is of finite", "presentation.", "By", "Limits, Lemma \\ref{limits-lemma-proper-limit-of-proper-finite-presentation}", "we can write", "$$", "Y = \\lim Y_\\lambda", "$$", "as a directed limit of schemes $Y_\\lambda$ proper and of finite presentation", "over $X$ such that the morphisms $Y \\to Y_\\lambda$ and the", "the transition morphisms are closed immersions. Observe that", "each $Y_\\lambda \\to X$ is surjective.", "By Limits, Lemma \\ref{limits-lemma-descend-opens}", "we can find a $\\lambda$ and quasi-compact opens", "$Y_{\\lambda, j} \\subset Y_\\lambda$, $j = 1, \\ldots, m$", "covering $Y_\\lambda$ and restricting to $Y_j$ in $Y$.", "Then $Y_j = \\lim Y_{\\lambda, j}$.", "After increasing $\\lambda$ we may assume $Y_{\\lambda, j}$", "is affine for all $j$, see", "Limits, Lemma \\ref{limits-lemma-limit-affine}.", "Finally, since $X_i \\to X$ is locally of finite presentation", "we can use the functorial characterization of morphisms", "which are locally of finite presentation", "(Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation})", "to find a $\\lambda$ such that for each $j$ there is", "a morphism $h_{\\lambda, j} : Y_{\\lambda, j} \\to X_{i_j}$", "whose restriction to $Y_j$ is the morphism $h_j$ chosen above.", "Thus $\\{Y_{\\lambda, j} \\to X\\}$ refines", "$\\{X_i \\to X\\}$ and the proof is complete." ], "refs": [ "flat-definition-h-covering", "topologies-definition-ph-covering", "limits-lemma-proper-limit-of-proper-finite-presentation", "limits-lemma-descend-opens", "limits-lemma-limit-affine", "limits-proposition-characterize-locally-finite-presentation" ], "ref_ids": [ 6220, 12544, 15090, 15041, 15043, 15127 ] } ], "ref_ids": [] }, { "id": 6142, "type": "theorem", "label": "flat-lemma-zariski-h", "categories": [ "flat" ], "title": "flat-lemma-zariski-h", "contents": [ "An fppf covering is a h covering. Hence syntomic, smooth, \\'etale,", "and Zariski coverings are h coverings as well." ], "refs": [], "proofs": [ { "contents": [ "This is true because in an fppf covering the morphisms are", "required to be locally of finite presentation and because", "fppf coverings are ph covering, see More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-fppf-ph}.", "The second statement follows from the first and", "Topologies, Lemma \\ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf}." ], "refs": [ "more-morphisms-lemma-fppf-ph", "topologies-lemma-zariski-etale-smooth-syntomic-fppf" ], "ref_ids": [ 13927, 12471 ] } ], "ref_ids": [] }, { "id": 6143, "type": "theorem", "label": "flat-lemma-surjective-proper-finite-presentation-h", "categories": [ "flat" ], "title": "flat-lemma-surjective-proper-finite-presentation-h", "contents": [ "Let $f : Y \\to X$ be a surjective proper morphism of schemes", "which is of finite presentation. Then $\\{Y \\to X\\}$ is an h covering." ], "refs": [], "proofs": [ { "contents": [ "Combine Topologies, Lemmas", "\\ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc-ph-V} and", "\\ref{topologies-lemma-surjective-proper-ph}." ], "refs": [ "topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc-ph-V", "topologies-lemma-surjective-proper-ph" ], "ref_ids": [ 12511, 12483 ] } ], "ref_ids": [] }, { "id": 6144, "type": "theorem", "label": "flat-lemma-refine-by-h", "categories": [ "flat" ], "title": "flat-lemma-refine-by-h", "contents": [ "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family", "of morphisms such that $f_i$ is locally of finite presentation for all $i$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\{T_i \\to T\\}_{i \\in I}$ is an h covering,", "\\item there is an h covering which refines $\\{T_i \\to T\\}_{i \\in I}$, and", "\\item $\\{\\coprod_{i \\in I} T_i \\to T\\}$ is an h covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from the analogous statement for ph coverings", "(Topologies, Lemma \\ref{topologies-lemma-refine-by-ph})", "or from the analogous statement for V coverings", "(Topologies, Lemma \\ref{topologies-lemma-refine-by-V})." ], "refs": [ "topologies-lemma-refine-by-ph", "topologies-lemma-refine-by-V" ], "ref_ids": [ 12484, 12509 ] } ], "ref_ids": [] }, { "id": 6145, "type": "theorem", "label": "flat-lemma-h", "categories": [ "flat" ], "title": "flat-lemma-h", "contents": [ "Let $T$ be a scheme.", "\\begin{enumerate}", "\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$", "is an h covering of $T$.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an h covering and for each", "$i$ we have an h covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then", "$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is an h covering.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an h covering", "and $T' \\to T$ is a morphism of schemes then", "$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is an h covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from the corresponding statement for", "either ph or V coverings", "(Topologies, Lemma \\ref{topologies-lemma-ph} or", "\\ref{topologies-lemma-V})", "and the fact that the class of morphisms which are locally", "of finite presentation is preserved under base change and", "composition." ], "refs": [ "topologies-lemma-ph", "topologies-lemma-V" ], "ref_ids": [ 12485, 12510 ] } ], "ref_ids": [] }, { "id": 6146, "type": "theorem", "label": "flat-lemma-h-induced", "categories": [ "flat" ], "title": "flat-lemma-h-induced", "contents": [ "Let $\\Sch_h$ be a big h site as in", "Definition \\ref{definition-big-h-site}.", "Let $T \\in \\Ob(\\Sch_h)$.", "Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary h covering of $T$.", "\\begin{enumerate}", "\\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site", "$\\Sch_h$ which refines $\\{T_i \\to T\\}_{i \\in I}$.", "\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard h covering, then", "it is tautologically equivalent to a covering of $\\Sch_h$.", "\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then", "it is tautologically equivalent to a covering of $\\Sch_h$.", "\\end{enumerate}" ], "refs": [ "flat-definition-big-h-site" ], "proofs": [ { "contents": [ "Omitted. Hint: this is exactly the same as the proof of", "Topologies, Lemma \\ref{topologies-lemma-ph-induced}." ], "refs": [ "topologies-lemma-ph-induced" ], "ref_ids": [ 12486 ] } ], "ref_ids": [ 6221 ] }, { "id": 6147, "type": "theorem", "label": "flat-lemma-verify-site-h", "categories": [ "flat" ], "title": "flat-lemma-verify-site-h", "contents": [ "Let $S$ be a scheme. Let $\\Sch_h$ be a big h", "site containing $S$. Then $(\\textit{Aff}/S)_h$ is a site." ], "refs": [], "proofs": [ { "contents": [ "Reasoning as in the proof of", "Topologies, Lemma \\ref{topologies-lemma-verify-site-etale}", "it suffices to show that the collection of standard h coverings", "satisfies properties (1), (2) and (3) of", "Sites, Definition \\ref{sites-definition-site}.", "This is clear since for example, given a standard h covering", "$\\{T_i \\to T\\}_{i\\in I}$ and for each", "$i$ a standard h covering $\\{T_{ij} \\to T_i\\}_{j \\in J_i}$, then", "$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a h covering", "(Lemma \\ref{lemma-h}), $\\bigcup_{i\\in I} J_i$ is finite and", "each $T_{ij}$ is affine. Thus $\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$", "is a standard h covering." ], "refs": [ "topologies-lemma-verify-site-etale", "sites-definition-site", "flat-lemma-h" ], "ref_ids": [ 12449, 8652, 6145 ] } ], "ref_ids": [] }, { "id": 6148, "type": "theorem", "label": "flat-lemma-fibre-products-h", "categories": [ "flat" ], "title": "flat-lemma-fibre-products-h", "contents": [ "Let $S$ be a scheme. Let $\\Sch_h$ be a big h", "site containing $S$. The underlying categories of the sites", "$\\Sch_h$, $(\\Sch/S)_h$, and $(\\textit{Aff}/S)_h$ have fibre products.", "In each case the obvious functor into the category $\\Sch$ of", "all schemes commutes with taking fibre products. The category", "$(\\Sch/S)_h$ has a final object, namely $S/S$." ], "refs": [], "proofs": [ { "contents": [ "For $\\Sch_h$ it is true by construction, see", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "Suppose we have $U \\to S$, $V \\to U$, $W \\to U$ morphisms", "of schemes with $U, V, W \\in \\Ob(\\Sch_h)$.", "The fibre product $V \\times_U W$ in $\\Sch_h$", "is a fibre product in $\\Sch$ and", "is the fibre product of $V/S$ with $W/S$ over $U/S$ in", "the category of all schemes over $S$, and hence also a", "fibre product in $(\\Sch/S)_h$.", "This proves the result for $(\\Sch/S)_h$.", "If $U, V, W$ are affine, so is $V \\times_U W$ and hence the", "result for $(\\textit{Aff}/S)_h$." ], "refs": [ "sets-lemma-what-is-in-it" ], "ref_ids": [ 8795 ] } ], "ref_ids": [] }, { "id": 6149, "type": "theorem", "label": "flat-lemma-affine-big-site-h", "categories": [ "flat" ], "title": "flat-lemma-affine-big-site-h", "contents": [ "Let $S$ be a scheme. Let $\\Sch_h$ be a big h", "site containing $S$.", "The functor $(\\textit{Aff}/S)_h \\to (\\Sch/S)_h$", "is cocontinuous and induces an equivalence of topoi from", "$\\Sh((\\textit{Aff}/S)_h)$ to", "$\\Sh((\\Sch/S)_h)$." ], "refs": [], "proofs": [ { "contents": [ "The notion of a special cocontinuous functor is introduced in", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}.", "Thus we have to verify assumptions (1) -- (5) of", "Sites, Lemma \\ref{sites-lemma-equivalence}.", "Denote the inclusion functor", "$u : (\\textit{Aff}/S)_h \\to (\\Sch/S)_h$.", "Being cocontinuous follows because any h covering of", "$T/S$, $T$ affine, can be refined by a standard h covering", "for example by Lemma \\ref{lemma-approximate-h-cover}. Hence (1) holds.", "We see $u$ is continuous simply because a standard h covering", "is a h covering.", "Hence (2) holds. Parts (3) and (4) follow immediately from the fact", "that $u$ is fully faithful. And finally condition (5) follows from the", "fact that every scheme has an affine open covering (which is", "a h covering)." ], "refs": [ "sites-definition-special-cocontinuous-functor", "sites-lemma-equivalence", "flat-lemma-approximate-h-cover" ], "ref_ids": [ 8672, 8578, 6141 ] } ], "ref_ids": [] }, { "id": 6150, "type": "theorem", "label": "flat-lemma-characterize-sheaf-h", "categories": [ "flat" ], "title": "flat-lemma-characterize-sheaf-h", "contents": [ "Let $\\mathcal{F}$ be a presheaf on $(\\Sch/S)_h$.", "Then $\\mathcal{F}$ is a sheaf if and only if", "\\begin{enumerate}", "\\item $\\mathcal{F}$ satisfies the sheaf condition for", "Zariski coverings, and", "\\item if $f : V \\to U$ is proper, surjective, and of finite presentation, then", "$\\mathcal{F}(U)$ maps bijectively to the equalizer", "of the two maps $\\mathcal{F}(V) \\to \\mathcal{F}(V \\times_U V)$.", "\\end{enumerate}", "Moreover, in the presence of (1) property (2) is equivalent to", "property", "\\begin{enumerate}", "\\item[(2')] the sheaf property for $\\{V \\to U\\}$ as in (2) with $U$ affine.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will show that if (1) and (2) hold, then $\\mathcal{F}$ is sheaf.", "Let $\\{T_i \\to T\\}$ be a covering in $(\\Sch/S)_h$.", "We will verify the sheaf condition for this covering.", "Let $s_i \\in \\mathcal{F}(T_i)$ be sections which restrict to the same", "section over $T_i \\times_T T_{i'}$. We will show that there exists a", "unique section $s \\in \\mathcal{F}(T)$ restricting to $s_i$ over $T_i$.", "Let $T = \\bigcup U_j$ be an affine open covering.", "By property (1) it suffices to produce sections $s_j \\in \\mathcal{F}(U_j)$", "which agree on $U_j \\cap U_{j'}$ in order to produce $s$.", "Consider the coverings $\\{T_i \\times_T U_j \\to U_j\\}$.", "Then $s_{ji} = s_i|_{T_i \\times_T U_j}$ are sections agreeing", "over $(T_i \\times_T U_j) \\times_{U_j} (T_{i'} \\times_T U_j)$.", "Choose a proper surjective morphism $V_j \\to U_j$ of finite presentation", "and a finite affine open covering $V_j = \\bigcup V_{jk}$", "such that $\\{V_{jk} \\to U_j\\}$ refines $\\{T_i \\times_T U_j \\to U_j\\}$.", "See Lemma \\ref{lemma-approximate-h-cover}.", "If $s_{jk} \\in \\mathcal{F}(V_{jk})$", "denotes the pullback of $s_{ji}$ to $V_{jk}$ by the", "implied morphisms, then we find that $s_{jk}$ glue to a section", "$s'_j \\in \\mathcal{F}(V_j)$. Using the agreement on overlaps", "once more, we find that $s'_j$ is in the equalizer of the two", "maps $\\mathcal{F}(V_j) \\to \\mathcal{F}(V_j \\times_{U_j} V_j)$.", "Hence by (2) we find that $s'_j$ comes from a unique section", "$s_j \\in \\mathcal{F}(U_j)$. We omit the verification that these", "sections $s_j$ have all the desired properties.", "\\medskip\\noindent", "Proof of the equivalence of (2) and (2') in the presence of (1).", "Suppose $V \\to U$ is a morphism of $(\\Sch/S)_h$ which is", "proper, surjective, and of finite presentation. Choose an", "affine open covering $U = \\bigcup U_i$ and set $V_i = V \\times_U U_i$.", "Then we see that $\\mathcal{F}(U) \\to \\mathcal{F}(V)$", "is injective because we know $\\mathcal{F}(U_i) \\to \\mathcal{F}(V_i)$", "is injective by (2') and we know $\\mathcal{F}(U) \\to \\prod \\mathcal{F}(U_i)$", "is injective by (1). Finally, suppose that we are given an", "$t \\in \\mathcal{F}(V)$ in the equalizer of the two maps", "$\\mathcal{F}(V) \\to \\mathcal{F}(V \\times_U V)$.", "Then $t|_{V_i}$ is in the equalizer of the two maps", "$\\mathcal{F}(V_i) \\to \\mathcal{F}(V_i \\times_{U_i} V_i)$", "for all $i$. Hence we obtain a unique section $s_i \\in \\mathcal{F}(U_i)$", "mapping to $t|_{V_i}$ for all $i$ by (2').", "We omit the verification that $s_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}$", "for all $i, j$; this uses the uniqueness property just shown.", "By the sheaf property for the covering $U = \\bigcup U_i$ we obtain", "a section $s \\in \\mathcal{F}(U)$. We omit the proof that $s$", "maps to $t$ in $\\mathcal{F}(V)$." ], "refs": [ "flat-lemma-approximate-h-cover" ], "ref_ids": [ 6141 ] } ], "ref_ids": [] }, { "id": 6151, "type": "theorem", "label": "flat-lemma-morphism-big-h", "categories": [ "flat" ], "title": "flat-lemma-morphism-big-h", "contents": [ "Let $\\Sch_h$ be a big h site.", "Let $f : T \\to S$ be a morphism in $\\Sch_h$.", "The functor", "$$", "u : (\\Sch/T)_h \\longrightarrow (\\Sch/S)_h,", "\\quad", "V/T \\longmapsto V/S", "$$", "is cocontinuous, and has a continuous right adjoint", "$$", "v : (\\Sch/S)_h \\longrightarrow (\\Sch/T)_h,", "\\quad", "(U \\to S) \\longmapsto (U \\times_S T \\to T).", "$$", "They induce the same morphism of topoi", "$$", "f_{big} :", "\\Sh((\\Sch/T)_h)", "\\longrightarrow", "\\Sh((\\Sch/S)_h)", "$$", "We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.", "We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.", "Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with", "fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "The functor $u$ is cocontinuous, continuous, and commutes with fibre products", "and equalizers. Hence", "Sites, Lemmas \\ref{sites-lemma-when-shriek} and", "\\ref{sites-lemma-preserve-equalizers}", "apply and we deduce the formula", "for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover,", "the functor $v$ is a right adjoint because given $U/T$ and $V/S$", "we have $\\Mor_S(u(U), V) = \\Mor_T(U, V \\times_S T)$", "as desired. Thus we may apply", "Sites, Lemmas \\ref{sites-lemma-have-functor-other-way} and", "\\ref{sites-lemma-have-functor-other-way-morphism} to get the", "formula for $f_{big, *}$." ], "refs": [ "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers", "sites-lemma-have-functor-other-way", "sites-lemma-have-functor-other-way-morphism" ], "ref_ids": [ 8545, 8546, 8549, 8550 ] } ], "ref_ids": [] }, { "id": 6152, "type": "theorem", "label": "flat-lemma-composition-h", "categories": [ "flat" ], "title": "flat-lemma-composition-h", "contents": [ "Given schemes $X$, $Y$, $Y$ in $(\\Sch/S)_h$", "and morphisms $f : X \\to Y$, $g : Y \\to Z$ we have", "$g_{big} \\circ f_{big} = (g \\circ f)_{big}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the simple description of pushforward", "and pullback for the functors on the big sites from", "Lemma \\ref{lemma-morphism-big-h}." ], "refs": [ "flat-lemma-morphism-big-h" ], "ref_ids": [ 6151 ] } ], "ref_ids": [] }, { "id": 6153, "type": "theorem", "label": "flat-lemma-limit-h-topology", "categories": [ "flat" ], "title": "flat-lemma-limit-h-topology", "contents": [ "Let $T$ be an affine scheme which is written as a limit", "$T = \\lim_{i \\in I} T_i$ of a directed inverse system of affine schemes.", "\\begin{enumerate}", "\\item Let $\\mathcal{V} = \\{V_j \\to T\\}_{j = 1, \\ldots, m}$ be a", "standard h covering of $T$, see Definition", "\\ref{definition-standard-h}.", "Then there exists an index $i$ and a standard h covering", "$\\mathcal{V}_i = \\{V_{i, j} \\to T_i\\}_{j = 1, \\ldots, m}$", "whose base change $T \\times_{T_i} \\mathcal{V}_i$ to $T$", "is isomorphic to $\\mathcal{V}$.", "\\item Let $\\mathcal{V}_i$, $\\mathcal{V}'_i$ be a pair of standard", "h coverings of $T_i$. If", "$f : T \\times_{T_i} \\mathcal{V}_i \\to T \\times_{T_i} \\mathcal{V}'_i$ is", "a morphism of coverings of $T$, then there exists an index", "$i' \\geq i$ and a morphism", "$f_{i'} : T_{i'} \\times_{T_i} \\mathcal{V} \\to", "T_{i'} \\times_{T_i} \\mathcal{V}'_i$", "whose base change to $T$ is $f$.", "\\item If", "$f, g : \\mathcal{V} \\to \\mathcal{V}'_i$", "are morphisms of standard h coverings of $T_i$ whose", "base changes $f_T, g_T$ to $T$ are equal then there exists an", "index $i' \\geq i$ such that $f_{T_{i'}} = g_{T_{i'}}$.", "\\end{enumerate}", "In other words, the category of standard h coverings of $T$ is", "the colimit over $I$ of the categories of standard h coverings of $T_i$." ], "refs": [ "flat-definition-standard-h" ], "proofs": [ { "contents": [ "By Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "the category of schemes of finite presentation over $T$ is the", "colimit over $I$ of the categories of finite presentation over $T_i$. By", "Limits, Lemma \\ref{limits-lemma-descend-affine-finite-presentation}", "the same is true for category of schemes which are affine and", "of finite presentation over $T$.", "To finish the proof of the lemma it suffices to show that if", "$\\{V_{j, i} \\to T_i\\}_{j = 1, \\ldots, m}$ is a finite family of", "finitely presented morphisms with $V_{j, i}$ affine, and the", "base change family $\\{T \\times_{T_i} V_{j, i} \\to T\\}$ is", "an h covering, then for some $i' \\geq i$ the family", "$\\{T_{i'} \\times_{T_i} V_{j, i} \\to T_{i'}\\}$ is an h covering.", "To see this we use Lemma \\ref{lemma-approximate-h-cover} to", "choose a finitely presented, proper, surjective", "morphism $Y \\to T$ and a finite affine open covering", "$Y = \\bigcup_{k = 1, \\ldots, n} Y_k$ such that", "$\\{Y_k \\to T\\}_{k = 1, \\ldots, n}$ refines", "$\\{T \\times_{T_i} V_{j, i} \\to T\\}$.", "Using the arguments above and", "Limits, Lemmas \\ref{limits-lemma-eventually-proper},", "\\ref{limits-lemma-descend-surjective}, and", "\\ref{limits-lemma-descend-opens}", "we can find an $i' \\geq i$ and a finitely presented, surjective, proper", "morphism $Y_{i'} \\to T_{i'}$ and an affine open covering", "$Y_{i'} = \\bigcup_{k = 1, \\ldots, n} Y_{i', k}$", "such that moreover $\\{Y_{i', k} \\to Y_{i'}\\}$ refines", "$\\{T_{i'} \\times_{T_i} V_{j, i} \\to T_{i'}\\}$.", "It follows that this last mentioned family is", "a h covering and the proof is complete." ], "refs": [ "limits-lemma-descend-finite-presentation", "limits-lemma-descend-affine-finite-presentation", "flat-lemma-approximate-h-cover", "limits-lemma-eventually-proper", "limits-lemma-descend-surjective", "limits-lemma-descend-opens" ], "ref_ids": [ 15077, 15057, 6141, 15089, 15069, 15041 ] } ], "ref_ids": [ 6222 ] }, { "id": 6154, "type": "theorem", "label": "flat-lemma-extend-sheaf-h", "categories": [ "flat" ], "title": "flat-lemma-extend-sheaf-h", "contents": [ "Let $S$ be a scheme contained in a big site $\\Sch_h$.", "Let $F : (\\Sch/S)_h^{opp} \\to \\textit{Sets}$ be an h sheaf satisfying", "property (b) of Topologies, Lemma \\ref{topologies-lemma-extend}", "with $\\mathcal{C} = (\\Sch/S)_h$.", "Then the extension $F'$ of $F$ to the category of all", "schemes over $S$ satisfies the sheaf condition for all h coverings", "and is limit preserving (Limits, Remark \\ref{limits-remark-limit-preserving})." ], "refs": [ "topologies-lemma-extend", "limits-remark-limit-preserving" ], "proofs": [ { "contents": [ "This is proven by the arguments given in the proofs of", "Topologies, Lemmas \\ref{topologies-lemma-extend-sheaf-general} and", "\\ref{topologies-lemma-extend-sheaf} using", "Lemmas \\ref{lemma-limit-h-topology} and \\ref{lemma-h-induced}.", "Details omitted." ], "refs": [ "topologies-lemma-extend-sheaf-general", "topologies-lemma-extend-sheaf", "flat-lemma-limit-h-topology", "flat-lemma-h-induced" ], "ref_ids": [ 12519, 12520, 6153, 6146 ] } ], "ref_ids": [ 12517, 15130 ] }, { "id": 6155, "type": "theorem", "label": "flat-lemma-blow-up-square-ph", "categories": [ "flat" ], "title": "flat-lemma-blow-up-square-ph", "contents": [ "Let $\\mathcal{F}$ be a sheaf on a site $(\\Sch/S)_{ph}$, see", "Topologies, Definition \\ref{topologies-definition-big-small-ph}.", "Then for any blow up square (\\ref{equation-blow-up-square})", "in the category $(\\Sch/S)_{ph}$ the diagram", "$$", "\\xymatrix{", "\\mathcal{F}(E) & \\mathcal{F}(X') \\ar[l] \\\\", "\\mathcal{F}(Z) \\ar[u] & \\mathcal{F}(X) \\ar[u] \\ar[l]", "}", "$$", "is cartesian in the category of sets." ], "refs": [ "topologies-definition-big-small-ph" ], "proofs": [ { "contents": [ "Since $Z \\amalg X' \\to X$ is a surjective proper morphism", "we see that $\\{Z \\amalg X' \\to X\\}$ is a ph covering", "(Topologies, Lemma \\ref{topologies-lemma-surjective-proper-ph}).", "We have", "$$", "(Z \\amalg X') \\times_X (Z \\amalg X') =", "Z \\amalg E \\amalg E \\amalg X' \\times_X X'", "$$", "Since $\\mathcal{F}$ is a Zariski sheaf we see that", "$\\mathcal{F}$ sends disjoint unions to products.", "Thus the sheaf condition for the covering", "$\\{Z \\amalg X' \\to X\\}$ says that", "$\\mathcal{F}(X) \\to \\mathcal{F}(Z) \\times \\mathcal{F}(X')$", "is injective with image the set of pairs $(t, s')$ such that", "(a) $t|_E = s'|_E$ and (b) $s'$ is in the equalizer of the two maps", "$\\mathcal{F}(X') \\to \\mathcal{F}(X' \\times_X X')$.", "Next, observe that the obvious morphism", "$$", "E \\times_Z E \\amalg X' \\longrightarrow X' \\times_X X'", "$$", "is a surjective proper morphism as $b$ induces", "an isomorphism $X' \\setminus E \\to X \\setminus Z$. We conclude that", "$\\mathcal{F}(X' \\times_X X') \\to", "\\mathcal{F}(E \\times_Z E) \\times \\mathcal{F}(X')$ is injective.", "It follows that (a) $\\Rightarrow$ (b) which means that the lemma is true." ], "refs": [ "topologies-lemma-surjective-proper-ph" ], "ref_ids": [ 12483 ] } ], "ref_ids": [ 12546 ] }, { "id": 6156, "type": "theorem", "label": "flat-lemma-thickening-ph", "categories": [ "flat" ], "title": "flat-lemma-thickening-ph", "contents": [ "Let $\\mathcal{F}$ be a sheaf on a site $(\\Sch/S)_{ph}$", "as in Topologies, Definition \\ref{topologies-definition-big-small-ph}.", "Let $X \\to X'$ be a morphism of $(\\Sch/S)_{ph}$ which is", "a thickening. Then", "$\\mathcal{F}(X') \\to \\mathcal{F}(X)$ is bijective." ], "refs": [ "topologies-definition-big-small-ph" ], "proofs": [ { "contents": [ "Observe that $X \\to X'$ is a proper surjective morphism of and", "$X \\times_{X'} X = X$.", "By the sheaf property for the ph covering $\\{X \\to X'\\}$", "(Topologies, Lemma", "\\ref{topologies-lemma-surjective-proper-ph})", "we conclude." ], "refs": [ "topologies-lemma-surjective-proper-ph" ], "ref_ids": [ 12483 ] } ], "ref_ids": [ 12546 ] }, { "id": 6157, "type": "theorem", "label": "flat-lemma-base-change-almost-blow-up", "categories": [ "flat" ], "title": "flat-lemma-base-change-almost-blow-up", "contents": [ "Consider an almost blow up square (\\ref{equation-almost-blow-up-square}).", "Let $Y \\to X$ be any morphism. Then the base change", "$$", "\\xymatrix{", "Y \\times_X E \\ar[d] \\ar[r] & Y \\times_X X' \\ar[d] \\\\", "Y \\times_X Z \\ar[r] & Y", "}", "$$", "is an almost blow up square too." ], "refs": [], "proofs": [ { "contents": [ "The morphism $Y \\times_X X' \\to Y$ is proper and of finite presentation", "by Morphisms, Lemmas \\ref{morphisms-lemma-base-change-proper} and", "\\ref{morphisms-lemma-base-change-finite-presentation}.", "The morphism $Y \\times_X Z \\to Y$ is a closed immersion", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-closed-immersion}) of", "finite presentation. The inverse image of $Y \\times_X Z$ in $Y \\times_X X'$", "is equal to the inverse image of $E$ in $Y \\times_X X'$ and hence is", "locally principal", "(Divisors, Lemma \\ref{divisors-lemma-pullback-locally-principal}).", "Let $X'' \\subset X'$, resp.\\ $Y'' \\subset Y \\times_X X'$ be the closed", "subscheme corresponding to the quasi-coherent ideal of sections of", "$\\mathcal{O}_{X'}$, resp.\\ $\\mathcal{O}_{Y \\times_Y X'}$", "supported on $E$, resp.\\ $Y \\times_X E$.", "Clearly, $Y'' \\subset Y \\times_X X''$ is the closed subscheme", "corresponding to the quasi-coherent ideal of sections of", "$\\mathcal{O}_{Y \\times_Y X''}$ supported on $Y \\times_X (E \\cap X'')$.", "Thus $Y''$ is the strict transform of $Y$ relative to the blowing up", "$X'' \\to X$, see", "Divisors, Definition \\ref{divisors-definition-strict-transform}.", "Thus by Divisors, Lemma \\ref{divisors-lemma-strict-transform}", "we see that $Y''$ is the blow up of $Y \\times_X Z$ on $Y$." ], "refs": [ "morphisms-lemma-base-change-proper", "morphisms-lemma-base-change-finite-presentation", "morphisms-lemma-base-change-closed-immersion", "divisors-lemma-pullback-locally-principal", "divisors-definition-strict-transform", "divisors-lemma-strict-transform" ], "ref_ids": [ 5409, 5240, 5128, 7935, 8113, 8065 ] } ], "ref_ids": [] }, { "id": 6158, "type": "theorem", "label": "flat-lemma-shrink-almost-blow-up", "categories": [ "flat" ], "title": "flat-lemma-shrink-almost-blow-up", "contents": [ "Consider an almost blow up square (\\ref{equation-almost-blow-up-square}).", "Let $W \\to X'$ be a closed immersion of finite presentation.", "The following are equivalent", "\\begin{enumerate}", "\\item $X' \\setminus E$ is scheme theoretically contained in $W$,", "\\item the blowup $X''$ of $X$ in $Z$ is scheme theoretically contained in $W$,", "\\item the diagram", "$$", "\\xymatrix{", "E \\cap W \\ar[d] \\ar[r] & W \\ar[d] \\\\", "Z \\ar[r] & X", "}", "$$", "is an almost blow up square. Here $E \\cap W$ is the", "scheme theoretic intersection.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Then the surjection $\\mathcal{O}_{X'} \\to \\mathcal{O}_W$", "is an isomorphism over the open $X' \\subset E$. Since the ideal sheaf", "of $X'' \\subset X'$ is the sections of $\\mathcal{O}_{X'}$ supported on $E$", "(by our definition of almost blow up squares)", "we conclude (2) is true. If (2) is true, then (3) holds.", "If (3) holds, then (1) holds because $X'' \\cap (X' \\setminus E)$", "is isomorphic to $X \\setminus Z$ which in turn is isomorphic", "to $X' \\setminus E$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6159, "type": "theorem", "label": "flat-lemma-blow-up-limit-almost-blow-up", "categories": [ "flat" ], "title": "flat-lemma-blow-up-limit-almost-blow-up", "contents": [ "Consider an almost blow up square (\\ref{equation-almost-blow-up-square})", "with $X$ quasi-compact and quasi-separated. Then the blowup $X''$ of $X$", "in $Z$ can be written as", "$$", "X'' = \\lim X'_i", "$$", "where the limit is over the directed system of closed subschemes", "$X'_i \\subset X'$ of finite presentation satisfying the equivalent", "conditions of Lemma \\ref{lemma-shrink-almost-blow-up}." ], "refs": [ "flat-lemma-shrink-almost-blow-up" ], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\subset \\mathcal{O}_{X'}$ be the quasi-coherent", "sheaf of ideals corresponding to $X''$. By", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-colimit-finite-type}", "we can write $\\mathcal{I}$ as the filtered colimit", "$\\mathcal{I} = \\colim \\mathcal{I}_i$ of its quasi-coherent", "submodules of finite type. Since these modules correspond", "$1$-to-$1$ to the closed subschemes $X'_i$ the proof is complete." ], "refs": [ "properties-lemma-quasi-coherent-colimit-finite-type" ], "ref_ids": [ 3020 ] } ], "ref_ids": [ 6158 ] }, { "id": 6160, "type": "theorem", "label": "flat-lemma-almost-blow-up-square", "categories": [ "flat" ], "title": "flat-lemma-almost-blow-up-square", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Let $Z \\subset X$", "be a closed subscheme cut out by a finite type quasi-coherent", "sheaf of ideals. Then there exists an almost blow up square as in", "(\\ref{equation-almost-blow-up-square})." ], "refs": [], "proofs": [ { "contents": [ "We may write $X = \\lim X_i$ as a directed limit of an inverse system", "of Noetherian schemes with affine transition morphisms, see", "Limits, Proposition \\ref{limits-proposition-approximate}.", "We can find an index $i$ and a closed immersion $Z_i \\to X_i$", "whose base change to $X$ is the closed immersion $Z \\to X$.", "See Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation} and", "\\ref{limits-lemma-descend-closed-immersion-finite-presentation}.", "Let $b_i : X'_i \\to X_i$ be the blowing up with center $Z_i$.", "This produces a blow up square", "$$", "\\xymatrix{", "E_i \\ar[r] \\ar[d] & X'_i \\ar[d]^{b_i} \\\\", "Z_i \\ar[r] & X_i", "}", "$$", "where all the morphisms are finite type morphisms of Noetherian schemes", "and hence of finite presentation. Thus this is an almost blow up square.", "By Lemma \\ref{lemma-base-change-almost-blow-up}", "the base change of this diagram to $X$ produces the desired", "almost blow up square." ], "refs": [ "limits-proposition-approximate", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-closed-immersion-finite-presentation", "flat-lemma-base-change-almost-blow-up" ], "ref_ids": [ 15126, 15077, 15060, 6157 ] } ], "ref_ids": [] }, { "id": 6161, "type": "theorem", "label": "flat-lemma-almost-blow-up-unique", "categories": [ "flat" ], "title": "flat-lemma-almost-blow-up-unique", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme and let $Z \\subset X$", "be a closed subscheme cut out by a finite type quasi-coherent sheaf of ideals.", "Suppose given almost blow up squares (\\ref{equation-almost-blow-up-square})", "$$", "\\xymatrix{", "E_k \\ar[r] \\ar[d] & X_k' \\ar[d] \\\\", "Z \\ar[r] & X", "}", "$$", "for $k = 1, 2$, then there exists an almost blow up square", "$$", "\\xymatrix{", "E \\ar[r] \\ar[d] & X' \\ar[d] \\\\", "Z \\ar[r] & X", "}", "$$", "and closed immersions $i_k : X' \\to X'_k$ over $X$", "with $E = i_k^{-1}(E_k)$." ], "refs": [], "proofs": [ { "contents": [ "Denote $X'' \\to X$ the blowing up of $Z$ in $X$.", "We view $X''$ as a closed subscheme of both $X'_1$ and $X'_2$.", "Write $X'' = \\lim X'_{1, i}$ as in", "Lemma \\ref{lemma-blow-up-limit-almost-blow-up}.", "By Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}", "there exists an $i$ and a morphism $h : X'_{1, i} \\to X'_2$ agreeing", "with the inclusions $X'' \\subset X'_{1, i}$ and $X'' \\subset X'_2$.", "By Limits, Lemma \\ref{limits-lemma-eventually-closed-immersion}", "the restriction of $h$ to $X'_{1, i'}$ is a closed immersion", "for some $i' \\geq i$. This finishes the proof." ], "refs": [ "flat-lemma-blow-up-limit-almost-blow-up", "limits-proposition-characterize-locally-finite-presentation", "limits-lemma-eventually-closed-immersion" ], "ref_ids": [ 6159, 15127, 15050 ] } ], "ref_ids": [] }, { "id": 6162, "type": "theorem", "label": "flat-lemma-flat-after-almost-blowing-up", "categories": [ "flat" ], "title": "flat-lemma-flat-after-almost-blowing-up", "contents": [ "Let $Y$ be a quasi-compact and quasi-separated scheme.", "Let $X$ be a scheme of finite presentation over $Y$.", "Let $V \\subset Y$ be a quasi-compact open such that", "$X_V \\to V$ is flat. Then there exist a commutative diagram", "$$", "\\xymatrix{", "E \\ar[ddd] \\ar[rd] & & & D \\ar[lll] \\ar[ddd] \\ar[ld] \\\\", "& Y' \\ar[d] & X' \\ar[l] \\ar[d] \\\\", "& Y & X \\ar[l] \\\\", "Z \\ar[ru] & & & T \\ar[lll] \\ar[lu]", "}", "$$", "whose right and left hand squares are almost blow up squares,", "whose lower and top squares are cartesian, such that", "$Z \\cap V = \\emptyset$, and", "such that $X' \\to Y'$ is flat (and of finite presentation)." ], "refs": [], "proofs": [ { "contents": [ "If $Y$ is a Noetherian scheme, then this lemma follows immediately", "from Lemma \\ref{lemma-flat-after-blowing-up}", "because in this case blow up squares are almost blow up squares", "(we also use that strict transforms are blow ups).", "The general case is reduced to the Noetherian case by", "absolute Noetherian approximation.", "\\medskip\\noindent", "We may write $Y = \\lim Y_i$ as a directed limit of an inverse system", "of Noetherian schemes with affine transition morphisms, see", "Limits, Proposition \\ref{limits-proposition-approximate}.", "We can find an index $i$ and a morphism $X_i \\to Y_i$", "of finite presentation whose base change to $Y$ is $X \\to Y$.", "See Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation}.", "After increasing $i$ we may assume $V$ is the inverse image of", "an open subscheme $V_i \\subset Y_i$, see", "Limits, Lemma \\ref{limits-lemma-descend-opens}.", "Finally, after increasing $i$ we may assume that $X_{i, V_i} \\to V_i$", "is flat, see", "Limits, Lemma \\ref{limits-lemma-descend-flat-finite-presentation}.", "By the Noetherian case, we may construct a diagram as in the", "lemma for $X_i \\to Y_i \\supset V_i$. The base change of this", "diagram by $Y \\to Y_i$ provides the solution. Use that base change", "preserves properties of morphisms, see Morphisms, Lemmas", "\\ref{morphisms-lemma-base-change-proper},", "\\ref{morphisms-lemma-base-change-finite-presentation},", "\\ref{morphisms-lemma-base-change-closed-immersion}, and", "\\ref{morphisms-lemma-base-change-flat}", "and that base change of an almost", "blow up square is an almost blow up square, see", "Lemma \\ref{lemma-base-change-almost-blow-up}." ], "refs": [ "flat-lemma-flat-after-blowing-up", "limits-proposition-approximate", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-opens", "limits-lemma-descend-flat-finite-presentation", "morphisms-lemma-base-change-proper", "morphisms-lemma-base-change-finite-presentation", "morphisms-lemma-base-change-closed-immersion", "morphisms-lemma-base-change-flat", "flat-lemma-base-change-almost-blow-up" ], "ref_ids": [ 6123, 15126, 15077, 15041, 15062, 5409, 5240, 5128, 5265, 6157 ] } ], "ref_ids": [] }, { "id": 6163, "type": "theorem", "label": "flat-lemma-blow-up-square-h", "categories": [ "flat" ], "title": "flat-lemma-blow-up-square-h", "contents": [ "Let $\\mathcal{F}$ be a sheaf on one of the sites $(\\Sch/S)_h$", "constructed in Definition \\ref{definition-big-small-h}.", "Then for any almost blow up square (\\ref{equation-almost-blow-up-square})", "in the category $(\\Sch/S)_h$ the diagram", "$$", "\\xymatrix{", "\\mathcal{F}(E) & \\mathcal{F}(X') \\ar[l] \\\\", "\\mathcal{F}(Z) \\ar[u] & \\mathcal{F}(X) \\ar[u] \\ar[l]", "}", "$$", "is cartesian in the category of sets." ], "refs": [ "flat-definition-big-small-h" ], "proofs": [ { "contents": [ "Since $Z \\amalg X' \\to X$ is a surjective proper morphism", "of finite presentation we see that $\\{Z \\amalg X' \\to X\\}$ is an h", "covering (Lemma \\ref{lemma-surjective-proper-finite-presentation-h}).", "We have", "$$", "(Z \\amalg X') \\times_X (Z \\amalg X') =", "Z \\amalg E \\amalg E \\amalg X' \\times_X X'", "$$", "Since $\\mathcal{F}$ is a Zariski sheaf we see that", "$\\mathcal{F}$ sends disjoint unions to products.", "Thus the sheaf condition for the covering", "$\\{Z \\amalg X' \\to X\\}$ says that", "$\\mathcal{F}(X) \\to \\mathcal{F}(Z) \\times \\mathcal{F}(X')$", "is injective with image the set of pairs $(t, s')$ such that", "(a) $t|_E = s'|_E$ and (b) $s'$ is in the equalizer of the two maps", "$\\mathcal{F}(X') \\to \\mathcal{F}(X' \\times_X X')$.", "Next, observe that the obvious morphism", "$$", "E \\times_Z E \\amalg X' \\longrightarrow X' \\times_X X'", "$$", "is a surjective proper morphism of finite presentation as $b$ induces", "an isomorphism $X' \\setminus E \\to X \\setminus Z$. We conclude that", "$\\mathcal{F}(X' \\times_X X') \\to", "\\mathcal{F}(E \\times_Z E) \\times \\mathcal{F}(X')$ is injective.", "It follows that (a) $\\Rightarrow$ (b) which means that the lemma is true." ], "refs": [ "flat-lemma-surjective-proper-finite-presentation-h" ], "ref_ids": [ 6143 ] } ], "ref_ids": [ 6223 ] }, { "id": 6164, "type": "theorem", "label": "flat-lemma-thickening-h", "categories": [ "flat" ], "title": "flat-lemma-thickening-h", "contents": [ "Let $\\mathcal{F}$ be a sheaf on one of the sites $(\\Sch/S)_h$", "constructed in Definition \\ref{definition-big-small-h}.", "Let $X \\to X'$ be a morphism of $(\\Sch/S)_h$ which is", "a thickening and of finite presentation. Then", "$\\mathcal{F}(X') \\to \\mathcal{F}(X)$ is bijective." ], "refs": [ "flat-definition-big-small-h" ], "proofs": [ { "contents": [ "First proof. Observe that $X \\to X'$ is a proper surjective morphism of", "finite presentation and $X \\times_{X'} X = X$.", "By the sheaf property for the h covering $\\{X \\to X'\\}$", "(Lemma \\ref{lemma-surjective-proper-finite-presentation-h})", "we conclude.", "\\medskip\\noindent", "Second proof (silly). The blow up of $X'$ in $X$ is the empty scheme.", "The reason is that the affine blowup algebra $A[\\frac{I}{a}]$", "(Algebra, Section \\ref{algebra-section-blow-up})", "is zero if $a$ is a nilpotent element of $A$. Details omitted.", "Hence we get an almost blow up square of the form", "$$", "\\xymatrix{", "\\emptyset \\ar[r] \\ar[d] & \\emptyset \\ar[d] \\\\", "X \\ar[r] & X'", "}", "$$", "Since $\\mathcal{F}$ is a sheaf we have that $\\mathcal{F}(\\emptyset)$", "is a singleton.", "Applying Lemma \\ref{lemma-blow-up-square-h} we get the conclusion." ], "refs": [ "flat-lemma-surjective-proper-finite-presentation-h", "flat-lemma-blow-up-square-h" ], "ref_ids": [ 6143, 6163 ] } ], "ref_ids": [ 6223 ] }, { "id": 6165, "type": "theorem", "label": "flat-lemma-refine-check-h", "categories": [ "flat" ], "title": "flat-lemma-refine-check-h", "contents": [ "Let $\\mathcal{F}$ be a presheaf on one of the sites $(\\Sch/S)_h$", "constructed in Definition \\ref{definition-big-small-h}.", "Then $\\mathcal{F}$ is a sheaf if and only if the following", "conditions are satisfied", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a sheaf for the Zariski topology,", "\\item given a morphism $f : X \\to Y$ of $(\\Sch/S)_h$ with $Y$ affine", "and $f$ surjective, flat, proper, and of finite presentation, then", "$\\mathcal{F}(Y)$ is the equalizer of the two maps", "$\\mathcal{F}(X) \\to \\mathcal{F}(X \\times_Y X)$,", "\\item $\\mathcal{F}$ turns an almost blow up square as in", "Example \\ref{example-one-generator} in the category $(\\Sch/S)_h$", "into a cartesian diagram of sets, and", "\\item $\\mathcal{F}$ turns an almost blow up square as in", "Example \\ref{example-two-generators} in the category $(\\Sch/S)_h$", "into a cartesian diagram of sets.", "\\end{enumerate}" ], "refs": [ "flat-definition-big-small-h" ], "proofs": [ { "contents": [ "By Proposition \\ref{proposition-check-h} it suffices to show that given", "an almost blow up square (\\ref{equation-almost-blow-up-square})", "with $X$ affine in the category $(\\Sch/S)_h$ the diagram", "$$", "\\xymatrix{", "\\mathcal{F}(E) & \\mathcal{F}(X') \\ar[l] \\\\", "\\mathcal{F}(Z) \\ar[u] & \\mathcal{F}(X) \\ar[u] \\ar[l]", "}", "$$", "is cartesian in the category of sets. The rough idea of the proof", "is to dominate the morphism by other almost blowup squares to which", "we can apply assumptions (3) and (4) locally.", "\\medskip\\noindent", "Suppose we have an almost blow up square", "(\\ref{equation-almost-blow-up-square}) in the category $(\\Sch/S)_h$,", "an open covering $X = \\bigcup U_i$, and open coverings", "$U_i \\cap U_j = \\bigcup U_{ijk}$ such that the diagrams", "$$", "\\vcenter{", "\\xymatrix{", "\\mathcal{F}(E \\cap b^{-1}(U_i)) & \\mathcal{F}(b^{-1}(U_i)) \\ar[l] \\\\", "\\mathcal{F}(Z \\cap U_i) \\ar[u] & \\mathcal{F}(U_i) \\ar[u] \\ar[l]", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "\\mathcal{F}(E \\cap b^{-1}(U_{ijk})) &", "\\mathcal{F}(b^{-1}(U_{ijk})) \\ar[l] \\\\", "\\mathcal{F}(Z \\cap U_{ijk}) \\ar[u] &", "\\mathcal{F}(U_{ijk}) \\ar[u] \\ar[l]", "}", "}", "$$", "are cartesian, then the same is true for", "$$", "\\xymatrix{", "\\mathcal{F}(E) & \\mathcal{F}(X') \\ar[l] \\\\", "\\mathcal{F}(Z) \\ar[u] & \\mathcal{F}(X) \\ar[u] \\ar[l]", "}", "$$", "This follows as $\\mathcal{F}$ is a sheaf in the Zariski topology.", "\\medskip\\noindent", "In particular, if we have a blow up square", "(\\ref{equation-almost-blow-up-square}) such that $b : X' \\to X$", "is a closed immersion and $Z$ is a locally principal closed", "subscheme, then we see that", "$\\mathcal{F}(X) = \\mathcal{F}(X') \\times_{\\mathcal{F}(E)} \\mathcal{F}(Z)$.", "Namely, affine locally on $X$ we obtain an almost blow up square as in (3).", "\\medskip\\noindent", "Let $Z \\subset X$, $E_k \\subset X'_k \\to X$, $E \\subset X' \\to X$, and", "$i_k : X' \\to X'_k$ be as in the statement of", "Lemma \\ref{lemma-almost-blow-up-unique}. Then", "$$", "\\xymatrix{", "E \\ar[d] \\ar[r] & X' \\ar[d] \\\\", "E_k \\ar[r] & X'_k", "}", "$$", "is an almost blow up square of the kind discussed in the previous", "paragraph. Thus", "$$", "\\mathcal{F}(X'_k) = \\mathcal{F}(X') \\times_{\\mathcal{F}(E)} \\mathcal{F}(E_k)", "$$", "for $k = 1, 2$ by the result of the previous paragraph. It follows that", "$$", "\\mathcal{F}(X) \\longrightarrow", "\\mathcal{F}(X'_k) \\times_{\\mathcal{F}(E_k)} \\mathcal{F}(Z)", "$$", "is bijective for $k = 1$ if and only if it is bijective for $k = 2$.", "Thus given a closed immersion $Z \\to X$ of finite presentation", "with $X$ quasi-compact and quasi-separated, whether or not", "$\\mathcal{F}(X) = \\mathcal{F}(X') \\times_{\\mathcal{F}(E)} \\mathcal{F}(Z)$", "is independent of the choice of the almost blow up square", "(\\ref{equation-almost-blow-up-square}) one chooses.", "(Moreover, by Lemma \\ref{lemma-almost-blow-up-square}", "there does indeed exist an almost", "blow up square for $Z \\subset X$.)", "\\medskip\\noindent", "Finally, consider an affine object $X$ of $(\\Sch/S)_h$", "and a closed immersion $Z \\to X$ of finite presentation.", "We will prove the desired property for the pair $(X, Z)$", "by induction on the number of generators $r$ for the ideal", "defining $Z$ in $X$. If the number of generators is $\\leq 2$,", "then we can choose our almost blow up square as", "in Example \\ref{example-two-generators}", "and we conclude by assumption (4).", "\\medskip\\noindent", "Induction step. Suppose $X = \\Spec(A)$ and $Z = \\Spec(A/(f_1, \\ldots, f_r))$", "with $r > 2$. Choose a blow up square", "(\\ref{equation-almost-blow-up-square})", "for the pair $(X, Z)$. Set $Z_1 = \\Spec(A/(f_1, f_2))$", "and let", "$$", "\\xymatrix{", "E_1 \\ar[d] \\ar[r] & Y \\ar[d] \\\\", "Z_1 \\ar[r] & X", "}", "$$", "be the almost blow up square constructed in", "Example \\ref{example-two-generators}.", "By Lemma \\ref{lemma-base-change-almost-blow-up} the base changes", "$$", "(I)", "\\vcenter{", "\\xymatrix{", "Y \\times_X E \\ar[r] \\ar[d] & Y \\times_X X' \\ar[d] \\\\", "Y \\times_X Z \\ar[r] & Y", "}", "}", "\\quad\\text{and}\\quad", "(II)", "\\vcenter{", "\\xymatrix{", "E \\ar[r] \\ar[d] & Z_1 \\times_X X' \\ar[d] \\\\", "Z \\ar[r] & Z_1", "}", "}", "$$", "are almost blow up squares. The ideal of $Z$ in $Z_1$ is generated", "by $r - 2$ elements. The ideal of $Y \\times_X Z$", "is generated by the pullbacks of $f_1, \\ldots, f_r$ to $Y$. Locally on $Y$", "the ideal generated by $f_1, f_2$ can be generated by", "one element, thus $Y \\times_X Z$ is affine locally on $Y$", "cut out by at most $r - 1$ elements.", "By induction hypotheses and the discussion above", "$$", "\\mathcal{F}(Y) =", "\\mathcal{F}(Y \\times_X X') \\times_{\\mathcal{F}(Y \\times_X E)}", "\\mathcal{F}(Y \\times_X Z)", "$$", "and", "$$", "\\mathcal{F}(Z_1) =", "\\mathcal{F}(Z_1 \\times_X X') \\times_{\\mathcal{F}(E)}", "\\mathcal{F}(Z)", "$$", "By assumption (4) we have", "$$", "\\mathcal{F}(X) =", "\\mathcal{F}(Y) \\times_{\\mathcal{F}(E_1)} \\mathcal{F}(Z_1)", "$$", "Now suppose we have a pair $(s', t)$ with $s' \\in \\mathcal{F}(X')$", "and $t \\in \\mathcal{F}(Z)$ with same restriction in $\\mathcal{F}(E)$.", "Then $(s'|{Z_1 \\times_X X'}, t)$ are the image of a unique element", "$t_1 \\in \\mathcal{F}(Z_1)$. Similarly,", "$(s'|_{Y \\times_X X'}, t|_{Y \\times_X Z})$ are the image of a", "unique element $s_Y \\in \\mathcal{F}(Y)$.", "We claim that $s_Y$ and $t_1$ restrict to the same element", "of $\\mathcal{F}(E_1)$. This is true because the almost blow up", "square", "$$", "\\xymatrix{", "E_1 \\times_X E \\ar[r] \\ar[d] & E_1 \\times_X X' \\ar[d] \\\\", "E_1 \\times_X Z \\ar[r] & E_1", "}", "$$", "is the base change of almost blow up square (I) via $E_1 \\to Y$ and", "the base change of almost blow up square (II) via $E_1 \\to Z_1$ and", "because the pairs of sections used to construct $s_Y$ and $t_1$ match.", "Thus by the third fibre product equality we see that there is", "a unique $s \\in \\mathcal{F}(X)$ mapping to $s_Y$ in $\\mathcal{F}(Y)$", "and to $t_1$ in $\\mathcal{F}(Z)$.", "We omit the verification that $s$ maps to $s'$ in $\\mathcal{F}(X')$", "and to $t$ in $\\mathcal{F}(Z)$; hint: use uniqueness of $s$ just", "constructed and work affine locally." ], "refs": [ "flat-proposition-check-h", "flat-lemma-almost-blow-up-unique", "flat-lemma-almost-blow-up-square", "flat-lemma-base-change-almost-blow-up" ], "ref_ids": [ 6204, 6161, 6160, 6157 ] } ], "ref_ids": [ 6223 ] }, { "id": 6166, "type": "theorem", "label": "flat-lemma-refine-check-h-stack", "categories": [ "flat" ], "title": "flat-lemma-refine-check-h-stack", "contents": [ "Let $p : \\mathcal{S} \\to (\\Sch/S)_h$ be a category fibred in groupoids.", "Then $\\mathcal{S}$ is a stack in groupoids if and only if the following", "conditions are satisfied", "\\begin{enumerate}", "\\item $\\mathcal{S}$ is a stack in groupoids for the Zariski topology,", "\\item given a morphism $f : X \\to Y$ of $(\\Sch/S)_h$ with $Y$ affine", "and $f$ surjective, flat, proper, and of finite presentation, then", "$$", "\\mathcal{S}_Y \\longrightarrow", "\\mathcal{S}_X \\times_{\\mathcal{S}_{X \\times_Y X}} \\mathcal{S}_X", "$$", "is an equivalence of categories,", "\\item for an almost blow up square as in", "Example \\ref{example-one-generator} or \\ref{example-two-generators}", "in the category $(\\Sch/S)_h$ the functor", "$$", "\\mathcal{S}_X \\longrightarrow", "\\mathcal{S}_Z \\times_{\\mathcal{S}_E} \\mathcal{S}_{X'}", "$$", "is an equivalence of categories.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma is a formal consequence of Lemma \\ref{lemma-refine-check-h}", "and our defnition of stacks in groupoids.", "For example, assume (1), (2), (3). To show that", "$\\mathcal{S}$ is a stack, we have", "to prove descent for morphisms and objects, see", "Stacks, Definition \\ref{stacks-definition-stack-in-groupoids}.", "\\medskip\\noindent", "If $x, y$ are objects of $\\mathcal{S}$ over an object $U$ of", "$(\\Sch/S)_h$, then our assumptions imply $\\mathit{Isom}(x, y)$", "is a presheaf on $(\\Sch/U)_h$ which satisfies", "(1), (2), (3), and (4) of Lemma \\ref{lemma-refine-check-h}", "and therefore is a sheaf. Some details omitted.", "\\medskip\\noindent", "Let $\\{U_i \\to U\\}_{i \\in I}$ be a covering of $(\\Sch/S)_h$.", "Let $(x_i, \\varphi_{ij})$ be a descent datum in $\\mathcal{S}$", "relative to the family $\\{U_i \\to U\\}_{i \\in I}$, see", "Stacks, Definition \\ref{stacks-definition-descent-data}.", "Consider the rule $F$ which to $V/U$ in $(\\Sch/U)_h$ associates", "the set of pairs $(y, \\psi_i)$ where $y$ is an object of $\\mathcal{S}_V$", "and $\\psi_i : y|_{U_i \\times_U V} \\to x_i|_{U_i \\times_U V}$", "is a morphism of $\\mathcal{S}$ over $U_i \\times_U V$", "such that", "$$", "\\varphi_{ij}|_{U_i \\times_U U_j \\times_U V}", "\\circ", "\\psi_i|_{U_i \\times_U U_j \\times_U V}", "=", "\\psi_j|_{U_i \\times_U U_j \\times_U V}", "$$", "up to isomorphism. Since we already have descent for morphisms, it is", "clear that $F(V/U)$ is either empty or a singleton set. On the other hand,", "we have $F(U_{i_0}/U)$ is nonempty because it contains", "$(x_{i_0}, \\varphi_{i_0i})$. Since our goal is to prove that $F(U/U)$", "is nonempty, it suffices to show that $F$ is a sheaf on $(\\Sch/U)_h$.", "To do this we may use the criterion of Lemma \\ref{lemma-refine-check-h}.", "However, our assumptions (1), (2), (3) imply (by drawing some", "commutative diagrams which we omit), that properties (1), (2), (3), and (4)", "of Lemma \\ref{lemma-refine-check-h} hold for $F$.", "\\medskip\\noindent", "We omit the verification that if $\\mathcal{S}$ is a stack in groupoids,", "then (1), (2), and (3) are satisfied." ], "refs": [ "flat-lemma-refine-check-h", "stacks-definition-stack-in-groupoids", "flat-lemma-refine-check-h", "stacks-definition-descent-data", "flat-lemma-refine-check-h", "flat-lemma-refine-check-h" ], "ref_ids": [ 6165, 8998, 6165, 8993, 6165, 6165 ] } ], "ref_ids": [] }, { "id": 6167, "type": "theorem", "label": "flat-lemma-funny-blow-up", "categories": [ "flat" ], "title": "flat-lemma-funny-blow-up", "contents": [ "Let $Z, X, X', E$ be an almost blow up square as in", "Example \\ref{example-two-generators}.", "Then $H^p(X', \\mathcal{O}_{X'}) = 0$ for $p > 0$ and", "$\\Gamma(X, \\mathcal{O}_X) \\to \\Gamma(X', \\mathcal{O}_{X'})$", "is a surjective map of rings whose kernel is an ideal of square zero." ], "refs": [], "proofs": [ { "contents": [ "First assume that $A = \\mathbf{Z}[f_1, f_2]$ is the polynomial ring.", "In this case our almost blow up square is the blowing up of $X = \\Spec(A)$", "in the closed subscheme $Z$ and in fact $X' \\subset \\mathbf{P}^1_X$", "is an effective Cartier divisor cut out by the global section", "$f_2T_0 - f_1 T_1$ of $\\mathcal{O}_{\\mathbf{P}^1_X}(1)$.", "Thus we have a resolution", "$$", "0 \\to", "\\mathcal{O}_{\\mathbf{P}^1_X}(-1) \\to", "\\mathcal{O}_{\\mathbf{P}^1_X} \\to", "\\mathcal{O}_{X'} \\to", "0", "$$", "Using the description of the cohomology given in", "Cohomology of Schemes, Section", "\\ref{coherent-section-cohomology-projective-space}", "it follows that in this case", "$\\Gamma(X, \\mathcal{O}_X) \\to \\Gamma(X', \\mathcal{O}_{X'})$", "is an isomorphism and $H^1(X', \\mathcal{O}_{X'}) = 0$.", "\\medskip\\noindent", "Next, we observe that any diagram as in Example \\ref{example-two-generators}", "is the base change of the diagram in the previous paragraph by the", "ring map $\\mathbf{Z}[f_1, f_2] \\to A$. Hence by More on Morphisms, Lemmas", "\\ref{more-morphisms-lemma-check-h1-fibre-zero},", "\\ref{more-morphisms-lemma-h1-fibre-zero}, and", "\\ref{more-morphisms-lemma-h1-fibre-zero-check-h0-kappa}", "we conclude that $H^1(X', \\mathcal{O}_{X'})$ is zero in general", "and the surjectivity of the map", "$H^0(X, \\mathcal{O}_X) \\to H^0(X', \\mathcal{O}_{X'})$ in general.", "\\medskip\\noindent", "Next, in the general case, let us study the kernel. If", "$a \\in A$ maps to zero, then looking on affine charts", "we see that", "$$", "a = (f_1x - f_2)(a_0 + a_1x + \\ldots + a_rx^r)\\text{ in }A[x]", "$$", "for some $r \\geq 0$ and $a_0, \\ldots, a_r \\in A$ and similarly", "$$", "a = (f_1 - f_2y)(b_0 + b_1y + \\ldots + b_s y^s)\\text{ in }A[y]", "$$", "for some $s \\geq 0$ and $b_0, \\ldots, b_s \\in A$. This means we", "have", "$$", "a = f_2 a_0,\\ f_1 a_0 = f_2 a_1,\\ \\ldots,\\ f_1 a_r = 0,", "\\ a = f_1 b_0,\\ f_2 b_0 = f_1 b_1,\\ \\ldots,\\ f_2 b_s = 0", "$$", "If $(a', r', a'_i, s', b'_j)$ is a second such system, then we have", "$$", "aa' = f_1f_2a_0b'_0 = f_1f_2a_1b'_1 = f_1f_2a_2b'_2 = \\ldots = 0", "$$", "as desired." ], "refs": [ "more-morphisms-lemma-check-h1-fibre-zero", "more-morphisms-lemma-h1-fibre-zero", "more-morphisms-lemma-h1-fibre-zero-check-h0-kappa" ], "ref_ids": [ 14069, 14070, 14072 ] } ], "ref_ids": [] }, { "id": 6168, "type": "theorem", "label": "flat-lemma-h-sheaf-colim-F", "categories": [ "flat" ], "title": "flat-lemma-h-sheaf-colim-F", "contents": [ "Let $p$ be a prime number. Let $S$ be a scheme over $\\mathbf{F}_p$.", "Let $(\\Sch/S)_h$ be a site as in Definition \\ref{definition-big-small-h}.", "There is a unique sheaf $\\mathcal{F}$ on $(\\Sch/S)_h$ such that", "$$", "\\mathcal{F}(X) = \\colim_F \\Gamma(X, \\mathcal{O}_X)", "$$", "for any quasi-compact and quasi-separated object $X$ of $(\\Sch/S)_h$." ], "refs": [ "flat-definition-big-small-h" ], "proofs": [ { "contents": [ "Denote $\\mathcal{F}$ the Zariski sheafification", "of the functor", "$$", "X \\longrightarrow \\colim_F \\Gamma(X, \\mathcal{O}_X)", "$$", "For quasi-compact and quasi-separated schemes $X$", "we have $\\mathcal{F}(X) = \\colim_F \\Gamma(X, \\mathcal{O}_X)$.", "by Sheaves, Lemma \\ref{sheaves-lemma-directed-colimits-sections}", "and the fact that $\\mathcal{O}$ is a sheaf for the Zariski topology.", "Thus it suffices to show that $\\mathcal{F}$ is a h sheaf.", "To prove this we check conditions (1), (2), (3), and (4) of", "Lemma \\ref{lemma-refine-check-h}.", "Condition (1) holds because we performed an (almost unnecessary)", "Zariski sheafification. Condition (2) holds because", "$\\mathcal{O}$ is an fppf sheaf (Descent, Lemma", "\\ref{descent-lemma-sheaf-condition-holds}) and if", "$A$ is the equalizer of two maps $B \\to C$ of $\\mathbf{F}_p$-algebras,", "then $\\colim_F A$ is the equalizer of the two maps", "$\\colim_F B \\to \\colim_F C$.", "\\medskip\\noindent", "We check condition (3). Let $A, f, J$ be as in", "Example \\ref{example-one-generator}.", "We have to show that", "$$", "\\colim_F A = \\colim_F A/J \\times_{\\colim_F A/fA + J} \\colim_F A/fA", "$$", "This reduces to the following algebra question: suppose $a', a'' \\in A$", "are such that $F^n(a' - a'') \\in fA + J$. Find $a \\in A$ and $m \\geq 0$", "such that $a - F^m(a') \\in J$ and $a - F^m(a'') \\in fA$ and show that", "the pair $(a, m)$ is uniquely determined up to a replacement of the", "form $(a, m) \\mapsto (F(a), m + 1)$.", "To do this just write $F^n(a' - a'') = f h + g$ with $h \\in A$ and $g \\in J$", "and set $a = F^n(a') - g = F^n(a'') + fh$ and set $m = n$.", "To see uniqueness, suppose $(a_1, m_1)$ is a second solution.", "By a replacement of the form given above we may assume $m = m_1$.", "Then we see that $a - a_1 \\in J$ and $a - a_1 \\in fA$.", "Since $J$ is annihilated by a power of $f$ we see that", "$a - a_1$ is a nilpotent element. Hence $F^k(a - a_1)$ is zero", "for some large $k$. Thus after doing more replacements we get", "$a = a_1$.", "\\medskip\\noindent", "We check condition (4). Let $X, X', Z, E$ be as in", "Example \\ref{example-two-generators}. By", "Lemma \\ref{lemma-funny-blow-up} we see that", "$$", "\\mathcal{F}(X) = \\colim_F \\Gamma(X, \\mathcal{O}_X)", "\\longrightarrow", "\\colim_F \\Gamma(X', \\mathcal{O}_{X'}) = \\mathcal{F}(X')", "$$", "is bijective. Since $E = \\mathbf{P}^1_Z$ in this case we also", "see that $\\mathcal{F}(Z) \\to \\mathcal{F}(E)$ is bijective.", "Thus the conclusion holds in this case as well." ], "refs": [ "sheaves-lemma-directed-colimits-sections", "flat-lemma-refine-check-h", "descent-lemma-sheaf-condition-holds", "flat-lemma-funny-blow-up" ], "ref_ids": [ 14526, 6165, 14621, 6167 ] } ], "ref_ids": [ 6223 ] }, { "id": 6169, "type": "theorem", "label": "flat-lemma-h-sheaf-lim-F", "categories": [ "flat" ], "title": "flat-lemma-h-sheaf-lim-F", "contents": [ "Let $p$ be a prime number. Let $S$ be a scheme over $\\mathbf{F}_p$.", "Let $(\\Sch/S)_h$ be a site as in Definition \\ref{definition-big-small-h}.", "The rule", "$$", "\\mathcal{F}(X) = \\lim_F \\Gamma(X, \\mathcal{O}_X)", "$$", "defines a sheaf on $(\\Sch/S)_h$." ], "refs": [ "flat-definition-big-small-h" ], "proofs": [ { "contents": [ "To prove $\\mathcal{F}$ is a sheaf, let's check conditions", "(1), (2), (3), and (4) of Lemma \\ref{lemma-refine-check-h}.", "Condition (1) holds because limits of sheaves are sheaves", "and $\\mathcal{O}$ is a Zariski sheaf. Condition (2) holds because", "$\\mathcal{O}$ is an fppf sheaf (Descent, Lemma", "\\ref{descent-lemma-sheaf-condition-holds}) and if $A$ is the equalizer", "of two maps $B \\to C$ of $\\mathbf{F}_p$-algebras, then $\\lim_F A$", "is the equalizer of the two maps $\\lim_F B \\to \\lim_F C$.", "\\medskip\\noindent", "We check condition (3). Let $A, f, J$ be as in", "Example \\ref{example-one-generator}.", "We have to show that", "\\begin{align*}", "\\lim_F A", "& \\to", "\\lim_F A/J \\times_{\\lim_F A/fA + J} \\lim_F A/fA \\\\", "& =", "\\lim_F (A/J \\times_{A/fA + J} A/fA) \\\\", "& =", "\\lim_F A/(fA \\cap J)", "\\end{align*}", "is bijective. Since $J$ is annihilated by a power of $f$ we see that", "$\\mathfrak a = fA \\cap J$ is a nilpotent ideal, i.e., there exists an", "$n$ such that $\\mathfrak a^n = 0$. It is straightforward", "to verify that in this case $\\lim_F A \\to \\lim_F A/\\mathfrak a$", "is bijective.", "\\medskip\\noindent", "We check condition (4). Let $X, X', Z, E$ be as in", "Example \\ref{example-two-generators}. By", "Lemma \\ref{lemma-funny-blow-up} and the same argument as above", "we see that", "$$", "\\mathcal{F}(X) = \\lim_F \\Gamma(X, \\mathcal{O}_X)", "\\longrightarrow", "\\lim_F \\Gamma(X', \\mathcal{O}_{X'}) = \\mathcal{F}(X')", "$$", "is bijective. Since $E = \\mathbf{P}^1_Z$ in this case we also", "see that $\\mathcal{F}(Z) \\to \\mathcal{F}(E)$ is bijective.", "Thus the conclusion holds in this case as well." ], "refs": [ "flat-lemma-refine-check-h", "descent-lemma-sheaf-condition-holds", "flat-lemma-funny-blow-up" ], "ref_ids": [ 6165, 14621, 6167 ] } ], "ref_ids": [ 6223 ] }, { "id": 6170, "type": "theorem", "label": "flat-lemma-weak-normalization-ph-sheaf", "categories": [ "flat" ], "title": "flat-lemma-weak-normalization-ph-sheaf", "contents": [ "Let $(\\Sch/S)_{ph}$ be a site as in", "Topologies, Definition \\ref{topologies-definition-big-small-ph}.", "The rule", "$$", "X \\longmapsto \\Gamma(X^{awn}, \\mathcal{O}_{X^{awn}})", "$$", "is a sheaf on $(\\Sch/S)_{ph}$." ], "refs": [ "topologies-definition-big-small-ph" ], "proofs": [ { "contents": [ "To prove $\\mathcal{F}$ is a sheaf, let's check conditions", "(1) and (2) of Topologies, Lemma \\ref{topologies-lemma-characterize-sheaf}.", "Condition (1) holds because formation of $X^{awn}$ commutes with", "open coverings, see Morphisms, Lemma \\ref{morphisms-lemma-seminormalization}", "and its proof.", "\\medskip\\noindent", "Let $\\pi : Y \\to X$ be a surjective proper morphism. We have to show", "that the equalizer of the two maps", "$$", "\\Gamma(Y^{awn}, \\mathcal{O}_{Y^{awn}}) \\to", "\\Gamma((Y \\times_X Y)^{awn}, \\mathcal{O}_{(Y \\times_X Y)^{awn}})", "$$", "is equal to $\\Gamma(X^{awn}, \\mathcal{O}_{X^{awn}})$. Let $f$ be", "an element of this equalizer. Then we consider the morphism", "$$", "f : Y^{awn} \\longrightarrow \\mathbf{A}^1_X", "$$", "Since $Y^{awn} \\to X$ is universally closed, the scheme theoretic", "image $Z$ of $f$ is a closed subscheme of $\\mathbf{A}^1_X$ proper over $X$", "and $f : Y^{awn} \\to Z$ is surjective.", "See Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-image-is-proper}.", "Thus $Z \\to X$ is finite", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-proper})", "and surjective.", "\\medskip\\noindent", "Let $k$ be a field and let $z_1, z_2 : \\Spec(k) \\to Z$ be two", "morphisms equalized by $Z \\to X$. We claim that $z_1 = z_2$.", "It suffices to show the images $\\lambda_i = z_i^*f \\in k$ agree", "(as the structure sheaf of $Z$ is generated by $f$", "over the structure sheaf of $X$). To see this we", "choose a field extension", "$K/k$ and morphisms $y_1, y_2 : \\Spec(K) \\to Y^{awn}$ such that", "$z_i \\circ (\\Spec(K) \\to \\Spec(k)) = f \\circ y_i$. This is possible", "by the surjectivity of the map $Y^{awn} \\to Z$. Choose an algebraically", "closed extension $\\Omega/k$ of very large cardinality.", "For any $k$-algebra maps $\\sigma_i : K \\to \\Omega$", "we obtain", "$$", "\\Spec(\\Omega)", "\\xrightarrow{\\sigma_1, \\sigma_2}", "\\Spec(K \\otimes_k K)", "\\xrightarrow{y_1, y_2}", "Y^{awn} \\times_X Y^{awn}", "$$", "Since the canonical morphism", "$(Y \\times_X Y)^{awn} \\to Y^{awn} \\times_X Y^{awn}$", "is a universal homeomorphism and since $\\Omega$ is algebraically closed,", "we can lift the composition above uniquely to a morphism", "$\\Spec(\\Omega) \\to (Y \\times_X Y)^{awn}$. Since $f$ is in the equalizer", "above, this proves that $\\sigma_1(\\lambda_1) = \\sigma_2(\\lambda_2)$.", "An easy lemma about field extensions shows that this implies", "$\\lambda_1 = \\lambda_2$; details omitted.", "\\medskip\\noindent", "We conclude that $Z \\to X$ is universally injective, i.e.,", "$Z \\to X$ is injective on points and induces", "purely inseparated residue field extensions", "(Morphisms, Lemma \\ref{morphisms-lemma-universally-injective}).", "All in all we conclude that $Z \\to X$ is a universal homeomorphism, see", "Morphisms, Lemma \\ref{morphisms-lemma-universal-homeomorphism}.", "\\medskip\\noindent", "Let $g : X^{awn} \\to Z$ be the map obtained from the universal property", "of $X^{awn}$. Then $Y^{awn} \\to X^{awn} \\to Z$ and $f : Y^{awn} \\to Z$", "are two morphisms over $X$. By the universal property of", "$Y^{awn} \\to Y$ the two corresponding morphisms", "$Y^{awn} \\to Y \\times_X Z$ over $Y$ have to be equal. This implies", "that $g \\circ \\pi^{wan} = f$ as morphisms into $\\mathbf{A}^1_X$", "and we conclude that $g \\in \\Gamma(X^{awn}, \\mathcal{O}_{X^{awn}})$", "is the element we were looking for." ], "refs": [ "topologies-lemma-characterize-sheaf", "morphisms-lemma-seminormalization", "morphisms-lemma-scheme-theoretic-image-is-proper", "morphisms-lemma-finite-proper", "morphisms-lemma-universally-injective", "morphisms-lemma-universal-homeomorphism" ], "ref_ids": [ 12490, 5470, 5414, 5445, 5167, 5454 ] } ], "ref_ids": [ 12546 ] }, { "id": 6171, "type": "theorem", "label": "flat-lemma-weak-normalization-h-sheaf", "categories": [ "flat" ], "title": "flat-lemma-weak-normalization-h-sheaf", "contents": [ "Let $S$ be a scheme. Choose a site $(\\Sch/S)_h$", "as in Definition \\ref{definition-big-small-h}.", "The rule", "$$", "X \\longmapsto \\Gamma(X^{awn}, \\mathcal{O}_{X^{awn}})", "$$", "is the sheafification of the ``structure sheaf'' $\\mathcal{O}$", "on $(\\Sch/S)_h$. Similarly for the ph topology." ], "refs": [ "flat-definition-big-small-h" ], "proofs": [ { "contents": [ "In Lemma \\ref{lemma-weak-normalization-ph-sheaf}", "we have seen that the rule $\\mathcal{F}$ of the lemma", "defines a sheaf in the ph topology and hence a fortiori", "a sheaf for the h topology. Clearly, there is a canonical map", "of presheaves of rings $\\mathcal{O} \\to \\mathcal{F}$.", "To finish the proof, it suffices to show", "\\begin{enumerate}", "\\item if $f \\in \\mathcal{O}(X)$ maps to zero in $\\mathcal{F}(X)$,", "then there is a h covering $\\{X_i \\to X\\}$ such that $f|_{X_i} = 0$, and", "\\item given $f \\in \\mathcal{F}(X)$ there is a h covering", "$\\{X_i \\to X\\}$ such that $f|_{X_i}$ is the image of $f_i \\in \\mathcal{O}(X_i)$.", "\\end{enumerate}", "Let $f$ be as in (1). Then $f|_{X^{awn}} = 0$. This means that $f$", "is locally nilpotent. Thus if $X' \\subset X$ is the closed subscheme", "cut out by $f$, then $X' \\to X$ is a surjective closed immersion of", "finite presentation. Hence $\\{X' \\to X\\}$ is the desired h covering.", "Let $f$ be as in (2). After replacing $X$ by the members of an", "affine open covering we may assume $X = \\Spec(A)$ is affine.", "Then $f \\in A^{awn}$, see", "Morphisms, Lemma \\ref{morphisms-lemma-seminormalization-ring}.", "By Morphisms, Lemma \\ref{morphisms-lemma-universal-homeo-limit}", "we can find a ring map $A \\to B$ of finite presentation", "such that $\\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism", "and such that $f$ is the image of an element $b \\in B$ under", "the canonical map $B \\to A^{awn}$. Then", "$\\{\\Spec(B) \\to \\Spec(A)\\}$ is an h covering and we conclude.", "The statement about the ph topology follows in the same manner", "(or it can be deduced from the statement for the h topology)." ], "refs": [ "flat-lemma-weak-normalization-ph-sheaf", "morphisms-lemma-seminormalization-ring", "morphisms-lemma-universal-homeo-limit" ], "ref_ids": [ 6170, 5469, 5465 ] } ], "ref_ids": [ 6223 ] }, { "id": 6172, "type": "theorem", "label": "flat-lemma-perfect-weankly-normal", "categories": [ "flat" ], "title": "flat-lemma-perfect-weankly-normal", "contents": [ "Let $p$ be a prime number. An $\\mathbf{F}_p$-algebra $A$ is", "absolutely weakly normal if and only if it is perfect." ], "refs": [], "proofs": [ { "contents": [ "It is immediate from condition (2)(b) in", "Morphisms, Definition \\ref{morphisms-definition-seminormal-ring}", "that if $A$ is absolutely weakly normal, then it is perfect.", "\\medskip\\noindent", "Assume $A$ is perfect. Suppose $x, y \\in A$ with $x^3 = y^2$.", "If $p > 3$ then we can write $p = 2n + 3m$ for some $n, m > 0$.", "Choose $a, b \\in A$ with $a^p = x$ and $b^p = y$.", "Setting $c = a^n b^m$ we have", "$$", "c^{2p} = x^{2n} y^{2m} = x^{2n + 3m} = x^p", "$$", "and hence $c^2 = x$. Similarly $c^3 = y$. If $p = 2$, then", "write $x = a^2$ to get $a^6 = y^2$ which implies $a^3 = y$.", "If $p = 3$, then write $y = a^3$ to get $x^3 = a^6$ which", "implies $x = a^2$.", "\\medskip\\noindent", "Suppose $x, y \\in A$ with $\\ell^\\ell x = y^\\ell$ for some prime", "number $\\ell$. If $\\ell \\not = p$, then $a = y/\\ell$ satsifies", "$a^\\ell = x$ and $\\ell a = y$. If $\\ell = p$, then", "$y = 0$ and $x = a^p$ for some $a$." ], "refs": [ "morphisms-definition-seminormal-ring" ], "ref_ids": [ 5575 ] } ], "ref_ids": [] }, { "id": 6173, "type": "theorem", "label": "flat-lemma-char-p", "categories": [ "flat" ], "title": "flat-lemma-char-p", "contents": [ "Let $p$ be a prime number.", "\\begin{enumerate}", "\\item If $A$ is an $\\mathbf{F}_p$-algebra, then $\\colim_F A = A^{awn}$.", "\\item If $S$ is a scheme over $\\mathbf{F}_p$, then the", "h sheafification of $\\mathcal{O}$ sends a quasi-compact", "and quasi-separated $X$ to $\\colim_F \\Gamma(X, \\mathcal{O}_X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Observe that $A \\to \\colim_F A$ induces a", "universal homeomorphism on spectra by", "Algebra, Lemma \\ref{algebra-lemma-p-ring-map}.", "Thus it suffices to show that $B = \\colim_F A$ is", "absolutely weakly normal, see", "Morphisms, Lemma \\ref{morphisms-lemma-seminormalization-ring}.", "Note that the ring map $F : B \\to B$ is an automorphism,", "in other words, $B$ is a perfect ring. Hence", "Lemma \\ref{lemma-perfect-weankly-normal} applies.", "\\medskip\\noindent", "Proof of (2). This follows from (1) and", "Lemmas \\ref{lemma-h-sheaf-colim-F} and", "\\ref{lemma-weak-normalization-h-sheaf}", "by looking affine locally." ], "refs": [ "algebra-lemma-p-ring-map", "morphisms-lemma-seminormalization-ring", "flat-lemma-perfect-weankly-normal", "flat-lemma-h-sheaf-colim-F", "flat-lemma-weak-normalization-h-sheaf" ], "ref_ids": [ 582, 5469, 6172, 6168, 6171 ] } ], "ref_ids": [] }, { "id": 6174, "type": "theorem", "label": "flat-lemma-colim-F-Vect", "categories": [ "flat" ], "title": "flat-lemma-colim-F-Vect", "contents": [ "Let $p$ be a prime number. Let $S$ be a quasi-compact and quasi-separated", "scheme over $\\mathbf{F}_p$. The category $\\colim_F \\textit{Vect}(S)$", "is equivalent to the category of finite locally free modules", "over the sheaf of rings $\\colim_F \\mathcal{O}_S$ on $S$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6175, "type": "theorem", "label": "flat-lemma-vector-bundle-I", "categories": [ "flat" ], "title": "flat-lemma-vector-bundle-I", "contents": [ "Let $p$ be a prime number. Consider an almost blowup square $X, X', Z, E$", "in characteristic $p$ as in Example \\ref{example-one-generator}.", "Then the functor", "$$", "\\colim_F \\textit{Vect}(X)", "\\longrightarrow", "\\colim_F \\textit{Vect}(Z)", "\\times_{\\colim_F \\textit{Vect}(E)}", "\\colim_F \\textit{Vect}(X')", "$$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Let $A, f, J$ be as in Example \\ref{example-one-generator}.", "Since all our schemes are affine and since we have internal", "Hom in the category of vector bundles, the fully faithfulness", "of the functor follows if we can show that", "$$", "\\colim P \\otimes_{A, F^N} A =", "\\colim P \\otimes_{A, F^N} A/J", "\\times_{\\colim P \\otimes_{A, F^N} A/fA + J}", "\\colim P \\otimes_{A, F^N} A/fA", "$$", "for a finite projective $A$-module $P$. After writing $P$ as a summand", "of a finite free module, this follows from the case where $P$ is finite", "free. This case immediately reduces to the case $P = A$. The case", "$P = A$ follows from Lemma \\ref{lemma-h-sheaf-colim-F}", "(in fact we proved this case directly in the proof of this lemma).", "\\medskip\\noindent", "Essential surjectivity. Here we obtain the following algebra problem.", "Suppose $P_1$ is a finite projective $A/J$-module,", "$P_2$ is a finite projective $A/fA$-module, and", "$$", "\\varphi :", "P_1 \\otimes_{A/J} A/fA + J", "\\longrightarrow", "P_2 \\otimes_{A/fA} A/fA + J", "$$", "is an isomorphism. Goal: show that there exists an $N$, a finite", "projective $A$-module $P$, an isomorphism", "$\\varphi_1 : P \\otimes_A A/J \\to P_1 \\otimes_{A/J, F^N} A/J$,", "and an isomorphism", "$\\varphi_2 : P \\otimes_A A/fA \\to P_2 \\otimes_{A/fA, F^N} A/fA$", "compatible with $\\varphi$ in an obvious manner.", "This can be seen as follows. First, observe that", "$$", "A/(J \\cap fA) = A/J \\times_{A/fA + J} A/fA", "$$", "Hence by More on Algebra, Lemma", "\\ref{more-algebra-lemma-finitely-presented-module-over-fibre-product}", "there is a finite projective module $P'$ over", "$A/(J \\cap fA)$ which comes with isomorphisms", "$\\varphi'_1 : P' \\otimes_A A/J \\to P_1$ and", "$\\varphi_2 : P' \\otimes_A A/fA \\to P_2$", "compatible with $\\varphi$. Since $J$ is a finitely generated ideal and", "$f$-power torsion we see that $J \\cap fA$ is a nilpotent", "ideal. Hence for some $N$ there is a factorization", "$$", "A \\xrightarrow{\\alpha} A/(J \\cap fA) \\xrightarrow{\\beta} A", "$$", "of $F^N$. Setting $P = P' \\otimes_{A/(J \\cap fA), \\beta} A$", "we conclude." ], "refs": [ "flat-lemma-h-sheaf-colim-F", "more-algebra-lemma-finitely-presented-module-over-fibre-product" ], "ref_ids": [ 6168, 9825 ] } ], "ref_ids": [] }, { "id": 6176, "type": "theorem", "label": "flat-lemma-vector-bundle-II", "categories": [ "flat" ], "title": "flat-lemma-vector-bundle-II", "contents": [ "Let $p$ be a prime number. Consider an almost blowup square $X, X', Z, E$", "in characteristic $p$ as in Example \\ref{example-two-generators}.", "Then the functor", "$$", "G :", "\\colim_F \\textit{Vect}(X)", "\\longrightarrow", "\\colim_F \\textit{Vect}(Z)", "\\times_{\\colim_F \\textit{Vect}(E)}", "\\colim_F \\textit{Vect}(X')", "$$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Fully faithfulness. Suppose that $(\\mathcal{E}, n)$ and", "$(\\mathcal{F}, m)$ are objects of $\\colim_F \\textit{Vect}(X)$.", "Let $(a, b) : G(\\mathcal{E}, n) \\to G(\\mathcal{F}, m)$", "be a morphism in the RHS. We may choose $N \\gg 0$ and", "think of $a$ as a map", "$a : F^{N - n, *}\\mathcal{E}|_Z \\to F^{N - m, *}\\mathcal{F}|_Z$", "and $b$ as a map", "$b : F^{N - n, *}\\mathcal{E}|_{X'} \\to F^{N - m, *}\\mathcal{F}|_{X'}$", "agreeing over $E$.", "Choose a finite affine open covering", "$X = X_1 \\cup \\ldots \\cup X_n$ such that $\\mathcal{E}|_{X_i}$", "and $\\mathcal{F}|_{X_i}$ are finite free $\\mathcal{O}_{X_i}$-modules.", "For each $i$ the base change", "$$", "\\xymatrix{", "E_i \\ar[r] \\ar[d] & X'_i \\ar[d] \\\\", "Z_i \\ar[r] & X_i", "}", "$$", "is another almost blow up square as in Example \\ref{example-two-generators}.", "For these squares we know that", "$$", "\\colim_F H^0(X_i, \\mathcal{O}_{X_i}) =", "\\colim_F H^0(Z_i, \\mathcal{O}_{Z_i})", "\\times_{\\colim_F H^0(E_i, \\mathcal{O}_{E_i})}", "\\colim_F H^0(X'_i, \\mathcal{O}_{X'_i})", "$$", "by Lemma \\ref{lemma-h-sheaf-colim-F} (see proof of the lemma).", "Hence after increasing $N$ we may assume", "the maps $a|_{Z_i}$ and $b|_{X'_i}$ come from", "maps $c_i : F^{N - n, *}\\mathcal{E}|_{X_i} \\to F^{N - m, *}\\mathcal{F}|_{X_i}$.", "After possibly increasing $N$ we may assume $c_i$ and $c_j$", "agree on $X_i \\cap X_j$. Thus these maps glue to give the", "desired morphism $(\\mathcal{E}, n) \\to (\\mathcal{F}, m)$", "in the LHS.", "\\medskip\\noindent", "Essential surjectivity. Let $(\\mathcal{F}, \\mathcal{G}, \\varphi)$ be a", "triple consisting of", "a finite locally free $\\mathcal{O}_Z$-module $\\mathcal{F}$,", "a finite locally free $\\mathcal{O}_{X'}$-module $\\mathcal{G}$, and", "an isomorphism $\\varphi : \\mathcal{F}|_E \\to \\mathcal{G}|_E$.", "We have to show that after replacing this triple by a Frobenius", "power pullback, it comes from a finite locally free $\\mathcal{O}_X$-module.", "\\medskip\\noindent", "Noetherian reduction; we urge the reader to skip this paragraph.", "Recall that $X = \\Spec(A)$ and $Z = \\Spec(A/(f_1, f_2))$,", "$X' = \\text{Proj}(A[T_0, T_1]/(f_2T_0 - f_1T_1))$, and", "$E = \\mathbf{P}^1_Z$. By Limits, Lemma", "\\ref{limits-lemma-descend-invertible-modules}", "we can find a finitely generated $\\mathbf{F}_p$-subalgebra", "$A_0 \\subset A$ containing $f_1$ and $f_2$ such that the triple", "$(\\mathcal{F}, \\mathcal{G}, \\varphi)$ descends to", "$X_0 = \\Spec(A_0)$ and $Z_0 = \\Spec(A_0/(f_1, f_2))$,", "$X_0' = \\text{Proj}(A_0[T_0, T_1]/(f_2T_0 - f_1T_1))$, and", "$E_0 = \\mathbf{P}^1_{Z_0}$. Thus we may assume our schemes", "are Noetherian.", "\\medskip\\noindent", "Assume $X$ is Noetherian. We may choose a finite affine open covering", "$X = X_1 \\cup \\ldots \\cup X_n$ such that $\\mathcal{F}|_{Z \\cap X_i}$ is free.", "Since we can glue objects of $\\colim_F \\textit{Vect}(X)$", "in the Zariski topology (Lemma \\ref{lemma-colim-F-Vect}), and", "since we already know", "fully faithfulness over $X_i$ and $X_i \\cap X_j$ (see first paragraph", "of the proof), it suffices to prove the existence over each $X_i$.", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $X$ is Noetherian and $\\mathcal{F} = \\mathcal{O}_Z^{\\oplus r}$.", "Using $\\varphi$ we get an isomorphism", "$\\mathcal{O}_E^{\\oplus r} \\to \\mathcal{G}|_E$.", "Let $I = (f_1, f_2) \\subset A$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_{X'}$", "be the ideal sheaf of $E$; it is globally generated by $f_1$ and $f_2$.", "For any $n$ there is a surjection", "$$", "(\\mathcal{I}^n/\\mathcal{I}^{n + 1})^{\\oplus r} =", "\\mathcal{I}^n/\\mathcal{I}^{n + 1} \\otimes_{\\mathcal{O}_E}", "\\mathcal{G}|_E \\longrightarrow", "\\mathcal{I}^n\\mathcal{G}/\\mathcal{I}^{n + 1}\\mathcal{G}", "$$", "Hence the first cohomology group of this module is zero.", "Here we use that $E = \\mathbf{P}^1_Z$ and hence its structure", "sheaf and in fact any globally generated quasi-coherent module", "has vanishing $H^1$. Compare with More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-globally-generated-vanishing}.", "Then using the short exact sequences", "$$", "0 \\to \\mathcal{I}^n\\mathcal{G}/\\mathcal{I}^{n + 1}\\mathcal{G} \\to", "\\mathcal{G}/\\mathcal{I}^{n + 1}\\mathcal{G} \\to", "\\mathcal{G}/\\mathcal{I}^n\\mathcal{G} \\to 0", "$$", "and induction, we see that", "$$", "\\lim H^0(X', \\mathcal{G}/\\mathcal{I}^n\\mathcal{G})", "\\to", "H^0(E, \\mathcal{G}|_E) = H^0(E, \\mathcal{O}_E^{\\oplus r}) =", "A/I^{\\oplus r}", "$$", "is surjective. By the theorem on formal functions", "(Cohomology of Schemes, Theorem \\ref{coherent-theorem-formal-functions})", "this implies that", "$$", "H^0(X', \\mathcal{G}) \\to", "H^0(E, \\mathcal{G}|_E) = H^0(E, \\mathcal{O}_E^{\\oplus r}) =", "A/I^{\\oplus r}", "$$", "is surjective. Thus we can choose a map", "$\\alpha : \\mathcal{O}_{X'}^{\\oplus r} \\to \\mathcal{G}$", "which is compatible with the given trivialization", "of $\\mathcal{G}|_E$. Thus $\\alpha$ is an isomorphism over", "an open neighbourhood of $E$ in $X'$. Thus every point", "of $Z$ has an affine open neighbourhood where", "we can solve the problem. Since $X' \\setminus E \\to X \\setminus Z$", "is an isomorphism, the same holds for points of $X$ not in $Z$.", "Thus another Zariski glueing argument finishes the proof." ], "refs": [ "flat-lemma-h-sheaf-colim-F", "limits-lemma-descend-invertible-modules", "flat-lemma-colim-F-Vect", "more-morphisms-lemma-globally-generated-vanishing", "coherent-theorem-formal-functions" ], "ref_ids": [ 6168, 15079, 6174, 14071, 3278 ] } ], "ref_ids": [] }, { "id": 6177, "type": "theorem", "label": "flat-lemma-trivial-fibres-dvr", "categories": [ "flat" ], "title": "flat-lemma-trivial-fibres-dvr", "contents": [ "Let $f : X \\to S$ be a proper morphism with geometrically connected fibres", "where $S$ is the spectrum of a discrete valuation ring. Denote $\\eta \\in S$", "the generic point and denote $X_n \\subset X$ the closed subscheme", "cutout by the $n$th power of a uniformizer on $S$.", "Then there exists", "an integer $n$ such that the following is true: any finite", "locally free $\\mathcal{O}_X$-module $\\mathcal{E}$", "such that $\\mathcal{E}|_{X_\\eta}$ and $\\mathcal{E}|_{X_n}$", "are free, is free." ], "refs": [], "proofs": [ { "contents": [ "We first reduce to the case where $X \\to S$ has a section. Say $S = \\Spec(A)$.", "Choose a closed point $\\xi$ of $X_\\eta$. Choose an extension", "of discrete valuation rings $A \\subset B$ such that the fraction field", "of $B$ is $\\kappa(\\xi)$. This is possible by Krull-Akizuki", "(Algebra, Lemma \\ref{algebra-lemma-integral-closure-Dedekind})", "and the fact that $\\kappa(\\xi)$ is a finite extension of the", "fraction field of $A$.", "By the valuative criterion of properness", "(Morphisms, Lemma \\ref{morphisms-lemma-characterize-proper})", "we get a $B$-valued point $\\tau : \\Spec(B) \\to X$", "which induces a section $\\sigma : \\Spec(B) \\to X_B$.", "For a finite locally free $\\mathcal{O}_X$-module $\\mathcal{E}$", "let $\\mathcal{E}_B$ be the pullback to the base change $X_B$.", "By flat base change", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology})", "we see that $H^0(X_B, \\mathcal{E}_B) = H^0(X, \\mathcal{E}) \\otimes_A B$.", "Thus if $\\mathcal{E}_B$ is free of rank $r$, then the sections in", "$H^0(X, \\mathcal{E})$ generate the free $B$-module", "$\\tau^*\\mathcal{E} = \\sigma^*\\mathcal{E}_B$.", "In particular, we can find $r$ global sections $s_1, \\ldots, s_r$", "of $\\mathcal{E}$ which generate $\\tau^*\\mathcal{E}$. Then", "$$", "s_1, \\ldots, s_r :", "\\mathcal{O}_X^{\\oplus r}", "\\longrightarrow", "\\mathcal{E}", "$$", "is a map of finite locally free $\\mathcal{O}_X$-modules of rank $r$", "and the pullback to $X_B$ is a map of free $\\mathcal{O}_{X_B}$-modules", "which restricts to an isomorphism in one point of each fibre.", "Taking the determinant we get a function", "$g \\in \\Gamma(X_\\eta, \\mathcal{O}_{X_B})$", "which is invertible in one point of each fibre.", "As the fibres are proper and connected, we see that $g$", "must be invertible (details omitted; hint: use Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}).", "Thus it suffices to prove the lemma for the base change $X_B \\to \\Spec(B)$.", "\\medskip\\noindent", "Assume we have a section $\\sigma : S \\to X$. Let $\\mathcal{E}$", "be a finite locally free $\\mathcal{O}_X$-module which is assumed", "free on the generic fibre and on $X_n$ (we will choose $n$ later).", "Choose an isomorphism $\\sigma^*\\mathcal{E} = \\mathcal{O}_S^{\\oplus r}$.", "Consider the map", "$$", "K = R\\Gamma(X, \\mathcal{E}) \\longrightarrow", "R\\Gamma(S, \\sigma^*\\mathcal{E}) = A^{\\oplus r}", "$$", "in $D(A)$. Arguing as above, we see $\\mathcal{E}$ is free if (and only if)", "the induced map $H^0(K) = H^0(X, \\mathcal{E}) \\to A^{\\oplus r}$ is surjective.", "\\medskip\\noindent", "Set $L = R\\Gamma(X, \\mathcal{O}_X^{\\oplus r})$ and observe that the", "corresponding map $L \\to A^{\\oplus r}$ has the desired property.", "Observe that $K \\otimes_A Q(A) \\cong L \\otimes_A Q(A)$", "by flat base change and the assumption that $\\mathcal{E}$", "is free on the generic fibre. Let $\\pi \\in A$ be a uniformizer. Observe that", "$$", "K \\otimes_A^\\mathbf{L} A/\\pi^m A =", "R\\Gamma(X, \\mathcal{E} \\xrightarrow{\\pi^m} \\mathcal{E})", "$$", "and similarly for $L$.", "Denote $\\mathcal{E}_{tors} \\subset \\mathcal{E}$ the coherent subsheaf of", "sections supported on the special fibre and similarly for other", "$\\mathcal{O}_X$-modules. Choose $k > 0$ such that", "$(\\mathcal{O}_X)_{tors} \\to \\mathcal{O}_X/\\pi^k \\mathcal{O}_X$", "is injective (Cohomology of Schemes, Lemma \\ref{coherent-lemma-Artin-Rees}).", "Since $\\mathcal{E}$ is locally free, we see", "that $\\mathcal{E}_{tors} \\subset \\mathcal{E}/\\pi^k\\mathcal{E}$.", "Then for $n \\geq m + k$ we have isomorphisms", "\\begin{align*}", "(\\mathcal{E} \\xrightarrow{\\pi^m} \\mathcal{E})", "& \\cong", "(\\mathcal{E}/\\pi^k\\mathcal{E} \\xrightarrow{\\pi^m}", "\\mathcal{E}/\\pi^{k + m}\\mathcal{E}) \\\\", "& \\cong", "(\\mathcal{O}_X^{\\oplus r}/\\pi^k\\mathcal{O}_X^{\\oplus r} \\xrightarrow{\\pi^m}", "\\mathcal{O}_X^{\\oplus r}/\\pi^{k + m}\\mathcal{O}_X^{\\oplus r}) \\\\", "& \\cong", "(\\mathcal{O}_X^{\\oplus r} \\xrightarrow{\\pi^m} \\mathcal{O}_X^{\\oplus r})", "\\end{align*}", "in $D(\\mathcal{O}_X)$. This determines an isomorphism", "$$", "K \\otimes_A^\\mathbf{L} A/\\pi^m A \\cong L \\otimes_A^\\mathbf{L} A/\\pi^m A", "$$", "in $D(A)$ (holds when $n \\geq m + k$). Observe that these isomorphisms", "are compatible with pulling back by $\\sigma$ hence in particular", "we conclude that", "$K \\otimes_A^\\mathbf{L} A/\\pi^m A \\to (A/\\pi^m A)^{\\oplus r}$", "defines an surjection on degree $0$ cohomology modules (as", "this is true for $L$).", "Since $A$ is a discrete valuation ring, we have", "$$", "K \\cong \\bigoplus H^i(K)[-i]", "\\quad\\text{and}\\quad", "L \\cong", "\\bigoplus H^i(L)[-i]", "$$", "in $D(A)$. See More on Algebra, Example", "\\ref{more-algebra-example-finite-injective-finite-global-dimension}.", "The cohomology groups $H^i(K) = H^i(X, \\mathcal{E})$ and", "$H^i(L) = H^i(X, \\mathcal{O}_X)^{\\oplus r}$", "are finite $A$-modules by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-over-affine-cohomology-finite}.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-generalized-valuation-ring-modules}", "these modules are direct sums of cyclic modules.", "We have seen above that the rank $\\beta_i$ of the", "free part of $H^i(K)$ and $H^i(L)$ are the same.", "Next, observe that", "$$", "H^i(L \\otimes_A^\\mathbf{L} A/\\pi^m A) =", "H^i(L)/\\pi^m H^i(L) \\oplus H^{i + 1}(L)[\\pi^m]", "$$", "and similarly for $K$. Let $e$ be the largest integer", "such that $A/\\pi^eA$ occurs as a summand of $H^i(X, \\mathcal{O}_X)$,", "or equivalently $H^i(L)$, for some $i$. Then taking $m = e + 1$", "we see that $H^i(L \\otimes_A^\\mathbf{L} A/\\pi^m A)$ is a direct sum of", "$\\beta_i$ copies of $A/\\pi^m A$ and some other cyclic modules", "each annihilated by $\\pi^e$. By the same reasoning for $K$", "and the isomorphism", "$K \\otimes_A^\\mathbf{L} A/\\pi^m A \\cong L \\otimes_A^\\mathbf{L} A/\\pi^m A$", "it follows that $H^i(K)$", "cannot have any cyclic summands of the form $A/\\pi^l A$", "with $l > e$. (It also follows that $K$ is isomorphic to $L$", "as an object of $D(A)$, but we won't need this.)", "Then the only way the map", "$$", "H^0(K \\otimes^\\mathbf{L}_A A/\\pi^{e + 1} A) =", "H^0(K)/\\pi^{e + 1}H^0(K) \\oplus H^1(K)[\\pi^{e + 1}]", "\\longrightarrow", "(A/\\pi^{e + 1} A)^{\\oplus r}", "$$", "is surjective, is if it is surjective on the", "first summand. This is what we wanted to show.", "(To be precise, the integer $n$ in the statement of", "the lemma, if there is a section $\\sigma$,", "should be equal to $k + e + 1$ where $k$ and $e$ are as above", "and depend only on $X$.)" ], "refs": [ "algebra-lemma-integral-closure-Dedekind", "morphisms-lemma-characterize-proper", "coherent-lemma-flat-base-change-cohomology", "varieties-lemma-proper-geometrically-reduced-global-sections", "coherent-lemma-Artin-Rees", "coherent-lemma-proper-over-affine-cohomology-finite", "more-algebra-lemma-generalized-valuation-ring-modules" ], "ref_ids": [ 1042, 5416, 3298, 10948, 3321, 3355, 10556 ] } ], "ref_ids": [] }, { "id": 6178, "type": "theorem", "label": "flat-lemma-trivial-over-dvrs", "categories": [ "flat" ], "title": "flat-lemma-trivial-over-dvrs", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module.", "Assume", "\\begin{enumerate}", "\\item $f$ is flat and proper and $\\mathcal{O}_S = f_*\\mathcal{O}_X$,", "\\item $S$ is a normal Noetherian scheme,", "\\item the pullback of $\\mathcal{E}$ to $X \\times_S \\Spec(\\mathcal{O}_{S, s})$", "is free for every codimension $1$ point $s \\in S$.", "\\end{enumerate}", "Then $\\mathcal{E}$ is isomorphic to the pullback of a finite", "locally free $\\mathcal{O}_S$-module." ], "refs": [], "proofs": [ { "contents": [ "We will prove the canonical map", "$$", "\\Phi : f^*f_*\\mathcal{E} \\longrightarrow \\mathcal{E}", "$$", "is an isomorphism. By flat base change (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology})", "and assumptions (1) and (3) we see that", "the pullback of this to $X \\times_S \\Spec(\\mathcal{O}_{S, s})$", "is an isomorphism for every codimension $1$ point $s \\in S$.", "By Divisors, Lemma \\ref{divisors-lemma-check-isomorphism-via-depth-and-ass}", "it suffices to prove that $\\text{depth}((f^*f_*\\mathcal{E})_x) \\geq 2$", "for any point $x \\in X$ mapping to a point $s \\in S$ of codimension $\\geq 2$.", "Since $f$ is flat and", "$(f^*f_*\\mathcal{E})_x = (f_*\\mathcal{E})_s \\otimes_{\\mathcal{O}_{S, s}}", "\\mathcal{O}_{X, x}$, it suffices to prove that", "$\\text{depth}((f_*\\mathcal{E})_s) \\geq 2$, see", "Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck}.", "Since $S$ is a normal Noetherian scheme", "and $\\dim(\\mathcal{O}_{S, s}) \\geq 2$", "we have $\\text{depth}(\\mathcal{O}_{S, s}) \\geq 2$, see", "Properties, Lemma \\ref{properties-lemma-criterion-normal}.", "Thus we get what we want from", "Divisors, Lemma \\ref{divisors-lemma-depth-pushforward}." ], "refs": [ "coherent-lemma-flat-base-change-cohomology", "divisors-lemma-check-isomorphism-via-depth-and-ass", "algebra-lemma-apply-grothendieck", "properties-lemma-criterion-normal", "divisors-lemma-depth-pushforward" ], "ref_ids": [ 3298, 7864, 1361, 2989, 7888 ] } ], "ref_ids": [] }, { "id": 6179, "type": "theorem", "label": "flat-lemma-fitting-ideals-complex", "categories": [ "flat" ], "title": "flat-lemma-fitting-ideals-complex", "contents": [ "Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be pseudo-coherent.", "For every $p, k \\in \\mathbf{Z}$ there is an finite type quasi-coherent", "sheaf of ideals $\\text{Fit}_{p, k}(E) \\subset \\mathcal{O}_X$", "with the following property: for $U \\subset X$ open", "such that $E|_U$ is isomorphic to", "$$", "\\ldots \\to", "\\mathcal{O}_U^{\\oplus n_{b - 2}}", "\\xrightarrow{d_{b - 2}}", "\\mathcal{O}_U^{\\oplus n_{b - 1}}", "\\xrightarrow{d_{b - 1}}", "\\mathcal{O}_U^{\\oplus n_b} \\to 0 \\to \\ldots", "$$", "the restriction $\\text{Fit}_{p, k}(E)|_U$ is generated by the", "minors of the matrix of $d_p$ of size", "$$", "- k + n_{p + 1} - n_{p + 2} + \\ldots + (-1)^{b - p + 1} n_b", "$$", "Convention: the ideal generated by $r \\times r$-minors", "is $\\mathcal{O}_U$ if $r \\leq 0$ and the ideal generated by", "$r \\times r$-minors where $r > \\min(n_p, n_{p + 1})$ is zero." ], "refs": [], "proofs": [ { "contents": [ "Observe that $E$ locally on $X$ has the shape as stated in the lemma, see", "More on Algebra, Section \\ref{more-algebra-section-pseudo-coherent},", "Cohomology, Section \\ref{cohomology-section-pseudo-coherent}, and", "Derived Categories of Schemes, Section \\ref{perfect-section-spell-out}.", "Thus it suffices to prove that the ideal of minors is independent", "of the chosen representative. To do this, it suffices to check", "in local rings. Over a local ring $(R, \\mathfrak m, \\kappa)$", "consider a bounded above complex", "$$", "F^\\bullet :", "\\ldots \\to", "R^{\\oplus n_{b - 2}}", "\\xrightarrow{d_{b - 2}}", "R^{\\oplus n_{b - 1}}", "\\xrightarrow{d_{b - 1}}", "R^{\\oplus n_b} \\to 0 \\to \\ldots", "$$", "Denote $\\text{Fit}_{k, p}(F^\\bullet) \\subset R$ the ideal generated", "by the minors of size $k - n_{p + 1} + n_{p + 2} - \\ldots + (-1)^{b - p} n_b$", "in the matrix of $d_p$. Suppose some matrix coefficient of some", "differential of $F^\\bullet$ is invertible. Then we pick a largest", "integer $i$ such that $d_i$ has an invertible matrix coefficient.", "By Algebra, Lemma \\ref{algebra-lemma-add-trivial-complex}", "the complex $F^\\bullet$ is isomorphic to a direct sum of a trivial complex", "$\\ldots \\to 0 \\to R \\to R \\to 0 \\to \\ldots$ with nonzero terms", "in degrees $i$ and $i + 1$ and a complex $(F')^\\bullet$.", "We leave it to the reader to see that", "$\\text{Fit}_{p, k}(F^\\bullet) = \\text{Fit}_{p, k}((F')^\\bullet)$;", "this is where the formula for the size of the minors is used.", "If $(F')^\\bullet$ has another differential with an invertible", "matrix coefficient, we do it again, etc.", "Continuing in this manner, we eventually reach a complex $(F^\\infty)^\\bullet$", "all of whose differentials have matrices with coefficients", "in $\\mathfrak m$. Here you may have to do an infinite number", "of steps, but for any cutoff only a finite number of these", "steps affect the complex in degrees $\\geq $ the cutoff.", "Thus the ``limit'' $(F^\\infty)^\\bullet$ is a well defined bounded", "above complex of finite free modules, comes equipped", "with a quasi-isomorphism $(F^\\infty)^\\bullet \\to F^\\bullet$", "into the complex we started with, and", "$\\text{Fit}_{p, k}(F^\\bullet) = \\text{Fit}_{p, k}((F^\\infty)^\\bullet)$.", "Since the complex $(F^\\infty)^\\bullet$ is unique up to isomorphism by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-pseudo-coherent-from-residue-field}", "the proof is complete." ], "refs": [ "algebra-lemma-add-trivial-complex", "more-algebra-lemma-lift-pseudo-coherent-from-residue-field" ], "ref_ids": [ 908, 10231 ] } ], "ref_ids": [] }, { "id": 6180, "type": "theorem", "label": "flat-lemma-blowup-complex", "categories": [ "flat" ], "title": "flat-lemma-blowup-complex", "contents": [ "Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect.", "Let $U \\subset X$ be a scheme theoretically dense open subscheme", "such that $H^i(E|_U)$ is finite locally free of constant rank $r_i$", "for all $i \\in \\mathbf{Z}$.", "Then there exists a $U$-admissible blowup $b : X' \\to X$ such that", "$H^i(Lb^*E)$ is a perfect $\\mathcal{O}_{X'}$-module", "of tor dimension $\\leq 1$ for all $i \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "We will construct and study the blowup affine locally. Namely, suppose that", "$V \\subset X$ is an affine open subscheme such that", "$E|_V$ can be represented by the complex", "$$", "\\mathcal{O}_V^{\\oplus n_a} \\xrightarrow{d_a}", "\\ldots \\xrightarrow{d_{b - 1}} \\mathcal{O}_V^{\\oplus n_b}", "$$", "Set $k_ i = r_{i + 1} - r_{i + 2} + \\ldots + (-1)^{b - i + 1}r_b$.", "A computation which we omit show that over $U \\cap V$ the rank of", "$d_i$ is", "$$", "\\rho_i = - k_i + n_{i + 1} - n_{i + 2} + \\ldots + (-1)^{b - i + 1}n_b", "$$", "in the sense that the cokernel of $d_i$ is finite locally", "free of rank $n_{i + 1} - \\rho_i$. Let", "$\\mathcal{I}_i \\subset \\mathcal{O}_V$ be the ideal generated by the minors", "of size $\\rho_i \\times \\rho_i$ in the matrix of $d_i$.", "\\medskip\\noindent", "On the one hand, comparing with Lemma \\ref{lemma-fitting-ideals-complex}", "we see the ideal $\\mathcal{I}_i$ corresponds to the global ideal", "$\\text{Fit}_{i, k_i}(E)$ which was shown to be", "independent of the choice of the complex representing $E|_V$.", "On the other hand, $\\mathcal{I}_i$ is the $(n_{i + 1} - \\rho_i)$th", "Fitting ideal of $\\Coker(d_i)$. Please keep this in mind.", "\\medskip\\noindent", "We let $b : X' \\to X$ be the blowing up in the product of the ideals", "$\\text{Fit}_{i, k_i}(E)$; this makes sense as locally on $X$", "almost all of these ideals are equal to the unit ideal (see above).", "This blowup dominates the blowups $b_i : X'_i \\to X$ in", "the ideals $\\text{Fit}_{i, k_i}(E)$, see", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-two-ideals}.", "By Divisors, Lemma \\ref{divisors-lemma-blowup-fitting-ideal}", "each $b_i$ is a $U$-admissible blowup. It follows that", "$b$ is a $U$-admissible blowup (tiny detail omitted; compare with", "the proof of Divisors, Lemma \\ref{divisors-lemma-dominate-admissible-blowups}).", "Finally, $U$ is still a scheme theoretically dense open subscheme of $X'$.", "Thus after replacing $X$ by $X'$ we end up in the situation", "discussed in the next paragraph.", "\\medskip\\noindent", "Assume $\\text{Fit}_{i, k_i}(E)$ is an invertible ideal for all $i$.", "Choose an affine open $V$ and a complex of finite free modules", "representing $E|_V$ as above. It follows from", "Divisors, Lemma \\ref{divisors-lemma-blowup-fitting-ideal}", "that $\\Coker(d_i)$ has tor dimension $\\leq 1$. Thus", "$\\Im(d_i)$ is finite locally free as the kernel of a map", "from a finite locally free module to a finitely presented", "module of tor dimension $\\leq 1$. Hence $\\Ker(d_i)$", "is finite locally free as well (same argument). Thus the short", "exact sequence", "$$", "0 \\to \\Im(d_{i - 1}) \\to \\Ker(d_i) \\to H^i(E)|_V \\to 0", "$$", "shows what we want and the proof is complete." ], "refs": [ "flat-lemma-fitting-ideals-complex", "divisors-lemma-blowing-up-two-ideals", "divisors-lemma-blowup-fitting-ideal", "divisors-lemma-dominate-admissible-blowups", "divisors-lemma-blowup-fitting-ideal" ], "ref_ids": [ 6179, 8062, 8078, 8073, 8078 ] } ], "ref_ids": [] }, { "id": 6181, "type": "theorem", "label": "flat-lemma-blowup-complex-integral", "categories": [ "flat" ], "title": "flat-lemma-blowup-complex-integral", "contents": [ "Let $X$ be an integral scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect.", "Then there exists a nonempty open $U \\subset X$", "such that $H^i(E|_U)$ is finite locally free of constant rank $r_i$", "for all $i \\in \\mathbf{Z}$ and there exists a $U$-admissible blowup", "$b : X' \\to X$ such that $H^i(Lb^*E)$ is a perfect", "$\\mathcal{O}_{X'}$-module of tor dimension $\\leq 1$ for all $i \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "We strongly urge the reader to find their own proof of the", "existence of $U$. Let $\\eta \\in X$ be the generic point.", "The restriction of $E$ to $\\eta$ is isomorphic in $D(\\kappa(\\eta))$ to a", "finite complex $V^\\bullet$ of finite dimensional vector spaces with zero", "differentials. Set $r_i = \\dim_{\\kappa(\\eta)} V^i$.", "Then the perfect object $E'$ in $D(\\mathcal{O}_X)$ represented by the complex", "with terms $\\mathcal{O}_X^{\\oplus r_i}$ and zero differentials", "becomes isomorphic to $E$ after pulling back to $\\eta$.", "Hence by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-descend-relatively-perfect}", "there is an open neighbourhood $U$ of $\\eta$ such that $E|_U$ and $E'|_U$", "are isomorphic. This proves the first assertion. The second follows from the", "first and Lemma \\ref{lemma-blowup-complex} as any nonempty open is", "scheme theoretically dense in the integral scheme $X$." ], "refs": [ "perfect-lemma-descend-relatively-perfect", "flat-lemma-blowup-complex" ], "ref_ids": [ 7082, 6180 ] } ], "ref_ids": [] }, { "id": 6182, "type": "theorem", "label": "flat-lemma-blowup-pd1-derived", "categories": [ "flat" ], "title": "flat-lemma-blowup-pd1-derived", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{F}$ be a perfect", "$\\mathcal{O}_X$-module of tor dimension $\\leq 1$. For any", "blowup $b : X' \\to X$ we have $Lb^*\\mathcal{F} = b^*\\mathcal{F}$", "and $b^*\\mathcal{F}$ is a perfect $\\mathcal{O}_X$-module", "of tor dimension $\\leq 1$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $X = \\Spec(A)$ is affine and we may assume the $A$-module $M$", "corresponding to $\\mathcal{F}$ has a presentation", "$$", "0 \\to A^{\\oplus m} \\to A^{\\oplus n} \\to M \\to 0", "$$", "Suppose $I \\subset A$ is an ideal and $a \\in I$. Recall that the", "affine blowup algebra $A[\\frac{I}{a}]$ is a subring of $A_a$.", "Since localization is exact we see that $A_a^{\\oplus m} \\to A_a^{\\oplus n}$", "is injective. Hence $A[\\frac{I}{a}]^{\\oplus m} \\to A[\\frac{I}{a}]^{\\oplus n}$", "is injective too. This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6183, "type": "theorem", "label": "flat-lemma-blowup-pd1", "categories": [ "flat" ], "title": "flat-lemma-blowup-pd1", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{F}$ be a perfect $\\mathcal{O}_X$-module", "of tor dimension $\\leq 1$. Let $U \\subset X$ be a scheme theoretically", "dense open such that $\\mathcal{F}|_U$ is finite locally free of constant", "rank $r$. Then there exists a $U$-admissible blowup $b : X' \\to X$ such that", "there is a canonical short exact sequence", "$$", "0 \\to \\mathcal{K} \\to b^*\\mathcal{F} \\to \\mathcal{Q} \\to 0", "$$", "where $\\mathcal{Q}$ is finite locally free of rank $r$ and", "$\\mathcal{K}$ is a perfect $\\mathcal{O}_X$-module", "of tor dimension $\\leq 1$ whose restriction to $U$ is zero." ], "refs": [], "proofs": [ { "contents": [ "Combine Divisors, Lemma \\ref{divisors-lemma-blowup-fitting-ideal} and", "Lemma \\ref{lemma-blowup-pd1-derived}." ], "refs": [ "divisors-lemma-blowup-fitting-ideal", "flat-lemma-blowup-pd1-derived" ], "ref_ids": [ 8078, 6182 ] } ], "ref_ids": [] }, { "id": 6184, "type": "theorem", "label": "flat-lemma-canonical-blowup-torsion-pd1", "categories": [ "flat" ], "title": "flat-lemma-canonical-blowup-torsion-pd1", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{F}$ be a perfect $\\mathcal{O}_X$-module", "of tor dimension $\\leq 1$. Let $U \\subset X$ be an open such that", "$\\mathcal{F}|_U = 0$. Then there is a $U$-admissible blowup", "$$", "b : X' \\to X", "$$", "such that $\\mathcal{F}' = b^*\\mathcal{F}$ is equipped with two canonical", "locally finite filtrations", "$$", "0 = F^0 \\subset F^1 \\subset F^2 \\subset \\ldots \\subset \\mathcal{F}'", "\\quad\\text{and}\\quad", "\\mathcal{F}' = F_1 \\supset F_2 \\supset F_3 \\supset \\ldots \\supset 0", "$$", "such that for each $n \\geq 1$ there is an effective Cartier divisor", "$D_n \\subset X'$ with the property that", "$$", "F^i/F^{i - 1}", "\\quad\\text{and}\\quad", "F_i/F_{i + 1}", "$$", "are finite locally free of rank $i$ on $D_i$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine open $V \\subset X$ such that there exists a presentation", "$$", "0 \\to \\mathcal{O}_V^{\\oplus n} \\xrightarrow{A} \\mathcal{O}_V^{\\oplus n} \\to", "\\mathcal{F} \\to 0", "$$", "for some $n$ and some matrix $A$. The ideal we are going to blowup in", "is the product of the Fitting ideals $\\text{Fit}_k(\\mathcal{F})$", "for $k \\geq 0$. This makes sense because in the affine", "situation above we see that $\\text{Fit}_k(\\mathcal{F})|_V = \\mathcal{O}_V$", "for $k > n$. It is clear that this is a $U$-admissible blowing up. By", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-two-ideals}", "we see that on $X'$ the ideals $\\text{Fit}_k(\\mathcal{F})$", "are invertible. Thus we reduce to the case discussed in the", "next paragraph.", "\\medskip\\noindent", "Assume $\\text{Fit}_k(\\mathcal{F})$ is an invertible ideal for", "$k \\geq 0$. If $E_k \\subset X$ is the effective", "Cartier divisor defined by $\\text{Fit}_k(\\mathcal{F})$", "for $k \\geq 0$, then the effective Cartier divisors", "$D_k$ in the statement of the lemma will satisfy", "$$", "E_k = D_{k + 1} + 2 D_{k + 2} + 3 D_{k + 3} + \\ldots", "$$", "This makes sense as the collection $D_k$ will be locally finite.", "Moreover, it uniquely determines the effective Cartier divisors $D_k$", "hence it suffices to construct $D_k$ locally.", "\\medskip\\noindent", "Choose an affine open $V \\subset X$ and presentation of $\\mathcal{F}|_V$", "as above. We will construct the divisors and filtrations by induction", "on the integer $n$ in the presentation. We set $D_k|_V = \\emptyset$", "for $k > n$ and we set $D_n|V = E_{n - 1}|_V$.", "After shrinking $V$ we may assume that", "$\\text{Fit}_{n - 1}(\\mathcal{F})|_V$ is generated by a", "single nonzerodivisor $f \\in \\Gamma(V, \\mathcal{O}_V)$.", "Since $\\text{Fit}_{n - 1}(\\mathcal{F})|_V$ is the ideal generated", "by the entries of $A$, we see that there is a matrix", "$A'$ in $\\Gamma(V, \\mathcal{O}_V)$ such that $A = fA'$.", "Define $\\mathcal{F}'$ on $V$ by the short exact sequence", "$$", "0 \\to \\mathcal{O}_V^{\\oplus n} \\xrightarrow{A'} \\mathcal{O}_V^{\\oplus n} \\to", "\\mathcal{F}' \\to 0", "$$", "Since the entries of $A'$ generate the unit ideal in", "$\\Gamma(V, \\mathcal{O}_V)$ we see that $\\mathcal{F}'$", "locally on $V$ has a presentation with $n$ decreased", "by $1$, see Algebra, Lemma \\ref{algebra-lemma-add-trivial-complex}.", "Further note that", "$f^{n - k}\\text{Fit}_k(\\mathcal{F}') = \\text{Fit}_k(\\mathcal{F})|_V$", "for $k = 0, \\ldots, n$. Hence $\\text{Fit}_k(\\mathcal{F}')$", "is an invertible ideal for all $k$. We conclude by induction", "that there exist effective Cartier divisors $D'_k \\subset V$", "such that $\\mathcal{F}'$ has two canonical filtrations as in the statement", "of the lemma. Then we set $D_k|_V = D'_k$ for $k = 1, \\ldots, n - 1$.", "Observe that the equalities between effective Cartier divisors", "displayed above hold with these choices. Finally, we come", "to the construction of the filtrations. Namely,", "we have short exact sequences", "$$", "0 \\to", "\\mathcal{O}_{D_n \\cap V}^{\\oplus n} \\to", "\\mathcal{F} \\to \\mathcal{F}' \\to 0", "\\quad\\text{and}\\quad", "0 \\to", "\\mathcal{F}' \\to \\mathcal{F} \\to", "\\mathcal{O}_{D_n \\cap V}^{\\oplus n} \\to 0", "$$", "coming from the two factorizations $A = A'f = f A'$ of $A$.", "These sequences are canonical because in the first one", "the submodule is $\\Ker(f : \\mathcal{F} \\to \\mathcal{F})$", "and in the second one the quotient module is", "$\\Coker(f : \\mathcal{F} \\to \\mathcal{F})$." ], "refs": [ "divisors-lemma-blowing-up-two-ideals", "algebra-lemma-add-trivial-complex" ], "ref_ids": [ 8062, 908 ] } ], "ref_ids": [] }, { "id": 6185, "type": "theorem", "label": "flat-lemma-blowup-map-pd1", "categories": [ "flat" ], "title": "flat-lemma-blowup-map-pd1", "contents": [ "Let $X$ be a scheme. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "be a homorphism of perfect $\\mathcal{O}_X$-modules of tor dimension $\\leq 1$.", "Let $U \\subset X$ be a scheme theoretically dense open", "such that $\\mathcal{F}|_U = 0$ and $\\mathcal{G}|_U = 0$.", "Then there is a $U$-admissible blowup $b : X' \\to X$ such that", "the kernel, image, and cokernel of $b^*\\varphi$ are", "perfect $\\mathcal{O}_{X'}$-modules of tor dimension $\\leq 1$." ], "refs": [], "proofs": [ { "contents": [ "The assumptions tell us that the object $(\\mathcal{F} \\to \\mathcal{G})$", "of $D(\\mathcal{O}_X)$ is perfect. Thus we get a $U$-admissible blowup that", "works for the cokernel and kernel by Lemmas \\ref{lemma-blowup-complex}", "and \\ref{lemma-blowup-pd1-derived} (to see what the complex looks", "like after pullback). The image is the kernel of the cokernel", "and hence is going to be perfect of tor dimension $\\leq 1$ as well." ], "refs": [ "flat-lemma-blowup-complex", "flat-lemma-blowup-pd1-derived" ], "ref_ids": [ 6180, 6182 ] } ], "ref_ids": [] }, { "id": 6186, "type": "theorem", "label": "flat-lemma-eta-stalks", "categories": [ "flat" ], "title": "flat-lemma-eta-stalks", "contents": [ "Let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor", "with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.", "Let $\\mathcal{F}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules", "such that $\\mathcal{F}^i$ is $\\mathcal{I}$-torsion free for all $i$.", "\\begin{enumerate}", "\\item For $x \\in X$ choose a generator $f \\in \\mathcal{I}_x$. Then", "the stalk of $\\eta_\\mathcal{I}\\mathcal{F}^\\bullet$ is canonically", "isomorphic to $\\eta_f\\mathcal{F}^\\bullet_x$.", "\\item If the $\\mathcal{F}^i$ are quasi-coherent $\\mathcal{O}_X$-modules,", "then so are the $(\\eta_\\mathcal{I}\\mathcal{F})^i$ and in this case", "if $U = \\Spec(A) \\subset X$ is affine open and $D \\cap U = V(f)$,", "then $\\eta_f(\\mathcal{F}^\\bullet(U))$ is canonically isomorphic", "to $(\\eta_\\mathcal{I}\\mathcal{F}^\\bullet)(U)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6187, "type": "theorem", "label": "flat-lemma-eta-first-property", "categories": [ "flat" ], "title": "flat-lemma-eta-first-property", "contents": [ "Let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor", "with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.", "Let $\\mathcal{F}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules", "such that $\\mathcal{F}^i$ is $\\mathcal{I}$-torsion free for all $i$.", "There is a canonical isomorphism", "$$", "\\mathcal{I}^{\\otimes i} \\otimes_{\\mathcal{O}_X}", "\\left(", "H^i(\\mathcal{F}^\\bullet)/H^i(\\mathcal{F}^\\bullet)[\\mathcal{I}]", "\\right)", "\\longrightarrow H^i(\\eta_\\mathcal{I}\\mathcal{F}^\\bullet)", "$$", "of cohomology sheaves." ], "refs": [], "proofs": [ { "contents": [ "Via Lemma \\ref{lemma-eta-stalks} this translates into the result of", "More on Algebra, Lemma \\ref{more-algebra-lemma-eta-first-property}." ], "refs": [ "flat-lemma-eta-stalks", "more-algebra-lemma-eta-first-property" ], "ref_ids": [ 6186, 10397 ] } ], "ref_ids": [] }, { "id": 6188, "type": "theorem", "label": "flat-lemma-eta-qis", "categories": [ "flat" ], "title": "flat-lemma-eta-qis", "contents": [ "Let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor", "with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.", "Let $\\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$ be a map of complexes", "of $\\mathcal{O}_X$-modules such that $\\mathcal{F}^i$ and $\\mathcal{G}^i$", "are $\\mathcal{I}$-torsion free for all $i$.", "Then the induced map", "$\\eta_\\mathcal{I}\\mathcal{F}^\\bullet \\to \\eta_\\mathcal{I}\\mathcal{G}^\\bullet$", "is a quasi-isomorphism too." ], "refs": [], "proofs": [ { "contents": [ "This is true because the isomorphisms of Lemma \\ref{lemma-eta-first-property}", "are compatible with maps of complexes." ], "refs": [ "flat-lemma-eta-first-property" ], "ref_ids": [ 6187 ] } ], "ref_ids": [] }, { "id": 6189, "type": "theorem", "label": "flat-lemma-eta-tensor-invertible", "categories": [ "flat" ], "title": "flat-lemma-eta-tensor-invertible", "contents": [ "Let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor", "with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.", "Let $\\mathcal{F}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules", "such that $\\mathcal{F}^i$ is $\\mathcal{I}$-torsion free for all $i$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Then $\\eta_\\mathcal{I}(\\mathcal{F}^\\bullet \\otimes \\mathcal{L}) =", "(\\eta_\\mathcal{I}\\mathcal{F}^\\bullet) \\otimes \\mathcal{L}$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the construction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6190, "type": "theorem", "label": "flat-lemma-complex-and-divisor-blowup", "categories": [ "flat" ], "title": "flat-lemma-complex-and-divisor-blowup", "contents": [ "In Situation \\ref{situation-complex-and-divisor} let $b : X' \\to X$", "be the blowing up of the product of the ideals $\\mathcal{J}_i$ from", "Remark \\ref{remark-complex-and-divisor-ideal}.", "Denote $D' = b^{-1}D$ with ideal sheaf", "$\\mathcal{I}' \\subset \\mathcal{O}_{X'}$. Then", "$$", "\\mathcal{Q}^\\bullet =", "\\eta_{\\mathcal{I}'}b^*\\mathcal{E}^\\bullet", "$$", "is a bounded complex of finite locally free $\\mathcal{O}_{X'}$-modules." ], "refs": [ "flat-remark-complex-and-divisor-ideal" ], "proofs": [ { "contents": [ "Recall that $D'$ is an effective Cartier divisor", "(Divisors, Lemma \\ref{divisors-lemma-blow-up-pullback-effective-Cartier}).", "Observe that $\\mathcal{J}_i$ pulls back to an invertible ideal", "sheaf on $X'$ as $X'$ dominates the blowing up in $\\mathcal{J}_i$, see", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-two-ideals}.", "By Remark \\ref{remark-complex-and-divisor-ideal} we may replace", "$X$ by $X'$ and assume $\\mathcal{J}_i$ is invertible for all $i$.", "Via Lemma \\ref{lemma-eta-stalks} we obtain the result from", "More on Algebra, Lemma \\ref{more-algebra-lemma-eta-locally-free}." ], "refs": [ "divisors-lemma-blow-up-pullback-effective-Cartier", "divisors-lemma-blowing-up-two-ideals", "flat-remark-complex-and-divisor-ideal", "flat-lemma-eta-stalks", "more-algebra-lemma-eta-locally-free" ], "ref_ids": [ 8061, 8062, 6237, 6186, 10401 ] } ], "ref_ids": [ 6237 ] }, { "id": 6191, "type": "theorem", "label": "flat-lemma-complex-and-divisor-eta-pull", "categories": [ "flat" ], "title": "flat-lemma-complex-and-divisor-eta-pull", "contents": [ "In Situation \\ref{situation-complex-and-divisor} let $f : Y \\to X$", "be a morphism of schemes such that the inverse image $f^{-1}D$", "is an effective Cartier divisor with ideal sheaf $\\mathcal{J}$.", "Assume $\\mathcal{J}_i$ as in Remark \\ref{remark-complex-and-divisor-ideal}", "is invertible for all $i$. Then $f^*(\\eta_\\mathcal{I}\\mathcal{E}^\\bullet) =", "\\eta_\\mathcal{J}(f^*\\mathcal{E}^\\bullet)$." ], "refs": [ "flat-remark-complex-and-divisor-ideal" ], "proofs": [ { "contents": [ "Follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-eta-base-change}", "via Lemma \\ref{lemma-eta-stalks}." ], "refs": [ "more-algebra-lemma-eta-base-change", "flat-lemma-eta-stalks" ], "ref_ids": [ 10402, 6186 ] } ], "ref_ids": [ 6237 ] }, { "id": 6192, "type": "theorem", "label": "flat-lemma-complex-and-divisor-blowup-base-change", "categories": [ "flat" ], "title": "flat-lemma-complex-and-divisor-blowup-base-change", "contents": [ "In Situation \\ref{situation-complex-and-divisor} let $f : Y \\to X$", "be a morphism of schemes such that the inverse image $f^{-1}D$", "is an effective Cartier divisor. Let $X' \\to X$ and $\\mathcal{Q}^\\bullet$,", "resp.\\ $Y' \\to Y$ and $\\mathcal{Q}_{Y'}^\\bullet$, be as constructed in", "Lemma \\ref{lemma-complex-and-divisor-blowup} for", "$D \\subset X$ and $\\mathcal{E}^\\bullet$,", "resp.\\ $f^{-1}D \\subset Y$ and $f^*\\mathcal{E}^\\bullet$.", "Then $Y'$ is the strict transform of $Y$ with respect to $X' \\to X$", "and $\\mathcal{Q}_{Y'}^\\bullet = (Y' \\to X')^*\\mathcal{Q}^\\bullet$." ], "refs": [ "flat-lemma-complex-and-divisor-blowup" ], "proofs": [ { "contents": [ "In Remark \\ref{remark-complex-and-divisor-ideal} we have seen that", "$\\mathcal{J}_i$ pulls back to the corresponding ideal on $Y$. Hence", "$Y'$ is the strict transform of $Y$ by", "Divisors, Lemma \\ref{divisors-lemma-strict-transform}.", "The final statement follows from", "Lemma \\ref{lemma-complex-and-divisor-eta-pull}", "applied to $Y' \\to X'$." ], "refs": [ "flat-remark-complex-and-divisor-ideal", "divisors-lemma-strict-transform", "flat-lemma-complex-and-divisor-eta-pull" ], "ref_ids": [ 6237, 8065, 6191 ] } ], "ref_ids": [ 6190 ] }, { "id": 6193, "type": "theorem", "label": "flat-lemma-complex-and-divisor-blowup-good", "categories": [ "flat" ], "title": "flat-lemma-complex-and-divisor-blowup-good", "contents": [ "In Situation \\ref{situation-complex-and-divisor} let $U \\subset X$", "be the maximal open subscheme over which the cohomology sheaves", "of $\\mathcal{E}^\\bullet$ are locally free. Then the blowing up", "$b : X' \\to X$", "of Lemma \\ref{lemma-complex-and-divisor-blowup} is an isomorphism", "over $U$." ], "refs": [ "flat-lemma-complex-and-divisor-blowup" ], "proofs": [ { "contents": [ "Over $U$ all of the modules $\\Im(d^i)$ and $\\Ker(d^i)$ are finite", "locally free, see for example the discussion in", "Remark \\ref{remark-when-you-have-a-complex}.", "Let $x \\in U$. Choose an open neighbourhood $x \\in V \\subset U$", "such that $\\mathcal{I}|_V$, $\\mathcal{E}^i$, $\\Ker(d^i)$, $\\Im(d^i)$,", "and $H^i(\\mathcal{E}^\\bullet)$", "are free and choose splittings for the short exact sequences", "$$", "0 \\to \\Im(d^i) \\to \\Ker(d^{i + 1}) \\to H^{i + 1}(\\mathcal{E}^\\bullet) \\to 0", "\\quad\\text{and}\\quad", "0 \\to \\Ker(d^i) \\to \\mathcal{E}^i \\to \\Im(d^i) \\to 0", "$$", "Then we see that our complex looks like", "$$", "\\ldots \\to", "\\mathcal{O}_V^{\\oplus n_{i - 1} + m_i + n_i} \\to", "\\mathcal{O}_V^{\\oplus n_i + m_{i + 1} + n_{i + 1}} \\to \\ldots", "$$", "where the map identifies the last $n_i$ summands with the first", "$n_i$ summands. Thus $\\mathcal{J}_i|V$ is the ideal generated by", "$f^{n_{i - 1} + m_i}$ where $f \\in \\mathcal{O}_X(V)$", "is a generator for $\\mathcal{I}|_V$. Thus over $V$ we are blowing up", "an invertible ideal, which produces the identity morphism", "(Divisors, Lemma \\ref{divisors-lemma-blow-up-effective-Cartier-divisor})." ], "refs": [ "flat-remark-when-you-have-a-complex", "divisors-lemma-blow-up-effective-Cartier-divisor" ], "ref_ids": [ 6235, 8057 ] } ], "ref_ids": [ 6190 ] }, { "id": 6194, "type": "theorem", "label": "flat-lemma-complex-and-divisor-blowup-T", "categories": [ "flat" ], "title": "flat-lemma-complex-and-divisor-blowup-T", "contents": [ "In Situation \\ref{situation-complex-and-divisor}. Let $b : X' \\to X$,", "$D' \\subset X'$, and $\\mathcal{Q}^\\bullet$ be as in", "Lemma \\ref{lemma-complex-and-divisor-blowup}.", "Let $U \\subset X$ be as in Lemma \\ref{lemma-complex-and-divisor-blowup-good}.", "Then there exists a closed immersion $T \\to D'$ of finite presentation", "with $D' \\cap b^{-1}(U) \\subset T$ scheme theoretically", "such that $\\mathcal{Q}^\\bullet|_T$ has finite locally free cohomology sheaves." ], "refs": [ "flat-lemma-complex-and-divisor-blowup", "flat-lemma-complex-and-divisor-blowup-good" ], "proofs": [ { "contents": [ "Arguing exactly as in the proof of Lemma \\ref{lemma-complex-and-divisor-blowup}", "we may replace $X$ by $X'$ and $U$ by $b^{-1}(U)$ and assume that the ideals", "$\\mathcal{J}_i$ are invertible for all $i$.", "\\medskip\\noindent", "Assume $\\mathcal{J}_i$ invertible for all $i$ so that $b = \\text{id}_X$", "and $\\mathcal{Q}^\\bullet = \\eta_{\\mathcal{I}}\\mathcal{E}^\\bullet$.", "Let $x \\in D \\cap U$ and choose a generator $f \\in \\mathcal{I}_x$.", "Since $H^i(\\mathcal{E}^\\bullet)_x$ is a finite free", "$\\mathcal{O}_{X, x}$-module for all $i$ (by choice of $U$),", "we see that", "$$", "\\Ker(d^i : \\mathcal{E}^i_x/f^2\\mathcal{E}^i_x \\to", "\\mathcal{E}^{i + 1}_x/f^2\\mathcal{E}^{i + 1}_x)", "\\to", "\\Ker(d^i : \\mathcal{E}^i_x/f\\mathcal{E}^i_x \\to", "\\mathcal{E}^{i + 1}_x/f\\mathcal{E}^{i + 1}_x)", "$$", "is surjective, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-vanishing-beta}.", "This means that if $X = \\Spec(A)$ is affine, then via", "Lemma \\ref{lemma-eta-stalks} we may apply", "More on Algebra, Lemma \\ref{more-algebra-lemma-eta-vanishing-beta}", "to get a closed subscheme $T \\subset D$ with all the desired", "properties (some details omitted).", "\\medskip\\noindent", "To glue this affine local construction, we remark that in the proof of ", "More on Algebra, Lemma \\ref{more-algebra-lemma-eta-vanishing-beta-plus}", "the ideal cutting out $T$ is constructed with a certain universal", "property. Namely, the result of", "More on Algebra, Lemma \\ref{more-algebra-lemma-eta-locally-free}", "tells us that the canonical maps", "$$", "c^i : \\mathcal{Q}^i \\to", "\\mathcal{I}^i\\mathcal{E}^i \\oplus \\mathcal{I}^{i + 1}\\mathcal{E}^{i + 1}", "$$", "are locally split. The closed subscheme $T \\subset D$ is characterized", "by the property that a morphism of schemes $g : W \\to D$ factors through", "$T$ if and only if $g^*\\mathcal{Q}^i$ is a direct sum of sheaves compatible", "with the map $g^*c^i$ for all $i$. Hence it is clear", "that the affine locally constructed closed subschemes glue." ], "refs": [ "flat-lemma-complex-and-divisor-blowup", "more-algebra-lemma-vanishing-beta", "flat-lemma-eta-stalks", "more-algebra-lemma-eta-vanishing-beta", "more-algebra-lemma-eta-vanishing-beta-plus", "more-algebra-lemma-eta-locally-free" ], "ref_ids": [ 6190, 10403, 6186, 10404, 10406, 10401 ] } ], "ref_ids": [ 6190, 6193 ] }, { "id": 6195, "type": "theorem", "label": "flat-lemma-complex-and-divisor-blowup-T-ranks", "categories": [ "flat" ], "title": "flat-lemma-complex-and-divisor-blowup-T-ranks", "contents": [ "In Situation \\ref{situation-complex-and-divisor}. Let $b : X' \\to X$,", "$D' \\subset X'$, and $\\mathcal{Q}^\\bullet$ be as in", "Lemma \\ref{lemma-complex-and-divisor-blowup}.", "Given integers $\\rho_i \\geq 0$ almost all zero, let $U' \\subset X$ be", "the maximal open subscheme where $H^i(\\mathcal{E}^\\bullet)$ is finite locally", "free of rank $\\rho_i$ for all $i$.", "Let $T \\subset D'$ be as in Lemma \\ref{lemma-complex-and-divisor-blowup-T}.", "Then there exists an open and closed subscheme $T' \\subset T$", "containing $D' \\cap b^{-1}(U')$ scheme theoretically", "such that $\\mathcal{Q}^\\bullet|_{T'}$ has finite locally free", "cohomology sheaves $H^i(\\mathcal{Q}^\\bullet|_{T'})$ of", "rank $\\rho_i$." ], "refs": [ "flat-lemma-complex-and-divisor-blowup", "flat-lemma-complex-and-divisor-blowup-T" ], "proofs": [ { "contents": [ "This is obvious." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 6190, 6194 ] }, { "id": 6196, "type": "theorem", "label": "flat-lemma-complex-and-divisor-derived", "categories": [ "flat" ], "title": "flat-lemma-complex-and-divisor-derived", "contents": [ "Let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor. Let", "$E \\in D(\\mathcal{O}_X)$ be a perfect object. Let $U \\subset X$ be the maximal", "open over which the cohomology sheaves $H^i(E)$ are locally free.", "There exists a proper morphism", "$b : X' \\longrightarrow X$ and an object $Q \\in D(\\mathcal{O}_{X'})$", "with the following properties", "\\begin{enumerate}", "\\item $D' = b^{-1}D$ is an effective Cartier divisor,", "\\item $Q = L\\eta_{\\mathcal{I}'}Lb^*E$ where $\\mathcal{I}'$", "is the ideal sheaf of $D'$,", "\\item $Q$ is a perfect object of $D(\\mathcal{O}_{X'})$,", "\\item there exists a closed immersion $T \\to D'$ of finite presentation", "with $D' \\cap b^{-1}(U) \\subset T$ scheme theoretically such that", "$Q|_T$ has finite locally free cohomology sheaves,", "\\item for any open subscheme $V \\subset X$ such that", "$E|_V$ can be represented by a bounded complex $\\mathcal{E}^\\bullet$", "of finite locally free $\\mathcal{O}_V$-modules, the base changes", "of $X' \\to X$, $Q$, $D'$, and $T$ to $V$ are given by the", "constructions of Lemmas \\ref{lemma-complex-and-divisor-blowup} and", "\\ref{lemma-complex-and-divisor-blowup-T}.", "\\end{enumerate}" ], "refs": [ "flat-lemma-complex-and-divisor-blowup", "flat-lemma-complex-and-divisor-blowup-T" ], "proofs": [ { "contents": [ "We first construct the morphism $b : X' \\to X$ by glueing the blowings up", "constructed over opens $V \\subset X$ as in (5). By", "Constructions, Lemma \\ref{constructions-lemma-relative-glueing}", "to do this it suffices to show that given $V \\subset X$ open", "and two bounded complexes $\\mathcal{E}^\\bullet$", "and $(\\mathcal{E}')^\\bullet$ of finite locally free $\\mathcal{O}_V$-modules", "representing $E|_V$ the resulting blowing ups are canonically isomorphic.", "To do this, it suffices, by the universal property of blowing up", "of Divisors, Lemma \\ref{divisors-lemma-universal-property-blowing-up},", "to show that the ideals $\\mathcal{J}_i$ and $\\mathcal{J}'_i$ from", "Remark \\ref{remark-complex-and-divisor-ideal}", "constructed using $\\mathcal{E}^\\bullet$ and $(\\mathcal{E}')^\\bullet$ ", "locally differ by multiplication by an invertible ideal.", "We will in fact show that they differ locally by a power of the", "ideal sheaf $\\mathcal{I}$ of $D$. By More on Algebra, Lemma", "\\ref{more-algebra-lemma-compare-representatives-perfect}", "working locally it suffices to prove the relationship when", "$$", "(\\mathcal{E}')^\\bullet =", "\\mathcal{E}^\\bullet \\oplus ( \\ldots \\to 0 \\to", "\\mathcal{O}_V \\xrightarrow{1} \\mathcal{O}_V \\to 0 \\to \\ldots)", "$$", "with the two summands $\\mathcal{O}_V$ placed in degrees $i$ and $i + 1$ say.", "Computing minors explicitly one finds that", "$\\mathcal{J}'_{i + 1} = \\mathcal{I}\\mathcal{J}_{i + 1}$", "and all other ideals stay the same.", "\\medskip\\noindent", "Thus we have the morphism $b : X' \\to X$ agreeing locally with the", "blowing ups in (5). Of course this immediately gives us the", "effective Cartier divisor $D' = b^{-1}D$, its invertible ideal sheaf", "$\\mathcal{I}'$ and the object $Q = L\\eta_{\\mathcal{I}'}Lb^*E$.", "See Remark \\ref{remark-Leta} for the construction of $L\\eta_{\\mathcal{I}'}$.", "Since the construction commutes with restricting to opens we find", "that $Q|_{V'}$ is represented by the complex", "$\\mathcal{Q}^\\bullet$ over the open $V' = b^{-1}(V)$ constructed", "using $\\mathcal{E}^\\bullet$ over $V$.", "\\medskip\\noindent", "To finish the proof it suffices to show that the closed subschemes", "$T_V \\subset V'$ constructed in Lemma \\ref{lemma-complex-and-divisor-blowup-T}", "glue. Again by relative glueing, it suffices to show that the construction", "of $T$ does not depend on the choice of the complex $\\mathcal{E}^\\bullet$", "representing $E|_V$. Again we reduce to the case where", "$$", "(\\mathcal{E}')^\\bullet =", "\\mathcal{E}^\\bullet \\oplus ( \\ldots \\to 0 \\to", "\\mathcal{O}_V \\xrightarrow{1} \\mathcal{O}_V \\to 0 \\to \\ldots)", "$$", "with the two summands $\\mathcal{O}_V$ placed in degrees $i$ and $i + 1$ say.", "Note that in this case $(\\mathcal{Q}')^\\bullet$ and $\\mathcal{Q}^\\bullet$", "differ as follows", "$$", "(\\mathcal{Q}')^\\bullet = \\mathcal{Q}^\\bullet \\oplus", "( \\ldots \\to 0 \\to", "(\\mathcal{I}')^{i + 1}|_{V'} \\xrightarrow{1}", "(\\mathcal{I}')^{i + 1}|_{V'} \\to 0 \\to \\ldots)", "$$", "In the proof of Lemma \\ref{lemma-complex-and-divisor-blowup-T}", "we defined $T \\subset D'$ as the largest closed subscheme of $D'$ such", "that $\\mathcal{Q}^i|_T$ is a direct sum of two parts compatible with", "the restriction to $T$ of the canonical split injective maps", "$$", "c^i :", "\\mathcal{Q}^i", "\\longrightarrow", "(\\mathcal{I}')^ib^*\\mathcal{E}^i \\oplus", "(\\mathcal{I}')^{i + 1}b^*\\mathcal{E}^{i + 1}", "$$", "for all $i$. The direct sum decomposition for $(\\mathcal{Q}')^\\bullet$", "in terms of $\\mathcal{Q}^\\bullet$ and the explicit complex", "$(\\mathcal{I}')^{i + 1}|_{V'} \\to (\\mathcal{I}')^{i + 1}|_{V'}$", "implies in a straightforward manner that $T$ plays the same", "role for $(\\mathcal{Q}')^\\bullet$ and the proof is complete." ], "refs": [ "constructions-lemma-relative-glueing", "divisors-lemma-universal-property-blowing-up", "flat-remark-complex-and-divisor-ideal", "more-algebra-lemma-compare-representatives-perfect", "flat-remark-Leta", "flat-lemma-complex-and-divisor-blowup-T", "flat-lemma-complex-and-divisor-blowup-T" ], "ref_ids": [ 12581, 8055, 6237, 10233, 6236, 6194, 6194 ] } ], "ref_ids": [ 6190, 6194 ] }, { "id": 6197, "type": "theorem", "label": "flat-lemma-graph-construction", "categories": [ "flat" ], "title": "flat-lemma-graph-construction", "contents": [ "The construction above has the following properties:", "\\begin{enumerate}", "\\item $b$ is an isomorphism over $\\mathbf{P}^1_U \\cup \\mathbf{A}^1_X$,", "\\item the restriction of $\\mathcal{Q}^\\bullet$ to $\\mathbf{A}^1_X$", "is equal to the pullback of $\\mathcal{E}^\\bullet$,", "\\item there exists a closed immersion $T \\to W_\\infty$ of finite presentation", "such that $\\infty(U) \\subset T$ scheme theoretically and", "such that $\\mathcal{Q}^\\bullet|_T$ has finite locally free cohomology", "sheaves.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the results in", "Section \\ref{section-blowup-complexes-II},", "especially Lemma \\ref{lemma-complex-and-divisor-blowup-T}." ], "refs": [ "flat-lemma-complex-and-divisor-blowup-T" ], "ref_ids": [ 6194 ] } ], "ref_ids": [] }, { "id": 6198, "type": "theorem", "label": "flat-proposition-existence-complete-at-x", "categories": [ "flat" ], "title": "flat-proposition-existence-complete-at-x", "contents": [ "Let $S$ be a scheme.", "Let $X$ be locally of finite type over $S$.", "Let $x \\in X$ be a point with image $s \\in S$.", "There exists a commutative diagram", "$$", "\\xymatrix{", "(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\", "(S, s) & (S', s') \\ar[l]", "}", "$$", "of pointed schemes such that the horizontal", "arrows are elementary \\'etale neighbourhoods", "and such that $g^*\\mathcal{F}/X'/S'$ has a complete", "d\\'evissage at $x$." ], "refs": [], "proofs": [ { "contents": [ "We prove this by induction on the integer", "$d = \\dim_x(\\text{Supp}(\\mathcal{F}_s))$.", "By", "Lemma \\ref{lemma-elementary-devissage-variant}", "there exists a diagram", "$$", "\\xymatrix{", "(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\", "(S, s) & (S', s') \\ar[l]", "}", "$$", "of pointed schemes such that the horizontal", "arrows are elementary \\'etale neighbourhoods", "and such that $g^*\\mathcal{F}/X'/S'$ has a one step d\\'evissage at $x'$.", "The local nature of the problem implies that we may replace", "$(X, x) \\to (S, s)$ by $(X', x') \\to (S', s')$. Thus after doing so", "we may assume that there exists a one step d\\'evissage", "$(Z_1, Y_1, i_1, \\pi_1, \\mathcal{G}_1)$ of $\\mathcal{F}/X/S$ at $x$.", "\\medskip\\noindent", "We apply", "Lemma \\ref{lemma-existence-alpha}", "to find a map", "$$", "\\alpha_1 :", "\\mathcal{O}_{Y_1}^{\\oplus r_1}", "\\longrightarrow", "\\pi_{1, *}\\mathcal{G}_1", "$$", "which induces an isomorphism of vector spaces over $\\kappa(\\xi_1)$", "where $\\xi_1 \\in Y_1$ is the unique generic point of the fibre of", "$Y_1$ over $s$. Moreover", "$\\dim_{y_1}(\\text{Supp}(\\Coker(\\alpha_1)_s)) < d$.", "It may happen that the stalk of $\\Coker(\\alpha_1)_s$", "at $y_1$ is zero. In this case we may shrink $Y_1$ by", "Lemma \\ref{lemma-shrink} (\\ref{item-shrink-on-Y})", "and assume that $\\Coker(\\alpha_1) = 0$ so we obtain a", "complete d\\'evissage of length zero.", "\\medskip\\noindent", "Assume now that the stalk of $\\Coker(\\alpha_1)_s$", "at $y_1$ is not zero. In this case, by induction, there exists a", "commutative diagram", "\\begin{equation}", "\\label{equation-overcome-this}", "\\vcenter{", "\\xymatrix{", "(Y_1, y_1) \\ar[d] & (Y'_1, y'_1) \\ar[l]^h \\ar[d] \\\\", "(S, s) & (S', s') \\ar[l]", "}", "}", "\\end{equation}", "of pointed schemes such that the horizontal", "arrows are elementary \\'etale neighbourhoods", "and such that $h^*\\Coker(\\alpha_1)/Y'_1/S'$ has a complete", "d\\'evissage", "$$", "(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k, z_k, y_k)_{k = 2, \\ldots, n}", "$$", "at $y'_1$. (In particular $i_2 : Z_2 \\to Y'_1$ is a closed immersion into", "$Y'_2$.) At this point we apply", "Lemma \\ref{lemma-elementary-etale-neighbourhood}", "to $S, X, \\mathcal{F}, x, s$, the system", "$(Z_1, Y_1, i_1, \\pi_1, \\mathcal{G}_1)$ and", "diagram (\\ref{equation-overcome-this}). We obtain a diagram", "$$", "\\xymatrix{", "& & (X'', x'') \\ar[lld] \\ar[d] & (Z''_1, z''_1) \\ar[l] \\ar[lld] \\ar[d] \\\\", "(X, x) \\ar[d] & (Z_1, z_1) \\ar[l] \\ar[d] &", "(S'', s'') \\ar[lld] & (Y''_1, y''_1) \\ar[lld] \\ar[l] \\\\", "(S, s) & (Y_1, y_1) \\ar[l]", "}", "$$", "with all the properties as listed in the referenced lemma.", "In particular $Y''_1 \\subset Y'_1 \\times_{S'} S''$. Set", "$X_1 = Y'_1 \\times_{S'} S''$ and let $\\mathcal{F}_1$ denote the", "pullback of $\\Coker(\\alpha_1)$. By", "Lemma \\ref{lemma-base-change-complete-at-x}", "the system", "\\begin{equation}", "\\label{equation-shrink-this}", "(Z_k \\times_{S'} S'',", "Y_k \\times_{S'} S'', i''_k, \\pi''_k, \\mathcal{G}''_k,", "\\alpha''_k, z''_k, y''_k)_{k = 2, \\ldots, n}", "\\end{equation}", "is a complete d\\'evissage of $\\mathcal{F}_1$", "to $X_1$. Again, the nature of the problem allows", "us to replace $(X, x) \\to (S, s)$ by $(X'', x'') \\to (S'', s'')$.", "In this we see that we may assume:", "\\begin{enumerate}", "\\item[(a)] There exists a one step d\\'evissage", "$(Z_1, Y_1, i_1, \\pi_1, \\mathcal{G}_1)$ of $\\mathcal{F}/X/S$ at $x$,", "\\item[(b)] there exists an $\\alpha_1 : \\mathcal{O}_{Y_1}^{\\oplus r_1}", "\\to \\pi_{1, *}\\mathcal{G}_1$ such that $\\alpha \\otimes \\kappa(\\xi_1)$", "is an isomorphism,", "\\item[(c)] $Y_1 \\subset X_1$ is open, $y_1 = x_1$, and", "$\\mathcal{F}_1|_{Y_1} \\cong \\Coker(\\alpha_1)$, and", "\\item[(d)] there exists a complete d\\'evissage", "$(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k, z_k, y_k)_{k = 2, \\ldots, n}$", "of $\\mathcal{F}_1/X_1/S$ at $x_1$.", "\\end{enumerate}", "To finish the proof all we have to do is shrink the one step d\\'evissage", "and the complete d\\'evissage such that they fit together to a complete", "d\\'evissage. (We suggest the reader do this on their own using", "Lemmas \\ref{lemma-shrink} and", "\\ref{lemma-shrink-complete}", "instead of reading the proof that follows.) Since $Y_1 \\subset X_1$", "is an open neighbourhood of $x_1$ we may apply", "Lemma \\ref{lemma-shrink-complete} (\\ref{item-shrink-on-X-complete})", "to find a standard shrinking $S', X'_1, Z'_2, Y'_2, \\ldots, Y'_n$", "of the datum (d) so that $X'_1 \\subset Y_1$. Note that $X'_1$ is also", "a standard open of the affine scheme $Y_1$. Next, we shrink the datum", "(a) as follows: first we shrink the base $S$ to $S'$, see", "Lemma \\ref{lemma-shrink} (\\ref{item-shrink-base}) and then", "we shrink the result to $S''$, $X''$, $Z''_1$, $Y''_1$ using", "Lemma \\ref{lemma-shrink} (\\ref{item-shrink-on-Y})", "such that eventually $Y''_1 = X'_1 \\times_S S''$ and $S'' \\subset S'$.", "Then we see that", "$$", "Z''_1, Y''_1, Z'_2 \\times_{S'} S'', Y'_2 \\times_{S'} S'', \\ldots,", "Y'_n \\times_{S'} S''", "$$", "gives the complete d\\'evissage we were looking for." ], "refs": [ "flat-lemma-elementary-devissage-variant", "flat-lemma-existence-alpha", "flat-lemma-shrink", "flat-lemma-elementary-etale-neighbourhood", "flat-lemma-base-change-complete-at-x", "flat-lemma-shrink", "flat-lemma-shrink-complete", "flat-lemma-shrink-complete", "flat-lemma-shrink", "flat-lemma-shrink" ], "ref_ids": [ 5991, 5996, 5994, 5995, 5998, 5994, 5999, 5999, 5994, 5994 ] } ], "ref_ids": [] }, { "id": 6199, "type": "theorem", "label": "flat-proposition-finite-type-flat-at-point", "categories": [ "flat" ], "title": "flat-proposition-finite-type-flat-at-point", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $x \\in X$ with image $s \\in S$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $\\mathcal{F}$ is of finite type, and", "\\item $\\mathcal{F}$ is flat at $x$ over $S$.", "\\end{enumerate}", "Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$", "and an open subscheme", "$$", "V \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})", "$$", "which contains the unique point of", "$X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ mapping to $x$", "such that the pullback of $\\mathcal{F}$ to $V$ is an $\\mathcal{O}_V$-module", "of finite presentation and flat over $\\mathcal{O}_{S', s'}$." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "This proof is longer but does not use the existence of a complete d\\'evissage.", "The problem is local around $x$ and $s$, hence we may assume that $X$", "and $S$ are affine. During the proof we will finitely many times replace", "$S$ by an elementary \\'etale neighbourhood of $(S, s)$. The goal is then to find", "(after such a replacement) an open", "$V \\subset X \\times_S \\Spec(\\mathcal{O}_{S, s})$ containing $x$", "such that $\\mathcal{F}|_V$ is flat over $S$ and finitely presented.", "Of course we may also replace $S$ by $\\Spec(\\mathcal{O}_{S, s})$", "at any point of the proof, i.e., we may assume $S$ is a local scheme.", "We will prove the proposition by induction on the integer", "$n = \\dim_x(\\text{Supp}(\\mathcal{F}_s))$.", "\\medskip\\noindent", "We can choose", "\\begin{enumerate}", "\\item elementary \\'etale neighbourhoods $g : (X', x') \\to (X, x)$,", "$e : (S', s') \\to (S, s)$,", "\\item a commutative diagram", "$$", "\\xymatrix{", "X \\ar[dd]_f & X' \\ar[dd] \\ar[l]^g & Z' \\ar[l]^i \\ar[d]^\\pi \\\\", "& & Y' \\ar[d]^h \\\\", "S & S' \\ar[l]_e & S' \\ar@{=}[l]", "}", "$$", "\\item a point $z' \\in Z'$ with $i(z') = x'$, $y' = \\pi(z')$, $h(y') = s'$,", "\\item a finite type quasi-coherent $\\mathcal{O}_{Z'}$-module $\\mathcal{G}$,", "\\end{enumerate}", "as in", "Lemma \\ref{lemma-elementary-devissage}.", "We are going to replace $S$ by $\\Spec(\\mathcal{O}_{S', s'})$, see", "remarks in first paragraph of the proof. Consider the diagram", "$$", "\\xymatrix{", "X_{\\mathcal{O}_{S', s'}} \\ar[ddr]_f &", "X'_{\\mathcal{O}_{S', s'}} \\ar[dd] \\ar[l]^g &", "Z'_{\\mathcal{O}_{S', s'}} \\ar[l]^i \\ar[d]^\\pi \\\\", "& & Y'_{\\mathcal{O}_{S', s'}} \\ar[dl]^h \\\\", "& \\Spec(\\mathcal{O}_{S', s'})", "}", "$$", "Here we have base changed the schemes $X', Z', Y'$ over $S'$ via", "$\\Spec(\\mathcal{O}_{S', s'}) \\to S'$ and the scheme $X$ over $S$ via", "$\\Spec(\\mathcal{O}_{S', s'}) \\to S$. It is still the case that", "$g$ is \\'etale, see", "Lemma \\ref{lemma-etale-at-point}.", "After replacing $X$ by $X_{\\mathcal{O}_{S', s'}}$,", "$X'$ by $X'_{\\mathcal{O}_{S', s'}}$,", "$Z'$ by $Z'_{\\mathcal{O}_{S', s'}}$, and", "$Y'$ by $Y'_{\\mathcal{O}_{S', s'}}$", "we may assume we have a diagram as", "Lemma \\ref{lemma-elementary-devissage}", "where in addition $S = S'$ is a local scheme with closed point $s$. By", "Lemmas \\ref{lemma-devissage-finite-presentation} and", "\\ref{lemma-devissage-flat}", "the result for $Y' \\to S$, the sheaf $\\pi_*\\mathcal{G}$, and the", "point $y'$ implies the result for $X \\to S$, $\\mathcal{F}$ and $x$.", "Hence we may assume that $S$ is local and $X \\to S$ is a smooth morphism", "of affines with geometrically irreducible fibres of dimension $n$.", "\\medskip\\noindent", "The base case of the induction: $n = 0$.", "As $X \\to S$ is smooth with geometrically", "irreducible fibres of dimension $0$ we see that $X \\to S$ is an open", "immersion, see", "Descent, Lemma \\ref{descent-lemma-universally-injective-etale-open-immersion}.", "As $S$ is local and the closed point is in the image of $X \\to S$", "we conclude that $X = S$. Thus we see that $\\mathcal{F}$ corresponds", "to a finite flat $\\mathcal{O}_{S, s}$ module. In this case the result", "follows from", "Algebra, Lemma \\ref{algebra-lemma-finite-flat-local}", "which tells us that $\\mathcal{F}$ is in fact finite free.", "\\medskip\\noindent", "The induction step. Assume the result holds whenever the dimension", "of the support in the closed fibre is $< n$. Write $S = \\Spec(A)$,", "$X = \\Spec(B)$ and $\\mathcal{F} = \\widetilde{N}$ for some $B$-module", "$N$. Note that $A$ is a local ring; denote its maximal ideal $\\mathfrak m$.", "Then $\\mathfrak p = \\mathfrak mB$ is the unique minimal prime lying over", "$\\mathfrak m$ as $X \\to S$ has geometrically irreducible fibres. Finally,", "let $\\mathfrak q \\subset B$ be the prime corresponding to $x$. By", "Lemma \\ref{lemma-induction-step}", "we can choose a map", "$$", "\\alpha : B^{\\oplus r} \\to N", "$$", "such that $\\kappa(\\mathfrak p)^{\\oplus r} \\to N \\otimes_B \\kappa(\\mathfrak p)$", "is an isomorphism. Moreover, as $N_{\\mathfrak q}$ is $A$-flat the lemma", "also shows that $\\alpha$ is injective and that", "$\\Coker(\\alpha)_{\\mathfrak q}$ is $A$-flat.", "Set $Q = \\Coker(\\alpha)$. Note that the support of $Q/\\mathfrak mQ$", "does not contain $\\mathfrak p$. Hence it is certainly the case that", "$\\dim_{\\mathfrak q}(\\text{Supp}(Q/\\mathfrak mQ)) < n$.", "Combining everything we know about $Q$ we see", "that the induction hypothesis applies to $Q$. It follows that there exists", "an elementary \\'etale morphism $(S', s) \\to (S, s)$ such that the conclusion", "holds for $Q \\otimes_A A'$ over $B \\otimes_A A'$ where", "$A' = \\mathcal{O}_{S', s'}$. After replacing $A$ by $A'$ we have an", "exact sequence", "$$", "0 \\to B^{\\oplus r} \\to N \\to Q \\to 0", "$$", "(here we use that $\\alpha$ is injective as mentioned above)", "of finite $B$-modules and we also get an element", "$g \\in B$, $g \\not \\in \\mathfrak q$ such that", "$Q_g$ is finitely presented over $B_g$ and flat over $A$. Since localization", "is exact we see that", "$$", "0 \\to B_g^{\\oplus r} \\to N_g \\to Q_g \\to 0", "$$", "is still exact. As $B_g$ and $Q_g$ are flat over $A$ we conclude that", "$N_g$ is flat over $A$, see", "Algebra, Lemma \\ref{algebra-lemma-flat-ses},", "and as $B_g$ and $Q_g$ are finitely presented over $B_g$ the same holds", "for $N_g$, see", "Algebra, Lemma \\ref{algebra-lemma-extension}." ], "refs": [ "flat-lemma-elementary-devissage", "flat-lemma-etale-at-point", "flat-lemma-elementary-devissage", "flat-lemma-devissage-finite-presentation", "flat-lemma-devissage-flat", "descent-lemma-universally-injective-etale-open-immersion", "algebra-lemma-finite-flat-local", "flat-lemma-induction-step", "algebra-lemma-flat-ses", "algebra-lemma-extension" ], "ref_ids": [ 5988, 5979, 5988, 5989, 5990, 14700, 797, 6015, 533, 330 ] } ], "ref_ids": [] }, { "id": 6200, "type": "theorem", "label": "flat-proposition-finite-presentation-flat-at-point", "categories": [ "flat" ], "title": "flat-proposition-finite-presentation-flat-at-point", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $x \\in X$ with image $s \\in S$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $\\mathcal{F}$ is of finite presentation, and", "\\item $\\mathcal{F}$ is flat at $x$ over $S$.", "\\end{enumerate}", "Then there exists a commutative diagram of pointed schemes", "$$", "\\xymatrix{", "(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\", "(S, s) & (S', s') \\ar[l]", "}", "$$", "whose horizontal arrows are elementary \\'etale neighbourhoods", "such that $X'$, $S'$ are affine and such that", "$\\Gamma(X', g^*\\mathcal{F})$ is a projective", "$\\Gamma(S', \\mathcal{O}_{S'})$-module." ], "refs": [], "proofs": [ { "contents": [ "By openness of flatness, see", "More on Morphisms, Theorem \\ref{more-morphisms-theorem-openness-flatness}", "we may replace $X$ by an open neighbourhood of $x$ and assume that", "$\\mathcal{F}$ is flat over $S$. Next, we apply", "Proposition \\ref{proposition-existence-complete-at-x}", "to find a diagram as in the statement of the proposition such", "that $g^*\\mathcal{F}/X'/S'$ has a complete d\\'evissage over $s'$.", "(In particular $S'$ and $X'$ are affine.) By", "Morphisms, Lemma \\ref{morphisms-lemma-flat-permanence}", "we see that $g^*\\mathcal{F}$ is flat over $S$ and by", "Lemma \\ref{lemma-etale-flat-up-down}", "we see that it is flat over $S'$. Via", "Remark \\ref{remark-same-notion}", "we deduce that", "$$", "\\Gamma(X', g^*\\mathcal{F})/", "\\Gamma(X', \\mathcal{O}_{X'})/", "\\Gamma(S', \\mathcal{O}_{S'})", "$$", "has a complete d\\'evissage over the prime of $\\Gamma(S', \\mathcal{O}_{S'})$", "corresponding to $s'$. Thus", "Lemma \\ref{lemma-complete-devissage-flat-finitely-presented-module}", "implies that the result of the proposition holds after replacing", "$S'$ by a standard open neighbourhood of $s'$." ], "refs": [ "more-morphisms-theorem-openness-flatness", "flat-proposition-existence-complete-at-x", "morphisms-lemma-flat-permanence", "flat-lemma-etale-flat-up-down", "flat-remark-same-notion", "flat-lemma-complete-devissage-flat-finitely-presented-module" ], "ref_ids": [ 13670, 6198, 5270, 5980, 6225, 6030 ] } ], "ref_ids": [] }, { "id": 6201, "type": "theorem", "label": "flat-proposition-flat-finite-type-finite-presentation-domain", "categories": [ "flat" ], "title": "flat-proposition-flat-finite-type-finite-presentation-domain", "contents": [ "Let $R$ be a domain. Let $R \\to S$ be a ring map of finite type.", "Let $M$ be a finite $S$-module.", "\\begin{enumerate}", "\\item If $S$ is flat over $R$, then $S$ is a finitely presented $R$-algebra.", "\\item If $M$ is flat as an $R$-module, then $M$ is finitely presented", "as an $S$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is a special case of", "Lemma \\ref{lemma-finite-type-flat-over-integral}.", "For Part (2) choose a surjection $R[x_1, \\ldots, x_n] \\to S$.", "By Lemma \\ref{lemma-finite-type-flat-algebra} we find that $M$", "is finitely presented as an $R[x_1, \\ldots, x_n]$-module.", "We conclude by Algebra, Lemma", "\\ref{algebra-lemma-finitely-presented-over-subring}." ], "refs": [ "flat-lemma-finite-type-flat-over-integral", "flat-lemma-finite-type-flat-algebra", "algebra-lemma-finitely-presented-over-subring" ], "ref_ids": [ 6045, 6043, 335 ] } ], "ref_ids": [] }, { "id": 6202, "type": "theorem", "label": "flat-proposition-finite-presentation-flat-pure-is-projective", "categories": [ "flat" ], "title": "flat-proposition-finite-presentation-flat-pure-is-projective", "contents": [ "Let $f : X \\to S$ be an affine, finitely presented morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of", "finite presentation, flat over $S$. Then the following", "are equivalent", "\\begin{enumerate}", "\\item $f_*\\mathcal{F}$ is locally projective on $S$, and", "\\item $\\mathcal{F}$ is pure relative to $S$.", "\\end{enumerate}", "In particular, given a ring map $A \\to B$ of finite presentation and", "a finitely presented $B$-module $N$ flat over $A$ we have:", "$N$ is projective as an $A$-module if and only if $\\widetilde{N}$", "on $\\Spec(B)$ is pure relative to $\\Spec(A)$." ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) is", "Lemma \\ref{lemma-affine-locally-projective-pure}.", "Assume $\\mathcal{F}$ is pure relative to $S$.", "Note that by", "Lemma \\ref{lemma-finite-type-flat-pure-along-fibre-is-universal}", "this implies $\\mathcal{F}$ remains pure after any base change. By", "Descent, Lemma \\ref{descent-lemma-locally-projective-descends}", "it suffices to prove $f_*\\mathcal{F}$ is fpqc locally projective on $S$.", "Pick $s \\in S$. We will prove that the restriction of", "$f_*\\mathcal{F}$ to an \\'etale neighbourhood of $s$ is locally projective.", "Namely, by", "Lemma \\ref{lemma-finite-presentation-flat-along-fibre},", "after replacing $S$ by an affine elementary \\'etale", "neighbourhood of $s$, we may assume there exists a diagram", "$$", "\\xymatrix{", "X \\ar[dr] & & X' \\ar[ll]^g \\ar[ld] \\\\", "& S &", "}", "$$", "of schemes affine and of finite presentation over $S$,", "where $g$ is \\'etale, $X_s \\subset g(X')$, and with", "$\\Gamma(X', g^*\\mathcal{F})$ a projective $\\Gamma(S, \\mathcal{O}_S)$-module.", "Note that in this case $g^*\\mathcal{F}$ is universally pure over $S$, see", "Lemma \\ref{lemma-affine-locally-projective-pure}.", "Hence by", "Lemma \\ref{lemma-criterion}", "we see that the open $g(X')$ contains the points of", "$\\text{Ass}_{X/S}(\\mathcal{F})$ lying over $\\Spec(\\mathcal{O}_{S, s})$.", "Set", "$$", "E = \\{t \\in S \\mid \\text{Ass}_{X_t}(\\mathcal{F}_t) \\subset g(X') \\}.", "$$", "By", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-relative-assassin-constructible}", "$E$ is a constructible subset of $S$. We have seen that", "$\\Spec(\\mathcal{O}_{S, s}) \\subset E$. By", "Morphisms, Lemma \\ref{morphisms-lemma-constructible-containing-open}", "we see that $E$ contains an open neighbourhood of $s$. Hence after", "replacing $S$ by an affine neighbourhood of $s$ we may assume that", "$\\text{Ass}_{X/S}(\\mathcal{F}) \\subset g(X')$.", "By", "Lemma \\ref{lemma-base-change-universally-flat}", "this means that", "$$", "\\Gamma(X, \\mathcal{F}) \\longrightarrow \\Gamma(X', g^*\\mathcal{F})", "$$", "is $\\Gamma(S, \\mathcal{O}_S)$-universally injective.", "By Algebra, Lemma \\ref{algebra-lemma-pure-submodule-ML}", "we conclude that $\\Gamma(X, \\mathcal{F})$ is Mittag-Leffler as an", "$\\Gamma(S, \\mathcal{O}_S)$-module. Since", "$\\Gamma(X, \\mathcal{F})$ is countably generated and flat as a", "$\\Gamma(S, \\mathcal{O}_S)$-module, we conclude", "it is projective by", "Algebra, Lemma \\ref{algebra-lemma-countgen-projective}." ], "refs": [ "flat-lemma-affine-locally-projective-pure", "flat-lemma-finite-type-flat-pure-along-fibre-is-universal", "descent-lemma-locally-projective-descends", "flat-lemma-finite-presentation-flat-along-fibre", "flat-lemma-affine-locally-projective-pure", "flat-lemma-criterion", "more-morphisms-lemma-relative-assassin-constructible", "morphisms-lemma-constructible-containing-open", "flat-lemma-base-change-universally-flat", "algebra-lemma-pure-submodule-ML", "algebra-lemma-countgen-projective" ], "ref_ids": [ 6064, 6067, 14618, 6031, 6064, 6066, 13812, 5251, 6005, 837, 850 ] } ], "ref_ids": [] }, { "id": 6203, "type": "theorem", "label": "flat-proposition-finite-type-injective-into-flat-mod-m", "categories": [ "flat" ], "title": "flat-proposition-finite-type-injective-into-flat-mod-m", "contents": [ "Let $A \\to B$ be a local ring homomorphism of local rings", "which is essentially of finite type. Let $M$ be a flat $A$-module,", "$N$ a finite $B$-module and $u : N \\to M$ an $A$-module map such that", "$\\overline{u} : N/\\mathfrak m_AN \\to M/\\mathfrak m_AM$ is injective.", "Then $u$ is $A$-universally injective, $N$ is of finite presentation over", "$B$, and $N$ is flat over $A$." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $B$ is the localization of a finitely presented", "$A$-algebra $B_0$ and that $N$ is the localization of a", "finitely presented $B_0$-module $M_0$, see", "Lemma \\ref{lemma-reduce-finite-type-injective-into-flat-mod-m}.", "By Lemma \\ref{lemma-generic-flatness-stratification}", "there exists a ``generic flatness stratification''", "for $\\widetilde{M_0}$ on $\\Spec(B_0)$ over $\\Spec(A)$.", "Translating back to $N$ we find a sequence of closed subschemes", "$$", "S = \\Spec(A) \\supset S_0 \\supset S_1 \\supset \\ldots \\supset S_t = \\emptyset", "$$", "with $S_i \\subset S$ cut out by a finitely generated ideal of $A$", "such that the pullback of $\\widetilde{N}$ to", "$\\Spec(B) \\times_S (S_i \\setminus S_{i + 1})$ is flat over", "$S_i \\setminus S_{i + 1}$. We will prove the proposition by", "induction on $t$ (the base case $t = 1$ will be proved in parallel", "with the other steps). Let $\\Spec(A/J_i)$ be the scheme theoretic", "closure of $S_i \\setminus S_{i + 1}$.", "\\medskip\\noindent", "{\\bf Claim 1.} $N/J_iN$ is flat over $A/J_i$. This is immediate for", "$i = t - 1$ and follows from the induction hypothesis for $i > 0$.", "Thus we may assume $t > 1$, $S_{t - 1} \\not = \\emptyset$, and", "$J_0 = 0$ and we have to prove that $N$ is flat. Let $J \\subset A$", "be the ideal defining $S_1$. By induction on $t$ again, we also", "have flatness modulo powers of $J$. Let $A^h$ be the henselization of $A$", "and let $B'$ be the localization of $B \\otimes_A A^h$ at the maximal ideal", "$\\mathfrak m_B \\otimes A^h + B \\otimes \\mathfrak m_{A^h}$. Then $B \\to B'$", "is faithfully flat. Set $N' = N \\otimes_B B'$. Note that $N'$", "is $A^h$-flat if and only if $N$ is $A$-flat. By", "Theorem \\ref{theorem-flattening-local} there is a smallest ideal", "$I \\subset A^h$ such that $N'/IN'$ is flat over $A^h/I$, and", "$I$ is finitely generated. By the above $I \\subset J^nA^h$ for", "all $n \\geq 1$. Let $S_i^h \\subset \\Spec(A^h)$ be the inverse image", "of $S_i \\subset \\Spec(A)$. By Lemma \\ref{lemma-closed-points-complement}", "we see that $V(I)$ contains the closed points of $U = \\Spec(A^h) - S_1^h$.", "By construction $N'$ is $A^h$-flat over $U$.", "By Lemma \\ref{lemma-make-smaller-flatness-ideal} we see that $N'/I_2N'$", "is flat over $A/I_2$, where", "$I_2 = \\Ker(I \\to \\Gamma(U, I/I^2))$. Hence $I = I_2$ by minimality", "of $I$. This implies that $I = I^2$ locally on $U$, i.e.,", "we have $I\\mathcal{O}_{U, u} = (0)$ or $I\\mathcal{O}_{U, u} = (1)$", "for all $u \\in U$. Since $V(I)$ contains the closed points of $U$", "we see that $I = 0$ on $U$. Since $U \\subset \\Spec(A^h)$ is scheme", "theoretically dense (because replaced $A$ by $A/J_0$ in the beginning", "of this paragraph), we see that $I = 0$. Thus $N'$ is $A^h$-flat", "and hence Claim 1 holds.", "\\medskip\\noindent", "We return to the situation as laid out before Claim 1. With", "$A^h$ the henselization of $A$, with $B'$ the localization", "of $B \\otimes_A A^h$ at the maximal ideal", "$\\mathfrak m_B \\otimes A^h + B \\otimes \\mathfrak m_{A^h}$, and with", "$N' = N \\otimes_B B'$ we now see that the flattening ideal $I \\subset A^h$", "of Theorem \\ref{theorem-flattening-local} is nilpotent.", "If $nil(A^h)$ denotes the ideal of nilpotent elements, then", "$nil(A^h) = nil(A) A^h$", "(More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-nil}).", "Hence there exists a finitely generated", "nilpotent ideal $I_0 \\subset A$ such that $N/I_0N$ is flat over $A/I_0$.", "\\medskip\\noindent", "{\\bf Claim 2.} For every prime ideal $\\mathfrak p \\subset A$", "the map", "$\\kappa(\\mathfrak p) \\otimes_A N \\to \\kappa(\\mathfrak p) \\otimes_A M$", "is injective. We say $\\mathfrak p$ is bad it this is false.", "Suppose that $C$ is a nonempty chain of bad primes and set", "$\\mathfrak p^* = \\bigcup_{\\mathfrak p \\in C} \\mathfrak p$.", "By Lemma \\ref{lemma-find-pure-spreadout}", "there is a finitely generated ideal", "$\\mathfrak a \\subset \\mathfrak p^*A_{\\mathfrak p^*}$", "such that there is a pure spreadout over $V(\\mathfrak a)$.", "If $\\mathfrak p^*$ were good, then it would follow from", "Lemma \\ref{lemma-properties-pure-spreadout}", "that the points of $V(\\mathfrak a)$ are good.", "However, since $\\mathfrak a$ is finitely generated and since", "$\\mathfrak p^*A_{\\mathfrak p^*} = \\bigcup_{\\mathfrak p \\in C}A_{\\mathfrak p^*}$", "we see that $V(\\mathfrak a)$ contains a $\\mathfrak p \\in C$, contradiction.", "Hence $\\mathfrak p^*$ is bad. By Zorn's lemma, if there exists a", "bad prime, there exists a maximal one, say $\\mathfrak p$.", "In other words, we may assume every $\\mathfrak p' \\supset \\mathfrak p$,", "$\\mathfrak p' \\not = \\mathfrak p$ is good.", "In this case we see that for every $f \\in A$, $f \\not \\in \\mathfrak p$", "the map $u \\otimes \\text{id}_{A/(\\mathfrak p + f)}$ is universally", "injective, see Lemma \\ref{lemma-universally-injective-if-flat}.", "Thus it suffices to show that $N/\\mathfrak p N$ is separated", "for the topology defined by the submodules $f(N/\\mathfrak pN)$.", "Since $B \\to B'$ is faithfully flat, it is enough to prove the", "same for the module $N'/\\mathfrak p N'$.", "By Lemma \\ref{lemma-flat-finite-type-local-colimit-free} and", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-content-exists-flat-Mittag-Leffler}", "elements of $N'/\\mathfrak pN'$ have content ideals in $A^h/\\mathfrak pA^h$.", "Thus it suffices to show that", "$\\bigcap_{f \\in A, f \\not \\in \\mathfrak p} f(A^h/\\mathfrak p A^h) = 0$.", "Then it suffices to show the same for", "$A^h/\\mathfrak q A^h$ for every prime $\\mathfrak q \\subset A^h$", "minimal over $\\mathfrak p A^h$. Because $A \\to A^h$ is the henselization,", "every $\\mathfrak q$ contracts to $\\mathfrak p$ and every", "$\\mathfrak q' \\supset \\mathfrak q$, $\\mathfrak q' \\not = \\mathfrak q$", "contracts to a prime $\\mathfrak p'$ which strictly contains $\\mathfrak p$.", "Thus we get the vanishing of the intersections from", "Lemma \\ref{lemma-big-intersection-is-zero}.", "\\medskip\\noindent", "At this point we can put everything together. Namely, using", "Claim 1 and Claim 2 we see that $N/I_0 N \\to M/I_0M$ is", "$A/I_0$-universally injective by", "Lemma \\ref{lemma-universally-injective-if-flat}.", "Then the diagrams", "$$", "\\xymatrix{", "N \\otimes_A (I_0^n/I_0^{n + 1}) \\ar[r] \\ar[d] &", "M \\otimes_A (I_0^n/I_0^{n + 1}) \\ar@{=}[d] \\\\", "I_0^n N /I_0^{n + 1} N \\ar[r] &", "I_0^n M /I_0^{n + 1} M", "}", "$$", "show that the left vertical arrows are injective. Hence by", "Algebra, Lemma \\ref{algebra-lemma-what-does-it-mean-again}", "we see that $N$ is flat. In a similar way the", "universal injectivity of $u$ can be reduced (even without", "proving flatness of $N$ first) to the one modulo $I_0$. This finishes", "the proof." ], "refs": [ "flat-lemma-reduce-finite-type-injective-into-flat-mod-m", "flat-lemma-generic-flatness-stratification", "flat-theorem-flattening-local", "flat-lemma-closed-points-complement", "flat-lemma-make-smaller-flatness-ideal", "flat-theorem-flattening-local", "more-algebra-lemma-henselization-nil", "flat-lemma-find-pure-spreadout", "flat-lemma-properties-pure-spreadout", "flat-lemma-universally-injective-if-flat", "flat-lemma-flat-finite-type-local-colimit-free", "more-algebra-lemma-content-exists-flat-Mittag-Leffler", "flat-lemma-big-intersection-is-zero", "flat-lemma-universally-injective-if-flat", "algebra-lemma-what-does-it-mean-again" ], "ref_ids": [ 6091, 6084, 5970, 6099, 6100, 5970, 10059, 6096, 6095, 6097, 6074, 9942, 6098, 6097, 891 ] } ], "ref_ids": [] }, { "id": 6204, "type": "theorem", "label": "flat-proposition-check-h", "categories": [ "flat" ], "title": "flat-proposition-check-h", "contents": [ "Let $\\mathcal{F}$ be a presheaf on one of the sites $(\\Sch/S)_h$", "constructed in Definition \\ref{definition-big-small-h}.", "Then $\\mathcal{F}$ is a sheaf if and only if the following", "conditions are satisfied", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a sheaf for the Zariski topology,", "\\item given a morphism $f : X \\to Y$ of $(\\Sch/S)_h$ with $Y$ affine", "and $f$ surjective, flat, proper, and of finite presentation, then", "$\\mathcal{F}(Y)$ is the equalizer of the two maps", "$\\mathcal{F}(X) \\to \\mathcal{F}(X \\times_Y X)$,", "\\item given an almost blow up square (\\ref{equation-almost-blow-up-square})", "with $X$ affine in the category $(\\Sch/S)_h$ the diagram", "$$", "\\xymatrix{", "\\mathcal{F}(E) & \\mathcal{F}(X') \\ar[l] \\\\", "\\mathcal{F}(Z) \\ar[u] & \\mathcal{F}(X) \\ar[u] \\ar[l]", "}", "$$", "is cartesian in the category of sets.", "\\end{enumerate}" ], "refs": [ "flat-definition-big-small-h" ], "proofs": [ { "contents": [ "Assume $\\mathcal{F}$ is a sheaf. Condition (1) holds because", "a Zariski covering is a h covering, see", "Lemma \\ref{lemma-zariski-h}.", "Condition (2) holds because for $f$ as in (2) we have that", "$\\{X \\to Y\\}$ is an fppf covering (this is clear)", "and hence an h covering, see Lemma \\ref{lemma-zariski-h}.", "Condition (3) holds by Lemma \\ref{lemma-blow-up-square-h}.", "\\medskip\\noindent", "Conversely, assume $\\mathcal{F}$ satisfies (1), (2), and (3).", "We will prove $\\mathcal{F}$ is a sheaf by applying", "Lemma \\ref{lemma-characterize-sheaf-h}. Consider", "a surjective, finitely presented, proper morphism", "$f : X \\to Y$ in $(\\Sch/S)_h$ with $Y$ affine. It suffices to show", "that $\\mathcal{F}(Y)$ is the equalizer of the two maps", "$\\mathcal{F}(X) \\to \\mathcal{F}(X \\times_Y X)$.", "\\medskip\\noindent", "First, assume that $f : X \\to Y$ is in addition a closed immersion", "(in other words, $f$ is a thickening). Then the blow up of $Y$ in $X$", "is the empty scheme and this produces an almost blow up square", "consisting with $\\emptyset, \\emptyset, X, Y$ at the vertices", "(compare with the second proof of Lemma \\ref{lemma-thickening-h}).", "Hence we see that condition (3) tells us that", "$$", "\\xymatrix{", "\\mathcal{F}(\\emptyset) & \\mathcal{F}(\\emptyset) \\ar[l] \\\\", "\\mathcal{F}(X) \\ar[u] & \\mathcal{F}(Y) \\ar[u] \\ar[l]", "}", "$$", "is cartesian in the category of sets. Since $\\mathcal{F}$ is a sheaf", "for the Zariski topology, we see that $\\mathcal{F}(\\emptyset)$", "is a singleton. Hence we see that $\\mathcal{F}(X) = \\mathcal{F}(Y)$.", "\\medskip\\noindent", "Interlude A: let $T \\to T'$ be a morphism of $(\\Sch/S)_h$ which is", "a thickening and of finite presentation. Then", "$\\mathcal{F}(T') \\to \\mathcal{F}(T)$ is bijective.", "Namely, choose an affine open covering $T' = \\bigcup T'_i$", "and let $T_i = T \\times_{T'} T'_i$ be the corresponding affine", "opens of $T$. Then we have $\\mathcal{F}(T'_i) \\to \\mathcal{F}(T_i)$", "is bijective for all $i$ by the result of the previous paragraph.", "Using the Zariski sheaf property we see that", "$\\mathcal{F}(T') \\to \\mathcal{F}(T)$ is injective. Repeating the", "argument we find that it is bijective. Minor details omitted.", "\\medskip\\noindent", "Interlude B: consider an almost blow up square", "(\\ref{equation-almost-blow-up-square}) in the category $(\\Sch/S)_h$.", "Then we claim the diagram", "$$", "\\xymatrix{", "\\mathcal{F}(E) & \\mathcal{F}(X') \\ar[l] \\\\", "\\mathcal{F}(Z) \\ar[u] & \\mathcal{F}(X) \\ar[u] \\ar[l]", "}", "$$", "is cartesian in the category of sets. This is a consequence of condition", "(3) as follows by choosing an affine open covering of $X$ and arguing", "as in Interlude A. We omit the details.", "\\medskip\\noindent", "Next, let $f : X \\to Y$ be a surjective, finitely presented, proper morphism", "in $(\\Sch/S)_h$ with $Y$ affine. Choose a generic flatness stratification", "$$", "Y \\supset Y_0 \\supset Y_1 \\supset \\ldots \\supset Y_t = \\emptyset", "$$", "as in Lemma \\ref{lemma-generic-flatness-stratification} for $f : X \\to Y$.", "We are going to use all the properties of the stratification", "without further mention. Set $X_0 = X \\times_Y Y_0$.", "By the Interlude B we have $\\mathcal{F}(Y_0) = \\mathcal{F}(Y)$,", "$\\mathcal{F}(X_0) = \\mathcal{F}(X)$, and", "$\\mathcal{F}(X_0 \\times_{Y_0} X_0) = \\mathcal{F}(X \\times_Y X)$.", "\\medskip\\noindent", "We are going to prove the result by induction on $t$. If $t = 1$", "then $X_0 \\to Y_0$ is surjective, proper, flat, and of finite presentation", "and we see that the result holds by property (2).", "For $t > 1$ we may replace $Y$ by $Y_0$ and $X$ by $X_0$", "(see above) and assume $Y = Y_0$.", "\\medskip\\noindent", "Consider the quasi-compact open subscheme", "$V = Y \\setminus Y_1 = Y_0 \\setminus Y_1$.", "Choose a diagram", "$$", "\\xymatrix{", "E \\ar[ddd] \\ar[rd] & & & D \\ar[lll] \\ar[ddd] \\ar[ld] \\\\", "& Y' \\ar[d] & X' \\ar[l] \\ar[d] \\\\", "& Y & X \\ar[l] \\\\", "Z \\ar[ru] & & & T \\ar[lll] \\ar[lu]", "}", "$$", "as in Lemma \\ref{lemma-flat-after-almost-blowing-up}", "for $f : X \\to Y \\supset V$. Then $f' : X' \\to Y'$ is flat and of", "finite presentation. Also $f'$ is proper (use", "Morphisms, Lemmas \\ref{morphisms-lemma-composition-proper} and", "\\ref{morphisms-lemma-image-proper-scheme-closed} to see this).", "Thus the image $W = f'(X') \\subset Y'$ is an open", "(Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}) and closed", "subscheme of $Y'$. Observe that $Y' \\setminus E$ is contained", "in $W$. By Lemma \\ref{lemma-shrink-almost-blow-up}", "this means we may replace $Y'$ by $W$", "in the above diagram. In other words, we may and do", "assume $f'$ is surjective. At this point we know that", "$$", "\\vcenter{", "\\xymatrix{", "\\mathcal{F}(E) & \\mathcal{F}(Y') \\ar[l] \\\\", "\\mathcal{F}(Z) \\ar[u] & \\mathcal{F}(Y) \\ar[u] \\ar[l]", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "\\mathcal{F}(D) & \\mathcal{F}(X') \\ar[l] \\\\", "\\mathcal{F}(T) \\ar[u] & \\mathcal{F}(X) \\ar[u] \\ar[l]", "}", "}", "$$", "are cartesian by Interlude B. Note that", "$Z \\cap Y_1 \\to Z$ is a thickening of finite", "presentation (as $Z$ is set theoretically contained in $Y_1$", "as a closed subscheme of $Y$ disjoint from $V$).", "Thus we obtain a filtration", "$$", "Z \\supset Z \\cap Y_1 \\supset Z \\cap Y_2 \\subset \\ldots \\subset", "Z \\cap Y_t = \\emptyset", "$$", "as above for the restriction $T = Z \\times_Y X \\to Z$ of $f$ to $T$.", "Thus by induction hypothesis we find that", "$\\mathcal{F}(Z) \\to \\mathcal{F}(T)$", "is an injective map of sets whose image is the equalizer", "of the two maps $\\mathcal{F}(T) \\to", "\\mathcal{F}(T \\times_Z T)$.", "\\medskip\\noindent", "Let $s \\in \\mathcal{F}(X)$ be in the equalizer of the two maps", "$\\mathcal{F}(X) \\to \\mathcal{F}(X \\times_Y X)$.", "By the above we see that the restriction $s|_T$", "comes from a unique element $t \\in \\mathcal{F}(Z)$", "and similarly that the restriction $s|_{X'}$", "comes from a unique element $t' \\in \\mathcal{F}(Y')$.", "Chasing sections using the restriction maps for $\\mathcal{F}$", "corresponding to the arrows in the huge commutative diagram above", "the reader finds that $t$ and $t'$ restrict to the same element of", "$\\mathcal{F}(E)$ because they restrict to the same element of", "$\\mathcal{F}(D)$ and we have (2); here we use that $D \\to E$ is", "surjective, flat, proper, and of finite presentation as the restriction", "of $X' \\to Y'$. Thus by the first of the two", "cartesian squares displayed above we get a unique section", "$u \\in \\mathcal{F}(Y)$", "restricting to $t$ and $t'$ on $Z$ and $Y'$.", "To see that $u$ restrict to $s$ on $X$ use the second diagram." ], "refs": [ "flat-lemma-zariski-h", "flat-lemma-zariski-h", "flat-lemma-blow-up-square-h", "flat-lemma-characterize-sheaf-h", "flat-lemma-thickening-h", "flat-lemma-generic-flatness-stratification", "flat-lemma-flat-after-almost-blowing-up", "morphisms-lemma-composition-proper", "morphisms-lemma-image-proper-scheme-closed", "morphisms-lemma-fppf-open", "flat-lemma-shrink-almost-blow-up" ], "ref_ids": [ 6142, 6142, 6163, 6150, 6164, 6084, 6162, 5408, 5411, 5267, 6158 ] } ], "ref_ids": [ 6223 ] }, { "id": 6205, "type": "theorem", "label": "flat-proposition-h-descent-vector-bundles-p", "categories": [ "flat" ], "title": "flat-proposition-h-descent-vector-bundles-p", "contents": [ "Let $p$ be a prime number. Let $S$ be a scheme in characteristic $p$.", "Then the category fibred in groupoids", "$$", "p : \\mathcal{S} \\longrightarrow (\\Sch/S)_h", "$$", "whose fibre category over $U$ is the category", "of finite locally free $\\colim_F \\mathcal{O}_U$-modules over $U$", "is a stack in groupoids. Moreover, if $U$ is quasi-compact", "and quasi-separated, then $\\mathcal{S}_U$ is $\\colim_F \\textit{Vect}(U)$." ], "refs": [], "proofs": [ { "contents": [ "The final assertion is the content of Lemma \\ref{lemma-colim-F-Vect}.", "To prove the proposition we will check conditions (1), (2), and (3) of", "Lemma \\ref{lemma-refine-check-h-stack}.", "\\medskip\\noindent", "Condition (1) holds because by definition we have glueing", "for the Zariski topology.", "\\medskip\\noindent", "To see condition (2), suppose that $f : X \\to Y$ is a surjective,", "flat, proper morphism of finite presentation over $S$ with $Y$ affine.", "Since $Y, X, X \\times_Y X$ are quasi-compact and quasi-separated,", "we can use the description of fibre categories given in the", "statement of the proposition. Then it is clearly enough to", "show that", "$$", "\\textit{Vect}(Y)", "\\longrightarrow", "\\textit{Vect}(X)", "\\times_{\\textit{Vect}(X \\times_Y X)}", "\\textit{Vect}(X)", "$$", "is an equivalence (as this will imply the same for the colimits).", "This follows immediately from fppf descent of finite locally free modules, see", "Descent, Proposition \\ref{descent-proposition-fpqc-descent-quasi-coherent} and", "Lemma \\ref{descent-lemma-finite-locally-free-descends}.", "\\medskip\\noindent", "Condition (3) is the content of", "Lemmas \\ref{lemma-vector-bundle-I} and \\ref{lemma-vector-bundle-II}." ], "refs": [ "flat-lemma-colim-F-Vect", "flat-lemma-refine-check-h-stack", "descent-proposition-fpqc-descent-quasi-coherent", "descent-lemma-finite-locally-free-descends", "flat-lemma-vector-bundle-I", "flat-lemma-vector-bundle-II" ], "ref_ids": [ 6174, 6166, 14753, 14617, 6175, 6176 ] } ], "ref_ids": [] }, { "id": 6238, "type": "theorem", "label": "curves-theorem-curves-rational-maps", "categories": [ "curves" ], "title": "curves-theorem-curves-rational-maps", "contents": [ "Let $k$ be a field. The following categories are canonically equivalent", "\\begin{enumerate}", "\\item The category of finitely generated field extensions $K/k$ of", "transcendence degree $1$.", "\\item The category of curves and dominant rational maps.", "\\item The category of normal projective curves and nonconstant morphisms.", "\\item The category of nonsingular projective curves and nonconstant morphisms.", "\\item The category of regular projective curves and nonconstant morphisms.", "\\item The category of normal proper curves and nonconstant morphisms.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence between categories (1) and (2) is the restriction of the", "equivalence of", "Varieties, Theorem \\ref{varieties-theorem-varieties-rational-maps}.", "Namely, a variety is a curve if and only if its function field has", "transcendence degree $1$, see for example", "Varieties, Lemma \\ref{varieties-lemma-dimension-locally-algebraic}.", "\\medskip\\noindent", "The categories in (3), (4), (5), and (6) are the same. First of all, the", "terms ``regular'' and ``nonsingular'' are synonyms, see", "Properties, Definition \\ref{properties-definition-regular}.", "Being normal and regular are the same thing for Noetherian", "$1$-dimensional schemes", "(Properties, Lemmas \\ref{properties-lemma-regular-normal} and", "\\ref{properties-lemma-normal-dimension-1-regular}). See", "Varieties, Lemma \\ref{varieties-lemma-regular-point-on-curve}", "for the case of curves. Thus (3) is the same as (5). Finally, (6)", "is the same as (3) by", "Varieties, Lemma \\ref{varieties-lemma-dim-1-proper-projective}.", "\\medskip\\noindent", "If $f : X \\to Y$ is a nonconstant morphism of nonsingular projective curves,", "then $f$ sends the generic point $\\eta$ of $X$ to the generic point $\\xi$ of", "$Y$. Hence we obtain a morphism", "$k(Y) = \\mathcal{O}_{Y, \\xi} \\to \\mathcal{O}_{X, \\eta} = k(X)$", "in the category (1). If two morphisms $f,g: X \\to Y$ gives the same morphism", "$k(Y) \\to k(X)$, then by the equivalence between (1) and (2),", "$f$ and $g$ are equivalent as rational maps, so $f=g$ by", "Lemma \\ref{lemma-extend-over-normal-curve}.", "Conversely, suppose that we have a map", "$k(Y) \\to k(X)$ in the category (1). Then we obtain a morphism $U \\to Y$", "for some nonempty open $U \\subset X$. By Lemma \\ref{lemma-extend-over-dvr}", "this extends to all of $X$ and we obtain a morphism in the category (5).", "Thus we see that there is a fully faithful functor (5)$\\to$(1).", "\\medskip\\noindent", "To finish the proof we have to show that every $K/k$ in (1)", "is the function field of a normal projective curve.", "We already know that $K = k(X)$ for some curve $X$.", "After replacing $X$ by its normalization", "(which is a variety birational to $X$)", "we may assume $X$ is normal", "(Varieties, Lemma \\ref{varieties-lemma-normalization-locally-algebraic}).", "Then we choose $X \\to \\overline{X}$ with", "$\\overline{X} \\setminus X = \\{x_1, \\ldots, x_n\\}$ as in", "Varieties, Lemma \\ref{varieties-lemma-reduced-dim-1-projective-completion}.", "Since $X$ is normal and since each", "of the local rings $\\mathcal{O}_{\\overline{X}, x_i}$ is normal", "we conclude that $\\overline{X}$ is a normal projective curve as desired.", "(Remark: We can also first compactify using", "Varieties, Lemma \\ref{varieties-lemma-dim-1-projective-completion}", "and then normalize using", "Varieties, Lemma \\ref{varieties-lemma-normalization-locally-algebraic}.", "Doing it this way we avoid using the somewhat tricky", "Morphisms, Lemma \\ref{morphisms-lemma-relative-normalization-normal-codim-1}.)" ], "refs": [ "varieties-theorem-varieties-rational-maps", "varieties-lemma-dimension-locally-algebraic", "properties-definition-regular", "properties-lemma-regular-normal", "properties-lemma-normal-dimension-1-regular", "varieties-lemma-regular-point-on-curve", "varieties-lemma-dim-1-proper-projective", "curves-lemma-extend-over-normal-curve", "curves-lemma-extend-over-dvr", "varieties-lemma-normalization-locally-algebraic", "varieties-lemma-reduced-dim-1-projective-completion", "varieties-lemma-dim-1-projective-completion", "varieties-lemma-normalization-locally-algebraic", "morphisms-lemma-relative-normalization-normal-codim-1" ], "ref_ids": [ 10900, 10989, 3075, 2977, 2990, 11118, 11099, 6240, 6239, 11013, 11101, 11100, 11013, 5511 ] } ], "ref_ids": [] }, { "id": 6239, "type": "theorem", "label": "curves-lemma-extend-over-dvr", "categories": [ "curves" ], "title": "curves-lemma-extend-over-dvr", "contents": [ "Let $k$ be a field. Let $X$ be a curve and $Y$ a proper variety.", "Let $U \\subset X$ be a nonempty open and let $f : U \\to Y$ be a morphism.", "If $x \\in X$ is a closed point such that $\\mathcal{O}_{X, x}$", "is a discrete valuation ring, then there exists an open", "$U \\subset U' \\subset X$ containing $x$ and a morphism of", "varieties $f' : U' \\to Y$ extending $f$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Morphisms, Lemma \\ref{morphisms-lemma-extend-across}." ], "refs": [ "morphisms-lemma-extend-across" ], "ref_ids": [ 5419 ] } ], "ref_ids": [] }, { "id": 6240, "type": "theorem", "label": "curves-lemma-extend-over-normal-curve", "categories": [ "curves" ], "title": "curves-lemma-extend-over-normal-curve", "contents": [ "Let $k$ be a field. Let $X$ be a normal curve and $Y$ a proper variety.", "The set of rational maps from $X$ to $Y$ is the same as the set", "of morphisms $X \\to Y$." ], "refs": [], "proofs": [ { "contents": [ "A rational map from $X$ to $Y$ can be extended to a morphism $X \\to Y$", "by Lemma \\ref{lemma-extend-over-dvr}", "as every local ring is a discrete valuation ring", "(for example by Varieties, Lemma \\ref{varieties-lemma-regular-point-on-curve}).", "Conversely, if two morphisms $f,g: X \\to Y$ are equivalent as rational maps,", "then $f = g$ by Morphisms, Lemma \\ref{morphisms-lemma-equality-of-morphisms}." ], "refs": [ "curves-lemma-extend-over-dvr", "varieties-lemma-regular-point-on-curve", "morphisms-lemma-equality-of-morphisms" ], "ref_ids": [ 6239, 11118, 5157 ] } ], "ref_ids": [] }, { "id": 6241, "type": "theorem", "label": "curves-lemma-flat", "categories": [ "curves" ], "title": "curves-lemma-flat", "contents": [ "Let $k$ be a field. Let $f : X \\to Y$ be a nonconstant morphism", "of curves over $k$. If $Y$ is normal, then $f$ is flat." ], "refs": [], "proofs": [ { "contents": [ "Pick $x \\in X$ mapping to $y \\in Y$. Then $\\mathcal{O}_{Y, y}$ is either a", "field or a discrete valuation ring", "(Varieties, Lemma \\ref{varieties-lemma-regular-point-on-curve}).", "Since $f$ is nonconstant it is dominant (as it must map the", "generic point of $X$ to the generic point of $Y$). This implies that", "$\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ is injective", "(Morphisms, Lemma \\ref{morphisms-lemma-dominant-between-integral}).", "Hence $\\mathcal{O}_{X, x}$ is torsion free as a $\\mathcal{O}_{Y, y}$-module", "and therefore $\\mathcal{O}_{X, x}$ is flat as a $\\mathcal{O}_{Y, y}$-module", "by More on Algebra, Lemma", "\\ref{more-algebra-lemma-valuation-ring-torsion-free-flat}." ], "refs": [ "varieties-lemma-regular-point-on-curve", "morphisms-lemma-dominant-between-integral", "more-algebra-lemma-valuation-ring-torsion-free-flat" ], "ref_ids": [ 11118, 5162, 9920 ] } ], "ref_ids": [] }, { "id": 6242, "type": "theorem", "label": "curves-lemma-finite", "categories": [ "curves" ], "title": "curves-lemma-finite", "contents": [ "Let $k$ be a field. Let $f : X \\to Y$ be a morphism of", "schemes over $k$. Assume", "\\begin{enumerate}", "\\item $Y$ is separated over $k$,", "\\item $X$ is proper of dimension $\\leq 1$ over $k$,", "\\item $f(Z)$ has at least two points for every irreducible", "component $Z \\subset X$ of dimension $1$.", "\\end{enumerate}", "Then $f$ is finite." ], "refs": [], "proofs": [ { "contents": [ "The morphism $f$ is proper by", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}.", "Thus $f(X)$ is closed and images of closed points are closed.", "Let $y \\in Y$ be the image of a closed point in $X$.", "Then $f^{-1}(\\{y\\})$ is a closed subset of $X$ not", "containing any of the generic points of irreducible components", "of dimension $1$ by condition (3). It follows that $f^{-1}(\\{y\\})$", "is finite. Hence $f$ is finite over an open neighbourhood of $y$", "by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood}", "(if $Y$ is Noetherian, then you can use the easier", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-finite-fibre-finite-in-neighbourhood}).", "Since we've seen above that there are enough of these points", "$y$, the proof is complete." ], "refs": [ "morphisms-lemma-image-proper-scheme-closed", "more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood", "coherent-lemma-proper-finite-fibre-finite-in-neighbourhood" ], "ref_ids": [ 5411, 13904, 3366 ] } ], "ref_ids": [] }, { "id": 6243, "type": "theorem", "label": "curves-lemma-extend-to-completion", "categories": [ "curves" ], "title": "curves-lemma-extend-to-completion", "contents": [ "Let $k$ be a field. Let $X \\to Y$ be a morphism of varieties", "with $Y$ proper and $X$ a curve.", "There exists a factorization $X \\to \\overline{X} \\to Y$", "where $X \\to \\overline{X}$ is an open immersion", "and $\\overline{X}$ is a projective curve." ], "refs": [], "proofs": [ { "contents": [ "This is clear from Lemma \\ref{lemma-extend-over-dvr}", "and Varieties, Lemma \\ref{varieties-lemma-reduced-dim-1-projective-completion}." ], "refs": [ "curves-lemma-extend-over-dvr", "varieties-lemma-reduced-dim-1-projective-completion" ], "ref_ids": [ 6239, 11101 ] } ], "ref_ids": [] }, { "id": 6244, "type": "theorem", "label": "curves-lemma-nonsingular-model-smooth", "categories": [ "curves" ], "title": "curves-lemma-nonsingular-model-smooth", "contents": [ "Let $k$ be a field. Let $X$ be a curve and let $Y$ be the nonsingular", "projective model of $X$. If $k$ is perfect, then $Y$ is a smooth", "projective curve." ], "refs": [], "proofs": [ { "contents": [ "See Varieties, Lemma \\ref{varieties-lemma-regular-point-on-curve}", "for example." ], "refs": [ "varieties-lemma-regular-point-on-curve" ], "ref_ids": [ 11118 ] } ], "ref_ids": [] }, { "id": 6245, "type": "theorem", "label": "curves-lemma-smooth-models", "categories": [ "curves" ], "title": "curves-lemma-smooth-models", "contents": [ "Let $k$ be a field. Let $X$ be a geometrically irreducible curve over $k$.", "For a field extension $K/k$ denote $Y_K$ a nonsingular projective model", "of $(X_K)_{red}$.", "\\begin{enumerate}", "\\item If $X$ is proper, then $Y_K$ is the normalization of $X_K$.", "\\item There exists $K/k$ finite purely inseparable such that $Y_K$ is smooth.", "\\item Whenever $Y_K$ is smooth\\footnote{Or even geometrically reduced.}", "we have $H^0(Y_K, \\mathcal{O}_{Y_K}) = K$.", "\\item Given a commutative diagram", "$$", "\\xymatrix{", "\\Omega & K' \\ar[l] \\\\", "K \\ar[u] & k \\ar[l] \\ar[u]", "}", "$$", "of fields such that $Y_K$ and $Y_{K'}$ are smooth, then", "$Y_\\Omega = (Y_K)_\\Omega = (Y_{K'})_\\Omega$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X'$ be a nonsingular projective model of $X$. Then $X'$ and", "$X$ have isomorphic nonempty open subschemes. In particular", "$X'$ is geometrically irreducible as $X$ is (some details omitted).", "Thus we may assume that $X$ is projective.", "\\medskip\\noindent", "Assume $X$ is proper. Then $X_K$ is proper and hence the normalization", "$(X_K)^\\nu$ is proper as a scheme finite over a proper scheme", "(Varieties, Lemma \\ref{varieties-lemma-normalization-locally-algebraic}", "and Morphisms, Lemmas \\ref{morphisms-lemma-finite-proper} and", "\\ref{morphisms-lemma-composition-proper}).", "On the other hand, $X_K$ is irreducible as $X$ is geometrically", "irreducible. Hence $X_K^\\nu$ is proper, normal, irreducible, and birational", "to $(X_K)_{red}$. This proves (1) because a proper curve is projective", "(Varieties, Lemma \\ref{varieties-lemma-dim-1-proper-projective}).", "\\medskip\\noindent", "Proof of (2). As $X$ is proper and we have (1), we can apply", "Varieties, Lemma \\ref{varieties-lemma-finite-extension-geometrically-normal}", "to find $K/k$ finite purely inseparable such that", "$Y_K$ is geometrically normal. Then $Y_K$ is geometrically regular", "as normal and regular are the same for curves", "(Properties, Lemma \\ref{properties-lemma-normal-dimension-1-regular}).", "Then $Y$ is a smooth variety by", "Varieties, Lemma \\ref{varieties-lemma-geometrically-regular-smooth}.", "\\medskip\\noindent", "If $Y_K$ is geometrically reduced, then $Y_K$ is geometrically", "integral (Varieties, Lemma \\ref{varieties-lemma-geometrically-integral})", "and we see that $H^0(Y_K, \\mathcal{O}_{Y_K}) = K$ by", "Varieties, Lemma \\ref{varieties-lemma-regular-functions-proper-variety}.", "This proves (3) because a smooth variety is geometrically reduced", "(even geometrically regular, see", "Varieties, Lemma \\ref{varieties-lemma-geometrically-regular-smooth}).", "\\medskip\\noindent", "If $Y_K$ is smooth, then for every extension $\\Omega/K$", "the base change $(Y_K)_\\Omega$ is smooth over $\\Omega$", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-smooth}).", "Hence it is clear that $Y_\\Omega = (Y_K)_\\Omega$. This proves (4)." ], "refs": [ "varieties-lemma-normalization-locally-algebraic", "morphisms-lemma-finite-proper", "morphisms-lemma-composition-proper", "varieties-lemma-dim-1-proper-projective", "varieties-lemma-finite-extension-geometrically-normal", "properties-lemma-normal-dimension-1-regular", "varieties-lemma-geometrically-regular-smooth", "varieties-lemma-geometrically-integral", "varieties-lemma-regular-functions-proper-variety", "varieties-lemma-geometrically-regular-smooth", "morphisms-lemma-base-change-smooth" ], "ref_ids": [ 11013, 5445, 5408, 11099, 11015, 2990, 10962, 10947, 11012, 10962, 5327 ] } ], "ref_ids": [] }, { "id": 6246, "type": "theorem", "label": "curves-lemma-linear-series", "categories": [ "curves" ], "title": "curves-lemma-linear-series", "contents": [ "Let $k$ be a field. Let $X$ be a nonsingular proper curve over $k$.", "Let $(\\mathcal{L}, V)$ be a $\\mathfrak g^r_d$ on $X$. Then", "there exists a morphism", "$$", "\\varphi : X \\longrightarrow \\mathbf{P}^r_k = \\text{Proj}(k[T_0, \\ldots, T_r])", "$$", "of varieties over $k$ and a map", "$\\alpha : \\varphi^*\\mathcal{O}_{\\mathbf{P}^r_k}(1) \\to \\mathcal{L}$", "such that $\\varphi^*T_0, \\ldots, \\varphi^*T_r$", "are sent to a basis of $V$ by $\\alpha$." ], "refs": [], "proofs": [ { "contents": [ "Let $s_0, \\ldots, s_r \\in V$ be a $k$-basis. Since $X$ is nonsingular", "the image $\\mathcal{L}' \\subset \\mathcal{L}$ of the map", "$s_0, \\ldots, s_r : \\mathcal{O}_X^{\\oplus r} \\to \\mathcal{L}$", "is an invertible $\\mathcal{O}_X$-module for example by", "Divisors, Lemma \\ref{divisors-lemma-torsion-free-over-regular-dim-1}.", "Then we use", "Constructions, Lemma \\ref{constructions-lemma-projective-space}", "to get a morphism", "$$", "\\varphi = \\varphi_{(\\mathcal{L}', (s_0, \\ldots, s_r))} :", "X \\longrightarrow \\mathbf{P}^r_k", "$$", "as in the statement of the lemma." ], "refs": [ "divisors-lemma-torsion-free-over-regular-dim-1", "constructions-lemma-projective-space" ], "ref_ids": [ 7912, 12621 ] } ], "ref_ids": [] }, { "id": 6247, "type": "theorem", "label": "curves-lemma-linear-series-trivial-existence", "categories": [ "curves" ], "title": "curves-lemma-linear-series-trivial-existence", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$.", "If $X$ has a $\\mathfrak g^r_d$, then $X$ has a $\\mathfrak g^s_d$ for", "all $0 \\leq s \\leq r$." ], "refs": [], "proofs": [ { "contents": [ "This is true because a vector space $V$ of dimension $r + 1$", "over $k$ has a linear subspace of dimension $s + 1$ for all", "$0 \\leq s \\leq r$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6248, "type": "theorem", "label": "curves-lemma-g1d", "categories": [ "curves" ], "title": "curves-lemma-g1d", "contents": [ "Let $k$ be a field. Let $X$ be a nonsingular proper curve over $k$.", "Let $(\\mathcal{L}, V)$ be a $\\mathfrak g^1_d$ on $X$. Then the morphism", "$\\varphi : X \\to \\mathbf{P}^1_k$ of Lemma \\ref{lemma-linear-series}", "either", "\\begin{enumerate}", "\\item is nonconstant and has degree $\\leq d$, or", "\\item factors through a closed point of $\\mathbf{P}^1_k$ and in this", "case $H^0(X, \\mathcal{O}_X) \\not = k$.", "\\end{enumerate}" ], "refs": [ "curves-lemma-linear-series" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-linear-series} we see that", "$\\mathcal{L}' = \\varphi^*\\mathcal{O}_{\\mathbf{P}^1_k}(1)$", "has a nonzero map $\\mathcal{L}' \\to \\mathcal{L}$.", "Hence by Varieties, Lemma \\ref{varieties-lemma-check-invertible-sheaf-trivial}", "we see that $0 \\leq \\deg(\\mathcal{L}') \\leq d$.", "If $\\deg(\\mathcal{L}') = 0$, then the same lemma tells us", "$\\mathcal{L}' \\cong \\mathcal{O}_X$ and since we have", "two linearly independent sections we find we are in case (2).", "If $\\deg(\\mathcal{L}') > 0$ then $\\varphi$ is nonconstant (since the", "pullback of an invertible module by a constant morphism is trivial).", "Hence", "$$", "\\deg(\\mathcal{L}') =", "\\deg(X/\\mathbf{P}^1_k) \\deg(\\mathcal{O}_{\\mathbf{P}^1_k}(1))", "$$", "by Varieties, Lemma \\ref{varieties-lemma-degree-pullback-map-proper-curves}.", "This finishes the proof as the degree of", "$\\mathcal{O}_{\\mathbf{P}^1_k}(1)$ is $1$." ], "refs": [ "curves-lemma-linear-series", "varieties-lemma-check-invertible-sheaf-trivial", "varieties-lemma-degree-pullback-map-proper-curves" ], "ref_ids": [ 6246, 11114, 11113 ] } ], "ref_ids": [ 6246 ] }, { "id": 6249, "type": "theorem", "label": "curves-lemma-grd-inequalities", "categories": [ "curves" ], "title": "curves-lemma-grd-inequalities", "contents": [ "Let $k$ be a field. Let $X$ be a proper curve over $k$ with", "$H^0(X, \\mathcal{O}_X) = k$. If $X$ has a $\\mathfrak g^r_d$, then", "$r \\leq d$. If equality holds, then $H^1(X, \\mathcal{O}_X) = 0$, i.e.,", "the genus of $X$ (Definition \\ref{definition-genus}) is $0$." ], "refs": [ "curves-definition-genus" ], "proofs": [ { "contents": [ "Let $(\\mathcal{L}, V)$ be a $\\mathfrak g^r_d$. Since this will only", "increase $r$, we may assume $V = H^0(X, \\mathcal{L})$. Choose a", "nonzero element $s \\in V$. Then the zero scheme of $s$ is an effective Cartier", "divisor $D \\subset X$, we have $\\mathcal{L} = \\mathcal{O}_X(D)$, and", "we have a short exact sequence", "$$", "0 \\to \\mathcal{O}_X \\to \\mathcal{L} \\to \\mathcal{L}|_D \\to 0", "$$", "see Divisors, Lemma \\ref{divisors-lemma-characterize-OD} and", "Remark \\ref{divisors-remark-ses-regular-section}.", "By Varieties, Lemma \\ref{varieties-lemma-degree-effective-Cartier-divisor}", "we have $\\deg(D) = \\deg(\\mathcal{L}) = d$.", "Since $D$ is an Artinian scheme we have", "$\\mathcal{L}|_D \\cong \\mathcal{O}_D$\\footnote{In our case this", "follows from Divisors, Lemma", "\\ref{divisors-lemma-finite-trivialize-invertible-upstairs}", "as $D \\to \\Spec(k)$ is finite.}.", "Thus", "$$", "\\dim_k H^0(D, \\mathcal{L}|_D) = \\dim_k H^0(D, \\mathcal{O}_D) = \\deg(D) = d", "$$", "On the other hand, by assumption", "$\\dim_k H^0(X, \\mathcal{O}_X) = 1$ and $\\dim H^0(X, \\mathcal{L}) = r + 1$.", "We conclude that $r + 1 \\leq 1 + d$, i.e., $r \\leq d$ as in the lemma.", "\\medskip\\noindent", "Assume equality holds. Then", "$H^0(X, \\mathcal{L}) \\to H^0(X, \\mathcal{L}|_D)$ is surjective.", "If we knew that $H^1(X, \\mathcal{L})$ was zero, then we would", "conclude that $H^1(X, \\mathcal{O}_X)$ is zero by the long exact", "cohomology sequence and the proof would", "be complete. Our strategy will be to replace $\\mathcal{L}$ by a", "large power which has vanishing. As $\\mathcal{L}|_D$ is the", "trivial invertible module (see above), we can", "find a section $t$ of $\\mathcal{L}$ whose restriction", "of $D$ generates $\\mathcal{L}|_D$.", "Consider the multiplication map", "$$", "\\mu :", "H^0(X, \\mathcal{L}) \\otimes_k H^0(X, \\mathcal{L})", "\\longrightarrow", "H^0(X, \\mathcal{L}^{\\otimes 2})", "$$", "and consider the short exact sequence", "$$", "0 \\to \\mathcal{L} \\xrightarrow{s}", "\\mathcal{L}^{\\otimes 2} \\to \\mathcal{L}^{\\otimes 2}|_D \\to 0", "$$", "Since $H^0(\\mathcal{L}) \\to H^0(\\mathcal{L}|_D)$ is surjective and since", "$t$ maps to a trivialization of $\\mathcal{L}|_D$ we see that", "$\\mu(H^0(X, \\mathcal{L}) \\otimes t)$ gives a subspace of", "$H^0(X, \\mathcal{L}^{\\otimes 2})$ surjecting onto the global sections of", "$\\mathcal{L}^{\\otimes 2}|_D$. Thus we see that", "$$", "\\dim H^0(X, \\mathcal{L}^{\\otimes 2}) = r + 1 + d = 2r + 1 =", "\\deg(\\mathcal{L}^{\\otimes 2}) + 1", "$$", "Ok, so $\\mathcal{L}^{\\otimes 2}$ has the same property as $\\mathcal{L}$, i.e.,", "that the dimension of the space of global sections is equal to the", "degree plus one. Since $\\mathcal{L}$ is ample", "(Varieties, Lemma \\ref{varieties-lemma-ample-curve})", "there exists some $n_0$ such that $\\mathcal{L}^{\\otimes n}$", "has vanishing $H^1$ for all $n \\geq n_0$", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-proper-ample}).", "Thus applying the argument above to $\\mathcal{L}^{\\otimes n}$", "with $n = 2^m$ for some sufficiently large $m$ we conclude the", "lemma is true." ], "refs": [ "divisors-lemma-characterize-OD", "varieties-lemma-degree-effective-Cartier-divisor", "divisors-lemma-finite-trivialize-invertible-upstairs", "varieties-lemma-ample-curve", "coherent-lemma-coherent-proper-ample" ], "ref_ids": [ 7944, 11111, 7963, 11116, 3343 ] } ], "ref_ids": [ 6355 ] }, { "id": 6250, "type": "theorem", "label": "curves-lemma-duality-dim-1", "categories": [ "curves" ], "title": "curves-lemma-duality-dim-1", "contents": [ "Let $X$ be a proper scheme of dimension $\\leq 1$ over a field $k$.", "There exists a dualizing complex $\\omega_X^\\bullet$ with the", "following properties", "\\begin{enumerate}", "\\item $H^i(\\omega_X^\\bullet)$ is nonzero only for $i = -1, 0$,", "\\item $\\omega_X = H^{-1}(\\omega_X^\\bullet)$", "is a coherent Cohen-Macaulay module whose support is the", "irreducible components of dimension $1$,", "\\item for $x \\in X$ closed, the module $H^0(\\omega_{X, x}^\\bullet)$", "is nonzero if and only if either", "\\begin{enumerate}", "\\item $\\dim(\\mathcal{O}_{X, x}) = 0$ or", "\\item $\\dim(\\mathcal{O}_{X, x}) = 1$", "and $\\mathcal{O}_{X, x}$ is not Cohen-Macaulay,", "\\end{enumerate}", "\\item for $K \\in D_\\QCoh(\\mathcal{O}_X)$ there are functorial", "isomorphisms\\footnote{This property", "characterizes $\\omega_X^\\bullet$ in $D_\\QCoh(\\mathcal{O}_X)$", "up to unique isomorphism by the Yoneda lemma. Since $\\omega_X^\\bullet$", "is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ in fact it suffices to consider", "$K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.}", "$$", "\\Ext^i_X(K, \\omega_X^\\bullet) = \\Hom_k(H^{-i}(X, K), k)", "$$", "compatible with shifts and distinguished triangles,", "\\item there are functorial isomorphisms", "$\\Hom(\\mathcal{F}, \\omega_X) = \\Hom_k(H^1(X, \\mathcal{F}), k)$", "for $\\mathcal{F}$ quasi-coherent on $X$,", "\\item if $X \\to \\Spec(k)$ is smooth of relative dimension $1$,", "then $\\omega_X \\cong \\Omega_{X/k}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $f : X \\to \\Spec(k)$ the structure morphism.", "We start with the relative dualizing complex", "$$", "\\omega_X^\\bullet = \\omega_{X/k}^\\bullet = a(\\mathcal{O}_{\\Spec(k)})", "$$", "as described in Duality for Schemes,", "Remark \\ref{duality-remark-relative-dualizing-complex}.", "Then property (4) holds by construction as $a$ is the right", "adjoint for $f_* : D_\\QCoh(\\mathcal{O}_X) \\to D(\\mathcal{O}_{\\Spec(k)})$.", "Since $f$ is proper we have", "$f^!(\\mathcal{O}_{\\Spec(k)}) = a(\\mathcal{O}_{\\Spec(k)})$ by", "definition, see", "Duality for Schemes, Section \\ref{duality-section-upper-shriek}.", "Hence $\\omega_X^\\bullet$ and $\\omega_X$ are as in", "Duality for Schemes, Example \\ref{duality-example-proper-over-local}", "and as in", "Duality for Schemes, Example \\ref{duality-example-equidimensional-over-field}.", "Parts (1) and (2) follow from", "Duality for Schemes, Lemma \\ref{duality-lemma-vanishing-good-dualizing}.", "For a closed point $x \\in X$ we see that $\\omega_{X, x}^\\bullet$ is a", "normalized dualizing complex over $\\mathcal{O}_{X, x}$, see", "Duality for Schemes, Lemma \\ref{duality-lemma-good-dualizing-normalized}.", "Assertion (3) then follows from", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-apply-CM}.", "Assertion (5) follows from", "Duality for Schemes, Lemma \\ref{duality-lemma-dualizing-module-proper-over-A}", "for coherent $\\mathcal{F}$ and in general by unwinding", "(4) for $K = \\mathcal{F}[0]$ and $i = -1$.", "Assertion (6) follows from Duality for Schemes,", "Lemma \\ref{duality-lemma-smooth-proper}." ], "refs": [ "duality-remark-relative-dualizing-complex", "duality-lemma-vanishing-good-dualizing", "duality-lemma-good-dualizing-normalized", "dualizing-lemma-apply-CM", "duality-lemma-dualizing-module-proper-over-A", "duality-lemma-smooth-proper" ], "ref_ids": [ 13649, 13584, 13576, 2875, 13585, 13550 ] } ], "ref_ids": [] }, { "id": 6251, "type": "theorem", "label": "curves-lemma-duality-dim-1-CM", "categories": [ "curves" ], "title": "curves-lemma-duality-dim-1-CM", "contents": [ "Let $X$ be a proper scheme over a field $k$ which is Cohen-Macaulay", "and equidimensional of dimension $1$. The module $\\omega_X$", "of Lemma \\ref{lemma-duality-dim-1} has the following properties", "\\begin{enumerate}", "\\item $\\omega_X$ is a dualizing module on $X$", "(Duality for Schemes, Section \\ref{duality-section-dualizing-module}),", "\\item $\\omega_X$ is a coherent Cohen-Macaulay module whose support is $X$,", "\\item there are functorial isomorphisms", "$\\Ext^i_X(K, \\omega_X[1]) = \\Hom_k(H^{-i}(X, K), k)$", "compatible with shifts for $K \\in D_\\QCoh(X)$,", "\\item there are functorial isomorphisms", "$\\Ext^{1 + i}(\\mathcal{F}, \\omega_X) = \\Hom_k(H^{-i}(X, \\mathcal{F}), k)$", "for $\\mathcal{F}$ quasi-coherent on $X$.", "\\end{enumerate}" ], "refs": [ "curves-lemma-duality-dim-1" ], "proofs": [ { "contents": [ "Recall from the proof of Lemma \\ref{lemma-duality-dim-1}", "that $\\omega_X$ is as in Duality for Schemes, Example", "\\ref{duality-example-proper-over-local} and hence is", "a dualizing module. The other statements follow from", "Lemma \\ref{lemma-duality-dim-1}", "and the fact that $\\omega_X^\\bullet = \\omega_X[1]$", "as $X$ is Cohen-Macualay (Duality for Schemes, Lemma", "\\ref{duality-lemma-dualizing-module-CM-scheme})." ], "refs": [ "curves-lemma-duality-dim-1", "curves-lemma-duality-dim-1", "duality-lemma-dualizing-module-CM-scheme" ], "ref_ids": [ 6250, 6250, 13586 ] } ], "ref_ids": [ 6250 ] }, { "id": 6252, "type": "theorem", "label": "curves-lemma-sanity-check-duality", "categories": [ "curves" ], "title": "curves-lemma-sanity-check-duality", "contents": [ "Let $X$ be a proper scheme of dimension $\\leq 1$ over a field $k$.", "Let $\\omega_X^\\bullet$ and $\\omega_X$ be as in Lemma \\ref{lemma-duality-dim-1}.", "\\begin{enumerate}", "\\item If $X \\to \\Spec(k)$ factors as $X \\to \\Spec(k') \\to \\Spec(k)$", "for some field $k'$, then $\\omega_X^\\bullet$ and $\\omega_X$", "satisfy properties (4), (5), (6) with $k$ replaced with $k'$.", "\\item If $K/k$ is a field extension, then the pullback of", "$\\omega_X^\\bullet$ and $\\omega_X$ to the base change $X_K$", "are as in Lemma \\ref{lemma-duality-dim-1} for the morphism", "$X_K \\to \\Spec(K)$.", "\\end{enumerate}" ], "refs": [ "curves-lemma-duality-dim-1", "curves-lemma-duality-dim-1" ], "proofs": [ { "contents": [ "Denote $f : X \\to \\Spec(k)$ the structure morphism.", "Assertion (1) really means that $\\omega_X^\\bullet$ and $\\omega_X$", "are as in Lemma \\ref{lemma-duality-dim-1} for the morphism", "$f' : X \\to \\Spec(k')$. In the proof of Lemma \\ref{lemma-duality-dim-1}", "we took $\\omega_X^\\bullet = a(\\mathcal{O}_{\\Spec(k)})$", "where $a$ be is the right adjoint of", "Duality for Schemes, Lemma", "\\ref{duality-lemma-twisted-inverse-image} for $f$.", "Thus we have to show", "$a(\\mathcal{O}_{\\Spec(k)}) \\cong a'(\\mathcal{O}_{\\Spec(k)})$", "where $a'$ be is the right adjoint of", "Duality for Schemes, Lemma", "\\ref{duality-lemma-twisted-inverse-image} for $f'$.", "Since $k' \\subset H^0(X, \\mathcal{O}_X)$ we see that $k'/k$ is a finite", "extension (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-over-affine-cohomology-finite}).", "By uniqueness of adjoints we have $a = a' \\circ b$ where", "$b$ is the right adjoint of Duality for Schemes, Lemma", "\\ref{duality-lemma-twisted-inverse-image} for $g : \\Spec(k') \\to \\Spec(k)$.", "Another way to say this: we have $f^! = (f')^! \\circ g^!$.", "Thus it suffices to show that $\\Hom_k(k', k) \\cong k'$ as", "$k'$-modules, see Duality for Schemes, Example", "\\ref{duality-example-affine-twisted-inverse-image}.", "This holds because these are $k'$-vector spaces of", "the same dimension (namely dimension $1$).", "\\medskip\\noindent", "Proof of (2). This holds because we have base change for $a$ by", "Duality for Schemes, Lemma \\ref{duality-lemma-more-base-change}.", "See discussion in Duality for Schemes, Remark", "\\ref{duality-remark-relative-dualizing-complex}." ], "refs": [ "curves-lemma-duality-dim-1", "curves-lemma-duality-dim-1", "duality-lemma-twisted-inverse-image", "duality-lemma-twisted-inverse-image", "coherent-lemma-proper-over-affine-cohomology-finite", "duality-lemma-twisted-inverse-image", "duality-lemma-more-base-change", "duality-remark-relative-dualizing-complex" ], "ref_ids": [ 6250, 6250, 13503, 13503, 3355, 13503, 13512, 13649 ] } ], "ref_ids": [ 6250, 6250 ] }, { "id": 6253, "type": "theorem", "label": "curves-lemma-closed-immersion-dim-1-CM", "categories": [ "curves" ], "title": "curves-lemma-closed-immersion-dim-1-CM", "contents": [ "Let $X$ be a proper scheme of dimension $\\leq 1$ over a field $k$.", "Let $i : Y \\to X$ be a closed immersion.", "Let $\\omega_X^\\bullet$, $\\omega_X$, $\\omega_Y^\\bullet$, $\\omega_Y$", "be as in Lemma \\ref{lemma-duality-dim-1}. Then", "\\begin{enumerate}", "\\item $\\omega_Y^\\bullet = R\\SheafHom(\\mathcal{O}_Y, \\omega_X^\\bullet)$,", "\\item $\\omega_Y = \\SheafHom(\\mathcal{O}_Y, \\omega_X)$ and", "$i_*\\omega_Y = \\SheafHom_{\\mathcal{O}_X}(i_*\\mathcal{O}_Y, \\omega_X)$.", "\\end{enumerate}" ], "refs": [ "curves-lemma-duality-dim-1" ], "proofs": [ { "contents": [ "Denote $g : Y \\to \\Spec(k)$ and $f : X \\to \\Spec(k)$ the structure morphisms.", "Then $g = f \\circ i$. Denote $a, b, c$ the right adjoint of", "Duality for Schemes, Lemma", "\\ref{duality-lemma-twisted-inverse-image} for $f, g, i$.", "Then $b = c \\circ a$ by uniqueness of right adjoints", "and because $Rg_* = Rf_* \\circ Ri_*$.", "In the proof of Lemma \\ref{lemma-duality-dim-1}", "we set", "$\\omega_X^\\bullet = a(\\mathcal{O}_{\\Spec(k)})$ and", "$\\omega_Y^\\bullet = b(\\mathcal{O}_{\\Spec(k)})$.", "Hence $\\omega_Y^\\bullet = c(\\omega_X^\\bullet)$", "which implies (1) by Duality for Schemes, Lemma", "\\ref{duality-lemma-twisted-inverse-image-closed}.", "Since $\\omega_X = H^{-1}(\\omega_X^\\bullet)$ and", "$\\omega_Y = H^{-1}(\\omega_Y^\\bullet)$ we conclude that", "$\\omega_Y = \\SheafHom(\\mathcal{O}_Y, \\omega_X)$.", "This implies", "$i_*\\omega_Y = \\SheafHom_{\\mathcal{O}_X}(i_*\\mathcal{O}_Y, \\omega_X)$", "by Duality for Schemes, Lemma", "\\ref{duality-lemma-sheaf-with-exact-support-ext}." ], "refs": [ "duality-lemma-twisted-inverse-image", "curves-lemma-duality-dim-1", "duality-lemma-twisted-inverse-image-closed", "duality-lemma-sheaf-with-exact-support-ext" ], "ref_ids": [ 13503, 6250, 13525, 13521 ] } ], "ref_ids": [ 6250 ] }, { "id": 6254, "type": "theorem", "label": "curves-lemma-closed-subscheme-reduced-gorenstein", "categories": [ "curves" ], "title": "curves-lemma-closed-subscheme-reduced-gorenstein", "contents": [ "Let $X$ be a proper scheme over a field $k$ which is", "Gorenstein, reduced, and equidimensional of dimension $1$.", "Let $i : Y \\to X$ be a reduced closed subscheme equidimensional", "of dimension $1$. Let $j : Z \\to X$ be the scheme theoretic", "closure of $X \\setminus Y$. Then", "\\begin{enumerate}", "\\item $Y$ and $Z$ are Cohen-Macaulay,", "\\item if $\\mathcal{I} \\subset \\mathcal{O}_X$,", "resp.\\ $\\mathcal{J} \\subset \\mathcal{O}_X$ is the ideal sheaf of", "$Y$, resp.\\ $Z$ in $X$, then", "$$", "\\mathcal{I} = i_*\\mathcal{I}'", "\\quad\\text{and}\\quad", "\\mathcal{J} = j_*\\mathcal{J}'", "$$", "where $\\mathcal{I}' \\subset \\mathcal{O}_Z$,", "resp.\\ $\\mathcal{J}' \\subset \\mathcal{O}_Y$ is the ideal sheaf", "of $Y \\cap Z$ in $Z$, resp.\\ $Y$,", "\\item $\\omega_Y = \\mathcal{J}'(i^*\\omega_X)$ and", "$i_*(\\omega_Y) = \\mathcal{J}\\omega_X$,", "\\item $\\omega_Z = \\mathcal{I}'(i^*\\omega_X)$ and", "$i_*(\\omega_Z) = \\mathcal{I}\\omega_X$,", "\\item we have the following short exact sequences", "\\begin{align*}", "0 \\to \\omega_X \\to i_*i^*\\omega_X \\oplus j_*j^*\\omega_X \\to", "\\mathcal{O}_{Y \\cap Z} \\to 0 \\\\", "0 \\to i_*\\omega_Y \\to \\omega_X \\to j_*j^*\\omega_X \\to 0 \\\\", "0 \\to j_*\\omega_Z \\to \\omega_X \\to i_*i^*\\omega_X \\to 0 \\\\", "0 \\to i_*\\omega_Y \\oplus j_*\\omega_Z \\to \\omega_X \\to", "\\mathcal{O}_{Y \\cap Z} \\to 0 \\\\", "0 \\to \\omega_Y \\to i^*\\omega_X \\to \\mathcal{O}_{Y \\cap Z} \\to 0 \\\\", "0 \\to \\omega_Z \\to j^*\\omega_X \\to \\mathcal{O}_{Y \\cap Z} \\to 0", "\\end{align*}", "\\end{enumerate}", "Here $\\omega_X$, $\\omega_Y$, $\\omega_Z$ are as in", "Lemma \\ref{lemma-duality-dim-1}." ], "refs": [ "curves-lemma-duality-dim-1" ], "proofs": [ { "contents": [ "A reduced $1$-dimensional Noetherian scheme is Cohen-Macaulay, so", "(1) is true. Since $X$ is reduced, we see that $X = Y \\cup Z$", "scheme theoretically. With notation as in", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-union}", "and by the statement of that lemma", "we have a short exact sequence", "$$", "0 \\to \\mathcal{O}_X \\to", "\\mathcal{O}_Y \\oplus \\mathcal{O}_Z \\to \\mathcal{O}_{Y \\cap Z} \\to 0", "$$", "Since $\\mathcal{J} = \\Ker(\\mathcal{O}_X \\to \\mathcal{O}_Z)$,", "$\\mathcal{J}' = \\Ker(\\mathcal{O}_Y \\to \\mathcal{O}_{Y \\cap Z})$,", "$\\mathcal{I} = \\Ker(\\mathcal{O}_X \\to \\mathcal{O}_Y)$, and", "$\\mathcal{I}' = \\Ker(\\mathcal{O}_Z \\to \\mathcal{O}_{Y \\cap Z})$", "a diagram chase implies (2).", "Observe that $\\mathcal{I} + \\mathcal{J}$ is the ideal sheaf", "of $Y \\cap Z$ and that $\\mathcal{I} \\cap \\mathcal{J} = 0$.", "Hence we have the following exact sequences", "\\begin{align*}", "0 \\to \\mathcal{O}_X \\to \\mathcal{O}_Y \\oplus \\mathcal{O}_Z \\to", "\\mathcal{O}_{Y \\cap Z} \\to 0 \\\\", "0 \\to \\mathcal{J} \\to \\mathcal{O}_X \\to \\mathcal{O}_Z \\to 0 \\\\", "0 \\to \\mathcal{I} \\to \\mathcal{O}_X \\to \\mathcal{O}_Y \\to 0 \\\\", "0 \\to \\mathcal{J} \\oplus \\mathcal{I} \\to \\mathcal{O}_X \\to", "\\mathcal{O}_{Y \\cap Z} \\to 0 \\\\", "0 \\to \\mathcal{J}' \\to \\mathcal{O}_Y \\to \\mathcal{O}_{Y \\cap Z} \\to 0 \\\\", "0 \\to \\mathcal{I}' \\to \\mathcal{O}_Z \\to \\mathcal{O}_{Y \\cap Z} \\to 0", "\\end{align*}", "Since $X$ is Gorenstein $\\omega_X$ is an invertible $\\mathcal{O}_X$-module", "(Duality for Schemes, Lemma \\ref{duality-lemma-gorenstein}).", "Since $Y \\cap Z$ has dimension $0$ we have", "$\\omega_X|_{Y \\cap Z} \\cong \\mathcal{O}_{Y \\cap Z}$.", "Thus if we prove (3) and (4), then we obtain the short exact", "sequences of the lemma by tensoring the above", "short exact sequence with the invertible module $\\omega_X$.", "By symmetry it suffices to prove (3) and by", "(2) it suffices to prove $i_*(\\omega_Y) = \\mathcal{J}\\omega_X$.", "\\medskip\\noindent", "We have", "$i_*\\omega_Y = \\SheafHom_{\\mathcal{O}_X}(i_*\\mathcal{O}_Y, \\omega_X)$", "by Lemma \\ref{lemma-closed-immersion-dim-1-CM}.", "Again using that $\\omega_X$ is invertible", "we finally conclude that it suffices to show", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{O}_X/\\mathcal{I}, \\mathcal{O}_X)$", "maps isomorphically to $\\mathcal{J}$ by evaluation at $1$.", "In other words, that $\\mathcal{J}$ is the annihilator of", "$\\mathcal{I}$. This follows from the above." ], "refs": [ "morphisms-lemma-scheme-theoretic-union", "duality-lemma-gorenstein", "curves-lemma-closed-immersion-dim-1-CM" ], "ref_ids": [ 5140, 13591, 6253 ] } ], "ref_ids": [ 6250 ] }, { "id": 6255, "type": "theorem", "label": "curves-lemma-euler", "categories": [ "curves" ], "title": "curves-lemma-euler", "contents": [ "Let $X$ be a proper scheme of dimension $\\leq 1$ over a field $k$.", "With $\\omega_X^\\bullet$ and $\\omega_X$ as in Lemma \\ref{lemma-duality-dim-1}", "we have", "$$", "\\chi(X, \\mathcal{O}_X) = \\chi(X, \\omega_X^\\bullet)", "$$", "If $X$ is Cohen-Macaulay and equidimensional of dimension $1$, then", "$$", "\\chi(X, \\mathcal{O}_X) = - \\chi(X, \\omega_X)", "$$" ], "refs": [ "curves-lemma-duality-dim-1" ], "proofs": [ { "contents": [ "We define the right hand side of the first formula as follows:", "$$", "\\chi(X, \\omega_X^\\bullet) =", "\\sum\\nolimits_{i \\in \\mathbf{Z}} (-1)^i\\dim_k H^i(X, \\omega_X^\\bullet)", "$$", "This is well defined because $\\omega_X^\\bullet$ is in", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$, but also because", "$$", "H^i(X, \\omega_X^\\bullet) =", "\\Ext^i(\\mathcal{O}_X, \\omega_X^\\bullet) =", "H^{-i}(X, \\mathcal{O}_X)", "$$", "which is always finite dimensional and nonzero only if $i = 0, -1$.", "This of course also proves the first formula. The second is a consequence", "of the first because $\\omega_X^\\bullet = \\omega_X[1]$ in the CM case, see", "Lemma \\ref{lemma-duality-dim-1-CM}." ], "refs": [ "curves-lemma-duality-dim-1-CM" ], "ref_ids": [ 6251 ] } ], "ref_ids": [ 6250 ] }, { "id": 6256, "type": "theorem", "label": "curves-lemma-rr", "categories": [ "curves" ], "title": "curves-lemma-rr", "contents": [ "Let $X$ be a proper scheme over a field $k$ which is Gorenstein and", "equidimensional of dimension $1$. Let $\\omega_X$ be as in", "Lemma \\ref{lemma-duality-dim-1}. Then", "\\begin{enumerate}", "\\item $\\omega_X$ is an invertible $\\mathcal{O}_X$-module,", "\\item $\\deg(\\omega_X) = -2\\chi(X, \\mathcal{O}_X)$,", "\\item for a locally free $\\mathcal{O}_X$-module $\\mathcal{E}$", "of constant rank we have", "$$", "\\chi(X, \\mathcal{E}) = \\deg(\\mathcal{E}) -", "\\textstyle{\\frac{1}{2}} \\text{rank}(\\mathcal{E}) \\deg(\\omega_X)", "$$", "and $\\dim_k(H^i(X, \\mathcal{E})) =", "\\dim_k(H^{1 - i}(X, \\mathcal{E}^\\vee \\otimes_{\\mathcal{O}_X} \\omega_X))$", "for all $i \\in \\mathbf{Z}$.", "\\end{enumerate}" ], "refs": [ "curves-lemma-duality-dim-1" ], "proofs": [ { "contents": [ "Recall that Gorenstein schemes are Cohen-Macaulay", "(Duality for Schemes, Lemma \\ref{duality-lemma-gorenstein-CM})", "and hence $\\omega_X$ is a dualizing module on $X$, see", "Lemma \\ref{lemma-duality-dim-1-CM}.", "It follows more or less from the definition of the Gorenstein property", "that the dualizing sheaf is invertible, see", "Duality for Schemes, Section \\ref{duality-section-gorenstein}.", "By (\\ref{equation-rr}) applied to $\\omega_X$ we have", "$$", "\\chi(X, \\omega_X) = ", "\\deg(c_1(\\omega_X) \\cap [X]_1) + \\chi(X, \\mathcal{O}_X)", "$$", "Combined with Lemma \\ref{lemma-euler} this gives", "$$", "2\\chi(X, \\mathcal{O}_X) = - \\deg(c_1(\\omega_X) \\cap [X]_1) = - \\deg(\\omega_X)", "$$", "the second equality by (\\ref{equation-degree-c1}). Putting this back into", "(\\ref{equation-rr}) for $\\mathcal{E}$ gives the displayed formula of the lemma.", "The symmetry in dimensions is a consequence of duality for $X$, see", "Remark \\ref{remark-rework-duality-locally-free}." ], "refs": [ "duality-lemma-gorenstein-CM", "curves-lemma-duality-dim-1-CM", "curves-lemma-euler", "curves-remark-rework-duality-locally-free" ], "ref_ids": [ 13589, 6251, 6255, 6362 ] } ], "ref_ids": [ 6250 ] }, { "id": 6257, "type": "theorem", "label": "curves-lemma-automatic", "categories": [ "curves" ], "title": "curves-lemma-automatic", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$", "and $H^0(X, \\mathcal{O}_X) = k$. Then $X$ is connected, Cohen-Macaulay,", "and equidimensional of dimension $1$." ], "refs": [], "proofs": [ { "contents": [ "Since $\\Gamma(X, \\mathcal{O}_X) = k$ has no nontrivial idempotents,", "we see that $X$ is connected. This already shows that $X$ is", "equidimensional of dimension $1$ (any irreducible component", "of dimension $0$ would be a connected component).", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be the maximal coherent submodule supported in closed points.", "Then $\\mathcal{I}$ exists", "(Divisors, Lemma \\ref{divisors-lemma-remove-embedded-points})", "and is globally generated", "(Varieties, Lemma \\ref{varieties-lemma-chi-tensor-finite}).", "Since $1 \\in \\Gamma(X, \\mathcal{O}_X)$ is not a section", "of $\\mathcal{I}$ we conclude that $\\mathcal{I} = 0$.", "Thus $X$ does not have embedded points", "(Divisors, Lemma \\ref{divisors-lemma-remove-embedded-points}).", "Thus $X$ has $(S_1)$ by", "Divisors, Lemma \\ref{divisors-lemma-S1-no-embedded}.", "Hence $X$ is Cohen-Macaulay." ], "refs": [ "divisors-lemma-remove-embedded-points", "varieties-lemma-chi-tensor-finite", "divisors-lemma-remove-embedded-points", "divisors-lemma-S1-no-embedded" ], "ref_ids": [ 7870, 11030, 7870, 7867 ] } ], "ref_ids": [] }, { "id": 6258, "type": "theorem", "label": "curves-lemma-vanishing", "categories": [ "curves" ], "title": "curves-lemma-vanishing", "contents": [ "In Situation \\ref{situation-Cohen-Macaulay-curve}. Given an exact sequence", "$$", "\\omega_X \\to \\mathcal{F} \\to \\mathcal{Q} \\to 0", "$$", "of coherent $\\mathcal{O}_X$-modules with $H^1(X, \\mathcal{Q}) = 0$", "(for example if $\\dim(\\text{Supp}(\\mathcal{Q})) = 0$), then", "either $H^1(X, \\mathcal{F}) = 0$ or", "$\\mathcal{F} = \\omega_X \\oplus \\mathcal{Q}$." ], "refs": [], "proofs": [ { "contents": [ "(The parenthetical statement follows from", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-support-dimension-0}.)", "Since $H^0(X, \\mathcal{O}_X) = k$ is dual to $H^1(X, \\omega_X)$", "(see Section \\ref{section-Riemann-Roch})", "we see that $\\dim H^1(X, \\omega_X) = 1$. The sheaf $\\omega_X$", "represents the functor", "$\\mathcal{F} \\mapsto \\Hom_k(H^1(X, \\mathcal{F}), k)$", "on the category of coherent $\\mathcal{O}_X$-modules", "(Duality for Schemes, Lemma", "\\ref{duality-lemma-dualizing-module-proper-over-A}).", "Consider an exact sequence as in the statement of the lemma", "and assume that $H^1(X, \\mathcal{F}) \\not = 0$. Since", "$H^1(X, \\mathcal{Q}) = 0$ we see that", "$H^1(X, \\omega_X) \\to H^1(X, \\mathcal{F})$ is an isomorphism.", "By the universal property of $\\omega_X$ stated above, we conclude there", "is a map $\\mathcal{F} \\to \\omega_X$ whose action on $H^1$ is the inverse", "of this isomorphism. The composition $\\omega_X \\to \\mathcal{F} \\to \\omega_X$", "is the identity (by the universal property) and the lemma is proved." ], "refs": [ "coherent-lemma-coherent-support-dimension-0", "duality-lemma-dualizing-module-proper-over-A" ], "ref_ids": [ 3317, 13585 ] } ], "ref_ids": [] }, { "id": 6259, "type": "theorem", "label": "curves-lemma-vanishing-twist", "categories": [ "curves" ], "title": "curves-lemma-vanishing-twist", "contents": [ "In Situation \\ref{situation-Cohen-Macaulay-curve}. Let", "$\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module which is", "globally generated and not isomorphic to $\\mathcal{O}_X$. Then", "$H^1(X, \\omega_X \\otimes \\mathcal{L}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "By duality as discussed in Section \\ref{section-Riemann-Roch} we have to", "show that $H^0(X, \\mathcal{L}^{\\otimes - 1}) = 0$. If not, then we can", "choose a global section $t$ of $\\mathcal{L}^{\\otimes - 1}$", "and a global section $s$ of $\\mathcal{L}$ such that $st \\not = 0$.", "However, then $st$ is a constant multiple of $1$, by our assumption", "that $H^0(X, \\mathcal{O}_X) = k$. It follows that", "$\\mathcal{L} \\cong \\mathcal{O}_X$, which is a contradiction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6260, "type": "theorem", "label": "curves-lemma-globally-generated", "categories": [ "curves" ], "title": "curves-lemma-globally-generated", "contents": [ "In Situation \\ref{situation-Cohen-Macaulay-curve}. Given an exact sequence", "$$", "\\omega_X \\to \\mathcal{F} \\to \\mathcal{Q} \\to 0", "$$", "of coherent $\\mathcal{O}_X$-modules with $\\dim(\\text{Supp}(\\mathcal{Q})) = 0$", "and $\\dim_k H^0(X, \\mathcal{Q}) \\geq 2$ and such that there is no nonzero", "submodule $\\mathcal{Q}' \\subset \\mathcal{F}$ such that", "$\\mathcal{Q}' \\to \\mathcal{Q}$ is injective.", "Then the submodule of $\\mathcal{F}$ generated by global", "sections surjects onto $\\mathcal{Q}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}' \\subset \\mathcal{F}$ be the submodule generated by", "global sections and the image of $\\omega_X \\to \\mathcal{F}$. Since", "$\\dim_k H^0(X, \\mathcal{Q}) \\geq 2$ and", "$\\dim_k H^1(X, \\omega_X) = \\dim_k H^0(X, \\mathcal{O}_X) = 1$,", "we see that $\\mathcal{F}' \\to \\mathcal{Q}$ is not zero and", "$\\omega_X \\to \\mathcal{F}'$ is not an isomorphism.", "Hence $H^1(X, \\mathcal{F}') = 0$ by Lemma \\ref{lemma-vanishing}", "and our assumption on $\\mathcal{F}$.", "Consider the short exact sequence", "$$", "0 \\to \\mathcal{F}' \\to \\mathcal{F} \\to", "\\mathcal{Q}/\\Im(\\mathcal{F}' \\to \\mathcal{Q}) \\to 0", "$$", "If the quotient on the right is nonzero, then we obtain a contradiction", "because then $H^0(X, \\mathcal{F})$ is bigger than $H^0(X, \\mathcal{F}')$." ], "refs": [ "curves-lemma-vanishing" ], "ref_ids": [ 6258 ] } ], "ref_ids": [] }, { "id": 6261, "type": "theorem", "label": "curves-lemma-globally-generated-curve", "categories": [ "curves" ], "title": "curves-lemma-globally-generated-curve", "contents": [ "In Situation \\ref{situation-Cohen-Macaulay-curve} assume that", "$X$ is integral. Let $0 \\to \\omega_X \\to \\mathcal{F} \\to \\mathcal{Q} \\to 0$", "be a short exact sequence of coherent $\\mathcal{O}_X$-modules with", "$\\mathcal{F}$ torsion free, $\\dim(\\text{Supp}(\\mathcal{Q})) = 0$,", "and $\\dim_k H^0(X, \\mathcal{Q}) \\geq 2$. Then $\\mathcal{F}$", "is globally generated." ], "refs": [], "proofs": [ { "contents": [ "Consider the submodule $\\mathcal{F}'$ generated by the global sections. By", "Lemma \\ref{lemma-globally-generated} we see that $\\mathcal{F}' \\to \\mathcal{Q}$", "is surjective, in particular $\\mathcal{F}' \\not = 0$. Since $X$ is a curve, we", "see that $\\mathcal{F}' \\subset \\mathcal{F}$ is an inclusion of rank $1$", "sheaves, hence $\\mathcal{Q}' = \\mathcal{F}/\\mathcal{F}'$ is supported in", "finitely many points. To get a contradiction, assume that", "$\\mathcal{Q}'$ is nonzero. Then we see that $H^1(X, \\mathcal{F}') \\not = 0$.", "Then we get a nonzero map $\\mathcal{F}' \\to \\omega_X$ by the universal", "property (Duality for Schemes, Lemma", "\\ref{duality-lemma-dualizing-module-proper-over-A}).", "The image of the composition $\\mathcal{F}' \\to \\omega_X \\to \\mathcal{F}$", "is generated by global sections, hence is inside of $\\mathcal{F}'$.", "Thus we get a nonzero self map $\\mathcal{F}' \\to \\mathcal{F}'$.", "Since $\\mathcal{F}'$ is torsion free of rank $1$ on a proper curve", "this has to be an automorphism (details omitted). But then this implies that", "$\\mathcal{F}'$ is contained in $\\omega_X \\subset \\mathcal{F}$", "contradicting the surjectivity of $\\mathcal{F}' \\to \\mathcal{Q}$." ], "refs": [ "curves-lemma-globally-generated", "duality-lemma-dualizing-module-proper-over-A" ], "ref_ids": [ 6260, 13585 ] } ], "ref_ids": [] }, { "id": 6262, "type": "theorem", "label": "curves-lemma-tensor-omega-with-globally-generated-invertible", "categories": [ "curves" ], "title": "curves-lemma-tensor-omega-with-globally-generated-invertible", "contents": [ "In Situation \\ref{situation-Cohen-Macaulay-curve}. Let $\\mathcal{L}$", "be a very ample invertible $\\mathcal{O}_X$-module with", "$\\deg(\\mathcal{L}) \\geq 2$. Then", "$\\omega_X \\otimes_{\\mathcal{O}_X} \\mathcal{L}$ is globally generated." ], "refs": [], "proofs": [ { "contents": [ "Assume $k$ is algebraically closed. Let $x \\in X$ be a closed point.", "Let $C_i \\subset X$ be the irreducible components and for each $i$", "let $x_i \\in C_i$ be the generic point. By", "Varieties, Lemma \\ref{varieties-lemma-very-ample-vanish-at-point}", "we can choose a section $s \\in H^0(X, \\mathcal{L})$ such that $s$", "vanishes at $x$ but not at $x_i$ for all $i$. The corresponding", "module map $s : \\mathcal{O}_X \\to \\mathcal{L}$ is injective with", "cokernel $\\mathcal{Q}$ supported in finitely many points and", "with $H^0(X, \\mathcal{Q}) \\geq 2$. Consider the corresponding", "exact sequence", "$$", "0 \\to \\omega_X \\to \\omega_X \\otimes \\mathcal{L} \\to", "\\omega_X \\otimes \\mathcal{Q} \\to 0", "$$", "By Lemma \\ref{lemma-globally-generated} we see that the module generated", "by global sections surjects onto $\\omega_X \\otimes \\mathcal{Q}$.", "Since $x$ was arbitrary this proves the lemma. Some details omitted.", "\\medskip\\noindent", "We will reduce the case where $k$ is not algebraically closed, to", "the algebraically closed field case. We suggest the reader skip", "the rest of the proof. Choose an algebraic closure $\\overline{k}$", "of $k$ and consider the base change $X_{\\overline{k}}$. Let us", "check that $X_{\\overline{k}} \\to \\Spec(\\overline{k})$ is an example", "of Situation \\ref{situation-Cohen-Macaulay-curve}. By flat base change", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology})", "we see that $H^0(X_{\\overline{k}}, \\mathcal{O}) = \\overline{k}$.", "The scheme $X_{\\overline{k}}$ is proper over $\\overline{k}$ (Morphisms,", "Lemma \\ref{morphisms-lemma-base-change-proper}) and", "equidimensional of dimension $1$", "(Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}).", "The pullback of $\\omega_X$ to $X_{\\overline{k}}$ is the dualizing", "module of $X_{\\overline{k}}$ by Lemma \\ref{lemma-sanity-check-duality}.", "The pullback of $\\mathcal{L}$ to $X_{\\overline{k}}$ is very ample", "(Morphisms, Lemma \\ref{morphisms-lemma-very-ample-base-change}).", "The degree of the pullback of $\\mathcal{L}$ to $X_{\\overline{k}}$", "is equal to the degree of $\\mathcal{L}$ on $X$ (Varieties, Lemma", "\\ref{varieties-lemma-degree-base-change}). Finally, we see that", "$\\omega_X \\otimes \\mathcal{L}$ is globally generated if and only", "if its base change is so", "(Varieties, Lemma \\ref{varieties-lemma-globally-generated-base-change}).", "In this way we see that the result follows from the result in the", "case of an algebraically closed ground field." ], "refs": [ "varieties-lemma-very-ample-vanish-at-point", "curves-lemma-globally-generated", "coherent-lemma-flat-base-change-cohomology", "morphisms-lemma-base-change-proper", "morphisms-lemma-dimension-fibre-after-base-change", "curves-lemma-sanity-check-duality", "morphisms-lemma-very-ample-base-change", "varieties-lemma-degree-base-change", "varieties-lemma-globally-generated-base-change" ], "ref_ids": [ 10995, 6260, 3298, 5409, 5279, 6252, 5390, 11104, 10994 ] } ], "ref_ids": [] }, { "id": 6263, "type": "theorem", "label": "curves-lemma-criterion-very-ample", "categories": [ "curves" ], "title": "curves-lemma-criterion-very-ample", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$", "and $H^0(X, \\mathcal{O}_X) = k$. Let $\\mathcal{L}$ be an invertible", "$\\mathcal{O}_X$-module. Assume", "\\begin{enumerate}", "\\item $\\mathcal{L}$ has a regular global section,", "\\item $H^1(X, \\mathcal{L}) = 0$, and", "\\item $\\mathcal{L}$ is ample.", "\\end{enumerate}", "Then $\\mathcal{L}^{\\otimes 6}$ is very ample on $X$ over $k$." ], "refs": [], "proofs": [ { "contents": [ "Let $s$ be a regular global section of $\\mathcal{L}$. Let", "$i : Z = Z(s) \\to X$ be the zero scheme of $s$, see", "Divisors, Section \\ref{divisors-section-effective-Cartier-invertible}.", "By condition (3) we see that $Z \\not = \\emptyset$ (small detail omitted).", "Consider the short exact sequence", "$$", "0 \\to \\mathcal{O}_X \\xrightarrow{s} \\mathcal{L} \\to", "i_*(\\mathcal{L}|_Z) \\to 0", "$$", "Tensoring with $\\mathcal{L}$ we obtain", "$$", "0 \\to \\mathcal{L} \\to \\mathcal{L}^{\\otimes 2} \\to", "i_*(\\mathcal{L}^{\\otimes 2}|_Z) \\to 0", "$$", "Observe that $Z$ has dimension $0$", "(Divisors, Lemma \\ref{divisors-lemma-effective-Cartier-makes-dimension-drop})", "and hence is the spectrum of an Artinian ring", "(Varieties, Lemma \\ref{varieties-lemma-algebraic-scheme-dim-0})", "hence $\\mathcal{L}|_Z \\cong \\mathcal{O}_Z$", "(Algebra, Lemma \\ref{algebra-lemma-locally-free-semi-local-free}).", "The short exact sequence also shows that", "$H^1(X, \\mathcal{L}^{\\otimes 2}) = 0$ (for example using", "Varieties, Lemma \\ref{varieties-lemma-chi-tensor-finite}", "to see vanishing in the spot on the right). Using induction", "on $n \\geq 1$ and the sequence", "$$", "0 \\to \\mathcal{L}^{\\otimes n} \\xrightarrow{s}", "\\mathcal{L}^{\\otimes n + 1} \\to", "i_*(\\mathcal{L}^{\\otimes n + 1}|_Z) \\to 0", "$$", "we see that $H^1(X, \\mathcal{L}^{\\otimes n}) = 0$ for $n > 0$", "and that there exists a global section $t_{n + 1}$", "of $\\mathcal{L}^{\\otimes n + 1}$ which gives a trivialization of", "$\\mathcal{L}^{\\otimes n + 1}|_Z \\cong \\mathcal{O}_Z$.", "\\medskip\\noindent", "Consider the multiplication map", "$$", "\\mu_n :", "H^0(X, \\mathcal{L}) \\otimes_k H^0(X, \\mathcal{L}^{\\otimes n})", "\\oplus", "H^0(X, \\mathcal{L}^{\\otimes 2}) \\otimes_k H^0(X, \\mathcal{L}^{\\otimes n - 1})", "\\longrightarrow", "H^0(X, \\mathcal{L}^{\\otimes n + 1})", "$$", "We claim this is surjective for $n \\geq 3$.", "To see this we consider the short exact sequence", "$$", "0 \\to \\mathcal{L}^{\\otimes n} \\xrightarrow{s}", "\\mathcal{L}^{\\otimes n + 1} \\to i_*(\\mathcal{L}^{\\otimes n + 1}|_Z) \\to 0", "$$", "The sections of $\\mathcal{L}^{\\otimes n + 1}$ coming from the left in this", "sequence are in the image of $\\mu_n$. On the other hand, since", "$H^0(\\mathcal{L}^{\\otimes 2}) \\to H^0(\\mathcal{L}^{\\otimes 2}|_Z)$", "is surjective (see above) and since $t_{n - 1}$ maps to a trivialization of", "$\\mathcal{L}^{\\otimes n - 1}|_Z$", "we see that $\\mu_n(H^0(X, \\mathcal{L}^{\\otimes 2}) \\otimes t_{n - 1})$", "gives a subspace", "of $H^0(X, \\mathcal{L}^{\\otimes n + 1})$ surjecting onto the global sections of", "$\\mathcal{L}^{\\otimes n + 1}|_Z$. This proves the claim.", "\\medskip\\noindent", "From the claim in the previous paragraph we conclude", "that the graded $k$-algebra", "$$", "S = \\bigoplus\\nolimits_{n \\geq 0} H^0(X, \\mathcal{L}^{\\otimes n})", "$$", "is generated in degrees $0, 1, 2, 3$ over $k$.", "Recall that $X = \\text{Proj}(S)$, see", "Morphisms, Lemma \\ref{morphisms-lemma-proper-ample-is-proj}.", "Thus $S^{(6)} = \\bigoplus_{n} S_{6n}$ is generated in degree $1$.", "This means that $\\mathcal{L}^{\\otimes 6}$ is very ample as desired." ], "refs": [ "divisors-lemma-effective-Cartier-makes-dimension-drop", "varieties-lemma-algebraic-scheme-dim-0", "algebra-lemma-locally-free-semi-local-free", "varieties-lemma-chi-tensor-finite", "morphisms-lemma-proper-ample-is-proj" ], "ref_ids": [ 7930, 10988, 799, 11030, 5434 ] } ], "ref_ids": [] }, { "id": 6264, "type": "theorem", "label": "curves-lemma-criterion-very-ample-bis", "categories": [ "curves" ], "title": "curves-lemma-criterion-very-ample-bis", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$", "and $H^0(X, \\mathcal{O}_X) = k$. Let $\\mathcal{L}$ be an invertible", "$\\mathcal{O}_X$-module. Assume", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is globally generated,", "\\item $H^1(X, \\mathcal{L}) = 0$, and", "\\item $\\mathcal{L}$ is ample.", "\\end{enumerate}", "Then $\\mathcal{L}^{\\otimes 2}$ is very ample on $X$ over $k$." ], "refs": [], "proofs": [ { "contents": [ "Choose basis $s_0, \\ldots, s_n$ of $H^0(X, \\mathcal{L}^{\\otimes 2})$", "over $k$. By property (1) we see that $\\mathcal{L}^{\\otimes 2}$", "is globally generated and we get a morphism", "$$", "\\varphi_{\\mathcal{L}^{\\otimes 2}, (s_0, \\ldots, s_n)} :", "X \\longrightarrow \\mathbf{P}^n_k", "$$", "See Constructions, Section \\ref{constructions-section-projective-space}.", "The lemma asserts that this morphism is a closed immersion.", "To check this we may replace $k$ by its algebraic closure, see", "Descent, Lemma \\ref{descent-lemma-descending-property-closed-immersion}.", "Thus we may assume $k$ is algebraically closed.", "\\medskip\\noindent", "Assume $k$ is algebraically closed. For each generic point $\\eta_i \\in X$", "let $V_i \\subset H^0(X, \\mathcal{L})$ be the $k$-subvector space of", "sections vanishing at $\\eta_i$. Since $\\mathcal{L}$ is globally generated,", "we see that $V_i \\not = H^0(X, \\mathcal{L})$. Since $X$ has only a", "finite number of irreducible components and $k$ is infinite, we can find", "$s \\in H^0(X, \\mathcal{L})$ nonvanishing at $\\eta_i$ for all $i$.", "Then $s$ is a regular section of $\\mathcal{L}$ (because $X$ is", "Cohen-Macaulay by Lemma \\ref{lemma-automatic} and hence $\\mathcal{L}$", "has no embedded associated points).", "\\medskip\\noindent", "In particular, all of the statements given in the proof of", "Lemma \\ref{lemma-criterion-very-ample} hold with this $s$.", "Moreover, as $\\mathcal{L}$ is globally generated, we can find", "a global section $t \\in H^0(X, \\mathcal{L})$ such that", "$t|_Z$ is nonvanishing (argue as above using the finite number", "of points of $Z$). Then in the proof of Lemma \\ref{lemma-criterion-very-ample}", "we can use $t$ to see that additionally the multiplication map", "$$", "\\mu_n :", "H^0(X, \\mathcal{L}) \\otimes_k H^0(X, \\mathcal{L}^{\\otimes 2})", "\\longrightarrow", "H^0(X, \\mathcal{L}^{\\otimes 3})", "$$", "is surjective. Thus", "$$", "S = \\bigoplus\\nolimits_{n \\geq 0} H^0(X, \\mathcal{L}^{\\otimes n})", "$$", "is generated in degrees $0, 1, 2$ over $k$. Arguing as in the", "proof of Lemma \\ref{lemma-criterion-very-ample} we find that", "$S^{(2)} = \\bigoplus_{n} S_{2n}$ is generated in degree $1$.", "This means that $\\mathcal{L}^{\\otimes 2}$ is very ample as desired.", "Some details omitted." ], "refs": [ "descent-lemma-descending-property-closed-immersion", "curves-lemma-automatic", "curves-lemma-criterion-very-ample", "curves-lemma-criterion-very-ample", "curves-lemma-criterion-very-ample" ], "ref_ids": [ 14684, 6257, 6263, 6263, 6263 ] } ], "ref_ids": [] }, { "id": 6265, "type": "theorem", "label": "curves-lemma-genus-base-change", "categories": [ "curves" ], "title": "curves-lemma-genus-base-change", "contents": [ "Let $k'/k$ be a field extension. Let $X$ be a proper scheme over $k$ having", "dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. Then $X_{k'}$ is a", "proper scheme over $k'$", "having dimension $1$ and $H^0(X_{k'}, \\mathcal{O}_{X_{k'}}) = k'$.", "Moreover the genus of $X_{k'}$ is equal to the genus of $X$." ], "refs": [], "proofs": [ { "contents": [ "The dimension of $X_{k'}$ is $1$ for example by", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}.", "The morphism $X_{k'} \\to \\Spec(k')$ is proper by", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-proper}.", "The equality $H^0(X_{k'}, \\mathcal{O}_{X_{k'}}) = k'$ follows from", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology}.", "The equality of the genus follows from the same lemma." ], "refs": [ "morphisms-lemma-dimension-fibre-after-base-change", "morphisms-lemma-base-change-proper", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 5279, 5409, 3298 ] } ], "ref_ids": [] }, { "id": 6266, "type": "theorem", "label": "curves-lemma-genus-gorenstein", "categories": [ "curves" ], "title": "curves-lemma-genus-gorenstein", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having", "dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. If $X$ is Gorenstein,", "then", "$$", "\\deg(\\omega_X) = 2g - 2", "$$", "where $g$ is the genus of $X$ and $\\omega_X$ is as in", "Lemma \\ref{lemma-duality-dim-1}." ], "refs": [ "curves-lemma-duality-dim-1" ], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-rr}." ], "refs": [ "curves-lemma-rr" ], "ref_ids": [ 6256 ] } ], "ref_ids": [ 6250 ] }, { "id": 6267, "type": "theorem", "label": "curves-lemma-genus-smooth", "categories": [ "curves" ], "title": "curves-lemma-genus-smooth", "contents": [ "Let $X$ be a smooth proper curve over a field $k$", "with $H^0(X, \\mathcal{O}_X) = k$. Then", "$$", "\\dim_k H^0(X, \\Omega_{X/k}) = g", "\\quad\\text{and}\\quad", "\\deg(\\Omega_{X/k}) = 2g - 2", "$$", "where $g$ is the genus of $X$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-duality-dim-1} we have $\\Omega_{X/k} = \\omega_X$.", "Hence the formulas hold by (\\ref{equation-genus}) and", "Lemma \\ref{lemma-genus-gorenstein}." ], "refs": [ "curves-lemma-duality-dim-1", "curves-lemma-genus-gorenstein" ], "ref_ids": [ 6250, 6266 ] } ], "ref_ids": [] }, { "id": 6268, "type": "theorem", "label": "curves-lemma-equation-plane-curve", "categories": [ "curves" ], "title": "curves-lemma-equation-plane-curve", "contents": [ "Let $Z \\subset \\mathbf{P}^2_k$ be a closed subscheme which", "is equidimensional of dimension $1$ and has no embedded points", "(equivalently $Z$ is Cohen-Macaulay).", "Then the ideal $I(Z) \\subset k[T_0, T_1, T_2]$ corresponding", "to $Z$ is principal." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Divisors, Lemma \\ref{divisors-lemma-equation-codim-1-in-projective-space}", "(see also Varieties, Lemma", "\\ref{varieties-lemma-equation-codim-1-in-projective-space}).", "The parenthetical statement follows from the fact that a", "$1$ dimensional Noetherian scheme is Cohen-Macaulay", "if and only if it has no embedded points, see", "Divisors, Lemma \\ref{divisors-lemma-noetherian-dim-1-CM-no-embedded-points}." ], "refs": [ "divisors-lemma-noetherian-dim-1-CM-no-embedded-points" ], "ref_ids": [ 7868 ] } ], "ref_ids": [] }, { "id": 6269, "type": "theorem", "label": "curves-lemma-plane-curve", "categories": [ "curves" ], "title": "curves-lemma-plane-curve", "contents": [ "Let $Z \\subset \\mathbf{P}^2_k$ be as in Lemma \\ref{lemma-equation-plane-curve}", "and let $I(Z) = (F)$ for some $F \\in k[T_0, T_1, T_2]$.", "Then $Z$ is a curve if and only if $F$ is irreducible." ], "refs": [], "proofs": [ { "contents": [ "If $F$ is reducible, say $F = F' F''$ then let $Z'$ be the closed subscheme", "of $\\mathbf{P}^2_k$ defined by $F'$. It is clear that $Z' \\subset Z$", "and that $Z' \\not = Z$. Since $Z'$ has dimension $1$ as well, we conclude", "that either $Z$ is not reduced, or that $Z$ is not irreducible.", "Conversely, write $Z = \\sum a_i D_i$ where $D_i$ are the irreducible", "components of $Z$, see", "Divisors, Lemmas \\ref{divisors-lemma-codim-1-part} and", "\\ref{divisors-lemma-codimension-1-is-effective-Cartier}.", "Let $F_i \\in k[T_0, T_1, T_2]$ be the homogeneous", "polynomial generating the ideal of $D_i$. Then it is clear that", "$F$ and $\\prod F_i^{a_i}$ cut out the same closed subscheme of", "$\\mathbf{P}^2_k$. Hence $F = \\lambda \\prod F_i^{a_i}$ for some", "$\\lambda \\in k^*$ because both generate the ideal of $Z$.", "Thus we see that if $F$ is irreducible, then $Z$ is", "a prime divisor, i.e., a curve." ], "refs": [ "divisors-lemma-codim-1-part", "divisors-lemma-codimension-1-is-effective-Cartier" ], "ref_ids": [ 7952, 7953 ] } ], "ref_ids": [] }, { "id": 6270, "type": "theorem", "label": "curves-lemma-genus-plane-curve", "categories": [ "curves" ], "title": "curves-lemma-genus-plane-curve", "contents": [ "Let $Z \\subset \\mathbf{P}^2_k$ be as in Lemma \\ref{lemma-equation-plane-curve}", "and let $I(Z) = (F)$ for some $F \\in k[T_0, T_1, T_2]$.", "Then $H^0(Z, \\mathcal{O}_Z) = k$ and the genus of $Z$ is", "$(d - 1)(d - 2)/2$ where $d = \\deg(F)$." ], "refs": [], "proofs": [ { "contents": [ "Let $S = k[T_0, T_1, T_2]$.", "There is an exact sequence of graded modules", "$$", "0 \\to S(-d) \\xrightarrow{F} S \\to S/(F) \\to 0", "$$", "Denote $i : Z \\to \\mathbf{P}^2_k$ the given closed immersion.", "Applying the exact functor $\\widetilde{\\ }$", "(Constructions, Lemma \\ref{constructions-lemma-proj-sheaves})", "we obtain", "$$", "0 \\to \\mathcal{O}_{\\mathbf{P}^2_k}(-d) \\to", "\\mathcal{O}_{\\mathbf{P}^2_k} \\to i_*\\mathcal{O}_Z \\to 0", "$$", "because $F$ generates the ideal of $Z$.", "Note that the cohomology groups of $\\mathcal{O}_{\\mathbf{P}^2_k}(-d)$ and", "$\\mathcal{O}_{\\mathbf{P}^2_k}$ are given in", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cohomology-projective-space-over-ring}.", "On the other hand, we have", "$H^q(Z, \\mathcal{O}_Z) = H^q(\\mathbf{P}^2_k, i_*\\mathcal{O}_Z)$ by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology}.", "Applying the long exact cohomology sequence", "we first obtain that", "$$", "k = H^0(\\mathbf{P}^2_k, \\mathcal{O}_{\\mathbf{P}^2_k}) \\longrightarrow", "H^0(Z, \\mathcal{O}_Z)", "$$", "is an isomorphism and next that the boundary map", "$$", "H^1(Z, \\mathcal{O}_Z) \\longrightarrow", "H^2(\\mathbf{P}^2_k, \\mathcal{O}_{\\mathbf{P}^2_k}(-d)) \\cong", "k[T_0, T_1, T_2]_{d - 3}", "$$", "is an isomorphism. Since it is easy to see that the dimension of this", "is $(d - 1)(d - 2)/2$ the proof is finished." ], "refs": [ "constructions-lemma-proj-sheaves", "coherent-lemma-cohomology-projective-space-over-ring", "coherent-lemma-relative-affine-cohomology" ], "ref_ids": [ 12594, 3304, 3284 ] } ], "ref_ids": [] }, { "id": 6271, "type": "theorem", "label": "curves-lemma-smooth-plane-curve-point-over-separable", "categories": [ "curves" ], "title": "curves-lemma-smooth-plane-curve-point-over-separable", "contents": [ "Let $Z \\subset \\mathbf{P}^2_k$ be as in Lemma \\ref{lemma-equation-plane-curve}", "and let $I(Z) = (F)$ for some $F \\in k[T_0, T_1, T_2]$.", "If $Z \\to \\Spec(k)$ is smooth in at least one point and $k$ is infinite, then", "there exists a closed point $z \\in Z$ contained in the smooth", "locus such that $\\kappa(z)/k$ is finite separable of degree", "at most $d$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $z' \\in Z$ is a point where $Z \\to \\Spec(k)$ is smooth.", "After renumbering the coordinates if necessary we may assume", "$z'$ is contained in $D_+(T_0)$. Set $f = F(1, x, y) \\in k[x, y]$.", "Then $Z \\cap D_+(X_0)$ is isomorphic to the spectrum of $k[x, y]/(f)$.", "Let $f_x, f_y$ be the partial derivatives of $f$ with respect to", "$x, y$. Since $z'$ is a smooth point of $Z/k$ we see that either", "$f_x$ or $f_y$ is nonzero in $z'$ (see discussion in", "Algebra, Section \\ref{algebra-section-smooth}).", "After renumbering the coordinates", "we may assume $f_y$ is not zero at $z'$. Hence there is a nonempty", "open subscheme $V \\subset Z \\cap D_{+}(X_0)$ such that the", "projection", "$$", "p : V \\longrightarrow \\Spec(k[x])", "$$", "is \\'etale. Because the degree of $f$ as a polynomial in $y$", "is at most $d$, we see that the degrees of the fibres of the", "projection $p$ are at most $d$ (see discussion in", "Morphisms, Section \\ref{morphisms-section-universally-bounded}).", "Moreover, as $p$ is \\'etale", "the image of $p$ is an open $U \\subset \\Spec(k[x])$.", "Finally, since $k$ is infinite, the set of $k$-rational points", "$U(k)$ of $U$ is infinite, in particular not empty. Pick any", "$t \\in U(k)$ and let $z \\in V$ be a point mapping to $t$.", "Then $z$ works." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6272, "type": "theorem", "label": "curves-lemma-genus-zero-pic", "categories": [ "curves" ], "title": "curves-lemma-genus-zero-pic", "contents": [ "Let $X$ be a proper curve over a field $k$ with $H^0(X, \\mathcal{O}_X) = k$.", "If $X$ has genus $0$, then every invertible $\\mathcal{O}_X$-module", "$\\mathcal{L}$ of degree $0$ is trivial." ], "refs": [], "proofs": [ { "contents": [ "Namely, we have $\\dim_k H^0(X, \\mathcal{L}) \\geq 0 + 1 - 0 = 1$", "by Riemann-Roch (Lemma \\ref{lemma-rr}), hence $\\mathcal{L}$ has a", "nonzero section, hence $\\mathcal{L} \\cong \\mathcal{O}_X$ by", "Varieties, Lemma \\ref{varieties-lemma-check-invertible-sheaf-trivial}." ], "refs": [ "curves-lemma-rr", "varieties-lemma-check-invertible-sheaf-trivial" ], "ref_ids": [ 6256, 11114 ] } ], "ref_ids": [] }, { "id": 6273, "type": "theorem", "label": "curves-lemma-genus-zero-positive-degree", "categories": [ "curves" ], "title": "curves-lemma-genus-zero-positive-degree", "contents": [ "Let $X$ be a proper curve over a field $k$ with $H^0(X, \\mathcal{O}_X) = k$.", "Assume $X$ has genus $0$. Let $\\mathcal{L}$ be an invertible", "$\\mathcal{O}_X$-module of degree $d > 0$. Then we have", "\\begin{enumerate}", "\\item $\\dim_k H^0(X, \\mathcal{L}) = d + 1$ and $\\dim_k H^1(X, \\mathcal{L}) = 0$,", "\\item $\\mathcal{L}$ is very ample and defines a closed immersion into", "$\\mathbf{P}^d_k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By definition of degree and genus we have", "$$", "\\dim_k H^0(X, \\mathcal{L}) - \\dim_k H^1(X, \\mathcal{L}) = d + 1", "$$", "Let $s$ be a nonzero section of $\\mathcal{L}$.", "Then the zero scheme of $s$ is an effective Cartier", "divisor $D \\subset X$, we have $\\mathcal{L} = \\mathcal{O}_X(D)$ and", "we have a short exact sequence", "$$", "0 \\to \\mathcal{O}_X \\to \\mathcal{L} \\to \\mathcal{L}|_D \\to 0", "$$", "see Divisors, Lemma \\ref{divisors-lemma-characterize-OD} and", "Remark \\ref{divisors-remark-ses-regular-section}.", "Since $H^1(X, \\mathcal{O}_X) = 0$ by assumption, we see that", "$H^0(X, \\mathcal{L}) \\to H^0(X, \\mathcal{L}|_D)$ is surjective.", "As $\\mathcal{L}|_D$ is generated by global sections", "(because $\\dim(D) = 0$, see", "Varieties, Lemma \\ref{varieties-lemma-chi-tensor-finite})", "we conclude that the invertible module $\\mathcal{L}$", "is generated by global sections.", "In fact, since $D$ is an Artinian scheme we have", "$\\mathcal{L}|_D \\cong \\mathcal{O}_D$\\footnote{In our case this", "follows from Divisors, Lemma", "\\ref{divisors-lemma-finite-trivialize-invertible-upstairs}", "as $D \\to \\Spec(k)$ is finite.} and hence we can", "find a section $t$ of $\\mathcal{L}$ whose restriction", "of $D$ generates $\\mathcal{L}|_D$.", "The short exact sequence also shows that $H^1(X, \\mathcal{L}) = 0$.", "\\medskip\\noindent", "For $n \\geq 1$ consider the multiplication map", "$$", "\\mu_n :", "H^0(X, \\mathcal{L}) \\otimes_k H^0(X, \\mathcal{L}^{\\otimes n})", "\\longrightarrow", "H^0(X, \\mathcal{L}^{\\otimes n + 1})", "$$", "We claim this is surjective. To see this we consider the short exact", "sequence", "$$", "0 \\to \\mathcal{L}^{\\otimes n} \\xrightarrow{s}", "\\mathcal{L}^{\\otimes n + 1} \\to \\mathcal{L}^{\\otimes n + 1}|_D \\to 0", "$$", "The sections of $\\mathcal{L}^{\\otimes n + 1}$ coming from the left in this", "sequence are in the image of $\\mu_n$. On the other hand, since", "$H^0(\\mathcal{L}) \\to H^0(\\mathcal{L}|_D)$ is surjective and since", "$t^n$ maps to a trivialization of $\\mathcal{L}^{\\otimes n}|_D$", "we see that $\\mu_n(H^0(X, \\mathcal{L}) \\otimes t^n)$ gives a subspace", "of $H^0(X, \\mathcal{L}^{\\otimes n + 1})$ surjecting onto the global sections of", "$\\mathcal{L}^{\\otimes n + 1}|_D$. This proves the claim.", "\\medskip\\noindent", "Observe that $\\mathcal{L}$ is ample by", "Varieties, Lemma \\ref{varieties-lemma-ample-curve}.", "Hence", "Morphisms, Lemma \\ref{morphisms-lemma-proper-ample-is-proj}", "gives an isomorphism", "$$", "X \\longrightarrow", "\\text{Proj}\\left(", "\\bigoplus\\nolimits_{n \\geq 0} H^0(X, \\mathcal{L}^{\\otimes n})\\right)", "$$", "Since the maps $\\mu_n$ are surjective for all $n \\geq 1$ we see that", "the graded algebra on the right hand side is a quotient of", "the symmetric algebra on $H^0(X, \\mathcal{L})$. Choosing a $k$-basis", "$s_0, \\ldots, s_d$ of $H^0(X, \\mathcal{L})$ we see that", "it is a quotient of a polynomial algebra in $d + 1$ variables.", "Since quotients of graded rings correspond to closed immersions", "of $\\text{Proj}$ (Constructions, Lemma", "\\ref{constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj})", "we find a closed immersion $X \\to \\mathbf{P}^d_k$. We omit the", "verification that this morphism is the morphism of", "Constructions, Lemma \\ref{constructions-lemma-projective-space}", "associated to the sections $s_0, \\ldots, s_d$ of $\\mathcal{L}$." ], "refs": [ "divisors-lemma-characterize-OD", "varieties-lemma-chi-tensor-finite", "divisors-lemma-finite-trivialize-invertible-upstairs", "varieties-lemma-ample-curve", "morphisms-lemma-proper-ample-is-proj", "constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj", "constructions-lemma-projective-space" ], "ref_ids": [ 7944, 11030, 7963, 11116, 5434, 12612, 12621 ] } ], "ref_ids": [] }, { "id": 6274, "type": "theorem", "label": "curves-lemma-genus-zero", "categories": [ "curves" ], "title": "curves-lemma-genus-zero", "contents": [ "Let $X$ be a proper curve over a field $k$ with $H^0(X, \\mathcal{O}_X) = k$.", "If $X$ is Gorenstein and has genus $0$, then $X$", "is isomorphic to a plane curve of degree $2$." ], "refs": [], "proofs": [ { "contents": [ "Consider the invertible sheaf $\\mathcal{L} = \\omega_X^{\\otimes -1}$ where", "$\\omega_X$ is as in Lemma \\ref{lemma-duality-dim-1}. Then", "$\\deg(\\omega_X) = -2$ by Lemma \\ref{lemma-genus-gorenstein}", "and hence $\\deg(\\mathcal{L}) = 2$. By", "Lemma \\ref{lemma-genus-zero-positive-degree}", "we conclude that choosing a basis $s_0, s_1, s_2$ of the $k$-vector", "space of global sections of $\\mathcal{L}$ we obtain a closed immersion", "$$", "\\varphi_{(\\mathcal{L}, (s_0, s_1, s_2))} :", "X \\longrightarrow \\mathbf{P}^2_k", "$$", "Thus $X$ is a plane curve of some degree $d$. Let $F \\in k[T_0, T_1, T_2]_d$", "be its equation (Lemma \\ref{lemma-equation-plane-curve}).", "Because the genus of $X$ is $0$ we see that $d$ is $1$ or $2$", "(Lemma \\ref{lemma-genus-plane-curve}). Observe that", "$F$ restricts to the zero section on $\\varphi(X)$ and hence", "$F(s_0, s_1, s_2)$ is the zero section of $\\mathcal{L}^{\\otimes 2}$.", "Because $s_0, s_1, s_2$ are linearly independent we see that $F$", "cannot be linear, i.e., $d = \\deg(F) \\geq 2$. Thus $d = 2$", "and the proof is complete." ], "refs": [ "curves-lemma-duality-dim-1", "curves-lemma-genus-gorenstein", "curves-lemma-genus-zero-positive-degree", "curves-lemma-genus-plane-curve" ], "ref_ids": [ 6250, 6266, 6273, 6270 ] } ], "ref_ids": [] }, { "id": 6275, "type": "theorem", "label": "curves-lemma-genus-zero-singular", "categories": [ "curves" ], "title": "curves-lemma-genus-zero-singular", "contents": [ "Let $X$ be a proper curve over a field $k$ with $H^0(X, \\mathcal{O}_X) = k$.", "Assume $X$ is singular and has genus $0$. Then there exists a diagram", "$$", "\\xymatrix{", "x' \\ar[d] \\ar[r] & X' \\ar[d]^\\nu \\ar[r] & \\Spec(k') \\ar[d] \\\\", "x \\ar[r] & X \\ar[r] & \\Spec(k)", "}", "$$", "where", "\\begin{enumerate}", "\\item $k'/k$ is a nontrivial finite extension,", "\\item $X' \\cong \\mathbf{P}^1_{k'}$,", "\\item $x'$ is a $k'$-rational point of $X'$,", "\\item $x$ is a $k$-rational point of $X$,", "\\item $X' \\setminus \\{x'\\} \\to X \\setminus \\{x\\}$ is an isomorphism,", "\\item $0 \\to \\mathcal{O}_X \\to \\nu_*\\mathcal{O}_{X'} \\to k'/k \\to 0$", "is a short exact sequence", "where $k'/k = \\kappa(x')/\\kappa(x)$ indicates the skyscraper sheaf", "on the point $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\nu : X' \\to X$ be the normalization of $X$, see", "Varieties, Sections \\ref{varieties-section-normalization} and", "\\ref{varieties-section-normalization-one-dimensional}.", "Since $X$ is singular $\\nu$ is not an isomorphism.", "Then $k' = H^0(X', \\mathcal{O}_{X'})$ is a finite extension of $k$", "(Varieties, Lemma \\ref{varieties-lemma-regular-functions-proper-variety}).", "The short exact sequence", "$$", "0 \\to \\mathcal{O}_X \\to \\nu_*\\mathcal{O}_{X'} \\to \\mathcal{Q} \\to 0", "$$", "and the fact that $\\mathcal{Q}$ is supported in finitely many", "closed points give us that", "\\begin{enumerate}", "\\item $H^1(X', \\mathcal{O}_{X'}) = 0$, i.e., $X'$ has genus $0$", "as a curve over $k'$,", "\\item there is a short exact sequence", "$0 \\to k \\to k' \\to H^0(X, \\mathcal{Q}) \\to 0$.", "\\end{enumerate}", "In particular $k'/k$ is a nontrivial extension.", "\\medskip\\noindent", "Next, we consider what is often called the {\\it conductor ideal}", "$$", "\\mathcal{I} = \\SheafHom_{\\mathcal{O}_X}(\\nu_*\\mathcal{O}_{X'}, \\mathcal{O}_X)", "$$", "This is a quasi-coherent $\\mathcal{O}_X$-module. We view $\\mathcal{I}$", "as an ideal in $\\mathcal{O}_X$ via the map $\\varphi \\mapsto \\varphi(1)$.", "Thus $\\mathcal{I}(U)$ is the set of $f \\in \\mathcal{O}_X(U)$ such that", "$f \\left(\\nu_*\\mathcal{O}_{X'}(U)\\right) \\subset \\mathcal{O}_X(U)$. In", "other words, the condition is that $f$ annihilates $\\mathcal{Q}$.", "In other words, there is a defining exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_X \\to", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{Q}, \\mathcal{Q})", "$$", "Let $U \\subset X$ be an affine open containing the support of $\\mathcal{Q}$.", "Then $V = \\mathcal{Q}(U) = H^0(X, \\mathcal{Q})$ is a $k$-vector space", "of dimension $n - 1$. The image of", "$\\mathcal{O}_X(U) \\to \\Hom_k(V, V)$ is a commutative subalgebra,", "hence has dimension $\\leq n - 1$ over $k$ (this is a property of", "commutative subalgebras of matrix algebras; details omitted).", "We conclude that we have a short exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_X \\to \\mathcal{A} \\to 0", "$$", "where $\\text{Supp}(\\mathcal{A}) = \\text{Supp}(\\mathcal{Q})$", "and $\\dim_k H^0(X, \\mathcal{A}) \\leq n - 1$.", "On the other hand, the description", "$\\mathcal{I} = \\SheafHom_{\\mathcal{O}_X}(\\nu_*\\mathcal{O}_{X'}, \\mathcal{O}_X)$", "provides $\\mathcal{I}$ with a $\\nu_*\\mathcal{O}_{X'}$-module structure", "such that the inclusion map $\\mathcal{I} \\to \\nu_*\\mathcal{O}_{X'}$", "is a $\\nu_*\\mathcal{O}_{X'}$-module map.", "We conclude that $\\mathcal{I} = \\nu_*\\mathcal{I}'$", "for some quasi-coherent sheaf of ideals", "$\\mathcal{I}' \\subset \\mathcal{O}_{X'}$, see", "Morphisms, Lemma \\ref{morphisms-lemma-affine-equivalence-modules}.", "Define $\\mathcal{A}'$ as the cokernel:", "$$", "0 \\to \\mathcal{I}' \\to \\mathcal{O}_{X'} \\to \\mathcal{A}' \\to 0", "$$", "Combining the exact sequences so far we obtain a short exact sequence", "$0 \\to \\mathcal{A} \\to \\nu_*\\mathcal{A}' \\to \\mathcal{Q} \\to 0$.", "Using the estimate above,", "combined with $\\dim_k H^0(X, \\mathcal{Q}) = n - 1$, gives", "$$", "\\dim_k H^0(X', \\mathcal{A}') =", "\\dim_k H^0(X, \\mathcal{A}) + \\dim_k H^0(X, \\mathcal{Q}) \\leq 2 n - 2", "$$", "However, since $X'$ is a curve over $k'$ we see that", "the left hand side is divisible by $n$", "(Varieties, Lemma \\ref{varieties-lemma-divisible}).", "As $\\mathcal{A}$ and $\\mathcal{A}'$ cannot be zero, we conclude that", "$\\dim_k H^0(X', \\mathcal{A}') = n$ which means that $\\mathcal{I}'$", "is the ideal sheaf of a $k'$-rational point $x'$.", "By Proposition \\ref{proposition-projective-line}", "we find $X' \\cong \\mathbf{P}^1_{k'}$.", "Going back to the equalities above, we conclude that", "$\\dim_k H^0(X, \\mathcal{A}) = 1$. This", "means that $\\mathcal{I}$ is the ideal sheaf of a", "$k$-rational point $x$. Then $\\mathcal{A} = \\kappa(x) = k$", "and $\\mathcal{A}' = \\kappa(x') = k'$ as skyscraper sheaves.", "Comparing the exact sequences given above,", "this immediately implies the result on structure sheaves", "as stated in the lemma." ], "refs": [ "varieties-lemma-regular-functions-proper-variety", "morphisms-lemma-affine-equivalence-modules", "varieties-lemma-divisible", "curves-proposition-projective-line" ], "ref_ids": [ 11012, 5174, 11112, 6350 ] } ], "ref_ids": [] }, { "id": 6276, "type": "theorem", "label": "curves-lemma-generically-etale", "categories": [ "curves" ], "title": "curves-lemma-generically-etale", "contents": [ "\\begin{slogan}", "A morphism of smooth curves is separable iff it is etale almost everywhere", "\\end{slogan}", "Let $k$ be a field. Let $f : X \\to Y$ be a morphism of smooth curves over $k$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\text{d}f : f^*\\Omega_{Y/k} \\to \\Omega_{X/k}$ is nonzero,", "\\item $\\Omega_{X/Y}$ is supported on a proper closed subset of $X$,", "\\item there exists a nonempty open $U \\subset X$ such that", "$f|_U : U \\to Y$ is unramified,", "\\item there exists a nonempty open $U \\subset X$ such that", "$f|_U : U \\to Y$ is \\'etale,", "\\item the extension $k(Y) \\subset k(X)$ of function fields is", "finite separable.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $X$ and $Y$ are smooth, the sheaves $\\Omega_{X/k}$ and", "$\\Omega_{Y/k}$ are invertible modules, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-omega-finite-locally-free}.", "Using the exact sequence", "$$", "f^*\\Omega_{Y/k} \\longrightarrow \\Omega_{X/k}", "\\longrightarrow \\Omega_{X/Y} \\longrightarrow 0", "$$", "of Morphisms, Lemma \\ref{morphisms-lemma-triangle-differentials}", "we see that (1) and (2) are equivalent and equivalent to the", "condition that $f^*\\Omega_{Y/k} \\to \\Omega_{X/k}$ is nonzero", "in the generic point. The equivalence of (2) and (3) follows", "from Morphisms, Lemma \\ref{morphisms-lemma-unramified-omega-zero}.", "The equivalence between (3) and (4) follows from", "Morphisms, Lemma \\ref{morphisms-lemma-flat-unramified-etale}", "and the fact that flatness is automatic", "(Lemma \\ref{lemma-flat}).", "To see the equivalence of (5) and (4)", "use Algebra, Lemma \\ref{algebra-lemma-smooth-at-generic-point}.", "Some details omitted." ], "refs": [ "morphisms-lemma-smooth-omega-finite-locally-free", "morphisms-lemma-triangle-differentials", "morphisms-lemma-unramified-omega-zero", "morphisms-lemma-flat-unramified-etale", "curves-lemma-flat", "algebra-lemma-smooth-at-generic-point" ], "ref_ids": [ 5334, 5313, 5343, 5373, 6241, 1228 ] } ], "ref_ids": [] }, { "id": 6277, "type": "theorem", "label": "curves-lemma-rh", "categories": [ "curves" ], "title": "curves-lemma-rh", "contents": [ "Let $f : X \\to Y$ be a morphism of smooth proper curves", "over a field $k$ which satisfies the equivalent conditions of", "Lemma \\ref{lemma-generically-etale}. If", "$k = H^0(Y, \\mathcal{O}_Y) = H^0(X, \\mathcal{O}_X)$", "and $X$ and $Y$ have genus $g_X$ and $g_Y$, then", "$$", "2g_X - 2 = (2g_Y - 2) \\deg(f) + \\deg(R)", "$$", "where $R \\subset X$ is the effective Cartier divisor cut out by", "the different of $f$." ], "refs": [ "curves-lemma-generically-etale" ], "proofs": [ { "contents": [ "See discussion above; we used", "Discriminants, Lemma", "\\ref{discriminant-lemma-discriminant-quasi-finite-morphism-smooth},", "Lemma \\ref{lemma-genus-smooth}, and", "Varieties, Lemmas \\ref{varieties-lemma-degree-tensor-product} and", "\\ref{varieties-lemma-degree-pullback-map-proper-curves}." ], "refs": [ "discriminant-lemma-discriminant-quasi-finite-morphism-smooth", "curves-lemma-genus-smooth", "varieties-lemma-degree-tensor-product", "varieties-lemma-degree-pullback-map-proper-curves" ], "ref_ids": [ 14996, 6267, 11109, 11113 ] } ], "ref_ids": [ 6276 ] }, { "id": 6278, "type": "theorem", "label": "curves-lemma-uniformizer-works", "categories": [ "curves" ], "title": "curves-lemma-uniformizer-works", "contents": [ "Let $X \\to \\Spec(k)$ be smooth of relative dimension $1$ at a closed", "point $x \\in X$. If $\\kappa(x)$ is separable over $k$, then for", "any uniformizer $s$ in the discrete valuation ring $\\mathcal{O}_{X, x}$", "the element $\\text{d}s$ freely generates $\\Omega_{X/k, x}$", "over $\\mathcal{O}_{X, x}$." ], "refs": [], "proofs": [ { "contents": [ "The ring $\\mathcal{O}_{X, x}$ is a discrete valuation ring by", "Algebra, Lemma \\ref{algebra-lemma-characterize-smooth-over-field}.", "Since $x$ is closed $\\kappa(x)$ is finite over $k$. Hence if", "$\\kappa(x)/k$ is separable, then any uniformizer $s$", "maps to a nonzero element of", "$\\Omega_{X/k, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)$ by", "Algebra, Lemma \\ref{algebra-lemma-computation-differential}.", "Since $\\Omega_{X/k, x}$ is free of rank $1$ over $\\mathcal{O}_{X, x}$", "the result follows." ], "refs": [ "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-computation-differential" ], "ref_ids": [ 1223, 1224 ] } ], "ref_ids": [] }, { "id": 6279, "type": "theorem", "label": "curves-lemma-rhe", "categories": [ "curves" ], "title": "curves-lemma-rhe", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-rh}. For a closed point", "$x \\in X$ let $d_x$ be the multiplicity of $x$ in $R$. Then", "$$", "2g_X - 2 = (2g_Y - 2) \\deg(f) + \\sum\\nolimits d_x [\\kappa(x) : k]", "$$", "Moreover, we have the following results", "\\begin{enumerate}", "\\item $d_x = \\text{length}_{\\mathcal{O}_{X, x}}(\\Omega_{X/Y, x})$,", "\\item $d_x \\geq e_x - 1$ where $e_x$ is the ramification index", "of $\\mathcal{O}_{X, x}$ over $\\mathcal{O}_{Y, y}$,", "\\item $d_x = e_x - 1$ if and only if $\\mathcal{O}_{X, x}$ is tamely", "ramified over $\\mathcal{O}_{Y, y}$.", "\\end{enumerate}" ], "refs": [ "curves-lemma-rh" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-rh} and the discussion above", "(which used", "Varieties, Lemma \\ref{varieties-lemma-algebraic-scheme-dim-0}", "and", "Algebra, Lemma \\ref{algebra-lemma-pushdown-module})", "it suffices to prove the results on the", "multiplicity $d_x$ of $x$ in $R$. Part (1) was proved", "in the discussion above. In the discussion above", "we proved (2) and (3) only in the case where $\\kappa(x)$ is", "separable over $k$.", "In the rest of the proof we give a uniform treatment", "of (2) and (3) using material on differents of", "quasi-finite Gorenstein morphisms.", "\\medskip\\noindent", "First, observe that $f$ is a quasi-finite Gorenstein morphism.", "This is true for example because", "$f$ is a flat quasi-finite morphism and $X$ is Gorenstein", "(see Duality for Schemes, Lemma", "\\ref{duality-lemma-flat-morphism-from-gorenstein-scheme})", "or because it was shown in the proof of", "Discriminants, Lemma", "\\ref{discriminant-lemma-discriminant-quasi-finite-morphism-smooth}", "(which we used above). Thus $\\omega_{X/Y}$ is invertible by", "Discriminants, Lemma \\ref{discriminant-lemma-gorenstein-quasi-finite}", "and the same remains true after replacing $X$ by opens and after", "performing a base change by some $Y' \\to Y$. We will use this", "below without further mention.", "\\medskip\\noindent", "Choose affine opens $U \\subset X$ and $V \\subset Y$", "such that $x \\in U$, $y \\in V$, $f(U) \\subset V$, and $x$ is the only", "point of $U$ lying over $y$. Write $U = \\Spec(A)$ and $V = \\Spec(B)$.", "Then $R \\cap U$ is the different of $f|_U : U \\to V$.", "By Discriminants, Lemma \\ref{discriminant-lemma-base-change-different}", "formation of the different commutes with arbitrary base change", "in our case. By our choice of $U$ and $V$ we have", "$$", "A \\otimes_B \\kappa(y) =", "\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{Y, y}} \\kappa(y) =", "\\mathcal{O}_{X, x}/(s^{e_x})", "$$", "where $e_x$ is the ramification index as in the statement of the lemma.", "Let $C = \\mathcal{O}_{X, x}/(s^{e_x})$ viewed as a finite algebra", "over $\\kappa(y)$. Let $\\mathfrak{D}_{C/\\kappa(y)}$ be the different", "of $C$ over $\\kappa(y)$ in the sense of", "Discriminants, Definition \\ref{discriminant-definition-different}.", "It suffices to show: $\\mathfrak{D}_{C/\\kappa(y)}$", "is nonzero if and only if the extension", "$\\mathcal{O}_{Y, y} \\subset \\mathcal{O}_{X, x}$ is tamely ramified", "and in the tamely ramified case $\\mathfrak{D}_{C/\\kappa(y)}$", "is equal to the ideal generated by $s^{e_x - 1}$ in $C$.", "Recall that tame ramification means exactly that $\\kappa(x)/\\kappa(y)$", "is separable and that the characteristic of $\\kappa(y)$ does not", "divide $e_x$. On the other hand, the different of $C/\\kappa(y)$ is nonzero", "if and only if $\\tau_{C/\\kappa(y)} \\in \\omega_{C/\\kappa(y)}$ is nonzero.", "Namely, since $\\omega_{C/\\kappa(y)}$ is an invertible $C$-module", "(as the base change of $\\omega_{A/B}$)", "it is free of rank $1$, say with generator $\\lambda$. Write", "$\\tau_{C/\\kappa(y)} = h\\lambda$ for some $h \\in C$. Then", "$\\mathfrak{D}_{C/\\kappa(y)} = (h) \\subset C$ whence the claim.", "By Discriminants, Lemma \\ref{discriminant-lemma-tau-nonzero}", "we have $\\tau_{C/\\kappa(y)} \\not = 0$", "if and only if $\\kappa(x)/\\kappa(y)$", "is separable and $e_x$ is prime to the characteristic.", "Finally, even if $\\tau_{C/\\kappa(y)}$ is nonzero, then", "it is still the case that $s \\tau_{C/\\kappa(y)} = 0$", "because $s\\tau_{C/\\kappa(y)} : C \\to \\kappa(y)$", "sends $c$ to the trace of the nilpotent operator $sc$ which is zero.", "Hence $sh = 0$, hence $h \\in (s^{e_x - 1})$ which proves", "that $\\mathfrak{D}_{C/\\kappa(y)} \\subset (s^{e_x - 1})$ always.", "Since $(s^{e_x - 1}) \\subset C$ is the smallest nonzero ideal,", "we have proved the final assertion." ], "refs": [ "curves-lemma-rh", "varieties-lemma-algebraic-scheme-dim-0", "algebra-lemma-pushdown-module", "duality-lemma-flat-morphism-from-gorenstein-scheme", "discriminant-lemma-discriminant-quasi-finite-morphism-smooth", "discriminant-lemma-gorenstein-quasi-finite", "discriminant-lemma-base-change-different", "discriminant-definition-different", "discriminant-lemma-tau-nonzero" ], "ref_ids": [ 6277, 10988, 639, 13599, 14996, 15001, 14977, 15005, 14961 ] } ], "ref_ids": [ 6277 ] }, { "id": 6280, "type": "theorem", "label": "curves-lemma-dominated-by-smooth", "categories": [ "curves" ], "title": "curves-lemma-dominated-by-smooth", "contents": [ "Let $k$ be a field. Let $f : X \\to Y$ be a surjective morphism", "of curves over $k$. If $X$ is smooth over $k$ and", "$Y$ is normal, then $Y$ is smooth over $k$." ], "refs": [], "proofs": [ { "contents": [ "Let $y \\in Y$. Pick $x \\in X$ mapping to $y$.", "By Varieties, Lemma \\ref{varieties-lemma-flat-under-smooth}", "it suffices to show that $f$ is flat at $x$.", "This follows from Lemma \\ref{lemma-flat}." ], "refs": [ "varieties-lemma-flat-under-smooth", "curves-lemma-flat" ], "ref_ids": [ 11010, 6241 ] } ], "ref_ids": [] }, { "id": 6281, "type": "theorem", "label": "curves-lemma-purely-inseparable", "categories": [ "curves" ], "title": "curves-lemma-purely-inseparable", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a", "nonconstant morphism of proper nonsingular curves over $k$.", "If the extension $k(Y) \\subset k(X)$ of function fields", "is purely inseparable, then there exists a factorization", "$$", "X = X_0 \\to X_1 \\to \\ldots \\to X_n = Y", "$$", "such that each $X_i$ is a proper nonsingular curve", "and $X_i \\to X_{i + 1}$ is a degree $p$", "morphism with $k(X_{i + 1}) \\subset k(X_i)$", "inseparable." ], "refs": [], "proofs": [ { "contents": [ "This follows from Theorem \\ref{theorem-curves-rational-maps}", "and the fact that a finite purely inseparable extension of fields", "can always be gotten as a sequence of (inseparable) extensions of degree $p$,", "see Fields, Lemma \\ref{fields-lemma-finite-purely-inseparable}." ], "refs": [ "curves-theorem-curves-rational-maps", "fields-lemma-finite-purely-inseparable" ], "ref_ids": [ 6238, 4481 ] } ], "ref_ids": [] }, { "id": 6282, "type": "theorem", "label": "curves-lemma-inseparable-deg-p-smooth", "categories": [ "curves" ], "title": "curves-lemma-inseparable-deg-p-smooth", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a", "nonconstant morphism of proper nonsingular curves over $k$.", "If $X$ is smooth and $k(Y) \\subset k(X)$ is inseparable of degree $p$,", "then there is a unique isomorphism $Y = X^{(p)}$ such that", "$f$ is $F_{X/k}$." ], "refs": [], "proofs": [ { "contents": [ "The relative frobenius morphism $F_{X/k} : X \\to X^{(p)}$", "is constructed in Varieties, Section \\ref{varieties-section-frobenius}.", "Observe that $X^{(p)}$ is a smooth proper curve over $k$", "as a base change of $X$. The morphism $F_{X/k}$ has degree $p$ by", "Varieties, Lemma \\ref{varieties-lemma-inseparable-deg-p-smooth}.", "Thus $k(X^{(p)})$ and $k(Y)$ are both subfields of $k(X)$", "with $[k(X) : k(Y)] = [k(X) : k(X^{(p)})] = p$. To prove the lemma", "it suffices to show that $k(Y) = k(X^{(p)})$ inside $k(X)$. See", "Theorem \\ref{theorem-curves-rational-maps}.", "\\medskip\\noindent", "Write $K = k(X)$. Consider the map $\\text{d} : K \\to \\Omega_{K/k}$.", "It follows from Lemma \\ref{lemma-generically-etale}", "that both $k(Y)$ is contained in the", "kernel of $\\text{d}$. By", "Varieties, Lemma \\ref{varieties-lemma-relative-frobenius-omega}", "we see that $k(X^{(p)})$ is in the kernel of $\\text{d}$.", "Since $X$ is a smooth curve we know that $\\Omega_{K/k}$", "is a vector space of dimension $1$ over $K$.", "Then More on Algebra, Lemma \\ref{more-algebra-lemma-p-basis}.", "implies that $\\Ker(\\text{d}) = kK^p$ and", "that $[K : kK^p] = p$.", "Thus $k(Y) = kK^p = k(X^{(p)})$ for reasons of degree." ], "refs": [ "varieties-lemma-inseparable-deg-p-smooth", "curves-theorem-curves-rational-maps", "curves-lemma-generically-etale", "varieties-lemma-relative-frobenius-omega", "more-algebra-lemma-p-basis" ], "ref_ids": [ 11054, 6238, 6276, 11051, 10068 ] } ], "ref_ids": [] }, { "id": 6283, "type": "theorem", "label": "curves-lemma-purely-inseparable-smooth", "categories": [ "curves" ], "title": "curves-lemma-purely-inseparable-smooth", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a", "nonconstant morphism of proper nonsingular curves over $k$.", "If $X$ is smooth and $k(Y) \\subset k(X)$ is purely inseparable,", "then there is a unique $n \\geq 0$ and a unique isomorphism $Y = X^{(p^n)}$", "such that $f$ is the $n$-fold relative Frobenius of $X/k$." ], "refs": [], "proofs": [ { "contents": [ "The $n$-fold relative Frobenius of $X/k$ is defined in", "Varieties, Remark \\ref{varieties-remark-n-fold-relative-frobenius}.", "The lemma follows by combining Lemmas \\ref{lemma-inseparable-deg-p-smooth}", "and \\ref{lemma-purely-inseparable}." ], "refs": [ "varieties-remark-n-fold-relative-frobenius", "curves-lemma-inseparable-deg-p-smooth", "curves-lemma-purely-inseparable" ], "ref_ids": [ 11168, 6282, 6281 ] } ], "ref_ids": [] }, { "id": 6284, "type": "theorem", "label": "curves-lemma-purely-inseparable-smooth-genus", "categories": [ "curves" ], "title": "curves-lemma-purely-inseparable-smooth-genus", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a", "nonconstant morphism of proper nonsingular curves over $k$.", "Assume", "\\begin{enumerate}", "\\item $X$ is smooth,", "\\item $H^0(X, \\mathcal{O}_X) = k$,", "\\item $k(X)/k(Y)$ is purely inseparable.", "\\end{enumerate}", "Then $Y$ is smooth, $H^0(Y, \\mathcal{O}_Y) = k$, and the genus of $Y$", "is equal to the genus of $X$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-purely-inseparable-smooth}", "we see that $Y = X^{(p^n)}$ is the base change of", "$X$ by $F_{\\Spec(k)}^n$. Thus $Y$ is smooth and the result on the", "cohomology and genus follows from", "Lemma \\ref{lemma-genus-base-change}." ], "refs": [ "curves-lemma-purely-inseparable-smooth", "curves-lemma-genus-base-change" ], "ref_ids": [ 6283, 6265 ] } ], "ref_ids": [] }, { "id": 6285, "type": "theorem", "label": "curves-lemma-inseparable-linear-system", "categories": [ "curves" ], "title": "curves-lemma-inseparable-linear-system", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $X$ be a smooth proper", "curve over $k$. Let $(\\mathcal{L}, V)$ be a $\\mathfrak g^r_d$ with $r \\geq 1$.", "Then one of the following two is true", "\\begin{enumerate}", "\\item there exists a $\\mathfrak g^1_d$ whose corresponding morphism", "$X \\to \\mathbf{P}^1_k$ (Lemma \\ref{lemma-linear-series})", "is generically \\'etale (i.e., is as in Lemma \\ref{lemma-generically-etale}), or", "\\item there exists a $\\mathfrak g^r_{d'}$ on $X^{(p)}$ where", "$d' \\leq d/p$.", "\\end{enumerate}" ], "refs": [ "curves-lemma-linear-series", "curves-lemma-generically-etale" ], "proofs": [ { "contents": [ "Pick two $k$-linearly independent elements $s, t \\in V$.", "Then $f = s/t$ is the rational function defining the morphism", "$X \\to \\mathbf{P}^1_k$ corresponding to the linear series", "$(\\mathcal{L}, ks + kt)$. If this morphism is not", "generically \\'etale, then $f \\in k(X^{(p)})$ by", "Proposition \\ref{proposition-unwind-morphism-smooth}.", "Now choose a basis $s_0, \\ldots, s_r$ of $V$ and let", "$\\mathcal{L}' \\subset \\mathcal{L}$ be the invertible sheaf", "generated by $s_0, \\ldots, s_r$. Set $f_i = s_i/s_0$ in $k(X)$.", "If for each pair $(s_0, s_i)$ we have $f_i \\in k(X^{(p)})$, then", "the morphism", "$$", "\\varphi = \\varphi_{(\\mathcal{L}', (s_0, \\ldots, s_r)} :", "X", "\\longrightarrow", "\\mathbf{P}^r_k = \\text{Proj}(k[T_0, \\ldots, T_r])", "$$", "factors through $X^{(p)}$ as this is true over the affine open", "$D_+(T_0)$ and we can extend the morphism over the affine part", "to the whole of the smooth curve $X^{(p)}$ by", "Lemma \\ref{lemma-extend-over-normal-curve}.", "Introducing notation, say we have the factorization", "$$", "X \\xrightarrow{F_{X/k}} X^{(p)} \\xrightarrow{\\psi} \\mathbf{P}^r_k", "$$", "of $\\varphi$. Then $\\mathcal{N} = \\psi^*\\mathcal{O}_{\\mathbf{P}^1_k}(1)$", "is an invertible $\\mathcal{O}_{X^{(p)}}$-module with", "$\\mathcal{L}' = F_{X/k}^*\\mathcal{N}$ and with", "$\\psi^*T_0, \\ldots, \\psi^*T_r$ $k$-linearly independent", "(as they pullback to $s_0, \\ldots, s_r$ on $X$).", "Finally, we have", "$$", "d = \\deg(\\mathcal{L}) \\geq \\deg(\\mathcal{L}') =", "\\deg(F_{X/k}) \\deg(\\mathcal{N}) = p \\deg(\\mathcal{N})", "$$", "as desired. Here we used Varieties, Lemmas", "\\ref{varieties-lemma-check-invertible-sheaf-trivial},", "\\ref{varieties-lemma-degree-pullback-map-proper-curves}, and", "\\ref{varieties-lemma-inseparable-deg-p-smooth}." ], "refs": [ "curves-proposition-unwind-morphism-smooth", "curves-lemma-extend-over-normal-curve", "varieties-lemma-check-invertible-sheaf-trivial", "varieties-lemma-degree-pullback-map-proper-curves", "varieties-lemma-inseparable-deg-p-smooth" ], "ref_ids": [ 6351, 6240, 11114, 11113, 11054 ] } ], "ref_ids": [ 6246, 6276 ] }, { "id": 6286, "type": "theorem", "label": "curves-lemma-point-over-separable-extension", "categories": [ "curves" ], "title": "curves-lemma-point-over-separable-extension", "contents": [ "Let $k$ be a field. Let $X$ be a smooth proper curve over $k$", "with $H^0(X, \\mathcal{O}_X) = k$ and genus $g \\geq 2$.", "Then there exists a closed point $x \\in X$ with", "$\\kappa(x)/k$ separable of degree $\\leq 2g - 2$." ], "refs": [], "proofs": [ { "contents": [ "Set $\\omega = \\Omega_{X/k}$. By", "Lemma \\ref{lemma-genus-smooth} this has degree $2g - 2$", "and has $g$ global sections. Thus we have a $\\mathfrak g^{g - 1}_{2g - 2}$.", "By the trivial Lemma \\ref{lemma-linear-series-trivial-existence}", "there exists a $\\mathfrak g^1_{2g - 2}$", "and by Lemma \\ref{lemma-g1d} we obtain a morphism", "$$", "\\varphi : X \\longrightarrow \\mathbf{P}^1_k", "$$", "of some degree $d \\leq 2g - 2$. Since $\\varphi$ is flat", "(Lemma \\ref{lemma-flat}) and finite", "(Lemma \\ref{lemma-finite})", "it is finite locally free of degree $d$", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}).", "Pick any rational point $t \\in \\mathbf{P}^1_k$", "and any point $x \\in X$ with $\\varphi(x) = t$.", "Then", "$$", "d \\geq [\\kappa(x) : \\kappa(t)] = [\\kappa(x) : k]", "$$", "for example by", "Morphisms, Lemmas \\ref{morphisms-lemma-finite-locally-free-universally-bounded}", "and \\ref{morphisms-lemma-characterize-universally-bounded}.", "Thus if $k$ is perfect (for example has characteristic zero", "or is finite) then the lemma is proved. Thus we reduce to the", "case discussed in the next paragraph.", "\\medskip\\noindent", "Assume that $k$ is an infinite field of characteristic $p > 0$.", "As above we will use that $X$ has a $\\mathfrak g^{g - 1}_{2g - 2}$.", "The smooth proper curve $X^{(p)}$ has the same genus as $X$.", "Hence its genus is $> 0$. We conclude that $X^{(p)}$ does not have a", "$\\mathfrak g^{g - 1}_d$ for any $d \\leq g - 1$ by", "Lemma \\ref{lemma-grd-inequalities}.", "Applying Lemma \\ref{lemma-inseparable-linear-system}", "to our $\\mathfrak g^{g - 1}_{2g - 2}$ (and noting that $2g - 2/p \\leq g - 1$)", "we conclude that possibility (2) does not occur. Hence we obtain a morphism", "$$", "\\varphi : X \\longrightarrow \\mathbf{P}^1_k", "$$", "which is generically \\'etale (in the sense of the lemma)", "and has degree $\\leq 2g - 2$. Let $U \\subset X$ be the nonempty", "open subscheme where $\\varphi$ is \\'etale. Then", "$\\varphi(U) \\subset \\mathbf{P}^1_k$ is a nonempty Zariski open", "and we can pick a $k$-rational point $t \\in \\varphi(U)$ as $k$ is infinite.", "Let $u \\in U$ be a point with $\\varphi(u) = t$.", "Then $\\kappa(u)/\\kappa(t)$ is separable", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}),", "$\\kappa(t) = k$, and $[\\kappa(u) : k] \\leq 2g - 2$ as before." ], "refs": [ "curves-lemma-genus-smooth", "curves-lemma-linear-series-trivial-existence", "curves-lemma-g1d", "curves-lemma-flat", "curves-lemma-finite", "morphisms-lemma-finite-flat", "morphisms-lemma-finite-locally-free-universally-bounded", "morphisms-lemma-characterize-universally-bounded", "curves-lemma-grd-inequalities", "curves-lemma-inseparable-linear-system", "morphisms-lemma-etale-over-field" ], "ref_ids": [ 6267, 6247, 6248, 6241, 6242, 5471, 5525, 5524, 6249, 6285, 5364 ] } ], "ref_ids": [] }, { "id": 6287, "type": "theorem", "label": "curves-lemma-ramification-to-algebraic-closure", "categories": [ "curves" ], "title": "curves-lemma-ramification-to-algebraic-closure", "contents": [ "Let $X$ be a smooth curve over a field $k$. Let", "$\\overline{x} \\in X_{\\overline{k}}$ be a closed", "point with image $x \\in X$. The ramification index of", "$\\mathcal{O}_{X, x} \\subset \\mathcal{O}_{X_{\\overline{k}}, \\overline{x}}$", "is the inseparable degree of $\\kappa(x)/k$." ], "refs": [], "proofs": [ { "contents": [ "After shrinking $X$ we may assume there is an \\'etale morphism", "$\\pi : X \\to \\mathbf{A}^1_k$, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-etale-over-affine-space}.", "Then we can consider the diagram of local rings", "$$", "\\xymatrix{", "\\mathcal{O}_{X_{\\overline{k}}, \\overline{x}} &", "\\mathcal{O}_{\\mathbf{A}^1_{\\overline{k}}, \\pi(\\overline{x})} \\ar[l] \\\\", "\\mathcal{O}_{X, x} \\ar[u] &", "\\mathcal{O}_{\\mathbf{A}^1_k, \\pi(x)} \\ar[l] \\ar[u]", "}", "$$", "The horizontal arrows have ramification index $1$ as they correspond to", "\\'etale morphisms. Moreover, the extension $\\kappa(x)/\\kappa(\\pi(x))$ is", "separable hence $\\kappa(x)$ and $\\kappa(\\pi(x))$ have the same", "inseparable degree over $k$.", "By multiplicativity of ramification indices it suffices to", "prove the result when $x$ is a point of the affine line.", "\\medskip\\noindent", "Assume $X = \\mathbf{A}^1_k$. In this case, the local ring of $X$ at $x$", "looks like", "$$", "\\mathcal{O}_{X, x} = k[t]_{(P)}", "$$", "where $P$ is an irreducible monic polynomial over $k$.", "Then $P(t) = Q(t^q)$ for some separable polynomial $Q \\in k[t]$, see", "Fields, Lemma \\ref{fields-lemma-irreducible-polynomials}.", "Observe that $\\kappa(x) = k[t]/(P)$ has inseparable degree $q$", "over $k$. On the other hand, over $\\overline{k}$ we can factor", "$Q(t) = \\prod (t - \\alpha_i)$ with $\\alpha_i$ pairwise distinct.", "Write $\\alpha_i = \\beta_i^q$ for some unique $\\beta_i \\in \\overline{k}$.", "Then our point $\\overline{x}$ corresponds to one of the $\\beta_i$", "and we conclude because the ramification index of", "$$", "k[t]_{(P)} \\longrightarrow \\overline{k}[t]_{(t - \\beta_i)}", "$$", "is indeed equal to $q$ as the uniformizer $P$ maps to", "$(t - \\beta_i)^q$ times a unit." ], "refs": [ "morphisms-lemma-smooth-etale-over-affine-space", "fields-lemma-irreducible-polynomials" ], "ref_ids": [ 5377, 4464 ] } ], "ref_ids": [] }, { "id": 6288, "type": "theorem", "label": "curves-lemma-complete-local-ring-pushout", "categories": [ "curves" ], "title": "curves-lemma-complete-local-ring-pushout", "contents": [ "In the situation above, let $Z = \\Spec(k')$ where $k'$ is a field and", "$Z' = \\Spec(k'_1 \\times \\ldots \\times k'_n)$ with $k'_i/k'$", "finite extensions of fields. Let $x \\in X$ be the image of $Z \\to X$", "and $x'_i \\in X'$ the image of $\\Spec(k'_i) \\to X'$.", "Then we have a fibre product diagram", "$$", "\\xymatrix{", "\\prod\\nolimits_{i = 1, \\ldots, n} k'_i &", "\\prod\\nolimits_{i = 1, \\ldots, n} \\mathcal{O}_{X', x'_i}^\\wedge \\ar[l] \\\\", "k' \\ar[u] &", "\\mathcal{O}_{X, x}^\\wedge \\ar[u] \\ar[l]", "}", "$$", "where the horizontal arrows are given by the maps to the residue fields." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine open neighbourhood $\\Spec(A)$ of $x$ in $X$.", "Let $\\Spec(A') \\subset X'$ be the inverse image. By construction", "we have a fibre product diagram", "$$", "\\xymatrix{", "\\prod\\nolimits_{i = 1, \\ldots, n} k'_i &", "A' \\ar[l] \\\\", "k' \\ar[u] &", "A \\ar[u] \\ar[l]", "}", "$$", "Since everything is finite over $A$ we see that the diagram remains", "a fibre product diagram after completion with respect to the", "maximal ideal $\\mathfrak m \\subset A$ corresponding to $x$", "(Algebra, Lemma \\ref{algebra-lemma-completion-flat}).", "Finally, apply Algebra, Lemma", "\\ref{algebra-lemma-completion-finite-extension}", "to identify the completion of $A'$." ], "refs": [ "algebra-lemma-completion-flat", "algebra-lemma-completion-finite-extension" ], "ref_ids": [ 870, 876 ] } ], "ref_ids": [] }, { "id": 6289, "type": "theorem", "label": "curves-lemma-no-in-between-over-k", "categories": [ "curves" ], "title": "curves-lemma-no-in-between-over-k", "contents": [ "Let $k$ be an algebraically closed field. Let $k \\subset A$ be a ring", "extension such that $A$ has exactly two $k$-sub algebras, then", "either $A = k \\times k$ or $A = k[\\epsilon]$." ], "refs": [], "proofs": [ { "contents": [ "The assumption means $k \\not = A$ and any subring $k \\subset C \\subset A$", "is equal to either $k$ or $A$. Let $t \\in A$, $t \\not \\in k$.", "Then $A$ is generated by $t$ over $k$. Hence $A = k[x]/I$ for some", "ideal $I$. If $I = (0)$, then we have the subalgebra $k[x^2]$", "which is not allowed. Otherwise $I$ is generated by a monic polynomial $P$.", "Write $P = \\prod_{i = 1}^d (t - a_i)$. If $d > 2$, then the subalgebra", "generated by $(t - a_1)(t - a_2)$ gives a contradiction.", "Thus $d = 2$. If $a_1 \\not = a_2$, then $A = k \\times k$,", "if $a_1 = a_2$, then $A = k[\\epsilon]$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6290, "type": "theorem", "label": "curves-lemma-factor-almost-isomorphism", "categories": [ "curves" ], "title": "curves-lemma-factor-almost-isomorphism", "contents": [ "Let $k$ be an algebraically closed field. Let $f : X' \\to X$ be a", "finite morphism algebraic $k$-schemes such that", "$\\mathcal{O}_X \\subset f_*\\mathcal{O}_{X'}$ and such that $f$ is an", "isomorphism away from a finite set of points. Then there is a factorization", "$$", "X' = X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "such that each $X_i \\to X_{i - 1}$ is either the glueing of", "two points or the squishing of a tangent vector", "(see Examples \\ref{example-glue-points} and", "\\ref{example-squish-tangent-vector})." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be the maximal open set over which $f$ is an isomorphism.", "Then $X \\setminus U = \\{x_1, \\ldots, x_n\\}$ with $x_i \\in X(k)$.", "We will consider factorizations $X' \\to Y \\to X$ of $f$ such that", "both morphisms are finite and", "$$", "\\mathcal{O}_X \\subset g_*\\mathcal{O}_Y \\subset f_*\\mathcal{O}_{X'}", "$$", "where $g : Y \\to X$ is the given morphism. By assumption", "$\\mathcal{O}_{X, x} \\to (f_*\\mathcal{O}_{X'})_x$ is an isomorphism", "onless $x = x_i$ for some $i$. Hence the cokernel", "$$", "f_*\\mathcal{O}_{X'}/\\mathcal{O}_X = \\bigoplus \\mathcal{Q}_i", "$$", "is a direct sum of skyscraper sheaves $\\mathcal{Q}_i$ supported at", "$x_1, \\ldots, x_n$.", "Because the displayed quotient is a coherent $\\mathcal{O}_X$-module,", "we conclude that $\\mathcal{Q}_i$ has finite length over", "$\\mathcal{O}_{X, x_i}$. Hence we can argue", "by induction on the sum of these lengths, i.e., the length of", "the whole cokernel.", "\\medskip\\noindent", "If $n > 1$, then we can define an $\\mathcal{O}_X$-subalgebra", "$\\mathcal{A} \\subset f_*\\mathcal{O}_{X'}$ by taking the inverse", "image of $\\mathcal{Q}_1$. This will give a nontrivial factorization", "and we win by induction.", "\\medskip\\noindent", "Assume $n = 1$. We abbreviate $x = x_1$. Consider the finite", "$k$-algebra extension", "$$", "A = \\mathcal{O}_{X, x} \\subset (f_*\\mathcal{O}_{X'})_x = B", "$$", "Note that $\\mathcal{Q} = \\mathcal{Q}_1$ is the skyscraper sheaf", "with value $B/A$.", "We have a $k$-subalgebra $A \\subset A + \\mathfrak m_A B \\subset B$.", "If both inclusions are strict, then we obtain a nontrivial", "factorization and we win by induction as above.", "If $A + \\mathfrak m_A B = B$, then $A = B$ by Nakayama, then", "$f$ is an isomorphism and there is nothing to prove.", "We conclude that we may assume $B = A + \\mathfrak m_A B$.", "Set $C = B/\\mathfrak m_A B$. If $C$ has more than $2$", "$k$-subalgebras, then we obtain a subalgebra between $A$", "and $B$ by taking the inverse image in $B$. Thus we may", "assume $C$ has exactly $2$ $k$-subalgebras. Thus $C = k \\times k$", "or $C = k[\\epsilon]$ by Lemma \\ref{lemma-no-in-between-over-k}.", "In this case $f$ is correspondingly the glueing two points or the", "squishing of a tangent vector." ], "refs": [ "curves-lemma-no-in-between-over-k" ], "ref_ids": [ 6289 ] } ], "ref_ids": [] }, { "id": 6291, "type": "theorem", "label": "curves-lemma-glue-points", "categories": [ "curves" ], "title": "curves-lemma-glue-points", "contents": [ "Let $k$ be an algebraically closed field. If $f : X' \\to X$ is the", "glueing of two points $a, b$ as in Example \\ref{example-glue-points}, then", "there is an exact sequence", "$$", "k^* \\to \\Pic(X) \\to \\Pic(X') \\to 0", "$$", "The first map is zero if $a$ and $b$ are on different", "connected components of $X'$ and injective", "if $X'$ is proper and $a$ and $b$ are on the same connected component of $X'$." ], "refs": [], "proofs": [ { "contents": [ "The map $\\Pic(X) \\to \\Pic(X')$ is surjective", "by Varieties, Lemma \\ref{varieties-lemma-surjective-pic-birational-finite}.", "Using the short exact sequence", "$$", "0 \\to \\mathcal{O}_X^* \\to f_*\\mathcal{O}_{X'}^*", "\\xrightarrow{ab^{-1}} x_*k^* \\to 0", "$$", "we obtain", "$$", "H^0(X', \\mathcal{O}_{X'}^*) \\xrightarrow{ab^{-1}} k^* \\to", "H^1(X, \\mathcal{O}_X^*) \\to H^1(X, f_*\\mathcal{O}_{X'}^*)", "$$", "We have $H^1(X, f_*\\mathcal{O}_{X'}^*) \\subset H^1(X', \\mathcal{O}_{X'}^*)$", "(for example by the Leray spectral sequence, see", "Cohomology, Lemma \\ref{cohomology-lemma-Leray}).", "Hence the kernel of $\\Pic(X) \\to \\Pic(X')$ is the", "cokernel of $ab^{-1} : H^0(X', \\mathcal{O}_{X'}^*) \\to k^*$.", "If $a$ and $b$ are on different connected components of $X'$,", "then $ab^{-1}$ is surjective.", "Because $k$ is algebraically closed any regular function on a", "reduced connected proper scheme over $k$ comes from an element of $k$, see", "Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}.", "Thus $ab^{-1}$ is zero if $X'$ is proper and $a$ and $b$ are on", "the same connected component." ], "refs": [ "varieties-lemma-surjective-pic-birational-finite", "cohomology-lemma-Leray", "varieties-lemma-proper-geometrically-reduced-global-sections" ], "ref_ids": [ 11072, 2070, 10948 ] } ], "ref_ids": [] }, { "id": 6292, "type": "theorem", "label": "curves-lemma-squish-tangent-vector", "categories": [ "curves" ], "title": "curves-lemma-squish-tangent-vector", "contents": [ "Let $k$ be an algebraically closed field. If $f : X' \\to X$ is the", "squishing of a tangent vector $\\vartheta$ as in", "Example \\ref{example-squish-tangent-vector}, then", "there is an exact sequence", "$$", "(k, +) \\to \\Pic(X) \\to \\Pic(X') \\to 0", "$$", "and the first map is injective if $X'$ is proper and reduced." ], "refs": [], "proofs": [ { "contents": [ "The map $\\Pic(X) \\to \\Pic(X')$ is surjective", "by Varieties, Lemma \\ref{varieties-lemma-surjective-pic-birational-finite}.", "Using the short exact sequence", "$$", "0 \\to \\mathcal{O}_X^* \\to f_*\\mathcal{O}_{X'}^*", "\\xrightarrow{\\vartheta} x_*k \\to 0", "$$", "of Example \\ref{example-squish-tangent-vector} we obtain", "$$", "H^0(X', \\mathcal{O}_{X'}^*) \\xrightarrow{\\vartheta} k \\to", "H^1(X, \\mathcal{O}_X^*) \\to H^1(X, f_*\\mathcal{O}_{X'}^*)", "$$", "We have $H^1(X, f_*\\mathcal{O}_{X'}^*) \\subset H^1(X', \\mathcal{O}_{X'}^*)$", "(for example by the Leray spectral sequence, see", "Cohomology, Lemma \\ref{cohomology-lemma-Leray}).", "Hence the kernel of $\\Pic(X) \\to \\Pic(X')$ is the", "cokernel of the map $\\vartheta : H^0(X', \\mathcal{O}_{X'}^*) \\to k$.", "Because $k$ is algebraically closed any regular function on a", "reduced connected proper scheme over $k$ comes from an element of $k$, see", "Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}.", "Thus the final statement of the lemma." ], "refs": [ "varieties-lemma-surjective-pic-birational-finite", "cohomology-lemma-Leray", "varieties-lemma-proper-geometrically-reduced-global-sections" ], "ref_ids": [ 11072, 2070, 10948 ] } ], "ref_ids": [] }, { "id": 6293, "type": "theorem", "label": "curves-lemma-multicross-algebra", "categories": [ "curves" ], "title": "curves-lemma-multicross-algebra", "contents": [ "Let $k$ be a separably closed field. Let $A$ be a $1$-dimensional", "reduced Nagata local $k$-algebra with residue field $k$. Then", "$$", "\\delta\\text{-invariant }A \\geq \\text{number of branches of }A - 1", "$$", "If equality holds, then $A^\\wedge$ is as in (\\ref{equation-multicross})." ], "refs": [], "proofs": [ { "contents": [ "Since the residue field of $A$ is separably closed, the number", "of branches of $A$ is equal to the number of geometric branches", "of $A$, see", "More on Algebra, Definition \\ref{more-algebra-definition-number-of-branches}.", "The inequality holds by", "Varieties, Lemma \\ref{varieties-lemma-delta-number-branches-inequality}.", "Assume equality holds.", "We may replace $A$ by the completion of $A$; this does", "not change the number of branches or the $\\delta$-invariant, see", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-one-dimensional-number-of-branches}", "and Varieties, Lemma \\ref{varieties-lemma-delta-same-after-completion}.", "Then $A$ is strictly henselian, see", "Algebra, Lemma \\ref{algebra-lemma-complete-henselian}.", "By Varieties, Lemma \\ref{varieties-lemma-delta-number-branches-inequality-sh}", "we see that $A$ is a wedge of complete discrete valuation rings.", "Each of these is isomorphic to $k[[t]]$ by Algebra, Lemma", "\\ref{algebra-lemma-regular-complete-containing-coefficient-field}.", "Hence $A$ is as in (\\ref{equation-multicross})." ], "refs": [ "more-algebra-definition-number-of-branches", "varieties-lemma-delta-number-branches-inequality", "more-algebra-lemma-one-dimensional-number-of-branches", "varieties-lemma-delta-same-after-completion", "algebra-lemma-complete-henselian", "varieties-lemma-delta-number-branches-inequality-sh", "algebra-lemma-regular-complete-containing-coefficient-field" ], "ref_ids": [ 10638, 11088, 10478, 11081, 1282, 11087, 1331 ] } ], "ref_ids": [] }, { "id": 6294, "type": "theorem", "label": "curves-lemma-multicross", "categories": [ "curves" ], "title": "curves-lemma-multicross", "contents": [ "Let $k$ be an algebraically closed field. Let $X$ be a reduced algebraic", "$1$-dimensional $k$-scheme. Let $x \\in X$. The following are equivalent", "\\begin{enumerate}", "\\item $x$ defines a multicross singularity,", "\\item the $\\delta$-invariant of $X$ at $x$ is the", "number of branches of $X$ at $x$ minus $1$,", "\\item there is a sequence of morphisms", "$U_n \\to U_{n - 1} \\to \\ldots \\to U_0 = U \\subset X$", "where $U$ is an open neighbourhood of $x$, where", "$U_n$ is nonsingular, and where each $U_i \\to U_{i - 1}$", "is the glueing of two points as in Example \\ref{example-glue-points}.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is Lemma \\ref{lemma-multicross-algebra}.", "\\medskip\\noindent", "Assume (3). We will argue by descending induction on $i$ that all singularities", "of $U_i$ are multicross. This is true for $U_n$ as $U_n$ has no singular points.", "If $U_i$ is gotten from $U_{i + 1}$ by glueing $a, b \\in U_{i + 1}$", "to a point $c \\in U_i$, then we see that", "$$", "\\mathcal{O}_{U_i, c}^\\wedge \\subset", "\\mathcal{O}_{U_{i + 1}, a}^\\wedge \\times \\mathcal{O}_{U_{i + 1}, b}^\\wedge", "$$", "is the set of elements having the same residue classes in $k$.", "Thus the number of branches at $c$ is the sum of the number of", "branches at $a$ and $b$, and the $\\delta$-invariant at $c$", "is the sum of the $\\delta$-invariants at $a$ and $b$ plus $1$", "(because the displayed inclusion has codimension $1$).", "This proves that (2) holds as desired.", "\\medskip\\noindent", "Assume the equivalent conditions (1) and (2). We may choose an open", "$U \\subset X$ such that $x$ is the only singular point of $U$.", "Then we apply Lemma \\ref{lemma-factor-almost-isomorphism} to", "the normalization morphism", "$$", "U^\\nu = U_n \\to U_{n - 1} \\to \\ldots \\to U_1 \\to U_0 = U", "$$", "All we have to do is show that in none of the steps we are", "squishing a tangent vector. Suppose $U_{i + 1} \\to U_i$ is the", "smallest $i$ such that this is the squishing of a tangent", "vector $\\theta$ at $u' \\in U_{i + 1}$ lying over $u \\in U_i$.", "Arguing as above, we see that $u_i$ is a multicross singularity", "(because the maps $U_i \\to \\ldots \\to U_0$ are glueing of", "pairs of points). But now the number of branches at $u'$ and $u$", "is the same and the $\\delta$-invariant of $U_i$ at $u$", "is $1$ bigger than the $\\delta$-invariant of $U_{i + 1}$ at $u'$.", "By Lemma \\ref{lemma-multicross-algebra}", "this implies that $u$ cannot be a multicross singularity which", "is a contradiction." ], "refs": [ "curves-lemma-multicross-algebra", "curves-lemma-factor-almost-isomorphism", "curves-lemma-multicross-algebra" ], "ref_ids": [ 6293, 6290, 6293 ] } ], "ref_ids": [] }, { "id": 6295, "type": "theorem", "label": "curves-lemma-multicross-gorenstein-is-nodal", "categories": [ "curves" ], "title": "curves-lemma-multicross-gorenstein-is-nodal", "contents": [ "Let $k$ be an algebraically closed field. Let $X$ be a reduced algebraic", "$1$-dimensional $k$-scheme. Let $x \\in X$ be a multicross singularity", "(Definition \\ref{definition-multicross}).", "If $X$ is Gorenstein, then $x$ is a node." ], "refs": [ "curves-definition-multicross" ], "proofs": [ { "contents": [ "The map $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X, x}^\\wedge$", "is flat and unramified in the sense that", "$\\kappa(x) = \\mathcal{O}_{X, x}^\\wedge/\\mathfrak m_x \\mathcal{O}_{X, x}^\\wedge$.", "(See More on Algebra, Section \\ref{more-algebra-section-permanence-completion}.)", "Thus $X$ is Gorenstein implies $\\mathcal{O}_{X, x}$ is Gorenstein, implies", "$\\mathcal{O}_{X, x}^\\wedge$ is Gorenstein by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-flat-under-gorenstein}.", "Thus it suffices to show that the ring $A$ in", "(\\ref{equation-multicross}) with $n \\geq 2$", "is Gorenstein if and only if $n = 2$.", "\\medskip\\noindent", "If $n = 2$, then $A = k[[x, y]]/(xy)$ is a complete intersection", "and hence Gorenstein. For example this follows from", "Duality for Schemes, Lemma \\ref{duality-lemma-gorenstein-lci}", "applied to $k[[x, y]] \\to A$ and the fact that the regular", "local ring $k[[x, y]]$ is Gorenstein by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-regular-gorenstein}.", "\\medskip\\noindent", "Assume $n > 2$. If $A$ where Gorenstein, then $A$ would be a", "dualizing complex over $A$", "(Duality for Schemes, Definition \\ref{duality-definition-gorenstein}).", "Then $R\\Hom(k, A)$ would be equal to $k[n]$ for some $n \\in \\mathbf{Z}$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-find-function}.", "It would follow that $\\Ext^1_A(k, A) \\cong k$", "or $\\Ext^1_A(k, A) = 0$ (depending on the value of $n$;", "in fact $n$ has to be $-1$ but it doesn't matter to us here).", "Using the exact sequence", "$$", "0 \\to \\mathfrak m_A \\to A \\to k \\to 0", "$$", "we find that", "$$", "\\Ext^1_A(k, A) = \\Hom_A(\\mathfrak m_A, A)/A", "$$", "where $A \\to \\Hom_A(\\mathfrak m_A, A)$ is given by", "$a \\mapsto (a' \\mapsto aa')$. Let $e_i \\in \\Hom_A(\\mathfrak m_A, A)$", "be the element that sends $(f_1, \\ldots, f_n) \\in \\mathfrak m_A$ to", "$(0, \\ldots, 0, f_i, 0, \\ldots, 0)$. The reader verifies easily", "that $e_1, \\ldots, e_{n - 1}$ are $k$-linearly independent in", "$\\Hom_A(\\mathfrak m_A, A)/A$. Thus", "$\\dim_k \\Ext^1_A(k, A) \\geq n - 1 \\geq 2$ which", "finishes the proof.", "(Observe that $e_1 + \\ldots + e_n$ is the image of $1$ under the map", "$A \\to \\Hom_A(\\mathfrak m_A, A)$.)" ], "refs": [ "dualizing-lemma-flat-under-gorenstein", "duality-lemma-gorenstein-lci", "dualizing-lemma-regular-gorenstein", "duality-definition-gorenstein", "dualizing-lemma-find-function" ], "ref_ids": [ 2885, 13592, 2880, 13642, 2856 ] } ], "ref_ids": [ 6357 ] }, { "id": 6296, "type": "theorem", "label": "curves-lemma-torsion-picard-smooth-projective", "categories": [ "curves" ], "title": "curves-lemma-torsion-picard-smooth-projective", "contents": [ "Let $k$ be an algebraically closed field.", "Let $X$ be a smooth projective curve of genus $g$ over $k$.", "\\begin{enumerate}", "\\item If $n \\geq 1$ is invertible in $k$, then", "$\\Pic(X)[n] \\cong (\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2g}$.", "\\item If the characteristic of $k$ is $p > 0$, then there exists", "an integer $0 \\leq f \\leq g$ such that", "$\\Pic(X)[p^m] \\cong (\\mathbf{Z}/p^m\\mathbf{Z})^{\\oplus f}$ for", "all $m \\geq 1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\Pic^0(X) \\subset \\Pic(X)$", "denote the subgroup of invertible sheaves of degree $0$.", "In other words, there is a short exact sequence", "$$", "0 \\to \\Pic^0(X) \\to \\Pic(X) \\xrightarrow{\\deg} \\mathbf{Z} \\to 0.", "$$", "The group $\\Pic^0(X)$ is the $k$-points of", "the group scheme $\\underline{\\Picardfunctor}^0_{X/k}$, see", "Picard Schemes of Curves, Lemma \\ref{pic-lemma-picard-pieces}.", "The same lemma tells us that $\\underline{\\Picardfunctor}^0_{X/k}$", "is a $g$-dimensional abelian variety over $k$ as defined in", "Groupoids, Definition \\ref{groupoids-definition-abelian-variety}.", "Thus we conclude by the results of", "Groupoids, Proposition \\ref{groupoids-proposition-review-abelian-varieties}." ], "refs": [ "pic-lemma-picard-pieces", "groupoids-definition-abelian-variety", "groupoids-proposition-review-abelian-varieties" ], "ref_ids": [ 12571, 9675, 9668 ] } ], "ref_ids": [] }, { "id": 6297, "type": "theorem", "label": "curves-lemma-torsion-picard-becomes-visible", "categories": [ "curves" ], "title": "curves-lemma-torsion-picard-becomes-visible", "contents": [ "Let $k$ be a field. Let $n$ be prime to the characteristic of $k$.", "Let $X$ be a smooth proper curve over $k$ with $H^0(X, \\mathcal{O}_X) = k$", "and of genus $g$.", "\\begin{enumerate}", "\\item If $g = 1$ then there exists a finite separable extension", "$k'/k$ such that $X_{k'}$ has a $k'$-rational point and", "$\\Pic(X_{k'})[n] \\cong (\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2}$.", "\\item If $g \\geq 2$ then there exists a finite separable extension", "$k'/k$ with $[k' : k] \\leq (2g - 2)(n^{2g})!$", "such that $X_{k'}$ has a $k'$-rational point and", "$\\Pic(X_{k'})[n] \\cong (\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2g}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $g \\geq 2$. First we may choose a finite separable extension", "of degree at most $2g - 2$ such that $X$ acquires a rational point.", "Thus we may assume $X$ has a $k$-rational point $x \\in X(k)$", "but now we have to prove the lemma with", "$[k' : k] \\leq \\leq (n^{2g})!$.", "Let $k \\subset k^{sep} \\subset \\overline{k}$ be a separable algebraic", "closure inside an algebraic closure.", "By Lemma \\ref{lemma-torsion-picard-smooth-projective} we have", "$$", "\\Pic(X_{\\overline{k}})[n] \\cong (\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2g}", "$$", "By Picard Schemes of Curves, Lemma \\ref{pic-lemma-torsion-descends}", "we conclude that", "$$", "\\Pic(X_{k^{sep}})[n] \\cong (\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2g}", "$$", "By Picard Schemes of Curves, Lemma \\ref{pic-lemma-torsion-descends}", "there is a continuous action", "$$", "\\text{Gal}(k^{sep}/k)", "\\longrightarrow", "\\text{Aut}(\\Pic(X_{k^{sep}})[n]", "$$", "and the lemma is true for the fixed field $k'$ of the kernel of this map.", "The kernel is open because the action is continuous which implies", "that $k'/k$ is finite. By Galois theory $\\text{Gal}(k'/k)$", "is the image of the displayed arrow. Since the permutation", "group of a set of cardinality $n^{2g}$ has cardinality $(n^{2g})!$", "we conclude by Galois theory that $[k' : k] \\leq (n^{2g})!$.", "(Of course this proves the lemma with the bound", "$|\\text{GL}_{2g}(\\mathbf{Z}/n\\mathbf{Z})|$, but all we want", "here is that there is some bound.)", "\\medskip\\noindent", "If the genus is $1$, then there is no upper bound on the degree of a", "finite separable field extension over which $X$ acquires a rational point", "(details omitted). Still, there is such an extension for example by", "Varieties, Lemma \\ref{varieties-lemma-smooth-separable-closed-points-dense}.", "The rest of the proof is the same as in the case of $g \\geq 2$." ], "refs": [ "curves-lemma-torsion-picard-smooth-projective", "pic-lemma-torsion-descends", "pic-lemma-torsion-descends", "varieties-lemma-smooth-separable-closed-points-dense" ], "ref_ids": [ 6296, 12573, 12573, 11007 ] } ], "ref_ids": [] }, { "id": 6298, "type": "theorem", "label": "curves-lemma-bound-geometric-genus", "categories": [ "curves" ], "title": "curves-lemma-bound-geometric-genus", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$.", "Then", "$$", "g_{geom}(X/k) = \\sum\\nolimits_{C \\subset X} g_{geom}(C/k)", "$$", "where the sum is over irreducible components $C \\subset X$ of dimension $1$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition and the fact that an irreducible", "component $\\overline{Z}$ of $X_{\\overline{k}}$ maps onto an", "irreducible component $Z$ of $X$", "(Varieties, Lemma \\ref{varieties-lemma-image-irreducible})", "of the same dimension", "(Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}", "applied to the generic point of $\\overline{Z}$)." ], "refs": [ "varieties-lemma-image-irreducible", "morphisms-lemma-dimension-fibre-after-base-change" ], "ref_ids": [ 10940, 5279 ] } ], "ref_ids": [] }, { "id": 6299, "type": "theorem", "label": "curves-lemma-geometric-genus-normalization", "categories": [ "curves" ], "title": "curves-lemma-geometric-genus-normalization", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$.", "Then", "\\begin{enumerate}", "\\item We have $g_{geom}(X/k) = g_{geom}(X_{red}/k)$.", "\\item If $X' \\to X$ is a birational proper morphism, then", "$g_{geom}(X'/k) = g_{geom}(X/k)$.", "\\item If $X^\\nu \\to X$ is the normalization morphism, then", "$g_{geom}(X^\\nu/k) = g_{geom}(X/k)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is immediate from Lemma \\ref{lemma-bound-geometric-genus}.", "If $X' \\to X$ is proper birational, then it is finite and", "an isomorphism over a dense open (see", "Varieties, Lemmas \\ref{varieties-lemma-finite-in-codim-1} and", "\\ref{varieties-lemma-modification-normal-iso-over-codimension-1}).", "Hence $X'_{\\overline{k}} \\to X_{\\overline{k}}$ is an isomorphism", "over a dense open. Thus the irreducible components of $X'_{\\overline{k}}$", "and $X_{\\overline{k}}$ are in bijective correspondence and the", "corresponding components have isomorphic function fields.", "In particular these components have isomorphic nonsingular projective models", "and hence have the same geometric genera.", "This proves (2).", "Part (3) follows from (1) and (2) and the fact that", "$X^\\nu \\to X_{red}$ is birational", "(Morphisms, Lemma \\ref{morphisms-lemma-normalization-birational})." ], "refs": [ "curves-lemma-bound-geometric-genus", "varieties-lemma-finite-in-codim-1", "varieties-lemma-modification-normal-iso-over-codimension-1", "morphisms-lemma-normalization-birational" ], "ref_ids": [ 6298, 10978, 10979, 5517 ] } ], "ref_ids": [] }, { "id": 6300, "type": "theorem", "label": "curves-lemma-genus-goes-down", "categories": [ "curves" ], "title": "curves-lemma-genus-goes-down", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme of dimension", "$\\leq 1$ over $k$. Let $f : Y \\to X$ be a finite morphism", "such that there exists a dense open $U \\subset X$ over", "which $f$ is a closed immersion. Then", "$$", "\\dim_k H^1(X, \\mathcal{O}_X) \\geq \\dim_k H^1(Y, \\mathcal{O}_Y)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Consider the exact sequence", "$$", "0 \\to \\mathcal{G} \\to \\mathcal{O}_X \\to f_*\\mathcal{O}_Y \\to \\mathcal{F} \\to 0", "$$", "of coherent sheaves on $X$.", "By assumption $\\mathcal{F}$ is supported in finitely many closed points", "and hence has vanishing higher cohomology", "(Varieties, Lemma \\ref{varieties-lemma-chi-tensor-finite}).", "On the other hand, we have $H^2(X, \\mathcal{G}) = 0$ by", "Cohomology, Proposition \\ref{cohomology-proposition-vanishing-Noetherian}.", "It follows formally that the induced map", "$H^1(X, \\mathcal{O}_X) \\to H^1(X, f_*\\mathcal{O}_Y)$", "is surjective. Since $H^1(X, f_*\\mathcal{O}_Y) = H^1(Y, \\mathcal{O}_Y)$", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology})", "we conclude the lemma holds." ], "refs": [ "varieties-lemma-chi-tensor-finite", "cohomology-proposition-vanishing-Noetherian", "coherent-lemma-relative-affine-cohomology" ], "ref_ids": [ 11030, 2246, 3284 ] } ], "ref_ids": [] }, { "id": 6301, "type": "theorem", "label": "curves-lemma-genus-normalization", "categories": [ "curves" ], "title": "curves-lemma-genus-normalization", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$.", "If $X' \\to X$ is a birational proper morphism, then", "$$", "\\dim_k H^1(X, \\mathcal{O}_X) \\geq \\dim_k H^1(X', \\mathcal{O}_{X'})", "$$", "If $X$ is reduced, $H^0(X, \\mathcal{O}_X) \\to H^0(X', \\mathcal{O}_{X'})$", "is surjective, and equality holds, then $X' = X$." ], "refs": [], "proofs": [ { "contents": [ "If $f : X' \\to X$ is proper birational, then it is finite and", "an isomorphism over a dense open (see", "Varieties, Lemmas \\ref{varieties-lemma-finite-in-codim-1} and", "\\ref{varieties-lemma-modification-normal-iso-over-codimension-1}).", "Thus the inequality by Lemma \\ref{lemma-genus-goes-down}.", "Assume $X$ is reduced. Then $\\mathcal{O}_X \\to f_*\\mathcal{O}_{X'}$", "is injective and we obtain a short exact sequence", "$$", "0 \\to \\mathcal{O}_X \\to f_*\\mathcal{O}_{X'} \\to \\mathcal{F} \\to 0", "$$", "Under the assumptions given in the second statement,", "we conclude from the long exact cohomology sequence that", "$H^0(X, \\mathcal{F}) = 0$. Then $\\mathcal{F} = 0$ because", "$\\mathcal{F}$ is generated by global sections", "(Varieties, Lemma \\ref{varieties-lemma-chi-tensor-finite}).", "and $\\mathcal{O}_X = f_*\\mathcal{O}_{X'}$. Since $f$ is affine", "this implies $X = X'$." ], "refs": [ "varieties-lemma-finite-in-codim-1", "varieties-lemma-modification-normal-iso-over-codimension-1", "curves-lemma-genus-goes-down", "varieties-lemma-chi-tensor-finite" ], "ref_ids": [ 10978, 10979, 6300, 11030 ] } ], "ref_ids": [] }, { "id": 6302, "type": "theorem", "label": "curves-lemma-bound-geometric-genus-curve", "categories": [ "curves" ], "title": "curves-lemma-bound-geometric-genus-curve", "contents": [ "Let $k$ be a field. Let $C$ be a proper curve over $k$.", "Set $\\kappa = H^0(C, \\mathcal{O}_C)$. Then", "$$", "[\\kappa : k]_s \\dim_\\kappa H^1(C, \\mathcal{O}_C) \\geq g_{geom}(C/k)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Varieties, Lemma \\ref{varieties-lemma-regular-functions-proper-variety}", "implies $\\kappa$ is a field and a finite extension of $k$.", "By Fields, Lemma \\ref{fields-lemma-separable-degree}", "we have $[\\kappa : k]_s = |\\Mor_k(\\kappa, \\overline{k})|$", "and hence $\\Spec(\\kappa \\otimes_k \\overline{k})$ has", "$[\\kappa : k]_s$ points each with residue field $\\overline{k}$.", "Thus", "$$", "C_{\\overline{k}} =", "\\bigcup\\nolimits_{t \\in \\Spec(\\kappa \\otimes_k \\overline{k})} C_t", "$$", "(set theoretic union). Here", "$C_t = C \\times_{\\Spec(\\kappa), t} \\Spec(\\overline{k})$ where", "we view $t$ as a $k$-algebra map $t : \\kappa \\to \\overline{k}$.", "The conclusion is that $g_{geom}(C/k) = \\sum_t g_{geom}(C_t/\\overline{k})$", "and the sum is over an index set of size $[\\kappa : k]_s$.", "We have", "$$", "H^0(C_t, \\mathcal{O}_{C_t}) = \\overline{k}", "\\quad\\text{and}\\quad", "\\dim_{\\overline{k}} H^1(C_t, \\mathcal{O}_{C_t}) =", "\\dim_\\kappa H^1(C, \\mathcal{O}_C)", "$$", "by cohomology and base change", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}).", "Observe that the normalization $C_t^\\nu$ is the disjoint", "union of the nonsingular projective models of the", "irreducible components of $C_t$", "(Morphisms, Lemma \\ref{morphisms-lemma-normalization-in-terms-of-components}).", "Hence $\\dim_{\\overline{k}} H^1(C_t^\\nu, \\mathcal{O}_{C_t^\\nu})$", "is equal to $g_{geom}(C_t/\\overline{k})$.", "By Lemma \\ref{lemma-genus-goes-down} we have", "$$", "\\dim_{\\overline{k}} H^1(C_t, \\mathcal{O}_{C_t}) \\geq", "\\dim_{\\overline{k}} H^1(C_t^\\nu, \\mathcal{O}_{C_t^\\nu})", "$$", "and this finishes the proof." ], "refs": [ "varieties-lemma-regular-functions-proper-variety", "fields-lemma-separable-degree", "coherent-lemma-flat-base-change-cohomology", "morphisms-lemma-normalization-in-terms-of-components", "curves-lemma-genus-goes-down" ], "ref_ids": [ 11012, 4483, 3298, 5516, 6300 ] } ], "ref_ids": [] }, { "id": 6303, "type": "theorem", "label": "curves-lemma-bound-torsion-simple", "categories": [ "curves" ], "title": "curves-lemma-bound-torsion-simple", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$.", "Let $\\ell$ be a prime number invertible in $k$. Then", "$$", "\\dim_{\\mathbf{F}_\\ell} \\Pic(X)[\\ell] \\leq", "\\dim_k H^1(X, \\mathcal{O}_X) + g_{geom}(X/k)", "$$", "where $g_{geom}(X/k)$ is as defined above." ], "refs": [], "proofs": [ { "contents": [ "The map $\\Pic(X) \\to \\Pic(X_{\\overline{k}})$", "is injective by Varieties, Lemma \\ref{varieties-lemma-change-fields-pic}.", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}", "$\\dim_k H^1(X, \\mathcal{O}_X)$ equals", "$\\dim_{\\overline{k}} H^1(X_{\\overline{k}}, \\mathcal{O}_{X_{\\overline{k}}})$.", "Hence we may assume $k$ is algebraically closed.", "\\medskip\\noindent", "Let $X_{red}$ be the reduction of $X$. Then the surjection", "$\\mathcal{O}_X \\to \\mathcal{O}_{X_{red}}$ induces a surjection", "$H^1(X, \\mathcal{O}_X) \\to H^1(X, \\mathcal{O}_{X_{red}})$", "because cohomology of quasi-coherent sheaves vanishes in degrees", "$\\geq 2$ by", "Cohomology, Proposition \\ref{cohomology-proposition-vanishing-Noetherian}.", "Since $X_{red} \\to X$ induces an isomorphism on irreducible", "components over $\\overline{k}$ and an isomorphism on", "$\\ell$-torsion in Picard groups", "(Picard Schemes of Curves, Lemma \\ref{pic-lemma-torsion-descends})", "we may replace $X$ by $X_{red}$. In this way we reduce to", "Proposition \\ref{proposition-torsion-picard-reduced-proper}." ], "refs": [ "varieties-lemma-change-fields-pic", "coherent-lemma-flat-base-change-cohomology", "cohomology-proposition-vanishing-Noetherian", "pic-lemma-torsion-descends", "curves-proposition-torsion-picard-reduced-proper" ], "ref_ids": [ 11027, 3298, 2246, 12573, 6352 ] } ], "ref_ids": [] }, { "id": 6304, "type": "theorem", "label": "curves-lemma-reduced-quotient-regular-ring-dim-2", "categories": [ "curves" ], "title": "curves-lemma-reduced-quotient-regular-ring-dim-2", "contents": [ "Let $(A, \\mathfrak m)$ be a regular local ring of dimension $2$.", "Let $I \\subset \\mathfrak m$ be an ideal.", "\\begin{enumerate}", "\\item If $A/I$ is reduced, then $I = (0)$, $I = \\mathfrak m$, or", "$I = (f)$ for some nonzero $f \\in \\mathfrak m$.", "\\item If $A/I$ has depth $1$, then $I = (f)$ for some nonzero", "$f \\in \\mathfrak m$. ", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $I \\not = 0$. Write $I = (f_1, \\ldots, f_r)$. As $A$ is a UFD", "(More on Algebra, Lemma \\ref{more-algebra-lemma-regular-local-UFD})", "we can write $f_i = fg_i$ where $f$ is the gcd of $f_1, \\ldots, f_r$.", "Thus the gcd of $g_1, \\ldots, g_r$ is $1$ which means that", "there is no height $1$ prime ideal over $g_1, \\ldots, g_r$.", "Since $\\dim(A) = 2$ this implies that", "$V(g_1, \\ldots, g_r) = \\{\\mathfrak m\\}$, i.e.,", "$\\mathfrak m = \\sqrt{(g_1, \\ldots, g_r)}$.", "\\medskip\\noindent", "Assume $A/I$ reduced, i.e., $I$ radical. If $f$ is a unit, then since $I$", "is radical we see that $I = \\mathfrak m$. If $f \\in \\mathfrak m$, then we", "see that $f^n$ maps to zero in $A/I$. Hence $f \\in I$ by reducedness and", "we conclude $I = (f)$.", "\\medskip\\noindent", "Assume $A/I$ has depth $1$. Then $\\mathfrak m$ is not an associated", "prime of $A/I$. Since the class of $f$ modulo $I$ is annihilated", "by $g_1, \\ldots, g_r$, this implies that the class of $f$ is zero", "in $A/I$. Thus $I = (f)$ as desired." ], "refs": [ "more-algebra-lemma-regular-local-UFD" ], "ref_ids": [ 10544 ] } ], "ref_ids": [] }, { "id": 6305, "type": "theorem", "label": "curves-lemma-nodal-algebraic", "categories": [ "curves" ], "title": "curves-lemma-nodal-algebraic", "contents": [ "Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian", "local $k$-algebra. The following are equivalent", "\\begin{enumerate}", "\\item $\\kappa/k$ is separable, $A$ is reduced,", "$\\dim_\\kappa(\\mathfrak m/\\mathfrak m^2) = 2$, and there exists a nondegenerate", "$q \\in \\text{Sym}^2_\\kappa(\\mathfrak m/\\mathfrak m^2)$", "which maps to zero in $\\mathfrak m^2/\\mathfrak m^3$,", "\\item $\\kappa/k$ is separable, $\\text{depth}(A) = 1$,", "$\\dim_\\kappa(\\mathfrak m/\\mathfrak m^2) = 2$, and there exists a nondegenerate", "$q \\in \\text{Sym}^2_\\kappa(\\mathfrak m/\\mathfrak m^2)$", "which maps to zero in $\\mathfrak m^2/\\mathfrak m^3$,", "\\item $\\kappa/k$ is separable,", "$A^\\wedge \\cong \\kappa[[x, y]]/(ax^2 + bxy + cy^2)$", "as a $k$-algebra where $ax^2 + bxy + cy^2$ is a nondegenerate quadratic form", "over $\\kappa$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (3). Then $A^\\wedge$ is reduced because $ax^2 + bxy + cy^2$", "is either irreducible or a product of two nonassociated prime elements.", "Hence $A \\subset A^\\wedge$ is reduced. It follows that (1) is true.", "\\medskip\\noindent", "Assume (1). Then $A$ cannot be Artinian, since it would not be reduced", "because $\\mathfrak m \\not = (0)$.", "Hence $\\dim(A) \\geq 1$, hence $\\text{depth}(A) \\geq 1$", "by Algebra, Lemma \\ref{algebra-lemma-criterion-reduced}.", "On the other hand $\\dim(A) = 2$ implies $A$ is regular", "which contradicts the existence of $q$ by", "Algebra, Lemma \\ref{algebra-lemma-regular-graded}.", "Thus $\\dim(A) \\leq 1$ and we conclude $\\text{depth}(A) = 1$", "by Algebra, Lemma \\ref{algebra-lemma-bound-depth}.", "It follows that (2) is true.", "\\medskip\\noindent", "Assume (2). Since the depth of $A$ is the same as the depth of $A^\\wedge$", "(More on Algebra, Lemma \\ref{more-algebra-lemma-completion-depth})", "and since the other conditions are insensitive to completion, we may", "assume that $A$ is complete. Choose $\\kappa \\to A$ as in", "More on Algebra, Lemma \\ref{more-algebra-lemma-lift-residue-field}.", "Since $\\dim_\\kappa(\\mathfrak m/\\mathfrak m^2) = 2$ we can choose", "$x_0, y_0 \\in \\mathfrak m$ which map to a basis.", "We obtain a continuous $\\kappa$-algebra map", "$$", "\\kappa[[x, y]] \\longrightarrow A", "$$", "by the rules $x \\mapsto x_0$ and $y \\mapsto y_0$. Let", "$q$ be the class of $ax_0^2 + bx_0y_0 + cy_0^2$ in", "$\\text{Sym}^2_\\kappa(\\mathfrak m/\\mathfrak m^2)$.", "Write $Q(x, y) = ax^2 + bxy + cy^2$ viewed as a polynomial", "in two variables. Then we see that", "$$", "Q(x_0, y_0) = ax_0^2 + bx_0y_0 + cy_0^2 =", "\\sum\\nolimits_{i + j = 3} a_{ij} x_0^iy_0^j", "$$", "for some $a_{ij}$ in $A$. We want to prove that we can", "increase the order of vanishing by changing our choice", "of $x_0$, $y_0$. Suppose that $x_1, y_1 \\in \\mathfrak m^2$.", "Then", "$$", "Q(x_0 + x_1, y_0 + y_1) = Q(x_0, y_0) +", "(2ax_0 + by_0)x_1 + (bx_0 + 2cy_0)y_1 \\bmod \\mathfrak m^4", "$$", "Nondegeneracy of $Q$ means exactly that $2ax_0 + by_0$ and $bx_0 + 2cy_0$", "are a $\\kappa$-basis for $\\mathfrak m/\\mathfrak m^2$, see discussion", "preceding the lemma. Hence we can", "certainly choose $x_1, y_1 \\in \\mathfrak m^2$ such that", "$Q(x_0 + x_1, y_0 + y_1) \\in \\mathfrak m^4$.", "Continuing in this fashion by induction", "we can find $x_i, y_i \\in \\mathfrak m^{i + 1}$", "such that", "$$", "Q(x_0 + x_1 + \\ldots + x_n, y_0 + y_1 + \\ldots + y_n) \\in \\mathfrak m^{n + 3}", "$$", "Since $A$ is complete we can set", "$x_\\infty = \\sum x_i$ and $y_\\infty = \\sum y_i$", "and we can consider the map $\\kappa[[x, y]] \\longrightarrow A$", "sending $x$ to $x_\\infty$ and $y$ to $y_\\infty$. This map", "induces a surjection $\\kappa[[x, y]]/(Q) \\longrightarrow A$ by", "Algebra, Lemma \\ref{algebra-lemma-completion-generalities}.", "By Lemma \\ref{lemma-reduced-quotient-regular-ring-dim-2}", "the kernel of $k[[x, y]] \\to A$ is principal.", "But the kernel cannot contain a proper divisor of $Q$", "as such a divisor would have degree $1$ in $x, y$", "and this would contradict $\\dim(\\mathfrak m/\\mathfrak m^2) = 2$.", "Hence $Q$ generates the kernel as desired." ], "refs": [ "algebra-lemma-criterion-reduced", "algebra-lemma-regular-graded", "algebra-lemma-bound-depth", "more-algebra-lemma-completion-depth", "more-algebra-lemma-lift-residue-field", "algebra-lemma-completion-generalities", "curves-lemma-reduced-quotient-regular-ring-dim-2" ], "ref_ids": [ 1310, 939, 770, 10043, 10023, 858, 6304 ] } ], "ref_ids": [] }, { "id": 6306, "type": "theorem", "label": "curves-lemma-2-branches-delta-1", "categories": [ "curves" ], "title": "curves-lemma-2-branches-delta-1", "contents": [ "Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be a", "Nagata local $k$-algebra. The following are equivalent", "\\begin{enumerate}", "\\item $k \\to A$ is as in Lemma \\ref{lemma-nodal-algebraic},", "\\item $\\kappa/k$ is separable, $A$ is reduced of dimension $1$,", "the $\\delta$-invariant of $A$ is $1$, and $A$ has $2$ geometric branches.", "\\end{enumerate}", "If this holds, then the integral closure $A'$ of $A$", "in its total ring of fractions has either $1$ or $2$", "maximal ideals $\\mathfrak m'$ and the extensions", "$\\kappa(\\mathfrak m')/k$ are separable." ], "refs": [ "curves-lemma-nodal-algebraic" ], "proofs": [ { "contents": [ "In both cases $A$ and $A^\\wedge$ are reduced. In case (2)", "because the completion of a reduced local Nagata ring is reduced", "(More on Algebra, Lemma \\ref{more-algebra-lemma-completion-reduced}).", "In both cases $A$ and $A^\\wedge$ have dimension $1$", "(More on Algebra, Lemma \\ref{more-algebra-lemma-completion-dimension}).", "The $\\delta$-invariant and the number of geometric branches of", "$A$ and $A^\\wedge$ agree by", "Varieties, Lemma \\ref{varieties-lemma-delta-same-after-completion}", "and", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-one-dimensional-number-of-branches}.", "Let $A'$ be the integral closure of $A$ in its total ring of fractions", "as in Varieties, Lemma \\ref{varieties-lemma-pre-delta-invariant}.", "By Varieties, Lemma \\ref{varieties-lemma-normalization-same-after-completion}", "we see that $A' \\otimes_A A^\\wedge$ plays the same role for $A^\\wedge$.", "Thus we may replace $A$ by $A^\\wedge$ and assume $A$ is complete.", "\\medskip\\noindent", "Assume (1) holds. It suffices to show that $A$ has two", "geometric branches and $\\delta$-invariant $1$.", "We may assume $A = \\kappa[[x, y]]/(ax^2 + bxy + cy^2)$ with", "$q = ax^2 + bxy + cy^2$ nondegenerate. There are two cases.", "\\medskip\\noindent", "Case I: $q$ splits over $\\kappa$. In this case we may after", "changing coordinates assume that $q = xy$. Then we see that", "$$", "A' = \\kappa[[x, y]]/(x) \\times \\kappa[[x, y]]/(y)", "$$", "\\medskip\\noindent", "Case II: $q$ does not split. In this case $c \\not = 0$ and", "nondegenerate means $b^2 - 4ac \\not = 0$. Hence", "$\\kappa' = \\kappa[t]/(a + bt + ct^2)$ is a degree $2$", "separable extension of $\\kappa$. Then $t = y/x$", "is integral over $A$ and we conclude that", "$$", "A' = \\kappa'[[x]]", "$$", "with $y$ mapping to $tx$ on the right hand side.", "\\medskip\\noindent", "In both cases one verifies by hand that the $\\delta$-invariant", "is $1$ and the number of geometric branches is $2$. In", "this way we see that (1) implies (2).", "Moreover we conclude that the final statement of the lemma holds.", "\\medskip\\noindent", "Assume (2) holds. ", "More on Algebra, Lemma \\ref{more-algebra-lemma-number-of-branches-1}", "implies $A'$ either has two maximal ideals or $A'$ has one maximal ideal", "and $[\\kappa(\\mathfrak m') : \\kappa]_s = 2$.", "\\medskip\\noindent", "Case I: $A'$ has two maximal ideals $\\mathfrak m'_1$, $\\mathfrak m'_2$", "with residue fields $\\kappa_1$, $\\kappa_2$.", "Since the $\\delta$-invariant is the length of $A'/A$ and", "since there is a surjection $A'/A \\to (\\kappa_1 \\times \\kappa_2)/\\kappa$", "we see that $\\kappa = \\kappa_1 = \\kappa_2$. Since $A$ is complete", "(and henselian by Algebra, Lemma \\ref{algebra-lemma-complete-henselian})", "and $A'$ is finite over $A$ we see that $A' = A_1 \\times A_2$", "(by Algebra, Lemma \\ref{algebra-lemma-finite-over-henselian}).", "Since $A'$ is a normal ring it follows that $A_1$ and $A_2$ are", "discrete valuation rings.", "Hence $A_1$ and $A_2$ are isomorphic to $\\kappa[[t]]$", "(as $k$-algebras) by", "More on Algebra, Lemma \\ref{more-algebra-lemma-power-series-over-residue-field}.", "Since the $\\delta$-invariant is $1$ we conclude that $A$", "is the wedge of $A_1$ and $A_2$", "(Varieties, Definition \\ref{varieties-definition-wedge}).", "It follows easily that $A \\cong \\kappa[[x, y]]/(xy)$.", "\\medskip\\noindent", "Case II: $A'$ has a single maximal ideal $\\mathfrak m'$ with residue", "field $\\kappa'$ and $[\\kappa' : \\kappa]_s = 2$. Arguing exactly", "as in Case I we see that $[\\kappa' : \\kappa] = 2$ and $\\kappa'$", "is separable over $\\kappa$. Since $A'$ is normal we see that", "$A'$ is isomorphic to $\\kappa'[[t]]$ (see reference above).", "Since $A'/A$ has length $1$ we conclude that", "$$", "A = \\{f \\in \\kappa'[[t]] \\mid f(0) \\in \\kappa\\}", "$$", "Then a simple computation shows that $A$ as in case (1)." ], "refs": [ "more-algebra-lemma-completion-reduced", "more-algebra-lemma-completion-dimension", "varieties-lemma-delta-same-after-completion", "more-algebra-lemma-one-dimensional-number-of-branches", "varieties-lemma-pre-delta-invariant", "varieties-lemma-normalization-same-after-completion", "more-algebra-lemma-number-of-branches-1", "algebra-lemma-complete-henselian", "algebra-lemma-finite-over-henselian", "more-algebra-lemma-power-series-over-residue-field", "varieties-definition-wedge" ], "ref_ids": [ 10047, 10042, 11081, 10478, 11078, 11080, 10469, 1282, 1277, 10024, 11159 ] } ], "ref_ids": [ 6305 ] }, { "id": 6307, "type": "theorem", "label": "curves-lemma-fitting-ideal-well-defined", "categories": [ "curves" ], "title": "curves-lemma-fitting-ideal-well-defined", "contents": [ "Let $k$ be a field. Let $A = k[[x_1, \\ldots, x_n]]$. Let", "$I = (f_1, \\ldots, f_m) \\subset A$ be an ideal. For any", "$r \\geq 0$ the ideal in $A/I$ generated by the $r \\times r$-minors", "of the matrix $(\\partial f_j/\\partial x_i)$ is independent", "of the choice of the generators of $I$ or the", "regular system of parameters $x_1, \\ldots, x_n$ of $A$." ], "refs": [], "proofs": [ { "contents": [ "The ``correct'' proof of this lemma is to prove that this ideal", "is the $(n - r)$th Fitting ideal of a module of continuous differentials", "of $A/I$ over $k$. Here is a direct proof.", "If $g_1, \\ldots g_l$ is a second set of generators of $I$, then", "we can write $g_s = \\sum a_{sj}f_j$ and we have the equality of matrices", "$$", "(\\partial g_s/\\partial x_i) = (a_{sj}) (\\partial f_j/\\partial x_i)", "+ (\\partial a_{sj}/\\partial x_i f_j)", "$$", "The final term is zero in $A/I$.", "By the Cauchy-Binet formula we see that the ideal of minors for the", "$g_s$ is contained in the ideal for the $f_j$. By symmetry", "these ideals are the same. If $y_1, \\ldots, y_n \\in \\mathfrak m_A$", "is a second regular system of parameters, then the matrix", "$(\\partial y_j/\\partial x_i)$", "is invertible and we can use the chain rule for differentiation.", "Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6308, "type": "theorem", "label": "curves-lemma-fitting-ideal", "categories": [ "curves" ], "title": "curves-lemma-fitting-ideal", "contents": [ "Let $k$ be a field. Let $A = k[[x_1, \\ldots, x_n]]$. Let", "$I = (f_1, \\ldots, f_m) \\subset \\mathfrak m_A$ be an ideal. The following", "are equivalent", "\\begin{enumerate}", "\\item $k \\to A/I$ is as in Lemma \\ref{lemma-nodal-algebraic},", "\\item $A/I$ is reduced and the", "$(n - 1) \\times (n - 1)$ minors of the matrix", "$(\\partial f_j/\\partial x_i)$ generate $I + \\mathfrak m_A$,", "\\item $\\text{depth}(A/I) = 1$ and the", "$(n - 1) \\times (n - 1)$ minors of the matrix", "$(\\partial f_j/\\partial x_i)$ generate $I + \\mathfrak m_A$.", "\\end{enumerate}" ], "refs": [ "curves-lemma-nodal-algebraic" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-fitting-ideal-well-defined}", "we may change our system of coordinates and the", "choice of generators during the proof.", "\\medskip\\noindent", "If (1) holds, then we may change coordinates such that", "$x_1, \\ldots, x_{n - 2}$ map to zero in $A/I$ and", "$A/I = k[[x_{n - 1}, x_n]]/(a x_{n - 1}^2 + b x_{n - 1}x_n + c x_n^2)$", "for some nondegenerate quadric $a x_{n - 1}^2 + b x_{n - 1}x_n + c x_n^2$.", "Then we can explicitly compute to show that both (2) and (3) are true.", "\\medskip\\noindent", "Assume the $(n - 1) \\times (n - 1)$ minors of the matrix", "$(\\partial f_j/\\partial x_i)$ generate $I + \\mathfrak m_A$.", "Suppose that for some $i$ and $j$ the partial derivative", "$\\partial f_j/\\partial x_i$ is a unit in $A$. Then", "we may use the system of parameters", "$f_j, x_1, \\ldots, x_{i - 1}, \\hat x_i, x_{i + 1}, \\ldots, x_n$", "and the generators", "$f_j, f_1, \\ldots, f_{j - 1}, \\hat f_j, f_{j + 1}, \\ldots, f_m$", "of $I$. Then we get a regular system of parameters $x_1, \\ldots, x_n$", "and generators $x_1, f_2, \\ldots, f_m$ of $I$.", "Next, we look for an $i \\geq 2$ and $j \\geq 2$ such that", "$\\partial f_j/\\partial x_i$ is a unit in $A$. If such a pair", "exists, then we can make a replacement as above and assume", "that we have a regular system of parameters", "$x_1, \\ldots, x_n$ and generators $x_1, x_2, f_3, \\ldots, f_m$ of $I$.", "Continuing, in finitely many steps we reach the situation where", "we have a regular system of parameters", "$x_1, \\ldots, x_n$ and generators", "$x_1, \\ldots, x_t, f_{t + 1}, \\ldots, f_m$ of $I$", "such that $\\partial f_j/\\partial x_i \\in \\mathfrak m_A$", "for all $i, j \\geq t + 1$.", "\\medskip\\noindent", "In this case the matrix of partial derivatives has the following", "block shape", "$$", "\\left(", "\\begin{matrix}", "I_{t \\times t} & * \\\\", "0 & \\mathfrak m_A", "\\end{matrix}", "\\right)", "$$", "Hence every $(n - 1) \\times (n - 1)$-minor is in $\\mathfrak m_A^{n - 1 - t}$.", "Note that $I \\not = \\mathfrak m_A$ otherwise the ideal of minors", "would contain $1$. It follows that $n - 1 - t \\leq 1$ because there", "is an element of $\\mathfrak m_A \\setminus \\mathfrak m_A^2 + I$ (otherwise", "$I = \\mathfrak m_A$ by Nakayama). Thus $t \\geq n - 2$.", "We have seen that $t \\not = n$ above and similarly if", "$t = n - 1$, then there is an invertible $(n - 1) \\times (n - 1)$-minor", "which is disallowed as well. Hence $t = n - 2$.", "Then $A/I$ is a quotient of $k[[x_{n - 1}, x_n]]$ and", "Lemma \\ref{lemma-reduced-quotient-regular-ring-dim-2}", "implies in both cases (2) and (3) that $I$ is generated by", "$x_1, \\ldots, x_{n - 2}, f$ for some $f = f(x_{n - 1}, x_n)$.", "In this case the condition on the minors exactly says that the quadratic", "term in $f$ is nondegenerate, i.e., $A/I$ is as in", "Lemma \\ref{lemma-nodal-algebraic}." ], "refs": [ "curves-lemma-fitting-ideal-well-defined", "curves-lemma-reduced-quotient-regular-ring-dim-2", "curves-lemma-nodal-algebraic" ], "ref_ids": [ 6307, 6304, 6305 ] } ], "ref_ids": [ 6305 ] }, { "id": 6309, "type": "theorem", "label": "curves-lemma-nodal", "categories": [ "curves" ], "title": "curves-lemma-nodal", "contents": [ "Let $k$ be a field. Let $X$ be a $1$-dimensional algebraic $k$-scheme.", "Let $x \\in X$ be a closed point. The following are equivalent", "\\begin{enumerate}", "\\item $x$ is a node,", "\\item $k \\to \\mathcal{O}_{X, x}$ is as in Lemma \\ref{lemma-nodal-algebraic},", "\\item any $\\overline{x} \\in X_{\\overline{k}}$ mapping to $x$ defines", "a nodal singularity,", "\\item $\\kappa(x)/k$ is separable, $\\mathcal{O}_{X, x}$ is reduced, and", "the first Fitting ideal of $\\Omega_{X/k}$ generates $\\mathfrak m_x$", "in $\\mathcal{O}_{X, x}$,", "\\item $\\kappa(x)/k$ is separable, $\\text{depth}(\\mathcal{O}_{X, x}) = 1$, and", "the first Fitting ideal of $\\Omega_{X/k}$ generates $\\mathfrak m_x$", "in $\\mathcal{O}_{X, x}$,", "\\item $\\kappa(x)/k$ is separable and $\\mathcal{O}_{X, x}$ is reduced, has", "$\\delta$-invariant $1$, and has $2$ geometric branches.", "\\end{enumerate}" ], "refs": [ "curves-lemma-nodal-algebraic" ], "proofs": [ { "contents": [ "First assume that $k$ is algebraically closed.", "In this case the equivalence of (1) and (3) is trivial.", "The equivalence of (1) and (3) with (2) holds because the only", "nondegenerate quadric in two variables is $xy$ up to change in", "coordinates. The equivalence of (1) and (6) is ", "Lemma \\ref{lemma-multicross-algebra}.", "After replacing $X$ by an affine neighbourhood", "of $x$, we may assume there is a closed immersion $X \\to \\mathbf{A}^n_k$", "mapping $x$ to $0$. Let $f_1, \\ldots, f_m \\in k[x_1, \\ldots, x_n]$", "be generators for the ideal $I$ of $X$ in $\\mathbf{A}^n_k$.", "Then $\\Omega_{X/k}$ corresponds to the $R = k[x_1, \\ldots, x_n]/I$-module", "$\\Omega_{R/k}$ which has a presentation", "$$", "R^{\\oplus m} \\xrightarrow{(\\partial f_j/\\partial x_i)}", "R^{\\oplus n} \\to \\Omega_{R/k} \\to 0", "$$", "(See Algebra, Sections \\ref{algebra-section-differentials} and", "\\ref{algebra-section-netherlander}.)", "The first Fitting ideal of $\\Omega_{R/k}$ is thus the ideal", "generated by the $(n - 1) \\times (n - 1)$-minors of the", "matrix $(\\partial f_j/\\partial x_i)$. Hence (2), (4), (5)", "are equivalent by Lemma \\ref{lemma-fitting-ideal} applied", "to the completion of $k[x_1, \\ldots, x_n] \\to R$", "at the maximal ideal $(x_1, \\ldots, x_n)$.", "\\medskip\\noindent", "Now assume $k$ is an arbitrary field.", "In cases (2), (4), (5), (6) the residue field $\\kappa(x)$ is", "separable over $k$. Let us show this holds as well", "in cases (1) and (3). Namely, let $Z \\subset X$ be the closed subscheme", "of $X$ defined by the first Fitting ideal of $\\Omega_{X/k}$.", "The formation of $Z$ commutes with field extension", "(Divisors, Lemma \\ref{divisors-lemma-base-change-and-fitting-ideal-omega}).", "If (1) or (3) is true, then there exists a point", "$\\overline{x}$ of $X_{\\overline{k}}$ such that $\\overline{x}$", "is an isolated point of multiplicity $1$ of $Z_{\\overline{k}}$ (as we have the", "equivalence of the conditions of the lemma over $\\overline{k}$).", "In particular $Z_{\\overline{x}}$ is geometrically reduced at $\\overline{x}$", "(because $\\overline{k}$ is algebraically closed). Hence", "$Z$ is geometrically reduced at $x$", "(Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced-upstairs}).", "In particular, $Z$ is reduced at $x$, hence $Z = \\Spec(\\kappa(x))$", "in a neighbourhood of $x$ and $\\kappa(x)$ is geometrically reduced", "over $k$. This means that $\\kappa(x)/k$ is separable", "(Algebra, Lemma \\ref{algebra-lemma-characterize-separable-field-extensions}).", "\\medskip\\noindent", "The argument of the previous paragraph shows that if (1) or (3) holds, then", "the first Fitting ideal of $\\Omega_{X/k}$ generates $\\mathfrak m_x$.", "Since $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X_{\\overline{k}}, \\overline{x}}$", "is flat and since $\\mathcal{O}_{X_{\\overline{k}}, \\overline{x}}$", "is reduced and has depth $1$, we see that (4) and (5) hold", "(use Algebra, Lemmas \\ref{algebra-lemma-descent-reduced} and", "\\ref{algebra-lemma-apply-grothendieck}).", "Conversely, (4) implies (5) by", "Algebra, Lemma \\ref{algebra-lemma-criterion-reduced}.", "If (5) holds, then $Z$ is geometrically reduced at $x$", "(because $\\kappa(x)/k$ separable and $Z$ is $x$ in a neighbourhood).", "Hence $Z_{\\overline{k}}$ is reduced at any point $\\overline{x}$", "of $X_{\\overline{k}}$ lying over $x$. In other words, the first", "fitting ideal of $\\Omega_{X_{\\overline{k}}/\\overline{k}}$ generates", "$\\mathfrak m_{\\overline{x}}$ in $\\mathcal{O}_{X_{\\overline{k}, \\overline{x}}}$.", "Moreover, since", "$\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X_{\\overline{k}}, \\overline{x}}$ is flat", "we see that $\\text{depth}(\\mathcal{O}_{X_{\\overline{k}}, \\overline{x}}) = 1$", "(see reference above).", "Hence (5) holds for $\\overline{x} \\in X_{\\overline{k}}$ and we", "conclude that (3) holds (because of the equivalence over algebraically", "closed fields). In this way we see that (1), (3), (4), (5)", "are equivalent.", "\\medskip\\noindent", "The equivalence of (2) and (6) follows from", "Lemma \\ref{lemma-2-branches-delta-1}.", "\\medskip\\noindent", "Finally, we prove the equivalence of (2) $=$ (6) with", "(1) $=$ (3) $=$ (4) $=$ (5). First we note that the geometric number", "of branches of $X$ at $x$ and the geometric number of branches", "of $X_{\\overline{k}}$ at $\\overline{x}$ are equal by", "Varieties, Lemma", "\\ref{varieties-lemma-geometric-branches-and-change-of-fields}.", "We conclude from the information available to us at this point", "that in all cases this number is equal to $2$.", "On the other hand, in case (1) it is clear that $X$ is geometrically", "reduced at $x$, and hence", "$$", "\\delta\\text{-invariant of }X\\text{ at }x \\leq", "\\delta\\text{-invariant of }X_{\\overline{k}}\\text{ at }\\overline{x}", "$$", "by Varieties, Lemma \\ref{varieties-lemma-delta-invariant-and-change-of-fields}.", "Since in case (1) the right hand side is $1$, this", "forces the $\\delta$-invariant of $X$ at $x$ to be $1$", "(because if it were zero, then $\\mathcal{O}_{X, x}$ would", "be a discrete valuation ring by", "Varieties, Lemma \\ref{varieties-lemma-delta-invariant-is-zero}", "which is unibranch, a contradiction). Thus (5) holds.", "Conversely, if (2) $=$ (5) is true, then assumptions (a), (b), (c) of", "Varieties, Lemma \\ref{varieties-lemma-geometrically-normal-in-codim-1}", "hold for $x \\in X$ by", "Lemma \\ref{lemma-2-branches-delta-1}. Thus", "Varieties, Lemma", "\\ref{varieties-lemma-delta-invariant-and-change-of-fields-better}", "applies and shows that we have equality in the above displayed inequality.", "We conclude that (5) holds for $\\overline{x} \\in X_{\\overline{k}}$", "and we are back in case (1) by the equivalence of the conditions", "over an algebraically closed field." ], "refs": [ "curves-lemma-multicross-algebra", "curves-lemma-fitting-ideal", "divisors-lemma-base-change-and-fitting-ideal-omega", "varieties-lemma-geometrically-reduced-upstairs", "algebra-lemma-characterize-separable-field-extensions", "algebra-lemma-descent-reduced", "algebra-lemma-apply-grothendieck", "algebra-lemma-criterion-reduced", "curves-lemma-2-branches-delta-1", "varieties-lemma-geometric-branches-and-change-of-fields", "varieties-lemma-delta-invariant-and-change-of-fields", "varieties-lemma-delta-invariant-is-zero", "varieties-lemma-geometrically-normal-in-codim-1", "curves-lemma-2-branches-delta-1", "varieties-lemma-delta-invariant-and-change-of-fields-better" ], "ref_ids": [ 6293, 6308, 7900, 10910, 569, 1371, 1361, 1310, 6306, 11085, 11082, 11079, 11017, 6306, 11083 ] } ], "ref_ids": [ 6305 ] }, { "id": 6310, "type": "theorem", "label": "curves-lemma-split-node", "categories": [ "curves" ], "title": "curves-lemma-split-node", "contents": [ "Let $k$ be a field. Let $X$ be a $1$-dimensional algebraic $k$-scheme.", "Let $x \\in X$ be a closed point. The following are equivalent", "\\begin{enumerate}", "\\item $x$ is a split node,", "\\item $x$ is a node and there are exactly two points $x_1, x_2$", "of the normalization $X^\\nu$ lying over $x$ with", "$k = \\kappa(x_1) = \\kappa(x_2)$,", "\\item $\\mathcal{O}_{X, x}^\\wedge \\cong k[[x, y]]/(xy)$ as a $k$-algebra, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from the discussion in", "Remark \\ref{remark-trivial-quadratic-extension}", "and Lemma \\ref{lemma-nodal}." ], "refs": [ "curves-remark-trivial-quadratic-extension", "curves-lemma-nodal" ], "ref_ids": [ 6365, 6309 ] } ], "ref_ids": [] }, { "id": 6311, "type": "theorem", "label": "curves-lemma-node-field-extension", "categories": [ "curves" ], "title": "curves-lemma-node-field-extension", "contents": [ "Let $K/k$ be an extension of fields. Let $X$ be a locally algebraic", "$k$-scheme of dimension $1$. Let $y \\in X_K$ be a point with image", "$x \\in X$. The following are equivalent", "\\begin{enumerate}", "\\item $x$ is a closed point of $X$ and a node, and", "\\item $y$ is a closed point of $Y$ and a node.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $x$ is a closed point of $X$, then $y$ is too (look at residue fields).", "But conversely, this need not be the case, i.e., it can happen that a", "closed point of $Y$ maps to a nonclosed point of $X$. However, in this", "case $y$ cannot be a node. Namely, then $X$ would be geometrically", "unibranch at $x$ (because $x$ would be a generic point of $X$ and", "$\\mathcal{O}_{X, x}$ would be Artinian and any Artinian local ring is", "geometrically unibranch), hence $Y$ is geometrically unibranch at $y$", "(Varieties, Lemma", "\\ref{varieties-lemma-geometrically-unibranch-and-change-of-fields}),", "which means that $y$ cannot be a node by Lemma \\ref{lemma-nodal}.", "Thus we may and do assume that both $x$ and $y$ are closed points.", "\\medskip\\noindent", "Choose algebraic closures $\\overline{k}$, $\\overline{K}$", "and a map $\\overline{k} \\to \\overline{K}$ extending the", "given map $k \\to K$. Using the equivalence of (1) and (3)", "in Lemma \\ref{lemma-nodal}", "we reduce to the case where $k$ and $K$ are algebraically closed.", "In this case we can argue as in the proof of", "Lemma \\ref{lemma-nodal} that the geometric number of branches", "and $\\delta$-invariants of $X$ at $x$ and $Y$ at $y$ are the same.", "Another argument can be given by choosing an isomorphism", "$k[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m) \\to \\mathcal{O}_{X, x}^\\wedge$", "of $k$-algebras as in Varieties, Lemma", "\\ref{varieties-lemma-complete-local-ring-structure-as-algebra}.", "By Varieties, Lemma \\ref{varieties-lemma-base-change-complete-local-ring}", "this gives an isomorphism", "$K[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m) \\to \\mathcal{O}_{Y, y}^\\wedge$", "of $K$-algebras. By definition we have to show that", "$$", "k[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m) \\cong k[[s, t]]/(st)", "$$", "if and only if", "$$", "K[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m) \\cong K[[s, t]]/(st)", "$$", "We encourage the reader to prove this for themselves.", "Since $k$ and $K$ are algebraically closed fields, this is the same as", "asking these rings to be as in Lemma \\ref{lemma-nodal-algebraic}.", "Via Lemma \\ref{lemma-fitting-ideal} this translates into a statement", "about the $(n - 1) \\times (n - 1)$-minors of the matrix", "$(\\partial g_j/\\partial x_i)$ which is clearly independent of the", "field used. We omit the details." ], "refs": [ "varieties-lemma-geometrically-unibranch-and-change-of-fields", "curves-lemma-nodal", "curves-lemma-nodal", "curves-lemma-nodal", "varieties-lemma-complete-local-ring-structure-as-algebra", "varieties-lemma-base-change-complete-local-ring", "curves-lemma-nodal-algebraic", "curves-lemma-fitting-ideal" ], "ref_ids": [ 11086, 6309, 6309, 6309, 10992, 10993, 6305, 6308 ] } ], "ref_ids": [] }, { "id": 6312, "type": "theorem", "label": "curves-lemma-node-etale-local", "categories": [ "curves" ], "title": "curves-lemma-node-etale-local", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic", "$k$-scheme of dimension $1$. Let $Y \\to X$ be an \\'etale morphism.", "Let $y \\in Y$ be a point with image $x \\in X$. The following are equivalent", "\\begin{enumerate}", "\\item $x$ is a closed point of $X$ and a node, and", "\\item $y$ is a closed point of $Y$ and a node.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-node-field-extension}", "we may base change to the algebraic closure of $k$.", "Then the residue fields of $x$ and $y$ are $k$.", "Hence the map $\\mathcal{O}_{X, x}^\\wedge \\to \\mathcal{O}_{Y, y}^\\wedge$", "is an isomorphism (for example by", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-characterize-etale-completions} or", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-unramified}).", "Thus the lemma is clear." ], "refs": [ "curves-lemma-node-field-extension", "etale-lemma-characterize-etale-completions", "more-algebra-lemma-flat-unramified" ], "ref_ids": [ 6311, 10705, 10050 ] } ], "ref_ids": [] }, { "id": 6313, "type": "theorem", "label": "curves-lemma-node-over-separable-extension", "categories": [ "curves" ], "title": "curves-lemma-node-over-separable-extension", "contents": [ "Let $k'/k$ be a finite separable field extension.", "Let $X$ be a locally algebraic $k'$-scheme of dimension $1$.", "Let $x \\in X$ be a closed point. The following are equivalent", "\\begin{enumerate}", "\\item $x$ is a node, and", "\\item $x$ is a node when $X$ viewed as a locally algebraic $k$-scheme.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from the characterization of nodes in", "Lemma \\ref{lemma-nodal}." ], "refs": [ "curves-lemma-nodal" ], "ref_ids": [ 6309 ] } ], "ref_ids": [] }, { "id": 6314, "type": "theorem", "label": "curves-lemma-nodal-lci", "categories": [ "curves" ], "title": "curves-lemma-nodal-lci", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme", "equidimensional of dimension $1$.", "The following are equivalent", "\\begin{enumerate}", "\\item the singularities of $X$ are at-worst-nodal, and", "\\item $X$ is a local complete intersection over $k$", "and the closed subscheme $Z \\subset X$ cut out by the", "first fitting ideal of $\\Omega_{X/k}$ is unramified over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We urge the reader to find their own proof of", "this lemma; what follows is just putting together earlier results", "and may hide what is really going on.", "\\medskip\\noindent", "Assume (2). Since $Z \\to \\Spec(k)$ is quasi-finite", "(Morphisms, Lemma \\ref{morphisms-lemma-unramified-quasi-finite})", "we see that the residue fields of points $x \\in Z$ are finite", "over $k$ (as well as separable) by", "Morphisms, Lemma \\ref{morphisms-lemma-residue-field-quasi-finite}.", "Hence each $x \\in Z$ is a closed point of $X$ by", "Morphisms, Lemma", "\\ref{morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre}.", "The local ring $\\mathcal{O}_{X, x}$ is Cohen-Macaulay by", "Algebra, Lemma \\ref{algebra-lemma-lci-CM}.", "Since $\\dim(\\mathcal{O}_{X, x}) = 1$ by dimension theory", "(Varieties, Section \\ref{varieties-section-algebraic-schemes}), we conclude", "that $\\text{depth}(\\mathcal{O}_{X, x})) = 1$. Thus $x$ is a node", "by Lemma \\ref{lemma-nodal}. If $x \\in X$, $x \\not \\in Z$, then", "$X \\to \\Spec(k)$ is smooth at $x$ by", "Divisors, Lemma \\ref{divisors-lemma-d-fitting-ideal-omega-smooth}.", "\\medskip\\noindent", "Assume (1). Under this assumption $X$ is geometrically reduced", "at every closed point (see", "Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced-upstairs}).", "Hence $X \\to \\Spec(k)$ is smooth on a dense open by", "Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced-dense-smooth-open}.", "Thus $Z$ is closed and consists of closed points.", "By Divisors, Lemma \\ref{divisors-lemma-d-fitting-ideal-omega-smooth}", "the morphism $X \\setminus Z \\to \\Spec(k)$ is smooth.", "Hence $X \\setminus Z$ is a local complete intersection by", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-syntomic}", "and the definition of a local complete intersection in", "Morphisms, Definition \\ref{morphisms-definition-syntomic}.", "By Lemma \\ref{lemma-nodal} for every point $x \\in Z$", "the local ring $\\mathcal{O}_{Z, x}$ is equal to $\\kappa(x)$", "and $\\kappa(x)$ is separable over $k$. Thus $Z \\to \\Spec(k)$", "is unramified (Morphisms, Lemma \\ref{morphisms-lemma-unramified-over-field}).", "Finally, Lemma \\ref{lemma-nodal} via part (3) of", "Lemma \\ref{lemma-nodal-algebraic}, shows that $\\mathcal{O}_{X, x}$", "is a complete intersection in the sense of", "Divided Power Algebra, Definition \\ref{dpa-definition-lci}.", "However, Divided Power Algebra, Lemma \\ref{dpa-lemma-check-lci-agrees}", "and Morphisms, Lemma \\ref{morphisms-lemma-local-complete-intersection}", "show that this agrees with the notion used to define a local", "complete intersection scheme over a field and the proof is complete." ], "refs": [ "morphisms-lemma-unramified-quasi-finite", "morphisms-lemma-residue-field-quasi-finite", "morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre", "algebra-lemma-lci-CM", "curves-lemma-nodal", "divisors-lemma-d-fitting-ideal-omega-smooth", "varieties-lemma-geometrically-reduced-upstairs", "varieties-lemma-geometrically-reduced-dense-smooth-open", "divisors-lemma-d-fitting-ideal-omega-smooth", "morphisms-lemma-smooth-syntomic", "morphisms-definition-syntomic", "curves-lemma-nodal", "morphisms-lemma-unramified-over-field", "curves-lemma-nodal", "curves-lemma-nodal-algebraic", "dpa-definition-lci", "dpa-lemma-check-lci-agrees", "morphisms-lemma-local-complete-intersection" ], "ref_ids": [ 5351, 5225, 5222, 1166, 6309, 7902, 10910, 11008, 7902, 5329, 5560, 6309, 5352, 6309, 6305, 1701, 1681, 5296 ] } ], "ref_ids": [] }, { "id": 6315, "type": "theorem", "label": "curves-lemma-facts-about-nodal-curves", "categories": [ "curves" ], "title": "curves-lemma-facts-about-nodal-curves", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme", "equidimensional of dimension $1$ whose singularities are at-worst-nodal.", "Then $X$ is Gorenstein and geometrically reduced." ], "refs": [], "proofs": [ { "contents": [ "The Gorenstein assertion follows from Lemma \\ref{lemma-nodal-lci}", "and Duality for Schemes, Lemma \\ref{duality-lemma-gorenstein-lci}.", "Or you can use that it suffices to check after passing to the", "algebraic closure (Duality for Schemes, Lemma", "\\ref{duality-lemma-gorenstein-base-change}), then use that", "a Noetherian local ring is Gorenstein if and only if its", "completion is so (by Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-flat-under-gorenstein}), and", "then prove that the local rings $k[[t]]$ and $k[[x, y]]/(xy)$", "are Gorenstein by hand.", "\\medskip\\noindent", "To see that $X$ is geometrically reduced, it suffices to show that", "$X_{\\overline{k}}$ is reduced (Varieties, Lemmas", "\\ref{varieties-lemma-perfect-reduced} and", "\\ref{varieties-lemma-geometrically-reduced}).", "But $X_{\\overline{k}}$ is a nodal curve over an", "algebraically closed field. Thus the complete local rings", "of $X_{\\overline{k}}$ are isomorphic to either", "$\\overline{k}[[t]]$ or $\\overline{k}[[x, y]]/(xy)$", "which are reduced as desired." ], "refs": [ "curves-lemma-nodal-lci", "duality-lemma-gorenstein-lci", "duality-lemma-gorenstein-base-change", "dualizing-lemma-flat-under-gorenstein", "varieties-lemma-perfect-reduced", "varieties-lemma-geometrically-reduced" ], "ref_ids": [ 6314, 13592, 13594, 2885, 10907, 10908 ] } ], "ref_ids": [] }, { "id": 6316, "type": "theorem", "label": "curves-lemma-closed-subscheme-nodal-curve", "categories": [ "curves" ], "title": "curves-lemma-closed-subscheme-nodal-curve", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme", "equidimensional of dimension $1$ whose singularities are at-worst-nodal.", "If $Y \\subset X$ is a reduced closed subscheme", "equidimensional of dimension $1$, then", "\\begin{enumerate}", "\\item the singularities of $Y$ are at-worst-nodal, and", "\\item if $Z \\subset X$ is the scheme theoretic closure of", "$X \\setminus Y$, then", "\\begin{enumerate}", "\\item the scheme theoretic intersection $Y \\cap Z$ is", "the disjoint union of spectra of finite separable extensions of $k$,", "\\item each point of $Y \\cap Z$ is a node of $X$, and", "\\item $Y \\to \\Spec(k)$ is smooth at every point of $Y \\cap Z$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $X$ and $Y$ are reduced and equidimensional of dimension $1$,", "we see that $Y$ is the scheme theoretic union of a subset of the", "irreducible components of $X$ (in a reduced ring $(0)$", "is the intersection of the minimal primes).", "Let $y \\in Y$ be a closed point.", "If $y$ is in the smooth locus of", "$X \\to \\Spec(k)$, then $y$ is on a unique irreducible component", "of $X$ and we see that $Y$ and $X$ agree in an open neighbourhood", "of $y$. Hence $Y \\to \\Spec(k)$ is smooth at $y$.", "If $y$ is a node of $X$ but still lies on a unique irreducible", "component of $X$, then $y$ is a node on $Y$ by the same argument.", "Suppose that $y$ lies on more than $1$ irreducible component of $X$.", "Since the number of geometric branches of $X$ at $y$ is $2$", "by Lemma \\ref{lemma-nodal},", "there can be at most $2$ irreducible components passing through $y$", "by Properties, Lemma", "\\ref{properties-lemma-number-of-branches-irreducible-components}.", "If $Y$ contains both of these, then again $Y = X$ in an open neighbourhood", "of $y$ and $y$ is a node of $Y$. Finally, assume $Y$ contains only one", "of the irreducible components. After replacing $X$ by an open", "neighbourhood of $x$ we may assume $Y$ is one of the two", "irreducble components and $Z$ is the other. By Properties, Lemma", "\\ref{properties-lemma-number-of-branches-irreducible-components}", "again we see that $X$ has two branches at $y$, i.e., the local ring", "$\\mathcal{O}_{X, y}$ has two branches and that these branches", "come from $\\mathcal{O}_{Y, y}$ and $\\mathcal{O}_{Z, y}$. Write", "$\\mathcal{O}_{X, y}^\\wedge \\cong \\kappa(y)[[u, v]]/(uv)$", "as in Remark \\ref{remark-trivial-quadratic-extension}.", "The field $\\kappa(y)$ is finite separable over $k$ by", "Lemma \\ref{lemma-nodal} for example.", "Thus, after possibly switching the roles of $u$ and $v$,", "the completion of the map", "$\\mathcal{O}_{X, y} \\to \\mathcal{O}_{Y, Y}$", "corresponds to $\\kappa(y)[[u, v]]/(uv) \\to \\kappa(y)[[u]]$", "and the completion of the map", "$\\mathcal{O}_{X, y} \\to \\mathcal{O}_{Y, Y}$", "corresponds to $\\kappa(y)[[u, v]]/(uv) \\to \\kappa(y)[[v]]$.", "The scheme theoretic intersection of $Y \\cap Z$ is cut out", "by the sum of their ideas which in the completion is $(u, v)$, i.e.,", "the maximal ideal. Thus (2)(a) and (2)(b) are clear.", "Finally, (2)(c) holds: the completion of $\\mathcal{O}_{Y, y}$", "is regular, hence $\\mathcal{O}_{Y, y}$ is regular", "(More on Algebra, Lemma \\ref{more-algebra-lemma-completion-regular})", "and $\\kappa(y)/k$ is separable, hence", "smoothness in an open neighbourhood", "by Algebra, Lemma \\ref{algebra-lemma-separable-smooth}." ], "refs": [ "curves-lemma-nodal", "properties-lemma-number-of-branches-irreducible-components", "properties-lemma-number-of-branches-irreducible-components", "curves-remark-trivial-quadratic-extension", "curves-lemma-nodal", "more-algebra-lemma-completion-regular", "algebra-lemma-separable-smooth" ], "ref_ids": [ 6309, 3000, 3000, 6365, 6309, 10045, 1225 ] } ], "ref_ids": [] }, { "id": 6317, "type": "theorem", "label": "curves-lemma-nodal-family", "categories": [ "curves" ], "title": "curves-lemma-nodal-family", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is flat, locally of finite presentation, every nonempty fibre", "$X_s$ is equidimensional of dimension $1$, and $X_s$ has", "at-worst-nodal singularities, and", "\\item $f$ is syntomic of relative dimension $1$ and the closed subscheme", "$\\text{Sing}(f) \\subset X$ defined by the first Fitting ideal of", "$\\Omega_{X/S}$ is unramified over $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that the formation of $\\text{Sing}(f)$ commutes with base", "change, see Divisors, Lemma", "\\ref{divisors-lemma-base-change-and-fitting-ideal-omega}.", "Thus the lemma follows from Lemma \\ref{lemma-nodal-lci},", "Morphisms, Lemma \\ref{morphisms-lemma-syntomic-flat-fibres}, and", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-etale-fibres}.", "(We also use the trivial", "Morphisms, Lemmas \\ref{morphisms-lemma-syntomic-locally-finite-presentation}", "and \\ref{morphisms-lemma-syntomic-flat}.)" ], "refs": [ "divisors-lemma-base-change-and-fitting-ideal-omega", "curves-lemma-nodal-lci", "morphisms-lemma-syntomic-flat-fibres", "morphisms-lemma-unramified-etale-fibres", "morphisms-lemma-syntomic-locally-finite-presentation", "morphisms-lemma-syntomic-flat" ], "ref_ids": [ 7900, 6314, 5298, 5353, 5293, 5294 ] } ], "ref_ids": [] }, { "id": 6318, "type": "theorem", "label": "curves-lemma-smooth-relative-dimension-1", "categories": [ "curves" ], "title": "curves-lemma-smooth-relative-dimension-1", "contents": [ "A smooth morphism of relative dimension $1$ is", "at-worst-nodal of relative dimension $1$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6319, "type": "theorem", "label": "curves-lemma-base-change-nodal-family", "categories": [ "curves" ], "title": "curves-lemma-base-change-nodal-family", "contents": [ "Let $f : X \\to S$ be at-worst-nodal of relative dimension $1$.", "Then the same is true for any base change of $f$." ], "refs": [], "proofs": [ { "contents": [ "This is true because the base change of a syntomic morphism", "is syntomic (Morphisms, Lemma \\ref{morphisms-lemma-base-change-syntomic}),", "the base change of a morphism of relative dimension $1$ has", "relative dimension $1$", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-relative-dimension-d}),", "the formation of $\\text{Sing}(f)$ commutes with base change", "(Divisors, Lemma", "\\ref{divisors-lemma-base-change-and-fitting-ideal-omega}), and", "the base change of an unramified morphism is unramified", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-unramified})." ], "refs": [ "morphisms-lemma-base-change-syntomic", "morphisms-lemma-base-change-relative-dimension-d", "divisors-lemma-base-change-and-fitting-ideal-omega", "morphisms-lemma-base-change-unramified" ], "ref_ids": [ 5291, 5284, 7900, 5346 ] } ], "ref_ids": [] }, { "id": 6320, "type": "theorem", "label": "curves-lemma-locus-where-nodal", "categories": [ "curves" ], "title": "curves-lemma-locus-where-nodal", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat and", "locally of finite presentation. Then there is a maximal open", "subscheme $U \\subset X$ such that $f|_U : U \\to S$ is", "at-worst-nodal of relative dimension $1$. Moreover, formation", "of $U$ commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-lci}", "we find that there is such an open where $f$ is syntomic.", "Hence we may assume that $f$ is a syntomic morphism.", "In particular $f$ is a Cohen-Macaulay morphism", "(Duality for Schemes, Lemmas \\ref{duality-lemma-lci-gorenstein} and", "\\ref{duality-lemma-gorenstein-CM-morphism}).", "Thus $X$ is a disjoint union of open and closed subschemes on which", "$f$ has given relative dimension, see Morphisms, Lemma", "\\ref{morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension}.", "This decomposition is preserved by arbitrary base change, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-relative-dimension-d}.", "Discarding all but one piece we may assume $f$ is syntomic of", "relative dimension $1$. Let $\\text{Sing}(f) \\subset X$ be the", "closed subscheem defined by the first fitting ideal of", "$\\Omega_{X/S}$. There is a maximal open subscheme", "$W \\subset \\text{Sing}(f)$ such that $W \\to S$ is unramified", "and its formation commutes with base change", "(Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-unramified}).", "Since also formation of $\\text{Sing}(f)$ commutes with base change", "(Divisors, Lemma", "\\ref{divisors-lemma-base-change-and-fitting-ideal-omega}),", "we see that", "$$", "U = (X \\setminus \\text{Sing}(f)) \\cup W", "$$", "is the maximal open subscheme of $X$ such that", "$f|_U : U \\to S$ is at-worst-nodal of relative dimension $1$", "and that formation of $U$ commutes with base change." ], "refs": [ "morphisms-lemma-set-points-where-fibres-lci", "duality-lemma-lci-gorenstein", "duality-lemma-gorenstein-CM-morphism", "morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension", "morphisms-lemma-base-change-relative-dimension-d", "morphisms-lemma-set-points-where-fibres-unramified", "divisors-lemma-base-change-and-fitting-ideal-omega" ], "ref_ids": [ 5299, 13597, 13596, 5286, 5284, 5356, 7900 ] } ], "ref_ids": [] }, { "id": 6321, "type": "theorem", "label": "curves-lemma-nodal-family-precompose-etale", "categories": [ "curves" ], "title": "curves-lemma-nodal-family-precompose-etale", "contents": [ "Let $f : X \\to S$ be at-worst-nodal of relative dimension $1$.", "If $Y \\to X$ is an \\'etale morphism, then the composition $g : Y \\to S$", "is at-worst-nodal of relative dimension $1$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $g$ is flat and locally of finite presentation as", "a composition of morphisms which are flat and locally of finite", "presentation (use", "Morphisms, Lemmas \\ref{morphisms-lemma-etale-locally-finite-presentation},", "\\ref{morphisms-lemma-etale-flat},", "\\ref{morphisms-lemma-composition-finite-presentation}, and", "\\ref{morphisms-lemma-composition-flat}).", "Thus it suffices to prove the fibres have at-worst-nodal singularities.", "This follows from Lemma \\ref{lemma-node-etale-local}", "(and the fact that the composition of an \\'etale morphism and", "a smooth morphism is smooth by", "Morphisms, Lemmas \\ref{morphisms-lemma-etale-smooth-unramified} and", "\\ref{morphisms-lemma-composition-smooth})." ], "refs": [ "morphisms-lemma-etale-locally-finite-presentation", "morphisms-lemma-etale-flat", "morphisms-lemma-composition-finite-presentation", "morphisms-lemma-composition-flat", "curves-lemma-node-etale-local", "morphisms-lemma-etale-smooth-unramified", "morphisms-lemma-composition-smooth" ], "ref_ids": [ 5368, 5369, 5239, 5263, 6312, 5362, 5326 ] } ], "ref_ids": [] }, { "id": 6322, "type": "theorem", "label": "curves-lemma-nodal-family-postcompose-etale", "categories": [ "curves" ], "title": "curves-lemma-nodal-family-postcompose-etale", "contents": [ "Let $S' \\to S$ be an \\'etale morphism of schemes.", "Let $f : X \\to S'$ be at-worst-nodal of relative dimension $1$.", "Then the composition $g : X \\to S$", "is at-worst-nodal of relative dimension $1$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $g$ is flat and locally of finite presentation as", "a composition of morphisms which are flat and locally of finite", "presentation (use", "Morphisms, Lemmas \\ref{morphisms-lemma-etale-locally-finite-presentation},", "\\ref{morphisms-lemma-etale-flat},", "\\ref{morphisms-lemma-composition-finite-presentation}, and", "\\ref{morphisms-lemma-composition-flat}).", "Thus it suffices to prove the fibres of $g$", "have at-worst-nodal singularities.", "This follows from Lemma \\ref{lemma-node-over-separable-extension}", "and the analogous result for smooth points." ], "refs": [ "morphisms-lemma-etale-locally-finite-presentation", "morphisms-lemma-etale-flat", "morphisms-lemma-composition-finite-presentation", "morphisms-lemma-composition-flat", "curves-lemma-node-over-separable-extension" ], "ref_ids": [ 5368, 5369, 5239, 5263, 6313 ] } ], "ref_ids": [] }, { "id": 6323, "type": "theorem", "label": "curves-lemma-nodal-family-etale-local-source", "categories": [ "curves" ], "title": "curves-lemma-nodal-family-etale-local-source", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $\\{U_i \\to X\\}$", "be an \\'etale covering. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is at-worst-nodal of relative dimension $1$,", "\\item each $U_i \\to S$ is at-worst-nodal of relative dimension $1$.", "\\end{enumerate}", "In other words, being at-worst-nodal of relative dimension $1$", "is \\'etale local on the source." ], "refs": [], "proofs": [ { "contents": [ "One direction we have seen in Lemma \\ref{lemma-nodal-family-precompose-etale}.", "For the other direction, observe that being locally of finite", "presentation, flat, or to have relative dimension $1$", "is \\'etale local on the source", "(Descent, Lemmas", "\\ref{descent-lemma-locally-finite-presentation-fppf-local-source},", "\\ref{descent-lemma-flat-fpqc-local-source}, and", "\\ref{descent-lemma-dimension-at-point}). Taking fibres we reduce", "to the case where $S$ is the spectrum of a field. In this case the", "result follows from Lemma \\ref{lemma-node-etale-local}", "(and the fact that being smooth is \\'etale local on the source by", "Descent, Lemma \\ref{descent-lemma-smooth-smooth-local-source})." ], "refs": [ "curves-lemma-nodal-family-precompose-etale", "descent-lemma-locally-finite-presentation-fppf-local-source", "descent-lemma-flat-fpqc-local-source", "descent-lemma-dimension-at-point", "curves-lemma-node-etale-local", "descent-lemma-smooth-smooth-local-source" ], "ref_ids": [ 6321, 14710, 14708, 14731, 6312, 14715 ] } ], "ref_ids": [] }, { "id": 6324, "type": "theorem", "label": "curves-lemma-nodal-family-fpqc-local-target", "categories": [ "curves" ], "title": "curves-lemma-nodal-family-fpqc-local-target", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $\\{U_i \\to S\\}$", "be an fpqc covering. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is at-worst-nodal of relative dimension $1$,", "\\item each $X \\times_S U_i \\to U_i$ is at-worst-nodal of relative", "dimension $1$.", "\\end{enumerate}", "In other words, being at-worst-nodal of relative dimension $1$", "is fpqc local on the target." ], "refs": [], "proofs": [ { "contents": [ "One direction we have seen in Lemma \\ref{lemma-base-change-nodal-family}.", "For the other direction, observe that being locally of finite", "presentation, flat, or to have relative dimension $1$", "is fpqc local on the target", "(Descent, Lemmas", "\\ref{descent-lemma-descending-property-locally-finite-presentation},", "\\ref{descent-lemma-descending-property-flat}, and", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}).", "Taking fibres we reduce", "to the case where $S$ is the spectrum of a field. In this case the", "result follows from Lemma \\ref{lemma-node-field-extension}", "(and the fact that being smooth is fpqc local on the target by", "Descent, Lemma \\ref{descent-lemma-descending-property-smooth})." ], "refs": [ "curves-lemma-base-change-nodal-family", "descent-lemma-descending-property-locally-finite-presentation", "descent-lemma-descending-property-flat", "morphisms-lemma-dimension-fibre-after-base-change", "curves-lemma-node-field-extension", "descent-lemma-descending-property-smooth" ], "ref_ids": [ 6319, 14676, 14680, 5279, 6311, 14692 ] } ], "ref_ids": [] }, { "id": 6325, "type": "theorem", "label": "curves-lemma-descend-nodal-family", "categories": [ "curves" ], "title": "curves-lemma-descend-nodal-family", "contents": [ "Let $S = \\lim S_i$ be a limit of a directed system of schemes", "with affine transition morphisms.", "Let $0 \\in I$ and let $f_0 : X_0 \\to Y_0$ be a morphism of schemes over $S_0$.", "Assume $S_0$, $X_0$, $Y_0$ are quasi-compact and quasi-separated.", "Let $f_i : X_i \\to Y_i$ be the base change of $f_0$ to $S_i$ and", "let $f : X \\to Y$ be the base change of $f_0$ to $S$.", "If", "\\begin{enumerate}", "\\item $f$ is at-worst-nodal of relative dimension $1$, and", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then there exists an $i \\geq 0$ such that $f_i$ is at-worst-nodal", "of relative dimension $1$." ], "refs": [], "proofs": [ { "contents": [ "By Limits, Lemma \\ref{limits-lemma-descend-syntomic}", "there exists an $i$ such that $f_i$ is syntomic.", "Then $X_i = \\coprod_{d \\geq 0} X_{i, d}$ is a disjoint union of", "open and closed subschemes such that $X_{i, d} \\to Y_i$", "has relative dimension $d$, see", "Morphisms, Lemma \\ref{morphisms-lemma-syntomic-relative-dimension}.", "Because of the behaviour of dimensions of fibres under base change given in", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}", "we see that $X \\to X_i$ maps into $X_{i, 1}$.", "Then there exists an $i' \\geq i$ such that $X_{i'} \\to X_i$", "maps into $X_{i, 1}$, see", "Limits, Lemma \\ref{limits-lemma-limit-contained-in-constructible}.", "Thus $f_{i'} : X_{i'} \\to Y_{i'}$ is syntomic of relative dimension $1$", "(by Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}", "again).", "Consider the morphism $\\text{Sing}(f_{i'}) \\to Y_{i'}$.", "We know that the base change to $Y$ is an unramified morphism.", "Hence by Limits, Lemma \\ref{limits-lemma-descend-unramified}", "we see that after increasing $i'$ the morphism", "$\\text{Sing}(f_{i'}) \\to Y_{i'}$ becomes unramified.", "This finishes the proof." ], "refs": [ "limits-lemma-descend-syntomic", "morphisms-lemma-syntomic-relative-dimension", "morphisms-lemma-dimension-fibre-after-base-change", "limits-lemma-limit-contained-in-constructible", "morphisms-lemma-dimension-fibre-after-base-change", "limits-lemma-descend-unramified" ], "ref_ids": [ 15070, 5301, 5279, 15040, 5279, 15059 ] } ], "ref_ids": [] }, { "id": 6326, "type": "theorem", "label": "curves-lemma-formal-local-structure-nodal-family", "categories": [ "curves" ], "title": "curves-lemma-formal-local-structure-nodal-family", "contents": [ "Let $f : T \\to S$ be a morphism of schemes. Let $t \\in T$", "with image $s \\in S$. Assume", "\\begin{enumerate}", "\\item $f$ is flat at $t$,", "\\item $\\mathcal{O}_{S, s}$ is Noetherian,", "\\item $f$ is locally of finite type,", "\\item $t$ is a split node of the fibre $T_s$.", "\\end{enumerate}", "Then there exists an $h \\in \\mathfrak m_s^\\wedge$ and an isomorphism", "$$", "\\mathcal{O}_{T, t}^\\wedge \\cong", "\\mathcal{O}_{S, s}^\\wedge[[x, y]]/(xy - h)", "$$", "of $\\mathcal{O}_{S, s}^\\wedge$-algebras." ], "refs": [], "proofs": [ { "contents": [ "We replace $S$ by $\\Spec(\\mathcal{O}_{S, s})$ and $T$ by the base change", "to $\\Spec(\\mathcal{O}_{S, s})$. Then $T$ is locally Noetherian and hence", "$\\mathcal{O}_{T, t}$ is Noetherian.", "Set $A = \\mathcal{O}_{S, s}^\\wedge$, $\\mathfrak m = \\mathfrak m_A$, and", "$B = \\mathcal{O}_{T, t}^\\wedge$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-completion}", "we see that $A \\to B$ is flat. Since", "$\\mathcal{O}_{T, t}/\\mathfrak m_s \\mathcal{O}_{T, t} = \\mathcal{O}_{T_s, t}$", "we see that $B/\\mathfrak m B = \\mathcal{O}_{T_s, t}^\\wedge$.", "By assumption (4) and Lemma \\ref{lemma-split-node}", "we conclude there exist $\\overline{u}, \\overline{v} \\in B/\\mathfrak m B$", "such that the map", "$$", "(A/\\mathfrak m)[[x, y]] \\longrightarrow B/\\mathfrak m B,\\quad", "x \\longmapsto \\overline{u},", "x \\longmapsto \\overline{v}", "$$", "is surjective with kernel $(xy)$.", "\\medskip\\noindent", "Assume we have $n \\geq 1$ and $u, v \\in B$ mapping to", "$\\overline{u}, \\overline{v}$ such that", "$$", "u v = h + \\delta", "$$", "for some $h \\in A$ and $\\delta \\in \\mathfrak m^nB$.", "We claim that there exist $u', v' \\in B$ with", "$u - u', v - v' \\in \\mathfrak m^n B$ such that", "$$", "u' v' = h' + \\delta'", "$$", "for some $h' \\in A$ and $\\delta' \\in \\mathfrak m^{n + 1}B$.", "To see this, write $\\delta = \\sum f_i b_i$ with", "$f_i \\in \\mathfrak m^n$ and $b_i \\in B$. Then write", "$b_i = a_i + u b_{i, 1} + v b_{i, 2} + \\delta_i$ with", "$a_i \\in A$, $b_{i, 1}, b_{i, 2} \\in B$ and $\\delta_i \\in \\mathfrak m B$.", "This is possible because the residue field of $B$ agrees with the", "residue field of $A$ and the images of $u$ and $v$ in $B/\\mathfrak m B$", "generate the maximal ideal. Then we set", "$$", "u' = u - \\sum b_{i, 2}f_i,\\quad", "v' = v - \\sum b_{i, 1}f_i", "$$", "and we obtain", "$$", "u'v' = h + \\delta - \\sum (b_{i, 1}u + b_{i, 2}v)f_i + \\sum", "c_{ij}f_if_j =", "h + \\sum a_if_i + \\sum f_i \\delta_i + \\sum c_{ij}f_if_j", "$$", "for some $c_{i, j} \\in B$.", "Thus we get a formula as above with $h' = h + \\sum a_if_i$", "and $\\delta' = \\sum f_i \\delta_i + \\sum c_{ij}f_if_j$.", "\\medskip\\noindent", "Arguing by induction and starting with any lifts $u_1, v_1 \\in B$", "of $\\overline{u}, \\overline{v}$ the result of the previous paragraph", "shows that we find a sequence of elements", "$u_n, v_n \\in B$ and $h_n \\in A$ such that", "$u_n - u_{n + 1} \\in \\mathfrak m^n B$,", "$v_n - v_{n + 1} \\in \\mathfrak m^n B$,", "$h_n - h_{n + 1} \\in \\mathfrak m^n$,", "and such that $u_n v_n - h_n \\in \\mathfrak m^n B$.", "Since $A$ and $B$ are complete we can set", "$u_\\infty = \\lim u_n$, $v_\\infty = \\lim v_n$, and", "$h_\\infty = \\lim h_n$, and then we obtain $u_\\infty v_\\infty = h_\\infty$", "in $B$. Thus we have an $A$-algebra map", "$$", "A[[x, y]]/(xy - h_\\infty) \\longrightarrow B", "$$", "sending $x$ to $u_\\infty$ and $v$ to $v_\\infty$.", "This is a map of flat $A$-algebras which is an", "isomorphism after dividing by $\\mathfrak m$.", "It is surjective modulo $\\mathfrak m$ and hence surjective", "by completeness and", "Algebra, Lemma \\ref{algebra-lemma-completion-generalities}.", "Then we can apply Algebra, Lemma \\ref{algebra-lemma-mod-injective}", "to conclude it is an isomorphism." ], "refs": [ "more-algebra-lemma-flat-completion", "curves-lemma-split-node", "algebra-lemma-completion-generalities", "algebra-lemma-mod-injective" ], "ref_ids": [ 10049, 6310, 858, 883 ] } ], "ref_ids": [] }, { "id": 6327, "type": "theorem", "label": "curves-lemma-etale-local-structure-nodal-family", "categories": [ "curves" ], "title": "curves-lemma-etale-local-structure-nodal-family", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Assume that", "$f$ is at-worst-nodal of relative dimension $1$. Let", "$x \\in X$ be a point which is a singular point of the", "fibre $X_s$. Then there exists a commutative diagram of schemes", "$$", "\\xymatrix{", "X \\ar[d] &", "U \\ar[rr] \\ar[l] \\ar[rd] & &", "W \\ar[r] \\ar[ld] &", "\\Spec(\\mathbf{Z}[u, v, a]/(uv - a)) \\ar[d] \\\\", "S & &", "V \\ar[ll] \\ar[rr] & & \\Spec(\\mathbf{Z}[a])", "}", "$$", "with $X \\leftarrow U$, $S \\leftarrow V$, and $U \\to W$ \\'etale morphisms,", "and with the right hand square cartesian, such that there exists", "a point $u \\in U$ mapping to $x$ in $X$." ], "refs": [], "proofs": [ { "contents": [ "We first use absolute Noetherian approximation to reduce to the", "case of schemes of finite type over $\\mathbf{Z}$.", "The question is local on $X$ and $S$. Hence we may assume that", "$X$ and $S$ are affine. Then we can write $S = \\Spec(R)$", "and write $R$ as a filtered colimit $R = \\colim R_i$", "of finite type $\\mathbf{Z}$-algebras.", "Using Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can find an $i$ and a morphism $f_i : X_i \\to \\Spec(R_i)$ whose base", "change to $S$ is $f$. After increasing $i$ we may assume that $f_i$", "is at-worst-nodal of relative dimension $1$, see", "Lemma \\ref{lemma-descend-nodal-family}.", "The image $x_i \\in X_i$ of $x$ will be a singular", "point of its fibre, for example because the formation of", "$\\text{Sing}(f)$ commutes with base change (Divisors, Lemma", "\\ref{divisors-lemma-base-change-and-fitting-ideal-omega}).", "If we can prove the lemma for $f_i : X_i \\to S_i$ and", "$x_i$, then the lemma follows for $f : X \\to S$ by base", "change. Thus we reduce to the case studied in the next", "paragraph.", "\\medskip\\noindent", "Assume $S$ is of finite type over $\\mathbf{Z}$. Let $s \\in S$ be the", "image of $x$. Recall that $\\kappa(x)$ is a finite separable extension", "of $\\kappa(s)$, for example because $\\text{Sing}(f) \\to S$", "is unramified or because $x$ is a node of the fibre $X_s$", "and we can apply Lemma \\ref{lemma-nodal}.", "Furthermore, let $\\kappa'/\\kappa(x)$ be the", "degree $2$ separable algebra associated to $\\mathcal{O}_{X_s, x}$ in", "Remark \\ref{remark-quadratic-extension}.", "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-realize-prescribed-residue-field-extension-etale}", "we can choose an \\'etale neighbourhood $(V, v) \\to (S, s)$", "such that the extension $\\kappa(v)/\\kappa(s)$ realizes either", "the extension $\\kappa(x)/\\kappa(s)$ in case", "$\\kappa' \\cong \\kappa(x) \\times \\kappa(x)$ or", "the extension $\\kappa'/\\kappa(s)$ if $\\kappa'$ is a field.", "After replacing $X$ by $X \\times_S V$ and $S$ by $V$", "we reduce to the situation described in the next paragraph.", "\\medskip\\noindent", "Assume $S$ is of finite type over $\\mathbf{Z}$ and", "$x \\in X_s$ is a split node, see Definition \\ref{definition-split-node}.", "By Lemma \\ref{lemma-formal-local-structure-nodal-family} we see that there", "exists an $\\mathcal{O}_{S, s}$-algebra isomorphism", "$$", "\\mathcal{O}_{X, x}^\\wedge \\cong", "\\mathcal{O}_{S, s}^\\wedge[[s, t]]/(st - h)", "$$", "for some $h \\in \\mathfrak m_s^\\wedge \\subset \\mathcal{O}_{S, s}^\\wedge$.", "In other words, if we consider the homomorphism", "$$", "\\sigma : \\mathbf{Z}[a] \\longrightarrow \\mathcal{O}_{S, s}^\\wedge", "$$", "sending $a$ to $h$, then there exists an $\\mathcal{O}_{S, s}$-algebra", "isomorphism", "$$", "\\mathcal{O}_{X, x}^\\wedge", "\\longrightarrow", "\\mathcal{O}_{Y_\\sigma, y_\\sigma}^\\wedge", "$$", "where", "$$", "Y_\\sigma = \\Spec(\\mathbf{Z}[u, v, t]/(uv - a))", "\\times_{\\Spec(\\mathbf{Z}[a]), \\sigma} \\Spec(\\mathcal{O}_{S, s}^\\wedge)", "$$", "and $y_\\sigma$ is the point of $Y_\\sigma$ lying over", "the closed point of $\\Spec(\\mathcal{O}_{S, s}^\\wedge)$", "and having coordinates $u, v$ equal to zero. Since", "$\\mathcal{O}_{S, s}$ is a G-ring by", "More on Algebra, Proposition \\ref{more-algebra-proposition-ubiquity-G-ring}", "we may apply More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-relative-map-approximation-pre}", "to conclude." ], "refs": [ "limits-lemma-descend-finite-presentation", "curves-lemma-descend-nodal-family", "divisors-lemma-base-change-and-fitting-ideal-omega", "curves-lemma-nodal", "curves-remark-quadratic-extension", "more-morphisms-lemma-realize-prescribed-residue-field-extension-etale", "curves-definition-split-node", "curves-lemma-formal-local-structure-nodal-family", "more-algebra-proposition-ubiquity-G-ring", "more-morphisms-lemma-relative-map-approximation-pre" ], "ref_ids": [ 15077, 6325, 7900, 6309, 6364, 13866, 6359, 6326, 10582, 13885 ] } ], "ref_ids": [] }, { "id": 6328, "type": "theorem", "label": "curves-lemma-h1-nonzero-degree-leq-2g-2", "categories": [ "curves" ], "title": "curves-lemma-h1-nonzero-degree-leq-2g-2", "contents": [ "In Situation \\ref{situation-Cohen-Macaulay-curve} assume $X$ is integral and", "has genus $g$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $Z \\subset X$ be a $0$-dimensional closed subscheme with ideal", "sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. If $H^1(X, \\mathcal{I}\\mathcal{L})$", "is nonzero, then", "$$", "\\deg(\\mathcal{L}) \\leq 2g - 2 + \\deg(Z)", "$$", "with strict inequality unless $\\mathcal{I}\\mathcal{L} \\cong \\omega_X$." ], "refs": [], "proofs": [ { "contents": [ "Any curve, e.g.\\ $X$, is Cohen-Macaulay.", "If $H^1(X, \\mathcal{I}\\mathcal{L})$ is nonzero, then there is a nonzero", "map $\\mathcal{I}\\mathcal{L} \\to \\omega_X$, see", "Lemma \\ref{lemma-duality-dim-1-CM}.", "Since $\\mathcal{I}\\mathcal{L}$ is torsion free, this map is injective.", "Since a field is Gorenstein and $X$ is reduced, we find", "that the Gorenstein locus $U \\subset X$ of $X$ is nonempty, see", "Duality for Schemes, Lemma \\ref{duality-lemma-gorenstein}.", "This lemma also tells us that $\\omega_X|_U$ is invertible.", "In this way we see we have a short exact sequence", "$$", "0 \\to \\mathcal{I}\\mathcal{L} \\to \\omega_X \\to \\mathcal{Q} \\to 0", "$$", "where the support of $\\mathcal{Q}$ is zero dimensional.", "Hence we have", "\\begin{align*}", "0 & \\leq \\dim \\Gamma(X, \\mathcal{Q})\\\\", "& =", "\\chi(\\mathcal{Q}) \\\\", "& =", "\\chi(\\omega_X) - \\chi(\\mathcal{I}\\mathcal{L}) \\\\", "& =", "\\chi(\\omega_X) - \\deg(\\mathcal{L}) - \\chi(\\mathcal{I}) \\\\", "& =", "2g - 2 - \\deg(\\mathcal{L}) + \\deg(Z)", "\\end{align*}", "by Lemmas \\ref{lemma-euler} and \\ref{lemma-rr}, by (\\ref{equation-genus}),", "and by Varieties, Lemmas \\ref{varieties-lemma-chi-tensor-finite}", "and \\ref{varieties-lemma-degree-on-proper-curve}. We have also used", "that $\\deg(Z) = \\dim_k \\Gamma(Z, \\mathcal{O}_Z) = \\chi(\\mathcal{O}_Z)$", "and the short exact sequence", "$0 \\to \\mathcal{I} \\to \\mathcal{O}_X \\to \\mathcal{O}_Z \\to 0$.", "The lemma follows." ], "refs": [ "curves-lemma-duality-dim-1-CM", "duality-lemma-gorenstein", "curves-lemma-euler", "curves-lemma-rr", "varieties-lemma-chi-tensor-finite", "varieties-lemma-degree-on-proper-curve" ], "ref_ids": [ 6251, 13591, 6255, 6256, 11030, 11107 ] } ], "ref_ids": [] }, { "id": 6329, "type": "theorem", "label": "curves-lemma-degree-more-than-2g-2", "categories": [ "curves" ], "title": "curves-lemma-degree-more-than-2g-2", "contents": [ "\\begin{reference}", "\\cite[Lemma 2]{Jongmin}", "\\end{reference}", "In Situation \\ref{situation-Cohen-Macaulay-curve}", "assume $X$ is integral and has genus $g$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $Z \\subset X$ be a $0$-dimensional closed subscheme with ideal", "sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.", "If $\\deg(\\mathcal{L}) > 2g - 2 + \\deg(Z)$, then", "$H^1(X, \\mathcal{I}\\mathcal{L}) = 0$ and one of the following possibilities", "occurs", "\\begin{enumerate}", "\\item $H^0(X, \\mathcal{I}\\mathcal{L}) \\not = 0$, or", "\\item $g = 0$ and $\\deg(\\mathcal{L}) = \\deg(Z) - 1$.", "\\end{enumerate}", "In case (2) if $Z = \\emptyset$, then $X \\cong \\mathbf{P}^1_k$ and $\\mathcal{L}$", "corresponds to $\\mathcal{O}_{\\mathbf{P}^1}(-1)$." ], "refs": [], "proofs": [ { "contents": [ "The vanishing of $H^1(X, \\mathcal{I}\\mathcal{L})$ follows from", "Lemma \\ref{lemma-h1-nonzero-degree-leq-2g-2}.", "If $H^0(X, \\mathcal{I}\\mathcal{L}) = 0$, then", "$\\chi(\\mathcal{I}\\mathcal{L}) = 0$. From the short exact", "sequence $0 \\to \\mathcal{I}\\mathcal{L} \\to \\mathcal{L} \\to \\mathcal{O}_Z \\to 0$", "we conclude $\\deg(\\mathcal{L}) = g - 1 + \\deg(Z)$.", "Thus $g - 1 + \\deg(Z) > 2g - 2 + \\deg(Z)$ which implies $g = 0$", "hence (2) holds. If $Z = \\emptyset$ in case (2),", "then $\\mathcal{L}^{-1}$ is an invertible sheaf of degree $1$.", "This implies there is an isomorphism $X \\to \\mathbf{P}^1_k$ and", "$\\mathcal{L}^{-1}$ is the pullback of $\\mathcal{O}_{\\mathbf{P}^1}(1)$ by", "Lemma \\ref{lemma-genus-zero-positive-degree}." ], "refs": [ "curves-lemma-h1-nonzero-degree-leq-2g-2", "curves-lemma-genus-zero-positive-degree" ], "ref_ids": [ 6328, 6273 ] } ], "ref_ids": [] }, { "id": 6330, "type": "theorem", "label": "curves-lemma-degree-more-than-2g", "categories": [ "curves" ], "title": "curves-lemma-degree-more-than-2g", "contents": [ "\\begin{reference}", "\\cite[Lemma 3]{Jongmin}", "\\end{reference}", "In Situation \\ref{situation-Cohen-Macaulay-curve}", "assume $X$ is integral and has genus $g$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "If $\\deg(\\mathcal{L}) \\geq 2g$, then $\\mathcal{L}$", "is globally generated." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset X$ be the closed subscheme cut out by the", "global sections of $\\mathcal{L}$. By Lemma \\ref{lemma-degree-more-than-2g-2}", "we see that $Z \\not = X$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be the ideal sheaf cutting out $Z$. Consider the short exact sequence", "$$", "0 \\to \\mathcal{I}\\mathcal{L}", "\\to \\mathcal{L} \\to \\mathcal{O}_Z \\to 0", "$$", "If $Z \\not = \\emptyset$, then", "$H^1(X, \\mathcal{I}\\mathcal{L})$ is nonzero", "as follows from the long exact sequence of cohomology.", "By Lemma \\ref{lemma-duality-dim-1-CM} this gives a", "nonzero and hence injective map", "$$", "\\mathcal{I}\\mathcal{L}", "\\longrightarrow", "\\omega_X", "$$", "In particular, we find an injective map", "$H^0(X, \\mathcal{L}) = H^0(X, \\mathcal{I}\\mathcal{L})", "\\to H^0(X, \\omega_X)$. This is impossible as", "$$", "\\dim_k H^0(X, \\mathcal{L}) = \\dim_k H^1(X, \\mathcal{L}) +", "\\deg(\\mathcal{L}) + 1 - g \\geq g + 1", "$$", "and $\\dim H^0(X, \\omega_X) = g$ by (\\ref{equation-genus})." ], "refs": [ "curves-lemma-degree-more-than-2g-2", "curves-lemma-duality-dim-1-CM" ], "ref_ids": [ 6329, 6251 ] } ], "ref_ids": [] }, { "id": 6331, "type": "theorem", "label": "curves-lemma-degree-more-than-2g-1-and-Z", "categories": [ "curves" ], "title": "curves-lemma-degree-more-than-2g-1-and-Z", "contents": [ "In Situation \\ref{situation-Cohen-Macaulay-curve}", "assume $X$ is integral and has genus $g$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $Z \\subset X$ be a nonempty $0$-dimensional closed subscheme.", "If $\\deg(\\mathcal{L}) \\geq 2g - 1 + \\deg(Z)$, then $\\mathcal{L}$", "is globally generated and $H^0(X, \\mathcal{L}) \\to H^0(X, \\mathcal{L}|_Z)$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Global generation by Lemma \\ref{lemma-degree-more-than-2g}.", "If $\\mathcal{I} \\subset \\mathcal{O}_X$ is the ideal sheaf", "of $Z$, then $H^1(X, \\mathcal{I}\\mathcal{L}) = 0$ by", "Lemma \\ref{lemma-h1-nonzero-degree-leq-2g-2}. Hence surjectivity." ], "refs": [ "curves-lemma-degree-more-than-2g", "curves-lemma-h1-nonzero-degree-leq-2g-2" ], "ref_ids": [ 6330, 6328 ] } ], "ref_ids": [] }, { "id": 6332, "type": "theorem", "label": "curves-lemma-vanishing-on-gorenstein", "categories": [ "curves" ], "title": "curves-lemma-vanishing-on-gorenstein", "contents": [ "\\begin{reference}", "Weak version of \\cite[Lemma 4]{Jongmin}", "\\end{reference}", "Let $k$ be a field. Let $X$ be a proper scheme over $k$", "which is reduced, connected, and of dimension $1$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $Z \\subset X$ be a $0$-dimensional closed subscheme with ideal", "sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.", "If $H^1(X, \\mathcal{I}\\mathcal{L}) \\not = 0$, then there exists", "a reduced connected closed subscheme $Y \\subset X$", "of dimension $1$ such that", "$$", "\\deg(\\mathcal{L}|_Y) \\leq -2\\chi(Y, \\mathcal{O}_Y) + \\deg(Z \\cap Y)", "$$", "where $Z \\cap Y$ is the scheme theoretic intersection." ], "refs": [], "proofs": [ { "contents": [ "If $H^1(X, \\mathcal{I}\\mathcal{L})$ is nonzero, then there is a nonzero map", "$\\varphi : \\mathcal{I}\\mathcal{L} \\to \\omega_X$, see", "Lemma \\ref{lemma-duality-dim-1-CM}. Let $Y \\subset X$", "be the union of the irreducible components $C$ of $X$ such that", "$\\varphi$ is nonzero in the generic point of $C$.", "Then $Y$ is a reduced closed subscheme.", "Let $\\mathcal{J} \\subset \\mathcal{O}_X$ be the ideal sheaf of $Y$.", "Since $\\mathcal{J}\\mathcal{I}\\mathcal{L}$", "has no embedded associated points", "(as a submodule of $\\mathcal{L}$) and as $\\varphi$ is zero", "in the generic points of the support of $\\mathcal{J}$", "(by choice of $Y$ and as $X$ is reduced), we find that", "$\\varphi$ factors as", "$$", "\\mathcal{I}\\mathcal{L} \\to", "\\mathcal{I}\\mathcal{L}/\\mathcal{J}\\mathcal{I}\\mathcal{L} \\to \\omega_X", "$$", "We can view $\\mathcal{I}\\mathcal{L}/\\mathcal{J}\\mathcal{I}\\mathcal{L}$", "as the pushforward of a coherent sheaf on $Y$ which by abuse of", "notation we indicate with the same symbol.", "Since $\\omega_Y = \\SheafHom(\\mathcal{O}_Y, \\omega_X)$", "by Lemma \\ref{lemma-closed-immersion-dim-1-CM}", "we find a map", "$$", "\\mathcal{I}\\mathcal{L}/", "\\mathcal{J}\\mathcal{I}\\mathcal{L} ", "\\to \\omega_Y", "$$", "of $\\mathcal{O}_Y$-modules which is injective in the generic points", "of $Y$. Let $\\mathcal{I}' \\subset \\mathcal{O}_Y$ be the ideal", "sheaf of $Z \\cap Y$. There is a map", "$\\mathcal{I}\\mathcal{L}/\\mathcal{J}\\mathcal{I}\\mathcal{L} \\to", "\\mathcal{I}'\\mathcal{L}|_Y$ whose kernel is supported in closed points.", "Since $\\omega_Y$ is a Cohen-Macaulay module, the map above", "factors through an injective map $\\mathcal{I}'\\mathcal{L}|_Y \\to", "\\omega_Y$. We see that we get", "an exact sequence", "$$", "0 \\to \\mathcal{I}'\\mathcal{L}|_Y \\to \\omega_Y \\to \\mathcal{Q} \\to 0", "$$", "of coherent sheaves on $Y$ where $\\mathcal{Q}$ is supported in dimension $0$", "(this uses that $\\omega_Y$ is an invertible module in the generic points", "of $Y$). We conclude that", "$$", "0 \\leq \\dim \\Gamma(Y, \\mathcal{Q}) =", "\\chi(\\mathcal{Q}) = \\chi(\\omega_Y) - \\chi(\\mathcal{I}'\\mathcal{L}) =", "-2\\chi(\\mathcal{O}_Y) - \\deg(\\mathcal{L}|_Y) + \\deg(Z \\cap Y)", "$$", "by Lemma \\ref{lemma-euler} and", "Varieties, Lemma \\ref{varieties-lemma-chi-tensor-finite}.", "If $Y$ is connected, then this proves the lemma.", "If not, then we repeat the last part of the argument", "for one of the connected components of $Y$." ], "refs": [ "curves-lemma-duality-dim-1-CM", "curves-lemma-closed-immersion-dim-1-CM", "curves-lemma-euler", "varieties-lemma-chi-tensor-finite" ], "ref_ids": [ 6251, 6253, 6255, 11030 ] } ], "ref_ids": [] }, { "id": 6333, "type": "theorem", "label": "curves-lemma-global-generation", "categories": [ "curves" ], "title": "curves-lemma-global-generation", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$", "which is reduced, connected, and of dimension $1$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Assume that for every reduced connected closed subscheme", "$Y \\subset X$ of dimension $1$ we have", "$$", "\\deg(\\mathcal{L}|_Y) \\geq 2\\dim_k H^1(Y, \\mathcal{O}_Y)", "$$", "Then $\\mathcal{L}$ is globally generated." ], "refs": [], "proofs": [ { "contents": [ "By induction on the number of irreducible components of $X$.", "If $X$ is irreducible, then the lemma holds by", "Lemma \\ref{lemma-degree-more-than-2g}", "applied to $X$ viewed as a scheme over the field", "$k' = H^0(X, \\mathcal{O}_X)$. Assume $X$ is not irreducible.", "Before we continue, if $k$ is finite, then we replace $k$", "by a purely transcendental extension $K$. This is allowed by", "Varieties, Lemmas", "\\ref{varieties-lemma-globally-generated-base-change},", "\\ref{varieties-lemma-degree-base-change},", "\\ref{varieties-lemma-geometrically-reduced-any-base-change}, and", "\\ref{varieties-lemma-bijection-irreducible-components},", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology},", "Lemma \\ref{lemma-sanity-check-duality} and the elementary fact that", "$K$ is geometrically integral over $k$.", "\\medskip\\noindent", "Assume that $\\mathcal{L}$ is not globally generated to get a contradiction.", "Then we may choose a coherent ideal sheaf", "$\\mathcal{I} \\subset \\mathcal{O}_X$ such that", "$H^0(X, \\mathcal{I}\\mathcal{L}) = H^0(X, \\mathcal{L})$", "and such that $\\mathcal{O}_X/\\mathcal{I}$ is nonzero with", "support of dimension $0$. For example, take $\\mathcal{I}$", "the ideal sheaf of any closed point in the common", "vanishing locus of the global sections of $\\mathcal{L}$.", "We consider the short exact sequence", "$$", "0 \\to \\mathcal{I}\\mathcal{L} \\to \\mathcal{L} \\to ", "\\mathcal{L}/\\mathcal{I}\\mathcal{L} \\to 0", "$$", "Since the support of $\\mathcal{L}/\\mathcal{I}\\mathcal{L}$", "has dimension $0$ we see that $\\mathcal{L}/\\mathcal{I}\\mathcal{L}$", "is generated by global sections", "(Varieties, Lemma \\ref{varieties-lemma-chi-tensor-finite}).", "From the short exact sequence,", "and the fact that $H^0(X, \\mathcal{I}\\mathcal{L}) = H^0(X, \\mathcal{L})$", "we get an injection", "$H^0(X, \\mathcal{L}/\\mathcal{I}\\mathcal{L}) \\to H^1(X, \\mathcal{I}\\mathcal{L})$.", "\\medskip\\noindent", "Recall that the $k$-vector space $H^1(X, \\mathcal{I}\\mathcal{L})$", "is dual to $\\Hom(\\mathcal{I}\\mathcal{L}, \\omega_X)$.", "Choose $\\varphi : \\mathcal{I}\\mathcal{L} \\to \\omega_X$.", "By Lemma \\ref{lemma-vanishing-on-gorenstein} we have", "$H^1(X, \\mathcal{L}) = 0$. Hence", "$$", "\\dim_k H^0(X, \\mathcal{I}\\mathcal{L}) = \\dim_k H^0(X, \\mathcal{L}) =", "\\deg(\\mathcal{L}) + \\chi(\\mathcal{O}_X) > \\dim_k H^1(X, \\mathcal{O}_X) =", "\\dim_k H^0(X, \\omega_X)", "$$", "We conclude that $\\varphi$ is not injective on global sections, in particular", "$\\varphi$ is not injective. For every generic point $\\eta \\in X$", "of an irreducible component of $X$ denote", "$V_\\eta \\subset \\Hom(\\mathcal{I}\\mathcal{L}, \\omega_X)$ the $k$-subvector", "space consisting of those $\\varphi$ which are zero at $\\eta$.", "Since every associated point of $\\mathcal{I}\\mathcal{L}$", "is a generic point of $X$, the above shows that", "$\\Hom(\\mathcal{I}\\mathcal{L}, \\omega_X) = \\bigcup V_\\eta$.", "As $X$ has finitely many generic points and $k$ is infinite, we conclude", "$\\Hom(\\mathcal{I}\\mathcal{L}, \\omega_X) = V_\\eta$ for some $\\eta$.", "Let $\\eta \\in C \\subset X$ be the corresponding irreducible component.", "Let $Y \\subset X$ be the union of the other irreducible components", "of $X$. Then $Y$ is a nonempty reduced closed subscheme not equal to $X$.", "Let $\\mathcal{J} \\subset \\mathcal{O}_X$ be the ideal sheaf of $Y$.", "Please keep in mind that the support of $\\mathcal{J}$ is $C$.", "\\medskip\\noindent", "Let $\\varphi : \\mathcal{I}\\mathcal{L} \\to \\omega_X$ be arbitrary.", "Since $\\mathcal{J}\\mathcal{I}\\mathcal{L}$", "has no embedded associated points", "(as a submodule of $\\mathcal{L}$) and as $\\varphi$ is zero", "in the generic point $\\eta$ of the support of $\\mathcal{J}$, we find that", "$\\varphi$ factors as", "$$", "\\mathcal{I}\\mathcal{L} \\to", "\\mathcal{I}\\mathcal{L}/\\mathcal{J}\\mathcal{I}\\mathcal{L} \\to \\omega_X", "$$", "We can view $\\mathcal{I}\\mathcal{L}/\\mathcal{J}\\mathcal{I}\\mathcal{L}$", "as the pushforward of a coherent sheaf on $Y$ which by abuse of", "notation we indicate with the same symbol.", "Since $\\omega_Y = \\SheafHom(\\mathcal{O}_Y, \\omega_X)$", "by Lemma \\ref{lemma-closed-immersion-dim-1-CM}", "we find a factorization", "$$", "\\mathcal{I}\\mathcal{L} \\to", "\\mathcal{I}\\mathcal{L}/", "\\mathcal{J}\\mathcal{I}\\mathcal{L} ", "\\xrightarrow{\\varphi'} \\omega_Y \\to \\omega_X", "$$", "of $\\varphi$. Let $\\mathcal{I}' \\subset \\mathcal{O}_Y$ be the", "image of $\\mathcal{I} \\subset \\mathcal{O}_X$. There is a surjective map", "$\\mathcal{I}\\mathcal{L}/\\mathcal{J}\\mathcal{I}\\mathcal{L} \\to", "\\mathcal{I}'\\mathcal{L}|_Y$ whose kernel is supported in closed points.", "Since $\\omega_Y$ is a Cohen-Macaulay module on $Y$, the map $\\varphi'$", "factors through a map", "$\\varphi'' : \\mathcal{I}'\\mathcal{L}|_Y \\to \\omega_Y$.", "Thus we have commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{I}\\mathcal{L} \\ar[r] \\ar[d] &", "\\mathcal{L} \\ar[r] \\ar[d] &", "\\mathcal{L}/\\mathcal{I}\\mathcal{L} \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{I}'\\mathcal{L}|_Y \\ar[r] &", "\\mathcal{L}|_Y \\ar[r] &", "\\mathcal{L}|_Y/\\mathcal{I}'\\mathcal{L}|_Y \\ar[r] &", "0", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "\\mathcal{I}\\mathcal{L} \\ar[r]_\\varphi \\ar[d] & \\omega_X \\\\", "\\mathcal{I}'\\mathcal{L}|_Y \\ar[r]^{\\varphi''} & \\omega_Y \\ar[u]", "}", "}", "$$", "Now we can finish the proof as follows:", "Since for every $\\varphi$ we have a $\\varphi''$ and since", "$\\omega_X \\in \\textit{Coh}(\\mathcal{O}_X)$", "represents the functor $\\mathcal{F} \\mapsto \\Hom_k(H^1(X, \\mathcal{F}), k)$,", "we find that", "$H^1(X, \\mathcal{I}\\mathcal{L}) \\to H^1(Y, \\mathcal{I}'\\mathcal{L}|_Y)$", "is injective. Since the boundary", "$H^0(X, \\mathcal{L}/\\mathcal{I}\\mathcal{L}) \\to H^1(X, \\mathcal{I}\\mathcal{L})$", "is injective, we conclude the composition", "$$", "H^0(X, \\mathcal{L}/\\mathcal{I}\\mathcal{L}) \\to", "H^0(X, \\mathcal{L}|_Y/\\mathcal{I}'\\mathcal{L}|_Y) \\to", "H^1(X, \\mathcal{I}'\\mathcal{L}|_Y)", "$$", "is injective. Since", "$\\mathcal{L}/\\mathcal{I}\\mathcal{L} \\to", "\\mathcal{L}|_Y/\\mathcal{I}'\\mathcal{L}|_Y$", "is a surjective map of coherent modules whose supports have", "dimension $0$, we see that the first map", "$H^0(X, \\mathcal{L}/\\mathcal{I}\\mathcal{L}) \\to", "H^0(X, \\mathcal{L}|_Y/\\mathcal{I}'\\mathcal{L}|_Y)$", "is surjective (and hence bijective).", "But by induction we have that $\\mathcal{L}|_Y$ is globally", "generated (if $Y$ is disconnected this still works of course)", "and hence the boundary map", "$$", "H^0(X, \\mathcal{L}|_Y/\\mathcal{I}'\\mathcal{L}|_Y) \\to", "H^1(X, \\mathcal{I}'\\mathcal{L}|_Y)", "$$", "cannot be injective.", "This contradiction finishes the proof." ], "refs": [ "curves-lemma-degree-more-than-2g", "varieties-lemma-globally-generated-base-change", "varieties-lemma-degree-base-change", "varieties-lemma-geometrically-reduced-any-base-change", "varieties-lemma-bijection-irreducible-components", "coherent-lemma-flat-base-change-cohomology", "curves-lemma-sanity-check-duality", "varieties-lemma-chi-tensor-finite", "curves-lemma-vanishing-on-gorenstein", "curves-lemma-closed-immersion-dim-1-CM" ], "ref_ids": [ 6330, 10994, 11104, 10911, 10934, 3298, 6252, 11030, 6332, 6253 ] } ], "ref_ids": [] }, { "id": 6334, "type": "theorem", "label": "curves-lemma-rational-tail-negative", "categories": [ "curves" ], "title": "curves-lemma-rational-tail-negative", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$", "and $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are", "at-worst-nodal. Let $C \\subset X$ be a rational tail", "(Example \\ref{example-rational-tail}). Then $\\deg(\\omega_X|_C) < 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $X' \\subset X$ be as in the example. Then we have a short exact", "sequence", "$$", "0 \\to \\omega_C \\to \\omega_X|_C \\to \\mathcal{O}_{C \\cap X'} \\to 0", "$$", "See Lemmas \\ref{lemma-closed-subscheme-reduced-gorenstein},", "\\ref{lemma-facts-about-nodal-curves}, and", "\\ref{lemma-closed-subscheme-nodal-curve}.", "With $k'$ as in the example we see that $\\deg(\\omega_C) = -2[k' : k]$", "as $C \\cong \\mathbf{P}^1_{k'}$ by Proposition \\ref{proposition-projective-line}", "and $\\deg(C \\cap X') = [k' : k]$.", "Hence $\\deg(\\omega_X|_C) = -[k' : k]$ which is negative." ], "refs": [ "curves-lemma-closed-subscheme-reduced-gorenstein", "curves-lemma-facts-about-nodal-curves", "curves-lemma-closed-subscheme-nodal-curve", "curves-proposition-projective-line" ], "ref_ids": [ 6254, 6315, 6316, 6350 ] } ], "ref_ids": [] }, { "id": 6335, "type": "theorem", "label": "curves-lemma-rational-tail-field-extension", "categories": [ "curves" ], "title": "curves-lemma-rational-tail-field-extension", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$", "and $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are", "at-worst-nodal. Let $C \\subset X$ be a rational tail", "(Example \\ref{example-rational-tail}).", "For any field extension $K/k$ the base change $C_K \\subset X_K$", "is a finite disjoint union of rational tails." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in C$ and $k' = \\kappa(x)$ be as in the example.", "Observe that $C \\cong \\mathbf{P}^1_{k'}$ by", "Proposition \\ref{proposition-projective-line}.", "Since $k'/k$ is finite separable, we see that", "$k' \\otimes_k K = K'_1 \\times \\ldots \\times K'_n$", "is a finite product of finite separable extensions $K'_i/K$.", "Set $C_i = \\mathbf{P}^1_{K'_i}$ and denote $x_i \\in C_i$", "the inverse image of $x$. Then $C_K = \\coprod C_i$ and", "$X'_K \\cap C_i = x_i$ as desired." ], "refs": [ "curves-proposition-projective-line" ], "ref_ids": [ 6350 ] } ], "ref_ids": [] }, { "id": 6336, "type": "theorem", "label": "curves-lemma-no-rational-tail", "categories": [ "curves" ], "title": "curves-lemma-no-rational-tail", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$", "and $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are", "at-worst-nodal. If $X$ does not have a rational tail", "(Example \\ref{example-rational-tail}),", "then for every reduced connected closed subscheme", "$Y \\subset X$, $Y \\not = X$ of dimension $1$ we have", "$\\deg(\\omega_X|_Y) \\geq \\dim_k H^1(Y, \\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "Let $Y \\subset X$ be as in the statement. Then $k' = H^0(Y, \\mathcal{O}_Y)$", "is a field and a finite extension of $k$ and $[k' : k]$", "divides all numerical invariants below associated to $Y$ and", "coherent sheaves on $Y$, see", "Varieties, Lemma \\ref{varieties-lemma-divisible}.", "Let $Z \\subset X$ be as in", "Lemma \\ref{lemma-closed-subscheme-reduced-gorenstein}.", "We will use the results of this lemma and of", "Lemmas \\ref{lemma-facts-about-nodal-curves} and", "\\ref{lemma-closed-subscheme-nodal-curve} without further mention.", "Then we get a short exact sequence", "$$", "0 \\to \\omega_Y \\to \\omega_X|_Y \\to \\mathcal{O}_{Y \\cap Z} \\to 0", "$$", "See Lemma \\ref{lemma-closed-subscheme-reduced-gorenstein}.", "We conclude that", "$$", "\\deg(\\omega_X|_Y) = \\deg(Y \\cap Z) + \\deg(\\omega_Y) =", "\\deg(Y \\cap Z) - 2\\chi(Y, \\mathcal{O}_Y)", "$$", "Hence, if the lemma is false, then", "$$", "2[k' : k] > \\deg(Y \\cap Z) + \\dim_k H^1(Y, \\mathcal{O}_Y)", "$$", "Since $Y \\cap Z$ is nonempty and by the divisiblity mentioned above,", "this can happen only if $Y \\cap Z$ is a single $k'$-rational point", "of the smooth locus of $Y$ and $H^1(Y, \\mathcal{O}_Y) = 0$.", "If $Y$ is irreducible, then this implies $Y$ is a rational tail.", "If $Y$ is reducible, then since $\\deg(\\omega_X|_Y) = -[k' : k]$", "we find there is some irreducible component $C$ of $Y$", "such that $\\deg(\\omega_X|_C) < 0$, see", "Varieties, Lemma \\ref{varieties-lemma-degree-in-terms-of-components}.", "Then the analysis above applied", "to $C$ gives that $C$ is a rational tail." ], "refs": [ "varieties-lemma-divisible", "curves-lemma-closed-subscheme-reduced-gorenstein", "curves-lemma-facts-about-nodal-curves", "curves-lemma-closed-subscheme-nodal-curve", "curves-lemma-closed-subscheme-reduced-gorenstein", "varieties-lemma-degree-in-terms-of-components" ], "ref_ids": [ 11112, 6254, 6315, 6316, 6254, 11108 ] } ], "ref_ids": [] }, { "id": 6337, "type": "theorem", "label": "curves-lemma-no-rational-tail-semiample-genus-geq-2", "categories": [ "curves" ], "title": "curves-lemma-no-rational-tail-semiample-genus-geq-2", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$", "and $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are", "at-worst-nodal. Assume $X$ does not have a rational tail", "(Example \\ref{example-rational-tail}). If", "\\begin{enumerate}", "\\item the genus of $X$ is $0$, then $X$ is isomorphic to an", "irreducible plane conic and $\\omega_X^{\\otimes -1}$ is very ample,", "\\item the genus of $X$ is $1$, then $\\omega_X \\cong \\mathcal{O}_X$,", "\\item the genus of $X$ is $\\geq 2$, then", "$\\omega_X^{\\otimes m}$ is globally generated for $m \\geq 2$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-facts-about-nodal-curves} we find that $X$ is", "Gorenstein, i.e., $\\omega_X$ is an invertible $\\mathcal{O}_X$-module.", "\\medskip\\noindent", "If the genus of $X$ is zero, then $\\deg(\\omega_X) < 0$, hence if", "$X$ has more than one irreducible component, we get a contradiction", "with Lemma \\ref{lemma-no-rational-tail}. In the irreducible case", "we see that $X$ is isomorphic to an irreducible plane conic and", "$\\omega_X^{\\otimes -1}$ is very ample by Lemma \\ref{lemma-genus-zero}.", "\\medskip\\noindent", "If the genus of $X$ is $1$, then $\\omega_X$ has a global section and", "$\\deg(\\omega_X|_C) = 0$ for all irreducible components.", "Namely, $\\deg(\\omega_X|_C) \\geq 0$ for all irreducible components $C$", "by Lemma \\ref{lemma-no-rational-tail}, the sum of these numbers is", "$0$ by Lemma \\ref{lemma-genus-gorenstein}, and we can apply", "Varieties, Lemma \\ref{varieties-lemma-degree-in-terms-of-components}.", "Then $\\omega_X \\cong \\mathcal{O}_X$ by", "Varieties, Lemma \\ref{varieties-lemma-no-sections-dual-nef}.", "\\medskip\\noindent", "Assume the genus $g$ of $X$ is greater than or equal to $2$.", "If $X$ is irreducible, then we are done by", "Lemma \\ref{lemma-degree-more-than-2g}.", "Assume $X$ reducible.", "By Lemma \\ref{lemma-no-rational-tail} the", "inequalities of Lemma \\ref{lemma-global-generation}", "hold for every $Y \\subset X$ as in the statement, except for", "$Y = X$. Analyzing the proof of Lemma \\ref{lemma-global-generation}", "we see that (in the reducible case) the only inequality", "used for $Y = X$ are", "$$", "\\deg(\\omega_X^{\\otimes m}) > -2 \\chi(\\mathcal{O}_X)", "\\quad\\text{and}\\quad", "\\deg(\\omega_X^{\\otimes m}) + \\chi(\\mathcal{O}_X) > \\dim_k H^1(X, \\mathcal{O}_X)", "$$", "Since these both hold under the assumption $g \\geq 2$ and $m \\geq 2$ we win." ], "refs": [ "curves-lemma-facts-about-nodal-curves", "curves-lemma-no-rational-tail", "curves-lemma-genus-zero", "curves-lemma-no-rational-tail", "curves-lemma-genus-gorenstein", "varieties-lemma-degree-in-terms-of-components", "varieties-lemma-no-sections-dual-nef", "curves-lemma-degree-more-than-2g", "curves-lemma-no-rational-tail", "curves-lemma-global-generation", "curves-lemma-global-generation" ], "ref_ids": [ 6315, 6336, 6274, 6336, 6266, 11108, 11115, 6330, 6336, 6333, 6333 ] } ], "ref_ids": [] }, { "id": 6338, "type": "theorem", "label": "curves-lemma-contracting-rational-tails", "categories": [ "curves" ], "title": "curves-lemma-contracting-rational-tails", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$", "with $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are", "at-worst-nodal. Consider a sequence", "$$", "X = X_0 \\to X_1 \\to \\ldots \\to X_n = X'", "$$", "of contractions of rational tails (Example \\ref{example-rational-tail})", "until none are left. Then", "\\begin{enumerate}", "\\item if the genus of $X$ is $0$, then $X'$ is an irreducible", "plane conic,", "\\item if the genus of $X$ is $1$, then $\\omega_{X'} \\cong \\mathcal{O}_X$,", "\\item if the genus of $X$ is $> 1$, then", "$\\omega_{X'}^{\\otimes m}$ is globally generated for $m \\geq 2$.", "\\end{enumerate}", "If the genus of $X$ is $\\geq 1$, then the morphism $X \\to X'$", "is independent of choices and formation of this morphism", "commutes with base field extensions." ], "refs": [], "proofs": [ { "contents": [ "We proceed by contracting rational tails until there are none", "left. Then we see that (1), (2), (3) hold by", "Lemma \\ref{lemma-no-rational-tail-semiample-genus-geq-2}.", "\\medskip\\noindent", "Uniqueness. To see that $f : X \\to X'$ is independent of the choices", "made, it suffices to show: any rational tail $C \\subset X$ is", "mapped to a point by $X \\to X'$; some details omitted.", "If not, then we can find a section", "$s \\in \\Gamma(X', \\omega_{X'}^{\\otimes 2})$ which does", "not vanish in the generic point of the irreducible component $f(C)$.", "Since in each of the contractions $X_i \\to X_{i + 1}$", "we have a section $X_{i + 1} \\to X_i$, there is a section", "$X' \\to X$ of $f$. Then we have an exact sequence", "$$", "0 \\to \\omega_{X'} \\to \\omega_X \\to \\omega_X|_{X''} \\to 0", "$$", "where $X'' \\subset X$ is the union of the irreducible components", "contracted by $f$. See Lemma \\ref{lemma-closed-subscheme-reduced-gorenstein}.", "Thus we get a map $\\omega_{X'}^{\\otimes 2} \\to \\omega_X^{\\otimes 2}$", "and we can take the image of $s$ to get a section of", "$\\omega_X^{\\otimes 2}$ not vanishing in the generic point of $C$.", "This is a contradiction with the fact that the restriction of", "$\\omega_X$ to a rational tail has negative degree", "(Lemma \\ref{lemma-rational-tail-negative}).", "\\medskip\\noindent", "The statement on base field extensions follows from", "Lemma \\ref{lemma-rational-tail-field-extension}. Some details omitted." ], "refs": [ "curves-lemma-no-rational-tail-semiample-genus-geq-2", "curves-lemma-closed-subscheme-reduced-gorenstein", "curves-lemma-rational-tail-negative", "curves-lemma-rational-tail-field-extension" ], "ref_ids": [ 6337, 6254, 6334, 6335 ] } ], "ref_ids": [] }, { "id": 6339, "type": "theorem", "label": "curves-lemma-rational-bridge-zero", "categories": [ "curves" ], "title": "curves-lemma-rational-bridge-zero", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$", "and $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are", "at-worst-nodal. Let $C \\subset X$ be a rational bridge", "(Example \\ref{example-rational-bridge}). Then $\\deg(\\omega_X|_C) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $X' \\subset X$ be as in the example. Then we have a short exact", "sequence", "$$", "0 \\to \\omega_C \\to \\omega_X|_C \\to \\mathcal{O}_{C \\cap X'} \\to 0", "$$", "See Lemmas \\ref{lemma-closed-subscheme-reduced-gorenstein},", "\\ref{lemma-facts-about-nodal-curves}, and", "\\ref{lemma-closed-subscheme-nodal-curve}.", "With $k''/k'/k$ as in the example we see that", "$\\deg(\\omega_C) = -2[k' : k]$ as $C$ has genus $0$", "(Lemma \\ref{lemma-rr})", "and $\\deg(C \\cap X') = [k'' : k] = 2[k' : k]$.", "Hence $\\deg(\\omega_X|_C) = 0$." ], "refs": [ "curves-lemma-closed-subscheme-reduced-gorenstein", "curves-lemma-facts-about-nodal-curves", "curves-lemma-closed-subscheme-nodal-curve", "curves-lemma-rr" ], "ref_ids": [ 6254, 6315, 6316, 6256 ] } ], "ref_ids": [] }, { "id": 6340, "type": "theorem", "label": "curves-lemma-rational-bridge-field-extension", "categories": [ "curves" ], "title": "curves-lemma-rational-bridge-field-extension", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$", "and $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are", "at-worst-nodal. Let $C \\subset X$ be a rational bridge", "(Example \\ref{example-rational-bridge}).", "For any field extension $K/k$ the base change $C_K \\subset X_K$", "is a finite disjoint union of rational bridges." ], "refs": [], "proofs": [ { "contents": [ "Let $k''/k'/k$ be as in the example.", "Since $k'/k$ is finite separable, we see that", "$k' \\otimes_k K = K'_1 \\times \\ldots \\times K'_n$", "is a finite product of finite separable extensions $K'_i/K$.", "The corresponding product decomposition", "$k'' \\otimes_k K = \\prod K''_i$ gives degree $2$", "separable algebra extensions $K''_i/K'_i$.", "Set $C_i = C_{K'_i}$. Then $C_K = \\coprod C_i$", "and therefore each $C_i$ has genus $0$ (viewed as a curve", "over $K'_i$), because $H^1(C_K, \\mathcal{O}_{C_K}) = 0$", "by flat base change.", "Finally, we have $X'_K \\cap C_i = \\Spec(K''_i)$ has degree $2$", "over $K'_i$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6341, "type": "theorem", "label": "curves-lemma-rational-bridge-canonical", "categories": [ "curves" ], "title": "curves-lemma-rational-bridge-canonical", "contents": [ "Let $c : X \\to Y$ be the contraction of a rational bridge", "(Example \\ref{example-rational-bridge}).", "Then $c^*\\omega_Y \\cong \\omega_X$." ], "refs": [], "proofs": [ { "contents": [ "You can prove this by direct computation, but we prefer to use the", "characterization of $\\omega_X$ as the coherent $\\mathcal{O}_X$-module", "which represents the functor", "$\\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Sets}$,", "$\\mathcal{F} \\mapsto \\Hom_k(H^1(X, \\mathcal{F}), k) =", "H^1(X, \\mathcal{F})^\\vee$, see", "Lemma \\ref{lemma-duality-dim-1-CM} or", "Duality for Schemes, Lemma", "\\ref{duality-lemma-dualizing-module-proper-over-A}.", "\\medskip\\noindent", "To be precise, denote $\\mathcal{C}_Y$ the category whose objects are", "invertible $\\mathcal{O}_Y$-modules and whose maps are", "$\\mathcal{O}_Y$-module homomorphisms. Denote $\\mathcal{C}_X$ the category", "whose objects are invertible $\\mathcal{O}_X$-modules $\\mathcal{L}$ with", "$\\mathcal{L}|_C \\cong \\mathcal{O}_C$ and whose maps are", "$\\mathcal{O}_Y$-module homomorphisms. We claim that the functor", "$$", "c^* : \\mathcal{C}_Y \\to \\mathcal{C}_X", "$$", "is an equivalence of categories. Namely, by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-bijection-on-Pic}", "it is essentially surjective. Then the projection formula", "(Cohomology, Lemma \\ref{cohomology-lemma-projection-formula})", "shows $c_*c^*\\mathcal{N} = \\mathcal{N}$ and hence $c^*$", "is an equivalence with quasi-inverse given by $c_*$.", "\\medskip\\noindent", "We claim $\\omega_X$ is an object of $\\mathcal{C}_X$. Namely, we have a", "short exact sequence", "$$", "0 \\to \\omega_C \\to \\omega_X|_C \\to \\mathcal{O}_{C \\cap X'} \\to 0", "$$", "See Lemma \\ref{lemma-closed-subscheme-reduced-gorenstein}.", "Taking degrees we find $\\deg(\\omega_X|_C) = 0$ (small detail omitted).", "Thus $\\omega_X|_C$ is trivial by Lemma \\ref{lemma-genus-zero-pic}", "and $\\omega_X$ is an object of $\\mathcal{C}_X$.", "\\medskip\\noindent", "Since $R^1c_*\\mathcal{O}_X = 0$ the projection formula shows that", "$R^1c_*c^*\\mathcal{N} = 0$ for $\\mathcal{N} \\in \\Ob(\\mathcal{C}_Y)$.", "Therefore the Leray spectral sequence", "(Cohomology, Lemma \\ref{cohomology-lemma-apply-Leray})", "the diagram", "$$", "\\xymatrix{", "\\mathcal{C}_Y \\ar[rr]_{c^*} \\ar[dr]_{H^1(Y, -)^\\vee} & &", "\\mathcal{C}_X \\ar[ld]^{H^1(X, -)^\\vee} \\\\", "& \\textit{Sets}", "}", "$$", "of categories and functors is commutative. Since", "$\\omega_Y \\in \\Ob(\\mathcal{C}_Y)$ represents the south-east arrow and", "$\\omega_X \\in \\Ob(\\mathcal{C}_X)$ represents the south-east arrow", "we conclude by the Yoneda lemma", "(Categories, Lemma \\ref{categories-lemma-yoneda})." ], "refs": [ "curves-lemma-duality-dim-1-CM", "duality-lemma-dualizing-module-proper-over-A", "more-morphisms-lemma-bijection-on-Pic", "cohomology-lemma-projection-formula", "curves-lemma-closed-subscheme-reduced-gorenstein", "curves-lemma-genus-zero-pic", "cohomology-lemma-apply-Leray", "categories-lemma-yoneda" ], "ref_ids": [ 6251, 13585, 14076, 2243, 6254, 6272, 2071, 12203 ] } ], "ref_ids": [] }, { "id": 6342, "type": "theorem", "label": "curves-lemma-no-rational-bridge-ample-genus-geq-2", "categories": [ "curves" ], "title": "curves-lemma-no-rational-bridge-ample-genus-geq-2", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$", "and $H^0(X, \\mathcal{O}_X) = k$. Assume", "\\begin{enumerate}", "\\item the singularities of $X$ are at-worst-nodal,", "\\item $X$ does not have a rational tail", "(Example \\ref{example-rational-tail}),", "\\item $X$ does not have a rational bridge", "(Example \\ref{example-rational-bridge}),", "\\item the genus $g$ of $X$ is $\\geq 2$.", "\\end{enumerate}", "Then $\\omega_X$ is ample." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that $\\deg(\\omega_X|_C) > 0$ for every irreducible", "component $C$ of $X$, see Varieties, Lemma", "\\ref{varieties-lemma-ampleness-in-terms-of-degrees-components}.", "If $X = C$ is irreducible, this follows from $g \\geq 2$", "and Lemma \\ref{lemma-genus-gorenstein}.", "Otherwise, set $k' = H^0(C, \\mathcal{O}_C)$. This", "is a field and a finite extension of $k$ and $[k' : k]$", "divides all numerical invariants below associated to $C$ and", "coherent sheaves on $C$, see", "Varieties, Lemma \\ref{varieties-lemma-divisible}.", "Let $X' \\subset X$ be the closure of $X \\setminus C$ as in", "Lemma \\ref{lemma-closed-subscheme-reduced-gorenstein}.", "We will use the results of this lemma and of", "Lemmas \\ref{lemma-facts-about-nodal-curves} and", "\\ref{lemma-closed-subscheme-nodal-curve} without further mention.", "Then we get a short exact sequence", "$$", "0 \\to \\omega_C \\to \\omega_X|_C \\to \\mathcal{O}_{C \\cap X'} \\to 0", "$$", "See Lemma \\ref{lemma-closed-subscheme-reduced-gorenstein}.", "We conclude that", "$$", "\\deg(\\omega_X|_C) = \\deg(C \\cap X') + \\deg(\\omega_C) =", "\\deg(C \\cap X') - 2\\chi(C, \\mathcal{O}_C)", "$$", "Hence, if the lemma is false, then", "$$", "2[k' : k] \\geq \\deg(C \\cap X') + 2\\dim_k H^1(C, \\mathcal{O}_C)", "$$", "Since $C \\cap X'$ is nonempty and by the divisiblity mentioned above,", "this can happen only if either", "\\begin{enumerate}", "\\item[(a)] $C \\cap X'$ is a single $k'$-rational point of $C$ and", "$H^1(C, \\mathcal{O}_C) = 0$, and", "\\item[(b)] $C \\cap X'$ has degree $2$ over $k'$ and", "$H^1(C, \\mathcal{O}_C) = 0$.", "\\end{enumerate}", "The first possibility means $C$ is a rational tail", "and the second that $C$ is a rational bridge.", "Since both are excluded the proof is complete." ], "refs": [ "varieties-lemma-ampleness-in-terms-of-degrees-components", "curves-lemma-genus-gorenstein", "varieties-lemma-divisible", "curves-lemma-closed-subscheme-reduced-gorenstein", "curves-lemma-facts-about-nodal-curves", "curves-lemma-closed-subscheme-nodal-curve", "curves-lemma-closed-subscheme-reduced-gorenstein" ], "ref_ids": [ 11117, 6266, 11112, 6254, 6315, 6316, 6254 ] } ], "ref_ids": [] }, { "id": 6343, "type": "theorem", "label": "curves-lemma-contracting-rational-bridges", "categories": [ "curves" ], "title": "curves-lemma-contracting-rational-bridges", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$", "with $H^0(X, \\mathcal{O}_X) = k$ having genus $g \\geq 2$.", "Assume the singularities of $X$ are at-worst-nodal and that", "$X$ has no rational tails. Consider a sequence", "$$", "X = X_0 \\to X_1 \\to \\ldots \\to X_n = X'", "$$", "of contractions of rational bridges", "(Example \\ref{example-rational-bridge}) until none are left.", "Then $\\omega_{X'}$ ample.", "The morphism $X \\to X'$ is independent of choices and", "formation of this morphism commutes with base field extensions." ], "refs": [], "proofs": [ { "contents": [ "We proceed by contracting rational bridges until there are none", "left. Then $\\omega_{X'}$ is ample by", "Lemma \\ref{lemma-no-rational-bridge-ample-genus-geq-2}.", "\\medskip\\noindent", "Denote $f : X \\to X'$ the composition. By", "Lemma \\ref{lemma-rational-bridge-canonical} and induction we see that", "$f^*\\omega_{X'} = \\omega_X$.", "We have $f_*\\mathcal{O}_X = \\mathcal{O}_{X'}$", "because this is true for contraction of a rational bridge.", "Thus the projection formula says that", "$f_*f^*\\mathcal{L} = \\mathcal{L}$ for all invertible", "$\\mathcal{O}_{X'}$-modules $\\mathcal{L}$.", "Hence", "$$", "\\Gamma(X', \\omega_{X'}^{\\otimes m}) = \\Gamma(X, \\omega_X^{\\otimes m})", "$$", "for all $m$. Since $X'$ is the Proj of the direct sum of these", "by Morphisms, Lemma \\ref{morphisms-lemma-proper-ample-is-proj}", "we conclude that the morphism $X \\to X'$ is completely canonical.", "\\medskip\\noindent", "Let $K/k$ be an extension of fields, then", "$\\omega_{X_K}$ is the pullback of $\\omega_X$", "(Lemma \\ref{lemma-sanity-check-duality}) and we have", "$\\Gamma(X, \\omega_X^{\\otimes m}) \\otimes_k K$", "is equal to", "$\\Gamma(X_K, \\omega_{X_K}^{\\otimes m})$", "by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}.", "Thus formation of $f : X \\to X'$ commutes with base change by", "$K/k$ by the arguments given above. Some details omitted." ], "refs": [ "curves-lemma-no-rational-bridge-ample-genus-geq-2", "curves-lemma-rational-bridge-canonical", "morphisms-lemma-proper-ample-is-proj", "curves-lemma-sanity-check-duality", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 6342, 6341, 5434, 6252, 3298 ] } ], "ref_ids": [] }, { "id": 6344, "type": "theorem", "label": "curves-lemma-contract-gorenstein-canonical", "categories": [ "curves" ], "title": "curves-lemma-contract-gorenstein-canonical", "contents": [ "Let $k$ be a field. Let $c : X \\to Y$ be a morphism of proper schemes over $k$", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{O}_Y = c_*\\mathcal{O}_X$ and $R^1c_*\\mathcal{O}_X = 0$,", "\\item $X$ and $Y$ are reduced, Gorenstein, and have dimension $1$,", "\\item $\\exists\\ m \\in \\mathbf{Z}$ with", "$H^1(X, \\omega_X^{\\otimes m}) = 0$ and $\\omega_X^{\\otimes m}$", "generated by global sections.", "\\end{enumerate}", "Then $c^*\\omega_Y \\cong \\omega_X$." ], "refs": [], "proofs": [ { "contents": [ "The fibres of $c$ are geometrically connected by", "More on Morphisms, Theorem", "\\ref{more-morphisms-theorem-stein-factorization-Noetherian}.", "In particular $c$ is surjective.", "There are finitely many closed points $y = y_1, \\ldots, y_r$ of $Y$ where", "$X_y$ has dimension $1$ and over $Y \\setminus \\{y_1, \\ldots, y_r\\}$", "the morphism $c$ is an isomorphism.", "Some details omitted; hint: outside of $\\{y_1, \\ldots, y_r\\}$", "the morphism $c$ is finite, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-characterize-finite}.", "\\medskip\\noindent", "Let us carefully construct a map $b : c^*\\omega_Y \\to \\omega_X$.", "Denote $f : X \\to \\Spec(k)$ and $g : Y \\to \\Spec(k)$ the structure", "morphisms. We have $f^!k = \\omega_X[1]$ and $g^!k = \\omega_Y[1]$, see", "Lemma \\ref{lemma-duality-dim-1} and its proof. Then", "$f^! = c^! \\circ g^!$ and hence $c^!\\omega_Y = \\omega_X$.", "Thus there is a functorial isomorphism", "$$", "\\Hom_{D(\\mathcal{O}_X)}(\\mathcal{F}, \\omega_X)", "\\longrightarrow", "\\Hom_{D(\\mathcal{O}_Y)}(Rc_*\\mathcal{F}, \\omega_Y)", "$$", "for coherent $\\mathcal{O}_X$-modules $\\mathcal{F}$ by definition", "of $c^!$\\footnote{As the restriction of the right adjoint of", "Duality for Schemes, Lemma \\ref{duality-lemma-twisted-inverse-image} to", "$D^+_\\QCoh(\\mathcal{O}_Y)$.}.", "This isomorphism is induced by a trace map $t : Rc_*\\omega_X \\to \\omega_Y$", "(the counit of the adjunction). By the projection formula", "(Cohomology, Lemma \\ref{cohomology-lemma-projection-formula})", "the canonical map $a : \\omega_Y \\to Rc_*c^*\\omega_Y$ is an isomorphism.", "Combining the above we see there is a canonical map", "$b : c^*\\omega_Y \\to \\omega_X$ such that", "$$", "t \\circ Rc_*(b) = a^{-1}", "$$", "In particular, if we restrict $b$ to $c^{-1}(Y \\setminus \\{y_1, \\ldots, y_r\\})$", "then it is an isomorphism (because it is a map between invertible modules", "whose composition with another gives the isomorphism $a^{-1}$).", "\\medskip\\noindent", "Choose $m \\in \\mathbf{Z}$ as in (3) consider the map", "$$", "b^{\\otimes m} :", "\\Gamma(Y, \\omega_Y^{\\otimes m})", "\\longrightarrow", "\\Gamma(X, \\omega_X^{\\otimes m})", "$$", "This map is injective because $Y$ is reduced and by the last", "property of $b$ mentioned in its construction.", "By Riemann-Roch (Lemma \\ref{lemma-rr}) we have", "$\\chi(X, \\omega_X^{\\otimes m}) =\\chi(Y, \\omega_Y^{\\otimes m})$.", "Thus", "$$", "\\dim_k \\Gamma(Y, \\omega_Y^{\\otimes m}) \\geq", "\\dim_k \\Gamma(X, \\omega_X^{\\otimes m}) = \\chi(X, \\omega_X^{\\otimes m})", "$$", "and we conclude $b^{\\otimes m}$ induces an isomorphism on global", "sections. So", "$b^{\\otimes m} : c^*\\omega_Y^{\\otimes m} \\to \\omega_X^{\\otimes m}$", "is surjective as generators of $\\omega_X^{\\otimes m}$ are in the image.", "Hence $b^{\\otimes m}$ is an isomorphism. Thus $b$ is an isomorphism." ], "refs": [ "more-morphisms-theorem-stein-factorization-Noetherian", "coherent-lemma-characterize-finite", "curves-lemma-duality-dim-1", "duality-lemma-twisted-inverse-image", "cohomology-lemma-projection-formula", "curves-lemma-rr" ], "ref_ids": [ 13674, 3365, 6250, 13503, 2243, 6256 ] } ], "ref_ids": [] }, { "id": 6345, "type": "theorem", "label": "curves-lemma-characterize-contraction-to-stable", "categories": [ "curves" ], "title": "curves-lemma-characterize-contraction-to-stable", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension", "$1$ with $H^0(X, \\mathcal{O}_X) = k$ having genus $g \\geq 2$.", "Assume the singularities of $X$ are at-worst-nodal.", "There is a unique morphism (up to unique isomorphism)", "$$", "c : X \\longrightarrow Y", "$$", "of schemes over $k$ having the following properties:", "\\begin{enumerate}", "\\item $Y$ is proper over $k$, $\\dim(Y) = 1$, the singularities of $Y$", "are at-worst-nodal,", "\\item $\\mathcal{O}_Y = c_*\\mathcal{O}_X$ and $R^1c_*\\mathcal{O}_X = 0$, and", "\\item $\\omega_Y$ is ample on $Y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Existence: A morphism with all the properties listed exists by", "combining Lemmas \\ref{lemma-contracting-rational-tails} and", "\\ref{lemma-contracting-rational-bridges} as discussed in the", "introduction to this section.", "Moreover, we see that it can be written as a composition", "$$", "X \\to X_1 \\to X_2 \\ldots \\to X_n \\to X_{n + 1} \\to \\ldots \\to X_{n + n'}", "$$", "where the first $n$ morphisms are contractions of rational tails", "and the last $n'$ morphisms are contractions of rational bridges.", "Note that property (2) holds for each contraction of a rational", "tail (Example \\ref{example-rational-tail}) and contraction of a", "rational bridge (Example \\ref{example-rational-bridge}).", "It is easy to see that this property is inherited by compositions of morphisms.", "\\medskip\\noindent", "Uniqueness: Let $c : X \\to Y$ be a morphism satisfying conditions", "(1), (2), and (3). We will show that there is a unique isomorphism", "$X_{n + n'} \\to Y$ compatible with the morphisms $X \\to X_{n + n'}$ and $c$.", "\\medskip\\noindent", "Before we start the proof we make some observations about $c$.", "We first observe that the fibres of $c$ are geometrically connected by", "More on Morphisms, Theorem", "\\ref{more-morphisms-theorem-stein-factorization-Noetherian}.", "In particular $c$ is surjective.", "For a closed point $y \\in Y$ the fibre $X_y$ satisfies", "$$", "H^1(X_y, \\mathcal{O}_{X_y}) = 0", "\\quad\\text{and}\\quad", "H^0(X_y, \\mathcal{O}_{X_y}) = \\kappa(y)", "$$", "The first equality by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-check-h1-fibre-zero}", "and the second by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-h1-fibre-zero-check-h0-kappa}.", "Thus either $X_y = x$ where $x$ is the unique point of $X$ mapping to", "$y$ and has the same residue field as $y$, or $X_y$ is a $1$-dimensional", "proper scheme over $\\kappa(y)$. Observe that in the second case", "$X_y$ is Cohen-Macaulay (Lemma \\ref{lemma-automatic}).", "However, since $X$ is reduced, we see that $X_y$ must be reduced", "at all of its generic points (details omitted), and hence $X_y$", "is reduced by Properties, Lemma \\ref{properties-lemma-criterion-reduced}.", "It follows that the singularities of $X_y$ are at-worst-nodal", "(Lemma \\ref{lemma-closed-subscheme-nodal-curve}).", "Note that the genus of $X_y$ is zero (see above).", "Finally, there are only a finite number of points $y$ where", "the fibre $X_y$ has dimension $1$, say", "$\\{y_1, \\ldots, y_r\\}$, and $c^{-1}(Y \\setminus \\{y_1, \\ldots, y_r\\})$", "maps isomorphically to $Y \\setminus \\{y_1, \\ldots, y_r\\}$ by $c$.", "Some details omitted; hint: outside of $\\{y_1, \\ldots, y_r\\}$", "the morphism $c$ is finite, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-characterize-finite}.", "\\medskip\\noindent", "Let $C \\subset X$ be a rational tail.", "We claim that $c$ maps $C$ to a point. Assume that this", "is not the case to get a contradiction. Then the image", "of $C$ is an irreducible component $D \\subset Y$.", "Recall that $H^0(C, \\mathcal{O}_C) = k'$ is a finite separable", "extension of $k$ and that $C$ has a $k'$-rational point $x$", "which is also the unique intersection of $C$ with the ``rest'' of $X$.", "We conclude from the general discussion above that", "$C \\setminus \\{x\\} \\subset c^{-1}(Y \\setminus \\{y_1, \\ldots, y_r\\})$", "maps isomorphically to an open $V$ of $D$. Let $y = c(x) \\in D$.", "Observe that $y$ is the only point of $D$ meeting the", "``rest'' of $Y$. If $y \\not \\in \\{y_1, \\ldots, y_r\\}$, then $C \\cong D$", "and it is clear that $D$ is a rational tail of $Y$ which is", "a contradiction with the ampleness of $\\omega_Y$", "(Lemma \\ref{lemma-rational-tail-negative}).", "Thus $y \\in \\{y_1, \\ldots, y_r\\}$ and $\\dim(X_y) = 1$.", "Then $x \\in X_y \\cap C$ and $x$ is a smooth point of $X_y$ and $C$", "(Lemma \\ref{lemma-closed-subscheme-nodal-curve}).", "If $y \\in D$ is a singular point of $D$, then $y$ is a node", "and then $Y = D$ (because there cannot be another component of", "$Y$ passing through $y$ by Lemma \\ref{lemma-closed-subscheme-nodal-curve}).", "Then $X = X_y \\cup C$ which means $g = 0$ because it is", "equal to the genus of $X_y$ by the discussion in", "Example \\ref{example-rational-tail}; a contradiction.", "If $y \\in D$ is a smooth point of $D$, then", "$C \\to D$ is an isomorphism (because the nonsingular projective", "model is unique and $C$ and $D$ are birational, see", "Section \\ref{section-curves-function-fields}). Then $D$ is", "a rational tail of $Y$ which is a contradiction with", "ampleness of $\\omega_Y$.", "\\medskip\\noindent", "Assume $n \\geq 1$. If $C \\subset X$ is the rational tail contracted", "by $X \\to X_1$, then we see that $C$ is mapped to a point of $Y$ by", "the previous paragraph. Hence $c : X \\to Y$ factors through $X \\to X_1$", "(because $X$ is the pushout of $C$ and $X_1$, see discussion in", "Example \\ref{example-rational-tail}).", "After replacing $X$ by $X_1$ we have decreased", "$n$. By induction we may assume $n = 0$, i.e., $X$ does not have", "a rational tail.", "\\medskip\\noindent", "Assume $n = 0$, i.e., $X$ does not have any rational tails.", "Then $\\omega_X^{\\otimes 2}$ and $\\omega_X^{\\otimes 3}$ are", "globally generated by Lemma \\ref{lemma-no-rational-tail-semiample-genus-geq-2}.", "It follows that $H^1(X, \\omega_X^{\\otimes 3}) = 0$ by", "Lemma \\ref{lemma-vanishing-twist}.", "By Lemma \\ref{lemma-contract-gorenstein-canonical} applied with $m = 3$", "we find that $c^*\\omega_Y \\cong \\omega_X$.", "We also have that $\\omega_X = (X \\to X_{n'})^*\\omega_{X_{n'}}$ by", "Lemma \\ref{lemma-rational-bridge-canonical} and induction.", "Applying the projection formula for both $c$ and", "$X \\to X_{n'}$ we conclude that", "$$", "\\Gamma(X_{n'}, \\omega_{X_{n'}}^{\\otimes m}) =", "\\Gamma(X, \\omega_X^{\\otimes m}) =", "\\Gamma(Y, \\omega_Y^{\\otimes m})", "$$", "for all $m$.", "Since $X_{n'}$ and $Y$ are the Proj of the direct sum of these", "by Morphisms, Lemma \\ref{morphisms-lemma-proper-ample-is-proj}", "we conclude that there is a canonical isomorphism $X_{n'} = Y$", "as desired. We omit the verification that this is the unique", "isomorphism making the diagram commute." ], "refs": [ "curves-lemma-contracting-rational-tails", "curves-lemma-contracting-rational-bridges", "more-morphisms-theorem-stein-factorization-Noetherian", "more-morphisms-lemma-check-h1-fibre-zero", "more-morphisms-lemma-h1-fibre-zero-check-h0-kappa", "curves-lemma-automatic", "properties-lemma-criterion-reduced", "curves-lemma-closed-subscheme-nodal-curve", "coherent-lemma-characterize-finite", "curves-lemma-rational-tail-negative", "curves-lemma-closed-subscheme-nodal-curve", "curves-lemma-closed-subscheme-nodal-curve", "curves-lemma-no-rational-tail-semiample-genus-geq-2", "curves-lemma-vanishing-twist", "curves-lemma-contract-gorenstein-canonical", "curves-lemma-rational-bridge-canonical", "morphisms-lemma-proper-ample-is-proj" ], "ref_ids": [ 6338, 6343, 13674, 14069, 14072, 6257, 2988, 6316, 3365, 6334, 6316, 6316, 6337, 6259, 6344, 6341, 5434 ] } ], "ref_ids": [] }, { "id": 6346, "type": "theorem", "label": "curves-lemma-tricanonical", "categories": [ "curves" ], "title": "curves-lemma-tricanonical", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$ with", "$H^0(X, \\mathcal{O}_X) = k$ having genus $g \\geq 2$. Assume the singularities", "of $X$ are at-worst-nodal and $\\omega_X$ is ample. Then", "$\\omega_X^{\\otimes 3}$ is very ample and $H^1(X, \\omega_X^{\\otimes 3}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Combining Varieties, Lemma", "\\ref{varieties-lemma-ampleness-in-terms-of-degrees-components} and", "Lemmas \\ref{lemma-rational-tail-negative} and \\ref{lemma-rational-bridge-zero}", "we see that $X$ contains no rational tails or bridges.", "Then we see that $\\omega_X^{\\otimes 3}$ is globally generated", "by Lemma \\ref{lemma-contracting-rational-tails}.", "Choose a $k$-basis $s_0, \\ldots, s_n$ of", "$H^0(X, \\omega_X^{\\otimes 3})$. We get a morphism", "$$", "\\varphi_{\\omega_X^{\\otimes 3}, (s_0, \\ldots, s_n)} :", "X \\longrightarrow \\mathbf{P}^n_k", "$$", "See Constructions, Section \\ref{constructions-section-projective-space}.", "The lemma asserts that this morphism is a closed immersion.", "To check this we may replace $k$ by its algebraic closure, see", "Descent, Lemma \\ref{descent-lemma-descending-property-closed-immersion}.", "Thus we may assume $k$ is algebraically closed.", "\\medskip\\noindent", "Assume $k$ is algebraically closed.", "We will use Varieties, Lemma", "\\ref{varieties-lemma-variant-separate-points-tangent-vectors}", "to prove the lemma.", "Let $Z \\subset X$ be a closed subscheme of degree $2$ over $Z$", "with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.", "We have to show that", "$$", "H^0(X, \\mathcal{L}) \\to H^0(Z, \\mathcal{L}|_Z)", "$$", "is surjective. Thus it suffices to show that", "$H^1(X, \\mathcal{I}\\mathcal{L}) = 0$.", "To do this we will use Lemma \\ref{lemma-vanishing-on-gorenstein}.", "Thus it suffices to show that", "$$", "3\\deg(\\omega_X|_Y) > -2\\chi(Y, \\mathcal{O}_Y) + \\deg(Z \\cap Y)", "$$", "for every reduced connected closed subscheme $Y \\subset X$.", "Since $k$ is algebraically closed and $Y$ connected and reduced", "we have $H^0(Y, \\mathcal{O}_Y) = k$ (Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}).", "Hence $\\chi(Y, \\mathcal{O}_Y) = 1 - \\dim H^1(Y, \\mathcal{O}_Y)$.", "Thus we have to show", "$$", "3\\deg(\\omega_X|_Y) > -2 + 2\\dim H^1(Y, \\mathcal{O}_Y) + \\deg(Z \\cap Y)", "$$", "which is true by Lemma \\ref{lemma-no-rational-tail}", "except possibly if $Y = X$ or if $\\deg(\\omega_X|_Y) = 0$.", "Since $\\omega_X$ is ample the second possibility does not", "occur (see first lemma cited in this proof). Finally, if", "$Y = X$ we can use Riemann-Roch (Lemma \\ref{lemma-rr})", "and the fact that $g \\geq 2$ to see that the inquality holds.", "The same argument with $Z = \\emptyset$ shows that", "$H^1(X, \\omega_X^{\\otimes 3}) = 0$." ], "refs": [ "varieties-lemma-ampleness-in-terms-of-degrees-components", "curves-lemma-rational-tail-negative", "curves-lemma-rational-bridge-zero", "curves-lemma-contracting-rational-tails", "descent-lemma-descending-property-closed-immersion", "varieties-lemma-variant-separate-points-tangent-vectors", "curves-lemma-vanishing-on-gorenstein", "varieties-lemma-proper-geometrically-reduced-global-sections", "curves-lemma-no-rational-tail", "curves-lemma-rr" ], "ref_ids": [ 11117, 6334, 6339, 6338, 14684, 10998, 6332, 10948, 6336, 6256 ] } ], "ref_ids": [] }, { "id": 6347, "type": "theorem", "label": "curves-lemma-smooth-vector-fields", "categories": [ "curves" ], "title": "curves-lemma-smooth-vector-fields", "contents": [ "Let $k$ be an algebraically closed field.", "Let $X$ be a smooth, proper, connected curve over $k$.", "Let $g$ be the genus of $X$.", "\\begin{enumerate}", "\\item If $g \\geq 2$, then $\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)$", "is zero,", "\\item if $g = 1$ and $D \\in \\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)$", "is nonzero, then $D$ does not fix any closed point of $X$, and", "\\item if $g = 0$ and $D \\in \\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)$", "is nonzero, then $D$ fixes at most $2$ closed points of $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that we have a universal $k$-derivation", "$d : \\mathcal{O}_X \\to \\Omega_{X/k}$ and hence $D = \\theta \\circ d$", "for some $\\mathcal{O}_X$-linear map $\\theta : \\Omega_{X/k} \\to \\mathcal{O}_X$.", "Recall that $\\Omega_{X/k} \\cong \\omega_X$, see", "Lemma \\ref{lemma-duality-dim-1}.", "By Riemann-Roch we have $\\deg(\\omega_X) = 2g - 2$", "(Lemma \\ref{lemma-rr}).", "Thus we see that $\\theta$ is forced to be zero", "if $g > 1$ by Varieties, Lemma", "\\ref{varieties-lemma-check-invertible-sheaf-trivial}.", "This proves part (1).", "If $g = 1$, then a nonzero $\\theta$ does not vanish anywhere and if", "$g = 0$, then a nonzero $\\theta$ vanishes in a divisor of degree $2$.", "Thus parts (2) and (3) follow if we show that", "vanishing of $\\theta$ at a closed point $x \\in X$ is", "equivalent to the statement that $D$ fixes $x$ (as defined above).", "Let $z \\in \\mathcal{O}_{X, x}$ be a uniformizer.", "Then $dz$ is a basis element for $\\Omega_{X, x}$, see", "Lemma \\ref{lemma-uniformizer-works}.", "Since $D(z) = \\theta(dz)$ we conclude." ], "refs": [ "curves-lemma-duality-dim-1", "curves-lemma-rr", "varieties-lemma-check-invertible-sheaf-trivial", "curves-lemma-uniformizer-works" ], "ref_ids": [ 6250, 6256, 11114, 6278 ] } ], "ref_ids": [] }, { "id": 6348, "type": "theorem", "label": "curves-lemma-nodal-vector-fields", "categories": [ "curves" ], "title": "curves-lemma-nodal-vector-fields", "contents": [ "Let $k$ be an algebraically closed field.", "Let $X$ be an at-worst-nodal, proper, connected", "$1$-dimensional scheme over $k$. Let $\\nu : X^\\nu \\to X$ be the normalization.", "Let $S \\subset X^\\nu$ be the set of points where $\\nu$ is not an", "isomorphism. Then", "$$", "\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) =", "\\{D' \\in \\text{Der}_k(\\mathcal{O}_{X^\\nu}, \\mathcal{O}_{X^\\nu}) \\mid", "D' \\text{ fixes every }x^\\nu \\in S\\}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$ be a node. Let $x', x'' \\in X^\\nu$ be the inverse images", "of $x$. (Every node is a split node since $k$ is algebriacally closed, see", "Definition \\ref{definition-split-node} and", "Lemma \\ref{lemma-split-node}.)", "Let $u \\in \\mathcal{O}_{X^\\nu, x'}$ and $v \\in \\mathcal{O}_{X^\\nu, x''}$", "be uniformizers. Observe that we have an exact sequence", "$$", "0 \\to \\mathcal{O}_{X, x} \\to", "\\mathcal{O}_{X^\\nu, x'} \\times \\mathcal{O}_{X^\\nu, x''} \\to k \\to 0", "$$", "This follows from Lemma \\ref{lemma-multicross}.", "Thus we can view $u$ and $v$ as elements of $\\mathcal{O}_{X, x}$", "with $uv = 0$.", "\\medskip\\noindent", "Let $D \\in \\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)$. Then", "$0 = D(uv) = vD(u) + uD(v)$. Since $(u)$ is annihilator of", "$v$ in $\\mathcal{O}_{X, x}$ and vice versa, we see that $D(u) \\in (u)$ and", "$D(v) \\in (v)$. As $\\mathcal{O}_{X^\\nu, x'} = k + (u)$", "we conclude that we can extend $D$ to $\\mathcal{O}_{X^\\nu, x'}$", "and moreover the extension fixes $x'$. This produces a $D'$", "in the right hand side of the equality. Conversely, given a", "$D'$ fixing $x'$ and $x''$ we find that $D'$ preserves", "the subring $\\mathcal{O}_{X, x} \\subset", "\\mathcal{O}_{X^\\nu, x'} \\times \\mathcal{O}_{X^\\nu, x''}$", "and this is how we go from right to left in the equality." ], "refs": [ "curves-definition-split-node", "curves-lemma-split-node", "curves-lemma-multicross" ], "ref_ids": [ 6359, 6310, 6294 ] } ], "ref_ids": [] }, { "id": 6349, "type": "theorem", "label": "curves-lemma-stable-vector-fields", "categories": [ "curves" ], "title": "curves-lemma-stable-vector-fields", "contents": [ "Let $k$ be an algebraically closed field.", "Let $X$ be an at-worst-nodal, proper, connected", "$1$-dimensional scheme over $k$. Assume the genus of $X$ is at least", "$2$ and that $X$ has no rational tails", "or bridges. Then", "$\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $D \\in \\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)$.", "Let $X^\\nu$ be the normalization of $X$.", "Let $D' \\in \\text{Der}_k(\\mathcal{O}_{X^\\nu}, \\mathcal{O}_{X^\\nu})$", "be the element corresponding to $D$ via Lemma \\ref{lemma-nodal-vector-fields}.", "Let $C \\subset X^\\nu$ be an irreducible component.", "If the genus of $C$ is $> 1$, then $D'|_{\\mathcal{O}_C} = 0$", "by Lemma \\ref{lemma-smooth-vector-fields} part (1).", "If the genus of $C$ is $1$, then there is at least one closed point $c$ of $C$", "which maps to a node on $X$ (since otherwise $X \\cong C$ would have genus $1$).", "By the correspondence this means that", "$D'|_{\\mathcal{O}_C}$ fixes $c$ hence is zero by", "Lemma \\ref{lemma-smooth-vector-fields} part (2).", "Finally, if the genus of $C$ is zero, then there are at least", "$3$ pairwise distinct closed points $c_1, c_2, c_3 \\in C$", "mapping to nodes in $X$, since otherwise either", "$X$ is $C$ with two points glued (two points of $C$ mapping", "to the same node), or", "$C$ is a rational bridge (two points mapping to different nodes of $X$), or", "$C$ is a rational tail (one point mapping to a node of $X$).", "These three possibilities are not permitted since", "$C$ has genus $\\geq 2$ and has no rational bridges, or rational tails.", "Whence $D'|_{\\mathcal{O}_C}$ fixes $c_1, c_2, c_3$ hence is zero by", "Lemma \\ref{lemma-smooth-vector-fields} part (3)." ], "refs": [ "curves-lemma-nodal-vector-fields", "curves-lemma-smooth-vector-fields", "curves-lemma-smooth-vector-fields", "curves-lemma-smooth-vector-fields" ], "ref_ids": [ 6348, 6347, 6347, 6347 ] } ], "ref_ids": [] }, { "id": 6350, "type": "theorem", "label": "curves-proposition-projective-line", "categories": [ "curves" ], "title": "curves-proposition-projective-line", "contents": [ "Let $k$ be a field. Let $X$ be a proper curve over $k$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X \\cong \\mathbf{P}^1_k$,", "\\item $X$ is smooth and geometrically irreducible over $k$,", "$X$ has genus $0$, and $X$ has an invertible module of odd degree,", "\\item $X$ is geometrically integral over $k$, $X$ has genus $0$,", "$X$ is Gorenstein, and $X$ has an invertible sheaf of odd degree,", "\\item $H^0(X, \\mathcal{O}_X) = k$, $X$ has genus $0$, $X$ is Gorenstein,", "and $X$ has an invertible sheaf of odd degree,", "\\item $X$ is geometrically integral over $k$, $X$ has genus $0$,", "and $X$ has an invertible $\\mathcal{O}_X$-module of degree $1$,", "\\item $H^0(X, \\mathcal{O}_X) = k$, $X$ has genus $0$,", "and $X$ has an invertible $\\mathcal{O}_X$-module of degree $1$,", "\\item $H^1(X, \\mathcal{O}_X) = 0$ and $X$ has an invertible", "$\\mathcal{O}_X$-module of degree $1$,", "\\item $H^1(X, \\mathcal{O}_X) = 0$ and $X$", "has closed points $x_1, \\ldots, x_n$ such that", "$\\mathcal{O}_{X, x_i}$ is normal and $\\gcd([\\kappa(x_i) : k]) = 1$, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will prove that each condition (2) -- (8) implies (1) and we omit", "the verification that (1) implies (2) -- (8).", "\\medskip\\noindent", "Assume (2). A smooth scheme over $k$ is geometrically reduced", "(Varieties, Lemma \\ref{varieties-lemma-smooth-geometrically-normal})", "and regular (Varieties, Lemma \\ref{varieties-lemma-smooth-regular}).", "Hence $X$ is Gorenstein (Duality for Schemes, Lemma", "\\ref{duality-lemma-regular-gorenstein}).", "Thus we reduce to (3).", "\\medskip\\noindent", "Assume (3). Since $X$ is geometrically integral over $k$ we have", "$H^0(X, \\mathcal{O}_X) = k$ by", "Varieties, Lemma \\ref{varieties-lemma-regular-functions-proper-variety}.", "and we reduce to (4).", "\\medskip\\noindent", "Assume (4). Since $X$ is Gorenstein the dualizing module", "$\\omega_X$ as in Lemma \\ref{lemma-duality-dim-1} has degree", "$\\deg(\\omega_X) = -2$ by Lemma \\ref{lemma-genus-gorenstein}.", "Combined with the assumed existence of an odd degree invertible", "module, we conclude there exists an invertible module of degree $1$.", "In this way we reduce to (6).", "\\medskip\\noindent", "Assume (5). Since $X$ is geometrically integral over $k$ we have", "$H^0(X, \\mathcal{O}_X) = k$ by", "Varieties, Lemma \\ref{varieties-lemma-regular-functions-proper-variety}.", "and we reduce to (6).", "\\medskip\\noindent", "Assume (6). Then $X \\cong \\mathbf{P}^1_k$ by", "Lemma \\ref{lemma-genus-zero-positive-degree}.", "\\medskip\\noindent", "Assume (7). Observe that $\\kappa = H^0(X, \\mathcal{O}_X)$ is a field", "finite over $k$ by", "Varieties, Lemma \\ref{varieties-lemma-regular-functions-proper-variety}.", "If $d = [\\kappa : k] > 1$, then every invertible sheaf has degree", "divisible by $d$ and there cannot be an invertible sheaf of degree $1$.", "Hence $d = 1$ and we reduce to case (6).", "\\medskip\\noindent", "Assume (8). Observe that $\\kappa = H^0(X, \\mathcal{O}_X)$ is a field", "finite over $k$ by", "Varieties, Lemma \\ref{varieties-lemma-regular-functions-proper-variety}.", "Since $\\kappa \\subset \\kappa(x_i)$ we see that $k = \\kappa$", "by the assumption on the gcd of the degrees. The same condition", "allows us to find integers $a_i$ such that", "$1 = \\sum a_i[\\kappa(x_i) : k]$. Because $x_i$ defines an", "effective Cartier divisor on $X$ by", "Varieties, Lemma \\ref{varieties-lemma-regular-point-on-curve}", "we can consider the invertible module $\\mathcal{O}_X(\\sum a_i x_i)$.", "By our choice of $a_i$ the degree of $\\mathcal{L}$ is $1$.", "Thus $X \\cong \\mathbf{P}^1_k$ by Lemma \\ref{lemma-genus-zero-positive-degree}." ], "refs": [ "varieties-lemma-smooth-geometrically-normal", "varieties-lemma-smooth-regular", "duality-lemma-regular-gorenstein", "varieties-lemma-regular-functions-proper-variety", "curves-lemma-duality-dim-1", "curves-lemma-genus-gorenstein", "varieties-lemma-regular-functions-proper-variety", "curves-lemma-genus-zero-positive-degree", "varieties-lemma-regular-functions-proper-variety", "varieties-lemma-regular-functions-proper-variety", "varieties-lemma-regular-point-on-curve", "curves-lemma-genus-zero-positive-degree" ], "ref_ids": [ 11005, 11004, 13590, 11012, 6250, 6266, 11012, 6273, 11012, 11012, 11118, 6273 ] } ], "ref_ids": [] }, { "id": 6351, "type": "theorem", "label": "curves-proposition-unwind-morphism-smooth", "categories": [ "curves" ], "title": "curves-proposition-unwind-morphism-smooth", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a", "nonconstant morphism of proper smooth curves over $k$.", "Then we can factor $f$ as", "$$", "X \\longrightarrow X^{(p^n)} \\longrightarrow Y", "$$", "where $X^{(p^n)} \\to Y$ is a nonconstant morphism of proper smooth curves", "inducing a separable field extension $k(X^{(p^n)})/k(Y)$, we have", "$$", "X^{(p^n)} = X \\times_{\\Spec(k), F_{\\Spec(k)}^n} \\Spec(k),", "$$", "and $X \\to X^{(p^n)}$ is the $n$-fold relative frobenius of $X$." ], "refs": [], "proofs": [ { "contents": [ "By Fields, Lemma \\ref{fields-lemma-separable-first}", "there is a subextension $k(X)/E/k(Y)$ such that", "$k(X)/E$ is purely inseparable and $E/k(Y)$ is separable.", "By Theorem \\ref{theorem-curves-rational-maps}", "this corresponds to a factorization", "$X \\to Z \\to Y$ of $f$ with $Z$ a nonsingular proper curve.", "Apply Lemma \\ref{lemma-purely-inseparable-smooth}", "to the morphism $X \\to Z$ to conclude." ], "refs": [ "fields-lemma-separable-first", "curves-theorem-curves-rational-maps", "curves-lemma-purely-inseparable-smooth" ], "ref_ids": [ 4482, 6238, 6283 ] } ], "ref_ids": [] }, { "id": 6352, "type": "theorem", "label": "curves-proposition-torsion-picard-reduced-proper", "categories": [ "curves" ], "title": "curves-proposition-torsion-picard-reduced-proper", "contents": [ "Let $k$ be an algebraically closed field. Let $X$ be a proper scheme over $k$", "which is reduced, connected, and has dimension $1$. Let $g$ be the genus", "of $X$ and let $g_{geom}$ be the sum of the geometric genera of the", "irreducible components of $X$. For any prime $\\ell$ different from", "the characteristic of $k$ we have", "$$", "\\dim_{\\mathbf{F}_\\ell} \\Pic(X)[\\ell]", "\\leq g + g_{geom}", "$$", "and equality holds if and only if all the singularities of $X$", "are multicross." ], "refs": [], "proofs": [ { "contents": [ "Let $\\nu : X^\\nu \\to X$ be the normalization", "(Varieties, Lemma \\ref{varieties-lemma-prepare-delta-invariant}).", "Choose a factorization", "$$", "X^\\nu = X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "as in Lemma \\ref{lemma-factor-almost-isomorphism}.", "Let us denote $h^0_i = \\dim_k H^0(X_i, \\mathcal{O}_{X_i})$", "and $h^1_i = \\dim_k H^1(X_i, \\mathcal{O}_{X_i})$.", "By Lemmas \\ref{lemma-glue-points} and \\ref{lemma-squish-tangent-vector}", "for each $n > i \\geq 0$ we have", "one of the following there possibilities", "\\begin{enumerate}", "\\item $X_i$ is obtained by glueing $a, b \\in X_{i + 1}$", "which are on different connected components: in this case", "$\\Pic(X_i) = \\Pic(X_{i + 1})$,", "$h^0_{i + 1} = h^0_i + 1$, $h^1_{i + 1} = h^1_i$,", "\\item $X_i$ is obtained by glueing $a, b \\in X_{i + 1}$", "which are on the same connected component: in this case", "there is a short exact sequence", "$$", "0 \\to k^* \\to \\Pic(X_i) \\to \\Pic(X_{i + 1}) \\to 0,", "$$", "and $h^0_{i + 1} = h^0_i$, $h^1_{i + 1} = h^1_i - 1$,", "\\item $X_i$ is obtained by squishing a tangent vector in $X_{i + 1}$:", "in this case there is a short exact sequence", "$$", "0 \\to (k, +) \\to \\Pic(X_i) \\to \\Pic(X_{i + 1}) \\to 0,", "$$", "and $h^0_{i + 1} = h^0_i$, $h^1_{i + 1} = h^1_i - 1$.", "\\end{enumerate}", "To prove the statements on dimensions of cohomology groups of the", "structure sheaf, use the exact sequences in", "Examples \\ref{example-glue-points} and \\ref{example-squish-tangent-vector}.", "Since $k$ is algebraically closed of characteristic prime to $\\ell$", "we see that $(k, +)$ and $k^*$ are $\\ell$-divisible and with", "$\\ell$-torsion $(k, +)[\\ell] = 0$ and $k^*[\\ell] \\cong \\mathbf{F}_\\ell$.", "Hence", "$$", "\\dim_{\\mathbf{F}_\\ell} \\Pic(X_{i + 1})[\\ell] -", "\\dim_{\\mathbf{F}_\\ell}\\Pic(X_i)[\\ell]", "$$", "is zero, except in case (2) where it is equal to $-1$.", "At the end of this process we get the normalization", "$X^\\nu = X_n$ which is a disjoint union of smooth projective", "curves over $k$. Hence we have", "\\begin{enumerate}", "\\item $h^1_n = g_{geom}$ and", "\\item $\\dim_{\\mathbf{F}_\\ell} \\Pic(X_n)[\\ell] = 2g_{geom}$.", "\\end{enumerate}", "The last equality by Lemma \\ref{lemma-torsion-picard-smooth-projective}.", "Since $g = h^1_0$ we see that the number of steps of type", "(2) and (3) is at most $h^1_0 - h^1_n = g - g_{geom}$.", "By our comptation of the differences in ranks we conclude that", "$$", "\\dim_{\\mathbf{F}_\\ell} \\Pic(X)[\\ell] \\leq", "g - g_{geom} + 2g_{geom} = g + g_{geom}", "$$", "and equality holds if and only if no steps of type (3) occur.", "This indeed means that all singularities of $X$ are multicross", "by Lemma \\ref{lemma-multicross}. Conversely, if all the singularities", "are multicross, then Lemma \\ref{lemma-multicross} guarantees that", "we can find a sequence $X^\\nu = X_n \\to \\ldots \\to X_0 = X$", "as above such that no steps of type (3) occur in the sequence", "and we find equality holds in the lemma (just glue the local sequences", "for each point to find one that works for all singular points of $x$;", "some details omitted)." ], "refs": [ "varieties-lemma-prepare-delta-invariant", "curves-lemma-factor-almost-isomorphism", "curves-lemma-glue-points", "curves-lemma-squish-tangent-vector", "curves-lemma-torsion-picard-smooth-projective", "curves-lemma-multicross", "curves-lemma-multicross" ], "ref_ids": [ 11090, 6290, 6291, 6292, 6296, 6294, 6294 ] } ], "ref_ids": [] }, { "id": 6366, "type": "theorem", "label": "etale-cohomology-theorem-sheafification", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-sheafification", "contents": [ "Let $\\mathcal{C}$ be a site and $\\mathcal{F}$ a presheaf on $\\mathcal{C}$.", "\\begin{enumerate}", "\\item The rule", "$$", "U \\mapsto \\mathcal{F}^+(U) :=", "\\colim_{\\mathcal{U} \\text{ covering of }U}", "\\check H^0(\\mathcal{U}, \\mathcal{F})", "$$", "is a presheaf. And the colimit is a directed one.", "\\item There is a canonical map of presheaves $\\mathcal{F} \\to \\mathcal{F}^+$.", "\\item If $\\mathcal{F}$ is a separated presheaf then $\\mathcal{F}^+$ is a sheaf", "and the map in (2) is injective.", "\\item $\\mathcal{F}^+$ is a separated presheaf.", "\\item $\\mathcal{F}^\\# = (\\mathcal{F}^+)^+$ is a sheaf, and the canonical", "map induces a functorial isomorphism", "$$", "\\Hom_{\\textit{PSh}(\\mathcal{C})}(\\mathcal{F}, \\mathcal{G}) =", "\\Hom_{\\Sh(\\mathcal{C})}(\\mathcal{F}^\\#, \\mathcal{G})", "$$", "for any $\\mathcal{G} \\in \\Sh(\\mathcal{C})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See Sites, Theorem \\ref{sites-theorem-plus}." ], "refs": [ "sites-theorem-plus" ], "ref_ids": [ 8492 ] } ], "ref_ids": [] }, { "id": 6367, "type": "theorem", "label": "etale-cohomology-theorem-enough-injectives", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-enough-injectives", "contents": [ "The category of abelian sheaves on a site is an abelian category", "which has enough injectives." ], "refs": [], "proofs": [ { "contents": [ "See", "Modules on Sites, Lemma \\ref{sites-modules-lemma-abelian-abelian} and", "Injectives, Theorem \\ref{injectives-theorem-sheaves-injectives}." ], "refs": [ "sites-modules-lemma-abelian-abelian", "injectives-theorem-sheaves-injectives" ], "ref_ids": [ 14139, 7765 ] } ], "ref_ids": [] }, { "id": 6368, "type": "theorem", "label": "etale-cohomology-theorem-descent-quasi-coherent", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-descent-quasi-coherent", "contents": [ "If $\\mathcal{V} = \\{T_i \\to T\\}_{i\\in I}$ is an fpqc covering, then all", "descent data for quasi-coherent sheaves with respect to $\\mathcal{V}$", "are effective." ], "refs": [], "proofs": [ { "contents": [ "See", "Descent, Proposition \\ref{descent-proposition-fpqc-descent-quasi-coherent}." ], "refs": [ "descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 14753 ] } ], "ref_ids": [] }, { "id": 6369, "type": "theorem", "label": "etale-cohomology-theorem-descent-modules", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-descent-modules", "contents": [ "If $A \\to B$ is faithfully flat then descent data with respect to $A\\to B$", "are effective." ], "refs": [], "proofs": [ { "contents": [ "See", "Descent, Proposition \\ref{descent-proposition-descent-module}.", "See also", "Descent, Remark \\ref{descent-remark-homotopy-equivalent-cosimplicial-algebras}", "for an alternative view of the proof." ], "refs": [ "descent-proposition-descent-module", "descent-remark-homotopy-equivalent-cosimplicial-algebras" ], "ref_ids": [ 14752, 14784 ] } ], "ref_ids": [] }, { "id": 6370, "type": "theorem", "label": "etale-cohomology-theorem-quasi-coherent", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-quasi-coherent", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{C}$ be a site. Assume that", "\\begin{enumerate}", "\\item the underlying category $\\mathcal{C}$ is a", "full subcategory of $\\Sch/S$,", "\\item any Zariski covering of $T \\in \\Ob(\\mathcal{C})$", "can be refined by a covering of $\\mathcal{C}$,", "\\item $S/S$ is an object of $\\mathcal{C}$,", "\\item every covering of $\\mathcal{C}$ is an fpqc covering of schemes.", "\\end{enumerate}", "Then the presheaf $\\mathcal{O}$ is a sheaf on $\\mathcal{C}$ and", "any quasi-coherent $\\mathcal{O}$-module on $(\\mathcal{C}, \\mathcal{O})$", "is of the form $\\mathcal{F}^a$ for some quasi-coherent sheaf", "$\\mathcal{F}$ on $S$." ], "refs": [], "proofs": [ { "contents": [ "After some formal arguments this is exactly Theorem", "\\ref{theorem-descent-quasi-coherent}. Details omitted. In", "Descent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent}", "we prove a more precise version of the theorem for the", "big Zariski, fppf, \\'etale, smooth, and syntomic sites of $S$,", "as well as the small Zariski and \\'etale sites of $S$." ], "refs": [ "etale-cohomology-theorem-descent-quasi-coherent", "descent-proposition-equivalence-quasi-coherent" ], "ref_ids": [ 6368, 14755 ] } ], "ref_ids": [] }, { "id": 6371, "type": "theorem", "label": "etale-cohomology-theorem-cech-derived", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-cech-derived", "contents": [ "On $\\textit{PAb}(\\mathcal{C})$ the functors $\\check{H}^p(\\mathcal{U}, -)$ are", "the right derived functors of $\\check{H}^0(\\mathcal{U}, -)$." ], "refs": [], "proofs": [ { "contents": [ "By the Lemma \\ref{lemma-hom-injective}, the functors", "$\\check H^p(\\mathcal{U}, -)$ are universal", "$\\delta$-functors since they are effaceable.", "So are the right derived functors of $\\check H^0(\\mathcal{U}, -)$. Since they", "agree in degree $0$, they agree by the universal property of universal", "$\\delta$-functors. For more details see", "Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-cech-cohomology-derived-presheaves}." ], "refs": [ "etale-cohomology-lemma-hom-injective", "sites-cohomology-lemma-cech-cohomology-derived-presheaves" ], "ref_ids": [ 6408, 4196 ] } ], "ref_ids": [] }, { "id": 6372, "type": "theorem", "label": "etale-cohomology-theorem-cech-ss", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-cech-ss", "contents": [ "Let $\\mathcal{C}$ be a site. For any covering", "$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ of $U \\in \\Ob(\\mathcal{C})$", "and any abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}$", "there is a spectral sequence", "$$", "E_2^{p, q}", "=", "\\check H^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F}))", "\\Rightarrow", "H^{p+q}(U, \\mathcal{F}),", "$$", "where $\\underline{H}^q(\\mathcal{F})$ is the abelian presheaf", "$V \\mapsto H^q(V, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Choose an injective resolution $\\mathcal{F}\\to \\mathcal{I}^\\bullet$ in", "$\\textit{Ab}(\\mathcal{C})$, and consider the double complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet)$", "and the maps", "$$", "\\xymatrix{", "\\Gamma(U, I^\\bullet) \\ar[r] &", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet) \\\\", "& \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\ar[u]", "}", "$$", "Here the horizontal map is the natural map", "$\\Gamma(U, I^\\bullet) \\to", "\\check{\\mathcal{C}}^0(\\mathcal{U}, \\mathcal{I}^\\bullet)$", "to the left column, and the vertical map is induced by", "$\\mathcal{F}\\to \\mathcal{I}^0$ and lands in the bottom row.", "By assumption, $\\mathcal{I}^\\bullet$ is a complex of injectives in", "$\\textit{Ab}(\\mathcal{C})$, hence by", "Lemma \\ref{lemma-forget-injectives}, it is a complex of injectives in", "$\\textit{PAb}(\\mathcal{C})$. Thus, the rows of the double complex are", "exact in positive degrees (Lemma \\ref{lemma-hom-injective}), and", "the kernel of $\\check{\\mathcal{C}}^0(\\mathcal{U}, \\mathcal{I}^\\bullet)", "\\to \\check{\\mathcal{C}}^1(\\mathcal{U}, \\mathcal{I}^\\bullet)$", "is equal to", "$\\Gamma(U, \\mathcal{I}^\\bullet)$, since $\\mathcal{I}^\\bullet$", "is a complex of sheaves. In particular, the cohomology of the total complex", "is the standard", "cohomology of the global sections functor $H^0(U, \\mathcal{F})$.", "\\medskip\\noindent", "For the vertical direction, the $q$th cohomology group of the $p$th column is", "$$", "\\prod_{i_0, \\ldots, i_p}", "H^q(U_{i_0} \\times_U \\ldots \\times_U U_{i_p}, \\mathcal{F})", "=", "\\prod_{i_0, \\ldots, i_p}", "\\underline{H}^q(\\mathcal{F})(U_{i_0} \\times_U \\ldots \\times_U U_{i_p})", "$$", "in the entry $E_1^{p, q}$. So this is a standard double complex spectral", "sequence, and the $E_2$-page is as prescribed. For more details see", "Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-cech-spectral-sequence}." ], "refs": [ "etale-cohomology-lemma-forget-injectives", "etale-cohomology-lemma-hom-injective", "sites-cohomology-lemma-cech-spectral-sequence" ], "ref_ids": [ 6409, 6408, 4202 ] } ], "ref_ids": [] }, { "id": 6373, "type": "theorem", "label": "etale-cohomology-theorem-zariski-fpqc-quasi-coherent", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-zariski-fpqc-quasi-coherent", "contents": [ "Let $S$ be a scheme and $\\mathcal{F}$ a quasi-coherent $\\mathcal{O}_S$-module.", "Let $\\mathcal{C}$ be either $(\\Sch/S)_\\tau$ for", "$\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$ or", "$S_\\etale$. Then", "$$", "H^p(S, \\mathcal{F}) = H^p_\\tau(S, \\mathcal{F}^a)", "$$", "for all $p \\geq 0$ where", "\\begin{enumerate}", "\\item the left hand side indicates the usual cohomology of the sheaf", "$\\mathcal{F}$ on the underlying topological space of the scheme $S$, and", "\\item the right hand side indicates cohomology", "of the abelian sheaf $\\mathcal{F}^a$ (see", "Proposition \\ref{proposition-quasi-coherent-sheaf-fpqc})", "on the site $\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [ "etale-cohomology-proposition-quasi-coherent-sheaf-fpqc" ], "proofs": [ { "contents": [ "We are going to show that", "$H^p(U, f^*\\mathcal{F}) = H^p_\\tau(U, \\mathcal{F}^a)$", "for any object $f : U \\to S$ of the site $\\mathcal{C}$.", "The result is true for $p = 0$ by the sheaf property.", "\\medskip\\noindent", "Assume that $U$ is affine. Then we want to prove that", "$H^p_\\tau(U, \\mathcal{F}^a) = 0$ for all $p > 0$. We use induction on $p$.", "\\begin{enumerate}", "\\item[p = 1]", "Pick $\\xi \\in H^1_\\tau(U, \\mathcal{F}^a)$.", "By Lemma \\ref{lemma-locality-cohomology},", "there exists an fpqc covering $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$", "such that $\\xi|_{U_i} = 0$ for all $i \\in I$. Up to refining", "$\\mathcal{U}$, we may assume that $\\mathcal{U}$ is a standard", "$\\tau$-covering. Applying the spectral sequence of", "Theorem \\ref{theorem-cech-ss},", "we see that $\\xi$ comes from a cohomology class", "$\\check \\xi \\in \\check H^1(\\mathcal{U}, \\mathcal{F}^a)$.", "Consider the covering $\\mathcal{V} = \\{\\coprod_{i\\in I} U_i \\to U\\}$. By", "Lemma \\ref{lemma-cech-complex},", "$\\check H^\\bullet(\\mathcal{U}, \\mathcal{F}^a) =", "\\check H^\\bullet(\\mathcal{V}, \\mathcal{F}^a)$.", "On the other hand, since $\\mathcal{V}$ is a covering of the form", "$\\{\\Spec(B) \\to \\Spec(A)\\}$ and $f^*\\mathcal{F} = \\widetilde{M}$", "for some $A$-module $M$, we see the {\\v C}ech complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{F})$", "is none other than the complex $(B/A)_\\bullet \\otimes_A M$.", "Now by Lemma \\ref{lemma-descent-modules},", "$H^p((B/A)_\\bullet \\otimes_A M) = 0$ for $p > 0$, hence $\\check \\xi = 0$", "and so $\\xi = 0$.", "\\item[p > 1]", "Pick $\\xi \\in H^p_\\tau(U, \\mathcal{F}^a)$. By", "Lemma \\ref{lemma-locality-cohomology},", "there exists an fpqc covering $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$", "such that $\\xi|_{U_i} = 0$ for all $i \\in I$. Up to refining", "$\\mathcal{U}$, we may assume that $\\mathcal{U}$ is a standard", "$\\tau$-covering. We apply the spectral sequence of", "Theorem \\ref{theorem-cech-ss}.", "Observe that the intersections $U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$", "are affine, so that by induction hypothesis the cohomology groups", "$$", "E_2^{p, q} = \\check H^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F}^a))", "$$", "vanish for all $0 < q < p$. We see that $\\xi$ must come from a", "$\\check \\xi \\in \\check H^p(\\mathcal{U}, \\mathcal{F}^a)$. Replacing", "$\\mathcal{U}$ with the covering $\\mathcal{V}$ containing only one morphism", "and using Lemma \\ref{lemma-descent-modules} again,", "we see that the {\\v C}ech cohomology class $\\check \\xi$ must be zero,", "hence $\\xi = 0$.", "\\end{enumerate}", "Next, assume that $U$ is separated. Choose an affine open covering", "$U = \\bigcup_{i \\in I} U_i$ of $U$. The family", "$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ is then an fpqc covering,", "and all the intersections", "$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ are affine", "since $U$ is separated. So all rows of the spectral sequence of", "Theorem \\ref{theorem-cech-ss}", "are zero, except the zeroth row. Therefore", "$$", "H^p_\\tau(U, \\mathcal{F}^a) =", "\\check H^p(\\mathcal{U}, \\mathcal{F}^a) =", "\\check H^p(\\mathcal{U}, \\mathcal{F}) = H^p(U, \\mathcal{F})", "$$", "where the last equality results from standard scheme theory, see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cech-cohomology-quasi-coherent}.", "\\medskip\\noindent", "The general case is technical and (to extend the proof as given here)", "requires a discussion about maps of spectral sequences, so we won't treat it.", "It follows from", "Descent, Proposition \\ref{descent-proposition-same-cohomology-quasi-coherent}", "(whose proof takes a slightly different approach) combined with", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-of-open}." ], "refs": [ "etale-cohomology-lemma-locality-cohomology", "etale-cohomology-theorem-cech-ss", "etale-cohomology-lemma-cech-complex", "etale-cohomology-lemma-descent-modules", "etale-cohomology-lemma-locality-cohomology", "etale-cohomology-theorem-cech-ss", "etale-cohomology-lemma-descent-modules", "etale-cohomology-theorem-cech-ss", "coherent-lemma-cech-cohomology-quasi-coherent", "descent-proposition-same-cohomology-quasi-coherent", "sites-cohomology-lemma-cohomology-of-open" ], "ref_ids": [ 6417, 6372, 6416, 6403, 6417, 6372, 6403, 6372, 3286, 14754, 4186 ] } ], "ref_ids": [ 6696 ] }, { "id": 6374, "type": "theorem", "label": "etale-cohomology-theorem-picard-group", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-picard-group", "contents": [ "For any scheme $X$ we have canonical identifications", "\\begin{align*}", "H_{fppf}^1(X, \\mathbf{G}_m) & = H^1_{syntomic}(X, \\mathbf{G}_m) \\\\", "& = H^1_{smooth}(X, \\mathbf{G}_m) \\\\", "& = H_\\etale^1(X, \\mathbf{G}_m) \\\\", "& = H^1_{Zar}(X, \\mathbf{G}_m) \\\\", "& = \\Pic(X) \\\\", "& = H^1(X, \\mathcal{O}_X^*)", "\\end{align*}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\tau$ be one of the topologies considered in", "Section \\ref{section-big-small}.", "By", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-h1-invertible}", "we see that", "$H^1_\\tau(X, \\mathbf{G}_m) =", "H^1_\\tau(X, \\mathcal{O}_\\tau^*) =", "\\Pic(\\mathcal{O}_\\tau)$", "where $\\mathcal{O}_\\tau$ is the structure sheaf of the site", "$(\\Sch/X)_\\tau$. Now an invertible $\\mathcal{O}_\\tau$-module", "is a quasi-coherent $\\mathcal{O}_\\tau$-module.", "By Theorem \\ref{theorem-quasi-coherent} or the more precise", "Descent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent}", "we see that $\\Pic(\\mathcal{O}_\\tau) = \\Pic(X)$.", "The last equality is proved in the same way." ], "refs": [ "sites-cohomology-lemma-h1-invertible", "etale-cohomology-theorem-quasi-coherent", "descent-proposition-equivalence-quasi-coherent" ], "ref_ids": [ 4185, 6370, 14755 ] } ], "ref_ids": [] }, { "id": 6375, "type": "theorem", "label": "etale-cohomology-theorem-standard-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-standard-etale", "contents": [ "A ring map $A \\to B$ is \\'etale at a prime $\\mathfrak q$ if and only if there", "exists $g \\in B$, $g \\not \\in \\mathfrak q$ such that $B_g$ is standard", "\\'etale over $A$." ], "refs": [], "proofs": [ { "contents": [ "See", "Algebra, Proposition \\ref{algebra-proposition-etale-locally-standard}." ], "refs": [ "algebra-proposition-etale-locally-standard" ], "ref_ids": [ 1427 ] } ], "ref_ids": [] }, { "id": 6376, "type": "theorem", "label": "etale-cohomology-theorem-exactness-stalks", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-exactness-stalks", "contents": [ "Let $S$ be a scheme.", "A map $a : \\mathcal{F} \\to \\mathcal{G}$ of sheaves of sets is injective", "(resp.\\ surjective) if and only if the map on stalks", "$a_{\\overline{s}} : \\mathcal{F}_{\\overline{s}} \\to \\mathcal{G}_{\\overline{s}}$", "is injective (resp.\\ surjective) for all geometric points of $S$.", "A sequence of abelian sheaves on $S_\\etale$ is exact", "if and only if it is exact on all stalks at geometric points of $S$." ], "refs": [], "proofs": [ { "contents": [ "The necessity of exactness on stalks follows from", "Lemma \\ref{lemma-stalk-exact}.", "For the converse, it suffices to show that a map of sheaves is surjective", "(respectively injective) if and only if it is surjective (respectively", "injective) on all stalks. We prove this in the case of surjectivity, and omit", "the proof in the case of injectivity.", "\\medskip\\noindent", "Let $\\alpha : \\mathcal{F} \\to \\mathcal{G}$ be a map of sheaves such", "that $\\mathcal{F}_{\\overline{s}} \\to \\mathcal{G}_{\\overline{s}}$", "is surjective for all geometric points. Fix", "$U \\in \\Ob(S_\\etale)$", "and $s \\in \\mathcal{G}(U)$. For every $u \\in U$ choose some", "$\\overline{u} \\to U$ lying over $u$ and an \\'etale neighborhood", "$(V_u , \\overline{v}_u) \\to (U, \\overline{u})$ such that", "$s|_{V_u} = \\alpha(s_{V_u})$ for some", "$s_{V_u} \\in \\mathcal{F}(V_u)$.", "This is possible since $\\alpha$ is surjective on", "stalks. Then $\\{V_u \\to U\\}_{u \\in U}$", "is an \\'etale covering on which the restrictions of $s$", "are in the image of the map $\\alpha$.", "Thus, $\\alpha$ is surjective, see", "Sites, Section \\ref{sites-section-sheaves-injective}." ], "refs": [ "etale-cohomology-lemma-stalk-exact" ], "ref_ids": [ 6425 ] } ], "ref_ids": [] }, { "id": 6377, "type": "theorem", "label": "etale-cohomology-theorem-quasi-finite-etale-locally", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-quasi-finite-etale-locally", "contents": [ "Let $A\\to B$ be finite type ring map and $\\mathfrak p \\subset A$ a prime", "ideal. Then there exist an \\'etale ring map $A \\to A'$ and a prime", "$\\mathfrak p' \\subset A'$ lying over $\\mathfrak p$ such that", "\\begin{enumerate}", "\\item", "$\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$,", "\\item", "$ B \\otimes_A A' = B_1\\times \\ldots \\times B_r \\times C$,", "\\item", "$ A'\\to B_i$ is finite and there exists a unique prime $q_i\\subset B_i$ lying", "over $\\mathfrak p'$, and", "\\item all irreducible components of the fibre", "$\\Spec(C \\otimes_{A'} \\kappa(\\mathfrak p'))$ of $C$ over $\\mathfrak p'$", "have dimension at least 1.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-etale-makes-quasi-finite-finite}, or", "see \\cite[Th\\'eor\\`eme 18.12.1]{EGA4}. For a slew of versions in terms of", "morphisms of schemes, see", "More on Morphisms, Section \\ref{more-morphisms-section-etale-localization}." ], "refs": [ "algebra-lemma-etale-makes-quasi-finite-finite" ], "ref_ids": [ 1247 ] } ], "ref_ids": [] }, { "id": 6378, "type": "theorem", "label": "etale-cohomology-theorem-hensel", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-hensel", "contents": [ "Complete local rings are henselian." ], "refs": [], "proofs": [ { "contents": [ "Newton's method. See", "Algebra, Lemma \\ref{algebra-lemma-complete-henselian}." ], "refs": [ "algebra-lemma-complete-henselian" ], "ref_ids": [ 1282 ] } ], "ref_ids": [] }, { "id": 6379, "type": "theorem", "label": "etale-cohomology-theorem-henselian", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-henselian", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. The following are equivalent:", "\\begin{enumerate}", "\\item $R$ is henselian,", "\\item for any $f\\in R[T]$ and any factorization $\\bar f = g_0 h_0$ in", "$\\kappa[T]$ with $\\gcd(g_0, h_0)=1$, there exists a factorization $f = gh$ in", "$R[T]$ with $\\bar g = g_0$ and $\\bar h = h_0$,", "\\item any finite $R$-algebra $S$ is isomorphic to a finite product of", "local rings finite over $R$,", "\\item any finite type $R$-algebra $A$ is isomorphic to a product", "$A \\cong A' \\times C$ where $A' \\cong A_1 \\times \\ldots \\times A_r$", "is a product of finite local $R$-algebras and all the irreducible", "components of $C \\otimes_R \\kappa$ have dimension at least 1,", "\\item if $A$ is an \\'etale $R$-algebra and $\\mathfrak n$ is a maximal ideal of", "$A$ lying over $\\mathfrak m$ such that $\\kappa \\cong A/\\mathfrak n$, then there", "exists an isomorphism $\\varphi : A \\cong R \\times A'$ such that", "$\\varphi(\\mathfrak n) = \\mathfrak m \\times A' \\subset R \\times A'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is just a subset of the results from", "Algebra, Lemma \\ref{algebra-lemma-characterize-henselian}.", "Note that part (5) above corresponds to part (8) of", "Algebra, Lemma \\ref{algebra-lemma-characterize-henselian}", "but is formulated slightly differently." ], "refs": [ "algebra-lemma-characterize-henselian", "algebra-lemma-characterize-henselian" ], "ref_ids": [ 1276, 1276 ] } ], "ref_ids": [] }, { "id": 6380, "type": "theorem", "label": "etale-cohomology-theorem-henselization", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-henselization", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring and", "$\\kappa\\subset\\kappa^{sep}$ a separable algebraic closure.", "There exist canonical flat local ring maps $R \\to R^h \\to R^{sh}$ where", "\\begin{enumerate}", "\\item $R^h$, $R^{sh}$ are filtered colimits of \\'etale $R$-algebras,", "\\item $R^h$ is henselian, $R^{sh}$ is strictly henselian,", "\\item $\\mathfrak m R^h$ (resp.\\ $\\mathfrak m R^{sh}$) is the", "maximal ideal of $R^h$ (resp.\\ $R^{sh}$), and", "\\item $\\kappa = R^h/\\mathfrak m R^h$, and", "$\\kappa^{sep} = R^{sh}/\\mathfrak m R^{sh}$ as extensions of $\\kappa$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The structure of $R^h$ and $R^{sh}$ is described in", "Algebra, Lemmas \\ref{algebra-lemma-henselization} and", "\\ref{algebra-lemma-strict-henselization}." ], "refs": [ "algebra-lemma-henselization", "algebra-lemma-strict-henselization" ], "ref_ids": [ 1294, 1295 ] } ], "ref_ids": [] }, { "id": 6381, "type": "theorem", "label": "etale-cohomology-theorem-fully-faithful", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-fully-faithful", "contents": [ "Let $X$, $Y$ be schemes. Let", "$$", "(g, g^\\#) :", "(\\Sh(X_\\etale), \\mathcal{O}_X)", "\\longrightarrow", "(\\Sh(Y_\\etale), \\mathcal{O}_Y)", "$$", "be a morphism of locally ringed topoi. Then there exists a", "unique morphism of schemes $f : X \\to Y$ such that", "$(g, g^\\#)$ is isomorphic to $(f_{small}, f_{small}^\\sharp)$.", "In other words, the construction", "$$", "\\Sch \\longrightarrow \\textit{Locally ringed topoi},", "\\quad", "X \\longrightarrow (X_\\etale, \\mathcal{O}_X)", "$$", "is fully faithful (morphisms up to $2$-isomorphisms on the right hand side)." ], "refs": [], "proofs": [ { "contents": [ "You can prove this theorem by carefuly adjusting the arguments of", "the proof of", "Lemma \\ref{lemma-morphism-ringed-etale-topoi-affines}", "to the global setting. However, we want to indicate how we", "can glue the result of that lemma to get a global morphism", "due to the rigidity provided by the result of", "Lemma \\ref{lemma-2-morphism}.", "Unfortunately, this is a bit messy.", "\\medskip\\noindent", "Let us prove existence when $Y$ is affine. In this case choose an", "affine open covering $X = \\bigcup U_i$. For each $i$ the inclusion", "morphism $j_i : U_i \\to X$ induces a morphism of locally ringed topoi", "$(j_{i, small}, j_{i, small}^\\sharp) :", "(\\Sh(U_{i, \\etale}), \\mathcal{O}_{U_i})", "\\to", "(\\Sh(X_\\etale), \\mathcal{O}_X)$", "by", "Lemma \\ref{lemma-morphism-locally-ringed}.", "We can compose this with $(g, g^\\sharp)$ to obtain a morphism", "of locally ringed topoi", "$$", "(g, g^\\sharp) \\circ (j_{i, small}, j_{i, small}^\\sharp) :", "(\\Sh(U_{i, \\etale}), \\mathcal{O}_{U_i})", "\\to", "(\\Sh(Y_\\etale), \\mathcal{O}_Y)", "$$", "see", "Modules on Sites,", "Lemma \\ref{sites-modules-lemma-composition-morphisms-locally-ringed-topoi}.", "By", "Lemma \\ref{lemma-morphism-ringed-etale-topoi-affines}", "there exists a unique morphism of schemes $f_i : U_i \\to Y$", "and a $2$-isomorphism", "$$", "t_i :", "(f_{i, small}, f_{i, small}^\\sharp)", "\\longrightarrow", "(g, g^\\sharp) \\circ (j_{i, small}, j_{i, small}^\\sharp).", "$$", "Set $U_{i, i'} = U_i \\cap U_{i'}$, and denote $j_{i, i'} : U_{i, i'} \\to U_i$", "the inclusion morphism. Since we have", "$j_i \\circ j_{i, i'} = j_{i'} \\circ j_{i', i}$", "we see that", "\\begin{align*}", "(g, g^\\sharp) \\circ", "(j_{i, small}, j_{i, small}^\\sharp) \\circ", "(j_{i, i', small}, j_{i, i', small}^\\sharp)", "= \\\\", "(g, g^\\sharp) \\circ", "(j_{i', small}, j_{i', small}^\\sharp) \\circ", "(j_{i', i, small}, j_{i', i, small}^\\sharp)", "\\end{align*}", "Hence by uniqueness (see", "Lemma \\ref{lemma-faithful})", "we conclude that", "$f_i \\circ j_{i, i'} = f_{i'} \\circ j_{i', i}$, in other words the", "morphisms of schemes $f_i = f \\circ j_i$ are the restrictions of a", "global morphism of schemes $f : X \\to Y$. Consider the diagram", "of $2$-isomorphisms (where we drop the components ${}^\\sharp$ to ease the", "notation)", "$$", "\\xymatrix{", "g \\circ j_{i, small} \\circ j_{i, i', small}", "\\ar[rr]^{t_i \\star \\text{id}_{j_{i, i', small}}}", "\\ar@{=}[d] & &", "f_{small} \\circ j_{i, small} \\circ j_{i, i', small} \\ar@{=}[d] \\\\", "g \\circ j_{i', small} \\circ j_{i', i, small}", "\\ar[rr]^{t_{i'} \\star \\text{id}_{j_{i', i, small}}} & &", "f_{small} \\circ j_{i', small} \\circ j_{i', i, small}", "}", "$$", "The notation $\\star$ indicates horizontal composition, see", "Categories, Definition \\ref{categories-definition-2-category}", "in general and", "Sites, Section \\ref{sites-section-2-category}", "for our particular case. By the result of", "Lemma \\ref{lemma-2-morphism}", "this diagram commutes. Hence for any sheaf $\\mathcal{G}$", "on $Y_\\etale$ the isomorphisms", "$t_i : f_{small}^{-1}\\mathcal{G}|_{U_i} \\to g^{-1}\\mathcal{G}|_{U_i}$", "agree over $U_{i, i'}$ and we obtain a global isomorphism", "$t : f_{small}^{-1}\\mathcal{G} \\to g^{-1}\\mathcal{G}$.", "It is clear that this isomorphism is functorial in $\\mathcal{G}$", "and is compatible with the maps $f_{small}^\\sharp$ and $g^\\sharp$", "(because it is compatible with these maps locally).", "This proves the theorem in case $Y$ is affine.", "\\medskip\\noindent", "In the general case, let $V \\subset Y$ be an affine open.", "Then $h_V$ is a subsheaf of the final sheaf $*$ on $Y_\\etale$.", "As $g$ is exact we see that $g^{-1}h_V$ is a subsheaf of the final", "sheaf on $X_\\etale$. Hence by", "Lemma \\ref{lemma-support-subsheaf-final}", "there exists an open subscheme $W \\subset X$ such that $g^{-1}h_V = h_W$. By", "Modules on Sites,", "Lemma \\ref{sites-modules-lemma-localize-morphism-locally-ringed-topoi}", "there exists a commutative diagram of morphisms of locally ringed", "topoi", "$$", "\\xymatrix{", "(\\Sh(W_\\etale), \\mathcal{O}_W) \\ar[r] \\ar[d]_{g'} &", "(\\Sh(X_\\etale), \\mathcal{O}_X) \\ar[d]^g \\\\", "(\\Sh(V_\\etale), \\mathcal{O}_V) \\ar[r] &", "(\\Sh(Y_\\etale), \\mathcal{O}_Y)", "}", "$$", "where the horizontal arrows are the localization morphisms", "(induced by the inclusion morphisms $V \\to Y$ and $W \\to X$)", "and where $g'$ is induced from $g$. By the result of the preceding", "paragraph we obtain a morphism of schemes $f' : W \\to V$ and", "a $2$-isomorphism", "$t : (f'_{small}, (f'_{small})^\\sharp) \\to (g', (g')^\\sharp)$.", "Exactly as before these morphisms $f'$ (for varying affine opens $V \\subset Y$)", "agree on overlaps by uniqueness, so we get a morphism $f : X \\to Y$.", "Moreover, the $2$-isomorphisms $t$ are compatible on overlaps by", "Lemma \\ref{lemma-2-morphism}", "again and we obtain a global $2$-isomorphism", "$(f_{small}, (f_{small})^\\sharp) \\to (g, (g)^\\sharp)$.", "as desired. Some details omitted." ], "refs": [ "etale-cohomology-lemma-morphism-ringed-etale-topoi-affines", "etale-cohomology-lemma-2-morphism", "etale-cohomology-lemma-morphism-locally-ringed", "sites-modules-lemma-composition-morphisms-locally-ringed-topoi", "etale-cohomology-lemma-morphism-ringed-etale-topoi-affines", "etale-cohomology-lemma-faithful", "categories-definition-2-category", "etale-cohomology-lemma-2-morphism", "etale-cohomology-lemma-support-subsheaf-final", "sites-modules-lemma-localize-morphism-locally-ringed-topoi", "etale-cohomology-lemma-2-morphism" ], "ref_ids": [ 6445, 6443, 6442, 14259, 6445, 6444, 12378, 6443, 6428, 14261, 6443 ] } ], "ref_ids": [] }, { "id": 6382, "type": "theorem", "label": "etale-cohomology-theorem-etale-topological", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-etale-topological", "contents": [ "Let $X$ and $Y$ be two schemes over a base scheme $S$. Let", "$S' \\to S$ be a universal homeomorphism.", "Denote $X'$ (resp.\\ $Y'$) the base change to $S'$.", "If $X$ is \\'etale over $S$, then the map", "$$", "\\Mor_S(Y, X) \\longrightarrow \\Mor_{S'}(Y', X')", "$$", "is bijective." ], "refs": [], "proofs": [ { "contents": [ "After base changing via $Y \\to S$, we may assume that $Y = S$.", "Thus we may and do assume both $X$ and $Y$ are \\'etale over $S$.", "In other words, the theorem states that the base change functor", "is a fully faithful functor from the category of schemes \\'etale", "over $S$ to the category of schemes \\'etale over $S'$.", "\\medskip\\noindent", "Consider the forgetful functor", "\\begin{equation}", "\\label{equation-descent-etale-forget}", "\\begin{matrix}", "\\text{descent data }(X', \\varphi')\\text{ relative to }S'/S \\\\", "\\text{ with }X'\\text{ \\'etale over }S'", "\\end{matrix}", "\\longrightarrow", "\\text{schemes }X'\\text{ \\'etale over }S'", "\\end{equation}", "We claim this functor is an equivalence. On the other hand, the", "functor", "\\begin{equation}", "\\label{equation-descent-etale}", "\\text{schemes }X\\text{ \\'etale over }S \\longrightarrow", "\\begin{matrix}", "\\text{descent data }(X', \\varphi')\\text{ relative to }S'/S \\\\", "\\text{ with }X'\\text{ \\'etale over }S'", "\\end{matrix}", "\\end{equation}", "is fully faithful by \\'Etale Morphisms, Lemma", "\\ref{etale-lemma-fully-faithful-cases}.", "Thus the claim implies the theorem.", "\\medskip\\noindent", "Proof of the claim.", "Recall that a universal homeomorphism is the same thing as an", "integral, universally injective, surjective morphism, see", "Morphisms, Lemma \\ref{morphisms-lemma-universal-homeomorphism}.", "In particular, the diagonal $\\Delta : S' \\to S' \\times_S S'$ is a thickening", "by Morphisms, Lemma \\ref{morphisms-lemma-universally-injective}.", "Thus by \\'Etale Morphisms, Theorem", "\\ref{etale-theorem-etale-topological}", "we see that given $X' \\to S'$ \\'etale there is a unique isomorphism", "$$", "\\varphi' : X' \\times_S S' \\to S' \\times_S X'", "$$", "of schemes \\'etale over $S' \\times_S S'$ which pulls back under", "$\\Delta$ to $\\text{id} : X' \\to X'$ over $S'$.", "Since $S' \\to S' \\times_S S' \\times_S S'$", "is a thickening as well (it is bijective and a closed immersion)", "we conclude that $(X', \\varphi')$ is a descent datum relative to $S'/S$.", "The canonical nature of the construction of $\\varphi'$ shows", "that it is compatible with morphisms between schemes \\'etale over $S'$.", "In other words, we obtain a quasi-inverse", "$X' \\mapsto (X', \\varphi')$ of the functor", "(\\ref{equation-descent-etale-forget}). This proves the claim and", "finishes the proof of the theorem." ], "refs": [ "etale-lemma-fully-faithful-cases", "morphisms-lemma-universal-homeomorphism", "morphisms-lemma-universally-injective", "etale-theorem-etale-topological" ], "ref_ids": [ 10718, 5454, 5167, 10693 ] } ], "ref_ids": [] }, { "id": 6383, "type": "theorem", "label": "etale-cohomology-theorem-topological-invariance", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-topological-invariance", "contents": [ "\\begin{reference}", "\\cite[IV Theorem 18.1.2]{EGA}", "\\end{reference}", "Let $f : X \\to Y$ be a morphism of schemes.", "Assume $f$ is integral, universally injective and surjective", "(i.e., $f$ is a universal homeomorphism, see", "Morphisms, Lemma \\ref{morphisms-lemma-universal-homeomorphism}).", "The functor", "$$", "V \\longmapsto V_X = X \\times_Y V", "$$", "defines an equivalence of categories", "$$", "\\{", "\\text{schemes }V\\text{ \\'etale over }Y", "\\}", "\\leftrightarrow", "\\{", "\\text{schemes }U\\text{ \\'etale over }X", "\\}", "$$" ], "refs": [ "morphisms-lemma-universal-homeomorphism" ], "proofs": [ { "contents": [ "[First proof]", "By Theorem \\ref{theorem-etale-topological}", "we see that the functor is fully faithful.", "It remains to show that the functor is essentially surjective.", "Let $U \\to X$ be an \\'etale morphism of schemes.", "\\medskip\\noindent", "Suppose that the result holds if $U$ and $Y$ are affine.", "In that case, we choose an affine open covering", "$U = \\bigcup U_i$ such that each $U_i$ maps", "into an affine open of $Y$. By assumption (affine case) we can", "find \\'etale morphisms $V_i \\to Y$ such that $X \\times_Y V_i \\cong U_i$", "as schemes over $X$. Let $V_{i, i'} \\subset V_i$", "be the open subscheme whose underlying topological space", "corresponds to $U_i \\cap U_{i'}$. Because we have isomorphisms", "$$", "X \\times_Y V_{i, i'} \\cong U_i \\cap U_{i'} \\cong X \\times_Y V_{i', i}", "$$", "as schemes over $X$ we see by fully faithfulness that we", "obtain isomorphisms", "$\\theta_{i, i'} : V_{i, i'} \\to V_{i', i}$ of schemes over $Y$.", "We omit the verification that these isomorphisms satisfy the", "cocycle condition of Schemes, Section \\ref{schemes-section-glueing-schemes}.", "Applying Schemes, Lemma \\ref{schemes-lemma-glue-schemes}", "we obtain a scheme $V \\to Y$ by", "glueing the schemes $V_i$ along the identifications $\\theta_{i, i'}$.", "It is clear that $V \\to Y$ is \\'etale and $X \\times_Y V \\cong U$", "by construction.", "\\medskip\\noindent", "Thus it suffices to show the lemma in case $U$ and $Y$ are affine.", "Recall that in the proof of Theorem \\ref{theorem-etale-topological}", "we showed that $U$ comes with a unique descent datum", "$(U, \\varphi)$ relative to $X/Y$. By", "\\'Etale Morphisms, Proposition \\ref{etale-proposition-effective}", "(which applies because $U \\to X$ is quasi-compact and separated", "as well as \\'etale by our reduction to the affine case)", "there exists an \\'etale morphism $V \\to Y$ such that", "$X \\times_Y V \\cong U$ and the proof is complete." ], "refs": [ "etale-cohomology-theorem-etale-topological", "schemes-lemma-glue-schemes", "etale-cohomology-theorem-etale-topological", "etale-proposition-effective" ], "ref_ids": [ 6382, 7687, 6382, 10733 ] } ], "ref_ids": [ 5454 ] }, { "id": 6384, "type": "theorem", "label": "etale-cohomology-theorem-colimit", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-colimit", "contents": [ "Let $X = \\lim_{i \\in I} X_i$ be a limit of a directed system of schemes", "with affine transition morphisms $f_{i'i} : X_{i'} \\to X_i$. We assume", "that $X_i$ is quasi-compact and quasi-separated for all $i \\in I$.", "Let $(\\mathcal{F}_i, \\varphi_{i'i})$ be a system of abelian sheaves", "on $(X_i, f_{i'i})$. Denote $f_i : X \\to X_i$ the projection and set", "$\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$. Then", "$$", "\\colim_{i\\in I} H_\\etale^p(X_i, \\mathcal{F}_i) = H_\\etale^p(X, \\mathcal{F}).", "$$", "for all $p \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-alternative} we can compute the cohomology", "of $\\mathcal{F}$ on $X_{affine, \\etale}$.", "Thus the result by a combination of", "Lemma \\ref{lemma-colimit-affine-sites}", "and", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-colimit}." ], "refs": [ "etale-cohomology-lemma-alternative", "etale-cohomology-lemma-colimit-affine-sites", "sites-cohomology-lemma-colimit" ], "ref_ids": [ 6413, 6471, 4226 ] } ], "ref_ids": [] }, { "id": 6385, "type": "theorem", "label": "etale-cohomology-theorem-higher-direct-images", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-higher-direct-images", "contents": [ "Let $f: X \\to S$ be a quasi-compact and quasi-separated morphism of schemes,", "$\\mathcal{F}$ an abelian sheaf on $X_\\etale$, and $\\overline{s}$ a", "geometric point of $S$ lying over $s \\in S$. Then", "$$", "\\left(R^nf_* \\mathcal{F}\\right)_{\\overline{s}} =", "H_\\etale^n( X \\times_S \\Spec(\\mathcal{O}_{S, s}^{sh}),", "p^{-1}\\mathcal{F})", "$$", "where $p : X \\times_S \\Spec(\\mathcal{O}_{S, s}^{sh}) \\to X$", "is the projection." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I}$ be the category of \\'etale neighborhoods of $\\overline{s}$", "on $S$. By Lemma \\ref{lemma-higher-direct-images}", "we have", "$$", "(R^nf_*\\mathcal{F})_{\\overline{s}} =", "\\colim_{(V, \\overline{v}) \\in \\mathcal{I}^{opp}}", "H_\\etale^n(X \\times_S V, \\mathcal{F}|_{X \\times_S V}).", "$$", "We may replace $\\mathcal{I}$ by the initial subcategory consisting", "of affine \\'etale neighbourhoods of $\\overline{s}$. Observe that", "$$", "\\Spec(\\mathcal{O}_{S, s}^{sh}) =", "\\lim_{(V, \\overline{v}) \\in \\mathcal{I}} V", "$$", "by Lemma \\ref{lemma-describe-etale-local-ring} and", "Limits, Lemma", "\\ref{limits-lemma-directed-inverse-system-affine-schemes-has-limit}.", "Since fibre products commute with limits we also obtain", "$$", "X \\times_S \\Spec(\\mathcal{O}_{S, s}^{sh}) =", "\\lim_{(V, \\overline{v}) \\in \\mathcal{I}} X \\times_S V", "$$", "We conclude by Lemma \\ref{lemma-directed-colimit-cohomology}." ], "refs": [ "etale-cohomology-lemma-higher-direct-images", "etale-cohomology-lemma-describe-etale-local-ring", "limits-lemma-directed-inverse-system-affine-schemes-has-limit", "etale-cohomology-lemma-directed-colimit-cohomology" ], "ref_ids": [ 6474, 6433, 15026, 6473 ] } ], "ref_ids": [] }, { "id": 6386, "type": "theorem", "label": "etale-cohomology-theorem-equivalence-sheaves-point", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-equivalence-sheaves-point", "contents": [ "Let $S = \\Spec(K)$ with $K$ a field.", "Let $\\overline{s}$ be a geometric point of $S$.", "Let $G = \\text{Gal}_{\\kappa(s)}$ denote the absolute Galois group.", "Taking stalks induces an equivalence of categories", "$$", "\\Sh(S_\\etale) \\longrightarrow G\\textit{-Sets},", "\\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}_{\\overline{s}}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let us construct the inverse to this functor. In", "Fundamental Groups, Lemma \\ref{pione-lemma-sheaves-point}", "we have seen that given a $G$-set $M$ there exists an \\'etale morphism", "$X \\to \\Spec(K)$", "such that $\\Mor_K(\\Spec(K^{sep}), X)$ is", "isomorphic to $M$ as a $G$-set. Consider the sheaf", "$\\mathcal{F}$ on $\\Spec(K)_\\etale$ defined by", "the rule $U \\mapsto \\Mor_K(U, X)$. This is a sheaf as the \\'etale", "topology is subcanonical. Then we see that", "$\\mathcal{F}_{\\overline{s}} = \\Mor_K(\\Spec(K^{sep}), X) = M$", "as $G$-sets (details omitted). This gives the inverse of the functor and", "we win." ], "refs": [ "pione-lemma-sheaves-point" ], "ref_ids": [ 4024 ] } ], "ref_ids": [] }, { "id": 6387, "type": "theorem", "label": "etale-cohomology-theorem-central-simple-algebra", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-central-simple-algebra", "contents": [ "Let $K$ be a field. For a unital, associative (not necessarily commutative)", "$K$-algebra $A$ the following are equivalent", "\\begin{enumerate}", "\\item $A$ is finite central simple $K$-algebra,", "\\item $A$ is a finite dimensional $K$-vector space, $K$ is the center of $A$,", "and $A$ has no nontrivial two-sided ideal,", "\\item there exists $d \\geq 1$ such that", "$A \\otimes_K \\bar K \\cong \\text{Mat}(d \\times d, \\bar K)$,", "\\item there exists $d \\geq 1$ such that", "$A \\otimes_K K^{sep} \\cong \\text{Mat}(d \\times d, K^{sep})$,", "\\item there exist $d \\geq 1$ and a finite Galois extension $K \\subset K'$", "such that", "$A \\otimes_{K'} K' \\cong \\text{Mat}(d \\times d, K')$,", "\\item there exist $n \\geq 1$ and a finite central skew field $D$", "over $K$ such that $A \\cong \\text{Mat}(n \\times n, D)$.", "\\end{enumerate}", "The integer $d$ is called the {\\it degree} of $A$." ], "refs": [], "proofs": [ { "contents": [ "This is a copy of", "Brauer Groups, Lemma \\ref{brauer-lemma-finite-central-simple-algebra}." ], "refs": [ "brauer-lemma-finite-central-simple-algebra" ], "ref_ids": [ 7847 ] } ], "ref_ids": [] }, { "id": 6388, "type": "theorem", "label": "etale-cohomology-theorem-brauer-delta", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-brauer-delta", "contents": [ "Let $K$ be a field with separable algebraic closure $K^{sep}$. The map", "$\\delta : \\text{Br}(K) \\to H^2(\\text{Gal}(K^{sep}/K), (K^{sep})^*)$", "defined above is a group isomorphism." ], "refs": [], "proofs": [ { "contents": [ "[Sketch of proof]", "To prove that $\\delta$ defines a group homomorphism, i.e., that", "$\\delta(A \\otimes_K B) = \\delta(A) + \\delta(B)$, one computes", "directly with cocycles.", "\\medskip\\noindent", "Injectivity of $\\delta$. In the abelian case ($d = 1$), one has the", "identification", "$$", "H^1(\\text{Gal}(K^{sep}/K), \\text{GL}_d(K^{sep})) =", "H_\\etale^1(\\Spec(K), \\text{GL}_d(\\mathcal{O}))", "$$", "the latter of which is trivial by fpqc descent. If this were true in the", "non-abelian case, this would readily imply injectivity of $\\delta$. (See", "\\cite{SGA4.5}.) Rather, to prove this, one can reinterpret $\\delta([A])$ as the", "obstruction to the existence of a $K$-vector space $V$ with a left $A$-module", "structure and such that $\\dim_K V = \\deg A$. In the case where $V$ exists, one", "has $A \\cong \\text{End}_K(V)$.", "\\medskip\\noindent", "For surjectivity, pick a", "cohomology class $\\xi \\in H^2(\\text{Gal}(K^{sep}/K), (K^{sep})^*)$,", "then there exists a finite Galois extension $K \\subset K' \\subset K^{sep}$", "such that $\\xi$ is the image of some", "$\\xi' \\in H^2(\\text{Gal}(K'|K), (K')^*)$. Then write", "down an explicit central simple algebra over $K$ using the data $K', \\xi'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6389, "type": "theorem", "label": "etale-cohomology-theorem-C1-brauer-group-zero", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-C1-brauer-group-zero", "contents": [ "Let $K$ be a $C_1$ field. Then $\\text{Br}(K) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $D$ be a finite dimensional division algebra over $K$ with center $K$. We", "have seen that", "$$", "D \\otimes_K K^{sep} \\cong \\text{Mat}_d(K^{sep})", "$$", "uniquely up to inner isomorphism. Hence the determinant $\\det :", "\\text{Mat}_d(K^{sep}) \\to K^{sep}$ is Galois invariant and descends to a", "homogeneous degree $d$ map", "$$", "\\det = N_\\text{red} : D \\longrightarrow K", "$$", "called the {\\it reduced norm}. Since $K$ is $C_1$, if $d > 1$, then there", "exists a nonzero $x \\in D$ with $N_\\text{red}(x) = 0$. This clearly implies", "that $x$ is not invertible, which is a contradiction. Hence $\\text{Br}(K) = 0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6390, "type": "theorem", "label": "etale-cohomology-theorem-tsen", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-tsen", "contents": [ "The function field of a variety of dimension $r$ over an algebraically closed", "field $k$ is $C_r$." ], "refs": [], "proofs": [ { "contents": [ "For projective space one can show directly that the field", "$k(x_1, \\ldots, x_r)$ is $C_r$ (exercise).", "\\medskip\\noindent", "General case. Without loss of generality, we may assume $X$ to be projective.", "Let $f \\in k(X)[T_1, \\ldots, T_n]_d$ with $0 < d^r < n$.", "Say the coefficients of $f$ are in $\\Gamma(X, \\mathcal{O}_X(H))$", "for some ample $H \\subset X$. Let", "$\\mathbf{\\alpha} = (\\alpha_1, \\ldots, \\alpha_n)$ with $\\alpha_i \\in \\Gamma(X,", "\\mathcal{O}_X(eH))$. Then $f(\\mathbf{\\alpha}) \\in \\Gamma(X,", "\\mathcal{O}_X((de + 1)H))$. Consider the system of equations $f(\\mathbf{\\alpha})", "=0$. Then by asymptotic Riemann-Roch", "(Varieties, Proposition \\ref{varieties-proposition-asymptotic-riemann-roch})", "there exists a $c > 0$ such that", "\\begin{itemize}", "\\item the number of variables is", "$n\\dim_k \\Gamma(X, \\mathcal{O}_X(eH)) \\sim n e^r c$, and", "\\item the number of equations is", "$\\dim_k \\Gamma(X, \\mathcal{O}_X((de + 1)H)) \\sim (de + 1)^r c$.", "\\end{itemize}", "Since $n > d^r$, there are more variables than equations. The equations are", "homogeneous hence there is a solution by", "Lemma \\ref{lemma-algebraically-closed-find-solutions}." ], "refs": [ "varieties-proposition-asymptotic-riemann-roch", "etale-cohomology-lemma-algebraically-closed-find-solutions" ], "ref_ids": [ 11140, 6513 ] } ], "ref_ids": [] }, { "id": 6391, "type": "theorem", "label": "etale-cohomology-theorem-fundamental-exact-sequence", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-fundamental-exact-sequence", "contents": [ "There is a short exact sequence of \\'etale sheaves on $X$", "$$", "0 \\longrightarrow", "\\mathbf{G}_{m, X} \\longrightarrow", "j_* \\mathbf{G}_{m, \\eta} \\longrightarrow", "\\bigoplus\\nolimits_{x \\in X_0} {i_x}_* \\underline{\\mathbf{Z}}", "\\longrightarrow 0.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi : U \\to X$ be an \\'etale morphism. Then by properties of", "\\'etale morphisms (Proposition \\ref{proposition-etale-morphisms}),", "$U = \\coprod_i U_i$ where each $U_i$ is a smooth curve mapping to $X$.", "The above sequence for $U$ is a product of the corresponding sequences", "for each $U_i$, so it suffices to treat the case where $U$ is connected,", "hence irreducible. In this case, there is a well known exact sequence", "$$", "1 \\longrightarrow", "\\Gamma(U, \\mathcal{O}_U^*) \\longrightarrow", "k(U)^* \\longrightarrow \\bigoplus\\nolimits_{y \\in U_0} \\mathbf{Z}_y.", "$$", "This amounts to a sequence", "$$", "0 \\longrightarrow \\Gamma(U, \\mathcal{O}_U^*) \\longrightarrow", "\\Gamma(\\eta \\times_X U, \\mathcal{O}_{\\eta \\times_X U}^*) \\longrightarrow", "\\bigoplus\\nolimits_{x \\in X_0}", "\\Gamma(x \\times_X U, \\underline{\\mathbf{Z}})", "$$", "which, unfolding definitions, is nothing but a sequence", "$$", "0 \\longrightarrow \\mathbf{G}_m(U) \\longrightarrow", "j_* \\mathbf{G}_{m, \\eta}(U) \\longrightarrow", "\\left(\\bigoplus\\nolimits_{x \\in X_0} {i_x}_* \\underline{\\mathbf{Z}}\\right)(U).", "$$", "This defines the maps in the Fundamental Exact Sequence and shows it is exact", "except possibly at the last step. To see surjectivity, let us recall that if", "$U$ is a nonsingular curve and $D$ is a divisor on $U$,", "then there exists a Zariski open covering $\\{U_j \\to U\\}$ of $U$", "such that $D|_{U_j} = \\text{div}(f_j)$ for some $f_j \\in k(U)^*$." ], "refs": [ "etale-cohomology-proposition-etale-morphisms" ], "ref_ids": [ 6697 ] } ], "ref_ids": [] }, { "id": 6392, "type": "theorem", "label": "etale-cohomology-theorem-vanishing-cohomology-Gm-curve", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-vanishing-cohomology-Gm-curve", "contents": [ "Let $X$ be a smooth curve over an algebraically closed field. Then", "$$", "H_\\etale^q(X, \\mathbf{G}_m) = 0 \\ \\ \\text{ for all } q \\geq 2.", "$$" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6393, "type": "theorem", "label": "etale-cohomology-theorem-gabber", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-gabber", "contents": [ "Let $(A, I)$ be a henselian pair. Set $X = \\Spec(A)$ and", "$Z = \\Spec(A/I)$. For any torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$", "we have $H^q_\\etale(X, \\mathcal{F}) = H^q_\\etale(Z, \\mathcal{F}|_Z)$." ], "refs": [], "proofs": [ { "contents": [ "The result holds for $q = 0$ by Lemma \\ref{lemma-h0-henselian-pair}.", "Let $q \\geq 1$. Suppose the result has been shown in all degrees $< q$.", "Let $\\mathcal{F}$ be a torsion abelian sheaf. Let", "$\\mathcal{F} \\to \\mathcal{F}'$", "be an injective map of torsion abelian sheaves (to be chosen later)", "with cokernel $\\mathcal{Q}$ so that we have the short exact sequence", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{Q} \\to 0", "$$", "of torsion abelian sheaves on $X_\\etale$. This gives a map of long exact", "cohomology sequences over $X$ and $Z$ part of which looks like", "$$", "\\xymatrix{", "H^{q - 1}_\\etale(X, \\mathcal{F}') \\ar[d] \\ar[r] &", "H^{q - 1}_\\etale(X, \\mathcal{Q}) \\ar[d] \\ar[r] &", "H^q_\\etale(X, \\mathcal{F}) \\ar[d] \\ar[r] &", "H^q_\\etale(X, \\mathcal{F}') \\ar[d] \\\\", "H^{q - 1}_\\etale(Z, \\mathcal{F}'|_Z) \\ar[r] &", "H^{q - 1}_\\etale(Z, \\mathcal{Q}|_Z) \\ar[r] &", "H^q_\\etale(Z, \\mathcal{F}|_Z) \\ar[r] &", "H^q_\\etale(Z, \\mathcal{F}'|_Z)", "}", "$$", "Using this commutative diagram of abelian groups with exact rows", "we will finish the proof.", "\\medskip\\noindent", "Injectivity for $\\mathcal{F}$. Let $\\xi$ be a nonzero element of", "$H^q_\\etale(X, \\mathcal{F})$. By", "Lemma \\ref{lemma-efface-cohomology-on-closed-by-finite-cover} applied with", "$Z = X$ (!) we can find $\\mathcal{F} \\subset \\mathcal{F}'$ such that", "$\\xi$ maps to zero to the right. Then $\\xi$ is the image of", "an element of $H^{q - 1}_\\etale(X, \\mathcal{Q})$ and bijectivity", "for $q - 1$ implies $\\xi$ does not map to zero in", "$H^q_\\etale(Z, \\mathcal{F}|_Z)$.", "\\medskip\\noindent", "Surjectivity for $\\mathcal{F}$. Let $\\xi$ be an element of", "$H^q_\\etale(Z, \\mathcal{F}|_Z)$. By", "Lemma \\ref{lemma-efface-cohomology-on-closed-by-finite-cover} applied with", "$Z = Z$ we can find $\\mathcal{F} \\subset \\mathcal{F}'$ such that", "$\\xi$ maps to zero to the right. Then $\\xi$ is the image of", "an element of $H^{q - 1}_\\etale(Z, \\mathcal{Q}|_Z)$ and bijectivity", "for $q - 1$ implies $\\xi$ is in the image of the vertical map." ], "refs": [ "etale-cohomology-lemma-h0-henselian-pair", "etale-cohomology-lemma-efface-cohomology-on-closed-by-finite-cover", "etale-cohomology-lemma-efface-cohomology-on-closed-by-finite-cover" ], "ref_ids": [ 6583, 6579, 6579 ] } ], "ref_ids": [] }, { "id": 6394, "type": "theorem", "label": "etale-cohomology-theorem-vanishing-affine-curves", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-vanishing-affine-curves", "contents": [ "If $k$ is an algebraically closed field, $X$ is a separated, finite type", "scheme of dimension $\\leq 1$ over $k$, and $\\mathcal{F}$ is a torsion", "abelian sheaf on $X_\\etale$, then", "\\begin{enumerate}", "\\item", "$H^q_\\etale(X, \\mathcal{F}) = 0$ for $q > 2$,", "\\item", "$H^q_\\etale(X, \\mathcal{F}) = 0$ for $q > 1$ if $X$ is affine,", "\\item", "$H^q_\\etale(X, \\mathcal{F}) = 0$ for $q > 1$ if $p = \\text{char}(k) > 0$", "and $\\mathcal{F}$ is $p$-power torsion,", "\\item", "$H^q_\\etale(X, \\mathcal{F})$ is finite if $\\mathcal{F}$ is", "constructible and torsion prime to $\\text{char}(k)$,", "\\item", "$H^q_\\etale(X, \\mathcal{F})$ is finite if $X$ is proper and", "$\\mathcal{F}$ constructible,", "\\item", "$H^q_\\etale(X, \\mathcal{F}) \\to", "H^q_\\etale(X_{k'}, \\mathcal{F}|_{X_{k'}})$ is an isomorphism", "for any extension $k \\subset k'$ of algebraically closed fields", "if $\\mathcal{F}$ is torsion prime to $\\text{char}(k)$,", "\\item", "$H^q_\\etale(X, \\mathcal{F}) \\to", "H^q_\\etale(X_{k'}, \\mathcal{F}|_{X_{k'}})$ is an isomorphism", "for any extension $k \\subset k'$ of algebraically closed fields", "if $X$ is proper,", "\\item", "$H^2_\\etale(X, \\mathcal{F}) \\to H^2_\\etale(U, \\mathcal{F})$", "is surjective for all $U \\subset X$ open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The theorem says that in Situation \\ref{situation-what-to-prove}", "statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold.", "Our first step is to replace $X$ by its reduction, which is permissible", "by Proposition \\ref{proposition-topological-invariance}.", "By Lemma \\ref{lemma-torsion-colimit-constructible} we can write", "$\\mathcal{F}$ as a filtered colimit of constructible abelian sheaves.", "Taking cohomology commutes with colimits, see Lemma \\ref{lemma-colimit}.", "Moreover, pullback via $X_{k'} \\to X$ commutes with colimits as a left", "adjoint. Thus it suffices to prove the statements for a constructible sheaf.", "\\medskip\\noindent", "In this paragraph we use Lemma \\ref{lemma-ses-statements} without further", "mention. Writing", "$\\mathcal{F} = \\mathcal{F}_1 \\oplus \\ldots \\oplus \\mathcal{F}_r$", "where $\\mathcal{F}_i$ is $\\ell_i$-primary for some prime $\\ell_i$, we may", "assume that $\\ell^n$ kills $\\mathcal{F}$ for some prime $\\ell$. Now consider", "the exact sequence", "$$", "0 \\to \\mathcal{F}[\\ell] \\to \\mathcal{F} \\to \\mathcal{F}/\\mathcal{F}[\\ell] \\to 0.", "$$", "Thus we see that it suffices to assume that $\\mathcal{F}$ is $\\ell$-torsion.", "This means that $\\mathcal{F}$ is a constructible sheaf of", "$\\mathbf{F}_\\ell$-vector spaces for some prime number $\\ell$.", "\\medskip\\noindent", "By definition this means there is a dense open $U \\subset X$", "such that $\\mathcal{F}|_U$ is finite locally constant sheaf of", "$\\mathbf{F}_\\ell$-vector spaces. Since $\\dim(X) \\leq 1$ we may", "assume, after shrinking $U$, that $U = U_1 \\amalg \\ldots \\amalg U_n$", "is a disjoint union of irreducible schemes (just remove the closed", "points which lie in the intersections of $\\geq 2$ components of $U$).", "Consider the short exact sequence", "$$", "0 \\to j_!j^{-1}\\mathcal{F} \\to \\mathcal{F} \\to", "\\bigoplus\\nolimits_{x \\in Z} i_{x *}M_x \\to 0", "$$", "where $Z = X \\setminus U$ and $M_x$ is a finite dimensional $\\mathbf{F}_\\ell$", "vector space, see", "Lemma \\ref{lemma-ses-associated-to-open}. Since the \\'etale cohomology", "of $i_{x *}M_x$ vanishes in degrees $\\geq 1$ and is equal to", "$M_x$ in degree $0$ it suffices to prove the theorem for", "$j_!j^{-1}\\mathcal{F}$ (argue exactly as in the proof of", "Lemma \\ref{lemma-even-easier}). Thus we reduce to the case", "$\\mathcal{F} = j_!\\mathcal{G}$ where", "$\\mathcal{G}$ is a finite locally constant sheaf of $\\mathbf{F}_\\ell$-vector", "spaces on $U$.", "\\medskip\\noindent", "Since we chose $U = U_1 \\amalg \\ldots \\amalg U_n$ with $U_i$ irreducible", "we have", "$$", "j_!\\mathcal{G} =", "j_{1!}(\\mathcal{G}|_{U_1}) \\oplus \\ldots \\oplus", "j_{n!}(\\mathcal{G}|_{U_n})", "$$", "where $j_i : U_i \\to X$ is the inclusion morphism.", "The case of $j_{i!}(\\mathcal{G}|_{U_i})$ is handled in", "Lemma \\ref{lemma-vanishing-easier}." ], "refs": [ "etale-cohomology-proposition-topological-invariance", "etale-cohomology-lemma-torsion-colimit-constructible", "etale-cohomology-lemma-colimit", "etale-cohomology-lemma-ses-statements", "etale-cohomology-lemma-ses-associated-to-open", "etale-cohomology-lemma-even-easier", "etale-cohomology-lemma-vanishing-easier" ], "ref_ids": [ 6699, 6539, 6472, 6586, 6526, 6588, 6590 ] } ], "ref_ids": [] }, { "id": 6395, "type": "theorem", "label": "etale-cohomology-theorem-vanishing-curves", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-vanishing-curves", "contents": [ "Let $X$ be a finite type, dimension $1$ scheme over an", "algebraically closed field $k$. Let $\\mathcal{F}$ be a torsion sheaf", "on $X_\\etale$. Then", "$$", "H_\\etale^q(X, \\mathcal{F}) = 0, \\quad \\forall q \\geq 3.", "$$", "If $X$ affine then also $H_\\etale^2(X, \\mathcal{F}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "If $X$ is separated, this follows immediately from the more precise", "Theorem \\ref{theorem-vanishing-affine-curves}.", "If $X$ is nonseparated, choose an affine open covering", "$X = X_1 \\cup \\ldots \\cup X_n$. By induction on $n$ we may assume", "the vanishing holds over $U = X_1 \\cup \\ldots \\cup X_{n - 1}$.", "Then Mayer-Vietoris (Lemma \\ref{lemma-mayer-vietoris}) gives", "$$", "H^2_\\etale(U, \\mathcal{F}) \\oplus H^2_\\etale(X_n, \\mathcal{F}) \\to", "H^2_\\etale(U \\cap X_n, \\mathcal{F}) \\to", "H^3_\\etale(X, \\mathcal{F}) \\to 0", "$$", "However, since $U \\cap X_n$ is an open of an affine scheme", "and hence affine by our dimension assumption, the group", "$H^2_\\etale(U \\cap X_n, \\mathcal{F})$ vanishes", "by Theorem \\ref{theorem-vanishing-affine-curves}." ], "refs": [ "etale-cohomology-theorem-vanishing-affine-curves", "etale-cohomology-lemma-mayer-vietoris", "etale-cohomology-theorem-vanishing-affine-curves" ], "ref_ids": [ 6394, 6469, 6394 ] } ], "ref_ids": [] }, { "id": 6396, "type": "theorem", "label": "etale-cohomology-theorem-smooth-base-change", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-smooth-base-change", "contents": [ "Consider a cartesian diagram of schemes", "$$", "\\xymatrix{", "X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\", "S & T \\ar[l]_g", "}", "$$", "where $f$ is smooth and $g$ quasi-compact and quasi-separated. Then", "$$", "f^{-1}R^qg_*\\mathcal{F} = R^qh_*e^{-1}\\mathcal{F}", "$$", "for any $q$ and any abelian sheaf $\\mathcal{F}$", "on $T_\\etale$ all of whose stalks at geometric points are torsion of", "orders invertible on $S$." ], "refs": [], "proofs": [ { "contents": [ "[First proof of smooth base change]", "This proof is very long but more direct (using less general theory)", "than the second proof given below.", "\\medskip\\noindent", "The theorem is local on $X_\\etale$. More precisely, suppose we have", "$U \\to X$ \\'etale such that $U \\to S$ factors as $U \\to V \\to S$", "with $V \\to S$ \\'etale. Then we can consider the cartesian square", "$$", "\\xymatrix{", "U \\ar[d]_{f'} & U \\times_X Y \\ar[l]^{h'} \\ar[d]^{e'} \\\\", "V & V \\times_S T \\ar[l]_{g'}", "}", "$$", "and setting $\\mathcal{F}' = \\mathcal{F}|_{V \\times_S T}$", "we have $f^{-1}R^qg_*\\mathcal{F}|_U = (f')^{-1}R^qg'_*\\mathcal{F}'$", "and $R^qh_*e^{-1}\\mathcal{F}|_U = R^qh'_*(e')^{-1}\\mathcal{F}'$", "(as follows from the compatibility of localization with morphisms of sites, see", "Sites, Lemma \\ref{sites-lemma-localize-morphism-strong} and", "and", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-restrict-direct-image-open}).", "Thus it suffices to produce an \\'etale covering of $X$ by", "$U \\to X$ and factorizations $U \\to V \\to S$", "as above such that the theorem holds for the diagram with", "$f'$, $h'$, $g'$, $e'$.", "\\medskip\\noindent", "By the local structure of smooth morphisms, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-etale-over-affine-space},", "we may assume $X$ and $S$ are affine and $X \\to S$", "factors through an \\'etale morphism $X \\to \\mathbf{A}^d_S$.", "If we have a tower of cartesian diagrams", "$$", "\\xymatrix{", "W \\ar[d]_i & Z \\ar[l]^j \\ar[d]^k \\\\", "X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\", "S & T \\ar[l]_g", "}", "$$", "and the theorem holds for the bottom and top squares, then", "the theorem holds for the outer rectangle; this is formal.", "Writing $X \\to S$ as the composition", "$$", "X \\to \\mathbf{A}^{d - 1}_S \\to \\mathbf{A}^{d - 2}_S \\to \\ldots \\to", "\\mathbf{A}^1_S \\to S", "$$", "we conclude that it suffices to prove the theorem when $X$ and $S$", "are affine and $X \\to S$ has relative dimension $1$.", "\\medskip\\noindent", "For every $n \\geq 1$ invertible on $S$, let $\\mathcal{F}[n]$", "be the subsheaf of sections of $\\mathcal{F}$ annihilated by $n$. Then", "$\\mathcal{F} = \\colim \\mathcal{F}[n]$ by our assumption on", "the stalks of $\\mathcal{F}$. The functors $e^{-1}$ and $f^{-1}$", "commute with colimits as they are left adjoints. The functors", "$R^qh_*$ and $R^qg_*$ commute with filtered colimits by", "Lemma \\ref{lemma-relative-colimit}.", "Thus it suffices to prove the theorem for $\\mathcal{F}[n]$.", "From now on we fix an integer $n$, we work with", "sheaves of $\\mathbf{Z}/n\\mathbf{Z}$-modules and", "we assume $S$ is a scheme over $\\Spec(\\mathbf{Z}[1/n])$.", "\\medskip\\noindent", "Next, we reduce to the case where $T$ is affine. Since $g$ is quasi-compact", "and quasi-separate and $S$ is affine, the scheme $T$ is quasi-compact and", "quasi-separated. Thus we can use the induction principle of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}.", "Hence it suffices to that if $T = W \\cup W'$", "is an open covering and the theorem holds for the squares", "$$", "\\xymatrix{", "X \\ar[d] & e^{-1}(W) \\ar[l]^i \\ar[d] \\\\", "S & W \\ar[l]_a", "}", "\\quad", "\\xymatrix{", "X \\ar[d] & e^{-1}(W') \\ar[l]^j \\ar[d] \\\\", "S & W' \\ar[l]_b", "}", "\\quad", "\\xymatrix{", "X \\ar[d] & e^{-1}(W \\cap W') \\ar[l]^-k \\ar[d] \\\\", "S & W \\cap W' \\ar[l]_c", "}", "$$", "then the theorem holds for the original diagram. To see this we consider the", "diagram", "$$", "\\xymatrix{", "f^{-1}R^{q - 1}c_*\\mathcal{F}|_{W \\cap W'} \\ar[d]_{\\cong} \\ar[r] &", "f^{-1}R^qg_*\\mathcal{F} \\ar[d] \\ar[r] &", "f^{-1}R^qa_*\\mathcal{F}|_W \\ar[d] \\oplus", "f^{-1}R^qb_*\\mathcal{F}|_{W'} \\ar[d]_{\\cong} \\\\", "R^qk_*e^{-1}\\mathcal{F}|_{e^{-1}(W \\cap W')} \\ar[r] &", "R^qh_*e^{-1}\\mathcal{F} \\ar[r] &", "R^qi_*e^{-1}\\mathcal{F}|_{e^{-1}(W)} \\oplus", "R^qj_*e^{-1}\\mathcal{F}|_{e^{-1}(W')}", "}", "$$", "whose rows are the long exact sequences of", "Lemma \\ref{lemma-relative-mayer-vietoris}.", "Thus the $5$-lemma gives the desired conclusion.", "\\medskip\\noindent", "Summarizing, we may assume $S$, $X$, $T$, and $Y$ affine,", "$\\mathcal{F}$ is $n$ torsion, $X \\to S$ is smooth of relative dimension $1$,", "and $S$ is a scheme over $\\mathbf{Z}[1/n]$.", "We will prove the theorem by induction on $q$. The base case $q = 0$", "is handled by Lemma \\ref{lemma-fppf-reduced-fibres-base-change-f-star}.", "Assume $q > 0$ and the theorem holds for all smaller degrees.", "Choose a short exact sequence", "$0 \\to \\mathcal{F} \\to \\mathcal{I} \\to \\mathcal{Q} \\to 0$", "where $\\mathcal{I}$ is an injective sheaf of $\\mathbf{Z}/n\\mathbf{Z}$-modules.", "Consider the induced diagram", "$$", "\\xymatrix{", "f^{-1}R^{q - 1}g_*\\mathcal{I} \\ar[d]_{\\cong} \\ar[r] &", "f^{-1}R^{q - 1}g_*\\mathcal{Q} \\ar[d]_{\\cong} \\ar[r] &", "f^{-1}R^qg_*\\mathcal{F} \\ar[d] \\ar[r] &", "0 \\ar[d] \\\\", "R^{q - 1}h_*e^{-1}\\mathcal{I} \\ar[r] &", "R^{q - 1}h_*e^{-1}\\mathcal{Q} \\ar[r] &", "R^qh_*e^{-1}\\mathcal{F} \\ar[r] &", "R^qh_*e^{-1}\\mathcal{I}", "}", "$$", "with exact rows. We have the zero in the right upper corner", "as $\\mathcal{I}$ is injective. The left two vertical arrows are", "isomorphisms by induction hypothesis. Thus it suffices to prove", "that $R^qh_*e^{-1}\\mathcal{I} = 0$.", "\\medskip\\noindent", "Write $S = \\Spec(A)$ and $T = \\Spec(B)$ and say the morphism", "$T \\to S$ is given by the ring map $A \\to B$. We can write", "$A \\to B = \\colim_{i \\in I} (A_i \\to B_i)$ as a filtered", "colimit of maps of rings of finite type over $\\mathbf{Z}[1/n]$", "(see Algebra, Lemma \\ref{algebra-lemma-limit-no-condition}).", "For $i \\in I$ we set $S_i = \\Spec(A_i)$ and $T_i = \\Spec(B_i)$.", "For $i$ large enough we can find a smooth morphism $X_i \\to S_i$", "of relative dimension $1$ such that $X = X_i \\times_{S_i} S$, see", "Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation},", "\\ref{limits-lemma-descend-smooth}, and", "\\ref{limits-lemma-descend-dimension-d}. Set $Y_i = X_i \\times_{S_i} T_i$", "to get squares", "$$", "\\xymatrix{", "X_i \\ar[d]_{f_i} & Y_i \\ar[l]^{h_i} \\ar[d]^{e_i} \\\\", "S_i & T_i \\ar[l]_{g_i}", "}", "$$", "Observe that $\\mathcal{I}_i = (T \\to T_i)_*\\mathcal{I}$", "is an injective sheaf of $\\mathbf{Z}/n\\mathbf{Z}$-modules on", "$T_i$, see Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-pushforward-injective-flat}.", "We have $\\mathcal{I} = \\colim (T \\to T_i)^{-1}\\mathcal{I}_i$", "by Lemma \\ref{lemma-linus-hamann}. Pulling back by $e$ we get", "$e^{-1}\\mathcal{I} = \\colim (Y \\to Y_i)^{-1}e_i^{-1}\\mathcal{I}_i$.", "By Lemma \\ref{lemma-relative-colimit-general} applied to the system", "of morphisms $Y_i \\to X_i$ with limit $Y \\to X$ we have", "$$", "R^qh_*e^{-1}\\mathcal{I} =", "\\colim (X \\to X_i)^{-1} R^qh_{i, *} e_i^{-1}\\mathcal{I}_i", "$$", "This reduces us to the case where $T$ and $S$ are affine of finite", "type over $\\mathbf{Z}[1/n]$.", "\\medskip\\noindent", "Summarizing, we have an integer $q \\geq 1$ such that the theorem holds", "in degrees $< q$, the schemes $S$ and $T$ affine of finite type", "type over $\\mathbf{Z}[1/n]$, we have", "$X \\to S$ smooth of relative dimension $1$ with $X$ affine, and", "$\\mathcal{I}$ is an injective sheaf of $\\mathbf{Z}/n\\mathbf{Z}$-modules", "and we have to show that $R^qh_*e^{-1}\\mathcal{I} = 0$.", "We will do this by induction on $\\dim(T)$.", "\\medskip\\noindent", "The base case is $T = \\emptyset$, i.e., $\\dim(T) < 0$.", "If you don't like this, you can take as your base case", "the case $\\dim(T) = 0$. In this case $T \\to S$ is finite", "(in fact even $T \\to \\Spec(\\mathbf{Z}[1/n])$ is finite as the", "target is Jacobson; details omitted), so", "$h$ is finite too and hence has", "vanishing higher direct images (see references below).", "\\medskip\\noindent", "Assume $\\dim(T) = d \\geq 0$ and we know the result for all situations where $T$", "has lower dimension. Pick $U$ affine and \\'etale over $X$ and a section $\\xi$", "of $R^qh_*q^{-1}\\mathcal{I}$ over $U$. We have to show", "that $\\xi$ is zero. Of course, we may replace $X$ by $U$", "(and correspondingly $Y$ by $U \\times_X Y$)", "and assume $\\xi \\in H^0(X, R^qh_*e^{-1}\\mathcal{I})$.", "Moreover, since $R^qh_*e^{-1}\\mathcal{I}$ is a sheaf,", "it suffices to prove that $\\xi$ is zero locally on $X$.", "Hence we may replace $X$ by the members of an \\'etale covering.", "In particular, using Lemma \\ref{lemma-higher-direct-images}", "we may assume that $\\xi$ is the image of an element", "$\\tilde \\xi \\in H^q(Y, e^{-1}\\mathcal{I})$.", "In terms of $\\tilde \\xi$ our task is to show that", "$\\tilde \\xi$ dies in $H^q(U_i \\times_X Y, e^{-1}\\mathcal{I})$", "for some \\'etale covering $\\{U_i \\to X\\}$.", "\\medskip\\noindent", "By More on Morphisms, Lemma \\ref{more-morphisms-lemma-cover-smooth-by-special}", "we may assume that $X \\to S$ factors as $X \\to V \\to S$", "where $V \\to S$ is \\'etale and $X \\to V$ is a smooth morphism", "of affine schemes of relative dimension $1$, has a section, and", "has geometrically connected fibres. Observe that", "$\\dim(V \\times_S T) \\leq \\dim(T) = d$", "for example by More on Algebra, Lemma", "\\ref{more-algebra-lemma-dimension-etale-extension}.", "Hence we may then replace $S$ by $V$ and $T$ by $V \\times_S T$", "(exactly as in the discussion in the first paragraph of the proof).", "Thus we may assume $X \\to S$ is smooth of relative dimension $1$,", "geometrically connected fibres, and has a section $\\sigma : S \\to X$.", "\\medskip\\noindent", "Let $\\pi : T' \\to T$ be a finite surjective morphism.", "We will use below that $\\dim(T') \\leq \\dim(T) = d$, see", "Algebra, Lemma \\ref{algebra-lemma-integral-dim-up}.", "Choose an injective map $\\pi^{-1}\\mathcal{I} \\to \\mathcal{I}'$", "into an injective sheaf of $\\mathbf{Z}/n\\mathbf{Z}$-modules.", "Then $\\mathcal{I} \\to \\pi_*\\mathcal{I}'$ is injective", "and hence has a splitting (as $\\mathcal{I}$ is an injective", "sheaf of $\\mathbf{Z}/n\\mathbf{Z}$-modules).", "Denote $\\pi' : Y' = Y \\times_T T' \\to Y$ the base change of $\\pi$", "and $e' : Y' \\to T'$ the base change of $e$. Picture", "$$", "\\xymatrix{", "X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e & Y' \\ar[l]^{\\pi'} \\ar[d]^{e'} \\\\", "S & T \\ar[l]_g & T' \\ar[l]_\\pi", "}", "$$", "By Proposition \\ref{proposition-finite-higher-direct-image-zero} and", "Lemma \\ref{lemma-finite-pushforward-commutes-with-base-change} we have", "$R\\pi'_*(e')^{-1}\\mathcal{I}' = e^{-1}\\pi_*\\mathcal{I}'$.", "Thus by the Leray spectral sequence", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-Leray})", "we have", "$$", "H^q(Y', (e')^{-1}\\mathcal{I}') =", "H^q(Y, e^{-1}\\pi_*\\mathcal{I}') \\supset H^q(Y, e^{-1}\\mathcal{I})", "$$", "and this remains true after base change by any $U \\to X$ \\'etale.", "Thus we may replace $T$ by $T'$, $\\mathcal{I}$ by $\\mathcal{I}'$", "and $\\tilde \\xi$ by its image in $H^q(Y', (e')^{-1}\\mathcal{I}')$.", "\\medskip\\noindent", "Suppose we have a factorization $T \\to S' \\to S$ where", "$\\pi : S' \\to S$ is finite.", "Setting $X' = S' \\times_S X$ we can consider the induced diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & X' \\ar[l]^{\\pi'} \\ar[d]^{f'} & Y \\ar[l]^{h'} \\ar[d]^e \\\\", "S & S' \\ar[l]_\\pi & T \\ar[l]_g", "}", "$$", "Since $\\pi'$ has vanishing higher direct images we see that", "$R^qh_*e^{-1}\\mathcal{I} = \\pi'_*R^qh'_*e^{-1}\\mathcal{I}$", "by the Leray spectral sequence. Hence", "$H^0(X, R^qh_*e^{-1}\\mathcal{I}) = H^0(X', R^qh'_*e^{-1}\\mathcal{I})$.", "Thus $\\xi$ is zero if and only if the corresponding section of", "$R^qh'_*e^{-1}\\mathcal{I}$ is zero\\footnote{This", "step can also be seen another way. Namely, we have to show that", "there is an \\'etale covering $\\{U_i \\to X\\}$ such that", "$\\tilde \\xi$ dies in $H^q(U_i \\times_X Y, e^{-1}\\mathcal{I})$.", "However, if we prove there is an \\'etale covering", "$\\{U'_j \\to X'\\}$ such that $\\tilde \\xi$ dies in", "$H^q(U'_i \\times_{X'} Y, e^{-1}\\mathcal{I})$, then by property (B)", "for $X' \\to X$ (Lemma \\ref{lemma-finite-B}) there exists an \\'etale covering", "$\\{U_i \\to X\\}$ such that $U_i \\times_X X'$ is a disjoint union", "of schemes over $X'$ each of which factors through $U'_j$ for some $j$.", "Thus we see that $\\tilde \\xi$ dies in $H^q(U_i \\times_X Y, e^{-1}\\mathcal{I})$", "as desired.}.", "Thus we may replace $S$ by $S'$ and $X$ by $X'$.", "Observe that $\\sigma : S \\to X$ base changes to $\\sigma' : S' \\to X'$", "and hence after this replacement it is still true that $X \\to S$", "has a section $\\sigma$ and geometrically connected fibres.", "\\medskip\\noindent", "We will use that $S$ and $T$ are Nagata schemes, see", "Algebra, Proposition \\ref{algebra-proposition-ubiquity-nagata}", "which will guarantee", "that various normalizations are finite, see", "Morphisms, Lemmas \\ref{morphisms-lemma-nagata-normalization-finite} and", "\\ref{morphisms-lemma-nagata-normalization}.", "In particular, we may first replace $T$ by its normalization", "and then replace $S$ by the normalization of $S$ in $T$.", "Then $T \\to S$ is a disjoint union of dominant", "morphisms of integral normal schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-normal-normalization}.", "Clearly we may argue one connnected component", "at a time, hence we may assume $T \\to S$ is a", "dominant morphism of integral normal schemes.", "\\medskip\\noindent", "Let $s \\in S$ and $t \\in T$ be the generic points. By", "Lemma \\ref{lemma-smooth-base-change-fields} there exist finite field", "extensions $K/\\kappa(t)$ and $k/\\kappa(s)$ such that $k$ is contained in $K$", "and a finite \\'etale Galois covering $Z \\to X_k$ with Galois group $G$", "of order dividing a power of $n$ split over $\\sigma(\\Spec(k))$", "such that $\\tilde \\xi$ maps to zero in $H^q(Z_K, e^{-1}\\mathcal{I}|_{Z_K})$.", "Let $T' \\to T$ be the normalization of $T$ in $\\Spec(K)$", "and let $S' \\to S$ be the normalization of $S$ in $\\Spec(k)$.", "Then we obtain a commutative diagram", "$$", "\\xymatrix{", "S' \\ar[d] & T' \\ar[l] \\ar[d] \\\\", "S & T \\ar[l]", "}", "$$", "whose vertical arrows are finite. By the arguments given above we", "may and do replace $S$ and $T$ by $S'$ and $T'$ (and correspondingly", "$X$ by $X \\times_S S'$ and $Y$ by $Y \\times_T T'$). After this replacement", "we conclude we have a finite \\'etale Galois covering", "$Z \\to X_s$ of the generic fibre of $X \\to S$", "with Galois group $G$ of order dividing a power of $n$", "split over $\\sigma(s)$ such that $\\tilde \\xi$ maps to zero in", "$H^q(Z_t, (Z_t \\to Y)^{-1}e^{-1}\\mathcal{I})$.", "Here $Z_t = Z \\times_S t = Z \\times_s t = Z \\times_{X_s} Y_t$.", "Since $n$ is invertible on $S$,", "by Fundamental Groups, Lemma \\ref{pione-lemma-extend-covering}", "we can find a finite \\'etale morphism $U \\to X$ whose restriction to $X_s$", "is $Z$.", "\\medskip\\noindent", "At this point we replace $X$ by $U$ and $Y$ by $U \\times_X Y$.", "After this replacement it may", "no longer be the case that the fibres of $X \\to S$ are geometrically", "connected (there still is a section but we won't use this), but what", "we gain is that after this replacement $\\tilde \\xi$ maps to zero", "in $H^q(Y_t, e^{-1}\\mathcal{I})$, i.e., $\\tilde \\xi$ restricts to", "zero on the generic fibre of $Y \\to T$.", "\\medskip\\noindent", "Recall that $t$ is the spectrum of the function field of $T$, i.e.,", "as a scheme $t$ is the limit of the nonempty affine open subschemes of $T$.", "By Lemma \\ref{lemma-directed-colimit-cohomology} we conclude there exists", "a nonempty open subscheme $V \\subset T$ such that $\\tilde \\xi$ maps to zero in", "$H^q(Y \\times_T V, e^{-1}\\mathcal{I}|_{Y \\times_T V})$.", "\\medskip\\noindent", "Denote $Z = T \\setminus V$. Consider the diagram", "$$", "\\xymatrix{", "Y \\times_T Z \\ar[d]_{e_Z} \\ar[r]_{i'} &", "Y \\ar[d]_e &", "Y \\times_T V \\ar[l]^{j'} \\ar[d]^{e_V} \\\\", "Z \\ar[r]^i &", "T &", "V \\ar[l]_j", "}", "$$", "Choose an injection $i^{-1}\\mathcal{I} \\to \\mathcal{I}'$", "into an injective sheaf of $\\mathbf{Z}/n\\mathbf{Z}$-modules on $Z$.", "Looking at stalks we see that the map", "$$", "\\mathcal{I} \\to j_*\\mathcal{I}|_V \\oplus i_*\\mathcal{I}'", "$$", "is injective and hence splits as $\\mathcal{I}$ is an injective sheaf", "of $\\mathbf{Z}/n\\mathbf{Z}$-modules. Thus it suffices to show that", "$\\tilde \\xi$ maps to zero in", "$$", "H^q(Y, e^{-1}j_*\\mathcal{I}|_V) \\oplus", "H^q(Y, e^{-1}i_*\\mathcal{I}')", "$$", "at least after replacing $X$ by the members of an \\'etale covering.", "Observe that", "$$", "e^{-1}j_*\\mathcal{I}|_V = j'_*e_V^{-1}\\mathcal{I}|_V,\\quad", "e^{-1}i_*\\mathcal{I}' = i'_*e_Z^{-1}\\mathcal{I}'", "$$", "By induction hypothesis on $q$ we see that", "$$", "R^aj'_*e_V^{-1}\\mathcal{I}|_V = 0, \\quad a = 1, \\ldots, q - 1", "$$", "By the Leray spectral sequence for $j'$ and the vanishing", "above it follows that", "$$", "H^q(Y, j'_*(e_V^{-1}\\mathcal{I}|_V))", "\\longrightarrow", "H^q(Y \\times_T V, e_V^{-1}\\mathcal{I}_V) =", "H^q(Y \\times_T V, e^{-1}\\mathcal{I}|_{Y \\times_T V})", "$$", "is injective. Thus the vanishing of the image of $\\tilde \\xi$", "in the first summand above because we know $\\tilde \\xi$ vanishes", "in $H^q(Y \\times_T V, e^{-1}\\mathcal{I}|_{Y \\times_T V})$.", "Since $\\dim(Z) < \\dim(T) = d$ by induction the image of $\\tilde \\xi$", "in the second summand", "$$", "H^q(Y, e^{-1}i_*\\mathcal{I}') =", "H^q(Y, i'_*e_Z^{-1}\\mathcal{I}') =", "H^q(Y \\times_T Z, e_Z^{-1}\\mathcal{I}')", "$$", "dies after replacing $X$ by the members of a suitable \\'etale covering.", "This finishes the proof of the smooth base change theorem." ], "refs": [ "sites-lemma-localize-morphism-strong", "sites-cohomology-lemma-restrict-direct-image-open", "morphisms-lemma-smooth-etale-over-affine-space", "etale-cohomology-lemma-relative-colimit", "coherent-lemma-induction-principle", "etale-cohomology-lemma-relative-mayer-vietoris", "etale-cohomology-lemma-fppf-reduced-fibres-base-change-f-star", "algebra-lemma-limit-no-condition", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-smooth", "limits-lemma-descend-dimension-d", "sites-cohomology-lemma-pushforward-injective-flat", "etale-cohomology-lemma-linus-hamann", "etale-cohomology-lemma-relative-colimit-general", "etale-cohomology-lemma-higher-direct-images", "more-morphisms-lemma-cover-smooth-by-special", "more-algebra-lemma-dimension-etale-extension", "algebra-lemma-integral-dim-up", "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-lemma-finite-pushforward-commutes-with-base-change", "sites-cohomology-lemma-Leray", "etale-cohomology-lemma-finite-B", "algebra-proposition-ubiquity-nagata", "morphisms-lemma-nagata-normalization-finite", "morphisms-lemma-nagata-normalization", "morphisms-lemma-normal-normalization", "etale-cohomology-lemma-smooth-base-change-fields", "pione-lemma-extend-covering", "etale-cohomology-lemma-directed-colimit-cohomology" ], "ref_ids": [ 8572, 4256, 5377, 6475, 3291, 6470, 6600, 1102, 15077, 15064, 15106, 4218, 6477, 6476, 6474, 13881, 10052, 984, 6703, 6481, 4220, 6452, 1431, 5510, 5520, 5508, 6611, 4125, 6473 ] } ], "ref_ids": [] }, { "id": 6397, "type": "theorem", "label": "etale-cohomology-theorem-proper-base-change", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-theorem-proper-base-change", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$ be", "a morphism of schemes. Set $X' = Y' \\times_Y X$", "and consider the cartesian diagram", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "Let $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$.", "Then the base change map", "$$", "g^{-1}Rf_*\\mathcal{F} \\longrightarrow Rf'_*(g')^{-1}\\mathcal{F}", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "In the terminology introduced above, this means that cohomology commutes", "with base change for every proper morphism of schemes. By", "Lemma \\ref{lemma-reduce-to-P1}", "it suffices to prove that cohomology commutes with base change", "for the morphism $\\mathbf{P}^1_S \\to S$ for every scheme $S$.", "\\medskip\\noindent", "Let $S$ be the spectrum of a strictly henselian local ring with closed", "point $s$. Set $X = \\mathbf{P}^1_S$ and $X_0 = X_s = \\mathbf{P}^1_s$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathbf{Z}/\\ell\\mathbf{Z}$-modules", "on $X_\\etale$. The key to our proof is that", "$$", "H^q_\\etale(X, \\mathcal{F}) = H^q_\\etale(X_0, \\mathcal{F}|_{X_0}).", "$$", "Namely, choose a resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$", "by injective sheaves of $\\mathbf{Z}/\\ell\\mathbf{Z}$-modules.", "Then $\\mathcal{I}^\\bullet|_{X_0}$ is a resolution of $\\mathcal{F}|_{X_0}$", "by right $H^0_\\etale(X_0, -)$-acyclic objects, see", "Lemma \\ref{lemma-efface-cohomology-on-fibre-by-finite-cover}.", "Leray's acyclicity lemma tells us the right hand side is computed by", "the complex $H^0_\\etale(X_0, \\mathcal{I}^\\bullet|_{X_0})$", "which is equal to $H^0_\\etale(X, \\mathcal{I}^\\bullet)$ by", "Lemma \\ref{lemma-h0-proper-over-henselian-local}. This complex", "computes the left hand side.", "\\medskip\\noindent", "Assume $S$ is general and $\\mathcal{F}$ is a sheaf of", "$\\mathbf{Z}/\\ell\\mathbf{Z}$-modules on $X_\\etale$.", "Let $\\overline{s} : \\Spec(k) \\to S$ be a geometric point", "of $S$ lying over $s \\in S$. We have", "$$", "(R^qf_*\\mathcal{F})_{\\overline{s}} =", "H^q_\\etale(\\mathbf{P}^1_{\\mathcal{O}_{S, \\overline{s}}^{sh}},", "\\mathcal{F}|_{\\mathbf{P}^1_{\\mathcal{O}_{S, \\overline{s}}^{sh}}}) =", "H^q_\\etale(\\mathbf{P}^1_{\\kappa(s)^{sep}},", "\\mathcal{F}|_{\\mathbf{P}^1_{\\kappa(s)^{sep}}})", "$$", "where $\\kappa(s)^{sep}$ is the residue field of", "$\\mathcal{O}_{S, \\overline{s}}^{sh}$, i.e., the separable algebraic", "closure of $\\kappa(s)$ in $k$.", "The first equality by Theorem \\ref{theorem-higher-direct-images}", "and the second equality by the displayed formula in the", "previous paragraph.", "\\medskip\\noindent", "Finally, consider any morphism of schemes $g : T \\to S$ where", "$S$ and $\\mathcal{F}$ are as above.", "Set $f' : \\mathbf{P}^1_T \\to T$ the projection and let", "$g' : \\mathbf{P}^1_T \\to \\mathbf{P}^1_S$ the morphism induced", "by $g$. Consider the base change map", "$$", "g^{-1}R^qf_*\\mathcal{F}", "\\longrightarrow", "R^qf'_*(g')^{-1}\\mathcal{F}", "$$", "Let $\\overline{t}$ be a geometric point of $T$ with image", "$\\overline{s} = g(\\overline{t})$. By our discussion", "above the map on stalks at $\\overline{t}$ is the map", "$$", "H^q_\\etale(\\mathbf{P}^1_{\\kappa(s)^{sep}},", "\\mathcal{F}|_{\\mathbf{P}^1_{\\kappa(s)^{sep}}})", "\\longrightarrow", "H^q_\\etale(\\mathbf{P}^1_{\\kappa(t)^{sep}},", "\\mathcal{F}|_{\\mathbf{P}^1_{\\kappa(t)^{sep}}})", "$$", "Since $\\kappa(s)^{sep} \\subset \\kappa(t)^{sep}$ this map is an", "isomorphism by Lemma \\ref{lemma-base-change-dim-1-separably-closed}.", "\\medskip\\noindent", "This proves cohomology commutes with base change for", "$\\mathbf{P}^1_S \\to S$ and sheaves of $\\mathbf{Z}/\\ell\\mathbf{Z}$-modules.", "In particular, for an injective sheaf of $\\mathbf{Z}/\\ell\\mathbf{Z}$-modules", "the higher direct images of any base change are zero.", "In other words, condition (2) of", "Lemma \\ref{lemma-proper-base-change-in-terms-of-injectives}", "holds and the proof is complete." ], "refs": [ "etale-cohomology-lemma-reduce-to-P1", "etale-cohomology-lemma-efface-cohomology-on-fibre-by-finite-cover", "etale-cohomology-lemma-h0-proper-over-henselian-local", "etale-cohomology-theorem-higher-direct-images", "etale-cohomology-lemma-base-change-dim-1-separably-closed", "etale-cohomology-lemma-proper-base-change-in-terms-of-injectives" ], "ref_ids": [ 6625, 6593, 6618, 6385, 6591, 6621 ] } ], "ref_ids": [] }, { "id": 6398, "type": "theorem", "label": "etale-cohomology-lemma-yoneda", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-yoneda", "contents": [ "\\begin{slogan}", "Morphisms between objects are in bijection with natural transformations", "between the functors they represent.", "\\end{slogan}", "Let $\\mathcal{C}$ be a category, and $X, Y \\in", "\\Ob(\\mathcal{C})$. There is a natural bijection", "$$", "\\begin{matrix}", "\\Mor_\\mathcal{C}(X, Y) &", "\\longrightarrow &", "\\Mor_{\\textit{PSh}(\\mathcal{C})} (h_X, h_Y) \\\\", "\\psi &", "\\longmapsto &", "h_\\psi = \\psi \\circ - : h_X \\to h_Y.", "\\end{matrix}", "$$" ], "refs": [], "proofs": [ { "contents": [ "See", "Categories, Lemma \\ref{categories-lemma-yoneda}." ], "refs": [ "categories-lemma-yoneda" ], "ref_ids": [ 12203 ] } ], "ref_ids": [] }, { "id": 6399, "type": "theorem", "label": "etale-cohomology-lemma-site-fpqc", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-site-fpqc", "contents": [ "The collection of fpqc coverings on the category of schemes", "satisfies the axioms of site." ], "refs": [], "proofs": [ { "contents": [ "See Topologies, Lemma \\ref{topologies-lemma-fpqc}." ], "refs": [ "topologies-lemma-fpqc" ], "ref_ids": [ 12498 ] } ], "ref_ids": [] }, { "id": 6400, "type": "theorem", "label": "etale-cohomology-lemma-fpqc-sheaves", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-fpqc-sheaves", "contents": [ "Let $\\mathcal{F}$ be a presheaf on $\\Sch/S$. Then", "$\\mathcal{F}$ satisfies the sheaf property for the fpqc topology", "if and only if", "\\begin{enumerate}", "\\item $\\mathcal{F}$ satisfies the sheaf property with respect to the", "Zariski topology, and", "\\item for every faithfully flat morphism $\\Spec(B) \\to \\Spec(A)$", "of affine schemes over $S$, the sheaf axiom holds for the covering", "$\\{\\Spec(B) \\to \\Spec(A)\\}$. Namely, this means that", "$$", "\\xymatrix{", "\\mathcal{F}(\\Spec(A)) \\ar[r] &", "\\mathcal{F}(\\Spec(B)) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\mathcal{F}(\\Spec(B \\otimes_A B))", "}", "$$", "is an equalizer diagram.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See Topologies, Lemma \\ref{topologies-lemma-sheaf-property-fpqc}." ], "refs": [ "topologies-lemma-sheaf-property-fpqc" ], "ref_ids": [ 12502 ] } ], "ref_ids": [] }, { "id": 6401, "type": "theorem", "label": "etale-cohomology-lemma-representable-sheaf-fpqc", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-representable-sheaf-fpqc", "contents": [ "Any representable presheaf on $\\Sch/S$ satisfies the", "sheaf condition for the fpqc topology." ], "refs": [], "proofs": [ { "contents": [ "See", "Descent, Lemma \\ref{descent-lemma-fpqc-universal-effective-epimorphisms}." ], "refs": [ "descent-lemma-fpqc-universal-effective-epimorphisms" ], "ref_ids": [ 14638 ] } ], "ref_ids": [] }, { "id": 6402, "type": "theorem", "label": "etale-cohomology-lemma-algebra-descent", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-algebra-descent", "contents": [ "If $A \\to B$ is faithfully flat, then the complex $(B/A)_\\bullet$ is exact in", "positive degrees, and $H^0((B/A)_\\bullet) = A$." ], "refs": [], "proofs": [ { "contents": [ "See Descent, Lemma \\ref{descent-lemma-ff-exact}." ], "refs": [ "descent-lemma-ff-exact" ], "ref_ids": [ 14598 ] } ], "ref_ids": [] }, { "id": 6403, "type": "theorem", "label": "etale-cohomology-lemma-descent-modules", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-descent-modules", "contents": [ "If $A \\to B$ is faithfully flat and $M$ is an $A$-module, then the", "complex $(B/A)_\\bullet \\otimes_A M$ is exact in positive degrees, and", "$H^0((B/A)_\\bullet \\otimes_A M) = M$." ], "refs": [], "proofs": [ { "contents": [ "See Descent, Lemma \\ref{descent-lemma-ff-exact}." ], "refs": [ "descent-lemma-ff-exact" ], "ref_ids": [ 14598 ] } ], "ref_ids": [] }, { "id": 6404, "type": "theorem", "label": "etale-cohomology-lemma-cech-presheaves", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cech-presheaves", "contents": [ "The functor $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, -)$", "is exact on the category $\\textit{PAb}(\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "This follows at once from the definition of a short exact sequence of", "presheaves. Namely, as the category of abelian presheaves is the category of", "functors on some category with values in $\\textit{Ab}$, it is automatically an", "abelian category: a sequence $\\mathcal{F}_1\\to \\mathcal{F}_2\\to \\mathcal{F}_3$", "is exact in $\\textit{PAb}$ if and only if for all", "$U \\in \\Ob(\\mathcal{C})$, the sequence", "$\\mathcal{F}_1(U) \\to \\mathcal{F}_2(U) \\to \\mathcal{F}_3(U)$ is exact in", "$\\textit{Ab}$. So the complex above is merely a product of short exact", "sequences in each degree. See also", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cech-exact-presheaves}." ], "refs": [ "sites-cohomology-lemma-cech-exact-presheaves" ], "ref_ids": [ 4191 ] } ], "ref_ids": [] }, { "id": 6405, "type": "theorem", "label": "etale-cohomology-lemma-yoneda-presheaf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-yoneda-presheaf", "contents": [ "For any presheaf $\\mathcal{F}$ on a category $\\mathcal{C}$ there is a", "functorial isomorphism", "$$", "\\Hom_{\\textit{PSh}(\\mathcal{C})}(h_U, \\mathcal{F}) =", "\\mathcal{F}(U).", "$$" ], "refs": [], "proofs": [ { "contents": [ "See Categories, Lemma \\ref{categories-lemma-yoneda}." ], "refs": [ "categories-lemma-yoneda" ], "ref_ids": [ 12203 ] } ], "ref_ids": [] }, { "id": 6406, "type": "theorem", "label": "etale-cohomology-lemma-cech-complex-describe", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cech-complex-describe", "contents": [ "The {\\v C}ech complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "can be described explicitly as follows", "\\begin{eqnarray*}", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "& = &", "\\left(", "\\prod_{i_0 \\in I}", "\\Hom_{\\textit{PAb}(\\mathcal{C})}(\\mathbf{Z}_{U_{i_0}}, \\mathcal{F}) \\to", "\\prod_{i_0, i_1 \\in I}", "\\Hom_{\\textit{PAb}(\\mathcal{C})}(", "\\mathbf{Z}_{U_{i_0} \\times_U U_{i_1}}, \\mathcal{F}) \\to \\ldots", "\\right) \\\\", "& = &", "\\Hom_{\\textit{PAb}(\\mathcal{C})}\\left(", "\\left(", "\\bigoplus_{i_0 \\in I} \\mathbf{Z}_{U_{i_0}} \\leftarrow", "\\bigoplus_{i_0, i_1 \\in I} \\mathbf{Z}_{U_{i_0} \\times_U U_{i_1}} \\leftarrow", "\\ldots", "\\right), \\mathcal{F}\\right)", "\\end{eqnarray*}" ], "refs": [], "proofs": [ { "contents": [ "This follows from the formula above. See", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cech-map-into}." ], "refs": [ "sites-cohomology-lemma-cech-map-into" ], "ref_ids": [ 4193 ] } ], "ref_ids": [] }, { "id": 6407, "type": "theorem", "label": "etale-cohomology-lemma-exact", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-exact", "contents": [ "The complex of abelian presheaves", "\\begin{align*}", "\\mathbf{Z}_\\mathcal{U}^\\bullet \\quad : \\quad", "\\bigoplus_{i_0 \\in I} \\mathbf{Z}_{U_{i_0}} \\leftarrow", "\\bigoplus_{i_0, i_1 \\in I} \\mathbf{Z}_{U_{i_0} \\times_U U_{i_1}} \\leftarrow", "\\bigoplus_{i_0, i_1, i_2 \\in I}", "\\mathbf{Z}_{U_{i_0} \\times_U U_{i_1} \\times_U U_{i_2}} \\leftarrow", "\\ldots", "\\end{align*}", "is exact in all degrees except $0$ in $\\textit{PAb}(\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "For any $V\\in \\Ob(\\mathcal{C})$ the complex of abelian groups", "$\\mathbf{Z}_\\mathcal{U}^\\bullet(V)$ is", "$$", "\\begin{matrix}", "\\mathbf{Z}\\left[", "\\coprod_{i_0\\in I} \\Mor_\\mathcal{C}(V, U_{i_0})\\right]", "\\leftarrow", "\\mathbf{Z}\\left[", "\\coprod_{i_0, i_1 \\in I}", "\\Mor_\\mathcal{C}(V, U_{i_0} \\times_U U_{i_1})\\right]", "\\leftarrow \\ldots = \\\\", "\\bigoplus_{\\varphi : V \\to U}", "\\left(", "\\mathbf{Z}\\left[\\coprod_{i_0 \\in I} \\Mor_\\varphi(V, U_{i_0})\\right]", "\\leftarrow", "\\mathbf{Z}\\left[\\coprod_{i_0, i_1\\in I} \\Mor_\\varphi(V, U_{i_0}) \\times", "\\Mor_\\varphi(V, U_{i_1})\\right]", "\\leftarrow", "\\ldots", "\\right)", "\\end{matrix}", "$$", "where", "$$", "\\Mor_{\\varphi}(V, U_i)", "=", "\\{ V \\to U_i \\text{ such that } V \\to U_i \\to U \\text{ equals } \\varphi \\}.", "$$", "Set $S_\\varphi = \\coprod_{i\\in I} \\Mor_\\varphi(V, U_i)$, so that", "$$", "\\mathbf{Z}_\\mathcal{U}^\\bullet(V)", "=", "\\bigoplus_{\\varphi : V \\to U}", "\\left(", "\\mathbf{Z}[S_\\varphi] \\leftarrow", "\\mathbf{Z}[S_\\varphi \\times S_\\varphi] \\leftarrow", "\\mathbf{Z}[S_\\varphi \\times S_\\varphi \\times S_\\varphi] \\leftarrow", "\\ldots \\right).", "$$", "Thus it suffices to show that for each $S = S_\\varphi$, the complex", "\\begin{align*}", "\\mathbf{Z}[S] \\leftarrow", "\\mathbf{Z}[S \\times S] \\leftarrow", "\\mathbf{Z}[S \\times S \\times S] \\leftarrow \\ldots", "\\end{align*}", "is exact in negative degrees. To see this, we can give an explicit homotopy.", "Fix $s\\in S$ and define $K: n_{(s_0, \\ldots, s_p)} \\mapsto n_{(s, s_0,", "\\ldots, s_p)}.$ One easily checks that $K$ is a nullhomotopy for the operator", "$$", "\\delta :", "\\eta_{(s_0, \\ldots, s_p)}", "\\mapsto", "\\sum\\nolimits_{i = 0}^p (-1)^p \\eta_{(s_0, \\ldots, \\hat s_i, \\ldots, s_p)}.", "$$", "See", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-homology-complex}", "for more details." ], "refs": [ "sites-cohomology-lemma-homology-complex" ], "ref_ids": [ 4194 ] } ], "ref_ids": [] }, { "id": 6408, "type": "theorem", "label": "etale-cohomology-lemma-hom-injective", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-hom-injective", "contents": [ "Let $\\mathcal{C}$ be a category. If $\\mathcal{I}$ is an injective object of", "$\\textit{PAb}(\\mathcal{C})$ and $\\mathcal{U}$ is a family of morphisms with", "fixed target in $\\mathcal{C}$, then $\\check H^p(\\mathcal{U}, \\mathcal{I}) = 0$", "for all $p > 0$." ], "refs": [], "proofs": [ { "contents": [ "The {\\v C}ech complex is the result of applying the functor", "$\\Hom_{\\textit{PAb}(\\mathcal{C})}(-, \\mathcal{I}) $ to the complex $", "\\mathbf{Z}^\\bullet_\\mathcal{U} $, i.e.,", "$$", "\\check H^p(\\mathcal{U}, \\mathcal{I}) = H^p", "(\\Hom_{\\textit{PAb}(\\mathcal{C})} (\\mathbf{Z}^\\bullet_\\mathcal{U},", "\\mathcal{I})).", "$$", "But we have just seen that $\\mathbf{Z}^\\bullet_\\mathcal{U}$ is exact in", "negative degrees, and the functor $\\Hom_{\\textit{PAb}(\\mathcal{C})}(-,", "\\mathcal{I})$ is exact, hence $\\Hom_{\\textit{PAb}(\\mathcal{C})}", "(\\mathbf{Z}^\\bullet_\\mathcal{U}, \\mathcal{I})$ is exact in positive degrees." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6409, "type": "theorem", "label": "etale-cohomology-lemma-forget-injectives", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-forget-injectives", "contents": [ "The forgetful functor $\\textit{Ab}(\\mathcal{C})\\to \\textit{PAb}(\\mathcal{C})$", "transforms injectives into injectives." ], "refs": [], "proofs": [ { "contents": [ "This is formal using the fact that the forgetful functor has a left adjoint,", "namely sheafification, which is an exact functor. For more details see", "Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-injective-abelian-sheaf-injective-presheaf}." ], "refs": [ "sites-cohomology-lemma-injective-abelian-sheaf-injective-presheaf" ], "ref_ids": [ 4197 ] } ], "ref_ids": [] }, { "id": 6410, "type": "theorem", "label": "etale-cohomology-lemma-tau-affine", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-tau-affine", "contents": [ "Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$.", "Any $\\tau$-covering of an affine scheme can be refined by a", "standard $\\tau$-covering." ], "refs": [], "proofs": [ { "contents": [ "See", "Topologies, Lemmas", "\\ref{topologies-lemma-fppf-affine},", "\\ref{topologies-lemma-syntomic-affine},", "\\ref{topologies-lemma-smooth-affine},", "\\ref{topologies-lemma-etale-affine}, and", "\\ref{topologies-lemma-zariski-affine}." ], "refs": [ "topologies-lemma-fppf-affine", "topologies-lemma-syntomic-affine", "topologies-lemma-smooth-affine", "topologies-lemma-etale-affine", "topologies-lemma-zariski-affine" ], "ref_ids": [ 12473, 12467, 12461, 12447, 12432 ] } ], "ref_ids": [] }, { "id": 6411, "type": "theorem", "label": "etale-cohomology-lemma-compare-cohomology-big-small", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-compare-cohomology-big-small", "contents": [ "Let $\\tau \\in \\{\\etale, Zariski\\}$.", "If $\\mathcal{F}$ is an abelian sheaf defined on", "$(\\Sch/S)_\\tau$, then", "the cohomology groups of $\\mathcal{F}$ over $S$ agree with the cohomology", "groups of $\\mathcal{F}|_{S_\\tau}$ over $S$." ], "refs": [], "proofs": [ { "contents": [ "By", "Topologies, Lemmas \\ref{topologies-lemma-at-the-bottom} and", "\\ref{topologies-lemma-at-the-bottom-etale}", "the functors $S_\\tau \\to (\\Sch/S)_\\tau$", "satisfy the hypotheses of", "Sites, Lemma \\ref{sites-lemma-bigger-site}.", "Hence our lemma follows from", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-bigger-site}." ], "refs": [ "topologies-lemma-at-the-bottom", "topologies-lemma-at-the-bottom-etale", "sites-lemma-bigger-site", "sites-cohomology-lemma-cohomology-bigger-site" ], "ref_ids": [ 12439, 12453, 8548, 4187 ] } ], "ref_ids": [] }, { "id": 6412, "type": "theorem", "label": "etale-cohomology-lemma-cohomology-enlarge-partial-universe", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomology-enlarge-partial-universe", "contents": [ "Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$.", "Let $S$ be a scheme.", "Let $(\\Sch/S)_\\tau$ and $(\\Sch'/S)_\\tau$ be two", "big $\\tau$-sites of $S$, and assume that the first is contained in the second.", "In this case", "\\begin{enumerate}", "\\item for any abelian sheaf $\\mathcal{F}'$ defined on $(\\Sch'/S)_\\tau$ and", "any object $U$ of $(\\Sch/S)_\\tau$ we have", "$$", "H^p_\\tau(U, \\mathcal{F}'|_{(\\Sch/S)_\\tau}) =", "H^p_\\tau(U, \\mathcal{F}')", "$$", "In words: the cohomology of $\\mathcal{F}'$ over $U$ computed in the bigger site", "agrees with the cohomology of $\\mathcal{F}'$ restricted to the smaller site", "over $U$.", "\\item for any abelian sheaf $\\mathcal{F}$ on $(\\Sch/S)_\\tau$ there is an", "abelian sheaf $\\mathcal{F}'$ on $(\\Sch/S)_\\tau'$ whose restriction to", "$(\\Sch/S)_\\tau$ is isomorphic to $\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Topologies, Lemma \\ref{topologies-lemma-change-alpha} the inclusion functor", "$(\\Sch/S)_\\tau \\to (\\Sch'/S)_\\tau$ satisfies the assumptions of", "Sites, Lemma \\ref{sites-lemma-bigger-site}. This implies (2) and (1)", "follows from", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-bigger-site}." ], "refs": [ "topologies-lemma-change-alpha", "sites-lemma-bigger-site", "sites-cohomology-lemma-cohomology-bigger-site" ], "ref_ids": [ 12516, 8548, 4187 ] } ], "ref_ids": [] }, { "id": 6413, "type": "theorem", "label": "etale-cohomology-lemma-alternative", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-alternative", "contents": [ "Let $S$ be a scheme. Let $S_{affine, \\etale}$ denote the", "full subcategory of $S_\\etale$", "whose objects are those $U/S \\in \\Ob(S_\\etale)$ with", "$U$ affine. A covering of $S_{affine, \\etale}$ will be a standard", "\\'etale covering, see", "Topologies, Definition \\ref{topologies-definition-standard-etale}.", "Then restriction", "$$", "\\mathcal{F} \\longmapsto \\mathcal{F}|_{S_{affine, \\etale}}", "$$", "defines an equivalence of topoi", "$\\Sh(S_\\etale) \\cong \\Sh(S_{affine, \\etale})$." ], "refs": [ "topologies-definition-standard-etale" ], "proofs": [ { "contents": [ "This you can show directly from the definitions, and is a good exercise.", "But it also follows immediately from", "Sites, Lemma \\ref{sites-lemma-equivalence}", "by checking that the inclusion functor", "$S_{affine, \\etale} \\to S_\\etale$", "is a special cocontinuous functor (see", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor})." ], "refs": [ "sites-lemma-equivalence", "sites-definition-special-cocontinuous-functor" ], "ref_ids": [ 8578, 8672 ] } ], "ref_ids": [ 12527 ] }, { "id": 6414, "type": "theorem", "label": "etale-cohomology-lemma-etale-topos-independent-partial-universe", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-etale-topos-independent-partial-universe", "contents": [ "Let $S$ be a scheme. The \\'etale topos of $S$ is independent", "(up to canonical equivalence) of the construction of the small", "\\'etale site in Definition \\ref{definition-tau-site}." ], "refs": [ "etale-cohomology-definition-tau-site" ], "proofs": [ { "contents": [ "We have to show, given two big \\'etale sites", "$\\Sch_\\etale$ and $\\Sch_\\etale'$ containing", "$S$, then $\\Sh(S_\\etale) \\cong \\Sh(S_\\etale')$", "with obvious notation. By Topologies, Lemma \\ref{topologies-lemma-contained-in}", "we may assume $\\Sch_\\etale \\subset \\Sch_\\etale'$.", "By Sets, Lemma \\ref{sets-lemma-what-is-in-it}", "any affine scheme \\'etale over $S$ is isomorphic to an object", "of both $\\Sch_\\etale$ and $\\Sch_\\etale'$.", "Thus the induced functor", "$S_{affine, \\etale} \\to S_{affine, \\etale}'$", "is an equivalence. Moreover, it is clear that both this functor", "and a quasi-inverse map transform standard \\'etale coverings into", "standard \\'etale coverings.", "Hence the result follows from Lemma \\ref{lemma-alternative}." ], "refs": [ "topologies-lemma-contained-in", "sets-lemma-what-is-in-it", "etale-cohomology-lemma-alternative" ], "ref_ids": [ 12515, 8795, 6413 ] } ], "ref_ids": [ 6727 ] }, { "id": 6415, "type": "theorem", "label": "etale-cohomology-lemma-alternative-zariski", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-alternative-zariski", "contents": [ "Let $S$ be a scheme. Let $S_{affine, Zar}$ denote the", "full subcategory of $S_{Zar}$ consisting of affine objects.", "A covering of $S_{affine, Zar}$ will be a standard", "Zariski covering, see", "Topologies, Definition \\ref{topologies-definition-standard-Zariski}.", "Then restriction", "$$", "\\mathcal{F} \\longmapsto \\mathcal{F}|_{S_{affine, Zar}}", "$$", "defines an equivalence of topoi", "$\\Sh(S_{Zar}) \\cong \\Sh(S_{affine, Zar})$." ], "refs": [ "topologies-definition-standard-Zariski" ], "proofs": [ { "contents": [ "Please skip the proof of this lemma. It follows immediately from", "Sites, Lemma \\ref{sites-lemma-equivalence} by checking that the", "inclusion functor $S_{affine, Zar} \\to S_{Zar}$", "is a special cocontinuous functor (see", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor})." ], "refs": [ "sites-lemma-equivalence", "sites-definition-special-cocontinuous-functor" ], "ref_ids": [ 8578, 8672 ] } ], "ref_ids": [ 12522 ] }, { "id": 6416, "type": "theorem", "label": "etale-cohomology-lemma-cech-complex", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cech-complex", "contents": [ "Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$.", "Let $S$ be a scheme.", "Let $\\mathcal{F}$ be an abelian sheaf on $(\\Sch/S)_\\tau$, or on", "$S_\\tau$ in case $\\tau = \\etale$, and let", "$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$", "be a standard $\\tau$-covering of this site.", "Let $V = \\coprod_{i \\in I} U_i$. Then", "\\begin{enumerate}", "\\item $V$ is an affine scheme,", "\\item $\\mathcal{V} = \\{V \\to U\\}$ is an fpqc covering", "and also a $\\tau$-covering unless $\\tau = Zariski$,", "\\item the {\\v C}ech complexes", "$\\check{\\mathcal{C}}^\\bullet (\\mathcal{U}, \\mathcal{F})$ and", "$\\check{\\mathcal{C}}^\\bullet (\\mathcal{V}, \\mathcal{F})$ agree.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The defintion of a standard $\\tau$-covering is given in", "Topologies, Definition", "\\ref{topologies-definition-standard-Zariski},", "\\ref{topologies-definition-standard-etale},", "\\ref{topologies-definition-standard-smooth},", "\\ref{topologies-definition-standard-syntomic}, and", "\\ref{topologies-definition-standard-fppf}.", "By definition each of the schemes", "$U_i$ is affine and $I$ is a finite set. Hence $V$ is an affine scheme.", "It is clear that $V \\to U$ is flat and surjective, hence", "$\\mathcal{V}$ is an fpqc covering, see", "Example \\ref{example-fpqc-coverings}.", "Excepting the Zariski case, the covering $\\mathcal{V}$", "is also a $\\tau$-covering, see", "Topologies, Definition", "\\ref{topologies-definition-etale-covering},", "\\ref{topologies-definition-smooth-covering},", "\\ref{topologies-definition-syntomic-covering}, and", "\\ref{topologies-definition-fppf-covering}.", "\\medskip\\noindent", "Note that $\\mathcal{U}$ is a refinement of $\\mathcal{V}$", "and hence there is a map of {\\v C}ech complexes", "$\\check{\\mathcal{C}}^\\bullet (\\mathcal{V}, \\mathcal{F}) \\to", "\\check{\\mathcal{C}}^\\bullet (\\mathcal{U}, \\mathcal{F})$, see", "Cohomology on Sites,", "Equation (\\ref{sites-cohomology-equation-map-cech-complexes}).", "Next, we observe that if $T = \\coprod_{j \\in J} T_j$ is a", "disjoint union of schemes in the site on which $\\mathcal{F}$ is defined", "then the family of morphisms with fixed target", "$\\{T_j \\to T\\}_{j \\in J}$ is a Zariski covering, and so", "\\begin{equation}", "\\label{equation-sheaf-coprod}", "\\mathcal{F}(T) =", "\\mathcal{F}(\\coprod\\nolimits_{j \\in J} T_j) =", "\\prod\\nolimits_{j \\in J} \\mathcal{F}(T_j)", "\\end{equation}", "by the sheaf condition of $\\mathcal{F}$.", "This implies the map of {\\v C}ech complexes above is an isomorphism", "in each degree because", "$$", "V \\times_U \\ldots \\times_U V", "=", "\\coprod\\nolimits_{i_0, \\ldots i_p} U_{i_0} \\times_U \\ldots \\times_U U_{i_p}", "$$", "as schemes." ], "refs": [ "topologies-definition-standard-Zariski", "topologies-definition-standard-etale", "topologies-definition-standard-smooth", "topologies-definition-standard-syntomic", "topologies-definition-standard-fppf", "topologies-definition-etale-covering", "topologies-definition-smooth-covering", "topologies-definition-syntomic-covering", "topologies-definition-fppf-covering" ], "ref_ids": [ 12522, 12527, 12532, 12536, 12540, 12526, 12531, 12535, 12539 ] } ], "ref_ids": [] }, { "id": 6417, "type": "theorem", "label": "etale-cohomology-lemma-locality-cohomology", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-locality-cohomology", "contents": [ "Let $\\mathcal{C}$ be a site, $\\mathcal{F}$ an abelian sheaf on $\\mathcal{C}$,", "$U$ an object of $\\mathcal{C}$, $p > 0$ an integer and $\\xi \\in", "H^p(U, \\mathcal{F})$. Then there exists a covering", "$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ of $U$ in $\\mathcal{C}$", "such that $\\xi |_{U_i} = 0$ for all $i \\in I$." ], "refs": [], "proofs": [ { "contents": [ "Choose an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$. Then", "$\\xi$ is represented by a cocycle $\\tilde{\\xi} \\in \\mathcal{I}^p(U)$", "with $d^p(\\tilde{\\xi}) = 0$. By assumption, the sequence", "$\\mathcal{I}^{p - 1} \\to \\mathcal{I}^p \\to \\mathcal{I}^{p + 1}$ in exact in", "$\\textit{Ab}(\\mathcal{C})$, which means that there exists a covering", "$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ such that", "$\\tilde{\\xi}|_{U_i} = d^{p - 1}(\\xi_i)$ for some", "$\\xi_i \\in \\mathcal{I}^{p-1}(U_i)$. Since", "the cohomology class $\\xi|_{U_i}$ is represented by the cocycle", "$\\tilde{\\xi}|_{U_i}$ which is a coboundary, it vanishes.", "For more details see", "Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-kill-cohomology-class-on-covering}." ], "refs": [ "sites-cohomology-lemma-kill-cohomology-class-on-covering" ], "ref_ids": [ 4188 ] } ], "ref_ids": [] }, { "id": 6418, "type": "theorem", "label": "etale-cohomology-lemma-etale-fpqc", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-etale-fpqc", "contents": [ "Any \\'etale covering is an fpqc covering." ], "refs": [], "proofs": [ { "contents": [ "(See also", "Topologies,", "Lemma \\ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc}.)", "Let $\\{\\varphi_i : U_i \\to U\\}_{i \\in I}$ be an \\'etale covering.", "Since an \\'etale morphism is flat, and the elements of the covering should", "cover its target, the property fp (faithfully flat) is satisfied.", "To check the property qc (quasi-compact), let $V \\subset U$ be an affine", "open, and write $\\varphi_i^{-1}(V) = \\bigcup_{j \\in J_i} V_{ij}$", "for some affine opens $V_{ij} \\subset U_i$. Since $\\varphi_i$ is open", "(as \\'etale morphisms are open), we see that", "$V = \\bigcup_{i\\in I} \\bigcup_{j \\in J_i} \\varphi_i(V_{ij})$", "is an open covering of $V$.", "Further, since $V$ is quasi-compact, this covering has a finite", "refinement." ], "refs": [ "topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc" ], "ref_ids": [ 12497 ] } ], "ref_ids": [] }, { "id": 6419, "type": "theorem", "label": "etale-cohomology-lemma-kummer-sequence", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-kummer-sequence", "contents": [ "If $n\\in \\mathcal{O}_S^*$ then", "$$", "0 \\to", "\\mu_{n, S} \\to", "\\mathbf{G}_{m, S} \\xrightarrow{(\\cdot)^n}", "\\mathbf{G}_{m, S} \\to 0", "$$", "is a short exact sequence of sheaves on both the small and", "big \\'etale site of $S$." ], "refs": [], "proofs": [ { "contents": [ "By definition the sheaf $\\mu_{n, S}$ is the kernel of the map", "$(\\cdot)^n$. Hence it suffices to show that the last map is surjective.", "Let $U$ be a scheme over $S$. Let", "$f \\in \\mathbf{G}_m(U) = \\Gamma(U, \\mathcal{O}_U^*)$.", "We need to show that we can find an \\'etale cover of", "$U$ over the members of which the restriction of $f$ is an $n$th power.", "Set", "$$", "U' =", "\\underline{\\Spec}_U(\\mathcal{O}_U[T]/(T^n-f))", "\\xrightarrow{\\pi}", "U.", "$$", "(See", "Constructions, Section \\ref{constructions-section-spec-via-glueing} or", "\\ref{constructions-section-spec}", "for a discussion of the relative spectrum.)", "Let $\\Spec(A) \\subset U$ be an affine open, and say $f|_{\\Spec(A)}$ corresponds", "to the unit $a \\in A^*$. Then $\\pi^{-1}(\\Spec(A)) = \\Spec(B)$ with", "$B = A[T]/(T^n - a)$. The ring map $A \\to B$ is finite free of rank $n$,", "hence it is faithfully flat, and hence we conclude that", "$\\Spec(B) \\to \\Spec(A)$ is surjective. Since this holds for every", "affine open in $U$ we conclude that $\\pi$ is surjective.", "In addition, $n$ and $T^{n - 1}$ are invertible in $B$, so", "$nT^{n-1} \\in B^*$ and the ring map $A \\to B$ is standard \\'etale,", "in particular \\'etale. Since this holds for every affine open of $U$", "we conclude that $\\pi$ is \\'etale. Hence", "$\\mathcal{U} = \\{\\pi : U' \\to U\\}$ is an \\'etale covering.", "Moreover, $f|_{U'} = (f')^n$ where $f'$ is the class of $T$", "in $\\Gamma(U', \\mathcal{O}_{U'}^*)$, so $\\mathcal{U}$ has the desired property." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6420, "type": "theorem", "label": "etale-cohomology-lemma-kummer-sequence-syntomic", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-kummer-sequence-syntomic", "contents": [ "For any $n \\in \\mathbf{N}$ the sequence", "$$", "0 \\to", "\\mu_{n, S} \\to", "\\mathbf{G}_{m, S} \\xrightarrow{(\\cdot)^n}", "\\mathbf{G}_{m, S} \\to 0", "$$", "is a short exact sequence of sheaves on the site", "$(\\Sch/S)_{fppf}$ and $(\\Sch/S)_{syntomic}$." ], "refs": [], "proofs": [ { "contents": [ "By definition the sheaf $\\mu_{n, S}$ is the kernel of the map", "$(\\cdot)^n$. Hence it suffices to show that the last map is surjective.", "Since the syntomic topology is weaker than the fppf topology, see", "Topologies, Lemma \\ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf},", "it suffices to prove this for the syntomic topology.", "Let $U$ be a scheme over $S$. Let", "$f \\in \\mathbf{G}_m(U) = \\Gamma(U, \\mathcal{O}_U^*)$.", "We need to show that we can find a syntomic cover of", "$U$ over the members of which the restriction of $f$ is an $n$th power.", "Set", "$$", "U' =", "\\underline{\\Spec}_U(\\mathcal{O}_U[T]/(T^n-f))", "\\xrightarrow{\\pi}", "U.", "$$", "(See", "Constructions, Section \\ref{constructions-section-spec-via-glueing} or", "\\ref{constructions-section-spec}", "for a discussion of the relative spectrum.)", "Let $\\Spec(A) \\subset U$ be an affine open, and say $f|_{\\Spec(A)}$ corresponds", "to the unit $a \\in A^*$. Then $\\pi^{-1}(\\Spec(A)) = \\Spec(B)$ with", "$B = A[T]/(T^n - a)$. The ring map $A \\to B$ is finite free of rank $n$,", "hence it is faithfully flat, and hence we conclude that", "$\\Spec(B) \\to \\Spec(A)$ is surjective. Since this holds for every", "affine open in $U$ we conclude that $\\pi$ is surjective.", "In addition, $B$ is a global relative complete intersection over $A$, so", "the ring map $A \\to B$ is standard syntomic,", "in particular syntomic. Since this holds for every affine open of $U$", "we conclude that $\\pi$ is syntomic. Hence", "$\\mathcal{U} = \\{\\pi : U' \\to U\\}$ is a syntomic covering.", "Moreover, $f|_{U'} = (f')^n$ where $f'$ is the class of $T$", "in $\\Gamma(U', \\mathcal{O}_{U'}^*)$, so $\\mathcal{U}$ has the desired property." ], "refs": [ "topologies-lemma-zariski-etale-smooth-syntomic-fppf" ], "ref_ids": [ 12471 ] } ], "ref_ids": [] }, { "id": 6421, "type": "theorem", "label": "etale-cohomology-lemma-describe-h1-mun", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-describe-h1-mun", "contents": [ "Let $S$ be a scheme. There is a canonical identification", "$$", "H_\\etale^1(S, \\mu_n) =", "\\text{group of pairs }(\\mathcal{L}, \\alpha)\\text{ up to isomorphism as above}", "$$", "if $n$ is invertible on $S$. In general we have", "$$", "H_{fppf}^1(S, \\mu_n) =", "\\text{group of pairs }(\\mathcal{L}, \\alpha)\\text{ up to isomorphism as above}.", "$$", "The same result holds with fppf replaced by syntomic." ], "refs": [], "proofs": [ { "contents": [ "We first prove the second isomorphism.", "Let $(\\mathcal{L}, \\alpha)$ be a pair as above.", "Choose an affine open covering $S = \\bigcup U_i$ such that", "$\\mathcal{L}|_{U_i} \\cong \\mathcal{O}_{U_i}$. Say $s_i \\in \\mathcal{L}(U_i)$", "is a generator. Then $\\alpha(s_i^{\\otimes n}) = f_i \\in \\mathcal{O}_S^*(U_i)$.", "Writing $U_i = \\Spec(A_i)$ we see there exists a global", "relative complete intersection $A_i \\to B_i = A_i[T]/(T^n - f_i)$", "such that $f_i$ maps to an $n$th power in $B_i$. In other words, setting", "$V_i = \\Spec(B_i)$ we obtain a syntomic covering", "$\\mathcal{V} = \\{V_i \\to S\\}_{i \\in I}$ and trivializations", "$\\varphi_i : (\\mathcal{L}, \\alpha)|_{V_i} \\to (\\mathcal{O}_{V_i}, 1)$.", "\\medskip\\noindent", "We will use this result (the existence of the covering $\\mathcal{V}$)", "to associate to this pair a cohomology class in", "$H^1_{syntomic}(S, \\mu_{n, S})$. We give two (equivalent) constructions.", "\\medskip\\noindent", "First construction: using {\\v C}ech cohomology.", "Over the double overlaps $V_i \\times_S V_j$ we have the isomorphism", "$$", "(\\mathcal{O}_{V_i \\times_S V_j}, 1)", "\\xrightarrow{\\text{pr}_0^*\\varphi_i^{-1}}", "(\\mathcal{L}|_{V_i \\times_S V_j}, \\alpha|_{V_i \\times_S V_j})", "\\xrightarrow{\\text{pr}_1^*\\varphi_j}", "(\\mathcal{O}_{V_i \\times_S V_j}, 1)", "$$", "of pairs. By (\\ref{equation-isomorphisms-pairs}) this is given by an", "element $\\zeta_{ij} \\in \\mu_n(V_i \\times_S V_j)$. We omit the verification", "that these $\\zeta_{ij}$'s give a $1$-cocycle, i.e., give", "an element $(\\zeta_{i_0i_1}) \\in \\check C(\\mathcal{V}, \\mu_n)$", "with $d(\\zeta_{i_0i_1}) = 0$. Thus its class is an element in", "$\\check H^1(\\mathcal{V}, \\mu_n)$ and by", "Theorem \\ref{theorem-cech-ss}", "it maps to a cohomology class in $H^1_{syntomic}(S, \\mu_{n, S})$.", "\\medskip\\noindent", "Second construction: Using torsors. Consider the presheaf", "$$", "\\mu_n(\\mathcal{L}, \\alpha) :", "U", "\\longmapsto", "\\mathit{Isom}_U((\\mathcal{O}_U, 1), (\\mathcal{L}, \\alpha)|_U)", "$$", "on $(\\Sch/S)_{syntomic}$.", "We may view this as a subpresheaf of", "$\\SheafHom_\\mathcal{O}(\\mathcal{O}, \\mathcal{L})$ (internal hom", "sheaf, see", "Modules on Sites, Section \\ref{sites-modules-section-internal-hom}).", "Since the conditions defining this subpresheaf are local, we see that it is", "a sheaf.", "By (\\ref{equation-isomorphisms-pairs}) this sheaf has a free action of", "the sheaf $\\mu_{n, S}$. Hence the only thing we have to check is that", "it locally has sections. This is true because of the existence of the", "trivializing cover $\\mathcal{V}$. Hence $\\mu_n(\\mathcal{L}, \\alpha)$", "is a $\\mu_{n, S}$-torsor and by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-torsors-h1}", "we obtain a corresponding element of $H^1_{syntomic}(S, \\mu_{n, S})$.", "\\medskip\\noindent", "Ok, now we have to still show the following", "\\begin{enumerate}", "\\item The two constructions give the same cohomology class.", "\\item Isomorphic pairs give rise to the same cohomology class.", "\\item The cohomology class of", "$(\\mathcal{L}, \\alpha) \\otimes (\\mathcal{L}', \\alpha')$", "is the sum of the cohomology classes of", "$(\\mathcal{L}, \\alpha)$ and $(\\mathcal{L}', \\alpha')$.", "\\item If the cohomology class is trivial, then the pair is trivial.", "\\item Any element of $H^1_{syntomic}(S, \\mu_{n, S})$ is the", "cohomology class of a pair.", "\\end{enumerate}", "We omit the proof of (1). Part (2) is clear from the second construction,", "since isomorphic torsors give the same cohomology classes.", "Part (3) is clear from the first construction, since the resulting", "{\\v C}ech classes add up. Part (4) is clear from the second construction", "since a torsor is trivial if and only if it has a global section, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-trivial-torsor}.", "\\medskip\\noindent", "Part (5) can be seen as follows (although a direct proof would be", "preferable). Suppose $\\xi \\in H^1_{syntomic}(S, \\mu_{n, S})$.", "Then $\\xi$ maps to an element", "$\\overline{\\xi} \\in H^1_{syntomic}(S, \\mathbf{G}_{m, S})$", "with $n \\overline{\\xi} = 0$. By", "Theorem \\ref{theorem-picard-group}", "we see that $\\overline{\\xi}$ corresponds to an invertible sheaf $\\mathcal{L}$", "whose $n$th tensor power is isomorphic to $\\mathcal{O}_S$.", "Hence there exists a pair $(\\mathcal{L}, \\alpha')$ whose cohomology", "class $\\xi'$ has the same image $\\overline{\\xi'}$ in", "$H^1_{syntomic}(S, \\mathbf{G}_{m, S})$. Thus it suffices to show", "that $\\xi - \\xi'$ is the class of a pair. By construction, and the", "long exact cohomology sequence above, we see that", "$\\xi - \\xi' = \\partial(f)$ for some $f \\in H^0(S, \\mathcal{O}_S^*)$.", "Consider the pair $(\\mathcal{O}_S, f)$. We omit the verification", "that the cohomology class of this pair is $\\partial(f)$, which", "finishes the proof of the first identification (with fppf replaced", "with syntomic).", "\\medskip\\noindent", "To see the first, note that if $n$ is invertible on $S$, then the", "covering $\\mathcal{V}$ constructed in the first part of the proof", "is actually an \\'etale covering (compare with the proof of", "Lemma \\ref{lemma-kummer-sequence}). The rest of the proof is independent", "of the topology, apart from the very last argument which uses that", "the Kummer sequence is exact, i.e., uses Lemma \\ref{lemma-kummer-sequence}." ], "refs": [ "etale-cohomology-theorem-cech-ss", "sites-cohomology-lemma-torsors-h1", "sites-cohomology-lemma-trivial-torsor", "etale-cohomology-theorem-picard-group", "etale-cohomology-lemma-kummer-sequence", "etale-cohomology-lemma-kummer-sequence" ], "ref_ids": [ 6372, 4182, 4181, 6374, 6419, 6419 ] } ], "ref_ids": [] }, { "id": 6422, "type": "theorem", "label": "etale-cohomology-lemma-cofinal-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cofinal-etale", "contents": [ "Let $S$ be a scheme, and let $\\overline{s}$ be a geometric point of $S$.", "The category of \\'etale neighborhoods is cofiltered. More precisely:", "\\begin{enumerate}", "\\item Let $(U_i, \\overline{u}_i)_{i = 1, 2}$ be two \\'etale neighborhoods of", "$\\overline{s}$ in $S$. Then there exists a third \\'etale neighborhood", "$(U, \\overline{u})$ and morphisms", "$(U, \\overline{u}) \\to (U_i, \\overline{u}_i)$, $i = 1, 2$.", "\\item Let $h_1, h_2: (U, \\overline{u}) \\to (U', \\overline{u}')$ be two", "morphisms between \\'etale neighborhoods of $\\overline{s}$. Then there exist an", "\\'etale neighborhood $(U'', \\overline{u}'')$ and a morphism", "$h : (U'', \\overline{u}'') \\to (U, \\overline{u})$", "which equalizes $h_1$ and $h_2$, i.e., such that", "$h_1 \\circ h = h_2 \\circ h$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For part (1), consider the fibre product $U = U_1 \\times_S U_2$.", "It is \\'etale over both $U_1$ and $U_2$ because \\'etale morphisms are", "preserved under base change, see", "Proposition \\ref{proposition-etale-morphisms}.", "The map $\\overline{s} \\to U$ defined by $(\\overline{u}_1, \\overline{u}_2)$", "gives it the structure of an \\'etale neighborhood mapping to both", "$U_1$ and $U_2$. For part (2), define $U''$ as the fibre product", "$$", "\\xymatrix{", "U'' \\ar[r] \\ar[d] & U \\ar[d]^{(h_1, h_2)} \\\\", "U' \\ar[r]^-\\Delta & U' \\times_S U'.", "}", "$$", "Since $\\overline{u}$ and $\\overline{u}'$ agree over $S$ with $\\overline{s}$,", "we see that $\\overline{u}'' = (\\overline{u}, \\overline{u}')$ is a geometric", "point of $U''$. In particular $U'' \\not = \\emptyset$.", "Moreover, since $U'$ is \\'etale over $S$, so is the fibre product", "$U'\\times_S U'$ (see", "Proposition \\ref{proposition-etale-morphisms}).", "Hence the vertical arrow $(h_1, h_2)$ is \\'etale by", "Remark \\ref{remark-etale-between-etale} above.", "Therefore $U''$ is \\'etale over $U'$ by base change, and hence also", "\\'etale over $S$ (because compositions of \\'etale morphisms are \\'etale).", "Thus $(U'', \\overline{u}'')$ is a solution to the problem." ], "refs": [ "etale-cohomology-proposition-etale-morphisms", "etale-cohomology-proposition-etale-morphisms", "etale-cohomology-remark-etale-between-etale" ], "ref_ids": [ 6697, 6697, 6774 ] } ], "ref_ids": [] }, { "id": 6423, "type": "theorem", "label": "etale-cohomology-lemma-geometric-lift-to-cover", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-geometric-lift-to-cover", "contents": [ "Let $S$ be a scheme.", "Let $\\overline{s}$ be a geometric point of $S$.", "Let $(U, \\overline{u})$ be an \\'etale neighborhood of $\\overline{s}$.", "Let $\\mathcal{U} = \\{\\varphi_i : U_i \\to U \\}_{i\\in I}$ be an \\'etale covering.", "Then there exist $i \\in I$ and $\\overline{u}_i : \\overline{s} \\to U_i$", "such that $\\varphi_i : (U_i, \\overline{u}_i) \\to (U, \\overline{u})$", "is a morphism of \\'etale neighborhoods." ], "refs": [], "proofs": [ { "contents": [ "As $U = \\bigcup_{i\\in I} \\varphi_i(U_i)$, the fibre product", "$\\overline{s} \\times_{\\overline{u}, U, \\varphi_i} U_i$ is not empty", "for some $i$. Then look at the cartesian diagram", "$$", "\\xymatrix{", "\\overline{s} \\times_{\\overline{u}, U, \\varphi_i} U_i", "\\ar[d]^{\\text{pr}_1} \\ar[r]_-{\\text{pr}_2} & U_i", "\\ar[d]^{\\varphi_i} \\\\", "\\Spec(k) = \\overline{s} \\ar@/^1pc/[u]^\\sigma", "\\ar[r]^-{\\overline{u}} & U", "}", "$$", "The projection $\\text{pr}_1$ is the base change of an \\'etale morphisms so it", "is \\'etale, see", "Proposition \\ref{proposition-etale-morphisms}.", "Therefore, $\\overline{s} \\times_{\\overline{u}, U, \\varphi_i} U_i$", "is a disjoint union of finite separable extensions of $k$, by", "Proposition \\ref{proposition-etale-morphisms}. Here", "$\\overline{s} = \\Spec(k)$. But $k$ is algebraically closed, so all", "these extensions are trivial, and there exists a section $\\sigma$ of", "$\\text{pr}_1$. The composition", "$\\text{pr}_2 \\circ \\sigma$ gives a map compatible with $\\overline{u}$." ], "refs": [ "etale-cohomology-proposition-etale-morphisms", "etale-cohomology-proposition-etale-morphisms" ], "ref_ids": [ 6697, 6697 ] } ], "ref_ids": [] }, { "id": 6424, "type": "theorem", "label": "etale-cohomology-lemma-stalk-gives-point", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-stalk-gives-point", "contents": [ "Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point of $S$.", "Consider the functor", "\\begin{align*}", "u : S_\\etale & \\longrightarrow \\textit{Sets}, \\\\", "U & \\longmapsto", "|U_{\\overline{s}}|", "=", "\\{\\overline{u} \\text{ such that }(U, \\overline{u})", "\\text{ is an \\'etale neighbourhood of }\\overline{s}\\}.", "\\end{align*}", "Here $|U_{\\overline{s}}|$ denotes the underlying set of the geometric fibre.", "Then $u$ defines a point $p$ of the site $S_\\etale$", "(Sites, Definition \\ref{sites-definition-point})", "and its associated stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_p$", "(Sites, Equation \\ref{sites-equation-stalk})", "is the functor $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{s}}$", "defined above." ], "refs": [ "sites-definition-point" ], "proofs": [ { "contents": [ "In the proof of", "Lemma \\ref{lemma-geometric-lift-to-cover}", "we have seen that the scheme $U_{\\overline{s}}$ is a disjoint union of", "schemes isomorphic to $\\overline{s}$. Thus we can also think of", "$|U_{\\overline{s}}|$ as the set of geometric points of $U$ lying over", "$\\overline{s}$, i.e., as the collection of morphisms", "$\\overline{u} : \\overline{s} \\to U$ fitting into the diagram of", "Definition \\ref{definition-geometric-point}.", "From this it follows that $u(S)$ is a singleton, and that", "$u(U \\times_V W) = u(U) \\times_{u(V)} u(W)$", "whenever $U \\to V$ and $W \\to V$ are morphisms in $S_\\etale$.", "And, given a covering $\\{U_i \\to U\\}_{i \\in I}$ in $S_\\etale$", "we see that $\\coprod u(U_i) \\to u(U)$ is surjective by", "Lemma \\ref{lemma-geometric-lift-to-cover}.", "Hence", "Sites, Proposition \\ref{sites-proposition-point-limits}", "applies, so $p$ is a point of the site $S_\\etale$.", "Finally, our functor $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{s}}$", "is given by exactly the same colimit as the functor", "$\\mathcal{F} \\mapsto \\mathcal{F}_p$ associated to $p$ in", "Sites, Equation \\ref{sites-equation-stalk}", "which proves the final assertion." ], "refs": [ "etale-cohomology-lemma-geometric-lift-to-cover", "etale-cohomology-definition-geometric-point", "etale-cohomology-lemma-geometric-lift-to-cover", "sites-proposition-point-limits" ], "ref_ids": [ 6423, 6735, 6423, 8642 ] } ], "ref_ids": [ 8675 ] }, { "id": 6425, "type": "theorem", "label": "etale-cohomology-lemma-stalk-exact", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-stalk-exact", "contents": [ "Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point of $S$.", "\\begin{enumerate}", "\\item The stalk functor", "$\\textit{PAb}(S_\\etale) \\to \\textit{Ab}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{s}}$", "is exact.", "\\item We have $(\\mathcal{F}^\\#)_{\\overline{s}} = \\mathcal{F}_{\\overline{s}}$", "for any presheaf of sets $\\mathcal{F}$ on $S_\\etale$.", "\\item The functor", "$\\textit{Ab}(S_\\etale) \\to \\textit{Ab}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{s}}$ is exact.", "\\item Similarly the functors", "$\\textit{PSh}(S_\\etale) \\to \\textit{Sets}$ and", "$\\Sh(S_\\etale) \\to \\textit{Sets}$ given by the stalk functor", "$\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$ are exact (see", "Categories, Definition \\ref{categories-definition-exact})", "and commute with arbitrary colimits.", "\\end{enumerate}" ], "refs": [ "categories-definition-exact" ], "proofs": [ { "contents": [ "Before we indicate how to prove this by direct arguments", "we note that the result follows from the general material in", "Modules on Sites, Section \\ref{sites-modules-section-stalks}.", "This is true because $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{s}}$", "comes from a point of the small \\'etale site of $S$, see", "Lemma \\ref{lemma-stalk-gives-point}.", "We will only give a direct proof of (1), (2) and (3), and omit", "a direct proof of (4).", "\\medskip\\noindent", "Exactness as a functor on $\\textit{PAb}(S_\\etale)$ is formal from the", "fact that directed colimits commute with all colimits and with finite", "limits. The identification of the stalks in (2) is via the map", "$$", "\\kappa :", "\\mathcal{F}_{\\overline{s}}", "\\longrightarrow", "(\\mathcal{F}^\\#)_{\\overline{s}}", "$$", "induced by the natural morphism $\\mathcal{F}\\to \\mathcal{F}^\\#$, see", "Theorem \\ref{theorem-sheafification}.", "We claim that this map is an isomorphism of abelian groups. We will show", "injectivity and omit the proof of surjectivity.", "\\medskip\\noindent", "Let $\\sigma\\in \\mathcal{F}_{\\overline{s}}$.", "There exists an \\'etale neighborhood", "$(U, \\overline{u})\\to (S, \\overline{s})$ such that $\\sigma$ is the image of some", "section $s \\in \\mathcal{F}(U)$. If $\\kappa(\\sigma) = 0$ in", "$(\\mathcal{F}^\\#)_{\\overline{s}}$ then there exists a morphism of \\'etale", "neighborhoods $(U', \\overline{u}')\\to (U, \\overline{u})$ such that", "$s|_{U'}$ is zero in $\\mathcal{F}^\\#(U')$. It follows there", "exists an \\'etale covering", "$\\{U_i'\\to U'\\}_{i\\in I}$ such that $s|_{U_i'}=0$ in", "$\\mathcal{F}(U_i')$ for all $i$. By Lemma \\ref{lemma-geometric-lift-to-cover}", "there exist $i \\in I$ and a morphism", "$\\overline{u}_i': \\overline{s} \\to U_i'$ such that", "$(U_i', \\overline{u}_i') \\to (U', \\overline{u}')\\to (U, \\overline{u})$", "are morphisms of \\'etale neighborhoods. Hence $\\sigma = 0$", "since $(U_i', \\overline{u}_i') \\to (U, \\overline{u})$", "is a morphism of \\'etale neighbourhoods such that", "we have $s|_{U'_i}=0$. This proves $\\kappa$ is injective.", "\\medskip\\noindent", "To show that the functor $\\textit{Ab}(S_\\etale) \\to \\textit{Ab}$ is", "exact, consider any short exact sequence in $\\textit{Ab}(S_\\etale)$:", "$", "0\\to \\mathcal{F}\\to \\mathcal{G}\\to \\mathcal H \\to 0.", "$", "This gives us the exact sequence of presheaves", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal H \\to", "\\mathcal H/^p\\mathcal{G} \\to 0,", "$$", "where $/^p$ denotes the quotient in $\\textit{PAb}(S_\\etale)$.", "Taking stalks at", "$\\overline{s}$, we see that $(\\mathcal H /^p\\mathcal{G})_{\\bar{s}} =", "(\\mathcal H /\\mathcal{G})_{\\bar{s}} = 0$, since the sheafification of", "$\\mathcal H/^p\\mathcal{G}$ is $0$.", "Therefore,", "$$", "0\\to \\mathcal{F}_{\\overline{s}} \\to \\mathcal{G}_{\\overline{s}} \\to", "\\mathcal{H}_{\\overline{s}} \\to 0 = (\\mathcal H/^p\\mathcal{G})_{\\overline{s}}", "$$", "is exact, since taking stalks is exact as a functor from presheaves." ], "refs": [ "etale-cohomology-lemma-stalk-gives-point", "etale-cohomology-theorem-sheafification", "etale-cohomology-lemma-geometric-lift-to-cover" ], "ref_ids": [ 6424, 6366, 6423 ] } ], "ref_ids": [ 12370 ] }, { "id": 6426, "type": "theorem", "label": "etale-cohomology-lemma-points-small-etale-site", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-points-small-etale-site", "contents": [ "Let $S$ be a scheme.", "\\begin{enumerate}", "\\item Let $p$ be a point of the small \\'etale site", "$S_\\etale$ of $S$ given by a functor", "$u : S_\\etale \\to \\textit{Sets}$.", "Then there exists a geometric point $\\overline{s}$ of $S$ such that", "$p$ is isomorphic to the point of $S_\\etale$ associated to", "$\\overline{s}$ in", "Lemma \\ref{lemma-stalk-gives-point}.", "\\item Let $p : \\Sh(pt) \\to \\Sh(S_\\etale)$ be a point", "of the small \\'etale topos of $S$. Then $p$ comes from a geometric point", "of $S$, i.e., the stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_p$", "is isomorphic to a stalk functor as defined in", "Definition \\ref{definition-stalk}.", "\\end{enumerate}" ], "refs": [ "etale-cohomology-lemma-stalk-gives-point", "etale-cohomology-definition-stalk" ], "proofs": [ { "contents": [ "By", "Sites, Lemma \\ref{sites-lemma-point-site-topos}", "there is a one to one correspondence between points of the site and points", "of the associated topos, hence it suffices to prove (1).", "By", "Sites, Proposition \\ref{sites-proposition-point-limits}", "the functor $u$ has the following properties:", "(a) $u(S) = \\{*\\}$, (b) $u(U \\times_V W) = u(U) \\times_{u(V)} u(W)$, and", "(c) if $\\{U_i \\to U\\}$ is an \\'etale covering, then", "$\\coprod u(U_i) \\to u(U)$ is surjective.", "In particular, if $U' \\subset U$ is an open subscheme,", "then $u(U') \\subset u(U)$. Moreover, by", "Sites, Lemma \\ref{sites-lemma-point-site-topos}", "we can write $u(U) = p^{-1}(h_U^\\#)$, in other words $u(U)$ is the", "stalk of the representable sheaf $h_U$. If", "$U = V \\amalg W$, then we see that $h_U = (h_V \\amalg h_W)^\\#$ and we get", "$u(U) = u(V) \\amalg u(W)$ since $p^{-1}$ is exact.", "\\medskip\\noindent", "Consider the restriction of $u$ to $S_{Zar}$. By", "Sites, Examples \\ref{sites-example-point-topological} and", "\\ref{sites-example-point-topology}", "there exists a unique point $s \\in S$ such that for $S' \\subset S$ open we", "have $u(S') = \\{*\\}$ if $s \\in S'$ and $u(S') = \\emptyset$ if $s \\not \\in S'$.", "Note that if $\\varphi : U \\to S$ is an object of $S_\\etale$ then", "$\\varphi(U) \\subset S$ is open (see", "Proposition \\ref{proposition-etale-morphisms})", "and $\\{U \\to \\varphi(U)\\}$ is an \\'etale covering. Hence we conclude that", "$u(U) = \\emptyset \\Leftrightarrow s \\in \\varphi(U)$.", "\\medskip\\noindent", "Pick a geometric point $\\overline{s} : \\overline{s} \\to S$ lying over $s$, see", "Definition \\ref{definition-geometric-point}", "for customary abuse of notation. Suppose that $\\varphi : U \\to S$ is an object", "of $S_\\etale$ with $U$ affine. Note that $\\varphi$ is separated, and", "that the fibre $U_s$ of $\\varphi$ over $s$ is an affine scheme over", "$\\Spec(\\kappa(s))$ which is the spectrum of a finite product of", "finite separable extensions $k_i$ of $\\kappa(s)$. Hence we may apply", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-etale-etale-local-technical}", "to get an \\'etale neighbourhood $(V, \\overline{v})$ of $(S, \\overline{s})$", "such that", "$$", "U \\times_S V = U_1 \\amalg \\ldots \\amalg U_n \\amalg W", "$$", "with $U_i \\to V$ an isomorphism and $W$ having no point lying over", "$\\overline{v}$. Thus we conclude that", "$$", "u(U) \\times u(V) =", "u(U \\times_S V) =", "u(U_1) \\amalg \\ldots \\amalg u(U_n) \\amalg u(W)", "$$", "and of course also $u(U_i) = u(V)$. After shrinking $V$ a bit we can", "assume that $V$ has exactly one point lying over $s$, and hence $W$ has no", "point lying over $s$. By the above this then gives $u(W) = \\emptyset$.", "Hence we obtain", "$$", "u(U) \\times u(V) =", "u(U_1) \\amalg \\ldots \\amalg u(U_n) =", "\\coprod\\nolimits_{i = 1, \\ldots, n} u(V)", "$$", "Note that $u(V) \\not = \\emptyset$ as $s$ is in the image of $V \\to S$.", "In particular, we see that in this situation $u(U)$ is a finite", "set with $n$ elements.", "\\medskip\\noindent", "Consider the limit", "$$", "\\lim_{(V, \\overline{v})} u(V)", "$$", "over the category of \\'etale neighbourhoods $(V, \\overline{v})$ of", "$\\overline{s}$. It is clear that we get the same value when taking", "the limit over the subcategory of $(V, \\overline{v})$ with $V$ affine.", "By the previous paragraph (applied with the roles of $V$ and $U$ switched)", "we see that in this case $u(V)$ is always a finite nonempty set.", "Moreover, the limit is cofiltered, see", "Lemma \\ref{lemma-cofinal-etale}.", "Hence by", "Categories, Section \\ref{categories-section-codirected-limits}", "the limit is nonempty. Pick an element $x$ from this limit.", "This means we obtain a $x_{V, \\overline{v}} \\in u(V)$ for", "every \\'etale neighbourhood $(V, \\overline{v})$ of $(S, \\overline{s})$", "such that for every morphism of \\'etale neighbourhoods", "$\\varphi : (V', \\overline{v}') \\to (V, \\overline{v})$ we have", "$u(\\varphi)(x_{V', \\overline{v}'}) = x_{V, \\overline{v}}$.", "\\medskip\\noindent", "We will use the choice of $x$ to construct a functorial bijective map", "$$", "c : |U_{\\overline{s}}| \\longrightarrow u(U)", "$$", "for $U \\in \\Ob(S_\\etale)$ which will conclude the proof. See", "Lemma \\ref{lemma-stalk-gives-point}", "and its proof for a description of $|U_{\\overline{s}}|$.", "First we claim that it suffices to construct the map for $U$ affine.", "We omit the proof of this claim.", "Assume $U \\to S$ in $S_\\etale$ with $U$ affine, and let", "$\\overline{u} : \\overline{s} \\to U$ be an element of $|U_{\\overline{s}}|$.", "Choose a $(V, \\overline{v})$ such that $U \\times_S V$ decomposes", "as in the third paragraph of the proof.", "Then the pair $(\\overline{u}, \\overline{v})$ gives a geometric point of", "$U \\times_S V$ lying over $\\overline{v}$ and determines one of the", "components $U_i$ of $U \\times_S V$. More precisely, there exists", "a section $\\sigma : V \\to U \\times_S V$ of the projection $\\text{pr}_U$", "such that $(\\overline{u}, \\overline{v}) = \\sigma \\circ \\overline{v}$. Set", "$c(\\overline{u}) = u(\\text{pr}_U)(u(\\sigma)(x_{V, \\overline{v}})) \\in u(U)$.", "We have to check this is independent of the choice of $(V, \\overline{v})$. By", "Lemma \\ref{lemma-cofinal-etale}", "the category of \\'etale neighbourhoods is cofiltered.", "Hence it suffice to show", "that given a morphism of \\'etale neighbourhood", "$\\varphi : (V', \\overline{v}') \\to (V, \\overline{v})$ and a choice of a", "section $\\sigma' : V' \\to U \\times_S V'$ of the projection such that", "$(\\overline{u}, \\overline{v'}) = \\sigma' \\circ \\overline{v}'$", "we have $u(\\sigma')(x_{V', \\overline{v}'}) = u(\\sigma)(x_{V, \\overline{v}})$.", "Consider the diagram", "$$", "\\xymatrix{", "V' \\ar[d]^{\\sigma'} \\ar[r]_\\varphi & V \\ar[d]^\\sigma \\\\", "U \\times_S V' \\ar[r]^{1 \\times \\varphi} &", "U \\times_S V", "}", "$$", "Now, it may not be the case that this diagram commutes. The reason is", "that the schemes $V'$ and $V$ may not be connected, and hence", "the decompositions used to construct $\\sigma'$ and $\\sigma$ above may", "not be unique. But we do know that", "$\\sigma \\circ \\varphi \\circ \\overline{v}' =", "(1 \\times \\varphi) \\circ \\sigma' \\circ \\overline{v}'$", "by construction. Hence, since $U \\times_S V$ is \\'etale over $S$,", "there exists an open neighbourhood", "$V'' \\subset V'$ of $\\overline{v'}$ such that the diagram does", "commute when restricted to $V''$, see", "Morphisms, Lemma \\ref{morphisms-lemma-value-at-one-point}.", "This means we may extend the diagram above to", "$$", "\\xymatrix{", "V'' \\ar[r] \\ar[d]^{\\sigma'|_{V''}} &", "V' \\ar[d]^{\\sigma'} \\ar[r]_\\varphi &", "V \\ar[d]^\\sigma \\\\", "U \\times_S V'' \\ar[r] &", "U \\times_S V' \\ar[r]^{1 \\times \\varphi} &", "U \\times_S V", "}", "$$", "such that the left square and the outer rectangle commute.", "Since $u$ is a functor this implies that", "$x_{V'', \\overline{v}'}$ maps to the same element in", "$u(U \\times_S V)$ no matter which route we take through the", "diagram. On the other hand, it maps to the elements", "$x_{V', \\overline{v}'}$ and $x_{V, \\overline{v}}$ in", "$u(V')$ and $u(V)$. This implies the desired equality", "$u(\\sigma')(x_{V', \\overline{v}'}) = u(\\sigma)(x_{V, \\overline{v}})$.", "\\medskip\\noindent", "In a similar manner one proves that the construction", "$c : |U_{\\overline{s}}| \\to u(U)$ is functorial in $U$;", "details omitted. And finally, by the results of the", "third paragraph it is clear that the map $c$ is bijective", "which ends the proof of the lemma." ], "refs": [ "sites-lemma-point-site-topos", "sites-proposition-point-limits", "sites-lemma-point-site-topos", "etale-cohomology-proposition-etale-morphisms", "etale-cohomology-definition-geometric-point", "etale-lemma-etale-etale-local-technical", "etale-cohomology-lemma-cofinal-etale", "etale-cohomology-lemma-stalk-gives-point", "etale-cohomology-lemma-cofinal-etale", "morphisms-lemma-value-at-one-point" ], "ref_ids": [ 8596, 8642, 8596, 6697, 6735, 10713, 6422, 6424, 6422, 5358 ] } ], "ref_ids": [ 6424, 6736 ] }, { "id": 6427, "type": "theorem", "label": "etale-cohomology-lemma-points-fppf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-points-fppf", "contents": [ "Let $S$ be a scheme. All of the following sites have enough points", "$S_{Zar}$, $S_\\etale$,", "$(\\Sch/S)_{Zar}$, $(\\textit{Aff}/S)_{Zar}$,", "$(\\Sch/S)_\\etale$, $(\\textit{Aff}/S)_\\etale$,", "$(\\Sch/S)_{smooth}$, $(\\textit{Aff}/S)_{smooth}$,", "$(\\Sch/S)_{syntomic}$, $(\\textit{Aff}/S)_{syntomic}$,", "$(\\Sch/S)_{fppf}$, and $(\\textit{Aff}/S)_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "For each of the big sites the associated topos is equivalent to the", "topos defined by the site $(\\textit{Aff}/S)_\\tau$, see", "Topologies, Lemmas \\ref{topologies-lemma-affine-big-site-Zariski},", "\\ref{topologies-lemma-affine-big-site-etale},", "\\ref{topologies-lemma-affine-big-site-smooth},", "\\ref{topologies-lemma-affine-big-site-syntomic}, and", "\\ref{topologies-lemma-affine-big-site-fppf}.", "The result for the sites $(\\textit{Aff}/S)_\\tau$ follows immediately", "from Deligne's result", "Sites, Lemma \\ref{sites-lemma-criterion-points}.", "\\medskip\\noindent", "The result for $S_{Zar}$ is clear. The result for $S_\\etale$", "either follows from (the proof of)", "Theorem \\ref{theorem-exactness-stalks}", "or from", "Lemma \\ref{lemma-alternative}", "and Deligne's result applied to $S_{affine, \\etale}$." ], "refs": [ "topologies-lemma-affine-big-site-Zariski", "topologies-lemma-affine-big-site-etale", "topologies-lemma-affine-big-site-smooth", "topologies-lemma-affine-big-site-syntomic", "topologies-lemma-affine-big-site-fppf", "sites-lemma-criterion-points", "etale-cohomology-theorem-exactness-stalks", "etale-cohomology-lemma-alternative" ], "ref_ids": [ 12436, 12451, 12463, 12469, 12477, 8616, 6376, 6413 ] } ], "ref_ids": [] }, { "id": 6428, "type": "theorem", "label": "etale-cohomology-lemma-support-subsheaf-final", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-support-subsheaf-final", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ be a subsheaf of the final", "object of the \\'etale topos of $S$ (see", "Sites, Example \\ref{sites-example-singleton-sheaf}).", "Then there exists a unique open", "$W \\subset S$ such that $\\mathcal{F} = h_W$." ], "refs": [], "proofs": [ { "contents": [ "The condition means that $\\mathcal{F}(U)$ is a singleton or", "empty for all $\\varphi : U \\to S$ in $\\Ob(S_\\etale)$.", "In particular local sections always glue. If", "$\\mathcal{F}(U) \\not = \\emptyset$, then", "$\\mathcal{F}(\\varphi(U)) \\not = \\emptyset$ because", "$\\{\\varphi : U \\to \\varphi(U)\\}$ is a covering.", "Hence we can take", "$W = \\bigcup_{\\varphi : U \\to S, \\mathcal{F}(U) \\not = \\emptyset} \\varphi(U)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6429, "type": "theorem", "label": "etale-cohomology-lemma-zero-over-image", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-zero-over-image", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{F}$ be an abelian sheaf on $S_\\etale$.", "Let $\\sigma \\in \\mathcal{F}(U)$ be a local section.", "There exists an open subset $W \\subset U$ such that", "\\begin{enumerate}", "\\item $W \\subset U$ is the largest Zariski open subset of $U$ such", "that $\\sigma|_W = 0$,", "\\item for every $\\varphi : V \\to U$ in $S_\\etale$ we have", "$$", "\\sigma|_V = 0 \\Leftrightarrow \\varphi(V) \\subset W,", "$$", "\\item for every geometric point $\\overline{u}$ of $U$ we have", "$$", "(U, \\overline{u}, \\sigma) = 0\\text{ in }\\mathcal{F}_{\\overline{s}}", "\\Leftrightarrow", "\\overline{u} \\in W", "$$", "where $\\overline{s} = (U \\to S) \\circ \\overline{u}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathcal{F}$ is a sheaf in the \\'etale topology the restriction of", "$\\mathcal{F}$ to $U_{Zar}$ is a sheaf on $U$ in the Zariski topology.", "Hence there exists a Zariski open $W$ having property (1), see", "Modules, Lemma \\ref{modules-lemma-support-section-closed}. Let", "$\\varphi : V \\to U$ be an arrow of $S_\\etale$. Note that", "$\\varphi(V) \\subset U$ is an open subset and that", "$\\{V \\to \\varphi(V)\\}$ is an \\'etale covering. Hence if", "$\\sigma|_V = 0$, then by the sheaf condition for $\\mathcal{F}$ we", "see that $\\sigma|_{\\varphi(V)} = 0$. This proves (2).", "To prove (3) we have to show that if $(U, \\overline{u}, \\sigma)$", "defines the zero element of $\\mathcal{F}_{\\overline{s}}$, then", "$\\overline{u} \\in W$. This is true because the assumption means", "there exists a morphism of \\'etale neighbourhoods", "$(V, \\overline{v}) \\to (U, \\overline{u})$ such that", "$\\sigma|_V = 0$. Hence by (2) we see that $V \\to U$ maps into $W$, and", "hence $\\overline{u} \\in W$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6430, "type": "theorem", "label": "etale-cohomology-lemma-support-section-closed", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-support-section-closed", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{F}$ be an abelian sheaf on $S_\\etale$.", "Let $U \\in \\Ob(S_\\etale)$ and $\\sigma \\in \\mathcal{F}(U)$.", "\\begin{enumerate}", "\\item The support of $\\sigma$ is closed in $U$.", "\\item The support of $\\sigma + \\sigma'$ is contained in the union of", "the supports of $\\sigma, \\sigma' \\in \\mathcal{F}(U)$.", "\\item If $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a map of", "abelian sheaves on $S_\\etale$, then the support of $\\varphi(\\sigma)$", "is contained in the support of $\\sigma \\in \\mathcal{F}(U)$.", "\\item The support of $\\mathcal{F}$ is the union of the images of the", "supports of all local sections of $\\mathcal{F}$.", "\\item If $\\mathcal{F} \\to \\mathcal{G}$ is surjective then the support", "of $\\mathcal{G}$ is a subset of the support of $\\mathcal{F}$.", "\\item If $\\mathcal{F} \\to \\mathcal{G}$ is injective then the support", "of $\\mathcal{F}$ is a subset of the support of $\\mathcal{G}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) holds by definition.", "Parts (2) and (3) hold because they holds for the restriction of", "$\\mathcal{F}$ and $\\mathcal{G}$ to $U_{Zar}$, see", "Modules, Lemma \\ref{modules-lemma-support-section-closed}.", "Part (4) is a direct consequence of", "Lemma \\ref{lemma-zero-over-image} part (3).", "Parts (5) and (6) follow from the other parts." ], "refs": [ "etale-cohomology-lemma-zero-over-image" ], "ref_ids": [ 6429 ] } ], "ref_ids": [] }, { "id": 6431, "type": "theorem", "label": "etale-cohomology-lemma-support-sheaf-rings-closed", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-support-sheaf-rings-closed", "contents": [ "The support of a sheaf of rings on $S_\\etale$ is closed." ], "refs": [], "proofs": [ { "contents": [ "This is true because (according to our conventions)", "a ring is $0$ if and only if", "$1 = 0$, and hence the support of a sheaf of rings", "is the support of the unit section." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6432, "type": "theorem", "label": "etale-cohomology-lemma-finite-over-henselian", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-finite-over-henselian", "contents": [ "If $R$ is henselian and $A$ is a finite $R$-algebra, then $A$ is a finite", "product of henselian local rings." ], "refs": [], "proofs": [ { "contents": [ "See", "Algebra, Lemma \\ref{algebra-lemma-finite-over-henselian}." ], "refs": [ "algebra-lemma-finite-over-henselian" ], "ref_ids": [ 1277 ] } ], "ref_ids": [] }, { "id": 6433, "type": "theorem", "label": "etale-cohomology-lemma-describe-etale-local-ring", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-describe-etale-local-ring", "contents": [ "\\begin{slogan}", "The stalk of the structure sheaf of a scheme", "in the etale topology is the strict henselization.", "\\end{slogan}", "Let $S$ be a scheme.", "Let $\\overline{s}$ be a geometric point of $S$ lying over $s \\in S$.", "Let $\\kappa = \\kappa(s)$ and let", "$\\kappa \\subset \\kappa^{sep} \\subset \\kappa(\\overline{s})$ denote", "the separable algebraic closure of $\\kappa$ in $\\kappa(\\overline{s})$.", "Then there is a canonical identification", "$$", "(\\mathcal{O}_{S, s})^{sh}", "\\cong", "\\mathcal{O}_{S, \\overline{s}}", "$$", "where the left hand side is the strict henselization of the local ring", "$\\mathcal{O}_{S, s}$ as described in", "Theorem \\ref{theorem-henselization}", "and right hand side is the stalk of the structure sheaf", "$\\mathcal{O}_S$ on $S_\\etale$ at", "the geometric point $\\overline{s}$." ], "refs": [ "etale-cohomology-theorem-henselization" ], "proofs": [ { "contents": [ "Let $\\Spec(A) \\subset S$ be an affine neighbourhood of $s$.", "Let $\\mathfrak p \\subset A$ be the prime ideal corresponding to $s$.", "With these choices we have canonical isomorphisms", "$\\mathcal{O}_{S, s} = A_{\\mathfrak p}$ and $\\kappa(s) = \\kappa(\\mathfrak p)$.", "Thus we have", "$\\kappa(\\mathfrak p) \\subset \\kappa^{sep} \\subset \\kappa(\\overline{s})$.", "Recall that", "$$", "\\mathcal{O}_{S, \\overline{s}} =", "\\colim_{(U, \\overline{u})} \\mathcal{O}(U)", "$$", "where the limit is over the \\'etale neighbourhoods of $(S, \\overline{s})$.", "A cofinal system is given by those \\'etale neighbourhoods $(U, \\overline{u})$", "such that $U$ is affine and $U \\to S$ factors through $\\Spec(A)$.", "In other words, we see that", "$$", "\\mathcal{O}_{S, \\overline{s}} = \\colim_{(B, \\mathfrak q, \\phi)} B", "$$", "where the colimit is over \\'etale $A$-algebras $B$ endowed with a prime", "$\\mathfrak q$ lying over $\\mathfrak p$ and a", "$\\kappa(\\mathfrak p)$-algebra map", "$\\phi : \\kappa(\\mathfrak q) \\to \\kappa(\\overline{s})$.", "Note that since $\\kappa(\\mathfrak q)$ is finite separable over", "$\\kappa(\\mathfrak p)$ the image of $\\phi$ is contained in $\\kappa^{sep}$.", "Via these translations the result of the lemma is equivalent", "to the result of", "Algebra, Lemma \\ref{algebra-lemma-strict-henselization-different}." ], "refs": [ "algebra-lemma-strict-henselization-different" ], "ref_ids": [ 1304 ] } ], "ref_ids": [ 6380 ] }, { "id": 6434, "type": "theorem", "label": "etale-cohomology-lemma-describe-henselization", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-describe-henselization", "contents": [ "Let $S$ be a scheme. Let $s \\in S$. Then we have", "$$", "\\mathcal{O}_{S, s}^h =", "\\colim_{(U, u)} \\mathcal{O}(U)", "$$", "where the colimit is over the filtered category of", "\\'etale neighbourhoods $(U, u)$ of $(S, s)$ such that", "$\\kappa(s) = \\kappa(u)$." ], "refs": [], "proofs": [ { "contents": [ "This lemma is a copy of", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-describe-henselization}." ], "refs": [ "more-morphisms-lemma-describe-henselization" ], "ref_ids": [ 13869 ] } ], "ref_ids": [] }, { "id": 6435, "type": "theorem", "label": "etale-cohomology-lemma-etale-site-locally-ringed", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-etale-site-locally-ringed", "contents": [ "Let $S$ be a scheme. The small \\'etale site $S_\\etale$ endowed with", "its structure sheaf $\\mathcal{O}_S$ is a locally ringed site, see", "Modules on Sites, Definition \\ref{sites-modules-definition-locally-ringed}." ], "refs": [ "sites-modules-definition-locally-ringed" ], "proofs": [ { "contents": [ "This follows because the stalks", "$\\mathcal{O}_{S, s}^{sh} = \\mathcal{O}_{S, \\overline{s}}$ are", "local, and because $S_\\etale$ has enough points, see", "Lemma \\ref{lemma-describe-etale-local-ring},", "Theorem \\ref{theorem-exactness-stalks},", "and", "Remarks \\ref{remarks-enough-points}.", "See", "Modules on Sites, Lemmas \\ref{sites-modules-lemma-locally-ringed-stalk} and", "\\ref{sites-modules-lemma-ringed-stalk-not-zero}", "for the fact that this implies the small \\'etale site is locally ringed." ], "refs": [ "etale-cohomology-lemma-describe-etale-local-ring", "etale-cohomology-theorem-exactness-stalks", "etale-cohomology-remarks-enough-points", "sites-modules-lemma-locally-ringed-stalk", "sites-modules-lemma-ringed-stalk-not-zero" ], "ref_ids": [ 6433, 6376, 6798, 14254, 14255 ] } ], "ref_ids": [ 14302 ] }, { "id": 6436, "type": "theorem", "label": "etale-cohomology-lemma-stalk-pullback", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-stalk-pullback", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "\\begin{enumerate}", "\\item The functor", "$f^{-1} : \\textit{Ab}(Y_\\etale) \\to \\textit{Ab}(X_\\etale)$", "is exact.", "\\item The functor", "$f^{-1} : \\Sh(Y_\\etale) \\to \\Sh(X_\\etale)$", "is exact, i.e., it commutes with finite limits and colimits, see", "Categories, Definition \\ref{categories-definition-exact}.", "\\item Let $\\overline{x} \\to X$ be a geometric point.", "Let $\\mathcal{G}$ be a sheaf on $Y_\\etale$.", "Then there is a canonical identification", "$$", "(f^{-1}\\mathcal{G})_{\\overline{x}} = \\mathcal{G}_{\\overline{y}}.", "$$", "where $\\overline{y} = f \\circ \\overline{x}$.", "\\item For any $V \\to Y$ \\'etale we have $f^{-1}h_V = h_{X \\times_Y V}$.", "\\end{enumerate}" ], "refs": [ "categories-definition-exact" ], "proofs": [ { "contents": [ "The exactness of $f^{-1}$ on sheaves of sets is a consequence of", "Sites, Proposition \\ref{sites-proposition-get-morphism}", "applied to our functor $u$ of Equation (\\ref{equation-functorial}).", "In fact the exactness of pullback is part of the definition of", "a morphism of topoi (or sites if you like). Thus we see (2) holds.", "It implies part (1) since given an abelian sheaf $\\mathcal{G}$ on", "$Y_\\etale$", "the underlying sheaf of sets of $f^{-1}\\mathcal{F}$ is the same", "as $f^{-1}$ of the underlying sheaf of sets of $\\mathcal{F}$, see", "Sites, Section \\ref{sites-section-sheaves-algebraic-structures}.", "See also", "Modules on Sites, Lemma \\ref{sites-modules-lemma-flat-pullback-exact}.", "In the literature (1) and (2) are sometimes deduced from (3) via", "Theorem \\ref{theorem-exactness-stalks}.", "\\medskip\\noindent", "Part (3) is a general fact about stalks of pullbacks, see", "Sites, Lemma \\ref{sites-lemma-point-morphism-sites}.", "We will also prove (3) directly as follows. Note that by", "Lemma \\ref{lemma-stalk-exact}", "taking stalks commutes with sheafification.", "Now recall that $f^{-1}\\mathcal{G}$ is the sheaf", "associated to the presheaf", "$$", "U \\longrightarrow \\colim_{U \\to X \\times_Y V} \\mathcal{G}(V),", "$$", "see Equation (\\ref{equation-pullback}).", "Thus we have", "\\begin{align*}", "(f^{-1}\\mathcal{G})_{\\overline{x}}", "& = \\colim_{(U, \\overline{u})} f^{-1}\\mathcal{G}(U) \\\\", "& = \\colim_{(U, \\overline{u})}", "\\colim_{a : U \\to X \\times_Y V} \\mathcal{G}(V) \\\\", "& = \\colim_{(V, \\overline{v})} \\mathcal{G}(V) \\\\", "& = \\mathcal{G}_{\\overline{y}}", "\\end{align*}", "in the third equality the pair $(U, \\overline{u})$ and the map", "$a : U \\to X \\times_Y V$ corresponds to the pair $(V, a \\circ \\overline{u})$.", "\\medskip\\noindent", "Part (4) can be proved in a similar manner by identifying the colimits", "which define $f^{-1}h_V$. Or you can use", "Yoneda's lemma (Categories, Lemma \\ref{categories-lemma-yoneda})", "and the functorial equalities", "$$", "\\Mor_{\\Sh(X_\\etale)}(f^{-1}h_V, \\mathcal{F}) =", "\\Mor_{\\Sh(Y_\\etale)}(h_V, f_*\\mathcal{F}) =", "f_*\\mathcal{F}(V) = \\mathcal{F}(X \\times_Y V)", "$$", "combined with the fact that representable presheaves are sheaves. See also", "Sites, Lemma \\ref{sites-lemma-pullback-representable-sheaf}", "for a completely general result." ], "refs": [ "sites-proposition-get-morphism", "sites-modules-lemma-flat-pullback-exact", "etale-cohomology-theorem-exactness-stalks", "sites-lemma-point-morphism-sites", "etale-cohomology-lemma-stalk-exact", "categories-lemma-yoneda", "sites-lemma-pullback-representable-sheaf" ], "ref_ids": [ 8641, 14223, 6376, 8603, 6425, 12203, 8524 ] } ], "ref_ids": [ 12370 ] }, { "id": 6437, "type": "theorem", "label": "etale-cohomology-lemma-where-sections-are-equal", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-where-sections-are-equal", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ be a sheaf of sets on $S_\\etale$.", "Let $s, t \\in \\mathcal{F}(S)$. Then there exists an open $W \\subset S$", "characterized by the following property: A morphism $f : T \\to S$", "factors through $W$ if and only if $s|_T = t|_T$ (restriction is", "pullback by $f_{small}$)." ], "refs": [], "proofs": [ { "contents": [ "Consider the presheaf which assigns to $U \\in \\Ob(S_\\etale)$ the empty set", "if $s|_U \\not = t|_U$ and a singleton else. It is clear that this is", "a subsheaf of the final object of $\\Sh(S_\\etale)$. By", "Lemma \\ref{lemma-support-subsheaf-final}", "we find an open $W \\subset S$ representing this presheaf.", "For a geometric point $\\overline{x}$ of $S$ we see that $\\overline{x} \\in W$", "if and only if the stalks of $s$ and $t$ at $\\overline{x}$ agree.", "By the description of stalks of pullbacks in", "Lemma \\ref{lemma-stalk-pullback}", "we see that $W$ has the desired property." ], "refs": [ "etale-cohomology-lemma-support-subsheaf-final", "etale-cohomology-lemma-stalk-pullback" ], "ref_ids": [ 6428, 6436 ] } ], "ref_ids": [] }, { "id": 6438, "type": "theorem", "label": "etale-cohomology-lemma-describe-pullback", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-describe-pullback", "contents": [ "Let $S$ be a scheme. Let $\\tau \\in \\{Zariski, \\etale\\}$. Consider the morphism", "$$", "\\pi_S : (\\Sch/S)_\\tau \\longrightarrow S_\\tau", "$$", "of Topologies, Lemma \\ref{topologies-lemma-at-the-bottom} or", "\\ref{topologies-lemma-at-the-bottom-etale}. Let $\\mathcal{F}$ be a sheaf on", "$S_\\tau$. Then $\\pi_S^{-1}\\mathcal{F}$ is given by the rule", "$$", "(\\pi_S^{-1}\\mathcal{F})(T) = \\Gamma(T_\\tau, f_{small}^{-1}\\mathcal{F})", "$$", "where $f : T \\to S$. Moreover, $\\pi_S^{-1}\\mathcal{F}$ satisfies the", "sheaf condition with respect to fpqc coverings." ], "refs": [ "topologies-lemma-at-the-bottom", "topologies-lemma-at-the-bottom-etale" ], "proofs": [ { "contents": [ "Observe that we have a morphism $i_f : \\Sh(T_\\tau) \\to \\Sh(\\Sch/S)_\\tau)$", "such that $\\pi_S \\circ i_f = f_{small}$ as morphisms", "$T_\\tau \\to S_\\tau$, see", "Topologies, Lemmas \\ref{topologies-lemma-put-in-T},", "\\ref{topologies-lemma-morphism-big-small},", "\\ref{topologies-lemma-put-in-T-etale}, and", "\\ref{topologies-lemma-morphism-big-small-etale}.", "Since pullback is transitive we see that", "$i_f^{-1} \\pi_S^{-1}\\mathcal{F} = f_{small}^{-1}\\mathcal{F}$ as desired.", "\\medskip\\noindent", "Let $\\{g_i : T_i \\to T\\}_{i \\in I}$ be an fpqc covering.", "The final statement means the following: Given a sheaf $\\mathcal{G}$", "on $T_\\tau$ and given sections", "$s_i \\in \\Gamma(T_i, g_{i, small}^{-1}\\mathcal{G})$ whose pullbacks", "to $T_i \\times_T T_j$ agree, there is a unique section $s$ of $\\mathcal{G}$", "over $T$ whose pullback to $T_i$ agrees with $s_i$.", "\\medskip\\noindent", "Let $V \\to T$ be an object of $T_\\tau$ and let $t \\in \\mathcal{G}(V)$.", "For every $i$ there is a largest open $W_i \\subset T_i \\times_T V$", "such that the pullbacks of $s_i$ and $t$ agree as sections of the pullback", "of $\\mathcal{G}$ to $W_i \\subset T_i \\times_T V$, see", "Lemma \\ref{lemma-where-sections-are-equal}.", "Because $s_i$ and $s_j$ agree over $T_i \\times_T T_j$ we find", "that $W_i$ and $W_j$ pullback to the same open over", "$T_i \\times_T T_j \\times_T V$. By", "Descent, Lemma \\ref{descent-lemma-open-fpqc-covering}", "we find an open $W \\subset V$ whose inverse image to $T_i \\times_T V$", "recovers $W_i$. ", "\\medskip\\noindent", "By construction of $g_{i, small}^{-1}\\mathcal{G}$ there exists", "a $\\tau$-covering $\\{T_{ij} \\to T_i\\}_{j \\in J_i}$, for each $j$ an", "open immersion or \\'etale morphism $V_{ij} \\to T$, a section", "$t_{ij} \\in \\mathcal{G}(V_{ij})$, and commutative diagrams", "$$", "\\xymatrix{", "T_{ij} \\ar[r] \\ar[d] & V_{ij} \\ar[d] \\\\", "T_i \\ar[r] & T", "}", "$$", "such that $s_i|_{T_{ij}}$ is the pullback of $t_{ij}$. In other words,", "after replacing the covering $\\{T_i \\to T\\}$ by $\\{T_{ij} \\to T\\}$", "we may assume there are factorizations $T_i \\to V_i \\to T$ with", "$V_i \\in \\Ob(T_\\tau)$ and sections $t_i \\in \\mathcal{G}(V_i)$", "pulling back to $s_i$ over $T_i$.", "By the result of the previous paragraph we find opens $W_i \\subset V_i$", "such that $t_i|_{W_i}$ ``agrees with'' every $s_j$ over $T_j \\times_T W_i$.", "Note that $T_i \\to V_i$ factors through $W_i$.", "Hence $\\{W_i \\to T\\}$ is a $\\tau$-covering and the lemma is proven." ], "refs": [ "topologies-lemma-put-in-T", "topologies-lemma-morphism-big-small", "topologies-lemma-put-in-T-etale", "topologies-lemma-morphism-big-small-etale", "etale-cohomology-lemma-where-sections-are-equal", "descent-lemma-open-fpqc-covering" ], "ref_ids": [ 12438, 12441, 12452, 12455, 6437, 14637 ] } ], "ref_ids": [ 12439, 12453 ] }, { "id": 6439, "type": "theorem", "label": "etale-cohomology-lemma-sections-upstairs", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-sections-upstairs", "contents": [ "Let $S$ be a scheme. Let $f : T \\to S$ be a morphism such that", "\\begin{enumerate}", "\\item $f$ is flat and quasi-compact, and", "\\item the geometric fibres of $f$ are connected.", "\\end{enumerate}", "Let $\\mathcal{F}$ be a sheaf on $S_\\etale$.", "Then $\\Gamma(S, \\mathcal{F}) = \\Gamma(T, f^{-1}_{small}\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "There is a canonical map", "$\\Gamma(S, \\mathcal{F}) \\to \\Gamma(T, f_{small}^{-1}\\mathcal{F})$.", "Since $f$ is surjective (because its fibres are connected) we see that", "this map is injective.", "\\medskip\\noindent", "To show that the map is surjective, let", "$\\alpha \\in \\Gamma(T, f_{small}^{-1}\\mathcal{F})$.", "Since $\\{T \\to S\\}$ is an fpqc covering we can use", "Lemma \\ref{lemma-describe-pullback} to see that suffices to prove that", "$\\alpha$ pulls back to the same section over $T \\times_S T$ by the", "two projections. Let $\\overline{s} \\to S$ be a geometric point.", "It suffices to show the agreement holds over $(T \\times_S T)_{\\overline{s}}$", "as every geometric point of $T \\times_S T$ is contained in one of", "these geometric fibres. In other words, we are trying to show that", "$\\alpha|_{T_{\\overline{s}}}$ pulls back to the same section over", "$$", "(T \\times_S T)_{\\overline{s}} =", "T_{\\overline{s}} \\times_{\\overline{s}} T_{\\overline{s}}", "$$", "by the two projections to $T_{\\overline{s}}$.", "However, since $\\mathcal{F}|_{T_{\\overline{s}}}$ is the", "pullback of $\\mathcal{F}|_{\\overline{s}}$ it is a constant sheaf", "with value $\\mathcal{F}_{\\overline{s}}$. Since $T_{\\overline{s}}$", "is connected by assumption, any section of a constant sheaf is constant.", "Hence $\\alpha|_{T_{\\overline{s}}}$ corresponds to an element", "of $\\mathcal{F}_{\\overline{s}}$. Thus the two pullbacks to", "$(T \\times_S T)_{\\overline{s}}$ both", "correspond to this same element and we conclude." ], "refs": [ "etale-cohomology-lemma-describe-pullback" ], "ref_ids": [ 6438 ] } ], "ref_ids": [] }, { "id": 6440, "type": "theorem", "label": "etale-cohomology-lemma-sections-upstairs-submersive", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-sections-upstairs-submersive", "contents": [ "Let $S$ be a scheme. Let $f : X \\to S$ be a morphism such that", "\\begin{enumerate}", "\\item $f$ is submersive, and", "\\item the geometric fibres of $f$ are connected.", "\\end{enumerate}", "Let $\\mathcal{F}$ be a sheaf on $S_\\etale$.", "Then $\\Gamma(S, \\mathcal{F}) = \\Gamma(X, f^{-1}_{small}\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "There is a canonical map", "$\\Gamma(S, \\mathcal{F}) \\to \\Gamma(X, f_{small}^{-1}\\mathcal{F})$.", "Since $f$ is surjective (because its fibres are connected) we see that", "this map is injective.", "\\medskip\\noindent", "To show that the map is surjective, let", "$\\tau \\in \\Gamma(X, f_{small}^{-1}\\mathcal{F})$.", "It suffices to find", "an \\'etale covering $\\{U_i \\to S\\}$ and", "sections $\\sigma_i \\in \\mathcal{F}(U_i)$", "such that $\\sigma_i$ pulls back to $\\tau|_{X \\times_S U_i}$.", "Namely, the injectivity shown above guarantees", "that $\\sigma_i$ and $\\sigma_j$ restrict to the same", "section of $\\mathcal{F}$ over $U_i \\times_S U_j$.", "Thus we obtain a unique section $\\sigma \\in \\mathcal{F}(S)$", "which restricts to $\\sigma_i$ over $U_i$.", "Then the pullback of $\\sigma$ to $X$ is $\\tau$", "because this is true locally.", "\\medskip\\noindent", "Let $\\overline{x}$ be a geometric point of $X$ with image $\\overline{s}$", "in $S$. Consider the image of $\\tau$ in the stalk", "$$", "(f_{small}^{-1}\\mathcal{F})_{\\overline{x}} = \\mathcal{F}_{\\overline{s}}", "$$", "See Lemma \\ref{lemma-stalk-pullback}.", "We can find an \\'etale neighbourhood $U \\to S$ of $\\overline{s}$", "and a section $\\sigma \\in \\mathcal{F}(U)$ mapping to this image", "in the stalk. Thus after replacing $S$ by $U$ and $X$ by $X \\times_S U$", "we may assume there exits a section $\\sigma$ of $\\mathcal{F}$ over $S$", "whose image in $(f_{small}^{-1}\\mathcal{F})_{\\overline{x}}$ is the", "same as $\\tau$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-where-sections-are-equal}", "there exists a maximal open $W \\subset X$ such that", "$f_{small}^{-1}\\sigma$ and $\\tau$ agree over $W$", "and the formation of $W$ commutes with further pullback.", "Observe that the pullback of $\\mathcal{F}$ to the", "geometric fibre $X_{\\overline{s}}$ is the pullback", "of $\\mathcal{F}_{\\overline{s}}$ viewed as a sheaf on", "$\\overline{s}$ by $X_{\\overline{s}} \\to \\overline{s}$.", "Hence we see that $\\tau$ and $\\sigma$ give sections", "of the constant sheaf with value $\\mathcal{F}_{\\overline{s}}$", "on $X_{\\overline{s}}$ which agree in one point. Since", "$X_{\\overline{s}}$ is connected by assumption, we conclude", "that $W$ contains $X_s$. The same argument for different", "geometric fibres shows that $W$ contains every fibre it meets.", "Since $f$ is submersive, we conclude that $W$ is the inverse", "image of an open neighbourhood of $s$ in $S$.", "This finishes the proof." ], "refs": [ "etale-cohomology-lemma-stalk-pullback", "etale-cohomology-lemma-where-sections-are-equal" ], "ref_ids": [ 6436, 6437 ] } ], "ref_ids": [] }, { "id": 6441, "type": "theorem", "label": "etale-cohomology-lemma-sections-base-field-extension", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-sections-base-field-extension", "contents": [ "Let $k \\subset K$ be an extension of fields with $k$ separably", "algebraically closed. Let $S$ be a scheme over $k$. Denote", "$p : S_K = S \\times_{\\Spec(k)} \\Spec(K) \\to S$ the projection.", "Let $\\mathcal{F}$ be a sheaf on $S_\\etale$.", "Then $\\Gamma(S, \\mathcal{F}) = \\Gamma(S_K, p^{-1}_{small}\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-sections-upstairs}. Namely, it is clear", "that $p$ is flat and quasi-compact as the base change of", "$\\Spec(K) \\to \\Spec(k)$. On the other hand, if $\\overline{s} : \\Spec(L) \\to S$", "is a geometric point, then the fibre of $p$ over $\\overline{s}$", "is the spectrum of $K \\otimes_k L$ which is irreducible hence connected by", "Algebra, Lemma \\ref{algebra-lemma-separably-closed-irreducible}." ], "refs": [ "etale-cohomology-lemma-sections-upstairs", "algebra-lemma-separably-closed-irreducible" ], "ref_ids": [ 6439, 588 ] } ], "ref_ids": [] }, { "id": 6442, "type": "theorem", "label": "etale-cohomology-lemma-morphism-locally-ringed", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-morphism-locally-ringed", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "The morphism of ringed sites $(f_{small}, f_{small}^\\sharp)$", "associated to $f$ is a morphism of locally ringed sites, see", "Modules on Sites,", "Definition \\ref{sites-modules-definition-morphism-locally-ringed-topoi}." ], "refs": [ "sites-modules-definition-morphism-locally-ringed-topoi" ], "proofs": [ { "contents": [ "Note that the assertion makes sense since we have seen that", "$(X_\\etale, \\mathcal{O}_{X_\\etale})$ and", "$(Y_\\etale, \\mathcal{O}_{Y_\\etale})$", "are locally ringed sites, see", "Lemma \\ref{lemma-etale-site-locally-ringed}.", "Moreover, we know that $X_\\etale$ has enough points, see", "Theorem \\ref{theorem-exactness-stalks}", "and", "Remarks \\ref{remarks-enough-points}.", "Hence it suffices to prove that $(f_{small}, f_{small}^\\sharp)$", "satisfies condition (3) of", "Modules on Sites,", "Lemma \\ref{sites-modules-lemma-locally-ringed-morphism}.", "To see this take a point $p$ of $X_\\etale$. By", "Lemma \\ref{lemma-points-small-etale-site}", "$p$ corresponds to a geometric point $\\overline{x}$ of $X$.", "By", "Lemma \\ref{lemma-stalk-pullback}", "the point $q = f_{small} \\circ p$ corresponds to the", "geometric point $\\overline{y} = f \\circ \\overline{x}$ of $Y$.", "Hence the assertion we have to prove is that the induced map", "of stalks", "$$", "\\mathcal{O}_{Y, \\overline{y}} \\longrightarrow \\mathcal{O}_{X, \\overline{x}}", "$$", "is a local ring map. Suppose that $a \\in \\mathcal{O}_{Y, \\overline{y}}$", "is an element of the left hand side which maps to an element of the maximal", "ideal of the right hand side. Suppose that $a$ is the equivalence class", "of a triple $(V, \\overline{v}, a)$ with $V \\to Y$ \\'etale,", "$\\overline{v} : \\overline{x} \\to V$ over $Y$, and $a \\in \\mathcal{O}(V)$.", "It maps to the equivalence class of", "$(X \\times_Y V, \\overline{x} \\times \\overline{v}, \\text{pr}_V^\\sharp(a))$", "in the local ring $\\mathcal{O}_{X, \\overline{x}}$. But it is clear that", "being in the maximal ideal means that pulling back $\\text{pr}_V^\\sharp(a)$", "to an element of $\\kappa(\\overline{x})$ gives zero. Hence also pulling back", "$a$ to $\\kappa(\\overline{x})$ is zero. Which means that $a$ lies in the", "maximal ideal of $\\mathcal{O}_{Y, \\overline{y}}$." ], "refs": [ "etale-cohomology-lemma-etale-site-locally-ringed", "etale-cohomology-theorem-exactness-stalks", "etale-cohomology-remarks-enough-points", "sites-modules-lemma-locally-ringed-morphism", "etale-cohomology-lemma-points-small-etale-site", "etale-cohomology-lemma-stalk-pullback" ], "ref_ids": [ 6435, 6376, 6798, 14258, 6426, 6436 ] } ], "ref_ids": [ 14304 ] }, { "id": 6443, "type": "theorem", "label": "etale-cohomology-lemma-2-morphism", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-2-morphism", "contents": [ "Let $X$, $Y$ be schemes. Let $f : X \\to Y$ be a morphism of schemes.", "Let $t$ be a $2$-morphism from $(f_{small}, f_{small}^\\sharp)$ to itself, see", "Modules on Sites,", "Definition \\ref{sites-modules-definition-2-morphism-ringed-topoi}.", "Then $t = \\text{id}$." ], "refs": [ "sites-modules-definition-2-morphism-ringed-topoi" ], "proofs": [ { "contents": [ "This means that $t : f^{-1}_{small} \\to f^{-1}_{small}$", "is a transformation of functors such that the diagram", "$$", "\\xymatrix{", "f_{small}^{-1}\\mathcal{O}_Y", "\\ar[rd]_{f_{small}^\\sharp} & &", "f_{small}^{-1}\\mathcal{O}_Y \\ar[ll]^t \\ar[ld]^{f_{small}^\\sharp} \\\\", "& \\mathcal{O}_X", "}", "$$", "is commutative. Suppose $V \\to Y$ is \\'etale with $V$ affine. By", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-affine-finite-type-over-S}", "we may choose an immersion $i : V \\to \\mathbf{A}^n_Y$ over $Y$.", "In terms of sheaves this means that $i$ induces an injection", "$h_i : h_V \\to \\prod_{j = 1, \\ldots, n} \\mathcal{O}_Y$ of sheaves.", "The base change $i'$ of $i$ to $X$ is an immersion", "(Schemes, Lemma \\ref{schemes-lemma-base-change-immersion}).", "Hence $i' : X \\times_Y V \\to \\mathbf{A}^n_X$ is an immersion, which", "in turn means that", "$h_{i'} : h_{X \\times_Y V} \\to \\prod_{j = 1, \\ldots, n} \\mathcal{O}_X$", "is an injection of sheaves.", "Via the identification $f_{small}^{-1}h_V = h_{X \\times_Y V}$ of", "Lemma \\ref{lemma-stalk-pullback}", "the map $h_{i'}$ is equal to", "$$", "\\xymatrix{", "f_{small}^{-1}h_V \\ar[r]^-{f^{-1}h_i} &", "\\prod_{j = 1, \\ldots, n} f_{small}^{-1}\\mathcal{O}_Y", "\\ar[r]^{\\prod f^\\sharp} &", "\\prod_{j = 1, \\ldots, n} \\mathcal{O}_X", "}", "$$", "(verification omitted). This means that the map", "$t : f_{small}^{-1}h_V \\to f_{small}^{-1}h_V$", "fits into the commutative diagram", "$$", "\\xymatrix{", "f_{small}^{-1}h_V \\ar[r]^-{f^{-1}h_i} \\ar[d]^t &", "\\prod_{j = 1, \\ldots, n} f_{small}^{-1}\\mathcal{O}_Y", "\\ar[r]^-{\\prod f^\\sharp} \\ar[d]^{\\prod t} &", "\\prod_{j = 1, \\ldots, n} \\mathcal{O}_X \\ar[d]^{\\text{id}}\\\\", "f_{small}^{-1}h_V \\ar[r]^-{f^{-1}h_i} &", "\\prod_{j = 1, \\ldots, n} f_{small}^{-1}\\mathcal{O}_Y", "\\ar[r]^-{\\prod f^\\sharp} &", "\\prod_{j = 1, \\ldots, n} \\mathcal{O}_X", "}", "$$", "The commutativity of the right square holds by our assumption on $t$", "explained above.", "Since the composition of the horizontal arrows is injective", "by the discussion above we conclude that the left vertical arrow", "is the identity map as well. Any sheaf of sets on", "$Y_\\etale$ admits a surjection from a (huge) coproduct of sheaves", "of the form $h_V$ with $V$ affine (combine", "Lemma \\ref{lemma-alternative}", "with", "Sites, Lemma \\ref{sites-lemma-sheaf-coequalizer-representable}).", "Thus we conclude that $t : f_{small}^{-1} \\to f_{small}^{-1}$", "is the identity transformation as desired." ], "refs": [ "morphisms-lemma-quasi-affine-finite-type-over-S", "schemes-lemma-base-change-immersion", "etale-cohomology-lemma-stalk-pullback", "etale-cohomology-lemma-alternative", "sites-lemma-sheaf-coequalizer-representable" ], "ref_ids": [ 5392, 7695, 6436, 6413, 8520 ] } ], "ref_ids": [ 14281 ] }, { "id": 6444, "type": "theorem", "label": "etale-cohomology-lemma-faithful", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-faithful", "contents": [ "Let $X$, $Y$ be schemes.", "Any two morphisms $a, b : X \\to Y$ of schemes", "for which there exists a $2$-isomorphism", "$(a_{small}, a_{small}^\\sharp) \\cong (b_{small}, b_{small}^\\sharp)$", "in the $2$-category of ringed topoi are equal." ], "refs": [], "proofs": [ { "contents": [ "Let us argue this carefuly since it is a bit confusing.", "Let $t : a_{small}^{-1} \\to b_{small}^{-1}$ be the $2$-isomorphism.", "Consider any open $V \\subset Y$. Note that $h_V$ is a subsheaf of the", "final sheaf $*$. Thus both $a_{small}^{-1}h_V = h_{a^{-1}(V)}$", "and $b_{small}^{-1}h_V = h_{b^{-1}(V)}$ are subsheaves of the final sheaf.", "Thus the isomorphism", "$$", "t : a_{small}^{-1}h_V = h_{a^{-1}(V)} \\to b_{small}^{-1}h_V = h_{b^{-1}(V)}", "$$", "has to be the identity, and $a^{-1}(V) = b^{-1}(V)$.", "It follows that $a$ and $b$ are equal on underlying topological spaces.", "Next, take a section $f \\in \\mathcal{O}_Y(V)$. This determines and", "is determined by a map of sheaves of sets", "$f : h_V \\to \\mathcal{O}_Y$.", "Pull this back and apply $t$ to get a commutative diagram", "$$", "\\xymatrix{", "h_{b^{-1}(V)} \\ar@{=}[r] &", "b_{small}^{-1}h_V \\ar[d]^{b_{small}^{-1}(f)} & &", "a_{small}^{-1}h_V \\ar[d]^{a_{small}^{-1}(f)} \\ar[ll]^t &", "h_{a^{-1}(V)} \\ar@{=}[l]", "\\\\", "& b_{small}^{-1}\\mathcal{O}_Y", "\\ar[rd]_{b^\\sharp} & &", "a_{small}^{-1}\\mathcal{O}_Y \\ar[ll]^t \\ar[ld]^{a^\\sharp} \\\\", "& & \\mathcal{O}_X", "}", "$$", "where the triangle is commutative by definition of a $2$-isomorphism in", "Modules on Sites, Section \\ref{sites-modules-section-2-category}.", "Above we have seen that the composition of the top horizontal", "arrows comes from the identity $a^{-1}(V) = b^{-1}(V)$.", "Thus the commutativity of the diagram tells us that", "$a_{small}^\\sharp(f) = b_{small}^\\sharp(f)$ in", "$\\mathcal{O}_X(a^{-1}(V)) = \\mathcal{O}_X(b^{-1}(V))$.", "Since this holds for every open $V$ and every $f \\in \\mathcal{O}_Y(V)$", "we conclude that $a = b$ as morphisms of schemes." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6445, "type": "theorem", "label": "etale-cohomology-lemma-morphism-ringed-etale-topoi-affines", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-morphism-ringed-etale-topoi-affines", "contents": [ "Let $X$, $Y$ be affine schemes.", "Let", "$$", "(g, g^\\#) :", "(\\Sh(X_\\etale), \\mathcal{O}_X)", "\\longrightarrow", "(\\Sh(Y_\\etale), \\mathcal{O}_Y)", "$$", "be a morphism of locally ringed topoi. Then there exists a", "unique morphism of schemes $f : X \\to Y$ such that", "$(g, g^\\#)$ is $2$-isomorphic to $(f_{small}, f_{small}^\\sharp)$,", "see", "Modules on Sites,", "Definition \\ref{sites-modules-definition-2-morphism-ringed-topoi}." ], "refs": [ "sites-modules-definition-2-morphism-ringed-topoi" ], "proofs": [ { "contents": [ "In this proof we write $\\mathcal{O}_X$ for the structure sheaf", "of the small \\'etale site $X_\\etale$, and similarly for", "$\\mathcal{O}_Y$. Say $Y = \\Spec(B)$ and $X = \\Spec(A)$. Since", "$B = \\Gamma(Y_\\etale, \\mathcal{O}_Y)$,", "$A = \\Gamma(X_\\etale, \\mathcal{O}_X)$", "we see that $g^\\sharp$ induces a ring map $\\varphi : B \\to A$.", "Let $f = \\Spec(\\varphi) : X \\to Y$ be the corresponding morphism", "of affine schemes. We will show this $f$ does the job.", "\\medskip\\noindent", "Let $V \\to Y$ be an affine scheme \\'etale over $Y$. Thus we may write", "$V = \\Spec(C)$ with $C$ an \\'etale $B$-algebra. We can write", "$$", "C = B[x_1, \\ldots, x_n]/(P_1, \\ldots, P_n)", "$$", "with $P_i$ polynomials such that $\\Delta = \\det(\\partial P_i/ \\partial x_j)$", "is invertible in $C$, see for example", "Algebra, Lemma \\ref{algebra-lemma-etale-standard-smooth}.", "If $T$ is a scheme over $Y$, then a $T$-valued point of $V$ is given by", "$n$ sections of $\\Gamma(T, \\mathcal{O}_T)$ which satisfy the polynomial", "equations $P_1 = 0, \\ldots, P_n = 0$. In other words, the sheaf $h_V$", "on $Y_\\etale$ is the equalizer of the two maps", "$$", "\\xymatrix{", "\\prod\\nolimits_{i = 1, \\ldots, n} \\mathcal{O}_Y", "\\ar@<1ex>[r]^a \\ar@<-1ex>[r]_b &", "\\prod\\nolimits_{j = 1, \\ldots, n} \\mathcal{O}_Y", "}", "$$", "where $b(h_1, \\ldots, h_n) = 0$ and", "$a(h_1, \\ldots, h_n) =", "(P_1(h_1, \\ldots, h_n), \\ldots, P_n(h_1, \\ldots, h_n))$.", "Since $g^{-1}$ is exact we conclude that the top row of the", "following solid commutative diagram is an equalizer diagram as well:", "$$", "\\xymatrix{", "g^{-1}h_V \\ar[r] \\ar@{..>}[d] &", "\\prod\\nolimits_{i = 1, \\ldots, n} g^{-1}\\mathcal{O}_Y", "\\ar@<1ex>[r]^{g^{-1}a} \\ar@<-1ex>[r]_{g^{-1}b} \\ar[d]^{\\prod g^\\sharp} &", "\\prod\\nolimits_{j = 1, \\ldots, n} g^{-1}\\mathcal{O}_Y \\ar[d]^{\\prod g^\\sharp}\\\\", "h_{X \\times_Y V} \\ar[r] &", "\\prod\\nolimits_{i = 1, \\ldots, n} \\mathcal{O}_X", "\\ar@<1ex>[r]^{a'} \\ar@<-1ex>[r]_{b'} &", "\\prod\\nolimits_{j = 1, \\ldots, n} \\mathcal{O}_X \\\\", "}", "$$", "Here $b'$ is the zero map and $a'$ is the map defined by the", "images $P'_i = \\varphi(P_i) \\in A[x_1, \\ldots, x_n]$ via the same", "rule", "$a'(h_1, \\ldots, h_n) =", "(P'_1(h_1, \\ldots, h_n), \\ldots, P'_n(h_1, \\ldots, h_n))$.", "that $a$ was defined by. The commutativity of the diagram follows from", "the fact that $\\varphi = g^\\sharp$ on global sections. The lower", "row is an equalizer diagram also, by exactly the same arguments as", "before since $X \\times_Y V$ is the affine scheme", "$\\Spec(A \\otimes_B C)$ and", "$A \\otimes_B C = A[x_1, \\ldots, x_n]/(P'_1, \\ldots, P'_n)$.", "Thus we obtain a unique dotted arrow", "$g^{-1}h_V \\to h_{X \\times_Y V}$ fitting into the diagram", "\\medskip\\noindent", "We claim that the map of sheaves $g^{-1}h_V \\to h_{X \\times_Y V}$", "is an isomorphism. Since the small \\'etale site of $X$ has enough points", "(Theorem \\ref{theorem-exactness-stalks})", "it suffices to prove this on stalks. Hence let $\\overline{x}$ be a", "geometric point of $X$, and denote $p$ the associate point of the", "small \\'etale topos of $X$. Set $q = g \\circ p$. This is a point of", "the small \\'etale topos of $Y$. By", "Lemma \\ref{lemma-points-small-etale-site}", "we see that $q$ corresponds to a geometric point $\\overline{y}$ of", "$Y$. Consider the map of stalks", "$$", "(g^\\sharp)_p :", "\\mathcal{O}_{Y, \\overline{y}} =", "\\mathcal{O}_{Y, q} =", "(g^{-1}\\mathcal{O}_Y)_p", "\\longrightarrow", "\\mathcal{O}_{X, p} =", "\\mathcal{O}_{X, \\overline{x}}", "$$", "Since $(g, g^\\sharp)$ is a morphism of {\\it locally} ringed topoi", "$(g^\\sharp)_p$ is a local ring homomorphism of strictly henselian", "local rings. Applying localization to the big commutative diagram above and", "Algebra, Lemma \\ref{algebra-lemma-strictly-henselian-solutions}", "we conclude that $(g^{-1}h_V)_p \\to (h_{X \\times_Y V})_p$ is an isomorphism", "as desired.", "\\medskip\\noindent", "We claim that the isomorphisms $g^{-1}h_V \\to h_{X \\times_Y V}$ are", "functorial. Namely, suppose that $V_1 \\to V_2$ is a morphism of affine", "schemes \\'etale over $Y$. Write", "$V_i = \\Spec(C_i)$ with", "$$", "C_i = B[x_{i, 1}, \\ldots, x_{i, n_i}]/(P_{i, 1}, \\ldots, P_{i, n_i})", "$$", "The morphism $V_1 \\to V_2$ is given by a $B$-algebra map $C_2 \\to C_1$", "which in turn is given by some polynomials", "$Q_j \\in B[x_{1, 1}, \\ldots, x_{1, n_1}]$ for $j = 1, \\ldots, n_2$.", "Then it is an easy matter to show that the diagram of sheaves", "$$", "\\xymatrix{", "h_{V_1} \\ar[d] \\ar[r] & \\prod_{i = 1, \\ldots, n_1} \\mathcal{O}_Y", "\\ar[d]^{Q_1, \\ldots, Q_{n_2}}\\\\", "h_{V_2} \\ar[r] & \\prod_{i = 1, \\ldots, n_2} \\mathcal{O}_Y", "}", "$$", "is commutative, and pulling back to $X_\\etale$ we obtain the", "solid commutative diagram", "$$", "\\xymatrix{", "g^{-1}h_{V_1} \\ar@{..>}[dd] \\ar[rrd] \\ar[r] &", "\\prod_{i = 1, \\ldots, n_1} g^{-1}\\mathcal{O}_Y", "\\ar[dd]^{g^\\sharp}", "\\ar[rrd]^{Q_1, \\ldots, Q_{n_2}} \\\\", "& & g^{-1}h_{V_2} \\ar@{..>}[dd] \\ar[r] &", "\\prod_{i = 1, \\ldots, n_2} g^{-1}\\mathcal{O}_Y", "\\ar[dd]^{g^\\sharp} \\\\", "h_{X \\times_Y V_1} \\ar[r] \\ar[rrd] &", "\\prod\\nolimits_{i = 1, \\ldots, n_1} \\mathcal{O}_X", "\\ar[rrd]^{Q'_1, \\ldots, Q'_{n_2}} \\\\", "& & h_{X \\times_Y V_2} \\ar[r] &", "\\prod\\nolimits_{i = 1, \\ldots, n_2} \\mathcal{O}_X", "}", "$$", "where $Q'_j \\in A[x_{1, 1}, \\ldots, x_{1, n_1}]$ is the image of", "$Q_j$ via $\\varphi$. Since the dotted arrows exist, make the", "two squares commute, and the horizontal arrows are injective", "we see that the whole diagram commutes. This proves functoriality", "(and also that the construction of $g^{-1}h_V \\to h_{X \\times_Y V}$", "is independent of the choice of the presentation, although we", "strictly speaking do not need to show this).", "\\medskip\\noindent", "At this point we are able to show that $f_{small, *} \\cong g_*$.", "Namely, let $\\mathcal{F}$ be a sheaf on $X_\\etale$. For every", "$V \\in \\Ob(X_\\etale)$ affine we have", "\\begin{align*}", "(g_*\\mathcal{F})(V)", "& =", "\\Mor_{\\Sh(Y_\\etale)}(h_V, g_*\\mathcal{F}) \\\\", "& =", "\\Mor_{\\Sh(X_\\etale)}(g^{-1}h_V, \\mathcal{F}) \\\\", "& =", "\\Mor_{\\Sh(X_\\etale)}(h_{X \\times_Y V}, \\mathcal{F}) \\\\", "& =", "\\mathcal{F}(X \\times_Y V) \\\\", "& =", "f_{small, *}\\mathcal{F}(V)", "\\end{align*}", "where in the third equality we use the isomorphism", "$g^{-1}h_V \\cong h_{X \\times_Y V}$ constructed above. These isomorphisms", "are clearly functorial in $\\mathcal{F}$ and functorial in $V$", "as the isomorphisms $g^{-1}h_V \\cong h_{X \\times_Y V}$ are functorial.", "Now any sheaf on $Y_\\etale$ is determined by the restriction", "to the subcategory of affine schemes", "(Lemma \\ref{lemma-alternative}),", "and hence we obtain an isomorphism of functors $f_{small, *} \\cong g_*$", "as desired.", "\\medskip\\noindent", "Finally, we have to check that, via the isomorphism", "$f_{small, *} \\cong g_*$ above, the maps $f_{small}^\\sharp$ and", "$g^\\sharp$ agree. By construction this is already the case for the", "global sections of $\\mathcal{O}_Y$, i.e., for the elements of $B$.", "We only need to check the result on", "sections over an affine $V$ \\'etale over $Y$ (by", "Lemma \\ref{lemma-alternative}", "again). Writing", "$V = \\Spec(C)$, $C = B[x_i]/(P_j)$ as before it suffices", "to check that the coordinate functions $x_i$ are mapped to", "the same sections of $\\mathcal{O}_X$ over $X \\times_Y V$.", "And this is exactly what it means that the diagram", "$$", "\\xymatrix{", "g^{-1}h_V \\ar[r] \\ar@{..>}[d] &", "\\prod\\nolimits_{i = 1, \\ldots, n} g^{-1}\\mathcal{O}_Y", "\\ar[d]^{\\prod g^\\sharp} \\\\", "h_{X \\times_Y V} \\ar[r] &", "\\prod\\nolimits_{i = 1, \\ldots, n} \\mathcal{O}_X", "}", "$$", "commutes. Thus the lemma is proved." ], "refs": [ "algebra-lemma-etale-standard-smooth", "etale-cohomology-theorem-exactness-stalks", "etale-cohomology-lemma-points-small-etale-site", "algebra-lemma-strictly-henselian-solutions", "etale-cohomology-lemma-alternative", "etale-cohomology-lemma-alternative" ], "ref_ids": [ 1230, 6376, 6426, 1285, 6413, 6413 ] } ], "ref_ids": [ 14281 ] }, { "id": 6446, "type": "theorem", "label": "etale-cohomology-lemma-property-A-implies", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-property-A-implies", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume (A).", "\\begin{enumerate}", "\\item", "$f_{small, *} :", "\\textit{Ab}(X_\\etale)", "\\to", "\\textit{Ab}(Y_\\etale)$", "reflects injections and surjections,", "\\item $f_{small}^{-1}f_{small, *}\\mathcal{F} \\to \\mathcal{F}$", "is surjective for any abelian sheaf $\\mathcal{F}$ on $X_\\etale$,", "\\item", "$f_{small, *} :", "\\textit{Ab}(X_\\etale)", "\\to", "\\textit{Ab}(Y_\\etale)$", "is faithful.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$.", "Let $U$ be an object of $X_\\etale$. By assumption we can find a", "covering $\\{W_i \\to U\\}$ in $X_\\etale$ such that each $W_i$ is", "an open and closed subscheme of $X \\times_Y V_i$ for some object", "$V_i$ of $Y_\\etale$. The sheaf condition shows that", "$$", "\\mathcal{F}(U) \\subset \\prod \\mathcal{F}(W_i)", "$$", "and that $\\mathcal{F}(W_i)$ is a direct summand of", "$\\mathcal{F}(X \\times_Y V_i) = f_{small, *}\\mathcal{F}(V_i)$.", "Hence it is clear that $f_{small, *}$ reflects injections.", "\\medskip\\noindent", "Next, suppose that $a : \\mathcal{G} \\to \\mathcal{F}$ is a map of", "abelian sheaves such that $f_{small, *}a$ is surjective. Let", "$s \\in \\mathcal{F}(U)$ with $U$ as above. With $W_i$, $V_i$ as", "above we see that it suffices to show that $s|_{W_i}$ is \\'etale locally", "the image of a section of $\\mathcal{G}$ under $a$. Since $\\mathcal{F}(W_i)$", "is a direct summand of $\\mathcal{F}(X \\times_Y V_i)$", "it suffices to show that for any $V \\in \\Ob(Y_\\etale)$", "any element $s \\in \\mathcal{F}(X \\times_Y V)$", "is \\'etale locally on $X \\times_Y V$ the image of a section of", "$\\mathcal{G}$ under $a$. Since", "$\\mathcal{F}(X \\times_Y V) = f_{small, *}\\mathcal{F}(V)$", "we see by assumption that there exists a covering $\\{V_j \\to V\\}$ such that", "$s$ is the image of", "$s_j \\in f_{small, *}\\mathcal{G}(V_j) = \\mathcal{G}(X \\times_Y V_j)$.", "This proves $f_{small, *}$ reflects surjections.", "\\medskip\\noindent", "Parts (2), (3) follow formally from part (1), see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-reflect-surjections}." ], "refs": [ "sites-modules-lemma-reflect-surjections" ], "ref_ids": [ 14161 ] } ], "ref_ids": [] }, { "id": 6447, "type": "theorem", "label": "etale-cohomology-lemma-locally-quasi-finite-A", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-locally-quasi-finite-A", "contents": [ "Let $f : X \\to Y$ be a separated locally quasi-finite morphism of schemes.", "Then property (A) above holds." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be an \\'etale morphism and $u \\in U$.", "The geometric statement (A) reduces directly to the case where $U$ and $Y$", "are affine schemes. Denote $x \\in X$ and $y \\in Y$ the", "images of $u$. Since $X \\to Y$ is locally quasi-finite, and $U \\to X$ is", "locally quasi-finite (see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-locally-quasi-finite})", "we see that $U \\to Y$ is locally quasi-finite (see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-quasi-finite}).", "Moreover both $X \\to Y$ and $U \\to Y$ are separated. Thus", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant}", "applies to both morphisms. This means we may pick an \\'etale neighbourhood", "$(V, v) \\to (Y, y)$ such that", "$$", "X \\times_Y V = W \\amalg R, \\quad", "U \\times_Y V = W' \\amalg R'", "$$", "and points $w \\in W$, $w' \\in W'$ such that", "\\begin{enumerate}", "\\item $W$, $R$ are open and closed in $X \\times_Y V$,", "\\item $W'$, $R'$ are open and closed in $U \\times_Y V$,", "\\item $W \\to V$ and $W' \\to V$ are finite,", "\\item $w$, $w'$ map to $v$,", "\\item $\\kappa(v) \\subset \\kappa(w)$ and $\\kappa(v) \\subset \\kappa(w')$", "are purely inseparable, and", "\\item no other point of $W$ or $W'$ maps to $v$.", "\\end{enumerate}", "Here is a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] & U \\times_Y V \\ar[l] \\ar[d] & W' \\amalg R' \\ar[d] \\ar[l] \\\\", "X \\ar[d] & X \\times_Y V \\ar[l] \\ar[d] & W \\amalg R \\ar[l] \\\\", "Y & V \\ar[l]", "}", "$$", "After shrinking $V$ we may assume that $W'$ maps into $W$:", "just remove the image the inverse image of $R$ in $W'$; this is", "a closed set (as $W' \\to V$ is finite) not containing $v$.", "Then $W' \\to W$ is finite because both $W \\to V$ and $W' \\to V$ are finite.", "Hence $W' \\to W$ is finite \\'etale, and there is exactly one point in the", "fibre over $w$ with $\\kappa(w) = \\kappa(w')$. Hence $W' \\to W$ is an", "isomorphism in an open neighbourhood $W^\\circ$ of $w$, see", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-finite-etale-one-point}.", "Since $W \\to V$ is finite the image of $W \\setminus W^\\circ$ is a closed", "subset $T$ of $V$ not containing $v$. Thus after replacing $V$ by", "$V \\setminus T$ we may assume that $W' \\to W$ is an isomorphism.", "Now the decomposition $X \\times_Y V = W \\amalg R$ and the morphism", "$W \\to U$ are as desired and we win." ], "refs": [ "morphisms-lemma-etale-locally-quasi-finite", "morphisms-lemma-composition-quasi-finite", "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant", "etale-lemma-finite-etale-one-point" ], "ref_ids": [ 5363, 5232, 13896, 10707 ] } ], "ref_ids": [] }, { "id": 6448, "type": "theorem", "label": "etale-cohomology-lemma-integral-A", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-integral-A", "contents": [ "Let $f : X \\to Y$ be an integral morphism of schemes.", "Then property (A) holds." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be \\'etale, and let $u \\in U$ be a point.", "We have to find $V \\to Y$ \\'etale, a disjoint union decomposition", "$X \\times_Y V = W \\amalg W'$ and an $X$-morphism $W \\to U$", "with $u$ in the image. We may shrink $U$ and $Y$ and assume", "$U$ and $Y$ are affine. In this case also $X$ is affine, since", "an integral morphism is affine by definition. Write $Y = \\Spec(A)$,", "$X = \\Spec(B)$ and $U = \\Spec(C)$. Then $A \\to B$ is an", "integral ring map, and $B \\to C$ is an \\'etale ring map. By", "Algebra, Lemma \\ref{algebra-lemma-etale}", "we can find a finite $A$-subalgebra $B' \\subset B$ and an \\'etale ring", "map $B' \\to C'$ such that $C = B \\otimes_{B'} C'$. Thus the question", "reduces to the \\'etale morphism", "$U' = \\Spec(C') \\to X' = \\Spec(B')$", "over the finite morphism $X' \\to Y$. In this case the result follows from", "Lemma \\ref{lemma-locally-quasi-finite-A}." ], "refs": [ "algebra-lemma-etale", "etale-cohomology-lemma-locally-quasi-finite-A" ], "ref_ids": [ 1231, 6447 ] } ], "ref_ids": [] }, { "id": 6449, "type": "theorem", "label": "etale-cohomology-lemma-when-push-pull-surjective", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-when-push-pull-surjective", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Denote", "$f_{small} :", "\\Sh(X_\\etale)", "\\to", "\\Sh(Y_\\etale)$", "the associated morphism of small \\'etale topoi. Assume at least one", "of the following", "\\begin{enumerate}", "\\item $f$ is integral, or", "\\item $f$ is separated and locally quasi-finite.", "\\end{enumerate}", "Then the functor", "$f_{small, *} :", "\\textit{Ab}(X_\\etale)", "\\to", "\\textit{Ab}(Y_\\etale)$", "has the following properties", "\\begin{enumerate}", "\\item the map", "$f_{small}^{-1}f_{small, *}\\mathcal{F} \\to \\mathcal{F}$", "is always surjective,", "\\item $f_{small, *}$ is faithful, and", "\\item $f_{small, *}$ reflects injections and surjections.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemmas \\ref{lemma-locally-quasi-finite-A},", "\\ref{lemma-integral-A}, and", "\\ref{lemma-property-A-implies}." ], "refs": [ "etale-cohomology-lemma-locally-quasi-finite-A", "etale-cohomology-lemma-integral-A", "etale-cohomology-lemma-property-A-implies" ], "ref_ids": [ 6447, 6448, 6446 ] } ], "ref_ids": [] }, { "id": 6450, "type": "theorem", "label": "etale-cohomology-lemma-property-B-implies", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-property-B-implies", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume (B) holds.", "Then the functor", "$f_{small, *} :", "\\Sh(X_\\etale)", "\\to", "\\Sh(Y_\\etale)$", "transforms surjections into surjections." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Sites, Lemma \\ref{sites-lemma-weaker}." ], "refs": [ "sites-lemma-weaker" ], "ref_ids": [ 8619 ] } ], "ref_ids": [] }, { "id": 6451, "type": "theorem", "label": "etale-cohomology-lemma-simplify-B", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-simplify-B", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Suppose", "\\begin{enumerate}", "\\item $V \\to Y$ is an \\'etale morphism of schemes,", "\\item $\\{U_i \\to X \\times_Y V\\}$ is an \\'etale covering, and", "\\item $v \\in V$ is a point.", "\\end{enumerate}", "Assume that for any such data there exists an \\'etale neighbourhood", "$(V', v') \\to (V, v)$, a disjoint union decomposition", "$X \\times_Y V' = \\coprod W'_i$, and morphisms $W'_i \\to U_i$", "over $X \\times_Y V$. Then property (B) holds." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6452, "type": "theorem", "label": "etale-cohomology-lemma-finite-B", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-finite-B", "contents": [ "Let $f : X \\to Y$ be a finite morphism of schemes.", "Then property (B) holds." ], "refs": [], "proofs": [ { "contents": [ "Consider $V \\to Y$ \\'etale, $\\{U_i \\to X \\times_Y V\\}$ an \\'etale covering, and", "$v \\in V$. We have to find a $V' \\to V$ and decomposition and maps as in", "Lemma \\ref{lemma-simplify-B}.", "We may shrink $V$ and $Y$, hence we may assume that $V$ and $Y$ are affine.", "Since $X$ is finite over $Y$, this also implies that $X$ is affine.", "During the proof we may (finitely often) replace $(V, v)$ by an", "\\'etale neighbourhood $(V', v')$ and correspondingly the covering", "$\\{U_i \\to X \\times_Y V\\}$ by $\\{V' \\times_V U_i \\to X \\times_Y V'\\}$.", "\\medskip\\noindent", "Since $X \\times_Y V \\to V$ is finite there exist finitely", "many (pairwise distinct) points $x_1, \\ldots, x_n \\in X \\times_Y V$", "mapping to $v$. We may apply", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant}", "to $X \\times_Y V \\to V$ and the points $x_1, \\ldots, x_n$ lying over", "$v$ and find an \\'etale neighbourhood $(V', v') \\to (V, v)$", "such that", "$$", "X \\times_Y V' = R \\amalg \\coprod T_a", "$$", "with $T_a \\to V'$ finite with exactly one point $p_a$ lying over $v'$", "and moreover $\\kappa(v') \\subset \\kappa(p_a)$ purely inseparable, and", "such that $R \\to V'$ has empty fibre over $v'$.", "Because $X \\to Y$ is finite, also $R \\to V'$ is finite. Hence after", "shrinking $V'$ we may assume that $R = \\emptyset$. Thus we may", "assume that $X \\times_Y V = X_1 \\amalg \\ldots \\amalg X_n$ with", "exactly one point $x_l \\in X_l$ lying over $v$ with moreover", "$\\kappa(v) \\subset \\kappa(x_l)$ purely inseparable. Note that this", "property is preserved under refinement of the \\'etale neighbourhood", "$(V, v)$.", "\\medskip\\noindent", "For each $l$ choose an $i_l$ and a point $u_l \\in U_{i_l}$ mapping to $x_l$.", "Now we apply property (A) for the finite morphism", "$X \\times_Y V \\to V$ and the \\'etale", "morphisms $U_{i_l} \\to X \\times_Y V$ and the points $u_l$.", "This is permissible by", "Lemma \\ref{lemma-integral-A}", "This gives produces an \\'etale neighbourhood $(V', v') \\to (V, v)$", "and decompositions", "$$", "X \\times_Y V' = W_l \\amalg R_l", "$$", "and $X$-morphisms $a_l : W_l \\to U_{i_l}$ whose image contains $u_{i_l}$.", "Here is a picture:", "$$", "\\xymatrix{", "& & & U_{i_l} \\ar[d] & \\\\", "W_l \\ar[rrru] \\ar[r] & W_l \\amalg R_l \\ar@{=}[r] &", "X \\times_Y V' \\ar[r] \\ar[d] &", "X \\times_Y V \\ar[r] \\ar[d] & X \\ar[d] \\\\", "& & V' \\ar[r] & V \\ar[r] & Y", "}", "$$", "After replacing $(V, v)$ by $(V', v')$ we conclude that each", "$x_l$ is contained in an open and closed neighbourhood $W_l$ such that", "the inclusion morphism $W_l \\to X \\times_Y V$ factors through", "$U_i \\to X \\times_Y V$ for some $i$. Replacing $W_l$ by $W_l \\cap X_l$", "we see that these open and closed sets are disjoint and moreover", "that $\\{x_1, \\ldots, x_n\\} \\subset W_1 \\cup \\ldots \\cup W_n$.", "Since $X \\times_Y V \\to V$ is finite we may shrink $V$ and assume that", "$X \\times_Y V = W_1 \\amalg \\ldots \\amalg W_n$ as desired." ], "refs": [ "etale-cohomology-lemma-simplify-B", "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant", "etale-cohomology-lemma-integral-A" ], "ref_ids": [ 6451, 13896, 6448 ] } ], "ref_ids": [] }, { "id": 6453, "type": "theorem", "label": "etale-cohomology-lemma-integral-B", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-integral-B", "contents": [ "Let $f : X \\to Y$ be an integral morphism of schemes.", "Then property (B) holds." ], "refs": [], "proofs": [ { "contents": [ "Consider $V \\to Y$ \\'etale, $\\{U_i \\to X \\times_Y V\\}$ an \\'etale covering, and", "$v \\in V$. We have to find a $V' \\to V$ and decomposition and maps as in", "Lemma \\ref{lemma-simplify-B}.", "We may shrink $V$ and $Y$, hence we may assume that $V$ and $Y$ are affine.", "Since $X$ is integral over $Y$, this also implies that $X$ and", "$X \\times_Y V$ are affine. We may refine the covering", "$\\{U_i \\to X \\times_Y V\\}$, and hence we may assume that", "$\\{U_i \\to X \\times_Y V\\}_{i = 1, \\ldots, n}$ is a standard \\'etale covering.", "Write $Y = \\Spec(A)$, $X = \\Spec(B)$,", "$V = \\Spec(C)$, and $U_i = \\Spec(B_i)$.", "Then $A \\to B$ is an integral ring map, and $B \\otimes_A C \\to B_i$ are", "\\'etale ring maps. By", "Algebra, Lemma \\ref{algebra-lemma-etale}", "we can find a finite $A$-subalgebra $B' \\subset B$ and an \\'etale ring", "map $B' \\otimes_A C \\to B'_i$ for $i = 1, \\ldots, n$", "such that $B_i = B \\otimes_{B'} B'_i$. Thus the question", "reduces to the \\'etale covering", "$\\{\\Spec(B'_i) \\to X' \\times_Y V\\}_{i = 1, \\ldots, n}$", "with $X' = \\Spec(B')$ finite over $Y$.", "In this case the result follows from", "Lemma \\ref{lemma-finite-B}." ], "refs": [ "etale-cohomology-lemma-simplify-B", "algebra-lemma-etale", "etale-cohomology-lemma-finite-B" ], "ref_ids": [ 6451, 1231, 6452 ] } ], "ref_ids": [] }, { "id": 6454, "type": "theorem", "label": "etale-cohomology-lemma-what-integral", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-what-integral", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume $f$ is integral (for example finite).", "Then", "\\begin{enumerate}", "\\item $f_{small, *}$ transforms surjections into surjections (on sheaves", "of sets and on abelian sheaves),", "\\item $f_{small}^{-1}f_{small, *}\\mathcal{F} \\to \\mathcal{F}$", "is surjective for any abelian sheaf $\\mathcal{F}$ on $X_\\etale$,", "\\item", "$f_{small, *} :", "\\textit{Ab}(X_\\etale)", "\\to", "\\textit{Ab}(Y_\\etale)$", "is faithful and reflects injections and surjections, and", "\\item", "$f_{small, *} :", "\\textit{Ab}(X_\\etale)", "\\to", "\\textit{Ab}(Y_\\etale)$", "is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (2), (3) we have seen in", "Lemma \\ref{lemma-when-push-pull-surjective}.", "Part (1) follows from", "Lemmas \\ref{lemma-integral-B} and \\ref{lemma-property-B-implies}.", "Part (4) is a consequence of part (1), see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-exactness}." ], "refs": [ "etale-cohomology-lemma-when-push-pull-surjective", "etale-cohomology-lemma-integral-B", "etale-cohomology-lemma-property-B-implies", "sites-modules-lemma-exactness" ], "ref_ids": [ 6449, 6453, 6450, 14162 ] } ], "ref_ids": [] }, { "id": 6455, "type": "theorem", "label": "etale-cohomology-lemma-property-C-implies", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-property-C-implies", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume (C) holds. Then the functor", "$f_{small, *} :", "\\Sh(X_\\etale)", "\\to", "\\Sh(Y_\\etale)$", "reflects injections and surjections." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Sites, Lemma \\ref{sites-lemma-cover-from-below}.", "We omit the verification that property (C) implies that the functor", "$Y_\\etale \\to X_\\etale$, $V \\mapsto X \\times_Y V$", "satisfies the assumption of", "Sites, Lemma \\ref{sites-lemma-cover-from-below}." ], "refs": [ "sites-lemma-cover-from-below", "sites-lemma-cover-from-below" ], "ref_ids": [ 8620, 8620 ] } ], "ref_ids": [] }, { "id": 6456, "type": "theorem", "label": "etale-cohomology-lemma-property-C-closed-implies", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-property-C-closed-implies", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume that", "for any $V \\to Y$ \\'etale we have that", "\\begin{enumerate}", "\\item $X \\times_Y V \\to V$ has property (C), and", "\\item $X \\times_Y V \\to V$ is closed.", "\\end{enumerate}", "Then the functor", "$Y_\\etale \\to X_\\etale$, $V \\mapsto X \\times_Y V$", "is almost cocontinuous, see", "Sites, Definition \\ref{sites-definition-almost-cocontinuous}." ], "refs": [ "sites-definition-almost-cocontinuous" ], "proofs": [ { "contents": [ "Let $V \\to Y$ be an object of $Y_\\etale$ and let", "$\\{U_i \\to X \\times_Y V\\}_{i \\in I}$ be a covering of $X_\\etale$.", "By assumption (1) for each $i$ we can find an \\'etale morphism", "$h_i : V_i \\to V$ and a surjective morphism $X \\times_Y V_i \\to U_i$", "over $X \\times_Y V$. Note that $\\bigcup h_i(V_i) \\subset V$ is an", "open set containing the closed set $Z = \\Im(X \\times_Y V \\to V)$.", "Let $h_0 : V_0 = V \\setminus Z \\to V$ be the open immersion.", "It is clear that $\\{V_i \\to V\\}_{i \\in I \\cup \\{0\\}}$ is an", "\\'etale covering such that for each $i \\in I \\cup \\{0\\}$ we have", "either $V_i \\times_Y X = \\emptyset$ (namely if $i = 0$), or", "$V_i \\times_Y X \\to V \\times_Y X$ factors through $U_i \\to X \\times_Y V$", "(if $i \\not = 0$). Hence the functor $Y_\\etale \\to X_\\etale$", "is almost cocontinuous." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8682 ] }, { "id": 6457, "type": "theorem", "label": "etale-cohomology-lemma-integral-homeo-onto-image-C", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-integral-homeo-onto-image-C", "contents": [ "Let $f : X \\to Y$ be an integral morphism of schemes which defines", "a homeomorphism of $X$ with a closed subset of $Y$.", "Then property (C) holds." ], "refs": [], "proofs": [ { "contents": [ "Let $g : U \\to X$ be an \\'etale morphism. We need to find an object", "$V \\to Y$ of $Y_\\etale$ and a surjective morphism $X \\times_Y V \\to U$", "over $X$. Suppose that for every $u \\in U$ we can find an object", "$V_u \\to Y$ of $Y_\\etale$ and a morphism $h_u : X \\times_Y V_u \\to U$", "over $X$ with $u \\in \\Im(h_u)$. Then we can take $V = \\coprod V_u$", "and $h = \\coprod h_u$ and we win. Hence given a point", "$u \\in U$ we find a pair $(V_u, h_u)$ as above. To do this we may", "shrink $U$ and assume that $U$ is affine. In this case", "$g : U \\to X$ is locally quasi-finite. Let", "$g^{-1}(g(\\{u\\})) = \\{u, u_2, \\ldots, u_n\\}$. Since there are no", "specializations $u_i \\leadsto u$ we may replace $U$ by an affine neighbourhood", "so that $g^{-1}(g(\\{u\\})) = \\{u\\}$.", "\\medskip\\noindent", "The image $g(U) \\subset X$ is open,", "hence $f(g(U))$ is locally closed in $Y$. Choose an open $V \\subset Y$ such", "that $f(g(U)) = f(X) \\cap V$. It follows that $g$ factors through", "$X \\times_Y V$ and that the resulting $\\{U \\to X \\times_Y V\\}$ is an \\'etale", "covering. Since $f$ has property (B) , see", "Lemma \\ref{lemma-integral-B},", "we see that there exists an \\'etale covering $\\{V_j \\to V\\}$ such that", "$X \\times_Y V_j \\to X \\times_Y V$ factor through $U$.", "This implies that $V' = \\coprod V_j$ is \\'etale over $Y$ and that there is a", "morphism $h : X \\times_Y V' \\to U$ whose image", "surjects onto $g(U)$. Since $u$ is the only point in its fibre it must", "be in the image of $h$ and we win." ], "refs": [ "etale-cohomology-lemma-integral-B" ], "ref_ids": [ 6453 ] } ], "ref_ids": [] }, { "id": 6458, "type": "theorem", "label": "etale-cohomology-lemma-integral-universally-injective", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-integral-universally-injective", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume that $f$ is", "universally injective and integral (for example a closed immersion).", "Then", "\\begin{enumerate}", "\\item", "$f_{small, *} :", "\\Sh(X_\\etale)", "\\to", "\\Sh(Y_\\etale)$", "reflects injections and surjections,", "\\item", "$f_{small, *} :", "\\Sh(X_\\etale)", "\\to", "\\Sh(Y_\\etale)$", "commutes with pushouts and coequalizers (and more generally", "finite connected colimits),", "\\item $f_{small, *}$ transforms surjections into surjections (on sheaves", "of sets and on abelian sheaves),", "\\item the map", "$f_{small}^{-1}f_{small, *}\\mathcal{F} \\to \\mathcal{F}$", "is surjective for any sheaf (of sets or of abelian groups)", "$\\mathcal{F}$ on $X_\\etale$,", "\\item the functor $f_{small, *}$ is faithful (on sheaves of sets and", "on abelian sheaves),", "\\item", "$f_{small, *} :", "\\textit{Ab}(X_\\etale)", "\\to", "\\textit{Ab}(Y_\\etale)$", "is exact, and", "\\item the functor", "$Y_\\etale \\to X_\\etale$, $V \\mapsto X \\times_Y V$ is", "almost cocontinuous.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By", "Lemmas \\ref{lemma-integral-A},", "\\ref{lemma-integral-B} and", "\\ref{lemma-integral-homeo-onto-image-C}", "we know that the morphism $f$ has properties (A), (B), and (C).", "Moreover, by", "Lemma \\ref{lemma-property-C-closed-implies}", "we know that the functor $Y_\\etale \\to X_\\etale$ is", "almost cocontinuous. Now we have", "\\begin{enumerate}", "\\item property (C) implies (1) by", "Lemma \\ref{lemma-property-C-implies},", "\\item almost continuous implies (2) by", "Sites, Lemma \\ref{sites-lemma-morphism-of-sites-almost-cocontinuous},", "\\item property (B) implies (3) by", "Lemma \\ref{lemma-property-B-implies}.", "\\end{enumerate}", "Properties (4), (5), and (6) follow formally from the first three, see", "Sites, Lemma \\ref{sites-lemma-exactness-properties}", "and", "Modules on Sites, Lemma \\ref{sites-modules-lemma-exactness}.", "Property (7) we saw above." ], "refs": [ "etale-cohomology-lemma-integral-A", "etale-cohomology-lemma-integral-B", "etale-cohomology-lemma-integral-homeo-onto-image-C", "etale-cohomology-lemma-property-C-closed-implies", "etale-cohomology-lemma-property-C-implies", "sites-lemma-morphism-of-sites-almost-cocontinuous", "etale-cohomology-lemma-property-B-implies", "sites-lemma-exactness-properties", "sites-modules-lemma-exactness" ], "ref_ids": [ 6448, 6453, 6457, 6456, 6455, 8624, 6450, 8618, 14162 ] } ], "ref_ids": [] }, { "id": 6459, "type": "theorem", "label": "etale-cohomology-lemma-closed-immersion-almost-full", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-closed-immersion-almost-full", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes.", "Let $U, U'$ be schemes \\'etale over $X$. Let $h : U_Z \\to U'_Z$", "be a morphism over $Z$. Then there exists a diagram", "$$", "\\xymatrix{", "U & W \\ar[l]_a \\ar[r]^b & U'", "}", "$$", "such that $a_Z : W_Z \\to U_Z$ is an isomorphism and $h = b_Z \\circ (a_Z)^{-1}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the scheme $M = U \\times_Y U'$. The graph $\\Gamma_h \\subset M_Z$", "of $h$ is open. This is true for example as $\\Gamma_h$ is the image of a", "section of the \\'etale morphism $\\text{pr}_{1, Z} : M_Z \\to U_Z$, see", "\\'Etale Morphisms, Proposition \\ref{etale-proposition-properties-sections}.", "Hence there exists an open subscheme $W \\subset M$ whose intersection with", "the closed subset $M_Z$ is $\\Gamma_h$. Set $a = \\text{pr}_1|_W$", "and $b = \\text{pr}_2|_W$." ], "refs": [ "etale-proposition-properties-sections" ], "ref_ids": [ 10726 ] } ], "ref_ids": [] }, { "id": 6460, "type": "theorem", "label": "etale-cohomology-lemma-closed-immersion-almost-essentially-surjective", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-closed-immersion-almost-essentially-surjective", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes.", "Let $V \\to Z$ be an \\'etale morphism of schemes.", "There exist \\'etale morphisms $U_i \\to X$ and morphisms", "$U_{i, Z} \\to V$ such that $\\{U_{i, Z} \\to V\\}$", "is a Zariski covering of $V$." ], "refs": [], "proofs": [ { "contents": [ "Since we only have to find a Zariski covering of $V$ consisting of schemes", "of the form $U_Z$ with $U$ \\'etale over $X$, we may Zariski localize on $X$", "and $V$. Hence we may assume $X$ and $V$ affine. In the affine case this is", "Algebra, Lemma \\ref{algebra-lemma-lift-etale}." ], "refs": [ "algebra-lemma-lift-etale" ], "ref_ids": [ 1238 ] } ], "ref_ids": [] }, { "id": 6461, "type": "theorem", "label": "etale-cohomology-lemma-stalk-pushforward-closed-immersion", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-stalk-pushforward-closed-immersion", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes.", "Let $\\mathcal{G}$ be a sheaf of sets on $Z_\\etale$.", "Let $\\overline{x}$ be a geometric point of $X$.", "Then", "$$", "(i_{small, *}\\mathcal{G})_{\\overline{x}} =", "\\left\\{", "\\begin{matrix}", "* & \\text{if} & \\overline{x} \\not \\in Z \\\\", "\\mathcal{G}_{\\overline{x}} & \\text{if} & \\overline{x} \\in Z", "\\end{matrix}", "\\right.", "$$", "where $*$ denotes a singleton set." ], "refs": [], "proofs": [ { "contents": [ "Note that $i_{small, *}\\mathcal{G}|_{U_\\etale} = *$ is the final", "object in the category of \\'etale sheaves on $U$, i.e., the sheaf", "which associates a singleton set to each scheme \\'etale over $U$.", "This explains the value of $(i_{small, *}\\mathcal{G})_{\\overline{x}}$", "if $\\overline{x} \\not \\in Z$.", "\\medskip\\noindent", "Next, suppose that $\\overline{x} \\in Z$. Note that", "$$", "(i_{small, *}\\mathcal{G})_{\\overline{x}}", "=", "\\colim_{(U, \\overline{u})} \\mathcal{G}(U_Z)", "$$", "and on the other hand", "$$", "\\mathcal{G}_{\\overline{x}}", "=", "\\colim_{(V, \\overline{v})} \\mathcal{G}(V).", "$$", "Let $\\mathcal{C}_1 = \\{(U, \\overline{u})\\}^{opp}$ be the opposite of the", "category of \\'etale neighbourhoods of $\\overline{x}$ in $X$, and let", "$\\mathcal{C}_2 = \\{(V, \\overline{v})\\}^{opp}$ be the opposite of the", "category of \\'etale neighbourhoods of $\\overline{x}$ in $Z$. The canonical map", "$$", "\\mathcal{G}_{\\overline{x}}", "\\longrightarrow", "(i_{small, *}\\mathcal{G})_{\\overline{x}}", "$$", "corresponds to the functor $F : \\mathcal{C}_1 \\to \\mathcal{C}_2$,", "$F(U, \\overline{u}) = (U_Z, \\overline{x})$. Now", "Lemmas \\ref{lemma-closed-immersion-almost-essentially-surjective} and", "\\ref{lemma-closed-immersion-almost-full}", "imply that $\\mathcal{C}_1$ is cofinal in $\\mathcal{C}_2$, see", "Categories, Definition \\ref{categories-definition-cofinal}.", "Hence it follows that the displayed arrow is an isomorphism, see", "Categories, Lemma \\ref{categories-lemma-cofinal}." ], "refs": [ "etale-cohomology-lemma-closed-immersion-almost-essentially-surjective", "etale-cohomology-lemma-closed-immersion-almost-full", "categories-definition-cofinal", "categories-lemma-cofinal" ], "ref_ids": [ 6460, 6459, 12361, 12217 ] } ], "ref_ids": [] }, { "id": 6462, "type": "theorem", "label": "etale-cohomology-lemma-monomorphism-big-push-pull", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-monomorphism-big-push-pull", "contents": [ "Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.", "Let $f : X \\to Y$ be a monomorphism of schemes.", "Then the canonical map", "$f_{big}^{-1}f_{big, *}\\mathcal{F} \\to \\mathcal{F}$", "is an isomorphism for any sheaf $\\mathcal{F}$ on", "$(\\Sch/X)_\\tau$." ], "refs": [], "proofs": [ { "contents": [ "In this case the functor $(\\Sch/X)_\\tau \\to (\\Sch/Y)_\\tau$", "is continuous, cocontinuous and fully faithful. Hence the result follows from", "Sites, Lemma \\ref{sites-lemma-back-and-forth}." ], "refs": [ "sites-lemma-back-and-forth" ], "ref_ids": [ 8547 ] } ], "ref_ids": [] }, { "id": 6463, "type": "theorem", "label": "etale-cohomology-lemma-closed-immersion-cover-from-below", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-closed-immersion-cover-from-below", "contents": [ "Let $f : X \\to Y$ be a closed immersion of schemes.", "Let $U \\to X$ be a syntomic (resp.\\ smooth, resp.\\ \\'etale) morphism.", "Then there exist syntomic (resp.\\ smooth, resp.\\ \\'etale) morphisms", "$V_i \\to Y$ and morphisms $V_i \\times_Y X \\to U$ such that", "$\\{V_i \\times_Y X \\to U\\}$ is a Zariski covering of $U$." ], "refs": [], "proofs": [ { "contents": [ "Let us prove the lemma when $\\tau = syntomic$.", "The question is local on $U$. Thus we may assume that $U$ is", "an affine scheme mapping into an affine of $Y$.", "Hence we reduce to proving the following case:", "$Y = \\Spec(A)$, $X = \\Spec(A/I)$, and", "$U = \\Spec(\\overline{B})$, where", "$A/I \\to \\overline{B}$ be a syntomic ring map.", "By Algebra, Lemma \\ref{algebra-lemma-lift-syntomic}", "we can find elements $\\overline{g}_i \\in \\overline{B}$", "such that", "$\\overline{B}_{\\overline{g}_i} = A_i/IA_i$ for certain syntomic ring maps", "$A \\to A_i$.", "This proves the lemma in the syntomic case.", "The proof of the smooth case is the same except it uses", "Algebra, Lemma \\ref{algebra-lemma-lift-smooth}.", "In the \\'etale case use", "Algebra, Lemma \\ref{algebra-lemma-lift-etale}." ], "refs": [ "algebra-lemma-lift-syntomic", "algebra-lemma-lift-smooth", "algebra-lemma-lift-etale" ], "ref_ids": [ 1188, 1203, 1238 ] } ], "ref_ids": [] }, { "id": 6464, "type": "theorem", "label": "etale-cohomology-lemma-prepare-closed-immersion-almost-cocontinuous", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-prepare-closed-immersion-almost-cocontinuous", "contents": [ "Let $f : X \\to Y$ be a closed immersion of schemes.", "Let $\\{U_i \\to X\\}$ be a syntomic (resp.\\ smooth, resp.\\ \\'etale) covering.", "There exists a syntomic (resp.\\ smooth, resp.\\ \\'etale) covering $\\{V_j \\to Y\\}$", "such that for each $j$, either $V_j \\times_Y X = \\emptyset$, or the", "morphism $V_j \\times_Y X \\to X$ factors through $U_i$ for some $i$." ], "refs": [], "proofs": [ { "contents": [ "For each $i$ we can choose syntomic (resp.\\ smooth, resp.\\ \\'etale) morphisms", "$g_{ij} : V_{ij} \\to Y$ and morphisms $V_{ij} \\times_Y X \\to U_i$ over $X$,", "such that $\\{V_{ij} \\times_Y X \\to U_i\\}$ are Zariski coverings, see", "Lemma \\ref{lemma-closed-immersion-cover-from-below}.", "This in particular implies that", "$\\bigcup_{ij} g_{ij}(V_{ij})$ contains the closed subset $f(X)$.", "Hence the family of syntomic (resp.\\ smooth, resp.\\ \\'etale) maps $g_{ij}$", "together with the open immersion $Y \\setminus f(X) \\to Y$ forms the desired", "syntomic (resp.\\ smooth, resp.\\ \\'etale) covering of $Y$." ], "refs": [ "etale-cohomology-lemma-closed-immersion-cover-from-below" ], "ref_ids": [ 6463 ] } ], "ref_ids": [] }, { "id": 6465, "type": "theorem", "label": "etale-cohomology-lemma-closed-immersion-almost-cocontinuous", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-closed-immersion-almost-cocontinuous", "contents": [ "Let $f : X \\to Y$ be a closed immersion of schemes.", "Let $\\tau \\in \\{syntomic, smooth, \\etale\\}$.", "The functor $V \\mapsto X \\times_Y V$ defines an almost", "cocontinuous functor (see", "Sites, Definition \\ref{sites-definition-almost-cocontinuous})", "$(\\Sch/Y)_\\tau \\to (\\Sch/X)_\\tau$ between", "big $\\tau$ sites." ], "refs": [ "sites-definition-almost-cocontinuous" ], "proofs": [ { "contents": [ "We have to show the following: given a morphism $V \\to Y$", "and any syntomic (resp.\\ smooth, resp.\\ \\'etale)", "covering $\\{U_i \\to X \\times_Y V\\}$, there exists a", "smooth (resp.\\ smooth, resp.\\ \\'etale) covering $\\{V_j \\to V\\}$", "such that for each $j$, either $X \\times_Y V_j$ is empty, or", "$X \\times_Y V_j \\to Z \\times_Y V$ factors through one of", "the $U_i$. This follows on applying", "Lemma \\ref{lemma-prepare-closed-immersion-almost-cocontinuous}", "above to the closed immersion $X \\times_Y V \\to V$." ], "refs": [ "etale-cohomology-lemma-prepare-closed-immersion-almost-cocontinuous" ], "ref_ids": [ 6464 ] } ], "ref_ids": [ 8682 ] }, { "id": 6466, "type": "theorem", "label": "etale-cohomology-lemma-closed-immersion-pushforward-exact", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-closed-immersion-pushforward-exact", "contents": [ "Let $f : X \\to Y$ be a closed immersion of schemes.", "Let $\\tau \\in \\{syntomic, smooth, \\etale\\}$.", "\\begin{enumerate}", "\\item The pushforward", "$f_{big, *} :", "\\Sh((\\Sch/X)_\\tau)", "\\to", "\\Sh((\\Sch/Y)_\\tau)$", "commutes with coequalizers and pushouts.", "\\item The pushforward", "$f_{big, *} :", "\\textit{Ab}((\\Sch/X)_\\tau)", "\\to", "\\textit{Ab}((\\Sch/Y)_\\tau)$", "is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Sites, Lemma \\ref{sites-lemma-morphism-of-sites-almost-cocontinuous},", "Modules on Sites,", "Lemma \\ref{sites-modules-lemma-morphism-ringed-sites-almost-cocontinuous},", "and", "Lemma \\ref{lemma-closed-immersion-almost-cocontinuous}", "above." ], "refs": [ "sites-lemma-morphism-of-sites-almost-cocontinuous", "sites-modules-lemma-morphism-ringed-sites-almost-cocontinuous", "etale-cohomology-lemma-closed-immersion-almost-cocontinuous" ], "ref_ids": [ 8624, 14163, 6465 ] } ], "ref_ids": [] }, { "id": 6467, "type": "theorem", "label": "etale-cohomology-lemma-exactness-lower-shriek", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-exactness-lower-shriek", "contents": [ "Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.", "Let $f : X \\to Y$ be a morphism of schemes. Let", "$$", "f_{big} :", "\\Sh((\\Sch/X)_\\tau)", "\\longrightarrow", "\\Sh((\\Sch/Y)_\\tau)", "$$", "be the corresponding morphism of topoi as in", "Topologies, Lemma", "\\ref{topologies-lemma-morphism-big},", "\\ref{topologies-lemma-morphism-big-etale},", "\\ref{topologies-lemma-morphism-big-smooth},", "\\ref{topologies-lemma-morphism-big-syntomic}, or", "\\ref{topologies-lemma-morphism-big-fppf}.", "\\begin{enumerate}", "\\item The functor", "$f_{big}^{-1} : \\textit{Ab}((\\Sch/Y)_\\tau) \\to \\textit{Ab}((\\Sch/X)_\\tau)$", "has a left adjoint", "$$", "f_{big!} : \\textit{Ab}((\\Sch/X)_\\tau) \\to \\textit{Ab}((\\Sch/Y)_\\tau)", "$$", "which is exact.", "\\item The functor", "$f_{big}^* :", "\\textit{Mod}((\\Sch/Y)_\\tau, \\mathcal{O})", "\\to", "\\textit{Mod}((\\Sch/X)_\\tau, \\mathcal{O})$", "has a left adjoint", "$$", "f_{big!} :", "\\textit{Mod}((\\Sch/X)_\\tau, \\mathcal{O})", "\\to", "\\textit{Mod}((\\Sch/Y)_\\tau, \\mathcal{O})", "$$", "which is exact.", "\\end{enumerate}", "Moreover, the two functors $f_{big!}$ agree on underlying sheaves", "of abelian groups." ], "refs": [ "topologies-lemma-morphism-big", "topologies-lemma-morphism-big-etale", "topologies-lemma-morphism-big-smooth", "topologies-lemma-morphism-big-syntomic", "topologies-lemma-morphism-big-fppf" ], "proofs": [ { "contents": [ "Recall that $f_{big}$ is the morphism of topoi associated to the", "continuous and cocontinuous functor", "$u : (\\Sch/X)_\\tau \\to (\\Sch/Y)_\\tau$, $U/X \\mapsto U/Y$.", "Moreover, we have $f_{big}^{-1}\\mathcal{O} = \\mathcal{O}$.", "Hence the existence of $f_{big!}$ follows from", "Modules on Sites, Lemma \\ref{sites-modules-lemma-g-shriek-adjoint},", "respectively", "Modules on Sites, Lemma \\ref{sites-modules-lemma-lower-shriek-modules}.", "Note that if $U$ is an object of $(\\Sch/X)_\\tau$ then the functor", "$u$ induces an equivalence of categories", "$$", "u' :", "(\\Sch/X)_\\tau/U", "\\longrightarrow", "(\\Sch/Y)_\\tau/U", "$$", "because both sides of the arrow are equal to $(\\Sch/U)_\\tau$.", "Hence the agreement of $f_{big!}$ on underlying abelian sheaves", "follows from the discussion in", "Modules on Sites, Remark \\ref{sites-modules-remark-when-shriek-equal}.", "The exactness of $f_{big!}$ follows from", "Modules on Sites, Lemma \\ref{sites-modules-lemma-exactness-lower-shriek}", "as the functor $u$ above which commutes with fibre products and equalizers." ], "refs": [ "sites-modules-lemma-g-shriek-adjoint", "sites-modules-lemma-lower-shriek-modules", "sites-modules-remark-when-shriek-equal", "sites-modules-lemma-exactness-lower-shriek" ], "ref_ids": [ 14164, 14262, 14312, 14165 ] } ], "ref_ids": [ 12440, 12454, 12464, 12470, 12478 ] }, { "id": 6468, "type": "theorem", "label": "etale-cohomology-lemma-compare-structure-sheaves", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-compare-structure-sheaves", "contents": [ "Let $X$ be a scheme. Let", "$\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.", "Let $\\mathcal{C}_1 \\subset \\mathcal{C}_2 \\subset (\\Sch/X)_\\tau$ be full", "subcategories with the following properties:", "\\begin{enumerate}", "\\item For an object $U/X$ of $\\mathcal{C}_t$,", "\\begin{enumerate}", "\\item if $\\{U_i \\to U\\}$ is a covering of $(\\Sch/X)_\\tau$, then", "$U_i/X$ is an object of $\\mathcal{C}_t$,", "\\item $U \\times \\mathbf{A}^1/X$ is an object of $\\mathcal{C}_t$.", "\\end{enumerate}", "\\item $X/X$ is an object of $\\mathcal{C}_t$.", "\\end{enumerate}", "We endow $\\mathcal{C}_t$ with the structure of a site whose coverings are", "exactly those coverings $\\{U_i \\to U\\}$ of $(\\Sch/X)_\\tau$ with", "$U \\in \\Ob(\\mathcal{C}_t)$. Then", "\\begin{enumerate}", "\\item[(a)] The functor $\\mathcal{C}_1 \\to \\mathcal{C}_2$", "is fully faithful, continuous, and cocontinuous.", "\\end{enumerate}", "Denote $g : \\Sh(\\mathcal{C}_1) \\to \\Sh(\\mathcal{C}_2)$ the corresponding", "morphism of topoi. Denote $\\mathcal{O}_t$ the restriction of $\\mathcal{O}$", "to $\\mathcal{C}_t$. Denote $g_!$ the functor of", "Modules on Sites, Definition \\ref{sites-modules-definition-g-shriek}.", "\\begin{enumerate}", "\\item[(b)] The canonical map $g_!\\mathcal{O}_1 \\to \\mathcal{O}_2$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [ "sites-modules-definition-g-shriek" ], "proofs": [ { "contents": [ "Assertion (a) is immediate from the definitions.", "In this proof all schemes are schemes over $X$ and all morphisms of", "schemes are morphisms of schemes over $X$. Note that $g^{-1}$ is", "given by restriction, so that for an object $U$ of $\\mathcal{C}_1$", "we have $\\mathcal{O}_1(U) = \\mathcal{O}_2(U) = \\mathcal{O}(U)$.", "Recall that $g_!\\mathcal{O}_1$ is the sheaf associated to the presheaf", "$g_{p!}\\mathcal{O}_1$ which associates to $V$ in $\\mathcal{C}_2$ the group", "$$", "\\colim_{V \\to U} \\mathcal{O}(U)", "$$", "where $U$ runs over the objects of $\\mathcal{C}_1$ and the colimit is", "taken in the category of abelian groups. Below we will use frequently", "that if", "$$", "V \\to U \\to U'", "$$", "are morphisms with $U, U' \\in \\Ob(\\mathcal{C}_1)$", "and if $f' \\in \\mathcal{O}(U')$ restricts to $f \\in \\mathcal{O}(U)$,", "then $(V \\to U, f)$ and $(V \\to U', f')$ define the same element of the", "colimit. Also, $g_!\\mathcal{O}_1 \\to \\mathcal{O}_2$ maps the element", "$(V \\to U, f)$ simply to the pullback of $f$ to $V$.", "\\medskip\\noindent", "Surjectivity. Let $V$ be a scheme and let $h \\in \\mathcal{O}(V)$.", "Then we obtain a morphism $V \\to X \\times \\mathbf{A}^1$ induced by $h$", "and the structure morphism $V \\to X$. Writing", "$\\mathbf{A}^1 = \\Spec(\\mathbf{Z}[x])$ we see the element", "$x \\in \\mathcal{O}(X \\times \\mathbf{A}^1)$ pulls", "back to $h$. Since $X \\times \\mathbf{A}^1$ is an object of $\\mathcal{C}_1$", "by assumptions (1)(b) and (2) we obtain the desired surjectivity.", "\\medskip\\noindent", "Injectivity. Let $V$ be a scheme. Let", "$s = \\sum_{i = 1, \\ldots, n} (V \\to U_i, f_i)$ be an element of the colimit", "displayed above. For any $i$ we can use the morphism", "$f_i : U_i \\to X \\times \\mathbf{A}^1$", "to see that $(V \\to U_i, f_i)$ defines the same element of the colimit as", "$(f_i : V \\to X \\times \\mathbf{A}^1, x)$. Then we can consider", "$$", "f_1 \\times \\ldots \\times f_n : V \\to X \\times \\mathbf{A}^n", "$$", "and we see that $s$ is equivalent in the colimit to", "$$", "\\sum\\nolimits_{i = 1, \\ldots, n}", "(f_1 \\times \\ldots \\times f_n : V \\to X \\times \\mathbf{A}^n, x_i) =", "(f_1 \\times \\ldots \\times f_n : V \\to X \\times \\mathbf{A}^n,", "x_1 + \\ldots + x_n)", "$$", "Now, if $x_1 + \\ldots + x_n$ restricts to zero on $V$, then we see", "that $f_1 \\times \\ldots \\times f_n$ factors through", "$X \\times \\mathbf{A}^{n - 1} = V(x_1 + \\ldots + x_n)$. Hence we see", "that $s$ is equivalent to zero in the colimit." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 14285 ] }, { "id": 6469, "type": "theorem", "label": "etale-cohomology-lemma-mayer-vietoris", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-mayer-vietoris", "contents": [ "Let $X$ be a scheme. Suppose that $X = U \\cup V$ is a", "union of two opens. For any abelian sheaf $\\mathcal{F}$ on $X_\\etale$", "there exists a long exact cohomology sequence", "$$", "\\begin{matrix}", "0 \\to", "H^0_\\etale(X, \\mathcal{F}) \\to", "H^0_\\etale(U, \\mathcal{F}) \\oplus H^0_\\etale(V, \\mathcal{F}) \\to", "H^0_\\etale(U \\cap V, \\mathcal{F}) \\phantom{\\to \\ldots} \\\\", "\\phantom{0} \\to H^1_\\etale(X, \\mathcal{F}) \\to", "H^1_\\etale(U, \\mathcal{F}) \\oplus H^1_\\etale(V, \\mathcal{F}) \\to", "H^1_\\etale(U \\cap V, \\mathcal{F}) \\to \\ldots", "\\end{matrix}", "$$", "This long exact sequence is functorial in $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Observe that if $\\mathcal{I}$ is an injective abelian sheaf, then", "$$", "0 \\to \\mathcal{I}(X) \\to \\mathcal{I}(U) \\oplus \\mathcal{I}(V) \\to", "\\mathcal{I}(U \\cap V) \\to 0", "$$", "is exact. This is true in the first and middle spots as $\\mathcal{I}$", "is a sheaf. It is true on the right, because", "$\\mathcal{I}(U) \\to \\mathcal{I}(U \\cap V)$ is surjective by", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-restriction-along-monomorphism-surjective}.", "Another way to prove it would be to show that the cokernel", "of the map $\\mathcal{I}(U) \\oplus \\mathcal{I}(V) \\to", "\\mathcal{I}(U \\cap V)$ is the first {\\v C}ech cohomology group", "of $\\mathcal{I}$ with respect to the covering", "$X = U \\cup V$ which vanishes by", "Lemmas \\ref{lemma-hom-injective} and \\ref{lemma-forget-injectives}.", "Thus, if $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ is an injective", "resolution, then", "$$", "0 \\to \\mathcal{I}^\\bullet(X) \\to", "\\mathcal{I}^\\bullet(U) \\oplus \\mathcal{I}^\\bullet(V) \\to", "\\mathcal{I}^\\bullet(U \\cap V) \\to 0", "$$", "is a short exact sequence of complexes and the associated long", "exact cohomology sequence is the sequence of the statement of the lemma." ], "refs": [ "sites-cohomology-lemma-restriction-along-monomorphism-surjective", "etale-cohomology-lemma-hom-injective", "etale-cohomology-lemma-forget-injectives" ], "ref_ids": [ 4212, 6408, 6409 ] } ], "ref_ids": [] }, { "id": 6470, "type": "theorem", "label": "etale-cohomology-lemma-relative-mayer-vietoris", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-relative-mayer-vietoris", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Suppose that $X = U \\cup V$", "is a union of two open subschemes. Denote", "$a = f|_U : U \\to Y$, $b = f|_V : V \\to Y$, and", "$c = f|_{U \\cap V} : U \\cap V \\to Y$.", "For every abelian sheaf $\\mathcal{F}$ on $X_\\etale$", "there exists a long exact sequence", "$$", "0 \\to", "f_*\\mathcal{F} \\to", "a_*(\\mathcal{F}|_U) \\oplus b_*(\\mathcal{F}|_V) \\to", "c_*(\\mathcal{F}|_{U \\cap V}) \\to", "R^1f_*\\mathcal{F} \\to \\ldots", "$$", "on $Y_\\etale$.", "This long exact sequence is functorial in $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ be an injective resolution", "of $\\mathcal{F}$ on $X_\\etale$. We claim that we", "get a short exact sequence of complexes", "$$", "0 \\to", "f_*\\mathcal{I}^\\bullet \\to", "a_*\\mathcal{I}^\\bullet|_U \\oplus b_*\\mathcal{I}^\\bullet|_V \\to", "c_*\\mathcal{I}^\\bullet|_{U \\cap V} \\to", "0.", "$$", "Namely, for any $W$ in $Y_\\etale$, and for any $n \\geq 0$ the", "corresponding sequence of groups of sections over $W$", "$$", "0 \\to", "\\mathcal{I}^n(W \\times_Y X) \\to", "\\mathcal{I}^n(W \\times_Y U)", "\\oplus \\mathcal{I}^n(W \\times_Y V) \\to", "\\mathcal{I}^n(W \\times_Y (U \\cap V)) \\to", "0", "$$", "was shown to be short exact in the proof of Lemma \\ref{lemma-mayer-vietoris}.", "The lemma follows by taking cohomology sheaves and using the fact that", "$\\mathcal{I}^\\bullet|_U$ is an injective resolution of $\\mathcal{F}|_U$", "and similarly for $\\mathcal{I}^\\bullet|_V$, $\\mathcal{I}^\\bullet|_{U \\cap V}$." ], "refs": [ "etale-cohomology-lemma-mayer-vietoris" ], "ref_ids": [ 6469 ] } ], "ref_ids": [] }, { "id": 6471, "type": "theorem", "label": "etale-cohomology-lemma-colimit-affine-sites", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-colimit-affine-sites", "contents": [ "Let $I$ be a directed set. Let $(X_i, f_{i'i})$ be an inverse", "system of schemes over $I$ with affine transition morphisms.", "Let $X = \\lim_{i \\in I} X_i$. With", "notation as in Lemma \\ref{lemma-alternative} we have", "$$", "X_{affine, \\etale} = \\colim (X_i)_{affine, \\etale}", "$$", "as sites in the sense of", "Sites, Lemma \\ref{sites-lemma-colimit-sites}." ], "refs": [ "etale-cohomology-lemma-alternative", "sites-lemma-colimit-sites" ], "proofs": [ { "contents": [ "Let us first prove this when $X$ and $X_i$ are quasi-compact and", "quasi-separated for all $i$ (as this is true in all cases of", "interest). In this case any object of", "$X_{affine, \\etale}$, resp.\\ $(X_i)_{affine, \\etale}$ is", "of finite presentation over $X$. Moreover, the", "category of schemes of finite presentation over $X$ is the", "colimit of the categories of schemes of finite presentation", "over $X_i$, see", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}.", "The same holds for the subcategories of affine objects \\'etale", "over $X$ by Limits, Lemmas", "\\ref{limits-lemma-limit-affine} and \\ref{limits-lemma-descend-etale}.", "Finally, if $\\{U^j \\to U\\}$ is a covering of $X_{affine, \\etale}$", "and if $U_i^j \\to U_i$ is morphism of affine schemes \\'etale over", "$X_i$ whose base change to $X$ is $U^j \\to U$, then we see that", "the base change of $\\{U^j_i \\to U_i\\}$ to some $X_{i'}$ is", "a covering for $i'$ large enough, see", "Limits, Lemma \\ref{limits-lemma-descend-surjective}.", "\\medskip\\noindent", "In the general case, let $U$ be an object of $X_{affine, \\etale}$.", "Then $U \\to X$ is \\'etale and separated (as $U$ is separated)", "but in general not quasi-compact. Still, $U \\to X$ is locally", "of finite presentation and hence by", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation-variant}", "there exists an $i$, a quasi-compact and quasi-separated scheme $U_i$, and", "a morphism $U_i \\to X_i$ which is locally of finite presentation", "whose base change to $X$ is $U \\to X$. Then $U = \\lim_{i' \\geq i} U_{i'}$", "where $U_{i'} = U_i \\times_{X_i} X_{i'}$.", "After increasing $i$ we may assume $U_i$ is affine, see", "Limits, Lemma \\ref{limits-lemma-limit-affine}.", "To check that $U_i \\to X_i$ is \\'etale for $i$ sufficiently large,", "choose a finite affine open covering $U_i = U_{i, 1} \\cup \\ldots \\cup U_{i, m}$", "such that $U_{i, j} \\to U_i \\to X_i$ maps into an affine open", "$W_{i, j} \\subset X_i$. Then we can apply", "Limits, Lemma \\ref{limits-lemma-descend-etale}", "to see that $U_{i, j} \\to W_{i, j}$ is \\'etale", "after possibly increasing $i$.", "In this way we see that the functor", "$\\colim (X_i)_{affine, \\etale} \\to X_{affine, \\etale}$", "is essentially surjective. Fully faithfulness follows", "directly from the already used", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation-variant}.", "The statement on coverings is proved in exactly the same", "manner as done in the first paragraph of the proof." ], "refs": [ "limits-lemma-descend-finite-presentation", "limits-lemma-limit-affine", "limits-lemma-descend-etale", "limits-lemma-descend-surjective", "limits-lemma-descend-finite-presentation-variant", "limits-lemma-limit-affine", "limits-lemma-descend-etale", "limits-lemma-descend-finite-presentation-variant" ], "ref_ids": [ 15077, 15043, 15065, 15069, 15081, 15043, 15065, 15081 ] } ], "ref_ids": [ 6413, 8532 ] }, { "id": 6472, "type": "theorem", "label": "etale-cohomology-lemma-colimit", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-colimit", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Let $I$", "be a directed set. Let $(\\mathcal{F}_i, \\varphi_{ij})$ be a system", "of abelian sheaves on $X_\\etale$ over $I$. Then", "$$", "\\colim_{i\\in I} H_\\etale^p(X, \\mathcal{F}_i) = H_\\etale^p(X,", "\\colim_{i\\in I} \\mathcal{F}_i).", "$$" ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Theorem \\ref{theorem-colimit}.", "We also sketch a direct proof.", "We prove it for all $X$ at the same time, by induction on $p$.", "\\begin{enumerate}", "\\item", "For any quasi-compact and quasi-separated scheme $X$ and any \\'etale covering", "$\\mathcal{U}$ of $X$, show that there exists a refinement", "$\\mathcal{V} = \\{V_j \\to X\\}_{j\\in J}$ with $J$ finite and each $V_j$", "quasi-compact and quasi-separated such that all", "$V_{j_0} \\times_X \\ldots \\times_X V_{j_p}$ are also", "quasi-compact and quasi-separated.", "\\item", "Using the previous step and the definition of colimits in the category of", "sheaves, show that the theorem holds for $p = 0$ and all $X$.", "\\item", "Using the locality of cohomology", "(Lemma \\ref{lemma-locality-cohomology}),", "the {\\v C}ech-to-cohomology spectral sequence", "(Theorem \\ref{theorem-cech-ss}) and the fact that the induction", "hypothesis applies to all", "$V_{j_0} \\times_X \\ldots \\times_X V_{j_p}$", "in the above situation, prove the induction step $p \\to p + 1$.", "\\end{enumerate}" ], "refs": [ "etale-cohomology-theorem-colimit", "etale-cohomology-lemma-locality-cohomology", "etale-cohomology-theorem-cech-ss" ], "ref_ids": [ 6384, 6417, 6372 ] } ], "ref_ids": [] }, { "id": 6473, "type": "theorem", "label": "etale-cohomology-lemma-directed-colimit-cohomology", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-directed-colimit-cohomology", "contents": [ "Let $A$ be a ring, $(I, \\leq)$ a directed set and $(B_i, \\varphi_{ij})$ a", "system of $A$-algebras. Set $B = \\colim_{i\\in I} B_i$. Let $X \\to \\Spec(A)$", "be a quasi-compact and quasi-separated morphism of schemes. Let", "$\\mathcal{F}$ an abelian sheaf on $X_\\etale$.", "Denote $Y_i = X \\times_{\\Spec(A)} \\Spec(B_i)$,", "$Y = X \\times_{\\Spec(A)} \\Spec(B)$,", "$\\mathcal{G}_i = (Y_i \\to X)^{-1}\\mathcal{F}$ and", "$\\mathcal{G} = (Y \\to X)^{-1}\\mathcal{F}$. Then", "$$", "H_\\etale^p(Y, \\mathcal{G}) =", "\\colim_{i\\in I} H_\\etale^p (Y_i, \\mathcal{G}_i).", "$$" ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Theorem \\ref{theorem-colimit}.", "We also outline a direct proof as follows.", "\\begin{enumerate}", "\\item Given $V \\to Y$ \\'etale with $V$ quasi-compact and", "quasi-separated, there exist $i\\in I$ and $V_i \\to Y_i$ such that", "$V = V_i \\times_{Y_i} Y$.", "If all the schemes considered were affine, this would correspond to the", "following algebra statement: if $B = \\colim B_i$ and $B \\to C$ is \\'etale,", "then there exist $i \\in I$ and $B_i \\to C_i$ \\'etale such that", "$C \\cong B \\otimes_{B_i} C_i$.", "This is proved in Algebra, Lemma \\ref{algebra-lemma-etale}.", "\\item In the situation of (1) show that", "$\\mathcal{G}(V) = \\colim_{i' \\geq i} \\mathcal{G}_{i'}(V_{i'})$", "where $V_{i'}$ is the base change of $V_i$ to $Y_{i'}$.", "\\item By (1), we see that for every \\'etale covering", "$\\mathcal{V} = \\{V_j \\to Y\\}_{j\\in J}$ with $J$ finite and the", "$V_j$s quasi-compact and quasi-separated, there exists $i \\in I$ and", "an \\'etale covering $\\mathcal{V}_i = \\{V_{ij} \\to Y_i\\}_{j \\in J}$", "such that $\\mathcal{V} \\cong \\mathcal{V}_i \\times_{Y_i} Y$.", "\\item Show that (2) and (3) imply", "$$", "\\check H^*(\\mathcal{V}, \\mathcal{G})=", "\\colim_{i\\in I} \\check H^*(\\mathcal{V}_i, \\mathcal{G}_i).", "$$", "\\item Cleverly use the {\\v C}ech-to-cohomology spectral sequence", "(Theorem \\ref{theorem-cech-ss}).", "\\end{enumerate}" ], "refs": [ "etale-cohomology-theorem-colimit", "algebra-lemma-etale", "etale-cohomology-theorem-cech-ss" ], "ref_ids": [ 6384, 1231, 6372 ] } ], "ref_ids": [] }, { "id": 6474, "type": "theorem", "label": "etale-cohomology-lemma-higher-direct-images", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-higher-direct-images", "contents": [ "Let $f: X\\to Y$ be a morphism of schemes and $\\mathcal{F}\\in", "\\textit{Ab}(X_\\etale)$. Then $R^pf_*\\mathcal{F}$ is the sheaf", "associated to the presheaf", "$$", "(V \\to Y) \\longmapsto H_\\etale^p(X \\times_Y V, \\mathcal{F}|_{X \\times_Y V}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "This lemma is valid for topological spaces, and the proof in this case is the", "same. See", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-higher-direct-images}", "for details." ], "refs": [ "sites-cohomology-lemma-higher-direct-images" ], "ref_ids": [ 4189 ] } ], "ref_ids": [] }, { "id": 6475, "type": "theorem", "label": "etale-cohomology-lemma-relative-colimit", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-relative-colimit", "contents": [ "Let $S$ be a scheme. Let $X = \\lim_{i \\in I} X_i$ be a limit of a", "directed system of schemes over $S$ with affine transition morphisms", "$f_{i'i} : X_{i'} \\to X_i$. We assume the structure morphisms", "$g_i : X_i \\to S$ and $g : X \\to S$ are quasi-compact and quasi-separated.", "Let $(\\mathcal{F}_i, \\varphi_{i'i})$ be a system of abelian sheaves", "on $(X_i, f_{i'i})$. Denote $f_i : X \\to X_i$ the projection and set", "$\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$. Then", "$$", "\\colim_{i\\in I} R^p g_{i, *} \\mathcal{F}_i = R^p g_* \\mathcal{F}", "$$", "for all $p \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Recall (Lemma \\ref{lemma-higher-direct-images})", "that $R^p g_{i, *} \\mathcal{F}_i$ is the sheaf associated to the", "presheaf $U \\mapsto H^p_\\etale(U \\times_S X_i, \\mathcal{F}_i)$", "and similarly for $R^pg_*\\mathcal{F}$. Moreover, the colimit of a", "system of sheaves is the sheafification of the colimit on the level", "of presheaves. Note that every object of $S_\\etale$ has a covering", "by quasi-compact and quasi-separated objects (e.g., affine schemes).", "Moreover, if $U$ is a quasi-compact and quasi-separated object,", "then we have", "$$", "\\colim H^p_\\etale(U \\times_S X_i, \\mathcal{F}_i) =", "H^p_\\etale(U \\times_S X, \\mathcal{F})", "$$", "by Theorem \\ref{theorem-colimit}. Thus the lemma follows." ], "refs": [ "etale-cohomology-lemma-higher-direct-images", "etale-cohomology-theorem-colimit" ], "ref_ids": [ 6474, 6384 ] } ], "ref_ids": [] }, { "id": 6476, "type": "theorem", "label": "etale-cohomology-lemma-relative-colimit-general", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-relative-colimit-general", "contents": [ "Let $I$ be a directed set. Let $g_i : X_i \\to S_i$ be an inverse system of", "morphisms of schemes over $I$. Assume $g_i$ is quasi-compact and", "quasi-separated and for $i' \\geq i$ the transition morphisms", "$f_{i'i} : X_{i'} \\to X_i$ and $h_{i'i} : S_{i'} \\to S_i$ are affine.", "Let $g : X \\to S$ be the limit of the morphisms $g_i$, see", "Limits, Section \\ref{limits-section-limits}.", "Denote $f_i : X \\to X_i$ and $h_i : S \\to S_i$ the projections.", "Let $(\\mathcal{F}_i, \\varphi_{i'i})$ be a system of sheaves", "on $(X_i, f_{i'i})$. Set $\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$. Then", "$$", "R^p g_* \\mathcal{F} =", "\\colim_{i \\in I} h_i^{-1}R^p g_{i, *} \\mathcal{F}_i", "$$", "for all $p \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "How is the map of the lemma constructed?", "For $i' \\geq i$ we have a commutative diagram", "$$", "\\xymatrix{", "X \\ar[r]_{f_{i'}} \\ar[d]_g &", "X_{i'} \\ar[r]_{f_{i'i}} \\ar[d]_{g_{i'}} &", "X_i \\ar[d]^{g_i} \\\\", "S \\ar[r]^{h_{i'}} &", "S_{i'} \\ar[r]^{h_{i'i}} &", "S_i", "}", "$$", "If we combine the base change map", "$h_{i'i}^{-1}Rg_{i, *}\\mathcal{F}_i \\to Rg_{i', *}f_{i'i}^{-1}\\mathcal{F}_i$", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-base-change-map-flat-case} or", "Remark \\ref{sites-cohomology-remark-base-change})", "with the map $Rg_{i', *}\\varphi_{i'i}$, then we obtain", "$\\psi_{i'i} : h_{i' i}^{-1} R^p g_{i, *} \\mathcal{F}_i \\to", "R^pg_{i', *} \\mathcal{F}_{i'}$. Similarly, using the left square", "in the diagram we obtain maps", "$\\psi_i : h_i^{-1}R^pg_{i, *}\\mathcal{F}_i \\to R^pg_*\\mathcal{F}$.", "The maps $h_{i'}^{-1}\\psi_{i'i}$ and $\\psi_i$ are the maps used in", "the statement of the lemma. For this to make sense, we have to check that", "$\\psi_{i''i} = \\psi_{i''i'} \\circ h_{i''i'}^{-1}\\psi_{i'i}$ and", "$\\psi_{i'} \\circ h_{i'}^{-1}\\psi_{i'i} = \\psi_i$; this follows", "from Cohomology on Sites, Remark", "\\ref{sites-cohomology-remark-compose-base-change-horizontal}.", "\\medskip\\noindent", "Proof of the equality. First proof using", "dimension shifting\\footnote{You can also use this method", "to produce the maps in the lemma.}. For any", "$U$ affine and \\'etale over $X$ by Theorem \\ref{theorem-colimit}", "we have", "$$", "g_*\\mathcal{F}(U) =", "H^0(U \\times_S X, \\mathcal{F}) =", "\\colim H^0(U_i \\times_{S_i} X_i, \\mathcal{F}_i) =", "\\colim g_{i, *}\\mathcal{F}_i(U_i)", "$$", "where the colimit is over $i$ large enough such that", "there exists an $i$ and $U_i$ affine \\'etale over $S_i$", "whose base change is $U$ over $S$ (see", "Lemma \\ref{lemma-colimit-affine-sites}).", "The right hand side is equal to", "$(\\colim h_i^{-1}g_{i, *}\\mathcal{F}_i)(U)$ by", "Sites, Lemma \\ref{sites-lemma-colimit}.", "This proves the lemma for $p = 0$.", "If $(\\mathcal{G}_i, \\varphi_{i'i})$ is a system with", "$\\mathcal{G} = \\colim f_i^{-1}\\mathcal{G}_i$", "such that $\\mathcal{G}_i$ is an injective abelian sheaf on $X_i$", "for all $i$, then for any $U$ affine and \\'etale over $X$ by", "Theorem \\ref{theorem-colimit} we have", "$$", "H^p(U \\times_S X, \\mathcal{G}) =", "\\colim H^p(U_i \\times_{S_i} X_i, \\mathcal{G}_i) = 0", "$$", "for $p > 0$ (same colimit as before). Hence $R^pg_*\\mathcal{G} = 0$", "and we get the result for $p > 0$ for such a system.", "In general we may choose a short exact sequence of systems", "$$", "0 \\to (\\mathcal{F}_i, \\varphi_{i'i}) \\to", "(\\mathcal{G}_i, \\varphi_{i'i}) \\to", "(\\mathcal{Q}_i, \\varphi_{i'i}) \\to 0", "$$", "where $(\\mathcal{G}_i, \\varphi_{i'i})$ is as above, see", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-colim-sites-injective}.", "By induction the lemma holds for $p - 1$ and by the above we have", "vanishing for $p$ and $(\\mathcal{G}_i, \\varphi_{i'i})$.", "Hence the result for $p$", "and $(\\mathcal{F}_i, \\varphi_{i'i})$ by the long exact sequence", "of cohomology.", "\\medskip\\noindent", "Second proof.", "Recall that $S_{affine, \\etale} = \\colim (S_i)_{affine, \\etale}$, see", "Lemma \\ref{lemma-colimit-affine-sites}. Thus if $U$ is an object of", "$S_{affine, \\etale}$, then we can write $U = U_i \\times_{S_i} S$", "for some $i$ and some $U_i$ in $(S_i)_{affine, \\etale}$ and", "$$", "(\\colim_{i \\in I} h_i^{-1}R^p g_{i, *} \\mathcal{F}_i)(U) =", "\\colim_{i' \\geq i} (R^p g_{i', *}\\mathcal{F}_{i'})(U_i \\times_{S_i} S_{i'})", "$$", "by Sites, Lemma \\ref{sites-lemma-colimit} and the construction of the", "transition maps in the system described above. Since", "$R^pg_{i', *}\\mathcal{F}_{i'}$ is the sheaf associated to the presheaf", "$U_{i'} \\mapsto H^p(U_{i'} \\times_{S_{i'}} X_{i'}, \\mathcal{F}_{i'})$", "and since $R^pg_*\\mathcal{F}$ is the sheaf associated to the presheaf", "$U \\mapsto H^p(U \\times_S X, \\mathcal{F})$", "(Lemma \\ref{lemma-higher-direct-images})", "we obtain a canonical commutative diagram", "$$", "\\xymatrix{", "\\colim_{i' \\geq i}", "H^p(U_i \\times_{S_i} X_{i'}, \\mathcal{F}_{i'}) \\ar[r] \\ar[d] &", "\\colim_{i' \\geq i}", "(R^p g_{i', *}\\mathcal{F}_{i'})(U_i \\times_{S_i} S_{i'}) \\ar[d] \\\\", "H^p(U \\times_S X, \\mathcal{F}) \\ar[r] &", "R^pg_*\\mathcal{F}(U)", "}", "$$", "Observe that the left hand vertical arrow is an isomorphism", "by Theorem \\ref{theorem-colimit}. We're trying to show that", "the right hand vertical arrow is an isomorphism. However, we", "already know that the source and target of this arrow", "are sheaves on $S_{affine, \\etale}$. Hence it suffices to", "show: (1) an element in the target, locally comes from an", "element in the source and (2) an element in the source", "which maps to zero in the target locally vanishes.", "Part (1) follows immediately from the above and the fact that", "the lower horizontal arrow comes from a map of presheaves", "which becomes an isomorphism after sheafification.", "For part (2), say $\\xi \\in \\colim_{i' \\geq i}", "(R^p g_{i', *}\\mathcal{F}_{i'})(U_i \\times_{S_i} S_{i'})$", "is in the kernel. Choose an $i' \\geq i$ and", "$\\xi_{i'} \\in (R^p g_{i', *}\\mathcal{F}_{i'})(U_i \\times_{S_i} S_{i'})$", "representing $\\xi$.", "Choose a standard \\'etale covering", "$\\{U_{i', k} \\to U_i \\times_{S_i} S_{i'}\\}_{k = 1, \\ldots, m}$", "such that $\\xi_{i'}|_{U_{i', k}}$ comes from", "$\\xi_{i', k} \\in H^p(U_{i', k} \\times_{S_{i'}} X_{i'}, \\mathcal{F}_{i'})$.", "Since it is enough to prove that $\\xi$ dies locally, we", "may replace $U$ by the members of the \\'etale", "covering $\\{U_{i', k} \\times_{S_{i'}} S \\to U = U_i \\times_{S_i} S\\}$.", "After this replacement we see that $\\xi$ is the image of", "an element $\\xi'$ of the group", "$\\colim_{i' \\geq i} H^p(U_i \\times_{S_i} X_{i'}, \\mathcal{F}_{i'})$", "in the diagram above. Since $\\xi'$ maps to zero in $R^pg_*\\mathcal{F}(U)$", "we can do another replacement and assume that $\\xi'$ maps", "to zero in $H^p(U \\times_S X, \\mathcal{F})$.", "However, since the left vertical arrow is an isomorphism", "we then conclude $\\xi' = 0$ hence $\\xi = 0$ as desired." ], "refs": [ "sites-cohomology-lemma-base-change-map-flat-case", "sites-cohomology-remark-base-change", "sites-cohomology-remark-compose-base-change-horizontal", "etale-cohomology-theorem-colimit", "etale-cohomology-lemma-colimit-affine-sites", "sites-lemma-colimit", "etale-cohomology-theorem-colimit", "sites-cohomology-lemma-colim-sites-injective", "etale-cohomology-lemma-colimit-affine-sites", "sites-lemma-colimit", "etale-cohomology-lemma-higher-direct-images", "etale-cohomology-theorem-colimit" ], "ref_ids": [ 4223, 4424, 4426, 6384, 6471, 8534, 6384, 4225, 6471, 8534, 6474, 6384 ] } ], "ref_ids": [] }, { "id": 6477, "type": "theorem", "label": "etale-cohomology-lemma-linus-hamann", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-linus-hamann", "contents": [ "Let $X = \\lim_{i \\in I} X_i$ be a directed limit of schemes", "with affine transition morphisms $f_{i'i}$ and projection morphisms", "$f_i : X \\to X_i$. Let $\\mathcal{F}$ be a sheaf on $X_\\etale$. Then", "\\begin{enumerate}", "\\item there are canonical maps", "$\\varphi_{i'i} : f_{i'i}^{-1}f_{i, *}\\mathcal{F} \\to f_{i', *}\\mathcal{F}$", "such that $(f_{i, *}\\mathcal{F}, \\varphi_{i'i})$ is a system of", "sheaves on $(X_i, f_{i'i})$ as in", "Definition \\ref{definition-inverse-system-sheaves}, and", "\\item $\\mathcal{F} = \\colim f_i^{-1}f_{i, *}\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [ "etale-cohomology-definition-inverse-system-sheaves" ], "proofs": [ { "contents": [ "Via Lemmas \\ref{lemma-alternative} and \\ref{lemma-colimit-affine-sites}", "this is a special case of", "Sites, Lemma \\ref{sites-lemma-colimit-push-pull}." ], "refs": [ "etale-cohomology-lemma-alternative", "etale-cohomology-lemma-colimit-affine-sites", "sites-lemma-colimit-push-pull" ], "ref_ids": [ 6413, 6471, 8535 ] } ], "ref_ids": [ 6745 ] }, { "id": 6478, "type": "theorem", "label": "etale-cohomology-lemma-compute-strangely", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-compute-strangely", "contents": [ "Let $I$ be a directed set. Let $g_i : X_i \\to S_i$ be an inverse system of", "morphisms of schemes over $I$. Assume $g_i$ is quasi-compact and", "quasi-separated and for $i' \\geq i$ the transition morphisms", "$X_{i'} \\to X_i$ and $S_{i'} \\to S_i$ are affine.", "Let $g : X \\to S$ be the limit of the morphisms $g_i$, see", "Limits, Section \\ref{limits-section-limits}.", "Denote $f_i : X \\to X_i$ and $h_i : S \\to S_i$ the projections.", "Let $\\mathcal{F}$ be an abelian sheaf on $X$. Then we have", "$$", "R^pg_*\\mathcal{F} = \\colim_{i \\in I} h_i^{-1}R^pg_{i, *}(f_{i, *}\\mathcal{F})", "$$" ], "refs": [], "proofs": [ { "contents": [ "Formal combination of Lemmas \\ref{lemma-relative-colimit-general}", "and \\ref{lemma-linus-hamann}." ], "refs": [ "etale-cohomology-lemma-relative-colimit-general", "etale-cohomology-lemma-linus-hamann" ], "ref_ids": [ 6476, 6477 ] } ], "ref_ids": [] }, { "id": 6479, "type": "theorem", "label": "etale-cohomology-lemma-prepare-leray", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-prepare-leray", "contents": [ "Let $f: X \\to Y$ be a morphism and $\\mathcal{I}$ an injective object of", "$\\textit{Ab}(X_\\etale)$. Let $V \\in \\Ob(Y_\\etale)$. Then", "\\begin{enumerate}", "\\item for any covering $\\mathcal{V} = \\{V_j\\to V\\}_{j \\in J}$ we have", "$\\check H^p(\\mathcal{V}, f_*\\mathcal{I}) = 0$ for all $p > 0$,", "\\item $f_*\\mathcal{I}$ is acyclic for the functor $\\Gamma(V, -)$, and", "\\item if $g : Y \\to Z$, then $f_*\\mathcal{I}$ is acyclic for $g_*$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, f_*\\mathcal{I}) =", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V} \\times_Y X, \\mathcal{I})$", "which has vanishing higher cohomology groups by Lemma \\ref{lemma-hom-injective}.", "This proves (1). The second statement follows as a sheaf which has", "vanishing higher {\\v C}ech cohomology groups for any covering has vanishing", "higher cohomology groups. This a wonderful exercise in using the", "{\\v C}ech-to-cohomology spectral sequence, but see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cech-vanish-collection}", "for details and a more precise and general statement.", "Part (3) is a consequence of (2) and the description of", "$R^pg_*$ in Lemma \\ref{lemma-higher-direct-images}." ], "refs": [ "etale-cohomology-lemma-hom-injective", "sites-cohomology-lemma-cech-vanish-collection", "etale-cohomology-lemma-higher-direct-images" ], "ref_ids": [ 6408, 4205, 6474 ] } ], "ref_ids": [] }, { "id": 6480, "type": "theorem", "label": "etale-cohomology-lemma-vanishing-etale-cohomology-strictly-henselian", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-vanishing-etale-cohomology-strictly-henselian", "contents": [ "Let $R$ be a strictly henselian local ring. Set $S = \\Spec(R)$ and let", "$\\overline{s}$ be its closed point. Then the global", "sections functor", "$\\Gamma(S, -) : \\textit{Ab}(S_\\etale) \\to \\textit{Ab}$", "is exact. In fact we have $\\Gamma(S, \\mathcal{F}) = \\mathcal{F}_{\\overline{s}}$", "for any sheaf of sets $\\mathcal{F}$. In particular", "$$", "\\forall p\\geq 1, \\quad H_\\etale^p(S, \\mathcal{F})=0", "$$", "for all $\\mathcal{F}\\in \\textit{Ab}(S_\\etale)$." ], "refs": [], "proofs": [ { "contents": [ "If we show that $\\Gamma(S, \\mathcal{F}) = \\mathcal{F}_{\\overline{s}}$", "then $\\Gamma(S, -)$ is exact as the stalk functor is exact.", "Let $(U, \\overline{u})$ be an \\'etale neighbourhood of $\\overline{s}$.", "Pick an affine open neighborhood $\\Spec(A)$ of $\\overline{u}$ in $U$.", "Then $R \\to A$ is \\'etale and $\\kappa(\\overline{s}) = \\kappa(\\overline{u})$.", "By Theorem \\ref{theorem-henselian} we see that $A \\cong R \\times A'$", "as an $R$-algebra compatible with maps to", "$\\kappa(\\overline{s}) = \\kappa(\\overline{u})$.", "Hence we get a section", "$$", "\\xymatrix{", "\\Spec(A) \\ar[r] & U \\ar[d]\\\\", "& S \\ar[ul]", "}", "$$", "It follows that in the system of \\'etale neighbourhoods of $\\overline{s}$", "the identity map $(S, \\overline{s}) \\to (S, \\overline{s})$ is cofinal.", "Hence $\\Gamma(S, \\mathcal{F}) = \\mathcal{F}_{\\overline{s}}$.", "The final statement of the lemma follows as the higher derived", "functors of an exact functor are zero, see", "Derived Categories, Lemma \\ref{derived-lemma-right-derived-exact-functor}." ], "refs": [ "etale-cohomology-theorem-henselian", "derived-lemma-right-derived-exact-functor" ], "ref_ids": [ 6379, 1845 ] } ], "ref_ids": [] }, { "id": 6481, "type": "theorem", "label": "etale-cohomology-lemma-finite-pushforward-commutes-with-base-change", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-finite-pushforward-commutes-with-base-change", "contents": [ "Consider a cartesian square", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "of schemes with $f$ a finite morphism.", "For any sheaf of sets $\\mathcal{F}$ on $X_\\etale$ we have", "$f'_*(g')^{-1}\\mathcal{F} = g^{-1}f_*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "In great generality there is a pullback map", "$g^{-1}f_*\\mathcal{F} \\to f'_*(g')^{-1}\\mathcal{F}$, see", "Sites, Section \\ref{sites-section-pullback}.", "It suffices to check on stalks (Theorem \\ref{theorem-exactness-stalks}).", "Let $\\overline{y}' : \\Spec(k) \\to Y'$ be a geometric point.", "We have", "\\begin{align*}", "(f'_*(g')^{-1}\\mathcal{F})_{\\overline{y}'}", "& =", "\\prod\\nolimits_{\\overline{x}' : \\Spec(k) \\to X',\\ f' \\circ \\overline{x}' =", "\\overline{y}'}", "((g')^{-1}\\mathcal{F})_{\\overline{x}'} \\\\", "& =", "\\prod\\nolimits_{\\overline{x}' : \\Spec(k) \\to X',\\ f' \\circ \\overline{x}' =", "\\overline{y}'} \\mathcal{F}_{g' \\circ \\overline{x}'} \\\\", "& =", "\\prod\\nolimits_{\\overline{x} : \\Spec(k) \\to X,\\ f \\circ \\overline{x} =", "g \\circ \\overline{y}'} \\mathcal{F}_{\\overline{x}} \\\\", "& =", "(f_*\\mathcal{F})_{g \\circ \\overline{y}'} \\\\", "& =", "(g^{-1}f_*\\mathcal{F})_{\\overline{y}'}", "\\end{align*}", "The first equality by", "Proposition \\ref{proposition-finite-higher-direct-image-zero}.", "The second equality by", "Lemma \\ref{lemma-stalk-pullback}.", "The third equality holds because the diagram is a cartesian square", "and hence the map", "$$", "\\{\\overline{x}' : \\Spec(k) \\to X',\\ f' \\circ \\overline{x}' =", "\\overline{y}'\\}", "\\longrightarrow", "\\{\\overline{x} : \\Spec(k) \\to X,\\ f \\circ \\overline{x} =", "g \\circ \\overline{y}'\\}", "$$", "sending $\\overline{x}'$ to $g' \\circ \\overline{x}'$ is a bijection.", "The fourth equality by", "Proposition \\ref{proposition-finite-higher-direct-image-zero}.", "The fifth equality by", "Lemma \\ref{lemma-stalk-pullback}." ], "refs": [ "etale-cohomology-theorem-exactness-stalks", "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-lemma-stalk-pullback", "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-lemma-stalk-pullback" ], "ref_ids": [ 6376, 6703, 6436, 6703, 6436 ] } ], "ref_ids": [] }, { "id": 6482, "type": "theorem", "label": "etale-cohomology-lemma-integral-pushforward-commutes-with-base-change", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-integral-pushforward-commutes-with-base-change", "contents": [ "Consider a cartesian square", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "of schemes with $f$ an integral morphism.", "For any sheaf of sets $\\mathcal{F}$ on $X_\\etale$ we have", "$f'_*(g')^{-1}\\mathcal{F} = g^{-1}f_*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $Y$ and hence we may assume $Y$ is affine.", "Then we can write $X = \\lim X_i$ with $f_i : X_i \\to Y$ finite", "(this is easy in the affine case, but see", "Limits, Lemma \\ref{limits-lemma-integral-limit-finite-and-finite-presentation}", "for a reference). Denote $p_{i'i} : X_{i'} \\to X_i$", "the transition morphisms and $p_i : X \\to X_i$ the projections.", "Setting $\\mathcal{F}_i = p_{i, *}\\mathcal{F}$ we obtain", "from Lemma \\ref{lemma-linus-hamann}", "a system $(\\mathcal{F}_i, \\varphi_{i'i})$", "with $\\mathcal{F} = \\colim p_i^{-1}\\mathcal{F}_i$.", "We get $f_*\\mathcal{F} = \\colim f_{i, *}\\mathcal{F}_i$", "from Lemma \\ref{lemma-relative-colimit}.", "Set $X'_i = Y' \\times_Y X_i$ with projections $f'_i$ and $g'_i$.", "Then $X' = \\lim X'_i$ as limits commute with limits.", "Denote $p'_i : X' \\to X'_i$ the projections. We have", "\\begin{align*}", "g^{-1}f_*\\mathcal{F}", "& =", "g^{-1} \\colim f_{i, *}\\mathcal{F}_i \\\\", "& =", "\\colim g^{-1}f_{i, *}\\mathcal{F}_i \\\\", "& =", "\\colim f'_{i, *}(g'_i)^{-1}\\mathcal{F}_i \\\\", "& =", "f'_*(\\colim (p'_i)^{-1}(g'_i)^{-1}\\mathcal{F}_i) \\\\", "& =", "f'_*(\\colim (g')^{-1}p_i^{-1}\\mathcal{F}_i) \\\\", "& =", "f'_*(g')^{-1} \\colim p_i^{-1}\\mathcal{F}_i \\\\", "& =", "f'_*(g')^{-1}\\mathcal{F}", "\\end{align*}", "as desired. For the first equality see above.", "For the second use that pullback commutes with colimits.", "For the third use the finite case, see", "Lemma \\ref{lemma-finite-pushforward-commutes-with-base-change}.", "For the fourth use Lemma \\ref{lemma-relative-colimit}.", "For the fifth use that $g'_i \\circ p'_i = p_i \\circ g'$.", "For the sixth use that pullback commutes with colimits.", "For the seventh use $\\mathcal{F} = \\colim p_i^{-1}\\mathcal{F}_i$." ], "refs": [ "limits-lemma-integral-limit-finite-and-finite-presentation", "etale-cohomology-lemma-linus-hamann", "etale-cohomology-lemma-relative-colimit", "etale-cohomology-lemma-finite-pushforward-commutes-with-base-change", "etale-cohomology-lemma-relative-colimit" ], "ref_ids": [ 15056, 6477, 6475, 6481, 6475 ] } ], "ref_ids": [] }, { "id": 6483, "type": "theorem", "label": "etale-cohomology-lemma-cohomological-descent-finite", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomological-descent-finite", "contents": [ "Let $f : X \\to Y$ be a surjective finite morphism of schemes.", "Set $f_n : X_n \\to Y$ equal to the $(n + 1)$-fold fibre product", "of $X$ over $Y$. For $\\mathcal{F} \\in \\textit{Ab}(Y_\\etale)$ set", "$\\mathcal{F}_n = f_{n, *}f_n^{-1}\\mathcal{F}$. There is an exact", "sequence", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{F}_0 \\to \\mathcal{F}_1 \\to", "\\mathcal{F}_2 \\to \\ldots", "$$", "on $X_\\etale$. Moreover, there is a spectral sequence", "$$", "E_1^{p, q} = H^q_\\etale(X_p, f_p^{-1}\\mathcal{F})", "$$", "converging to $H^{p + q}(Y_\\etale, \\mathcal{F})$.", "This spectral sequence is functorial in $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "If we prove the first statement of the lemma, then we obtain a spectral", "sequence with $E_1^{p, q} = H^q_\\etale(Y, \\mathcal{F})$ converging", "to $H^{p + q}(Y_\\etale, \\mathcal{F})$, see", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "On the other hand, since", "$R^if_{p, *}f_p^{-1}\\mathcal{F} = 0$ for $i > 0$", "(Proposition \\ref{proposition-finite-higher-direct-image-zero})", "we get", "$$", "H^q_\\etale(X_p, f_p^{-1}\\mathcal{F}) =", "H^q_\\etale(Y, f_{p, *}f_p^{-1} \\mathcal{F}) =", "H^q_\\etale(Y, \\mathcal{F}_p)", "$$", "by Proposition \\ref{proposition-leray}", "and we get the spectral sequence of the lemma.", "\\medskip\\noindent", "To prove the first statement of the lemma, observe that", "$X_n$ forms a simplicial scheme over $Y$, see", "Simplicial, Example \\ref{simplicial-example-fibre-products-simplicial-object}.", "Observe moreover, that for each of the projections", "$d_j : X_{n + 1} \\to X_n$ there is a map", "$d_j^{-1} f_n^{-1}\\mathcal{F} \\to f_{n + 1}^{-1}\\mathcal{F}$.", "These maps induce maps", "$$", "\\delta_j : \\mathcal{F}_n \\to \\mathcal{F}_{n + 1}", "$$", "for $j = 0, \\ldots, n + 1$. We use the alternating sum of these maps", "to define the differentials $\\mathcal{F}_n \\to \\mathcal{F}_{n + 1}$.", "Similarly, there is a canonical augmentation $\\mathcal{F} \\to \\mathcal{F}_0$,", "namely this is just the canonical map $\\mathcal{F} \\to f_*f^{-1}\\mathcal{F}$.", "To check that this sequence of sheaves is an exact complex it suffices", "to check on stalks at geometric points (Theorem \\ref{theorem-exactness-stalks}).", "Thus we let $\\overline{y} : \\Spec(k) \\to Y$ be a geometric point. Let", "$E = \\{\\overline{x} : \\Spec(k) \\to X \\mid f(\\overline{x}) = \\overline{y}\\}$.", "Then $E$ is a finite nonempty set and we see that", "$$", "(\\mathcal{F}_n)_{\\overline{y}} =", "\\bigoplus\\nolimits_{e \\in E^{n + 1}} \\mathcal{F}_{\\overline{y}}", "$$", "by Proposition \\ref{proposition-finite-higher-direct-image-zero}", "and Lemma \\ref{lemma-stalk-pullback}.", "Thus we have to see that given an abelian group $M$ the sequence", "$$", "0 \\to M \\to \\bigoplus\\nolimits_{e \\in E} M \\to", "\\bigoplus\\nolimits_{e \\in E^2} M \\to \\ldots", "$$", "is exact. Here the first map is the diagonal map and the map", "$\\bigoplus_{e \\in E^{n + 1}} M \\to \\bigoplus_{e \\in E^{n + 2}} M$", "is the alternating sum of the maps induced by the $(n + 2)$", "projections $E^{n + 2} \\to E^{n + 1}$. This can be shown directly", "or deduced by applying Simplicial, Lemma", "\\ref{simplicial-lemma-fibre-products-simplicial-object-w-section}", "to the map $E \\to \\{*\\}$." ], "refs": [ "derived-lemma-two-ss-complex-functor", "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-proposition-leray", "etale-cohomology-theorem-exactness-stalks", "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-lemma-stalk-pullback" ], "ref_ids": [ 1871, 6703, 6702, 6376, 6703, 6436 ] } ], "ref_ids": [] }, { "id": 6484, "type": "theorem", "label": "etale-cohomology-lemma-global-sections-point", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-global-sections-point", "contents": [ "Assumptions and notations as in", "Theorem \\ref{theorem-equivalence-sheaves-point}.", "There is a functorial bijection", "$$", "\\Gamma(S, \\mathcal{F}) = (\\mathcal{F}_{\\overline{s}})^G", "$$" ], "refs": [ "etale-cohomology-theorem-equivalence-sheaves-point" ], "proofs": [ { "contents": [ "We can prove this using formal arguments and the result of", "Theorem \\ref{theorem-equivalence-sheaves-point}", "as follows. Given a sheaf $\\mathcal{F}$ corresponding to", "the $G$-set $M = \\mathcal{F}_{\\overline{s}}$ we have", "\\begin{eqnarray*}", "\\Gamma(S, \\mathcal{F}) & = &", "\\Mor_{\\Sh(S_\\etale)}(h_{\\Spec(K)}, \\mathcal{F})", "\\\\", "& = & \\Mor_{G\\textit{-Sets}}(\\{*\\}, M) \\\\", "& = & M^G", "\\end{eqnarray*}", "Here the first identification is explained in", "Sites, Sections \\ref{sites-section-presheaves} and", "\\ref{sites-section-representable-sheaves},", "the second results from", "Theorem \\ref{theorem-equivalence-sheaves-point}", "and the third is clear. We will also give a direct proof\\footnote{For", "the doubting Thomases out there.}.", "\\medskip\\noindent", "Suppose that $t \\in \\Gamma(S, \\mathcal{F})$ is a global section.", "Then the triple $(S, \\overline{s}, t)$ defines an element of", "$\\mathcal{F}_{\\overline{s}}$ which is clearly invariant under the", "action of $G$. Conversely, suppose that $(U, \\overline{u}, t)$", "defines an element of $\\mathcal{F}_{\\overline{s}}$ which is invariant.", "Then we may shrink $U$ and assume $U = \\Spec(L)$ for some", "finite separable field extension of $K$, see", "Proposition \\ref{proposition-etale-morphisms}.", "In this case the map $\\mathcal{F}(U) \\to \\mathcal{F}_{\\overline{s}}$", "is injective, because for any morphism of \\'etale neighbourhoods", "$(U', \\overline{u}') \\to (U, \\overline{u})$ the restriction map", "$\\mathcal{F}(U) \\to \\mathcal{F}(U')$ is injective since $U' \\to U$", "is a covering of $S_\\etale$.", "After enlarging $L$ a bit we may assume $K \\subset L$ is a finite", "Galois extension. At this point we use that", "$$", "\\Spec(L) \\times_{\\Spec(K)} \\Spec(L)", "=", "\\coprod\\nolimits_{\\sigma \\in \\text{Gal}(L/K)} \\Spec(L)", "$$", "where the maps $\\Spec(L) \\to \\Spec(L \\otimes_K L)$", "come from the ring maps $a \\otimes b \\mapsto a\\sigma(b)$. Hence we", "see that the condition that $(U, \\overline{u}, t)$ is invariant", "under all of $G$ implies that $t \\in \\mathcal{F}(\\Spec(L))$", "maps to the same element of", "$\\mathcal{F}(\\Spec(L) \\times_{\\Spec(K)} \\Spec(L))$", "via restriction by either projection (this uses the injectivity mentioned", "above; details omitted). Hence the sheaf condition of $\\mathcal{F}$", "for the \\'etale covering $\\{\\Spec(L) \\to \\Spec(K)\\}$ kicks", "in and we conclude that $t$ comes from a unique section of $\\mathcal{F}$", "over $\\Spec(K)$." ], "refs": [ "etale-cohomology-theorem-equivalence-sheaves-point", "etale-cohomology-theorem-equivalence-sheaves-point", "etale-cohomology-proposition-etale-morphisms" ], "ref_ids": [ 6386, 6386, 6697 ] } ], "ref_ids": [ 6386 ] }, { "id": 6485, "type": "theorem", "label": "etale-cohomology-lemma-modules-abelian", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-modules-abelian", "contents": [ "Let $G$ be a topological group. Let $R$ be a ring.", "For every $i \\geq 0$ the diagram", "$$", "\\xymatrix{", "\\text{Mod}_{R, G} \\ar[rr]_{H^i(G, -)} \\ar[d] & &", "\\text{Mod}_R \\ar[d] \\\\", "\\text{Mod}_G \\ar[rr]^{H^i(G, -)} & &", "\\textit{Ab}", "}", "$$", "whose vertical arrows are the forgetful functors is commutative." ], "refs": [], "proofs": [ { "contents": [ "Let us denote the forgetful functor $F : \\text{Mod}_{R, G} \\to \\text{Mod}_G$.", "Then $F$ has a left adjoint $H : \\text{Mod}_G \\to \\text{Mod}_{R, G}$", "given by $H(M) = M \\otimes_\\mathbf{Z} R$. Observe that every object of", "$\\text{Mod}_G$ is a quotient of a direct sum of modules of the form", "$\\mathbf{Z}[G/U]$ where $U \\subset G$ is an open subgroup.", "Here $\\mathbf{Z}[G/U]$ denotes the $G$-modules of", "finite $\\mathbf{Z}$-linear combinations", "of right $U$ congruence classes in $G$ endowed with left $G$-action.", "Thus every bounded above complex in $\\text{Mod}_G$ is quasi-isomorphic", "to a bounded above complex in $\\text{Mod}_G$ whose underlying", "terms are flat $\\mathbf{Z}$-modules", "(Derived Categories, Lemma \\ref{derived-lemma-subcategory-left-resolution}).", "Thus it is clear that $LH$ exists on $D^-(\\text{Mod}_G)$ and is computed by", "evaluating $H$ on any complex whose terms are flat $\\mathbf{Z}$-modules;", "this follows from", "Derived Categories, Lemma \\ref{derived-lemma-subcategory-left-acyclics} and", "Proposition \\ref{derived-proposition-enough-acyclics}.", "We conclude from Derived Categories, Lemma", "\\ref{derived-lemma-pre-derived-adjoint-functors}", "that", "$$", "\\text{Ext}^i(\\mathbf{Z}, F(M)) = \\text{Ext}^i(R, M)", "$$", "for $M$ in $\\textit{Mod}_{R, G}$.", "Observe that $H^0(G, -) = \\Hom(\\mathbf{Z}, -)$ on", "$\\text{Mod}_G$ where $\\mathbf{Z}$ denotes the $G$-module", "with trivial action. Hence", "$H^i(G, -) = \\text{Ext}^i(\\mathbf{Z}, -)$ on $\\text{Mod}_G$.", "Similarly we have $H^i(G, -) = \\text{Ext}^i(R, -)$ on", "$\\text{Mod}_{R, G}$. Combining everything we see that the lemma is true." ], "refs": [ "derived-lemma-subcategory-left-resolution", "derived-lemma-subcategory-left-acyclics", "derived-proposition-enough-acyclics", "derived-lemma-pre-derived-adjoint-functors" ], "ref_ids": [ 1835, 1838, 1962, 1906 ] } ], "ref_ids": [] }, { "id": 6486, "type": "theorem", "label": "etale-cohomology-lemma-ext-modules-hom", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-ext-modules-hom", "contents": [ "Let $G$ be a topological group. Let $R$ be a ring.", "Let $M$, $N$ be $R\\text{-}G$-modules. If $M$ is finite projective", "as an $R$-module, then", "$\\text{Ext}^i(M, N) = H^i(G, M^\\vee \\otimes_R N)$ (for notation", "see proof)." ], "refs": [], "proofs": [ { "contents": [ "The module $M^\\vee = \\Hom_R(M, R)$ endowed with the contragredient", "action of $G$. Namely $(g \\cdot \\lambda)(m) = \\lambda(g^{-1} \\cdot m)$", "for $g \\in G$, $\\lambda \\in M^\\vee$, $m \\in M$. The action of $G$ on", "$M^\\vee \\otimes_R N$ is the diagonal one, i.e., given by", "$g \\cdot (\\lambda \\otimes n) = g \\cdot \\lambda \\otimes g \\cdot n$.", "Note that for a third $R\\text{-}G$-module $E$ we have", "$\\Hom(E, M^\\vee \\otimes_R N) = \\Hom(M \\otimes_R E, N)$.", "Namely, this is true on the level of $R$-modules by", "Algebra, Lemmas \\ref{algebra-lemma-hom-from-tensor-product} and", "\\ref{algebra-lemma-evaluation-map-iso-finite-projective}", "and the definitions of $G$-actions are chosen such that it", "remains true for $R\\text{-}G$-modules. It follows that", "$M^\\vee \\otimes_R N$ is an injective $R\\text{-}G$-module", "if $N$ is an injective $R\\text{-}G$-module. Hence if", "$N \\to N^\\bullet$ is an injective resolution, then", "$M^\\vee \\otimes_R N \\to M^\\vee \\otimes_R N^\\bullet$", "is an injective resolution. Then", "$$", "\\Hom(M, N^\\bullet) = \\Hom(R, M^\\vee \\otimes_R N^\\bullet) =", "(M^\\vee \\otimes_R N^\\bullet)^G", "$$", "Since the left hand side computes $\\text{Ext}^i(M, N)$ and the right", "hand side computes $H^i(G, M^\\vee \\otimes_R N)$ the proof is complete." ], "refs": [ "algebra-lemma-hom-from-tensor-product", "algebra-lemma-evaluation-map-iso-finite-projective" ], "ref_ids": [ 362, 801 ] } ], "ref_ids": [] }, { "id": 6487, "type": "theorem", "label": "etale-cohomology-lemma-finite-dim-group-cohomology", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-finite-dim-group-cohomology", "contents": [ "Let $G$ be a topological group. Let $k$ be a field.", "Let $V$ be a $k\\text{-}G$-module.", "If $G$ is topologically finitely generated and", "$\\dim_k(V) < \\infty$, then $\\dim_k H^1(G, V) < \\infty$." ], "refs": [], "proofs": [ { "contents": [ "Let $g_1, \\ldots, g_r \\in G$ be elements which topologically generate $G$,", "i.e., this means that the subgroup generated by $g_1, \\ldots, g_r$ is dense.", "By Lemma \\ref{lemma-ext-modules-hom}", "we see that $H^1(G, V)$ is the $k$-vector space of extensions", "$$", "0 \\to V \\to E \\to k \\to 0", "$$", "of $k\\text{-}G$-modules. Choose $e \\in E$ mapping to $1 \\in k$.", "Write", "$$", "g_i \\cdot e = v_i + e", "$$", "for some $v_i \\in V$. This is possible because $g_i \\cdot 1 = 1$.", "We claim that the list of elements $v_1, \\ldots, v_r \\in V$", "determine the isomorphism class of the extension $E$.", "Once we prove this the lemma follows as this means that our", "Ext vector space is isomorphic to a subquotient of the $k$-vector", "space $V^{\\oplus r}$; some details omitted.", "Since $E$ is an object of the category defined in", "Definition \\ref{definition-G-module-continuous}", "we know there is an open subgroup $U$ such that", "$u \\cdot e = e$ for all $u \\in U$.", "Now pick any $g \\in G$. Then $gU$ contains a word $w$ in", "the elements $g_1, \\ldots, g_r$.", "Say $gu = w$. Since the element $w \\cdot e$ is determined by", "$v_1, \\ldots, v_r$, we see that $g \\cdot e = (gu) \\cdot e = w \\cdot e$", "is too." ], "refs": [ "etale-cohomology-lemma-ext-modules-hom", "etale-cohomology-definition-G-module-continuous" ], "ref_ids": [ 6486, 6747 ] } ], "ref_ids": [] }, { "id": 6488, "type": "theorem", "label": "etale-cohomology-lemma-profinite-group-cohomology-torsion", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-profinite-group-cohomology-torsion", "contents": [ "Let $G$ be a profinite topological group.", "Then", "\\begin{enumerate}", "\\item $H^i(G, M)$ is torsion for $i > 0$ and any $G$-module $M$, and", "\\item $H^i(G, M) = 0$ if $M$ is a $\\mathbf{Q}$-vector space.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). By dimension shifting we see that it suffices", "to show that $H^1(G, M)$ is torsion for every $G$-module $M$.", "Choose an exact sequence $0 \\to M \\to I \\to N \\to 0$ with $I$", "an injective object of the category of $G$-modules.", "Then any element of $H^1(G, M)$ is the image of an element", "$y \\in N^G$. Choose $x \\in I$ mapping to $y$.", "The stabilizer $U \\subset G$ of $x$ is open, hence", "has finite index $r$. Let $g_1, \\ldots, g_r \\in G$ be a system", "of representatives for $G/U$. Then $\\sum g_i(x)$ is an invariant", "element of $I$ which maps to $ry$. Thus $r$ kills the element", "of $H^1(G, M)$ we started with. Part (2) follows as then", "$H^i(G, M)$ is both a $\\mathbf{Q}$-vector space and torsion." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6489, "type": "theorem", "label": "etale-cohomology-lemma-equivalence-abelian-sheaves-point", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-equivalence-abelian-sheaves-point", "contents": [ "Let $S = \\Spec(K)$ with $K$ a field.", "Let $\\overline{s}$ be a geometric point of $S$.", "Let $G = \\text{Gal}_{\\kappa(s)}$ denote the absolute Galois group.", "The stalk functor induces an equivalence of categories", "$$", "\\textit{Ab}(S_\\etale) \\longrightarrow \\text{Mod}_G,", "\\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}_{\\overline{s}}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "In", "Theorem \\ref{theorem-equivalence-sheaves-point}", "we have seen the equivalence between sheaves of sets and $G$-sets.", "The current lemma follows formally from this as an abelian sheaf is just", "a sheaf of sets endowed with a commutative group law, and a $G$-module", "is just a $G$-set endowed with a commutative group law." ], "refs": [ "etale-cohomology-theorem-equivalence-sheaves-point" ], "ref_ids": [ 6386 ] } ], "ref_ids": [] }, { "id": 6490, "type": "theorem", "label": "etale-cohomology-lemma-compare-cohomology-point", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-compare-cohomology-point", "contents": [ "Notation and assumptions as in", "Lemma \\ref{lemma-equivalence-abelian-sheaves-point}.", "Let $\\mathcal{F}$ be an abelian sheaf on $\\Spec(K)_\\etale$", "which corresponds to the $G$-module $M$.", "Then", "\\begin{enumerate}", "\\item in $D(\\textit{Ab})$ we have a canonical isomorphism", "$R\\Gamma(S, \\mathcal{F}) = R\\Gamma_G(M)$,", "\\item $H_\\etale^0(S, \\mathcal{F}) = M^G$, and", "\\item $H_\\etale^q(S, \\mathcal{F}) = H^q(G, M)$.", "\\end{enumerate}" ], "refs": [ "etale-cohomology-lemma-equivalence-abelian-sheaves-point" ], "proofs": [ { "contents": [ "Combine", "Lemma \\ref{lemma-equivalence-abelian-sheaves-point}", "with", "Lemma \\ref{lemma-global-sections-point}." ], "refs": [ "etale-cohomology-lemma-equivalence-abelian-sheaves-point", "etale-cohomology-lemma-global-sections-point" ], "ref_ids": [ 6489, 6484 ] } ], "ref_ids": [ 6489 ] }, { "id": 6491, "type": "theorem", "label": "etale-cohomology-lemma-all-modules-quasi-coherent", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-all-modules-quasi-coherent", "contents": [ "Let $R$ be a local ring of dimension $0$. Let $S = \\Spec(R)$.", "Then every $\\mathcal{O}_S$-module on $S_\\etale$ is quasi-coherent." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be an $\\mathcal{O}_S$-module on $S_\\etale$.", "We have to show that $\\mathcal{F}$ is determined by the", "$R$-module $M = \\Gamma(S, \\mathcal{F})$.", "More precisely, if $\\pi : X \\to S$ is \\'etale we have to", "show that $\\Gamma(X, \\mathcal{F}) = \\Gamma(X, \\pi^*\\widetilde{M})$.", "\\medskip\\noindent", "Let $\\mathfrak m \\subset R$ be the maximal ideal and let", "$\\kappa$ be the residue field. By", "Algebra, Lemma \\ref{algebra-lemma-local-dimension-zero-henselian}", "the local ring $R$ is henselian. If $X \\to S$ is \\'etale,", "then the underlying topological space of $X$ is discrete", "by Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}", "and hence $X$ is a disjoint union of affine schemes", "each having one point. Moreover, if $X = \\Spec(A)$ is affine and", "has one point, then $R \\to A$ is finite \\'etale by", "Algebra, Lemma \\ref{algebra-lemma-mop-up}.", "We have to show that $\\Gamma(X, \\mathcal{F}) = M \\otimes_R A$", "in this case.", "\\medskip\\noindent", "The functor $A \\mapsto A/\\mathfrak m A$ defines an equivalence of", "the category of finite \\'etale $R$-algebras", "with the category of finite separable $\\kappa$-algebras by", "Algebra, Lemma \\ref{algebra-lemma-henselian-cat-finite-etale}.", "Let us first consider the case where $A/\\mathfrak m A$", "is a Galois extension of $\\kappa$ with Galois group $G$.", "For each $\\sigma \\in G$ let $\\sigma : A \\to A$ denote the", "corresponding automorphism of $A$ over $R$.", "Let $N = \\Gamma(X, \\mathcal{F})$.", "Then $\\Spec(\\sigma) : X \\to X$ is an automorphism over $S$", "and hence pullback by this defines a map $\\sigma : N \\to N$", "which is a $\\sigma$-linear map: $\\sigma(an) = \\sigma(a) \\sigma(n)$", "for $a \\in A$ and $n \\in N$.", "We will apply Galois descent to the quasi-coherent module", "$\\widetilde{N}$ on $X$ endowed with the isomorphisms", "coming from the action on $\\sigma$ on $N$. See Descent, Lemma", "\\ref{descent-lemma-galois-descent-more-general}.", "This lemma tells us there is an isomorphism $N = N^G \\otimes_R A$.", "On the other hand, it is clear that $N^G = M$ by the sheaf property", "for $\\mathcal{F}$. Thus the required isomorphism holds.", "\\medskip\\noindent", "The general case (with $A$ local and finite \\'etale over $R$)", "is deduced from the Galois case as follows. Choose $A \\to B$", "finite \\'etale such that $B$ is local with residue field", "Galois over $\\kappa$. Let $G = \\text{Aut}(B/R) = \\text{Gal}(\\kappa_B/\\kappa)$.", "Let $H \\subset G$ be the Galois group corresponding to the", "Galois extension $\\kappa_B/\\kappa_A$. Then as above one", "shows that $\\Gamma(X, \\mathcal{F}) = \\Gamma(\\Spec(B), \\mathcal{F})^H$.", "By the result for Galois extensions (used twice) we get", "$$", "\\Gamma(X, \\mathcal{F}) = (M \\otimes_R B)^H = M \\otimes_R A", "$$", "as desired." ], "refs": [ "algebra-lemma-local-dimension-zero-henselian", "morphisms-lemma-etale-over-field", "algebra-lemma-mop-up", "algebra-lemma-henselian-cat-finite-etale", "descent-lemma-galois-descent-more-general" ], "ref_ids": [ 1283, 5364, 1278, 1280, 14611 ] } ], "ref_ids": [] }, { "id": 6492, "type": "theorem", "label": "etale-cohomology-lemma-brauer-inverse", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-brauer-inverse", "contents": [ "Let $A$ be a finite central simple algebra over $K$. Then", "$$", "\\begin{matrix}", "A \\otimes_K A^{opp} & \\longrightarrow & \\text{End}_K(A) \\\\", "\\ a \\otimes a' & \\longmapsto & (x \\mapsto a x a')", "\\end{matrix}", "$$", "is an isomorphism of algebras over $K$." ], "refs": [], "proofs": [ { "contents": [ "See", "Brauer Groups, Lemma \\ref{brauer-lemma-inverse}." ], "refs": [ "brauer-lemma-inverse" ], "ref_ids": [ 7837 ] } ], "ref_ids": [] }, { "id": 6493, "type": "theorem", "label": "etale-cohomology-lemma-central-simple-algebra-pgln", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-central-simple-algebra-pgln", "contents": [ "\\begin{slogan}", "Central simple algebras are classified by Galois cohomology of PGL.", "\\end{slogan}", "Let $K$ be a field and let $K^{sep}$ be a separable algebraic closure.", "Then the set of isomorphism classes of central simple algebras of degree", "$d$ over $K$ is in bijection with the non-abelian cohomology", "$H^1(\\text{Gal}(K^{sep}/K), \\text{PGL}_d(K^{sep}))$." ], "refs": [], "proofs": [ { "contents": [ "[Sketch of proof.]", "The Skolem-Noether theorem (see", "Brauer Groups, Theorem \\ref{brauer-theorem-skolem-noether})", "implies that for any field $L$ the group", "$\\text{Aut}_{L\\text{-Algebras}}(\\text{Mat}_d(L))$", "equals $\\text{PGL}_d(L)$. By", "Theorem \\ref{theorem-central-simple-algebra}, we see that", "central simple algebras of degree $d$ correspond", "to forms of the $K$-algebra $\\text{Mat}_d(K)$.", "Combined we see that isomorphism classes of degree $d$ central", "simple algebras correspond to elements of", "$H^1(\\text{Gal}(K^{sep}/K), \\text{PGL}_d(K^{sep}))$.", "For more details on twisting, see for example", "\\cite{SilvermanEllipticCurves}." ], "refs": [ "brauer-theorem-skolem-noether", "etale-cohomology-theorem-central-simple-algebra" ], "ref_ids": [ 7823, 6387 ] } ], "ref_ids": [] }, { "id": 6494, "type": "theorem", "label": "etale-cohomology-lemma-end-unique-up-to-invertible", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-end-unique-up-to-invertible", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ and $\\mathcal{G}$ be finite locally", "free sheaves of $\\mathcal{O}_S$-modules of positive rank. If there", "exists an isomorphism", "$\\SheafHom_{\\mathcal{O}_S}(\\mathcal{F}, \\mathcal{F}) \\cong", "\\SheafHom_{\\mathcal{O}_S}(\\mathcal{G}, \\mathcal{G})$ of", "$\\mathcal{O}_S$-algebras, then there exists an invertible sheaf", "$\\mathcal{L}$ on $S$ such that", "$\\mathcal{F} \\otimes_{\\mathcal{O}_S} \\mathcal{L} \\cong \\mathcal{G}$", "and such that this isomorphism induces the given isomorphism of", "endomorphism algebras." ], "refs": [], "proofs": [ { "contents": [ "Fix an isomorphism", "$\\SheafHom_{\\mathcal{O}_S}(\\mathcal{F}, \\mathcal{F}) \\to", "\\SheafHom_{\\mathcal{O}_S}(\\mathcal{G}, \\mathcal{G})$.", "Consider the sheaf $\\mathcal{L} \\subset \\SheafHom(\\mathcal{F}, \\mathcal{G})$", "generated as an $\\mathcal{O}_S$-module by the local isomorphisms", "$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ such that conjugation by", "$\\varphi$ is the given isomorphism of endomorphism algebras.", "A local calculation (reducing to the case that $\\mathcal{F}$ and $\\mathcal{G}$", "are finite free and $S$ is affine) shows that $\\mathcal{L}$ is invertible.", "Another local calculation shows that the evaluation map", "$$", "\\mathcal{F} \\otimes_{\\mathcal{O}_S} \\mathcal{L} \\longrightarrow \\mathcal{G}", "$$", "is an isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6495, "type": "theorem", "label": "etale-cohomology-lemma-annihilated-by-degree", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-annihilated-by-degree", "contents": [ "\\begin{reference}", "Argument taken from \\cite{Saltman-torsion}.", "\\end{reference}", "Let $S$ be a scheme. Let $\\mathcal{A}$ be an Azumaya algebra which is", "locally free of rank $d^2$ over $S$. Then the class", "of $\\mathcal{A}$ in the Brauer group of $S$ is annihilated by $d$." ], "refs": [], "proofs": [ { "contents": [ "Choose an \\'etale covering $\\{U_i \\to S\\}$ and choose isomorphisms", "$\\mathcal{A}|_{U_i} \\to \\SheafHom(\\mathcal{F}_i, \\mathcal{F}_i)$", "for some locally free $\\mathcal{O}_{U_i}$-modules $\\mathcal{F}_i$", "of rank $d$. (We may assume $\\mathcal{F}_i$ is free.) Consider the", "composition", "$$", "p_i : \\mathcal{F}_i^{\\otimes d} \\to", "\\wedge^d(\\mathcal{F}_i) \\to \\mathcal{F}_i^{\\otimes d}", "$$", "The first arrow is the usual projection and the second arrow is", "the isomorphism of the top exterior power of $\\mathcal{F}_i$ with", "the submodule of sections of $\\mathcal{F}_i^{\\otimes d}$ which transform", "according to the sign character under the action of the symmetric group", "on $d$ letters. Then $p_i^2 = d! p_i$ and the rank of $p_i$ is $1$.", "Using the given isomorphism", "$\\mathcal{A}|_{U_i} \\to \\SheafHom(\\mathcal{F}_i, \\mathcal{F}_i)$", "and the canonical isomorphism", "$$", "\\SheafHom(\\mathcal{F}_i, \\mathcal{F}_i)^{\\otimes d} =", "\\SheafHom(\\mathcal{F}_i^{\\otimes d}, \\mathcal{F}_i^{\\otimes d})", "$$", "we may think of $p_i$ as a section of $\\mathcal{A}^{\\otimes d}$", "over $U_i$. We claim that $p_i|_{U_i \\times_S U_j} = p_j|_{U_i \\times_S U_j}$", "as sections of $\\mathcal{A}^{\\otimes d}$. Namely, applying", "Lemma \\ref{lemma-end-unique-up-to-invertible}", "we obtain an invertible sheaf $\\mathcal{L}_{ij}$ and a canonical isomorphism", "$$", "\\mathcal{F}_i|_{U_i \\times_S U_j} \\otimes \\mathcal{L}_{ij}", "\\longrightarrow", "\\mathcal{F}_j|_{U_i \\times_S U_j}.", "$$", "Using this isomorphism we see that $p_i$ maps to $p_j$.", "Since $\\mathcal{A}^{\\otimes d}$ is a sheaf on $S_\\etale$", "(Proposition \\ref{proposition-quasi-coherent-sheaf-fpqc}) we find a canonical", "global section $p \\in \\Gamma(S, \\mathcal{A}^{\\otimes d})$. A local calculation", "shows that", "$$", "\\mathcal{H} =", "\\Im(\\mathcal{A}^{\\otimes d} \\to \\mathcal{A}^{\\otimes d}, f \\mapsto fp)", "$$", "is a locally free module of rank $d^d$ and that (left) multiplication", "by $\\mathcal{A}^{\\otimes d}$ induces an isomorphism", "$\\mathcal{A}^{\\otimes d} \\to \\SheafHom(\\mathcal{H}, \\mathcal{H})$.", "In other words, $\\mathcal{A}^{\\otimes d}$ is the trivial element", "of the Brauer group of $S$ as desired." ], "refs": [ "etale-cohomology-lemma-end-unique-up-to-invertible", "etale-cohomology-proposition-quasi-coherent-sheaf-fpqc" ], "ref_ids": [ 6494, 6696 ] } ], "ref_ids": [] }, { "id": 6496, "type": "theorem", "label": "etale-cohomology-lemma-vanishing-affine-char-p-p", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-vanishing-affine-char-p-p", "contents": [ "Let $p$ be a prime. Let $S$ be a scheme of characteristic $p$.", "\\begin{enumerate}", "\\item If $S$ is affine, then", "$H_\\etale^q(S, \\underline{\\mathbf{Z}/p\\mathbf{Z}}) = 0$ for all", "$q \\geq 2$.", "\\item If $S$ is a quasi-compact and quasi-separated scheme of", "dimension $d$, then $H_\\etale^q(S, \\underline{\\mathbf{Z}/p\\mathbf{Z}}) = 0$", "for all $q \\geq 2 + d$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that the \\'etale cohomology of the structure sheaf is equal", "to its cohomology on the underlying topological space", "(Theorem \\ref{theorem-zariski-fpqc-quasi-coherent}).", "The first statement follows from the Artin-Schreier exact sequence", "and the vanishing of cohomology of the structure sheaf on an affine", "scheme (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}).", "The second statement follows by the same argument from", "the vanishing of Cohomology, Proposition", "\\ref{cohomology-proposition-cohomological-dimension-spectral}", "and the fact that $S$ is a spectral space", "(Properties, Lemma", "\\ref{properties-lemma-quasi-compact-quasi-separated-spectral})." ], "refs": [ "etale-cohomology-theorem-zariski-fpqc-quasi-coherent", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "cohomology-proposition-cohomological-dimension-spectral", "properties-lemma-quasi-compact-quasi-separated-spectral" ], "ref_ids": [ 6373, 3282, 2247, 2941 ] } ], "ref_ids": [] }, { "id": 6497, "type": "theorem", "label": "etale-cohomology-lemma-F-1", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-F-1", "contents": [ "Let $k$ be an algebraically closed field of characteristic $p > 0$.", "Let $V$ be a finite dimensional $k$-vector space. Let $F : V \\to V$", "be a frobenius linear map, i.e., an additive map such that", "$F(\\lambda v) = \\lambda^p F(v)$ for all $\\lambda \\in k$ and $v \\in V$.", "Then $F - 1 : V \\to V$ is surjective with kernel a finite dimensional", "$\\mathbf{F}_p$-vector space of dimension $\\leq \\dim_k(V)$." ], "refs": [], "proofs": [ { "contents": [ "If $F = 0$, then the statement holds. If we have a filtration of $V$ by", "$F$-stable subvector spaces such that the statement holds for each", "graded piece, then it holds for $(V, F)$. Combining these two remarks", "we may assume the kernel of $F$ is zero.", "\\medskip\\noindent", "Choose a basis $v_1, \\ldots, v_n$ of $V$ and write", "$F(v_i) = \\sum a_{ij} v_j$. Observe that $v = \\sum \\lambda_i v_i$", "is in the kernel if and only if $\\sum \\lambda_i^p a_{ij} v_j = 0$.", "Since $k$ is algebraically closed this implies the matrix $(a_{ij})$", "is invertible. Let $(b_{ij})$ be its inverse. Then to see that $F - 1$", "is surjective we pick $w = \\sum \\mu_i v_i \\in V$ and we try to solve", "$$", "(F - 1)(\\sum \\lambda_iv_i) =", "\\sum \\lambda_i^p a_{ij} v_j - \\sum \\lambda_j v_j = \\sum \\mu_j v_j", "$$", "This is equivalent to", "$$", "\\sum \\lambda_j^p v_j - \\sum b_{ij} \\lambda_i v_j = \\sum b_{ij} \\mu_i v_j", "$$", "in other words", "$$", "\\lambda_j^p - \\sum b_{ij} \\lambda_i = \\sum b_{ij} \\mu_i,", "\\quad j = 1, \\ldots, \\dim(V).", "$$", "The algebra", "$$", "A = k[x_1, \\ldots, x_n]/", "(x_j^p - \\sum b_{ij} x_i - \\sum b_{ij} \\mu_i)", "$$", "is standard smooth over $k$", "(Algebra, Definition \\ref{algebra-definition-standard-smooth})", "because the matrix $(b_{ij})$ is invertible and the partial derivatives", "of $x_j^p$ are zero. A basis of $A$ over $k$ is the set of monomials", "$x_1^{e_1} \\ldots x_n^{e_n}$ with $e_i < p$, hence $\\dim_k(A) = p^n$.", "Since $k$ is algebraically closed we see that $\\Spec(A)$ has exactly", "$p^n$ points. It follows that $F - 1$ is surjective and every fibre", "has $p^n$ points, i.e., the kernel of $F - 1$ is a group with $p^n$ elements." ], "refs": [ "algebra-definition-standard-smooth" ], "ref_ids": [ 1535 ] } ], "ref_ids": [] }, { "id": 6498, "type": "theorem", "label": "etale-cohomology-lemma-top-cohomology-coherent", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-top-cohomology-coherent", "contents": [ "Let $X$ be a separated scheme of finite type over a field $k$.", "Let $\\mathcal{F}$ be a coherent sheaf of $\\mathcal{O}_X$-modules.", "Then $\\dim_k H^d(X, \\mathcal{F}) < \\infty$ where $d = \\dim(X)$." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by induction on $d$. The case $d = 0$ holds because", "in that case $X$ is the spectrum of a finite dimensional $k$-algebra $A$", "(Varieties, Lemma \\ref{varieties-lemma-algebraic-scheme-dim-0})", "and every coherent sheaf $\\mathcal{F}$ corresponds to a finite $A$-module", "$M = H^0(X, \\mathcal{F})$ which has $\\dim_k M < \\infty$.", "\\medskip\\noindent", "Assume $d > 0$ and the result has been shown for separated schemes", "of finite type of dimension $< d$. The scheme $X$ is Noetherian. Consider", "the property $\\mathcal{P}$ of coherent sheaves on $X$ defined by the rule", "$$", "\\mathcal{P}(\\mathcal{F}) \\Leftrightarrow", "\\dim_k H^d(X, \\mathcal{F}) < \\infty", "$$", "We are going to use the result of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-property-initial}", "to prove that $\\mathcal{P}$ holds for every coherent sheaf on $X$.", "\\medskip\\noindent", "Let", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0", "$$", "be a short exact sequence of coherent sheaves on $X$.", "Consider the long exact sequence of cohomology", "$$", "H^d(X, \\mathcal{F}_1) \\to", "H^d(X, \\mathcal{F}) \\to", "H^d(X, \\mathcal{F}_2)", "$$", "Thus if $\\mathcal{P}$ holds for $\\mathcal{F}_1$ and $\\mathcal{F}_2$,", "then it holds for $\\mathcal{F}$.", "\\medskip\\noindent", "Let $Z \\subset X$ be an integral closed subscheme. Let $\\mathcal{I}$", "be a coherent sheaf of ideals on $Z$. To finish the proof we have to show", "that $H^d(X, i_*\\mathcal{I}) = H^d(Z, \\mathcal{I})$ is finite dimensional.", "If $\\dim(Z) < d$, then the result holds because the cohomology group", "will be zero (Cohomology, Proposition", "\\ref{cohomology-proposition-vanishing-Noetherian}).", "In this way we reduce to the situation discussed in the following paragraph.", "\\medskip\\noindent", "Assume $X$ is a variety of dimension $d$ and ", "$\\mathcal{F} = \\mathcal{I}$ is a coherent ideal sheaf. In this", "case we have a short exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_X \\to i_*\\mathcal{O}_Z \\to 0", "$$", "where $i : Z \\to X$ is the closed subscheme defined by $\\mathcal{I}$.", "By induction hypothesis we see that", "$H^{d - 1}(Z, \\mathcal{O}_Z) = H^{d - 1}(X, i_*\\mathcal{O}_Z)$ is", "finite dimensional. Thus we see that it suffices to prove the result", "for the structure sheaf.", "\\medskip\\noindent", "We can apply Chow's lemma", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-chow-Noetherian})", "to the morphism $X \\to \\Spec(k)$. Thus we get a diagram", "$$", "\\xymatrix{", "X \\ar[rd]_g & X' \\ar[d]^-{g'} \\ar[l]^\\pi \\ar[r]_i & \\mathbf{P}^n_k \\ar[dl] \\\\", "& \\Spec(k) &", "}", "$$", "as in the statement of Chow's lemma. Also, let $U \\subset X$ be", "the dense open subscheme such that $\\pi^{-1}(U) \\to U$ is an isomorphism.", "We may assume $X'$ is a variety as well, see", "Cohomology of Schemes, Remark \\ref{coherent-remark-chow-Noetherian}.", "The morphism $i' = (i, \\pi) : X' \\to \\mathbf{P}^n_X$ is", "a closed immersion (loc. cit.). Hence", "$$", "\\mathcal{L} = i^*\\mathcal{O}_{\\mathbf{P}^n_k}(1) \\cong", "(i')^*\\mathcal{O}_{\\mathbf{P}^n_X}(1)", "$$", "is $\\pi$-relatively ample (for example by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-ample-on-finite-type}).", "Hence by Cohomology of Schemes, Lemma \\ref{coherent-lemma-kill-by-twisting}", "there exists an $n \\geq 0$ such that", "$R^p\\pi_*\\mathcal{L}^{\\otimes n} = 0$ for all $p > 0$.", "Set $\\mathcal{G} = \\pi_*\\mathcal{L}^{\\otimes n}$.", "Choose any nonzero global section $s$ of $\\mathcal{L}^{\\otimes n}$.", "Since $\\mathcal{G} = \\pi_*\\mathcal{L}^{\\otimes n}$, the section $s$", "corresponds to section of $\\mathcal{G}$, i.e., a map", "$\\mathcal{O}_X \\to \\mathcal{G}$.", "Since $s|_U \\not = 0$ as $X'$ is a variety and $\\mathcal{L}$", "invertible, we see that $\\mathcal{O}_X|_U \\to \\mathcal{G}|_U$", "is nonzero. As $\\mathcal{G}|_U = \\mathcal{L}^{\\otimes n}|_{\\pi^{-1}(U)}$", "is invertible we conclude that we have a short exact sequence", "$$", "0 \\to \\mathcal{O}_X \\to \\mathcal{G} \\to \\mathcal{Q} \\to 0", "$$", "where $\\mathcal{Q}$ is coherent and supported on a proper", "closed subscheme of $X$. Arguing as before using our induction", "hypothesis, we see that it", "suffices to prove $\\dim H^d(X, \\mathcal{G}) < \\infty$.", "\\medskip\\noindent", "By the Leray spectral sequence", "(Cohomology, Lemma \\ref{cohomology-lemma-apply-Leray})", "we see that $H^d(X, \\mathcal{G}) = H^d(X', \\mathcal{L}^{\\otimes n})$.", "Let $\\overline{X}' \\subset \\mathbf{P}^n_k$ be the closure", "of $X'$. Then $\\overline{X}'$ is a projective variety of dimension $d$", "over $k$ and $X' \\subset \\overline{X}'$ is a dense open.", "The invertible sheaf $\\mathcal{L}$ is the restriction of", "$\\mathcal{O}_{\\overline{X}'}(n)$ to $X$. By", "Cohomology, Proposition", "\\ref{cohomology-proposition-cohomological-dimension-spectral}", "the map", "$$", "H^d(\\overline{X}', \\mathcal{O}_{\\overline{X}'}(n))", "\\longrightarrow", "H^d(X', \\mathcal{L}^{\\otimes n})", "$$", "is surjective. Since the cohomology group on the left has", "finite dimension by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-projective}", "the proof is complete." ], "refs": [ "varieties-lemma-algebraic-scheme-dim-0", "coherent-lemma-property-initial", "cohomology-proposition-vanishing-Noetherian", "coherent-lemma-chow-Noetherian", "coherent-remark-chow-Noetherian", "morphisms-lemma-characterize-ample-on-finite-type", "coherent-lemma-kill-by-twisting", "cohomology-lemma-apply-Leray", "cohomology-proposition-cohomological-dimension-spectral", "coherent-lemma-coherent-projective" ], "ref_ids": [ 10988, 3330, 2246, 3354, 3406, 5397, 3344, 2071, 2247, 3338 ] } ], "ref_ids": [] }, { "id": 6499, "type": "theorem", "label": "etale-cohomology-lemma-vanishing-variety-char-p-p", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-vanishing-variety-char-p-p", "contents": [ "Let $X$ be separated of finite type over an algebraically closed", "field $k$ of characteristic $p > 0$. Then", "$H_\\etale^q(X, \\underline{\\mathbf{Z}/p\\mathbf{Z}}) = 0$ for", "$q \\geq dim(X) + 1$." ], "refs": [], "proofs": [ { "contents": [ "Let $d = \\dim(X)$. By the vanishing established in", "Lemma \\ref{lemma-vanishing-affine-char-p-p}", "it suffices to show that", "$H_\\etale^{d + 1}(X, \\underline{\\mathbf{Z}/p\\mathbf{Z}}) = 0$.", "By Lemma \\ref{lemma-top-cohomology-coherent} we see that", "$H^d(X, \\mathcal{O}_X)$ is a finite dimensional $k$-vector space.", "Hence the long exact cohomology sequence associated to the", "Artin-Schreier sequence ends with", "$$", "H^d(X, \\mathcal{O}_X) \\xrightarrow{F - 1}", "H^d(X, \\mathcal{O}_X) \\to H^{d + 1}_\\etale(X, \\mathbf{Z}/p\\mathbf{Z}) \\to 0", "$$", "By Lemma \\ref{lemma-F-1} the map $F - 1$ in this sequence is surjective.", "This proves the lemma." ], "refs": [ "etale-cohomology-lemma-vanishing-affine-char-p-p", "etale-cohomology-lemma-top-cohomology-coherent", "etale-cohomology-lemma-F-1" ], "ref_ids": [ 6496, 6498, 6497 ] } ], "ref_ids": [] }, { "id": 6500, "type": "theorem", "label": "etale-cohomology-lemma-finiteness-proper-variety-char-p-p", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-finiteness-proper-variety-char-p-p", "contents": [ "Let $X$ be a proper scheme over an algebraically closed", "field $k$ of characteristic $p > 0$. Then", "\\begin{enumerate}", "\\item $H_\\etale^q(X, \\underline{\\mathbf{Z}/p\\mathbf{Z}})$", "is a finite $\\mathbf{Z}/p\\mathbf{Z}$-module for all $q$, and", "\\item $H^q_\\etale(X, \\underline{\\mathbf{Z}/p\\mathbf{Z}}) \\to", "H^q_\\etale(X_{k'}, \\underline{\\mathbf{Z}/p\\mathbf{Z}}))$", "is an isomorphism if $k \\subset k'$ is an extension of algebraically", "closed fields.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-over-affine-cohomology-finite})", "and the comparison of cohomology of", "Theorem \\ref{theorem-zariski-fpqc-quasi-coherent} the cohomology groups", "$H^q_\\etale(X, \\mathbf{G}_a) = H^q(X, \\mathcal{O}_X)$ are", "finite dimensional $k$-vector spaces. Hence by", "Lemma \\ref{lemma-F-1} the long exact cohomology sequence", "associated to the Artin-Schreier sequence, splits into", "short exact sequences", "$$", "0 \\to H_\\etale^q(X, \\underline{\\mathbf{Z}/p\\mathbf{Z}}) \\to", "H^q(X, \\mathcal{O}_X) \\xrightarrow{F - 1} H^q(X, \\mathcal{O}_X) \\to 0", "$$", "and moreover the $\\mathbf{F}_p$-dimension of the cohomology groups", "$H_\\etale^q(X, \\underline{\\mathbf{Z}/p\\mathbf{Z}})$ is equal to the", "$k$-dimension of the vector space $H^q(X, \\mathcal{O}_X)$.", "This proves the first statement. The second statement follows", "as $H^q(X, \\mathcal{O}_X) \\otimes_k k' \\to H^q(X_{k'}, \\mathcal{O}_{X_{k'}})$", "is an isomorphism by flat base change", "(Cohomology of Schemes,", "Lemma \\ref{coherent-lemma-flat-base-change-cohomology})." ], "refs": [ "coherent-lemma-proper-over-affine-cohomology-finite", "etale-cohomology-theorem-zariski-fpqc-quasi-coherent", "etale-cohomology-lemma-F-1", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 3355, 6373, 6497, 3298 ] } ], "ref_ids": [] }, { "id": 6501, "type": "theorem", "label": "etale-cohomology-lemma-pullback-locally-constant", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-pullback-locally-constant", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. If $\\mathcal{G}$ is a", "locally constant sheaf of sets, abelian groups, or $\\Lambda$-modules", "on $Y_\\etale$, the same is true for $f^{-1}\\mathcal{G}$", "on $X_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "Holds for any morphism of topoi, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-pullback-locally-constant}." ], "refs": [ "sites-modules-lemma-pullback-locally-constant" ], "ref_ids": [ 14270 ] } ], "ref_ids": [] }, { "id": 6502, "type": "theorem", "label": "etale-cohomology-lemma-pushforward-locally-constant", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-pushforward-locally-constant", "contents": [ "Let $f : X \\to Y$ be a finite \\'etale morphism of schemes.", "If $\\mathcal{F}$ is a (finite) locally constant sheaf of sets,", "(finite) locally constant sheaf of abelian groups, or", "(finite type) locally constant sheaf of $\\Lambda$-modules", "on $X_\\etale$, the same is true for $f_*\\mathcal{F}$", "on $Y_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "The construction of $f_*$ commutes with \\'etale localization.", "A finite \\'etale morphism is locally isomorphic to a disjoint union", "of isomorphisms, see", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-finite-etale-etale-local}.", "Thus the lemma says that if $\\mathcal{F}_i$, $i = 1, \\ldots, n$", "are (finite) locally constant sheaves of sets, then", "$\\prod_{i = 1, \\ldots, n} \\mathcal{F}_i$ is too.", "This is clear. Similarly for sheaves of abelian groups and modules." ], "refs": [ "etale-lemma-finite-etale-etale-local" ], "ref_ids": [ 10714 ] } ], "ref_ids": [] }, { "id": 6503, "type": "theorem", "label": "etale-cohomology-lemma-characterize-finite-locally-constant", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-characterize-finite-locally-constant", "contents": [ "Let $X$ be a scheme and $\\mathcal{F}$ a sheaf of sets on $X_\\etale$.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is finite locally constant, and", "\\item $\\mathcal{F} = h_U$ for some finite \\'etale morphism $U \\to X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "A finite \\'etale morphism is locally isomorphic to a disjoint union", "of isomorphisms, see", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-finite-etale-etale-local}.", "Thus (2) implies (1). Conversely, if $\\mathcal{F}$ is finite locally", "constant, then there exists an \\'etale covering $\\{X_i \\to X\\}$ such that", "$\\mathcal{F}|_{X_i}$ is representable by $U_i \\to X_i$ finite \\'etale.", "Arguing exactly as in the proof of", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves}", "we obtain a descent datum for schemes $(U_i, \\varphi_{ij})$ relative to", "$\\{X_i \\to X\\}$ (details omitted). This descent datum is effective for", "example by Descent, Lemma \\ref{descent-lemma-affine}", "and the resulting morphism of schemes $U \\to X$ is finite \\'etale", "by Descent, Lemmas \\ref{descent-lemma-descending-property-finite} and", "\\ref{descent-lemma-descending-property-etale}." ], "refs": [ "etale-lemma-finite-etale-etale-local", "descent-lemma-descent-data-sheaves", "descent-lemma-affine", "descent-lemma-descending-property-finite", "descent-lemma-descending-property-etale" ], "ref_ids": [ 10714, 14751, 14748, 14688, 14694 ] } ], "ref_ids": [] }, { "id": 6504, "type": "theorem", "label": "etale-cohomology-lemma-morphism-locally-constant", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-morphism-locally-constant", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map", "of locally constant sheaves of sets on $X_\\etale$.", "If $\\mathcal{F}$ is finite locally constant, there exists an", "\\'etale covering $\\{U_i \\to X\\}$ such that", "$\\varphi|_{U_i}$ is the map of constant sheaves associated to", "a map of sets.", "\\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map", "of locally constant sheaves of abelian groups on $X_\\etale$.", "If $\\mathcal{F}$ is finite locally constant, there exists an \\'etale", "covering $\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is the map of", "constant abelian sheaves associated to a map of abelian groups.", "\\item Let $\\Lambda$ be a ring.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map", "of locally constant sheaves of $\\Lambda$-modules on $X_\\etale$.", "If $\\mathcal{F}$ is of finite type, then there exists an \\'etale covering", "$\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is the map of constant", "sheaves of $\\Lambda$-modules associated to a map of $\\Lambda$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This holds on any site, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-morphism-locally-constant}." ], "refs": [ "sites-modules-lemma-morphism-locally-constant" ], "ref_ids": [ 14271 ] } ], "ref_ids": [] }, { "id": 6505, "type": "theorem", "label": "etale-cohomology-lemma-kernel-finite-locally-constant", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-kernel-finite-locally-constant", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item The category of finite locally constant sheaves of sets", "is closed under finite limits and colimits inside $\\Sh(X_\\etale)$.", "\\item The category of finite locally constant abelian sheaves is a", "weak Serre subcategory of $\\textit{Ab}(X_\\etale)$.", "\\item Let $\\Lambda$ be a Noetherian ring. The category of", "finite type, locally constant sheaves of $\\Lambda$-modules on", "$X_\\etale$ is a weak Serre subcategory of", "$\\textit{Mod}(X_\\etale, \\Lambda)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This holds on any site, see", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-kernel-finite-locally-constant}." ], "refs": [ "sites-modules-lemma-kernel-finite-locally-constant" ], "ref_ids": [ 14273 ] } ], "ref_ids": [] }, { "id": 6506, "type": "theorem", "label": "etale-cohomology-lemma-tensor-product-locally-constant", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-tensor-product-locally-constant", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a ring.", "The tensor product of two locally constant sheaves of $\\Lambda$-modules", "on $X_\\etale$ is a locally constant sheaf of $\\Lambda$-modules." ], "refs": [], "proofs": [ { "contents": [ "This holds on any site, see", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-tensor-product-locally-constant}." ], "refs": [ "sites-modules-lemma-tensor-product-locally-constant" ], "ref_ids": [ 14274 ] } ], "ref_ids": [] }, { "id": 6507, "type": "theorem", "label": "etale-cohomology-lemma-connected-locally-constant", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-connected-locally-constant", "contents": [ "Let $X$ be a connected scheme. Let $\\Lambda$ be a ring and let", "$\\mathcal{F}$ be a locally constant sheaf of $\\Lambda$-modules.", "Then there exists a $\\Lambda$-module $M$ and an \\'etale covering", "$\\{U_i \\to X\\}$ such that $\\mathcal{F}|_{U_i} \\cong \\underline{M}|_{U_i}$." ], "refs": [], "proofs": [ { "contents": [ "Choose an \\'etale covering", "$\\{U_i \\to X\\}$ such that $\\mathcal{F}|_{U_i}$ is constant, say", "$\\mathcal{F}|_{U_i} \\cong \\underline{M_i}_{U_i}$.", "Observe that $U_i \\times_X U_j$ is empty if $M_i$ is not isomorphic", "to $M_j$.", "For each $\\Lambda$-module $M$ let $I_M = \\{i \\in I \\mid M_i \\cong M\\}$.", "As \\'etale morphisms are open we see that", "$U_M = \\bigcup_{i \\in I_M} \\Im(U_i \\to X)$", "is an open subset of $X$. Then $X = \\coprod U_M$ is a disjoint", "open covering of $X$. As $X$ is connected only one $U_M$ is nonempty", "and the lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6508, "type": "theorem", "label": "etale-cohomology-lemma-locally-constant-on-connected", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-locally-constant-on-connected", "contents": [ "Let $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point of $X$.", "\\begin{enumerate}", "\\item There is an equivalence of categories", "$$", "\\left\\{", "\\begin{matrix}", "\\text{finite locally constant}\\\\", "\\text{sheaves of sets on }X_\\etale", "\\end{matrix}", "\\right\\}", "\\longleftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{finite }\\pi_1(X, \\overline{x})\\text{-sets}", "\\end{matrix}", "\\right\\}", "$$", "\\item There is an equivalence of categories", "$$", "\\left\\{", "\\begin{matrix}", "\\text{finite locally constant}\\\\", "\\text{sheaves of abelian groups on }X_\\etale", "\\end{matrix}", "\\right\\}", "\\longleftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{finite }\\pi_1(X, \\overline{x})\\text{-modules}", "\\end{matrix}", "\\right\\}", "$$", "\\item Let $\\Lambda$ be a finite ring. There is an equivalence of categories", "$$", "\\left\\{", "\\begin{matrix}", "\\text{finite type, locally constant}\\\\", "\\text{sheaves of }\\Lambda\\text{-modules on }X_\\etale", "\\end{matrix}", "\\right\\}", "\\longleftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{finite }\\pi_1(X, \\overline{x})\\text{-modules endowed}\\\\", "\\text{with commuting }\\Lambda\\text{-module structure}", "\\end{matrix}", "\\right\\}", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We observe that $\\pi_1(X, \\overline{x})$ is a profinite", "topological group, see Fundamental Groups, Definition", "\\ref{pione-definition-fundamental-group}.", "The left hand categories are defined in", "Section \\ref{section-locally-constant}.", "The notation used in the right hand categories is taken from", "Fundamental Groups, Definition \\ref{pione-definition-G-set-continuous}", "for sets and", "Definition \\ref{definition-G-module-continuous} for abelian groups.", "This explains the notation.", "\\medskip\\noindent", "Assertion (1) follows from", "Lemma \\ref{lemma-characterize-finite-locally-constant}", "and Fundamental Groups, Theorem \\ref{pione-theorem-fundamental-group}.", "Parts (2) and (3) follow immediately from this by endowing the underlying", "(sheaves of) sets with additional structure. For example, a finite", "locally constant sheaf of abelian groups on $X_\\etale$ is the same thing", "as a finite locally constant sheaf of sets $\\mathcal{F}$", "together with a map $+ : \\mathcal{F} \\times \\mathcal{F} \\to \\mathcal{F}$", "satisfying the usual axioms. The equivalence in (1) sends products", "to products and hence sends $+$ to an addition on the corresponding", "finite $\\pi_1(X, \\overline{x})$-set. Since $\\pi_1(X, \\overline{x})$-modules", "are the same thing as $\\pi_1(X, \\overline{x})$-sets with a compatible", "abelian group structure we obtain (2). Part (3) is proved in", "exactly the same way." ], "refs": [ "pione-definition-fundamental-group", "pione-definition-G-set-continuous", "etale-cohomology-definition-G-module-continuous", "etale-cohomology-lemma-characterize-finite-locally-constant", "pione-theorem-fundamental-group" ], "ref_ids": [ 4143, 4141, 6747, 6503, 4021 ] } ], "ref_ids": [] }, { "id": 6509, "type": "theorem", "label": "etale-cohomology-lemma-pullback-filtered", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-pullback-filtered", "contents": [ "Let $S$ be a connected scheme. Let $\\ell$ be a prime number. Let", "$\\mathcal{F}$ be a finite type, locally constant sheaf of", "$\\mathbf{F}_\\ell$-vector spaces on $S_\\etale$.", "Then there exists a finite \\'etale morphism", "$f : T \\to S$ of degree prime to $\\ell$ such that $f^{-1}\\mathcal{F}$", "has a finite filtration whose successive quotients are", "$\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}_T$." ], "refs": [], "proofs": [ { "contents": [ "Choose a geometric point $\\overline{s}$ of $S$.", "Via the equivalence of Lemma \\ref{lemma-locally-constant-on-connected}", "the sheaf $\\mathcal{F}$ corresponds to a finite dimensional", "$\\mathbf{F}_\\ell$-vector space $V$ with a continuous", "$\\pi_1(S, \\overline{s})$-action.", "Let $G \\subset \\text{Aut}(V)$ be the image of the homomorphism", "$\\rho : \\pi_1(S, \\overline{s}) \\to \\text{Aut}(V)$ giving the action.", "Observe that $G$ is finite.", "The surjective continuous homomorphism", "$\\overline{\\rho} : \\pi_1(S, \\overline{s}) \\to G$", "corresponds to a Galois object $Y \\to S$ of", "$\\textit{F\\'Et}_S$ with automorphism group $G = \\text{Aut}(Y/S)$, see", "Fundamental Groups, Section \\ref{pione-section-finite-etale-under-galois}.", "Let $H \\subset G$ be an $\\ell$-Sylow subgroup.", "We claim that $T = Y/H \\to S$ works. Namely, let $\\overline{t} \\in T$", "be a geometric point over $\\overline{s}$. The image of", "$\\pi_1(T, \\overline{t}) \\to \\pi_1(S, \\overline{s})$", "is $(\\overline{\\rho})^{-1}(H)$ as follows from the functorial", "nature of fundamental groups. Hence the action of $\\pi_1(T, \\overline{t})$", "on $V$ corresponding to $f^{-1}\\mathcal{F}$ is through", "the map $\\pi_1(T, \\overline{t}) \\to H$, see", "Remark \\ref{remark-functorial-locally-constant-on-connected}. As", "$H$ is a finite $\\ell$-group, the irreducible constituents of the", "representation $\\rho|_{\\pi_1(T, \\overline{t})}$", "are each trivial of rank $1$ (this is a simple lemma on", "representation theory of finite groups; insert future reference here).", "Via the equivalence of", "Lemma \\ref{lemma-locally-constant-on-connected}", "this means $f^{-1}\\mathcal{F}$ is a successive extension of", "constant sheaves with value $\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}_T$.", "Moreover the degree of $T = Y/H \\to S$ is prime to $\\ell$", "as it is equal to the index of $H$ in $G$." ], "refs": [ "etale-cohomology-lemma-locally-constant-on-connected", "etale-cohomology-remark-functorial-locally-constant-on-connected", "etale-cohomology-lemma-locally-constant-on-connected" ], "ref_ids": [ 6508, 6789, 6508 ] } ], "ref_ids": [] }, { "id": 6510, "type": "theorem", "label": "etale-cohomology-lemma-nonvanishing-inherited", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-nonvanishing-inherited", "contents": [ "Let $\\ell$ be a prime number and $n$ an integer $> 0$.", "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $X = \\lim_{i \\in I} X_i$ be the limit of a", "directed system of $S$-schemes each $X_i \\to S$", "being finite \\'etale of constant degree relatively prime to $\\ell$.", "The following are equivalent:", "\\begin{enumerate}", "\\item there exists an $\\ell$-power torsion sheaf", "$\\mathcal{G}$ on $S$ such that $H_\\etale^n(S, \\mathcal{G}) \\neq 0$ and", "\\item there exists an $\\ell$-power torsion sheaf $\\mathcal{F}$ on $X$", "such that $H_\\etale^n(X, \\mathcal{F}) \\neq 0$.", "\\end{enumerate}", "In fact, given", "$\\mathcal{G}$ we can take $\\mathcal{F} = g^{-1}\\mathcal{F}$", "and given", "$\\mathcal{F}$ we can take $\\mathcal{G} = g_*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Let $g : X \\to S$ and $g_i : X_i \\to S$ denote the structure morphisms.", "Fix an $\\ell$-power torsion sheaf $\\mathcal{G}$ on $S$", "with $H^n_\\etale(S, \\mathcal{G}) \\not = 0$.", "The system given by $\\mathcal{G}_i = g_i^{-1}\\mathcal{G}$", "satisify the conditions of Theorem \\ref{theorem-colimit}", "with colimit sheaf given by $g^{-1}\\mathcal{G}$. This tells ", "us that:", "$$", "\\colim_{i\\in I} H^n_\\etale(X_i, g_i^{-1}\\mathcal{G}) =", "H^n_\\etale(X, \\mathcal{G})", "$$", "By virtue of the $g_i$ being finite \\'etale morphism of degree prime", "to $\\ell$ we can apply ``la m\\'ethode de la trace'' and we find", "the maps", "$$", "H^n_\\etale(S, \\mathcal{G}) \\to H^n_\\etale(X_i, g_i^{-1}\\mathcal{G})", "$$", "are all injective (and compatible with the transition maps).", "See Section \\ref{section-trace-method}. Thus, the colimit is non-zero, i.e.,", "$H^n(X,g^{-1}\\mathcal{G}) \\neq 0$, giving us the desired result with ", "$\\mathcal{F} = g^{-1}\\mathcal{G}$.", "\\medskip\\noindent", "Conversely, suppose given an $\\ell$-power torsion sheaf $\\mathcal{F}$ on $X$", "with $H^n_\\etale(X, \\mathcal{F}) \\not = 0$. We note that since the $g_i$", "are finite morphisms the higher direct images vanish", "(Proposition \\ref{proposition-finite-higher-direct-image-zero}).", "Then, by applying Lemma \\ref{lemma-relative-colimit}", "we may also conclude the same for $g$.", "The vanishing of the higher direct images tells us that", "$H^n_\\etale(X, \\mathcal{F}) = H^n(S, g_*\\mathcal{F}) \\neq 0$", "by Leray (Proposition \\ref{proposition-leray})", "giving us what we want with $\\mathcal{G} = g_*\\mathcal{F}$." ], "refs": [ "etale-cohomology-theorem-colimit", "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-lemma-relative-colimit", "etale-cohomology-proposition-leray" ], "ref_ids": [ 6384, 6703, 6475, 6702 ] } ], "ref_ids": [] }, { "id": 6511, "type": "theorem", "label": "etale-cohomology-lemma-reduce-to-l-group", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-reduce-to-l-group", "contents": [ "Let $\\ell$ be a prime number and $n$ an integer $> 0$.", "Let $K$ be a field with $G = Gal(K^{sep}/K)$ and let", "$H \\subset G$ be a maximal pro-$\\ell$ subgroup with $L/K$", "being the corresponding field extension. Then", "$H^n_\\etale(\\Spec(K), \\mathcal{F}) = 0$ for all", "$\\ell$-power torsion $\\mathcal{F}$ if and only if", "$H^n_\\etale(\\Spec(L), \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Write $L = \\bigcup L_i$ as the union of its finite subextensions over $K$.", "Our choice of $H$ implies that $[L_i : K]$ is prime to $\\ell$.", "Thus $\\Spec(L) = \\lim_{i \\in I} \\Spec(L_i)$ as in", "Lemma \\ref{lemma-nonvanishing-inherited}.", "Thus we may replace $K$ by $L$ and assume that", "the absolute Galois group $G$ of $K$ is a", "profinite pro-$\\ell$ group.", "\\medskip\\noindent", "Assume $H^n(\\Spec(K), \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}) = 0$.", "Let $\\mathcal{F}$ be an $\\ell$-power torsion sheaf on $\\Spec(K)_\\etale$.", "We will show that $H^n_\\etale(\\Spec(K), \\mathcal{F}) = 0$.", "By the correspondence specified in", "Lemma \\ref{lemma-equivalence-abelian-sheaves-point}", "our sheaf $\\mathcal{F}$ corresponds to an $\\ell$-power torsion", "$G$-module $M$. Any finite set of elements $x_1, \\ldots, x_m \\in M$", "must be fixed by an open subgroup $U$ by continuity.", "Let $M'$ be the module spanned by the orbits of $x_1, \\ldots, x_m$.", "This is a finite abelian $\\ell$-group", "as each $x_i$ is killed by a power of $\\ell$", "and the orbits are finite. Since $M$ is the filtered colimit of", "these submodules $M'$, we see that $\\mathcal{F}$ is the filtered", "colimit of the corresponding subsheaves $\\mathcal{F}' \\subset \\mathcal{F}$.", "Applying Theorem \\ref{theorem-colimit} to this colimit, we reduce", "to the case where $\\mathcal{F}$ is a finite locally constant sheaf.", "\\medskip\\noindent", "Let $M$ be a finite abelian $\\ell$-group with a continuous action", "of the profinite pro-$\\ell$ group $G$. Then there is a $G$-invariant", "filtration", "$$", "0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_r = M", "$$", "such that $M_{i + 1}/M_i \\cong \\mathbf{Z}/\\ell \\mathbf{Z}$ with", "trivial $G$-action (this is a simple lemma on representation", "theory of finite groups; insert future reference here).", "Thus the corresponding sheaf $\\mathcal{F}$ has a filtration", "$$", "0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset \\ldots \\subset", "\\mathcal{F}_r = \\mathcal{F}", "$$", "with successive quotients isomorphic to", "$\\underline{\\mathbf{Z}/\\ell \\mathbf{Z}}$.", "Thus by induction and the long exact cohomology", "sequence we conclude." ], "refs": [ "etale-cohomology-lemma-nonvanishing-inherited", "etale-cohomology-lemma-equivalence-abelian-sheaves-point", "etale-cohomology-theorem-colimit" ], "ref_ids": [ 6510, 6489, 6384 ] } ], "ref_ids": [] }, { "id": 6512, "type": "theorem", "label": "etale-cohomology-lemma-reduce-to-l-group-higher", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-reduce-to-l-group-higher", "contents": [ "Let $\\ell$ be a prime number and $n$ an integer $> 0$.", "Let $K$ be a field with $G = Gal(K^{sep}/K)$ and let", "$H \\subset G$ be a maximal pro-$\\ell$ subgroup ", "with $L/K$ being the corresponding field extension.", "Then $H^q_\\etale(\\Spec(K),\\mathcal{F}) = 0$ for $q \\geq n$ and all", "$\\ell$-torsion sheaves $\\mathcal{F}$ if and only if", "$H^n_\\etale(\\Spec(L), \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "The forward direction is trivial, so we need only prove the reverse direction. ", "We proceed by induction on $q$. The case of $q = n$ is", "Lemma \\ref{lemma-reduce-to-l-group}. Now let ", "$\\mathcal{F}$ be an $\\ell$-power torsion sheaf on $\\Spec(K)$.", "Let $f : \\Spec(K^{sep}) \\rightarrow \\Spec(K)$", "be the inclusion of a geometric point.", "Then consider the exact sequence:", "$$", "0 \\rightarrow \\mathcal{F} \\xrightarrow{res}", "f_* f^{-1} \\mathcal{F} \\rightarrow f_* f^{-1} \\mathcal{F}/\\mathcal{F} ", "\\rightarrow 0", "$$", "Note that $K^{sep}$ may be written as the filtered colimit of finite ", "separable extensions. Thus $f$", "is the limit of a directed system of finite \\'etale morphisms.", "We may, as was seen in the proof of", "Lemma \\ref{lemma-nonvanishing-inherited}, conclude that $f$ has ", "vanishing higher direct images. Thus, we may express the higher cohomology of ", "$f_* f^{-1} \\mathcal{F}$ as the higher cohomology on the geometric point which ", "clearly vanishes. Hence, as everything here is still $\\ell$-torsion, we may use ", "the inductive hypothesis in conjunction with the long-exact cohomology sequence ", "to conclude the result for $q + 1$." ], "refs": [ "etale-cohomology-lemma-reduce-to-l-group", "etale-cohomology-lemma-nonvanishing-inherited" ], "ref_ids": [ 6511, 6510 ] } ], "ref_ids": [] }, { "id": 6513, "type": "theorem", "label": "etale-cohomology-lemma-algebraically-closed-find-solutions", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-algebraically-closed-find-solutions", "contents": [ "Let $k$ be an algebraically closed field. Let", "$f_1, \\ldots, f_s \\in k[T_1, \\ldots, T_n]$", "be homogeneous polynomials of degree $d_1, \\ldots, d_s$ with $d_i", "> 0$. If $s < n$, then $f_1 = \\ldots = f_s = 0$ have a common nontrivial", "solution." ], "refs": [], "proofs": [ { "contents": [ "This follows from dimension theory, for example in the form of", "Varieties, Lemma \\ref{varieties-lemma-intersection-in-affine-space}", "applied $s - 1$ times." ], "refs": [ "varieties-lemma-intersection-in-affine-space" ], "ref_ids": [ 11034 ] } ], "ref_ids": [] }, { "id": 6514, "type": "theorem", "label": "etale-cohomology-lemma-curve-brauer-zero", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-curve-brauer-zero", "contents": [ "Let $C$ be a curve over an algebraically closed field $k$. Then", "the Brauer group of the function field of $C$ is zero:", "$\\text{Br}(k(C)) = 0$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from Tsen's theorem,", "Theorem \\ref{theorem-tsen} and", "Theorem \\ref{theorem-C1-brauer-group-zero}." ], "refs": [ "etale-cohomology-theorem-tsen", "etale-cohomology-theorem-C1-brauer-group-zero" ], "ref_ids": [ 6390, 6389 ] } ], "ref_ids": [] }, { "id": 6515, "type": "theorem", "label": "etale-cohomology-lemma-cohomology-Gm-function-field-curve", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomology-Gm-function-field-curve", "contents": [ "Let $k$ be an algebraically closed field and $k \\subset K$ a field extension", "of transcendence degree 1. Then for all $q \\geq 1$,", "$H_\\etale^q(\\Spec(K), \\mathbf{G}_m) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Recall that", "$H_\\etale^q(\\Spec(K), \\mathbf{G}_m) = H^q(\\text{Gal}(K^{sep}/K), (K^{sep})^*)$", "by Lemma \\ref{lemma-compare-cohomology-point}.", "Thus by Proposition \\ref{proposition-serre-galois}", "it suffices to show that if $K \\subset K'$ is a finite field extension, then", "$\\text{Br}(K') = 0$. Now observe that $K' = \\colim K''$, where $K''$ runs", "over the finitely generated subextensions of $k$ contained in $K'$ of", "transcendence degree $1$.", "Note that $\\text{Br}(K') = \\colim \\text{Br}(K'')$ which reduces us", "to a finitely generated field extension $K''/k$ of transcendence", "degree $1$. Such a field is the function field of a curve over $k$,", "hence has trivial Brauer group by", "Lemma \\ref{lemma-curve-brauer-zero}." ], "refs": [ "etale-cohomology-lemma-compare-cohomology-point", "etale-cohomology-proposition-serre-galois", "etale-cohomology-lemma-curve-brauer-zero" ], "ref_ids": [ 6490, 6704, 6514 ] } ], "ref_ids": [] }, { "id": 6516, "type": "theorem", "label": "etale-cohomology-lemma-higher-direct-jstar-Gm", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-higher-direct-jstar-Gm", "contents": [ "For any $q \\geq 1$, $R^q j_*\\mathbf{G}_{m, \\eta} = 0$." ], "refs": [], "proofs": [ { "contents": [ "We need to show that $(R^q j_*\\mathbf{G}_{m, \\eta})_{\\bar x} = 0$ for every", "geometric point $\\bar x$ of $X$.", "\\medskip\\noindent", "Assume that $\\bar x$ lies over a closed point $x$ of $X$. Let $\\Spec(A)$", "be an affine open neighbourhood of $x$ in $X$, and $K$ the fraction field", "of $A$. Then", "$$", "\\Spec(\\mathcal{O}^{sh}_{X, \\bar x}) \\times_X \\eta =", "\\Spec(\\mathcal{O}^{sh}_{X, \\bar x} \\otimes_A K).", "$$", "The ring $\\mathcal{O}^{sh}_{X, \\bar x} \\otimes_A K$ is a localization of", "the discrete valuation ring $\\mathcal{O}^{sh}_{X, \\bar x}$, so it is either", "$\\mathcal{O}^{sh}_{X, \\bar x}$ again, or its fraction field", "$K^{sh}_{\\bar x}$. But since some local uniformizer gets inverted, it must", "be the latter. Hence", "$$", "(R^q j_*\\mathbf{G}_{m, \\eta})_{(X, \\bar x)} = H_\\etale^q(\\Spec", "K^{sh}_{\\bar x}, \\mathbf{G}_m).", "$$", "Now recall that $\\mathcal{O}^{sh}_{X, \\bar x} =", "\\colim_{(U, \\bar u) \\to \\bar x} \\mathcal{O}(U) = \\colim_{A \\subset B} B$", "where $A \\to B$ is \\'etale, hence $K^{sh}_{\\bar x}$ is an algebraic", "extension of $K = k(X)$, and we may apply", "Lemma \\ref{lemma-cohomology-Gm-function-field-curve}", "to get the vanishing.", "\\medskip\\noindent", "Assume that $\\bar x = \\bar \\eta$ lies over the generic point $\\eta$ of $X$ (in", "fact, this case is superfluous). Then $\\mathcal{O}_{X, \\bar \\eta} =", "\\kappa(\\eta)^{sep}$ and thus", "\\begin{eqnarray*}", "(R^q j_*\\mathbf{G}_{m, \\eta})_{\\bar \\eta}", "& = &", "H_\\etale^q(\\Spec(\\kappa(\\eta)^{sep}) \\times_X \\eta, \\mathbf{G}_m) \\\\", "& = & H_\\etale^q (\\Spec(\\kappa(\\eta)^{sep}), \\mathbf{G}_m) \\\\", "& = & 0 \\ \\ \\text{ for } q \\geq 1", "\\end{eqnarray*}", "since the corresponding Galois group is trivial." ], "refs": [ "etale-cohomology-lemma-cohomology-Gm-function-field-curve" ], "ref_ids": [ 6515 ] } ], "ref_ids": [] }, { "id": 6517, "type": "theorem", "label": "etale-cohomology-lemma-cohomology-jstar-Gm", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomology-jstar-Gm", "contents": [ "For all $p \\geq 1$, $H_\\etale^p(X, j_*\\mathbf{G}_{m, \\eta}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "The Leray spectral sequence reads", "$$", "E_2^{p, q} = H_\\etale^p(X, R^qj_*\\mathbf{G}_{m, \\eta}) \\Rightarrow", "H_\\etale^{p+q}(\\eta, \\mathbf{G}_{m, \\eta}),", "$$", "which vanishes for $p+q \\geq 1$ by", "Lemma \\ref{lemma-cohomology-Gm-function-field-curve}. Taking", "$q = 0$, we get the desired vanishing." ], "refs": [ "etale-cohomology-lemma-cohomology-Gm-function-field-curve" ], "ref_ids": [ 6515 ] } ], "ref_ids": [] }, { "id": 6518, "type": "theorem", "label": "etale-cohomology-lemma-cohomology-istar-Z", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomology-istar-Z", "contents": [ "For all $q \\geq 1$, $H_\\etale^q(X, \\bigoplus_{x \\in X_0} {i_x}_*", "\\underline{\\mathbf{Z}}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "For $X$ quasi-compact and quasi-separated, cohomology commutes with colimits,", "so it suffices to show the vanishing of $H_\\etale^q(X, {i_x}_*", "\\underline{\\mathbf{Z}})$. But then the inclusion $i_x$ of a closed point is", "finite so $R^p {i_x}_* \\underline{\\mathbf{Z}} = 0$ for all $p \\geq 1$ by", "Proposition \\ref{proposition-finite-higher-direct-image-zero}.", "Applying the Leray spectral sequence, we see that", "$H_\\etale^q(X, {i_x}_* \\underline{\\mathbf{Z}}) =", "H_\\etale^q(x, \\underline{\\mathbf{Z}})$.", "Finally, since $x$ is the spectrum of an", "algebraically closed field, all higher cohomology on $x$ vanishes." ], "refs": [ "etale-cohomology-proposition-finite-higher-direct-image-zero" ], "ref_ids": [ 6703 ] } ], "ref_ids": [] }, { "id": 6519, "type": "theorem", "label": "etale-cohomology-lemma-cohomology-smooth-projective-curve", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomology-smooth-projective-curve", "contents": [ "Let $X$ be a smooth projective curve of genus $g$ over an", "algebraically closed field $k$ and let $n \\geq 1$ be invertible in $k$.", "Then there are canonical identifications", "$$", "H_\\etale^q(X, \\mu_n) =", "\\left\\{", "\\begin{matrix}", "\\mu_n(k) & \\text{ if }q = 0, \\\\", "\\Pic^0(X)[n] & \\text{ if }q = 1, \\\\", "\\mathbf{Z}/n\\mathbf{Z} & \\text{ if }q = 2, \\\\", "0 & \\text{ if }q \\geq 3.", "\\end{matrix}", "\\right.", "$$", "Since $\\mu_n \\cong \\underline{\\mathbf{Z}/n\\mathbf{Z}}$, this gives", "(noncanonical) identifications", "$$", "H_\\etale^q(X, \\underline{\\mathbf{Z}/n\\mathbf{Z}}) \\cong", "\\left\\{", "\\begin{matrix}", "\\mathbf{Z}/n\\mathbf{Z} & \\text{ if }q = 0, \\\\", "(\\mathbf{Z}/n\\mathbf{Z})^{2g} & \\text{ if }q = 1, \\\\", "\\mathbf{Z}/n\\mathbf{Z} & \\text{ if }q = 2, \\\\", "0 & \\text{ if }q \\geq 3.", "\\end{matrix}", "\\right.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Theorems \\ref{theorem-picard-group} and", "\\ref{theorem-vanishing-cohomology-Gm-curve}", "determine the \\'etale cohomology of $\\mathbf{G}_m$ on $X$", "in terms of the Picard group of $X$.", "The Kummer sequence $0\\to \\mu_{n, X} \\to \\mathbf{G}_{m, X}", "\\to \\mathbf{G}_{m, X}\\to 0$ (Lemma \\ref{lemma-kummer-sequence})", "then gives us the long exact cohomology sequence", "$$", "\\xymatrix{", "0 \\ar[r] & \\mu_n(k) \\ar[r] &", "k^* \\ar[r]^{(\\cdot)^n} &", "k^* \\ar@(rd, ul)[rdllllr] \\\\", "& H_\\etale^1(X, \\mu_n) \\ar[r] &", "\\Pic(X) \\ar[r]^{(\\cdot)^n} &", "\\Pic(X) \\ar@(rd, ul)[rdllllr] \\\\", "& H_\\etale^2(X, \\mu_n) \\ar[r] & 0 \\ar[r] & 0 \\ldots", "}", "$$", "The $n$th power map $k^* \\to k^*$ is surjective since $k$ is algebraically", "closed. So we need to compute the kernel and cokernel of the map", "$n : \\Pic(X) \\to \\Pic(X)$. Consider the commutative diagram with", "exact rows", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Pic^0(X) \\ar[r] \\ar@{>>}[d]^{(\\cdot)^n} &", "\\Pic(X) \\ar[r]_-\\deg \\ar[d]^{(\\cdot)^n} &", "\\mathbf{Z} \\ar[r] \\ar@{^{(}->}[d]^n & 0 \\\\", "0 \\ar[r] &", "\\Pic^0(X) \\ar[r] &", "\\Pic(X) \\ar[r]^-\\deg &", "\\mathbf{Z} \\ar[r] & 0", "}", "$$", "The group $\\Pic^0(X)$ is the $k$-points of", "the group scheme $\\underline{\\Picardfunctor}^0_{X/k}$, see", "Picard Schemes of Curves, Lemma \\ref{pic-lemma-picard-pieces}.", "The same lemma tells us that $\\underline{\\Picardfunctor}^0_{X/k}$", "is a $g$-dimensional abelian variety over $k$ as defined in", "Groupoids, Definition \\ref{groupoids-definition-abelian-variety}.", "Hence the left vertical map is surjective by", "Groupoids, Proposition \\ref{groupoids-proposition-review-abelian-varieties}.", "Applying the snake lemma gives canonical identifications as stated", "in the lemma.", "\\medskip\\noindent", "To get the noncanonical identifications of the lemma we need to", "show the kernel of $n : \\Pic^0(X) \\to \\Pic^0(X)$", "is isomorphic to $(\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2g}$.", "This is also part of Groupoids, Proposition", "\\ref{groupoids-proposition-review-abelian-varieties}." ], "refs": [ "etale-cohomology-theorem-picard-group", "etale-cohomology-theorem-vanishing-cohomology-Gm-curve", "etale-cohomology-lemma-kummer-sequence", "pic-lemma-picard-pieces", "groupoids-definition-abelian-variety", "groupoids-proposition-review-abelian-varieties", "groupoids-proposition-review-abelian-varieties" ], "ref_ids": [ 6374, 6392, 6419, 12571, 9675, 9668, 9668 ] } ], "ref_ids": [] }, { "id": 6520, "type": "theorem", "label": "etale-cohomology-lemma-pullback-on-h2-curve", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-pullback-on-h2-curve", "contents": [ "Let $\\pi : X \\to Y$ be a nonconstant morphism of smooth projective curves", "over an algebraically closed field $k$ and let $n \\geq 1$ be invertible in $k$.", "The map", "$$", "\\pi^* : H^2_\\etale(Y, \\mu_n) \\longrightarrow H^2_\\etale(X, \\mu_n)", "$$", "is given by multiplication by the degree of $\\pi$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the statement makes sense as we have identified both", "cohomology groups $ H^2_\\etale(Y, \\mu_n)$ and $H^2_\\etale(X, \\mu_n)$", "with $\\mathbf{Z}/n\\mathbf{Z}$ in", "Lemma \\ref{lemma-cohomology-smooth-projective-curve}.", "In fact, if $\\mathcal{L}$ is a line bundle of degree $1$", "on $Y$ with class $[\\mathcal{L}] \\in H^1_\\etale(Y, \\mathbf{G}_m)$,", "then the coboundary of $[\\mathcal{L}]$ is the generator of", "$H^2_\\etale(Y, \\mu_n)$. Here the coboundary is the coboundary", "of the long exact sequence of cohomology associated to the Kummer", "sequence. Thus the result of the lemma follows from the fact that", "the degree of the line bundle $\\pi^*\\mathcal{L}$ on $X$ is $\\deg(\\pi)$.", "Some details omitted." ], "refs": [ "etale-cohomology-lemma-cohomology-smooth-projective-curve" ], "ref_ids": [ 6519 ] } ], "ref_ids": [] }, { "id": 6521, "type": "theorem", "label": "etale-cohomology-lemma-vanishing-cohomology-mu-smooth-curve", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-vanishing-cohomology-mu-smooth-curve", "contents": [ "Let $X$ be an affine smooth curve over an algebraically closed field $k$ and", "$n\\in k^*$. Then", "\\begin{enumerate}", "\\item", "$H_\\etale^0(X, \\mu_n) = \\mu_n(k)$;", "\\item", "$H_\\etale^1(X, \\mu_n) \\cong", "\\left(\\mathbf{Z}/n\\mathbf{Z}\\right)^{2g+r-1}$, where", "$r$ is the number of points in $\\bar X - X$ for some smooth projective", "compactification $\\bar X$ of $X$, and", "\\item", "for all $q\\geq 2$, $H_\\etale^q(X, \\mu_n) = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $X = \\bar X - \\{ x_1, \\ldots, x_r\\}$. Then $\\Pic(X) =", "\\Pic(\\bar X)/ R$, where $R$ is the subgroup generated by", "$\\mathcal{O}_{\\bar X}(x_i)$, $1 \\leq i \\leq r$. Since $r \\geq 1$, we see that", "$\\Pic^0(\\bar X) \\to \\Pic(X)$ is surjective, hence $\\Pic(X)$ is", "divisible. Applying the Kummer sequence, we get (1) and (3). For (2), recall", "that", "\\begin{align*}", "H_\\etale^1(X, \\mu_n)", "& =", "\\{(\\mathcal L, \\alpha) |", "\\mathcal L \\in \\Pic(X),", "\\alpha : \\mathcal{L}^{\\otimes n} \\to \\mathcal{O}_X\\}/\\cong \\\\", "& =", "\\{(\\bar{\\mathcal L},\\ D,\\ \\bar \\alpha)\\}/\\tilde{R}", "\\end{align*}", "where $\\bar{\\mathcal L} \\in \\Pic^0(\\bar X)$, $D$ is a divisor on $\\bar X$", "supported on $\\left\\{x_1, \\ldots, x_r\\right\\}$ and $ \\bar{\\alpha}:", "\\bar{\\mathcal L}^{\\otimes n} \\cong \\mathcal{O}_{\\bar{X}}(D)$ is an isomorphism.", "Note that $D$ must have degree 0. Further $\\tilde{R}$ is the subgroup of", "triples of the form $(\\mathcal{O}_{\\bar X}(D'), n D', 1^{\\otimes n})$ where", "$D'$ is supported on $\\left\\{x_1, \\ldots, x_r\\right\\}$ and has degree 0. Thus,", "we get an exact sequence", "$$", "0 \\longrightarrow", "H_\\etale^1(\\bar X, \\mu_n) \\longrightarrow", "H_\\etale^1(X, \\mu_n) \\longrightarrow", "\\bigoplus_{i = 1}^r \\mathbf{Z}/n\\mathbf{Z}", "\\xrightarrow{\\ \\sum\\ }", "\\mathbf{Z}/n\\mathbf{Z} \\longrightarrow 0", "$$", "where the middle map sends the class of a triple $(\\bar{ \\mathcal L}, D, \\bar", "\\alpha)$ with $D = \\sum_{i = 1}^r a_i (x_i)$ to the $r$-tuple", "$(a_i)_{i = 1}^r$. It now suffices to use", "Lemma \\ref{lemma-cohomology-smooth-projective-curve}", "to count ranks." ], "refs": [ "etale-cohomology-lemma-cohomology-smooth-projective-curve" ], "ref_ids": [ 6519 ] } ], "ref_ids": [] }, { "id": 6522, "type": "theorem", "label": "etale-cohomology-lemma-jshriek-open", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-jshriek-open", "contents": [ "Let $j : U \\to X$ be an open immersion of schemes. For any", "abelian sheaf $\\mathcal{F}$ on $U_\\etale$, the adjunction mappings", "$j^{-1}j_*\\mathcal{F} \\to \\mathcal{F}$ and", "$\\mathcal{F} \\to j^{-1}j_!\\mathcal{F}$ are isomorphisms.", "In fact, $j_!\\mathcal{F}$ is the unique abelian sheaf on $X_\\etale$", "whose restriction to $U$ is $\\mathcal{F}$ and whose stalks at", "geometric points of $X \\setminus U$ are zero." ], "refs": [], "proofs": [ { "contents": [ "We encourage the reader to prove the first statement by working through the", "definitions, but here we just use that it is a special case of the very general", "Modules on Sites, Lemma \\ref{sites-modules-lemma-restrict-back}.", "For the second statement, observe that if $\\mathcal{G}$ is an abelian sheaf", "on $X_\\etale$ whose restriction to $U$ is $\\mathcal{F}$, then we obtain", "by adjointness a map $j_!\\mathcal{F} \\to \\mathcal{G}$. This map", "is then an isomorphism at stalks of geometric points of $U$ by", "Proposition \\ref{proposition-describe-jshriek}.", "Thus if $\\mathcal{G}$ has vanishing stalks at geometric points", "of $X \\setminus U$, then $j_!\\mathcal{F} \\to \\mathcal{G}$ is an", "isomorphism by Theorem \\ref{theorem-exactness-stalks}." ], "refs": [ "sites-modules-lemma-restrict-back", "etale-cohomology-proposition-describe-jshriek", "etale-cohomology-theorem-exactness-stalks" ], "ref_ids": [ 14173, 6705, 6376 ] } ], "ref_ids": [] }, { "id": 6523, "type": "theorem", "label": "etale-cohomology-lemma-shriek-base-change", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-shriek-base-change", "contents": [ "Let $f: Y \\to X$ be a morphism of schemes. Let $j: V \\to X$ be an \\'etale", "morphism. Consider the fibre product", "$$", "\\xymatrix{", "V' = Y \\times_X V \\ar[d]_{f'} \\ar[r]_-{j'} & Y \\ar[d]^f \\\\", "V \\ar[r]^j & X", "}", "$$", "Then we have $j'_! f'^{-1} = f^{-1} j_!$ on abelian sheaves and on", "sheaves of modules." ], "refs": [], "proofs": [ { "contents": [ "This is true because $j'_! f'^{-1}$ is left adjoint to", "$f'_* (j')^{-1}$ and $f^{-1} j_!$ is left adjoint to $j^{-1}f_*$.", "Further $f'_* (j')^{-1} = j^{-1}f_*$ because $f_*$ commutes with", "\\'etale localization (by construction). In fact, the lemma holds very generally", "in the setting of a morphism of sites, see", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-localize-morphism-ringed-sites}." ], "refs": [ "sites-modules-lemma-localize-morphism-ringed-sites" ], "ref_ids": [ 14174 ] } ], "ref_ids": [] }, { "id": 6524, "type": "theorem", "label": "etale-cohomology-lemma-shriek-into-star-separated-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-shriek-into-star-separated-etale", "contents": [ "Let $j : U \\to X$ be separated and \\'etale. Then there is a functorial", "injective map $j_!\\mathcal{F} \\to j_*\\mathcal{F}$", "on abelian sheaves and sheaves of $\\Lambda$-modules." ], "refs": [], "proofs": [ { "contents": [ "We prove this in the case of abelian sheaves. Let us construct a canonical map", "$$", "j_{p!}\\mathcal{F} \\to j_*\\mathcal{F}", "$$", "of abelian presheaves on $X_\\etale$ for any abelian sheaf", "$\\mathcal{F}$ on $U_\\etale$ where $j_{p!}$ is as in", "(\\ref{equation-j-p-shriek}). Sheafification of this map will", "be the desired map $j_!\\mathcal{F} \\to j_*\\mathcal{F}$.", "Evaluating both sides on $V \\to X$ \\'etale we obtain", "$$", "j_{p!}\\mathcal{F}(V) =", "\\bigoplus\\nolimits_{\\varphi : V \\to U} \\mathcal{F}(V \\xrightarrow{\\varphi} U)", "\\quad\\text{and}\\quad", "j_*\\mathcal{F}(V) = \\mathcal{F}(V \\times_X U)", "$$", "For each $\\varphi$ we have an open and closed immersion", "$$", "\\Gamma_\\varphi = (1, \\varphi) : V \\longrightarrow V \\times_X U", "$$", "over $U$. It is open as it is a morphism between schemes \\'etale", "over $U$ and it is closed as it is a section of a scheme separated", "over $V$ (Schemes, Lemma \\ref{schemes-lemma-section-immersion}).", "Thus for a section $s_\\varphi \\in \\mathcal{F}(V \\xrightarrow{\\varphi} U)$", "there exists a unique section $s'_\\varphi$ in $\\mathcal{F}(V \\times_X U)$", "which pulls back to $s_\\varphi$ by $\\Gamma_\\varphi$", "and which restricts to zero on the complement of the image of $\\Gamma_\\varphi$.", "\\medskip\\noindent", "To show that our map is injective suppose that", "$\\sum_{i = 1, \\ldots, n} s_{\\varphi_i}$ is an element of", "$j_{p!}\\mathcal{F}(V)$ in the formula above maps to zero", "in $j_*\\mathcal{F}(V)$. Our task is to show that", "$\\sum_{i = 1, \\ldots, n} s_{\\varphi_i}$ restricts to zero", "on the members of an \\'etale covering of $V$.", "Looking at all pairwise equalizers", "(which are open and closed in $V$) of the morphisms", "$\\varphi_i : V \\to U$ and working locally on $V$, we", "may assume the images of the morphisms", "$\\Gamma_{\\varphi_1}, \\ldots, \\Gamma_{\\varphi_n}$", "are pairwise disjoint. Since our assumption is that", "$\\sum_{i = 1, \\ldots, n} s'_{\\varphi_i} = 0$", "we then immediately conclude that $s'_{\\varphi_i} = 0$", "for each $i$ (by the disjointness of the supports of these", "sections), whence $s_{\\varphi_i} = 0$ for all $i$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6525, "type": "theorem", "label": "etale-cohomology-lemma-shriek-equals-star-finite-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-shriek-equals-star-finite-etale", "contents": [ "Let $j : U \\to X$ be finite and \\'etale. Then the map", "$j_! \\to j_*$ of Lemma \\ref{lemma-shriek-into-star-separated-etale}", "is an isomorphism", "on abelian sheaves and sheaves of $\\Lambda$-modules." ], "refs": [ "etale-cohomology-lemma-shriek-into-star-separated-etale" ], "proofs": [ { "contents": [ "\\medskip\\noindent", "It suffices to check $j_!\\mathcal{F} \\to j_*\\mathcal{F}$", "is an isomorphism \\'etale locally on $X$.", "Thus we may assume $U \\to X$ is a finite disjoint union", "of isomorphisms, see", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-finite-etale-etale-local}.", "We omit the proof in this case." ], "refs": [ "etale-lemma-finite-etale-etale-local" ], "ref_ids": [ 10714 ] } ], "ref_ids": [ 6524 ] }, { "id": 6526, "type": "theorem", "label": "etale-cohomology-lemma-ses-associated-to-open", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-ses-associated-to-open", "contents": [ "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme and let", "$U \\subset X$ be the complement. Denote $i : Z \\to X$ and $j : U \\to X$", "the inclusion morphisms. For every abelian sheaf $\\mathcal{F}$ on $X_\\etale$", "there is a canonical short exact sequence", "$$", "0 \\to j_!j^{-1}\\mathcal{F} \\to \\mathcal{F} \\to i_*i^{-1}\\mathcal{F} \\to 0", "$$", "on $X_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "We obtain the maps by the adjointness properties of the functors", "involved. For a geometric point $\\overline{x}$ in $X$ we have either", "$\\overline{x} \\in U$ in which case the map on the left hand side", "is an isomorphism on stalks and the stalk of $i_*i^{-1}\\mathcal{F}$", "is zero or $\\overline{x} \\in Z$ in which case the map on the right hand side", "is an isomorphism on stalks and the stalk of $j_!j^{-1}\\mathcal{F}$", "is zero. Here we have used the description of stalks of", "Lemma \\ref{lemma-stalk-pushforward-closed-immersion} and", "Proposition \\ref{proposition-describe-jshriek}." ], "refs": [ "etale-cohomology-lemma-stalk-pushforward-closed-immersion", "etale-cohomology-proposition-describe-jshriek" ], "ref_ids": [ 6461, 6705 ] } ], "ref_ids": [] }, { "id": 6527, "type": "theorem", "label": "etale-cohomology-lemma-constructible-quasi-compact-quasi-separated", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-constructible-quasi-compact-quasi-separated", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\mathcal{F}$", "be a sheaf of sets on $X_\\etale$. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is constructible,", "\\item there exists an open covering $X = \\bigcup U_i$ such that", "$\\mathcal{F}|_{U_i}$ is constructible, and", "\\item there exists a partition $X = \\bigcup X_i$ by constructible", "locally closed subschemes such that $\\mathcal{F}|_{X_i}$ is finite", "locally constant.", "\\end{enumerate}", "A similar statement holds for abelian sheaves and sheaves of", "$\\Lambda$-modules if $\\Lambda$ is Noetherian." ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2).", "\\medskip\\noindent", "Assume (2). For every $x \\in X$ we can find an $i$ and an affine open", "neighbourhood $V_x \\subset U_i$ of $x$. Hence we can find a finite", "affine open covering $X = \\bigcup V_j$ such that for each $j$ there", "exists a finite decomposition $V_j = \\coprod V_{j, k}$ by locally closed", "constructible subsets such that $\\mathcal{F}|_{V_{j, k}}$ is finite", "locally constant. By Topology, Lemma", "\\ref{topology-lemma-quasi-compact-open-immersion-constructible-image}", "each $V_{j, k}$ is constructible as a subset of $X$.", "By Topology, Lemma", "\\ref{topology-lemma-constructible-partition-refined-by-stratification}", "we can find a finite stratification $X = \\coprod X_l$ with constructible", "locally closed strata such that each", "$V_{j, k}$ is a union of $X_l$. Thus (3) holds.", "\\medskip\\noindent", "Assume (3) holds. Let $U \\subset X$ be an affine open.", "Then $U \\cap X_i$ is a constructible locally closed subset of $U$", "(for example by Properties, Lemma \\ref{properties-lemma-locally-constructible})", "and $U = \\coprod U \\cap X_i$ is a partition of $U$ as in", "Definition \\ref{definition-constructible}. Thus (1) holds." ], "refs": [ "topology-lemma-quasi-compact-open-immersion-constructible-image", "topology-lemma-constructible-partition-refined-by-stratification", "properties-lemma-locally-constructible", "etale-cohomology-definition-constructible" ], "ref_ids": [ 8256, 8335, 2938, 6756 ] } ], "ref_ids": [] }, { "id": 6528, "type": "theorem", "label": "etale-cohomology-lemma-constructible-constructible", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-constructible-constructible", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $\\mathcal{F}$ be a sheaf of sets, abelian groups,", "$\\Lambda$-modules (with $\\Lambda$ Noetherian) on $X_\\etale$.", "If there exist constructible locally closed subschemes $T_i \\subset X$", "such that (a) $X = \\bigcup T_j$ and (b) $\\mathcal{F}|_{T_j}$ is", "constructible, then $\\mathcal{F}$ is constructible." ], "refs": [], "proofs": [ { "contents": [ "First, we can assume the covering is finite as $X$ is quasi-compact", "in the spectral topology", "(Topology, Lemma \\ref{topology-lemma-constructible-hausdorff-quasi-compact}", "and", "Properties, Lemma", "\\ref{properties-lemma-quasi-compact-quasi-separated-spectral}).", "Observe that each $T_i$ is a quasi-compact and quasi-separated", "scheme in its own right (because it is constructible in $X$; details omitted).", "Thus we can find a finite partition $T_i = \\coprod T_{i, j}$ into", "locally closed constructible parts of $T_i$ such that", "$\\mathcal{F}|_{T_{i, j}}$ is finite locally constant", "(Lemma \\ref{lemma-constructible-quasi-compact-quasi-separated}).", "By Topology, Lemma \\ref{topology-lemma-constructible-in-constructible}", "we see that $T_{i, j}$ is a constructible locally closed subscheme of $X$.", "Then we can apply Topology, Lemma", "\\ref{topology-lemma-constructible-partition-refined-by-stratification}", "to $X = \\bigcup T_{i, j}$ to find the desired partition of $X$." ], "refs": [ "topology-lemma-constructible-hausdorff-quasi-compact", "properties-lemma-quasi-compact-quasi-separated-spectral", "etale-cohomology-lemma-constructible-quasi-compact-quasi-separated", "topology-lemma-constructible-in-constructible", "topology-lemma-constructible-partition-refined-by-stratification" ], "ref_ids": [ 8303, 2941, 6527, 8263, 8335 ] } ], "ref_ids": [] }, { "id": 6529, "type": "theorem", "label": "etale-cohomology-lemma-constructible-local", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-constructible-local", "contents": [ "Let $X$ be a scheme. Checking constructibility of a sheaf", "of sets, abelian groups, $\\Lambda$-modules (with $\\Lambda$ Noetherian)", "can be done Zariski locally on $X$." ], "refs": [], "proofs": [ { "contents": [ "The statement means if $X = \\bigcup U_i$ is an open covering", "such that $\\mathcal{F}|_{U_i}$ is constructible, then $\\mathcal{F}$", "is constructible. If $U \\subset X$ is affine open, then", "$U = \\bigcup U \\cap U_i$ and $\\mathcal{F}|_{U \\cap U_i}$ is constructible", "(it is trivial that the restriction of a constructible sheaf to", "an open is constructible). It follows from", "Lemma \\ref{lemma-constructible-quasi-compact-quasi-separated}", "that $\\mathcal{F}|_U$ is constructible, i.e., a suitable partition", "of $U$ exists." ], "refs": [ "etale-cohomology-lemma-constructible-quasi-compact-quasi-separated" ], "ref_ids": [ 6527 ] } ], "ref_ids": [] }, { "id": 6530, "type": "theorem", "label": "etale-cohomology-lemma-pullback-constructible", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-pullback-constructible", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. If $\\mathcal{F}$ is a", "constructible sheaf of sets, abelian groups, or $\\Lambda$-modules", "(with $\\Lambda$ Noetherian) on $Y_\\etale$, the same", "is true for $f^{-1}\\mathcal{F}$ on $X_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-constructible-local} this reduces to the case", "where $X$ and $Y$ are affine. By", "Lemma \\ref{lemma-constructible-quasi-compact-quasi-separated}", "it suffices to find a finite partition of $X$ by constructible", "locally closed subschemes such that $f^{-1}\\mathcal{F}$ is finite locally", "constant on each of them.", "To find it we just pull back the partition of $Y$ adapted to", "$\\mathcal{F}$ and use", "Lemma \\ref{lemma-pullback-locally-constant}." ], "refs": [ "etale-cohomology-lemma-constructible-local", "etale-cohomology-lemma-constructible-quasi-compact-quasi-separated", "etale-cohomology-lemma-pullback-locally-constant" ], "ref_ids": [ 6529, 6527, 6501 ] } ], "ref_ids": [] }, { "id": 6531, "type": "theorem", "label": "etale-cohomology-lemma-constructible-abelian", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-constructible-abelian", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item The category of constructible sheaves of sets", "is closed under finite limits and colimits inside $\\Sh(X_\\etale)$.", "\\item The category of constructible abelian sheaves is a", "weak Serre subcategory of $\\textit{Ab}(X_\\etale)$.", "\\item Let $\\Lambda$ be a Noetherian ring. The category of", "constructible sheaves of $\\Lambda$-modules on", "$X_\\etale$ is a weak Serre subcategory of", "$\\textit{Mod}(X_\\etale, \\Lambda)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We prove (3). We will use the criterion of", "Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}.", "Suppose that $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "is a map of constructible sheaves of $\\Lambda$-modules.", "We have to show that $\\mathcal{K} = \\Ker(\\varphi)$ and", "$\\mathcal{Q} = \\Coker(\\varphi)$ are constructible. ", "Similarly, suppose that", "$0 \\to \\mathcal{F} \\to \\mathcal{E} \\to \\mathcal{G} \\to 0$", "is a short exact sequence of sheaves of $\\Lambda$-modules", "with $\\mathcal{F}$, $\\mathcal{G}$ constructible. We have to show", "that $\\mathcal{E}$ is constructible.", "In both cases we can replace $X$ with the members of an", "affine open covering. Hence we may assume $X$ is affine.", "Then we may further replace $X$ by the members of a finite", "partition of $X$ by constructible locally closed subschemes", "on which $\\mathcal{F}$ and $\\mathcal{G}$ are of finite type and", "locally constant. Thus we may apply", "Lemma \\ref{lemma-kernel-finite-locally-constant} to conclude.", "\\medskip\\noindent", "The proofs of (1) and (2) are very similar and are omitted." ], "refs": [ "homology-lemma-characterize-weak-serre-subcategory", "etale-cohomology-lemma-kernel-finite-locally-constant" ], "ref_ids": [ 12046, 6505 ] } ], "ref_ids": [] }, { "id": 6532, "type": "theorem", "label": "etale-cohomology-lemma-tensor-product-constructible", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-tensor-product-constructible", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.", "The tensor product of two constructible sheaves of $\\Lambda$-modules", "on $X_\\etale$ is a constructible sheaf of $\\Lambda$-modules." ], "refs": [], "proofs": [ { "contents": [ "The question immediately reduces to the case where $X$ is affine.", "Since any two partitions of $X$ with constructible locally", "closed strata have a common refinement of the same type and", "since pullbacks commute with tensor product we reduce to", "Lemma \\ref{lemma-tensor-product-locally-constant}." ], "refs": [ "etale-cohomology-lemma-tensor-product-locally-constant" ], "ref_ids": [ 6506 ] } ], "ref_ids": [] }, { "id": 6533, "type": "theorem", "label": "etale-cohomology-lemma-support-constructible", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-support-constructible", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "\\begin{enumerate}", "\\item Let $\\mathcal{F} \\to \\mathcal{G}$ be a map of constructible", "sheaves of sets on $X_\\etale$. Then the set of points $x \\in X$", "where $\\mathcal{F}_{\\overline{x}} \\to \\mathcal{G}_{\\overline{x}}$", "is surjective, resp.\\ injective, resp.\\ is isomorphic to a given map", "of sets, is constructible in $X$.", "\\item Let $\\mathcal{F}$ be a constructible abelian sheaf on $X_\\etale$.", "The support of $\\mathcal{F}$ is constructible.", "\\item Let $\\Lambda$ be a Noetherian ring.", "Let $\\mathcal{F}$ be a constructible sheaf of $\\Lambda$-modules on $X_\\etale$.", "The support of $\\mathcal{F}$ is constructible.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1).", "Let $X = \\coprod X_i$ be a partition of $X$ by locally closed constructible", "subschemes such that both $\\mathcal{F}$ and $\\mathcal{G}$ are", "finite locally constant over the parts (use", "Lemma \\ref{lemma-constructible-quasi-compact-quasi-separated}", "for both $\\mathcal{F}$ and $\\mathcal{G}$ and choose a common", "refinement). Then apply Lemma \\ref{lemma-morphism-locally-constant}", "to the restriction of the map to each part.", "\\medskip\\noindent", "The proof of (2) and (3) is omitted." ], "refs": [ "etale-cohomology-lemma-constructible-quasi-compact-quasi-separated", "etale-cohomology-lemma-morphism-locally-constant" ], "ref_ids": [ 6527, 6504 ] } ], "ref_ids": [] }, { "id": 6534, "type": "theorem", "label": "etale-cohomology-lemma-colimit-constructible", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-colimit-constructible", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Let", "$\\mathcal{F} = \\colim_{i \\in I} \\mathcal{F}_i$ be a filtered", "colimit of sheaves of sets, abelian sheaves, or sheaves of modules.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ and $\\mathcal{F}_i$ are constructible sheaves of", "sets, then the ind-object $\\mathcal{F}_i$ is essentially constant with", "value $\\mathcal{F}$.", "\\item If $\\mathcal{F}$ and $\\mathcal{F}_i$ are constructible sheaves of", "abelian groups, then the ind-object $\\mathcal{F}_i$ is essentially constant", "with value $\\mathcal{F}$.", "\\item Let $\\Lambda$ be a Noetherian ring.", "If $\\mathcal{F}$ and $\\mathcal{F}_i$ are constructible sheaves of", "$\\Lambda$-modules, then the ind-object $\\mathcal{F}_i$ is essentially constant", "with value $\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). We will use without further mention that finite limits", "and colimits of constructible sheaves are constructible", "(Lemma \\ref{lemma-kernel-finite-locally-constant}).", "For each $i$ let $T_i \\subset X$ be the set of points $x \\in X$", "where $\\mathcal{F}_{i, \\overline{x}} \\to \\mathcal{F}_{\\overline{x}}$", "is not surjective. Because $\\mathcal{F}_i$ and $\\mathcal{F}$ are", "constructible $T_i$ is a constructible subset of $X$", "(Lemma \\ref{lemma-support-constructible}).", "Since the stalks of $\\mathcal{F}$ are finite", "and since $\\mathcal{F} = \\colim_{i \\in I} \\mathcal{F}_i$ we see", "that for all $x \\in X$ we have $x \\not \\in T_i$ for $i$ large enough.", "Since $X$ is a spectral space by Properties, Lemma", "\\ref{properties-lemma-quasi-compact-quasi-separated-spectral}", "the constructible topology on $X$ is quasi-compact by", "Topology, Lemma \\ref{topology-lemma-constructible-hausdorff-quasi-compact}.", "Thus $T_i = \\emptyset$ for $i$ large enough. Thus", "$\\mathcal{F}_i \\to \\mathcal{F}$ is surjective for $i$ large enough.", "Assume now that $\\mathcal{F}_i \\to \\mathcal{F}$ is surjective for all $i$.", "Choose $i \\in I$. For $i' \\geq i$ denote $S_{i'} \\subset X$ the set of", "points $x$ such that the number of elements in", "$\\Im(\\mathcal{F}_{i, \\overline{x}} \\to \\mathcal{F}_{\\overline{x}})$", "is equal to the number of elements in", "$\\Im(\\mathcal{F}_{i, \\overline{x}} \\to \\mathcal{F}_{i', \\overline{x}})$.", "Because $\\mathcal{F}_i$, $\\mathcal{F}_{i'}$ and $\\mathcal{F}$ are", "constructible $S_{i'}$ is a constructible subset of $X$", "(details omitted; hint: use Lemma \\ref{lemma-support-constructible}).", "Since the stalks of $\\mathcal{F}_i$ and $\\mathcal{F}$", "are finite and since $\\mathcal{F} = \\colim_{i' \\geq i} \\mathcal{F}_{i'}$", "we see that for all $x \\in X$ we have $x \\not \\in S_{i'}$ for $i'$", "large enough. By the same argument as above we can find a large $i'$ such", "that $S_{i'} = \\emptyset$. Thus $\\mathcal{F}_i \\to \\mathcal{F}_{i'}$", "factors through $\\mathcal{F}$ as desired.", "\\medskip\\noindent", "Proof of (2). Observe that a constructible abelian sheaf is a constructible", "sheaf of sets. Thus case (2) follows from (1).", "\\medskip\\noindent", "Proof of (3). We will use without further mention that the category of", "constructible sheaves of $\\Lambda$-modules is abelian", "(Lemma \\ref{lemma-kernel-finite-locally-constant}).", "For each $i$ let $\\mathcal{Q}_i$ be the cokernel of the map", "$\\mathcal{F}_i \\to \\mathcal{F}$. The support $T_i$ of $\\mathcal{Q}_i$", "is a constructible subset of $X$ as $\\mathcal{Q}_i$ is constructible", "(Lemma \\ref{lemma-support-constructible}).", "Since the stalks of $\\mathcal{F}$ are finite $\\Lambda$-modules", "and since $\\mathcal{F} = \\colim_{i \\in I} \\mathcal{F}_i$ we see", "that for all $x \\in X$ we have $x \\not \\in T_i$ for $i$ large enough.", "Since $X$ is a spectral space by Properties, Lemma", "\\ref{properties-lemma-quasi-compact-quasi-separated-spectral}", "the constructible topology on $X$ is quasi-compact by", "Topology, Lemma \\ref{topology-lemma-constructible-hausdorff-quasi-compact}.", "Thus $T_i = \\emptyset$ for $i$ large enough. This proves the first", "assertion. For the second, assume now that", "$\\mathcal{F}_i \\to \\mathcal{F}$ is surjective for all $i$.", "Choose $i \\in I$. For $i' \\geq i$ denote $\\mathcal{K}_{i'}$ the", "image of $\\Ker(\\mathcal{F}_i \\to \\mathcal{F})$ in $\\mathcal{F}_{i'}$.", "The support $S_{i'}$ of $\\mathcal{K}_{i'}$", "is a constructible subset of $X$ as $\\mathcal{K}_{i'}$ is constructible.", "Since the stalks of $\\Ker(\\mathcal{F}_i \\to \\mathcal{F})$", "are finite $\\Lambda$-modules and since", "$\\mathcal{F} = \\colim_{i' \\geq i} \\mathcal{F}_{i'}$ we see", "that for all $x \\in X$ we have $x \\not \\in S_{i'}$ for $i'$ large enough.", "By the same argument as above we can find a large $i'$ such", "that $S_{i'} = \\emptyset$. Thus $\\mathcal{F}_i \\to \\mathcal{F}_{i'}$", "factors through $\\mathcal{F}$ as desired." ], "refs": [ "etale-cohomology-lemma-kernel-finite-locally-constant", "etale-cohomology-lemma-support-constructible", "properties-lemma-quasi-compact-quasi-separated-spectral", "topology-lemma-constructible-hausdorff-quasi-compact", "etale-cohomology-lemma-support-constructible", "etale-cohomology-lemma-kernel-finite-locally-constant", "etale-cohomology-lemma-support-constructible", "properties-lemma-quasi-compact-quasi-separated-spectral", "topology-lemma-constructible-hausdorff-quasi-compact" ], "ref_ids": [ 6505, 6533, 2941, 8303, 6533, 6505, 6533, 2941, 8303 ] } ], "ref_ids": [] }, { "id": 6535, "type": "theorem", "label": "etale-cohomology-lemma-etale-stratified-finite", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-etale-stratified-finite", "contents": [ "Let $U \\to X$ be an \\'etale morphism of quasi-compact and quasi-separated", "schemes (for example an \\'etale morphism of Noetherian schemes). Then there", "exists a partition $X = \\coprod_i X_i$ by constructible locally closed", "subschemes such that $X_i \\times_X U \\to X_i$ is finite \\'etale for all $i$." ], "refs": [], "proofs": [ { "contents": [ "If $U \\to X$ is separated, then this is", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-stratify-flat-fp-qf}.", "In general, we may assume $X$ is affine. Choose a finite affine open", "covering $U = \\bigcup U_j$. Apply the previous case to all the morphisms", "$U_j \\to X$ and $U_j \\cap U_{j'} \\to X$ and choose a common", "refinement $X = \\coprod X_i$ of the resulting partitions.", "After refining the partition further we may assume $X_i$ affine as well.", "Fix $i$ and set $V = U \\times_X X_i$. The morphisms", "$V_j = U_j \\times_X X_i \\to X_i$ and", "$V_{jj'} = (U_j \\cap U_{j'}) \\times_X X_i \\to X_i$ are finite \\'etale.", "Hence $V_j$ and $V_{jj'}$ are affine schemes and $V_{jj'} \\subset V_j$", "is closed as well as open (since $V_{jj'} \\to X_i$ is proper, so", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}", "applies). Then $V = \\bigcup V_j$ is separated because", "$\\mathcal{O}(V_j) \\to \\mathcal{O}(V_{jj'})$ is surjective, see", "Schemes, Lemma \\ref{schemes-lemma-characterize-separated}.", "Thus the previous case applies to $V \\to X_i$ and we can further", "refine the partition if needed (it actually isn't but we don't", "need this)." ], "refs": [ "more-morphisms-lemma-stratify-flat-fp-qf", "morphisms-lemma-image-proper-scheme-closed", "schemes-lemma-characterize-separated" ], "ref_ids": [ 13909, 5411, 7710 ] } ], "ref_ids": [] }, { "id": 6536, "type": "theorem", "label": "etale-cohomology-lemma-generically-finite", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-generically-finite", "contents": [ "Let $f: X \\to Y$ be a morphism of schemes which is quasi-compact,", "quasi-separated, and locally of finite type. If $\\eta$ is a generic point", "of an irreducible component of $Y$ such that $f^{-1}(\\eta)$ is finite, then", "there exists an open $V \\subset Y$ containing $\\eta$ such that", "$f^{-1}(V) \\to V$ is finite." ], "refs": [], "proofs": [ { "contents": [ "This is Morphisms, Lemma \\ref{morphisms-lemma-generically-finite}." ], "refs": [ "morphisms-lemma-generically-finite" ], "ref_ids": [ 5487 ] } ], "ref_ids": [] }, { "id": 6537, "type": "theorem", "label": "etale-cohomology-lemma-decompose-quasi-finite-morphism", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-decompose-quasi-finite-morphism", "contents": [ "Let $f : Y \\to X$ be a quasi-finite and finitely presented", "morphism of affine schemes.", "\\begin{enumerate}", "\\item There exists a surjective morphism of affine schemes $X' \\to X$ and a", "closed subscheme $Z' \\subset Y' = X' \\times_X Y$ such that", "\\begin{enumerate}", "\\item $Z' \\subset Y'$ is a thickening, and", "\\item $Z' \\to X'$ is a finite \\'etale morphism.", "\\end{enumerate}", "\\item There exists a finite partition $X = \\coprod X_i$ by", "locally closed, constructible, affine strata, and surjective finite locally", "free morphisms $X'_i \\to X_i$ such that the reduction of", "$Y'_i = X'_i \\times_X Y \\to X'_i$ is isomorphic to", "$\\coprod_{j = 1}^{n_i} (X'_i)_{red} \\to (X'_i)_{red}$ for some $n_i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Setting $X' = \\coprod X'_i$ we see that (2) implies (1).", "Write $X = \\Spec(A)$ and $Y = \\Spec(B)$. Write $A$ as a filtered colimit", "of finite type $\\mathbf{Z}$-algebras $A_i$. Since $B$ is an $A$-algebra of", "finite presentation, we see that there exists $0 \\in I$ and a", "finite type ring map $A_0 \\to B_0$ such that $B = \\colim B_i$ with", "$B_i = A_i \\otimes_{A_0} B_0$, see", "Algebra, Lemma \\ref{algebra-lemma-colimit-category-fp-algebras}.", "For $i$ sufficiently large we see that $A_i \\to B_i$ is", "quasi-finite, see Limits, Lemma \\ref{limits-lemma-descend-quasi-finite}.", "Thus we reduce to the case of finite type algebras over $\\mathbf{Z}$,", "in particular we reduce to the Noetherian case. (Details omitted.)", "\\medskip\\noindent", "Assume $X$ and $Y$ Noetherian. In this case any locally closed", "subset of $X$ is constructible. By Lemma \\ref{lemma-generically-finite}", "and Noetherian induction we see that", "there is a finite partition $X = \\coprod X_i$ of $X$", "by locally closed strata such that $Y \\times_X X_i \\to X_i$ is finite.", "We can refine this partition to get affine strata.", "Thus after replacing $X$ by $X' = \\coprod X_i$ we may assume", "$Y \\to X$ is finite.", "\\medskip\\noindent", "Assume $X$ and $Y$ Noetherian and $Y \\to X$ finite.", "Suppose that we can prove (2) after base change by a surjective,", "flat, quasi-finite morphism $U \\to X$. Thus we have a partition", "$U = \\coprod U_i$ and finite locally free morphisms $U'_i \\to U_i$", "such that $U'_i \\times_X Y \\to U'_i$ is isomorphic to", "$\\coprod_{j = 1}^{n_i} (U'_i)_{red} \\to (U'_i)_{red}$ for some $n_i$.", "Then, by the argument in the previous paragraph, we can find a", "partition $X = \\coprod X_j$ with locally closed affine strata such that", "$X_j \\times_X U_i \\to X_j$ is finite for all $i, j$. By", "Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}", "each $X_j \\times_X U_i \\to X_j$ is finite locally free.", "Hence $X_j \\times_X U'_i \\to X_j$ is finite locally free", "(Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-locally-free}).", "It follows that $X = \\coprod X_j$ and $X_j' = \\coprod_i X_j \\times_X U'_i$", "is a solution for $Y \\to X$. Thus it suffices to prove", "the result (in the Noetherian case) after a surjective flat quasi-finite", "base change.", "\\medskip\\noindent", "Applying Morphisms, Lemma \\ref{morphisms-lemma-massage-finite}", "we see we may assume that $Y$ is a closed subscheme of an", "affine scheme $Z$ which is (set theoretically) a finite union", "$Z = \\bigcup_{i \\in I} Z_i$ of closed subschemes mapping isomorphically", "to $X$. In this case we will find a finite partition of $X = \\coprod X_j$", "with affine locally closed strata that works (in other words $X'_j = X_j$).", "Set $T_i = Y \\cap Z_i$. This is a closed subscheme of $X$.", "As $X$ is Noetherian we can find a finite partition of $X = \\coprod X_j$", "by affine locally closed subschemes, such that each", "$X_j \\times_X T_i$ is (set theoretically) a union of strata $X_j \\times_X Z_i$.", "Replacing $X$ by $X_j$ we see that we may assume $I = I_1 \\amalg I_2$", "with $Z_i \\subset Y$ for $i \\in I_1$ and $Z_i \\cap Y = \\emptyset$ for", "$i \\in I_2$. Replacing $Z$ by $\\bigcup_{i \\in I_1} Z_i$ we see that we", "may assume $Y = Z$.", "Finally, we can replace $X$ again by the members of a partition", "as above such that for every $i, i' \\subset I$ the intersection", "$Z_i \\cap Z_{i'}$ is either empty or (set theoretically) equal", "to $Z_i$ and $Z_{i'}$. This clearly means that $Y$ is (set theoretically)", "equal to a disjoint union of the $Z_i$ which is what we wanted to show." ], "refs": [ "algebra-lemma-colimit-category-fp-algebras", "limits-lemma-descend-quasi-finite", "etale-cohomology-lemma-generically-finite", "morphisms-lemma-finite-flat", "morphisms-lemma-composition-finite-locally-free", "morphisms-lemma-massage-finite" ], "ref_ids": [ 1097, 15105, 6536, 5471, 5472, 5475 ] } ], "ref_ids": [] }, { "id": 6538, "type": "theorem", "label": "etale-cohomology-lemma-jshriek-constructible", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-jshriek-constructible", "contents": [ "Let $j : U \\to X$ be an \\'etale morphism of quasi-compact and", "quasi-separated schemes.", "\\begin{enumerate}", "\\item The sheaf $h_U$ is a constructible sheaf of sets.", "\\item The sheaf $j_!\\underline{M}$ is a constructible abelian sheaf", "for a finite abelian group $M$.", "\\item If $\\Lambda$ is a Noetherian ring and $M$ is a finite $\\Lambda$-module,", "then $j_!\\underline{M}$ is a constructible sheaf of $\\Lambda$-modules", "on $X_\\etale$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-etale-stratified-finite} there is a partition", "$\\coprod_i X_i$ such that $\\pi_i : j^{-1}(X_i) \\to X_i$ is finite \\'etale.", "The restriction of $h_U$ to $X_i$ is $h_{j^{-1}(X_i)}$ which is finite", "locally constant by Lemma \\ref{lemma-characterize-finite-locally-constant}.", "For cases (2) and (3) we note that", "$$", "j_!(\\underline{M})|_{X_i} =", "\\pi_{i!}(\\underline{M}) =", "\\pi_{i*}(\\underline{M})", "$$", "by Lemmas \\ref{lemma-shriek-base-change} and", "\\ref{lemma-shriek-equals-star-finite-etale}.", "Thus it suffices to show the lemma for $\\pi : Y \\to X$ finite \\'etale.", "This is Lemma \\ref{lemma-pushforward-locally-constant}." ], "refs": [ "etale-cohomology-lemma-etale-stratified-finite", "etale-cohomology-lemma-characterize-finite-locally-constant", "etale-cohomology-lemma-shriek-base-change", "etale-cohomology-lemma-shriek-equals-star-finite-etale", "etale-cohomology-lemma-pushforward-locally-constant" ], "ref_ids": [ 6535, 6503, 6523, 6525, 6502 ] } ], "ref_ids": [] }, { "id": 6539, "type": "theorem", "label": "etale-cohomology-lemma-torsion-colimit-constructible", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-torsion-colimit-constructible", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$.", "Then $\\mathcal{F}$ is a filtered colimit of constructible", "sheaves of sets.", "\\item Let $\\mathcal{F}$ be a torsion abelian sheaf on $X_\\etale$.", "Then $\\mathcal{F}$ is a filtered colimit of constructible abelian sheaves.", "\\item Let $\\Lambda$ be a Noetherian ring and $\\mathcal{F}$ a sheaf", "of $\\Lambda$-modules on $X_\\etale$. Then", "$\\mathcal{F}$ is a filtered colimit of constructible sheaves of", "$\\Lambda$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{B}$ be the collection of quasi-compact and quasi-separated", "objects of $X_\\etale$. By Modules on Sites,", "Lemma \\ref{sites-modules-lemma-module-filtered-colimit-constructibles}", "any sheaf of sets is a filtered colimit of sheaves of the form", "$$", "\\text{Coequalizer}\\left(", "\\xymatrix{", "\\coprod\\nolimits_{j = 1, \\ldots, m} h_{V_j}", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\coprod\\nolimits_{i = 1, \\ldots, n} h_{U_i}", "}", "\\right)", "$$", "with $V_j$ and $U_i$ quasi-compact and quasi-separated objects", "of $X_\\etale$. By", "Lemmas \\ref{lemma-jshriek-constructible} and \\ref{lemma-constructible-abelian}", "these coequalizers are constructible. This proves (1).", "\\medskip\\noindent", "Let $\\Lambda$ be a Noetherian ring.", "By Modules on Sites,", "Lemma \\ref{sites-modules-lemma-module-filtered-colimit-constructibles}", "$\\Lambda$-modules $\\mathcal{F}$ is a filtered colimit", "of modules of the form", "$$", "\\Coker\\left(", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} j_{V_j!}\\underline{\\Lambda}_{V_j}", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\underline{\\Lambda}_{U_i}", "\\right)", "$$", "with $V_j$ and $U_i$ quasi-compact and quasi-separated objects", "of $X_\\etale$. By", "Lemmas \\ref{lemma-jshriek-constructible} and \\ref{lemma-constructible-abelian}", "these cokernels are constructible. This proves (3).", "\\medskip\\noindent", "Proof of (2). First write $\\mathcal{F} = \\bigcup \\mathcal{F}[n]$ where", "$\\mathcal{F}[n]$ is the $n$-torsion subsheaf. Then we can view", "$\\mathcal{F}[n]$ as a sheaf of $\\mathbf{Z}/n\\mathbf{Z}$-modules", "and apply (3)." ], "refs": [ "sites-modules-lemma-module-filtered-colimit-constructibles", "etale-cohomology-lemma-jshriek-constructible", "etale-cohomology-lemma-constructible-abelian", "sites-modules-lemma-module-filtered-colimit-constructibles", "etale-cohomology-lemma-jshriek-constructible", "etale-cohomology-lemma-constructible-abelian" ], "ref_ids": [ 14218, 6538, 6531, 14218, 6538, 6531 ] } ], "ref_ids": [] }, { "id": 6540, "type": "theorem", "label": "etale-cohomology-lemma-check-constructible", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-check-constructible", "contents": [ "Let $f : X \\to Y$ be a surjective morphism of quasi-compact and", "quasi-separated schemes.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be a sheaf of sets on $Y_\\etale$. Then $\\mathcal{F}$", "is constructible if and only if $f^{-1}\\mathcal{F}$ is constructible.", "\\item Let $\\mathcal{F}$ be an abelian sheaf on $Y_\\etale$. Then $\\mathcal{F}$", "is constructible if and only if $f^{-1}\\mathcal{F}$ is constructible.", "\\item Let $\\Lambda$ be a Noetherian ring.", "Let $\\mathcal{F}$ be sheaf of $\\Lambda$-modules on $Y_\\etale$.", "Then $\\mathcal{F}$ is constructible if and only if $f^{-1}\\mathcal{F}$", "is constructible.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "One implication follows from Lemma \\ref{lemma-pullback-constructible}.", "For the converse, assume $f^{-1}\\mathcal{F}$ is constructible.", "Write $\\mathcal{F} = \\colim \\mathcal{F}_i$ as a", "filtered colimit of constructible sheaves (of sets, abelian groups, or modules)", "using Lemma \\ref{lemma-torsion-colimit-constructible}.", "Since $f^{-1}$ is a left adjoint it commutes with colimits", "(Categories, Lemma \\ref{categories-lemma-adjoint-exact}) and we see that", "$f^{-1}\\mathcal{F} = \\colim f^{-1}\\mathcal{F}_i$.", "By Lemma \\ref{lemma-colimit-constructible} we see that", "$f^{-1}\\mathcal{F}_i \\to f^{-1}\\mathcal{F}$", "is surjective for all $i$ large enough.", "Since $f$ is surjective we conclude (by looking at stalks using", "Lemma \\ref{lemma-stalk-pullback} and", "Theorem \\ref{theorem-exactness-stalks})", "that $\\mathcal{F}_i \\to \\mathcal{F}$ is surjective for all $i$ large enough.", "Thus $\\mathcal{F}$ is the quotient of a constructible sheaf $\\mathcal{G}$.", "Applying the argument once more to", "$\\mathcal{G} \\times_\\mathcal{F} \\mathcal{G}$ or", "the kernel of $\\mathcal{G} \\to \\mathcal{F}$", "we conclude using that $f^{-1}$ is exact and that the category of", "constructible sheaves (of sets, abelian groups, or modules) is", "preserved under finite (co)limits or (co)kernels inside", "$\\Sh(Y_\\etale)$, $\\Sh(X_\\etale)$, $\\textit{Ab}(Y_\\etale)$,", "$\\textit{Ab}(X_\\etale)$, $\\textit{Mod}(Y_\\etale, \\Lambda)$, and", "$\\textit{Mod}(X_\\etale, \\Lambda)$, see", "Lemma \\ref{lemma-constructible-abelian}." ], "refs": [ "etale-cohomology-lemma-pullback-constructible", "etale-cohomology-lemma-torsion-colimit-constructible", "categories-lemma-adjoint-exact", "etale-cohomology-lemma-colimit-constructible", "etale-cohomology-lemma-stalk-pullback", "etale-cohomology-theorem-exactness-stalks", "etale-cohomology-lemma-constructible-abelian" ], "ref_ids": [ 6530, 6539, 12249, 6534, 6436, 6376, 6531 ] } ], "ref_ids": [] }, { "id": 6541, "type": "theorem", "label": "etale-cohomology-lemma-pushforward-constructible", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-pushforward-constructible", "contents": [ "Let $f : X \\to Y$ be a finite \\'etale morphism of schemes. Let $\\Lambda$ be a", "Noetherian ring. If $\\mathcal{F}$ is a constructible sheaf of sets,", "constructible sheaf of abelian groups, or constructible sheaf of", "$\\Lambda$-modules on $X_\\etale$, the same is true for", "$f_*\\mathcal{F}$ on $Y_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-constructible-local} it suffices to check this", "Zariski locally on $Y$ and by Lemma \\ref{lemma-check-constructible}", "we may replace $Y$ by an \\'etale cover (the construction of $f_*$", "commutes with \\'etale localization). A finite \\'etale morphism is", "\\'etale locally isomorphic to a disjoint union of isomorphisms, see", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-finite-etale-etale-local}.", "Thus, in the case of sheaves of sets, the lemma says that if", "$\\mathcal{F}_i$, $i = 1, \\ldots, n$ are constructible sheaves of sets, then", "$\\prod_{i = 1, \\ldots, n} \\mathcal{F}_i$ is too.", "This is clear. Similarly for sheaves of abelian groups and modules." ], "refs": [ "etale-cohomology-lemma-constructible-local", "etale-cohomology-lemma-check-constructible", "etale-lemma-finite-etale-etale-local" ], "ref_ids": [ 6529, 6540, 10714 ] } ], "ref_ids": [] }, { "id": 6542, "type": "theorem", "label": "etale-cohomology-lemma-category-constructible-sets", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-category-constructible-sets", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. The category of", "constructible sheaves of sets is the full subcategory of $\\Sh(X_\\etale)$", "consisting of sheaves $\\mathcal{F}$ which are coequalizers", "$$", "\\xymatrix{", "\\mathcal{F}_1", "\\ar@<1ex>[r] \\ar@<-1ex>[r]", "&", "\\mathcal{F}_0 \\ar[r]", "&", "\\mathcal{F}}", "$$", "such that $\\mathcal{F}_i$, $i = 0, 1$ is a finite coproduct of sheaves of", "the form $h_U$ with $U$ a quasi-compact and quasi-separated", "object of $X_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-torsion-colimit-constructible}", "we have seen that sheaves of this form are constructible.", "For the converse, suppose that for every constructible sheaf of", "sets $\\mathcal{F}$ we can find a surjection $\\mathcal{F}_0 \\to \\mathcal{F}$", "with $\\mathcal{F}_0$ as in the lemma. Then we find our surjection", "$\\mathcal{F}_1 \\to \\mathcal{F}_0 \\times_\\mathcal{F} \\mathcal{F}_0$", "because the latter is constructible by Lemma \\ref{lemma-constructible-abelian}.", "\\medskip\\noindent", "By Topology, Lemma", "\\ref{topology-lemma-constructible-partition-refined-by-stratification}", "we may choose a finite stratification", "$X = \\coprod_{i \\in I} X_i$ such that $\\mathcal{F}$ is finite locally", "constant on each stratum. We will prove the result by induction on", "the cardinality of $I$. Let $i \\in I$ be a minimal element in the", "partial ordering of $I$. Then $X_i \\subset X$ is closed.", "By induction, there exist finitely many quasi-compact and quasi-separated", "objects $U_\\alpha$ of $(X \\setminus X_i)_\\etale$ and a surjective", "map $\\coprod h_{U_\\alpha} \\to \\mathcal{F}|_{X \\setminus X_i}$.", "These determine a map", "$$", "\\coprod h_{U_\\alpha} \\to \\mathcal{F}", "$$", "which is surjective after restricting to $X \\setminus X_i$. By", "Lemma \\ref{lemma-characterize-finite-locally-constant}", "we see that $\\mathcal{F}|_{X_i} = h_V$ for some scheme $V$ finite \\'etale", "over $X_i$. Let $\\overline{v}$ be a geometric point of $V$ lying", "over $\\overline{x} \\in X_i$. We may think of $\\overline{v}$ as an", "element of the stalk $\\mathcal{F}_{\\overline{x}} = V_{\\overline{x}}$.", "Thus we can find an \\'etale neighbourhood $(U, \\overline{u})$", "of $\\overline{x}$ and a section $s \\in \\mathcal{F}(U)$ whose stalk at", "$\\overline{x}$ gives $\\overline{v}$. Thinking of $s$ as a map", "$s : h_U \\to \\mathcal{F}$, restricting to $X_i$ we obtain a morphism", "$s|_{X_i} : U \\times_X X_i \\to V$ over $X_i$ which maps $\\overline{u}$", "to $\\overline{v}$. Since $V$ is quasi-compact (finite over the closed", "subscheme $X_i$ of the quasi-compact scheme $X$) a finite number", "$s^{(1)}, \\ldots, s^{(m)}$ of these sections of $\\mathcal{F}$ over", "$U^{(1)}, \\ldots, U^{(m)}$ will determine a jointly", "surjective map", "$$", "\\coprod s^{(j)}|_{X_i} : \\coprod U^{(j)} \\times_X X_i \\longrightarrow V", "$$", "Then we obtain the surjection", "$$", "\\coprod h_{U_\\alpha} \\amalg \\coprod h_{U^{(j)}} \\to \\mathcal{F}", "$$", "as desired." ], "refs": [ "etale-cohomology-lemma-torsion-colimit-constructible", "etale-cohomology-lemma-constructible-abelian", "topology-lemma-constructible-partition-refined-by-stratification", "etale-cohomology-lemma-characterize-finite-locally-constant" ], "ref_ids": [ 6539, 6531, 8335, 6503 ] } ], "ref_ids": [] }, { "id": 6543, "type": "theorem", "label": "etale-cohomology-lemma-category-constructible-modules", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-category-constructible-modules", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\Lambda$", "be a Noetherian ring. The category of constructible sheaves of", "$\\Lambda$-modules is exactly the category of modules of the form", "$$", "\\Coker\\left(", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} j_{V_j!}\\underline{\\Lambda}_{V_j}", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\underline{\\Lambda}_{U_i}", "\\right)", "$$", "with $V_j$ and $U_i$ quasi-compact and quasi-separated objects of", "$X_\\etale$. In fact, we can even assume $U_i$ and $V_j$ affine." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-torsion-colimit-constructible}", "we have seen modules of this form are constructible. Since the", "category of constructible modules is abelian", "(Lemma \\ref{lemma-constructible-abelian})", "it suffices to prove that given a constructible module $\\mathcal{F}$", "there is a surjection", "$$", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\underline{\\Lambda}_{U_i}", "\\longrightarrow \\mathcal{F}", "$$", "for some affine objects $U_i$ in $X_\\etale$. By", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-module-filtered-colimit-constructibles}", "there is a surjection", "$$", "\\Psi :", "\\bigoplus\\nolimits_{i \\in I} j_{U_i!}\\underline{\\Lambda}_{U_i}", "\\longrightarrow", "\\mathcal{F}", "$$", "with $U_i$ affine and the direct sum over a possibly infinite", "index set $I$. For every finite subset $I' \\subset I$ set", "$$", "T_{I'} = \\text{Supp}(\\Coker(", "\\bigoplus\\nolimits_{i \\in I'} j_{U_i!}\\underline{\\Lambda}_{U_i}", "\\longrightarrow \\mathcal{F}))", "$$", "By the very definition of constructible sheaves, the set $T_{I'}$", "is a constructible subset of $X$. We want to show that $T_{I'} = \\emptyset$", "for some $I'$. Since every stalk $\\mathcal{F}_{\\overline{x}}$ is", "a finite type $\\Lambda$-module and since $\\Psi$ is surjective, for", "every $x \\in X$ there is an $I'$ such that $x \\not \\in T_{I'}$.", "In other words we have", "$\\emptyset = \\bigcap_{I' \\subset I\\text{ finite}} T_{I'}$. Since", "$X$ is a spectral space by Properties, Lemma", "\\ref{properties-lemma-quasi-compact-quasi-separated-spectral}", "the constructible topology on $X$ is quasi-compact by", "Topology, Lemma \\ref{topology-lemma-constructible-hausdorff-quasi-compact}.", "Thus $T_{I'} = \\emptyset$ for some $I' \\subset I$ finite", "as desired." ], "refs": [ "etale-cohomology-lemma-torsion-colimit-constructible", "etale-cohomology-lemma-constructible-abelian", "sites-modules-lemma-module-filtered-colimit-constructibles", "properties-lemma-quasi-compact-quasi-separated-spectral", "topology-lemma-constructible-hausdorff-quasi-compact" ], "ref_ids": [ 6539, 6531, 14218, 2941, 8303 ] } ], "ref_ids": [] }, { "id": 6544, "type": "theorem", "label": "etale-cohomology-lemma-category-constructible-abelian", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-category-constructible-abelian", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. The category of", "constructible abelian sheaves is exactly the category of abelian", "sheaves of the form", "$$", "\\Coker\\left(", "\\bigoplus\\nolimits_{j = 1, \\ldots, m}", "j_{V_j!}\\underline{\\mathbf{Z}/m_j\\mathbf{Z}}_{V_j}", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n}", "j_{U_i!}\\underline{\\mathbf{Z}/n_i\\mathbf{Z}}_{U_i}", "\\right)", "$$", "with $V_j$ and $U_i$ quasi-compact and quasi-separated objects of", "$X_\\etale$ and $m_j$, $n_i$ positive integers.", "In fact, we can even assume $U_i$ and $V_j$ affine." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-category-constructible-modules}", "applied with $\\Lambda = \\mathbf{Z}/n\\mathbf{Z}$", "and the fact that, since $X$ is quasi-compact, every constructible", "abelian sheaf is annihilated by some positive integer $n$ (details omitted)." ], "refs": [ "etale-cohomology-lemma-category-constructible-modules" ], "ref_ids": [ 6543 ] } ], "ref_ids": [] }, { "id": 6545, "type": "theorem", "label": "etale-cohomology-lemma-constructible-is-compact", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-constructible-is-compact", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\Lambda$ be a", "Noetherian ring. Let $\\mathcal{F}$ be a constructible sheaf of sets, abelian", "groups, or $\\Lambda$-modules on $X_\\etale$. Let", "$\\mathcal{G} = \\colim \\mathcal{G}_i$ be a filtered colimit of sheaves of", "sets, abelian groups, or $\\Lambda$-modules. Then", "$$", "\\Mor(\\mathcal{F}, \\mathcal{G}) = \\colim \\Mor(\\mathcal{F}, \\mathcal{G}_i)", "$$", "in the category of sheaves of sets, abelian groups, or $\\Lambda$-modules on", "$X_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "The case of sheaves of sets. By Lemma \\ref{lemma-category-constructible-sets}", "it suffices to prove the lemma for $h_U$ where $U$ is a quasi-compact", "and quasi-separated object of $X_\\etale$. Recall that", "$\\Mor(h_U, \\mathcal{G}) = \\mathcal{G}(U)$. Hence the result", "follows from Sites, Lemma \\ref{sites-lemma-directed-colimits-sections}.", "\\medskip\\noindent", "In the case of abelian sheaves or sheaves of modules, the result follows", "in the same way using", "Lemmas \\ref{lemma-category-constructible-abelian} and", "\\ref{lemma-category-constructible-modules}.", "For the case of abelian sheaves, we add that", "$\\Mor(j_{U!}\\underline{\\mathbf{Z}/n\\mathbf{Z}}, \\mathcal{G})$", "is equal to the $n$-torsion elements of $\\mathcal{G}(U)$." ], "refs": [ "etale-cohomology-lemma-category-constructible-sets", "sites-lemma-directed-colimits-sections", "etale-cohomology-lemma-category-constructible-abelian", "etale-cohomology-lemma-category-constructible-modules" ], "ref_ids": [ 6542, 8531, 6544, 6543 ] } ], "ref_ids": [] }, { "id": 6546, "type": "theorem", "label": "etale-cohomology-lemma-finite-pushforward-constructible", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-finite-pushforward-constructible", "contents": [ "Let $f : X \\to Y$ be a finite and finitely presented morphism of schemes.", "Let $\\Lambda$ be a Noetherian ring. If $\\mathcal{F}$ is a constructible", "sheaf of sets, abelian groups, or $\\Lambda$-modules on $X_\\etale$,", "then $f_*\\mathcal{F}$ is too." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove this when $X$ and $Y$ are affine by", "Lemma \\ref{lemma-constructible-local}.", "By", "Lemmas \\ref{lemma-finite-pushforward-commutes-with-base-change} and", "\\ref{lemma-check-constructible} we may base change to any", "affine scheme surjective over $X$. By", "Lemma \\ref{lemma-decompose-quasi-finite-morphism}", "this reduces us to the case of a finite \\'etale morphism", "(because a thickening leads to an equivalence of \\'etale topoi", "and even small \\'etale sites, see", "Theorem \\ref{theorem-topological-invariance}).", "The finite \\'etale case is", "Lemma \\ref{lemma-pushforward-constructible}." ], "refs": [ "etale-cohomology-lemma-constructible-local", "etale-cohomology-lemma-finite-pushforward-commutes-with-base-change", "etale-cohomology-lemma-check-constructible", "etale-cohomology-lemma-decompose-quasi-finite-morphism", "etale-cohomology-theorem-topological-invariance", "etale-cohomology-lemma-pushforward-constructible" ], "ref_ids": [ 6529, 6481, 6540, 6537, 6383, 6541 ] } ], "ref_ids": [] }, { "id": 6547, "type": "theorem", "label": "etale-cohomology-lemma-category-is-colimit", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-category-is-colimit", "contents": [ "Let $X = \\lim_{i \\in I} X_i$ be a limit of a directed", "system of schemes with affine transition morphisms.", "We assume that $X_i$ is quasi-compact and quasi-separated", "for all $i \\in I$.", "\\begin{enumerate}", "\\item The category of constructible sheaves of sets on $X_\\etale$", "is the colimit of the categories of constructible sheaves of sets", "on $(X_i)_\\etale$.", "\\item The category of constructible abelian sheaves on $X_\\etale$", "is the colimit of the categories of constructible abelian sheaves", "on $(X_i)_\\etale$.", "\\item Let $\\Lambda$ be a Noetherian ring. The category of constructible", "sheaves of $\\Lambda$-modules on $X_\\etale$ is the colimit of the", "categories of constructible sheaves of $\\Lambda$-modules on $(X_i)_\\etale$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Denote $f_i : X \\to X_i$ the projection maps.", "There are 3 parts to the proof corresponding to ``faithful'',", "``fully faithful'', and ``essentially surjective''.", "\\medskip\\noindent", "Faithful. Choose $0 \\in I$ and let $\\mathcal{F}_0$, $\\mathcal{G}_0$ be", "constructible sheaves on $X_0$. Suppose that", "$a, b : \\mathcal{F}_0 \\to \\mathcal{G}_0$ are maps such that", "$f_0^{-1}a = f_0^{-1}b$. Let $E \\subset X_0$ be the set", "of points $x \\in X_0$ such that $a_{\\overline{x}} = b_{\\overline{x}}$.", "By Lemma \\ref{lemma-support-constructible} the subset", "$E \\subset X_0$ is constructible. By assumption $X \\to X_0$ maps into $E$.", "By Limits, Lemma \\ref{limits-lemma-limit-contained-in-constructible}", "we find an $i \\geq 0$ such that $X_i \\to X_0$ maps into $E$.", "Hence $f_{i0}^{-1}a = f_{i0}^{-1}b$.", "\\medskip\\noindent", "Fully faithful. Choose $0 \\in I$ and let $\\mathcal{F}_0$, $\\mathcal{G}_0$ be", "constructible sheaves on $X_0$. Suppose that", "$a : f_0^{-1}\\mathcal{F}_0 \\to f_0^{-1}\\mathcal{G}_0$ is a map.", "We claim there is an $i$ and a map", "$a_i : f_{i0}^{-1}\\mathcal{F}_0 \\to f_{i0}^{-1}\\mathcal{G}_0$", "which pulls back to $a$ on $X$.", "By Lemma \\ref{lemma-category-constructible-sets}", "we can replace $\\mathcal{F}_0$", "by a finite coproduct of sheaves represented by quasi-compact", "and quasi-separated objects of $(X_0)_\\etale$.", "Thus we have to show: If $U_0 \\to X_0$ is such an object", "of $(X_0)_\\etale$, then", "$$", "f_0^{-1}\\mathcal{G}(U) = \\colim_{i \\geq 0} f_{i0}^{-1}\\mathcal{G}(U_i)", "$$", "where $U = X \\times_{X_0} U_0$ and $U_i = X_i \\times_{X_0} U_0$.", "This is a special case of Theorem \\ref{theorem-colimit}.", "\\medskip\\noindent", "Essentially surjective. We have to show every constructible $\\mathcal{F}$", "on $X$ is isomorphic to $f_i^{-1}\\mathcal{F}$ for some constructible", "$\\mathcal{F}_i$ on $X_i$. Applying", "Lemma \\ref{lemma-category-constructible-sets}", "and using the results of the previous two paragraphs, we see that", "it suffices to prove this for $h_U$ for some quasi-compact", "and quasi-separated object $U$ of $X_\\etale$.", "In this case we have to show that $U$ is the base change of", "a quasi-compact and quasi-separated scheme \\'etale over $X_i$ for some $i$.", "This follows from", "Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation} and", "\\ref{limits-lemma-descend-etale}.", "\\medskip\\noindent", "Proof of (3). The argument is very similar to the argument for", "sheaves of sets, but using", "Lemma \\ref{lemma-category-constructible-modules}", "instead of", "Lemma \\ref{lemma-category-constructible-sets}. Details omitted.", "Part (2) follows from part (3) because every constructible abelian", "sheaf over a quasi-compact scheme is a constructible sheaf of", "$\\mathbf{Z}/n\\mathbf{Z}$-modules for some $n$." ], "refs": [ "etale-cohomology-lemma-support-constructible", "limits-lemma-limit-contained-in-constructible", "etale-cohomology-lemma-category-constructible-sets", "etale-cohomology-theorem-colimit", "etale-cohomology-lemma-category-constructible-sets", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-etale", "etale-cohomology-lemma-category-constructible-modules", "etale-cohomology-lemma-category-constructible-sets" ], "ref_ids": [ 6533, 15040, 6542, 6384, 6542, 15077, 15065, 6543, 6542 ] } ], "ref_ids": [] }, { "id": 6548, "type": "theorem", "label": "etale-cohomology-lemma-irreducible-subsheaf-constant-zero", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-irreducible-subsheaf-constant-zero", "contents": [ "Let $X$ be an irreducible scheme with generic point $\\eta$.", "\\begin{enumerate}", "\\item Let $S' \\subset S$ be an inclusion of sets. If we have", "$\\underline{S'} \\subset \\mathcal{G} \\subset \\underline{S}$", "in $\\Sh(X_\\etale)$ and $S' = \\mathcal{G}_{\\overline{\\eta}}$, then", "$\\mathcal{G} = \\underline{S'}$.", "\\item Let $A' \\subset A$ be an inclusion of abelian groups. If we have", "$\\underline{A'} \\subset \\mathcal{G} \\subset \\underline{A}$", "in $\\textit{Ab}(X_\\etale)$ and $A' = \\mathcal{G}_{\\overline{\\eta}}$, then", "$\\mathcal{G} = \\underline{A'}$.", "\\item Let $M' \\subset M$ be an inclusion of modules over a ring $\\Lambda$.", "If we have $\\underline{M'} \\subset \\mathcal{G} \\subset \\underline{M}$", "in $\\textit{Mod}(X_\\etale, \\underline{\\Lambda})$", "and $M' = \\mathcal{G}_{\\overline{\\eta}}$, then", "$\\mathcal{G} = \\underline{M'}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is true because for every \\'etale morphism $U \\to X$", "with $U \\not = \\emptyset$ the point $\\eta$ is in the image." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6549, "type": "theorem", "label": "etale-cohomology-lemma-push-constant-sheaf-from-open", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-push-constant-sheaf-from-open", "contents": [ "Let $X$ be an integral normal scheme with function field $K$.", "Let $E$ be a set.", "\\begin{enumerate}", "\\item Let $g : \\Spec(K) \\to X$ be the inclusion of the generic point.", "Then $g_*\\underline{E} = \\underline{E}$.", "\\item Let $j : U \\to X$ be the inclusion of a nonempty open. Then", "$j_*\\underline{E} = \\underline{E}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $x \\in X$ be a point. Let $\\mathcal{O}_{X, \\overline{x}}$", "be a strict henselization of $\\mathcal{O}_{X, x}$. ", "By More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-normal}", "we see that $\\mathcal{O}_{X, \\overline{x}}$ is a normal domain.", "Hence $\\Spec(K) \\times_X \\Spec(\\mathcal{O}_{X, \\overline{x}})$", "is irreducible. It follows that", "the stalk $(g_*\\underline{E}_{\\underline{x}}$ is equal to $E$,", "see Theorem \\ref{theorem-higher-direct-images}.", "\\medskip\\noindent", "Proof of (2). Since $g$ factors through $j$ there is a map", "$j_*\\underline{E} \\to g_*\\underline{E}$. This map is injective because", "for every scheme $V$ \\'etale over $X$ the set $\\Spec(K) \\times_X V$", "is dense in $U \\times_X V$. On the other hand, we have a map", "$\\underline{E} \\to j_*\\underline{E}$ and we conclude." ], "refs": [ "more-algebra-lemma-henselization-normal", "etale-cohomology-theorem-higher-direct-images" ], "ref_ids": [ 10060, 6385 ] } ], "ref_ids": [] }, { "id": 6550, "type": "theorem", "label": "etale-cohomology-lemma-zero-in-generic-point", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-zero-in-generic-point", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Let", "$\\eta \\in X$ be a generic point of an irreducible component of $X$.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be a torsion abelian sheaf on $X_\\etale$", "whose stalk $\\mathcal{F}_{\\overline{\\eta}}$ is zero.", "Then $\\mathcal{F} = \\colim \\mathcal{F}_i$ is a filtered colimit of", "constructible abelian sheaves $\\mathcal{F}_i$ such that for each $i$", "the support of $\\mathcal{F}_i$ is contained", "in a closed subscheme not containing $\\eta$.", "\\item Let $\\Lambda$ be a Noetherian ring and $\\mathcal{F}$ a sheaf", "of $\\Lambda$-modules on $X_\\etale$ whose stalk", "$\\mathcal{F}_{\\overline{\\eta}}$ is zero. Then", "$\\mathcal{F} = \\colim \\mathcal{F}_i$", "is a filtered colimit of constructible sheaves of", "$\\Lambda$-modules $\\mathcal{F}_i$ such that for each $i$", "the support of $\\mathcal{F}_i$ is contained in a closed subscheme", "not containing $\\eta$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). We can write $\\mathcal{F} = \\colim_{i \\in I} \\mathcal{F}_i$", "with $\\mathcal{F}_i$ constructible abelian by", "Lemma \\ref{lemma-torsion-colimit-constructible}.", "Choose $i \\in I$. Since $\\mathcal{F}|_\\eta$ is zero by assumption, we", "see that there exists an $i'(i) \\geq i$ such that", "$\\mathcal{F}_i|_\\eta \\to \\mathcal{F}_{i'(i)}|_\\eta$ is zero, see", "Lemma \\ref{lemma-colimit-constructible}.", "Then $\\mathcal{G}_i = \\Im(\\mathcal{F}_i \\to \\mathcal{F}_{i'(i)})$", "is a constructible abelian sheaf (Lemma \\ref{lemma-constructible-abelian})", "whose stalk at $\\eta$ is zero.", "Hence the support $E_i$ of $\\mathcal{G}_i$ is a constructible", "subset of $X$ not containing $\\eta$. Since", "$\\eta$ is a generic point of an irreducible component of", "$X$, we see that $\\eta \\not \\in Z_i = \\overline{E_i}$ by", "Topology, Lemma \\ref{topology-lemma-generic-point-in-constructible}.", "Define a new directed set $I'$ by using the set $I$ with", "ordering defined by the rule", "$i_1$ is bigger or equal to $i_2$ if and only if $i_1 \\geq i'(i_2)$.", "Then the sheaves $\\mathcal{G}_i$ form a system over $I'$", "with colimit $\\mathcal{F}$ and the proof is complete.", "\\medskip\\noindent", "The proof in case (2) is exactly the same and we omit it." ], "refs": [ "etale-cohomology-lemma-torsion-colimit-constructible", "etale-cohomology-lemma-colimit-constructible", "etale-cohomology-lemma-constructible-abelian", "topology-lemma-generic-point-in-constructible" ], "ref_ids": [ 6539, 6534, 6531, 8266 ] } ], "ref_ids": [] }, { "id": 6551, "type": "theorem", "label": "etale-cohomology-lemma-constructible-over-noetherian-noetherian", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-constructible-over-noetherian-noetherian", "contents": [ "Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring.", "Consider inclusions", "$$", "\\mathcal{F}_1 \\subset \\mathcal{F}_2 \\subset \\mathcal{F}_3 \\subset \\ldots", "\\subset \\mathcal{F}", "$$", "in the category of sheaves of sets, abelian groups, or $\\Lambda$-modules.", "If $\\mathcal{F}$ is constructible, then for some $n$", "we have $\\mathcal{F}_n = \\mathcal{F}_{n + 1} = \\mathcal{F}_{n + 2} = \\ldots$." ], "refs": [], "proofs": [ { "contents": [ "By Proposition \\ref{proposition-constructible-over-noetherian}", "we see that $\\mathcal{F}_i$ and $\\colim \\mathcal{F}_i$ are constructible.", "Then the lemma follows from", "Lemma \\ref{lemma-colimit-constructible}." ], "refs": [ "etale-cohomology-proposition-constructible-over-noetherian", "etale-cohomology-lemma-colimit-constructible" ], "ref_ids": [ 6706, 6534 ] } ], "ref_ids": [] }, { "id": 6552, "type": "theorem", "label": "etale-cohomology-lemma-constructible-maps-into-constant", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-constructible-maps-into-constant", "contents": [ "Let $X$ be a Noetherian scheme.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be a constructible sheaf of sets on $X_\\etale$.", "There exist an injective map of sheaves", "$$", "\\mathcal{F} \\longrightarrow", "\\prod\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{E_i}", "$$", "where $f_i : Y_i \\to X$ is a finite morphism and $E_i$ is a finite set.", "\\item Let $\\mathcal{F}$ be a constructible abelian sheaf on $X_\\etale$.", "There exist an injective map of abelian sheaves", "$$", "\\mathcal{F} \\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{M_i}", "$$", "where $f_i : Y_i \\to X$ is a finite morphism and", "$M_i$ is a finite abelian group.", "\\item Let $\\Lambda$ be a Noetherian ring.", "Let $\\mathcal{F}$ be a constructible sheaf of $\\Lambda$-modules on $X_\\etale$.", "There exist an injective map of sheaves of modules", "$$", "\\mathcal{F} \\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{M_i}", "$$", "where $f_i : Y_i \\to X$ is a finite morphism and", "$M_i$ is a finite $\\Lambda$-module.", "\\end{enumerate}", "Moreover, we may assume each $Y_i$ is irreducible, reduced, maps onto", "an irreducible and reduced closed subscheme $Z_i \\subset X$ such that", "$Y_i \\to Z_i$ is finite \\'etale over a nonempty open of $Z_i$." ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Because we have the ascending chain condition for", "subsheaves of $\\mathcal{F}$", "(Lemma \\ref{lemma-constructible-over-noetherian-noetherian}), it", "suffices to show that for every point $x \\in X$ we", "can find a map $\\varphi : \\mathcal{F} \\to f_*\\underline{E}$ where", "$f : Y \\to X$ is finite and $E$ is a finite set such that", "$\\varphi_{\\overline{x}} : \\mathcal{F}_{\\overline{x}} \\to", "(f_*S)_{\\overline{x}}$ is injective.", "(This argument can be avoided by picking a partition of $X$ as in", "Lemma \\ref{lemma-constructible-quasi-compact-quasi-separated}", "and constructing a $Y_i \\to X$ for each irreducible component", "of each part.)", "Let $Z \\subset X$ be the induced reduced scheme structure", "(Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme})", "on $\\overline{\\{x\\}}$.", "Since $\\mathcal{F}$ is constructible, there is a finite separable", "extension $K/\\kappa(x)$ such that", "$\\mathcal{F}|_{\\Spec(K)}$ is the constant sheaf with value $E$", "for some finite set $E$. Let $Y \\to Z$ be the normalization", "of $Z$ in $\\Spec(K)$.", "By Morphisms, Lemma \\ref{morphisms-lemma-normal-normalization}", "we see that $Y$ is a normal integral scheme.", "As $K/\\kappa(x)$ is a finite extension, it is clear that $K$ is the function", "field of $Y$. Denote $g : \\Spec(K) \\to Y$ the inclusion.", "The map $\\mathcal{F}|_{\\Spec(K)} \\to \\underline{E}$ is adjoint", "to a map $\\mathcal{F}|_Y \\to g_*\\underline{E} = \\underline{E}$", "(Lemma \\ref{lemma-push-constant-sheaf-from-open}).", "This in turn is adjoint to a map", "$\\varphi : \\mathcal{F} \\to f_*\\underline{E}$.", "Observe that the stalk of $\\varphi$ at a geometric point", "$\\overline{x}$ is injective: we may take a lift $\\overline{y} \\in Y$", "of $\\overline{x}$ and the commutative diagram", "$$", "\\xymatrix{", "\\mathcal{F}_{\\overline{x}} \\ar@{=}[r] \\ar[d] &", "(\\mathcal{F}|_Y)_{\\overline{y}} \\ar@{=}[d] \\\\", "(f_*\\underline{E})_{\\overline{x}} \\ar[r] &", "\\underline{E}_{\\overline{y}}", "}", "$$", "proves the injectivity. We are not yet done, however, as the", "morphism $f : Y \\to Z$ is integral but in general not", "finite\\footnote{If $X$ is a Nagata scheme, for example of finite", "type over a field, then $Y \\to Z$ is finite.}.", "\\medskip\\noindent", "To fix the problem stated in the last sentence of the previous paragraph,", "we write $Y = \\lim_{i \\in I} Y_i$ with $Y_i$ irreducible, integral, and", "finite over $Z$. Namely, apply Properties, Lemma", "\\ref{properties-lemma-integral-algebra-directed-colimit-finite}", "to $f_*\\mathcal{O}_Y$ viewed as a sheaf of $\\mathcal{O}_Z$-algebras", "and apply the functor $\\underline{\\Spec}_Z$.", "Then $f_*\\underline{E} = \\colim f_{i, *}\\underline{E}$", "by Lemma \\ref{lemma-relative-colimit}.", "By Lemma \\ref{lemma-constructible-is-compact} the map", "$\\mathcal{F} \\to f_*\\underline{E}$", "factors through $f_{i, *}\\underline{E}$ for some $i$.", "Since $Y_i \\to Z$ is a finite morphism of integral schemes", "and since the function field extension", "induced by this morphism is finite separable, we see that the", "morphism is finite \\'etale over a nonempty open of $Z$ (use", "Algebra, Lemma \\ref{algebra-lemma-smooth-at-generic-point}; details omitted).", "This finishes the proof of (1).", "\\medskip\\noindent", "The proofs of (2) and (3) are identical to the proof of (1)." ], "refs": [ "etale-cohomology-lemma-constructible-over-noetherian-noetherian", "etale-cohomology-lemma-constructible-quasi-compact-quasi-separated", "schemes-definition-reduced-induced-scheme", "morphisms-lemma-normal-normalization", "etale-cohomology-lemma-push-constant-sheaf-from-open", "properties-lemma-integral-algebra-directed-colimit-finite", "etale-cohomology-lemma-relative-colimit", "etale-cohomology-lemma-constructible-is-compact", "algebra-lemma-smooth-at-generic-point" ], "ref_ids": [ 6551, 6527, 7745, 5508, 6549, 3030, 6475, 6545, 1228 ] } ], "ref_ids": [] }, { "id": 6553, "type": "theorem", "label": "etale-cohomology-lemma-constructible-maps-into-constant-general", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-constructible-maps-into-constant-general", "contents": [ "\\begin{reference}", "\\cite[Exposee IX, Proposition 2.14]{SGA4}", "\\end{reference}", "Let $X$ be a quasi-compact and quasi-separated scheme.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be a constructible sheaf of sets on $X_\\etale$.", "There exist an injective map of sheaves", "$$", "\\mathcal{F} \\longrightarrow", "\\prod\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{E_i}", "$$", "where $f_i : Y_i \\to X$ is a finite and finitely presented morphism and", "$E_i$ is a finite set.", "\\item Let $\\mathcal{F}$ be a constructible abelian sheaf on $X_\\etale$.", "There exist an injective map of abelian sheaves", "$$", "\\mathcal{F} \\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{M_i}", "$$", "where $f_i : Y_i \\to X$ is a finite and finitely presented morphism and", "$M_i$ is a finite abelian group.", "\\item Let $\\Lambda$ be a Noetherian ring.", "Let $\\mathcal{F}$ be a constructible sheaf of $\\Lambda$-modules on $X_\\etale$.", "There exist an injective map of sheaves of modules", "$$", "\\mathcal{F} \\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{M_i}", "$$", "where $f_i : Y_i \\to X$ is a finite and finitely presented morphism and", "$M_i$ is a finite $\\Lambda$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will reduce this lemma to the Noetherian case by absolute Noetherian", "approximation. Namely, by", "Limits, Proposition \\ref{limits-proposition-approximate}", "we can write $X = \\lim_{t \\in T} X_t$ with each $X_t$ of finite type over", "$\\Spec(\\mathbf{Z})$ and with affine transition morphisms. By", "Lemma \\ref{lemma-category-is-colimit}", "the category of constructible sheaves (of sets, abelian groups, or", "$\\Lambda$-modules) on $X_\\etale$ is the colimit of the corresponding", "categories for $X_t$. Thus our constructible sheaf $\\mathcal{F}$", "is the pullback of a similar constructible sheaf $\\mathcal{F}_t$", "over $X_t$ for some $t$. Then we apply the Noetherian case", "(Lemma \\ref{lemma-constructible-maps-into-constant})", "to find an injection", "$$", "\\mathcal{F}_t \\longrightarrow", "\\prod\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{E_i}", "\\quad\\text{or}\\quad", "\\mathcal{F}_t \\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{M_i}", "$$", "over $X_t$ for some finite morphisms $f_i : Y_i \\to X_t$.", "Since $X_t$ is Noetherian the morphisms $f_i$ are of finite presentation.", "Since pullback is exact and since formation of $f_{i, *}$ commutes", "with base change", "(Lemma \\ref{lemma-finite-pushforward-commutes-with-base-change}), we conclude." ], "refs": [ "limits-proposition-approximate", "etale-cohomology-lemma-category-is-colimit", "etale-cohomology-lemma-constructible-maps-into-constant", "etale-cohomology-lemma-finite-pushforward-commutes-with-base-change" ], "ref_ids": [ 15126, 6547, 6552, 6481 ] } ], "ref_ids": [] }, { "id": 6554, "type": "theorem", "label": "etale-cohomology-lemma-support-in-subset", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-support-in-subset", "contents": [ "Let $X$ be a Noetherian scheme. Let $E \\subset X$ be a subset closed", "under specialization.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be a torsion abelian sheaf on $X_\\etale$", "whose support is contained in $E$. Then $\\mathcal{F} = \\colim \\mathcal{F}_i$", "is a filtered colimit of constructible abelian sheaves $\\mathcal{F}_i$", "such that for each $i$ the support of $\\mathcal{F}_i$ is contained in", "a closed subset contained in $E$.", "\\item Let $\\Lambda$ be a Noetherian ring and $\\mathcal{F}$ a sheaf", "of $\\Lambda$-modules on $X_\\etale$ whose support is contained in $E$.", "Then $\\mathcal{F} = \\colim \\mathcal{F}_i$", "is a filtered colimit of constructible sheaves of", "$\\Lambda$-modules $\\mathcal{F}_i$ such that for each $i$", "the support of $\\mathcal{F}_i$ is contained in a closed subset", "contained in $E$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). We can write $\\mathcal{F} = \\colim_{i \\in I} \\mathcal{F}_i$", "with $\\mathcal{F}_i$ constructible abelian by", "Lemma \\ref{lemma-torsion-colimit-constructible}.", "By Proposition \\ref{proposition-constructible-over-noetherian}", "the image $\\mathcal{F}'_i \\subset \\mathcal{F}$", "of the map $\\mathcal{F}_i \\to \\mathcal{F}$ is constructible.", "Thus $\\mathcal{F} = \\colim \\mathcal{F}'_i$ and the support", "of $\\mathcal{F}'_i$ is contained in $E$.", "Since the support of $\\mathcal{F}'_i$ is constructible", "(by our definition of constructible sheaves), we see", "that its closure is also contained in $E$, see for example", "Topology, Lemma", "\\ref{topology-lemma-constructible-stable-specialization-closed}.", "\\medskip\\noindent", "The proof in case (2) is exactly the same and we omit it." ], "refs": [ "etale-cohomology-lemma-torsion-colimit-constructible", "etale-cohomology-proposition-constructible-over-noetherian", "topology-lemma-constructible-stable-specialization-closed" ], "ref_ids": [ 6539, 6706, 8307 ] } ], "ref_ids": [] }, { "id": 6555, "type": "theorem", "label": "etale-cohomology-lemma-restrict-and-shriek-from-etale-c", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-restrict-and-shriek-from-etale-c", "contents": [ "Let $\\Lambda$ be a Noetherian ring.", "If $j : U \\to X$ is an \\'etale morphism of schemes, then", "\\begin{enumerate}", "\\item $K|_U \\in D_c(U_\\etale, \\Lambda)$ if $K \\in D_c(X_\\etale, \\Lambda)$, and", "\\item $j_!M \\in D_c(X_\\etale, \\Lambda)$ if $M \\in D_c(U_\\etale, \\Lambda)$ and", "the morphism $j$ is quasi-compact and quasi-separated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first assertion is clear. The second follows from the fact", "that $j_!$ is exact and Lemma \\ref{lemma-jshriek-constructible}." ], "refs": [ "etale-cohomology-lemma-jshriek-constructible" ], "ref_ids": [ 6538 ] } ], "ref_ids": [] }, { "id": 6556, "type": "theorem", "label": "etale-cohomology-lemma-pullback-c", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-pullback-c", "contents": [ "Let $\\Lambda$ be a Noetherian ring.", "Let $f : X \\to Y$ be a morphism of schemes. If $K \\in D_c(Y_\\etale, \\Lambda)$", "then $Lf^*K \\in D_c(X_\\etale, \\Lambda)$." ], "refs": [], "proofs": [ { "contents": [ "This follows as $f^{-1} = f^*$ is exact and", "Lemma \\ref{lemma-pullback-constructible}." ], "refs": [ "etale-cohomology-lemma-pullback-constructible" ], "ref_ids": [ 6530 ] } ], "ref_ids": [] }, { "id": 6557, "type": "theorem", "label": "etale-cohomology-lemma-one-constructible", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-one-constructible", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $\\Lambda$ be a Noetherian ring. Let $K \\in D(X_\\etale, \\Lambda)$", "and $b \\in \\mathbf{Z}$ such that $H^b(K)$ is constructible.", "Then there exist a sheaf $\\mathcal{F}$ which is a finite direct sum", "of $j_{U!}\\underline{\\Lambda}$ with $U \\in \\Ob(X_\\etale)$ affine and", "a map $\\mathcal{F}[-b] \\to K$ in $D(X_\\etale, \\Lambda)$", "inducing a surjection $\\mathcal{F} \\to H^b(K)$." ], "refs": [], "proofs": [ { "contents": [ "Represent $K$ by a complex $\\mathcal{K}^\\bullet$ of sheaves of", "$\\Lambda$-modules. Consider the surjection", "$$", "\\Ker(\\mathcal{K}^b \\to \\mathcal{K}^{b + 1})", "\\longrightarrow", "H^b(K)", "$$", "By Modules on Sites, Lemma", "\\ref{sites-modules-lemma-module-quotient-direct-sum}", "we may choose a surjection", "$\\bigoplus_{i \\in I} j_{U_i!} \\underline{\\Lambda} \\to", "\\Ker(\\mathcal{K}^b \\to \\mathcal{K}^{b + 1})$", "with $U_i$ affine. For $I' \\subset I$ finite, denote", "$\\mathcal{H}_{I'} \\subset H^b(K)$ the image of", "$\\bigoplus_{i \\in I'} j_{U_i!} \\underline{\\Lambda}$. By", "Lemma \\ref{lemma-colimit-constructible}", "we see that $\\mathcal{H}_{I'} = H^b(K)$ for some $I' \\subset I$ finite.", "The lemma follows taking", "$\\mathcal{F} = \\bigoplus_{i \\in I'} j_{U_i!} \\underline{\\Lambda}$." ], "refs": [ "sites-modules-lemma-module-quotient-direct-sum", "etale-cohomology-lemma-colimit-constructible" ], "ref_ids": [ 14217, 6534 ] } ], "ref_ids": [] }, { "id": 6558, "type": "theorem", "label": "etale-cohomology-lemma-bounded-above-c", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-bounded-above-c", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $\\Lambda$ be a Noetherian ring. Let $K \\in D^-(X_\\etale, \\Lambda)$. Then", "the following are equivalent", "\\begin{enumerate}", "\\item $K$ is in $D_c(X_\\etale, \\Lambda)$,", "\\item $K$ can be represented by a bounded above complex", "whose terms are finite direct sums of $j_{U!}\\underline{\\Lambda}$", "with $U \\in \\Ob(X_\\etale)$ affine,", "\\item $K$ can be represented by a bounded above complex", "of flat constructible sheaves of $\\Lambda$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (2) implies (3) and that (3) implies (1).", "Assume $K$ is in $D_c^-(X_\\etale, \\Lambda)$.", "Say $H^i(K) = 0$ for $i > b$. By induction on $a$", "we will construct a complex $\\mathcal{F}^a \\to \\ldots \\to \\mathcal{F}^b$", "such that each $\\mathcal{F}^i$ is a finite direct sum", "of $j_{U!}\\underline{\\Lambda}$ with $U \\in \\Ob(X_\\etale)$ affine", "and a map $\\mathcal{F}^\\bullet \\to K$ which induces an isomorphism", "$H^i(\\mathcal{F}^\\bullet) \\to H^i(K)$ for $i > a$ and a surjection", "$H^a(\\mathcal{F}^\\bullet) \\to H^a(K)$.", "For $a = b$ this can be done by Lemma \\ref{lemma-one-constructible}.", "Given such a datum choose a distinguished triangle", "$$", "\\mathcal{F}^\\bullet \\to K \\to L \\to \\mathcal{F}^\\bullet[1]", "$$", "Then we see that $H^i(L) = 0$ for $i \\geq a$. Choose", "$\\mathcal{F}^{a - 1}[-a +1] \\to L$ as in", "Lemma \\ref{lemma-one-constructible}. The composition", "$\\mathcal{F}^{a - 1}[-a +1] \\to L \\to \\mathcal{F}^\\bullet$", "corresponds to a map $\\mathcal{F}^{a - 1} \\to \\mathcal{F}^a$", "such that the composition with $\\mathcal{F}^a \\to \\mathcal{F}^{a + 1}$", "is zero. By TR4 we obtain a map", "$$", "(\\mathcal{F}^{a - 1} \\to \\ldots \\to \\mathcal{F}^b) \\to K", "$$", "in $D(X_\\etale, \\Lambda)$. This finishes the induction step and the", "proof of the lemma." ], "refs": [ "etale-cohomology-lemma-one-constructible", "etale-cohomology-lemma-one-constructible" ], "ref_ids": [ 6557, 6557 ] } ], "ref_ids": [] }, { "id": 6559, "type": "theorem", "label": "etale-cohomology-lemma-tensor-c", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-tensor-c", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.", "Let $K, L \\in D_c^-(X_\\etale, \\Lambda)$. Then", "$K \\otimes_\\Lambda^\\mathbf{L} L$ is in $D_c^-(X_\\etale, \\Lambda)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-bounded-above-c} and", "\\ref{lemma-tensor-product-constructible}." ], "refs": [ "etale-cohomology-lemma-bounded-above-c", "etale-cohomology-lemma-tensor-product-constructible" ], "ref_ids": [ 6558, 6532 ] } ], "ref_ids": [] }, { "id": 6560, "type": "theorem", "label": "etale-cohomology-lemma-when-ctf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-when-ctf", "contents": [ "Let $\\Lambda$ be a Noetherian ring. Let $X$ be a quasi-compact", "and quasi-separated scheme. Let $K \\in D(X_\\etale, \\Lambda)$. The following", "are equivalent", "\\begin{enumerate}", "\\item $K \\in D_{ctf}(X_\\etale, \\Lambda)$, and", "\\item $K$ can be represented by a finite complex of constructible", "flat sheaves of $\\Lambda$-modules.", "\\end{enumerate}", "In fact, if $K$ has tor amplitude in $[a, b]$ then we can represent", "$K$ by a complex $\\mathcal{F}^a \\to \\ldots \\to \\mathcal{F}^b$ with", "$\\mathcal{F}^p$ a constructible flat sheaf of $\\Lambda$-modules." ], "refs": [], "proofs": [ { "contents": [ "It is clear that a finite complex of constructible", "flat sheaves of $\\Lambda$-modules has finite tor dimension.", "It is also clear that it is an object of $D_c(X_\\etale, \\Lambda)$.", "Thus we see that (2) implies (1).", "\\medskip\\noindent", "Assume (1). Choose $a, b \\in \\mathbf{Z}$ such that", "$H^i(K \\otimes_\\Lambda^\\mathbf{L} \\mathcal{G}) = 0$ if", "$i \\not \\in [a, b]$ for all sheaves of $\\Lambda$-modules $\\mathcal{G}$.", "We will prove the final assertion holds by induction on $b - a$. If", "$a = b$, then $K = H^a(K)[-a]$ is a flat constructible sheaf", "and the result holds. Next, assume $b > a$. Represent $K$", "by a complex $\\mathcal{K}^\\bullet$ of sheaves of $\\Lambda$-modules.", "Consider the surjection", "$$", "\\Ker(\\mathcal{K}^b \\to \\mathcal{K}^{b + 1})", "\\longrightarrow", "H^b(K)", "$$", "By Lemma \\ref{lemma-category-constructible-modules}", "we can find finitely many affine schemes $U_i$ \\'etale over $X$ and a", "surjection $\\bigoplus j_{U_i!}\\underline{\\Lambda}_{U_i} \\to H^b(K)$.", "After replacing $U_i$ by standard \\'etale coverings $\\{U_{ij} \\to U_i\\}$", "we may assume this surjection lifts to a map", "$\\mathcal{F} = \\bigoplus j_{U_i!}\\underline{\\Lambda}_{U_i} \\to", "\\Ker(\\mathcal{K}^b \\to \\mathcal{K}^{b + 1})$.", "This map determines a distinguished triangle", "$$", "\\mathcal{F}[-b] \\to K \\to L \\to \\mathcal{F}[-b + 1]", "$$", "in $D(X_\\etale, \\Lambda)$. Since $D_{ctf}(X_\\etale, \\Lambda)$ is a triangulated", "subcategory we see that $L$ is in it too. In fact $L$ has", "tor amplitude in $[a, b - 1]$ as $\\mathcal{F}$ surjects onto", "$H^b(K)$ (details omitted). By induction hypothesis we can find", "a finite complex $\\mathcal{F}^a \\to \\ldots \\to \\mathcal{F}^{b - 1}$", "of flat constructible sheaves of $\\Lambda$-modules representing $L$.", "The map $L \\to \\mathcal{F}[-b + 1]$ corresponds to a map", "$\\mathcal{F}^b \\to \\mathcal{F}$ annihilating the image", "of $\\mathcal{F}^{b - 1} \\to \\mathcal{F}^b$. Then it follows", "from axiom TR3 that $K$ is represented by the complex", "$$", "\\mathcal{F}^a \\to \\ldots \\to \\mathcal{F}^{b - 1} \\to \\mathcal{F}^b", "$$", "which finishes the proof." ], "refs": [ "etale-cohomology-lemma-category-constructible-modules" ], "ref_ids": [ 6543 ] } ], "ref_ids": [] }, { "id": 6561, "type": "theorem", "label": "etale-cohomology-lemma-restrict-and-shriek-from-etale-ctf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-restrict-and-shriek-from-etale-ctf", "contents": [ "Let $\\Lambda$ be a Noetherian ring.", "If $j : U \\to X$ is an \\'etale morphism of schemes, then", "\\begin{enumerate}", "\\item $K|_U \\in D_{ctf}(U_\\etale, \\Lambda)$ if", "$K \\in D_{ctf}(X_\\etale, \\Lambda)$, and", "\\item $j_!M \\in D_{ctf}(X_\\etale, \\Lambda)$ if", "$M \\in D_{ctf}(U_\\etale, \\Lambda)$ and", "the morphism $j$ is quasi-compact and quasi-separated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Perhaps the easiest way to prove this lemma is to reduce to the", "case where $X$ is affine and then apply Lemma \\ref{lemma-when-ctf}", "to translate it into a statement about finite complexes", "of flat constructible sheaves of $\\Lambda$-modules", "where the result follows from Lemma \\ref{lemma-jshriek-constructible}." ], "refs": [ "etale-cohomology-lemma-when-ctf", "etale-cohomology-lemma-jshriek-constructible" ], "ref_ids": [ 6560, 6538 ] } ], "ref_ids": [] }, { "id": 6562, "type": "theorem", "label": "etale-cohomology-lemma-pullback-ctf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-pullback-ctf", "contents": [ "Let $\\Lambda$ be a Noetherian ring. Let $f : X \\to Y$ be a morphism of schemes.", "If $K \\in D_{ctf}(Y_\\etale, \\Lambda)$ then", "$Lf^*K \\in D_{ctf}(X_\\etale, \\Lambda)$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-when-ctf} to reduce this to a question", "about finite complexes of flat constructible sheaves of $\\Lambda$-modules.", "Then the statement follows as $f^{-1} = f^*$ is exact and", "Lemma \\ref{lemma-pullback-constructible}." ], "refs": [ "etale-cohomology-lemma-when-ctf", "etale-cohomology-lemma-pullback-constructible" ], "ref_ids": [ 6560, 6530 ] } ], "ref_ids": [] }, { "id": 6563, "type": "theorem", "label": "etale-cohomology-lemma-connected-ctf-locally-constant", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-connected-ctf-locally-constant", "contents": [ "Let $X$ be a connected scheme. Let $\\Lambda$ be a Noetherian ring.", "Let $K \\in D_{ctf}(X_\\etale, \\Lambda)$ have locally constant cohomology sheaves.", "Then there exists a finite complex of finite projective $\\Lambda$-modules", "$M^\\bullet$ and an \\'etale covering $\\{U_i \\to X\\}$ such that", "$K|_{U_i} \\cong \\underline{M^\\bullet}|_{U_i}$ in $D(U_{i, \\etale}, \\Lambda)$." ], "refs": [], "proofs": [ { "contents": [ "Choose an \\'etale covering $\\{U_i \\to X\\}$ such that $K|_{U_i}$", "is constant, say $K|_{U_i} \\cong \\underline{M_i^\\bullet}_{U_i}$", "for some finite complex of finite $\\Lambda$-modules $M_i^\\bullet$.", "See Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-locally-constant}.", "Observe that $U_i \\times_X U_j$ is empty if $M_i^\\bullet$", "is not isomorphic to $M_j^\\bullet$ in $D(\\Lambda)$.", "For each complex of $\\Lambda$-modules $M^\\bullet$ let", "$I_{M^\\bullet} =", "\\{i \\in I \\mid M_i^\\bullet \\cong M^\\bullet\\text{ in }D(\\Lambda)\\}$.", "As \\'etale morphisms are open we see that", "$U_{M^\\bullet} = \\bigcup_{i \\in I_{M^\\bullet}} \\Im(U_i \\to X)$", "is an open subset of $X$. Then $X = \\coprod U_{M^\\bullet}$ is a disjoint", "open covering of $X$. As $X$ is connected only one $U_{M^\\bullet}$", "is nonempty. As $K$ is in $D_{ctf}(X_\\etale, \\Lambda)$ we see that $M^\\bullet$", "is a perfect complex of $\\Lambda$-modules, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-perfect}.", "Hence we may assume $M^\\bullet$ is a finite complex of finite projective", "$\\Lambda$-modules." ], "refs": [ "sites-cohomology-lemma-locally-constant", "more-algebra-lemma-perfect" ], "ref_ids": [ 4405, 10212 ] } ], "ref_ids": [] }, { "id": 6564, "type": "theorem", "label": "etale-cohomology-lemma-torsion-cohomology", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-torsion-cohomology", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is a torsion abelian sheaf on $X_\\etale$, then", "$H^n_\\etale(X, \\mathcal{F})$ is a torsion abelian group for all $n$.", "\\item If $K$ in $D^+(X_\\etale)$ has torsion cohomology sheaves, then", "$H^n_\\etale(X, K)$ is a torsion abelian group for all $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove (1) we write $\\mathcal{F} = \\bigcup \\mathcal{F}[n]$ where", "$\\mathcal{F}[d]$ is the $d$-torsion subsheaf. By", "Lemma \\ref{lemma-colimit} we have", "$H^n_\\etale(X, \\mathcal{F}) = \\colim H^n_\\etale(X, \\mathcal{F}[d])$.", "This proves (1) as $H^n_\\etale(X, \\mathcal{F}[d])$ is annihilated by $d$.", "\\medskip\\noindent", "To prove (2) we can use the spectral sequence", "$E_2^{p, q} = H^p_\\etale(X, H^q(K))$ converging to $H^n_\\etale(X, K)$", "(Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor})", "and the result for sheaves." ], "refs": [ "etale-cohomology-lemma-colimit", "derived-lemma-two-ss-complex-functor" ], "ref_ids": [ 6472, 1871 ] } ], "ref_ids": [] }, { "id": 6565, "type": "theorem", "label": "etale-cohomology-lemma-torsion-direct-image", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-torsion-direct-image", "contents": [ "Let $f : X \\to Y$ be a quasi-compact and quasi-separated", "morphism of schemes.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is a torsion abelian sheaf on $X_\\etale$, then", "$R^nf_*\\mathcal{F}$ is a torsion abelian sheaf on $Y_\\etale$ for all $n$.", "\\item If $K$ in $D^+(X_\\etale)$ has torsion cohomology sheaves, then", "$Rf_*K$ is an object of $D^+(Y_\\etale)$ whose cohomology sheaves are", "torsion abelian sheaves.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Recall that $R^nf_*\\mathcal{F}$ is the sheaf associated", "to the presheaf $V \\mapsto H^n_\\etale(X \\times_Y V, \\mathcal{F})$", "on $Y_\\etale$. See Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-higher-direct-images}.", "If we choose $V$ affine, then $X \\times_Y V$ is", "quasi-compact and quasi-separated because $f$ is, hence", "we can apply Lemma \\ref{lemma-torsion-cohomology} to see that", "$H^n_\\etale(X \\times_Y V, \\mathcal{F})$ is torsion.", "\\medskip\\noindent", "Proof of (2). Recall that $R^nf_*K$ is the sheaf associated", "to the presheaf $V \\mapsto H^n_\\etale(X \\times_Y V, K)$", "on $Y_\\etale$. See Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-unbounded-describe-higher-direct-images}.", "If we choose $V$ affine, then $X \\times_Y V$ is", "quasi-compact and quasi-separated because $f$ is, hence", "we can apply Lemma \\ref{lemma-torsion-cohomology} to see that", "$H^n_\\etale(X \\times_Y V, K)$ is torsion." ], "refs": [ "sites-cohomology-lemma-higher-direct-images", "etale-cohomology-lemma-torsion-cohomology", "sites-cohomology-lemma-unbounded-describe-higher-direct-images", "etale-cohomology-lemma-torsion-cohomology" ], "ref_ids": [ 4189, 6564, 4258, 6564 ] } ], "ref_ids": [] }, { "id": 6566, "type": "theorem", "label": "etale-cohomology-lemma-sections-with-support-acyclic", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-sections-with-support-acyclic", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes.", "Let $\\mathcal{I}$ be an injective abelian sheaf on $X_\\etale$.", "Then $\\mathcal{H}_Z(\\mathcal{I})$ is an injective abelian sheaf", "on $Z_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "Observe that for any abelian sheaf $\\mathcal{G}$ on $Z_\\etale$", "we have", "$$", "\\Hom_Z(\\mathcal{G}, \\mathcal{H}_Z(\\mathcal{F})) =", "\\Hom_X(i_*\\mathcal{G}, \\mathcal{F})", "$$", "because after all any section of $i_*\\mathcal{G}$ has support in $Z$.", "Since $i_*$ is exact (Section \\ref{section-closed-immersions}) and as", "$\\mathcal{I}$ is injective on $X_\\etale$ we conclude that", "$\\mathcal{H}_Z(\\mathcal{I})$ is injective on $Z_\\etale$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6567, "type": "theorem", "label": "etale-cohomology-lemma-cohomology-with-support-sheaf-on-support", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomology-with-support-sheaf-on-support", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes.", "Let $\\mathcal{G}$ be an injective abelian sheaf on $Z_\\etale$.", "Then $\\mathcal{H}^p_Z(i_*\\mathcal{G}) = 0$ for $p > 0$." ], "refs": [], "proofs": [ { "contents": [ "This is true because the functor $i_*$ is exact and transforms", "injective abelian sheaves into injective abelian sheaves", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-pushforward-injective-flat})." ], "refs": [ "sites-cohomology-lemma-pushforward-injective-flat" ], "ref_ids": [ 4218 ] } ], "ref_ids": [] }, { "id": 6568, "type": "theorem", "label": "etale-cohomology-lemma-cohomology-with-support-triangle", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomology-with-support-triangle", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes.", "Let $j : U \\to X$ be the inclusion of the complement of $Z$.", "Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$.", "There is a distinguished triangle", "$$", "i_*R\\mathcal{H}_Z(\\mathcal{F}) \\to \\mathcal{F} \\to Rj_*(\\mathcal{F}|_U) \\to", "i_*R\\mathcal{H}_Z(\\mathcal{F})[1]", "$$", "in $D(X_\\etale)$. This produces an exact sequence", "$$", "0 \\to i_*\\mathcal{H}_Z(\\mathcal{F}) \\to \\mathcal{F} \\to", "j_*(\\mathcal{F}|_U) \\to i_*\\mathcal{H}^1_Z(\\mathcal{F}) \\to 0", "$$", "and isomorphisms", "$R^pj_*(\\mathcal{F}|_U) \\cong i_*\\mathcal{H}^{p + 1}_Z(\\mathcal{F})$", "for $p \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "To get the distinguished triangle, choose an injective resolution", "$\\mathcal{F} \\to \\mathcal{I}^\\bullet$. Then we obtain a short exact", "sequence of complexes", "$$", "0 \\to", "i_*\\mathcal{H}_Z(\\mathcal{I}^\\bullet) \\to \\mathcal{I}^\\bullet", "\\to j_*(\\mathcal{I}^\\bullet|_U) \\to 0", "$$", "by the discussion above. Thus the distinguished triangle by", "Derived Categories, Section \\ref{derived-section-canonical-delta-functor}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6569, "type": "theorem", "label": "etale-cohomology-lemma-complexes-with-support-on-closed", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-complexes-with-support-on-closed", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes.", "The map $Ri_{small, *} = i_{small, *} : D(Z_\\etale) \\to D(X_\\etale)$", "induces an equivalence $D(Z_\\etale) \\to D_Z(X_\\etale)$ with quasi-inverse", "$$", "i_{small}^{-1}|_{D_Z(X_\\etale)} = R\\mathcal{H}_Z|_{D_Z(X_\\etale)}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Recall that $i_{small}^{-1}$ and $i_{small, *}$ is an adjoint pair of", "exact functors such that $i_{small}^{-1}i_{small, *}$ is isomorphic to", "the identify functor on abelian sheaves. See", "Proposition \\ref{proposition-closed-immersion-pushforward} and", "Lemma \\ref{lemma-stalk-pullback}. Thus", "$i_{small, *} : D(Z_\\etale) \\to D_Z(X_\\etale)$ is fully faithful and", "$i_{small}^{-1}$ determines", "a left inverse. On the other hand, suppose that $K$ is an object of", "$D_Z(X_\\etale)$ and consider the adjunction map", "$K \\to i_{small, *}i_{small}^{-1}K$.", "Using exactness of $i_{small, *}$ and $i_{small}^{-1}$", "this induces the adjunction maps", "$H^n(K) \\to i_{small, *}i_{small}^{-1}H^n(K)$ on cohomology sheaves.", "Since these cohomology", "sheaves are supported on $Z$ we see these adjunction maps are isomorphisms", "and we conclude that $D(Z_\\etale) \\to D_Z(X_\\etale)$ is an equivalence.", "\\medskip\\noindent", "To finish the proof we have to show that $R\\mathcal{H}_Z(K) = i_{small}^{-1}K$", "if $K$ is an object of $D_Z(X_\\etale)$. To do this we can use that", "$K = i_{small, *}i_{small}^{-1}K$", "as we've just proved this is the case. Then we", "can choose a K-injective representative $\\mathcal{I}^\\bullet$ for", "$i_{small}^{-1}K$.", "Since $i_{small, *}$ is the right adjoint to the exact functor", "$i_{small}^{-1}$, the", "complex $i_{small, *}\\mathcal{I}^\\bullet$ is K-injective", "(Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}).", "We see that $R\\mathcal{H}_Z(K)$ is computed by", "$\\mathcal{H}_Z(i_{small, *}\\mathcal{I}^\\bullet) = \\mathcal{I}^\\bullet$", "as desired." ], "refs": [ "etale-cohomology-proposition-closed-immersion-pushforward", "etale-cohomology-lemma-stalk-pullback", "derived-lemma-adjoint-preserve-K-injectives" ], "ref_ids": [ 6700, 6436, 1915 ] } ], "ref_ids": [] }, { "id": 6570, "type": "theorem", "label": "etale-cohomology-lemma-cohomology-with-support-quasi-coherent", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomology-with-support-quasi-coherent", "contents": [ "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module", "and denote $\\mathcal{F}^a$ the associated quasi-coherent sheaf", "on the small \\'etale site of $X$", "(Proposition \\ref{proposition-quasi-coherent-sheaf-fpqc}). Then", "\\begin{enumerate}", "\\item $H^q_Z(X, \\mathcal{F})$ agrees with $H^q_Z(X_\\etale, \\mathcal{F}^a)$,", "\\item if the complement of $Z$ is retrocompact in $X$, then", "$i_*\\mathcal{H}^q_Z(\\mathcal{F}^a)$ is a quasi-coherent sheaf of", "$\\mathcal{O}_X$-modules equal to $(i_*\\mathcal{H}^q_Z(\\mathcal{F}))^a$.", "\\end{enumerate}" ], "refs": [ "etale-cohomology-proposition-quasi-coherent-sheaf-fpqc" ], "proofs": [ { "contents": [ "Let $j : U \\to X$ be the inclusion of the complement of $Z$.", "The statement (1) on cohomology groups follows from the long", "exact sequences for cohomology with supports and the agreements", "$H^q(X_\\etale, \\mathcal{F}^a) = H^q(X, \\mathcal{F})$ and", "$H^q(U_\\etale, \\mathcal{F}^a) = H^q(U, \\mathcal{F})$, see", "Theorem \\ref{theorem-zariski-fpqc-quasi-coherent}.", "If $j : U \\to X$ is a quasi-compact morphism, i.e., if $U \\subset X$", "is retrocompact, then $R^qj_*$ transforms quasi-coherent sheaves", "into quasi-coherent sheaves", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images})", "and commutes with taking associated", "sheaf on \\'etale sites", "(Descent, Lemma \\ref{descent-lemma-higher-direct-images-small-etale}).", "We conclude by applying", "Lemma \\ref{lemma-cohomology-with-support-triangle}." ], "refs": [ "etale-cohomology-theorem-zariski-fpqc-quasi-coherent", "coherent-lemma-quasi-coherence-higher-direct-images", "descent-lemma-higher-direct-images-small-etale", "etale-cohomology-lemma-cohomology-with-support-triangle" ], "ref_ids": [ 6373, 3295, 14628, 6568 ] } ], "ref_ids": [ 6696 ] }, { "id": 6571, "type": "theorem", "label": "etale-cohomology-lemma-local-rings-strictly-henselian", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-local-rings-strictly-henselian", "contents": [ "Let $S$ be a scheme all of whose local rings are strictly henselian.", "Then for any abelian sheaf $\\mathcal{F}$ on $S_\\etale$ we have", "$H^i(S_\\etale, \\mathcal{F}) = H^i(S_{Zar}, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\epsilon : S_\\etale \\to S_{Zar}$ be the morphism of sites given", "by the inclusion functor. The Zariski sheaf $R^p\\epsilon_*\\mathcal{F}$", "is the sheaf associated to the presheaf $U \\mapsto H^p_\\etale(U, \\mathcal{F})$.", "Thus the stalk at $x \\in X$ is", "$\\colim H^p_\\etale(U, \\mathcal{F}) =", "H^p_\\etale(\\Spec(\\mathcal{O}_{X, x}), \\mathcal{G}_x)$", "where $\\mathcal{G}_x$ denotes the pullback of $\\mathcal{F}$", "to $\\Spec(\\mathcal{O}_{X, x})$, see", "Lemma \\ref{lemma-directed-colimit-cohomology}.", "Thus the higher direct images of $R^p\\epsilon_*\\mathcal{F}$ are", "zero by", "Lemma \\ref{lemma-vanishing-etale-cohomology-strictly-henselian}", "and we conclude by the Leray spectral sequence." ], "refs": [ "etale-cohomology-lemma-directed-colimit-cohomology", "etale-cohomology-lemma-vanishing-etale-cohomology-strictly-henselian" ], "ref_ids": [ 6473, 6480 ] } ], "ref_ids": [] }, { "id": 6572, "type": "theorem", "label": "etale-cohomology-lemma-affine-only-closed-points", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-affine-only-closed-points", "contents": [ "Let $S$ be an affine scheme such that", "(1) all points are closed, and (2) all residue fields are separably", "algebraically closed. Then", "for any abelian sheaf $\\mathcal{F}$ on $S_\\etale$ we have", "$H^i(S_\\etale, \\mathcal{F}) = 0$ for $i > 0$." ], "refs": [], "proofs": [ { "contents": [ "Condition (1) implies that the underlying topological space", "of $S$ is profinite, see", "Algebra, Lemma \\ref{algebra-lemma-ring-with-only-minimal-primes}.", "Thus the higher cohomology groups of an abelian sheaf on the topological", "space $S$ (i.e., Zariski cohomology) is trivial, see", "Cohomology, Lemma \\ref{cohomology-lemma-vanishing-for-profinite}.", "The local rings are strictly henselian by", "Algebra, Lemma \\ref{algebra-lemma-local-dimension-zero-henselian}.", "Thus \\'etale cohomology of $S$ is computed by Zariski cohomology", "by Lemma \\ref{lemma-local-rings-strictly-henselian}", "and the proof is done." ], "refs": [ "algebra-lemma-ring-with-only-minimal-primes", "cohomology-lemma-vanishing-for-profinite", "algebra-lemma-local-dimension-zero-henselian", "etale-cohomology-lemma-local-rings-strictly-henselian" ], "ref_ids": [ 426, 2092, 1283, 6571 ] } ], "ref_ids": [] }, { "id": 6573, "type": "theorem", "label": "etale-cohomology-lemma-normal-scheme-with-alg-closed-function-field", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-normal-scheme-with-alg-closed-function-field", "contents": [ "Let $X$ be an integral normal scheme with separably closed", "function field.", "\\begin{enumerate}", "\\item A separated \\'etale morphism $U \\to X$ is a", "disjoint union of open immersions.", "\\item All local rings of $X$ are strictly henselian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a normal domain whose fraction field is separably algebraically", "closed. Let $R \\to A$ be an \\'etale ring map. Then", "$A \\otimes_R K$ is as a $K$-algebra a finite product", "$\\prod_{i = 1, \\ldots, n} K$ of copies of $K$. Let $e_i$, $i = 1, \\ldots, n$", "be the corresponding idempotents of $A \\otimes_R K$. Since $A$ is normal", "(Algebra, Lemma \\ref{algebra-lemma-normal-goes-up})", "the idempotents $e_i$ are in $A$", "(Algebra, Lemma \\ref{algebra-lemma-normal-ring-integrally-closed}).", "Hence $A = \\prod Ae_i$ and we may assume $A \\otimes_R K = K$.", "Since $A \\subset A \\otimes_R K = K$ (by flatness of $R \\to A$ and", "since $R \\subset K$) we conclude that $A$ is a domain.", "By the same argument we conclude that", "$A \\otimes_R A \\subset (A \\otimes_R A) \\otimes_R K = K$.", "It follows that the map $A \\otimes_R A \\to A$ is", "injective as well as surjective. Thus $R \\to A$ defines an", "open immersion by", "Morphisms, Lemma \\ref{morphisms-lemma-universally-injective}", "and", "\\'Etale Morphisms, Theorem \\ref{etale-theorem-etale-radicial-open}.", "\\medskip\\noindent", "Let $f : U \\to X$ be a separated \\'etale morphism. Let $\\eta \\in X$", "be the generic point and let $f^{-1}(\\{\\eta\\}) = \\{\\xi_i\\}_{i \\in I}$.", "The result of the previous paragraph shows the following:", "For any affine open $U' \\subset U$ whose image in $X$ is contained in", "an affine we have $U' = \\coprod_{i \\in I} U'_i$ where $U'_i$", "is the set of point of $U'$ which are specializations of $\\xi_i$.", "Moreover, the morphism $U'_i \\to X$ is an open immersion.", "It follows that $U_i = \\overline{\\{\\xi_i\\}}$ is an open and closed", "subscheme of $U$ and that $U_i \\to X$ is locally on the source", "an isomorphism. By Morphisms,", "Lemma \\ref{morphisms-lemma-distinct-local-rings}", "the fact that $U_i \\to X$ is separated, implies that", "$U_i \\to X$ is injective and we conclude that $U_i \\to X$", "is an open immersion, i.e., (1) holds.", "\\medskip\\noindent", "Part (2) follows from part (1) and the description of the strict", "henselization of $\\mathcal{O}_{X, x}$ as the local ring at $\\overline{x}$", "on the \\'etale site of $X$ (Lemma \\ref{lemma-describe-etale-local-ring}).", "It can also be proved directly, see", "Fundamental Groups, Lemma", "\\ref{pione-lemma-normal-local-domain-separablly-closed-fraction-field}." ], "refs": [ "algebra-lemma-normal-goes-up", "algebra-lemma-normal-ring-integrally-closed", "morphisms-lemma-universally-injective", "etale-theorem-etale-radicial-open", "morphisms-lemma-distinct-local-rings", "etale-cohomology-lemma-describe-etale-local-ring", "pione-lemma-normal-local-domain-separablly-closed-fraction-field" ], "ref_ids": [ 1368, 511, 5167, 10692, 5479, 6433, 4058 ] } ], "ref_ids": [] }, { "id": 6574, "type": "theorem", "label": "etale-cohomology-lemma-Rf-star-zero-normal-with-alg-closed-function-field", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-Rf-star-zero-normal-with-alg-closed-function-field", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes where $X$ is an integral", "normal scheme with separably closed function field. Then", "$R^qf_*\\underline{M} = 0$ for $q > 0$ and any abelian group $M$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $R^qf_*\\underline{M}$ is the sheaf associated", "to the presheaf $V \\mapsto H^q_\\etale(V \\times_Y X, M)$ on $Y_\\etale$, see", "Lemma \\ref{lemma-higher-direct-images}.", "If $V$ is affine, then $V \\times_Y X \\to X$ is separated and \\'etale.", "Hence $V \\times_Y X = \\coprod U_i$ is a disjoint union of open", "subschemes $U_i$ of $X$, see", "Lemma \\ref{lemma-normal-scheme-with-alg-closed-function-field}.", "By Lemma \\ref{lemma-local-rings-strictly-henselian}", "we see that $H^q_\\etale(U_i, M)$ is equal to", "$H^q_{Zar}(U_i, M)$. This vanishes by", "Cohomology, Lemma \\ref{cohomology-lemma-irreducible-constant-cohomology-zero}." ], "refs": [ "etale-cohomology-lemma-higher-direct-images", "etale-cohomology-lemma-normal-scheme-with-alg-closed-function-field", "etale-cohomology-lemma-local-rings-strictly-henselian", "cohomology-lemma-irreducible-constant-cohomology-zero" ], "ref_ids": [ 6474, 6573, 6571, 2085 ] } ], "ref_ids": [] }, { "id": 6575, "type": "theorem", "label": "etale-cohomology-lemma-closed-of-affine-normal-scheme-with-alg-closed-function-field", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-closed-of-affine-normal-scheme-with-alg-closed-function-field", "contents": [ "Let $X$ be an affine integral normal scheme with separably closed", "function field. Let $Z \\subset X$ be a closed subscheme. Let", "$V \\to Z$ be an \\'etale morphism with $V$ affine. Then $V$ is a finite", "disjoint union of open subschemes of $Z$. If $V \\to Z$ is", "surjective and finite \\'etale, then $V \\to Z$ has a section." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-lift-etale}", "we can lift $V$ to an affine scheme $U$ \\'etale over $X$.", "Apply Lemma \\ref{lemma-normal-scheme-with-alg-closed-function-field}", "to $U \\to X$ to get the first statement.", "\\medskip\\noindent", "The final statement is a consequence of the first.", "Let $V = \\coprod_{i = 1, \\ldots, n} V_i$ be a finite", "decomposition into open and", "closed subschemes with $V_i \\to Z$ an open immersion.", "As $V \\to Z$ is finite we see that $V_i \\to Z$ is also closed.", "Let $U_i \\subset Z$ be the image. Then we have a decomposition", "into open and closed subschemes", "$$", "Z =", "\\coprod\\nolimits_{(A, B)}", "\\bigcap\\nolimits_{i \\in A} U_i \\cap", "\\bigcap\\nolimits_{i \\in B} U_i^c", "$$", "where the disjoint union is over $\\{1, \\ldots, n\\} = A \\amalg B$", "where $A$ has at least one element.", "Each of the strata is contained in a single $U_i$ and", "we find our section." ], "refs": [ "algebra-lemma-lift-etale", "etale-cohomology-lemma-normal-scheme-with-alg-closed-function-field" ], "ref_ids": [ 1238, 6573 ] } ], "ref_ids": [] }, { "id": 6576, "type": "theorem", "label": "etale-cohomology-lemma-gabber-for-h1-absolutely-algebraically-closed", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-gabber-for-h1-absolutely-algebraically-closed", "contents": [ "Let $X$ be a normal integral affine scheme with separably closed", "function field. Let $Z \\subset X$ be a closed subscheme.", "For any finite abelian group $M$ we have $H^1_\\etale(Z, \\underline{M}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "By Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-torsors-h1}", "an element of $H^1_\\etale(Z, \\underline{M})$ corresponds to a", "$\\underline{M}$-torsor $\\mathcal{F}$ on $Z_\\etale$.", "Such a torsor is clearly a finite locally constant sheaf.", "Hence $\\mathcal{F}$ is representable by a scheme $V$ finite", "\\'etale over $Z$, Lemma \\ref{lemma-characterize-finite-locally-constant}.", "Of course $V \\to Z$ is surjective as a torsor is locally trivial.", "Since $V \\to Z$ has a section by", "Lemma \\ref{lemma-closed-of-affine-normal-scheme-with-alg-closed-function-field}", "we are done." ], "refs": [ "sites-cohomology-lemma-torsors-h1", "etale-cohomology-lemma-characterize-finite-locally-constant", "etale-cohomology-lemma-closed-of-affine-normal-scheme-with-alg-closed-function-field" ], "ref_ids": [ 4182, 6503, 6575 ] } ], "ref_ids": [] }, { "id": 6577, "type": "theorem", "label": "etale-cohomology-lemma-gabber-for-absolutely-algebraically-closed", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-gabber-for-absolutely-algebraically-closed", "contents": [ "Let $X$ be a normal integral affine scheme with separably closed", "function field. Let $Z \\subset X$ be a closed subscheme.", "For any finite abelian group $M$ we have", "$H^q_\\etale(Z, \\underline{M}) = 0$ for $q \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "We have seen that the result is true for $H^1$ in", "Lemma \\ref{lemma-gabber-for-h1-absolutely-algebraically-closed}.", "We will prove the result for $q \\geq 2$ by induction on $q$.", "Let $\\xi \\in H^q_\\etale(Z, \\underline{M})$.", "\\medskip\\noindent", "Let $X = \\Spec(R)$. Let $I \\subset R$ be the set of elements", "$f \\in R$ such that $\\xi|_{Z \\cap D(f)} = 0$.", "All local rings of $Z$ are strictly henselian", "by Lemma \\ref{lemma-normal-scheme-with-alg-closed-function-field}", "and", "Algebra, Lemma \\ref{algebra-lemma-quotient-strict-henselization}.", "Hence \\'etale cohomology on $Z$ or open subschemes of $Z$", "is equal to Zariski cohomology, see", "Lemma \\ref{lemma-local-rings-strictly-henselian}.", "In particular $\\xi$ is Zariski locally trivial.", "It follows that for every prime $\\mathfrak p$ of $R$ there", "exists an $f \\in I$ with $f \\not \\in \\mathfrak p$.", "Thus if we can show that $I$ is an ideal, then $1 \\in I$ and", "we're done. It is clear that $f \\in I$, $r \\in R$ implies", "$rf \\in I$. Thus we now assume that $f, g \\in I$ and we show that", "$f + g \\in I$. Note that", "$$", "D(f + g) \\cap Z = D(f(f + g)) \\cap Z \\cup D(g(f + g)) \\cap Z", "$$", "By Mayer-Vietoris (Cohomology, Lemma \\ref{cohomology-lemma-mayer-vietoris}", "which applies as \\'etale cohomology on open subschemes of $Z$ equals", "Zariski cohomology) we have an exact sequence", "$$", "\\xymatrix{", "H^{q - 1}_\\etale(D(fg(f + g)) \\cap Z, \\underline{M}) \\ar[d] \\\\", "H^q_\\etale(D(f + g) \\cap Z, \\underline{M}) \\ar[d] \\\\", "H^q_\\etale(D(f(f + g)) \\cap Z, \\underline{M}) \\oplus", "H^q_\\etale(D(g(f + g)) \\cap Z, \\underline{M})", "}", "$$", "and the result follows as the first group is zero by induction." ], "refs": [ "etale-cohomology-lemma-gabber-for-h1-absolutely-algebraically-closed", "etale-cohomology-lemma-normal-scheme-with-alg-closed-function-field", "algebra-lemma-quotient-strict-henselization", "etale-cohomology-lemma-local-rings-strictly-henselian", "cohomology-lemma-mayer-vietoris" ], "ref_ids": [ 6576, 6573, 1307, 6571, 2042 ] } ], "ref_ids": [] }, { "id": 6578, "type": "theorem", "label": "etale-cohomology-lemma-integral-cover-trivial-cohomology", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-integral-cover-trivial-cohomology", "contents": [ "Let $X$ be an affine scheme.", "\\begin{enumerate}", "\\item There exists an integral surjective morphism $X' \\to X$ such that for", "every closed subscheme $Z' \\subset X'$, every finite abelian group $M$, and", "every $q \\geq 1$ we have $H^q_\\etale(Z', \\underline{M}) = 0$.", "\\item For any closed subscheme $Z \\subset X$, finite abelian group $M$,", "$q \\geq 1$, and $\\xi \\in H^q_\\etale(Z, \\underline{M})$ there exists a", "finite surjective morphism $X' \\to X$ of finite presentation such that", "$\\xi$ pulls back to zero in $H^q_\\etale(X' \\times_X Z, \\underline{M})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $X = \\Spec(A)$. Write $A = \\mathbf{Z}[x_i]/J$ for some ideal $J$.", "Let $R$ be the integral closure of $\\mathbf{Z}[x_i]$ in an algebraic", "closure of the fraction field of $\\mathbf{Z}[x_i]$. Let", "$A' = R/JR$ and set $X' = \\Spec(A')$. This gives an example as in (1) by", "Lemma \\ref{lemma-gabber-for-absolutely-algebraically-closed}.", "\\medskip\\noindent", "Proof of (2). Let $X' \\to X$ be the integral surjective morphism we found", "above. Certainly, $\\xi$ maps to zero in", "$H^q_\\etale(X' \\times_X Z, \\underline{M})$. We may write $X'$ as a", "limit $X' = \\lim X'_i$ of schemes finite and of finite presentation", "over $X$; this is easy to do in our current affine case, but it", "is a special case of the more general Limits, Lemma", "\\ref{limits-lemma-integral-limit-finite-and-finite-presentation}.", "By Lemma \\ref{lemma-directed-colimit-cohomology}", "we see that $\\xi$ maps to zero in $H^q_\\etale(X'_i \\times_X Z, \\underline{M})$", "for some $i$ large enough." ], "refs": [ "etale-cohomology-lemma-gabber-for-absolutely-algebraically-closed", "limits-lemma-integral-limit-finite-and-finite-presentation", "etale-cohomology-lemma-directed-colimit-cohomology" ], "ref_ids": [ 6577, 15056, 6473 ] } ], "ref_ids": [] }, { "id": 6579, "type": "theorem", "label": "etale-cohomology-lemma-efface-cohomology-on-closed-by-finite-cover", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-efface-cohomology-on-closed-by-finite-cover", "contents": [ "Let $X$ be an affine scheme. Let $\\mathcal{F}$ be a torsion abelian sheaf", "on $X_\\etale$. Let $Z \\subset X$ be a closed subscheme. Let", "$\\xi \\in H^q_\\etale(Z, \\mathcal{F}|_Z)$ for some $q > 0$.", "Then there exists an injective map $\\mathcal{F} \\to \\mathcal{F}'$", "of torsion abelian sheaves on $X_\\etale$ such that", "the image of $\\xi$ in $H^q_\\etale(Z, \\mathcal{F}'|_Z)$ is zero." ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-torsion-colimit-constructible} and \\ref{lemma-colimit}", "we can find a map $\\mathcal{G} \\to \\mathcal{F}$ with $\\mathcal{G}$", "a constructible abelian sheaf and $\\xi$ coming from an element $\\zeta$ of", "$H^q_\\etale(Z, \\mathcal{G}|_Z)$. Suppose we can find an injective map", "$\\mathcal{G} \\to \\mathcal{G}'$ of torsion abelian sheaves on $X_\\etale$", "such that the image of $\\zeta$ in $H^q_\\etale(Z, \\mathcal{G}'|_Z)$ is zero.", "Then we can take $\\mathcal{F}'$ to be the pushout", "$$", "\\mathcal{F}' = \\mathcal{G}' \\amalg_{\\mathcal{G}} \\mathcal{F}", "$$", "and we conclude the result of the lemma holds. (Observe that restriction", "to $Z$ is exact, so commutes with finite limits and colimits and moreover", "it commutes with arbitrary colimits as a left adjoint to pushforward.)", "Thus we may assume $\\mathcal{F}$ is constructible.", "\\medskip\\noindent", "Assume $\\mathcal{F}$ is constructible. By", "Lemma \\ref{lemma-constructible-maps-into-constant-general}", "it suffices to prove the result when $\\mathcal{F}$", "is of the form $f_*\\underline{M}$ where $M$ is a finite abelian group", "and $f : Y \\to X$ is a finite morphism of finite presentation", "(such sheaves are still constructible by", "Lemma \\ref{lemma-finite-pushforward-constructible}", "but we won't need this).", "Since formation of $f_*$ commutes with any base change", "(Lemma \\ref{lemma-finite-pushforward-commutes-with-base-change})", "we see that the restriction of $f_*\\underline{M}$ to $Z$ is", "equal to the pushforward of $\\underline{M}$ via", "$Y \\times_X Z \\to Z$. By the Leray spectral sequence", "(Proposition \\ref{proposition-leray})", "and vanishing of higher direct images", "(Proposition \\ref{proposition-finite-higher-direct-image-zero}),", "we find", "$$", "H^q_\\etale(Z, f_*\\underline{M}|_Z) = H^q_\\etale(Y \\times_X Z, \\underline{M}).", "$$", "By Lemma \\ref{lemma-integral-cover-trivial-cohomology}", "we can find a finite surjective morphism $Y' \\to Y$ of finite presentation", "such that $\\xi$ maps to zero in $H^q(Y' \\times_X Z, \\underline{M})$.", "Denoting $f' : Y' \\to X$ the composition $Y' \\to Y \\to X$ we claim", "the map", "$$", "f_*\\underline{M} \\longrightarrow f'_*\\underline{M}", "$$", "is injective which finishes the proof by what was said above.", "To see the desired injectivity we can look at stalks. Namely,", "if $\\overline{x} : \\Spec(k) \\to X$ is a geometric point, then", "$$", "(f_*\\underline{M})_{\\overline{x}} =", "\\bigoplus\\nolimits_{f(\\overline{y}) = \\overline{x}} M", "$$", "by Proposition \\ref{proposition-finite-higher-direct-image-zero}", "and similarly for the other sheaf.", "Since $Y' \\to Y$ is surjective and finite we see that", "the induced map on geometric points lifting $\\overline{x}$ is", "surjective too and we conclude." ], "refs": [ "etale-cohomology-lemma-torsion-colimit-constructible", "etale-cohomology-lemma-colimit", "etale-cohomology-lemma-constructible-maps-into-constant-general", "etale-cohomology-lemma-finite-pushforward-constructible", "etale-cohomology-lemma-finite-pushforward-commutes-with-base-change", "etale-cohomology-proposition-leray", "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-lemma-integral-cover-trivial-cohomology", "etale-cohomology-proposition-finite-higher-direct-image-zero" ], "ref_ids": [ 6539, 6472, 6553, 6546, 6481, 6702, 6703, 6578, 6703 ] } ], "ref_ids": [] }, { "id": 6580, "type": "theorem", "label": "etale-cohomology-lemma-gabber-h0", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-gabber-h0", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $i : Z \\to X$ be a closed immersion. Assume that", "\\begin{enumerate}", "\\item for any sheaf $\\mathcal{F}$ on $X_{Zar}$ the map", "$\\Gamma(X, \\mathcal{F}) \\to \\Gamma(Z, i^{-1}\\mathcal{F})$", "is bijective, and", "\\item for any finite morphism $X' \\to X$ assumption (1) holds", "for $Z \\times_X X' \\to X'$.", "\\end{enumerate}", "Then for any sheaf $\\mathcal{F}$ on $X_\\etale$ we have", "$\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, i^{-1}_{small}\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a sheaf on $X_\\etale$. There is a canonical", "(base change) map", "$$", "i^{-1}(\\mathcal{F}|_{X_{Zar}})", "\\longrightarrow", "(i_{small}^{-1}\\mathcal{F})|_{Z_{Zar}}", "$$", "of sheaves on $Z_{Zar}$. We will show this map is injective by looking", "at stalks. The stalk on the left hand side at $z \\in Z$", "is the stalk of $\\mathcal{F}|_{X_{Zar}}$ at $z$. The stalk on the right", "hand side is the colimit over all elementary \\'etale neighbourhoods", "$(U, u) \\to (X, z)$ such that $U \\times_X Z \\to Z$ has a section over", "a neighbourhood of $z$. As \\'etale morphisms are open, the image of", "$U \\to X$ is an open neighbourhood $U_0$ of $z$ in $X$. The map", "$\\mathcal{F}(U_0) \\to \\mathcal{F}(U)$ is injective by the sheaf", "condition for $\\mathcal{F}$ with respect to the \\'etale covering $U \\to U_0$.", "Taking the colimit over all $U$ and $U_0$ we obtain injectivity on stalks.", "\\medskip\\noindent", "It follows from this and assumption (1) that the map", "$\\Gamma(X, \\mathcal{F}) \\to \\Gamma(Z, i^{-1}_{small}\\mathcal{F})$", "is injective. By (2) the same thing is true on all $X'$ finite over $X$.", "\\medskip\\noindent", "Let $s \\in \\Gamma(Z, i^{-1}_{small}\\mathcal{F})$. By construction of", "$i^{-1}_{small}\\mathcal{F}$ there exists an \\'etale covering", "$\\{V_j \\to Z\\}$, \\'etale morphisms $U_j \\to X$, sections", "$s_j \\in \\mathcal{F}(U_j)$ and morphisms $V_j \\to U_j$ over $X$", "such that $s|_{V_j}$ is the pullback of $s_j$.", "Observe that every nonempty closed subscheme $T \\subset X$ meets $Z$", "by assumption (1) applied to the sheaf $(T \\to X)_*\\underline{\\mathbf{Z}}$", "for example. Thus we see that $\\coprod U_j \\to X$ is surjective.", "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-there-is-a-scheme-integral-over}", "we can find a finite surjective morphism $X' \\to X$", "such that $X' \\to X$ Zariski locally factors through $\\coprod U_j \\to X$.", "It follows that $s|_{Z'}$ Zariski locally comes", "from a section of $\\mathcal{F}|_{X'}$. In other words,", "$s|_{Z'}$ comes from $t' \\in \\Gamma(X', \\mathcal{F}|_{X'})$", "by assumption (2).", "By injectivity we conclude that the two pullbacks of $t'$ to", "$X' \\times_X X'$ are the same (after all this is true for", "the pullbacks of $s$ to $Z' \\times_Z Z'$). Hence we conclude", "$t'$ comes from a section of $\\mathcal{F}$ over $X$ by", "Remark \\ref{remark-cohomological-descent-finite}." ], "refs": [ "more-morphisms-lemma-there-is-a-scheme-integral-over", "etale-cohomology-remark-cohomological-descent-finite" ], "ref_ids": [ 13912, 6786 ] } ], "ref_ids": [] }, { "id": 6581, "type": "theorem", "label": "etale-cohomology-lemma-connected-topological", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-connected-topological", "contents": [ "Let $Z \\subset X$ be a closed subset of a topological space $X$.", "Assume", "\\begin{enumerate}", "\\item $X$ is a spectral space", "(Topology, Definition \\ref{topology-definition-spectral-space}), and", "\\item for $x \\in X$ the intersection $Z \\cap \\overline{\\{x\\}}$", "is connected (in particular nonempty).", "\\end{enumerate}", "If $Z = Z_1 \\amalg Z_2$ with $Z_i$ closed in $Z$,", "then there exists a decomposition $X = X_1 \\amalg X_2$ with", "$X_i$ closed in $X$ and $Z_i = Z \\cap X_i$." ], "refs": [ "topology-definition-spectral-space" ], "proofs": [ { "contents": [ "Observe that $Z_i$ is quasi-compact. Hence the set of points", "$W_i$ specializing to $Z_i$ is closed in the constructible topology", "by Topology, Lemma \\ref{topology-lemma-make-spectral-space}.", "Assumption (2) implies that $X = W_1 \\amalg W_2$.", "Let $x \\in \\overline{W_1}$. By Topology, Lemma", "\\ref{topology-lemma-constructible-stable-specialization-closed} part (1)", "there exists a specialization $x_1 \\leadsto x$ with $x_1 \\in W_1$.", "Thus $\\overline{\\{x\\}} \\subset \\overline{\\{x_1\\}}$ and we see", "that $x \\in W_1$. In other words, setting $X_i = W_i$ does the job." ], "refs": [ "topology-lemma-make-spectral-space", "topology-lemma-constructible-stable-specialization-closed" ], "ref_ids": [ 8323, 8307 ] } ], "ref_ids": [ 8370 ] }, { "id": 6582, "type": "theorem", "label": "etale-cohomology-lemma-h0-topological", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-h0-topological", "contents": [ "Let $Z \\subset X$ be a closed subset of a topological space $X$.", "Assume", "\\begin{enumerate}", "\\item $X$ is a spectral space", "(Topology, Definition \\ref{topology-definition-spectral-space}), and", "\\item for $x \\in X$ the intersection $Z \\cap \\overline{\\{x\\}}$", "is connected (in particular nonempty).", "\\end{enumerate}", "Then for any sheaf $\\mathcal{F}$ on $X$ we have", "$\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, \\mathcal{F}|_Z)$." ], "refs": [ "topology-definition-spectral-space" ], "proofs": [ { "contents": [ "If $x \\leadsto x'$ is a specialization of points, then there is a", "canonical map $\\mathcal{F}_{x'} \\to \\mathcal{F}_x$ compatible with", "sections over opens and functorial in $\\mathcal{F}$. Since every point", "of $X$ specializes to a point of $Z$ it follows that", "$\\Gamma(X, \\mathcal{F}) \\to \\Gamma(Z, \\mathcal{F}|_Z)$ is injective.", "The difficult part is to show that it is surjective.", "\\medskip\\noindent", "Denote $\\mathcal{B}$ be the set of all quasi-compact opens of $X$.", "Write $\\mathcal{F}$ as a filtered colimit $\\mathcal{F} = \\colim \\mathcal{F}_i$", "where each $\\mathcal{F}_i$ is as in", "Modules, Equation (\\ref{modules-equation-towards-constructible-sets}).", "See Modules, Lemma \\ref{modules-lemma-filtered-colimit-constructibles}.", "Then $\\mathcal{F}|_Z = \\colim \\mathcal{F}_i|_Z$ as restriction to $Z$", "is a left adjoint (Categories, Lemma \\ref{categories-lemma-adjoint-exact} and", "Sheaves, Lemma \\ref{sheaves-lemma-f-map}).", "By Sheaves, Lemma \\ref{sheaves-lemma-directed-colimits-sections}", "the functors $\\Gamma(X, -)$ and $\\Gamma(Z, -)$ commute with filtered colimits.", "Hence we may assume our sheaf $\\mathcal{F}$ is as in", "Modules, Equation (\\ref{modules-equation-towards-constructible-sets}).", "\\medskip\\noindent", "Suppose that we have an embedding $\\mathcal{F} \\subset \\mathcal{G}$.", "Then we have", "$$", "\\Gamma(X, \\mathcal{F}) =", "\\Gamma(Z, \\mathcal{F}|_Z) \\cap \\Gamma(X, \\mathcal{G})", "$$", "where the intersection takes place in $\\Gamma(Z, \\mathcal{G}|_Z)$.", "This follows from the first remark of the proof because we can check", "whether a global section of $\\mathcal{G}$ is in $\\mathcal{F}$ by looking", "at the stalks and because every point of $X$ specializes to a point of $Z$.", "\\medskip\\noindent", "By Modules, Lemma \\ref{modules-lemma-constructible-in-constant}", "there is an injection $\\mathcal{F} \\to \\prod (Z_i \\to X)_*\\underline{S_i}$", "where the product is finite, $Z_i \\subset X$ is closed, and $S_i$ is finite.", "Thus it suffices to prove surjectivity for the sheaves", "$(Z_i \\to X)_*\\underline{S_i}$. Observe that", "$$", "\\Gamma(X, (Z_i \\to X)_*\\underline{S_i}) = \\Gamma(Z_i, \\underline{S_i})", "\\quad\\text{and}\\quad", "\\Gamma(X, (Z_i \\to X)_*\\underline{S_i}|_Z) =", "\\Gamma(Z \\cap Z_i, \\underline{S_i})", "$$", "Moreover, conditions (1) and (2) are inherited by $Z_i$; this is clear", "for (2) and follows from", "Topology, Lemma \\ref{topology-lemma-spectral-sub} for (1). Thus it", "suffices to prove the lemma in the case of a (finite) constant sheaf.", "This case is a restatement of Lemma \\ref{lemma-connected-topological}", "which finishes the proof." ], "refs": [ "modules-lemma-filtered-colimit-constructibles", "categories-lemma-adjoint-exact", "sheaves-lemma-f-map", "sheaves-lemma-directed-colimits-sections", "modules-lemma-constructible-in-constant", "topology-lemma-spectral-sub", "etale-cohomology-lemma-connected-topological" ], "ref_ids": [ 13284, 12249, 14509, 14526, 13286, 8306, 6581 ] } ], "ref_ids": [ 8370 ] }, { "id": 6583, "type": "theorem", "label": "etale-cohomology-lemma-h0-henselian-pair", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-h0-henselian-pair", "contents": [ "Let $(A, I)$ be a henselian pair. Set $X = \\Spec(A)$ and", "$Z = \\Spec(A/I)$. For any sheaf $\\mathcal{F}$ on $X_\\etale$", "we have $\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, \\mathcal{F}|_Z)$." ], "refs": [], "proofs": [ { "contents": [ "Recall that the spectrum of any ring is a spectral space, see", "Algebra, Lemma \\ref{algebra-lemma-spec-spectral}. By", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-irreducible-henselian-pair-connected}", "we see that $\\overline{\\{x\\}} \\cap Z$ is connected for every $x \\in X$.", "By Lemma \\ref{lemma-h0-topological} we see that the statement", "is true for sheaves on $X_{Zar}$. For any finite morphism $X' \\to X$", "we have $X' = \\Spec(A')$ and $Z \\times_X X' = \\Spec(A'/IA')$", "with $(A', IA')$ a henselian pair, see More on Algebra, Lemma", "\\ref{more-algebra-lemma-integral-over-henselian-pair}", "and we get the same statement for sheaves on $(X')_{Zar}$.", "Thus we can apply Lemma \\ref{lemma-gabber-h0} to conclude." ], "refs": [ "algebra-lemma-spec-spectral", "more-algebra-lemma-irreducible-henselian-pair-connected", "etale-cohomology-lemma-h0-topological", "more-algebra-lemma-integral-over-henselian-pair", "etale-cohomology-lemma-gabber-h0" ], "ref_ids": [ 423, 9870, 6582, 9863, 6580 ] } ], "ref_ids": [] }, { "id": 6584, "type": "theorem", "label": "etale-cohomology-lemma-vanishing-restriction-injective", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-vanishing-restriction-injective", "contents": [ "Let $X$ be a scheme with affine diagonal which can be covered by", "$n + 1$ affine opens. Let $Z \\subset X$ be a closed subscheme.", "Let $\\mathcal{A}$ be a torsion sheaf of rings on $X_\\etale$", "and let $\\mathcal{I}$ be an injective sheaf of $\\mathcal{A}$-modules", "on $X_\\etale$.", "Then $H^q_\\etale(Z, \\mathcal{I}|_Z) = 0$ for $q > n$." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by induction on $n$. If $n = 0$, then $X$ is affine.", "Say $X = \\Spec(A)$ and $Z = \\Spec(A/I)$. Let $A^h$ be the filtered colimit", "of \\'etale $A$-algebras $B$ such that $A/I \\to B/IB$ is an isomorphism.", "Then $(A^h, IA^h)$ is a henselian pair and $A/I = A^h/IA^h$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization}", "and its proof. Set $X^h = \\Spec(A^h)$.", "By Theorem \\ref{theorem-gabber}", "we see that", "$$", "H^q_\\etale(Z, \\mathcal{I}|_Z) = H^q_\\etale(X^h, \\mathcal{I}|_{X^h})", "$$", "By Theorem \\ref{theorem-colimit} we have", "$$", "H^q_\\etale(X^h, \\mathcal{I}|_{X^h}) =", "\\colim_{A \\to B} H^q_\\etale(\\Spec(B), \\mathcal{I}|_{\\Spec(B)})", "$$", "where the colimit is over the $A$-algebras $B$ as above.", "Since the morphisms $\\Spec(B) \\to \\Spec(A)$ are \\'etale,", "the restriction $\\mathcal{I}|_{\\Spec(B)}$ is an injective", "sheaf of $\\mathcal{A}|_{\\Spec(B)}$-modules", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-of-open}).", "Thus the cohomology groups on the right are zero and we get the", "result in this case.", "\\medskip\\noindent", "Induction step. We can use Mayer-Vietoris to do the induction step.", "Namely, suppose that $X = U \\cup V$ where $U$ is a union of $n$ affine", "opens and $V$ is affine. Then, using that the diagonal of $X$ is affine,", "we see that $U \\cap V$ is the union of $n$ affine opens. Mayer-Vietoris", "gives an exact sequence", "$$", "H^{q - 1}_\\etale(U \\cap V \\cap Z, \\mathcal{I}|_Z) \\to", "H^q_\\etale(Z, \\mathcal{I}|_Z) \\to", "H^q_\\etale(U \\cap Z, \\mathcal{I}|_Z) \\oplus", "H^q_\\etale(V \\cap Z, \\mathcal{I}|_Z)", "$$", "and by our induction hypothesis we obtain vanishing for $q > n$ as desired." ], "refs": [ "more-algebra-lemma-henselization", "etale-cohomology-theorem-gabber", "etale-cohomology-theorem-colimit", "sites-cohomology-lemma-cohomology-of-open" ], "ref_ids": [ 9871, 6393, 6384, 4186 ] } ], "ref_ids": [] }, { "id": 6585, "type": "theorem", "label": "etale-cohomology-lemma-constant-smooth-statements", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-constant-smooth-statements", "contents": [ "In Situation \\ref{situation-what-to-prove}", "assume $X$ is smooth and $\\mathcal{F} = \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$", "for some prime number $\\ell$. Then statements", "(\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold", "for $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is smooth, we see that $X$ is a finite disjoint union of", "smooth curves. Hence we may assume $X$ is a smooth curve.", "\\medskip\\noindent", "Case I: $\\ell$ different from the characteristic of $k$.", "This case follows from", "Lemma \\ref{lemma-cohomology-smooth-projective-curve}", "(projective case) and", "Lemma \\ref{lemma-vanishing-cohomology-mu-smooth-curve}", "(affine case). Statement (\\ref{item-base-change-prime-to-p})", "on cohomology and extension of algebraically closed ground", "field follows from the fact that the genus $g$ and the number", "of ``punctures'' $r$ do not change when passing from $k$ to $k'$.", "Statement (\\ref{item-surjective}) follows as $H^2_\\etale(U, \\mathcal{F})$", "is zero as soon as $U \\not = X$, because then $U$ is affine", "(Varieties, Lemmas \\ref{varieties-lemma-proper-minus-point} and", "\\ref{varieties-lemma-curve-affine-projective}).", "\\medskip\\noindent", "Case II: $\\ell$ is equal to the characteristic of $k$.", "Vanishing by Lemma \\ref{lemma-vanishing-variety-char-p-p}.", "Statements (\\ref{item-finite-proper}) and (\\ref{item-base-change-proper})", "follow from", "Lemma \\ref{lemma-finiteness-proper-variety-char-p-p}." ], "refs": [ "etale-cohomology-lemma-cohomology-smooth-projective-curve", "etale-cohomology-lemma-vanishing-cohomology-mu-smooth-curve", "varieties-lemma-proper-minus-point", "varieties-lemma-curve-affine-projective", "etale-cohomology-lemma-vanishing-variety-char-p-p", "etale-cohomology-lemma-finiteness-proper-variety-char-p-p" ], "ref_ids": [ 6519, 6521, 11097, 11102, 6499, 6500 ] } ], "ref_ids": [] }, { "id": 6586, "type": "theorem", "label": "etale-cohomology-lemma-ses-statements", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-ses-statements", "contents": [ "Let $k$ be an algebraically closed field. Let $X$ be a separated finite", "type scheme over $k$ of dimension $\\leq 1$. Let", "$0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0$", "be a short exact sequence of torsion abelian sheaves on $X$.", "If statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold", "for $\\mathcal{F}_1$ and $\\mathcal{F}_2$, then they hold", "for $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "This is mostly immediate from the definitions and the long exact sequence", "of cohomology. Also observe that $\\mathcal{F}$ is constructible", "(resp.\\ of torsion prime to the characteristic of $k$) if and only if", "both $\\mathcal{F}_1$ and $\\mathcal{F}_2$ are constructible", "(resp.\\ of torsion prime to the characteristic of $k$). See", "Proposition \\ref{proposition-constructible-over-noetherian}.", "Some details omitted." ], "refs": [ "etale-cohomology-proposition-constructible-over-noetherian" ], "ref_ids": [ 6706 ] } ], "ref_ids": [] }, { "id": 6587, "type": "theorem", "label": "etale-cohomology-lemma-finite-pushforward-statements", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-finite-pushforward-statements", "contents": [ "Let $k$ be an algebraically closed field. Let $f : X \\to Y$ be a", "finite morphism of separated finite type schemes over $k$ of", "dimension $\\leq 1$. Let $\\mathcal{F}$ be a torsion abelian sheaf on $X$.", "If statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold", "for $\\mathcal{F}$, then they hold for $f_*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Follows from the vanishing of the higher direct images $R^qf_*$", "(Proposition \\ref{proposition-finite-higher-direct-image-zero}),", "the Leray spectral sequence (Proposition \\ref{proposition-leray}),", "and the fact that formation of $f_*$ commutes with arbitrary base change", "(Lemma \\ref{lemma-finite-pushforward-commutes-with-base-change})." ], "refs": [ "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-proposition-leray", "etale-cohomology-lemma-finite-pushforward-commutes-with-base-change" ], "ref_ids": [ 6703, 6702, 6481 ] } ], "ref_ids": [] }, { "id": 6588, "type": "theorem", "label": "etale-cohomology-lemma-even-easier", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-even-easier", "contents": [ "In Situation \\ref{situation-what-to-prove} assume $X$ is smooth.", "Let $j : U \\to X$ an open immersion. Let $\\ell$ be a prime number.", "Let $\\mathcal{F} = j_!\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$.", "Then statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold", "for $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the short exact sequence", "$$", "0 \\longrightarrow j_!\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}_U", "\\longrightarrow \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}_X \\longrightarrow", "\\bigoplus\\nolimits_{x \\in X \\setminus U}", "{i_x}_*(\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}})", "\\longrightarrow 0.", "$$", "Statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold", "for $\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$ by", "Lemma \\ref{lemma-constant-smooth-statements}.", "Since the inclusion morphisms $i_x : x \\to X$ are finite", "and since $x$ is the spectrum of an algebraically closed field,", "we see that $H^q_\\etale(X, i_{x*}\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}})$", "is zero for $q > 0$ and equal to $\\mathbf{Z}/\\ell\\mathbf{Z}$", "for $q = 0$. Thus we get from the long exact cohomology sequence", "$$", "\\xymatrix{", "0 \\ar[r] &", "H^0_\\etale(X, \\mathcal{F}) \\ar[r] &", "H^0(X, \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}_X) \\ar[r] &", "\\bigoplus\\nolimits_{x \\in X \\setminus U} \\mathbf{Z}/\\ell\\mathbf{Z}", "\\ar@(rd, ul)[rdllllr] \\\\", " & H^1_\\etale(X, \\mathcal{F}) \\ar[r] &", "H^1_\\etale(X, \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}_X) \\ar[r] & 0", "}", "$$", "and $H^q_\\etale(X, \\mathcal{F}) =", "H^q_\\etale(X, \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}_X)$", "for $q \\geq 2$. Each of the statements ", "(\\ref{item-vanishing}) -- (\\ref{item-surjective}) follows by inspection." ], "refs": [ "etale-cohomology-lemma-constant-smooth-statements" ], "ref_ids": [ 6585 ] } ], "ref_ids": [] }, { "id": 6589, "type": "theorem", "label": "etale-cohomology-lemma-somewhat-easier", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-somewhat-easier", "contents": [ "In Situation \\ref{situation-what-to-prove} assume $X$ reduced.", "Let $j : U \\to X$ an open immersion. Let $\\ell$ be a prime number", "and $\\mathcal{F} = j_! \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$.", "Then statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold", "for $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "The difference with Lemma \\ref{lemma-even-easier} is that here we do not", "assume $X$ is smooth. Let $\\nu : X' \\to X$ be the normalization", "morphism which is finite as varieties are Nagata schemes.", "Let $j' : U' \\to X'$ be the inverse image of $U$.", "By Lemma \\ref{lemma-even-easier} the result holds for", "$j'_!\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$.", "By Lemma \\ref{lemma-finite-pushforward-statements}", "the result holds for $\\nu_*j'_!\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$.", "In general it won't be true that", "$\\nu_*j'_!\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$ is equal to", "$j_!\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$, but there will be a canonical", "injective map", "$$", "j_!\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}} \\longrightarrow", "\\nu_*j'_!\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}", "$$", "whose cokernel is of the form $\\bigoplus_{x \\in Z} i_{x *}M_x$", "where $Z \\subset X$ is a finite set of closed points and $M_x$", "is a finite dimensional $\\mathbf{F}_\\ell$-vector space for each", "$x \\in Z$. We obtain a short exact sequence", "$$", "0 \\to j_!\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}} \\to", "\\nu_*j'_!\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}} \\to", "\\bigoplus\\nolimits_{x \\in Z} i_{x *}M_x \\to 0", "$$", "and we can argue exactly as in the proof of Lemma \\ref{lemma-even-easier}", "to finish the argument. Some details omitted." ], "refs": [ "etale-cohomology-lemma-even-easier", "etale-cohomology-lemma-even-easier", "etale-cohomology-lemma-finite-pushforward-statements", "etale-cohomology-lemma-even-easier" ], "ref_ids": [ 6588, 6588, 6587, 6588 ] } ], "ref_ids": [] }, { "id": 6590, "type": "theorem", "label": "etale-cohomology-lemma-vanishing-easier", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-vanishing-easier", "contents": [ "In Situation \\ref{situation-what-to-prove} assume $X$ reduced.", "Let $j : U \\to X$ an open immersion with $U$ connected. Let", "$\\ell$ be a prime number. Let $\\mathcal{G}$ a finite locally", "constant sheaf of $\\mathbf{F}_\\ell$-vector spaces on $U$. Let", "$\\mathcal{F} = j_!\\mathcal{G}$. Then statements", "(\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold for $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Let $f : V \\to U$ be a finite \\'etale morphism of degree prime to $\\ell$", "as in Lemma \\ref{lemma-pullback-filtered}. The discussion in", "Section \\ref{section-trace-method} gives maps", "$$", "\\mathcal{G} \\to f_*f^{-1}\\mathcal{G} \\to \\mathcal{G}", "$$", "whose composition is an isomorphism. Hence it suffices to prove the", "lemma with $\\mathcal{F} = j_!f_*f^{-1}\\mathcal{G}$.", "By Zariski's Main theorem", "(More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite})", "we can choose a diagram", "$$", "\\xymatrix{", "V \\ar[r]_{j'} \\ar[d]_f & Y \\ar[d]^{\\overline{f}} \\\\", "U \\ar[r]^j & X", "}", "$$", "with $\\overline{f} : Y \\to X$ finite and $j'$ an open immersion", "with dense image. We may replace $Y$ by its reduction (this does", "not change $V$ as $V$ is reduced being \\'etale over $U$).", "Since $f$ is finite we have $V = U \\times_X Y$. By", "Lemma \\ref{lemma-finite-pushforward-commutes-with-base-change}", "we have", "$$", "j^{-1}\\overline{f}_*j'_!f^{-1}\\mathcal{G} =", "f_*(j')^{-1}j'_!f^{-1}\\mathcal{G} =", "f_*f^{-1}\\mathcal{G}", "$$", "because $(j')^{-1}j'_! = \\text{id}$ by Lemma \\ref{lemma-jshriek-open}.", "Hence there is a canonical map", "$j_!f_*f^{-1}\\mathcal{G} \\to \\overline{f}_*j'_!f^{-1}\\mathcal{G}$", "which is an isomorphism over $U$ (by the same lemma). However, the stalk", "of $\\overline{f}_*j'_!f^{-1}\\mathcal{G}$ at a geometric point $\\overline{x}$", "of $X \\setminus U$ are zero by", "Proposition \\ref{proposition-finite-higher-direct-image-zero}", "and the fact that the stalks of $j'_!f^{-1}\\mathcal{F}$ are", "zero at all geometric points $\\overline{y}$ of $Y$ lying over $\\overline{x}$", "(by Lemma \\ref{lemma-jshriek-open}).", "Hence $j_!f_*f^{-1}\\mathcal{G} \\to \\overline{f}_*j'_!f^{-1}\\mathcal{G}$", "is an isomorphism, again by Lemma \\ref{lemma-jshriek-open}.", "By Lemma \\ref{lemma-finite-pushforward-statements} it suffices to", "prove the lemma for $j'_!f^{-1}\\mathcal{G}$.", "The existence of the filtration given by", "Lemma \\ref{lemma-pullback-filtered},", "the fact that $j'_!$ is exact, and", "Lemma \\ref{lemma-ses-statements}", "reduces us to the case", "$\\mathcal{F} = j'_!\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$", "which is Lemma \\ref{lemma-somewhat-easier}." ], "refs": [ "etale-cohomology-lemma-pullback-filtered", "more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "etale-cohomology-lemma-finite-pushforward-commutes-with-base-change", "etale-cohomology-lemma-jshriek-open", "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-lemma-jshriek-open", "etale-cohomology-lemma-jshriek-open", "etale-cohomology-lemma-finite-pushforward-statements", "etale-cohomology-lemma-pullback-filtered", "etale-cohomology-lemma-ses-statements", "etale-cohomology-lemma-somewhat-easier" ], "ref_ids": [ 6509, 13901, 6481, 6522, 6703, 6522, 6522, 6587, 6509, 6586, 6589 ] } ], "ref_ids": [] }, { "id": 6591, "type": "theorem", "label": "etale-cohomology-lemma-base-change-dim-1-separably-closed", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-dim-1-separably-closed", "contents": [ "Let $k \\subset k'$ be an extension of separably closed fields.", "Let $X$ be a proper scheme over $k$ of dimension $\\leq 1$.", "Let $\\mathcal{F}$ be a torsion abelian sheaf on $X$.", "Then the map $H^q_\\etale(X, \\mathcal{F}) \\to", "H^q_\\etale(X_{k'}, \\mathcal{F}|_{X_{k'}})$ is an isomorphism", "for $q \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "We have seen this for algebraically closed fields in", "Theorem \\ref{theorem-vanishing-affine-curves}.", "Given $k \\subset k'$ as in the statement of the lemma we can", "choose a diagram", "$$", "\\xymatrix{", "k' \\ar[r] & \\overline{k}' \\\\", "k \\ar[u] \\ar[r] & \\overline{k} \\ar[u]", "}", "$$", "where $k \\subset \\overline{k}$ and $k' \\subset \\overline{k}'$ are", "the algebraic closures. Since $k$ and $k'$ are separably closed", "the field extensions", "$k \\subset \\overline{k}$ and $k' \\subset \\overline{k}'$", "are algebraic and purely inseparable. In this case the morphisms", "$X_{\\overline{k}} \\to X$ and $X_{\\overline{k}'} \\to X_{k'}$", "are universal homeomorphisms. Thus the cohomology of $\\mathcal{F}$", "may be computed on $X_{\\overline{k}}$ and the cohomology", "of $\\mathcal{F}|_{X_{k'}}$ may be computed on $X_{\\overline{k}'}$,", "see Proposition \\ref{proposition-topological-invariance}.", "Hence we deduce the general case from the case of algebraically", "closed fields." ], "refs": [ "etale-cohomology-theorem-vanishing-affine-curves", "etale-cohomology-proposition-topological-invariance" ], "ref_ids": [ 6394, 6699 ] } ], "ref_ids": [] }, { "id": 6592, "type": "theorem", "label": "etale-cohomology-lemma-proper-over-henselian-and-h1", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-proper-over-henselian-and-h1", "contents": [ "Let $A$ be a henselian local ring. Let $X$ be a proper scheme over $A$", "with closed fibre $X_0$. Let $M$ be a finite abelian group.", "Then $H^1_\\etale(X, \\underline{M}) = H^1_\\etale(X_0, \\underline{M})$." ], "refs": [], "proofs": [ { "contents": [ "By Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-torsors-h1}", "an element of $H^1_\\etale(X, \\underline{M})$ corresponds to a", "$\\underline{M}$-torsor $\\mathcal{F}$ on $X_\\etale$.", "Such a torsor is clearly a finite locally constant sheaf.", "Hence $\\mathcal{F}$ is representable by a scheme $V$ finite", "\\'etale over $X$, Lemma \\ref{lemma-characterize-finite-locally-constant}.", "Conversely, a scheme $V$ finite \\'etale over $X$ with an $M$-action", "which turns it into an $M$-torsor over $X$ gives rise to a cohomology", "class. The same translation between cohomology classes over $X_0$ and", "torsors finite \\'etale over $X_0$ holds. Thus the lemma", "is a consequence of the equivalence of categories of", "Fundamental Groups, Lemma", "\\ref{pione-lemma-finite-etale-on-proper-over-henselian}." ], "refs": [ "sites-cohomology-lemma-torsors-h1", "etale-cohomology-lemma-characterize-finite-locally-constant", "pione-lemma-finite-etale-on-proper-over-henselian" ], "ref_ids": [ 4182, 6503, 4044 ] } ], "ref_ids": [] }, { "id": 6593, "type": "theorem", "label": "etale-cohomology-lemma-efface-cohomology-on-fibre-by-finite-cover", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-efface-cohomology-on-fibre-by-finite-cover", "contents": [ "Let $A$ be a henselian local ring. Let $X = \\mathbf{P}^1_A$.", "Let $X_0 \\subset X$ be the closed fibre. Let $\\ell$ be a prime", "number. Let $\\mathcal{I}$ be an injective sheaf of", "$\\mathbf{Z}/\\ell\\mathbf{Z}$-modules on $X_\\etale$. Then", "$H^q_\\etale(X_0, \\mathcal{I}|_{X_0}) = 0$ for $q > 0$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $X$ is a separated scheme which can be covered by $2$", "affine opens. Hence for $q > 1$ this follows from Gabber's affine", "variant of the proper base change theorem, see", "Lemma \\ref{lemma-vanishing-restriction-injective}.", "Thus we may assume $q = 1$. Let", "$\\xi \\in H^1_\\etale(X_0, \\mathcal{I}|_{X_0})$.", "Goal: show that $\\xi$ is $0$.", "By Lemmas \\ref{lemma-torsion-colimit-constructible} and", "\\ref{lemma-colimit} we can find a map $\\mathcal{F} \\to \\mathcal{I}$", "with $\\mathcal{F}$ a constructible sheaf of", "$\\mathbf{Z}/\\ell\\mathbf{Z}$-modules", "and $\\xi$ coming from an element $\\zeta$ of", "$H^1_\\etale(X_0, \\mathcal{F}|_{X_0})$. Suppose we have an injective map", "$\\mathcal{F} \\to \\mathcal{F}'$ of sheaves of", "$\\mathbf{Z}/\\ell\\mathbf{Z}$-modules on $X_\\etale$.", "Since $\\mathcal{I}$ is injective we can extend the given map", "$\\mathcal{F} \\to \\mathcal{I}$ to a map $\\mathcal{F}' \\to \\mathcal{I}$.", "In this situation we may replace $\\mathcal{F}$ by $\\mathcal{F}'$", "and $\\zeta$ by the image of $\\zeta$ in $H^1_\\etale(X_0, \\mathcal{F}'|_{X_0})$.", "Also, if $\\mathcal{F} = \\mathcal{F}_1 \\oplus \\mathcal{F}_2$ is a direct sum,", "then we may replace $\\mathcal{F}$ by $\\mathcal{F}_i$", "and $\\zeta$ by the image of $\\zeta$ in $H^1_\\etale(X_0, \\mathcal{F}_i|_{X_0})$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-constructible-maps-into-constant-general}", "and the remarks above we may assume $\\mathcal{F}$", "is of the form $f_*\\underline{M}$ where $M$ is a finite", "$\\mathbf{Z}/\\ell\\mathbf{Z}$-module", "and $f : Y \\to X$ is a finite morphism of finite presentation", "(such sheaves are still constructible by", "Lemma \\ref{lemma-finite-pushforward-constructible}", "but we won't need this).", "Since formation of $f_*$ commutes with any base change", "(Lemma \\ref{lemma-finite-pushforward-commutes-with-base-change})", "we see that the restriction of $f_*\\underline{M}$ to $X_0$ is", "equal to the pushforward of $\\underline{M}$ via the induced morphism", "$Y_0 \\to X_0$ of special fibres. By the Leray spectral sequence", "(Proposition \\ref{proposition-leray})", "and vanishing of higher direct images", "(Proposition \\ref{proposition-finite-higher-direct-image-zero}),", "we find", "$$", "H^1_\\etale(X_0, f_*\\underline{M}|_{X_0}) = H^1_\\etale(Y_0, \\underline{M}).", "$$", "Since $Y \\to \\Spec(A)$ is proper we can use", "Lemma \\ref{lemma-proper-over-henselian-and-h1} to see that", "the $H^1_\\etale(Y_0, \\underline{M})$ is equal to", "$H^1_\\etale(Y, \\underline{M})$. Thus we see that our cohomology", "class $\\zeta$ lifts to a cohomology class", "$$", "\\tilde\\zeta \\in H^1_\\etale(Y, \\underline{M}) = H^1_\\etale(X, f_*\\underline{M})", "$$", "However, $\\tilde \\zeta$ maps to zero in", "$H^1_\\etale(X, \\mathcal{I})$ as $\\mathcal{I}$ is injective", "and by commutativity of", "$$", "\\xymatrix{", "H^1_\\etale(X, f_*\\underline{M}) \\ar[r] \\ar[d] &", "H^1_\\etale(X, \\mathcal{I}) \\ar[d] \\\\", "H^1_\\etale(X_0, (f_*\\underline{M})|_{X_0}) \\ar[r] &", "H^1_\\etale(X_0, \\mathcal{I}|_{X_0})", "}", "$$", "we conclude that the image $\\xi$ of $\\zeta$ is zero as well." ], "refs": [ "etale-cohomology-lemma-vanishing-restriction-injective", "etale-cohomology-lemma-torsion-colimit-constructible", "etale-cohomology-lemma-colimit", "etale-cohomology-lemma-constructible-maps-into-constant-general", "etale-cohomology-lemma-finite-pushforward-constructible", "etale-cohomology-lemma-finite-pushforward-commutes-with-base-change", "etale-cohomology-proposition-leray", "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-lemma-proper-over-henselian-and-h1" ], "ref_ids": [ 6584, 6539, 6472, 6553, 6546, 6481, 6702, 6703, 6592 ] } ], "ref_ids": [] }, { "id": 6594, "type": "theorem", "label": "etale-cohomology-lemma-base-change-local", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-local", "contents": [ "Consider a cartesian diagram of schemes", "$$", "\\xymatrix{", "X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\", "S & T \\ar[l]_g", "}", "$$", "and a sheaf $\\mathcal{F}$ on $T_\\etale$. Let $\\{U_i \\to X\\}$ be an", "\\'etale covering such that $U_i \\to S$", "factors as $U_i \\to V_i \\to S$ with $V_i \\to S$ \\'etale", "and consider the cartesian diagrams", "$$", "\\xymatrix{", "U_i \\ar[d]_{f_i} & U_i \\times_X Y \\ar[l]^{h_i} \\ar[d]^{e_i} \\\\", "V_i & V_i \\times_S T \\ar[l]_{g_i}", "}", "$$", "Set $\\mathcal{F}_i = \\mathcal{F}|_{V_i \\times_S T}$.", "\\begin{enumerate}", "\\item If $f_i^{-1}g_{i, *}\\mathcal{F}_i = h_{i, *}e_i^{-1}\\mathcal{F}_i$", "for all $i$, then", "$f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$.", "\\item If $\\mathcal{F}$ is an abelian sheaf and", "$f_i^{-1}R^qg_{i, *}\\mathcal{F}_i = R^qh_{i, *}e_i^{-1}\\mathcal{F}_i$", "for all $i$, then", "$f^{-1}R^qg_*\\mathcal{F} = R^qh_*e^{-1}\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have", "$f^{-1}R^qg_*\\mathcal{F}|_{U_i} = f_i^{-1}R^qg_{i, *}\\mathcal{F}_i$", "and", "$R^qh_*e^{-1}\\mathcal{F}|_{U_i} = R^qh_{i, *}e_i^{-1}\\mathcal{F}_i$", "as follows from the compatibility of localization with", "morphisms of sites, see", "Sites, Lemma \\ref{sites-lemma-localize-morphism-strong}", "and", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-restrict-direct-image-open}." ], "refs": [ "sites-lemma-localize-morphism-strong", "sites-cohomology-lemma-restrict-direct-image-open" ], "ref_ids": [ 8572, 4256 ] } ], "ref_ids": [] }, { "id": 6595, "type": "theorem", "label": "etale-cohomology-lemma-base-change-compose", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-compose", "contents": [ "Consider a tower of cartesian diagrams of schemes", "$$", "\\xymatrix{", "W \\ar[d]_i & Z \\ar[l]^j \\ar[d]^k \\\\", "X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\", "S & T \\ar[l]_g", "}", "$$", "Let $K$ in $D(T_\\etale)$. If", "$$", "f^{-1}Rg_*K \\to Rh_*e^{-1}K", "\\quad\\text{and}\\quad", "i^{-1}Rh_*e^{-1}K \\to Rj_*k^{-1}e^{-1}K", "$$", "are isomorphisms, then", "$(f \\circ i)^{-1}Rg_*K \\to Rj_*(e \\circ k)^{-1}K$", "is an isomorphism.", "Similarly, if $\\mathcal{F}$ is an abelian sheaf on $T_\\etale$ and if", "$$", "f^{-1}R^qg_*\\mathcal{F} \\to R^qh_*e^{-1}\\mathcal{F}", "\\quad\\text{and}\\quad", "i^{-1}R^qh_*e^{-1}\\mathcal{F} \\to R^qj_*k^{-1}e^{-1}\\mathcal{F}", "$$", "are isomorphisms, then", "$(f \\circ i)^{-1}R^qg_*\\mathcal{F} \\to R^qj_*(e \\circ k)^{-1}\\mathcal{F}$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is formal, provided one checks that the composition of these", "base change maps is the base change maps for the outer rectangle, see", "Cohomology on Sites, Remark", "\\ref{sites-cohomology-remark-compose-base-change-horizontal}." ], "refs": [ "sites-cohomology-remark-compose-base-change-horizontal" ], "ref_ids": [ 4426 ] } ], "ref_ids": [] }, { "id": 6596, "type": "theorem", "label": "etale-cohomology-lemma-base-change-Rf-star-colim", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-Rf-star-colim", "contents": [ "Let $I$ be a directed set. Consider an inverse system of", "cartesian diagrams of schemes", "$$", "\\xymatrix{", "X_i \\ar[d]_{f_i} & Y_i \\ar[l]^{h_i} \\ar[d]^{e_i} \\\\", "S_i & T_i \\ar[l]_{g_i}", "}", "$$", "with affine transition morphisms and with $g_i$ quasi-compact and", "quasi-separated. Set $X = \\lim X_i$,", "$S = \\lim S_i$, $T = \\lim T_i$ and $Y = \\lim Y_i$ to", "obtain the cartesian diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\", "S & T \\ar[l]_g", "}", "$$", "Let $(\\mathcal{F}_i, \\varphi_{i'i})$ be a system of sheaves on", "$(T_i)$ as in Definition \\ref{definition-inverse-system-sheaves}. Set", "$\\mathcal{F} = \\colim p_i^{-1}\\mathcal{F}_i$ on $T$", "where $p_i : T \\to T_i$ is the projection.", "Then we have the following", "\\begin{enumerate}", "\\item If $f_i^{-1}g_{i, *}\\mathcal{F}_i = h_{i, *}e_i^{-1}\\mathcal{F}_i$", "for all $i$, then", "$f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$.", "\\item If $\\mathcal{F}_i$ is an abelian sheaf for all $i$ and", "$f_i^{-1}R^qg_{i, *}\\mathcal{F}_i = R^qh_{i, *}e_i^{-1}\\mathcal{F}_i$", "for all $i$, then", "$f^{-1}R^qg_*\\mathcal{F} = R^qh_*e^{-1}\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [ "etale-cohomology-definition-inverse-system-sheaves" ], "proofs": [ { "contents": [ "We prove (2) and we omit the proof of (1). We will use without further", "mention that pullback of sheaves commutes with colimits as it is a", "left adjoint. Observe that $h_i$ is quasi-compact and quasi-separated as a", "base change of $g_i$.", "Denoting $q_i : Y \\to Y_i$ the projections, observe that", "$e^{-1}\\mathcal{F} = \\colim e^{-1}p_i^{-1}\\mathcal{F}_i =", "\\colim q_i^{-1}e_i^{-1}\\mathcal{F}_i$.", "By Lemma \\ref{lemma-relative-colimit-general}", "this gives", "$$", "R^qh_*e^{-1}\\mathcal{F} = \\colim r_i^{-1}R^qh_{i, *}e_i^{-1}\\mathcal{F}_i", "$$", "where $r_i : X \\to X_i$ is the projection.", "Similarly, we have", "$$", "f^{-1}Rg_*\\mathcal{F} =", "f^{-1}\\colim s_i^{-1}R^qg_{i, *}\\mathcal{F}_i =", "\\colim r_i^{-1}f_i^{-1}R^qg_{i, *}\\mathcal{F}_i", "$$", "where $s_i : S \\to S_i$ is the projection. The lemma follows." ], "refs": [ "etale-cohomology-lemma-relative-colimit-general" ], "ref_ids": [ 6476 ] } ], "ref_ids": [ 6745 ] }, { "id": 6597, "type": "theorem", "label": "etale-cohomology-lemma-base-change-f-star-general-stalks", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-f-star-general-stalks", "contents": [ "Consider a cartesian diagram of schemes", "$$", "\\xymatrix{", "X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\", "S & T \\ar[l]_g", "}", "$$", "where $g : T \\to S$ is quasi-compact and quasi-separated.", "Let $\\mathcal{F}$ be an", "abelian sheaf on $T_\\etale$. Let $q \\geq 0$. The following are equivalent", "\\begin{enumerate}", "\\item For every geometric point $\\overline{x}$ of $X$ with image", "$\\overline{s} = f(\\overline{x})$ we have", "$$", "H^q(\\Spec(\\mathcal{O}^{sh}_{X, \\overline{x}}) \\times_S T, \\mathcal{F})", "=", "H^q(\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}}) \\times_S T, \\mathcal{F})", "$$", "\\item $f^{-1}R^qg_*\\mathcal{F} \\to R^qh_*e^{-1}\\mathcal{F}$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $Y = X \\times_S T$ we have", "$\\Spec(\\mathcal{O}^{sh}_{X, \\overline{x}}) \\times_X Y =", "\\Spec(\\mathcal{O}^{sh}_{X, \\overline{x}}) \\times_S T$. Thus", "the map in (1) is the map of stalks at $\\overline{x}$ for the map", "in (2) by Theorem \\ref{theorem-higher-direct-images} (and", "Lemma \\ref{lemma-stalk-pullback}).", "Thus the result by Theorem \\ref{theorem-exactness-stalks}." ], "refs": [ "etale-cohomology-theorem-higher-direct-images", "etale-cohomology-lemma-stalk-pullback", "etale-cohomology-theorem-exactness-stalks" ], "ref_ids": [ 6385, 6436, 6376 ] } ], "ref_ids": [] }, { "id": 6598, "type": "theorem", "label": "etale-cohomology-lemma-check-stalks-better", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-check-stalks-better", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\overline{x}$ be a geometric point of $X$ with image $\\overline{s}$ in $S$.", "Let $\\Spec(K) \\to \\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})$", "be a morphism with $K$ a separably closed field. Let $\\mathcal{F}$ be an", "abelian sheaf on $\\Spec(K)_\\etale$. Let $q \\geq 0$. The following are", "equivalent", "\\begin{enumerate}", "\\item", "$H^q(\\Spec(\\mathcal{O}^{sh}_{X, \\overline{x}}) \\times_S \\Spec(K), \\mathcal{F}) =", "H^q(\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}}) \\times_S \\Spec(K), \\mathcal{F})$", "\\item", "$H^q(\\Spec(\\mathcal{O}^{sh}_{X, \\overline{x}})", "\\times_{\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})} \\Spec(K), \\mathcal{F}) =", "H^q(\\Spec(K), \\mathcal{F})$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\Spec(K) \\times_S \\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})$", "is the spectrum of a filtered colimit of \\'etale algebras over $K$.", "Since $K$ is separably closed, each \\'etale $K$-algebra", "is a finite product of copies of $K$. Thus we can write", "$$", "\\Spec(K) \\times_S \\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}}) =", "\\lim_{i \\in I} \\coprod\\nolimits_{a \\in A_i} \\Spec(K)", "$$", "as a cofiltered limit where each term is a disjoint union", "of copies of $\\Spec(K)$ over a finite set $A_i$.", "Note that $A_i$ is nonempty as we are given", "$\\Spec(K) \\to \\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})$.", "It follows that", "\\begin{align*}", "\\Spec(\\mathcal{O}^{sh}_{X, \\overline{x}}) \\times_S \\Spec(K)", "& =", "\\Spec(\\mathcal{O}^{sh}_{X, \\overline{x}})", "\\times_{\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})}", "\\left(", "\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})", "\\times_S \\Spec(K)\\right) \\\\", "& =", "\\colim_{i \\in I} \\coprod\\nolimits_{a \\in A_i}", "\\Spec(\\mathcal{O}^{sh}_{X, \\overline{x}})", "\\times_{\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})} \\Spec(K)", "\\end{align*}", "Since taking cohomology in our setting commutes with limits", "of schemes (Theorem \\ref{theorem-colimit}) we conclude." ], "refs": [ "etale-cohomology-theorem-colimit" ], "ref_ids": [ 6384 ] } ], "ref_ids": [] }, { "id": 6599, "type": "theorem", "label": "etale-cohomology-lemma-base-change-f-star-general", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-f-star-general", "contents": [ "Consider the cartesian diagram of schemes", "$$", "\\xymatrix{", "X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\", "S & T \\ar[l]_g", "}", "$$", "Assume that $f$ is flat and every object $U$ of $X_\\etale$ has", "a covering $\\{U_i \\to U\\}$ such that $U_i \\to S$", "factors as $U_i \\to V_i \\to S$ with $V_i \\to S$", "\\'etale and $U_i \\to V_i$ quasi-compact with", "geometrically connected fibres.", "Then for any sheaf $\\mathcal{F}$ of sets on $T_\\etale$ we have", "$f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be an \\'etale morphism such that $U \\to S$ factors as", "$U \\to V \\to S$ with $V \\to S$ \\'etale and $U \\to V$ quasi-compact", "with geometrically connected fibres. Observe that $U \\to V$ is flat", "(More on Flatness, Lemma \\ref{flat-lemma-etale-flat-up-down}).", "We claim that", "\\begin{align*}", "f^{-1}g_*\\mathcal{F}(U)", "& = g_*\\mathcal{F}(V) \\\\", "& = \\mathcal{F}(V \\times_S T) \\\\", "& = e^{-1}\\mathcal{F}(U \\times_X Y) \\\\", "& = h_*e^{-1}\\mathcal{F}(U)", "\\end{align*}", "Namely, thinking of $U$ as an object of $X_\\etale$ and", "$V$ as an object of $S_\\etale$ we see that the first equality", "follows from Lemma \\ref{lemma-sections-upstairs}\\footnote{Strictly", "speaking, we are also using that the restriction of $f^{-1}g_*\\mathcal{F}$", "to $U_\\etale$ is the pullback via $U \\to V$ of the restriction of", "$g_*\\mathcal{F}$ to $V_\\etale$. See", "Sites, Lemma \\ref{sites-lemma-localize-morphism-strong}.}.", "Thinking of $V \\times_S T$ as an object of $T_\\etale$", "the second equality follows from the definition of $g_*$.", "Observe that $U \\times_X Y = U \\times_S T$ (because $Y = X \\times_S T$)", "and hence $U \\times_X Y \\to V \\times_S T$", "has geometrically connected fibres as a base change of $U \\to V$.", "Thinking of $U \\times_X Y$ as an object of $Y_\\etale$, we see that", "the third equality follows from Lemma \\ref{lemma-sections-upstairs}", "as before. Finally, the fourth equality follows from the definition", "of $h_*$.", "\\medskip\\noindent", "Since by assumption every object of $X_\\etale$ has an \\'etale", "covering to which the argument of the previous paragraph applies", "we see that the lemma is true." ], "refs": [ "flat-lemma-etale-flat-up-down", "etale-cohomology-lemma-sections-upstairs", "sites-lemma-localize-morphism-strong", "etale-cohomology-lemma-sections-upstairs" ], "ref_ids": [ 5980, 6439, 8572, 6439 ] } ], "ref_ids": [] }, { "id": 6600, "type": "theorem", "label": "etale-cohomology-lemma-fppf-reduced-fibres-base-change-f-star", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-fppf-reduced-fibres-base-change-f-star", "contents": [ "Consider a cartesian diagram of schemes", "$$", "\\xymatrix{", "X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\", "S & T \\ar[l]_g", "}", "$$", "where $f$ is flat and locally of finite presentation", "with geometrically reduced fibres.", "Then $f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$", "for any sheaf $\\mathcal{F}$ on $T_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-base-change-f-star-general} with", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-cover-by-geometrically-connected}." ], "refs": [ "etale-cohomology-lemma-base-change-f-star-general", "more-morphisms-lemma-cover-by-geometrically-connected" ], "ref_ids": [ 6599, 13915 ] } ], "ref_ids": [] }, { "id": 6601, "type": "theorem", "label": "etale-cohomology-lemma-base-change-f-star-field", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-f-star-field", "contents": [ "Consider the cartesian diagrams of schemes", "$$", "\\xymatrix{", "X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\", "S & T \\ar[l]_g", "}", "$$", "Assume that $S$ is the spectrum of a separably closed field.", "Then $f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$", "for any sheaf $\\mathcal{F}$ on $T_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "We may work locally on $X$. Hence we may assume $X$ is affine.", "Then we can write $X$ as a cofiltered limit of affine schemes of", "finite type over $S$. By Lemma \\ref{lemma-base-change-Rf-star-colim}", "we may assume that $X$ is of finite type over $S$.", "Then Lemma \\ref{lemma-base-change-f-star-general}", "applies because any scheme of finite", "type over a separably closed field is a finite disjoint", "union of connected and geometrically connected schemes", "(see Varieties, Lemma", "\\ref{varieties-lemma-separably-closed-field-connected-components})." ], "refs": [ "etale-cohomology-lemma-base-change-Rf-star-colim", "etale-cohomology-lemma-base-change-f-star-general", "varieties-lemma-separably-closed-field-connected-components" ], "ref_ids": [ 6596, 6599, 10918 ] } ], "ref_ids": [] }, { "id": 6602, "type": "theorem", "label": "etale-cohomology-lemma-base-change-f-star-valuation", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-f-star-valuation", "contents": [ "Consider a cartesian diagram of schemes", "$$", "\\xymatrix{", "X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\", "S & T \\ar[l]_g", "}", "$$", "Assume that", "\\begin{enumerate}", "\\item $f$ is flat and open,", "\\item the residue fields of $S$ are separably algebraically closed,", "\\item given an \\'etale morphism $U \\to X$ with $U$ affine", "we can write $U$ as a finite disjoint union of open subschemes", "of $X$ (for example if $X$ is a normal integral scheme", "with separably closed function field),", "\\item any nonempty open of a fibre $X_s$ of $f$ is connected", "(for example if $X_s$ is irreducible or empty).", "\\end{enumerate}", "Then for any sheaf $\\mathcal{F}$ of sets on $T_\\etale$ we have", "$f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: the assumptions almost trivially imply", "the condition of Lemma \\ref{lemma-base-change-f-star-general}.", "The for example in part (3) follows from", "Lemma \\ref{lemma-normal-scheme-with-alg-closed-function-field}." ], "refs": [ "etale-cohomology-lemma-base-change-f-star-general", "etale-cohomology-lemma-normal-scheme-with-alg-closed-function-field" ], "ref_ids": [ 6599, 6573 ] } ], "ref_ids": [] }, { "id": 6603, "type": "theorem", "label": "etale-cohomology-lemma-fppf-reduced-fibres-pullback-products", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-fppf-reduced-fibres-pullback-products", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat and", "locally of finite presentation with geometrically reduced fibres.", "Then $f^{-1} : \\Sh(S_\\etale) \\to \\Sh(X_\\etale)$ commutes", "with products." ], "refs": [], "proofs": [ { "contents": [ "Let $I$ be a set and let $\\mathcal{G}_i$ be a sheaf on $S_\\etale$", "for $i \\in I$.", "Let $U \\to X$ be an \\'etale morphism such that $U \\to S$ factors as", "$U \\to V \\to S$ with $V \\to S$ \\'etale and $U \\to V$ flat of finite", "presentation with geometrically connected fibres. Then we have", "\\begin{align*}", "f^{-1}(\\prod \\mathcal{G}_i)(U)", "& =", "(\\prod \\mathcal{G}_i)(V) \\\\", "& =", "\\prod \\mathcal{G}_i(V) \\\\", "& =", "\\prod f^{-1}\\mathcal{G}_i(U) \\\\", "& =", "(\\prod f^{-1}\\mathcal{G}_i)(U)", "\\end{align*}", "where we have used Lemma \\ref{lemma-sections-upstairs}", "in the first and third equality", "(we are also using that the restriction of $f^{-1}\\mathcal{G}$", "to $U_\\etale$ is the pullback via $U \\to V$ of the restriction of", "$\\mathcal{G}$ to $V_\\etale$, see", "Sites, Lemma \\ref{sites-lemma-localize-morphism-strong}).", "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-cover-by-geometrically-connected}", "every object $U$ of $X_\\etale$ has an \\'etale covering", "$\\{U_i \\to U\\}$ such that the discussion in the previous", "paragraph applies to $U_i$. The lemma follows." ], "refs": [ "etale-cohomology-lemma-sections-upstairs", "sites-lemma-localize-morphism-strong", "more-morphisms-lemma-cover-by-geometrically-connected" ], "ref_ids": [ 6439, 8572, 13915 ] } ], "ref_ids": [] }, { "id": 6604, "type": "theorem", "label": "etale-cohomology-lemma-base-change-f-star", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-f-star", "contents": [ "Let $f : X \\to S$ be a flat morphism of schemes such", "that for every geometric point $\\overline{x}$ of $X$ the map", "$$", "\\mathcal{O}_{S, f(\\overline{x})}^{sh}", "\\longrightarrow", "\\mathcal{O}_{X, \\overline{x}}^{sh}", "$$", "has geometrically connected fibres. Then for every", "cartesian diagram of schemes", "$$", "\\xymatrix{", "X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\", "S & T \\ar[l]_g", "}", "$$", "with $g$ quasi-compact and quasi-separated we have", "$f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$", "for any sheaf $\\mathcal{F}$ of sets on $T_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "It suffices to check equality on stalks, see", "Theorem \\ref{theorem-exactness-stalks}.", "By Theorem \\ref{theorem-higher-direct-images} we have", "$$", "(h_*e^{-1}\\mathcal{F})_{\\overline{x}} =", "\\Gamma(\\Spec(\\mathcal{O}_{X, \\overline{x}}^{sh}) \\times_X Y, e^{-1}\\mathcal{F})", "$$", "and we have similarly", "$$", "(f^{-1}g_*^{-1}\\mathcal{F})_{\\overline{x}} =", "(g_*^{-1}\\mathcal{F})_{f(\\overline{x})} =", "\\Gamma(\\Spec(\\mathcal{O}_{S, f(\\overline{x})}^{sh}) \\times_S T, \\mathcal{F})", "$$", "These sets are equal by an application of Lemma \\ref{lemma-sections-upstairs}", "to the morphism", "$$", "\\Spec(\\mathcal{O}_{X, \\overline{x}}^{sh}) \\times_X Y", "\\longrightarrow", "\\Spec(\\mathcal{O}_{S, f(\\overline{x})}^{sh}) \\times_S T", "$$", "which is a base change of", "$\\Spec(\\mathcal{O}_{X, \\overline{x}}^{sh}) \\to", "\\Spec(\\mathcal{O}_{S, f(\\overline{x})}^{sh})$", "because $Y = X \\times_S T$." ], "refs": [ "etale-cohomology-theorem-exactness-stalks", "etale-cohomology-theorem-higher-direct-images", "etale-cohomology-lemma-sections-upstairs" ], "ref_ids": [ 6376, 6385, 6439 ] } ], "ref_ids": [] }, { "id": 6605, "type": "theorem", "label": "etale-cohomology-lemma-base-change-q-injective", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-q-injective", "contents": [ "With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds}", "assume for some $q \\geq 1$ we have $BC(f, n, q - 1)$. Then", "for every commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & X' \\ar[l] \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\", "S & S' \\ar[l] & T \\ar[l]_g", "}", "$$", "with $X' = X \\times_S S'$ and $Y = X' \\times_{S'} T$ and", "$g$ quasi-compact and quasi-separated, and every abelian sheaf", "$\\mathcal{F}$ on $T_\\etale$ annihilated by $n$", "\\begin{enumerate}", "\\item the base change map", "$(f')^{-1}R^qg_*\\mathcal{F}\\to R^qh_*e^{-1}\\mathcal{F}$", "is injective,", "\\item if $\\mathcal{F} \\subset \\mathcal{G}$ where $\\mathcal{G}$", "on $T_\\etale$ is annihilated by $n$, then", "$$", "\\Coker\\left(", "(f')^{-1}R^qg_*\\mathcal{F}\\to R^qh_*e^{-1}\\mathcal{F}", "\\right)", "\\subset", "\\Coker\\left(", "(f')^{-1}R^qg_*\\mathcal{G}\\to R^qh_*e^{-1}\\mathcal{G}", "\\right)", "$$", "\\item if in (2) the sheaf $\\mathcal{G}$ is an injective sheaf", "of $\\mathbf{Z}/n\\mathbf{Z}$-modules, then", "$$", "\\Coker\\left((f')^{-1}R^qg_*\\mathcal{F}\\to R^qh_*e^{-1}\\mathcal{F} \\right)", "\\subset R^qh_*e^{-1}\\mathcal{G}", "$$", "\\end{enumerate}" ], "refs": [ "etale-cohomology-remark-base-change-holds" ], "proofs": [ { "contents": [ "Choose a short exact sequence", "$0 \\to \\mathcal{F} \\to \\mathcal{I} \\to \\mathcal{Q} \\to 0$", "where $\\mathcal{I}$ is an injective sheaf of $\\mathbf{Z}/n\\mathbf{Z}$-modules.", "Consider the induced diagram", "$$", "\\xymatrix{", "(f')^{-1}R^{q - 1}g_*\\mathcal{I} \\ar[d]_{\\cong} \\ar[r] &", "(f')^{-1}R^{q - 1}g_*\\mathcal{Q} \\ar[d]_{\\cong} \\ar[r] &", "(f')^{-1}R^qg_*\\mathcal{F} \\ar[d] \\ar[r] &", "0 \\ar[d] \\\\", "R^{q - 1}h_*e^{-1}\\mathcal{I} \\ar[r] &", "R^{q - 1}h_*e^{-1}\\mathcal{Q} \\ar[r] &", "R^qh_*e^{-1}\\mathcal{F} \\ar[r] &", "R^qh_*e^{-1}\\mathcal{I}", "}", "$$", "with exact rows. We have the zero in the right upper corner", "as $\\mathcal{I}$ is injective. The left two vertical arrows are", "isomorphisms by $BC(f, n, q - 1)$. We conclude that part (1) holds.", "The above also shows that", "$$", "\\Coker\\left(", "(f')^{-1}R^qg_*\\mathcal{F}\\to R^qh_*e^{-1}\\mathcal{F}", "\\right)", "\\subset", "R^qh_*e^{-1}\\mathcal{I}", "$$", "hence part (3) holds. To prove (2) choose", "$\\mathcal{F} \\subset \\mathcal{G} \\subset \\mathcal{I}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 6795 ] }, { "id": 6606, "type": "theorem", "label": "etale-cohomology-lemma-base-change-q-integral-top", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-q-integral-top", "contents": [ "With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds}", "assume for some $q \\geq 1$ we have $BC(f, n, q - 1)$. Consider", "commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "X \\ar[d]_f &", "X' \\ar[d]_{f'} \\ar[l] &", "Y \\ar[l]^h \\ar[d]^e &", "Y' \\ar[l]^{\\pi'} \\ar[d]^{e'} \\\\", "S &", "S' \\ar[l] &", "T \\ar[l]_g &", "T' \\ar[l]_\\pi", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "X' \\ar[d]_{f'} & & Y' \\ar[ll]^{h' = h \\circ \\pi'} \\ar[d]^{e'} \\\\", "S' & & T' \\ar[ll]_{g' = g \\circ \\pi}", "}", "}", "$$", "where all squares are cartesian, $g$ quasi-compact and quasi-separated, and", "$\\pi$ is integral surjective. Let $\\mathcal{F}$ be an abelian sheaf", "on $T_\\etale$ annihilated by $n$ and set $\\mathcal{F}' = \\pi^{-1}\\mathcal{F}$.", "If the base change map", "$$", "(f')^{-1}R^qg'_*\\mathcal{F}' \\longrightarrow R^qh'_*(e')^{-1}\\mathcal{F}'", "$$", "is an isomorphism, then the base change map", "$(f')^{-1}R^qg_*\\mathcal{F} \\to R^qh_*e^{-1}\\mathcal{F}$", "is an isomorphism." ], "refs": [ "etale-cohomology-remark-base-change-holds" ], "proofs": [ { "contents": [ "Observe that $\\mathcal{F} \\to \\pi_*\\pi^{-1}\\mathcal{F}'$ is injective", "as $\\pi$ is surjective (check on stalks). Thus by", "Lemma \\ref{lemma-base-change-q-injective}", "we see that it suffices to show that the base change map", "$$", "(f')^{-1}R^qg_*\\pi_*\\mathcal{F}'", "\\longrightarrow", "R^qh_*e^{-1}\\pi_*\\mathcal{F}'", "$$", "is an isomorphism. This follows from the assumption because", "we have $R^qg_*\\pi_*\\mathcal{F}' = R^qg'_*\\mathcal{F}'$,", "we have $e^{-1}\\pi_*\\mathcal{F}' =\\pi'_*(e')^{-1}\\mathcal{F}'$, and", "we have $R^qh_*\\pi'_*(e')^{-1}\\mathcal{F}' = R^qh'_*(e')^{-1}\\mathcal{F}'$.", "This follows from Lemmas", "\\ref{lemma-integral-pushforward-commutes-with-base-change} and", "\\ref{lemma-what-integral} and the relative leray spectral sequence", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-relative-Leray})." ], "refs": [ "etale-cohomology-lemma-base-change-q-injective", "etale-cohomology-lemma-integral-pushforward-commutes-with-base-change", "etale-cohomology-lemma-what-integral", "sites-cohomology-lemma-relative-Leray" ], "ref_ids": [ 6605, 6482, 6454, 4222 ] } ], "ref_ids": [ 6795 ] }, { "id": 6607, "type": "theorem", "label": "etale-cohomology-lemma-base-change-q-integral-bottom", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-q-integral-bottom", "contents": [ "With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds}", "assume for some $q \\geq 1$ we have $BC(f, n, q - 1)$. Consider", "commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "X \\ar[d]_f &", "X' \\ar[d]_{f'} \\ar[l] &", "X'' \\ar[l]^{\\pi'} \\ar[d]_{f''} &", "Y \\ar[l]^{h'} \\ar[d]^e \\\\", "S &", "S' \\ar[l] &", "S'' \\ar[l]_\\pi &", "T \\ar[l]_{g'}", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "X' \\ar[d]_{f'} & & Y \\ar[ll]^{h = h' \\circ \\pi'} \\ar[d]^e \\\\", "S' & & T \\ar[ll]_{g = g' \\circ \\pi}", "}", "}", "$$", "where all squares are cartesian, $g'$ quasi-compact and quasi-separated, and", "$\\pi$ is integral. Let $\\mathcal{F}$ be an abelian sheaf", "on $T_\\etale$ annihilated by $n$. If the base change map", "$$", "(f')^{-1}R^qg_*\\mathcal{F} \\longrightarrow R^qh_*e^{-1}\\mathcal{F}", "$$", "is an isomorphism, then the base change map", "$(f'')^{-1}R^qg'_*\\mathcal{F} \\to R^qh'_*e^{-1}\\mathcal{F}$", "is an isomorphism." ], "refs": [ "etale-cohomology-remark-base-change-holds" ], "proofs": [ { "contents": [ "Since $\\pi$ and $\\pi'$ are integral we have $R\\pi_* = \\pi_*$ and", "$R\\pi'_* = \\pi'_*$, see Lemma \\ref{lemma-what-integral}.", "We also have $(f')^{-1}\\pi_* = \\pi'_*(f'')^{-1}$. Thus we see that", "$\\pi'_*(f'')^{-1}R^qg'_*\\mathcal{F} = (f')^{-1}R^qg_*\\mathcal{F}$", "and", "$\\pi'_*R^qh'_*e^{-1}\\mathcal{F} = R^qh_*e^{-1}\\mathcal{F}$.", "Thus the assumption means that our map becomes an", "isomorphism after applying the functor $\\pi'_*$.", "Hence we see that it is an isomorphism by Lemma \\ref{lemma-what-integral}." ], "refs": [ "etale-cohomology-lemma-what-integral", "etale-cohomology-lemma-what-integral" ], "ref_ids": [ 6454, 6454 ] } ], "ref_ids": [ 6795 ] }, { "id": 6608, "type": "theorem", "label": "etale-cohomology-lemma-formal-argument", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-formal-argument", "contents": [ "Let $T$ be a quasi-compact and quasi-separated scheme.", "Let $P$ be a property for quasi-compact and quasi-separated", "schemes over $T$. Assume", "\\begin{enumerate}", "\\item If $T'' \\to T'$ is a thickening of quasi-compact and", "quasi-separated schemes over $T$, then $P(T'')$ if and only if $P(T')$.", "\\item If $T' = \\lim T_i$ is a limit of an inverse system of", "quasi-compact and quasi-separated schemes over $T$ with affine", "transition morphisms and $P(T_i)$ holds for all $i$, then", "$P(T')$ holds.", "\\item If $Z \\subset T'$ is a closed subscheme with", "quasi-compact complement $V \\subset T'$ and $P(T')$ holds,", "then either $P(V)$ or $P(Z)$ holds.", "\\end{enumerate}", "Then $P(T)$ implies $P(\\Spec(K))$ for some morphism $\\Spec(K) \\to T$", "where $K$ is a field." ], "refs": [], "proofs": [ { "contents": [ "Consider the set $\\mathfrak T$ of closed subschemes $T' \\subset T$", "such that $P(T')$. By assumption (2) this set has a minimal element,", "say $T'$. By assumption (1) we see that $T'$ is reduced.", "Let $\\eta \\in T'$ be the generic point of an irreducible", "component of $T'$. Then $\\eta = \\Spec(K)$ for some field", "$K$ and $\\eta = \\lim V$ where the limit is over the affine", "open subschemes $V \\subset T'$ containing $\\eta$.", "By assumption (3) and the minimality of $T'$ we see", "that $P(V)$ holds for all these $V$. Hence $P(\\eta)$", "by (2) and the proof is complete." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6609, "type": "theorem", "label": "etale-cohomology-lemma-base-change-does-not-hold-pre", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-does-not-hold-pre", "contents": [ "With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds}", "assume for some $q \\geq 1$ we have that $BC(f, n, q - 1)$ is true, but", "$BC(f, n, q)$ is not. Then there exist a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & X' \\ar[d]_{f'} \\ar[l] & Y \\ar[l]^h \\ar[d]^e \\\\", "S & S' \\ar[l] & \\Spec(K) \\ar[l]_g", "}", "$$", "where $X' = X \\times_S S'$, $Y = X' \\times_{S'} \\Spec(K)$,", "$K$ is a field, and $\\mathcal{F}$ is an abelian sheaf", "on $\\Spec(K)$ annihilated by $n$ such that", "$(f')^{-1}R^qg_*\\mathcal{F} \\to R^qh_*e^{-1}\\mathcal{F}$", "is not an isomorphism." ], "refs": [ "etale-cohomology-remark-base-change-holds" ], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & X' \\ar[l] \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\", "S & S' \\ar[l] & T \\ar[l]_g", "}", "$$", "with $X' = X \\times_S S'$ and $Y = X' \\times_{S'} T$ and", "$g$ quasi-compact and quasi-separated, and an abelian sheaf", "$\\mathcal{F}$ on $T_\\etale$ annihilated by $n$ such that", "the base change map", "$(f')^{-1}R^qg_*\\mathcal{F} \\to R^qh_*e^{-1}\\mathcal{F}$", "is not an isomorphism. Of course we may and do replace $S'$", "by an affine open of $S'$; this implies that $T$ is quasi-compact", "and quasi-separated. By Lemma \\ref{lemma-base-change-q-injective} we see", "$(f')^{-1}R^qg_*\\mathcal{F} \\to R^qh_*e^{-1}\\mathcal{F}$", "is injective. Pick a geometric point $\\overline{x}$ of $X'$", "and an element $\\xi$ of $(R^qh_*q^{-1}\\mathcal{F})_{\\overline{x}}$", "which is not in the image of the map", "$((f')^{-1}R^qg_*\\mathcal{F})_{\\overline{x}} \\to", "(R^qh_*e^{-1}\\mathcal{F})_{\\overline{x}}$.", "\\medskip\\noindent", "Consider a morphism $\\pi : T' \\to T$ with $T'$ quasi-compact", "and quasi-separated and denote $\\mathcal{F}' = \\pi^{-1}\\mathcal{F}$.", "Denote $\\pi' : Y' = Y \\times_T T' \\to Y$ the base change of $\\pi$", "and $e' : Y' \\to T'$ the base change of $e$. Picture", "$$", "\\vcenter{", "\\xymatrix{", "X' \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e & Y' \\ar[l]^{\\pi'} \\ar[d]^{e'} \\\\", "S' & T \\ar[l]_g & T' \\ar[l]_\\pi", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "X' \\ar[d]_{f'} & & Y' \\ar[ll]^{h' = h \\circ \\pi'} \\ar[d]^{e'} \\\\", "S' & & T' \\ar[ll]_{g' = g \\circ \\pi}", "}", "}", "$$", "Using pullback maps we obtain a canonical commutative diagram", "$$", "\\xymatrix{", "(f')^{-1}R^qg_*\\mathcal{F} \\ar[r] \\ar[d] &", "(f')^{-1}R^qg'_*\\mathcal{F}' \\ar[d] \\\\", "R^qh_*e^{-1}\\mathcal{F} \\ar[r] &", "R^qh'_*(e')^{-1}\\mathcal{F}'", "}", "$$", "of abelian sheaves on $X'$. Let $P(T')$ be the property", "\\begin{itemize}", "\\item The image $\\xi'$ of $\\xi$ in", "$(Rh'_*(e')^{-1}\\mathcal{F}')_{\\overline{x}}$ is", "not in the image of the map", "$(f^{-1}R^qg'_*\\mathcal{F}')_{\\overline{x}} \\to", "(R^qh'_*(e')^{-1}\\mathcal{F}')_{\\overline{x}}$.", "\\end{itemize}", "We claim that hypotheses (1), (2), and (3) of", "Lemma \\ref{lemma-formal-argument} hold for $P$", "which proves our lemma.", "\\medskip\\noindent", "Condition (1) of Lemma \\ref{lemma-formal-argument}", "holds for $P$ because the \\'etale topology of a scheme", "and a thickening of the scheme is the same. See", "Proposition \\ref{proposition-topological-invariance}.", "\\medskip\\noindent", "Suppose that $I$ is a directed set and that $T_i$", "is an inverse system over $I$ of quasi-compact and quasi-separated", "schemes over $T$ with affine transition morphisms.", "Set $T' = \\lim T_i$. Denote $\\mathcal{F}'$ and $\\mathcal{F}_i$", "the pullback of $\\mathcal{F}$ to $T'$, resp.\\ $T_i$. Consider", "the diagrams", "$$", "\\vcenter{", "\\xymatrix{", "X \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e & Y_i \\ar[l]^{\\pi_i'} \\ar[d]^{e_i} \\\\", "S & T \\ar[l]_g & T_i \\ar[l]_{\\pi_i}", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "X \\ar[d]_{f'} & & Y_i \\ar[ll]^{h_i = h \\circ \\pi_i'} \\ar[d]^{e_i} \\\\", "S & & T_i \\ar[ll]_{g_i = g \\circ \\pi_i}", "}", "}", "$$", "as in the previous paragraph. It is clear that $\\mathcal{F}'$ on", "$T'$ is the colimit of the pullbacks of $\\mathcal{F}_i$ to $T'$", "and that $(e')^{-1}\\mathcal{F}'$ is the colimit of the pullbacks", "of $e_i^{-1}\\mathcal{F}_i$ to $Y'$.", "By Lemma \\ref{lemma-relative-colimit-general}", "we have", "$$", "R^qh'_*(e')^{-1}\\mathcal{F}' = \\colim R^qh_{i, *}e_i^{-1}\\mathcal{F}_i", "\\quad\\text{and}\\quad", "(f')^{-1}R^qg'_*\\mathcal{F}' = \\colim (f')^{-1}R^qg_{i, *}\\mathcal{F}_i", "$$", "It follows that if $P(T_i)$ is true for all $i$, then", "$P(T')$ holds. Thus condition (2) of Lemma \\ref{lemma-formal-argument}", "holds for $P$.", "\\medskip\\noindent", "The most interesting is condition (3) of Lemma \\ref{lemma-formal-argument}.", "Assume $T'$ is a quasi-compact and quasi-separated scheme over $T$", "such that $P(T')$ is true.", "Let $Z \\subset T'$ be a closed subscheme with complement $V \\subset T'$", "quasi-compact. Consider the diagram", "$$", "\\xymatrix{", "Y' \\times_{T'} Z \\ar[d]_{e_Z} \\ar[r]_{i'} &", "Y' \\ar[d]_{e'} &", "Y' \\times_{T'} V \\ar[l]^{j'} \\ar[d]^{e_V} \\\\", "Z \\ar[r]^i &", "T' &", "V \\ar[l]_j", "}", "$$", "Choose an injective map $j^{-1}\\mathcal{F}' \\to \\mathcal{J}$", "where $\\mathcal{J}$ is an injective sheaf of $\\mathbf{Z}/n\\mathbf{Z}$-modules", "on $V$. Looking at stalks we see that the map", "$$", "\\mathcal{F}' \\to \\mathcal{G} = j_*\\mathcal{J} \\oplus i_*i^{-1}\\mathcal{F}'", "$$", "is injective. Thus $\\xi'$ maps to a nonzero element of", "\\begin{align*}", "& \\Coker\\left(", "((f')^{-1}R^qg'_*\\mathcal{G})_{\\overline{x}}", "\\to", "(R^qh'_*(e')^{-1}\\mathcal{G})_{\\overline{x}}", "\\right) = \\\\", "&", "\\Coker\\left(", "((f')^{-1}R^qg'_*j_*\\mathcal{J})_{\\overline{x}}", "\\to", "(R^qh'_*(e')^{-1}j_*\\mathcal{J})_{\\overline{x}}", "\\right) \\oplus \\\\", "& \\Coker\\left(", "((f')^{-1}R^qg'_*i_*i^{-1}\\mathcal{F}')_{\\overline{x}}", "\\to", "(R^qh'_*(e')^{-1}i_*i^{-1}\\mathcal{F}')_{\\overline{x}}", "\\right)", "\\end{align*}", "by part (2) of Lemma \\ref{lemma-base-change-q-injective}.", "If $\\xi'$ does not map to zero in the second summand, then", "we use", "$$", "(f')^{-1}R^qg'_*i_*i^{-1}\\mathcal{F}' =", "(f')^{-1}R^q(g' \\circ i)_*i^{-1}\\mathcal{F}'", "$$", "(because $Ri_* = i_*$ by", "Proposition \\ref{proposition-finite-higher-direct-image-zero}) and", "$$", "R^qh'_*(e')^{-1}i_*i^{-1}\\mathcal{F} =", "R^qh'_*i'_*e_Z^{-1}i^{-1}\\mathcal{F} =", "R^q(h' \\circ i')_*e_Z^{-1}i^{-1}\\mathcal{F}'", "$$", "(first equality by", "Lemma \\ref{lemma-finite-pushforward-commutes-with-base-change}", "and the second because", "$Ri'_* = i'_*$ by", "Proposition \\ref{proposition-finite-higher-direct-image-zero})", "to we see that we have $P(Z)$.", "Finally, suppose $\\xi'$ does not map to zero in the first summand.", "We have", "$$", "(e')^{-1}j_*\\mathcal{J} = j'_*e_V^{-1}\\mathcal{J}", "\\quad\\text{and}\\quad", "R^aj'_*e_V^{-1}\\mathcal{J} = 0, \\quad a = 1, \\ldots, q - 1", "$$", "by $BC(f, n, q - 1)$ applied to the diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & Y' \\ar[l] \\ar[d]_{e'} & Y \\ar[l]^{j'} \\ar[d]^{e_V} \\\\", "S & T' \\ar[l] & V \\ar[l]_j", "}", "$$", "and the fact that $\\mathcal{J}$ is injective.", "By the relative Leray spectral sequence for $h' \\circ j'$", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-relative-Leray})", "we deduce that", "$$", "R^qh'_*(e')^{-1}j_*\\mathcal{J} =", "R^qh'_*j'_*e_V^{-1}\\mathcal{J}", "\\longrightarrow", "R^q(h' \\circ j')_* e_V^{-1}\\mathcal{J}", "$$", "is injective. Thus $\\xi$ maps to a nonzero element of", "$(R^q(h' \\circ j')_* e_V^{-1}\\mathcal{J})_{\\overline{x}}$.", "Applying part (3) of Lemma \\ref{lemma-base-change-q-injective}", "to the injection $j^{-1}\\mathcal{F}' \\to \\mathcal{J}$", "we conclude that $P(V)$ holds." ], "refs": [ "etale-cohomology-lemma-base-change-q-injective", "etale-cohomology-lemma-formal-argument", "etale-cohomology-lemma-formal-argument", "etale-cohomology-proposition-topological-invariance", "etale-cohomology-lemma-relative-colimit-general", "etale-cohomology-lemma-formal-argument", "etale-cohomology-lemma-formal-argument", "etale-cohomology-lemma-base-change-q-injective", "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-lemma-finite-pushforward-commutes-with-base-change", "etale-cohomology-proposition-finite-higher-direct-image-zero", "sites-cohomology-lemma-relative-Leray", "etale-cohomology-lemma-base-change-q-injective" ], "ref_ids": [ 6605, 6608, 6608, 6699, 6476, 6608, 6608, 6605, 6703, 6481, 6703, 4222, 6605 ] } ], "ref_ids": [ 6795 ] }, { "id": 6610, "type": "theorem", "label": "etale-cohomology-lemma-base-change-does-not-hold", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-does-not-hold", "contents": [ "With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds}", "assume for some $q \\geq 1$ we have that", "$BC(f, n, q - 1)$ is true, but $BC(f, n, q)$ is not.", "Then there exist a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & X' \\ar[d] \\ar[l] & Y \\ar[l]^h \\ar[d] \\\\", "S & S' \\ar[l] & \\Spec(K) \\ar[l]", "}", "$$", "with both squares cartesian, where", "\\begin{enumerate}", "\\item $S'$ is affine, integral, and normal with algebraically", "closed function field,", "\\item $K$ is algebraically closed and $\\Spec(K) \\to S'$", "is dominant (in other words $K$ is an extension of", "the function field of $S'$)", "\\end{enumerate}", "and there exists an integer $d | n$", "such that $R^qh_*(\\mathbf{Z}/d\\mathbf{Z})$ is nonzero." ], "refs": [ "etale-cohomology-remark-base-change-holds" ], "proofs": [ { "contents": [ "First choose a diagram and $\\mathcal{F}$ as in", "Lemma \\ref{lemma-base-change-does-not-hold-pre}.", "We may and do assume $S'$ is affine (this is obvious, but", "see proof of the lemma in case of doubt).", "By Lemma \\ref{lemma-base-change-q-integral-top}", "we may assume $K$ is algebraically closed.", "Then $\\mathcal{F}$ corresponds to a $\\mathbf{Z}/n\\mathbf{Z}$-module.", "Such a modules is a direct sum of copies of $\\mathbf{Z}/d\\mathbf{Z}$", "for varying $d | n$ hence we may assume $\\mathcal{F}$ is", "constant with value $\\mathbf{Z}/d\\mathbf{Z}$.", "By Lemma \\ref{lemma-base-change-q-integral-bottom}", "we may replace $S'$ by the normalization", "of $S'$ in $\\Spec(K)$ which finishes the proof." ], "refs": [ "etale-cohomology-lemma-base-change-does-not-hold-pre", "etale-cohomology-lemma-base-change-q-integral-top", "etale-cohomology-lemma-base-change-q-integral-bottom" ], "ref_ids": [ 6609, 6606, 6607 ] } ], "ref_ids": [ 6795 ] }, { "id": 6611, "type": "theorem", "label": "etale-cohomology-lemma-smooth-base-change-fields", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-smooth-base-change-fields", "contents": [ "Let $K/k$ be an extension of fields. Let $X$ be a smooth affine curve", "over $k$ with a rational point $x \\in X(k)$. Let $\\mathcal{F}$ be an abelian", "sheaf on $\\Spec(K)$ annihilated by an integer $n$ invertible in $k$.", "Let $q > 0$ and", "$$", "\\xi \\in H^q(X_K, (X_K \\to \\Spec(K))^{-1}\\mathcal{F})", "$$", "There exist", "\\begin{enumerate}", "\\item finite extensions $K'/K$ and $k'/k$ with $k' \\subset K'$,", "\\item a finite \\'etale Galois cover $Z \\to X_{k'}$ with group $G$", "\\end{enumerate}", "such that the order of $G$ divides a power of $n$, such that", "$Z \\to X_{k'}$ is split over $x_{k'}$, and", "such that $\\xi$ dies in $H^q(Z_{K'}, (Z_{K'} \\to \\Spec(K))^{-1}\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "For $q > 1$ we know that $\\xi$ dies in", "$H^q(X_{\\overline{K}}, (X_{\\overline{K}} \\to \\Spec(K))^{-1}\\mathcal{F})$", "(Theorem \\ref{theorem-vanishing-affine-curves}).", "By Lemma \\ref{lemma-directed-colimit-cohomology} we see that", "this means there is a finite extension $K'/K$ such that", "$\\xi$ dies in $H^q(X_{K'}, (X_{K'} \\to \\Spec(K))^{-1}\\mathcal{F})$.", "Thus we can take $k' = k$ and $Z = X$ in this case.", "\\medskip\\noindent", "Assume $q = 1$. Recall that $\\mathcal{F}$ corresponds to a", "discrete module $M$ with continuous $\\text{Gal}_K$-action, see", "Lemma \\ref{lemma-equivalence-abelian-sheaves-point}.", "Since $M$ is $n$-torsion, it is the uninon of finite", "$\\text{Gal}_K$-stable subgroups. Thus", "we reduce to the case where $M$ is a finite abelian group annihilated by $n$,", "see Lemma \\ref{lemma-colimit}. After replacing $K$ by a finite extension", "we may assume that the action of $\\text{Gal}_K$ on $M$ is trivial.", "Thus we may assume $\\mathcal{F} = \\underline{M}$ is the constant", "sheaf with value a finite abelian group $M$ annihilated by $n$.", "\\medskip\\noindent", "We can write $M$ as a direct sum of cyclic groups.", "Any two finite \\'etale Galois coverings whose Galois groups", "have order invertible in $k$, can be dominated by a third one whose", "Galois group has order invertible in $k$", "(Fundamental Groups, Section \\ref{pione-section-finite-etale-under-galois}).", "Thus it suffices to prove the lemma when", "$M = \\mathbf{Z}/d\\mathbf{Z}$ where $d | n$.", "\\medskip\\noindent", "Assume $M = \\mathbf{Z}/d\\mathbf{Z}$ where $d | n$.", "In this case $\\overline{\\xi} = \\xi|_{X_{\\overline{K}}}$ is an element of", "$$", "H^1(X_{\\overline{k}}, \\mathbf{Z}/d\\mathbf{Z}) =", "H^1(X_{\\overline{K}}, \\mathbf{Z}/d\\mathbf{Z})", "$$", "See Theorem \\ref{theorem-vanishing-affine-curves}.", "This group classifies $\\mathbf{Z}/d\\mathbf{Z}$-torsors, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-torsors-h1}.", "The torsor corresponding to $\\overline{\\xi}$ (viewed as a sheaf on", "$X_{\\overline{k}, \\etale}$) in turn gives rise to a finite \\'etale morphism", "$T \\to X_{\\overline{k}}$ endowed an action of $\\mathbf{Z}/d\\mathbf{Z}$", "transitive on the fibre of $T$ over $x_{\\overline{k}}$, see", "Lemma \\ref{lemma-characterize-finite-locally-constant}.", "Choose a connected component $T' \\subset T$ (if $\\overline{\\xi}$ has", "order $d$, then $T$ is already connected).", "Then $T' \\to X_{\\overline{k}}$ is a finite \\'etale Galois", "cover whose Galois group is a subgroup $G \\subset \\mathbf{Z}/d\\mathbf{Z}$", "(small detail omitted). Moreover the element $\\overline{\\xi}$ maps to zero", "under the map $H^1(X_{\\overline{k}}, \\mathbf{Z}/d\\mathbf{Z}) \\to", "H^1(T', \\mathbf{Z}/d\\mathbf{Z})$ as this is one", "of the defining properties of $T$.", "\\medskip\\noindent", "Next, we use a limit argument to choose a finite extension $k'/k$", "contained in $\\overline{k}$ such that $T' \\to X_{\\overline{k}}$", "descends to a finite \\'etale Galois cover $Z \\to X_{k'}$ with group $G$.", "See Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation},", "\\ref{limits-lemma-descend-finite-finite-presentation}, and", "\\ref{limits-lemma-descend-etale}.", "After increasing $k'$ we may assume that $Z$ splits over $x_{k'}$.", "The image of $\\xi$ in", "$H^1(Z_{\\overline{K}}, \\mathbf{Z}/d\\mathbf{Z})$ is zero by construction.", "Thus by Lemma \\ref{lemma-directed-colimit-cohomology}", "we can find a finite subextension $\\overline{K}/K'/K$", "containing $k'$ such that $\\xi$ dies in $H^1(Z_{K'}, \\mathbf{Z}/d\\mathbf{Z})$", "and this finishes the proof." ], "refs": [ "etale-cohomology-theorem-vanishing-affine-curves", "etale-cohomology-lemma-directed-colimit-cohomology", "etale-cohomology-lemma-equivalence-abelian-sheaves-point", "etale-cohomology-lemma-colimit", "etale-cohomology-theorem-vanishing-affine-curves", "sites-cohomology-lemma-torsors-h1", "etale-cohomology-lemma-characterize-finite-locally-constant", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-finite-finite-presentation", "limits-lemma-descend-etale", "etale-cohomology-lemma-directed-colimit-cohomology" ], "ref_ids": [ 6394, 6473, 6489, 6472, 6394, 4182, 6503, 15077, 15058, 15065, 6473 ] } ], "ref_ids": [] }, { "id": 6612, "type": "theorem", "label": "etale-cohomology-lemma-smooth-base-change-general", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-smooth-base-change-general", "contents": [ "Let $S$ be a scheme. Let $S' = \\lim S_i$ be a directed inverse", "limit of schemes $S_i$ smooth over $S$ with affine transition", "morphisms. Let $f : X \\to S$ be quas-compact and quasi-separated", "and form the fibre square", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "Then", "$$", "g^{-1}Rf_*E = R(f')_*(g')^{-1}E", "$$", "for any $E \\in D^+(X_\\etale)$ whose cohomology sheaves $H^q(E)$", "have stalks which are torsion of orders invertible on $S$." ], "refs": [], "proofs": [ { "contents": [ "Consider the spectral sequences", "$$", "E_2^{p, q} = R^pf_*H^q(E)", "\\quad\\text{and}\\quad", "{E'}_2^{p, q} = R^pf'_*H^q((g')^{-1}E) = R^pf'_*(g')^{-1}H^q(E)", "$$", "converging to $R^nf_*E$ and $R^nf'_*(g')^{-1}E$.", "These spectral sequences are constructed in", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "Combining the smooth base change theorem", "(Theorem \\ref{theorem-smooth-base-change})", "with Lemma \\ref{lemma-base-change-Rf-star-colim} we see that", "$$", "g^{-1}R^pf_*H^q(E) = R^p(f')_*(g')^{-1}H^q(E)", "$$", "Combining all of the above we get the lemma." ], "refs": [ "derived-lemma-two-ss-complex-functor", "etale-cohomology-theorem-smooth-base-change", "etale-cohomology-lemma-base-change-Rf-star-colim" ], "ref_ids": [ 1871, 6396, 6596 ] } ], "ref_ids": [] }, { "id": 6613, "type": "theorem", "label": "etale-cohomology-lemma-base-change-field-extension", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-field-extension", "contents": [ "Let $L/K$ be an extension of fields. Let $g : T \\to S$ be a quasi-compact", "and quasi-separated morphism of schemes over $K$. Denote", "$g_L : T_L \\to S_L$ the base change of $g$ to $\\Spec(L)$.", "Let $E \\in D^+(T_\\etale)$ have cohomology sheaves whose stalks", "are torsion of orders invertible in $K$. Let $E_L$ be the", "pullback of $E$ to $(T_L)_\\etale$. Then", "$Rg_{L, *}E_L$ is the pullback of $Rg_*E$ to $S_L$." ], "refs": [], "proofs": [ { "contents": [ "If $L/K$ is separable, then $L$ is a filtered colimit of smooth", "$K$-algebras, see", "Algebra, Lemma \\ref{algebra-lemma-colimit-syntomic}.", "Thus the lemma in this case follows immediately from", "Lemma \\ref{lemma-smooth-base-change-general}.", "In the general case, let $K'$ and $L'$ be the perfect closures", "(Algebra, Definition \\ref{algebra-definition-perfection})", "of $K$ and $L$. Then $\\Spec(K') \\to \\Spec(K)$ and", "$\\Spec(L') \\to \\Spec(L)$ are universal homeomorphisms", "as $K'/K$ and $L'/L$ are purely inseparable", "(see Algebra, Lemma \\ref{algebra-lemma-p-ring-map}).", "Thus we have $(T_{K'})_\\etale = T_\\etale$,", "$(S_{K'})_\\etale = S_\\etale$,", "$(T_{L'})_\\etale = (T_L)\\etale$, and", "$(S_{L'})_\\etale = (S_L)_\\etale$ by", "the topological invariance of \\'etale cohomology, see", "Proposition \\ref{proposition-topological-invariance}.", "This reduces the lemma to the case of the field", "extension $L'/K'$ which is separable (by definition of", "perfect fields, see Algebra, Definition \\ref{algebra-definition-perfect})." ], "refs": [ "algebra-lemma-colimit-syntomic", "etale-cohomology-lemma-smooth-base-change-general", "algebra-definition-perfection", "algebra-lemma-p-ring-map", "etale-cohomology-proposition-topological-invariance", "algebra-definition-perfect" ], "ref_ids": [ 1323, 6612, 1463, 582, 6699, 1462 ] } ], "ref_ids": [] }, { "id": 6614, "type": "theorem", "label": "etale-cohomology-lemma-smooth-base-change-separably-closed", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-smooth-base-change-separably-closed", "contents": [ "Let $K/k$ be an extension of separably closed fields. Let $X$", "be a quasi-compact and quasi-separated scheme over $k$.", "Let $E \\in D^+(X_\\etale)$ have cohomology sheaves whose stalks", "are torsion of orders invertible in $k$. Then", "\\begin{enumerate}", "\\item the maps $H^q_\\etale(X, E) \\to H^q_\\etale(X_K, E|_{X_K})$", "are isomorphisms, and", "\\item $E \\to R(X_K \\to X)_*E|_{X_K}$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). First let $\\overline{k}$ and $\\overline{K}$", "be the algebraic closures", "of $k$ and $K$. The morphisms $\\Spec(\\overline{k}) \\to \\Spec(k)$ and", "$\\Spec(\\overline{K}) \\to \\Spec(K)$ are universal homeomorphisms", "as $\\overline{k}/k$ and $\\overline{K}/K$ are purely inseparable", "(see Algebra, Lemma \\ref{algebra-lemma-p-ring-map}).", "Thus $H^q_\\etale(X, \\mathcal{F}) =", "H^q_\\etale(X_{\\overline{k}}, \\mathcal{F}_{X_{\\overline{k}}})$ by", "the topological invariance of \\'etale cohomology, see", "Proposition \\ref{proposition-topological-invariance}.", "Similarly for $X_K$ and $X_{\\overline{K}}$.", "Thus we may assume $k$ and $K$ are algebraically closed.", "In this case $K$ is a limit of smooth $k$-algebras, see", "Algebra, Lemma \\ref{algebra-lemma-colimit-syntomic}.", "We conclude our lemma is a special case of", "Theorem \\ref{theorem-smooth-base-change} as reformulated in", "Lemma \\ref{lemma-smooth-base-change-general}.", "\\medskip\\noindent", "Proof of (2). For any quasi-compact and quasi-separated $U$ in $X_\\etale$", "the above shows that the restriction of the map", "$E \\to R(X_K \\to X)_*E|_{X_K}$ determines an isomorphism on cohomology.", "Since every object of $X_\\etale$ has an \\'etale covering by such $U$", "this proves the desired statement." ], "refs": [ "algebra-lemma-p-ring-map", "etale-cohomology-proposition-topological-invariance", "algebra-lemma-colimit-syntomic", "etale-cohomology-theorem-smooth-base-change", "etale-cohomology-lemma-smooth-base-change-general" ], "ref_ids": [ 582, 6699, 1323, 6396, 6612 ] } ], "ref_ids": [] }, { "id": 6615, "type": "theorem", "label": "etale-cohomology-lemma-base-change-does-not-hold-post", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-does-not-hold-post", "contents": [ "With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds}", "assume $n$ is invertible on $S$ and that for some $q \\geq 1$", "we have that $BC(f, n, q - 1)$ is true, but $BC(f, n, q)$ is not.", "Then there exist a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & X' \\ar[d] \\ar[l] & Y \\ar[l]^h \\ar[d] \\\\", "S & S' \\ar[l] & \\Spec(K) \\ar[l]", "}", "$$", "with both squares cartesian, where $S'$ is affine, integral, and normal", "with algebraically closed function field $K$ and there exists an integer", "$d | n$ such that $R^qh_*(\\mathbf{Z}/d\\mathbf{Z})$ is nonzero." ], "refs": [ "etale-cohomology-remark-base-change-holds" ], "proofs": [ { "contents": [ "First choose a diagram and $\\mathcal{F}$ as in", "Lemma \\ref{lemma-base-change-does-not-hold}.", "We may and do assume $S'$ is affine (this is obvious, but", "see proof of the lemma in case of doubt).", "Let $K'$ be the function field of $S'$", "and let $Y' = X' \\times_{S'} \\Spec(K')$ to get the diagram", "$$", "\\xymatrix{", "X \\ar[d]_f &", "X' \\ar[d] \\ar[l] &", "Y' \\ar[l]^{h'} \\ar[d] &", "Y \\ar[l] \\ar[d] \\\\", "S &", "S' \\ar[l] &", "\\Spec(K') \\ar[l] &", "\\Spec(K) \\ar[l]", "}", "$$", "By Lemma \\ref{lemma-smooth-base-change-separably-closed}", "the total direct image $R(Y \\to Y')_*\\mathbf{Z}/d\\mathbf{Z}$", "is isomorphic to $\\mathbf{Z}/d\\mathbf{Z}$ in $D(Y'_\\etale)$; here", "we use that $n$ is invertible on $S$.", "Thus $Rh'_*\\mathbf{Z}/d\\mathbf{Z} = Rh_*\\mathbf{Z}/d\\mathbf{Z}$", "by the relative Leray spectral sequence. This finishes the proof." ], "refs": [ "etale-cohomology-lemma-base-change-does-not-hold", "etale-cohomology-lemma-smooth-base-change-separably-closed" ], "ref_ids": [ 6610, 6614 ] } ], "ref_ids": [ 6795 ] }, { "id": 6616, "type": "theorem", "label": "etale-cohomology-lemma-zariski-h0-proper-over-henselian-pair", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-zariski-h0-proper-over-henselian-pair", "contents": [ "Let $(A, I)$ be a henselian pair. Let $f : X \\to \\Spec(A)$ be a proper morphism", "of schemes. Let $Z = X \\times_{\\Spec(A)} \\Spec(A/I)$. For any", "sheaf $\\mathcal{F}$ on the topological space associated to $X$ we", "have $\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, \\mathcal{F}|_Z)$." ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-h0-topological} to prove this. First observe", "that the underlying topological space of $X$ is spectral by Properties, Lemma", "\\ref{properties-lemma-quasi-compact-quasi-separated-spectral}.", "Let $Y \\subset X$ be an irreducible closed subscheme. To finish the proof", "we show that $Y \\cap Z = Y \\times_{\\Spec(A)} \\Spec(A/I)$ is connected.", "Replacing $X$ by $Y$", "we may assume that $X$ is irreducible and we have to show that $Z$", "is connected. Let $X \\to \\Spec(B) \\to \\Spec(A)$ be the Stein factorization", "of $f$ (More on Morphisms, Theorem", "\\ref{more-morphisms-theorem-stein-factorization-general}).", "Then $A \\to B$ is integral and $(B, IB)$ is a henselian pair", "(More on Algebra, Lemma \\ref{more-algebra-lemma-integral-over-henselian-pair}).", "Thus we may assume the fibres of $X \\to \\Spec(A)$ are geometrically", "connected. On the other hand, the image $T \\subset \\Spec(A)$ of $f$", "is irreducible and closed as $X$ is proper over $A$. Hence $T \\cap V(I)$", "is connected by More on Algebra, Lemma", "\\ref{more-algebra-lemma-irreducible-henselian-pair-connected}.", "Now $Y \\times_{\\Spec(A)} \\Spec(A/I) \\to T \\cap V(I)$", "is a surjective closed map with connected fibres.", "The result now follows from Topology, Lemma", "\\ref{topology-lemma-connected-fibres-quotient-topology-connected-components}." ], "refs": [ "etale-cohomology-lemma-h0-topological", "properties-lemma-quasi-compact-quasi-separated-spectral", "more-morphisms-theorem-stein-factorization-general", "more-algebra-lemma-integral-over-henselian-pair", "more-algebra-lemma-irreducible-henselian-pair-connected", "topology-lemma-connected-fibres-quotient-topology-connected-components" ], "ref_ids": [ 6582, 2941, 13675, 9863, 9870, 8207 ] } ], "ref_ids": [] }, { "id": 6617, "type": "theorem", "label": "etale-cohomology-lemma-h0-proper-over-henselian-pair", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-h0-proper-over-henselian-pair", "contents": [ "Let $(A, I)$ be a henselian pair. Let $f : X \\to \\Spec(A)$ be a proper morphism", "of schemes. Let $i : Z \\to X$ be the closed immersion of", "$X \\times_{\\Spec(A)} \\Spec(A/I)$ into $X$. For any", "sheaf $\\mathcal{F}$ on $X_\\etale$ we", "have $\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, i_{small}^{-1}\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-gabber-h0} and", "\\ref{lemma-zariski-h0-proper-over-henselian-pair}", "and the fact that any scheme finite over $X$ is proper over $\\Spec(A)$." ], "refs": [ "etale-cohomology-lemma-gabber-h0", "etale-cohomology-lemma-zariski-h0-proper-over-henselian-pair" ], "ref_ids": [ 6580, 6616 ] } ], "ref_ids": [] }, { "id": 6618, "type": "theorem", "label": "etale-cohomology-lemma-h0-proper-over-henselian-local", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-h0-proper-over-henselian-local", "contents": [ "Let $A$ be a henselian local ring. Let $f : X \\to \\Spec(A)$", "be a proper morphism of schemes. Let $X_0 \\subset X$ be the fibre of", "$f$ over the closed point. For any sheaf $\\mathcal{F}$ on $X_\\etale$ we", "have $\\Gamma(X, \\mathcal{F}) = \\Gamma(X_0, \\mathcal{F}|_{X_0})$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-h0-proper-over-henselian-pair}." ], "refs": [ "etale-cohomology-lemma-h0-proper-over-henselian-pair" ], "ref_ids": [ 6617 ] } ], "ref_ids": [] }, { "id": 6619, "type": "theorem", "label": "etale-cohomology-lemma-proper-pushforward-stalk", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-proper-pushforward-stalk", "contents": [ "Let $f : X \\to S$ be a proper morphism of schemes. Let", "$\\overline{s} \\to S$ be a geometric point.", "For any sheaf $\\mathcal{F}$ on $X_\\etale$", "the canonical map", "$$", "(f_*\\mathcal{F})_{\\overline{s}} \\longrightarrow", "\\Gamma(X_{\\overline{s}}, \\mathcal{F}_{\\overline{s}})", "$$", "is bijective." ], "refs": [], "proofs": [ { "contents": [ "By Theorem \\ref{theorem-higher-direct-images} (for sheaves of sets)", "we have", "$$", "(f_*\\mathcal{F})_{\\overline{s}} =", "\\Gamma(X \\times_S \\Spec(\\mathcal{O}_{S, \\overline{s}}^{sh}),", "p_{small}^{-1}\\mathcal{F})", "$$", "where $p : X \\times_S \\Spec(\\mathcal{O}_{S, \\overline{s}}^{sh}) \\to X$", "is the projection. Since the residue field of the strictly henselian", "local ring $\\mathcal{O}_{S, \\overline{s}}^{sh}$ is $\\kappa(s)^{sep}$", "we conclude from the discussion above the lemma and", "Lemma \\ref{lemma-h0-proper-over-henselian-local}." ], "refs": [ "etale-cohomology-theorem-higher-direct-images", "etale-cohomology-lemma-h0-proper-over-henselian-local" ], "ref_ids": [ 6385, 6618 ] } ], "ref_ids": [] }, { "id": 6620, "type": "theorem", "label": "etale-cohomology-lemma-proper-base-change-f-star", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-proper-base-change-f-star", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$", "be a morphism of schemes. Set $X' = Y' \\times_Y X$ with projections", "$f' : X' \\to Y'$ and $g' : X' \\to X$. Let $\\mathcal{F}$ be any sheaf on", "$X_\\etale$. Then $g^{-1}f_*\\mathcal{F} = f'_*(g')^{-1}\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "There is a canonical map $g^{-1}f_*\\mathcal{F} \\to f'_*(g')^{-1}\\mathcal{F}$.", "Namely, it is adjoint to the map", "$$", "f_*\\mathcal{F} \\longrightarrow", "g_*f'_*(g')^{-1}\\mathcal{F} = f_*g'_*(g')^{-1}\\mathcal{F}", "$$", "which is $f_*$ applied to the canonical map", "$\\mathcal{F} \\to g'_*(g')^{-1}\\mathcal{F}$. To check this map is an", "isomorphism we can compute what happens on stalks.", "Let $y' : \\Spec(k) \\to Y'$ be a geometric point with image $y$ in $Y$.", "By Lemma \\ref{lemma-proper-pushforward-stalk} the stalks are", "$\\Gamma(X'_{y'}, \\mathcal{F}_{y'})$ and $\\Gamma(X_y, \\mathcal{F}_y)$", "respectively. Here the sheaves $\\mathcal{F}_y$ and $\\mathcal{F}_{y'}$", "are the pullbacks of $\\mathcal{F}$ by the projections $X_y \\to X$", "and $X'_{y'} \\to X$. Thus we see that the groups agree by", "Lemma \\ref{lemma-sections-base-field-extension}. We omit the", "verification that this isomorphism is compatible with our map." ], "refs": [ "etale-cohomology-lemma-proper-pushforward-stalk", "etale-cohomology-lemma-sections-base-field-extension" ], "ref_ids": [ 6619, 6441 ] } ], "ref_ids": [] }, { "id": 6621, "type": "theorem", "label": "etale-cohomology-lemma-proper-base-change-in-terms-of-injectives", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-proper-base-change-in-terms-of-injectives", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item cohomology commutes with base change for $f$ (see above),", "\\item for every prime number $\\ell$ and every injective", "sheaf of $\\mathbf{Z}/\\ell\\mathbf{Z}$-modules $\\mathcal{I}$", "on $X_\\etale$ and every diagram (\\ref{equation-base-change-diagram})", "where $X' = Y' \\times_Y X$ the sheaves", "$R^qf'_*(g')^{-1}\\mathcal{I}$ are zero for $q > 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2). Conversely, assume (2) and let", "$\\mathcal{F}$ be a torsion abelian sheaf on $X_\\etale$. Let $Y' \\to Y$", "be a morphism of schemes and let $X' = Y' \\times_Y X$", "with projections $g' : X' \\to X$ and $f' : X' \\to Y'$ as in", "diagram (\\ref{equation-base-change-diagram}).", "We want to show the maps of sheaves", "$$", "g^{-1}R^qf_*\\mathcal{F} \\longrightarrow R^qf'_*(g')^{-1}\\mathcal{F}", "$$", "are isomorphisms for all $q \\geq 0$.", "\\medskip\\noindent", "For every $n \\geq 1$, let $\\mathcal{F}[n]$ be the subsheaf of sections", "of $\\mathcal{F}$ annihilated by $n$. Then", "$\\mathcal{F} = \\colim \\mathcal{F}[n]$.", "The functors $g^{-1}$ and $(g')^{-1}$ commute with arbitrary colimits", "(as left adjoints). Taking higher direct images along $f$ or $f'$", "commutes with filtered colimits by Lemma \\ref{lemma-relative-colimit}.", "Hence we see that", "$$", "g^{-1}R^qf_*\\mathcal{F} = \\colim g^{-1}R^qf_*\\mathcal{F}[n]", "\\quad\\text{and}\\quad", "R^qf'_*(g')^{-1}\\mathcal{F} =", "\\colim R^qf'_*(g')^{-1}\\mathcal{F}[n]", "$$", "Thus it suffices to prove the result in case $\\mathcal{F}$ is", "annihilated by a positive integer $n$.", "\\medskip\\noindent", "If $n = \\ell n'$ for some prime number $\\ell$, then we obtain a short", "exact sequence", "$$", "0 \\to \\mathcal{F}[\\ell] \\to \\mathcal{F} \\to", "\\mathcal{F}/\\mathcal{F}[\\ell] \\to 0", "$$", "Observe that $\\mathcal{F}/\\mathcal{F}[\\ell]$ is annihilated by $n'$.", "Moreover, if the result holds for both $\\mathcal{F}[\\ell]$ and", "$\\mathcal{F}/\\mathcal{F}[\\ell]$, then the result holds by", "the long exact sequence of higher direct images (and the $5$ lemma).", "In this way we reduce to the case that $\\mathcal{F}$ is annihilated", "by a prime number $\\ell$.", "\\medskip\\noindent", "Assume $\\mathcal{F}$ is annihilated by a prime number $\\ell$.", "Choose an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$", "in $D(X_\\etale, \\mathbf{Z}/\\ell\\mathbf{Z})$. Applying assumption", "(2) and Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "we see that", "$$", "f'_*(g')^{-1}\\mathcal{I}^\\bullet", "$$", "computes $Rf'_*(g')^{-1}\\mathcal{F}$. We conclude by applying", "Lemma \\ref{lemma-proper-base-change-f-star}." ], "refs": [ "etale-cohomology-lemma-relative-colimit", "derived-lemma-leray-acyclicity", "etale-cohomology-lemma-proper-base-change-f-star" ], "ref_ids": [ 6475, 1844, 6620 ] } ], "ref_ids": [] }, { "id": 6622, "type": "theorem", "label": "etale-cohomology-lemma-sandwich", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-sandwich", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms of schemes. Assume", "\\begin{enumerate}", "\\item cohomology commutes with base change for $f$,", "\\item cohomology commutes with base change for $g \\circ f$, and", "\\item $f$ is surjective.", "\\end{enumerate}", "Then cohomology commutes with base change for $g$." ], "refs": [], "proofs": [ { "contents": [ "We will use the equivalence of", "Lemma \\ref{lemma-proper-base-change-in-terms-of-injectives}", "without further mention. Let $\\ell$ be a prime number.", "Let $\\mathcal{I}$ be an injective sheaf of", "$\\mathbf{Z}/\\ell\\mathbf{Z}$-modules on $Y_\\etale$.", "Choose an injective map of sheaves $f^{-1}\\mathcal{I} \\to \\mathcal{J}$", "where $\\mathcal{J}$ is an injective sheaf of", "$\\mathbf{Z}/\\ell\\mathbf{Z}$-modules on $Z_\\etale$.", "Since $f$ is surjective the map $\\mathcal{I} \\to f_*\\mathcal{J}$", "is injective (look at stalks in geometric points).", "Since $\\mathcal{I}$ is injective we see that $\\mathcal{I}$", "is a direct summand of $f_*\\mathcal{J}$. Thus it suffices", "to prove the desired vanishing for $f_*\\mathcal{J}$.", "\\medskip\\noindent", "Let $Z' \\to Z$ be a morphism of schemes and set", "$Y' = Z' \\times_Z Y$ and $X' = Z' \\times_Z X = Y' \\times_ Y X$.", "Denote $a : X' \\to X$, $b : Y' \\to Y$, and $c : Z' \\to Z$ the", "projections. Similarly for $f' : X' \\to Y'$ and $g' : Y' \\to Z'$.", "By Lemma \\ref{lemma-proper-base-change-f-star} we have", "$b^{-1}f_*\\mathcal{J} = f'_*a^{-1}\\mathcal{J}$.", "On the other hand, we know that $R^qf'_*a^{-1}\\mathcal{J}$ and", "$R^q(g' \\circ f')_*a^{-1}\\mathcal{J}$ are zero for $q > 0$.", "Using the spectral sequence", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-relative-Leray})", "$$", "R^pg'_* R^qf'_* a^{-1}\\mathcal{J} \\Rightarrow", "R^{p + q}(g' \\circ f')_* a^{-1}\\mathcal{J}", "$$", "we conclude that", "$ R^pg'_*(b^{-1}f_*\\mathcal{J}) = R^pg'_*(f'_*a^{-1}\\mathcal{J}) = 0$", "for $p > 0$ as desired." ], "refs": [ "etale-cohomology-lemma-proper-base-change-in-terms-of-injectives", "etale-cohomology-lemma-proper-base-change-f-star", "sites-cohomology-lemma-relative-Leray" ], "ref_ids": [ 6621, 6620, 4222 ] } ], "ref_ids": [] }, { "id": 6623, "type": "theorem", "label": "etale-cohomology-lemma-composition", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-composition", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms of schemes. Assume", "\\begin{enumerate}", "\\item cohomology commutes with base change for $f$, and", "\\item cohomology commutes with base change for $g$.", "\\end{enumerate}", "Then cohomology commutes with base change for $g \\circ f$." ], "refs": [], "proofs": [ { "contents": [ "We will use the equivalence of", "Lemma \\ref{lemma-proper-base-change-in-terms-of-injectives}", "without further mention. Let $\\ell$ be a prime number.", "Let $\\mathcal{I}$ be an injective sheaf of", "$\\mathbf{Z}/\\ell\\mathbf{Z}$-modules on $X_\\etale$.", "Then $f_*\\mathcal{I}$ is an injective sheaf of", "$\\mathbf{Z}/\\ell\\mathbf{Z}$-modules on $Y_\\etale$", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-pushforward-injective-flat}).", "The result follows formally from this, but we will also", "spell it out.", "\\medskip\\noindent", "Let $Z' \\to Z$ be a morphism of schemes and set", "$Y' = Z' \\times_Z Y$ and $X' = Z' \\times_Z X = Y' \\times_ Y X$.", "Denote $a : X' \\to X$, $b : Y' \\to Y$, and $c : Z' \\to Z$ the", "projections. Similarly for $f' : X' \\to Y'$ and $g' : Y' \\to Z'$.", "By Lemma \\ref{lemma-proper-base-change-f-star} we have", "$b^{-1}f_*\\mathcal{I} = f'_*a^{-1}\\mathcal{I}$.", "On the other hand, we know that $R^qf'_*a^{-1}\\mathcal{I}$ and", "$R^q(g')_*b^{-1}f_*\\mathcal{I}$ are zero for $q > 0$.", "Using the spectral sequence", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-relative-Leray})", "$$", "R^pg'_* R^qf'_* a^{-1}\\mathcal{I} \\Rightarrow", "R^{p + q}(g' \\circ f')_* a^{-1}\\mathcal{I}", "$$", "we conclude that $R^p(g' \\circ f')_*a^{-1}\\mathcal{I} = 0$ for", "$p > 0$ as desired." ], "refs": [ "etale-cohomology-lemma-proper-base-change-in-terms-of-injectives", "sites-cohomology-lemma-pushforward-injective-flat", "etale-cohomology-lemma-proper-base-change-f-star", "sites-cohomology-lemma-relative-Leray" ], "ref_ids": [ 6621, 4218, 6620, 4222 ] } ], "ref_ids": [] }, { "id": 6624, "type": "theorem", "label": "etale-cohomology-lemma-finite", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-finite", "contents": [ "\\begin{slogan}", "Proper base change for \\'etale cohomology holds for finite morphisms.", "\\end{slogan}", "Let $f : X \\to Y$ be a finite morphism of schemes.", "Then cohomology commutes with base change for $f$." ], "refs": [], "proofs": [ { "contents": [ "Observe that a finite morphism is proper, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}.", "Moreover, the base change of a finite morphism is finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite}.", "Thus the result follows from", "Lemma \\ref{lemma-proper-base-change-in-terms-of-injectives}", "combined with", "Proposition \\ref{proposition-finite-higher-direct-image-zero}." ], "refs": [ "morphisms-lemma-finite-proper", "morphisms-lemma-base-change-finite", "etale-cohomology-lemma-proper-base-change-in-terms-of-injectives", "etale-cohomology-proposition-finite-higher-direct-image-zero" ], "ref_ids": [ 5445, 5440, 6621, 6703 ] } ], "ref_ids": [] }, { "id": 6625, "type": "theorem", "label": "etale-cohomology-lemma-reduce-to-P1", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-reduce-to-P1", "contents": [ "To prove that cohomology commutes with base change for", "every proper morphism of schemes it suffices to prove it", "holds for the morphism $\\mathbf{P}^1_S \\to S$ for every scheme $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes.", "Let $Y = \\bigcup Y_i$ be an affine open covering", "and set $X_i = f^{-1}(Y_i)$. If we can prove", "cohomology commutes with base change for $X_i \\to Y_i$,", "then cohomology commutes with base change for $f$.", "Namely, the formation of the higher direct images", "commutes with Zariski (and even \\'etale) localization", "on the base, see", "Lemma \\ref{lemma-higher-direct-images}.", "Thus we may assume $Y$ is affine.", "\\medskip\\noindent", "Let $Y$ be an affine scheme and let $X \\to Y$ be a proper morphism.", "By Chow's lemma there exists a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rd] & X' \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_Y \\ar[dl] \\\\", "& Y &", "}", "$$", "where $X' \\to \\mathbf{P}^n_Y$ is an immersion, and", "$\\pi : X' \\to X$ is proper and surjective, see", "Limits, Lemma \\ref{limits-lemma-chow-finite-type}.", "Since $X \\to Y$ is proper, we find that $X' \\to Y$ is proper", "(Morphisms, Lemma \\ref{morphisms-lemma-composition-proper}).", "Hence $X' \\to \\mathbf{P}^n_Y$ is a closed immersion", "(Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}).", "It follows that $X' \\to X \\times_Y \\mathbf{P}^n_Y = \\mathbf{P}^n_X$", "is a closed immersion (as an immersion with closed image).", "\\medskip\\noindent", "By Lemma \\ref{lemma-sandwich}", "it suffices to prove cohomology commutes with base change for", "$\\pi$ and $X' \\to Y$. These morphisms both factor as a closed", "immersion followed by a projection $\\mathbf{P}^n_S \\to S$ (for some $S$).", "By Lemma \\ref{lemma-finite} the result holds for closed", "immersions (as closed immersions are finite).", "By Lemma \\ref{lemma-composition} it suffices to prove the", "result for projections $\\mathbf{P}^n_S \\to S$.", "\\medskip\\noindent", "For every $n \\geq 1$ there is a finite surjective morphism", "$$", "\\mathbf{P}^1_S \\times_S \\ldots \\times_S \\mathbf{P}^1_S", "\\longrightarrow", "\\mathbf{P}^n_S", "$$", "given on coordinates by", "$$", "((x_1 : y_1), (x_2 : y_2), \\ldots, (x_n : y_n))", "\\longmapsto", "(F_0 : \\ldots : F_n)", "$$", "where $F_0, \\ldots, F_n$ in $x_1, \\ldots, y_n$", "are the polynomials with integer coefficients such that", "$$", "\\prod (x_i t + y_i) = F_0 t^n + F_1 t^{n - 1} + \\ldots + F_n", "$$", "Applying", "Lemmas \\ref{lemma-sandwich}, \\ref{lemma-finite}, and \\ref{lemma-composition}", "one more time we conclude that the lemma is true." ], "refs": [ "etale-cohomology-lemma-higher-direct-images", "limits-lemma-chow-finite-type", "morphisms-lemma-composition-proper", "morphisms-lemma-image-proper-scheme-closed", "etale-cohomology-lemma-sandwich", "etale-cohomology-lemma-finite", "etale-cohomology-lemma-composition", "etale-cohomology-lemma-sandwich", "etale-cohomology-lemma-finite", "etale-cohomology-lemma-composition" ], "ref_ids": [ 6474, 15087, 5408, 5411, 6622, 6624, 6623, 6622, 6624, 6623 ] } ], "ref_ids": [] }, { "id": 6626, "type": "theorem", "label": "etale-cohomology-lemma-proper-base-change", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-proper-base-change", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$ be", "a morphism of schemes. Set $X' = Y' \\times_Y X$ and denote", "$f' : X' \\to Y'$ and $g' : X' \\to X$ the projections.", "Let $E \\in D^+(X_\\etale)$ have torsion cohomology sheaves.", "Then the base change map (\\ref{equation-base-change})", "$g^{-1}Rf_*E \\to Rf'_*(g')^{-1}E$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is a simple consequence of the proper base change theorem", "(Theorem \\ref{theorem-proper-base-change}) using the spectral", "sequences", "$$", "E_2^{p, q} = R^pf_*H^q(E)", "\\quad\\text{and}\\quad", "{E'}_2^{p, q} = R^pf'_*(g')^{-1}H^q(E)", "$$", "converging to $R^nf_*E$ and $R^nf'_*(g')^{-1}E$.", "The spectral sequences are constructed in", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "Some details omitted." ], "refs": [ "etale-cohomology-theorem-proper-base-change", "derived-lemma-two-ss-complex-functor" ], "ref_ids": [ 6397, 1871 ] } ], "ref_ids": [] }, { "id": 6627, "type": "theorem", "label": "etale-cohomology-lemma-proper-base-change-stalk", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-proper-base-change-stalk", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes. Let $\\overline{y} \\to Y$", "be a geometric point.", "\\begin{enumerate}", "\\item For a torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have", "$(R^nf_*\\mathcal{F})_{\\overline{y}} =", "H^n_\\etale(X_{\\overline{y}}, \\mathcal{F}_{\\overline{y}})$.", "\\item For $E \\in D^+(X_\\etale)$ with torsion cohomology sheaves we have", "$(R^nf_*E)_{\\overline{y}} = H^n_\\etale(X_{\\overline{y}}, E_{\\overline{y}})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In the statement, $\\mathcal{F}_{\\overline{y}}$ denotes the pullback", "of $\\mathcal{F}$ to the scheme theoretic fibre", "$X_{\\overline{y}} = \\overline{y} \\times_Y X$.", "Since pulling back by $\\overline{y} \\to Y$ produces the", "stalk of $\\mathcal{F}$, the first statement of the lemma", "is a special case of Theorem \\ref{theorem-proper-base-change}.", "The second one is a special case of Lemma \\ref{lemma-proper-base-change}." ], "refs": [ "etale-cohomology-theorem-proper-base-change", "etale-cohomology-lemma-proper-base-change" ], "ref_ids": [ 6397, 6626 ] } ], "ref_ids": [] }, { "id": 6628, "type": "theorem", "label": "etale-cohomology-lemma-base-change-separably-closed", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-base-change-separably-closed", "contents": [ "Let $K/k$ be an extension of separably closed fields.", "Let $X$ be a proper scheme over $k$.", "Let $\\mathcal{F}$ be a torsion abelian sheaf on $X_\\etale$.", "Then the map $H^q_\\etale(X, \\mathcal{F}) \\to", "H^q_\\etale(X_K, \\mathcal{F}|_{X_K})$ is an isomorphism", "for $q \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Looking at stalks we see that", "this is a special case of", "Theorem \\ref{theorem-proper-base-change}." ], "refs": [ "etale-cohomology-theorem-proper-base-change" ], "ref_ids": [ 6397 ] } ], "ref_ids": [] }, { "id": 6629, "type": "theorem", "label": "etale-cohomology-lemma-cohomological-dimension-proper", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomological-dimension-proper", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes", "all of whose fibres have dimension $\\leq n$.", "Then for any abelian torsion sheaf $\\mathcal{F}$ on $X_\\etale$", "we have $R^qf_*\\mathcal{F} = 0$ for $q > 2n$." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by induction on $n$ for all proper morphisms.", "\\medskip\\noindent", "If $n = 0$, then $f$ is a finite morphism", "(More on Morphisms, Lemma \\ref{more-morphisms-lemma-characterize-finite})", "and the", "result is true by Proposition \\ref{proposition-finite-higher-direct-image-zero}.", "\\medskip\\noindent", "If $n > 0$, then using Lemma \\ref{lemma-proper-base-change-stalk}", "we see that it suffices to prove $H^i_\\etale(X, \\mathcal{F}) = 0$", "for $i > 2n$ and $X$ a proper scheme, $\\dim(X) \\leq n$", "over an algebraically closed field $k$", "and $\\mathcal{F}$ is a torsion abelian sheaf on $X$.", "\\medskip\\noindent", "If $n = 1$ this follows from Theorem \\ref{theorem-vanishing-curves}.", "Assume $n > 1$. By Proposition \\ref{proposition-topological-invariance}", "we may replace $X$ by its reduction.", "Let $\\nu : X^\\nu \\to X$ be the normalization.", "This is a surjective birational finite morphism", "(see Varieties, Lemma \\ref{varieties-lemma-normalization-locally-algebraic})", "and hence an isomorphism over a dense open $U \\subset X$", "(Morphisms, Lemma \\ref{morphisms-lemma-birational-birational}).", "Then we see that", "$c : \\mathcal{F} \\to \\nu_*\\nu^{-1}\\mathcal{F}$ is injective", "(as $\\nu$ is surjective) and an isomorphism over $U$.", "Denote $i : Z \\to X$ the inclusion of the complement of $U$.", "Since $U$ is dense in $X$ we have $\\dim(Z) < \\dim(X) = n$.", "By Proposition \\ref{proposition-closed-immersion-pushforward} have", "$\\Coker(c) = i_*\\mathcal{G}$ for some", "abelian torsion sheaf $\\mathcal{G}$ on $Z_\\etale$.", "Then $H^q_\\etale(X, \\Coker(c)) = H^q_\\etale(Z, \\mathcal{F})$", "(by Proposition \\ref{proposition-finite-higher-direct-image-zero}", "and the Leray spectral sequence) and by induction hypothesis we conclude that", "the cokernel of $c$ has cohomology in degrees $\\leq 2(n - 1)$.", "Thus it suffices to prove the result for $\\nu_*\\nu^{-1}\\mathcal{F}$.", "As $\\nu$ is finite this reduces us to showing", "that $H^i_\\etale(X^\\nu, \\nu^{-1}\\mathcal{F})$ is", "zero for $i > 2n$. This case is treated in the next paragraph.", "\\medskip\\noindent", "Assume $X$ is integral normal proper scheme over $k$ of dimension $n$.", "Choose a nonconstant rational function $f$ on $X$. The graph", "$X' \\subset X \\times \\mathbf{P}^1_k$ of $f$ sits into a diagram", "$$", "X \\xleftarrow{b} X' \\xrightarrow{f} \\mathbf{P}^1_k", "$$", "Observe that $b$ is an isomorphism over an open subscheme", "$U \\subset X$ whose complement is a closed subscheme", "$Z \\subset X$ of codimension $\\geq 2$. Namely, $U$ is the", "domain of definition of $f$ which contains all codimension $1$", "points of $X$, see", "Morphisms, Lemmas \\ref{morphisms-lemma-rational-map-from-reduced-to-separated}", "and \\ref{morphisms-lemma-extend-across}", "(combined with Serre's criterion for normality, see", "Properties, Lemma \\ref{properties-lemma-criterion-normal}).", "Moreover the fibres of $b$ have dimension $\\leq 1$ (as closed subschemes", "of $\\mathbf{P}^1$). Hence $R^ib_*b^{-1}\\mathcal{F}$ is nonzero only", "if $i \\in \\{0, 1, 2\\}$ by induction. Choose a distinguished triangle", "$$", "\\mathcal{F} \\to Rb_*b^{-1}\\mathcal{F} \\to Q \\to \\mathcal{F}[1]", "$$", "Using that $\\mathcal{F} \\to b_*b^{-1}\\mathcal{F}$ is injective", "as before and using what we just said, we see that $Q$ has nonzero", "cohomology sheaves only in degrees $0, 1, 2$ sitting on $Z$.", "Moreover, these cohomology sheaves are torsion by", "Lemma \\ref{lemma-torsion-direct-image}.", "By induction we see that $H^i(X, Q)$ is zero for", "$i > 2 + 2\\dim(Z) \\leq 2 + 2(n - 2) = 2n - 2$. Thus it suffices", "to prove that $H^i(X', b^{-1}\\mathcal{F}) = 0$ for", "$i > 2n$. At this point we use the morphism", "$$", "f : X' \\to \\mathbf{P}^1_k", "$$", "whose fibres have dimension $< n$. Hence by induction we see that", "$R^if_*b^{-1}\\mathcal{F} = 0$ for $i > 2(n - 1)$.", "We conclude by the Leray spectral seqence", "$$", "H^i(\\mathbf{P}^1_k, R^jf_*b^{-1}\\mathcal{F})", "\\Rightarrow", "H^{i + j}(X', b^{-1}\\mathcal{F})", "$$", "and the fact that $\\dim(\\mathbf{P}^1_k) = 1$." ], "refs": [ "more-morphisms-lemma-characterize-finite", "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-lemma-proper-base-change-stalk", "etale-cohomology-theorem-vanishing-curves", "etale-cohomology-proposition-topological-invariance", "varieties-lemma-normalization-locally-algebraic", "morphisms-lemma-birational-birational", "etale-cohomology-proposition-closed-immersion-pushforward", "etale-cohomology-proposition-finite-higher-direct-image-zero", "morphisms-lemma-rational-map-from-reduced-to-separated", "morphisms-lemma-extend-across", "properties-lemma-criterion-normal", "etale-cohomology-lemma-torsion-direct-image" ], "ref_ids": [ 13903, 6703, 6627, 6395, 6699, 11013, 5484, 6700, 6703, 5480, 5419, 2989, 6565 ] } ], "ref_ids": [] }, { "id": 6630, "type": "theorem", "label": "etale-cohomology-lemma-proper-base-change-mod-n", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-proper-base-change-mod-n", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$ be", "a morphism of schemes. Set $X' = Y' \\times_Y X$ and denote", "$f' : X' \\to Y'$ and $g' : X' \\to X$ the projections.", "Let $n \\geq 1$ be an integer.", "Let $E \\in D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$.", "Then the base change map (\\ref{equation-base-change})", "$g^{-1}Rf_*E \\to Rf'_*(g')^{-1}E$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "It is enough to prove this when $Y$ and $Y'$ are quasi-compact.", "By Morphisms, Lemma", "\\ref{morphisms-lemma-morphism-finite-type-bounded-dimension}", "we see that the dimension of the fibres of", "$f : X \\to Y$ and $f' : X' \\to Y'$ are bounded. Thus", "Lemma \\ref{lemma-cohomological-dimension-proper} implies that", "$$", "f_* :", "\\textit{Mod}(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})", "\\longrightarrow", "\\textit{Mod}(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})", "$$", "and", "$$", "f'_* :", "\\textit{Mod}(X'_\\etale, \\mathbf{Z}/n\\mathbf{Z})", "\\longrightarrow", "\\textit{Mod}(Y'_\\etale, \\mathbf{Z}/n\\mathbf{Z})", "$$", "have finite cohomological dimension in the sense of", "Derived Categories, Lemma \\ref{derived-lemma-unbounded-right-derived}.", "Choose a K-injective complex $\\mathcal{I}^\\bullet$", "of $\\mathbf{Z}/n\\mathbf{Z}$-modules each of whose", "terms $\\mathcal{I}^n$ is an injective sheaf of", "$\\mathbf{Z}/n\\mathbf{Z}$-modules representing $E$.", "See Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "By the usual proper base change theorem we find", "that $R^qf'_*(g')^{-1}\\mathcal{I}^n = 0$ for $q > 0$, see", "Theorem \\ref{theorem-proper-base-change}.", "Hence we conclude by ", "Derived Categories, Lemma \\ref{derived-lemma-unbounded-right-derived}", "that we may compute $Rf'_*(g')^{-1}E$ by the complex", "$f'_*(g')^{-1}\\mathcal{I}^\\bullet$. Another application", "of the usual proper base change theorem shows that", "this is equal to $g^{-1}f_*\\mathcal{I}^\\bullet$ as desired." ], "refs": [ "morphisms-lemma-morphism-finite-type-bounded-dimension", "etale-cohomology-lemma-cohomological-dimension-proper", "derived-lemma-unbounded-right-derived", "injectives-theorem-K-injective-embedding-grothendieck", "etale-cohomology-theorem-proper-base-change", "derived-lemma-unbounded-right-derived" ], "ref_ids": [ 5281, 6629, 1917, 7768, 6397, 1917 ] } ], "ref_ids": [] }, { "id": 6631, "type": "theorem", "label": "etale-cohomology-lemma-pull-out-constant", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-pull-out-constant", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $E \\in D^+(X_\\etale)$ and $K \\in D^+(\\mathbf{Z})$.", "Then", "$$", "R\\Gamma(X, E \\otimes_\\mathbf{Z}^\\mathbf{L} \\underline{K}) =", "R\\Gamma(X, E) \\otimes_\\mathbf{Z}^\\mathbf{L} K", "$$" ], "refs": [], "proofs": [ { "contents": [ "Say $H^i(E) = 0$ for $i \\geq a$ and $H^j(K) = 0$ for $j \\geq b$.", "We may represent $K$ by a bounded below complex", "$K^\\bullet$ of torsion free $\\mathbf{Z}$-modules.", "(Choose a K-flat complex $L^\\bullet$ representing $K$ and then", "take $K^\\bullet = \\tau_{\\geq b - 1}L^\\bullet$. This works because", "$\\mathbf{Z}$ has global dimension $1$. See", "More on Algebra, Lemma \\ref{more-algebra-lemma-last-one-flat}.)", "We may represent $E$ by a bounded below complex", "$\\mathcal{E}^\\bullet$.", "Then $E \\otimes_\\mathbf{Z}^\\mathbf{L} \\underline{K}$ is", "represented by", "$$", "\\text{Tot}(\\mathcal{E}^\\bullet \\otimes_\\mathbf{Z} \\underline{K}^\\bullet)", "$$", "Using distinguished triangles", "$$", "\\sigma_{\\geq -b + n + 1}K^\\bullet", "\\to K^\\bullet \\to", "\\sigma_{\\leq -b + n}K^\\bullet", "$$", "and the trivial vanishing", "$$", "H^n(X,", "\\text{Tot}(\\mathcal{E}^\\bullet \\otimes_\\mathbf{Z}", "\\sigma_{\\geq -a + n + 1}\\underline{K}^\\bullet) = 0", "$$", "and", "$$", "H^n(R\\Gamma(X, E)", "\\otimes_\\mathbf{Z}^\\mathbf{L} \\sigma_{\\geq -a + n + 1}K^\\bullet) = 0", "$$", "we reduce to the case where $K^\\bullet$ is a bounded complex of", "flat $\\mathbf{Z}$-modules. Repeating the argument we reduce to the", "case where $K^\\bullet$ is equal to a single flat $\\mathbf{Z}$-module", "sitting in some degree. Next, using the stupid", "trunctions for $\\mathcal{E}^\\bullet$", "we reduce in exactly the same manner to the case where", "$\\mathcal{E}^\\bullet$ is a single abelian sheaf sitting", "in some degree. Thus it suffices to show that", "$$", "H^n(X, \\mathcal{E} \\otimes_\\mathbf{Z} \\underline{M}) =", "H^n(X, \\mathcal{E}) \\otimes_\\mathbf{Z} M", "$$", "when $M$ is a flat $\\mathbf{Z}$-module and $\\mathcal{E}$", "is an abelian sheaf on $X$. In this case we", "write $M$ is a filtered colimit of finite free $\\mathbf{Z}$-modules", "(Lazard's theorem, see Algebra, Theorem \\ref{algebra-theorem-lazard}).", "By Theorem \\ref{theorem-colimit} this reduces us to the case of finite free", "$\\mathbf{Z}$-module $M$ in which case the result is trivially true." ], "refs": [ "more-algebra-lemma-last-one-flat", "algebra-theorem-lazard", "etale-cohomology-theorem-colimit" ], "ref_ids": [ 10169, 318, 6384 ] } ], "ref_ids": [] }, { "id": 6632, "type": "theorem", "label": "etale-cohomology-lemma-projection-formula-proper", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-projection-formula-proper", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes.", "Let $E \\in D^+(X_\\etale)$ have torsion cohomology sheaves.", "Let $K \\in D^+(Y_\\etale)$. Then", "$$", "Rf_*E \\otimes_\\mathbf{Z}^\\mathbf{L} K =", "Rf_*(E \\otimes_\\mathbf{Z}^\\mathbf{L} f^{-1}K)", "$$", "in $D^+(Y_\\etale)$." ], "refs": [], "proofs": [ { "contents": [ "There is a canonical map from left to right by", "Cohomology on Sites, Section \\ref{sites-cohomology-section-projection-formula}.", "We will check the equality on stalks.", "Recall that computing derived tensor products commutes with pullbacks.", "See Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-pullback-tensor-product}.", "Thus we have", "$$", "(E \\otimes_\\mathbf{Z}^\\mathbf{L} f^{-1}K)_{\\overline{x}}", "=", "E_{\\overline{x}} \\otimes_\\mathbf{Z}^\\mathbf{L} K_{\\overline{y}}", "$$", "where $\\overline{y}$ is the image of $\\overline{x}$ in $Y$.", "Since $\\mathbf{Z}$ has global dimension $1$ we see that", "this complex has vanishing cohomology in degree $< - 1 + a + b$", "if $H^i(E) = 0$ for $i \\geq a$ and $H^j(K) = 0$ for $j \\geq b$.", "Moreover, since $H^i(E)$ is a torsion abelian", "sheaf for each $i$, the same is true for the cohomology", "sheaves of the complex $E \\otimes_\\mathbf{Z}^\\mathbf{L} K$.", "Namely, we have", "$$", "(E \\otimes_\\mathbf{Z}^\\mathbf{L} f^{-1}K)", "\\otimes_{\\mathbf{Z}}^\\mathbf{L} \\mathbf{Q} =", "(E \\otimes_\\mathbf{Z}^\\mathbf{L} \\mathbf{Q})", "\\otimes_{\\mathbf{Q}}^\\mathbf{L}", "(f^{-1}K \\otimes_{\\mathbf{Z}}^\\mathbf{L} \\mathbf{Q})", "$$", "which is zero in the derived category.", "In this way we see that Lemma \\ref{lemma-proper-base-change-stalk}", "applies to both sides to see that it suffices to show", "$$", "R\\Gamma(X_{\\overline{y}},", "E|_{X_{\\overline{y}}} \\otimes_\\mathbf{Z}^\\mathbf{L}", "(X_{\\overline{y}} \\to \\overline{y})^{-1}K_{\\overline{y}}) =", "R\\Gamma(X_{\\overline{y}},", "E|_{X_{\\overline{y}}}) \\otimes_\\mathbf{Z}^\\mathbf{L} K_{\\overline{y}}", "$$", "This is shown in Lemma \\ref{lemma-pull-out-constant}." ], "refs": [ "sites-cohomology-lemma-pullback-tensor-product", "etale-cohomology-lemma-proper-base-change-stalk", "etale-cohomology-lemma-pull-out-constant" ], "ref_ids": [ 4244, 6627, 6631 ] } ], "ref_ids": [] }, { "id": 6633, "type": "theorem", "label": "etale-cohomology-lemma-cd-limit", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cd-limit", "contents": [ "Let $X = \\lim X_i$ be a directed limit of a system of", "quasi-compact and quasi-separated schemes with affine", "transition morphisms. Then $\\text{cd}(X) \\leq \\max \\text{cd}(X_i)$." ], "refs": [], "proofs": [ { "contents": [ "Denote $f_i : X \\to X_i$ the projections.", "Let $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$.", "Then we have $\\mathcal{F} = \\lim f_i^{-1}f_{i, *}\\mathcal{F}$", "by Lemma \\ref{lemma-linus-hamann}.", "Thus $H^q_\\etale(X, \\mathcal{F}) =", "\\colim H^q_\\etale(X_i, f_{i, *}\\mathcal{F})$", "by Theorem \\ref{theorem-colimit}.", "The lemma follows." ], "refs": [ "etale-cohomology-lemma-linus-hamann", "etale-cohomology-theorem-colimit" ], "ref_ids": [ 6477, 6384 ] } ], "ref_ids": [] }, { "id": 6634, "type": "theorem", "label": "etale-cohomology-lemma-cd-curve-over-field", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cd-curve-over-field", "contents": [ "Let $K$ be a field. Let $X$ be a $1$-dimensional", "affine scheme of finite type over $K$. Then", "$\\text{cd}(X) \\leq 1 + \\text{cd}(K)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$.", "Consider the Leray spectral sequence for the morphism", "$f : X \\to \\Spec(K)$. We obtain", "$$", "E_2^{p, q} = H^p(\\Spec(K), R^qf_*\\mathcal{F})", "$$", "converging to $H^{p + q}_\\etale(X, \\mathcal{F})$.", "The stalk of $R^qf_*\\mathcal{F}$ at a geometric point", "$\\Spec(\\overline{K}) \\to \\Spec(K)$ is the cohomology of the", "pullback of $\\mathcal{F}$ to $X_{\\overline{K}}$.", "Hence it vanishes in degrees $\\geq 2$ by", "Theorem \\ref{theorem-vanishing-affine-curves}." ], "refs": [ "etale-cohomology-theorem-vanishing-affine-curves" ], "ref_ids": [ 6394 ] } ], "ref_ids": [] }, { "id": 6635, "type": "theorem", "label": "etale-cohomology-lemma-cd-field-extension", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cd-field-extension", "contents": [ "Let $L/K$ be a field extension. Then we have", "$\\text{cd}(L) \\leq \\text{cd}(K) + \\text{trdeg}_K(L)$." ], "refs": [], "proofs": [ { "contents": [ "If $\\text{trdeg}_K(L) = \\infty$, then this is clear.", "If not then we can find a sequence of extensions", "$L= L_r/L_{r - 1}/ \\ldots /L_1/L_0 = K$ such that", "$\\text{trdeg}_{L_i}(L_{i + 1}) = 1$ and $r = \\text{trdeg}_K(L)$.", "Hence it suffices to prove the lemma in the case that $r = 1$.", "In this case we can write $L = \\colim A_i$", "as a filtered colimit of its finite type $K$-subalgebras.", "By Lemma \\ref{lemma-cd-limit} it suffices to prove that", "$\\text{cd}(A_i) \\leq 1 + \\text{cd}(K)$. This follows", "from Lemma \\ref{lemma-cd-curve-over-field}." ], "refs": [ "etale-cohomology-lemma-cd-limit", "etale-cohomology-lemma-cd-curve-over-field" ], "ref_ids": [ 6633, 6634 ] } ], "ref_ids": [] }, { "id": 6636, "type": "theorem", "label": "etale-cohomology-lemma-strictly-henselian", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-strictly-henselian", "contents": [ "Let $K$ be a field. Let $X$ be a scheme of finite type over $K$.", "Let $x \\in X$. Set $a = \\text{trdeg}_K(\\kappa(x))$", "and $d = \\dim_x(X)$. Then there is a map", "$$", "K(t_1, \\ldots, t_a)^{sep} \\longrightarrow \\mathcal{O}_{X, x}^{sh}", "$$", "such that", "\\begin{enumerate}", "\\item the residue field of $\\mathcal{O}_{X, x}^{sh}$ is a purely inseparable", "extension of $K(t_1, \\ldots, t_a)^{sep}$,", "\\item $\\mathcal{O}_{X, x}^{sh}$ is a filtered colimit of finite", "type $K(t_1, \\ldots, t_a)^{sep}$-algebras of dimension $\\leq d - a$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may assume $X$ is affine. By Noether normalization, after possibly", "shrinking $X$ again, we can choose a finite morphism", "$\\pi : X \\to \\mathbf{A}^d_K$, see", "Algebra, Lemma \\ref{algebra-lemma-Noether-normalization-at-point}.", "Since $\\kappa(x)$ is a finite extension of the residue field of $\\pi(x)$,", "this residue field has transcendence degree $a$ over $K$ as well.", "Thus we can find a finite morphism $\\pi' : \\mathbf{A}^d_K \\to \\mathbf{A}^d_K$", "such that $\\pi'(\\pi(x))$ corresponds to the generic point of the linear", "subspace $\\mathbf{A}^a_K \\subset \\mathbf{A}^d_K$ given by setting", "the last $d - a$ coordinates equal to zero. Hence the composition", "$$", "X \\xrightarrow{\\pi' \\circ \\pi} \\mathbf{A}^d_K \\xrightarrow{p} \\mathbf{A}^a_K", "$$", "of $\\pi' \\circ \\pi$ and the projection $p$ onto the first $a$ coordinates", "maps $x$ to the generic point $\\eta \\in \\mathbf{A}^a_K$. The induced map", "$$", "K(t_1, \\ldots, t_a)^{sep} =", "\\mathcal{O}_{\\mathbf{A}^a_k, \\eta}^{sh}", "\\longrightarrow \\mathcal{O}_{X, x}^{sh}", "$$", "on \\'etale local rings satisfies (1) since it is clear that the residue field", "of $\\mathcal{O}_{X, x}^{sh}$ is an algebraic extension of the separably closed", "field $K(t_1, \\ldots, t_a)^{sep}$. On the other hand, if $X = \\Spec(B)$,", "then $\\mathcal{O}_{X, x}^{sh} = \\colim B_j$ is a filtered colimit", "of \\'etale $B$-algebras $B_j$. Observe that $B_j$ is quasi-finite over", "$K[t_1, \\ldots, t_d]$ as $B$ is finite over $K[t_1, \\ldots, t_d]$.", "We may similarly write", "$K(t_1, \\ldots, t_a)^{sep} = \\colim A_i$ as a filtered colimit", "of \\'etale $K[t_1, \\ldots, t_a]$-algebras. For every $i$ we can", "find an $j$ such that", "$A_i \\to K(t_1, \\ldots, t_a)^{sep} \\to \\mathcal{O}_{X, x}^{sh}$", "factors through a map $\\psi_{i, j} : A_i \\to B_j$. Then $B_j$ is quasi-finite", "over $A_i[t_{a + 1}, \\ldots, t_d]$. Hence", "$$", "B_{i, j} = B_j \\otimes_{\\psi_{i, j}, A_i} K(t_1, \\ldots, t_a)^{sep}", "$$", "has dimension $\\leq d - a$ as it is quasi-finite over", "$K(t_1, \\ldots, t_a)^{sep}[t_{a + 1}, \\ldots, t_d]$.", "The proof of (2) is now finished as $\\mathcal{O}_{X, x}^{sh}$ is a", "filtered colimit\\footnote{Let $R$ be a ring.", "Let $A = \\colim_{i \\in I} A_i$", "be a filtered colimit of finitely presented $R$-algebras.", "Let $B = \\colim_{j \\in J} B_j$ be a filtered colimit of $R$-algebras.", "Let $A \\to B$ be an $R$-algebra map.", "Assume that for all $i \\in I$ there is a $j \\in J$ and an $R$-algebra map", "$\\psi_{i, j} : A_i \\to B_j$.", "Say $(i', j', \\psi_{i', j'}) \\geq (i, j, \\psi_{i, j})$ if", "$i' \\geq i$, $j' \\geq j$, and $\\psi_{i, j}$ and $\\psi_{i', j'}$", "are compatible. Then the collection of triples forms a", "directed set and $B = \\colim B_j \\otimes_{\\psi_{i, j} A_i} A$.}", "of the algebras $B_{i, j}$. Some details omitted." ], "refs": [ "algebra-lemma-Noether-normalization-at-point" ], "ref_ids": [ 1002 ] } ], "ref_ids": [] }, { "id": 6637, "type": "theorem", "label": "etale-cohomology-lemma-interlude-II", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-interlude-II", "contents": [ "Let $K$ be a field. Let $X$ be an affine scheme of finite type over $K$.", "Let $E_a \\subset X$ be the set of points", "$x \\in X$ with $\\text{trdeg}_K(\\kappa(x)) \\leq a$.", "Let $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$", "whose support is contained in $E_a$. Then", "$H^b_\\etale(X, \\mathcal{F}) = 0$ for $b > a + \\text{cd}(K)$." ], "refs": [], "proofs": [ { "contents": [ "We can write $\\mathcal{F} = \\colim \\mathcal{F}_i$", "with $\\mathcal{F}_i$ a torsion abelian sheaf", "supported on a closed subscheme $Z_i$ contained in $E_a$, see", "Lemma \\ref{lemma-support-in-subset}.", "Then Proposition \\ref{proposition-cd-affine} gives", "the desired vanishing for $\\mathcal{F}_i$. Details omitted;", "hints: first use Proposition \\ref{proposition-closed-immersion-pushforward}", "to write $\\mathcal{F}_i$ as the pushforward of a sheaf on $Z_i$,", "use the vanishing for this sheaf on $Z_i$, and use", "the Leray spectral sequence for $Z_i \\to Z$ to get the vanishing", "for $\\mathcal{F}_i$. Finally, we", "conclude by Theorem \\ref{theorem-colimit}." ], "refs": [ "etale-cohomology-lemma-support-in-subset", "etale-cohomology-proposition-cd-affine", "etale-cohomology-proposition-closed-immersion-pushforward", "etale-cohomology-theorem-colimit" ], "ref_ids": [ 6554, 6707, 6700, 6384 ] } ], "ref_ids": [] }, { "id": 6638, "type": "theorem", "label": "etale-cohomology-lemma-interlude-I", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-interlude-I", "contents": [ "Let $f : X \\to Y$ be an affine morphism of schemes of finite", "type over a field $K$. Let $E_a(X)$ be the set of points $x \\in X$", "with $\\text{trdeg}_K(\\kappa(x)) \\leq a$.", "Let $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$", "whose support is contained in $E_a$. Then", "$R^qf_*\\mathcal{F}$ has support contained in", "$E_{a - q}(Y)$." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $Y$ hence we can assume $Y$ is affine.", "Then $X$ is affine too and we can choose a diagram", "$$", "\\xymatrix{", "X \\ar[d]_f \\ar[r]_i & \\mathbf{A}^{n + m}_K \\ar[d]^{\\text{pr}} \\\\", "Y \\ar[r]^j & \\mathbf{A}^n_K", "}", "$$", "where the horizontal arrows are closed immersions and the vertical", "arrow on the right is the projection (details omitted).", "Then $j_*R^qf_*\\mathcal{F} = R^q\\text{pr}_*i_*\\mathcal{F}$", "by the vanishing of the higher direct images of $i$ and $j$, see", "Proposition \\ref{proposition-finite-higher-direct-image-zero}.", "Moreover, the description of the stalks of $j_*$ in the proposition", "shows that it suffices to prove the vanishing for $j_*R^qf_*\\mathcal{F}$.", "Thus we may assume $f$ is the projection", "morphism $\\text{pr} : \\mathbf{A}^{n + m}_K \\to \\mathbf{A}^n_K$", "and an abelian torsion sheaf $\\mathcal{F}$ on $\\mathbf{A}^{n + m}_K$", "satisfying the assumption in the statement of the lemma.", "\\medskip\\noindent", "Let $y$ be a point in $\\mathbf{A}^n_K$.", "By Theorem \\ref{theorem-higher-direct-images} we have", "$$", "(R^q\\text{pr}_*\\mathcal{F})_{\\overline{y}} =", "H^q(\\mathbf{A}^{n + m}_K \\times_{A^n_K} \\Spec(\\mathcal{O}_{Y, y}^{sh}),", "\\mathcal{F}) =", "H^q(\\mathbf{A}^m_{\\mathcal{O}_{Y, y}^{sh}}, \\mathcal{F})", "$$", "Say $b = \\text{trdeg}_K(\\kappa(y))$. From Lemma \\ref{lemma-strictly-henselian}", "we get an embedding", "$$", "L = K(t_1, \\ldots, t_b)^{sep} \\longrightarrow \\mathcal{O}_{Y, y}^{sh}", "$$", "Write $\\mathcal{O}_{Y, y}^{sh} = \\colim B_i$ as the filtered", "colimit of finite type $L$-subalgebras $B_i \\subset \\mathcal{O}_{Y, y}^{sh}$", "containing the ring $K[T_1, \\ldots, T_n]$ of regular functions", "on $\\mathbf{A}^n_K$. Then we get", "$$", "\\mathbf{A}^m_{\\mathcal{O}_{Y, y}^{sh}} =", "\\lim \\mathbf{A}^m_{B_i}", "$$", "If $z \\in \\mathbf{A}^m_{B_i}$ is a point in the support of", "$\\mathcal{F}$, then the image $x$ of $z$ in $\\mathbf{A}^{m + n}_K$", "satisfies $\\text{trdeg}_K(\\kappa(x)) \\leq a$ by our assumption on", "$\\mathcal{F}$ in the lemma.", "Since $\\mathcal{O}_{Y, y}^{sh}$ is a filtered colimit of", "\\'etale algebras over $K[T_1, \\ldots, T_n]$ and since", "$B_i \\subset \\mathcal{O}_{Y, y}^{sh}$", "we see that $\\kappa(z)/\\kappa(x)$ is algebraic", "(some details omitted). Then $\\text{trdeg}_K(\\kappa(z)) \\leq a$", "and hence $\\text{trdeg}_L(\\kappa(z)) \\leq a - b$.", "By Lemma \\ref{lemma-interlude-II} we see that", "$$", "H^q(\\mathbf{A}^m_{B_i}, \\mathcal{F}) = 0\\text{ for }q > a - b", "$$", "Thus by Theorem \\ref{theorem-colimit} we get", "$(Rf_*\\mathcal{F})_{\\overline{y}} = 0$ for $q > a - b$", "as desired." ], "refs": [ "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-theorem-higher-direct-images", "etale-cohomology-lemma-strictly-henselian", "etale-cohomology-lemma-interlude-II", "etale-cohomology-theorem-colimit" ], "ref_ids": [ 6703, 6385, 6636, 6637, 6384 ] } ], "ref_ids": [] }, { "id": 6639, "type": "theorem", "label": "etale-cohomology-lemma-finite-cd", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-finite-cd", "contents": [ "Let $K$ be a field.", "\\begin{enumerate}", "\\item If $f : X \\to Y$ is a morphism of finite type schemes over $K$,", "then $\\text{cd}(f) < \\infty$.", "\\item If $\\text{cd}(K) < \\infty$, then $\\text{cd}(X) < \\infty$", "for any finite type scheme $X$ over $K$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). We may assume $Y$ is affine. We will use the", "induction principle of ", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}", "to prove this. If $X$ is affine too, then the result holds by", "Lemma \\ref{lemma-interlude-I}. Thus it suffices to show that if", "$X = U \\cup V$ and the result is true for $U \\to Y$, $V \\to Y$, and", "$U \\cap V \\to Y$, then it is true for $f$. This follows from the", "relative Mayer-Vietoris sequence, see", "Lemma \\ref{lemma-relative-mayer-vietoris}.", "\\medskip\\noindent", "Proof of (2). We will use the induction principle of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}", "to prove this. If $X$ is affine, then the result holds by", "Proposition \\ref{proposition-cd-affine}. Thus it suffices to show that if", "$X = U \\cup V$ and the result is true for $U$, $V$, and", "$U \\cap V $, then it is true for $X$. This follows from the", "Mayer-Vietoris sequence, see", "Lemma \\ref{lemma-mayer-vietoris}." ], "refs": [ "coherent-lemma-induction-principle", "etale-cohomology-lemma-interlude-I", "etale-cohomology-lemma-relative-mayer-vietoris", "coherent-lemma-induction-principle", "etale-cohomology-proposition-cd-affine", "etale-cohomology-lemma-mayer-vietoris" ], "ref_ids": [ 3291, 6638, 6470, 3291, 6707, 6469 ] } ], "ref_ids": [] }, { "id": 6640, "type": "theorem", "label": "etale-cohomology-lemma-finite-cd-mod-n-direct-sums", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-finite-cd-mod-n-direct-sums", "contents": [ "Cohomology and direct sums. Let $n \\geq 1$ be an integer.", "\\begin{enumerate}", "\\item Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism", "of schemes with $\\text{cd}(f) < \\infty$. Then the functor", "$$", "Rf_* :", "D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})", "\\longrightarrow", "D(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})", "$$", "commutes with direct sums.", "\\item Let $X$ be a quasi-compact and quasi-separated scheme with", "$\\text{cd}(X) < \\infty$. Then the functor", "$$", "R\\Gamma(X, -) :", "D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})", "\\longrightarrow", "D(\\mathbf{Z}/n\\mathbf{Z})", "$$", "commutes with direct sums.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Since $\\text{cd}(f) < \\infty$ we see that", "$$", "f_* :", "\\textit{Mod}(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})", "\\longrightarrow", "\\textit{Mod}(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})", "$$", "has finite cohomological dimension in the sense of", "Derived Categories, Lemma \\ref{derived-lemma-unbounded-right-derived}.", "Let $I$ be a set and for $i \\in I$ let $E_i$ be", "an object of $D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$.", "Choose a K-injective complex $\\mathcal{I}_i^\\bullet$", "of $\\mathbf{Z}/n\\mathbf{Z}$-modules each of whose", "terms $\\mathcal{I}_i^n$ is an injective sheaf of", "$\\mathbf{Z}/n\\mathbf{Z}$-modules representing $E_i$.", "See Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "Then $\\bigoplus E_i$ is represented by the complex", "$\\bigoplus \\mathcal{I}_i^\\bullet$ (termwise direct sum), see", "Injectives, Lemma \\ref{injectives-lemma-derived-products}.", "By Lemma \\ref{lemma-relative-colimit} we have", "$$", "R^qf_*(\\bigoplus \\mathcal{I}_i^n) =", "\\bigoplus R^qf_*(\\mathcal{I}_i^n) = 0", "$$", "for $q > 0$ and any $n$. Hence we conclude by ", "Derived Categories, Lemma \\ref{derived-lemma-unbounded-right-derived}", "that we may compute $Rf_*(\\bigoplus E_i)$ by the complex", "$$", "f_*(\\bigoplus \\mathcal{I}_i^\\bullet) =", "\\bigoplus f_*(\\mathcal{I}_i^\\bullet)", "$$", "(equality again by Lemma \\ref{lemma-relative-colimit}) which", "represents $\\bigoplus Rf_*E_i$ by the already used", "Injectives, Lemma \\ref{injectives-lemma-derived-products}.", "\\medskip\\noindent", "Proof of (2). This is identical to the proof of (1)", "and we omit it." ], "refs": [ "derived-lemma-unbounded-right-derived", "injectives-theorem-K-injective-embedding-grothendieck", "injectives-lemma-derived-products", "etale-cohomology-lemma-relative-colimit", "derived-lemma-unbounded-right-derived", "etale-cohomology-lemma-relative-colimit", "injectives-lemma-derived-products" ], "ref_ids": [ 1917, 7768, 7795, 6475, 1917, 6475, 7795 ] } ], "ref_ids": [] }, { "id": 6641, "type": "theorem", "label": "etale-cohomology-lemma-proper-mod-n-direct-sums", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-proper-mod-n-direct-sums", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes. Let $n \\geq 1$", "be an integer. Then the functor", "$$", "Rf_* :", "D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})", "\\longrightarrow", "D(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})", "$$", "commutes with direct sums." ], "refs": [], "proofs": [ { "contents": [ "It is enough to prove this when $Y$ is quasi-compact. By", "Morphisms, Lemma \\ref{morphisms-lemma-morphism-finite-type-bounded-dimension}", "we see that the dimension of the fibres of", "$f : X \\to Y$ is bounded.", "Thus Lemma \\ref{lemma-cohomological-dimension-proper}", "implies that $\\text{cd}(f) < \\infty$. Hence the", "result by Lemma \\ref{lemma-finite-cd-mod-n-direct-sums}." ], "refs": [ "morphisms-lemma-morphism-finite-type-bounded-dimension", "etale-cohomology-lemma-cohomological-dimension-proper", "etale-cohomology-lemma-finite-cd-mod-n-direct-sums" ], "ref_ids": [ 5281, 6629, 6640 ] } ], "ref_ids": [] }, { "id": 6642, "type": "theorem", "label": "etale-cohomology-lemma-pull-out-constant-mod-n", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-pull-out-constant-mod-n", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme", "such that $\\text{cd}(X) < \\infty$. Let $n \\geq 1$ be an integer.", "Let $E \\in D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$ and", "$K \\in D(\\mathbf{Z}/n\\mathbf{Z})$.", "Then", "$$", "R\\Gamma(X, E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} \\underline{K}) =", "R\\Gamma(X, E) \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} K", "$$" ], "refs": [], "proofs": [ { "contents": [ "There is a canonical map from left to right by", "Cohomology on Sites, Section \\ref{sites-cohomology-section-projection-formula}.", "Let $T(K)$ be the property that the statement of the", "lemma holds for $K \\in D(\\mathbf{Z}/n\\mathbf{Z})$.", "We will check conditions (1), (2), and (3) of", "More on Algebra, Remark \\ref{more-algebra-remark-P-resolution}", "hold for $T$ to conclude.", "Property (1) holds because both sides of the equality", "commute with direct sums, see", "Lemma \\ref{lemma-finite-cd-mod-n-direct-sums}.", "Property (2) holds because we are comparing exact functors", "between triangulated categories and we can use", "Derived Categories, Lemma \\ref{derived-lemma-third-isomorphism-triangle}.", "Property (3) says the lemma holds when $K = \\mathbf{Z}/n\\mathbf{Z}[k]$", "for any shift $k \\in \\mathbf{Z}$ and this is obvious." ], "refs": [ "more-algebra-remark-P-resolution", "etale-cohomology-lemma-finite-cd-mod-n-direct-sums", "derived-lemma-third-isomorphism-triangle" ], "ref_ids": [ 10653, 6640, 1759 ] } ], "ref_ids": [] }, { "id": 6643, "type": "theorem", "label": "etale-cohomology-lemma-projection-formula-proper-mod-n", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-projection-formula-proper-mod-n", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes. Let $n \\geq 1$ be", "an integer. Let $E \\in D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$ and", "$K \\in D(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})$. Then", "$$", "Rf_*E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} K =", "Rf_*(E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} f^{-1}K)", "$$", "in $D(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})$." ], "refs": [], "proofs": [ { "contents": [ "There is a canonical map from left to right by", "Cohomology on Sites, Section \\ref{sites-cohomology-section-projection-formula}.", "We will check the equality on stalks at $\\overline{y}$.", "By the proper base change (in the form of", "Lemma \\ref{lemma-proper-base-change-mod-n} where $Y' = \\overline{y}$)", "this reduces to the case where $Y$ is the spectrum of", "an algebraically closed field.", "This is shown in Lemma \\ref{lemma-pull-out-constant-mod-n}", "where we use that $\\text{cd}(X) < \\infty$ by", "Lemma \\ref{lemma-cohomological-dimension-proper}." ], "refs": [ "etale-cohomology-lemma-proper-base-change-mod-n", "etale-cohomology-lemma-pull-out-constant-mod-n", "etale-cohomology-lemma-cohomological-dimension-proper" ], "ref_ids": [ 6630, 6642, 6629 ] } ], "ref_ids": [] }, { "id": 6644, "type": "theorem", "label": "etale-cohomology-lemma-kunneth-one-proper", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-kunneth-one-proper", "contents": [ "Let $k$ be a separably closed field. Let $X$ be a proper scheme over $k$.", "Let $Y$ be a quasi-compact and quasi-separated scheme over $k$.", "\\begin{enumerate}", "\\item If $E \\in D^+(X_\\etale)$ has torsion cohomology sheaves and", "$K \\in D^+(Y_\\etale)$, then", "$$", "R\\Gamma(X \\times_{\\Spec(k)} Y,", "\\text{pr}_1^{-1}E", "\\otimes_\\mathbf{Z}^\\mathbf{L}", "\\text{pr}_2^{-1}K", ")", "=", "R\\Gamma(X, E)", "\\otimes_\\mathbf{Z}^\\mathbf{L}", "R\\Gamma(Y, K)", "$$", "\\item If $n \\geq 1$ is an integer, $Y$ is of finite type over $k$,", "$E \\in D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$, and", "$K \\in D(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})$, then", "$$", "R\\Gamma(X \\times_{\\Spec(k)} Y,", "\\text{pr}_1^{-1}E", "\\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L}", "\\text{pr}_2^{-1}K", ")", "=", "R\\Gamma(X, E)", "\\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L}", "R\\Gamma(Y, K)", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). By Lemma \\ref{lemma-projection-formula-proper} we have", "$$", "R\\text{pr}_{2, *}(", "\\text{pr}_1^{-1}E", "\\otimes_\\mathbf{Z}^\\mathbf{L}", "\\text{pr}_2^{-1}K) =", "R\\text{pr}_{2, *}(\\text{pr}_1^{-1}E)", "\\otimes_\\mathbf{Z}^\\mathbf{L}", "K", "$$", "By proper base change (in the form of Lemma \\ref{lemma-proper-base-change})", "this is equal to the object", "$$", "\\underline{R\\Gamma(X, E)}", "\\otimes_\\mathbf{Z}^\\mathbf{L}", "K", "$$", "of $D(Y_\\etale)$. Taking $R\\Gamma(Y, -)$ on this object reproduces the", "left hand side of the equality in (1) by the Leray spectral sequence", "for $\\text{pr}_2$. Thus we conclude by", "Lemma \\ref{lemma-pull-out-constant}.", "\\medskip\\noindent", "Proof of (2). This is exactly the same as the proof of (1)", "except that we use Lemmas \\ref{lemma-projection-formula-proper-mod-n},", "\\ref{lemma-proper-base-change-mod-n}, and", "\\ref{lemma-pull-out-constant-mod-n} as well as", "$\\text{cd}(Y) < \\infty$ by Lemma \\ref{lemma-finite-cd}." ], "refs": [ "etale-cohomology-lemma-projection-formula-proper", "etale-cohomology-lemma-proper-base-change", "etale-cohomology-lemma-pull-out-constant", "etale-cohomology-lemma-projection-formula-proper-mod-n", "etale-cohomology-lemma-proper-base-change-mod-n", "etale-cohomology-lemma-pull-out-constant-mod-n", "etale-cohomology-lemma-finite-cd" ], "ref_ids": [ 6632, 6626, 6631, 6643, 6630, 6642, 6639 ] } ], "ref_ids": [] }, { "id": 6645, "type": "theorem", "label": "etale-cohomology-lemma-supported-in-closed-points", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-supported-in-closed-points", "contents": [ "Let $K$ be a separably closed field. Let $X$ be a scheme of finite", "type over $K$. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$", "whose support is contained in the set of closed points of $X$.", "Then $H^q(X, \\mathcal{F}) = 0$ for $q > 0$ and $\\mathcal{F}$", "is globally generated." ], "refs": [], "proofs": [ { "contents": [ "(If $\\mathcal{F}$ is torsion, then the vanishing follows immediately", "from Lemma \\ref{lemma-interlude-II}.)", "By Lemma \\ref{lemma-support-in-subset} we can write $\\mathcal{F}$", "as a filtered colimit of constructible sheaves $\\mathcal{F}_i$", "of $\\mathbf{Z}$-modules whose supports $Z_i \\subset X$ are finite", "sets of closed points. By", "Proposition \\ref{proposition-closed-immersion-pushforward}", "such a sheaf is of the form $(Z_i \\to X)_*\\mathcal{G}_i$", "where $\\mathcal{G}_i$ is a sheaf on $Z_i$. As $K$ is separably closed,", "the scheme $Z_i$ is a finite disjoint union of spectra of separably closed", "fields. Recall that $H^q(Z_i, \\mathcal{G}_i) = H^q(X, \\mathcal{F}_i)$", "by the Leray spectral sequence for $Z_i \\to X$ and vanising of higher", "direct images for this morphism", "(Proposition \\ref{proposition-finite-higher-direct-image-zero}).", "By Lemmas \\ref{lemma-equivalence-abelian-sheaves-point} and", "\\ref{lemma-compare-cohomology-point}", "we see that $H^q(Z_i, \\mathcal{G}_i)$ is zero for $q > 0$", "and that $H^0(Z_i, \\mathcal{G}_i)$ generates $\\mathcal{G}_i$.", "We conclude the vanishing of $H^q(X, \\mathcal{F}_i)$ for $q > 0$", "and that $\\mathcal{F}_i$ is generated by global sections.", "By Theorem \\ref{theorem-colimit} we see that", "$H^q(X, \\mathcal{F}) = 0$ for $q > 0$.", "The proof is now done because a", "filtered colimit of globally generated sheaves of abelian groups", "is globally generated (details omitted)." ], "refs": [ "etale-cohomology-lemma-interlude-II", "etale-cohomology-lemma-support-in-subset", "etale-cohomology-proposition-closed-immersion-pushforward", "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-lemma-equivalence-abelian-sheaves-point", "etale-cohomology-lemma-compare-cohomology-point", "etale-cohomology-theorem-colimit" ], "ref_ids": [ 6637, 6554, 6700, 6703, 6489, 6490, 6384 ] } ], "ref_ids": [] }, { "id": 6646, "type": "theorem", "label": "etale-cohomology-lemma-vanishing-closed-points", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-vanishing-closed-points", "contents": [ "Let $K$ be a separably closed field. Let $X$ be a scheme of finite", "type over $K$. Let $Q \\in D(X_\\etale)$. Assume that $Q_{\\overline{x}}$", "is nonzero only if $x$ is a closed point of $X$. Then", "$$", "Q = 0 \\Leftrightarrow H^i(X, Q) = 0 \\text{ for all }i", "$$" ], "refs": [], "proofs": [ { "contents": [ "The implication from left to right is trivial. Thus we need", "to prove the reverse implication.", "\\medskip\\noindent", "Assume $Q$ is bounded below; this cases suffices for almost all", "applications. If $Q$ is not zero, then we can look at the smallest $i$", "such that the cohomology sheaf $H^i(Q)$ is nonzero.", "By Lemma \\ref{lemma-supported-in-closed-points} we have", "$H^i(X, Q) = H^0(X, H^i(Q)) \\not = 0$ and we conclude.", "\\medskip\\noindent", "General case. Let $\\mathcal{B} \\subset \\Ob(X_\\etale)$ be the", "quasi-compact objects. By Lemma \\ref{lemma-supported-in-closed-points}", "the assumptions of Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-cohomology-over-U-trivial}", "are satisfied. We conclude that $H^q(U, Q) = H^0(U, H^q(Q))$", "for all $U \\in \\mathcal{B}$. In particular, this holds for $U = X$.", "Thus the conclusion by Lemma \\ref{lemma-supported-in-closed-points}", "as $Q$ is zero in $D(X_\\etale)$ if and only if $H^q(Q)$ is zero", "for all $q$." ], "refs": [ "etale-cohomology-lemma-supported-in-closed-points", "etale-cohomology-lemma-supported-in-closed-points", "sites-cohomology-lemma-cohomology-over-U-trivial", "etale-cohomology-lemma-supported-in-closed-points" ], "ref_ids": [ 6645, 6645, 4274, 6645 ] } ], "ref_ids": [] }, { "id": 6647, "type": "theorem", "label": "etale-cohomology-lemma-kunneth-localize-on-X", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-kunneth-localize-on-X", "contents": [ "Let $K$ be a field. Let $j : U \\to X$ be an open immersion of", "schemes of finite type over $K$. Let $Y$ be a scheme of finite type", "over $K$. Consider the diagram", "$$", "\\xymatrix{", "Y \\times_{\\Spec(K)} X \\ar[d]_q &", "Y \\times_{\\Spec(K)} U \\ar[l]^h \\ar[d]^p \\\\", "X & U \\ar[l]_j", "}", "$$", "Then the base change map $q^{-1}Rj_*\\mathcal{F} \\to Rh_*p^{-1}\\mathcal{F}$", "is an isomorphism for $\\mathcal{F}$ an abelian sheaf on $U_\\etale$", "whose stalks are torsion of orders invertible in $K$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\mathcal{F} = \\colim \\mathcal{F}[n]$ where the colimit", "is over the multiplicative system of integers invertible in $K$.", "Since cohomology commutes with filtered colimits in our situation", "(for a precise reference see Lemma \\ref{lemma-base-change-Rf-star-colim}),", "it suffices to prove the lemma for $\\mathcal{F}[n]$. Thus we may", "assume $\\mathcal{F}$ is a sheaf of $\\mathbf{Z}/n\\mathbf{Z}$-modules", "for some $n$ invertible in $K$ (we will use this at the very end of", "the proof). In the proof we use the short hand $X \\times_K Y$ for the fibre", "product over $\\Spec(K)$.", "We will prove the lemma by induction on $\\dim(X) + \\dim(Y)$.", "The lemma is trivial if $\\dim(X) \\leq 0$, since in this case", "$U$ is an open and closed subscheme of $X$.", "Choose a point $z \\in X \\times_K Y$. We will show", "the stalk at $\\overline{z}$ is an isomorphism.", "\\medskip\\noindent", "Suppose that $z \\mapsto x \\in X$ and assume $\\text{trdeg}_K(\\kappa(x)) > 0$.", "Set $X' = \\Spec(\\mathcal{O}_{X, x}^{sh})$", "and denote $U' \\subset X'$ the inverse image of $U$.", "Consider the base change", "$$", "\\xymatrix{", "Y \\times_K X' \\ar[d]_{q'} &", "Y \\times_K U' \\ar[l]^{h'} \\ar[d]^{p'} \\\\", "X' & U' \\ar[l]_{j'}", "}", "$$", "of our diagram by $X' \\to X$. Observe that $X' \\to X$ is a filtered", "colimit of \\'etale morphisms. By smooth base change in the form of", "Lemma \\ref{lemma-smooth-base-change-general} the pullback of", "$q^{-1}Rj_*\\mathcal{F} \\to Rh_*p^{-1}\\mathcal{F}$ to $X'$ to", "$Y \\times_K X'$ is the map", "$(q')^{-1}Rj'_*\\mathcal{F}' \\to Rj'_*(p')^{-1}\\mathcal{F}'$ where", "$\\mathcal{F}'$ is the pullback of $\\mathcal{F}$ to $U'$.", "(In this step it would suffice to use \\'etale base change which is", "an essentially trivial result.) So it suffices to show", "that $(q')^{-1}Rj'_*\\mathcal{F}' \\to Rj'_*(p')^{-1}\\mathcal{F}'$", "is an isomorphism in order to prove that our original", "map is an isomorphism on stalks at $\\overline{z}$.", "By Lemma \\ref{lemma-strictly-henselian}", "there is a separably closed field $L/K$ such that", "$X' = \\lim X_i$ with $X_i$ affine of finite type over $L$", "and $\\dim(X_i) < \\dim(X)$. For $i$ large enough there", "exists an open $U_i \\subset X_i$ restricting to $U'$ in $X'$.", "We may apply the induction hypothesis to the diagram", "$$", "\\vcenter{", "\\xymatrix{", "Y \\times_K X_i \\ar[d]_{q_i} &", "Y \\times_K U_i \\ar[l]^{h_i} \\ar[d]^{p_i} \\\\", "X_i & U_i \\ar[l]_{j_i}", "}", "}", "\\quad\\text{equal to}\\quad", "\\vcenter{", "\\xymatrix{", "Y_L \\times_L X_i \\ar[d]_{q_i} &", "Y_L \\times_L U_i \\ar[l]^{h_i} \\ar[d]^{p_i} \\\\", "X_i & U_i \\ar[l]_{j_i}", "}", "}", "$$", "over the field $L$ and the pullback of $\\mathcal{F}$ to these diagrams.", "By Lemma \\ref{lemma-base-change-Rf-star-colim} we conclude that the map", "$(q')^{-1}Rj'_*\\mathcal{F}' \\to Rj'_*(p')^{-1}\\mathcal{F}$ is an isomorphism.", "\\medskip\\noindent", "Suppose that $z \\mapsto y \\in Y$ and assume $\\text{trdeg}_K(\\kappa(y)) > 0$.", "Let $Y' = \\Spec(\\mathcal{O}_{X, x}^{sh})$.", "By Lemma \\ref{lemma-strictly-henselian} there is a separably closed field", "$L/K$ such that $Y' = \\lim Y_i$ with $Y_i$ affine of finite type over $L$", "and $\\dim(Y_i) < \\dim(Y)$. In particular $Y'$ is a scheme over $L$.", "Denote with a subscript $L$ the base change from schemes over $K$", "to schemes over $L$. Consider the commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "Y' \\times_K X \\ar[d]_f &", "Y' \\times_K U \\ar[l]^{h'} \\ar[d]^{f'} \\\\", "Y \\times_K X \\ar[d]_q &", "Y \\times_K U \\ar[l]^h \\ar[d]^p \\\\", "X & U \\ar[l]_j", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "Y' \\times_L X_L \\ar[d]_{q'} &", "Y' \\times_L U_L \\ar[l]^{h'} \\ar[d]^{p'} \\\\", "X_L \\ar[d] &", "U_L \\ar[l]^{j_L} \\ar[d] \\\\", "X & U \\ar[l]_j", "}", "}", "$$", "and observe the top and bottom rows are the same on the left and the right.", "By smooth base change we see that", "$f^{-1}Rh_*p^{-1}\\mathcal{F} = Rh'_*(f')^{-1}p^{-1}\\mathcal{F}$", "(similarly to the previous paragraph).", "By smooth base change for $\\Spec(L) \\to \\Spec(K)$", "(Lemma \\ref{lemma-base-change-field-extension})", "we see that $Rj_{L, *}\\mathcal{F}_L$ is the pullback of", "$Rj_*\\mathcal{F}$ to $X_L$. Combining these two observations,", "we conclude that it suffices to prove the base change map", "for the upper square in the diagram on the right", "is an isomorphism in order to prove that our original", "map is an isomorphism on stalks at $\\overline{z}$\\footnote{Here we", "use that a ``vertical composition'' of base change maps is a base change", "map as explained in Cohomology on Sites, Remark", "\\ref{sites-cohomology-remark-compose-base-change}.}.", "Then using that $Y' = \\lim Y_i$ and argueing exactly", "as in the previous paragraph we see that the induction", "hypothesis forces our map over $Y' \\times_K X$ to be", "an isomorphism.", "\\medskip\\noindent", "Thus any counter example with $\\dim(X) + \\dim(Y)$ minimal", "would only have nonisomorphisms", "$q^{-1}Rj_*\\mathcal{F} \\to Rh_*p^{-1}\\mathcal{F}$", "on stalks at closed points of $X \\times_K Y$ (because a point", "$z$ of $X \\times_K Y$ is a closed point if and only if", "both the image of $z$ in $X$ and in $Y$ are closed).", "Since it is enough to prove the isomorphism locally,", "we may assume $X$ and $Y$ are affine. However, then", "we can choose an open dense immersion $Y \\to Y'$ with $Y'$", "projective. (Choose a closed immersion $Y \\to \\mathbf{A}^n_K$", "and let $Y'$ be the scheme theoretic closure of $Y$ in", "$\\mathbf{P}^n_K$.) Then $\\dim(Y') = \\dim(Y)$ and hence", "we get a ``minimal'' counter example with $Y$ projective over $K$.", "In the next paragraph we show that this can't happen.", "\\medskip\\noindent", "Consider a diagram as in the statement of the lemma", "such that $q^{-1}Rj_*\\mathcal{F} \\to Rh_*p^{-1}\\mathcal{F}$", "is an isomorphism at all non-closed points of $X \\times_K Y$", "and such that $Y$ is projective.", "The restriction of the map to $(X \\times_K Y)_{K^{sep}}$", "is the corresponding map for the diagram of the lemma", "base changed to $K^{sep}$. Thus we may and do assume $K$", "is separably algebraically closed. Choose a distinguished triangle", "$$", "q^{-1}Rj_*\\mathcal{F} \\to Rh_*p^{-1}\\mathcal{F} \\to Q \\to", "(q^{-1}Rj_*\\mathcal{F})[1]", "$$", "in $D((X \\times_K Y)_\\etale)$. Since $Q$ is supported in", "closed points we see that it suffices to prove", "$H^i(X \\times_K Y, Q) = 0$ for all $i$, see", "Lemma \\ref{lemma-vanishing-closed-points}.", "Thus it suffices to prove that", "$q^{-1}Rj_*\\mathcal{F} \\to Rh_*p^{-1}\\mathcal{F}$", "induces an isomorphism on cohomology. Recall that $\\mathcal{F}$", "is annihilated by $n$ invertible in $K$.", "By the K\\\"unneth formula of Lemma \\ref{lemma-kunneth-one-proper}", "we have", "\\begin{align*}", "R\\Gamma(X \\times_K Y, q^{-1}Rj_*\\mathcal{F})", "&=", "R\\Gamma(X, Rj_*\\mathcal{F})", "\\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L}", "R\\Gamma(Y, \\mathbf{Z}/n\\mathbf{Z}) \\\\", "& =", "R\\Gamma(U, \\mathcal{F})", "\\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L}", "R\\Gamma(Y, \\mathbf{Z}/n\\mathbf{Z})", "\\end{align*}", "and", "$$", "R\\Gamma(X \\times_K Y, Rh_*p^{-1}\\mathcal{F}) =", "R\\Gamma(U \\times_K Y, p^{-1}\\mathcal{F}) =", "R\\Gamma(U, \\mathcal{F})", "\\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L}", "R\\Gamma(Y, \\mathbf{Z}/n\\mathbf{Z})", "$$", "This finishes the proof." ], "refs": [ "etale-cohomology-lemma-base-change-Rf-star-colim", "etale-cohomology-lemma-smooth-base-change-general", "etale-cohomology-lemma-strictly-henselian", "etale-cohomology-lemma-base-change-Rf-star-colim", "etale-cohomology-lemma-strictly-henselian", "etale-cohomology-lemma-base-change-field-extension", "sites-cohomology-remark-compose-base-change", "etale-cohomology-lemma-vanishing-closed-points", "etale-cohomology-lemma-kunneth-one-proper" ], "ref_ids": [ 6596, 6612, 6636, 6596, 6636, 6613, 4425, 6646, 6644 ] } ], "ref_ids": [] }, { "id": 6648, "type": "theorem", "label": "etale-cohomology-lemma-punctual-base-change", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-punctual-base-change", "contents": [ "Let $K$ be a field. For any commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l] \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\", "\\Spec(K) & S' \\ar[l] & T \\ar[l]_g", "}", "$$", "of schemes over $K$ with", "$X' = X \\times_{\\Spec(K)} S'$ and $Y = X' \\times_{S'} T$ and", "$g$ quasi-compact and quasi-separated, and every abelian sheaf", "$\\mathcal{F}$ on $T_\\etale$ whose stalks are torsion of orders", "invertible in $K$ the base change map", "$$", "(f')^{-1}Rg_*\\mathcal{F}", "\\longrightarrow", "Rh_*e^{-1}\\mathcal{F}", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$, hence we may assume $X$ is affine.", "By Limits, Lemma \\ref{limits-lemma-relative-approximation}", "we can write $X = \\lim X_i$ as a cofiltered limit with affine", "transition morphisms of schemes $X_i$ of finite type over $K$.", "Denote $X'_i = X_i \\times_{\\Spec(K)} S'$ and $Y_i = X'_i \\times_{S'} T$.", "By Lemma \\ref{lemma-base-change-Rf-star-colim}", "it suffices to prove the statement for the squares", "with corners $X_i, Y_i, S_i, T_i$.", "Thus we may assume $X$ is of finite type over $K$.", "Similarly, we may write", "$\\mathcal{F} = \\colim \\mathcal{F}[n]$ where the colimit", "is over the multiplicative system of integers invertible in $K$.", "The same lemma used above reduces us to the case where", "$\\mathcal{F}$ is a sheaf of $\\mathbf{Z}/n\\mathbf{Z}$-modules", "for some $n$ invertible in $K$.", "\\medskip\\noindent", "We may replace $K$ by its algebraic closure $\\overline{K}$.", "Namely, formation of direct image commutes with base change", "to $\\overline{K}$ according to", "Lemma \\ref{lemma-base-change-field-extension} (works for both", "$g$ and $h$). And it suffices to prove the agreement after", "restriction to $X'_{\\overline{K}}$. Next, we may replace", "$X$ by its reduction as we have the topological invariance of \\'etale", "cohomology, see", "Proposition \\ref{proposition-topological-invariance}.", "After this replacement the morphism $X \\to \\Spec(K)$", "is flat, finite presentation, with geometrically reduced fibres", "and the same is true for any base change, in particular for $X' \\to S'$.", "Hence $(f')^{-1}g_*\\mathcal{F} \\to Rh_*e^{-1}\\mathcal{F}$", "is an isomorphism by Lemma \\ref{lemma-fppf-reduced-fibres-base-change-f-star}.", "\\medskip\\noindent", "At this point we may apply", "Lemma \\ref{lemma-base-change-does-not-hold-post}", "to see that it suffices to prove: given a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & X' \\ar[d] \\ar[l] & Y \\ar[l]^h \\ar[d] \\\\", "\\Spec(K) & S' \\ar[l] & \\Spec(L) \\ar[l]", "}", "$$", "with both squares cartesian, where $S'$ is affine, integral, and normal", "with algebraically closed function field $K$, then", "$R^qh_*(\\mathbf{Z}/d\\mathbf{Z})$ is zero for $q > 0$ and $d | n$.", "Observe that this vanishing is equivalent to the statement that", "$$", "(f')^{-1}R^q(\\Spec(L) \\to S')_*\\mathbf{Z}/d\\mathbf{Z}", "\\longrightarrow", "R^qh_*\\mathbf{Z}/d\\mathbf{Z}", "$$", "is an isomorphism, because the left hand side is zero for example by", "Lemma \\ref{lemma-Rf-star-zero-normal-with-alg-closed-function-field}.", "\\medskip\\noindent", "Write $S' = \\Spec(B)$ so that $L$ is the fraction field of $B$.", "Write $B = \\bigcup_{i \\in I} B_i$ as the union of its finite type", "$K$-subalgebras $B_i$. Let $J$ be the set of pairs $(i, g)$ where", "$i \\in I$ and $g \\in B_i$ nonzero with ordering", "$(i', g') \\geq (i, g)$ if and only if $i' \\geq i$ and", "$g$ maps to an invertible element of $(B_{i'})_{g'}$.", "Then $L = \\colim_{(i, g) \\in J} (B_i)_g$.", "For $j = (i, g) \\in J$ set $S_j = \\Spec(B_i)$", "and $U_j = \\Spec((B_i)_g)$.", "Then", "$$", "\\vcenter{", "\\xymatrix{", "X' \\ar[d] & Y \\ar[l]^h \\ar[d] \\\\", "S' & \\Spec(L) \\ar[l]", "}", "}", "\\quad\\text{is the colimit of}\\quad", "\\vcenter{", "\\xymatrix{", "X \\times_K S_j \\ar[d] & X \\times_K U_j \\ar[l]^{h_j} \\ar[d] \\\\", "S_j & U_j \\ar[l]", "}", "}", "$$", "Thus we may apply Lemma \\ref{lemma-base-change-Rf-star-colim}", "to see that it suffices to prove", "base change holds in the diagrams on the right which is what we", "proved in Lemma \\ref{lemma-kunneth-localize-on-X}." ], "refs": [ "limits-lemma-relative-approximation", "etale-cohomology-lemma-base-change-Rf-star-colim", "etale-cohomology-lemma-base-change-field-extension", "etale-cohomology-proposition-topological-invariance", "etale-cohomology-lemma-fppf-reduced-fibres-base-change-f-star", "etale-cohomology-lemma-base-change-does-not-hold-post", "etale-cohomology-lemma-Rf-star-zero-normal-with-alg-closed-function-field", "etale-cohomology-lemma-base-change-Rf-star-colim", "etale-cohomology-lemma-kunneth-localize-on-X" ], "ref_ids": [ 15055, 6596, 6613, 6699, 6600, 6615, 6574, 6596, 6647 ] } ], "ref_ids": [] }, { "id": 6649, "type": "theorem", "label": "etale-cohomology-lemma-punctual-base-change-upgrade", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-punctual-base-change-upgrade", "contents": [ "Let $K$ be a field. Let $n \\geq 1$ be invertible in $K$.", "Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l]^p \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\", "\\Spec(K) & S' \\ar[l] & T \\ar[l]_g", "}", "$$", "of schemes with", "$X' = X \\times_{\\Spec(K)} S'$ and $Y = X' \\times_{S'} T$ and", "$g$ quasi-compact and quasi-separated. The canonical map", "$$", "p^{-1}E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} (f')^{-1}Rg_*F", "\\longrightarrow", "Rh_*(h^{-1}p^{-1}E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} e^{-1}F)", "$$", "is an isomorphism if $E$ in $D^+(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$", "has tor amplitude in $[a, \\infty]$ for some $a \\in \\mathbf{Z}$ and", "$F$ in $D^+(T_\\etale, \\mathbf{Z}/n\\mathbf{Z})$." ], "refs": [], "proofs": [ { "contents": [ "This lemma is a generalization of Lemma \\ref{lemma-punctual-base-change}", "to objects of the derived category; the assertion of our lemma is true because", "in Lemma \\ref{lemma-punctual-base-change} the scheme $X$ over $K$", "is arbitrary. We strongly urge the reader to skip the laborious proof", "(alternative: read only the last paragraph).", "\\medskip\\noindent", "We may represent $E$ by a bounded below K-flat complex", "$\\mathcal{E}^\\bullet$ consisting of flat $\\mathbf{Z}/n\\mathbf{Z}$-modules.", "See Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-bounded-below-tor-amplitude}.", "Choose an integer $b$ such that $H^i(F) = 0$ for $i < b$.", "Choose a large integer $N$ and consider the short exact sequence", "$$", "0 \\to \\sigma_{\\geq N + 1}\\mathcal{E}^\\bullet \\to", "\\mathcal{E}^\\bullet \\to", "\\sigma_{\\leq N}\\mathcal{E}^\\bullet \\to 0", "$$", "of stupid truncations. This produces a distinguished triangle", "$E'' \\to E \\to E' \\to E''[1]$ in $D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$.", "For fixed $F$ both sides of the arrow", "in the statement of the lemma are exact functors in $E$. Observe that", "$$", "p^{-1}E'' \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} (f')^{-1}Rg_*F", "\\quad\\text{and}\\quad", "Rh_*(h^{-1}p^{-1}E'' \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} e^{-1}F)", "$$", "are sitting in degrees $\\geq N + b$. Hence, if we can prove the lemma", "for the object $E'$, then we see that the lemma holds in degrees", "$\\leq N + b$ and we will conclude. Some details omitted.", "Thus we may assume $E$ is represented", "by a bounded complex of flat $\\mathbf{Z}/n\\mathbf{Z}$-modules.", "Doing another argument of the same nature, we may assume", "$E$ is given by a single flat $\\mathbf{Z}/n\\mathbf{Z}$-module", "$\\mathcal{E}$.", "\\medskip\\noindent", "Next, we use the same arguments for the variable $F$", "to reduce to the case where $F$ is given by a single", "sheaf of $\\mathbf{Z}/n\\mathbf{Z}$-modules $\\mathcal{F}$.", "Say $\\mathcal{F}$ is annihilated by an integer $m | n$.", "If $\\ell$ is a prime number dividing $m$ and $m > \\ell$,", "then we can look at the short exact sequence", "$0 \\to \\mathcal{F}[\\ell] \\to \\mathcal{F} \\to", "\\mathcal{F}/\\mathcal{F}[\\ell] \\to 0$", "and reduce to smaller $m$. This finally reduces us to", "the case where $\\mathcal{F}$ is annihilated by a prime", "number $\\ell$ dividing $n$.", "In this case observe that", "$$", "p^{-1}\\mathcal{E}", "\\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L}", "(f')^{-1}Rg_*\\mathcal{F}", "=", "p^{-1}(\\mathcal{E}/\\ell \\mathcal{E})", "\\otimes_{\\mathbf{F}_\\ell}^\\mathbf{L}", "(f')^{-1}Rg_*\\mathcal{F}", "$$", "by the flatness of $\\mathcal{E}$. Similarly for the other term.", "This reduces us to the case where we are working with sheaves", "of $\\mathbf{F}_\\ell$-vector spaces which is discussed", "\\medskip\\noindent", "Assume $\\ell$ is a prime number invertible in $K$.", "Assume $\\mathcal{E}$, $\\mathcal{F}$ are sheaves of", "$\\mathbf{F}_\\ell$-vector spaces on $X_\\etale$ and $T_\\etale$.", "We want to show that", "$$", "p^{-1}\\mathcal{E} \\otimes_{\\mathbf{F}_\\ell} (f')^{-1}R^qg_*\\mathcal{F}", "\\longrightarrow", "R^qh_*(h^{-1}p^{-1}\\mathcal{E} \\otimes_{\\mathbf{F}_\\ell} e^{-1}\\mathcal{F})", "$$", "is an isomorphism for every $q \\geq 0$. This question is local on $X$", "hence we may assume $X$ is affine. We can write $\\mathcal{E}$", "as a filtered colimit of constructible sheaves of", "$\\mathbf{F}_\\ell$-vector spaces on $X_\\etale$, see", "Lemma \\ref{lemma-torsion-colimit-constructible}.", "Since tensor products commute", "with filtered colimits and since", "higher direct images do too (Lemma \\ref{lemma-relative-colimit})", "we may assume $\\mathcal{E}$ is a constructible sheaf of", "$\\mathbf{F}_\\ell$-vector spaces on $X_\\etale$.", "Then we can choose an integer $m$ and finite and finitely presented morphisms", "$\\pi_i : X_i \\to X$, $i = 1, \\ldots, m$", "such that there is an injective map", "$$", "\\mathcal{E} \\to", "\\bigoplus\\nolimits_{i = 1, \\ldots, m}", "\\pi_{i, *}\\mathbf{F}_\\ell", "$$", "See Lemma \\ref{lemma-constructible-maps-into-constant-general}.", "Observe that the direct sum is a constructible sheaf as well", "(Lemma \\ref{lemma-finite-pushforward-constructible}).", "Thus the cokernel is constructible too", "(Lemma \\ref{lemma-constructible-abelian}).", "By dimension shifting, i.e., induction on $q$,", "on the category of constructible sheaves of", "$\\mathbf{F}_\\ell$-vector spaces on $X_\\etale$, it suffices to prove the", "result for the sheaves $\\pi_{i, *}\\mathbf{F}_\\ell$", "(details omitted; hint: start with proving injectivity for $q = 0$", "for all constructible $\\mathcal{E}$).", "To prove this case we extend the diagram of the lemma to", "$$", "\\xymatrix{", "X_i \\ar[d]^{\\pi_i} &", "X'_i \\ar[l]^{p_i} \\ar[d]^{\\pi'_i} &", "Y_i \\ar[l]^{h_i} \\ar[d]^{\\rho_i} \\\\", "X \\ar[d] & X' \\ar[l]^p \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\", "\\Spec(K) & S' \\ar[l] & T \\ar[l]_g", "}", "$$", "with all squares cartesian. In the equations below we are", "going to use that $R\\pi_{i, *} = \\pi_{i, *}$ and similarly", "for $\\pi'_i$, $\\rho_i$, we are going to use the Leray spectral sequence,", "we are going to use", "Lemma \\ref{lemma-finite-pushforward-commutes-with-base-change}, and", "we are going to use", "Lemma \\ref{lemma-projection-formula-proper-mod-n}", "(although this lemma is almost trivial for finite morphisms) for", "$\\pi_i$, $\\pi'_i$, $\\rho_i$.", "Doing so we see that", "\\begin{align*}", "p^{-1}\\pi_{i, *}\\mathbf{F}_\\ell", "\\otimes_{\\mathbf{F}_\\ell} (f')^{-1}R^qg_*\\mathcal{F}", "& =", "\\pi'_{i, *}\\mathbf{F}_\\ell", "\\otimes_{\\mathbf{F}_\\ell} (f')^{-1}R^qg_*\\mathcal{F} \\\\", "& =", "\\pi'_{i, *}((\\pi'_i)^{-1} (f')^{-1}R^qg_*\\mathcal{F})", "\\end{align*}", "Similarly, we have", "\\begin{align*}", "R^qh_*(h^{-1}p^{-1} \\pi_{i, *}\\mathbf{F}_\\ell", "\\otimes_{\\mathbf{F}_\\ell} e^{-1}\\mathcal{F})", "& =", "R^qh_*(\\rho_{i, *}\\mathbf{F}_\\ell", "\\otimes_{\\mathbf{F}_\\ell} e^{-1}\\mathcal{F}) \\\\", "& =", "R^qh_*(\\rho_i^{-1}e^{-1}\\mathcal{F}) \\\\", "& =", "\\pi'_{i, *}R^qh_{i, *} \\rho_i^{-1}e^{-1}\\mathcal{F})", "\\end{align*}", "Simce", "$R^qh_{i, *} \\rho_i^{-1}e^{-1}\\mathcal{F} = ", "(\\pi'_i)^{-1} (f')^{-1}R^qg_*\\mathcal{F}$ by", "Lemma \\ref{lemma-punctual-base-change}", "we conclude." ], "refs": [ "etale-cohomology-lemma-punctual-base-change", "etale-cohomology-lemma-punctual-base-change", "sites-cohomology-lemma-bounded-below-tor-amplitude", "etale-cohomology-lemma-torsion-colimit-constructible", "etale-cohomology-lemma-relative-colimit", "etale-cohomology-lemma-constructible-maps-into-constant-general", "etale-cohomology-lemma-finite-pushforward-constructible", "etale-cohomology-lemma-constructible-abelian", "etale-cohomology-lemma-finite-pushforward-commutes-with-base-change", "etale-cohomology-lemma-projection-formula-proper-mod-n", "etale-cohomology-lemma-punctual-base-change" ], "ref_ids": [ 6648, 6648, 4374, 6539, 6475, 6553, 6546, 6531, 6481, 6643, 6648 ] } ], "ref_ids": [] }, { "id": 6650, "type": "theorem", "label": "etale-cohomology-lemma-punctual-base-change-upgrade-unbounded", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-punctual-base-change-upgrade-unbounded", "contents": [ "Let $K$ be a field. Let $n \\geq 1$ be invertible in $K$.", "Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l]^p \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\", "\\Spec(K) & S' \\ar[l] & T \\ar[l]_g", "}", "$$", "of schemes of finite type over $K$ with", "$X' = X \\times_{\\Spec(K)} S'$ and $Y = X' \\times_{S'} T$.", "The canonical map", "$$", "p^{-1}E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} (f')^{-1}Rg_*F", "\\longrightarrow", "Rh_*(h^{-1}p^{-1}E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} e^{-1}F)", "$$", "is an isomorphism for $E$ in $D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$", "and $F$ in $D(T_\\etale, \\mathbf{Z}/n\\mathbf{Z})$." ], "refs": [], "proofs": [ { "contents": [ "We will reduce this to", "Lemma \\ref{lemma-punctual-base-change-upgrade}", "using that our functors commute with direct sums.", "We suggest the reader skip the proof.", "Recall that derived tensor product commutes with", "direct sums. Recall that (derived) pullback commutes with direct sums.", "Recall that $Rh_*$ and $Rg_*$ commute with direct sums, see", "Lemmas \\ref{lemma-finite-cd} and", "\\ref{lemma-finite-cd-mod-n-direct-sums}", "(this is where we use our schemes are of finite type", "over $K$).", "\\medskip\\noindent", "To finish the proof we can argue as follows.", "First we write $E = \\text{hocolim} \\tau_{\\leq N} E$.", "Since our functors commute with direct sums, they commute", "with homotopy colimits. Hence it suffices to prove", "the lemma for $E$ bounded above. Similarly for $F$ we", "may assume $F$ is bounded above.", "Then we can represent $E$ by a bounded above complex", "$\\mathcal{E}^\\bullet$ of sheaves of $\\mathbf{Z}/n\\mathbf{Z}$-modules.", "Then", "$$", "\\mathcal{E}^\\bullet = \\colim \\sigma_{\\geq -N}\\mathcal{E}^\\bullet", "$$", "(stupid truncations).", "Thus we may assume $\\mathcal{E}^\\bullet$ is a bounded", "complex of sheaves of $\\mathbf{Z}/n\\mathbf{Z}$-modules.", "For $F$ we choose a bounded above complex of", "flat(!) sheaves of $\\mathbf{Z}/n\\mathbf{Z}$-modules.", "Then we reduce to the case where $F$ is represented", "by a bounded complex of flat sheaves of $\\mathbf{Z}/n\\mathbf{Z}$-modules.", "At this point Lemma \\ref{lemma-punctual-base-change-upgrade}", "kicks in and we conclude." ], "refs": [ "etale-cohomology-lemma-punctual-base-change-upgrade", "etale-cohomology-lemma-finite-cd", "etale-cohomology-lemma-finite-cd-mod-n-direct-sums", "etale-cohomology-lemma-punctual-base-change-upgrade" ], "ref_ids": [ 6649, 6639, 6640, 6649 ] } ], "ref_ids": [] }, { "id": 6651, "type": "theorem", "label": "etale-cohomology-lemma-kunneth", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-kunneth", "contents": [ "Let $k$ be a separably closed field. Let $X$ and $Y$ be", "finite type schemes over $k$. Let $n \\geq 1$ be an integer", "invertible in $k$. Then for", "$E \\in D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$ and", "$K \\in D(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})$", "we have", "$$", "R\\Gamma(X \\times_{\\Spec(k)} Y,", "\\text{pr}_1^{-1}E", "\\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L}", "\\text{pr}_2^{-1}K", ")", "=", "R\\Gamma(X, E)", "\\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L}", "R\\Gamma(Y, K)", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-punctual-base-change-upgrade-unbounded} we have", "$$", "R\\text{pr}_{1, *}(", "\\text{pr}_1^{-1}E", "\\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L}", "\\text{pr}_2^{-1}K) =", "E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L}", "\\underline{R\\Gamma(Y, K)}", "$$", "We conclude by", "Lemma \\ref{lemma-pull-out-constant-mod-n}", "which we may use because", "$\\text{cd}(X) < \\infty$ by Lemma \\ref{lemma-finite-cd}." ], "refs": [ "etale-cohomology-lemma-punctual-base-change-upgrade-unbounded", "etale-cohomology-lemma-pull-out-constant-mod-n", "etale-cohomology-lemma-finite-cd" ], "ref_ids": [ 6650, 6642, 6639 ] } ], "ref_ids": [] }, { "id": 6652, "type": "theorem", "label": "etale-cohomology-lemma-check-zar", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-check-zar", "contents": [ "In the situation above let $K$ be an object of $D^+(X_{affine, chaotic})$.", "Then $K$ is in the essential image of the (fully faithful) functor", "$R\\epsilon_* ; D(X_{affine, Zar}) \\to D(X_{affine, chaotic})$ if and only", "if the following two conditions hold", "\\begin{enumerate}", "\\item $R\\Gamma(\\emptyset, K)$ is zero in $D(\\textit{Ab})$, and", "\\item if $U = V \\cup W$ with $U, V, W \\subset X$ affine open and", "$V, W \\subset U$ standard open", "(Algebra, Definition \\ref{algebra-definition-Zariski-topology}), then", "the map $c^K_{U, V, W, V \\cap W}$ of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-c-square}", "is a quasi-isomorphism.", "\\end{enumerate}" ], "refs": [ "algebra-definition-Zariski-topology", "sites-cohomology-lemma-c-square" ], "proofs": [ { "contents": [ "Except for a snafu having to do with the empty set,", "this follows from the very general Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-descent-squares} whose hypotheses hold by", "Schemes, Lemma \\ref{schemes-lemma-sheaf-on-affines} and", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-descent-squares-helper}.", "\\medskip\\noindent", "To get around the snafu, denote $X_{affine, almost-chaotic}$", "the site where the empty object $\\emptyset$ has two coverings,", "namely, $\\{\\emptyset \\to \\emptyset\\}$ and the empty covering", "(see Sites, Example \\ref{sites-example-site-topological} for a", "discussion). Then we have morphisms of sites", "$$", "X_{affine, Zar} \\to X_{affine, almost-chaotic} \\to X_{affine, chaotic}", "$$", "The argument above works for the first arrow. Then we leave it", "to the reader to see that an object $K$ of $D^+(X_{affine, chaotic})$", "is in the essential image of the (fully faithful) functor", "$D(X_{affine, almost-chaotic}) \\to D(X_{affine, chaotic})$ if and only", "if $R\\Gamma(\\emptyset, K)$ is zero in $D(\\textit{Ab})$." ], "refs": [ "sites-cohomology-lemma-descent-squares", "schemes-lemma-sheaf-on-affines", "sites-cohomology-lemma-descent-squares-helper" ], "ref_ids": [ 4294, 7677, 4295 ] } ], "ref_ids": [ 1445, 4282 ] }, { "id": 6653, "type": "theorem", "label": "etale-cohomology-lemma-compare-injectives", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-compare-injectives", "contents": [ "Let $S$ be a scheme. Let $T$ be an object of $(\\Sch/S)_\\etale$.", "\\begin{enumerate}", "\\item If $\\mathcal{I}$ is injective in $\\textit{Ab}((\\Sch/S)_\\etale)$, then", "\\begin{enumerate}", "\\item $i_f^{-1}\\mathcal{I}$ is injective in $\\textit{Ab}(T_\\etale)$,", "\\item $\\mathcal{I}|_{S_\\etale}$ is injective in $\\textit{Ab}(S_\\etale)$,", "\\end{enumerate}", "\\item If $\\mathcal{I}^\\bullet$ is a K-injective complex", "in $\\textit{Ab}((\\Sch/S)_\\etale)$, then", "\\begin{enumerate}", "\\item $i_f^{-1}\\mathcal{I}^\\bullet$ is a K-injective complex in", "$\\textit{Ab}(T_\\etale)$,", "\\item $\\mathcal{I}^\\bullet|_{S_\\etale}$ is a K-injective complex in", "$\\textit{Ab}(S_\\etale)$,", "\\end{enumerate}", "\\end{enumerate}", "The corresponding statements for modules do not hold." ], "refs": [], "proofs": [ { "contents": [ "Parts (1)(b) and (2)(b)", "follow formally from the fact that the restriction functor", "$\\pi_{S, *} = i_S^{-1}$ is a right adjoint of the exact functor", "$\\pi_S^{-1}$, see", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives} and", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}.", "\\medskip\\noindent", "Parts (1)(a) and (2)(a) can be seen in two ways. First proof: We can use", "that $i_f^{-1}$ is a right adjoint of the exact functor $i_{f, !}$.", "This functor is constructed in", "Topologies, Lemma \\ref{topologies-lemma-put-in-T-etale}", "for sheaves of sets and for abelian sheaves in", "Modules on Sites, Lemma \\ref{sites-modules-lemma-g-shriek-adjoint}.", "It is shown in Modules on Sites, Lemma", "\\ref{sites-modules-lemma-exactness-lower-shriek} that it is exact.", "Second proof. We can use that $i_f = i_T \\circ f_{big}$ as is shown", "in Topologies, Lemma \\ref{topologies-lemma-morphism-big-small-etale}.", "Since $f_{big}$ is a localization, we see that pullback by it", "preserves injectives and K-injectives, see", "Cohomology on Sites, Lemmas \\ref{sites-cohomology-lemma-cohomology-of-open} and", "\\ref{sites-cohomology-lemma-restrict-K-injective-to-open}.", "Then we apply the already proved parts (1)(b) and (2)(b)", "to the functor $i_T^{-1}$ to conclude.", "\\medskip\\noindent", "Let $S = \\Spec(\\mathbf{Z})$ and consider the map", "$2 : \\mathcal{O}_S \\to \\mathcal{O}_S$. This is an injective map", "of $\\mathcal{O}_S$-modules on $S_\\etale$. However, the pullback", "$\\pi_S^*(2) : \\mathcal{O} \\to \\mathcal{O}$ is not injective", "as we see by evaluating on $\\Spec(\\mathbf{F}_2)$. Now choose", "an injection $\\alpha : \\mathcal{O} \\to \\mathcal{I}$ into an injective", "$\\mathcal{O}$-module $\\mathcal{I}$ on $(\\Sch/S)_\\etale$.", "Then consider the diagram", "$$", "\\xymatrix{", "\\mathcal{O}_S \\ar[d]_2 \\ar[rr]_{\\alpha|_{S_\\etale}} & &", "\\mathcal{I}|_{S_\\etale} \\\\", "\\mathcal{O}_S \\ar@{..>}[rru]", "}", "$$", "Then the dotted arrow cannot exist in the category of $\\mathcal{O}_S$-modules", "because it would mean", "(by adjunction) that the injective map $\\alpha$ factors through", "the noninjective map $\\pi_S^*(2)$ which cannot be the case.", "Thus $\\mathcal{I}|_{S_\\etale}$ is not an injective $\\mathcal{O}_S$-module." ], "refs": [ "homology-lemma-adjoint-preserve-injectives", "derived-lemma-adjoint-preserve-K-injectives", "topologies-lemma-put-in-T-etale", "sites-modules-lemma-g-shriek-adjoint", "sites-modules-lemma-exactness-lower-shriek", "topologies-lemma-morphism-big-small-etale", "sites-cohomology-lemma-cohomology-of-open", "sites-cohomology-lemma-restrict-K-injective-to-open" ], "ref_ids": [ 12116, 1915, 12452, 14164, 14165, 12455, 4186, 4253 ] } ], "ref_ids": [] }, { "id": 6654, "type": "theorem", "label": "etale-cohomology-lemma-compare-higher-direct-image", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-compare-higher-direct-image", "contents": [ "Let $f : T \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item For $K$ in $D((\\Sch/T)_\\etale)$ we have", "$", "(Rf_{big, *}K)|_{S_\\etale} = Rf_{small, *}(K|_{T_\\etale})", "$", "in $D(S_\\etale)$.", "\\item For $K$ in $D((\\Sch/T)_\\etale, \\mathcal{O})$ we have", "$", "(Rf_{big, *}K)|_{S_\\etale} = Rf_{small, *}(K|_{T_\\etale})", "$", "in $D(\\textit{Mod}(S_\\etale, \\mathcal{O}_S))$.", "\\end{enumerate}", "More generally, let $g : S' \\to S$ be an object of $(\\Sch/S)_\\etale$.", "Consider the fibre product", "$$", "\\xymatrix{", "T' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "Then", "\\begin{enumerate}", "\\item[(3)] For $K$ in $D((\\Sch/T)_\\etale)$ we have", "$i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$", "in $D(S'_\\etale)$.", "\\item[(4)] For $K$ in $D((\\Sch/T)_\\etale, \\mathcal{O})$ we have", "$i_g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$", "in $D(\\textit{Mod}(S'_\\etale, \\mathcal{O}_{S'}))$.", "\\item[(5)] For $K$ in $D((\\Sch/T)_\\etale)$ we have", "$g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)$", "in $D((\\Sch/S')_\\etale)$.", "\\item[(6)] For $K$ in $D((\\Sch/T)_\\etale, \\mathcal{O})$ we have", "$g_{big}^*(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^*K)$", "in $D(\\textit{Mod}(S'_\\etale, \\mathcal{O}_{S'}))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from", "Lemma \\ref{lemma-compare-injectives}", "and (\\ref{equation-compare-big-small})", "on choosing a K-injective complex of abelian sheaves representing $K$.", "\\medskip\\noindent", "Part (3) follows from Lemma \\ref{lemma-compare-injectives}", "and Topologies, Lemma", "\\ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale}", "on choosing a K-injective complex of abelian sheaves representing $K$.", "\\medskip\\noindent", "Part (5) is Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-localize-cartesian-square}.", "\\medskip\\noindent", "Part (6) is Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-localize-cartesian-square-modules}.", "\\medskip\\noindent", "Part (2) can be proved as follows. Above we have seen", "that $\\pi_S \\circ f_{big} = f_{small} \\circ \\pi_T$ as morphisms", "of ringed sites. Hence we obtain", "$R\\pi_{S, *} \\circ Rf_{big, *} = Rf_{small, *} \\circ R\\pi_{T, *}$", "by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-derived-pushforward-composition}.", "Since the restriction functors $\\pi_{S, *}$ and $\\pi_{T, *}$", "are exact, we conclude.", "\\medskip\\noindent", "Part (4) follows from part (6) and part (2) applied to $f' : T' \\to S'$." ], "refs": [ "etale-cohomology-lemma-compare-injectives", "etale-cohomology-lemma-compare-injectives", "topologies-lemma-morphism-big-small-cartesian-diagram-etale", "sites-cohomology-lemma-localize-cartesian-square", "sites-cohomology-lemma-localize-cartesian-square-modules", "sites-cohomology-lemma-derived-pushforward-composition" ], "ref_ids": [ 6653, 6653, 12457, 4263, 4264, 4250 ] } ], "ref_ids": [] }, { "id": 6655, "type": "theorem", "label": "etale-cohomology-lemma-compare-cohomology", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-compare-cohomology", "contents": [ "Let $f : T \\to S$ be a morphism of schemes. Then", "\\begin{enumerate}", "\\item For $K$ in $D(S_\\etale)$ we have", "$H^n_\\etale(S, \\pi_S^{-1}K) = H^n(S_\\etale, K)$.", "\\item For $K$ in $D(S_\\etale, \\mathcal{O}_S)$ we have", "$H^n_\\etale(S, L\\pi_S^*K) = H^n(S_\\etale, K)$.", "\\item For $K$ in $D(S_\\etale)$ we have", "$H^n_\\etale(T, \\pi_S^{-1}K) = H^n(T_\\etale, f_{small}^{-1}K)$.", "\\item For $K$ in $D(S_\\etale, \\mathcal{O}_S)$ we have", "$H^n_\\etale(T, L\\pi_S^*K) = H^n(T_\\etale, Lf_{small}^*K)$.", "\\item For $M$ in $D((\\Sch/S)_\\etale)$ we have", "$H^n_\\etale(T, M) = H^n(T_\\etale, i_f^{-1}M)$.", "\\item For $M$ in $D((\\Sch/S)_\\etale, \\mathcal{O})$ we have", "$H^n_\\etale(T, M) = H^n(T_\\etale, i_f^*M)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove (5) represent $M$ by a K-injective complex of abelian sheaves", "and apply Lemma \\ref{lemma-compare-injectives}", "and work out the definitions. Part (3) follows from", "this as $i_f^{-1}\\pi_S^{-1} = f_{small}^{-1}$. Part (1) is a special", "case of (3).", "\\medskip\\noindent", "Part (6) follows from the very general Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-pullback-same-cohomology}. Then part", "(4) follows because $Lf_{small}^* = i_f^* \\circ L\\pi_S^*$.", "Part (2) is a special case of (4)." ], "refs": [ "etale-cohomology-lemma-compare-injectives", "sites-cohomology-lemma-pullback-same-cohomology" ], "ref_ids": [ 6653, 4339 ] } ], "ref_ids": [] }, { "id": 6656, "type": "theorem", "label": "etale-cohomology-lemma-cohomological-descent-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomological-descent-etale", "contents": [ "Let $S$ be a scheme. For $K \\in D(S_\\etale)$ the map", "$$", "K \\longrightarrow R\\pi_{S, *}\\pi_S^{-1}K", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is true because both $\\pi_S^{-1}$ and $\\pi_{S, *} = i_S^{-1}$", "are exact functors and the composition $\\pi_{S, *} \\circ \\pi_S^{-1}$", "is the identity functor." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6657, "type": "theorem", "label": "etale-cohomology-lemma-compare-higher-direct-image-proper", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-compare-higher-direct-image-proper", "contents": [ "Let $f : T \\to S$ be a proper morphism of schemes. Then we have", "\\begin{enumerate}", "\\item $\\pi_S^{-1} \\circ f_{small, *} = f_{big, *} \\circ \\pi_T^{-1}$", "as functors $\\Sh(T_\\etale) \\to \\Sh((\\Sch/S)_\\etale)$,", "\\item $\\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_T^{-1}K$", "for $K$ in $D^+(T_\\etale)$ whose cohomology sheaves are torsion,", "\\item $\\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_T^{-1}K$", "for $K$ in $D(T_\\etale, \\mathbf{Z}/n\\mathbf{Z})$, and", "\\item $\\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_T^{-1}K$", "for all $K$ in $D(T_\\etale)$ if $f$ is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $\\mathcal{F}$ be a sheaf on $T_\\etale$.", "Let $g : S' \\to S$ be an object of $(\\Sch/S)_\\etale$. Consider the", "fibre product", "$$", "\\xymatrix{", "T' \\ar[r]_{f'} \\ar[d]_{g'} & S' \\ar[d]^g \\\\", "T \\ar[r]^f & S", "}", "$$", "Then we have", "$$", "(f_{big, *}\\pi_T^{-1}\\mathcal{F})(S') =", "(\\pi_T^{-1}\\mathcal{F})(T') =", "((g'_{small})^{-1}\\mathcal{F})(T') =", "(f'_{small, *}(g'_{small})^{-1}\\mathcal{F})(S')", "$$", "the second equality by Lemma \\ref{lemma-describe-pullback}.", "On the other hand", "$$", "(\\pi_S^{-1}f_{small, *}\\mathcal{F})(S') =", "(g_{small}^{-1}f_{small, *}\\mathcal{F})(S')", "$$", "again by Lemma \\ref{lemma-describe-pullback}.", "Hence by proper base change for sheaves of sets", "(Lemma \\ref{lemma-proper-base-change-f-star})", "we conclude the two sets are canonically isomorphic.", "The isomorphism is compatible with restriction mappings", "and defines an isomorphism", "$\\pi_S^{-1}f_{small, *}\\mathcal{F} = f_{big, *}\\pi_T^{-1}\\mathcal{F}$.", "Thus an isomorphism of functors", "$\\pi_S^{-1} \\circ f_{small, *} = f_{big, *} \\circ \\pi_T^{-1}$.", "\\medskip\\noindent", "Proof of (2). There is a canonical base change map", "$\\pi_S^{-1}Rf_{small, *}K \\to Rf_{big, *}\\pi_T^{-1}K$", "for any $K$ in $D(T_\\etale)$, see", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change}.", "To prove it is an isomorphism, it suffices to prove the pull back of", "the base change map by $i_g : \\Sh(S'_\\etale) \\to \\Sh((\\Sch/S)_\\etale)$", "is an isomorphism for any object $g : S' \\to S$ of $(\\Sch/S)_\\etale$.", "Let $T', g', f'$ be as in the previous paragraph.", "The pullback of the base change map is", "\\begin{align*}", "g_{small}^{-1}Rf_{small, *}K", "& =", "i_g^{-1}\\pi_S^{-1}Rf_{small, *}K \\\\", "& \\to", "i_g^{-1}Rf_{big, *}\\pi_T^{-1}K \\\\", "& =", "Rf'_{small, *}(i_{g'}^{-1}\\pi_T^{-1}K) \\\\", "& =", "Rf'_{small, *}((g'_{small})^{-1}K)", "\\end{align*}", "where we have used $\\pi_S \\circ i_g = g_{small}$,", "$\\pi_T \\circ i_{g'} = g'_{small}$, and", "Lemma \\ref{lemma-compare-higher-direct-image}.", "This map is an isomorphism by the proper base change theorem", "(Lemma \\ref{lemma-proper-base-change}) provided $K$ is bounded", "below and the cohomology sheaves of $K$ are torsion.", "\\medskip\\noindent", "The proof of part (3) is the same as the proof of part (2), except", "we use Lemma \\ref{lemma-proper-base-change-mod-n}", "instead of Lemma \\ref{lemma-proper-base-change}.", "\\medskip\\noindent", "Proof of (4). If $f$ is finite, then the functors", "$f_{small, *}$ and $f_{big, *}$ are exact. This follows", "from Proposition \\ref{proposition-finite-higher-direct-image-zero}", "for $f_{small}$. Since any base change $f'$ of $f$ is finite too,", "we conclude from Lemma \\ref{lemma-compare-higher-direct-image} part (3)", "that $f_{big, *}$ is exact too (as the higher derived functors are zero).", "Thus this case follows from part (1)." ], "refs": [ "etale-cohomology-lemma-describe-pullback", "etale-cohomology-lemma-describe-pullback", "etale-cohomology-lemma-proper-base-change-f-star", "sites-cohomology-remark-base-change", "etale-cohomology-lemma-compare-higher-direct-image", "etale-cohomology-lemma-proper-base-change", "etale-cohomology-lemma-proper-base-change-mod-n", "etale-cohomology-lemma-proper-base-change", "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-lemma-compare-higher-direct-image" ], "ref_ids": [ 6438, 6438, 6620, 4424, 6654, 6626, 6630, 6626, 6703, 6654 ] } ], "ref_ids": [] }, { "id": 6658, "type": "theorem", "label": "etale-cohomology-lemma-describe-pullback-pi-fppf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-describe-pullback-pi-fppf", "contents": [ "With notation as above.", "Let $\\mathcal{F}$ be a sheaf on $S_\\etale$. The rule", "$$", "(\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets},\\quad", "(f : X \\to S) \\longmapsto \\Gamma(X, f_{small}^{-1}\\mathcal{F})", "$$", "is a sheaf and a fortiori a sheaf on $(\\Sch/S)_\\etale$.", "In fact this sheaf is equal to", "$\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_\\etale$ and", "$\\epsilon_S^{-1}\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "The statement about the \\'etale topology is the content", "of Lemma \\ref{lemma-describe-pullback}. To finish the proof it", "suffices to show that $\\pi_S^{-1}\\mathcal{F}$ is a sheaf for the fppf", "topology. This is shown in Lemma \\ref{lemma-describe-pullback}", "as well." ], "refs": [ "etale-cohomology-lemma-describe-pullback", "etale-cohomology-lemma-describe-pullback" ], "ref_ids": [ 6438, 6438 ] } ], "ref_ids": [] }, { "id": 6659, "type": "theorem", "label": "etale-cohomology-lemma-push-pull-fppf-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-push-pull-fppf-etale", "contents": [ "With notation as above.", "Let $f : X \\to Y$ be a morphism of $(\\Sch/S)_{fppf}$.", "Then there are commutative diagrams of topoi", "$$", "\\xymatrix{", "\\Sh((\\Sch/X)_{fppf}) \\ar[rr]_{f_{big, fppf}} \\ar[d]_{\\epsilon_X} & &", "\\Sh((\\Sch/Y)_{fppf}) \\ar[d]^{\\epsilon_Y} \\\\", "\\Sh((\\Sch/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & &", "\\Sh((\\Sch/Y)_\\etale)", "}", "$$", "and", "$$", "\\xymatrix{", "\\Sh((\\Sch/X)_{fppf}) \\ar[rr]_{f_{big, fppf}} \\ar[d]_{a_X} & &", "\\Sh((\\Sch/Y)_{fppf}) \\ar[d]^{a_Y} \\\\", "\\Sh(X_\\etale) \\ar[rr]^{f_{small}} & &", "\\Sh(Y_\\etale)", "}", "$$", "with $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$." ], "refs": [], "proofs": [ { "contents": [ "The commutativity of the diagrams follows from the discussion in", "Topologies, Section \\ref{topologies-section-change-topologies}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6660, "type": "theorem", "label": "etale-cohomology-lemma-proper-push-pull-fppf-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-proper-push-pull-fppf-etale", "contents": [ "In Lemma \\ref{lemma-push-pull-fppf-etale} if $f$ is proper, then we have", "$a_Y^{-1} \\circ f_{small, *} = f_{big, fppf, *} \\circ a_X^{-1}$." ], "refs": [ "etale-cohomology-lemma-push-pull-fppf-etale" ], "proofs": [ { "contents": [ "You can prove this by repeating the proof of", "Lemma \\ref{lemma-compare-higher-direct-image-proper} part (1);", "we will instead deduce the result from this.", "As $\\epsilon_{Y, *}$ is the identity functor on underlying presheaves,", "it reflects isomorphisms. The description", "in Lemma \\ref{lemma-describe-pullback-pi-fppf}", "shows that $\\epsilon_{Y, *} \\circ a_Y^{-1} = \\pi_Y^{-1}$", "and similarly for $X$. To show that the canonical map", "$a_Y^{-1}f_{small, *}\\mathcal{F} \\to f_{big, fppf, *}a_X^{-1}\\mathcal{F}$", "is an isomorphism, it suffices to show that", "\\begin{align*}", "\\pi_Y^{-1}f_{small, *}\\mathcal{F}", "& =", "\\epsilon_{Y, *}a_Y^{-1}f_{small, *}\\mathcal{F} \\\\", "& \\to ", "\\epsilon_{Y, *}f_{big, fppf, *}a_X^{-1}\\mathcal{F} \\\\", "& =", "f_{big, \\etale, *} \\epsilon_{X, *}a_X^{-1}\\mathcal{F} \\\\", "& =", "f_{big, \\etale, *}\\pi_X^{-1}\\mathcal{F}", "\\end{align*}", "is an isomorphism. This is part", "(1) of Lemma \\ref{lemma-compare-higher-direct-image-proper}." ], "refs": [ "etale-cohomology-lemma-compare-higher-direct-image-proper", "etale-cohomology-lemma-describe-pullback-pi-fppf", "etale-cohomology-lemma-compare-higher-direct-image-proper" ], "ref_ids": [ 6657, 6658, 6657 ] } ], "ref_ids": [ 6659 ] }, { "id": 6661, "type": "theorem", "label": "etale-cohomology-lemma-descent-sheaf-fppf-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-descent-sheaf-fppf-etale", "contents": [ "In Lemma \\ref{lemma-push-pull-fppf-etale} assume", "$f$ is flat, locally of finite presentation, and surjective.", "Then the functor", "$$", "\\Sh(Y_\\etale) \\longrightarrow", "\\left\\{", "(\\mathcal{G}, \\mathcal{H}, \\alpha)", "\\middle|", "\\begin{matrix}", "\\mathcal{G} \\in \\Sh(X_\\etale),\\ \\mathcal{H} \\in \\Sh((\\Sch/Y)_{fppf}), \\\\", "\\alpha : a_X^{-1}\\mathcal{G} \\to f_{big, fppf}^{-1}\\mathcal{H}", "\\text{ an isomorphism}", "\\end{matrix}", "\\right\\}", "$$", "sending $\\mathcal{F}$ to", "$(f_{small}^{-1}\\mathcal{F}, a_Y^{-1}\\mathcal{F}, can)$ is an equivalence." ], "refs": [ "etale-cohomology-lemma-push-pull-fppf-etale" ], "proofs": [ { "contents": [ "The functor $a_X^{-1}$ is fully faithful (as $a_{X, *}a_X^{-1} = \\text{id}$ by", "Lemma \\ref{lemma-describe-pullback-pi-fppf}). Hence the forgetful functor", "$(\\mathcal{G}, \\mathcal{H}, \\alpha) \\mapsto \\mathcal{H}$ identifies the", "category of triples with a full subcategory of $\\Sh((\\Sch/Y)_{fppf})$.", "Moreover, the functor $a_Y^{-1}$ is fully faithful, hence the functor", "in the lemma is fully faithful as well.", "\\medskip\\noindent", "Suppose that we have an \\'etale covering $\\{Y_i \\to Y\\}$.", "Let $f_i : X_i \\to Y_i$ be the base change of $f$.", "Denote $f_{ij} = f_i \\times f_j : X_i \\times_X X_j \\to Y_i \\times_Y Y_j$.", "Claim: if the lemma is true for $f_i$ and $f_{ij}$ for all $i, j$, then", "the lemma is true for $f$. To see this, note that the given \\'etale covering", "determines an \\'etale covering of the final object in each of", "the four sites $Y_\\etale, X_\\etale, (\\Sch/Y)_{fppf}, (\\Sch/X)_{fppf}$.", "Thus the category of sheaves is equivalent to the category of", "glueing data for this covering", "(Sites, Lemma \\ref{sites-lemma-mapping-property-glue})", "in each of the four cases. A huge commutative diagram of", "categories then finishes the proof of the claim. We omit the details.", "The claim shows that we may work \\'etale locally on $Y$.", "\\medskip\\noindent", "Note that $\\{X \\to Y\\}$ is an fppf covering. Working \\'etale locally on $Y$,", "we may assume there exists a morphism $s : X' \\to X$ such that the composition", "$f' = f \\circ s : X' \\to Y$ is surjective finite locally free, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-dominate-fppf-etale-locally}.", "Claim: if the lemma is true for $f'$, then it is true for $f$.", "Namely, given a triple $(\\mathcal{G}, \\mathcal{H}, \\alpha)$", "for $f$, we can pullback by $s$ to get a triple", "$(s_{small}^{-1}\\mathcal{G}, \\mathcal{H}, s_{big, fppf}^{-1}\\alpha)$", "for $f'$. A solution for this triple gives a sheaf", "$\\mathcal{F}$ on $Y_\\etale$ with $a_Y^{-1}\\mathcal{F} = \\mathcal{H}$.", "By the first paragraph of the proof this means the triple is", "in the essential image. This reduces us to", "the case described in the next paragraph.", "\\medskip\\noindent", "Assume $f$ is surjective finite locally free.", "Let $(\\mathcal{G}, \\mathcal{H}, \\alpha)$ be a triple.", "In this case consider the triple", "$$", "(\\mathcal{G}_1, \\mathcal{H}_1, \\alpha_1) =", "(f_{small}^{-1}f_{small, *}\\mathcal{G},", "f_{big, fppf, *}f_{big, fppf}^{-1}\\mathcal{H}, \\alpha_1)", "$$", "where $\\alpha_1$ comes from the identifications", "\\begin{align*}", "a_X^{-1}f_{small}^{-1}f_{small, *}\\mathcal{G}", "& =", "f_{big, fppf}^{-1}a_Y^{-1}f_{small, *}\\mathcal{G} \\\\", "& =", "f_{big, fppf}^{-1}f_{big, fppf, *}a_X^{-1}\\mathcal{G} \\\\", "& \\to", "f_{big, fppf}^{-1}f_{big, fppf, *}f_{big, fppf}^{-1}\\mathcal{H}", "\\end{align*}", "where the third equality is Lemma \\ref{lemma-proper-push-pull-fppf-etale}", "and the arrow is given by $\\alpha$.", "This triple is in the image of our functor because", "$\\mathcal{F}_1 = f_{small, *}\\mathcal{F}$ is a solution", "(to see this use Lemma \\ref{lemma-proper-push-pull-fppf-etale} again;", "details omitted). There is a canonical map of triples", "$$", "(\\mathcal{G}, \\mathcal{H}, \\alpha)", "\\to", "(\\mathcal{G}_1, \\mathcal{H}_1, \\alpha_1)", "$$", "which uses the unit $\\text{id} \\to f_{big, fppf, *}f_{big, fppf}^{-1}$", "on the second entry (it is enough to prescribe morphisms on the", "second entry by the first paragraph of the proof). Since", "$\\{f : X \\to Y\\}$ is an fppf covering the map", "$\\mathcal{H} \\to \\mathcal{H}_1$ is injective (details omitted).", "Set", "$$", "\\mathcal{G}_2 = \\mathcal{G}_1 \\amalg_\\mathcal{G} \\mathcal{G}_1\\quad", "\\mathcal{H}_2 = \\mathcal{H}_1 \\amalg_\\mathcal{H} \\mathcal{H}_1", "$$", "and let $\\alpha_2$ be the induced isomorphism (pullback functors", "are exact, so this makes sense). Then $\\mathcal{H}$ is the", "equalizer of the two maps $\\mathcal{H}_1 \\to \\mathcal{H}_2$.", "Repeating the arguments above for the triple", "$(\\mathcal{G}_2, \\mathcal{H}_2, \\alpha_2)$", "we find an injective morphism of triples", "$$", "(\\mathcal{G}_2, \\mathcal{H}_2, \\alpha_2)", "\\to", "(\\mathcal{G}_3, \\mathcal{H}_3, \\alpha_3)", "$$", "such that this last triple is in the image of our functor.", "Say it corresponds to $\\mathcal{F}_3$ in $\\Sh(Y_\\etale)$.", "By fully faithfulness we obtain two maps", "$\\mathcal{F}_1 \\to \\mathcal{F}_3$ and we can let", "$\\mathcal{F}$ be the equalizer of these two maps.", "By exactness of the pullback functors involved we", "find that $a_Y^{-1}\\mathcal{F} = \\mathcal{H}$ as desired." ], "refs": [ "etale-cohomology-lemma-describe-pullback-pi-fppf", "sites-lemma-mapping-property-glue", "more-morphisms-lemma-dominate-fppf-etale-locally", "etale-cohomology-lemma-proper-push-pull-fppf-etale", "etale-cohomology-lemma-proper-push-pull-fppf-etale" ], "ref_ids": [ 6658, 8565, 13921, 6660, 6660 ] } ], "ref_ids": [ 6659 ] }, { "id": 6662, "type": "theorem", "label": "etale-cohomology-lemma-compare-fppf-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-compare-fppf-etale", "contents": [ "Consider the comparison morphism", "$\\epsilon : (\\Sch/S)_{fppf} \\to (\\Sch/S)_\\etale$.", "Let $\\mathcal{P}$ denote the class of finite morphisms of schemes.", "For $X$ in $(\\Sch/S)_\\etale$ denote", "$\\mathcal{A}'_X \\subset \\textit{Ab}((\\Sch/X)_\\etale)$", "the full subcategory consisting of sheaves of the form", "$\\pi_X^{-1}\\mathcal{F}$ with $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$.", "Then Cohomology on Sites, Properties", "(\\ref{sites-cohomology-item-base-change-P}),", "(\\ref{sites-cohomology-item-restriction-A}),", "(\\ref{sites-cohomology-item-A-sheaf}),", "(\\ref{sites-cohomology-item-A-and-P}), and", "(\\ref{sites-cohomology-item-refine-tau-by-P})", "of Cohomology on Sites, Situation", "\\ref{sites-cohomology-situation-compare} hold." ], "refs": [], "proofs": [ { "contents": [ "We first show that $\\mathcal{A}'_X \\subset \\textit{Ab}((\\Sch/X)_\\etale)$", "is a weak Serre subcategory by checking conditions (1), (2), (3), and (4)", "of Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}.", "Parts (1), (2), (3) are immediate as $\\pi_X^{-1}$ is exact and", "fully faithful for example by Lemma \\ref{lemma-cohomological-descent-etale}. If", "$0 \\to \\pi_X^{-1}\\mathcal{F} \\to \\mathcal{G} \\to \\pi_X^{-1}\\mathcal{F}' \\to 0$", "is a short exact sequence in $\\textit{Ab}((\\Sch/X)_\\etale)$", "then $0 \\to \\mathcal{F} \\to \\pi_{X, *}\\mathcal{G} \\to \\mathcal{F}' \\to 0$", "is exact by Lemma \\ref{lemma-cohomological-descent-etale}.", "Hence $\\mathcal{G} = \\pi_X^{-1}\\pi_{X, *}\\mathcal{G}$ is in", "$\\mathcal{A}'_X$ which checks the final condition.", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-base-change-P}) holds", "by the existence of fibre products of schemes", "and the fact that the base change of a finite morphism of", "schemes is a finite morphism of schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite}.", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-restriction-A})", "follows from the commutative diagram (3) in", "Topologies, Lemma \\ref{topologies-lemma-morphism-big-small-etale}.", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-A-sheaf}) is", "Lemma \\ref{lemma-describe-pullback-pi-fppf}.", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-A-and-P}) holds by", "Lemma \\ref{lemma-compare-higher-direct-image-proper} part (4).", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-refine-tau-by-P})", "is implied by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-dominate-fppf-etale-locally}." ], "refs": [ "homology-lemma-characterize-weak-serre-subcategory", "etale-cohomology-lemma-cohomological-descent-etale", "etale-cohomology-lemma-cohomological-descent-etale", "morphisms-lemma-base-change-finite", "topologies-lemma-morphism-big-small-etale", "etale-cohomology-lemma-describe-pullback-pi-fppf", "etale-cohomology-lemma-compare-higher-direct-image-proper", "more-morphisms-lemma-dominate-fppf-etale-locally" ], "ref_ids": [ 12046, 6656, 6656, 5440, 12455, 6658, 6657, 13921 ] } ], "ref_ids": [] }, { "id": 6663, "type": "theorem", "label": "etale-cohomology-lemma-V-C-all-n-etale-fppf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-V-C-all-n-etale-fppf", "contents": [ "With notation as above.", "\\begin{enumerate}", "\\item For $X \\in \\Ob((\\Sch/S)_{fppf})$ and an abelian sheaf $\\mathcal{F}$", "on $X_\\etale$ we have", "$\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$", "and $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$.", "\\item For a finite morphism $f : X \\to Y$ in $(\\Sch/S)_{fppf}$", "and abelian sheaf $\\mathcal{F}$ on $X$ we have", "$a_Y^{-1}(R^if_{small, *}\\mathcal{F}) =", "R^if_{big, fppf, *}(a_X^{-1}\\mathcal{F})$", "for all $i$.", "\\item For a scheme $X$ and $K$ in $D^+(X_\\etale)$ the map", "$\\pi_X^{-1}K \\to R\\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism.", "\\item For a finite morphism $f : X \\to Y$ of schemes", "and $K$ in $D^+(X_\\etale)$ we have", "$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$.", "\\item For a proper morphism $f : X \\to Y$ of schemes", "and $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves we have", "$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-compare-fppf-etale} the lemmas in", "Cohomology on Sites, Section \\ref{sites-cohomology-section-compare-general}", "all apply to our current setting. To translate the results", "observe that the category $\\mathcal{A}_X$ of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-A}", "is the essential image of", "$a_X^{-1} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}((\\Sch/X)_{fppf})$.", "\\medskip\\noindent", "Part (1) is equivalent to $(V_n)$ for all $n$ which holds by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-C-all-n-general}.", "\\medskip\\noindent", "Part (2) follows by applying $\\epsilon_Y^{-1}$ to the conclusion of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-C-general}.", "\\medskip\\noindent", "Part (3) follows from Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-V-C-all-n-general} part (1)", "because $\\pi_X^{-1}K$ is in $D^+_{\\mathcal{A}'_X}((\\Sch/X)_\\etale)$", "and $a_X^{-1} = \\epsilon_X^{-1} \\circ a_X^{-1}$.", "\\medskip\\noindent", "Part (4) follows from Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-V-C-all-n-general} part (2)", "for the same reason.", "\\medskip\\noindent", "Part (5). We use that", "\\begin{align*}", "R\\epsilon_{Y, *}Rf_{big, fppf, *}a_X^{-1}K", "& =", "Rf_{big, \\etale, *}R\\epsilon_{X, *}a_X^{-1}K \\\\", "& =", "Rf_{big, \\etale, *}\\pi_X^{-1}K \\\\", "& =", "\\pi_Y^{-1}Rf_{small, *}K \\\\", "& =", "R\\epsilon_{Y, *} a_Y^{-1}Rf_{small, *}K", "\\end{align*}", "The first equality by the commutative diagram in", "Lemma \\ref{lemma-push-pull-fppf-etale}", "and Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-derived-pushforward-composition}.", "The second equality is (3). The third is", "Lemma \\ref{lemma-compare-higher-direct-image-proper} part (2).", "The fourth is (3) again. Thus the base change map", "$a_Y^{-1}(Rf_{small, *}K) \\to Rf_{big, fppf, *}(a_X^{-1}K)$", "induces an isomorphism", "$$", "R\\epsilon_{Y, *}a_Y^{-1}Rf_{small, *}K \\to", "R\\epsilon_{Y, *}Rf_{big, fppf, *}a_X^{-1}K", "$$", "The proof is finished by the following remark: a map", "$\\alpha : a_Y^{-1}L \\to M$ with $L$ in $D^+(Y_\\etale)$", "and $M$ in $D^+((\\Sch/Y)_{fppf})$ such that $R\\epsilon_{Y, *}\\alpha$", "is an isomorphism, is an isomorphism. Namely, ", "we show by induction on $i$ that $H^i(\\alpha)$ is an isomorphism.", "This is true for all sufficiently small $i$.", "If it holds for $i \\leq i_0$, then we see that", "$R^j\\epsilon_{Y, *}H^i(M) = 0$ for $j > 0$ and $i \\leq i_0$", "by (1) because $H^i(M) = a_Y^{-1}H^i(L)$ in this range.", "Hence $\\epsilon_{Y, *}H^{i_0 + 1}(M) = H^{i_0 + 1}(R\\epsilon_{Y, *}M)$", "by a spectral sequence argument.", "Thus $\\epsilon_{Y, *}H^{i_0 + 1}(M) = \\pi_Y^{-1}H^{i_0 + 1}(L) =", "\\epsilon_{Y, *}a_Y^{-1}H^{i_0 + 1}(L)$.", "This implies $H^{i_0 + 1}(\\alpha)$ is an isomorphism", "(because $\\epsilon_{Y, *}$ reflects isomorphisms as it is the", "identity on underlying presheaves) as desired." ], "refs": [ "etale-cohomology-lemma-compare-fppf-etale", "sites-cohomology-lemma-A", "sites-cohomology-lemma-V-C-all-n-general", "sites-cohomology-lemma-V-implies-C-general", "sites-cohomology-lemma-V-C-all-n-general", "sites-cohomology-lemma-V-C-all-n-general", "etale-cohomology-lemma-push-pull-fppf-etale", "sites-cohomology-lemma-derived-pushforward-composition", "etale-cohomology-lemma-compare-higher-direct-image-proper" ], "ref_ids": [ 6662, 4296, 4302, 4297, 4302, 4302, 6659, 4250, 6657 ] } ], "ref_ids": [] }, { "id": 6664, "type": "theorem", "label": "etale-cohomology-lemma-cohomological-descent-etale-fppf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomological-descent-etale-fppf", "contents": [ "Let $X$ be a scheme. For $K \\in D^+(X_\\etale)$ the map", "$$", "K \\longrightarrow Ra_{X, *}a_X^{-1}K", "$$", "is an isomorphism with $a_X : \\Sh((\\Sch/X)_{fppf}) \\to \\Sh(X_\\etale)$", "as above." ], "refs": [], "proofs": [ { "contents": [ "We first reduce the statement to the case where", "$K$ is given by a single abelian sheaf. Namely, represent $K$", "by a bounded below complex $\\mathcal{F}^\\bullet$. By the case of a", "sheaf we see that", "$\\mathcal{F}^n = a_{X, *} a_X^{-1} \\mathcal{F}^n$", "and that the sheaves $R^qa_{X, *}a_X^{-1}\\mathcal{F}^n$", "are zero for $q > 0$. By Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "applied to $a_X^{-1}\\mathcal{F}^\\bullet$", "and the functor $a_{X, *}$ we conclude. From now on assume $K = \\mathcal{F}$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-describe-pullback-pi-fppf} we have", "$a_{X, *}a_X^{-1}\\mathcal{F} = \\mathcal{F}$. Thus it suffices to show that", "$R^qa_{X, *}a_X^{-1}\\mathcal{F} = 0$ for $q > 0$.", "For this we can use $a_X = \\epsilon_X \\circ \\pi_X$ and", "the Leray spectral sequence", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-relative-Leray}).", "By Lemma \\ref{lemma-V-C-all-n-etale-fppf}", "we have $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$", "and", "$\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$.", "By Lemma \\ref{lemma-cohomological-descent-etale} we have", "$R^j\\pi_{X, *}(\\pi_X^{-1}\\mathcal{F}) = 0$ for $j > 0$.", "This concludes the proof." ], "refs": [ "derived-lemma-leray-acyclicity", "etale-cohomology-lemma-describe-pullback-pi-fppf", "sites-cohomology-lemma-relative-Leray", "etale-cohomology-lemma-V-C-all-n-etale-fppf", "etale-cohomology-lemma-cohomological-descent-etale" ], "ref_ids": [ 1844, 6658, 4222, 6663, 6656 ] } ], "ref_ids": [] }, { "id": 6665, "type": "theorem", "label": "etale-cohomology-lemma-compare-cohomology-etale-fppf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-compare-cohomology-etale-fppf", "contents": [ "For a scheme $X$ and $a_X : \\Sh((\\Sch/X)_{fppf}) \\to \\Sh(X_\\etale)$", "as above:", "\\begin{enumerate}", "\\item $H^q(X_\\etale, \\mathcal{F}) = H^q_{fppf}(X, a_X^{-1}\\mathcal{F})$", "for an abelian sheaf $\\mathcal{F}$ on $X_\\etale$,", "\\item $H^q(X_\\etale, K) = H^q_{fppf}(X, a_X^{-1}K)$ for $K \\in D^+(X_\\etale)$.", "\\end{enumerate}", "Example: if $A$ is an abelian group, then", "$H^q_\\etale(X, \\underline{A}) = H^q_{fppf}(X, \\underline{A})$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-cohomological-descent-etale-fppf}", "by Cohomology on Sites, Remark \\ref{sites-cohomology-remark-before-Leray}." ], "refs": [ "etale-cohomology-lemma-cohomological-descent-etale-fppf", "sites-cohomology-remark-before-Leray" ], "ref_ids": [ 6664, 4423 ] } ], "ref_ids": [] }, { "id": 6666, "type": "theorem", "label": "etale-cohomology-lemma-review-quasi-coherent", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-review-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_S$-module on $S_\\etale$.", "\\begin{enumerate}", "\\item The rule", "$$", "\\mathcal{F}^a : (\\Sch/S)_\\etale \\longrightarrow \\textit{Ab},\\quad", "(f : T \\to S) \\longmapsto \\Gamma(T, f_{small}^*\\mathcal{F})", "$$", "satisfies the sheaf condition for fppf and a fortiori \\'etale coverings,", "\\item $\\mathcal{F}^a = \\pi_S^*\\mathcal{F}$ on $(\\Sch/S)_\\etale$,", "\\item $\\mathcal{F}^a = a_S^*\\mathcal{F}$ on $(\\Sch/S)_{fppf}$,", "\\item the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ defines", "an equivalence between quasi-coherent $\\mathcal{O}_S$-modules", "and quasi-coherent modules on", "$((\\Sch/S)_\\etale, \\mathcal{O})$,", "\\item the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ defines", "an equivalence between quasi-coherent $\\mathcal{O}_S$-modules", "and quasi-coherent modules on", "$((\\Sch/S)_{fppf}, \\mathcal{O})$,", "\\item we have $\\epsilon_{S, *}a_S^*\\mathcal{F} = \\pi_S^*\\mathcal{F}$", "and $a_{S, *}a_S^*\\mathcal{F} = \\mathcal{F}$,", "\\item we have $R^i\\epsilon_{S, *}(a_S^*\\mathcal{F}) = 0$", "and $R^ia_{S, *}(a_S^*\\mathcal{F}) = 0$ for $i > 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We urge the reader to find their own proof of these results", "based on the material in", "Descent, Section \\ref{descent-section-quasi-coherent-sheaves}.", "\\medskip\\noindent", "We first explain why the notation in this lemma is consistent with our", "earlier use of the notation $\\mathcal{F}^a$ in", "Sections \\ref{section-quasi-coherent} and", "\\ref{section-cohomology-quasi-coherent}", "and in", "Descent, Section \\ref{descent-section-quasi-coherent-sheaves}.", "Namely, we know by", "Descent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent}", "that there exists a quasi-coherent module", "$\\mathcal{F}_0$ on the scheme $S$ (in other words on the small", "Zariski site) such that $\\mathcal{F}$ is the restriction of the", "rule", "$$", "\\mathcal{F}_0^a : (\\Sch/S)_\\etale \\longrightarrow \\textit{Ab},\\quad", "(f : T \\to S) \\longmapsto \\Gamma(T, f^*\\mathcal{F})", "$$", "to the subcategory $S_\\etale \\subset (\\Sch/S)_\\etale$", "where here $f^*$ denotes usual pullback of sheaves of modules on schemes.", "Since $\\mathcal{F}_0^a$ is pullback by the morphism of ringed", "sites", "$$", "((\\Sch/S)_\\etale, \\mathcal{O}) \\longrightarrow (S_{Zar}, \\mathcal{O}_{S_{Zar}})", "$$", "by Descent, Remark \\ref{descent-remark-change-topologies-ringed-sites}", "it follows immediately (from composition of pullbacks) that", "$\\mathcal{F}^a = \\mathcal{F}_0^a$. This proves the sheaf property", "even for fpqc coverings by", "Descent, Lemma \\ref{descent-lemma-sheaf-condition-holds} (see also", "Proposition \\ref{proposition-quasi-coherent-sheaf-fpqc}).", "Then (2) and (3) follow", "again by Descent, Remark \\ref{descent-remark-change-topologies-ringed-sites}", "and (4) and (5) follow from", "Descent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent}", "(see also the meta result", "Theorem \\ref{theorem-quasi-coherent}).", "\\medskip\\noindent", "Part (6) is immediate from the description of the sheaf", "$\\mathcal{F}^a = \\pi_S^*\\mathcal{F} = a_S^*\\mathcal{F}$.", "\\medskip\\noindent", "For any abelian $\\mathcal{H}$ on $(\\Sch/S)_{fppf}$ the", "higher direct image $R^p\\epsilon_{S, *}\\mathcal{H}$ is the sheaf", "associated to the presheaf $U \\mapsto H^p_{fppf}(U, \\mathcal{H})$", "on $(\\Sch/S)_\\etale$. See", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-higher-direct-images}.", "Hence to prove", "$R^p\\epsilon_{S, *}a_S^*\\mathcal{F} = R^p\\epsilon_{S, *}\\mathcal{F}^a = 0$", "for $p > 0$ it suffices to show that any scheme $U$ over $S$", "has an \\'etale covering $\\{U_i \\to U\\}_{i \\in I}$ such that", "$H^p_{fppf}(U_i, \\mathcal{F}^a) = 0$ for $p > 0$.", "If we take an open covering by affines, then the required", "vanishing follows from comparison with usual cohomology", "(Descent, Proposition \\ref{descent-proposition-same-cohomology-quasi-coherent}", "or", "Theorem \\ref{theorem-zariski-fpqc-quasi-coherent})", "and the vanishing of cohomology of quasi-coherent sheaves", "on affine schemes afforded by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "\\medskip\\noindent", "To show that $R^pa_{S, *}a_S^{-1}\\mathcal{F} = R^pa_{S, *}\\mathcal{F}^a = 0$", "for $p > 0$ we argue in exactly the same manner. This finishes the proof." ], "refs": [ "descent-proposition-equivalence-quasi-coherent", "descent-remark-change-topologies-ringed-sites", "descent-lemma-sheaf-condition-holds", "etale-cohomology-proposition-quasi-coherent-sheaf-fpqc", "descent-remark-change-topologies-ringed-sites", "descent-proposition-equivalence-quasi-coherent", "etale-cohomology-theorem-quasi-coherent", "sites-cohomology-lemma-higher-direct-images", "descent-proposition-same-cohomology-quasi-coherent", "etale-cohomology-theorem-zariski-fpqc-quasi-coherent", "coherent-lemma-quasi-coherent-affine-cohomology-zero" ], "ref_ids": [ 14755, 14793, 14621, 6696, 14793, 14755, 6370, 4189, 14754, 6373, 3282 ] } ], "ref_ids": [] }, { "id": 6667, "type": "theorem", "label": "etale-cohomology-lemma-cohomological-descent-etale-fppf-modules", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomological-descent-etale-fppf-modules", "contents": [ "Let $S$ be a scheme. For $\\mathcal{F}$ a quasi-coherent", "$\\mathcal{O}_S$-module on $S_\\etale$ the maps", "$$", "\\pi_S^*\\mathcal{F} \\longrightarrow R\\epsilon_{S, *}(a_S^*\\mathcal{F})", "\\quad\\text{and}\\quad", "\\mathcal{F} \\longrightarrow Ra_{S, *}(a_S^*\\mathcal{F})", "$$", "are isomorphisms with", "$a_S : \\Sh((\\Sch/S)_{fppf}) \\to \\Sh(S_\\etale)$ as above." ], "refs": [], "proofs": [ { "contents": [ "This is an immediate consequence of", "parts (6) and (7) of", "Lemma \\ref{lemma-review-quasi-coherent}." ], "refs": [ "etale-cohomology-lemma-review-quasi-coherent" ], "ref_ids": [ 6666 ] } ], "ref_ids": [] }, { "id": 6668, "type": "theorem", "label": "etale-cohomology-lemma-describe-pullback-pi-ph", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-describe-pullback-pi-ph", "contents": [ "With notation as above.", "Let $\\mathcal{F}$ be a sheaf on $S_\\etale$. The rule", "$$", "(\\Sch/S)_{ph} \\longrightarrow \\textit{Sets},\\quad", "(f : X \\to S) \\longmapsto \\Gamma(X, f_{small}^{-1}\\mathcal{F})", "$$", "is a sheaf and a fortiori a sheaf on $(\\Sch/S)_\\etale$.", "In fact this sheaf is equal to", "$\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_\\etale$ and", "$\\epsilon_S^{-1}\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_{ph}$." ], "refs": [], "proofs": [ { "contents": [ "The statement about the \\'etale topology is the content", "of Lemma \\ref{lemma-describe-pullback}. To finish the proof it", "suffices to show that $\\pi_S^{-1}\\mathcal{F}$ is a sheaf for the ph", "topology. By Topologies, Lemma \\ref{topologies-lemma-characterize-sheaf}", "it suffices to show that given a proper surjective morphism", "$V \\to U$ of schemes over $S$ we have an equalizer diagram", "$$", "\\xymatrix{", "(\\pi_S^{-1}\\mathcal{F})(U) \\ar[r] &", "(\\pi_S^{-1}\\mathcal{F})(V) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "(\\pi_S^{-1}\\mathcal{F})(V \\times_U V)", "}", "$$", "Set $\\mathcal{G} = \\pi_S^{-1}\\mathcal{F}|_{U_\\etale}$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "V \\times_U V \\ar[r] \\ar[rd]_g \\ar[d] & V \\ar[d]^f \\\\", "V \\ar[r]^f & U", "}", "$$", "We have", "$$", "(\\pi_S^{-1}\\mathcal{F})(V) = \\Gamma(V, f^{-1}\\mathcal{G}) =", "\\Gamma(U, f_*f^{-1}\\mathcal{G})", "$$", "where we use $f_*$ and $f^{-1}$ to denote functorialities between", "small \\'etale sites. Second, we have", "$$", "(\\pi_S^{-1}\\mathcal{F})(V \\times_U V) =", "\\Gamma(V \\times_U V, g^{-1}\\mathcal{G}) =", "\\Gamma(U, g_*g^{-1}\\mathcal{G})", "$$", "The two maps in the equalizer diagram come from the two maps", "$$", "f_*f^{-1}\\mathcal{G} \\longrightarrow g_*g^{-1}\\mathcal{G}", "$$", "Thus it suffices to prove $\\mathcal{G}$ is", "the equalizer of these two maps of sheaves.", "Let $\\overline{u}$ be a geometric point of $U$. Set", "$\\Omega = \\mathcal{G}_{\\overline{u}}$.", "Taking stalks at $\\overline{u}$ by", "Lemma \\ref{lemma-proper-pushforward-stalk}", "we obtain the two maps", "$$", "H^0(V_{\\overline{u}}, \\underline{\\Omega}) \\longrightarrow", "H^0((V \\times_U V)_{\\overline{u}}, \\underline{\\Omega}) =", "H^0(V_{\\overline{u}} \\times_{\\overline{u}} V_{\\overline{u}},", "\\underline{\\Omega})", "$$", "where $\\underline{\\Omega}$ indicates the constant sheaf with value", "$\\Omega$. Of course these maps are the pullback by the projection maps.", "Then it is clear that the sections coming from pullback", "by projection onto the first factor are constant on the fibres of", "the first projection, and sections coming from pullback", "by projection onto the first factor are constant on the fibres of", "the first projection. The sections in the intersection of the images", "of these pullback maps are constant on all of", "$V_{\\overline{u}} \\times_{\\overline{u}} V_{\\overline{u}}$, i.e.,", "these come from elements of $\\Omega$ as desired." ], "refs": [ "etale-cohomology-lemma-describe-pullback", "topologies-lemma-characterize-sheaf", "etale-cohomology-lemma-proper-pushforward-stalk" ], "ref_ids": [ 6438, 12490, 6619 ] } ], "ref_ids": [] }, { "id": 6669, "type": "theorem", "label": "etale-cohomology-lemma-push-pull-ph-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-push-pull-ph-etale", "contents": [ "With notation as above.", "Let $f : X \\to Y$ be a morphism of $(\\Sch/S)_{ph}$.", "Then there are commutative diagrams of topoi", "$$", "\\xymatrix{", "\\Sh((\\Sch/X)_{ph}) \\ar[rr]_{f_{big, ph}} \\ar[d]_{\\epsilon_X} & &", "\\Sh((\\Sch/Y)_{ph}) \\ar[d]^{\\epsilon_Y} \\\\", "\\Sh((\\Sch/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & &", "\\Sh((\\Sch/Y)_\\etale)", "}", "$$", "and", "$$", "\\xymatrix{", "\\Sh((\\Sch/X)_{ph}) \\ar[rr]_{f_{big, ph}} \\ar[d]_{a_X} & &", "\\Sh((\\Sch/Y)_{ph}) \\ar[d]^{a_Y} \\\\", "\\Sh(X_\\etale) \\ar[rr]^{f_{small}} & &", "\\Sh(Y_\\etale)", "}", "$$", "with $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$." ], "refs": [], "proofs": [ { "contents": [ "The commutativity of the diagrams follows from the discussion in", "Topologies, Section \\ref{topologies-section-change-topologies}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6670, "type": "theorem", "label": "etale-cohomology-lemma-proper-push-pull-ph-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-proper-push-pull-ph-etale", "contents": [ "In Lemma \\ref{lemma-push-pull-ph-etale} if $f$ is proper, then we have", "$a_Y^{-1} \\circ f_{small, *} = f_{big, ph, *} \\circ a_X^{-1}$." ], "refs": [ "etale-cohomology-lemma-push-pull-ph-etale" ], "proofs": [ { "contents": [ "You can prove this by repeating the proof of", "Lemma \\ref{lemma-compare-higher-direct-image-proper} part (1);", "we will instead deduce the result from this.", "As $\\epsilon_{Y, *}$ is the identity functor on underlying presheaves,", "it reflects isomorphisms. The description", "in Lemma \\ref{lemma-describe-pullback-pi-ph}", "shows that $\\epsilon_{Y, *} \\circ a_Y^{-1} = \\pi_Y^{-1}$", "and similarly for $X$. To show that the canonical map", "$a_Y^{-1}f_{small, *}\\mathcal{F} \\to f_{big, ph, *}a_X^{-1}\\mathcal{F}$", "is an isomorphism, it suffices to show that", "\\begin{align*}", "\\pi_Y^{-1}f_{small, *}\\mathcal{F}", "& =", "\\epsilon_{Y, *}a_Y^{-1}f_{small, *}\\mathcal{F} \\\\", "& \\to ", "\\epsilon_{Y, *}f_{big, ph, *}a_X^{-1}\\mathcal{F} \\\\", "& =", "f_{big, \\etale, *} \\epsilon_{X, *}a_X^{-1}\\mathcal{F} \\\\", "& =", "f_{big, \\etale, *}\\pi_X^{-1}\\mathcal{F}", "\\end{align*}", "is an isomorphism. This is part", "(1) of Lemma \\ref{lemma-compare-higher-direct-image-proper}." ], "refs": [ "etale-cohomology-lemma-compare-higher-direct-image-proper", "etale-cohomology-lemma-describe-pullback-pi-ph", "etale-cohomology-lemma-compare-higher-direct-image-proper" ], "ref_ids": [ 6657, 6668, 6657 ] } ], "ref_ids": [ 6669 ] }, { "id": 6671, "type": "theorem", "label": "etale-cohomology-lemma-compare-ph-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-compare-ph-etale", "contents": [ "Consider the comparison morphism", "$\\epsilon : (\\Sch/S)_{ph} \\to (\\Sch/S)_\\etale$.", "Let $\\mathcal{P}$ denote the class of proper morphisms of schemes.", "For $X$ in $(\\Sch/S)_\\etale$ denote", "$\\mathcal{A}'_X \\subset \\textit{Ab}((\\Sch/X)_\\etale)$", "the full subcategory consisting of sheaves of the form", "$\\pi_X^{-1}\\mathcal{F}$ where $\\mathcal{F}$ is a", "torsion abelian sheaf on $X_\\etale$", "Then Cohomology on Sites, Properties", "(\\ref{sites-cohomology-item-base-change-P}),", "(\\ref{sites-cohomology-item-restriction-A}),", "(\\ref{sites-cohomology-item-A-sheaf}),", "(\\ref{sites-cohomology-item-A-and-P}), and", "(\\ref{sites-cohomology-item-refine-tau-by-P})", "of Cohomology on Sites, Situation", "\\ref{sites-cohomology-situation-compare} hold." ], "refs": [], "proofs": [ { "contents": [ "We first show that $\\mathcal{A}'_X \\subset \\textit{Ab}((\\Sch/X)_\\etale)$", "is a weak Serre subcategory by checking conditions (1), (2), (3), and (4)", "of Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}.", "Parts (1), (2), (3) are immediate as $\\pi_X^{-1}$ is exact and", "fully faithful for example by Lemma \\ref{lemma-cohomological-descent-etale}. If", "$0 \\to \\pi_X^{-1}\\mathcal{F} \\to \\mathcal{G} \\to \\pi_X^{-1}\\mathcal{F}' \\to 0$", "is a short exact sequence in $\\textit{Ab}((\\Sch/X)_\\etale)$", "then $0 \\to \\mathcal{F} \\to \\pi_{X, *}\\mathcal{G} \\to \\mathcal{F}' \\to 0$", "is exact by Lemma \\ref{lemma-cohomological-descent-etale}.", "In particular we see that $\\pi_{X, *}\\mathcal{G}$ is an abelian", "torsion sheaf on $X_\\etale$.", "Hence $\\mathcal{G} = \\pi_X^{-1}\\pi_{X, *}\\mathcal{G}$ is in", "$\\mathcal{A}'_X$ which checks the final condition.", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-base-change-P}) holds", "by the existence of fibre products of schemes", "and the fact that the base change of a proper morphism of", "schemes is a proper morphism of schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-proper}.", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-restriction-A})", "follows from the commutative diagram (3) in", "Topologies, Lemma \\ref{topologies-lemma-morphism-big-small-etale}.", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-A-sheaf}) is", "Lemma \\ref{lemma-describe-pullback-pi-ph}.", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-A-and-P}) holds by", "Lemma \\ref{lemma-compare-higher-direct-image-proper} part (2)", "and the fact that $R^if_{small}\\mathcal{F}$", "is torsion if $\\mathcal{F}$ is an abelian torsion", "sheaf on $X_\\etale$, see Lemma \\ref{lemma-torsion-direct-image}.", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-refine-tau-by-P})", "follows from More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-dominate-fppf-etale-locally}", "combined with the fact that a finite morphism is proper", "and a surjective proper morphism defines a ph covering, see", "Topologies, Lemma \\ref{topologies-lemma-surjective-proper-ph}." ], "refs": [ "homology-lemma-characterize-weak-serre-subcategory", "etale-cohomology-lemma-cohomological-descent-etale", "etale-cohomology-lemma-cohomological-descent-etale", "morphisms-lemma-base-change-proper", "topologies-lemma-morphism-big-small-etale", "etale-cohomology-lemma-describe-pullback-pi-ph", "etale-cohomology-lemma-compare-higher-direct-image-proper", "etale-cohomology-lemma-torsion-direct-image", "more-morphisms-lemma-dominate-fppf-etale-locally", "topologies-lemma-surjective-proper-ph" ], "ref_ids": [ 12046, 6656, 6656, 5409, 12455, 6668, 6657, 6565, 13921, 12483 ] } ], "ref_ids": [] }, { "id": 6672, "type": "theorem", "label": "etale-cohomology-lemma-V-C-all-n-etale-ph", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-V-C-all-n-etale-ph", "contents": [ "With notation as above.", "\\begin{enumerate}", "\\item For $X \\in \\Ob((\\Sch/S)_{ph})$ and an abelian torsion sheaf $\\mathcal{F}$", "on $X_\\etale$ we have", "$\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$", "and $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$.", "\\item For a proper morphism $f : X \\to Y$ in $(\\Sch/S)_{ph}$", "and abelian torsion sheaf $\\mathcal{F}$ on $X$ we have", "$a_Y^{-1}(R^if_{small, *}\\mathcal{F}) =", "R^if_{big, ph, *}(a_X^{-1}\\mathcal{F})$", "for all $i$.", "\\item For a scheme $X$ and $K$ in $D^+(X_\\etale)$ with torsion", "cohomology sheaves the map", "$\\pi_X^{-1}K \\to R\\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism.", "\\item For a proper morphism $f : X \\to Y$ of schemes", "and $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves we have", "$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, ph, *}(a_X^{-1}K)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-compare-ph-etale} the lemmas in", "Cohomology on Sites, Section \\ref{sites-cohomology-section-compare-general}", "all apply to our current setting. To translate the results", "observe that the category $\\mathcal{A}_X$ of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-A}", "is the full subcategory of $\\textit{Ab}((\\Sch/X)_{ph})$", "consisting of sheaves of the form $a_X^{-1}\\mathcal{F}$", "where $\\mathcal{F}$ is an abelian torsion sheaf on $X_\\etale$.", "\\medskip\\noindent", "Part (1) is equivalent to $(V_n)$ for all $n$ which holds by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-C-all-n-general}.", "\\medskip\\noindent", "Part (2) follows by applying $\\epsilon_Y^{-1}$ to the conclusion of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-C-general}.", "\\medskip\\noindent", "Part (3) follows from Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-V-C-all-n-general} part (1)", "because $\\pi_X^{-1}K$ is in $D^+_{\\mathcal{A}'_X}((\\Sch/X)_\\etale)$", "and $a_X^{-1} = \\epsilon_X^{-1} \\circ a_X^{-1}$.", "\\medskip\\noindent", "Part (4) follows from Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-V-C-all-n-general} part (2)", "for the same reason." ], "refs": [ "etale-cohomology-lemma-compare-ph-etale", "sites-cohomology-lemma-A", "sites-cohomology-lemma-V-C-all-n-general", "sites-cohomology-lemma-V-implies-C-general", "sites-cohomology-lemma-V-C-all-n-general", "sites-cohomology-lemma-V-C-all-n-general" ], "ref_ids": [ 6671, 4296, 4302, 4297, 4302, 4302 ] } ], "ref_ids": [] }, { "id": 6673, "type": "theorem", "label": "etale-cohomology-lemma-cohomological-descent-etale-ph", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomological-descent-etale-ph", "contents": [ "Let $X$ be a scheme. For $K \\in D^+(X_\\etale)$ with torsion cohomology", "sheaves the map", "$$", "K \\longrightarrow Ra_{X, *}a_X^{-1}K", "$$", "is an isomorphism with $a_X : \\Sh((\\Sch/X)_{ph}) \\to \\Sh(X_\\etale)$ as above." ], "refs": [], "proofs": [ { "contents": [ "We first reduce the statement to the case where", "$K$ is given by a single abelian sheaf. Namely, represent $K$", "by a bounded below complex $\\mathcal{F}^\\bullet$ of torsion", "abelian sheaves. This is possible by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-torsion}. By the case of a", "sheaf we see that", "$\\mathcal{F}^n = a_{X, *} a_X^{-1} \\mathcal{F}^n$", "and that the sheaves $R^qa_{X, *}a_X^{-1}\\mathcal{F}^n$", "are zero for $q > 0$. By Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "applied to $a_X^{-1}\\mathcal{F}^\\bullet$", "and the functor $a_{X, *}$ we conclude. From now on assume $K = \\mathcal{F}$", "where $\\mathcal{F}$ is a torsion abelian sheaf.", "\\medskip\\noindent", "By Lemma \\ref{lemma-describe-pullback-pi-ph} we have", "$a_{X, *}a_X^{-1}\\mathcal{F} = \\mathcal{F}$. Thus it suffices to show that", "$R^qa_{X, *}a_X^{-1}\\mathcal{F} = 0$ for $q > 0$.", "For this we can use $a_X = \\epsilon_X \\circ \\pi_X$ and", "the Leray spectral sequence", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-relative-Leray}).", "By Lemma \\ref{lemma-V-C-all-n-etale-ph}", "we have $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$", "and $\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$.", "By Lemma \\ref{lemma-cohomological-descent-etale} we have", "$R^j\\pi_{X, *}(\\pi_X^{-1}\\mathcal{F}) = 0$ for $j > 0$.", "This concludes the proof." ], "refs": [ "sites-cohomology-lemma-torsion", "derived-lemma-leray-acyclicity", "etale-cohomology-lemma-describe-pullback-pi-ph", "sites-cohomology-lemma-relative-Leray", "etale-cohomology-lemma-V-C-all-n-etale-ph", "etale-cohomology-lemma-cohomological-descent-etale" ], "ref_ids": [ 4252, 1844, 6668, 4222, 6672, 6656 ] } ], "ref_ids": [] }, { "id": 6674, "type": "theorem", "label": "etale-cohomology-lemma-compare-cohomology-etale-ph", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-compare-cohomology-etale-ph", "contents": [ "For a scheme $X$ and $a_X : \\Sh((\\Sch/X)_{ph}) \\to \\Sh(X_\\etale)$", "as above:", "\\begin{enumerate}", "\\item $H^q(X_\\etale, \\mathcal{F}) = H^q_{ph}(X, a_X^{-1}\\mathcal{F})$", "for a torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$,", "\\item $H^q(X_\\etale, K) = H^q_{ph}(X, a_X^{-1}K)$", "for $K \\in D^+(X_\\etale)$ with torsion cohomology sheaves.", "\\end{enumerate}", "Example: if $A$ is a torsion abelian group, then", "$H^q_\\etale(X, \\underline{A}) = H^q_{ph}(X, \\underline{A})$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-cohomological-descent-etale-ph}", "by Cohomology on Sites, Remark \\ref{sites-cohomology-remark-before-Leray}." ], "refs": [ "etale-cohomology-lemma-cohomological-descent-etale-ph", "sites-cohomology-remark-before-Leray" ], "ref_ids": [ 6673, 4423 ] } ], "ref_ids": [] }, { "id": 6675, "type": "theorem", "label": "etale-cohomology-lemma-describe-pullback-pi-h", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-describe-pullback-pi-h", "contents": [ "With notation as above.", "Let $\\mathcal{F}$ be a sheaf on $S_\\etale$. The rule", "$$", "(\\Sch/S)_h \\longrightarrow \\textit{Sets},\\quad", "(f : X \\to S) \\longmapsto \\Gamma(X, f_{small}^{-1}\\mathcal{F})", "$$", "is a sheaf and a fortiori a sheaf on $(\\Sch/S)_\\etale$.", "In fact this sheaf is equal to", "$\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_\\etale$ and", "$\\epsilon_S^{-1}\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_h$." ], "refs": [], "proofs": [ { "contents": [ "The statement about the \\'etale topology is the content", "of Lemma \\ref{lemma-describe-pullback}. To finish the proof it", "suffices to show that $\\pi_S^{-1}\\mathcal{F}$ is a sheaf for the h", "topology. However, in Lemma \\ref{lemma-describe-pullback-pi-ph}", "we have shown that", "$\\pi_S^{-1}\\mathcal{F}$ is a sheaf even in the stronger ph", "topology." ], "refs": [ "etale-cohomology-lemma-describe-pullback", "etale-cohomology-lemma-describe-pullback-pi-ph" ], "ref_ids": [ 6438, 6668 ] } ], "ref_ids": [] }, { "id": 6676, "type": "theorem", "label": "etale-cohomology-lemma-push-pull-h-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-push-pull-h-etale", "contents": [ "With notation as above.", "Let $f : X \\to Y$ be a morphism of $(\\Sch/S)_h$.", "Then there are commutative diagrams of topoi", "$$", "\\xymatrix{", "\\Sh((\\Sch/X)_h) \\ar[rr]_{f_{big, h}} \\ar[d]_{\\epsilon_X} & &", "\\Sh((\\Sch/Y)_h) \\ar[d]^{\\epsilon_Y} \\\\", "\\Sh((\\Sch/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & &", "\\Sh((\\Sch/Y)_\\etale)", "}", "$$", "and", "$$", "\\xymatrix{", "\\Sh((\\Sch/X)_h) \\ar[rr]_{f_{big, h}} \\ar[d]_{a_X} & &", "\\Sh((\\Sch/Y)_h) \\ar[d]^{a_Y} \\\\", "\\Sh(X_\\etale) \\ar[rr]^{f_{small}} & &", "\\Sh(Y_\\etale)", "}", "$$", "with $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$." ], "refs": [], "proofs": [ { "contents": [ "The commutativity of the diagrams follows similarly to what was said in", "Topologies, Section \\ref{topologies-section-change-topologies}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6677, "type": "theorem", "label": "etale-cohomology-lemma-proper-push-pull-h-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-proper-push-pull-h-etale", "contents": [ "In Lemma \\ref{lemma-push-pull-h-etale} if $f$ is proper, then we have", "$a_Y^{-1} \\circ f_{small, *} = f_{big, h, *} \\circ a_X^{-1}$." ], "refs": [ "etale-cohomology-lemma-push-pull-h-etale" ], "proofs": [ { "contents": [ "You can prove this by repeating the proof of", "Lemma \\ref{lemma-compare-higher-direct-image-proper} part (1);", "we will instead deduce the result from this.", "As $\\epsilon_{Y, *}$ is the identity functor on underlying presheaves,", "it reflects isomorphisms. The description", "in Lemma \\ref{lemma-describe-pullback-pi-h}", "shows that $\\epsilon_{Y, *} \\circ a_Y^{-1} = \\pi_Y^{-1}$", "and similarly for $X$. To show that the canonical map", "$a_Y^{-1}f_{small, *}\\mathcal{F} \\to f_{big, h, *}a_X^{-1}\\mathcal{F}$", "is an isomorphism, it suffices to show that", "\\begin{align*}", "\\pi_Y^{-1}f_{small, *}\\mathcal{F}", "& =", "\\epsilon_{Y, *}a_Y^{-1}f_{small, *}\\mathcal{F} \\\\", "& \\to ", "\\epsilon_{Y, *}f_{big, h, *}a_X^{-1}\\mathcal{F} \\\\", "& =", "f_{big, \\etale, *} \\epsilon_{X, *}a_X^{-1}\\mathcal{F} \\\\", "& =", "f_{big, \\etale, *}\\pi_X^{-1}\\mathcal{F}", "\\end{align*}", "is an isomorphism. This is part", "(1) of Lemma \\ref{lemma-compare-higher-direct-image-proper}." ], "refs": [ "etale-cohomology-lemma-compare-higher-direct-image-proper", "etale-cohomology-lemma-describe-pullback-pi-h", "etale-cohomology-lemma-compare-higher-direct-image-proper" ], "ref_ids": [ 6657, 6675, 6657 ] } ], "ref_ids": [ 6676 ] }, { "id": 6678, "type": "theorem", "label": "etale-cohomology-lemma-compare-h-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-compare-h-etale", "contents": [ "Consider the comparison morphism $\\epsilon : (\\Sch/S)_h \\to (\\Sch/S)_\\etale$.", "Let $\\mathcal{P}$ denote the class of proper morphisms.", "For $X$ in $(\\Sch/S)_\\etale$ denote", "$\\mathcal{A}'_X \\subset \\textit{Ab}((\\Sch/X)_\\etale)$", "the full subcategory consisting of sheaves of the form", "$\\pi_X^{-1}\\mathcal{F}$ where $\\mathcal{F}$ is a", "torsion abelian sheaf on $X_\\etale$", "Then Cohomology on Sites, Properties", "(\\ref{sites-cohomology-item-base-change-P}),", "(\\ref{sites-cohomology-item-restriction-A}),", "(\\ref{sites-cohomology-item-A-sheaf}),", "(\\ref{sites-cohomology-item-A-and-P}), and", "(\\ref{sites-cohomology-item-refine-tau-by-P})", "of Cohomology on Sites, Situation", "\\ref{sites-cohomology-situation-compare} hold." ], "refs": [], "proofs": [ { "contents": [ "We first show that $\\mathcal{A}'_X \\subset \\textit{Ab}((\\Sch/X)_\\etale)$", "is a weak Serre subcategory by checking conditions (1), (2), (3), and (4)", "of Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}.", "Parts (1), (2), (3) are immediate as $\\pi_X^{-1}$ is exact and", "fully faithful for example by Lemma \\ref{lemma-cohomological-descent-etale}. If", "$0 \\to \\pi_X^{-1}\\mathcal{F} \\to \\mathcal{G} \\to \\pi_X^{-1}\\mathcal{F}' \\to 0$", "is a short exact sequence in $\\textit{Ab}((\\Sch/X)_\\etale)$", "then $0 \\to \\mathcal{F} \\to \\pi_{X, *}\\mathcal{G} \\to \\mathcal{F}' \\to 0$", "is exact by Lemma \\ref{lemma-cohomological-descent-etale}.", "In particular we see that $\\pi_{X, *}\\mathcal{G}$ is an abelian", "torsion sheaf on $X_\\etale$.", "Hence $\\mathcal{G} = \\pi_X^{-1}\\pi_{X, *}\\mathcal{G}$ is in", "$\\mathcal{A}'_X$ which checks the final condition.", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-base-change-P}) holds", "by the existence of fibre products of schemes,", "the fact that the base change of a proper morphism of", "schemes is a proper morphism of schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-proper}, and", "the fact that the base change of a morphism of finite presentation", "is a morphism of finite presentation, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-presentation}.", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-restriction-A})", "follows from the commutative diagram (3) in", "Topologies, Lemma \\ref{topologies-lemma-morphism-big-small-etale}.", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-A-sheaf}) is", "Lemma \\ref{lemma-describe-pullback-pi-h}.", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-A-and-P}) holds by", "Lemma \\ref{lemma-compare-higher-direct-image-proper} part (2)", "and the fact that $R^if_{small}\\mathcal{F}$", "is torsion if $\\mathcal{F}$ is an abelian torsion", "sheaf on $X_\\etale$, see Lemma \\ref{lemma-torsion-direct-image}.", "\\medskip\\noindent", "Cohomology on Sites, Property (\\ref{sites-cohomology-item-refine-tau-by-P})", "is implied by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-dominate-fppf-etale-locally}", "combined with the fact that a surjective finite locally free morphism", "is surjective, proper, and of finite presentation and hence", "defines a h-covering by More on Flatness, Lemma", "\\ref{flat-lemma-surjective-proper-finite-presentation-h}." ], "refs": [ "homology-lemma-characterize-weak-serre-subcategory", "etale-cohomology-lemma-cohomological-descent-etale", "etale-cohomology-lemma-cohomological-descent-etale", "morphisms-lemma-base-change-proper", "morphisms-lemma-base-change-finite-presentation", "topologies-lemma-morphism-big-small-etale", "etale-cohomology-lemma-describe-pullback-pi-h", "etale-cohomology-lemma-compare-higher-direct-image-proper", "etale-cohomology-lemma-torsion-direct-image", "more-morphisms-lemma-dominate-fppf-etale-locally", "flat-lemma-surjective-proper-finite-presentation-h" ], "ref_ids": [ 12046, 6656, 6656, 5409, 5240, 12455, 6675, 6657, 6565, 13921, 6143 ] } ], "ref_ids": [] }, { "id": 6679, "type": "theorem", "label": "etale-cohomology-lemma-V-C-all-n-etale-h", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-V-C-all-n-etale-h", "contents": [ "With notation as above.", "\\begin{enumerate}", "\\item For $X \\in \\Ob((\\Sch/S)_{h})$ and an abelian torsion sheaf $\\mathcal{F}$", "on $X_\\etale$ we have", "$\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$", "and $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$.", "\\item For a proper morphism $f : X \\to Y$ in $(\\Sch/S)_h$", "and abelian torsion sheaf $\\mathcal{F}$ on $X$ we have", "$a_Y^{-1}(R^if_{small, *}\\mathcal{F}) =", "R^if_{big, h, *}(a_X^{-1}\\mathcal{F})$", "for all $i$.", "\\item For a scheme $X$ and $K$ in $D^+(X_\\etale)$ with torsion", "cohomology sheaves the map", "$\\pi_X^{-1}K \\to R\\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism.", "\\item For a proper morphism $f : X \\to Y$ of schemes", "and $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves we have", "$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, h, *}(a_X^{-1}K)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-compare-h-etale} the lemmas in", "Cohomology on Sites, Section \\ref{sites-cohomology-section-compare-general}", "all apply to our current setting. To translate the results", "observe that the category $\\mathcal{A}_X$ of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-A}", "is the full subcategory of $\\textit{Ab}((\\Sch/X)_h)$", "consisting of sheaves of the form $a_X^{-1}\\mathcal{F}$", "where $\\mathcal{F}$ is an abelian torsion sheaf on $X_\\etale$.", "\\medskip\\noindent", "Part (1) is equivalent to $(V_n)$ for all $n$ which holds by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-C-all-n-general}.", "\\medskip\\noindent", "Part (2) follows by applying $\\epsilon_Y^{-1}$ to the conclusion of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-C-general}.", "\\medskip\\noindent", "Part (3) follows from Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-V-C-all-n-general} part (1)", "because $\\pi_X^{-1}K$ is in $D^+_{\\mathcal{A}'_X}((\\Sch/X)_\\etale)$", "and $a_X^{-1} = \\epsilon_X^{-1} \\circ a_X^{-1}$.", "\\medskip\\noindent", "Part (4) follows from Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-V-C-all-n-general} part (2)", "for the same reason." ], "refs": [ "etale-cohomology-lemma-compare-h-etale", "sites-cohomology-lemma-A", "sites-cohomology-lemma-V-C-all-n-general", "sites-cohomology-lemma-V-implies-C-general", "sites-cohomology-lemma-V-C-all-n-general", "sites-cohomology-lemma-V-C-all-n-general" ], "ref_ids": [ 6678, 4296, 4302, 4297, 4302, 4302 ] } ], "ref_ids": [] }, { "id": 6680, "type": "theorem", "label": "etale-cohomology-lemma-cohomological-descent-etale-h", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-cohomological-descent-etale-h", "contents": [ "Let $X$ be a scheme. For $K \\in D^+(X_\\etale)$ with torsion cohomology", "sheaves the map", "$$", "K \\longrightarrow Ra_{X, *}a_X^{-1}K", "$$", "is an isomorphism with $a_X : \\Sh((\\Sch/X)_h) \\to \\Sh(X_\\etale)$ as above." ], "refs": [], "proofs": [ { "contents": [ "We first reduce the statement to the case where", "$K$ is given by a single abelian sheaf. Namely, represent $K$", "by a bounded below complex $\\mathcal{F}^\\bullet$ of torsion", "abelian sheaves. This is possible by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-torsion}. By the case of a", "sheaf we see that", "$\\mathcal{F}^n = a_{X, *} a_X^{-1} \\mathcal{F}^n$", "and that the sheaves $R^qa_{X, *}a_X^{-1}\\mathcal{F}^n$", "are zero for $q > 0$. By Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "applied to $a_X^{-1}\\mathcal{F}^\\bullet$", "and the functor $a_{X, *}$ we conclude. From now on assume $K = \\mathcal{F}$", "where $\\mathcal{F}$ is a torsion abelian sheaf.", "\\medskip\\noindent", "By Lemma \\ref{lemma-describe-pullback-pi-h} we have", "$a_{X, *}a_X^{-1}\\mathcal{F} = \\mathcal{F}$. Thus it suffices to show that", "$R^qa_{X, *}a_X^{-1}\\mathcal{F} = 0$ for $q > 0$.", "For this we can use $a_X = \\epsilon_X \\circ \\pi_X$ and", "the Leray spectral sequence", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-relative-Leray}).", "By Lemma \\ref{lemma-V-C-all-n-etale-h}", "we have $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$", "and $\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$.", "By Lemma \\ref{lemma-cohomological-descent-etale} we have", "$R^j\\pi_{X, *}(\\pi_X^{-1}\\mathcal{F}) = 0$ for $j > 0$.", "This concludes the proof." ], "refs": [ "sites-cohomology-lemma-torsion", "derived-lemma-leray-acyclicity", "etale-cohomology-lemma-describe-pullback-pi-h", "sites-cohomology-lemma-relative-Leray", "etale-cohomology-lemma-V-C-all-n-etale-h", "etale-cohomology-lemma-cohomological-descent-etale" ], "ref_ids": [ 4252, 1844, 6675, 4222, 6679, 6656 ] } ], "ref_ids": [] }, { "id": 6681, "type": "theorem", "label": "etale-cohomology-lemma-compare-cohomology-etale-h", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-compare-cohomology-etale-h", "contents": [ "For a scheme $X$ and $a_X : \\Sh((\\Sch/X)_h) \\to \\Sh(X_\\etale)$", "as above:", "\\begin{enumerate}", "\\item $H^q(X_\\etale, \\mathcal{F}) = H^q_h(X, a_X^{-1}\\mathcal{F})$", "for a torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$,", "\\item $H^q(X_\\etale, K) = H^q_h(X, a_X^{-1}K)$", "for $K \\in D^+(X_\\etale)$ with torsion cohomology sheaves.", "\\end{enumerate}", "Example: if $A$ is a torsion abelian group, then", "$H^q_\\etale(X, \\underline{A}) = H^q_h(X, \\underline{A})$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-cohomological-descent-etale-h}", "by Cohomology on Sites, Remark \\ref{sites-cohomology-remark-before-Leray}." ], "refs": [ "etale-cohomology-lemma-cohomological-descent-etale-h", "sites-cohomology-remark-before-Leray" ], "ref_ids": [ 6680, 4423 ] } ], "ref_ids": [] }, { "id": 6682, "type": "theorem", "label": "etale-cohomology-lemma-glue-etale-sheaf-section", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-glue-etale-sheaf-section", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes which has a section. Then the", "functor", "$$", "\\Sh(Y_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}", "$$", "sending $\\mathcal{G}$ in $\\Sh(Y_\\etale)$ to the canonical descent datum", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "This is formal and depends only on functoriality of the pullback", "functors. We omit the details. Hint: If $s : Y \\to X$ is a section,", "then a quasi-inverse is the functor sending $(\\mathcal{F}, \\varphi)$", "to $s_{small}^{-1}\\mathcal{F}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6683, "type": "theorem", "label": "etale-cohomology-lemma-glue-etale-sheaf-integral-surjective", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-glue-etale-sheaf-integral-surjective", "contents": [ "Let $f : X \\to Y$ be a surjective integral morphism of schemes.", "The functor", "$$", "\\Sh(Y_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "In this proof we drop the subscript ${}_{small}$ from our pullback", "and pushforward functors.", "Denote $X_1 = X \\times_Y X$ and denote $f_1 : X_1 \\to Y$ the", "morphism $f \\circ \\text{pr}_0 = f \\circ \\text{pr}_1$.", "Let $(\\mathcal{F}, \\varphi)$ be a descent datum for $\\{X \\to Y\\}$.", "Let us set $\\mathcal{F}_1 = \\text{pr}_0^{-1}\\mathcal{F}$.", "We may think of $\\varphi$ as defining an isomorphism", "$\\mathcal{F}_1 \\to \\text{pr}_1^{-1}\\mathcal{F}$.", "We claim that the rule which sends a descent datum", "$(\\mathcal{F}, \\varphi)$", "to the sheaf", "$$", "\\mathcal{G} =", "\\text{Equalizer}\\left(", "\\xymatrix{", "f_*\\mathcal{F}", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "f_{1, *}\\mathcal{F}_1", "}", "\\right)", "$$", "is a quasi-inverse to the functor in the statement of the lemma.", "The first of the two arrows comes from the map", "$$", "f_*\\mathcal{F} \\to", "f_*\\text{pr}_{0, *}\\text{pr}_0^{-1}\\mathcal{F} =", "f_{1, *}\\mathcal{F}_1", "$$", "and the second arrow comes from the map", "$$", "f_*\\mathcal{F} \\to", "f_* \\text{pr}_{1, *}\\text{pr}_1^{-1}\\mathcal{F}", "\\xleftarrow{\\varphi}", "f_* \\text{pr}_{0, *} \\text{pr}_0^{-1}\\mathcal{F} =", "f_{1, *}\\mathcal{F}_1", "$$", "where the arrow pointing left is invertible.", "To prove this works we have to show", "that the canonical map $f^{-1}\\mathcal{G} \\to \\mathcal{F}$", "is an isomorphism; details omitted. In order to prove this it", "suffices to check after pulling back by any collection of morphisms", "$\\Spec(k) \\to Y$ where $k$ is an algebraically closed field.", "Namely, the corresponing base changes $X_k \\to X$ are jointly", "surjective and we can check whether a map of sheaves on $X_\\etale$", "is an isomorphism by looking at stalks on geometric points, see", "Theorem \\ref{theorem-exactness-stalks}.", "By Lemma \\ref{lemma-integral-pushforward-commutes-with-base-change}", "the construction of $\\mathcal{G}$ from the descent datum", "$(\\mathcal{F}, \\varphi)$ commutes with any base change.", "Thus we may assume $Y$ is the spectrum of an algebraically", "closed point (note that base change preserves the properties", "of the morphism $f$, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-surjective}", "and \\ref{morphisms-lemma-base-change-finite}).", "In this case the morphism $X \\to Y$ has a section, so we", "know that the functor is an equivalence by", "Lemma \\ref{lemma-glue-etale-sheaf-section}.", "However, the reader may show that the functor is an equivalence", "if and only if the construction above is a quasi-inverse;", "details omitted. This finishes the proof." ], "refs": [ "etale-cohomology-theorem-exactness-stalks", "etale-cohomology-lemma-integral-pushforward-commutes-with-base-change", "morphisms-lemma-base-change-surjective", "morphisms-lemma-base-change-finite" ], "ref_ids": [ 6376, 6482, 5165, 5440 ] } ], "ref_ids": [] }, { "id": 6684, "type": "theorem", "label": "etale-cohomology-lemma-glue-etale-sheaf-proper-surjective", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-glue-etale-sheaf-proper-surjective", "contents": [ "Let $f : X \\to Y$ be a surjective proper morphism of schemes.", "The functor", "$$", "\\Sh(Y_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "The exact same proof as given in", "Lemma \\ref{lemma-glue-etale-sheaf-integral-surjective}", "works, except the appeal to", "Lemma \\ref{lemma-integral-pushforward-commutes-with-base-change}", "should be replaced by an appeal to", "Lemma \\ref{lemma-proper-base-change-f-star}." ], "refs": [ "etale-cohomology-lemma-glue-etale-sheaf-integral-surjective", "etale-cohomology-lemma-integral-pushforward-commutes-with-base-change", "etale-cohomology-lemma-proper-base-change-f-star" ], "ref_ids": [ 6683, 6482, 6620 ] } ], "ref_ids": [] }, { "id": 6685, "type": "theorem", "label": "etale-cohomology-lemma-glue-etale-sheaf-check-after-base-change", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-glue-etale-sheaf-check-after-base-change", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $Z \\to Y$ be a surjective", "integral morphism of schemes or a surjective proper morphism of schemes.", "If the functors", "$$", "\\Sh(Z_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{X \\times_Y Z \\to Z\\}", "$$", "and", "$$", "\\Sh((Z \\times_Y Z)_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }", "\\{X \\times_Y (Z \\times_Y Z) \\to Z \\times_Y Z\\}", "$$", "are equivalences of categories, then", "$$", "\\Sh(Y_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}", "$$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Formal consequence of the definitions and", "Lemmas \\ref{lemma-glue-etale-sheaf-integral-surjective} and", "\\ref{lemma-glue-etale-sheaf-proper-surjective}. Details omitted." ], "refs": [ "etale-cohomology-lemma-glue-etale-sheaf-integral-surjective", "etale-cohomology-lemma-glue-etale-sheaf-proper-surjective" ], "ref_ids": [ 6683, 6684 ] } ], "ref_ids": [] }, { "id": 6686, "type": "theorem", "label": "etale-cohomology-lemma-glue-etale-sheaf-fppf-cover", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-glue-etale-sheaf-fppf-cover", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes which is", "surjective, flat, locally of finite presentation.", "The functor", "$$", "\\Sh(Y_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "Exactly as in the proof of", "Lemma \\ref{lemma-glue-etale-sheaf-integral-surjective}", "we claim a quasi-inverse is given by the functor sending", "$(\\mathcal{F}, \\varphi)$ to", "$$", "\\mathcal{G} =", "\\text{Equalizer}\\left(", "\\xymatrix{", "f_*\\mathcal{F}", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "f_{1, *}\\mathcal{F}_1", "}", "\\right)", "$$", "and in order to prove this it suffices to show that", "$f^{-1}\\mathcal{G} \\to \\mathcal{F}$ is an isomorphism.", "This we may check locally, hence we may and do assume $Y$", "is affine. Then we can find a finite surjective morphism", "$Z \\to Y$ such that there exists an open covering", "$Z = \\bigcup W_i$ such that $W_i \\to Y$ factors through $X$.", "See", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-dominate-fppf-finite}.", "Applying Lemma", "\\ref{lemma-glue-etale-sheaf-check-after-base-change}", "we see that it suffices to prove", "the lemma after replacing $Y$ by $Z$ and $Z \\times_Y Z$ and $f$", "by its base change. Thus we may assume $f$ has sections Zariski locally.", "Of course, using that the problem is local on $Y$ we reduce", "to the case where we have a section which is", "Lemma \\ref{lemma-glue-etale-sheaf-section}." ], "refs": [ "etale-cohomology-lemma-glue-etale-sheaf-integral-surjective", "more-morphisms-lemma-dominate-fppf-finite", "etale-cohomology-lemma-glue-etale-sheaf-check-after-base-change" ], "ref_ids": [ 6683, 13926, 6685 ] } ], "ref_ids": [] }, { "id": 6687, "type": "theorem", "label": "etale-cohomology-lemma-glue-etale-sheaf-fppf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-glue-etale-sheaf-fppf", "contents": [ "Let $\\{f_i : X_i \\to X\\}$ be an fppf covering of schemes.", "The functor", "$$", "\\Sh(X_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{f_i : X_i \\to X\\}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "We have Lemma \\ref{lemma-glue-etale-sheaf-fppf-cover}", "for the morphism $f : \\coprod X_i \\to X$.", "Then a formal argument shows that descent data for $f$", "are the same thing as descent data for the covering, compare", "with Descent, Lemma \\ref{descent-lemma-family-is-one}.", "Details omitted." ], "refs": [ "etale-cohomology-lemma-glue-etale-sheaf-fppf-cover", "descent-lemma-family-is-one" ], "ref_ids": [ 6686, 14732 ] } ], "ref_ids": [] }, { "id": 6688, "type": "theorem", "label": "etale-cohomology-lemma-glue-etale-sheaf-modification", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-glue-etale-sheaf-modification", "contents": [ "Let $f : X' \\to X$ be a proper morphism of schemes. Let $i : Z \\to X$", "be a closed immersion. Set $E = Z \\times_X X'$. Picture", "$$", "\\xymatrix{", "E \\ar[d]_g \\ar[r]_j & X' \\ar[d]^f \\\\", "Z \\ar[r]^i & X", "}", "$$", "If $f$ is an isomorphism over $X \\setminus Z$, then the functor", "$$", "\\Sh(X_\\etale)", "\\longrightarrow", "\\Sh(X'_\\etale) \\times_{\\Sh(E_\\etale)} \\Sh(Z_\\etale)", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "We will work with the $2$-fibre product category as constructed in", "Categories, Example \\ref{categories-example-2-fibre-product-categories}.", "The functor sends $\\mathcal{F}$ to the triple", "$(f^{-1}\\mathcal{F}, i^{-1}\\mathcal{F}, c)$ where", "$c : j^{-1}f^{-1}\\mathcal{F} \\to g^{-1}i^{-1}\\mathcal{F}$", "is the canonical isomorphism. We will construct a quasi-inverse functor. Let", "$(\\mathcal{F}', \\mathcal{G}, \\alpha)$ be an object", "of the right hand side of the arrow.", "We obtain an isomorphism", "$$", "i^{-1}f_*\\mathcal{F}' = g_*j^{-1}\\mathcal{F}'", "\\xrightarrow{g_*\\alpha}", "g_*g^{-1}\\mathcal{G}", "$$", "The first equality is Lemma \\ref{lemma-proper-base-change-f-star}.", "Using this we obtain maps", "$i_*\\mathcal{G} \\to i_*g_*g^{-1}\\mathcal{G}$", "and $f'_*\\mathcal{F}' \\to i_*g_*g^{-1}\\mathcal{G}$. We set", "$$", "\\mathcal{F} = f_*\\mathcal{F}' \\times_{i_*g_*g^{-1}\\mathcal{G}} i_*\\mathcal{G}", "$$", "and we claim that $\\mathcal{F}$ is an object of the left hand side", "of the arrow whose image in the right hand side is isomorphic to", "the triple we started out with. Let us compute the stalk of $\\mathcal{F}$", "at a geometric point $\\overline{x}$ of $X$.", "\\medskip\\noindent", "If $\\overline{x}$ is not", "in $Z$, then on the one hand $\\overline{x}$ comes from a unique", "geometric point $\\overline{x}'$ of $X'$ and", "$\\mathcal{F}'_{\\overline{x}'} = (f_*\\mathcal{F}')_{\\overline{x}}$", "and on the other hand we have $(i_*\\mathcal{G})_{\\overline{x}}$", "and $(i_*g_*g^{-1}\\mathcal{G})_{\\overline{x}}$ are singletons.", "Hence we see that $\\mathcal{F}_{\\overline{x}}$ equals", "$\\mathcal{F}'_{\\overline{x}'}$.", "\\medskip\\noindent", "If $\\overline{x}$ is in $Z$, i.e., $\\overline{x}$ is the image of", "a geometric point $\\overline{z}$ of $Z$, then we obtain", "$(i_*\\mathcal{G})_{\\overline{x}} = \\mathcal{G}_{\\overline{z}}$", "and", "$$", "(i_*g_*g^{-1}\\mathcal{G})_{\\overline{x}} =", "(g_*g^{-1}\\mathcal{G})_{\\overline{z}} =", "\\Gamma(E_{\\overline{z}}, g^{-1}\\mathcal{G}|_{E_{\\overline{z}}})", "$$", "(by the proper base change for pushforward used above)", "and similarly", "$$", "(f_*\\mathcal{F}')_{\\overline{x}} =", "\\Gamma(X'_{\\overline{x}}, \\mathcal{F}'|_{X'_{\\overline{x}}})", "$$", "Since we have the identification", "$E_{\\overline{z}} = X'_{\\overline{x}}$ and since $\\alpha$", "defines an isomorphism between the sheaves", "$\\mathcal{F}'|_{X'_{\\overline{x}}}$ and", "$g^{-1}\\mathcal{G}|_{E_{\\overline{z}}}$", "we conclude that we get", "$$", "\\mathcal{F}_{\\overline{x}} = \\mathcal{G}_{\\overline{z}}", "$$", "in this case.", "\\medskip\\noindent", "To finish the proof, we observe that there are canonical maps", "$i^{-1}\\mathcal{F} \\to \\mathcal{G}$ and $f^{-1}\\mathcal{F} \\to \\mathcal{F}'$", "compatible with $\\alpha$ which on stalks produce the isomorphisms", "we saw above. We omit the careful construction of these maps." ], "refs": [ "etale-cohomology-lemma-proper-base-change-f-star" ], "ref_ids": [ 6620 ] } ], "ref_ids": [] }, { "id": 6689, "type": "theorem", "label": "etale-cohomology-lemma-h-descent-etale-sheaves", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-h-descent-etale-sheaves", "contents": [ "Let $S$ be a scheme. Then the category fibred in groupoids", "$$", "p : \\mathcal{S} \\longrightarrow (\\Sch/S)_h", "$$", "whose fibre category over $U$ is the category $\\Sh(U_\\etale)$", "of sheaves on the small \\'etale site of $U$ is a stack in groupoids." ], "refs": [], "proofs": [ { "contents": [ "To prove the lemma we will check conditions (1), (2), and (3) of", "More on Flatness, Lemma \\ref{flat-lemma-refine-check-h-stack}.", "\\medskip\\noindent", "Condition (1) holds because we have glueing for sheaves (and", "Zariski coverings are \\'etale coverings). See", "Sites, Lemma \\ref{sites-lemma-glue-sheaves}.", "\\medskip\\noindent", "To see condition (2), suppose that $f : X \\to Y$ is a surjective,", "flat, proper morphism of finite presentation over $S$ with $Y$ affine.", "Then we have descent for $\\{X \\to Y\\}$ by either", "Lemma \\ref{lemma-glue-etale-sheaf-fppf-cover} or", "Lemma \\ref{lemma-glue-etale-sheaf-proper-surjective}.", "\\medskip\\noindent", "Condition (3) follows immediately from the more general", "Lemma \\ref{lemma-glue-etale-sheaf-modification}." ], "refs": [ "flat-lemma-refine-check-h-stack", "sites-lemma-glue-sheaves", "etale-cohomology-lemma-glue-etale-sheaf-fppf-cover", "etale-cohomology-lemma-glue-etale-sheaf-proper-surjective", "etale-cohomology-lemma-glue-etale-sheaf-modification" ], "ref_ids": [ 6166, 8564, 6686, 6684, 6688 ] } ], "ref_ids": [] }, { "id": 6690, "type": "theorem", "label": "etale-cohomology-lemma-blow-up-square-cohomological-descent", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-blow-up-square-cohomological-descent", "contents": [ "Let $X$ be a scheme and let $Z \\subset X$ be a closed subscheme", "cut out by a quasi-coherent ideal of finite type. Consider the", "corresponding blow up square", "$$", "\\xymatrix{", "E \\ar[d]_\\pi \\ar[r]_j & X' \\ar[d]^b \\\\", "Z \\ar[r]^i & X", "}", "$$", "For $K \\in D^+(X_\\etale)$ with torsion cohomology sheaves", "we have a distinguished triangle", "$$", "K \\to Ri_*(K|_Z) \\oplus Rb_*(K|_{X'}) \\to Rc_*(K|_E) \\to K[1]", "$$", "in $D(X_\\etale)$ where $c = i \\circ \\pi = b \\circ j$." ], "refs": [], "proofs": [ { "contents": [ "The notation $K|_{X'}$ stands for $b_{small}^{-1}K$.", "Choose a bounded below complex $\\mathcal{F}^\\bullet$", "of abelian sheaves representing $K$. Observe that", "$i_*(\\mathcal{F}^\\bullet|_Z)$ represents $Ri_*(K|_Z)$", "because $i_*$ is exact", "(Proposition \\ref{proposition-finite-higher-direct-image-zero}).", "Choose a quasi-isomorphism", "$b_{small}^{-1}\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$", "where $\\mathcal{I}^\\bullet$ is a bounded below complex of injective", "abelian sheaves on $X'_\\etale$. This map is adjoint to a map", "$\\mathcal{F}^\\bullet \\to b_*(\\mathcal{I}^\\bullet)$ and", "$b_*(\\mathcal{I}^\\bullet)$ represents $Rb_*(K|_{X'})$.", "We have $\\pi_*(\\mathcal{I}^\\bullet|_E) = (b_*\\mathcal{I}^\\bullet)|_Z$", "by Lemma \\ref{lemma-proper-base-change-f-star} and by", "Lemma \\ref{lemma-proper-base-change} this complex represents", "$R\\pi_*(K|_E)$. Hence the map", "$$", "Ri_*(K|_Z) \\oplus Rb_*(K|_{X'}) \\to Rc_*(K|_E)", "$$", "is represented by the surjective map of bounded below complexes", "$$", "i_*(\\mathcal{F}^\\bullet|_Z) \\oplus", "b_*(\\mathcal{I}^\\bullet)", "\\to", "i_*\\left(b_*(\\mathcal{I}^\\bullet)|_Z\\right)", "$$", "To get our distinguished triangle it suffices to show that", "the canonical map", "$\\mathcal{F}^\\bullet \\to i_*(\\mathcal{F}^\\bullet|_Z) \\oplus", "b_*(\\mathcal{I}^\\bullet)$", "maps quasi-isomorphically onto the kernel of the map", "of complexes displayed above (namely a short exact sequence", "of complexes determines a distinguished triangle in the derived", "category, see", "Derived Categories, Section \\ref{derived-section-canonical-delta-functor}).", "We may check this on stalks at a geometric point $\\overline{x}$ of $X$.", "If $\\overline{x}$ is not in $Z$, then $X' \\to X$ is an isomorphism", "over an open neighbourhood of $\\overline{x}$. Thus, if $\\overline{x}'$", "denotes the corresponding geometric point of $X'$ in this case, then", "we have to show that", "$$", "\\mathcal{F}^\\bullet_{\\overline{x}} \\to \\mathcal{I}^\\bullet_{\\overline{x}'}", "$$", "is a quasi-isomorphism. This is true by our choice of $\\mathcal{I}^\\bullet$.", "If $\\overline{x}$ is in $Z$, then", "$b_(\\mathcal{I}^\\bullet)_{\\overline{x}} \\to ", "i_*\\left(b_*(\\mathcal{I}^\\bullet)|_Z\\right)_{\\overline{x}}$", "is an isomorphism of complexes of abelian groups. Hence the", "kernel is equal to", "$i_*(\\mathcal{F}^\\bullet|_Z)_{\\overline{x}} =", "\\mathcal{F}^\\bullet_{\\overline{x}}$ as desired." ], "refs": [ "etale-cohomology-proposition-finite-higher-direct-image-zero", "etale-cohomology-lemma-proper-base-change-f-star", "etale-cohomology-lemma-proper-base-change" ], "ref_ids": [ 6703, 6620, 6626 ] } ], "ref_ids": [] }, { "id": 6691, "type": "theorem", "label": "etale-cohomology-lemma-blow-up-square-etale-cohomology", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-blow-up-square-etale-cohomology", "contents": [ "Let $X$ be a scheme and let $K \\in D^+(X_\\etale)$ have", "torsion cohomology sheaves. Let $Z \\subset X$ be a closed subscheme", "cut out by a quasi-coherent ideal of finite type. Consider the", "corresponding blow up square", "$$", "\\xymatrix{", "E \\ar[d] \\ar[r] & X' \\ar[d]^b \\\\", "Z \\ar[r] & X", "}", "$$", "Then there is a canonical long exact sequence", "$$", "H^p_\\etale(X, K) \\to", "H^p_\\etale(X', K|_{X'}) \\oplus", "H^p_\\etale(Z, K|_Z) \\to", "H^p_\\etale(E, K|_E) \\to", "H^{p + 1}_\\etale(X, K)", "$$" ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "This follows immediately from", "Lemma \\ref{lemma-blow-up-square-cohomological-descent}", "and the fact that", "$$", "R\\Gamma(X, Rb_*(K|_{X'})) = R\\Gamma(X', K|_{X'})", "$$", "(see Cohomology on Sites, Section \\ref{sites-cohomology-section-leray})", "and similarly for the others." ], "refs": [ "etale-cohomology-lemma-blow-up-square-cohomological-descent" ], "ref_ids": [ 6690 ] } ], "ref_ids": [] }, { "id": 6692, "type": "theorem", "label": "etale-cohomology-lemma-blow-up-square-equivalence", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-blow-up-square-equivalence", "contents": [ "Let $X$ be a scheme and let $Z \\subset X$ be a closed subscheme", "cut out by a quasi-coherent ideal of finite type. Consider the", "corresponding blow up square", "$$", "\\xymatrix{", "E \\ar[d]_\\pi \\ar[r]_j & X' \\ar[d]^b \\\\", "Z \\ar[r]^i & X", "}", "$$", "Suppose given", "\\begin{enumerate}", "\\item an object $K'$ of $D^+(X'_\\etale)$ with torsion cohomology sheaves,", "\\item an object $L$ of $D^+(Z_\\etale)$ with torsion cohomology sheaves, and", "\\item an isomorphism $\\gamma : K'|_E \\to L|_E$.", "\\end{enumerate}", "Then there exists an object $K$ of $D^+(X_\\etale)$", "and isomorphisms $f : K|_{X'} \\to K'$, $g : K|_Z \\to L$ such", "that $\\gamma = g|_E \\circ f^{-1}|_E$.", "Moreover, given", "\\begin{enumerate}", "\\item an object $M$ of $D^+(X_\\etale)$ with torsion cohomology sheaves,", "\\item a morphism $\\alpha : K' \\to M|_{X'}$ of $D(X'_\\etale)$,", "\\item a morphism $\\beta : L \\to M|_Z$ of $D(Z_\\etale)$,", "\\end{enumerate}", "such that", "$$", "\\alpha|_E = \\beta|_E \\circ \\gamma.", "$$", "Then there exists a morphism $M \\to K$ in $D(X_\\etale)$", "whose restriction to $X'$ is $a \\circ f$", "and whose restriction to $Z$ is $b \\circ g$." ], "refs": [], "proofs": [ { "contents": [ "If $K$ exists, then Lemma \\ref{lemma-blow-up-square-cohomological-descent}", "tells us a distinguished triangle that it fits in. Thus we simply choose", "a distinguished triangle", "$$", "K \\to Ri_*(L) \\oplus Rb_*(K') \\to Rc_*(L|_E) \\to K[1]", "$$", "where $c = i \\circ \\pi = b \\circ j$. Here the map $Ri_*(L) \\to Rc_*(L|_E)$", "is $Ri_*$ applied to the adjunction mapping $E \\to R\\pi_*(L|_E)$.", "The map $Rb_*(K') \\to Rc_*(L|_E)$ is the composition of the canonical map", "$Rb_*(K') \\to Rc_*(K'|_E)) = R$ and $Rc_*(\\gamma)$.", "The maps $g$ and $f$ of the statement of the lemma are the adjoints", "of these maps. If we restrict this distinguished triangle to $Z$", "then the map $Rb_*(K) \\to Rc_*(L|_E)$ becomes an isomorphism", "by the base change theorem (Lemma \\ref{lemma-proper-base-change}) and hence", "the map $g : K|_Z \\to L$ is an isomorphism.", "Looking at the distinguished triangle we see that $f : K|_{X'} \\to K'$", "is an isomorphism over $X' \\setminus E = X \\setminus Z$.", "Moreover, we have $\\gamma \\circ f|_E = g|_E$ by construction.", "Then since $\\gamma$ and $g$ are isomorphisms we conclude", "that $f$ induces isomorphisms on stalks at geometric points of $E$", "as well. Thus $f$ is an isomorphism.", "\\medskip\\noindent", "For the final statement, we may replace $K'$ by $K|_{X'}$,", "$L$ by $K|_Z$, and $\\gamma$ by the canonical identification.", "Observe that $\\alpha$ and $\\beta$ induce a commutative square", "$$", "\\xymatrix{", "K \\ar[r] \\ar@{..>}[d] &", "Ri_*(K|_Z) \\oplus Rb_*(K|_{X'}) \\ar[r] \\ar[d]_{\\beta \\oplus \\alpha} &", "Rc_*(K|_E) \\ar[r] \\ar[d]_{\\alpha|_E} &", "K[1] \\ar@{..>}[d] \\\\", "M \\ar[r] &", "Ri_*(M|_Z) \\oplus Rb_*(M|_{X'}) \\ar[r] &", "Rc_*(M|_E) \\ar[r] &", "M[1]", "}", "$$", "Thus by the axioms of a derived category we get a dotted", "arrow producing a morphism of distinguished triangles." ], "refs": [ "etale-cohomology-lemma-blow-up-square-cohomological-descent", "etale-cohomology-lemma-proper-base-change" ], "ref_ids": [ 6690, 6626 ] } ], "ref_ids": [] }, { "id": 6693, "type": "theorem", "label": "etale-cohomology-lemma-blow-up-square-h", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-blow-up-square-h", "contents": [ "With notation as above, if $K$ is in the essential image", "of $R\\epsilon_*$, then the maps $c^K_{X, Z, X', E}$ of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-c-square}", "are quasi-isomorphisms." ], "refs": [ "sites-cohomology-lemma-c-square" ], "proofs": [ { "contents": [ "Denote ${}^\\#$ sheafification in the h topology.", "We have seen in More on Flatness, Lemma \\ref{flat-lemma-blow-up-square-h}", "that $h_X^\\# = h_Z^\\# \\amalg_{h_E^\\#} h_{X'}^\\#$. On the other hand,", "the map $h_E^\\# \\to h_{X'}^\\#$ is injective as $E \\to X'$ is a", "monomorphism. Thus this lemma is a special case of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-descent-squares-helper}", "(which itself is a formal consequence of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-square-triangle})." ], "refs": [ "flat-lemma-blow-up-square-h", "sites-cohomology-lemma-descent-squares-helper", "sites-cohomology-lemma-square-triangle" ], "ref_ids": [ 6163, 4295, 4284 ] } ], "ref_ids": [ 4282 ] }, { "id": 6694, "type": "theorem", "label": "etale-cohomology-lemma-refine-check-h", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-refine-check-h", "contents": [ "Let $K$ be an object of $D^+((\\Sch/S)_{fppf})$. Then $K$ is in the", "essential image of $R\\epsilon_* : D((\\Sch/S)_h) \\to D((\\Sch/S)_{fppf})$", "if and only if $c^K_{X, X', Z, E}$ is a quasi-isomorphism", "for every almost blow up square as in", "More on Flatness, Examples \\ref{flat-example-one-generator} and", "\\ref{flat-example-two-generators}." ], "refs": [], "proofs": [ { "contents": [ "We prove this by applying", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-descent-squares}", "whose hypotheses hold by", "Lemma \\ref{lemma-blow-up-square-h} and", "More on Flatness, Lemma \\ref{flat-lemma-refine-check-h}" ], "refs": [ "sites-cohomology-lemma-descent-squares", "etale-cohomology-lemma-blow-up-square-h", "flat-lemma-refine-check-h" ], "ref_ids": [ 4294, 6693, 6165 ] } ], "ref_ids": [] }, { "id": 6695, "type": "theorem", "label": "etale-cohomology-lemma-h-sheaf-colim-F", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-lemma-h-sheaf-colim-F", "contents": [ "Let $p$ be a prime number. Let $S$ be a scheme over $\\mathbf{F}_p$.", "Consider the sheaf $\\mathcal{O}^{perf} = \\colim_F \\mathcal{O}$", "on $(\\Sch/S)_{fppf}$. Then $\\mathcal{O}^{perf}$ is in the essential", "image of $R\\epsilon_* : D((\\Sch/S)_h) \\to D((\\Sch/S)_{fppf})$." ], "refs": [], "proofs": [ { "contents": [ "We prove this using the criterion of Lemma \\ref{lemma-refine-check-h}.", "Before check the conditions, we note that for a", "quasi-compact and quasi-separated object $X$ of", "$(\\Sch/S)_{fppf}$ we have", "$$", "H^i_{fppf}(X, \\mathcal{O}^{perf}) = \\colim_F H^i_{fppf}(X, \\mathcal{O})", "$$", "See Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-colim-works-over-collection}.", "We will also use that $H^i_{fppf}(X, \\mathcal{O}) = H^i(X, \\mathcal{O})$, see", "Descent, Proposition \\ref{descent-proposition-same-cohomology-quasi-coherent}.", "\\medskip\\noindent", "Let $A, f, J$ be as in", "More on Flatness, Example \\ref{flat-example-one-generator}", "and consider the associated almost blow up square.", "Since $X$, $X'$, $Z$, $E$ are affine, we have no", "higher cohomology of $\\mathcal{O}$. Hence we only", "have to check that", "$$", "0 \\to", "\\mathcal{O}^{perf}(X) \\to", "\\mathcal{O}^{perf}(X') \\oplus \\mathcal{O}^{perf}(Z) \\to", "\\mathcal{O}^{perf}(E) \\to 0", "$$", "is a short exact sequence. This was shown in (the proof of)", "More on Flatness, Lemma \\ref{flat-lemma-h-sheaf-colim-F}.", "\\medskip\\noindent", "Let $X, X', Z, E$ be as in", "More on Flatness, Example \\ref{flat-example-two-generators}.", "Since $X$ and $Z$ are affine we have", "$H^p(X, \\mathcal{O}_X) = H^p(Z, \\mathcal{O}_X) = 0$ for $p > 0$.", "By More on Flatness, Lemma \\ref{flat-lemma-funny-blow-up}", "we have $H^p(X', \\mathcal{O}_{X'}) = 0$ for $p > 0$.", "Since $E = \\mathbf{P}^1_Z$ and $Z$ is affine we also have", "$H^p(E, \\mathcal{O}_E) = 0$ for $p > 0$. As in the previous", "paragraph we reduce to checking that", "$$", "0 \\to", "\\mathcal{O}^{perf}(X) \\to", "\\mathcal{O}^{perf}(X') \\oplus \\mathcal{O}^{perf}(Z) \\to", "\\mathcal{O}^{perf}(E) \\to 0", "$$", "is a short exact sequence. This was shown in (the proof of)", "More on Flatness, Lemma \\ref{flat-lemma-h-sheaf-colim-F}." ], "refs": [ "etale-cohomology-lemma-refine-check-h", "sites-cohomology-lemma-colim-works-over-collection", "descent-proposition-same-cohomology-quasi-coherent", "flat-lemma-h-sheaf-colim-F", "flat-lemma-funny-blow-up", "flat-lemma-h-sheaf-colim-F" ], "ref_ids": [ 6694, 4224, 14754, 6168, 6167, 6168 ] } ], "ref_ids": [] }, { "id": 6696, "type": "theorem", "label": "etale-cohomology-proposition-quasi-coherent-sheaf-fpqc", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-proposition-quasi-coherent-sheaf-fpqc", "contents": [ "For any quasi-coherent sheaf $\\mathcal{F}$ on $S$ the presheaf", "$$", "\\begin{matrix}", "\\mathcal{F}^a : & \\Sch/S & \\to & \\textit{Ab}\\\\", "& (f: T \\to S) & \\mapsto & \\Gamma(T, f^*\\mathcal{F})", "\\end{matrix}", "$$", "is an $\\mathcal{O}$-module which satisfies the sheaf condition for the", "fpqc topology." ], "refs": [], "proofs": [ { "contents": [ "This is proved in", "Descent, Lemma \\ref{descent-lemma-sheaf-condition-holds}.", "We indicate the proof here. As established in", "Lemma \\ref{lemma-fpqc-sheaves},", "it is enough to check the sheaf property", "on Zariski coverings and faithfully flat morphisms of affine schemes. The", "sheaf property for Zariski coverings is standard scheme theory, since", "$\\Gamma(U, i^\\ast \\mathcal{F}) = \\mathcal{F}(U)$ when", "$i : U \\hookrightarrow S$ is an open immersion.", "\\medskip\\noindent", "For $\\left\\{\\Spec(B)\\to \\Spec(A)\\right\\}$ with $A\\to B$ faithfully", "flat and", "$\\mathcal{F}|_{\\Spec(A)} = \\widetilde{M}$", "this corresponds to the fact that", "$M = H^0\\left((B/A)_\\bullet \\otimes_A M \\right)$, i.e., that", "\\begin{align*}", "0 \\to M \\to B \\otimes_A M \\to B \\otimes_A B \\otimes_A M", "\\end{align*}", "is exact by", "Lemma \\ref{lemma-descent-modules}." ], "refs": [ "descent-lemma-sheaf-condition-holds", "etale-cohomology-lemma-fpqc-sheaves", "etale-cohomology-lemma-descent-modules" ], "ref_ids": [ 14621, 6400, 6403 ] } ], "ref_ids": [] }, { "id": 6697, "type": "theorem", "label": "etale-cohomology-proposition-etale-morphisms", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-proposition-etale-morphisms", "contents": [ "Facts on \\'etale morphisms.", "\\begin{enumerate}", "\\item Let $k$ be a field. A morphism of schemes $U \\to \\Spec(k)$ is", "\\'etale if and only if $U \\cong \\coprod_{i \\in I} \\Spec(k_i)$", "such that for each $i \\in I$", "the ring $k_i$ is a field which is a finite separable extension of $k$.", "\\item Let $\\varphi : U \\to S$ be a morphism of schemes. The following", "conditions are equivalent:", "\\begin{enumerate}", "\\item $\\varphi$ is \\'etale,", "\\item $\\varphi$ is locally finitely presented, flat, and all its fibres are", "\\'etale,", "\\item $\\varphi$ is flat, unramified and locally of finite presentation.", "\\end{enumerate}", "\\item A ring map $A \\to B$ is \\'etale if and only if", "$B \\cong A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$", "such that $\\Delta = \\det \\left( \\frac{\\partial f_i}{\\partial x_j} \\right)$", "is invertible in $B$.", "\\item The base change of an \\'etale morphism is \\'etale.", "\\item Compositions of \\'etale morphisms are \\'etale.", "\\item Fibre products and products of \\'etale morphisms are \\'etale.", "\\item An \\'etale morphism has relative dimension 0.", "\\item Let $Y \\to X$ be an \\'etale morphism.", "If $X$ is reduced (respectively regular) then so is $Y$.", "\\item \\'Etale morphisms are open.", "\\item If $X \\to S$ and $Y \\to S$ are \\'etale, then any", "$S$-morphism $X \\to Y$ is also \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have proved these facts (and more) in the preceding chapters.", "Here is a list of references:", "(1) Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}.", "(2) Morphisms, Lemmas \\ref{morphisms-lemma-etale-flat-etale-fibres}", "and \\ref{morphisms-lemma-flat-unramified-etale}.", "(3) Algebra, Lemma \\ref{algebra-lemma-etale-standard-smooth}.", "(4) Morphisms, Lemma \\ref{morphisms-lemma-base-change-etale}.", "(5) Morphisms, Lemma \\ref{morphisms-lemma-composition-etale}.", "(6) Follows formally from (4) and (5).", "(7) Morphisms, Lemmas \\ref{morphisms-lemma-etale-locally-quasi-finite}", "and \\ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}.", "(8) See Algebra, Lemmas \\ref{algebra-lemma-reduced-goes-up} and", "\\ref{algebra-lemma-Rk-goes-up}, see also more results of this kind", "in \\'Etale Morphisms, Section \\ref{etale-section-properties-permanence}.", "(9) See Morphisms, Lemma \\ref{morphisms-lemma-fppf-open} and", "\\ref{morphisms-lemma-etale-flat}.", "(10) See Morphisms, Lemma \\ref{morphisms-lemma-etale-permanence}." ], "refs": [ "morphisms-lemma-etale-over-field", "morphisms-lemma-etale-flat-etale-fibres", "morphisms-lemma-flat-unramified-etale", "algebra-lemma-etale-standard-smooth", "morphisms-lemma-base-change-etale", "morphisms-lemma-composition-etale", "morphisms-lemma-etale-locally-quasi-finite", "morphisms-lemma-locally-quasi-finite-rel-dimension-0", "algebra-lemma-reduced-goes-up", "algebra-lemma-Rk-goes-up", "morphisms-lemma-fppf-open", "morphisms-lemma-etale-flat", "morphisms-lemma-etale-permanence" ], "ref_ids": [ 5364, 5365, 5373, 1230, 5361, 5360, 5363, 5287, 1366, 1364, 5267, 5369, 5375 ] } ], "ref_ids": [] }, { "id": 6698, "type": "theorem", "label": "etale-cohomology-proposition-cohomology-restrict-small-site", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-proposition-cohomology-restrict-small-site", "contents": [ "Let $S$ be a scheme and $\\mathcal{F}$ an abelian sheaf on", "$(\\Sch/S)_\\etale$.", "Then $\\mathcal{F}|_{S_\\etale}$ is a sheaf on $S_\\etale$ and", "$$", "H^p_\\etale(S, \\mathcal{F}|_{S_\\etale}) =", "H^p_\\etale(S, \\mathcal{F})", "$$", "for all $p \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-compare-cohomology-big-small}." ], "refs": [ "etale-cohomology-lemma-compare-cohomology-big-small" ], "ref_ids": [ 6411 ] } ], "ref_ids": [] }, { "id": 6699, "type": "theorem", "label": "etale-cohomology-proposition-topological-invariance", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-proposition-topological-invariance", "contents": [ "Let $X_0 \\to X$ be a universal homeomorphism of schemes", "(for example the closed immersion defined by a nilpotent sheaf of ideals).", "Then", "\\begin{enumerate}", "\\item the \\'etale sites $X_\\etale$ and $(X_0)_\\etale$ are isomorphic,", "\\item the \\'etale topoi $\\Sh(X_\\etale)$ and $\\Sh((X_0)_\\etale)$", "are equivalent, and", "\\item $H^q_\\etale(X, \\mathcal{F}) = H^q_\\etale(X_0, \\mathcal{F}|_{X_0})$", "for all $q$ and", "for any abelian sheaf $\\mathcal{F}$ on $X_\\etale$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of categories $X_\\etale \\to (X_0)_\\etale$ is", "given by Theorem \\ref{theorem-topological-invariance}. We omit", "the proof that under this equivalence the \\'etale coverings correspond.", "Hence (1) holds. Parts (2) and (3) follow formally from (1)." ], "refs": [ "etale-cohomology-theorem-topological-invariance" ], "ref_ids": [ 6383 ] } ], "ref_ids": [] }, { "id": 6700, "type": "theorem", "label": "etale-cohomology-proposition-closed-immersion-pushforward", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-proposition-closed-immersion-pushforward", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes.", "\\begin{enumerate}", "\\item The functor", "$$", "i_{small, *} :", "\\Sh(Z_\\etale)", "\\longrightarrow", "\\Sh(X_\\etale)", "$$", "is fully faithful and its essential image is those sheaves of sets", "$\\mathcal{F}$ on $X_\\etale$ whose restriction to $X \\setminus Z$ is", "isomorphic to $*$, and", "\\item the functor", "$$", "i_{small, *} :", "\\textit{Ab}(Z_\\etale)", "\\longrightarrow", "\\textit{Ab}(X_\\etale)", "$$", "is fully faithful and its essential image is those abelian sheaves on", "$X_\\etale$ whose support is contained in $Z$.", "\\end{enumerate}", "In both cases $i_{small}^{-1}$ is a left inverse to the functor", "$i_{small, *}$." ], "refs": [], "proofs": [ { "contents": [ "Let's discuss the case of sheaves of sets.", "For any sheaf $\\mathcal{G}$ on $Z$ the morphism", "$i_{small}^{-1}i_{small, *}\\mathcal{G} \\to \\mathcal{G}$", "is an isomorphism by", "Lemma \\ref{lemma-stalk-pushforward-closed-immersion}", "(and", "Theorem \\ref{theorem-exactness-stalks}).", "This implies formally that $i_{small, *}$ is fully faithful, see", "Sites, Lemma \\ref{sites-lemma-exactness-properties}.", "It is clear that $i_{small, *}\\mathcal{G}|_{U_\\etale} \\cong *$", "where $U = X \\setminus Z$. Conversely, suppose that $\\mathcal{F}$", "is a sheaf of sets on $X$ such that $\\mathcal{F}|_{U_\\etale} \\cong *$.", "Consider the adjunction mapping", "$$", "\\mathcal{F} \\longrightarrow i_{small, *}i_{small}^{-1}\\mathcal{F}", "$$", "Combining", "Lemmas \\ref{lemma-stalk-pushforward-closed-immersion} and", "\\ref{lemma-stalk-pullback}", "we see that it is an isomorphism. This finishes the proof of (1).", "The proof of (2) is identical." ], "refs": [ "etale-cohomology-lemma-stalk-pushforward-closed-immersion", "etale-cohomology-theorem-exactness-stalks", "sites-lemma-exactness-properties", "etale-cohomology-lemma-stalk-pushforward-closed-immersion", "etale-cohomology-lemma-stalk-pullback" ], "ref_ids": [ 6461, 6376, 8618, 6461, 6436 ] } ], "ref_ids": [] }, { "id": 6701, "type": "theorem", "label": "etale-cohomology-proposition-integral-universally-injective-pushforward", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-proposition-integral-universally-injective-pushforward", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes which is integral", "and universally injective.", "\\begin{enumerate}", "\\item The functor", "$$", "f_{small, *} :", "\\Sh(X_\\etale)", "\\longrightarrow", "\\Sh(Y_\\etale)", "$$", "is fully faithful and its essential image is those sheaves of sets", "$\\mathcal{F}$ on $Y_\\etale$ whose restriction to $Y \\setminus f(X)$ is", "isomorphic to $*$, and", "\\item the functor", "$$", "f_{small, *} :", "\\textit{Ab}(X_\\etale)", "\\longrightarrow", "\\textit{Ab}(Y_\\etale)", "$$", "is fully faithful and its essential image is those abelian sheaves on", "$Y_\\etale$ whose support is contained in $f(X)$.", "\\end{enumerate}", "In both cases $f_{small}^{-1}$ is a left inverse to the functor", "$f_{small, *}$." ], "refs": [], "proofs": [ { "contents": [ "We may factor $f$ as", "$$", "\\xymatrix{", "X \\ar[r]^h & Z \\ar[r]^i & Y", "}", "$$", "where $h$ is integral, universally injective and surjective", "and $i : Z \\to Y$ is a closed immersion.", "Apply", "Proposition \\ref{proposition-closed-immersion-pushforward}", "to $i$ and apply", "Theorem \\ref{theorem-topological-invariance}", "to $h$." ], "refs": [ "etale-cohomology-proposition-closed-immersion-pushforward", "etale-cohomology-theorem-topological-invariance" ], "ref_ids": [ 6700, 6383 ] } ], "ref_ids": [] }, { "id": 6702, "type": "theorem", "label": "etale-cohomology-proposition-leray", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-proposition-leray", "contents": [ "Let $f: X \\to Y$ be a morphism of schemes and $\\mathcal{F}$ an \\'etale sheaf on", "$X$. Then there is a spectral sequence", "$$", "E_2^{p, q} = H_\\etale^p(Y, R^qf_*\\mathcal{F}) \\Rightarrow", "H_\\etale^{p+q}(X, \\mathcal{F}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "See Lemma \\ref{lemma-prepare-leray} and see", "Derived Categories, Section", "\\ref{derived-section-composition-right-derived-functors}." ], "refs": [ "etale-cohomology-lemma-prepare-leray" ], "ref_ids": [ 6479 ] } ], "ref_ids": [] }, { "id": 6703, "type": "theorem", "label": "etale-cohomology-proposition-finite-higher-direct-image-zero", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-proposition-finite-higher-direct-image-zero", "contents": [ "Let $f : X \\to Y$ be a finite morphism of schemes.", "\\begin{enumerate}", "\\item For any geometric point $\\overline{y} : \\Spec(k) \\to Y$ we have", "$$", "(f_*\\mathcal{F})_{\\overline{y}} =", "\\prod\\nolimits_{\\overline{x} : \\Spec(k) \\to X,\\ f(\\overline{x}) =", "\\overline{y}} \\mathcal{F}_{\\overline{x}}.", "$$", "for $\\mathcal{F}$ in $\\Sh(X_\\etale)$ and", "$$", "(f_*\\mathcal{F})_{\\overline{y}} =", "\\bigoplus\\nolimits_{\\overline{x} : \\Spec(k) \\to X,\\ f(\\overline{x}) =", "\\overline{y}} \\mathcal{F}_{\\overline{x}}.", "$$", "for $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$.", "\\item For any $q \\geq 1$ we have $R^q f_*\\mathcal{F} = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X_{\\overline{y}}^{sh}$ denote the fiber product", "$X \\times_Y \\Spec(\\mathcal{O}_{Y, \\overline{y}}^{sh})$.", "By Theorem \\ref{theorem-higher-direct-images}", "the stalk of $R^qf_*\\mathcal{F}$ at $\\overline{y}$ is computed by", "$H_\\etale^q(X_{\\overline{y}}^{sh}, \\mathcal{F})$.", "Since $f$ is finite, $X_{\\bar y}^{sh}$ is finite over", "$\\Spec(\\mathcal{O}_{Y, \\overline{y}}^{sh})$, thus", "$X_{\\bar y}^{sh} = \\Spec(A)$ for some ring $A$", "finite over $\\mathcal{O}_{Y, \\bar y}^{sh}$.", "Since the latter is strictly henselian,", "Lemma \\ref{lemma-finite-over-henselian}", "implies that $A$ is a finite product of henselian local rings", "$A = A_1 \\times \\ldots \\times A_r$. Since the residue field of", "$\\mathcal{O}_{Y, \\overline{y}}^{sh}$ is separably closed the", "same is true for each $A_i$. Hence $A_i$ is strictly henselian.", "This implies that $X_{\\overline{y}}^{sh} = \\coprod_{i = 1}^r \\Spec(A_i)$.", "The vanishing of", "Lemma \\ref{lemma-vanishing-etale-cohomology-strictly-henselian}", "implies that $(R^qf_*\\mathcal{F})_{\\overline{y}} = 0$ for $q > 0$", "which implies (2) by Theorem \\ref{theorem-exactness-stalks}.", "Part (1) follows from the corresponding statement of", "Lemma \\ref{lemma-vanishing-etale-cohomology-strictly-henselian}." ], "refs": [ "etale-cohomology-theorem-higher-direct-images", "etale-cohomology-lemma-finite-over-henselian", "etale-cohomology-lemma-vanishing-etale-cohomology-strictly-henselian", "etale-cohomology-theorem-exactness-stalks", "etale-cohomology-lemma-vanishing-etale-cohomology-strictly-henselian" ], "ref_ids": [ 6385, 6432, 6480, 6376, 6480 ] } ], "ref_ids": [] }, { "id": 6704, "type": "theorem", "label": "etale-cohomology-proposition-serre-galois", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-proposition-serre-galois", "contents": [ "\\begin{reference}", "\\cite[Chapter II, Section 3, Proposition 5]{SerreGaloisCohomology}", "\\end{reference}", "Let $K$ be a field with separable algebraic closure $K^{sep}$.", "Assume that for any finite extension $K'$ of $K$ we have", "$\\text{Br}(K') = 0$. Then", "\\begin{enumerate}", "\\item $H^q(\\text{Gal}(K^{sep}/K), (K^{sep})^*) = 0$", "for all $q \\geq 1$, and", "\\item $H^q(\\text{Gal}(K^{sep}/K), M) = 0$", "for any torsion $\\text{Gal}(K^{sep}/K)$-module $M$ and any $q \\geq 2$,", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Set $p = \\text{char}(K)$.", "By Lemma \\ref{lemma-compare-cohomology-point},", "Theorem \\ref{theorem-brauer-delta},", "and Example \\ref{example-sheaves-point}", "the proposition is equivalent to showing that if ", "$H^2(\\Spec(K'),\\mathbf{G}_m|_{\\Spec(K')_\\etale}) = 0$", "for all finite extensions $K'/K$ then:", "\\begin{itemize}", "\\item $H^q(\\Spec(K),\\mathbf{G}_m|_{\\Spec(K)_\\etale}) = 0$", "for all $q \\geq 1$, and", "\\item $H^q(\\Spec(K),\\mathcal{F}) = 0$", "for any torsion sheaf $\\mathcal{F}$ and any $q \\geq 2$.", "\\end{itemize}", "We prove the second part first.", "Since $\\mathcal{F}$ is a torsion sheaf, we may use the ", "$\\ell$-primary decomposition as well as the compatibility of", "cohomology with colimits (i.e, direct sums, see Theorem \\ref{theorem-colimit})", "to reduce to showing $H^q(\\Spec(K),\\mathcal{F}) = 0$, $q \\geq 2$", "for all $\\ell$-power torsion sheaves for every prime $\\ell$. This ", "allows us to analyze each prime individually.", "\\medskip\\noindent", "Suppose that $\\ell \\neq p$. For any extension $K'/K$", "consider the Kummer sequence (Lemma \\ref{lemma-kummer-sequence})", "$$", "0 \\to", "\\mu_{\\ell, \\Spec{K'}} \\to", "\\mathbf{G}_{m, \\Spec{K'}} \\xrightarrow{(\\cdot)^{\\ell}}", "\\mathbf{G}_{m, \\Spec{K'}} \\to 0", "$$", "Since $H^q(\\Spec{K'},\\mathbf{G}_m|_{\\Spec(K')_\\etale}) = 0$", "for $q = 2$ by assumption and for $q = 1$ by", "Theorem \\ref{theorem-picard-group} combined with", "$\\Pic(K) = (0)$. Thus, by the long-exact cohomology ", "sequence we may conclude that $H^2(\\Spec{K'}, \\mu_\\ell) = 0$ for ", "any separable $K'/K$. Now let $H$ be a maximal pro-$\\ell$ subgroup", "of the absolute Galois group of $K$ and let $L$ be the ", "corresponding extension. We can write $L$ as the colimit of finite extensions, ", "applying Theorem \\ref{theorem-colimit} to this colimit we see that", "$H^2(\\Spec(L), \\mu_\\ell) = 0$.", "Now $\\mu_\\ell$ must be the constant sheaf. If it weren't, that would imply", "there exists a Galois extension of degree relatively prime to ", "$\\ell$ of $L$ which is not true by definition of $L$ (namely, the extension", "one gets by adjoining the $\\ell$th roots of unity to $L$).", "Hence, via Lemma \\ref{lemma-reduce-to-l-group-higher},", "we conclude the result for $\\ell \\neq p$. ", "\\medskip\\noindent", "Now suppose that $\\ell = p$. We consider the", "Artin-Schrier exact sequence (Section \\ref{section-artin-schreier})", "$$", "0 \\longrightarrow \\underline{\\mathbf{Z}/p\\mathbf{Z}}_{\\Spec{K}} \\longrightarrow", "\\mathbf{G}_{a, \\Spec{K}} \\xrightarrow{F-1} \\mathbf{G}_{a, \\Spec{K}} ", "\\longrightarrow 0", "$$", "where $F - 1$ is the map $x \\mapsto x^p - x$. Then note that the higher ", "Cohomology of $\\mathbf{G}_{a, \\Spec{K}}$ vanishes, by", "Remark \\ref{remark-special-case-fpqc-cohomology-quasi-coherent} and the ", "vanishing of the higher cohomology of the structure sheaf of an affine scheme", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}).", "Note this can be applied to any field of ", "characteristic $p$. In particular, we can apply it to the field extension $L$ ", "defined by a maximal pro-$p$ subgroup $H$. This allows us to conclude ", "$H^n(\\Spec{L}, \\underline{\\mathbf{Z}/p\\mathbf{Z}}_{\\Spec{L}}) = 0$", "for $n \\geq 2$, from which the result follows for $\\ell = p$, by", "Lemma \\ref{lemma-reduce-to-l-group-higher}.", "\\medskip\\noindent", "To finish the proof we still have to show that", "$H^q(\\text{Gal}(K^{sep}/K), (K^{sep})^*) = 0$ for all $q \\geq 1$.", "Set $G = \\text{Gal}(K^{sep}/K)$ and set $M = (K^{sep})^*$", "viewed as a $G$-module. We have already shown (above) that", "$H^1(G, M) = 0$ and $H^2(G, M) = 0$. Consider the exact sequence", "$$", "0 \\to A \\to M \\to M \\otimes \\mathbf{Q} \\to B \\to 0", "$$", "of $G$-modules. By the above we have $H^i(G, A) = 0$", "and $H^i(G, B) = 0$ for $i > 1$ since $A$ and $B$ are", "torsion $G$-modules. By", "Lemma \\ref{lemma-profinite-group-cohomology-torsion}", "we have $H^i(G, M \\otimes \\mathbf{Q}) = 0$ for $i > 0$.", "It is a pleasant exercise to see that this implies that", "$H^i(G, M) = 0$ also for $i \\geq 3$." ], "refs": [ "etale-cohomology-lemma-compare-cohomology-point", "etale-cohomology-theorem-brauer-delta", "etale-cohomology-theorem-colimit", "etale-cohomology-lemma-kummer-sequence", "etale-cohomology-theorem-picard-group", "etale-cohomology-theorem-colimit", "etale-cohomology-lemma-reduce-to-l-group-higher", "etale-cohomology-remark-special-case-fpqc-cohomology-quasi-coherent", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "etale-cohomology-lemma-reduce-to-l-group-higher", "etale-cohomology-lemma-profinite-group-cohomology-torsion" ], "ref_ids": [ 6490, 6388, 6384, 6419, 6374, 6384, 6512, 6771, 3282, 6512, 6488 ] } ], "ref_ids": [] }, { "id": 6705, "type": "theorem", "label": "etale-cohomology-proposition-describe-jshriek", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-proposition-describe-jshriek", "contents": [ "Let $j : U \\to X$ be an \\'etale morphism of schemes.", "Let $\\mathcal{F}$ in $\\textit{Ab}(U_\\etale)$.", "If $\\overline{x} : \\Spec(k) \\to X$ is a geometric point of $X$, then ", "$$", "(j_!\\mathcal{F})_{\\overline{x}} =", "\\bigoplus\\nolimits_{\\overline{u} : \\Spec(k) \\to U,\\ j(\\overline{u}) =", "\\overline{x}} \\mathcal{F}_{\\bar{u}}.", "$$", "In particular, $j_!$ is an exact functor." ], "refs": [], "proofs": [ { "contents": [ "Exactness of $j_!$ is very general, see Modules on Sites, ", "Lemma \\ref{sites-modules-lemma-extension-by-zero-exact}.", "Of course it does also follow from the description of stalks.", "The formula for the stalk follows from", "Modules on Sites, Lemma \\ref{sites-modules-lemma-stalk-j-shriek}", "and the description of points of the small \\'etale site", "in terms of geometric points, see Lemma \\ref{lemma-points-small-etale-site}.", "\\medskip\\noindent", "For later use we note that the isomorphism", "\\begin{align*}", "(j_!\\mathcal{F})_{\\overline{x}}", "& =", "(j_{p!}\\mathcal{F})_{\\overline{x}} \\\\", "& =", "\\colim_{(V, \\overline{v})} j_{p!}\\mathcal{F}(V) \\\\", "& =", "\\colim_{(V, \\overline{v})}", "\\bigoplus\\nolimits_{\\varphi : V \\to U}", "\\mathcal{F}(V \\xrightarrow{\\varphi} U) \\\\", "& \\to", "\\bigoplus\\nolimits_{\\overline{u} : \\Spec(k) \\to U,\\ j(\\overline{u}) =", "\\overline{x}} \\mathcal{F}_{\\bar{u}}.", "\\end{align*}", "constructed in Modules on Sites, Lemma \\ref{sites-modules-lemma-stalk-j-shriek}", "sends $(V, \\overline{v}, \\varphi, s)$ to the class of $s$ in the stalk", "of $\\mathcal{F}$ at $\\overline{u} = \\varphi(\\overline{v})$." ], "refs": [ "sites-modules-lemma-extension-by-zero-exact", "sites-modules-lemma-stalk-j-shriek", "etale-cohomology-lemma-points-small-etale-site", "sites-modules-lemma-stalk-j-shriek" ], "ref_ids": [ 14170, 14248, 6426, 14248 ] } ], "ref_ids": [] }, { "id": 6706, "type": "theorem", "label": "etale-cohomology-proposition-constructible-over-noetherian", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-proposition-constructible-over-noetherian", "contents": [ "Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring.", "\\begin{enumerate}", "\\item Any sub or quotient sheaf of a constructible sheaf of sets", "is constructible.", "\\item The category of constructible abelian sheaves on $X_\\etale$ is a", "(strong) Serre subcategory of $\\textit{Ab}(X_\\etale)$. In particular,", "every sub and quotient sheaf of a constructible abelian sheaf", "on $X_\\etale$ is constructible.", "\\item The category of constructible sheaves of $\\Lambda$-modules", "on $X_\\etale$ is a (strong) Serre subcategory of", "$\\textit{Mod}(X_\\etale, \\Lambda)$. In particular, every submodule", "and quotient module of a constructible sheaf of $\\Lambda$-modules", "on $X_\\etale$ is constructible.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $\\mathcal{G} \\subset \\mathcal{F}$ with $\\mathcal{F}$", "a constructible sheaf of sets on $X_\\etale$. Let $\\eta \\in X$ be a generic", "point of an irreducible component of $X$. By Noetherian induction", "it suffices to find an open neighbourhood $U$ of $\\eta$ such that", "$\\mathcal{G}|_U$ is locally constant. To do this we may replace $X$", "by an \\'etale neighbourhood of $\\eta$.", "Hence we may assume $\\mathcal{F}$ is constant and $X$ is irreducible.", "\\medskip\\noindent", "Say $\\mathcal{F} = \\underline{S}$ for some finite set $S$.", "Then $S' = \\mathcal{G}_{\\overline{\\eta}} \\subset S$", "say $S' = \\{s_1, \\ldots, s_t\\}$.", "Pick an \\'etale neighbourhood $(U, \\overline{u})$ of $\\overline{\\eta}$", "and sections $\\sigma_1, \\ldots, \\sigma_t \\in \\mathcal{G}(U)$ which map to", "$s_i$ in $\\mathcal{G}_{\\overline{\\eta}} \\subset S$.", "Since $\\sigma_i$ maps to an element", "$s_i \\in S' \\subset S = \\Gamma(X, \\mathcal{F})$", "we see that the two pullbacks of $\\sigma_i$ to $U \\times_X U$", "are the same as sections of $\\mathcal{G}$. By the sheaf condition", "for $\\mathcal{G}$ we find that $\\sigma_i$ comes from a section", "of $\\mathcal{G}$ over the open $\\Im(U \\to X)$ of $X$.", "Shrinking $X$ we may assume", "$\\underline{S'} \\subset \\mathcal{G} \\subset \\underline{S}$.", "Then we see that $\\underline{S'} = \\mathcal{G}$ by", "Lemma \\ref{lemma-irreducible-subsheaf-constant-zero}.", "\\medskip\\noindent", "Let $\\mathcal{F} \\to \\mathcal{Q}$ be a surjection with $\\mathcal{F}$", "a constructible sheaf of sets on $X_\\etale$. Then set", "$\\mathcal{G} = \\mathcal{F} \\times_\\mathcal{Q} \\mathcal{F}$.", "By the first part of the proof we see that $\\mathcal{G}$ is", "constructible as a subsheaf of $\\mathcal{F} \\times \\mathcal{F}$.", "This in turn implies that $\\mathcal{Q}$ is constructible, see", "Lemma \\ref{lemma-constructible-abelian}.", "\\medskip\\noindent", "Proof of (3). we already know that constructible sheaves of modules", "form a weak Serre subcategory, see Lemma \\ref{lemma-constructible-abelian}.", "Thus it suffices to show the statement on submodules.", "\\medskip\\noindent", "Let $\\mathcal{G} \\subset \\mathcal{F}$ be a submodule of a", "constructible sheaf of $\\Lambda$-modules on $X_\\etale$. Let $\\eta \\in X$", "be a generic point of an irreducible component of $X$. By Noetherian induction", "it suffices to find an open neighbourhood $U$ of $\\eta$ such that", "$\\mathcal{G}|_U$ is locally constant. To do this we may replace $X$", "by an \\'etale neighbourhood of $\\eta$. Hence we may assume $\\mathcal{F}$", "is constant and $X$ is irreducible.", "\\medskip\\noindent", "Say $\\mathcal{F} = \\underline{M}$ for some finite $\\Lambda$-module $M$.", "Then $M' = \\mathcal{G}_{\\overline{\\eta}} \\subset M$. Pick finitely", "many elements $s_1, \\ldots, s_t$ generating $M'$ as a $\\Lambda$-module.", "(This is possible as $\\Lambda$ is Noetherian and $M$ is finite.)", "Pick an \\'etale neighbourhood $(U, \\overline{u})$ of $\\overline{\\eta}$", "and sections $\\sigma_1, \\ldots, \\sigma_t \\in \\mathcal{G}(U)$ which map to", "$s_i$ in $\\mathcal{G}_{\\overline{\\eta}} \\subset M$.", "Since $\\sigma_i$ maps to an element", "$s_i \\in M' \\subset M = \\Gamma(X, \\mathcal{F})$", "we see that the two pullbacks of $\\sigma_i$ to $U \\times_X U$", "are the same as sections of $\\mathcal{G}$. By the sheaf condition", "for $\\mathcal{G}$ we find that $\\sigma_i$ comes from a section", "of $\\mathcal{G}$ over the open $\\Im(U \\to X)$ of $X$.", "Shrinking $X$ we may assume", "$\\underline{M'} \\subset \\mathcal{G} \\subset \\underline{M}$.", "Then we see that $\\underline{M'} = \\mathcal{G}$ by", "Lemma \\ref{lemma-irreducible-subsheaf-constant-zero}.", "\\medskip\\noindent", "Proof of (2). This follows in the usual manner from (3). Details", "omitted." ], "refs": [ "etale-cohomology-lemma-irreducible-subsheaf-constant-zero", "etale-cohomology-lemma-constructible-abelian", "etale-cohomology-lemma-constructible-abelian", "etale-cohomology-lemma-irreducible-subsheaf-constant-zero" ], "ref_ids": [ 6548, 6531, 6531, 6548 ] } ], "ref_ids": [] }, { "id": 6707, "type": "theorem", "label": "etale-cohomology-proposition-cd-affine", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-proposition-cd-affine", "contents": [ "Let $K$ be a field. Let $X$ be an affine scheme of finite type over $K$.", "Then we have $\\text{cd}(X) \\leq \\dim(X) + \\text{cd}(K)$." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by induction on $\\dim(X)$.", "Let $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$.", "\\medskip\\noindent", "The case $\\dim(X) = 0$. In this case the structure morphism", "$f : X \\to \\Spec(K)$ is finite. Hence we see that $R^if_*\\mathcal{F} = 0$", "for $i > 0$, see Proposition \\ref{proposition-finite-higher-direct-image-zero}.", "Thus $H^i_\\etale(X, \\mathcal{F}) = H^i_\\etale(\\Spec(K), f_*\\mathcal{F})$", "by the Leray spectral sequence for $f$", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-Leray})", "and the result is clear.", "\\medskip\\noindent", "The case $\\dim(X) = 1$. This is Lemma \\ref{lemma-cd-curve-over-field}.", "\\medskip\\noindent", "Assume $d = \\dim(X) > 1$ and the proposition holds for finite type", "affine schemes of dimension $< d$ over fields.", "By Noether normalization, see for example", "Varieties, Lemma \\ref{varieties-lemma-noether-normalization-affine},", "there exists a finite morphism $f : X \\to \\mathbf{A}^d_K$.", "Recall that $R^if_*\\mathcal{F} = 0$ for $i > 0$ by", "Proposition \\ref{proposition-finite-higher-direct-image-zero}.", "By the Leray spectral sequence for $f$", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-Leray})", "we conclude that it suffices to prove the result for $\\pi_*\\mathcal{F}$", "on $\\mathbf{A}^d_K$.", "\\medskip\\noindent", "Interlude I. Let $j : X \\to Y$ be an open immersion of smooth", "$d$-dimensional varieties over $K$ (not necessarily affine)", "whose complement is the support of an effective Cartier divisor $D$.", "The sheaves $R^qj_*\\mathcal{F}$ for $q > 0$ are supported on $D$.", "We claim that $(R^qj_*\\mathcal{F})_{\\overline{y}} = 0$", "for $a = \\text{trdeg}_K(\\kappa(y)) > d - q$. Namely, by", "Theorem \\ref{theorem-higher-direct-images} we have", "$$", "(R^qj_*\\mathcal{F})_{\\overline{y}} =", "H^q(\\Spec(\\mathcal{O}_{Y, y}^{sh}) \\times_Y X, \\mathcal{F})", "$$", "Choose a local equation $f \\in \\mathfrak m_y = \\mathcal{O}_{Y, y}$", "for $D$. Then we have", "$$", "\\Spec(\\mathcal{O}_{Y, y}^{sh}) \\times_Y X =", "\\Spec(\\mathcal{O}_{Y, y}^{sh}[1/f])", "$$", "Using Lemma \\ref{lemma-strictly-henselian} we get an embedding", "$$", "K(t_1, \\ldots, t_a)^{sep}(x) =", "K(t_1, \\ldots, t_a)^{sep}[x]_{(x)}[1/x]", "\\longrightarrow", "\\mathcal{O}_{Y, y}^{sh}[1/f]", "$$", "Since the transcendence degree over $K$ of the fraction field of", "$\\mathcal{O}_{Y, y}^{sh}$ is $d$, we see that $\\mathcal{O}_{Y, y}^{sh}[1/f]$", "is a filtered colimit of $(d - a - 1)$-dimensional finite type algebras over", "the field $K(t_1, \\ldots, t_a)^{sep}(x)$ which itself has cohomological", "dimension $1$ by Lemma \\ref{lemma-cd-field-extension}. Thus by induction", "hypothesis and Lemma \\ref{lemma-cd-limit}", "we obtain the desired vanishing.", "\\medskip\\noindent", "Interlude II. Let $Z$ be a smooth variety over $K$ of dimension $d - 1$.", "Let $E_a \\subset Z$ be the set of points $z \\in Z$ with", "$\\text{trdeg}_K(\\kappa(z)) \\leq a$. Observe that $E_a$ is closed", "under specialization, see", "Varieties, Lemma \\ref{varieties-lemma-dimension-locally-algebraic}.", "Suppose that $\\mathcal{G}$ is a torsion abelian sheaf on $Z$", "whose support is contained in $E_a$. Then we claim that", "$H^b_\\etale(Z, \\mathcal{G}) = 0$ for $b > a + \\text{cd}(K)$.", "Namely, we can write $\\mathcal{G} = \\colim \\mathcal{G}_i$", "with $\\mathcal{G}_i$ a torsion abelian sheaf", "supported on a closed subscheme $Z_i$ contained in $E_a$, see", "Lemma \\ref{lemma-support-in-subset}.", "Then the induction hypothesis kicks in to imply", "the desired vanishing for $\\mathcal{G}_i$\\footnote{Here", "we first use Proposition \\ref{proposition-closed-immersion-pushforward}", "to write $\\mathcal{G}_i$ as the pushforward of a sheaf on $Z_i$,", "the induction hypothesis gives the vanishing for this sheaf on $Z_i$, and", "the Leray spectral sequence for $Z_i \\to Z$ gives the vanishing", "for $\\mathcal{G}_i$.}. Finally, we", "conclude by Theorem \\ref{theorem-colimit}.", "\\medskip\\noindent", "Consider the commutative diagram", "$$", "\\xymatrix{", "\\mathbf{A}^d_K \\ar[rd]_f \\ar[rr]_-j & &", "\\mathbf{P}^1_K \\times_K \\mathbf{A}^{d - 1}_K \\ar[ld]^g \\\\", "& \\mathbf{A}^{d - 1}_K", "}", "$$", "Observe that $j$ is an open immersion of smooth $d$-dimensional", "varieties whose complement is an effective Cartier divisor $D$. Thus", "we may use the results obtained in interlude I.", "We are going to study the relative Leray spectral sequence", "$$", "E_2^{p, q} = R^pg_*R^qj_*\\mathcal{F} \\Rightarrow R^{p + q}f_*\\mathcal{F}", "$$", "Since $R^qj_*\\mathcal{F}$ for $q > 0$ is supported on $D$ and since", "$g|_D : D \\to \\mathbf{A}^{d - 1}_K$ is an isomorphism, we find", "$R^pg_*R^qj_*\\mathcal{F} = 0$ for $p > 0$ and $q > 0$. Moreover, we have", "$R^qj_*\\mathcal{F} = 0$ for $q > d$. On the other hand, $g$ is a", "proper morphism of relative dimension $1$. Hence by", "Lemma \\ref{lemma-cohomological-dimension-proper}", "we see that $R^pg_*j_*\\mathcal{F} = 0$ for $p > 2$. Thus the $E_2$-page", "of the spectral sequence looks like this", "$$", "\\begin{matrix}", "g_*R^dj_*\\mathcal{F} & 0 & 0 \\\\", "\\ldots & \\ldots & \\ldots \\\\", "g_*R^2j_*\\mathcal{F} & 0 & 0 \\\\", "g_*R^1j_*\\mathcal{F} & 0 & 0 \\\\", "g_*j_*\\mathcal{F} & R^1g_*j_*\\mathcal{F} & R^2g_*j_*\\mathcal{F}", "\\end{matrix}", "$$", "We conclude that $R^qf_*\\mathcal{F} = g_*R^qj_*\\mathcal{F}$ for $q > 2$.", "By interlude I we see that the support of $R^qf_*\\mathcal{F}$ for $q > 2$", "is contained in the set of points of $\\mathbf{A}^{d - 1}_K$", "whose residue field has transcendence degree $\\leq d - q$.", "By interlude II", "$$", "H^p(\\mathbf{A}^{d - 1}_K, R^qf_*\\mathcal{F}) = 0", "\\text{ for }p > d - q + \\text{cd}(K)\\text{ and }q > 2", "$$", "On the other hand, by Theorem \\ref{theorem-higher-direct-images}", "we have $R^2f_*\\mathcal{F}_{\\overline{\\eta}} =", "H^2(\\mathbf{A}^1_{\\overline{\\eta}}, \\mathcal{F}) = 0$", "(vanishing by the case of dimension $1$) where $\\eta$ is the", "generic point of $\\mathbf{A}^{d - 1}_K$.", "Hence by interlude II again we see", "$$", "H^p(\\mathbf{A}^{d - 1}_K, R^2f_*\\mathcal{F}) = 0", "\\text{ for }p > d - 2 + \\text{cd}(K)", "$$", "Finally, we have", "$$", "H^p(\\mathbf{A}^{d - 1}_K, R^qf_*\\mathcal{F}) = 0", "\\text{ for }p > d - 1 + \\text{cd}(K)\\text{ and }q = 0, 1", "$$", "by induction hypothesis. Combining everything we just said", "with the Leray spectral sequence", "$H^p(\\mathbf{A}^{d - 1}_K, R^qf_*\\mathcal{F}) \\Rightarrow", "H^{p + q}(\\mathbf{A}^d_K, \\mathcal{F})$ we conclude." ], "refs": [ "etale-cohomology-proposition-finite-higher-direct-image-zero", "sites-cohomology-lemma-Leray", "etale-cohomology-lemma-cd-curve-over-field", "varieties-lemma-noether-normalization-affine", "etale-cohomology-proposition-finite-higher-direct-image-zero", "sites-cohomology-lemma-Leray", "etale-cohomology-theorem-higher-direct-images", "etale-cohomology-lemma-strictly-henselian", "etale-cohomology-lemma-cd-field-extension", "etale-cohomology-lemma-cd-limit", "varieties-lemma-dimension-locally-algebraic", "etale-cohomology-lemma-support-in-subset", "etale-cohomology-proposition-closed-immersion-pushforward", "etale-cohomology-theorem-colimit", "etale-cohomology-lemma-cohomological-dimension-proper", "etale-cohomology-theorem-higher-direct-images" ], "ref_ids": [ 6703, 4220, 6634, 10981, 6703, 4220, 6385, 6636, 6635, 6633, 10989, 6554, 6700, 6384, 6629, 6385 ] } ], "ref_ids": [] }, { "id": 6708, "type": "theorem", "label": "etale-cohomology-proposition-check-h", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-proposition-check-h", "contents": [ "Let $K$ be an object of $D^+((\\Sch/S)_{fppf})$.", "Then $K$ is in the essential image of", "$R\\epsilon_* : D((\\Sch/S)_h) \\to D((\\Sch/S)_{fppf})$", "if and only if $c^K_{X, X', Z, E}$ is a quasi-isomorphism", "for every almost blow up square (\\ref{equation-almost-blow-up-square})", "in $(\\Sch/S)_h$ with $X$ affine." ], "refs": [], "proofs": [ { "contents": [ "We prove this by applying", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-descent-squares}", "whose hypotheses hold by", "Lemma \\ref{lemma-blow-up-square-h} and", "More on Flatness, Proposition \\ref{flat-proposition-check-h}." ], "refs": [ "sites-cohomology-lemma-descent-squares", "etale-cohomology-lemma-blow-up-square-h", "flat-proposition-check-h" ], "ref_ids": [ 4294, 6693, 6204 ] } ], "ref_ids": [] }, { "id": 6709, "type": "theorem", "label": "etale-cohomology-proposition-h-cohomology-structure-sheaf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-proposition-h-cohomology-structure-sheaf", "contents": [ "Let $p$ be a prime number. Let $S$ be a quasi-compact and quasi-separated", "scheme over $\\mathbf{F}_p$. Then", "$$", "H^i((\\Sch/S)_h, \\mathcal{O}^h) =", "\\colim_F H^i(S, \\mathcal{O})", "$$", "Here on the left hand side by $\\mathcal{O}^h$ we mean", "the h sheafification of the structure sheaf." ], "refs": [], "proofs": [ { "contents": [ "This is just a reformulation of Lemma \\ref{lemma-h-sheaf-colim-F}.", "Recall that", "$\\mathcal{O}^h = \\mathcal{O}^{perf} = \\colim_F \\mathcal{O}$, see", "More on Flatness, Lemma \\ref{flat-lemma-char-p}.", "By Lemma \\ref{lemma-h-sheaf-colim-F} we see that", "$\\mathcal{O}^{perf}$ viewed as an object of $D((\\Sch/S)_{fppf})$", "is of the form $R\\epsilon_*K$ for some $K \\in D((\\Sch/S)_h)$.", "Then $K = \\epsilon^{-1}\\mathcal{O}^{perf}$ which is actually", "equal to $\\mathcal{O}^{perf}$ because $\\mathcal{O}^{perf}$ is an h sheaf. See", "Cohomology on Sites, Section \\ref{sites-cohomology-section-compare}.", "Hence $R\\epsilon_*\\mathcal{O}^{perf} = \\mathcal{O}^{perf}$", "(with apologies for the confusing notation).", "Thus the lemma now follows from Leray", "$$", "R\\Gamma_h(S, \\mathcal{O}^{perf}) =", "R\\Gamma_{fppf}(S, R\\epsilon_*\\mathcal{O}^{perf}) =", "R\\Gamma_{fppf}(S, \\mathcal{O}^{perf})", "$$", "and the fact that", "$$", "H^i_{fppf}(S, \\mathcal{O}^{perf}) =", "H^i_{fppf}(S, \\colim_F \\mathcal{O}) =", "\\colim_F H^i_{fppf}(S, \\mathcal{O})", "$$", "as $S$ is quasi-compact and quasi-separated", "(see proof of Lemma \\ref{lemma-h-sheaf-colim-F})." ], "refs": [ "etale-cohomology-lemma-h-sheaf-colim-F", "flat-lemma-char-p", "etale-cohomology-lemma-h-sheaf-colim-F", "etale-cohomology-lemma-h-sheaf-colim-F" ], "ref_ids": [ 6695, 6173, 6695, 6695 ] } ], "ref_ids": [] }, { "id": 6799, "type": "theorem", "label": "equiv-theorem-bondal-van-den-bergh", "categories": [ "equiv" ], "title": "equiv-theorem-bondal-van-den-bergh", "contents": [ "\\begin{reference}", "\\cite[Theorem A.1]{BvdB}", "\\end{reference}", "Let $X$ be a projective scheme over a field $k$.", "Let $F : D_{perf}(\\mathcal{O}_X)^{opp} \\to \\text{Vect}_k$", "be a $k$-linear cohomological functor such that", "$$", "\\sum\\nolimits_{n \\in \\mathbf{Z}} \\dim_k F(E[n]) < \\infty", "$$", "for all $E \\in D_{perf}(\\mathcal{O}_X)$. Then $F$ is isomorphic to a functor", "of the form $E \\mapsto \\Hom_X(E, K)$ for some", "$K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "The derived category $D_\\QCoh(\\mathcal{O}_X)$ has direct sums,", "is compactly generated, and $D_{perf}(\\mathcal{O}_X)$ is the full subcategory", "of compact objects, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-direct-sums},", "Theorem \\ref{perfect-theorem-bondal-van-den-Bergh}, and", "Proposition \\ref{perfect-proposition-compact-is-perfect}.", "By Lemma \\ref{lemma-van-den-bergh} we may assume", "$F(E) = \\Hom_X(E, K)$ for some $K \\in \\Ob(D_\\QCoh(\\mathcal{O}_X))$.", "Then it follows that $K$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$", "by Lemma \\ref{lemma-characterize-dbcoh-projective}." ], "refs": [ "perfect-lemma-quasi-coherence-direct-sums", "perfect-theorem-bondal-van-den-Bergh", "perfect-proposition-compact-is-perfect", "equiv-lemma-van-den-bergh", "equiv-lemma-characterize-dbcoh-projective" ], "ref_ids": [ 6937, 6935, 7111, 6810, 6808 ] } ], "ref_ids": [] }, { "id": 6800, "type": "theorem", "label": "equiv-theorem-fully-faithful", "categories": [ "equiv" ], "title": "equiv-theorem-fully-faithful", "contents": [ "\\begin{reference}", "\\cite[Theorem 2.2]{Orlov-K3}; this is shown in \\cite{Noah}", "without the assumption that $X$ be projective", "\\end{reference}", "Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$", "with $X$ projective over $k$. Any $k$-linear fully faithful exact ", "functor $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$", "is a Fourier-Mukai functor for some kernel in", "$D_{perf}(\\mathcal{O}_{X \\times Y})$." ], "refs": [], "proofs": [ { "contents": [ "Let $F'$ be the Fourier-Mukai functor which is a sibling of $F$", "as in Lemma \\ref{lemma-fully-faithful}.", "By Proposition \\ref{proposition-siblings-isomorphic} we have $F \\cong F'$", "provided we can show that $\\textit{Coh}(\\mathcal{O}_X)$ has enough", "negative objects. However, if $X = \\Spec(k)$ for example, then", "this isn't true. Thus we first decompose $X = \\coprod X_i$", "into its connected (and irreducible) components and we", "argue that it suffices to prove the result for each of the", "(fully faithful) composition functors", "$$", "F_i :", "D_{perf}(\\mathcal{O}_{X_i}) \\to", "D_{perf}(\\mathcal{O}_X) \\to", "D_{perf}(\\mathcal{O}_Y)", "$$", "Details omitted. Thus we may assume $X$ is irreducible.", "\\medskip\\noindent", "The case $\\dim(X) = 0$. Here $X$ is the spectrum of a finite (separable)", "extension $k'/k$ and hence $D_{perf}(\\mathcal{O}_X)$", "is equivalent to the category", "of graded $k'$-vector spaces such that $\\mathcal{O}_X$ corresponds to the", "trivial $1$-dimensional vector space in degree $0$.", "It is straightforward to see that any two", "siblings $F, F' : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$", "are isomorphic. Namely, we are given an isomorphism", "$F(\\mathcal{O}_X) \\cong F'(\\mathcal{O}_X)$", "compatible the action of the $k$-algebra", "$k' = \\text{End}_{D_{perf}(\\mathcal{O}_X)}(\\mathcal{O}_X)$", "which extends canonically to an isomorphism on any graded $k'$-vector space.", "\\medskip\\noindent", "The case $\\dim(X) > 0$. Here $X$ is a projective smooth", "variety of dimension $> 1$. Let $\\mathcal{F}$ be a coherent", "$\\mathcal{O}_X$-module. We have to show there exists a", "coherent module $\\mathcal{N}$ such that", "\\begin{enumerate}", "\\item there is a surjection $\\mathcal{N} \\to \\mathcal{F}$,", "\\item $\\Ext^q(\\mathcal{N}, \\mathcal{F}) = 0$ for $q > 0$,", "\\item $\\Hom(\\mathcal{F}, \\mathcal{N}) = 0$.", "\\end{enumerate}", "Choose an ample invertible $\\mathcal{O}_X$-module $\\mathcal{L}$.", "We claim that $\\mathcal{N} = (\\mathcal{L}^{\\otimes n})^{\\oplus r}$", "will work for $n \\ll 0$ and $r$ large enough.", "Condition (1) follows from", "Properties, Proposition \\ref{properties-proposition-characterize-ample}.", "Condition (2) follows from", "$\\Ext^q(\\mathcal{L}^{\\otimes n}, \\mathcal{F}) =", "H^q(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes -n})$ and", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-vanshing-gives-ample}.", "Finally, we have", "$$", "\\Hom(\\mathcal{F}, \\mathcal{L}^{\\otimes n}) =", "H^0(X, \\SheafHom(\\mathcal{F}, \\mathcal{L}^{\\otimes n})) =", "H^0(X, \\SheafHom(\\mathcal{F}, \\mathcal{O}_X) \\otimes \\mathcal{L}^{\\otimes n})", "$$", "Since the dual $\\SheafHom(\\mathcal{F}, \\mathcal{O}_X)$ is torsion free, this", "this vanishes for $n \\ll 0$ by Varieties, Lemma ", "\\ref{varieties-lemma-vanishin-h0-negative}. This finishes the proof." ], "refs": [ "equiv-lemma-fully-faithful", "equiv-proposition-siblings-isomorphic", "properties-proposition-characterize-ample", "coherent-lemma-vanshing-gives-ample", "varieties-lemma-vanishin-h0-negative" ], "ref_ids": [ 6856, 6872, 3067, 3346, 11133 ] } ], "ref_ids": [] }, { "id": 6801, "type": "theorem", "label": "equiv-theorem-countable", "categories": [ "equiv" ], "title": "equiv-theorem-countable", "contents": [ "\\begin{reference}", "Slight improvement of \\cite{AT}", "\\end{reference}", "Let $K$ be an algebraically closed field. Let $\\mathbf{X}$ be a smooth proper", "scheme over $K$. There are at most countably many isomorphism classes", "of smooth proper schemes $\\mathbf{Y}$ over $K$ which are derived", "equivalent to $\\mathbf{X}$." ], "refs": [], "proofs": [ { "contents": [ "Choose a countable set $I$ and for $i \\in I$ systems", "$(S_i/K, X_i \\to S_i, Y_i \\to S_i, M_i)$ satisfying properties", "(1), (2), (3), and (4) of Lemma \\ref{lemma-countable-equivs}.", "Pick $i \\in I$ and set $S = S_i$, $X = X_i$, $Y = Y_i$, and", "$M = M_i$. Clearly it suffice to show that", "the set of isomorphism classes of fibres $Y_s$ for $s \\in S(K)$", "such that $X_s \\cong \\mathbf{X}$ is countable.", "This we prove in the next paragraph.", "\\medskip\\noindent", "Let $S$ be a finite type scheme over $K$, let $X \\to S$ and $Y \\to S$", "be proper smooth morphisms, and let $M \\in D_{perf}(\\mathcal{O}_{X \\times_S Y})$", "be the Fourier-Mukai kernel of a relative equivalence from $X$", "to $Y$ over $S$. We will show", "the set of isomorphism classes of fibres $Y_s$ for $s \\in S(K)$", "such that $X_s \\cong \\mathbf{X}$ is countable.", "By Lemma \\ref{lemma-countable-isos} applied", "to the families $\\mathbf{X} \\times S \\to S$ and $X \\to S$", "there exists a countable set $I$ and for $i \\in I$ a pair", "$(S_i \\to S, h_i)$ with the following properties", "\\begin{enumerate}", "\\item $S_i \\to S$ is a morphism of finite type, set", "$X_i = X \\times_S S_i$,", "\\item $h_i : \\mathbf{X} \\times S_i \\to X_i$", "is an isomorphism over $S_i$, and", "\\item for any closed point $s \\in S(K)$ if $\\mathbf{X} \\cong X_s$", "over $K = \\kappa(s)$ then $s$ is in the image of $S_i \\to S$", "for some $i$.", "\\end{enumerate}", "Set $Y_i = Y \\times_S S_i$. Denote", "$M_i \\in D_{perf}(\\mathcal{O}_{X_i \\times_{S_i} Y_i})$", "the pullback of $M$. By Lemma \\ref{lemma-base-change-rek}", "$M_i$ is the Fourier-Mukai kernel of a relative equivalence from", "$X_i$ to $Y_i$ over $S_i$. Since $I$ is countable, by", "property (3) it suffices to prove that", "the set of isomorphism classes of fibres $Y_{i, s}$ for $s \\in S_i(K)$", "is countable.", "In fact, this number is finite by", "Lemma \\ref{lemma-no-deformations-better}", "and the proof is complete." ], "refs": [ "equiv-lemma-countable-equivs", "equiv-lemma-countable-isos", "equiv-lemma-base-change-rek", "equiv-lemma-no-deformations-better" ], "ref_ids": [ 6871, 6870, 6860, 6866 ] } ], "ref_ids": [] }, { "id": 6802, "type": "theorem", "label": "equiv-lemma-Serre-functor-exists", "categories": [ "equiv" ], "title": "equiv-lemma-Serre-functor-exists", "contents": [ "Let $k$ be a field. Let $\\mathcal{T}$ be a $k$-linear", "triangulated category such that $\\dim_k \\Hom_\\mathcal{T}(X, Y) < \\infty$", "for all $X, Y \\in \\Ob(\\mathcal{T})$. The following are equivalent", "\\begin{enumerate}", "\\item there exists a $k$-linear equivalence", "$S : \\mathcal{T} \\to \\mathcal{T}$ and $k$-linear isomorphisms", "$c_{X, Y} : \\Hom_\\mathcal{T}(X, Y) \\to \\Hom_\\mathcal{T}(Y, S(X))^\\vee$", "functorial in $X, Y \\in \\Ob(\\mathcal{T})$,", "\\item for every $X \\in \\Ob(\\mathcal{T})$", "the functor $Y \\mapsto \\Hom_\\mathcal{T}(X, Y)^\\vee$", "is representable and the functor $Y \\mapsto \\Hom_\\mathcal{T}(Y, X)^\\vee$", "is corepresentable.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Condition (1) implies (2) since given $(S, c)$ and $X \\in \\Ob(\\mathcal{T})$", "the object $S(X)$ represents the functor", "$Y \\mapsto \\Hom_\\mathcal{T}(X, Y)^\\vee$ and the object $S^{-1}(X)$ corepresents", "the functor $Y \\mapsto \\Hom_\\mathcal{T}(Y, X)^\\vee$.", "\\medskip\\noindent", "Assume (2). We will repeatedly use the Yoneda lemma, see", "Categories, Lemma \\ref{categories-lemma-yoneda}.", "For every $X$ denote $S(X)$ the object representing the", "functor $Y \\mapsto \\Hom_\\mathcal{T}(X, Y)^\\vee$. Given", "$\\varphi : X \\to X'$, we obtain a unique arrow $S(\\varphi) : S(X) \\to S(X')$", "determined by the corresponding transformation of functors", "$\\Hom_\\mathcal{T}(X, -)^\\vee \\to \\Hom_\\mathcal{T}(X', -)^\\vee$.", "Thus $S$ is a functor and we obtain the isomorphisms $c_{X, Y}$", "by construction. It remains to show that $S$ is an equivalence.", "For every $X$ denote $S'(X)$ the object corepresenting the", "functor $Y \\mapsto \\Hom_\\mathcal{T}(Y, X)^\\vee$. Arguing as", "above we find that $S'$ is a functor. We claim that $S'$", "is quasi-inverse to $S$. To see this observe that", "$$", "\\Hom_\\mathcal{T}(X, Y) = \\Hom_\\mathcal{T}(Y, S(X))^\\vee =", "\\Hom_\\mathcal{T}(S'(S(X)), Y)", "$$", "bifunctorially, i.e., we find $S' \\circ S \\cong \\text{id}_\\mathcal{T}$.", "Similarly, we have", "$$", "\\Hom_\\mathcal{T}(Y, X) = \\Hom_\\mathcal{T}(S'(X), Y)^\\vee =", "\\Hom_\\mathcal{T}(Y, S(S'(X)))", "$$", "and we find $S \\circ S' \\cong \\text{id}_\\mathcal{T}$." ], "refs": [ "categories-lemma-yoneda" ], "ref_ids": [ 12203 ] } ], "ref_ids": [] }, { "id": 6803, "type": "theorem", "label": "equiv-lemma-Serre-functor", "categories": [ "equiv" ], "title": "equiv-lemma-Serre-functor", "contents": [ "In the situation of Definition \\ref{definition-Serre-functor}.", "If a Serre functor exists, then it is unique up to unique isomorphism and", "it is an exact functor of triangulated categories." ], "refs": [ "equiv-definition-Serre-functor" ], "proofs": [ { "contents": [ "Given a Serre functor $S$ the object $S(X)$ represents", "the functor $Y \\mapsto \\Hom_\\mathcal{T}(X, Y)^\\vee$.", "Thus the object $S(X)$ together with the functorial identification", "$\\Hom_\\mathcal{T}(X, Y)^\\vee = \\Hom_\\mathcal{T}(Y, S(X))$", "is determined up to unique isomorphism by the Yoneda lemma", "(Categories, Lemma \\ref{categories-lemma-yoneda}).", "Moreover, for $\\varphi : X \\to X'$, the arrow $S(\\varphi) : S(X) \\to S(X')$", "is uniquely determined by the corresponding transformation of functors", "$\\Hom_\\mathcal{T}(X, -)^\\vee \\to \\Hom_\\mathcal{T}(X', -)^\\vee$.", "\\medskip\\noindent", "For objects $X, Y$ of $\\mathcal{T}$ we have", "\\begin{align*}", "\\Hom(Y, S(X)[1])^\\vee", "& =", "\\Hom(Y[-1], S(X))^\\vee \\\\", "& =", "\\Hom(X, Y[-1]) \\\\", "& =", "\\Hom(X[1], Y) \\\\", "& =", "\\Hom(Y, S(X[1]))^\\vee", "\\end{align*}", "By the Yoneda lemma we conclude that there is a unique isomorphism", "$S(X[1]) \\to S(X)[1]$ inducing the isomorphism from top left to bottom right.", "Since each of the isomorphisms above is functorial in both $X$ and $Y$", "we find that this defines an isomorphism of functors", "$S \\circ [1] \\to [1] \\circ S$.", "\\medskip\\noindent", "Let $(A, B, C, f, g, h)$ be a distinguished triangle in $\\mathcal{T}$.", "We have to show that the triangle $(S(A), S(B), S(C), S(f), S(g), S(h))$", "is distinguished. Here we use the canonical isomorphism $S(A[1]) \\to S(A)[1]$", "constructed above to identify the target $S(A[1])$ of $S(h)$ with $S(A)[1]$.", "We first observe that for any $X$ in $\\mathcal{T}$", "the triangle $(S(A), S(B), S(C), S(f), S(g), S(h))$ induces", "a long exact sequence", "$$", "\\ldots \\to", "\\Hom(X, S(A)) \\to", "\\Hom(X, S(B)) \\to", "\\Hom(X, S(C)) \\to", "\\Hom(X, S(A)[1]) \\to \\ldots", "$$", "of finite dimensional $k$-vector spaces. Namely, this sequence is", "$k$-linear dual of the sequence", "$$", "\\ldots \\leftarrow", "\\Hom(A, X) \\leftarrow", "\\Hom(B, X) \\leftarrow", "\\Hom(C, X) \\leftarrow", "\\Hom(A[1], X) \\leftarrow", "\\ldots", "$$", "which is exact by Derived Categories, Lemma", "\\ref{derived-lemma-representable-homological}.", "Next, we choose a distinguished triangle $(S(A), E, S(C), i, p, S(h))$", "which is possible by axioms TR1 and TR2. We want to construct the dotted", "arrow making following diagram commute", "$$", "\\xymatrix{", "S(C)[-1] \\ar[r]_-{S(h[-1])} &", "S(A) \\ar[r]_{S(f)} &", "S(B) \\ar[r]_{S(g)} &", "S(C) \\ar[r]_{S(h)} &", "S(A)[1] \\\\", "S(C)[-1] \\ar[r]^-{S(h[-1])} \\ar@{=}[u] &", "S(A) \\ar[r]^i \\ar@{=}[u] &", "E \\ar[r]^p \\ar@{..>}[u]^\\varphi &", "S(C) \\ar[r]^{S(h)} \\ar@{=}[u] &", "S(A)[1] \\ar@{=}[u]", "}", "$$", "Namely, if we have $\\varphi$, then we claim for any $X$ the resulting", "map $\\Hom(X, E) \\to \\Hom(X, S(B))$ will be an isomorphism of $k$-vector", "spaces. Namely, we will obtain a commutative diagram", "$$", "\\xymatrix{", "\\Hom(X, S(C)[-1]) \\ar[r] &", "\\Hom(X, S(A)) \\ar[r] &", "\\Hom(X, S(B)) \\ar[r] &", "\\Hom(X, S(C)) \\ar[r] &", "\\Hom(X, S(A)[1]) \\\\", "\\Hom(X, S(C)[-1]) \\ar[r] \\ar@{=}[u] &", "\\Hom(X, S(A)) \\ar[r] \\ar@{=}[u] &", "\\Hom(X, E) \\ar[r] \\ar[u]^\\varphi &", "\\Hom(X, S(C)) \\ar[r] \\ar@{=}[u] &", "\\Hom(X, S(A)[1]) \\ar@{=}[u]", "}", "$$", "with exact rows (see above) and we can apply the 5 lemma", "(Homology, Lemma \\ref{homology-lemma-five-lemma}) to see", "that the middle arrow is an isomorphism. By the Yoneda lemma", "we conclude that $\\varphi$ is an isomorphism.", "To find $\\varphi$ consider the following diagram", "$$", "\\xymatrix{", "\\Hom(E, S(C)) \\ar[r] &", "\\Hom(S(A), S(C)) \\\\", "\\Hom(E, S(B)) \\ar[u] \\ar[r] &", "\\Hom(S(A), S(B)) \\ar[u]", "}", "$$", "The elements $p$ and $S(f)$ in positions $(0, 1)$ and", "$(1, 0)$ define a cohomology class $\\xi$ in the total complex", "of this double complex. The existence of $\\varphi$ is", "equivalent to whether $\\xi$ is zero. If we take $k$-linear duals", "of this and we use the defining property of $S$ we obtain", "$$", "\\xymatrix{", "\\Hom(C, E) \\ar[d] &", "\\Hom(C, S(A)) \\ar[l] \\ar[d] \\\\", "\\Hom(B, E) &", "\\Hom(B, S(A)) \\ar[l]", "}", "$$", "Since both $A \\to B \\to C$ and $S(A) \\to E \\to S(C)$ are distinguished", "triangles, we know by TR3 that given elements $\\alpha \\in \\Hom(C, E)$", "and $\\beta \\in \\Hom(B, S(A))$ mapping to the same element in", "$\\Hom(B, E)$, there exists an element in $\\Hom(C, S(A))$ mapping", "to both $\\alpha$ and $\\beta$. In other words, the cohomology of", "the total complex associated to this double complex is zero in degree", "$1$, i.e., the degree corresponding to $\\Hom(C, E) \\oplus \\Hom(B, S(A))$.", "Taking duals the same must be true for the previous one which concludes", "the proof." ], "refs": [ "categories-lemma-yoneda", "derived-lemma-representable-homological", "homology-lemma-five-lemma" ], "ref_ids": [ 12203, 1758, 12030 ] } ], "ref_ids": [ 6874 ] }, { "id": 6804, "type": "theorem", "label": "equiv-lemma-Serre-functor-Gorenstein-proper", "categories": [ "equiv" ], "title": "equiv-lemma-Serre-functor-Gorenstein-proper", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is Gorenstein.", "Consider the complex $\\omega_X^\\bullet$ of", "Duality for Schemes, Lemmas \\ref{duality-lemma-duality-proper-over-field}.", "Then the functor", "$$", "S : D_{perf}(\\mathcal{O}_X) \\longrightarrow D_{perf}(\\mathcal{O}_X),\\quad", "K \\longmapsto S(K) = \\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K", "$$", "is a Serre functor." ], "refs": [ "duality-lemma-duality-proper-over-field" ], "proofs": [ { "contents": [ "The statement make sense because $\\dim \\Hom_X(K, L) < \\infty$", "for $K, L \\in D_{perf}(\\mathcal{O}_X)$ by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-ext-finite}.", "Since $X$ is Gorenstein the dualizing complex $\\omega_X^\\bullet$", "is an invertible object of $D(\\mathcal{O}_X)$, see", "Duality for Schemes, Lemma \\ref{duality-lemma-gorenstein}.", "In particular, locally on $X$ the complex $\\omega_X^\\bullet$", "has one nonzero cohomology sheaf which is an invertible module, see", "Cohomology, Lemma \\ref{cohomology-lemma-invertible-derived}.", "Thus $S(K)$ lies in $D_{perf}(\\mathcal{O}_X)$.", "On the other hand, the invertibility of $\\omega_X^\\bullet$", "clearly implies that $S$ is a self-equivalence of $D_{perf}(\\mathcal{O}_X)$.", "Finally, we have to find an isomorphism", "$$", "c_{K, L} : \\Hom_X(K, L) \\longrightarrow", "\\Hom_X(L, \\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K)^\\vee", "$$", "bifunctorially in $K, L$. To do this we use the canonical isomorphisms", "$$", "\\Hom_X(K, L) = H^0(X, L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K^\\vee)", "$$", "and", "$$", "\\Hom_X(L, \\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K) =", "H^0(X, ", "\\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L} L^\\vee)", "$$", "given in Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "Since $(L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K^\\vee)^\\vee =", "(K^\\vee)^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L^\\vee$", "and since there is a canonical isomorphism $K \\to (K^\\vee)^\\vee$", "we find these $k$-vector spaces are canonically dual by", "Duality for Schemes, Lemma", "\\ref{duality-lemma-duality-proper-over-field-perfect}.", "This produces the isomorphisms $c_{K, L}$.", "We omit the proof that these isomorphisms are functorial." ], "refs": [ "perfect-lemma-ext-finite", "duality-lemma-gorenstein", "cohomology-lemma-invertible-derived", "cohomology-lemma-dual-perfect-complex", "duality-lemma-duality-proper-over-field-perfect" ], "ref_ids": [ 6988, 13591, 2239, 2233, 13607 ] } ], "ref_ids": [ 13606 ] }, { "id": 6805, "type": "theorem", "label": "equiv-lemma-perfect-for-R", "categories": [ "equiv" ], "title": "equiv-lemma-perfect-for-R", "contents": [ "With $k$, $n$, and $R$ as above, for an object $K$ of $D(R)$", "the following are equivalent", "\\begin{enumerate}", "\\item $\\sum_{i \\in \\mathbf{Z}} \\dim_k H^i(K) < \\infty$, and", "\\item $K$ is a compact object.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $K$ is a compact object, then $K$ can be represented by a complex", "$M^\\bullet$ which is finite projective as a graded $R$-module, see", "Differential Graded Algebra, Lemma \\ref{dga-lemma-compact}.", "Since $\\dim_k R < \\infty$ we conclude $\\sum \\dim_k M^i < \\infty$", "and a fortiori $\\sum \\dim_k H^i(M^\\bullet) < \\infty$.", "(One can also easily deduce this implication from the easier", "Differential Graded Algebra, Proposition \\ref{dga-proposition-compact}.)", "\\medskip\\noindent", "Assume $K$ satisfies (1). Consider the distinguished triangle", "of trunctions $\\tau_{\\leq m}K \\to K \\to \\tau_{\\geq m + 1}K$, see", "Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}.", "It is clear that both $\\tau_{\\leq m}K$ and $\\tau_{\\geq m + 1} K$", "satisfy (1). If we can show both are compact, then so is $K$, see", "Derived Categories, Lemma \\ref{derived-lemma-compact-objects-subcategory}.", "Hence, arguing on the number of nonzero cohomology modules of $K$", "we may assume $H^i(K)$ is nonzero only for one $i$.", "Shifting, we may assume $K$ is given by the complex", "consisting of a single finite dimensional $R$-module $M$ sitting", "in degree $0$.", "\\medskip\\noindent", "Since $\\dim_k(M) < \\infty$ we see that $M$ is Artinian as an $R$-module.", "Thus it suffices to show that every simple $R$-module represents a", "compact object of $D(R)$. Observe that", "$$", "I =", "\\left(", "\\begin{matrix}", "0 & S_1 & S_2 & \\ldots & \\ldots \\\\", "0 & 0 & S_1 & \\ldots & \\ldots\\\\", "0 & 0 & 0 & \\ldots & \\ldots \\\\", "\\ldots & \\ldots & \\ldots & \\ldots & \\ldots \\\\", "0 & \\ldots & \\ldots & \\ldots & 0", "\\end{matrix}", "\\right)", "$$", "is a nilpotent two sided ideal of $R$ and that $R/I$", "is a commutative $k$-algebra isomorphic to a product of $n + 1$ copies of", "$k$ (placed along the diagonal in the matrix, i.e., $R/I$ can be lifted", "to a $k$-subalgebra of $R$). It follows that $R$ has exactly $n + 1$", "isomorphism classes of simple modules $M_0, \\ldots, M_n$ (sitting along", "the diagonal). Consider the right $R$-module $P_i$ of row vectors", "$$", "P_i =", "\\left(", "\\begin{matrix}", "0 &", "\\ldots &", "0 &", "S_0 &", "\\ldots &", "S_{i - 1} &", "S_i", "\\end{matrix}", "\\right)", "$$", "with obvious multiplication $P_i \\times R \\to P_i$. Then we see that", "$R \\cong P_0 \\oplus \\ldots \\oplus P_n$ as a right $R$-module. Since clearly", "$R$ is a compact object of $D(R)$, we conclude each $P_i$ is a compact", "object of $D(R)$. (We of course also conclude each $P_i$ is projective", "as an $R$-module, but this isn't what we have to show in this proof.)", "Clearly, $P_0 = M_0$ is the first of our simple $R$-modules.", "For $P_1$ we have a short exact sequence", "$$", "0 \\to P_0^{\\oplus n + 1} \\to P_1 \\to M_1 \\to 0", "$$", "which proves that $M_1$ fits into a distinguished triangle whose", "other members are compact objects and hence $M_1$ is a compact", "object of $D(R)$. More generally, there exists a short exact sequence", "$$", "0 \\to C_i \\to P_i \\to M_i \\to 0", "$$", "where $C_i$ is a finite dimensional $R$-module whose simple constituents", "are isomorphic to $M_j$ for $j < i$. By induction, we first conclude that", "$C_i$ determines a compact object of $D(R)$ whereupon we conclude that $M_i$", "does too as desired." ], "refs": [ "dga-lemma-compact", "dga-proposition-compact", "derived-remark-truncation-distinguished-triangle", "derived-lemma-compact-objects-subcategory" ], "ref_ids": [ 13122, 13132, 2016, 1940 ] } ], "ref_ids": [] }, { "id": 6806, "type": "theorem", "label": "equiv-lemma-coherent-on-projective-space", "categories": [ "equiv" ], "title": "equiv-lemma-coherent-on-projective-space", "contents": [ "Let $k$ be a field. Let $n \\geq 0$. Let", "$K \\in D_\\QCoh(\\mathcal{O}_{\\mathbf{P}^n_k})$.", "The following are equivalent", "\\begin{enumerate}", "\\item $K$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_{\\mathbf{P}^n_k})$,", "\\item $\\sum_{i \\in \\mathbf{Z}}", "\\dim_k H^i(\\mathbf{P}^n_k, E \\otimes^\\mathbf{L} K) < \\infty$", "for each perfect object $E$ of", "$D(\\mathcal{O}_{\\mathbf{P}^n_k})$,", "\\item $\\sum_{i \\in \\mathbf{Z}}", "\\dim_k \\Ext^i_{\\mathbf{P}^n_k}(E, K) < \\infty$", "for each perfect object $E$ of $D(\\mathcal{O}_{\\mathbf{P}^n_k})$,", "\\item $\\sum_{i \\in \\mathbf{Z}} \\dim_k H^i(\\mathbf{P}^n_k,", "K \\otimes^\\mathbf{L} \\mathcal{O}_{\\mathbf{P}^n_k}(d)) < \\infty$", "for $d = 0, 1, \\ldots, n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (2) and (3) are equivalent by", "Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "If (1) is true, then for $E$ perfect the derived tensor product", "$E \\otimes^\\mathbf{L} K$ is in", "$D^b_{\\textit{Coh}}(\\mathcal{O}_{\\mathbf{P}^n_k})$", "and we see that (2) holds by ", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-direct-image-coherent}.", "It is clear that (2) implies (4) as $\\mathcal{O}_{\\mathbf{P}^n_k}(d)$", "can be viewed", "as a perfect object of the derived category of $\\mathbf{P}^n_k$.", "Thus it suffices to prove that (4) implies (1).", "\\medskip\\noindent", "Assume (4). Let $R$ be as in Lemma \\ref{lemma-perfect-for-R}.", "Let $P = \\bigoplus_{d = 0, \\ldots, n} \\mathcal{O}_{\\mathbf{P}^n_k}(-d)$.", "Recall that $R = \\text{End}_{\\mathbf{P}^n_k}(P)$ whereas all other", "self-Exts of $P$ are zero and that $P$ determines an equivalence", "$- \\otimes^\\mathbf{L} P : D(R) \\to D_\\QCoh(\\mathcal{O}_{\\mathbf{P}^n_k})$", "by Derived Categories of Schemes, Lemma \\ref{perfect-lemma-Pn-module-category}.", "Say $K$ corresponds to $L$ in $D(R)$. Then", "\\begin{align*}", "H^i(L)", "& =", "\\Ext^i_{D(R)}(R, L) \\\\", "& =", "\\Ext^i_{\\mathbf{P}^n_k}(P, K) \\\\", "& =", "H^i(\\mathbf{P}^n_k, K \\otimes P^\\vee) \\\\", "& =", "\\bigoplus\\nolimits_{d = 0, \\ldots, n}", "H^i(\\mathbf{P}^n_k, K \\otimes \\mathcal{O}(d))", "\\end{align*}", "by Differential Graded Algebra, Lemma", "\\ref{dga-lemma-upgrade-tensor-with-complex-derived}", "(and the fact that $- \\otimes^\\mathbf{L} P$ is an equivalence)", "and Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "Thus our assumption (4) implies that $L$ satisfies condition (2) of", "Lemma \\ref{lemma-perfect-for-R} and hence is a compact object of $D(R)$.", "Therefore $K$ is a compact object of", "$D_\\QCoh(\\mathcal{O}_{\\mathbf{P}^n_k})$.", "Thus $K$ is perfect by", "Derived Categories of Schemes, Proposition", "\\ref{perfect-proposition-compact-is-perfect}.", "Since $D_{perf}(\\mathcal{O}_{\\mathbf{P}^n_k}) =", "D^b_{\\textit{Coh}}(\\mathcal{O}_{\\mathbf{P}^n_k})$", "by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-perfect-on-regular}", "we conclude (1) holds." ], "refs": [ "cohomology-lemma-dual-perfect-complex", "perfect-lemma-direct-image-coherent", "equiv-lemma-perfect-for-R", "perfect-lemma-Pn-module-category", "dga-lemma-upgrade-tensor-with-complex-derived", "cohomology-lemma-dual-perfect-complex", "equiv-lemma-perfect-for-R", "perfect-proposition-compact-is-perfect", "perfect-lemma-perfect-on-regular" ], "ref_ids": [ 2233, 6984, 6805, 7021, 13117, 2233, 6805, 7111, 6989 ] } ], "ref_ids": [] }, { "id": 6807, "type": "theorem", "label": "equiv-lemma-finiteness", "categories": [ "equiv" ], "title": "equiv-lemma-finiteness", "contents": [ "Let $X$ be a scheme proper over a field $k$. Let", "$K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ and let $E$ in $D(\\mathcal{O}_X)$", "be perfect. Then", "$\\sum_{i \\in \\mathbf{Z}} \\dim_k \\Ext^i_X(E, K) < \\infty$." ], "refs": [], "proofs": [ { "contents": [ "This follows for example by combining", "Derived Categories of Schemes, Lemmas \\ref{perfect-lemma-ext-finite} and", "\\ref{perfect-lemma-ext-from-perfect-into-bounded-QCoh}.", "Alternative proof: combine", "Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-perfect-on-noetherian} and", "\\ref{perfect-lemma-direct-image-coherent}." ], "refs": [ "perfect-lemma-ext-finite", "perfect-lemma-ext-from-perfect-into-bounded-QCoh", "perfect-lemma-perfect-on-noetherian", "perfect-lemma-direct-image-coherent" ], "ref_ids": [ 6988, 7018, 6987, 6984 ] } ], "ref_ids": [] }, { "id": 6808, "type": "theorem", "label": "equiv-lemma-characterize-dbcoh-projective", "categories": [ "equiv" ], "title": "equiv-lemma-characterize-dbcoh-projective", "contents": [ "\\begin{reference}", "\\cite[Lemma 7.46]{Rouquier-dimensions} and implicit in", "\\cite[Theorem A.1]{BvdB}", "\\end{reference}", "Let $X$ be a projective scheme over a field $k$. Let", "$K \\in \\Ob(D_\\QCoh(\\mathcal{O}_X))$. The following are equivalent", "\\begin{enumerate}", "\\item $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$, and", "\\item $\\sum_{i \\in \\mathbf{Z}} \\dim_k \\Ext^i_X(E, K) < \\infty$", "for all perfect $E$ in $D(\\mathcal{O}_X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) follows from", "Lemma \\ref{lemma-finiteness}.", "\\medskip\\noindent", "Assume (2).", "Choose a closed immersion $i : X \\to \\mathbf{P}^n_k$. It suffices to show", "that $Ri_*K$ is in $D^b_{\\textit{Coh}}(\\mathbf{P}^n_k)$ since a quasi-coherent", "module $\\mathcal{F}$ on $X$ is coherent, resp.\\ zero if and only if", "$i_*\\mathcal{F}$ is coherent, resp.\\ zero. For a perfect object $E$", "of $D(\\mathcal{O}_{\\mathbf{P}^n_k})$, $Li^*E$ is a perfect object of", "$D(\\mathcal{O}_X)$ and", "$$", "\\Ext^q_{\\mathbf{P}^n_k}(E, Ri_*K) = \\Ext^q_X(Li^*E, K)", "$$", "Hence by our assumption we see that", "$\\sum_{q \\in \\mathbf{Z}} \\dim_k \\Ext^q_{\\mathbf{P}^n_k}(E, Ri_*K) < \\infty$.", "We conclude by Lemma \\ref{lemma-coherent-on-projective-space}." ], "refs": [ "equiv-lemma-finiteness", "equiv-lemma-coherent-on-projective-space" ], "ref_ids": [ 6807, 6806 ] } ], "ref_ids": [] }, { "id": 6809, "type": "theorem", "label": "equiv-lemma-maps-from-compact-filtered", "categories": [ "equiv" ], "title": "equiv-lemma-maps-from-compact-filtered", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let", "$\\mathcal{D}' \\subset \\mathcal{D}$ be a full triangulated subcategory. Let", "$X \\in \\Ob(\\mathcal{D})$. The category of arrows $E \\to X$ with", "$E \\in \\Ob(\\mathcal{D}')$ is filtered." ], "refs": [], "proofs": [ { "contents": [ "We check the conditions of", "Categories, Definition \\ref{categories-definition-directed}.", "The category is nonempty because it contains $0 \\to X$.", "If $E_i \\to X$, $i = 1, 2$ are objects, then $E_1 \\oplus E_2 \\to X$", "is an object and there are morphisms $(E_i \\to X) \\to (E_1 \\oplus E_2 \\to X)$.", "Finally, suppose that $a, b : (E \\to X) \\to (E' \\to X)$ are morphisms.", "Choose a distinguished triangle $E \\xrightarrow{a - b} E' \\to E''$", "in $\\mathcal{D}'$. By Axiom TR3 we obtain a morphism of triangles", "$$", "\\xymatrix{", "E \\ar[r]_{a - b} \\ar[d] &", "E' \\ar[d] \\ar[r] & E'' \\ar[d] \\\\", "0 \\ar[r] &", "X \\ar[r] &", "X", "}", "$$", "and we find that the resulting arrow $(E' \\to X) \\to (E'' \\to X)$", "equalizes $a$ and $b$." ], "refs": [ "categories-definition-directed" ], "ref_ids": [ 12363 ] } ], "ref_ids": [] }, { "id": 6810, "type": "theorem", "label": "equiv-lemma-van-den-bergh", "categories": [ "equiv" ], "title": "equiv-lemma-van-den-bergh", "contents": [ "\\begin{reference}", "\\cite[Lemma 2.14]{CKN}", "\\end{reference}", "Let $k$ be a field. Let $\\mathcal{D}$ be a $k$-linear triangulated category", "which has direct sums and is compactly generated.", "Denote $\\mathcal{D}_c$ the full", "subcategory of compact objects. Let $H : \\mathcal{D}_c^{opp} \\to \\text{Vect}_k$", "be a $k$-linear cohomological functor such that", "$\\dim_k H(X) < \\infty$ for all $X \\in \\Ob(\\mathcal{D}_c)$.", "Then $H$ is isomorphic to the functor $X \\mapsto \\Hom(X, Y)$", "for some $Y \\in \\Ob(\\mathcal{D})$." ], "refs": [], "proofs": [ { "contents": [ "We will use Derived Categories, Lemma", "\\ref{derived-lemma-compact-objects-subcategory} without further mention.", "Denote $G : \\mathcal{D}_c \\to \\text{Vect}_k$ the $k$-linear homological", "functor which sends $X$ to $H(X)^\\vee$. For any object $Y$ of $\\mathcal{D}$", "we set", "$$", "G'(Y) = \\colim_{X \\to Y, X \\in \\Ob(\\mathcal{D}_c)} G(X)", "$$", "The colimit is filtered by Lemma \\ref{lemma-maps-from-compact-filtered}.", "We claim that $G'$ is a $k$-linear homological functor,", "the restriction of $G'$ to $\\mathcal{D}_c$ is $G$, and $G'$", "sends direct sums to direct sums.", "\\medskip\\noindent", "Namely, suppose that $Y_1 \\to Y_2 \\to Y_3$ is a distinguished triangle.", "Let $\\xi \\in G'(Y_2)$ map to zero in $G'(Y_3)$. Since the colimit is", "filtered $\\xi$ is represented by some $X \\to Y_2$ with", "$X \\in \\Ob(\\mathcal{D}_c)$ and $g \\in G(X)$.", "The fact that $\\xi$ maps to zero in $G'(Y_3)$ means the composition", "$X \\to Y_2 \\to Y_3$ factors as $X \\to X' \\to Y_3$ with $X' \\in \\mathcal{D}_c$", "and $g$ mapping to zero in $G(X')$. Choose a distinguished", "triangle $X'' \\to X \\to X'$. Then $X'' \\in \\Ob(\\mathcal{D}_c)$.", "Since $G$ is homological we find that $g$ is the image of some", "$g'' \\in G'(X'')$. By Axiom TR3 the maps $X \\to Y_2$ and $X' \\to Y_3$ fit into", "a morphism of distinguished triangles", "$(X'' \\to X \\to X') \\to (Y_1 \\to Y_2 \\to Y_3)$", "and we find that indeed $\\xi$ is the image of the", "element of $G'(Y_1)$ represented by $X'' \\to Y_1$ and $g'' \\in G(X'')$.", "\\medskip\\noindent", "If $Y \\in \\Ob(\\mathcal{D}_c)$, then $\\text{id} : Y \\to Y$ is the final", "object in the category of arrows $X \\to Y$ with $X \\in \\Ob(\\mathcal{D}_c)$.", "Hence we see that $G'(Y) = G(Y)$ in this case and the", "statement on restriction holds. Let $Y = \\bigoplus_{i \\in I} Y_i$", "be a direct sum. Let $a : X \\to Y$ with $X \\in \\Ob(\\mathcal{D}_c)$", "and $g \\in G(X)$ represent an element $\\xi$ of $G'(Y)$.", "The morphism $a : X \\to Y$ can be uniquely written as a sum of morphisms", "$a_i : X \\to Y_i$ almost all zero as $X$ is a compact object of $\\mathcal{D}$.", "Let $I' = \\{i \\in I \\mid a_i \\not = 0\\}$. Then we can factor", "$a$ as the composition", "$$", "X \\xrightarrow{(1, \\ldots, 1)}", "\\bigoplus\\nolimits_{i \\in I'} X", "\\xrightarrow{\\bigoplus_{i \\in I'} a_i}", "\\bigoplus\\nolimits_{i \\in I} Y_i = Y", "$$", "We conclude that $\\xi = \\sum_{i \\in I'} \\xi_i$", "is the sum of the images of the elements", "$\\xi_i \\in G'(Y_i)$ corresponding to $a_i : X \\to Y_i$", "and $g \\in G(X)$. Hence $\\bigoplus G'(Y_i) \\to G'(Y)$", "is surjective. We omit the (trivial) verification that it is injective.", "\\medskip\\noindent", "It follows that the functor $Y \\mapsto G'(Y)^\\vee$ is cohomological", "and sends direct sums to direct products. Hence by Brown representability,", "see Derived Categories, Proposition \\ref{derived-proposition-brown}", "we conclude that there exists a $Y \\in \\Ob(\\mathcal{D})$", "and an isomorphism", "$G'(Z)^\\vee = \\Hom(Z, Y)$ functorially in $Z$.", "For $X \\in \\Ob(\\mathcal{D}_c)$ we have", "$G'(X)^\\vee = G(X)^\\vee = (H(X)^\\vee)^\\vee = H(X)$", "because $\\dim_k H(X) < \\infty$ and the proof is complete." ], "refs": [ "derived-lemma-compact-objects-subcategory", "equiv-lemma-maps-from-compact-filtered", "derived-proposition-brown" ], "ref_ids": [ 1940, 6809, 1966 ] } ], "ref_ids": [] }, { "id": 6811, "type": "theorem", "label": "equiv-lemma-trace-map", "categories": [ "equiv" ], "title": "equiv-lemma-trace-map", "contents": [ "\\begin{reference}", "The proof given here follows the argument given in", "\\cite[Remark 3.4]{MS}", "\\end{reference}", "Let $f : X' \\to X$ be a proper birational morphism of integral Noetherian", "schemes with $X$ regular. The map $\\mathcal{O}_X \\to Rf_*\\mathcal{O}_{X'}$", "canonically splits in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Set $E = Rf_*\\mathcal{O}_{X'}$ in $D(\\mathcal{O}_X)$.", "Observe that $E$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-direct-image-coherent}.", "By ", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-perfect-on-regular}", "we find that $E$ is a perfect object of $D(\\mathcal{O}_X)$.", "Since $\\mathcal{O}_{X'}$ is a sheaf of algebras, we have the", "relative cup product $\\mu : E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E \\to E$", "by Cohomology, Remark \\ref{cohomology-remark-cup-product}.", "Let $\\sigma : E \\otimes E^\\vee \\to E^\\vee \\otimes E$ be the commutativity", "constraint on the symmetric monoidal category $D(\\mathcal{O}_X)$", "(Cohomology, Lemma \\ref{cohomology-lemma-symmetric-monoidal-derived}).", "Denote $\\eta : \\mathcal{O}_X \\to E \\otimes E^\\vee$ and", "$\\epsilon : E^\\vee \\otimes E \\to \\mathcal{O}_X$ the maps", "constructed in Cohomology, Example \\ref{cohomology-example-dual-derived}.", "Then we can consider the map", "$$", "E \\xrightarrow{\\eta \\otimes 1} E \\otimes E^\\vee \\otimes E", "\\xrightarrow{\\sigma \\otimes 1} E^\\vee \\otimes E \\otimes E", "\\xrightarrow{1 \\otimes \\mu} E^\\vee \\otimes E", "\\xrightarrow{\\epsilon} \\mathcal{O}_X", "$$", "We claim that this map is a one sided inverse to the map in the", "statement of the lemma. To see this it suffices to show that", "the composition $\\mathcal{O}_X \\to \\mathcal{O}_X$ is the identity", "map. This we may do in the generic point of $X$ (or on an open", "subscheme of $X$ over which $f$ is an isomorphism). In this", "case $E = \\mathcal{O}_X$ and $\\mu$ is the usual multiplication map", "and the result is clear." ], "refs": [ "perfect-lemma-direct-image-coherent", "perfect-lemma-perfect-on-regular", "cohomology-remark-cup-product", "cohomology-lemma-symmetric-monoidal-derived" ], "ref_ids": [ 6984, 6989, 2272, 2234 ] } ], "ref_ids": [] }, { "id": 6812, "type": "theorem", "label": "equiv-lemma-characterize-dbcoh-proper-regular", "categories": [ "equiv" ], "title": "equiv-lemma-characterize-dbcoh-proper-regular", "contents": [ "Let $X$ be a proper scheme over a field $k$ which is regular. Let", "$K \\in \\Ob(D_\\QCoh(\\mathcal{O}_X))$. The following are equivalent", "\\begin{enumerate}", "\\item $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X) = D_{perf}(\\mathcal{O}_X)$, and", "\\item $\\sum_{i \\in \\mathbf{Z}} \\dim_k \\Ext^i_X(E, K) < \\infty$", "for all perfect $E$ in $D(\\mathcal{O}_X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equality in (1) holds by Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-perfect-on-regular}.", "The implication (1) $\\Rightarrow$ (2) follows from", "Lemma \\ref{lemma-finiteness}.", "The implication (2) $\\Rightarrow$ (1) follows from", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-characterize-relatively-perfect}." ], "refs": [ "perfect-lemma-perfect-on-regular", "equiv-lemma-finiteness", "more-morphisms-lemma-characterize-relatively-perfect" ], "ref_ids": [ 6989, 6807, 14062 ] } ], "ref_ids": [] }, { "id": 6813, "type": "theorem", "label": "equiv-lemma-bondal-van-den-bergh", "categories": [ "equiv" ], "title": "equiv-lemma-bondal-van-den-bergh", "contents": [ "Let $X$ be a proper scheme over a field $k$ which is regular.", "\\begin{enumerate}", "\\item Let $F : D_{perf}(\\mathcal{O}_X)^{opp} \\to \\text{Vect}_k$", "be a $k$-linear cohomological functor such that", "$$", "\\sum\\nolimits_{n \\in \\mathbf{Z}} \\dim_k F(E[n]) < \\infty", "$$", "for all $E \\in D_{perf}(\\mathcal{O}_X)$. Then $F$ is isomorphic to a functor", "of the form $E \\mapsto \\Hom_X(E, K)$ for some $K \\in D_{perf}(\\mathcal{O}_X)$.", "\\item Let $G : D_{perf}(\\mathcal{O}_X) \\to \\text{Vect}_k$", "be a $k$-linear homological functor such that", "$$", "\\sum\\nolimits_{n \\in \\mathbf{Z}} \\dim_k G(E[n]) < \\infty", "$$", "for all $E \\in D_{perf}(\\mathcal{O}_X)$. Then $G$ is isomorphic to a functor", "of the form $E \\mapsto \\Hom_X(K, E)$ for some $K \\in D_{perf}(\\mathcal{O}_X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). The derived category $D_\\QCoh(\\mathcal{O}_X)$ has direct sums,", "is compactly generated, and $D_{perf}(\\mathcal{O}_X)$ is the full subcategory", "of compact objects, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-direct-sums},", "Theorem \\ref{perfect-theorem-bondal-van-den-Bergh}, and", "Proposition \\ref{perfect-proposition-compact-is-perfect}.", "By Lemma \\ref{lemma-van-den-bergh} we may assume", "$F(E) = \\Hom_X(E, K)$ for some $K \\in \\Ob(D_\\QCoh(\\mathcal{O}_X))$.", "Then it follows that $K$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$", "by Lemma \\ref{lemma-characterize-dbcoh-proper-regular}.", "\\medskip\\noindent", "Proof of (2). Consider the contravariant functor $E \\mapsto E^\\vee$", "on $D_{perf}(\\mathcal{O}_X)$, see", "Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "This functor is an exact anti-self-equivalence of $D_{perf}(\\mathcal{O}_X)$.", "Hence we may apply part (1) to the functor $F(E) = G(E^\\vee)$ to find", "$K \\in D_{perf}(\\mathcal{O}_X)$ such that $G(E^\\vee) = \\Hom_X(E, K)$.", "It follows that $G(E) = \\Hom_X(E^\\vee, K) = \\Hom_X(K^\\vee, E)$", "and we conclude that taking $K^\\vee$ works." ], "refs": [ "perfect-lemma-quasi-coherence-direct-sums", "perfect-theorem-bondal-van-den-Bergh", "perfect-proposition-compact-is-perfect", "equiv-lemma-van-den-bergh", "equiv-lemma-characterize-dbcoh-proper-regular", "cohomology-lemma-dual-perfect-complex" ], "ref_ids": [ 6937, 6935, 7111, 6810, 6812, 2233 ] } ], "ref_ids": [] }, { "id": 6814, "type": "theorem", "label": "equiv-lemma-always-right-adjoints", "categories": [ "equiv" ], "title": "equiv-lemma-always-right-adjoints", "contents": [ "Let $k$ be a field. Let $X$ and $Y$ be proper schemes over $k$.", "If $X$ is regular, then $k$-linear any exact functor", "$F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$", "has an exact right adjoint and an exact left adjoint." ], "refs": [], "proofs": [ { "contents": [ "If an adjoint exists it is an exact functor by the very general", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-is-exact}.", "\\medskip\\noindent", "Let us prove the existence of a right adjoint.", "To see existence, it suffices to show that for", "$M \\in D_{perf}(\\mathcal{O}_Y)$ the contravariant functor", "$K \\mapsto \\Hom_Y(F(K), M)$ is representable.", "This functor is contravariant, $k$-linear, and cohomological.", "Hence by Lemma \\ref{lemma-bondal-van-den-bergh} part (1)", "it suffices to show that", "$$", "\\sum\\nolimits_{i \\in \\mathbf{Z}} \\dim_k \\Ext^i_Y(F(K), M) < \\infty", "$$", "This follows from Lemma \\ref{lemma-finiteness}.", "\\medskip\\noindent", "For the existence of the left adjoint we argue in the same", "manner using part (2) of Lemma \\ref{lemma-bondal-van-den-bergh}." ], "refs": [ "derived-lemma-adjoint-is-exact", "equiv-lemma-bondal-van-den-bergh", "equiv-lemma-finiteness", "equiv-lemma-bondal-van-den-bergh" ], "ref_ids": [ 1792, 6813, 6807, 6813 ] } ], "ref_ids": [] }, { "id": 6815, "type": "theorem", "label": "equiv-lemma-fourier-Mukai-QCoh", "categories": [ "equiv" ], "title": "equiv-lemma-fourier-Mukai-QCoh", "contents": [ "Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$.", "Let $K \\in D(\\mathcal{O}_{X \\times_S Y})$.", "The corresponding Fourier-Mukai functor $\\Phi_K$ sends", "$D_\\QCoh(\\mathcal{O}_X)$ into $D_\\QCoh(\\mathcal{O}_Y)$", "if $K$ is in $D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$ and $X \\to S$ is", "quasi-compact and quasi-separated." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that derived pullback preserves", "$D_\\QCoh$", "(Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-pullback}),", "derived tensor products preserve $D_\\QCoh$", "(Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-tensor-product}),", "the projection $\\text{pr}_2 : X \\times_S Y \\to Y$ is", "quasi-compact and quasi-separated", "(Schemes, Lemmas", "\\ref{schemes-lemma-quasi-compact-preserved-base-change} and", "\\ref{schemes-lemma-separated-permanence}), and", "total direct image along a quasi-separated and quasi-compact", "morphism preserves $D_\\QCoh$", "(Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-direct-image})." ], "refs": [ "perfect-lemma-quasi-coherence-pullback", "perfect-lemma-quasi-coherence-tensor-product", "schemes-lemma-quasi-compact-preserved-base-change", "schemes-lemma-separated-permanence", "perfect-lemma-quasi-coherence-direct-image" ], "ref_ids": [ 6944, 6945, 7698, 7714, 6946 ] } ], "ref_ids": [] }, { "id": 6816, "type": "theorem", "label": "equiv-lemma-compose-fourier-mukai", "categories": [ "equiv" ], "title": "equiv-lemma-compose-fourier-mukai", "contents": [ "Let $S$ be a scheme. Let $X, Y, Z$ be schemes over $S$. Assume", "$X \\to S$, $Y \\to S$, and $Z \\to S$ are quasi-compact and quasi-separated.", "Let $K \\in D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$.", "Let $K' \\in D_\\QCoh(\\mathcal{O}_{Y \\times_S Z})$.", "Consider the Fourier-Mukai functors", "$\\Phi_K : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$", "and $\\Phi_{K'} : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_Z)$.", "If $X$ and $Z$ are tor independent over $S$ and $Y \\to S$ is flat,", "then", "$$", "\\Phi_{K'} \\circ \\Phi_K = \\Phi_{K''} :", "D_\\QCoh(\\mathcal{O}_X)", "\\longrightarrow", "D_\\QCoh(\\mathcal{O}_Z)", "$$", "where", "$$", "K'' = R\\text{pr}_{13, *}(", "L\\text{pr}_{12}^*K", "\\otimes_{\\mathcal{O}_{X \\times_S Y \\times_S Z}}^\\mathbf{L}", "L\\text{pr}_{23}^*K')", "$$", "in $D_\\QCoh(\\mathcal{O}_{X \\times_S Z})$." ], "refs": [], "proofs": [ { "contents": [ "The statement makes sense by Lemma \\ref{lemma-fourier-Mukai-QCoh}.", "We are going to use", "Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-quasi-coherence-pullback},", "\\ref{perfect-lemma-quasi-coherence-tensor-product}, and", "\\ref{perfect-lemma-quasi-coherence-direct-image}", "and Schemes, Lemmas", "\\ref{schemes-lemma-quasi-compact-preserved-base-change} and", "\\ref{schemes-lemma-separated-permanence}", "without further mention.", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-base-change-tor-independent}", "we see that $X \\times_S Y$ and $Y \\times_S Z$ are tor independent", "over $Y$. This means that we have base change for the cartesian diagram", "$$", "\\xymatrix{", "X \\times_S Y \\times_S Z \\ar[d] \\ar[r] &", "Y \\times_S Z \\ar[d]^{p^{YZ}_Y} \\\\", "X \\times_S Y \\ar[r]^{p^{XY}_Y} & Y", "}", "$$", "for complexes with quasi-coherent cohomology sheaves, see", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-compare-base-change}.", "Abbreviating $p^* = Lp^*$, $p_* = Rp_*$ and $\\otimes = \\otimes^\\mathbf{L}$", "we have for $M \\in D_\\QCoh(\\mathcal{O}_X)$ the sequence of equalities", "\\begin{align*}", "\\Phi_{K'}(\\Phi_K(M))", "& =", "p^{YZ}_{Z, *}(p^{YZ, *}_Y p^{XY}_{Y, *}(p^{XY, *}_X M \\otimes K) \\otimes K') \\\\", "& =", "p^{YZ}_{Z, *}(\\text{pr}_{23, *} \\text{pr}_{12}^*(p^{XY, *}_X M \\otimes K)", "\\otimes K') \\\\", "& =", "p^{YZ}_{Z, *}(\\text{pr}_{23, *}(\\text{pr}_1^*M \\otimes \\text{pr}_{12}^*K)", "\\otimes K') \\\\", "& =", "p^{YZ}_{Z, *}(\\text{pr}_{23, *}(\\text{pr}_1^*M \\otimes \\text{pr}_{12}^*K", "\\otimes \\text{pr}_{23}^*K')) \\\\", "& =", "\\text{pr}_{3, *}(\\text{pr}_1^*M \\otimes \\text{pr}_{12}^*K", "\\otimes \\text{pr}_{23}^*K') \\\\", "& =", "p^{XZ}_{Z, *}\\text{pr}_{13, *}(\\text{pr}_1^*M \\otimes \\text{pr}_{12}^*K", "\\otimes \\text{pr}_{23}^*K') \\\\", "& =", "p^{XZ}_{Z, *} (p^{XZ, *}_X M \\otimes \\text{pr}_{13, *}(\\text{pr}_{12}^*K", "\\otimes \\text{pr}_{23}^*K'))", "\\end{align*}", "as desired. Here we have used the remark on base change in the", "second equality and we have use Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-cohomology-base-change} in the $4$th and", "last equality." ], "refs": [ "equiv-lemma-fourier-Mukai-QCoh", "perfect-lemma-quasi-coherence-pullback", "perfect-lemma-quasi-coherence-tensor-product", "perfect-lemma-quasi-coherence-direct-image", "schemes-lemma-quasi-compact-preserved-base-change", "schemes-lemma-separated-permanence", "perfect-lemma-flat-base-change-tor-independent", "perfect-lemma-compare-base-change", "perfect-lemma-cohomology-base-change" ], "ref_ids": [ 6815, 6944, 6945, 6946, 7698, 7714, 7027, 7028, 7025 ] } ], "ref_ids": [] }, { "id": 6817, "type": "theorem", "label": "equiv-lemma-fourier-mukai", "categories": [ "equiv" ], "title": "equiv-lemma-fourier-mukai", "contents": [ "Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$.", "Let $K \\in D(\\mathcal{O}_{X \\times_S Y})$.", "The corresponding Fourier-Mukai functor $\\Phi_K$ sends", "$D_{perf}(\\mathcal{O}_X)$ into $D_{perf}(\\mathcal{O}_Y)$ if at least", "one of the following conditions is satisfied:", "\\begin{enumerate}", "\\item $S$ is Noetherian, $X \\to S$ and $Y \\to S$ are of finite type,", "$K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_{X \\times_S Y})$, the support of $H^i(K)$", "is proper over $Y$ for all $i$, and $K$ has finite tor dimension", "as an object of $D(\\text{pr}_2^{-1}\\mathcal{O}_Y)$,", "\\item $X \\to S$ is of finite presentation and $K$ can be represented", "by a bounded complex $\\mathcal{K}^\\bullet$ of finitely presented", "$\\mathcal{O}_{X \\times_S Y}$-modules, flat over $Y$, with support", "proper over $Y$,", "\\item $X \\to S$ is a proper flat morphism of finite presentation", "and $K$ is perfect,", "\\item $S$ is Noetherian, $X \\to S$ is flat and proper, and $K$ is perfect", "\\item $X \\to S$ is a proper flat morphism of finite presentation", "and $K$ is $Y$-perfect,", "\\item $S$ is Noetherian, $X \\to S$ is flat and proper, and $K$ is", "$Y$-perfect.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $M$ is perfect on $X$, then $L\\text{pr}_1^*M$", "is perfect on $X \\times_S Y$, see", "Cohomology, Lemma \\ref{cohomology-lemma-perfect-pullback}.", "We will use this without further mention below.", "We will also use that if $X \\to S$ is of finite type, or proper, or", "flat, or of finite presentation, then the same thing is true for", "the base change $\\text{pr}_2 : X \\times_S Y \\to Y$, see", "Morphisms, Lemmas", "\\ref{morphisms-lemma-base-change-finite-type},", "\\ref{morphisms-lemma-base-change-proper},", "\\ref{morphisms-lemma-base-change-flat}, and", "\\ref{morphisms-lemma-base-change-finite-presentation}.", "\\medskip\\noindent", "Part (1) follows from", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-perfect-direct-image}", "combined with", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-perfect-on-noetherian}.", "\\medskip\\noindent", "Part (2) follows from", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-base-change-tensor-perfect}.", "\\medskip\\noindent", "Part (3) follows from", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image-general}.", "\\medskip\\noindent", "Part (4) follows from part (3) and the fact that a finite type", "morphism of Noetherian schemes is of finite presentation by Morphisms, Lemma", "\\ref{morphisms-lemma-noetherian-finite-type-finite-presentation}.", "\\medskip\\noindent", "Part (5) follows from", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-derived-pushforward-rel-perfect} combined with", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-perfect-relatively-perfect}.", "\\medskip\\noindent", "Part (6) follows from part (5) in the same way that part (4) follows from", "part (3)." ], "refs": [ "cohomology-lemma-perfect-pullback", "morphisms-lemma-base-change-finite-type", "morphisms-lemma-base-change-proper", "morphisms-lemma-base-change-flat", "morphisms-lemma-base-change-finite-presentation", "perfect-lemma-perfect-direct-image", "perfect-lemma-perfect-on-noetherian", "perfect-lemma-base-change-tensor-perfect", "perfect-lemma-flat-proper-perfect-direct-image-general", "morphisms-lemma-noetherian-finite-type-finite-presentation", "perfect-lemma-derived-pushforward-rel-perfect", "perfect-lemma-perfect-relatively-perfect" ], "ref_ids": [ 2225, 5200, 5409, 5265, 5240, 7043, 6987, 7052, 7054, 5245, 7083, 7079 ] } ], "ref_ids": [] }, { "id": 6818, "type": "theorem", "label": "equiv-lemma-fourier-mukai-Coh", "categories": [ "equiv" ], "title": "equiv-lemma-fourier-mukai-Coh", "contents": [ "Let $S$ be a Noetherian scheme. Let $X$ and $Y$ be schemes of finite type", "over $S$. Let $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_{X \\times_S Y})$.", "The corresponding Fourier-Mukai functor $\\Phi_K$ sends", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ into $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$", "if at least one of the following conditions is satisfied:", "\\begin{enumerate}", "\\item the support of $H^i(K)$ is proper over $Y$ for all $i$, and $K$", "has finite tor dimension as an object of $D(\\text{pr}_1^{-1}\\mathcal{O}_X)$,", "\\item $K$ can be represented by a bounded complex $\\mathcal{K}^\\bullet$", "of coherent $\\mathcal{O}_{X \\times_S Y}$-modules, flat over $X$, with support", "proper over $Y$,", "\\item the support of $H^i(K)$ is proper over $Y$ for all $i$", "and $X$ is a regular scheme,", "\\item $K$ is perfect, the support of $H^i(K)$ is proper over $Y$ for all $i$,", "and $Y \\to S$ is flat.", "\\end{enumerate}", "Furthermore in each case the support condition is automatic", "if $X \\to S$ is proper." ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.", "In each case we will use Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-direct-image-coherent} to show that", "$$", "\\Phi_K(M) = R\\text{pr}_{2, *}(", "L\\text{pr}_1^*M", "\\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L}", "K)", "$$", "is in $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$. The derived tensor product", "$L\\text{pr}_1^*M \\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} K$", "is a pseudo-coherent object of $D(\\mathcal{O}_{X \\times_S Y})$", "(by", "Cohomology, Lemma \\ref{cohomology-lemma-pseudo-coherent-pullback},", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-identify-pseudo-coherent-noetherian}, and", "Cohomology, Lemma \\ref{cohomology-lemma-tensor-pseudo-coherent})", "whence has coherent cohomology sheaves (by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-identify-pseudo-coherent-noetherian} again).", "In each case the supports of the cohomology sheaves", "$H^i(L\\text{pr}_1^*M \\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} K)$", "is proper over $Y$ as these supports are contained in the", "union of the supports of the $H^i(K)$. Hence in each case", "it suffices to prove that this tensor product is bounded below.", "\\medskip\\noindent", "Case (1). By Cohomology, Lemma \\ref{cohomology-lemma-variant-derived-pullback}", "we have", "$$", "L\\text{pr}_1^*M", "\\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L}", "K", "\\cong", "\\text{pr}_1^{-1}M", "\\otimes_{\\text{pr}_1^{-1}\\mathcal{O}_X}^\\mathbf{L}", "K", "$$", "with obvious notation. Hence the assumption on tor dimension", "and the fact that $M$ has only a finite number of nonzero", "cohomology sheaves, implies the bound we want.", "\\medskip\\noindent", "Case (2) follows because here the assumption implies that $K$ has", "finite tor dimension as an object of $D(\\text{pr}_1^{-1}\\mathcal{O}_X)$", "hence the argument in the previous paragraph applies.", "\\medskip\\noindent", "In Case (3) it is also the case that $K$ has finite tor dimension", "as an object of $D(\\text{pr}_1^{-1}\\mathcal{O}_X)$. Namely, choose", "affine opens $U = \\Spec(A)$ and $V = \\Spec(B)$ of $X$ and $Y$ mapping into the", "affine open $W = \\Spec(R)$ of $S$. Then", "$K|_{U \\times V}$ is given by a bounded complex of finite", "$A \\otimes_R B$-modules $M^\\bullet$. Since $A$ is a regular ring", "of finite dimension we see that each $M^i$ has finite projective dimension", "as an $A$-module (Algebra, Lemma", "\\ref{algebra-lemma-finite-gl-dim-finite-dim-regular})", "and hence finite tor dimension as an $A$-module.", "Thus $M^\\bullet$ has finite tor dimension as a complex of $A$-modules", "(More on Algebra, Lemma", "\\ref{more-algebra-lemma-complex-finite-tor-dimension-modules}).", "Since $X \\times Y$ is quasi-compact we conclude there exist $[a, b]$", "such that for every point $z \\in X \\times Y$ the stalk $K_z$", "has tor amplitude in $[a, b]$ over $\\mathcal{O}_{X, \\text{pr}_1(z)}$.", "This implies $K$ has bounded tor dimension as an object of", "$D(\\text{pr}_1^{-1}\\mathcal{O}_X)$, see", "Cohomology, Lemma \\ref{cohomology-lemma-tor-amplitude-stalk}.", "We conclude as in the previous to paragraphs.", "\\medskip\\noindent", "Case (4). With notation as above, the ring map $R \\to B$ is flat.", "Hence the ring map $A \\to A \\otimes_R B$ is flat. Hence any projective", "$A \\otimes_R B$-module is $A$-flat. Thus any perfect complex of", "$A \\otimes_R B$-modules has finite tor dimension as a complex", "of $A$-modules and we conclude as before." ], "refs": [ "perfect-lemma-direct-image-coherent", "cohomology-lemma-pseudo-coherent-pullback", "perfect-lemma-identify-pseudo-coherent-noetherian", "cohomology-lemma-tensor-pseudo-coherent", "perfect-lemma-identify-pseudo-coherent-noetherian", "cohomology-lemma-variant-derived-pullback", "algebra-lemma-finite-gl-dim-finite-dim-regular", "more-algebra-lemma-complex-finite-tor-dimension-modules", "cohomology-lemma-tor-amplitude-stalk" ], "ref_ids": [ 6984, 2206, 6976, 2208, 6976, 2119, 980, 10175, 2217 ] } ], "ref_ids": [] }, { "id": 6819, "type": "theorem", "label": "equiv-lemma-fourier-mukai-right-adjoint", "categories": [ "equiv" ], "title": "equiv-lemma-fourier-mukai-right-adjoint", "contents": [ "\\begin{reference}", "Compare with discussion in \\cite{Rizzardo}.", "\\end{reference}", "Let $X \\to S$ and $Y \\to S$ be morphisms of quasi-compact and quasi-separated", "schemes. Let $\\Phi : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$", "be a Fourier-Mukai functor with pseudo-coherent kernel", "$K \\in D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$.", "Let $a : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$", "be the right adjoint to $R\\text{pr}_{2, *}$, see", "Duality for Schemes, Lemma \\ref{duality-lemma-twisted-inverse-image}.", "Denote", "$$", "K' = (Y \\times_S X \\to X \\times_S Y)^*", "R\\SheafHom_{\\mathcal{O}_{X \\times_S Y}}(K, a(\\mathcal{O}_Y)) \\in", "D_\\QCoh(\\mathcal{O}_{Y \\times_S X})", "$$", "and denote $\\Phi' : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$", "the corresponding Fourier-Mukai transform. There is a canonical map", "$$", "\\Hom_X(M, \\Phi'(N)) \\longrightarrow \\Hom_Y(\\Phi(M), N)", "$$", "functorial in $M$ in $D_\\QCoh(\\mathcal{O}_X)$ and $N$ in", "$D_\\QCoh(\\mathcal{O}_Y)$ which is an isomorphism if", "\\begin{enumerate}", "\\item $N$ is perfect, or", "\\item $K$ is perfect and $X \\to S$ is proper flat and of finite presentation.", "\\end{enumerate}" ], "refs": [ "duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-fourier-Mukai-QCoh} we obtain a functor $\\Phi$", "as in the statement. Observe that $a(\\mathcal{O}_Y)$ is in", "$D^+_\\QCoh(\\mathcal{O}_{X \\times_S Y})$ by Duality for Schemes,", "Lemma \\ref{duality-lemma-twisted-inverse-image-bounded-below}.", "Hence for $K$ pseudo-coherent we have", "$K' \\in D_\\QCoh(\\mathcal{O}_{Y \\times_S X})$", "by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-internal-hom}", "we we obtain $\\Phi'$ as indicated.", "\\medskip\\noindent", "We abbreviate", "$\\otimes^\\mathbf{L} = \\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L}$", "and", "$\\SheafHom = R\\SheafHom_{\\mathcal{O}_{X \\times_S Y}}$.", "Let $M$ be in $D_\\QCoh(\\mathcal{O}_X)$ and let", "$N$ be in $D_\\QCoh(\\mathcal{O}_Y)$. We have", "\\begin{align*}", "\\Hom_Y(\\Phi(M), N)", "& =", "\\Hom_Y(R\\text{pr}_{2, *}(L\\text{pr}_1^*M \\otimes^\\mathbf{L} K), N) \\\\", "& =", "\\Hom_{X \\times_S Y}(L\\text{pr}_1^*M \\otimes^\\mathbf{L} K, a(N)) \\\\", "& =", "\\Hom_{X \\times_S Y}(L\\text{pr}_1^*M,", "R\\SheafHom(K, a(N))) \\\\", "& =", "\\Hom_X(M, R\\text{pr}_{1, *}R\\SheafHom(K, a(N)))", "\\end{align*}", "where we have used Cohomology, Lemmas \\ref{cohomology-lemma-internal-hom}", "and \\ref{cohomology-lemma-adjoint}. There are canonical maps", "$$", "L\\text{pr}_2^*N \\otimes^\\mathbf{L} R\\SheafHom(K, a(\\mathcal{O}_Y))", "\\xrightarrow{\\alpha}", "R\\SheafHom(K, L\\text{pr}_2^*N \\otimes^\\mathbf{L} a(\\mathcal{O}_Y))", "\\xrightarrow{\\beta}", "R\\SheafHom(K, a(N))", "$$", "Here $\\alpha$ is", "Cohomology, Lemma \\ref{cohomology-lemma-internal-hom-diagonal-better}", "and $\\beta$ is Duality for Schemes, Equation", "(\\ref{duality-equation-compare-with-pullback}).", "Combining all of these arrows we obtain the functorial displayed", "arrow in the statement of the lemma.", "\\medskip\\noindent", "The arrow $\\alpha$ is an isomorphism by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-internal-hom-evaluate-tensor-isomorphism}", "as soon as either $K$ or $N$ is perfect.", "The arrow $\\beta$ is an isomorphism if $N$ is perfect by", "Duality for Schemes, Lemma \\ref{duality-lemma-compare-with-pullback-perfect}", "or in general if $X \\to S$ is", "flat proper of finite presentation by", "Duality for Schemes, Lemma", "\\ref{duality-lemma-compare-with-pullback-flat-proper}." ], "refs": [ "equiv-lemma-fourier-Mukai-QCoh", "duality-lemma-twisted-inverse-image-bounded-below", "perfect-lemma-quasi-coherence-internal-hom", "cohomology-lemma-internal-hom", "cohomology-lemma-adjoint", "cohomology-lemma-internal-hom-diagonal-better", "perfect-lemma-internal-hom-evaluate-tensor-isomorphism", "duality-lemma-compare-with-pullback-perfect", "duality-lemma-compare-with-pullback-flat-proper" ], "ref_ids": [ 6815, 13504, 6981, 2183, 2121, 2187, 6982, 13515, 13535 ] } ], "ref_ids": [ 13503 ] }, { "id": 6820, "type": "theorem", "label": "equiv-lemma-fourier-mukai-left-adjoint", "categories": [ "equiv" ], "title": "equiv-lemma-fourier-mukai-left-adjoint", "contents": [ "\\begin{reference}", "Compare with discussion in \\cite{Rizzardo}.", "\\end{reference}", "Let $S$ be a Noetherian scheme. Let $Y \\to S$ be a flat proper", "Gorenstein morphism and let $X \\to S$ be a finite type morphism.", "Denote $\\omega^\\bullet_{Y/S}$ the relative dualizing complex of", "$Y$ over $S$. Let $\\Phi : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$", "be a Fourier-Mukai functor with perfect kernel", "$K \\in D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$. Denote", "$$", "K' = (Y \\times_S X \\to X \\times_S Y)^*(K^\\vee", "\\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L}", "L\\text{pr}_2^*\\omega^\\bullet_{Y/S})", "\\in", "D_\\QCoh(\\mathcal{O}_{Y \\times_S X})", "$$", "and denote $\\Phi' : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$", "the corresponding Fourier-Mukai transform. There is a canonical", "isomorphism", "$$", "\\Hom_Y(N, \\Phi(M)) \\longrightarrow \\Hom_X(\\Phi'(N), M)", "$$", "functorial in $M$ in $D_\\QCoh(\\mathcal{O}_X)$ and $N$ in", "$D_\\QCoh(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-fourier-Mukai-QCoh} we obtain a functor $\\Phi$", "as in the statement.", "\\medskip\\noindent", "Observe that formation of the relative dualizing complex commutes", "with base change in our setting, see Duality for Schemes,", "Remark \\ref{duality-remark-relative-dualizing-complex}.", "Thus $L\\text{pr}_2^*\\omega^\\bullet_{Y/S} = \\omega^\\bullet_{X \\times_S Y/X}$.", "Moreover, we observe that $\\omega^\\bullet_{Y/S}$ is an", "invertible object of the derived category, see Duality for Schemes, Lemma", "\\ref{duality-lemma-affine-flat-Noetherian-gorenstein}, and a fortiori", "perfect.", "\\medskip\\noindent", "To actually prove the lemma we're going to cheat. Namely, we will", "show that if we replace the roles of $X$ and $Y$ and $K$ and $K'$", "then these are as in Lemma \\ref{lemma-fourier-mukai-right-adjoint}", "and we get the result. It is clear that $K'$ is perfect as a", "tensor product of perfect objects so that the discussion in", "Lemma \\ref{lemma-fourier-mukai-right-adjoint} applies to it.", "To show that the procedure of", "Lemma \\ref{lemma-fourier-mukai-right-adjoint} applied to $K'$", "on $Y \\times_S X$ produces a complex isomorphic to $K$ it suffices", "(details omitted) to show that", "$$", "R\\SheafHom(R\\SheafHom(K, \\omega^\\bullet_{X \\times_S Y/X}),", "\\omega^\\bullet_{X \\times_S Y/X}) = K", "$$", "This is clear because $K$ is perfect and $\\omega^\\bullet_{X \\times_S Y/X}$", "is invertible; details omitted. Thus", "Lemma \\ref{lemma-fourier-mukai-right-adjoint} produces a map", "$$", "\\Hom_Y(N, \\Phi(M)) \\longrightarrow \\Hom_X(\\Phi'(N), M)", "$$", "functorial in $M$ in $D_\\QCoh(\\mathcal{O}_X)$ and $N$ in", "$D_\\QCoh(\\mathcal{O}_Y)$ which is an isomorphism because", "$K'$ is perfect. This finishes the proof." ], "refs": [ "equiv-lemma-fourier-Mukai-QCoh", "duality-remark-relative-dualizing-complex", "duality-lemma-affine-flat-Noetherian-gorenstein", "equiv-lemma-fourier-mukai-right-adjoint", "equiv-lemma-fourier-mukai-right-adjoint", "equiv-lemma-fourier-mukai-right-adjoint", "equiv-lemma-fourier-mukai-right-adjoint" ], "ref_ids": [ 6815, 13649, 13602, 6819, 6819, 6819, 6819 ] } ], "ref_ids": [] }, { "id": 6821, "type": "theorem", "label": "equiv-lemma-fourier-mukai-flat-proper-over-noetherian", "categories": [ "equiv" ], "title": "equiv-lemma-fourier-mukai-flat-proper-over-noetherian", "contents": [ "Let $S$ be a Noetherian scheme.", "\\begin{enumerate}", "\\item For $X$, $Y$ proper and flat over $S$ and $K$ in", "$D_{perf}(\\mathcal{O}_{X \\times_S Y})$ we obtain a Fourier-Mukai functor", "$\\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$.", "\\item For $X$, $Y$, $Z$ proper and flat over $S$, $K \\in", "D_{perf}(\\mathcal{O}_{X \\times_S Y})$, $K' \\in", "D_{perf}(\\mathcal{O}_{Y \\times_S Z})$ the composition", "$\\Phi_{K'} \\circ \\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Z)$", "is equal to $\\Phi_{K''}$ with $K'' \\in D_{perf}(\\mathcal{O}_{X \\times_S Z})$", "computed as in Lemma \\ref{lemma-compose-fourier-mukai},", "\\item For $X$, $Y$, $K$, $\\Phi_K$ as in (1) if $X \\to S$ is Gorenstein, then", "$\\Phi_{K'} : D_{perf}(\\mathcal{O}_Y) \\to D_{perf}(\\mathcal{O}_X)$ is a right", "adjoint to $\\Phi_K$ where $K' \\in D_{perf}(\\mathcal{O}_{Y \\times_S X})$", "is the pullback of $L\\text{pr}_1^*\\omega_{X/S}^\\bullet", "\\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} K^\\vee$ by", "$Y \\times_S X \\to X \\times_S Y$.", "\\item For $X$, $Y$, $K$, $\\Phi_K$ as in (1) if $Y \\to S$ is Gorenstein, then", "$\\Phi_{K''} : D_{perf}(\\mathcal{O}_Y) \\to D_{perf}(\\mathcal{O}_X)$ is a left", "adjoint to $\\Phi_K$ where $K'' \\in D_{perf}(\\mathcal{O}_{Y \\times_S X})$", "is the pullback of $L\\text{pr}_2^*\\omega_{Y/S}^\\bullet", "\\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} K^\\vee$ by", "$Y \\times_S X \\to X \\times_S Y$.", "\\end{enumerate}" ], "refs": [ "equiv-lemma-compose-fourier-mukai" ], "proofs": [ { "contents": [ "Part (1) is immediate from Lemma \\ref{lemma-fourier-mukai} part (4).", "\\medskip\\noindent", "Part (2) follows from Lemma \\ref{lemma-compose-fourier-mukai} and the", "fact that", "$K'' = R\\text{pr}_{13, *}(", "L\\text{pr}_{12}^*K", "\\otimes_{\\mathcal{O}_{X \\times_S Y \\times_S Z}}^\\mathbf{L}", "L\\text{pr}_{23}^*K')$ is perfect for example by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image}.", "\\medskip\\noindent", "The adjointness in part (3) on all complexes with quasi-coherent cohomology", "sheaves follows from Lemma \\ref{lemma-fourier-mukai-right-adjoint} with", "$K'$ equal to the pullback of", "$R\\SheafHom_{\\mathcal{O}_{X \\times_S Y}}(K, a(\\mathcal{O}_Y))$", "by $Y \\times_S X \\to X \\times_S Y$ where $a$ is the right adjoint", "to $R\\text{pr}_{2, *} : D_\\QCoh(\\mathcal{O}_{X \\times_S Y}) \\to", "D_\\QCoh(\\mathcal{O}_Y)$. Denote $f : X \\to S$ the structure morphism of $X$.", "Since $f$ is proper the functor", "$f^! : D_\\QCoh^+(\\mathcal{O}_S) \\to D_\\QCoh^+(\\mathcal{O}_X)$", "is the restriction to $D_\\QCoh^+(\\mathcal{O}_S)$", "of the right adjoint to", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_S)$, see", "Duality for Schemes, Section \\ref{duality-section-upper-shriek}.", "Hence the relative dualizing complex $\\omega_{X/S}^\\bullet$ as defined in", "Duality for Schemes, Remark", "\\ref{duality-remark-relative-dualizing-complex}", "is equal to $\\omega_{X/S}^\\bullet = f^!\\mathcal{O}_S$.", "Since formation of the relative dualizing complex", "commutes with base change (see Duality for Schemes, Remark", "\\ref{duality-remark-relative-dualizing-complex}) we see that", "$a(\\mathcal{O}_Y) = L\\text{pr}_1^*\\omega_{X/S}^\\bullet$.", "Thus", "$$", "R\\SheafHom_{\\mathcal{O}_{X \\times_S Y}}(K, a(\\mathcal{O}_Y))", "\\cong", "L\\text{pr}_1^*\\omega_{X/S}^\\bullet", "\\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} K^\\vee", "$$", "by Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "Finally, since $X \\to S$ is assumed Gorenstein the relative dualizing complex", "is invertible: this follows from Duality for Schemes, Lemma", "\\ref{duality-lemma-affine-flat-Noetherian-gorenstein}.", "We conclude that $\\omega_{X/S}^\\bullet$ is perfect", "(Cohomology, Lemma \\ref{cohomology-lemma-invertible-derived})", "and hence $K'$ is perfect.", "Therefore $\\Phi_{K'}$ does indeed map $D_{perf}(\\mathcal{O}_Y)$", "into $D_{perf}(\\mathcal{O}_X)$ which finishes the proof of (3).", "\\medskip\\noindent", "The proof of (4) is the same as the proof of (3) except one uses", "Lemma \\ref{lemma-fourier-mukai-left-adjoint} instead of", "Lemma \\ref{lemma-fourier-mukai-right-adjoint}." ], "refs": [ "equiv-lemma-fourier-mukai", "equiv-lemma-compose-fourier-mukai", "perfect-lemma-flat-proper-perfect-direct-image", "equiv-lemma-fourier-mukai-right-adjoint", "duality-remark-relative-dualizing-complex", "duality-remark-relative-dualizing-complex", "cohomology-lemma-dual-perfect-complex", "duality-lemma-affine-flat-Noetherian-gorenstein", "cohomology-lemma-invertible-derived", "equiv-lemma-fourier-mukai-left-adjoint", "equiv-lemma-fourier-mukai-right-adjoint" ], "ref_ids": [ 6817, 6816, 7046, 6819, 13649, 13649, 2233, 13602, 2239, 6820, 6819 ] } ], "ref_ids": [ 6816 ] }, { "id": 6822, "type": "theorem", "label": "equiv-lemma-on-product", "categories": [ "equiv" ], "title": "equiv-lemma-on-product", "contents": [ "Let $R$ be a Noetherian ring. Let $X$, $Y$ be finite type schemes over $R$", "having the resolution property. For any coherent", "$\\mathcal{O}_{X \\times_R Y}$-module $\\mathcal{F}$ there exist", "a surjection $\\mathcal{E} \\boxtimes \\mathcal{G} \\to \\mathcal{F}$", "where $\\mathcal{E}$ is a finite locally free $\\mathcal{O}_X$-module", "and $\\mathcal{G}$ is a finite locally free $\\mathcal{O}_Y$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ and $V \\subset Y$ be affine open subschemes. Let", "$\\mathcal{I} \\subset \\mathcal{O}_X$ be the ideal sheaf of the", "reduced induced closed subscheme structure on $X \\setminus U$.", "Similarly, let $\\mathcal{I}' \\subset \\mathcal{O}_Y$ be the ideal sheaf of the", "reduced induced closed subscheme structure on $Y \\setminus V$.", "Then the ideal sheaf", "$$", "\\mathcal{J} = \\Im(\\text{pr}_1^*\\mathcal{I} \\otimes_{\\mathcal{O}_{X \\times_R Y}}", "\\text{pr}_2^*\\mathcal{I}' \\to \\mathcal{O}_{X \\times_R Y})", "$$", "satisfies $V(\\mathcal{J}) = X \\times_R Y \\setminus U \\times_R V$.", "For any section $s \\in \\mathcal{F}(U \\times_R V)$ we can find an integer", "$n > 0$ and a map $\\mathcal{J}^n \\to \\mathcal{F}$ whose restriction to", "$U \\times_R V$ gives $s$, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-homs-over-open}.", "By assumption we can choose surjections", "$\\mathcal{E} \\to \\mathcal{I}$ and $\\mathcal{G} \\to \\mathcal{I}'$.", "These produce corresponding surjections", "$$", "\\mathcal{E} \\boxtimes \\mathcal{G} \\to \\mathcal{J}", "\\quad\\text{and}\\quad", "\\mathcal{E}^{\\otimes n} \\boxtimes \\mathcal{G}^{\\otimes n} \\to \\mathcal{J}^n", "$$", "and hence a map", "$\\mathcal{E}^{\\otimes n} \\boxtimes \\mathcal{G}^{\\otimes n} \\to \\mathcal{F}$", "whose image contains the section $s$ over $U \\times_R V$.", "Since we can cover $X \\times_R Y$ by a finite number of affine opens", "of the form $U \\times_R V$ and since $\\mathcal{F}|_{U \\times_R V}$", "is generated by finitely many sections (Properties, Lemma", "\\ref{properties-lemma-finite-type-module})", "we conclude that there exists a surjection", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, N}", "\\mathcal{E}_j^{\\otimes n_j} \\boxtimes \\mathcal{G}_j^{\\otimes n_j}", "\\to \\mathcal{F}", "$$", "where $\\mathcal{E}_j$ is finite locally free on $X$ and", "$\\mathcal{G}_j$ is finite locally free on $Y$.", "Setting $\\mathcal{E} = \\bigoplus \\mathcal{E}_j^{\\otimes n_j}$", "and $\\mathcal{G} = \\bigoplus \\mathcal{G}_j^{\\otimes n_j}$", "we conclude that the lemma is true." ], "refs": [ "coherent-lemma-homs-over-open", "properties-lemma-finite-type-module" ], "ref_ids": [ 3322, 3002 ] } ], "ref_ids": [] }, { "id": 6823, "type": "theorem", "label": "equiv-lemma-on-product-general", "categories": [ "equiv" ], "title": "equiv-lemma-on-product-general", "contents": [ "Let $R$ be a ring. Let $X$, $Y$ be quasi-compact and quasi-separated", "schemes over $R$ having the resolution property. For any finite", "type quasi-coherent $\\mathcal{O}_{X \\times_R Y}$-module $\\mathcal{F}$", "there exist a surjection $\\mathcal{E} \\boxtimes \\mathcal{G} \\to \\mathcal{F}$", "where $\\mathcal{E}$ is a finite locally free $\\mathcal{O}_X$-module", "and $\\mathcal{G}$ is a finite locally free $\\mathcal{O}_Y$-module." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-on-product} by a limit argument.", "We urge the reader to skip the proof.", "Since $X \\times_R Y$ is a closed subscheme of $X \\times_\\mathbf{Z} Y$", "it is harmless if we replace $R$ by $\\mathbf{Z}$.", "We can write $\\mathcal{F}$ as the quotient of", "a finitely presented $\\mathcal{O}_{X \\times_R Y}$-module by", "Properties, Lemma", "\\ref{properties-lemma-finite-directed-colimit-surjective-maps}.", "Hence we may assume $\\mathcal{F}$ is of", "finite presentation. Next we can write $X = \\lim X_i$", "with $X_i$ of finite presentation over $\\mathbf{Z}$ and similarly", "$Y = \\lim Y_j$, see Limits, Proposition \\ref{limits-proposition-approximate}.", "Then $\\mathcal{F}$ will descend to $\\mathcal{F}_{ij}$ on some $X_i \\times_R Y_j$", "(Limits, Lemma \\ref{limits-lemma-descend-modules-finite-presentation}) and", "so does the property of having the resolution property", "(Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-resolution-property-descends}).", "Then we apply Lemma \\ref{lemma-on-product}", "to $\\mathcal{F}_{ij}$ and we pullback." ], "refs": [ "equiv-lemma-on-product", "properties-lemma-finite-directed-colimit-surjective-maps", "limits-proposition-approximate", "limits-lemma-descend-modules-finite-presentation", "perfect-lemma-resolution-property-descends", "equiv-lemma-on-product" ], "ref_ids": [ 6822, 3025, 15126, 15078, 7091, 6822 ] } ], "ref_ids": [] }, { "id": 6824, "type": "theorem", "label": "equiv-lemma-diagonal-resolution", "categories": [ "equiv" ], "title": "equiv-lemma-diagonal-resolution", "contents": [ "Let $R$ be a Noetherian ring. Let $X$ be a separated finite type scheme", "over $R$ which has the resolution property. Set", "$\\mathcal{O}_\\Delta = \\Delta_*(\\mathcal{O}_X)$ where", "$\\Delta : X \\to X \\times_R X$ is the diagonal of $X/k$.", "There exists a resolution", "$$", "\\ldots \\to", "\\mathcal{E}_2 \\boxtimes \\mathcal{G}_2 \\to", "\\mathcal{E}_1 \\boxtimes \\mathcal{G}_1 \\to", "\\mathcal{E}_0 \\boxtimes \\mathcal{G}_0 \\to", "\\mathcal{O}_\\Delta \\to 0", "$$", "where each $\\mathcal{E}_i$ and $\\mathcal{G}_i$ is a finite locally", "free $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is separated, the diagonal morphism $\\Delta$ is a closed", "immersion and hence $\\mathcal{O}_\\Delta$ is a coherent", "$\\mathcal{O}_{X \\times_R X}$-module (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-i-star-equivalence}).", "Thus the lemma follows immediately from Lemma \\ref{lemma-on-product}." ], "refs": [ "coherent-lemma-i-star-equivalence", "equiv-lemma-on-product" ], "ref_ids": [ 3315, 6822 ] } ], "ref_ids": [] }, { "id": 6825, "type": "theorem", "label": "equiv-lemma-Ext-0-regular", "categories": [ "equiv" ], "title": "equiv-lemma-Ext-0-regular", "contents": [ "Let $X$ be a regular Noetherian scheme of dimension $d < \\infty$. Then", "\\begin{enumerate}", "\\item for $\\mathcal{F}$, $\\mathcal{G}$ coherent $\\mathcal{O}_X$-modules", "we have $\\Ext^n_X(\\mathcal{F}, \\mathcal{G}) = 0$ for $n > d$, and", "\\item for $K, L \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ and $a \\in \\mathbf{Z}$", "if $H^i(K) = 0$ for $i < a + d$ and $H^i(L) = 0$ for $i \\geq a$ then", "$\\Hom_X(K, L) = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove (1) we use the spectral sequence", "$$", "H^p(X, \\SheafExt^q(\\mathcal{F}, \\mathcal{G})) \\Rightarrow", "\\Ext^{p + q}_X(\\mathcal{F}, \\mathcal{G})", "$$", "of Cohomology, Section \\ref{cohomology-section-ext}. Recall that", "taking $\\SheafExt$ of coherent modules agrees with the algebra", "construction affine locally, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-internal-hom}.", "Let $x \\in X$ be a point. A regular local ring is Gorenstein, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-regular-gorenstein}.", "Hence $\\mathcal{O}_{X, x}[d_x]$ where $d_x = \\dim(\\mathcal{O}_{X, x})$", "is a normalized dualizing complex for $\\mathcal{O}_{X, x}$. It follows that", "the modules $\\SheafExt^q(\\mathcal{F}, \\mathcal{G})$ have support", "of dimension at most $d - q$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-sitting-in-degrees}.", "Hence we have", "$H^p(X, \\SheafExt^q(\\mathcal{F}, \\mathcal{G})) = 0$ for $p > d - q$", "by Cohomology, Proposition \\ref{cohomology-proposition-vanishing-Noetherian}.", "This proves (1).", "\\medskip\\noindent", "Proof of (2).", "We may use induction on the number of nonzero cohomology sheaves", "of $K$ and $L$. The case where these numbers are $0, 1$ follows", "from (1). If the number of nonzero cohomology sheaves of $K$", "is $> 1$, then we let $i \\in \\mathbf{Z}$ be minimal such that", "$H^i(K)$ is nonzero. We obtain a distinguished triangle", "$$", "H^i(K)[-i] \\to K \\to \\tau_{\\geq i + 1}K", "$$", "(Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle})", "and we get the vanishing of $\\Hom(K, L)$ from the vanishing", "of $\\Hom(H^i(K)[-i], L)$ and $\\Hom(\\tau_{\\geq i + 1}K, L)$", "by Derived Categories, Lemma \\ref{derived-lemma-representable-homological}.", "Simlarly if $L$ has more than one nonzero cohomology sheaf." ], "refs": [ "perfect-lemma-quasi-coherence-internal-hom", "dualizing-lemma-regular-gorenstein", "dualizing-lemma-sitting-in-degrees", "cohomology-proposition-vanishing-Noetherian", "derived-remark-truncation-distinguished-triangle", "derived-lemma-representable-homological" ], "ref_ids": [ 6981, 2880, 2861, 2246, 2016, 1758 ] } ], "ref_ids": [] }, { "id": 6826, "type": "theorem", "label": "equiv-lemma-split-complex-regular", "categories": [ "equiv" ], "title": "equiv-lemma-split-complex-regular", "contents": [ "Let $X$ be a regular Noetherian scheme of dimension $d < \\infty$.", "Let $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ and $a \\in \\mathbf{Z}$.", "If $H^i(K) = 0$ for $a < i < a + d$, then", "$K = \\tau_{\\leq a}K \\oplus \\tau_{\\geq a + d}K$." ], "refs": [], "proofs": [ { "contents": [ "We have $\\tau_{\\leq a}K = \\tau_{\\leq a + d - 1}K$ by the assumed", "vanishing of cohomology sheaves. By Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}", "we have a distinguished triangle", "$$", "\\tau_{\\leq a}K \\to K \\to \\tau_{\\geq a + d}K \\xrightarrow{\\delta}", "(\\tau_{\\leq a}K)[1]", "$$", "By Derived Categories, Lemma \\ref{derived-lemma-split} it", "suffices to show that the morphism $\\delta$ is zero.", "This follows from Lemma \\ref{lemma-Ext-0-regular}." ], "refs": [ "derived-remark-truncation-distinguished-triangle", "derived-lemma-split", "equiv-lemma-Ext-0-regular" ], "ref_ids": [ 2016, 1766, 6825 ] } ], "ref_ids": [] }, { "id": 6827, "type": "theorem", "label": "equiv-lemma-diagonal-trick", "categories": [ "equiv" ], "title": "equiv-lemma-diagonal-trick", "contents": [ "Let $k$ be a field. Let $X$ be a quasi-compact separated smooth scheme over $k$.", "There exist finite locally free $\\mathcal{O}_X$-modules", "$\\mathcal{E}$ and $\\mathcal{G}$ such that", "$$", "\\mathcal{O}_\\Delta \\in \\langle \\mathcal{E} \\boxtimes \\mathcal{G} \\rangle", "$$", "in $D(\\mathcal{O}_{X \\times X})$ where the notation is as in", "Derived Categories, Section \\ref{derived-section-generators}." ], "refs": [], "proofs": [ { "contents": [ "Recall that $X$ is regular by", "Varieties, Lemma \\ref{varieties-lemma-smooth-regular}.", "Hence $X$ has the resolution property by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-regular-resolution-property}.", "Hence we may choose a resolution as in Lemma \\ref{lemma-diagonal-resolution}.", "Say $\\dim(X) = d$. Since $X \\times X$ is smooth over $k$ it is regular.", "Hence $X \\times X$ is a regular Noetherian scheme with", "$\\dim(X \\times X) = 2d$. The object", "$$", "K = (\\mathcal{E}_{2d} \\boxtimes \\mathcal{G}_{2d} \\to", "\\ldots \\to", "\\mathcal{E}_0 \\boxtimes \\mathcal{G}_0)", "$$", "of $D_{perf}(\\mathcal{O}_{X \\times X})$", "has cohomology sheaves $\\mathcal{O}_\\Delta$", "in degree $0$ and $\\Ker(\\mathcal{E}_{2d} \\boxtimes \\mathcal{G}_{2d} \\to", "\\mathcal{E}_{2d-1} \\boxtimes \\mathcal{G}_{2d-1})$ in degree $-2d$ and zero", "in all other degrees.", "Hence by Lemma \\ref{lemma-split-complex-regular} we see that", "$\\mathcal{O}_\\Delta$ is a summand of $K$ in", "$D_{perf}(\\mathcal{O}_{X \\times X})$.", "Clearly, the object $K$ is in", "$$", "\\left\\langle", "\\bigoplus\\nolimits_{i = 0, \\ldots, 2d} \\mathcal{E}_i \\boxtimes \\mathcal{G}_i", "\\right\\rangle", "\\subset", "\\left\\langle", "\\left(\\bigoplus\\nolimits_{i = 0, \\ldots, 2d} \\mathcal{E}_i\\right)", "\\boxtimes", "\\left(\\bigoplus\\nolimits_{i = 0, \\ldots, 2d} \\mathcal{G}_i\\right)", "\\right\\rangle", "$$", "which finishes the proof. (The reader may consult", "Derived Categories, Lemmas \\ref{derived-lemma-generated-by-E-explicit} and", "\\ref{derived-lemma-in-cone-n} to see that our object is contained in this", "category.)" ], "refs": [ "varieties-lemma-smooth-regular", "perfect-lemma-regular-resolution-property", "equiv-lemma-diagonal-resolution", "equiv-lemma-split-complex-regular", "derived-lemma-generated-by-E-explicit", "derived-lemma-in-cone-n" ], "ref_ids": [ 11004, 7090, 6824, 6826, 1935, 1933 ] } ], "ref_ids": [] }, { "id": 6828, "type": "theorem", "label": "equiv-lemma-smooth-proper-strong-generator", "categories": [ "equiv" ], "title": "equiv-lemma-smooth-proper-strong-generator", "contents": [ "Let $k$ be a field. Let $X$ be a scheme proper and smooth over $k$.", "Then $D_{perf}(\\mathcal{O}_X)$", "has a strong generator." ], "refs": [], "proofs": [ { "contents": [ "Using Lemma \\ref{lemma-diagonal-trick} choose finite locally free", "$\\mathcal{O}_X$-modules $\\mathcal{E}$ and $\\mathcal{G}$ such that", "$\\mathcal{O}_\\Delta \\in \\langle \\mathcal{E} \\boxtimes \\mathcal{G} \\rangle$", "in $D(\\mathcal{O}_{X \\times X})$. We claim that $\\mathcal{G}$", "is a strong generator for $D_{perf}(\\mathcal{O}_X)$. With notation as in", "Derived Categories, Section \\ref{derived-section-operate-on-full}", "choose $m, n \\geq 1$ such that", "$$", "\\mathcal{O}_\\Delta \\in", "smd(add(\\mathcal{E} \\boxtimes \\mathcal{G}[-m, m])^{\\star n})", "$$", "This is possible by Derived Categories, Lemma", "\\ref{derived-lemma-find-smallest-containing-E}.", "Let $K$ be an object of $D_{perf}(\\mathcal{O}_X)$. Since", "$L\\text{pr}_1^*K \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L} -$", "is an exact functor and since", "$$", "L\\text{pr}_1^*K \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L}", "(\\mathcal{E} \\boxtimes \\mathcal{G}) =", "(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{E}) \\boxtimes \\mathcal{G}", "$$", "we conclude from", "Derived Categories, Remark \\ref{derived-remark-operations-functor} that", "$$", "L\\text{pr}_1^*K", "\\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L}", "\\mathcal{O}_\\Delta", "\\in", "smd(add(", "(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{E})", "\\boxtimes \\mathcal{G}[-m, m])^{\\star n})", "$$", "Applying the exact functor $R\\text{pr}_{2, *}$ and observing that", "$$", "R\\text{pr}_{2, *}", "\\left((K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{E}) \\boxtimes", "\\mathcal{G}\\right) =", "R\\Gamma(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{E})", "\\otimes_k \\mathcal{G}", "$$", "by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-cohomology-base-change} we conclude that", "$$", "K = R\\text{pr}_{2, *}(L\\text{pr}_1^*K", "\\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L} \\mathcal{O}_\\Delta)", "\\in", "smd(add(R\\Gamma(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{E})", "\\otimes_k \\mathcal{G}[-m, m])^{\\star n})", "$$", "The equality follows from the discussion in", "Example \\ref{example-diagonal-fourier-mukai}.", "Since $K$ is perfect, there exist $a \\leq b$ such that", "$H^i(X, K)$ is nonzero only for $i \\in [a, b]$. Since $X$ is proper,", "each $H^i(X, K)$ is finite dimensional. We conclude that", "the right hand side is contained in", "$smd(add(\\mathcal{G}[-m + a, m + b])^{\\star n})$ which is", "itself contained in $\\langle \\mathcal{G} \\rangle_n$ by one of the", "references given above. This finishes the proof." ], "refs": [ "equiv-lemma-diagonal-trick", "derived-lemma-find-smallest-containing-E", "derived-remark-operations-functor", "perfect-lemma-cohomology-base-change" ], "ref_ids": [ 6827, 1936, 2028, 7025 ] } ], "ref_ids": [] }, { "id": 6829, "type": "theorem", "label": "equiv-lemma-diagonal-trick-proper", "categories": [ "equiv" ], "title": "equiv-lemma-diagonal-trick-proper", "contents": [ "Let $k$ be a field. Let $X$ be a proper smooth scheme over $k$.", "There exists integers $m, n \\geq 1$ and a finite locally free", "$\\mathcal{O}_X$-module $\\mathcal{G}$ such that every coherent", "$\\mathcal{O}_X$-module is contained in $smd(add(\\mathcal{G}[-m, m])^{\\star n})$", "with notation as in Derived Categories, Section", "\\ref{derived-section-operate-on-full}." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-smooth-proper-strong-generator}", "we have shown that there exist $m', n \\geq 1$ such that for any", "coherent $\\mathcal{O}_X$-module $\\mathcal{F}$,", "$$", "\\mathcal{F} \\in smd(add(\\mathcal{G}[-m' + a, m' + b])^{\\star n})", "$$", "for any $a \\leq b$ such that $H^i(X, \\mathcal{F})$ is nonzero only", "for $i \\in [a, b]$. Thus we can take $a = 0$ and $b = \\dim(X)$.", "Taking $m = \\max(m', m' + b)$ finishes the proof." ], "refs": [ "equiv-lemma-smooth-proper-strong-generator" ], "ref_ids": [ 6828 ] } ], "ref_ids": [] }, { "id": 6830, "type": "theorem", "label": "equiv-lemma-boundedness", "categories": [ "equiv" ], "title": "equiv-lemma-boundedness", "contents": [ "Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$.", "Let $\\mathcal{A}$ be an abelian category. Let", "$H : D_{perf}(\\mathcal{O}_X) \\to \\mathcal{A}$ be a homological", "functor (Derived Categories, Definition \\ref{derived-definition-homological})", "such that for all $K$ in $D_{perf}(\\mathcal{O}_X)$ the object", "$H^i(K)$ is nonzero for only a finite number of $i \\in \\mathbf{Z}$.", "Then there exists an integer $m \\geq 1$ such that", "$H^i(\\mathcal{F}) = 0$ for any coherent $\\mathcal{O}_X$-module", "$\\mathcal{F}$ and $i \\not \\in [-m, m]$.", "Similarly for cohomological functors." ], "refs": [ "derived-definition-homological" ], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-diagonal-trick-proper} with", "Derived Categories, Lemma \\ref{derived-lemma-forward-cone-n}." ], "refs": [ "equiv-lemma-diagonal-trick-proper", "derived-lemma-forward-cone-n" ], "ref_ids": [ 6829, 1934 ] } ], "ref_ids": [ 1971 ] }, { "id": 6831, "type": "theorem", "label": "equiv-lemma-bounded-fibres", "categories": [ "equiv" ], "title": "equiv-lemma-bounded-fibres", "contents": [ "Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$.", "Let $K_0 \\to K_1 \\to K_2 \\to \\ldots$ be a system of objects", "of $D_{perf}(\\mathcal{O}_{X \\times Y})$ and $m \\geq 0$ an integer such that", "\\begin{enumerate}", "\\item $H^q(K_i)$ is nonzero only for $q \\leq m$,", "\\item for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ with", "$\\dim(\\text{Supp}(\\mathcal{F})) = 0$ the object", "$$", "R\\text{pr}_{2, *}(", "\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}}^\\mathbf{L}", "K_n)", "$$", "has vanshing cohomology sheaves in degrees outside", "$[-m, m] \\cup [-m - n, m - n]$ and for $n > 2m$ the transition maps", "induce isomorphisms on cohomology sheaves in degrees in $[-m, m]$.", "\\end{enumerate}", "Then $K_n$ has vanshing cohomology sheaves in degrees outside", "$[-m, m] \\cup [-m - n, m - n]$ and for $n > 2m$ the", "transition maps induce isomorphisms on cohomology sheaves in degrees in", "$[-m, m]$. Moreover, if $X$ and $Y$ are smooth over $k$, then for $n$", "large enough we find $K_n = K \\oplus C_n$ in", "$D_{perf}(\\mathcal{O}_{X \\times Y})$", "where $K$ has cohomology only indegrees $[-m, m]$ and $C_n$ only in", "degrees $[-m - n, m - n]$ and the transition maps", "define isomorphisms between various copies of $K$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z$ be the scheme theoretic support of an $\\mathcal{F}$ as in (2).", "Then $Z \\to \\Spec(k)$ is finite, hence $Z \\times Y \\to Y$ is finite.", "It follows that for an object $M$ of $D_\\QCoh(\\mathcal{O}_{X \\times Y})$", "with cohomology sheaves supported on $Z \\times Y$ we have", "$H^i(R\\text{pr}_{2, *}(M)) = \\text{pr}_{2, *}H^i(M)$ and the functor", "$\\text{pr}_{2, *}$ is faithful on quasi-coherent modules supported", "on $Z \\times Y$; details omitted. Hence we see that the objects", "$$", "\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}}^\\mathbf{L} K_n", "$$", "in $D_{perf}(\\mathcal{O}_{X \\times Y})$ have vanshing cohomology sheaves", "outside $[-m, m] \\cup [-m - n, m - n]$ and for $n > 2m$ the transition maps", "induce isomorphisms on cohomology sheaves in $[-m, m]$.", "Let $z \\in X \\times Y$ be a closed point mapping to the closed point", "$x \\in X$. Then we know that", "$$", "K_{n, z} \\otimes_{\\mathcal{O}_{X \\times Y, z}}^\\mathbf{L}", "\\mathcal{O}_{X \\times Y, z}/\\mathfrak m_x^t\\mathcal{O}_{X \\times Y, z}", "$$", "has nonzero cohomology only in the intervals", "$[-m, m] \\cup [-m - n, m - n]$.", "We conclude by More on Algebra, Lemma", "\\ref{more-algebra-lemma-kollar-kovacs-pseudo-coherent}", "that $K_{n, z}$ only has nonzero cohomology", "in degrees $[-m, m] \\cup [-m - n, m - n]$. Since this holds for all", "closed points of $X \\times Y$, we conclude $K_n$ only has nonzero", "cohomology sheaves in degrees $[-m, m] \\cup [-m - n, m - n]$.", "In exactly the same way we see that the maps $K_n \\to K_{n + 1}$", "are isomorphisms on cohomology sheaves in degrees $[-m, m]$", "for $n > 2m$.", "\\medskip\\noindent", "If $X$ and $Y$ are smooth over $k$, then $X \\times Y$ is smooth", "over $k$ and hence regular by", "Varieties, Lemma \\ref{varieties-lemma-smooth-regular}.", "Thus we will obtain the direct sum decomposition of $K_n$", "as soon as $n > 2m + \\dim(X \\times Y)$ from", "Lemma \\ref{lemma-split-complex-regular}. The final statement", "is clear from this." ], "refs": [ "more-algebra-lemma-kollar-kovacs-pseudo-coherent", "varieties-lemma-smooth-regular", "equiv-lemma-split-complex-regular" ], "ref_ids": [ 10420, 11004, 6826 ] } ], "ref_ids": [] }, { "id": 6832, "type": "theorem", "label": "equiv-lemma-functor-quasi-coherent-from-affine", "categories": [ "equiv" ], "title": "equiv-lemma-functor-quasi-coherent-from-affine", "contents": [ "Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$ with $X$ affine.", "There is an equivalence of categories between", "\\begin{enumerate}", "\\item the category of $R$-linear functors", "$F : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$", "which are right exact and commute with arbitrary direct sums, and", "\\item the category $\\QCoh(\\mathcal{O}_{X \\times_R Y})$", "\\end{enumerate}", "given by sending $\\mathcal{K}$ to the functor $F$ in (\\ref{equation-FM-QCoh})." ], "refs": [], "proofs": [ { "contents": [ "First we observe that since $\\text{pr}_2 : X \\times_R Y Y$ is affine", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-affine}) the functor", "$\\text{pr}_{2, *}$ is exact (see for example Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-relative-affine-vanishing}). Hence the functor", "(\\ref{equation-FM-QCoh}) is right exact in this case.", "\\medskip\\noindent", "Let us construct the quasi-inverse to the construction. Let $F$ be", "as in (1). Say $X = \\Spec(A)$. Consider the quasi-coherent", "$\\mathcal{O}_Y$-module $\\mathcal{G} = F(\\mathcal{O}_X)$.", "Every element $a \\in A$ induces an endomorphism of $\\mathcal{G}$", "and this defines an $R$-linear map", "$A \\to \\text{End}_{\\mathcal{O}_Y}(\\mathcal{G})$. Hence we see that", "$\\mathcal{G}$ is a sheaf of modules over", "$$", "A \\otimes_R \\mathcal{O}_Y = \\text{pr}_{2, *}\\mathcal{O}_{X \\times_R Y}", "$$", "By Morphisms, Lemma \\ref{morphisms-lemma-affine-equivalence-modules}", "we find that there is a unique", "quasi-coherent module $\\mathcal{K}$ on $X \\times_R Y$ such that", "$\\mathcal{G} = \\text{pr}_{2, *}\\mathcal{K}$ compatible with action", "of $A$ and $\\mathcal{O}_Y$. Commutation with direct sums shows that", "$F(\\bigoplus_{i \\in I} \\mathcal{O}_X) = \\bigoplus_{i \\in I} \\mathcal{G}$.", "Finally, since $X = \\Spec(A)$ for every quasi-coherent $\\mathcal{O}_X$-module", "$\\mathcal{F}$ we can choose an exact sequence", "$$", "\\bigoplus\\nolimits_{j \\in J} \\mathcal{O}_X \\to", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{O}_X \\to \\mathcal{F} \\to 0", "$$", "This leads to an exact sequence", "$$", "\\bigoplus\\nolimits_{j \\in J} \\mathcal{K} \\to", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{K} \\to", "\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_R Y}} \\mathcal{K} \\to 0", "$$", "which using the exact functor $\\text{pr}_{2, *}$ gives the exact sequence", "$$", "\\bigoplus\\nolimits_{j \\in J} \\mathcal{G} \\to", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{G} \\to", "\\text{pr}_{2, *}(\\text{pr}_1^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X \\times_R Y}} \\mathcal{K})", "\\to 0", "$$", "which as $F$ commutes with direct sums we may rewrite as", "$$", "F(\\bigoplus\\nolimits_{j \\in J} \\mathcal{O}_X) \\to", "F(\\bigoplus\\nolimits_{i \\in I} \\mathcal{O}_X) \\to", "\\text{pr}_{2, *}(\\text{pr}_1^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X \\times_R Y}} \\mathcal{K})", "\\to 0", "$$", "By right exactness of $F$ we conclude $F$ is isomorphic to", "the functor (\\ref{equation-FM-QCoh})." ], "refs": [ "morphisms-lemma-base-change-affine", "coherent-lemma-relative-affine-vanishing", "morphisms-lemma-affine-equivalence-modules" ], "ref_ids": [ 5176, 3283, 5174 ] } ], "ref_ids": [] }, { "id": 6833, "type": "theorem", "label": "equiv-lemma-functor-quasi-coherent-from-affine-compose", "categories": [ "equiv" ], "title": "equiv-lemma-functor-quasi-coherent-from-affine-compose", "contents": [ "In Lemma \\ref{lemma-functor-quasi-coherent-from-affine} let $F$", "correspond to $\\mathcal{K}$ in $\\QCoh(\\mathcal{O}_{X \\times_R Y})$.", "We have", "\\begin{enumerate}", "\\item If $f : X' \\to X$ is an affine morphism, then $F \\circ f_*$", "corresponds to $(f \\times \\text{id}_Y)^*\\mathcal{K}$.", "\\item If $g : Y' \\to Y$ is a quasi-compact and quasi-separated flat", "morphism, then $g^* \\circ F$ corresponds to", "$(\\text{id}_X \\times g)^*\\mathcal{K}$.", "\\item If $j : V \\to Y$ is an open immersion, then $j^* \\circ F$", "corresponds to $\\mathcal{K}|_{X \\times_R V}$.", "\\end{enumerate}" ], "refs": [ "equiv-lemma-functor-quasi-coherent-from-affine" ], "proofs": [ { "contents": [ "For part (1) let $\\mathcal{F}'$ be a quasi-coherent module on $X'$.", "With obvious notation we have", "\\begin{align*}", "\\text{pr}_{2, *}(\\text{pr}_1^*f_*\\mathcal{F}'", "\\otimes_{\\mathcal{O}_{X \\times_R Y}} \\mathcal{K})", "& =", "\\text{pr}_{2, *}((f \\times \\text{id}_Y)_*", "(\\text{pr}'_1)^*\\mathcal{F}'", "\\otimes_{\\mathcal{O}_{X \\times_R Y}} \\mathcal{K}) \\\\", "& =", "\\text{pr}_{2, *}(f \\times \\text{id}_Y)_*", "\\left((\\text{pr}'_1)^*\\mathcal{F}'", "\\otimes_{\\mathcal{O}_{X' \\times_R Y}}", "(f \\times \\text{id}_Y)^*\\mathcal{K})\\right) \\\\", "& =", "\\text{pr}'_{2, *}((\\text{pr}'_1)^*\\mathcal{F}'", "\\otimes_{\\mathcal{O}_{X' \\times_R Y}} (f \\times \\text{id}_Y)^*\\mathcal{K})", "\\end{align*}", "Here the first equality is affine base change, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-affine-base-change}.", "The second equality hold by Remark \\ref{remark-affine-morphism}.", "The third equality is functoriality of pushforwards for modules.", "For part (2) we have", "$$", "g^*\\text{pr}_{2, *}(\\text{pr}_1^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X \\times_R Y}} \\mathcal{K}) =", "\\text{pr}'_{2, *}((\\text{pr}'_1)^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X \\times_R Y'}}", "(\\text{id}_X \\times g)^*\\mathcal{K})", "$$", "by flat base change, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}.", "For part (3) we only have to remark that formation of", "$\\text{pr}_2$ commutes with localization on the target." ], "refs": [ "coherent-lemma-affine-base-change", "equiv-remark-affine-morphism", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 3297, 6880, 3298 ] } ], "ref_ids": [ 6832 ] }, { "id": 6834, "type": "theorem", "label": "equiv-lemma-coh-noetherian-from-affine-flat", "categories": [ "equiv" ], "title": "equiv-lemma-coh-noetherian-from-affine-flat", "contents": [ "In Lemma \\ref{lemma-functor-quasi-coherent-from-affine}", "if $F$ is an exact functor, then the corresponding object", "$\\mathcal{K}$ of $\\QCoh(\\mathcal{O}_{X \\times_R Y})$ is flat over $X$." ], "refs": [ "equiv-lemma-functor-quasi-coherent-from-affine" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-functor-quasi-coherent-from-affine-compose}", "we may assume $Y$ is affine. In this case we can translate the statement", "into algebra as follows: Given a ring $R$ and $R$-algebras $A$, $B$", "for an $A \\otimes_R B$-module $K$ the functor", "$\\text{Mod}_A \\to \\text{Mod}_B, M \\mapsto M \\otimes_A K$", "is exact if and only if $K$ is flat as an $A$-module.", "This is obvious." ], "refs": [ "equiv-lemma-functor-quasi-coherent-from-affine-compose" ], "ref_ids": [ 6833 ] } ], "ref_ids": [ 6832 ] }, { "id": 6835, "type": "theorem", "label": "equiv-lemma-functor-quasi-coherent-from-affine-diagonal", "categories": [ "equiv" ], "title": "equiv-lemma-functor-quasi-coherent-from-affine-diagonal", "contents": [ "Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$. Assume $X$ is", "quasi-compact and that the diagonal morphism of $X$ is affine.", "There is an equivalence of categories between", "\\begin{enumerate}", "\\item the category of $R$-linear exact functors", "$F : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$", "which commute with arbitrary direct sums, and", "\\item the full subcategory of $\\QCoh(\\mathcal{O}_{X \\times_R Y})$ consisting", "of $\\mathcal{K}$ such that", "\\begin{enumerate}", "\\item $\\mathcal{K}$ is flat over $X$,", "\\item for $\\mathcal{F} \\in \\QCoh(\\mathcal{O}_X)$ we have", "$R^q\\text{pr}_{2, *}(\\text{pr}_1^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X \\times_R Y}} \\mathcal{K}) = 0$ for $q > 0$.", "\\end{enumerate}", "\\end{enumerate}", "given by sending $\\mathcal{K}$ to the functor $F$ in (\\ref{equation-FM-QCoh})." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{K}$ be as in (2). The functor $F$ in", "(\\ref{equation-FM-QCoh}) commutes with direct sums.", "Since by (1) (a) the modules $\\mathcal{K}$ is $X$-flat,", "we see that given a short exact", "sequence $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "we obtain a short exact sequence", "$$", "0 \\to", "\\text{pr}_1^*\\mathcal{F}_1 \\otimes_{\\mathcal{O}_{X \\times_R Y}} \\mathcal{K} \\to", "\\text{pr}_1^*\\mathcal{F}_2 \\otimes_{\\mathcal{O}_{X \\times_R Y}} \\mathcal{K} \\to", "\\text{pr}_1^*\\mathcal{F}_3 \\otimes_{\\mathcal{O}_{X \\times_R Y}} \\mathcal{K} \\to", "0", "$$", "Since by (2)(b) the higher direct image $R^1\\text{pr}_{2, *}$", "on the first term is zero, we conclude that", "$0 \\to F(\\mathcal{F}_1) \\to F(\\mathcal{F}_2) \\to F(\\mathcal{F}_3) \\to 0$", "and we see that $F$ is as in (1).", "\\medskip\\noindent", "Let us construct the quasi-inverse to the construction. Let $F$ be", "as in (1). Choose an affine open covering $X = \\bigcup_{i = 1, \\ldots, n} U_i$.", "Since the diagonal of $X$ is affine, we see that the intersections", "$U_{i_0 \\ldots i_p} = U_{i_0} \\cap \\ldots \\cap U_{i_p}$ are affine", "and that the inclusion morphisms", "$j_{i_0 \\ldots i_p} : U_{i_0 \\ldots i_p} \\to X$", "are affine. See Morphisms, Lemma \\ref{morphisms-lemma-affine-permanence}.", "In particular, the composition", "$$", "\\QCoh(\\mathcal{O}_{U_{i_0 \\ldots i_p}})", "\\xrightarrow{j_{i_0 \\ldots i_p *}}", "\\QCoh(\\mathcal{O}_X) \\xrightarrow{F}", "\\QCoh(\\mathcal{O}_Y)", "$$", "is an exact functor commuting with direct sums as a composition of such", "functors. By Lemmas \\ref{lemma-functor-quasi-coherent-from-affine} and", "\\ref{lemma-coh-noetherian-from-affine-flat}", "this functor is given by a quasi-coherent module", "$\\mathcal{K}_{i_0 \\ldots i_p}$ on $U_{i_0 \\ldots i_p} \\times_R Y$", "flat over $U_{i_0 \\ldots i_p}$. Since", "$$", "\\QCoh(\\mathcal{O}_{U_{i_0 \\ldots i_p i_{p + 1}}})", "\\xrightarrow{(U_{i_0 \\ldots i_p i_{p + 1}} \\to U_{i_0 \\ldots i_p})_*}", "\\QCoh(\\mathcal{O}_{U_{i_0 \\ldots i_p}})", "\\xrightarrow{j_{i_0 \\ldots i_p *}}", "\\QCoh(\\mathcal{O}_X)", "$$", "is equal to $j_{i_0 \\ldots i_p i_{p + 1} *}$ we conclude from", "Lemma \\ref{lemma-functor-quasi-coherent-from-affine-compose}", "and the equivalence of categories of the already used", "Lemma \\ref{lemma-functor-quasi-coherent-from-affine}", "that we obtain identifications", "$$", "\\mathcal{K}_{i_0 \\ldots i_p i_{p + 1}} =", "\\mathcal{K}_{i_0 \\ldots i_p}|_{U_{i_0 \\ldots i_p i_{p + 1}} \\times_R Y}", "$$", "which satisfy the usual compatibilites for glueing.", "In other words, there exists a unique", "$\\mathcal{K} \\in \\QCoh(\\mathcal{O}_{X \\times_R Y})$", "flat over $X$ which restricts to each $\\mathcal{K}_{i_0 \\ldots i_p}$", "on $U_{i_0 \\ldots i_p} \\times_R Y$ compatible with these identifications.", "For every quasi-coherent $\\mathcal{O}_X$-module", "we have the sheafified {\\v C}ech complex", "$$", "0 \\to \\mathcal{F} \\to", "\\bigoplus\\nolimits_{i_0} j_{i_0 *}\\mathcal{F}|_{U_{i_0}} \\to", "\\bigoplus\\nolimits_{i_0i_1} j_{i_0 i_1 *}\\mathcal{F}|_{U_{i_0 i_1}} \\to", "\\ldots", "$$", "which is exact.", "See Cohomology, Lemma \\ref{cohomology-lemma-covering-resolution}.", "Applying the exact functor $F$ we find that $F(\\mathcal{F})$", "maps quasi-isomorphically to the relative {\\v C}ech complex with terms", "$$", "\\bigoplus\\nolimits_{i_0 \\ldots i_p}", "(U_{i_0 \\ldots i_p} \\times_R Y \\to Y)_*(", "\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_R Y}} \\mathcal{K}", ")|_{U_{i_0 \\ldots i_p} \\times_R Y}", "$$", "Since this {\\v C}ech complex computes the pushfoward and", "higher direct images of", "$\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_R Y}} \\mathcal{K}$", "by $\\text{pr}_2$ (by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-separated-case-relative-cech})", "we conclude $F$ and $\\mathcal{K}$ correspond and", "that we have property (2)(b)." ], "refs": [ "morphisms-lemma-affine-permanence", "equiv-lemma-functor-quasi-coherent-from-affine", "equiv-lemma-coh-noetherian-from-affine-flat", "equiv-lemma-functor-quasi-coherent-from-affine-compose", "equiv-lemma-functor-quasi-coherent-from-affine", "cohomology-lemma-covering-resolution", "coherent-lemma-separated-case-relative-cech" ], "ref_ids": [ 5179, 6832, 6834, 6833, 6832, 2097, 3301 ] } ], "ref_ids": [] }, { "id": 6836, "type": "theorem", "label": "equiv-lemma-persistence-exactness", "categories": [ "equiv" ], "title": "equiv-lemma-persistence-exactness", "contents": [ "Let $R$, $X$, $Y$, and $\\mathcal{K}$ be as in", "Lemma \\ref{lemma-functor-quasi-coherent-from-affine-diagonal} part (2).", "Then for any scheme $T$ over $R$ we have", "$$", "R^q\\text{pr}_{13, *}(\\text{pr}_{12}^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{T \\times_R X \\times_R Y}}", "\\text{pr}_{23}^*\\mathcal{K}) = 0", "$$", "for $\\mathcal{F}$ quasi-coherent on $T \\times_R X$ and $q > 0$." ], "refs": [ "equiv-lemma-functor-quasi-coherent-from-affine-diagonal" ], "proofs": [ { "contents": [ "The question is local on $T$ hence we may assume $T$ is affine.", "In this case we can consider the diagram", "$$", "\\xymatrix{", "T \\times_R X \\ar[d] &", "T \\times_R X \\times_R Y \\ar[d] \\ar[l] \\ar[r] &", "T \\times_R Y \\ar[d] \\\\", "X &", "X \\times_R Y \\ar[l] \\ar[r] &", "Y", "}", "$$", "whose vertical arrows are affine. In particular the pushforward along", "$T \\times_R Y \\to Y$ is faithful and exact. Chasing around in the diagram", "using that higher direct images along affine morphisms vanish we see that", "it suffices to prove", "$$", "R^q\\text{pr}_{2, *}(", "\\text{pr}_{23, *}(\\text{pr}_{12}^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{T \\times_R X \\times_R Y}}", "\\text{pr}_{23}^*\\mathcal{K})) =", "R^q\\text{pr}_{2, *}(", "\\text{pr}_{23, *}(\\text{pr}_{12}^*\\mathcal{F})", "\\otimes_{\\mathcal{O}_{X \\times_R Y}}", "\\mathcal{K}))", "$$", "is zero which is true by assumption on $\\mathcal{K}$.", "The equality holds by Remark \\ref{remark-affine-morphism}." ], "refs": [ "equiv-remark-affine-morphism" ], "ref_ids": [ 6880 ] } ], "ref_ids": [ 6835 ] }, { "id": 6837, "type": "theorem", "label": "equiv-lemma-functor-quasi-coherent-from-separated", "categories": [ "equiv" ], "title": "equiv-lemma-functor-quasi-coherent-from-separated", "contents": [ "In Lemma \\ref{lemma-functor-quasi-coherent-from-affine-diagonal}", "let $F$ and $\\mathcal{K}$ correspond. If $X$ is separated and", "flat over $R$, then there is a surjection", "$\\mathcal{O}_X \\boxtimes F(\\mathcal{O}_X) \\to \\mathcal{K}$." ], "refs": [ "equiv-lemma-functor-quasi-coherent-from-affine-diagonal" ], "proofs": [ { "contents": [ "Let $\\Delta : X \\to X \\times_R X$ be the diagonal morphism and", "set $\\mathcal{O}_\\Delta = \\Delta_*\\mathcal{O}_X$.", "Since $\\Delta$ is a closed immersion have a short exact sequence", "$$", "0 \\to \\mathcal{I} \\to ", "\\mathcal{O}_{X \\times_R X} \\to \\mathcal{O}_\\Delta \\to 0", "$$", "Since $\\mathcal{K}$ is flat over $X$, the pullback", "$\\text{pr}_{23}^*\\mathcal{K}$ to $X \\times_R X \\times_R Y$", "is flat over $X \\times_R X$ and we obtain a short exact sequence", "$$", "0 \\to ", "\\text{pr}_{12}^*\\mathcal{I}", "\\otimes", "\\text{pr}_{23}^*\\mathcal{K} \\to", "\\text{pr}_{12}^*\\mathcal{O}_{X \\times_R X}", "\\otimes", "\\text{pr}_{23}^*\\mathcal{K} \\to", "\\text{pr}_{12}^*\\mathcal{O}_\\Delta", "\\otimes", "\\text{pr}_{23}^*\\mathcal{K} \\to 0", "$$", "on $X \\times_R X \\times_R Y$. Thus, by Lemma \\ref{lemma-persistence-exactness}", "we obtain a surjection", "$$", "\\text{pr}_{13, *}(", "\\text{pr}_{12}^*\\mathcal{O}_{X \\times_R X}", "\\otimes", "\\text{pr}_{23}^*\\mathcal{K})", "\\to", "\\text{pr}_{13, *}(", "\\text{pr}_{12}^*\\mathcal{O}_\\Delta", "\\otimes", "\\text{pr}_{23}^*\\mathcal{K})", "$$", "By flat base change (", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology})", "the source of this arrow is equal to $\\mathcal{O}_X \\boxtimes F(\\mathcal{O}_X)$.", "On the other hand the target is equal to", "$$", "\\text{pr}_{13, *}(", "\\text{pr}_{12}^*\\mathcal{O}_\\Delta", "\\otimes", "\\text{pr}_{23}^*\\mathcal{K}) =", "\\text{pr}_{13, *} (\\Delta \\times \\text{id}_Y)_* \\mathcal{K} =", "\\mathcal{K}", "$$", "which finishes the proof. The first equality holds for example by", "Cohomology, Lemma \\ref{cohomology-lemma-projection-formula-closed-immersion}", "and the fact that $\\text{pr}_{12}^*\\mathcal{O}_\\Delta =", "(\\Delta \\times \\text{id}_Y)_*\\mathcal{O}_{X \\times_R Y}$." ], "refs": [ "equiv-lemma-persistence-exactness", "coherent-lemma-flat-base-change-cohomology", "cohomology-lemma-projection-formula-closed-immersion" ], "ref_ids": [ 6836, 3298, 2245 ] } ], "ref_ids": [ 6835 ] }, { "id": 6838, "type": "theorem", "label": "equiv-lemma-functor-coherent", "categories": [ "equiv" ], "title": "equiv-lemma-functor-coherent", "contents": [ "Let $X$ and $Y$ be Noetherian schemes. Let", "$F : \\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(\\mathcal{O}_Y)$", "be a functor. Then $F$ extends uniquely to a functor", "$\\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$", "which commutes with filtered colimits.", "If $F$ is additive, then its extension commutes with arbitrary direct sums.", "If $F$ is exact, left exact, or right exact, so is its extension." ], "refs": [], "proofs": [ { "contents": [ "The existence and uniqueness of the extension is a general fact, see", "Categories, Lemma \\ref{categories-lemma-extend-functor-by-colim}.", "To see that the lemma applies observe that coherent modules", "are of finite presentation", "(Modules, Lemma \\ref{modules-lemma-coherent-finite-presentation}) and hence", "categorically compact objects of $\\textit{Mod}(\\mathcal{O}_X)$ by", "Modules, Lemma \\ref{modules-lemma-finite-presentation-quasi-compact-colimit}.", "Finally, every quasi-coherent module is a filtered colimit", "of coherent ones for example by", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-colimit-finite-type}.", "\\medskip\\noindent", "Assume $F$ is additive. If $\\mathcal{F} = \\bigoplus_{j \\in J} \\mathcal{H}_j$", "with $\\mathcal{H}_j$ quasi-coherent, then", "$\\mathcal{F} = \\colim_{J' \\subset J\\text{ finite}}", "\\bigoplus_{j \\in J'} \\mathcal{H}_j$.", "Denoting the extension of $F$ also by $F$ we obtain", "\\begin{align*}", "F(\\mathcal{F})", "& =", "\\colim_{J' \\subset J\\text{ finite}}", "F(\\bigoplus\\nolimits_{j \\in J'} \\mathcal{H}_j) \\\\", "& =", "\\colim_{J' \\subset J\\text{ finite}}", "\\bigoplus\\nolimits_{j \\in J'} F(\\mathcal{H}_j) \\\\", "& =", "\\bigoplus\\nolimits_{j \\in J} F(\\mathcal{H}_j)", "\\end{align*}", "Thus $F$ commutes with arbitrary direct sums.", "\\medskip\\noindent", "Suppose $0 \\to \\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{F}'' \\to 0$", "is a short exact sequence of quasi-coherent $\\mathcal{O}_X$-modules.", "Then we write $\\mathcal{F}' = \\bigcup \\mathcal{F}'_i$ as the", "union of its coherent submodules, see", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-colimit-finite-type}.", "Denote $\\mathcal{F}''_i \\subset \\mathcal{F}''$ the image of $\\mathcal{F}'_i$", "and denote $\\mathcal{F}_i = \\mathcal{F} \\cap \\mathcal{F}'_i =", "\\Ker(\\mathcal{F}'_i \\to \\mathcal{F}''_i)$. Then it is clear that", "$\\mathcal{F} = \\bigcup \\mathcal{F}_i$ and", "$\\mathcal{F}'' = \\bigcup \\mathcal{F}''_i$", "and that we have short exact sequences", "$$", "0 \\to \\mathcal{F}_i \\to \\mathcal{F}_i' \\to \\mathcal{F}_i'' \\to 0", "$$", "Since the extension commutes with filtered colimits we have", "$F(\\mathcal{F}) = \\colim_{i \\in I} F(\\mathcal{F}_i)$,", "$F(\\mathcal{F}') = \\colim_{i \\in I} F(\\mathcal{F}'_i)$, and", "$F(\\mathcal{F}'') = \\colim_{i \\in I} F(\\mathcal{F}''_i)$.", "Since filtered colimits are exact", "(Modules, Lemma \\ref{modules-lemma-limits-colimits}) we", "conclude that exactness properties of $F$ are inherited by", "its extension." ], "refs": [ "categories-lemma-extend-functor-by-colim", "modules-lemma-coherent-finite-presentation", "modules-lemma-finite-presentation-quasi-compact-colimit", "properties-lemma-quasi-coherent-colimit-finite-type", "properties-lemma-quasi-coherent-colimit-finite-type", "modules-lemma-limits-colimits" ], "ref_ids": [ 12254, 13254, 13252, 3020, 3020, 13222 ] } ], "ref_ids": [] }, { "id": 6839, "type": "theorem", "label": "equiv-lemma-characterize-finite", "categories": [ "equiv" ], "title": "equiv-lemma-characterize-finite", "contents": [ "Let $f : V \\to X$ be a quasi-finite separated morphism of Noetherian", "schemes. If there exists a coherent $\\mathcal{O}_V$-module $\\mathcal{K}$", "whose support is $V$ such that $f_*\\mathcal{K}$ is coherent and", "$R^qf_*\\mathcal{K} = 0$, then $f$ is finite." ], "refs": [], "proofs": [ { "contents": [ "By Zariski's main theorem we can find an open immersion", "$j : V \\to Y$ over $X$ with $\\pi : Y \\to X$ finite, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite}.", "Since $\\pi$ is affine the functor $\\pi_*$ is exact and faithful", "on the category of coherent $\\mathcal{O}_X$-modules.", "Hence we see that $j_*\\mathcal{K}$ is coherent and", "that $R^qj_*\\mathcal{K}$ is zero for $q > 0$.", "In other words, we reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $f$ is an open immersion. We may replace $X$ by the", "scheme theoretic closure of $V$. Assume $X \\setminus V$ is nonempty", "to get a contradiction. Choose a generic point $\\xi \\in X \\setminus V$", "of an irreducible component of $X \\setminus V$. Looking at the situation", "after base change by $\\Spec(\\mathcal{O}_{X, \\xi}) \\to X$ using flat base", "change and using", "Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}", "we reduce to the algebra problem discussed in the next paragraph.", "\\medskip\\noindent", "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $M$ be a finite", "$A$-module whose support is $\\Spec(A)$. Then $H^i_\\mathfrak m(A) \\not = 0$", "for some $i$. This is true by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth}", "and the fact that $M$ is not zero hence has finite depth." ], "refs": [ "more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "dualizing-lemma-depth" ], "ref_ids": [ 13901, 9729, 2826 ] } ], "ref_ids": [] }, { "id": 6840, "type": "theorem", "label": "equiv-lemma-functor-coherent-over-field", "categories": [ "equiv" ], "title": "equiv-lemma-functor-coherent-over-field", "contents": [ "Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with", "$X$ separated. There is an equivalence of categories between", "\\begin{enumerate}", "\\item the category of $k$-linear exact functors", "$F : \\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(\\mathcal{O}_Y)$, and", "\\item the category of coherent $\\mathcal{O}_{X \\times Y}$-modules", "$\\mathcal{K}$ which are flat over $X$ and have support finite over $Y$", "\\end{enumerate}", "given by sending $\\mathcal{K}$ to the restriction of the functor", "(\\ref{equation-FM-QCoh}) to $\\textit{Coh}(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{K}$ be as in (2). By", "Lemma \\ref{lemma-functor-quasi-coherent-from-affine-diagonal}", "the functor $F$ given by (\\ref{equation-FM-QCoh}) is exact and $k$-linear.", "Moreover, $F$ sends $\\textit{Coh}(\\mathcal{O}_X)$ into", "$\\textit{Coh}(\\mathcal{O}_Y)$ for example by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-support-proper-over-base-pushforward}.", "\\medskip\\noindent", "Let us construct the quasi-inverse to the construction. Let $F$ be", "as in (1). By Lemma \\ref{lemma-functor-coherent} we can extend $F$", "to a $k$-linear exact functor on the", "categories of quasi-coherent modules which commutes with arbitrary direct sums.", "By Lemma \\ref{lemma-functor-quasi-coherent-from-affine-diagonal}", "the extension corresponds to a unique quasi-coherent module", "$\\mathcal{K}$, flat over $X$, such that", "$R^q\\text{pr}_{2, *}(\\text{pr}_1^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X \\times Y}} \\mathcal{K}) = 0$ for $q > 0$", "for all quasi-coherent $\\mathcal{O}_X$-modules $\\mathcal{F}$.", "Since $F(\\mathcal{O}_X)$ is a coherent $\\mathcal{O}_Y$-module, we", "conclude from Lemma \\ref{lemma-functor-quasi-coherent-from-separated}", "that $\\mathcal{K}$ is coherent.", "\\medskip\\noindent", "For a closed point $x \\in X$ denote $\\mathcal{O}_x$ the skyscraper sheaf", "at $x$ with value the residue field of $x$. We have", "$$", "F(\\mathcal{O}_x) =", "\\text{pr}_{2, *}(\\text{pr}_1^*\\mathcal{O}_x \\otimes \\mathcal{K}) =", "(x \\times Y \\to Y)_*(\\mathcal{K}|_{x \\times Y})", "$$", "Since $x \\times Y \\to Y$ is finite, we see that the pushforward along", "this morphism is faithful. Hence if $y \\in Y$ is in the image of the", "support of $\\mathcal{K}|_{x \\times Y}$, then $y$ is in the support of", "$F(\\mathcal{O}_x)$.", "\\medskip\\noindent", "Let $Z \\subset X \\times Y$ be the scheme theoretic support $Z$ of", "$\\mathcal{K}$, see", "Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-support}.", "We first prove that $Z \\to Y$ is quasi-finite, by proving that its fibres", "over closed points are finite. Namely, if the fibre of $Z \\to Y$ over a", "closed point $y \\in Y$ has dimension $> 0$, then we can find infinitely", "many pairwise distinct closed points $x_1, x_2, \\ldots$ in the image of", "$Z_y \\to X$. Since we have a surjection", "$\\mathcal{O}_X \\to \\bigoplus_{i = 1, \\ldots, n} \\mathcal{O}_{x_i}$", "we obtain a surjection", "$$", "F(\\mathcal{O}_X) \\to \\bigoplus\\nolimits_{i = 1, \\ldots, n} F(\\mathcal{O}_{x_i})", "$$", "By what we said above, the point $y$ is in the support of each", "of the coherent modules $F(\\mathcal{O}_{x_i})$. Since $F(\\mathcal{O}_X)$", "is a coherent module, this will lead to a contradiction because", "the stalk of $F(\\mathcal{O}_X)$ at $y$ will be generated by $< n$ elements", "if $n$ is large enough. Hence $Z \\to Y$ is quasi-finite.", "Since $\\text{pr}_{2, *}\\mathcal{K}$ is coherent and", "$R^q\\text{pr}_{2, *}\\mathcal{K} = 0$ for $q > 0$ we conclude", "that $Z \\to Y$ is finite by Lemma \\ref{lemma-characterize-finite}." ], "refs": [ "equiv-lemma-functor-quasi-coherent-from-affine-diagonal", "coherent-lemma-support-proper-over-base-pushforward", "equiv-lemma-functor-coherent", "equiv-lemma-functor-quasi-coherent-from-affine-diagonal", "equiv-lemma-functor-quasi-coherent-from-separated", "morphisms-definition-scheme-theoretic-support", "equiv-lemma-characterize-finite" ], "ref_ids": [ 6835, 3394, 6838, 6835, 6837, 5538, 6839 ] } ], "ref_ids": [] }, { "id": 6841, "type": "theorem", "label": "equiv-lemma-pushforward-invertible-pre", "categories": [ "equiv" ], "title": "equiv-lemma-pushforward-invertible-pre", "contents": [ "Let $f : X \\to Y$ be a finite type separated morphism of schemes. Let", "$\\mathcal{F}$ be a finite type quasi-coherent module on $X$", "with support finite over $Y$", "and with $\\mathcal{L} = f_*\\mathcal{F}$ an invertible $\\mathcal{O}_X$-module.", "Then there exists a section $s : Y \\to X$ such that", "$\\mathcal{F} \\cong s_*\\mathcal{L}$." ], "refs": [], "proofs": [ { "contents": [ "Looking affine locally this translates into the following algebra problem.", "Let $A \\to B$ be a ring map and let $N$ be a $B$-module which is", "invertible as an $A$-module. Then the annihilator $J$ of $N$ in $B$", "has the property that $A \\to B/J$ is an isomorphism. We omit the details." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6842, "type": "theorem", "label": "equiv-lemma-pushforward-invertible", "categories": [ "equiv" ], "title": "equiv-lemma-pushforward-invertible", "contents": [ "Let $f : X \\to Y$ be a finite type separated morphism of schemes with a section", "$s : Y \\to X$. Let $\\mathcal{F}$ be a finite type quasi-coherent module", "on $X$, set theoretically supported on $s(Y)$ with", "$\\mathcal{L} = f_*\\mathcal{F}$", "an invertible $\\mathcal{O}_X$-module. If $Y$ is reduced, then", "$\\mathcal{F} \\cong s_*\\mathcal{L}$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-pushforward-invertible-pre}", "there exists a section $s' : Y \\to X$ such that", "$\\mathcal{F} = s'_*\\mathcal{L}$. Since $s'(Y)$ and $s(Y)$", "have the same underlying closed subset", "and since both are reduced closed subschemes of $X$, they have to be equal.", "Hence $s = s'$ and the lemma holds." ], "refs": [ "equiv-lemma-pushforward-invertible-pre" ], "ref_ids": [ 6841 ] } ], "ref_ids": [] }, { "id": 6843, "type": "theorem", "label": "equiv-lemma-equivalence-coherent-over-field", "categories": [ "equiv" ], "title": "equiv-lemma-equivalence-coherent-over-field", "contents": [ "\\begin{reference}", "Weak version of the result in \\cite{Gabriel}", "stating that the category of quasi-coherent modules", "determines the isomorphism class of a scheme.", "\\end{reference}", "Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with", "$X$ separated and $Y$ reduced. If there is a $k$-linear equivalence", "$F : \\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(\\mathcal{O}_Y)$", "of categories, then there is an isomorphism $f : Y \\to X$", "over $k$ and an invertible $\\mathcal{O}_Y$-module $\\mathcal{L}$", "such that $F(\\mathcal{F}) = f^*\\mathcal{F} \\otimes \\mathcal{L}$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-functor-coherent-over-field} we obtain a coherent", "$\\mathcal{O}_{X \\times Y}$-module $\\mathcal{K}$ which is flat", "over $X$ with support finite over $Y$ such that $F$ is given by", "the restriction of the functor", "(\\ref{equation-FM-QCoh}) to $\\textit{Coh}(\\mathcal{O}_X)$.", "If we can show that $F(\\mathcal{O}_X)$ is an invertible $\\mathcal{O}_Y$-module,", "then by Lemma \\ref{lemma-pushforward-invertible-pre}", "we see that $\\mathcal{K} = s_*\\mathcal{L}$", "for some section $s : Y \\to X \\times Y$ of $\\text{pr}_2$ and some", "invertible $\\mathcal{O}_Y$-module $\\mathcal{L}$.", "This will show that $F$ has the form indicated with", "$f = \\text{pr}_1 \\circ s$. Some details omitted.", "\\medskip\\noindent", "It remains to show that $F(\\mathcal{O}_X)$ is invertible. We only", "sketch the proof and we omit some of the details.", "For a closed point $x \\in X$ we denote", "$\\mathcal{O}_x$ in $\\textit{Coh}(\\mathcal{O}_X)$", "the skyscraper sheaf at $x$ with value $\\kappa(x)$.", "First we observe that the only simple objects", "of the category $\\textit{Coh}(\\mathcal{O}_X)$", "are these skyscraper sheaves $\\mathcal{O}_x$.", "The same is true for $Y$. Hence for every closed point $y \\in Y$", "there exists a closed point $x \\in X$ such that", "$\\mathcal{O}_y \\cong F(\\mathcal{O}_x)$. Moreover, looking at endomorphisms", "we find that $\\kappa(x) \\cong \\kappa(y)$ as finite extensions of $k$.", "Then", "$$", "\\Hom_Y(F(\\mathcal{O}_X), \\mathcal{O}_y) \\cong", "\\Hom_Y(F(\\mathcal{O}_X), F(\\mathcal{O}_x)) \\cong", "\\Hom_X(\\mathcal{O}_X, \\mathcal{O}_x) \\cong \\kappa(x) \\cong \\kappa(y)", "$$", "This implies that the stalk of the coherent $\\mathcal{O}_Y$-module", "$F(\\mathcal{O}_X)$ at $y \\in Y$ can be generated by $1$ generator", "(and no less) for each closed point $y \\in Y$. It follows immediately", "that $F(\\mathcal{O}_X)$ is locally generated by $1$ element (and no less)", "and since $Y$ is reduced this indeed tells us it is an invertible module." ], "refs": [ "equiv-lemma-functor-coherent-over-field", "equiv-lemma-pushforward-invertible-pre" ], "ref_ids": [ 6840, 6841 ] } ], "ref_ids": [] }, { "id": 6844, "type": "theorem", "label": "equiv-lemma-sibling-fully-faithful", "categories": [ "equiv" ], "title": "equiv-lemma-sibling-fully-faithful", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{D}$ be a", "triangulated category. Let", "$F, F' : D^b(\\mathcal{A}) \\longrightarrow \\mathcal{D}$", "be exact functors of triangulated categories. Assume", "\\begin{enumerate}", "\\item the functors $F \\circ i$ and $F' \\circ i$ are isomorphic", "where $i : \\mathcal{A} \\to D^b(\\mathcal{A})$ is the inclusion functor, and", "\\item for all $X, Y \\in \\Ob(\\mathcal{A})$ we have", "$\\Ext^q_\\mathcal{D}(F(X), F(Y)) = 0$ for $q < 0$ (for example", "if $F$ is fully faithful).", "\\end{enumerate}", "Then $F$ and $F'$ are siblings." ], "refs": [], "proofs": [ { "contents": [ "Let $K \\in D^b(\\mathcal{A})$. We will show $F(K)$ is isomorphic to $F'(K)$.", "We can represent $K$ by a bounded complex $A^\\bullet$ of objects of", "$\\mathcal{A}$. After replacing $K$ by a translation we may", "assume $A^i = 0$ for $i > 0$. Choose $n \\geq 0$ such that $A^{-i} = 0$", "for $i > n$. The objects", "$$", "M_i = (A^{-i} \\to \\ldots \\to A^0)[-i],\\quad i = 0, \\ldots, n", "$$", "form a Postnikov system in $D^b(\\mathcal{A})$ for the complex", "$A^\\bullet = A^{-n} \\to \\ldots \\to A^0$ in $D^b(\\mathcal{A})$.", "See Derived Categories, Example \\ref{derived-example-key-postnikov}.", "Since both $F$ and $F'$ are exact functors of triangulated categories both", "$$", "F(M_i)", "\\quad\\text{and}\\quad", "F'(M_i)", "$$", "form a Postnikov system in $\\mathcal{D}$ for the complex", "$$", "F(A^{-n}) \\to \\ldots \\to F(A^0) =", "F'(A^{-n}) \\to \\ldots \\to F'(A^0)", "$$", "Since all negative $\\Ext$s between these objects vanish by assumption", "we conclude by uniqueness of Postnikov systems", "(Derived Categories, Lemma \\ref{derived-lemma-existence-postnikov-system})", "that $F(K) = F(M_n[n]) \\cong F'(M_n[n]) = F'(K)$." ], "refs": [ "derived-lemma-existence-postnikov-system" ], "ref_ids": [ 1952 ] } ], "ref_ids": [] }, { "id": 6845, "type": "theorem", "label": "equiv-lemma-sibling-faithful", "categories": [ "equiv" ], "title": "equiv-lemma-sibling-faithful", "contents": [ "Let $F$ and $F'$ be siblings as in Definition \\ref{definition-siblings}.", "Then", "\\begin{enumerate}", "\\item if $F$ is essentially surjective, then $F'$ is essentially", "surjective,", "\\item if $F$ is fully faithful, then $F'$ is fully faithful.", "\\end{enumerate}" ], "refs": [ "equiv-definition-siblings" ], "proofs": [ { "contents": [ "Part (1) is immediate from property (2) for siblings.", "\\medskip\\noindent", "Assume $F$ is fully faithful. Denote $\\mathcal{D}' \\subset \\mathcal{D}$", "the essential image of $F$ so that $F : D^b(\\mathcal{A}) \\to \\mathcal{D}'$", "is an equivalence. Since the functor $F'$ factors through $\\mathcal{D}'$", "by property (2) for siblings, we can consider the functor", "$H = F^{-1} \\circ F' : D^b(\\mathcal{A}) \\to D^b(\\mathcal{A})$.", "Observe that $H$ is a sibling of the identity functor.", "Since it suffices to prove that $H$ is fully faithful,", "we reduce to the problem discussed in the next paragraph.", "\\medskip\\noindent", "Set $\\mathcal{D} = D^b(\\mathcal{A})$. We have to show a sibling", "$F : \\mathcal{D} \\to \\mathcal{D}$ of the identity functor is fully faithful.", "Denote $a_X : X \\to F(X)$ the functorial isomorphism for", "$X \\in \\Ob(\\mathcal{A})$ given to us by Definition \\ref{definition-siblings}.", "For any $K$ in $\\mathcal{D}$ and distinguished triangle", "$K_1 \\to K_2 \\to K_3$ of $\\mathcal{D}$", "if the maps", "$$", "F : \\Hom(K, K_i[n]) \\to \\Hom(F(K), F(K_i[n]))", "$$", "are isomorphisms for all $n \\in \\mathbf{Z}$ and $i = 1, 3$, then the", "same is true for $i = 2$ and all $n \\in \\mathbf{Z}$. This uses the", "$5$-lemma Homology, Lemma \\ref{homology-lemma-five-lemma} and", "Derived Categories, Lemma \\ref{derived-lemma-representable-homological};", "details omitted. Similarly, if the maps", "$$", "F : \\Hom(K_i[n], K) \\to \\Hom(F(K_i[n]), F(K))", "$$", "are isomorphisms for all $n \\in \\mathbf{Z}$ and $i = 1, 3$, then the", "same is true for $i = 2$ and all $n \\in \\mathbf{Z}$. Using the canonical", "truncations and induction on the number of nonzero cohomology objects,", "we see that it is enough to show", "$$", "F : \\Ext^q(X, Y) \\to \\Ext^q(F(X), F(Y))", "$$", "is bijective for all $X, Y \\in \\Ob(\\mathcal{A})$ and all $q \\in \\mathbf{Z}$.", "Since $F$ is a sibling of $\\text{id}$ we have $F(X) \\cong X$ and", "$F(Y) \\cong Y$ hence the right hand side is zero for $q < 0$.", "The case $q = 0$ is OK by our assumption that $F$ is a sibling of", "the identity functor. It remains to prove the cases $q > 0$.", "\\medskip\\noindent", "The case $q = 1$: Injectivity. An element $\\xi$ of $\\Ext^1(X, Y)$", "gives rise to a distinguished triangle", "$$", "Y \\to E \\to X \\xrightarrow{\\xi} Y[1]", "$$", "Observe that $E \\in \\Ob(\\mathcal{A})$. Since $F$ is a sibling of the", "identity functor we obtain a commutative diagram", "$$", "\\xymatrix{", "E \\ar[d] \\ar[r] & X \\ar[d] \\\\", "F(E) \\ar[r] & F(X)", "}", "$$", "whose vertical arrows are the isomorphisms $a_E$ and $a_X$.", "By TR3 the distinguished triangle associated to $\\xi$ we started", "with is isomorphic to the distinguished triangle", "$$", "F(Y) \\to F(E) \\to F(X) \\xrightarrow{F(\\xi)} F(Y[1]) = F(Y)[1]", "$$", "Thus $\\xi = 0$ if and only if $F(\\xi)$ is zero, i.e., we see that", "$F : \\Ext^1(X, Y) \\to \\Ext^1(F(X), F(Y))$ is injective.", "\\medskip\\noindent", "The case $q = 1$: Surjectivity. Let $\\theta$ be an element of", "$\\Ext^1(F(X), F(Y))$. This defines an extension of $F(X)$ by $F(Y)$", "in $\\mathcal{A}$ which we may write as $F(E)$", "as $F$ is a sibling of the identity functor. We thus get a distinguished", "triangle", "$$", "F(Y) \\xrightarrow{F(\\alpha)} F(E)", "\\xrightarrow{F(\\beta)} F(X)", "\\xrightarrow{\\theta} F(Y[1]) = F(Y)[1]", "$$", "for some morphisms $\\alpha : Y \\to E$ and $\\beta : E \\to X$.", "Since $F$ is a sibling of the identity functor, the sequence", "$0 \\to Y \\to E \\to X \\to 0$", "is a short exact sequence in $\\mathcal{A}$! Hence we obtain a", "distinguished triangle", "$$", "Y \\xrightarrow{\\alpha} E \\xrightarrow{\\beta} X \\xrightarrow{\\delta} Y[1]", "$$", "for some morphism $\\delta : X \\to Y[1]$. Applying the exact functor", "$F$ we obtain the distinguished triangle", "$$", "F(Y) \\xrightarrow{F(\\alpha)} F(E) \\xrightarrow{F(\\beta)} F(X)", "\\xrightarrow{F(\\delta)} F(Y)[1]", "$$", "Arguing as above, we see that these triangles are isomorphic.", "Hence there exists a commutative diagram", "$$", "\\xymatrix{", "F(X) \\ar[d]^\\gamma \\ar[r]_{F(\\delta)} & F(Y[1]) \\ar[d]_\\epsilon \\\\", "F(X) \\ar[r]^\\theta & F(Y[1])", "}", "$$", "for some isomorphisms $\\gamma$, $\\epsilon$ (we can say more but we won't", "need more information). We may write $\\gamma = F(\\gamma')$ and", "$\\epsilon = F(\\epsilon')$. Then we have", "$\\theta = F(\\epsilon' \\circ \\delta \\circ (\\gamma')^{-1})$", "and we see the surjectivity holds.", "\\medskip\\noindent", "The case $q > 1$: surjectivity. Using Yoneda extensions, see", "Derived Categories, Section \\ref{derived-section-ext}, we find that for any", "element $\\xi$ in $\\Ext^q(F(X), F(Y))$ we can find", "$F(X) = B_0, B_1, \\ldots, B_{q - 1}, B_q = F(Y) \\in \\Ob(\\mathcal{A})$ and", "elements", "$$", "\\xi_i \\in \\Ext^1(B_{i - 1}, B_i)", "$$", "such that $\\xi$ is the composition $\\xi_q \\circ \\ldots \\circ \\xi_1$.", "Write $B_i = F(A_i)$ (of course we have $A_i = B_i$ but we don't", "need to use this) so that", "$$", "\\xi_i = F(\\eta_i) \\in \\Ext^1(F(A_{i - 1}), F(A_i))", "\\quad\\text{with}\\quad", "\\eta_i \\in \\Ext^1(A_{i - 1}, A_i)", "$$", "by surjectivity for $q = 1$. Then $\\eta = \\eta_q \\circ \\ldots \\circ \\eta_1$", "is an element of $\\Ext^q(X, Y)$ with $F(\\eta) = \\xi$.", "\\medskip\\noindent", "The case $q > 1$: injectivity. An element $\\xi$ of $\\Ext^q(X, Y)$", "gives rise to a distinguished triangle", "$$", "Y[q - 1] \\to E \\to X \\xrightarrow{\\xi} Y[q]", "$$", "Applying $F$ we obtain a distinguished triangle", "$$", "F(Y)[q - 1] \\to F(E) \\to F(X) \\xrightarrow{F(\\xi)} F(Y)[q]", "$$", "If $F(\\xi) = 0$, then $F(E) \\cong F(Y)[q - 1] \\oplus F(X)$", "in $\\mathcal{D}$, see", "Derived Categories, Lemma \\ref{derived-lemma-split}.", "Since $F$ is a sibling of the identity functor we have", "$E \\cong F(E)$ and hence", "$$", "E \\cong F(E) \\cong F(Y)[q - 1] \\oplus F(X) \\cong Y[q - 1] \\oplus X", "$$", "In other words, $E$ is isomorphic to the", "direct sum of its cohomology objects. This implies that the", "initial distinguished triangle is split, i.e., $\\xi = 0$." ], "refs": [ "equiv-definition-siblings", "homology-lemma-five-lemma", "derived-lemma-representable-homological", "derived-lemma-split" ], "ref_ids": [ 6876, 12030, 1758, 1766 ] } ], "ref_ids": [ 6876 ] }, { "id": 6846, "type": "theorem", "label": "equiv-lemma-get-fully-faithful", "categories": [ "equiv" ], "title": "equiv-lemma-get-fully-faithful", "contents": [ "\\begin{reference}", "Variant of \\cite[Lemma 2.15]{Orlov-K3}", "\\end{reference}", "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of", "triangulated categories. Let $S \\subset \\Ob(\\mathcal{D})$ be", "a set of objects. Assume", "\\begin{enumerate}", "\\item $F$ has both right and left adjoints,", "\\item for $K \\in \\mathcal{D}$ if $\\Hom(E, K[i]) = 0$ for all", "$E \\in S$ and $i \\in \\mathbf{Z}$ then $K = 0$,", "\\item for $K \\in \\mathcal{D}$ if $\\Hom(K, E[i]) = 0$ for all", "$E \\in S$ and $i \\in \\mathbf{Z}$ then $K = 0$,", "\\item the map $\\Hom(E, E'[i]) \\to \\Hom(F(E), F(E')[i])$ induced by $F$", "is bijective for all $E, E' \\in S$ and $i \\in \\mathbf{Z}$.", "\\end{enumerate}", "Then $F$ is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "Denote $F_r$ and $F_l$ the right and left adjoints of $F$. For", "$E \\in S$ choose a distinguished triangle", "$$", "E \\to F_r(F(E)) \\to C \\to E[1]", "$$", "where the first arrow is the unit of the adjunction. For $E' \\in S$ we have", "$$", "\\Hom(E', F_r(F(E))[i]) = \\Hom(F(E'), F(E)[i]) = \\Hom(E', E[i])", "$$", "The last equality holds by assumption (4).", "Hence applying the homological functor $\\Hom(E', -)$", "(Derived Categories, Lemma \\ref{derived-lemma-representable-homological})", "to the distinguished triangle above we conclude that $\\Hom(E', C[i]) = 0$", "for all $i \\in \\mathbf{Z}$ and $E' \\in S$. By assumption (2) we conclude", "that $C = 0$ and $E = F_r(F(E))$.", "\\medskip\\noindent", "For $K \\in \\Ob(\\mathcal{D})$ choose a distinguished triangle", "$$", "F_l(F(K)) \\to K \\to C \\to F_l(F(K))[1]", "$$", "where the first arrow is the counit of the adjunction. For $E \\in S$", "we have", "$$", "\\Hom(F_l(F(K)), E[i]) = \\Hom(F(K), F(E)[i]) =", "\\Hom(K, F_r(F(E))[i]) = \\Hom(K, E[i])", "$$", "where the last equality holds by the result of the first paragraph.", "Thus we conclude as before that $\\Hom(C, E[i]) = 0$ for all $E \\in S$", "and $i \\in \\mathbf{Z}$. Hence $C = 0$ by assumption (3).", "Thus $F$ is fully faithful by Categories, Lemma", "\\ref{categories-lemma-adjoint-fully-faithful}." ], "refs": [ "derived-lemma-representable-homological", "categories-lemma-adjoint-fully-faithful" ], "ref_ids": [ 1758, 12248 ] } ], "ref_ids": [] }, { "id": 6847, "type": "theorem", "label": "equiv-lemma-duality-at-point", "categories": [ "equiv" ], "title": "equiv-lemma-duality-at-point", "contents": [ "Let $k$ be a field. Let $X$ be a scheme of finite type over $k$ which", "is regular. Let $x \\in X$ be a closed point. For a coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ supported at $x$ choose", "a coherent $\\mathcal{O}_X$-module $\\mathcal{F}'$ supported at $x$", "such that $\\mathcal{F}_x$ and $\\mathcal{F}'_x$ are Matlis dual.", "Then there is an isomorphism", "$$", "\\Hom_X(\\mathcal{F}, M) =", "H^0(X, M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}'[-d_x])", "$$", "where $d_x = \\dim(\\mathcal{O}_{X, x})$", "functorial in $M$ in $D_{perf}(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathcal{F}$ is supported at $x$ we have", "$$", "\\Hom_X(\\mathcal{F}, M) =", "\\Hom_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x, M_x)", "$$", "and similarly we have", "$$", "H^0(X, M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}'[-d_x]) =", "\\text{Tor}^{\\mathcal{O}_{X, x}}_{d_x}(M_x, \\mathcal{F}'_x)", "$$", "Thus it suffices to show that given a Noetherian regular local ring $A$", "of dimension $d$ and a finite length $A$-module $N$, if", "$N'$ is the Matlis dual to $N$, then there exists a functorial isomorphism", "$$", "\\Hom_A(N, K) = \\text{Tor}^A_d(K, N')", "$$", "for $K$ in $D_{perf}(A)$. We can write the left hand side as", "$H^0(R\\Hom_A(N, A) \\otimes_A^\\mathbf{L} K)$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-dual-perfect-complex}", "and the fact that $N$ determines a perfect object of $D(A)$.", "Hence the formula holds because", "$$", "R\\Hom_A(N, A) = R\\Hom_A(N, A[d])[-d] = N'[-d]", "$$", "by Dualizing Complexes, Lemma \\ref{dualizing-lemma-dualizing-finite-length}", "and the fact that $A[d]$ is a normalized dualizing complex over $A$", "($A$ is Gorenstein by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-regular-gorenstein})." ], "refs": [ "more-algebra-lemma-dual-perfect-complex", "dualizing-lemma-dualizing-finite-length", "dualizing-lemma-regular-gorenstein" ], "ref_ids": [ 10224, 2860, 2880 ] } ], "ref_ids": [] }, { "id": 6848, "type": "theorem", "label": "equiv-lemma-orthogonal-point-sheaf", "categories": [ "equiv" ], "title": "equiv-lemma-orthogonal-point-sheaf", "contents": [ "Let $k$ be a field. Let $X$ be a scheme of finite type over $k$ which", "is regular. Let $x \\in X$ be a closed point and denote $\\mathcal{O}_x$", "the skyscraper sheaf at $x$ with value $\\kappa(x)$. Let $K$ in", "$D_{perf}(\\mathcal{O}_X)$.", "\\begin{enumerate}", "\\item If $\\Ext^i_X(\\mathcal{O}_x, K) = 0$ then there exists an open", "neighbourhood $U$ of $x$ such that $H^{i - d_x}(K)|_U = 0$ where", "$d_x = \\dim(\\mathcal{O}_{X, x})$.", "\\item If $\\Hom_X(\\mathcal{O}_x, K[i]) = 0$ for all", "$i \\in \\mathbf{Z}$, then $K$ is zero in an open neighbourhood of $x$.", "\\item If $\\Ext^i_X(K, \\mathcal{O}_x) = 0$ then there exists an open", "neighbourhood $U$ of $x$ such that $H^i(K^\\vee)|_U = 0$.", "\\item If $\\Hom_X(K, \\mathcal{O}_x[i]) = 0$ for all", "$i \\in \\mathbf{Z}$, then $K$ is zero in an open neighbourhood of $x$.", "\\item If $H^i(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_x) = 0$", "then there exists an open neighbourhood $U$ of $x$ such that", "$H^i(K)|_U = 0$.", "\\item If $H^i(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_x) = 0$", "for $i \\in \\mathbf{Z}$ then $K$ is zero in an", "open neighbourhood of $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $H^i(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_x)$", "is equal to $K_x \\otimes_{\\mathcal{O}_{X, x}}^\\mathbf{L} \\kappa(x)$.", "Hence part (5) follows from More on Algebra, Lemma", "\\ref{more-algebra-lemma-cut-complex-in-two}.", "Part (6) follows from part (5).", "Part (1) follows from part (5), Lemma \\ref{lemma-duality-at-point}, and the", "fact that the Matlis dual of $\\kappa(x)$ is $\\kappa(x)$.", "Part (2) follows from part (1).", "Part (3) follows from part (5) and the fact that", "$\\Ext^i(K, \\mathcal{O}_x) =", "H^i(X, K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_x)$ by", "Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "Part (4) follows from part (3) and the fact that $K \\cong (K^\\vee)^\\vee$", "by the lemma just cited." ], "refs": [ "more-algebra-lemma-cut-complex-in-two", "equiv-lemma-duality-at-point", "cohomology-lemma-dual-perfect-complex" ], "ref_ids": [ 10238, 6847, 2233 ] } ], "ref_ids": [] }, { "id": 6849, "type": "theorem", "label": "equiv-lemma-get-fully-faithful-geometric", "categories": [ "equiv" ], "title": "equiv-lemma-get-fully-faithful-geometric", "contents": [ "Let $k$ be a field. Let $X$ and $Y$ be proper schemes over $k$.", "Assume $X$ is regular. Then a $k$-linear exact functor", "$F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$", "is fully faithful if and only if", "for any closed points $x, x' \\in X$ the maps", "$$", "F : \\Ext^i_X(\\mathcal{O}_x, \\mathcal{O}_{x'})", "\\longrightarrow", "\\Ext^i_Y(F(\\mathcal{O}_x), F(\\mathcal{O}_{x'}))", "$$", "are isomorphisms for all $i \\in \\mathbf{Z}$.", "Here $\\mathcal{O}_x$ is the skyscraper sheaf at $x$ with value $\\kappa(x)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-always-right-adjoints} the functor $F$", "has both a left and a right adjoint. Thus we may apply the criterion", "of Lemma \\ref{lemma-get-fully-faithful}", "because assumptions (2) and (3) of that lemma", "follow from Lemma \\ref{lemma-orthogonal-point-sheaf}." ], "refs": [ "equiv-lemma-always-right-adjoints", "equiv-lemma-get-fully-faithful", "equiv-lemma-orthogonal-point-sheaf" ], "ref_ids": [ 6814, 6846, 6848 ] } ], "ref_ids": [] }, { "id": 6850, "type": "theorem", "label": "equiv-lemma-noah-pre", "categories": [ "equiv" ], "title": "equiv-lemma-noah-pre", "contents": [ "\\begin{reference}", "Email from Noah Olander of Jun 9, 2020", "\\end{reference}", "Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$.", "Let $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_X)$", "be a $k$-linear exact functor. Assume for every coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ with $\\dim(\\text{Supp}(\\mathcal{F})) = 0$", "there is an isomorphism $\\mathcal{F} \\cong F(\\mathcal{F})$.", "Then $F$ is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-get-fully-faithful-geometric} it suffices to show", "that the maps", "$$", "F : \\Ext^i_X(\\mathcal{O}_x, \\mathcal{O}_{x'})", "\\longrightarrow", "\\Ext^i_Y(F(\\mathcal{O}_x), F(\\mathcal{O}_{x'}))", "$$", "are isomorphisms for all $i \\in \\mathbf{Z}$ and all closed points", "$x, x' \\in X$. By assumption, the source and the target are isomorphic.", "If $x \\not = x'$, then both sides are zero and the result is true.", "If $x = x'$, then it suffices to prove that the map is either injective", "or surjective. For $i < 0$ both sides are zero and the result is true.", "For $i = 0$ any nonzero map $\\alpha : \\mathcal{O}_x \\to \\mathcal{O}_x$ of", "$\\mathcal{O}_X$-modules is an isomorphism. Hence $F(\\alpha)$ is an", "isomorphism too and so $F(\\alpha)$ is nonzero. Thus the result for $i = 0$.", "For $i = 1$ a nonzero element $\\xi$ in $\\Ext^1(\\mathcal{O}_x, \\mathcal{O}_x)$", "corresponds to a nonsplit short exact sequence", "$$", "0 \\to \\mathcal{O}_x \\to \\mathcal{F} \\to \\mathcal{O}_x \\to 0", "$$", "Since $F(\\mathcal{F}) \\cong \\mathcal{F}$ we see that $F(\\mathcal{F})$", "is a nonsplit extension of $\\mathcal{O}_x$ by $\\mathcal{O}_x$ as well.", "Since $\\mathcal{O}_x \\cong F(\\mathcal{O}_x)$ is a simple", "$\\mathcal{O}_X$-module and $\\mathcal{F} \\cong F(\\mathcal{F})$ has", "length $2$, we see that in the distinguished triangle", "$$", "F(\\mathcal{O}_x) \\to F(\\mathcal{F}) \\to F(\\mathcal{O}_x)", "\\xrightarrow{F(\\xi)} F(\\mathcal{O}_x)[1]", "$$", "the first two arrows must form a short exact sequence which must be", "isomorphic to the above short exact sequence and hence is nonsplit.", "It follows that $F(\\xi)$ is nonzero and we conclude for $i = 1$.", "For $i > 1$ composition of ext classes defines a surjection", "$$", "\\Ext^1(F(\\mathcal{O}_x), F(\\mathcal{O}_x)) \\otimes \\ldots \\otimes", "\\Ext^1(F(\\mathcal{O}_x), F(\\mathcal{O}_x))", "\\longrightarrow", "\\Ext^i(F(\\mathcal{O}_x), F(\\mathcal{O}_x))", "$$", "See Duality for Schemes, Lemma \\ref{duality-lemma-regular-ideal-ext}.", "Hence surjectivity in degree $1$ implies surjectivity for $i > 0$.", "This finishes the proof." ], "refs": [ "equiv-lemma-get-fully-faithful-geometric", "duality-lemma-regular-ideal-ext" ], "ref_ids": [ 6849, 13547 ] } ], "ref_ids": [] }, { "id": 6851, "type": "theorem", "label": "equiv-lemma-exact-functor-preserving-Coh", "categories": [ "equiv" ], "title": "equiv-lemma-exact-functor-preserving-Coh", "contents": [ "Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with", "$X$ separated. Let", "$F : D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$", "be a $k$-linear exact functor sending", "$\\textit{Coh}(\\mathcal{O}_X) \\subset D^b_{\\textit{Coh}}(\\mathcal{O}_X)$", "into", "$\\textit{Coh}(\\mathcal{O}_Y) \\subset D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$.", "Then there exists a Fourier-Mukai functor", "$F' : D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$", "whose kernel is a coherent $\\mathcal{O}_{X \\times Y}$-module $\\mathcal{K}$", "flat over $X$ and with support finite over $Y$ which is a sibling of $F$." ], "refs": [], "proofs": [ { "contents": [ "Denote $H : \\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(\\mathcal{O}_Y)$", "the restriction of $F$. Since $F$ is an exact functor of triangulated", "categories, we see that $H$ is an exact functor of abelian categories.", "Of course $H$ is $k$-linear as $F$ is. By", "Lemma \\ref{lemma-functor-coherent-over-field}", "we obtain a coherent $\\mathcal{O}_{X \\times Y}$-module", "$\\mathcal{K}$ which is flat over $X$ and has support finite over $Y$.", "Let $F'$ be the Fourier-Mukai functor defined using $\\mathcal{K}$", "so that $F'$ restricts to $H$ on $ \\textit{Coh}(\\mathcal{O}_X)$.", "The functor $F'$ sends $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$", "into $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ by", "Lemma \\ref{lemma-fourier-mukai-Coh}.", "Observe that $F$ and $F'$ satisfy the first and second", "condition of Lemma \\ref{lemma-sibling-fully-faithful} and hence are siblings." ], "refs": [ "equiv-lemma-functor-coherent-over-field", "equiv-lemma-fourier-mukai-Coh", "equiv-lemma-sibling-fully-faithful" ], "ref_ids": [ 6840, 6818, 6844 ] } ], "ref_ids": [] }, { "id": 6852, "type": "theorem", "label": "equiv-lemma-preserves-Coh", "categories": [ "equiv" ], "title": "equiv-lemma-preserves-Coh", "contents": [ "Let $k$ be a field. Let $X$ be a separated scheme of finite type over $k$ which", "is regular. Let $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_X)$", "be a $k$-linear exact functor. Assume for every coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ with $\\dim(\\text{Supp}(\\mathcal{F})) = 0$", "there is an isomorphism of $k$-vector spaces", "$$", "\\Hom_X(\\mathcal{F}, M) = \\Hom_X(\\mathcal{F}, F(M))", "$$", "functorial in $M$ in $D_{perf}(\\mathcal{O}_X)$. Then there exists an", "automorphism $f : X \\to X$ over $k$ which induces the identity on the", "underlying topological space\\footnote{This often forces $f$", "to be the identity, see Lemma \\ref{lemma-automorphism}.} and an", "invertible $\\mathcal{O}_X$-module $\\mathcal{L}$", "such that $F$ and $F'(M) = f^*M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{L}$", "are siblings." ], "refs": [ "equiv-lemma-automorphism" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-duality-at-point} we conclude that for every", "coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ whose support is a", "closed point there are isomorphisms", "$$", "H^0(X, M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{F}) =", "H^0(X, F(M) \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{F})", "$$", "functorial in $M$.", "\\medskip\\noindent", "Let $x \\in X$ be a closed point and apply the above with", "$\\mathcal{F} = \\mathcal{O}_x$ the skyscraper sheaf with value", "$\\kappa(x)$ at $x$. We find", "$$", "\\dim_{\\kappa(x)} ", "\\text{Tor}^{\\mathcal{O}_{X, x}}_p(M_x, \\kappa(x)) =", "\\dim_{\\kappa(x)} ", "\\text{Tor}^{\\mathcal{O}_{X, x}}_p(F(M)_x, \\kappa(x))", "$$", "for all $p \\in \\mathbf{Z}$. In particular, if", "$H^i(M) = 0$ for $i > 0$, then $H^i(F(M)) = 0$ for $i > 0$", "by Lemma \\ref{lemma-orthogonal-point-sheaf}.", "\\medskip\\noindent", "If $\\mathcal{E}$ is locally free of rank $r$, then", "$F(\\mathcal{E})$ is locally free of rank $r$. This is", "true because a perfect complex $K$ over $\\mathcal{O}_{X, x}$", "with", "$$", "\\dim_{\\kappa(x)} \\text{Tor}^{\\mathcal{O}_{X, x}}_i(K, \\kappa(x)) =", "\\left\\{", "\\begin{matrix}", "r & \\text{if} & i = 0 \\\\", "0 & \\text{if} & i \\not = 0", "\\end{matrix}", "\\right.", "$$", "is equal to a free module of rank $r$ placed in degree $0$. See", "for example More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-perfect-from-residue-field}.", "\\medskip\\noindent", "If $M$ is supported on a closed subscheme $Z \\subset X$, then", "$F(M)$ is also supported on $Z$. This is clear because", "we will have $M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_x = 0$", "for $x \\not \\in Z$ and hence the same will be true for $F(M)$", "and hence we get the conclusion from", "Lemma \\ref{lemma-orthogonal-point-sheaf}.", "\\medskip\\noindent", "In particular $F(\\mathcal{O}_x)$ is supported at $\\{x\\}$.", "Let $i \\in \\mathbf{Z}$ be the minimal integer such that", "$H^i(\\mathcal{O}_x) \\not = 0$. We know that $i \\leq 0$.", "If $i < 0$, then there is a morphism", "$\\mathcal{O}_x[-i] \\to F(\\mathcal{O}_x)$", "which contradicts the fact that all morphisms", "$\\mathcal{O}_x[-i] \\to \\mathcal{O}_x$ are zero.", "Thus $F(\\mathcal{O}_x) = \\mathcal{H}[0]$ where", "$\\mathcal{H}$ is a skyscraper sheaf at $x$.", "\\medskip\\noindent", "Let $\\mathcal{G}$ be a coherent $\\mathcal{O}_X$-module with", "$\\dim(\\text{Supp}(\\mathcal{G})) = 0$. Then there exists a", "filtration", "$$", "0 = \\mathcal{G}_0 \\subset \\mathcal{G}_1 \\subset \\ldots \\subset", "\\mathcal{G}_n = \\mathcal{G}", "$$", "such that for $n \\geq i \\geq 1$ the quotient $\\mathcal{G}_i/\\mathcal{G}_{i - 1}$", "is isomorphic to $\\mathcal{O}_{x_i}$ for some closed point $x_i \\in X$.", "Then we get distinguished triangles", "$$", "F(\\mathcal{G}_{i - 1}) \\to F(\\mathcal{G}_i) \\to F(\\mathcal{O}_{x_i})", "$$", "and using induction we find that $F(\\mathcal{G}_i)$ is a", "coherent sheaf placed in degree $0$.", "\\medskip\\noindent", "Let $\\mathcal{G}$ be a coherent $\\mathcal{O}_X$-module. We know that", "$H^i(F(\\mathcal{G})) = 0$ for $i > 0$. To get a contradiction assume", "that $H^i(F(\\mathcal{G}))$ is nonzero for some $i < 0$. We choose", "$i$ minimal with this property so that we have a morphism", "$H^i(F(\\mathcal{G}))[-i] \\to F(\\mathcal{G})$ in $D_{perf}(\\mathcal{O}_X)$.", "Choose a closed point $x \\in X$ in the support of $H^i(F(\\mathcal{G}))$.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-kollar-kovacs-pseudo-coherent}", "there exists an $n > 0$ such that", "$$", "H^i(F(\\mathcal{G}))_x \\otimes_{\\mathcal{O}_{X, x}}", "\\mathcal{O}_{X, x}/\\mathfrak m_x^n", "\\longrightarrow", "\\text{Tor}^{\\mathcal{O}_{X, x}}_{-i}(F(\\mathcal{G})_x,", "\\mathcal{O}_{X, x}/\\mathfrak m_x^n)", "$$", "is nonzero. Next, we take $m \\geq 1$ and we consider the short", "exact sequence", "$$", "0 \\to \\mathfrak m_x^m \\mathcal{G} \\to \\mathcal{G} \\to", "\\mathcal{G}/\\mathfrak m_x^m\\mathcal{G} \\to 0", "$$", "By the above we know that $F(\\mathcal{G}/\\mathfrak m_x^m\\mathcal{G})$", "is a sheaf placed in degree $0$. Hence", "$H^i(F(\\mathfrak m_x^m \\mathcal{G})) \\to H^i(F(\\mathcal{G}))$", "is an isomorphism. Consider the commutative diagram", "$$", "\\xymatrix{", "H^i(F(\\mathfrak m_x^m\\mathcal{G}))_x \\otimes_{\\mathcal{O}_{X, x}}", "\\mathcal{O}_{X, x}/\\mathfrak m_x^n \\ar[r] \\ar[d] &", "\\text{Tor}^{\\mathcal{O}_{X, x}}_{-i}(F(\\mathfrak m_x^m\\mathcal{G})_x,", "\\mathcal{O}_{X, x}/\\mathfrak m_x^n) \\ar[d] \\\\", "H^i(F(\\mathcal{G}))_x \\otimes_{\\mathcal{O}_{X, x}}", "\\mathcal{O}_{X, x}/\\mathfrak m_x^n \\ar[r] &", "\\text{Tor}^{\\mathcal{O}_{X, x}}_{-i}(F(\\mathcal{G})_x,", "\\mathcal{O}_{X, x}/\\mathfrak m_x^n)", "}", "$$", "Since the left vertical arrow is an isomorphism and the bottom arrow", "is nonzero, we conclude that", "the right vertical arrow is nonzero for all $m \\geq 1$.", "On the other hand, by the first paragraph of the proof,", "we know this arrow is isomorphic to the arrow", "$$", "\\text{Tor}^{\\mathcal{O}_{X, x}}_{-i}(\\mathfrak m_x^m\\mathcal{G}_x,", "\\mathcal{O}_{X, x}/\\mathfrak m_x^n)", "\\longrightarrow", "\\text{Tor}^{\\mathcal{O}_{X, x}}_{-i}(\\mathcal{G}_x,", "\\mathcal{O}_{X, x}/\\mathfrak m_x^n)", "$$", "However, this arrow is zero for $m \\gg n$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-tor-annihilated}", "which is the contradiction we're looking for.", "\\medskip\\noindent", "Thus we know that $F$ preserves coherent modules. By", "Lemma \\ref{lemma-exact-functor-preserving-Coh}", "we find $F$ is a sibling to the Fourier-Mukai functor $F'$ given by", "a coherent $\\mathcal{O}_{X \\times X}$-module $\\mathcal{K}$", "flat over $X$ via $\\text{pr}_1$ and finite over $X$ via $\\text{pr}_2$.", "Since $F(\\mathcal{O}_X)$ is an invertible $\\mathcal{O}_X$-module", "$\\mathcal{L}$ placed in degree $0$ we see that", "$$", "\\mathcal{L} \\cong F(\\mathcal{O}_X) \\cong F'(\\mathcal{O}_X) \\cong", "\\text{pr}_{2, *}\\mathcal{K}", "$$", "Thus by Lemma \\ref{lemma-pushforward-invertible-pre} there", "is a morphism $s : X \\to X \\times X$ with $\\text{pr}_2 \\circ s = \\text{id}_X$", "such that $\\mathcal{K} = s_*\\mathcal{L}$. Set $f = \\text{pr}_1 \\circ s$.", "Then we have", "\\begin{align*}", "F'(M)", "& =", "R\\text{pr}_{2, *}(L\\text{pr}_1^*K \\otimes \\mathcal{K}) \\\\", "& =", "R\\text{pr}_{2, *}(L\\text{pr}_1^*M \\otimes s_*\\mathcal{L}) \\\\", "& =", "R\\text{pr}_{2, *}(Rs_*(Lf^*M \\otimes \\mathcal{L})) \\\\", "& =", "Lf^*M \\otimes \\mathcal{L}", "\\end{align*}", "where we have used", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-cohomology-base-change}", "in the third step.", "Since for all closed points $x \\in X$ the module $F(\\mathcal{O}_x)$", "is supported at $x$, we see that $f$ induces the identity on the", "underlying topological space of $X$. We still have to show that", "$f$ is an isomorphism which we will do in the next paragraph.", "\\medskip\\noindent", "Let $x \\in X$ be a closed point.", "For $n \\geq 1$ denote $\\mathcal{O}_{x, n}$ the skyscaper", "sheaf at $x$ with value $\\mathcal{O}_{X, x}/\\mathfrak m_x^n$.", "We have", "$$", "\\Hom_X(\\mathcal{O}_{x, m}, \\mathcal{O}_{x, n}) \\cong", "\\Hom_X(\\mathcal{O}_{x, m}, F(\\mathcal{O}_{x, n})) \\cong", "\\Hom_X(\\mathcal{O}_{x, m}, f^*\\mathcal{O}_{x, n} \\otimes \\mathcal{L})", "$$", "functorially with respect to $\\mathcal{O}_X$-module homomorphisms", "between the $\\mathcal{O}_{x, n}$. (The first isomorphism exists", "by assumption and the second isomorphism because $F$ and $F'$ are siblings.)", "For $m \\geq n$ we have $\\mathcal{O}_{X, x}/\\mathfrak m^n =", "\\Hom_X(\\mathcal{O}_{x, m}, \\mathcal{O}_{x, n})$", "via the action on $\\mathcal{O}_{x, n}$", "we conclude that $f^\\sharp : \\mathcal{O}_{X, x}/\\mathfrak m_x^n \\to", "\\mathcal{O}_{X, x}/\\mathfrak m_x^n$ is bijective for all $n$.", "Thus $f$ induces isomorphisms on complete local rings at closed", "points and hence is \\'etale", "(\\'Etale Morphisms, Lemma \\ref{etale-lemma-characterize-etale-completions}).", "Looking at closed points we see that", "$\\Delta_f : X \\to X \\times_{f, X, f} X$ (which is an open immersion", "as $f$ is \\'etale) is bijective hence an isomorphism.", "Hence $f$ is a monomorphism. Finally, we conclude $f$ is an isomorphism", "as Descent, Lemma", "\\ref{descent-lemma-flat-surjective-quasi-compact-monomorphism-isomorphism}", "tells us it is an open immersion." ], "refs": [ "equiv-lemma-duality-at-point", "equiv-lemma-orthogonal-point-sheaf", "more-algebra-lemma-lift-perfect-from-residue-field", "equiv-lemma-orthogonal-point-sheaf", "more-algebra-lemma-kollar-kovacs-pseudo-coherent", "more-algebra-lemma-tor-annihilated", "equiv-lemma-exact-functor-preserving-Coh", "equiv-lemma-pushforward-invertible-pre", "perfect-lemma-cohomology-base-change", "etale-lemma-characterize-etale-completions", "descent-lemma-flat-surjective-quasi-compact-monomorphism-isomorphism" ], "ref_ids": [ 6847, 6848, 10232, 6848, 10420, 10435, 6851, 6841, 7025, 10705, 14699 ] } ], "ref_ids": [ 6853 ] }, { "id": 6853, "type": "theorem", "label": "equiv-lemma-automorphism", "categories": [ "equiv" ], "title": "equiv-lemma-automorphism", "contents": [ "Let $X$ be a reduced scheme of finite type over a field $k$. Let $f : X \\to X$", "be an automorphism over $k$ which induces the identity map on the underlying", "topological space of $X$. Then", "\\begin{enumerate}", "\\item $f^*\\mathcal{F} \\cong \\mathcal{F}$ for every coherent", "$\\mathcal{O}_X$-module, and", "\\item if $\\dim(Z) > 0$ for every irreducible component $Z \\subset X$,", "then $f$ is the identity.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from part (2) and the fact that the connected components", "of $X$ of dimension $0$ are spectra of fields.", "\\medskip\\noindent", "Let $Z \\subset X$ be an irreducible component viewed as an integral closed", "subscheme. Clearly $f(Z) \\subset Z$ and $f|_Z : Z \\to Z$ is an automorphism", "over $k$ which induces the identity map on the underlying topological space", "of $Z$. Since $X$ is reduced, it suffices to show that the arrows", "$f|_Z : Z \\to Z$ are the identity. This reduces us to the case discussed", "in the next paragraph.", "\\medskip\\noindent", "Assume $X$ is irreducible of dimension $> 0$. Choose a nonempty", "affine open $U \\subset X$. Since $f(U) \\subset U$ and since", "$U \\subset X$ is scheme theoretically dense it suffices to", "prove that $f|_U : U \\to U$ is the identity.", "\\medskip\\noindent", "Assume $X = \\Spec(A)$ is affine, irreducible, of dimension $> 0$", "and $k$ is an infinite field. Let $g \\in A$ be nonconstant. The set", "$$", "S = \\bigcup\\nolimits_{\\lambda \\in k} V(g - \\lambda)", "$$", "is dense in $X$ because it is the inverse image of the dense subset", "$\\mathbf{A}^1_k(k)$ by the nonconstant morphism $g : X \\to \\mathbf{A}^1_k$.", "If $x \\in S$, then the image $g(x)$ of $g$ in $\\kappa(x)$", "is in the image of $k \\to \\kappa(x)$. Hence", "$f^\\sharp : \\kappa(x) \\to \\kappa(x)$ fixes $g(x)$.", "Thus the image of $f^\\sharp(g)$ in $\\kappa(x)$ is equal to $g(x)$.", "We conclude that", "$$", "S \\subset V(g - f^\\sharp(g))", "$$", "and since $X$ is reduced and $S$ is dense we conclude $g=f^\\sharp(g)$.", "This proves $f^\\sharp = \\text{id}_A$ as $A$ is generated as a $k$-algebra", "by elements $g$ as above (details omitted; hint: the set of constant", "functions is a finite dimensional $k$-subvector space of $A$).", "We conclude that $f = \\text{id}_X$.", "\\medskip\\noindent", "Assume $X = \\Spec(A)$ is affine, irreducible, of dimension $> 0$", "and $k$ is a finite field. If for every $1$-dimensional integral", "closed subscheme $C \\subset X$ the restriction $f|_C : C \\to C$", "is the identity, then $f$ is the identity. This reduces us to the", "case where $X$ is a curve. A curve over a finite field has a", "finite automorphism group (details omitted). Hence $f$ has finite", "order, say $n$. Then we pick $g : X \\to \\mathbf{A}^1_k$", "nonconstant as above and we consider", "$$", "S = \\{x \\in X\\text{ closed such that }[\\kappa(g(x)) : k]", "\\text{ is prime to }n\\}", "$$", "Arguing as before we find that $S$ is dense in $X$. Since", "for $x \\in X$ closed the map", "$f^\\sharp : \\kappa(x) \\to \\kappa(x)$ is an", "automorphism of order dividing $n$", "we see that for $x \\in S$ this automorphism", "acts trivially on the subfield generated by", "the image of $g$ in $\\kappa(x)$. Thus we conclude that", "$S \\subset V(g - f^\\sharp(g))$ and we win as before." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6854, "type": "theorem", "label": "equiv-lemma-noah", "categories": [ "equiv" ], "title": "equiv-lemma-noah", "contents": [ "\\begin{reference}", "Email from Noah Olander of Jun 8, 2020", "\\end{reference}", "Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$.", "Let $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_X)$", "be a $k$-linear exact functor. Assume for every coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ with $\\dim(\\text{Supp}(\\mathcal{F})) = 0$", "there is an isomorphism $\\mathcal{F} \\cong F(\\mathcal{F})$.", "Then there exists an automorphism $f : X \\to X$ over $k$", "which induces the identity on the", "underlying topological space\\footnote{This often forces $f$", "to be the identity, see Lemma \\ref{lemma-automorphism}.} and an", "invertible $\\mathcal{O}_X$-module $\\mathcal{L}$", "such that $F$ and $F'(M) = f^*M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{L}$", "are siblings." ], "refs": [ "equiv-lemma-automorphism" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-noah-pre} the functor $F$ is fully faithful.", "We claim that Lemma \\ref{lemma-preserves-Coh} applies to $F$.", "Namely, for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "with $\\dim(\\text{Supp}(\\mathcal{F})) = 0$ there is an isomorphism of", "$k$-vector spaces", "$$", "\\Hom_X(\\mathcal{F}, M) = \\Hom_X(F(\\mathcal{F}), F(M))", "\\cong \\Hom_X(\\mathcal{F}, F(M))", "$$", "functorial in $M$ in $D_{perf}(\\mathcal{O}_X)$. The first equality because", "$F$ is fully faithful." ], "refs": [ "equiv-lemma-noah-pre", "equiv-lemma-preserves-Coh" ], "ref_ids": [ 6850, 6852 ] } ], "ref_ids": [ 6853 ] }, { "id": 6855, "type": "theorem", "label": "equiv-lemma-two-functors", "categories": [ "equiv" ], "title": "equiv-lemma-two-functors", "contents": [ "Let $k$ be a field. Let $X$, $Y$ be smooth proper schemes over $k$.", "Let $F, G : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$", "be $k$-linear exact functors such that", "\\begin{enumerate}", "\\item $F(\\mathcal{F}) \\cong G(\\mathcal{F})$ for any coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ with $\\dim(\\text{Supp}(\\mathcal{F})) = 0$,", "\\item $F$ is fully faithful, and", "\\item $G$ is a Fourier-Mukai functor whose kernel is in", "$D_{perf}(\\mathcal{O}_{X \\times Y})$.", "\\end{enumerate}", "Then there exists a Fourier-Mukai functor", "$F' : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$", "whose kernel is in $D_{perf}(\\mathcal{O}_{X \\times Y})$", "such that $F$ and $F'$ are siblings." ], "refs": [], "proofs": [ { "contents": [ "Recall that $F$ has both adjoints, see", "Lemma \\ref{lemma-always-right-adjoints}. In particular", "the essential image $\\mathcal{A} \\subset D_{perf}(\\mathcal{O}_Y)$ of $F$", "satisfies the equivalent conditions of", "Derived Categories, Lemma \\ref{derived-lemma-right-adjoint}.", "We claim that $G$ factors through $\\mathcal{A}$.", "Since $\\mathcal{A} = {}^\\perp(\\mathcal{A}^\\perp)$ by", "Derived Categories, Lemma \\ref{derived-lemma-right-adjoint}", "it suffices to show that $\\Hom_Y(G(M), N) = 0$ for", "all $M$ in $D_{perf}(\\mathcal{O}_X)$ and $N \\in \\mathcal{A}^\\perp$.", "We have", "$$", "\\Hom_Y(G(M), N) = \\Hom_X(M, G_r(N))", "$$", "where $G_r$ is the right adjoint to $G$. Since", "$G(\\mathcal{F}) \\cong F(\\mathcal{F})$ for $\\mathcal{F}$ as in (1)", "we see that $\\Hom_X(\\mathcal{F}, G_r(N)) = 0$ by the same formula", "and the fact that $N$ is in the right orthogonal to the essential", "image $\\mathcal{A}$ of $F$. Of course, the", "same vanishing holds for $\\Hom_X(\\mathcal{F}, G_r(N)[i])$", "for any $i \\in \\mathbf{Z}$. Thus $G_r(N) = 0$ by", "Lemma \\ref{lemma-orthogonal-point-sheaf}", "and the claim holds.", "\\medskip\\noindent", "Apply Lemma \\ref{lemma-noah} to the functor $H = F^{-1} \\circ G$", "which makes sense because the essential image of $G$ is contained", "in the essential image of $F$ by the previous paragraph and because", "$F$ is fully faithful. We obtain an automorphism $f : X \\to X$", "and an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ such that", "the functor $H' : K \\mapsto f^*K \\otimes \\mathcal{L}$", "is a sibling of $H$. In particular", "$H$ is an auto-equivalence by Lemma \\ref{lemma-sibling-faithful}", "and $H$ induces an auto-equivalence of", "$\\textit{Coh}(\\mathcal{O}_X)$ (as this is true for its sibling functor $H'$).", "Thus the quasi-inverses $H^{-1}$ and $(H')^{-1}$ exist, are siblings", "(small detail omitted), and $(H')^{-1}$ sends $M$ to", "$(f^{-1})^*(M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{L}^{\\otimes -1})$", "which is a Fourier-Mukai functor (details omitted).", "Then of course $F = G \\circ H^{-1}$ is a sibling of", "$G \\circ (H')^{-1}$. Since compositions of Fourier-Mukai", "functors are Fourier-Mukai by", "Lemma \\ref{lemma-compose-fourier-mukai}", "we conclude." ], "refs": [ "equiv-lemma-always-right-adjoints", "derived-lemma-right-adjoint", "derived-lemma-right-adjoint", "equiv-lemma-orthogonal-point-sheaf", "equiv-lemma-noah", "equiv-lemma-sibling-faithful", "equiv-lemma-compose-fourier-mukai" ], "ref_ids": [ 6814, 1947, 1947, 6848, 6854, 6845, 6816 ] } ], "ref_ids": [] }, { "id": 6856, "type": "theorem", "label": "equiv-lemma-fully-faithful", "categories": [ "equiv" ], "title": "equiv-lemma-fully-faithful", "contents": [ "Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$.", "Given a $k$-linear, exact, fully faithful functor", "$F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$", "there exists a Fourier-Mukai functor", "$F' : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ whose kernel", "is in $D_{perf}(\\mathcal{O}_{X \\times Y})$ which is a sibling to $F$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-two-functors} to $F$ and the functor", "$G$ constructed above." ], "refs": [ "equiv-lemma-two-functors" ], "ref_ids": [ 6855 ] } ], "ref_ids": [] }, { "id": 6857, "type": "theorem", "label": "equiv-lemma-uniqueness", "categories": [ "equiv" ], "title": "equiv-lemma-uniqueness", "contents": [ "Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$.", "Let $K \\in D_{perf}(\\mathcal{O}_{X \\times X})$. If the Fourier-Mukai", "functor $\\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_X)$", "is isomorphic to the identity functor, then", "$K \\cong \\Delta_*\\mathcal{O}_X$ in $_{perf}(\\mathcal{O}_{X \\times X})$." ], "refs": [], "proofs": [ { "contents": [ "Let $i$ be the minimal integer such that the cohomology sheaf $H^i(K)$ is", "nonzero. Let $\\mathcal{E}$ and $\\mathcal{G}$ be finite locally free", "$\\mathcal{O}_X$-modules. Then", "\\begin{align*}", "H^i(X \\times X, K \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L}", "(\\mathcal{E} \\boxtimes \\mathcal{G}))", "& =", "H^i(X, R\\text{pr}_{2, *}(K \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L}", "(\\mathcal{E} \\boxtimes \\mathcal{G}))) \\\\", "& =", "H^i(X, \\Phi_K(\\mathcal{E}) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}) \\\\", "& \\cong", "H^i(X, \\mathcal{E} \\otimes \\mathcal{G})", "\\end{align*}", "which is zero if $i < 0$. On the other hand, we can choose", "$\\mathcal{E}$ and $\\mathcal{G}$ such that there is a surjection", "$\\mathcal{E}^\\vee \\boxtimes \\mathcal{G}^\\vee \\to H^i(K)$", "by Lemma \\ref{lemma-on-product}.", "In this case the left hand side of the equalities is nonzero.", "Hence we conclude that $H^i(K) = 0$ for $i < 0$.", "\\medskip\\noindent", "Let $i$ be the maximal integer such that $H^i(K)$ is nonzero.", "The same argument with $\\mathcal{E}$ and $\\mathcal{G}$", "support of dimension $0$ shows that $i \\leq 0$.", "Hence we conclude that $K$ is given by a single coherent", "$\\mathcal{O}_{X \\times X}$-module $\\mathcal{K}$ sitting in degree $0$.", "\\medskip\\noindent", "Since $R\\text{pr}_{2, *}(\\text{pr}_1^*\\mathcal{F} \\otimes \\mathcal{K})$", "is $\\mathcal{F}$, by taking $\\mathcal{F}$ supported at closed points", "we see that the support of $\\mathcal{K}$ is finite over $X$ via", "$\\text{pr}_2$. Since $R\\text{pr}_{2, *}(\\mathcal{K}) \\cong \\mathcal{O}_X$", "we conclude by Lemma \\ref{lemma-pushforward-invertible-pre}", "that $\\mathcal{K} = s_*\\mathcal{O}_X$ for some section $s : X \\to X \\times X$", "of the second projection. Then $\\Phi_K(M) = f^*M$ where", "$f = \\text{pr}_1 \\circ s$ and this can happen only if $s$", "is the diagonal morphism as desired." ], "refs": [ "equiv-lemma-on-product", "equiv-lemma-pushforward-invertible-pre" ], "ref_ids": [ 6822, 6841 ] } ], "ref_ids": [] }, { "id": 6858, "type": "theorem", "label": "equiv-lemma-base-change-is-functor", "categories": [ "equiv" ], "title": "equiv-lemma-base-change-is-functor", "contents": [ "Let $S' \\to S$ be a morphism of schemes.", "The rule which sends", "\\begin{enumerate}", "\\item a smooth proper scheme $X$ over $S$ to $X' = S' \\times_S X$, and", "\\item the isomorphism class of an object $K$", "of $D_{perf}(\\mathcal{O}_{X \\times_S Y})$ to the isomorphism class of", "$L(X' \\times_{S'} Y' \\to X \\times_S Y)^*K$", "in $D_{perf}(\\mathcal{O}_{X' \\times_{S'} Y'})$", "\\end{enumerate}", "is a functor from the category defined for $S$ to the category", "defined for $S'$." ], "refs": [], "proofs": [ { "contents": [ "To see this suppose we have $X, Y, Z$ and", "$K \\in D_{perf}(\\mathcal{O}_{X \\times_S Y})$ and", "$M \\in D_{perf}(\\mathcal{O}_{Y \\times_S Z})$.", "Denote", "$K' \\in D_{perf}(\\mathcal{O}_{X' \\times_{S'} Y'})$ and", "$M' \\in D_{perf}(\\mathcal{O}_{Y' \\times_{S'} Z'})$", "their pullbacks as in the statement of the lemma.", "The diagram", "$$", "\\xymatrix{", "X' \\times_{S'} Y' \\times_{S'} Z' \\ar[r] \\ar[d]_{\\text{pr}'_{13}} &", "X \\times_S Y \\times_S Z \\ar[d]^{\\text{pr}_{13}} \\\\", "X' \\times_{S'} Z' \\ar[r] &", "X \\times_S Z", "}", "$$", "is cartesian and $\\text{pr}_{13}$ is proper and smooth.", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image-general}", "we see that the derived pullback by the lower horizontal", "arrow of the composition", "$$", "R\\text{pr}_{13, *}(", "L\\text{pr}_{12}^*K", "\\otimes_{\\mathcal{O}_{X \\times_S Y \\times_S Z}}^\\mathbf{L}", "L\\text{pr}_{23}^*M)", "$$", "indeed is (canonically) isomorphic to", "$$", "R\\text{pr}'_{13, *}(", "L(\\text{pr}'_{12})^*K'", "\\otimes_{\\mathcal{O}_{X' \\times_{S'} Y' \\times_{S'} Z'}}^\\mathbf{L}", "L(\\text{pr}'_{23})^*M')", "$$", "as desired. Some details omitted." ], "refs": [ "perfect-lemma-flat-proper-perfect-direct-image-general" ], "ref_ids": [ 7054 ] } ], "ref_ids": [] }, { "id": 6859, "type": "theorem", "label": "equiv-lemma-equivalences-rek", "categories": [ "equiv" ], "title": "equiv-lemma-equivalences-rek", "contents": [ "With notation as in Definition \\ref{definition-relative-equivalence-kernel}", "let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$", "to $Y$ over $S$. Then the corresponding Fourier-Mukai functors", "$\\Phi_K : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$", "(Lemma \\ref{lemma-fourier-Mukai-QCoh})", "and $\\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$", "(Lemma \\ref{lemma-fourier-mukai})", "are equivalences." ], "refs": [ "equiv-definition-relative-equivalence-kernel", "equiv-lemma-fourier-Mukai-QCoh", "equiv-lemma-fourier-mukai" ], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-compose-fourier-mukai} and", "Example \\ref{example-diagonal-fourier-mukai}." ], "refs": [ "equiv-lemma-compose-fourier-mukai" ], "ref_ids": [ 6816 ] } ], "ref_ids": [ 6878, 6815, 6817 ] }, { "id": 6860, "type": "theorem", "label": "equiv-lemma-base-change-rek", "categories": [ "equiv" ], "title": "equiv-lemma-base-change-rek", "contents": [ "With notation as in Definition \\ref{definition-relative-equivalence-kernel}", "let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$", "to $Y$ over $S$. Let $S_1 \\to S$ be a morphism of schemes. Let", "$X_1 = S_1 \\times_S X$ and $Y_1 = S_1 \\times_S Y$. Then the pullback", "$K_1 = L(X_1 \\times_{S_1} Y_1 \\to X \\times_S Y)^*K$ is", "the Fourier-Mukai kernel of a relative equivalence from $X_1$", "to $Y_1$ over $S_1$." ], "refs": [ "equiv-definition-relative-equivalence-kernel" ], "proofs": [ { "contents": [ "Let $K' \\in D_{perf}(\\mathcal{O}_{Y \\times_S X})$ be the object assumed to", "exist in Definition \\ref{definition-relative-equivalence-kernel}.", "Denote $K'_1$ the pullback of $K'$ by", "$Y_1 \\times_{S_1} X_1 \\to Y \\times_S X$.", "Then it suffices to prove that we have", "$$", "\\Delta_{X_1/S_1, *}\\mathcal{O}_X \\cong", "R\\text{pr}_{13, *}(L\\text{pr}_{12}^*K_1", "\\otimes_{\\mathcal{O}_{X_1 \\times_{S_1} Y_1 \\times_{S_1} X_1}}^\\mathbf{L}", "L\\text{pr}_{23}^*K_1')", "$$", "in $D(\\mathcal{O}_{X_1 \\times_{S_1} X_1})$ and similarly for the other", "condition. Since", "$$", "\\xymatrix{", "X_1 \\times_{S_1} Y_1 \\times_{S_1} X_1 \\ar[r] \\ar[d]_{\\text{pr}_{13}} &", "X \\times_S Y \\times_S X \\ar[d]^{\\text{pr}_{13}} \\\\", "X_1 \\times_{S_1} X_1 \\ar[r] &", "X \\times_S X", "}", "$$", "is cartesian it suffices by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image-general}", "to prove that", "$$", "\\Delta_{X_1/S_1, *}\\mathcal{O}_{X_1}", "\\cong", "L(X_1 \\times_{S_1} X_1 \\to X \\times_S X)^*\\Delta_{X/S, *}\\mathcal{O}_X ", "$$", "This in turn will be true if $X$ and $X_1 \\times_{S_1} X_1$ are tor", "independent over $X \\times_S X$, see", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-compare-base-change}.", "This tor independence can be seen directly but also follows from", "the more general More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-case-of-tor-independence} applied to the square", "with corners $X, X, X, S$ and its base change by $S_1 \\to S$." ], "refs": [ "equiv-definition-relative-equivalence-kernel", "perfect-lemma-flat-proper-perfect-direct-image-general", "perfect-lemma-compare-base-change", "more-morphisms-lemma-case-of-tor-independence" ], "ref_ids": [ 6878, 7054, 7028, 14057 ] } ], "ref_ids": [ 6878 ] }, { "id": 6861, "type": "theorem", "label": "equiv-lemma-descend-rek", "categories": [ "equiv" ], "title": "equiv-lemma-descend-rek", "contents": [ "Let $S = \\lim_{i \\in I} S_i$ be a limit of a directed system of schemes", "with affine transition morphisms $g_{i'i} : S_{i'} \\to S_i$.", "We assume that $S_i$ is quasi-compact and quasi-separated for all $i \\in I$.", "Let $0 \\in I$. Let $X_0 \\to S_0$ and $Y_0 \\to S_0$ be smooth proper morphisms.", "We set $X_i = S_i \\times_{S_0} X_0$ for $i \\geq 0$", "and $X = S \\times_{S_0} X_0$ and similarly for $Y_0$. If $K$ is the", "Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$", "then for some $i \\geq 0$ there exists a", "Fourier-Mukai kernel of a relative equivalence from $X_i$ to $Y_i$ over $S_i$." ], "refs": [], "proofs": [ { "contents": [ "Let $K' \\in D_{perf}(\\mathcal{O}_{Y \\times_S X})$ be the object assumed to", "exist in Definition \\ref{definition-relative-equivalence-kernel}.", "Since $X \\times_S Y = \\lim X_i \\times_{S_i} Y_i$ there exists an", "$i$ and objects $K_i$ and $K'_i$ in", "$D_{perf}(\\mathcal{O}_{Y_i \\times_{S_i} X_i})$", "whose pullbacks to $Y \\times_S X$ give $K$ and $K'$.", "See Derived Categories of Schemes, Lemma \\ref{perfect-lemma-descend-perfect}.", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image-general}", "the object", "$$", "R\\text{pr}_{13, *}(L\\text{pr}_{12}^*K_i", "\\otimes_{\\mathcal{O}_{X_i \\times_{S_i} Y_i \\times_{S_i} X_i}}^\\mathbf{L}", "L\\text{pr}_{23}^*K_i')", "$$", "is perfect and its pullback to $X \\times_S X$ is equal to", "$$", "R\\text{pr}_{13, *}(L\\text{pr}_{12}^*K", "\\otimes_{\\mathcal{O}_{X \\times_S Y \\times_S X}}^\\mathbf{L}", "L\\text{pr}_{23}^*K') \\cong \\Delta_{X/S, *}\\mathcal{O}_X", "$$", "See proof of Lemma \\ref{lemma-base-change-rek}.", "On the other hand, since $X_i \\to S$ is smooth and separated the", "object", "$$", "\\Delta_{i, *}\\mathcal{O}_{X_i}", "$$", "of $D(\\mathcal{O}_{X_i \\times_{S_i} X_i})$ is also perfect", "(by More on Morphisms, Lemmas", "\\ref{more-morphisms-lemma-smooth-diagonal-perfect} and", "\\ref{more-morphisms-lemma-perfect-proper-perfect-direct-image}) and", "its pullback to $X \\times_S X$ is equal to", "$$", "\\Delta_{X/S, *}\\mathcal{O}_X", "$$", "See proof of Lemma \\ref{lemma-base-change-rek}. Thus by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-descend-perfect}", "after increasing $i$ we may assume that", "$$", "\\Delta_{i, *}\\mathcal{O}_{X_i} \\cong", "R\\text{pr}_{13, *}(L\\text{pr}_{12}^*K_i", "\\otimes_{\\mathcal{O}_{X_i \\times_{S_i} Y_i \\times_{S_i} X_i}}^\\mathbf{L}", "L\\text{pr}_{23}^*K_i')", "$$", "as desired. The same works for the roles of $K$ and $K'$ reversed." ], "refs": [ "equiv-definition-relative-equivalence-kernel", "perfect-lemma-descend-perfect", "perfect-lemma-flat-proper-perfect-direct-image-general", "equiv-lemma-base-change-rek", "more-morphisms-lemma-smooth-diagonal-perfect", "more-morphisms-lemma-perfect-proper-perfect-direct-image", "equiv-lemma-base-change-rek", "perfect-lemma-descend-perfect" ], "ref_ids": [ 6878, 7051, 7054, 6860, 14016, 13997, 6860, 7051 ] } ], "ref_ids": [] }, { "id": 6862, "type": "theorem", "label": "equiv-lemma-deform-koszul", "categories": [ "equiv" ], "title": "equiv-lemma-deform-koszul", "contents": [ "Let $(R, \\mathfrak m, \\kappa) \\to (A, \\mathfrak n, \\lambda)$", "be a flat local ring homorphism of local rings", "which is essentially of finite presentation.", "Let $\\overline{f}_1, \\ldots, \\overline{f}_r \\in \\mathfrak n/\\mathfrak m A", "\\subset A/\\mathfrak m A$ be a regular sequence. Let $K \\in D(A)$. Assume", "\\begin{enumerate}", "\\item $K$ is perfect,", "\\item $K \\otimes_A^\\mathbf{L} A/\\mathfrak m A$ is isomorphic in", "$D(A/\\mathfrak m A)$ to the", "Koszul complex on $\\overline{f}_1, \\ldots, \\overline{f}_r$.", "\\end{enumerate}", "Then $K$ is isomorphic in $D(A)$ to a Koszul complex on a regular sequence", "$f_1, \\ldots, f_r \\in A$ lifting the given elements", "$\\overline{f}_1, \\ldots, \\overline{f}_r$. Moreover, $A/(f_1, \\ldots, f_r)$", "is flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "Let us use chain complexes in the proof of this lemma.", "The Koszul complex $K_\\bullet(\\overline{f}_1, \\ldots, \\overline{f}_r)$", "is defined in More on Algebra, Definition", "\\ref{more-algebra-definition-koszul-complex}.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-lift-complex-stably-frees}", "we can represent $K$ by a complex", "$$", "K_\\bullet :", "A \\to A^{\\oplus r} \\to \\ldots \\to A^{\\oplus r} \\to A", "$$", "whose tensor product with $A/\\mathfrak mA$ is equal (!)", "to $K_\\bullet(\\overline{f}_1, \\ldots, \\overline{f}_r)$.", "Denote $f_1, \\ldots, f_r \\in A$ the components of the", "arrow $A^{\\oplus r} \\to A$. These $f_i$ are lifts of the", "$\\overline{f}_i$. By Algebra, Lemma", "\\ref{algebra-lemma-grothendieck-regular-sequence-general}", "$f_1, \\ldots, f_r$ form a regular sequence in $A$ and $A/(f_1, \\ldots, f_r)$", "is flat over $R$. Let $J = (f_1, \\ldots, f_r) \\subset A$.", "Consider the diagram", "$$", "\\xymatrix{", "K_\\bullet \\ar[rd] \\ar@{..>}[rr]_{\\varphi_\\bullet} & &", "K_\\bullet(f_1, \\ldots, f_r) \\ar[ld] \\\\", "& A/J", "}", "$$", "Since $f_1, \\ldots, f_r$ is a regular sequence the south-west arrow", "is a quasi-isomorphism (see", "More on Algebra, Lemma \\ref{more-algebra-lemma-regular-koszul-regular}).", "Hence we can find the dotted arrow making the", "diagram commute for example by", "Algebra, Lemma \\ref{algebra-lemma-compare-resolutions}.", "Reducing modulo $\\mathfrak m$ we obtain a commutative diagram", "$$", "\\xymatrix{", "K_\\bullet(\\overline{f}_1, \\ldots, \\overline{f}_r)", "\\ar[rd] \\ar[rr]_{\\overline{\\varphi}_\\bullet} & &", "K_\\bullet(\\overline{f}_1, \\ldots, \\overline{f}_r) \\ar[ld] \\\\", "& (A/\\mathfrak m A)/(\\overline{f}_1, \\ldots, \\overline{f}_r)", "}", "$$", "by our choice of $K_\\bullet$. Thus $\\overline{\\varphi}$ is an isomorphism", "in the derived category $D(A/\\mathfrak m A)$. It follows that", "$\\overline{\\varphi} \\otimes_{A/\\mathfrak m A}^\\mathbf{L} \\lambda$", "is an isomorphism. Since $\\overline{f}_i \\in \\mathfrak n / \\mathfrak m A$", "we see that", "$$", "\\text{Tor}_i^{A/\\mathfrak m A}(", "K_\\bullet(\\overline{f}_1, \\ldots, \\overline{f}_r), \\lambda)", "=", "K_i(\\overline{f}_1, \\ldots, \\overline{f}_r) \\otimes_{A/\\mathfrak m A} \\lambda", "$$", "Hence $\\varphi_i \\bmod \\mathfrak n$ is invertible.", "Since $A$ is local this means that $\\varphi_i$ is an", "isomorphism and the proof is complete." ], "refs": [ "more-algebra-definition-koszul-complex", "more-algebra-lemma-lift-complex-stably-frees", "algebra-lemma-grothendieck-regular-sequence-general", "more-algebra-lemma-regular-koszul-regular", "algebra-lemma-compare-resolutions" ], "ref_ids": [ 10606, 10230, 1112, 9973, 763 ] } ], "ref_ids": [] }, { "id": 6863, "type": "theorem", "label": "equiv-lemma-limit-arguments", "categories": [ "equiv" ], "title": "equiv-lemma-limit-arguments", "contents": [ "Let $R \\to S$ be a finite type flat ring map of Noetherian rings.", "Let $\\mathfrak q \\subset S$ be a prime ideal lying over", "$\\mathfrak p \\subset R$. Let $K \\in D(S)$ be perfect.", "Let $f_1, \\ldots, f_r \\in \\mathfrak q S_\\mathfrak q$", "be a regular sequence such that $S_\\mathfrak q/(f_1, \\ldots, f_r)$", "is flat over $R$ and such that", "$K \\otimes_S^\\mathbf{L} S_\\mathfrak q$ is isomorphic to the", "Koszul complex on $f_1, \\ldots, f_r$. Then there exists a", "$g \\in S$, $g \\not \\in \\mathfrak q$ such that", "\\begin{enumerate}", "\\item $f_1, \\ldots, f_r$ are the images of", "$f'_1, \\ldots, f'_r \\in S_g$,", "\\item $f'_1, \\ldots, f'_r$ form a regular sequence in $S_g$,", "\\item $S_g/(f'_1, \\ldots, f'_r)$ is flat over $R$,", "\\item $K \\otimes_S^\\mathbf{L} S_g$ is isomorphic to the", "Koszul complex on $f_1, \\ldots, f_r$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We can find $g \\in S$, $g \\not \\in \\mathfrak q$ with property (1) by", "the definition of localizations. After replacing $g$ by", "$gg'$ for some $g' \\in S$, $g' \\not \\in \\mathfrak q$", "we may assume (2) holds, see", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-neighbourhood}.", "By Algebra, Theorem \\ref{algebra-theorem-openness-flatness}", "we find that $S_g/(f'_1, \\ldots, f'_r)$ is flat over $R$", "in an open neighbourhood of $\\mathfrak q$.", "Hence after once more replacing $g$ by $gg'$ for some", "$g' \\in S$, $g' \\not \\in \\mathfrak q$ we may assume (3) holds as well.", "Finally, we get (4) for a further replacement by", "More on Algebra, Lemma \\ref{more-algebra-lemma-colimit-perfect-complexes}." ], "refs": [ "algebra-lemma-regular-sequence-in-neighbourhood", "algebra-theorem-openness-flatness", "more-algebra-lemma-colimit-perfect-complexes" ], "ref_ids": [ 741, 326, 10226 ] } ], "ref_ids": [] }, { "id": 6864, "type": "theorem", "label": "equiv-lemma-isomorphism-in-neighbourhood", "categories": [ "equiv" ], "title": "equiv-lemma-isomorphism-in-neighbourhood", "contents": [ "Let $S$ be a Noetherian scheme. Let $s \\in S$.", "Let $p : X \\to Y$ be a morphism of schemes over $S$.", "Assume", "\\begin{enumerate}", "\\item $Y \\to S$ and $X \\to S$ proper,", "\\item $X$ is flat over $S$,", "\\item $X_s \\to Y_s$ an isomorphism.", "\\end{enumerate}", "Then there exists an open neighbourhood $U \\subset S$ of $s$", "such that the base change $X_U \\to Y_U$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The morphism $p$ is proper by Morphisms, Lemma", "\\ref{morphisms-lemma-closed-immersion-proper}.", "By Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-finite-fibre-finite-in-neighbourhood}", "there is an open $Y_s \\subset V \\subset Y$ such that", "$p|_{p^{-1}(V)} : p^{-1}(V) \\to V$ is finite.", "By More on Morphisms, Theorem", "\\ref{more-morphisms-theorem-criterion-flatness-fibre-Noetherian}", "there is an open $X_s \\subset U \\subset X$ such that", "$p|_U : U \\to Y$ is flat. After removing the images of", "$X \\setminus U$ and $Y \\setminus V$ (which are closed subsets", "not containing $s$) we may assume $p$ is flat and finite.", "Then $p$ is open (Morphisms, Lemma \\ref{morphisms-lemma-fppf-open})", "and $Y_s \\subset p(X) \\subset Y$ hence after shrinking $S$", "we may assume $p$ is surjective.", "As $p_s : X_s \\to Y_s$ is an isomorphism, the map", "$$", "p^\\sharp : \\mathcal{O}_Y \\longrightarrow p_*\\mathcal{O}_X", "$$", "of coherent $\\mathcal{O}_Y$-modules ($p$ is finite)", "becomes an isomorphism after pullback by $i : Y_s \\to Y$", "(by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-affine-base-change} for example).", "By Nakayama's lemma, this implies that", "$\\mathcal{O}_{Y, y} \\to (p_*\\mathcal{O}_X)_y$ is surjective", "for all $y \\in Y_s$. Hence there is an open $Y_s \\subset V \\subset Y$", "such that $p^\\sharp|_V$ is surjective", "(Modules, Lemma \\ref{modules-lemma-finite-type-surjective-on-stalk}).", "Hence after shrinking $S$ once more we may assume", "$p^\\sharp$ is surjective which means that $p$ is a closed", "immersion (as $p$ is already finite).", "Thus now $p$ is a surjective flat closed immersion", "of Noetherian schemes and hence an isomorphism, see", "Morphisms, Section \\ref{morphisms-section-flat-closed-immersions}." ], "refs": [ "morphisms-lemma-closed-immersion-proper", "coherent-lemma-proper-finite-fibre-finite-in-neighbourhood", "more-morphisms-theorem-criterion-flatness-fibre-Noetherian", "morphisms-lemma-fppf-open", "coherent-lemma-affine-base-change", "modules-lemma-finite-type-surjective-on-stalk" ], "ref_ids": [ 5410, 3366, 13671, 5267, 3297, 13238 ] } ], "ref_ids": [] }, { "id": 6865, "type": "theorem", "label": "equiv-lemma-no-deformations", "categories": [ "equiv" ], "title": "equiv-lemma-no-deformations", "contents": [ "Let $k$ be a field. Let $S$ be a finite type scheme over $k$", "with $k$-rational point $s$. Let $Y \\to S$ be a smooth proper morphism.", "Let $X = Y_s \\times S \\to S$ be the constant family with fibre", "$Y_s$. Let $K$ be the Fourier-Mukai kernel of a relative equivalence", "from $X$ to $Y$ over $S$. Assume the restriction", "$$", "L(Y_s \\times_S Y_s \\to X \\times_S Y)^*K \\cong ", "\\Delta_{Y_s/k, *} \\mathcal{O}_{Y_s}", "$$", "in $D(\\mathcal{O}_{Y_s \\times Y_s})$. Then there is an open neighbourhood", "$s \\in U \\subset S$ such that $Y|_U$ is isomorphic to $Y_s \\times U$ over $U$." ], "refs": [], "proofs": [ { "contents": [ "Denote $i : Y_s \\times Y_s = X_s \\times Y_s \\to X \\times_S Y$", "the natural closed immersion. (We will write $Y_s$ and not $X_s$", "for the fibre of $X$ over $s$ from now on.) Let", "$z \\in Y_s \\times Y_s = (X \\times_S Y)_s \\subset X \\times_S Y$", "be a closed point. As indicated we think of $z$ both as a closed point", "of $Y_s \\times Y_s$ as well as a closed point of $X \\times_S Y$.", "\\medskip\\noindent", "Case I: $z \\not \\in \\Delta_{Y_s/k}(Y_s)$. Denote $\\mathcal{O}_z$", "the coherent $\\mathcal{O}_{Y_s \\times Y_s}$-module supported at $z$", "whose value is $\\kappa(z)$. Then $i_*\\mathcal{O}_z$ is the", "coherent $\\mathcal{O}_{X \\times_S Y}$-module supported at $z$", "whose value is $\\kappa(z)$. Our assumption means that", "$$", "K \\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} i_*\\mathcal{O}_z =", "Li^*K \\otimes_{\\mathcal{O}_{Y_s \\times Y_s}}^\\mathbf{L} \\mathcal{O}_z = 0", "$$", "Hence by Lemma \\ref{lemma-orthogonal-point-sheaf}", "we find an open neighbourhood $U(z) \\subset X \\times_S Y$ of $z$", "such that $K|_{U(z)} = 0$. In this case we set $Z(z) = \\emptyset$", "as closed subscheme of $U(z)$.", "\\medskip\\noindent", "Case II: $z \\in \\Delta_{Y_s/k}(Y_s)$. Since $Y_s$ is smooth over $k$", "we know that $\\Delta_{Y_s/k} : Y_s \\to Y_s \\times Y_s$ is a", "regular immersion, see More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-smooth-diagonal-perfect}.", "Choose a regular sequence $\\overline{f}_1, \\ldots, \\overline{f}_r \\in", "\\mathcal{O}_{Y_s \\times Y_s, z}$ cutting out the ideal sheaf of", "$\\Delta_{Y_s/k}(Y_s)$. Since a regular sequence is Koszul-regular", "(More on Algebra, Lemma \\ref{more-algebra-lemma-regular-koszul-regular})", "our assumption means that", "$$", "K_z \\otimes_{\\mathcal{O}_{X \\times_S Y, z}}^\\mathbf{L}", "\\mathcal{O}_{Y_s \\times Y_s, z}", "\\in D(\\mathcal{O}_{Y_s \\times Y_s, z})", "$$", "is represented by the Koszul complex on", "$\\overline{f}_1, \\ldots, \\overline{f}_r$ over", "$\\mathcal{O}_{Y_s \\times Y_s, z}$.", "By Lemma \\ref{lemma-deform-koszul} applied to", "$\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X \\times_S Y, z}$", "we conclude that $K_z \\in D(\\mathcal{O}_{X \\times_S Y, z})$ is", "represented by the Koszul complex on a regular sequence", "$f_1, \\ldots, f_r \\in \\mathcal{O}_{X \\times_S Y, z}$", "lifting the regular sequence", "$\\overline{f}_1, \\ldots, \\overline{f}_r$", "such that moreover $\\mathcal{O}_{X \\times_S Y}/(f_1, \\ldots, f_r)$", "is flat over $\\mathcal{O}_{S, s}$.", "By some limit arguments (Lemma \\ref{lemma-limit-arguments})", "we conclude that there exists an affine open neighbourhood", "$U(z) \\subset X \\times_S Y$ of $z$ and a closed subscheme", "$Z(z) \\subset U(z)$ such that", "\\begin{enumerate}", "\\item $Z(z) \\to U(z)$ is a regular closed immersion,", "\\item $K|_{U(z)}$ is quasi-isomorphic to $\\mathcal{O}_{Z(z)}$,", "\\item $Z(z) \\to S$ is flat,", "\\item $Z(z)_s = \\Delta_{Y_s/k}(Y_s) \\cap U(z)_s$", "as closed subschemes of $U(z)_s$.", "\\end{enumerate}", "\\noindent", "By property (2), for $z, z' \\in Y_s \\times Y_s$, we", "find that $Z(z) \\cap U(z') = Z(z') \\cap U(z)$ as closed subschemes.", "Hence we obtain an open neighbourhood", "$$", "U = \\bigcup\\nolimits_{z \\in Y_s \\times Y_s\\text{ closed}} U(z)", "$$", "of $Y_s \\times Y_s$ in $X \\times_S Y$ and a closed subscheme $Z \\subset U$", "such that (1) $Z \\to U$ is a regular closed immersion,", "(2) $Z \\to S$ is flat, and (3) $Z_s = \\Delta_{Y_s/k}(Y_s)$.", "Since $X \\times_S Y \\to S$ is proper, after replacing $S$", "by an open neighbourhood of $s$ we may assume $U = X \\times_S Y$.", "Since the projections $Z_s \\to Y_s$ and $Z_s \\to X_s$", "are isomorphisms, we conclude that after shrinking $S$", "we may assume $Z \\to Y$ and $Z \\to X$ are isomorphisms, see", "Lemma \\ref{lemma-isomorphism-in-neighbourhood}.", "This finishes the proof." ], "refs": [ "equiv-lemma-orthogonal-point-sheaf", "more-morphisms-lemma-smooth-diagonal-perfect", "more-algebra-lemma-regular-koszul-regular", "equiv-lemma-deform-koszul", "equiv-lemma-limit-arguments", "equiv-lemma-isomorphism-in-neighbourhood" ], "ref_ids": [ 6848, 14016, 9973, 6862, 6863, 6864 ] } ], "ref_ids": [] }, { "id": 6866, "type": "theorem", "label": "equiv-lemma-no-deformations-better", "categories": [ "equiv" ], "title": "equiv-lemma-no-deformations-better", "contents": [ "Let $k$ be an algebraically closed field. Let $X$", "be a smooth proper scheme over $k$.", "Let $f : Y \\to S$ be a smooth proper morphism with $S$ of finite type over $k$.", "Let $K$ be the Fourier-Mukai kernel of a relative equivalence", "from $X \\times S$ to $Y$ over $S$. Then $S$ can be covered by", "open subschemes $U$ such that there is a $U$-isomorphism", "$f^{-1}(U) \\cong Y_0 \\times U$ for some $Y_0$ proper and smooth over $k$." ], "refs": [], "proofs": [ { "contents": [ "Choose a closed point $s \\in S$. Since $k$ is algebraically closed", "this is a $k$-rational point. Set $Y_0 = Y_s$. The restriction", "$K_0$ of $K$ to $X \\times Y_0$ is the Fourier-Mukai kernel of a", "relative equivalence from $X$ to $Y_0$ over $\\Spec(k)$ by", "Lemma \\ref{lemma-base-change-rek}. Let $K'_0$ in", "$D_{perf}(\\mathcal{O}_{Y_0 \\times X})$ be the ", "object assumed to", "exist in Definition \\ref{definition-relative-equivalence-kernel}.", "Then $K'_0$ is the Fourier-Mukai kernel of a", "relative equivalence from $Y_0$ to $X$ over $\\Spec(k)$", "by the symmetry inherent in", "Definition \\ref{definition-relative-equivalence-kernel}.", "Hence by", "Lemma \\ref{lemma-base-change-rek}", "we see that the pullback", "$$", "M = (Y_0 \\times X \\times S \\to Y_0 \\times X)^*K'_0", "$$", "on $(Y_0 \\times S) \\times_S (X \\times S) = Y_0 \\times X \\times S$", "is the Fourier-Mukai kernel of a", "relative equivalence from $Y_0 \\times S$ to $X \\times S$ over $S$.", "Now consider the kernel", "$$", "K_{new} =", "R\\text{pr}_{13, *}(L\\text{pr}_{12}^*M", "\\otimes_{\\mathcal{O}_{(Y_0 \\times S) \\times_S (X \\times S)", "\\times_S Y}}^\\mathbf{L}", "L\\text{pr}_{23}^*K)", "$$", "on $(Y_0 \\times S) \\times_S Y$. This is the Fourier-Mukai kernel of a", "relative equivalence from $Y_0 \\times S$ to $Y$ over $S$ since it is", "the composition of two invertible arrows in", "the category constructed in", "Section \\ref{section-category-Fourier-Mukai-kernels}.", "Moreover, this composition passes through base change", "(Lemma \\ref{lemma-base-change-is-functor}).", "Hence we see that the pullback of $K_{new}$ to", "$((Y_0 \\times S) \\times_S Y)_s = Y_0 \\times Y_0$", "is equal to the composition of $K_0$ and $K'_0$", "and hence equal to the identity in this category.", "In other words, we have", "$$", "L(Y_0 \\times Y_0 \\to (Y_0 \\times S) \\times_S Y)^*K_{new}", "\\cong", "\\Delta_{Y_0/k, *}\\mathcal{O}_{Y_0}", "$$", "Thus by Lemma \\ref{lemma-no-deformations} we conclude that $Y \\to S$", "is isomorphic to $Y_0 \\times S$ in an open neighbourhood of $s$.", "This finishes the proof." ], "refs": [ "equiv-lemma-base-change-rek", "equiv-definition-relative-equivalence-kernel", "equiv-definition-relative-equivalence-kernel", "equiv-lemma-base-change-rek", "equiv-lemma-base-change-is-functor", "equiv-lemma-no-deformations" ], "ref_ids": [ 6860, 6878, 6878, 6860, 6858, 6865 ] } ], "ref_ids": [] }, { "id": 6867, "type": "theorem", "label": "equiv-lemma-countable-finite-type", "categories": [ "equiv" ], "title": "equiv-lemma-countable-finite-type", "contents": [ "Let $R$ be a countable Noetherian ring. Then the category of schemes of finite", "type over $R$ is countable." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6868, "type": "theorem", "label": "equiv-lemma-countable-abelian", "categories": [ "equiv" ], "title": "equiv-lemma-countable-abelian", "contents": [ "Let $\\mathcal{A}$ be a countable abelian category.", "Then $D^b(\\mathcal{A})$ is countable." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove the statement for $D(\\mathcal{A})$ as the others", "are full subcategories of this one. Since every object in $D(\\mathcal{A})$", "is a complex of objects of $\\mathcal{A}$ it is immediate that the set of", "isomorphism classes of objects of $D^b(\\mathcal{A})$ is countable.", "Moreover, for bounded complexes $A^\\bullet$ and $B^\\bullet$ of $\\mathcal{A}$", "it is clear that $\\Hom_{K^b(\\mathcal{A})}(A^\\bullet, B^\\bullet)$ is countable.", "We have", "$$", "\\Hom_{D^b(\\mathcal{A})}(A^\\bullet, B^\\bullet) =", "\\colim_{s : (A')^\\bullet \\to A^\\bullet", "\\text{ qis and }(A')^\\bullet\\text{ bounded}}", "\\Hom_{K^b(\\mathcal{A})}((A')^\\bullet, B^\\bullet)", "$$", "by Derived Categories, Lemma \\ref{derived-lemma-bounded-derived}.", "Thus this is a countable set as a countable colimit of" ], "refs": [ "derived-lemma-bounded-derived" ], "ref_ids": [ 1813 ] } ], "ref_ids": [] }, { "id": 6869, "type": "theorem", "label": "equiv-lemma-countable-perfect", "categories": [ "equiv" ], "title": "equiv-lemma-countable-perfect", "contents": [ "Let $X$ be a scheme of finite type over a countable Noetherian ring.", "Then the categories $D_{perf}(\\mathcal{O}_X)$ and", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ are countable." ], "refs": [], "proofs": [ { "contents": [ "Observe that $X$ is Noetherian by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}.", "Hence $D_{perf}(\\mathcal{O}_X)$ is a full subcategory of", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-perfect-on-noetherian}.", "Thus it suffices to prove", "the result for $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.", "Recall that", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X) = D^b(\\textit{Coh}(\\mathcal{O}_X))$", "by", "Derived Categories of Schemes, Proposition \\ref{perfect-proposition-DCoh}.", "Hence by Lemma \\ref{lemma-countable-abelian}", "it suffices to prove that $\\textit{Coh}(\\mathcal{O}_X)$ is", "countable. This we omit." ], "refs": [ "morphisms-lemma-finite-type-noetherian", "perfect-lemma-perfect-on-noetherian", "perfect-proposition-DCoh", "equiv-lemma-countable-abelian" ], "ref_ids": [ 5202, 6987, 7110, 6868 ] } ], "ref_ids": [] }, { "id": 6870, "type": "theorem", "label": "equiv-lemma-countable-isos", "categories": [ "equiv" ], "title": "equiv-lemma-countable-isos", "contents": [ "Let $K$ be an algebraically closed field.", "Let $S$ be a finite type scheme over $K$.", "Let $X \\to S$ and $Y \\to S$ be finite type morphisms.", "There exists a countable set $I$ and for $i \\in I$ a pair", "$(S_i \\to S, h_i)$ with the following properties", "\\begin{enumerate}", "\\item $S_i \\to S$ is a morphism of finite type, set", "$X_i = X \\times_S S_i$ and $Y_i = Y \\times_S S_i$,", "\\item $h_i : X_i \\to Y_i$ is an isomorphism over $S_i$, and", "\\item for any closed point $s \\in S(K)$ if $X_s \\cong Y_s$", "over $K = \\kappa(s)$ then $s$ is in the image of $S_i \\to S$", "for some $i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The field $K$ is the filtered union of its countable subfields.", "Dually, $\\Spec(K)$ is the cofiltered limit of the spectra", "of the countable subfields of $K$.", "Hence Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "guarantees that we can find a countable subfield", "$k$ and morphisms $X_0 \\to S_0$ and $Y_0 \\to S_0$", "of schemes of finite type over $k$ such that", "$X \\to S$ and $Y \\to S$ are the base changes of these.", "\\medskip\\noindent", "By Lemma \\ref{lemma-countable-finite-type} there is a countable set $I$ and", "pairs $(S_{0, i} \\to S_0, h_{0, i})$ such that", "\\begin{enumerate}", "\\item $S_{0, i} \\to S_0$ is a morphism of finite type, set", "$X_{0, i} = X_0 \\times_{S_0} S_{0, i}$ and", "$Y_{0, i} = Y_0 \\times_{S_0} S_{0, i}$,", "\\item $h_{0, i} : X_{0, i} \\to Y_{0, i}$ is an isomorphism over $S_{0, i}$.", "\\end{enumerate}", "such that every pair $(T \\to S_0, h_T)$ with $T \\to S_0$ of finite type", "and $h_T : X_0 \\times_{S_0} T \\to Y_0 \\times_{S_0} T$ an isomorphism", "is isomorphic to one of these.", "Denote $(S_i \\to S, h_i)$ the base change of $(S_{0, i} \\to S_0, h_{0, i})$", "by $\\Spec(K) \\to \\Spec(k)$.", "We claim this works.", "\\medskip\\noindent", "Let $s \\in S(K)$ and let $h_s : X_s \\to Y_s$ be an isomorphism over", "$K = \\kappa(s)$. We can write $K$ as the filtered union of its", "finitely generated $k$-subalgebras. Hence by", "Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation} and", "Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can find such a finitely generated $k$-subalgebra", "$K \\supset A \\supset k$ such that", "\\begin{enumerate}", "\\item there is a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[d]_s \\ar[r] &", "\\Spec(A) \\ar[d]^{s'} \\\\", "S \\ar[r] &", "S_0}", "$$", "for some morphism $s' : \\Spec(A) \\to S_0$ over $k$,", "\\item $h_s$ is the base change of an isomorphism", "$h_{s'} : X_0 \\times_{S_0, s'} \\Spec(A) \\to", "X_0 \\times_{S_0, s'} \\Spec(A)$ over $A$.", "\\end{enumerate}", "Of course, then $(s' : \\Spec(A) \\to S_0, h_{s'})$ is isomorphic", "to the pair $(S_{0, i} \\to S_0, h_{0, i})$ for some $i \\in I$.", "This concludes the proof because the commutative diagram", "in (1) shows that $s$ is in the image of", "the base change of $s'$ to $\\Spec(K)$." ], "refs": [ "limits-lemma-descend-finite-presentation", "equiv-lemma-countable-finite-type", "limits-proposition-characterize-locally-finite-presentation", "limits-lemma-descend-finite-presentation" ], "ref_ids": [ 15077, 6867, 15127, 15077 ] } ], "ref_ids": [] }, { "id": 6871, "type": "theorem", "label": "equiv-lemma-countable-equivs", "categories": [ "equiv" ], "title": "equiv-lemma-countable-equivs", "contents": [ "Let $K$ be an algebraically closed field. There exists a countable set $I$", "and for $i \\in I$ a pair $(S_i/K, X_i \\to S_i, Y_i \\to S_i, M_i)$", "with the following properties", "\\begin{enumerate}", "\\item $S_i$ is a scheme of finite type over $K$,", "\\item $X_i \\to S_i$ and $Y_i \\to S_i$ are proper smooth", "morphisms of schemes,", "\\item $M_i \\in D_{perf}(\\mathcal{O}_{X_i \\times_{S_i} Y_i})$", "is the Fourier-Mukai kernel of a relative equivalence from", "$X_i$ to $Y_i$ over $S_i$, and", "\\item for any smooth proper schemes $X$ and $Y$ over $K$", "such that there is a $K$-linear exact equivalence", "$D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$", "there exists an $i \\in I$ and a $s \\in S_i(K)$", "such that $X \\cong (X_i)_s$ and $Y \\cong (Y_i)_s$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a countable subfield $k \\subset K$ for example the prime field.", "By Lemmas \\ref{lemma-countable-finite-type} and \\ref{lemma-countable-perfect}", "there exists a countable set of isomorphism classes of systems", "over $k$ satisfying parts (1), (2), (3) of the lemma.", "Thus we can choose a countable set", "$I$ and for each $i \\in I$ such a system", "$$", "(S_{0, i}/k, X_{0, i} \\to S_{0, i}, Y_{0, i} \\to S_{0, i}, M_{0, i})", "$$", "over $k$ such that each isomorphism class occurs at least once.", "Denote $(S_i/K, X_i \\to S_i, Y_i \\to S_i, M_i)$ the base change", "of the displayed system to $K$. This system has properties (1), (2), (3),", "see Lemma \\ref{lemma-base-change-rek}. Let us prove property (4).", "\\medskip\\noindent", "Consider smooth proper schemes $X$ and $Y$ over $K$", "such that there is a $K$-linear exact equivalence", "$F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$.", "By Proposition \\ref{proposition-equivalence}", "we may assume that there exists an object", "$M \\in D_{perf}(\\mathcal{O}_{X \\times Y})$", "such that $F = \\Phi_M$ is the corresponding Fourier-Mukai functor.", "By Lemma \\ref{lemma-fourier-mukai-flat-proper-over-noetherian}", "there is an $M'$ in $D_{perf}(\\mathcal{O}_{Y \\times X})$", "such that $\\Phi_{M'}$ is the right adjoint to $\\Phi_M$.", "Since $\\Phi_M$ is an equivalence, this means that", "$\\Phi_{M'}$ is the quasi-inverse to $\\Phi_M$.", "By Lemma \\ref{lemma-fourier-mukai-flat-proper-over-noetherian}", "we see that the Fourier-Mukai functors defined by the objects", "$$", "A = R\\text{pr}_{13, *}(", "L\\text{pr}_{12}^*M", "\\otimes_{\\mathcal{O}_{X \\times Y \\times X}}^\\mathbf{L}", "L\\text{pr}_{23}^*M')", "$$", "in $D_{perf}(\\mathcal{O}_{X \\times X})$ and", "$$", "B = R\\text{pr}_{13, *}(", "L\\text{pr}_{12}^*M'", "\\otimes_{\\mathcal{O}_{Y \\times X \\times Y}}^\\mathbf{L}", "L\\text{pr}_{23}^*M)", "$$", "in $D_{perf}(\\mathcal{O}_{Y \\times Y})$ ", "are isomorphic to", "$\\text{id} : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_X)$", "and", "$\\text{id} : D_{perf}(\\mathcal{O}_Y) \\to D_{perf}(\\mathcal{O}_Y)$", "Hence", "$A \\cong \\Delta_{X/K, *}\\mathcal{O}_X$ and", "$B \\cong \\Delta_{Y/K, *}\\mathcal{O}_Y$", "by Lemma \\ref{lemma-uniqueness}. Hence we see that $M$ is the", "Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$", "over $K$ by definition.", "\\medskip\\noindent", "We can write $K$ as the filtered colimit of its finite type", "$k$-subalgebras $A \\subset K$. By", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can find $X_0, Y_0$ of finite type over $A$ whose", "base changes to $K$ produces $X$ and $Y$.", "By Limits, Lemmas", "\\ref{limits-lemma-eventually-proper} and \\ref{limits-lemma-descend-smooth}", "after enlarging $A$ we may assume $X_0$ and $Y_0$", "are smooth and proper over $A$.", "By Lemma \\ref{lemma-descend-rek}", "after enlarging $A$ we may assume $M$ is the pullback of", "some $M_0 \\in D_{perf}(\\mathcal{O}_{X_0 \\times_{\\Spec(A)} Y_0})$", "which is the Fourier-Mukai kernel of a relative equivalence", "from $X_0$ to $Y_0$ over $\\Spec(A)$.", "Thus we see that $(S_0/k, X_0 \\to S_0, Y_0 \\to S_0, M_0)$", "is isomorphic to", "$(S_{0, i}/k, X_{0, i} \\to S_{0, i}, Y_{0, i} \\to S_{0, i}, M_{0, i})$", "for some $i \\in I$.", "Since $S_i = S_{0, i} \\times_{\\Spec(k)} \\Spec(K)$", "we conclude that (4) is true with $s : \\Spec(K) \\to S_i$", "induced by the morphism $\\Spec(K) \\to \\Spec(A) \\cong S_{0, i}$", "we get from $A \\subset K$." ], "refs": [ "equiv-lemma-countable-finite-type", "equiv-lemma-countable-perfect", "equiv-lemma-base-change-rek", "equiv-proposition-equivalence", "equiv-lemma-fourier-mukai-flat-proper-over-noetherian", "equiv-lemma-fourier-mukai-flat-proper-over-noetherian", "equiv-lemma-uniqueness", "limits-lemma-descend-finite-presentation", "limits-lemma-eventually-proper", "limits-lemma-descend-smooth", "equiv-lemma-descend-rek" ], "ref_ids": [ 6867, 6869, 6860, 6873, 6821, 6821, 6857, 15077, 15089, 15064, 6861 ] } ], "ref_ids": [] }, { "id": 6872, "type": "theorem", "label": "equiv-proposition-siblings-isomorphic", "categories": [ "equiv" ], "title": "equiv-proposition-siblings-isomorphic", "contents": [ "\\begin{reference}", "\\cite[Proposition 2.16]{Orlov-K3}", "\\end{reference}", "Let $F$ and $F'$ be siblings as in Definition \\ref{definition-siblings}.", "Assume that $F$ is fully faithful and that $\\mathcal{A}$ has enough", "negative objects (see above). Then $F$ and $F'$ are isomorphic functors." ], "refs": [ "equiv-definition-siblings" ], "proofs": [ { "contents": [ "By part (2) of Definition \\ref{definition-siblings} the image of the functor", "$F'$ is contained in the essential image of the functor $F$. Hence", "the functor $H = F^{-1} \\circ F'$ is a sibling of the identity functor.", "This reduces us to the case described in the next paragraph.", "\\medskip\\noindent", "Let $\\mathcal{D} = D^b(\\mathcal{A})$. We have to show a sibling", "$F : \\mathcal{D} \\to \\mathcal{D}$ of the identity functor is", "isomorphic to the identity functor. Given an object $X$ of $\\mathcal{D}$", "let us say $X$ has {\\it width} $w = w(X)$ if $w \\geq 0$ is minimal", "such that there exists an integer $a \\in \\mathbf{Z}$ with $H^i(X) = 0$", "for $i \\not \\in [a, a + w - 1]$. Since $F$ is a sibling of the identity", "and since $F \\circ [n] = [n] \\circ F$ we are aready given isomorphisms", "$$", "c_X : X \\to F(X)", "$$", "for $w(X) \\leq 1$ compatible with shifts. Moreover, if $X = A[-a]$ and", "$X' = A'[-a]$ for some $A, A' \\in \\Ob(\\mathcal{A})$ then for any morphism", "$f : X \\to X'$ the diagram", "\\begin{equation}", "\\label{equation-to-show}", "\\vcenter{", "\\xymatrix{", "X \\ar[d]_{c_X} \\ar[r]_f &", "X' \\ar[d]^{c_{X'}} \\\\", "F(X) \\ar[r]^{F(f)} &", "F(X')", "}", "}", "\\end{equation}", "is commutative.", "\\medskip\\noindent", "Next, let us show that for any morphism $f : X \\to X'$ with", "$w(X), w(X') \\leq 1$ the diagram (\\ref{equation-to-show}) commutes.", "If $X$ or $X'$ is zero, this is clear. If not then we can write", "$X = A[-a]$ and $X' = A'[-a']$ for unique $A, A'$ in $\\mathcal{A}$", "and $a, a' \\in \\mathbf{Z}$. The case $a = a'$ was discussed above.", "If $a' > a$, then $f = 0$ (Derived Categories, Lemma", "\\ref{derived-lemma-negative-exts}) and the result is clear.", "If $a' < a$ then $f$ corresponds to an element", "$\\xi \\in \\Ext^q(A, A')$ with $q = a - a'$. Using Yoneda extensions, see", "Derived Categories, Section \\ref{derived-section-ext}, we can find", "$A = A_0, A_1, \\ldots, A_{q - 1}, A_q = A' \\in \\Ob(\\mathcal{A})$ and", "elements", "$$", "\\xi_i \\in \\Ext^1(A_{i - 1}, A_i)", "$$", "such that $\\xi$ is the composition $\\xi_q \\circ \\ldots \\circ \\xi_1$.", "In other words, setting $X_i = A_i[-a + i]$", "we obtain morphisms", "$$", "X = X_0 \\xrightarrow{f_1} X_1 \\to \\ldots \\to X_{q - 1}", "\\xrightarrow{f_q} X_q = X'", "$$", "whose compostion is $f$. Since the commutativity of (\\ref{equation-to-show})", "for $f_1, \\ldots, f_q$ implies it for $f$, this reduces us to the case $q = 1$.", "In this case after shifting we may assume we have a distinguished triangle", "$$", "A' \\to E \\to A \\xrightarrow{f} A'[1]", "$$", "Observe that $E$ is an object of $\\mathcal{A}$. Consider the following", "diagram", "$$", "\\xymatrix{", "E \\ar[d]_{c_E} \\ar[r] &", "A \\ar[d]_{c_A} \\ar[r]_f &", "A'[1] \\ar[d]^{c_{A'}[1]}", "\\ar@{..>}@<-1ex>[d]_\\gamma \\ar@{..>}[ld]^\\epsilon \\ar[r] &", "E[1] \\ar[d]^{c_E[1]} \\\\", "F(E) \\ar[r] &", "F(A) \\ar[r]^{F(f)} &", "F(A')[1] \\ar[r] &", "F(E)[1]", "}", "$$", "whose rows are distinguished triangles.", "The square on the right commutes already but we don't yet know that", "the middle square does. By the axioms of a triangulated category", "we can find a morphism $\\delta$ which does make the diagram commute.", "Then $\\gamma - c_{A'}[1]$ composed with", "$F(A')[1] \\to F(E)[1]$ is zero hence we", "can find $\\epsilon : A'[1] \\to F(A)$ such that", "$\\gamma - c_{A'}[1] = F(f) \\circ \\epsilon$. However, any arrow", "$A'[1] \\to F(A)$ is zero as it is a negative ext class", "between objects of $\\mathcal{A}$. Hence $\\gamma = c_{A'}[1]$", "and we conclude the middle square commutes too which is what we", "wanted to show.", "\\medskip\\noindent", "To finish the proof we are going to argue by induction on $w$", "that there exist isomorphisms $c_X : X \\to F(X)$ for all", "$X$ with $w(X) \\leq w$ compatible with all morphisms between", "such objects. The base case $w = 1$ was shown above. Assume", "we know the result for some $w \\geq 1$.", "\\medskip\\noindent", "Let $X$ be an object with $w(X) = w + 1$. Pick $a \\in \\mathbf{Z}$ with", "$H^i(X) = 0$ for $i \\not \\in [a, a + w]$. Set $b = a + w$ so that", "$H^b(X)$ is nonzero. Pick $N$ in $\\mathcal{A}$ such that there exists", "a surjection $N \\to H^b(X)$, such that $\\Hom(H^b(X), N) = 0$", "and such that $\\Ext^q(N, H^i(X)) = 0$ for $i \\in \\mathbf{Z}$ and $q > 0$.", "This is possible because $\\mathcal{A}$ has enough negative objects", "by appplying the definition to $\\bigoplus H^i(X)$.", "By the vanishing of Exts we can lift the surjection", "$N \\to H^b(X)$ to a morphism $N[-b] \\to X$; details omitted.", "Let us call a morphism $N[-b] \\to X$ constructed in this manner a", "{\\it good morphism}. Given a good morphism $N[-b] \\to X$", "choose a distinguished diagram", "$$", "N[-b] \\to X \\to Y \\to N[-b + 1]", "$$", "Computing the long exact cohomology sequence we find", "$w(Y) \\leq w$. Hence by induction we find the solid arrows", "in the following diagram", "$$", "\\xymatrix{", "N[-b] \\ar[r] \\ar[d]_{c_N[-b]} &", "X \\ar[r] \\ar@{..>}[d]_{c_{N[-b] \\to X}} &", "Y \\ar[r] \\ar[d]^{c_Y} &", "N[-b + 1] \\ar[d]^{c_N[-b + 1]} \\\\", "F(N)[-b] \\ar[r] &", "F(X) \\ar[r] &", "F(Y) \\ar[r] &", "F(N)[-b + 1]", "}", "$$", "We obtain the dotted arrow $c_{N[-b] \\to X}$.", "By Derived Categories, Lemma \\ref{derived-lemma-uniqueness-third-arrow}", "the dotted arrow is unique because $\\Hom(X, F(N)[-b]) \\cong \\Hom(X, N[-b]) = 0$", "by our choice of $N$. In fact, $c_{N[-b] \\to X}$ is the unique dotted", "arrow making the square with vertices $X, Y, F(X), F(Y)$ commute.", "Our goal is to show that $c_{N[-b] \\to X}$ is independent of", "the choice of good morphism $N[-b] \\to X$ and that the diagrams", "(\\ref{equation-to-show}) commute.", "\\medskip\\noindent", "Independence of the choice of good morphism. Given two good morphisms", "$N[-b] \\to X$ and $N'[-b] \\to X$ we get another good morphism, namely", "$(N \\oplus N')[-b] \\to X$. Thus we may assume $N'[-b] \\to X$ factors", "as $N'[-b] \\to N[-b] \\to X$ for some morphism $N' \\to N$.", "Choose distinguished triangles $N[-b] \\to X \\to Y \\to N[-b + 1]$ and", "$N'[-b] \\to X \\to Y' \\to N'[-b + 1]$. By axiom TR3 we can find", "a morphism $g : Y' \\to Y$ which joint with $\\text{id}_X$ and $N' \\to N$", "forms a morphism of triangles. Since we have", "(\\ref{equation-to-show}) for $g$ we conclude that", "$$", "(F(X) \\to F(Y)) \\circ c_{N'[-b] \\to X} = (F(X) \\to F(Y)) \\circ c_{N[-b] \\to X}", "$$", "The uniqueness of $c_{N[-b] \\to X}$ pointed out in the construction", "above now shows that $c_{N'[-b] \\to X} = c_{N[-b] \\to X}$.", "\\medskip\\noindent", "Let $f : X \\to X'$ be a morphism of objects with $w(X) \\leq w + 1$", "and $w(X') \\leq w + 1$. Choose $a \\leq b \\leq a + w$ such that", "$H^i(X) = 0$ for $i \\not \\in [a, b]$ and", "$a' \\leq b' \\leq a' + w$ such that $H^i(X') = 0$ for", "$i \\not \\in [a', b']$. We will use induction on", "$(b' - a') + (b - a)$ to show this. (The base case", "is when this number is zero which is OK because $w \\geq 1$.)", "We distinguish two cases.", "\\medskip\\noindent", "Case I: $b' < b$. In this case we choose a good morphism", "$N[-b] \\to X$ such that in addition $\\Ext^q(N, H^i(X')) = 0$", "for $q > 0$ and all $i$. Choose a distuiguished triangle", "$N[-b] \\to X \\to Y \\to N[-b + 1]$. Since", "$\\Hom(N[-b], X') = 0$ by our choice of $N$ and we find that $f$ factors", "as $X \\to Y \\to X'$. Since $H^i(Y)$ is nonzero only for $i \\in [a, b - 1]$", "we see by induction that (\\ref{equation-to-show}) commutes for", "$Y \\to X'$. The diagram (\\ref{equation-to-show}) commutes for", "$X \\to Y$ by construction if $w(X) = w + 1$ and by our first", "induction hypothesis if $w(X) \\leq w$.", "Hence (\\ref{equation-to-show}) commutes for $f$.", "\\medskip\\noindent", "Case II: $b' \\geq b$. In this case we choose a good morphism", "$N'[-b'] \\to X'$ such that $\\Hom(H^{b'}(X), N') = 0$ (this is", "relevant only if $b' = b$). We choose a distinguished triangle", "$N'[-b'] \\to X' \\to Y' \\to N'[-b' + 1]$. Since", "$\\Hom(X, X') \\to \\Hom(X, Y')$ is injective by our choice of $N'$", "(details omitted) the same is true for", "$\\Hom(X, F(X')) \\to \\Hom(X, F(Y'))$.", "Hence it suffices in this case to check that", "(\\ref{equation-to-show}) commutes for the composition $X \\to Y'$", "of the morphisms $X \\to X' \\to Y'$.", "Since $H^i(Y')$ is nonzero only for $i \\in [a', b' - 1]$", "we conclude by induction hypothesis." ], "refs": [ "equiv-definition-siblings", "derived-lemma-negative-exts", "derived-lemma-uniqueness-third-arrow" ], "ref_ids": [ 6876, 1893, 1763 ] } ], "ref_ids": [ 6876 ] }, { "id": 6873, "type": "theorem", "label": "equiv-proposition-equivalence", "categories": [ "equiv" ], "title": "equiv-proposition-equivalence", "contents": [ "Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$.", "If $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$", "is a $k$-linear exact equivalence of triangulated categories then", "there exists a Fourier-Mukai functor", "$F' : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ whose", "kernel is in $D_{perf}(\\mathcal{O}_{X \\times Y})$", "which is an equivalence and a sibling of $F$." ], "refs": [], "proofs": [ { "contents": [ "The functor $F'$ of Lemma \\ref{lemma-fully-faithful}", "is an equivalence by Lemma \\ref{lemma-sibling-faithful}." ], "refs": [ "equiv-lemma-fully-faithful", "equiv-lemma-sibling-faithful" ], "ref_ids": [ 6856, 6845 ] } ], "ref_ids": [] }, { "id": 6882, "type": "theorem", "label": "stacks-more-morphisms-theorem-chow-finite-type", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-theorem-chow-finite-type", "contents": [ "\\begin{reference}", "This is a result due to Ofer Gabber, see", "\\cite[Theorem 1.1]{olsson_proper}", "\\end{reference}", "Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack", "to an algebraic space. Assume", "\\begin{enumerate}", "\\item $Y$ is quasi-compact and quasi-separated,", "\\item $f$ is separated of finite type.", "\\end{enumerate}", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{X} \\ar[rd] & X \\ar[l] \\ar[d] \\ar[r] & \\overline{X} \\ar[ld] \\\\", "& Y", "}", "$$", "where $X \\to \\mathcal{X}$ is proper surjective,", "$X \\to \\overline{X}$ is an open immersion, and", "$\\overline{X} \\to Y$ is proper morphism of algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "The rough idea is to use that $\\mathcal{X}$ has a dense open", "which is a gerbe (Morphisms of Stacks, Proposition", "\\ref{stacks-morphisms-proposition-open-stratum})", "and appeal to Lemma \\ref{lemma-make-section}.", "The reason this does not work is that the open may not be", "quasi-compact and one runs into technical problems. Thus", "we first do a (standard) reduction to the Noetherian case.", "\\medskip\\noindent", "First we choose a closed immersion", "$\\mathcal{X} \\to \\mathcal{X}'$ where $\\mathcal{X}'$", "is an algebraic stack separated and of finite type over $Y$.", "See Limits of Stacks, Lemma", "\\ref{stacks-limits-lemma-separated-closed-in-finite-presentation}.", "Clearly it suffices to prove the theorem for", "$\\mathcal{X}'$, hence we may assume $\\mathcal{X} \\to Y$", "is separated and of finite presentation.", "\\medskip\\noindent", "Assume $\\mathcal{X} \\to Y$ is separated and of finite presentation.", "By Limits of Spaces, Proposition \\ref{spaces-limits-proposition-approximate}", "we can write $Y = \\lim Y_i$ as the directed limit of a system", "of Noetherian algebraic spaces with affine transition morphisms.", "By Limits of Stacks, Lemma \\ref{stacks-limits-lemma-descend-a-stack-down}", "there is an $i$ and a morphism $\\mathcal{X}_i \\to Y_i$ of finite presentation", "from an algebraic stack to $Y_i$ such that", "$\\mathcal{X} = Y \\times_{Y_i} \\mathcal{X}_i$.", "After increasing $i$ we may assume that $\\mathcal{X}_i \\to Y_i$", "is separated, see Limits of Stacks, Lemma", "\\ref{stacks-limits-lemma-eventually-separated}.", "Then it suffices to prove the theorem", "for $\\mathcal{X}_i \\to Y_i$. This reduces us to the case discussed", "in the next paragraph.", "\\medskip\\noindent", "Assume $Y$ is Noetherian. We may replace $\\mathcal{X}$ by its", "reduction (Properties of Stacks, Definition", "\\ref{stacks-properties-definition-reduced-induced-stack}).", "This reduces us to the case discussed", "in the next paragraph.", "\\medskip\\noindent", "Assume $Y$ is Noetherian and $\\mathcal{X}$ is reduced.", "Since $\\mathcal{X} \\to Y$ is separated and $Y$ quasi-separated,", "we see that $\\mathcal{X}$ is quasi-separated as an algebraic stack.", "Hence the inertia $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$", "is quasi-compact. Thus by Morphisms of Stacks, Proposition", "\\ref{stacks-morphisms-proposition-open-stratum}", "there exists a dense open substack $\\mathcal{V} \\subset \\mathcal{X}$", "which is a gerbe. Let $\\mathcal{V} \\to V$ be the morphism", "which expresses $\\mathcal{V}$ as a gerbe over the algebraic space $V$.", "See", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-gerbe-over-iso-classes}", "for a construction of $\\mathcal{V} \\to V$.", "This construction in particular shows that the morphism", "$\\mathcal{V} \\to Y$ factors as $\\mathcal{V} \\to V \\to Y$.", "Picture", "$$", "\\xymatrix{", "\\mathcal{V} \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\", "V \\ar[r] & Y", "}", "$$", "Since the morphism $\\mathcal{V} \\to V$ is surjective, flat, and", "of finite presentation", "(Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-gerbe-fppf})", "and since $\\mathcal{V} \\to Y$ is locally of finite presentation,", "it follows that $V \\to Y$ is locally of finite presentation", "(Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-flat-finite-presentation-permanence}).", "Note that $\\mathcal{V} \\to V$ is a universal homeomorphism", "(Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-gerbe-bijection-points}).", "Since $\\mathcal{V}$ is quasi-compact (see", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-locally-closed-in-noetherian})", "we see that $V$ is quasi-compact.", "Finally, since $\\mathcal{V} \\to Y$ is separated the same is true", "for $V \\to Y$ by", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-check-separated-on-ui-cover}", "applied to $\\mathcal{V} \\to V \\to Y$", "(whose assumptions are satisfied as we've already seen).", "\\medskip\\noindent", "All of the above means that the assumptions of", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-embedding-into-affine-over-qs}", "apply to the morphism $V \\to Y$. Thus we can find a dense open", "subspace $V' \\subset V$ and an immersion $V' \\to \\mathbf{P}^n_Y$", "over $Y$. Clearly we may replace $V$ by $V'$ and $\\mathcal{V}$", "by the inverse image of $V'$ in $\\mathcal{V}$ (recall that", "$|\\mathcal{V}| = |V|$ as we've seen above).", "Thus we may assume we have a diagram", "$$", "\\xymatrix{", "\\mathcal{V} \\ar[rr] \\ar[d] & & \\mathcal{X} \\ar[d] \\\\", "V \\ar[r] & \\mathbf{P}^n_Y \\ar[r] & Y", "}", "$$", "where the arrow $V \\to \\mathbf{P}^n_Y$ is an immersion.", "Let $\\mathcal{X}'$ be the scheme theoretic image of the morphism", "$$", "j : \\mathcal{V} \\longrightarrow \\mathbf{P}^n_Y \\times_Y \\mathcal{X}", "$$", "and let $Y'$ be the scheme theoretic image of the morphism", "$V \\to \\mathbf{P}^n_Y$. We obtain a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{V} \\ar[r] \\ar[d] &", "\\mathcal{X}' \\ar[r] \\ar[d] &", "\\mathbf{P}^n_Y \\times_Y \\mathcal{X} \\ar[d] \\ar[r] &", "\\mathcal{X} \\ar[d] \\\\", "V \\ar[r] &", "Y' \\ar[r] &", "\\mathbf{P}^n_Y \\ar[r] &", "Y", "}", "$$", "(See Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-factor-factor}).", "We claim that $\\mathcal{V} = V \\times_{Y'} \\mathcal{X}'$ and that", "Lemma \\ref{lemma-make-section} applies to the morphism $\\mathcal{X}' \\to Y'$", "and the open subspace $V \\subset Y'$. If the claim is true, then we obtain", "$$", "\\xymatrix{", "\\overline{X} \\ar[rd]_{\\overline{g}} &", "X \\ar[l] \\ar[d]_g \\ar[r]_h & \\mathcal{X}' \\ar[ld]^f \\\\", "& Y'", "}", "$$", "with $X \\to \\overline{X}$ an open immersion, $\\overline{g}$ and $h$ proper,", "and such that $|V|$ is contained in the image of $|g|$.", "Then the composition $X \\to \\mathcal{X}' \\to \\mathcal{X}$ is", "proper (as a composition of proper morphisms) and its image", "contains $|\\mathcal{V}|$, hence this composition is surjective.", "As well, $\\overline{X} \\to Y' \\to Y$ is proper as a composition", "of proper morphisms.", "\\medskip\\noindent", "The last step is to prove the claim.", "Observe that $\\mathcal{X}' \\to Y'$ is separated and of finite type,", "that $Y'$ is quasi-compact and quasi-separated, and that $V$ is quasi-compact", "(we omit checking all the details completely).", "Next, we observe that", "$b : \\mathcal{X}' \\to \\mathcal{X}$ is an isomorphism over", "$\\mathcal{V}$ by Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-scheme-theoretic-image-of-partial-section}.", "In particular $\\mathcal{V}$ is identified with an open substack", "of $\\mathcal{X}'$.", "The morphism $j$ is quasi-compact", "(source is quasi-compact and target is quasi-separated), so formation", "of the scheme theoretic image of $j$ commutes with flat base change by", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-existence-plus-flat-base-change}.", "In particular we see that $V \\times_{Y'} \\mathcal{X}'$ is the", "scheme theoretic image of", "$\\mathcal{V} \\to V \\times_{Y'} \\mathcal{X}'$.", "However, by Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-universally-closed-permanence}", "the image of $|\\mathcal{V}| \\to |V \\times_{Y'} \\mathcal{X}'|$", "is closed (use that $\\mathcal{V} \\to V$ is a universal homeomorphism", "as we've seen above and hence is universally closed).", "Also the image is dense (combine what we just said with", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-topology-scheme-theoretic-image})", "we conclude $|\\mathcal{V}| = |V \\times_{Y'} \\mathcal{X}'|$.", "Thus $\\mathcal{V} \\to V \\times_{Y'} \\mathcal{X}'$ is an", "isomorphism and the proof of the claim is complete." ], "refs": [ "stacks-morphisms-proposition-open-stratum", "stacks-limits-lemma-separated-closed-in-finite-presentation", "spaces-limits-proposition-approximate", "stacks-limits-lemma-descend-a-stack-down", "stacks-limits-lemma-eventually-separated", "stacks-properties-definition-reduced-induced-stack", "stacks-morphisms-proposition-open-stratum", "stacks-morphisms-lemma-gerbe-over-iso-classes", "stacks-morphisms-lemma-gerbe-fppf", "stacks-morphisms-lemma-flat-finite-presentation-permanence", "stacks-morphisms-lemma-gerbe-bijection-points", "stacks-morphisms-lemma-locally-closed-in-noetherian", "stacks-morphisms-lemma-check-separated-on-ui-cover", "spaces-limits-lemma-embedding-into-affine-over-qs", "stacks-morphisms-lemma-factor-factor", "stacks-morphisms-lemma-existence-plus-flat-base-change", "stacks-morphisms-lemma-universally-closed-permanence", "stacks-morphisms-lemma-topology-scheme-theoretic-image" ], "ref_ids": [ 7600, 15023, 4656, 15021, 15020, 8922, 7600, 7516, 7522, 7510, 7525, 7431, 7515, 4620, 7567, 7568, 7563, 7569 ] } ], "ref_ids": [] }, { "id": 6883, "type": "theorem", "label": "stacks-more-morphisms-theorem-keel-mori", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-theorem-keel-mori", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Assume", "$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite.", "Then there exists a uniform categorical moduli space", "$$", "f : \\mathcal{X} \\longrightarrow M", "$$", "and $f$ is separated, quasi-compact, and a universal homeomorphism." ], "refs": [], "proofs": [ { "contents": [ "We choose a set $I$\\footnote{The reader who is still keeping", "track of set theoretic issues should make sure $I$ is not too large.}", "and for $i \\in I$ a morphism of algebraic stacks", "$g_i : \\mathcal{X}_i \\to \\mathcal{X}$ as in", "Lemma \\ref{lemma-etale-local-finite-inertia}; we will use all", "of the properties listed in this lemma without further mention.", "Let", "$$", "f_i : \\mathcal{X}_i \\to M_i", "$$", "be as in Lemma \\ref{lemma-well-nigh-affine-moduli-space}.", "Consider the stacks", "$$", "\\mathcal{X}_{ij} = \\mathcal{X}_i \\times_{g_i, \\mathcal{X}, g_j} \\mathcal{X}_j", "$$", "for $i, j \\in I$. The projections $\\mathcal{X}_{ij} \\to \\mathcal{X}_i$", "and $\\mathcal{X}_{ij} \\to \\mathcal{X}_j$ are separated", "by Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-base-change-separated},", "\\'etale by Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-base-change-etale},", "and induce isomorphisms on automorphism groups", "(as in Morphisms of Stacks, Remark", "\\ref{stacks-morphisms-remark-identify-automorphism-groups}) by", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-base-change-stabilizer-preserving}.", "Thus we may apply Lemma \\ref{lemma-etale-separated-over-well-nigh-affine}", "to find a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{X}_i \\ar[d]_{f_i} &", "\\mathcal{X}_{ij} \\ar[d]_{f_{ij}} \\ar[l] \\ar[r] &", "\\mathcal{X}_j \\ar[d]_{f_j} \\\\", "M_i &", "M_{ij} \\ar[l] \\ar[r] &", "M_j", "}", "$$", "with cartesian squares where $M_{ij} \\to M_i$ and $M_{ij} \\to M_j$", "are separated \\'etale morphisms of schemes; here we also use that $f_i$", "is a uniform categorical quotient by", "Lemma \\ref{lemma-moduli-space-finite-affine}.", "Claim:", "$$", "\\coprod M_{ij} \\longrightarrow \\coprod M_i \\times \\coprod M_i", "$$", "is an \\'etale equivalence relation.", "\\medskip\\noindent", "Proof of the claim. Set $R = \\coprod M_{ij}$ and $U = \\coprod M_i$.", "We have already seen that $t : R \\to U$ and $s : R \\to U$ are \\'etale.", "Let us construct a morphism $c : R \\times_{s, U, t} R \\to R$", "compatible with $\\text{pr}_{13} : U \\times U \\times U \\to U \\times U$.", "Namely, for $i, j, k \\in I$ we consider", "$$", "\\mathcal{X}_{ijk} =", "\\mathcal{X}_i \\times_{g_i, \\mathcal{X}, g_j} \\mathcal{X}_j", "\\times_{g_j, \\mathcal{X}, g_k} \\mathcal{X}_k =", "\\mathcal{X}_{ij} \\times_{\\mathcal{X}_j} \\mathcal{X}_{jk}", "$$", "Arguing exactly as in the previous paragraph,", "we find that $M_{ijk} = M_{ij} \\times_{M_j} M_{jk}$", "is a categorical moduli space for $\\mathcal{X}_{ijk}$.", "In particular, there is a canonical morphism", "$M_{ijk} = M_{ij} \\times_{M_j} M_{jk} \\to M_{ik}$", "coming from the projection $\\mathcal{X}_{ijk} \\to \\mathcal{X}_{ik}$.", "Putting these morphisms together we obtain the morphism $c$.", "In a similar fashion we construct a morphism $e : U \\to R$", "compatible with $\\Delta : U \\to U \\times U$ and", "$i : R \\to R$ compatible with the flip $U \\times U \\to U \\times U$.", "Let $k$ be an algebraically closed field. Then", "$$", "\\Mor(\\Spec(k), \\mathcal{X}_i) \\to \\Mor(\\Spec(k), M_i) = M_i(k)", "$$", "is bijective on isomorphism classes and the same remains true after any", "base change by a morphism $M' \\to M$. This follows from our choice", "of $f_i$ and Morphisms of Stacks, Lemmas", "\\ref{stacks-morphisms-lemma-universally-injective} and", "\\ref{stacks-morphisms-lemma-universally-injective-point}.", "By construction of $2$-fibred products the diagram", "$$", "\\xymatrix{", "\\Mor(\\Spec(k), \\mathcal{X}_{ij}) \\ar[d] \\ar[r] &", "\\Mor(\\Spec(k), \\mathcal{X}_j) \\ar[d] \\\\", "\\Mor(\\Spec(k), \\mathcal{X}_i) \\ar[r] &", "\\Mor(\\Spec(k), \\mathcal{X})", "}", "$$", "is a fibre product of categories. By our choice of $g_i$ the", "functors in this diagram induce bijections on automorphism groups.", "It follows that this diagram induces a fibre product diagram", "on sets of isomorphism classes! Thus we see that", "$$", "R(k) = U(k) \\times_{|\\Mor(\\Spec(k), \\mathcal{X})|} U(k)", "$$", "where $|\\Mor(\\Spec(k), \\mathcal{X})|$ denotes the set", "of isomorphism classes.", "In particular, for any algebraically closed field $k$", "the map on $k$-valued point is an equivalence relation.", "We conclude the claim holds by", "Groupoids, Lemma \\ref{groupoids-lemma-etale-equivalence-relation}.", "\\medskip\\noindent", "Let $M = U/R$ be the algebraic space which is the quotient of the above", "\\'etale equivalence relation, see", "Spaces, Theorem \\ref{spaces-theorem-presentation}.", "There is a canonical morphism $f : \\mathcal{X} \\to M$", "fitting into commutative diagrams", "\\begin{equation}", "\\label{equation-fundamental-diagram}", "\\xymatrix{", "\\mathcal{X}_i \\ar[r]_{g_i} \\ar[d]_{f_i} & \\mathcal{X} \\ar[d]^f \\\\", "M_i \\ar[r] & M", "}", "\\end{equation}", "Namely, such a morphism $f$ is given by a functor", "$$", "f : \\Mor(T, \\mathcal{X}) \\longrightarrow \\Mor(T, M)", "$$", "for any scheme $T$ compatible with base change. Let $a : T \\to \\mathcal{X}$", "be an object of the left hand side. We obtain an \\'etale covering", "$\\{T_i \\to T\\}$ with $T_i = \\mathcal{X}_i \\times_\\mathcal{X} T$", "and morphisms $a_i : T_i \\to \\mathcal{X}_i$. Then we get", "$b_i = f_i \\circ a_i : T_i \\to M_i$. Since", "$T_i \\times_T T_j = \\mathcal{X}_{ij} \\times_\\mathcal{X} T$", "we moreover get a morphism $a_{ij} : T_i \\times_T T_j \\to \\mathcal{X}_{ij}$.", "Setting $b_{ij} = f_{ij} \\circ a_{ij}$ we find that", "$b_i \\times b_j$ factors through the monomorphism", "$M_{ij} \\to M_i \\times M_j$. Hence the morphisms", "$$", "T_i \\xrightarrow{b_i} M_i \\to M", "$$", "agree on $T_i \\times_T T_j$. As $M$ is a sheaf for the \\'etale", "topology, we see that these morphisms glue to a unique", "morphism $b = f(a) : T \\to M$. We omit the verification that", "this construction is compatible with base change and we omit", "the verification that the diagrams", "(\\ref{equation-fundamental-diagram}) commute.", "\\medskip\\noindent", "Claim: the diagrams (\\ref{equation-fundamental-diagram}) are cartesian.", "To see this we study the induced morphism", "$$", "h_i : \\mathcal{X}_i \\longrightarrow M_i \\times_M \\mathcal{X}", "$$", "This is a morphism of stacks \\'etale over $\\mathcal{X}$", "and hence $h_i$ is \\'etale (Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-etale-permanence}).", "Since $g_i$ is separated, we see $h_i$ is separated", "(use Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-compose-after-separated} and the fact", "seen above that the diagonal of $\\mathcal{X}$ is separated).", "The morphism $h_i$ induces isomorphisms on automorphism groups", "(Morphisms of Stacks, Remark", "\\ref{stacks-morphisms-remark-identify-automorphism-groups})", "as this is true for $g_i$. For an algebraically closed field $k$", "the diagram", "$$", "\\xymatrix{", "\\Mor(\\Spec(k), M_i \\times_M \\mathcal{X}) \\ar[r] \\ar[d] &", "\\Mor(\\Spec(k), \\mathcal{X}) \\ar[d] \\\\", "M_i(k) \\ar[r] &", "M(k)", "}", "$$", "is a catesian diagram of categories and the top arrow", "induces bijections on automorphism groups.", "On the other hand, we have", "$$", "M(k) = U(k)/R(k) = U(k)/", "U(k) \\times_{|\\Mor(\\Spec(k), \\mathcal{X})|} U(k) =", "|\\Mor(\\Spec(k), \\mathcal{X})|", "$$", "by what we said above. Thus the right vertical arrow in the", "cartesian diagram above is a bijection on isomorphism classes.", "We conclude that", "$|\\Mor(\\Spec(k), M_i \\times_M \\mathcal{X})| \\to M_i(k)$ is bijective.", "Review: $h_i$ is a separated, \\'etale, induces isomorphisms on", "automorphism groups (as in Morphisms of Stacks, Remark", "\\ref{stacks-morphisms-remark-identify-automorphism-groups}), and", "induces an equivalence on fibre categories over algebraically closed fields.", "Hence it is an isomorphism by Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-etale-iso}.", "\\medskip\\noindent", "From the claim we get in particular the following:", "we have a surjective \\'etale morphism $U \\to M$", "such that the base change of $f$ is separated, quasi-compact,", "and a universal homeomorphism. It follows that $f$ is separated,", "quasi-compact, and a universal homeomorphism.", "See Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-check-separated-covering},", "\\ref{stacks-morphisms-lemma-check-quasi-compact-covering}, and", "\\ref{stacks-morphisms-lemma-check-universal-homeomorphism-covering}", "\\medskip\\noindent", "To finish the proof we have to show that $f : \\mathcal{X} \\to M$", "is a uniform categorical moduli space.", "To prove this it suffices to show that given a flat morphism", "$M' \\to M$ of algebraic spaces, the base change", "$$", "M' \\times_M \\mathcal{X} \\longrightarrow M'", "$$", "is a categorical moduli space. Thus we consider a morphism", "$$", "\\theta : M' \\times_M \\mathcal{X} \\longrightarrow E", "$$", "where $E$ is an algebraic space. For each $i$ we know that", "$f_i$ is a uniform categorical moduli space. Hence we obtain", "$$", "\\xymatrix{", "M' \\times_M \\mathcal{X}_i \\ar[d] \\ar[r] &", "M' \\times_M \\mathcal{X} \\ar[d]^\\theta \\\\", "M' \\times_M M_i \\ar[r]^{\\psi_i} &", "E", "}", "$$", "Since $\\{M' \\times_M M_i \\to M'\\}$ is an \\'etale covering,", "to obtain the desired morphism $\\psi : M' \\to E$ it suffices", "to show that $\\psi_i$ and $\\psi_j$ agree over", "$M' \\times_M M_i \\times_M M_j = M' \\times_M M_{ij}$.", "This follows easily from the fact that", "$f_{ij} : \\mathcal{X}_{ij} =", "\\mathcal{X}_i \\times_\\mathcal{X} \\mathcal{X}_j \\to M_{ij}$ is a uniform", "categorical quotient; details omitted.", "Then finally one shows that $\\psi$ fits into the commutative diagram", "$$", "\\xymatrix{", "M' \\times_M \\mathcal{X} \\ar[d] \\ar[rd]^\\theta \\\\", "M' \\ar[r]^\\psi &", "E", "}", "$$", "because ``$\\{M' \\times_M \\mathcal{X}_i \\to M' \\times_M \\mathcal{X}\\}$", "is an \\'etale covering'' and the morphisms $\\psi_i$ fit into the", "corresponding commutative diagrams by construction.", "This finishes the proof of the Keel-Mori theorem." ], "refs": [ "stacks-more-morphisms-lemma-etale-local-finite-inertia", "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space", "stacks-morphisms-lemma-base-change-separated", "stacks-morphisms-lemma-base-change-etale", "stacks-morphisms-remark-identify-automorphism-groups", "stacks-morphisms-lemma-base-change-stabilizer-preserving", "stacks-more-morphisms-lemma-etale-separated-over-well-nigh-affine", "stacks-more-morphisms-lemma-moduli-space-finite-affine", "stacks-morphisms-lemma-universally-injective", "stacks-morphisms-lemma-universally-injective-point", "groupoids-lemma-etale-equivalence-relation", "spaces-theorem-presentation", "stacks-morphisms-lemma-etale-permanence", "stacks-morphisms-lemma-compose-after-separated", "stacks-morphisms-remark-identify-automorphism-groups", "stacks-morphisms-remark-identify-automorphism-groups", "stacks-morphisms-lemma-etale-iso", "stacks-morphisms-lemma-check-separated-covering", "stacks-morphisms-lemma-check-quasi-compact-covering", "stacks-morphisms-lemma-check-universal-homeomorphism-covering" ], "ref_ids": [ 6922, 6918, 7398, 7548, 7637, 7595, 6921, 6920, 7450, 7451, 9580, 8124, 7551, 7406, 7637, 7637, 7597, 7399, 7430, 7457 ] } ], "ref_ids": [] }, { "id": 6884, "type": "theorem", "label": "stacks-more-morphisms-lemma-thickening", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-thickening", "contents": [ "Let $i : \\mathcal{X} \\to \\mathcal{X}'$ be a morphism of algebraic stacks.", "The following are equivalent", "\\begin{enumerate}", "\\item $i$ is a thickening of algebraic stacks (abuse of language as above), and", "\\item $i$ is representable by algebraic spaces and", "is a thickening in the sense of Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}.", "\\end{enumerate}", "In this case $i$ is a closed immersion and a universal homeomorphism." ], "refs": [], "proofs": [ { "contents": [ "By More on Morphisms of Spaces, Lemmas", "\\ref{spaces-more-morphisms-lemma-descending-property-thickening} and", "\\ref{spaces-more-morphisms-lemma-base-change-thickening}", "the property $P$ that a morphism of algebraic spaces is a", "(first order) thickening is fpqc local on the base and stable under base", "change. Thus the discussion in Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms} indeed applies.", "Having said this the equivalence of (1) and (2) follows from", "the fact that $P = P_1 + P_2$ where $P_1$ is the property of being", "a closed immersion and $P_2$ is the property of being surjective.", "(Strictly speaking, the reader should also consult", "More on Morphisms of Spaces, Definition", "\\ref{spaces-more-morphisms-definition-thickening},", "Properties of Stacks, Definition \\ref{stacks-properties-definition-immersion}", "and the discussion following, Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-surjective-representable},", "Properties of Stacks, Section \\ref{stacks-properties-section-surjective}", "to see that all the concepts all match up.)", "The final assertion is clear from the foregoing." ], "refs": [ "spaces-more-morphisms-lemma-descending-property-thickening", "spaces-more-morphisms-lemma-base-change-thickening", "spaces-more-morphisms-definition-thickening", "stacks-properties-definition-immersion", "spaces-morphisms-lemma-surjective-representable" ], "ref_ids": [ 54, 52, 283, 8920, 4724 ] } ], "ref_ids": [] }, { "id": 6885, "type": "theorem", "label": "stacks-more-morphisms-lemma-base-change-thickening", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-base-change-thickening", "contents": [ "Let $\\mathcal{Y} \\subset \\mathcal{Y}'$ be a thickening of algebraic stacks.", "Let $\\mathcal{X}' \\to \\mathcal{Y}'$ be a morphism of algebraic stacks", "and set $\\mathcal{X} = \\mathcal{Y} \\times_{\\mathcal{Y}'} \\mathcal{X}'$.", "Then", "$(\\mathcal{X} \\subset \\mathcal{X}') \\to (\\mathcal{Y} \\subset \\mathcal{Y}')$", "is a morphism of thickenings. If $\\mathcal{Y} \\subset \\mathcal{Y}'$ is a first", "order thickening, then $\\mathcal{X} \\subset \\mathcal{X}'$ is a first", "order thickening." ], "refs": [], "proofs": [ { "contents": [ "See discussion above, Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}, and", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-base-change-thickening}." ], "refs": [ "spaces-more-morphisms-lemma-base-change-thickening" ], "ref_ids": [ 52 ] } ], "ref_ids": [] }, { "id": 6886, "type": "theorem", "label": "stacks-more-morphisms-lemma-composition-thickening", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-composition-thickening", "contents": [ "If $\\mathcal{X} \\subset \\mathcal{X}'$ and $\\mathcal{X}' \\subset \\mathcal{X}''$", "are thickenings of algebraic stacks, then so is", "$\\mathcal{X} \\subset \\mathcal{X}''$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above, Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}, and", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-composition-thickening}" ], "refs": [ "spaces-more-morphisms-lemma-composition-thickening" ], "ref_ids": [ 53 ] } ], "ref_ids": [] }, { "id": 6887, "type": "theorem", "label": "stacks-more-morphisms-lemma-reduced-diagonal", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-reduced-diagonal", "contents": [ "Let $(f, f') : (\\mathcal{X} \\subset \\mathcal{X}') \\to", "(\\mathcal{Y} \\subset \\mathcal{Y}')$ be a morphism of thickenings", "of algebraic stacks. Then", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\to", "\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{X}'$", "is a thickening and the canonical diagram", "$$", "\\xymatrix{", "\\mathcal{X} \\ar[r]_-\\Delta \\ar[d] &", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\ar[d] \\\\", "\\mathcal{X}' \\ar[r]^-{\\Delta'} &", "\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{X}'", "}", "$$", "is cartesian." ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathcal{X} \\to \\mathcal{Y}'$ factors through the closed", "substack $\\mathcal{Y}$ we see that", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} =", "\\mathcal{X} \\times_{\\mathcal{Y}'} \\mathcal{X}$.", "Hence", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\to", "\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{X}'$", "is isomorphic to the composition", "$$", "\\mathcal{X} \\times_{\\mathcal{Y}'} \\mathcal{X} \\to", "\\mathcal{X} \\times_{\\mathcal{Y}'} \\mathcal{X}' \\to", "\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{X}'", "$$", "both of which are thickenings as base changes of thickenings", "(Lemma \\ref{lemma-base-change-thickening}).", "Hence so is the composition", "(Lemma \\ref{lemma-composition-thickening}).", "Since $\\mathcal{X} \\to \\mathcal{X}'$ is a monomorphism,", "the final statement of the lemma follows from", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-monomorphism-diagonal}", "applied to $\\mathcal{X} \\to \\mathcal{X}' \\to \\mathcal{Y}'$." ], "refs": [ "stacks-more-morphisms-lemma-base-change-thickening", "stacks-more-morphisms-lemma-composition-thickening", "stacks-properties-lemma-monomorphism-diagonal" ], "ref_ids": [ 6885, 6886, 8881 ] } ], "ref_ids": [] }, { "id": 6888, "type": "theorem", "label": "stacks-more-morphisms-lemma-thickening-diagonals", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-thickening-diagonals", "contents": [ "Let $(f, f') : (\\mathcal{X} \\subset \\mathcal{X}') \\to", "(\\mathcal{Y} \\subset \\mathcal{Y}')$ be a morphism of thickenings", "of algebraic stacks.", "Let $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$ and", "$\\Delta' : \\mathcal{X}' \\to \\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{X}'$", "be the corresponding diagonal morphisms.", "Then each property from the following list is satisfied by $\\Delta$ if", "and only if it is satisfied by $\\Delta'$:", "(a) representable by schemes, (b) affine, (c) surjective, (d) quasi-compact,", "(e) universally closed, (f) integral, (g) quasi-separated, (h) separated,", "(i) universally injective, (j) universally open, (k) locally quasi-finite,", "(l) finite, (m) unramified, (n) monomorphism, (o) immersion,", "(p) closed immersion, and (q) proper." ], "refs": [], "proofs": [ { "contents": [ "Observe that", "$$", "(\\Delta, \\Delta') :", "(\\mathcal{X} \\subset \\mathcal{X}')", "\\longrightarrow", "(\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\subset", "\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{X}')", "$$", "is a morphism of thickenings (Lemma \\ref{lemma-reduced-diagonal}).", "Moreover $\\Delta$ and $\\Delta'$ are", "representable by algebraic spaces by", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-properties-diagonal}.", "Hence, via the discussion in", "Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}", "the lemma follows for cases (a), (b), (c), (d),", "(e), (f), (g), (h), (i), and (j) by using", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-thicken-property-morphisms}.", "\\medskip\\noindent", "Lemma \\ref{lemma-reduced-diagonal} tells us that", "$\\mathcal{X} = (\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X})", "\\times_{(\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{X}')} \\mathcal{X}'$.", "Moreover, $\\Delta$ and $\\Delta'$ are locally of finite type by", "the aforementioned", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-properties-diagonal}.", "Hence the result for cases (k), (l), (m), (n), (o), (p), and (q) by using", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-properties-that-extend-over-thickenings}." ], "refs": [ "stacks-more-morphisms-lemma-reduced-diagonal", "stacks-morphisms-lemma-properties-diagonal", "spaces-more-morphisms-lemma-thicken-property-morphisms", "stacks-more-morphisms-lemma-reduced-diagonal", "stacks-morphisms-lemma-properties-diagonal", "spaces-more-morphisms-lemma-properties-that-extend-over-thickenings" ], "ref_ids": [ 6887, 7392, 55, 6887, 7392, 57 ] } ], "ref_ids": [] }, { "id": 6889, "type": "theorem", "label": "stacks-more-morphisms-lemma-thickening-properties", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-thickening-properties", "contents": [ "\\begin{reference}", "\\cite[Theorem 2.2.5]{Conrad-moduli}", "\\end{reference}", "Let $\\mathcal{X} \\subset \\mathcal{X}'$ be a thickening of algebraic", "stacks. Then", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is an algebraic space if and only if $\\mathcal{X}'$", "is an algebraic space,", "\\item $\\mathcal{X}$ is a scheme if and only if $\\mathcal{X}'$ is a scheme,", "\\item $\\mathcal{X}$ is DM if and only if $\\mathcal{X}'$ is DM,", "\\item $\\mathcal{X}$ is quasi-DM if and only if $\\mathcal{X}'$ is quasi-DM,", "\\item $\\mathcal{X}$ is separated if and only if $\\mathcal{X}'$ is separated,", "\\item $\\mathcal{X}$ is quasi-separated if and only if $\\mathcal{X}'$ is", "quasi-separated, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In each case we reduce to a question about the diagonal and then", "we use Lemma \\ref{lemma-thickening-diagonals} applied to the", "morphism of thickenings", "$$", "(\\mathcal{X} \\subset \\mathcal{X}') \\to", "\\left(\\Spec(\\mathbf{Z}) \\subset \\Spec(\\mathbf{Z})\\right)", "$$", "We do this after viewing", "$\\mathcal{X} \\subset \\mathcal{X}'$ as a thickening of algebraic stacks", "over $\\Spec(\\mathbf{Z})$ via", "Algebraic Stacks, Definition \\ref{algebraic-definition-viewed-as}.", "\\medskip\\noindent", "Case (1). An algebraic stack is an algebraic space if and only if its", "diagonal is a monomorphism, see", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-hierarchy}", "(this also follows immediately from Algebraic Stacks,", "Proposition \\ref{algebraic-proposition-algebraic-stack-no-automorphisms}).", "\\medskip\\noindent", "Case (2). By (1) we may assume that $\\mathcal{X}$ and $\\mathcal{X}'$", "are algebraic spaces and then we can use", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-thickening-scheme}.", "\\medskip\\noindent", "Case (3) -- (6). Each of these cases corresponds to a condition", "on the diagonal, see Morphisms of Stacks, Definitions", "\\ref{stacks-morphisms-definition-separated} and", "\\ref{stacks-morphisms-definition-absolute-separated}." ], "refs": [ "stacks-more-morphisms-lemma-thickening-diagonals", "algebraic-definition-viewed-as", "stacks-morphisms-lemma-hierarchy", "algebraic-proposition-algebraic-stack-no-automorphisms", "spaces-more-morphisms-lemma-thickening-scheme", "stacks-morphisms-definition-separated", "stacks-morphisms-definition-absolute-separated" ], "ref_ids": [ 6888, 8489, 7418, 8480, 49, 7601, 7602 ] } ], "ref_ids": [] }, { "id": 6890, "type": "theorem", "label": "stacks-more-morphisms-lemma-thicken-property-morphisms", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-thicken-property-morphisms", "contents": [ "Let $(f, f') : (\\mathcal{X} \\subset \\mathcal{X}') \\to", "(\\mathcal{Y} \\subset \\mathcal{Y}')$", "be a morphism of thickenings of algebraic stacks. Then", "\\begin{enumerate}", "\\item $f$ is an affine morphism if and only if $f'$ is an affine morphism,", "\\item $f$ is a surjective morphism if and only if $f'$ is a surjective morphism,", "\\item $f$ is quasi-compact if and only if $f'$ quasi-compact,", "\\item $f$ is universally closed if and only if $f'$ is universally closed,", "\\item $f$ is integral if and only if $f'$ is integral,", "\\item $f$ is universally injective if and only if $f'$ is universally injective,", "\\item $f$ is universally open if and only if $f'$ is universally open,", "\\item $f$ is quasi-DM if and only if $f'$ is quasi-DM,", "\\item $f$ is DM if and only if $f'$ is DM,", "\\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,", "\\item $f$ is representable if and only if $f'$ is representable,", "\\item $f$ is representable by algebraic spaces if and only if $f'$ is", "representable by algebraic spaces,", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-thickening} the morphisms $\\mathcal{X} \\to \\mathcal{X}'$", "and $\\mathcal{Y} \\to \\mathcal{Y}'$ are universal homeomorphisms. Thus any", "condition on $|f| : |\\mathcal{X}| \\to |\\mathcal{Y}|$ is equivalent with", "the corresponding condition on $|f'| : |\\mathcal{X}'| \\to |\\mathcal{Y}'|$", "and the same is true after arbitrary base change by a morphism", "$\\mathcal{Z}' \\to \\mathcal{Y}'$. This proves that", "(2), (3), (4), (6), (7) hold.", "\\medskip\\noindent", "In cases (8), (9), (10), (12) we can translate the conditions on", "$f$ and $f'$ into conditions on the diagonals $\\Delta$ and $\\Delta'$", "as in Lemma \\ref{lemma-thickening-diagonals}. See", "Morphisms of Stacks, Definition \\ref{stacks-morphisms-definition-separated} and", "Lemma \\ref{stacks-morphisms-lemma-hierarchy}.", "Hence these cases follow from Lemma \\ref{lemma-thickening-diagonals}.", "\\medskip\\noindent", "Proof of (11). If $f'$ is representable, then so is $f$, because", "for a scheme $T$ and a morphism $T \\to \\mathcal{Y}$ we have", "$\\mathcal{X} \\times_\\mathcal{Y} T =", "\\mathcal{X} \\times_{\\mathcal{X}'} (\\mathcal{X}' \\times_{\\mathcal{Y}'} T)$", "and $\\mathcal{X} \\to \\mathcal{X}'$ is a closed immersion (hence representable).", "Conversely, assume $f$ is representable, and let $T' \\to \\mathcal{Y}'$", "be a morphism where $T'$ is a scheme. Then", "$$", "\\mathcal{X} \\times_{\\mathcal{Y}}", "(\\mathcal{Y} \\times_{\\mathcal{Y}'} T') =", "\\mathcal{X} \\times_{\\mathcal{X}'}", "(\\mathcal{X}' \\times_{\\mathcal{Y}'} T') \\to", "\\mathcal{X}' \\times_{\\mathcal{Y}'} T'", "$$", "is a thickening (by Lemma \\ref{lemma-base-change-thickening})", "and the source is a scheme. Hence the target is a scheme by", "Lemma \\ref{lemma-thickening-properties}.", "\\medskip\\noindent", "In cases (1) and (5) if either $f$ or $f'$ has the stated property,", "then both $f$ and $f'$ are representable by (11). In this case", "choose an algebraic space $V'$ and a surjective smooth morphism", "$V' \\to \\mathcal{Y}'$. Set $V = \\mathcal{Y} \\times_{\\mathcal{Y}'} V'$,", "$U' = \\mathcal{X}' \\times_{\\mathcal{Y}'} V'$, and", "$U = \\mathcal{X} \\times_{\\mathcal{Y}'} V'$. Then the desired", "results follow from the corresponding results for", "the morphism $(U \\subset U') \\to (V \\subset V')$ of thickenings", "of algebraic spaces via the principle of Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}.", "See More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-thicken-property-morphisms}", "for the corresponding results in the case of algebraic spaces." ], "refs": [ "stacks-more-morphisms-lemma-thickening", "stacks-more-morphisms-lemma-thickening-diagonals", "stacks-morphisms-definition-separated", "stacks-morphisms-lemma-hierarchy", "stacks-more-morphisms-lemma-thickening-diagonals", "stacks-more-morphisms-lemma-base-change-thickening", "stacks-more-morphisms-lemma-thickening-properties", "stacks-properties-lemma-check-property-covering", "spaces-more-morphisms-lemma-thicken-property-morphisms" ], "ref_ids": [ 6884, 6888, 7601, 7418, 6888, 6885, 6889, 8859, 55 ] } ], "ref_ids": [] }, { "id": 6891, "type": "theorem", "label": "stacks-more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "(\\mathcal{X} \\subset \\mathcal{X}') \\ar[rr]_{(f, f')} \\ar[rd] & &", "(\\mathcal{Y} \\subset \\mathcal{Y}') \\ar[ld] \\\\", "& (\\mathcal{B} \\subset \\mathcal{B}')", "}", "$$", "of thickenings of algebraic stacks. Assume", "\\begin{enumerate}", "\\item $\\mathcal{Y}' \\to \\mathcal{B}'$ is locally of finite type,", "\\item $\\mathcal{X}' \\to \\mathcal{B}'$ is", "flat and locally of finite presentation,", "\\item $f$ is flat, and", "\\item $\\mathcal{X} = \\mathcal{B} \\times_{\\mathcal{B}'} \\mathcal{X}'$ and", "$\\mathcal{Y} = \\mathcal{B} \\times_{\\mathcal{B}'} \\mathcal{Y}'$.", "\\end{enumerate}", "Then $f'$ is flat and for all $y' \\in |\\mathcal{Y}'|$ in the image of $|f'|$", "the morphism $\\mathcal{Y}' \\to \\mathcal{B}'$ is flat at $y'$." ], "refs": [], "proofs": [ { "contents": [ "Choose an algebraic space $U'$ and a surjective smooth morphism", "$U' \\to \\mathcal{B}'$.", "Choose an algebraic space $V'$ and a surjective smooth morphism", "$V' \\to U' \\times_{\\mathcal{B}'} \\mathcal{Y}'$.", "Choose an algebraic space $W'$ and a surjective smooth morphism", "$W' \\to V' \\times_{\\mathcal{Y}'} \\mathcal{X}'$. Let $U, V, W$", "be the base change of $U', V', W'$ by $\\mathcal{B} \\to \\mathcal{B}'$.", "Then flatness of $f'$ is equivalent to flatness of $W' \\to V'$ and", "we are given that $W \\to V$ is flat. Hence we may apply the lemma", "in the case of algebraic spaces to the diagram", "$$", "\\xymatrix{", "(W \\subset W') \\ar[rr] \\ar[rd] & & (V \\subset V') \\ar[ld] \\\\", "& (U \\subset U')", "}", "$$", "of thickenings of algebraic spaces. See", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft}.", "The statement about flatness of $\\mathcal{Y}'/\\mathcal{B}'$ at points in the", "image of $|f'|$ follows in the same manner." ], "refs": [ "spaces-more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft" ], "ref_ids": [ 104 ] } ], "ref_ids": [] }, { "id": 6892, "type": "theorem", "label": "stacks-more-morphisms-lemma-deform-property-fp-over-ft", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-deform-property-fp-over-ft", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "(\\mathcal{X} \\subset \\mathcal{X}') \\ar[rr]_{(f, f')} \\ar[rd] & &", "(\\mathcal{Y} \\subset \\mathcal{Y}') \\ar[ld] \\\\", "& (\\mathcal{B} \\subset \\mathcal{B}')", "}", "$$", "of thickenings of algebraic stacks.", "Assume $\\mathcal{Y}' \\to \\mathcal{B}'$ locally of finite type,", "$\\mathcal{X}' \\to \\mathcal{B}'$ flat and locally of finite presentation,", "$\\mathcal{X} = \\mathcal{B} \\times_{\\mathcal{B}'} \\mathcal{X}'$, and", "$\\mathcal{Y} = \\mathcal{B} \\times_{\\mathcal{B}'} \\mathcal{Y}'$. Then", "\\begin{enumerate}", "\\item $f$ is flat if and only if $f'$ is flat,", "\\label{item-flat-fp-over-ft}", "\\item $f$ is an isomorphism if and only if $f'$ is an isomorphism,", "\\label{item-isomorphism-fp-over-ft}", "\\item $f$ is an open immersion if and only if $f'$ is an open immersion,", "\\label{item-open-immersion-fp-over-ft}", "\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,", "\\label{item-monomorphism-fp-over-ft}", "\\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,", "\\label{item-quasi-finite-fp-over-ft}", "\\item $f$ is syntomic if and only if $f'$ is syntomic,", "\\label{item-syntomic-fp-over-ft}", "\\item $f$ is smooth if and only if $f'$ is smooth,", "\\label{item-smooth-fp-over-ft}", "\\item $f$ is unramified if and only if $f'$ is unramified,", "\\label{item-unramified-fp-over-ft}", "\\item $f$ is \\'etale if and only if $f'$ is \\'etale,", "\\label{item-etale-fp-over-ft}", "\\item $f$ is finite if and only if $f'$ is finite, and", "\\label{item-finite-fp-over-ft}", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In case (\\ref{item-flat-fp-over-ft}) this follows from", "Lemma \\ref{lemma-flatness-morphism-thickenings-fp-over-ft}.", "\\medskip\\noindent", "In cases", "(\\ref{item-syntomic-fp-over-ft}), (\\ref{item-smooth-fp-over-ft})", "this can be proved by the method used in the proof of", "Lemma \\ref{lemma-flatness-morphism-thickenings-fp-over-ft}.", "Namely, choose an algebraic space $U'$ and a surjective smooth morphism", "$U' \\to \\mathcal{B}'$.", "Choose an algebraic space $V'$ and a surjective smooth morphism", "$V' \\to U' \\times_{\\mathcal{B}'} \\mathcal{Y}'$.", "Choose an algebraic space $W'$ and a surjective smooth morphism", "$W' \\to V' \\times_{\\mathcal{Y}'} \\mathcal{X}'$. Let $U, V, W$", "be the base change of $U', V', W'$ by $\\mathcal{B} \\to \\mathcal{B}'$.", "Then the property for $f$, resp.\\ $f'$", "is equivalent to the property for of $W' \\to V'$, resp.\\ $W \\to V$.", "Hence we may apply the lemma in the case of algebraic spaces to the", "diagram", "$$", "\\xymatrix{", "(W \\subset W') \\ar[rr] \\ar[rd] & & (V \\subset V') \\ar[ld] \\\\", "& (U \\subset U')", "}", "$$", "of thickenings of algebraic spaces. See", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-deform-property-fp-over-ft}.", "\\medskip\\noindent", "In cases (\\ref{item-unramified-fp-over-ft}) and (\\ref{item-etale-fp-over-ft})", "we first see that the assumption for $f$ or $f'$ implies that both", "$f$ and $f'$ are DM morphisms of algebraic stacks, see", "Lemma \\ref{lemma-thicken-property-morphisms}. Then we can choose", "an algebraic space $U'$ and a surjective smooth morphism", "$U' \\to \\mathcal{B}'$.", "Choose an algebraic space $V'$ and a surjective smooth morphism", "$V' \\to U' \\times_{\\mathcal{B}'} \\mathcal{Y}'$.", "Choose an algebraic space $W'$ and a surjective \\'etale(!) morphism", "$W' \\to V' \\times_{\\mathcal{Y}'} \\mathcal{X}'$. Let $U, V, W$", "be the base change of $U', V', W'$ by $\\mathcal{B} \\to \\mathcal{B}'$.", "Then $W \\to V \\times_\\mathcal{Y} \\mathcal{X}$ is surjective", "\\'etale as well. Hence the property for $f$, resp.\\ $f'$", "is equivalent to the property for of $W' \\to V'$, resp.\\ $W \\to V$.", "Hence we may apply the lemma in the case of algebraic spaces to the", "diagram", "$$", "\\xymatrix{", "(W \\subset W') \\ar[rr] \\ar[rd] & & (V \\subset V') \\ar[ld] \\\\", "& (U \\subset U')", "}", "$$", "of thickenings of algebraic spaces. See", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-deform-property-fp-over-ft}.", "\\medskip\\noindent", "In cases (\\ref{item-isomorphism-fp-over-ft}),", "(\\ref{item-open-immersion-fp-over-ft}),", "(\\ref{item-monomorphism-fp-over-ft}),", "(\\ref{item-finite-fp-over-ft})", "we first conclude by Lemma \\ref{lemma-thicken-property-morphisms}", "that $f$ and $f'$ are representable by algebraic spaces. Thus we may choose", "an algebraic space $U'$ and a surjective smooth morphism", "$U' \\to \\mathcal{B}'$,", "an algebraic space $V'$ and a surjective smooth morphism", "$V' \\to U' \\times_{\\mathcal{B}'} \\mathcal{Y}'$, and then", "$W' = V' \\times_{\\mathcal{Y}'} \\mathcal{X}'$ will be an algebraic space.", "Let $U, V, W$ be the base change of", "$U', V', W'$ by $\\mathcal{B} \\to \\mathcal{B}'$.", "Then $W = V \\times_\\mathcal{Y} \\mathcal{X}$ as well.", "Then we have to see that $W' \\to V'$ is an", "isomorphism, resp.\\ an open immersion, resp.\\ a monomorphism,", "resp.\\ finite, if and only if $W \\to V$ has the same property.", "See Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}.", "Thus we conclude by applying the results", "for algebraic spaces as above.", "\\medskip\\noindent", "In the case (\\ref{item-quasi-finite-fp-over-ft}) we first", "observe that $f$ and $f'$ are locally of finite type by", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-finite-type-permanence}.", "On the other hand, the morphism $f$ is quasi-DM if and only if", "$f'$ is by", "Lemma \\ref{lemma-thicken-property-morphisms}.", "The last thing to check to see if $f$ or $f'$ is locally quasi-finite", "(Morphisms of Stacks, Definition", "\\ref{stacks-morphisms-definition-locally-quasi-finite})", "is a condition on underlying topological spaces", "which holds for $f$ if and only if it holds for $f'$ by", "the discussion in the first paragraph of the proof." ], "refs": [ "stacks-more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft", "stacks-more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft", "spaces-more-morphisms-lemma-deform-property-fp-over-ft", "stacks-more-morphisms-lemma-thicken-property-morphisms", "spaces-more-morphisms-lemma-deform-property-fp-over-ft", "stacks-more-morphisms-lemma-thicken-property-morphisms", "stacks-properties-lemma-check-property-covering", "stacks-morphisms-lemma-finite-type-permanence", "stacks-more-morphisms-lemma-thicken-property-morphisms", "stacks-morphisms-definition-locally-quasi-finite" ], "ref_ids": [ 6891, 6891, 105, 6890, 105, 6890, 8859, 7465, 6890, 7616 ] } ], "ref_ids": [] }, { "id": 6893, "type": "theorem", "label": "stacks-more-morphisms-lemma-morphisms-lifts-etale", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-morphisms-lifts-etale", "contents": [ "For any morphism (\\ref{equation-morphism}) the map $f' : V' \\to U'$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Namely $f : V \\to U$ is \\'etale as a morphism in $W_{spaces, \\etale}$", "and we can apply", "Lemma \\ref{lemma-deform-property-fp-over-ft} because $U' \\to \\mathcal{X}'$", "and $V' \\to \\mathcal{X}'$ are smooth and", "$U = \\mathcal{X} \\times_{\\mathcal{X}'} U'$ and", "$V = \\mathcal{X} \\times_{\\mathcal{X}'} V'$." ], "refs": [ "stacks-more-morphisms-lemma-deform-property-fp-over-ft" ], "ref_ids": [ 6892 ] } ], "ref_ids": [] }, { "id": 6894, "type": "theorem", "label": "stacks-more-morphisms-lemma-gerbe-of-lifts-fibred", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-gerbe-of-lifts-fibred", "contents": [ "The category $p : \\mathcal{C} \\to W_{spaces, \\etale}$ constructed", "in Remark \\ref{remark-gerbe-of-lifts} is fibred in groupoids." ], "refs": [ "stacks-more-morphisms-remark-gerbe-of-lifts" ], "proofs": [ { "contents": [ "We claim the fibre categories of $p$ are groupoids.", "If $(f, f', \\gamma')$ as in (\\ref{equation-morphism})", "is a morphism such that $f : U \\to V$ is an isomorphism, then", "$f'$ is an isomorphism by Lemma \\ref{lemma-deform-property-fp-over-ft}", "and hence $(f, f', \\gamma')$ is an isomorphism.", "\\medskip\\noindent", "Consider a morphism $f : V \\to U$ in $W_{spaces, \\etale}$", "and an object $\\xi = (U, U', a, i, x', \\alpha)$", "of $\\mathcal{C}$ over $U$. We are going to construct the ``pullback''", "$f^*\\xi$ over $V$. ", "Namely, set $b = a \\circ f$. Let $f' : V' \\to U'$ be the \\'etale morphism", "whose restriction to $V$ is $f$ (More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-topological-invariance}).", "Denote $j : V \\to V'$ the corresponding thickening.", "Let $y' = x' \\circ f'$ and $\\gamma = \\text{id} : x' \\circ f' \\to y'$.", "Set", "$$", "\\beta = \\alpha \\star \\text{id}_f :", "x \\circ b = x \\circ a \\circ f \\to", "x' \\circ i \\circ f = x' \\circ f' \\circ j = y' \\circ j", "$$", "It is clear that $(f, f', \\gamma) : (V, V', b, j, y', \\beta) \\to", "(U, U', a, i, x', \\alpha)$ is a morphism as in", "(\\ref{equation-morphism}). The morphisms $(f, f', \\gamma)$", "so constructed are strongly cartesian", "(Categories, Definition \\ref{categories-definition-cartesian-over-C}).", "We omit the detailed proof, but essentially the reason is that", "given a morphism", "$(g, g', \\epsilon) : (Y, Y', c, k, z', \\delta) \\to (U, U', a, i, x', \\alpha)$", "in $\\mathcal{C}$ such that $g$ factors as $g = f \\circ h$ for some", "$h : Y \\to V$, then we get a unique factorization $g' = f' \\circ h'$", "from More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-topological-invariance}", "and after that one can produce the necessary $\\zeta$ such that", "$(h, h', \\zeta) : (Y, Y', c, k, z', \\delta) \\to", "(V, V', b, j, y', \\beta)$ is a morphism of $\\mathcal{C}$", "with $(g, g', \\epsilon) = (f, f', \\gamma) \\circ (h, h', \\zeta)$.", "\\medskip\\noindent", "Therefore $p : \\mathcal{C} \\to W_\\etale$ is a fibred", "category (Categories, Definition \\ref{categories-definition-fibred-category}).", "Combined with the fact that the fibre categories are groupoids", "seen above we conclude that $p : \\mathcal{C} \\to W_\\etale$", "is fibred in groupoids by Categories, Lemma", "\\ref{categories-lemma-fibred-groupoids}." ], "refs": [ "stacks-more-morphisms-lemma-deform-property-fp-over-ft", "spaces-more-morphisms-lemma-topological-invariance", "categories-definition-cartesian-over-C", "spaces-more-morphisms-lemma-topological-invariance", "categories-definition-fibred-category", "categories-lemma-fibred-groupoids" ], "ref_ids": [ 6892, 45, 12387, 45, 12388, 12294 ] } ], "ref_ids": [ 6933 ] }, { "id": 6895, "type": "theorem", "label": "stacks-more-morphisms-lemma-gerbe-of-lifts-stack", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-gerbe-of-lifts-stack", "contents": [ "The category $p : \\mathcal{C} \\to W_{spaces, \\etale}$ constructed", "in Remark \\ref{remark-gerbe-of-lifts} is a stack in groupoids." ], "refs": [ "stacks-more-morphisms-remark-gerbe-of-lifts" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-gerbe-of-lifts-fibred} we see the first condition", "of Stacks, Definition \\ref{stacks-definition-stack-in-groupoids} holds.", "As is customary we check descent of objects and we leave it to the reader", "to check descent of morphisms. Thus suppose we have $a : U \\to W$", "in $W_{spaces, \\etale}$, a covering $\\{U_k \\to U\\}_{k \\in K}$ in", "$W_{spaces, \\etale}$, objects $\\xi_k = (U_k, U'_k, a_k, i_k, x'_k, \\alpha_k)$", "of $\\mathcal{C}$ over $U_k$, and morphisms", "$$", "\\varphi_{kk'} = (f_{kk'}, f'_{kk'}, \\gamma_{kk'}) :", "\\xi_k|_{U_k \\times_U U_{k'}} \\to", "\\xi_{k'}|_{U_k \\times_U U_{k'}}", "$$", "between restrictions satisfying the cocycle condition. In order to prove", "effectivity we may first refine the covering. Hence we may assume each", "$U_k$ is a scheme (even an affine scheme if you like). Let us write", "$$", "\\xi_k|_{U_k \\times_U U_{k'}} =", "(U_k \\times_U U_{k'}, U'_{kk'}, a_{kk'}, x'_{kk'}, \\alpha_{kk'})", "$$", "Then we get an \\'etale (by Lemma \\ref{lemma-morphisms-lifts-etale}) morphism", "$s_{kk'} : U'_{kk'} \\to U'_k$", "as the second component of the morphism", "$\\xi_k|_{U_k \\times_U U_{k'}} \\to \\xi_k$ of $\\mathcal{C}$.", "Similarly we obtain an \\'etale morphism $t_{kk'} : U'_{kk'} \\to U'_{k'}$", "by looking at the second component of the composition", "$$", "\\xi_k|_{U_k \\times_U U_{k'}} \\xrightarrow{\\varphi_{kk'}}", "\\xi_{k'}|_{U_k \\times_U U_{k'}} \\to \\xi_{k'}", "$$", "We claim that", "$$", "j :", "\\coprod\\nolimits_{(k, k') \\in K \\times K} U'_{kk'}", "\\xrightarrow{(\\coprod s_{kk'}, \\coprod t_{kk'})}", "(\\coprod\\nolimits_{k \\in K} U'_k) \\times (\\coprod\\nolimits_{k \\in K} U'_k)", "$$", "is an \\'etale equivalence relation. First, we have already seen", "that the components $s, t$ of the displayed morphism are \\'etale.", "The base change of the morphism $j$ by", "$(\\coprod U_k) \\times (\\coprod U_k) \\to (\\coprod U'_k) \\times (\\coprod U'_k)$", "is a monomorphism because it is the map", "$$", "\\coprod\\nolimits_{(k, k') \\in K \\times K} U_k \\times_U U_{k'}", "\\longrightarrow", "(\\coprod\\nolimits_{k \\in K} U_k) \\times (\\coprod\\nolimits_{k \\in K} U_k)", "$$", "Hence $j$ is a monomorphism by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-properties-that-extend-over-thickenings}.", "Finally, symmetry of the relation $j$ comes from the fact that", "$\\varphi_{kk'}^{-1}$ is the ``flip'' of $\\varphi_{k'k}$ (see", "Stacks, Remarks \\ref{stacks-remarks-definition-descent-datum})", "and transitivity comes from the cocycle condition (details omitted).", "Thus the quotient of $\\coprod U'_k$ by $j$ is an algebraic space $U'$", "(Spaces, Theorem \\ref{spaces-theorem-presentation}).", "Above we have already shown that there is a thickening", "$i : U \\to U'$ as we saw that the restriction of $j$ on", "$\\coprod U_k$ gives $(\\coprod U_k) \\times_U (\\coprod U_k)$.", "Finally, if we temporarily view the $1$-morphisms", "$x'_k : U'_k \\to \\mathcal{X}'$ as objects of the stack", "$\\mathcal{X}'$ over $U'_k$ then we see that these come endowed with a", "descent datum with respect to the \\'etale covering", "$\\{U'_k \\to U'\\}$ given by the third component $\\gamma_{kk'}$", "of the morphisms $\\varphi_{kk'}$ in $\\mathcal{C}$.", "Since $\\mathcal{X}'$ is a stack", "this descent datum is effective and translating back we obtain", "a $1$-morphism $x' : U' \\to \\mathcal{X}'$ such that the compositions", "$U'_k \\to U' \\to \\mathcal{X}'$ come equipped with isomorphisms to $x'_k$", "compatible with $\\gamma_{kk'}$. This means that the morphisms", "$\\alpha_k : x \\circ a_k \\to x'_k \\circ i_k$ glue to a morphism", "$\\alpha : x \\circ a \\to x' \\circ i$. Then $\\xi = (U, U', a, i, x', \\alpha)$", "is the desired object over $U$." ], "refs": [ "stacks-more-morphisms-lemma-gerbe-of-lifts-fibred", "stacks-definition-stack-in-groupoids", "stacks-more-morphisms-lemma-morphisms-lifts-etale", "more-morphisms-lemma-properties-that-extend-over-thickenings", "stacks-remarks-definition-descent-datum", "spaces-theorem-presentation" ], "ref_ids": [ 6894, 8998, 6893, 13685, 9009, 8124 ] } ], "ref_ids": [ 6933 ] }, { "id": 6896, "type": "theorem", "label": "stacks-more-morphisms-lemma-etale-local-lifts", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-etale-local-lifts", "contents": [ "Let $\\mathcal{X} \\subset \\mathcal{X}'$ be a thickening of algebraic stacks.", "Let $W$ be an algebraic space and let $W \\to \\mathcal{X}$ be a smooth morphism.", "There exists an \\'etale covering $\\{W_i \\to W\\}_{i \\in I}$ and for each $i$", "a cartesian diagram", "$$", "\\xymatrix{", "W_i \\ar[r] \\ar[d] & W_i' \\ar[d] \\\\", "\\mathcal{X} \\ar[r] & \\mathcal{X}'", "}", "$$", "with $W_i' \\to \\mathcal{X}'$ smooth." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U'$ and a surjective smooth morphism $U' \\to \\mathcal{X}'$.", "As usual we set $U = \\mathcal{X} \\times_{\\mathcal{X}'} U'$. Then", "$U \\to \\mathcal{X}$ is a surjective smooth morphism. Therefore the base change", "$$", "V = W \\times_{\\mathcal{X}} U \\longrightarrow W", "$$", "is a surjective smooth morphism of algebraic spaces.", "By Topologies on Spaces, Lemma", "\\ref{spaces-topologies-lemma-etale-dominates-smooth}", "we can find an \\'etale covering $\\{W_i \\to W\\}$ such", "that $W_i \\to W$ factors through $V \\to W$.", "After covering $W_i$ by affines (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-cover-by-union-affines})", "we may assume each $W_i$ is affine. We may and do replace $W$ by $W_i$", "which reduces us to the situation discussed in the next paragraph.", "\\medskip\\noindent", "Assume $W$ is affine and the given morphism $W \\to \\mathcal{X}$ factors", "through $U$. Picture", "$$", "W \\xrightarrow{i} U \\to \\mathcal{X}", "$$", "Since $W$ and $U$ are smooth over $\\mathcal{X}$ we see that", "$i$ is locally of finite type (Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-finite-type-permanence}).", "After replacing $U$ by $\\mathbf{A}^n_U$ we may assume", "that $i$ is an immersion, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-affine-finite-type-over-S}.", "By Morphisms of Stacks,", "Lemma \\ref{stacks-morphisms-lemma-lci-permanence}", "the morphism $i$ is a local complete intersection.", "Hence $i$ is a Koszul-regular immersion (as defined in", "Divisors, Definition \\ref{divisors-definition-regular-immersion})", "by More on Morphisms, Lemma \\ref{more-morphisms-lemma-lci}.", "\\medskip\\noindent", "We may still replace $W$ by an affine open covering.", "For every point $w \\in W$ we can choose an affine open", "$U'_w \\subset U'$ such that if $U_w \\subset U$ is the", "corresponding affine open, then $w \\in i^{-1}(U_w)$ and", "$i^{-1}(U_w) \\to U_w$ is a closed immersion cut out", "by a Koszul-regular sequence", "$f_1, \\ldots, f_r \\in \\Gamma(U_w, \\mathcal{O}_{U_w})$.", "This follows from the definition of Koszul-regular immersions", "and Divisors, Lemma \\ref{divisors-lemma-regular-ideal-sheaf-scheme}.", "Set $W_w = i^{-1}(U_w)$; this is an affine open neighbourhood", "of $w \\in W$.", "Choose lifts $f'_1, \\ldots, f'_r \\in \\Gamma(U'_w, \\mathcal{O}_{U'_w})$", "of $f_1, \\ldots, f_r$. This is possible as $U_w \\to U'_w$", "is a closed immersion of affine schemes.", "Let $W'_w \\subset U'_w$ be the closed subscheme cut out by", "$f'_1, \\ldots, f'_r$.", "We claim that $W'_w \\to \\mathcal{X}'$ is smooth.", "The claim finishes the proof as", "$W_w = \\mathcal{X} \\times_{\\mathcal{X}'} W'_w$", "by construction.", "\\medskip\\noindent", "To check the claim it suffices to check that the base change", "$W'_w \\times_{\\mathcal{X}'} X' \\to X'$ is smooth for every", "affine scheme $X'$ smooth over $\\mathcal{X}'$. Choose an", "\\'etale morphism", "$$", "Y' \\to U'_w \\times_{\\mathcal{X}'} X'", "$$", "with $Y'$ affine. Because $U'_w \\times_{\\mathcal{X}'} X'$ is covered", "by the images of such morphisms, it is enough to show that the closed", "subscheme $Z'$ of $Y'$ cut out by $f'_1, \\ldots, f'_r$ is smooth over $X'$.", "Picture", "$$", "\\xymatrix{", "Z' \\ar[r] \\ar[d] & Y' \\ar[d] \\\\", "W'_w \\times_{\\mathcal{X}'} X' \\ar[d] \\ar[r] &", "U'_w \\times_{\\mathcal{X}'} X' \\ar[d] \\ar[r] & X' \\\\", "W'_w = V(f'_1, \\ldots, f'_r) \\ar[r] & U'_w", "}", "$$", "Set $X = \\mathcal{X} \\times_{\\mathcal{X}'} X'$,", "$Y = X \\times_{X'} Y' = \\mathcal{X} \\times_{\\mathcal{X}'} Y'$, and", "$Z = Y \\times_{Y'} Z' = X \\times_{X'} Z' =", "\\mathcal{X} \\times_{\\mathcal{X}'} Z'$.", "Then $(Z \\subset Z') \\to (Y \\subset Y') \\subset (X \\subset X')$", "are (cartesian) morphisms of thickenings of affine schemes and", "we are given that $Z \\to X$ and $Y' \\to X'$ are smooth.", "Finally, the sequence of functions $f'_1, \\ldots, f'_r$", "map to a Koszul-regular sequence in $\\Gamma(Y', \\mathcal{O}_{Y'})$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-koszul-regular-flat-base-change}", "because $Y' \\to U'_w$ is smooth and hence flat.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-cut-by-koszul}", "(and the fact that Koszul-regular sequences are quasi-regular sequences", "by More on Algebra, Lemmas \\ref{more-algebra-lemma-regular-koszul-regular},", "\\ref{more-algebra-lemma-koszul-regular-H1-regular}, and", "\\ref{more-algebra-lemma-H1-regular-quasi-regular})", "we conclude that $Z' \\to X'$ is smooth as desired." ], "refs": [ "spaces-topologies-lemma-etale-dominates-smooth", "spaces-properties-lemma-cover-by-union-affines", "stacks-morphisms-lemma-finite-type-permanence", "morphisms-lemma-quasi-affine-finite-type-over-S", "stacks-morphisms-lemma-lci-permanence", "divisors-definition-regular-immersion", "more-morphisms-lemma-lci", "divisors-lemma-regular-ideal-sheaf-scheme", "more-algebra-lemma-koszul-regular-flat-base-change", "more-algebra-lemma-cut-by-koszul", "more-algebra-lemma-regular-koszul-regular", "more-algebra-lemma-koszul-regular-H1-regular", "more-algebra-lemma-H1-regular-quasi-regular" ], "ref_ids": [ 3653, 11830, 7465, 5392, 7591, 8099, 14001, 7987, 9976, 9994, 9973, 9974, 9977 ] } ], "ref_ids": [] }, { "id": 6897, "type": "theorem", "label": "stacks-more-morphisms-lemma-etale-local-lifts-isomorphic", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-etale-local-lifts-isomorphic", "contents": [ "Let $\\mathcal{X} \\subset \\mathcal{X}'$ be a thickening of algebraic stacks.", "Consider a commutative diagram", "$$", "\\xymatrix{", "W'' \\ar[d]_{x''} & W \\ar[l] \\ar[r] \\ar[d]_x & W' \\ar[d]^{x'} \\\\", "\\mathcal{X}' & \\mathcal{X} \\ar[l] \\ar[r] & \\mathcal{X}'", "}", "$$", "with cartesian squares where $W', W, W''$ are algebraic spaces and", "the vertical arrows are smooth. Then there exist", "\\begin{enumerate}", "\\item an \\'etale covering $\\{f'_k : W'_k \\to W'\\}_{k \\in K}$,", "\\item \\'etale morphisms $f''_k : W'_k \\to W''$, and", "\\item $2$-morphisms $\\gamma_k : x'' \\circ f''_k \\to x' \\circ f'_k$", "\\end{enumerate}", "such that (a) $(f'_k)^{-1}(W) = (f''_k)^{-1}(W)$, (b)", "$f'_k|_{(f'_k)^{-1}(W)} = f''_k|_{(f''_k)^{-1}(W)}$, and", "(c) pulling back $\\gamma_k$ to the closed subscheme of (a)", "agrees with the $2$-morphism given by the commutativity of", "the initial diagram over $W$." ], "refs": [], "proofs": [ { "contents": [ "Denote $i : W \\to W'$ and $i'' : W \\to W''$ the given thickenings.", "The commutativity of the diagram in the statement of the lemma", "means there is a $2$-morphism $\\delta : x' \\circ i' \\to x'' \\circ i''$", "This is the $2$-morphism referred to in part (c) of the statement.", "Consider the algebraic space", "$$", "I' = W' \\times_{x', \\mathcal{X}', x''} W''", "$$", "with projections $p' : I' \\to W'$ and $q' : I' \\to W''$.", "Observe that there is a ``universal'' $2$-morphism", "$\\gamma : x' \\circ p' \\to x'' \\circ q'$ (we will use this later).", "The choice of $\\delta$ defines a morphism", "$$", "\\xymatrix{", "W \\ar[rr]_\\delta & & I' \\ar[ld]^{p'} \\ar[rd]_{q'} \\\\", "& W' & & W''", "}", "$$", "such that the compositions $W \\to I' \\to W'$ and $W \\to I' \\to W''$", "are $i : W \\to W'$ and $i' : W \\to W''$.", "Since $x''$ is smooth, the morphism $p' : I' \\to W'$ is smooth", "as a base change of $x''$.", "\\medskip\\noindent", "Suppose we can find an \\'etale covering $\\{f'_k : W'_k \\to W'\\}$", "and morphisms $\\delta_k : W'_k \\to I'$ such that the restriction", "of $\\delta_k$ to $W_k = (f'_k)^{-1}$ is equal to $\\delta \\circ f_k$", "where $f_k = f'_k|_{W_k}$. Picture", "$$", "\\xymatrix{", "W_k \\ar[r]^{f_k} \\ar[d] & W \\ar[r]^\\delta & I' \\ar[d]^{p'} \\\\", "W'_k \\ar[rr]^{f'_k} \\ar[rru]^{\\delta_k} & & W'", "}", "$$", "In other words, we want to be able to extend the given section", "$\\delta : W \\to I'$ of $p'$ to a section over $W'$ after possibly", "replacing $W'$ by an \\'etale covering.", "\\medskip\\noindent", "If this is true, then we can set $f''_k = q' \\circ \\delta_k$", "and $\\gamma_k = \\gamma \\star \\text{id}_{\\delta_k}$ (more succinctly", "$\\gamma_k = \\delta_k^*\\gamma$). Namely, the only thing left to show", "at this is that the morphism $f''_k$ is \\'etale.", "By construction the morphism $x' \\circ p'$ is $2$-isomorphic", "to $x'' \\circ q'$. Hence $x'' \\circ f''_k$ is $2$-isomorphic", "to $x' \\circ f'_k$. We conclude that the composition", "$$", "W'_k \\xrightarrow{f''_k} W'' \\xrightarrow{x''} \\mathcal{X}'", "$$", "is smooth because $x' \\circ f'_k$ is so.", "As $f_k$ is \\'etale we conclude $f''_k$ is \\'etale", "by Lemma \\ref{lemma-deform-property-fp-over-ft}.", "\\medskip\\noindent", "If the thickening is a first order thickening, then we can", "choose any \\'etale covering $\\{W'_k \\to W'\\}$ with $W_k'$ affine.", "Namely, since $p'$ is smooth we see that $p'$ is formally smooth by the", "infinitesimal lifting criterion (More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-smooth-formally-smooth}).", "As $W_k$ is affine and as $W_k \\to W'_k$ is a first order thickening", "(as a base change of $\\mathcal{X} \\to \\mathcal{X}'$, see", "Lemma \\ref{lemma-base-change-thickening}) we get $\\delta_k$ as", "desired.", "\\medskip\\noindent", "In the general case the existence of the covering and the morphisms", "$\\delta_k$ follows from More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-smooth-strong-lift}." ], "refs": [ "stacks-more-morphisms-lemma-deform-property-fp-over-ft", "spaces-more-morphisms-lemma-smooth-formally-smooth", "stacks-more-morphisms-lemma-base-change-thickening", "spaces-more-morphisms-lemma-smooth-strong-lift" ], "ref_ids": [ 6892, 110, 6885, 111 ] } ], "ref_ids": [] }, { "id": 6898, "type": "theorem", "label": "stacks-more-morphisms-lemma-gerbe-of-lifts", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-gerbe-of-lifts", "contents": [ "The category $p : \\mathcal{C} \\to W_{spaces, \\etale}$ constructed", "in Remark \\ref{remark-gerbe-of-lifts} is a gerbe." ], "refs": [ "stacks-more-morphisms-remark-gerbe-of-lifts" ], "proofs": [ { "contents": [ "In Lemma \\ref{lemma-gerbe-of-lifts-stack}", "we have seen that it is a stack in groupoids.", "Thus it remains to check conditions (2) and (3) of", "Stacks, Definition \\ref{stacks-definition-gerbe}.", "Condition (2) follows from", "Lemma \\ref{lemma-etale-local-lifts}.", "Condition (3) follows from", "Lemma \\ref{lemma-etale-local-lifts-isomorphic}." ], "refs": [ "stacks-more-morphisms-lemma-gerbe-of-lifts-stack", "stacks-definition-gerbe", "stacks-more-morphisms-lemma-etale-local-lifts", "stacks-more-morphisms-lemma-etale-local-lifts-isomorphic" ], "ref_ids": [ 6895, 9003, 6896, 6897 ] } ], "ref_ids": [ 6933 ] }, { "id": 6899, "type": "theorem", "label": "stacks-more-morphisms-lemma-gerbe-of-lifts-first-order", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-gerbe-of-lifts-first-order", "contents": [ "In Remark \\ref{remark-gerbe-of-lifts} assume $\\mathcal{X} \\subset \\mathcal{X}'$", "is a first order thickening. Then", "\\begin{enumerate}", "\\item the automorphism sheaves of objects of the gerbe", "$p : \\mathcal{C} \\to W_{spaces, \\etale}$ constructed", "in Remark \\ref{remark-gerbe-of-lifts} are abelian, and", "\\item the sheaf of groups $\\mathcal{G}$ constructed in", "Stacks, Lemma \\ref{stacks-lemma-gerbe-abelian-auts}", "is a quasi-coherent $\\mathcal{O}_W$-module.", "\\end{enumerate}" ], "refs": [ "stacks-more-morphisms-remark-gerbe-of-lifts", "stacks-more-morphisms-remark-gerbe-of-lifts", "stacks-lemma-gerbe-abelian-auts" ], "proofs": [ { "contents": [ "We will prove both statements at the same time. Namely, given", "an object $\\xi = (U, U', a, i, x', \\alpha)$ we will endow", "$\\mathit{Aut}(\\xi)$ with the structure of a", "quasi-coherent $\\mathcal{O}_U$-module on $U_{spaces, \\etale}$ and", "we will show that this structure is compatible with pullbacks.", "This will be sufficient by glueing of sheaves", "(Sites, Section \\ref{sites-section-glueing-sheaves})", "and the construction of $\\mathcal{G}$ in the proof of", "Stacks, Lemma \\ref{stacks-lemma-gerbe-abelian-auts}", "as the glueing of the automorphism sheaves $\\mathit{Aut}(\\xi)$", "and the fact that it suffices to check a module is", "quasi-coherent after going to an \\'etale covering", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-characterize-quasi-coherent}).", "\\medskip\\noindent", "We will describe the sheaf $\\mathit{Aut}(\\xi)$ using the", "same method as used in the proof of", "Lemma \\ref{lemma-etale-local-lifts-isomorphic}.", "Consider the algebraic space", "$$", "I' = U' \\times_{x', \\mathcal{X}', x'} U'", "$$", "with projections $p' : I' \\to U'$ and $q' : I' \\to U'$.", "Over $I'$ there is a universal $2$-morphism", "$\\gamma : x' \\circ p' \\to x' \\circ q'$.", "The identity $x' \\to x'$ defines a diagonal morphism", "$$", "\\xymatrix{", "U' \\ar[rr]_{\\Delta'} & & I' \\ar[ld]^{p'} \\ar[rd]_{q'} \\\\", "& U' & & U'", "}", "$$", "such that the compositions $U' \\to I' \\to U'$ and $U' \\to I' \\to U'$", "are the identity morphisms. We will denote the base change of", "$U', I', p', q', \\Delta'$ to $\\mathcal{X}$ by $U, I, p, q, \\Delta$.", "Since $W' \\to \\mathcal{X}'$ is smooth, we see that $p' : I' \\to U'$", "is smooth as a base change.", "\\medskip\\noindent", "A section of $\\mathit{Aut}(\\xi)$ over $U$ is a morphism $\\delta' : U' \\to I'$", "such that $\\delta'|_U = \\Delta$ and such that", "$p' \\circ \\delta' = \\text{id}_{U'}$. To be explicit,", "$(\\text{id}_U, q' \\circ \\delta', (\\delta')^*\\gamma) : \\xi \\to \\xi$", "is a formula for the corresponding automorphism.", "More generally, if", "$f : V \\to U$ is an \\'etale morphism, then there is a thickening", "$j : V \\to V'$ and an \\'etale morphism", "$f' : V' \\to U'$ whose restriction to $V$ is $f$ and $f^*\\xi$", "corresponds to $(V, V', a \\circ f, j, x' \\circ f', f^*\\alpha)$, see proof of", "Lemma \\ref{lemma-gerbe-of-lifts-fibred}.", " a section of $\\mathit{Aut}(\\xi)$ over $V$ is a morphism", "$\\delta' : V' \\to I'$", "such that $\\delta'|_V = \\Delta \\circ f$", "and $p' \\circ \\delta' = f'$\\footnote{A formula for the corresponding", "automorphism is $(\\text{id}_V, h', (\\delta')^*\\gamma)$.", "Here $h' : V' \\to V'$ is the unique (iso)morphism such that", "$h'|_V = \\text{id}_V$ and such that", "$$", "\\xymatrix{", "V' \\ar[r]_{h'} \\ar[rd]_{q' \\circ \\delta'} & V' \\ar[d]^{f'} \\\\", "& U'", "}", "$$", "commutes. Uniqueness and existence of $h'$ by topological invariance of", "the \\'etale site, see More on Morphisms of Spaces,", "Theorem \\ref{spaces-more-morphisms-theorem-topological-invariance}.", "The reader may feel we should instead look at morphisms", "$\\delta'' : V' \\to V' \\times_{\\mathcal{X}'} V'$ with", "$\\delta'' \\circ j = \\Delta_{V'/\\mathcal{X}'}$ and", "$\\text{pr}_1 \\circ \\delta'' = \\text{id}_{V'}$.", "This would be fine too: as $V' \\times_{\\mathcal{X}'} V' \\to I'$ is \\'etale,", "the same topological invariance tells us that sending", "$\\delta''$ to $\\delta' = (V' \\times_{\\mathcal{X}'} V' \\to I') \\circ \\delta''$", "is a bijection between the two", "sets of morphisms.}.", "\\medskip\\noindent", "We conclude that $\\mathit{Aut}(\\xi)$ as a sheaf of sets agrees with", "the sheaf defined in", "More on Morphisms of Spaces, Remark", "\\ref{spaces-more-morphisms-remark-another-special-case}", "for the thickenings $(U \\subset U')$ and $(I \\subset I')$ over", "$(U \\subset U')$ via $\\text{id}_{U'}$ and $p'$.", "The diagonal $\\Delta'$ is a section of this sheaf and by", "acting on this section using More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-action-sheaf}", "we get an isomorphism", "\\begin{equation}", "\\label{equation-isomorphism}", "\\SheafHom_{\\mathcal{O}_U}(\\Delta^*\\Omega_{I/U}, \\mathcal{C}_{U/U'})", "\\longrightarrow", "\\mathit{Aut}(\\xi)", "\\end{equation}", "on $U_{spaces, \\etale}$. There three things left to check", "\\begin{enumerate}", "\\item the construction of (\\ref{equation-isomorphism})", "commutes with \\'etale localization,", "\\item $\\SheafHom_{\\mathcal{O}_U}(\\Delta^*\\Omega_{I/U}, \\mathcal{C}_{U/U'})$", "is a quasi-coherent module on $U$,", "\\item the composition in $\\mathit{Aut}(\\xi)$ corresponds", "to addition of sections in this quasi-coherent module.", "\\end{enumerate}", "We will check these in order.", "\\medskip\\noindent", "To see (1) we have to show that if $f : V \\to U$ is \\'etale,", "then (\\ref{equation-isomorphism}) constructed using $\\xi$ over $U$,", "restricts to the map (\\ref{equation-isomorphism})", "$$", "\\SheafHom_{\\mathcal{O}_V}(", "\\Delta_V^*\\Omega_{V \\times_\\mathcal{X} V/V}, \\mathcal{C}_{V/V'}) \\to", "\\mathit{Aut}(\\xi|_V)", "$$", "constructed using $\\xi|_V$ over $V$ on $V_{spaces, \\etale}$.", "This follows from the discussion in the footnote above", "and More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-action-by-derivations-etale-localization}.", "\\medskip\\noindent", "Proof of (2). Since $p'$ is smooth, the morphism $I \\to U$ is smooth,", "and hence the relative module of differentials $\\Omega_{I/U}$", "is finite locally free (More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-smooth-omega-finite-locally-free}).", "On the other hand, $\\mathcal{C}_{U/U'}$ is quasi-coherent", "(More on Morphisms of Spaces, Definition", "\\ref{spaces-more-morphisms-definition-conormal-sheaf}).", "By Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-properties-quasi-coherent}", "we conclude.", "\\medskip\\noindent", "Proof of (3). There exists a morphism $c' : I' \\times_{p', U', q'} I' \\to I'$", "such that $(U', I', p', q', c')$ is a groupoid in algebraic spaces", "with identity $\\Delta'$. See", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack} for example.", "Composition in $\\mathit{Aut}(\\xi)$ is induced by the morphism", "$c'$ as follows. Suppose we have two morphisms", "$$", "\\delta'_1, \\delta'_2 : U' \\longrightarrow I'", "$$", "corresponding to sections of $\\mathit{Aut}(\\xi)$ over $U$ as above,", "in other words, we have $\\delta'_i|U = \\Delta_U$ and", "$p' \\circ \\delta'_i = \\text{id}_{U'}$. Then the composition", "in $\\mathit{Aut}(\\xi)$ is", "$$", "\\delta'_1 \\circ \\delta'_2 = c'(\\delta'_1 \\circ q' \\circ \\delta'_2, \\delta'_2)", "$$", "We omit the detailed verification\\footnote{The reader can see immediately", "that it is necessary to precompose", "$\\delta'_1$ by $q' \\circ \\delta'_2$ to get a well defined $U'$-valued", "point of the fibre product $I' \\times_{p', U', q'} I'$.}.", "Thus we are in the situation described in", "More on Groupoids in Spaces, Section", "\\ref{spaces-more-groupoids-section-groupoid-sections}", "and the desired result follows from", "More on Groupoids in Spaces, Lemma", "\\ref{spaces-more-groupoids-lemma-composition-is-addition}." ], "refs": [ "stacks-lemma-gerbe-abelian-auts", "spaces-properties-lemma-characterize-quasi-coherent", "stacks-more-morphisms-lemma-etale-local-lifts-isomorphic", "stacks-more-morphisms-lemma-gerbe-of-lifts-fibred", "spaces-more-morphisms-theorem-topological-invariance", "spaces-more-morphisms-remark-another-special-case", "spaces-more-morphisms-lemma-action-sheaf", "spaces-more-morphisms-lemma-action-by-derivations-etale-localization", "spaces-more-morphisms-lemma-smooth-omega-finite-locally-free", "spaces-more-morphisms-definition-conormal-sheaf", "spaces-properties-lemma-properties-quasi-coherent", "algebraic-lemma-map-space-into-stack", "spaces-more-groupoids-lemma-composition-is-addition" ], "ref_ids": [ 8979, 11911, 6897, 6894, 9, 308, 99, 100, 44, 279, 11912, 8473, 13175 ] } ], "ref_ids": [ 6933, 6933, 8979 ] }, { "id": 6900, "type": "theorem", "label": "stacks-more-morphisms-lemma-inf-quasi-coherent", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-inf-quasi-coherent", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack over a scheme $S$.", "Assume $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally", "of finite presentation. Let $A \\to B$ be a flat $S$-algebra homomorphism.", "Let $x$ be an object of $\\mathcal{X}$ over $A$ and set $y = x|_B$.", "Then $\\text{Inf}_x(M) \\otimes_A B = \\text{Inf}_y(M \\otimes_A B)$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\text{Inf}_x(M)$ is the set of automorphisms of the", "trivial deformation of $x$ to $A[M]$ which induce the identity", "automorphism of $x$ over $A$. The trivial deformation is", "the pullback of $x$ to $\\Spec(A[M])$ via $\\Spec(A[M]) \\to \\Spec(A)$.", "Let $G \\to \\Spec(A)$ be the automorphism group algebraic space of $x$", "(this exists because $\\mathcal{X}$ is an algebraic space).", "Let $e : \\Spec(A) \\to G$ be the neutral element.", "The discussion in More on Morphisms of Spaces, Section", "\\ref{spaces-more-morphisms-section-action-by-derivations}", "gives", "$$", "\\text{Inf}_x(M) = \\Hom_A(e^*\\Omega_{G/A}, M)", "$$", "By the same token", "$$", "\\text{Inf}_y(M \\otimes_A B) = \\Hom_B(e_B^*\\Omega_{G_B/B}, M \\otimes_A B)", "$$", "Since $G \\to \\Spec(A)$ is locally of finite presentation by", "assumption, we see that $\\Omega_{G/A}$ is locally of finite", "presentation, see", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-finite-presentation-differentials}.", "Hence $e^*\\Omega_{G/A}$ is a finitely presented $A$-module.", "Moreover, $\\Omega_{G_B/B}$ is the pullback of $\\Omega_{G/A}$ by", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-base-change-differentials}.", "Therefore $e_B^*\\Omega_{G_B/B} = e^*\\Omega_{G/A} \\otimes_A B$.", "we conclude by More on Algebra, Lemma", "\\ref{more-algebra-lemma-pseudo-coherence-and-base-change-ext}." ], "refs": [ "spaces-more-morphisms-lemma-finite-presentation-differentials", "spaces-more-morphisms-lemma-base-change-differentials", "more-algebra-lemma-pseudo-coherence-and-base-change-ext" ], "ref_ids": [ 43, 40, 10165 ] } ], "ref_ids": [] }, { "id": 6901, "type": "theorem", "label": "stacks-more-morphisms-lemma-sheaf-of-infinitesimal-lifts", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-sheaf-of-infinitesimal-lifts", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack over a base scheme $S$.", "Assume $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally", "of finite presentation. Let $(A' \\to A, x)$ be a deformation situation.", "Then the functor", "$$", "F : B' \\longmapsto", "\\{\\text{lifts of }x|_{B' \\otimes_{A'} A}\\text{ to } B'\\}/\\text{isomorphisms}", "$$", "is a sheaf on the site $(\\textit{Aff}/\\Spec(A'))_{fppf}$ of", "Topologies, Definition \\ref{topologies-definition-big-small-fppf}." ], "refs": [ "topologies-definition-big-small-fppf" ], "proofs": [ { "contents": [ "Let $\\{T'_i \\to T'\\}_{i = 1, \\ldots n}$ be a standard fppf covering", "of affine schemes over $A'$. Write $T' = \\Spec(B')$. As usual denote", "$$", "T'_{i_0 \\ldots i_p} =", "T'_{i_0} \\times_{T'} \\ldots \\times_{T'} T'_{i_p} = \\Spec(B'_{i_0 \\ldots i_p})", "$$", "where the ring is a suitable tensor product.", "Set $B = B' \\otimes_{A'} A$ and", "$B_{i_0 \\ldots i_p} = B'_{i_0 \\ldots i_p} \\otimes_{A'} A$.", "Denote $y = x|_B$ and $y_{i_0 \\ldots i_p} = x|_{B_{i_0 \\ldots i_p}}$.", "Let $\\gamma_i \\in F(B'_i)$ such that $\\gamma_{i_0}$ and $\\gamma_{i_1}$", "map to the same element of $F(B'_{i_0i_1})$.", "We have to find a unique $\\gamma \\in F(B')$ mapping to", "$\\gamma_i$ in $F(B'_i)$.", "\\medskip\\noindent", "Choose an actual object $y'_i$ of $\\textit{Lift}(y_i, B'_i)$", "in the isomorphism class $\\gamma_i$.", "Choose isomorphisms", "$\\varphi_{i_0i_1} : y'_{i_0}|_{B'_{i_0i_1}} \\to y'_{i_1}|_{B'_{i_0i_1}}$", "in the category $\\textit{Lift}(y_{i_0i_1}, B'_{i_0i_1})$.", "If the maps $\\varphi_{i_0i_1}$ satisfy the cocycle condition,", "then we obtain our object $\\gamma$ because $\\mathcal{X}$ is", "a stack in the fppf topology. The cocycle condition is that the", "composition", "$$", "y'_{i_0}|_{B'_{i_0i_1i_2}}", "\\xrightarrow{\\varphi_{i_0i_1}|_{B'_{i_0i_1i_2}}}", "y'_{i_1}|_{B'_{i_0i_1i_2}}", "\\xrightarrow{\\varphi_{i_1i_2}|_{B'_{i_0i_1i_2}}}", "y'_{i_2}|_{B'_{i_0i_1i_2}}", "\\xrightarrow{\\varphi_{i_2i_0}|_{B'_{i_0i_1i_2}}}", "y'_{i_0}|_{B'_{i_0i_1i_2}}", "$$", "is the identity. If not, then these maps give elements", "$$", "\\delta_{i_0i_1i_2} \\in", "\\text{Inf}_{y_{i_0i_1i_2}}(J_{i_0i_1i_2}) =", "\\text{Inf}_y(J) \\otimes_B B_{i_0i_1i_2}", "$$", "Here $J = \\Ker(B' \\to B)$ and", "$J_{i_0 \\ldots i_p} = \\Ker(B'_{i_0 \\ldots i_p} \\to B_{i_0 \\ldots i_p})$.", "The equality in the displayed equation holds by", "Lemma \\ref{lemma-inf-quasi-coherent} applied to", "$B' \\to B'_{i_0 \\ldots i_p}$ and $y$ and $y_{i_0 \\ldots i_p}$,", "the flatness of the maps $B' \\to B'_{i_0 \\ldots i_p}$", "which also guarantees that", "$J_{i_0 \\ldots i_p} = J \\otimes_{B'} B'_{i_0 \\ldots i_p}$.", "A computation (omitted) shows that $\\delta_{i_0i_1i_2}$ gives", "a $2$-cocycle in the {\\v C}ech complex", "$$", "\\prod \\text{Inf}_y(J) \\otimes_B B_{i_0} \\to", "\\prod \\text{Inf}_y(J) \\otimes_B B_{i_0i_1} \\to", "\\prod \\text{Inf}_y(J) \\otimes_B B_{i_0i_1i_2} \\to \\ldots", "$$", "By Descent, Lemma \\ref{descent-lemma-standard-covering-Cech-quasi-coherent}", "this complex is acyclic in positive degrees and has $H^0 = \\text{Inf}_y(J)$.", "Since $\\text{Inf}_{y_{i_0i_1}}(J_{i_0i_1})$ acts on", "morphisms (Artin's Axioms, Remark \\ref{artin-remark-automorphisms})", "this means we can modify our choice of $\\varphi_{i_0i_1}$", "to get to the case where $\\delta_{i_0i_1i_2} = 0$.", "\\medskip\\noindent", "Uniqueness. We still have to show there is at most one $\\gamma$ restricting", "to $\\gamma_i$ for all $i$. Suppose we have objects $y', z'$ of", "$\\textit{Lift}(y, B')$ and isomorphisms $\\psi_i : y'|_{B'_i} \\to z'|_{B'_i}$", "in $\\textit{Lift}(y_i, B'_i)$. Then we can consider", "$$", "\\psi_{i_1}^{-1} \\circ \\psi_{i_0} \\in", "\\text{Inf}_{y_{i_0i_1}}(J_{i_0i_1}) =", "\\text{Inf}_y(J) \\otimes_B B_{i_0i_1}", "$$", "Arguing as before, the obstruction to finding an isomorphism between", "$y'$ and $z'$ over $B'$ is an element in the $H^1$ of the", "{\\v C}ech complex displayed above which is zero." ], "refs": [ "stacks-more-morphisms-lemma-inf-quasi-coherent", "descent-lemma-standard-covering-Cech-quasi-coherent", "artin-remark-automorphisms" ], "ref_ids": [ 6900, 14625, 11432 ] } ], "ref_ids": [ 12542 ] }, { "id": 6902, "type": "theorem", "label": "stacks-more-morphisms-lemma-T-quasi-coherent", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-T-quasi-coherent", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack over a scheme $S$ whose", "structure morphism $\\mathcal{X} \\to S$ is locally of finite presentation.", "Let $A \\to B$ be a flat $S$-algebra homomorphism.", "Let $x$ be an object of $\\mathcal{X}$ over $A$.", "Then $T_x(M) \\otimes_A B = T_y(M \\otimes_A B)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective smooth morphism $U \\to \\mathcal{X}$.", "We first reduce the lemma to the case where $x$ lifts to $U$.", "Recall that $T_x(M)$ is the set of isomorphism classes of lifts of $x$", "to $A[M]$. Therefore", "Lemma \\ref{lemma-sheaf-of-infinitesimal-lifts}\\footnote{This lemma applies:", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_S \\mathcal{X}$", "is locally of finite presentation by", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-finite-presentation-permanence}", "and the assumption that $\\mathcal{X} \\to S$ is locally of finite presentation.", "Therefore $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally", "of finite presentation as a base change of $\\Delta$.}", "says that the rule", "$$", "A_1 \\mapsto T_{x|_{A_1}}(M \\otimes_A A_1)", "$$", "is a sheaf on the small \\'etale site of $\\Spec(A)$; the tensor product", "is needed to make $A[M] \\to A_1[M \\otimes_A A_1]$ a flat ring map.", "We may choose a faithfully flat \\'etale ring map $A \\to A_1$", "such that $x|_{A_1}$ lifts to a morphism $u_1 : \\Spec(A_1) \\to U$, see", "for example Sheaves on Stacks, Lemma", "\\ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}.", "Write $A_2 = A_1 \\otimes_A A_1$ and set $B_1 = B \\otimes_A A_1$", "and $B_2 = B \\otimes_A A_2$. Consider the diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "T_y(M \\otimes_A B) \\ar[r] &", "T_{y|_{B_1}}(M \\otimes_A B_1) \\ar[r] &", "T_{y|_{B_2}}(M \\otimes_A B_2) \\\\", "0 \\ar[r] &", "T_x(M) \\ar[r] \\ar[u] &", "T_{x|_{A_1}}(M \\otimes_A A_1) \\ar[r] \\ar[u] &", "T_{x|_{A_2}}(M \\otimes_A A_2) \\ar[u]", "}", "$$", "The rows are exact by the sheaf condition. We have", "$M \\otimes_A B_i = (M \\otimes_A A_i) \\otimes_{A_i} B_i$.", "Thus if we prove the result for the middle and right vertical arrow, then", "the result follows. This reduces us to the case discussed in", "the next paragraph.", "\\medskip\\noindent", "Assume that $x$ is the image of a morphism $u : \\Spec(A) \\to U$.", "Observe that $T_u(M) \\to T_x(M)$ is surjective since", "$U \\to \\mathcal{X}$ is smooth and representable by", "algebraic spaces, see Criteria for Representability, Lemma", "\\ref{criteria-lemma-representable-by-spaces-formally-smooth}", "(see discussion preceding it for explanation) and", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-smooth-formally-smooth}.", "Set $R = U \\times_\\mathcal{X} U$. Recall that we obtain", "a groupoid $(U, R, s, t, c, e, i)$ in algebraic spaces with", "$\\mathcal{X} = [U/R]$. By", "Artin's Axioms, Lemma \\ref{artin-lemma-ses-inf-and-T}", "we have an exact sequence", "$$", "T_{e \\circ u}(M) \\to T_u(M) \\oplus T_u(M) \\to T_x(M) \\to 0", "$$", "where the zero on the right was shown above. A similar sequence", "holds for the base change to $B$. Thus the result we want follows if", "we can prove the result of the lemma for", "$T_u(M)$ and $T_{e \\circ u}(M)$. This reduces us to the case discussed", "in the next paragraph.", "\\medskip\\noindent", "Assume that $\\mathcal{X} = X$ is an algebraic space locally of finite", "presentation over $S$. Then we have", "$$", "T_x(M) = \\Hom_A(x^*\\Omega_{X/S}, M)", "$$", "by the discussion in More on Morphisms of Spaces, Section", "\\ref{spaces-more-morphisms-section-action-by-derivations}.", "By the same token", "$$", "T_y(M \\otimes_A B) = \\Hom_B(y^*\\Omega_{X/S}, M \\otimes_A B)", "$$", "Since $X \\to S$ is locally of finite presentation, we see that", "$\\Omega_{X/S}$ is locally of finite presentation, see", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-finite-presentation-differentials}.", "Hence $x^*\\Omega_{X/S}$ is a finitely presented $A$-module.", "Clearly, we have $y^*\\Omega_{X/S} = x^*\\Omega_{X/S} \\otimes_A B$.", "we conclude by More on Algebra, Lemma", "\\ref{more-algebra-lemma-pseudo-coherence-and-base-change-ext}." ], "refs": [ "stacks-more-morphisms-lemma-sheaf-of-infinitesimal-lifts", "stacks-morphisms-lemma-finite-presentation-permanence", "stacks-sheaves-lemma-surjective-flat-locally-finite-presentation", "criteria-lemma-representable-by-spaces-formally-smooth", "spaces-more-morphisms-lemma-smooth-formally-smooth", "artin-lemma-ses-inf-and-T", "spaces-more-morphisms-lemma-finite-presentation-differentials", "more-algebra-lemma-pseudo-coherence-and-base-change-ext" ], "ref_ids": [ 6901, 7504, 11606, 3107, 110, 11389, 43, 10165 ] } ], "ref_ids": [] }, { "id": 6903, "type": "theorem", "label": "stacks-more-morphisms-lemma-local-lift-enough", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-local-lift-enough", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack over a scheme $S$ whose", "structure morphism $\\mathcal{X} \\to S$ is locally of finite presentation.", "Let $(A' \\to A, x)$ be a deformation situation. If there exists a", "faithfully flat finitely presented $A'$-algebra $B'$ and an", "object $y'$ of $\\mathcal{X}$ over $B'$ lifting $x|_{B' \\otimes_{A'} A}$,", "then there exists an object $x'$ over $A'$ lifting $x$." ], "refs": [], "proofs": [ { "contents": [ "Let $I = \\Ker(A' \\to A)$. Set $B'_1 = B' \\otimes_{A'} B'$ and", "$B'_2 = B' \\otimes_{A'} B' \\otimes_{A'} B'$. Let", "$J = IB'$, $J_1 = IB'_1$, $J_2 = IB'_2$ and", "$B = B'/J$, $B_1 = B'_1/J_1$, $B_2 = B'_2/J_2$.", "Set $y = x|_B$, $y_1 = x|_{B_1}$, $y_2 = x|_{B_2}$.", "Let $F$ be the fppf sheaf of", "Lemma \\ref{lemma-sheaf-of-infinitesimal-lifts}", "(which applies, see footnote in the proof of", "Lemma \\ref{lemma-T-quasi-coherent}).", "Thus we have an equalizer diagram", "$$", "\\xymatrix{", "F(A') \\ar[r] &", "F(B') \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "F(B'_1)", "}", "$$", "On the other hand, we have $F(B') = \\text{Lift}(y, B')$,", "$F(B'_1) = \\text{Lift}(y_1, B'_1)$, and $F(B'_2) = \\text{Lift}(y_2, B'_2)$", "in the terminology from Artin's Axioms, Section \\ref{artin-section-inf}.", "These sets are nonempty and are (canonically) principal homogeneous spaces for", "$T_y(J)$, $T_{y_1}(J_1)$, $T_{y_2}(J_2)$, see", "Artin's Axioms, Lemma \\ref{artin-lemma-properties-lift-RS-star}.", "Thus the difference of the two images of $y'$ in", "$F(B'_1)$ is an element", "$$", "\\delta_1 \\in T_{y_1}(J_1) = T_x(I) \\otimes_A B_1", "$$", "The equality in the displayed equation holds by", "Lemma \\ref{lemma-T-quasi-coherent} applied to", "$A' \\to B'_1$ and $x$ and $y_1$, the flatness of $A' \\to B'_1$", "which also guarantees that $J_1 = I \\otimes_{A'} B'_1$. We have", "similar equalities for $B'$ and $B'_2$.", "A computation (omitted) shows that $\\delta_1$ gives", "a $1$-cocycle in the {\\v C}ech complex", "$$", "T_x(I) \\otimes_A B \\to", "T_x(I) \\otimes_A B_1 \\to", "T_x(I) \\otimes_A B_2 \\to \\ldots", "$$", "By Descent, Lemma \\ref{descent-lemma-standard-covering-Cech-quasi-coherent}", "this complex is acyclic in positive degrees and has $H^0 = T_x(I)$.", "Thus we may choose an element in $T_x(I) \\otimes_A B = T_y(J)$", "whose boundary is $\\delta_1$. Replacing $y'$ by the result", "of this element acting on it, we find a new choice $y'$ with $\\delta_1 = 0$.", "Thus $y'$ maps to the same element under the two maps", "$F(B') \\to F(B'_1)$ and we obtain an element o $F(A')$ by", "the sheaf condition." ], "refs": [ "stacks-more-morphisms-lemma-sheaf-of-infinitesimal-lifts", "stacks-more-morphisms-lemma-T-quasi-coherent", "artin-lemma-properties-lift-RS-star", "stacks-more-morphisms-lemma-T-quasi-coherent", "descent-lemma-standard-covering-Cech-quasi-coherent" ], "ref_ids": [ 6901, 6902, 11388, 6902, 14625 ] } ], "ref_ids": [] }, { "id": 6904, "type": "theorem", "label": "stacks-more-morphisms-lemma-reformulate-formal-smoothness", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-reformulate-formal-smoothness", "contents": [ "A morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks is", "formally smooth (Definition \\ref{definition-formally-smooth})", "if and only if for every diagram (\\ref{equation-diagram}) and $\\gamma$", "the category of dotted arrows is nonempty." ], "refs": [ "stacks-more-morphisms-definition-formally-smooth" ], "proofs": [ { "contents": [ "Translation between different languages omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 6930 ] }, { "id": 6905, "type": "theorem", "label": "stacks-more-morphisms-lemma-lift-to-smooth", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-lift-to-smooth", "contents": [ "Let $T \\to T'$ be a first order thickening of affine schemes.", "Let $\\mathcal{X}'$ be an algebraic stack over $T'$", "whose structure morphism $\\mathcal{X}' \\to T'$ is smooth.", "Let $x : T \\to \\mathcal{X}'$ be a morphism", "over $T'$. Then there exists a morphsm $x' : T' \\to \\mathcal{X}'$", "over $T'$ with $x'|_T = x$." ], "refs": [], "proofs": [ { "contents": [ "We may apply the result of Lemma \\ref{lemma-local-lift-enough}.", "Thus it suffices to construct a smooth surjective morphism", "$W' \\to T'$ with $W'$ affine such that", "$x|_{T \\times_{W'} T'}$ lifts to $W'$.", "(We urge the reader to find their own proof of this fact", "using the analogous result for algebraic spaces already", "established.) We choose a", "scheme $U'$ and a surjective smooth morphism $U' \\to \\mathcal{X}'$.", "Observe that $U' \\to T'$ is smooth and that the projection", "$T \\times_{\\mathcal{X}'} U' \\to T$ is surjective smooth.", "Choose an affine scheme $W$ and an \\'etale morphism", "$W \\to T \\times_{\\mathcal{X}'} U'$ such that $W \\to T$", "is surjective. Then $W \\to T$ is a smooth morphism of", "affine schemes. After replacing $W$ by a disjoint union of", "principal affine opens, we may assume there exists a", "smooth morphism of affines $W' \\to T'$ such that", "$W = T \\times_{T'} W'$, see Algebra, Lemma \\ref{algebra-lemma-lift-smooth}.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-smooth-formally-smooth}", "we can find a morphism $W' \\to U'$ over $T'$ lifting", "the given morphism $W \\to U'$. This finishes the proof." ], "refs": [ "stacks-more-morphisms-lemma-local-lift-enough", "algebra-lemma-lift-smooth", "spaces-more-morphisms-lemma-smooth-formally-smooth" ], "ref_ids": [ 6903, 1203, 110 ] } ], "ref_ids": [] }, { "id": 6906, "type": "theorem", "label": "stacks-more-morphisms-lemma-smooth-formally-smooth", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-smooth-formally-smooth", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is smooth.", "\\item The morphism $f$ is locally of finite presentation and", "formally smooth.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f$ is smooth. Then $f$ is locally of finite presentation by", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-smooth-locally-finite-presentation}.", "Hence it suffices given a diagram (\\ref{equation-diagram})", "and a $\\gamma : y \\circ i \\to f \\circ x$ to find a dotted", "arrow (see Lemma \\ref{lemma-reformulate-formal-smoothness}).", "Forming fibre products we obtain", "$$", "\\xymatrix{", "T \\ar[d] \\ar[r] &", "T' \\times_\\mathcal{Y} \\mathcal{X} \\ar[d] \\ar[r] &", "\\mathcal{X} \\ar[d] \\\\", "T' \\ar[r] & T' \\ar[r] & \\mathcal{Y}", "}", "$$", "Thus we see it is sufficient to find a dotted arrow in the", "left square. Since $T' \\times_\\mathcal{Y} \\mathcal{X} \\to T'$", "is smooth", "(Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-base-change-smooth})", "existence of a dotted arrow in the left square is guaranteed by", "Lemma \\ref{lemma-lift-to-smooth}.", "\\medskip\\noindent", "Conversely, suppose that $f$ is locally of finite presentation and", "formally smooth. Choose a scheme $U$ and a surjective smooth morphism", "$U \\to \\mathcal{X}$. Then $a : U \\to \\mathcal{X}$ and $b : U \\to \\mathcal{Y}$", "are representable by algebraic spaces and locally of finite presentation", "(use Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-composition-finite-presentation}", "and the fact seen above", "that a smooth morphism is locally of finite presentation).", "We will apply the general principle of", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-representable-transformations-property-implication}", "with as input the equivalence of More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-smooth-formally-smooth}", "and simultaneously use the translation of", "Criteria for Representability, Lemma", "\\ref{criteria-lemma-representable-by-spaces-formally-smooth}.", "We first apply this to $a$ to see that $a$ is", "formally smooth on objects. Next, we use that $f$ is", "formally smooth on objects by assumption", "(see Lemma \\ref{lemma-reformulate-formal-smoothness})", "and Criteria for Representability, Lemma", "\\ref{criteria-lemma-composition-formally-smooth}", "to see that $b = f \\circ a$ is formally smooth on objects.", "Then we apply the principle once more to conclude that $b$ is smooth.", "This means that $f$ is smooth by the definition of smoothness", "for morphisms of algebraic stacks and the proof is complete." ], "refs": [ "stacks-morphisms-lemma-smooth-locally-finite-presentation", "stacks-more-morphisms-lemma-reformulate-formal-smoothness", "stacks-morphisms-lemma-base-change-smooth", "stacks-more-morphisms-lemma-lift-to-smooth", "stacks-morphisms-lemma-composition-finite-presentation", "algebraic-lemma-representable-transformations-property-implication", "spaces-more-morphisms-lemma-smooth-formally-smooth", "criteria-lemma-representable-by-spaces-formally-smooth", "stacks-more-morphisms-lemma-reformulate-formal-smoothness", "criteria-lemma-composition-formally-smooth" ], "ref_ids": [ 7542, 6904, 7540, 6905, 7500, 8459, 110, 3107, 6904, 3106 ] } ], "ref_ids": [] }, { "id": 6907, "type": "theorem", "label": "stacks-more-morphisms-lemma-flatten-stack", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-flatten-stack", "contents": [ "Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack", "to an algebraic space. Let $V \\subset Y$ be an open subspace. Assume", "\\begin{enumerate}", "\\item $Y$ is quasi-compact and quasi-separated,", "\\item $f$ is of finite type and quasi-separated,", "\\item $V$ is quasi-compact, and", "\\item $\\mathcal{X}_V$ is flat and locally of finite presentation over $V$.", "\\end{enumerate}", "Then there exists a $V$-admissible blowup $Y' \\to Y$", "and a closed substack $\\mathcal{X}' \\subset \\mathcal{X}_{Y'}$", "with $\\mathcal{X}'_V = \\mathcal{X}_V$ such that", "$\\mathcal{X}' \\to Y'$ is flat and of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\mathcal{X}$ is quasi-compact.", "Choose an affine scheme $U$ and a surjective smooth", "morphism $U \\to \\mathcal{X}$.", "Let $R = U \\times_\\mathcal{X} U$ so that we obtain", "a groupoid $(U, R, s, t, c)$ in algebraic spaces over $Y$ with", "$\\mathcal{X} = [U/R]$", "(Algebraic Stacks, Lemma \\ref{algebraic-lemma-stack-presentation}).", "We may apply", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-flat-after-blowing-up}", "to $U \\to Y$ and the open $V \\subset Y$.", "Thus we obtain a $V$-admissible blowup $Y' \\to Y$", "such that the strict transform $U' \\subset U_{Y'}$", "is flat and of finite presentation over $Y'$.", "Let $R' \\subset R_{Y'}$ be the strict transform of $R$.", "Since $s$ and $t$ are smooth (and in particular flat)", "it follows from", "Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-strict-transform-flat}", "that we have cartesian diagrams", "$$", "\\vcenter{", "\\xymatrix{", "R' \\ar[r] \\ar[d] & R_{Y'} \\ar[d]^{s_{Y'}} \\\\", "U' \\ar[r] & U_{Y'}", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "R' \\ar[r] \\ar[d] & R_{Y'} \\ar[d]^{t_{Y'}} \\\\", "U' \\ar[r] & U_{Y'}", "}", "}", "$$", "In other words, $U'$ is an $R_{Y'}$-invariant closed subspace", "of $U_{Y'}$. Thus $U'$ defines a closed substack", "$\\mathcal{X}' \\subset \\mathcal{X}_{Y'}$ by", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-substacks-presentation}.", "The morphism $\\mathcal{X}' \\to Y'$ is flat and", "locally of finite presentation because this is", "true for $U' \\to Y'$. On the other hand,", "we already know $\\mathcal{X}' \\to Y'$ is quasi-compact and quasi-separated", "(by our assumptions on $f$ and because this is true for closed immersions).", "This finishes the proof." ], "refs": [ "algebraic-lemma-stack-presentation", "spaces-more-morphisms-lemma-flat-after-blowing-up", "spaces-divisors-lemma-strict-transform-flat", "stacks-properties-lemma-substacks-presentation" ], "ref_ids": [ 8474, 190, 13002, 8889 ] } ], "ref_ids": [] }, { "id": 6908, "type": "theorem", "label": "stacks-more-morphisms-lemma-finite-cover-factor", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-finite-cover-factor", "contents": [ "Let $Y$ be a quasi-compact and quasi-separated algebraic space.", "Let $V \\subset Y$ be a quasi-compact open. Let $f : \\mathcal{X} \\to V$", "be surjective, flat, and locally of finite presentation.", "Then there exists a finite surjective morphism $g : Y' \\to Y$ such that", "$V' = g^{-1}(V) \\to Y$ factors Zariski locally through $f$." ], "refs": [], "proofs": [ { "contents": [ "We first prove this when $Y$ is a scheme.", "We may choose a scheme $U$ and a surjective smooth morphism", "$U \\to \\mathcal{X}$. Then $\\{U \\to V\\}$ is an fppf covering of schemes.", "By More on Morphisms, Lemma \\ref{more-morphisms-lemma-dominate-fppf-finite}", "there exists a finite surjective morphism", "$V' \\to V$ such that $V' \\to V$ factors Zariski locally", "through $U$. By", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-extend-finite-surjective-morphisms}", "we can find a finite surjective morphism $Y' \\to Y$", "whose restriction to $V$ is $V' \\to V$ as desired.", "\\medskip\\noindent", "If $Y$ is an algebraic space, then we see the lemma is true by", "first doing a finite base change by a finite surjective morphism", "$Y' \\to Y$ where $Y'$ is a scheme. See", "Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-there-is-a-scheme-finite-over}." ], "refs": [ "more-morphisms-lemma-dominate-fppf-finite", "more-morphisms-lemma-extend-finite-surjective-morphisms", "spaces-limits-proposition-there-is-a-scheme-finite-over" ], "ref_ids": [ 13926, 13924, 4659 ] } ], "ref_ids": [] }, { "id": 6909, "type": "theorem", "label": "stacks-more-morphisms-lemma-make-section", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-make-section", "contents": [ "Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack", "to an algebraic space. Let $V \\subset Y$ be an open subspace.", "Assume", "\\begin{enumerate}", "\\item $f$ is separated and of finite type,", "\\item $Y$ is quasi-compact and quasi-separated,", "\\item $V$ is quasi-compact, and", "\\item $\\mathcal{X}_V$ is a gerbe over $V$.", "\\end{enumerate}", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "\\overline{Z} \\ar[rd]_{\\overline{g}} &", "Z \\ar[l]^j \\ar[d]_g \\ar[r]_h & \\mathcal{X} \\ar[ld]^f \\\\", "& Y", "}", "$$", "with $j$ an open immersion, $\\overline{g}$ and $h$ proper,", "and such that $|V|$ is contained in the image of $|g|$." ], "refs": [], "proofs": [ { "contents": [ "Suppose we have a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[d]_{f'} \\ar[r] & \\mathcal{X} \\ar[d]^f \\\\", "Y' \\ar[r] & Y", "}", "$$", "and a quasi-compact open $V' \\subset Y'$, such that", "$Y' \\to Y$ is a proper morphism of algebraic spaces,", "$\\mathcal{X}' \\to \\mathcal{X}$ is a proper morphism of algebraic stacks,", "$V' \\subset Y'$ maps surjectively onto $V$, and", "$\\mathcal{X}'_{V'}$ is a gerbe over $V'$.", "Then it suffices to prove the lemma for the pair", "$(f' : \\mathcal{X}' \\to Y', V')$. Some details omitted.", "\\medskip\\noindent", "Overall strategy of the proof. We will reduce", "to the case where the image of $f$ is open and $f$", "has a section over this open by repeatedly applying the", "above remark. Each step is straightforward, but there are", "quite a few of them which makes the proof a bit involved.", "\\medskip\\noindent", "Using Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-there-is-a-scheme-finite-over}", "we reduce to the case where $Y$ is a scheme.", "(Let $Y' \\to Y$ be a finite surjective morphism where $Y'$ is", "a scheme. Set $\\mathcal{X}' = \\mathcal{X}_{Y'}$ and apply", "the initial remark of the proof.)", "\\medskip\\noindent", "Using Lemma \\ref{lemma-flatten-stack}", "(and Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-gerbe-fppf}", "to see that a gerbe is flat and locally of finite presentation)", "we reduce to the case where $f$ is flat and of finite presentation.", "\\medskip\\noindent", "Since $f$ is flat and locally of finite", "presentation, we see that the image of $|f|$ is an open $W \\subset Y$.", "Since $\\mathcal{X}$ is quasi-compact (as $f$ is of finite type", "and $Y$ is quasi-compact) we see that $W$ is quasi-compact.", "By Lemma \\ref{lemma-finite-cover-factor}", "we can find a finite surjective morphism $g : Y' \\to Y$", "such that $g^{-1}(W) \\to Y$ factors Zariski locally", "through $\\mathcal{X} \\to Y$.", "After replacing $Y$ by $Y'$ and $\\mathcal{X}$ by", "$\\mathcal{X} \\times_Y Y'$ we reduce to the situation", "described in the next paragraph.", "\\medskip\\noindent", "Assume there exists $n \\geq 0$, quasi-compact opens", "$W_i \\subset Y$, $i = 1, \\ldots, n$, and", "morphisms $x_i : W_i \\to \\mathcal{X}$ such that", "(a) $f \\circ x_i = \\text{id}_{W_i}$,", "(b) $W = \\bigcup_{i = 1, \\ldots, n} W_i$ contains $V$, and", "(c) $W$ is the image of $|f|$.", "We will use induction on $n$. The base case is $n = 0$: this", "implies $V = \\emptyset$ and in this case we can take", "$\\overline{Z} = \\emptyset$.", "If $n > 0$, then for $i = 1, \\ldots, n$", "consider the reduced closed subschemes", "$Y_i$ with underlying topological space", "$Y \\setminus W_i$. Consider the finite morphism", "$$", "Y' = Y \\amalg \\coprod\\nolimits_{i = 1, \\ldots, n} Y_i \\longrightarrow Y", "$$", "and the quasi-compact open", "$$", "V' = (W_1 \\cap \\ldots \\cap W_n \\cap V) \\amalg", "\\coprod_{i = 1, \\ldots, n} (V \\cap Y_i).", "$$", "By the initial remark of the proof, if we can prove the lemma for the pairs", "$$", "(\\mathcal{X} \\to Y, W_1 \\cap \\ldots \\cap W_n \\cap V)", "\\quad\\text{and}\\quad", "(\\mathcal{X} \\times_Y Y_i \\to Y_i, V \\cap Y_i),\\quad", "i = 1, \\ldots, n", "$$", "then the result is true. Here we use the settheoretic equality", "$V = (W_1 \\cap \\ldots \\cap W_n \\cap V) \\cup", "\\bigcup\\nolimits_{i = 1, \\ldots n} (V \\cap Y_i)$.", "The induction hypothesis applies to the second type of", "pairs above. Hence we reduce to the situation described in", "the next paragraph.", "\\medskip\\noindent", "Assume there exists $n \\geq 0$, quasi-compact opens", "$W_i \\subset Y$, $i = 1, \\ldots, n$, and", "morphisms $x_i : W_i \\to \\mathcal{X}$ such that", "(a) $f \\circ x_i = \\text{id}_{W_i}$,", "(b) $W = \\bigcup_{i = 1, \\ldots, n} W_i$ contains $V$,", "(c) $W$ is the image of $|f|$, and", "(d) $V \\subset W_1 \\cap \\ldots \\cap W_n$.", "The morphisms", "$$", "T_{ij} = \\mathit{Isom}_\\mathcal{X}(x_i|_{W_i \\cap W_j \\cap V},", "x_j|_{W_i \\cap W_j \\cap V}) \\longrightarrow W_i \\cap W_j \\cap V", "$$", "are surjective, flat, and locally of finite presentation", "(Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-gerbe-isom-fppf}).", "We apply Lemma \\ref{lemma-finite-cover-factor}", "to each quasi-compact open $W_i \\cap W_j \\cap V$ and the morphisms", "$T_{ij} \\to W_i \\cap W_j \\cap V$ to get finite surjective morphisms", "$Y'_{ij} \\to Y$. After replacing $Y$ by the fibre product of all", "of the $Y'_{ij}$ over $Y$ we reduce to the situation described", "in the next paragraph.", "\\medskip\\noindent", "Assume there exists $n \\geq 0$, quasi-compact opens", "$W_i \\subset Y$, $i = 1, \\ldots, n$, and", "morphisms $x_i : W_i \\to \\mathcal{X}$ such that", "(a) $f \\circ x_i = \\text{id}_{W_i}$,", "(b) $W = \\bigcup_{i = 1, \\ldots, n} W_i$ contains $V$,", "(c) $W$ is the image of $|f|$,", "(d) $V \\subset W_1 \\cap \\ldots \\cap W_n$, and", "(e) $x_i$ and $x_j$ are Zariski locally isomorphic over $W_i \\cap W_j \\cap V$.", "Let $y \\in V$ be arbitrary.", "Suppose that we can find a quasi-compact open neighbourhood", "$y \\in V_y \\subset V$ such that the lemma is true for", "the pair $(\\mathcal{X} \\to Y, V_y)$, say with solution", "$\\overline{Z}_y, Z_y, \\overline{g}_y, g_y, h_y$.", "Since $V$ is quasi-compact, we can find a finite number", "$y_1, \\ldots, y_m$ such that $V = V_{y_1} \\cup \\ldots \\cup V_{y_m}$.", "Then it follows that setting", "$$", "\\overline{Z} = \\coprod \\overline{Z}_{y_j},\\quad", "Z = \\coprod Z_{y_j},\\quad", "\\overline{g} = \\coprod \\overline{g}_{y_j},\\quad", "g = \\coprod g_{y_j},\\quad", "h = \\coprod h_{y_j}", "$$", "is a solution for the lemma. Given $y$ by condition (e)", "we can choose a quasi-compact open neighbourhood $y \\in V_y \\subset V$", "and isomorphisms $\\varphi_i : x_1|_{V_y} \\to x_i|_{V_y}$ for", "$i = 2, \\ldots, n$. Set $\\varphi_{ij} = \\varphi_j \\circ \\varphi_i^{-1}$.", "This leads us to the situation described in the next paragraph.", "\\medskip\\noindent", "Assume there exists $n \\geq 0$, quasi-compact opens", "$W_i \\subset Y$, $i = 1, \\ldots, n$, and", "morphisms $x_i : W_i \\to \\mathcal{X}$ such that", "(a) $f \\circ x_i = \\text{id}_{W_i}$,", "(b) $W = \\bigcup_{i = 1, \\ldots, n} W_i$ contains $V$,", "(c) $W$ is the image of $|f|$,", "(d) $V \\subset W_1 \\cap \\ldots \\cap W_n$, and", "(f) there are isomorphisms $\\varphi_{ij} : x_i|_V \\to x_j|_V$", "satisfying $\\varphi_{jk} \\circ \\varphi_{ij} = \\varphi_{ik}$.", "The morphisms", "$$", "I_{ij} = \\mathit{Isom}_\\mathcal{X}(x_i|_{W_i \\cap W_j},", "x_j|_{W_i \\cap W_j}) \\longrightarrow W_i \\cap W_j", "$$", "are proper because $f$ is separated", "(Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-separated-implies-isom}).", "Observe that $\\varphi_{ij}$ defines a section $V \\to I_{ij}$", "of $I_{ij} \\to W_i \\cap W_j$ over $V$.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-get-section-after-blowup}", "we can find $V$-admissible blowups", "$p_{ij} : Y_{ij} \\to Y$ such that $s_{ij}$", "extends to $p_{ij}^{-1}(W_i \\cap W_j)$.", "After replacing $Y$ by the fibre product of all the $Y_{ij}$", "over $Y$ we get to the situation described in the next paragraph.", "\\medskip\\noindent", "Assume there exists $n \\geq 0$, quasi-compact opens", "$W_i \\subset Y$, $i = 1, \\ldots, n$, and", "morphisms $x_i : W_i \\to \\mathcal{X}$ such that", "(a) $f \\circ x_i = \\text{id}_{W_i}$,", "(b) $W = \\bigcup_{i = 1, \\ldots, n} W_i$ contains $V$,", "(c) $W$ is the image of $|f|$,", "(d) $V \\subset W_1 \\cap \\ldots \\cap W_n$, and", "(g) there are isomorphisms", "$\\varphi_{ij} : x_i|_{W_i \\cap W_j} \\to x_j|_{W_i \\cap W_j}$", "satisfying", "$$", "\\varphi_{jk}|_V \\circ \\varphi_{ij}|_V = \\varphi_{ik}|_V.", "$$", "After replacing $Y$ by another $V$-admissible blowup if necessary", "we may assume that $V$ is dense and scheme theoretically dense", "in $Y$ and hence in any open subspace of $Y$ containing $V$.", "After such a replacement we conclude that", "$$", "\\varphi_{jk}|_{W_i \\cap W_j \\cap W_k} \\circ", "\\varphi_{ij}|_{W_i \\cap W_j \\cap W_k} =", "\\varphi_{ik}|_{W_i \\cap W_j \\cap W_k}", "$$", "by appealing to Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-equality-of-morphisms}", "and the fact that $I_{ik} \\to W_i \\cap W_j$ is proper", "(hence separated).", "Of course this means that $(x_i, \\varphi_{ij})$", "is a desent datum and we obtain a morphism", "$x : W \\to \\mathcal{X}$ agreeing with $x_i$ over $W_i$", "because $\\mathcal{X}$ is a stack.", "Since $x$ is a section of the separated morphism", "$\\mathcal{X} \\to W$ we see that $x$ is proper", "(Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-section-immersion}).", "Thus the lemma now holds with $\\overline{Z} = Y$,", "$Z = W$, $\\overline{g} = \\text{id}_Y$, $g = \\text{id}_W$,", "$h = x$." ], "refs": [ "spaces-limits-proposition-there-is-a-scheme-finite-over", "stacks-more-morphisms-lemma-flatten-stack", "stacks-morphisms-lemma-gerbe-fppf", "stacks-more-morphisms-lemma-finite-cover-factor", "stacks-morphisms-lemma-gerbe-isom-fppf", "stacks-more-morphisms-lemma-finite-cover-factor", "stacks-morphisms-lemma-separated-implies-isom", "spaces-morphisms-lemma-equality-of-morphisms" ], "ref_ids": [ 4659, 6907, 7522, 6908, 7523, 6908, 7421, 4791 ] } ], "ref_ids": [] }, { "id": 6910, "type": "theorem", "label": "stacks-more-morphisms-lemma-check-separated-dvr", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-check-separated-dvr", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Assume", "\\begin{enumerate}", "\\item $\\mathcal{Y}$ is locally Noetherian,", "\\item $f$ is locally of finite type and quasi-separated,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\", "\\Spec(A) \\ar[r]^y \\ar@{-->}[ru] & \\mathcal{Y}", "}", "$$", "where $A$ is a discrete valuation ring and $K$ its fraction field", "and any $2$-arrow $\\gamma : y \\circ j \\to f \\circ x$ the category", "of dotted arrows (Morphisms of Stacks, Definition", "\\ref{stacks-morphisms-definition-fill-in-diagram})", "is either empty or a setoid with exactly one isomorphism class.", "\\end{enumerate}", "Then $f$ is separated." ], "refs": [ "stacks-morphisms-definition-fill-in-diagram" ], "proofs": [ { "contents": [ "To prove that $f$ is separated we have to show that", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "is proper. We already know that $\\Delta$ is representable by", "algebraic spaces, locally of finite type (Morphisms of Stacks,", "Lemma \\ref{stacks-morphisms-lemma-properties-diagonal}) and", "quasi-compact and quasi-separated (by definition of $f$ being quasi-separated).", "Choose a scheme $U$ and a surjective smooth morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$.", "Set", "$$", "V = \\mathcal{X} \\times_{\\Delta, \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}} U", "$$", "It suffices to show that the morphism of algebraic spaces $V \\to U$", "is proper (Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}).", "Observe that $U$ is locally Noetherian", "(use Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-locally-finite-type-locally-noetherian}", "and the fact that $U \\to \\mathcal{Y}$ is locally of finite type)", "and $V \\to U$", "is of finite type and quasi-separated (as the base change", "of $\\Delta$ and properties of $\\Delta$ listed above). Applying", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-check-proper-dvr}", "it suffices to show: Given a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_v \\ar[d]_j &", "V \\ar[d]^g \\ar[r] &", "\\mathcal{X} \\ar[d]^\\Delta \\\\", "\\Spec(A) \\ar[r]^u \\ar@{-->}[ru] \\ar@{..>}[rru] &", "U \\ar[r] &", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}", "}", "$$", "where $A$ is a discrete valuation ring and $K$ its fraction field,", "there is a unique dashed arrow making the diagram commute.", "By Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-cat-dotted-arrows-base-change}", "the categories of dashed and dotted arrows are equivalent.", "Assumption (3) implies there is a unique dotted arrow up to", "isomorphism, see Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-helper-diagonal}. We conclude", "there is a unique dashed arrow as desired." ], "refs": [ "stacks-morphisms-lemma-properties-diagonal", "stacks-properties-lemma-check-property-covering", "stacks-morphisms-lemma-locally-finite-type-locally-noetherian", "spaces-cohomology-lemma-check-proper-dvr", "stacks-morphisms-lemma-cat-dotted-arrows-base-change", "stacks-morphisms-lemma-helper-diagonal" ], "ref_ids": [ 7392, 8859, 7462, 11329, 7573, 7583 ] } ], "ref_ids": [ 7628 ] }, { "id": 6911, "type": "theorem", "label": "stacks-more-morphisms-lemma-refined-valuative-criterion-proper", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-refined-valuative-criterion-proper", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $h : \\mathcal{U} \\to \\mathcal{X}$", "be morphisms of algebraic stacks. Assume that $\\mathcal{Y}$ is", "locally Noetherian, that $f$ and $h$ are of finite type,", "that $f$ is separated, and that the image of", "$|h| : |\\mathcal{U}| \\to |\\mathcal{X}|$ is dense in $|\\mathcal{X}|$.", "If given any $2$-commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_-u \\ar[d]_j & \\mathcal{U} \\ar[r]_h & \\mathcal{X} \\ar[d]^f \\\\", "\\Spec(A) \\ar[rr]^-y & & \\mathcal{Y}", "}", "$$", "where $A$ is a discrete valuation ring with field of fractions $K$", "and $\\gamma : y \\circ j \\to f \\circ h \\circ u$ there", "exist an extension $K'/K$ of fields, a valuation ring $A' \\subset K'$", "dominating $A$ such that the category of dotted arrows for the", "induced diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r]_-{x'} \\ar[d]_{j'} & \\mathcal{X} \\ar[d]^f \\\\", "\\Spec(A') \\ar[r]^-{y'} \\ar@{..>}[ru] & \\mathcal{Y}", "}", "$$", "with induced $2$-arrow $\\gamma' : y' \\circ j' \\to f \\circ x'$", "is nonempty (Morphisms of Stacks, Definition", "\\ref{stacks-morphisms-definition-fill-in-diagram}), then $f$ is proper." ], "refs": [ "stacks-morphisms-definition-fill-in-diagram" ], "proofs": [ { "contents": [ "It suffices to prove that $f$ is universally closed.", "Let $V \\to \\mathcal{Y}$ be a smooth morphism where $V$ is an affine scheme.", "By Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-points-cartesian}", "the image $I$ of", "$|\\mathcal{U} \\times_\\mathcal{Y} V| \\to |\\mathcal{X} \\times_\\mathcal{Y} V|$", "is the inverse image of the image of $|h|$. Since", "$|\\mathcal{X} \\times_\\mathcal{Y} V| \\to |\\mathcal{X}|$ is open", "(Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-fppf-open})", "we conclude that $I$ is dense in $|\\mathcal{X} \\times_\\mathcal{Y} V|$.", "Also since the category of dotted arrows behaves well with respect", "to base change (Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-cat-dotted-arrows-base-change})", "the assumption on existence of dotted arrows (after extension)", "is inherited by the morphisms", "$\\mathcal{U} \\times_\\mathcal{Y} V \\to \\mathcal{X} \\times_\\mathcal{Y} V \\to V$.", "Therefore the assumptions of the lemma are", "satisfied for the morphisms", "$\\mathcal{U} \\times_\\mathcal{Y} V \\to \\mathcal{X} \\times_\\mathcal{Y} V \\to V$.", "Hence we may assume $\\mathcal{Y}$ is an affine scheme.", "\\medskip\\noindent", "Assume $\\mathcal{Y} = Y$ is an affine scheme.", "(From now on we no longer have to worry about the", "$2$-arrows $\\gamma$ and $\\gamma'$, see Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-cat-dotted-arrows-independent}.)", "Then $\\mathcal{U}$ is quasi-compact. Choose an affine scheme $U$ and a", "surjective smooth morphism $U \\to \\mathcal{U}$.", "Then we may and do replace $\\mathcal{U}$ by $U$.", "Thus we may assume that $\\mathcal{U}$ is an affine scheme.", "\\medskip\\noindent", "Assume $\\mathcal{Y} = Y$ and $\\mathcal{U} = U$ are affine schemes.", "By Chow's lemma (Theorem \\ref{theorem-chow-finite-type})", "we can choose a surjective proper morphism $X \\to \\mathcal{X}$", "where $X$ is an algebraic space. We will use below that $X \\to Y$ is separated", "as a composition of separated morphisms. Consider the", "algebraic space $W = X \\times_\\mathcal{X} U$. The projection morphism", "$W \\to X$ is of finite type.", "We may replace $X$ by the scheme theoretic image of", "$W \\to X$ and hence we may assume that the image of $|W|$ in $|X|$", "is dense in $|X|$ (here we use that the image of $|h|$", "is dense in $|\\mathcal{X}|$, so after this replacement, the", "morphism $X \\to \\mathcal{X}$ is still surjective).", "We claim that for every solid commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & W \\ar[r] & X \\ar[d] \\\\", "\\Spec(A) \\ar[rr] \\ar@{..>}[rru] & & Y", "}", "$$", "where $A$ is a discrete valuation ring with field of fractions $K$, there", "exists a dotted arrow making the diagram commute. First, it is enough to", "prove there exists a dotted arrow after replacing $K$ by an extension", "and $A$ by a valuation ring in this extension dominating $A$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-push-down-solution}.", "By the assumption of the lemma we get an extension $K'/K$ and a valuation", "ring $A' \\subset K'$ dominating $A$ and", "an arrow $\\Spec(A') \\to \\mathcal{X}$ lifting the composition", "$\\Spec(A') \\to \\Spec(A) \\to Y$ and compatible with the composition", "$\\Spec(K') \\to \\Spec(K) \\to W \\to X$. Because $X \\to \\mathcal{X}$", "is proper, we can use the valuative criterion of properness", "(Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-criterion-proper})", "to find an extension $K''/K'$ and a valuation ring $A'' \\subset K''$", "dominating $A'$ and a morphism $\\Spec(A'') \\to X$ lifting the composition", "$\\Spec(A'') \\to \\Spec(A') \\to \\mathcal{X}$ and compatible with the composition", "$\\Spec(K'') \\to \\Spec(K') \\to \\Spec(K) \\to X$.", "Then $K''/K$ and $A'' \\subset K''$ and the morphism $\\Spec(A'') \\to X$", "is a solution to the problem posed above and the claim holds.", "\\medskip\\noindent", "The claim implies the morphism $X \\to Y$ is proper by the", "case of the lemma for algebraic spaces", "(Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-refined-valuative-criterion-proper}).", "By Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-image-proper-is-proper}", "we conclude that $\\mathcal{X} \\to Y$ is proper as desired." ], "refs": [ "stacks-properties-lemma-points-cartesian", "stacks-morphisms-lemma-fppf-open", "stacks-morphisms-lemma-cat-dotted-arrows-base-change", "stacks-morphisms-lemma-cat-dotted-arrows-independent", "stacks-more-morphisms-theorem-chow-finite-type", "spaces-morphisms-lemma-push-down-solution", "stacks-morphisms-lemma-criterion-proper", "spaces-limits-lemma-refined-valuative-criterion-proper", "stacks-morphisms-lemma-image-proper-is-proper" ], "ref_ids": [ 8864, 7513, 7573, 7572, 6882, 4926, 7588, 4645, 7564 ] } ], "ref_ids": [ 7628 ] }, { "id": 6912, "type": "theorem", "label": "stacks-more-morphisms-lemma-refined-valuative-criterion-separated", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-refined-valuative-criterion-separated", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $h : \\mathcal{U} \\to \\mathcal{X}$", "be morphisms of algebraic stacks. Assume that $\\mathcal{Y}$ is", "locally Noetherian, that $f$ is locally of finite type and quasi-separated,", "that $h$ is of finite type, and that the image of", "$|h| : |\\mathcal{U}| \\to |\\mathcal{X}|$ is dense in $|\\mathcal{X}|$.", "If given any $2$-commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_-u \\ar[d]_j & \\mathcal{U} \\ar[r]_h & \\mathcal{X} \\ar[d]^f \\\\", "\\Spec(A) \\ar[rr]^-y \\ar@{..>}[rru] & & \\mathcal{Y}", "}", "$$", "where $A$ is a discrete valuation ring with field of fractions $K$", "and $\\gamma : y \\circ j \\to f \\circ h \\circ u$, the category", "of dotted arrows is either empty or a setoid with exactly", "one isomorphism class, then $f$ is separated." ], "refs": [], "proofs": [ { "contents": [ "We have to prove $\\Delta$ is a proper morphism.", "Assume first that $\\Delta$ is separated. Then we may apply", "Lemma \\ref{lemma-refined-valuative-criterion-proper}", "to the morphisms $\\mathcal{U} \\to \\mathcal{X}$ and", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$.", "Observe that $\\Delta$ is quasi-compact as $f$ is quasi-separated.", "Of course $\\Delta$ is locally of finite type (true for any", "diagonal morphism, see Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-properties-diagonal}).", "Finally, suppose given a $2$-commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_-u \\ar[d]_j &", "\\mathcal{U} \\ar[r]_h &", "\\mathcal{X} \\ar[d]^\\Delta \\\\", "\\Spec(A) \\ar[rr]^-y \\ar@{..>}[rru] & &", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}", "}", "$$", "where $A$ is a discrete valuation ring with field of fractions $K$", "and $\\gamma : y \\circ j \\to \\Delta \\circ h \\circ u$.", "By Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-helper-diagonal}", "and the assumption in the lemma", "we find there exists a unique dotted arrow.", "This proves the last assumption of", "Lemma \\ref{lemma-refined-valuative-criterion-proper}", "holds and the result follows.", "\\medskip\\noindent", "In the general case, it suffices to prove $\\Delta$ is separated", "since then we'll be back in the previous case. In fact, we claim", "that the assumptions of the lemma hold for", "$$", "\\mathcal{U} \\to \\mathcal{X}", "\\quad\\text{and}\\quad", "\\Delta :", "\\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}", "$$", "Namely, since $\\Delta$ is representable by algebraic spaces, the", "category of dotted arrows for a diagram as in the previous paragraph", "is a setoid (see for example", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-cat-dotted-arrows}).", "The argument in the preceding paragraph shows these categories", "are either empty or have one isomorphism class.", "Thus $\\Delta$ is separated." ], "refs": [ "stacks-more-morphisms-lemma-refined-valuative-criterion-proper", "stacks-morphisms-lemma-properties-diagonal", "stacks-morphisms-lemma-helper-diagonal", "stacks-more-morphisms-lemma-refined-valuative-criterion-proper", "stacks-morphisms-lemma-cat-dotted-arrows" ], "ref_ids": [ 6911, 7392, 7583, 6911, 7571 ] } ], "ref_ids": [] }, { "id": 6913, "type": "theorem", "label": "stacks-more-morphisms-lemma-quotient-compare", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-quotient-compare", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces with", "$s, t : R \\to U$ flat and locally of finite presentation.", "Consider the algebraic stack $\\mathcal{X} = [U/R]$.", "Given an algebraic space $Y$ there is a $1$-to-$1$ correspondence between", "morphisms $f : \\mathcal{X} \\to Y$ and $R$-invariant morphisms", "$\\phi : U \\to Y$." ], "refs": [], "proofs": [ { "contents": [ "Criteria for Representability, Theorem", "\\ref{criteria-theorem-flat-groupoid-gives-algebraic-stack}", "tells us $\\mathcal{X}$ is an algebraic stack.", "Given a morphism $f : \\mathcal{X} \\to Y$ we let $\\phi : U \\to Y$ be", "the composition $U \\to \\mathcal{X} \\to Y$. Since $R = U \\times_\\mathcal{X} U$", "(Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-2-cartesian})", "it is immediate that $\\phi$ is $R$-invariant.", "Conversely, if $\\phi : U \\to Y$ is an $R$-invariant morphism towards", "an algebraic space, we obtain a morphism", "$f : \\mathcal{X} \\to Y$ by", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-2-coequalizer}.", "You can also construct $f$ from $\\phi$ using the explicit description of", "the quotient stack in", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-objects}." ], "refs": [ "criteria-theorem-flat-groupoid-gives-algebraic-stack", "spaces-groupoids-lemma-quotient-stack-2-cartesian", "spaces-groupoids-lemma-quotient-stack-2-coequalizer", "spaces-groupoids-lemma-quotient-stack-objects" ], "ref_ids": [ 3094, 9324, 9327, 9328 ] } ], "ref_ids": [] }, { "id": 6914, "type": "theorem", "label": "stacks-more-morphisms-lemma-categorical-quotient-compare", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-categorical-quotient-compare", "contents": [ "With assumption and notation as in Lemma \\ref{lemma-quotient-compare}.", "Then $f$ is a (uniform) categorical moduli space", "if and only if $\\phi$ is a (uniform) categorical quotient.", "Similarly for moduli spaces in a full subcategory." ], "refs": [ "stacks-more-morphisms-lemma-quotient-compare" ], "proofs": [ { "contents": [ "It is immediate from the $1$-to-$1$ correspondence established in", "Lemma \\ref{lemma-quotient-compare} that $f$ is a categorical moduli space", "if and only if $\\phi$ is a categorical quotient", "(Quotients of Groupoids, Definition", "\\ref{groupoids-quotients-definition-categorical}).", "If $Y' \\to Y$ is a morphism, then", "$U' = Y' \\times_Y U \\to Y' \\times_Y \\mathcal{X} = \\mathcal{X}'$", "is a surjective, flat, locally finitely presented morphism", "as a base change of $U \\to \\mathcal{X}$", "(Criteria for Representability, Lemma", "\\ref{criteria-lemma-flat-quotient-flat-presentation}).", "And $R' = Y' \\times_Y R$ is equal to $U' \\times_{\\mathcal{X}'} U'$", "by transitivity of fibre products.", "Hence $\\mathcal{X}' = [U'/R']$, see", "Algebraic Stacks, Remark \\ref{algebraic-remark-flat-fp-presentation}.", "Thus the base change of our situation to $Y'$ is another situation", "as in the statement of the lemma. From this it immediately", "follows that $f$ is a uniform categorical moduli space", "if and only if $\\phi$ is a uniform categorical quotient." ], "refs": [ "stacks-more-morphisms-lemma-quotient-compare", "groupoids-quotients-definition-categorical", "criteria-lemma-flat-quotient-flat-presentation", "algebraic-remark-flat-fp-presentation" ], "ref_ids": [ 6913, 13460, 3137, 8491 ] } ], "ref_ids": [ 6913 ] }, { "id": 6915, "type": "theorem", "label": "stacks-more-morphisms-lemma-check-uniform-categorical-quotient-on-affines", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-check-uniform-categorical-quotient-on-affines", "contents": [ "Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack", "to an algebraic space. If for every affine scheme $Y'$ and flat", "morphism $Y' \\to Y$ the base change", "$f' : Y' \\times_Y \\mathcal{X} \\to Y'$ is a categorical moduli space,", "then $f$ is a uniform categorical moduli space." ], "refs": [], "proofs": [ { "contents": [ "Choose an \\'etale covering $\\{Y_i \\to Y\\}$ where $Y_i$ is an affine scheme.", "For each $i$ and $j$ choose a affine open covering", "$Y_i \\times_Y Y_j = \\bigcup Y_{ijk}$.", "Set $\\mathcal{X}_i = Y_i \\times_Y \\mathcal{X}$ and", "$\\mathcal{X}_{ijk} = Y_{ijk} \\times_Y \\mathcal{X}$.", "Let $g : \\mathcal{X} \\to W$ be a morphism towards", "an algebraic space. Then we consider the diagram", "$$", "\\xymatrix{", "\\mathcal{X}_i \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\ar[r]_g & W \\\\", "Y_i \\ar[r] \\ar@{..>}[rru] & Y", "}", "$$", "The assumption that $\\mathcal{X}_i \\to Y_i$ is a categorical moduli", "space, produces a unique dotted arrow $h_i : Y_i \\to W$.", "The assumption that $\\mathcal{X}_{ijk} \\to Y_{ijk}$ is a categorical", "moduli space, implies the restriction of $h_i$ and $h_j$ to", "$Y_{ijk}$ are equal. Hence $h_i$ and $h_j$ agree on $Y_i \\times_Y Y_j$.", "Since $Y = \\coprod Y_i / \\coprod Y_i \\times_Y Y_j$", "(by Spaces, Section \\ref{spaces-section-presentations}) we conclude", "that there is a unique morphism $Y \\to W$ through which $g$ factors.", "Thus $f$ is a categorical moduli space. The same argument applies", "after a flat base change, hence $f$ is a uniform categorical moduli space." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6916, "type": "theorem", "label": "stacks-more-morphisms-lemma-well-nigh-affine", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-well-nigh-affine", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is well-nigh affine, and", "\\item there exists a groupoid scheme $(U, R, s, t, c)$ with $U$ and", "$R$ affine and $s, t : R \\to U$ finite locally free such that", "$\\mathcal{X} = [U/R]$.", "\\end{enumerate}", "If true then $\\mathcal{X}$ is quasi-compact, quasi-DM, and separated." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{X}$ is well-nigh affine. Choose an affine scheme $U$", "and a surjective, flat, finite, and finitely presented morphism", "$U \\to \\mathcal{X}$. Set $R = U \\times_\\mathcal{X} U$. Then we", "obtain a groupoid $(U, R, s, t, c)$ in algebraic spaces and an", "isomorphism $[U/R] \\to \\mathcal{X}$, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack}", "and Remark \\ref{algebraic-remark-flat-fp-presentation}.", "Since $s, t : R \\to U$ are ", "flat, finite, and finitely presented morphisms", "(as base changes of $U \\to \\mathcal{X})$ we see that", "$s, t$ are finite locally free", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}).", "This implies that $R$ is affine (as finite morphisms are affine)", "and hence (2) holds.", "\\medskip\\noindent", "Suppose that we have a groupoid scheme $(U, R, s, t, c)$ with $U$ and", "$R$ are affine and $s, t : R \\to U$ finite locally free.", "Set $\\mathcal{X} = [U/R]$. Then $\\mathcal{X}$ is an algebraic stack", "by Criteria for Representability, Theorem", "\\ref{criteria-theorem-flat-groupoid-gives-algebraic-stack} (strictly speaking", "we don't need this here, but it can't be stressed enough that this is true).", "The morphism $U \\to \\mathcal{X}$ is surjective, flat, and locally", "of finite presentation by", "Criteria for Representability, Lemma", "\\ref{criteria-lemma-flat-quotient-flat-presentation}.", "Thus we can check whether $U \\to \\mathcal{X}$ is finite by", "checking whether the projection $U \\times_\\mathcal{X} U \\to U$", "has this property, see Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}.", "Since $U \\times_\\mathcal{X} U = R$ by", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-2-cartesian}", "we see that this is true. Thus $\\mathcal{X}$ is well-nigh affine.", "\\medskip\\noindent", "Proof of the final statement. We see that $\\mathcal{X}$", "is quasi-compact by Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-quasi-compact-stack}.", "We see that $\\mathcal{X} = [U/R]$ is quasi-DM and separated by", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-properties-diagonal-from-presentation}." ], "refs": [ "algebraic-lemma-map-space-into-stack", "algebraic-remark-flat-fp-presentation", "morphisms-lemma-finite-flat", "criteria-theorem-flat-groupoid-gives-algebraic-stack", "criteria-lemma-flat-quotient-flat-presentation", "stacks-properties-lemma-check-property-covering", "spaces-groupoids-lemma-quotient-stack-2-cartesian", "stacks-properties-lemma-quasi-compact-stack", "stacks-morphisms-lemma-properties-diagonal-from-presentation" ], "ref_ids": [ 8473, 8491, 5471, 3094, 3137, 8859, 9324, 8873, 7475 ] } ], "ref_ids": [] }, { "id": 6917, "type": "theorem", "label": "stacks-more-morphisms-lemma-affine-over-well-nigh-affine", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-affine-over-well-nigh-affine", "contents": [ "Let the algebraic stack $\\mathcal{X}$ be well-nigh affine.", "\\begin{enumerate}", "\\item If $\\mathcal{X}$ is an algebraic space, then it is affine.", "\\item If $\\mathcal{X}' \\to \\mathcal{X}$ is an affine morphism", "of algebraic stacks, then $\\mathcal{X}'$ is well-nigh affine.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from immediately from", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-affine}.", "However, this is overkill, since (1) also follows from", "Lemma \\ref{lemma-well-nigh-affine} combined with", "Groupoids, Proposition", "\\ref{groupoids-proposition-finite-flat-equivalence}.", "\\medskip\\noindent", "To prove (2) we choose an affine scheme $U$ and a", "surjective, flat, finite, and finitely presented morphism $U \\to \\mathcal{X}$.", "Then $U' = \\mathcal{X}' \\times_\\mathcal{X} U$ admits an affine", "morphism to $U$ (Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-base-change-affine}).", "Therefore $U'$ is an affine scheme. Of course", "$U' \\to \\mathcal{X}'$ is surjective, flat, finite, and finitely presented", "as a base change of $U \\to \\mathcal{X}$." ], "refs": [ "spaces-limits-lemma-affine", "stacks-more-morphisms-lemma-well-nigh-affine", "groupoids-proposition-finite-flat-equivalence", "stacks-morphisms-lemma-base-change-affine" ], "ref_ids": [ 4626, 6916, 9669, 7433 ] } ], "ref_ids": [] }, { "id": 6918, "type": "theorem", "label": "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space", "contents": [ "Let the algebraic stack $\\mathcal{X}$ be well-nigh affine. There exists", "a uniform categorical moduli space", "$$", "f : \\mathcal{X} \\longrightarrow M", "$$", "in the category of affine schemes. Moreover", "$f$ is separated, quasi-compact, and a universal homeomorphism." ], "refs": [], "proofs": [ { "contents": [ "Write $\\mathcal{X} = [U/R]$ with $(U, R, s, t, c)$ as in", "Lemma \\ref{lemma-well-nigh-affine}. Let $C$ be the ring of", "$R$-invariant functions on $U$, see", "Groupoids, Section \\ref{groupoids-section-finite-flat}.", "We set $M = \\Spec(C)$. The $R$-invariant morphism", "$U \\to M$ corresponds to a morphism $f : \\mathcal{X} \\to M$ by", "Lemma \\ref{lemma-quotient-compare}.", "The characterization of morphisms into affine schemes given in", "Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}", "immediately guarantees that $\\phi : U \\to M$ is a categorical", "quotient in the category of affine schemes. Hence $f$ is a", "categorical moduli space in the category of affine schemes", "(Lemma \\ref{lemma-categorical-quotient-compare}).", "\\medskip\\noindent", "Since $\\mathcal{X}$ is separated by Lemma \\ref{lemma-well-nigh-affine}", "we find that $f$ is separated by Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-compose-after-separated}.", "\\medskip\\noindent", "Since $U \\to \\mathcal{X}$ is surjective and since $U \\to M$ is quasi-compact,", "we see that $f$ is quasi-compact by Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-surjection-from-quasi-compact}.", "\\medskip\\noindent", "By Groupoids, Lemma \\ref{groupoids-lemma-integral-over-invariants}", "the composition", "$$", "U \\to \\mathcal{X} \\to M", "$$", "is an integral morphism of affine schemes. In particular, it is", "universally closed", "(Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed}).", "Since $U \\to \\mathcal{X}$ is surjective, it follows that $\\mathcal{X} \\to M$", "is universally closed (Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-image-proper-is-proper}).", "To conclude that $\\mathcal{X} \\to M$ is a universal homeomorphism,", "it is enough to show that it is universally bijective, i.e.,", "surjective and universally injective.", "\\medskip\\noindent", "We have $|\\mathcal{X}| = |U|/|R|$ by", "Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-points-presentation}.", "Thus $|f|$ is surjective and even bijective", "by Groupoids, Lemma \\ref{groupoids-lemma-points}.", "\\medskip\\noindent", "Let $C \\to C'$ be a ring map. Let $(U', R', s', t', c')$ be", "the base change of $(U, R, s, t, c)$ by $M' = \\Spec(C') \\to M$.", "Setting $\\mathcal{X}' = [U'/R']$, we observe that", "$M' \\times_M \\mathcal{X} = \\mathcal{X}'$ by", "Quotients of Groupoids, Lemma", "\\ref{groupoids-quotients-lemma-base-change-quotient-stack}.", "Let $C^1$ be the ring of $R'$-invariant functions on $U'$.", "Set $M^1 = \\Spec(C^1)$ and consider the diagram", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[d]^{f'} \\ar[r] & \\mathcal{X} \\ar[dd]^f \\\\", "M^1 \\ar[d] \\\\", "M' \\ar[r] & M", "}", "$$", "By Groupoids, Lemma \\ref{groupoids-lemma-invariants-base-change} and", "Algebra, Lemma \\ref{algebra-lemma-universally-bijective}", "the morphism $M^1 \\to M'$ is a homeomorphism.", "On the other hand, the previous paragraph applied to", "$(U', R', s', t', c')$ shows that $|f'|$ is bijective.", "We conclude that $f$ induces a bijection on points after any", "base change by an affine scheme. Thus $f$ is universally injective", "by Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-universally-injective-local}.", "\\medskip\\noindent", "Finally, we still have to show that $f$ is a uniform moduli space", "in the category of affine schemes. This follows from the discussion", "above and the fact that if the", "ring map $C \\to C'$ is flat, then $C' \\to C^1$ is an isomorphism", "by Groupoids, Lemma \\ref{groupoids-lemma-invariants-base-change}." ], "refs": [ "stacks-more-morphisms-lemma-well-nigh-affine", "stacks-more-morphisms-lemma-quotient-compare", "schemes-lemma-morphism-into-affine", "stacks-more-morphisms-lemma-categorical-quotient-compare", "stacks-more-morphisms-lemma-well-nigh-affine", "stacks-morphisms-lemma-compose-after-separated", "stacks-morphisms-lemma-surjection-from-quasi-compact", "groupoids-lemma-integral-over-invariants", "morphisms-lemma-integral-universally-closed", "stacks-morphisms-lemma-image-proper-is-proper", "stacks-morphisms-lemma-points-presentation", "groupoids-lemma-points", "groupoids-quotients-lemma-base-change-quotient-stack", "groupoids-lemma-invariants-base-change", "algebra-lemma-universally-bijective", "stacks-morphisms-lemma-universally-injective-local", "groupoids-lemma-invariants-base-change" ], "ref_ids": [ 6916, 6913, 7655, 6914, 6916, 7406, 7426, 9659, 5441, 7564, 7476, 9661, 13441, 9660, 586, 7452, 9660 ] } ], "ref_ids": [] }, { "id": 6919, "type": "theorem", "label": "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space-etale", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space-etale", "contents": [ "Let $h : \\mathcal{X}' \\to \\mathcal{X}$ be a morphism of algebraic stacks.", "Assume $\\mathcal{X}'$ and $\\mathcal{X}$ are well-nigh affine,", "$h$ is \\'etale, and $h$ induces isomorphisms on automorphism groups", "(Morphisms of Stacks, Remark", "\\ref{stacks-morphisms-remark-identify-automorphism-groups}).", "Then there exists a cartesian diagram", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[d] \\ar[r] & \\mathcal{X} \\ar[d] \\\\", "M' \\ar[r] & M", "}", "$$", "where $M' \\to M$ is \\'etale and", "the vertical arrows are the moduli spaces constructed in", "Lemma \\ref{lemma-well-nigh-affine-moduli-space}." ], "refs": [ "stacks-morphisms-remark-identify-automorphism-groups", "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space" ], "proofs": [ { "contents": [ "Observe that $h$ is representable by algebraic spaces by", "Morphisms of Stacks, Lemmas", "\\ref{stacks-morphisms-lemma-aut-iso-unramified} and", "\\ref{stacks-morphisms-lemma-stabilizer-preserving}.", "Choose an affine scheme $U$ and a", "surjective, flat, finite, and finitely presented morphism $U \\to \\mathcal{X}$.", "Then $U' = \\mathcal{X}' \\times_\\mathcal{X} U$ is an algebraic", "space with a finite (in particular affine) morphism $U' \\to \\mathcal{X}'$.", "By Lemma \\ref{lemma-affine-over-well-nigh-affine}", "we conclude that $U'$ is affine.", "Setting $R = U \\times_\\mathcal{X} U$ and $R' = U' \\times_{\\mathcal{X}'} U'$", "we obtain groupoids $(U, R, s, t, c)$ and $(U', R', s', t', c')$", "such that $\\mathcal{X} = [U/R]$ and $\\mathcal{X}' = [U'/R']$,", "see proof of Lemma \\ref{lemma-well-nigh-affine}.", "we see that the diagrams", "$$", "\\xymatrix{", "R' \\ar[d]_{s'} \\ar[r]_f & R \\ar[d]^s \\\\", "U' \\ar[r]^f & U", "}", "\\quad", "\\quad", "\\xymatrix{", "R' \\ar[d]_{t'} \\ar[r]_f & R \\ar[d]^t \\\\", "U' \\ar[r]^f & U", "}", "\\quad", "\\quad", "\\xymatrix{", "G' \\ar[d] \\ar[r]_f & G \\ar[d] \\\\", "U' \\ar[r]^f & U", "}", "$$", "are cartesian where $G$ and $G'$ are the stabilizer group schemes.", "This follows for the first two by transitivity of fibre products", "and for the last one this follows because it is the pullback of the", "isomorphism $\\mathcal{I}_{\\mathcal{X}'} \\to", "\\mathcal{X}' \\times_\\mathcal{X} \\mathcal{I}_\\mathcal{X}$", "(by the already used Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-aut-iso-unramified}).", "Recall that $M$, resp.\\ $M'$ was constructed in", "Lemma \\ref{lemma-well-nigh-affine-moduli-space}", "as the spectrum of the ring of $R$-invariant functions on $U$,", "resp.\\ the ring of $R'$-invariant functions on $U'$.", "Thus $M' \\to M$ is \\'etale and $U' = M' \\times_M U$", "by Groupoids, Lemma \\ref{groupoids-lemma-etale}.", "It follows that $R' = M' \\times_M U$, in other words", "the groupoid $(U', R', s', t', c')$ is the base change of", "$(U, R, s, t, c)$ by $M' \\to M$.", "This implies that the diagram in the lemma is", "cartesian by", "Quotients of Groupoids, Lemma", "\\ref{groupoids-quotients-lemma-base-change-quotient-stack}." ], "refs": [ "stacks-morphisms-lemma-aut-iso-unramified", "stacks-morphisms-lemma-stabilizer-preserving", "stacks-more-morphisms-lemma-affine-over-well-nigh-affine", "stacks-more-morphisms-lemma-well-nigh-affine", "stacks-morphisms-lemma-aut-iso-unramified", "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space", "groupoids-lemma-etale", "groupoids-quotients-lemma-base-change-quotient-stack" ], "ref_ids": [ 7593, 7592, 6917, 6916, 7593, 6918, 9662, 13441 ] } ], "ref_ids": [ 7637, 6918 ] }, { "id": 6920, "type": "theorem", "label": "stacks-more-morphisms-lemma-moduli-space-finite-affine", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-moduli-space-finite-affine", "contents": [ "Let the algebraic stack $\\mathcal{X}$ be well-nigh affine. The morphism", "$$", "f : \\mathcal{X} \\longrightarrow M", "$$", "of Lemma \\ref{lemma-well-nigh-affine-moduli-space}", "is a uniform categorical moduli space." ], "refs": [ "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space" ], "proofs": [ { "contents": [ "We already know that $M$ is a uniform categorical moduli space", "in the category of affine schemes. By", "Lemma \\ref{lemma-check-uniform-categorical-quotient-on-affines}", "it suffices to show that the base change", "$f' : M' \\times_M \\mathcal{X} \\to M'$", "is a categorical moduli space for any flat morphism", "$M' \\to M$ of affine schemes.", "Observe that $\\mathcal{X}' = M' \\times_M \\mathcal{X}$ is well-nigh affine by", "Lemma \\ref{lemma-affine-over-well-nigh-affine}.", "This after replacing $\\mathcal{X}$ by $\\mathcal{X}'$", "and $M$ by $M'$, we reduce to proving $f$ is a categorical", "moduli space.", "\\medskip\\noindent", "Let $g : \\mathcal{X} \\to Y$ be a morphism where $Y$ is an algebraic space.", "We have to show that $g = h \\circ f$ for a unique morphism $h : M \\to Y$.", "\\medskip\\noindent", "Uniqueness. Suppose we have two morphisms $h_i : M \\to Y$ with", "$g = h_1 \\circ f = h_2 \\circ f$. Let $M' \\subset M$ be the equalizer", "of $h_1$ and $h_2$. Then $M' \\to M$ is a monomorphism and", "$f : \\mathcal{X} \\to M$ factors through $M'$. Thus $M' \\to M$", "is a universal homeomorphism. We conclude $M'$ is affine", "(Morphisms, Lemma \\ref{morphisms-lemma-universal-homeomorphism}).", "But then since $f : \\mathcal{X} \\to M$", "is a categorical moduli space in the category of affine schemes,", "we see $M' = M$.", "\\medskip\\noindent", "Existence. Below we will show that for every $p \\in M$ there exists", "a cartesian square", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\", "M' \\ar[r] & M", "}", "$$", "with $M' \\to M$ an \\'etale morphism of affines and $p$ in the image such that", "the composition $\\mathcal{X}' \\to \\mathcal{X} \\to Y$ factors through $M'$.", "This means we can construct the map $h : M \\to Y$ \\'etale locally on $M$.", "Since $Y$ is a sheaf for the \\'etale topology and by the uniqueness shown", "above, this is enough (small detail omitted).", "\\medskip\\noindent", "Let $y \\in |Y|$ be the image of $p$.", "Let $(V, v) \\to (Y, y)$ be an \\'etale morphism with $V$ affine.", "Consider $\\mathcal{X}' = V \\times_Y \\mathcal{X}$.", "Observe that $\\mathcal{X}' \\to \\mathcal{X}$ is separated and \\'etale", "as the base change of $V \\to Y$. Moreover, $\\mathcal{X}' \\to \\mathcal{X}$", "induces isomorphisms on automorphism groups", "(Morphisms of Stacks, Remark", "\\ref{stacks-morphisms-remark-identify-automorphism-groups})", "as this is true for", "$V \\to Y$, see Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-base-change-stabilizer-preserving}.", "Choose a presentation $\\mathcal{X} = [U/R]$", "as in Lemma \\ref{lemma-well-nigh-affine}.", "Set $U' = \\mathcal{X}' \\times_\\mathcal{X} U = V \\times_Y U$", "and choose $u' \\in U'$ mapping to $p$ and $v$ (possible", "by Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}).", "Since $U' \\to U$ is separated and \\'etale we see that", "every finite set of points of $U'$ is contained in an affine open, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-over-affine}.", "On the other hand, the morphism $U' \\to \\mathcal{X}'$ is", "surjective, finite, flat, and locally of finite presentation.", "Setting $R' = U' \\times_{\\mathcal{X}'} U'$ we see", "that $s', t' : R' \\to U'$ are finite locally free.", "By Groupoids, Lemma \\ref{groupoids-lemma-find-invariant-affine}", "there exists an $R'$-invariant affine open subscheme $U'' \\subset U'$", "containing $u'$.", "Let $\\mathcal{X}'' \\subset \\mathcal{X}'$ be", "the corresponding open substack. Then $\\mathcal{X}''$ is", "well-nigh affine. By Lemma \\ref{lemma-well-nigh-affine-moduli-space-etale}", "we obtain a cartesian square", "$$", "\\xymatrix{", "\\mathcal{X}'' \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\", "M'' \\ar[r] & M", "}", "$$", "with $M'' \\to M$ \\'etale. Since $\\mathcal{X}'' \\to M''$ is", "a categorical moduli space in the category of affine schemes", "we obtain a morphism $M'' \\to V$ such that the composition", "$\\mathcal{X}'' \\to \\mathcal{X}' \\to V$ is equal to the composition", "$\\mathcal{X}'' \\to M'' \\to V$. This proves our claim and finishes", "the proof." ], "refs": [ "stacks-more-morphisms-lemma-check-uniform-categorical-quotient-on-affines", "stacks-more-morphisms-lemma-affine-over-well-nigh-affine", "morphisms-lemma-universal-homeomorphism", "stacks-morphisms-remark-identify-automorphism-groups", "stacks-morphisms-lemma-base-change-stabilizer-preserving", "stacks-more-morphisms-lemma-well-nigh-affine", "spaces-properties-lemma-points-cartesian", "more-morphisms-lemma-separated-locally-quasi-finite-over-affine", "groupoids-lemma-find-invariant-affine", "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space-etale" ], "ref_ids": [ 6915, 6917, 5454, 7637, 7595, 6916, 11819, 13906, 9664, 6919 ] } ], "ref_ids": [ 6918 ] }, { "id": 6921, "type": "theorem", "label": "stacks-more-morphisms-lemma-etale-separated-over-well-nigh-affine", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-etale-separated-over-well-nigh-affine", "contents": [ "Let $h : \\mathcal{X}' \\to \\mathcal{X}$ be a morphism of algebraic stacks.", "Assume $\\mathcal{X}$ is well-nigh affine, $h$ is \\'etale, $h$ is separated,", "and $h$ induces isomorphisms on automorphism groups", "(Morphisms of Stacks, Remark", "\\ref{stacks-morphisms-remark-identify-automorphism-groups}).", "Then there exists a cartesian diagram", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[d] \\ar[r] & \\mathcal{X} \\ar[d] \\\\", "M' \\ar[r] & M", "}", "$$", "where $M' \\to M$ is a separated \\'etale morphism of schemes and", "$\\mathcal{X} \\to M$ is the moduli space constructed in", "Lemma \\ref{lemma-well-nigh-affine-moduli-space}." ], "refs": [ "stacks-morphisms-remark-identify-automorphism-groups", "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space" ], "proofs": [ { "contents": [ "Choose an affine scheme $U$ and a surjective, flat, finite, and", "locally finitely presented morphism $U \\to \\mathcal{X}$.", "Since $h$ is representable by algebraic spaces", "(Morphisms of Stacks, Lemmas", "\\ref{stacks-morphisms-lemma-aut-iso-unramified} and", "\\ref{stacks-morphisms-lemma-stabilizer-preserving})", "we see that $U' = \\mathcal{X}' \\times_\\mathcal{X} U$ is", "an algebraic space. Since $U' \\to U$ is separated and \\'etale,", "we see that $U'$ is a scheme and that every finite set of points", "of $U'$ is contained in an affine open, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable}", "and", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-over-affine}.", "Setting $R' = U' \\times_{\\mathcal{X}'} U'$ we see", "that $s', t' : R' \\to U'$ are finite locally free.", "By Groupoids, Lemma \\ref{groupoids-lemma-find-invariant-affine}", "there exists an open covering $U' = \\bigcup U'_i$ by", "$R'$-invariant affine open subschemes $U'_i \\subset U'$.", "Let $\\mathcal{X}'_i \\subset \\mathcal{X}'$ be the corresponding", "open substacks. These are well-nigh affine as $U'_i \\to \\mathcal{X}'_i$", "is surjective, flat, finite and of finite presentation. By", "Lemma \\ref{lemma-well-nigh-affine-moduli-space-etale}", "we obtain cartesian diagrams", "$$", "\\xymatrix{", "\\mathcal{X}'_i \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\", "M'_i \\ar[r] & M", "}", "$$", "with $M'_i \\to M$ an \\'etale morphism of affine schemes", "and vertical arrows as in", "Lemma \\ref{lemma-well-nigh-affine-moduli-space}.", "Observe that", "$\\mathcal{X}'_{ij} = \\mathcal{X}'_i \\cap \\mathcal{X}'_j$", "is an open subspace of $\\mathcal{X}'_i$ and $\\mathcal{X}'_j$.", "Hence we get corresponding open subschemes", "$V_{ij} \\subset M'_i$ and $V_{ji} \\subset M'_j$.", "By the result of", "Lemma \\ref{lemma-moduli-space-finite-affine}", "we see that both", "$\\mathcal{X}'_{ij} \\to V_{ij}$ and", "$\\mathcal{X}'_{ji} \\to V_{ji}$ are categorical moduli spaces!", "Thus we get a unique isomorphism $\\varphi_{ij} : V_{ij} \\to V_{ji}$", "such that", "$$", "\\xymatrix{", "\\mathcal{X}'_i \\ar[d] & &", "\\mathcal{X}'_i \\cap \\mathcal{X}'_j \\ar[rr] \\ar[ll] \\ar[ld] \\ar[rd] & &", "\\mathcal{X}'_j \\ar[d] \\\\", "M'_i &", "V_{ij} \\ar[l] \\ar[rr]^{\\varphi_{ij}} & &", "V_{ji} \\ar[r] &", "M'_j", "}", "$$", "is commutative. These isomorphisms satisfy the cocyclce condition of", "Schemes, Section \\ref{schemes-section-glueing-schemes} by a computation", "(and another application of the previous lemma) which we omit.", "Thus we can glue the affine schemes in to scheme $M'$, see", "Schemes, Lemma \\ref{schemes-lemma-glue}.", "Let us identify the $M'_i$ with their image in $M'$.", "We claim there is a morphism $\\mathcal{X}' \\to M'$ fitting into", "cartesian diagrams", "$$", "\\xymatrix{", "\\mathcal{X}'_i \\ar[r] \\ar[d] & \\mathcal{X}' \\ar[d] \\\\", "M'_i \\ar[r] & M'", "}", "$$", "This is clear from the description of the morphisms into the glued scheme $M'$", "in Schemes, Lemma \\ref{schemes-lemma-glue} and the fact that to give a morphism", "$\\mathcal{X}' \\to M'$ is the same thing as given a morphism $T \\to M'$", "for any morphism $T \\to \\mathcal{X}'$.", "Similarly, there is a morphism $M' \\to M$ restricting to the", "given morphisms $M'_i \\to M$ on $M'_i$.", "The morphism $M' \\to M$ is \\'etale (being \\'etale on the members of an", "\\'etale covering) and the fibre product property holds as it can", "be checked on members of the (affine) open covering $M' = \\bigcup M'_i$.", "Finally, $M' \\to M$ is separated because the composition", "$U' \\to \\mathcal{X}' \\to M'$ is surjective and universally closed", "and we can apply Morphisms, Lemma", "\\ref{morphisms-lemma-image-universally-closed-separated}." ], "refs": [ "stacks-morphisms-lemma-aut-iso-unramified", "stacks-morphisms-lemma-stabilizer-preserving", "spaces-morphisms-lemma-locally-quasi-finite-separated-representable", "more-morphisms-lemma-separated-locally-quasi-finite-over-affine", "groupoids-lemma-find-invariant-affine", "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space-etale", "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space", "stacks-more-morphisms-lemma-moduli-space-finite-affine", "schemes-lemma-glue", "schemes-lemma-glue", "morphisms-lemma-image-universally-closed-separated" ], "ref_ids": [ 7593, 7592, 4972, 13906, 9664, 6919, 6918, 6920, 7686, 7686, 5415 ] } ], "ref_ids": [ 7637, 6918 ] }, { "id": 6922, "type": "theorem", "label": "stacks-more-morphisms-lemma-etale-local-finite-inertia", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-etale-local-finite-inertia", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Assume", "$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite.", "Then there exist a set $I$ and for $i \\in I$ a morphism of algebraic stacks", "$$", "g_i : \\mathcal{X}_i \\longrightarrow \\mathcal{X}", "$$", "with the following properties", "\\begin{enumerate}", "\\item $|\\mathcal{X}| = \\bigcup |g_i|(|\\mathcal{X}_i|)$,", "\\item $\\mathcal{X}_i$ is well-nigh affine,", "\\item $\\mathcal{I}_{\\mathcal{X}_i} \\to", "\\mathcal{X}_i \\times_\\mathcal{X} \\mathcal{I}_\\mathcal{X}$", "is an isomorphism, and", "\\item $g_i : \\mathcal{X}_i \\to \\mathcal{X}$ is representable", "by algebraic spaces, separated, and \\'etale,", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For any $x \\in |\\mathcal{X}|$ we can choose", "$g : \\mathcal{U} \\to \\mathcal{X}$, $\\mathcal{U} = [U/R]$, and $u$ as in", "Morphisms of Stacks,", "Lemma \\ref{stacks-morphisms-lemma-etale-local-quasi-DM-at-x}.", "Then by", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-stabilizer-preserving-unramified}", "we see that there exists an open substack", "$\\mathcal{U}' \\subset \\mathcal{U}$ containing $u$", "such that $\\mathcal{I}_{\\mathcal{U}'} \\to", "\\mathcal{U}' \\times_\\mathcal{X} \\mathcal{I}_\\mathcal{X}$", "is an isomorphism.", "Let $U' \\subset U$ be the $R$-invariant open corresponding to", "the open substack $\\mathcal{U}'$.", "Let $u' \\in U'$ be a point of $U'$ mapping to $u$.", "Observe that $t(s^{-1}(\\{u'\\}))$ is finite as $s : R \\to U$ is finite.", "By Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-affine}", "and Groupoids, Lemma \\ref{groupoids-lemma-find-invariant-affine}", "we can find an $R$-invariant affine open $U'' \\subset U'$", "containing $u'$. Let $R''$ be the restriction of $R$ to $U''$.", "Then $\\mathcal{U}'' = [U''/R'']$ is an open substack of", "$\\mathcal{U}'$ containing $u$, is well-nigh affine,", "$\\mathcal{I}_{\\mathcal{U}''} \\to", "\\mathcal{U}'' \\times_\\mathcal{X} \\mathcal{I}_\\mathcal{X}$", "is an isomorphism, and $\\mathcal{U}'' \\to \\mathcal{X}$", "and is representable by algebraic spaces and \\'etale.", "Finally, $\\mathcal{U}'' \\to \\mathcal{X}$ is separated as", "$\\mathcal{U}''$ is separated (Lemma \\ref{lemma-well-nigh-affine})", "the diagonal of $\\mathcal{X}$ is separated", "(Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-diagonal-diagonal})", "and separatedness follows from Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-compose-after-separated}.", "Since the point $x \\in |\\mathcal{X}|$ is arbitrary the proof is complete." ], "refs": [ "stacks-morphisms-lemma-etale-local-quasi-DM-at-x", "stacks-morphisms-lemma-stabilizer-preserving-unramified", "properties-lemma-ample-finite-set-in-affine", "groupoids-lemma-find-invariant-affine", "stacks-more-morphisms-lemma-well-nigh-affine", "stacks-morphisms-lemma-diagonal-diagonal", "stacks-morphisms-lemma-compose-after-separated" ], "ref_ids": [ 7537, 7594, 3062, 9664, 6916, 7416, 7406 ] } ], "ref_ids": [] }, { "id": 6923, "type": "theorem", "label": "stacks-more-morphisms-lemma-etale-separated-over-keel-mori", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-etale-separated-over-keel-mori", "contents": [ "Let $h : \\mathcal{X}' \\to \\mathcal{X}$ be a morphism of algebraic stacks.", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite,", "\\item $h$ is \\'etale, separated, and induces isomorphisms on", "automorphism groups (Morphisms of Stacks, Remark", "\\ref{stacks-morphisms-remark-identify-automorphism-groups}).", "\\end{enumerate}", "Then there exists a cartesian diagram", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[d] \\ar[r] &", "\\mathcal{X} \\ar[d] \\\\", "M' \\ar[r] &", "M", "}", "$$", "where $M' \\to M$ is a separated \\'etale morphism of algebraic spaces and", "the vertical arrows are the moduli spaces constructed in", "Theorem \\ref{theorem-keel-mori}." ], "refs": [ "stacks-morphisms-remark-identify-automorphism-groups", "stacks-more-morphisms-theorem-keel-mori" ], "proofs": [ { "contents": [ "By Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-aut-iso-unramified}", "we see that", "$\\mathcal{I}_{\\mathcal{X}'} \\to", "\\mathcal{X}' \\times_\\mathcal{X} \\mathcal{I}_\\mathcal{X}$", "is an isomorphism. Hence $\\mathcal{I}_{\\mathcal{X}'} \\to \\mathcal{X}'$", "is finite as a base change of $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$.", "Let $f' : \\mathcal{X}' \\to M'$ and $f : \\mathcal{X} \\to M$ be as in", "Theorem \\ref{theorem-keel-mori}.", "We obtain a commutative diagram as in the lemma because", "$f'$ is categorical moduli space.", "Choose $I$ and $g'_i : \\mathcal{X}'_i \\to \\mathcal{X}'$ as in", "Lemma \\ref{lemma-etale-local-finite-inertia}.", "Observe that $g_i = h \\circ g'_i$", "is \\'etale, separated, and induces isomorphisms on", "automorphism groups (Morphisms of Stacks, Remark", "\\ref{stacks-morphisms-remark-identify-automorphism-groups}).", "Let $f'_i : \\mathcal{X}'_i \\to M'_i$ be as in", "Lemma \\ref{lemma-well-nigh-affine-moduli-space}.", "In the proof of Theorem \\ref{theorem-keel-mori}", "we have seen that the diagrams", "$$", "\\xymatrix{", "\\mathcal{X}'_i \\ar[d]_{f'_i} \\ar[r]_{g'_i} &", "\\mathcal{X}' \\ar[d]^{f'} \\\\", "M'_i \\ar[r] &", "M'", "}", "\\quad\\text{and}\\quad", "\\xymatrix{", "\\mathcal{X}'_i \\ar[d]_{f'_i} \\ar[r]_{g_i} &", "\\mathcal{X} \\ar[d]^f \\\\", "M'_i \\ar[r] &", "M", "}", "$$", "are cartesian and that $M'_i \\to M'$ and $M'_i \\to M$ are \\'etale", "(this also follows directly from the properties of the morphisms", "$g'_i, g_i, f', f'_i, f$ listed sofar by arguing in exactly the same way).", "This first implies that $M' \\to M$ is \\'etale and second that", "the diagram in the lemma is cartesian. We still need to show", "that $M' \\to M$ is separated. To do this we contemplate the", "diagram", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[r] \\ar[d] &", "\\mathcal{X}' \\times_\\mathcal{X} \\mathcal{X}' \\ar[d] \\\\", "M' \\ar[r] &", "M' \\times_M M'", "}", "$$", "The top horizontal arrow is universally closed as", "$\\mathcal{X}' \\to \\mathcal{X}$ is separated.", "The vertical arrows are as in Theorem \\ref{theorem-keel-mori}", "(as flat base changes of $\\mathcal{X} \\to M$)", "hence universal homeomorphisms. Thus the lower horizontal", "arrow is universally closed. This (combined with it being an \\'etale", "monomorphism of algebraic spaces) proves it is a closed immersion as desired." ], "refs": [ "stacks-morphisms-lemma-aut-iso-unramified", "stacks-more-morphisms-theorem-keel-mori", "stacks-more-morphisms-lemma-etale-local-finite-inertia", "stacks-morphisms-remark-identify-automorphism-groups", "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space", "stacks-more-morphisms-theorem-keel-mori", "stacks-more-morphisms-theorem-keel-mori" ], "ref_ids": [ 7593, 6883, 6922, 7637, 6918, 6883, 6883 ] } ], "ref_ids": [ 7637, 6883 ] }, { "id": 6924, "type": "theorem", "label": "stacks-more-morphisms-lemma-keel-mori-finite-type", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-keel-mori-finite-type", "contents": [ "Let $p : \\mathcal{X} \\to Y$ be a morphism of an algebraic stack to an", "algebraic space. Assume", "\\begin{enumerate}", "\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite,", "\\item $Y$ is locally Noetherian, and", "\\item $p$ is locally of finite type.", "\\end{enumerate}", "Let $f : \\mathcal{X} \\to M$ be the moduli space constructed in", "Theorem \\ref{theorem-keel-mori}.", "Then $M \\to Y$ is locally of finite type." ], "refs": [ "stacks-more-morphisms-theorem-keel-mori" ], "proofs": [ { "contents": [ "Since $f$ is a uniform categorical moduli space we obtain the", "morphism $M \\to Y$. It suffices to check that $M \\to Y$", "is locally of finite type \\'etale locally on $M$ and $Y$.", "Since $f$ is a uniform categorical moduli space, we", "may first replace $Y$ by an affine scheme \\'etale over $Y$.", "Next, we may choose $I$ and $g_i : \\mathcal{X}_i \\to \\mathcal{X}$", "as in Lemma \\ref{lemma-etale-local-finite-inertia}.", "Then by Lemma \\ref{lemma-etale-separated-over-keel-mori}", "we reduce to the case $\\mathcal{X} = \\mathcal{X}_i$.", "In other words, we may assume $\\mathcal{X}$ is well-nigh affine.", "In this case we have $Y = \\Spec(D)$, we have", "$\\mathcal{X} = [U/R]$ with $U = \\Spec(A)$ and", "$M = \\Spec(C)$ where $C \\subset A$ is the set of $R$-invariant", "functions on $U$. See", "Lemmas \\ref{lemma-well-nigh-affine} and", "\\ref{lemma-well-nigh-affine-moduli-space}.", "Then $A_0$ is Noetherian and $A_0 \\to A$ is of finite type.", "Moreover $A$ is integral over $C$ by", "Groupoids, Lemma \\ref{groupoids-lemma-integral-over-invariants},", "hence finite over $C$", "(being of finite type over $A_0$).", "Thus we may finally apply", "Algebra, Lemma \\ref{algebra-lemma-Artin-Tate}", "to conclude." ], "refs": [ "stacks-more-morphisms-lemma-etale-local-finite-inertia", "stacks-more-morphisms-lemma-etale-separated-over-keel-mori", "stacks-more-morphisms-lemma-well-nigh-affine", "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space", "groupoids-lemma-integral-over-invariants", "algebra-lemma-Artin-Tate" ], "ref_ids": [ 6922, 6923, 6916, 6918, 9659, 629 ] } ], "ref_ids": [ 6883 ] }, { "id": 6925, "type": "theorem", "label": "stacks-more-morphisms-lemma-keel-mori-diagonal", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-keel-mori-diagonal", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Assume", "$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite.", "Let $f : \\mathcal{X} \\to M$ be the moduli space constructed in", "Theorem \\ref{theorem-keel-mori}.", "\\begin{enumerate}", "\\item If $\\mathcal{X}$ is quasi-separated, then $M$ is quasi-separated.", "\\item If $\\mathcal{X}$ is separated, then $M$ is separated.", "\\item Add more here, for example relative versions of the above.", "\\end{enumerate}" ], "refs": [ "stacks-more-morphisms-theorem-keel-mori" ], "proofs": [ { "contents": [ "To prove this consider the following diagram", "$$", "\\xymatrix{", "\\mathcal{X} \\ar[d]_f \\ar[r]_{\\Delta_\\mathcal{X}} &", "\\mathcal{X} \\times \\mathcal{X} \\ar[d]^{f \\times f} \\\\", "M \\ar[r]^{\\Delta_M} &", "M \\times M", "}", "$$", "Since $f$ is a universal homeomorphism, we see that", "$f \\times f$ is a universal homeomorphism.", "\\medskip\\noindent", "If $\\mathcal{X}$ is separated, then $\\Delta_\\mathcal{X}$ is proper,", "hence $\\Delta_\\mathcal{X}$ is universally closed, hence", "$\\Delta_M$ is universally closed, hence $M$ is separated", "by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-separated-diagonal-proper}.", "\\medskip\\noindent", "If $\\mathcal{X}$ is quasi-separated, then $\\Delta_\\mathcal{X}$ is", "quasi-compact, hence $\\Delta_M$ is quasi-compact, hence $M$ is", "quasi-separated." ], "refs": [ "spaces-morphisms-lemma-separated-diagonal-proper" ], "ref_ids": [ 4923 ] } ], "ref_ids": [ 6883 ] }, { "id": 6926, "type": "theorem", "label": "stacks-more-morphisms-lemma-keel-mori-proper", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-lemma-keel-mori-proper", "contents": [ "Let $p : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack", "to an algebraic space. Assume", "\\begin{enumerate}", "\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite,", "\\item $p$ is proper, and", "\\item $Y$ is locally Noetherian.", "\\end{enumerate}", "Let $f : \\mathcal{X} \\to M$ be the moduli space constructed in", "Theorem \\ref{theorem-keel-mori}. Then $M \\to Y$ is proper." ], "refs": [ "stacks-more-morphisms-theorem-keel-mori" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-keel-mori-finite-type}", "we see that $M \\to Y$ is locally of finite type.", "By Lemma \\ref{lemma-keel-mori-diagonal} we see that", "$M \\to Y$ is separated.", "Of course $M \\to Y$ is quasi-compact and universally closed", "as these are topological properties and $\\mathcal{X} \\to Y$", "has these properties and $\\mathcal{X} \\to M$ is a universal", "homeomorphism." ], "refs": [ "stacks-more-morphisms-lemma-keel-mori-finite-type", "stacks-more-morphisms-lemma-keel-mori-diagonal" ], "ref_ids": [ 6924, 6925 ] } ], "ref_ids": [ 6883 ] }, { "id": 6927, "type": "theorem", "label": "stacks-more-morphisms-proposition-affine-smooth-lift-to-first-order", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-proposition-affine-smooth-lift-to-first-order", "contents": [ "\\begin{reference}", "Email of Matthew Emerton dated April 27, 2016.", "\\end{reference}", "Let $\\mathcal{X} \\subset \\mathcal{X}'$ be a first order thickening", "of algebraic stacks. Let $W$ be an affine scheme and let", "$W \\to \\mathcal{X}$ be a smooth morphism. Then there exists", "a cartesian diagram", "$$", "\\xymatrix{", "W \\ar[d] \\ar[r] & W' \\ar[d] \\\\", "\\mathcal{X} \\ar[r] & \\mathcal{X}'", "}", "$$", "with $W' \\to \\mathcal{X}'$ smooth and $W'$ affine." ], "refs": [], "proofs": [ { "contents": [ "Consider the category $p : \\mathcal{C} \\to W_{spaces, \\etale}$", "introduced in Remark \\ref{remark-gerbe-of-lifts}.", "The proposition states that there exists an object of $\\mathcal{C}$", "lying over $W$. Namely, if we have such an object", "$(W, W', a, i, y', \\alpha)$ then $W = \\mathcal{X} \\times_{\\mathcal{X}'} W'$.", "Hence $W \\to W'$ is a thickening of algebraic spaces so", "$W'$ is affine by", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-thickening-scheme}", "and More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-thickening-affine-scheme}.", "\\medskip\\noindent", "Lemma \\ref{lemma-gerbe-of-lifts} tells us $\\mathcal{C}$ is a gerbe over", "$W_{spaces, \\etale}$. This means we can \\'etale locally find a solution and", "these local solutions are \\'etale locally isomorphic;", "this part does not require the assumption that the thickening is first order.", "By Lemma \\ref{lemma-gerbe-of-lifts-first-order}", "the automorphism sheaves of objects of our gerbe are abelian and", "fit together to form a quasi-coherent module $\\mathcal{G}$", "on $W_{spaces, \\etale}$. We will verify conditions (1) and (2)", "of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-existence}", "to conclude the existence of an object of $\\mathcal{C}$ lying over $W$.", "Condition (1) is true: the \\'etale coverings $\\{W_i \\to W\\}$", "with each $W_i$ affine are cofinal in the collection of all coverings.", "For such a covering $W_i$ and $W_i \\times_W W_j$ are affine", "and $H^1(W_i, \\mathcal{G})$ and $H^1(W_i \\times_W W_j, \\mathcal{G})$", "are zero: the cohomology of a quasi-coherent module over an affine", "algebraic space is zero for example by Cohomology of Spaces, Proposition", "\\ref{spaces-cohomology-proposition-vanishing}.", "Finally, condition (2) is that $H^2(W, \\mathcal{G}) = 0$", "for our quasi-coherent sheaf $\\mathcal{G}$ which again follows", "from Cohomology of Spaces, Proposition", "\\ref{spaces-cohomology-proposition-vanishing}.", "This finishes the proof." ], "refs": [ "stacks-more-morphisms-remark-gerbe-of-lifts", "spaces-more-morphisms-lemma-thickening-scheme", "more-morphisms-lemma-thickening-affine-scheme", "stacks-more-morphisms-lemma-gerbe-of-lifts", "stacks-more-morphisms-lemma-gerbe-of-lifts-first-order", "sites-cohomology-lemma-existence", "spaces-cohomology-proposition-vanishing", "spaces-cohomology-proposition-vanishing" ], "ref_ids": [ 6933, 49, 13678, 6898, 6899, 4206, 11345, 11345 ] } ], "ref_ids": [] }, { "id": 6934, "type": "theorem", "label": "perfect-theorem-approximation", "categories": [ "perfect" ], "title": "perfect-theorem-approximation", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Then approximation by perfect complexes holds on $X$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the induction principle of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}", "and Lemmas \\ref{lemma-induction-step} and \\ref{lemma-approximation-affine}." ], "refs": [ "coherent-lemma-induction-principle", "perfect-lemma-induction-step", "perfect-lemma-approximation-affine" ], "ref_ids": [ 3291, 7008, 7007 ] } ], "ref_ids": [] }, { "id": 6935, "type": "theorem", "label": "perfect-theorem-bondal-van-den-Bergh", "categories": [ "perfect" ], "title": "perfect-theorem-bondal-van-den-Bergh", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. The category", "$D_\\QCoh(\\mathcal{O}_X)$ can be generated by a single", "perfect object. More precisely, there exists a perfect object", "$P$ of $D(\\mathcal{O}_X)$ such that for ", "$E \\in D_\\QCoh(\\mathcal{O}_X)$ the following are equivalent", "\\begin{enumerate}", "\\item $E = 0$, and", "\\item $\\Hom_{D(\\mathcal{O}_X)}(P[n], E) = 0$ for all $n \\in \\mathbf{Z}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will prove this using the induction principle of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}.", "\\medskip\\noindent", "If $X$ is affine, then $\\mathcal{O}_X$ is a perfect generator.", "This follows from Lemma \\ref{lemma-affine-compare-bounded}.", "\\medskip\\noindent", "Assume that $X = U \\cup V$ is an open covering with $U$ quasi-compact", "such that the theorem holds for $U$ and $V$ is an affine open.", "Let $P$ be a perfect object of $D(\\mathcal{O}_U)$ which is a generator", "for $D_\\QCoh(\\mathcal{O}_U)$. Using", "Lemma \\ref{lemma-direct-summand-of-a-restriction} we may", "choose a perfect object", "$Q$ of $D(\\mathcal{O}_X)$ whose restriction to $U$ is a direct sum one", "of whose summands is $P$. Say $V = \\Spec(A)$. Let $Z = X \\setminus U$.", "This is a closed subset of $V$ with $V \\setminus Z$ quasi-compact.", "Choose $f_1, \\ldots, f_r \\in A$ such that", "$Z = V(f_1, \\ldots, f_r)$. Let $K \\in D(\\mathcal{O}_V)$ be the perfect", "object corresponding to the Koszul complex on $f_1, \\ldots, f_r$ over $A$.", "Note that since $K$ is supported on $Z \\subset V$ closed, the pushforward", "$K' = R(V \\to X)_*K$ is a perfect object of $D(\\mathcal{O}_X)$ whose", "restriction to $V$ is $K$ (see", "Cohomology, Lemma \\ref{cohomology-lemma-pushforward-perfect}).", "We claim that $Q \\oplus K'$ is a generator for", "$D_\\QCoh(\\mathcal{O}_X)$.", "\\medskip\\noindent", "Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$ such that", "there are no nontrivial maps from any shift of $Q \\oplus K'$ into $E$.", "By Cohomology, Lemma \\ref{cohomology-lemma-pushforward-restriction}", "we have $K' = R(V \\to X)_! K$ and hence", "$$", "\\Hom_{D(\\mathcal{O}_X)}(K'[n], E) = \\Hom_{D(\\mathcal{O}_V)}(K[n], E|_V)", "$$", "Thus by Lemma \\ref{lemma-orthogonal-koszul-complex} the vanishing of", "these groups implies that $E|_V$ is isomorphic to", "$R(U \\cap V \\to V)_*E|_{U \\cap V}$. This implies that $E = R(U \\to X)_*E|_U$", "(small detail omitted). If this is the case then", "$$", "\\Hom_{D(\\mathcal{O}_X)}(Q[n], E) = \\Hom_{D(\\mathcal{O}_U)}(Q|_U[n], E|_U)", "$$", "which contains $\\Hom_{D(\\mathcal{O}_U)}(P[n], E|_U)$ as a direct summand.", "Thus by our choice of $P$ the vanishing of these groups implies that $E|_U$", "is zero. Whence $E$ is zero." ], "refs": [ "coherent-lemma-induction-principle", "perfect-lemma-affine-compare-bounded", "perfect-lemma-direct-summand-of-a-restriction", "cohomology-lemma-pushforward-perfect", "cohomology-lemma-pushforward-restriction", "perfect-lemma-orthogonal-koszul-complex" ], "ref_ids": [ 3291, 6941, 7009, 2229, 2148, 7010 ] } ], "ref_ids": [] }, { "id": 6936, "type": "theorem", "label": "perfect-theorem-DQCoh-is-Ddga", "categories": [ "perfect" ], "title": "perfect-theorem-DQCoh-is-Ddga", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Then there exist a differential graded algebra $(E, \\text{d})$", "with only a finite number of nonzero cohomology groups $H^i(E)$", "such that $D_\\QCoh(\\mathcal{O}_X)$ is equivalent", "to $D(E, \\text{d})$." ], "refs": [], "proofs": [ { "contents": [ "Let $K^\\bullet$ be a K-injective complex of $\\mathcal{O}$-modules which", "is perfect and generates $D_\\QCoh(\\mathcal{O}_X)$. Such a", "thing exists by Theorem \\ref{theorem-bondal-van-den-Bergh}", "and the existence of K-injective resolutions. We will show the", "theorem holds with", "$$", "(E, \\text{d}) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, K^\\bullet)", "$$", "where $\\text{Comp}^{dg}(\\mathcal{O}_X)$ is the differential graded category", "of complexes of $\\mathcal{O}$-modules. Please see", "Differential Graded Algebra, Section \\ref{dga-section-variant-base-change}.", "Since $K^\\bullet$ is K-injective we", "have", "\\begin{equation}", "\\label{equation-E-is-OK}", "H^n(E) = \\Ext^n_{D(\\mathcal{O}_X)}(K^\\bullet, K^\\bullet)", "\\end{equation}", "for all $n \\in \\mathbf{Z}$. Only a finite number of these Exts", "are nonzero by Lemma \\ref{lemma-ext-from-perfect-into-bounded-QCoh}.", "Consider the functor", "$$", "- \\otimes_E^\\mathbf{L} K^\\bullet :", "D(E, \\text{d}) \\longrightarrow D(\\mathcal{O}_X)", "$$", "of", "Differential Graded Algebra, Lemma", "\\ref{dga-lemma-tensor-with-complex-derived}.", "Since $K^\\bullet$ is perfect, it defines a compact object of", "$D(\\mathcal{O}_X)$, see Proposition \\ref{proposition-compact-is-perfect}.", "Combined with (\\ref{equation-E-is-OK}) the functor above is fully", "faithful as follows from", "Differential Graded Algebra, Lemmas", "\\ref{dga-lemma-fully-faithful-in-compact-case}. It has a right adjoint", "$$", "R\\Hom(K^\\bullet, - ) : D(\\mathcal{O}_X) \\longrightarrow D(E, \\text{d})", "$$", "by Differential Graded Algebra, Lemmas", "\\ref{dga-lemma-tensor-with-complex-hom-adjoint}", "which is a left quasi-inverse functor by generalities on adjoint", "functors. On the other hand, it follows from", "Lemma \\ref{lemma-tensor-with-QCoh-complex} that we obtain", "$$", "- \\otimes_E^\\mathbf{L} K^\\bullet :", "D(E, \\text{d}) \\longrightarrow D_\\QCoh(\\mathcal{O}_X)", "$$", "and by our choice of $K^\\bullet$ as a generator of", "$D_\\QCoh(\\mathcal{O}_X)$ the kernel of the adjoint", "restricted to $D_\\QCoh(\\mathcal{O}_X)$ is zero.", "A formal argument shows that we obtain the desired equivalence, see", "Derived Categories, Lemma", "\\ref{derived-lemma-fully-faithful-adjoint-kernel-zero}." ], "refs": [ "perfect-theorem-bondal-van-den-Bergh", "perfect-lemma-ext-from-perfect-into-bounded-QCoh", "dga-lemma-tensor-with-complex-derived", "perfect-proposition-compact-is-perfect", "dga-lemma-fully-faithful-in-compact-case", "dga-lemma-tensor-with-complex-hom-adjoint", "perfect-lemma-tensor-with-QCoh-complex", "derived-lemma-fully-faithful-adjoint-kernel-zero" ], "ref_ids": [ 6935, 7018, 13116, 7111, 13119, 13118, 7017, 1793 ] } ], "ref_ids": [] }, { "id": 6937, "type": "theorem", "label": "perfect-lemma-quasi-coherence-direct-sums", "categories": [ "perfect" ], "title": "perfect-lemma-quasi-coherence-direct-sums", "contents": [ "Let $X$ be a scheme. Then $D_\\QCoh(\\mathcal{O}_X)$", "has direct sums." ], "refs": [], "proofs": [ { "contents": [ "By Injectives, Lemma \\ref{injectives-lemma-derived-products}", "the derived category $D(\\mathcal{O}_X)$ has direct sums and", "they are computed by taking termwise direct sums of any representatives.", "Thus it is clear that the cohomology sheaf of a direct sum is the", "direct sum of the cohomology sheaves as taking direct sums is", "an exact functor (in any Grothendieck abelian category). The lemma", "follows as the direct sum of quasi-coherent sheaves is quasi-coherent, see", "Schemes, Section \\ref{schemes-section-quasi-coherent}." ], "refs": [ "injectives-lemma-derived-products" ], "ref_ids": [ 7795 ] } ], "ref_ids": [] }, { "id": 6938, "type": "theorem", "label": "perfect-lemma-Rlim-quasi-coherent", "categories": [ "perfect" ], "title": "perfect-lemma-Rlim-quasi-coherent", "contents": [ "Let $X$ be a scheme. Let $(K_n)$ be an inverse system of", "$D_\\QCoh(\\mathcal{O}_X)$ with derived limit", "$K = R\\lim K_n$ in $D(\\mathcal{O}_X)$. Assume $H^q(K_{n + 1}) \\to H^q(K_n)$", "is surjective for all $q \\in \\mathbf{Z}$ and $n \\geq 1$.", "Then", "\\begin{enumerate}", "\\item $H^q(K) = \\lim H^q(K_n)$,", "\\item $R\\lim H^q(K_n) = \\lim H^q(K_n)$, and", "\\item for every affine open $U \\subset X$ we have", "$H^p(U, \\lim H^q(K_n)) = 0$ for $p > 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{B}$ be the set of affine opens of $X$.", "Since $H^q(K_n)$ is quasi-coherent we have $H^p(U, H^q(K_n)) = 0$", "for $U \\in \\mathcal{B}$ by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "Moreover, the maps $H^0(U, H^q(K_{n + 1})) \\to H^0(U, H^q(K_n))$", "are surjective for $U \\in \\mathcal{B}$ by", "Schemes, Lemma \\ref{schemes-lemma-equivalence-quasi-coherent}.", "Part (1) follows from Cohomology, Lemma", "\\ref{cohomology-lemma-derived-limit-suitable-system}", "whose conditions we have just verified.", "Parts (2) and (3) follow from", "Cohomology, Lemma \\ref{cohomology-lemma-inverse-limit-is-derived-limit}." ], "refs": [ "coherent-lemma-quasi-coherent-affine-cohomology-zero", "schemes-lemma-equivalence-quasi-coherent", "cohomology-lemma-derived-limit-suitable-system", "cohomology-lemma-inverse-limit-is-derived-limit" ], "ref_ids": [ 3282, 7664, 2169, 2162 ] } ], "ref_ids": [] }, { "id": 6939, "type": "theorem", "label": "perfect-lemma-nice-K-injective", "categories": [ "perfect" ], "title": "perfect-lemma-nice-K-injective", "contents": [ "Let $X$ be a scheme. Let $E$ be an object of", "$D_\\QCoh(\\mathcal{O}_X)$. Then the canonical map", "$E \\to R\\lim \\tau_{\\geq -n}E$ is an isomorphism\\footnote{In particular,", "$E$ has a K-injective representative as in", "Cohomology, Lemma \\ref{cohomology-lemma-K-injective}.}." ], "refs": [ "cohomology-lemma-K-injective" ], "proofs": [ { "contents": [ "Denote $\\mathcal{H}^i = H^i(E)$ the $i$th cohomology sheaf of $E$.", "Let $\\mathcal{B}$ be the set of affine open subsets of $X$. Then", "$H^p(U, \\mathcal{H}^i) = 0$ for all $p > 0$, all $i \\in \\mathbf{Z}$,", "and all $U \\in \\mathcal{B}$, see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "Thus the lemma follows from", "Cohomology, Lemma \\ref{cohomology-lemma-is-limit-dimension}." ], "refs": [ "coherent-lemma-quasi-coherent-affine-cohomology-zero", "cohomology-lemma-is-limit-dimension" ], "ref_ids": [ 3282, 2167 ] } ], "ref_ids": [ 2170 ] }, { "id": 6940, "type": "theorem", "label": "perfect-lemma-application-nice-K-injective", "categories": [ "perfect" ], "title": "perfect-lemma-application-nice-K-injective", "contents": [ "Let $X$ be a scheme. Let $F : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Ab}$", "be an additive functor and $N \\geq 0$ an integer. Assume that", "\\begin{enumerate}", "\\item $F$ commutes with countable direct products,", "\\item $R^pF(\\mathcal{F}) = 0$ for all $p \\geq N$ and $\\mathcal{F}$", "quasi-coherent.", "\\end{enumerate}", "Then for $E \\in D_\\QCoh(\\mathcal{O}_X)$", "\\begin{enumerate}", "\\item $H^i(RF(\\tau_{\\leq a}E)) \\to H^i(RF(E))$ is an isomorphism", "for $i \\leq a$,", "\\item $H^i(RF(E)) \\to H^i(RF(\\tau_{\\geq b - N + 1}E))$ is an isomorphism", "for $i \\geq b$,", "\\item if $H^i(E) = 0$ for $i \\not \\in [a, b]$ for some", "$-\\infty \\leq a \\leq b \\leq \\infty$, then $H^i(RF(E)) = 0$", "for $i \\not \\in [a, b + N - 1]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Statement (1) is", "Derived Categories, Lemma \\ref{derived-lemma-negative-vanishing}.", "\\medskip\\noindent", "Proof of statement (2). Write $E_n = \\tau_{\\geq -n}E$. We have", "$E = R\\lim E_n$, see Lemma \\ref{lemma-nice-K-injective}. Thus", "$RF(E) = R\\lim RF(E_n)$ in $D(\\textit{Ab})$ by Injectives, Lemma", "\\ref{injectives-lemma-RF-commutes-with-Rlim}. Thus for every $i \\in \\mathbf{Z}$", "we have a short exact sequence", "$$", "0 \\to R^1\\lim H^{i - 1}(RF(E_n)) \\to H^i(RF(E)) \\to \\lim H^i(RF(E_n)) \\to 0", "$$", "see More on Algebra, Remark", "\\ref{more-algebra-remark-compare-derived-limit}.", "To prove (2) we will show that the term on the left is zero", "and that the term on the right equals $H^i(RF(E_{-b + N - 1})$", "for any $b$ with $i \\geq b$.", "\\medskip\\noindent", "For every $n$ we have a distinguished triangle", "$$", "H^{-n}(E)[n] \\to E_n \\to E_{n - 1} \\to H^{-n}(E)[n + 1]", "$$", "(Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle})", "in $D(\\mathcal{O}_X)$. Since $H^{-n}(E)$ is quasi-coherent we have", "$$", "H^i(RF(H^{-n}(E)[n])) = R^{i + n}F(H^{-n}(E)) = 0", "$$", "for $i + n \\geq N$ and", "$$", "H^i(RF(H^{-n}(E)[n + 1])) = R^{i + n + 1}F(H^{-n}(E)) = 0", "$$", "for $i + n + 1 \\geq N$. We conclude that", "$$", "H^i(RF(E_n)) \\to H^i(RF(E_{n - 1}))", "$$", "is an isomorphism for $n \\geq N - i$. Thus the systems $H^i(RF(E_n))$ all", "satisfy the ML condition and the $R^1\\lim$ term in our short exact", "sequence is zero (see discussion in", "More on Algebra, Section \\ref{more-algebra-section-Rlim}).", "Moreover, the system $H^i(RF(E_n))$ is constant starting", "with $n = N - i - 1$ as desired.", "\\medskip\\noindent", "Proof of (3). Under the assumption on $E$ we have", "$\\tau_{\\leq a - 1}E = 0$ and we get the vanishing", "of $H^i(RF(E))$ for $i \\leq a - 1$ from (1).", "Similarly, we have $\\tau_{\\geq b + 1}E = 0$ and hence", "we get the vanishing of $H^i(RF(E))$ for $i \\geq b + n$ from", "part (2)." ], "refs": [ "derived-lemma-negative-vanishing", "perfect-lemma-nice-K-injective", "injectives-lemma-RF-commutes-with-Rlim", "more-algebra-remark-compare-derived-limit", "derived-remark-truncation-distinguished-triangle" ], "ref_ids": [ 1839, 6939, 7796, 10658, 2016 ] } ], "ref_ids": [] }, { "id": 6941, "type": "theorem", "label": "perfect-lemma-affine-compare-bounded", "categories": [ "perfect" ], "title": "perfect-lemma-affine-compare-bounded", "contents": [ "Let $X = \\Spec(A)$ be an affine scheme. All the functors in the diagram", "$$", "\\xymatrix{", "D(\\QCoh(\\mathcal{O}_X)) \\ar[rr]_{(\\ref{equation-compare})}", "& &", "D_\\QCoh(\\mathcal{O}_X) \\ar[ld]^{R\\Gamma(X, -)} \\\\", "& D(A) \\ar[lu]^{\\widetilde{\\ \\ }}", "}", "$$", "are equivalences of triangulated categories. Moreover, for $E$ in", "$D_\\QCoh(\\mathcal{O}_X)$ we have $H^0(X, E) = H^0(X, H^0(E))$." ], "refs": [], "proofs": [ { "contents": [ "The functor $R\\Gamma(X, -)$ gives a functor", "$D(\\mathcal{O}_X) \\to D(A)$ and hence by restriction a functor", "\\begin{equation}", "\\label{equation-back}", "R\\Gamma(X, -) : D_\\QCoh(\\mathcal{O}_X) \\longrightarrow D(A).", "\\end{equation}", "We will show this functor is quasi-inverse to (\\ref{equation-compare})", "via the equivalence between quasi-coherent modules on $X$ and", "the category of $A$-modules.", "\\medskip\\noindent", "Elucidation. Denote $(Y, \\mathcal{O}_Y)$ the one point space with sheaf", "of rings given by $A$. Denote", "$\\pi : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "the obvious morphism of ringed spaces.", "Then $R\\Gamma(X, -)$ can be identified with $R\\pi_*$ and the functor", "(\\ref{equation-compare}) via the equivalence", "$\\textit{Mod}(\\mathcal{O}_Y) = \\text{Mod}_A = \\QCoh(\\mathcal{O}_X)$", "can be identified with $L\\pi^* = \\pi^* = \\widetilde{\\ }$ (see", "Modules, Lemma \\ref{modules-lemma-construct-quasi-coherent-sheaves} and", "Schemes, Lemmas \\ref{schemes-lemma-compare-constructions} and", "\\ref{schemes-lemma-equivalence-quasi-coherent}). Thus the functors", "$$", "\\xymatrix{", "D(A) \\ar@<1ex>[r] & D(\\mathcal{O}_X) \\ar@<1ex>[l]", "}", "$$", "are adjoint (by Cohomology, Lemma \\ref{cohomology-lemma-adjoint}). In", "particular we obtain canonical adjunction mappings", "$$", "a : \\widetilde{R\\Gamma(X, E)} \\longrightarrow E", "$$", "for $E$ in $D(\\mathcal{O}_X)$ and", "$$", "b : M^\\bullet \\longrightarrow R\\Gamma(X, \\widetilde{M^\\bullet})", "$$", "for $M^\\bullet$ a complex of $A$-modules.", "\\medskip\\noindent", "Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. We may apply", "Lemma \\ref{lemma-application-nice-K-injective}", "to the functor $F(-) = \\Gamma(X, -)$", "with $N = 1$ by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "Hence", "$$", "H^0(R\\Gamma(X, E)) = H^0(R\\Gamma(X, \\tau_{\\geq 0}E)) = \\Gamma(X, H^0(E))", "$$", "(the last equality by definition of the canonical truncation).", "Using this we will show that the adjunction mappings $a$ and $b$", "induce isomorphisms $H^0(a)$ and $H^0(b)$. Thus $a$ and $b$", "are quasi-isomorphisms (as the statement is invariant under shifts)", "and the lemma is proved.", "\\medskip\\noindent", "In both cases we use that $\\widetilde{\\ }$ is an exact functor", "(Schemes, Lemma \\ref{schemes-lemma-spec-sheaves}). Namely, this", "implies that", "$$", "H^0\\left(\\widetilde{R\\Gamma(X, E)}\\right) =", "\\widetilde{H^0(R\\Gamma(X, E))} =", "\\widetilde{\\Gamma(X, H^0(E))}", "$$", "which is equal to $H^0(E)$ because $H^0(E)$ is quasi-coherent. Thus", "$H^0(a)$ is an isomorphism. For the other direction we have", "$$", "H^0(R\\Gamma(X, \\widetilde{M^\\bullet})) =", "\\Gamma(X, H^0(\\widetilde{M^\\bullet})) =", "\\Gamma(X, \\widetilde{H^0(M^\\bullet)}) =", "H^0(M^\\bullet)", "$$", "which proves that $H^0(b)$ is an isomorphism." ], "refs": [ "modules-lemma-construct-quasi-coherent-sheaves", "schemes-lemma-compare-constructions", "schemes-lemma-equivalence-quasi-coherent", "cohomology-lemma-adjoint", "perfect-lemma-application-nice-K-injective", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "schemes-lemma-spec-sheaves" ], "ref_ids": [ 13245, 7660, 7664, 2121, 6940, 3282, 7651 ] } ], "ref_ids": [] }, { "id": 6942, "type": "theorem", "label": "perfect-lemma-affine-K-flat", "categories": [ "perfect" ], "title": "perfect-lemma-affine-K-flat", "contents": [ "Let $X = \\Spec(A)$ be an affine scheme. If $K^\\bullet$ is a K-flat", "complex of $A$-modules, then $\\widetilde{K^\\bullet}$ is a K-flat", "complex of $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "By More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-K-flat}", "we see that $K^\\bullet \\otimes_A A_\\mathfrak p$ is a K-flat complex", "of $A_\\mathfrak p$-modules for every $\\mathfrak p \\in \\Spec(A)$.", "Hence we conclude from", "Cohomology, Lemma \\ref{cohomology-lemma-check-K-flat-stalks}", "(and", "Schemes, Lemma \\ref{schemes-lemma-spec-sheaves})", "that $\\widetilde{K^\\bullet}$ is K-flat." ], "refs": [ "more-algebra-lemma-base-change-K-flat", "cohomology-lemma-check-K-flat-stalks", "schemes-lemma-spec-sheaves" ], "ref_ids": [ 10124, 2104, 7651 ] } ], "ref_ids": [] }, { "id": 6943, "type": "theorem", "label": "perfect-lemma-quasi-coherence-pushforward", "categories": [ "perfect" ], "title": "perfect-lemma-quasi-coherence-pushforward", "contents": [ "If $f : X \\to Y$ is a morphism of affine schemes given by the ring map", "$A \\to B$, then the diagram", "$$", "\\xymatrix{", "D(B) \\ar[d] \\ar[r] & D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\", "D(A) \\ar[r] & D_\\QCoh(\\mathcal{O}_Y)", "}", "$$", "commutes." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-affine-compare-bounded}", "using that $R\\Gamma(Y, Rf_*K) = R\\Gamma(X, K)$ by", "Cohomology, Lemma \\ref{cohomology-lemma-Leray-unbounded}." ], "refs": [ "perfect-lemma-affine-compare-bounded", "cohomology-lemma-Leray-unbounded" ], "ref_ids": [ 6941, 2138 ] } ], "ref_ids": [] }, { "id": 6944, "type": "theorem", "label": "perfect-lemma-quasi-coherence-pullback", "categories": [ "perfect" ], "title": "perfect-lemma-quasi-coherence-pullback", "contents": [ "Let $f : Y \\to X$ be a morphism of schemes.", "\\begin{enumerate}", "\\item The functor $Lf^*$ sends $D_\\QCoh(\\mathcal{O}_X)$", "into $D_\\QCoh(\\mathcal{O}_Y)$.", "\\item If $X$ and $Y$ are affine and $f$ is given by the ring map", "$A \\to B$, then the diagram", "$$", "\\xymatrix{", "D(B) \\ar[r] & D_\\QCoh(\\mathcal{O}_Y) \\\\", "D(A) \\ar[r] \\ar[u]^{- \\otimes_A^\\mathbf{L} B} &", "D_\\QCoh(\\mathcal{O}_X) \\ar[u]_{Lf^*}", "}", "$$", "commutes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We first prove the diagram", "$$", "\\xymatrix{", "D(B) \\ar[r] & D(\\mathcal{O}_Y) \\\\", "D(A) \\ar[r] \\ar[u]^{- \\otimes_A^\\mathbf{L} B} &", "D(\\mathcal{O}_X) \\ar[u]_{Lf^*}", "}", "$$", "commutes. This is clear from Lemma \\ref{lemma-affine-K-flat} and", "the constructions of the functors in question. To see (1) let", "$E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. To see that", "$Lf^*E$ has quasi-coherent cohomology sheaves we may work locally on $X$.", "Note that $Lf^*$ is compatible with restricting to open subschemes.", "Hence we can assume that $f$ is a morphism of affine schemes as in (2).", "Then we can apply Lemma \\ref{lemma-affine-compare-bounded} to see that", "$E$ comes from a complex of $A$-modules. By the commutativity of the first", "diagram of the proof the same holds for $Lf^*E$ and we conclude (1) is true." ], "refs": [ "perfect-lemma-affine-K-flat", "perfect-lemma-affine-compare-bounded" ], "ref_ids": [ 6942, 6941 ] } ], "ref_ids": [] }, { "id": 6945, "type": "theorem", "label": "perfect-lemma-quasi-coherence-tensor-product", "categories": [ "perfect" ], "title": "perfect-lemma-quasi-coherence-tensor-product", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item For objects $K, L$ of $D_\\QCoh(\\mathcal{O}_X)$", "the derived tensor product $K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} L$ is in", "$D_\\QCoh(\\mathcal{O}_X)$.", "\\item If $X = \\Spec(A)$ is affine then", "$$", "\\widetilde{M^\\bullet} \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\widetilde{K^\\bullet}", "=", "\\widetilde{M^\\bullet \\otimes_A^\\mathbf{L} K^\\bullet}", "$$", "for any pair of complexes of $A$-modules $K^\\bullet$, $M^\\bullet$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equality of (2) follows immediately from Lemma \\ref{lemma-affine-K-flat}", "and the construction of the derived tensor product.", "To see (1) let $K, L$ be objects of $D_\\QCoh(\\mathcal{O}_X)$.", "To check that $K \\otimes^\\mathbf{L} L$ is in", "$D_\\QCoh(\\mathcal{O}_X)$ we may work locally on $X$, hence", "we may assume $X = \\Spec(A)$ is affine. By", "Lemma \\ref{lemma-affine-compare-bounded} we may represent", "$K$ and $L$ by complexes of $A$-modules. Then part (2) implies", "the result." ], "refs": [ "perfect-lemma-affine-K-flat", "perfect-lemma-affine-compare-bounded" ], "ref_ids": [ 6942, 6941 ] } ], "ref_ids": [] }, { "id": 6946, "type": "theorem", "label": "perfect-lemma-quasi-coherence-direct-image", "categories": [ "perfect" ], "title": "perfect-lemma-quasi-coherence-direct-image", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume that $f$ is quasi-separated and quasi-compact.", "\\begin{enumerate}", "\\item The functor $Rf_*$ sends $D_\\QCoh(\\mathcal{O}_X)$", "into $D_\\QCoh(\\mathcal{O}_S)$.", "\\item If $S$ is quasi-compact, there exists an integer $N = N(X, S, f)$", "such that for an object $E$ of $D_\\QCoh(\\mathcal{O}_X)$", "with $H^m(E) = 0$ for $m > 0$ we have", "$H^m(Rf_*E) = 0$ for $m \\geq N$.", "\\item In fact, if $S$ is quasi-compact we can find $N = N(X, S, f)$", "such that for every morphism of schemes $S' \\to S$", "the same conclusion holds for the functor $R(f')_*$", "where $f' : X' \\to S'$ is the base change of $f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. To prove (1) we have to", "show that $Rf_*E$ has quasi-coherent cohomology sheaves. The question is local", "on $S$, hence we may assume $S$ is quasi-compact. Pick $N = N(X, S, f)$ as in", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images}.", "Thus $R^pf_*\\mathcal{F} = 0$ for all quasi-coherent $\\mathcal{O}_X$-modules", "$\\mathcal{F}$ and all $p \\geq N$ and the same remains true after base change.", "\\medskip\\noindent", "First, assume $E$ is bounded below. We will show (1) and (2) and (3) hold", "for such $E$ with our choice of $N$. In this case we can for example use the", "spectral sequence", "$$", "R^pf_*H^q(E) \\Rightarrow R^{p + q}f_*E", "$$", "(Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}),", "the quasi-coherence of $R^pf_*H^q(E)$, and the vanishing of $R^pf_*H^q(E)$", "for $p \\geq N$ to see that (1), (2), and (3) hold in this case.", "\\medskip\\noindent", "Next we prove (2) and (3). Say $H^m(E) = 0$ for $m > 0$.", "Let $U \\subset S$ be affine open. By Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images-application}", "and our choice of $N$", "we have $H^p(f^{-1}(U), \\mathcal{F}) = 0$ for $p \\geq N$", "and any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$.", "Hence we may apply Lemma \\ref{lemma-application-nice-K-injective}", "to the functor $\\Gamma(f^{-1}(U), -)$ to see that", "$$", "R\\Gamma(U, Rf_*E) = R\\Gamma(f^{-1}(U), E)", "$$", "has vanishing cohomology in degrees $\\geq N$. Since this holds for", "all $U \\subset S$ affine open we conclude that $H^m(Rf_*E) = 0$", "for $m \\geq N$.", "\\medskip\\noindent", "Next, we prove (1) in the general case. Recall that there is a", "distinguished triangle", "$$", "\\tau_{\\leq -n - 1}E \\to E \\to \\tau_{\\geq -n}E \\to", "(\\tau_{\\leq -n - 1}E)[1]", "$$", "in $D(\\mathcal{O}_X)$, see Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}.", "By (2) we see that $Rf_*\\tau_{\\leq -n - 1}E$", "has vanishing cohomology sheaves in degrees $\\geq -n + N$.", "Thus, given an integer $q$ we see that $R^qf_*E$ is equal", "to $R^qf_*\\tau_{\\geq -n}E$ for some $n$ and the result", "above applies." ], "refs": [ "coherent-lemma-quasi-coherence-higher-direct-images", "derived-lemma-two-ss-complex-functor", "coherent-lemma-quasi-coherence-higher-direct-images-application", "perfect-lemma-application-nice-K-injective", "derived-remark-truncation-distinguished-triangle" ], "ref_ids": [ 3295, 1871, 3296, 6940, 2016 ] } ], "ref_ids": [] }, { "id": 6947, "type": "theorem", "label": "perfect-lemma-acyclicity-lemma", "categories": [ "perfect" ], "title": "perfect-lemma-acyclicity-lemma", "contents": [ "Let $f : X \\to S$ be a quasi-separated and quasi-compact morphism", "of schemes. Let $\\mathcal{F}^\\bullet$ be a complex of quasi-coherent", "$\\mathcal{O}_X$-modules each of which is right acyclic for $f_*$.", "Then $f_*\\mathcal{F}^\\bullet$ represents $Rf_*\\mathcal{F}^\\bullet$", "in $D(\\mathcal{O}_S)$." ], "refs": [], "proofs": [ { "contents": [ "There is always a canonical map", "$f_*\\mathcal{F}^\\bullet \\to Rf_*\\mathcal{F}^\\bullet$.", "Our task is to show that this is an isomorphism on cohomology sheaves.", "As the statement is invariant under shifts it suffices to show that", "$H^0(f_*(\\mathcal{F}^\\bullet)) \\to R^0f_*\\mathcal{F}^\\bullet$", "is an isomorphism. The statement is local on $S$ hence we", "may assume $S$ affine. By", "Lemma \\ref{lemma-quasi-coherence-direct-image}", "we have $R^0f_*\\mathcal{F}^\\bullet = R^0f_*\\tau_{\\geq -n}\\mathcal{F}^\\bullet$", "for all sufficiently large $n$. Thus we may assume $\\mathcal{F}^\\bullet$", "bounded below. As each $\\mathcal{F}^n$ is right $f_*$-acyclic by", "assumption we see that $f_*\\mathcal{F}^\\bullet \\to Rf_*\\mathcal{F}^\\bullet$", "is a quasi-isomorphism by Leray's acyclicity lemma (Derived Categories, Lemma", "\\ref{derived-lemma-leray-acyclicity})." ], "refs": [ "perfect-lemma-quasi-coherence-direct-image", "derived-lemma-leray-acyclicity" ], "ref_ids": [ 6946, 1844 ] } ], "ref_ids": [] }, { "id": 6948, "type": "theorem", "label": "perfect-lemma-acyclicity-lemma-global", "categories": [ "perfect" ], "title": "perfect-lemma-acyclicity-lemma-global", "contents": [ "Let $X$ be a quasi-separated and quasi-compact scheme.", "Let $\\mathcal{F}^\\bullet$ be a complex of quasi-coherent", "$\\mathcal{O}_X$-modules each of which is right acyclic for $\\Gamma(X, -)$.", "Then $\\Gamma(X, \\mathcal{F}^\\bullet)$ represents", "$R\\Gamma(X, \\mathcal{F}^\\bullet)$ in $D(\\Gamma(X, \\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-acyclicity-lemma} to the canonical morphism", "$X \\to \\Spec(\\Gamma(X, \\mathcal{O}_X))$. Some details omitted." ], "refs": [ "perfect-lemma-acyclicity-lemma" ], "ref_ids": [ 6947 ] } ], "ref_ids": [] }, { "id": 6949, "type": "theorem", "label": "perfect-lemma-spectral-sequence", "categories": [ "perfect" ], "title": "perfect-lemma-spectral-sequence", "contents": [ "Let $X$ be a quasi-separated and quasi-compact scheme. For any object", "$K$ of $D_\\QCoh(\\mathcal{O}_X)$ the spectral sequence", "$$", "E_2^{i, j} = H^i(X, H^j(K)) \\Rightarrow H^{i + j}(X, K)", "$$", "of Cohomology, Example \\ref{cohomology-example-spectral-sequence}", "is bounded and converges." ], "refs": [], "proofs": [ { "contents": [ "By the construction of the spectral sequence via", "Cohomology, Lemma \\ref{cohomology-lemma-spectral-sequence-filtered-object}", "using the filtration given by $\\tau_{\\leq -p}K$, we see that", "suffices to show that given $n \\in \\mathbf{Z}$ we have", "$$", "H^n(X, \\tau_{\\leq -p}K) = 0 \\text{ for } p \\gg 0", "$$", "and", "$$", "H^n(X, K) = H^n(X, \\tau_{\\leq -p}K) \\text{ for } p \\ll 0", "$$", "The first follows from Lemma \\ref{lemma-application-nice-K-injective}", "applied with $F = \\Gamma(X, -)$ and the bound in", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images}.", "The second holds whenever", "$-p \\leq n$ for any ringed space $(X, \\mathcal{O}_X)$ and any", "$K \\in D(\\mathcal{O}_X)$." ], "refs": [ "cohomology-lemma-spectral-sequence-filtered-object", "perfect-lemma-application-nice-K-injective", "coherent-lemma-quasi-coherence-higher-direct-images" ], "ref_ids": [ 2124, 6940, 3295 ] } ], "ref_ids": [] }, { "id": 6950, "type": "theorem", "label": "perfect-lemma-quasi-coherence-pushforward-direct-sums", "categories": [ "perfect" ], "title": "perfect-lemma-quasi-coherence-pushforward-direct-sums", "contents": [ "Let $f : X \\to S$ be a quasi-separated and quasi-compact morphism of", "schemes. Then", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_S)$", "commutes with direct sums." ], "refs": [], "proofs": [ { "contents": [ "Let $E_i$ be a family of objects of $D_\\QCoh(\\mathcal{O}_X)$", "and set $E = \\bigoplus E_i$. We want to show that the map", "$$", "\\bigoplus Rf_*E_i \\longrightarrow Rf_*E", "$$", "is an isomorphism. We will show it induces an isomorphism on", "cohomology sheaves in degree $0$ which will imply the lemma.", "Choose an integer $N$ as in Lemma \\ref{lemma-quasi-coherence-direct-image}.", "Then $R^0f_*E = R^0f_*\\tau_{\\geq -N}E$ and", "$R^0f_*E_i = R^0f_*\\tau_{\\geq -N}E_i$ by the lemma cited. Observe that", "$\\tau_{\\geq -N}E = \\bigoplus \\tau_{\\geq -N}E_i$.", "Thus we may assume all of the $E_i$ have vanishing cohomology", "sheaves in degrees $< -N$. Next we use the spectral sequences", "$$", "R^pf_*H^q(E) \\Rightarrow R^{p + q}f_*E", "\\quad\\text{and}\\quad", "R^pf_*H^q(E_i) \\Rightarrow R^{p + q}f_*E_i", "$$", "(Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor})", "to reduce to the case of a direct sum of quasi-coherent sheaves.", "This case is handled by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-colimit-cohomology}." ], "refs": [ "perfect-lemma-quasi-coherence-direct-image", "derived-lemma-two-ss-complex-functor", "coherent-lemma-colimit-cohomology" ], "ref_ids": [ 6946, 1871, 3300 ] } ], "ref_ids": [] }, { "id": 6951, "type": "theorem", "label": "perfect-lemma-pushforward-affine-morphism", "categories": [ "perfect" ], "title": "perfect-lemma-pushforward-affine-morphism", "contents": [ "Let $f : X \\to S$ be an affine morphism of schemes. Let $\\mathcal{F}^\\bullet$", "be a complex of quasi-coherent $\\mathcal{O}_X$-modules. Then", "$f_*\\mathcal{F}^\\bullet = Rf_*\\mathcal{F}^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-acyclicity-lemma} with", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-vanishing}.", "An alternative proof is to work affine locally on $S$", "and use Lemma \\ref{lemma-quasi-coherence-pushforward}." ], "refs": [ "perfect-lemma-acyclicity-lemma", "coherent-lemma-relative-affine-vanishing", "perfect-lemma-quasi-coherence-pushforward" ], "ref_ids": [ 6947, 3283, 6943 ] } ], "ref_ids": [] }, { "id": 6952, "type": "theorem", "label": "perfect-lemma-affine-morphism", "categories": [ "perfect" ], "title": "perfect-lemma-affine-morphism", "contents": [ "Let $f : X \\to S$ be an affine morphism of schemes.", "Then", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_S)$", "reflects isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "The statement means that a morphism $\\alpha : E \\to F$ of", "$D_\\QCoh(\\mathcal{O}_X)$ is an isomorphism if", "$Rf_*\\alpha$ is an isomorphism. We may check this on cohomology sheaves.", "In particular, the question is local on $S$. Hence we may assume $S$", "and therefore $X$ is affine. In this case the statement is clear from", "the description of the derived categories", "$D_\\QCoh(\\mathcal{O}_X)$ and", "$D_\\QCoh(\\mathcal{O}_S)$ given in", "Lemma \\ref{lemma-affine-compare-bounded}.", "Some details omitted." ], "refs": [ "perfect-lemma-affine-compare-bounded" ], "ref_ids": [ 6941 ] } ], "ref_ids": [] }, { "id": 6953, "type": "theorem", "label": "perfect-lemma-affine-morphism-pull-push", "categories": [ "perfect" ], "title": "perfect-lemma-affine-morphism-pull-push", "contents": [ "Let $f : X \\to S$ be an affine morphism of schemes.", "For $E$ in $D_\\QCoh(\\mathcal{O}_S)$ we have", "$Rf_* Lf^* E = E \\otimes^\\mathbf{L}_{\\mathcal{O}_S} f_*\\mathcal{O}_X$." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is affine the map $f_*\\mathcal{O}_X \\to Rf_*\\mathcal{O}_X$", "is an isomorphism", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-vanishing}).", "There is a canonical map $E \\otimes^\\mathbf{L} f_*\\mathcal{O}_X =", "E \\otimes^\\mathbf{L} Rf_*\\mathcal{O}_X \\to Rf_* Lf^* E$", "adjoint to the map", "$$", "Lf^*(E \\otimes^\\mathbf{L} Rf_*\\mathcal{O}_X) =", "Lf^*E \\otimes^\\mathbf{L} Lf^*Rf_*\\mathcal{O}_X \\longrightarrow", "Lf^* E \\otimes^\\mathbf{L} \\mathcal{O}_X = Lf^* E", "$$", "coming from $1 : Lf^*E \\to Lf^*E$ and the canonical map", "$Lf^*Rf_*\\mathcal{O}_X \\to \\mathcal{O}_X$. To check the map so constructed", "is an isomorphism we may work locally on $S$. Hence we may assume", "$S$ and therefore $X$ is affine. In this case the statement is clear from", "the description of the derived categories", "$D_\\QCoh(\\mathcal{O}_X)$ and", "$D_\\QCoh(\\mathcal{O}_S)$ and the functor $Lf^*$ given in", "Lemmas \\ref{lemma-affine-compare-bounded} and", "\\ref{lemma-quasi-coherence-pullback}.", "Some details omitted." ], "refs": [ "coherent-lemma-relative-affine-vanishing", "perfect-lemma-affine-compare-bounded", "perfect-lemma-quasi-coherence-pullback" ], "ref_ids": [ 3283, 6941, 6944 ] } ], "ref_ids": [] }, { "id": 6954, "type": "theorem", "label": "perfect-lemma-affine-morphism-equivalence", "categories": [ "perfect" ], "title": "perfect-lemma-affine-morphism-equivalence", "contents": [ "Let $f : X \\to Y$ be an affine morphism of schemes. Then $f_*$ induces", "an equivalence", "$$", "\\Phi : D_\\QCoh(\\mathcal{O}_X) \\longrightarrow D_\\QCoh(f_*\\mathcal{O}_X)", "$$", "whose composition with $D_\\QCoh(f_*\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$", "is $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $Rf_*$ is computed on an object $K \\in D_\\QCoh(\\mathcal{O}_X)$", "by choosing a K-injective complex $\\mathcal{I}^\\bullet$ of", "$\\mathcal{O}_X$-modules representing $K$ and taking $f_*\\mathcal{I}^\\bullet$.", "Thus we let $\\Phi(K)$ be the complex $f_*\\mathcal{I}^\\bullet$", "viewed as a complex of $f_*\\mathcal{O}_X$-modules.", "Denote $g : (X, \\mathcal{O}_X) \\to (Y, f_*\\mathcal{O}_X)$ the", "obvious morphism of ringed spaces. Then $g$ is a flat morphism of", "ringed spaces (see below for a description of the stalks) and", "$\\Phi$ is the restriction of $Rg_*$ to $D_\\QCoh(\\mathcal{O}_X)$.", "We claim that $Lg^*$ is a quasi-inverse. First, observe that", "$Lg^*$ sends $D_\\QCoh(f_*\\mathcal{O}_X)$ into $D_\\QCoh(\\mathcal{O}_X)$", "because $g^*$ transforms quasi-coherent modules into quasi-coherent", "modules (Modules, Lemma \\ref{modules-lemma-pullback-quasi-coherent}).", "To finish the proof it suffices to show that", "the adjunction mappings", "$$", "Lg^*\\Phi(K) = Lg^*Rg_*K \\to K", "\\quad\\text{and}\\quad", "M \\to Rg_*Lg^*M = \\Phi(Lg^*M)", "$$", "are isomorphisms for $K \\in D_\\QCoh(\\mathcal{O}_X)$ and", "$M \\in D_\\QCoh(f_*\\mathcal{O}_X)$. This is a local question, hence", "we may assume $Y$ and therefore $X$ are affine.", "\\medskip\\noindent", "Assume $Y = \\Spec(B)$ and $X = \\Spec(A)$. Let", "$\\mathfrak p = x \\in \\Spec(A) = X$ be a point mapping to", "$\\mathfrak q = y \\in \\Spec(B) = Y$. Then", "$(f_*\\mathcal{O}_X)_y = A_\\mathfrak q$ and $\\mathcal{O}_{X, x} = A_\\mathfrak p$", "hence $g$ is flat. Hence $g^*$ is exact and $H^i(Lg^*M) = g^*H^i(M)$", "for any $M$ in $D(f_*\\mathcal{O}_X)$.", "For $K \\in D_\\QCoh(\\mathcal{O}_X)$ we see that", "$$", "H^i(\\Phi(K)) = H^i(Rf_*K) = f_*H^i(K)", "$$", "by the vanishing of higher direct images", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-vanishing})", "and Lemma \\ref{lemma-application-nice-K-injective} (small detail omitted).", "Thus it suffice to show that", "$$", "g^*g_*\\mathcal{F} \\to \\mathcal{F}", "\\quad\\text{and}\\quad", "\\mathcal{G} \\to g_*g^*\\mathcal{F}", "$$", "are isomorphisms where $\\mathcal{F}$ is", "a quasi-coherent $\\mathcal{O}_X$-module and $\\mathcal{G}$ is", "a quasi-coherent $f_*\\mathcal{O}_X$-module. This follows from", "Morphisms, Lemma \\ref{morphisms-lemma-affine-equivalence-modules}." ], "refs": [ "modules-lemma-pullback-quasi-coherent", "coherent-lemma-relative-affine-vanishing", "perfect-lemma-application-nice-K-injective", "morphisms-lemma-affine-equivalence-modules" ], "ref_ids": [ 13244, 3283, 6940, 5174 ] } ], "ref_ids": [] }, { "id": 6955, "type": "theorem", "label": "perfect-lemma-support-quasi-coherent", "categories": [ "perfect" ], "title": "perfect-lemma-support-quasi-coherent", "contents": [ "Let $X$ be a scheme. Let $T \\subset X$ be a closed subset such that", "$X \\setminus T$ is a retrocompact open of $X$. Let $i : T \\to X$ be", "the inclusion.", "\\begin{enumerate}", "\\item For $E$ in $D_\\QCoh(\\mathcal{O}_X)$ we have", "$i_*R\\mathcal{H}_T(E)$ in $D_{\\QCoh, T}(\\mathcal{O}_X)$.", "\\item The functor", "$i_* \\circ R\\mathcal{H}_T : D_\\QCoh(\\mathcal{O}_X) \\to", "D_{\\QCoh, T}(\\mathcal{O}_X)$ is right adjoint to the inclusion functor", "$D_{\\QCoh, T}(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Set $U = X \\setminus T$ and denote $j : U \\to X$ the inclusion. By", "Cohomology, Lemma \\ref{cohomology-lemma-triangle-sections-with-support-sheaves}", "there is a distinguished triangle", "$$", "i_*R\\mathcal{H}_T(E) \\to E \\to Rj_*(E|_U) \\to i_*R\\mathcal{H}_Z(E)[1]", "$$", "in $D(\\mathcal{O}_X)$. By Lemma \\ref{lemma-quasi-coherence-direct-image}", "the complex $Rj_*(E|_U)$ has quasi-coherent cohomology sheaves", "(this is where we use that $U$ is retrocompact in $X$).", "Thus we see that (1) is true. Part (2) follows from this and", "the adjointness of functors in", "Cohomology, Lemma \\ref{cohomology-lemma-complexes-with-support-on-closed}." ], "refs": [ "cohomology-lemma-triangle-sections-with-support-sheaves", "perfect-lemma-quasi-coherence-direct-image", "cohomology-lemma-complexes-with-support-on-closed" ], "ref_ids": [ 2155, 6946, 2151 ] } ], "ref_ids": [] }, { "id": 6956, "type": "theorem", "label": "perfect-lemma-support-direct-sums", "categories": [ "perfect" ], "title": "perfect-lemma-support-direct-sums", "contents": [ "Let $X$ be a scheme. Let $T \\subset X$ be a closed subset such that", "$X \\setminus T$ is a retrocompact open of $X$. Then for a family of", "objects $E_i$, $i \\in I$ of $D_\\QCoh(\\mathcal{O}_X)$ we have", "$R\\mathcal{H}_T(\\bigoplus E_i) = \\bigoplus R\\mathcal{H}_T(E_i)$." ], "refs": [], "proofs": [ { "contents": [ "Set $U = X \\setminus T$ and denote $j : U \\to X$ the inclusion. By", "Cohomology, Lemma \\ref{cohomology-lemma-triangle-sections-with-support-sheaves}", "there is a distinguished triangle", "$$", "i_*R\\mathcal{H}_T(E) \\to E \\to Rj_*(E|_U) \\to i_*R\\mathcal{H}_Z(E)[1]", "$$", "in $D(\\mathcal{O}_X)$ for any $E$ in $D(\\mathcal{O}_X)$. The functor", "$E \\mapsto Rj_*(E|_U)$ commutes with direct sums on $D_\\QCoh(\\mathcal{O}_X)$", "by Lemma \\ref{lemma-quasi-coherence-pushforward-direct-sums}.", "It follows that the same is true for the functor $i_* \\circ R\\mathcal{H}_T$", "(details omitted). Since $i_* : D(i^{-1}\\mathcal{O}_X) \\to D_T(\\mathcal{O}_X)$", "is an equivalence", "(Cohomology, Lemma \\ref{cohomology-lemma-complexes-with-support-on-closed})", "we conclude." ], "refs": [ "cohomology-lemma-triangle-sections-with-support-sheaves", "perfect-lemma-quasi-coherence-pushforward-direct-sums", "cohomology-lemma-complexes-with-support-on-closed" ], "ref_ids": [ 2155, 6950, 2151 ] } ], "ref_ids": [] }, { "id": 6957, "type": "theorem", "label": "perfect-lemma-extended-alternating-zero", "categories": [ "perfect" ], "title": "perfect-lemma-extended-alternating-zero", "contents": [ "With $X$, $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$, and", "$\\mathcal{F}$ as in Remark \\ref{remark-support-c-equations}", "the complex (\\ref{equation-extended-alternating}) restricts to an acyclic", "complex over $X \\setminus Z$." ], "refs": [ "perfect-remark-support-c-equations" ], "proofs": [ { "contents": [ "Let $W \\subset X \\setminus Z$ be an open subset. Evaluating the complex", "of sheaves (\\ref{equation-extended-alternating}) on $W$ we obtain the", "complex", "$$", "\\mathcal{F}(W) \\to \\bigoplus\\nolimits_{i_0} \\mathcal{F}(U_{i_0} \\cap W) \\to", "\\bigoplus\\nolimits_{i_0 < i_1} \\mathcal{F}(U_{i_0i_1} \\cap W) \\to \\ldots", "$$", "In other words, we obtain the extended ordered {\\v C}ech complex", "for the covering $W = \\bigcup U_i \\cap W$ and the standard", "ordering on $\\{1, \\ldots, c\\}$, see", "Cohomology, Section \\ref{cohomology-section-alternating-cech}.", "By Cohomology, Lemma \\ref{cohomology-lemma-alternating-cech-trivial}", "this complex is homotopic to zero as soon as $W$ is contained in", "$V(f_i)$ for some $1 \\leq i \\leq c$. This finishes the proof." ], "refs": [ "cohomology-lemma-alternating-cech-trivial" ], "ref_ids": [ 2096 ] } ], "ref_ids": [ 7122 ] }, { "id": 6958, "type": "theorem", "label": "perfect-lemma-extended-alternating-represented", "categories": [ "perfect" ], "title": "perfect-lemma-extended-alternating-represented", "contents": [ "With $X$, $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$, and", "$\\mathcal{F}$ as in Remark \\ref{remark-support-c-equations}.", "If $\\mathcal{F}$ is quasi-coherent, then the complex", "(\\ref{equation-extended-alternating}) represents", "$i_* R\\mathcal{H}_Z(\\mathcal{F})$ in $D_Z(\\mathcal{O}_X)$." ], "refs": [ "perfect-remark-support-c-equations" ], "proofs": [ { "contents": [ "Let us denote $\\mathcal{F}^\\bullet$ the complex", "(\\ref{equation-extended-alternating}).", "The statement of the lemma means that the map", "$\\mathcal{F}^\\bullet \\to i_*R\\mathcal{H}_Z(\\mathcal{F})$", "of Remark \\ref{remark-extended-alternating-map-to-support}", "is an isomorphism. Since $\\mathcal{F}^\\bullet$ is in", "$D_Z(\\mathcal{O}_X)$ (see remark cited), we see that", "$i_*R\\mathcal{H}_Z(\\mathcal{F}^\\bullet) = \\mathcal{F}^\\bullet$", "by Cohomology, Lemma \\ref{cohomology-lemma-complexes-with-support-on-closed}.", "The morphism $U_{i_0 \\ldots i_p} \\to X$ is affine", "as it is given over affine opens of $X$ by inverting the function", "$f_{i_0} \\ldots f_{i_p}$. Thus we see that", "$$", "\\mathcal{F}_{i_0 \\ldots i_p} =", "(U_{i_0 \\ldots i_p} \\to X)_*\\mathcal{F}|_{U_{i_0 \\ldots i_p}} =", "R(U_{i_0 \\ldots i_p} \\to X)_*\\mathcal{F}|_{U_{i_0 \\ldots i_p}}", "$$", "by Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-vanishing}", "and the assumption that $\\mathcal{F}$ is quasi-coherent. We conclude that", "$R\\mathcal{H}_Z(\\mathcal{F}_{i_0 \\ldots i_p}) = 0$ by Cohomology, Lemma", "\\ref{cohomology-lemma-sections-support-in-closed-disjoint-open}.", "Thus $i_*R\\mathcal{H}_Z(\\mathcal{F}^p) = 0$ for $p > 0$.", "Putting everything together we obtain", "$$", "\\mathcal{F}^\\bullet = i_*R\\mathcal{H}_Z(\\mathcal{F}^\\bullet) =", "i_*R\\mathcal{H}_Z(\\mathcal{F})", "$$", "as desired." ], "refs": [ "perfect-remark-extended-alternating-map-to-support", "cohomology-lemma-complexes-with-support-on-closed", "coherent-lemma-relative-affine-vanishing", "cohomology-lemma-sections-support-in-closed-disjoint-open" ], "ref_ids": [ 7123, 2151, 3283, 2156 ] } ], "ref_ids": [ 7122 ] }, { "id": 6959, "type": "theorem", "label": "perfect-lemma-supported-trivial-vanishing", "categories": [ "perfect" ], "title": "perfect-lemma-supported-trivial-vanishing", "contents": [ "Let $X$ be a scheme. Let $T \\subset X$ be a closed subset which can", "locally be cut out by at most $c$ elements of the structure sheaf.", "Then $\\mathcal{H}^i_Z(\\mathcal{F}) = 0$ for $i > c$ and any", "quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the local description of", "$R\\mathcal{H}_T(\\mathcal{F})$ given in", "Lemma \\ref{lemma-extended-alternating-represented}." ], "refs": [ "perfect-lemma-extended-alternating-represented" ], "ref_ids": [ 6958 ] } ], "ref_ids": [] }, { "id": 6960, "type": "theorem", "label": "perfect-lemma-supported-vanishing", "categories": [ "perfect" ], "title": "perfect-lemma-supported-vanishing", "contents": [ "Let $X$ be a scheme. Let $T \\subset X$ be a closed subset which can", "locally be cut out by a Koszul regular sequence having $c$ elements.", "Then $\\mathcal{H}^i_Z(\\mathcal{F}) = 0$ for $i \\not = c$ for every", "flat, quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "By the description of $R\\mathcal{H}_Z(\\mathcal{F})$ given in", "Lemma \\ref{lemma-extended-alternating-represented} this boils", "down to the following algebra statement: given a ring $R$,", "a Koszul regular sequence $f_1, \\ldots, f_c \\in R$, and a flat", "$R$-module $M$, the extended alternating {\\v C}ech complex", "$M \\to \\bigoplus\\nolimits_{i_0} M_{f_{i_0}} \\to", "\\bigoplus\\nolimits_{i_0 < i_1} M_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to M_{f_1 \\ldots f_c}$", "from More on Algebra, Section \\ref{more-algebra-section-alternating-cech}", "only has cohomology in degree $c$. By More on Algebra, Lemma", "\\ref{more-algebra-lemma-vanishing-extended-alternating-koszul}", "we obtain the desired vanishing for the extended alternating", "{\\v C}ech complex of $R$. Since the complex for $M$ is obtained", "by tensoring this with the flat $R$-module $M$", "(More on Algebra, Lemma", "\\ref{more-algebra-lemma-extended-alternating-form-module})", "we conclude." ], "refs": [ "perfect-lemma-extended-alternating-represented", "more-algebra-lemma-vanishing-extended-alternating-koszul", "more-algebra-lemma-extended-alternating-form-module" ], "ref_ids": [ 6958, 9989, 9968 ] } ], "ref_ids": [] }, { "id": 6961, "type": "theorem", "label": "perfect-lemma-supported-map-determinant", "categories": [ "perfect" ], "title": "perfect-lemma-supported-map-determinant", "contents": [ "With $X$, $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$, and", "$\\mathcal{F}$ as in Remark \\ref{remark-support-c-equations}.", "Let $a_{ji} \\in \\Gamma(X, \\mathcal{O}_X)$ for $1 \\leq i, j \\leq c$", "and set $g_j = \\sum_{i = 1, \\ldots, c} a_{ji}f_i$. Assume $g_1, \\ldots, g_c$", "scheme theoretically cut out $Z$. If $\\mathcal{F}$ is quasi-coherent, then", "$$", "c_{f_1, \\ldots, f_c} = \\det(a_{ji}) c_{g_1, \\ldots, g_c}", "$$", "where $c_{f_1, \\ldots, f_c}$ and $c_{g_1, \\ldots, g_c}$ are as in", "Remark \\ref{remark-supported-map-c-equations}." ], "refs": [ "perfect-remark-support-c-equations", "perfect-remark-supported-map-c-equations" ], "proofs": [ { "contents": [ "We will prove that $c_{f_1, \\ldots, f_c}(s) =", "\\det(a_{ij}) c_{g_1, \\ldots, g_c}(s)$ as global sections of", "$\\mathcal{H}_Z(\\mathcal{F})$ for any $s \\in \\mathcal{F}(X)$.", "This is sufficient since we then obtain the same result for", "section over any open subscheme of $X$.", "To do this, for $1 \\leq i_0 < \\ldots < i_p \\leq c$", "and $1 \\leq j_0 < \\ldots < j_q \\leq c$ we denote", "$U_{i_0 \\ldots i_p} \\subset X$,", "$V_{j_0 \\ldots j_q} \\subset X$, and", "$W_{i_0 \\ldots i_p, j_0 \\ldots j_q} \\subset X$", "the open subscheme where $f_{i_0} \\ldots f_{i_p}$ is invertible,", "$g_{j_0} \\ldots g_{j_q}$ is invertible, and where", "$f_{i_0} \\ldots f_{i_p}g_{j_0} \\ldots g_{j_q}$ is invertible.", "We denote $\\mathcal{F}_{i_0 \\ldots i_p}$,", "resp.\\ $\\mathcal{F}'_{j_0 \\ldots j_q}$", "$\\mathcal{F}''_{i_0 \\ldots i_p, j_0 \\ldots j_q}$", "the pushforward to $X$ of the restriction of $\\mathcal{F}$ to", "$U_{i_0 \\ldots i_p}$, resp.\\ $V_{j_0 \\ldots j_q}$,", "resp.\\ $W_{i_0 \\ldots i_p, j_0 \\ldots j_q}$.", "Then we obtain three extended alternating {\\v C}ech complexes", "$$", "\\mathcal{F}^\\bullet :", "\\mathcal{F} \\to \\bigoplus\\nolimits_{i_0} \\mathcal{F}_{i_0} \\to", "\\bigoplus\\nolimits_{i_0 < i_1} \\mathcal{F}_{i_0i_1} \\to \\ldots", "$$", "and", "$$", "(\\mathcal{F}')^\\bullet :", "\\mathcal{F} \\to \\bigoplus\\nolimits_{j_0} \\mathcal{F}'_{j_0} \\to", "\\bigoplus\\nolimits_{j_0 < j_1} \\mathcal{F}'_{j_0j_1} \\to \\ldots", "$$", "and", "$$", "(\\mathcal{F}'')^\\bullet :", "\\mathcal{F} \\to", "\\bigoplus\\nolimits_{i_0} \\mathcal{F}_{i_0} \\oplus", "\\bigoplus\\nolimits_{j_0} \\mathcal{F}'_{j_0} \\to", "\\bigoplus\\nolimits_{i_0 < i_1} \\mathcal{F}_{i_0i_1} \\oplus", "\\bigoplus\\nolimits_{i_0, j_0} \\mathcal{F}''_{i_0, j_0} \\oplus", "\\bigoplus\\nolimits_{j_0 < j_1} \\mathcal{F}'_{j_0j_1} \\to", "\\ldots", "$$", "whose differentials are those used in defining", "(\\ref{equation-extended-alternating}).", "There are maps of complexes", "$$", "(\\mathcal{F}'')^\\bullet \\to \\mathcal{F}^\\bullet", "\\quad\\text{and}\\quad", "(\\mathcal{F}'')^\\bullet \\to (\\mathcal{F}')^\\bullet", "$$", "given by the projection maps on the terms (and hence inducing", "the identity map in degree $0$). Observe that by", "Lemma \\ref{lemma-extended-alternating-represented}", "each of these complexes represents", "$i_*R\\mathcal{H}_Z(\\mathcal{F})$ and these maps", "represent the identity on this object. Thus it suffices", "to find an element", "$$", "\\sigma \\in H^c((\\mathcal{F}'')^\\bullet(X))", "$$", "mapping to $c_{f_1, \\ldots, f_c}(s)$ and $\\det(a_{ji})c_{g_1, \\ldots, g_c}(s)$", "by these two maps. It turns out we can explicitly give a cocycle", "for $\\sigma$. Namely, we take", "$$", "\\sigma_{1 \\ldots c} = \\frac{s}{f_1 \\ldots f_c} \\in \\mathcal{F}_{1 \\ldots c}(X)", "\\quad\\text{and}\\quad", "\\sigma'_{1 \\ldots c} = \\frac{\\det(a_{ji})s}{g_1 \\ldots g_c} \\in", "\\mathcal{F}'_{1 \\ldots c}(X)", "$$", "and we take", "$$", "\\sigma_{i_0 \\ldots i_p, j_0 \\ldots j_{c - p - 2}} =", "\\frac{\\lambda(i_0 \\ldots i_p, j_0 \\ldots j_{c - p - 2})s}%", "{f_{i_0} \\ldots f_{i_p}g_{j_0} \\ldots g_{j_{c - p - 2}}}", "\\in \\mathcal{F}''_{i_0 \\ldots i_p, j_0 \\ldots j_{c - p - 2}}(X)", "$$", "where $\\lambda(i_0 \\ldots i_p, j_0 \\ldots j_{c - p - 2})$", "is the coefficient of $e_1 \\wedge \\ldots \\wedge e_c$ in", "the formal expresssion", "$$", "e_{i_0} \\wedge \\ldots \\wedge e_{i_p} \\wedge", "(a_{j_01} e_1 + \\ldots + a_{j_0c}e_c) \\wedge \\ldots \\wedge", "(a_{j_{c - p - 2}1} e_1 + \\ldots + a_{j_{c - p - 2}c}e_c)", "$$", "To verify that $\\sigma$ is a cocycle, we have to show for", "$1 \\leq i_0 < \\ldots < i_p \\leq c$ and", "$1 \\leq j_0 < \\ldots < j_{c - p - 1} \\leq c$", "that we have", "\\begin{align*}", "0 & =", "\\sum\\nolimits_{a = 0, \\ldots, p} (-1)^a", "f_{i_a} \\lambda(i_0 \\ldots \\hat i_a \\ldots i_p, j_0 \\ldots j_{c - p - 1}) \\\\", "& +", "\\sum\\nolimits_{b = 0, \\ldots, c - p - 1} (-1)^{p + b + 1}g_{j_b}", "\\lambda(i_0 \\ldots i_p, j_0 \\ldots \\hat j_b \\ldots j_{c - p - 1})", "\\end{align*}", "The easiest way to see this is perhaps to argue that the formal expression", "$$", "\\xi = e_{i_0} \\wedge \\ldots \\wedge e_{i_p} \\wedge", "(a_{j_01} e_1 + \\ldots + a_{j_0c}e_c) \\wedge \\ldots \\wedge", "(a_{j_{c - p - 1}1} e_1 + \\ldots + a_{j_{c - p - 1}c}e_c)", "$$", "is $0$ as it is an element of the $(c + 1)$st wedge power of the free module", "on $e_1, \\ldots, e_c$ and that the expression above is the image of", "$\\xi$ under the Koszul differential sending $e_i \\to f_i$. Some details", "omitted." ], "refs": [ "perfect-lemma-extended-alternating-represented" ], "ref_ids": [ 6958 ] } ], "ref_ids": [ 7122, 7124 ] }, { "id": 6962, "type": "theorem", "label": "perfect-lemma-supported-map-global", "categories": [ "perfect" ], "title": "perfect-lemma-supported-map-global", "contents": [ "Let $X$ be a scheme. Let $Z \\to X$ be a closed immersion of finite presentation", "whose conormal sheaf $\\mathcal{C}_{Z/X}$ is locally free of rank $c$.", "Then there is a canonical map", "$$", "c :", "\\wedge^c(\\mathcal{C}_{Z/X})^\\vee \\otimes_{\\mathcal{O}_Z} i^*\\mathcal{F}", "\\longrightarrow", "\\mathcal{H}_Z^c(\\mathcal{F})", "$$", "functorial in the quasi-coherent module $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Follows from the construction in", "Remark \\ref{remark-supported-map-c-equations}", "and the independence of the choice", "of generators of the ideal sheaf shown in", "Lemma \\ref{lemma-supported-map-determinant}.", "Some details omitted." ], "refs": [ "perfect-remark-supported-map-c-equations", "perfect-lemma-supported-map-determinant" ], "ref_ids": [ 7124, 6961 ] } ], "ref_ids": [] }, { "id": 6963, "type": "theorem", "label": "perfect-lemma-affine-pushforward", "categories": [ "perfect" ], "title": "perfect-lemma-affine-pushforward", "contents": [ "Let $f : X \\to Y$ be an affine morphism of schemes.", "Then $f_*$ defines a derived functor", "$f_* : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$.", "This functor has the property that", "$$", "\\xymatrix{", "D(\\QCoh(\\mathcal{O}_X)) \\ar[d]_{f_*} \\ar[r] &", "D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\", "D(\\QCoh(\\mathcal{O}_Y)) \\ar[r] &", "D_\\QCoh(\\mathcal{O}_Y)", "}", "$$", "commutes." ], "refs": [], "proofs": [ { "contents": [ "The functor", "$f_* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$", "is exact, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-vanishing}.", "Hence $f_*$ defines a derived functor", "$f_* : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$", "by simply applying $f_*$ to any representative complex, see", "Derived Categories, Lemma \\ref{derived-lemma-right-derived-exact-functor}.", "The diagram commutes by Lemma \\ref{lemma-pushforward-affine-morphism}." ], "refs": [ "coherent-lemma-relative-affine-vanishing", "derived-lemma-right-derived-exact-functor", "perfect-lemma-pushforward-affine-morphism" ], "ref_ids": [ 3283, 1845, 6951 ] } ], "ref_ids": [] }, { "id": 6964, "type": "theorem", "label": "perfect-lemma-flat-pushforward-coherator", "categories": [ "perfect" ], "title": "perfect-lemma-flat-pushforward-coherator", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume $f$ is", "quasi-compact, quasi-separated, and flat. Then, denoting", "$$", "\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))", "$$", "the right derived functor of", "$f_* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$", "we have $RQ_Y \\circ Rf_* = \\Phi \\circ RQ_X$." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by showing that $RQ_Y \\circ Rf_*$ and $\\Phi \\circ RQ_X$", "are right adjoint to the same functor", "$D(\\QCoh(\\mathcal{O}_Y)) \\to D(\\mathcal{O}_X)$.", "\\medskip\\noindent", "Since $f$ is quasi-compact and quasi-separated, we see that", "$f_*$ preserves quasi-coherence, see", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}.", "Recall that $\\QCoh(\\mathcal{O}_X)$ is a Grothendieck abelian category", "(Properties, Proposition \\ref{properties-proposition-coherator}).", "Hence any $K$ in $D(\\QCoh(\\mathcal{O}_X))$", "can be represented by a K-injective complex $\\mathcal{I}^\\bullet$", "of $\\QCoh(\\mathcal{O}_X)$, see", "Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "Then we can define $\\Phi(K) = f_*\\mathcal{I}^\\bullet$.", "\\medskip\\noindent", "Since $f$ is flat, the functor $f^*$ is exact. Hence $f^*$ defines", "$f^* : D(\\mathcal{O}_Y) \\to D(\\mathcal{O}_X)$ and also", "$f^* : D(\\QCoh(\\mathcal{O}_Y)) \\to D(\\QCoh(\\mathcal{O}_X))$.", "The functor $f^* = Lf^* : D(\\mathcal{O}_Y) \\to D(\\mathcal{O}_X)$", "is left adjoint to", "$Rf_* : D(\\mathcal{O}_X) \\to D(\\mathcal{O}_Y)$,", "see Cohomology, Lemma \\ref{cohomology-lemma-adjoint}.", "Similarly, the functor", "$f^* : D(\\QCoh(\\mathcal{O}_Y)) \\to D(\\QCoh(\\mathcal{O}_X))$", "is left adjoint to", "$\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$", "by Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors}.", "\\medskip\\noindent", "Let $A$ be an object of $D(\\QCoh(\\mathcal{O}_Y))$ and", "$E$ an object of $D(\\mathcal{O}_X)$. Then", "\\begin{align*}", "\\Hom_{D(\\QCoh(\\mathcal{O}_Y))}(A, RQ_Y(Rf_*E))", "& =", "\\Hom_{D(\\mathcal{O}_Y)}(A, Rf_*E) \\\\", "& =", "\\Hom_{D(\\mathcal{O}_X)}(f^*A, E) \\\\", "& =", "\\Hom_{D(\\QCoh(\\mathcal{O}_X))}(f^*A, RQ_X(E)) \\\\", "& =", "\\Hom_{D(\\QCoh(\\mathcal{O}_Y))}(A, \\Phi(RQ_X(E)))", "\\end{align*}", "This implies what we want." ], "refs": [ "schemes-lemma-push-forward-quasi-coherent", "properties-proposition-coherator", "injectives-theorem-K-injective-embedding-grothendieck", "cohomology-lemma-adjoint", "derived-lemma-derived-adjoint-functors" ], "ref_ids": [ 7730, 3066, 7768, 2121, 1907 ] } ], "ref_ids": [] }, { "id": 6965, "type": "theorem", "label": "perfect-lemma-affine-coherator", "categories": [ "perfect" ], "title": "perfect-lemma-affine-coherator", "contents": [ "Let $X = \\Spec(A)$ be an affine scheme. Then", "\\begin{enumerate}", "\\item $Q_X : \\textit{Mod}(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_X)$", "is the functor", "which sends $\\mathcal{F}$ to the quasi-coherent $\\mathcal{O}_X$-module", "associated to the $A$-module $\\Gamma(X, \\mathcal{F})$,", "\\item $RQ_X : D(\\mathcal{O}_X) \\to D(\\QCoh(\\mathcal{O}_X))$", "is the functor which sends $E$ to the complex of quasi-coherent", "$\\mathcal{O}_X$-modules associated to the object $R\\Gamma(X, E)$ of $D(A)$,", "\\item restricted to $D_\\QCoh(\\mathcal{O}_X)$ the functor", "$RQ_X$ defines a quasi-inverse to (\\ref{equation-compare}).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The functor $Q_X$ is the functor", "$$", "\\mathcal{F} \\mapsto \\widetilde{\\Gamma(X, \\mathcal{F})}", "$$", "by Schemes, Lemma \\ref{schemes-lemma-compare-constructions}.", "This immediately implies (1) and (2). The third assertion", "follows from (the proof of)", "Lemma \\ref{lemma-affine-compare-bounded}." ], "refs": [ "schemes-lemma-compare-constructions", "perfect-lemma-affine-compare-bounded" ], "ref_ids": [ 7660, 6941 ] } ], "ref_ids": [] }, { "id": 6966, "type": "theorem", "label": "perfect-lemma-argument-proves", "categories": [ "perfect" ], "title": "perfect-lemma-argument-proves", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Suppose that", "for every affine open $U \\subset X$ the right derived functor", "$$", "\\Phi : D(\\QCoh(\\mathcal{O}_U)) \\to D(\\QCoh(\\mathcal{O}_X))", "$$", "of the left exact functor", "$j_* : \\QCoh(\\mathcal{O}_U) \\to \\QCoh(\\mathcal{O}_X)$", "fits into a commutative diagram", "$$", "\\xymatrix{", "D(\\QCoh(\\mathcal{O}_U)) \\ar[d]_\\Phi \\ar[r]_{i_U} &", "D_\\QCoh(\\mathcal{O}_U) \\ar[d]^{Rj_*} \\\\", "D(\\QCoh(\\mathcal{O}_X)) \\ar[r]^{i_X} &", "D_\\QCoh(\\mathcal{O}_X)", "}", "$$", "Then the functor (\\ref{equation-compare})", "$$", "D(\\QCoh(\\mathcal{O}_X))", "\\longrightarrow", "D_\\QCoh(\\mathcal{O}_X)", "$$", "is an equivalence with quasi-inverse given by $RQ_X$." ], "refs": [], "proofs": [ { "contents": [ "Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$ and", "let $A$ be an object of $D(\\QCoh(\\mathcal{O}_X))$.", "We have to show that the adjunction maps", "$$", "RQ_X(i_X(A)) \\to A", "\\quad\\text{and}\\quad", "E \\to i_X(RQ_X(E))", "$$", "are isomorphisms. Consider the hypothesis", "$H_n$: the adjunction maps above are isomorphisms", "whenever $E$ and $i_X(A)$ are supported", "(Definition \\ref{definition-supported-on})", "on a closed subset of $X$ which", "is contained in the union of $n$ affine opens of $X$.", "We will prove $H_n$ by induction on $n$.", "\\medskip\\noindent", "Base case: $n = 0$. In this case $E = 0$, hence the map", "$E \\to i_X(RQ_X(E))$ is an isomorphism. Similarly $i_X(A) = 0$.", "Thus the cohomology sheaves of $i_X(A)$ are zero. Since the inclusion", "functor $\\QCoh(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X)$", "is fully faithful and exact, we conclude that the cohomology", "objects of $A$ are zero, i.e., $A = 0$ and", "$RQ_X(i_X(A)) \\to A$ is an isomorphism as well.", "\\medskip\\noindent", "Induction step. Suppose that $E$ and $i_X(A)$ are supported on a", "closed subset $T$ of $X$ contained in $U_1 \\cup \\ldots \\cup U_n$", "with $U_i \\subset X$ affine open. Set $U = U_n$.", "Consider the distinguished triangles", "$$", "A \\to \\Phi(A|_U) \\to A' \\to A[1]", "\\quad\\text{and}\\quad", "E \\to Rj_*(E|_U) \\to E' \\to E[1]", "$$", "where $\\Phi$ is as in the statement of the lemma.", "Note that $E \\to Rj_*(E|_U)$ is a quasi-isomorphism over $U = U_n$.", "Since $i_X \\circ \\Phi = Rj_* \\circ i_U$ by assumption", "and since $i_X(A)|_U = i_U(A|_U)$", "we see that $i_X(A) \\to i_X(\\Phi(A|_U))$ is a quasi-isomorphism over $U$.", "Hence $i_X(A')$ and $E'$ are supported on the closed", "subset $T \\setminus U$ of $X$ which is contained in", "$U_1 \\cup \\ldots \\cup U_{n - 1}$.", "By induction hypothesis the statement is true for $A'$ and $E'$. By", "Derived Categories, Lemma \\ref{derived-lemma-third-isomorphism-triangle}", "it suffices to prove the maps", "$$", "RQ_X(i_X(\\Phi(A|_U))) \\to \\Phi(A|_U)", "\\quad\\text{and}\\quad", "Rj_*(E|_U) \\to i_X(RQ_X(Rj_*E|_U))", "$$", "are isomorphisms. By assumption and by", "Lemma \\ref{lemma-flat-pushforward-coherator}", "(the inclusion morphism $j : U \\to X$ is flat, quasi-compact, and", "quasi-separated) we have", "$$", "RQ_X(i_X(\\Phi(A|_U))) = RQ_X(Rj_*(i_U(A|_U))) = \\Phi(RQ_U(i_U(A|_U)))", "$$", "and", "$$", "i_X(RQ_X(Rj_*(E|_U))) = i_X(\\Phi(RQ_U(E|_U))) = Rj_*(i_U(RQ_U(E|_U)))", "$$", "Finally, the maps", "$$", "RQ_U(i_U(A|_U)) \\to A|_U", "\\quad\\text{and}\\quad", "E|_U \\to i_U(RQ_U(E|_U))", "$$", "are isomorphisms by Lemma \\ref{lemma-affine-coherator}. The result follows." ], "refs": [ "perfect-definition-supported-on", "derived-lemma-third-isomorphism-triangle", "perfect-lemma-flat-pushforward-coherator", "perfect-lemma-affine-coherator" ], "ref_ids": [ 7115, 1759, 6964, 6965 ] } ], "ref_ids": [] }, { "id": 6967, "type": "theorem", "label": "perfect-lemma-direct-image-coherator", "categories": [ "perfect" ], "title": "perfect-lemma-direct-image-coherator", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume $X$ and $Y$ are quasi-compact and have affine diagonal.", "Then, denoting", "$$", "\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))", "$$", "the right derived functor of", "$f_* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$", "the diagram", "$$", "\\xymatrix{", "D(\\QCoh(\\mathcal{O}_X)) \\ar[d]_\\Phi \\ar[r] &", "D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\", "D(\\QCoh(\\mathcal{O}_Y)) \\ar[r] &", "D_\\QCoh(\\mathcal{O}_Y)", "}", "$$", "is commutative." ], "refs": [], "proofs": [ { "contents": [ "Observe that the horizontal arrows in the diagram are", "equivalences of categories by", "Proposition \\ref{proposition-quasi-compact-affine-diagonal}.", "Hence we can identify these categories (and similarly for", "other quasi-compact schemes with affine diagonal).", "The statement of the lemma is that the canonical map", "$\\Phi(K) \\to Rf_*(K)$ is an isomorphism for all $K$ in", "$D(\\QCoh(\\mathcal{O}_X))$. Note that if $K_1 \\to K_2 \\to K_3 \\to K_1[1]$", "is a distinguished triangle in $D(\\QCoh(\\mathcal{O}_X))$ and", "the statement is true for two-out-of-three, then it is true", "for the third.", "\\medskip\\noindent", "Let $U \\subset X$ be an affine open. Since the diagonal of $X$ is affine,", "the inclusion morphism $j : U \\to X$", "is affine (Morphisms, Lemma \\ref{morphisms-lemma-affine-permanence}).", "Similarly, the composition $g = f \\circ j : U \\to Y$ is affine.", "Let $\\mathcal{I}^\\bullet$ be a K-injective complex in $\\QCoh(\\mathcal{O}_U)$.", "Since $j_* : \\QCoh(\\mathcal{O}_U) \\to \\QCoh(\\mathcal{O}_X)$", "has an exact left adjoint", "$j^* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_U)$", "we see that $j_*\\mathcal{I}^\\bullet$ is a K-injective complex", "in $\\QCoh(\\mathcal{O}_X)$, see", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}.", "It follows that", "$$", "\\Phi(j_*\\mathcal{I}^\\bullet) =", "f_*j_*\\mathcal{I}^\\bullet =", "g_*\\mathcal{I}^\\bullet", "$$", "By Lemma \\ref{lemma-affine-pushforward} we see that", "$j_*\\mathcal{I}^\\bullet$ represents $Rj_*\\mathcal{I}^\\bullet$ and", "$g_*\\mathcal{I}^\\bullet$ represents $Rg_*\\mathcal{I}^\\bullet$.", "On the other hand, we have $Rf_* \\circ Rj_* = Rg_*$.", "Hence $f_*j_*\\mathcal{I}^\\bullet$ represents $Rf_*(j_*\\mathcal{I}^\\bullet)$.", "We conclude that the lemma is true for any complex", "of the form $j_*\\mathcal{G}^\\bullet$ with $\\mathcal{G}^\\bullet$", "a complex of quasi-coherent modules on $U$. (Note that if", "$\\mathcal{G}^\\bullet \\to \\mathcal{I}^\\bullet$ is a quasi-isomorphism,", "then $j_*\\mathcal{G}^\\bullet \\to j_*\\mathcal{I}^\\bullet$ is a", "quasi-isomorphism as well since $j_*$ is an exact functor", "on quasi-coherent modules.)", "\\medskip\\noindent", "Let $\\mathcal{F}^\\bullet$ be a complex of quasi-coherent", "$\\mathcal{O}_X$-modules. Let $T \\subset X$ be a closed subset", "such that the support of $\\mathcal{F}^p$ is contained in $T$", "for all $p$. We will use induction on the minimal number $n$", "of affine opens $U_1, \\ldots, U_n$ such that", "$T \\subset U_1 \\cup \\ldots \\cup U_n$. The base case $n = 0$ is trivial.", "If $n \\geq 1$, then set $U = U_1$ and denote $j : U \\to X$ the", "open immersion as above. We consider the map of complexes", "$c : \\mathcal{F}^\\bullet \\to j_*j^*\\mathcal{F}^\\bullet$.", "We obtain two short exact sequences of complexes:", "$$", "0 \\to \\Ker(c) \\to \\mathcal{F}^\\bullet \\to \\Im(c) \\to 0", "$$", "and", "$$", "0 \\to \\Im(c) \\to j_*j^*\\mathcal{F}^\\bullet \\to \\Coker(c) \\to 0", "$$", "The complexes $\\Ker(c)$ and $\\Coker(c)$ are supported", "on $T \\setminus U \\subset U_2 \\cup \\ldots \\cup U_n$ and the result", "holds for them by induction. The result holds for", "$j_*j^*\\mathcal{F}^\\bullet$ by the discussion in the preceding", "paragraph. We conclude by looking at the distinguished triangles", "associated to the short exact sequences and using the initial", "remark of the proof." ], "refs": [ "perfect-proposition-quasi-compact-affine-diagonal", "morphisms-lemma-affine-permanence", "derived-lemma-adjoint-preserve-K-injectives", "perfect-lemma-affine-pushforward" ], "ref_ids": [ 7107, 5179, 1915, 6963 ] } ], "ref_ids": [] }, { "id": 6968, "type": "theorem", "label": "perfect-lemma-injective-quasi-coherent-sheaf-Noetherian", "categories": [ "perfect" ], "title": "perfect-lemma-injective-quasi-coherent-sheaf-Noetherian", "contents": [ "Let $X$ be a Noetherian scheme. Let $\\mathcal{J}$ be an injective", "object of $\\QCoh(\\mathcal{O}_X)$. Then $\\mathcal{J}$", "is a flasque sheaf of $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be an open subset and let $s \\in \\mathcal{J}(U)$", "be a section. Let $\\mathcal{I} \\subset X$ be the quasi-coherent sheaf", "of ideals defining the reduced induced scheme structure on $X \\setminus U$", "(see Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme}).", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-homs-over-open}", "the section $s$ corresponds to a map $\\sigma : \\mathcal{I}^n \\to \\mathcal{J}$", "for some $n$. As $\\mathcal{J}$ is an injective object of", "$\\QCoh(\\mathcal{O}_X)$ we can extend $\\sigma$ to a map", "$\\tilde s : \\mathcal{O}_X \\to \\mathcal{J}$. Then $\\tilde s$ corresponds", "to a global section of $\\mathcal{J}$ restricting to $s$." ], "refs": [ "schemes-definition-reduced-induced-scheme", "coherent-lemma-homs-over-open" ], "ref_ids": [ 7745, 3322 ] } ], "ref_ids": [] }, { "id": 6969, "type": "theorem", "label": "perfect-lemma-Noetherian-pushforward", "categories": [ "perfect" ], "title": "perfect-lemma-Noetherian-pushforward", "contents": [ "Let $f : X \\to Y$ be a morphism of Noetherian schemes.", "Then $f_*$ on quasi-coherent sheaves has a right derived", "extension", "$\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$", "such that the diagram", "$$", "\\xymatrix{", "D(\\QCoh(\\mathcal{O}_X)) \\ar[d]_{\\Phi} \\ar[r] &", "D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\", "D(\\QCoh(\\mathcal{O}_Y)) \\ar[r] &", "D_\\QCoh(\\mathcal{O}_Y)", "}", "$$", "commutes." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ and $Y$ are Noetherian schemes the morphism is quasi-compact", "and quasi-separated (see", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian-quasi-separated}", "and", "Schemes, Remark \\ref{schemes-remark-quasi-compact-and-quasi-separated}).", "Thus $f_*$ preserve quasi-coherence, see", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}.", "Next, let $K$ be an object of $D(\\QCoh(\\mathcal{O}_X))$.", "Since $\\QCoh(\\mathcal{O}_X)$ is a Grothendieck abelian category", "(Properties, Proposition \\ref{properties-proposition-coherator}), we can", "represent $K$ by a K-injective complex $\\mathcal{I}^\\bullet$", "such that each $\\mathcal{I}^n$ is an injective object of", "$\\QCoh(\\mathcal{O}_X)$, see", "Injectives, Theorem", "\\ref{injectives-theorem-K-injective-embedding-grothendieck}.", "Thus we see that the functor $\\Phi$ is defined by setting", "$$", "\\Phi(K) = f_*\\mathcal{I}^\\bullet", "$$", "where the right hand side is viewed as an object of", "$D(\\QCoh(\\mathcal{O}_Y))$. To finish the proof of the lemma", "it suffices to show that the canonical map", "$$", "f_*\\mathcal{I}^\\bullet \\longrightarrow Rf_*\\mathcal{I}^\\bullet", "$$", "is an isomorphism in $D(\\mathcal{O}_Y)$. To see this by", "Lemma \\ref{lemma-acyclicity-lemma}", "it suffices to show that $\\mathcal{I}^n$ is right", "$f_*$-acyclic for all $n \\in \\mathbf{Z}$.", "This is true because $f_*\\mathcal{I}^n$ is flasque by", "Lemma \\ref{lemma-injective-quasi-coherent-sheaf-Noetherian}", "and flasque modules are right $f_*$-acyclic by", "Cohomology, Lemma \\ref{cohomology-lemma-flasque-acyclic-pushforward}." ], "refs": [ "properties-lemma-locally-Noetherian-quasi-separated", "schemes-remark-quasi-compact-and-quasi-separated", "schemes-lemma-push-forward-quasi-coherent", "properties-proposition-coherator", "injectives-theorem-K-injective-embedding-grothendieck", "perfect-lemma-acyclicity-lemma", "perfect-lemma-injective-quasi-coherent-sheaf-Noetherian", "cohomology-lemma-flasque-acyclic-pushforward" ], "ref_ids": [ 2953, 7763, 7730, 3066, 7768, 6947, 6968, 2066 ] } ], "ref_ids": [] }, { "id": 6970, "type": "theorem", "label": "perfect-lemma-alternating-cech-complex", "categories": [ "perfect" ], "title": "perfect-lemma-alternating-cech-complex", "contents": [ "In Situation \\ref{situation-complex}. Let $M$ be an $A$-module and", "denote $\\mathcal{F}$ the associated $\\mathcal{O}_X$-module. Then", "there is a canonical isomorphism of complexes", "$$", "\\colim_e \\Hom_A(I^\\bullet(f_1^e, \\ldots, f_r^e), M)", "\\longrightarrow", "\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})", "$$", "functorial in $M$." ], "refs": [], "proofs": [ { "contents": [ "Recall that the alternating {\\v C}ech complex is the subcomplex", "of the usual {\\v C}ech complex given by alternating cochains, see", "Cohomology, Section \\ref{cohomology-section-alternating-cech}.", "As usual we view a $p$-cochain in", "$\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "as an alternating function $s$ on $\\{1, \\ldots, r\\}^{p + 1}$", "whose value $s_{i_0\\ldots i_p}$ at $(i_0, \\ldots, i_p)$ lies in", "$M_{f_{i_0}\\ldots f_{i_p}} = \\mathcal{F}(U_{i_0\\ldots i_p})$.", "On the other hand, a $p$-cochain $t$ in", "$\\Hom_A(I^\\bullet(f_1^e, \\ldots, f_r^e), M)$", "is given by a map $t : \\wedge^{p + 1}(A^{\\oplus r}) \\to M$.", "Write $[i] \\in A^{\\oplus r}$ for the $i$th basis element and", "write", "$$", "[i_0, \\ldots, i_p] = [i_0] \\wedge \\ldots \\wedge [i_p]", "\\in \\wedge^{p + 1}(A^{\\oplus r})", "$$", "Then we send $t$ as above to $s$ with", "$$", "s_{i_0\\ldots i_p} = \\frac{t([i_0, \\ldots, i_p])}{f_{i_0}^e\\ldots f_{i_p}^e}", "$$", "It is clear that $s$ so defined is an alternating cochain.", "The construction of this map is compatible with the transition maps", "of the system as the transition map", "$$", "I^\\bullet(f_1^e, \\ldots, f_r^e) \\leftarrow", "I^\\bullet(f_1^{e + 1}, \\ldots, f_r^{e + 1}),", "$$", "of the (\\ref{equation-system}) sends $[i_0, \\ldots, i_p]$", "to $f_{i_0}\\ldots f_{i_p}[i_0, \\ldots, i_p]$.", "It is clear from the description of the localizations", "$M_{f_{i_0}\\ldots f_{i_p}}$ in", "Algebra, Lemma \\ref{algebra-lemma-localization-colimit}", "that these maps define an isomorphism of cochain modules in degree $p$", "in the limit. To finish the proof we have to show that the map", "is compatible with differentials. To see this recall that", "\\begin{align*}", "d(s)_{i_0\\ldots i_{p + 1}}", "& =", "\\sum\\nolimits_{j = 0}^{p + 1} (-1)^j", "s_{i_0\\ldots \\hat i_j \\ldots i_p} \\\\", "& = ", "\\sum\\nolimits_{j = 0}^{p + 1} (-1)^j", "\\frac{t([i_0, \\ldots, \\hat i_j, \\ldots i_{p + 1}])}", "{f_{i_0}^e\\ldots \\hat f_{i_j}^e \\ldots f_{i_{p + 1}}^e}", "\\end{align*}", "On the other hand, we have", "\\begin{align*}", "\\frac{d(t)([i_0, \\ldots, i_{p + 1}])}{f_{i_0}^e\\ldots f_{i_{p + 1}}^e}", "& =", "\\frac{t(d[i_0, \\ldots, i_{p + 1}])}{f_{i_0}^e\\ldots f_{i_{p + 1}}^e} \\\\", "& =", "\\frac{\\sum_j (-1)^j f_{i_j}^e t([i_0, \\ldots, \\hat i_j, \\ldots i_{p + 1}])}", "{f_{i_0}^e \\ldots f_{i_{p + 1}}^e}", "\\end{align*}", "The two formulas agree by inspection." ], "refs": [ "algebra-lemma-localization-colimit" ], "ref_ids": [ 348 ] } ], "ref_ids": [] }, { "id": 6971, "type": "theorem", "label": "perfect-lemma-alternating-cech-complex-complex", "categories": [ "perfect" ], "title": "perfect-lemma-alternating-cech-complex-complex", "contents": [ "In Situation \\ref{situation-complex}. Let $M^\\bullet$ be a", "complex of $A$-modules and", "denote $\\mathcal{F}^\\bullet$ the associated complex of", "$\\mathcal{O}_X$-modules. Then", "there is a canonical isomorphism of complexes", "$$", "\\colim_e \\text{Tot}(\\Hom_A(I^\\bullet(f_1^e, \\ldots, f_r^e), M^\\bullet))", "\\longrightarrow", "\\text{Tot}(\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))", "$$", "functorial in $M^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-alternating-cech-complex}", "and our conventions for taking associated total complexes." ], "refs": [ "perfect-lemma-alternating-cech-complex" ], "ref_ids": [ 6970 ] } ], "ref_ids": [] }, { "id": 6972, "type": "theorem", "label": "perfect-lemma-alternating-cech-complex-complex-computes-cohomology", "categories": [ "perfect" ], "title": "perfect-lemma-alternating-cech-complex-complex-computes-cohomology", "contents": [ "In Situation \\ref{situation-complex}. Let $\\mathcal{F}^\\bullet$", "be a complex of quasi-coherent $\\mathcal{O}_X$-modules. Then", "there is a canonical isomorphism", "$$", "\\text{Tot}(\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))", "\\longrightarrow", "R\\Gamma(U, \\mathcal{F}^\\bullet)", "$$", "in $D(A)$ functorial in $\\mathcal{F}^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{B}$ be the set of affine opens of $U$. Since the higher", "cohomology groups of a quasi-coherent module on an affine scheme are zero", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero})", "this is a special case of", "Cohomology, Lemma \\ref{cohomology-lemma-alternating-cech-complex-complex-ss}." ], "refs": [ "coherent-lemma-quasi-coherent-affine-cohomology-zero", "cohomology-lemma-alternating-cech-complex-complex-ss" ], "ref_ids": [ 3282, 2173 ] } ], "ref_ids": [] }, { "id": 6973, "type": "theorem", "label": "perfect-lemma-represent-cohomology-class-on-closed", "categories": [ "perfect" ], "title": "perfect-lemma-represent-cohomology-class-on-closed", "contents": [ "In Situation \\ref{situation-complex}. Let $E$ be an object of", "$D_\\QCoh(\\mathcal{O}_X)$.", "Assume that $H^i(E)|_U = 0$ for $i = - r + 1, \\ldots, 0$.", "Then given $s \\in H^0(X, E)$ there exists an $e \\geq 0$ and", "a morphism $K_e \\to E$ such that $s$ is in the image of", "$H^0(X, K_e) \\to H^0(X, E)$." ], "refs": [], "proofs": [ { "contents": [ "Since $U$ is covered by $r$ affine opens we have $H^j(U, \\mathcal{F}) = 0$", "for $j \\geq r$ and any quasi-coherent module", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-vanishing-nr-affines}).", "By Lemma \\ref{lemma-application-nice-K-injective} we see that $H^0(U, E)$", "is equal to $H^0(U, \\tau_{\\geq -r + 1}E)$. There is", "a spectral sequence", "$$", "H^j(U, H^i(\\tau_{\\geq -r + 1}E)) \\Rightarrow H^{i + j}(U, \\tau_{\\geq -N}E)", "$$", "see Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "Hence $H^0(U, E) = 0$ by our assumed vanishing of cohomology sheaves of $E$.", "We conclude that $s|_U = 0$.", "Think of $s$ as a morphism $\\mathcal{O}_X \\to E$ in $D(\\mathcal{O}_X)$.", "By Proposition \\ref{proposition-represent-cohomology-class-on-open}", "the composition $I_e \\to \\mathcal{O}_X \\to E$ is zero for some $e$.", "By the distinguished triangle $I_e \\to \\mathcal{O}_X \\to K_e \\to I_e[1]$", "we obtain a morphism $K_e \\to E$ such that $s$ is the composition", "$\\mathcal{O}_X \\to K_e \\to E$." ], "refs": [ "coherent-lemma-vanishing-nr-affines", "perfect-lemma-application-nice-K-injective", "derived-lemma-two-ss-complex-functor", "perfect-proposition-represent-cohomology-class-on-open" ], "ref_ids": [ 3292, 6940, 1871, 7109 ] } ], "ref_ids": [] }, { "id": 6974, "type": "theorem", "label": "perfect-lemma-pseudo-coherent", "categories": [ "perfect" ], "title": "perfect-lemma-pseudo-coherent", "contents": [ "Let $X$ be a scheme. If $E$ is an $m$-pseudo-coherent", "object of $D(\\mathcal{O}_X)$, then $H^i(E)$ is a quasi-coherent", "$\\mathcal{O}_X$-module for $i > m$ and $H^m(E)$ is a quotient", "of a quasi-coherent $\\mathcal{O}_X$-module.", "If $E$ is pseudo-coherent, then $E$ is an object of", "$D_\\QCoh(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Locally on $X$ there exists a strictly perfect complex $\\mathcal{E}^\\bullet$", "such that $H^i(E)$ is isomorphic to $H^i(\\mathcal{E}^\\bullet)$ for $i > m$", "and $H^m(E)$ is a quotient of $H^m(\\mathcal{E}^\\bullet)$. The sheaves", "$\\mathcal{E}^i$ are direct summands of finite free modules,", "hence quasi-coherent. The lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6975, "type": "theorem", "label": "perfect-lemma-pseudo-coherent-affine", "categories": [ "perfect" ], "title": "perfect-lemma-pseudo-coherent-affine", "contents": [ "Let $X = \\Spec(A)$ be an affine scheme. Let $M^\\bullet$ be a", "complex of $A$-modules and let $E$ be the corresponding object", "of $D(\\mathcal{O}_X)$. Then $E$ is an $m$-pseudo-coherent", "(resp.\\ pseudo-coherent) as an object of $D(\\mathcal{O}_X)$", "if and only if $M^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)", "as a complex of $A$-modules." ], "refs": [], "proofs": [ { "contents": [ "It is immediate from the definitions that if $M^\\bullet$ is", "$m$-pseudo-coherent, so is $E$. To prove the converse, assume", "$E$ is $m$-pseudo-coherent. As $X = \\Spec(A)$ is quasi-compact with", "a basis for the topology given by standard opens, we can find a standard", "open covering $X = D(f_1) \\cup \\ldots \\cup D(f_n)$ and strictly", "perfect complexes $\\mathcal{E}_i^\\bullet$ on $D(f_i)$ and", "maps $\\alpha_i : \\mathcal{E}_i^\\bullet \\to E|_{U_i}$ inducing", "isomorphisms on $H^j$ for $j > m$ and surjections on $H^m$.", "By Cohomology, Lemma \\ref{cohomology-lemma-local-actual}", "after refining the open covering", "we may assume $\\alpha_i$ is given by a map of complexes", "$\\mathcal{E}_i^\\bullet \\to \\widetilde{M^\\bullet}|_{U_i}$", "for each $i$. By Modules, Lemma", "\\ref{modules-lemma-direct-summand-of-locally-free-is-locally-free}", "the terms $\\mathcal{E}_i^n$ are finite locally free modules.", "Hence after refining the open covering we may assume each", "$\\mathcal{E}_i^n$ is a finite free $\\mathcal{O}_{U_i}$-module.", "From the definition it follows that $M^\\bullet_{f_i}$ is", "an $m$-pseudo-coherent complex of $A_{f_i}$-modules.", "We conclude by applying", "More on Algebra, Lemma \\ref{more-algebra-lemma-glue-pseudo-coherent}.", "\\medskip\\noindent", "The case ``pseudo-coherent'' follows from the fact that $E$ is", "pseudo-coherent if and only if $E$ is $m$-pseudo-coherent for", "all $m$ (by definition) and the same is true for $M^\\bullet$", "by More on Algebra, Lemma \\ref{more-algebra-lemma-pseudo-coherent}." ], "refs": [ "cohomology-lemma-local-actual", "modules-lemma-direct-summand-of-locally-free-is-locally-free", "more-algebra-lemma-glue-pseudo-coherent", "more-algebra-lemma-pseudo-coherent" ], "ref_ids": [ 2202, 13266, 10157, 10148 ] } ], "ref_ids": [] }, { "id": 6976, "type": "theorem", "label": "perfect-lemma-identify-pseudo-coherent-noetherian", "categories": [ "perfect" ], "title": "perfect-lemma-identify-pseudo-coherent-noetherian", "contents": [ "Let $X$ be a Noetherian scheme. Let $E$ be an object of", "$D_\\QCoh(\\mathcal{O}_X)$. For $m \\in \\mathbf{Z}$ the", "following are equivalent", "\\begin{enumerate}", "\\item $H^i(E)$ is coherent for $i \\geq m$ and zero for $i \\gg 0$, and", "\\item $E$ is $m$-pseudo-coherent.", "\\end{enumerate}", "In particular, $E$ is pseudo-coherent if and only if $E$ is an object", "of $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "As $X$ is quasi-compact we see that in both (1) and (2) the object $E$", "is bounded above. Thus the question is local on $X$ and we may assume", "$X$ is affine. Say $X = \\Spec(A)$ for some Noetherian ring $A$.", "In this case $E$ corresponds to a complex of $A$-modules $M^\\bullet$", "by Lemma \\ref{lemma-affine-compare-bounded}. By", "Lemma \\ref{lemma-pseudo-coherent-affine}", "we see that $E$ is $m$-pseudo-coherent if and only if $M^\\bullet$", "is $m$-pseudo-coherent. On the other hand, $H^i(E)$ is coherent", "if and only if $H^i(M^\\bullet)$ is a finite $A$-module", "(Properties, Lemma \\ref{properties-lemma-finite-type-module}).", "Thus the result follows from More on Algebra, Lemma", "\\ref{more-algebra-lemma-Noetherian-pseudo-coherent}." ], "refs": [ "perfect-lemma-affine-compare-bounded", "perfect-lemma-pseudo-coherent-affine", "properties-lemma-finite-type-module", "more-algebra-lemma-Noetherian-pseudo-coherent" ], "ref_ids": [ 6941, 6975, 3002, 10160 ] } ], "ref_ids": [] }, { "id": 6977, "type": "theorem", "label": "perfect-lemma-tor-dimension-affine", "categories": [ "perfect" ], "title": "perfect-lemma-tor-dimension-affine", "contents": [ "Let $X = \\Spec(A)$ be an affine scheme. Let $M^\\bullet$ be a", "complex of $A$-modules and let $E$ be the corresponding object", "of $D(\\mathcal{O}_X)$. Then", "\\begin{enumerate}", "\\item $E$ has tor amplitude in $[a, b]$ if and only if $M^\\bullet$", "has tor amplitude in $[a, b]$.", "\\item $E$ has finite tor dimension if and only if $M^\\bullet$", "has finite tor dimension.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) follows trivially from part (1). In the proof of (1) we will", "use the equivalence $D(A) = D_\\QCoh(X)$ of", "Lemma \\ref{lemma-affine-compare-bounded}", "without further mention.", "Assume $M^\\bullet$ has tor amplitude in $[a, b]$. Then $K^\\bullet$", "is isomorphic in $D(A)$ to a complex $K^\\bullet$ of flat $A$-modules", "with $K^i = 0$ for $i \\not \\in [a, b]$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-tor-amplitude}.", "Then $E$ is isomorphic to $\\widetilde{K^\\bullet}$. Since each", "$\\widetilde{K^i}$ is a flat $\\mathcal{O}_X$-module, we see", "that $E$ has tor amplitude in $[a, b]$ by", "Cohomology, Lemma \\ref{cohomology-lemma-tor-amplitude}.", "\\medskip\\noindent", "Assume that $E$ has tor amplitude in $[a, b]$. Then $E$ is bounded", "whence $M^\\bullet$ is in $K^-(A)$. Thus we may replace $M^\\bullet$", "by a bounded above complex of $A$-modules. We may even choose", "a projective resolution and assume that $M^\\bullet$ is a bounded above", "complex of free $A$-modules. Then for any $A$-module $N$ we have", "$$", "E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\widetilde{N}", "\\cong", "\\widetilde{M^\\bullet} \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\widetilde{N}", "\\cong", "\\widetilde{M^\\bullet \\otimes_A N}", "$$", "in $D(\\mathcal{O}_X)$. Thus the vanishing of cohomology sheaves of", "the left hand side implies $M^\\bullet$ has tor amplitude in $[a, b]$." ], "refs": [ "perfect-lemma-affine-compare-bounded", "more-algebra-lemma-tor-amplitude", "cohomology-lemma-tor-amplitude" ], "ref_ids": [ 6941, 10170, 2215 ] } ], "ref_ids": [] }, { "id": 6978, "type": "theorem", "label": "perfect-lemma-tor-dimension-rel-affine", "categories": [ "perfect" ], "title": "perfect-lemma-tor-dimension-rel-affine", "contents": [ "Let $f : X \\to S$ be a morphism of affine schemes corresponding", "to the ring map $R \\to A$. Let $M^\\bullet$ be a", "complex of $A$-modules and let $E$ be the corresponding object", "of $D(\\mathcal{O}_X)$. Then", "\\begin{enumerate}", "\\item $E$ as an object of $D(f^{-1}\\mathcal{O}_S)$ has tor amplitude in", "$[a, b]$ if and only if $M^\\bullet$ has tor amplitude in $[a, b]$", "as an object of $D(R)$.", "\\item $E$ locally has finite tor dimension as an object of", "$D(f^{-1}\\mathcal{O}_S)$ if and only if $M^\\bullet$", "has finite tor dimension as an object of $D(R)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider a prime $\\mathfrak q \\subset A$ lying over $\\mathfrak p \\subset R$.", "Let $x \\in X$ and $s = f(x) \\in S$ be the corresponding points.", "Then $(f^{-1}\\mathcal{O}_S)_x = \\mathcal{O}_{S, s} = R_\\mathfrak p$", "and $E_x = M^\\bullet_\\mathfrak q$. Keeping this in mind we can see", "the equivalence as follows.", "\\medskip\\noindent", "If $M^\\bullet$ has tor amplitude in $[a, b]$ as a complex of $R$-modules,", "then the same is true for the localization of $M^\\bullet$ at any prime of $A$.", "Then we conclude by", "Cohomology, Lemma \\ref{cohomology-lemma-tor-amplitude-stalk}", "that $E$ has tor amplitude in $[a, b]$ as a complex of sheaves", "of $f^{-1}\\mathcal{O}_S$-modules.", "Conversely, assume that $E$ has tor amplitude in $[a, b]$", "as an object of $D(f^{-1}\\mathcal{O}_S)$.", "We conclude (using the last cited lemma) that", "$M^\\bullet_\\mathfrak q$ has tor amplitude in $[a, b]$", "as a complex of $R_\\mathfrak p$-modules for every", "prime $\\mathfrak q \\subset A$ lying over $\\mathfrak p \\subset R$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-tor-amplitude-localization}", "we find that $M^\\bullet$ has tor amplitude in $[a, b]$", "as a complex of $R$-modules.", "This finishes the proof of (1).", "\\medskip\\noindent", "Since $X$ is quasi-compact, if $E$ locally has finite tor dimension", "as a complex of $f^{-1}\\mathcal{O}_S$-modules, then actually $E$", "has tor amplitude in $[a, b]$ for some $a, b$ as a complex of", "$f^{-1}\\mathcal{O}_S$-modules. Thus (2) follows from (1)." ], "refs": [ "cohomology-lemma-tor-amplitude-stalk", "more-algebra-lemma-tor-amplitude-localization" ], "ref_ids": [ 2217, 10182 ] } ], "ref_ids": [] }, { "id": 6979, "type": "theorem", "label": "perfect-lemma-tor-qc-qs", "categories": [ "perfect" ], "title": "perfect-lemma-tor-qc-qs", "contents": [ "Let $X$ be a quasi-separated scheme. Let $E$ be an object", "of $D_\\QCoh(\\mathcal{O}_X)$. Let $a \\leq b$. The", "following are equivalent", "\\begin{enumerate}", "\\item $E$ has tor amplitude in $[a, b]$, and", "\\item for all $\\mathcal{F}$ in $\\QCoh(\\mathcal{O}_X)$", "we have $H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}) = 0$", "for $i \\not \\in [a, b]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2). Assume (2). Let $U \\subset X$ be", "an affine open. As $X$ is quasi-separated the morphism $j : U \\to X$", "is quasi-compact and separated, hence $j_*$ transforms quasi-coherent", "modules into quasi-coherent modules", "(Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}).", "Thus the functor", "$\\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_U)$", "is essentially surjective. It follows that condition (2)", "implies the vanishing of", "$H^i(E|_U \\otimes_{\\mathcal{O}_U}^\\mathbf{L} \\mathcal{G})$", "for $i \\not \\in [a, b]$ for all quasi-coherent $\\mathcal{O}_U$-modules", "$\\mathcal{G}$. Write $U = \\Spec(A)$ and let $M^\\bullet$ be the", "complex of $A$-modules corresponding to $E|_U$ by", "Lemma \\ref{lemma-affine-compare-bounded}.", "We have just shown that $M^\\bullet \\otimes_A^\\mathbf{L} N$", "has vanishing cohomology groups outside the range $[a, b]$,", "in other words $M^\\bullet$ has tor amplitude in $[a, b]$.", "By Lemma \\ref{lemma-tor-dimension-affine}", "we conclude that $E|_U$ has tor amplitude in $[a, b]$.", "This proves the lemma." ], "refs": [ "schemes-lemma-push-forward-quasi-coherent", "perfect-lemma-affine-compare-bounded", "perfect-lemma-tor-dimension-affine" ], "ref_ids": [ 7730, 6941, 6977 ] } ], "ref_ids": [] }, { "id": 6980, "type": "theorem", "label": "perfect-lemma-perfect-affine", "categories": [ "perfect" ], "title": "perfect-lemma-perfect-affine", "contents": [ "Let $X = \\Spec(A)$ be an affine scheme. Let $M^\\bullet$ be a", "complex of $A$-modules and let $E$ be the corresponding object", "of $D(\\mathcal{O}_X)$. Then $E$ is a perfect object of $D(\\mathcal{O}_X)$", "if and only if $M^\\bullet$ is perfect as an object of $D(A)$." ], "refs": [], "proofs": [ { "contents": [ "This is a logical consequence of", "Lemmas \\ref{lemma-pseudo-coherent-affine} and", "\\ref{lemma-tor-dimension-affine},", "Cohomology, Lemma \\ref{cohomology-lemma-perfect}, and", "More on Algebra, Lemma \\ref{more-algebra-lemma-perfect}." ], "refs": [ "perfect-lemma-pseudo-coherent-affine", "perfect-lemma-tor-dimension-affine", "cohomology-lemma-perfect", "more-algebra-lemma-perfect" ], "ref_ids": [ 6975, 6977, 2224, 10212 ] } ], "ref_ids": [] }, { "id": 6981, "type": "theorem", "label": "perfect-lemma-quasi-coherence-internal-hom", "categories": [ "perfect" ], "title": "perfect-lemma-quasi-coherence-internal-hom", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item If $L$ is in $D^+_\\QCoh(\\mathcal{O}_X)$ and", "$K$ in $D(\\mathcal{O}_X)$ is pseudo-coherent, then", "$R\\SheafHom(K, L)$ is in $D_\\QCoh(\\mathcal{O}_X)$", "and locally bounded below.", "\\item If $L$ is in $D_\\QCoh(\\mathcal{O}_X)$ and", "$K$ in $D(\\mathcal{O}_X)$ is perfect, then", "$R\\SheafHom(K, L)$ is in $D_\\QCoh(\\mathcal{O}_X)$.", "\\item If $X = \\Spec(A)$ is affine and $K, L \\in D(A)$ then", "$$", "R\\SheafHom(\\widetilde{K}, \\widetilde{L}) = \\widetilde{R\\Hom_A(K, L)}", "$$", "in the following two cases", "\\begin{enumerate}", "\\item $K$ is pseudo-coherent and $L$ is bounded below,", "\\item $K$ is perfect and $L$ arbitrary.", "\\end{enumerate}", "\\item If $X = \\Spec(A)$ and $K, L$ are in $D(A)$, then the $n$th", "cohomology sheaf of $R\\SheafHom(\\widetilde{K}, \\widetilde{L})$", "is the sheaf associated to the presheaf", "$$", "X \\supset D(f) \\longmapsto \\Ext^n_{A_f}(K \\otimes_A A_f, L \\otimes_A A_f)", "$$", "for $f \\in A$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The construction of the internal hom in the derived category of", "$\\mathcal{O}_X$ commutes with localization (see", "Cohomology, Section \\ref{cohomology-section-internal-hom}).", "Hence to prove (1) and (2) we may replace $X$ by an affine open.", "By Lemmas \\ref{lemma-affine-compare-bounded},", "\\ref{lemma-pseudo-coherent-affine}, and", "\\ref{lemma-perfect-affine}", "in order to prove (1) and (2) it suffices to prove (3).", "\\medskip\\noindent", "Part (3) follows from the computation of the", "internal hom of Cohomology, Lemma", "\\ref{cohomology-lemma-Rhom-complex-of-direct-summands-finite-free}", "by representing $K$ by a bounded above (resp.\\ finite) complex of", "finite projective $A$-modules and $L$ by a bounded below", "(resp.\\ arbitrary) complex of $A$-modules.", "\\medskip\\noindent", "To prove (4) recall that on any ringed space the $n$th cohomology sheaf of", "$R\\SheafHom(A, B)$ is the sheaf associated to the presheaf", "$$", "U \\mapsto \\Hom_{D(U)}(A|_U, B|_U[n]) =", "\\Ext^n_{D(\\mathcal{O}_U)}(A|_U, B|_U)", "$$", "See Cohomology, Section \\ref{cohomology-section-internal-hom}.", "On the other hand, the restriction of $\\widetilde{K}$ to a principal", "open $D(f)$ is the image of $K \\otimes_A A_f$ and similarly for $L$.", "Hence (4) follows from the equivalence of categories of", "Lemma \\ref{lemma-affine-compare-bounded}." ], "refs": [ "perfect-lemma-affine-compare-bounded", "perfect-lemma-pseudo-coherent-affine", "perfect-lemma-perfect-affine", "cohomology-lemma-Rhom-complex-of-direct-summands-finite-free", "perfect-lemma-affine-compare-bounded" ], "ref_ids": [ 6941, 6975, 6980, 2204, 6941 ] } ], "ref_ids": [] }, { "id": 6982, "type": "theorem", "label": "perfect-lemma-internal-hom-evaluate-tensor-isomorphism", "categories": [ "perfect" ], "title": "perfect-lemma-internal-hom-evaluate-tensor-isomorphism", "contents": [ "Let $X$ be a scheme. Let $K, L, M$ be objects of $D_\\QCoh(\\mathcal{O}_X)$.", "The map", "$$", "K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(M, L)", "\\longrightarrow", "R\\SheafHom(M, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)", "$$", "of Cohomology, Lemma \\ref{cohomology-lemma-internal-hom-diagonal-better}", "is an isomorphism in the following cases", "\\begin{enumerate}", "\\item $M$ perfect, or", "\\item $K$ is perfect, or", "\\item $M$ is pseudo-coherent, $L \\in D^+(\\mathcal{O}_X)$, and $K$ has finite", "tor dimension.", "\\end{enumerate}" ], "refs": [ "cohomology-lemma-internal-hom-diagonal-better" ], "proofs": [ { "contents": [ "Lemma \\ref{lemma-quasi-coherence-internal-hom}", "reduces cases (1) and (3) to the affine case which is treated in", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism}.", "(You also have to use Lemmas \\ref{lemma-pseudo-coherent-affine},", "\\ref{lemma-perfect-affine}, and \\ref{lemma-tor-dimension-affine}", "to do the translation into algebra.)", "If $K$ is perfect but no other assumptions are made, then we", "do not know that either side of the arrow is in $D_\\QCoh(\\mathcal{O}_X)$", "but the result is still true because we can work locally and reduce", "to the case that $K$ is a finite complex of finite free modules", "in which case it is clear." ], "refs": [ "perfect-lemma-quasi-coherence-internal-hom", "more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism", "perfect-lemma-pseudo-coherent-affine", "perfect-lemma-perfect-affine", "perfect-lemma-tor-dimension-affine" ], "ref_ids": [ 6981, 10415, 6975, 6980, 6977 ] } ], "ref_ids": [ 2187 ] }, { "id": 6983, "type": "theorem", "label": "perfect-lemma-coh-to-qcoh", "categories": [ "perfect" ], "title": "perfect-lemma-coh-to-qcoh", "contents": [ "Let $X$ be a Noetherian scheme. Then the functor", "$$", "D^-(\\textit{Coh}(\\mathcal{O}_X))", "\\longrightarrow", "D^-_{\\textit{Coh}(\\mathcal{O}_X)}(\\QCoh(\\mathcal{O}_X))", "$$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\textit{Coh}(\\mathcal{O}_X) \\subset \\QCoh(\\mathcal{O}_X)$", "is a Serre subcategory, see", "Homology, Definition \\ref{homology-definition-serre-subcategory} and", "Lemma \\ref{homology-lemma-characterize-serre-subcategory} and", "Cohomology of Schemes, Lemmas", "\\ref{coherent-lemma-coherent-abelian-Noetherian} and", "\\ref{coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient}.", "On the other hand, if $\\mathcal{G} \\to \\mathcal{F}$ is a surjection", "from a quasi-coherent $\\mathcal{O}_X$-module to a coherent", "$\\mathcal{O}_X$-module, then there exists a coherent submodule", "$\\mathcal{G}' \\subset \\mathcal{G}$ which surjects onto $\\mathcal{F}$.", "Namely, we can write $\\mathcal{G}$ as the filtered union of its coherent", "submodules by", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-colimit-finite-type}", "and then one of these will do the job.", "Thus the lemma follows from", "Derived Categories, Lemma \\ref{derived-lemma-fully-faithful-embedding}." ], "refs": [ "homology-definition-serre-subcategory", "homology-lemma-characterize-serre-subcategory", "coherent-lemma-coherent-abelian-Noetherian", "coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient", "properties-lemma-quasi-coherent-colimit-finite-type", "derived-lemma-fully-faithful-embedding" ], "ref_ids": [ 12146, 12045, 3309, 3310, 3020, 1849 ] } ], "ref_ids": [] }, { "id": 6984, "type": "theorem", "label": "perfect-lemma-direct-image-coherent", "categories": [ "perfect" ], "title": "perfect-lemma-direct-image-coherent", "contents": [ "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes", "which is locally of finite type. Let $E$ be an object of", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ such that the support of $H^i(E)$", "is proper over $S$ for all $i$.", "Then $Rf_*E$ is an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_S)$." ], "refs": [], "proofs": [ { "contents": [ "Consider the spectral sequence", "$$", "R^pf_*H^q(E) \\Rightarrow R^{p + q}f_*E", "$$", "see Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "By assumption and", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-support-proper-over-base-pushforward}", "the sheaves $R^pf_*H^q(E)$ are coherent. Hence", "$R^{p + q}f_*E$ is coherent, i.e., $Rf_*E \\in D_{\\textit{Coh}}(\\mathcal{O}_S)$.", "Boundedness from below is trivial. Boundedness from above", "follows from", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images}", "or from", "Lemma \\ref{lemma-quasi-coherence-direct-image}." ], "refs": [ "derived-lemma-two-ss-complex-functor", "coherent-lemma-support-proper-over-base-pushforward", "coherent-lemma-quasi-coherence-higher-direct-images", "perfect-lemma-quasi-coherence-direct-image" ], "ref_ids": [ 1871, 3394, 3295, 6946 ] } ], "ref_ids": [] }, { "id": 6985, "type": "theorem", "label": "perfect-lemma-direct-image-coherent-bdd-below", "categories": [ "perfect" ], "title": "perfect-lemma-direct-image-coherent-bdd-below", "contents": [ "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes", "which is locally of finite type. Let $E$ be an object of", "$D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ such that the support of $H^i(E)$", "is proper over $S$ for all $i$.", "Then $Rf_*E$ is an object of $D^+_{\\textit{Coh}}(\\mathcal{O}_S)$." ], "refs": [], "proofs": [ { "contents": [ "The proof is the same as the proof of", "Lemma \\ref{lemma-direct-image-coherent}.", "You can also deduce it from", "Lemma \\ref{lemma-direct-image-coherent}", "by considering what the exact functor $Rf_*$ does to", "the distinguished triangles", "$\\tau_{\\leq a}E \\to E \\to \\tau_{\\geq a + 1}E \\to \\tau_{\\leq a}E[1]$." ], "refs": [ "perfect-lemma-direct-image-coherent", "perfect-lemma-direct-image-coherent" ], "ref_ids": [ 6984, 6984 ] } ], "ref_ids": [] }, { "id": 6986, "type": "theorem", "label": "perfect-lemma-coherent-internal-hom", "categories": [ "perfect" ], "title": "perfect-lemma-coherent-internal-hom", "contents": [ "Let $X$ be a locally Noetherian scheme. If $L$ is in", "$D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and $K$ in", "$D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, then", "$R\\SheafHom(K, L)$ is in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove this when $X$ is the spectrum of", "a Noetherian ring $A$.", "By Lemma \\ref{lemma-identify-pseudo-coherent-noetherian}", "we see that $K$ is pseudo-coherent.", "Then we can use Lemma \\ref{lemma-quasi-coherence-internal-hom}", "to translate the problem into the following algebra problem:", "for $L \\in D^+_{\\textit{Coh}}(A)$ and $K$ in $D^-_{\\textit{Coh}}(A)$, then", "$R\\Hom_A(K, L)$ is in $D^+_{\\textit{Coh}}(A)$.", "Since $L$ is bounded below and $K$ is bounded below there is a", "convergent spectral sequence", "$$", "\\Ext^p_A(K, H^q(L)) \\Rightarrow \\text{Ext}^{p + q}_A(K, L)", "$$", "and there are convergent spectral sequences", "$$", "\\Ext^i_A(H^{-j}(K), H^q(L)) \\Rightarrow \\text{Ext}^{i + j}_A(K, H^q(L))", "$$", "See Injectives, Remarks \\ref{injectives-remark-spectral-sequences-ext}", "and \\ref{injectives-remark-spectral-sequences-ext-variant}.", "This finishes the proof as the modules $\\Ext^p_A(M, N)$", "are finite for finite $A$-modules $M$, $N$ by", "Algebra, Lemma \\ref{algebra-lemma-ext-noetherian}." ], "refs": [ "perfect-lemma-identify-pseudo-coherent-noetherian", "perfect-lemma-quasi-coherence-internal-hom", "injectives-remark-spectral-sequences-ext", "injectives-remark-spectral-sequences-ext-variant", "algebra-lemma-ext-noetherian" ], "ref_ids": [ 6976, 6981, 7819, 7821, 768 ] } ], "ref_ids": [] }, { "id": 6987, "type": "theorem", "label": "perfect-lemma-perfect-on-noetherian", "categories": [ "perfect" ], "title": "perfect-lemma-perfect-on-noetherian", "contents": [ "Let $X$ be a Noetherian scheme. Let $E$ in $D(\\mathcal{O}_X)$ be perfect.", "Then", "\\begin{enumerate}", "\\item $E$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$,", "\\item if $L$ is in $D_{\\textit{Coh}}(\\mathcal{O}_X)$ then", "$E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ and", "$R\\SheafHom_{\\mathcal{O}_X}(E, L)$ are in", "$D_{\\textit{Coh}}(\\mathcal{O}_X)$,", "\\item if $L$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ then", "$E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ and", "$R\\SheafHom_{\\mathcal{O}_X}(E, L)$ are in", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$,", "\\item if $L$ is in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ then", "$E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ and", "$R\\SheafHom_{\\mathcal{O}_X}(E, L)$ are in", "$D^+_{\\textit{Coh}}(\\mathcal{O}_X)$,", "\\item if $L$ is in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$ then", "$E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ and", "$R\\SheafHom_{\\mathcal{O}_X}(E, L)$ are in", "$D^-_{\\textit{Coh}}(\\mathcal{O}_X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is quasi-compact, each of these statements can be checked", "over the members of any open covering of $X$.", "Thus we may assume $E$ is represented by", "a bounded complex $\\mathcal{E}^\\bullet$ of finite free modules, see", "Cohomology, Lemma \\ref{cohomology-lemma-perfect-on-locally-ringed}.", "In this case each of the statements is clear as both", "$R\\SheafHom_{\\mathcal{O}_X}(E, L)$", "and $E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ can be computed on the", "level of complexes using $\\mathcal{E}^\\bullet$, see", "Cohomology, Lemmas \\ref{cohomology-lemma-Rhom-strictly-perfect} and", "\\ref{cohomology-lemma-bounded-flat-K-flat}. Some details omitted." ], "refs": [ "cohomology-lemma-perfect-on-locally-ringed", "cohomology-lemma-Rhom-strictly-perfect", "cohomology-lemma-bounded-flat-K-flat" ], "ref_ids": [ 2222, 2203, 2109 ] } ], "ref_ids": [] }, { "id": 6988, "type": "theorem", "label": "perfect-lemma-ext-finite", "categories": [ "perfect" ], "title": "perfect-lemma-ext-finite", "contents": [ "Let $A$ be a Noetherian ring. Let $X$ be a proper scheme over $A$.", "For $L$ in", "$D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and $K$ in", "$D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, the $A$-modules", "$\\Ext_{\\mathcal{O}_X}^n(K, L)$ are finite." ], "refs": [], "proofs": [ { "contents": [ "Recall that", "$$", "\\Ext_{\\mathcal{O}_X}^n(K, L) =", "H^n(X, R\\SheafHom_{\\mathcal{O}_X}(K, L)) =", "H^n(\\Spec(A), Rf_*R\\SheafHom_{\\mathcal{O}_X}(K, L))", "$$", "see Cohomology, Lemma \\ref{cohomology-lemma-section-RHom-over-U}", "and Cohomology, Section \\ref{cohomology-section-Leray}.", "Thus the result follows from", "Lemmas \\ref{lemma-coherent-internal-hom} and", "\\ref{lemma-direct-image-coherent-bdd-below}." ], "refs": [ "perfect-lemma-coherent-internal-hom", "perfect-lemma-direct-image-coherent-bdd-below" ], "ref_ids": [ 6986, 6985 ] } ], "ref_ids": [] }, { "id": 6989, "type": "theorem", "label": "perfect-lemma-perfect-on-regular", "categories": [ "perfect" ], "title": "perfect-lemma-perfect-on-regular", "contents": [ "Let $X$ be a Noetherian regular scheme of finite dimension. Then", "every object of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ is perfect", "and conversely every perfect object of $D(\\mathcal{O}_X)$", "is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Combine More on Algebra, Lemma \\ref{more-algebra-lemma-regular-perfect}", "with Lemma \\ref{lemma-perfect-affine}." ], "refs": [ "more-algebra-lemma-regular-perfect", "perfect-lemma-perfect-affine" ], "ref_ids": [ 10223, 6980 ] } ], "ref_ids": [] }, { "id": 6990, "type": "theorem", "label": "perfect-lemma-tor-amplitude-descends", "categories": [ "perfect" ], "title": "perfect-lemma-tor-amplitude-descends", "contents": [ "Let $f : X \\to Y$ be a surjective flat morphism of schemes", "(or more generally locally ringed spaces).", "Let $E \\in D(\\mathcal{O}_Y)$. Let $a, b \\in \\mathbf{Z}$.", "Then $E$ has tor-amplitude in $[a, b]$ if and only if", "$Lf^*E$ has tor-amplitude in $[a, b]$." ], "refs": [], "proofs": [ { "contents": [ "Pullback always preserves tor-amplitude, see", "Cohomology, Lemma \\ref{cohomology-lemma-tor-amplitude-pullback}.", "We may check tor-amplitude in $[a, b]$ on stalks, see", "Cohomology, Lemma \\ref{cohomology-lemma-tor-amplitude-stalk}.", "A flat local ring homomorphism is faithfully flat by", "Algebra, Lemma \\ref{algebra-lemma-local-flat-ff}.", "Thus the result follows from", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-flat-descent-tor-amplitude}." ], "refs": [ "cohomology-lemma-tor-amplitude-pullback", "cohomology-lemma-tor-amplitude-stalk", "algebra-lemma-local-flat-ff", "more-algebra-lemma-flat-descent-tor-amplitude" ], "ref_ids": [ 2216, 2217, 537, 10184 ] } ], "ref_ids": [] }, { "id": 6991, "type": "theorem", "label": "perfect-lemma-pseudo-coherent-descends-fpqc", "categories": [ "perfect" ], "title": "perfect-lemma-pseudo-coherent-descends-fpqc", "contents": [ "Let $\\{f_i : X_i \\to X\\}$ be an fpqc covering of schemes. Let", "$E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$.", "Then $E$ is $m$-pseudo-coherent if and only if each", "$Lf_i^*E$ is $m$-pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "Pullback always preserves $m$-pseudo-coherence, see", "Cohomology, Lemma \\ref{cohomology-lemma-pseudo-coherent-pullback}.", "Conversely, assume that $Lf_i^*E$ is $m$-pseudo-coherent for all $i$.", "Let $U \\subset X$ be an affine open. It suffices to prove that", "$E|_U$ is $m$-pseudo-coherent. Since $\\{f_i : X_i \\to X\\}$ is an", "fpqc covering, we can find finitely many affine open $V_j \\subset X_{a(j)}$", "such that $f_{a(j)}(V_j) \\subset U$ and $U = \\bigcup f_{a(j)}(V_j)$.", "Set $V = \\coprod V_i$.", "Thus we may replace $X$ by $U$ and $\\{f_i : X_i \\to X\\}$ by", "$\\{V \\to U\\}$ and assume that $X$ is affine and our covering", "is given by a single surjective flat morphism $\\{f : Y \\to X\\}$", "of affine schemes. In this case the result follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-descent-pseudo-coherent}", "via Lemmas \\ref{lemma-affine-compare-bounded} and", "\\ref{lemma-pseudo-coherent-affine}." ], "refs": [ "cohomology-lemma-pseudo-coherent-pullback", "more-algebra-lemma-flat-descent-pseudo-coherent", "perfect-lemma-affine-compare-bounded", "perfect-lemma-pseudo-coherent-affine" ], "ref_ids": [ 2206, 10158, 6941, 6975 ] } ], "ref_ids": [] }, { "id": 6992, "type": "theorem", "label": "perfect-lemma-pseudo-coherent-descends-fppf", "categories": [ "perfect" ], "title": "perfect-lemma-pseudo-coherent-descends-fppf", "contents": [ "Let $\\{f_i : X_i \\to X\\}$ be an fppf covering of schemes. Let", "$E \\in D(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$.", "Then $E$ is $m$-pseudo-coherent if and only if each", "$Lf_i^*E$ is $m$-pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "Pullback always preserves $m$-pseudo-coherence, see", "Cohomology, Lemma \\ref{cohomology-lemma-pseudo-coherent-pullback}.", "Conversely, assume that $Lf_i^*E$ is $m$-pseudo-coherent for all $i$.", "Let $U \\subset X$ be an affine open. It suffices to prove that", "$E|_U$ is $m$-pseudo-coherent. Since $\\{f_i : X_i \\to X\\}$ is an", "fppf covering, we can find finitely many affine open $V_j \\subset X_{a(j)}$", "such that $f_{a(j)}(V_j) \\subset U$ and $U = \\bigcup f_{a(j)}(V_j)$.", "Set $V = \\coprod V_i$.", "Thus we may replace $X$ by $U$ and $\\{f_i : X_i \\to X\\}$ by", "$\\{V \\to U\\}$ and assume that $X$ is affine and our covering", "is given by a single surjective flat morphism $\\{f : Y \\to X\\}$", "of finite presentation.", "\\medskip\\noindent", "Since $f$ is flat the derived functor $Lf^*$ is just given by $f^*$ and $f^*$", "is exact. Hence $H^i(Lf^*E) = f^*H^i(E)$. Since $Lf^*E$ is $m$-pseudo-coherent,", "we see that $Lf^*E \\in D^-(\\mathcal{O}_Y)$. Since $f$ is surjective and flat,", "we see that $E \\in D^-(\\mathcal{O}_X)$. Let $i \\in \\mathbf{Z}$ be the largest", "integer such that $H^i(E)$ is nonzero. If $i < m$, then we are done. Otherwise,", "$f^*H^i(E)$ is a finite type $\\mathcal{O}_Y$-module by", "Cohomology, Lemma \\ref{cohomology-lemma-finite-cohomology}.", "Then by Descent, Lemma \\ref{descent-lemma-finite-type-descends-fppf}", "the $\\mathcal{O}_X$-module $H^i(E)$ is of finite type.", "Thus, after replacing $X$ by the members of a finite affine open covering,", "we may assume there exists a map", "$$", "\\alpha : \\mathcal{O}_X^{\\oplus n}[-i] \\longrightarrow E", "$$", "such that $H^i(\\alpha)$ is a surjection. Let $C$ be the cone of $\\alpha$", "in $D(\\mathcal{O}_X)$. Pulling back to $Y$ and using", "Cohomology, Lemma \\ref{cohomology-lemma-cone-pseudo-coherent}", "we find that $Lf^*C$ is $m$-pseudo-coherent. Moreover $H^j(C) = 0$", "for $j \\geq i$. Thus by induction on $i$ we see that $C$ is", "$m$-pseudo-coherent. Using", "Cohomology, Lemma \\ref{cohomology-lemma-cone-pseudo-coherent}", "again we conclude." ], "refs": [ "cohomology-lemma-pseudo-coherent-pullback", "cohomology-lemma-finite-cohomology", "descent-lemma-finite-type-descends-fppf", "cohomology-lemma-cone-pseudo-coherent", "cohomology-lemma-cone-pseudo-coherent" ], "ref_ids": [ 2206, 2212, 14613, 2207, 2207 ] } ], "ref_ids": [] }, { "id": 6993, "type": "theorem", "label": "perfect-lemma-perfect-descends-fpqc", "categories": [ "perfect" ], "title": "perfect-lemma-perfect-descends-fpqc", "contents": [ "Let $\\{f_i : X_i \\to X\\}$ be an fpqc covering of schemes. Let", "$E \\in D(\\mathcal{O}_X)$. Then $E$ is perfect", "if and only if each $Lf_i^*E$ is perfect." ], "refs": [], "proofs": [ { "contents": [ "Pullback always preserves perfect complexes, see", "Cohomology, Lemma \\ref{cohomology-lemma-perfect-pullback}.", "Conversely, assume that $Lf_i^*E$ is perfect for all $i$.", "Then the cohomology sheaves of each $Lf_i^*E$ are quasi-coherent, see", "Lemma \\ref{lemma-pseudo-coherent}", "and", "Cohomology, Lemma \\ref{cohomology-lemma-perfect}.", "Since the morphisms $f_i$ is flat we see that $H^p(Lf_i^*E) = f_i^*H^p(E)$.", "Thus the cohomology sheaves of $E$ are quasi-coherent by", "Descent, Proposition \\ref{descent-proposition-fpqc-descent-quasi-coherent}.", "Having said this the lemma follows formally from", "Cohomology, Lemma \\ref{cohomology-lemma-perfect}", "and", "Lemmas \\ref{lemma-tor-amplitude-descends} and", "\\ref{lemma-pseudo-coherent-descends-fpqc}." ], "refs": [ "cohomology-lemma-perfect-pullback", "perfect-lemma-pseudo-coherent", "cohomology-lemma-perfect", "descent-proposition-fpqc-descent-quasi-coherent", "cohomology-lemma-perfect", "perfect-lemma-tor-amplitude-descends", "perfect-lemma-pseudo-coherent-descends-fpqc" ], "ref_ids": [ 2225, 6974, 2224, 14753, 2224, 6990, 6991 ] } ], "ref_ids": [] }, { "id": 6994, "type": "theorem", "label": "perfect-lemma-closed-push-pseudo-coherent", "categories": [ "perfect" ], "title": "perfect-lemma-closed-push-pseudo-coherent", "contents": [ "Let $i : Z \\to X$ be a morphism of ringed spaces such that", "$i$ is a closed immersion of underlying topological spaces and such that", "$i_*\\mathcal{O}_Z$ is pseudo-coherent as an $\\mathcal{O}_X$-module.", "Let $E \\in D(\\mathcal{O}_Z)$. Then $E$ is $m$-pseudo-coherent", "if and only if $Ri_*E$ is $m$-pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "Throughout this proof we will use that $i_*$ is an exact functor, and", "hence that $Ri_* = i_*$, see Modules, Lemma \\ref{modules-lemma-i-star-exact}.", "\\medskip\\noindent", "Assume $E$ is $m$-pseudo-coherent. Let $x \\in X$. We will find a neighbourhood", "of $x$ such that $i_*E$ is $m$-pseudo-coherent on it. If $x \\not \\in Z$", "then this is clear. Thus we may assume $x \\in Z$. We will use", "that $U \\cap Z$ for $x \\in U \\subset X$ open form a fundamental system of", "neighbourhoods of $x$ in $Z$. After shrinking $X$ we may assume $E$ is", "bounded above. We will argue by induction on", "the largest integer $p$ such that $H^p(E)$ is nonzero. If $p < m$, then", "there is nothing to prove. If $p \\geq m$, then $H^p(E)$ is an", "$\\mathcal{O}_Z$-module of finite type, see", "Cohomology, Lemma \\ref{cohomology-lemma-finite-cohomology}.", "Thus we may choose, after shrinking $X$, a map", "$\\mathcal{O}_Z^{\\oplus n}[-p] \\to E$ which induces a surjection", "$\\mathcal{O}_Z^{\\oplus n} \\to H^p(E)$. Choose a distinguished triangle", "$$", "\\mathcal{O}_Z^{\\oplus n}[-p] \\to E \\to C \\to \\mathcal{O}_Z^{\\oplus n}[-p + 1]", "$$", "We see that $H^j(C) = 0$ for $j \\geq p$ and that $C$ is $m$-pseudo-coherent", "by Cohomology, Lemma \\ref{cohomology-lemma-cone-pseudo-coherent}.", "By induction we see that $i_*C$ is $m$-pseudo-coherent on $X$.", "Since $i_*\\mathcal{O}_Z$ is $m$-pseudo-coherent on $X$ as well, we conclude", "from the distinguished triangle", "$$", "i_*\\mathcal{O}_Z^{\\oplus n}[-p] \\to i_*E \\to i_*C \\to", "i_*\\mathcal{O}_Z^{\\oplus n}[-p + 1]", "$$", "and ", "Cohomology, Lemma \\ref{cohomology-lemma-cone-pseudo-coherent}", "that $i_*E$ is $m$-pseudo-coherent.", "\\medskip\\noindent", "Assume that $i_*E$ is $m$-pseudo-coherent. Let $z \\in Z$.", "We will find a neighbourhood of $z$ such that $E$", "is $m$-pseudo-coherent on it. We will use", "that $U \\cap Z$ for $z \\in U \\subset X$ open form a fundamental system of", "neighbourhoods of $z$ in $Z$. After shrinking $X$ we may assume $i_*E$", "and hence $E$ is bounded above. We will argue by induction on", "the largest integer $p$ such that $H^p(E)$ is nonzero. If $p < m$, then", "there is nothing to prove. If $p \\geq m$, then $H^p(i_*E) = i_*H^p(E)$", "is an $\\mathcal{O}_X$-module of finite type, see", "Cohomology, Lemma \\ref{cohomology-lemma-finite-cohomology}.", "Choose a complex $\\mathcal{E}^\\bullet$ of $\\mathcal{O}_Z$-modules", "representing $E$. We may choose, after shrinking $X$,", "a map $\\alpha : \\mathcal{O}_X^{\\oplus n}[-p] \\to i_*\\mathcal{E}^\\bullet$", "which induces a surjection", "$\\mathcal{O}_X^{\\oplus n} \\to i_*H^p(\\mathcal{E}^\\bullet)$.", "By adjunction we find a map", "$\\alpha : \\mathcal{O}_Z^{\\oplus n}[-p] \\to \\mathcal{E}^\\bullet$", "which induces a surjection", "$\\mathcal{O}_Z^{\\oplus n} \\to H^p(\\mathcal{E}^\\bullet)$.", "Choose a distinguished triangle", "$$", "\\mathcal{O}_Z^{\\oplus n}[-p] \\to E \\to C \\to \\mathcal{O}_Z^{\\oplus n}[-p + 1]", "$$", "We see that $H^j(C) = 0$ for $j \\geq p$. From the distinguished triangle", "$$", "i_*\\mathcal{O}_Z^{\\oplus n}[-p] \\to i_*E \\to i_*C \\to", "i_*\\mathcal{O}_Z^{\\oplus n}[-p + 1]", "$$", "the fact that $i_*\\mathcal{O}_Z$ is pseudo-coherent", "and ", "Cohomology, Lemma \\ref{cohomology-lemma-cone-pseudo-coherent}", "we conclude that $i_*C$ is $m$-pseudo-coherent.", "By induction we conclude that $C$ is $m$-pseudo-coherent.", "By Cohomology, Lemma \\ref{cohomology-lemma-cone-pseudo-coherent}", "again we conclude that $E$ is $m$-pseudo-coherent." ], "refs": [ "modules-lemma-i-star-exact", "cohomology-lemma-finite-cohomology", "cohomology-lemma-cone-pseudo-coherent", "cohomology-lemma-cone-pseudo-coherent", "cohomology-lemma-finite-cohomology", "cohomology-lemma-cone-pseudo-coherent", "cohomology-lemma-cone-pseudo-coherent" ], "ref_ids": [ 13232, 2212, 2207, 2207, 2212, 2207, 2207 ] } ], "ref_ids": [] }, { "id": 6995, "type": "theorem", "label": "perfect-lemma-finite-push-pseudo-coherent", "categories": [ "perfect" ], "title": "perfect-lemma-finite-push-pseudo-coherent", "contents": [ "Let $f : X \\to Y$ be a finite morphism of schemes such that", "$f_*\\mathcal{O}_X$ is pseudo-coherent as an", "$\\mathcal{O}_Y$-module\\footnote{This means that $f$ is pseudo-coherent, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-finite-pseudo-coherent}.}.", "Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. Then $E$ is $m$-pseudo-coherent", "if and only if $Rf_*E$ is $m$-pseudo-coherent." ], "refs": [ "more-morphisms-lemma-finite-pseudo-coherent" ], "proofs": [ { "contents": [ "This is a translation of", "More on Algebra, Lemma \\ref{more-algebra-lemma-finite-push-pseudo-coherent}", "into the language of schemes. To do the translation, use", "Lemmas \\ref{lemma-affine-compare-bounded} and", "\\ref{lemma-pseudo-coherent-affine}." ], "refs": [ "more-algebra-lemma-finite-push-pseudo-coherent", "perfect-lemma-affine-compare-bounded", "perfect-lemma-pseudo-coherent-affine" ], "ref_ids": [ 10154, 6941, 6975 ] } ], "ref_ids": [ 13981 ] }, { "id": 6996, "type": "theorem", "label": "perfect-lemma-lift-quasi-coherent", "categories": [ "perfect" ], "title": "perfect-lemma-lift-quasi-coherent", "contents": [ "Let $X$ be a scheme and let $j : U \\to X$ be a quasi-compact", "open immersion. The functors", "$$", "D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_U)", "\\quad\\text{and}\\quad", "D^+_\\QCoh(\\mathcal{O}_X) \\to D^+_\\QCoh(\\mathcal{O}_U)", "$$", "are essentially surjective. If $X$ is quasi-compact, then the functors", "$$", "D^-_\\QCoh(\\mathcal{O}_X) \\to D^-_\\QCoh(\\mathcal{O}_U)", "\\quad\\text{and}\\quad", "D^b_\\QCoh(\\mathcal{O}_X) \\to D^b_\\QCoh(\\mathcal{O}_U)", "$$", "are essentially surjective." ], "refs": [], "proofs": [ { "contents": [ "The argument preceding the lemma applies for the first case because $Rj_*$", "maps $D_\\QCoh(\\mathcal{O}_U)$ into $D_\\QCoh(\\mathcal{O}_X)$", "by Lemma \\ref{lemma-quasi-coherence-direct-image}.", "It is clear that $Rj_*$ maps", "$D^+_\\QCoh(\\mathcal{O}_U)$ into", "$D^+_\\QCoh(\\mathcal{O}_X)$", "which implies the statement on bounded below complexes.", "Finally, Lemma \\ref{lemma-quasi-coherence-direct-image}", "guarantees that $Rj_*$ maps", "$D^-_\\QCoh(\\mathcal{O}_U)$ into", "$D^-_\\QCoh(\\mathcal{O}_X)$", "if $X$ is quasi-compact. Combining these two we obtain the last statement." ], "refs": [ "perfect-lemma-quasi-coherence-direct-image", "perfect-lemma-quasi-coherence-direct-image" ], "ref_ids": [ 6946, 6946 ] } ], "ref_ids": [] }, { "id": 6997, "type": "theorem", "label": "perfect-lemma-lift-coherent", "categories": [ "perfect" ], "title": "perfect-lemma-lift-coherent", "contents": [ "Let $X$ be a Noetherian scheme and let $j : U \\to X$ be an open immersion.", "The functor", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_U)$", "is essentially surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $K$ be an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_U)$.", "By Proposition \\ref{proposition-DCoh} we can represent $K$ by a bounded", "complex $\\mathcal{F}^\\bullet$ of coherent $\\mathcal{O}_U$-modules.", "Say $\\mathcal{F}^i = 0$ for $i \\not \\in [a, b]$ for some $a \\leq b$.", "Since $j$ is quasi-compact and separated, the terms of the bounded complex", "$j_*\\mathcal{F}^\\bullet$ are quasi-coherent modules on $X$, see", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}.", "We inductively pick a coherent submodule", "$\\mathcal{G}^i \\subset j_*\\mathcal{F}^i$ as follows.", "For $i = a$ we pick any coherent submodule", "$\\mathcal{G}^a \\subset j_*\\mathcal{F}^a$ whose restriction", "to $U$ is $\\mathcal{F}^a$. This is possible by", "Properties, Lemma \\ref{properties-lemma-extend}.", "For $i > a$ we first pick any coherent submodule", "$\\mathcal{H}^i \\subset j_*\\mathcal{F}^i$", "whose restriction to $U$ is $\\mathcal{F}^i$", "and then we set", "$\\mathcal{G}^i = \\Im(\\mathcal{H}^i \\oplus \\mathcal{G}^{i - 1}", "\\to j_*\\mathcal{F}^i)$. It is clear that", "$\\mathcal{G}^\\bullet \\subset j_*\\mathcal{F}^\\bullet$", "is a bounded complex of coherent $\\mathcal{O}_X$-modules", "whose restriction to $U$ is $\\mathcal{F}^\\bullet$ as desired." ], "refs": [ "perfect-proposition-DCoh", "schemes-lemma-push-forward-quasi-coherent", "properties-lemma-extend" ], "ref_ids": [ 7110, 7730, 3019 ] } ], "ref_ids": [] }, { "id": 6998, "type": "theorem", "label": "perfect-lemma-lift-pseudo-coherent", "categories": [ "perfect" ], "title": "perfect-lemma-lift-pseudo-coherent", "contents": [ "Let $X$ be an affine scheme and let $U \\subset X$ be a quasi-compact", "open subscheme. For any pseudo-coherent object $E$ of $D(\\mathcal{O}_U)$", "there exists a bounded above complex of finite free $\\mathcal{O}_X$-modules ", "whose restriction to $U$ is isomorphic to $E$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-pseudo-coherent} we see that $E$ is an object of", "$D_\\QCoh(\\mathcal{O}_U)$. By", "Lemma \\ref{lemma-lift-quasi-coherent}", "we may assume $E = E'|U$ for some object $E'$ of", "$D_\\QCoh(\\mathcal{O}_X)$.", "Write $X = \\Spec(A)$. By Lemma \\ref{lemma-affine-compare-bounded}", "we can find a complex $M^\\bullet$ of $A$-modules whose associated", "complex of $\\mathcal{O}_X$-modules is a representative of $E'$.", "\\medskip\\noindent", "Choose $f_1, \\ldots, f_r \\in A$ such that $U = D(f_1) \\cup \\ldots \\cup D(f_r)$.", "By Lemma \\ref{lemma-pseudo-coherent-affine} the complexes", "$M^\\bullet_{f_j}$ are pseudo-coherent complexes of $A_{f_j}$-modules.", "Let $n$ be an integer. Assume we have a map of complexes", "$\\alpha : F^\\bullet \\to M^\\bullet$ where $F^\\bullet$ is", "bounded above, $F^i = 0$ for $i < n$, each $F^i$ is a finite free", "$R$-module, such that", "$$", "H^i(\\alpha_{f_j}) : H^i(F^\\bullet_{f_j}) \\to H^i(M^\\bullet_{f_j})", "$$", "is an isomorphism for $i > n$ and surjective for $i = n$. Picture", "$$", "\\xymatrix{", "& F^n \\ar[r] \\ar[d]^\\alpha & F^{n + 1} \\ar[d]^\\alpha \\ar[r] & \\ldots \\\\", "M^{n-1} \\ar[r] & M^n \\ar[r] & M^{n + 1} \\ar[r] & \\ldots", "}", "$$", "Since each $M^\\bullet_{f_j}$ has vanishing cohomology", "in large degrees we can find such a map for $n \\gg 0$.", "By induction on $n$ we are going to extend this to a map", "of complexes $F^\\bullet \\to M^\\bullet$", "such that $H^i(\\alpha_{f_j})$ is an isomorphism", "for all $i$. The lemma will follow by taking $\\widetilde{F^\\bullet}$.", "\\medskip\\noindent", "The induction step will be to extend the diagram", "above by adding $F^{n - 1}$. Let $C^\\bullet$ be the cone on $\\alpha$", "(Derived Categories, Definition \\ref{derived-definition-cone}).", "The long exact sequence of cohomology shows that", "$H^i(C^\\bullet_{f_j}) = 0$ for $i \\geq n$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-cone-pseudo-coherent}", "we see that $C^\\bullet_{f_j}$ is $(n - 1)$-pseudo-coherent. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-finite-cohomology}", "we see that $H^{-1}(C^\\bullet_{f_j})$ is a finite $A_{f_j}$-module.", "Choose a finite free $A$-module $F^{n - 1}$ and an $A$-module", "$\\beta : F^{n - 1} \\to C^{-1}$ such that the composition", "$F^{n - 1} \\to C^{n - 1} \\to C^n$ is zero and such that", "$F^{n - 1}_{f_j}$ surjects onto $H^{n - 1}(C^\\bullet_{f_j})$.", "(Some details omitted; hint: clear denominators.)", "Since $C^{n - 1} = M^{n - 1} \\oplus F^n$", "we can write $\\beta = (\\alpha^{n - 1}, -d^{n - 1})$. The vanishing of the", "composition $F^{n - 1} \\to C^{n - 1} \\to C^n$ implies", "these maps fit into a morphism of complexes", "$$", "\\xymatrix{", "& F^{n - 1} \\ar[d]^{\\alpha^{n - 1}} \\ar[r]_{d^{n - 1}} &", "F^n \\ar[r] \\ar[d]^\\alpha &", "F^{n + 1} \\ar[d]^\\alpha \\ar[r] & \\ldots \\\\", "\\ldots \\ar[r] &", "M^{n - 1} \\ar[r] & M^n \\ar[r] & M^{n + 1} \\ar[r] & \\ldots", "}", "$$", "Moreover, these maps define a morphism of distinguished triangles", "$$", "\\xymatrix{", "(F^n \\to \\ldots) \\ar[r] \\ar[d] &", "(F^{n-1} \\to \\ldots) \\ar[r] \\ar[d] &", "F^{n-1} \\ar[r] \\ar[d]_\\beta &", "(F^n \\to \\ldots)[1] \\ar[d] \\\\", "(F^n \\to \\ldots) \\ar[r] &", "M^\\bullet \\ar[r] &", "C^\\bullet \\ar[r] &", "(F^n \\to \\ldots)[1]", "}", "$$", "Hence our choice of $\\beta$ implies that the map of complexes", "$(F^{-1} \\to \\ldots) \\to M^\\bullet$ induces an isomorphism on", "cohomology localized at $f_j$ in degrees $\\geq n$ and a surjection", "in degree $-1$. This finishes the proof of the lemma." ], "refs": [ "perfect-lemma-pseudo-coherent", "perfect-lemma-lift-quasi-coherent", "perfect-lemma-affine-compare-bounded", "perfect-lemma-pseudo-coherent-affine", "derived-definition-cone", "more-algebra-lemma-cone-pseudo-coherent", "more-algebra-lemma-finite-cohomology" ], "ref_ids": [ 6974, 6996, 6941, 6975, 1978, 10145, 10146 ] } ], "ref_ids": [] }, { "id": 6999, "type": "theorem", "label": "perfect-lemma-vanishing-ext", "categories": [ "perfect" ], "title": "perfect-lemma-vanishing-ext", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $E \\in D^b_\\QCoh(\\mathcal{O}_X)$.", "There exists an integer $n_0 > 0$ such that", "$\\Ext^n_{D(\\mathcal{O}_X)}(\\mathcal{E}, E) = 0$", "for every finite locally free", "$\\mathcal{O}_X$-module $\\mathcal{E}$ and every $n \\geq n_0$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\Ext^n_{D(\\mathcal{O}_X)}(\\mathcal{E}, E) =", "\\Hom_{D(\\mathcal{O}_X)}(\\mathcal{E}, E[n])$. We have", "Mayer-Vietoris for morphisms in the derived category, see", "Cohomology, Lemma \\ref{cohomology-lemma-mayer-vietoris-hom}.", "Thus if $X = U \\cup V$ and the result of the lemma holds", "for $E|_U$, $E|_V$, and $E|_{U \\cap V}$ for some bound $n_0$,", "then the result holds for $E$ with bound $n_0 + 1$.", "Thus it suffices to prove the lemma when $X$ is affine, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}.", "\\medskip\\noindent", "Assume $X = \\Spec(A)$ is affine. Choose a complex of $A$-modules", "$M^\\bullet$ whose associated complex of quasi-coherent modules", "represents $E$, see Lemma \\ref{lemma-affine-compare-bounded}.", "Write $\\mathcal{E} = \\widetilde{P}$ for some $A$-module $P$.", "Since $\\mathcal{E}$ is finite locally free, we see that $P$", "is a finite projective $A$-module. We have", "\\begin{align*}", "\\Hom_{D(\\mathcal{O}_X)}(\\mathcal{E}, E[n])", "& = ", "\\Hom_{D(A)}(P, M^\\bullet[n]) \\\\", "& =", "\\Hom_{K(A)}(P, M^\\bullet[n]) \\\\", "& =", "\\Hom_A(P, H^n(M^\\bullet))", "\\end{align*}", "The first equality by Lemma \\ref{lemma-affine-compare-bounded},", "the second equality by", "Derived Categories, Lemma", "\\ref{derived-lemma-morphisms-from-projective-complex}, and", "the final equality because $\\Hom_A(P, -)$ is an exact functor.", "As $E$ and hence $M^\\bullet$ is bounded", "we get zero for all sufficiently large $n$." ], "refs": [ "cohomology-lemma-mayer-vietoris-hom", "coherent-lemma-induction-principle", "perfect-lemma-affine-compare-bounded", "perfect-lemma-affine-compare-bounded", "derived-lemma-morphisms-from-projective-complex" ], "ref_ids": [ 2145, 3291, 6941, 6941, 1862 ] } ], "ref_ids": [] }, { "id": 7000, "type": "theorem", "label": "perfect-lemma-lift-perfect-complex-plus-locally-free", "categories": [ "perfect" ], "title": "perfect-lemma-lift-perfect-complex-plus-locally-free", "contents": [ "Let $X$ be an affine scheme. Let $U \\subset X$ be a quasi-compact open.", "For every perfect object $E$ of $D(\\mathcal{O}_U)$ there exists an integer", "$r$ and a finite locally free sheaf $\\mathcal{F}$ on $U$ such that", "$\\mathcal{F}[-r] \\oplus E$ is the restriction of a perfect object of", "$D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Say $X = \\Spec(A)$. Recall that a perfect complex is", "pseudo-coherent, see", "Cohomology, Lemma \\ref{cohomology-lemma-perfect}.", "By Lemma \\ref{lemma-lift-pseudo-coherent} we can find a bounded above complex", "$\\mathcal{F}^\\bullet$ of finite free $A$-modules such that $E$ is", "isomorphic to $\\mathcal{F}^\\bullet|_U$ in $D(\\mathcal{O}_U)$.", "By Cohomology, Lemma \\ref{cohomology-lemma-perfect} and since", "$U$ is quasi-compact, we see that $E$ has finite tor dimension, say", "$E$ has tor amplitude in $[a, b]$. Pick $r < a$ and set", "$$", "\\mathcal{F} = \\Ker(\\mathcal{F}^{r} \\to \\mathcal{F}^{r + 1})", "= \\Im(\\mathcal{F}^{r - 1} \\to \\mathcal{F}^r).", "$$", "Since $E$ has tor amplitude in $[a, b]$ we see that $\\mathcal{F}|_U$ is", "flat (Cohomology, Lemma \\ref{cohomology-lemma-last-one-flat}).", "Hence $\\mathcal{F}|_U$ is flat and of finite presentation, thus finite", "locally free (Properties, Lemma \\ref{properties-lemma-finite-locally-free}).", "It follows that", "$$", "(\\mathcal{F} \\to \\mathcal{F}^r \\to \\mathcal{F}^{r + 1} \\to \\ldots )|_U", "$$", "is a strictly perfect complex on $U$ representing $E$.", "We obtain a distinguished triangle", "$$", "\\mathcal{F}|_U[- r - 1] \\to E \\to", "(\\mathcal{F}^r \\to \\mathcal{F}^{r + 1} \\to \\ldots )|_U \\to", "\\mathcal{F}|_U[- r]", "$$", "Note that $(\\mathcal{F}^r \\to \\mathcal{F}^{r + 1} \\to \\ldots )$ is", "a perfect complex on $X$. To finish the proof it suffices to pick $r$", "such that the map", "$\\mathcal{F}|_U[- r - 1] \\to E$ is zero in $D(\\mathcal{O}_U)$, see", "Derived Categories, Lemma \\ref{derived-lemma-split}. By", "Lemma \\ref{lemma-vanishing-ext} this holds if $r \\ll 0$." ], "refs": [ "cohomology-lemma-perfect", "perfect-lemma-lift-pseudo-coherent", "cohomology-lemma-perfect", "cohomology-lemma-last-one-flat", "properties-lemma-finite-locally-free", "derived-lemma-split", "perfect-lemma-vanishing-ext" ], "ref_ids": [ 2224, 6998, 2224, 2214, 3014, 1766, 6999 ] } ], "ref_ids": [] }, { "id": 7001, "type": "theorem", "label": "perfect-lemma-lift-map", "categories": [ "perfect" ], "title": "perfect-lemma-lift-map", "contents": [ "Let $X$ be an affine scheme. Let $U \\subset X$ be a quasi-compact open.", "Let $E, E'$ be objects of $D_\\QCoh(\\mathcal{O}_X)$ with $E$ perfect.", "For every map $\\alpha : E|_U \\to E'|_U$ there exist maps", "$$", "E \\xleftarrow{\\beta} E_1 \\xrightarrow{\\gamma} E'", "$$", "of perfect complexes on $X$ such that $\\beta : E_1 \\to E$ restricts to an", "isomorphism on $U$ and such that $\\alpha = \\gamma|_U \\circ \\beta|_U^{-1}$.", "Moreover we can assume $E_1 = E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} I$", "for some perfect complex $I$ on $X$." ], "refs": [], "proofs": [ { "contents": [ "Write $X = \\Spec(A)$. Write $U = D(f_1) \\cup \\ldots \\cup D(f_r)$. Choose", "finite complex of finite projective $A$-modules $M^\\bullet$ representing", "$E$ (Lemma \\ref{lemma-perfect-affine}). Choose a complex of $A$-modules", "$(M')^\\bullet$ representing $E'$ (Lemma \\ref{lemma-affine-compare-bounded}).", "In this case the complex $H^\\bullet = \\Hom_A(M^\\bullet, (M')^\\bullet)$", "is a complex of $A$-modules whose associated complex of quasi-coherent", "$\\mathcal{O}_X$-modules represents $R\\SheafHom(E, E')$, see", "Cohomology, Lemma \\ref{cohomology-lemma-Rhom-strictly-perfect}.", "Then $\\alpha$ determines an element $s$ of $H^0(U, R\\SheafHom(E, E'))$, see", "Cohomology, Lemma \\ref{cohomology-lemma-section-RHom-over-U}.", "There exists an $e$ and a map", "$$", "\\xi : I^\\bullet(f_1^e, \\ldots, f_r^e) \\to \\Hom_A(M^\\bullet, (M')^\\bullet)", "$$", "corresponding to $s$, see", "Proposition \\ref{proposition-represent-cohomology-class-on-open}.", "Letting $E_1$ be the object corresponding to", "complex of quasi-coherent $\\mathcal{O}_X$-modules", "associated to", "$$", "\\text{Tot}(I^\\bullet(f_1^e, \\ldots, f_r^e) \\otimes_A M^\\bullet)", "$$", "we obtain $E_1 \\to E$ using the canonical map", "$I^\\bullet(f_1^e, \\ldots, f_r^e) \\to A$ and $E_1 \\to E'$", "using $\\xi$ and", "Cohomology, Lemma \\ref{cohomology-lemma-section-RHom-over-U}." ], "refs": [ "perfect-lemma-perfect-affine", "perfect-lemma-affine-compare-bounded", "cohomology-lemma-Rhom-strictly-perfect", "perfect-proposition-represent-cohomology-class-on-open" ], "ref_ids": [ 6980, 6941, 2203, 7109 ] } ], "ref_ids": [] }, { "id": 7002, "type": "theorem", "label": "perfect-lemma-lift-perfect-complex-plus-shift", "categories": [ "perfect" ], "title": "perfect-lemma-lift-perfect-complex-plus-shift", "contents": [ "Let $X$ be an affine scheme. Let $U \\subset X$ be a quasi-compact open.", "For every perfect object $F$ of $D(\\mathcal{O}_U)$", "the object $F \\oplus F[1]$ is the restriction of", "a perfect object of $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-lift-perfect-complex-plus-locally-free}", "we can find a perfect object $E$ of $D(\\mathcal{O}_X)$", "such that $E|_U = \\mathcal{F}[r] \\oplus F$ for some finite locally", "free $\\mathcal{O}_U$-module $\\mathcal{F}$.", "By Lemma \\ref{lemma-lift-map} we can find a morphism of", "perfect complexes $\\alpha : E_1 \\to E$ such that $(E_1)|_U \\cong E|_U$", "and such that $\\alpha|_U$ is the map", "$$", "\\left(", "\\begin{matrix}", "\\text{id}_{\\mathcal{F}[r]} & 0 \\\\", "0 & 0", "\\end{matrix}", "\\right)", ":", "\\mathcal{F}[r] \\oplus F \\to \\mathcal{F}[r] \\oplus F", "$$", "Then the cone on $\\alpha$ is a solution." ], "refs": [ "perfect-lemma-lift-perfect-complex-plus-locally-free", "perfect-lemma-lift-map" ], "ref_ids": [ 7000, 7001 ] } ], "ref_ids": [] }, { "id": 7003, "type": "theorem", "label": "perfect-lemma-perfect-into-support-on-T", "categories": [ "perfect" ], "title": "perfect-lemma-perfect-into-support-on-T", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $f \\in \\Gamma(X, \\mathcal{O}_X)$.", "For any morphism $\\alpha : E \\to E'$ in", "$D_\\QCoh(\\mathcal{O}_X)$ such that", "\\begin{enumerate}", "\\item $E$ is perfect, and", "\\item $E'$ is supported on $T = V(f)$", "\\end{enumerate}", "there exists an $n \\geq 0$ such that $f^n \\alpha = 0$." ], "refs": [], "proofs": [ { "contents": [ "We have Mayer-Vietoris for morphisms in the derived category, see", "Cohomology, Lemma \\ref{cohomology-lemma-mayer-vietoris-hom}.", "Thus if $X = U \\cup V$ and the result of the lemma holds", "for $f|_U$, $f|_V$, and $f|_{U \\cap V}$, then the result holds for $f$.", "Thus it suffices to prove the lemma when $X$ is affine, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}.", "\\medskip\\noindent", "Let $X = \\Spec(A)$. Then $f \\in A$. We will", "use the equivalence $D(A) = D_\\QCoh(X)$ of", "Lemma \\ref{lemma-affine-compare-bounded}", "without further mention.", "Represent $E$ by a finite complex of finite projective $A$-modules", "$P^\\bullet$. This is possible by Lemma \\ref{lemma-perfect-affine}.", "Let $t$ be the largest integer such that $P^t$ is nonzero.", "The distinguished triangle", "$$", "P^t[-t] \\to P^\\bullet \\to \\sigma_{\\leq t - 1}P^\\bullet \\to P^t[-t + 1]", "$$", "shows that by induction on the length of the complex $P^\\bullet$", "we can reduce to the case where $P^\\bullet$ has a single nonzero term.", "This and the shift functor reduces us to the case where $P^\\bullet$", "consists of a single finite projective $A$-module $P$ in degree $0$.", "Represent $E'$ by a complex $M^\\bullet$ of $A$-modules.", "Then $\\alpha$ corresponds to a map $P \\to H^0(M^\\bullet)$.", "Since the module $H^0(M^\\bullet)$ is supported on $V(f)$ by assumption (2)", "we see that every element of $H^0(M^\\bullet)$ is annihilated by a power", "of $f$. Since $P$ is a finite $A$-module the map", "$f^n\\alpha : P \\to H^0(M^\\bullet)$ is zero for some $n$ as desired." ], "refs": [ "cohomology-lemma-mayer-vietoris-hom", "coherent-lemma-induction-principle", "perfect-lemma-affine-compare-bounded", "perfect-lemma-perfect-affine" ], "ref_ids": [ 2145, 3291, 6941, 6980 ] } ], "ref_ids": [] }, { "id": 7004, "type": "theorem", "label": "perfect-lemma-lift-perfect-complex-plus-shift-support", "categories": [ "perfect" ], "title": "perfect-lemma-lift-perfect-complex-plus-shift-support", "contents": [ "Let $X$ be an affine scheme. Let $T \\subset X$ be a closed subset", "such that $X \\setminus T$ is quasi-compact. Let $U \\subset X$ be a", "quasi-compact open. For every perfect object $F$ of $D(\\mathcal{O}_U)$", "supported on $T \\cap U$ the object $F \\oplus F[1]$ is the restriction of", "a perfect object $E$ of $D(\\mathcal{O}_X)$ supported in $T$." ], "refs": [], "proofs": [ { "contents": [ "Say $T = V(g_1, \\ldots, g_s)$. After replacing $g_j$ by a power we", "may assume multiplication by $g_j$ is zero on $F$, see", "Lemma \\ref{lemma-perfect-into-support-on-T}. Choose $E$ as in", "Lemma \\ref{lemma-lift-perfect-complex-plus-shift}.", "Note that $g_j : E \\to E$ restricts to zero on $U$.", "Choose a distinguished triangle", "$$", "E \\xrightarrow{g_1} E \\to C_1 \\to E[1]", "$$", "By Derived Categories, Lemma \\ref{derived-lemma-split}", "the object $C_1$ restricts to", "$F \\oplus F[1] \\oplus F[1] \\oplus F[2]$ on $U$.", "Moreover, $g_1 : C_1 \\to C_1$ has square zero by", "Derived Categories, Lemma \\ref{derived-lemma-third-map-square-zero}.", "Namely, the diagram", "$$", "\\xymatrix{", "E \\ar[r] \\ar[d]_0 & C_1 \\ar[d]_{g_1} \\ar[r] & E[1] \\ar[d]_0 \\\\", "E \\ar[r] & C_1 \\ar[r] & E[1]", "}", "$$", "is commutative since the compositions $E \\xrightarrow{g_1} E \\to C_1$ and", "$C_1 \\to E[1] \\xrightarrow{g_1} E[1]$ are zero. Continuing, setting", "$C_{i + 1}$ equal to the cone of the map $g_i : C_i \\to C_i$ we obtain", "a perfect complex $C_s$ on $X$ supported on $T$", "whose restriction to $U$ gives", "$$", "F \\oplus F[1]^{\\oplus s} \\oplus F[2]^{\\oplus {s \\choose 2}}", "\\oplus \\ldots \\oplus F[s]", "$$", "Choose morphisms of perfect complexes $\\beta : C' \\to C_s$", "and $\\gamma : C' \\to C_s$ as in Lemma \\ref{lemma-lift-map}", "such that $\\beta|_U$ is an isomorphism and such that", "$\\gamma|_U \\circ \\beta|_U^{-1}$ is the morphism", "$$", "F \\oplus F[1]^{\\oplus s} \\oplus F[2]^{\\oplus {s \\choose 2}}", "\\oplus \\ldots \\oplus F[s]", "\\to", "F \\oplus F[1]^{\\oplus s} \\oplus F[2]^{\\oplus {s \\choose 2}}", "\\oplus \\ldots \\oplus F[s]", "$$", "which is the identity on all summands except for $F$ where it is zero.", "By Lemma \\ref{lemma-lift-map} we also have", "$C' = C_s \\otimes^\\mathbf{L} I$ for some perfect complex", "$I$ on $X$. Hence the nullity of $g_j^2\\text{id}_{C_s}$ implies the", "same thing for $C'$. Thus $C'$ is supported on $T$ as well.", "Then $\\text{Cone}(\\gamma)$ is a solution." ], "refs": [ "perfect-lemma-perfect-into-support-on-T", "perfect-lemma-lift-perfect-complex-plus-shift", "derived-lemma-split", "derived-lemma-third-map-square-zero", "perfect-lemma-lift-map", "perfect-lemma-lift-map" ], "ref_ids": [ 7003, 7002, 1766, 1760, 7001, 7001 ] } ], "ref_ids": [] }, { "id": 7005, "type": "theorem", "label": "perfect-lemma-lift-map-from-perfect-complex-with-support", "categories": [ "perfect" ], "title": "perfect-lemma-lift-map-from-perfect-complex-with-support", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $U \\subset X$ be a quasi-compact open. Let $T \\subset X$", "be a closed subset with $X \\setminus T$ retro-compact in $X$.", "Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$.", "Let $\\alpha : P \\to E|_U$ be a map where $P$ is a perfect object of", "$D(\\mathcal{O}_U)$ supported on $T \\cap U$. Then there exists a map", "$\\beta : R \\to E$ where $R$ is a perfect object of $D(\\mathcal{O}_X)$", "supported on $T$ such that $P$ is a direct summand of $R|_U$ in", "$D(\\mathcal{O}_U)$ compatible $\\alpha$ and $\\beta|_U$." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is quasi-compact there exists an integer $m$ such that", "$X = U \\cup V_1 \\cup \\ldots \\cup V_m$ for some affine opens $V_j$ of $X$.", "Arguing by induction on $m$ we see that we may assume $m = 1$. In other", "words, we may assume that $X = U \\cup V$ with $V$ affine. By", "Lemma \\ref{lemma-lift-perfect-complex-plus-shift-support}", "we can choose a perfect object $Q$ in $D(\\mathcal{O}_V)$", "supported on $T \\cap V$ and an isomorphism", "$Q|_{U \\cap V} \\to (P \\oplus P[1])|_{U \\cap V}$.", "By Lemma \\ref{lemma-lift-map} we can replace $Q$ by", "$Q \\otimes^\\mathbf{L} I$ (still supported on $T \\cap V$)", "and assume that the map", "$$", "Q|_{U \\cap V} \\to (P \\oplus P[1])|_{U \\cap V}", "\\longrightarrow P|_{U \\cap V}", "\\longrightarrow", "E|_{U \\cap V}", "$$", "lifts to $Q \\to E|_V$. By", "Cohomology, Lemma \\ref{cohomology-lemma-glue}", "we find an morphism $a : R \\to E$ of $D(\\mathcal{O}_X)$", "such that $a|_U$ is isomorphic to $P \\oplus P[1] \\to E|_U$", "and $a|_V$ isomorphic to $Q \\to E|_V$.", "Thus $R$ is perfect and supported on $T$ as desired." ], "refs": [ "perfect-lemma-lift-perfect-complex-plus-shift-support", "perfect-lemma-lift-map", "cohomology-lemma-glue" ], "ref_ids": [ 7004, 7001, 2191 ] } ], "ref_ids": [] }, { "id": 7006, "type": "theorem", "label": "perfect-lemma-open", "categories": [ "perfect" ], "title": "perfect-lemma-open", "contents": [ "Let $X$ be a scheme. Let $U \\subset X$ be an open subscheme.", "Let $(T, E, m)$ be a triple as in", "Definition \\ref{definition-approximation-holds}.", "If", "\\begin{enumerate}", "\\item $T \\subset U$,", "\\item approximation holds for $(T, E|_U, m)$, and", "\\item the sheaves $H^i(E)$ for $i \\geq m$ are supported on $T$,", "\\end{enumerate}", "then approximation holds for $(T, E, m)$." ], "refs": [ "perfect-definition-approximation-holds" ], "proofs": [ { "contents": [ "Let $j : U \\to X$ be the inclusion morphism.", "If $P \\to E|_U$ is an approximation of the triple $(T, E|_U, m)$", "over $U$, then $j_!P = Rj_*P \\to j_!(E|_U) \\to E$ is an approximation", "of $(T, E, m)$ over $X$.", "See Cohomology, Lemmas \\ref{cohomology-lemma-pushforward-restriction} and", "\\ref{cohomology-lemma-pushforward-perfect}." ], "refs": [ "cohomology-lemma-pushforward-restriction", "cohomology-lemma-pushforward-perfect" ], "ref_ids": [ 2148, 2229 ] } ], "ref_ids": [ 7116 ] }, { "id": 7007, "type": "theorem", "label": "perfect-lemma-approximation-affine", "categories": [ "perfect" ], "title": "perfect-lemma-approximation-affine", "contents": [ "Let $X$ be an affine scheme. Then approximation holds for every", "triple $(T, E, m)$ as in Definition \\ref{definition-approximation-holds}", "such that there exists an integer $r \\geq 0$ with", "\\begin{enumerate}", "\\item $E$ is $m$-pseudo-coherent,", "\\item $H^i(E)$ is supported on $T$ for $i \\geq m - r + 1$,", "\\item $X \\setminus T$ is the union of $r$ affine opens.", "\\end{enumerate}", "In particular, approximation by perfect complexes holds for affine schemes." ], "refs": [ "perfect-definition-approximation-holds" ], "proofs": [ { "contents": [ "Say $X = \\Spec(A)$. Write $T = V(f_1, \\ldots, f_r)$.", "(The case $r = 0$, i.e., $T = X$ follows immediately from", "Lemma \\ref{lemma-pseudo-coherent-affine} and the definitions.)", "Let $(T, E, m)$ be a triple as in the lemma.", "Let $t$ be the largest integer such that $H^t(E)$ is nonzero.", "We will proceed by induction on $t$. The base case is $t < m$; in", "this case the result is trivial. Now suppose that $t \\geq m$. By", "Cohomology, Lemma \\ref{cohomology-lemma-finite-cohomology}", "the sheaf $H^t(E)$ is of finite type. Since it is quasi-coherent", "it is generated by finitely many sections", "(Properties, Lemma \\ref{properties-lemma-finite-type-module}).", "For every $s \\in \\Gamma(X, H^t(E)) = H^t(X, E)$", "(see proof of Lemma \\ref{lemma-affine-compare-bounded})", "we can find an $e > 0$ and a morphism $K_e[-t] \\to E$", "such that $s$ is in the image of", "$H^0(K_e) = H^t(K_e[-t]) \\to H^t(E)$, see", "Lemma \\ref{lemma-represent-cohomology-class-on-closed}.", "Taking a finite direct sum of these maps we obtain a map", "$P \\to E$ where $P$ is a perfect complex supported on $T$,", "where $H^i(P) = 0$ for $i > t$, and where $H^t(P) \\to E$ is", "surjective. Choose a distinguished triangle", "$$", "P \\to E \\to E' \\to P[1]", "$$", "Then $E'$ is $m$-pseudo-coherent", "(Cohomology, Lemma \\ref{cohomology-lemma-cone-pseudo-coherent}),", "$H^i(E') = 0$ for $i \\geq t$, and", "$H^i(E')$ is supported on $T$ for $i \\geq m - r + 1$.", "By induction we find an approximation $P' \\to E'$", "of $(T, E', m)$. Fit the composition $P' \\to E' \\to P[1]$", "into a distinguished triangle $P \\to P'' \\to P' \\to P[1]$", "and extend the morphisms $P' \\to E'$ and $P[1] \\to P[1]$ into", "a morphism of distinguished triangles", "$$", "\\xymatrix{", "P \\ar[r] \\ar[d] & P'' \\ar[d] \\ar[r] & P' \\ar[d] \\ar[r] & P[1] \\ar[d] \\\\", "P \\ar[r] & E \\ar[r] & E' \\ar[r] & P[1]", "}", "$$", "using TR3. Then $P''$ is a perfect complex", "(Cohomology, Lemma \\ref{cohomology-lemma-two-out-of-three-perfect})", "supported on $T$.", "An easy diagram chase shows that $P'' \\to E$ is the desired", "approximation." ], "refs": [ "perfect-lemma-pseudo-coherent-affine", "cohomology-lemma-finite-cohomology", "properties-lemma-finite-type-module", "perfect-lemma-affine-compare-bounded", "perfect-lemma-represent-cohomology-class-on-closed", "cohomology-lemma-cone-pseudo-coherent", "cohomology-lemma-two-out-of-three-perfect" ], "ref_ids": [ 6975, 2212, 3002, 6941, 6973, 2207, 2226 ] } ], "ref_ids": [ 7116 ] }, { "id": 7008, "type": "theorem", "label": "perfect-lemma-induction-step", "categories": [ "perfect" ], "title": "perfect-lemma-induction-step", "contents": [ "Let $X$ be a scheme. Let $X = U \\cup V$ be an open covering", "with $U$ quasi-compact, $V$ affine, and $U \\cap V$ quasi-compact.", "If approximation by perfect complexes holds on $U$,", "then approximation holds on $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $T \\subset X$ be a closed subset with $X \\setminus T$ retro-compact", "in $X$. Let $r_U$ be the integer of Definition \\ref{definition-approximation}", "adapted to the pair $(U, T \\cap U)$.", "Set $T' = T \\setminus U$. Note that", "$T' \\subset V$ and that $V \\setminus T' = (X \\setminus T) \\cap U \\cap V$", "is quasi-compact by our assumption on $T$.", "Let $r'$ be the number of affines needed to cover $V \\setminus T'$.", "We claim that $r = \\max(r_U, r')$ works for the pair $(X, T)$.", "\\medskip\\noindent", "To see this choose a triple $(T, E, m)$ such that $E$ is", "$(m - r)$-pseudo-coherent and $H^i(E)$ is supported on $T$ for", "$i \\geq m - r$. Let $t$ be the largest integer such that", "$H^t(E)|_U$ is nonzero. (Such an integer exists as $U$ is quasi-compact", "and $E|_U$ is $(m - r)$-pseudo-coherent.)", "We will prove that $E$ can be approximated by induction on $t$.", "\\medskip\\noindent", "Base case: $t \\leq m - r'$. This means that $H^i(E)$ is supported", "on $T'$ for $i \\geq m - r'$. Hence", "Lemma \\ref{lemma-approximation-affine}", "guarantees the existence of an approximation", "$P \\to E|_V$ of $(T', E|_V, m)$ on $V$.", "Applying Lemma \\ref{lemma-open} we see that", "$(T', E, m)$ can be approximated. Such an approximation", "is also an approximation of $(T, E, m)$.", "\\medskip\\noindent", "Induction step. Choose an approximation $P \\to E|_U$", "of $(T \\cap U, E|_U, m)$. This in particular gives a surjection", "$H^t(P) \\to H^t(E|_U)$. By", "Lemma \\ref{lemma-lift-perfect-complex-plus-shift-support}", "we can choose a perfect object $Q$ in $D(\\mathcal{O}_V)$", "supported on $T \\cap V$ and an isomorphism", "$Q|_{U \\cap V} \\to (P \\oplus P[1])|_{U \\cap V}$.", "By Lemma \\ref{lemma-lift-map} we can replace $Q$ by", "$Q \\otimes^\\mathbf{L} I$", "and assume that the map", "$$", "Q|_{U \\cap V} \\to (P \\oplus P[1])|_{U \\cap V}", "\\longrightarrow P|_{U \\cap V}", "\\longrightarrow", "E|_{U \\cap V}", "$$", "lifts to $Q \\to E|_V$. By", "Cohomology, Lemma \\ref{cohomology-lemma-glue}", "we find an morphism $a : R \\to E$ of $D(\\mathcal{O}_X)$", "such that $a|_U$ is isomorphic to $P \\oplus P[1] \\to E|_U$", "and $a|_V$ isomorphic to $Q \\to E|_V$.", "Thus $R$ is perfect and supported on $T$", "and the map $H^t(R) \\to H^t(E)$ is surjective on restriction to $U$.", "Choose a distinguished triangle", "$$", "R \\to E \\to E' \\to R[1]", "$$", "Then $E'$ is $(m - r)$-pseudo-coherent", "(Cohomology, Lemma \\ref{cohomology-lemma-cone-pseudo-coherent}),", "$H^i(E')|_U = 0$ for $i \\geq t$, and", "$H^i(E')$ is supported on $T$ for $i \\geq m - r$.", "By induction we find an approximation $R' \\to E'$", "of $(T, E', m)$. Fit the composition $R' \\to E' \\to R[1]$", "into a distinguished triangle $R \\to R'' \\to R' \\to R[1]$", "and extend the morphisms $R' \\to E'$ and $R[1] \\to R[1]$ into", "a morphism of distinguished triangles", "$$", "\\xymatrix{", "R \\ar[r] \\ar[d] & R'' \\ar[d] \\ar[r] & R' \\ar[d] \\ar[r] & R[1] \\ar[d] \\\\", "R \\ar[r] & E \\ar[r] & E' \\ar[r] & R[1]", "}", "$$", "using TR3. Then $R''$ is a perfect complex", "(Cohomology, Lemma \\ref{cohomology-lemma-two-out-of-three-perfect})", "supported on $T$.", "An easy diagram chase shows that $R'' \\to E$ is the desired", "approximation." ], "refs": [ "perfect-definition-approximation", "perfect-lemma-approximation-affine", "perfect-lemma-open", "perfect-lemma-lift-perfect-complex-plus-shift-support", "perfect-lemma-lift-map", "cohomology-lemma-glue", "cohomology-lemma-cone-pseudo-coherent", "cohomology-lemma-two-out-of-three-perfect" ], "ref_ids": [ 7117, 7007, 7006, 7004, 7001, 2191, 2207, 2226 ] } ], "ref_ids": [] }, { "id": 7009, "type": "theorem", "label": "perfect-lemma-direct-summand-of-a-restriction", "categories": [ "perfect" ], "title": "perfect-lemma-direct-summand-of-a-restriction", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $U$ be a quasi-compact open subscheme.", "Let $P$ be a perfect object of $D(\\mathcal{O}_U)$.", "Then $P$ is a direct summand of the restriction of a perfect", "object of $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-lift-map-from-perfect-complex-with-support}." ], "refs": [ "perfect-lemma-lift-map-from-perfect-complex-with-support" ], "ref_ids": [ 7005 ] } ], "ref_ids": [] }, { "id": 7010, "type": "theorem", "label": "perfect-lemma-orthogonal-koszul-complex", "categories": [ "perfect" ], "title": "perfect-lemma-orthogonal-koszul-complex", "contents": [ "\\begin{reference}", "\\cite[Proposition 6.1]{Bokstedt-Neeman}", "\\end{reference}", "In Situation \\ref{situation-complex} denote $j : U \\to X$ the open", "immersion and let $K$ be the perfect object of $D(\\mathcal{O}_X)$", "corresponding to the Koszul complex on $f_1, \\ldots, f_r$ over $A$.", "For $E \\in D_\\QCoh(\\mathcal{O}_X)$ the following are equivalent", "\\begin{enumerate}", "\\item $E = Rj_*(E|_U)$, and", "\\item $\\Hom_{D(\\mathcal{O}_X)}(K[n], E) = 0$ for all $n \\in \\mathbf{Z}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a distinguished triangle $E \\to Rj_*(E|_U) \\to N \\to E[1]$.", "Observe that", "$$", "\\Hom_{D(\\mathcal{O}_X)}(K[n], Rj_*(E|_U)) =", "\\Hom_{D(\\mathcal{O}_U)}(K|_U[n], E) = 0", "$$", "for all $n$ as $K|_U = 0$. Thus it suffices to prove the result for", "$N$. In other words, we may assume that $E$ restricts to zero on $U$.", "Observe that there are distinguished triangles", "$$", "K^\\bullet(f_1^{e_1}, \\ldots, f_i^{e'_i}, \\ldots, f_r^{e_r}) \\to", "K^\\bullet(f_1^{e_1}, \\ldots, f_i^{e'_i + e''_i}, \\ldots, f_r^{e_r}) \\to", "K^\\bullet(f_1^{e_1}, \\ldots, f_i^{e''_i}, \\ldots, f_r^{e_r}) \\to \\ldots", "$$", "of Koszul complexes, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-koszul-mult}.", "Hence if $\\Hom_{D(\\mathcal{O}_X)}(K[n], E) = 0$ for all $n \\in \\mathbf{Z}$", "then the same thing is true for the $K$ replaced by", "$K_e$ as in Lemma \\ref{lemma-represent-cohomology-class-on-closed}.", "Thus our lemma follows immediately from that one and the fact that $E$", "is determined by the complex of $A$-modules $R\\Gamma(X, E)$, see", "Lemma \\ref{lemma-affine-compare-bounded}." ], "refs": [ "more-algebra-lemma-koszul-mult", "perfect-lemma-represent-cohomology-class-on-closed", "perfect-lemma-affine-compare-bounded" ], "ref_ids": [ 9965, 6973, 6941 ] } ], "ref_ids": [] }, { "id": 7011, "type": "theorem", "label": "perfect-lemma-generator-with-support", "categories": [ "perfect" ], "title": "perfect-lemma-generator-with-support", "contents": [ "\\begin{reference}", "\\cite[Theorem 6.8]{Rouquier-dimensions}", "\\end{reference}", "Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \\subset X$ be a", "closed subset such that $X \\setminus T$ is quasi-compact. With notation", "as above, the category $D_{\\QCoh, T}(\\mathcal{O}_X)$ is generated by a", "single perfect object." ], "refs": [], "proofs": [ { "contents": [ "We will prove this using the induction principle of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}.", "\\medskip\\noindent", "Assume $X = \\Spec(A)$ is affine. In this case there exist", "$f_1, \\ldots, f_r \\in A$ such that $T = V(f_1, \\ldots, f_r)$.", "Let $K$ be the Koszul complex on $f_1, \\ldots, f_r$ as in", "Lemma \\ref{lemma-orthogonal-koszul-complex}.", "Then $K$ is a perfect object with cohomology supported on", "$T$ and hence a perfect object of $D_{\\QCoh, T}(\\mathcal{O}_X)$.", "On the other hand, if $E \\in D_{\\QCoh, T}(\\mathcal{O}_X)$ and", "$\\Hom(K, E[n]) = 0$ for all $n$, then", "Lemma \\ref{lemma-orthogonal-koszul-complex}", "tells us that $E = Rj_*(E|_{X \\setminus T}) = 0$.", "Hence $K$ generates $D_{\\QCoh, T}(\\mathcal{O}_X)$,", "(by our definition of generators of triangulated categories in", "Derived Categories, Definition \\ref{derived-definition-generators}).", "\\medskip\\noindent", "Assume that $X = U \\cup V$ is an open covering with $V$ affine and", "$U$ quasi-compact such that the lemma holds for $U$.", "Let $P$ be a perfect object of $D(\\mathcal{O}_U)$ supported on $T \\cap U$", "which is a generator for $D_{\\QCoh, T \\cap U}(\\mathcal{O}_U)$. Using", "Lemma \\ref{lemma-lift-map-from-perfect-complex-with-support}", "we may choose a perfect object $Q$ of $D(\\mathcal{O}_X)$ supported on $T$", "whose restriction to $U$ is a direct sum one of whose summands is $P$.", "Write $V = \\Spec(B)$. Let $Z = X \\setminus U$. Then $Z$ is a closed subset", "of $V$ such that $V \\setminus Z$ is quasi-compact. As $X$ is quasi-separated,", "it follows that $Z \\cap T$ is a closed subset of $V$ such that", "$W = V \\setminus (Z \\cap T)$ is quasi-compact. Thus we can choose", "$g_1, \\ldots, g_s \\in B$ such that $Z \\cap T = V(g_1, \\ldots, g_r)$.", "Let $K \\in D(\\mathcal{O}_V)$ be the perfect object corresponding to the", "Koszul complex on $g_1, \\ldots, g_s$ over $B$. Note that since $K$ is", "supported on $(Z \\cap T) \\subset V$ closed, the pushforward", "$K' = R(V \\to X)_*K$ is a perfect object of $D(\\mathcal{O}_X)$ whose", "restriction to $V$ is $K$ (see", "Cohomology, Lemma \\ref{cohomology-lemma-pushforward-perfect}).", "We claim that $Q \\oplus K'$ is a generator for", "$D_{\\QCoh, T}(\\mathcal{O}_X)$.", "\\medskip\\noindent", "Let $E$ be an object of $D_{\\QCoh, T}(\\mathcal{O}_X)$ such that", "there are no nontrivial maps from any shift of $Q \\oplus K'$ into $E$.", "By Cohomology, Lemma \\ref{cohomology-lemma-pushforward-restriction}", "we have $K' = R(V \\to X)_! K$ and hence", "$$", "\\Hom_{D(\\mathcal{O}_X)}(K'[n], E) = \\Hom_{D(\\mathcal{O}_V)}(K[n], E|_V)", "$$", "Thus by Lemma \\ref{lemma-orthogonal-koszul-complex} we have", "$E|_V = Rj_*E|_W$ where $j : W \\to V$ is the inclusion. Picture", "$$", "\\xymatrix{", "W \\ar[r]_j & V & Z \\cap T \\ar[l] \\ar[d] \\\\", "U \\cap V \\ar[u]^{j'} \\ar[ru]_{j''} & & Z \\ar[lu]", "}", "$$", "Since $E$ is supported on $T$ we see that $E|_W$ is supported on", "$T \\cap W = T \\cap U \\cap V$ which is closed in $W$.", "We conclude that", "$$", "E|_V = Rj_*(E|_W) = Rj_*(Rj'_*(E|_{U \\cap V})) = Rj''_*(E|_{U \\cap V})", "$$", "where the second equality is part (1) of", "Cohomology, Lemma \\ref{cohomology-lemma-pushforward-restriction}.", "This implies that $E = R(U \\to X)_*E|_U$ (small detail omitted). If", "this is the case then", "$$", "\\Hom_{D(\\mathcal{O}_X)}(Q[n], E) = \\Hom_{D(\\mathcal{O}_U)}(Q|_U[n], E|_U)", "$$", "which contains $\\Hom_{D(\\mathcal{O}_U)}(P[n], E|_U)$ as a direct summand.", "Thus by our choice of $P$ the vanishing of these groups implies that $E|_U$", "is zero. Whence $E$ is zero." ], "refs": [ "coherent-lemma-induction-principle", "perfect-lemma-orthogonal-koszul-complex", "perfect-lemma-orthogonal-koszul-complex", "derived-definition-generators", "perfect-lemma-lift-map-from-perfect-complex-with-support", "cohomology-lemma-pushforward-perfect", "cohomology-lemma-pushforward-restriction", "perfect-lemma-orthogonal-koszul-complex", "cohomology-lemma-pushforward-restriction" ], "ref_ids": [ 3291, 7010, 7010, 2003, 7005, 2229, 2148, 7010, 2148 ] } ], "ref_ids": [] }, { "id": 7012, "type": "theorem", "label": "perfect-lemma-nonzero-some-cohomology", "categories": [ "perfect" ], "title": "perfect-lemma-nonzero-some-cohomology", "contents": [ "Let $X$ be a scheme and $\\mathcal{L}$ an ample invertible", "$\\mathcal{O}_X$-module. If $K$ is a nonzero object of", "$D_\\QCoh(\\mathcal{O}_X)$, then for some $n \\geq 0$ and $p \\in \\mathbf{Z}$", "the cohomology group", "$H^p(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{L}^{\\otimes n})$", "is nonzero." ], "refs": [], "proofs": [ { "contents": [ "Recall that as $X$ has an ample invertible sheaf, it is quasi-compact", "and separated (Properties, Definition \\ref{properties-definition-ample} and", "Lemma \\ref{properties-lemma-affine-s-opens-cover-quasi-separated}).", "Thus we may apply", "Proposition \\ref{proposition-quasi-compact-affine-diagonal}", "and represent $K$ by a complex $\\mathcal{F}^\\bullet$ of", "quasi-coherent modules. Pick any $p$ such that", "$\\mathcal{H}^p = \\Ker(\\mathcal{F}^p \\to \\mathcal{F}^{p + 1})/", "\\Im(\\mathcal{F}^{p - 1} \\to \\mathcal{F}^p)$ is nonzero.", "Choose a point $x \\in X$ such that the stalk $\\mathcal{H}^p_x$ is", "nonzero. Choose an $n \\geq 0$ and $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$", "such that $X_s$ is an affine open neighbourhood of $x$.", "Choose $\\tau \\in \\mathcal{H}^p(X_s)$ which maps to a nonzero", "element of the stalk $\\mathcal{H}^p_x$; this is possible", "as $\\mathcal{H}^p$ is quasi-coherent and $X_s$ is affine.", "Since taking sections over $X_s$ is an exact functor on", "quasi-coherent modules, we can find a section $\\tau' \\in \\mathcal{F}^p(X_s)$", "mapping to zero in $\\mathcal{F}^{p + 1}(X_s)$ and mapping to", "$\\tau$ in $\\mathcal{H}^p(X_s)$. By", "Properties, Lemma \\ref{properties-lemma-invert-s-sections}", "there exists an $m$ such that $\\tau' \\otimes s^{\\otimes m}$", "is the image of a section", "$\\tau'' \\in \\Gamma(X, \\mathcal{F}^p \\otimes \\mathcal{L}^{\\otimes mn})$.", "Applying the same lemma once more, we find $l \\geq 0$ such that", "$\\tau'' \\otimes s^{\\otimes l}$ maps to zero in", "$\\mathcal{F}^{p + 1} \\otimes \\mathcal{L}^{\\otimes (m + l)n}$.", "Then $\\tau''$ gives a nonzero class in", "$H^p(X, K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{L}^{(m + l)n})$", "as desired." ], "refs": [ "properties-definition-ample", "properties-lemma-affine-s-opens-cover-quasi-separated", "perfect-proposition-quasi-compact-affine-diagonal", "properties-lemma-invert-s-sections" ], "ref_ids": [ 3088, 3045, 7107, 3005 ] } ], "ref_ids": [] }, { "id": 7013, "type": "theorem", "label": "perfect-lemma-construct-the-next-one", "categories": [ "perfect" ], "title": "perfect-lemma-construct-the-next-one", "contents": [ "Let $A$ be a ring. Let $X = \\mathbf{P}^n_A$. For every $a \\in \\mathbf{Z}$", "there exists an exact complex", "$$", "0 \\to \\mathcal{O}_X(a) \\to \\ldots", "\\to \\mathcal{O}_X(a + i)^{\\oplus {n + 1 \\choose i}} \\to", "\\ldots \\to \\mathcal{O}_X(a + n + 1) \\to 0", "$$", "of vector bundles on $X$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\mathbf{P}^n_A$ is $\\text{Proj}(A[X_0, \\ldots, X_n])$, see", "Constructions, Definition \\ref{constructions-definition-projective-space}.", "Consider the Koszul complex", "$$", "K_\\bullet = K_\\bullet(A[X_0, \\ldots, X_n], X_0, \\ldots, X_n)", "$$", "over $S = A[X_0, \\ldots, X_n]$ on $X_0, \\ldots, X_n$.", "Since $X_0, \\ldots, X_n$ is clearly a regular sequence in the", "polynomial ring $S$, we see that", "(More on Algebra, Lemma \\ref{more-algebra-lemma-regular-koszul-regular})", "that the Koszul complex $K_\\bullet$ is exact, except in degree $0$", "where the cohomology is $S/(X_0, \\ldots, X_n)$.", "Note that $K_\\bullet$ becomes a complex of graded modules if we", "put the generators of $K_i$ in degree $+i$. In other words an", "exact complex", "$$", "0 \\to S(-n - 1) \\to \\ldots \\to S(-n - 1 + i)^{\\oplus {n \\choose i}} \\to \\ldots", "\\to S \\to S/(X_0, \\ldots, X_n) \\to 0", "$$", "Applying the exact functor $\\tilde{\\ }$ functor of Constructions, ", "Lemma \\ref{constructions-lemma-proj-sheaves} and using that", "the last term is in the kernel of this functor,", "we obtain the exact complex", "$$", "0 \\to \\mathcal{O}_X(-n - 1) \\to \\ldots", "\\to \\mathcal{O}_X(-n - 1 + i)^{\\oplus {n + 1 \\choose i}} \\to", "\\ldots \\to \\mathcal{O}_X \\to 0", "$$", "Twisting by the invertible sheaves $\\mathcal{O}_X(n + a)$", "we get the exact complexes of the lemma." ], "refs": [ "constructions-definition-projective-space", "more-algebra-lemma-regular-koszul-regular", "constructions-lemma-proj-sheaves" ], "ref_ids": [ 12664, 9973, 12594 ] } ], "ref_ids": [] }, { "id": 7014, "type": "theorem", "label": "perfect-lemma-generator-P1", "categories": [ "perfect" ], "title": "perfect-lemma-generator-P1", "contents": [ "Let $A$ be a ring. Let $X = \\mathbf{P}^n_A$. Then", "$$", "E =", "\\mathcal{O}_X \\oplus \\mathcal{O}_X(-1) \\oplus \\ldots \\oplus \\mathcal{O}_X(-n)", "$$", "is a generator", "(Derived Categories, Definition \\ref{derived-definition-generators})", "of $D_\\QCoh(X)$." ], "refs": [ "derived-definition-generators" ], "proofs": [ { "contents": [ "Let $K \\in D_\\QCoh(\\mathcal{O}_X)$. Assume", "$\\Hom(E, K[p]) = 0$ for all $p \\in \\mathbf{Z}$.", "We have to show that $K = 0$.", "By Derived Categories, Lemma", "\\ref{derived-lemma-right-orthogonal}", "we see that $\\Hom(E', K[p])$ is zero for all $E' \\in \\langle E \\rangle$", "and $p \\in \\mathbf{Z}$.", "By Lemma \\ref{lemma-construct-the-next-one}", "applied with $a = -n - 1$", "we see that $\\mathcal{O}_X(-n - 1) \\in \\langle E \\rangle$", "because it is quasi-isomorphic to a finite complex", "whose terms are finite direct sums of summands of $E$.", "Repeating the argument with $a = -n - 2$ we see that", "$\\mathcal{O}_X(-n - 2) \\in \\langle E \\rangle$.", "Arguing by induction we find that $\\mathcal{O}_X(-m) \\in \\langle E \\rangle$", "for all $m \\geq 0$.", "Since", "$$", "\\Hom(\\mathcal{O}_X(-m), K[p]) =", "H^p(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_X(m)) =", "H^p(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_X(1)^{\\otimes m})", "$$", "we conclude that $K = 0$ by Lemma \\ref{lemma-nonzero-some-cohomology}.", "(This also uses that $\\mathcal{O}_X(1)$ is an ample", "invertible sheaf on $X$ which follows from", "Properties, Lemma \\ref{properties-lemma-open-in-proj-ample}.)" ], "refs": [ "derived-lemma-right-orthogonal", "perfect-lemma-construct-the-next-one", "perfect-lemma-nonzero-some-cohomology", "properties-lemma-open-in-proj-ample" ], "ref_ids": [ 1937, 7013, 7012, 3050 ] } ], "ref_ids": [ 2003 ] }, { "id": 7015, "type": "theorem", "label": "perfect-lemma-compact-is-perfect-with-support", "categories": [ "perfect" ], "title": "perfect-lemma-compact-is-perfect-with-support", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $T \\subset X$ be a closed subset such that $X \\setminus T$", "is quasi-compact. An object of $D_{\\QCoh, T}(\\mathcal{O}_X)$ is compact", "if and only if it is perfect as an object of $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "We observe that $D_{\\QCoh, T}(\\mathcal{O}_X)$ is a triangulated", "category with direct sums by the remark preceding the lemma.", "By Proposition \\ref{proposition-compact-is-perfect}", "the perfect objects define compact objects of $D(\\mathcal{O}_X)$", "hence a fortiori of any subcategory preserved under taking direct", "sums. For the converse we will use there exists a generator", "$E \\in D_{\\QCoh, T}(\\mathcal{O}_X)$ which is a perfect complex", "of $\\mathcal{O}_X$-modules, see", "Lemma \\ref{lemma-generator-with-support}.", "Hence by the above, $E$ is compact. Then it follows from", "Derived Categories, Proposition", "\\ref{derived-proposition-generator-versus-classical-generator}", "that $E$ is a classical generator of the full subcategory", "of compact objects of $D_{\\QCoh, T}(\\mathcal{O}_X)$.", "Thus any compact object can be constructed out of $E$ by", "a finite sequence of operations consisting of", "(a) taking shifts, (b) taking finite direct sums, (c) taking cones, and", "(d) taking direct summands. Each of these operations preserves", "the property of being perfect and the result follows." ], "refs": [ "perfect-proposition-compact-is-perfect", "perfect-lemma-generator-with-support", "derived-proposition-generator-versus-classical-generator" ], "ref_ids": [ 7111, 7011, 1965 ] } ], "ref_ids": [] }, { "id": 7016, "type": "theorem", "label": "perfect-lemma-map-from-pseudo-coherent-to-complex-with-support", "categories": [ "perfect" ], "title": "perfect-lemma-map-from-pseudo-coherent-to-complex-with-support", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \\subset X$", "be a closed subset such that $U = X \\setminus T$ is quasi-compact.", "Let $\\alpha : P \\to E$ be a morphism of $D_\\QCoh(\\mathcal{O}_X)$ with", "either", "\\begin{enumerate}", "\\item $P$ is perfect and $E$ supported on $T$, or", "\\item $P$ pseudo-coherent, $E$ supported on $T$, and $E$ bounded below.", "\\end{enumerate}", "Then there exists a perfect complex of $\\mathcal{O}_X$-modules $I$", "and a map $I \\to \\mathcal{O}_X[0]$ such that", "$I \\otimes^\\mathbf{L} P \\to E$ is zero and such that", "$I|_U \\to \\mathcal{O}_U[0]$ is an", "isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Set $\\mathcal{D} = D_{\\QCoh, T}(\\mathcal{O}_X)$. In both cases the complex", "$K = R\\SheafHom(P, E)$ is an object of $\\mathcal{D}$. See", "Lemma \\ref{lemma-quasi-coherence-internal-hom} for quasi-coherence.", "It is clear that $K$ is supported on $T$ as formation of $R\\SheafHom$", "commutes with restriction to opens.", "The map $\\alpha$ defines an element of", "$H^0(K) = \\Hom_{D(\\mathcal{O}_X)}(\\mathcal{O}_X[0], K)$.", "Then it suffices to prove the result for the map", "$\\alpha : \\mathcal{O}_X[0] \\to K$.", "\\medskip\\noindent", "Let $E \\in \\mathcal{D}$ be a perfect generator, see", "Lemma \\ref{lemma-generator-with-support}. Write", "$$", "K = \\text{hocolim} K_n", "$$", "as in Derived Categories, Lemma \\ref{derived-lemma-write-as-colimit}", "using the generator $E$. Since the functor $\\mathcal{D} \\to D(\\mathcal{O}_X)$", "commutes with direct sums, we see that $K = \\text{hocolim} K_n$", "holds in $D(\\mathcal{O}_X)$. Since $\\mathcal{O}_X$ is a compact", "object of $D(\\mathcal{O}_X)$ we find an $n$ and a morphism", "$\\alpha_n : \\mathcal{O}_X \\to K_n$ which gives rise to $\\alpha$, see", "Derived Categories, Lemma \\ref{derived-lemma-commutes-with-countable-sums}.", "By Derived Categories, Lemma \\ref{derived-lemma-factor-through}", "applied to the morphism $\\mathcal{O}_X[0] \\to K_n$ in the ambient", "category $D(\\mathcal{O}_X)$ we see that $\\alpha_n$ factors as", "$\\mathcal{O}_X[0] \\to Q \\to K_n$ where $Q$ is an object", "of $\\langle E \\rangle$. We conclude that $Q$ is a perfect complex", "supported on $T$.", "\\medskip\\noindent", "Choose a distinguished triangle", "$$", "I \\to \\mathcal{O}_X[0] \\to Q \\to I[1]", "$$", "By construction $I$ is perfect, the map $I \\to \\mathcal{O}_X[0]$", "restricts to an isomorphism over $U$, and the composition", "$I \\to K$ is zero as $\\alpha$ factors through $Q$.", "This proves the lemma." ], "refs": [ "perfect-lemma-quasi-coherence-internal-hom", "perfect-lemma-generator-with-support", "derived-lemma-write-as-colimit", "derived-lemma-commutes-with-countable-sums", "derived-lemma-factor-through" ], "ref_ids": [ 6981, 7011, 1941, 1924, 1942 ] } ], "ref_ids": [] }, { "id": 7017, "type": "theorem", "label": "perfect-lemma-tensor-with-QCoh-complex", "categories": [ "perfect" ], "title": "perfect-lemma-tensor-with-QCoh-complex", "contents": [ "Let $X$ be a scheme. Let $K^\\bullet$ be a complex of $\\mathcal{O}_X$-modules", "whose cohomology sheaves are quasi-coherent. Let", "$(E, d) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, K^\\bullet)$", "be the endomorphism differential graded algebra. Then the functor", "$$", "- \\otimes_E^\\mathbf{L} K^\\bullet :", "D(E, \\text{d}) \\longrightarrow D(\\mathcal{O}_X)", "$$", "of", "Differential Graded Algebra, Lemma", "\\ref{dga-lemma-tensor-with-complex-derived}", "has image contained in $D_\\QCoh(\\mathcal{O}_X)$." ], "refs": [ "dga-lemma-tensor-with-complex-derived" ], "proofs": [ { "contents": [ "Let $P$ be a differential graded $E$-module with property (P)", "and let $F_\\bullet$ be a filtration on $P$ as in", "Differential Graded Algebra, Section \\ref{dga-section-P-resolutions}.", "Then we have", "$$", "P \\otimes_E K^\\bullet = \\text{hocolim}\\ F_iP \\otimes_E K^\\bullet", "$$", "Each of the $F_iP$ has a finite filtration whose graded pieces", "are direct sums of $E[k]$. The result follows easily." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13116 ] }, { "id": 7018, "type": "theorem", "label": "perfect-lemma-ext-from-perfect-into-bounded-QCoh", "categories": [ "perfect" ], "title": "perfect-lemma-ext-from-perfect-into-bounded-QCoh", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $K$ be a perfect object of $D(\\mathcal{O}_X)$. Then", "\\begin{enumerate}", "\\item there exist integers $a \\leq b$ such that", "$\\Hom_{D(\\mathcal{O}_X)}(K, L) = 0$ for $L \\in D_\\QCoh(\\mathcal{O}_X)$", "with $H^i(L) = 0$ for $i \\in [a, b]$, and", "\\item if $L$ is bounded, then $\\Ext^n_{D(\\mathcal{O}_X)}(K, L)$", "is zero for all but finitely many $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) follows from (1) as $\\Ext^n_{D(\\mathcal{O}_X)}(K, L) =", "\\Hom_{D(\\mathcal{O}_X)}(K, L[n])$. We prove (1).", "Since $K$ is perfect we have", "$$", "\\Hom_{D(\\mathcal{O}_X)}(K, L) =", "H^0(X, K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)", "$$", "where $K^\\vee$ is the ``dual'' perfect complex to $K$, see", "Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "Note that $K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$", "is in $D_\\QCoh(X)$ by", "Lemmas \\ref{lemma-quasi-coherence-tensor-product} and", "\\ref{lemma-pseudo-coherent} (to see that a perfect complex", "has quasi-coherent cohomology sheaves). Say $K^\\vee$ has", "tor amplitude in $[a, b]$. Then the spectral sequence", "$$", "E_1^{p, q} = H^p(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} H^q(L))", "\\Rightarrow", "H^{p + q}(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)", "$$", "shows that $H^j(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)$", "is zero if $H^q(L) = 0$ for $q \\in [j - b, j - a]$.", "Let $N$ be the integer $d$ of Cohomology of Schemes,", "Lemma \\ref{coherent-lemma-vanishing-nr-affines-quasi-separated}.", "Then $H^0(X, K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)$", "vanishes if the cohomology sheaves", "$$", "H^{-N}(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L),", "\\ H^{-N + 1}(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L),", "\\ \\ldots,", "\\ H^0(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)", "$$", "are zero. Namely, by the lemma cited and", "Lemma \\ref{lemma-application-nice-K-injective}, we have", "$$", "H^0(X, K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) =", "H^0(X, \\tau_{\\geq -N}(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L))", "$$", "and by the vanishing of cohomology sheaves, this is equal to", "$H^0(X, \\tau_{\\geq 1}(K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L))$", "which is zero by Derived Categories, Lemma", "\\ref{derived-lemma-negative-vanishing}.", "It follows that $\\Hom_{D(\\mathcal{O}_X)}(K, L)$ is zero if", "$H^i(L) = 0$ for $i \\in [-b - N, -a]$." ], "refs": [ "cohomology-lemma-dual-perfect-complex", "perfect-lemma-quasi-coherence-tensor-product", "perfect-lemma-pseudo-coherent", "coherent-lemma-vanishing-nr-affines-quasi-separated", "perfect-lemma-application-nice-K-injective", "derived-lemma-negative-vanishing" ], "ref_ids": [ 2233, 6945, 6974, 3294, 6940, 1839 ] } ], "ref_ids": [] }, { "id": 7019, "type": "theorem", "label": "perfect-lemma-pseudo-coherent-hocolim", "categories": [ "perfect" ], "title": "perfect-lemma-pseudo-coherent-hocolim", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $K \\in D(\\mathcal{O}_X)$. The following are equivalent", "\\begin{enumerate}", "\\item $K$ is pseudo-coherent, and", "\\item $K = \\text{hocolim} K_n$ where", "$K_n$ is perfect and $\\tau_{\\geq -n}K_n \\to \\tau_{\\geq -n}K$", "is an isomorphism for all $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is true on any ringed space.", "Namely, assume (2) holds. Recall that a perfect object of the derived", "category is pseudo-coherent, see", "Cohomology, Lemma \\ref{cohomology-lemma-perfect}.", "Then it follows from the definitions that", "$\\tau_{\\geq -n}K_n$ is $(-n + 1)$-pseudo-coherent", "and hence $\\tau_{\\geq -n}K$ is $(-n + 1)$-pseudo-coherent,", "hence $K$ is $(-n + 1)$-pseudo-coherent. This is true for", "all $n$, hence $K$ is pseudo-coherent, see", "Cohomology, Definition \\ref{cohomology-definition-pseudo-coherent}.", "\\medskip\\noindent", "Assume (1). We start by choosing an approximation", "$K_1 \\to K$ of $(X, K, -2)$ by a perfect complex $K_1$, see", "Definitions \\ref{definition-approximation-holds} and", "\\ref{definition-approximation} and", "Theorem \\ref{theorem-approximation}.", "Suppose by induction we have", "$$", "K_1 \\to K_2 \\to \\ldots \\to K_n \\to K", "$$", "with $K_i$ perfect such that", "such that $\\tau_{\\geq -i}K_i \\to \\tau_{\\geq -i}K$ is an isomorphism", "for all $1 \\leq i \\leq n$. Then we pick $a \\leq b$ as in", "Lemma \\ref{lemma-ext-from-perfect-into-bounded-QCoh}", "for the perfect object $K_n$. Choose an approximation", "$K_{n + 1} \\to K$ of $(X, K, \\min(a - 1, -n - 1))$.", "Choose a distinguished triangle", "$$", "K_{n + 1} \\to K \\to C \\to K_{n + 1}[1]", "$$", "Then we see that $C \\in D_\\QCoh(\\mathcal{O}_X)$ has", "$H^i(C) = 0$ for $i \\geq a$. Thus by our choice of $a, b$", "we see that $\\Hom_{D(\\mathcal{O}_X)}(K_n, C) = 0$.", "Hence the composition $K_n \\to K \\to C$ is zero. Hence by", "Derived Categories, Lemma \\ref{derived-lemma-representable-homological}", "we can factor $K_n \\to K$ through $K_{n + 1}$", "proving the induction step.", "\\medskip\\noindent", "We still have to prove that $K = \\text{hocolim} K_n$.", "This follows by an application of", "Derived Categories, Lemma \\ref{derived-lemma-cohomology-of-hocolim}", "to the functors", "$H^i( - ) : D(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X)$", "and our choice of $K_n$." ], "refs": [ "cohomology-lemma-perfect", "cohomology-definition-pseudo-coherent", "perfect-definition-approximation-holds", "perfect-definition-approximation", "perfect-theorem-approximation", "perfect-lemma-ext-from-perfect-into-bounded-QCoh", "derived-lemma-representable-homological", "derived-lemma-cohomology-of-hocolim" ], "ref_ids": [ 2224, 2258, 7116, 7117, 6934, 7018, 1758, 1923 ] } ], "ref_ids": [] }, { "id": 7020, "type": "theorem", "label": "perfect-lemma-pseudo-coherent-hocolim-with-support", "categories": [ "perfect" ], "title": "perfect-lemma-pseudo-coherent-hocolim-with-support", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $T \\subset X$ be a closed subset such that $X \\setminus T$", "is quasi-compact. Let $K \\in D(\\mathcal{O}_X)$ supported on $T$.", "The following are equivalent", "\\begin{enumerate}", "\\item $K$ is pseudo-coherent, and", "\\item $K = \\text{hocolim} K_n$ where", "$K_n$ is perfect, supported on $T$, and", "$\\tau_{\\geq -n}K_n \\to \\tau_{\\geq -n}K$ is an isomorphism for all $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The proof of this lemma is exactly the same as the proof of", "Lemma \\ref{lemma-pseudo-coherent-hocolim}", "except that in the choice of the approximations we use", "the triples $(T, K, m)$." ], "refs": [ "perfect-lemma-pseudo-coherent-hocolim" ], "ref_ids": [ 7019 ] } ], "ref_ids": [] }, { "id": 7021, "type": "theorem", "label": "perfect-lemma-Pn-module-category", "categories": [ "perfect" ], "title": "perfect-lemma-Pn-module-category", "contents": [ "\\begin{reference}", "\\cite{Beilinson}", "\\end{reference}", "Let $A$ be a ring. Let $X = \\mathbf{P}^n_A = \\text{Proj}(S)$", "where $S = A[X_0, \\ldots, X_n]$. With", "$P$ as in (\\ref{equation-generator-Pn}) and", "$R$ as in (\\ref{equation-algebra-for-Pn})", "the functor", "$$", "- \\otimes_R^\\mathbf{L} P : D(R) \\longrightarrow D_\\QCoh(\\mathcal{O}_X)", "$$", "is an $A$-linear equivalence of triangulated categories sending $R$", "to $P$." ], "refs": [], "proofs": [ { "contents": [ "To prove that our functor is fully faithful it suffices to prove", "that $\\Ext^i_X(P, P)$ is zero for $i \\not = 0$ and equal", "to $R$ for $i = 0$, see", "Differential Graded Algebra, Lemma", "\\ref{dga-lemma-fully-faithful-in-compact-case}.", "As in the proof of", "Lemma \\ref{lemma-ext-from-perfect-into-bounded-QCoh}", "we see that", "$$", "\\Ext^i_X(P, P) = H^i(X, P^\\wedge \\otimes P) =", "\\bigoplus\\nolimits_{0 \\leq a, b \\leq n} H^i(X, \\mathcal{O}_X(a - b))", "$$", "By the computation of cohomology of projective space", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cohomology-projective-space-over-ring})", "we find that", "these $\\Ext$-groups are zero unless $i = 0$.", "For $i = 0$ we recover $R$ because this is how we defined $R$", "in (\\ref{equation-algebra-for-Pn}).", "By Differential Graded Algebra, Lemma", "\\ref{dga-lemma-tensor-with-complex-hom-adjoint}", "our functor has a right adjoint, namely", "$R\\Hom(P, -) : D_\\QCoh(\\mathcal{O}_X) \\to D(R)$.", "Since $P$ is a generator for $D_\\QCoh(\\mathcal{O}_X)$ by", "Lemma \\ref{lemma-generator-P1}", "we see that the kernel of $R\\Hom(P, -)$ is zero.", "Hence our functor is an equivalence of triangulated", "categories by Derived Categories, Lemma", "\\ref{derived-lemma-fully-faithful-adjoint-kernel-zero}." ], "refs": [ "dga-lemma-fully-faithful-in-compact-case", "perfect-lemma-ext-from-perfect-into-bounded-QCoh", "coherent-lemma-cohomology-projective-space-over-ring", "dga-lemma-tensor-with-complex-hom-adjoint", "perfect-lemma-generator-P1", "derived-lemma-fully-faithful-adjoint-kernel-zero" ], "ref_ids": [ 13119, 7018, 3304, 13118, 7014, 1793 ] } ], "ref_ids": [] }, { "id": 7022, "type": "theorem", "label": "perfect-lemma-better-coherator", "categories": [ "perfect" ], "title": "perfect-lemma-better-coherator", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "The inclusion functor $D_\\QCoh(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$", "has a right adjoint $DQ_X$." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "We will use the induction principle as in", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}", "to prove this. If $D(\\QCoh(\\mathcal{O}_X)) \\to D_\\QCoh(\\mathcal{O}_X)$", "is an equivalence, then the lemma is true because the functor", "$RQ_X$ of Section \\ref{section-coherator} is a right adjoint to the functor", "$D(\\QCoh(\\mathcal{O}_X)) \\to D(\\mathcal{O}_X)$.", "In particular, our lemma is true for affine schemes, see", "Lemma \\ref{lemma-affine-coherator}.", "Thus we see that it suffices to show: if $X = U \\cup V$", "is a union of two quasi-compact opens and the", "lemma holds for $U$, $V$, and $U \\cap V$, then the lemma holds for $X$.", "\\medskip\\noindent", "The adjoint exists if and only if for every object $K$ of", "$D(\\mathcal{O}_X)$ we can find a distinguished triangle", "$$", "E' \\to E \\to K \\to E'[1]", "$$", "in $D(\\mathcal{O}_X)$", "such that $E'$ is in $D_\\QCoh(\\mathcal{O}_X)$ and such that", "$\\Hom(M, K) = 0$ for all $M$ in $D_\\QCoh(\\mathcal{O}_X)$. See", "Derived Categories, Lemma \\ref{derived-lemma-right-adjoint}.", "Consider the distinguished triangle", "$$", "E \\to Rj_{U, *}E|_U \\oplus Rj_{V, *}E|_V \\to", "Rj_{U \\cap V, *}E|_{U \\cap V} \\to E[1]", "$$", "in $D(\\mathcal{O}_X)$ of", "Cohomology, Lemma \\ref{cohomology-lemma-exact-sequence-j-star}.", "By Derived Categories, Lemma \\ref{derived-lemma-prepare-adjoint}", "it suffices to construct the desired distinguished triangles", "for $Rj_{U, *}E|_U$, $Rj_{V, *}E|_V$, and", "$Rj_{U \\cap V, *}E|_{U \\cap V}$. This reduces us to the statement", "discussed in the next paragraph.", "\\medskip\\noindent", "Let $j : U \\to X$ be an open immersion corresponding with $U$ a quasi-compact", "open for which the lemma is true. Let $L$ be an object of $D(\\mathcal{O}_U)$.", "Then there exists a distinguished triangle", "$$", "E' \\to Rj_*L \\to K \\to E'[1]", "$$", "in $D(\\mathcal{O}_X)$", "such that $E'$ is in $D_\\QCoh(\\mathcal{O}_X)$ and such that", "$\\Hom(M, K) = 0$ for all $M$ in $D_\\QCoh(\\mathcal{O}_X)$.", "To see this we choose a distinguished triangle", "$$", "L' \\to L \\to Q \\to L'[1]", "$$", "in $D(\\mathcal{O}_U)$ such that $L'$ is in $D_\\QCoh(\\mathcal{O}_U)$", "and such that $\\Hom(N, Q) = 0$ for all $N$ in $D_\\QCoh(\\mathcal{O}_U)$.", "This is possible because the statement in", "Derived Categories, Lemma \\ref{derived-lemma-right-adjoint}", "is an if and only if.", "We obtain a distinguished triangle", "$$", "Rj_*L' \\to Rj_*L \\to Rj_*Q \\to Rj_*L'[1]", "$$", "in $D(\\mathcal{O}_X)$. Observe that $Rj_*L'$ is in $D_\\QCoh(\\mathcal{O}_X)$", "by Lemma \\ref{lemma-quasi-coherence-direct-image}.", "On the other hand, if $M$ in $D_\\QCoh(\\mathcal{O}_X)$, then", "$$", "\\Hom(M, Rj_*Q) = \\Hom(Lj^*M, Q) = 0", "$$", "because $Lj^*M$ is in $D_\\QCoh(\\mathcal{O}_U)$ by", "Lemma \\ref{lemma-quasi-coherence-pullback}.", "This finishes the proof." ], "refs": [ "coherent-lemma-induction-principle", "perfect-lemma-affine-coherator", "derived-lemma-right-adjoint", "cohomology-lemma-exact-sequence-j-star", "derived-lemma-prepare-adjoint", "derived-lemma-right-adjoint", "perfect-lemma-quasi-coherence-direct-image", "perfect-lemma-quasi-coherence-pullback" ], "ref_ids": [ 3291, 6965, 1947, 2144, 1946, 1947, 6946, 6944 ] } ], "ref_ids": [] }, { "id": 7023, "type": "theorem", "label": "perfect-lemma-pushforward-better-coherator", "categories": [ "perfect" ], "title": "perfect-lemma-pushforward-better-coherator", "contents": [ "Let $f : X \\to Y$ be a quasi-compact and quasi-separated", "morphism of schemes. If the right adjoints $DQ_X$ and $DQ_Y$", "of the inclusion functors $D_\\QCoh \\to D$ exist for $X$ and $Y$, then", "$$", "Rf_* \\circ DQ_X = DQ_Y \\circ Rf_*", "$$" ], "refs": [], "proofs": [ { "contents": [ "The statement makes sense because $Rf_*$ sends", "$D_\\QCoh(\\mathcal{O}_X)$ into $D_\\QCoh(\\mathcal{O}_Y)$ by", "Lemma \\ref{lemma-quasi-coherence-direct-image}.", "The statement is true because $Lf^*$ similarly maps", "$D_\\QCoh(\\mathcal{O}_Y)$ into $D_\\QCoh(\\mathcal{O}_X)$", "(Lemma \\ref{lemma-quasi-coherence-pullback})", "and hence both $Rf_* \\circ DQ_X$ and $DQ_Y \\circ Rf_*$", "are right adjoint to $Lf^* : D_\\QCoh(\\mathcal{O}_Y) \\to D(\\mathcal{O}_X)$." ], "refs": [ "perfect-lemma-quasi-coherence-direct-image", "perfect-lemma-quasi-coherence-pullback" ], "ref_ids": [ 6946, 6944 ] } ], "ref_ids": [] }, { "id": 7024, "type": "theorem", "label": "perfect-lemma-boundedness-better-coherator", "categories": [ "perfect" ], "title": "perfect-lemma-boundedness-better-coherator", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. The functor", "$DQ_X$ of Lemma \\ref{lemma-better-coherator}", "has the following boundedness property:", "there exists an integer $N = N(X)$ such that, if", "$K$ in $D(\\mathcal{O}_X)$ with", "$H^i(U, K) = 0$ for $U$ affine open in $X$ and $i \\not \\in [a, b]$, then", "the cohomology sheaves $H^i(DQ_X(K))$ are zero for", "$i \\not \\in [a, b + N]$." ], "refs": [ "perfect-lemma-better-coherator" ], "proofs": [ { "contents": [ "We will prove this using the induction principle of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}.", "\\medskip\\noindent", "If $X$ is affine, then the lemma is true with $N = 0$ because then", "$RQ_X = DQ_X$ is given by taking the complex of", "quasi-coherent sheaves associated to $R\\Gamma(X, K)$.", "See Lemmas \\ref{lemma-affine-compare-bounded} and \\ref{lemma-affine-coherator}.", "\\medskip\\noindent", "Asssume $U, V$ are quasi-compact open in $X$", "and the lemma holds for $U$, $V$, and $U \\cap V$.", "Say with integers $N(U)$, $N(V)$, and $N(U \\cap V)$.", "Now suppose $K$ is in $D(\\mathcal{O}_X)$ with", "$H^i(W, K) = 0$ for all affine open $W \\subset X$ and all $i \\not \\in [a, b]$.", "Then $K|_U$, $K|_V$, $K|_{U \\cap V}$ have the same property.", "Hence we see that $RQ_U(K|_U)$ and $RQ_V(K|_V)$ and", "$RQ_{U \\cap V}(K|_{U \\cap V})$ have vanishing cohomology", "sheaves outside the inverval $[a, b + \\max(N(U), N(V), N(U \\cap V))$.", "Since the functors $Rj_{U, *}$, $Rj_{V, *}$, $Rj_{U \\cap V, *}$", "have finite cohomological dimension on $D_\\QCoh$ by", "Lemma \\ref{lemma-quasi-coherence-direct-image}", "we see that there exists an $N$ such that", "$Rj_{U, *}DQ_U(K|_U)$, $Rj_{V, *}DQ_V(K|_V)$, and", "$Rj_{U \\cap V, *}DQ_{U \\cap V}(K|_{U \\cap V})$ have vanishing", "cohomology sheaves outside the interval $[a, b + N]$.", "Then finally we conclude by the distinguished triangle", "of Remark \\ref{remark-explain-consequence}." ], "refs": [ "coherent-lemma-induction-principle", "perfect-lemma-affine-compare-bounded", "perfect-lemma-affine-coherator", "perfect-lemma-quasi-coherence-direct-image", "perfect-remark-explain-consequence" ], "ref_ids": [ 3291, 6941, 6965, 6946, 7133 ] } ], "ref_ids": [ 7022 ] }, { "id": 7025, "type": "theorem", "label": "perfect-lemma-cohomology-base-change", "categories": [ "perfect" ], "title": "perfect-lemma-cohomology-base-change", "contents": [ "Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism", "of schemes. For $E$ in $D_\\QCoh(\\mathcal{O}_X)$ and", "$K$ in $D_\\QCoh(\\mathcal{O}_Y)$ the map", "$$", "Rf_*(E) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} K", "\\longrightarrow", "Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*K)", "$$", "defined in", "Cohomology, Equation (\\ref{cohomology-equation-projection-formula-map})", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "To check the map is an isomorphism we may work locally on $Y$.", "Hence we reduce to the case that $Y$ is affine.", "\\medskip\\noindent", "Suppose that $K = \\bigoplus K_i$ is a direct", "sum of some complexes $K_i \\in D_\\QCoh(\\mathcal{O}_Y)$.", "If the statement holds for each $K_i$, then it holds for $K$.", "Namely, the functors $Lf^*$ and $\\otimes^\\mathbf{L}$ preserve", "direct sums by construction and $Rf_*$ commutes with direct sums", "(for complexes with quasi-coherent cohomology sheaves) by", "Lemma \\ref{lemma-quasi-coherence-pushforward-direct-sums}.", "Moreover, suppose that $K \\to L \\to M \\to K[1]$ is a distinguished", "triangle in $D_\\QCoh(Y)$. Then if the statement of the", "lemma holds for two of $K, L, M$, then it holds for the third", "(as the functors involved are exact functors of triangulated categories).", "\\medskip\\noindent", "Assume $Y$ affine, say $Y = \\Spec(A)$. The functor", "$\\widetilde{\\ } : D(A) \\to D_\\QCoh(\\mathcal{O}_Y)$ is an equivalence", "(Lemma \\ref{lemma-affine-compare-bounded}).", "Let $T$ be the property for $K \\in D(A)$ that", "the statement of the lemma holds for $\\widetilde{K}$.", "The discussion above and", "More on Algebra, Remark \\ref{more-algebra-remark-P-resolution}", "shows that it suffices to prove $T$ holds for $A[k]$.", "This finishes the proof, as the statement of the lemma", "is clear for shifts of the structure sheaf." ], "refs": [ "perfect-lemma-quasi-coherence-pushforward-direct-sums", "perfect-lemma-affine-compare-bounded", "more-algebra-remark-P-resolution" ], "ref_ids": [ 6950, 6941, 10653 ] } ], "ref_ids": [] }, { "id": 7026, "type": "theorem", "label": "perfect-lemma-tor-independent", "categories": [ "perfect" ], "title": "perfect-lemma-tor-independent", "contents": [ "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ and $Y$ are tor independent over $S$, and", "\\item for every affine opens $U \\subset X$, $V \\subset Y$, $W \\subset S$", "with $f(U) \\subset W$ and $g(V) \\subset W$ the rings", "$\\mathcal{O}_X(U)$ and $\\mathcal{O}_Y(V)$ are tor independent over", "$\\mathcal{O}_S(W)$.", "\\item there exists an affine open overing $S = \\bigcup W_i$ and", "for each $i$ affine open coverings $f^{-1}(W_i) = \\bigcup U_{ij}$", "and $g^{-1}(W_i) = \\bigcup V_{ik}$ such that the rings", "$\\mathcal{O}_X(U_{ij})$ and $\\mathcal{O}_Y(V_{ik})$ are tor independent over", "$\\mathcal{O}_S(W_i)$ for all $i, j, k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: use More on Algebra, Lemma", "\\ref{more-algebra-lemma-tor-independent}." ], "refs": [ "more-algebra-lemma-tor-independent" ], "ref_ids": [ 10143 ] } ], "ref_ids": [] }, { "id": 7027, "type": "theorem", "label": "perfect-lemma-flat-base-change-tor-independent", "categories": [ "perfect" ], "title": "perfect-lemma-flat-base-change-tor-independent", "contents": [ "Let $X \\to S$ and $Y \\to S$ be morphisms of schemes. Let $S' \\to S$ be a", "morphism of schemes and denote $X' = X \\times_S S'$", "and $Y' = Y \\times_S S'$.", "If $X$ and $Y$ are tor independent over $S$ and $S' \\to S$ is flat,", "then $X'$ and $Y'$ are tor independent over $S'$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: use Lemma \\ref{lemma-tor-independent} and", "on affine opens use More on Algebra, Lemma", "\\ref{more-algebra-lemma-flat-base-change-tor-independent}." ], "refs": [ "perfect-lemma-tor-independent", "more-algebra-lemma-flat-base-change-tor-independent" ], "ref_ids": [ 7026, 10141 ] } ], "ref_ids": [] }, { "id": 7028, "type": "theorem", "label": "perfect-lemma-compare-base-change", "categories": [ "perfect" ], "title": "perfect-lemma-compare-base-change", "contents": [ "Let $g : S' \\to S$ be a morphism of schemes.", "Let $f : X \\to S$ be quasi-compact and quasi-separated.", "Consider the base change diagram", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "S' \\ar[r]^g &", "S", "}", "$$", "If $X$ and $S'$ are Tor independent over $S$, then for all", "$E \\in D_\\QCoh(\\mathcal{O}_X)$ we have", "$Rf'_*L(g')^*E = Lg^*Rf_*E$." ], "refs": [], "proofs": [ { "contents": [ "For any object $E$ of $D(\\mathcal{O}_X)$ we can use", "Cohomology, Remark \\ref{cohomology-remark-base-change} to get a", "canonical base change map $Lg^*Rf_*E \\to Rf'_*L(g')^*E$. To check this", "is an isomorphism we may work locally on $S'$. Hence we may assume", "$g : S' \\to S$ is a morphism of affine schemes. In particular, $g$", "is affine and it suffices to show that", "$$", "Rg_*Lg^*Rf_*E \\to Rg_*Rf'_*L(g')^*E = Rf_*(Rg'_* L(g')^* E)", "$$", "is an isomorphism, see Lemma \\ref{lemma-affine-morphism}", "(and use Lemmas \\ref{lemma-quasi-coherence-pullback},", "\\ref{lemma-quasi-coherence-tensor-product}, and", "\\ref{lemma-quasi-coherence-direct-image}", "to see that the objects $Rf'_*L(g')^*E$ and $Lg^*Rf_*E$", "have quasi-coherent cohomology sheaves). Note that $g'$ is", "affine as well (Morphisms, Lemma \\ref{morphisms-lemma-base-change-affine}).", "By Lemma \\ref{lemma-affine-morphism-pull-push} the map becomes a map", "$$", "Rf_*E \\otimes_{\\mathcal{O}_S}^\\mathbf{L} g_*\\mathcal{O}_{S'}", "\\longrightarrow", "Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} g'_*\\mathcal{O}_{X'})", "$$", "Observe that $g'_*\\mathcal{O}_{X'} = f^*g_*\\mathcal{O}_{S'}$. Thus by", "Lemma \\ref{lemma-cohomology-base-change} it suffices to prove that", "$Lf^*g_*\\mathcal{O}_{S'} = f^*g_*\\mathcal{O}_{S'}$. This follows from our", "assumption that $X$ and $S'$ are Tor independent over $S$. Namely, to", "check it we may work locally on $X$, hence we may also assume $X$ is affine.", "Say $X = \\Spec(A)$, $S = \\Spec(R)$ and $S' = \\Spec(R')$. Our assumption", "implies that $A$ and $R'$ are Tor independent over $R$", "(More on Algebra, Lemma \\ref{more-algebra-lemma-tor-independent}), i.e.,", "$\\text{Tor}_i^R(A, R') = 0$ for $i > 0$. In other words", "$A \\otimes_R^\\mathbf{L} R' = A \\otimes_R R'$ which exactly means", "that $Lf^*g_*\\mathcal{O}_{S'} = f^*g_*\\mathcal{O}_{S'}$", "(use Lemma \\ref{lemma-quasi-coherence-pullback})." ], "refs": [ "cohomology-remark-base-change", "perfect-lemma-affine-morphism", "perfect-lemma-quasi-coherence-pullback", "perfect-lemma-quasi-coherence-tensor-product", "perfect-lemma-quasi-coherence-direct-image", "morphisms-lemma-base-change-affine", "perfect-lemma-affine-morphism-pull-push", "perfect-lemma-cohomology-base-change", "more-algebra-lemma-tor-independent", "perfect-lemma-quasi-coherence-pullback" ], "ref_ids": [ 2269, 6952, 6944, 6945, 6946, 5176, 6953, 7025, 10143, 6944 ] } ], "ref_ids": [] }, { "id": 7029, "type": "theorem", "label": "perfect-lemma-affine-morphism-and-hom-out-of-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-affine-morphism-and-hom-out-of-perfect", "contents": [ "Consider a cartesian square", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "of quasi-compact and quasi-separated schemes. Assume $g$ and $f$", "Tor independent and $S = \\Spec(R)$, $S' = \\Spec(R')$ affine. For", "$M, K \\in D(\\mathcal{O}_X)$ the canonical map", "$$", "R\\Hom_X(M, K) \\otimes^\\mathbf{L}_R R'", "\\longrightarrow", "R\\Hom_{X'}(L(g')^*M, L(g')^*K)", "$$", "in $D(R')$ is an isomorphism in the following two cases", "\\begin{enumerate}", "\\item $M \\in D(\\mathcal{O}_X)$ is perfect and $K \\in D_\\QCoh(X)$, or", "\\item $M \\in D(\\mathcal{O}_X)$ is pseudo-coherent,", "$K \\in D_\\QCoh^+(X)$, and $R'$ has finite tor dimension over $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "There is a canonical map", "$R\\Hom_X(M, K) \\to R\\Hom_{X'}(L(g')^*M, L(g')^*K)$", "in $D(\\Gamma(X, \\mathcal{O}_X))$ of global hom complexes, see", "Cohomology, Section \\ref{cohomology-section-global-RHom}.", "Restricting scalars we can view this as a map in $D(R)$.", "Then we can use the adjointness of restriction and", "$- \\otimes_R^\\mathbf{L} R'$ to get the displayed map of the lemma.", "Having defined the map it suffices to prove it is an isomorphism", "in the derived category of abelian groups.", "\\medskip\\noindent", "The right hand side is equal to", "$$", "R\\Hom_X(M, R(g')_*L(g')^*K) =", "R\\Hom_X(M, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} g'_*\\mathcal{O}_{X'})", "$$", "by Lemma \\ref{lemma-affine-morphism-pull-push}. In both cases the complex", "$R\\SheafHom(M, K)$ is an object of $D_\\QCoh(\\mathcal{O}_X)$ by", "Lemma \\ref{lemma-quasi-coherence-internal-hom}. There is a natural map", "$$", "R\\SheafHom(M, K) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} g'_*\\mathcal{O}_{X'}", "\\longrightarrow", "R\\SheafHom(M, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} g'_*\\mathcal{O}_{X'})", "$$", "which is an isomorphism in both cases by", "Lemma \\ref{lemma-internal-hom-evaluate-tensor-isomorphism}.", "To see that this lemma applies in case (2) we note that", "$g'_*\\mathcal{O}_{X'} = Rg'_*\\mathcal{O}_{X'} =", "Lf^*g_*\\mathcal{O}_X$ the second equality by", "Lemma \\ref{lemma-compare-base-change}.", "Using Lemma \\ref{lemma-tor-dimension-affine} and", "Cohomology, Lemma \\ref{cohomology-lemma-tor-amplitude-pullback}", "we conclude that $g'_*\\mathcal{O}_{X'}$ has finite Tor dimension.", "Hence, in both cases by replacing $K$ by $R\\SheafHom(M, K)$ we reduce", "to proving", "$$", "R\\Gamma(X, K) \\otimes^\\mathbf{L}_A A' \\longrightarrow", "R\\Gamma(X, K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} g'_*\\mathcal{O}_{X'})", "$$", "is an isomorphism.", "Note that the left hand side is equal to $R\\Gamma(X', L(g')^*K)$", "by Lemma \\ref{lemma-affine-morphism-pull-push}.", "Hence the result follows from", "Lemma \\ref{lemma-compare-base-change}." ], "refs": [ "perfect-lemma-affine-morphism-pull-push", "perfect-lemma-quasi-coherence-internal-hom", "perfect-lemma-internal-hom-evaluate-tensor-isomorphism", "perfect-lemma-compare-base-change", "perfect-lemma-tor-dimension-affine", "cohomology-lemma-tor-amplitude-pullback", "perfect-lemma-affine-morphism-pull-push", "perfect-lemma-compare-base-change" ], "ref_ids": [ 6953, 6981, 6982, 7028, 6977, 2216, 6953, 7028 ] } ], "ref_ids": [] }, { "id": 7030, "type": "theorem", "label": "perfect-lemma-tor-independence-and-tor-amplitude", "categories": [ "perfect" ], "title": "perfect-lemma-tor-independence-and-tor-amplitude", "contents": [ "Consider a cartesian square of schemes", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "Assume $g$ and $f$ Tor independent.", "\\begin{enumerate}", "\\item If $E \\in D(\\mathcal{O}_X)$ has tor amplitude", "in $[a, b]$ as a complex of $f^{-1}\\mathcal{O}_S$-modules,", "then $L(g')^*E$ has tor amplitude", "in $[a, b]$ as a complex of $f^{-1}\\mathcal{O}_{S'}$-modules.", "\\item If $\\mathcal{G}$ is an $\\mathcal{O}_X$-module flat", "over $S$, then $L(g')^*\\mathcal{G} = (g')^*\\mathcal{G}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We can compute tor dimension at stalks, see", "Cohomology, Lemma \\ref{cohomology-lemma-tor-amplitude-stalk}.", "If $x' \\in X'$ with image $x \\in X$, then", "$$", "(L(g')^*E)_{x'} =", "E_x \\otimes_{\\mathcal{O}_{X, x}}^\\mathbf{L} \\mathcal{O}_{X', x'}", "$$", "Let $s' \\in S'$ and $s \\in S$ be the image of $x'$ and $x$.", "Since $X$ and $S'$ are tor independent over $S$, we can apply", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-comparison}", "to see that the right hand side of the displayed formula is equal to", "$E_x \\otimes_{\\mathcal{O}_{S, s}}^\\mathbf{L} \\mathcal{O}_{S', s'}$", "in $D(\\mathcal{O}_{S', s'})$.", "Thus (1) follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-pull-tor-amplitude}.", "To see (2) observe that flatness of $\\mathcal{G}$ is equivalent to", "the condition that $\\mathcal{G}[0]$ has tor amplitude in $[0, 0]$.", "Applying (1) we conclude." ], "refs": [ "cohomology-lemma-tor-amplitude-stalk", "more-algebra-lemma-base-change-comparison", "more-algebra-lemma-pull-tor-amplitude" ], "ref_ids": [ 2217, 10139, 10180 ] } ], "ref_ids": [] }, { "id": 7031, "type": "theorem", "label": "perfect-lemma-compare-base-change-closed-immersion", "categories": [ "perfect" ], "title": "perfect-lemma-compare-base-change-closed-immersion", "contents": [ "Consider a cartesian diagram of schemes", "$$", "\\xymatrix{", "Z' \\ar[r]_{i'} \\ar[d]_g & X' \\ar[d]^f \\\\", "Z \\ar[r]^i & X", "}", "$$", "where $i$ is a closed immersion. If $Z$ and $X'$ are", "tor independent over $X$, then $Ri'_* \\circ Lg^* = Lf^* \\circ Ri_*$", "as functors $D(\\mathcal{O}_Z) \\to D(\\mathcal{O}_{X'})$." ], "refs": [], "proofs": [ { "contents": [ "Note that the lemma is supposed to hold for all $K \\in D(\\mathcal{O}_Z)$.", "Observe that $i_*$ and $i'_*$ are exact functors and hence", "$Ri_*$ and $Ri'_*$ are computed by applying $i_*$ and $i'_*$", "to any representatives. Thus the base change map", "$$", "Lf^*(Ri_*(K)) \\longrightarrow Ri'_*(Lg^*(K))", "$$", "on stalks at a point $z' \\in Z'$ with image $z \\in Z$ is given by", "$$", "K_z \\otimes_{\\mathcal{O}_{X, z}}^\\mathbf{L} \\mathcal{O}_{X', z'}", "\\longrightarrow", "K_z \\otimes_{\\mathcal{O}_{Z, z}}^\\mathbf{L} \\mathcal{O}_{Z', z'}", "$$", "This map is an isomorphism by", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-comparison}", "and the assumed tor independence." ], "refs": [ "more-algebra-lemma-base-change-comparison" ], "ref_ids": [ 10139 ] } ], "ref_ids": [] }, { "id": 7032, "type": "theorem", "label": "perfect-lemma-kunneth", "categories": [ "perfect" ], "title": "perfect-lemma-kunneth", "contents": [ "In the situation above, if $a$ and $b$ are quasi-compact and quasi-separated", "and $X$ and $Y$ are tor-independent over $S$, then (\\ref{equation-kunneth})", "is an isomorphism for $K \\in D_\\QCoh(\\mathcal{O}_X)$ and", "$M \\in D_\\QCoh(\\mathcal{O}_Y)$. If in addition $S = \\Spec(A)$ is affine,", "then the map (\\ref{equation-kunneth-global}) is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "This follows from the following sequence of isomorphisms", "\\begin{align*}", "Rf_*(Lp^*K \\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} Lq^*M)", "& =", "Ra_*Rp_*(Lp^*K \\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} Lq^*M) \\\\", "& =", "Ra_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Rp_*Lq^*M) \\\\", "& =", "Ra_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} La^*Rb_*M) \\\\", "& =", "Ra_*K \\otimes_{\\mathcal{O}_S}^\\mathbf{L} Rb_*M", "\\end{align*}", "The first equality holds because $f = a \\circ p$. The second equality", "by Lemma \\ref{lemma-cohomology-base-change}. The third equality by", "Lemma \\ref{lemma-compare-base-change}. The fourth equality by", "Lemma \\ref{lemma-cohomology-base-change}.", "We omit the verification that the composition of these isomorphisms", "is the same as the map (\\ref{equation-kunneth}).", "If $S$ is affine, then the source and target of the arrow", "(\\ref{equation-kunneth-global}) are the result of applying", "$R\\Gamma(S, -)$ to the source and target of (\\ref{equation-kunneth})", "and we obtain the final statement; details omitted." ], "refs": [ "perfect-lemma-cohomology-base-change", "perfect-lemma-compare-base-change", "perfect-lemma-cohomology-base-change" ], "ref_ids": [ 7025, 7028, 7025 ] } ], "ref_ids": [] }, { "id": 7033, "type": "theorem", "label": "perfect-lemma-cohomology-de-rham-base-change", "categories": [ "perfect" ], "title": "perfect-lemma-cohomology-de-rham-base-change", "contents": [ "Let $a : X \\to S$ be a quasi-compact and quasi-separated morphism", "of schemes. Let $\\mathcal{F}^\\bullet$ be a locally bounded", "complex of $a^{-1}\\mathcal{O}_S$-modules. Assume for all $n \\in \\mathbf{Z}$", "the sheaf $\\mathcal{F}^n$ is a flat $a^{-1}\\mathcal{O}_S$-module and", "$\\mathcal{F}^n$ has the structure of a quasi-coherent $\\mathcal{O}_X$-module", "compatible with the given $a^{-1}\\mathcal{O}_S$-module structure (but the", "differentials in the complex $\\mathcal{F}^\\bullet$ need not", "be $\\mathcal{O}_X$-linear). Then the following hold", "\\begin{enumerate}", "\\item $Ra_*\\mathcal{F}^\\bullet$ is locally bounded,", "\\item $Ra_*\\mathcal{F}^\\bullet$ is in $D_\\QCoh(\\mathcal{O}_S)$,", "\\item $Ra_*\\mathcal{F}^\\bullet$ locally has finite tor dimension,", "\\item $\\mathcal{G} \\otimes_{\\mathcal{O}_S}^\\mathbf{L} Ra_*\\mathcal{F}^\\bullet =", "Ra_*(a^{-1}\\mathcal{G} \\otimes_{a^{-1}\\mathcal{O}_S} \\mathcal{F}^\\bullet)$", "for $\\mathcal{G} \\in \\QCoh(\\mathcal{O}_S)$, and", "\\item $K \\otimes_{\\mathcal{O}_S}^\\mathbf{L} Ra_*\\mathcal{F}^\\bullet =", "Ra_*(a^{-1}K \\otimes_{a^{-1}\\mathcal{O}_S}^\\mathbf{L} \\mathcal{F}^\\bullet)$", "for $K \\in D_\\QCoh(\\mathcal{O}_S)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1), (2), (3) are local on $S$ hence we may and do assume $S$", "is affine. Since $a$ is quasi-compact we conclude that $X$ is quasi-compact.", "Since $\\mathcal{F}^\\bullet$ is locally bounded, we conclude that", "$\\mathcal{F}^\\bullet$ is bounded.", "\\medskip\\noindent", "For (1) and (2) we can use the first spectral sequence", "$R^pa_*\\mathcal{F}^q \\Rightarrow R^{p + q}a_*\\mathcal{F}^\\bullet$ of", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "Combining Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images}", "and Homology, Lemma \\ref{homology-lemma-biregular-ss-converges}", "we conclude.", "\\medskip\\noindent", "Let us prove (3) by the induction principle of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}.", "Namely, for a quasi-compact open of $U$ of $X$ consider the", "condition that $R(a|_U)_*(\\mathcal{F}^\\bullet|_U)$ has", "finite tor dimension. If $U, V$ are quasi-compact open in", "$X$, then we have a relative Mayer-Vietoris distinguished triangle", "$$", "R(a|_{U \\cup V})_*\\mathcal{F}^\\bullet|_{U \\cup V} \\to", "R(a|_U)_*\\mathcal{F}^\\bullet|_U \\oplus", "R(a|_V)_*\\mathcal{F}^\\bullet|_V \\to", "R(a|_{U \\cap V})_*\\mathcal{F}^\\bullet|_{U \\cap V} \\to", "$$", "by Cohomology, Lemma \\ref{cohomology-lemma-unbounded-relative-mayer-vietoris}.", "By the behaviour of tor amplitude in distinguished triangles", "(see Cohomology, Lemma \\ref{cohomology-lemma-cone-tor-amplitude})", "we see that if we know the result for $U$, $V$, $U \\cap V$, then", "the result holds for $U \\cup V$. This reduces us to the case where", "$X$ is affine. In this case we have", "$$", "Ra_*\\mathcal{F}^\\bullet = a_*\\mathcal{F}^\\bullet", "$$", "by Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "and the vanishing of higher direct images of quasi-coherent modules", "under an affine morphism", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-vanishing}).", "Since $\\mathcal{F}^n$ is $S$-flat by assumption and $X$ affine, the modules", "$a_*\\mathcal{F}^n$ are flat for all $n$. Hence $a_*\\mathcal{F}^\\bullet$", "is a bounded complex of flat $\\mathcal{O}_S$-modules and hence has", "finite tor dimension.", "\\medskip\\noindent", "Proof of part (5). Denote", "$a' : (X, a^{-1}\\mathcal{O}_S) \\to (S, \\mathcal{O}_S)$", "the obvious flat morphism of ringed spaces. Part (5) says that", "$$", "K \\otimes_{\\mathcal{O}_S}^\\mathbf{L} Ra'_*\\mathcal{F}^\\bullet =", "Ra'_*(L(a')^*K \\otimes_{a^{-1}\\mathcal{O}_S}^\\mathbf{L}", "\\mathcal{F}^\\bullet)", "$$", "Thus", "Cohomology, Equation (\\ref{cohomology-equation-projection-formula-map})", "gives a functorial map from the left to the right and we want to show", "this map is an isomorphism.", "This question is local on $S$ hence we may and do assume $S$", "is affine. The rest of the proof is {\\it exactly} the same as the", "proof of Lemma \\ref{lemma-cohomology-base-change} except that we have", "to show that the functor", "$K \\mapsto Ra'_*(L(a')^*K \\otimes_{a^{-1}\\mathcal{O}_S}^\\mathbf{L}", "\\mathcal{F}^\\bullet)$ commutes with direct sums.", "This is where we will use $\\mathcal{F}^n$ has the structure", "of a quasi-coherent $\\mathcal{O}_X$-module. Namely, observe that", "$K \\mapsto L(a')^*K", "\\otimes_{a^{-1}\\mathcal{O}_S}^\\mathbf{L} \\mathcal{F}^\\bullet$", "commutes with arbitrary direct sums. Next, if", "$\\mathcal{F}^\\bullet$ consists of a single quasi-coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}^\\bullet = \\mathcal{F}^n[-n]$", "then we have $L(a')^*G", "\\otimes_{a^{-1}\\mathcal{O}_S}^\\mathbf{L} \\mathcal{F}^\\bullet =", "La^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}^n[-n]$, see", "Cohomology, Lemma \\ref{cohomology-lemma-variant-derived-pullback}.", "Hence in this case the commutation with direct sums follows from", "Lemma \\ref{lemma-quasi-coherence-pushforward-direct-sums}.", "Now, in general, since $S$ is affine (hence $X$ quasi-compact)", "and $\\mathcal{F}^\\bullet$ is locally bounded, we see that", "$$", "\\mathcal{F}^\\bullet = (\\mathcal{F}^a \\to \\ldots \\to \\mathcal{F}^b)", "$$", "is bounded. Arguing by induction on $b - a$ and considering the", "distinguished triangle", "$$", "\\mathcal{F}^b[-b] \\to (\\mathcal{F}^a \\to \\ldots \\to \\mathcal{F}^b)", "\\to (\\mathcal{F}^a \\to \\ldots \\to \\mathcal{F}^{b - 1}) \\to", "\\mathcal{F}^b[-b + 1]", "$$", "the proof of this part is finished. Some details omitted.", "\\medskip\\noindent", "Proof of part (4). Let $a' : (X, a^{-1}\\mathcal{O}_S) \\to (S, \\mathcal{O}_S)$", "be as above. Since $\\mathcal{F}^\\bullet$ is a locally bounded", "complex of flat $a^{-1}\\mathcal{O}_S$-modules we see the complex", "$a^{-1}\\mathcal{G} \\otimes_{a^{-1}\\mathcal{O}_S} \\mathcal{F}^\\bullet$", "represents $L(a')^*\\mathcal{G}", "\\otimes_{a^{-1}\\mathcal{O}_S}^\\mathbf{L}", "\\mathcal{F}^\\bullet$ in $D(a^{-1}\\mathcal{O}_S)$. Hence (4)", "follows from (5)." ], "refs": [ "derived-lemma-two-ss-complex-functor", "coherent-lemma-quasi-coherence-higher-direct-images", "homology-lemma-biregular-ss-converges", "coherent-lemma-induction-principle", "cohomology-lemma-unbounded-relative-mayer-vietoris", "cohomology-lemma-cone-tor-amplitude", "derived-lemma-leray-acyclicity", "coherent-lemma-relative-affine-vanishing", "perfect-lemma-cohomology-base-change", "cohomology-lemma-variant-derived-pullback", "perfect-lemma-quasi-coherence-pushforward-direct-sums" ], "ref_ids": [ 1871, 3295, 12101, 3291, 2147, 2218, 1844, 3283, 7025, 2119, 6950 ] } ], "ref_ids": [] }, { "id": 7034, "type": "theorem", "label": "perfect-lemma-K-flat", "categories": [ "perfect" ], "title": "perfect-lemma-K-flat", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes with $Y = \\Spec(A)$ affine.", "Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be a finite affine open covering", "such that all the finite intersections", "$U_{i_0 \\ldots i_p} = U_{i_0} \\cap \\ldots \\cap U_{i_p}$", "are affine. Let $\\mathcal{F}^\\bullet$ be a bounded complex of", "$f^{-1}\\mathcal{O}_Y$-modules. Assume for all $n \\in \\mathbf{Z}$", "the sheaf $\\mathcal{F}^n$ is a flat $f^{-1}\\mathcal{O}_Y$-module and", "$\\mathcal{F}^n$ has the structure of a quasi-coherent $\\mathcal{O}_X$-module", "compatible with the given $p^{-1}\\mathcal{O}_Y$-module structure (but the", "differentials in the complex $\\mathcal{F}^\\bullet$ need not", "be $\\mathcal{O}_X$-linear). Then the complex", "$\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))$", "is K-flat as a complex of $A$-modules." ], "refs": [], "proofs": [ { "contents": [ "We may write", "$$", "\\mathcal{F}^\\bullet = (\\mathcal{F}^a \\to \\ldots \\to \\mathcal{F}^b)", "$$", "Arguing by induction on $b - a$ and considering the distinguished triangle", "$$", "\\mathcal{F}^b[-b] \\to (\\mathcal{F}^a \\to \\ldots \\to \\mathcal{F}^b)", "\\to (\\mathcal{F}^a \\to \\ldots \\to \\mathcal{F}^{b - 1}) \\to", "\\mathcal{F}^b[-b + 1]", "$$", "and using", "More on Algebra, Lemma \\ref{more-algebra-lemma-K-flat-two-out-of-three}", "we reduce to the case where $\\mathcal{F}^\\bullet$ consists of a", "single quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "placed in degree $0$. In this case the {\\v C}ech complex", "for $\\mathcal{F}$ and $\\mathcal{U}$ is homotopy equivalent to the", "alternating {\\v C}ech complex, see", "Cohomology, Lemma \\ref{cohomology-lemma-alternating-usual}.", "Since $U_{i_0 \\ldots i_p}$ is always affine, we see that", "$\\mathcal{F}(U_{i_0 \\ldots i_p})$ is $A$-flat.", "Hence", "$\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "is a bounded complex of flat $A$-modules and hence K-flat", "by More on Algebra, Lemma", "\\ref{more-algebra-lemma-derived-tor-quasi-isomorphism}." ], "refs": [ "more-algebra-lemma-K-flat-two-out-of-three", "cohomology-lemma-alternating-usual", "more-algebra-lemma-derived-tor-quasi-isomorphism" ], "ref_ids": [ 10126, 2095, 10128 ] } ], "ref_ids": [] }, { "id": 7035, "type": "theorem", "label": "perfect-lemma-kunneth-single-sheaf", "categories": [ "perfect" ], "title": "perfect-lemma-kunneth-single-sheaf", "contents": [ "In the situation above the map (\\ref{equation-kunneth-single-sheaves}) is an", "isomorphism if $S$ is affine, $\\mathcal{F}$ and $\\mathcal{G}$ are $S$-flat and", "quasi-coherent and $X$ and $Y$ are quasi-compact with affine diagonal." ], "refs": [], "proofs": [ { "contents": [ "We strongly urge the reader to read the proof of", "Varieties, Lemma \\ref{varieties-lemma-kunneth} first.", "Choose finite affine open coverings", "$\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ and", "$\\mathcal{V} : Y = \\bigcup_{j \\in J} V_j$.", "This determines an affine open covering", "$\\mathcal{W} : X \\times_S Y = \\bigcup_{(i, j) \\in I \\times J} U_i \\times_S V_j$.", "Note that $\\mathcal{W}$ is a refinement of", "$\\text{pr}_1^{-1}\\mathcal{U}$ and of $\\text{pr}_2^{-1}\\mathcal{V}$.", "Thus by the discussion in Cohomology, Section", "\\ref{cohomology-section-cech-cohomology-of-complexes}", "we obtain maps", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W}, p^*\\mathcal{F})", "\\quad\\text{and}\\quad", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G})", "\\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W}, q^*\\mathcal{G})", "$$", "well defined up to homotopy and compatible with pullback maps on cohomology.", "In Cohomology, Equation (\\ref{cohomology-equation-needs-signs})", "we have constructed a map of complexes", "$$", "\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W}, p^*\\mathcal{F})", "\\otimes_A", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W}, q^*\\mathcal{G}))", "\\longrightarrow", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W},", "p^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_S Y}}", "q^*\\mathcal{G})", "$$", "which is compatible with the cup product on cohomology by", "Cohomology, Lemma \\ref{cohomology-lemma-diagrams-commute}.", "Combining the above we obtain a map of complexes", "\\begin{equation}", "\\label{equation-kunneth-on-cech}", "\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\otimes_A", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G}))", "\\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W},", "p^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X \\times_S Y}}", "q^*\\mathcal{G})", "\\end{equation}", "We claim this is the map in the statement of the lemma, i.e.,", "the source and target of this arrow are the same as the source", "and target of (\\ref{equation-kunneth-single-sheaves}). Namely, by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}", "and", "Cohomology, Lemma \\ref{cohomology-lemma-cech-complex-complex-computes}", "the canonical maps", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\to", "R\\Gamma(X, \\mathcal{F}),", "\\quad", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G})", "\\to", "R\\Gamma(Y, \\mathcal{G})", "$$", "and", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W},", "p^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_S Y}} q^*\\mathcal{G})", "\\to", "R\\Gamma(X \\times_S Y,", "p^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_S Y}}", "q^*\\mathcal{G})", "$$", "are isomorphisms. On the other hand, the complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "is K-flat by Lemma \\ref{lemma-K-flat} and we conclude that", "$\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\otimes_A", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G}))$", "represents the derived tensor product", "$R\\Gamma(X, \\mathcal{F}) \\otimes_A^\\mathbf{L} R\\Gamma(Y, \\mathcal{G})$", "as claimed.", "\\medskip\\noindent", "We still have to show that (\\ref{equation-kunneth-on-cech})", "is a quasi-isomorphism. We will do this using dimension shifting.", "Set $d(\\mathcal{F}) = \\max \\{d \\mid H^d(X, \\mathcal{F}) \\not = 0\\}$.", "Assume $d(\\mathcal{F}) > 0$. Set $U = \\coprod\\nolimits_{i \\in I} U_i$.", "This is an affine scheme as $I$ is finite. Denote", "$j : U \\to X$ the morphism which is the inclusion $U_i \\to X$", "on each $U_i$. Since the diagonal of $X$ is affine, the morphism", "$j$ is affine, see", "Morphisms, Lemma \\ref{morphisms-lemma-affine-permanence}.", "It follows that $\\mathcal{F}' = j_*j^*\\mathcal{F}$ is $S$-flat, see", "Morphisms, Lemma \\ref{morphisms-lemma-pushforward-flat-affine}.", "It also follows that $d(\\mathcal{F}') = 0$ by combining", "Cohomology of Schemes, Lemmas", "\\ref{coherent-lemma-relative-affine-cohomology} and", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "For all $x \\in X$ we have $\\mathcal{F}_x \\to \\mathcal{F}'_x$", "is the inclusion of a direct summand: if $x \\in U_i$,", "then $\\mathcal{F}' \\to (U_i \\to X)_*\\mathcal{F}|_{U_i}$", "gives a splitting. We conclude that", "$\\mathcal{F} \\to \\mathcal{F}'$ is injective and", "$\\mathcal{F}'' = \\mathcal{F}'/\\mathcal{F}$", "is $S$-flat as well. The short exact sequence", "$0 \\to \\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{F}'' \\to 0$", "of flat quasi-coherent $\\mathcal{O}_X$-modules", "produces a short exact sequence of complexes", "$$", "0 \\to", "\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\otimes_A", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G})) \\to", "\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}')", "\\otimes_A", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G})) \\to", "\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}'')", "\\otimes_A", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G})) \\to 0", "$$", "and a short exact sequence of complexes", "$$", "0 \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W},", "p^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X \\times_S Y}}", "q^*\\mathcal{G}) \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W},", "p^*\\mathcal{F}'", "\\otimes_{\\mathcal{O}_{X \\times_S Y}}", "q^*\\mathcal{G}) \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W},", "p^*\\mathcal{F}''", "\\otimes_{\\mathcal{O}_{X \\times_S Y}}", "q^*\\mathcal{G}) \\to 0", "$$", "Moreover, the maps (\\ref{equation-kunneth-on-cech}) between these are", "compatible with these short exact sequences. Hence it suffices to prove", "(\\ref{equation-kunneth-on-cech})", "is an isomorphism for $\\mathcal{F}'$ and $\\mathcal{F}''$. Finally,", "we have $d(\\mathcal{F}'') < d(\\mathcal{F})$.", "In this way we reduce to the case $d(\\mathcal{F}) = 0$.", "\\medskip\\noindent", "Arguing in the same fashion for $\\mathcal{G}$ we find that we", "may assume that both $\\mathcal{F}$ and $\\mathcal{G}$", "have nonzero cohomology only in degree $0$.", "Observe that this means that $\\Gamma(X, \\mathcal{F})$", "is quasi-isomorphic to the $K$-flat complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "of $A$-modules sitting in degrees $\\geq 0$.", "It follows that $\\Gamma(X, \\mathcal{F})$ is a flat $A$-module", "(because we can compute higher Tor's against this module", "by tensoring with the Cech complex).", "Let $V \\subset Y$ be an affine open. Consider the affine open covering", "$\\mathcal{U}_V : X \\times_S V = \\bigcup_{i \\in I} U_i \\times_S V$.", "It is immediate that", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\otimes_A \\mathcal{G}(V) =", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}_V,", "p^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}}", "q^*\\mathcal{G})", "$$", "(equality of complexes). By the flatness of $\\mathcal{G}(V)$", "over $A$ we see that", "$\\Gamma(X, \\mathcal{F}) \\otimes_A \\mathcal{G}(V) \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\otimes_A \\mathcal{G}(V)$ is a quasi-isomorphism.", "Since the sheafification of", "$V \\mapsto \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}_V,", "p^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}}", "q^*\\mathcal{G})$ represents", "$Rq_*(p^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}} q^*\\mathcal{G})$", "by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-separated-case-relative-cech}", "we conclude that", "$$", "Rq_*(p^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}} q^*\\mathcal{G})", "\\cong", "\\Gamma(X, \\mathcal{F}) \\otimes_A \\mathcal{G}", "$$", "on $Y$ where the notation on the right hand side indicates the module", "$$", "b^*\\widetilde{\\Gamma(X, \\mathcal{F})} \\otimes_{\\mathcal{O}_Y} \\mathcal{G}", "$$", "Using the Leray spectral sequence for $q$ we find", "$$", "H^n(X \\times_S Y, p^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}}", "q^*\\mathcal{G}) =", "H^n(Y,", "b^*\\widetilde{\\Gamma(X, \\mathcal{F})} \\otimes_{\\mathcal{O}_Y} \\mathcal{G})", "$$", "Using Lemma \\ref{lemma-cohomology-base-change} for the morphism", "$b : Y \\to S = \\Spec(A)$ and using that $\\Gamma(X, \\mathcal{F})$", "is $A$-flat we conclude that", "$H^n(X \\times_S Y, p^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}}", "q^*\\mathcal{G})$ is zero for $n > 0$ and isomorphic to", "$H^0(X, \\mathcal{F}) \\otimes_A H^0(Y, \\mathcal{G})$ for $n = 0$.", "Of course, here we also use that $\\mathcal{G}$ only has", "cohomology in degree $0$.", "This finishes the proof (except that we should check that the", "isomorphism is indeed given by cup product in degree $0$; we omit", "the verification)." ], "refs": [ "varieties-lemma-kunneth", "cohomology-lemma-diagrams-commute", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "cohomology-lemma-cech-complex-complex-computes", "perfect-lemma-K-flat", "morphisms-lemma-affine-permanence", "morphisms-lemma-pushforward-flat-affine", "coherent-lemma-relative-affine-cohomology", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "coherent-lemma-separated-case-relative-cech", "perfect-lemma-cohomology-base-change" ], "ref_ids": [ 11023, 2130, 3282, 2099, 7034, 5179, 5261, 3284, 3282, 3301, 7025 ] } ], "ref_ids": [] }, { "id": 7036, "type": "theorem", "label": "perfect-lemma-kunneth-special", "categories": [ "perfect" ], "title": "perfect-lemma-kunneth-special", "contents": [ "In the situation above the cup product (\\ref{equation-de-rham-kunneth})", "is an isomorphism in $D(A)$ if the following assumptions hold", "\\begin{enumerate}", "\\item $S = \\Spec(A)$ is affine,", "\\item $X$ and $Y$ are quasi-compact with affine diagonal,", "\\item $\\mathcal{F}^\\bullet$ is bounded,", "\\item $\\mathcal{G}^\\bullet$ is bounded below,", "\\item $\\mathcal{F}^n$ is $S$-flat, and", "\\item $\\mathcal{G}^m$ is $S$-flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use the notation $\\mathcal{A}_{X/S}$ and $\\mathcal{A}_{Y/S}$", "introduced in Morphisms, Remark", "\\ref{morphisms-remark-base-change-differential-operators}.", "Suppose that we have maps of complexes", "$$", "\\mathcal{F}_1^\\bullet \\to", "\\mathcal{F}_2^\\bullet \\to", "\\mathcal{F}_3^\\bullet \\to", "\\mathcal{F}_1^\\bullet[1]", "$$", "in the category $\\mathcal{A}_{X/S}$.", "Then by the functoriality of the cup product", "we obtain a commutative diagram", "$$", "\\xymatrix{", "R\\Gamma(X, \\mathcal{F}_1^\\bullet)", "\\otimes_A^\\mathbf{L}", "R\\Gamma(Y, \\mathcal{G}^\\bullet)", "\\ar[r] \\ar[d] &", "R\\Gamma(X \\times_S Y,", "\\text{Tot}(\\mathcal{F}_1^\\bullet \\boxtimes \\mathcal{G}^\\bullet)) \\ar[d] \\\\", "R\\Gamma(X, \\mathcal{F}_2^\\bullet)", "\\otimes_A^\\mathbf{L}", "R\\Gamma(Y, \\mathcal{G}^\\bullet)", "\\ar[r] \\ar[d] &", "R\\Gamma(X \\times_S Y,", "\\text{Tot}(\\mathcal{F}_2^\\bullet \\boxtimes \\mathcal{G}^\\bullet)) \\ar[d] \\\\", "R\\Gamma(X, \\mathcal{F}_3^\\bullet)", "\\otimes_A^\\mathbf{L}", "R\\Gamma(Y, \\mathcal{G}^\\bullet)", "\\ar[r] \\ar[d] &", "R\\Gamma(X \\times_S Y,", "\\text{Tot}(\\mathcal{F}_3^\\bullet \\boxtimes \\mathcal{G}^\\bullet)) \\ar[d] \\\\", "R\\Gamma(X, \\mathcal{F}_1^\\bullet[1])", "\\otimes_A^\\mathbf{L}", "R\\Gamma(Y, \\mathcal{G}^\\bullet)", "\\ar[r] &", "R\\Gamma(X \\times_S Y,", "\\text{Tot}(\\mathcal{F}_1^\\bullet[1] \\boxtimes \\mathcal{G}^\\bullet))", "}", "$$", "If the original maps form a distinguished triangle in the homotopy category", "of $\\mathcal{A}_{X/S}$, then", "the columns of this diagram form distinguished triangles in $D(A)$.", "\\medskip\\noindent", "In the situation of the lemma,", "suppose that $\\mathcal{F}^n = 0$ for $n < i$. Then we may consider the", "termwise split short exact sequence of complexes", "$$", "0 \\to \\sigma_{\\geq i + 1}\\mathcal{F}^\\bullet \\to", "\\mathcal{F}^\\bullet \\to \\mathcal{F}^i[-i] \\to 0", "$$", "where the truncation is as in", "Homology, Section \\ref{homology-section-truncations}.", "This produces the distinguished triangle", "$$", "\\sigma_{\\geq i + 1}\\mathcal{F}^\\bullet \\to", "\\mathcal{F}^\\bullet \\to", "\\mathcal{F}^i[-i] \\to", "(\\sigma_{\\geq i + 1}\\mathcal{F}^\\bullet)[1]", "$$", "in the homotopy category of $\\mathcal{A}_{X/S}$", "where the final arrow is given by the boundary map", "$\\mathcal{F}^i \\to \\mathcal{F}^{i + 1}$.", "It follows from the discussion above that it suffices to prove the lemma for", "$\\mathcal{F}^i[-i]$ and $\\sigma_{\\geq i + 1}\\mathcal{F}^\\bullet$.", "Since $\\sigma_{\\geq i + 1}\\mathcal{F}^\\bullet$ has fewer nonzero", "terms, by induction, if we can prove the lemma if $\\mathcal{F}^\\bullet$ is", "nonzero only in single degree, then the lemma follows.", "Thus we may assume $\\mathcal{F}^\\bullet$ is nonzero only in one degree.", "\\medskip\\noindent", "Assume $\\mathcal{F}^\\bullet$ is the complex which has an $S$-flat quasi-coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ sitting in degree $0$ and is zero in", "other degrees. Observe that $R\\Gamma(X, \\mathcal{F})$ has finite", "tor dimension by Lemma \\ref{lemma-cohomology-de-rham-base-change} for example.", "Say it has tor amplitude in $[i, j]$.", "Pick $N \\gg 0$ and consider the distinguished triangle", "$$", "\\sigma_{\\geq N + 1}\\mathcal{G}^\\bullet \\to", "\\mathcal{G}^\\bullet \\to", "\\sigma_{\\leq N}\\mathcal{G}^\\bullet \\to", "(\\sigma_{\\geq N + 1}\\mathcal{G}^\\bullet)[1]", "$$", "in the homotopy category of $\\mathcal{A}_{Y/S}$. Now observe that both", "$$", "R\\Gamma(X, \\mathcal{F})", "\\otimes_A^\\mathbf{L}", "R\\Gamma(Y, \\sigma_{\\geq N + 1}\\mathcal{G}^\\bullet)", "\\quad\\text{and}\\quad", "R\\Gamma(X \\times_S Y,", "\\text{Tot}(\\mathcal{F} \\boxtimes \\sigma_{\\geq N + 1}\\mathcal{G}^\\bullet))", "$$", "have vanishing cohomology in degrees $\\leq N + i$. Thus, using the arguments", "given above, if we want to prove our statement in a given degree, then we may", "assume $\\mathcal{G}^\\bullet$ is bounded.", "Repeating the arguments above one more time we may also assume", "$\\mathcal{G}^\\bullet$ is nonzero only in one degree.", "This case is handled by Lemma \\ref{lemma-kunneth-single-sheaf}." ], "refs": [ "morphisms-remark-base-change-differential-operators", "perfect-lemma-cohomology-de-rham-base-change", "perfect-lemma-kunneth-single-sheaf" ], "ref_ids": [ 5599, 7033, 7035 ] } ], "ref_ids": [] }, { "id": 7037, "type": "theorem", "label": "perfect-lemma-kunneth-Ext", "categories": [ "perfect" ], "title": "perfect-lemma-kunneth-Ext", "contents": [ "In the situation above, assume $a$ and $b$ are quasi-compact and", "quasi-separated and $X$ and $Y$ are tor independent over $S$.", "If $K$ is perfect, $K' \\in D_\\QCoh(\\mathcal{O}_X)$, $M$ is perfect, and", "$M' \\in D_\\QCoh(\\mathcal{O}_Y)$, then (\\ref{equation-kunneth-ext})", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "In this case we have $R\\SheafHom(K, K') = K' \\otimes^\\mathbf{L} K^\\vee$,", "$R\\SheafHom(M, M') = M' \\otimes^\\mathbf{L} M^\\vee$, and", "$$", "R\\SheafHom(K \\boxtimes M, K' \\boxtimes M') =", "(K' \\otimes^\\mathbf{L} K^\\vee) \\boxtimes", "(M' \\otimes^\\mathbf{L} M^\\vee)", "$$", "See Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}", "and we also use that being perfect is preserved by pullback", "and by tensor products.", "Hence this case follows from Lemma \\ref{lemma-kunneth}.", "(We omit the verification that with these identifications", "we obtain the same map.)" ], "refs": [ "cohomology-lemma-dual-perfect-complex", "perfect-lemma-kunneth" ], "ref_ids": [ 2233, 7032 ] } ], "ref_ids": [] }, { "id": 7038, "type": "theorem", "label": "perfect-lemma-single-complex-base-change-condition", "categories": [ "perfect" ], "title": "perfect-lemma-single-complex-base-change-condition", "contents": [ "Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "S' \\ar[r]^g &", "S", "}", "$$", "be a cartesian diagram of schemes. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$", "and let $L(g')^*K \\to K'$ be a map in $D_\\QCoh(\\mathcal{O}_{X'})$.", "The following are equivalent", "\\begin{enumerate}", "\\item for any $x' \\in X'$ and $i \\in \\mathbf{Z}$ the map (\\ref{equation-bc})", "is an isomorphism,", "\\item for $U \\subset X$, $V' \\subset S'$ affine open both mapping", "into the affine open $V \\subset S$ with $U' = V' \\times_V U$", "the composition", "$$", "R\\Gamma(U, K) \\otimes_{\\mathcal{O}_S(U)}^\\mathbf{L} \\mathcal{O}_{S'}(V')", "\\to", "R\\Gamma(U, K) \\otimes_{\\mathcal{O}_X(U)}^\\mathbf{L} \\mathcal{O}_{X'}(U')", "\\to", "R\\Gamma(U', K')", "$$", "is an isomorphism in $D(\\mathcal{O}_{S'}(V'))$, and", "\\item there is a set $I$ of quadruples $U_i, V_i', V_i, U_i'$, $i \\in I$", "as in (2) with $X' = \\bigcup U'_i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The second arrow in (2) comes from the equality", "$$", "R\\Gamma(U, K) \\otimes_{\\mathcal{O}_X(U)}^\\mathbf{L} \\mathcal{O}_{X'}(U') =", "R\\Gamma(U', L(g')^*K)", "$$", "of Lemma \\ref{lemma-quasi-coherence-pullback} and the given arrow", "$L(g')^*K \\to K'$. The first arrow of (2) is", "More on Algebra, Equation (\\ref{more-algebra-equation-comparison-map}).", "It is clear that (2) implies (3). Observe that (1) is local on $X'$.", "Therefore it suffices to show that if $X$, $S$, $S'$, $X'$ are affine, then", "(1) is equivalent to the condition that", "$$", "R\\Gamma(X, K) \\otimes_{\\mathcal{O}_S(S)}^\\mathbf{L} \\mathcal{O}_{S'}(S')", "\\to", "R\\Gamma(X, K) \\otimes_{\\mathcal{O}_X(X)}^\\mathbf{L} \\mathcal{O}_{X'}(X')", "\\to", "R\\Gamma(X', K')", "$$", "is an isomorphism in $D(\\mathcal{O}_{S'}(S'))$. Say", "$S = \\Spec(R)$, $X = \\Spec(A)$, $S' = \\Spec(R')$, $X' = \\Spec(A')$,", "$K$ corresponds to the complex $M^\\bullet$ of $A$-modules, and", "$K'$ corresponds to the complex $N^\\bullet$ of $A'$-modules.", "Note that $A' = A \\otimes_R R'$. The condition above is that the composition", "$$", "M^\\bullet \\otimes_R^\\mathbf{L} R' \\to", "M^\\bullet \\otimes_A^\\mathbf{L} A' \\to", "N^\\bullet", "$$", "is an isomorphism in $D(R')$. Equivalently, it is that for all", "$i \\in \\mathbf{Z}$ the map", "$$", "H^i(M^\\bullet \\otimes_R^\\mathbf{L} R') \\to", "H^i(M^\\bullet \\otimes_A^\\mathbf{L} A') \\to", "H^i(N^\\bullet)", "$$", "is an isomorphism. Observe that this is a map of $A \\otimes_R R'$-modules,", "i.e., of $A'$-modules. On the other hand, (1) is the requirement", "that for compatible primes", "$\\mathfrak q' \\subset A'$, $\\mathfrak q \\subset A$,", "$\\mathfrak p' \\subset R'$, $\\mathfrak p \\subset R$", "the composition", "$$", "H^i(M^\\bullet_\\mathfrak q \\otimes_{R_\\mathfrak p}^\\mathbf{L} R'_{\\mathfrak p'})", "\\otimes_{(A_\\mathfrak q \\otimes_{R_\\mathfrak p} R'_{\\mathfrak p'})}", "A'_{\\mathfrak q'} \\to", "H^i(M^\\bullet_{\\mathfrak q}", "\\otimes_{A_\\mathfrak q}^\\mathbf{L} A'_{\\mathfrak q'})", "\\to H^i(N^\\bullet_{\\mathfrak q'})", "$$", "is an isomorphism. Since", "$$", "H^i(M^\\bullet_\\mathfrak q \\otimes_{R_\\mathfrak p}^\\mathbf{L} R'_{\\mathfrak p'})", "\\otimes_{(A_\\mathfrak q \\otimes_{R_\\mathfrak p} R'_{\\mathfrak p'})}", "A'_{\\mathfrak q'} =", "H^i(M^\\bullet \\otimes_R^\\mathbf{L} R') \\otimes_{A'} A'_{\\mathfrak q'}", "$$", "is the localization at $\\mathfrak q'$,", "we see that these two conditions are equivalent by", "Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}." ], "refs": [ "perfect-lemma-quasi-coherence-pullback", "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 6944, 410 ] } ], "ref_ids": [] }, { "id": 7039, "type": "theorem", "label": "perfect-lemma-single-complex-base-change", "categories": [ "perfect" ], "title": "perfect-lemma-single-complex-base-change", "contents": [ "Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "S' \\ar[r]^g &", "S", "}", "$$", "be a cartesian diagram of schemes. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$", "and let $L(g')^*K \\to K'$ be a map in $D_\\QCoh(\\mathcal{O}_{X'})$.", "If", "\\begin{enumerate}", "\\item the equivalent conditions of", "Lemma \\ref{lemma-single-complex-base-change-condition} hold, and", "\\item $f$ is quasi-compact and quasi-separated,", "\\end{enumerate}", "then the composition $Lg^*Rf_*K \\to Rf'_*L(g')^*K \\to Rf'_*K'$", "is an isomorphism." ], "refs": [ "perfect-lemma-single-complex-base-change-condition" ], "proofs": [ { "contents": [ "We could prove this using the same method as in the proof of", "Lemma \\ref{lemma-compare-base-change} but instead we will prove", "it using the induction principle and relative Mayer-Vietoris.", "\\medskip\\noindent", "To check the map is an isomorphism we may work locally on $S'$.", "Hence we may assume $g : S' \\to S$ is a morphism of affine schemes.", "In particular $X$ is a quasi-compact and quasi-separated scheme.", "We will use the induction principle of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}", "to prove that for any quasi-compact open $U \\subset X$ the similarly", "constructed map $Lg^*R(U \\to S)_*K|_U \\to R(U' \\to S')_*K'|_{U'}$", "is an isomorphism. Here $U' = (g')^{-1}(U)$.", "\\medskip\\noindent", "If $U \\subset X$ is an affine open, then we find that the result is", "true by assumption, see", "Lemma \\ref{lemma-single-complex-base-change-condition} part (2)", "and the translation into algebra afforded to us", "by Lemmas \\ref{lemma-affine-compare-bounded} and", "\\ref{lemma-quasi-coherence-pullback}.", "\\medskip\\noindent", "The induction step. Suppose that $X = U \\cup V$ is an open covering", "with $U$, $V$, $U \\cap V$", "quasi-compact such that the result holds for $U$, $V$, and $U \\cap V$.", "Denote $a = f|_U$, $b = f|_V$ and $c = f|_{U \\cap V}$.", "Let $a' : U' \\to S'$, $b' : V' \\to S'$ and $c' : U' \\cap V' \\to S'$", "be the base changes of $a$, $b$, and $c$.", "Using the distinguished triangles from relative Mayer-Vietoris", "(Cohomology, Lemma \\ref{cohomology-lemma-unbounded-relative-mayer-vietoris})", "we obtain a commutative diagram", "$$", "\\xymatrix{", "Lg^*Rf_*K \\ar[r] \\ar[d] &", "Rf'_* K' \\ar[d] \\\\", "Lg^*Ra_* K|_U \\oplus", "Lg^*Rb_* K|_V \\ar[r] \\ar[d] &", "Ra'_* K'|_{U'} \\oplus", "Rb'_* K'|_{V'} \\ar[d] \\\\", "Lg^*Rc_* K|_{U \\cap V} \\ar[r] \\ar[d] &", "Rc'_* K'|_{U' \\cap V'} \\ar[d] \\\\", "Lg^*Rf_* K[1] \\ar[r] &", "Rf'_* K'[1]", "}", "$$", "Since the 2nd and 3rd horizontal arrows are isomorphisms so is the first", "(Derived Categories, Lemma \\ref{derived-lemma-third-isomorphism-triangle})", "and the proof of the lemma is finished." ], "refs": [ "perfect-lemma-compare-base-change", "coherent-lemma-induction-principle", "perfect-lemma-single-complex-base-change-condition", "perfect-lemma-affine-compare-bounded", "perfect-lemma-quasi-coherence-pullback", "cohomology-lemma-unbounded-relative-mayer-vietoris", "derived-lemma-third-isomorphism-triangle" ], "ref_ids": [ 7028, 3291, 7038, 6941, 6944, 2147, 1759 ] } ], "ref_ids": [ 7038 ] }, { "id": 7040, "type": "theorem", "label": "perfect-lemma-single-complex-base-change-condition-inherited", "categories": [ "perfect" ], "title": "perfect-lemma-single-complex-base-change-condition-inherited", "contents": [ "Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "S' \\ar[r]^g &", "S", "}", "$$", "be a cartesian diagram of schemes. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$", "and let $L(g')^*K \\to K'$ be a map in $D_\\QCoh(\\mathcal{O}_{X'})$.", "If the equivalent conditions of", "Lemma \\ref{lemma-single-complex-base-change-condition} hold, then", "\\begin{enumerate}", "\\item for $E \\in D_\\QCoh(\\mathcal{O}_X)$ the equivalent", "conditions of Lemma \\ref{lemma-single-complex-base-change-condition} hold", "for $L(g')^*(E \\otimes^\\mathbf{L} K) \\to L(g')^*E \\otimes^\\mathbf{L} K'$,", "\\item if $E$ in $D(\\mathcal{O}_X)$ is perfect the equivalent conditions of", "Lemma \\ref{lemma-single-complex-base-change-condition} hold for", "$L(g')^*R\\SheafHom(E, K) \\to R\\SheafHom(L(g')^*E, K')$, and", "\\item if $K$ is bounded below and $E$ in $D(\\mathcal{O}_X)$", "pseudo-coherent the equivalent conditions of", "Lemma \\ref{lemma-single-complex-base-change-condition} hold for", "$L(g')^*R\\SheafHom(E, K) \\to R\\SheafHom(L(g')^*E, K')$.", "\\end{enumerate}" ], "refs": [ "perfect-lemma-single-complex-base-change-condition", "perfect-lemma-single-complex-base-change-condition", "perfect-lemma-single-complex-base-change-condition", "perfect-lemma-single-complex-base-change-condition" ], "proofs": [ { "contents": [ "The statement makes sense as the complexes involved have quasi-coherent", "cohomology sheaves by Lemmas", "\\ref{lemma-quasi-coherence-pullback},", "\\ref{lemma-quasi-coherence-tensor-product}, and", "\\ref{lemma-quasi-coherence-internal-hom} and", "Cohomology, Lemmas \\ref{cohomology-lemma-pseudo-coherent-pullback} and", "\\ref{cohomology-lemma-perfect-pullback}.", "Having said this, we can check the maps (\\ref{equation-bc})", "are isomorphisms in case (1) by computing the source and target", "of (\\ref{equation-bc}) using the transitive property of tensor product, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-triple-tensor-product}.", "The map in (2) and (3) is the composition", "$$", "L(g')^*R\\SheafHom(E, K) \\to R\\SheafHom(L(g')^*E, L(g')^*K)", "\\to R\\SheafHom(L(g')^*E, K')", "$$", "where the first arrow is", "Cohomology, Remark \\ref{cohomology-remark-prepare-fancy-base-change}", "and the second arrow comes from the given map $L(g')^*K \\to K'$.", "To prove the maps (\\ref{equation-bc}) are isomorphisms one represents", "$E_x$ by a bounded complex of finite projective $\\mathcal{O}_{X. x}$-modules", "in case (2) or by a bounded above complex of finite free modules in case (3)", "and computes the source and target of the arrow.", "Some details omitted." ], "refs": [ "perfect-lemma-quasi-coherence-pullback", "perfect-lemma-quasi-coherence-tensor-product", "perfect-lemma-quasi-coherence-internal-hom", "cohomology-lemma-pseudo-coherent-pullback", "cohomology-lemma-perfect-pullback", "more-algebra-lemma-triple-tensor-product", "cohomology-remark-prepare-fancy-base-change" ], "ref_ids": [ 6944, 6945, 6981, 2206, 2225, 10134, 2282 ] } ], "ref_ids": [ 7038, 7038, 7038, 7038 ] }, { "id": 7041, "type": "theorem", "label": "perfect-lemma-base-change-tensor", "categories": [ "perfect" ], "title": "perfect-lemma-base-change-tensor", "contents": [ "Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of", "schemes. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $\\mathcal{G}^\\bullet$", "be a bounded above complex of quasi-coherent", "$\\mathcal{O}_X$-modules flat over $S$. Then formation of", "$$", "Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)", "$$", "commutes with arbitrary base change (see proof for precise statement)." ], "refs": [], "proofs": [ { "contents": [ "The statement means the following. Let $g : S' \\to S$ be a morphism of", "schemes and consider the base change diagram", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "S' \\ar[r]^g &", "S", "}", "$$", "in other words $X' = S' \\times_S X$. The lemma asserts that", "$$", "Lg^*Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)", "\\longrightarrow", "Rf'_*\\left(", "L(g')^*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{X'}} (g')^*\\mathcal{G}^\\bullet", "\\right)", "$$", "is an isomorphism. Observe that on the right hand side we do {\\bf not}", "use the derived pullback on $\\mathcal{G}^\\bullet$.", "To prove this, we apply Lemmas \\ref{lemma-single-complex-base-change} and", "\\ref{lemma-single-complex-base-change-condition-inherited} to see that it", "suffices to prove the canonical map", "$$", "L(g')^*\\mathcal{G}^\\bullet \\to (g')^*\\mathcal{G}^\\bullet", "$$", "satisfies the equivalent conditions of", "Lemma \\ref{lemma-single-complex-base-change-condition}.", "This follows by checking the condition on stalks, where it", "immediately follows from the fact that", "$\\mathcal{G}^\\bullet_x \\otimes_{\\mathcal{O}_{S, s}} \\mathcal{O}_{S', s'}$", "computes the derived tensor product by our assumptions on the complex", "$\\mathcal{G}^\\bullet$." ], "refs": [ "perfect-lemma-single-complex-base-change", "perfect-lemma-single-complex-base-change-condition-inherited", "perfect-lemma-single-complex-base-change-condition" ], "ref_ids": [ 7039, 7040, 7038 ] } ], "ref_ids": [] }, { "id": 7042, "type": "theorem", "label": "perfect-lemma-base-change-RHom", "categories": [ "perfect" ], "title": "perfect-lemma-base-change-RHom", "contents": [ "Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes.", "Let $E$ be an object of $D(\\mathcal{O}_X)$.", "Let $\\mathcal{G}^\\bullet$ be a complex of", "quasi-coherent $\\mathcal{O}_X$-modules. If", "\\begin{enumerate}", "\\item $E$ is perfect, $\\mathcal{G}^\\bullet$ is a bounded above,", "and $\\mathcal{G}^n$ is flat over $S$, or", "\\item $E$ is pseudo-coherent, $\\mathcal{G}^\\bullet$ is bounded,", "and $\\mathcal{G}^n$ is flat over $S$,", "\\end{enumerate}", "then formation of", "$$", "Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet)", "$$", "commutes with arbitrary base change (see proof for precise statement)." ], "refs": [], "proofs": [ { "contents": [ "The statement means the following. Let $g : S' \\to S$ be a morphism of", "schemes and consider the base change diagram", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "S' \\ar[r]^g &", "S", "}", "$$", "in other words $X' = S' \\times_S X$. The lemma asserts that", "$$", "Lg^*Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet)", "\\longrightarrow", "R(f')_*R\\SheafHom(L(g')^*E, (g')^*\\mathcal{G}^\\bullet)", "$$", "is an isomorphism. Observe that on the right hand side we do {\\bf not}", "use the derived pullback on $\\mathcal{G}^\\bullet$. To prove this, we apply", "Lemmas \\ref{lemma-single-complex-base-change} and", "\\ref{lemma-single-complex-base-change-condition-inherited} to see that it", "suffices to prove the canonical map", "$$", "L(g')^*\\mathcal{G}^\\bullet \\to (g')^*\\mathcal{G}^\\bullet", "$$", "satisfies the equivalent conditions of", "Lemma \\ref{lemma-single-complex-base-change-condition}.", "This was shown in the proof of Lemma \\ref{lemma-base-change-tensor}." ], "refs": [ "perfect-lemma-single-complex-base-change", "perfect-lemma-single-complex-base-change-condition-inherited", "perfect-lemma-single-complex-base-change-condition", "perfect-lemma-base-change-tensor" ], "ref_ids": [ 7039, 7040, 7038, 7041 ] } ], "ref_ids": [] }, { "id": 7043, "type": "theorem", "label": "perfect-lemma-perfect-direct-image", "categories": [ "perfect" ], "title": "perfect-lemma-perfect-direct-image", "contents": [ "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes", "which is locally of finite type. Let $E \\in D(\\mathcal{O}_X)$ such that", "\\begin{enumerate}", "\\item $E \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$,", "\\item the support of $H^i(E)$ is proper over $S$ for all $i$, and", "\\item $E$ has finite tor dimension as an object of $D(f^{-1}\\mathcal{O}_S)$.", "\\end{enumerate}", "Then $Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-direct-image-coherent} we see that $Rf_*E$ is an object of", "$D^b_{\\textit{Coh}}(\\mathcal{O}_S)$. Hence $Rf_*E$ is pseudo-coherent", "(Lemma \\ref{lemma-identify-pseudo-coherent-noetherian}).", "Hence it suffices to show that $Rf_*E$ has finite tor dimension, see", "Cohomology, Lemma \\ref{cohomology-lemma-perfect}.", "By Lemma \\ref{lemma-tor-qc-qs} it suffices to check that", "$Rf_*(E) \\otimes_{\\mathcal{O}_S}^\\mathbf{L} \\mathcal{F}$", "has universally bounded cohomology for all quasi-coherent", "sheaves $\\mathcal{F}$ on $S$. Bounded from above is clear as $Rf_*(E)$", "is bounded from above. Let $T \\subset X$ be the union of the supports", "of $H^i(E)$ for all $i$. Then $T$ is proper over $S$ by assumptions", "(1) and (2), see Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-union-closed-proper-over-base}.", "In particular there exists a quasi-compact open", "$X' \\subset X$ containing $T$. Setting $f' = f|_{X'}$ we have", "$Rf_*(E) = Rf'_*(E|_{X'})$ because $E$ restricts to zero on $X \\setminus T$.", "Thus we may replace $X$ by $X'$ and assume $f$ is quasi-compact.", "Moreover, $f$ is quasi-separated by Morphisms, Lemma", "\\ref{morphisms-lemma-finite-type-Noetherian-quasi-separated}. Now", "$$", "Rf_*(E) \\otimes_{\\mathcal{O}_S}^\\mathbf{L} \\mathcal{F} =", "Rf_*\\left(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*\\mathcal{F}\\right) =", "Rf_*\\left(E \\otimes_{f^{-1}\\mathcal{O}_S}^\\mathbf{L} f^{-1}\\mathcal{F}\\right)", "$$", "by", "Lemma \\ref{lemma-cohomology-base-change}", "and", "Cohomology, Lemma \\ref{cohomology-lemma-variant-derived-pullback}.", "By assumption (3) the complex", "$E \\otimes_{f^{-1}\\mathcal{O}_S}^\\mathbf{L} f^{-1}\\mathcal{F}$", "has cohomology sheaves in a", "given finite range, say $[a, b]$. Then $Rf_*$ of it", "has cohomology in the range $[a, \\infty)$ and we win." ], "refs": [ "perfect-lemma-direct-image-coherent", "perfect-lemma-identify-pseudo-coherent-noetherian", "cohomology-lemma-perfect", "perfect-lemma-tor-qc-qs", "coherent-lemma-union-closed-proper-over-base", "morphisms-lemma-finite-type-Noetherian-quasi-separated", "perfect-lemma-cohomology-base-change", "cohomology-lemma-variant-derived-pullback" ], "ref_ids": [ 6984, 6976, 2224, 6979, 3390, 5203, 7025, 2119 ] } ], "ref_ids": [] }, { "id": 7044, "type": "theorem", "label": "perfect-lemma-tensor-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-tensor-perfect", "contents": [ "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes", "which is locally of finite type. Let $E \\in D(\\mathcal{O}_X)$ be perfect.", "Let $\\mathcal{G}^\\bullet$ be a bounded complex of coherent", "$\\mathcal{O}_X$-modules flat over $S$ with support proper over $S$.", "Then $K = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)$", "is a perfect object of $D(\\mathcal{O}_S)$." ], "refs": [], "proofs": [ { "contents": [ "The object $K$ is perfect by Lemma \\ref{lemma-perfect-direct-image}.", "We check the lemma applies: Locally $E$ is isomorphic to a finite complex", "of finite free $\\mathcal{O}_X$-modules. Hence locally", "$E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet$ is isomorphic", "to a finite complex whose terms are of the form", "$$", "\\bigoplus\\nolimits_{i = a, \\ldots, b} (\\mathcal{G}^i)^{\\oplus r_i}", "$$", "for some integers $a, b, r_a, \\ldots, r_b$. This immediately implies the", "cohomology sheaves $H^i(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G})$", "are coherent. The hypothesis on the tor dimension also follows as", "$\\mathcal{G}^i$ is flat over $f^{-1}\\mathcal{O}_S$." ], "refs": [ "perfect-lemma-perfect-direct-image" ], "ref_ids": [ 7043 ] } ], "ref_ids": [] }, { "id": 7045, "type": "theorem", "label": "perfect-lemma-ext-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-ext-perfect", "contents": [ "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes", "which is locally of finite type. Let $E \\in D(\\mathcal{O}_X)$ be perfect.", "Let $\\mathcal{G}^\\bullet$ be a bounded complex of coherent", "$\\mathcal{O}_X$-modules flat over $S$ with support proper over $S$.", "Then $K = Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet)$ is a perfect object of", "$D(\\mathcal{O}_S)$." ], "refs": [], "proofs": [ { "contents": [ "Since $E$ is a perfect complex there exists a dual perfect complex", "$E^\\vee$, see Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "Observe that $R\\SheafHom(E, \\mathcal{G}^\\bullet) =", "E^\\vee \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet$.", "Thus the perfectness of $K$ follows from Lemma \\ref{lemma-tensor-perfect}." ], "refs": [ "cohomology-lemma-dual-perfect-complex", "perfect-lemma-tensor-perfect" ], "ref_ids": [ 2233, 7044 ] } ], "ref_ids": [] }, { "id": 7046, "type": "theorem", "label": "perfect-lemma-flat-proper-perfect-direct-image", "categories": [ "perfect" ], "title": "perfect-lemma-flat-proper-perfect-direct-image", "contents": [ "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a flat proper", "morphism of schemes. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Then", "$Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$." ], "refs": [], "proofs": [ { "contents": [ "We claim that Lemma \\ref{lemma-perfect-direct-image} applies.", "Conditions (1) and (2) are immediate. Condition (3) is local", "on $X$. Thus we may assume $X$ and $S$ affine and $E$", "represented by a strictly perfect complex of $\\mathcal{O}_X$-modules.", "Since $\\mathcal{O}_X$ is flat as a sheaf of $f^{-1}\\mathcal{O}_S$-modules", "we find that condition (3) is satisfied." ], "refs": [ "perfect-lemma-perfect-direct-image" ], "ref_ids": [ 7043 ] } ], "ref_ids": [] }, { "id": 7047, "type": "theorem", "label": "perfect-lemma-compute-tensor-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-compute-tensor-perfect", "contents": [ "Assumptions and notation as in Lemma \\ref{lemma-tensor-perfect}.", "Then there are functorial isomorphisms", "$$", "H^i(S, K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F})", "\\longrightarrow", "H^i(X, E \\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}))", "$$", "for $\\mathcal{F}$ quasi-coherent on $S$", "compatible with boundary maps (see proof)." ], "refs": [ "perfect-lemma-tensor-perfect" ], "proofs": [ { "contents": [ "We have", "$$", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*\\mathcal{F} =", "\\mathcal{G}^\\bullet \\otimes_{f^{-1}\\mathcal{O}_S}^\\mathbf{L} f^{-1}\\mathcal{F} =", "\\mathcal{G}^\\bullet \\otimes_{f^{-1}\\mathcal{O}_S} f^{-1}\\mathcal{F} =", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}", "$$", "the first equality by", "Cohomology, Lemma \\ref{cohomology-lemma-variant-derived-pullback},", "the second as $\\mathcal{G}^n$ is a flat $f^{-1}\\mathcal{O}_S$-module, and", "the third by definition of pullbacks. Hence we obtain", "\\begin{align*}", "H^i(X, E \\otimes^\\mathbf{L}_{\\mathcal{O}_X}", "(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}))", "& =", "H^i(X, E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*\\mathcal{F}) \\\\", "& =", "H^i(S,", "Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet", "\\otimes^\\mathbf{L}_{\\mathcal{O}_X} Lf^*\\mathcal{F})) \\\\", "& =", "H^i(S, Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)", "\\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}) \\\\", "& =", "H^i(S, K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}) ", "\\end{align*}", "The first equality by the above, the second by Leray", "(Cohomology, Lemma \\ref{cohomology-lemma-before-Leray}), and", "the third equality by Lemma \\ref{lemma-cohomology-base-change}.", "The statement on boundary maps means the following: Given a short", "exact sequence $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "of quasi-coherent $\\mathcal{O}_S$-modules, the isomorphisms fit into", "commutative diagrams", "$$", "\\xymatrix{", "H^i(S, K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}_3)", "\\ar[r] \\ar[d]_\\delta &", "H^i(X, E \\otimes^\\mathbf{L}_{\\mathcal{O}_X}", "(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_3))", "\\ar[d]^\\delta \\\\", "H^{i + 1}(S, K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}_1)", "\\ar[r] &", "H^{i + 1}(X, E \\otimes^\\mathbf{L}_{\\mathcal{O}_X}", "(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_1))", "}", "$$", "where the boundary maps come from the distinguished triangle", "$$", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}_1 \\to", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}_2 \\to", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}_3 \\to", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}_1[1]", "$$", "and the distinguished triangle in $D(\\mathcal{O}_X)$ associated to", "the short exact sequence", "$$", "0 \\to", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_1 \\to", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_2 \\to", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_3 \\to 0", "$$", "of complexes of $\\mathcal{O}_X$-modules.", "This sequence is exact because $\\mathcal{G}^n$ is flat over $S$.", "We omit the verification of the commutativity of the displayed diagram." ], "refs": [ "cohomology-lemma-variant-derived-pullback", "cohomology-lemma-before-Leray", "perfect-lemma-cohomology-base-change" ], "ref_ids": [ 2119, 2068, 7025 ] } ], "ref_ids": [ 7044 ] }, { "id": 7048, "type": "theorem", "label": "perfect-lemma-compute-ext-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-compute-ext-perfect", "contents": [ "Assumptions and notation as in Lemma \\ref{lemma-ext-perfect}.", "Then there are functorial isomorphisms", "$$", "H^i(S, K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F})", "\\longrightarrow", "\\Ext^i_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "$$", "for $\\mathcal{F}$ quasi-coherent on $S$", "compatible with boundary maps (see proof)." ], "refs": [ "perfect-lemma-ext-perfect" ], "proofs": [ { "contents": [ "As in the proof of Lemma \\ref{lemma-ext-perfect} let $E^\\vee$ be the", "dual perfect complex and recall that", "$K = Rf_*(E^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)$.", "Since we also have", "$$", "\\Ext^i_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "=", "H^i(X, E^\\vee \\otimes^\\mathbf{L}_{\\mathcal{O}_X}", "(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}))", "$$", "by construction of $E^\\vee$, the existence of the isomorphisms follows", "from Lemma \\ref{lemma-compute-tensor-perfect} applied to $E^\\vee$", "and $\\mathcal{G}^\\bullet$.", "The statement on boundary maps means the following: Given a short", "exact sequence $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "then the isomorphisms fit into commutative diagrams", "$$", "\\xymatrix{", "H^i(S, K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}_3)", "\\ar[r] \\ar[d]_\\delta &", "\\Ext^i_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_3) \\ar[d]^\\delta \\\\", "H^{i + 1}(S, K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}_1)", "\\ar[r] &", "\\Ext^{i + 1}_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_1)", "}", "$$", "where the boundary maps come from the distinguished triangle", "$$", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}_1 \\to", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}_2 \\to", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}_3 \\to", "K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}_1[1]", "$$", "and the distinguished triangle in $D(\\mathcal{O}_X)$ associated to", "the short exact sequence", "$$", "0 \\to", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_1 \\to", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_2 \\to", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}_3 \\to 0", "$$", "of complexes.", "This sequence is exact because $\\mathcal{G}$ is flat over $S$.", "We omit the verification of the commutativity of the displayed diagram." ], "refs": [ "perfect-lemma-ext-perfect", "perfect-lemma-compute-tensor-perfect" ], "ref_ids": [ 7045, 7047 ] } ], "ref_ids": [ 7045 ] }, { "id": 7049, "type": "theorem", "label": "perfect-lemma-compute-ext", "categories": [ "perfect" ], "title": "perfect-lemma-compute-ext", "contents": [ "Let $f : X \\to S$ be a morphism of schemes, $E \\in D(\\mathcal{O}_X)$", "and $\\mathcal{G}^\\bullet$ a complex of $\\mathcal{O}_X$-modules.", "Assume", "\\begin{enumerate}", "\\item $S$ is Noetherian,", "\\item $f$ is locally of finite type,", "\\item $E \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$,", "\\item $\\mathcal{G}^\\bullet$ is a bounded complex of", "coherent $\\mathcal{O}_X$-modules flat over $S$ with support proper over $S$.", "\\end{enumerate}", "Then the following two statements are true", "\\begin{enumerate}", "\\item[(A)] for every $m \\in \\mathbf{Z}$ there exists a perfect object $K$", "of $D(\\mathcal{O}_S)$ and functorial maps", "$$", "\\alpha^i_\\mathcal{F} :", "\\Ext^i_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "\\longrightarrow", "H^i(S, K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F})", "$$", "for $\\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps", "(see proof) such that $\\alpha^i_\\mathcal{F}$ is an isomorphism for $i \\leq m$", "\\item[(B)] there exists a pseudo-coherent $L \\in D(\\mathcal{O}_S)$", "and functorial isomorphisms", "$$", "\\Ext^i_{\\mathcal{O}_S}(L, \\mathcal{F}) \\longrightarrow", "\\Ext^i_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "$$", "for $\\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (A).", "Suppose $\\mathcal{G}^i$ is nonzero only for $i \\in [a, b]$.", "We may replace $X$ by a quasi-compact open neighbourhood of", "the union of the supports of $\\mathcal{G}^i$.", "Hence we may assume $X$ is Noetherian.", "In this case $X$ and $f$ are quasi-compact and quasi-separated.", "Choose an approximation $P \\to E$ by a perfect complex $P$ of", "$(X, E, -m - 1 + a)$", "(possible by Theorem \\ref{theorem-approximation}).", "Then the induced map", "$$", "\\Ext^i_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "\\longrightarrow", "\\Ext^i_{\\mathcal{O}_X}(P,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "$$", "is an isomorphism for $i \\leq m$. Namely, the kernel, resp.\\ cokernel of this", "map is a quotient, resp.\\ submodule of", "$$", "\\Ext^i_{\\mathcal{O}_X}(C,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "\\quad\\text{resp.}\\quad", "\\Ext^{i + 1}_{\\mathcal{O}_X}(C,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "$$", "where $C$ is the cone of $P \\to E$. Since $C$ has vanishing cohomology", "sheaves in degrees $\\geq -m - 1 + a$ these $\\Ext$-groups are zero", "for $i \\leq m + 1$ by", "Derived Categories, Lemma \\ref{derived-lemma-negative-exts}.", "This reduces us to the case that", "$E$ is a perfect complex which is Lemma \\ref{lemma-compute-ext-perfect}.", "The statement on boundaries is explained in the proof of", "Lemma \\ref{lemma-compute-ext-perfect}.", "\\medskip\\noindent", "Proof of (B). As in the proof of (A) we may assume $X$ is Noetherian.", "Observe that $E$ is pseudo-coherent by", "Lemma \\ref{lemma-identify-pseudo-coherent-noetherian}.", "By Lemma \\ref{lemma-pseudo-coherent-hocolim} we can write", "$E = \\text{hocolim} E_n$ with $E_n$ perfect and $E_n \\to E$ inducing", "an isomorphism on truncations $\\tau_{\\geq -n}$. Let $E_n^\\vee$", "be the dual perfect complex", "(Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}).", "We obtain an inverse system $\\ldots \\to E_3^\\vee \\to E_2^\\vee \\to E_1^\\vee$", "of perfect objects. This in turn gives rise to an inverse system", "$$", "\\ldots \\to K_3 \\to K_2 \\to K_1\\quad\\text{with}\\quad", "K_n = Rf_*(E_n^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)", "$$", "perfect on $S$, see Lemma \\ref{lemma-tensor-perfect}.", "By Lemma \\ref{lemma-compute-ext-perfect} and its proof and", "by the arguments in the previous paragraph (with $P = E_n$)", "for any quasi-coherent $\\mathcal{F}$ on $S$ we have", "functorial canonical maps", "$$", "\\xymatrix{", "& \\Ext^i_{\\mathcal{O}_X}(E,", "\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})", "\\ar[ld] \\ar[rd] \\\\", "H^i(S, K_{n + 1} \\otimes_{\\mathcal{O}_S}^\\mathbf{L} \\mathcal{F})", "\\ar[rr] & &", "H^i(S, K_n \\otimes_{\\mathcal{O}_S}^\\mathbf{L} \\mathcal{F})", "}", "$$", "which are isomorphisms for $i \\leq n + a$.", "Let $L_n = K_n^\\vee$ be the dual perfect complex.", "Then we see that $L_1 \\to L_2 \\to L_3 \\to \\ldots$", "is a system of perfect objects in $D(\\mathcal{O}_S)$", "such that for any quasi-coherent $\\mathcal{F}$ on $S$", "the maps", "$$", "\\Ext^i_{\\mathcal{O}_S}(L_{n + 1}, \\mathcal{F})", "\\longrightarrow", "\\Ext^i_{\\mathcal{O}_S}(L_n, \\mathcal{F})", "$$", "are isomorphisms for $i \\leq n + a - 1$. This implies that", "$L_n \\to L_{n + 1}$ induces an isomorphism on truncations", "$\\tau_{\\geq -n - a + 2}$ (hint: take cone of $L_n \\to L_{n + 1}$", "and look at its last nonvanishing cohomology sheaf).", "Thus $L = \\text{hocolim} L_n$ is pseudo-coherent, see", "Lemma \\ref{lemma-pseudo-coherent-hocolim}. The mapping property", "of homotopy colimits gives that", "$\\Ext^i_{\\mathcal{O}_S}(L, \\mathcal{F}) =", "\\Ext^i_{\\mathcal{O}_S}(L_n, \\mathcal{F})$", "for $i \\leq n + a - 3$ which finishes the proof." ], "refs": [ "perfect-theorem-approximation", "derived-lemma-negative-exts", "perfect-lemma-compute-ext-perfect", "perfect-lemma-compute-ext-perfect", "perfect-lemma-identify-pseudo-coherent-noetherian", "perfect-lemma-pseudo-coherent-hocolim", "cohomology-lemma-dual-perfect-complex", "perfect-lemma-tensor-perfect", "perfect-lemma-compute-ext-perfect", "perfect-lemma-pseudo-coherent-hocolim" ], "ref_ids": [ 6934, 1893, 7048, 7048, 6976, 7019, 2233, 7044, 7048, 7019 ] } ], "ref_ids": [] }, { "id": 7050, "type": "theorem", "label": "perfect-lemma-descend-homomorphisms", "categories": [ "perfect" ], "title": "perfect-lemma-descend-homomorphisms", "contents": [ "In Situation \\ref{situation-descent}.", "Let $E_0$ and $K_0$ be objects of", "$D(\\mathcal{O}_{S_0})$.", "Set $E_i = Lf_{i0}^*E_0$ and $K_i = Lf_{i0}^*K_0$ for $i \\geq 0$", "and set $E = Lf_0^*E_0$ and $K = Lf_0^*K_0$. Then the map", "$$", "\\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{S_i})}(E_i, K_i)", "\\longrightarrow", "\\Hom_{D(\\mathcal{O}_S)}(E, K)", "$$", "is an isomorphism if either", "\\begin{enumerate}", "\\item $E_0$ is perfect and $K_0 \\in D_\\QCoh(\\mathcal{O}_{S_0})$, or", "\\item $E_0$ is pseudo-coherent and", "$K_0 \\in D_\\QCoh(\\mathcal{O}_{S_0})$ has finite tor dimension.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For every open $U_0 \\subset S_0$ consider the condition $P$ that the canonical", "map", "$$", "\\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{U_i})}(E_i|_{U_i}, K_i|_{U_i})", "\\longrightarrow", "\\Hom_{D(\\mathcal{O}_U)}(E|_U, K|_U)", "$$", "is an isomorphism, where $U = f_0^{-1}(U_0)$ and $U_i = f_{i0}^{-1}(U_0)$.", "We will prove $P$ holds for all quasi-compact opens $U_0$", "by the induction principle of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}.", "Condition (2) of this lemma follows immediately from Mayer-Vietoris", "for hom in the derived category, see", "Cohomology, Lemma \\ref{cohomology-lemma-mayer-vietoris-hom}.", "Thus it suffices to prove the lemma when $S_0$ is affine.", "\\medskip\\noindent", "Assume $S_0$ is affine. Say $S_0 = \\Spec(A_0)$, $S_i = \\Spec(A_i)$, and", "$S = \\Spec(A)$. We will use Lemma \\ref{lemma-affine-compare-bounded}", "without further mention.", "\\medskip\\noindent", "In case (1) the object $E_0^\\bullet$ corresponds to a finite complex", "of finite projective $A_0$-modules, see Lemma \\ref{lemma-perfect-affine}.", "We may represent the object $K_0$ by a K-flat complex $K_0^\\bullet$", "of $A_0$-modules. In this situation we are trying to prove", "$$", "\\colim_{i \\geq 0} \\Hom_{D(A_i)}(E_0^\\bullet \\otimes_{A_0} A_i,", "K_0^\\bullet \\otimes_{A_0} A_i)", "\\longrightarrow", "\\Hom_{D(A)}(E_0^\\bullet \\otimes_{A_0} A, K_0^\\bullet \\otimes_{A_0} A)", "$$", "Because $E_0^\\bullet$ is a bounded above complex of projective modules", "we can rewrite this as", "$$", "\\colim_{i \\geq 0} \\Hom_{K(A_0)}(E_0^\\bullet,", "K_0^\\bullet \\otimes_{A_0} A_i)", "\\longrightarrow", "\\Hom_{K(A_0)}(E_0^\\bullet, K_0^\\bullet \\otimes_{A_0} A)", "$$", "Since there are only a finite number of nonzero modules", "$E_0^n$ and since these are all finitely presented modules, this", "map is an isomorphism.", "\\medskip\\noindent", "In case (2) the object $E_0$ corresponds to a", "bounded above complex $E_0^\\bullet$ of finite free $A_0$-modules,", "see Lemma \\ref{lemma-pseudo-coherent-affine}.", "We may represent $K_0$ by a finite complex $K_0^\\bullet$", "of flat $A_0$-modules, see Lemma \\ref{lemma-tor-dimension-affine}", "and", "More on Algebra, Lemma \\ref{more-algebra-lemma-tor-amplitude}.", "In particular $K_0^\\bullet$ is K-flat and we can argue as before", "to arrive at the map", "$$", "\\colim_{i \\geq 0} \\Hom_{K(A_0)}(E_0^\\bullet,", "K_0^\\bullet \\otimes_{A_0} A_i)", "\\longrightarrow", "\\Hom_{K(A_0)}(E_0^\\bullet, K_0^\\bullet \\otimes_{A_0} A)", "$$", "It is clear that this map is an isomorphism (only a finite number of", "terms are involved since $K_0^\\bullet$ is bounded)." ], "refs": [ "coherent-lemma-induction-principle", "cohomology-lemma-mayer-vietoris-hom", "perfect-lemma-affine-compare-bounded", "perfect-lemma-perfect-affine", "perfect-lemma-pseudo-coherent-affine", "perfect-lemma-tor-dimension-affine", "more-algebra-lemma-tor-amplitude" ], "ref_ids": [ 3291, 2145, 6941, 6980, 6975, 6977, 10170 ] } ], "ref_ids": [] }, { "id": 7051, "type": "theorem", "label": "perfect-lemma-descend-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-descend-perfect", "contents": [ "In Situation \\ref{situation-descent} the category of perfect", "objects of $D(\\mathcal{O}_S)$ is the colimit of the categories", "of perfect objects of $D(\\mathcal{O}_{S_i})$." ], "refs": [], "proofs": [ { "contents": [ "For every open $U_0 \\subset S_0$ consider the condition $P$ that", "the functor", "$$", "\\colim_{i \\geq 0} D_{perf}(\\mathcal{O}_{U_i})", "\\longrightarrow", "D_{perf}(\\mathcal{O}_U)", "$$", "is an equivalence where ${}_{perf}$ indicates the full subcategory of", "perfect objects and where $U = f_0^{-1}(U_0)$ and $U_i = f_{i0}^{-1}(U_0)$.", "We will prove $P$ holds for all quasi-compact opens $U_0$", "by the induction principle of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}.", "First, we observe that we already know the functor is fully faithful", "by Lemma \\ref{lemma-descend-homomorphisms}. Thus it suffices to prove", "essential surjectivity.", "\\medskip\\noindent", "We first check condition (2) of the induction principle. Thus suppose", "that we have $S_0 = U_0 \\cup V_0$ and that $P$ holds for", "$U_0$, $V_0$, and $U_0 \\cap V_0$. Let $E$ be a perfect object", "of $D(\\mathcal{O}_S)$. We can find $i \\geq 0$ and $E_{U, i}$ perfect on $U_i$", "and $E_{V, i}$ perfect on $V_i$ whose pullback to $U$ and $V$ are isomorphic", "to $E|_U$ and $E|_V$. Denote", "$$", "a : E_{U, i} \\to (Rf_{i, *}E)|_{U_i}", "\\quad\\text{and}\\quad", "b : E_{V, i} \\to (Rf_{i, *}E)|_{V_i}", "$$", "the maps adjoint to the isomorphisms $Lf_i^*E_{U, i} \\to E|_U$", "and $Lf_i^*E_{V, i} \\to E|_V$.", "By fully faithfulness, after increasing $i$,", "we can find an isomorphism", "$c : E_{U, i}|_{U_i \\cap V_i} \\to E_{V, i}|_{U_i \\cap V_i}$", "which pulls back to the identifications ", "$$", "Lf_i^*E_{U, i}|_{U \\cap V} \\to E|_{U \\cap V} \\to Lf_i^*E_{V, i}|_{U \\cap V}.", "$$", "Apply Cohomology, Lemma \\ref{cohomology-lemma-glue}", "to get an object $E_i$ on $S_i$ and a map $d : E_i \\to Rf_{i, *}E$", "which restricts to the maps $a$ and $b$ over $U_i$ and $V_i$.", "Then it is clear that $E_i$ is perfect and that", "$d$ is adjoint to an isomorphism $Lf_i^*E_i \\to E$.", "\\medskip\\noindent", "Finally, we check condition (1) of the induction principle, in other", "words, we check the lemma holds when $S_0$ is affine.", "Say $S_0 = \\Spec(A_0)$, $S_i = \\Spec(A_i)$, and", "$S = \\Spec(A)$. Using Lemmas \\ref{lemma-affine-compare-bounded}", "and \\ref{lemma-perfect-affine} we see that we have to show that", "$$", "D_{perf}(A) = \\colim D_{perf}(A_i)", "$$", "This is clear from the fact that perfect complexes over rings are", "given by finite complexes of finite projective (hence finitely presented)", "modules. See More on Algebra, Lemma", "\\ref{more-algebra-lemma-colimit-perfect-complexes} for details." ], "refs": [ "coherent-lemma-induction-principle", "perfect-lemma-descend-homomorphisms", "cohomology-lemma-glue", "perfect-lemma-affine-compare-bounded", "perfect-lemma-perfect-affine", "more-algebra-lemma-colimit-perfect-complexes" ], "ref_ids": [ 3291, 7050, 2191, 6941, 6980, 10226 ] } ], "ref_ids": [] }, { "id": 7052, "type": "theorem", "label": "perfect-lemma-base-change-tensor-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-base-change-tensor-perfect", "contents": [ "Let $f : X \\to S$ be a morphism of finite presentation.", "Let $E \\in D(\\mathcal{O}_X)$ be a perfect object. Let $\\mathcal{G}^\\bullet$", "be a bounded complex of finitely presented $\\mathcal{O}_X$-modules,", "flat over $S$, with support proper over $S$. Then", "$$", "K = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)", "$$", "is a perfect object of $D(\\mathcal{O}_S)$ and its formation", "commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "The statement on base change is Lemma \\ref{lemma-base-change-tensor}.", "Thus it suffices to show that $K$ is a perfect object. If $S$ is", "Noetherian, then this follows from", "Lemma \\ref{lemma-tensor-perfect}.", "We will reduce to this case by Noetherian approximation.", "We encourage the reader to skip the rest of this proof.", "\\medskip\\noindent", "The question is local on $S$, hence we may assume $S$ is affine.", "Say $S = \\Spec(R)$. We write $R = \\colim R_i$ as a filtered colimit", "of Noetherian rings $R_i$. By Limits, Lemma", "\\ref{limits-lemma-descend-finite-presentation}", "there exists an $i$ and a scheme $X_i$ of finite presentation over $R_i$", "whose base change to $R$ is $X$. By", "Limits, Lemma \\ref{limits-lemma-descend-modules-finite-presentation}", "we may assume after increasing $i$, that there exists a bounded", "complex of finitely presented $\\mathcal{O}_{X_i}$-modules", "$\\mathcal{G}_i^\\bullet$ whose pullback to $X$ is $\\mathcal{G}^\\bullet$.", "After increasing $i$ we may assume $\\mathcal{G}_i^n$ is flat over $R_i$, see", "Limits, Lemma \\ref{limits-lemma-descend-module-flat-finite-presentation}.", "After increasing $i$ we may assume the support of $\\mathcal{G}_i^n$", "is proper over $R_i$, see", "Limits, Lemma \\ref{limits-lemma-eventually-proper-support}", "and Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-module-support-proper-over-base}.", "Finally, by Lemma \\ref{lemma-descend-perfect}", "we may, after increasing $i$, assume there exists a perfect", "object $E_i$ of $D(\\mathcal{O}_{X_i})$ whose pullback to", "$X$ is $E$. Applying Lemma \\ref{lemma-tensor-perfect}", "to $X_i \\to \\Spec(R_i)$, $E_i$, $\\mathcal{G}_i^\\bullet$ and using the", "base change property already shown we obtain the result." ], "refs": [ "perfect-lemma-base-change-tensor", "perfect-lemma-tensor-perfect", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-modules-finite-presentation", "limits-lemma-descend-module-flat-finite-presentation", "limits-lemma-eventually-proper-support", "coherent-lemma-module-support-proper-over-base", "perfect-lemma-descend-perfect", "perfect-lemma-tensor-perfect" ], "ref_ids": [ 7041, 7044, 15077, 15078, 15080, 15093, 3391, 7051, 7044 ] } ], "ref_ids": [] }, { "id": 7053, "type": "theorem", "label": "perfect-lemma-base-change-tensor-pseudo-coherent", "categories": [ "perfect" ], "title": "perfect-lemma-base-change-tensor-pseudo-coherent", "contents": [ "Let $f : X \\to S$ be a morphism of finite presentation.", "Let $E \\in D(\\mathcal{O}_X)$ be a pseudo-coherent object.", "Let $\\mathcal{G}^\\bullet$ be a bounded above complex of", "finitely presented $\\mathcal{O}_X$-modules, flat over $S$,", "with support proper over $S$. Then", "$$", "K = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)", "$$", "is a pseudo-coherent object of $D(\\mathcal{O}_S)$ and its formation", "commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "The statement on base change is Lemma \\ref{lemma-base-change-tensor}.", "Thus it suffices to show that $K$ is a pseudo-coherent object.", "This will follow from Lemma \\ref{lemma-base-change-tensor-perfect}", "by approximation by perfect complexes. We encourage the reader to", "skip the rest of the proof.", "\\medskip\\noindent", "The question is local on $S$, hence we may assume $S$ is affine.", "Then $X$ is quasi-compact and quasi-separated. Moreover, there", "exists an integer $N$ such that total direct image", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_S)$", "has cohomological dimension $N$ as explained in", "Lemma \\ref{lemma-quasi-coherence-direct-image}.", "Choose an integer $b$ such that $\\mathcal{G}^i = 0$ for $i > b$.", "It suffices to show that $K$ is $m$-pseudo-coherent for", "every $m$. Choose an approximation $P \\to E$ by a perfect complex $P$", "of $(X, E, m - N - 1 - b)$. This is possible by", "Theorem \\ref{theorem-approximation}.", "Choose a distinguished triangle", "$$", "P \\to E \\to C \\to P[1]", "$$", "in $D_\\QCoh(\\mathcal{O}_X)$. The cohomology sheaves of $C$ are zero", "in degrees $\\geq m - N - 1 - b$. Hence the cohomology sheaves of", "$C \\otimes^\\mathbf{L} \\mathcal{G}^\\bullet$ are zero in degrees", "$\\geq m - N - 1$. Thus the cohomology sheaves of", "$Rf_*(C \\otimes^\\mathbf{L} \\mathcal{G}^\\bullet)$", "are zero in degrees $\\geq m - 1$.", "Hence", "$$", "Rf_*(P \\otimes^\\mathbf{L} \\mathcal{G}^\\bullet) \\to", "Rf_*(E \\otimes^\\mathbf{L} \\mathcal{G}^\\bullet)", "$$", "is an isomorphism on cohomology sheaves in degrees $\\geq m$.", "Next, suppose that $H^i(P) = 0$ for $i > a$. Then", "$", "P \\otimes^\\mathbf{L} \\sigma_{\\geq m - N - 1 - a}\\mathcal{G}^\\bullet", "\\longrightarrow", "P \\otimes^\\mathbf{L} \\mathcal{G}^\\bullet", "$", "is an isomorphism on cohomology sheaves in degrees $\\geq m - N - 1$.", "Thus again we find that", "$$", "Rf_*(P \\otimes^\\mathbf{L} \\sigma_{\\geq m - N - 1 - a}\\mathcal{G}^\\bullet) \\to", "Rf_*(P \\otimes^\\mathbf{L} \\mathcal{G}^\\bullet)", "$$", "is an isomorphism on cohomology sheaves in degrees $\\geq m$.", "By Lemma \\ref{lemma-base-change-tensor-perfect} the source", "is a perfect complex.", "We conclude that $K$ is $m$-pseudo-coherent as desired." ], "refs": [ "perfect-lemma-base-change-tensor", "perfect-lemma-base-change-tensor-perfect", "perfect-lemma-quasi-coherence-direct-image", "perfect-theorem-approximation", "perfect-lemma-base-change-tensor-perfect" ], "ref_ids": [ 7041, 7052, 6946, 6934, 7052 ] } ], "ref_ids": [] }, { "id": 7054, "type": "theorem", "label": "perfect-lemma-flat-proper-perfect-direct-image-general", "categories": [ "perfect" ], "title": "perfect-lemma-flat-proper-perfect-direct-image-general", "contents": [ "Let $S$ be a scheme. Let $f : X \\to S$ be a proper", "morphism of finite presentation.", "\\begin{enumerate}", "\\item Let $E \\in D(\\mathcal{O}_X)$ be perfect and $f$ flat. Then", "$Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$ and its formation", "commutes with arbitrary base change.", "\\item Let $\\mathcal{G}$ be an $\\mathcal{O}_X$-module of finite presentation,", "flat over $S$. Then $Rf_*\\mathcal{G}$ is a perfect object of", "$D(\\mathcal{O}_S)$ and its formation commutes with arbitrary base change.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Special cases of", "Lemma \\ref{lemma-base-change-tensor-perfect} applied with", "(1) $\\mathcal{G}^\\bullet$ equal to $\\mathcal{O}_X$ in degree $0$", "and (2) $E = \\mathcal{O}_X$ and $\\mathcal{G}^\\bullet$ consisting", "of $\\mathcal{G}$ sitting in degree $0$." ], "refs": [ "perfect-lemma-base-change-tensor-perfect" ], "ref_ids": [ 7052 ] } ], "ref_ids": [] }, { "id": 7055, "type": "theorem", "label": "perfect-lemma-flat-proper-pseudo-coherent-direct-image-general", "categories": [ "perfect" ], "title": "perfect-lemma-flat-proper-pseudo-coherent-direct-image-general", "contents": [ "Let $S$ be a scheme. Let $f : X \\to S$ be a flat proper", "morphism of finite presentation. Let $E \\in D(\\mathcal{O}_X)$", "be pseudo-coherent. Then $Rf_*E$ is a pseudo-coherent object of", "$D(\\mathcal{O}_S)$ and its formation commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "Special case of", "Lemma \\ref{lemma-base-change-tensor-pseudo-coherent} applied with", "$\\mathcal{G}^\\bullet$ equal to $\\mathcal{O}_X$ in degree $0$." ], "refs": [ "perfect-lemma-base-change-tensor-pseudo-coherent" ], "ref_ids": [ 7053 ] } ], "ref_ids": [] }, { "id": 7056, "type": "theorem", "label": "perfect-lemma-pullback-and-limits", "categories": [ "perfect" ], "title": "perfect-lemma-pullback-and-limits", "contents": [ "Let $R$ be a ring. Let $X$ be a scheme and let", "$f : X \\to \\Spec(R)$ be proper, flat, and", "of finite presentation. Let $(M_n)$ be an inverse", "system of $R$-modules with surjective transition maps.", "Then the canonical map", "$$", "\\mathcal{O}_X \\otimes_R (\\lim M_n)", "\\longrightarrow", "\\lim \\mathcal{O}_X \\otimes_R M_n", "$$", "induces an isomorphism from the source to $DQ_X$ applied to the target." ], "refs": [], "proofs": [ { "contents": [ "The statement means that for any object $E$ of", "$D_\\QCoh(\\mathcal{O}_X)$ the induced map", "$$", "\\Hom(E, \\mathcal{O}_X \\otimes_R (\\lim M_n))", "\\longrightarrow", "\\Hom(E, \\lim \\mathcal{O}_X \\otimes_R M_n)", "$$", "is an isomorphism. Since $D_\\QCoh(\\mathcal{O}_X)$ has", "a perfect generator (Theorem \\ref{theorem-bondal-van-den-Bergh})", "it suffices to check this for perfect $E$.", "By Lemma \\ref{lemma-Rlim-quasi-coherent} we have", "$\\lim \\mathcal{O}_X \\otimes_R M_n = R\\lim \\mathcal{O}_X \\otimes_R M_n$.", "The exact functor", "$R\\Hom_X(E, -) : D_\\QCoh(\\mathcal{O}_X) \\to D(R)$", "of Cohomology, Section \\ref{cohomology-section-global-RHom}", "commutes with products and hence with derived limits, whence", "$$", "R\\Hom_X(E, \\lim \\mathcal{O}_X \\otimes_R M_n) =", "R\\lim R\\Hom_X(E, \\mathcal{O}_X \\otimes_R M_n)", "$$", "Let $E^\\vee$ be the dual perfect complex, see", "Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "We have", "$$", "R\\Hom_X(E, \\mathcal{O}_X \\otimes_R M_n) =", "R\\Gamma(X, E^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*M_n) =", "R\\Gamma(X, E^\\vee) \\otimes_R^\\mathbf{L} M_n", "$$", "by Lemma \\ref{lemma-cohomology-base-change}.", "From Lemma \\ref{lemma-flat-proper-perfect-direct-image-general}", "we see $R\\Gamma(X, E^\\vee)$ is a perfect complex of $R$-modules.", "In particular it is a pseudo-coherent complex and by", "More on Algebra, Lemma \\ref{more-algebra-lemma-pseudo-coherent-tensor-limit}", "we obtain", "$$", "R\\lim R\\Gamma(X, E^\\vee) \\otimes_R^\\mathbf{L} M_n =", "R\\Gamma(X, E^\\vee) \\otimes_R^\\mathbf{L} \\lim M_n", "$$", "as desired." ], "refs": [ "perfect-theorem-bondal-van-den-Bergh", "perfect-lemma-Rlim-quasi-coherent", "cohomology-lemma-dual-perfect-complex", "perfect-lemma-cohomology-base-change", "perfect-lemma-flat-proper-perfect-direct-image-general", "more-algebra-lemma-pseudo-coherent-tensor-limit" ], "ref_ids": [ 6935, 6938, 2233, 7025, 7054, 10436 ] } ], "ref_ids": [] }, { "id": 7057, "type": "theorem", "label": "perfect-lemma-base-change-RHom-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-base-change-RHom-perfect", "contents": [ "Let $f : X \\to S$ be a morphism of finite presentation.", "Let $E \\in D(\\mathcal{O}_X)$ be a perfect object. Let $\\mathcal{G}^\\bullet$", "be a bounded complex of finitely presented $\\mathcal{O}_X$-modules,", "flat over $S$, with support proper over $S$. Then", "$$", "K = Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet)", "$$", "is a perfect object of $D(\\mathcal{O}_S)$ and its formation", "commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "The statement on base change is Lemma \\ref{lemma-base-change-RHom}.", "Thus it suffices to show that $K$ is a perfect object. If $S$ is", "Noetherian, then this follows from", "Lemma \\ref{lemma-ext-perfect}.", "We will reduce to this case by Noetherian approximation.", "We encourage the reader to skip the rest of this proof.", "\\medskip\\noindent", "The question is local on $S$, hence we may assume $S$ is affine.", "Say $S = \\Spec(R)$. We write $R = \\colim R_i$ as a filtered colimit", "of Noetherian rings $R_i$. By Limits, Lemma", "\\ref{limits-lemma-descend-finite-presentation}", "there exists an $i$ and a scheme $X_i$ of finite presentation over $R_i$", "whose base change to $R$ is $X$. By", "Limits, Lemma \\ref{limits-lemma-descend-modules-finite-presentation}", "we may assume after increasing $i$, that there exists a bounded complex", "of finitely presented $\\mathcal{O}_{X_i}$-modules $\\mathcal{G}_i^\\bullet$", "whose pullback to $X$ is $\\mathcal{G}^\\bullet$. After increasing $i$", "we may assume $\\mathcal{G}_i^n$ is flat over $R_i$, see", "Limits, Lemma \\ref{limits-lemma-descend-module-flat-finite-presentation}.", "After increasing $i$ we may assume the support of $\\mathcal{G}_i^n$", "is proper over $R_i$, see", "Limits, Lemma \\ref{limits-lemma-eventually-proper-support} and", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-module-support-proper-over-base}.", "Finally, by Lemma \\ref{lemma-descend-perfect}", "we may, after increasing $i$, assume there exists a perfect", "object $E_i$ of $D(\\mathcal{O}_{X_i})$ whose pullback to", "$X$ is $E$. Applying Lemma \\ref{lemma-ext-perfect}", "to $X_i \\to \\Spec(R_i)$, $E_i$, $\\mathcal{G}_i^\\bullet$ and using the", "base change property already shown we obtain the result." ], "refs": [ "perfect-lemma-base-change-RHom", "perfect-lemma-ext-perfect", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-modules-finite-presentation", "limits-lemma-descend-module-flat-finite-presentation", "limits-lemma-eventually-proper-support", "coherent-lemma-module-support-proper-over-base", "perfect-lemma-descend-perfect", "perfect-lemma-ext-perfect" ], "ref_ids": [ 7042, 7045, 15077, 15078, 15080, 15093, 3391, 7051, 7045 ] } ], "ref_ids": [] }, { "id": 7058, "type": "theorem", "label": "perfect-lemma-jump-loci", "categories": [ "perfect" ], "title": "perfect-lemma-jump-loci", "contents": [ "Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be pseudo-coherent", "(for example perfect). For any $i \\in \\mathbf{Z}$ consider the function", "$$", "\\beta_i : X \\longrightarrow \\{0, 1, 2, \\ldots\\},\\quad", "x \\longmapsto", "\\dim_{\\kappa(x)}", "H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\kappa(x))", "$$", "Then we have", "\\begin{enumerate}", "\\item formation of $\\beta_i$ commutes with arbitrary base change,", "\\item the functions $\\beta_i$ are upper semi-continuous, and", "\\item the level sets of $\\beta_i$ are locally constructible in $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider a morphism of schemes $f : Y \\to X$ and a point $y \\in Y$.", "Let $x$ be the image of $y$ and consider the commutative diagram", "$$", "\\xymatrix{", "y \\ar[r]_j \\ar[d]_g & Y \\ar[d]^f \\\\", "x \\ar[r]^i & X", "}", "$$", "Then we see that $Lg^* \\circ Li^* = Lj^* \\circ Lf^*$. This implies that", "the function $\\beta'_i$ associated to the pseudo-coherent complex $Lf^*E$", "is the pullback of the function $\\beta_i$, in a formula:", "$\\beta'_i = \\beta_i \\circ f$. This is the meaning of (1).", "\\medskip\\noindent", "Fix $i$ and let $x \\in X$. It is enough to prove (2) and (3)", "holds in an open neighbourhood of $x$, hence we may assume $X$ affine.", "Then we can represent $E$ by a bounded above complex $\\mathcal{F}^\\bullet$", "of finite free modules (Lemma \\ref{lemma-lift-pseudo-coherent}).", "Then $P = \\sigma_{\\geq i - 1}\\mathcal{F}^\\bullet$ is a perfect object", "and $P \\to E$ induces an isomorphism", "$$", "H^i(P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\kappa(x')) \\to", "H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\kappa(x'))", "$$", "for all $x' \\in X$. Thus we may assume $E$ is perfect. In this case", "by More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-perfect-from-residue-field}", "there exists an affine open neighbourhood $U$ of $x$ and", "$a \\leq b$ such that $E|_U$ is represented by a complex", "$$", "\\ldots \\to 0 \\to \\mathcal{O}_U^{\\oplus \\beta_a(x)}", "\\to \\mathcal{O}_U^{\\oplus \\beta_{a + 1}(x)} \\to", "\\ldots \\to", "\\mathcal{O}_U^{\\oplus \\beta_{b - 1}(x)} \\to", "\\mathcal{O}_U^{\\oplus \\beta_b(x)} \\to 0", "\\to \\ldots", "$$", "(This also uses earlier results to turn the problem into algebra, for example", "Lemmas \\ref{lemma-affine-compare-bounded} and", "\\ref{lemma-perfect-affine}.)", "It follows immediately that $\\beta_i(x') \\leq \\beta_i(x)$", "for all $x' \\in U$. This proves that $\\beta_i$ is upper", "semi-continuous.", "\\medskip\\noindent", "To prove (3) we may assume that $X$ is affine and", "$E$ is given by a complex of finite", "free $\\mathcal{O}_X$-modules (for example by arguing as in the previous", "paragraph, or by using Cohomology, Lemma", "\\ref{cohomology-lemma-perfect-on-locally-ringed}).", "Thus we have to show that given a complex", "$$", "\\mathcal{O}_X^{\\oplus a} \\to", "\\mathcal{O}_X^{\\oplus b} \\to", "\\mathcal{O}_X^{\\oplus c}", "$$", "the function associated to a point $x \\in X$ the dimension of the cohomology", "of $\\kappa_x^{\\oplus a} \\to \\kappa_x^{\\oplus b} \\to \\kappa_x^{\\oplus c}$", "in the middle has constructible level sets. Let", "$A \\in \\text{Mat}(a \\times b, \\Gamma(X, \\mathcal{O}_X))$ be the matrix", "of the first arrow. The rank of the image of $A$ in", "$\\text{Mat}(a \\times b, \\kappa(x))$ is equal to $r$ if all", "$(r + 1) \\times (r + 1)$-minors of $A$ vanish at $x$ and there is some", "$r \\times r$-minor of $A$ which does not vanish at $x$. Thus the set", "of points where the rank is $r$ is a constructible locally closed set.", "Arguing similarly for the second arrow and putting everything together", "we obtain the desired result." ], "refs": [ "perfect-lemma-lift-pseudo-coherent", "more-algebra-lemma-lift-perfect-from-residue-field", "perfect-lemma-affine-compare-bounded", "perfect-lemma-perfect-affine", "cohomology-lemma-perfect-on-locally-ringed" ], "ref_ids": [ 6998, 10232, 6941, 6980, 2222 ] } ], "ref_ids": [] }, { "id": 7059, "type": "theorem", "label": "perfect-lemma-chi-locally-constant", "categories": [ "perfect" ], "title": "perfect-lemma-chi-locally-constant", "contents": [ "Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect.", "The function", "$$", "\\chi_E : X \\longrightarrow \\mathbf{Z},\\quad", "x \\longmapsto \\sum (-1)^i", "\\dim_{\\kappa(x)} H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\kappa(x))", "$$", "is locally constant on $X$." ], "refs": [], "proofs": [ { "contents": [ "By Cohomology, Lemma", "\\ref{cohomology-lemma-perfect-on-locally-ringed}", "we see that we can, locally on $X$, represent $E$ by a finite", "complex $\\mathcal{E}^\\bullet$ of finite free $\\mathcal{O}_X$-modules.", "On such an open the function $\\chi_E$ is constant with value", "$\\sum (-1)^i \\text{rank}(\\mathcal{E}^i)$." ], "refs": [ "cohomology-lemma-perfect-on-locally-ringed" ], "ref_ids": [ 2222 ] } ], "ref_ids": [] }, { "id": 7060, "type": "theorem", "label": "perfect-lemma-open-where-cohomology-in-degree-i-rank-r", "categories": [ "perfect" ], "title": "perfect-lemma-open-where-cohomology-in-degree-i-rank-r", "contents": [ "Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect.", "Given $i, r \\in \\mathbf{Z}$, there exists an", "open subscheme $U \\subset X$ characterized by the following", "\\begin{enumerate}", "\\item $E|_U \\cong H^i(E|_U)[-i]$ and $H^i(E|_U)$ is a locally free", "$\\mathcal{O}_U$-module of rank $r$,", "\\item a morphism $f : Y \\to X$ factors through $U$ if and only if", "$Lf^*E$ is isomorphic to a locally free module of rank $r$", "placed in degree $i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\beta_j : X \\to \\{0, 1, 2, \\ldots\\}$ for $j \\in \\mathbf{Z}$", "be the functions of Lemma \\ref{lemma-jump-loci}. Then the set", "$$", "W = \\{x \\in X \\mid \\beta_j(x) \\leq 0\\text{ for all }j \\not = i\\}", "$$", "is open in $X$ and its formation commutes with pullback to any", "$Y$ over $X$. This follows from the lemma using that", "apriori in a neighbourhood of any point only a finite number", "of the $\\beta_j$ are nonzero. Thus we may replace $X$ by $W$", "and assume that $\\beta_j(x) = 0$ for all $x \\in X$ and all $j \\not = i$.", "In this case $H^i(E)$ is a finite locally free module and", "$E \\cong H^i(E)[-i]$, see for example ", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-perfect-from-residue-field}.", "Thus $X$ is the disjoint union of the open subschemes where the", "rank of $H^i(E)$ is fixed and we win." ], "refs": [ "perfect-lemma-jump-loci", "more-algebra-lemma-lift-perfect-from-residue-field" ], "ref_ids": [ 7058, 10232 ] } ], "ref_ids": [] }, { "id": 7061, "type": "theorem", "label": "perfect-lemma-locally-closed-where-H0-locally-free", "categories": [ "perfect" ], "title": "perfect-lemma-locally-closed-where-H0-locally-free", "contents": [ "Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect", "of tor-amplitude in $[a, b]$ for some $a, b \\in \\mathbf{Z}$.", "Let $r \\geq 0$.", "Then there exists a locally closed subscheme $j : Z \\to X$", "characterized by the following", "\\begin{enumerate}", "\\item $H^a(Lj^*E)$ is a locally free $\\mathcal{O}_Z$-module of rank $r$, and", "\\item a morphism $f : Y \\to X$ factors through $Z$", "if and only if for all morphisms $g : Y' \\to Y$ the", "$\\mathcal{O}_{Y'}$-module $H^a(L(f \\circ g)^*E)$ is locally free", "of rank $r$.", "\\end{enumerate}", "Moreover, $j : Z \\to X$ is of finite presentation and we have", "\\begin{enumerate}", "\\item[(3)] if $f : Y \\to X$ factors as $Y \\xrightarrow{g} Z \\to X$, then", "$H^a(Lf^*E) = g^*H^a(Lj^*E)$,", "\\item[(4)] if $\\beta_a(x) \\leq r$ for all $x \\in X$, then", "$j$ is a closed immersion and given $f : Y \\to X$ the following", "are equivalent", "\\begin{enumerate}", "\\item $f : Y \\to X$ factors through $Z$,", "\\item $H^0(Lf^*E)$ is a locally free $\\mathcal{O}_Y$-module of rank $r$,", "\\end{enumerate}", "and if $r = 1$ these are also equivalent to", "\\begin{enumerate}", "\\item[(c)] $\\mathcal{O}_Y \\to \\SheafHom_{\\mathcal{O}_Y}(H^0(Lf^*E), H^0(Lf^*E))$", "is injective.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First, let $U \\subset X$ be the locally constructible", "open subscheme where the function", "$\\beta_a$ of Lemma \\ref{lemma-jump-loci} has values $\\leq r$.", "Let $f : Y \\to X$ be as in (2). Then for any $y \\in Y$", "we have $\\beta_a(Lf^*E) = r$ hence $y$ maps into $U$ by", "Lemma \\ref{lemma-jump-loci}. Hence $f$ as in (2) factors through $U$.", "Thus we may replace $X$ by $U$ and assume that", "$\\beta_a(x) \\in \\{0, 1, \\ldots, r\\}$ for all $x \\in X$.", "We will show that in this case", "there is a closed subscheme $Z \\subset X$ cut out by a finite type", "quasi-coherent ideal characterized", "by the equivalence of (4) (a), (b) and (4)(c) if $r = 1$", "and that (3) holds.", "This will finish the proof because it will a fortiori show", "that morphisms as in (2) factor through $Z$.", "\\medskip\\noindent", "If $x \\in X$ and $\\beta_a(x) < r$, then there is an", "open neighbourhood of $x$ where $\\beta_a < r$", "(Lemma \\ref{lemma-jump-loci}). In this way we", "see that set theoretically at least $Z$ is a closed subset.", "\\medskip\\noindent", "To get a scheme theoretic structure, consider a point $x \\in X$", "with $\\beta_a(x) = r$. Set $\\beta = \\beta_{a + 1}(x)$.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-perfect-from-residue-field}", "there exists an affine open neighbourhood $U$ of $x$", "such that $K|_U$ is represented by a complex", "$$", "\\ldots \\to 0 \\to \\mathcal{O}_U^{\\oplus r}", "\\xrightarrow{(f_{ij})} \\mathcal{O}_U^{\\oplus \\beta} \\to", "\\ldots \\to", "\\mathcal{O}_U^{\\oplus \\beta_{b - 1}(x)} \\to", "\\mathcal{O}_U^{\\oplus \\beta_b(x)} \\to 0", "\\to \\ldots", "$$", "(This also uses earlier results to turn the problem into algebra, for example", "Lemmas \\ref{lemma-affine-compare-bounded} and", "\\ref{lemma-perfect-affine}.) Now, if $g : Y \\to U$ is any morphism", "of schemes such that $g^\\sharp(f_{ij})$ is nonzero for some pair $i, j$, then", "$H^0(Lg^*E)$ is not a locally free $\\mathcal{O}_Y$-module of rank $r$.", "See More on Algebra, Lemma \\ref{more-algebra-lemma-coker-injective-free}.", "Trivially $H^0(Lg^*E)$ is a locally free $\\mathcal{O}_Y$-module if", "$g^\\sharp(f_{ij}) = 0$ for all $i, j$. Thus we see that over $U$ the", "closed subscheme cut out by all $f_{ij}$ satisfies", "(3) and we have the equivalence of (4)(a) and (b).", "The characterization of $Z$ shows that the locally constructed patches", "glue (details omitted). Finally, if $r = 1$ then", "(4)(c) is equivalent to (4)(b) because in this case locally", "$H^0(Lg^*E) \\subset \\mathcal{O}_Y$ is the annihilator of the ideal generated", "by the elements $g^\\sharp(f_{ij})$." ], "refs": [ "perfect-lemma-jump-loci", "perfect-lemma-jump-loci", "perfect-lemma-jump-loci", "more-algebra-lemma-lift-perfect-from-residue-field", "perfect-lemma-affine-compare-bounded", "perfect-lemma-perfect-affine", "more-algebra-lemma-coker-injective-free" ], "ref_ids": [ 7058, 7058, 7058, 10232, 6941, 6980, 9892 ] } ], "ref_ids": [] }, { "id": 7062, "type": "theorem", "label": "perfect-lemma-jump-loci-geometric", "categories": [ "perfect" ], "title": "perfect-lemma-jump-loci-geometric", "contents": [ "Let $f : X \\to S$ be a flat, proper morphism of finite presentation.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation,", "flat over $S$. For fixed $i \\in \\mathbf{Z}$ consider the function", "$$", "\\beta_i : S \\to \\{0, 1, 2, \\ldots\\},\\quad", "s \\longmapsto \\dim_{\\kappa(s)} H^i(X_s, \\mathcal{F}_s)", "$$", "Then we have", "\\begin{enumerate}", "\\item formation of $\\beta_i$ commutes with arbitrary base change,", "\\item the functions $\\beta_i$ are upper semi-continuous, and", "\\item the level sets of $\\beta_i$ are locally constructible in $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By cohomology and base change (more precisely by", "Lemma \\ref{lemma-flat-proper-perfect-direct-image-general})", "the object $K = Rf_*\\mathcal{F}$ is a perfect object of the derived", "category of $S$ whose formation commutes with arbitrary base change.", "In particular we have", "$$", "H^i(X_s, \\mathcal{F}_s) = H^i(K \\otimes_{\\mathcal{O}_S}^\\mathbf{L} \\kappa(s))", "$$", "Thus the lemma follows from", "Lemma \\ref{lemma-jump-loci}." ], "refs": [ "perfect-lemma-flat-proper-perfect-direct-image-general", "perfect-lemma-jump-loci" ], "ref_ids": [ 7054, 7058 ] } ], "ref_ids": [] }, { "id": 7063, "type": "theorem", "label": "perfect-lemma-chi-locally-constant-geometric", "categories": [ "perfect" ], "title": "perfect-lemma-chi-locally-constant-geometric", "contents": [ "Let $f : X \\to S$ be a flat, proper morphism of finite presentation.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation,", "flat over $S$. The function", "$$", "s \\longmapsto \\chi(X_s, \\mathcal{F}_s)", "$$", "is locally constant on $S$. Formation of this function commutes with", "base change." ], "refs": [], "proofs": [ { "contents": [ "By cohomology and base change (more precisely by", "Lemma \\ref{lemma-flat-proper-perfect-direct-image-general})", "the object $K = Rf_*\\mathcal{F}$ is a perfect object of the derived", "category of $S$ whose formation commutes with arbitrary base change.", "Thus we have to show the map", "$$", "s \\longmapsto \\sum (-1)^i \\dim_{\\kappa(s)}", "H^i(K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\kappa(s))", "$$", "is locally constant on $S$. This is Lemma \\ref{lemma-chi-locally-constant}." ], "refs": [ "perfect-lemma-flat-proper-perfect-direct-image-general", "perfect-lemma-chi-locally-constant" ], "ref_ids": [ 7054, 7059 ] } ], "ref_ids": [] }, { "id": 7064, "type": "theorem", "label": "perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric", "categories": [ "perfect" ], "title": "perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric", "contents": [ "Let $f : X \\to S$ be a flat, proper morphism of finite presentation.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation,", "flat over $S$. Fix $i, r \\in \\mathbf{Z}$.", "Then there exists an open subscheme", "$U \\subset S$ with the following property:", "A morphism $T \\to S$ factors through $U$ if and only if", "$Rf_{T, *}\\mathcal{F}_T$ is isomorphic to a", "finite locally free module of rank $r$ placed in degree $i$." ], "refs": [], "proofs": [ { "contents": [ "By cohomology and base change (more precisely by", "Lemma \\ref{lemma-flat-proper-perfect-direct-image-general})", "the object $K = Rf_*\\mathcal{F}$ is a perfect object of the derived", "category of $S$ whose formation commutes with arbitrary base change.", "Thus this lemma follows immediately from", "Lemma \\ref{lemma-open-where-cohomology-in-degree-i-rank-r}." ], "refs": [ "perfect-lemma-flat-proper-perfect-direct-image-general", "perfect-lemma-open-where-cohomology-in-degree-i-rank-r" ], "ref_ids": [ 7054, 7060 ] } ], "ref_ids": [] }, { "id": 7065, "type": "theorem", "label": "perfect-lemma-vanishing-implies-locally-free", "categories": [ "perfect" ], "title": "perfect-lemma-vanishing-implies-locally-free", "contents": [ "Let $f : X \\to S$ be a morphism of finite presentation.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation,", "flat over $S$ with support proper over $S$. If $R^if_*\\mathcal{F} = 0$", "for $i > 0$, then $f_*\\mathcal{F}$ is locally free and its formation", "commutes with arbitrary base change (see proof for explanation)." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-base-change-tensor-perfect}", "the object $E = Rf_*\\mathcal{F}$ of $D(\\mathcal{O}_S)$", "is perfect and its formation commutes with arbitrary base change,", "in the sense that $Rf'_*(g')^*\\mathcal{F} = Lg^*E$", "for any cartesian diagram", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "S' \\ar[r]^g &", "S", "}", "$$", "of schemes.", "Since there is never any cohomology in degrees $< 0$, we see that", "$E$ (locally) has tor-amplitude in $[0, b]$ for some $b$.", "If $H^i(E) = R^if_*\\mathcal{F} = 0$ for $i > 0$,", "then $E$ has tor amplitude in $[0, 0]$. Whence", "$E = H^0(E)[0]$. We conclude $H^0(E) = f_*\\mathcal{F}$", "is finite locally free by", "More on Algebra, Lemma \\ref{more-algebra-lemma-perfect}", "(and the characterization of finite projective modules", "in Algebra, Lemma \\ref{algebra-lemma-finite-projective}).", "Commutation with base change means that", "$g^*f_*\\mathcal{F} = f'_*(g')^*\\mathcal{F}$ for", "a diagram as above and it follows from the already", "established commutation of base change for $E$." ], "refs": [ "perfect-lemma-base-change-tensor-perfect", "more-algebra-lemma-perfect", "algebra-lemma-finite-projective" ], "ref_ids": [ 7052, 10212, 795 ] } ], "ref_ids": [] }, { "id": 7066, "type": "theorem", "label": "perfect-lemma-proper-flat-h0", "categories": [ "perfect" ], "title": "perfect-lemma-proper-flat-h0", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Assume", "\\begin{enumerate}", "\\item $f$ is proper, flat, and of finite presentation, and", "\\item for all $s \\in S$ we have $\\kappa(s) = H^0(X_s, \\mathcal{O}_{X_s})$.", "\\end{enumerate}", "Then we have", "\\begin{enumerate}", "\\item[(a)] $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and", "this holds after any base change,", "\\item[(b)] locally on $S$ we have", "$$", "Rf_*\\mathcal{O}_X = \\mathcal{O}_S \\oplus P", "$$", "in $D(\\mathcal{O}_S)$", "where $P$ is perfect of tor amplitude in $[1, \\infty)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By cohomology and base change", "(Lemma \\ref{lemma-flat-proper-perfect-direct-image-general})", "the complex $E = Rf_*\\mathcal{O}_X$", "is perfect and its formation commutes with arbitrary base change.", "This first implies that $E$ has tor aplitude in $[0, \\infty)$.", "Second, it implies that for $s \\in S$ we have", "$H^0(E \\otimes^\\mathbf{L} \\kappa(s)) =", "H^0(X_s, \\mathcal{O}_{X_s}) = \\kappa(s)$.", "It follows that the map $\\mathcal{O}_S \\to Rf_*\\mathcal{O}_X = E$", "induces an isomorphism", "$\\mathcal{O}_S \\otimes \\kappa(s) \\to H^0(E \\otimes^\\mathbf{L} \\kappa(s))$.", "Hence $H^0(E) \\otimes \\kappa(s) \\to H^0(E \\otimes^\\mathbf{L} \\kappa(s))$", "is surjective and we may apply", "More on Algebra, Lemma \\ref{more-algebra-lemma-better-cut-complex-in-two}", "to see that, after replacing $S$ by an affine open neighbourhood of $s$,", "we have a decomposition $E = H^0(E) \\oplus \\tau_{\\geq 1}E$", "with $\\tau_{\\geq 1}E$ perfect of tor amplitude in $[1, \\infty)$.", "Since $E$ has tor amplitude in $[0, \\infty)$ we find that", "$H^0(E)$ is a flat $\\mathcal{O}_S$-module.", "It follows that $H^0(E)$ is a flat, perfect $\\mathcal{O}_S$-module,", "hence finite locally free, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-perfect}", "(and the fact that finite projective modules are finite locally free by", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}).", "It follows that the map $\\mathcal{O}_S \\to H^0(E)$ is", "an isomorphism as we can check this after tensoring with", "residue fields (Algebra, Lemma \\ref{algebra-lemma-cokernel-flat})." ], "refs": [ "perfect-lemma-flat-proper-perfect-direct-image-general", "more-algebra-lemma-better-cut-complex-in-two", "more-algebra-lemma-perfect", "algebra-lemma-finite-projective", "algebra-lemma-cokernel-flat" ], "ref_ids": [ 7054, 10236, 10212, 795, 804 ] } ], "ref_ids": [] }, { "id": 7067, "type": "theorem", "label": "perfect-lemma-proper-flat-geom-red-connected", "categories": [ "perfect" ], "title": "perfect-lemma-proper-flat-geom-red-connected", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Assume", "\\begin{enumerate}", "\\item $f$ is proper, flat, and of finite presentation, and", "\\item the geometric fibres of $f$ are reduced and connected.", "\\end{enumerate}", "Then $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this holds", "after any base change." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-proper-flat-h0}", "it suffices to show that $\\kappa(s) = H^0(X_s, \\mathcal{O}_{X_s})$", "for all $s \\in S$. This follows from", "Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}", "and the fact that $X_s$ is geometrically connected and geometrically reduced." ], "refs": [ "perfect-lemma-proper-flat-h0", "varieties-lemma-proper-geometrically-reduced-global-sections" ], "ref_ids": [ 7066, 10948 ] } ], "ref_ids": [] }, { "id": 7068, "type": "theorem", "label": "perfect-lemma-proper-idempotent-on-fibre", "categories": [ "perfect" ], "title": "perfect-lemma-proper-idempotent-on-fibre", "contents": [ "Let $f : X \\to S$ be a proper morphism of schemes. Let $s \\in S$", "and let $e \\in H^0(X_s, \\mathcal{O}_{X_s})$ be an idempotent.", "Then $e$ is in the image of the map", "$(f_*\\mathcal{O}_X)_s \\to H^0(X_s, \\mathcal{O}_{X_s})$." ], "refs": [], "proofs": [ { "contents": [ "Let $X_s = T_1 \\amalg T_2$ be the disjoint union decomposition", "with $T_1$ and $T_2$ nonempty and open and closed in $X_s$", "corresponding to $e$, i.e., such that $e$ is identitically $1$", "on $T_1$ and identically $0$ on $T_2$.", "\\medskip\\noindent", "Assume $S$ is Noetherian. We will use the theorem on formal functions", "in the form of Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-formal-functions-stalk}.", "It tells us that", "$$", "(f_*\\mathcal{O}_X)_s^\\wedge = \\lim_n H^0(X_n, \\mathcal{O}_{X_n})", "$$", "where $X_n$ is the $n$th infinitesimal neighbourhood of $X_s$.", "Since the underlying topological space of $X_n$ is equal to that", "of $X_s$ we obtain for all $n $ a disjoint union decomposition of schemes", "$X_n = T_{1, n} \\amalg T_{2, n}$ where the underlying topological space", "of $T_{i, n}$ is $T_i$ for $i = 1, 2$. This means", "$H^0(X_n, \\mathcal{O}_{X_n})$ contains a nontrivial idempotent $e_n$,", "namely the function which is identically $1$ on $T_{1, n}$ and", "identically $0$ on $T_{2, n}$. It is clear that $e_{n + 1}$", "restricts to $e_n$ on $X_n$. Hence $e_\\infty = \\lim e_n$", "is a nontrivial idempotent of the limit. Thus $e_\\infty$", "is an element of the completion of $(f_*\\mathcal{O}_X)_s$", "mapping to $e$ in $H^0(X_s, \\mathcal{O}_{X_s})$.", "Since the map $(f_*\\mathcal{O}_X)_s^\\wedge \\to H^0(X_s, \\mathcal{O}_{X_s})$", "factors through", "$(f_*\\mathcal{O}_X)^\\wedge_s / \\mathfrak m_s (f_*\\mathcal{O}_X)_s^\\wedge =", "(f_*\\mathcal{O}_X)_s / \\mathfrak m_s (f_*\\mathcal{O}_X)_s$", "(Algebra, Lemma \\ref{algebra-lemma-hathat-finitely-generated})", "we conclude that $e$ is in the image of the", "map $(f_*\\mathcal{O}_X)_s \\to H^0(X_s, \\mathcal{O}_{X_s})$", "as desired.", "\\medskip\\noindent", "General case: we reduce the general case to the Noetherian case by", "limit arguments. We urge the reader to skip the proof.", "We may replace $S$ by an affine open neighbourhood of $s$.", "Thus we may and do assume that $S$ is affine. By Limits, Lemma", "\\ref{limits-lemma-proper-limit-of-proper-finite-presentation-noetherian}", "we can write $(f : X \\to S) = \\lim (f_i : X_i \\to S_i)$ with $f_i$", "proper and $S_i$ Noetherian. Denote $s_i \\in S_i$ the image of $s$.", "Then $s = \\lim s_i$, see", "Limits, Lemma \\ref{limits-lemma-inverse-limit-irreducibles}.", "Then $X_s = X \\times_S s = \\lim X_i \\times_{S_i} s_i = \\lim X_{i, s_i}$", "because limits commute with limits", "(Categories, Lemma \\ref{categories-lemma-colimits-commute}).", "Hence $e$ is the image of some idempotent", "$e_i \\in H^0(X_{i, s_i}, \\mathcal{O}_{X_{i, s_i}})$", "by Limits, Lemma \\ref{limits-lemma-descend-section}.", "By the Noetherian case there is an element $\\tilde e_i$", "in the stalk $(f_{i, *}\\mathcal{O}_{X_i})_{s_i}$ mapping", "to $e_i$. Taking the pullback of $\\tilde e_i$ we get an", "element $\\tilde e$ of $(f_*\\mathcal{O}_X)_s$ mapping", "to $e$ and the proof is complete." ], "refs": [ "coherent-lemma-formal-functions-stalk", "algebra-lemma-hathat-finitely-generated", "limits-lemma-proper-limit-of-proper-finite-presentation-noetherian", "limits-lemma-inverse-limit-irreducibles", "categories-lemma-colimits-commute" ], "ref_ids": [ 3362, 859, 15091, 15035, 12212 ] } ], "ref_ids": [] }, { "id": 7069, "type": "theorem", "label": "perfect-lemma-proper-flat-geom-red", "categories": [ "perfect" ], "title": "perfect-lemma-proper-flat-geom-red", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $s \\in S$. Assume", "\\begin{enumerate}", "\\item $f$ is proper, flat, and of finite presentation, and", "\\item the fibre $X_s$ is geometrically reduced.", "\\end{enumerate}", "Then, after replacing $S$ by an open neighbourhood of $s$, there", "exists a direct sum decomposition", "$Rf_*\\mathcal{O}_X = f_*\\mathcal{O}_X \\oplus P$", "in $D(\\mathcal{O}_S)$ where $f_*\\mathcal{O}_X$ is a finite \\'etale", "$\\mathcal{O}_S$-algebra and", "$P$ is a perfect of tor amplitude in $[1, \\infty)$." ], "refs": [], "proofs": [ { "contents": [ "The proof of this lemma is similar to the proof of", "Lemma \\ref{lemma-proper-flat-h0}", "which we suggest the reader read first.", "By cohomology and base change", "(Lemma \\ref{lemma-flat-proper-perfect-direct-image-general})", "the complex $E = Rf_*\\mathcal{O}_X$", "is perfect and its formation commutes with arbitrary base change.", "This first implies that $E$ has tor aplitude in $[0, \\infty)$.", "\\medskip\\noindent", "We claim that after replacing $S$ by an open neighbourhood of", "$s$ we can find a direct sum decomposition", "$E = H^0(E) \\oplus \\tau_{\\geq 1}E$ in $D(\\mathcal{O}_S)$", "with $\\tau_{\\geq 1}E$ of tor amplitude in $[1, \\infty)$.", "Assume the claim is true for now and assume we've made the replacement", "so we have the direct sum decomposition.", "Since $E$ has tor amplitude in $[0, \\infty)$ we find that", "$H^0(E)$ is a flat $\\mathcal{O}_S$-module.", "Hence $H^0(E)$ is a flat, perfect $\\mathcal{O}_S$-module,", "hence finite locally free, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-perfect}", "(and the fact that finite projective modules are finite locally free by", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}).", "Of course $H^0(E) = f_*\\mathcal{O}_X$ is an $\\mathcal{O}_S$-algebra.", "By cohomology and base change we obtain", "$H^0(E) \\otimes \\kappa(s) = H^0(X_s, \\mathcal{O}_{X_s})$.", "By Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}", "and the assumption that $X_s$ is geometrically reduced, we", "see that $\\kappa(s) \\to H^0(E) \\otimes \\kappa(s)$", "is finite \\'etale. By Morphisms, Lemma", "\\ref{morphisms-lemma-set-points-where-fibres-etale}", "applied to the finite locally free morphism", "$\\underline{\\Spec}_S(H^0(E)) \\to S$,", "we conclude that after shrinking $S$ the $\\mathcal{O}_S$-algebra", "$H^0(E)$ is finite \\'etale.", "\\medskip\\noindent", "It remains to prove the claim. For this it suffices to prove that the map", "$$", "(f_*\\mathcal{O}_X)_s", "\\longrightarrow", "H^0(X_s, \\mathcal{O}_{X_s}) = H^0(E \\otimes^\\mathbf{L} \\kappa(s))", "$$", "is surjective, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-better-cut-complex-in-two}.", "Choose a flat local ring homomorphism $\\mathcal{O}_{S, s} \\to A$", "such that the residue field $k$ of $A$ is algebraically closed, see", "Algebra, Lemma \\ref{algebra-lemma-flat-local-given-residue-field}.", "By flat base change (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology})", "we get $H^0(X_A, \\mathcal{O}_{X_A}) =", "(f_*\\mathcal{O}_X)_s \\otimes_{\\mathcal{O}_{S, s}} A$", "and $H^0(X_k, \\mathcal{O}_{X_k}) =", "H^0(X_s, \\mathcal{O}_{X_s}) \\otimes_{\\kappa(s)} k$.", "Hence it suffices to prove that", "$H^0(X_A, \\mathcal{O}_{X_A}) \\to H^0(X_k, \\mathcal{O}_{X_k})$", "is surjective. Since $X_k$ is a reduced proper scheme over $k$", "and since $k$ is algebraically closed, we see that", "$H^0(X_k, \\mathcal{O}_{X_k})$ is a finite product of copies", "of $k$ by the already used Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}.", "Since by Lemma \\ref{lemma-proper-idempotent-on-fibre}", "the idempotents of this $k$-algebra are in the image", "of $H^0(X_A, \\mathcal{O}_{X_A}) \\to H^0(X_k, \\mathcal{O}_{X_k})$ we conclude." ], "refs": [ "perfect-lemma-proper-flat-h0", "perfect-lemma-flat-proper-perfect-direct-image-general", "more-algebra-lemma-perfect", "algebra-lemma-finite-projective", "varieties-lemma-proper-geometrically-reduced-global-sections", "morphisms-lemma-set-points-where-fibres-etale", "more-algebra-lemma-better-cut-complex-in-two", "algebra-lemma-flat-local-given-residue-field", "coherent-lemma-flat-base-change-cohomology", "varieties-lemma-proper-geometrically-reduced-global-sections", "perfect-lemma-proper-idempotent-on-fibre" ], "ref_ids": [ 7066, 7054, 10212, 795, 10948, 5374, 10236, 1324, 3298, 10948, 7068 ] } ], "ref_ids": [] }, { "id": 7070, "type": "theorem", "label": "perfect-lemma-cohomology-over-coherent-ring", "categories": [ "perfect" ], "title": "perfect-lemma-cohomology-over-coherent-ring", "contents": [ "Let $R$ be a coherent ring. Let $X$ be a scheme of finite presentation over $R$.", "Let $\\mathcal{G}$ be an $\\mathcal{O}_X$-module of finite presentation,", "flat over $R$, with support proper over $R$. Then", "$H^i(X, \\mathcal{G})$ is a coherent $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-base-change-tensor-perfect} with", "More on Algebra, Lemmas \\ref{more-algebra-lemma-coherent-pseudo-coherent} and", "\\ref{more-algebra-lemma-perfect}." ], "refs": [ "perfect-lemma-base-change-tensor-perfect", "more-algebra-lemma-coherent-pseudo-coherent", "more-algebra-lemma-perfect" ], "ref_ids": [ 7052, 10161, 10212 ] } ], "ref_ids": [] }, { "id": 7071, "type": "theorem", "label": "perfect-lemma-countable-cohomology", "categories": [ "perfect" ], "title": "perfect-lemma-countable-cohomology", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$", "such that the cohomology sheaves $H^i(K)$ have countable", "sets of sections over affine opens. Then for any quasi-compact open", "$U \\subset X$ and any perfect object $E$ in $D(\\mathcal{O}_X)$", "the sets", "$$", "H^i(U, K \\otimes^\\mathbf{L} E),\\quad \\Ext^i(E|_U, K|_U)", "$$", "are countable." ], "refs": [], "proofs": [ { "contents": [ "Using Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}", "we see that it suffices to prove the result", "for the groups $H^i(U, K \\otimes^\\mathbf{L} E)$.", "We will use the induction principle to prove the lemma, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}.", "\\medskip\\noindent", "First we show that it holds when $U = \\Spec(A)$ is affine. Namely, we can", "represent $K$ by a complex of $A$-modules $K^\\bullet$ and $E$ by a", "finite complex of finite projective $A$-modules $P^\\bullet$.", "See Lemmas \\ref{lemma-affine-compare-bounded} and", "\\ref{lemma-perfect-affine}", "and our definition of perfect complexes of $A$-modules", "(More on Algebra, Definition \\ref{more-algebra-definition-perfect}).", "Then $(E \\otimes^\\mathbf{L} K)|_U$ is represented by", "the total complex associated to the double complex", "$P^\\bullet \\otimes_A K^\\bullet$", "(Lemma \\ref{lemma-quasi-coherence-tensor-product}).", "Using induction on the length of the complex", "$P^\\bullet$ (or using a suitable spectral sequence)", "we see that it suffices to show that", "$H^i(P^a \\otimes_A K^\\bullet)$ is countable for each $a$.", "Since $P^a$ is a direct summand of $A^{\\oplus n}$ for", "some $n$ this follows from the assumption that", "the cohomology group $H^i(K^\\bullet)$ is countable.", "\\medskip\\noindent", "To finish the proof it suffices to show: if $U = V \\cup W$", "and the result holds for $V$, $W$, and $V \\cap W$, then", "the result holds for $U$. This is an immediate consquence", "of the Mayer-Vietoris sequence, see", "Cohomology, Lemma \\ref{cohomology-lemma-unbounded-mayer-vietoris}." ], "refs": [ "cohomology-lemma-dual-perfect-complex", "coherent-lemma-induction-principle", "perfect-lemma-affine-compare-bounded", "perfect-lemma-perfect-affine", "more-algebra-definition-perfect", "perfect-lemma-quasi-coherence-tensor-product", "cohomology-lemma-unbounded-mayer-vietoris" ], "ref_ids": [ 2233, 3291, 6941, 6980, 10628, 6945, 2146 ] } ], "ref_ids": [] }, { "id": 7072, "type": "theorem", "label": "perfect-lemma-countable", "categories": [ "perfect" ], "title": "perfect-lemma-countable", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme such that", "the sets of sections of $\\mathcal{O}_X$ over affine opens are countable.", "Let $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. The", "following are equivalent", "\\begin{enumerate}", "\\item $K = \\text{hocolim} E_n$ with $E_n$ a perfect object of", "$D(\\mathcal{O}_X)$, and", "\\item the cohomology sheaves $H^i(K)$ have countable", "sets of sections over affine opens.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (1) is true, then (2) is true because homotopy colimits commutes", "with taking cohomology sheaves", "(by Derived Categories, Lemma \\ref{derived-lemma-cohomology-of-hocolim})", "and because a perfect complex is", "locally isomorphic to a finite complex of finite free $\\mathcal{O}_X$-modules", "and therefore satisfies (2) by assumption on $X$.", "\\medskip\\noindent", "Assume (2).", "Choose a K-injective complex $\\mathcal{K}^\\bullet$ representing $K$.", "Choose a perfect generator $E$ of $D_\\QCoh(\\mathcal{O}_X)$ and", "represent it by a K-injective complex $\\mathcal{I}^\\bullet$.", "According to Theorem \\ref{theorem-DQCoh-is-Ddga}", "and its proof there is an equivalence", "of triangulated categories $F : D_\\QCoh(\\mathcal{O}_X) \\to D(A, \\text{d})$", "where $(A, \\text{d})$ is the differential graded algebra", "$$", "(A, \\text{d}) =", "\\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}", "(\\mathcal{I}^\\bullet, \\mathcal{I}^\\bullet)", "$$", "which maps $K$ to the differential graded module", "$$", "M = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}", "(\\mathcal{I}^\\bullet, \\mathcal{K}^\\bullet)", "$$", "Note that $H^i(A) = \\Ext^i(E, E)$ and", "$H^i(M) = \\Ext^i(E, K)$.", "Moreover, since $F$ is an equivalence it and its quasi-inverse commute", "with homotopy colimits.", "Therefore, it suffices to write $M$ as a homotopy colimit", "of compact objects of $D(A, \\text{d})$.", "By Differential Graded Algebra, Lemma \\ref{dga-lemma-countable}", "it suffices show that $\\Ext^i(E, E)$ and", "$\\Ext^i(E, K)$ are countable for each $i$.", "This follows from Lemma \\ref{lemma-countable-cohomology}." ], "refs": [ "derived-lemma-cohomology-of-hocolim", "perfect-theorem-DQCoh-is-Ddga", "dga-lemma-countable", "perfect-lemma-countable-cohomology" ], "ref_ids": [ 1923, 6936, 13129, 7071 ] } ], "ref_ids": [] }, { "id": 7073, "type": "theorem", "label": "perfect-lemma-computing-sections-as-colim", "categories": [ "perfect" ], "title": "perfect-lemma-computing-sections-as-colim", "contents": [ "Let $A$ be a ring. Let $X$ be a scheme of finite presentation over $A$.", "Let $f : U \\to X$ be a flat morphism of finite presentation. Then", "\\begin{enumerate}", "\\item there exists an inverse system of perfect objects $L_n$ of", "$D(\\mathcal{O}_X)$ such that", "$$", "R\\Gamma(U, Lf^*K) = \\text{hocolim}\\ R\\Hom_X(L_n, K)", "$$", "in $D(A)$ functorially in $K$ in $D_\\QCoh(\\mathcal{O}_X)$, and", "\\item there exists a system of perfect objects $E_n$ of", "$D(\\mathcal{O}_X)$ such that", "$$", "R\\Gamma(U, Lf^*K) = \\text{hocolim}\\ R\\Gamma(X, E_n \\otimes^\\mathbf{L} K)", "$$", "in $D(A)$ functorially in $K$ in $D_\\QCoh(\\mathcal{O}_X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-cohomology-base-change} we have", "$$", "R\\Gamma(U, Lf^*K) = R\\Gamma(X, Rf_*\\mathcal{O}_U \\otimes^\\mathbf{L} K)", "$$", "functorially in $K$. Observe that $R\\Gamma(X, -)$ commutes with", "homotopy colimits because it commutes with direct sums by", "Lemma \\ref{lemma-quasi-coherence-pushforward-direct-sums}.", "Similarly, $- \\otimes^\\mathbf{L} K$ commutes with derived colimits", "because $- \\otimes^\\mathbf{L} K$ commutes with direct sums", "(because direct sums in $D(\\mathcal{O}_X)$", "are given by direct sums of representing complexes).", "Hence to prove (2) it suffices to write", "$Rf_*\\mathcal{O}_U = \\text{hocolim} E_n$ for a system of", "perfect objects $E_n$ of $D(\\mathcal{O}_X)$. Once this is done", "we obtain (1) by setting $L_n = E_n^\\vee$, see", "Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "\\medskip\\noindent", "Write $A = \\colim A_i$ with $A_i$ of finite type over $\\mathbf{Z}$.", "By Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can find an $i$ and morphisms $U_i \\to X_i \\to \\Spec(A_i)$", "of finite presentation whose base change to $\\Spec(A)$ recovers", "$U \\to X \\to \\Spec(A)$.", "After increasing $i$ we may assume that $f_i : U_i \\to X_i$ is", "flat, see Limits, Lemma \\ref{limits-lemma-descend-flat-finite-presentation}.", "By Lemma \\ref{lemma-compare-base-change}", "the derived pullback of $Rf_{i, *}\\mathcal{O}_{U_i}$", "by $g : X \\to X_i$ is equal to $Rf_*\\mathcal{O}_U$.", "Since $Lg^*$ commutes with derived colimits, it suffices", "to prove what we want for $f_i$. Hence we may assume that", "$U$ and $X$ are of finite type over $\\mathbf{Z}$.", "\\medskip\\noindent", "Assume $f : U \\to X$ is a morphism of schemes of finite type over $\\mathbf{Z}$.", "To finish the proof we will show that $Rf_*\\mathcal{O}_U$ is a homotopy", "colimit of perfect complexes. To see this we apply Lemma \\ref{lemma-countable}.", "Thus it suffices to show that $R^if_*\\mathcal{O}_U$", "has countable sets of sections over affine opens.", "This follows from Lemma \\ref{lemma-countable-cohomology}", "applied to the structure sheaf." ], "refs": [ "perfect-lemma-cohomology-base-change", "perfect-lemma-quasi-coherence-pushforward-direct-sums", "cohomology-lemma-dual-perfect-complex", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-flat-finite-presentation", "perfect-lemma-compare-base-change", "perfect-lemma-countable", "perfect-lemma-countable-cohomology" ], "ref_ids": [ 7025, 6950, 2233, 15077, 15062, 7028, 7072, 7071 ] } ], "ref_ids": [] }, { "id": 7074, "type": "theorem", "label": "perfect-lemma-pseudo-coherent-over-algebra", "categories": [ "perfect" ], "title": "perfect-lemma-pseudo-coherent-over-algebra", "contents": [ "Let $A$ be a ring. Let $R$ be a (possibly noncommutative) $A$-algebra", "which is finite free as an $A$-module. Then any object $M$ of $D(R)$", "which is pseudo-coherent in $D(A)$ can be represented by a", "bounded above complex of finite free (right) $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "Choose a complex $M^\\bullet$ of right $R$-modules representing $M$.", "Since $M$ is pseudo-coherent we have $H^i(M) = 0$ for large enough $i$.", "Let $m$ be the smallest index such that $H^m(M)$ is nonzero.", "Then $H^m(M)$ is a finite $A$-module by", "More on Algebra, Lemma \\ref{more-algebra-lemma-finite-cohomology}.", "Thus we can choose a finite free $R$-module $F^m$ and a map", "$F^m \\to M^m$ such that $F^m \\to M^m \\to M^{m + 1}$ is zero", "and such that $F^m \\to H^m(M)$ is surjective.", "Picture:", "$$", "\\xymatrix{", "& F^m \\ar[d]^\\alpha \\ar[r] & 0 \\ar[d] \\ar[r] & \\ldots \\\\", "M^{m - 1} \\ar[r] & M^m \\ar[r] & M^{m + 1} \\ar[r] & \\ldots", "}", "$$", "By descending induction on $n \\leq m$ we are going to construct", "finite free $R$-modules $F^i$ for $i \\geq n$, differentials", "$d^i : F^i \\to F^{i + 1}$ for $i \\geq n$, maps $\\alpha : F^i \\to K^i$", "compatible with differentials, such that", "(1) $H^i(\\alpha)$ is an isomorphism for $i > n$ and surjective for $i = n$, and", "(2) $F^i = 0$ for $i > m$. Picture", "$$", "\\xymatrix{", "& F^n \\ar[r] \\ar[d]^\\alpha & F^{n + 1} \\ar[d]^\\alpha \\ar[r] & \\ldots \\ar[r]", "& F^i \\ar[d]^\\alpha \\ar[r] & 0 \\ar[d] \\ar[r] & \\ldots \\\\", "M^{n - 1} \\ar[r] & M^n \\ar[r] & M^{n + 1} \\ar[r] & \\ldots \\ar[r] &", "M^i \\ar[r] & M^{i + 1} \\ar[r] & \\ldots", "}", "$$", "The base case is $n = m$ which we've done above.", "Induction step. Let $C^\\bullet$ be the cone on $\\alpha$", "(Derived Categories, Definition \\ref{derived-definition-cone}).", "The long exact sequence", "of cohomology shows that $H^i(C^\\bullet) = 0$ for $i \\geq n$.", "Observe that $F^\\bullet$ is pseudo-coherent as a complex of $A$-modules", "because $R$ is finite free as an $A$-module. Hence", "by More on Algebra, Lemma \\ref{more-algebra-lemma-cone-pseudo-coherent}", "we see that $C^\\bullet$ is $(n - 1)$-pseudo-coherent as a complex of", "$A$-modules. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-finite-cohomology}", "we see that $H^{n - 1}(C^\\bullet)$ is a finite $A$-module.", "Choose a finite free $R$-module $F^{n - 1}$ and a map", "$\\beta : F^{n - 1} \\to C^{n - 1}$ such that the composition", "$F^{n - 1} \\to C^{n - 1} \\to C^n$ is zero and such that $F^{n - 1}$", "surjects onto $H^{n - 1}(C^\\bullet)$. Since", "$C^{n - 1} = M^{n - 1} \\oplus F^n$", "we can write $\\beta = (\\alpha^{n - 1}, -d^{n - 1})$. The vanishing of the", "composition $F^{n - 1} \\to C^{n - 1} \\to C^n$ implies", "these maps fit into a morphism of complexes", "$$", "\\xymatrix{", "& F^{n - 1} \\ar[d]^{\\alpha^{n - 1}} \\ar[r]_{d^{n - 1}} &", "F^n \\ar[r] \\ar[d]^\\alpha &", "F^{n + 1} \\ar[d]^\\alpha \\ar[r] & \\ldots \\\\", "\\ldots \\ar[r] &", "M^{n - 1} \\ar[r] & M^n \\ar[r] & M^{n + 1} \\ar[r] & \\ldots", "}", "$$", "Moreover, these maps define a morphism of distinguished triangles", "$$", "\\xymatrix{", "(F^n \\to \\ldots) \\ar[r] \\ar[d] &", "(F^{n - 1} \\to \\ldots) \\ar[r] \\ar[d] &", "F^{n - 1} \\ar[r] \\ar[d]_\\beta &", "(F^n \\to \\ldots)[1] \\ar[d] \\\\", "(F^n \\to \\ldots) \\ar[r] &", "M^\\bullet \\ar[r] &", "C^\\bullet \\ar[r] &", "(F^n \\to \\ldots)[1]", "}", "$$", "Hence our choice of $\\beta$ implies that the map of complexes", "$(F^{n - 1} \\to \\ldots) \\to M^\\bullet$ induces an isomorphism on", "cohomology in degrees $\\geq n$ and a surjection in degree $n - 1$.", "This finishes the proof of the lemma." ], "refs": [ "more-algebra-lemma-finite-cohomology", "derived-definition-cone", "more-algebra-lemma-cone-pseudo-coherent", "more-algebra-lemma-finite-cohomology" ], "ref_ids": [ 10146, 1978, 10145, 10146 ] } ], "ref_ids": [] }, { "id": 7075, "type": "theorem", "label": "perfect-lemma-pseudo-coherent-on-projective-space", "categories": [ "perfect" ], "title": "perfect-lemma-pseudo-coherent-on-projective-space", "contents": [ "Let $A$ be a ring. Let $n \\geq 0$. Let", "$K \\in D_\\QCoh(\\mathcal{O}_{\\mathbf{P}^n_A})$.", "The following are equivalent", "\\begin{enumerate}", "\\item $K$ is pseudo-coherent,", "\\item $R\\Gamma(\\mathbf{P}^n_A, E \\otimes^\\mathbf{L} K)$ is a pseudo-coherent", "object of $D(A)$ for each pseudo-coherent object $E$ of", "$D(\\mathcal{O}_{\\mathbf{P}^n_A})$,", "\\item $R\\Gamma(\\mathbf{P}^n_A, E \\otimes^\\mathbf{L} K)$ is a pseudo-coherent", "object of $D(A)$ for each perfect object $E$ of", "$D(\\mathcal{O}_{\\mathbf{P}^n_A})$,", "\\item $R\\Hom_{\\mathbf{P}^n_A}(E, K)$ is a pseudo-coherent", "object of $D(A)$ for each perfect object $E$ of", "$D(\\mathcal{O}_{\\mathbf{P}^n_A})$,", "\\item $R\\Gamma(\\mathbf{P}^n_A,", "K \\otimes^\\mathbf{L} \\mathcal{O}_{\\mathbf{P}^n_A}(d))$ is pseudo-coherent", "object of $D(A)$ for $d = 0, 1, \\ldots, n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that", "$$", "R\\Hom_{\\mathbf{P}^n_A}(E, K) =", "R\\Gamma(\\mathbf{P}^n_A, R\\SheafHom_{\\mathcal{O}_{\\mathbf{P}^n_A}}(E, K))", "$$", "by definition, see Cohomology, Section \\ref{cohomology-section-global-RHom}.", "Thus parts (4) and (3) are equivalent by", "Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "\\medskip\\noindent", "Since every perfect complex is pseudo-coherent, it is clear that", "(2) implies (3).", "\\medskip\\noindent", "Assume (1) holds. Then $E \\otimes^\\mathbf{L} K$ is pseudo-coherent", "for every pseudo-coherent $E$, see", "Cohomology, Lemma \\ref{cohomology-lemma-tensor-pseudo-coherent}.", "By Lemma \\ref{lemma-flat-proper-pseudo-coherent-direct-image-general}", "the direct image of such a pseudo-coherent complex is pseudo-coherent", "and we see that (2) is true.", "\\medskip\\noindent", "Part (3) implies (5) because we can take $E = \\mathcal{O}_{\\mathbf{P}^n_A}(d)$", "for $d = 0, 1, \\ldots, n$.", "\\medskip\\noindent", "To finish the proof we have to show that (5) implies (1).", "Let $P$ be as in (\\ref{equation-generator-Pn}) and", "$R$ as in (\\ref{equation-algebra-for-Pn}).", "By Lemma \\ref{lemma-Pn-module-category} we have an equivalence", "$$", "- \\otimes^\\mathbf{L}_R P :", "D(R) \\longrightarrow D_\\QCoh(\\mathcal{O}_{\\mathbf{P}^n_A})", "$$", "Let $M \\in D(R)$ be an object such that $M \\otimes^\\mathbf{L} P = K$.", "By Differential Graded Algebra, Lemma", "\\ref{dga-lemma-upgrade-tensor-with-complex-derived}", "there is an isomorphism", "$$", "R\\Hom(R, M) = R\\Hom_{\\mathbf{P}^n_A}(P, K)", "$$", "in $D(A)$. Arguing as above we obtain", "$$", "R\\Hom_{\\mathbf{P}^n_A}(P, K) = R\\Gamma(\\mathbf{P}^n_A,", "R\\SheafHom_{\\mathcal{O}_{\\mathbf{P}^n_A}}(E, K)) =", "R\\Gamma(\\mathbf{P}^n_A,", "P^\\vee \\otimes^\\mathbf{L}_{\\mathcal{O}_{\\mathbf{P}^n_A}} K).", "$$", "Using that $P^\\vee$ is the direct sum of", "$\\mathcal{O}_{\\mathbf{P}^n_A}(d)$ for $d = 0, 1, \\ldots, n$", "and (5) we conclude $R\\Hom(R, M)$ is pseudo-coherent as a complex", "of $A$-modules. Of course $M = R\\Hom(R, M)$ in $D(A)$.", "Thus $M$ is pseudo-coherent as a complex of $A$-modules.", "By Lemma \\ref{lemma-pseudo-coherent-over-algebra}", "we may represent $M$ by a bounded above complex $F^\\bullet$", "of finite free $R$-modules. Then", "$F^\\bullet = \\bigcup_{p \\geq 0} \\sigma_{\\geq p}F^\\bullet$", "is a filtration which shows that $F^\\bullet$ is a differential", "graded $R$-module with property (P), see", "Differential Graded Algebra, Section \\ref{dga-section-P-resolutions}.", "Hence $K = M \\otimes^\\mathbf{L}_R P$ is represented", "by $F^\\bullet \\otimes_R P$ (follows from the construction of the", "derived tensor functor, see for example the proof of", "Differential Graded Algebra, Lemma \\ref{dga-lemma-tensor-with-complex-derived}).", "Since $F^\\bullet \\otimes_R P$", "is a bounded above complex whose terms are direct sums of", "copies of $P$ we conclude that the lemma is true." ], "refs": [ "cohomology-lemma-dual-perfect-complex", "cohomology-lemma-tensor-pseudo-coherent", "perfect-lemma-flat-proper-pseudo-coherent-direct-image-general", "perfect-lemma-Pn-module-category", "dga-lemma-upgrade-tensor-with-complex-derived", "perfect-lemma-pseudo-coherent-over-algebra", "dga-lemma-tensor-with-complex-derived" ], "ref_ids": [ 2233, 2208, 7055, 7021, 13117, 7074, 13116 ] } ], "ref_ids": [] }, { "id": 7076, "type": "theorem", "label": "perfect-lemma-perfect-enough", "categories": [ "perfect" ], "title": "perfect-lemma-perfect-enough", "contents": [ "Let $A$ be a ring. Let $X$ be a scheme over $A$ which is quasi-compact", "and quasi-separated. Let $K \\in D^-_\\QCoh(\\mathcal{O}_X)$.", "If $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent", "in $D(A)$ for every perfect $E$ in $D(\\mathcal{O}_X)$,", "then $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent", "in $D(A)$ for every pseudo-coherent $E$ in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "There exists an integer $N$ such that", "$R\\Gamma(X, -) : D_\\QCoh(\\mathcal{O}_X) \\to D(A)$", "has cohomological dimension $N$ as explained in", "Lemma \\ref{lemma-quasi-coherence-direct-image}.", "Let $b \\in \\mathbf{Z}$ be such that $H^i(K) = 0$ for $i > b$.", "Let $E$ be pseudo-coherent on $X$.", "It suffices to show that $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$", "is $m$-pseudo-coherent for every $m$.", "Choose an approximation $P \\to E$ by a perfect complex $P$", "of $(X, E, m - N - 1 - b)$. This is possible by", "Theorem \\ref{theorem-approximation}.", "Choose a distinguished triangle", "$$", "P \\to E \\to C \\to P[1]", "$$", "in $D_\\QCoh(\\mathcal{O}_X)$. The cohomology sheaves of $C$ are zero", "in degrees $\\geq m - N - 1 - b$. Hence the cohomology", "sheaves of $C \\otimes^\\mathbf{L} K$ are zero in degrees $\\geq m - N - 1$.", "Thus the cohomology of $R\\Gamma(X, C \\otimes^\\mathbf{L} K)$", "are zero in degrees $\\geq m - 1$. Hence", "$$", "R\\Gamma(X, P \\otimes^\\mathbf{L} K) \\to R\\Gamma(X, E \\otimes^\\mathbf{L} K)", "$$", "is an isomorphism on cohomology in degrees $\\geq m$.", "By assumption the source is pseudo-coherent.", "We conclude that $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$", "is $m$-pseudo-coherent as desired." ], "refs": [ "perfect-lemma-quasi-coherence-direct-image", "perfect-theorem-approximation" ], "ref_ids": [ 6946, 6934 ] } ], "ref_ids": [] }, { "id": 7077, "type": "theorem", "label": "perfect-lemma-affine-locally-rel-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-affine-locally-rel-perfect", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat and", "locally of finite presentation. Let $E$ be an object of", "$D_\\QCoh(\\mathcal{O}_X)$. The following are equivalent", "\\begin{enumerate}", "\\item $E$ is $S$-perfect,", "\\item for any affine open $U \\subset X$ mapping into an affine open", "$V \\subset S$ the complex $R\\Gamma(U, E)$ is $\\mathcal{O}_S(V)$-perfect.", "\\item there exists an affine open covering $S = \\bigcup V_i$", "and for each $i$ an affine open covering $f^{-1}(V_i) = \\bigcup U_{ij}$", "such that the complex $R\\Gamma(U_{ij}, E)$ is $\\mathcal{O}_S(V_i)$-perfect.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Being pseudo-coherent is a local property and", "``locally having finite tor dimension'' is a local property.", "Hence this lemma immediately reduces to the statement:", "if $X$ and $S$ are affine, then $E$ is $S$-perfect", "if and only if $K = R\\Gamma(X, E)$ is $\\mathcal{O}_S(S)$-perfect.", "Say $X = \\Spec(A)$, $S = \\Spec(R)$ and $E$ corresponds to", "$K \\in D(A)$, i.e., $K = R\\Gamma(X, E)$, see", "Lemma \\ref{lemma-affine-compare-bounded}.", "\\medskip\\noindent", "Observe that $K$ is $R$-perfect if and only if $K$ is", "pseudo-coherent and has finite tor dimension as a complex", "of $R$-modules (More on Algebra, Definition", "\\ref{more-algebra-definition-relatively-perfect}).", "By Lemma \\ref{lemma-pseudo-coherent-affine}", "we see that $E$ is pseudo-coherent if and only if", "$K$ is pseudo-coherent.", "By Lemma \\ref{lemma-tor-dimension-rel-affine} we see that", "$E$ has finite tor dimension over $f^{-1}\\mathcal{O}_S$", "if and only if $K$ has finite tor dimension as a complex of", "$R$-modules." ], "refs": [ "perfect-lemma-affine-compare-bounded", "more-algebra-definition-relatively-perfect", "perfect-lemma-pseudo-coherent-affine", "perfect-lemma-tor-dimension-rel-affine" ], "ref_ids": [ 6941, 10632, 6975, 6978 ] } ], "ref_ids": [] }, { "id": 7078, "type": "theorem", "label": "perfect-lemma-triangulated", "categories": [ "perfect" ], "title": "perfect-lemma-triangulated", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is", "flat and locally of finite presentation. The full subcategory", "of $D(\\mathcal{O}_X)$ consisting of $S$-perfect objects is", "a saturated\\footnote{Derived Categories, Definition", "\\ref{derived-definition-saturated}.} triangulated subcategory." ], "refs": [ "derived-definition-saturated" ], "proofs": [ { "contents": [ "This follows from Cohomology, Lemmas", "\\ref{cohomology-lemma-cone-pseudo-coherent},", "\\ref{cohomology-lemma-summands-pseudo-coherent},", "\\ref{cohomology-lemma-cone-tor-amplitude}, and", "\\ref{cohomology-lemma-summands-tor-amplitude}." ], "refs": [ "cohomology-lemma-cone-pseudo-coherent", "cohomology-lemma-summands-pseudo-coherent", "cohomology-lemma-cone-tor-amplitude", "cohomology-lemma-summands-tor-amplitude" ], "ref_ids": [ 2207, 2209, 2218, 2220 ] } ], "ref_ids": [ 1974 ] }, { "id": 7079, "type": "theorem", "label": "perfect-lemma-perfect-relatively-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-perfect-relatively-perfect", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat and locally", "of finite presentation. A perfect object of $D(\\mathcal{O}_X)$ is $S$-perfect.", "If $K, M \\in D(\\mathcal{O}_X)$, then $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M$", "is $S$-perfect if $K$ is perfect and $M$ is $S$-perfect." ], "refs": [], "proofs": [ { "contents": [ "First proof: reduce to the affine case using", "Lemma \\ref{lemma-affine-locally-rel-perfect}", "and then apply More on Algebra, Lemma", "\\ref{more-algebra-lemma-perfect-relatively-perfect}." ], "refs": [ "perfect-lemma-affine-locally-rel-perfect", "more-algebra-lemma-perfect-relatively-perfect" ], "ref_ids": [ 7077, 10289 ] } ], "ref_ids": [] }, { "id": 7080, "type": "theorem", "label": "perfect-lemma-base-change-relatively-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-base-change-relatively-perfect", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat and", "locally of finite presentation.", "Let $g : S' \\to S$ be a morphism of schemes. Set $X' = S' \\times_S X$", "and denote $g' : X' \\to X$ the projection.", "If $K \\in D(\\mathcal{O}_X)$ is $S$-perfect, then $L(g')^*K$", "is $S'$-perfect." ], "refs": [], "proofs": [ { "contents": [ "First proof: reduce to the affine case using", "Lemma \\ref{lemma-affine-locally-rel-perfect}", "and then apply More on Algebra, Lemma", "\\ref{more-algebra-lemma-base-change-relatively-perfect}.", "\\medskip\\noindent", "Second proof: $L(g')^*K$ is pseudo-coherent by", "Cohomology, Lemma \\ref{cohomology-lemma-pseudo-coherent-pullback}", "and the bounded tor dimension property follows from", "Lemma \\ref{lemma-tor-independence-and-tor-amplitude}." ], "refs": [ "perfect-lemma-affine-locally-rel-perfect", "more-algebra-lemma-base-change-relatively-perfect", "cohomology-lemma-pseudo-coherent-pullback", "perfect-lemma-tor-independence-and-tor-amplitude" ], "ref_ids": [ 7077, 10291, 2206, 7030 ] } ], "ref_ids": [] }, { "id": 7081, "type": "theorem", "label": "perfect-lemma-relative-descend-homomorphisms", "categories": [ "perfect" ], "title": "perfect-lemma-relative-descend-homomorphisms", "contents": [ "In Situation \\ref{situation-relative-descent}.", "Let $K_0$ and $L_0$ be objects of $D(\\mathcal{O}_{X_0})$.", "Set $K_i = Lf_{i0}^*K_0$ and $L_i = Lf_{i0}^*L_0$ for $i \\geq 0$", "and set $K = Lf_0^*K_0$ and $L = Lf_0^*L_0$. Then the map", "$$", "\\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{X_i})}(K_i, L_i)", "\\longrightarrow", "\\Hom_{D(\\mathcal{O}_X)}(K, L)", "$$", "is an isomorphism if $K_0$ is pseudo-coherent and", "$L_0 \\in D_\\QCoh(\\mathcal{O}_{X_0})$ has (locally)", "finite tor dimension as an object of", "$D((X_0 \\to S_0)^{-1}\\mathcal{O}_{S_0})$" ], "refs": [], "proofs": [ { "contents": [ "For every quasi-compact open $U_0 \\subset X_0$ consider the", "condition $P$ that", "$$", "\\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{U_i})}(K_i|_{U_i}, L_i|_{U_i})", "\\longrightarrow", "\\Hom_{D(\\mathcal{O}_U)}(K|_U, L|_U)", "$$", "is an isomorphism where $U = f_0^{-1}(U_0)$ and $U_i = f_{i0}^{-1}(U_0)$.", "If $P$ holds for $U_0$, $V_0$ and $U_0 \\cap V_0$, then it holds", "for $U_0 \\cup V_0$ by Mayer-Vietoris", "for hom in the derived category, see", "Cohomology, Lemma \\ref{cohomology-lemma-mayer-vietoris-hom}.", "\\medskip\\noindent", "Denote $\\pi_0 : X_0 \\to S_0$ the given morphism.", "Then we can first consider $U_0 = \\pi_0^{-1}(W_0)$ with", "$W_0 \\subset S_0$ quasi-compact open. By the induction principle of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}", "applied to quasi-compact opens of $S_0$", "and the remark above, we find that it is enough to prove", "$P$ for $U_0 = \\pi_0^{-1}(W_0)$ with $W_0$ affine.", "In other words, we have reduced to the case where $S_0$ is affine.", "Next, we apply the induction principle again, this time to all", "quasi-compact and quasi-separated opens of $X_0$, to reduce to the", "case where $X_0$ is affine as well.", "\\medskip\\noindent", "If $X_0$ and $S_0$ are affine, the result follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-colimit-relatively-perfect}.", "Namely, by Lemmas \\ref{lemma-pseudo-coherent} and", "\\ref{lemma-affine-compare-bounded}", "the statement is translated into computations of homs in the", "derived categories of modules. Then", "Lemma \\ref{lemma-pseudo-coherent-affine}", "shows that the complex of modules corresponding to $K_0$", "is pseudo-coherent. And", "Lemma \\ref{lemma-tor-dimension-rel-affine}", "shows that the complex of modules corresponding to $L_0$", "has finite tor dimension over $\\mathcal{O}_{S_0}(S_0)$.", "Thus the assumptions of", "More on Algebra, Lemma \\ref{more-algebra-lemma-colimit-relatively-perfect}", "are satisfied and we win." ], "refs": [ "cohomology-lemma-mayer-vietoris-hom", "coherent-lemma-induction-principle", "more-algebra-lemma-colimit-relatively-perfect", "perfect-lemma-pseudo-coherent", "perfect-lemma-affine-compare-bounded", "perfect-lemma-pseudo-coherent-affine", "perfect-lemma-tor-dimension-rel-affine", "more-algebra-lemma-colimit-relatively-perfect" ], "ref_ids": [ 2145, 3291, 10293, 6974, 6941, 6975, 6978, 10293 ] } ], "ref_ids": [] }, { "id": 7082, "type": "theorem", "label": "perfect-lemma-descend-relatively-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-descend-relatively-perfect", "contents": [ "In Situation \\ref{situation-relative-descent} the category of", "$S$-perfect objects of $D(\\mathcal{O}_X)$ is the colimit of the categories", "of $S_i$-perfect objects of $D(\\mathcal{O}_{X_i})$." ], "refs": [], "proofs": [ { "contents": [ "For every quasi-compact open $U_0 \\subset X_0$ consider the condition $P$", "that the functor", "$$", "\\colim_{i \\geq 0} D_{S_i\\text{-perfect}}(\\mathcal{O}_{U_i})", "\\longrightarrow", "D_{S\\text{-perfect}}(\\mathcal{O}_U)", "$$", "is an equivalence where $U = f_0^{-1}(U_0)$ and $U_i = f_{i0}^{-1}(U_0)$.", "We observe that we already know this functor is fully faithful", "by Lemma \\ref{lemma-relative-descend-homomorphisms}. Thus it suffices to prove", "essential surjectivity.", "\\medskip\\noindent", "Suppose that $P$ holds for quasi-compact opens $U_0$, $V_0$ of $X_0$.", "We claim that $P$ holds for $U_0 \\cup V_0$. We will use the notation", "$U_i = f_{i0}^{-1}U_0$, $U = f_0^{-1}U_0$, $V_i = f_{i0}^{-1}V_0$,", "and $V = f_0^{-1}V_0$ and we will abusively use the symbol", "$f_i$ for all the morphisms $U \\to U_i$, $V \\to V_i$,", "$U \\cap V \\to U_i \\cap V_i$, and $U \\cup V \\to U_i \\cup V_i$.", "Suppose $E$ is an $S$-perfect object of $D(\\mathcal{O}_{U \\cup V})$.", "Goal: show $E$ is in the essential image of the functor.", "By assumption,", "we can find $i \\geq 0$, an $S_i$-perfect object $E_{U, i}$ on $U_i$,", "an $S_i$-perfect object $E_{V, i}$ on $V_i$, and", "isomorphisms $Lf_i^*E_{U, i} \\to E|_U$ and $Lf_i^*E_{V, i} \\to E|_V$.", "Let", "$$", "a : E_{U, i} \\to (Rf_{i, *}E)|_{U_i}", "\\quad\\text{and}\\quad", "b : E_{V, i} \\to (Rf_{i, *}E)|_{V_i}", "$$", "the maps adjoint to the isomorphisms $Lf_i^*E_{U, i} \\to E|_U$", "and $Lf_i^*E_{V, i} \\to E|_V$.", "By fully faithfulness, after increasing $i$,", "we can find an isomorphism", "$c : E_{U, i}|_{U_i \\cap V_i} \\to E_{V, i}|_{U_i \\cap V_i}$", "which pulls back to the identifications ", "$$", "Lf_i^*E_{U, i}|_{U \\cap V} \\to E|_{U \\cap V} \\to Lf_i^*E_{V, i}|_{U \\cap V}.", "$$", "Apply Cohomology, Lemma \\ref{cohomology-lemma-glue}", "to get an object $E_i$ on $U_i \\cup V_i$ and a map $d : E_i \\to Rf_{i, *}E$", "which restricts to the maps $a$ and $b$ over $U_i$ and $V_i$.", "Then it is clear that $E_i$ is $S_i$-perfect (because being", "relatively perfect is a local property) and that", "$d$ is adjoint to an isomorphism $Lf_i^*E_i \\to E$.", "\\medskip\\noindent", "By exactly the same argument as used in", "the proof of Lemma \\ref{lemma-relative-descend-homomorphisms}", "using the induction principle", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle})", "we reduce to the case where both $X_0$ and $S_0$", "are affine. (First work with opens in $S_0$ to reduce to", "$S_0$ affine, then work with opens in $X_0$ to reduce to", "$X_0$ affine.) In the affine case the result follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-colimit-relatively-perfect}.", "The translation into algebra is done by", "Lemma \\ref{lemma-affine-locally-rel-perfect}." ], "refs": [ "perfect-lemma-relative-descend-homomorphisms", "cohomology-lemma-glue", "perfect-lemma-relative-descend-homomorphisms", "coherent-lemma-induction-principle", "more-algebra-lemma-colimit-relatively-perfect", "perfect-lemma-affine-locally-rel-perfect" ], "ref_ids": [ 7081, 2191, 7081, 3291, 10293, 7077 ] } ], "ref_ids": [] }, { "id": 7083, "type": "theorem", "label": "perfect-lemma-derived-pushforward-rel-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-derived-pushforward-rel-perfect", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat, proper, and", "of finite presentation. Let $E \\in D(\\mathcal{O}_X)$ be $S$-perfect.", "Then $Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$", "and its formation commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "The statement on base change is Lemma \\ref{lemma-base-change-tensor}", "(with $\\mathcal{G}^\\bullet$ equal to $\\mathcal{O}_X$ in degree $0$).", "Thus it suffices to show that $Rf_*E$ is a perfect object. We will reduce", "to the case where $S$ is Noetherian affine by a limit argument.", "\\medskip\\noindent", "The question is local on $S$, hence we may assume $S$ is affine.", "Say $S = \\Spec(R)$. We write $R = \\colim R_i$ as a filtered colimit", "of Noetherian rings $R_i$. By Limits, Lemma", "\\ref{limits-lemma-descend-finite-presentation}", "there exists an $i$ and a scheme $X_i$ of finite presentation over $R_i$", "whose base change to $R$ is $X$. By", "Limits, Lemmas \\ref{limits-lemma-eventually-proper} and", "\\ref{limits-lemma-descend-flat-finite-presentation}", "we may assume $X_i$ is proper and flat over $R_i$.", "By Lemma \\ref{lemma-descend-relatively-perfect}", "we may assume there exists a $R_i$-perfect object $E_i$ of", "$D(\\mathcal{O}_{X_i})$ whose pullback to $X$ is $E$.", "Applying Lemma \\ref{lemma-perfect-direct-image}", "to $X_i \\to \\Spec(R_i)$ and $E_i$ and using the", "base change property already shown we obtain the result." ], "refs": [ "perfect-lemma-base-change-tensor", "limits-lemma-descend-finite-presentation", "limits-lemma-eventually-proper", "limits-lemma-descend-flat-finite-presentation", "perfect-lemma-descend-relatively-perfect", "perfect-lemma-perfect-direct-image" ], "ref_ids": [ 7041, 15077, 15089, 15062, 7082, 7043 ] } ], "ref_ids": [] }, { "id": 7084, "type": "theorem", "label": "perfect-lemma-compute-ext-rel-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-compute-ext-rel-perfect", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $E, K \\in D(\\mathcal{O}_X)$.", "Assume", "\\begin{enumerate}", "\\item $S$ is quasi-compact and quasi-separated,", "\\item $f$ is proper, flat, and of finite presentation,", "\\item $E$ is $S$-perfect,", "\\item $K$ is pseudo-coherent.", "\\end{enumerate}", "Then there exists a pseudo-coherent $L \\in D(\\mathcal{O}_S)$ such that", "$$", "Rf_*R\\SheafHom(K, E) = R\\SheafHom(L, \\mathcal{O}_S)", "$$", "and the same is true after arbitrary base change: given", "$$", "\\vcenter{", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "S' \\ar[r]^g &", "S", "}", "}", "\\quad\\quad", "\\begin{matrix}", "\\text{cartesian, then we have } \\\\", "Rf'_*R\\SheafHom(L(g')^*K, L(g')^*E) \\\\", "= R\\SheafHom(Lg^*L, \\mathcal{O}_{S'})", "\\end{matrix}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since $S$ is quasi-compact and quasi-separated, the same is true for $X$.", "By Lemma \\ref{lemma-pseudo-coherent-hocolim} we can write", "$K = \\text{hocolim} K_n$ with $K_n$ perfect and $K_n \\to K$ inducing", "an isomorphism on truncations $\\tau_{\\geq -n}$. Let $K_n^\\vee$", "be the dual perfect complex", "(Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}).", "We obtain an inverse system $\\ldots \\to K_3^\\vee \\to K_2^\\vee \\to K_1^\\vee$", "of perfect objects. By Lemma \\ref{lemma-perfect-relatively-perfect}", "we see that $K_n^\\vee \\otimes_{\\mathcal{O}_X} E$ is $S$-perfect.", "Thus we may apply Lemma \\ref{lemma-derived-pushforward-rel-perfect}", "to $K_n^\\vee \\otimes_{\\mathcal{O}_X} E$ and we obtain an inverse system", "$$", "\\ldots \\to M_3 \\to M_2 \\to M_1", "$$", "of perfect complexes on $S$ with", "$$", "M_n = Rf_*(K_n^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) =", "Rf_*R\\SheafHom(K_n, E)", "$$", "Moreover, the formation of these complexes commutes with any", "base change, namely $Lg^*M_n =", "Rf'_*((L(g')^*K_n)^\\vee \\otimes_{\\mathcal{O}_{X'}}^\\mathbf{L} L(g')^*E) =", "Rf'_*R\\SheafHom(L(g')^*K_n, L(g')^*E)$.", "\\medskip\\noindent", "As $K_n \\to K$ induces an isomorphism on $\\tau_{\\geq -n}$, we see that", "$K_n \\to K_{n + 1}$ induces an isomorphism on $\\tau_{\\geq -n}$.", "It follows that $K_{n + 1}^\\vee \\to K_n^\\vee$", "induces an isomorphism on $\\tau_{\\leq n}$ as", "$K_n^\\vee = R\\SheafHom(K_n, \\mathcal{O}_X)$.", "Suppose that $E$ has tor amplitude in $[a, b]$ as a complex", "of $f^{-1}\\mathcal{O}_Y$-modules. Then the same is true after", "any base change, see Lemma \\ref{lemma-tor-independence-and-tor-amplitude}.", "We find that", "$K_{n + 1}^\\vee \\otimes_{\\mathcal{O}_X} E \\to", "K_n^\\vee \\otimes_{\\mathcal{O}_X} E$", "induces an isomorphism on $\\tau_{\\leq n + a}$", "and the same is true after any base change.", "Applying the right derived functor $Rf_*$", "we conclude the maps $M_{n + 1} \\to M_n$", "induce isomorphisms on $\\tau_{\\leq n + a}$", "and the same is true after any base change.", "Choose a distinguished triangle", "$$", "M_{n + 1} \\to M_n \\to C_n \\to M_{n + 1}[1]", "$$", "Take $S'$ equal to the spectrum of the residue field at a point", "$s \\in S$ and pull back to see that", "$C_n \\otimes_{\\mathcal{O}_S}^\\mathbf{L} \\kappa(s)$", "has nonzero cohomology only in degrees $\\geq n + a$. By", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-perfect-from-residue-field}", "we see that the perfect complex $C_n$ has tor amplitude in", "$[n + a, m_n]$ for some integer $m_n$.", "In particular, the dual perfect complex $C_n^\\vee$ has tor amplitude in", "$[-m_n, -n - a]$.", "\\medskip\\noindent", "Let $L_n = M_n^\\vee$ be the dual perfect complex. The", "conclusion from the discussion in the previous paragraph is that", "$L_n \\to L_{n + 1}$ induces isomorphisms on $\\tau_{\\geq -n - a}$.", "Thus $L = \\text{hocolim} L_n$ is pseudo-coherent, see", "Lemma \\ref{lemma-pseudo-coherent-hocolim}.", "Since we have", "$$", "R\\SheafHom(K, E) = R\\SheafHom(\\text{hocolim} K_n, E) =", "R\\lim R\\SheafHom(K_n, E) = R\\lim K_n^\\vee \\otimes_{\\mathcal{O}_X} E", "$$", "(Cohomology, Lemma \\ref{cohomology-lemma-colim-and-lim-of-duals})", "and since $R\\lim$ commutes with $Rf_*$ we find that", "$$", "Rf_*R\\SheafHom(K, E) = R\\lim M_n = R\\lim R\\SheafHom(L_n, \\mathcal{O}_S) =", "R\\SheafHom(L, \\mathcal{O}_S)", "$$", "This proves the formula over $S$. Since the construction of $M_n$ is", "compatible with base chance, the formula continues to hold after", "any base change." ], "refs": [ "perfect-lemma-pseudo-coherent-hocolim", "cohomology-lemma-dual-perfect-complex", "perfect-lemma-perfect-relatively-perfect", "perfect-lemma-derived-pushforward-rel-perfect", "perfect-lemma-tor-independence-and-tor-amplitude", "more-algebra-lemma-lift-perfect-from-residue-field", "perfect-lemma-pseudo-coherent-hocolim", "cohomology-lemma-colim-and-lim-of-duals" ], "ref_ids": [ 7019, 2233, 7079, 7083, 7030, 10232, 7019, 2236 ] } ], "ref_ids": [] }, { "id": 7085, "type": "theorem", "label": "perfect-lemma-bounded-on-fibres", "categories": [ "perfect" ], "title": "perfect-lemma-bounded-on-fibres", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat and", "locally of finite presentation. Let $E$ be a pseudo-coherent", "object of $D(\\mathcal{O}_X)$. The following are equivalent", "\\begin{enumerate}", "\\item $E$ is $S$-perfect, and", "\\item $E$ is locally bounded below and for every point $s \\in S$", "the object $L(X_s \\to X)^*E$ of $D(\\mathcal{O}_{X_s})$", "is locally bounded below.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since everything is local we immediately reduce to the", "case that $X$ and $S$ are affine, see Lemma", "\\ref{lemma-affine-locally-rel-perfect}.", "Say $X \\to S$ corresponds to $\\Spec(A) \\to \\Spec(R)$ and", "$E$ corresponds to $K$ in $D(A)$. If $s$ corresponds to", "the prime $\\mathfrak p \\subset R$, then $L(X_s \\to X)^*E$", "corresponds to $K \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p)$", "as $R \\to A$ is flat, see for example", "Lemma \\ref{lemma-compare-base-change}.", "Thus we see that our lemma follows from the corresponding algebra", "result, see More on Algebra, Lemma", "\\ref{more-algebra-lemma-bounded-on-fibres-relatively-perfect}." ], "refs": [ "perfect-lemma-affine-locally-rel-perfect", "perfect-lemma-compare-base-change", "more-algebra-lemma-bounded-on-fibres-relatively-perfect" ], "ref_ids": [ 7077, 7028, 10296 ] } ], "ref_ids": [] }, { "id": 7086, "type": "theorem", "label": "perfect-lemma-resolution-property-ample", "categories": [ "perfect" ], "title": "perfect-lemma-resolution-property-ample", "contents": [ "Let $X$ be a scheme. If $X$ has an ample invertible $\\mathcal{O}_X$-module,", "then $X$ has the resolution property." ], "refs": [], "proofs": [ { "contents": [ "Immediate consquence of Properties, Proposition", "\\ref{properties-proposition-characterize-ample}." ], "refs": [ "properties-proposition-characterize-ample" ], "ref_ids": [ 3067 ] } ], "ref_ids": [] }, { "id": 7087, "type": "theorem", "label": "perfect-lemma-resolution-property-ample-relative", "categories": [ "perfect" ], "title": "perfect-lemma-resolution-property-ample-relative", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume", "\\begin{enumerate}", "\\item $Y$ is quasi-compact and quasi-separated and has the resolution property,", "\\item there exists an $f$-ample invertible module on $X$.", "\\end{enumerate}", "Then $X$ has the resolution property." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\mathcal{L}$ be an $f$-ample invertible module.", "Choose an affine open covering $Y = V_1 \\cup \\ldots \\cup V_m$.", "Set $U_j = f^{-1}(V_j)$. By Properties, Proposition", "\\ref{properties-proposition-characterize-ample}", "for each $j$ we know there exists finitely many maps", "$s_{j, i} : \\mathcal{L}^{\\otimes n_{j, i}}|_{U_j} \\to \\mathcal{F}|_{U_j}$", "which are jointly surjective. Consider the quasi-coherent", "$\\mathcal{O}_Y$-modules", "$$", "\\mathcal{H}_n =", "f_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n})", "$$", "We may think of $s_{j, i}$ as a section over $V_j$ of the sheaf", "$\\mathcal{H}_{-n_{j, i}}$. Suppose we can find finite locally", "free $\\mathcal{O}_Y$-modules $\\mathcal{E}_{i, j}$ and maps", "$\\mathcal{E}_{i, j} \\to \\mathcal{H}_{-n_{j, i}}$ such that", "$s_{j, i}$ is in the image. Then the corresponding maps", "$$", "f^*\\mathcal{E}_{i, j}", "\\otimes_{\\mathcal{O}_X}", "\\mathcal{L}^{\\otimes n_{i, j}} \\longrightarrow \\mathcal{F}", "$$", "are going to be jointly surjective and the lemma is proved. By", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-colimit-finite-type}", "for each $i, j$ we can find a finite", "type quasi-coherent submodule", "$\\mathcal{H}'_{i, j} \\subset \\mathcal{H}_{-n_{j, i}}$", "which contains the section $s_{i, j}$ over $V_j$.", "Thus using the resolution property of $Y$ to get surjections", "$\\mathcal{E}_{i, j} \\to \\mathcal{H}'_{j, i}$ and we conclude." ], "refs": [ "properties-proposition-characterize-ample", "properties-lemma-quasi-coherent-colimit-finite-type" ], "ref_ids": [ 3067, 3020 ] } ], "ref_ids": [] }, { "id": 7088, "type": "theorem", "label": "perfect-lemma-resolution-property-goes-up-affine", "categories": [ "perfect" ], "title": "perfect-lemma-resolution-property-goes-up-affine", "contents": [ "Let $f : X \\to Y$ be an affine morphism of schemes with $Y$ quasi-compact", "and quasi-separated. If $Y$ has the resolution property, so does $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.", "The pushforward $f_*\\mathcal{F}$ is quasi-coherent, see", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}.", "The adjunction map $f^*f_*\\mathcal{F} \\to \\mathcal{F}$ is surjective;", "this follows from Schemes, Lemma \\ref{schemes-lemma-widetilde-pullback}", "after restricting to $f^{-1}(V)$ for $V \\subset Y$ affine open.", "Write $f_*\\mathcal{F} = \\colim \\mathcal{G}_i$ as a filtered colimit", "with $\\mathcal{G}_i$ quasi-coherent $\\mathcal{O}_Y$-modules of finite type, see", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-colimit-finite-type}.", "Then we see that $\\colim f^*\\mathcal{G}_i \\to \\mathcal{F}$ is surjective.", "Since $\\mathcal{F}$ is of finite type and $X$ is quasi-compact,", "we conclude that for some $i$ the map $f^*\\mathcal{G}_i \\to \\mathcal{F}$", "is surjective (details omitted; look at generators on affines).", "Hence if $\\mathcal{G}_i$ is a quotient of a finite locally free", "$\\mathcal{O}_Y$-module, then $\\mathcal{F}$ is a quotient of the", "pullback which is a finite locally free $\\mathcal{O}_X$-module." ], "refs": [ "schemes-lemma-push-forward-quasi-coherent", "schemes-lemma-widetilde-pullback", "properties-lemma-quasi-coherent-colimit-finite-type" ], "ref_ids": [ 7730, 7662, 3020 ] } ], "ref_ids": [] }, { "id": 7089, "type": "theorem", "label": "perfect-lemma-resolution-property-finite-number", "categories": [ "perfect" ], "title": "perfect-lemma-resolution-property-finite-number", "contents": [ "Let $X$ be a scheme. Suppose given", "\\begin{enumerate}", "\\item a finite affine open covering $X = U_1 \\cup \\ldots \\cup U_m$", "\\item finite type quasi-coherent ideals $\\mathcal{I}_j$", "with $V(\\mathcal{I}_j) = X \\setminus U_j$", "\\end{enumerate}", "Then $X$ has the resolution property if and only if $\\mathcal{I}_j$", "is the quotient of a finite locally free $\\mathcal{O}_X$-module", "for $j = 1, \\ldots, m$." ], "refs": [], "proofs": [ { "contents": [ "One direction of the lemma is trivial. For the other, ", "say $\\mathcal{E}_j \\to \\mathcal{I}_j$ is a surjection with", "$\\mathcal{E}_j$ finite locally free. In the next paragraph, we", "reduce to the Noetherian case; we suggest the reader skip it.", "\\medskip\\noindent", "The first observation is that $U_j \\cap U_{j'}$ is quasi-compact", "as the complement of the zero scheme of the quasi-coherent finite type ideal", "$\\mathcal{I}_{j'}|{U_j}$ on the affine scheme $U_j$, see", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-finite-type-ideals}.", "Hence $X$ is quasi-compact and quasi-separated, see", "Schemes, Lemma \\ref{schemes-lemma-characterize-quasi-separated}.", "By Limits, Proposition \\ref{limits-proposition-approximate}", "we can write $X = \\lim X_i$ as the limit of a direct system of", "Noetherian schemes with affine transition morphisms. For each $j$", "we can find an $i$ and a finite type quasi-coherent ideal sheaf", "$\\mathcal{I}_{i, j} \\subset \\mathcal{O}_{X_j}$", "pulling back to $\\mathcal{I}_j$, see", "Limits, Lemma \\ref{limits-lemma-descend-invertible-modules}.", "Denoting $U_{i, j} \\subset X_i$ the complementary opens, we", "may assume these are affine for all $i, j$, see", "Limits, Lemma \\ref{limits-lemma-limit-affine}.", "Similarly, we may assume the maps $\\mathcal{E}_j \\to \\mathcal{I}_j$", "are the pullbacks of surjections", "$\\mathcal{E}_{i, j} \\to \\mathcal{I}_{i, j}$", "with $\\mathcal{E}_{i, j}$ finite locally free on $X_i$, see", "Limits, Lemmas \\ref{limits-lemma-descend-invertible-modules}", "and \\ref{limits-lemma-descend-modules-finite-presentation}.", "Using this and Lemma \\ref{lemma-resolution-property-goes-up-affine}", "we reduce to the case of a Noetherian scheme.", "\\medskip\\noindent", "Assume $X$ is Noetherian. For every coherent module $\\mathcal{F}$", "we can choose a finite list of sections $s_{jk} \\in \\mathcal{F}(U_j)$,", "$k = 1, \\ldots, e_j$ which generate the restriction of $\\mathcal{F}$ to $U_j$.", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-homs-over-open}", "we can extend $s_{jk}$ to a map", "$s'_{jk} : \\mathcal{I}_i^{n_{jk}} \\to \\mathcal{F}$ for some $n_{jk} \\geq 1$.", "Then we can consider the compositions", "$$", "\\mathcal{E}_j^{\\otimes n_{jk}} \\to \\mathcal{I}_j^{n_{jk}} \\to \\mathcal{F}", "$$", "to conclude." ], "refs": [ "properties-lemma-quasi-coherent-finite-type-ideals", "schemes-lemma-characterize-quasi-separated", "limits-proposition-approximate", "limits-lemma-descend-invertible-modules", "limits-lemma-limit-affine", "limits-lemma-descend-invertible-modules", "limits-lemma-descend-modules-finite-presentation", "perfect-lemma-resolution-property-goes-up-affine", "coherent-lemma-homs-over-open" ], "ref_ids": [ 3033, 7709, 15126, 15079, 15043, 15079, 15078, 7088, 3322 ] } ], "ref_ids": [] }, { "id": 7090, "type": "theorem", "label": "perfect-lemma-regular-resolution-property", "categories": [ "perfect" ], "title": "perfect-lemma-regular-resolution-property", "contents": [ "Let $X$ be a quasi-compact, regular scheme with affine diagonal.", "Then $X$ has the resolution property." ], "refs": [], "proofs": [ { "contents": [ "Observe that $X$ is a finite disjoint union of integral schemes", "(Properties, Lemmas", "\\ref{properties-lemma-regular-normal} and", "\\ref{properties-lemma-normal-Noetherian}).", "Thus we may assume that $X$ is integral as well as Noetherian,", "regular, and having affine diagonal.", "Choose an affine open covering $X = U_1 \\cup \\ldots \\cup U_m$.", "We may and do assume $U_j$ nonempty for all $j$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-regular-local-UFD}", "the local rings of $X$ are UFDs and hence by Divisors, Lemma", "\\ref{divisors-lemma-complement-open-affine-effective-cartier-divisor-bis}", "we can find an effective Cartier divisors $D_j \\subset X$", "whose complement is $U_j$. Then the ideal sheaf of $D_j$ is", "invertible, hence a finite locally free module and we conclude", "that $X$ has the resolution property by", "Lemma \\ref{lemma-resolution-property-finite-number}." ], "refs": [ "properties-lemma-regular-normal", "properties-lemma-normal-Noetherian", "more-algebra-lemma-regular-local-UFD", "divisors-lemma-complement-open-affine-effective-cartier-divisor-bis", "perfect-lemma-resolution-property-finite-number" ], "ref_ids": [ 2977, 2970, 10544, 7962, 7089 ] } ], "ref_ids": [] }, { "id": 7091, "type": "theorem", "label": "perfect-lemma-resolution-property-descends", "categories": [ "perfect" ], "title": "perfect-lemma-resolution-property-descends", "contents": [ "Let $X = \\lim X_i$ be a limit of a direct system of quasi-compact", "and quasi-separated schemes with affine transition morphisms.", "Then $X$ has the resolution property if and only if $X_i$ has", "the resolution properties for some $i$." ], "refs": [], "proofs": [ { "contents": [ "If $X_i$ has the resolution property, then $X$ does by", "Lemma \\ref{lemma-resolution-property-goes-up-affine}.", "Assume $X$ has the resolution property.", "Choose a finite affine open covering", "$X = U_1 \\cup \\ldots \\cup U_m$.", "For each $j$ choose a finite type quasi-coherent sheaf", "of ideals $\\mathcal{I}_j \\subset \\mathcal{O}_X$", "such that $X \\setminus V(\\mathcal{I}_j) = U_j$, see", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-finite-type-ideals}.", "Choose finite locally free $\\mathcal{O}_X$-modules", "and surjections $\\mathcal{E}_j \\to \\mathcal{I}_j$.", "By Limits, Lemmas \\ref{limits-lemma-descend-invertible-modules} and", "\\ref{limits-lemma-descend-modules-finite-presentation}", "we can find an $i$ and finite locally free", "$\\mathcal{O}_{X_i}$-modules $\\mathcal{E}_{i, j}$", "and maps $\\mathcal{E}_{i, j} \\to \\mathcal{O}_{X_i}$", "whose base changes to $X$ recover the maps", "$\\mathcal{E}_j \\to \\mathcal{I}_j$, $j = 1, \\ldots, m$.", "Denote $\\mathcal{I}_{i, j} \\subset \\mathcal{O}_{X_i}$", "the image of these maps.", "Set $U_{i, j} = X_i \\setminus V(\\mathcal{I}_{i, j})$.", "After increasing $i$ we may assume $U_{i, j}$ is affine, see", "Limits, Lemma \\ref{limits-lemma-limit-affine}.", "Then we conclude that $X_i$ has the resolution property", "by Lemma \\ref{lemma-resolution-property-finite-number}." ], "refs": [ "perfect-lemma-resolution-property-goes-up-affine", "properties-lemma-quasi-coherent-finite-type-ideals", "limits-lemma-descend-invertible-modules", "limits-lemma-descend-modules-finite-presentation", "limits-lemma-limit-affine", "perfect-lemma-resolution-property-finite-number" ], "ref_ids": [ 7088, 3033, 15079, 15078, 15043, 7089 ] } ], "ref_ids": [] }, { "id": 7092, "type": "theorem", "label": "perfect-lemma-resolution-property-affine-diagonal", "categories": [ "perfect" ], "title": "perfect-lemma-resolution-property-affine-diagonal", "contents": [ "\\begin{reference}", "Special case of \\cite[Proposition 1.3]{totaro_resolution}.", "\\end{reference}", "Let $X$ be a quasi-compact and quasi-separated scheme with", "the resolution property. Then $X$ has affine diagonal." ], "refs": [], "proofs": [ { "contents": [ "Combining Limits, Proposition \\ref{limits-proposition-approximate}", "and Lemma \\ref{lemma-resolution-property-descends}", "this reduces to the case where $X$ is Noetherian (small detail omitted).", "Assume $X$ is Noetherian.", "Recall that $X \\times X$ is covered by the affine opens", "$U \\times V$ for affine opens $U$, $V$ of $X$, see", "Schemes, Section \\ref{schemes-section-fibre-products}.", "Hence to show that the diagonal $\\Delta : X \\to X \\times X$", "is affine, it suffices to show that $U \\cap V = \\Delta^{-1}(U \\times V)$", "is affine for all affine opens $U$, $V$ of $X$, see", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-affine}.", "In particular, it suffices to show that the inclusion morphism", "$j : U \\to X$ is affine if $U$ is an affine open of $X$.", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-criterion-affine-morphism}", "it suffices to show that $R^1j_*\\mathcal{G} = 0$ for any", "quasi-coherent $\\mathcal{O}_U$-module $\\mathcal{G}$.", "By Proposition \\ref{proposition-Noetherian} (this is where we use", "that we've reduced to the Noetherian case)", "we can represent $Rj_*\\mathcal{G}$ by a complex", "$\\mathcal{H}^\\bullet$ of quasi-coherent $\\mathcal{O}_X$-modules.", "Assume", "$$", "H^1(\\mathcal{H}^\\bullet) =", "\\Ker(\\mathcal{H}^1 \\to \\mathcal{H}^2)/\\Im(\\mathcal{H}^0 \\to \\mathcal{H}^1)", "$$", "is nonzero in order to get a contradiction. Then we can find a coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ and a map", "$$", "\\mathcal{F} \\longrightarrow", "\\Ker(\\mathcal{H}^1 \\to \\mathcal{H}^2)", "$$", "such that the composition with the projection onto $H^1(\\mathcal{H}^\\bullet)$", "is nonzero. Namely, we can write $\\Ker(\\mathcal{H}^1 \\to \\mathcal{H}^2)$", "as the filtered union of its coherent submodules by", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-colimit-finite-type}", "and then one of these will do the job.", "Next, we choose a finite locally free $\\mathcal{O}_X$-module", "$\\mathcal{E}$ and a surjection $\\mathcal{E} \\to \\mathcal{F}$ using", "the resolution property of $X$.", "This produces a map in the derived category", "$$", "\\mathcal{E}[-1] \\longrightarrow Rj_*\\mathcal{G}", "$$", "which is nonzero on cohomology sheaves and hence nonzero in $D(\\mathcal{O}_X)$.", "By adjunction, this is the same thing as a map", "$$", "j^*\\mathcal{E}[-1] \\to \\mathcal{G}", "$$", "nonzero in $D(\\mathcal{O}_U)$. Since $\\mathcal{E}$ is finite locally", "free this is the same thing as a nonzero element of", "$$", "H^1(U, j^*\\mathcal{E}^\\vee \\otimes_{\\mathcal{O}_U} \\mathcal{G})", "$$", "where", "$\\mathcal{E}^\\vee = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}, \\mathcal{O}_X)$", "is the dual finite locally free module. However, this group is zero by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}", "which is the desired contradiction.", "(If in doubt about the step using duals, please see the more general", "Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.)" ], "refs": [ "limits-proposition-approximate", "perfect-lemma-resolution-property-descends", "morphisms-lemma-characterize-affine", "coherent-lemma-criterion-affine-morphism", "perfect-proposition-Noetherian", "properties-lemma-quasi-coherent-colimit-finite-type", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "cohomology-lemma-dual-perfect-complex" ], "ref_ids": [ 15126, 7091, 5172, 3290, 7108, 3020, 3282, 2233 ] } ], "ref_ids": [] }, { "id": 7093, "type": "theorem", "label": "perfect-lemma-construct-strictly-perfect", "categories": [ "perfect" ], "title": "perfect-lemma-construct-strictly-perfect", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme with the", "resolution property.", "Let $\\mathcal{F}^\\bullet$ be a bounded below complex of quasi-coherent", "$\\mathcal{O}_X$-modules representing a perfect object of", "$D(\\mathcal{O}_X)$. Then there exists a bounded complex", "$\\mathcal{E}^\\bullet$ of finite locally free $\\mathcal{O}_X$-modules", "and a quasi-isomorphism $\\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Let $a, b \\in \\mathbf{Z}$ be integers such that $\\mathcal{F}^\\bullet$", "has tor amplitude in $[a, b]$ and such that $\\mathcal{F}^n = 0$ for", "$n < a$. The existence of such a pair of integers", "follows from Cohomology, Lemma \\ref{cohomology-lemma-perfect}", "and the fact that $X$ is quasi-compact.", "If $b < a$, then $\\mathcal{F}^\\bullet$ is zero in the", "derived category and the lemma holds.", "We will prove by induction on", "$b - a \\geq 0$ that there exists a complex", "$\\mathcal{E}^a \\to \\ldots \\to \\mathcal{E}^b$", "with $\\mathcal{E}^i$ finite locally free", "and a quasi-isomorphism $\\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$.", "\\medskip\\noindent", "The base case is the case $b - a = 0$. In this case", "$H^b(\\mathcal{F}^\\bullet) = H^a(\\mathcal{F}^\\bullet) =", "\\Ker(\\mathcal{F}^a \\to \\mathcal{F}^{a + 1})$", "is finite locally free. Namely, it is a finitely presented", "$\\mathcal{O}_X$-module of tor dimension $0$ and hence finite", "locally free. See Cohomology, Lemmas \\ref{cohomology-lemma-perfect} and", "\\ref{cohomology-lemma-finite-cohomology} and", "Properties, Lemma \\ref{properties-lemma-finite-locally-free}.", "Thus we can take $\\mathcal{E}^\\bullet$ to be", "$H^b(\\mathcal{F}^\\bullet)$ sitting in degree $b$.", "The rest of the proof is dedicated to the induction step.", "\\medskip\\noindent", "Assume $b > a$. Observe that", "$$", "H^b(\\mathcal{F}^\\bullet) =", "\\Ker(\\mathcal{F}^b \\to \\mathcal{F}^{b + 1})/", "\\Im(\\mathcal{F}^{b - 1} \\to \\mathcal{F}^b)", "$$", "is a finite type quasi-coherent $\\mathcal{O}_X$-module, see", "Cohomology, Lemmas \\ref{cohomology-lemma-perfect} and", "\\ref{cohomology-lemma-finite-cohomology}. Then we can find a coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ and a map", "$$", "\\mathcal{F} \\longrightarrow", "\\Ker(\\mathcal{F}^b \\to \\mathcal{F}^{b + 1})", "$$", "such that the composition with the projection onto", "$H^b(\\mathcal{F}^\\bullet)$ is surjective.", "Namely, we can write $\\Ker(\\mathcal{F}^b \\to \\mathcal{F}^{b + 1})$", "as the filtered union of its coherent submodules by", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-colimit-finite-type}", "and then one of these will do the job.", "Next, we choose a finite locally free $\\mathcal{O}_X$-module", "$\\mathcal{E}^b$ and a surjection $\\mathcal{E}^b \\to \\mathcal{F}$ using", "the resolution property of $X$. Consider the map of complexes", "$$", "\\alpha : \\mathcal{E}^b[-b] \\to \\mathcal{F}^\\bullet", "$$", "and its cone $C(\\alpha)^\\bullet$, see", "Derived Categories, Definition \\ref{derived-definition-cone}.", "We observe that $C(\\alpha)^\\bullet$ is nonzero only in degrees", "$\\geq a$, has tor amplitude in $[a, b]$ by", "Cohomology, Lemma \\ref{cohomology-lemma-cone-tor-amplitude},", "and has $H^b(C(\\alpha)^\\bullet) = 0$ by construction.", "Thus we actually find that $C(\\alpha)^\\bullet$ has tor amplitude", "in $[a, b - 1]$. Hence the induction hypothesis applies to", "$C(\\alpha)^\\bullet$ and we find a map of complexes", "$$", "(\\mathcal{E}^a \\to \\ldots \\to \\mathcal{E}^{b - 1})", "\\longrightarrow", "C(\\alpha)^\\bullet", "$$", "with properties as stated in the induction hypothesis. Unwinding", "the definition of the cone this gives a commutative diagram", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "\\mathcal{E}^{b - 2} \\ar[r] \\ar[d] &", "\\mathcal{E}^{b - 1} \\ar[r] \\ar[d] &", "0 \\ar[r] \\ar[d] &", "\\ldots \\\\", "\\ldots \\ar[r] &", "\\mathcal{F}^{b - 2} \\ar[r] &", "\\mathcal{F}^{b - 1} \\oplus \\mathcal{E}^b \\ar[r] &", "\\mathcal{F}^b \\ar[r] &", "\\ldots", "}", "$$", "It is clear that we obtain a map of complexes", "$(\\mathcal{E}^a \\to \\ldots \\to \\mathcal{E}^b) \\to \\mathcal{F}^\\bullet$.", "We omit the verification that this map is a quasi-isomorphism." ], "refs": [ "cohomology-lemma-perfect", "cohomology-lemma-perfect", "cohomology-lemma-finite-cohomology", "properties-lemma-finite-locally-free", "cohomology-lemma-perfect", "cohomology-lemma-finite-cohomology", "properties-lemma-quasi-coherent-colimit-finite-type", "derived-definition-cone", "cohomology-lemma-cone-tor-amplitude" ], "ref_ids": [ 2224, 2224, 2212, 3014, 2224, 2212, 3020, 1978, 2218 ] } ], "ref_ids": [] }, { "id": 7094, "type": "theorem", "label": "perfect-lemma-resolution-property-perfect-complex", "categories": [ "perfect" ], "title": "perfect-lemma-resolution-property-perfect-complex", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme with the", "resolution property. Then every perfect object of $D(\\mathcal{O}_X)$", "can be represented by a bounded complex of finite locally free", "$\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "Let $E$ be a perfect object of $D(\\mathcal{O}_X)$.", "By Lemma \\ref{lemma-resolution-property-affine-diagonal}", "we see that $X$ has affine diagonal. Hence by", "Proposition \\ref{proposition-quasi-compact-affine-diagonal}", "we can represent $E$ by a complex $\\mathcal{F}^\\bullet$", "of quasi-coherent $\\mathcal{O}_X$-modules.", "Observe that $E$ is in $D^b(\\mathcal{O}_X)$ because", "$X$ is quasi-compact. Hence $\\tau_{\\geq n}\\mathcal{F}^\\bullet$", "is a bounded below complex of quasi-coherent $\\mathcal{O}_X$-modules", "which represents $E$ if $n \\ll 0$. Thus we may apply", "Lemma \\ref{lemma-construct-strictly-perfect} to conclude." ], "refs": [ "perfect-lemma-resolution-property-affine-diagonal", "perfect-proposition-quasi-compact-affine-diagonal", "perfect-lemma-construct-strictly-perfect" ], "ref_ids": [ 7092, 7107, 7093 ] } ], "ref_ids": [] }, { "id": 7095, "type": "theorem", "label": "perfect-lemma-resolution-property-map-perfect-complex", "categories": [ "perfect" ], "title": "perfect-lemma-resolution-property-map-perfect-complex", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme with the", "resolution property. Let $\\mathcal{E}^\\bullet$ and $\\mathcal{F}^\\bullet$", "be finite complexes of finite locally free $\\mathcal{O}_X$-modules.", "Then any", "$\\alpha \\in \\Hom_{D(\\mathcal{O}_X)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$", "can be represented by a diagram", "$$", "\\mathcal{E}^\\bullet \\leftarrow \\mathcal{G}^\\bullet \\to \\mathcal{F}^\\bullet", "$$", "where $\\mathcal{G}^\\bullet$ is a bounded complex of finite locally free", "$\\mathcal{O}_X$-modules and where $\\mathcal{G}^\\bullet \\to \\mathcal{E}^\\bullet$", "is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-resolution-property-affine-diagonal}", "we see that $X$ has affine diagonal. Hence by", "Proposition \\ref{proposition-quasi-compact-affine-diagonal}", "we can represent $\\alpha$ by a diagram", "$$", "\\mathcal{E}^\\bullet \\leftarrow \\mathcal{H}^\\bullet \\to \\mathcal{F}^\\bullet", "$$", "where $\\mathcal{H}^\\bullet$ is a complex of quasi-coherent", "$\\mathcal{O}_X$-modules and where", "$\\mathcal{H}^\\bullet \\to \\mathcal{E}^\\bullet$", "is a quasi-isomorphism. For $n \\ll 0$ the maps", "$\\mathcal{H}^\\bullet \\to \\mathcal{E}^\\bullet$ and", "$\\mathcal{H}^\\bullet \\to \\mathcal{F}^\\bullet$", "factor through the quasi-isomorphism", "$\\mathcal{H}^\\bullet \\to \\tau_{\\geq n}\\mathcal{H}^\\bullet$", "simply because $\\mathcal{E}^\\bullet$ and $\\mathcal{F}^\\bullet$", "are bounded complexes. Thus we may replace $\\mathcal{H}^\\bullet$", "by $\\tau_{\\geq n}\\mathcal{H}^\\bullet$ and assume that $\\mathcal{H}^\\bullet$", "is bounded below.", "Then we may apply", "Lemma \\ref{lemma-construct-strictly-perfect} to conclude." ], "refs": [ "perfect-lemma-resolution-property-affine-diagonal", "perfect-proposition-quasi-compact-affine-diagonal", "perfect-lemma-construct-strictly-perfect" ], "ref_ids": [ 7092, 7107, 7093 ] } ], "ref_ids": [] }, { "id": 7096, "type": "theorem", "label": "perfect-lemma-resolution-property-homotopy-map-perfect-complex", "categories": [ "perfect" ], "title": "perfect-lemma-resolution-property-homotopy-map-perfect-complex", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme with the", "resolution property. Let $\\mathcal{E}^\\bullet$ and $\\mathcal{F}^\\bullet$", "be finite complexes of finite locally free $\\mathcal{O}_X$-modules.", "Let $\\alpha^\\bullet, \\beta^\\bullet :\\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$", "be two maps of complexes defining the same map in $D(\\mathcal{O}_X)$.", "Then there exists a quasi-isomorphism", "$\\gamma^\\bullet : \\mathcal{G}^\\bullet \\to \\mathcal{E}^\\bullet$", "where $\\mathcal{G}^\\bullet$ is a bounded complex of finite locally free", "$\\mathcal{O}_X$-modules", "such that $\\alpha^\\bullet \\circ \\gamma^\\bullet$ and", "$\\beta^\\bullet \\circ \\gamma^\\bullet$ are homotopic maps of complexes." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-resolution-property-affine-diagonal}", "we see that $X$ has affine diagonal. Hence by", "Proposition \\ref{proposition-quasi-compact-affine-diagonal}", "(and the definition of the derived category)", "there exists a quasi-isomorphism", "$\\gamma^\\bullet : \\mathcal{G}^\\bullet \\to \\mathcal{E}^\\bullet$", "where $\\mathcal{G}^\\bullet$ is a complex of quasi-coherent", "$\\mathcal{O}_X$-modules", "such that $\\alpha^\\bullet \\circ \\gamma^\\bullet$ and", "$\\beta^\\bullet \\circ \\gamma^\\bullet$ are homotopic maps of complexes.", "Choose a homotopy $h^i : \\mathcal{G}^i \\to \\mathcal{F}^{i - 1}$", "witnessing this fact. Choose $n \\ll 0$. Then the map", "$\\gamma^\\bullet$ factors canonically over the quotient", "map $\\mathcal{G}^\\bullet \\to \\tau_{\\geq n}\\mathcal{G}^\\bullet$", "as $\\mathcal{E}^\\bullet$ is bounded below. For the exact same", "reason the maps $h^i$ will factor over the surjections", "$\\mathcal{G}^i \\to (\\tau_{\\geq n}\\mathcal{G})^i$.", "Hence we see that we may replace $\\mathcal{G}^\\bullet$", "by $\\tau_{\\geq n}\\mathcal{G}^\\bullet$.", "Then we may apply Lemma \\ref{lemma-construct-strictly-perfect} to conclude." ], "refs": [ "perfect-lemma-resolution-property-affine-diagonal", "perfect-proposition-quasi-compact-affine-diagonal", "perfect-lemma-construct-strictly-perfect" ], "ref_ids": [ 7092, 7107, 7093 ] } ], "ref_ids": [] }, { "id": 7097, "type": "theorem", "label": "perfect-lemma-Noetherian-Kprime", "categories": [ "perfect" ], "title": "perfect-lemma-Noetherian-Kprime", "contents": [ "Let $X$ be a Noetherian scheme. Then", "$$", "K_0(\\textit{Coh}(\\mathcal{O}_X)) =", "K_0(D^b(\\textit{Coh}(\\mathcal{O}_X)) =", "K_0(D^b_{\\textit{Coh}}(\\mathcal{O}_X))", "$$" ], "refs": [], "proofs": [ { "contents": [ "The first equality is", "Derived Categories, Lemma \\ref{derived-lemma-K-bounded-derived}.", "The second equality holds because", "$D^b(\\textit{Coh}(\\mathcal{O}_X)) = D^b_{\\textit{Coh}}(\\mathcal{O}_X)$", "by Proposition \\ref{proposition-DCoh}." ], "refs": [ "derived-lemma-K-bounded-derived", "perfect-proposition-DCoh" ], "ref_ids": [ 1898, 7110 ] } ], "ref_ids": [] }, { "id": 7098, "type": "theorem", "label": "perfect-lemma-K-agrees-affine", "categories": [ "perfect" ], "title": "perfect-lemma-K-agrees-affine", "contents": [ "Let $X = \\Spec(R)$ be an affine scheme. Then $K_0(X) = K_0(R)$", "and if $R$ is Noetherian then $K'_0(X) = K'_0(R)$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $K'_0(R)$ and $K_0(R)$ have been defined in", "Algebra, Section \\ref{algebra-section-K-groups}.", "\\medskip\\noindent", "By More on Algebra, Lemma \\ref{more-algebra-lemma-perfect-to-K-group-universal}", "we have $K_0(R) = K_0(D_{perf}(R))$.", "By Lemmas \\ref{lemma-perfect-affine} and \\ref{lemma-affine-compare-bounded}", "we have $D_{perf}(R) = D_{perf}(\\mathcal{O}_X)$.", "This proves the equality $K_0(R) = K_0(X)$.", "\\medskip\\noindent", "The equality $K'_0(R) = K'_0(X)$ holds because", "$\\textit{Coh}(\\mathcal{O}_X)$ is equivalent to the category", "of finite $R$-modules by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-coherent-Noetherian}. Moreover it is", "clear that $K'_0(R)$ is the zeroth K-group of the category", "of finite $R$-modules from the definitions." ], "refs": [ "more-algebra-lemma-perfect-to-K-group-universal", "perfect-lemma-perfect-affine", "perfect-lemma-affine-compare-bounded", "coherent-lemma-coherent-Noetherian" ], "ref_ids": [ 10542, 6980, 6941, 3308 ] } ], "ref_ids": [] }, { "id": 7099, "type": "theorem", "label": "perfect-lemma-Kprime-K", "categories": [ "perfect" ], "title": "perfect-lemma-Kprime-K", "contents": [ "Let $X$ be a Noetherian regular scheme of finite dimension. Then", "the map $K_0(X) \\to K'_0(X)$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-perfect-on-regular}", "and our construction of the map $K_0(X) \\to K'_0(X)$ above." ], "refs": [ "perfect-lemma-perfect-on-regular" ], "ref_ids": [ 6989 ] } ], "ref_ids": [] }, { "id": 7100, "type": "theorem", "label": "perfect-lemma-K-is-old-K", "categories": [ "perfect" ], "title": "perfect-lemma-K-is-old-K", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme with the", "resolution property. Then the map $K_0(\\textit{Vect}(X)) \\to K_0(X)$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This lemma will follow in a straightforward manner from", "Lemmas \\ref{lemma-resolution-property-perfect-complex},", "\\ref{lemma-resolution-property-map-perfect-complex}, and", "\\ref{lemma-resolution-property-homotopy-map-perfect-complex}", "whose results we will use without further mention.", "Let us construct an inverse map", "$$", "c : K_0(X) = K_0(D_{perf}(\\mathcal{O}_X)) \\longrightarrow", "K_0(\\textit{Vect}(X))", "$$", "Namely, any object of $D_{perf}(\\mathcal{O}_X)$ can be represented", "by a bounded complex $\\mathcal{E}^\\bullet$ of finite locally free", "$\\mathcal{O}_X$-modules. Then we set", "$$", "c([\\mathcal{E}^\\bullet]) = \\sum (-1)^i[\\mathcal{E}^i]", "$$", "Of course we have to show that this is well defined. For the moment", "we view $c$ as a map defined on bounded complexes of finite locally free", "$\\mathcal{O}_X$-modules.", "\\medskip\\noindent", "Suppose that $\\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$ is a surjective map", "of bounded complexes of finite locally free $\\mathcal{O}_X$-modules.", "Let $\\mathcal{K}^\\bullet$ be the kernel. Then we obtain short exact", "sequences of $\\mathcal{O}_X$-modules", "$$", "0 \\to \\mathcal{K}^n \\to \\mathcal{E}^n \\to \\mathcal{F}^n \\to 0", "$$", "which are locally split because $\\mathcal{F}^n$ is finite locally free.", "Hence $\\mathcal{K}^\\bullet$ is also a bounded complex of finite", "locally free $\\mathcal{O}_X$-modules and we have", "$c(\\mathcal{E}^\\bullet) = c(\\mathcal{K}^\\bullet) + c(\\mathcal{F}^\\bullet)$", "in $K_0(\\textit{Vect}(X))$.", "\\medskip\\noindent", "Suppose given a bounded complex $\\mathcal{E}^\\bullet$", "of finite locally free $\\mathcal{O}_X$-modules which is acyclic.", "Say $\\mathcal{E}^n = 0$ for $n \\not \\in [a, b]$. Then we", "can break $\\mathcal{E}^\\bullet$ into short exact sequences", "$$", "\\begin{matrix}", "0 \\to \\mathcal{E}^a \\to \\mathcal{E}^{a + 1} \\to \\mathcal{F}^{a + 1} \\to 0,\\\\", "0 \\to \\mathcal{F}^{a + 1} \\to \\mathcal{E}^{a + 2} \\to", "\\mathcal{F}^{a + 3} \\to 0, \\\\", "\\ldots \\\\", "0 \\to \\mathcal{F}^{b - 3} \\to \\mathcal{E}^{b - 2} \\to", "\\mathcal{F}^{b - 2} \\to 0, \\\\", "0 \\to \\mathcal{F}^{b - 2} \\to \\mathcal{E}^{b - 1} \\to \\mathcal{E}^b \\to 0", "\\end{matrix}", "$$", "Arguing by descending induction we see that", "$\\mathcal{F}^{b - 2}, \\ldots, \\mathcal{F}^{a + 1}$", "are finite locally free $\\mathcal{O}_X$-modules, and", "$$", "c(\\mathcal{E}^\\bullet) = \\sum (-1)[\\mathcal{E}^n] =", "\\sum (-1)^n([\\mathcal{F}^{n - 1}] + [\\mathcal{F}^n]) = 0", "$$", "Thus our construction gives zero on acyclic complexes.", "\\medskip\\noindent", "It follows from the results of the preceding two paragraphs that $c$", "is well defined. Namely, suppose the bounded complexes", "$\\mathcal{E}^\\bullet$ and $\\mathcal{F}^\\bullet$ of finite locally free", "$\\mathcal{O}_X$-modules represent the same object of $D(\\mathcal{O}_X)$.", "Then we can find quasi-isomorphisms", "$a : \\mathcal{G}^\\bullet \\to \\mathcal{E}^\\bullet$ and", "$b : \\mathcal{G}^\\bullet \\to \\mathcal{F}^\\bullet$", "with $\\mathcal{G}^\\bullet$ bounded complex of finite locally free", "$\\mathcal{O}_X$-modules.", "We obtain a short exact sequence of complexes", "$$", "0 \\to \\mathcal{E}^\\bullet \\to C(a)^\\bullet \\to \\mathcal{G}^\\bullet[1] \\to 0", "$$", "see Derived Categories, Definition \\ref{derived-definition-cone}.", "Since $a$ is a quasi-isomorphism, the cone $C(a)^\\bullet$ is", "acyclic (this follows for example from the discussion in", "Derived Categories, Section \\ref{derived-section-canonical-delta-functor}).", "Hence", "$$", "0 = c(C(f)^\\bullet) = c(\\mathcal{E}^\\bullet) + c(\\mathcal{G}^\\bullet[1]) =", "c(\\mathcal{E}^\\bullet) - c(\\mathcal{G}^\\bullet)", "$$", "as desired. The same argument using $b$ shows that", "$0 = c(\\mathcal{F}^\\bullet) - c(\\mathcal{G}^\\bullet)$.", "Hence we find that $c(\\mathcal{E}^\\bullet) = c(\\mathcal{F}^\\bullet)$", "and $c$ is well defined.", "\\medskip\\noindent", "A similar argument using the cone on a map", "$\\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$", "of bounded complexes of finite locally free $\\mathcal{O}_X$-modules", "shows that $c(Y) = c(X) + c(Z)$ if $X \\to Y \\to Z$ is a distinguished triangle", "in $D_{perf}(\\mathcal{O}_X)$. Details omitted.", "Thus we get the desired homomorphism", "of abelian groups $c : K_0(X) \\to K_0(\\textit{Vect}(X))$.", "\\medskip\\noindent", "It is clear that the composition", "$K_0(\\textit{Vect}(X)) \\to K_0(X) \\to K_0(\\textit{Vect}(X))$", "is the identity. On the other hand, let $\\mathcal{E}^\\bullet$", "be a bounded complex of finite locally free $\\mathcal{O}_X$-modules.", "Then the the existence of the distinguished triangles", "of ``stupid truncations''", "(see Homology, Section \\ref{homology-section-truncations})", "$$", "\\sigma_{\\geq n}\\mathcal{E}^\\bullet \\to", "\\sigma_{\\geq n - 1}\\mathcal{E}^\\bullet \\to", "\\mathcal{E}^{n - 1}[-n + 1] \\to", "(\\sigma_{\\geq n}\\mathcal{E}^\\bullet)[1]", "$$", "and induction show that", "$$", "[\\mathcal{E}^\\bullet] = \\sum (-1)^i[\\mathcal{E}^i[0]]", "$$", "in $K_0(X) = K_0(D_{perf}(\\mathcal{O}_X))$ with apologies for the notation.", "Hence the map $K_0(\\textit{Vect}(X)) \\to K_0(D_{perf}(\\mathcal{O}_X)) = K_0(X)$", "is surjective which finishes the proof." ], "refs": [ "perfect-lemma-resolution-property-perfect-complex", "perfect-lemma-resolution-property-map-perfect-complex", "perfect-lemma-resolution-property-homotopy-map-perfect-complex", "derived-definition-cone" ], "ref_ids": [ 7094, 7095, 7096, 1978 ] } ], "ref_ids": [] }, { "id": 7101, "type": "theorem", "label": "perfect-lemma-projection-formula", "categories": [ "perfect" ], "title": "perfect-lemma-projection-formula", "contents": [ "Let $f : X \\to Y$ be a proper morphism of locally Noetherian schemes.", "Then we have $f_*(\\alpha \\cdot f^*\\beta) = f_*\\alpha \\cdot \\beta$", "for $\\alpha \\in K'_0(X)$ and $\\beta \\in K_0(Y)$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-cohomology-base-change}, the discussion in", "Remark \\ref{remark-pushforward-K}, and the definition of the product", "$K'_0(X) \\times K_0(X) \\to K'_0(X)$ in Remark \\ref{remark-K-ring}." ], "refs": [ "perfect-lemma-cohomology-base-change", "perfect-remark-pushforward-K", "perfect-remark-K-ring" ], "ref_ids": [ 7025, 7141, 7140 ] } ], "ref_ids": [] }, { "id": 7102, "type": "theorem", "label": "perfect-lemma-determinant-two-term-complexes", "categories": [ "perfect" ], "title": "perfect-lemma-determinant-two-term-complexes", "contents": [ "Let $X$ be a scheme. There is a functor", "$$", "\\det :", "\\left\\{", "\\begin{matrix}", "\\text{category of perfect complexes} \\\\", "\\text{with tor amplitude in }[-1, 0] \\\\", "\\text{morphisms are isomorphisms}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{category of invertible modules} \\\\", "\\text{morphisms are isomorphisms}", "\\end{matrix}", "\\right\\}", "$$", "In addition, given a rank $0$ perfect object $L$ of $D(\\mathcal{O}_X)$ with", "tor-amplitude in $[-1, 0]$ there is a canonical element", "$\\delta(L) \\in \\Gamma(X, \\det(L))$ such that for any isomorphism", "$a : L \\to K$ in $D(\\mathcal{O}_X)$ we have $\\det(a)(\\delta(L)) = \\delta(K)$.", "Moreover, the construction is affine locally given by the construction", "of More on Algebra, Section \\ref{more-algebra-section-determinants-complexes}." ], "refs": [], "proofs": [ { "contents": [ "Let $L$ be an object of the left hand side. If $\\Spec(A) = U \\subset X$", "is an affine open, then $L|_U$ corresponds to a perfect complex $L^\\bullet$", "of $A$-modules with tor-amplitude in $[-1, 0]$, see", "Lemmas \\ref{lemma-affine-compare-bounded},", "\\ref{lemma-tor-dimension-affine}, and", "\\ref{lemma-perfect-affine}.", "Then we can consider the invertible $A$-module $\\det(L^\\bullet)$ constructed in", "More on Algebra, Lemma \\ref{more-algebra-lemma-determinant-two-term-complexes}.", "If $\\Spec(B) = V \\subset U$ is another affine open contained in $U$,", "then $\\det(L^\\bullet) \\otimes_A B = \\det(L^\\bullet \\otimes_A B)$", "and hence this construction is compatible with restriction mappings", "(see Lemma \\ref{lemma-quasi-coherence-pullback} and note $A \\to B$ is flat).", "Thus we can glue these invertible modules to obtain an invertible module", "$\\det(L)$ on $X$. The functoriality and canonical sections", "are constructed in exactly the same manner. Details omitted." ], "refs": [ "perfect-lemma-affine-compare-bounded", "perfect-lemma-tor-dimension-affine", "perfect-lemma-perfect-affine", "more-algebra-lemma-determinant-two-term-complexes", "perfect-lemma-quasi-coherence-pullback" ], "ref_ids": [ 6941, 6977, 6980, 10549, 6944 ] } ], "ref_ids": [] }, { "id": 7103, "type": "theorem", "label": "perfect-lemma-orthogonal-koszul-first-variant", "categories": [ "perfect" ], "title": "perfect-lemma-orthogonal-koszul-first-variant", "contents": [ "In Situation \\ref{situation-complex} denote $j : U \\to X$ the open", "immersion and let $K$ be the perfect object of $D(\\mathcal{O}_X)$", "corresponding to the Koszul complex on $f_1, \\ldots, f_r$ over $A$.", "Let $E \\in D_\\QCoh(\\mathcal{O}_X)$ and $a \\in \\mathbf{Z}$.", "Consider the following conditions", "\\begin{enumerate}", "\\item The canonical map $\\tau_{\\geq a}E \\to \\tau_{\\geq a} Rj_*(E|_U)$", "is an isomorphism.", "\\item We have $\\Hom_{D(\\mathcal{O}_X)}(K[-n], E) = 0$ for all $n \\geq a$.", "\\end{enumerate}", "Then (2) implies (1) and (1) implies (2) with $a$ replaced by $a + 1$." ], "refs": [], "proofs": [ { "contents": [ "Choose a distinguished triangle $N \\to E \\to Rj_*(E|_U) \\to N[1]$.", "Then (1) implies $\\tau_{\\geq a + 1} N = 0$ and (1) is implied by", "$\\tau_{\\geq a}N = 0$. Observe that", "$$", "\\Hom_{D(\\mathcal{O}_X)}(K[-n], Rj_*(E|_U)) =", "\\Hom_{D(\\mathcal{O}_U)}(K|_U[-n], E) = 0", "$$", "for all $n$ as $K|_U = 0$. Thus (2) is equivalent to", "$\\Hom_{D(\\mathcal{O}_X)}(K[-n], N) = 0$ for all $n \\geq a$. ", "Observe that there are distinguished triangles", "$$", "K^\\bullet(f_1^{e_1}, \\ldots, f_i^{e'_i}, \\ldots, f_r^{e_r}) \\to", "K^\\bullet(f_1^{e_1}, \\ldots, f_i^{e'_i + e''_i}, \\ldots, f_r^{e_r}) \\to", "K^\\bullet(f_1^{e_1}, \\ldots, f_i^{e''_i}, \\ldots, f_r^{e_r}) \\to \\ldots", "$$", "of Koszul complexes, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-koszul-mult}. Hence", "$\\Hom_{D(\\mathcal{O}_X)}(K[-n], N) = 0$ for all $n \\geq a$", "is equivalent to", "$\\Hom_{D(\\mathcal{O}_X)}(K_e[-n], N) = 0$ for all $n \\geq a$ and", "all $e \\geq 1$ with $K_e$ as in", "Lemma \\ref{lemma-represent-cohomology-class-on-closed}.", "Since $N|_U = 0$, that lemma implies that this in turn is equivalent to", "$H^n(X, N) = 0$ for $n \\geq a$. We conclude that (2) is equivalent", "to $\\tau_{\\geq a}N = 0$ since $N$ is determined by the complex of", "$A$-modules $R\\Gamma(X, N)$, see Lemma \\ref{lemma-affine-compare-bounded}.", "Thus we find that our lemma is true." ], "refs": [ "more-algebra-lemma-koszul-mult", "perfect-lemma-represent-cohomology-class-on-closed", "perfect-lemma-affine-compare-bounded" ], "ref_ids": [ 9965, 6973, 6941 ] } ], "ref_ids": [] }, { "id": 7104, "type": "theorem", "label": "perfect-lemma-orthogonal-koszul-second-variant", "categories": [ "perfect" ], "title": "perfect-lemma-orthogonal-koszul-second-variant", "contents": [ "In Situation \\ref{situation-complex} denote $j : U \\to X$ the open", "immersion and let $K$ be the perfect object of $D(\\mathcal{O}_X)$", "corresponding to the Koszul complex on $f_1, \\ldots, f_r$ over $A$.", "Let $E \\in D_\\QCoh(\\mathcal{O}_X)$ and $a \\in \\mathbf{Z}$. Consider", "the following conditions", "\\begin{enumerate}", "\\item The canonical map $\\tau_{\\leq a}E \\to \\tau_{\\leq a} Rj_*(E|_U)$", "is an isomorphism, and", "\\item $\\Hom_{D(\\mathcal{O}_X)}(K[-n], E) = 0$ for all $n \\leq a$.", "\\end{enumerate}", "Then (2) implies (1) and (1) implies (2) with $a$ replaced by $a - 1$." ], "refs": [], "proofs": [ { "contents": [ "Choose a distinguished triangle $E \\to Rj_*(E|_U) \\to N \\to E[1]$. Then (1)", "implies $\\tau_{\\leq a - 1}N = 0$ and (1) is implied by $\\tau_{\\leq a}N = 0$.", "Observe that", "$$", "\\Hom_{D(\\mathcal{O}_X)}(K[-n], Rj_*(E|_U)) =", "\\Hom_{D(\\mathcal{O}_U)}(K|_U[-n], E) = 0", "$$", "for all $n$ as $K|_U = 0$. Thus (2) is equivalent to", "$\\Hom_{D(\\mathcal{O}_X)}(K[-n], N) = 0$ for all $n \\leq a$. ", "Observe that there are distinguished triangles", "$$", "K^\\bullet(f_1^{e_1}, \\ldots, f_i^{e'_i}, \\ldots, f_r^{e_r}) \\to", "K^\\bullet(f_1^{e_1}, \\ldots, f_i^{e'_i + e''_i}, \\ldots, f_r^{e_r}) \\to", "K^\\bullet(f_1^{e_1}, \\ldots, f_i^{e''_i}, \\ldots, f_r^{e_r}) \\to \\ldots", "$$", "of Koszul complexes, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-koszul-mult}. Hence", "$\\Hom_{D(\\mathcal{O}_X)}(K[-n], N) = 0$ for all $n \\leq a$", "is equivalent to", "$\\Hom_{D(\\mathcal{O}_X)}(K_e[-n], N) = 0$ for all $n \\leq a$ and all $e \\geq 1$", "with $K_e$ as in Lemma \\ref{lemma-represent-cohomology-class-on-closed}.", "Since $N|_U = 0$, that lemma implies that this in turn is equivalent to", "$H^n(X, N) = 0$ for $n \\leq a$. We conclude that (2) is equivalent to", "$\\tau_{\\leq a}N = 0$ since $N$ is determined by the complex of", "$A$-modules $R\\Gamma(X, N)$, see Lemma \\ref{lemma-affine-compare-bounded}.", "Thus we find that our lemma is true." ], "refs": [ "more-algebra-lemma-koszul-mult", "perfect-lemma-represent-cohomology-class-on-closed", "perfect-lemma-affine-compare-bounded" ], "ref_ids": [ 9965, 6973, 6941 ] } ], "ref_ids": [] }, { "id": 7105, "type": "theorem", "label": "perfect-lemma-bounded-truncation", "categories": [ "perfect" ], "title": "perfect-lemma-bounded-truncation", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $P \\in D_{perf}(\\mathcal{O}_X)$ and $E \\in D_{\\QCoh}(\\mathcal{O}_X)$. ", "Let $a \\in \\mathbf{Z}$. The following are equivalent", "\\begin{enumerate}", "\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], E) = 0$ for $i \\gg 0$, and", "\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{\\geq a} E) = 0$ for $i \\gg 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Using the triangle $ \\tau_{< a} E \\to E \\to \\tau_{\\geq a} E \\to$", "we see that the equivalence follows if we can show", "$$", "\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{< a} E) =", "\\Hom_{D(\\mathcal{O}_X)}(P, (\\tau_{< a} E)[i]) = 0 ", "$$", "for $i \\gg 0$. As $P$ is perfect this is true by", "Lemma \\ref{lemma-ext-from-perfect-into-bounded-QCoh}." ], "refs": [ "perfect-lemma-ext-from-perfect-into-bounded-QCoh" ], "ref_ids": [ 7018 ] } ], "ref_ids": [] }, { "id": 7106, "type": "theorem", "label": "perfect-lemma-bounded-below-truncation", "categories": [ "perfect" ], "title": "perfect-lemma-bounded-below-truncation", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Let", "$P \\in D_{perf}(\\mathcal{O}_X)$ and $E \\in D_{\\QCoh}(\\mathcal{O}_X)$.", "Let $a \\in \\mathbf{Z}$. The following are equivalent", "\\begin{enumerate}", "\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], E) = 0$ for $i \\ll 0$, and", "\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{\\leq a} E) = 0$ for $i \\ll 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Using the triangle $ \\tau_{\\leq a} E \\to E \\to \\tau_{> a} E \\to$", "we see that the equivalence follows if we can show", "$$", "\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{> a} E) =", "\\Hom_{D(\\mathcal{O}_X)}(P, (\\tau_{> a} E)[i]) = 0", "$$", "for $i \\ll 0$. As $P$ is perfect this is true by", "Lemma \\ref{lemma-ext-from-perfect-into-bounded-QCoh}." ], "refs": [ "perfect-lemma-ext-from-perfect-into-bounded-QCoh" ], "ref_ids": [ 7018 ] } ], "ref_ids": [] }, { "id": 7107, "type": "theorem", "label": "perfect-proposition-quasi-compact-affine-diagonal", "categories": [ "perfect" ], "title": "perfect-proposition-quasi-compact-affine-diagonal", "contents": [ "Let $X$ be a quasi-compact scheme with affine diagonal.", "Then the functor (\\ref{equation-compare})", "$$", "D(\\QCoh(\\mathcal{O}_X))", "\\longrightarrow", "D_\\QCoh(\\mathcal{O}_X)", "$$", "is an equivalence with quasi-inverse given by $RQ_X$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be an affine open. Then the morphism", "$U \\to X$ is affine by", "Morphisms, Lemma \\ref{morphisms-lemma-affine-permanence}.", "Thus the assumption of Lemma \\ref{lemma-argument-proves}", "holds by Lemma \\ref{lemma-affine-pushforward} and we win." ], "refs": [ "morphisms-lemma-affine-permanence", "perfect-lemma-argument-proves", "perfect-lemma-affine-pushforward" ], "ref_ids": [ 5179, 6966, 6963 ] } ], "ref_ids": [] }, { "id": 7108, "type": "theorem", "label": "perfect-proposition-Noetherian", "categories": [ "perfect" ], "title": "perfect-proposition-Noetherian", "contents": [ "Let $X$ be a Noetherian scheme. Then the functor (\\ref{equation-compare})", "$$", "D(\\QCoh(\\mathcal{O}_X))", "\\longrightarrow", "D_\\QCoh(\\mathcal{O}_X)", "$$", "is an equivalence with quasi-inverse given by $RQ_X$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-argument-proves} and", "Lemma \\ref{lemma-Noetherian-pushforward}." ], "refs": [ "perfect-lemma-argument-proves", "perfect-lemma-Noetherian-pushforward" ], "ref_ids": [ 6966, 6969 ] } ], "ref_ids": [] }, { "id": 7109, "type": "theorem", "label": "perfect-proposition-represent-cohomology-class-on-open", "categories": [ "perfect" ], "title": "perfect-proposition-represent-cohomology-class-on-open", "contents": [ "In Situation \\ref{situation-complex}. For every object $E$", "of $D_\\QCoh(\\mathcal{O}_X)$ the map", "(\\ref{equation-comparison}) is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-affine-compare-bounded} we may assume that $E$", "is given by a complex of quasi-coherent sheaves $\\mathcal{F}^\\bullet$.", "Let $M^\\bullet = \\Gamma(X, \\mathcal{F}^\\bullet)$ be the corresponding", "complex of $A$-modules. By", "Lemmas \\ref{lemma-alternating-cech-complex-complex} and", "\\ref{lemma-alternating-cech-complex-complex-computes-cohomology}", "we have quasi-isomorphisms", "$$", "\\colim_e \\text{Tot}(\\Hom_A(I^\\bullet(f_1^e, \\ldots, f_r^e), M^\\bullet))", "\\longrightarrow", "\\text{Tot}(\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))", "\\longrightarrow", "R\\Gamma(U, \\mathcal{F}^\\bullet)", "$$", "Taking $H^0$ on both sides we obtain", "$$", "\\colim_e \\Hom_{D(A)}(I^\\bullet(f_1^e, \\ldots, f_r^e), M^\\bullet)", "=", "H^0(U, E)", "$$", "Since $\\Hom_{D(A)}(I^\\bullet(f_1^e, \\ldots, f_r^e), M^\\bullet) =", "\\Hom_{D(\\mathcal{O}_X)}(I_e, E)$ by", "Lemma \\ref{lemma-affine-compare-bounded} the lemma follows." ], "refs": [ "perfect-lemma-affine-compare-bounded", "perfect-lemma-alternating-cech-complex-complex", "perfect-lemma-alternating-cech-complex-complex-computes-cohomology", "perfect-lemma-affine-compare-bounded" ], "ref_ids": [ 6941, 6971, 6972, 6941 ] } ], "ref_ids": [] }, { "id": 7110, "type": "theorem", "label": "perfect-proposition-DCoh", "categories": [ "perfect" ], "title": "perfect-proposition-DCoh", "contents": [ "Let $X$ be a Noetherian scheme. Then the functors", "$$", "D^-(\\textit{Coh}(\\mathcal{O}_X))", "\\longrightarrow", "D^-_{\\textit{Coh}}(\\mathcal{O}_X)", "\\quad\\text{and}\\quad", "D^b(\\textit{Coh}(\\mathcal{O}_X))", "\\longrightarrow", "D^b_{\\textit{Coh}}(\\mathcal{O}_X)", "$$", "are equivalences." ], "refs": [], "proofs": [ { "contents": [ "Consider the commutative diagram", "$$", "\\xymatrix{", "D^-(\\textit{Coh}(\\mathcal{O}_X)) \\ar[r] \\ar[d] &", "D^-_{\\textit{Coh}}(\\mathcal{O}_X) \\ar[d] \\\\", "D^-(\\QCoh(\\mathcal{O}_X)) \\ar[r] &", "D^-_\\QCoh(\\mathcal{O}_X)", "}", "$$", "By Lemma \\ref{lemma-coh-to-qcoh} the left vertical arrow is fully faithful.", "By Proposition \\ref{proposition-Noetherian} the bottom arrow is an equivalence.", "By construction the right vertical arrow is fully faithful.", "We conclude that the top horizontal arrow is fully faithful.", "If $K$ is an object of $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$", "then the object $K'$ of $D^-(\\QCoh(\\mathcal{O}_X))$ which corresponds", "to it by Proposition \\ref{proposition-Noetherian} will have", "coherent cohomology sheaves. Hence $K'$ is in the essential", "image of the left vertical arrow by Lemma \\ref{lemma-coh-to-qcoh}", "and we find that the top horizontal arrow is essentially surjective.", "This finishes the proof for the bounded above case. The bounded", "case follows immediately from the bounded above case." ], "refs": [ "perfect-lemma-coh-to-qcoh", "perfect-proposition-Noetherian", "perfect-proposition-Noetherian", "perfect-lemma-coh-to-qcoh" ], "ref_ids": [ 6983, 7108, 7108, 6983 ] } ], "ref_ids": [] }, { "id": 7111, "type": "theorem", "label": "perfect-proposition-compact-is-perfect", "categories": [ "perfect" ], "title": "perfect-proposition-compact-is-perfect", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "An object of $D_\\QCoh(\\mathcal{O}_X)$ is compact", "if and only if it is perfect." ], "refs": [], "proofs": [ { "contents": [ "If $K$ is a perfect object of $D(\\mathcal{O}_X)$ with dual", "$K^\\vee$ (Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex})", "we have", "$$", "\\Hom_{D(\\mathcal{O}_X)}(K, M) =", "H^0(X, K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)", "$$", "functorially in $M$. Since $K^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} -$", "commutes with direct sums and since $H^0(X, -)$ commutes with direct", "sums on $D_\\QCoh(\\mathcal{O}_X)$ by", "Lemma \\ref{lemma-quasi-coherence-pushforward-direct-sums}", "we conclude that $K$ is compact in $D_\\QCoh(\\mathcal{O}_X)$.", "\\medskip\\noindent", "Conversely, let $K$ be a compact object of $D_\\QCoh(\\mathcal{O}_X)$.", "To show that $K$ is perfect, it suffices to show that", "$K|_U$ is perfect for every affine open $U \\subset X$, see", "Cohomology, Lemma \\ref{cohomology-lemma-perfect-independent-representative}.", "Observe that $j : U \\to X$ is a quasi-compact and separated morphism.", "Hence", "$Rj_* : D_\\QCoh(\\mathcal{O}_U) \\to D_\\QCoh(\\mathcal{O}_X)$", "commutes with direct sums, see", "Lemma \\ref{lemma-quasi-coherence-pushforward-direct-sums}.", "Thus the adjointness of restriction to $U$ and $Rj_*$ implies that", "$K|_U$ is a compact object of $D_\\QCoh(\\mathcal{O}_U)$.", "Hence we reduce to the case that $X$ is affine.", "\\medskip\\noindent", "Assume $X = \\Spec(A)$ is affine. By Lemma \\ref{lemma-affine-compare-bounded}", "the problem is translated into the same problem for $D(A)$.", "For $D(A)$ the result is", "More on Algebra, Proposition \\ref{more-algebra-proposition-perfect-is-compact}." ], "refs": [ "cohomology-lemma-dual-perfect-complex", "perfect-lemma-quasi-coherence-pushforward-direct-sums", "cohomology-lemma-perfect-independent-representative", "perfect-lemma-quasi-coherence-pushforward-direct-sums", "perfect-lemma-affine-compare-bounded", "more-algebra-proposition-perfect-is-compact" ], "ref_ids": [ 2233, 6950, 2221, 6950, 6941, 10585 ] } ], "ref_ids": [] }, { "id": 7112, "type": "theorem", "label": "perfect-proposition-perfect-resolution-property", "categories": [ "perfect" ], "title": "perfect-proposition-perfect-resolution-property", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme with the", "resolution property. Denote", "\\begin{enumerate}", "\\item $\\mathcal{A}$ the additive category of finite locally free", "$\\mathcal{O}_X$-modules,", "\\item $K^b(\\mathcal{A})$ the homotopy category of bounded complexes", "in $\\mathcal{A}$, see", "Derived Categories, Section \\ref{derived-section-homotopy}, and", "\\item $D_{perf}(\\mathcal{O}_X)$ the strictly full, saturated,", "triangulated subcategory of $D(\\mathcal{O}_X)$ consisting of", "perfect objects.", "\\end{enumerate}", "With this notation the obvious functor", "$$", "K^b(\\mathcal{A}) \\longrightarrow D_{perf}(\\mathcal{O}_X)", "$$", "is an exact functor of trianglated categories which factors through an", "equivalence $S^{-1}K^b(\\mathcal{A}) \\to D_{perf}(\\mathcal{O}_X)$", "of triangulated categories", "where $S$ is the saturated multiplicative system of quasi-isomorphisms", "in $K^b(\\mathcal{A})$." ], "refs": [], "proofs": [ { "contents": [ "If you can parse the statement of the proposition, then please skip this", "first paragraph. For some of the definitions used, please see", "Derived Categories, Definition", "\\ref{derived-definition-triangulated-subcategory}", "(triangulated subcategory),", "Derived Categories, Definition \\ref{derived-definition-saturated}", "(saturated triangulated subcategory),", "Derived Categories, Definition \\ref{derived-definition-localization}", "(multiplicative system compatible with the triangulated structure), and", "Categories,", "Definition \\ref{categories-definition-saturated-multiplicative-system}", "(saturated multiplicative system).", "Observe that $D_{perf}(\\mathcal{O}_X)$ is a saturated triangulated subcategory", "of $D(\\mathcal{O}_X)$ by", "Cohomology, Lemmas \\ref{cohomology-lemma-two-out-of-three-perfect} and", "\\ref{cohomology-lemma-summands-perfect}. Also, note that", "$K^b(\\mathcal{A})$ is a triangulated category, see", "Derived Categories, Lemma", "\\ref{derived-lemma-bounded-triangulated-subcategories}.", "\\medskip\\noindent", "It is clear that the functor sends distinguished triangles to", "distinguished triangles, i.e., is exact. Then $S$ is a saturated", "multiplicative system compatible with the triangulated structure", "on $K^b(\\mathcal{A})$ by", "Derived Categories, Lemma \\ref{derived-lemma-triangle-functor-localize}.", "Hence the localization $S^{-1}K^b(\\mathcal{A})$ exists and is", "a triangulated category by", "Derived Categories, Proposition", "\\ref{derived-proposition-construct-localization}.", "We get an exact factorization", "$S^{-1}K^b(\\mathcal{A}) \\to D_{perf}(\\mathcal{O}_X)$ by", "Derived Categories, Lemma", "\\ref{derived-lemma-universal-property-localization}.", "By Lemmas \\ref{lemma-resolution-property-perfect-complex},", "\\ref{lemma-resolution-property-map-perfect-complex}, and", "\\ref{lemma-resolution-property-homotopy-map-perfect-complex}", "this functor is an equivalence. Then finally the functor", "$S^{-1}K^b(\\mathcal{A}) \\to D_{perf}(\\mathcal{O}_X)$", "is an equivalence of triangulated categories (in the sense that", "distinguished triangles correspond) by", "Derived Categories, Lemma \\ref{derived-lemma-exact-equivalence}." ], "refs": [ "derived-definition-triangulated-subcategory", "derived-definition-saturated", "derived-definition-localization", "categories-definition-saturated-multiplicative-system", "cohomology-lemma-two-out-of-three-perfect", "cohomology-lemma-summands-perfect", "derived-lemma-bounded-triangulated-subcategories", "derived-lemma-triangle-functor-localize", "derived-proposition-construct-localization", "derived-lemma-universal-property-localization", "perfect-lemma-resolution-property-perfect-complex", "perfect-lemma-resolution-property-map-perfect-complex", "perfect-lemma-resolution-property-homotopy-map-perfect-complex", "derived-lemma-exact-equivalence" ], "ref_ids": [ 1970, 1974, 1973, 12376, 2226, 2228, 1807, 1779, 1959, 1781, 7094, 7095, 7096, 1773 ] } ], "ref_ids": [] }, { "id": 7113, "type": "theorem", "label": "perfect-proposition-detecting-bounded-above", "categories": [ "perfect" ], "title": "perfect-proposition-detecting-bounded-above", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Let", "$G \\in D_{perf}(\\mathcal{O}_X)$ be a perfect complex which generates ", "$D_\\QCoh (\\mathcal{O}_X)$. Let $E \\in D_\\QCoh (\\mathcal{O}_X)$.", "The following are equivalent", "\\begin{enumerate}", "\\item $E \\in D^-_\\QCoh (\\mathcal{O}_X)$,", "\\item $\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = 0$ for $i \\gg 0$,", "\\item $\\Ext^i_X(G, E) = 0$ for $i \\gg 0$,", "\\item $R\\Hom_X(G, E)$ is in $D^-(\\mathbf{Z})$,", "\\item $H^i(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$", "for $i \\gg 0$,", "\\item $R\\Gamma(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$", "is in $D^-(\\mathbf{Z})$,", "\\item for every perfect object $P$ of $D(\\mathcal{O}_X)$", "\\begin{enumerate}", "\\item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and", "\\item $H^i(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\gg 0$,", "\\item $R\\Gamma(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$", "is in $D^-(\\mathbf{Z})$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Since", "$\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = \\Hom_{D(\\mathcal{O}_X)}(G, E[i])$", "we see that this is zero for $i \\gg 0$ by", "Lemma \\ref{lemma-ext-from-perfect-into-bounded-QCoh}. This proves", "that (1) implies (2).", "\\medskip\\noindent", "Parts (2), (3), (4) are equivalent by the discussion in", "Cohomology, Section \\ref{cohomology-section-global-RHom}.", "Part (5) and (6) are equivalent as $H^i(X, -) = H^i(R\\Gamma(X, -))$", "by definition. The equivalent conditions (2), (3), (4) are", "equivalent to the equivalent conditions (5), (6) by", "Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}", "and the fact that $(G[-i])^\\vee = G^\\vee[i]$.", "\\medskip\\noindent", "It is clear that (7) implies (2). Conversely, ", "let us prove that the equivalent conditions (2) -- (6) imply (7).", "Recall that $G$ is a classical generator for $D_{perf}(\\mathcal{O}_X)$ by", "Remark \\ref{remark-classical-generator}.", "For $P \\in D_{perf}(\\mathcal{O}_X)$ let $T(P)$ be the assertion that", "$R\\Hom_X(P, E)$ is in $D^-(\\mathbf{Z})$.", "Clearly, $T$ is inherited by direct sums,", "satisfies the 2-out-of-three property for distinguished", "triangles, is inherited by direct summands, and is perserved by shifts.", "Hence by Derived Categories, Remark \\ref{derived-remark-check-on-generator}", "we see that (4) implies $T$ holds on all of $D_{perf}(\\mathcal{O}_X)$.", "The same argument works for all other properties, except that for property", "(7)(b) and (7)(c) we also use that $P \\mapsto P^\\vee$ is a self", "equivalence of $D_{perf}(\\mathcal{O}_X)$. Small detail omitted.", "\\medskip\\noindent", "We will prove the equivalent conditions (2) -- (7) imply (1)", "using the induction principle of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}.", "\\medskip\\noindent", "First, we prove (2) -- (7) $\\Rightarrow$ (1) if $X$ is affine.", "Set $P = \\mathcal{O}_X[0]$. From (7) we obtain $H^i (X, E) = 0$ for $i \\gg 0$.", "Hence (1) follows since $E$ is determined by $R\\Gamma (X, E)$,", "see Lemma \\ref{lemma-affine-compare-bounded}.", "\\medskip\\noindent", "Now assume $X = U \\cup V$ with $U$ a quasi-compact open of $X$ and", "$V$ an affine open, and assume the implication (2) -- (7) $\\Rightarrow$ (1)", "is known for the schemes $U$, $V$, and $U \\cap V$.", "Suppose $E \\in D_\\QCoh(\\mathcal{O}_X)$ satisfies (2) -- (7).", "By Lemma \\ref{lemma-direct-summand-of-a-restriction} and", "Theorem \\ref{theorem-bondal-van-den-Bergh} there exists a perfect", "complex $Q$ on $X$ such that $Q|_U$ generates $D_\\QCoh (\\mathcal{O}_U)$. ", "Let $f_1, \\dots , f_r \\in \\Gamma (V, \\mathcal{O}_V)$", "be such that $V \\setminus U = V(f_1, \\dots , f_r)$ as subsets of", "$V$. Let $K \\in D_{perf}(\\mathcal{O}_V)$ be the object", "corresponding to the Koszul complex on $f_1, \\dots , f_r$.", "Let $K' \\in D_{perf}(\\mathcal{O}_X)$ be", "\\begin{equation}", "\\label{equation-detecting-bounded-above}", "K' = R (V \\to X)_* K = R (V \\to X)_! K,", "\\end{equation}", "see Cohomology, Lemmas \\ref{cohomology-lemma-pushforward-restriction} and", "\\ref{cohomology-lemma-pushforward-perfect}. This is a perfect", "complex on $X$ supported on the closed set $X \\setminus U \\subset V$", "and isomorphic to $K$ on $V$. By assumption, we know", "$R\\Hom_{\\mathcal{O}_X}(Q, E)$ and", "$R\\Hom_{\\mathcal{O}_X}(K', E)$ are bounded above.", "\\medskip\\noindent", "By the second description of $K'$ in (\\ref{equation-detecting-bounded-above})", "we have", "$$", "\\Hom_{D(\\mathcal{O}_V)}(K[-i], E|_V) = \\Hom_{D(\\mathcal{O}_X)}(K'[-i], E) = 0", "$$", "for $i \\gg 0$. Therefore, we may apply", "Lemma \\ref{lemma-orthogonal-koszul-first-variant} to $E|_V$ to", "obtain an integer $a$ such that", "$\\tau_{\\geq a}(E|_V) = \\tau_{\\geq a} R (U \\cap V \\to V)_* (E|_{U \\cap V})$.", "Then $\\tau_{\\geq a} E = \\tau_{\\geq a} R (U \\to X)_* (E |_U)$", "(check that the canonical map is an isomorphism after restricting to", "$U$ and to $V$). Hence using Lemma \\ref{lemma-bounded-truncation}", "twice we see that", "$$", "\\Hom_{D(\\mathcal{O}_U)}(Q|_U [-i], E|_U) =", "\\Hom_{D(\\mathcal{O}_X)}(Q[-i], R (U \\to X)_* (E|_U)) = 0", "$$", "for $i \\gg 0$. Since the Proposition holds for $U$ and the generator", "$Q|_U$, we have $E|_U \\in D^-_\\QCoh(\\mathcal{O}_U)$. But then since", "the functor $R (U \\to X)_*$ preserves $D^-_\\QCoh$ ", "(by Lemma \\ref{lemma-quasi-coherence-direct-image}), we get", "$\\tau_{\\geq a}E \\in D^-_\\QCoh(\\mathcal{O}_X)$. Thus ", "$E \\in D^-_\\QCoh (\\mathcal{O}_X)$." ], "refs": [ "perfect-lemma-ext-from-perfect-into-bounded-QCoh", "cohomology-lemma-dual-perfect-complex", "perfect-remark-classical-generator", "derived-remark-check-on-generator", "coherent-lemma-induction-principle", "perfect-lemma-affine-compare-bounded", "perfect-lemma-direct-summand-of-a-restriction", "perfect-theorem-bondal-van-den-Bergh", "cohomology-lemma-pushforward-restriction", "cohomology-lemma-pushforward-perfect", "perfect-lemma-orthogonal-koszul-first-variant", "perfect-lemma-bounded-truncation", "perfect-lemma-quasi-coherence-direct-image" ], "ref_ids": [ 7018, 2233, 7129, 2030, 3291, 6941, 7009, 6935, 2148, 2229, 7103, 7105, 6946 ] } ], "ref_ids": [] }, { "id": 7114, "type": "theorem", "label": "perfect-proposition-detecting-bounded-below", "categories": [ "perfect" ], "title": "perfect-proposition-detecting-bounded-below", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $G \\in D_{perf}(\\mathcal{O}_X)$", "be a perfect complex which generates $D_\\QCoh (\\mathcal{O}_X)$. Let", "$E \\in D_\\QCoh (\\mathcal{O}_X)$. The following are equivalent", "\\begin{enumerate}", "\\item $E \\in D^+_\\QCoh (\\mathcal{O}_X)$,", "\\item $\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = 0$ for $i \\ll 0$,", "\\item $\\Ext^i_X(G, E) = 0$ for $i \\ll 0$,", "\\item $R\\Hom_X(G, E)$ is in $D^+(\\mathbf{Z})$,", "\\item $H^i(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$", "for $i \\ll 0$,", "\\item $R\\Gamma(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$", "is in $D^+(\\mathbf{Z})$,", "\\item for every perfect object $P$ of $D(\\mathcal{O}_X)$", "\\begin{enumerate}", "\\item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and", "\\item $H^i(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\ll 0$,", "\\item $R\\Gamma(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$", "is in $D^+(\\mathbf{Z})$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Since", "$\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = \\Hom_{D(\\mathcal{O}_X)}(G, E[i])$", "we see that this is zero for $i \\ll 0$ by", "Lemma \\ref{lemma-ext-from-perfect-into-bounded-QCoh}. This proves", "that (1) implies (2).", "\\medskip\\noindent", "Parts (2), (3), (4) are equivalent by the discussion in", "Cohomology, Section \\ref{cohomology-section-global-RHom}.", "Part (5) and (6) are equivalent as $H^i(X, -) = H^i(R\\Gamma(X, -))$", "by definition. The equivalent conditions (2), (3), (4) are", "equivalent to the equivalent conditions (5), (6) by", "Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}", "and the fact that $(G[-i])^\\vee = G^\\vee[i]$.", "\\medskip\\noindent", "It is clear that (7) implies (2). Conversely, ", "let us prove that the equivalent conditions (2) -- (6) imply (7).", "Recall that $G$ is a classical generator for $D_{perf}(\\mathcal{O}_X)$ by", "Remark \\ref{remark-classical-generator}.", "For $P \\in D_{perf}(\\mathcal{O}_X)$ let $T(P)$ be the assertion that", "$R\\Hom_X(P, E)$ is in $D^+(\\mathbf{Z})$.", "Clearly, $T$ is inherited by direct sums,", "satisfies the 2-out-of-three property for distinguished", "triangles, is inherited by direct summands, and is perserved by shifts.", "Hence by Derived Categories, Remark \\ref{derived-remark-check-on-generator}", "we see that (4) implies $T$ holds on all of $D_{perf}(\\mathcal{O}_X)$.", "The same argument works for all other properties, except that for property", "(7)(b) and (7)(c) we also use that $P \\mapsto P^\\vee$ is a self", "equivalence of $D_{perf}(\\mathcal{O}_X)$. Small detail omitted.", "\\medskip\\noindent", "We will prove the equivalent conditions (2) -- (7) imply (1)", "using the induction principle of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}.", "\\medskip\\noindent", "First, we prove (2) -- (7) $\\Rightarrow$ (1) if $X$ is affine.", "Let $P = \\mathcal{O}_X[0]$. From (7) we obtain $H^i (X, E) = 0$", "for $i \\ll 0$. Hence (1) follows since $E$ is", "determined by $R\\Gamma (X, E)$, see Lemma \\ref{lemma-affine-compare-bounded}.", "\\medskip\\noindent", "Now assume $X = U \\cup V$ with $U$ a quasi-compact open of $X$ and", "$V$ an affine open, and assume the implication (2) -- (7) $\\Rightarrow$ (1)", "is known for the schemes $U$, $V$, and $U \\cap V$.", "Suppose $E \\in D_\\QCoh(\\mathcal{O}_X)$ satisfies (2) -- (7).", "By Lemma \\ref{lemma-direct-summand-of-a-restriction} and", "Theorem \\ref{theorem-bondal-van-den-Bergh} there exists a perfect", "complex $Q$ on $X$ such that $Q|_U$ generates $D_\\QCoh (\\mathcal{O}_U)$. ", "Let $f_1, \\dots , f_r \\in \\Gamma (V, \\mathcal{O}_V)$", "be such that $V \\setminus U = V(f_1, \\dots , f_r)$ as subsets of", "$V$. Let $K \\in D_{perf}(\\mathcal{O}_V)$ be the object", "corresponding to the Koszul complex on $f_1, \\dots , f_r$.", "Let $K' \\in D_{perf}(\\mathcal{O}_X)$ be", "\\begin{equation}", "\\label{equation-detecting-bounded-below}", "K' = R (V \\to X)_* K = R (V \\to X)_! K,", "\\end{equation}", "see Cohomology, Lemmas \\ref{cohomology-lemma-pushforward-restriction} and", "\\ref{cohomology-lemma-pushforward-perfect}. This is a perfect", "complex on $X$ supported on the closed set $X \\setminus U \\subset V$", "and isomorphic to $K$ on $V$. By assumption, we know", "$R\\Hom_{\\mathcal{O}_X}(Q, E)$ and", "$R\\Hom_{\\mathcal{O}_X}(K', E)$ are bounded below.", "\\medskip\\noindent", "By the second description of $K'$ in (\\ref{equation-detecting-bounded-below})", "we have", "$$", "\\Hom_{D(\\mathcal{O}_V)}(K[-i], E|_V) = \\Hom_{D(\\mathcal{O}_X)}(K'[-i], E) = 0", "$$", "for $i \\ll 0$. Therefore, we may apply", "Lemma \\ref{lemma-orthogonal-koszul-second-variant} to $E|_V$ to", "obtain an integer $a$ such that", "$\\tau_{\\leq a}(E|_V) = \\tau_{\\leq a} R(U \\cap V \\to V)_*(E|_{U \\cap V})$.", "Then $\\tau_{\\leq a} E = \\tau_{\\leq a} R(U \\to X)_*(E|_U)$", "(check that the canonical map is an isomorphism after restricting to", "$U$ and to $V$). Hence using Lemma \\ref{lemma-bounded-below-truncation}", "twice we see that", "$$", "\\Hom_{D(\\mathcal{O}_U)}(Q|_U [-i], E|_U) =", "\\Hom_{D(\\mathcal{O}_X)}(Q[-i], R (U \\to X)_* (E|_U)) = 0", "$$", "for $i \\ll 0$. Since the Proposition holds for $U$ and the generator", "$Q|_U$, we have $E|_U \\in D^+_\\QCoh (\\mathcal{O}_U)$. But then since", "the functor $R(U \\to X)_*$ preserves bounded below objects", "(see Cohomology, Section \\ref{cohomology-section-derived-functors}) we get", "$\\tau_{\\leq a} E \\in D^+_\\QCoh(\\mathcal{O}_X)$. Thus ", "$E \\in D^+_\\QCoh (\\mathcal{O}_X)$." ], "refs": [ "perfect-lemma-ext-from-perfect-into-bounded-QCoh", "cohomology-lemma-dual-perfect-complex", "perfect-remark-classical-generator", "derived-remark-check-on-generator", "coherent-lemma-induction-principle", "perfect-lemma-affine-compare-bounded", "perfect-lemma-direct-summand-of-a-restriction", "perfect-theorem-bondal-van-den-Bergh", "cohomology-lemma-pushforward-restriction", "cohomology-lemma-pushforward-perfect", "perfect-lemma-orthogonal-koszul-second-variant", "perfect-lemma-bounded-below-truncation" ], "ref_ids": [ 7018, 2233, 7129, 2030, 3291, 6941, 7009, 6935, 2148, 2229, 7104, 7106 ] } ], "ref_ids": [] }, { "id": 7144, "type": "theorem", "label": "spaces-flat-theorem-finite-type-flat", "categories": [ "spaces-flat" ], "title": "spaces-flat-theorem-finite-type-flat", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume", "\\begin{enumerate}", "\\item $X \\to Y$ is locally of finite presentation,", "\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type, and", "\\item the set of weakly associated points of $Y$ is locally finite in $Y$.", "\\end{enumerate}", "Then $U = \\{x \\in |X| : \\mathcal{F}\\text{ flat at }x\\text{ over }Y\\}$", "is open in $X$ and $\\mathcal{F}|_U$ is an $\\mathcal{O}_U$-module", "of finite presentation and flat over $Y$." ], "refs": [], "proofs": [ { "contents": [ "Condition (3) means that if $V \\to Y$ is a surjective \\'etale morphism", "where $V$ is a scheme, then the weakly associated points of $V$", "are locally finite on the scheme $V$. (Recall that the weakly associated", "points of $V$ are exactly the inverse image of the weakly associated", "points of $Y$ by Divisors on Spaces, Definition", "\\ref{spaces-divisors-definition-weakly-associated}.)", "Having said this the question", "is \\'etale local on $X$ and $Y$, hence we may assume $X$ and $Y$", "are schemes. Thus the result follows from", "More on Flatness, Theorem \\ref{flat-theorem-finite-type-flat}." ], "refs": [ "spaces-divisors-definition-weakly-associated", "flat-theorem-finite-type-flat" ], "ref_ids": [ 13013, 5968 ] } ], "ref_ids": [] }, { "id": 7145, "type": "theorem", "label": "spaces-flat-theorem-check-flatness-at-associated-points", "categories": [ "spaces-flat" ], "title": "spaces-flat-theorem-check-flatness-at-associated-points", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "which is locally of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.", "Let $x \\in |X|$ with image $y \\in |Y|$.", "Set $F = f^{-1}(\\{y\\}) \\subset |X|$.", "Consider the conditions", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is flat at $x$ over $Y$, and", "\\item for every $x' \\in F \\cap \\text{Ass}_{X/Y}(\\mathcal{F})$ which", "specializes to $x$ we have that $\\mathcal{F}$ is flat at $x'$ over $Y$.", "\\end{enumerate}", "Then we always have (2) $\\Rightarrow$ (1). If $X$ and $Y$ are", "decent, then (1) $\\Rightarrow$ (2)." ], "refs": [], "proofs": [ { "contents": [ "Assume (2).", "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to V \\times_Y X$.", "Choose a point $u \\in U$ mapping to $x$. Let $v \\in V$ be the image of $u$.", "We will deduce the result from the corresponding result for", "$\\mathcal{F}|_U = (U \\to X)^*\\mathcal{F}$ and the point $u$.", "$U_v$. This works because $\\text{Ass}_{U/V}(\\mathcal{F}|_U) \\cap |U_v|$", "is equal to $\\text{Ass}_{U_v}(\\mathcal{F}|_{U_v})$ and equal to the inverse", "image of $F \\cap \\text{Ass}_{X/Y}(\\mathcal{F})$.", "Since the map $|U_v| \\to F$ is continuous we see that", "specializations in $|U_v|$ map to specializations in $F$,", "hence condition (2) is inherited by $U \\to V$,", "$\\mathcal{F}|_U$, and the point $u$.", "Thus More on Flatness, Theorem", "\\ref{flat-theorem-check-flatness-at-associated-points} applies", "and we conclude that (1) holds.", "\\medskip\\noindent", "If $Y$ is decent, then we can represent", "$y$ by a quasi-compact monomorphism $\\Spec(k) \\to Y$", "(by definition of decent spaces, see", "Decent Spaces, Definition \\ref{decent-spaces-definition-very-reasonable}).", "Then $F = |X_k|$, see", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-topology-fibre}.", "If in addition $X$ is decent (or more generally if $f$ is decent, see", "Decent Spaces, Definition \\ref{decent-spaces-definition-relative-conditions}", "and Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-property-for-morphism-out-of-property}),", "then $X_y$ is a decent space too. Furthermore, specializations in", "$F$ can be lifted to specializations", "in $U_v \\to X_y$, see", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-specialization}.", "Having said this it is clear that the reverse implication", "holds, because it holds in the case of schemes." ], "refs": [ "flat-theorem-check-flatness-at-associated-points", "decent-spaces-definition-very-reasonable", "decent-spaces-lemma-topology-fibre", "decent-spaces-definition-relative-conditions", "decent-spaces-lemma-property-for-morphism-out-of-property", "decent-spaces-lemma-decent-specialization" ], "ref_ids": [ 5971, 9562, 9525, 9567, 9514, 9494 ] } ], "ref_ids": [] }, { "id": 7146, "type": "theorem", "label": "spaces-flat-theorem-flattening-map", "categories": [ "spaces-flat" ], "title": "spaces-flat-theorem-flattening-map", "contents": [ "In Situation \\ref{situation-iso} assume", "\\begin{enumerate}", "\\item $f$ is of finite presentation,", "\\item $\\mathcal{F}$ is of finite presentation, flat over $B$, and", "pure relative to $B$, and", "\\item $u$ is surjective.", "\\end{enumerate}", "Then $F_{iso}$ is representable by a closed immersion $Z \\to B$.", "Moreover $Z \\to S$ is of finite presentation if $\\mathcal{G}$ is", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{K} = \\Ker(u)$ and denote $v : \\mathcal{K} \\to \\mathcal{F}$", "the inclusion. By Lemma \\ref{lemma-relate-zero-iso} we see that", "$F_{u, iso} = F_{v, zero}$. By Lemma \\ref{lemma-F-zero-closed-pure}", "applied to $v$ we see that $F_{u, iso} = F_{v, zero}$ is representable", "by a closed subspace of $B$. Note that $\\mathcal{K}$ is of finite type", "if $\\mathcal{G}$ is of finite presentation, see", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation}.", "Hence we also obtain the final statement of the lemma." ], "refs": [ "spaces-flat-lemma-relate-zero-iso", "spaces-flat-lemma-F-zero-closed-pure", "sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation" ], "ref_ids": [ 7174, 7181, 14187 ] } ], "ref_ids": [] }, { "id": 7147, "type": "theorem", "label": "spaces-flat-theorem-flat-dimension-n-representable", "categories": [ "spaces-flat" ], "title": "spaces-flat-theorem-flat-dimension-n-representable", "contents": [ "In Situation \\ref{situation-flat-dimension-n}.", "Assume moreover that $f$ is of finite presentation, that", "$\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation,", "and that $\\mathcal{F}$ is pure relative to $Y$.", "Then $F_n$ is an algebraic space and", "$F_n \\to Y$ is a monomorphism of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "The functor $F_n$ is a sheaf for the fppf topology by", "Lemma \\ref{lemma-flat-dimension-n}.", "Since $F_n \\to Y$ is a monomorphism of sheaves on", "$(\\Sch/S)_{fppf}$ we see that $\\Delta : F_n \\to F_n \\times F_n$", "is the pullback of the diagonal $\\Delta_Y : Y \\to Y \\times_S Y$.", "Hence the representability of $\\Delta_Y$ implies the same", "thing for $F_n$. Therefore it suffices to prove that", "there exists a scheme $W$ over $S$ and a surjective \\'etale morphism", "$W \\to F_n$.", "\\medskip\\noindent", "To construct $W \\to F_n$ choose an \\'etale covering $\\{Y_i \\to Y\\}$", "with $Y_i$ a scheme. Let $X_i = X \\times_Y Y_i$ and let", "$\\mathcal{F}_i$ be the pullback of $\\mathcal{F}$ to $X_i$.", "Then $\\mathcal{F}_i$ is pure relative to $Y_i$ either by definition", "or by Lemma \\ref{lemma-quasi-finite-base-change}.", "The other assumptions of the theorem are preserved as well.", "Finally, the restriction of $F_n$ to $Y_i$ is", "the functor $F_n$ corresponding to $X_i \\to Y_i$ and $\\mathcal{F}_i$.", "Hence it suffices to show: Given $\\mathcal{F}$ and $f : X \\to Y$", "as in the statement of the theorem where $Y$ is a scheme, the", "functor $F_n$ is representable by a scheme $Z_n$ and", "$Z_n \\to Y$ is a monomorphism of finite presentation.", "\\medskip\\noindent", "Observe that a monomorphism of finite presentation is", "separated and quasi-finite (Morphisms, Lemma", "\\ref{morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}).", "Hence combining", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves},", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend}", ", and", "Descent, Lemmas \\ref{descent-lemma-descending-property-monomorphism} and", "\\ref{descent-lemma-descending-property-finite-presentation}", "we see that the question is local for the \\'etale topology on $Y$.", "\\medskip\\noindent", "In particular the situation is local for the Zariski topology on $Y$", "and we may assume that $Y$ is affine. In this case the dimension of the", "fibres of $f$ is bounded above, hence we see that $F_n$ is representable", "for $n$ large enough. Thus we may use descending induction on $n$.", "Suppose that we know $F_{n + 1}$ is representable by a monomorphism", "$Z_{n + 1} \\to Y$ of finite presentation. Consider the base change", "$X_{n + 1} = Z_{n + 1} \\times_Y X$ and the pullback $\\mathcal{F}_{n + 1}$", "of $\\mathcal{F}$ to $X_{n + 1}$. The morphism $Z_{n + 1} \\to Y$ is", "quasi-finite as it is a monomorphism of finite presentation, hence", "Lemma \\ref{lemma-quasi-finite-base-change}", "implies that $\\mathcal{F}_{n + 1}$ is pure relative to $Z_{n + 1}$.", "Since $F_n$ is a subfunctor of $F_{n + 1}$ we conclude that in order", "to prove the result for $F_n$ it suffices to prove the result for the", "corresponding functor for the situation", "$\\mathcal{F}_{n + 1}/X_{n + 1}/Z_{n + 1}$.", "In this way we reduce to proving the result for $F_n$ in case", "$Y_{n + 1} = Y$, i.e., we may assume that $\\mathcal{F}$ is flat", "in dimensions $\\geq n + 1$ over $Y$.", "\\medskip\\noindent", "Fix $n$ and assume $\\mathcal{F}$ is flat in dimensions $\\geq n + 1$", "over the affine scheme $Y$.", "To finish the proof we have to show that $F_n$ is representable", "by a monomorphism $Z_n \\to S$ of finite presentation.", "Since the question is local in the \\'etale topology on $Y$ it suffices to", "show that for every $y \\in Y$ there exists an \\'etale neighbourhood", "$(Y', y') \\to (Y, y)$ such that the result holds after base change to $Y'$.", "Thus by", "Lemma \\ref{lemma-existence-complete}", "we may assume there exist \\'etale morphisms $h_j : W_j \\to X$,", "$j = 1, \\ldots, m$ such that for each $j$ there exists a complete", "d\\'evissage of $\\mathcal{F}_j/W_j/Y$ over $y$,", "where $\\mathcal{F}_j$ is the pullback of $\\mathcal{F}$ to $W_j$", "and such that $|X_y| \\subset \\bigcup h_j(W_j)$. Since", "$h_j$ is \\'etale, by", "Lemma \\ref{lemma-pre-flat-dimension-n}", "the sheaves $\\mathcal{F}_j$ are still flat over in", "dimensions $\\geq n + 1$ over $Y$.", "Set $W = \\bigcup h_j(W_j)$, which is a quasi-compact open of $X$.", "As $\\mathcal{F}$ is pure along $X_y$ we see that", "$$", "E = \\{t \\in |Y| : \\text{Ass}_{X_t}(\\mathcal{F}_t) \\subset W \\}.", "$$", "contains all generalizations of $y$. By", "Divisors on Spaces,", "Lemma \\ref{spaces-divisors-lemma-relative-assassin-constructible}", "$E$ is a constructible subset of $Y$. We have seen that", "$\\Spec(\\mathcal{O}_{Y, y}) \\subset E$. By", "Morphisms, Lemma \\ref{morphisms-lemma-constructible-containing-open}", "we see that $E$ contains an open neighbourhood of $y$.", "Hence after shrinking $Y$ we may assume that $E = Y$.", "It follows from", "Lemma \\ref{lemma-localize-flat-dimension-n}", "that it suffices to prove the lemma for the functor $F_n$ associated to", "$X = \\coprod W_j$ and $\\mathcal{F} = \\coprod \\mathcal{F}_j$.", "If $F_{j, n}$ denotes the functor for $W_j \\to Y$ and the sheaf", "$\\mathcal{F}_j$ we see that $F_n = \\prod F_{j, n}$. Hence it suffices", "to prove each $F_{j, n}$ is representable by some monomorphism", "$Z_{j, n} \\to Y$ of finite presentation, since then", "$$", "Z_n = Z_{1, n} \\times_Y \\ldots \\times_Y Z_{m, n}", "$$", "Thus we have reduced the theorem to the special case handled in", "More on Flatness, Lemma \\ref{flat-lemma-flat-dimension-n-representable}." ], "refs": [ "spaces-flat-lemma-flat-dimension-n", "spaces-flat-lemma-quasi-finite-base-change", "morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "descent-lemma-descent-data-sheaves", "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "descent-lemma-descending-property-monomorphism", "descent-lemma-descending-property-finite-presentation", "spaces-flat-lemma-quasi-finite-base-change", "spaces-flat-lemma-existence-complete", "spaces-flat-lemma-pre-flat-dimension-n", "spaces-divisors-lemma-relative-assassin-constructible", "morphisms-lemma-constructible-containing-open", "spaces-flat-lemma-localize-flat-dimension-n", "flat-lemma-flat-dimension-n-representable" ], "ref_ids": [ 7187, 7154, 5235, 14751, 13949, 14696, 14678, 7154, 7158, 7186, 12927, 5251, 7188, 6105 ] } ], "ref_ids": [] }, { "id": 7148, "type": "theorem", "label": "spaces-flat-theorem-existence", "categories": [ "spaces-flat" ], "title": "spaces-flat-theorem-existence", "contents": [ "In Situation \\ref{situation-existence}", "there exists a finitely presented $\\mathcal{O}_X$-module", "$\\mathcal{F}$, flat over $A$, with support proper over $A$,", "such that", "$\\mathcal{F}_n = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X_n}$", "for all $n$ compatibly with the maps $\\varphi_n$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemmas \\ref{lemma-compute-what-it-should-be},", "\\ref{lemma-compute-against-perfect},", "\\ref{lemma-relative-pseudo-coherence},", "\\ref{lemma-compute-over-affine},", "\\ref{lemma-finitely-presented}, and", "\\ref{lemma-proper-support}", "to get an open subspace $W \\subset X$ containing all points", "lying over $\\Spec(A_n)$", "and a finitely presented $\\mathcal{O}_W$-module $\\mathcal{F}$", "whose support is proper over $A$ with", "$\\mathcal{F}_n = \\mathcal{F} \\otimes_{\\mathcal{O}_W} \\mathcal{O}_{X_n}$", "for all $n \\geq 1$. (This makes sense as $X_n \\subset W$.)", "By Lemma \\ref{lemma-proper-pure} we see that $\\mathcal{F}$", "is universally pure relative to $\\Spec(A)$.", "By Theorem \\ref{theorem-flat-dimension-n-representable}", "(for explanation, see Lemma \\ref{lemma-when-universal-flattening})", "there exists a universal flattening $S' \\to \\Spec(A)$", "of $\\mathcal{F}$ and moreover the morphism $S' \\to \\Spec(A)$", "is a monomorphism of finite presentation.", "In particular $S'$ is a scheme (this follows from the proof", "of the theorem but it also follows a postoriori by", "Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}).", "Since the base change of $\\mathcal{F}$ to $\\Spec(A_n)$", "is $\\mathcal{F}_n$ we find that $\\Spec(A_n) \\to \\Spec(A)$", "factors (uniquely) through $S'$ for each $n$.", "By More on Flatness, Lemma \\ref{flat-lemma-monomorphism-isomorphism}", "we see that $S' = \\Spec(A)$.", "This means that $\\mathcal{F}$ is flat over $A$.", "Finally, since the scheme theoretic support $Z$ of $\\mathcal{F}$", "is proper over $\\Spec(A)$, the morphism $Z \\to X$ is closed.", "Hence the pushforward $(W \\to X)_*\\mathcal{F}$ is supported", "on $W$ and has all the desired properties." ], "refs": [ "spaces-flat-lemma-compute-what-it-should-be", "spaces-flat-lemma-compute-against-perfect", "spaces-flat-lemma-relative-pseudo-coherence", "spaces-flat-lemma-compute-over-affine", "spaces-flat-lemma-finitely-presented", "spaces-flat-lemma-proper-support", "spaces-flat-lemma-proper-pure", "spaces-flat-theorem-flat-dimension-n-representable", "spaces-flat-lemma-when-universal-flattening", "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "flat-lemma-monomorphism-isomorphism" ], "ref_ids": [ 7190, 7191, 7192, 7193, 7194, 7195, 7157, 7147, 7189, 4983, 6113 ] } ], "ref_ids": [] }, { "id": 7149, "type": "theorem", "label": "spaces-flat-theorem-existence-derived", "categories": [ "spaces-flat" ], "title": "spaces-flat-theorem-existence-derived", "contents": [ "In Situation \\ref{situation-existence-derived}", "there exists a pseudo-coherent $K$ in $D(\\mathcal{O}_X)$", "such that $K_n = K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_{X_n}$", "for all $n$ compatibly with the maps $\\varphi_n$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemmas \\ref{lemma-compute-what-it-should-be-derived},", "\\ref{lemma-compute-against-perfect-derived},", "\\ref{lemma-relative-pseudo-coherence-derived}", "to get a pseudo-coherent object $K$ of $D(\\mathcal{O}_X)$.", "Choosing affine $U$ in Lemma", "\\ref{lemma-compute-over-affine-derived}", "it follows immediately that $K$ restricts to $K_n$ over $X_n$." ], "refs": [ "spaces-flat-lemma-compute-what-it-should-be-derived", "spaces-flat-lemma-compute-against-perfect-derived", "spaces-flat-lemma-relative-pseudo-coherence-derived", "spaces-flat-lemma-compute-over-affine-derived" ], "ref_ids": [ 7196, 7197, 7198, 7199 ] } ], "ref_ids": [] }, { "id": 7150, "type": "theorem", "label": "spaces-flat-lemma-impure-limit", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-impure-limit", "contents": [ "In Situation \\ref{situation-pre-pure}.", "Let $(g : T \\to S, t' \\leadsto t, \\xi)$ be an impurity of", "$\\mathcal{F}$ above $y$. Assume $T = \\lim_{i \\in I} T_i$ is a directed limit", "of affine schemes over $Y$. Then for some $i$ the triple", "$(T_i \\to Y, t'_i \\leadsto t_i, \\xi_i)$ is an impurity of", "$\\mathcal{F}$ above $y$." ], "refs": [], "proofs": [ { "contents": [ "The notation in the statement means this: Let $p_i : T \\to T_i$", "be the projection morphisms, let $t_i = p_i(t)$ and $t'_i = p_i(t')$.", "Finally $\\xi_i \\in |X_{T_i}|$ is the image of $\\xi$. By", "Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-base-change-relative-assassin}", "we have $\\xi_i \\in \\text{Ass}_{X_{T_i}/T_i}(\\mathcal{F}_{T_i})$.", "Thus the only point is to show that", "$t_i \\not \\in f_{T_i}(\\overline{\\{\\xi_i\\}})$ for some $i$.", "\\medskip\\noindent", "Let $Z_i \\subset X_{T_i}$ be the reduced induced scheme structure", "on $\\overline{\\{\\xi_i\\}} \\subset |X_{T_i}|$", "and let $Z \\subset X_T$ be the reduced induced scheme structure on", "$\\overline{\\{\\xi\\}} \\subset |X_T|$.", "Then $Z = \\lim Z_i$ by", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-inverse-limit-irreducibles}", "(the lemma applies because each $X_{T_i}$ is decent).", "Choose a field $k$ and a morphism $\\Spec(k) \\to T$ whose image is $t$.", "Then", "$$", "\\emptyset =", "Z \\times_T \\Spec(k) = (\\lim Z_i) \\times_{(\\lim T_i)} \\Spec(k)", "= \\lim Z_i \\times_{T_i} \\Spec(k)", "$$", "because limits commute with fibred products (limits commute with limits).", "Each $Z_i \\times_{T_i} \\Spec(k)$ is quasi-compact because $X_{T_i} \\to T_i$", "is of finite type and hence $Z_i \\to T_i$ is of finite type.", "Hence $Z_i \\times_{T_i} \\Spec(k)$ is empty for some $i$ by", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-limit-nonempty}.", "Since the image of the composition $\\Spec(k) \\to T \\to T_i$ is $t_i$", "we obtain what we want." ], "refs": [ "spaces-divisors-lemma-base-change-relative-assassin", "spaces-limits-lemma-inverse-limit-irreducibles", "spaces-limits-lemma-limit-nonempty" ], "ref_ids": [ 12924, 4573, 4572 ] } ], "ref_ids": [] }, { "id": 7151, "type": "theorem", "label": "spaces-flat-lemma-flat-ascent-impurity", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-flat-ascent-impurity", "contents": [ "In Situation \\ref{situation-pre-pure}.", "Let $(Y_1, y_1) \\to (Y, y)$ be a morphism of pointed", "algebraic spaces over $S$. Assume $Y_1 \\to Y$ is flat at $y_1$.", "If $(T \\to Y, t' \\leadsto t, \\xi)$ is an impurity of", "$\\mathcal{F}$ above $y$, then there exists an impurity", "$(T_1 \\to Y_1, t_1' \\leadsto t_1, \\xi_1)$ of the pullback", "$\\mathcal{F}_1$ of $\\mathcal{F}$ to $X_1 = Y_1 \\times_Y X$", "over $y_1$ such that $T_1$ is \\'etale over $Y_1 \\times_Y T$." ], "refs": [], "proofs": [ { "contents": [ "Choose an \\'etale morphism $T_1 \\to Y_1 \\times_Y T$ where $T_1$", "is a scheme and let $t_1 \\in T_1$ be a point mapping to $y_1$ and $t$.", "It is possible to find a pair $(T_1, t_1)$ like this by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}.", "The morphism of schemes $T_1 \\to T$ is flat at $t_1$", "(use Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-flat}", "and the definition of flat morphisms of algebraic spaces)", "there exists a specialization $t'_1 \\leadsto t_1$ lying over", "$t' \\leadsto t$, see", "Morphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat}.", "Choose a point $\\xi_1 \\in |X_{T_1}|$ mapping to $t'_1$", "and $\\xi$ with $\\xi_1 \\in \\text{Ass}_{X_{T_1}/T_1}(\\mathcal{F}_{T_1})$.", "point of $\\Spec(\\kappa(t'_1) \\otimes_{\\kappa(t')} \\kappa(\\xi))$.", "This is possible by", "Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-base-change-relative-assassin}.", "As the closure $Z_1$ of $\\{\\xi_1\\}$ in $|X_{T_1}|$ maps into the", "closure of $\\{\\xi\\}$ in $|X_T|$ we conclude that", "the image of $Z_1$ in $|T_1|$ cannot contain $t_1$.", "Hence $(T_1 \\to Y_1, t'_1 \\leadsto t_1, \\xi_1)$", "is an impurity of $\\mathcal{F}_1$ above $Y_1$." ], "refs": [ "spaces-properties-lemma-points-cartesian", "spaces-morphisms-lemma-base-change-flat", "morphisms-lemma-generalizations-lift-flat", "spaces-divisors-lemma-base-change-relative-assassin" ], "ref_ids": [ 11819, 4853, 5266, 12924 ] } ], "ref_ids": [] }, { "id": 7152, "type": "theorem", "label": "spaces-flat-lemma-pure-along-X-y", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-pure-along-X-y", "contents": [ "In Situation \\ref{situation-pre-pure}. Let $\\overline{y}$ be a geometric", "point lying over $y$. Let $\\mathcal{O} = \\mathcal{O}_{Y, \\overline{y}}$", "be the \\'etale local ring of $Y$ at $\\overline{y}$. Denote", "$Y^{sh} = \\Spec(\\mathcal{O})$, $X^{sh} = X \\times_Y Y^{sh}$, and", "$\\mathcal{F}^{sh}$ the pullback of $\\mathcal{F}$ to $X^{sh}$.", "The following are equivalent", "\\begin{enumerate}", "\\item there exists an impurity", "$(Y^{sh} \\to Y, y' \\leadsto \\overline{y}, \\xi)$", "of $\\mathcal{F}$ above $y$,", "\\item every point of $\\text{Ass}_{X^{sh}/Y^{sh}}(\\mathcal{F}^{sh})$", "specializes to a point of the closed fibre $X_{\\overline{y}}$,", "\\item there exists an impurity $(T \\to Y, t' \\leadsto t, \\xi)$", "of $\\mathcal{F}$ above $y$ such that $(T, t) \\to (Y, y)$ is an", "\\'etale neighbourhood, and", "\\item there exists an impurity $(T \\to Y, t' \\leadsto t, \\xi)$", "of $\\mathcal{F}$ above $y$ such that $T \\to Y$ is quasi-finite at $t$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "That parts (1) and (2) are equivalent is immediate from the definition.", "\\medskip\\noindent", "Recall that $\\mathcal{O} = \\mathcal{O}_{Y, \\overline{y}}$", "is the filtered colimit of $\\mathcal{O}(V)$ over the category", "of \\'etale neighbourhoods $(V, \\overline{v}) \\to (Y, \\overline{y})$", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-cofinal-etale}).", "Moreover, it suffices to consider affine \\'etale neighbourhoods $V$.", "Hence $Y^{sh} = \\Spec(\\mathcal{O}) = \\lim \\Spec(\\mathcal{O}(V)) = \\lim V$.", "Thus we see that (1) implies (3) by Lemma \\ref{lemma-impure-limit}.", "\\medskip\\noindent", "Since an \\'etale morphism is locally quasi-finite", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-etale-locally-quasi-finite})", "we see that (3) implies (4).", "\\medskip\\noindent", "Finally, assume (4). After replacing $T$ by an open neighbourhood of $t$", "we may assume $T \\to Y$ is locally quasi-finite.", "By Lemma \\ref{lemma-flat-ascent-impurity}", "we find an impurity", "$(T_1 \\to Y^{sh}, t_1' \\leadsto t_1, \\xi_1)$", "with $T_1 \\to T \\times_Y Y^{sh}$", "\\'etale. Since an \\'etale morphism is locally quasi-finite", "and using Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-quasi-finite} and", "Morphisms, Lemma \\ref{morphisms-lemma-composition-quasi-finite}", "we see that $T_1 \\to Y^{sh}$ is locally quasi-finite.", "As $\\mathcal{O}$ is strictly henselian, we can apply More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point}", "to see that after replacing $T_1$ by an open and closed neighbourhood", "of $t_1$ we may assume that $T_1 \\to Y^{sh} = \\Spec(\\mathcal{O})$", "is finite. Let $\\theta \\in |X^{sh}|$ be the image of", "$\\xi_1$ and let $y' \\in \\Spec(\\mathcal{O})$ be the image", "of $t_1'$. By Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-base-change-relative-assassin}", "we see that $\\theta \\in \\text{Ass}_{X^{sh}/Y^{sh}}(\\mathcal{F}^{sh})$.", "Since $\\pi : X_{T_1} \\to X^{sh}$", "is finite, it induces a closed map $|X_{T_1}| \\to |X^{sh}|$.", "Hence the image of $\\overline{\\{\\xi_1\\}}$ is $\\overline{\\{\\theta\\}}$.", "It follows that $(Y^{sh} \\to Y, y' \\leadsto \\overline{y}, \\theta)$", "is an impurity of $\\mathcal{F}$ above $y$ and the proof is complete." ], "refs": [ "spaces-properties-lemma-cofinal-etale", "spaces-flat-lemma-impure-limit", "spaces-morphisms-lemma-etale-locally-quasi-finite", "spaces-flat-lemma-flat-ascent-impurity", "spaces-morphisms-lemma-base-change-quasi-finite", "morphisms-lemma-composition-quasi-finite", "more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point", "spaces-divisors-lemma-base-change-relative-assassin" ], "ref_ids": [ 11870, 7150, 4908, 7151, 4832, 5232, 13892, 12924 ] } ], "ref_ids": [] }, { "id": 7153, "type": "theorem", "label": "spaces-flat-lemma-base-change-universally", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-base-change-universally", "contents": [ "In Situation \\ref{situation-pre-pure}.", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is universally pure above $y$, and", "\\item for every morphism $(Y', y') \\to (Y, y)$ of pointed algebraic spaces", "the pullback $\\mathcal{F}_{Y'}$ is pure above $y'$.", "\\end{enumerate}", "In particular, $\\mathcal{F}$ is universally pure relative to $Y$ if and", "only if every base change $\\mathcal{F}_{Y'}$ of $\\mathcal{F}$ is", "pure relative to $Y'$." ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7154, "type": "theorem", "label": "spaces-flat-lemma-quasi-finite-base-change", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-quasi-finite-base-change", "contents": [ "In Situation \\ref{situation-pre-pure}.", "Let $(Y', y') \\to (Y, y)$ be a morphism of pointed algebraic spaces.", "If $Y' \\to Y$ is quasi-finite at $y'$ and $\\mathcal{F}$ is pure above $y$,", "then $\\mathcal{F}_{Y'}$ is pure above $y'$." ], "refs": [], "proofs": [ { "contents": [ "It $(T \\to Y', t' \\leadsto t, \\xi)$ is an impurity of", "$\\mathcal{F}_{Y'}$ above $y'$ with $T \\to Y'$ quasi-finite at $t$,", "then $(T \\to Y, t' \\to t, \\xi)$ is an impurity of $\\mathcal{F}$", "above $y$ with $T \\to Y$ quasi-finite at $t$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-composition-quasi-finite}.", "Hence the lemma follows immediately from the definition of purity." ], "refs": [ "spaces-morphisms-lemma-composition-quasi-finite" ], "ref_ids": [ 4831 ] } ], "ref_ids": [] }, { "id": 7155, "type": "theorem", "label": "spaces-flat-lemma-flat-descend-pure", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-flat-descend-pure", "contents": [ "In Situation \\ref{situation-pre-pure}.", "Let $(Y_1, y_1) \\to (Y, y)$ be a morphism of pointed algebraic spaces.", "Assume $Y_1 \\to Y$ is flat at $y_1$.", "\\begin{enumerate}", "\\item If $\\mathcal{F}_{Y_1}$ is pure above $y_1$,", "then $\\mathcal{F}$ is pure above $y$.", "\\item If $\\mathcal{F}_{Y_1}$ is universally pure above $y_1$,", "then $\\mathcal{F}$ is universally pure above $y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is true because impurities go up along a flat base change, see", "Lemma \\ref{lemma-flat-ascent-impurity}. For example", "part (1) follows because by any impurity $(T \\to Y, t' \\leadsto t, \\xi)$", "of $\\mathcal{F}$ above $y$ with $T \\to Y$ quasi-finite at $t$", "by the lemma leads to an impurity", "$(T_1 \\to Y_1, t_1' \\leadsto t_1, \\xi_1)$ of the pullback", "$\\mathcal{F}_1$ of $\\mathcal{F}$ to $X_1 = Y_1 \\times_Y X$", "over $y_1$ such that $T_1$ is \\'etale over $Y_1 \\times_Y T$.", "Hence $T_1 \\to Y_1$ is quasi-finite at $t_1$ because", "\\'etale morphisms are locally quasi-finite and compositions", "of locally quasi-finite morphisms are locally quasi-finite", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-etale-locally-quasi-finite} and", "\\ref{spaces-morphisms-lemma-composition-quasi-finite}).", "Similarly for part (2)." ], "refs": [ "spaces-flat-lemma-flat-ascent-impurity", "spaces-morphisms-lemma-etale-locally-quasi-finite", "spaces-morphisms-lemma-composition-quasi-finite" ], "ref_ids": [ 7151, 4908, 4831 ] } ], "ref_ids": [] }, { "id": 7156, "type": "theorem", "label": "spaces-flat-lemma-supported-on-closed", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-supported-on-closed", "contents": [ "In Situation \\ref{situation-pre-pure}. Let $i : Z \\to X$ be a closed immersion", "and assume that $\\mathcal{F} = i_*\\mathcal{G}$ for some", "finite type, quasi-coherent sheaf $\\mathcal{G}$ on $Z$.", "Then $\\mathcal{G}$ is (universally) pure above $y$", "if and only if $\\mathcal{F}$ is (universally) pure above $y$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-relative-weak-assassin-finite}." ], "refs": [ "spaces-divisors-lemma-relative-weak-assassin-finite" ], "ref_ids": [ 12926 ] } ], "ref_ids": [] }, { "id": 7157, "type": "theorem", "label": "spaces-flat-lemma-proper-pure", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-proper-pure", "contents": [ "In Situation \\ref{situation-pre-pure}.", "\\begin{enumerate}", "\\item If the support of $\\mathcal{F}$ is proper over $Y$, then", "$\\mathcal{F}$ is universally pure relative to $Y$.", "\\item If $f$ is proper, then", "$\\mathcal{F}$ is universally pure relative to $Y$.", "\\item If $f$ is proper, then $X$ is universally pure relative to $Y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First we reduce (1) to (2). Namely, let $Z \\subset X$ be the", "scheme theoretic support of $\\mathcal{F}$", "(Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-scheme-theoretic-support}). Let $i : Z \\to X$", "be the corresponding closed immersion and write", "$\\mathcal{F} = i_*\\mathcal{G}$ for some finite type quasi-coherent", "$\\mathcal{O}_Z$-module $\\mathcal{G}$.", "In case (1) $Z \\to Y$ is proper by assumption.", "Thus by Lemma \\ref{lemma-supported-on-closed} case (1) reduces to case (2).", "\\medskip\\noindent", "Assume $f$ is proper.", "Let $(g : T \\to Y, t' \\leadsto t, \\xi)$ be an impurity of $\\mathcal{F}$", "above $y$. Since $f$ is proper, it is universally closed. Hence", "$f_T : X_T \\to T$ is closed. Since $f_T(\\xi) = t'$ this implies that", "$t \\in f(\\overline{\\{\\xi\\}})$ which is a contradiction." ], "refs": [ "spaces-morphisms-definition-scheme-theoretic-support", "spaces-flat-lemma-supported-on-closed" ], "ref_ids": [ 4993, 7156 ] } ], "ref_ids": [] }, { "id": 7158, "type": "theorem", "label": "spaces-flat-lemma-existence-complete", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-existence-complete", "contents": [ "Let $S$ be a scheme.", "Let $X \\to Y$ be a finite type morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Let $y \\in |Y|$ be a point. There exists an \\'etale morphism", "$(Y', y') \\to (Y, y)$ with $Y'$ an affine scheme and \\'etale morphisms", "$h_i : W_i \\to X_{Y'}$, $i = 1, \\ldots, n$ such that for each", "$i$ there exists a complete d\\'evissage of $\\mathcal{F}_i/W_i/Y'$ over $y'$,", "where $\\mathcal{F}_i$ is the pullback of $\\mathcal{F}$ to $W_i$", "and such that $|(X_{Y'})_{y'}| \\subset \\bigcup h_i(W_i)$." ], "refs": [], "proofs": [ { "contents": [ "The question is \\'etale local on $Y$ hence we may assume $Y$", "is an affine scheme. Then $X$ is quasi-compact, hence we can", "choose an affine scheme $X'$ and a surjective \\'etale morphism", "$X' \\to X$. Then we may apply", "More on Flatness, Lemma \\ref{flat-lemma-existence-complete}", "to $X' \\to Y$, $(X' \\to Y)^*\\mathcal{F}$, and $y$ to", "get what we want." ], "refs": [ "flat-lemma-existence-complete" ], "ref_ids": [ 6000 ] } ], "ref_ids": [] }, { "id": 7159, "type": "theorem", "label": "spaces-flat-lemma-open-in-fibre-where-flat", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-open-in-fibre-where-flat", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "which is locally of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.", "Let $y \\in |Y|$ and $F = f^{-1}(\\{y\\}) \\subset |X|$. Then the set", "$$", "\\{x \\in F \\mid \\mathcal{F} \\text{ flat over }Y\\text{ at }x\\}", "$$", "is open in $F$." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$, a point $v \\in V$, and an \\'etale morphism $V \\to Y$", "mapping $v$ to $y$. Choose a scheme $U$ and a surjective \\'etale", "morphism $U \\to V \\times_Y X$. Then $|U_v| \\to F$ is an open continuous", "map of topological spaces as $|U| \\to |X|$ is continuous and open.", "Hence the result follows from the case of schemes which is", "More on Flatness, Lemma \\ref{flat-lemma-open-in-fibre-where-flat}." ], "refs": [ "flat-lemma-open-in-fibre-where-flat" ], "ref_ids": [ 6017 ] } ], "ref_ids": [] }, { "id": 7160, "type": "theorem", "label": "spaces-flat-lemma-bourbaki-finite-type-general-base-at-point", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-bourbaki-finite-type-general-base-at-point", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is", "locally of finite type. Let $x \\in |X|$ with image $y \\in |Y|$.", "Let $\\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$.", "Let $\\mathcal{G}$ be a quasi-coherent sheaf on $Y$.", "If $\\mathcal{F}$ is flat at $x$ over $Y$, then", "$$", "x \\in \\text{WeakAss}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G})", "\\Leftrightarrow", "y \\in \\text{WeakAss}_Y(\\mathcal{G})", "\\text{ and }", "x \\in \\text{Ass}_{X/Y}(\\mathcal{F}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_g & V \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "where $U$ and $V$ are schemes and the vertical arrows are surjective \\'etale.", "Choose $u \\in U$ mapping to $x$. Let $\\mathcal{E} = \\mathcal{F}|_U$", "and $\\mathcal{H} = \\mathcal{G}|_V$.", "Let $v \\in V$ be the image of $u$. Then", "$x \\in \\text{WeakAss}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G})$", "if and only if", "$u \\in \\text{WeakAss}_X(\\mathcal{E} \\otimes_{\\mathcal{O}_X} g^*\\mathcal{H})$", "by Divisors on Spaces, Definition", "\\ref{spaces-divisors-definition-weakly-associated}.", "Similarly, $y \\in \\text{WeakAss}_Y(\\mathcal{G})$ if and only if", "$v \\in \\text{WeakAss}_V(\\mathcal{H})$.", "Finally, we have $x \\in \\text{Ass}_{X/Y}(\\mathcal{F})$ if and only if", "$u \\in \\text{Ass}_{U_v}(\\mathcal{E}|_{U_v})$ by", "Divisors on Spaces, Definition", "\\ref{spaces-divisors-definition-relative-weak-assassin}.", "Observe that flatness of $\\mathcal{F}$ at $x$ is", "equivalent to flatness of $\\mathcal{E}$ at $u$, see", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-flat-module}.", "The equivalence for $g : U \\to V$, $\\mathcal{E}$, $\\mathcal{H}$, $u$, and $v$", "is More on Flatness, Lemma", "\\ref{flat-lemma-bourbaki-finite-type-general-base-at-point}." ], "refs": [ "spaces-divisors-definition-weakly-associated", "spaces-divisors-definition-relative-weak-assassin", "spaces-morphisms-definition-flat-module", "flat-lemma-bourbaki-finite-type-general-base-at-point" ], "ref_ids": [ 13013, 13015, 5008, 6040 ] } ], "ref_ids": [] }, { "id": 7161, "type": "theorem", "label": "spaces-flat-lemma-bourbaki-finite-type-general-base", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-bourbaki-finite-type-general-base", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ which is locally of finite type.", "Let $\\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$", "which is flat over $Y$. Let $\\mathcal{G}$ be a quasi-coherent sheaf on $Y$.", "Then we have", "$$", "\\text{WeakAss}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}) =", "\\text{Ass}_{X/Y}(\\mathcal{F}) \\cap", "|f|^{-1}(\\text{WeakAss}_Y(\\mathcal{G}))", "$$" ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Lemma \\ref{lemma-bourbaki-finite-type-general-base-at-point}." ], "refs": [ "spaces-flat-lemma-bourbaki-finite-type-general-base-at-point" ], "ref_ids": [ 7160 ] } ], "ref_ids": [] }, { "id": 7162, "type": "theorem", "label": "spaces-flat-lemma-finite-type-flat-along-fibre-free-variant", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-finite-type-flat-along-fibre-free-variant", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $y \\in |Y|$. Set $F = f^{-1}(\\{y\\}) \\subset |X|$. Assume that", "\\begin{enumerate}", "\\item $f$ is of finite type,", "\\item $\\mathcal{F}$ is of finite type, and", "\\item $\\mathcal{F}$ is flat over $Y$ at all $x \\in F$.", "\\end{enumerate}", "Then there exists an \\'etale morphism $(Y', y') \\to (Y, y)$", "where $Y'$ is a scheme and a commutative diagram of algebraic spaces", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\", "Y & \\Spec(\\mathcal{O}_{Y', y'}) \\ar[l]", "}", "$$", "such that $X' \\to X \\times_Y \\Spec(\\mathcal{O}_{Y', y'})$", "is \\'etale, $|X'_{y'}| \\to F$ is surjective, $X'$ is affine,", "and $\\Gamma(X', g^*\\mathcal{F})$ is a free $\\mathcal{O}_{Y', y'}$-module." ], "refs": [], "proofs": [ { "contents": [ "Choose an \\'etale morphism $(Y', y') \\to (Y, y)$ where $Y'$ is an", "affine scheme. Then $X \\times_Y Y'$ is quasi-compact.", "Choose an affine scheme $X'$ and a surjective \\'etale morphism", "$X' \\to X \\times_Y Y'$. Picture", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\", "Y & Y' \\ar[l]", "}", "$$", "Then $\\mathcal{F}' = g^*\\mathcal{F}$ is flat over $Y'$ at all", "points of $X'_{y'}$, see Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-module-flat}.", "Hence we can apply the lemma in the case of schemes", "(More on Flatness, Lemma", "\\ref{flat-lemma-finite-type-flat-along-fibre-free-variant})", "to the morphism", "$X' \\to Y'$, the quasi-coherent sheaf $g^*\\mathcal{F}$, and the point $y'$.", "This gives an \\'etale morphism $(Y'', y'') \\to (Y', y')$ and a commutative", "diagram", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l]^g \\ar[d] & X'' \\ar[l]^{g'} \\ar[d] \\\\", "Y & Y' \\ar[l] & \\Spec(\\mathcal{O}_{Y'', y''}) \\ar[l]", "}", "$$", "To get what we want we take $(Y'', y'') \\to (Y, y)$", "and $g \\circ g' : X'' \\to X$." ], "refs": [ "spaces-morphisms-lemma-base-change-module-flat", "flat-lemma-finite-type-flat-along-fibre-free-variant" ], "ref_ids": [ 4863, 6037 ] } ], "ref_ids": [] }, { "id": 7163, "type": "theorem", "label": "spaces-flat-lemma-check-along-closed-fibre", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-check-along-closed-fibre", "contents": [ "Let $S$ be a local scheme with closed point $s$.", "Let $f : X \\to S$ be a morphism from an algebraic space $X$ to $S$", "which is locally of finite type.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Assume that", "\\begin{enumerate}", "\\item every point of $\\text{Ass}_{X/S}(\\mathcal{F})$ specializes", "to a point of the closed fibre $X_s$\\footnote{For example this holds if", "$f$ is finite type and $\\mathcal{F}$ is pure along $X_s$, or", "if $f$ is proper.},", "\\item $\\mathcal{F}$ is flat over $S$ at every point of $X_s$.", "\\end{enumerate}", "Then $\\mathcal{F}$ is flat over $S$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the fact that it suffices to check for", "flatness at points of the relative assassin of $\\mathcal{F}$", "over $S$ by", "Theorem \\ref{theorem-check-flatness-at-associated-points}." ], "refs": [ "spaces-flat-theorem-check-flatness-at-associated-points" ], "ref_ids": [ 7145 ] } ], "ref_ids": [] }, { "id": 7164, "type": "theorem", "label": "spaces-flat-lemma-finite-presentation-flat-along-fibre", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-finite-presentation-flat-along-fibre", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $y \\in |Y|$. Set $F = f^{-1}(\\{y\\}) \\subset |X|$. Assume that", "\\begin{enumerate}", "\\item $f$ is of finite presentation,", "\\item $\\mathcal{F}$ is of finite presentation, and", "\\item $\\mathcal{F}$ is flat over $Y$ at all $x \\in F$.", "\\end{enumerate}", "Then there exists a commutative diagram of algebraic spaces", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\", "Y & Y' \\ar[l]_h", "}", "$$", "such that $h$ and $g$ are \\'etale, there is a point", "$y' \\in |Y'|$ mapping to $y$, we have $F \\subset g(|X'|)$,", "the algebraic spaces $X'$, $Y'$ are affine, and", "$\\Gamma(X', g^*\\mathcal{F})$ is a projective", "$\\Gamma(Y', \\mathcal{O}_{Y'})$-module." ], "refs": [], "proofs": [ { "contents": [ "As formulated this lemma immmediately reduces", "to the case of schemes, which is", "More on Flatness, Lemma", "\\ref{flat-lemma-finite-presentation-flat-along-fibre}." ], "refs": [ "flat-lemma-finite-presentation-flat-along-fibre" ], "ref_ids": [ 6031 ] } ], "ref_ids": [] }, { "id": 7165, "type": "theorem", "label": "spaces-flat-lemma-associated-point-specializes", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-associated-point-specializes", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space", "locally of finite type over $S$.", "Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module.", "Let $s \\in S$ such that $\\mathcal{F}$ is flat over $S$ at all points of $X_s$.", "Let $x' \\in \\text{Ass}_{X/S}(\\mathcal{F})$. If", "the closure of $\\{x'\\}$ in $|X|$ meets $|X_s|$, then the closure", "meets $\\text{Ass}_{X/S}(\\mathcal{F}) \\cap |X_s|$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $|X_s| \\subset |X|$ is the set of points of $|X|$", "lying over $s \\in S$, see", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-topology-fibre}.", "Let $t \\in |X_s|$ be a specialization of $x'$ in $|X|$.", "Choose an affine scheme $U$ and a point $u \\in U$ and", "an \\'etale morphism $\\varphi : U \\to X$ mapping $u$ to $t$.", "By Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-specialization}", "we can choose a specialization $u' \\leadsto u$", "with $u'$ mapping to $x'$. Set $g = f \\circ \\varphi$.", "Observe that $s' = g(u') = f(x')$ specializes to $s$.", "By our definition of $\\text{Ass}_{X/S}(\\mathcal{F})$", "we have $u' \\in \\text{Ass}_{U/S}(\\varphi^*\\mathcal{F})$.", "By the schemes version of this lemma", "(More on Flatness, Lemma \\ref{flat-lemma-associated-point-specializes})", "we see that there is a specialization $u' \\leadsto u$ with", "$u \\in \\text{Ass}_{U_s}(\\varphi^*\\mathcal{F}_s) =", "\\text{Ass}_{U/S}(\\varphi^*\\mathcal{F}) \\cap U_s$.", "Hence $x = \\varphi(u) \\in \\text{Ass}_{X/S}(\\mathcal{F})$", "lies over $s$ and the lemma is proved." ], "refs": [ "decent-spaces-lemma-topology-fibre", "decent-spaces-lemma-decent-specialization", "flat-lemma-associated-point-specializes" ], "ref_ids": [ 9525, 9494, 6065 ] } ], "ref_ids": [] }, { "id": 7166, "type": "theorem", "label": "spaces-flat-lemma-contains-relative-ass-after-base-change", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-contains-relative-ass-after-base-change", "contents": [ "Let $Y$ be an algebraic space over a scheme $S$. Let $g : X' \\to X$ be a", "morphism of algebraic spaces over $Y$ with $X$ locally of finite type over $Y$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "If $\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$, then for any morphism", "$Z \\to Y$ we have $\\text{Ass}_{X_Z/Z}(\\mathcal{F}_Z) \\subset g_Z(|X'_Z|)$." ], "refs": [], "proofs": [ { "contents": [ "By Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}", "the map $|X'_Z| \\to |X_Z| \\times_{|X|} |X'|$ is surjective as", "$X'_Z$ is equal to $X_Z \\times_X X'$.", "By Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-base-change-relative-assassin}", "the map $|X_Z| \\to |X|$ sends $\\text{Ass}_{X_Z/Z}(\\mathcal{F}_Z)$", "into $\\text{Ass}_{X/Y}(\\mathcal{F})$. The lemma follows." ], "refs": [ "spaces-properties-lemma-points-cartesian", "spaces-divisors-lemma-base-change-relative-assassin" ], "ref_ids": [ 11819, 12924 ] } ], "ref_ids": [] }, { "id": 7167, "type": "theorem", "label": "spaces-flat-lemma-pure-on-top", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-pure-on-top", "contents": [ "Let $Y$ be an algebraic space over a scheme $S$. Let $g : X' \\to X$ be an", "\\'etale morphism of algebraic spaces over $Y$. Assume the structure", "morphisms $X' \\to Y$ and $X \\to Y$ are decent and of finite type.", "Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module.", "Let $y \\in |Y|$. Set $F = f^{-1}(\\{y\\}) \\subset |X|$.", "\\begin{enumerate}", "\\item If $\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$", "and $g^*\\mathcal{F}$ is (universally) pure above $y$, then", "$\\mathcal{F}$ is (universally) pure above $y$.", "\\item If $\\mathcal{F}$ is pure above $y$, $g(|X'|)$ contains $F$, and", "$Y$ is affine local with closed point $y$, then", "$\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$.", "\\item If $\\mathcal{F}$ is pure above $y$, $\\mathcal{F}$ is flat", "at all points of $F$, $g(|X'|)$ contains", "$\\text{Ass}_{X/Y}(\\mathcal{F}) \\cap F$, and $Y$ is affine local", "with closed point $y$, then", "$\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$.", "\\item Add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The assumptions on $X \\to Y$ and $X' \\to Y$ guarantee that", "we may apply the material in Sections \\ref{section-impure} and", "\\ref{section-pure}", "to these morphisms and the sheaves $\\mathcal{F}$ and $g^*\\mathcal{F}$.", "Since $g$ is \\'etale we see that", "$\\text{Ass}_{X'/Y}(g^*\\mathcal{F})$", "is the inverse image of $\\text{Ass}_{X/Y}(\\mathcal{F})$", "and the same remains true after base change.", "\\medskip\\noindent", "Proof of (1). Assume $\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$.", "Suppose that $(T \\to Y, t' \\leadsto t, \\xi)$", "is an impurity of $\\mathcal{F}$ above $y$. Since", "$\\text{Ass}_{X_T/T}(\\mathcal{F}_T) \\subset g_T(|X'_T|)$ by", "Lemma \\ref{lemma-contains-relative-ass-after-base-change}", "we can choose", "a point $\\xi' \\in |X'_T|$ mapping to $\\xi$. By the above we see", "that $(T \\to Y, t' \\leadsto t, \\xi')$ is an impurity of", "$g^*\\mathcal{F}$ above $y'$. This implies (1) is true.", "\\medskip\\noindent", "Proof of (2). This follows from the fact that $g(|X'|)$ is open", "in $|X|$ and the fact that by purity every point of", "$\\text{Ass}_{X/Y}(\\mathcal{F})$ specializes to a point of $F$.", "\\medskip\\noindent", "Proof of (3). This follows from the fact that $g(|X'|)$ is open", "in $|X|$ and the fact that by purity combined with", "Lemma \\ref{lemma-associated-point-specializes} every point of", "$\\text{Ass}_{X/Y}(\\mathcal{F})$ specializes to a point of", "$\\text{Ass}_{X/Y}(\\mathcal{F}) \\cap F$." ], "refs": [ "spaces-flat-lemma-contains-relative-ass-after-base-change", "spaces-flat-lemma-associated-point-specializes" ], "ref_ids": [ 7166, 7165 ] } ], "ref_ids": [] }, { "id": 7168, "type": "theorem", "label": "spaces-flat-lemma-finite-type-flat-pure-along-fibre-is-universal", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-finite-type-flat-pure-along-fibre-is-universal", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $y \\in |Y|$.", "Assume", "\\begin{enumerate}", "\\item $f$ is decent and of finite type,", "\\item $\\mathcal{F}$ is of finite type,", "\\item $\\mathcal{F}$ is flat over $Y$ at all points lying over $y$, and", "\\item $\\mathcal{F}$ is pure above $y$.", "\\end{enumerate}", "Then $\\mathcal{F}$ is universally pure above $y$." ], "refs": [], "proofs": [ { "contents": [ "Consider the morphism $\\Spec(\\mathcal{O}_{Y, \\overline{y}}) \\to Y$.", "This is a flat morphism from the spectrum of a stricly henselian", "local ring which maps the closed point to $y$.", "By Lemma \\ref{lemma-flat-descend-pure} we reduce to the case", "described in the next paragraph.", "\\medskip\\noindent", "Assume $Y$ is the spectrum of a strictly henselian local ring $R$", "with closed point $y$.", "By Lemma \\ref{lemma-finite-type-flat-along-fibre-free-variant}", "there exists an \\'etale morphism $g : X' \\to X$ with", "$g(|X'|) \\supset |X_y|$, with $X'$ affine, and with", "$\\Gamma(X', g^*\\mathcal{F})$ a free $R$-module.", "Then $g^*\\mathcal{F}$ is universally pure relative to $Y$, see", "More on Flatness, Lemma \\ref{flat-lemma-affine-locally-projective-pure}.", "Hence it suffices to prove that", "$g(|X'|)$ contains $\\text{Ass}_{X/Y}(\\mathcal{F})$", "by Lemma \\ref{lemma-pure-on-top} part (1).", "This in turn follows from", "Lemma \\ref{lemma-pure-on-top} part (2)." ], "refs": [ "spaces-flat-lemma-flat-descend-pure", "spaces-flat-lemma-finite-type-flat-along-fibre-free-variant", "flat-lemma-affine-locally-projective-pure", "spaces-flat-lemma-pure-on-top", "spaces-flat-lemma-pure-on-top" ], "ref_ids": [ 7155, 7162, 6064, 7167, 7167 ] } ], "ref_ids": [] }, { "id": 7169, "type": "theorem", "label": "spaces-flat-lemma-finite-type-flat-pure-is-universal", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-finite-type-flat-pure-is-universal", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a decent, finite type morphism of algebraic", "spaces over $S$.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Assume $\\mathcal{F}$ is flat over $Y$. In this case", "$\\mathcal{F}$ is pure relative to $Y$ if and only if $\\mathcal{F}$", "is universally pure relative to $Y$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Lemma \\ref{lemma-finite-type-flat-pure-along-fibre-is-universal}", "and the definitions." ], "refs": [ "spaces-flat-lemma-finite-type-flat-pure-along-fibre-is-universal" ], "ref_ids": [ 7168 ] } ], "ref_ids": [] }, { "id": 7170, "type": "theorem", "label": "spaces-flat-lemma-universally-separating", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-universally-separating", "contents": [ "Let $Y$ be an algebraic space over a scheme $S$.", "Let $g : X' \\to X$ be a flat morphism of algebraic spaces over $Y$", "with $X$ locally of finite type over $Y$.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module", "which is flat over $Y$. If $\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$", "then the canonical map", "$$", "\\mathcal{F} \\longrightarrow g_*g^*\\mathcal{F}", "$$", "is injective, and remains injective after any base change." ], "refs": [], "proofs": [ { "contents": [ "The final assertion means that $\\mathcal{F}_Z \\to (g_Z)_*g_Z^*\\mathcal{F}_Z$", "is injective for any morphism $Z \\to Y$. Since the assumption on", "the relative assassin is preserved by base change", "(Lemma \\ref{lemma-contains-relative-ass-after-base-change})", "it suffices to prove the injectivity of the displayed arrow.", "\\medskip\\noindent", "Let $\\mathcal{K} = \\Ker(\\mathcal{F} \\to g_*g^*\\mathcal{F})$.", "Our goal is to prove that $\\mathcal{K} = 0$.", "In order to do this it suffices to prove that", "$\\text{WeakAss}_X(\\mathcal{K}) = \\emptyset$, see", "Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-weakly-ass-zero}.", "We have", "$\\text{WeakAss}_X(\\mathcal{K}) \\subset \\text{WeakAss}_X(\\mathcal{F})$, see", "Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-ses-weakly-ass}.", "As $\\mathcal{F}$ is flat we see from", "Lemma \\ref{lemma-bourbaki-finite-type-general-base}", "that $\\text{WeakAss}_X(\\mathcal{F}) \\subset \\text{Ass}_{X/Y}(\\mathcal{F})$.", "By assumption any point $x$ of $\\text{Ass}_{X/Y}(\\mathcal{F})$", "is the image of some $x' \\in |X'|$. Since $g$ is flat the", "local ring map", "$\\mathcal{O}_{X, \\overline{x}} \\to \\mathcal{O}_{X', \\overline{x}'}$", "is faithfully flat, hence the map", "$$", "\\mathcal{F}_{\\overline{x}}", "\\longrightarrow", "(g^*\\mathcal{F})_{\\overline{x}'} =", "\\mathcal{F}_{\\overline{x}} \\otimes_{\\mathcal{O}_{X, \\overline{x}}}", "\\mathcal{O}_{X', \\overline{x}'}", "$$", "is injective (see", "Algebra, Lemma \\ref{algebra-lemma-faithfully-flat-universally-injective}).", "Since the displayed arrow factors through", "$\\mathcal{F}_{\\overline{x}} \\to (g_*g^*\\mathcal{F})_{\\overline{x}}$,", "we conclude that", "$\\mathcal{K}_{\\overline{x}} = 0$. Hence $x$ cannot be a weakly associated", "point of $\\mathcal{K}$ and we win." ], "refs": [ "spaces-flat-lemma-contains-relative-ass-after-base-change", "spaces-divisors-lemma-weakly-ass-zero", "spaces-divisors-lemma-ses-weakly-ass", "spaces-flat-lemma-bourbaki-finite-type-general-base", "algebra-lemma-faithfully-flat-universally-injective" ], "ref_ids": [ 7166, 12905, 12904, 7161, 814 ] } ], "ref_ids": [] }, { "id": 7171, "type": "theorem", "label": "spaces-flat-lemma-iso-sheaf", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-iso-sheaf", "contents": [ "In Situation \\ref{situation-iso}.", "Each of the functors $F_{iso}$, $F_{inj}$, $F_{surj}$, $F_{zero}$", "satisfies the sheaf property for the fpqc topology." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{T_i \\to T\\}_{i \\in I}$ be an fpqc covering of schemes over $B$.", "Set $X_i = X_{T_i} = X \\times_S T_i$ and $u_i = u_{T_i}$.", "Note that $\\{X_i \\to X_T\\}_{i \\in I}$ is an fpqc covering of $X_T$, see", "Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-fpqc}.", "In particular, for every $x \\in |X_T|$ there exists an $i \\in I$", "and an $x_i \\in |X_i|$ mapping to $x$. Since", "$\\mathcal{O}_{X_T, \\overline{x}} \\to \\mathcal{O}_{X_i, \\overline{x_i}}$", "is flat, hence faithfully flat (see", "Morphisms of Spaces, Section \\ref{spaces-morphisms-section-flat}).", "we conclude that $(u_i)_{x_i}$ is injective, surjective, bijective, or zero", "if and only if $(u_T)_x$ is injective, surjective, bijective, or zero.", "The lemma follows." ], "refs": [ "spaces-topologies-lemma-fpqc" ], "ref_ids": [ 3678 ] } ], "ref_ids": [] }, { "id": 7172, "type": "theorem", "label": "spaces-flat-lemma-iso-go-up", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-iso-go-up", "contents": [ "In Situation \\ref{situation-iso} let $X' \\to X$ be a flat morphism", "of algebraic spaces. Denote $u' : \\mathcal{F}' \\to \\mathcal{G}'$", "the pullback of $u$ to $X'$. Denote $F'_{iso}$, $F'_{inj}$, $F'_{surj}$,", "$F'_{zero}$ the functors on $\\Sch/B$ associated to $u'$.", "\\begin{enumerate}", "\\item If $\\mathcal{G}$ is of finite type and the image of $|X'| \\to |X|$", "contains the support of $\\mathcal{G}$, then $F_{surj} = F'_{surj}$", "and $F_{zero} = F'_{zero}$.", "\\item If $\\mathcal{F}$ is of finite type and the image of $|X'| \\to |X|$", "contains the support of $\\mathcal{F}$, then $F_{inj} = F'_{inj}$", "and $F_{zero} = F'_{zero}$.", "\\item If $\\mathcal{F}$ and $\\mathcal{G}$ are of finite type and the image of", "$|X'| \\to |X|$ contains the supports of $\\mathcal{F}$ and $\\mathcal{G}$,", "then $F_{iso} = F'_{iso}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "let $v : \\mathcal{H} \\to \\mathcal{E}$ be a map of quasi-coherent", "modules on an algebraic space $Y$ and let $\\varphi : Y' \\to Y$ be a", "surjective flat morphism of algebraic spaces, then $v$ is", "an isomorphism, injective, surjective, or zero if and only if $\\varphi^*v$ is", "an isomorphism, injective, surjective, or zero. Namely,", "for every $y \\in |Y|$ there exists a $y' \\in |Y'|$ and the map", "of local rings", "$\\mathcal{O}_{Y, \\overline{y}} \\to \\mathcal{O}_{Y', \\overline{y'}}$", "is faithfully flat (see", "Morphisms of Spaces, Section \\ref{spaces-morphisms-section-flat}).", "Of course, to check for injectivity or being zero it suffices to look", "at the points in the support of $\\mathcal{H}$, and to check for", "surjectivity it suffices to look at points in the support of $\\mathcal{E}$.", "Moreover, under the finite type assumptions as in the statement of", "the lemma, taking the supports commutes with base change, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-support-finite-type}.", "Thus the lemma is clear." ], "refs": [ "spaces-morphisms-lemma-support-finite-type" ], "ref_ids": [ 4777 ] } ], "ref_ids": [] }, { "id": 7173, "type": "theorem", "label": "spaces-flat-lemma-iso-limits", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-iso-limits", "contents": [ "In Situation \\ref{situation-iso}.", "\\begin{enumerate}", "\\item If $\\mathcal{G}$ is of finite type and the scheme theoretic support", "of $\\mathcal{G}$ is quasi-compact over $B$, then $F_{surj}$ is limit", "preserving.", "\\item If $\\mathcal{F}$ of finite type and the scheme theoretic support", "of $\\mathcal{F}$ is quasi-compact over $B$, then", "$F_{zero}$ is limit preserving.", "\\item If $\\mathcal{F}$ is of finite type,", "$\\mathcal{G}$ is of finite presentation, and the", "scheme theoretic supports of $\\mathcal{F}$ and $\\mathcal{G}$ are", "quasi-compact over $B$, then $F_{iso}$ is limit preserving.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $i : Z \\to X$ be the scheme theoretic support of", "$\\mathcal{G}$ and think of $\\mathcal{G}$ as a finite type quasi-coherent", "module on $Z$. We may replace $X$ by $Z$ and $u$ by the map", "$i^*\\mathcal{F} \\to \\mathcal{G}$ (details omitted). Hence we may assume", "$f$ is quasi-compact and $\\mathcal{G}$ of finite type.", "Let $T = \\lim_{i \\in I} T_i$ be a directed limit of affine $B$-schemes", "and assume that $u_T$ is surjective.", "Set $X_i = X_{T_i} = X \\times_S T_i$ and", "$u_i = u_{T_i} : \\mathcal{F}_i = \\mathcal{F}_{T_i}", "\\to \\mathcal{G}_i = \\mathcal{G}_{T_i}$.", "To prove (1) we have to show that $u_i$ is surjective for some $i$.", "Pick $0 \\in I$ and replace $I$ by $\\{i \\mid i \\geq 0\\}$.", "Since $f$ is quasi-compact we see $X_0$ is quasi-compact.", "Hence we may choose a surjective \\'etale morphism $\\varphi_0 : W_0 \\to X_0$", "where $W_0$ is an affine scheme. Set $W = W_0 \\times_{T_0} T$", "and $W_i = W_0 \\times_{T_0} T_i$ for $i \\geq 0$. These", "are affine schemes endowed", "with a surjective \\'etale morphisms $\\varphi : W \\to X_T$ and", "$\\varphi_i : W_i \\to X_i$. Note that $W = \\lim W_i$.", "Hence $\\varphi^*u_T$ is surjective and it suffices to prove that", "$\\varphi_i^*u_i$ is surjective for some $i$. Thus we have reduced", "the problem to the affine case which is", "Algebra, Lemma \\ref{algebra-lemma-module-map-property-in-colimit} part (2).", "\\medskip\\noindent", "Proof of (2). Assume $\\mathcal{F}$ is of finite type with scheme theoretic", "support $Z \\subset B$ quasi-compact over $B$. Let $T = \\lim_{i \\in I} T_i$", "be a directed limit of affine $B$-schemes and assume that $u_T$ is zero.", "Set $X_i = T_i \\times_B X$ and denote $u_i : \\mathcal{F}_i \\to \\mathcal{G}_i$", "the pullback. Choose $0 \\in I$ and replace $I$ by", "$\\{i \\mid i \\geq 0\\}$. Set $Z_0 = Z \\times_X X_0$. By", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-support-finite-type}", "the support of $\\mathcal{F}_i$ is $|Z_0|$. Since $|Z_0|$ is quasi-compact", "we can find an affine scheme $W_0$ and an \\'etale morphism $W_0 \\to X_0$", "such that $|Z_0| \\subset \\Im(|W_0| \\to |X_0|)$.", "Set $W = W_0 \\times_{T_0} T$ and $W_i = W_0 \\times_{T_0} T_i$ for $i \\geq 0$.", "These are affine schemes endowed", "with \\'etale morphisms $\\varphi : W \\to X_T$ and", "$\\varphi_i : W_i \\to X_i$. Note that $W = \\lim W_i$", "and that the support of $\\mathcal{F}_T$ and $\\mathcal{F}_i$", "is contained in the image of $|W| \\to |X_T|$ and $|W_i| \\to |X_i|$.", "Now $\\varphi^*u_T$ is injective and it suffices to prove that", "$\\varphi_i^*u_i$ is injective for some $i$.", "Thus we have reduced the problem to the affine case which is", "Algebra, Lemma \\ref{algebra-lemma-module-map-property-in-colimit} part (1).", "\\medskip\\noindent", "Proof of (3). This can be proven in exactly the same manner as in the", "previous two paragraphs using", "Algebra, Lemma \\ref{algebra-lemma-module-map-property-in-colimit} part (3).", "We can also deduce it from (1) and (2) as follows.", "Let $T = \\lim_{i \\in I} T_i$ be a directed limit of affine $B$-schemes", "and assume that $u_T$ is an isomorphism. By part (1) there exists", "an $0 \\in I$ such that $u_{T_0}$ is surjective. Set", "$\\mathcal{K} = \\Ker(u_{T_0})$ and consider the map of quasi-coherent", "modules $v : \\mathcal{K} \\to \\mathcal{F}_{T_0}$. For $i \\geq 0$ the base", "change $v_{T_i}$ is zero if and only if $u_i$ is an isomorphism. Moreover,", "$v_T$ is zero. Since $\\mathcal{G}_{T_0}$", "is of finite presentation, $\\mathcal{F}_{T_0}$ is of finite type, and", "$u_{T_0}$ is surjective we conclude that $\\mathcal{K}$ is of finite type", "(Modules on Sites, Lemma", "\\ref{sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation}).", "It is clear that the support of $\\mathcal{K}$ is contained in the", "support of $\\mathcal{F}_{T_0}$ which is quasi-compact over $T_0$.", "Hence we can apply part (2) to see that $v_{T_i}$ is zero for some $i$." ], "refs": [ "algebra-lemma-module-map-property-in-colimit", "spaces-morphisms-lemma-support-finite-type", "algebra-lemma-module-map-property-in-colimit", "algebra-lemma-module-map-property-in-colimit", "sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation" ], "ref_ids": [ 1094, 4777, 1094, 1094, 14187 ] } ], "ref_ids": [] }, { "id": 7174, "type": "theorem", "label": "spaces-flat-lemma-relate-zero-iso", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-relate-zero-iso", "contents": [ "In Situation \\ref{situation-iso} suppose given an exact sequence", "$$", "\\mathcal{F} \\xrightarrow{u} \\mathcal{G} \\xrightarrow{v} \\mathcal{H} \\to 0", "$$", "Then we have $F_{v, iso} = F_{u, zero}$ with obvious notation." ], "refs": [], "proofs": [ { "contents": [ "Since pullback is right exact we see that", "$\\mathcal{F}_T \\to \\mathcal{G}_T \\to \\mathcal{H}_T \\to 0$", "is exact for every scheme $T$ over $B$. Hence $u_T$ is", "surjective if and only if $v_T$ is an isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7175, "type": "theorem", "label": "spaces-flat-lemma-relate-zero-affine", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-relate-zero-affine", "contents": [ "In Situation \\ref{situation-iso} suppose given an affine morphism", "$i : Z \\to X$ and a quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{H}$", "such that $\\mathcal{G} = i_*\\mathcal{H}$. Let", "$v : i^*\\mathcal{F} \\to \\mathcal{H}$ be the map adjoint to $u$.", "Then", "\\begin{enumerate}", "\\item $F_{v, zero} = F_{u, zero}$, and", "\\item if $i$ is a closed immersion, then $F_{v, surj} = F_{u, surj}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme over $B$. Denote $i_T : Z_T \\to X_T$", "the base change of $i$ and $\\mathcal{H}_T$ the pullback of $\\mathcal{H}$", "to $Z_T$. Observe that $(i^*\\mathcal{F})_T = i_T^*\\mathcal{F}_T$", "and $i_{T, *}\\mathcal{H}_T = (i_*\\mathcal{H})_T$.", "The first statement follows from commutativity of pullbacks", "and the second from Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-affine-base-change}.", "Hence we see that $u_T$ and $v_T$ are adjoint maps as well.", "Thus $u_T = 0$ if and only if $v_T = 0$. This proves (1).", "In case (2) we see that $u_T$ is surjective if and only if", "$v_T$ is surjective because $u_T$ factors as", "$$", "\\mathcal{F}_T \\to", "i_{T, *}i_T^*\\mathcal{F}_T \\xrightarrow{i_{T, *}v_T} i_{T, *}\\mathcal{H}_T", "$$", "and the fact that $i_{T, *}$ is an exact functor", "fully faithfully embedding the category of quasi-coherent modules on", "$Z_T$ into the category of quasi-coherent $\\mathcal{O}_{X_T}$-modules.", "See Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-i-star-equivalence}." ], "refs": [ "spaces-cohomology-lemma-affine-base-change", "spaces-morphisms-lemma-i-star-equivalence" ], "ref_ids": [ 11295, 4771 ] } ], "ref_ids": [] }, { "id": 7176, "type": "theorem", "label": "spaces-flat-lemma-relate-zero-affine-push", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-relate-zero-affine-push", "contents": [ "In Situation \\ref{situation-iso} suppose given an affine morphism", "$g : X \\to X'$. Set $u' = f_*u : f_*\\mathcal{F} \\to f_*\\mathcal{G}$.", "Then $F_{u, iso} = F_{u', iso}$, $F_{u, inj} = F_{u', inj}$,", "$F_{u, surj} = F_{u', surj}$, and $F_{u, zero} = F_{u', zero}$." ], "refs": [], "proofs": [ { "contents": [ "By Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-affine-base-change}", "we have $g_{T, *}u_T = u'_T$.", "Moreover, $g_{T, *} : \\QCoh(\\mathcal{O}_{X_T}) \\to \\QCoh(\\mathcal{O}_X)$", "is a faithful, exact functor reflecting isomorphisms, injective maps,", "and surjective maps." ], "refs": [ "spaces-cohomology-lemma-affine-base-change" ], "ref_ids": [ 11295 ] } ], "ref_ids": [] }, { "id": 7177, "type": "theorem", "label": "spaces-flat-lemma-flat", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-flat", "contents": [ "In Situation \\ref{situation-flat}.", "\\begin{enumerate}", "\\item The functor $F_{flat}$ satisfies the sheaf property for the fpqc topology.", "\\item If $f$ is quasi-compact and locally of finite presentation", "and $\\mathcal{F}$ is of finite presentation, then the functor", "$F_{flat}$ is limit preserving.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from the following statement: If $T' \\to T$ is a surjective", "flat morphism of algebraic spaces over $Y$, then", "$\\mathcal{F}_{T'}$ is flat over $T'$ if and only if", "$\\mathcal{F}_T$ is flat over $T$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-module-flat}.", "Part (2) follows from", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-descend-flat}", "if $f$ is also quasi-separated (i.e., $f$ is of finite presentation).", "For the general case, first reduce to the case where the", "base is affine and then cover $X$ by finitely many affines", "to reduce to the quasi-separated case. Details omitted." ], "refs": [ "spaces-morphisms-lemma-base-change-module-flat", "spaces-limits-lemma-descend-flat" ], "ref_ids": [ 4863, 4595 ] } ], "ref_ids": [] }, { "id": 7178, "type": "theorem", "label": "spaces-flat-lemma-F-zero-somewhat-closed", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-F-zero-somewhat-closed", "contents": [ "In Situation \\ref{situation-somewhat-closed}. Let $T \\to S$", "be a quasi-compact morphism of schemes such that the base change $u_T$ is", "zero. Then exists a closed subscheme $Z \\subset S$ such that", "(a) $T \\to S$ factors through $Z$ and (b) the base change $u_Z$ is zero.", "If $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module and", "the scheme theoretic support of $\\mathcal{F}$ is quasi-compact,", "then we can take $Z \\to S$ of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be a surjective \\'etale morphism of algebraic spaces", "where $U = \\coprod U_i$ is a disjoint union of affine schemes (see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-cover-by-union-affines}).", "By Lemma \\ref{lemma-iso-go-up} we see that we may", "replace $X$ by $U$. In other words, we may assume that $X = \\coprod X_i$", "is a disjoint union of affine schemes $X_i$. Suppose that we can prove", "the lemma for $u_i = u|_{X_i}$. Then we find a closed subscheme", "$Z_i \\subset S$ such that $T \\to S$ factors through $Z_i$ and", "$u_{i, Z_i}$ is zero. If", "$Z_i = \\Spec(R/I_i) \\subset \\Spec(R) = S$, then taking", "$Z = \\Spec(R/\\sum I_i)$ works. Thus we may assume that", "$X = \\Spec(A)$ is affine.", "\\medskip\\noindent", "Choose a finite affine open covering $T = T_1 \\cup \\ldots \\cup T_m$.", "It is clear that we may replace $T$ by $\\coprod_{j = 1, \\ldots, m} T_j$.", "Hence we may assume $T$ is affine. Say $T = \\Spec(R')$.", "Let $u : M \\to N$ be the homomorphisms of $A$-modules", "corresponding to $u : \\mathcal{F} \\to \\mathcal{G}$.", "Then $N$ is a flat $R$-module as $\\mathcal{G}$ is flat over $S$.", "The assumption of the lemma means that the composition", "$$", "M \\otimes_R R' \\to N \\otimes_R R'", "$$", "is zero. Let $z \\in M$. By Lazard's theorem", "(Algebra, Theorem \\ref{algebra-theorem-lazard}) and the fact", "that $\\otimes$ commutes with colimits we can find free $R$-module", "$F_z$, an element $\\tilde z \\in F_z$, and a map $F_z \\to N$ such that", "$u(z)$ is the image of $\\tilde z$ and $\\tilde z$ maps to zero in", "$F_z \\otimes_R R'$. Choose a basis $\\{e_{z, \\alpha}\\}$ of $F_z$ and write", "$\\tilde z = \\sum f_{z, \\alpha} e_{z, \\alpha}$ with $f_{z, \\alpha} \\in R$.", "Let $I \\subset R$ be the ideal generated by the elements $f_{z, \\alpha}$", "with $z$ ranging over all elements of $M$.", "By construction $I$ maps to zero in $R'$ and the elements $\\tilde z$", "map to zero in $F_z/IF_z$ whence in $N/IN$. Thus $Z = \\Spec(R/I)$", "is a solution to the problem in this case.", "\\medskip\\noindent", "Assume $\\mathcal{F}$ is of finite type with quasi-compact scheme", "theoretic support. Write $Z = \\Spec(R/I)$.", "Write $I = \\bigcup I_\\lambda$ as a filtered union of finitely generated", "ideals. Set $Z_\\lambda = \\Spec(R/I_\\lambda)$, so $Z = \\colim Z_\\lambda$.", "Since $u_Z$ is zero, we see that $u_{Z_\\lambda}$ is zero", "for some $\\lambda$ by Lemma \\ref{lemma-iso-limits}.", "This finishes the proof of the lemma." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-flat-lemma-iso-go-up", "algebra-theorem-lazard", "spaces-flat-lemma-iso-limits" ], "ref_ids": [ 11830, 7172, 318, 7173 ] } ], "ref_ids": [] }, { "id": 7179, "type": "theorem", "label": "spaces-flat-lemma-F-zero-module-map", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-F-zero-module-map", "contents": [ "Let $A$ be a ring. Let $u : M \\to N$ be a map of $A$-modules.", "If $N$ is projective as an $A$-module, then there exists an ideal", "$I \\subset A$ such that for any ring map $\\varphi : A \\to B$", "the following are equivalent", "\\begin{enumerate}", "\\item $u \\otimes 1 : M \\otimes_A B \\to N \\otimes_A B$ is zero, and", "\\item $\\varphi(I) = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As $N$ is projective we can find a projective $A$-module $C$", "such that $F = N \\oplus C$ is a free $R$-module.", "By replacing $u$ by $u \\oplus 1 : F = M \\oplus C \\to N \\oplus C$", "we see that we may assume $N$ is free. In this case let $I$ be", "the ideal of $A$ generated by coefficients of all the elements of", "$\\Im(u)$ with respect to some (fixed) basis of $N$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7180, "type": "theorem", "label": "spaces-flat-lemma-F-zero-somewhat-closed-points", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-F-zero-somewhat-closed-points", "contents": [ "In Situation \\ref{situation-somewhat-closed}.", "Let $T \\subset S$ be a subset. Let $s \\in S$ be in the closure of $T$.", "For $t \\in T$, let $u_t$ be the pullback of $u$ to $X_t$", "and let $u_s$ be the pullback of $u$ to $X_s$.", "If $X$ is locally of finite presentation over $S$,", "$\\mathcal{G}$ is of finite presentation\\footnote{It would", "suffice if $X$ is locally of finite type over $S$", "and $\\mathcal{G}$ is finitely presented relative to $S$,", "but this notion hasn't yet been defined in the setting", "of algebraic spaces. The definition for schemes is", "given in More on Morphisms, Section", "\\ref{more-morphisms-section-finite-type-finite-presentation}.}, and", "$u_t = 0$ for all $t \\in T$, then $u_s = 0$." ], "refs": [], "proofs": [ { "contents": [ "To check whether $u_s$ is zero, is \\'etale local on the fibre $X_s$.", "Hence we may pick a point $x \\in |X_s| \\subset |X|$ and check", "in an \\'etale neighbourhood. Choose", "$$", "\\xymatrix{", "(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\", "(S, s) & (S', s') \\ar[l]", "}", "$$", "as in Proposition \\ref{proposition-finite-presentation-flat-at-point}.", "Let $T' \\subset S'$ be the inverse image of $T$. Observe that", "$s'$ is in the closure of $T'$ because $S' \\to S$ is open.", "Hence we reduce to the algebra problem described in the", "next paragraph.", "\\medskip\\noindent", "We have an $R$-module map $u : M \\to N$ such that $N$ is projective", "as an $R$-module and such that", "$u_t : M \\otimes_R \\kappa(t) \\to N \\otimes_R \\kappa(t)$", "is zero for each $t \\in T$. Problem: show that $u_s = 0$.", "Let $I \\subset R$ be the ideal defined in Lemma \\ref{lemma-F-zero-module-map}.", "Then $I$ maps to zero in $\\kappa(t)$ for all $t \\in T$.", "Hence $T \\subset V(I)$. Since $s$ is in the closure of $T$", "we see that $s \\in V(I)$. Hence $u_s = 0$." ], "refs": [ "spaces-flat-proposition-finite-presentation-flat-at-point", "spaces-flat-lemma-F-zero-module-map" ], "ref_ids": [ 7200, 7179 ] } ], "ref_ids": [] }, { "id": 7181, "type": "theorem", "label": "spaces-flat-lemma-F-zero-closed-pure", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-F-zero-closed-pure", "contents": [ "In Situation \\ref{situation-iso}. Assume", "\\begin{enumerate}", "\\item $f$ is of finite presentation, and", "\\item $\\mathcal{G}$ is of finite presentation,", "flat over $B$, and pure relative to $B$.", "\\end{enumerate}", "Then $F_{zero}$ is an algebraic space and $F_{zero} \\to B$", "is a closed immersion. If $\\mathcal{F}$ is of finite type, then", "$F_{zero} \\to B$ is of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-type-flat-pure-is-universal}", "the module $\\mathcal{G}$ is universally pure relative to $B$.", "In order to prove that $F_{zero}$ is an algebraic space,", "it suffices to show that $F_{zero} \\to B$ is representable, see", "Spaces, Lemma \\ref{spaces-lemma-representable-over-space}.", "Let $B' \\to B$ be a morphism where $B'$ is a scheme and let", "$u' : \\mathcal{F}' \\to \\mathcal{G}'$ be the pullback of $u$ to $X' = X_{B'}$.", "Then the associated functor $F'_{zero}$ equals $F_{zero} \\times_B B'$.", "This reduces us to the case that $B$ is a scheme.", "\\medskip\\noindent", "Assume $B$ is a scheme. We will show that $F_{zero}$ is representable", "by a closed subscheme of $B$. By Lemma \\ref{lemma-iso-sheaf} and", "Descent, Lemmas \\ref{descent-lemma-closed-immersion} and", "\\ref{descent-lemma-descent-data-sheaves}", "the question is local for the \\'etale topology on $B$. Let $b \\in B$.", "We first replace $B$ by an affine neighbourhood of $b$.", "Choose a diagram", "$$", "\\xymatrix{", "X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\", "B & B' \\ar[l]", "}", "$$", "and $b' \\in B'$ mapping to $b \\in B$", "as in Lemma \\ref{lemma-finite-presentation-flat-along-fibre}.", "As we are working \\'etale locally, we may replace", "$B$ by $B'$ and assume that we have a diagram", "$$", "\\xymatrix{", "X \\ar[rd] & & X' \\ar[ll]^g \\ar[ld] \\\\", "& B", "}", "$$", "with $B$ and $X'$ affine such that $\\Gamma(X', g^*\\mathcal{G})$", "is a projective $\\Gamma(B, \\mathcal{O}_B)$-module and", "$g(|X'|) \\supset |X_b|$. Let $U \\subset X$ be the open subspace", "with $|U| = g(|X'|)$. By", "Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-relative-assassin-constructible} the set", "$$", "E = \\{t \\in B : \\text{Ass}_{X_t}(\\mathcal{G}_t) \\subset |U_t|\\} =", "\\{t \\in B : \\text{Ass}_{X/B}(\\mathcal{G}) \\cap |X_t| \\subset |U_t|\\}", "$$", "is constructible in $B$. By Lemma \\ref{lemma-pure-on-top} part (2)", "we see that $E$ contains $\\Spec(\\mathcal{O}_{B, b})$. By", "Morphisms, Lemma \\ref{morphisms-lemma-constructible-containing-open}", "we see that $E$ contains an open neighbourhood of $b$. Hence after", "replacing $B$ by a smaller affine neighbourhood of $b$ we may assume that", "$\\text{Ass}_{X/B}(\\mathcal{G}) \\subset g(|X'|)$.", "\\medskip\\noindent", "From Lemma \\ref{lemma-universally-separating}", "it follows that $u : \\mathcal{F} \\to \\mathcal{G}$ is injective if and only if", "$g^*u : g^*\\mathcal{F} \\to g^*\\mathcal{G}$ is injective, and the same remains", "true after any base change. Hence we have reduced to the case where,", "in addition to the assumptions in the theorem, $X \\to B$ is a morphism of", "affine schemes and $\\Gamma(X, \\mathcal{G})$ is a projective", "$\\Gamma(B, \\mathcal{O}_B)$-module. This case follows immediately from", "Lemma \\ref{lemma-F-zero-module-map}.", "\\medskip\\noindent", "We still have to show that $F_{zero} \\to B$ is of finite presentation if", "$\\mathcal{F}$ is of finite type. This follows from", "Lemma \\ref{lemma-iso-limits} combined with", "Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-characterize-locally-finite-presentation}." ], "refs": [ "spaces-flat-lemma-finite-type-flat-pure-is-universal", "spaces-lemma-representable-over-space", "spaces-flat-lemma-iso-sheaf", "descent-lemma-closed-immersion", "descent-lemma-descent-data-sheaves", "spaces-flat-lemma-finite-presentation-flat-along-fibre", "spaces-divisors-lemma-relative-assassin-constructible", "spaces-flat-lemma-pure-on-top", "morphisms-lemma-constructible-containing-open", "spaces-flat-lemma-universally-separating", "spaces-flat-lemma-F-zero-module-map", "spaces-flat-lemma-iso-limits", "spaces-limits-proposition-characterize-locally-finite-presentation" ], "ref_ids": [ 7169, 8156, 7171, 14749, 14751, 7164, 12927, 7167, 5251, 7170, 7179, 7173, 4655 ] } ], "ref_ids": [] }, { "id": 7182, "type": "theorem", "label": "spaces-flat-lemma-F-zero-closed-proper", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-F-zero-closed-proper", "contents": [ "In Situation \\ref{situation-iso}. Assume", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $\\mathcal{G}$ is an $\\mathcal{O}_X$-module of finite presentation", "flat over $B$,", "\\item the support of $\\mathcal{G}$ is proper over $B$.", "\\end{enumerate}", "Then the functor $F_{zero}$ is an algebraic space and $F_{zero} \\to B$", "is a closed immersion. If $\\mathcal{F}$ is of finite type, then", "$F_{zero} \\to B$ is of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "If $f$ is of finite presentation, then this follows immediately from", "Lemmas \\ref{lemma-F-zero-closed-pure} and \\ref{lemma-proper-pure}.", "This is the only case of interest and we urge the reader to skip", "the rest of the proof, which deals with the possibility (allowed", "by the assumptions in this lemma)", "that $f$ is not quasi-separated or quasi-compact.", "\\medskip\\noindent", "Let $i : Z \\to X$ be the closed subspace cut out by the zeroth", "fitting ideal of $\\mathcal{G}$", "(Divisors on Spaces, Section", "\\ref{spaces-divisors-section-fitting-ideals}).", "Then $Z \\to B$ is proper by assumption (see", "Derived Categories of Spaces, Section", "\\ref{spaces-perfect-section-proper-over-base}).", "On the other hand $i$ is of finite presentation", "(Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-fitting-ideal-of-finitely-presented} and", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-closed-immersion-finite-presentation}).", "There exists a quasi-coherent $\\mathcal{O}_Z$-module", "$\\mathcal{H}$ of finite type with $i_*\\mathcal{H} = \\mathcal{G}$", "(Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-on-subscheme-cut-out-by-Fit-0}).", "In fact $\\mathcal{H}$ is of finite presentation as an $\\mathcal{O}_Z$-module", "by Algebra, Lemma \\ref{algebra-lemma-finitely-presented-over-subring}", "(details omitted).", "Then $F_{zero}$ is the same as the functor $F_{zero}$", "for the map $i^*\\mathcal{F} \\to \\mathcal{H}$ adjoint to $u$, see", "Lemma \\ref{lemma-relate-zero-affine}.", "The sheaf $\\mathcal{H}$ is flat relative to $B$ because", "the same is true for $\\mathcal{G}$ (check on stalks; details omitted).", "Moreover, note that if $\\mathcal{F}$ is of finite type,", "then $i^*\\mathcal{F}$ is of finite type.", "Hence we have reduced the lemma to the case", "discussed in the first paragraph of the proof." ], "refs": [ "spaces-flat-lemma-F-zero-closed-pure", "spaces-flat-lemma-proper-pure", "spaces-divisors-lemma-fitting-ideal-of-finitely-presented", "spaces-morphisms-lemma-closed-immersion-finite-presentation", "spaces-divisors-lemma-on-subscheme-cut-out-by-Fit-0", "algebra-lemma-finitely-presented-over-subring", "spaces-flat-lemma-relate-zero-affine" ], "ref_ids": [ 7181, 7157, 12929, 4849, 12930, 335, 7175 ] } ], "ref_ids": [] }, { "id": 7183, "type": "theorem", "label": "spaces-flat-lemma-F-iso-closed", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-F-iso-closed", "contents": [ "In Situation \\ref{situation-iso}. Assume", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $\\mathcal{F}$ is locally of finite presentation and flat over $B$,", "\\item the support of $\\mathcal{F}$ is proper over $B$, and", "\\item $u$ is surjective.", "\\end{enumerate}", "Then the functor $F_{iso}$ is an algebraic space and $F_{iso} \\to B$", "is a closed immersion. If $\\mathcal{G}$ is of finite presentation, then", "$F_{iso} \\to B$ is of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{K} = \\Ker(u)$ and denote $v : \\mathcal{K} \\to \\mathcal{F}$", "the inclusion. By Lemma \\ref{lemma-relate-zero-iso} we see that", "$F_{u, iso} = F_{v, zero}$. By Lemma \\ref{lemma-F-zero-closed-proper}", "applied to $v$ we see that $F_{u, iso} = F_{v, zero}$ is representable", "by a closed subspace of $B$. Note that $\\mathcal{K}$ is of finite type", "if $\\mathcal{G}$ is of finite presentation, see", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation}.", "Hence we also obtain the final statement of the lemma." ], "refs": [ "spaces-flat-lemma-relate-zero-iso", "spaces-flat-lemma-F-zero-closed-proper", "sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation" ], "ref_ids": [ 7174, 7182, 14187 ] } ], "ref_ids": [] }, { "id": 7184, "type": "theorem", "label": "spaces-flat-lemma-F-surj-open", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-F-surj-open", "contents": [ "In Situation \\ref{situation-iso}. Assume", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $\\mathcal{G}$ is of finite type,", "\\item the support of $\\mathcal{G}$ is proper over $B$.", "\\end{enumerate}", "Then $F_{surj}$ is an algebraic space and $F_{surj} \\to B$", "is an open immersion." ], "refs": [], "proofs": [ { "contents": [ "Consider $\\Coker(u)$. Observe that", "$\\Coker(u_T) = \\Coker(u)_T$ for any $T/B$.", "Note that formation of the support of a finite type", "quasi-coherent module commutes with pullback", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-support-covering}).", "Hence $F_{surj}$ is representable by the open subspace of $B$", "corresponding to the open set", "$$", "|B| \\setminus |f|(\\text{Supp}(\\Coker(u)))", "$$", "see Properties of Spaces, Lemma \\ref{spaces-properties-lemma-open-subspaces}.", "This is an open because $|f|$ is closed on $\\text{Supp}(\\mathcal{G})$", "and $\\text{Supp}(\\Coker(u))$ is a closed subset of", "$\\text{Supp}(\\mathcal{G})$." ], "refs": [ "spaces-morphisms-lemma-support-covering", "spaces-properties-lemma-open-subspaces" ], "ref_ids": [ 4776, 11823 ] } ], "ref_ids": [] }, { "id": 7185, "type": "theorem", "label": "spaces-flat-lemma-freebie", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-freebie", "contents": [ "Let $S$ be the spectrum of a henselian local ring with closed point $s$.", "Let $X \\to S$ be a morphism of algebraic spaces which is", "locally of finite type.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "Let $E \\subset |X_s|$ be a subset. There exists a closed subscheme", "$Z \\subset S$ with the following property: for any morphism of pointed", "schemes $(T, t) \\to (S, s)$ the following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}_T$ is flat over $T$ at all points of", "$|X_t|$ which map to a point of $E \\subset |X_s|$, and", "\\item $\\Spec(\\mathcal{O}_{T, t}) \\to S$ factors through $Z$.", "\\end{enumerate}", "Moreover, if $X \\to S$ is locally of finite presentation,", "$\\mathcal{F}$ is of finite presentation, and $E \\subset |X_s|$ is", "closed and quasi-compact, then $Z \\to S$ is of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and an \\'etale morphism $\\varphi : U \\to X$.", "Let $E' \\subset |U_s|$ be the inverse image of $E$. If", "$E' \\to E$ is surjective, then condition (1) is equivalent to:", "$(\\varphi^*\\mathcal{F})_T$ is flat over $T$ at all points of", "$|U_t|$ which map to a point of $E' \\subset |U_t|$.", "Choosing $\\varphi$ to be surjective, we reduced to the case of schemes which is", "More on Flatness, Lemma \\ref{flat-lemma-freebie}.", "If $E$ is closed and quasi-compact, then we may choose $U$ to be", "affine such that $E' \\to E$ is surjective. Then $E'$ is closed", "and quasi-compact and the final statement follows from the", "final statement of", "More on Flatness, Lemma \\ref{flat-lemma-freebie}." ], "refs": [ "flat-lemma-freebie", "flat-lemma-freebie" ], "ref_ids": [ 6089, 6089 ] } ], "ref_ids": [] }, { "id": 7186, "type": "theorem", "label": "spaces-flat-lemma-pre-flat-dimension-n", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-pre-flat-dimension-n", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces which is", "locally of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite", "type. Let $n \\geq 0$. The following are equivalent", "\\begin{enumerate}", "\\item for some commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_\\varphi \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with surjective, \\'etale vertical arrows where $U$ and $V$ are", "schemes, the sheaf $\\varphi^*\\mathcal{F}$ is flat over $V$", "in dimensions $\\geq n$ (More on Flatness, Definition", "\\ref{flat-definition-flat-dimension-n}),", "\\item for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_\\varphi \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with \\'etale vertical arrows where $U$ and $V$ are schemes,", "the sheaf $\\varphi^*\\mathcal{F}$ is flat over $V$ in dimensions $\\geq n$, and", "\\item for $x \\in |X|$ such that $\\mathcal{F}$ is not flat at $x$", "over $Y$ the transcendence degree of $x/f(x)$ is $< n$ (Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-dimension-fibre}).", "\\end{enumerate}", "If this is true, then it remains true after any base change $Y' \\to Y$." ], "refs": [ "flat-definition-flat-dimension-n", "spaces-morphisms-definition-dimension-fibre" ], "proofs": [ { "contents": [ "Suppose that we have a diagram as in (1). Then the equivalence of", "the conditions in More on Flatness, Lemma \\ref{flat-lemma-pre-flat-dimension-n}", "shows that (1) and (3) are equivalent. But condition (3) is inherited", "by $\\varphi^*\\mathcal{F}$ for any $U \\to V$ as in (2).", "Whence we see that (3) implies (2) by the result for schemes again.", "The result for schemes also implies the statement on base change." ], "refs": [ "flat-lemma-pre-flat-dimension-n" ], "ref_ids": [ 6081 ] } ], "ref_ids": [ 6217, 5009 ] }, { "id": 7187, "type": "theorem", "label": "spaces-flat-lemma-flat-dimension-n", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-flat-dimension-n", "contents": [ "In Situation \\ref{situation-flat-dimension-n}.", "\\begin{enumerate}", "\\item The functor $F_n$ satisfies the sheaf property for the fpqc topology.", "\\item If $f$ is quasi-compact and locally of finite presentation", "and $\\mathcal{F}$ is of finite presentation, then the functor $F_n$ is", "limit preserving.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Suppose that $\\{T_i \\to T\\}$ is an fpqc covering of", "a scheme $T$ over $Y$. We have to show that if $F_n(T_i)$ is nonempty", "for all $i$, then $F_n(T)$ is nonempty.", "Choose a diagram as in part (1) of Lemma \\ref{lemma-pre-flat-dimension-n}.", "Denote $F'_n$ the corresponding functor for", "$\\varphi^*\\mathcal{F}$ and the morphism $U \\to V$.", "By More on Flatness, Lemma \\ref{flat-lemma-flat-dimension-n}", "we have the sheaf property for $F'_n$.", "Thus we get the sheaf property for $F_n$ because", "for $T \\to Y$ we have $F_n(T) = F'_n(V \\times_Y T)$", "by Lemma \\ref{lemma-pre-flat-dimension-n}", "and because $\\{V \\times_Y T_i \\to V \\times_Y T\\}$", "is an fpqc covering.", "\\medskip\\noindent", "Proof of (2). Suppose that $T = \\lim_{i \\in I} T_i$ is a filtered limit", "of affine schemes $T_i$ over $Y$ and assume that $F_n(T)$ is nonempty.", "We have to show that $F_n(T_i)$ is nonempty for some $i$.", "Choose a diagram as in", "part (1) of Lemma \\ref{lemma-pre-flat-dimension-n}.", "Fix $i \\in I$ and choose an affine open $W_i \\subset V \\times_Y T_i$", "mapping surjectively onto $T_i$. For $i' \\geq i$ let $W_{i'}$", "be the inverse image of $W_i$ in $V \\times_Y T_{i'}$ and", "let $W \\subset V \\times_Y T$ be the inverse image of $W_i$.", "Then $W = \\lim_{i' \\geq i} W_i$ is a filtered limit of affine", "schemes over $V$. By", "Lemma \\ref{lemma-pre-flat-dimension-n} again", "it suffices to show that $F'_n(W_{i'})$ is nonempty for", "some $i' \\geq i$. But we know that $F'_n(W)$ is nonempty", "because of our assumption that $F_n(T) = F'_n(V \\times_Y T)$", "is nonempty. Thus we can apply", "More on Flatness, Lemma \\ref{flat-lemma-flat-dimension-n}", "to conclude." ], "refs": [ "spaces-flat-lemma-pre-flat-dimension-n", "flat-lemma-flat-dimension-n", "spaces-flat-lemma-pre-flat-dimension-n", "spaces-flat-lemma-pre-flat-dimension-n", "spaces-flat-lemma-pre-flat-dimension-n", "flat-lemma-flat-dimension-n" ], "ref_ids": [ 7186, 6082, 7186, 7186, 7186, 6082 ] } ], "ref_ids": [] }, { "id": 7188, "type": "theorem", "label": "spaces-flat-lemma-localize-flat-dimension-n", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-localize-flat-dimension-n", "contents": [ "In Situation \\ref{situation-flat-dimension-n}.", "Let $h : X' \\to X$ be an \\'etale morphism.", "Set $\\mathcal{F}' = h^*\\mathcal{F}$ and $f' = f \\circ h$.", "Let $F_n'$ be (\\ref{equation-flat-dimension-n})", "associated to $(f' : X' \\to Y, \\mathcal{F}')$.", "Then $F_n$ is a subfunctor of $F_n'$ and if", "$h(X') \\supset \\text{Ass}_{X/Y}(\\mathcal{F})$, then $F_n = F'_n$." ], "refs": [], "proofs": [ { "contents": [ "Choose $U \\to X$, $V \\to Y$, $U \\to V$ as in part (1) of", "Lemma \\ref{lemma-pre-flat-dimension-n}. Choose a surjective", "\\'etale morphism $U' \\to U \\times_X X'$ where $U'$ is a scheme.", "Then we have the lemma for the two functors", "$F_{U, n}$ and $F_{U', n}$ determined by $U' \\to U$ and $\\mathcal{F}|_U$", "over $V$, see", "More on Flatness, Lemma \\ref{flat-lemma-localize-flat-dimension-n}.", "On the other hand, Lemma \\ref{lemma-pre-flat-dimension-n}", "tells us that given $T \\to Y$ we have", "$F_n(T) = F_{U, n}(V \\times_Y T)$", "and", "$F'_n(T) = F_{U', n}(V \\times_Y T)$.", "This proves the lemma." ], "refs": [ "spaces-flat-lemma-pre-flat-dimension-n", "flat-lemma-localize-flat-dimension-n", "spaces-flat-lemma-pre-flat-dimension-n" ], "ref_ids": [ 7186, 6103, 7186 ] } ], "ref_ids": [] }, { "id": 7189, "type": "theorem", "label": "spaces-flat-lemma-when-universal-flattening", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-when-universal-flattening", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item If $f$ is of finite presentation, $\\mathcal{F}$ is an", "$\\mathcal{O}_X$-module of finite presentation, and $\\mathcal{F}$ is", "pure relative to $Y$, then there exists a universal flattening", "$Y' \\to Y$ of $\\mathcal{F}$. Moreover $Y' \\to Y$ is a monomorphism", "of finite presentation.", "\\item If $f$ is of finite presentation and $X$ is pure relative to $Y$,", "then there exists a universal flattening $Y' \\to Y$ of $X$.", "Moreover $Y' \\to Y$ is a monomorphism of finite presentation.", "\\item If $f$ is proper and of finite presentation and $\\mathcal{F}$ is an", "$\\mathcal{O}_X$-module of finite presentation, then there exists a", "universal flattening $Y' \\to Y$ of $\\mathcal{F}$. Moreover $Y' \\to Y$ is", "a monomorphism of finite presentation.", "\\item If $f$ is proper and of finite presentation", "then there exists a universal flattening $Y' \\to Y$ of $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "These statements follow immediately from", "Theorem \\ref{theorem-flat-dimension-n-representable}", "applied to $F_0 = F_{flat}$", "and the fact that if $f$ is proper then $\\mathcal{F}$ is automatically", "pure over the base, see", "Lemma \\ref{lemma-proper-pure}." ], "refs": [ "spaces-flat-theorem-flat-dimension-n-representable", "spaces-flat-lemma-proper-pure" ], "ref_ids": [ 7147, 7157 ] } ], "ref_ids": [] }, { "id": 7190, "type": "theorem", "label": "spaces-flat-lemma-compute-what-it-should-be", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-compute-what-it-should-be", "contents": [ "In Situation \\ref{situation-existence} consider", "$$", "K = R\\lim_{D_\\QCoh(\\mathcal{O}_X)}(\\mathcal{F}_n) =", "DQ_X(R\\lim_{D(\\mathcal{O}_X)}\\mathcal{F}_n)", "$$", "Then $K$ is in $D^b_{\\QCoh}(\\mathcal{O}_X)$ and in fact", "$K$ has nonzero cohomology sheaves only in degrees $\\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Special case of", "Derived Categories of Spaces, Example", "\\ref{spaces-perfect-example-inverse-limit-quasi-coherent}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7191, "type": "theorem", "label": "spaces-flat-lemma-compute-against-perfect", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-compute-against-perfect", "contents": [ "In Situation \\ref{situation-existence} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be}. For any perfect", "object $E$ of $D(\\mathcal{O}_X)$ we have", "\\begin{enumerate}", "\\item $M = R\\Gamma(X, K \\otimes^\\mathbf{L} E)$ is a perfect object of $D(A)$", "and there is a canonical isomorphism", "$R\\Gamma(X_n, \\mathcal{F}_n \\otimes^\\mathbf{L} E|_{X_n}) =", "M \\otimes_A^\\mathbf{L} A_n$", "in $D(A_n)$,", "\\item $N = R\\Hom_X(E, K)$ is a perfect object of $D(A)$", "and there is a canonical isomorphism", "$R\\Hom_{X_n}(E|_{X_n}, \\mathcal{F}_n) = N \\otimes_A^\\mathbf{L} A_n$", "in $D(A_n)$.", "\\end{enumerate}", "In both statements $E|_{X_n}$ denotes the derived pullback", "of $E$ to $X_n$." ], "refs": [ "spaces-flat-lemma-compute-what-it-should-be" ], "proofs": [ { "contents": [ "Proof of (2). Write $E_n = E|_{X_n}$ and", "$N_n = R\\Hom_{X_n}(E_n, \\mathcal{F}_n)$.", "Recall that $R\\Hom_{X_n}(-, -)$ is equal to", "$R\\Gamma(X_n, R\\SheafHom(-, -))$, see", "Cohomology on Sites, Section \\ref{sites-cohomology-section-global-RHom}.", "Hence by Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-base-change-RHom-perfect}", "we see that $N_n$ is a perfect object of $D(A_n)$", "whose formation commutes with base change. Thus the maps", "$N_n \\otimes_{A_n}^\\mathbf{L} A_{n - 1} \\to N_{n - 1}$", "coming from $\\varphi_n$ are isomorphisms.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-Rlim-perfect-gives-perfect}", "we find that $R\\lim N_n$ is perfect and", "that its base change back to $A_n$ recovers $N_n$.", "On the other hand, the exact functor", "$R\\Hom_X(E, -) : D_\\QCoh(\\mathcal{O}_X) \\to D(A)$", "of triangulated categories commutes with products", "and hence with derived limits, whence", "$$", "R\\Hom_X(E, K) =", "R\\lim R\\Hom_X(E, \\mathcal{F}_n) =", "R\\lim R\\Hom_X(E_n, \\mathcal{F}_n) =", "R\\lim N_n", "$$", "This proves (2). To see that (1) holds, translate it into (2)", "using Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-dual-perfect-complex}." ], "refs": [ "spaces-perfect-lemma-base-change-RHom-perfect", "more-algebra-lemma-Rlim-perfect-gives-perfect", "sites-cohomology-lemma-dual-perfect-complex" ], "ref_ids": [ 2742, 10409, 4390 ] } ], "ref_ids": [ 7190 ] }, { "id": 7192, "type": "theorem", "label": "spaces-flat-lemma-relative-pseudo-coherence", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-relative-pseudo-coherence", "contents": [ "In Situation \\ref{situation-existence} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be}. Then $K$", "is pseudo-coherent relative to $A$." ], "refs": [ "spaces-flat-lemma-compute-what-it-should-be" ], "proofs": [ { "contents": [ "Combinging Lemma \\ref{lemma-compute-against-perfect} and", "Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-perfect-enough}", "we see that $R\\Gamma(X, K \\otimes^\\mathbf{L} E)$", "is pseudo-coherent in $D(A)$ for all pseudo-coherent", "$E$ in $D(\\mathcal{O}_X)$. Thus the lemma follows from", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-characterize-pseudo-coherent}." ], "refs": [ "spaces-flat-lemma-compute-against-perfect", "spaces-perfect-lemma-perfect-enough", "spaces-more-morphisms-lemma-characterize-pseudo-coherent" ], "ref_ids": [ 7191, 2741, 257 ] } ], "ref_ids": [ 7190 ] }, { "id": 7193, "type": "theorem", "label": "spaces-flat-lemma-compute-over-affine", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-compute-over-affine", "contents": [ "In Situation \\ref{situation-existence} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be}. For any", "\\'etale morphism $U \\to X$ with $U$ quasi-compact and quasi-separated we have", "$$", "R\\Gamma(U, K) \\otimes_A^\\mathbf{L} A_n =", "R\\Gamma(U_n, \\mathcal{F}_n)", "$$", "in $D(A_n)$ where $U_n = U \\times_X X_n$." ], "refs": [ "spaces-flat-lemma-compute-what-it-should-be" ], "proofs": [ { "contents": [ "Fix $n$. By Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-computing-sections-as-colim}", "there exists a system of perfect complexes $E_m$", "on $X$ such that", "$R\\Gamma(U, K) = \\text{hocolim} R\\Gamma(X, K \\otimes^\\mathbf{L} E_m)$.", "In fact, this formula holds not just for $K$ but for every object of", "$D_\\QCoh(\\mathcal{O}_X)$.", "Applying this to $\\mathcal{F}_n$", "we obtain", "\\begin{align*}", "R\\Gamma(U_n, \\mathcal{F}_n)", "& =", "R\\Gamma(U, \\mathcal{F}_n) \\\\", "& =", "\\text{hocolim}_m R\\Gamma(X, \\mathcal{F}_n \\otimes^\\mathbf{L} E_m) \\\\", "& =", "\\text{hocolim}_m R\\Gamma(X_n, \\mathcal{F}_n \\otimes^\\mathbf{L} E_m|_{X_n})", "\\end{align*}", "Using Lemma \\ref{lemma-compute-against-perfect}", "and the fact that $- \\otimes_A^\\mathbf{L} A_n$", "commutes with homotopy colimits we obtain the result." ], "refs": [ "spaces-perfect-lemma-computing-sections-as-colim", "spaces-flat-lemma-compute-against-perfect" ], "ref_ids": [ 2753, 7191 ] } ], "ref_ids": [ 7190 ] }, { "id": 7194, "type": "theorem", "label": "spaces-flat-lemma-finitely-presented", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-finitely-presented", "contents": [ "In Situation \\ref{situation-existence} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be}.", "Denote $X_0 \\subset |X|$ the closed subset", "consisting of points lying over the closed subset", "$\\Spec(A_1) = \\Spec(A_2) = \\ldots$ of $\\Spec(A)$.", "There exists an open subspace $W \\subset X$ containing $X_0$", "such that", "\\begin{enumerate}", "\\item $H^i(K)|_W$ is zero unless $i = 0$,", "\\item $\\mathcal{F} = H^0(K)|_W$ is of finite presentation, and", "\\item $\\mathcal{F}_n = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X_n}$.", "\\end{enumerate}" ], "refs": [ "spaces-flat-lemma-compute-what-it-should-be" ], "proofs": [ { "contents": [ "Fix $n \\geq 1$. By construction there is a canonical map", "$K \\to \\mathcal{F}_n$ in $D_\\QCoh(\\mathcal{O}_X)$", "and hence a canonical map $H^0(K) \\to \\mathcal{F}_n$", "of quasi-coherent sheaves. This explains the meaning of part (3).", "\\medskip\\noindent", "Let $x \\in X_0$ be a point. We will find an open neighbourhood $W$", "of $x$ such that (1), (2), and (3) are true. Since $X_0$ is quasi-compact", "this will prove the lemma. Let $U \\to X$ be an \\'etale morphism", "with $U$ affine and $u \\in U$ a point mapping to $x$. Since $|U| \\to |X|$", "is open it suffices to find an open neighbourhood of $u$ in $U$", "where (1), (2), and (3) are true. Say $U = \\Spec(B)$.", "Choose a surjection $P \\to B$ with $P$ smooth over $A$.", "By Lemma \\ref{lemma-relative-pseudo-coherence}", "and the definition of relative pseudo-coherence", "there exists a bounded above complex $F^\\bullet$", "of finite free $P$-modules representing", "$Ri_*K$ where $i : U \\to \\Spec(P)$ is the closed", "immersion induced by the presentation.", "Let $M_n$ be the $B$-module corresponding to $\\mathcal{F}_n|_U$.", "By Lemma \\ref{lemma-compute-over-affine}", "$$", "H^i(F^\\bullet \\otimes_A A_n) =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & i \\not = 0 \\\\", "M_n & \\text{if} & i = 0", "\\end{matrix}", "\\right.", "$$", "Let $i$ be the maximal index such that $F^i$ is nonzero.", "If $i \\leq 0$, then (1), (2), and (3) are true.", "If not, then $i > 0$ and we see that the rank of the map", "$$", "F^{i - 1} \\to F^i", "$$", "in the point $u$ is maximal. Hence in an open neighbourhood", "of $u$ inside $\\Spec(P)$ the rank is maximal. Thus after replacing", "$P$ by a principal localization we may assume that the displayed", "map is surjective. Since $F^i$ is finite free we may choose", "a splitting $F^{i - 1} = F' \\oplus F^i$. Then we may", "replace $F^\\bullet$ by the complex", "$$", "\\ldots \\to F^{i - 2} \\to F' \\to 0 \\to \\ldots", "$$", "and we win by induction on $i$." ], "refs": [ "spaces-flat-lemma-relative-pseudo-coherence", "spaces-flat-lemma-compute-over-affine" ], "ref_ids": [ 7192, 7193 ] } ], "ref_ids": [ 7190 ] }, { "id": 7195, "type": "theorem", "label": "spaces-flat-lemma-proper-support", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-proper-support", "contents": [ "In Situation \\ref{situation-existence} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be}. Let $W \\subset X$", "be as in Lemma \\ref{lemma-finitely-presented}.", "Set $\\mathcal{F} = H^0(K)|_W$. Then, after possibly shrinking the open $W$,", "the support of $\\mathcal{F}$ is proper over $A$." ], "refs": [ "spaces-flat-lemma-compute-what-it-should-be", "spaces-flat-lemma-finitely-presented" ], "proofs": [ { "contents": [ "Fix $n \\geq 1$. Let $I_n = \\Ker(A \\to A_n)$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-limit-henselian}", "the pair $(A, I_n)$ is henselian.", "Let $Z \\subset W$ be the scheme theoretic support of $\\mathcal{F}$.", "This is a closed subspace as $\\mathcal{F}$ is of finite presentation.", "By part (3) of Lemma \\ref{lemma-finitely-presented}", "we see that $Z \\times_{\\Spec(A)} \\Spec(A_n)$", "is equal to the support of $\\mathcal{F}_n$ and hence", "proper over $\\Spec(A/I)$.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-split-off-proper-part-henselian}", "we can write $Z = Z_1 \\amalg Z_2$ with $Z_1, Z_2$ open and", "closed in $Z$, with $Z_1$ proper", "over $A$, and with $Z_1 \\times_{\\Spec(A)} \\Spec(A/I_n)$", "equal to the support of $\\mathcal{F}_n$.", "In other words, $|Z_2|$ does not meet $X_0$.", "Hence after replacing $W$ by $W \\setminus Z_2$ we obtain the lemma." ], "refs": [ "more-algebra-lemma-limit-henselian", "spaces-flat-lemma-finitely-presented", "spaces-more-morphisms-lemma-split-off-proper-part-henselian" ], "ref_ids": [ 9858, 7194, 183 ] } ], "ref_ids": [ 7190, 7194 ] }, { "id": 7196, "type": "theorem", "label": "spaces-flat-lemma-compute-what-it-should-be-derived", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-compute-what-it-should-be-derived", "contents": [ "In Situation \\ref{situation-existence-derived} consider", "$$", "K = R\\lim_{D_\\QCoh(\\mathcal{O}_X)}(K_n) =", "DQ_X(R\\lim_{D(\\mathcal{O}_X)} K_n)", "$$", "Then $K$ is in $D^-_{\\QCoh}(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "The functor $DQ_X$ exists because $X$ is quasi-compact and", "quasi-separated, see Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-better-coherator}.", "Since $DQ_X$ is a right adjoint it commutes with products", "and therefore with derived limits. Hence the equality", "in the statement of the lemma.", "\\medskip\\noindent", "By Derived Categories of Spaces,", "Lemma \\ref{spaces-perfect-lemma-boundedness-better-coherator}", "the functor $DQ_X$ has bounded cohomological dimension.", "Hence it suffices to show that $R\\lim K_n \\in D^-(\\mathcal{O}_X)$.", "To see this, let $U \\to X$ be \\'etale with $U$ affine.", "Then there is a canonical exact sequence", "$$", "0 \\to", "R^1\\lim H^{m - 1}(U, K_n) \\to H^m(U, R\\lim K_n) \\to", "\\lim H^m(U, K_n) \\to 0", "$$", "by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-RGamma-commutes-with-Rlim}.", "Since $U$ is affine and $K_n$ is pseudo-coherent (and hence has", "quasi-coherent cohomology sheaves by", "Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-pseudo-coherent})", "we see that $H^m(U, K_n) = H^m(K_n)(U)$ by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-affine-compare-bounded}.", "Thus we conclude that it suffices to show that $K_n$", "is bounded above independent of $n$.", "\\medskip\\noindent", "Since $K_n$ is pseudo-coherent we have $K_n \\in D^-(\\mathcal{O}_{X_n})$.", "Suppose that $a_n$ is maximal such that $H^{a_n}(K_n)$ is nonzero.", "Of course $a_1 \\leq a_2 \\leq a_3 \\leq \\ldots$.", "Note that $H^{a_n}(K_n)$ is an", "$\\mathcal{O}_{X_n}$-module of finite presentation", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-finite-cohomology}).", "We have $H^{a_n}(K_{n - 1}) =", "H^{a_n}(K_n) \\otimes_{\\mathcal{O}_{X_n}} \\mathcal{O}_{X_{n - 1}}$.", "Since $X_{n - 1} \\to X_n$ is a thickening, it follows from", "Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK}) that if", "$H^{a_n}(K_n) \\otimes_{\\mathcal{O}_{X_n}} \\mathcal{O}_{X_{n - 1}}$", "is zero, then $H^{a_n}(K_n)$ is zero too (argue by checking on stalks", "for example; small detail omitted).", "Thus $a_{n - 1} = a_n$ for all $n$ and we conclude." ], "refs": [ "spaces-perfect-lemma-better-coherator", "spaces-perfect-lemma-boundedness-better-coherator", "sites-cohomology-lemma-RGamma-commutes-with-Rlim", "spaces-perfect-lemma-pseudo-coherent", "perfect-lemma-affine-compare-bounded", "sites-cohomology-lemma-finite-cohomology", "algebra-lemma-NAK" ], "ref_ids": [ 2715, 2717, 4266, 2696, 6941, 4371, 401 ] } ], "ref_ids": [] }, { "id": 7197, "type": "theorem", "label": "spaces-flat-lemma-compute-against-perfect-derived", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-compute-against-perfect-derived", "contents": [ "In Situation \\ref{situation-existence-derived} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be-derived}. For any perfect", "object $E$ of $D(\\mathcal{O}_X)$ the cohomology", "$$", "M = R\\Gamma(X, K \\otimes^\\mathbf{L} E)", "$$", "is a pseudo-coherent object of $D(A)$ and there is a canonical isomorphism", "$$", "R\\Gamma(X_n, K_n \\otimes^\\mathbf{L} E|_{X_n}) = M \\otimes_A^\\mathbf{L} A_n", "$$", "in $D(A_n)$. Here $E|_{X_n}$ denotes the derived pullback of $E$ to $X_n$." ], "refs": [ "spaces-flat-lemma-compute-what-it-should-be-derived" ], "proofs": [ { "contents": [ "Write $E_n = E|_{X_n}$ and", "$M_n = R\\Gamma(X_n, K_n \\otimes^\\mathbf{L} E|_{X_n})$.", "By Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-flat-proper-pseudo-coherent-direct-image-general}", "we see that $M_n$ is a pseudo-coherent object of $D(A_n)$", "whose formation commutes with base change. Thus the maps", "$M_n \\otimes_{A_n}^\\mathbf{L} A_{n - 1} \\to M_{n - 1}$", "coming from $\\varphi_n$ are isomorphisms. By", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-Rlim-pseudo-coherent-gives-pseudo-coherent}", "we find that $R\\lim M_n$ is pseudo-coherent and", "that its base change back to $A_n$ recovers $M_n$.", "On the other hand, the exact functor", "$R\\Gamma(X, -) : D_\\QCoh(\\mathcal{O}_X) \\to D(A)$", "of triangulated categories commutes with products", "and hence with derived limits, whence", "$$", "R\\Gamma(X, E \\otimes^\\mathbf{L} K) =", "R\\lim R\\Gamma(X, E \\otimes^\\mathbf{L} K_n) =", "R\\lim R\\Gamma(X_n, E_n \\otimes^\\mathbf{L} K_n) =", "R\\lim M_n", "$$", "as desired." ], "refs": [ "spaces-perfect-lemma-flat-proper-pseudo-coherent-direct-image-general", "more-algebra-lemma-Rlim-pseudo-coherent-gives-pseudo-coherent" ], "ref_ids": [ 2739, 10407 ] } ], "ref_ids": [ 7196 ] }, { "id": 7198, "type": "theorem", "label": "spaces-flat-lemma-relative-pseudo-coherence-derived", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-relative-pseudo-coherence-derived", "contents": [ "In Situation \\ref{situation-existence-derived} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be-derived}. Then $K$", "is pseudo-coherent on $X$." ], "refs": [ "spaces-flat-lemma-compute-what-it-should-be-derived" ], "proofs": [ { "contents": [ "Combinging Lemma \\ref{lemma-compute-against-perfect-derived} and", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-perfect-enough}", "we see that $R\\Gamma(X, K \\otimes^\\mathbf{L} E)$", "is pseudo-coherent in $D(A)$ for all pseudo-coherent", "$E$ in $D(\\mathcal{O}_X)$. Thus it follows from", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-characterize-pseudo-coherent}", "that $K$ is pseudo-coherent relative to $A$.", "Since $X$ is of flat and of finite presentation", "over $A$, this is the same as being pseudo-coherent on $X$, see", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-relative-pseudo-coherent-is-moot}." ], "refs": [ "spaces-flat-lemma-compute-against-perfect-derived", "spaces-perfect-lemma-perfect-enough", "spaces-more-morphisms-lemma-characterize-pseudo-coherent", "spaces-more-morphisms-lemma-relative-pseudo-coherent-is-moot" ], "ref_ids": [ 7197, 2741, 257, 224 ] } ], "ref_ids": [ 7196 ] }, { "id": 7199, "type": "theorem", "label": "spaces-flat-lemma-compute-over-affine-derived", "categories": [ "spaces-flat" ], "title": "spaces-flat-lemma-compute-over-affine-derived", "contents": [ "In Situation \\ref{situation-existence-derived} let $K$ be as in", "Lemma \\ref{lemma-compute-what-it-should-be-derived}. For any", "\\'etale morphism $U \\to X$ with $U$ quasi-compact and quasi-separated we have", "$$", "R\\Gamma(U, K) \\otimes_A^\\mathbf{L} A_n =", "R\\Gamma(U_n, K_n)", "$$", "in $D(A_n)$ where $U_n = U \\times_X X_n$." ], "refs": [ "spaces-flat-lemma-compute-what-it-should-be-derived" ], "proofs": [ { "contents": [ "Fix $n$. By Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-computing-sections-as-colim}", "there exists a system of perfect complexes $E_m$", "on $X$ such that", "$R\\Gamma(U, K) = \\text{hocolim} R\\Gamma(X, K \\otimes^\\mathbf{L} E_m)$.", "In fact, this formula holds not just for $K$ but for every object of", "$D_\\QCoh(\\mathcal{O}_X)$.", "Applying this to $K_n$", "we obtain", "\\begin{align*}", "R\\Gamma(U_n, K_n)", "& =", "R\\Gamma(U, K_n) \\\\", "& =", "\\text{hocolim}_m R\\Gamma(X, K_n \\otimes^\\mathbf{L} E_m) \\\\", "& =", "\\text{hocolim}_m R\\Gamma(X_n, K_n \\otimes^\\mathbf{L} E_m|_{X_n})", "\\end{align*}", "Using Lemma \\ref{lemma-compute-against-perfect-derived}", "and the fact that $- \\otimes_A^\\mathbf{L} A_n$", "commutes with homotopy colimits we obtain the result." ], "refs": [ "spaces-perfect-lemma-computing-sections-as-colim", "spaces-flat-lemma-compute-against-perfect-derived" ], "ref_ids": [ 2753, 7197 ] } ], "ref_ids": [ 7196 ] }, { "id": 7200, "type": "theorem", "label": "spaces-flat-proposition-finite-presentation-flat-at-point", "categories": [ "spaces-flat" ], "title": "spaces-flat-proposition-finite-presentation-flat-at-point", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $x \\in |X|$ with image $y \\in |Y|$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $\\mathcal{F}$ is of finite presentation, and", "\\item $\\mathcal{F}$ is flat at $x$ over $Y$.", "\\end{enumerate}", "Then there exists a commutative diagram of pointed schemes", "$$", "\\xymatrix{", "(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\", "(Y, y) & (Y', y') \\ar[l]", "}", "$$", "whose horizontal arrows are \\'etale such that $X'$, $Y'$", "are affine and such that", "$\\Gamma(X', g^*\\mathcal{F})$ is a projective", "$\\Gamma(Y', \\mathcal{O}_{Y'})$-module." ], "refs": [], "proofs": [ { "contents": [ "As formulated this proposition immmediately reduces", "to the case of schemes, which is", "More on Flatness, Proposition", "\\ref{flat-proposition-finite-presentation-flat-at-point}." ], "refs": [ "flat-proposition-finite-presentation-flat-at-point" ], "ref_ids": [ 6200 ] } ], "ref_ids": [] }, { "id": 7206, "type": "theorem", "label": "spaces-chow-lemma-delta-is-dimension", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-delta-is-dimension", "contents": [ "In Situation \\ref{situation-setup} assume $B$ is Jacobson", "and that $\\delta(b) = 0$ for every closed point $b$ of $|B|$.", "Let $X/B$ be good. If $Z \\subset X$ is an integral closed subspace", "with generic point $\\xi \\in |Z|$, then the following integers are the same:", "\\begin{enumerate}", "\\item $\\delta(\\xi) = \\delta_{X/B}(\\xi)$,", "\\item $\\dim(|Z|)$,", "\\item $\\text{codim}(\\{z\\}, |Z|)$ for $z \\in |Z|$ closed,", "\\item the dimension of the local ring of $Z$ at $z$ for", "$z \\in |Z|$ closed, and", "\\item $\\dim(\\mathcal{O}_{Z, \\overline{z}})$ for $z \\in |Z|$ closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X$, $Z$, $\\xi$ be as in the lemma.", "Since $X$ is locally of finite type over $B$ we see that $X$ is Jacobson, see", "Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-Jacobson-universally-Jacobson}.", "Hence $X_{\\text{ft-pts}} \\subset |X|$ is the set of closed points", "by Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-Jacobson-ft-pts}.", "Given a chain $T_0 \\supset \\ldots \\supset T_e$", "of irreducible closed subsets of $|Z|$ we have", "$T_e \\cap X_{\\text{ft-pts}}$ nonempty by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-enough-finite-type-points}.", "Thus we can always assume such a chain ends", "with $T_e = \\{z\\}$ for some $z \\in |Z|$ closed.", "It follows that $\\dim(Z) = \\sup_z \\text{codim}(\\{z\\}, |Z|)$", "where $z$ runs over the closed points of $|Z|$.", "We have $\\text{codim}(\\{z\\}, Z) = \\delta(\\xi) - \\delta(z)$", "by Topology, Lemma \\ref{topology-lemma-dimension-function-catenary}.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-type-points-morphism}", "the image of $z$ is a finite type point of $B$, i.e.,", "a closed point of $|B|$. By", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-jacobson-finite-type-points}", "the transcendence degree of $z/b$ is $0$.", "We conclude that $\\delta(z) = \\delta(b) = 0$ by assumption.", "Thus we obtain equality", "$$", "\\dim(|Z|) = \\text{codim}(\\{z\\}, Z) = \\delta(\\xi)", "$$", "for all $z \\in |Z|$ closed. Finally, we have that", "$\\text{codim}(\\{z\\}, Z)$ is equal to the dimension of the", "local ring of $Z$ at $z$ by", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-codimension-local-ring}", "which in turn is equal to", "$\\dim(\\mathcal{O}_{Z, \\overline{z}})$ by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-dimension-local-ring}." ], "refs": [ "decent-spaces-lemma-Jacobson-universally-Jacobson", "decent-spaces-lemma-decent-Jacobson-ft-pts", "spaces-morphisms-lemma-enough-finite-type-points", "topology-lemma-dimension-function-catenary", "spaces-morphisms-lemma-finite-type-points-morphism", "spaces-morphisms-lemma-jacobson-finite-type-points", "decent-spaces-lemma-codimension-local-ring", "spaces-properties-lemma-dimension-local-ring" ], "ref_ids": [ 9547, 9549, 4826, 8291, 4824, 4868, 9532, 11886 ] } ], "ref_ids": [] }, { "id": 7207, "type": "theorem", "label": "spaces-chow-lemma-length", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-length", "contents": [ "Let $S$ be a scheme and let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $x \\in |X|$. Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$.", "The following are equivalent", "\\begin{enumerate}", "\\item", "$\\text{length}_{\\mathcal{O}_{X, \\overline{x}}} \\mathcal{F}_{\\overline{x}} = d$", "\\item for some \\'etale morphism $U \\to X$ with $U$ a scheme", "and $u \\in U$ mapping to $x$ we have", "$\\text{length}_{\\mathcal{O}_{U, u}} (\\mathcal{F}|_U)_u = d$", "\\item for any \\'etale morphism $U \\to X$ with $U$ a scheme", "and $u \\in U$ mapping to $x$ we have", "$\\text{length}_{\\mathcal{O}_{U, u}} (\\mathcal{F}|_U)_u = d$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ and $u \\in U$ be as in (2) or (3). Then we know that", "$\\mathcal{O}_{X, \\overline{x}}$ is the strict henselization of", "$\\mathcal{O}_{U, u}$ and that", "$$", "\\mathcal{F}_{\\overline{x}} =", "(\\mathcal{F}|_U)_u \\otimes_{\\mathcal{O}_{U, u}} \\mathcal{O}_{X, \\overline{x}}", "$$", "See Properties of Spaces,", "Lemmas \\ref{spaces-properties-lemma-describe-etale-local-ring} and", "\\ref{spaces-properties-lemma-stalk-quasi-coherent}.", "Thus the equality of the lengths follows from", "Algebra, Lemma \\ref{algebra-lemma-pullback-module}", "the fact that $\\mathcal{O}_{U, u} \\to \\mathcal{O}_{X, \\overline{x}}$", "is flat and the fact that", "$\\mathcal{O}_{X, \\overline{x}}/\\mathfrak m_u\\mathcal{O}_{X, \\overline{x}}$", "is equal to the residue field of $\\mathcal{O}_{X, \\overline{x}}$.", "These facts about strict henselizations can be found in", "More on Algebra, Lemma \\ref{more-algebra-lemma-dumb-properties-henselization}." ], "refs": [ "spaces-properties-lemma-describe-etale-local-ring", "spaces-properties-lemma-stalk-quasi-coherent", "algebra-lemma-pullback-module", "more-algebra-lemma-dumb-properties-henselization" ], "ref_ids": [ 11884, 11909, 640, 10055 ] } ], "ref_ids": [] }, { "id": 7208, "type": "theorem", "label": "spaces-chow-lemma-length-closed-immersion", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-length-closed-immersion", "contents": [ "Let $S$ be a scheme. Let $i : Y \\to X$ be a closed immersion of", "algebraic spaces over $S$. Let $\\mathcal{G}$ be a quasi-coherent", "$\\mathcal{O}_Y$-module. Let $y \\in |Y|$ with image $x \\in |X|$.", "Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$. The following are", "equivalent", "\\begin{enumerate}", "\\item $\\mathcal{G}$ has length $d$ at $y$, and", "\\item $i_*\\mathcal{G}$ has length $d$ at $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose an \\'etale morphism $f : U \\to X$ with $U$ a scheme", "and $u \\in U$ mapping to $x$. Set $V = Y \\times_X U$.", "Denote $g : V \\to Y$ and $j : V \\to U$ the projections.", "Then $j : V \\to U$ is a closed immersion and there is a unique", "point $v \\in V$ mapping to $y \\in |Y|$ and $u \\in U$", "(use Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}", "and Spaces, Lemma \\ref{spaces-lemma-base-change-immersions}).", "We have $j_*(\\mathcal{G}|_V) = (i_*\\mathcal{G})|_U$ as modules", "on the scheme $V$ and $j_*$ the ``usual'' pushforward of modules", "for the morphism of schemes $j$, see discussion surrounding", "Cohomology of Spaces, Equation", "(\\ref{spaces-cohomology-equation-representable-higher-direct-image}).", "In this way we reduce to the case of schemes: if $i : Y \\to X$", "is a closed immersion of schemes, then", "$$", "(i_*\\mathcal{G})_x = \\mathcal{G}_y", "$$", "as modules over $\\mathcal{O}_{X, x}$ where the module structure", "on the right hand side is given by the surjection", "$i_y^\\sharp : \\mathcal{O}_{X, x} \\to \\mathcal{O}_{Y, y}$.", "Thus equality by", "Algebra, Lemma \\ref{algebra-lemma-length-independent}." ], "refs": [ "spaces-properties-lemma-points-cartesian", "spaces-lemma-base-change-immersions", "algebra-lemma-length-independent" ], "ref_ids": [ 11819, 8161, 633 ] } ], "ref_ids": [] }, { "id": 7209, "type": "theorem", "label": "spaces-chow-lemma-length-finite", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-length-finite", "contents": [ "Let $S$ be a scheme and let $X$ be a", "locally Noetherian algebraic space over $S$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Let $x \\in |X|$. The following are equivalent", "\\begin{enumerate}", "\\item for some \\'etale morphism $U \\to X$ with $U$ a scheme", "and $u \\in U$ mapping to $x$ we have $u$ is a generic point", "of an irreducible component of $\\text{Supp}(\\mathcal{F}|_U)$,", "\\item for any \\'etale morphism $U \\to X$ with $U$ a scheme", "and $u \\in U$ mapping to $x$ we have $u$ is a generic point", "of an irreducible component of $\\text{Supp}(\\mathcal{F}|_U)$,", "\\item the length of $\\mathcal{F}$ at $x$ is finite and nonzero.", "\\end{enumerate}", "If $X$ is decent (equivalently quasi-separated) then these are", "also equivalent to", "\\begin{enumerate}", "\\item[(4)] $x$ is a generic point of an irreducible component of", "$\\text{Supp}(\\mathcal{F})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f : U \\to X$ is an \\'etale morphism with $U$ a scheme", "and $u \\in U$ maps to $x$. Then $\\mathcal{F}|_U = f^*\\mathcal{F}$", "is a coherent $\\mathcal{O}_U$-module on the locally Noetherian", "scheme $U$ and in particular $(\\mathcal{F}|_U)_u$ is a finite", "$\\mathcal{O}_{U, u}$-module, see Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-coherent-Noetherian}", "and Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-coherent-Noetherian}.", "Recall that the support of $\\mathcal{F}|_U$ is a closed subset of $U$", "(Morphisms, Lemma \\ref{morphisms-lemma-support-finite-type}) and", "that the support of $(\\mathcal{F}|_U)_u$ is the pullback", "of the support of $\\mathcal{F}|_U$ by the morphism", "$\\Spec(\\mathcal{O}_{U, u}) \\to U$. Thus $u$ is a generic point", "of an irreducible component of $\\text{Supp}(\\mathcal{F}|_U)$", "if and only if the support of $(\\mathcal{F}|_U)_u$ is equal", "to the maximal ideal of $\\mathcal{O}_{U, u}$.", "Now the equivalence of (1), (2), (3) follows from", "by Algebra, Lemma \\ref{algebra-lemma-support-point}.", "\\medskip\\noindent", "If $X$ is decent we choose an \\'etale morphism $f : U \\to X$ and a point", "$u \\in U$ mapping to $x$. The support of $\\mathcal{F}$ pulls back to", "the support of $\\mathcal{F}|_U$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-support-finite-type}.", "Also, specializations $x' \\leadsto x$ in $|X|$ lift to specializations", "$u' \\leadsto u$ in $U$ and any nontrivial specialization $u' \\leadsto u$ in $U$", "maps to a nontrivial specialization $f(u') \\leadsto f(u)$ in $|X|$, see", "Decent Spaces, Lemmas", "\\ref{decent-spaces-lemma-decent-specialization} and", "\\ref{decent-spaces-lemma-decent-no-specializations-map-to-same-point}.", "Using that $|X|$ and $U$ are sober topological spaces", "(Decent Spaces, Proposition \\ref{decent-spaces-proposition-reasonable-sober}", "and", "Schemes, Lemma \\ref{schemes-lemma-scheme-sober}) we conclude", "$x$ is a generic point of the support of $\\mathcal{F}$", "if and only if $u$ is a generic point of the support of", "$\\mathcal{F}|_U$. We conclude (4) is equivalent to (1).", "\\medskip\\noindent", "The parenthetical statement follows from Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-locally-Noetherian-decent-quasi-separated}." ], "refs": [ "spaces-cohomology-lemma-coherent-Noetherian", "coherent-lemma-coherent-Noetherian", "morphisms-lemma-support-finite-type", "algebra-lemma-support-point", "spaces-morphisms-lemma-support-finite-type", "decent-spaces-lemma-decent-specialization", "decent-spaces-lemma-decent-no-specializations-map-to-same-point", "decent-spaces-proposition-reasonable-sober", "schemes-lemma-scheme-sober", "decent-spaces-lemma-locally-Noetherian-decent-quasi-separated" ], "ref_ids": [ 11297, 3308, 5143, 693, 4777, 9494, 9493, 9559, 7672, 9506 ] } ], "ref_ids": [] }, { "id": 7210, "type": "theorem", "label": "spaces-chow-lemma-point-of-max-dimension", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-point-of-max-dimension", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $T \\subset |X|$ be a closed subset and $t \\in T$.", "If $\\dim_\\delta(T) \\leq k$ and $\\delta(t) = k$, then", "$t$ is a generic point of an irreducible component of $T$." ], "refs": [], "proofs": [ { "contents": [ "We know $t$ is contained in an irreducible component $T' \\subset T$.", "Let $t' \\in T'$ be the generic point. Then", "$k \\geq \\delta(t') \\geq \\delta(t)$. Since $\\delta$ is a dimension", "function we see that $t = t'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7211, "type": "theorem", "label": "spaces-chow-lemma-reformulate-coeff-coherent", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-reformulate-coeff-coherent", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module", "with $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$.", "Let $Z$ be an integral closed subspace of $X$ with $\\dim_\\delta(Z) = k$.", "Let $\\xi \\in |Z|$ be the generic point.", "Then the coefficient of $Z$ in $[\\mathcal{F}]_k$", "is the length of $\\mathcal{F}$ at $\\xi$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $|Z|$ is an irreducible component of $\\text{Supp}(\\mathcal{F})$", "if and only if $\\xi \\in \\text{Supp}(\\mathcal{F})$, see", "Lemma \\ref{lemma-point-of-max-dimension}.", "Moreover, the length of $\\mathcal{F}$ at $\\xi$ is zero if", "$\\xi \\not \\in \\text{Supp}(\\mathcal{F})$. Combining this", "with Definition \\ref{definition-cycle-associated-to-coherent-sheaf}", "we conclude." ], "refs": [ "spaces-chow-lemma-point-of-max-dimension", "spaces-chow-definition-cycle-associated-to-coherent-sheaf" ], "ref_ids": [ 7210, 7285 ] } ], "ref_ids": [] }, { "id": 7212, "type": "theorem", "label": "spaces-chow-lemma-cycle-closed-coherent", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-cycle-closed-coherent", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $Y \\subset X$ be a closed subspace.", "If $\\dim_\\delta(Y) \\leq k$, then $[Y]_k = [i_*\\mathcal{O}_Y]_k$", "where $i : Y \\to X$ is the inclusion morphism." ], "refs": [], "proofs": [ { "contents": [ "Let $Z$ be an integral closed subspace of $X$ with $\\dim_\\delta(Z) = k$.", "If $Z \\not \\subset Y$ the $Z$ has coefficient zero in both", "$[Y]_k$ and $[i_*\\mathcal{O}_Y]_k$. If $Z \\subset Y$, then", "the generic point of $Z$ may be viewed as a point $y \\in |Y|$", "whose image $x \\in |X|$. Then the coefficient of $Z$ in", "$[Y]_k$ is the length of $\\mathcal{O}_Y$ at $y$ and the", "coefficient of $Z$ in $[i_*\\mathcal{O}_Y]_k$ is the length", "of $i_*\\mathcal{O}_Y$ at $x$. Thus the equality of the coefficients", "follows from Lemma \\ref{lemma-length-closed-immersion}." ], "refs": [ "spaces-chow-lemma-length-closed-immersion" ], "ref_ids": [ 7208 ] } ], "ref_ids": [] }, { "id": 7213, "type": "theorem", "label": "spaces-chow-lemma-additivity-sheaf-cycle", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-additivity-sheaf-cycle", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0$", "be a short exact sequence of coherent $\\mathcal{O}_X$-modules.", "Assume that the $\\delta$-dimension of the supports", "of $\\mathcal{F}$, $\\mathcal{G}$, and $\\mathcal{H}$ are $\\leq k$.", "Then $[\\mathcal{G}]_k = [\\mathcal{F}]_k + [\\mathcal{H}]_k$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z$ be an integral closed subspace of $X$ with $\\dim_\\delta(Z) = k$.", "It suffices to show that the coefficients of", "$Z$ in $[\\mathcal{G}]_k$, $[\\mathcal{F}]_k$, and $[\\mathcal{H}]_k$", "satisfy the corresponding additivity. By", "Lemma \\ref{lemma-reformulate-coeff-coherent}", "it suffices to show", "$$", "\\text{the length of }\\mathcal{G}\\text{ at }x =", "\\text{the length of }\\mathcal{F}\\text{ at }x +", "\\text{the length of }\\mathcal{H}\\text{ at }x", "$$", "for any $x \\in |X|$. Looking at Definition \\ref{definition-length-at-x}", "this follows immediately from additivity of lengths, see", "Algebra, Lemma \\ref{algebra-lemma-length-additive}." ], "refs": [ "spaces-chow-lemma-reformulate-coeff-coherent", "spaces-chow-definition-length-at-x", "algebra-lemma-length-additive" ], "ref_ids": [ 7211, 7283, 631 ] } ], "ref_ids": [] }, { "id": 7214, "type": "theorem", "label": "spaces-chow-lemma-proper-image", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-proper-image", "contents": [ "In Situation \\ref{situation-setup} let $X,Y/B$ be good and let $f : X \\to Y$", "be a morphism over $B$. If $Z \\subset X$ is an integral closed subspace, then", "there exists a unique integral closed subspace $Z' \\subset Y$ such that there", "is a commutative diagram", "$$", "\\xymatrix{", "Z \\ar[r] \\ar[d] & X \\ar[d]^f \\\\", "Z' \\ar[r] & Y", "}", "$$", "with $Z \\to Z'$ dominant. If $f$ is proper, then $Z \\to Z'$ is proper", "and surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\xi \\in |Z|$ be the generic point. Let $Z' \\subset Y$ be the integral", "closed subspace whose generic point is $\\xi' = f(\\xi)$, see", "Remark \\ref{remark-integral}. Since $\\xi \\in |f^{-1}(Z')| = |f|^{-1}(|Z'|)$", "by Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}", "and since $Z$ is the reduced with $|Z| = \\overline{\\{\\xi\\}}$", "we see that $Z \\subset f^{-1}(Z')$ as closed subspaces of $X$ (see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-map-into-reduction}).", "Thus we obtain our morphism $Z \\to Z'$.", "This morphism is dominant as the generic point of $Z$", "maps to the generic point of $Z'$. Uniqueness of $Z'$ is clear.", "If $f$ is proper, then $Z \\to Y$ is proper as a composition", "of proper morphisms (Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-proper} and", "\\ref{spaces-morphisms-lemma-closed-immersion-proper}).", "Then we conclude that $Z \\to Z'$ is proper by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-permanence}.", "Surjectivity then follows as the image of a proper morphism is closed." ], "refs": [ "spaces-chow-remark-integral", "spaces-properties-lemma-points-cartesian", "spaces-properties-lemma-map-into-reduction", "spaces-morphisms-lemma-base-change-proper", "spaces-morphisms-lemma-closed-immersion-proper", "spaces-morphisms-lemma-universally-closed-permanence" ], "ref_ids": [ 7298, 11819, 11847, 4917, 4919, 4920 ] } ], "ref_ids": [] }, { "id": 7215, "type": "theorem", "label": "spaces-chow-lemma-equal-dimension", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-equal-dimension", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good and let", "$f : X \\to Y$ be a morphism over $B$.", "Assume $X$, $Y$ integral and $\\dim_\\delta(X) = \\dim_\\delta(Y)$.", "Then either $f$ factors through a proper closed subspace", "of $Y$, or $f$ is dominant and the extension of function fields", "$R(X) / R(Y)$ is finite." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-proper-image} there is a unique integral closed", "subspace $Z \\subset Y$ such that $f$ factors through a dominant", "morphism $X \\to Z$. Then $Z = Y$ if and only if", "$\\dim_\\delta(Z) = \\dim_\\delta(Y)$. On the other hand, by", "our construction of dimension functions (see Situation \\ref{situation-setup})", "we have $\\dim_\\delta(X) = \\dim_\\delta(Z) + r$ where $r$ the", "transcendence degree of the extension $R(X)/R(Z)$. Combining this with", "Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-finite-degree}", "the lemma follows." ], "refs": [ "spaces-chow-lemma-proper-image", "spaces-over-fields-lemma-finite-degree" ], "ref_ids": [ 7214, 12829 ] } ], "ref_ids": [] }, { "id": 7216, "type": "theorem", "label": "spaces-chow-lemma-quasi-compact-locally-finite", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-quasi-compact-locally-finite", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Let $f : X \\to Y$ be a morphism over $B$.", "Assume $f$ is quasi-compact, and $\\{T_i\\}_{i \\in I}$ is a locally", "finite collection of closed subsets of $|X|$.", "Then $\\{\\overline{|f|(T_i)}\\}_{i \\in I}$ is a locally finite", "collection of closed subsets of $|Y|$." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset |Y|$ be a quasi-compact open subset.", "Then $|f|^{-1}(V) \\subset |X|$ is quasi-compact", "by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-is-quasi-compact}.", "Hence the set", "$\\{i \\in I : T_i \\cap |f|^{-1}(V) \\not = \\emptyset \\}$", "is finite by a simple topological argument which we omit.", "Since this is the same as the set", "$$", "\\{i \\in I : |f|(T_i) \\cap V \\not = \\emptyset \\} =", "\\{i \\in I : \\overline{|f|(T_i)} \\cap V \\not = \\emptyset \\}", "$$", "the lemma is proved." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-is-quasi-compact" ], "ref_ids": [ 4737 ] } ], "ref_ids": [] }, { "id": 7217, "type": "theorem", "label": "spaces-chow-lemma-compose-pushforward", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-compose-pushforward", "contents": [ "In Situation \\ref{situation-setup} let $X, Y, Z/B$ be good.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms over $B$.", "Then $g_* \\circ f_* = (g \\circ f)_*$ as maps $Z_k(X) \\to Z_k(Z)$." ], "refs": [], "proofs": [ { "contents": [ "Let $W \\subset X$ be an integral closed subspace of dimension $k$.", "Consider the integral closed subspaces", "$W' \\subset Y$ and $W'' \\subset Z$", "we get by applying Lemma \\ref{lemma-proper-image}", "to $f$ and $W$ and then to $g$ and $W'$.", "Then $W \\to W'$ and $W' \\to W''$ are surjective and proper.", "We have to show that $g_*(f_*[W]) = (f \\circ g)_*[W]$.", "If $\\dim_\\delta(W'') < k$, then both sides are zero.", "If $\\dim_\\delta(W'') = k$, then we see $W \\to W'$ and $W' \\to W''$", "both satisfy the hypotheses of Lemma \\ref{lemma-equal-dimension}.", "Hence", "$$", "g_*(f_*[W]) = \\deg(W/W')\\deg(W'/W'')[W''],", "\\quad", "(f \\circ g)_*[W] = \\deg(W/W'')[W''].", "$$", "Then we can apply", "Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-degree-composition}", "to conclude." ], "refs": [ "spaces-chow-lemma-proper-image", "spaces-chow-lemma-equal-dimension", "spaces-over-fields-lemma-degree-composition" ], "ref_ids": [ 7214, 7215, 12830 ] } ], "ref_ids": [] }, { "id": 7218, "type": "theorem", "label": "spaces-chow-lemma-cycle-push-sheaf", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-cycle-push-sheaf", "contents": [ "In Situation \\ref{situation-setup} let $f : X \\to Y$ be a proper morphism", "of good algebraic spaces over $B$.", "\\begin{enumerate}", "\\item Let $Z \\subset X$ be a closed subspace with $\\dim_\\delta(Z) \\leq k$.", "Then", "$$", "f_*[Z]_k = [f_*{\\mathcal O}_Z]_k.", "$$", "\\item Let $\\mathcal{F}$ be a coherent sheaf on $X$ such that", "$\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$. Then", "$$", "f_*[\\mathcal{F}]_k = [f_*{\\mathcal F}]_k.", "$$", "\\end{enumerate}", "Note that the statement makes sense since $f_*\\mathcal{F}$ and", "$f_*\\mathcal{O}_Z$ are coherent $\\mathcal{O}_Y$-modules by", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-pushforward-coherent}." ], "refs": [ "spaces-cohomology-lemma-proper-pushforward-coherent" ], "proofs": [ { "contents": [ "Part (1) follows from (2) and Lemma \\ref{lemma-cycle-closed-coherent}.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Assume that $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$.", "By Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-coherent-support-closed}", "there exists a closed immersion $i : Z \\to X$ and a coherent", "$\\mathcal{O}_Z$-module $\\mathcal{G}$ such that", "$i_*\\mathcal{G} \\cong \\mathcal{F}$ and such that the support", "of $\\mathcal{F}$ is $Z$. Let $Z' \\subset Y$ be the scheme theoretic image", "of $f|_Z : Z \\to Y$, see Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-scheme-theoretic-image}.", "Consider the commutative diagram", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[d]_{f|_Z} &", "X \\ar[d]^f \\\\", "Z' \\ar[r]^{i'} & Y", "}", "$$", "of algebraic spaces over $B$. Observe that $f|_Z$ is surjective", "(follows from Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image}", "and the fact that $|f|$ is closed) and proper", "(follows from Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-proper},", "\\ref{spaces-morphisms-lemma-closed-immersion-proper}, and", "\\ref{spaces-morphisms-lemma-universally-closed-permanence}).", "We have $f_*\\mathcal{F} = f_*i_*\\mathcal{G} = i'_*(f|_Z)_*\\mathcal{G}$", "by going around the diagram in two ways. Suppose we know the result holds", "for closed immersions and for $f|_Z$. Then we see that", "$$", "f_*[\\mathcal{F}]_k = f_*i_*[\\mathcal{G}]_k", "= (i')_*(f|_Z)_*[\\mathcal{G}]_k =", "(i')_*[(f|_Z)_*\\mathcal{G}]_k =", "[(i')_*(f|_Z)_*\\mathcal{G}]_k = [f_*\\mathcal{F}]_k", "$$", "as desired. The case of a closed immersion follows from", "Lemma \\ref{lemma-length-closed-immersion} and the definitions.", "Thus we have reduced to the case where", "$\\dim_\\delta(X) \\leq k$ and $f : X \\to Y$ is proper and surjective.", "\\medskip\\noindent", "Assume $\\dim_\\delta(X) \\leq k$ and $f : X \\to Y$ is proper and surjective.", "For every irreducible component $Z \\subset Y$", "with generic point $\\eta$ there exists a point $\\xi \\in X$ such", "that $f(\\xi) = \\eta$. Hence $\\delta(\\eta) \\leq \\delta(\\xi) \\leq k$.", "Thus we see that in the expressions", "$$", "f_*[\\mathcal{F}]_k = \\sum n_Z[Z],", "\\quad", "\\text{and}", "\\quad", "[f_*\\mathcal{F}]_k = \\sum m_Z[Z].", "$$", "whenever $n_Z \\not = 0$, or $m_Z \\not = 0$ the integral closed", "subspace $Z$ is actually an irreducible component of $Y$ of", "$\\delta$-dimension $k$ (see Lemma \\ref{lemma-point-of-max-dimension}).", "Pick such an integral closed subspace $Z \\subset Y$ and denote $\\eta$", "its generic point.", "Note that for any $\\xi \\in X$ with $f(\\xi) = \\eta$ we have $\\delta(\\xi) \\geq k$", "and hence $\\xi$ is a generic point of an irreducible component", "of $X$ of $\\delta$-dimension $k$ as well", "(see Lemma \\ref{lemma-point-of-max-dimension}).", "By Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-finite-in-codim-1}", "there exists an open subspace $\\eta \\in V \\subset Y$", "such that $f^{-1}(V) \\to V$ is finite.", "Since $\\eta$ is a generic point of an irreducible component of", "$|Y|$ we may assume $V$ is an affine scheme, see", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}.", "Replacing $Y$ by $V$ and $X$ by $f^{-1}(V)$ we reduce to the", "case where $Y$ is affine, and $f$ is finite.", "In particular $X$ and $Y$ are schemes and we reduce to", "the corresponding result for schemes, see", "Chow Homology, Lemma \\ref{chow-lemma-cycle-push-sheaf}", "(applied with $S = Y$)." ], "refs": [ "spaces-chow-lemma-cycle-closed-coherent", "spaces-cohomology-lemma-coherent-support-closed", "spaces-morphisms-definition-scheme-theoretic-image", "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image", "spaces-morphisms-lemma-base-change-proper", "spaces-morphisms-lemma-closed-immersion-proper", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-chow-lemma-length-closed-immersion", "spaces-chow-lemma-point-of-max-dimension", "spaces-chow-lemma-point-of-max-dimension", "spaces-over-fields-lemma-finite-in-codim-1", "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme", "chow-lemma-cycle-push-sheaf" ], "ref_ids": [ 7212, 11302, 4994, 4780, 4917, 4919, 4920, 7208, 7210, 7210, 12822, 11917, 5676 ] } ], "ref_ids": [ 11331 ] }, { "id": 7219, "type": "theorem", "label": "spaces-chow-lemma-flat-inverse-image-dimension", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-flat-inverse-image-dimension", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Let $f : X \\to Y$ be a morphism over $B$.", "Assume $f$ is flat of relative dimension $r$.", "For any closed subset $T \\subset |Y|$ we have", "$$", "\\dim_\\delta(|f|^{-1}(T)) = \\dim_\\delta(T) + r.", "$$", "provided $|f|^{-1}(T)$ is nonempty.", "If $Z \\subset Y$ is an integral closed subscheme and", "$Z' \\subset f^{-1}(Z)$ is an irreducible component, then", "$Z'$ dominates $Z$ and $\\dim_\\delta(Z') = \\dim_\\delta(Z) + r$." ], "refs": [], "proofs": [ { "contents": [ "Since the $\\delta$-dimension of a closed subset is the supremum of", "the $\\delta$-dimensions of the irreducible components, it suffices", "to prove the final statement. We may replace $Y$ by the", "integral closed subscheme $Z$ and $X$ by $f^{-1}(Z) = Z \\times_Y X$.", "Hence we may assume $Z = Y$ is integral and $f$ is a flat morphism", "of relative dimension $r$. Since $Y$ is locally Noetherian the", "morphism $f$ which is locally of finite type,", "is actually locally of finite presentation. Hence", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-fppf-open}", "applies and we see that $f$ is open.", "Let $\\xi \\in X$ be a generic point of an irreducible component", "of $X$. By the openness of $f$ we see that $f(\\xi)$ is the", "generic point $\\eta$ of $Z = Y$. Thus $Z'$ dominates $Z = Y$.", "Finally, we see that $\\xi$ and $\\eta$ are in the schematic", "locus of $X$ and $Y$ by", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}.", "Since $\\xi$ is a generic point of $X$ we see that", "$\\mathcal{O}_{X, \\xi} = \\mathcal{O}_{X_\\eta, \\xi}$ has only one", "prime ideal and hence has dimension $0$ (we may use usual", "local rings as $\\xi$ and $\\eta$ are in the schematic loci", "of $X$ and $Y$). Thus by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-dimension-fibre-at-a-point}", "(and the definition of morphisms of given relative dimension)", "we conclude that the transcendence", "degree of $\\kappa(\\xi)$ over $\\kappa(\\eta)$ is $r$.", "In other words, $\\delta(\\xi) = \\delta(\\eta) + r$ as desired." ], "refs": [ "spaces-morphisms-lemma-fppf-open", "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme", "spaces-morphisms-lemma-dimension-fibre-at-a-point" ], "ref_ids": [ 4855, 11917, 4870 ] } ], "ref_ids": [] }, { "id": 7220, "type": "theorem", "label": "spaces-chow-lemma-inverse-image-locally-finite", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-inverse-image-locally-finite", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Let $f : X \\to Y$ be a morphism over $B$.", "Assume $\\{T_i\\}_{i \\in I}$ is a locally", "finite collection of closed subsets of $|Y|$.", "Then $\\{|f|^{-1}(T_i)\\}_{i \\in I}$ is a locally finite", "collection of closed subsets of $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset |X|$ be a quasi-compact open subset.", "Since the image $|f|(U) \\subset |Y|$ is a quasi-compact subset", "there exists a quasi-compact open $V \\subset |Y|$ such that", "$|f|(U) \\subset V$. Note that", "$$", "\\{i \\in I : |f|^{-1}(T_i) \\cap U \\not = \\emptyset \\}", "\\subset", "\\{i \\in I : T_i \\cap V \\not = \\emptyset \\}.", "$$", "Since the right hand side is finite by assumption we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7221, "type": "theorem", "label": "spaces-chow-lemma-exact-sequence-open", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-exact-sequence-open", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $U \\subset X$ be an open subspace. Let $Y$ be the", "reduced closed subspace of $X$ with $|Y| = |X| \\setminus |U|$", "and denote $i : Y \\to X$ the inclusion morphism.", "For every $k \\in \\mathbf{Z}$ the sequence", "$$", "\\xymatrix{", "Z_k(Y) \\ar[r]^{i_*} & Z_k(X) \\ar[r]^{j^*} & Z_k(U) \\ar[r] & 0", "}", "$$", "is an exact complex of abelian groups." ], "refs": [], "proofs": [ { "contents": [ "Surjectivity of $j^*$ we saw above.", "First assume $X$ is quasi-compact. Then $Z_k(X)$ is a free $\\mathbf{Z}$-module", "with basis given by the elements $[Z]$ where $Z \\subset X$ is integral", "closed of $\\delta$-dimension $k$. Such a basis element maps", "either to the basis element $[Z \\cap U]$ of $Z_k(U)$", "or to zero if $Z \\subset Y$.", "Hence the lemma is clear in this case. The general case is similar", "and the proof is omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7222, "type": "theorem", "label": "spaces-chow-lemma-etale-pullback", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-etale-pullback", "contents": [ "In Situation \\ref{situation-setup} let $f : X \\to Y$ be an \\'etale", "morphism of good algebraic spaces over $B$. If $Z \\subset Y$ is an integral", "closed subspace, then $f^*[Z] = \\sum [Z']$ where the sum is over the", "irreducible components (Remark \\ref{remark-irreducible-component})", "of $f^{-1}(Z)$." ], "refs": [ "spaces-chow-remark-irreducible-component" ], "proofs": [ { "contents": [ "The meaning of the lemma is that the coefficient of $[Z']$ is $1$.", "This follows from the fact that $f^{-1}(Z)$ is a reduced algebraic space", "because it is \\'etale over the integral algebraic space $Z$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 7299 ] }, { "id": 7223, "type": "theorem", "label": "spaces-chow-lemma-compose-flat-pullback", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-compose-flat-pullback", "contents": [ "In Situation \\ref{situation-setup} let $X, Y, Z/B$ be good.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be flat morphisms of relative dimensions", "$r$ and $s$ over $B$. Then $g \\circ f$ is flat of relative dimension", "$r + s$ and", "$$", "f^* \\circ g^* = (g \\circ f)^*", "$$", "as maps $Z_k(Z) \\to Z_{k + r + s}(X)$." ], "refs": [], "proofs": [ { "contents": [ "The composition is flat of relative dimension $r + s$ by", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-dimension-fibre-at-a-point-additive} and", "\\ref{spaces-morphisms-lemma-composition-flat}.", "Suppose that", "\\begin{enumerate}", "\\item $A \\subset Z$ is a closed integral subspace of $\\delta$-dimension $k$,", "\\item $A' \\subset Y$ is a closed integral subspace of $\\delta$-dimension", "$k + s$ with $A' \\subset g^{-1}(A)$, and", "\\item $A'' \\subset Y$ is a closed integral subspace of $\\delta$-dimension", "$k + s + r$ with $A'' \\subset f^{-1}(W')$.", "\\end{enumerate}", "We have to show that the coefficient $n$ of $[A'']$ in $(g \\circ f)^*[A]$", "agrees with the coefficient $m$ of $[A'']$ in $f^*(g^*[A])$. We may choose", "a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\ar[r] & W \\ar[d] \\\\", "X \\ar[r] & Y \\ar[r] & Z", "}", "$$", "where $U, V, W$ are schemes, the vertical arrows are \\'etale, and", "there exist points $u \\in U$, $v \\in V$, $w \\in W$ such that", "$u \\mapsto v \\mapsto w$ and such that $u, v, w$ map to the generic", "points of $A'', A', A$. (Details omitted.)", "Then we have flat local ring homorphisms", "$\\mathcal{O}_{W, w} \\to \\mathcal{O}_{V, v}$,", "$\\mathcal{O}_{V, v} \\to \\mathcal{O}_{U, u}$,", "and repeatedly using Lemma \\ref{lemma-length}", "we find", "$$", "n = \\text{length}_{\\mathcal{O}_{U, u}}(", "\\mathcal{O}_{U, u}/\\mathfrak m_w\\mathcal{O}_{U, u})", "$$", "and", "$$", "m =", "\\text{length}_{\\mathcal{O}_{V, v}}(", "\\mathcal{O}_{V, v}/\\mathfrak m_w\\mathcal{O}_{V, v})", "\\text{length}_{\\mathcal{O}_{U, u}}(", "\\mathcal{O}_{U, u}/\\mathfrak m_v\\mathcal{O}_{U, u})", "$$", "Hence the equality follows from", "Algebra, Lemma \\ref{algebra-lemma-pullback-transitive}." ], "refs": [ "spaces-morphisms-lemma-dimension-fibre-at-a-point-additive", "spaces-morphisms-lemma-composition-flat", "spaces-chow-lemma-length", "algebra-lemma-pullback-transitive" ], "ref_ids": [ 4871, 4852, 7207, 641 ] } ], "ref_ids": [] }, { "id": 7224, "type": "theorem", "label": "spaces-chow-lemma-pullback-coherent", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-pullback-coherent", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Let $f : X \\to Y$ be a flat morphism of relative dimension $r$.", "\\begin{enumerate}", "\\item Let $Z \\subset Y$ be a closed subspace with", "$\\dim_\\delta(Z) \\leq k$. Then we have", "$\\dim_\\delta(f^{-1}(Z)) \\leq k + r$", "and $[f^{-1}(Z)]_{k + r} = f^*[Z]_k$ in $Z_{k + r}(X)$.", "\\item Let $\\mathcal{F}$ be a coherent sheaf on $Y$ with", "$\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$.", "Then we have $\\dim_\\delta(\\text{Supp}(f^*\\mathcal{F})) \\leq k + r$", "and", "$$", "f^*[{\\mathcal F}]_k = [f^*{\\mathcal F}]_{k+r}", "$$", "in $Z_{k + r}(X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from part (2) by Lemma \\ref{lemma-cycle-closed-coherent}", "and the fact that $f^*\\mathcal{O}_Z = \\mathcal{O}_{f^{-1}(Z)}$.", "\\medskip\\noindent", "Proof of (2).", "As $X$, $Y$ are locally Noetherian we may apply", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-coherent-Noetherian}", "to see", "that $\\mathcal{F}$ is of finite type, hence $f^*\\mathcal{F}$ is", "of finite type (Modules on Sites, Lemma", "\\ref{sites-modules-lemma-local-pullback}),", "hence $f^*\\mathcal{F}$ is coherent", "(Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-coherent-Noetherian}", "again).", "Thus the lemma makes sense. Let $W \\subset Y$ be an integral closed", "subspace of $\\delta$-dimension $k$, and let $W' \\subset X$ be", "an integral closed subspace of dimension $k + r$ mapping into $W$", "under $f$. We have to show that the coefficient $n$ of", "$[W']$ in $f^*[{\\mathcal F}]_k$ agrees with the coefficient", "$m$ of $[W']$ in $[f^*{\\mathcal F}]_{k+r}$. We may choose", "a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U, V$ are schemes, the vertical arrows are \\'etale, and", "there exist points $u \\in U$, $v \\in V$ such that", "$u \\mapsto v$ and such that $u, v$ map to the generic", "points of $W', W$. (Details omitted.)", "Consider the stalk $M = (\\mathcal{F}|_V)_v$ as an $\\mathcal{O}_{V, v}$-module.", "(Note that $M$ has finite length by our dimension assumptions, but we", "actually do not need to verify this. See Lemma \\ref{lemma-length-finite}.)", "We have", "$(f^*\\mathcal{F}|_U)_u = \\mathcal{O}_{U, u} \\otimes_{\\mathcal{O}_{V, v}} M$.", "Thus we see that", "$$", "n = \\text{length}_{\\mathcal{O}_{U, u}}", "(\\mathcal{O}_{U, u} \\otimes_{\\mathcal{O}_{V, v}} M)", "\\quad", "\\text{and}", "\\quad", "m = \\text{length}_{\\mathcal{O}_{V, v}}(M)", "\\text{length}_{\\mathcal{O}_{V, v}}(", "\\mathcal{O}_{U, u}/\\mathfrak m_v \\mathcal{O}_{U, u})", "$$", "Thus the equality follows from", "Algebra, Lemma \\ref{algebra-lemma-pullback-module}." ], "refs": [ "spaces-chow-lemma-cycle-closed-coherent", "spaces-cohomology-lemma-coherent-Noetherian", "sites-modules-lemma-local-pullback", "spaces-cohomology-lemma-coherent-Noetherian", "spaces-chow-lemma-length-finite", "algebra-lemma-pullback-module" ], "ref_ids": [ 7212, 11297, 14186, 11297, 7209, 640 ] } ], "ref_ids": [] }, { "id": 7225, "type": "theorem", "label": "spaces-chow-lemma-flat-pullback-proper-pushforward", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-flat-pullback-proper-pushforward", "contents": [ "In Situation \\ref{situation-setup} let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "be a fibre product diagram of good algebraic spaces over $B$.", "Assume $f : X \\to Y$ proper and $g : Y' \\to Y$ flat of relative dimension $r$.", "Then also $f'$ is proper and $g'$ is flat of relative dimension $r$.", "For any $k$-cycle $\\alpha$ on $X$ we have", "$$", "g^*f_*\\alpha = f'_*(g')^*\\alpha", "$$", "in $Z_{k + r}(Y')$." ], "refs": [], "proofs": [ { "contents": [ "The assertion that $f'$ is proper follows from", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-proper}.", "The assertion that $g'$ is flat of relative dimension $r$ follows from", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}", "and \\ref{spaces-morphisms-lemma-base-change-flat}.", "It suffices to prove the equality of cycles when $\\alpha = [W]$", "for some integral closed subspace $W \\subset X$ of $\\delta$-dimension $k$.", "Note that in this case we have $\\alpha = [\\mathcal{O}_W]_k$, see", "Lemma \\ref{lemma-cycle-closed-coherent}.", "By Lemmas \\ref{lemma-cycle-push-sheaf} and", "\\ref{lemma-pullback-coherent} it therefore suffices", "to show that $f'_*(g')^*\\mathcal{O}_W$ is isomorphic to", "$g^*f_*\\mathcal{O}_W$. This follows from cohomology and", "base change, see Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-flat-base-change-cohomology}." ], "refs": [ "spaces-morphisms-lemma-base-change-proper", "spaces-morphisms-lemma-dimension-fibre-after-base-change", "spaces-morphisms-lemma-base-change-flat", "spaces-chow-lemma-cycle-closed-coherent", "spaces-chow-lemma-cycle-push-sheaf", "spaces-chow-lemma-pullback-coherent", "spaces-cohomology-lemma-flat-base-change-cohomology" ], "ref_ids": [ 4917, 4872, 4853, 7212, 7218, 7224, 11296 ] } ], "ref_ids": [] }, { "id": 7226, "type": "theorem", "label": "spaces-chow-lemma-finite-flat", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-finite-flat", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Let $f : X \\to Y$ be a finite locally free morphism", "of degree $d$ (see", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-finite-locally-free}).", "Then $f$ is both proper and flat of relative dimension $0$, and", "$$", "f_*f^*\\alpha = d\\alpha", "$$", "for every $\\alpha \\in Z_k(Y)$." ], "refs": [ "spaces-morphisms-definition-finite-locally-free" ], "proofs": [ { "contents": [ "A finite locally free morphism is flat and finite by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-flat},", "and a finite morphism is proper", "by Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-proper}.", "We omit showing that a finite", "morphism has relative dimension $0$. Thus the formula makes sense.", "To prove it, let $Z \\subset Y$ be an integral closed subscheme", "of $\\delta$-dimension $k$. It suffices to prove the formula", "for $\\alpha = [Z]$. Since the base change of a finite locally free", "morphism is finite locally free", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-finite-locally-free})", "we see that $f_*f^*\\mathcal{O}_Z$ is a finite locally free sheaf of", "rank $d$ on $Z$. Thus clearly $f_*f^*\\mathcal{O}_Z$ has length $d$", "at the generic point of $Z$. Hence", "$$", "f_*f^*[Z] = f_*f^*[\\mathcal{O}_Z]_k =", "[f_*f^*\\mathcal{O}_Z]_k = d[Z]", "$$", "where we have used Lemmas \\ref{lemma-pullback-coherent} and", "\\ref{lemma-cycle-push-sheaf}." ], "refs": [ "spaces-morphisms-lemma-finite-flat", "spaces-morphisms-lemma-finite-proper", "spaces-morphisms-lemma-base-change-finite-locally-free", "spaces-chow-lemma-pullback-coherent", "spaces-chow-lemma-cycle-push-sheaf" ], "ref_ids": [ 4954, 4946, 4953, 7224, 7218 ] } ], "ref_ids": [ 5018 ] }, { "id": 7227, "type": "theorem", "label": "spaces-chow-lemma-divisor-delta-dimension", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-divisor-delta-dimension", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good. Assume $X$ is", "integral.", "\\begin{enumerate}", "\\item If $Z \\subset X$ is an integral closed subspace, then", "the following are equivalent:", "\\begin{enumerate}", "\\item $Z$ is a prime divisor,", "\\item $|Z|$ has codimension $1$ in $|X|$, and", "\\item $\\dim_\\delta(Z) = \\dim_\\delta(X) - 1$.", "\\end{enumerate}", "\\item If $Z$ is an irreducible component of an effective Cartier", "divisor on $X$, then $\\dim_\\delta(Z) = \\dim_\\delta(X) - 1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from the definition of a prime divisor", "(Spaces over Fields, Definition", "\\ref{spaces-over-fields-definition-Weil-divisor}),", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-codimension-local-ring},", "and the definition of a dimension function", "(Topology, Definition \\ref{topology-definition-dimension-function}).", "\\medskip\\noindent", "Let $D \\subset X$ be an effective Cartier divisor. Let $Z \\subset D$", "be an irreducible component and let $\\xi \\in |Z|$ be the generic point.", "Choose an \\'etale neighbourhood $(U, u) \\to (X, \\xi)$ where $U = \\Spec(A)$", "and $D \\times_X U$ is cut out by a nonzerodivisor $f \\in A$, see", "Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-characterize-effective-Cartier-divisor}.", "Then $u$ is a generic point of $V(f)$ by", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-generic-points}.", "Hence $\\mathcal{O}_{U, u}$ has dimension $1$ by", "Krull's Hauptidealsatz (Algebra, Lemma \\ref{algebra-lemma-minimal-over-1}).", "Thus $\\xi$ is a codimension $1$ point on $X$ and $Z$ is a prime", "divisor as desired." ], "refs": [ "spaces-over-fields-definition-Weil-divisor", "decent-spaces-lemma-codimension-local-ring", "topology-definition-dimension-function", "spaces-divisors-lemma-characterize-effective-Cartier-divisor", "decent-spaces-lemma-decent-generic-points", "algebra-lemma-minimal-over-1" ], "ref_ids": [ 12886, 9532, 8367, 12935, 9531, 683 ] } ], "ref_ids": [] }, { "id": 7228, "type": "theorem", "label": "spaces-chow-lemma-etale-pullback-principal-divisor", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-etale-pullback-principal-divisor", "contents": [ "In Situation \\ref{situation-setup} let $f : X \\to Y$ be an \\'etale", "morphism of good algebraic spaces over $B$. Assume $Y$ is integral.", "Let $g \\in R(Y)^*$. As cycles on $X$ we have", "$$", "f^*(\\text{div}_Y(g)) =", "\\sum\\nolimits_{X'} (X' \\to X)_*\\text{div}_{X'}(g \\circ f|_{X'})", "$$", "where the sum is over the irreducible components of $X$", "(Remark \\ref{remark-irreducible-component})." ], "refs": [ "spaces-chow-remark-irreducible-component" ], "proofs": [ { "contents": [ "The map $|X| \\to |Y|$ is open. The set of irreducible components of $|X|$", "is locally finite in $|X|$. We conclude that $f|_{X'} : X' \\to Y$", "is dominant for every irreducible component $X' \\subset X$.", "Thus $g \\circ f|_{X'}$ is defined (Morphisms of Spaces, Section", "\\ref{spaces-morphisms-section-rational-maps}), hence", "$\\text{div}_{X'}(g \\circ f|_{X'})$ is defined. Moreover, the", "sum is locally finite and we find that the right hand side indeed", "is a cycle on $X$. The left hand side is defined by", "Definition \\ref{definition-flat-pullback}", "and the fact that an \\'etale morphism is flat of relative dimension $0$.", "\\medskip\\noindent", "Since $f$ is \\'etale we see that $\\delta_X(x) = \\delta_y(f(x))$", "for all $x \\in |X|$. Thus if $\\dim_\\delta(Y) = n$, then $\\dim_\\delta(X') = n$", "for every irreducible component $X'$ of $X$ (since generic points of $X$", "map to the generic point of $Y$, see above). Thus both left", "and right hand side are $(n - 1)$-cycles.", "\\medskip\\noindent", "Let $Z \\subset X$ be an integral closed subspace with $\\dim_\\delta(Z) = n - 1$.", "To prove the equality, we need to show that the coefficients", "of $Z$ are the same. Let $Z' \\subset Y$ be the integral closed", "subspace constructed in Lemma \\ref{lemma-proper-image}.", "Then $\\dim_\\delta(Z') = n - 1$ too. Let $\\xi \\in |Z|$ be the generic point.", "Then $\\xi' = f(\\xi) \\in |Z'|$ is the generic point.", "Consider the commutative diagram", "$$", "\\xymatrix{", "\\Spec(\\mathcal{O}_{X, \\xi}^h) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(\\mathcal{O}_{Y, \\xi'}^h) \\ar[r] & Y", "}", "$$", "of Decent Spaces, Remark", "\\ref{decent-spaces-remark-functoriality-henselian-local-ring}.", "We have to be slightly careful as the reduced Noetherian local rings", "$\\mathcal{O}_{X, \\xi}^h$ and $\\mathcal{O}_{Y, \\xi'}^h$ need not be domains.", "Thus we work with total rings of fractions $Q(-)$ rather than fraction fields.", "By definition, to get the coefficient of $Z'$ in $\\text{div}_Y(g)$", "we write the image of $g$ in $Q(\\mathcal{O}_{Y, \\xi'}^h)$", "as $a/b$ with $a, b \\in \\mathcal{O}_{Y, \\xi'}^h$ nonzerodivisors", "and we take", "$$", "\\text{ord}_{Z'}(g) =", "\\text{length}_{\\mathcal{O}_{Y, \\xi'}^h}", "(\\mathcal{O}_{Y, \\xi'}^h/a \\mathcal{O}_{Y, \\xi'}^h) -", "\\text{length}_{\\mathcal{O}_{Y, \\xi'}^h}", "(\\mathcal{O}_{Y, \\xi'}^h/b \\mathcal{O}_{Y, \\xi'}^h)", "$$", "Observe that the coefficient of $Z$ in $f^*\\text{div}_Y(G)$", "is the same integer, see Lemma \\ref{lemma-etale-pullback}.", "Suppose that $\\xi \\in X'$. Then we can consider the maps", "$$", "\\mathcal{O}_{Y, \\xi'}^h \\to", "\\mathcal{O}_{X, \\xi}^h \\to", "\\mathcal{O}_{X', \\xi}^h", "$$", "The first arrow is flat and the second arrow is a surjective", "map of reduced local Noetherian rings of dimension $1$.", "Therefore both these maps send nonzerodivisors to nonzerodivisors", "and we conclude the coefficient of $Z'$ in $\\text{div}_{X'}(g \\circ f|_{X'})$", "is", "$$", "\\text{ord}_Z(g \\circ f|_{X'}) =", "\\text{length}_{\\mathcal{O}_{X', \\xi}^h}", "(\\mathcal{O}_{X', \\xi}^h/a \\mathcal{O}_{X', \\xi}^h) -", "\\text{length}_{\\mathcal{O}_{Y, \\xi'}^h}", "(\\mathcal{O}_{X', \\xi}^h/b \\mathcal{O}_{X', \\xi}^h)", "$$", "by the same prescription as above. Thus it suffices to show", "$$", "\\text{length}_{\\mathcal{O}_{Y, \\xi'}^h}", "(\\mathcal{O}_{Y, \\xi'}^h/a \\mathcal{O}_{Y, \\xi'}^h) =", "\\sum\\nolimits_{\\xi \\in |X'|}", "\\text{length}_{\\mathcal{O}_{X', \\xi}^h}", "(\\mathcal{O}_{X', \\xi}^h/a \\mathcal{O}_{X', \\xi}^h)", "$$", "First, since the ring map $\\mathcal{O}_{Y, \\xi'}^h \\to", "\\mathcal{O}_{X, \\xi}^h$ is flat and unramified, we have", "$$", "\\text{length}_{\\mathcal{O}_{Y, \\xi'}^h}", "(\\mathcal{O}_{Y, \\xi'}^h/a \\mathcal{O}_{Y, \\xi'}^h) =", "\\text{length}_{\\mathcal{O}_{X, \\xi}^h}", "(\\mathcal{O}_{X, \\xi}^h/a \\mathcal{O}_{X, \\xi}^h)", "$$", "by Algebra, Lemma \\ref{algebra-lemma-pullback-module}.", "Let $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ be the nonmaximal", "primes of $\\mathcal{O}_{X, \\xi}^h$ and set", "$R_j = \\mathcal{O}_{X, \\xi}^h/\\mathfrak q_j$.", "For $X'$ as above, denote $J(X') \\subset \\{1, \\ldots, t\\}$", "the set of indices such that $\\mathfrak q_j$ corresponds", "to a point of $X'$, i.e., such that under the surjection", "$\\mathcal{O}_{X, \\xi}^h \\to \\mathcal{O}_{X', \\xi}$", "the prime $\\mathfrak q_j$ corresponds to a prime", "of $\\mathcal{O}_{X', \\xi}$.", "By Chow Homology, Lemma \\ref{chow-lemma-additivity-divisors-restricted}", "we get", "$$", "\\text{length}_{\\mathcal{O}_{X, \\xi}^h}", "(\\mathcal{O}_{X, \\xi}^h/a \\mathcal{O}_{X, \\xi}^h) =", "\\sum\\nolimits_j \\text{length}_{R_j}(R_j/a R_j)", "$$", "and", "$$", "\\text{length}_{\\mathcal{O}_{X', \\xi}^h}", "(\\mathcal{O}_{X', \\xi}^h/a \\mathcal{O}_{X', \\xi}^h) =", "\\sum\\nolimits_{j \\in J(X')} \\text{length}_{R_j}(R_j/a R_j)", "$$", "Thus the result of the lemma holds because", "$\\{1, \\ldots, t\\}$ is the disjoint union of the sets $J(X')$:", "each point of codimension $0$ on $X$ lies on a unique $X'$." ], "refs": [ "spaces-chow-definition-flat-pullback", "spaces-chow-lemma-proper-image", "decent-spaces-remark-functoriality-henselian-local-ring", "spaces-chow-lemma-etale-pullback", "algebra-lemma-pullback-module", "chow-lemma-additivity-divisors-restricted" ], "ref_ids": [ 7287, 7214, 9575, 7222, 640, 5653 ] } ], "ref_ids": [ 7299 ] }, { "id": 7229, "type": "theorem", "label": "spaces-chow-lemma-proper-pushforward-alteration", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-proper-pushforward-alteration", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Assume $X$, $Y$ are integral and $n = \\dim_\\delta(X) = \\dim_\\delta(Y)$.", "Let $p : X \\to Y$ be a dominant proper morphism.", "Let $f \\in R(X)^*$. Set", "$$", "g = \\text{Nm}_{R(X)/R(Y)}(f).", "$$", "Then we have", "$p_*\\text{div}(f) = \\text{div}(g)$." ], "refs": [], "proofs": [ { "contents": [ "We are going to deduce this from the case of schemes by \\'etale localization.", "Let $Z \\subset Y$ be an integral closed subspace of $\\delta$-dimension", "$n - 1$. We want to show that the coefficient of $[Z]$ in", "$p_*\\text{div}(f)$ and $\\text{div}(g)$ are equal. Apply", "Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-finite-in-codim-1}", "to the morphism $p : X \\to Y$ and the generic point $\\xi \\in |Z|$.", "We find that we may replace $Y$ by an open subspace containing $\\xi$", "and assume that $p : X \\to Y$ is finite. Pick an \\'etale neighbourhood", "$(V, v) \\to (Y, \\xi)$ where $V$ is an affine scheme.", "By Lemma \\ref{lemma-etale-pullback} it suffices to prove the equality", "of cycles after pulling back to $V$.", "Set $U = V \\times_Y X$ and consider the commutative diagram", "$$", "\\xymatrix{", "U \\ar[r]_a \\ar[d]_{p'} & X \\ar[d]^p \\\\", "V \\ar[r]^b & Y", "}", "$$", "Let $V_j \\subset V$, $j = 1, \\ldots, m$ be the irreducible components", "of $V$. For each $i$, let $U_{j, i}$, $i = 1, \\ldots, n_j$ be the", "irreducible components of $U$ dominating $V_j$. Denote", "$p'_{j, i} : U_{j, i} \\to V_j$ the restriction of $p' : U \\to V$.", "By the case of schemes", "(Chow Homology, Lemma \\ref{chow-lemma-proper-pushforward-alteration})", "we see that", "$$", "p'_{j, i, *}\\text{div}_{U_{j, i}}(f_{j, i}) = \\text{div}_{V_j}(g_{j, i})", "$$", "where $f_{j, i}$ is the restriction of $f$ to $U_{j, i}$ and", "$g_{j, i}$ is the norm of $f_{j, i}$ along the finite extension", "$R(U_{j, i})/R(V_j)$. We have", "\\begin{align*}", "b^* p_*\\text{div}_X(f)", "& =", "p'_* a^* \\text{div}_X(f) \\\\", "& =", "p'_*\\left(\\sum\\nolimits_{j, i}", "(U_{j, i} \\to U)_*\\text{div}_{U_{j, i}}(f_{j, i})\\right) \\\\", "& =", "\\sum\\nolimits_{j, i}", "(V_j \\to V)_*p'_{j, i, *}\\text{div}_{U_{j, i}}(f_{j, i}) \\\\", "& =", "\\sum\\nolimits_j", "(V_j \\to V)_*\\left(\\sum\\nolimits_i \\text{div}_{V_j}(g_{j, i})\\right) \\\\", "& =", "\\sum\\nolimits_j (V_j \\to V)_*\\text{div}_{V_j}(\\prod\\nolimits_i g_{j, i})", "\\end{align*}", "by Lemmas", "\\ref{lemma-flat-pullback-proper-pushforward},", "\\ref{lemma-etale-pullback-principal-divisor}, and", "\\ref{lemma-compose-pushforward}.", "To finish the proof, using Lemma \\ref{lemma-etale-pullback-principal-divisor}", "again, it suffices to show that", "$$", "g \\circ b|_{V_j} = \\prod\\nolimits_i g_{j, i}", "$$", "as elements of the function field of $V_j$. In terms of fields", "this is the following statement: let $L/K$ be a finite extension.", "Let $M/K$ be a finite separable extension. Write", "$M \\otimes_K L = \\prod M_i$. Then for $t \\in L$ with images", "$t_i \\in M_i$ the image of", "$\\text{Norm}_{L/K}(t)$ in $M$ is $\\prod \\text{Norm}_{M_i/M}(t_i)$.", "We omit the proof." ], "refs": [ "spaces-over-fields-lemma-finite-in-codim-1", "spaces-chow-lemma-etale-pullback", "chow-lemma-proper-pushforward-alteration", "spaces-chow-lemma-flat-pullback-proper-pushforward", "spaces-chow-lemma-etale-pullback-principal-divisor", "spaces-chow-lemma-compose-pushforward", "spaces-chow-lemma-etale-pullback-principal-divisor" ], "ref_ids": [ 12822, 7222, 5687, 7225, 7228, 7217, 7228 ] } ], "ref_ids": [] }, { "id": 7230, "type": "theorem", "label": "spaces-chow-lemma-restrict-to-open", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-restrict-to-open", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $U \\subset X$ be an open subspace. Let $Y$ be the", "reduced closed subspace of $X$ with $|Y| = |X| \\setminus |U|$", "and denote $i : Y \\to X$ the inclusion morphism. Let $k \\in \\mathbf{Z}$.", "Suppose $\\alpha, \\beta \\in Z_k(X)$. If $\\alpha|_U \\sim_{rat} \\beta|_U$", "then there exist a cycle $\\gamma \\in Z_k(Y)$ such that", "$$", "\\alpha \\sim_{rat} \\beta + i_*\\gamma.", "$$", "In other words, the sequence", "$$", "\\xymatrix{", "\\CH_k(Y) \\ar[r]^{i_*} & \\CH_k(X) \\ar[r]^{j^*} & \\CH_k(U) \\ar[r] & 0", "}", "$$", "is an exact complex of abelian groups." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{W_j\\}_{j \\in J}$ be a locally finite collection of integral closed", "subspaces of $U$ of $\\delta$-dimension $k + 1$, and let $f_j \\in R(W_j)^*$", "be elements such that $(\\alpha - \\beta)|_U = \\sum (i_j)_*\\text{div}(f_j)$", "as in the definition. Let $W_j' \\subset X$ be the corresponding integral", "closed subspace of $X$, i.e., having the same generic point as $W_j$.", "Suppose that $V \\subset X$ is a quasi-compact open.", "Then also $V \\cap U$ is quasi-compact open in $U$ as $V$ is Noetherian.", "Hence the set $\\{j \\in J \\mid W_j \\cap V \\not = \\emptyset\\}", "= \\{j \\in J \\mid W'_j \\cap V \\not = \\emptyset\\}$", "is finite since $\\{W_j\\}$ is locally finite. In other words we see that", "$\\{W'_j\\}$ is also locally finite. Since $R(W_j) = R(W'_j)$ we see", "that", "$$", "\\alpha - \\beta - \\sum (i'_j)_*\\text{div}(f_j)", "$$", "is a cycle on $X$ whose restriction to $U$ is zero.", "The lemma follows by applying Lemma \\ref{lemma-exact-sequence-open}." ], "refs": [ "spaces-chow-lemma-exact-sequence-open" ], "ref_ids": [ 7221 ] } ], "ref_ids": [] }, { "id": 7231, "type": "theorem", "label": "spaces-chow-lemma-prepare-flat-pullback-rational-equivalence", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-prepare-flat-pullback-rational-equivalence", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Assume $Y$ integral with $\\dim_\\delta(Y) = k$.", "Let $f : X \\to Y$ be a flat morphism of", "relative dimension $r$. Then for $g \\in R(Y)^*$ we have", "$$", "f^*\\text{div}_Y(g) =", "\\sum m_{X', X} (X' \\to X)_*\\text{div}_{X'}(g \\circ f|_{X'})", "$$", "as $(k + r - 1)$-cycles on $X$ where the sum is over the irreducible", "components $X'$ of $X$ and $m_{X', X}$ is the multiplicity of $X'$ in $X$." ], "refs": [], "proofs": [ { "contents": [ "Observe that any irreducible component of $X$ dominates $Y$", "(Lemma \\ref{lemma-flat-inverse-image-dimension})", "and hence the composition $g \\circ f|_{X'}$ is defined", "(Morphisms of Spaces, Section \\ref{spaces-morphisms-section-rational-maps}).", "We will reduce this to the case of schemes. Choose a scheme $V$ and a", "surjective \\'etale morphism $V \\to Y$. Choose a scheme $U$ and a surjective", "\\'etale morphism $U \\to V \\times_Y X$. Picture", "$$", "\\xymatrix{", "U \\ar[r]_a \\ar[d]_h & X \\ar[d]^f \\\\", "V \\ar[r]^b & Y", "}", "$$", "Since $a$ is surjective and \\'etale it follows from", "Lemma \\ref{lemma-etale-pullback} that it suffices to prove the", "equality of cycles after pulling back by $a$.", "We can use Lemma \\ref{lemma-etale-pullback-principal-divisor} to write", "$$", "b^*\\text{div}_Y(g) = \\sum (V' \\to V)_*\\text{div}_{V'}(g \\circ b|_{V'})", "$$", "where the sum is over the irreducible components $V'$ of $V$.", "Using Lemma \\ref{lemma-flat-pullback-proper-pushforward} we find", "$$", "h^*b^*\\text{div}_Y(g) =", "\\sum (V' \\times_V U \\to U)_*(h')^*\\text{div}_{V'}(g \\circ b|_{V'})", "$$", "where $h' : V' \\times_V U \\to V'$ is the projection.", "We may apply the lemma in the case of schemes", "(Chow Homology, Lemma", "\\ref{chow-lemma-prepare-flat-pullback-rational-equivalence})", "to the morphism $h' : V' \\times_V U \\to V'$ to see that we have", "$$", "(h')^*\\text{div}_{V'}(g \\circ b|_{V'}) =", "\\sum", "m_{U', V' \\times_V U}", "(U' \\to V' \\times_V U)_*\\text{div}_{U'}(g \\circ b|_{V'} \\circ h'|_{U'})", "$$", "where the sum is over the irreducible components $U'$ of $V' \\times_V U$.", "Each $U'$ occurring in this sum is an irreducible component of $U$", "and conversely every irreducible component $U'$ of $U$ is an", "irreducible component of $V' \\times_V U$", "for a unique irreducible component $V' \\subset V$.", "Given an irreducible component $U' \\subset U$,", "denote $\\overline{a(U')} \\subset X$ the ``image'' in $X$", "(Lemma \\ref{lemma-proper-image}); this", "is an irreducible component of $X$ for example by", "Lemma \\ref{lemma-flat-inverse-image-dimension}.", "The muplticity $m_{U', V' \\times_V U}$ is equal to", "the multiplicity $m_{\\overline{a(U')}, X}$.", "This follows from the equality $h^*a^*[Y] = b^*f^*[Y]$", "(Lemma \\ref{lemma-compose-flat-pullback}), the definitions, and", "Lemma \\ref{lemma-etale-pullback}.", "Combining all of what we just said we obtain", "$$", "a^*f^*\\text{div}_Y(g) =", "h^*b^*\\text{div}_Y(g) =", "\\sum m_{\\overline{a(U')}, X}", "(U' \\to U)_*\\text{div}_{U'}(g \\circ (f \\circ a)|_{U'})", "$$", "Next, we analyze what happens with the right hand side of the", "formula in the statement of the lemma if we pullback by $a$.", "First, we use Lemma \\ref{lemma-flat-pullback-proper-pushforward}", "to get", "$$", "a^*\\sum m_{X', X} (X' \\to X)_*\\text{div}_{X'}(g \\circ f|_{X'}) =", "\\sum m_{X', X} (X' \\times_X U \\to U)_*(a')^*\\text{div}_{X'}(g \\circ f|_{X'})", "$$", "where $a' : X' \\times_X U \\to X'$ is the projection.", "By Lemma \\ref{lemma-etale-pullback-principal-divisor}", "we get", "$$", "(a')^*\\text{div}_{X'}(g \\circ f|_{X'}) =", "\\sum (U' \\to X' \\times_X U)_*\\text{div}_{U'}(g \\circ (f \\circ a)|_{U'})", "$$", "where the sum is over the irreducible components $U'$ of $X' \\times_X U$.", "These $U'$ are irreducible components of $U$ and in fact are exactly the", "irreducible components of $U$ such that $\\overline{a(U')} = X'$.", "Comparing with what we obtained above we conclude." ], "refs": [ "spaces-chow-lemma-flat-inverse-image-dimension", "spaces-chow-lemma-etale-pullback", "spaces-chow-lemma-etale-pullback-principal-divisor", "spaces-chow-lemma-flat-pullback-proper-pushforward", "chow-lemma-prepare-flat-pullback-rational-equivalence", "spaces-chow-lemma-proper-image", "spaces-chow-lemma-flat-inverse-image-dimension", "spaces-chow-lemma-compose-flat-pullback", "spaces-chow-lemma-etale-pullback", "spaces-chow-lemma-flat-pullback-proper-pushforward", "spaces-chow-lemma-etale-pullback-principal-divisor" ], "ref_ids": [ 7219, 7222, 7228, 7225, 5692, 7214, 7219, 7223, 7222, 7225, 7228 ] } ], "ref_ids": [] }, { "id": 7232, "type": "theorem", "label": "spaces-chow-lemma-flat-pullback-rational-equivalence", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-flat-pullback-rational-equivalence", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Let $f : X \\to Y$ be a flat morphism of relative dimension $r$.", "Let $\\alpha \\sim_{rat} \\beta$ be rationally equivalent $k$-cycles", "on $Y$. Then $f^*\\alpha \\sim_{rat} f^*\\beta$ as $(k + r)$-cycles on $X$." ], "refs": [], "proofs": [ { "contents": [ "What do we have to show? Well, suppose we are given a collection", "$$", "i_j : W_j \\longrightarrow Y", "$$", "of closed immersions, with each $W_j$ integral of $\\delta$-dimension $k + 1$", "and rational functions $g_j \\in R(W_j)^*$. Moreover, assume that", "the collection $\\{|i_j|(|W_j|)\\}_{j \\in J}$ is locally finite in $|Y|$.", "Then we have to show that", "$$", "f^*(\\sum i_{j, *}\\text{div}(g_j)) = \\sum f^*i_{j, *}\\text{div}(g_j)", "$$", "is rationally equivalent to zero on $X$. The sum on the right", "makes sense by Lemma \\ref{lemma-inverse-image-locally-finite}.", "\\medskip\\noindent", "Consider the fibre products", "$$", "i'_j : W'_j = W_j \\times_Y X \\longrightarrow X.", "$$", "and denote $f_j : W'_j \\to W_j$ the first projection.", "By Lemma \\ref{lemma-flat-pullback-proper-pushforward}", "we can write the sum above as", "$$", "\\sum i'_{j, *}(f_j^*\\text{div}(g_j))", "$$", "By Lemma \\ref{lemma-prepare-flat-pullback-rational-equivalence}", "we see that each $f_j^*\\text{div}(g_j)$ is rationally equivalent", "to zero on $W'_j$. Hence each $i'_{j, *}(f_j^*\\text{div}(g_j))$", "is rationally equivalent to zero. Then the same is true for", "the displayed sum by the discussion in", "Remark \\ref{remark-infinite-sums-rational-equivalences}." ], "refs": [ "spaces-chow-lemma-inverse-image-locally-finite", "spaces-chow-lemma-flat-pullback-proper-pushforward", "spaces-chow-lemma-prepare-flat-pullback-rational-equivalence", "spaces-chow-remark-infinite-sums-rational-equivalences" ], "ref_ids": [ 7220, 7225, 7231, 7302 ] } ], "ref_ids": [] }, { "id": 7233, "type": "theorem", "label": "spaces-chow-lemma-proper-pushforward-rational-equivalence", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-proper-pushforward-rational-equivalence", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Let $p : X \\to Y$ be a proper morphism.", "Suppose $\\alpha, \\beta \\in Z_k(X)$ are rationally equivalent.", "Then $p_*\\alpha$ is rationally equivalent to $p_*\\beta$." ], "refs": [], "proofs": [ { "contents": [ "What do we have to show? Well, suppose we are given a collection", "$$", "i_j : W_j \\longrightarrow X", "$$", "of closed immersions, with each $W_j$ integral of $\\delta$-dimension $k + 1$", "and rational functions $f_j \\in R(W_j)^*$.", "Moreover, assume that", "the collection $\\{i_j(W_j)\\}_{j \\in J}$ is locally finite on $X$.", "Then we have to show that", "$$", "p_*\\left(\\sum i_{j, *}\\text{div}(f_j)\\right)", "$$", "is rationally equivalent to zero on $X$.", "\\medskip\\noindent", "Note that the sum is equal to", "$$", "\\sum p_*i_{j, *}\\text{div}(f_j).", "$$", "Let $W'_j \\subset Y$ be the integral closed subspace which is the", "image of $p \\circ i_j$, see Lemma \\ref{lemma-proper-image}.", "The collection $\\{W'_j\\}$ is locally finite", "in $Y$ by Lemma \\ref{lemma-quasi-compact-locally-finite}.", "Hence it suffices to show, for a given $j$, that either", "$p_*i_{j, *}\\text{div}(f_j) = 0$ or that it", "is equal to $i'_{j, *}\\text{div}(g_j)$ for some $g_j \\in R(W'_j)^*$.", "\\medskip\\noindent", "The arguments above therefore reduce us to the case of a since", "integral closed subspace $W \\subset X$ of $\\delta$-dimension $k + 1$.", "Let $f \\in R(W)^*$. Let $W' = p(W)$ as above.", "We get a commutative diagram of morphisms", "$$", "\\xymatrix{", "W \\ar[r]_i \\ar[d]_{p'} & X \\ar[d]^p \\\\", "W' \\ar[r]^{i'} & Y", "}", "$$", "Note that $p_*i_*\\text{div}(f) = i'_*(p')_*\\text{div}(f)$ by", "Lemma \\ref{lemma-compose-pushforward}. As explained above", "we have to show that $(p')_*\\text{div}(f)$", "is the divisor of a rational function on $W'$ or zero.", "There are three cases to distinguish.", "\\medskip\\noindent", "The case $\\dim_\\delta(W') < k$. In this case automatically", "$(p')_*\\text{div}(f) = 0$ and there is nothing to prove.", "\\medskip\\noindent", "The case $\\dim_\\delta(W') = k$. Let us show that $(p')_*\\text{div}(f) = 0$", "in this case. Since $(p')_*\\text{div}(f)$ is a $k$-cycle, we see", "that $(p')_*\\text{div}(f) = n[W']$ for some $n \\in \\mathbf{Z}$.", "In order to prove that $n = 0$ we may replace $W'$ by", "a nonempty open subspace. In particular, we may and", "do assume that $W'$ is a scheme. Let $\\eta \\in W'$ be the generic point.", "Let $K = \\kappa(\\eta) = R(W')$ be the function field.", "Consider the base change diagram", "$$", "\\xymatrix{", "W_\\eta \\ar[r] \\ar[d]_c & W \\ar[d]^{p'} \\\\", "\\Spec(K) \\ar[r]^\\eta & W'", "}", "$$", "Observe that $c$ is proper. Also $|W_\\eta|$ has dimension $1$:", "use Decent Spaces, Lemma \\ref{decent-spaces-lemma-topology-fibre}", "to identify $|W_\\eta|$ as the subspace of $|W|$ of points mapping to $\\eta$", "and note that since $\\dim_\\delta(W) = k + 1$ and $\\delta(\\eta) = k$", "points of $W_\\eta$ must have $\\delta$-value either $k$ or $k + 1$.", "Thus the local rings have dimension $\\leq 1$ by", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-codimension-local-ring}.", "By Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-codim-1-point-in-schematic-locus}", "we find that $W_\\eta$ is a scheme.", "Since $\\Spec(K)$ is the limit of the nonempty affine open subschemes of $W'$", "we conclude that we may assume that $W$ is a scheme by", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-limit-is-scheme}.", "Then finally by the case of schemes", "(Chow Homology, Lemma \\ref{chow-lemma-proper-pushforward-rational-equivalence})", "we find that $n = 0$.", "\\medskip\\noindent", "The case $\\dim_\\delta(W') = k + 1$. In this case", "Lemma \\ref{lemma-proper-pushforward-alteration} applies,", "and we see that indeed $p'_*\\text{div}(f) = \\text{div}(g)$", "for some $g \\in R(W')^*$ as desired." ], "refs": [ "spaces-chow-lemma-proper-image", "spaces-chow-lemma-quasi-compact-locally-finite", "spaces-chow-lemma-compose-pushforward", "decent-spaces-lemma-topology-fibre", "decent-spaces-lemma-codimension-local-ring", "spaces-over-fields-lemma-codim-1-point-in-schematic-locus", "spaces-limits-lemma-limit-is-scheme", "chow-lemma-proper-pushforward-rational-equivalence", "spaces-chow-lemma-proper-pushforward-alteration" ], "ref_ids": [ 7214, 7216, 7217, 9525, 9532, 12845, 4579, 5694, 7229 ] } ], "ref_ids": [] }, { "id": 7234, "type": "theorem", "label": "spaces-chow-lemma-compute-c1", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-compute-c1", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Assume $X$ is integral and $n = \\dim_\\delta(X)$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{L})$ be a nonzero global section.", "Then", "$$", "\\text{div}_\\mathcal{L}(s) = [Z(s)]_{n - 1}", "$$", "in $Z_{n - 1}(X)$ and", "$$", "c_1(\\mathcal{L}) \\cap [X] = [Z(s)]_{n - 1}", "$$", "in $\\CH_{n - 1}(X)$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset X$ be an integral closed subspace of $\\delta$-dimension $n - 1$.", "Let $\\xi \\in |Z|$ be its generic point. To prove the first equality", "we compare the coefficients of $Z$ on both sides. Choose an elementary", "\\'etale neighbourhood $(U, u) \\to (X, \\xi)$, see", "Decent Spaces, Section", "\\ref{decent-spaces-section-residue-fields-henselian-local-rings}", "and recall that $\\mathcal{O}_{X, \\xi}^h = \\mathcal{O}_{U, u}^h$", "in this case. After replacing $U$ by an open neighbourhood", "of $u$ we may assume there is a trivializing section", "$s_U$ of $\\mathcal{L}|_U$. Write $s|_U = f s_U$ for some", "$f \\in \\Gamma(U, \\mathcal{O}_U)$. Then $Z \\times_X U$ is", "equal to $V(f)$ as a closed subscheme of $U$, see", "Divisors on Spaces, Definition \\ref{spaces-divisors-definition-zero-scheme-s}.", "As in Spaces over Fields, Section \\ref{spaces-over-fields-section-c1}", "denote $\\mathcal{L}_\\xi$ the pullback of $\\mathcal{L}$", "under the canonical morphism", "$c_\\xi : \\Spec(\\mathcal{O}_{X, \\xi}^h) \\to X$.", "Denote $s_\\xi$ the pullback of $s_U$; it is a trivialization of", "$\\mathcal{L}_\\xi$. Then we see that", "$c_\\xi^*(s) = fs_\\xi$. The coefficient of $Z$ in $[Z(s)]_{n - 1}$", "is by definition", "$$", "\\text{length}_{\\mathcal{O}_{U, u}}(\\mathcal{O}_{U, u}/f\\mathcal{O}_{U, u})", "$$", "Since $\\mathcal{O}_{U, u} \\to \\mathcal{O}_{X, \\xi}^h$", "is flat and identifies residue fields this is equal to", "$$", "\\text{length}_{\\mathcal{O}_{X, \\xi}^h}", "(\\mathcal{O}_{X, \\xi}^h/f\\mathcal{O}_{X, \\xi}^h)", "$$", "by Algebra, Lemma \\ref{algebra-lemma-pullback-module}.", "This final quantity is equal to $\\text{ord}_{Z, \\mathcal{L}}(s)$ by", "Spaces over Fields, Definition", "\\ref{spaces-over-fields-definition-order-vanishing-meromorphic}, i.e.,", "to the coefficient of $Z$ in", "$\\text{div}_\\mathcal{L}(s)$", "as desired." ], "refs": [ "spaces-divisors-definition-zero-scheme-s", "algebra-lemma-pullback-module", "spaces-over-fields-definition-order-vanishing-meromorphic" ], "ref_ids": [ 13021, 640, 12890 ] } ], "ref_ids": [] }, { "id": 7235, "type": "theorem", "label": "spaces-chow-lemma-Gm-torsor", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-Gm-torsor", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "The morphism", "$$", "q :", "T = \\underline{\\Spec}\\left(", "\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{L}^{\\otimes n}\\right)", "\\longrightarrow", "X", "$$", "has the following properties:", "\\begin{enumerate}", "\\item $q$ is surjective, smooth, affine, of relative dimension $1$,", "\\item there is an isomorphism $\\alpha : q^*\\mathcal{L} \\cong \\mathcal{O}_T$,", "\\item formation of $(q : T \\to X, \\alpha)$ commutes with base change,", "\\item $q^* : Z_k(X) \\to Z_{k + 1}(T)$ is injective,", "\\item if $Z \\subset X$ is an integral closed subspace, then", "$q^{-1}(Z) \\subset T$ is an integral closed subspace,", "\\item if $Z \\subset X$ is a closed subspace of $X$", "of $\\delta$-dimension $\\leq k$, then $q^{-1}(Z)$ is a closed subspace of $T$", "of $\\delta$-dimension $\\leq k + 1$ and", "$q^*[Z]_k = [q^{-1}(Z)]_{k + 1}$,", "\\item if $\\xi' \\in |T|$ is the generic point of the fibre of $|T| \\to |X|$", "over $\\xi$, then the ring map", "$\\mathcal{O}_{X, \\xi}^h \\to \\mathcal{O}_{T, \\xi'}^h$ is flat,", "we have $\\mathfrak m_{\\xi'}^h = \\mathfrak m_\\xi^h \\mathcal{O}_{T, \\xi'}^h$, and", "the residue field extension is purely transcendental of", "transcendence degree $1$, and", "\\item add more here as needed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be an \\'etale morphism such that $\\mathcal{L}|_U$", "is trivial. Then $T \\times_X U \\to U$ is isomorphic to the projection", "morphism $\\mathbf{G}_m \\times U \\to U$, where", "$\\mathbf{G}_m$ is the multipliciative group scheme, see", "Groupoids, Example \\ref{groupoids-example-multiplicative-group}.", "Thus (1) is clear.", "\\medskip\\noindent", "To see (2) observe that", "$q_*q^*\\mathcal{L} =", "\\bigoplus_{n \\in \\mathbf{Z}} \\mathcal{L}^{\\otimes n + 1}$.", "Thus there is an obvious isomorphism", "$q_*q^*\\mathcal{L} \\to q_*\\mathcal{O}_T$", "of $q_*\\mathcal{O}_T$-modules. By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-affine-equivalence-modules}", "this determines an isomorphism $q^*\\mathcal{L} \\to \\mathcal{O}_T$.", "\\medskip\\noindent", "Part (3) holds because forming the relative spectrum commutes", "with arbitrary base change and the same thing is clearly true", "for the isomorphism $\\alpha$.", "\\medskip\\noindent", "Part (4) follows immediately from (1) and the definitions.", "\\medskip\\noindent", "Part (5) follows from the fact that if $Z$ is an integral algebraic", "space, then $\\mathbf{G}_m \\times Z$ is an integral algebraic space.", "\\medskip\\noindent", "Part (6) follows from the fact that lengths are preserved:", "if $(A, \\mathfrak m)$ is a local ring and $B = A[x]_{\\mathfrak m A[x]}$", "and if $M$ is an $A$-module, then", "$\\text{length}_A(M) = \\text{length}_B(M \\otimes_A B)$.", "This implies that if $\\mathcal{F}$ is a coherent", "$\\mathcal{O}_X$-module and $\\xi \\in |X|$ with $\\xi' \\in |T|$", "the generic point of the fibre over $\\xi$, then", "the length of $\\mathcal{F}$ at $\\xi$ is the same as the", "length of $q^*\\mathcal{F}$ at $\\xi'$.", "Tracing through the definitions this gives (6) and more.", "\\medskip\\noindent", "The map in part (7) comes from Decent Spaces, Remark", "\\ref{decent-spaces-remark-functoriality-henselian-local-ring}.", "However, in our case we have", "$$", "\\Spec(\\mathcal{O}_{X, \\xi}^h) \\times_X T =", "\\mathbf{G}_m \\times \\Spec(\\mathcal{O}_{X, \\xi}^h) =", "\\Spec(\\mathcal{O}_{X, \\xi}^h[t, t^{-1}])", "$$", "and $\\xi'$ corresponds to the generic point of the special", "fibre of this over $\\Spec(\\mathcal{O}_{X, \\xi}^h)$. Thus", "$\\mathcal{O}_{T, \\xi'}^h$ is the henselization of the localization", "of $\\mathcal{O}_{X, \\xi}^h[t, t^{-1}]$ at the corresponding prime.", "Part (7) follows from this and some commutative algebra; details omitted." ], "refs": [ "spaces-morphisms-lemma-affine-equivalence-modules", "decent-spaces-remark-functoriality-henselian-local-ring" ], "ref_ids": [ 4803, 9575 ] } ], "ref_ids": [] }, { "id": 7236, "type": "theorem", "label": "spaces-chow-lemma-Gm-torsor-divisor-meromorphic-section", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-Gm-torsor-divisor-meromorphic-section", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Assume $X$ is integral. Let $s$ be a nonzero meromorphic", "section of $\\mathcal{L}$. Let $q : T \\to X$ be the morphism", "of Lemma \\ref{lemma-Gm-torsor}. Then", "$$", "q^*\\text{div}_\\mathcal{L}(s) = \\text{div}_T(q^*(s))", "$$", "where we view the pullback $q^*(s)$", "as a nonzero meromorphic function on $T$", "using the isomorphism $q^*\\mathcal{L} \\to \\mathcal{O}_T$" ], "refs": [ "spaces-chow-lemma-Gm-torsor" ], "proofs": [ { "contents": [ "Observe that $\\text{div}_T(q^*(s)) = \\text{div}_{\\mathcal{O}_T}(q^*(s))$", "by the compatibility between the constructions given in", "Spaces over Fields, Sections", "\\ref{spaces-over-fields-section-Weil-divisors} and", "\\ref{spaces-over-fields-section-c1}.", "We will show the agreement with $\\text{div}_{\\mathcal{O}_T}(q^*(s))$", "in this proof.", "We will use all the properties of $q : T \\to X$ stated in", "Lemma \\ref{lemma-Gm-torsor} without further mention.", "Let $Z \\subset T$ be a prime divisor.", "Then either $Z \\to X$ is dominant or $Z = q^{-1}(Z')$ for some", "prime divisor $Z' \\subset X$.", "If $Z \\to X$ is dominant, then the coefficient of $Z$ in", "either side of the equality of the lemma is zero.", "Thus we may assume $Z = q^{-1}(Z')$ where $Z' \\subset X$", "is a prime divisor.", "Let $\\xi' \\in |Z'|$ and $\\xi \\in |Z|$ be the generic points.", "Then we obtain a commutative diagram", "$$", "\\xymatrix{", "\\Spec(\\mathcal{O}_{T, \\xi}^h) \\ar[r]_-{c_\\xi} \\ar[d]_h & T \\ar[d]^q \\\\", "\\Spec(\\mathcal{O}_{X, \\xi'}^h) \\ar[r]^-{c_{\\xi'}} & X", "}", "$$", "see Decent Spaces, Remark", "\\ref{decent-spaces-remark-functoriality-henselian-local-ring}.", "Choose a trivialization $s_{\\xi'}$ of", "$\\mathcal{L}_{\\xi'} = c_{\\xi'}^*\\mathcal{L}$.", "Then we can use the pullback $s_\\xi$ of $s_{\\xi'}$ via $h$", "as our trivialization of $\\mathcal{L}_\\xi = c_\\xi^* q^*\\mathcal{L}$.", "Write $s/s_{\\xi'} = a/b$ for $a, b \\in \\mathcal{O}_{X, \\xi'}$ nonzerodivisors.", "By definition the coefficient of $Z'$ in", "$\\text{div}_\\mathcal{L}(s)$ is", "$$", "\\text{length}_{\\mathcal{O}_{X, \\xi'}^h}(", "\\mathcal{O}_{X, \\xi'}^h/a \\mathcal{O}_{X, \\xi'}^h)", "-", "\\text{length}_{\\mathcal{O}_{X, \\xi'}^h}(", "\\mathcal{O}_{X, \\xi'}^h/b \\mathcal{O}_{X, \\xi'}^h)", "$$", "Since $Z = q^{-1}(Z')$, this is also the coefficient of $Z$ in", "$q^*\\text{div}_\\mathcal{L}(s)$. Since", "$\\mathcal{O}_{X, \\xi'}^h \\to \\mathcal{O}_{T, \\xi}^h$", "is flat the elements $a, b$ map to nonzerodivisors in", "$\\mathcal{O}_{T, \\xi}^h$. Thus $q^*(s)/s_\\xi = a/b$ in the total quotient", "ring of $\\mathcal{O}_{T, \\xi}^h$. By definition the coefficient of", "$Z$ in $\\text{div}_T(q^*(s))$ is", "$$", "\\text{length}_{\\mathcal{O}_{T, \\xi}^h}(", "\\mathcal{O}_{T, \\xi}^h/a \\mathcal{O}_{T, \\xi}^h)", "-", "\\text{length}_{\\mathcal{O}_{T, \\xi}^h}(", "\\mathcal{O}_{T, \\xi}^h/b \\mathcal{O}_{T, \\xi}^h)", "$$", "The proof is finished because these lengths are the same as before", "by Algebra, Lemma \\ref{algebra-lemma-pullback-module} and the", "fact that $\\mathfrak m_\\xi^h = \\mathfrak m_{\\xi'}^h\\mathcal{O}_{T, \\xi}^h$", "shown in Lemma \\ref{lemma-Gm-torsor}." ], "refs": [ "spaces-chow-lemma-Gm-torsor", "decent-spaces-remark-functoriality-henselian-local-ring", "algebra-lemma-pullback-module", "spaces-chow-lemma-Gm-torsor" ], "ref_ids": [ 7235, 9575, 640, 7235 ] } ], "ref_ids": [ 7235 ] }, { "id": 7237, "type": "theorem", "label": "spaces-chow-lemma-c1-cap-additive", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-c1-cap-additive", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{L}$, $\\mathcal{N}$ be an invertible sheaves on $X$.", "Then", "$$", "c_1(\\mathcal{L}) \\cap \\alpha + c_1(\\mathcal{N}) \\cap \\alpha =", "c_1(\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}) \\cap \\alpha", "$$", "in $\\CH_k(X)$ for every $\\alpha \\in Z_{k - 1}(X)$. Moreover,", "$c_1(\\mathcal{O}_X) \\cap \\alpha = 0$ for all $\\alpha$." ], "refs": [], "proofs": [ { "contents": [ "The additivity follows directly from", "Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-c1-additive}", "and the definitions. To see that $c_1(\\mathcal{O}_X) \\cap \\alpha = 0$", "consider the section $1 \\in \\Gamma(X, \\mathcal{O}_X)$. This restricts", "to an everywhere nonzero section on any integral closed subspace", "$W \\subset X$. Hence $c_1(\\mathcal{O}_X) \\cap [W] = 0$ as desired." ], "refs": [ "spaces-over-fields-lemma-c1-additive" ], "ref_ids": [ 12838 ] } ], "ref_ids": [] }, { "id": 7238, "type": "theorem", "label": "spaces-chow-lemma-prepare-geometric-cap", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-prepare-geometric-cap", "contents": [ "In Situation \\ref{situation-setup} let $Y/B$ be good.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_Y$-module.", "Let $s \\in \\Gamma(Y, \\mathcal{L})$ be a regular section and", "assume $\\dim_\\delta(Y) \\leq k + 1$.", "Write $[Y]_{k + 1} = \\sum n_i[Y_i]$ where $Y_i \\subset Y$ are the", "irreducible components of $Y$ of $\\delta$-dimension $k + 1$.", "Set $s_i = s|_{Y_i} \\in \\Gamma(Y_i, \\mathcal{L}|_{Y_i})$. Then", "\\begin{equation}", "\\label{equation-equal-as-cycles}", "[Z(s)]_k = \\sum n_i[Z(s_i)]_k", "\\end{equation}", "as $k$-cycles on $Y$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi : V \\to Y$ be a surjective \\'etale morphism where $V$", "is a scheme. It suffices to prove the equality after pulling back", "by $\\varphi$, see Lemma \\ref{lemma-etale-pullback}.", "That same lemma tells us that", "$\\varphi^*[Y_i] = [\\varphi^{-1}(Y_i)] = \\sum [V_{i, j}]$", "where $V_{i, j}$ are the irreducible components of $V$ lying over $Y_i$.", "Hence if we first apply the case of schemes", "(Chow Homology, Lemma \\ref{chow-lemma-prepare-geometric-cap}) to", "$\\varphi^*s_i$ on $Y_i \\times_Y V$ we", "find that $\\varphi^*[Z(s_i)]_k = [Z(\\varphi^*s_i)] =", "\\sum [Z(s_{i, j})]_k$ where $s_{i, j}$ is the pullback", "of $s$ to $V_{i, j}$. Applying the case of schemes to $\\varphi^*s$ we get", "$$", "\\varphi^*[Z(s)]_k =", "[Z(\\varphi^*s)]_k =", "\\sum n_i[Z(s_{i, j})]_k", "$$", "by our remark on multiplicities above.", "Combining all of the above the proof is complete." ], "refs": [ "spaces-chow-lemma-etale-pullback", "chow-lemma-prepare-geometric-cap" ], "ref_ids": [ 7222, 5706 ] } ], "ref_ids": [] }, { "id": 7239, "type": "theorem", "label": "spaces-chow-lemma-geometric-cap", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-geometric-cap", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $Y \\subset X$ be a closed subscheme with", "$\\dim_\\delta(Y) \\leq k + 1$ and let $s \\in \\Gamma(Y, \\mathcal{L}|_Y)$", "be a regular section. Then", "$$", "c_1(\\mathcal{L}) \\cap [Y]_{k + 1} = [Z(s)]_k", "$$", "in $\\CH_k(X)$." ], "refs": [], "proofs": [ { "contents": [ "Write", "$$", "[Y]_{k + 1} = \\sum n_i[Y_i]", "$$", "where $Y_i \\subset Y$ are the irreducible components of", "$Y$ of $\\delta$-dimension $k + 1$ and $n_i > 0$.", "By assumption the restriction", "$s_i = s|_{Y_i} \\in \\Gamma(Y_i, \\mathcal{L}|_{Y_i})$ is not", "zero, and hence is a regular section. By Lemma \\ref{lemma-compute-c1}", "we see that $[Z(s_i)]_k$ represents $c_1(\\mathcal{L}|_{Y_i})$.", "Hence by definition", "$$", "c_1(\\mathcal{L}) \\cap [Y]_{k + 1} = \\sum n_i[Z(s_i)]_k", "$$", "Thus the result follows from Lemma \\ref{lemma-prepare-geometric-cap}." ], "refs": [ "spaces-chow-lemma-compute-c1", "spaces-chow-lemma-prepare-geometric-cap" ], "ref_ids": [ 7234, 7238 ] } ], "ref_ids": [] }, { "id": 7240, "type": "theorem", "label": "spaces-chow-lemma-prepare-flat-pullback-cap-c1", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-prepare-flat-pullback-cap-c1", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Let $f : X \\to Y$ be a flat morphism of relative dimension $r$.", "Let $\\mathcal{L}$ be an invertible sheaf on $Y$.", "Assume $Y$ is integral and $n = \\dim_\\delta(Y)$.", "Let $s$ be a nonzero meromorphic section of $\\mathcal{L}$.", "Then we have", "$$", "f^*\\text{div}_\\mathcal{L}(s) = \\sum n_i\\text{div}_{f^*\\mathcal{L}|_{X_i}}(s_i)", "$$", "in $Z_{n + r - 1}(X)$. Here the sum is over the irreducible", "components $X_i \\subset X$ of $\\delta$-dimension $n + r$,", "the section $s_i = f|_{X_i}^*(s)$ is the pullback of $s$, and", "$n_i = m_{X_i, X}$ is the multiplicity of $X_i$ in $X$." ], "refs": [], "proofs": [ { "contents": [ "Using sleight of hand we will deduce this from", "Lemma \\ref{lemma-prepare-flat-pullback-rational-equivalence}.", "(An alternative is to redo the proof of that lemma in the", "setting of meromorphic sections of invertible modules.)", "Namely, let $q : T \\to Y$ be the morphism of", "Lemma \\ref{lemma-Gm-torsor} constructed using $\\mathcal{L}$ on $Y$.", "We will use all the properties of $T$ stated in this lemma.", "Consider the fibre product diagram", "$$", "\\xymatrix{", "T' \\ar[r]_{q'} \\ar[d]_h & X \\ar[d]^f \\\\", "T \\ar[r]^q & Y", "}", "$$", "Then $q' : T' \\to X$ is the morphism constructed using $f^*\\mathcal{L}$", "on $X$. Then it suffices to prove", "$$", "(q')^*f^*\\text{div}_\\mathcal{L}(s) =", "\\sum n_i (q')^*\\text{div}_{f^*\\mathcal{L}|_{X_i}}(s_i)", "$$", "Observe that $T'_i = q^{-1}(X_i)$ are the irreducible components of $T'$", "and that $n_i$ is the multiplicity of $T'_i$ in $T'$.", "The left hand side is equal to", "$$", "h^*q^*\\text{div}_\\mathcal{L}(s) = h^*\\text{div}_T(q^*(s))", "$$", "by Lemma \\ref{lemma-Gm-torsor-divisor-meromorphic-section}", "(and Lemma \\ref{lemma-compose-flat-pullback}).", "On the other hand, denoting $q'_i : T'_i \\to X_i$", "the restriction of $q'$ we find that", "Lemma \\ref{lemma-Gm-torsor-divisor-meromorphic-section}", "also tells us the right hand side is equal to", "$$", "\\sum n_i \\text{div}_{T_i}((q'_i)^*(s_i))", "$$", "In these two formulas the expressions $q^*(s)$ and $(q'_i)^*(s_i)$", "represent the rational functions corresponding to the pulled back", "meromorphic sections of $q^*\\mathcal{L}$ and $(q'_i)^*f^*\\mathcal{L}|_{X_i}$", "via the isomorphism $\\alpha : q^*\\mathcal{L} \\to \\mathcal{O}_T$", "and its pullbacks to spaces over $T$. With this convention it is", "clear that $(q'_i)^*(s_i)$ is the composition of the rational function", "$q^*(s)$ on $T$ and the morphism $h|_{T'_i} : T'_i \\to T$.", "Thus Lemma \\ref{lemma-prepare-flat-pullback-rational-equivalence}", "exactly says that", "$$", "h^*\\text{div}_T(q^*(s)) = \\sum n_i \\text{div}_{T_i}((q'_i)^*(s_i))", "$$", "as desired." ], "refs": [ "spaces-chow-lemma-prepare-flat-pullback-rational-equivalence", "spaces-chow-lemma-Gm-torsor", "spaces-chow-lemma-compose-flat-pullback", "spaces-chow-lemma-prepare-flat-pullback-rational-equivalence" ], "ref_ids": [ 7231, 7235, 7223, 7231 ] } ], "ref_ids": [] }, { "id": 7241, "type": "theorem", "label": "spaces-chow-lemma-flat-pullback-cap-c1", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-flat-pullback-cap-c1", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Let $f : X \\to Y$ be a flat morphism of relative dimension $r$.", "Let $\\mathcal{L}$ be an invertible sheaf on $Y$.", "Let $\\alpha$ be a $k$-cycle on $Y$.", "Then", "$$", "f^*(c_1(\\mathcal{L}) \\cap \\alpha) = c_1(f^*\\mathcal{L}) \\cap f^*\\alpha", "$$", "in $\\CH_{k + r - 1}(X)$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\alpha = \\sum n_i[W_i]$. We will show that", "$$", "f^*(c_1(\\mathcal{L}) \\cap [W_i]) = c_1(f^*\\mathcal{L}) \\cap f^*[W_i]", "$$", "in $\\CH_{k + r - 1}(X)$ by producing a rational equivalence", "on the closed subspace $f^{-1}(W_i)$ of $X$.", "By the discussion in", "Remark \\ref{remark-infinite-sums-rational-equivalences}", "this will prove the equality of the lemma is true.", "\\medskip\\noindent", "Let $W \\subset Y$ be an integral closed subspace of $\\delta$-dimension $k$.", "Consider the closed subspace $W' = f^{-1}(W) = W \\times_Y X$", "so that we have the fibre product diagram", "$$", "\\xymatrix{", "W' \\ar[r] \\ar[d]_h & X \\ar[d]^f \\\\", "W \\ar[r] & Y", "}", "$$", "We have to show that", "$f^*(c_1(\\mathcal{L}) \\cap [W]) = c_1(f^*\\mathcal{L}) \\cap f^*[W]$.", "Choose a nonzero meromorphic section $s$ of $\\mathcal{L}|_W$.", "Let $W'_i \\subset W'$ be the irreducible components of", "$\\delta$-dimension $k + r$. Write $[W']_{k + r} = \\sum n_i[W'_i]$", "with $n_i$ the multiplicity of $W'_i$ in $W'$ as per definition.", "So $f^*[W] = \\sum n_i[W'_i]$ in $Z_{k + r}(X)$.", "Since each $W'_i \\to W$ is dominant we", "see that $s_i = s|_{W'_i}$ is a nonzero meromorphic section for", "each $i$. By Lemma \\ref{lemma-prepare-flat-pullback-cap-c1}", "we have the following equality of cycles", "$$", "h^*\\text{div}_{\\mathcal{L}|_W}(s) =", "\\sum n_i\\text{div}_{f^*\\mathcal{L}|_{W'_i}}(s_i)", "$$", "in $Z_{k + r - 1}(W')$. This finishes the proof since", "the left hand side is a cycle on $W'$ which pushes to", "$f^*(c_1(\\mathcal{L}) \\cap [W])$ in $\\CH_{k + r - 1}(X)$", "and the right hand side is a cycle on $W'$ which pushes to", "$c_1(f^*\\mathcal{L}) \\cap f^*[W]$ in $\\CH_{k + r - 1}(X)$." ], "refs": [ "spaces-chow-remark-infinite-sums-rational-equivalences", "spaces-chow-lemma-prepare-flat-pullback-cap-c1" ], "ref_ids": [ 7302, 7240 ] } ], "ref_ids": [] }, { "id": 7242, "type": "theorem", "label": "spaces-chow-lemma-equal-c1-as-cycles", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-equal-c1-as-cycles", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Let $f : X \\to Y$ be a proper morphism.", "Let $\\mathcal{L}$ be an invertible sheaf on $Y$.", "Assume $X$, $Y$ integral, $f$ dominant, and $\\dim_\\delta(X) = \\dim_\\delta(Y)$.", "Let $s$ be a nonzero meromorphic section $s$ of $\\mathcal{L}$ on $Y$.", "Then", "$$", "f_*\\left(\\text{div}_{f^*\\mathcal{L}}(f^*s)\\right) =", "[R(X) : R(Y)]\\text{div}_\\mathcal{L}(s).", "$$", "as cycles on $Y$. In particular", "$$", "f_*(c_1(f^*\\mathcal{L}) \\cap [X]) = c_1(\\mathcal{L}) \\cap f_*[Y].", "$$" ], "refs": [], "proofs": [ { "contents": [ "The last equation follows from the first since $f_*[X] = [R(X) : R(Y)][Y]$ by", "definition. Proof of the first equaltion. Let $q : T \\to Y$ be the morphism of", "Lemma \\ref{lemma-Gm-torsor} constructed using $\\mathcal{L}$ on $Y$. We will use", "all the properties of $T$ stated in this lemma. Consider the fibre product", "diagram", "$$", "\\xymatrix{", "T' \\ar[r]_{q'} \\ar[d]_h & X \\ar[d]^f \\\\", "T \\ar[r]^q & Y", "}", "$$", "Then $q' : T' \\to X$ is the morphism constructed using $f^*\\mathcal{L}$", "on $X$. It suffices to prove the equality after pulling back to $T'$.", "The left hand side pulls back to", "\\begin{align*}", "q^*f_*\\left(\\text{div}_{f^*\\mathcal{L}}(f^*s)\\right)", "& =", "h_*(q')^*\\text{div}_{f^*\\mathcal{L}}(f^*s) \\\\", "& =", "h_*\\text{div}_{(q')^*f^*\\mathcal{L}}((q')^*f^*s) \\\\", "& =", "h_*\\text{div}_{h^*q^*\\mathcal{L}}(h^*q^*s)", "\\end{align*}", "The first equality by Lemma \\ref{lemma-flat-pullback-proper-pushforward}.", "The second by Lemma \\ref{lemma-prepare-flat-pullback-cap-c1}", "using that $T'$ is integral. The third because the displayed diagram commutes.", "The right hand side pulls back to", "$$", "[R(X) : R(Y)]q^*\\text{div}_\\mathcal{L}(s) =", "[R(T') : R(T)]\\text{div}_{q^*\\mathcal{L}}(q^*s)", "$$", "This follows from Lemma \\ref{lemma-prepare-flat-pullback-cap-c1},", "the fact that $T$ is integral, and the equality", "$[R(T') : R(T)] = [R(X) : R(Y)]$ whose proof we omit", "(it follows from Lemma \\ref{lemma-flat-pullback-proper-pushforward}", "but that would be a silly way to prove the equality).", "Thus it suffices to prove the lemma for $h : T' \\to T$, the", "invertible module $q^\\mathcal{L}$ and the section $q^*s$.", "Since $q^*\\mathcal{L} = \\mathcal{O}_T$", "we reduce to the case where $\\mathcal{L} \\cong \\mathcal{O}$", "discussed in the next paragraph.", "\\medskip\\noindent", "Assume that $\\mathcal{L} = \\mathcal{O}_Y$. In this case $s$", "corresponds to a rational function $g \\in R(Y)$. Using the", "embedding $R(Y) \\subset R(X)$ we may think of $g$ as a rational", "on $X$ and we are simply trying to prove", "$$", "f_*\\left(\\text{div}_X(g)\\right) = [R(X) : R(Y)]\\text{div}_Y(g).", "$$", "Comparing with the result of Lemma \\ref{lemma-proper-pushforward-alteration}", "we see this true since $\\text{Nm}_{R(X)/R(Y)}(g) = g^{[R(X) : R(Y)]}$", "as $g \\in R(Y)^*$." ], "refs": [ "spaces-chow-lemma-Gm-torsor", "spaces-chow-lemma-flat-pullback-proper-pushforward", "spaces-chow-lemma-prepare-flat-pullback-cap-c1", "spaces-chow-lemma-prepare-flat-pullback-cap-c1", "spaces-chow-lemma-flat-pullback-proper-pushforward", "spaces-chow-lemma-proper-pushforward-alteration" ], "ref_ids": [ 7235, 7225, 7240, 7240, 7225, 7229 ] } ], "ref_ids": [] }, { "id": 7243, "type": "theorem", "label": "spaces-chow-lemma-pushforward-cap-c1", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-pushforward-cap-c1", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Let $p : X \\to Y$ be a proper morphism.", "Let $\\alpha \\in Z_{k + 1}(X)$.", "Let $\\mathcal{L}$ be an invertible sheaf on $Y$.", "Then", "$$", "p_*(c_1(p^*\\mathcal{L}) \\cap \\alpha) = c_1(\\mathcal{L}) \\cap p_*\\alpha", "$$", "in $\\CH_k(Y)$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $p$ has the property that for every integral", "closed subspace $W \\subset X$ the map $p|_W : W \\to Y$", "is a closed immersion. Then, by definition of capping", "with $c_1(\\mathcal{L})$ the lemma holds.", "\\medskip\\noindent", "We will use this remark to reduce to a special case. Namely,", "write $\\alpha = \\sum n_i[W_i]$ with $n_i \\not = 0$ and $W_i$ pairwise", "distinct. Let $W'_i \\subset Y$ be the ``image'' of $W_i$ as", "in Lemma \\ref{lemma-proper-image}. Consider the diagram", "$$", "\\xymatrix{", "X' = \\coprod W_i \\ar[r]_-q \\ar[d]_{p'} & X \\ar[d]^p \\\\", "Y' = \\coprod W'_i \\ar[r]^-{q'} & Y.", "}", "$$", "Since $\\{W_i\\}$ is locally finite on $X$, and $p$ is proper", "we see that $\\{W'_i\\}$ is locally finite on $Y$ and that", "$q, q', p'$ are also proper morphisms.", "We may think of $\\sum n_i[W_i]$ also as a $k$-cycle", "$\\alpha' \\in Z_k(X')$. Clearly $q_*\\alpha' = \\alpha$.", "We have", "$q_*(c_1(q^*p^*\\mathcal{L}) \\cap \\alpha')", "= c_1(p^*\\mathcal{L}) \\cap q_*\\alpha'$", "and", "$(q')_*(c_1((q')^*\\mathcal{L}) \\cap p'_*\\alpha') =", "c_1(\\mathcal{L}) \\cap q'_*p'_*\\alpha'$ by the initial", "remark of the proof. Hence it suffices to prove the lemma", "for the morphism $p'$ and the cycle $\\sum n_i[W_i]$.", "Clearly, this means we may assume $X$, $Y$ integral,", "$f : X \\to Y$ dominant and $\\alpha = [X]$.", "In this case the result follows from", "Lemma \\ref{lemma-equal-c1-as-cycles}." ], "refs": [ "spaces-chow-lemma-proper-image", "spaces-chow-lemma-equal-c1-as-cycles" ], "ref_ids": [ 7214, 7242 ] } ], "ref_ids": [] }, { "id": 7244, "type": "theorem", "label": "spaces-chow-lemma-key-formula", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-key-formula", "contents": [ "In the situation above the cycle", "$$", "\\sum", "(Z_i \\to X)_*\\left(", "\\text{ord}_{B_i}(f_i) \\text{div}_{\\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) -", "\\text{ord}_{B_i}(g_i) \\text{div}_{\\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) \\right)", "$$", "is equal to the cycle", "$$", "\\sum (Z_i \\to X)_*\\text{div}(\\partial_{B_i}(f_i, g_i))", "$$" ], "refs": [], "proofs": [ { "contents": [ "The strategy of the proof will be: first reduce to the case where", "$\\mathcal{L}$ and $\\mathcal{N}$ are trivial invertible modules,", "then change our choices of local trivializations, and then finally", "use \\'etale localization to reduce to the case of schemes\\footnote{It", "is possible that a shorter proof can be given by immediately", "applying \\'etale localization.}.", "\\medskip\\noindent", "First step. Let $q : T \\to X$ be the morphism constructed in", "Lemma \\ref{lemma-Gm-torsor}. We will use all properties stated in", "that lemma without further mention. In particular, it suffices to", "show that the cycles are equal after pulling back by $q$.", "Denote $s'$ and $t'$ the pullbacks of $s$ and $t$ to meromorphic sections", "of $q^*\\mathcal{L}$ and $q^*\\mathcal{N}$.", "Denote $Z'_i = q^{-1}(Z_i)$, denote $\\xi'_i \\in |Z'_i|$ the generic point,", "denote $B'_i = \\mathcal{O}_{T, \\xi'_i}^h$, denote", "$\\mathcal{L}_{\\xi'_i}$ and $\\mathcal{N}_{\\xi'_i}$ the pullbacks", "of $\\mathcal{L}$ and $\\mathcal{N}$ to $\\Spec(B'_i)$.", "Recall that we have commutative diagrams", "$$", "\\xymatrix{", "\\Spec(B'_i) \\ar[r]_-{c_{\\xi'_i}} \\ar[d] & T \\ar[d]^q \\\\", "\\Spec(B_i) \\ar[r]^-{c_{\\xi_i}} & X", "}", "$$", "see Decent Spaces, Remark", "\\ref{decent-spaces-remark-functoriality-henselian-local-ring}.", "Denote $s'_i$ and $t'_i$ the pullbacks of $s_i$ and $t_i$ which", "are generators of $\\mathcal{L}_{\\xi'_i}$ and $\\mathcal{N}_{\\xi'_i}$.", "Then we have", "$$", "s' = f'_i s'_i", "\\quad\\text{and}\\quad", "t' = g'_i t'_i", "$$", "where $f'_i$ and $g'_i$ are the images of $f_i, g_i$", "under the map $Q(B_i) \\to Q(B'_i)$ induced by $B_i \\to B'_i$.", "By Algebra, Lemma \\ref{algebra-lemma-pullback-module} we have", "$$", "\\text{ord}_{B_i}(f_i) = \\text{ord}_{B'_i}(f'_i)", "\\quad\\text{and}\\quad", "\\text{ord}_{B_i}(g_i) = \\text{ord}_{B'_i}(g'_i)", "$$", "By Lemma \\ref{lemma-prepare-flat-pullback-cap-c1} applied", "to $q : Z'_i \\to Z_i$ we have", "$$", "q^*\\text{div}_{\\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) =", "\\text{div}_{q^*\\mathcal{N}|_{Z'_i}}(t'_i|_{Z'_i})", "\\quad\\text{and}\\quad", "q^*\\text{div}_{\\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) =", "\\text{div}_{q^*\\mathcal{L}|_{Z'_i}}(s'_i|_{Z'_i})", "$$", "This already shows that the first cycle in the statement of the", "lemma pulls back to the corresponding cycle for", "$s', t', Z'_i, s'_i, t'_i$. To see the same is true for the", "second, note that by", "Chow Homology, Lemma \\ref{chow-lemma-tame-symbol-formally-smooth} we have", "$$", "\\partial_{B_i}(f_i, g_i) \\mapsto", "\\partial_{B'_i}(f'_i, g'_i)", "\\quad\\text{via}\\quad", "\\kappa(\\xi_i) \\to \\kappa(\\xi'_i)", "$$", "Hence the same lemma as before shows that", "$$", "q^*\\text{div}(\\partial_{B_i}(f_i, g_i)) =", "\\text{div}(\\partial_{B'_i}(f'_i, g'_i))", "$$", "Since $q^*\\mathcal{L} \\cong \\mathcal{O}_T$ we find that it suffices", "to prove the equality in case $\\mathcal{L}$ is trivial.", "Exchanging the roles of $\\mathcal{L}$ and $\\mathcal{N}$ we", "see that we may similarly assume $\\mathcal{N}$ is trivial.", "This finishes the proof of the first step.", "\\medskip\\noindent", "Second step. Assume $\\mathcal{L} = \\mathcal{O}_X$ and", "$\\mathcal{N} = \\mathcal{O}_X$. Denote $1$ the trivializing", "section of $\\mathcal{L}$. Then $s_i = u \\cdot 1$", "for some unit $u \\in B_i$. Let us examine what happens", "if we replace $s_i$ by $1$. Then $f_i$ gets replaced by $u f_i$.", "Thus the first part of the first expression of the lemma is unchanged", "and in the second part we add", "$$", "\\text{ord}_{B_i}(g_i)\\text{div}(u|_{Z_i})", "$$", "where $u|_{Z_i}$ is the image of $u$ in the residue field by", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-divisor-meromorphic-well-defined}", "and in the second expression we add", "$$", "\\text{div}(\\partial_{B_i}(u, g_i))", "$$", "by bi-linearity of the tame symbol. These terms agree by the property", "of the tame symbol given in", "Chow Homology, Equation (\\ref{chow-item-normalization}).", "\\medskip\\noindent", "Let $Y \\subset X$ be an integral closed subspace with $\\dim_\\delta(Y) = n - 2$.", "To show that the coefficients of $Y$ of the two cycles of the lemma", "is the same, we may do a replacement of $s_i$ by $1$ as in the previous", "paragraph. In exactly the same way one shows that we may do a replacement", "of $t_i$ by $1$. Since there are only a finite number of $Z_i$", "such that $Y \\subset Z_i$ we may assume $s_i = 1$ and $t_i = 1$", "for all these $Z_i$.", "\\medskip\\noindent", "Third step. Here we prove the coefficients of $Y$ in the cycles", "of the lemma agree for an integral closed subspace $Y$ with", "$\\dim_\\delta(Y) = n - 2$ such that moreover", "$\\mathcal{L} = \\mathcal{O}_X$ and $\\mathcal{N} = \\mathcal{O}_X$", "and $s_i = 1$ and $t_i = 1$ for all $Z_i$ such that $Y \\subset Z_i$.", "After replacing $X$ by a smaller open subspace we may", "in fact assume that $s_i$ and $t_i$ are equal to $1$ for all $i$.", "In this case the first cycle is zero. Our task is to show that", "the coefficient of $Y$ in the second cycle is zero as well.", "\\medskip\\noindent", "First, since $\\mathcal{L} = \\mathcal{O}_X$ and $\\mathcal{N} = \\mathcal{O}_X$", "we may and do think of $s, t$ as rational functions $f, g$ on $X$.", "Since $s_i$ and $t_i$ are equal to $1$ we find that $f_i$, resp.\\ $g_i$", "is the image of $f$, resp.\\ $g$ in $Q(B_i)$ for all $i$.", "Let $\\zeta \\in |Y|$ be the generic point. Choose an", "\\'etale neighbourhood", "$$", "(U, u) \\longrightarrow (X, \\zeta)", "$$", "and denote $Y' = \\overline{\\{u\\}} \\subset U$.", "Since an \\'etale morphism is flat, we can pullback $f$ and $g$", "to regular meromorphic functions on $U$ which we will also denote", "$f$ and $g$.", "For every prime divisor $Y \\subset Z \\subset X$ the scheme $Z \\times_X U$", "is a union of prime divisors of $U$. Conversely, given a prime divisor", "$Y' \\subset Z' \\subset U$, there is a prime divisor", "$Y \\subset Z \\subset X$ such that $Z'$ is a component of $Z \\times_X U$.", "Given such a pair $(Z, Z')$ the ring map", "$$", "\\mathcal{O}_{X, \\xi}^h \\to \\mathcal{O}_{U, \\xi'}^h", "$$", "is \\'etale (in fact it is finite \\'etale). Hence we find that", "$$", "\\partial_{\\mathcal{O}_{X, \\xi}^h}(f, g) \\mapsto", "\\partial_{\\mathcal{O}_{U, \\xi'}^h}(f, g)", "\\quad\\text{via}\\quad", "\\kappa(\\xi) \\to \\kappa(\\xi')", "$$", "by Chow Homology, Lemma \\ref{chow-lemma-tame-symbol-formally-smooth}.", "Thus Lemma \\ref{lemma-etale-pullback-principal-divisor} applies to show", "$$", "(Z \\times_X U \\to Z)^*\\text{div}_Z(\\partial_{\\mathcal{O}_{X, \\xi}^h}(f, g))", "=", "\\sum\\nolimits_{Z' \\subset Z \\times_X U}", "\\text{div}_{Z'}(\\partial_{\\mathcal{O}_{U, \\xi'}^h}(f, g))", "$$", "Since flat pullback commutes with pushforward along closed", "immersions (Lemma \\ref{lemma-flat-pullback-proper-pushforward})", "we see that it suffices to prove that the coefficient", "of $Y'$ in", "$$", "\\sum\\nolimits_{Z' \\subset U} ", "(Z' \\to U)_*\\text{div}_{Z'}(\\partial_{\\mathcal{O}_{U, \\xi'}^h}(f, g))", "$$", "is zero.", "\\medskip\\noindent", "Let $A = \\mathcal{O}_{U, u}$. Then $f, g \\in Q(A)^*$.", "Thus we can write $f = a/b$ and $g = c/d$ with", "$a, b, c, d \\in A$ nonzerodivisors.", "The coefficient of $Y'$ in the expression above is", "$$", "\\sum\\nolimits_{\\mathfrak q \\subset A\\text{ height }1}", "\\text{ord}_{A/\\mathfrak q}(\\partial_{A_\\mathfrak q}(f, g))", "$$", "By bilinearity of $\\partial_A$ it suffices to prove", "$$", "\\sum\\nolimits_{\\mathfrak q \\subset A\\text{ height }1}", "\\text{ord}_{A/\\mathfrak q}(\\partial_{A_\\mathfrak q}(a, c))", "$$", "is zero and similarly for the other pairs $(a, d)$, $(b, c)$, and", "$(b, d)$. This is true by", "Chow Homology, Lemma \\ref{chow-lemma-key-nonzerodivisors}." ], "refs": [ "spaces-chow-lemma-Gm-torsor", "decent-spaces-remark-functoriality-henselian-local-ring", "algebra-lemma-pullback-module", "spaces-chow-lemma-prepare-flat-pullback-cap-c1", "chow-lemma-tame-symbol-formally-smooth", "spaces-over-fields-lemma-divisor-meromorphic-well-defined", "chow-lemma-tame-symbol-formally-smooth", "spaces-chow-lemma-etale-pullback-principal-divisor", "spaces-chow-lemma-flat-pullback-proper-pushforward", "chow-lemma-key-nonzerodivisors" ], "ref_ids": [ 7235, 9575, 640, 7240, 5663, 12837, 5663, 7228, 7225, 5665 ] } ], "ref_ids": [] }, { "id": 7245, "type": "theorem", "label": "spaces-chow-lemma-commutativity-on-integral", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-commutativity-on-integral", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Assume $X$ integral and $\\dim_\\delta(X) = n$.", "Let $\\mathcal{L}$, $\\mathcal{N}$ be invertible on $X$.", "Choose a nonzero meromorphic section $s$ of $\\mathcal{L}$", "and a nonzero meromorphic section $t$ of $\\mathcal{N}$.", "Set $\\alpha = \\text{div}_\\mathcal{L}(s)$ and", "$\\beta = \\text{div}_\\mathcal{N}(t)$.", "Then", "$$", "c_1(\\mathcal{N}) \\cap \\alpha", "=", "c_1(\\mathcal{L}) \\cap \\beta", "$$", "in $\\CH_{n - 2}(X)$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the key Lemma \\ref{lemma-key-formula}", "and the discussion preceding it." ], "refs": [ "spaces-chow-lemma-key-formula" ], "ref_ids": [ 7244 ] } ], "ref_ids": [] }, { "id": 7246, "type": "theorem", "label": "spaces-chow-lemma-factors", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-factors", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{L}$ be invertible on $X$.", "The operation $\\alpha \\mapsto c_1(\\mathcal{L}) \\cap \\alpha$", "factors through rational equivalence to give an operation", "$$", "c_1(\\mathcal{L}) \\cap - : \\CH_{k + 1}(X) \\to \\CH_k(X)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha \\in Z_{k + 1}(X)$, and $\\alpha \\sim_{rat} 0$.", "We have to show that $c_1(\\mathcal{L}) \\cap \\alpha$", "as defined in Definition \\ref{definition-cap-c1} is zero.", "By Definition \\ref{definition-rational-equivalence} there", "exists a locally finite family $\\{W_j\\}$ of integral closed", "subspaces with $\\dim_\\delta(W_j) = k + 2$ and rational functions", "$f_j \\in R(W_j)^*$ such that", "$$", "\\alpha = \\sum (i_j)_*\\text{div}_{W_j}(f_j)", "$$", "Note that $p : \\coprod W_j \\to X$ is a proper morphism,", "and hence $\\alpha = p_*\\alpha'$ where $\\alpha' \\in Z_{k + 1}(\\coprod W_j)$", "is the sum of the principal divisors $\\text{div}_{W_j}(f_j)$.", "By Lemma \\ref{lemma-pushforward-cap-c1} we have", "$c_1(\\mathcal{L}) \\cap \\alpha = p_*(c_1(p^*\\mathcal{L}) \\cap \\alpha')$.", "Hence it suffices to show that each", "$c_1(\\mathcal{L}|_{W_j}) \\cap \\text{div}_{W_j}(f_j)$ is zero.", "In other words we may assume that $X$ is integral and", "$\\alpha = \\text{div}_X(f)$ for some $f \\in R(X)^*$.", "\\medskip\\noindent", "Assume $X$ is integral and $\\alpha = \\text{div}_X(f)$ for some $f \\in R(X)^*$.", "We can think of $f$ as a regular meromorphic section of the invertible", "sheaf $\\mathcal{N} = \\mathcal{O}_X$. Choose a meromorphic section", "$s$ of $\\mathcal{L}$ and denote $\\beta = \\text{div}_\\mathcal{L}(s)$.", "By Lemma \\ref{lemma-commutativity-on-integral}", "we conclude that", "$$", "c_1(\\mathcal{L}) \\cap \\alpha = c_1(\\mathcal{O}_X) \\cap \\beta.", "$$", "However, by Lemma \\ref{lemma-c1-cap-additive} we see that the right hand side", "is zero in $\\CH_k(X)$ as desired." ], "refs": [ "spaces-chow-definition-cap-c1", "spaces-chow-definition-rational-equivalence", "spaces-chow-lemma-pushforward-cap-c1", "spaces-chow-lemma-commutativity-on-integral", "spaces-chow-lemma-c1-cap-additive" ], "ref_ids": [ 7291, 7289, 7243, 7245, 7237 ] } ], "ref_ids": [] }, { "id": 7247, "type": "theorem", "label": "spaces-chow-lemma-cap-commutative", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-cap-commutative", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{L}$, $\\mathcal{N}$ be invertible on $X$.", "For any $\\alpha \\in \\CH_{k + 2}(X)$ we have", "$$", "c_1(\\mathcal{L}) \\cap c_1(\\mathcal{N}) \\cap \\alpha", "=", "c_1(\\mathcal{N}) \\cap c_1(\\mathcal{L}) \\cap \\alpha", "$$", "as elements of $\\CH_k(X)$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\alpha = \\sum m_j[Z_j]$ for some locally finite", "collection of integral closed subspaces $Z_j \\subset X$", "with $\\dim_\\delta(Z_j) = k + 2$.", "Consider the proper morphism $p : \\coprod Z_j \\to X$.", "Set $\\alpha' = \\sum m_j[Z_j]$ as a $(k + 2)$-cycle on", "$\\coprod Z_j$. By several applications of", "Lemma \\ref{lemma-pushforward-cap-c1} we see that", "$c_1(\\mathcal{L}) \\cap c_1(\\mathcal{N}) \\cap \\alpha", "= p_*(c_1(p^*\\mathcal{L}) \\cap c_1(p^*\\mathcal{N}) \\cap \\alpha')$", "and", "$c_1(\\mathcal{N}) \\cap c_1(\\mathcal{L}) \\cap \\alpha", "= p_*(c_1(p^*\\mathcal{N}) \\cap c_1(p^*\\mathcal{L}) \\cap \\alpha')$.", "Hence it suffices to prove the formula in case $X$ is integral", "and $\\alpha = [X]$. In this case the result follows", "from Lemma \\ref{lemma-commutativity-on-integral} and the definitions." ], "refs": [ "spaces-chow-lemma-pushforward-cap-c1", "spaces-chow-lemma-commutativity-on-integral" ], "ref_ids": [ 7243, 7245 ] } ], "ref_ids": [] }, { "id": 7248, "type": "theorem", "label": "spaces-chow-lemma-support-cap-effective-Cartier", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-support-cap-effective-Cartier", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good. Let", "$(\\mathcal{L}, s, i : D \\to X)$ be as in", "Definition \\ref{definition-gysin-homomorphism}. Let $\\alpha$ be a", "$(k + 1)$-cycle on $X$. Then $i_*i^*\\alpha = c_1(\\mathcal{L}) \\cap \\alpha$", "in $\\CH_k(X)$. In particular, if $D$ is an effective Cartier divisor, then", "$D \\cdot \\alpha = c_1(\\mathcal{O}_X(D)) \\cap \\alpha$." ], "refs": [ "spaces-chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "Write $\\alpha = \\sum n_j[W_j]$ where $i_j : W_j \\to X$ are integral closed", "subspaces with $\\dim_\\delta(W_j) = k$.", "Since $D$ is the vanishing locus of $s$ we see that", "$D \\cap W_j$ is the vanishing locus", "of the restriction $s|_{W_j}$. Hence for each $j$ such that", "$W_j \\not \\subset D$ we have", "$c_1(\\mathcal{L}) \\cap [W_j] = [D \\cap W_j]_k$", "by Lemma \\ref{lemma-geometric-cap}. So we have", "$$", "c_1(\\mathcal{L}) \\cap \\alpha", "=", "\\sum\\nolimits_{W_j \\not \\subset D} n_j[D \\cap W_j]_k", "+", "\\sum\\nolimits_{W_j \\subset D}", "n_j i_{j, *}(c_1(\\mathcal{L})|_{W_j}) \\cap [W_j])", "$$", "in $\\CH_k(X)$ by Definition \\ref{definition-cap-c1}.", "The right hand side matches (termwise) the pushforward of the class", "$i^*\\alpha$ on $D$ from Definition \\ref{definition-gysin-homomorphism}.", "Hence we win." ], "refs": [ "spaces-chow-lemma-geometric-cap", "spaces-chow-definition-cap-c1", "spaces-chow-definition-gysin-homomorphism" ], "ref_ids": [ 7239, 7291, 7292 ] } ], "ref_ids": [ 7292 ] }, { "id": 7249, "type": "theorem", "label": "spaces-chow-lemma-closed-in-X-gysin", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-closed-in-X-gysin", "contents": [ "In Situation \\ref{situation-setup}. Let $f : X' \\to X$ be a proper morphism", "of good algebraic spaces over $B$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in", "Definition \\ref{definition-gysin-homomorphism}.", "Form the diagram", "$$", "\\xymatrix{", "D' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\", "D \\ar[r]^i & X", "}", "$$", "as in Remark \\ref{remark-pullback-pairs}.", "For any $(k + 1)$-cycle $\\alpha'$ on $X'$ we have", "$i^*f_*\\alpha' = g_*(i')^*\\alpha'$ in $\\CH_k(D)$", "(this makes sense as $f_*$ is defined on the level of cycles)." ], "refs": [ "spaces-chow-definition-gysin-homomorphism", "spaces-chow-remark-pullback-pairs" ], "proofs": [ { "contents": [ "Suppose $\\alpha = [W']$ for some integral closed subspace", "$W' \\subset X'$. Let $W \\subset X$ be the ``image'' of $W'$", "as in Lemma \\ref{lemma-proper-image}. In case $W' \\not \\subset D'$,", "then $W \\not \\subset D$ and we see that", "$$", "[W' \\cap D']_k = \\text{div}_{\\mathcal{L}'|_{W'}}({s'|_{W'}})", "\\quad\\text{and}\\quad", "[W \\cap D]_k = \\text{div}_{\\mathcal{L}|_W}(s|_W)", "$$", "and hence $f_*$ of the first cycle equals the second cycle by", "Lemma \\ref{lemma-equal-c1-as-cycles}. Hence the", "equality holds as cycles. In case $W' \\subset D'$, then", "$W \\subset D$ and $f_*(c_1(\\mathcal{L}|_{W'}) \\cap [W'])$", "is equal to $c_1(\\mathcal{L}|_W) \\cap [W]$ in $\\CH_k(W)$ by the second", "assertion of Lemma \\ref{lemma-equal-c1-as-cycles}.", "By Remark \\ref{remark-infinite-sums-rational-equivalences}", "the result follows for general $\\alpha'$." ], "refs": [ "spaces-chow-lemma-proper-image", "spaces-chow-lemma-equal-c1-as-cycles", "spaces-chow-lemma-equal-c1-as-cycles", "spaces-chow-remark-infinite-sums-rational-equivalences" ], "ref_ids": [ 7214, 7242, 7242, 7302 ] } ], "ref_ids": [ 7292, 7304 ] }, { "id": 7250, "type": "theorem", "label": "spaces-chow-lemma-gysin-flat-pullback", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-gysin-flat-pullback", "contents": [ "In Situation \\ref{situation-setup}. Let $f : X' \\to X$", "be a flat morphism of relative dimension $r$ of", "good algebraic spaces over $B$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in", "Definition \\ref{definition-gysin-homomorphism}. Form the diagram", "$$", "\\xymatrix{", "D' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\", "D \\ar[r]^i & X", "}", "$$", "as in Remark \\ref{remark-pullback-pairs}.", "For any $(k + 1)$-cycle $\\alpha$ on $X$ we have", "$(i')^*f^*\\alpha = g^*i^*\\alpha'$ in $\\CH_{k + r}(D)$", "(this makes sense as $f^*$ is defined on the level of cycles)." ], "refs": [ "spaces-chow-definition-gysin-homomorphism", "spaces-chow-remark-pullback-pairs" ], "proofs": [ { "contents": [ "Suppose $\\alpha = [W]$ for some integral closed subspace", "$W \\subset X$. Let $W' = f^{-1}(W) \\subset X'$. In case $W \\not \\subset D$,", "then $W' \\not \\subset D'$ and we see that", "$$", "W' \\cap D' = g^{-1}(W \\cap D)", "$$", "as closed subspaces of $D'$. Hence the", "equality holds as cycles, see Lemma \\ref{lemma-pullback-coherent}.", "In case $W \\subset D$, then $W' \\subset D'$ and $W' = g^{-1}(W)$", "with $[W']_{k + 1 + r} = g^*[W]$ and equality holds in", "$\\CH_{k + r}(D')$ by Lemma \\ref{lemma-flat-pullback-cap-c1}.", "By Remark \\ref{remark-infinite-sums-rational-equivalences}", "the result follows for general $\\alpha'$." ], "refs": [ "spaces-chow-lemma-pullback-coherent", "spaces-chow-lemma-flat-pullback-cap-c1", "spaces-chow-remark-infinite-sums-rational-equivalences" ], "ref_ids": [ 7224, 7241, 7302 ] } ], "ref_ids": [ 7292, 7304 ] }, { "id": 7251, "type": "theorem", "label": "spaces-chow-lemma-easy-gysin", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-easy-gysin", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $(\\mathcal{L}, s, i : D \\to X)$ be as in", "Definition \\ref{definition-gysin-homomorphism}.", "Let $Z \\subset X$ be a closed subscheme such", "that $\\dim_\\delta(Z) \\leq k + 1$ and such that", "$D \\cap Z$ is an effective Cartier divisor on $Z$. Then", "$i^*([Z]_{k + 1}) = [D \\cap Z]_k$." ], "refs": [ "spaces-chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "The assumption means that $s|_Z$ is a regular section of $\\mathcal{L}|_Z$.", "Thus $D \\cap Z = Z(s)$ and we get", "$$", "[D \\cap Z]_k = \\sum n_i [Z(s_i)]_k", "$$", "as cycles where $s_i = s|_{Z_i}$, the $Z_i$ are the irreducible components", "of $\\delta$-dimension $k + 1$, and $[Z]_{k + 1} = \\sum n_i[Z_i]$.", "See Lemma \\ref{lemma-prepare-geometric-cap}.", "We have $D \\cap Z_i = Z(s_i)$. Comparing with the definition", "of the gysin map we conclude." ], "refs": [ "spaces-chow-lemma-prepare-geometric-cap" ], "ref_ids": [ 7238 ] } ], "ref_ids": [ 7292 ] }, { "id": 7252, "type": "theorem", "label": "spaces-chow-lemma-gysin-factors-general", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-gysin-factors-general", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Assume $X$ integral and $n = \\dim_\\delta(X)$.", "Let $i : D \\to X$ be an effective Cartier divisor.", "Let $\\mathcal{N}$ be an invertible $\\mathcal{O}_X$-module", "and let $t$ be a nonzero meromorphic section of $\\mathcal{N}$.", "Then $i^*\\text{div}_\\mathcal{N}(t) = c_1(\\mathcal{N}) \\cap [D]_{n - 1}$", "in $\\CH_{n - 2}(D)$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\text{div}_\\mathcal{N}(t) = \\sum \\text{ord}_{Z_i, \\mathcal{N}}(t)[Z_i]$", "for some integral closed subspaces $Z_i \\subset X$ of $\\delta$-dimension", "$n - 1$. We may assume that the family $\\{Z_i\\}$ is locally", "finite, that $t \\in \\Gamma(U, \\mathcal{N}|_U)$ is a generator", "where $U = X \\setminus \\bigcup Z_i$, and that every irreducible component", "of $D$ is one of the $Z_i$, see", "Spaces over Fields, Lemmas", "\\ref{spaces-over-fields-lemma-components-locally-finite},", "\\ref{spaces-over-fields-lemma-divisor-locally-finite}, and", "\\ref{spaces-over-fields-lemma-divisor-meromorphic-locally-finite}.", "\\medskip\\noindent", "Set $\\mathcal{L} = \\mathcal{O}_X(D)$. Denote", "$s \\in \\Gamma(X, \\mathcal{O}_X(D)) = \\Gamma(X, \\mathcal{L})$", "the canonical section. We will apply the discussion of", "Section \\ref{section-key} to our current situation.", "For each $i$ let $\\xi_i \\in |Z_i|$ be its generic point. Let", "$B_i = \\mathcal{O}_{X, \\xi_i}^h$. For each $i$ we pick generators", "$s_i$ of $\\mathcal{L}_{\\xi_i}$ and $t_i$ of $\\mathcal{N}_{\\xi_i}$", "over $B_i$ but we insist that we pick $s_i = s$ if $Z_i \\not \\subset D$.", "Write $s = f_i s_i$ and $t = g_i t_i$ with $f_i, g_i \\in B_i$.", "Then $\\text{ord}_{Z_i, \\mathcal{N}}(t) = \\text{ord}_{B_i}(g_i)$.", "On the other hand, we have $f_i \\in B_i$ and", "$$", "[D]_{n - 1} = \\sum \\text{ord}_{B_i}(f_i)[Z_i]", "$$", "because of our choices of $s_i$. We claim that", "$$", "i^*\\text{div}_\\mathcal{N}(t) =", "\\sum \\text{ord}_{B_i}(g_i) \\text{div}_{\\mathcal{L}|_{Z_i}}(s_i|_{Z_i})", "$$", "as cycles. More precisely, the right hand side is a cycle", "representing the left hand side. Namely, this is clear by our", "formula for $\\text{div}_\\mathcal{N}(t)$ and the fact that", "$\\text{div}_{\\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) = [Z(s_i|_{Z_i})]_{n - 2} =", "[Z_i \\cap D]_{n - 2}$ when $Z_i \\not \\subset D$ because in", "that case $s_i|_{Z_i} = s|_{Z_i}$ is a regular section, see", "Lemma \\ref{lemma-compute-c1}. Similarly,", "$$", "c_1(\\mathcal{N}) \\cap [D]_{n - 1} =", "\\sum \\text{ord}_{B_i}(f_i) \\text{div}_{\\mathcal{N}|_{Z_i}}(t_i|_{Z_i})", "$$", "The key formula (Lemma \\ref{lemma-key-formula}) gives the equality", "$$", "\\sum \\left(", "\\text{ord}_{B_i}(f_i) \\text{div}_{\\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) -", "\\text{ord}_{B_i}(g_i) \\text{div}_{\\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) \\right) =", "\\sum \\text{div}_{Z_i}(\\partial_{B_i}(f_i, g_i))", "$$", "of cycles. If $Z_i \\not \\subset D$, then $f_i = 1$ and hence", "$\\text{div}_{Z_i}(\\partial_{B_i}(f_i, g_i)) = 0$. Thus we get a rational", "equivalence between our specific cycles representing", "$i^*\\text{div}_\\mathcal{N}(t)$ and $c_1(\\mathcal{N}) \\cap [D]_{n - 1}$", "on $D$. This finishes the proof." ], "refs": [ "spaces-over-fields-lemma-components-locally-finite", "spaces-over-fields-lemma-divisor-locally-finite", "spaces-over-fields-lemma-divisor-meromorphic-locally-finite", "spaces-chow-lemma-compute-c1", "spaces-chow-lemma-key-formula" ], "ref_ids": [ 12831, 12834, 12836, 7234, 7244 ] } ], "ref_ids": [] }, { "id": 7253, "type": "theorem", "label": "spaces-chow-lemma-gysin-factors", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-gysin-factors", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $(\\mathcal{L}, s, i : D \\to X)$ be as in", "Definition \\ref{definition-gysin-homomorphism}.", "The Gysin homomorphism factors through rational equivalence to", "give a map $i^* : \\CH_{k + 1}(X) \\to \\CH_k(D)$." ], "refs": [ "spaces-chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "Let $\\alpha \\in Z_{k + 1}(X)$ and assume that $\\alpha \\sim_{rat} 0$.", "This means there exists a locally finite collection of integral", "closed subspaces $W_j \\subset X$ of $\\delta$-dimension $k + 2$", "and $f_j \\in R(W_j)^*$ such that", "$\\alpha = \\sum i_{j, *}\\text{div}_{W_j}(f_j)$.", "Set $X' = \\coprod W_i$ and consider the diagram", "$$", "\\xymatrix{", "D' \\ar[d]_q \\ar[r]_{i'} & X' \\ar[d]^p \\\\", "D \\ar[r]^i & X", "}", "$$", "of Remark \\ref{remark-pullback-pairs}. Since $X' \\to X$ is proper", "we see that $i^*p_* = q_*(i')^*$ by Lemma \\ref{lemma-closed-in-X-gysin}.", "As we know that $q_*$ factors through rational equivalence", "(Lemma \\ref{lemma-proper-pushforward-rational-equivalence}), it suffices", "to prove the result for $\\alpha' = \\sum \\text{div}_{W_j}(f_j)$", "on $X'$. Clearly this reduces us to the case where $X$ is integral", "and $\\alpha = \\text{div}(f)$ for some $f \\in R(X)^*$.", "\\medskip\\noindent", "Assume $X$ is integral and $\\alpha = \\text{div}(f)$ for some $f \\in R(X)^*$.", "If $X = D$, then we see that $i^*\\alpha$ is equal", "to $c_1(\\mathcal{L}) \\cap \\alpha$.", "This is rationally equivalent to zero by Lemma \\ref{lemma-factors}.", "If $D \\not = X$, then we see that $i^*\\text{div}_X(f)$ is equal to", "$c_1(\\mathcal{O}_D) \\cap [D]_{n - 1}$ in $\\CH_k(D)$ by", "Lemma \\ref{lemma-gysin-factors-general}. Of course", "capping with $c_1(\\mathcal{O}_D)$ is the zero map." ], "refs": [ "spaces-chow-remark-pullback-pairs", "spaces-chow-lemma-closed-in-X-gysin", "spaces-chow-lemma-proper-pushforward-rational-equivalence", "spaces-chow-lemma-factors", "spaces-chow-lemma-gysin-factors-general" ], "ref_ids": [ 7304, 7249, 7233, 7246, 7252 ] } ], "ref_ids": [ 7292 ] }, { "id": 7254, "type": "theorem", "label": "spaces-chow-lemma-gysin-commutes-cap-c1", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-gysin-commutes-cap-c1", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $(\\mathcal{L}, s, i : D \\to X)$", "be a triple as in Definition \\ref{definition-gysin-homomorphism}.", "Let $\\mathcal{N}$ be an invertible $\\mathcal{O}_X$-module.", "Then $i^*(c_1(\\mathcal{N}) \\cap \\alpha) = c_1(i^*\\mathcal{N}) \\cap i^*\\alpha$", "in $\\CH_{k - 2}(D)$ for all $\\alpha \\in \\CH_k(Z)$." ], "refs": [ "spaces-chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "With exactly the same proof as in Lemma \\ref{lemma-gysin-factors}", "this follows from Lemmas", "\\ref{lemma-pushforward-cap-c1},", "\\ref{lemma-cap-commutative}, and", "\\ref{lemma-gysin-factors-general}." ], "refs": [ "spaces-chow-lemma-gysin-factors", "spaces-chow-lemma-pushforward-cap-c1", "spaces-chow-lemma-cap-commutative", "spaces-chow-lemma-gysin-factors-general" ], "ref_ids": [ 7253, 7243, 7247, 7252 ] } ], "ref_ids": [ 7292 ] }, { "id": 7255, "type": "theorem", "label": "spaces-chow-lemma-gysin-commutes-gysin", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-gysin-commutes-gysin", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $(\\mathcal{L}, s, i : D \\to X)$ and", "$(\\mathcal{L}', s', i' : D' \\to X)$ be two triples as in", "Definition \\ref{definition-gysin-homomorphism}. Then the diagram", "$$", "\\xymatrix{", "\\CH_k(X) \\ar[r]_{i^*} \\ar[d]_{(i')^*} & \\CH_{k - 1}(D) \\ar[d] \\\\", "\\CH_{k - 1}(D') \\ar[r] & \\CH_{k - 2}(D \\cap D')", "}", "$$", "commutes where each of the maps is a gysin map." ], "refs": [ "spaces-chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "Denote $j : D \\cap D' \\to D$ and $j' : D \\cap D' \\to D'$ the closed", "immersions corresponding to $(\\mathcal{L}|_{D'}, s|_{D'}$ and", "$(\\mathcal{L}'_D, s|_D)$. We have to show that", "$(j')^*i^*\\alpha = j^* (i')^*\\alpha$ for all $\\alpha \\in \\CH_k(X)$.", "Let $W \\subset X$ be an integral closed subscheme of dimension $k$.", "We will prove the equality in case $\\alpha = [W]$.", "The general case will then follow from the observation in", "Remark \\ref{remark-infinite-sums-rational-equivalences}", "(and the specific shape of our rational equivalence produced below).", "We will deduce the equality for $\\alpha = [W]$ from the key formula.", "\\medskip\\noindent", "We let $\\sigma$ be a nonzero meromorphic section of $\\mathcal{L}|_W$", "which we require to be equal to $s|_W$ if $W \\not \\subset D$.", "We let $\\sigma'$ be a nonzero meromorphic section of $\\mathcal{L}'|_W$", "which we require to be equal to $s'|_W$ if $W \\not \\subset D'$.", "Write", "$$", "\\text{div}_{\\mathcal{L}|_W}(\\sigma) =", "\\sum \\text{ord}_{Z_i, \\mathcal{L}|_W}(\\sigma)[Z_i] = \\sum n_i[Z_i]", "$$", "and similarly", "$$", "\\text{div}_{\\mathcal{L}'|_W}(\\sigma') =", "\\sum \\text{ord}_{Z_i, \\mathcal{L}'|_W}(\\sigma')[Z_i] = \\sum n'_i[Z_i]", "$$", "as in the discussion in Section \\ref{section-key}.", "Then we see that $Z_i \\subset D$ if $n_i \\not = 0$ and", "$Z'_i \\subset D'$ if $n'_i \\not = 0$. For each $i$, let $\\xi_i \\in |Z_i|$", "be the generic point. As in Section \\ref{section-key} we choose", "for each $i$ an element", "$\\sigma_i \\in \\mathcal{L}_{\\xi_i}$, resp.\\ $\\sigma'_i \\in \\mathcal{L}'_{\\xi_i}$", "which generates over $B_i = \\mathcal{O}_{W, \\xi_i}^h$", "and which is equal to the image of", "$s$, resp.\\ $s'$ if $Z_i \\not \\subset D$, resp.\\ $Z_i \\not \\subset D'$.", "Write $\\sigma = f_i \\sigma_i$ and $\\sigma' = f'_i\\sigma'_i$ so that", "$n_i = \\text{ord}_{B_i}(f_i)$ and", "$n'_i = \\text{ord}_{B_i}(f'_i)$.", "From our definitions it follows that", "$$", "(j')^*i^*[W] =", "\\sum \\text{ord}_{B_i}(f_i) \\text{div}_{\\mathcal{L}'|_{Z_i}}(\\sigma'_i|_{Z_i})", "$$", "as cycles and", "$$", "j^*(i')^*[W] =", "\\sum \\text{ord}_{B_i}(f'_i) \\text{div}_{\\mathcal{L}|_{Z_i}}(\\sigma_i|_{Z_i})", "$$", "The key formula (Lemma \\ref{lemma-key-formula}) now gives the equality", "$$", "\\sum \\left(", "\\text{ord}_{B_i}(f_i) \\text{div}_{\\mathcal{L}'|_{Z_i}}(\\sigma'_i|_{Z_i}) -", "\\text{ord}_{B_i}(f'_i) \\text{div}_{\\mathcal{L}|_{Z_i}}(\\sigma_i|_{Z_i})", "\\right) =", "\\sum \\text{div}_{Z_i}(\\partial_{B_i}(f_i, f'_i))", "$$", "of cycles. Note that $\\text{div}_{Z_i}(\\partial_{B_i}(f_i, f'_i)) = 0$ if", "$Z_i \\not \\subset D \\cap D'$ because in this case either $f_i = 1$", "or $f'_i = 1$. Thus we get a rational equivalence between our specific", "cycles representing $(j')^*i^*[W]$ and $j^*(i')^*[W]$ on $D \\cap D' \\cap W$." ], "refs": [ "spaces-chow-remark-infinite-sums-rational-equivalences", "spaces-chow-lemma-key-formula" ], "ref_ids": [ 7302, 7244 ] } ], "ref_ids": [ 7292 ] }, { "id": 7256, "type": "theorem", "label": "spaces-chow-lemma-relative-effective-cartier", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-relative-effective-cartier", "contents": [ "In Situation \\ref{situation-setup}. Let $X, Y/B$ be good.", "Let $p : X \\to Y$ be a flat morphism of relative dimension $r$.", "Let $i : D \\to X$ be a relative effective Cartier divisor", "(Divisors on Spaces, Definition", "\\ref{spaces-divisors-definition-relative-effective-Cartier-divisor}).", "Let $\\mathcal{L} = \\mathcal{O}_X(D)$.", "For any $\\alpha \\in \\CH_{k + 1}(Y)$ we have", "$$", "i^*p^*\\alpha = (p|_D)^*\\alpha", "$$", "in $\\CH_{k + r}(D)$ and", "$$", "c_1(\\mathcal{L}) \\cap p^*\\alpha = i_* ((p|_D)^*\\alpha)", "$$", "in $\\CH_{k + r}(X)$." ], "refs": [ "spaces-divisors-definition-relative-effective-Cartier-divisor" ], "proofs": [ { "contents": [ "Let $W \\subset Y$ be an integral closed subspace of $\\delta$-dimension", "$k + 1$. By Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-relative-Cartier}", "we see that $D \\cap p^{-1}W$ is an effective", "Cartier divisor on $p^{-1}W$. By Lemma \\ref{lemma-easy-gysin}", "we get the first equality in", "$$", "i^*[p^{-1}W]_{k + r + 1} =", "[D \\cap p^{-1}W]_{k + r} =", "[(p|_D)^{-1}(W)]_{k + r}.", "$$", "and the second because $D \\cap p^{-1}(W) = (p|_D)^{-1}(W)$ as algebraic spaces.", "Since by definition $p^*[W] = [p^{-1}W]_{k + r + 1}$ we see that", "$i^*p^*[W] = (p|_D)^*[W]$ as cycles. If $\\alpha = \\sum m_j[W_j]$ is a", "general $k + 1$ cycle, then we get", "$i^*\\alpha = \\sum m_j i^*p^*[W_j] = \\sum m_j(p|_D)^*[W_j]$ as cycles.", "This proves then first equality. To deduce the second from the", "first apply Lemma \\ref{lemma-support-cap-effective-Cartier}." ], "refs": [ "spaces-divisors-lemma-relative-Cartier", "spaces-chow-lemma-easy-gysin", "spaces-chow-lemma-support-cap-effective-Cartier" ], "ref_ids": [ 12954, 7251, 7248 ] } ], "ref_ids": [ 13022 ] }, { "id": 7257, "type": "theorem", "label": "spaces-chow-lemma-pullback-affine-fibres-surjective", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-pullback-affine-fibres-surjective", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Let $f : X \\to Y$ be a quasi-compact flat morphism over $B$", "of relative dimension $r$. Assume that for every $y \\in Y$ we have", "$X_y \\cong \\mathbf{A}^r_{\\kappa(y)}$.", "Then $f^* : \\CH_k(Y) \\to \\CH_{k + r}(X)$ is surjective for all", "$k \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha \\in \\CH_{k + r}(X)$. Write $\\alpha = \\sum m_j[W_j]$ with", "$m_j \\not = 0$ and $W_j$ pairwise distinct integral closed subspaces of", "$\\delta$-dimension $k + r$. Then the family $\\{W_j\\}$", "is locally finite in $X$. Let $Z_j \\subset Y$ be the integral", "closed subspace such that we obtain a dominant morphism $W_j \\to Z_j$", "as in Lemma \\ref{lemma-proper-image}. For any quasi-compact open", "$V \\subset Y$ we see that $f^{-1}(V) \\cap W_j$", "is nonempty only for finitely many $j$. Hence the", "collection $Z_j$ of closures of images is a locally finite collection", "of integral closed subspaces of $Y$.", "\\medskip\\noindent", "Consider the fibre product diagrams", "$$", "\\xymatrix{", "f^{-1}(Z_j) \\ar[r] \\ar[d]_{f_j} & X \\ar[d]^f \\\\", "Z_j \\ar[r] & Y", "}", "$$", "Suppose that $[W_j] \\in Z_{k + r}(f^{-1}(Z_j))$", "is rationally equivalent to $f_j^*\\beta_j$ for some", "$k$-cycle $\\beta_j \\in \\CH_k(Z_j)$. Then", "$\\beta = \\sum m_j \\beta_j$ will be a $k$-cycle on $Y$", "and $f^*\\beta = \\sum m_j f_j^*\\beta_j$ will be rationally", "equivalent to $\\alpha$ (see", "Remark \\ref{remark-infinite-sums-rational-equivalences}).", "This reduces us to the case $Y$ integral, and", "$\\alpha = [W]$ for some integral closed subscheme", "of $X$ dominating $Y$. In particular we may", "assume that $d = \\dim_\\delta(Y) < \\infty$.", "\\medskip\\noindent", "Hence we can use induction on $d = \\dim_\\delta(Y)$.", "If $d < k$, then $\\CH_{k + r}(X) = 0$ and the lemma holds;", "this is the base case of the induction.", "Consider a nonempty open $V \\subset Y$.", "Suppose that we can show that $\\alpha|_{f^{-1}(V)} = f^*\\beta$", "for some $\\beta \\in Z_k(V)$. By Lemma \\ref{lemma-exact-sequence-open}", "we see that", "$\\beta = \\beta'|_V$ for some $\\beta' \\in Z_k(Y)$.", "By the exact sequence", "$\\CH_k(f^{-1}(Y \\setminus V)) \\to \\CH_k(X) \\to \\CH_k(f^{-1}(V))$", "of Lemma \\ref{lemma-restrict-to-open}", "we see that $\\alpha - f^*\\beta'$ comes from", "a cycle $\\alpha' \\in \\CH_{k + r}(f^{-1}(Y \\setminus V))$.", "Since $\\dim_\\delta(Y \\setminus V) < d$ we win by", "induction on $d$.", "\\medskip\\noindent", "In particular, by replacing $Y$ by a suitable open we may assume", "$Y$ is a scheme with generic point $\\eta$. The isomorphism", "$Y_\\eta \\cong \\mathbf{A}^r_\\eta$ extends to an isomorphism", "over a nonempty open $V \\subset Y$, see", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-descend-finite-presentation}.", "This reduces us to the case of schemes which is", "Chow Homology, Lemma \\ref{chow-lemma-pullback-affine-fibres-surjective}." ], "refs": [ "spaces-chow-lemma-proper-image", "spaces-chow-remark-infinite-sums-rational-equivalences", "spaces-chow-lemma-exact-sequence-open", "spaces-chow-lemma-restrict-to-open", "spaces-limits-lemma-descend-finite-presentation", "chow-lemma-pullback-affine-fibres-surjective" ], "ref_ids": [ 7214, 7302, 7221, 7230, 4598, 5726 ] } ], "ref_ids": [] }, { "id": 7258, "type": "theorem", "label": "spaces-chow-lemma-linebundle", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-linebundle", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let", "$$", "p :", "L = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{L}))", "\\longrightarrow", "X", "$$", "be the associated vector bundle over $X$.", "Then $p^* : \\CH_k(X) \\to \\CH_{k + 1}(L)$ is an isomorphism for all $k$." ], "refs": [], "proofs": [ { "contents": [ "For surjectivity see Lemma \\ref{lemma-pullback-affine-fibres-surjective}.", "Let $o : X \\to L$ be the zero section of $L \\to X$, i.e., the morphism", "corresponding to the surjection $\\text{Sym}^*(\\mathcal{L}) \\to \\mathcal{O}_X$", "which maps $\\mathcal{L}^{\\otimes n}$ to zero for all $n > 0$.", "Then $p \\circ o = \\text{id}_X$ and $o(X)$ is an effective", "Cartier divisor on $L$. Hence by Lemma \\ref{lemma-relative-effective-cartier}", "we see that $o^* \\circ p^* = \\text{id}$ and we conclude that $p^*$ is", "injective too." ], "refs": [ "spaces-chow-lemma-pullback-affine-fibres-surjective", "spaces-chow-lemma-relative-effective-cartier" ], "ref_ids": [ 7257, 7256 ] } ], "ref_ids": [] }, { "id": 7259, "type": "theorem", "label": "spaces-chow-lemma-cap-c1-bivariant", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-cap-c1-bivariant", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Then the rule that to $f : X' \\to X$ assigns", "$c_1(f^*\\mathcal{L}) \\cap - : \\CH_k(X') \\to \\CH_{k - 1}(X')$", "is a bivariant class of degree $1$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-factors},", "\\ref{lemma-pushforward-cap-c1},", "\\ref{lemma-flat-pullback-cap-c1}, and", "\\ref{lemma-gysin-commutes-cap-c1}." ], "refs": [ "spaces-chow-lemma-factors", "spaces-chow-lemma-pushforward-cap-c1", "spaces-chow-lemma-flat-pullback-cap-c1", "spaces-chow-lemma-gysin-commutes-cap-c1" ], "ref_ids": [ 7246, 7243, 7241, 7254 ] } ], "ref_ids": [] }, { "id": 7260, "type": "theorem", "label": "spaces-chow-lemma-flat-pullback-bivariant", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-flat-pullback-bivariant", "contents": [ "In Situation \\ref{situation-setup} let $f : X \\to Y$ be a morphism", "of good algebraic spaces over $B$ which is flat of relative dimension $r$.", "Then the rule that to $Y' \\to Y$ assigns", "$(f')^* : \\CH_k(Y') \\to \\CH_{k + r}(X')$ where $X' = X \\times_Y Y'$", "is a bivariant class of degree $-r$." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemmas \\ref{lemma-flat-pullback-rational-equivalence},", "\\ref{lemma-compose-flat-pullback},", "\\ref{lemma-flat-pullback-proper-pushforward}, and", "\\ref{lemma-gysin-flat-pullback}." ], "refs": [ "spaces-chow-lemma-flat-pullback-rational-equivalence", "spaces-chow-lemma-compose-flat-pullback", "spaces-chow-lemma-flat-pullback-proper-pushforward", "spaces-chow-lemma-gysin-flat-pullback" ], "ref_ids": [ 7232, 7223, 7225, 7250 ] } ], "ref_ids": [] }, { "id": 7261, "type": "theorem", "label": "spaces-chow-lemma-gysin-bivariant", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-gysin-bivariant", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $(\\mathcal{L}, s, i : D \\to X)$ be a triple as in", "Definition \\ref{definition-gysin-homomorphism}.", "Then the rule that to $f : X' \\to X$ assigns", "$(i')^* : \\CH_k(X') \\to \\CH_{k - 1}(D')$ where $D' = D \\times_X X'$", "is a bivariant class of degree $1$." ], "refs": [ "spaces-chow-definition-gysin-homomorphism" ], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-gysin-factors},", "\\ref{lemma-closed-in-X-gysin},", "\\ref{lemma-gysin-flat-pullback}, and", "\\ref{lemma-gysin-commutes-gysin}." ], "refs": [ "spaces-chow-lemma-gysin-factors", "spaces-chow-lemma-closed-in-X-gysin", "spaces-chow-lemma-gysin-flat-pullback", "spaces-chow-lemma-gysin-commutes-gysin" ], "ref_ids": [ 7253, 7249, 7250, 7255 ] } ], "ref_ids": [ 7292 ] }, { "id": 7262, "type": "theorem", "label": "spaces-chow-lemma-push-proper-bivariant", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-push-proper-bivariant", "contents": [ "In Situation \\ref{situation-setup} let $f : X \\to Y$ and", "$g : Y \\to Z$ be morphisms of good algebraic spaces over $B$.", "Let $c \\in A^p(X \\to Z)$ and assume $f$ is proper.", "Then the rule that to $X' \\to X$ assigns", "$\\alpha \\longmapsto f_*(c \\cap \\alpha)$", "is a bivariant class of degree $p$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-compose-pushforward},", "\\ref{lemma-flat-pullback-proper-pushforward}, and", "\\ref{lemma-closed-in-X-gysin}." ], "refs": [ "spaces-chow-lemma-compose-pushforward", "spaces-chow-lemma-flat-pullback-proper-pushforward", "spaces-chow-lemma-closed-in-X-gysin" ], "ref_ids": [ 7217, 7225, 7249 ] } ], "ref_ids": [] }, { "id": 7263, "type": "theorem", "label": "spaces-chow-lemma-c1-center", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-c1-center", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Then $c_1(\\mathcal{L}) \\in A^1(X)$ commutes with every", "element $c \\in A^p(X)$." ], "refs": [], "proofs": [ { "contents": [ "Let $p : L \\to X$ be as in Lemma \\ref{lemma-linebundle} and let $o : X \\to L$", "be the zero section. Observe that $p^*\\mathcal{L}^{\\otimes -1}$ has a", "canonical section whose vanishing locus is exactly the", "effective Cartier divisor $o(X)$. Let $\\alpha \\in \\CH_k(X)$. Then we see that", "$$", "p^*(c_1(\\mathcal{L}^{\\otimes -1}) \\cap \\alpha) =", "c_1(p^*\\mathcal{L}^{\\otimes -1}) \\cap p^*\\alpha =", "o_* o^* p^*\\alpha", "$$", "by Lemmas \\ref{lemma-flat-pullback-cap-c1} and", "\\ref{lemma-relative-effective-cartier}.", "Since $c$ is a bivariant class we have", "\\begin{align*}", "p^*(c \\cap c_1(\\mathcal{L}^{\\otimes -1}) \\cap \\alpha)", "& =", "c \\cap p^*(c_1(\\mathcal{L}^{\\otimes -1}) \\cap \\alpha) \\\\", "& =", "c \\cap o_* o^* p^*\\alpha \\\\", "& =", "o_* o^* p^*(c \\cap \\alpha) \\\\", "& =", "p^*(c_1(\\mathcal{L}^{\\otimes -1}) \\cap c \\cap \\alpha)", "\\end{align*}", "(last equality by the above applied to $c \\cap \\alpha$).", "Since $p^*$ is injective by a lemma cited above we get that", "$c_1(\\mathcal{L}^{\\otimes -1})$", "is in the center of $A^*(X)$. This proves the lemma." ], "refs": [ "spaces-chow-lemma-linebundle", "spaces-chow-lemma-flat-pullback-cap-c1", "spaces-chow-lemma-relative-effective-cartier" ], "ref_ids": [ 7258, 7241, 7256 ] } ], "ref_ids": [] }, { "id": 7264, "type": "theorem", "label": "spaces-chow-lemma-bivariant-zero", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-bivariant-zero", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good. Let $c \\in A^p(X)$.", "Then $c$ is zero if and only if $c \\cap [Y] = 0$ in $\\CH_*(Y)$", "for every integral algebraic space $Y$ locally of finite type over $X$." ], "refs": [], "proofs": [ { "contents": [ "The if direction is clear. For the converse, assume that $c \\cap [Y] = 0$ in", "$\\CH_*(Y)$ for every integral algebraic space $Y$ locally of finite type", "over $X$.", "Let $X' \\to X$ be locally of finite type. Let $\\alpha \\in \\CH_k(X')$.", "Write $\\alpha = \\sum n_i [Y_i]$ with $Y_i \\subset X'$ a locally finite", "collection of integral closed subschemes of $\\delta$-dimension $k$.", "Then we see that $\\alpha$ is pushforward of the cycle", "$\\alpha' = \\sum n_i[Y_i]$ on $X'' = \\coprod Y_i$ under the", "proper morphism $X'' \\to X'$. By the properties of bivariant", "classes it suffices to prove that $c \\cap \\alpha' = 0$ in $\\CH_{k - p}(X'')$.", "We have $\\CH_{k - p}(X'') = \\prod \\CH_{k - p}(Y_i)$ as follows immediately", "from the definitions. The projection maps", "$\\CH_{k - p}(X'') \\to \\CH_{k - p}(Y_i)$", "are given by flat pullback. Since capping with $c$ commutes with", "flat pullback, we see that it suffices to show that $c \\cap [Y_i]$", "is zero in $\\CH_{k - p}(Y_i)$ which is true by assumption." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7265, "type": "theorem", "label": "spaces-chow-lemma-cap-projective-bundle", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-cap-projective-bundle", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module", "$\\mathcal{E}$ of rank $r$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$", "be the projective bundle associated to $\\mathcal{E}$.", "For any $\\alpha \\in \\CH_k(X)$ the element", "$$", "\\pi_*\\left(", "c_1(\\mathcal{O}_P(1))^s \\cap \\pi^*\\alpha", "\\right)", "\\in", "\\CH_{k + r - 1 - s}(X)", "$$", "is $0$ if $s < r - 1$ and is equal to $\\alpha$ when $s = r - 1$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset X$ be an integral closed subspace of $\\delta$-dimension $k$.", "We will prove the lemma for $\\alpha = [Z]$. We omit the argument", "deducing the general case from this special case; hint: argue as in", "Remark \\ref{remark-infinite-sums-rational-equivalences}.", "\\medskip\\noindent", "Let $P_Z = P \\times_X Z$ be the base change; of course", "$\\pi_Z : P_Z \\to Z$ is the projective bundle associated to $\\mathcal{E}|_Z$", "and $\\mathcal{O}_P(1)$ pulls back to the corresponding", "invertible module on $P_Z$. Since $c_1(\\mathcal{O}_P(1) \\cap -$, and", "$\\pi^*$ are bivariant classes by Lemmas", "\\ref{lemma-cap-c1-bivariant} and \\ref{lemma-flat-pullback-bivariant}", "we see that", "$$", "\\pi_*\\left(", "c_1(\\mathcal{O}_P(1))^s \\cap \\pi^*[Z]", "\\right)", "=", "(Z \\to X)_*\\pi_{Z, *}\\left(", "c_1(\\mathcal{O}_{P_Z}(1))^s \\cap \\pi_Z^*[Z]", "\\right)", "$$", "Hence it suffices to prove the lemma in case $X$ is integral", "and $\\alpha = [X]$.", "\\medskip\\noindent", "Assume $X$ is integral, $\\dim_\\delta(X) = k$, and $\\alpha = [X]$.", "Note that $\\pi^*[X] = [P]$ as $P$ is integral of", "$\\delta$-dimension $r - 1$. If $s < r - 1$, then by construction", "$c_1(\\mathcal{O}_P(1))^s \\cap [P]$ a $(k + r - 1 - s)$-cycle.", "Hence the pushforward of this cycle is zero for dimension reasons.", "\\medskip\\noindent", "Let $s = r - 1$. By the argument given above we see that", "$\\pi_*(c_1(\\mathcal{O}_P(1))^s \\cap [P]) = n [X]$", "for some $n \\in \\mathbf{Z}$. We want to show that $n = 1$.", "For the same dimension reasons as above it suffices to prove this", "result after replacing $X$ by a dense open.", "Thus we may assume $X$ is a scheme and the result follows", "from Chow Homology, Lemma \\ref{chow-lemma-cap-projective-bundle}." ], "refs": [ "spaces-chow-remark-infinite-sums-rational-equivalences", "spaces-chow-lemma-cap-c1-bivariant", "spaces-chow-lemma-flat-pullback-bivariant", "chow-lemma-cap-projective-bundle" ], "ref_ids": [ 7302, 7259, 7260, 5742 ] } ], "ref_ids": [] }, { "id": 7266, "type": "theorem", "label": "spaces-chow-lemma-chow-ring-projective-bundle", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-chow-ring-projective-bundle", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module", "$\\mathcal{E}$ of rank $r$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$", "be the projective bundle associated to $\\mathcal{E}$.", "The map", "$$", "\\bigoplus\\nolimits_{i = 0}^{r - 1}", "\\CH_{k + i}(X)", "\\longrightarrow", "\\CH_{k + r - 1}(P),", "$$", "$$", "(\\alpha_0, \\ldots, \\alpha_{r-1})", "\\longmapsto", "\\pi^*\\alpha_0 +", "c_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\alpha_1", "+ \\ldots +", "c_1(\\mathcal{O}_P(1))^{r - 1} \\cap \\pi^*\\alpha_{r-1}", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Fix $k \\in \\mathbf{Z}$. We first show the map is injective.", "Suppose that $(\\alpha_0, \\ldots, \\alpha_{r - 1})$ is an element", "of the left hand side that maps to zero.", "By Lemma \\ref{lemma-cap-projective-bundle} we see that", "$$", "0 = \\pi_*(\\pi^*\\alpha_0 +", "c_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\alpha_1", "+ \\ldots +", "c_1(\\mathcal{O}_P(1))^{r - 1} \\cap \\pi^*\\alpha_{r-1})", "= \\alpha_{r - 1}", "$$", "Next, we see that", "$$", "0 = \\pi_*(c_1(\\mathcal{O}_P(1)) \\cap (\\pi^*\\alpha_0 +", "c_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\alpha_1", "+ \\ldots +", "c_1(\\mathcal{O}_P(1))^{r - 2} \\cap \\pi^*\\alpha_{r - 2}))", "= \\alpha_{r - 2}", "$$", "and so on. Hence the map is injective.", "\\medskip\\noindent", "To prove the map is surjective, we will argue exactly", "as in the proof of Lemma \\ref{lemma-pullback-affine-fibres-surjective}", "to reduce to the case of schemes.", "We urge the reader to skip the proof.", "\\medskip\\noindent", "Let $\\beta \\in \\CH_{k + r - 1}(P)$. Write $\\beta = \\sum m_j[W_j]$ with", "$m_j \\not = 0$ and $W_j$ pairwise distinct integral closed subspaces of", "$\\delta$-dimension $k + r$. Then the family $\\{W_j\\}$", "is locally finite in $P$. Let $Z_j \\subset X$ be the ``image''", "of $W_j$ as in Lemma \\ref{lemma-proper-image}. For any quasi-compact open", "$U \\subset X$ we see that $\\pi^{-1}(U) \\cap W_j$", "is nonempty only for finitely many $j$. Hence the", "collection $Z_j$ of images is a locally finite collection", "of integral closed subspaces of $X$.", "\\medskip\\noindent", "Consider the fibre product diagrams", "$$", "\\xymatrix{", "P_j \\ar[r] \\ar[d]_{\\pi_j} & P \\ar[d]^\\pi \\\\", "Z_j \\ar[r] & X", "}", "$$", "Suppose that $[W_j] \\in Z_{k + r - 1}(P_j)$", "is rationally equivalent to", "$$", "\\pi_j^*\\alpha_{j, 0} +", "c_1(\\mathcal{O}(1)) \\cap \\pi_j^*\\alpha_{j, 1} +", "\\ldots +", "c_1(\\mathcal{O}(1))^{r - 1} \\cap \\pi_j^*\\alpha_{j, r - 1}", "$$", "for some $(k + i)$-cycle $\\alpha_{j, i} \\in \\CH_{k + i}(Z_j)$. Then", "$\\alpha_i = \\sum m_j \\beta_{j, i}$ will be a $(k + i)$-cycle on $X$", "and", "$$", "\\pi^*\\alpha_0 +", "c_1(\\mathcal{O}(1)) \\cap \\pi^*\\alpha_1 +", "\\ldots +", "c_1(\\mathcal{O}(1))^{r - 1} \\cap \\pi^*\\alpha_{r - 1}", "$$", "will be rationally equivalent to $\\beta$ (see", "Remark \\ref{remark-infinite-sums-rational-equivalences}).", "This reduces us to the case $X$ integral, and", "$\\alpha = [W]$ for some integral closed subscheme", "of $P$ dominating $X$. In particular we may", "assume that $d = \\dim_\\delta(X) < \\infty$.", "\\medskip\\noindent", "Hence we can use induction on $d = \\dim_\\delta(X)$.", "If $d < k$, then $\\CH_{k + r - 1}(X) = 0$ and the lemma holds;", "this is the base case of the induction.", "Consider a nonempty open $U \\subset X$.", "Suppose that we can show that", "$$", "\\beta|_{\\pi^{-1}(U)} =", "\\pi^*\\alpha_0 +", "c_1(\\mathcal{O}(1)) \\cap \\pi^*\\alpha_1 +", "\\ldots +", "c_1(\\mathcal{O}(1))^{r - 1} \\cap \\pi^*\\alpha_{r - 1}", "$$", "for some $\\alpha_i \\in Z_{k + i}(U)$.", "By Lemma \\ref{lemma-exact-sequence-open} we see that", "$\\alpha_i = \\alpha'_i|_U$ for some $\\alpha'_i \\in Z_{k + i}(X)$.", "By the exact sequences", "$\\CH_{k + i}(\\pi^{-1}(X \\setminus U)) \\to \\CH_{k + i}(P) \\to", "\\CH_{k + i}(\\pi^{-1}(U))$", "of Lemma \\ref{lemma-restrict-to-open}", "we see that", "$$", "\\beta -", "\\left(\\pi^*\\alpha'_0 +", "c_1(\\mathcal{O}(1)) \\cap \\pi^*\\alpha'_1 +", "\\ldots +", "c_1(\\mathcal{O}(1))^{r - 1} \\cap \\pi^*\\alpha'_{r - 1}\\right)", "$$", "comes from a cycle $\\beta' \\in \\CH_{k + r}(\\pi^{-1}(X \\setminus U))$.", "Since $\\dim_\\delta(X \\setminus U) < d$ we win by", "induction on $d$.", "\\medskip\\noindent", "In particular, by replacing $X$ by a suitable open we may assume", "$X$ is a scheme and we have reduced our problem to", "Chow Homology, Lemma \\ref{chow-lemma-chow-ring-projective-bundle}." ], "refs": [ "spaces-chow-lemma-cap-projective-bundle", "spaces-chow-lemma-pullback-affine-fibres-surjective", "spaces-chow-lemma-proper-image", "spaces-chow-remark-infinite-sums-rational-equivalences", "spaces-chow-lemma-exact-sequence-open", "spaces-chow-lemma-restrict-to-open", "chow-lemma-chow-ring-projective-bundle" ], "ref_ids": [ 7265, 7257, 7214, 7302, 7221, 7230, 5743 ] } ], "ref_ids": [] }, { "id": 7267, "type": "theorem", "label": "spaces-chow-lemma-vectorbundle", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-vectorbundle", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.", "Let", "$$", "p :", "E = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{E}))", "\\longrightarrow", "X", "$$", "be the associated vector bundle over $X$.", "Then $p^* : \\CH_k(X) \\to \\CH_{k + r}(E)$ is an isomorphism for all $k$." ], "refs": [], "proofs": [ { "contents": [ "(For the case of linebundles, see Lemma \\ref{lemma-linebundle}.)", "For surjectivity see Lemma \\ref{lemma-pullback-affine-fibres-surjective}.", "Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$", "be the projective space bundle associated", "to the finite locally free sheaf $\\mathcal{E} \\oplus \\mathcal{O}_X$.", "Let $s \\in \\Gamma(P, \\mathcal{O}_P(1))$ correspond to the global", "section $(0, 1) \\in \\Gamma(X, \\mathcal{E} \\oplus \\mathcal{O}_X)$.", "Let $D = Z(s) \\subset P$. Note that", "$(\\pi|_D : D \\to X , \\mathcal{O}_P(1)|_D)$", "is the projective space bundle associated", "to $\\mathcal{E}$. We denote $\\pi_D = \\pi|_D$ and", "$\\mathcal{O}_D(1) = \\mathcal{O}_P(1)|_D$.", "Moreover, $D$ is an effective", "Cartier divisor on $P$. Hence $\\mathcal{O}_P(D) = \\mathcal{O}_P(1)$", "(see Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-characterize-OD}).", "Also there is an isomorphism", "$E \\cong P \\setminus D$. Denote $j : E \\to P$ the", "corresponding open immersion.", "For injectivity we use that the kernel of", "$$", "j^* :", "\\CH_{k + r}(P)", "\\longrightarrow", "\\CH_{k + r}(E)", "$$", "are the cycles supported in the effective Cartier divisor $D$,", "see Lemma \\ref{lemma-restrict-to-open}. So if $p^*\\alpha = 0$, then", "$\\pi^*\\alpha = i_*\\beta$ for some $\\beta \\in \\CH_{k + r}(D)$.", "By Lemma \\ref{lemma-chow-ring-projective-bundle} we may write", "$$", "\\beta = \\pi_D^*\\beta_0 +", "\\ldots + c_1(\\mathcal{O}_D(1))^{r - 1} \\cap \\pi_D^* \\beta_{r - 1}.", "$$", "for some $\\beta_i \\in \\CH_{k + i}(X)$.", "By Lemmas \\ref{lemma-relative-effective-cartier}", "and \\ref{lemma-pushforward-cap-c1}", "this implies", "$$", "\\pi^*\\alpha = i_*\\beta =", "c_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\beta_0 +", "\\ldots +", "c_1(\\mathcal{O}_D(1))^r \\cap \\pi^*\\beta_{r - 1}.", "$$", "Since the rank of $\\mathcal{E} \\oplus \\mathcal{O}_X$ is $r + 1$", "this contradicts Lemma \\ref{lemma-pushforward-cap-c1} unless all", "$\\alpha$ and all $\\beta_i$ are zero." ], "refs": [ "spaces-chow-lemma-linebundle", "spaces-chow-lemma-pullback-affine-fibres-surjective", "spaces-divisors-lemma-characterize-OD", "spaces-chow-lemma-restrict-to-open", "spaces-chow-lemma-chow-ring-projective-bundle", "spaces-chow-lemma-relative-effective-cartier", "spaces-chow-lemma-pushforward-cap-c1", "spaces-chow-lemma-pushforward-cap-c1" ], "ref_ids": [ 7258, 7257, 12948, 7230, 7266, 7256, 7243, 7243 ] } ], "ref_ids": [] }, { "id": 7268, "type": "theorem", "label": "spaces-chow-lemma-segre-classes", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-segre-classes", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.", "Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$ be the projective space", "bundle associated to $\\mathcal{E}$. For every", "morphism $X' \\to X$ of good algebraic spaces over $B$", "there are unique maps", "$$", "c_i(\\mathcal{E}) \\cap - : \\CH_k(X') \\longrightarrow \\CH_{k - i}(X'),\\quad", "i = 0, \\ldots, r", "$$", "such that for $\\alpha \\in \\CH_k(X')$ we have", "$c_0(\\mathcal{E}) \\cap \\alpha = \\alpha$ and", "$$", "\\sum\\nolimits_{i = 0, \\ldots, r}", "(-1)^i c_1(\\mathcal{O}_{P'}(1))^i \\cap", "(\\pi')^*\\left(c_{r - i}(\\mathcal{E}) \\cap \\alpha\\right) = 0", "$$", "where $\\pi' : P' \\to X'$ is the base change of $\\pi$.", "Moreover, these maps define a bivariant class", "$c_i(\\mathcal{E})$ of degree $i$ on $X$." ], "refs": [], "proofs": [ { "contents": [ "Uniqueness and existence of the maps $c_i(\\mathcal{E}) \\cap -$", "follows immediately from Lemma \\ref{lemma-chow-ring-projective-bundle}", "and the given description of $c_0(\\mathcal{E})$. For every $i \\in \\mathbf{Z}$", "the rule which to every morphism $X' \\to X$ of good algebraic spaces", "over $B$ assigns the map", "$$", "t_i(\\mathcal{E}) \\cap - :", "\\CH_k(X') \\longrightarrow \\CH_{k - i}(X'),\\quad", "\\alpha \\longmapsto", "\\pi'_*(c_1(\\mathcal{O}_{P'}(1))^{r - 1 + i} \\cap (\\pi')^*\\alpha)", "$$", "is a bivariant class\\footnote{Up to signs these are", "the Segre classes of $\\mathcal{E}$.} by Lemmas \\ref{lemma-cap-c1-bivariant},", "\\ref{lemma-flat-pullback-bivariant}, and", "\\ref{lemma-push-proper-bivariant}.", "By Lemma \\ref{lemma-cap-projective-bundle} we have", "$t_i(\\mathcal{E}) = 0$ for $i < 0$ and $t_0(\\mathcal{E}) = 1$.", "Applying pushforward to the equation in the statement of the lemma", "we find from Lemma \\ref{lemma-cap-projective-bundle} that", "$$", "(-1)^r t_1(\\mathcal{E}) + (-1)^{r - 1}c_1(\\mathcal{E}) = 0", "$$", "In particular we find that $c_1(\\mathcal{E})$ is a bivariant class.", "If we multiply the equation in the statement of the lemma by", "$c_1(\\mathcal{O}_{P'}(1))$ and push the result forward to $X'$", "we find", "$$", "(-1)^r t_2(\\mathcal{E}) +", "(-1)^{r - 1} t_1(\\mathcal{E}) \\cap c_1(\\mathcal{E}) +", "(-1)^{r - 2} c_2(\\mathcal{E}) = 0", "$$", "As before we conclude that $c_2(\\mathcal{E})$ is a bivariant class.", "And so on." ], "refs": [ "spaces-chow-lemma-chow-ring-projective-bundle", "spaces-chow-lemma-cap-c1-bivariant", "spaces-chow-lemma-flat-pullback-bivariant", "spaces-chow-lemma-push-proper-bivariant", "spaces-chow-lemma-cap-projective-bundle", "spaces-chow-lemma-cap-projective-bundle" ], "ref_ids": [ 7266, 7259, 7260, 7262, 7265, 7265 ] } ], "ref_ids": [] }, { "id": 7269, "type": "theorem", "label": "spaces-chow-lemma-first-chern-class", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-first-chern-class", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "The first Chern class of $\\mathcal{L}$ on $X$ of", "Definition \\ref{definition-chern-classes}", "is equal to the bivariant class of Lemma \\ref{lemma-cap-c1-bivariant}." ], "refs": [ "spaces-chow-definition-chern-classes", "spaces-chow-lemma-cap-c1-bivariant" ], "proofs": [ { "contents": [ "Namely, in this case $P = \\mathbf{P}(\\mathcal{L}) = X$ with", "$\\mathcal{O}_P(1) = \\mathcal{L}$ by our normalization of the", "projective bundle, see Section \\ref{section-projective-space-bundle-formula}.", "Hence the equation in Lemma \\ref{lemma-segre-classes}", "reads", "$$", "(-1)^0 c_1(\\mathcal{L})^0 \\cap c^{new}_1(\\mathcal{L}) \\cap \\alpha +", "(-1)^1 c_1(\\mathcal{L})^1 \\cap c^{new}_0(\\mathcal{L}) \\cap \\alpha = 0", "$$", "where $c_i^{new}(\\mathcal{L})$ is as in", "Definition \\ref{definition-chern-classes}.", "Since $c_0^{new}(\\mathcal{L}) = 1$ and $c_1(\\mathcal{L})^0 = 1$", "we conclude." ], "refs": [ "spaces-chow-lemma-segre-classes", "spaces-chow-definition-chern-classes" ], "ref_ids": [ 7268, 7295 ] } ], "ref_ids": [ 7295, 7259 ] }, { "id": 7270, "type": "theorem", "label": "spaces-chow-lemma-cap-commutative-chern", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-cap-commutative-chern", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $r$.", "Then $c_j(\\mathcal{L}) \\in A^j(X)$ commutes with every", "element $c \\in A^p(X)$. In particular, if $\\mathcal{F}$ is a", "second locally free $\\mathcal{O}_X$-module on $X$ of rank $s$, then", "$$", "c_i(\\mathcal{E}) \\cap c_j(\\mathcal{F}) \\cap \\alpha", "=", "c_j(\\mathcal{F}) \\cap c_i(\\mathcal{E}) \\cap \\alpha", "$$", "as elements of $\\CH_{k - i - j}(X)$ for all $\\alpha \\in \\CH_k(X)$." ], "refs": [], "proofs": [ { "contents": [ "Let $X' \\to X$ be a morphism of good algebraic spaces over $B$.", "Let $\\alpha \\in \\CH_k(X')$. Write $\\alpha_j = c_j(\\mathcal{E}) \\cap \\alpha$, so", "$\\alpha_0 = \\alpha$. By Lemma \\ref{lemma-segre-classes} we have", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_{P'}(1))^i \\cap", "(\\pi')^*(\\alpha_{r - i}) = 0", "$$", "in the chow group of the projective bundle", "$(\\pi' : P' \\to X', \\mathcal{O}_{P'}(1))$", "associated to $(X' \\to X)^*\\mathcal{E}$.", "Applying $c \\cap -$ and using Lemma \\ref{lemma-c1-center}", "and the properties of bivariant classes we obtain", "$$", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_{P'}(1))^i \\cap", "\\pi^*(c \\cap \\alpha_{r - i}) = 0", "$$", "in the Chow group of $P'$. Hence we see that $c \\cap \\alpha_j$ is", "equal to $c_j(\\mathcal{E}) \\cap (c \\cap \\alpha)$ by the uniqueness in", "Lemma \\ref{lemma-segre-classes}. This proves the lemma." ], "refs": [ "spaces-chow-lemma-segre-classes", "spaces-chow-lemma-c1-center", "spaces-chow-lemma-segre-classes" ], "ref_ids": [ 7268, 7263, 7268 ] } ], "ref_ids": [] }, { "id": 7271, "type": "theorem", "label": "spaces-chow-lemma-chern-classes-E-tensor-L", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-chern-classes-E-tensor-L", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{E}$ be a finite locally free sheaf of", "rank $r$ on $X$. Let $\\mathcal{L}$ be an invertible", "sheaf on $X$. Then we have", "\\begin{equation}", "\\label{equation-twist}", "c_i({\\mathcal E} \\otimes {\\mathcal L})", "=", "\\sum\\nolimits_{j = 0}^i", "\\binom{r - i + j}{j} c_{i - j}({\\mathcal E}) c_1({\\mathcal L})^j", "\\end{equation}", "in $A^*(X)$." ], "refs": [], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Chow Homology, Lemma \\ref{chow-lemma-chern-classes-E-tensor-L}", "replacing the lemmas used there by Lemmas \\ref{lemma-bivariant-zero} and", "\\ref{lemma-segre-classes}." ], "refs": [ "chow-lemma-chern-classes-E-tensor-L", "spaces-chow-lemma-bivariant-zero", "spaces-chow-lemma-segre-classes" ], "ref_ids": [ 5753, 7264, 7268 ] } ], "ref_ids": [] }, { "id": 7272, "type": "theorem", "label": "spaces-chow-lemma-get-rid-of-trivial-subbundle", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-get-rid-of-trivial-subbundle", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{E}$, $\\mathcal{F}$ be finite locally free sheaves", "on $X$ of ranks $r$, $r - 1$ which fit into a short", "exact sequence", "$$", "0 \\to \\mathcal{O}_X \\to \\mathcal{E} \\to \\mathcal{F} \\to 0", "$$", "Then we have", "$$", "c_r(\\mathcal{E}) = 0, \\quad", "c_j(\\mathcal{E}) = c_j(\\mathcal{F}), \\quad j = 0, \\ldots, r - 1", "$$", "in $A^*(X)$." ], "refs": [], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Chow Homology, Lemma \\ref{chow-lemma-get-rid-of-trivial-subbundle}", "replacing the lemmas used there by", "Lemmas \\ref{lemma-bivariant-zero},", "\\ref{lemma-relative-effective-cartier},", "\\ref{lemma-pushforward-cap-c1}, and", "\\ref{lemma-segre-classes}." ], "refs": [ "chow-lemma-get-rid-of-trivial-subbundle", "spaces-chow-lemma-bivariant-zero", "spaces-chow-lemma-relative-effective-cartier", "spaces-chow-lemma-pushforward-cap-c1", "spaces-chow-lemma-segre-classes" ], "ref_ids": [ 5754, 7264, 7256, 7243, 7268 ] } ], "ref_ids": [] }, { "id": 7273, "type": "theorem", "label": "spaces-chow-lemma-additivity-invertible-subsheaf", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-additivity-invertible-subsheaf", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{E}$, $\\mathcal{F}$ be finite locally free sheaves", "on $X$ of ranks $r$, $r - 1$ which fit into a short", "exact sequence", "$$", "0 \\to \\mathcal{L} \\to \\mathcal{E} \\to \\mathcal{F} \\to 0", "$$", "where $\\mathcal{L}$ is an invertible sheaf.", "Then", "$$", "c(\\mathcal{E}) = c(\\mathcal{L}) c(\\mathcal{F})", "$$", "in $A^*(X)$." ], "refs": [], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Chow Homology, Lemma \\ref{chow-lemma-additivity-invertible-subsheaf}", "replacing the lemmas used there by", "Lemmas", "\\ref{lemma-get-rid-of-trivial-subbundle} and", "\\ref{lemma-chern-classes-E-tensor-L}." ], "refs": [ "chow-lemma-additivity-invertible-subsheaf", "spaces-chow-lemma-get-rid-of-trivial-subbundle", "spaces-chow-lemma-chern-classes-E-tensor-L" ], "ref_ids": [ 5755, 7272, 7271 ] } ], "ref_ids": [] }, { "id": 7274, "type": "theorem", "label": "spaces-chow-lemma-additivity-chern-classes", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-additivity-chern-classes", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Suppose that $\\mathcal{E}$ sits in an exact sequence", "$$", "0", "\\to", "\\mathcal{E}_1", "\\to", "\\mathcal{E}", "\\to", "\\mathcal{E}_2", "\\to", "0", "$$", "of finite locally free sheaves $\\mathcal{E}_i$ of rank $r_i$.", "The total Chern classes satisfy", "$$", "c(\\mathcal{E}) = c(\\mathcal{E}_1) c(\\mathcal{E}_2)", "$$", "in $A^*(X)$." ], "refs": [], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Chow Homology, Lemma \\ref{chow-lemma-additivity-chern-classes}", "replacing the lemmas used there by", "Lemmas \\ref{lemma-bivariant-zero},", "\\ref{lemma-additivity-invertible-subsheaf}, and", "\\ref{lemma-segre-classes}." ], "refs": [ "chow-lemma-additivity-chern-classes", "spaces-chow-lemma-bivariant-zero", "spaces-chow-lemma-additivity-invertible-subsheaf", "spaces-chow-lemma-segre-classes" ], "ref_ids": [ 5756, 7264, 7273, 7268 ] } ], "ref_ids": [] }, { "id": 7275, "type": "theorem", "label": "spaces-chow-lemma-chern-filter-by-linebundles", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-chern-filter-by-linebundles", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let ${\\mathcal L}_i$, $i = 1, \\ldots, r$ be invertible", "$\\mathcal{O}_X$-modules.", "Let $\\mathcal{E}$ be a locally free rank", "$\\mathcal{O}_X$-module endowed with a filtration", "$$", "0 = \\mathcal{E}_0 \\subset \\mathcal{E}_1 \\subset \\mathcal{E}_2", "\\subset \\ldots \\subset \\mathcal{E}_r = \\mathcal{E}", "$$", "such that $\\mathcal{E}_i/\\mathcal{E}_{i - 1} \\cong \\mathcal{L}_i$.", "Set $c_1({\\mathcal L}_i) = x_i$. Then", "$$", "c(\\mathcal{E})", "=", "\\prod\\nolimits_{i = 1}^r (1 + x_i)", "$$", "in $A^*(X)$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-additivity-invertible-subsheaf} and induction." ], "refs": [ "spaces-chow-lemma-additivity-invertible-subsheaf" ], "ref_ids": [ 7273 ] } ], "ref_ids": [] }, { "id": 7276, "type": "theorem", "label": "spaces-chow-lemma-splitting-principle", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-splitting-principle", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{E}_i$ be a finite collection of locally free", "$\\mathcal{O}_X$-modules of rank $r_i$. There exists a projective", "flat morphism $\\pi : P \\to X$ of relative dimension $d$ such that", "\\begin{enumerate}", "\\item for any morphism $f : Y \\to X$ of good algebraic spaces", "over $B$ the map", "$\\pi_Y^* : \\CH_*(Y) \\to \\CH_{* + d}(Y \\times_X P)$ is injective, and", "\\item each $\\pi^*\\mathcal{E}_i$ has a filtration", "whose successive quotients $\\mathcal{L}_{i, 1}, \\ldots, \\mathcal{L}_{i, r_i}$", "are invertible ${\\mathcal O}_P$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We prove this by induction on the integer $r = \\sum r_i$.", "If $r = 0$ we can take $\\pi = \\text{id}_X$.", "If $r_i = 1$ for all $i$, then we can also take $\\pi = \\text{id}_X$.", "Assume that $r_{i_0} > 1$ for some $i_0$.", "Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$", "be the projective bundle associated to $\\mathcal{E}_{i_0}$.", "The canonical map $\\pi^*\\mathcal{E}_{i_0} \\to \\mathcal{O}_P(1)$", "is surjective and hence its kernel $\\mathcal{E}'_{i_0}$", "is finite locally free of rank $r_{i_0} - 1$.", "Observe that $\\pi_Y^*$ is injective for any", "morphism $f : Y \\to X$ of good algebraic spaces", "over $B$, see Lemma \\ref{lemma-chow-ring-projective-bundle}.", "Thus it suffices to prove the lemma for", "$P$ and the locally free sheaves $\\pi^*\\mathcal{E}_i$.", "However, because we have the subbundle", "$\\mathcal{E}_{i_0} \\subset \\pi^*\\mathcal{E}_{i_0}$", "with invertible quotient, it now suffices to prove", "the lemma for the collection", "$\\{\\mathcal{E}_i\\}_{i \\not = i_0} \\cup \\{\\mathcal{E}'_{i_0}\\}$.", "This decreases $r$ by $1$ and we win by induction hypothesis." ], "refs": [ "spaces-chow-lemma-chow-ring-projective-bundle" ], "ref_ids": [ 7266 ] } ], "ref_ids": [] }, { "id": 7277, "type": "theorem", "label": "spaces-chow-lemma-chern-classes-dual", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-chern-classes-dual", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module", "with dual $\\mathcal{E}^\\vee$. Then", "$$", "c_i(\\mathcal{E}^\\vee) = (-1)^i c_i(\\mathcal{E})", "$$", "in $A^i(X)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a morphism $\\pi : P \\to X$ as in", "Lemma \\ref{lemma-splitting-principle}.", "By the injectivity of $\\pi^*$ (after any base change)", "it suffices to prove the relation between", "the Chern classes of $\\mathcal{E}$ and $\\mathcal{E}^\\vee$", "after pulling back to $P$. Thus we may assume there", "exist invertible $\\mathcal{O}_X$-modules", "${\\mathcal L}_i$, $i = 1, \\ldots, r$", "and a filtration", "$$", "0 = \\mathcal{E}_0 \\subset \\mathcal{E}_1 \\subset \\mathcal{E}_2", "\\subset \\ldots \\subset \\mathcal{E}_r = \\mathcal{E}", "$$", "such that $\\mathcal{E}_i/\\mathcal{E}_{i - 1} \\cong \\mathcal{L}_i$.", "Then we obtain the dual filtration", "$$", "0 = \\mathcal{E}_r^\\perp \\subset \\mathcal{E}_1^\\perp \\subset \\mathcal{E}_2^\\perp", "\\subset \\ldots \\subset \\mathcal{E}_0^\\perp = \\mathcal{E}^\\vee", "$$", "such that $\\mathcal{E}_{i - 1}^\\perp/\\mathcal{E}_i^\\perp \\cong", "\\mathcal{L}_i^{\\otimes -1}$.", "Set $x_i = c_1(\\mathcal{L}_i)$.", "Then $c_1(\\mathcal{L}_i^{\\otimes -1}) = - x_i$", "by Lemma \\ref{lemma-c1-cap-additive}.", "By Lemma \\ref{lemma-chern-filter-by-linebundles}", "we have", "$$", "c(\\mathcal{E}) = \\prod\\nolimits_{i = 1}^r (1 + x_i)", "\\quad\\text{and}\\quad", "c(\\mathcal{E}^\\vee) = \\prod\\nolimits_{i = 1}^r (1 - x_i)", "$$", "in $A^*(X)$. The result follows from a formal computation", "which we omit." ], "refs": [ "spaces-chow-lemma-splitting-principle", "spaces-chow-lemma-c1-cap-additive", "spaces-chow-lemma-chern-filter-by-linebundles" ], "ref_ids": [ 7276, 7237, 7275 ] } ], "ref_ids": [] }, { "id": 7278, "type": "theorem", "label": "spaces-chow-lemma-chern-classes-tensor-product", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-chern-classes-tensor-product", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{E}$ and $\\mathcal{F}$ be a finite locally free", "$\\mathcal{O}_X$-modules of ranks $r$ and $s$. Then we have", "$$", "c_1(\\mathcal{E} \\otimes \\mathcal{F})", "=", "r c_1(\\mathcal{F}) + s c_1(\\mathcal{E})", "$$", "$$", "c_2(\\mathcal{E} \\otimes \\mathcal{F})", "=", "r^2 c_2(\\mathcal{F}) +", "rs c_1(\\mathcal{F})c_1(\\mathcal{E}) +", "s^2 c_2(\\mathcal{E})", "$$", "and so on (see proof)." ], "refs": [], "proofs": [ { "contents": [ "Arguing exactly as in the proof of Lemma \\ref{lemma-chern-classes-dual}", "we may assume we have", "invertible $\\mathcal{O}_X$-modules", "${\\mathcal L}_i$, $i = 1, \\ldots, r$", "${\\mathcal N}_i$, $i = 1, \\ldots, s$", "filtrations", "$$", "0 = \\mathcal{E}_0 \\subset \\mathcal{E}_1 \\subset \\mathcal{E}_2", "\\subset \\ldots \\subset \\mathcal{E}_r = \\mathcal{E}", "\\quad\\text{and}\\quad", "0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset \\mathcal{F}_2", "\\subset \\ldots \\subset \\mathcal{F}_s = \\mathcal{F}", "$$", "such that $\\mathcal{E}_i/\\mathcal{E}_{i - 1} \\cong \\mathcal{L}_i$", "and such that $\\mathcal{F}_j/\\mathcal{F}_{j - 1} \\cong \\mathcal{N}_j$.", "Ordering pairs $(i, j)$ lexicographically", "we obtain a filtration", "$$", "0 \\subset \\ldots \\subset", "\\mathcal{E}_i \\otimes \\mathcal{F}_j", "+", "\\mathcal{E}_{i - 1} \\otimes \\mathcal{F}", "\\subset \\ldots \\subset \\mathcal{E} \\otimes \\mathcal{F}", "$$", "with successive quotients", "$$", "\\mathcal{L}_1 \\otimes \\mathcal{N}_1,", "\\mathcal{L}_1 \\otimes \\mathcal{N}_2,", "\\ldots,", "\\mathcal{L}_1 \\otimes \\mathcal{N}_s,", "\\mathcal{L}_2 \\otimes \\mathcal{N}_1,", "\\ldots,", "\\mathcal{L}_r \\otimes \\mathcal{N}_s", "$$", "By Lemma \\ref{lemma-chern-filter-by-linebundles}", "we have", "$$", "c(\\mathcal{E}) = \\prod (1 + x_i),", "\\quad", "c(\\mathcal{F}) = \\prod (1 + y_j),", "\\quad\\text{and}\\quad", "c(\\mathcal{F}) = \\prod (1 + x_i + y_j),", "$$", "in $A^*(X)$. The result follows from a formal computation", "which we omit." ], "refs": [ "spaces-chow-lemma-chern-classes-dual", "spaces-chow-lemma-chern-filter-by-linebundles" ], "ref_ids": [ 7277, 7275 ] } ], "ref_ids": [] }, { "id": 7279, "type": "theorem", "label": "spaces-chow-lemma-spell-out-degree-zero-cycle", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-spell-out-degree-zero-cycle", "contents": [ "Let $k$ be a field. Let $X$ be a proper algebraic space over $k$.", "Let $\\alpha = \\sum n_i[Z_i]$ be in $Z_0(X)$. Then", "$$", "\\deg(\\alpha) = \\sum n_i\\deg(Z_i)", "$$", "where $\\deg(Z_i)$ is the degree of $Z_i \\to \\Spec(k)$, i.e.,", "$\\deg(Z_i) = \\dim_k \\Gamma(Z_i, \\mathcal{O}_{Z_i})$." ], "refs": [], "proofs": [ { "contents": [ "This is the definition of proper pushforward", "(Definition \\ref{definition-proper-pushforward})." ], "refs": [ "spaces-chow-definition-proper-pushforward" ], "ref_ids": [ 7286 ] } ], "ref_ids": [] }, { "id": 7280, "type": "theorem", "label": "spaces-chow-lemma-degrees-and-numerical-intersections", "categories": [ "spaces-chow" ], "title": "spaces-chow-lemma-degrees-and-numerical-intersections", "contents": [ "Let $k$ be a field. Let $X$ be a proper algebraic space over $k$.", "Let $Z \\subset X$ be a closed subspace of dimension $d$.", "Let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible", "$\\mathcal{O}_X$-modules. Then", "$$", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) =", "\\deg(", "c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_1) \\cap [Z]_d)", "$$", "where the left hand side is defined in", "Spaces over Fields, Definition", "\\ref{spaces-over-fields-definition-intersection-number}." ], "refs": [ "spaces-over-fields-definition-intersection-number" ], "proofs": [ { "contents": [ "Let $Z_i \\subset Z$, $i = 1, \\ldots, t$ be the irreducible components", "of dimension $d$. Let $m_i$ be the multiplicity of $Z_i$ in $Z$. Then", "$[Z]_d = \\sum m_i[Z_i]$ and", "$c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_d) \\cap [Z]_d$", "is the sum of the cycles", "$m_i c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_d) \\cap [Z_i]$.", "Since we have a similar decomposition for", "$(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$ by", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-numerical-polynomial-leading-term}", "it suffices to prove the lemma in case $Z = X$", "is a proper integral algebraic space over $k$.", "\\medskip\\noindent", "By Chow's lemma there exists a proper morphism $f : X' \\to X$", "which is an isomorphism over a dense open $U \\subset X$", "such that $X'$ is a scheme. See More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-chow-noetherian-separated}.", "Then $X'$ is a proper scheme over $k$. After replacing $X'$", "by the scheme theoretic closure of $f^{-1}(U)$", "we may assume that $X'$ is integral. Then", "$$", "(f^*\\mathcal{L}_1 \\cdots f^*\\mathcal{L}_d \\cdot X') =", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot X)", "$$", "by Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-intersection-number-and-pullback}", "and we have", "$$", "f_*(c_1(f^*\\mathcal{L}_1) \\cap \\ldots \\cap c_1(f^*\\mathcal{L}_d) \\cap [Y]) =", "c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_d) \\cap [X]", "$$", "by Lemma \\ref{lemma-pushforward-cap-c1}. Thus we may replace $X$ by $X'$", "and assume that $X$ is a proper scheme over $k$. This case", "was proven in Chow Homology, Lemma", "\\ref{chow-lemma-degrees-and-numerical-intersections}." ], "refs": [ "spaces-over-fields-lemma-numerical-polynomial-leading-term", "spaces-more-morphisms-lemma-chow-noetherian-separated", "spaces-over-fields-lemma-intersection-number-and-pullback", "spaces-chow-lemma-pushforward-cap-c1", "chow-lemma-degrees-and-numerical-intersections" ], "ref_ids": [ 12877, 198, 12881, 7243, 5760 ] } ], "ref_ids": [ 12899 ] }, { "id": 7307, "type": "theorem", "label": "sdga-theorem-qis-into-dg-injective", "categories": [ "sdga" ], "title": "sdga-theorem-qis-into-dg-injective", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$", "be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$.", "For every differential graded $\\mathcal{A}$-module $\\mathcal{M}$ there", "exists a quasi-isomorphism $\\mathcal{M} \\to \\mathcal{I}$", "where $\\mathcal{I}$ is a graded injective and K-injective", "differential graded $\\mathcal{A}$-module. Moreover, the", "construction is functorial in $\\mathcal{M}$." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ and $\\mathcal{M}_r \\to \\mathcal{M}'_r$ be a set of", "morphisms of $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ found in", "Lemma \\ref{lemma-better-set-of-monos}.", "Let $M$ with transformation $\\text{id} \\to M$", "be as constructed in Lemma \\ref{lemma-functor-set-of-monos}", "using $R$ and $\\mathcal{M}_r \\to \\mathcal{M}'_r$.", "By transfinite induction on $\\alpha$ we define a sequence of functors", "$M_\\alpha$ and natural transformations $M_\\beta \\to M_\\alpha$", "for $\\alpha < \\beta$ by setting", "\\begin{enumerate}", "\\item $M_0 = \\text{id}$,", "\\item $M_{\\alpha + 1} = M \\circ M_\\alpha$ with natural transformation", "$M_\\beta \\to M_{\\alpha + 1}$ for $\\beta < \\alpha + 1$", "coming from the already constructed $M_\\beta \\to M_\\alpha$ and the", "maps $M_\\alpha \\to M \\circ M_\\alpha$ coming from $\\text{id} \\to M$, and", "\\item $M_\\alpha = \\colim_{\\beta < \\alpha} M_\\beta$ if $\\alpha$", "is a limit ordinal with the coprojections as transformations", "$M_\\beta \\to M_\\alpha$ for $\\alpha < \\beta$.", "\\end{enumerate}", "Observe that for every differential graded $\\mathcal{A}$-module the maps", "$\\mathcal{M} \\to M_\\beta(\\mathcal{M}) \\to M_\\alpha(\\mathcal{M})$", "are injective quasi-isomorphisms (as filtered colimits are exact).", "\\medskip\\noindent", "Recall that $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ is a Grothendieck", "abelian category. Thus by", "Injectives, Proposition \\ref{injectives-proposition-objects-are-small}", "(applied to the direct sum of $\\mathcal{M}_r$ for all $r \\in R$)", "there is a limit ordinal $\\alpha$ such that $\\mathcal{M}_r$ is $\\alpha$-small", "with respect to injections for every $r \\in R$.", "We claim that $\\mathcal{M} \\to M_\\alpha(\\mathcal{M})$ is", "the desired functorial embedding of $\\mathcal{M}$ into a", "graded injective K-injective module.", "\\medskip\\noindent", "Namely, any map $\\mathcal{M}_r \\to M_\\alpha(\\mathcal{M})$", "factors through $M_\\beta(\\mathcal{M})$ for some $\\beta < \\alpha$.", "However, by the construction of $M$ we see that this means", "that $\\mathcal{M}_r \\to M_{\\beta + 1}(\\mathcal{M}) = M(M_\\beta(\\mathcal{M}))$", "factors through $\\mathcal{M}'_r$. Since", "$M_\\beta(\\mathcal{M}) \\subset M_{\\beta + 1}(\\mathcal{M})", "\\subset M_\\alpha(\\mathcal{M})$ we get the desired factorizaton", "into $M_\\alpha(\\mathcal{M})$. We conclude by our choice of", "$R$ and $\\mathcal{M}_r \\to \\mathcal{M}'_r$ in", "Lemma \\ref{lemma-better-set-of-monos}." ], "refs": [ "sdga-lemma-better-set-of-monos", "sdga-lemma-functor-set-of-monos", "injectives-proposition-objects-are-small", "sdga-lemma-better-set-of-monos" ], "ref_ids": [ 7340, 7341, 7807, 7340 ] } ], "ref_ids": [] }, { "id": 7308, "type": "theorem", "label": "sdga-lemma-gm-abelian", "categories": [ "sdga" ], "title": "sdga-lemma-gm-abelian", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a graded $\\mathcal{O}$-algebra.", "The category $\\text{Mod}_\\mathcal{A}$ is an abelian category", "with the following properties", "\\begin{enumerate}", "\\item $\\text{Mod}_\\mathcal{A}$ has arbitrary direct sums,", "\\item $\\text{Mod}_\\mathcal{A}$ has arbitrary colimits,", "\\item filtered colimit in $\\text{Mod}_\\mathcal{A}$ are exact,", "\\item $\\text{Mod}_\\mathcal{A}$ has arbitrary products,", "\\item $\\text{Mod}_\\mathcal{A}$ has arbitrary limits.", "\\end{enumerate}", "The functor", "$$", "\\text{Mod}_\\mathcal{A} \\longrightarrow \\textit{Mod}(\\mathcal{O}),\\quad", "\\mathcal{M} \\longmapsto \\mathcal{M}^n", "$$", "sending a graded $\\mathcal{A}$-module to its $n$th term commutes", "with all limits and colimits." ], "refs": [], "proofs": [ { "contents": [ "Let us denote", "$\\text{gr}^n : \\text{Mod}_\\mathcal{A} \\to \\textit{Mod}(\\mathcal{O})$", "the functor in the statement of the lemma.", "Consider a homomorphism $f : \\mathcal{M} \\to \\mathcal{N}$", "of graded $\\mathcal{A}$-modules. The kernel", "and cokernel of $f$ as maps of graded $\\mathcal{O}$-modules", "are additionally endowed with multiplication maps as in", "Definition \\ref{definition-gm}. Hence these are also", "the kernel and cokernel in $\\text{Mod}_\\mathcal{A}$.", "Thus $\\text{Mod}_\\mathcal{A}$ is an abelian category", "and taking kernels and cokernels commutes with $\\text{gr}^n$.", "\\medskip\\noindent", "To prove the existence of limits and colimits it is sufficient", "to prove the existence of products and direct sums, see", "Categories, Lemmas \\ref{categories-lemma-limits-products-equalizers} and", "\\ref{categories-lemma-colimits-coproducts-coequalizers}.", "The same lemmas show that", "proving the commutation of limits and colimits with $\\text{gr}^n$", "follows if $\\text{gr}^n$ commutes with direct sums and products.", "\\medskip\\noindent", "Let $\\mathcal{M}_t$, $t \\in T$ be a set of graded $\\mathcal{A}$-modules.", "Then we can consider the graded $\\mathcal{A}$-module whose degree $n$", "term is $\\bigoplus_{t \\in T} \\mathcal{M}_t^n$ (with obvious multiplication", "maps). The reader easily verifies that this is a direct sum in", "$\\text{Mod}_\\mathcal{A}$. Similarly for products.", "\\medskip\\noindent", "Observe that $\\text{gr}^n$ is an exact functor for all $n$ and that", "a complex $\\mathcal{M}_1 \\to \\mathcal{M}_2 \\to \\mathcal{M}_3$", "of $\\text{Mod}_\\mathcal{A}$ is exact if and only if", "$\\text{gr}^n\\mathcal{M}_1 \\to \\text{gr}^n\\mathcal{M}_2 \\to", "\\text{gr}^n\\mathcal{M}_3$ is exact in $\\textit{Mod}(\\mathcal{O})$", "for all $n$. Hence we conclude that (3) holds as filtered", "colimits are exact in $\\textit{Mod}(\\mathcal{O})$;", "it is a Grothendieck abelian category, see", "Cohomology on Sites, Section \\ref{sites-cohomology-section-unbounded}." ], "refs": [ "sdga-definition-gm", "categories-lemma-limits-products-equalizers", "categories-lemma-colimits-coproducts-coequalizers" ], "ref_ids": [ 7370, 12213, 12214 ] } ], "ref_ids": [] }, { "id": 7309, "type": "theorem", "label": "sdga-lemma-tensor-hom-adjunction-gr", "categories": [ "sdga" ], "title": "sdga-lemma-tensor-hom-adjunction-gr", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$", "and $\\mathcal{B}$ be a sheaves of graded algebras on", "$(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{M}$ be a right", "graded $\\mathcal{A}$-module. Let $\\mathcal{N}$ be a", "graded $(\\mathcal{A}, \\mathcal{B})$-bimodule. Let $\\mathcal{L}$", "be a right graded $\\mathcal{B}$-module. With conventions as above", "we have", "$$", "\\Hom_{\\text{Mod}_\\mathcal{B}^{gr}}(", "\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}, \\mathcal{L}) =", "\\Hom_{\\text{Mod}_\\mathcal{A}^{gr}}(", "\\mathcal{M}, \\SheafHom_\\mathcal{B}^{gr}(\\mathcal{N}, \\mathcal{L}))", "$$", "and", "$$", "\\SheafHom_\\mathcal{B}^{gr}(", "\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}, \\mathcal{L}) =", "\\SheafHom_\\mathcal{A}^{gr}(", "\\mathcal{M}, \\SheafHom_\\mathcal{B}^{gr}(\\mathcal{N}, \\mathcal{L}))", "$$", "functorially in $\\mathcal{M}$, $\\mathcal{N}$, $\\mathcal{L}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This follows by interpreting both sides as", "$\\mathcal{A}$-bilinear graded maps", "$\\psi : \\mathcal{M} \\times \\mathcal{N} \\to \\mathcal{L}$", "which are $\\mathcal{B}$-linear on the right." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7310, "type": "theorem", "label": "sdga-lemma-adjunction-push-pull-gr", "categories": [ "sdga" ], "title": "sdga-lemma-adjunction-push-pull-gr", "contents": [ "In the situation above we have", "$$", "\\Hom_{\\text{Mod}_\\mathcal{B}^{gr}}(", "\\mathcal{N}, f_*\\mathcal{M}) =", "\\Hom_{\\text{Mod}_\\mathcal{A}^{gr}}(", "f^*\\mathcal{N}, \\mathcal{M})", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints: First prove that $f^{-1}$ and $f_*$ are adjoint", "as functors between $\\text{Mod}_\\mathcal{B}$ and", "$\\text{Mod}_{f^{-1}\\mathcal{B}}$ using the adjunction between", "$f^{-1}$ and $f_*$ on sheaves of abelian groups.", "Next, use the adjunction between base change and restriction", "given in Section \\ref{section-graded-bimodules}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7311, "type": "theorem", "label": "sdga-lemma-extension-by-zero-graded", "categories": [ "sdga" ], "title": "sdga-lemma-extension-by-zero-graded", "contents": [ "In the situation above we have", "$$", "\\Hom_{\\text{Mod}_\\mathcal{A}^{gr}}(", "j_!\\mathcal{M}, \\mathcal{N}) =", "\\Hom_{\\text{Mod}_{\\mathcal{A}_U}^{gr}}(", "\\mathcal{M}, j^*\\mathcal{N})", "$$" ], "refs": [], "proofs": [ { "contents": [ "By the discussion in", "Modules on Sites, Section \\ref{sites-modules-section-localize}", "the functors $j_!$ and $j^*$ on $\\mathcal{O}$-modules are adjoint.", "Thus if we only look at the $\\mathcal{O}$-module structures", "we know that", "$$", "\\Hom_{\\text{Mod}_\\mathcal{O}^{gr}}(", "j_!\\mathcal{M}, \\mathcal{N}) =", "\\Hom_{\\text{Mod}_{\\mathcal{O}_U}^{gr}}(", "\\mathcal{M}, j^*\\mathcal{N})", "$$", "Then one has to check that these identifications map the", "$\\mathcal{A}$-module maps on the left hand side to the", "$\\mathcal{A}_U$-module maps on the right hand side.", "To check this, given $\\mathcal{O}_U$-linear maps", "$f^n : \\mathcal{M}^n \\to j^*\\mathcal{N}^{n + d}$", "corresponding to $\\mathcal{O}$-linear maps", "$g^n : j_!\\mathcal{M}^n \\to \\mathcal{N}^{n + d}$", "it suffices to show that", "$$", "\\xymatrix{", "\\mathcal{M}^n \\otimes_{\\mathcal{O}_U} \\mathcal{A}_U^m", "\\ar[r]_{f^n \\otimes 1} \\ar[d] &", "j^*\\mathcal{N}^{n + d} \\otimes_{\\mathcal{O}_U} \\mathcal{A}_U^m \\ar[d] \\\\", "\\mathcal{M}^{n + m} \\ar[r]^{f^{n + m}} &", "j^*\\mathcal{N}^{n + m + d}", "}", "$$", "commutes if and only if", "$$", "\\xymatrix{", "j_!\\mathcal{M}^n \\otimes_\\mathcal{O} \\mathcal{A}^m", "\\ar[r]_{g^n \\otimes 1} \\ar[d] &", "\\mathcal{N}^{n + d} \\otimes_\\mathcal{O} \\mathcal{A}_U^m \\ar[d] \\\\", "j_!\\mathcal{M}^{n + m} \\ar[r]^{g^{n + m}} &", "\\mathcal{N}^{n + m + d}", "}", "$$", "commutes. However, we know that", "\\begin{align*}", "\\Hom_{\\mathcal{O}_U}(\\mathcal{M}^n \\otimes_{\\mathcal{O}_U} \\mathcal{A}_U^m,", "j^*\\mathcal{N}^{n + d + m})", "& =", "\\Hom_\\mathcal{O}(j_!(\\mathcal{M}^n \\otimes_{\\mathcal{O}_U} \\mathcal{A}_U^m),", "\\mathcal{N}^{n + d + m}) \\\\", "& =", "\\Hom_\\mathcal{O}(j_!\\mathcal{M}^n \\otimes_\\mathcal{O} \\mathcal{A}^m,", "\\mathcal{N}^{n + d + m})", "\\end{align*}", "by the already used", "Modules on Sites, Lemma \\ref{sites-modules-lemma-j-shriek-and-tensor}.", "We omit the verification that shows that the obstruction to the", "commutativity of the first diagram in the first group maps", "to the obstruction to the commutativity of the second diagram", "in the last group." ], "refs": [ "sites-modules-lemma-j-shriek-and-tensor" ], "ref_ids": [ 14197 ] } ], "ref_ids": [] }, { "id": 7312, "type": "theorem", "label": "sdga-lemma-tensor-with-extension-by-zero", "categories": [ "sdga" ], "title": "sdga-lemma-tensor-with-extension-by-zero", "contents": [ "In the situation above, let $\\mathcal{M}$ be a right graded", "$\\mathcal{A}_U$-module and let $\\mathcal{N}$ be a left graded", "$\\mathcal{A}$-module. Then", "$$", "j_!\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N} =", "j_!(\\mathcal{M} \\otimes_{\\mathcal{A}_U} \\mathcal{N}|_U)", "$$", "as graded $\\mathcal{O}$-modules functorially in $\\mathcal{M}$", "and $\\mathcal{N}$." ], "refs": [], "proofs": [ { "contents": [ "Recall that the degree $n$ component of", "$j_!\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}$ is the cokernel of", "the canonical map", "$$", "\\bigoplus\\nolimits_{r + s + t = n}", "j_!\\mathcal{M}^r \\otimes_\\mathcal{O}", "\\mathcal{A}^s \\otimes_\\mathcal{O}", "\\mathcal{N}^t", "\\longrightarrow", "\\bigoplus\\nolimits_{p + q = n}", "j_!\\mathcal{M}^p \\otimes_\\mathcal{O} \\mathcal{N}^q", "$$", "See Section \\ref{section-tensor-product}.", "By Modules on Sites, Lemma \\ref{sites-modules-lemma-j-shriek-and-tensor}", "this is the same thing as the cokernel of", "$$", "\\bigoplus\\nolimits_{r + s + t = n}", "j_!(\\mathcal{M}^r \\otimes_{\\mathcal{O}_U}", "\\mathcal{A}^s|_U \\otimes_{\\mathcal{O}_U}", "\\mathcal{N}^t|_U)", "\\longrightarrow", "\\bigoplus\\nolimits_{p + q = n}", "j_!(\\mathcal{M}^p \\otimes_{\\mathcal{O}_U} \\mathcal{N}^q|_U)", "$$", "and we win. An alternative proof would be to redo the Yoneda", "argument given in the proof of the lemma cited above." ], "refs": [ "sites-modules-lemma-j-shriek-and-tensor" ], "ref_ids": [ 14197 ] } ], "ref_ids": [] }, { "id": 7313, "type": "theorem", "label": "sdga-lemma-gm-grothendieck-abelian", "categories": [ "sdga" ], "title": "sdga-lemma-gm-grothendieck-abelian", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a graded $\\mathcal{O}$-algebra.", "The category $\\text{Mod}_\\mathcal{A}$ is a Grothendieck abelian category." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-gm-abelian} and the definition of a Grothendieck", "abelian category", "(Injectives, Definition \\ref{injectives-definition-grothendieck-conditions})", "it suffices to", "show that $\\text{Mod}_\\mathcal{A}$ has a generator. We claim that", "$$", "\\mathcal{G} = \\bigoplus\\nolimits_{k, U} j_{U!}\\mathcal{A}_U[k]", "$$", "is a generator where the sum is over all objects $U$ of $\\mathcal{C}$", "and $k \\in \\mathbf{Z}$. Indeed, given a graded $\\mathcal{A}$-module", "$\\mathcal{M}$ if there are no nonzero maps from $\\mathcal{G}$ to $\\mathcal{M}$,", "then we see that for all $k$ and $U$ we have", "$$", "\\Hom_{\\text{Mod}_\\mathcal{A}}(j_{U!}\\mathcal{A}_U[k], \\mathcal{M}) =", "\\Hom_{\\text{Mod}_{\\mathcal{A}_U}}(\\mathcal{A}_U[k], \\mathcal{M}|_U) =", "\\Gamma(U, \\mathcal{M}^{-k})", "$$", "is equal to zero. Hence $\\mathcal{M}$ is zero." ], "refs": [ "sdga-lemma-gm-abelian", "injectives-definition-grothendieck-conditions" ], "ref_ids": [ 7308, 7810 ] } ], "ref_ids": [] }, { "id": 7314, "type": "theorem", "label": "sdga-lemma-dgm-abelian", "categories": [ "sdga" ], "title": "sdga-lemma-dgm-abelian", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$ be a differential graded $\\mathcal{O}$-algebra.", "The category $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ is an abelian category", "with the following properties", "\\begin{enumerate}", "\\item $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ has arbitrary direct sums,", "\\item $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ has arbitrary colimits,", "\\item filtered colimit in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ are exact,", "\\item $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ has arbitrary products,", "\\item $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ has arbitrary limits.", "\\end{enumerate}", "The forgetful functor", "$$", "\\text{Mod}_{(\\mathcal{A}, \\text{d})}", "\\longrightarrow", "\\text{Mod}_\\mathcal{A}", "$$", "sending a differential graded $\\mathcal{A}$-module to its underlying", "graded module commutes with all limits and colimits." ], "refs": [], "proofs": [ { "contents": [ "Let us denote", "$F : \\text{Mod}_{(\\mathcal{A}, \\text{d})} \\to \\text{Mod}_\\mathcal{A}$", "the functor in the statement of the lemma. Observe that", "the category $\\text{Mod}_\\mathcal{A}$ has properties (1) -- (5), see", "Lemma \\ref{lemma-gm-abelian}.", "\\medskip\\noindent", "Consider a homomorphism $f : \\mathcal{M} \\to \\mathcal{N}$", "of graded $\\mathcal{A}$-modules. The kernel", "and cokernel of $f$ as maps of graded $\\mathcal{A}$-modules", "are additionally endowed with differentials as in", "Definition \\ref{definition-dgm}. Hence these are also", "the kernel and cokernel in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$.", "Thus $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ is an abelian category", "and taking kernels and cokernels commutes with $F$.", "\\medskip\\noindent", "To prove the existence of limits and colimits it is sufficient", "to prove the existence of products and direct sums, see", "Categories, Lemmas \\ref{categories-lemma-limits-products-equalizers} and", "\\ref{categories-lemma-colimits-coproducts-coequalizers}.", "The same lemmas show that", "proving the commutation of limits and colimits with $F$", "follows if $F$ commutes with direct sums and products.", "\\medskip\\noindent", "Let $\\mathcal{M}_t$, $t \\in T$ be a set of differential", "graded $\\mathcal{A}$-modules. Then we can consider the direct", "sum $\\bigoplus \\mathcal{M}_t$ as a graded $\\mathcal{A}$-module.", "Since the direct sum of graded modules is done termwise, it is", "clear that $\\bigoplus \\mathcal{M}_t$ comes endowed with a differential.", "The reader easily verifies that this is a direct sum in", "$\\text{Mod}_{(\\mathcal{A}, \\text{d})}$. Similarly for products.", "\\medskip\\noindent", "Observe that $F$ is an exact functor and that", "a complex $\\mathcal{M}_1 \\to \\mathcal{M}_2 \\to \\mathcal{M}_3$", "of $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ is exact if and only if", "$F(\\mathcal{M}_1) \\to F(\\mathcal{M}_2) \\to F(\\mathcal{M}_3)$", "is exact in $\\text{Mod}_\\mathcal{A}$. Hence we conclude that (3)", "holds as filtered colimits are exact in", "$\\textit{Mod}_\\mathcal{A})$." ], "refs": [ "sdga-lemma-gm-abelian", "sdga-definition-dgm", "categories-lemma-limits-products-equalizers", "categories-lemma-colimits-coproducts-coequalizers" ], "ref_ids": [ 7308, 7373, 12213, 12214 ] } ], "ref_ids": [] }, { "id": 7315, "type": "theorem", "label": "sdga-lemma-what-makes-a-bimodule-dg", "categories": [ "sdga" ], "title": "sdga-lemma-what-makes-a-bimodule-dg", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$", "and $\\mathcal{B}$ be a sheaves of differential graded algebras on", "$(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{N}$ be a right differential", "graded $\\mathcal{B}$-module. There is a $1$-to-$1$ correspondence", "between $(\\mathcal{A}, \\mathcal{B})$-bimodule structures on", "$\\mathcal{N}$ compatible with the given", "differential graded $\\mathcal{B}$-module structure and homomorphisms", "$$", "\\mathcal{A}", "\\longrightarrow", "\\SheafHom^{dg}_\\mathcal{B}(\\mathcal{N}, \\mathcal{N})", "$$", "of differential graded $\\mathcal{O}$-algebras." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7316, "type": "theorem", "label": "sdga-lemma-tensor-hom-adjunction-dg", "categories": [ "sdga" ], "title": "sdga-lemma-tensor-hom-adjunction-dg", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$", "and $\\mathcal{B}$ be a sheaves of differential graded algebras on", "$(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{M}$ be a right", "differential graded $\\mathcal{A}$-module. Let $\\mathcal{N}$ be a", "differential graded $(\\mathcal{A}, \\mathcal{B})$-bimodule. Let $\\mathcal{L}$", "be a right differential graded $\\mathcal{B}$-module. With conventions as above", "we have", "$$", "\\Hom_{\\text{Mod}_{(\\mathcal{B}, \\text{d})}^{dg}}(", "\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}, \\mathcal{L}) =", "\\Hom_{\\text{Mod}_{(\\mathcal{A}, \\text{d})}^{dg}}(", "\\mathcal{M}, \\SheafHom_\\mathcal{B}^{dg}(\\mathcal{N}, \\mathcal{L}))", "$$", "and", "$$", "\\SheafHom_\\mathcal{B}^{dg}(", "\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}, \\mathcal{L}) =", "\\SheafHom_\\mathcal{A}^{dg}(", "\\mathcal{M}, \\SheafHom_\\mathcal{B}^{dg}(\\mathcal{N}, \\mathcal{L}))", "$$", "functorially in $\\mathcal{M}$, $\\mathcal{N}$, $\\mathcal{L}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: On the graded level we have seen this is true in", "Lemma \\ref{lemma-tensor-hom-adjunction-gr}. Thus it suffices", "to check the isomorphisms are compatible with differentials", "which can be done by a computation on the level of local sections." ], "refs": [ "sdga-lemma-tensor-hom-adjunction-gr" ], "ref_ids": [ 7309 ] } ], "ref_ids": [] }, { "id": 7317, "type": "theorem", "label": "sdga-lemma-adjunction-push-pull-dg", "categories": [ "sdga" ], "title": "sdga-lemma-adjunction-push-pull-dg", "contents": [ "In the situation above we have", "$$", "\\Hom_{\\text{Mod}_{(\\mathcal{B}, \\text{d})}^{dg}}(", "\\mathcal{N}, f_*\\mathcal{M}) =", "\\Hom_{\\text{Mod}_{(\\mathcal{A}, \\text{d})}^{dg}}(", "f^*\\mathcal{N}, \\mathcal{M})", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints: This is true for the underlying graded categories", "by Lemma \\ref{lemma-adjunction-push-pull-gr}. A calculation shows", "that these isomorphisms are compatible with differentials." ], "refs": [ "sdga-lemma-adjunction-push-pull-gr" ], "ref_ids": [ 7310 ] } ], "ref_ids": [] }, { "id": 7318, "type": "theorem", "label": "sdga-lemma-extension-by-zero-dg", "categories": [ "sdga" ], "title": "sdga-lemma-extension-by-zero-dg", "contents": [ "In the situation above we have", "$$", "\\Hom_{\\text{Mod}_{(\\mathcal{A}, \\text{d})}^{dg}}(", "j_!\\mathcal{M}, \\mathcal{N}) =", "\\Hom_{\\text{Mod}_{(\\mathcal{A}_U, \\text{d})}^{dg}}(", "\\mathcal{M}, j^*\\mathcal{N})", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: We have seen in Lemma \\ref{lemma-extension-by-zero-graded}", "that the lemma is true on graded level. Thus all that needs to be", "checked is that the resulting isomorphism is compatible with differentials." ], "refs": [ "sdga-lemma-extension-by-zero-graded" ], "ref_ids": [ 7311 ] } ], "ref_ids": [] }, { "id": 7319, "type": "theorem", "label": "sdga-lemma-tensor-with-extension-by-zero-dg", "categories": [ "sdga" ], "title": "sdga-lemma-tensor-with-extension-by-zero-dg", "contents": [ "In the situation above, let $\\mathcal{M}$ be a right differential graded", "$\\mathcal{A}_U$-module and let $\\mathcal{N}$ be a left differential graded", "$\\mathcal{A}$-module. Then", "$$", "j_!\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N} =", "j_!(\\mathcal{M} \\otimes_{\\mathcal{A}_U} \\mathcal{N}|_U)", "$$", "as complexes of $\\mathcal{O}$-modules", "functorially in $\\mathcal{M}$ and $\\mathcal{N}$." ], "refs": [], "proofs": [ { "contents": [ "As graded modules, this follows from", "Lemma \\ref{lemma-tensor-with-extension-by-zero}.", "We omit the verification that this isomorphism", "is compatible with differentials." ], "refs": [ "sdga-lemma-tensor-with-extension-by-zero" ], "ref_ids": [ 7312 ] } ], "ref_ids": [] }, { "id": 7320, "type": "theorem", "label": "sdga-lemma-homotopy-direct-sums", "categories": [ "sdga" ], "title": "sdga-lemma-homotopy-direct-sums", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$.", "The homotopy category $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$", "has direct sums and products." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Just use the direct sums and products as in", "Lemma \\ref{lemma-dgm-abelian}. This works because we saw that", "these functors commute with the forgetful functor to the category", "of graded $\\mathcal{A}$-modules and because $\\prod$ and $\\bigoplus$", "are exact functors on the category of families of abelian groups." ], "refs": [ "sdga-lemma-dgm-abelian" ], "ref_ids": [ 7314 ] } ], "ref_ids": [] }, { "id": 7321, "type": "theorem", "label": "sdga-lemma-axioms-AB", "categories": [ "sdga" ], "title": "sdga-lemma-axioms-AB", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$.", "The differential graded category", "$\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})}$", "satisfies axioms (A) and (B) of", "Differential Graded Algebra, Section \\ref{dga-section-review}." ], "refs": [], "proofs": [ { "contents": [ "Suppose given differential graded $\\mathcal{A}$-modules", "$\\mathcal{M}$ and $\\mathcal{N}$. Consider the", "differential graded $\\mathcal{A}$-module $\\mathcal{M} \\oplus \\mathcal{N}$", "defined in the obvious manner. Then the coprojections", "$i : \\mathcal{M} \\to \\mathcal{M} \\oplus \\mathcal{N}$ and", "$j : \\mathcal{N} \\to \\mathcal{M} \\oplus \\mathcal{N}$ and the", "projections", "$p : \\mathcal{M} \\oplus \\mathcal{N} \\to \\mathcal{N}$ and", "$q : \\mathcal{M} \\oplus \\mathcal{N} \\to \\mathcal{M}$", "are morphisms of differential graded $\\mathcal{A}$-modules.", "Hence $i, j, p, q$ are homogeneous", "of degree $0$ and closed, i.e., $\\text{d}(i) = 0$, etc.", "Thus this direct sum is a differential graded sum in the sense of", "Differential Graded Algebra, Definition \\ref{dga-definition-dg-direct-sum}.", "This proves axiom (A).", "\\medskip\\noindent", "Axiom (B) was shown in Section \\ref{section-shift-dg}." ], "refs": [ "dga-definition-dg-direct-sum" ], "ref_ids": [ 13157 ] } ], "ref_ids": [] }, { "id": 7322, "type": "theorem", "label": "sdga-lemma-axiom-C", "categories": [ "sdga" ], "title": "sdga-lemma-axiom-C", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$.", "The differential graded category", "$\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})}$", "satisfies axiom (C) formulated in", "Differential Graded Algebra, Situation \\ref{dga-situation-ABC}." ], "refs": [], "proofs": [ { "contents": [ "Let $f : \\mathcal{K} \\to \\mathcal{L}$", "be a homomorphism of differential graded $\\mathcal{A}$-modules.", "By the above we have an admissible short exact sequence", "$$", "0 \\to \\mathcal{L} \\to C(f) \\to \\mathcal{K}[1] \\to 0", "$$", "To finish the proof we have to show that the boundary map", "$$", "\\delta : \\mathcal{K}[1] \\to \\mathcal{L}[1]", "$$", "associated to this (see discussion above) is equal to $f[1]$.", "For the section $s : \\mathcal{K}[1] \\to C(f)$ we use in degree", "$n$ the embeddding $\\mathcal{K}^{n + 1} \\to C(f)^n$. Then", "in degree $n$ the map $\\pi$ is given by the projections", "$C(f)^n \\to \\mathcal{L}^n$. Then finally we have to compute", "$$", "\\delta = \\pi \\circ \\text{d}_{C(f)} \\circ s", "$$", "(see discussion above). In matrix notation this is equal to", "$$", "\\left(", "\\begin{matrix}", "1 & 0", "\\end{matrix}", "\\right)", "\\left(", "\\begin{matrix}", "\\text{d}_\\mathcal{L} & f \\\\", "0 & -\\text{d}_\\mathcal{K}", "\\end{matrix}", "\\right)", "\\left(", "\\begin{matrix}", "0 \\\\", "1", "\\end{matrix}", "\\right) = f", "$$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7323, "type": "theorem", "label": "sdga-lemma-dgm-grothendieck-abelian", "categories": [ "sdga" ], "title": "sdga-lemma-dgm-grothendieck-abelian", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$ be a differential graded $\\mathcal{O}$-algebra.", "The category $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$", "is a Grothendieck abelian category." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-dgm-abelian} and the definition of a Grothendieck", "abelian category", "(Injectives, Definition \\ref{injectives-definition-grothendieck-conditions})", "it suffices to", "show that $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$", "has a generator. For every object $U$ of $\\mathcal{C}$ we denote", "$C_U$ the cone on the identity map $\\mathcal{A}_U \\to \\mathcal{A}_U$", "as in Remark \\ref{remark-cone-identity}. We claim that", "$$", "\\mathcal{G} = \\bigoplus\\nolimits_{k, U} j_{U!}C_U[k]", "$$", "is a generator where the sum is over all objects $U$ of $\\mathcal{C}$", "and $k \\in \\mathbf{Z}$. Indeed, given a", "differential graded $\\mathcal{A}$-module $\\mathcal{M}$", "if there are no nonzero maps from $\\mathcal{G}$ to $\\mathcal{M}$,", "then we see that for all $k$ and $U$ we have", "\\begin{align*}", "\\Hom_{\\text{Mod}_\\mathcal{A}}(j_{U!}C_U[k], \\mathcal{M}) \\\\", "& =", "\\Hom_{\\text{Mod}_{\\mathcal{A}_U}}(C_U[k], \\mathcal{M}|_U) \\\\", "& =", "\\{(x, y) \\in \\mathcal{M}^{-k}(U) \\times \\mathcal{M}^{-k - 1}(U) \\mid", "x = \\text{d}(y)\\}", "\\end{align*}", "is equal to zero. Hence $\\mathcal{M}$ is zero." ], "refs": [ "sdga-lemma-dgm-abelian", "injectives-definition-grothendieck-conditions", "sdga-remark-cone-identity" ], "ref_ids": [ 7314, 7810, 7385 ] } ], "ref_ids": [] }, { "id": 7324, "type": "theorem", "label": "sdga-lemma-supply-good", "categories": [ "sdga" ], "title": "sdga-lemma-supply-good", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. Let $U \\in \\Ob(\\mathcal{C})$.", "Then $j_!\\mathcal{A}_U$ is a good differential graded", "$\\mathcal{A}$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{N}$ be a left graded $\\mathcal{A}$-module.", "By Lemma \\ref{lemma-tensor-with-extension-by-zero} we have", "$$", "j_!\\mathcal{A}_U \\otimes_\\mathcal{A} \\mathcal{N} =", "j_!(\\mathcal{A}_U \\otimes_{\\mathcal{A}_U} \\mathcal{N}|_U) =", "j_!(\\mathcal{N}_U)", "$$", "as graded modules.", "Since both restriction to $U$ and $j_!$ are exact this proves", "condition (1). The same argument works for (2) using", "Lemma \\ref{lemma-tensor-with-extension-by-zero-dg}.", "\\medskip\\noindent", "Consider a morphism $(f, f^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}')", "\\to (\\Sh(\\mathcal{C}), \\mathcal{O})$", "of ringed topoi, a differential graded $\\mathcal{O}'$-algebra", "$\\mathcal{A}'$, and a map $\\varphi : f^{-1}\\mathcal{A} \\to \\mathcal{A}'$", "of differential graded $f^{-1}\\mathcal{O}$-algebras.", "We have to show that", "$$", "f^*j_!\\mathcal{A}_U =", "f^{-1}j_!\\mathcal{A}_U \\otimes_{f^{-1}\\mathcal{A}} \\mathcal{A}'", "$$", "satisfies (1) and (2) for the ringed topos", "$(\\Sh(\\mathcal{C}'), \\mathcal{O}')$ endowed with the", "sheaf of differential graded $\\mathcal{O}'$-algebras", "$\\mathcal{A}'$. To prove this we may replace", "$(\\Sh(\\mathcal{C}), \\mathcal{O})$ and", "$(\\Sh(\\mathcal{C}'), \\mathcal{O}')$ by equivalent ringed topoi.", "Thus by", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites}", "we may assume that $f$ comes from a morphism of sites", "$f : \\mathcal{C} \\to \\mathcal{C}'$ given by the continuous", "functor $u : \\mathcal{C} \\to \\mathcal{C}'$.", "In this case, set $U' = u(U)$ and denote", "$j' : \\Sh(\\mathcal{C}'/U') \\to \\Sh(\\mathcal{C}')$ the", "corresponding localization morphism.", "We obtain a commutative square of morphisms of ringed topoi", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}'/U'), \\mathcal{O}'_{U'})", "\\ar[rr]_{(j', (j')^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} & &", "(\\Sh(\\mathcal{C}'), \\mathcal{O}')", "\\ar[d]^{(f, f^\\sharp)} \\\\", "(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)", "\\ar[rr]^{(j, j^\\sharp)} & &", "(\\Sh(\\mathcal{C}), \\mathcal{O}).", "}", "$$", "and we have $f'_*(j')^{-1} = j^{-1}f_*$. See", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-localize-morphism-ringed-sites}.", "By uniqueness of adjoints we obtain", "$f^{-1}j_! = j'_!(f')^{-1}$. Thus we obtain", "\\begin{align*}", "f^*j_!\\mathcal{A}_U", "& =", "f^{-1}j_!\\mathcal{A}_U \\otimes_{f^{-1}\\mathcal{A}} \\mathcal{A}' \\\\", "& =", "j'_!(f')^{-1}\\mathcal{A}_U \\otimes_{f^{-1}\\mathcal{A}} \\mathcal{A}' \\\\", "& =", "j'_!\\left(", "(f')^{-1}\\mathcal{A}_U \\otimes_{f^{-1}\\mathcal{A}|_{U'}}", "\\mathcal{A}'|_{U'}\\right) \\\\", "& =", "j'_!\\mathcal{A}'_{U'}", "\\end{align*}", "The first equation is the definition of the pullback of $j_!\\mathcal{A}_U$", "to a differential graded module over $\\mathcal{A}'$.", "The second equation because $f^{-1}j_! = j'_!(f')^{-1}$.", "The third equation by Lemma \\ref{lemma-tensor-with-extension-by-zero-dg}", "applied to the ringed site $(\\mathcal{C}', f^{-1}\\mathcal{O})$ with", "sheaf of differential graded algebras $f^{-1}\\mathcal{A}$ and with", "differential graded modules $(f')^{-1}\\mathcal{A}_U$ on $\\mathcal{C}'/U'$", "and $\\mathcal{A}'$ on $\\mathcal{C}'$.", "The fourth equation holds because of course we have", "$(f')^{-1}\\mathcal{A}_U = f^{-1}\\mathcal{A}|_{U'}$.", "Hence we see that the pullback is another module of the", "same kind and we've proven conditions (1) and (2) for it above." ], "refs": [ "sdga-lemma-tensor-with-extension-by-zero", "sdga-lemma-tensor-with-extension-by-zero-dg", "sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites", "sites-modules-lemma-localize-morphism-ringed-sites", "sdga-lemma-tensor-with-extension-by-zero-dg" ], "ref_ids": [ 7312, 7319, 14145, 14174, 7319 ] } ], "ref_ids": [] }, { "id": 7325, "type": "theorem", "label": "sdga-lemma-good-admissible-ses", "categories": [ "sdga" ], "title": "sdga-lemma-good-admissible-ses", "contents": [ "et $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. Let", "$0 \\to \\mathcal{P} \\to \\mathcal{P}' \\to \\mathcal{P}'' \\to 0$", "be an admissible short exact sequence of differential graded", "$\\mathcal{A}$-modules. If two-out-of-three of these modules", "are good, so is the third." ], "refs": [], "proofs": [ { "contents": [ "For condition (1) this is immediate as the sequence is a direct sum", "at the graded level. For condition (2) note that for any", "left differential graded $\\mathcal{A}$-module, the sequence", "$$", "0 \\to ", "\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N} \\to", "\\mathcal{P}' \\otimes_\\mathcal{A} \\mathcal{N} \\to", "\\mathcal{P}'' \\otimes_\\mathcal{A} \\mathcal{N} \\to 0", "$$", "is an admissible short exact sequence of differential graded", "$\\mathcal{O}$-modules (since forgetting the differential", "the tensor product is just taken in the category of graded modules).", "Hence if two out of three are exact as complexes of $\\mathcal{O}$-modules,", "so is the third. Finally, the same argument shows that given", "a morphism $(f, f^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}')", "\\to (\\Sh(\\mathcal{C}), \\mathcal{O})$", "of ringed topoi, a differential graded $\\mathcal{O}'$-algebra", "$\\mathcal{A}'$, and a map $\\varphi : f^{-1}\\mathcal{A} \\to \\mathcal{A}'$", "of differential graded $f^{-1}\\mathcal{O}$-algebras", "we have that", "$$", "0 \\to f^*\\mathcal{P} \\to f^*\\mathcal{P}' \\to f^*\\mathcal{P}'' \\to 0", "$$", "is an admissible short exact sequence of differential graded", "$\\mathcal{A}'$-modules and the same argument as above applies here." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7326, "type": "theorem", "label": "sdga-lemma-good-direct-sum", "categories": [ "sdga" ], "title": "sdga-lemma-good-direct-sum", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. An arbitrary", "direct sum of good differential graded $\\mathcal{A}$-modules", "is good. A filtered colimit of good differential graded", "$\\mathcal{A}$-modules is good." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: direct sums and filtered colimits", "commute with tensor products and with pullbacks." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7327, "type": "theorem", "label": "sdga-lemma-good-quotient", "categories": [ "sdga" ], "title": "sdga-lemma-good-quotient", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{M}$", "be a differential graded $\\mathcal{A}$-module. There exists a homomorphism", "$\\mathcal{P} \\to \\mathcal{M}$ of differential graded $\\mathcal{A}$-modules", "with the following properties", "\\begin{enumerate}", "\\item $\\mathcal{P} \\to \\mathcal{M}$ is surjective,", "\\item $\\Ker(\\text{d}_\\mathcal{P}) \\to \\Ker(\\text{d}_\\mathcal{M})$", "is surjective, and", "\\item $\\mathcal{P}$ is good.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider triples $(U, k, x)$ where $U$ is an object of $\\mathcal{C}$,", "$k \\in \\mathbf{Z}$, and $x$ is a section of $\\mathcal{M}^k$ over $U$", "with $\\text{d}_\\mathcal{M}(x) = 0$. Then we obtain a unique morphism", "of differential graded $\\mathcal{A}_U$-modules", "$\\varphi_x : \\mathcal{A}_U[-k] \\to \\mathcal{M}|_U$", "mapping $1$ to $x$. This is adjoint to a morphism", "$\\psi_x : j_{U!}\\mathcal{A}_U[-k] \\to \\mathcal{M}$.", "Observe that $1 \\in \\mathcal{A}_U(U)$ corresponds to", "a section $1 \\in j_{U!}\\mathcal{A}_U[-k](U)$ of degree $k$", "whose differential is zero and which is mapped to $x$ by $\\psi_x$.", "Thus if we consider the map", "$$", "\\bigoplus\\nolimits_{(U, k, x)} j_{U!}\\mathcal{A}_U[-k]", "\\longrightarrow", "\\mathcal{M}", "$$", "then we will have conditions (2) and (3). Namely, the objects", "$j_{U!}\\mathcal{A}_U[-k]$ are good (Lemma \\ref{lemma-supply-good})", "and any direct sum of good objects is good (Lemma \\ref{lemma-good-direct-sum}).", "\\medskip\\noindent", "Next, consider triples $(U, k, x)$ where $U$ is an object of $\\mathcal{C}$,", "$k \\in \\mathbf{Z}$, and $x$ is a section of $\\mathcal{M}^k$ (not necessarily", "annihilated by the differential). Then we can consider", "the cone $C_U$ on the identity map $\\mathcal{A}_U \\to \\mathcal{A}_U$", "as in Remark \\ref{remark-cone-identity}. The element $x$ will determine a map", "$\\varphi_x : C_U[-k - 1] \\to \\mathcal{A}_U$, see", "Remark \\ref{remark-cone-identity}. Now, since we have", "an admissible short exact sequence", "$$", "0 \\to \\mathcal{A}_U \\to C_U \\to \\mathcal{A}_U[1] \\to 0", "$$", "we conclude that $j_{U!}C_U$ is a good module by", "Lemma \\ref{lemma-good-admissible-ses} and", "the already used Lemma \\ref{lemma-supply-good}.", "As above we conclude that the direct sum of the maps", "$\\psi_x : j_{U!}C_U \\to \\mathcal{M}$ adjoint to the $\\varphi_x$", "$$", "\\bigoplus\\nolimits_{(U, k, x)} j_{U!}C_U \\longrightarrow \\mathcal{M}", "$$", "is surjective. Taking the direct sum with the map produced", "in the first paragraph we conclude." ], "refs": [ "sdga-lemma-supply-good", "sdga-lemma-good-direct-sum", "sdga-remark-cone-identity", "sdga-remark-cone-identity", "sdga-lemma-good-admissible-ses", "sdga-lemma-supply-good" ], "ref_ids": [ 7324, 7326, 7385, 7385, 7325, 7324 ] } ], "ref_ids": [] }, { "id": 7328, "type": "theorem", "label": "sdga-lemma-free-graded-module-good", "categories": [ "sdga" ], "title": "sdga-lemma-free-graded-module-good", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a differential graded $\\mathcal{A}$-algebra.", "Let $\\mathcal{S}$ be a sheaf of graded sets on $\\mathcal{C}$.", "Then the free graded module $\\mathcal{A}[\\mathcal{S}]$", "on $\\mathcal{S}$ endowed with differential as in", "Remark \\ref{remark-sheaf-graded-sets}", "is a good differential graded $\\mathcal{A}$-module." ], "refs": [ "sdga-remark-sheaf-graded-sets" ], "proofs": [ { "contents": [ "Let $\\mathcal{N}$ be a left graded $\\mathcal{A}$-module.", "Then we have", "$$", "\\mathcal{A}[\\mathcal{S}] \\otimes_\\mathcal{A} \\mathcal{N} =", "\\mathcal{O}[\\mathcal{S}] \\otimes_\\mathcal{O} \\mathcal{N} =", "\\mathcal{N}[\\mathcal{S}]", "$$", "where $\\mathcal{N}[\\mathcal{S}$ is the", "graded $\\mathcal{O}$-module whose degree $n$ part", "is the sheaf associated to the presheaf", "$$", "U \\longmapsto", "\\bigoplus\\nolimits_{s \\in \\mathcal{S}(U)} s \\cdot \\mathcal{N}^{n - \\deg(s)}(U)", "$$", "It is clear that $\\mathcal{N} \\to \\mathcal{N}[\\mathcal{S}]$", "is an exact functor, hence $\\mathcal{A}[\\mathcal{S}$ is flat", "as a graded $\\mathcal{A}$-module. Next, suppose that $\\mathcal{N}$", "is a differential graded left $\\mathcal{A}$-module. Then we have", "$$", "H^*(\\mathcal{A}[\\mathcal{S}] \\otimes_\\mathcal{A} \\mathcal{N}) =", "H^*(\\mathcal{O}[\\mathcal{S}] \\otimes_\\mathcal{O} \\mathcal{N})", "$$", "as graded sheaves of $\\mathcal{O}$-modules, which by the flatness", "(over $\\mathcal{O})$ is equal to", "$$", "H^*(\\mathcal{N})[\\mathcal{S}]", "$$", "as a graded $\\mathcal{O}$-module. Hence if $\\mathcal{N}$ is acyclic,", "then $\\mathcal{A}[\\mathcal{S}] \\otimes_\\mathcal{A} \\mathcal{N}$", "is acyclic.", "\\medskip\\noindent", "Finally, consider a morphism", "$(f, f^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}')", "\\to (\\Sh(\\mathcal{C}), \\mathcal{O})$", "of ringed topoi, a differential graded $\\mathcal{O}'$-algebra", "$\\mathcal{A}'$, and a map $\\varphi : f^{-1}\\mathcal{A} \\to \\mathcal{A}'$", "of differential graded $f^{-1}\\mathcal{O}$-algebras.", "Then it is straightforward to see that", "$$", "f^*\\mathcal{A}[\\mathcal{S}] = \\mathcal{A}'[f^{-1}\\mathcal{S}]", "$$", "which finishes the proof that our module is good." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 7386 ] }, { "id": 7329, "type": "theorem", "label": "sdga-lemma-resolve", "categories": [ "sdga" ], "title": "sdga-lemma-resolve", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{M}$", "be a differential graded $\\mathcal{A}$-module. There exists a homomorphism", "$\\mathcal{P} \\to \\mathcal{M}$ of differential graded $\\mathcal{A}$-modules", "with the following properties", "\\begin{enumerate}", "\\item $\\mathcal{P} \\to \\mathcal{M}$ is a quasi-isomorphism, and", "\\item $\\mathcal{P}$ is good.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "Let $\\mathcal{S}_0$ be the sheaf of graded sets", "(Remark \\ref{remark-sheaf-graded-sets})", "whose degree $n$ part is $\\Ker(\\text{d}_\\mathcal{M}^n)$.", "Consider the homomorphism of differential graded modules", "$$", "\\mathcal{P}_0 = \\mathcal{A}[\\mathcal{S}_0] \\longrightarrow \\mathcal{M}", "$$", "where the left hand side is as in Remark \\ref{remark-sheaf-graded-sets}", "and the map sends a local section $s$ of $\\mathcal{S}_0$", "to the corresponding local section of $\\mathcal{M}^{\\deg(s)}$", "(which is in the kernel of the differential, so our map is", "a map of differential graded modules indeed). By construction the", "induced maps on cohomology sheaves $H^n(\\mathcal{P}_0) \\to H^n(\\mathcal{M})$", "are surjective. We are going to inductively construct maps", "$$", "\\mathcal{P}_0 \\to \\mathcal{P}_1 \\to \\mathcal{P}_2 \\to \\ldots \\to \\mathcal{M}", "$$", "Observe that of course $H^*(\\mathcal{P}_i) \\to H^*(\\mathcal{M})$", "will be surjective for all $i$.", "Given $\\mathcal{P}_i \\to \\mathcal{M}$ denote $\\mathcal{S}_{i + 1}$", "the sheaf of graded sets whose degree $n$ part is", "$$", "\\Ker(\\text{d}_{\\mathcal{P}_i}^{n + 1})", "\\times_{\\mathcal{M}^{n + 1}, \\text{d}}", "\\mathcal{M}^n", "$$", "Then we set", "$$", "\\mathcal{P}_{i + 1} = \\mathcal{P}_i \\oplus \\mathcal{A}[\\mathcal{S}_{i + 1}]", "$$", "as graded $\\mathcal{A}$-module with differential and map to", "$\\mathcal{M}$ defined as follows", "\\begin{enumerate}", "\\item for local sections of $\\mathcal{P}_i$ use the differential on", "$\\mathcal{P}_i$ and the given map to $\\mathcal{M}$,", "\\item for a local section $s = (p, m)$ of $\\mathcal{S}_{i + 1}$ we set", "$\\text{d}(s)$ equal to $p$ viewed as a section of $\\mathcal{P}_i$", "of degree $\\deg(s) + 1$ and we map $s$ to $m$ in $\\mathcal{M}$, and", "\\item extend the differential uniquely so that the Leibniz rule holds.", "\\end{enumerate}", "This makes sense because $\\text{d}(m)$ is the image of $p$ and", "$\\text{d}(p) = 0$.", "Finally, we set $\\mathcal{P} = \\colim \\mathcal{P}_i$ with the", "induced map to $\\mathcal{M}$.", "\\medskip\\noindent", "The map $\\mathcal{P} \\to \\mathcal{M}$ is a quasi-isomorphism:", "we have $H^n(\\mathcal{P}) = \\colim H^n(\\mathcal{P}_i)$", "and for each $i$ the map $H^n(\\mathcal{P}_i) \\to H^n(\\mathcal{M})$", "is surjective with kernel annihilated by the map", "$H^n(\\mathcal{P}_i) \\to H^n(\\mathcal{P}_{i + 1})$ by construction.", "Each $\\mathcal{P}_i$ is good because $\\mathcal{P}_0$ is good", "by Lemma \\ref{lemma-free-graded-module-good} and each $\\mathcal{P}_{i + 1}$", "is in the middle of the admissible short exact sequence", "$0 \\to \\mathcal{P}_i \\to \\mathcal{P}_{i + 1} \\to", "\\mathcal{A}[\\mathcal{S}_{i + 1}] \\to 0$", "whose outer terms are good by induction. Hence", "$\\mathcal{P}_{i + 1}$ is good by", "Lemma \\ref{lemma-good-admissible-ses}.", "Finally, we conclude that $\\mathcal{P}$ is good by", "Lemma \\ref{lemma-good-direct-sum}." ], "refs": [ "sdga-remark-sheaf-graded-sets", "sdga-remark-sheaf-graded-sets", "sdga-lemma-free-graded-module-good", "sdga-lemma-good-admissible-ses", "sdga-lemma-good-direct-sum" ], "ref_ids": [ 7386, 7386, 7328, 7325, 7326 ] } ], "ref_ids": [] }, { "id": 7330, "type": "theorem", "label": "sdga-lemma-acyclic-good", "categories": [ "sdga" ], "title": "sdga-lemma-acyclic-good", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{P}$ be a good", "acyclic right differential graded $\\mathcal{A}$-module.", "\\begin{enumerate}", "\\item for any differential graded left $\\mathcal{A}$-module", "$\\mathcal{N}$ the tensor product", "$\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}$ is acyclic,", "\\item for any morphism $(f, f^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}')", "\\to (\\Sh(\\mathcal{C}), \\mathcal{O})$", "of ringed topoi and any differential graded $\\mathcal{O}'$-algebra", "$\\mathcal{A}'$ and any map $\\varphi : f^{-1}\\mathcal{A} \\to \\mathcal{A}'$", "of differential graded $f^{-1}\\mathcal{O}$-algebras", "the pullback $f^*\\mathcal{P}$ is acyclic and good.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). By Lemma \\ref{lemma-resolve} we can choose a good", "left differential graded $\\mathcal{Q}$ and a quasi-isomorphism", "$\\mathcal{Q} \\to \\mathcal{N}$. Then", "$\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{Q}$", "is acyclic because $\\mathcal{Q}$ is good.", "Let $\\mathcal{N}'$ be the cone on the map", "$\\mathcal{Q} \\to \\mathcal{N}$. Then", "$\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}'$ is acyclic because", "$\\mathcal{P}$ is good and because $\\mathcal{N}'$ is acyclic", "(as the cone on a quasi-isomorphism). We have a distinguished", "triangle", "$$", "\\mathcal{Q} \\to \\mathcal{N} \\to \\mathcal{N}' \\to \\mathcal{Q}[1]", "$$", "in $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$ by our construction", "of the triangulated structure. Since", "$\\mathcal{P} \\otimes_\\mathcal{A} -$ sends distinguished", "triangles to distinguished triangles, we obtain a distinguished", "triangle", "$$", "\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{Q} \\to", "\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N} \\to", "\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}' \\to", "\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{Q}[1]", "$$", "in $K(\\textit{Mod}(\\mathcal{O}))$. Thus we conclude.", "\\medskip\\noindent", "Proof of (2). Observe that $f^*\\mathcal{P}$ is good by our definition", "of good modules. Recall that", "$f^*\\mathcal{P} = f^{-1}\\mathcal{P} \\otimes_{f^{-1}\\mathcal{A}} \\mathcal{A}'$.", "Then $f^{-1}\\mathcal{P}$ is a good acyclic (because $f^{-1}$ is exact)", "differential graded $f^{-1}\\mathcal{A}$-module. Hence", "we see that $f^*\\mathcal{P}$ is acyclic by part (1)." ], "refs": [ "sdga-lemma-resolve" ], "ref_ids": [ 7329 ] } ], "ref_ids": [] }, { "id": 7331, "type": "theorem", "label": "sdga-lemma-dg-hull", "categories": [ "sdga" ], "title": "sdga-lemma-dg-hull", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. The forgetful functor", "$F : \\text{Mod}_{(\\mathcal{A}, \\text{d})} \\to \\text{Mod}_\\mathcal{A}$", "has a left adjoint $G : \\text{Mod}_\\mathcal{A} \\to", "\\text{Mod}_{(\\mathcal{A}, \\text{d})}$." ], "refs": [], "proofs": [ { "contents": [ "To prove the existence of $G$ we can use the adjoint functor theorem, see", "Categories, Theorem \\ref{categories-theorem-adjoint-functor} (observe that", "we have switched the roles of $F$ and $G$). The exactness conditions on", "$F$ are satisfied by Lemma \\ref{lemma-dgm-abelian}. The set theoretic", "condition can be seen as follows: suppose given a graded $\\mathcal{A}$-module", "$\\mathcal{N}$. Then for any map", "$$", "\\varphi : \\mathcal{N} \\longrightarrow F(\\mathcal{M})", "$$", "we can consider the smallest differential graded $\\mathcal{A}$-submodule", "$\\mathcal{M}' \\subset \\mathcal{M}$ with", "$\\Im(\\varphi) \\subset F(\\mathcal{M}')$.", "It is clear that $\\mathcal{M}'$ is the image of the map", "of graded $\\mathcal{A}$-modules", "$$", "\\mathcal{N} \\oplus", "\\mathcal{N}[-1] \\otimes_\\mathcal{O} \\mathcal{A}", "\\longrightarrow", "\\mathcal{M}", "$$", "defined by", "$$", "(n, \\sum n_i \\otimes a_i) \\longmapsto", "\\varphi(n) + \\sum \\text{d}(\\varphi(n_i)) a_i", "$$", "because the image of this map is easily seen to be a differential graded", "submodule of $\\mathcal{M}$.", "Thus the number of possible isomorphism classes of these $\\mathcal{M}'$", "is bounded and we conclude." ], "refs": [ "categories-theorem-adjoint-functor", "sdga-lemma-dgm-abelian" ], "ref_ids": [ 12200, 7314 ] } ], "ref_ids": [] }, { "id": 7332, "type": "theorem", "label": "sdga-lemma-dg-hull-acyclic", "categories": [ "sdga" ], "title": "sdga-lemma-dg-hull-acyclic", "contents": [ "The functors $F, G$ of Lemma \\ref{lemma-dg-hull} have", "the following properties. Given a graded $\\mathcal{A}$-module", "$\\mathcal{N}$ we have", "\\begin{enumerate}", "\\item the counit $\\mathcal{N} \\to F(G(\\mathcal{N}))$ is injective,", "\\item the map $\\overline{\\text{d}} : \\mathcal{N} \\to", "\\Coker(\\mathcal{N} \\to F(G(\\mathcal{N})))[1]$ is an isomorphism, and", "\\item $G(\\mathcal{N})$ is an acyclic differential graded $\\mathcal{A}$-module.", "\\end{enumerate}" ], "refs": [ "sdga-lemma-dg-hull" ], "proofs": [ { "contents": [ "We observe that property (3) is a consequence of properties (1) and (2).", "Namely, if $s$ is a nonzero local section of $F(G(\\mathcal{N}))$", "with $\\text{d}(s) = 0$, then $s$ cannot be in the image of", "$\\mathcal{N} \\to F(G(\\mathcal{N}))$. Hence we can write the image", "$\\overline{s}$ of $s$ in the cokernel as", "$\\overline{\\text{d}}(s')$ for some local section $s'$ of $\\mathcal{N}$.", "Then we see that $s = \\text{d}(s')$ because the difference", "$s - \\text{d}(s')$ is still in the kernel of $\\text{d}$", "and is contained in the image of the counit.", "\\medskip\\noindent", "Let us write temporarily $\\mathcal{A}_{gr}$, respectively $\\mathcal{A}_{dg}$", "the sheaf $\\mathcal{A}$ viewed as a (right) graded module over itself,", "respectively as a (right) differential graded module over itself.", "The most important case of the lemma is to understand what", "is $G(\\mathcal{A}_{gr})$. Of course $G(\\mathcal{A}_{gr})$ is the object", "of $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ representing the", "functor", "$$", "\\mathcal{M} \\longmapsto", "\\Hom_{\\text{Mod}_\\mathcal{A}}(\\mathcal{A}_{gr}, F(\\mathcal{M})) =", "\\Gamma(\\mathcal{C}, \\mathcal{M})", "$$", "By Remark \\ref{remark-cone-identity} we see that this functor represented", "by $C[-1]$ where $C$ is the cone on the identity of $\\mathcal{A}_{dg}$.", "We have a short exact sequence", "$$", "0 \\to \\mathcal{A}_{dg}[-1] \\to C[-1] \\to \\mathcal{A}_{dg} \\to 0", "$$", "in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ which is split by the", "counit $\\mathcal{A}_{gr} \\to F(C[-1])$ in $\\text{Mod}_\\mathcal{A}$.", "Thus $G(\\mathcal{A}_{gr})$ satisfies properties (1) and (2).", "\\medskip\\noindent", "Let $U$ be an object of $\\mathcal{C}$. Denote", "$j_U : \\mathcal{C}/U \\to \\mathcal{C}$ the localization morphism.", "Denote $\\mathcal{A}_U$ the restriction of $\\mathcal{A}$ to $U$.", "We will use the notation $\\mathcal{A}_{U, gr}$ to denote", "$\\mathcal{A}_U$ viewed as a graded $\\mathcal{A}_U$-module.", "Denote $F_U : \\text{Mod}_{(\\mathcal{A}_U, \\text{d})} \\to", "\\text{Mod}_{\\mathcal{A}_U}$ the forgetful functor and denote", "$G_U$ its adjoint. Then we have the commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "\\text{Mod}_{(\\mathcal{A}, \\text{d})} \\ar[d]_{j_U^*} \\ar[r]_F &", "\\text{Mod}_\\mathcal{A} \\ar[d]^{j_U^*} \\\\", "\\text{Mod}_{(\\mathcal{A}_U, \\text{d})} \\ar[r]^{F_U} &", "\\text{Mod}_{\\mathcal{A}_U}", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "\\text{Mod}_{(\\mathcal{A}_U, \\text{d})} \\ar[r]_{F_U} \\ar[d]_{j_{U!}} &", "\\text{Mod}_{\\mathcal{A}_U} \\ar[d]^{j_{U!}} \\\\", "\\text{Mod}_{(\\mathcal{A}, \\text{d})} \\ar[r]^F &", "\\text{Mod}_\\mathcal{A}", "}", "}", "$$", "by the construction of $j^*_U$ and $j_{U!}$ in", "Sections \\ref{section-functoriality-graded},", "\\ref{section-functoriality-dg},", "\\ref{section-localize-graded}, and", "\\ref{section-localize-dg}.", "By uniqueness of adjoints we obtain $j_{U!} \\circ G_U = G \\circ j_{U!}$.", "Since $j_{U!}$ is an exact functor, we see that the properties", "(1) and (2) for the counit", "$\\mathcal{A}_{U, gr} \\to F_U(G_U(\\mathcal{A}_{U, gr}))$", "which we've seen in the previous part of the proof", "imply properties (1) and (2) for the counit", "$j_{U!}\\mathcal{A}_{U, gr} \\to F(G(j_{U!}\\mathcal{A}_{U, gr})) =", "j_{U!}F_U(G_U(\\mathcal{A}_{U, gr}))$.", "\\medskip\\noindent", "In the proof of Lemma \\ref{lemma-gm-grothendieck-abelian}", "we have seen that any object", "of $\\text{Mod}_\\mathcal{A}$ is a quotient of a direct sum", "of copies of $j_{U!}\\mathcal{A}_{U, gr}$. Since $G$ is a left", "adjoint, we see that $G$ commutes with direct sums. Thus", "properties (1) and (2) hold for direct sums of objects", "for which they hold. Thus we see that every object $\\mathcal{N}$", "of $\\text{Mod}_\\mathcal{A}$ fits into an exact sequence", "$$", "\\mathcal{N}_1 \\to \\mathcal{N}_0 \\to \\mathcal{N} \\to 0", "$$", "such that (1) and (2) hold for $\\mathcal{N}_1$ and $\\mathcal{N}_0$.", "We leave it to the reader to deduce (1) and (2) for", "$\\mathcal{N}$ using that $G$ is right exact." ], "refs": [ "sdga-remark-cone-identity", "sdga-lemma-gm-grothendieck-abelian" ], "ref_ids": [ 7385, 7313 ] } ], "ref_ids": [ 7331 ] }, { "id": 7333, "type": "theorem", "label": "sdga-lemma-characterize-injectives", "categories": [ "sdga" ], "title": "sdga-lemma-characterize-injectives", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$", "be a sheaf of graded algebras on $(\\mathcal{C}, \\mathcal{O})$.", "There exists a set $T$ and for each $t \\in T$ an injective map", "$\\mathcal{N}_t \\to \\mathcal{N}'_t$ of graded $\\mathcal{A}$-modules", "such that an object $\\mathcal{I}$ of $\\text{Mod}_\\mathcal{A}$", "is injective if and only if for every solid diagram", "$$", "\\xymatrix{", "\\mathcal{N}_t \\ar[r] \\ar[d] & \\mathcal{I} \\\\", "\\mathcal{N}'_t \\ar@{..>}[ru]", "}", "$$", "a dotted arrow exists in $\\text{Mod}_\\mathcal{A}$ making the diagram commute." ], "refs": [], "proofs": [ { "contents": [ "This is true in any Grothendieck abelian category, see", "Injectives, Lemma \\ref{injectives-lemma-characterize-injective}.", "By Lemma \\ref{lemma-gm-grothendieck-abelian} the category", "$\\text{Mod}_\\mathcal{A}$ is a Grothendieck abelian category." ], "refs": [ "injectives-lemma-characterize-injective", "sdga-lemma-gm-grothendieck-abelian" ], "ref_ids": [ 7787, 7313 ] } ], "ref_ids": [] }, { "id": 7334, "type": "theorem", "label": "sdga-lemma-product-graded-injective", "categories": [ "sdga" ], "title": "sdga-lemma-product-graded-injective", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. Let $T$ be a set and for", "each $t \\in T$ let $\\mathcal{I}_t$ be a graded injective", "diffential graded $\\mathcal{A}$-module. Then", "$\\prod \\mathcal{I}_t$ is a graded injective differential", "graded $\\mathcal{A}$-module." ], "refs": [], "proofs": [ { "contents": [ "This is true because products of injectives are injectives, see", "Homology, Lemma \\ref{homology-lemma-product-injectives}, and", "because products in", "$\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ are compatible with", "products in $\\text{Mod}_\\mathcal{A}$ via the forgetful functor." ], "refs": [ "homology-lemma-product-injectives" ], "ref_ids": [ 12113 ] } ], "ref_ids": [] }, { "id": 7335, "type": "theorem", "label": "sdga-lemma-characterize-graded-injectives-in-dg", "categories": [ "sdga" ], "title": "sdga-lemma-characterize-graded-injectives-in-dg", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$", "be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$.", "There exists a set $T$ and for each $t \\in T$ an injective map", "$\\mathcal{M}_t \\to \\mathcal{M}'_t$ of", "acyclic differential graded $\\mathcal{A}$-modules", "such that for an object $\\mathcal{I}$ of $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$", "the following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{I}$ is graded injective, and", "\\item for every solid diagram", "$$", "\\xymatrix{", "\\mathcal{M}_t \\ar[r] \\ar[d] & \\mathcal{I} \\\\", "\\mathcal{M}'_t \\ar@{..>}[ru]", "}", "$$", "a dotted arrow exists in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$", "making the diagram commute.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $T$ and $\\mathcal{N}_t \\to \\mathcal{N}'_t$ be as in", "Lemma \\ref{lemma-characterize-injectives}.", "Denote $F : \\text{Mod}_{(\\mathcal{A}, \\text{d})} \\to \\text{Mod}_\\mathcal{A}$", "the forgetful functor.", "Let $G$ be the left adjoint functor to", "$F$ as in Lemma \\ref{lemma-dg-hull}. Set", "$$", "\\mathcal{M}_t = G(\\mathcal{N}_t) \\to", "G(\\mathcal{N}'_t) = \\mathcal{M}'_t", "$$", "This is an injective map of acyclic differential graded", "$\\mathcal{A}$-modules by Lemma \\ref{lemma-dg-hull-acyclic}.", "Since $G$ is the left adjoint to $F$ we see that", "there exists a dotted arrow in the diagram", "$$", "\\xymatrix{", "\\mathcal{M}_t \\ar[r] \\ar[d] & \\mathcal{I} \\\\", "\\mathcal{M}'_t \\ar@{..>}[ru]", "}", "$$", "if and only if there exists a dotted arrow in the diagram", "$$", "\\xymatrix{", "\\mathcal{N}_t \\ar[r] \\ar[d] & F(\\mathcal{I}) \\\\", "\\mathcal{N}'_t \\ar@{..>}[ru]", "}", "$$", "Hence the result follows from the choice of our", "collection of arrows $\\mathcal{N}_t \\to \\mathcal{N}_t'$." ], "refs": [ "sdga-lemma-characterize-injectives", "sdga-lemma-dg-hull", "sdga-lemma-dg-hull-acyclic" ], "ref_ids": [ 7333, 7331, 7332 ] } ], "ref_ids": [] }, { "id": 7336, "type": "theorem", "label": "sdga-lemma-small-acyclics", "categories": [ "sdga" ], "title": "sdga-lemma-small-acyclics", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. There exists a set $S$ and for each $s$", "an acyclic differential graded $\\mathcal{A}$-module $\\mathcal{M}_s$ such", "that for every nonzero acyclic differential graded $\\mathcal{A}$-module", "$\\mathcal{M}$ there is an $s \\in S$ and an injective map", "$\\mathcal{M}_s \\to \\mathcal{M}$ in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$." ], "refs": [], "proofs": [ { "contents": [ "Before we start recall that our conventions guarantee the site $\\mathcal{C}$", "has a set of objects and morphisms and a set $\\text{Cov}(\\mathcal{C})$", "of coverings.", "If $\\mathcal{F}$", "is a differential graded $\\mathcal{A}$-module, let us define $|\\mathcal{F}|$", "to be the sum of the cardinality of", "$$", "\\coprod\\nolimits_{(U, n)} \\mathcal{F}^n(U)", "$$", "as $U$ ranges over the objects of $\\mathcal{C}$ and $n \\in \\mathbf{Z}$.", "Choose an infinite cardinal $\\kappa$ bigger than", "the cardinals $|\\Ob(\\mathcal{C})|$, $|\\text{Arrows}(\\mathcal{C})|$,", "$|\\text{Cov}(\\mathcal{C})|$, $\\sup |I|$ for", "$\\{U_i \\to U\\}_{i \\in I} \\in \\text{Cov}(\\mathcal{C})$,", "and $|\\mathcal{A}|$.", "\\medskip\\noindent", "Let $\\mathcal{F} \\subset \\mathcal{M}$ be an inclusion of", "differential graded $\\mathcal{A}$-modules.", "Suppose given a set $K$ and for each $k \\in K$ a triple", "$(U_k, n_k, x_k)$ consisting of an object $U_k$ of $\\mathcal{C}$,", "integer $n_k$, and a section $x_k \\in \\mathcal{M}^{n_k}(U_k)$.", "Then we can consider the smallest differential graded", "$\\mathcal{A}$-submodule $\\mathcal{F}' \\subset \\mathcal{M}$", "containing $\\mathcal{F}$ and the sections $x_k$ for $k \\in K$.", "We can describe", "$$", "(\\mathcal{F}')^n(U) \\subset \\mathcal{M}^n(U)", "$$", "as the set of elements $x \\in \\mathcal{M}^n(U)$ such that there", "exists $\\{f_i : U_i \\to U\\}_{i \\in I} \\in \\text{Cov}(\\mathcal{C})$ such that", "for each $i \\in I$ there is a finite set $T_i$ and morphisms", "$g_t : U_i \\to U_{k_t}$", "$$", "f_i^*x = y_i +", "\\sum\\nolimits_{t \\in T_i} a_{it}g_t^*x_{k_t} + b_{it}g_t^*\\text{d}(x_{k_t})", "$$", "for some section $y_i \\in \\mathcal{F}^n(U)$ and", "sections $a_{it} \\in \\mathcal{A}^{n - n_{k_t}}(U_i)$", "and $b_{it} \\in \\mathcal{A}^{n - n_{k_t} - 1}(U_i)$.", "(Details omitted; hints: these sections are certainly in $\\mathcal{F}'$", "and you show conversely that this rule defines a differential", "graded $\\mathcal{A}$-submodule.)", "It follows from this description that", "$|\\mathcal{F}'| \\leq \\max(|\\mathcal{F}|, |K|, \\kappa)$.", "\\medskip\\noindent", "Let $\\mathcal{M}$ be a nonzero acyclic differential graded", "$\\mathcal{A}$-module. Then we can find an integer $n$ and a", "nonzero section $x$ of $\\mathcal{M}^n$ over some object $U$", "of $\\mathcal{C}$. Let", "$$", "\\mathcal{F}_0 \\subset \\mathcal{M}", "$$", "be the smallest differential graded $\\mathcal{A}$-submodule", "containing $x$. By the previous paragraph we have", "$|\\mathcal{F}_0| \\leq \\kappa$. By induction, given", "$\\mathcal{F}_0, \\ldots, \\mathcal{F}_n$ define", "$\\mathcal{F}_{n + 1}$ as follows. Consider the set", "$$", "L = \\{(U, n, x)\\}", "\\{U_i \\to U\\}_{i \\in I}, (x_i)_{i \\in I})\\}", "$$", "of triples where $U$ is an object of $\\mathcal{C}$, $n \\in \\mathbf{Z}$,", "and $x \\in \\mathcal{F}_n(U)$ with $\\text{d}(x) = 0$. Since", "$\\mathcal{M}$ is acyclic for each triple $l = (U_l, n_l, x_l) \\in L$", "we can choose", "$\\{(U_{l, i} \\to U_l\\}_{i \\in I_l} \\in \\text{Cov}(\\mathcal{C})$ and", "$x_{l, i} \\in \\mathcal{M}^{n_l - 1}(U_{l, i})$ such that", "$\\text{d}(x_{l, i}) = x|_{U_{l, i}}$. Then we set", "$$", "K = \\{(U_{l, i}, n_l - 1, x_{l, i}) \\mid l \\in L, i \\in I_l\\}", "$$", "and we let $\\mathcal{F}_{n + 1}$ be the smallest differential", "graded $\\mathcal{A}$-submodule of $\\mathcal{M}$ containing", "$\\mathcal{F}_n$ and the sections $x_{l, i}$.", "Since $|K| \\leq \\max(\\kappa, |\\mathcal{F}_n|)$", "we conclude that $|\\mathcal{F}_{n + 1}| \\leq \\kappa$ by induction.", "\\medskip\\noindent", "By construction the inclusion $\\mathcal{F}_n \\to \\mathcal{F}_{n + 1}$", "induces the zero map on cohomology sheaves. Hence we see that", "$\\mathcal{F} = \\bigcup \\mathcal{F}_n$ is a nonzero acyclic submodule", "with $|\\mathcal{F}| \\leq \\kappa$. Since there is only a set", "of isomorphism classes of differential graded $\\mathcal{A}$-modules", "$\\mathcal{F}$ with $|\\mathcal{F}|$ bounded, we conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7337, "type": "theorem", "label": "sdga-lemma-product-K-injective", "categories": [ "sdga" ], "title": "sdga-lemma-product-K-injective", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. Let $T$ be a set and for", "each $t \\in T$ let $\\mathcal{I}_t$ be a K-injective", "diffential graded $\\mathcal{A}$-module. Then", "$\\prod \\mathcal{I}_t$ is a K-injective differential", "graded $\\mathcal{A}$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{K}$ be an acyclic differential graded $\\mathcal{A}$-module.", "Then we have", "$$", "\\Hom_{\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})}}(\\mathcal{K},", "\\prod\\nolimits_{t \\in T} \\mathcal{I}_t)", "=", "\\prod\\nolimits_{t \\in T}", "\\Hom_{\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})}}(\\mathcal{K}, \\mathcal{I}_t)", "$$", "because taking products in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$", "commutes with the forgetful functor to graded $\\mathcal{A}$-modules.", "Since taking products is an exact functor on the category of abelian groups", "we conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7338, "type": "theorem", "label": "sdga-lemma-first-property-dg-injective", "categories": [ "sdga" ], "title": "sdga-lemma-first-property-dg-injective", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$", "be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$.", "Let $\\mathcal{I}$ be a K-injective and graded injective", "object of $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$.", "For every solid diagram in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$", "$$", "\\xymatrix{", "\\mathcal{M} \\ar[r]_a \\ar[d]_b & \\mathcal{I} \\\\", "\\mathcal{M}' \\ar@{..>}[ru]", "}", "$$", "where $b$ is injective and $\\mathcal{M}$ is acyclic", "a dotted arrow exists making the diagram commute." ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathcal{M}$ is acyclic and $\\mathcal{I}$ is K-injective,", "there exists a graded $\\mathcal{A}$-module map", "$h : \\mathcal{M} \\to \\mathcal{I}$ of degree $-1$", "such that $a = \\text{d}(h)$. Since $\\mathcal{I}$ is graded injective", "and $b$ is injective, there exists a graded $\\mathcal{A}$-module", "map $h' : \\mathcal{M}' \\to \\mathcal{I}$ of degree $-1$", "such that $h = h' \\circ b$. Then we can take $a' = \\text{d}(h')$", "as the dotted arrow." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7339, "type": "theorem", "label": "sdga-lemma-second-property-dg-injective", "categories": [ "sdga" ], "title": "sdga-lemma-second-property-dg-injective", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on", "$(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{I}$ be a", "K-injective and graded injective", "object of $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$.", "For every solid diagram in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$", "$$", "\\xymatrix{", "\\mathcal{M} \\ar[r]_a \\ar[d]_b & \\mathcal{I} \\\\", "\\mathcal{M}' \\ar@{..>}[ru]", "}", "$$", "where $b$ is a quasi-isomorphism a dotted arrow exists making the", "diagram commute up to homotopy." ], "refs": [], "proofs": [ { "contents": [ "After replacing $\\mathcal{M}'$ by the direct sum of $\\mathcal{M}'$", "and the cone on the identity on $\\mathcal{M}$ (which is acyclic)", "we may assume $b$ is also injective. Then the cokernel $\\mathcal{Q}$", "of $b$ is acyclic. Thus we see that", "$$", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{Q}, \\mathcal{I}) =", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{Q}, \\mathcal{I})[1] = 0", "$$", "as $\\mathcal{I}$ is K-injective. As $\\mathcal{I}$ is graded injective", "by Remark \\ref{remark-why-graded-injective}", "we see that", "$$", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}', \\mathcal{I})", "\\longrightarrow", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}, \\mathcal{I})", "$$", "is bijective and the proof is complete." ], "refs": [ "sdga-remark-why-graded-injective" ], "ref_ids": [ 7387 ] } ], "ref_ids": [] }, { "id": 7340, "type": "theorem", "label": "sdga-lemma-better-set-of-monos", "categories": [ "sdga" ], "title": "sdga-lemma-better-set-of-monos", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$", "be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$.", "There exists a set $R$ and for each $r \\in R$ an injective map", "$\\mathcal{M}_r \\to \\mathcal{M}'_r$ of", "acyclic differential graded $\\mathcal{A}$-modules", "such that for an object $\\mathcal{I}$ of $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$", "the following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{I}$ is K-injective and graded injective, and", "\\item for every solid diagram", "$$", "\\xymatrix{", "\\mathcal{M}_r \\ar[r] \\ar[d] & \\mathcal{I} \\\\", "\\mathcal{M}'_r \\ar@{..>}[ru]", "}", "$$", "a dotted arrow exists in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$", "making the diagram commute.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $T$ and $\\mathcal{M}_t \\to \\mathcal{M}'_t$ be as in", "Lemma \\ref{lemma-characterize-graded-injectives-in-dg}.", "Let $S$ and $\\mathcal{M}_s$ be as in", "Lemma \\ref{lemma-small-acyclics}.", "Choose an injective map $\\mathcal{M}_s \\to \\mathcal{M}'_s$", "of acyclic differential graded $\\mathcal{A}$-modules", "which is homotopic to zero. This is possible because we", "may take $\\mathcal{M}'_s$ to be the cone on the identity;", "in that case it is even true that the identity on", "$\\mathcal{M}'_s$ is homotopic to zero, see", "Differential Graded Algebra, Lemma \\ref{dga-lemma-id-cone-null} which", "applies by the discussion in Section \\ref{section-conclude-triangulated}.", "We claim that $R = T \\coprod S$ with the given maps works.", "\\medskip\\noindent", "The implication (1) $\\Rightarrow$ (2) holds by", "Lemma \\ref{lemma-first-property-dg-injective}.", "\\medskip\\noindent", "Assume (2). First, by Lemma \\ref{lemma-characterize-graded-injectives-in-dg}", "we see that $\\mathcal{I}$ is graded injective. Next, let", "$\\mathcal{M}$ be an acyclic differential graded $\\mathcal{A}$-module.", "We have to show that", "$$", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}, \\mathcal{I}) = 0", "$$", "The proof will be exactly the same as the proof of", "Injectives, Lemma \\ref{injectives-lemma-characterize-K-injective}.", "\\medskip\\noindent", "We are going to construct by induction on the ordinal $\\alpha$", "an acyclic differential graded submodule", "$\\mathcal{K}_\\alpha \\subset \\mathcal{M}$ as follows.", "For $\\alpha = 0$ we set $\\mathcal{K}_0 = 0$. For $\\alpha > 0$", "we proceed as follows:", "\\begin{enumerate}", "\\item If $\\alpha = \\beta + 1$ and $\\mathcal{K}_\\beta = \\mathcal{M}$", "then we choose $\\mathcal{K}_\\alpha = \\mathcal{K}_\\beta$.", "\\item If $\\alpha = \\beta + 1$ and $\\mathcal{K}_\\beta \\not = \\mathcal{M}$", "then $\\mathcal{M}/\\mathcal{K}_\\beta$ is a nonzero acyclic", "differential graded $\\mathcal{A}$-module.", "We choose a differential graded $\\mathcal{A}$ submodule", "$\\mathcal{N}_\\alpha \\subset \\mathcal{M}/\\mathcal{K}_\\beta$", "isomorphic to $\\mathcal{M}_s$ for some $s \\in S$, see", "Lemma \\ref{lemma-small-acyclics}.", "Finally, we let $\\mathcal{K}_\\alpha \\subset \\mathcal{M}$", "be the inverse image of $\\mathcal{N}_\\alpha$.", "\\item If $\\alpha$ is a limit ordinal we set", "$\\mathcal{K}_\\beta = \\colim \\mathcal{K}_\\alpha$.", "\\end{enumerate}", "It is clear that $\\mathcal{M} = \\mathcal{K}_\\alpha$ for a suitably large", "ordinal $\\alpha$. We will prove that", "$$", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{K}_\\alpha, \\mathcal{I})", "$$", "is zero by transfinite induction on $\\alpha$. It holds for $\\alpha = 0$", "since $\\mathcal{K}_0$ is zero. Suppose it holds for $\\beta$ and", "$\\alpha = \\beta + 1$. In case (1) of the list above the result is clear.", "In case (2) there is a short exact sequence", "$$", "0 \\to \\mathcal{K}_\\beta \\to \\mathcal{K}_\\alpha \\to \\mathcal{N}_\\alpha \\to 0", "$$", "By Remark \\ref{remark-why-graded-injective}", "and since we've seen that $\\mathcal{I}$ is graded", "injective, we obtain an exact sequence", "$$", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{K}_\\beta, \\mathcal{I})", "\\to", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{K}_\\alpha, \\mathcal{I})", "\\to", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{N}_\\alpha, \\mathcal{I})", "$$", "By induction the term on the left is zero. By assumption (2)", "the term on the right is zero: any map $\\mathcal{M}_s \\to \\mathcal{I}$", "factors through $\\mathcal{M}'_s$ and hence is homotopic to zero.", "Thus the middle group is zero too. Finally, suppose that $\\alpha$ is a", "limit ordinal. Because we also have", "$\\mathcal{K}_\\alpha = \\colim \\mathcal{K}_\\alpha$ as graded", "$\\mathcal{A}$-modules we see that", "$$", "\\Hom_{\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})}}", "(\\mathcal{K}_\\alpha, \\mathcal{I})", "= \\lim_{\\beta < \\alpha}", "\\Hom_{\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})}}", "(\\mathcal{K}_\\beta, \\mathcal{I})", "$$", "as complexes of abelian groups. The cohomology groups of these", "complexes compute morphisms in $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$", "between shifts. The transition maps in the system of complexes", "are surjective by Remark \\ref{remark-why-graded-injective}", "because $\\mathcal{I}$ is graded injective.", "Moreover, for a limit ordinal $\\beta \\leq \\alpha$", "we have equality of limit and value. Thus we may apply", "Homology, Lemma \\ref{homology-lemma-ML-over-ordinals}", "to conclude." ], "refs": [ "sdga-lemma-characterize-graded-injectives-in-dg", "sdga-lemma-small-acyclics", "dga-lemma-id-cone-null", "sdga-lemma-first-property-dg-injective", "sdga-lemma-characterize-graded-injectives-in-dg", "injectives-lemma-characterize-K-injective", "sdga-lemma-small-acyclics", "sdga-remark-why-graded-injective", "sdga-remark-why-graded-injective", "homology-lemma-ML-over-ordinals" ], "ref_ids": [ 7335, 7336, 13078, 7338, 7335, 7790, 7336, 7387, 7387, 12129 ] } ], "ref_ids": [] }, { "id": 7341, "type": "theorem", "label": "sdga-lemma-functor-set-of-monos", "categories": [ "sdga" ], "title": "sdga-lemma-functor-set-of-monos", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$", "be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$.", "Let $R$ be a set and for each $r \\in R$ let an injective map", "$\\mathcal{M}_r \\to \\mathcal{M}'_r$ of", "acyclic differential graded $\\mathcal{A}$-modules be given.", "There exists a functor $M : \\text{Mod}_{(\\mathcal{A}, \\text{d})} \\to", "\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ and a natural transformation", "$j : \\text{id} \\to M$ such that", "\\begin{enumerate}", "\\item $j_\\mathcal{M} : \\mathcal{M} \\to M(\\mathcal{M})$ is injective", "and a quasi-isomorphism,", "\\item for every solid diagram", "$$", "\\xymatrix{", "\\mathcal{M}_r \\ar[r] \\ar[d] & \\mathcal{M} \\ar[d]^{j_\\mathcal{M}} \\\\", "\\mathcal{M}'_r \\ar@{..>}[r] & M(\\mathcal{M})", "}", "$$", "a dotted arrow exists in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$", "making the diagram commute.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We define $M(\\mathcal{M})$ as the pushout in the following diagram", "$$", "\\xymatrix{", "\\bigoplus_{(r, \\varphi)} \\mathcal{M}_r \\ar[r] \\ar[d] &", "\\mathcal{M} \\ar[d] \\\\", "\\bigoplus_{(r, \\varphi)} \\mathcal{M}'_r \\ar[r] &", "M(\\mathcal{M})", "}", "$$", "where the direct sum is over all pairs $(r, \\varphi)$", "with $r \\in R$ and $\\varphi \\in", "\\Hom_{\\text{Mod}_{(\\mathcal{A}, \\text{d})}}(\\mathcal{M}_r, \\mathcal{M})$.", "Since the pushout of an injective map is injective, we see that", "$\\mathcal{M} \\to M(\\mathcal{M})$ is injective.", "Since the cokernel of the left vertical arrow is acyclic,", "we see that the (isomorphic) cokernel of $\\mathcal{M} \\to M(\\mathcal{M})$", "is acyclic, hence $\\mathcal{M} \\to M(\\mathcal{M})$", "is a quasi-isomorphism. Property (2) holds by construction.", "We omit the verification that", "this procedure can be turned into a functor." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7342, "type": "theorem", "label": "sdga-lemma-cohomology-homological", "categories": [ "sdga" ], "title": "sdga-lemma-cohomology-homological", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. The functor ", "$H^0 : \\text{Mod}_{(\\mathcal{A}, \\text{d})} \\to \\textit{Mod}(\\mathcal{O})$", "of Section \\ref{section-modules} factors through a", "functor", "$$", "H^0 : K(\\text{Mod}_{(\\mathcal{A}, \\text{d})}) \\to \\textit{Mod}(\\mathcal{O})", "$$", "which is homological in the sense of", "Derived Categories, Definition \\ref{derived-definition-homological}." ], "refs": [ "derived-definition-homological" ], "proofs": [ { "contents": [ "It follows immediately from the definitions that there is", "a commutative diagram", "$$", "\\xymatrix{", "\\text{Mod}_{(\\mathcal{A}, \\text{d})} \\ar[r] \\ar[d] &", "K(\\text{Mod}_{(\\mathcal{A}, \\text{d})}) \\ar[d] \\\\", "\\text{Comp}(\\mathcal{O}) \\ar[r] &", "K(\\textit{Mod}(\\mathcal{O}))", "}", "$$", "Since $H^0(\\mathcal{M})$ is defined as the zeroth cohomology", "sheaf of the underlying complex of $\\mathcal{O}$-modules of $\\mathcal{M}$", "the lemma follows from the case of complexes of $\\mathcal{O}$-modules", "which is a special case of", "Derived Categories, Lemma \\ref{derived-lemma-cohomology-homological}." ], "refs": [ "derived-lemma-cohomology-homological" ], "ref_ids": [ 1810 ] } ], "ref_ids": [ 1971 ] }, { "id": 7343, "type": "theorem", "label": "sdga-lemma-acyclics", "categories": [ "sdga" ], "title": "sdga-lemma-acyclics", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. The full subcategory $\\text{Ac}$", "of the homotopy category $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$", "consisting of acyclic modules is a strictly full saturated", "triangulated subcategory of $K(\\text{Mod}_{(A, \\text{d})})$." ], "refs": [], "proofs": [ { "contents": [ "Of course an object $\\mathcal{M}$ of $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$", "is in $\\text{Ac}$ if and only if $H^i(\\mathcal{M}) = H^0(\\mathcal{M}[i])$", "is zero for all $i$. The lemma follows from this,", "Lemma \\ref{lemma-cohomology-homological}, and", "Derived Categories, Lemma \\ref{derived-lemma-homological-functor-kernel}.", "See also Derived Categories, Definitions \\ref{derived-definition-saturated}", "and \\ref{derived-definition-triangulated-subcategory} and", "Lemma \\ref{derived-lemma-triangulated-subcategory}." ], "refs": [ "sdga-lemma-cohomology-homological", "derived-lemma-homological-functor-kernel", "derived-definition-saturated", "derived-definition-triangulated-subcategory", "derived-lemma-triangulated-subcategory" ], "ref_ids": [ 7342, 1785, 1974, 1970, 1771 ] } ], "ref_ids": [] }, { "id": 7344, "type": "theorem", "label": "sdga-lemma-qis", "categories": [ "sdga" ], "title": "sdga-lemma-qis", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$.", "Consider the subclass", "$\\text{Qis} \\subset \\text{Arrows}(K(\\text{Mod}_{(A, \\text{d})}))$", "consisting of quasi-isomorphisms. This is a saturated multiplicative", "system compatible with the triangulated structure on", "$K(\\text{Mod}_{(A, \\text{d})})$." ], "refs": [], "proofs": [ { "contents": [ "Observe that if $f , g : \\mathcal{M} \\to \\mathcal{N}$ are morphisms", "of $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ which are homotopic,", "then $f$ is a quasi-isomorphism if and only if $g$ is a quasi-isomorphism.", "Namely, the maps $H^i(f) = H^0(f[i])$ and $H^i(g) = H^0(g[i])$ are", "the same by Lemma \\ref{lemma-cohomology-homological}. Thus it is", "unambiguous to say that a morphism of the homotopy category", "$K(\\text{Mod}_{(A, \\text{d})})$ is a quasi-isomorphism.", "For definitions of ``multiplicative system'', ``saturated'', and", "``compatible with the triangulated structure'' see", "Derived Categories, Definition \\ref{derived-definition-localization}", "and", "Categories, Definitions \\ref{categories-definition-multiplicative-system}", "and \\ref{categories-definition-saturated-multiplicative-system}.", "\\medskip\\noindent", "To actually prove the lemma consider the composition", "of exact functors of triangulated categories", "$$", "K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})", "\\longrightarrow", "K(\\textit{Mod}(\\mathcal{O}))", "\\longrightarrow", "D(\\mathcal{O})", "$$", "and observe that a morphism $f : \\mathcal{M} \\to \\mathcal{N}$", "of $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$ is in $\\text{Qis}$", "if and only if it maps to an isomorphism in $D(\\mathcal{O})$.", "Thus the lemma follows from Derived Categories, Lemma", "\\ref{derived-lemma-triangle-functor-localize}." ], "refs": [ "sdga-lemma-cohomology-homological", "derived-definition-localization", "categories-definition-multiplicative-system", "categories-definition-saturated-multiplicative-system", "derived-lemma-triangle-functor-localize" ], "ref_ids": [ 7342, 1973, 12373, 12376, 1779 ] } ], "ref_ids": [] }, { "id": 7345, "type": "theorem", "label": "sdga-lemma-kernel-localization", "categories": [ "sdga" ], "title": "sdga-lemma-kernel-localization", "contents": [ "In Definition \\ref{definition-derived-category}", "the kernel of the localization functor", "$Q : K(\\text{Mod}_{(\\mathcal{A}, \\text{d})}) \\to D(\\mathcal{A}, \\text{d})$", "is the category $\\text{Ac}$ of Lemma \\ref{lemma-acyclics}." ], "refs": [ "sdga-definition-derived-category", "sdga-lemma-acyclics" ], "proofs": [ { "contents": [ "This is immediate from", "Derived Categories, Lemma \\ref{derived-lemma-kernel-localization}", "and the fact that $0 \\to \\mathcal{M}$ is a quasi-isomorphism", "if and only if $\\mathcal{M}$ is acyclic." ], "refs": [ "derived-lemma-kernel-localization" ], "ref_ids": [ 1782 ] } ], "ref_ids": [ 7380, 7343 ] }, { "id": 7346, "type": "theorem", "label": "sdga-lemma-H0-over-D", "categories": [ "sdga" ], "title": "sdga-lemma-H0-over-D", "contents": [ "In Definition \\ref{definition-derived-category} the functor", "$H^0 : K(\\text{Mod}_{(\\mathcal{A}, \\text{d})}) \\to", "\\textit{Mod}(\\mathcal{O})$ factors through a homological functor", "$H^0 : D(\\mathcal{A}, \\text{d}) \\to \\textit{Mod}(\\mathcal{O})$." ], "refs": [ "sdga-definition-derived-category" ], "proofs": [ { "contents": [ "Follows immediately from", "Derived Categories, Lemma \\ref{derived-lemma-universal-property-localization}." ], "refs": [ "derived-lemma-universal-property-localization" ], "ref_ids": [ 1781 ] } ], "ref_ids": [ 7380 ] }, { "id": 7347, "type": "theorem", "label": "sdga-lemma-hom-derived", "categories": [ "sdga" ], "title": "sdga-lemma-hom-derived", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$.", "Let $\\mathcal{M}$ and $\\mathcal{N}$ be differential graded", "$\\mathcal{A}$-modules. Let $\\mathcal{N} \\to \\mathcal{I}$ be a", "quasi-isomorphism with $\\mathcal{I}$ a graded injective and", "K-injective differential graded $\\mathcal{A}$-module. Then", "$$", "\\Hom_{D(\\mathcal{A}, \\text{d})}(\\mathcal{M}, \\mathcal{N}) =", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}, \\mathcal{I})", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathcal{N} \\to \\mathcal{I}$ is a quasi-isomorphism", "we see that", "$$", "\\Hom_{D(\\mathcal{A}, \\text{d})}(\\mathcal{M}, \\mathcal{N}) =", "\\Hom_{D(\\mathcal{A}, \\text{d})}(\\mathcal{M}, \\mathcal{I})", "$$", "In the discussion preceding Definition \\ref{definition-derived-category}", "we found, using Lemma \\ref{lemma-second-property-dg-injective},", "that any morphism $\\mathcal{M} \\to \\mathcal{I}$", "in $D(\\mathcal{A}, \\text{d})$ can be represented by a morphism", "$f : \\mathcal{M} \\to \\mathcal{I}$ in", "$K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$.", "Now, if $f, f' : \\mathcal{M} \\to \\mathcal{I}$ are two morphism in", "$K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$, then they define", "the same morphism in $D(\\mathcal{A}, \\text{d})$ if and only", "if there exists a quasi-isomorphism $g : \\mathcal{I} \\to \\mathcal{K}$", "in $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$", "such that $g \\circ f = g \\circ f'$, see", "Categories, Lemma \\ref{categories-lemma-equality-morphisms-left-localization}.", "However, by Lemma \\ref{lemma-second-property-dg-injective} there", "exists a map", "$h : \\mathcal{K} \\to \\mathcal{I}$", "such that $h \\circ g = \\text{id}_\\mathcal{I}$ in", "in $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$.", "Thus $g \\circ f = g \\circ f'$ implies $f = f'$ and", "the proof is complete." ], "refs": [ "sdga-definition-derived-category", "sdga-lemma-second-property-dg-injective", "categories-lemma-equality-morphisms-left-localization", "sdga-lemma-second-property-dg-injective" ], "ref_ids": [ 7380, 7339, 12257, 7339 ] } ], "ref_ids": [] }, { "id": 7348, "type": "theorem", "label": "sdga-lemma-derived-products", "categories": [ "sdga" ], "title": "sdga-lemma-derived-products", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. Then", "\\begin{enumerate}", "\\item $D(\\mathcal{A}, \\text{d})$ has both direct sums and products,", "\\item direct sums are obtained by taking direct sums of differential graded", "$\\mathcal{A}$-modules,", "\\item products are obtained by taking products of", "K-injective differential graded modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use that $\\text{Mod}_{(A, \\text{d})}$ is an abelian category", "with arbitrary direct sums and products, and that these give rise", "to direct sums and products in $K(\\text{Mod}_{(A, \\text{d})})$.", "See Lemmas \\ref{lemma-dgm-abelian} and \\ref{lemma-homotopy-direct-sums}.", "\\medskip\\noindent", "Let $\\mathcal{M}_j$ be a family of differential graded $\\mathcal{A}$-modules.", "Consider the direct sum $\\mathcal{M} = \\bigoplus \\mathcal{M}_j$", "as a differential graded $\\mathcal{A}$-module.", "For a differential graded $\\mathcal{A}$-module", "$\\mathcal{N}$ choose a quasi-isomorphism", "$\\mathcal{N} \\to \\mathcal{I}$ where", "$\\mathcal{I}$ is graded injective and K-injective as a", "differential graded $\\mathcal{A}$-module. See", "Theorem \\ref{theorem-qis-into-dg-injective}.", "Using Lemma \\ref{lemma-hom-derived} we have", "\\begin{align*}", "\\Hom_{D(\\mathcal{A}, \\text{d})}(\\mathcal{M}, \\mathcal{N})", "& =", "\\Hom_{K(\\mathcal{A}, \\text{d})}(\\mathcal{M}, \\mathcal{I}) \\\\", "& =", "\\prod \\Hom_{K(\\mathcal{A}, \\text{d})}(\\mathcal{M}_j, \\mathcal{I}) \\\\", "& =", "\\prod \\Hom_{D(\\mathcal{A}, \\text{d})}(\\mathcal{M}_j, \\mathcal{I})", "\\end{align*}", "whence the existence of direct sums in $D(A, \\text{d})$ as given in", "part (2) of the lemma.", "\\medskip\\noindent", "Let $\\mathcal{M}_j$ be a family of differential graded $\\mathcal{A}$-modules.", "For each $j$ choose a quasi-isomorphism", "$\\mathcal{M} \\to \\mathcal{I}_j$ where", "$\\mathcal{I}_j$ is graded injective and K-injective as a", "differential graded $\\mathcal{A}$-module.", "Consider the product $\\mathcal{I} = \\prod \\mathcal{I}_j$", "of differential graded $\\mathcal{A}$-modules.", "By Lemmas \\ref{lemma-product-K-injective} and", "\\ref{lemma-product-graded-injective} we see that", "$\\mathcal{I}$ is graded injective and K-injective as a", "differential graded $\\mathcal{A}$-module.", "For a differential graded $\\mathcal{A}$-module", "$\\mathcal{N}$ using Lemma \\ref{lemma-hom-derived} we have", "\\begin{align*}", "\\Hom_{D(\\mathcal{A}, \\text{d})}(\\mathcal{N}, \\mathcal{I})", "& =", "\\Hom_{K(\\mathcal{A}, \\text{d})}(\\mathcal{N}, \\mathcal{I}) \\\\", "& =", "\\prod \\Hom_{K(\\mathcal{A}, \\text{d})}(\\mathcal{N}, \\mathcal{I}_j) \\\\", "& =", "\\prod \\Hom_{D(\\mathcal{A}, \\text{d})}(\\mathcal{N}, \\mathcal{M}_j)", "\\end{align*}", "whence the existence of products in $D(\\mathcal{A}, \\text{d})$ as given in", "part (3) of the lemma." ], "refs": [ "sdga-lemma-dgm-abelian", "sdga-lemma-homotopy-direct-sums", "sdga-theorem-qis-into-dg-injective", "sdga-lemma-hom-derived", "sdga-lemma-product-K-injective", "sdga-lemma-product-graded-injective", "sdga-lemma-hom-derived" ], "ref_ids": [ 7314, 7320, 7307, 7347, 7337, 7334, 7347 ] } ], "ref_ids": [] }, { "id": 7349, "type": "theorem", "label": "sdga-lemma-derived-canonical-delta-functor", "categories": [ "sdga" ], "title": "sdga-lemma-derived-canonical-delta-functor", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. The localization functor", "$\\text{Mod}_{(\\mathcal{A}, \\text{d})} \\to D(\\mathcal{A}, \\text{d})$", "has the natural structure of a $\\delta$-functor, with", "$$", "\\delta_{\\mathcal{K} \\to \\mathcal{L} \\to \\mathcal{M}} = - p \\circ q^{-1}", "$$", "with $p$ and $q$ as explained above." ], "refs": [], "proofs": [ { "contents": [ "We have already seen that this choice leads to a distinguished", "triangle whenever given a short exact sequence of complexes.", "We have to show functoriality of this construction, see", "Derived Categories, Definition \\ref{derived-definition-delta-functor}.", "This follows from Differential Graded Algebra, Lemma \\ref{dga-lemma-cone}", "(which we may use by the discussion in", "Section \\ref{section-conclude-triangulated}) with a bit of", "work. Compare with", "Derived Categories, Lemma \\ref{derived-lemma-derived-canonical-delta-functor}." ], "refs": [ "derived-definition-delta-functor", "dga-lemma-cone", "derived-lemma-derived-canonical-delta-functor" ], "ref_ids": [ 1972, 13077, 1814 ] } ], "ref_ids": [] }, { "id": 7350, "type": "theorem", "label": "sdga-lemma-homotopy-colimit", "categories": [ "sdga" ], "title": "sdga-lemma-homotopy-colimit", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. Let", "$\\mathcal{M}_n$ be a system of differential graded $\\mathcal{A}$-modules.", "Then the derived colimit $\\text{hocolim} \\mathcal{M}_n$ in", "$D(\\mathcal{A}, \\text{d})$ is represented", "by the differential graded module $\\colim \\mathcal{M}_n$." ], "refs": [], "proofs": [ { "contents": [ "Set $\\mathcal{M} = \\colim \\mathcal{M}_n$.", "We have an exact sequence of differential graded $\\mathcal{A}$-modules", "$$", "0 \\to \\bigoplus \\mathcal{M}_n \\to \\bigoplus \\mathcal{M}_n \\to \\mathcal{M} \\to 0", "$$", "by Derived Categories, Lemma \\ref{derived-lemma-compute-colimit}", "(applied the underlying complexes of $\\mathcal{O}$-modules).", "The direct sums are direct sums in $D(\\mathcal{A}, \\text{d})$ by", "Lemma \\ref{lemma-derived-products}.", "Thus the result follows from the definition", "of derived colimits in Derived Categories,", "Definition \\ref{derived-definition-derived-colimit}", "and the fact that a short exact sequence of complexes", "gives a distinguished triangle", "(Lemma \\ref{lemma-derived-canonical-delta-functor})." ], "refs": [ "derived-lemma-compute-colimit", "sdga-lemma-derived-products", "derived-definition-derived-colimit", "sdga-lemma-derived-canonical-delta-functor" ], "ref_ids": [ 1921, 7348, 2001, 7349 ] } ], "ref_ids": [] }, { "id": 7351, "type": "theorem", "label": "sdga-lemma-derived-tensor-product", "categories": [ "sdga" ], "title": "sdga-lemma-derived-tensor-product", "contents": [ "In the situation above, the functor (\\ref{equation-pullback})", "composed with the localization functor", "$K(\\text{Mod}_{(\\mathcal{A}', \\text{d})}) \\to D(\\mathcal{A}', \\text{d})$", "has a left derived extension", "$D(\\mathcal{B}, \\text{d}) \\to D(\\mathcal{A}', \\text{d})$ whose", "value on a good right differential graded $\\mathcal{B}$-module", "$\\mathcal{P}$ is $f^*\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}$." ], "refs": [], "proofs": [ { "contents": [ "Recall that for any (right) differential graded $\\mathcal{B}$-module", "$\\mathcal{M}$ there exists a quasi-isomorphism $\\mathcal{P} \\to \\mathcal{M}$", "with $\\mathcal{P}$ a good differential graded $\\mathcal{B}$-module.", "See Lemma \\ref{lemma-resolve}.", "Hence by Derived Categories, Lemma \\ref{derived-lemma-find-existence-computes}", "it suffices to show that given a quasi-isomorphism", "$\\mathcal{P} \\to \\mathcal{P}'$ of good differential graded", "$\\mathcal{B}$-modules the induced map", "$$", "f^*\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}", "\\longrightarrow", "f^*\\mathcal{P}' \\otimes_\\mathcal{A} \\mathcal{N}", "$$", "is a quasi-isomorphism. The cone $\\mathcal{P}''$ on", "$\\mathcal{P} \\to \\mathcal{P}'$ is a good differential graded", "$\\mathcal{A}$-module by Lemma \\ref{lemma-good-admissible-ses}.", "Since we have a distinguished triangle", "$$", "\\mathcal{P} \\to \\mathcal{P}' \\to \\mathcal{P}'' \\to \\mathcal{P}[1]", "$$", "in $K(\\text{Mod}_{(\\mathcal{B}, \\text{d})})$ we obtain a distinguished", "triangle", "$$", "f^*\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N} \\to", "f^*\\mathcal{P}' \\otimes_\\mathcal{A} \\mathcal{N} \\to", "f^*\\mathcal{P}'' \\otimes_\\mathcal{A} \\mathcal{N} \\to", "f^*\\mathcal{P}[1] \\otimes_\\mathcal{A} \\mathcal{N}", "$$", "in $K(\\text{Mod}_{(\\mathcal{A}', \\text{d})})$. By", "Lemma \\ref{lemma-acyclic-good}", "the differential graded module", "$f^*\\mathcal{P}'' \\otimes_\\mathcal{A} \\mathcal{N}$", "is acyclic and the proof is complete." ], "refs": [ "sdga-lemma-resolve", "derived-lemma-find-existence-computes", "sdga-lemma-good-admissible-ses", "sdga-lemma-acyclic-good" ], "ref_ids": [ 7329, 1832, 7325, 7330 ] } ], "ref_ids": [] }, { "id": 7352, "type": "theorem", "label": "sdga-lemma-compose-pullback-tensor", "categories": [ "sdga" ], "title": "sdga-lemma-compose-pullback-tensor", "contents": [ "In Lemma \\ref{lemma-derived-tensor-product} the functor", "$D(\\mathcal{B}, \\text{d}) \\to D(\\mathcal{A}', \\text{d})$ is equal to", "$\\mathcal{M} \\mapsto", "Lf^*\\mathcal{M} \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}$." ], "refs": [ "sdga-lemma-derived-tensor-product" ], "proofs": [ { "contents": [ "Immediate from the fact that we can compute these functors", "by representing objects by good differential graded modules", "and because $f^*\\mathcal{P}$ is a good differential graded", "$\\mathcal{A}$-module if $\\mathcal{P}$ is a good differential", "graded $\\mathcal{B}$-module." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 7351 ] }, { "id": 7353, "type": "theorem", "label": "sdga-lemma-compose-pullback", "categories": [ "sdga" ], "title": "sdga-lemma-compose-pullback", "contents": [ "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O})", "\\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ and", "$(g, g^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}')", "\\to (\\Sh(\\mathcal{C}''), \\mathcal{O}'')$", "be morphisms of ringed topoi. Let $\\mathcal{A}$, $\\mathcal{A}'$, and", "$\\mathcal{A}''$ be a differential graded $\\mathcal{O}$-algebra,", "$\\mathcal{O}'$-algebra, and $\\mathcal{O}''$-algebra. Let", "$\\varphi : \\mathcal{A}' \\to f_*\\mathcal{A}$ and", "$\\varphi' : \\mathcal{A}'' \\to g_*\\mathcal{A}'$", "be a homomorphism of differential graded $\\mathcal{O}'$-algebras", "and $\\mathcal{O}''$-algebras.", "Then we have $L(g \\circ f)^* = Lf^* \\circ Lg^* :", "D(\\mathcal{A}'', \\text{d}) \\to D(\\mathcal{A}, \\text{d})$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the fact that we can compute these functors", "by representing objects by good differential graded modules", "and because $f^*\\mathcal{P}$ is a good differential", "graded $\\mathcal{A}'$-module of $\\mathcal{P}$ is a good", "differential graded $\\mathcal{A}$-module." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7354, "type": "theorem", "label": "sdga-lemma-tensor-symmetry", "categories": [ "sdga" ], "title": "sdga-lemma-tensor-symmetry", "contents": [ "In the situation above, if $\\mathcal{N} \\to \\mathcal{N}'$ is an isomorphism", "on cohomology sheaves, then $t$ is an isomorphism of functors", "$(- \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}) \\to", "(- \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}')$." ], "refs": [], "proofs": [ { "contents": [ "It is enough to show that", "$\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N} \\to", "\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}'$", "is an isomorphism on cohomology sheaves for any good differential", "graded $\\mathcal{A}$-module $\\mathcal{P}$.", "To do this, let $\\mathcal{N}''$ be the cone on the map", "$\\mathcal{N} \\to \\mathcal{N}'$ as a left differential graded", "$\\mathcal{A}$-module, see Definition \\ref{definition-cone}.", "(To be sure, $\\mathcal{N}''$ is a bimodule too but we", "don't need this.) By functoriality of the tensor construction", "(it is a functor of differential graded categories)", "we see that $\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}''$", "is the cone (as a complex of $\\mathcal{O}$-modules) on the map", "$\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N} \\to", "\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}'$.", "Hence it suffices to show that", "$\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}''$", "is acyclic. This follows from the fact that $\\mathcal{P}$", "is good and the fact that $\\mathcal{N}''$ is acyclic", "as a cone on a quasi-isomorphism." ], "refs": [ "sdga-definition-cone" ], "ref_ids": [ 7377 ] } ], "ref_ids": [] }, { "id": 7355, "type": "theorem", "label": "sdga-lemma-good-on-other-side", "categories": [ "sdga" ], "title": "sdga-lemma-good-on-other-side", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$\\mathcal{A}$, $\\mathcal{B}$ be differential graded $\\mathcal{O}$-algebras.", "Let $\\mathcal{N}$ be a differential graded", "$(\\mathcal{A}, \\mathcal{B})$-bimodule. If $\\mathcal{N}$ is good", "as a left differential graded $\\mathcal{A}$-module, then", "we have $\\mathcal{M} \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N} =", "\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}$ for all", "differential graded $\\mathcal{A}$-modules $\\mathcal{M}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{P} \\to \\mathcal{M}$ be a quasi-isomorphism where", "$\\mathcal{P}$ is a good (right) differential graded $\\mathcal{A}$-module.", "To prove the lemma we have to show that", "$\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N} \\to", "\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}$", "is a quasi-isomorphism. The cone $C$ on the map", "$\\mathcal{P} \\to \\mathcal{M}$ is an acyclic right differential", "graded $\\mathcal{A}$-module. Hence", "$C \\otimes_\\mathcal{A} \\mathcal{N}$ is acyclic as $\\mathcal{N}$", "is assumed good as a left differential graded $\\mathcal{A}$-module.", "Since $C \\otimes_\\mathcal{A} \\mathcal{N}$ is the cone on the", "maps $\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N} \\to", "\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}$ as a complex", "of $\\mathcal{O}$-modules we conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7356, "type": "theorem", "label": "sdga-lemma-compose-tensor", "categories": [ "sdga" ], "title": "sdga-lemma-compose-tensor", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$\\mathcal{A}$, $\\mathcal{A}'$, $\\mathcal{A}''$ be differential graded", "$\\mathcal{O}$-algebras. Let $\\mathcal{N}$ and $\\mathcal{N}'$ be a", "differential graded $(\\mathcal{A}, \\mathcal{A}')$-bimodule", "and $(\\mathcal{A}', \\mathcal{A}'')$-bimodule. Assume", "that the canonical map", "$$", "\\mathcal{N} \\otimes_{\\mathcal{A}'}^\\mathbf{L} \\mathcal{N}'", "\\longrightarrow", "\\mathcal{N} \\otimes_{\\mathcal{A}'} \\mathcal{N}'", "$$", "in $D(\\mathcal{A}'', \\text{d})$ is a quasi-isomorphism.", "Then we have", "$$", "(\\mathcal{M}", "\\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N})", "\\otimes_{\\mathcal{A}'}^\\mathbf{L} \\mathcal{N}'", "=", "\\mathcal{M}", "\\otimes_\\mathcal{A}^\\mathbf{L}", "(\\mathcal{N} \\otimes_{\\mathcal{A}'} \\mathcal{N}')", "$$", "as functors $D(\\mathcal{A}, \\text{d}) \\to D(\\mathcal{A}'', \\text{d})$." ], "refs": [], "proofs": [ { "contents": [ "Choose a good differential graded $\\mathcal{A}$-module", "$\\mathcal{P}$ and a quasi-isomorphism $\\mathcal{P} \\to \\mathcal{M}$, see", "Lemma \\ref{lemma-resolve}. Then", "$$", "\\mathcal{M}", "\\otimes_\\mathcal{A}^\\mathbf{L}", "(\\mathcal{N} \\otimes_{\\mathcal{A}'} \\mathcal{N}') =", "\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}", "\\otimes_{\\mathcal{A}'} \\mathcal{N}'", "$$", "and we have", "$$", "(\\mathcal{M}", "\\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N})", "\\otimes_{\\mathcal{A}'}^\\mathbf{L} \\mathcal{N}' =", "(\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N})", "\\otimes_{\\mathcal{A}'}^\\mathbf{L} \\mathcal{N}'", "$$", "Thus we have to show the canonical map", "$$", "(\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N})", "\\otimes_{\\mathcal{A}'}^\\mathbf{L} \\mathcal{N}'", "\\longrightarrow", "\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}", "\\otimes_{\\mathcal{A}'} \\mathcal{N}'", "$$", "is a quasi-isomorphism. Choose a quasi-isomorphism", "$\\mathcal{Q} \\to \\mathcal{N}'$ where $\\mathcal{Q}$", "is a good left differential graded $\\mathcal{A}'$-module", "(Lemma \\ref{lemma-resolve}).", "By Lemma \\ref{lemma-good-on-other-side} the map", "above as a map in the derived category of $\\mathcal{O}$-modules is the map", "$$", "\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}", "\\otimes_{\\mathcal{A}'} \\mathcal{Q}", "\\longrightarrow", "\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}", "\\otimes_{\\mathcal{A}'} \\mathcal{N}'", "$$", "Since $\\mathcal{N} \\otimes_{\\mathcal{A}'} \\mathcal{Q} \\to", "\\mathcal{N} \\otimes_{\\mathcal{A}'} \\mathcal{N}'$ is a quasi-isomorphism", "by assumption and $\\mathcal{P}$ is a good differential graded", "$\\mathcal{A}$-module this map is an quasi-isomorphism by", "Lemma \\ref{lemma-tensor-symmetry} (the left and right hand side", "compute $\\mathcal{P} \\otimes_\\mathcal{A}^\\mathbf{L} (\\mathcal{N}", "\\otimes_{\\mathcal{A}'} \\mathcal{Q})$ and", "$\\mathcal{P} \\otimes_\\mathcal{A}^\\mathbf{L} (\\mathcal{N}", "\\otimes_{\\mathcal{A}'} \\mathcal{N}')$ or you can just repeat", "the argument in the proof of the lemma)." ], "refs": [ "sdga-lemma-resolve", "sdga-lemma-resolve", "sdga-lemma-good-on-other-side", "sdga-lemma-tensor-symmetry" ], "ref_ids": [ 7329, 7329, 7355, 7354 ] } ], "ref_ids": [] }, { "id": 7357, "type": "theorem", "label": "sdga-lemma-right-derived", "categories": [ "sdga" ], "title": "sdga-lemma-right-derived", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on", "$(\\mathcal{C}, \\mathcal{O})$. Then any exact functor", "$$", "T : K(\\text{Mod}_{(\\mathcal{A}, \\text{d})}) \\longrightarrow \\mathcal{D}", "$$", "of triangulated categories has a right derived extension", "$RT : D(\\mathcal{A}, \\text{d}) \\to \\mathcal{D}$", "whose value on a graded injective and K-injective", "differential graded $\\mathcal{A}$-module $\\mathcal{I}$", "is $T(\\mathcal{I})$." ], "refs": [], "proofs": [ { "contents": [ "By Theorem \\ref{theorem-qis-into-dg-injective}", "for any (right) differential graded $\\mathcal{A}$-module", "$\\mathcal{M}$ there exists a quasi-isomorphism", "$\\mathcal{M} \\to \\mathcal{I}$ where $\\mathcal{I}$", "is a graded injective and K-injective", "differential graded $\\mathcal{A}$-module.", "Hence by Derived Categories, Lemma \\ref{derived-lemma-find-existence-computes}", "it suffices to show that given a quasi-isomorphism", "$\\mathcal{I} \\to \\mathcal{I}'$ of differential graded", "$\\mathcal{A}$-modules which are both", "graded injective and K-injective", "then $T(\\mathcal{I}) \\to T(\\mathcal{I}')$ is an isomorphism.", "This is true because the map $\\mathcal{I} \\to \\mathcal{I}'$", "is an isomorphism in $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$ as follows", "for example from Lemma \\ref{lemma-hom-derived} (or one can", "deduce it from Lemma \\ref{lemma-second-property-dg-injective})." ], "refs": [ "sdga-theorem-qis-into-dg-injective", "derived-lemma-find-existence-computes", "sdga-lemma-hom-derived", "sdga-lemma-second-property-dg-injective" ], "ref_ids": [ 7307, 1832, 7347, 7339 ] } ], "ref_ids": [] }, { "id": 7358, "type": "theorem", "label": "sdga-lemma-derived-adjoint-tensor-hom", "categories": [ "sdga" ], "title": "sdga-lemma-derived-adjoint-tensor-hom", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$\\mathcal{A}$, $\\mathcal{B}$ be differential graded $\\mathcal{O}$-algebras.", "Let $\\mathcal{N}$ be a differential graded", "$(\\mathcal{A}, \\mathcal{B})$-bimodule. Then", "$$", "R\\SheafHom_\\mathcal{B}(\\mathcal{N}, -) :", "D(\\mathcal{B}, \\text{d})", "\\longrightarrow", "D(\\mathcal{A}, \\text{d})", "$$", "is right adjoint to", "$$", "- \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N} :", "D(\\mathcal{A}, \\text{d})", "\\longrightarrow", "D(\\mathcal{B}, \\text{d})", "$$" ], "refs": [], "proofs": [ { "contents": [ "This follows from Derived Categories, Lemma", "\\ref{derived-lemma-pre-derived-adjoint-functors-general}", "and Lemma \\ref{lemma-tensor-hom-adjunction-dg}." ], "refs": [ "derived-lemma-pre-derived-adjoint-functors-general", "sdga-lemma-tensor-hom-adjunction-dg" ], "ref_ids": [ 1905, 7316 ] } ], "ref_ids": [] }, { "id": 7359, "type": "theorem", "label": "sdga-lemma-derived-adjoint-push-pull", "categories": [ "sdga" ], "title": "sdga-lemma-derived-adjoint-push-pull", "contents": [ "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi. Let $\\mathcal{A}$ be a differential", "graded $\\mathcal{O}_\\mathcal{C}$-algebra. Let $\\mathcal{B}$ be a", "differential graded $\\mathcal{O}_\\mathcal{D}$-algebra. Let", "$\\varphi : \\mathcal{B} \\to f_*\\mathcal{A}$ be a homomorphism", "of differential graded $\\mathcal{O}_\\mathcal{D}$-algebras.", "Then", "$$", "Rf_* : ", "D(\\mathcal{A}, \\text{d})", "\\longrightarrow", "D(\\mathcal{B}, \\text{d})", "$$", "is right adjoint to", "$$", "Lf^* :", "D(\\mathcal{B}, \\text{d})", "\\longrightarrow", "D(\\mathcal{A}, \\text{d})", "$$" ], "refs": [], "proofs": [ { "contents": [ "This follows from Derived Categories, Lemma", "\\ref{derived-lemma-pre-derived-adjoint-functors-general}", "and Lemma \\ref{lemma-adjunction-push-pull-dg}." ], "refs": [ "derived-lemma-pre-derived-adjoint-functors-general", "sdga-lemma-adjunction-push-pull-dg" ], "ref_ids": [ 1905, 7317 ] } ], "ref_ids": [] }, { "id": 7360, "type": "theorem", "label": "sdga-lemma-compose-pushforward-hom", "categories": [ "sdga" ], "title": "sdga-lemma-compose-pushforward-hom", "contents": [ "In the situation above, denote $RT : D(\\mathcal{A}', \\text{d}) \\to", "D(\\mathcal{B}, \\text{d})$ the right derived extension of", "(\\ref{equation-pushforward}). Then we have", "$$", "RT(\\mathcal{M}) = Rf_* R\\SheafHom(\\mathcal{N}, \\mathcal{M})", "$$", "functorially in $\\mathcal{M}$." ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-tensor-hom-adjunction-dg} and", "\\ref{lemma-adjunction-push-pull-dg} the functor", "(\\ref{equation-pushforward}) is right adjoint to", "the functor (\\ref{equation-pullback}). By Derived Categories, Lemma", "\\ref{derived-lemma-pre-derived-adjoint-functors-general}", "the functor $RT$ is right adjoint to the functor", "of Lemma \\ref{lemma-derived-tensor-product} which is equal to", "$Lf^*(-) \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}$ by", "Lemma \\ref{lemma-compose-pullback-tensor}.", "By Lemmas \\ref{lemma-derived-adjoint-tensor-hom} and", "\\ref{lemma-derived-adjoint-push-pull} the functor", "$Lf^*(-) \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}$", "is left adjoint to", "$Rf_* R\\SheafHom(\\mathcal{N}, -)$", "Thus we conclude by uniqueness of adjoints." ], "refs": [ "sdga-lemma-tensor-hom-adjunction-dg", "sdga-lemma-adjunction-push-pull-dg", "derived-lemma-pre-derived-adjoint-functors-general", "sdga-lemma-derived-tensor-product", "sdga-lemma-compose-pullback-tensor", "sdga-lemma-derived-adjoint-tensor-hom", "sdga-lemma-derived-adjoint-push-pull" ], "ref_ids": [ 7316, 7317, 1905, 7351, 7352, 7358, 7359 ] } ], "ref_ids": [] }, { "id": 7361, "type": "theorem", "label": "sdga-lemma-compose-pushforward", "categories": [ "sdga" ], "title": "sdga-lemma-compose-pushforward", "contents": [ "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O})", "\\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ and", "$(g, g^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}')", "\\to (\\Sh(\\mathcal{C}''), \\mathcal{O}'')$", "be morphisms of ringed topoi. Let $\\mathcal{A}$, $\\mathcal{A}'$, and", "$\\mathcal{A}''$ be a differential graded $\\mathcal{O}$-algebra,", "$\\mathcal{O}'$-algebra, and $\\mathcal{O}''$-algebra. Let", "$\\varphi : \\mathcal{A}' \\to f_*\\mathcal{A}$ and", "$\\varphi' : \\mathcal{A}'' \\to g_*\\mathcal{A}'$", "be a homomorphism of differential graded $\\mathcal{O}'$-algebras", "and $\\mathcal{O}''$-algebras.", "Then we have $R(g \\circ f)_* = Rg_* \\circ Rf_* :", "D(\\mathcal{A}, \\text{d}) \\to D(\\mathcal{A}'', \\text{d})$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemmas \\ref{lemma-compose-pullback} and", "\\ref{lemma-derived-adjoint-push-pull}", "and uniqueness of adjoints." ], "refs": [ "sdga-lemma-compose-pullback", "sdga-lemma-derived-adjoint-push-pull" ], "ref_ids": [ 7353, 7359 ] } ], "ref_ids": [] }, { "id": 7362, "type": "theorem", "label": "sdga-lemma-compose-hom", "categories": [ "sdga" ], "title": "sdga-lemma-compose-hom", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$\\mathcal{A}$, $\\mathcal{A}'$, $\\mathcal{A}''$ be differential graded", "$\\mathcal{O}$-algebras. Let $\\mathcal{N}$ and $\\mathcal{N}'$ be a", "differential graded $(\\mathcal{A}, \\mathcal{A}')$-bimodule", "and $(\\mathcal{A}', \\mathcal{A}'')$-bimodule. Assume", "that the canonical map", "$$", "\\mathcal{N} \\otimes_{\\mathcal{A}'}^\\mathbf{L} \\mathcal{N}'", "\\longrightarrow", "\\mathcal{N} \\otimes_{\\mathcal{A}'} \\mathcal{N}'", "$$", "in $D(\\mathcal{A}'', \\text{d})$ is a quasi-isomorphism.", "Then we have", "$$", "R\\SheafHom_{\\mathcal{A}''}", "(\\mathcal{N} \\otimes_{\\mathcal{A}'} \\mathcal{N}', -)", "=", "R\\SheafHom_{\\mathcal{A}'}(\\mathcal{N},", "R\\SheafHom_{\\mathcal{A}''}(\\mathcal{N}', -))", "$$", "as functors $D(\\mathcal{A}'', \\text{d}) \\to D(\\mathcal{A}, \\text{d})$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemmas \\ref{lemma-compose-tensor} and", "\\ref{lemma-derived-adjoint-tensor-hom}", "and uniqueness of adjoints." ], "refs": [ "sdga-lemma-compose-tensor", "sdga-lemma-derived-adjoint-tensor-hom" ], "ref_ids": [ 7356, 7358 ] } ], "ref_ids": [] }, { "id": 7363, "type": "theorem", "label": "sdga-lemma-pushforward-agrees", "categories": [ "sdga" ], "title": "sdga-lemma-pushforward-agrees", "contents": [ "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi. Let $\\mathcal{A}$ be a differential", "graded $\\mathcal{O}_\\mathcal{C}$-algebra. Let $\\mathcal{B}$ be a", "differential graded $\\mathcal{O}_\\mathcal{D}$-algebra. Let", "$\\varphi : \\mathcal{B} \\to f_*\\mathcal{A}$ be a homomorphism", "of differential graded $\\mathcal{O}_\\mathcal{D}$-algebras.", "The diagram", "$$", "\\xymatrix{", "D(\\mathcal{A}, \\text{d}) \\ar[d]_{Rf_*} \\ar[rr]_{forget} & &", "D(\\mathcal{O}_\\mathcal{C}) \\ar[d]^{Rf_*} \\\\", "D(\\mathcal{B}, \\text{d}) \\ar[rr]^{forget} & &", "D(\\mathcal{O}_\\mathcal{D})", "}", "$$", "commutes." ], "refs": [], "proofs": [ { "contents": [ "Besides identifying some categories, this lemma follows immediately", "from Lemma \\ref{lemma-compose-pushforward}.", "\\medskip\\noindent", "We may view $(\\mathcal{O}_\\mathcal{C}, 0)$ as a differential graded", "$\\mathcal{O}_\\mathcal{C}$-algebra by placing $\\mathcal{O}_\\mathcal{C}$", "in degree $0$ and endowing it with the zero differential. It is clear", "that we have", "$$", "\\text{Mod}_{(\\mathcal{O}_\\mathcal{C}, 0)} =", "\\text{Comp}(\\mathcal{O}_\\mathcal{C})", "\\quad\\text{and}\\quad", "D(\\mathcal{O}_\\mathcal{C}, 0) = D(\\mathcal{O}_\\mathcal{C})", "$$", "Via this identification the forgetful functor", "$\\text{Mod}_{(\\mathcal{A}, \\text{d})} \\to", "\\text{Comp}(\\mathcal{O}_\\mathcal{C})$", "is the ``pushforward'' $\\text{id}_{\\mathcal{C}, *}$", "defined in Section \\ref{section-functoriality-dg}", "corresponding to the identity morphism", "$\\text{id}_\\mathcal{C} : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to", "(\\mathcal{C}, \\mathcal{O}_\\mathcal{C})$ of ringed topoi and the", "map $(\\mathcal{O}_\\mathcal{C}, 0) \\to (\\mathcal{A}, \\text{d})$", "of differential graded $\\mathcal{O}_\\mathcal{C}$-algebras.", "Since $\\text{id}_{\\mathcal{C}, *}$ is exact, we immediately see that", "$$", "R\\text{id}_{\\mathcal{C}, *} = forget :", "D(\\mathcal{A}, \\text{d}) \\longrightarrow", "D(\\mathcal{O}_\\mathcal{C}, 0) = D(\\mathcal{O}_\\mathcal{C})", "$$", "The exact same reasoning shows that", "$$", "R\\text{id}_{\\mathcal{D}, *} = forget :", "D(\\mathcal{B}, \\text{d}) \\longrightarrow", "D(\\mathcal{O}_\\mathcal{D}, 0) = D(\\mathcal{O}_\\mathcal{D})", "$$", "Moreover, the construction of", "$Rf_* : D(\\mathcal{O}_\\mathcal{C}) \\to D(\\mathcal{O}_\\mathcal{D})$", "of Cohomology on Sites, Section \\ref{sites-cohomology-section-unbounded}", "agrees with the construction of", "$Rf_* : D(\\mathcal{O}_\\mathcal{C}, 0) \\to D(\\mathcal{O}_\\mathcal{D}, 0)$", "in Definition \\ref{definition-pushforward}", "as both functors are defined as the right derived extension of", "pushforward on underlying complexes of modules.", "By Lemma \\ref{lemma-compose-pushforward} we see that both", "$Rf_* \\circ R\\text{id}_{\\mathcal{C}, *}$ and", "$R\\text{id}_{\\mathcal{D}, *} \\circ Rf_*$ are the derived functors", "of $f_* \\circ forget = forget \\circ f_*$ and hence equal", "by uniqueness of adjoints." ], "refs": [ "sdga-lemma-compose-pushforward", "sdga-definition-pushforward", "sdga-lemma-compose-pushforward" ], "ref_ids": [ 7361, 7382, 7361 ] } ], "ref_ids": [] }, { "id": 7364, "type": "theorem", "label": "sdga-lemma-cohomology-ext", "categories": [ "sdga" ], "title": "sdga-lemma-cohomology-ext", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a differential graded $\\mathcal{O}$-algebra.", "Let $\\mathcal{M}$ be a differential graded $\\mathcal{A}$-module.", "Let $n \\in \\mathbf{Z}$. We have", "$$", "H^n(\\mathcal{C}, \\mathcal{M}) =", "\\Hom_{D(\\mathcal{A}, \\text{d})}(\\mathcal{A}, \\mathcal{M}[n])", "$$", "where on the left hand side we have the cohomology of $\\mathcal{M}$", "viewed as a complex of $\\mathcal{O}$-modules." ], "refs": [], "proofs": [ { "contents": [ "To prove the formula, observe that", "$$", "R\\Gamma(\\mathcal{C}, \\mathcal{M}) = \\Gamma(\\mathcal{C}, \\mathcal{I})", "$$", "where $\\mathcal{M} \\to \\mathcal{I}$ is a quasi-isomorphism", "to a graded injective and K-injective differential graded", "$\\mathcal{A}$-module $\\mathcal{I}$ (combine", "Lemmas \\ref{lemma-right-derived} and \\ref{lemma-pushforward-agrees}).", "By Lemma \\ref{lemma-hom-derived} we have", "$$", "\\Hom_{D(\\mathcal{A}, \\text{d})}(\\mathcal{A}, \\mathcal{M}[n]) =", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}, \\mathcal{I}[n]) =", "H^0(\\Gamma(\\mathcal{C}, \\mathcal{I}[n])) =", "H^n(\\Gamma(\\mathcal{C}, \\mathcal{I}))", "$$", "Combining these two results we obtain our equality." ], "refs": [ "sdga-lemma-right-derived", "sdga-lemma-pushforward-agrees", "sdga-lemma-hom-derived" ], "ref_ids": [ 7357, 7363, 7347 ] } ], "ref_ids": [] }, { "id": 7365, "type": "theorem", "label": "sdga-lemma-qis-equivalence", "categories": [ "sdga" ], "title": "sdga-lemma-qis-equivalence", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "If $\\varphi : \\mathcal{A} \\to \\mathcal{B}$ is a homomorphism of", "differential graded $\\mathcal{O}$-algebras which induces an", "isomorphism on cohomology sheaves, then", "$$", "D(\\mathcal{A}, \\text{d}) \\longrightarrow D(\\mathcal{B}, \\text{d}), \\quad", "\\mathcal{M}", "\\longmapsto", "\\mathcal{M} \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{B}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "Recall that the restriction functor", "$$", "\\text{Mod}^{dg}_{(\\mathcal{B}, \\text{d})} \\to", "\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})},\\quad", "\\mathcal{N} \\mapsto res_\\varphi \\mathcal{N}", "$$", "is a right adjoint to", "$$", "\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})} \\to", "\\text{Mod}^{dg}_{(\\mathcal{B}, \\text{d})},\\quad", "\\mathcal{M} \\mapsto \\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{B}", "$$", "See Section \\ref{section-dg-bimodules}. Since restriction", "sends quasi-isomorphisms to quasi-isomorphisms, we see that", "it trivially has a left derived extension (given by", "restriction). This functor will be right adjoint to", "$- \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{B}$ by", "Derived Categories, Lemma", "\\ref{derived-lemma-pre-derived-adjoint-functors-general}.", "The adjunction map", "$$", "\\mathcal{M} \\to", "res_\\varphi(\\mathcal{M} \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{B})", "$$", "is an isomorphism in $D(\\mathcal{A}, \\text{d})$ by our assumption", "that $\\mathcal{A} \\to \\mathcal{B}$ is a quasi-isomorphism of", "(left) differential graded $\\mathcal{A}$-modules. In particular,", "the functor of the lemma is fully faithful, see", "Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}.", "It is clear that the kernel of the restriction functor", "$D(\\mathcal{B}, \\text{d}) \\to D(\\mathcal{A}, \\text{d})$", "is zero. Thus we conclude by Derived Categories, Lemma", "\\ref{derived-lemma-fully-faithful-adjoint-kernel-zero}." ], "refs": [ "derived-lemma-pre-derived-adjoint-functors-general", "categories-lemma-adjoint-fully-faithful", "derived-lemma-fully-faithful-adjoint-kernel-zero" ], "ref_ids": [ 1905, 12248, 1793 ] } ], "ref_ids": [] }, { "id": 7366, "type": "theorem", "label": "sdga-lemma-special-good", "categories": [ "sdga" ], "title": "sdga-lemma-special-good", "contents": [ "In the situation above the differential graded $\\mathcal{O}$-algebra", "$$", "\\mathcal{A} = \\colim \\mathcal{A}_i", "$$", "has the following property: for any morphism", "$(f, f^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}')", "\\to (\\Sh(\\mathcal{C}), \\mathcal{O})$", "of ringed topoi, the pullback $f^*\\mathcal{A}$", "is flat as a graded $\\mathcal{O}'$-module and", "is K-flat as a complex of $\\mathcal{O}'$-modules." ], "refs": [], "proofs": [ { "contents": [ "Observe that $f^*\\mathcal{A} = \\colim f^*\\mathcal{A}_i$", "and that", "$$", "f^*\\mathcal{A}_i = \\mathcal{O}'\\langle", "f^{-1}\\mathcal{S}_0 \\amalg \\ldots \\amalg f^{-1}\\mathcal{S}_i\\rangle", "$$", "with differential given by the inductive procedure above using", "$f^{-1}\\delta_{i + 1}$. Thus it suffices to prove that $\\mathcal{A}$", "is flat as a graded $\\mathcal{O}$-module and", "is K-flat as a complex of $\\mathcal{O}$-modules.", "For this it suffices to prove that each $\\mathcal{A}_i$", "is flat as a graded $\\mathcal{O}$-module and", "is K-flat as a complex of $\\mathcal{O}$-modules, compare with", "Lemma \\ref{lemma-good-direct-sum}.", "\\medskip\\noindent", "For $i \\geq 1$", "write $\\mathcal{S} = \\mathcal{S}_0 \\amalg \\ldots \\amalg \\mathcal{S}_i$", "so that we have $\\mathcal{A}_i = \\mathcal{O}\\langle \\mathcal{S} \\rangle$", "as a graded $\\mathcal{O}$-algebra. We are going to construct a filtration", "of this algebra by differential graded $\\mathcal{O}$-submodules.", "\\medskip\\noindent", "Set $W = \\mathbf{Z}_{\\geq 0}^{i + 1}$ considered with lexicographical", "ordering. Namely, given $w = (w_0, \\ldots w_i)$ and", "$w' = (w'_0, \\ldots, w'_i)$ in $W$ we say", "$$", "w > w' \\Leftrightarrow", "\\exists j,\\ 0 \\leq j \\leq i :", "w_i = w'_i,\\ w_{i - 1} = w'_{i - 1},\\ \\ldots ,", "\\ w_{j + 1} = w'_{j + 1},\\ w_j > w'_j", "$$", "and so on. Suppose given a section", "$s = s_1 \\cdot \\ldots \\cdot s_r$ of", "$\\mathcal{S} \\times \\ldots \\times \\mathcal{S}$", "over $U$. We say that the {\\it weight of $s$ is defined}", "if we have $s_a \\in \\mathcal{S}_{j_a}(U)$ for a unique", "$0 \\leq j_a \\leq i$. In this case we define the weight", "$$", "w(s) = (w_0(s), \\ldots, w_i(s)) \\in W,\\quad", "w_j(s) = |\\{a \\mid j_a = j\\}|", "$$", "The weight of any section of $\\mathcal{S} \\times \\ldots \\times \\mathcal{S}$", "is defined locally. The reader checks easily that we obtain a disjoint union", "decompostion", "$$", "\\mathcal{S} \\times \\ldots \\times \\mathcal{S} =", "\\coprod\\nolimits_{w \\in W} \\left(", "\\mathcal{S} \\times \\ldots \\times \\mathcal{S}\\right)_w", "$$", "into the subsheaves of sections of a given weight. Of course", "only $w \\in W$ with $\\sum_{0 \\leq j \\leq i} w_j = r$ show up for a given $r$.", "We correspondingly obtain a decomposition", "$$", "\\mathcal{A}_i = \\mathcal{O} \\oplus", "\\bigoplus\\nolimits_{r \\geq 1}", "\\bigoplus\\nolimits_{w \\in W}", "\\mathcal{O}[\\left(\\mathcal{S} \\times \\ldots \\times \\mathcal{S}\\right)_w]", "$$", "The rest of the proof relies on the following trivial observation:", "given $r$, $w$ and local section $s = s_1 \\cdot \\ldots \\cdot s_r$ of", "$\\left(\\mathcal{S} \\times \\ldots \\times \\mathcal{S}\\right)_w$", "we have", "$$", "\\text{d}(s) \\text{ is a local section of } \\mathcal{O} \\oplus", "\\bigoplus\\nolimits_{r' \\geq 1}", "\\bigoplus\\nolimits_{w' \\in W,\\ w' < w}", "\\mathcal{O}[\\left(\\mathcal{S} \\times \\ldots \\times \\mathcal{S}\\right)_{w'}]", "$$", "The reason is that in each of the expressions", "$$", "(-1)^{\\deg(s_1) + \\ldots + \\deg(s_{a - 1})}", "s_1 \\cdot \\ldots s_{a - 1} \\cdot \\delta(s_a) \\cdot", "s_{a + 1} \\cdot \\ldots \\cdot s_r", "$$", "whose sum give the element $\\text{d}(s)$ the element $\\delta(s_a)$", "is locally a $\\mathcal{O}$-linear combination of elements", "$s'_1 \\cdot \\ldots \\cdot s'_{r'}$ with $s'_{a'}$ in", "$\\mathcal{S}_{j'_a}$ for some $0 \\leq j'_{a'} < j_a$ where $j_a$ is such that", "$s_a$ is section of $\\mathcal{S}_{j_a}$.", "\\medskip\\noindent", "What this means is the following. Suppose for $w \\in W$ we set", "$$", "F_w \\mathcal{A}_i = \\mathcal{O} \\oplus \\bigoplus\\nolimits_{r \\geq 1}", "\\bigoplus\\nolimits_{w' \\in W,\\ w' \\leq w}", "\\mathcal{O}[\\left(\\mathcal{S} \\times \\ldots \\times \\mathcal{S}\\right)_{w'}]", "$$", "By the observation above this is a differential graded $\\mathcal{O}$-submodule.", "We get admissible short exact sequences", "$$", "0 \\to \\colim_{w' < w} F_{w'}\\mathcal{A}_i \\to", "F_w\\mathcal{A}_i \\to", "\\bigoplus\\nolimits_{r \\geq 1}", "\\mathcal{O}[\\left(\\mathcal{S} \\times \\ldots \\times \\mathcal{S}\\right)_w]", "\\to 0", "$$", "of differential graded $\\mathcal{A}$-modules where the differential", "on the right hand side is zero.", "\\medskip\\noindent", "Now we finish the proof by transfinite induction over the ordered", "set $W$. The differential graded complex $F_0\\mathcal{A}_0$ is", "the summand $\\mathcal{O}$ and this is K-flat and graded flat.", "For $w \\in W$ if the result is true for $F_{w'}\\mathcal{A}_i$", "for $w' < w$, then by Lemmas \\ref{lemma-good-direct-sum},", "\\ref{lemma-good-admissible-ses}, and \\ref{lemma-free-graded-module-good}", "we obtain the result for $w$. Finally, we have", "$\\mathcal{A}_i = \\colim_{w \\in W} F_w\\mathcal{A}_i$ and", "we conclude." ], "refs": [ "sdga-lemma-good-direct-sum", "sdga-lemma-good-direct-sum", "sdga-lemma-good-admissible-ses", "sdga-lemma-free-graded-module-good" ], "ref_ids": [ 7326, 7326, 7325, 7328 ] } ], "ref_ids": [] }, { "id": 7367, "type": "theorem", "label": "sdga-lemma-good-dga", "categories": [ "sdga" ], "title": "sdga-lemma-good-dga", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{B}, \\text{d})$ be a differential graded $\\mathcal{O}$-algebra.", "There exists a quasi-isomorphism of differential graded $\\mathcal{O}$-algebras", "$(\\mathcal{A}, \\text{d}) \\to (\\mathcal{B}, \\text{d})$ such that", "$\\mathcal{A}$ is graded flat and K-flat as a complex of $\\mathcal{O}$-modules", "and such that the same is true after pullback by any morphism of", "ringed topoi." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the first proof of", "Lemma \\ref{lemma-resolve} but now working with free graded", "algebras instead of free graded modules.", "\\medskip\\noindent", "We will construct $\\mathcal{A} = \\colim \\mathcal{A}_i$ as in", "Lemma \\ref{lemma-special-good} by constructing", "$$", "\\mathcal{A}_0 \\to \\mathcal{A}_1 \\to \\mathcal{A}_2 \\to \\ldots \\to \\mathcal{B}", "$$", "Let $\\mathcal{S}_0$ be the sheaf of graded sets", "(Remark \\ref{remark-sheaf-graded-sets})", "whose degree $n$ part is $\\Ker(\\text{d}_\\mathcal{B}^n)$.", "Consider the homomorphism of differential", "graded modules", "$$", "\\mathcal{A}_0 = \\mathcal{O}\\langle \\mathcal{S}_0 \\rangle", "\\longrightarrow", "\\mathcal{B}", "$$", "where map sends a local section $s$ of $\\mathcal{S}_0$", "to the corresponding local section of $\\mathcal{A}^{\\deg(s)}$", "(which is in the kernel of the differential, so our map is", "a map of differential graded algebras indeed). By construction the", "induced maps on cohomology sheaves $H^n(\\mathcal{A}_0) \\to H^n(\\mathcal{B})$", "are surjective and hence the same will remain true for all $i$.", "\\medskip\\noindent", "Induction step of the construction. Given $\\mathcal{A}_i \\to \\mathcal{B}$", "denote $\\mathcal{S}_{i + 1}$ the sheaf of graded sets whose degree $n$ part is", "$$", "\\Ker(\\text{d}_{\\mathcal{A}_i}^{n + 1})", "\\times_{\\mathcal{B}^{n + 1}, \\text{d}}", "\\mathcal{B}^n", "$$", "This comes equipped with a canonical map", "$$", "\\delta_{i + 1} : \\mathcal{S}_{i + 1} \\longrightarrow", "\\mathcal{A}_i", "$$", "whose image is contained in the kernel of $\\text{d}_{\\mathcal{A}_i}$", "by construction. Hence $\\mathcal{A}_{i + 1} = \\mathcal{O}\\langle", "\\mathcal{S}_0 \\amalg \\ldots \\mathcal{S}_{i + 1}\\rangle$ has a differential", "exteding the differential on $\\mathcal{A}_i$, see discussion at the", "start of this section. The map from $\\mathcal{A}_{i + 1}$ to", "$\\mathcal{B}$ is the unique map of graded algebras which", "restricts to the given map on $\\mathcal{A}_i$ and sends", "a local section $s = (a, b)$ of $\\mathcal{S}_{i + 1}$", "to $b$ in $\\mathcal{B}$. This is compatible with differentials", "exactly because $\\text{d}(b)$ is the image of $a$ in $\\mathcal{B}$.", "\\medskip\\noindent", "The map $\\mathcal{A} \\to \\mathcal{B}$ is a quasi-isomorphism:", "we have $H^n(\\mathcal{A}) = \\colim H^n(\\mathcal{A}_i)$", "and for each $i$ the map $H^n(\\mathcal{A}_i) \\to H^n(\\mathcal{B})$", "is surjective with kernel annihilated by the map", "$H^n(\\mathcal{A}_i) \\to H^n(\\mathcal{A}_{i + 1})$ by construction.", "Finally, the flatness condition for $\\mathcal{A}$ where shown in", "Lemma \\ref{lemma-special-good}." ], "refs": [ "sdga-lemma-resolve", "sdga-lemma-special-good", "sdga-remark-sheaf-graded-sets", "sdga-lemma-special-good" ], "ref_ids": [ 7329, 7366, 7386, 7366 ] } ], "ref_ids": [] }, { "id": 7368, "type": "theorem", "label": "sdga-proposition-homotopy-category-triangulated", "categories": [ "sdga" ], "title": "sdga-proposition-homotopy-category-triangulated", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$.", "The homotopy category $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$", "is a triangulated category where", "\\begin{enumerate}", "\\item the shift functors are those constructed in", "Section \\ref{section-shift-dg},", "\\item the distinghuished triangles are those triangles", "in $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$ which are", "isomorphic as a triangle to a triangle", "$$", "\\mathcal{K} \\to \\mathcal{L} \\to \\mathcal{N}", "\\xrightarrow{\\delta} \\mathcal{K}[1],\\quad\\quad", "\\delta = \\pi \\circ \\text{d}_\\mathcal{L} \\circ s", "$$", "constructed from an admissible short exact sequence", "$0 \\to \\mathcal{K} \\to \\mathcal{L} \\to \\mathcal{N} \\to 0$", "in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ above.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})}) =", "K(\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})})$, see", "Section \\ref{section-homotopy}.", "Having said this, the proposition", "follows from Lemmas \\ref{lemma-axioms-AB} and \\ref{lemma-axiom-C}", "and", "Differential Graded Algebra, Proposition", "\\ref{dga-proposition-ABC-homotopy-category-triangulated}." ], "refs": [ "sdga-lemma-axioms-AB", "sdga-lemma-axiom-C", "dga-proposition-ABC-homotopy-category-triangulated" ], "ref_ids": [ 7321, 7322, 13131 ] } ], "ref_ids": [] }, { "id": 7388, "type": "theorem", "label": "stacks-morphisms-theorem-quasi-DM", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-theorem-quasi-DM", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is quasi-DM, and", "\\item there exists a scheme $W$ and a surjective, flat, locally finitely", "presented, locally quasi-finite morphism $W \\to \\mathcal{X}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is", "Lemma \\ref{lemma-properties-covering-imply-diagonal}.", "Assume (1).", "Let $x \\in |\\mathcal{X}|$ be a finite type point. We will produce a scheme", "over $\\mathcal{X}$ which ``works'' in a neighbourhood of $x$. At the end", "of the proof we will take the disjoint union of all of these to conclude.", "\\medskip\\noindent", "Let $U$ be an affine scheme, $U \\to \\mathcal{X}$ a smooth morphism, and", "$u \\in U$ a closed point which maps to $x$, see", "Lemma \\ref{lemma-point-finite-type}.", "Denote $u = \\Spec(\\kappa(u))$ as usual. Consider the following", "commutative diagram", "$$", "\\xymatrix{", "u \\ar[d] & R \\ar[l] \\ar[d] \\\\", "U \\ar[d] & F \\ar[d] \\ar[l]^p \\\\", "\\mathcal{X} & u \\ar[l]", "}", "$$", "with both squares fibre product squares, in particular", "$R = u \\times_\\mathcal{X} u$. In the proof of", "Lemma \\ref{lemma-point-finite-type-monomorphism}", "we have seen that $(u, R, s, t, c)$ is a groupoid in algebraic spaces", "with $s, t$ locally of finite type. Let $G \\to u$ be the stabilizer group", "algebraic space (see", "Groupoids in Spaces, Definition", "\\ref{spaces-groupoids-definition-stabilizer-groupoid}).", "Note that", "$$", "G = R \\times_{(u \\times u)} u =", "(u \\times_\\mathcal{X} u) \\times_{(u \\times u)} u =", "\\mathcal{X} \\times_{\\mathcal{X} \\times \\mathcal{X}} u.", "$$", "As $\\mathcal{X}$ is quasi-DM we see that", "$G$ is locally quasi-finite over $u$. By", "More on Groupoids in Spaces, Lemma", "\\ref{spaces-more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer}", "we have $\\dim(R) = 0$.", "\\medskip\\noindent", "Let $e : u \\to R$ be the identity of the groupoid. Thus both compositions", "$u \\to R \\to u$ are equal to the identity morphism of $u$.", "Note that $R \\subset F$ is a closed", "subspace as $u \\subset U$ is a closed subscheme. Hence we can also think", "of $e$ as a point of $F$. Consider the maps of \\'etale local rings", "$$", "\\mathcal{O}_{U, u}", "\\xrightarrow{p^\\sharp}", "\\mathcal{O}_{F, \\overline{e}}", "\\longrightarrow", "\\mathcal{O}_{R, \\overline{e}}", "$$", "Note that $\\mathcal{O}_{R, \\overline{e}}$ has dimension $0$ by the result", "of the first paragraph. On the other hand, the kernel of the second arrow is", "$p^\\sharp(\\mathfrak m_u)\\mathcal{O}_{F, \\overline{e}}$ as", "$R$ is cut out in $F$ by $\\mathfrak m_u$. Thus we see that", "$$", "\\mathfrak m_{\\overline{z}} =", "\\sqrt{p^\\sharp(\\mathfrak m_u)\\mathcal{O}_{F, \\overline{e}}}", "$$", "On the other hand, as the morphism $U \\to \\mathcal{X}$ is smooth", "we see that $F \\to u$ is a smooth morphism of algebraic spaces.", "This means that $F$ is a regular algebraic space", "(Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-smooth-regular}).", "Hence $\\mathcal{O}_{F, \\overline{e}}$ is a regular local ring", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-regular}).", "Note that a regular local ring is Cohen-Macaulay", "(Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}).", "Let $d = \\dim(\\mathcal{O}_{F, \\overline{e}})$. By", "Algebra, Lemma \\ref{algebra-lemma-find-sequence-image-regular}", "we can find $f_1, \\ldots, f_d \\in \\mathcal{O}_{U, u}$ whose images", "$\\varphi(f_1), \\ldots, \\varphi(f_d)$ form a regular sequence", "in $\\mathcal{O}_{F, \\overline{z}}$. By", "Lemma \\ref{lemma-slice}", "after shrinking $U$ we may assume that", "$Z = V(f_1, \\ldots, f_d) \\to \\mathcal{X}$ is flat and", "locally of finite presentation. Note that by construction", "$F_Z = Z \\times_\\mathcal{X} u$ is a closed subspace of", "$F = U \\times_\\mathcal{X} u$, that $e$ is a point of this closed subspace,", "and that", "$$", "\\dim(\\mathcal{O}_{F_Z, \\overline{e}}) = 0.", "$$", "By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-dimension-fibre-at-a-point}", "it follows that $\\dim_e(F_Z) = 0$ because the transcendence degree", "of $e$ relative to $u$ is zero. Hence it follows from", "Lemma \\ref{lemma-quasi-finite-at-point}", "that after possibly shrinking $U$ the morphism $Z \\to \\mathcal{X}$", "is locally quasi-finite.", "\\medskip\\noindent", "We conclude that for every finite type point $x$ of $\\mathcal{X}$ there", "exists a locally quasi-finite, flat, locally finitely presented", "morphism $f_x : Z_x \\to \\mathcal{X}$ with $x$ in the image of $|f_x|$.", "Set $W = \\coprod_x Z_x$ and $f = \\coprod f_x$. Then $f$ is flat, locally", "of finite presentation, and locally quasi-finite. In particular the", "image of $|f|$ is open, see", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-topology-points}.", "By construction the image contains all finite type points of $\\mathcal{X}$,", "hence $f$ is surjective by", "Lemma \\ref{lemma-enough-finite-type-points} (and", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-characterize-surjective})." ], "refs": [ "stacks-morphisms-lemma-properties-covering-imply-diagonal", "stacks-morphisms-lemma-point-finite-type", "stacks-morphisms-lemma-point-finite-type-monomorphism", "spaces-groupoids-definition-stabilizer-groupoid", "spaces-more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer", "spaces-over-fields-lemma-smooth-regular", "spaces-properties-lemma-regular", "algebra-lemma-regular-ring-CM", "algebra-lemma-find-sequence-image-regular", "stacks-morphisms-lemma-slice", "spaces-morphisms-lemma-dimension-fibre-at-a-point", "stacks-morphisms-lemma-quasi-finite-at-point", "stacks-properties-lemma-topology-points", "stacks-morphisms-lemma-enough-finite-type-points", "stacks-properties-lemma-characterize-surjective" ], "ref_ids": [ 7408, 7466, 7471, 9349, 13188, 12872, 11894, 941, 930, 7477, 4870, 7478, 8867, 7470, 8865 ] } ], "ref_ids": [] }, { "id": 7389, "type": "theorem", "label": "stacks-morphisms-theorem-DM", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-theorem-DM", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is DM,", "\\item $\\mathcal{X}$ is Deligne-Mumford, and", "\\item there exists a scheme $W$ and a surjective \\'etale", "morphism $W \\to \\mathcal{X}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that (3) is the definition of (2), see", "Algebraic Stacks, Definition \\ref{algebraic-definition-deligne-mumford}.", "The implication (3) $\\Rightarrow$ (1) is", "Lemma \\ref{lemma-properties-covering-imply-diagonal}.", "Assume (1). Let $x \\in |\\mathcal{X}|$ be a finite type point.", "We will produce a scheme over $\\mathcal{X}$ which ``works'' in a", "neighbourhood of $x$. At the end of the proof we will take the disjoint", "union of all of these to conclude.", "\\medskip\\noindent", "By", "Lemma \\ref{lemma-point-finite-type-monomorphism}", "the residual gerbe $\\mathcal{Z}_x$ of $\\mathcal{X}$ at $x$ exists and", "$\\mathcal{Z}_x \\to \\mathcal{X}$ is locally of finite type. By", "Lemma \\ref{lemma-separation-properties-residual-gerbe}", "the algebraic stack $\\mathcal{Z}_x$ is DM. By", "Lemma \\ref{lemma-DM-residual-gerbe}", "there exists a field $k$ and a surjective \\'etale morphism", "$z : \\Spec(k) \\to \\mathcal{Z}_x$.", "In particular the composition $x : \\Spec(k) \\to \\mathcal{X}$", "is locally of finite type (by", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-composition-finite-type} and", "\\ref{spaces-morphisms-lemma-etale-locally-finite-type}).", "\\medskip\\noindent", "Pick a scheme $U$ and a smooth morphism $U \\to \\mathcal{X}$ such that", "$x$ is in the image of $|U| \\to |\\mathcal{X}|$.", "Consider the following fibre square", "$$", "\\xymatrix{", "U \\ar[d] & F \\ar[l] \\ar[d] \\\\", "\\mathcal{X} & \\Spec(k) \\ar[l]_-x", "}", "$$", "in other words $F = U \\times_{\\mathcal{X}, x} \\Spec(k)$. By", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-points-cartesian}", "we see that $F$ is nonempty.", "As $\\mathcal{Z}_x \\to \\mathcal{X}$ is a monomorphism we have", "$$", "\\Spec(k) \\times_{z, \\mathcal{Z}_x, z} \\Spec(k)", "=", "\\Spec(k) \\times_{x, \\mathcal{X}, x} \\Spec(k)", "$$", "with \\'etale projection maps to $\\Spec(k)$ by construction of $z$.", "Since", "$$", "F \\times_U F =", "(\\Spec(k) \\times_\\mathcal{X} \\Spec(k))", "\\times_{\\Spec(k)} F", "$$", "we see that the projections maps $F \\times_U F \\to F$ are \\'etale as well.", "It follows that $\\Delta_{F/U} : F \\to F \\times_U F$ is \\'etale (see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-etale-permanence}).", "By", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-etale-universally-injective-open}", "this implies that $\\Delta_{F/U}$ is an open immersion, which finally", "implies by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-diagonal-unramified-morphism}", "that $F \\to U$ is unramified.", "\\medskip\\noindent", "Pick a nonempty affine scheme $V$ and an \\'etale morphism $V \\to F$.", "(This could be avoided by working directly with $F$, but it seems easier", "to explain what's going on by doing so.) Picture", "$$", "\\xymatrix{", "U \\ar[d] & F \\ar[l] \\ar[d] & V \\ar[l] \\ar[ld] \\\\", "\\mathcal{X} & \\Spec(k) \\ar[l]_-x", "}", "$$", "Then $V \\to \\Spec(k)$ is a smooth morphism of schemes and $V \\to U$ is an", "unramified morphism of schemes (see", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-composition-smooth} and", "\\ref{spaces-morphisms-lemma-composition-unramified}).", "Pick a closed point $v \\in V$ with $k \\subset \\kappa(v)$ finite separable, see", "Varieties, Lemma \\ref{varieties-lemma-smooth-separable-closed-points-dense}.", "Let $u \\in U$ be the image point. The local ring", "$\\mathcal{O}_{V, v}$ is regular (see", "Varieties, Lemma \\ref{varieties-lemma-smooth-regular})", "and the local ring homomorphism", "$$", "\\varphi : \\mathcal{O}_{U, u} \\longrightarrow \\mathcal{O}_{V, v}", "$$", "coming from the morphism $V \\to U$ is such that", "$\\varphi(\\mathfrak m_u)\\mathcal{O}_{V, v} = \\mathfrak m_v$, see", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-at-point}.", "Hence we can find $f_1, \\ldots, f_d \\in \\mathcal{O}_{U, u}$", "such that the images $\\varphi(f_1), \\ldots, \\varphi(f_d)$", "form a basis for $\\mathfrak m_v/\\mathfrak m_v^2$ over $\\kappa(v)$.", "Since $\\mathcal{O}_{V, v}$ is a regular local ring this implies", "that $\\varphi(f_1), \\ldots, \\varphi(f_d)$ form a regular sequence", "in $\\mathcal{O}_{V, v}$ (see", "Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}).", "After replacing $U$ by an open neighbourhood of $u$ we may assume", "$f_1, \\ldots, f_d \\in \\Gamma(U, \\mathcal{O}_U)$. After replacing", "$U$ by a possibly even smaller open neighbourhood of $u$ we may", "assume that $V(f_1, \\ldots, f_d) \\to \\mathcal{X}$ is flat and", "locally of finite presentation, see", "Lemma \\ref{lemma-slice}.", "By construction", "$$", "V(f_1, \\ldots, f_d) \\times_\\mathcal{X} \\Spec(k)", "\\longleftarrow", "V(f_1, \\ldots, f_d) \\times_\\mathcal{X} V", "$$", "is \\'etale and $V(f_1, \\ldots, f_d) \\times_\\mathcal{X} V$", "is the closed subscheme $T \\subset V$ cut out by $f_1|_V, \\ldots, f_d|_V$.", "Hence by construction $v \\in T$ and", "$$", "\\mathcal{O}_{T, v} =", "\\mathcal{O}_{V, v}/(\\varphi(f_1), \\ldots, \\varphi(f_d)) = \\kappa(v)", "$$", "a finite separable extension of $k$. It follows that $T \\to \\Spec(k)$", "is unramified at $v$, see", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-at-point}.", "By definition of an unramified morphism of algebraic spaces this means that", "$V(f_1, \\ldots, f_d) \\times_\\mathcal{X} \\Spec(k) \\to \\Spec(k)$", "is unramified at the image of $v$ in", "$V(f_1, \\ldots, f_d) \\times_\\mathcal{X} \\Spec(k)$.", "Applying", "Lemma \\ref{lemma-etale-at-point}", "we see that on shrinking $U$ to yet another open neighbourhood of $u$", "the morphism $V(f_1, \\ldots, f_d) \\to \\mathcal{X}$ is \\'etale.", "\\medskip\\noindent", "We conclude that for every finite type point $x$ of $\\mathcal{X}$ there", "exists an \\'etale morphism $f_x : W_x \\to \\mathcal{X}$ with $x$ in the", "image of $|f_x|$. Set $W = \\coprod_x W_x$ and $f = \\coprod f_x$. Then $f$", "is \\'etale. In particular the image of $|f|$ is open, see", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-topology-points}.", "By construction the image contains all finite type points of $\\mathcal{X}$,", "hence $f$ is surjective by", "Lemma \\ref{lemma-enough-finite-type-points} (and", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-characterize-surjective})." ], "refs": [ "algebraic-definition-deligne-mumford", "stacks-morphisms-lemma-properties-covering-imply-diagonal", "stacks-morphisms-lemma-point-finite-type-monomorphism", "stacks-morphisms-lemma-separation-properties-residual-gerbe", "stacks-morphisms-lemma-DM-residual-gerbe", "spaces-morphisms-lemma-composition-finite-type", "spaces-morphisms-lemma-etale-locally-finite-type", "stacks-properties-lemma-points-cartesian", "spaces-morphisms-lemma-etale-permanence", "spaces-morphisms-lemma-etale-universally-injective-open", "spaces-morphisms-lemma-diagonal-unramified-morphism", "spaces-morphisms-lemma-composition-smooth", "spaces-morphisms-lemma-composition-unramified", "varieties-lemma-smooth-separable-closed-points-dense", "varieties-lemma-smooth-regular", "morphisms-lemma-unramified-at-point", "algebra-lemma-regular-ring-CM", "stacks-morphisms-lemma-slice", "morphisms-lemma-unramified-at-point", "stacks-morphisms-lemma-etale-at-point", "stacks-properties-lemma-topology-points", "stacks-morphisms-lemma-enough-finite-type-points", "stacks-properties-lemma-characterize-surjective" ], "ref_ids": [ 8485, 7408, 7471, 7410, 7479, 4814, 4912, 8864, 4914, 4973, 4902, 4886, 4896, 11007, 11004, 5355, 941, 7477, 5355, 7480, 8867, 7470, 8865 ] } ], "ref_ids": [] }, { "id": 7390, "type": "theorem", "label": "stacks-morphisms-lemma-isom-locally-finite-type", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-isom-locally-finite-type", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Let $T$ be a scheme and let $x, y$ be objects of the fibre category of", "$\\mathcal{X}$ over $T$. Then the morphism", "$\\mathit{Isom}_\\mathcal{X}(x, y) \\to T$ is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "By", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-stack-presentation}", "we may assume that $\\mathcal{X} = [U/R]$ for some smooth", "groupoid in algebraic spaces.", "By", "Descent on Spaces,", "Lemma \\ref{spaces-descent-lemma-descending-property-locally-finite-type}", "it suffices to check the property fppf locally on $T$.", "Thus we may assume that $x, y$ come from morphisms", "$x', y' : T \\to U$. By", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-morphisms}", "we see that in this case", "$\\mathit{Isom}_\\mathcal{X}(x, y) = T \\times_{(y', x'), U \\times_S U} R$.", "Hence it suffices to prove that $R \\to U \\times_S U$ is", "locally of finite type. This follows from the fact that the composition", "$s : R \\to U \\times_S U \\to U$ is smooth (hence locally of finite type, see", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-smooth-locally-finite-presentation} and", "\\ref{spaces-morphisms-lemma-finite-presentation-finite-type})", "and", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-permanence-finite-type}." ], "refs": [ "algebraic-lemma-stack-presentation", "spaces-descent-lemma-descending-property-locally-finite-type", "spaces-groupoids-lemma-quotient-stack-morphisms", "spaces-morphisms-lemma-smooth-locally-finite-presentation", "spaces-morphisms-lemma-finite-presentation-finite-type", "spaces-morphisms-lemma-permanence-finite-type" ], "ref_ids": [ 8474, 9389, 9323, 4889, 4842, 4818 ] } ], "ref_ids": [] }, { "id": 7391, "type": "theorem", "label": "stacks-morphisms-lemma-isom-pseudo-torsor-aut", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-isom-pseudo-torsor-aut", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Let $T$ be a scheme and let $x, y$ be objects of the fibre category of", "$\\mathcal{X}$ over $T$. Then", "\\begin{enumerate}", "\\item $\\mathit{Isom}_\\mathcal{X}(y, y)$ is a group algebraic space", "over $T$, and", "\\item $\\mathit{Isom}_\\mathcal{X}(x, y)$ is a pseudo torsor for", "$\\mathit{Isom}_\\mathcal{X}(y, y)$ over $T$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See", "Groupoids in Spaces,", "Definitions \\ref{spaces-groupoids-definition-group-space} and", "\\ref{spaces-groupoids-definition-pseudo-torsor}.", "The lemma follows immediately from the fact that $\\mathcal{X}$ is a", "stack in groupoids." ], "refs": [ "spaces-groupoids-definition-group-space", "spaces-groupoids-definition-pseudo-torsor" ], "ref_ids": [ 9341, 9344 ] } ], "ref_ids": [] }, { "id": 7392, "type": "theorem", "label": "stacks-morphisms-lemma-properties-diagonal", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-properties-diagonal", "contents": [ "\\begin{slogan}", "Diagonals of morphisms of algebraic stacks are representable by", "algebraic spaces and locally of finite type.", "\\end{slogan}", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Then", "\\begin{enumerate}", "\\item $\\Delta_f$ is representable by algebraic spaces,", "and", "\\item $\\Delta_f$ is locally of finite type.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme and let", "$a : T \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "be a morphism. By definition of the fibre product and the", "$2$-Yoneda lemma the morphism $a$ is given by a triple", "$a = (x, x', \\alpha)$ where $x, x'$ are objects of $\\mathcal{X}$", "over $T$, and $\\alpha : f(x) \\to f(x')$ is a morphism in the fibre", "category of $\\mathcal{Y}$ over $T$. By definition of an algebraic", "stack the sheaves $\\mathit{Isom}_\\mathcal{X}(x, x')$ and", "$\\mathit{Isom}_\\mathcal{Y}(f(x), f(x'))$ are algebraic spaces", "over $T$. In this language $\\alpha$ defines a section of the morphism", "$\\mathit{Isom}_\\mathcal{Y}(f(x), f(x')) \\to T$. A $T'$-valued point of", "$\\mathcal{X} \\times_{\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}, a} T$", "for $T' \\to T$ a scheme over $T$ is the same thing as an isomorphism", "$x|_{T'} \\to x'|_{T'}$ whose image under $f$ is $\\alpha|_{T'}$.", "Thus we see that", "\\begin{equation}", "\\label{equation-diagonal}", "\\vcenter{", "\\xymatrix{", "\\mathcal{X} \\times_{\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}, a} T", "\\ar[d] \\ar[r] &", "\\mathit{Isom}_\\mathcal{X}(x, x') \\ar[d] \\\\", "T\\ar[r]^-\\alpha &", "\\mathit{Isom}_\\mathcal{Y}(f(x), f(x'))", "}", "}", "\\end{equation}", "is a fibre square of sheaves over $T$. In particular we see that", "$\\mathcal{X} \\times_{\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}, a} T$", "is an algebraic space which proves part (1) of the lemma.", "\\medskip\\noindent", "To prove the second statement we have to show that the left", "vertical arrow of Diagram (\\ref{equation-diagonal}) is locally", "of finite type. By", "Lemma \\ref{lemma-isom-locally-finite-type}", "the algebraic space $\\mathit{Isom}_\\mathcal{X}(x, x')$ and", "is locally of finite type over $T$. Hence the right vertical arrow of", "Diagram (\\ref{equation-diagonal}) is locally of finite type, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-permanence-finite-type}.", "We conclude by", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-base-change-finite-type}." ], "refs": [ "stacks-morphisms-lemma-isom-locally-finite-type", "spaces-morphisms-lemma-permanence-finite-type", "spaces-morphisms-lemma-base-change-finite-type" ], "ref_ids": [ 7390, 4818, 4815 ] } ], "ref_ids": [] }, { "id": 7393, "type": "theorem", "label": "stacks-morphisms-lemma-properties-diagonal-representable", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-properties-diagonal-representable", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "which is representable by algebraic spaces. Then", "\\begin{enumerate}", "\\item $\\Delta_f$ is representable", "(by schemes),", "\\item $\\Delta_f$ is locally of finite type,", "\\item $\\Delta_f$ is a monomorphism,", "\\item $\\Delta_f$ is separated, and", "\\item $\\Delta_f$ is locally quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have already seen in", "Lemma \\ref{lemma-properties-diagonal}", "that $\\Delta_f$ is representable by algebraic", "spaces. Hence the statements (2) -- (5) make sense, see", "Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}.", "Also", "Lemma \\ref{lemma-properties-diagonal}", "guarantees (2) holds.", "Let $T \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$ be a morphism", "and contemplate Diagram (\\ref{equation-diagonal}). By", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids}", "the right vertical arrow is injective as a map of sheaves, i.e., a", "monomorphism of algebraic spaces. Hence also the morphism", "$T \\times_{\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}} \\mathcal{X} \\to T$", "is a monomorphism. Thus (3) holds. We already know that", "$T \\times_{\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}} \\mathcal{X} \\to T$", "is locally of finite type. Thus", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}", "allows us to conclude that", "$T \\times_{\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}} \\mathcal{X} \\to T$", "is locally quasi-finite and separated. This proves (4) and (5).", "Finally,", "Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}", "implies that", "$T \\times_{\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}} \\mathcal{X}$", "is a scheme which proves (1)." ], "refs": [ "stacks-morphisms-lemma-properties-diagonal", "stacks-morphisms-lemma-properties-diagonal", "algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids", "spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme" ], "ref_ids": [ 7392, 7392, 8442, 4838, 4983 ] } ], "ref_ids": [] }, { "id": 7394, "type": "theorem", "label": "stacks-morphisms-lemma-representable-separated-diagonal-closed", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-representable-separated-diagonal-closed", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "representable by algebraic spaces. Then the following are equivalent", "\\begin{enumerate}", "\\item $f$ is separated,", "\\item $\\Delta_f$ is a closed immersion,", "\\item $\\Delta_f$ is proper, or", "\\item $\\Delta_f$ is universally closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The statements", "``$f$ is separated'',", "``$\\Delta_f$ is a closed immersion'',", "``$\\Delta_f$ is universally closed'', and", "``$\\Delta_f$ is proper''", "refer to the notions defined in", "Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}.", "Choose a scheme $V$ and a surjective smooth morphism $V \\to \\mathcal{Y}$.", "Set $U = \\mathcal{X} \\times_\\mathcal{Y} V$ which is an algebraic", "space by assumption, and the morphism $U \\to \\mathcal{X}$ is surjective", "and smooth. By", "Categories, Lemma \\ref{categories-lemma-base-change-diagonal}", "and", "Properties of Stacks,", "Lemma \\ref{stacks-properties-lemma-check-property-covering}", "we see that for any property $P$ (as in that lemma) we have:", "$\\Delta_f$ has $P$ if and only if $\\Delta_{U/V} : U \\to U \\times_V U$ has $P$.", "Hence the equivalence of (2), (3) and (4) follows from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-separated-diagonal-proper}", "applied to $U \\to V$.", "Moreover, if (1) holds, then $U \\to V$ is separated and we see that", "$\\Delta_{U/V}$ is a closed immersion, i.e., (2) holds.", "Finally, assume (2) holds. Let $T$ be a scheme, and $a : T \\to \\mathcal{Y}$", "a morphism. Set $T' = \\mathcal{X} \\times_\\mathcal{Y} T$. To prove", "(1) we have to show that the morphism of algebraic spaces $T' \\to T$", "is separated. Using", "Categories, Lemma \\ref{categories-lemma-base-change-diagonal}", "once more we see that $\\Delta_{T'/T}$ is the base change of", "$\\Delta_f$. Hence our assumption (2) implies that $\\Delta_{T'/T}$", "is a closed immersion, hence $T' \\to T$ is separated as desired." ], "refs": [ "categories-lemma-base-change-diagonal", "stacks-properties-lemma-check-property-covering", "spaces-morphisms-lemma-separated-diagonal-proper", "categories-lemma-base-change-diagonal" ], "ref_ids": [ 12279, 8859, 4923, 12279 ] } ], "ref_ids": [] }, { "id": 7395, "type": "theorem", "label": "stacks-morphisms-lemma-representable-quasi-separated-diagonal-quasi-compact", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-representable-quasi-separated-diagonal-quasi-compact", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "representable by algebraic spaces. Then the following are equivalent", "\\begin{enumerate}", "\\item $f$ is quasi-separated,", "\\item $\\Delta_f$ is quasi-compact, or", "\\item $\\Delta_f$ is of finite type.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The statements", "``$f$ is quasi-separated'',", "``$\\Delta_f$ is quasi-compact'', and", "``$\\Delta_f$ is of finite type''", "refer to the notions defined in", "Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}.", "Note that (2) and (3) are equivalent in view of the fact that", "$\\Delta_f$ is locally of finite type by", "Lemma \\ref{lemma-properties-diagonal-representable}", "(and", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-representable-transformations-property-implication}).", "Choose a scheme $V$ and a surjective smooth morphism $V \\to \\mathcal{Y}$.", "Set $U = \\mathcal{X} \\times_\\mathcal{Y} V$ which is an algebraic", "space by assumption, and the morphism $U \\to \\mathcal{X}$ is surjective", "and smooth. By", "Categories, Lemma \\ref{categories-lemma-base-change-diagonal}", "and", "Properties of Stacks,", "Lemma \\ref{stacks-properties-lemma-check-property-covering}", "we see that we have: $\\Delta_f$ is quasi-compact if and only if", "$\\Delta_{U/V} : U \\to U \\times_V U$ is quasi-compact.", "If (1) holds, then $U \\to V$ is quasi-separated and we see that", "$\\Delta_{U/V}$ is quasi-compact, i.e., (2) holds.", "Assume (2) holds. Let $T$ be a scheme, and $a : T \\to \\mathcal{Y}$", "a morphism. Set $T' = \\mathcal{X} \\times_\\mathcal{Y} T$. To prove", "(1) we have to show that the morphism of algebraic spaces $T' \\to T$", "is quasi-separated. Using", "Categories, Lemma \\ref{categories-lemma-base-change-diagonal}", "once more we see that $\\Delta_{T'/T}$ is the base change of", "$\\Delta_f$. Hence our assumption (2) implies that $\\Delta_{T'/T}$", "is quasi-compact, hence $T' \\to T$ is quasi-separated as desired." ], "refs": [ "stacks-morphisms-lemma-properties-diagonal-representable", "algebraic-lemma-representable-transformations-property-implication", "categories-lemma-base-change-diagonal", "stacks-properties-lemma-check-property-covering", "categories-lemma-base-change-diagonal" ], "ref_ids": [ 7393, 8459, 12279, 8859, 12279 ] } ], "ref_ids": [] }, { "id": 7396, "type": "theorem", "label": "stacks-morphisms-lemma-representable-locally-separated-diagonal-immersion", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-representable-locally-separated-diagonal-immersion", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "representable by algebraic spaces. Then the following are equivalent", "\\begin{enumerate}", "\\item $f$ is locally separated, and", "\\item $\\Delta_f$ is an immersion.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The statements ``$f$ is locally separated'', and ``$\\Delta_f$ is an immersion''", "refer to the notions defined in", "Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}.", "Proof omitted. Hint: Argue as in the proofs of", "Lemmas \\ref{lemma-representable-separated-diagonal-closed} and", "\\ref{lemma-representable-quasi-separated-diagonal-quasi-compact}." ], "refs": [ "stacks-morphisms-lemma-representable-separated-diagonal-closed", "stacks-morphisms-lemma-representable-quasi-separated-diagonal-quasi-compact" ], "ref_ids": [ 7394, 7395 ] } ], "ref_ids": [] }, { "id": 7397, "type": "theorem", "label": "stacks-morphisms-lemma-trivial-implications", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-trivial-implications", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "\\begin{enumerate}", "\\item If $f$ is separated, then $f$ is quasi-separated.", "\\item If $f$ is DM, then $f$ is quasi-DM.", "\\item If $f$ is representable by algebraic spaces, then $f$ is DM.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To see (1) note that a proper morphism of algebraic spaces is quasi-compact", "and quasi-separated, see", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-proper}.", "To see (2) note that an unramified morphism of algebraic spaces is locally", "quasi-finite, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-unramified-quasi-finite}.", "Finally (3) follows from Lemma \\ref{lemma-properties-diagonal-representable}." ], "refs": [ "spaces-morphisms-definition-proper", "spaces-morphisms-lemma-unramified-quasi-finite", "stacks-morphisms-lemma-properties-diagonal-representable" ], "ref_ids": [ 5015, 4900, 7393 ] } ], "ref_ids": [] }, { "id": 7398, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-separated", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-separated", "contents": [ "All of the separation axioms listed in", "Definition \\ref{definition-separated}", "are stable under base change." ], "refs": [ "stacks-morphisms-definition-separated" ], "proofs": [ { "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and", "$\\mathcal{Y}' \\to \\mathcal{Y}$ be morphisms of algebraic stacks.", "Let $f' : \\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Y}'$", "be the base change of $f$ by $\\mathcal{Y}' \\to \\mathcal{Y}$.", "Then $\\Delta_{f'}$ is the base change of $\\Delta_f$ by the morphism", "$\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{X}' \\to", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$, see", "Categories, Lemma \\ref{categories-lemma-base-change-diagonal}.", "By the results of", "Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}", "each of the properties of the diagonal used in", "Definition \\ref{definition-separated}", "is stable under base change. Hence the lemma is true." ], "refs": [ "categories-lemma-base-change-diagonal", "stacks-morphisms-definition-separated" ], "ref_ids": [ 12279, 7601 ] } ], "ref_ids": [ 7601 ] }, { "id": 7399, "type": "theorem", "label": "stacks-morphisms-lemma-check-separated-covering", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-check-separated-covering", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $W \\to \\mathcal{Y}$ be a surjective, flat, and locally of finite", "presentation where $W$ is an algebraic space. If the base change", "$W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ has one of the separation properties", "of Definition \\ref{definition-separated}", "then so does $f$." ], "refs": [ "stacks-morphisms-definition-separated" ], "proofs": [ { "contents": [ "Denote $g : W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ the base change.", "Then $\\Delta_g$ is the base change of $\\Delta_f$ by the morphism", "$q : W \\times_\\mathcal{Y} (\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X})", "\\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$. Since $q$ is the base", "change of $W \\to \\mathcal{Y}$ we see that $q$ is representable by algebraic", "spaces, surjective, flat, and locally of finite presentation. Hence the", "result follows from", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-weak-covering}." ], "refs": [ "stacks-properties-lemma-check-property-weak-covering" ], "ref_ids": [ 8860 ] } ], "ref_ids": [ 7601 ] }, { "id": 7400, "type": "theorem", "label": "stacks-morphisms-lemma-change-of-base-separated", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-change-of-base-separated", "contents": [ "Let $S$ be a scheme. The property of being", "quasi-DM over $S$, quasi-separated over $S$, or separated over $S$ (see", "Definition \\ref{definition-absolute-separated})", "is stable under change of base scheme, see", "Algebraic Stacks, Definition \\ref{algebraic-definition-change-of-base}." ], "refs": [ "stacks-morphisms-definition-absolute-separated", "algebraic-definition-change-of-base" ], "proofs": [ { "contents": [ "Follows immediately from", "Lemma \\ref{lemma-base-change-separated}." ], "refs": [ "stacks-morphisms-lemma-base-change-separated" ], "ref_ids": [ 7398 ] } ], "ref_ids": [ 7602, 8490 ] }, { "id": 7401, "type": "theorem", "label": "stacks-morphisms-lemma-fibre-product-after-map", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-fibre-product-after-map", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Z}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$", "and $\\mathcal{Z} \\to \\mathcal{T}$ be morphisms of algebraic stacks.", "Consider the induced morphism", "$i : \\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to", "\\mathcal{X} \\times_\\mathcal{T} \\mathcal{Y}$.", "Then", "\\begin{enumerate}", "\\item $i$ is representable by algebraic spaces and locally of finite type,", "\\item if $\\Delta_{\\mathcal{Z}/\\mathcal{T}}$ is quasi-separated, then", "$i$ is quasi-separated,", "\\item if $\\Delta_{\\mathcal{Z}/\\mathcal{T}}$ is separated, then", "$i$ is separated,", "\\item if $\\mathcal{Z} \\to \\mathcal{T}$ is DM,", "then $i$ is unramified,", "\\item if $\\mathcal{Z} \\to \\mathcal{T}$ is quasi-DM,", "then $i$ is locally quasi-finite,", "\\item if $\\mathcal{Z} \\to \\mathcal{T}$ is separated, then $i$ is proper, and", "\\item if $\\mathcal{Z} \\to \\mathcal{T}$ is quasi-separated, then", "$i$ is quasi-compact and quasi-separated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The following diagram", "$$", "\\xymatrix{", "\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\ar[r]_i \\ar[d] &", "\\mathcal{X} \\times_\\mathcal{T} \\mathcal{Y} \\ar[d] \\\\", "\\mathcal{Z} \\ar[r]^-{\\Delta_{\\mathcal{Z}/\\mathcal{T}}} \\ar[r] &", "\\mathcal{Z} \\times_\\mathcal{T} \\mathcal{Z}", "}", "$$", "is a $2$-fibre product diagram, see", "Categories, Lemma \\ref{categories-lemma-fibre-product-after-map}.", "Hence $i$ is the base change of the", "diagonal morphism $\\Delta_{\\mathcal{Z}/\\mathcal{T}}$. Thus the lemma follows", "from", "Lemma \\ref{lemma-properties-diagonal},", "and the material in", "Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}." ], "refs": [ "categories-lemma-fibre-product-after-map", "stacks-morphisms-lemma-properties-diagonal" ], "ref_ids": [ 12278, 7392 ] } ], "ref_ids": [] }, { "id": 7402, "type": "theorem", "label": "stacks-morphisms-lemma-semi-diagonal", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-semi-diagonal", "contents": [ "Let $\\mathcal{T}$ be an algebraic stack. Let $g : \\mathcal{X} \\to \\mathcal{Y}$", "be a morphism of algebraic stacks over $\\mathcal{T}$. Consider the graph", "$i : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{T} \\mathcal{Y}$ of $g$. Then", "\\begin{enumerate}", "\\item $i$ is representable by algebraic spaces and locally of finite type,", "\\item if $\\mathcal{Y} \\to \\mathcal{T}$ is DM, then $i$ is unramified,", "\\item if $\\mathcal{Y} \\to \\mathcal{T}$ is quasi-DM, then $i$ is locally", "quasi-finite,", "\\item if $\\mathcal{Y} \\to \\mathcal{T}$ is separated, then $i$ is proper, and", "\\item if $\\mathcal{Y} \\to \\mathcal{T}$ is quasi-separated, then $i$ is", "quasi-compact and quasi-separated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-fibre-product-after-map}", "applied to the morphism", "$\\mathcal{X} = \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Y} \\to", "\\mathcal{X} \\times_\\mathcal{T} \\mathcal{Y}$." ], "refs": [ "stacks-morphisms-lemma-fibre-product-after-map" ], "ref_ids": [ 7401 ] } ], "ref_ids": [] }, { "id": 7403, "type": "theorem", "label": "stacks-morphisms-lemma-section-immersion", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-section-immersion", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{T}$ be a morphism of algebraic stacks.", "Let $s : \\mathcal{T} \\to \\mathcal{X}$ be a morphism such that", "$f \\circ s$ is $2$-isomorphic to $\\text{id}_\\mathcal{T}$. Then", "\\begin{enumerate}", "\\item $s$ is representable by algebraic spaces and locally of finite type,", "\\item if $f$ is DM, then $s$ is unramified,", "\\item if $f$ is quasi-DM, then $s$ is locally quasi-finite,", "\\item if $f$ is separated, then $s$ is proper, and", "\\item if $f$ is quasi-separated, then $s$ is quasi-compact and quasi-separated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-semi-diagonal} applied to", "$g = s$ and $\\mathcal{Y} = \\mathcal{T}$ in which case", "$i : \\mathcal{T} \\to \\mathcal{T} \\times_\\mathcal{T} \\mathcal{X}$", "is $2$-isomorphic to $s$." ], "refs": [ "stacks-morphisms-lemma-semi-diagonal" ], "ref_ids": [ 7402 ] } ], "ref_ids": [] }, { "id": 7404, "type": "theorem", "label": "stacks-morphisms-lemma-composition-separated", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-separated", "contents": [ "All of the separation axioms listed in", "Definition \\ref{definition-separated}", "are stable under composition of morphisms." ], "refs": [ "stacks-morphisms-definition-separated" ], "proofs": [ { "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and", "$g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks", "to which the axiom in question applies.", "The diagonal $\\Delta_{\\mathcal{X}/\\mathcal{Z}}$ is the composition", "$$", "\\mathcal{X} \\longrightarrow", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\longrightarrow", "\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{X}.", "$$", "Our separation axiom is defined by requiring the diagonal", "to have some property $\\mathcal{P}$. By", "Lemma \\ref{lemma-fibre-product-after-map}", "above we see that the second arrow also has this property.", "Hence the lemma follows since the composition of", "morphisms which are representable by algebraic spaces with property", "$\\mathcal{P}$ also is a morphism with property $\\mathcal{P}$, see", "our general discussion in", "Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}", "and", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-composition-unramified},", "\\ref{spaces-morphisms-lemma-composition-quasi-finite},", "\\ref{spaces-morphisms-lemma-composition-proper},", "\\ref{spaces-morphisms-lemma-composition-quasi-compact}, and", "\\ref{spaces-morphisms-lemma-composition-separated}." ], "refs": [ "stacks-morphisms-lemma-fibre-product-after-map", "spaces-morphisms-lemma-composition-unramified", "spaces-morphisms-lemma-composition-quasi-finite", "spaces-morphisms-lemma-composition-proper", "spaces-morphisms-lemma-composition-quasi-compact", "spaces-morphisms-lemma-composition-separated" ], "ref_ids": [ 7401, 4896, 4831, 4918, 4739, 4718 ] } ], "ref_ids": [ 7601 ] }, { "id": 7405, "type": "theorem", "label": "stacks-morphisms-lemma-separated-over-separated", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-separated-over-separated", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "over the base scheme $S$.", "\\begin{enumerate}", "\\item If $\\mathcal{Y}$ is DM over $S$ and $f$ is DM,", "then $\\mathcal{X}$ is DM over $S$.", "\\item If $\\mathcal{Y}$ is quasi-DM over $S$ and $f$ is quasi-DM,", "then $\\mathcal{X}$ is quasi-DM over $S$.", "\\item If $\\mathcal{Y}$ is separated over $S$ and $f$ is separated,", "then $\\mathcal{X}$ is separated over $S$.", "\\item If $\\mathcal{Y}$ is quasi-separated over $S$ and $f$ is quasi-separated,", "then $\\mathcal{X}$ is quasi-separated over $S$.", "\\item If $\\mathcal{Y}$ is DM and $f$ is DM,", "then $\\mathcal{X}$ is DM.", "\\item If $\\mathcal{Y}$ is quasi-DM and $f$ is quasi-DM,", "then $\\mathcal{X}$ is quasi-DM.", "\\item If $\\mathcal{Y}$ is separated and $f$ is separated,", "then $\\mathcal{X}$ is separated.", "\\item If $\\mathcal{Y}$ is quasi-separated and $f$ is quasi-separated,", "then $\\mathcal{X}$ is quasi-separated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1), (2), (3), and (4) follow immediately from", "Lemma \\ref{lemma-composition-separated}", "and", "Definition \\ref{definition-absolute-separated}.", "For (5), (6), (7), and (8) think of $\\mathcal{X}$ and $\\mathcal{Y}$ as", "algebraic stacks over $\\Spec(\\mathbf{Z})$ and apply", "Lemma \\ref{lemma-composition-separated}.", "Details omitted." ], "refs": [ "stacks-morphisms-lemma-composition-separated", "stacks-morphisms-definition-absolute-separated", "stacks-morphisms-lemma-composition-separated" ], "ref_ids": [ 7404, 7602, 7404 ] } ], "ref_ids": [] }, { "id": 7406, "type": "theorem", "label": "stacks-morphisms-lemma-compose-after-separated", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-compose-after-separated", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and", "$g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks.", "\\begin{enumerate}", "\\item If $g \\circ f$ is DM then so is $f$.", "\\item If $g \\circ f$ is quasi-DM then so is $f$.", "\\item If $g \\circ f$ is separated and $\\Delta_g$ is separated, then", "$f$ is separated.", "\\item If $g \\circ f$ is quasi-separated and", "$\\Delta_g$ is quasi-separated, then $f$ is quasi-separated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the factorization", "$$", "\\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{X}", "$$", "of the diagonal morphism of $g \\circ f$. Both morphisms are representable by", "algebraic spaces, see", "Lemmas \\ref{lemma-properties-diagonal} and", "\\ref{lemma-fibre-product-after-map}.", "Hence for any scheme $T$ and morphism", "$T \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "we get morphisms of algebraic spaces", "$$", "A = \\mathcal{X} \\times_{(\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{X})} T", "\\longrightarrow", "B = (\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X})", "\\times_{(\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{X})} T", "\\longrightarrow", "T.", "$$", "If $g \\circ f$ is DM (resp.\\ quasi-DM), then the composition $A \\to T$", "is unramified (resp.\\ locally quasi-finite). Hence (1) (resp.\\ (2))", "follows on applying", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-permanence-unramified}", "(resp.", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-permanence-quasi-finite}).", "This proves (1) and (2).", "\\medskip\\noindent", "Proof of (4). Assume $g \\circ f$ is quasi-separated and $\\Delta_g$ is", "quasi-separated. Consider the factorization", "$$", "\\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{X}", "$$", "of the diagonal morphism of $g \\circ f$. Both morphisms are", "representable by algebraic spaces and the second one is quasi-separated, see", "Lemmas \\ref{lemma-properties-diagonal} and", "\\ref{lemma-fibre-product-after-map}.", "Hence for any scheme $T$ and morphism", "$T \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "we get morphisms of algebraic spaces", "$$", "A = \\mathcal{X} \\times_{(\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{X})} T", "\\longrightarrow", "B = (\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X})", "\\times_{(\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{X})} T", "\\longrightarrow", "T", "$$", "such that $B \\to T$ is quasi-separated.", "The composition $A \\to T$ is quasi-compact and quasi-separated", "as we have assumed that $g \\circ f$ is quasi-separated.", "Hence $A \\to B$ is quasi-separated by", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-compose-after-separated}.", "And $A \\to B$ is quasi-compact by", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-quasi-compact-permanence}.", "Thus $f$ is quasi-separated.", "\\medskip\\noindent", "Proof of (3). Assume $g \\circ f$ is separated and $\\Delta_g$ is", "separated. Consider the factorization", "$$", "\\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{X}", "$$", "of the diagonal morphism of $g \\circ f$. Both morphisms are", "representable by algebraic spaces and the second one is separated, see", "Lemmas \\ref{lemma-properties-diagonal} and", "\\ref{lemma-fibre-product-after-map}.", "Hence for any scheme $T$ and morphism", "$T \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "we get morphisms of algebraic spaces", "$$", "A = \\mathcal{X} \\times_{(\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{X})} T", "\\longrightarrow", "B = (\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X})", "\\times_{(\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{X})} T", "\\longrightarrow", "T", "$$", "such that $B \\to T$ is separated.", "The composition $A \\to T$ is proper as we have assumed that", "$g \\circ f$ is quasi-separated. Hence $A \\to B$ is proper by", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-universally-closed-permanence}", "which means that $f$ is separated." ], "refs": [ "stacks-morphisms-lemma-properties-diagonal", "stacks-morphisms-lemma-fibre-product-after-map", "spaces-morphisms-lemma-permanence-unramified", "spaces-morphisms-lemma-permanence-quasi-finite", "stacks-morphisms-lemma-properties-diagonal", "stacks-morphisms-lemma-fibre-product-after-map", "spaces-morphisms-lemma-compose-after-separated", "spaces-morphisms-lemma-quasi-compact-permanence", "stacks-morphisms-lemma-properties-diagonal", "stacks-morphisms-lemma-fibre-product-after-map", "spaces-morphisms-lemma-universally-closed-permanence" ], "ref_ids": [ 7392, 7401, 4904, 4836, 7392, 7401, 4720, 4743, 7392, 7401, 4920 ] } ], "ref_ids": [] }, { "id": 7407, "type": "theorem", "label": "stacks-morphisms-lemma-separated-implies-morphism-separated", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-separated-implies-morphism-separated", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack over the base scheme $S$.", "\\begin{enumerate}", "\\item", "$\\mathcal{X}$ is DM $\\Leftrightarrow$", "$\\mathcal{X}$ is DM over $S$.", "\\item", "$\\mathcal{X}$ is quasi-DM $\\Leftrightarrow$", "$\\mathcal{X}$ is quasi-DM over $S$.", "\\item If $\\mathcal{X}$ is separated, then", "$\\mathcal{X}$ is separated over $S$.", "\\item If $\\mathcal{X}$ is quasi-separated, then", "$\\mathcal{X}$ is quasi-separated over $S$.", "\\end{enumerate}", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "over the base scheme $S$.", "\\begin{enumerate}", "\\item[(5)] If $\\mathcal{X}$ is DM over $S$, then $f$ is DM.", "\\item[(6)] If $\\mathcal{X}$ is quasi-DM over $S$, then $f$ is quasi-DM.", "\\item[(7)] If $\\mathcal{X}$ is separated over $S$ and", "$\\Delta_{\\mathcal{Y}/S}$ is separated, then $f$ is separated.", "\\item[(8)] If $\\mathcal{X}$ is quasi-separated over $S$ and", "$\\Delta_{\\mathcal{Y}/S}$ is quasi-separated, then $f$ is quasi-separated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (5), (6), (7), and (8) follow immediately from", "Lemma \\ref{lemma-compose-after-separated}", "and", "Spaces, Definition \\ref{spaces-definition-separated}.", "To prove (3) and (4) think of $X$ and $Y$ as algebraic stacks over", "$\\Spec(\\mathbf{Z})$ and apply", "Lemma \\ref{lemma-compose-after-separated}.", "Similarly, to prove (1) and (2), think of $\\mathcal{X}$ as an algebraic", "stack over $\\Spec(\\mathbf{Z})$ consider the", "morphisms", "$$", "\\mathcal{X} \\longrightarrow", "\\mathcal{X} \\times_S \\mathcal{X} \\longrightarrow", "\\mathcal{X} \\times_{\\Spec(\\mathbf{Z})} \\mathcal{X}", "$$", "Both arrows are representable by algebraic spaces.", "The second arrow is unramified and locally quasi-finite as the base change of", "the immersion $\\Delta_{S/\\mathbf{Z}}$. Hence the composition is", "unramified (resp.\\ locally quasi-finite) if and only if the first arrow", "is unramified (resp.\\ locally quasi-finite), see", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-composition-unramified} and", "\\ref{spaces-morphisms-lemma-permanence-unramified}", "(resp.\\ Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-composition-quasi-finite} and", "\\ref{spaces-morphisms-lemma-permanence-quasi-finite})." ], "refs": [ "stacks-morphisms-lemma-compose-after-separated", "spaces-definition-separated", "stacks-morphisms-lemma-compose-after-separated", "spaces-morphisms-lemma-composition-unramified", "spaces-morphisms-lemma-permanence-unramified", "spaces-morphisms-lemma-composition-quasi-finite", "spaces-morphisms-lemma-permanence-quasi-finite" ], "ref_ids": [ 7406, 8181, 7406, 4896, 4904, 4831, 4836 ] } ], "ref_ids": [] }, { "id": 7408, "type": "theorem", "label": "stacks-morphisms-lemma-properties-covering-imply-diagonal", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-properties-covering-imply-diagonal", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Let $W$ be an algebraic space, and let $f : W \\to \\mathcal{X}$", "be a surjective, flat, locally finitely presented morphism.", "\\begin{enumerate}", "\\item If $f$ is unramified (i.e., \\'etale, i.e., $\\mathcal{X}$", "is Deligne-Mumford), then $\\mathcal{X}$ is DM.", "\\item If $f$ is locally quasi-finite, then $\\mathcal{X}$ is quasi-DM.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that if $f$ is unramified, then it is \\'etale by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-unramified-flat-lfp-etale}.", "This explains the parenthetical remark in (1).", "Assume $f$ is unramified (resp.\\ locally quasi-finite). We have to show that", "$\\Delta_\\mathcal{X} : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$", "is unramified (resp.\\ locally quasi-finite). Note that", "$W \\times W \\to \\mathcal{X} \\times \\mathcal{X}$ is also", "surjective, flat, and locally of finite presentation. Hence it suffices to", "show that", "$$", "W \\times_{\\mathcal{X} \\times \\mathcal{X}, \\Delta_\\mathcal{X}} \\mathcal{X}", "=", "W \\times_\\mathcal{X} W", "\\longrightarrow", "W \\times W", "$$", "is unramified (resp.\\ locally quasi-finite), see", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}.", "By assumption the morphism $\\text{pr}_i : W \\times_\\mathcal{X} W \\to W$", "is unramified (resp.\\ locally quasi-finite). Hence", "the displayed arrow is unramified (resp.\\ locally quasi-finite) by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-permanence-unramified}", "(resp.\\ Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-permanence-quasi-finite})." ], "refs": [ "spaces-morphisms-lemma-unramified-flat-lfp-etale", "stacks-properties-lemma-check-property-covering", "spaces-morphisms-lemma-permanence-unramified", "spaces-morphisms-lemma-permanence-quasi-finite" ], "ref_ids": [ 4915, 8859, 4904, 4836 ] } ], "ref_ids": [] }, { "id": 7409, "type": "theorem", "label": "stacks-morphisms-lemma-monomorphism-separated", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-monomorphism-separated", "contents": [ "A monomorphism of algebraic stacks is separated and DM.", "The same is true for immersions of algebraic stacks." ], "refs": [], "proofs": [ { "contents": [ "If $f : \\mathcal{X} \\to \\mathcal{Y}$ is a monomorphism of algebraic stacks,", "then $\\Delta_f$ is an isomorphism, see", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-monomorphism}.", "Since an isomorphism of algebraic spaces is proper and unramified we", "see that $f$ is separated and DM. The second assertion follows from the", "first as an immersion is a monomorphism, see", "Properties of Stacks,", "Lemma \\ref{stacks-properties-lemma-immersion-monomorphism}." ], "refs": [ "stacks-properties-lemma-monomorphism", "stacks-properties-lemma-immersion-monomorphism" ], "ref_ids": [ 8879, 8885 ] } ], "ref_ids": [] }, { "id": 7410, "type": "theorem", "label": "stacks-morphisms-lemma-separation-properties-residual-gerbe", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-separation-properties-residual-gerbe", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$.", "Assume the residual gerbe $\\mathcal{Z}_x$ of $\\mathcal{X}$ at $x$ exists.", "If $\\mathcal{X}$ is DM, resp.\\ quasi-DM, resp.\\ separated,", "resp.\\ quasi-separated, then so is $\\mathcal{Z}_x$." ], "refs": [], "proofs": [ { "contents": [ "This is true because $\\mathcal{Z}_x \\to \\mathcal{X}$ is a monomorphism", "hence DM and separated by", "Lemma \\ref{lemma-monomorphism-separated}.", "Apply", "Lemma \\ref{lemma-separated-over-separated}", "to conclude." ], "refs": [ "stacks-morphisms-lemma-monomorphism-separated", "stacks-morphisms-lemma-separated-over-separated" ], "ref_ids": [ 7409, 7405 ] } ], "ref_ids": [] }, { "id": 7411, "type": "theorem", "label": "stacks-morphisms-lemma-inertia", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-inertia", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Then the inertia stack", "$\\mathcal{I}_\\mathcal{X}$ is an algebraic stack as well.", "The morphism", "$$", "\\mathcal{I}_\\mathcal{X} \\longrightarrow \\mathcal{X}", "$$", "is representable by algebraic spaces and locally of finite type.", "More generally, let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism", "of algebraic stacks. Then the relative inertia", "$\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$ is an algebraic stack and the", "morphism", "$$", "\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\longrightarrow \\mathcal{X}", "$$", "is representable by algebraic spaces and locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "By", "Categories, Lemma \\ref{categories-lemma-inertia-fibred-category}", "there are equivalences", "$$", "\\mathcal{I}_\\mathcal{X} \\to", "\\mathcal{X} \\times_{\\Delta, \\mathcal{X} \\times_S \\mathcal{X}, \\Delta}", "\\mathcal{X}", "\\quad\\text{and}\\quad", "\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to", "\\mathcal{X}", "\\times_{\\Delta, \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}, \\Delta}", "\\mathcal{X}", "$$", "which shows that the inertia stacks are algebraic stacks.", "Let $T \\to \\mathcal{X}$ be a morphism given by", "the object $x$ of the fibre category of $\\mathcal{X}$ over $T$.", "Then we get a $2$-fibre product square", "$$", "\\xymatrix{", "\\mathit{Isom}_\\mathcal{X}(x, x) \\ar[d] \\ar[r] &", "\\mathcal{I}_\\mathcal{X} \\ar[d] \\\\", "T \\ar[r]^x & \\mathcal{X}", "}", "$$", "This follows immediately from the definition of $\\mathcal{I}_\\mathcal{X}$.", "Since $\\mathit{Isom}_\\mathcal{X}(x, x)$ is always an algebraic space", "locally of finite type over $T$ (see", "Lemma \\ref{lemma-isom-locally-finite-type})", "we conclude that $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is representable", "by algebraic spaces and locally of finite type. Finally, for", "the relative inertia we get", "$$", "\\vcenter{", "\\xymatrix{", "\\mathit{Isom}_\\mathcal{X}(x, x) \\ar[d] &", "K \\ar[l] \\ar[d] \\ar[r] &", "\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\ar[d] \\\\", "\\mathit{Isom}_\\mathcal{Y}(f(x), f(x)) &", "T \\ar[l]_-e \\ar[r]^x & \\mathcal{X}", "}", "}", "$$", "with both squares $2$-fibre products. This follows from", "Categories, Lemma \\ref{categories-lemma-relative-inertia-as-fibre-product}.", "The left vertical arrow is a morphism of algebraic spaces locally of finite", "type over $T$, and hence is locally of finite type, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-permanence-finite-type}.", "Thus $K$ is an algebraic space and $K \\to T$ is locally of finite type.", "This proves the assertion on the relative inertia." ], "refs": [ "categories-lemma-inertia-fibred-category", "stacks-morphisms-lemma-isom-locally-finite-type", "categories-lemma-relative-inertia-as-fibre-product", "spaces-morphisms-lemma-permanence-finite-type" ], "ref_ids": [ 12292, 7390, 12293, 4818 ] } ], "ref_ids": [] }, { "id": 7412, "type": "theorem", "label": "stacks-morphisms-lemma-isom-pseudo-torsor-aut-over-space", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-isom-pseudo-torsor-aut-over-space", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $Z$ be an algebraic space and let $x_i : Z \\to \\mathcal{X}$, $i = 1, 2$", "be morphisms. Then", "\\begin{enumerate}", "\\item $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x_2, x_2)$", "is a group algebraic space over $Z$,", "\\item there is an exact sequence of groups", "$$", "0 \\to \\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x_2, x_2)", "\\to \\mathit{Isom}_\\mathcal{X}(x_2, x_2)", "\\to \\mathit{Isom}_\\mathcal{Y}(f \\circ x_2, f \\circ x_2)", "$$", "\\item there is a map of algebraic spaces", "$", "\\mathit{Isom}_\\mathcal{X}(x_1, x_2)", "\\to \\mathit{Isom}_\\mathcal{Y}(f \\circ x_1, f \\circ x_2)", "$", "such that for any $2$-morphism $\\alpha : f \\circ x_1 \\to f \\circ x_2$", "we obtain a cartesian diagram", "$$", "\\xymatrix{", "\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}^\\alpha(x_1, x_2) \\ar[d] \\ar[r] &", "Z \\ar[d]^\\alpha \\\\", "\\mathit{Isom}_\\mathcal{X}(x_1, x_2) \\ar[r] &", "\\mathit{Isom}_\\mathcal{Y}(f \\circ x_1, f \\circ x_2)", "}", "$$", "\\item for any $2$-morphism $\\alpha : f \\circ x_1 \\to f \\circ x_2$ the", "algebraic space $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}^\\alpha(x_1, x_2)$", "is a pseudo torsor for $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x_2, x_2)$", "over $Z$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from Definition \\ref{definition-isom}.", "Part (2) comes from the exact sequence (\\ref{equation-exact-sequence-isom})", "\\'etale locally on $Z$. Part (3) can be seen by unwinding the definitions.", "Locally on $Z$ in the \\'etale topology part (4) reduces to", "part (2) of Lemma \\ref{lemma-isom-pseudo-torsor-aut}." ], "refs": [ "stacks-morphisms-definition-isom", "stacks-morphisms-lemma-isom-pseudo-torsor-aut" ], "ref_ids": [ 7603, 7391 ] } ], "ref_ids": [] }, { "id": 7413, "type": "theorem", "label": "stacks-morphisms-lemma-cartesian-square-inertia", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-cartesian-square-inertia", "contents": [ "Let $\\pi : \\mathcal{X} \\to \\mathcal{Y}$ and", "$f : \\mathcal{Y}' \\to \\mathcal{Y}$ be morphisms of algebraic stacks.", "Set $\\mathcal{X}' = \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Y}'$.", "Then both squares in the diagram", "$$", "\\xymatrix{", "\\mathcal{I}_{\\mathcal{X}'/\\mathcal{Y}'} \\ar[r]", "\\ar[d]_{", "\\text{Categories, Equation}\\ (\\ref{categories-equation-functorial})", "} &", "\\mathcal{X}' \\ar[r]_{\\pi'} \\ar[d] & \\mathcal{Y}' \\ar[d]^f \\\\", "\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\ar[r] &", "\\mathcal{X} \\ar[r]^\\pi & \\mathcal{Y}", "}", "$$", "are fibre product squares." ], "refs": [], "proofs": [ { "contents": [ "The inertia stack $\\mathcal{I}_{\\mathcal{X}'/\\mathcal{Y}'}$ is defined as the", "category of pairs $(x', \\alpha')$ where $x'$ is an object of $\\mathcal{X}'$", "and $\\alpha'$ is an automorphism of $x'$ with $\\pi'(\\alpha') = \\text{id}$, see", "Categories, Section \\ref{categories-section-inertia}.", "Suppose that $x'$ lies over the scheme $U$ and maps to the object", "$x$ of $\\mathcal{X}$. By the construction of the $2$-fibre product in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}", "we see that $x' = (U, x, y', \\beta)$ where $y'$ is an object of $\\mathcal{Y}'$", "over $U$ and $\\beta$ is an isomorphism", "$\\beta : \\pi(x) \\to f(y')$ in the fibre category of $\\mathcal{Y}$ over $U$.", "By the very construction of the $2$-fibre product the automorphism $\\alpha'$", "is a pair $(\\alpha, \\gamma)$ where $\\alpha$ is an automorphism of $x$ over $U$", "and $\\gamma$ is an automorphism of $y'$ over $U$ such that", "$\\alpha$ and $\\gamma$ are compatible via $\\beta$. The condition", "$\\pi'(\\alpha') = \\text{id}$ signifies that $\\gamma = \\text{id}$", "whereupon the condition that $\\alpha, \\beta, \\gamma$ are compatible", "is exactly the condition $\\pi(\\alpha) = \\text{id}$, i.e., means", "exactly that $(x, \\alpha)$ is an object of", "$\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$.", "In this way we see that the left square is a fibre product square", "(some details omitted)." ], "refs": [ "categories-lemma-2-product-categories-over-C" ], "ref_ids": [ 12280 ] } ], "ref_ids": [] }, { "id": 7414, "type": "theorem", "label": "stacks-morphisms-lemma-monomorphism-cartesian-square-inertia", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-monomorphism-cartesian-square-inertia", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a monomorphism of algebraic stacks.", "Then the diagram", "$$", "\\xymatrix{", "\\mathcal{I}_\\mathcal{X} \\ar[r] \\ar[d] &", "\\mathcal{X} \\ar[d] \\\\", "\\mathcal{I}_\\mathcal{Y} \\ar[r] &", "\\mathcal{Y}", "}", "$$", "is a fibre product square." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the fact that $f$ is fully faithful (see", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-monomorphism})", "and the definition of the inertia in", "Categories, Section \\ref{categories-section-inertia}.", "Namely, an object of $\\mathcal{I}_\\mathcal{X}$ over a scheme $T$ is", "the same thing as a pair $(x, \\alpha)$ consisting of an object", "$x$ of $\\mathcal{X}$ over $T$ and a morphism $\\alpha : x \\to x$ in", "the fibre category of $\\mathcal{X}$ over $T$. As $f$ is fully faithful", "we see that $\\alpha$ is the same thing as a morphism", "$\\beta : f(x) \\to f(x)$ in the fibre category of $\\mathcal{Y}$ over $T$.", "Hence we can think of objects of $\\mathcal{I}_\\mathcal{X}$ over $T$", "as triples $((y, \\beta), x, \\gamma)$ where $y$ is an object of", "$\\mathcal{Y}$ over $T$, $\\beta : y \\to y$ in $\\mathcal{Y}_T$ and", "$\\gamma : y \\to f(x)$ is an isomorphism over $T$, i.e., an object", "of $\\mathcal{I}_\\mathcal{Y} \\times_\\mathcal{Y} \\mathcal{X}$ over $T$." ], "refs": [ "stacks-properties-lemma-monomorphism" ], "ref_ids": [ 8879 ] } ], "ref_ids": [] }, { "id": 7415, "type": "theorem", "label": "stacks-morphisms-lemma-presentation-inertia", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-presentation-inertia", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $[U/R] \\to \\mathcal{X}$", "be a presentation. Let $G/U$ be the stabilizer group algebraic space", "associated to the groupoid $(U, R, s, t, c)$. Then", "$$", "\\xymatrix{", "G \\ar[d] \\ar[r] & U \\ar[d] \\\\", "\\mathcal{I}_\\mathcal{X} \\ar[r] & \\mathcal{X}", "}", "$$", "is a fibre product diagram." ], "refs": [], "proofs": [ { "contents": [ "Immediate from", "Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-2-cartesian-inertia}." ], "refs": [ "spaces-groupoids-lemma-2-cartesian-inertia" ], "ref_ids": [ 9333 ] } ], "ref_ids": [] }, { "id": 7416, "type": "theorem", "label": "stacks-morphisms-lemma-diagonal-diagonal", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-diagonal-diagonal", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "\\begin{enumerate}", "\\item", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$", "is separated,", "\\item $\\Delta_{f, 1} = \\Delta_f :", "\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "is separated, and", "\\item $\\Delta_{f, 2} = e :", "\\mathcal{X} \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$", "is a closed immersion.", "\\end{enumerate}", "\\item", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$", "is quasi-separated,", "\\item $\\Delta_{f, 1} = \\Delta_f :", "\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "is quasi-separated, and", "\\item $\\Delta_{f, 2} = e :", "\\mathcal{X} \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$", "is a quasi-compact.", "\\end{enumerate}", "\\item", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$", "is locally separated,", "\\item $\\Delta_{f, 1} = \\Delta_f :", "\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "is locally separated, and", "\\item $\\Delta_{f, 2} = e :", "\\mathcal{X} \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$", "is an immersion.", "\\end{enumerate}", "\\item", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$", "is unramified,", "\\item $f$ is DM.", "\\end{enumerate}", "\\item", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$", "is locally quasi-finite,", "\\item $f$ is quasi-DM.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1), (2), and (3).", "Choose an algebraic space $U$ and a surjective smooth morphism", "$U \\to \\mathcal{X}$. Then", "$G = U \\times_\\mathcal{X} \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$", "is an algebraic space over $U$ (Lemma \\ref{lemma-inertia}).", "In fact, $G$ is a group algebraic space over $U$", "by the group law on relative", "inertia constructed in Remark \\ref{remark-inertia-is-group-in-spaces}.", "Moreover, $G \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$", "is surjective and smooth as a base change of $U \\to \\mathcal{X}$.", "Finally, the base change of", "$e : \\mathcal{X} \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$", "by $G \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$", "is the identity $U \\to G$ of $G/U$.", "Thus the equivalence of (a) and (c) follows from", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-group-scheme-separated}.", "Since $\\Delta_{f, 2}$ is the diagonal of $\\Delta_f$ we have", "(b) $\\Leftrightarrow$ (c) by definition.", "\\medskip\\noindent", "Proof of (4) and (5). Recall that (4)(b) means $\\Delta_f$ is", "unramified and (5)(b) means that $\\Delta_f$ is locally quasi-finite.", "Choose a scheme $Z$ and a morphism", "$a : Z \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$.", "Then $a = (x_1, x_2, \\alpha)$ where $x_i : Z \\to \\mathcal{X}$", "and $\\alpha : f \\circ x_1 \\to f \\circ x_2$ is a $2$-morphism.", "Recall that", "$$", "\\vcenter{", "\\xymatrix{", "\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}^\\alpha(x_1, x_2)", "\\ar[d] \\ar[r] &", "Z \\ar[d] \\\\", "\\mathcal{X} \\ar[r]^{\\Delta_f} &", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x_2, x_2)", "\\ar[d] \\ar[r] &", "Z \\ar[d]^{x_2} \\\\", "\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\ar[r] &", "\\mathcal{X}", "}", "}", "$$", "are cartesian squares. By Lemma \\ref{lemma-isom-pseudo-torsor-aut-over-space}", "the", "algebraic space $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}^\\alpha(x_1, x_2)$", "is a pseudo torsor for $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x_2, x_2)$", "over $Z$. Thus the equivalences in (4) and (5) follow from", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-pseudo-torsor-implications}." ], "refs": [ "stacks-morphisms-lemma-inertia", "stacks-morphisms-remark-inertia-is-group-in-spaces", "spaces-groupoids-lemma-group-scheme-separated", "stacks-morphisms-lemma-isom-pseudo-torsor-aut-over-space", "spaces-groupoids-lemma-pseudo-torsor-implications" ], "ref_ids": [ 7411, 7632, 9289, 7412, 9295 ] } ], "ref_ids": [] }, { "id": 7417, "type": "theorem", "label": "stacks-morphisms-lemma-second-diagonal", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-second-diagonal", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent:", "\\begin{enumerate}", "\\item the morphism $f$ is representable by algebraic spaces,", "\\item the second diagonal of $f$ is an isomorphism,", "\\item the group stack $ \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$", "is trivial over $\\mathcal X$, and", "\\item for a scheme $T$ and a morphism $x : T \\to \\mathcal{X}$", "the kernel of $\\mathit{Isom}_\\mathcal{X}(x, x) \\to", "\\mathit{Isom}_\\mathcal{Y}(f(x), f(x))$ is trivial.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We first prove the equivalence of (1) and (2).", "Namely, $f$ is representable by algebraic spaces if and only if $f$ is", "faithful, see", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}.", "On the other hand, $f$ is faithful if and only if for every object $x$", "of $\\mathcal{X}$ over a scheme $T$ the functor $f$ induces an injection", "$\\mathit{Isom}_\\mathcal{X}(x, x) \\to", "\\mathit{Isom}_\\mathcal{Y}(f(x), f(x))$,", "which happens if and only if the kernel $K$ is trivial, which happens if and", "only if $e : T \\to K$ is an isomorphism for every $x : T \\to \\mathcal{X}$.", "Since $K = T \\times_{x, \\mathcal{X}} \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$", "as discussed above, this proves the equivalence of (1) and (2). To prove", "the equivalence of (2) and (3), by the discussion above, it suffices to", "note that a group stack is trivial if and only if its identity section", "is an isomorphism. Finally, the equivalence of (3) and (4) follows", "from the definitions: in the proof of Lemma \\ref{lemma-inertia}", "we have seen that the kernel in (4) corresponds to the fibre product", "$T \\times_{x, \\mathcal{X}} \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$ over $T$." ], "refs": [ "algebraic-lemma-characterize-representable-by-algebraic-spaces", "stacks-morphisms-lemma-inertia" ], "ref_ids": [ 8469, 7411 ] } ], "ref_ids": [] }, { "id": 7418, "type": "theorem", "label": "stacks-morphisms-lemma-hierarchy", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-hierarchy", "contents": [ "A morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks is", "\\begin{enumerate}", "\\item a monomorphism if and only if $\\Delta_{f, 1}$ is an isomorphism, and", "\\item representable by algebraic spaces if and only if $\\Delta_{f, 1}$", "is a monomorphism.", "\\end{enumerate}", "Moreover, the second diagonal $\\Delta_{f, 2}$ is always a monomorphism." ], "refs": [], "proofs": [ { "contents": [ "Recall from Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-monomorphism}", "that a morphism of algebraic stacks is a monomorphism", "if and only if its diagonal is an isomorphism of stacks.", "Thus Lemma \\ref{lemma-second-diagonal}", "can be rephrased as saying that a morphism is", "representable by algebraic spaces if the diagonal", "is a monomorphism. In particular, it shows that condition", "(3) of Lemma \\ref{lemma-properties-diagonal-representable}", "is actually an if and only if, i.e., a morphism of algebraic stacks", "is representable by algebraic spaces if and only if", "its diagonal is a monomorphism." ], "refs": [ "stacks-properties-lemma-monomorphism", "stacks-morphisms-lemma-second-diagonal", "stacks-morphisms-lemma-properties-diagonal-representable" ], "ref_ids": [ 8879, 7417, 7393 ] } ], "ref_ids": [] }, { "id": 7419, "type": "theorem", "label": "stacks-morphisms-lemma-first-diagonal-separated-second-diagonal-closed", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-first-diagonal-separated-second-diagonal-closed", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Then", "\\begin{enumerate}", "\\item $\\Delta_{f, 1}$ separated $\\Leftrightarrow$", "$\\Delta_{f, 2}$ closed immersion $\\Leftrightarrow$", "$\\Delta_{f, 2}$ proper $\\Leftrightarrow$", "$\\Delta_{f, 2}$ universally closed,", "\\item $\\Delta_{f, 1}$ quasi-separated $\\Leftrightarrow$", "$\\Delta_{f, 2}$ finite type $\\Leftrightarrow$ $\\Delta_{f, 2}$ quasi-compact,", "and", "\\item $\\Delta_{f, 1}$ locally separated $\\Leftrightarrow$", "$\\Delta_{f, 2}$ immersion.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemmas \\ref{lemma-representable-separated-diagonal-closed},", "\\ref{lemma-representable-quasi-separated-diagonal-quasi-compact}, and", "\\ref{lemma-representable-locally-separated-diagonal-immersion}", "applied to $\\Delta_{f, 1}$." ], "refs": [ "stacks-morphisms-lemma-representable-separated-diagonal-closed", "stacks-morphisms-lemma-representable-quasi-separated-diagonal-quasi-compact", "stacks-morphisms-lemma-representable-locally-separated-diagonal-immersion" ], "ref_ids": [ 7394, 7395, 7396 ] } ], "ref_ids": [] }, { "id": 7420, "type": "theorem", "label": "stacks-morphisms-lemma-definition-separated", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-definition-separated", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Then", "\\begin{enumerate}", "\\item $f$ is separated if and only if $\\Delta_{f, 1}$ and $\\Delta_{f, 2}$", "are universally closed, and", "\\item $f$ is quasi-separated if and only if $\\Delta_{f, 1}$ and $\\Delta_{f, 2}$", "are quasi-compact.", "\\item $f$ is quasi-DM if and only if $\\Delta_{f, 1}$ and $\\Delta_{f, 2}$", "are locally quasi-finite.", "\\item $f$ is DM if and only if $\\Delta_{f, 1}$ and $\\Delta_{f, 2}$", "are unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Assume that $\\Delta_{f, 2}$ and $\\Delta_{f, 1}$ are", "universally closed. Then $\\Delta_{f, 1}$ is separated and universally", "closed by", "Lemma \\ref{lemma-first-diagonal-separated-second-diagonal-closed}.", "By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-universally-closed-quasi-compact}", "and", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-representable-transformations-property-implication}", "we see that $\\Delta_{f, 1}$ is quasi-compact.", "Hence it is quasi-compact, separated, universally closed and locally of", "finite type (by", "Lemma \\ref{lemma-properties-diagonal})", "so proper. This proves ``$\\Leftarrow$'' of (1).", "The proof of the implication in the other direction is omitted.", "\\medskip\\noindent", "Proof of (2). This follows immediately from", "Lemma \\ref{lemma-first-diagonal-separated-second-diagonal-closed}.", "\\medskip\\noindent", "Proof of (3). This follows from the fact that $\\Delta_{f, 2}$ is always locally", "quasi-finite by", "Lemma \\ref{lemma-properties-diagonal-representable}", "applied to $\\Delta_f = \\Delta_{f, 1}$.", "\\medskip\\noindent", "Proof of (4). This follows from the fact that $\\Delta_{f, 2}$ is always", "unramified as", "Lemma \\ref{lemma-properties-diagonal-representable}", "applied to $\\Delta_f = \\Delta_{f, 1}$ shows that", "$\\Delta_{f, 2}$ is locally of finite type and a monomorphism.", "See", "More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-universally-injective-unramified}." ], "refs": [ "stacks-morphisms-lemma-first-diagonal-separated-second-diagonal-closed", "spaces-morphisms-lemma-universally-closed-quasi-compact", "algebraic-lemma-representable-transformations-property-implication", "stacks-morphisms-lemma-properties-diagonal", "stacks-morphisms-lemma-first-diagonal-separated-second-diagonal-closed", "stacks-morphisms-lemma-properties-diagonal-representable", "stacks-morphisms-lemma-properties-diagonal-representable", "spaces-more-morphisms-lemma-universally-injective-unramified" ], "ref_ids": [ 7419, 4749, 8459, 7392, 7419, 7393, 7393, 73 ] } ], "ref_ids": [] }, { "id": 7421, "type": "theorem", "label": "stacks-morphisms-lemma-separated-implies-isom", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-separated-implies-isom", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a separated", "(resp.\\ quasi-separated, resp.\\ quasi-DM, resp.\\ DM)", "morphism of algebraic stacks. Then", "\\begin{enumerate}", "\\item given algebraic spaces $T_i$, $i = 1, 2$ and morphisms", "$x_i : T_i \\to \\mathcal{X}$, with $y_i = f \\circ x_i$ the morphism", "$$", "T_1 \\times_{x_1, \\mathcal{X}, x_2} T_2 \\longrightarrow", "T_1 \\times_{y_1, \\mathcal{Y}, y_2} T_2", "$$", "is proper (resp.\\ quasi-compact and quasi-separated,", "resp.\\ locally quasi-finite, resp.\\ unramified),", "\\item given an algebraic space $T$ and morphisms", "$x_i : T \\to \\mathcal{X}$, $i = 1, 2$, with $y_i = f \\circ x_i$ the morphism", "$$", "\\mathit{Isom}_\\mathcal{X}(x_1, x_2) \\longrightarrow", "\\mathit{Isom}_\\mathcal{Y}(y_1, y_2)", "$$", "is proper (resp.\\ quasi-compact and quasi-separated,", "resp.\\ locally quasi-finite, resp.\\ unramified).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Observe that the diagram", "$$", "\\xymatrix{", "T_1 \\times_{x_1, \\mathcal{X}, x_2} T_2 \\ar[d] \\ar[r] &", "T_1 \\times_{y_1, \\mathcal{Y}, y_2} T_2 \\ar[d] \\\\", "\\mathcal{X} \\ar[r] & \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}", "}", "$$", "is cartesian. Hence this follows from the fact that $f$ is separated", "(resp.\\ quasi-separated, resp.\\ quasi-DM, resp.\\ DM)", "if and only if the diagonal is proper", "(resp.\\ quasi-compact and quasi-separated,", "resp.\\ locally quasi-finite, resp.\\ unramified).", "\\medskip\\noindent", "Proof of (2). This is true because", "$$", "\\mathit{Isom}_\\mathcal{X}(x_1, x_2) =", "(T \\times_{x_1, \\mathcal{X}, x_2} T) \\times_{T \\times T, \\Delta_T} T", "$$", "hence the morphism in (2) is a base change of the morphism in (1)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7422, "type": "theorem", "label": "stacks-morphisms-lemma-characterize-representable-quasi-compact", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-characterize-representable-quasi-compact", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "which is representable by algebraic spaces. The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is quasi-compact (as in Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}), and", "\\item for every quasi-compact algebraic stack $\\mathcal{Z}$", "and any morphism $\\mathcal{Z} \\to \\mathcal{Y}$ the algebraic stack", "$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1), and let $\\mathcal{Z} \\to \\mathcal{Y}$", "be a morphism of algebraic stacks with $\\mathcal{Z}$ quasi-compact. By", "Properties of Stacks,", "Lemma \\ref{stacks-properties-lemma-quasi-compact-stack}", "there exists a quasi-compact scheme $U$ and a surjective smooth", "morphism $U \\to \\mathcal{Z}$. Since $f$ is representable by algebraic", "spaces and quasi-compact we see by definition that", "$U \\times_\\mathcal{Y} \\mathcal{X}$ is an algebraic space, and that", "$U \\times_\\mathcal{Y} \\mathcal{X} \\to U$ is quasi-compact.", "Hence $U \\times_Y X$ is a quasi-compact algebraic space.", "The morphism", "$U \\times_\\mathcal{Y} \\mathcal{X} \\to", "\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$", "is smooth and surjective (as the base change of the smooth", "and surjective morphism $U \\to \\mathcal{Z}$).", "Hence $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$", "is quasi-compact by another application of", "Properties of Stacks,", "Lemma \\ref{stacks-properties-lemma-quasi-compact-stack}", "\\medskip\\noindent", "Assume (2). Let $Z \\to \\mathcal{Y}$ be a morphism, where $Z$ is a scheme.", "We have to show that the morphism of algebraic spaces", "$p : Z \\times_\\mathcal{Y} \\mathcal{X} \\to Z$ is quasi-compact.", "Let $U \\subset Z$ be affine open. Then", "$p^{-1}(U) = U \\times_\\mathcal{Y} \\mathcal{Z}$", "and the algebraic space $U \\times_\\mathcal{Y} \\mathcal{Z}$", "is quasi-compact by assumption (2). Hence $p$ is quasi-compact, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-quasi-compact-local}." ], "refs": [ "stacks-properties-lemma-quasi-compact-stack", "stacks-properties-lemma-quasi-compact-stack", "spaces-morphisms-lemma-quasi-compact-local" ], "ref_ids": [ 8873, 8873, 4742 ] } ], "ref_ids": [] }, { "id": 7423, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-quasi-compact", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-quasi-compact", "contents": [ "The base change of a quasi-compact morphism of algebraic stacks", "by any morphism of algebraic stacks is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7424, "type": "theorem", "label": "stacks-morphisms-lemma-composition-quasi-compact", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-quasi-compact", "contents": [ "The composition of a pair of quasi-compact morphisms of algebraic stacks", "is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7425, "type": "theorem", "label": "stacks-morphisms-lemma-closed-immersion-quasi-compact", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-closed-immersion-quasi-compact", "contents": [ "A closed immersion of algebraic stacks is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that immersions are always representable and", "the corresponding fact for closed immersion of algebraic spaces." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7426, "type": "theorem", "label": "stacks-morphisms-lemma-surjection-from-quasi-compact", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-surjection-from-quasi-compact", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{X} \\ar[rr]_f \\ar[rd]_p & &", "\\mathcal{Y} \\ar[dl]^q \\\\", "& \\mathcal{Z}", "}", "$$", "be a $2$-commutative diagram of morphisms of algebraic stacks.", "If $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{T}$ be a quasi-compact algebraic stack, and let", "$\\mathcal{T} \\to \\mathcal{Z}$ be a morphism. By", "Properties of Stacks,", "Lemma \\ref{stacks-properties-lemma-base-change-surjective}", "the morphism", "$\\mathcal{T} \\times_\\mathcal{Z} \\mathcal{X} \\to", "\\mathcal{T} \\times_\\mathcal{Z} \\mathcal{Y}$", "is surjective and by assumption", "$\\mathcal{T} \\times_\\mathcal{Z} \\mathcal{X}$", "is quasi-compact. Hence", "$\\mathcal{T} \\times_\\mathcal{Z} \\mathcal{Y}$", "is quasi-compact by", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-quasi-compact-stack}." ], "refs": [ "stacks-properties-lemma-base-change-surjective", "stacks-properties-lemma-quasi-compact-stack" ], "ref_ids": [ 8870, 8873 ] } ], "ref_ids": [] }, { "id": 7427, "type": "theorem", "label": "stacks-morphisms-lemma-quasi-compact-permanence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-quasi-compact-permanence", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and", "$g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks.", "If $g \\circ f$ is quasi-compact and $g$ is quasi-separated", "then $f$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "This is true because $f$ equals the composition", "$(1, f) : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to", "\\mathcal{Y}$.", "The first map is quasi-compact by", "Lemma \\ref{lemma-section-immersion}", "because it is a section of the quasi-separated morphism", "$\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to \\mathcal{X}$", "(a base change of $g$, see", "Lemma \\ref{lemma-base-change-separated}).", "The second map is quasi-compact as it is the base change of $f$, see", "Lemma \\ref{lemma-base-change-quasi-compact}.", "And compositions of quasi-compact", "morphisms are quasi-compact, see Lemma \\ref{lemma-composition-quasi-compact}." ], "refs": [ "stacks-morphisms-lemma-base-change-separated", "stacks-morphisms-lemma-base-change-quasi-compact", "stacks-morphisms-lemma-composition-quasi-compact" ], "ref_ids": [ 7398, 7423, 7424 ] } ], "ref_ids": [] }, { "id": 7428, "type": "theorem", "label": "stacks-morphisms-lemma-quasi-compact-quasi-separated-permanence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-quasi-compact-quasi-separated-permanence", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "\\begin{enumerate}", "\\item If $\\mathcal{X}$ is quasi-compact and $\\mathcal{Y}$ is", "quasi-separated, then $f$ is quasi-compact.", "\\item If $\\mathcal{X}$ is quasi-compact and quasi-separated and $\\mathcal{Y}$", "is quasi-separated, then $f$ is quasi-compact and quasi-separated.", "\\item A fibre product of quasi-compact and quasi-separated algebraic stacks", "is quasi-compact and quasi-separated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from", "Lemma \\ref{lemma-quasi-compact-permanence}.", "Part (2) follows from (1) and", "Lemma \\ref{lemma-compose-after-separated}.", "For (3) let $\\mathcal{X} \\to \\mathcal{Y}$ and $\\mathcal{Z} \\to \\mathcal{Y}$", "be morphisms of quasi-compact and quasi-separated algebraic stacks.", "Then $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$", "is quasi-compact and quasi-separated as a base change of", "$\\mathcal{X} \\to \\mathcal{Y}$ using (2) and", "Lemmas \\ref{lemma-base-change-quasi-compact} and", "\\ref{lemma-base-change-separated}.", "Hence $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$", "is quasi-compact and quasi-separated as", "an algebraic stack quasi-compact and quasi-separated over", "$\\mathcal{Z}$, see", "Lemmas \\ref{lemma-separated-over-separated} and", "\\ref{lemma-composition-quasi-compact}." ], "refs": [ "stacks-morphisms-lemma-quasi-compact-permanence", "stacks-morphisms-lemma-compose-after-separated", "stacks-morphisms-lemma-base-change-quasi-compact", "stacks-morphisms-lemma-base-change-separated", "stacks-morphisms-lemma-separated-over-separated", "stacks-morphisms-lemma-composition-quasi-compact" ], "ref_ids": [ 7427, 7406, 7423, 7398, 7405, 7424 ] } ], "ref_ids": [] }, { "id": 7429, "type": "theorem", "label": "stacks-morphisms-lemma-reach-points-scheme-theoretic-image", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-reach-points-scheme-theoretic-image", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact morphism of", "algebraic stacks. Let $y \\in |\\mathcal{Y}|$ be a point in the closure", "of the image of $|f|$. There exists a valuation ring $A$ with", "fraction field $K$ and a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\", "\\Spec(A) \\ar[r] & \\mathcal{Y}", "}", "$$", "such that the closed point of $\\Spec(A)$ maps to $y$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $V$ and a point $v \\in V$ and a smooth morphism", "$V \\to \\mathcal{Y}$ sending $v$ to $y$. Consider the base change diagram", "$$", "\\xymatrix{", "V \\times_\\mathcal{Y} \\mathcal{X} \\ar[r] \\ar[d]_g & \\mathcal{X} \\ar[d]^f \\\\", "V \\ar[r] & \\mathcal{Y}", "}", "$$", "Recall that $|V \\times_\\mathcal{Y} \\mathcal{X}| \\to", "|V| \\times_{|\\mathcal{Y}|} |\\mathcal{X}|$ is surjective", "(Properties of Stacks, Lemma \\ref{stacks-properties-lemma-points-cartesian}).", "Because $|V| \\to |\\mathcal{Y}|$ is open", "(Properties of Stacks, Lemma \\ref{stacks-properties-lemma-topology-points})", "we conclude that $v$ is in the closure of the image of $|g|$.", "Thus it suffices to prove the lemma for the quasi-compact morphism $g$", "(Lemma \\ref{lemma-base-change-quasi-compact}) which we do in the next", "paragraph.", "\\medskip\\noindent", "Assume $\\mathcal{Y} = Y$ is an affine scheme. Then $\\mathcal{X}$", "is quasi-compact as $f$ is quasi-compact", "(Definition \\ref{definition-quasi-compact}).", "Choose an affine scheme $W$ and a surjective smooth morphism", "$W \\to \\mathcal{X}$. Then the image of $|f|$ is the image", "of $W \\to Y$.", "By Morphisms, Lemma \\ref{morphisms-lemma-reach-points-scheme-theoretic-image}", "we can choose a diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & W \\ar[d] \\ar[r] & \\mathcal{X} \\ar[d] \\\\", "\\Spec(A) \\ar[r] & Y \\ar[r] & Y", "}", "$$", "such that the closed point of $\\Spec(A)$ maps to $y$.", "Composing with $W \\to \\mathcal{X}$ we obtain a solution." ], "refs": [ "stacks-properties-lemma-points-cartesian", "stacks-properties-lemma-topology-points", "stacks-morphisms-lemma-base-change-quasi-compact", "stacks-morphisms-definition-quasi-compact", "morphisms-lemma-reach-points-scheme-theoretic-image" ], "ref_ids": [ 8864, 8867, 7423, 7604, 5147 ] } ], "ref_ids": [] }, { "id": 7430, "type": "theorem", "label": "stacks-morphisms-lemma-check-quasi-compact-covering", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-check-quasi-compact-covering", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $W \\to \\mathcal{Y}$ be surjective, flat, and locally of finite", "presentation where $W$ is an algebraic space. If the base change", "$W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is quasi-compact, then", "$f$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Assume $W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is quasi-compact.", "Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a morphism with $\\mathcal{Z}$", "a quasi-compact algebraic stack. Choose a scheme $U$ and a surjective", "smooth morphism", "$U \\to W \\times_\\mathcal{Y} \\mathcal{Z}$.", "Since $U \\to \\mathcal{Z}$ is flat, surjective, and locally of finite", "presentation and $\\mathcal{Z}$ is quasi-compact, we can find a", "quasi-compact open subscheme $U' \\subset U$ such that", "$U' \\to \\mathcal{Z}$ is surjective.", "Then", "$U' \\times_\\mathcal{Y} \\mathcal{X} =", "U' \\times_W \\times_W (W \\times \\mathcal{Y} \\mathcal{X})$", "is quasi-compact by assumption and surjects onto", "$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$.", "Hence $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$", "is quasi-compact as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7431, "type": "theorem", "label": "stacks-morphisms-lemma-locally-closed-in-noetherian", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-locally-closed-in-noetherian", "contents": [ "Let $j : \\mathcal{X} \\to \\mathcal{Y}$ be an immersion of algebraic stacks.", "\\begin{enumerate}", "\\item If $\\mathcal{Y}$ is locally Noetherian, then", "$\\mathcal{X}$ is locally Noetherian and $j$ is quasi-compact.", "\\item If $\\mathcal{Y}$ is Noetherian, then $\\mathcal{X}$ is Noetherian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective smooth morphism $V \\to \\mathcal{Y}$.", "Then $U = \\mathcal{X} \\times_\\mathcal{Y} V$ is a scheme and", "$V \\to U$ is an immersion, see", "Properties of Stacks, Definition \\ref{stacks-properties-definition-immersion}.", "Recall that $\\mathcal{Y}$ is locally Noetherian if and only if $V$", "is locally Noetherian. In this case $U$ is locally Noetherian too", "(Morphisms, Lemmas \\ref{morphisms-lemma-immersion-locally-finite-type} and", "\\ref{morphisms-lemma-finite-type-noetherian}) and $U \\to V$ is quasi-compact", "(Properties, Lemma \\ref{properties-lemma-immersion-into-noetherian}).", "This shows that $j$ is quasi-compact", "(Lemma \\ref{lemma-check-quasi-compact-covering})", "and that $\\mathcal{X}$ is locally Noetherian.", "Finally, if $\\mathcal{Y}$ is Noetherian, then we see from the above", "that $\\mathcal{X}$ is quasi-compact and locally Noetherian.", "To finish the proof observe that $j$ is separated and hence", "$\\mathcal{X}$ is quasi-separated because $\\mathcal{Y}$ is so by", "Lemma \\ref{lemma-separated-over-separated}." ], "refs": [ "stacks-properties-definition-immersion", "morphisms-lemma-immersion-locally-finite-type", "morphisms-lemma-finite-type-noetherian", "properties-lemma-immersion-into-noetherian", "stacks-morphisms-lemma-check-quasi-compact-covering", "stacks-morphisms-lemma-separated-over-separated" ], "ref_ids": [ 8920, 5201, 5202, 2952, 7430, 7405 ] } ], "ref_ids": [] }, { "id": 7432, "type": "theorem", "label": "stacks-morphisms-lemma-Noetherian-topology", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-Noetherian-topology", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "\\begin{enumerate}", "\\item If $\\mathcal{X}$ is locally Noetherian then $|\\mathcal{X}|$", "is a locally Noetherian topological space.", "\\item If $\\mathcal{X}$ is quasi-compact and locally Noetherian, then", "$|\\mathcal{X}|$ is a Noetherian topological space.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{X}$ is locally Noetherian.", "Choose a scheme $U$ and a surjective smooth morphism", "$U \\to \\mathcal{X}$. As $\\mathcal{X}$ is locally Noetherian", "we see that $U$ is locally Noetherian. By", "Properties, Lemma \\ref{properties-lemma-Noetherian-topology}", "this means that $|U|$ is a locally Noetherian topological space.", "Since $|U| \\to |\\mathcal{X}|$ is open and surjective we conclude that", "$|\\mathcal{X}|$ is locally Noetherian by", "Topology, Lemma \\ref{topology-lemma-image-Noetherian}.", "This proves (1). If $\\mathcal{X}$ is quasi-compact and locally Noetherian,", "then $|\\mathcal{X}|$ is quasi-compact and locally Noetherian. Hence", "$|\\mathcal{X}|$ is Noetherian by Topology, Lemma", "\\ref{topology-lemma-quasi-compact-locally-Noetherian-Noetherian}." ], "refs": [ "properties-lemma-Noetherian-topology", "topology-lemma-image-Noetherian", "topology-lemma-quasi-compact-locally-Noetherian-Noetherian" ], "ref_ids": [ 2954, 8221, 8240 ] } ], "ref_ids": [] }, { "id": 7433, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-affine", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-affine", "contents": [ "Let $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $\\mathcal{Z} \\to \\mathcal{Y}$ be an affine morphism of algebraic", "stacks. Then $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}$", "is an affine morphism of algebraic stacks." ], "refs": [], "proofs": [ { "contents": [ "This follows from the discussion in", "Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7434, "type": "theorem", "label": "stacks-morphisms-lemma-composition-affine", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-affine", "contents": [ "Compositions of affine morphisms of algebraic stacks are affine." ], "refs": [], "proofs": [ { "contents": [ "This follows from the discussion in", "Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}", "and", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-composition-affine}." ], "refs": [ "spaces-morphisms-lemma-composition-affine" ], "ref_ids": [ 4799 ] } ], "ref_ids": [] }, { "id": 7435, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-integral", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-integral", "contents": [ "Let $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $\\mathcal{Z} \\to \\mathcal{Y}$ be an integral (or finite)", "morphism of algebraic stacks. Then", "$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}$", "is an integral (or finite) morphism of algebraic stacks." ], "refs": [], "proofs": [ { "contents": [ "This follows from the discussion in", "Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7436, "type": "theorem", "label": "stacks-morphisms-lemma-composition-integral", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-integral", "contents": [ "Compositions of integral, resp.\\ finite morphisms of algebraic stacks", "are integral, resp.\\ finite." ], "refs": [], "proofs": [ { "contents": [ "This follows from the discussion in", "Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}", "and", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-composition-integral}." ], "refs": [ "spaces-morphisms-lemma-composition-integral" ], "ref_ids": [ 4941 ] } ], "ref_ids": [] }, { "id": 7437, "type": "theorem", "label": "stacks-morphisms-lemma-characterize-representable-universally-open", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-characterize-representable-universally-open", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of", "algebraic stacks which is representable by algebraic spaces.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally open (as in Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}), and", "\\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$", "the morphism of topological spaces", "$|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$ is open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1), and let $\\mathcal{Z} \\to \\mathcal{Y}$ be as in (2).", "Choose a scheme $V$ and a surjective smooth morphism $V \\to \\mathcal{Z}$.", "By assumption the morphism $V \\times_\\mathcal{Y} \\mathcal{X} \\to V$", "of algebraic spaces is universally open, in particular the map", "$|V \\times_\\mathcal{Y} \\mathcal{X}| \\to |V|$ is open. By", "Properties of Stacks, Section \\ref{stacks-properties-section-points}", "in the commutative diagram", "$$", "\\xymatrix{", "|V \\times_\\mathcal{Y} \\mathcal{X}| \\ar[r] \\ar[d] &", "|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\ar[d] \\\\", "|V| \\ar[r] & |\\mathcal{Z}|", "}", "$$", "the horizontal arrows are open and surjective, and moreover", "$$", "|V \\times_\\mathcal{Y} \\mathcal{X}| \\longrightarrow", "|V| \\times_{|\\mathcal{Z}|} |\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}|", "$$", "is surjective. Hence as the left vertical arrow is open it follows that", "the right vertical arrow is open. This proves (2).", "The implication (2) $\\Rightarrow$ (1) follows from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7438, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-universally-open", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-universally-open", "contents": [ "The base change of a universally open morphism of algebraic stacks", "by any morphism of algebraic stacks is universally open." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7439, "type": "theorem", "label": "stacks-morphisms-lemma-composition-universally-open", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-universally-open", "contents": [ "The composition of a pair of (universally) open morphisms of", "algebraic stacks is (universally) open." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7440, "type": "theorem", "label": "stacks-morphisms-lemma-characterize-representable-universally-submersive", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-characterize-representable-universally-submersive", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of", "algebraic stacks which is representable by algebraic spaces.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally submersive (as in Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}), and", "\\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$", "the morphism of topological spaces", "$|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$ is submersive.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1), and let $\\mathcal{Z} \\to \\mathcal{Y}$ be as in (2).", "Choose a scheme $V$ and a surjective smooth morphism $V \\to \\mathcal{Z}$.", "By assumption the morphism $V \\times_\\mathcal{Y} \\mathcal{X} \\to V$", "of algebraic spaces is universally submersive, in particular the map", "$|V \\times_\\mathcal{Y} \\mathcal{X}| \\to |V|$ is submersive. By", "Properties of Stacks, Section \\ref{stacks-properties-section-points}", "in the commutative diagram", "$$", "\\xymatrix{", "|V \\times_\\mathcal{Y} \\mathcal{X}| \\ar[r] \\ar[d] &", "|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\ar[d] \\\\", "|V| \\ar[r] & |\\mathcal{Z}|", "}", "$$", "the horizontal arrows are open and surjective, and moreover", "$$", "|V \\times_\\mathcal{Y} \\mathcal{X}| \\longrightarrow", "|V| \\times_{|\\mathcal{Z}|} |\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}|", "$$", "is surjective. Hence as the left vertical arrow is submersive it follows that", "the right vertical arrow is submersive. This proves (2).", "The implication (2) $\\Rightarrow$ (1) follows from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7441, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-universally-submersive", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-universally-submersive", "contents": [ "The base change of a universally submersive morphism of algebraic stacks", "by any morphism of algebraic stacks is universally submersive." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7442, "type": "theorem", "label": "stacks-morphisms-lemma-composition-universally-submersive", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-universally-submersive", "contents": [ "The composition of a pair of (universally) submersive morphisms of", "algebraic stacks is (universally) submersive." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7443, "type": "theorem", "label": "stacks-morphisms-lemma-characterize-representable-universally-closed", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-characterize-representable-universally-closed", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of", "algebraic stacks which is representable by algebraic spaces.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally closed (as in Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}), and", "\\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$", "the morphism of topological spaces", "$|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$ is closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1), and let $\\mathcal{Z} \\to \\mathcal{Y}$ be as in (2).", "Choose a scheme $V$ and a surjective smooth morphism $V \\to \\mathcal{Z}$.", "By assumption the morphism $V \\times_\\mathcal{Y} \\mathcal{X} \\to V$", "of algebraic spaces is universally closed, in particular the map", "$|V \\times_\\mathcal{Y} \\mathcal{X}| \\to |V|$ is closed. By", "Properties of Stacks, Section \\ref{stacks-properties-section-points}", "in the commutative diagram", "$$", "\\xymatrix{", "|V \\times_\\mathcal{Y} \\mathcal{X}| \\ar[r] \\ar[d] &", "|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\ar[d] \\\\", "|V| \\ar[r] & |\\mathcal{Z}|", "}", "$$", "the horizontal arrows are open and surjective, and moreover", "$$", "|V \\times_\\mathcal{Y} \\mathcal{X}| \\longrightarrow", "|V| \\times_{|\\mathcal{Z}|} |\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}|", "$$", "is surjective. Hence as the left vertical arrow is closed it follows that", "the right vertical arrow is closed. This proves (2).", "The implication (2) $\\Rightarrow$ (1) follows from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7444, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-universally-closed", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-universally-closed", "contents": [ "The base change of a universally closed morphism of algebraic stacks", "by any morphism of algebraic stacks is universally closed." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7445, "type": "theorem", "label": "stacks-morphisms-lemma-composition-universally-closed", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-universally-closed", "contents": [ "The composition of a pair of (universally) closed morphisms of", "algebraic stacks is (universally) closed." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7446, "type": "theorem", "label": "stacks-morphisms-lemma-universally-closed-local", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-universally-closed-local", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally closed,", "\\item for every scheme $Z$ and every morphism $Z \\to \\mathcal{Y}$", "the projection $|Z \\times_\\mathcal{Y} \\mathcal{X}| \\to |Z|$", "is closed,", "\\item for every affine scheme $Z$ and every morphism $Z \\to \\mathcal{Y}$", "the projection $|Z \\times_\\mathcal{Y} \\mathcal{X}| \\to |Z|$ is", "closed, and", "\\item there exists an algebraic space $V$ and a surjective smooth morphism", "$V \\to \\mathcal{Y}$ such that $V \\times_\\mathcal{Y} \\mathcal{X} \\to V$", "is a universally closed morphism of algebraic stacks.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We omit the proof that (1) implies (2), and that (2) implies (3).", "\\medskip\\noindent", "Assume (3). Choose a surjective smooth morphism $V \\to \\mathcal{Y}$.", "We are going to show that $V \\times_\\mathcal{Y} \\mathcal{X} \\to V$", "is a universally closed morphism of algebraic stacks.", "Let $\\mathcal{Z} \\to V$ be a morphism from an algebraic stack to $V$.", "Let $W \\to \\mathcal{Z}$ be a surjective smooth morphism where", "$W = \\coprod W_i$ is a disjoint union of affine schemes.", "Then we have the following commutative diagram", "$$", "\\xymatrix{", "\\coprod_i |W_i \\times_\\mathcal{Y} \\mathcal{X}| \\ar@{=}[r] \\ar[d] &", "|W \\times_\\mathcal{Y} \\mathcal{X}| \\ar[r] \\ar[d] &", "|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\ar[d] \\ar@{=}[r] &", "|\\mathcal{Z} \\times_V (V \\times_\\mathcal{Y} \\mathcal{X})| \\ar[ld] \\\\", "\\coprod |W_i| \\ar@{=}[r] &", "|W| \\ar[r] &", "|\\mathcal{Z}|", "}", "$$", "We have to show the south-east arrow is closed. The middle horizontal", "arrows are surjective and open", "(Properties of Stacks, Lemma \\ref{stacks-properties-lemma-topology-points}).", "By assumption (3), and the fact that", "$W_i$ is affine we see that the left vertical arrows are closed. Hence", "it follows that the right vertical arrow is closed.", "\\medskip\\noindent", "Assume (4). We will show that $f$ is universally closed.", "Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Consider the diagram", "$$", "\\xymatrix{", "|(V \\times_\\mathcal{Y} \\mathcal{Z})", "\\times_V (V \\times_\\mathcal{Y} \\mathcal{X})| \\ar@{=}[r] \\ar[rd] &", "|V \\times_\\mathcal{Y} \\mathcal{X}| \\ar[r] \\ar[d] &", "|Z \\times_\\mathcal{Y} \\mathcal{X}| \\ar[d] \\\\", " &", "|V \\times_\\mathcal{Y} \\mathcal{Z}| \\ar[r] &", "|\\mathcal{Z}|", "}", "$$", "The south-west arrow is closed by assumption. The horizontal arrows are", "surjective and open because the corresponding morphisms of", "algebraic stacks are surjective and smooth (see reference above).", "It follows that the right vertical arrow is closed." ], "refs": [ "stacks-properties-lemma-topology-points" ], "ref_ids": [ 8867 ] } ], "ref_ids": [] }, { "id": 7447, "type": "theorem", "label": "stacks-morphisms-lemma-characterize-representable-universally-injective", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-characterize-representable-universally-injective", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of", "algebraic stacks which is representable by algebraic spaces.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally injective (as in Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}), and", "\\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$", "the map $|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$", "is injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1), and let $\\mathcal{Z} \\to \\mathcal{Y}$ be as in (2).", "Choose a scheme $V$ and a surjective smooth morphism $V \\to \\mathcal{Z}$.", "By assumption the morphism $V \\times_\\mathcal{Y} \\mathcal{X} \\to V$", "of algebraic spaces is universally injective, in particular the map", "$|V \\times_\\mathcal{Y} \\mathcal{X}| \\to |V|$ is injective. By", "Properties of Stacks, Section \\ref{stacks-properties-section-points}", "in the commutative diagram", "$$", "\\xymatrix{", "|V \\times_\\mathcal{Y} \\mathcal{X}| \\ar[r] \\ar[d] &", "|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\ar[d] \\\\", "|V| \\ar[r] & |\\mathcal{Z}|", "}", "$$", "the horizontal arrows are open and surjective, and moreover", "$$", "|V \\times_\\mathcal{Y} \\mathcal{X}| \\longrightarrow", "|V| \\times_{|\\mathcal{Z}|} |\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}|", "$$", "is surjective. Hence as the left vertical arrow is injective it follows that", "the right vertical arrow is injective. This proves (2).", "The implication (2) $\\Rightarrow$ (1) follows from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7448, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-universally-injective", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-universally-injective", "contents": [ "The base change of a universally injective morphism of algebraic stacks", "by any morphism of algebraic stacks is universally injective." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7449, "type": "theorem", "label": "stacks-morphisms-lemma-composition-universally-injective", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-universally-injective", "contents": [ "The composition of a pair of universally injective morphisms of", "algebraic stacks is universally injective." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7450, "type": "theorem", "label": "stacks-morphisms-lemma-universally-injective", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-universally-injective", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally injective,", "\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "is surjective, and", "\\item for an algebraically closed field, for", "$x_1, x_2 : \\Spec(k) \\to \\mathcal{X}$, and for a $2$-arrow", "$\\beta : f \\circ x_1 \\to f \\circ x_2$ there is a", "$2$-arrow $\\alpha : x_1 \\to x_2$ with", "$\\beta = \\text{id}_f \\star \\alpha$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "(1) $\\Rightarrow$ (2). If $f$ is universally injective, then the", "first projection", "$|\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{X}|$", "is injective, which implies that $|\\Delta|$ is surjective.", "\\medskip\\noindent", "(2) $\\Rightarrow$ (1). Assume $\\Delta$ is surjective. Then any base change", "of $\\Delta$ is surjective (see Properties of Stacks, Section", "\\ref{stacks-properties-section-surjective}).", "Since the diagonal of a base change", "of $f$ is a base change of $\\Delta$, we see that it suffices", "to show that $|\\mathcal{X}| \\to |\\mathcal{Y}|$ is injective.", "If not, then by Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-points-cartesian}", "we find that the first projection", "$|\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{X}|$", "is not injective. Of course this means that $|\\Delta|$ is not", "surjective.", "\\medskip\\noindent", "(3) $\\Rightarrow$ (2). Let", "$t \\in |\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}|$.", "Then we can represent $t$ by a morphism", "$t : \\Spec(k) \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "with $k$ an algebraically closed field.", "By our construction of $2$-fibre products we can represent", "$t$ by $(x_1, x_2, \\beta)$ where $x_1, x_2 : \\Spec(k) \\to \\mathcal{X}$", "and $\\beta : f \\circ x_1 \\to f \\circ x_2$ is a $2$-morphism.", "Then (3) implies that there is a $2$-morphism", "$\\alpha : x_1 \\to x_2$ mapping to $\\beta$.", "This exactly means that $\\Delta(x_1) = (x_1, x_1, \\text{id})$", "is isomorphic to $t$. Hence (2) holds.", "\\medskip\\noindent", "(2) $\\Rightarrow$ (3). Let $x_1, x_2 : \\Spec(k) \\to \\mathcal{X}$", "be morphisms with $k$ an algebraically closed field. Let", "$\\beta : f \\circ x_1 \\to f \\circ x_2$ be a $2$-morphism.", "As in the previous paragraph, we obtain a morphism", "$t = (x_1, x_2, \\beta) :", "\\Spec(k) \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$.", "By Lemma \\ref{lemma-properties-diagonal}", "$$", "T = \\mathcal{X}", "\\times_{\\Delta, \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}, t} \\Spec(k)", "$$", "is an algebraic space locally of finite type over $\\Spec(k)$.", "Condition (2) implies that $T$ is nonempty. Then since $k$ is", "algebraically closed, there is a $k$-point in $T$.", "Unwinding the definitions this means there is a morphism", "$\\alpha : x_1 \\to x_2$ in $\\Mor(\\Spec(k), \\mathcal{X})$", "such that $\\beta = \\text{id}_f \\star \\alpha$." ], "refs": [ "stacks-properties-lemma-points-cartesian", "stacks-morphisms-lemma-properties-diagonal" ], "ref_ids": [ 8864, 7392 ] } ], "ref_ids": [] }, { "id": 7451, "type": "theorem", "label": "stacks-morphisms-lemma-universally-injective-point", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-universally-injective-point", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a universally injective", "morphism of algebraic stacks. Let $y : \\Spec(k) \\to \\mathcal{Y}$", "be a morphism where $k$ is an algebraically closed field.", "If $y$ is in the image of $|\\mathcal{X}| \\to |\\mathcal{Y}|$,", "then there is a morphism $x : \\Spec(k) \\to \\mathcal{X}$", "with $y = f \\circ x$." ], "refs": [], "proofs": [ { "contents": [ "We first remark this lemma is not a triviality, because the assumption that", "$y$ is in the image of $|f|$ means only that we can lift", "$y$ to a morphism into $\\mathcal{X}$ after possibly replacing", "$k$ by an extension field. To prove the lemma we may base change", "$f$ by $y$, hence we may assume we have a nonempty algebraic stack", "$\\mathcal{X}$ and a universally injective morphism", "$\\mathcal{X} \\to \\Spec(k)$ and we want to find a $k$-valued point", "of $\\mathcal{X}$. We may replace $\\mathcal{X}$ by its reduction.", "We may choose a field $k'$ and a surjective, flat, locally finite type morphism", "$\\Spec(k') \\to \\mathcal{X}$, see", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-unique-point}.", "Since $\\mathcal{X} \\to \\Spec(k)$ is universally injective, we find that", "$$", "\\Spec(k') \\times_\\mathcal{X} \\Spec(k') \\to \\Spec(k' \\otimes_k k')", "$$", "is surjective as the base change of the surjective morphism", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_{\\Spec(k)} \\mathcal{X}$", "(Lemma \\ref{lemma-universally-injective}).", "Since $k$ is algebraically closed $k' \\otimes_k k'$ is a domain", "(Algebra, Lemma", "\\ref{algebra-lemma-geometrically-integral-any-integral-base-change}).", "Let $\\xi \\in \\Spec(k') \\times_\\mathcal{X} \\Spec(k')$", "be a point mapping to the generic point of $\\Spec(k' \\otimes_k k')$.", "Let $U$ be the reduced induced closed subscheme structure on", "the connected component of $\\Spec(k') \\times_\\mathcal{X} \\Spec(k')$", "containing $\\xi$. Then the two projections $U \\to \\Spec(k')$", "are locally of finite type, as this was true for the projections", "$\\Spec(k') \\times_\\mathcal{X} \\Spec(k') \\to \\Spec(k')$", "as base changes of the morphism $\\Spec(k') \\to \\mathcal{X}$.", "Applying", "Varieties, Proposition \\ref{varieties-proposition-unique-base-field}", "we find that the integral closures of the two images", "of $k'$ in $\\Gamma(U, \\mathcal{O}_U)$ are equal.", "Looking in $\\kappa(\\xi)$ means that any element of the form", "$\\lambda \\otimes 1$ is algebraically dependend on", "the subfield", "$$", "1 \\otimes k' \\subset ", "(\\text{fraction field of }k' \\otimes_k k') \\subset", "\\kappa(\\xi).", "$$", "Since $k$ is algebraically closed, this is only possible", "if $k' = k$ and the proof is complete." ], "refs": [ "stacks-properties-lemma-unique-point", "stacks-morphisms-lemma-universally-injective", "algebra-lemma-geometrically-integral-any-integral-base-change", "varieties-proposition-unique-base-field" ], "ref_ids": [ 8901, 7450, 607, 11137 ] } ], "ref_ids": [] }, { "id": 7452, "type": "theorem", "label": "stacks-morphisms-lemma-universally-injective-local", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-universally-injective-local", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is universally injective,", "\\item for every affine scheme $Z$ and any morphism", "$Z \\to \\mathcal{Y}$ the morphism $Z \\times_\\mathcal{Y} \\mathcal{X} \\to Z$", "is universally injective, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) is immediate.", "Assume (2) holds. We will show that", "$\\Delta_f : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "is surjective, which implies (1) by Lemma \\ref{lemma-universally-injective}.", "Consider an affine scheme $V$ and a smooth morphism", "$V \\to \\mathcal{Y}$. Since", "$g : V \\times_\\mathcal{Y} \\mathcal{X} \\to V$", "is universally injective by (2), we see that", "$\\Delta_g$ is surjective.", "However, $\\Delta_g$ is the base change of $\\Delta_f$", "by the smooth morphism $V \\to \\mathcal{Y}$.", "Since the collection of these morphisms $V \\to \\mathcal{Y}$", "are jointly surjective, we conclude $\\Delta_f$ is surjective." ], "refs": [ "stacks-morphisms-lemma-universally-injective" ], "ref_ids": [ 7450 ] } ], "ref_ids": [] }, { "id": 7453, "type": "theorem", "label": "stacks-morphisms-lemma-check-universally-injective-covering", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-check-universally-injective-covering", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $W \\to \\mathcal{Y}$ be surjective, flat, and locally of finite", "presentation where $W$ is an algebraic space. If the base change", "$W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is universally injective,", "then $f$ is universally injective." ], "refs": [], "proofs": [ { "contents": [ "Observe that the diagonal $\\Delta_g$ of the morphism", "$g : W \\times_\\mathcal{Y} \\mathcal{X} \\to W$", "is the base change of $\\Delta_f$ by $W \\to \\mathcal{Y}$.", "Hence if $\\Delta_g$ is surjective, then so is $\\Delta_f$", "by Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}.", "Thus the lemma follows from the characterization (2)", "in Lemma \\ref{lemma-universally-injective}." ], "refs": [ "stacks-properties-lemma-check-property-covering", "stacks-morphisms-lemma-universally-injective" ], "ref_ids": [ 8859, 7450 ] } ], "ref_ids": [] }, { "id": 7454, "type": "theorem", "label": "stacks-morphisms-lemma-characterize-representable-universal-homeomorphism", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-characterize-representable-universal-homeomorphism", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of", "algebraic stacks which is representable by algebraic spaces.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is a universal homeomorphism (Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}), and", "\\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$", "the map of topological spaces", "$|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$ is", "a homeomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1), and let $\\mathcal{Z} \\to \\mathcal{Y}$ be as in (2).", "Choose a scheme $V$ and a surjective smooth morphism $V \\to \\mathcal{Z}$.", "By assumption the morphism $V \\times_\\mathcal{Y} \\mathcal{X} \\to V$", "of algebraic spaces is a universal homeomorphism, in particular the map", "$|V \\times_\\mathcal{Y} \\mathcal{X}| \\to |V|$ is a homeomorphism. By", "Properties of Stacks, Section \\ref{stacks-properties-section-points}", "in the commutative diagram", "$$", "\\xymatrix{", "|V \\times_\\mathcal{Y} \\mathcal{X}| \\ar[r] \\ar[d] &", "|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\ar[d] \\\\", "|V| \\ar[r] & |\\mathcal{Z}|", "}", "$$", "the horizontal arrows are open and surjective, and moreover", "$$", "|V \\times_\\mathcal{Y} \\mathcal{X}| \\longrightarrow", "|V| \\times_{|\\mathcal{Z}|} |\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}|", "$$", "is surjective. Hence as the left vertical arrow is a homeomorphism", "it follows that the right vertical arrow is a homeomorphism. This proves (2).", "The implication (2) $\\Rightarrow$ (1) follows from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7455, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-universal-homeomorphism", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-universal-homeomorphism", "contents": [ "The base change of a universal homeomorphism of algebraic stacks", "by any morphism of algebraic stacks is a universal homeomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7456, "type": "theorem", "label": "stacks-morphisms-lemma-composition-universal-homeomorphism", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-universal-homeomorphism", "contents": [ "The composition of a pair of universal homeomorphisms of", "algebraic stacks is a universal homeomorphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7457, "type": "theorem", "label": "stacks-morphisms-lemma-check-universal-homeomorphism-covering", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-check-universal-homeomorphism-covering", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $W \\to \\mathcal{Y}$ be surjective, flat, and locally of finite", "presentation where $W$ is an algebraic space. If the base change", "$W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is a universal homeomorphism,", "then $f$ is a universal homeomorphism." ], "refs": [], "proofs": [ { "contents": [ "Assume $g : W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is a universal", "homeomorphism. Then $g$ is universally injective, hence $f$ is", "universally injective by", "Lemma \\ref{lemma-check-universally-injective-covering}.", "On the other hand, let $\\mathcal{Z} \\to \\mathcal{Y}$", "be a morphism with $\\mathcal{Z}$ an algebraic stack.", "Choose a scheme $U$ and a surjective", "smooth morphism $U \\to W \\times_\\mathcal{Y} \\mathcal{Z}$.", "Consider the diagram", "$$", "\\xymatrix{", "W \\times_\\mathcal{Y} \\mathcal{X} \\ar[d]^g &", "U \\times_\\mathcal{Y} \\mathcal{X} \\ar[d] \\ar[l] \\ar[r] &", "\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\ar[d] \\\\", "W &", "U \\ar[l] \\ar[r] &", "\\mathcal{Z}", "}", "$$", "The middle vertical arrow induces a homeomorphism", "on topological space by assumption on $g$.", "The morphism $U \\to \\mathcal{Z}$ and", "$U \\times_\\mathcal{Y} \\mathcal{X} \\to", "\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$", "are surjective, flat, and locally of finite presentation", "hence induce open maps on topological spaces.", "We conclude that", "$|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$", "is open. Surjectivity is easy to prove; we omit the proof." ], "refs": [ "stacks-morphisms-lemma-check-universally-injective-covering" ], "ref_ids": [ 7453 ] } ], "ref_ids": [] }, { "id": 7458, "type": "theorem", "label": "stacks-morphisms-lemma-local-source-target", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-local-source-target", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces", "which is smooth local on the source-and-target.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Consider commutative diagrams", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "\\mathcal{X} \\ar[r]^f & \\mathcal{Y}", "}", "$$", "where $U$ and $V$ are algebraic spaces and the vertical arrows are smooth.", "The following are equivalent", "\\begin{enumerate}", "\\item for any diagram as above such that in addition", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ is smooth", "the morphism $h$ has property $\\mathcal{P}$, and", "\\item for some diagram as above with $a : U \\to \\mathcal{X}$ surjective", "the morphism $h$ has property $\\mathcal{P}$.", "\\end{enumerate}", "If $\\mathcal{X}$ and $\\mathcal{Y}$ are representable by algebraic spaces,", "then this is also equivalent to $f$ (as a morphism of algebraic spaces)", "having property $\\mathcal{P}$. If $\\mathcal{P}$ is also preserved under", "any base change, and fppf local on the base, then for morphisms $f$", "which are representable by algebraic spaces this", "is also equivalent to $f$ having property $\\mathcal{P}$ in the sense", "of", "Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}." ], "refs": [], "proofs": [ { "contents": [ "Let us prove the implication (1) $\\Rightarrow$ (2). Pick an algebraic", "space $V$ and a surjective and smooth morphism $V \\to \\mathcal{Y}$.", "Pick an algebraic space $U$ and a surjective and smooth morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$. Note that $U \\to \\mathcal{X}$", "is surjective and smooth as well, as a composition of the base change", "$\\mathcal{X} \\times_\\mathcal{Y} V \\to \\mathcal{X}$ and the chosen", "map $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$. Hence we obtain a", "diagram as in (1). Thus if (1) holds, then $h : U \\to V$ has property", "$\\mathcal{P}$, which means that (2) holds as $U \\to \\mathcal{X}$ is surjective.", "\\medskip\\noindent", "Conversely, assume (2) holds and let $U, V, a, b, h$ be as in (2).", "Next, let $U', V', a', b', h'$ be any diagram as in (1).", "Picture", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_h & V \\ar[d] \\\\", "\\mathcal{X} \\ar[r]^f & \\mathcal{Y}", "}", "\\quad\\quad", "\\xymatrix{", "U' \\ar[d] \\ar[r]_{h'} & V' \\ar[d] \\\\", "\\mathcal{X} \\ar[r]^f & \\mathcal{Y}", "}", "$$", "To show that (2) implies (1) we have to prove that $h'$ has $\\mathcal{P}$.", "To do this consider the commutative diagram", "$$", "\\xymatrix{", "U \\ar[dd]^h &", "U \\times_\\mathcal{X} U' \\ar[d] \\ar[l] \\ar@/^6ex/[dd]^{(h, h')} \\ar[r] &", "U' \\ar[dd]^{h'} \\\\", "& U \\times_\\mathcal{Y} V' \\ar[lu] \\ar[d] & \\\\", "V &", "V \\times_\\mathcal{Y} V' \\ar[l] \\ar[r] &", "V'", "}", "$$", "of algebraic spaces. Note that the horizontal arrows are", "smooth as base changes of the smooth morphisms", "$V \\to \\mathcal{Y}$, $V' \\to \\mathcal{Y}$, $U \\to \\mathcal{X}$, and", "$U' \\to \\mathcal{X}$. Note that", "$$", "\\xymatrix{", "U \\times_\\mathcal{X} U' \\ar[d] \\ar[r] & U' \\ar[d] \\\\", "U \\times_\\mathcal{Y} V' \\ar[r] & \\mathcal{X} \\times_\\mathcal{Y} V'", "}", "$$", "is cartesian, hence the left vertical arrow is smooth as", "$U', V', a', b', h'$ is as in (1).", "Since $\\mathcal{P}$ is smooth local on the target by", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-local-source-target-implies} part (2)", "we see", "that the base change $U \\times_\\mathcal{Y} V' \\to V \\times_\\mathcal{Y} V'$", "has $\\mathcal{P}$. Since $\\mathcal{P}$ is smooth local on the source by", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-local-source-target-implies} part (1)", "we can precompose by the smooth morphism", "$U \\times_\\mathcal{X} U' \\to U \\times_\\mathcal{Y} V'$ and", "conclude $(h, h')$ has $\\mathcal{P}$.", "Since $V \\times_\\mathcal{Y} V' \\to V'$ is smooth we conclude", "$U \\times_\\mathcal{X} U' \\to V'$ has $\\mathcal{P}$ by", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-local-source-target-implies} part (3).", "Finally, since $U \\times_X U' \\to U'$", "is surjective and smooth and $\\mathcal{P}$ is smooth local", "on the source (same lemma) we conclude", "that $h'$ has $\\mathcal{P}$. This finishes the proof of the equivalence", "of (1) and (2).", "\\medskip\\noindent", "If $\\mathcal{X}$ and $\\mathcal{Y}$ are representable, then", "Descent on Spaces,", "Lemma \\ref{spaces-descent-lemma-local-source-target-characterize}", "applies which shows that (1) and (2) are equivalent to $f$ having", "$\\mathcal{P}$.", "\\medskip\\noindent", "Finally, suppose $f$ is representable, and $U, V, a, b, h$ are", "as in part (2) of the lemma, and that $\\mathcal{P}$ is preserved under", "arbitrary base change. We have to show that for any scheme", "$Z$ and morphism $Z \\to \\mathcal{X}$ the base change", "$Z \\times_\\mathcal{Y} \\mathcal{X} \\to Z$", "has property $\\mathcal{P}$. Consider the diagram", "$$", "\\xymatrix{", "Z \\times_\\mathcal{Y} U \\ar[d] \\ar[r] &", "Z \\times_\\mathcal{Y} V \\ar[d] \\\\", "Z \\times_\\mathcal{Y} \\mathcal{X} \\ar[r] &", "Z", "}", "$$", "Note that the top horizontal arrow is a base change of $h$ and", "hence has property $\\mathcal{P}$. The left vertical arrow is smooth", "and surjective and the right vertical arrow is smooth. Thus", "Descent on Spaces,", "Lemma \\ref{spaces-descent-lemma-local-source-target-characterize}", "kicks in and shows that $Z \\times_\\mathcal{Y} \\mathcal{X} \\to Z$", "has property $\\mathcal{P}$." ], "refs": [ "spaces-descent-lemma-local-source-target-implies", "spaces-descent-lemma-local-source-target-implies", "spaces-descent-lemma-local-source-target-implies", "spaces-descent-lemma-local-source-target-characterize", "spaces-descent-lemma-local-source-target-characterize" ], "ref_ids": [ 9427, 9427, 9427, 9428, 9428 ] } ], "ref_ids": [] }, { "id": 7459, "type": "theorem", "label": "stacks-morphisms-lemma-composition-finite-type", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-finite-type", "contents": [ "The composition of finite type morphisms is of finite type.", "The same holds for locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Remark \\ref{remark-composition}", "with", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-composition-finite-type}." ], "refs": [ "stacks-morphisms-remark-composition", "spaces-morphisms-lemma-composition-finite-type" ], "ref_ids": [ 7633, 4814 ] } ], "ref_ids": [] }, { "id": 7460, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-finite-type", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-finite-type", "contents": [ "A base change of a finite type morphism is finite type.", "The same holds for locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Remark \\ref{remark-base-change}", "with", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-finite-type}." ], "refs": [ "stacks-morphisms-remark-base-change", "spaces-morphisms-lemma-base-change-finite-type" ], "ref_ids": [ 7634, 4815 ] } ], "ref_ids": [] }, { "id": 7461, "type": "theorem", "label": "stacks-morphisms-lemma-immersion-locally-finite-type", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-immersion-locally-finite-type", "contents": [ "An immersion is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "Combine Remark \\ref{remark-implication} with", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-immersion-locally-finite-type}." ], "refs": [ "stacks-morphisms-remark-implication", "spaces-morphisms-lemma-immersion-locally-finite-type" ], "ref_ids": [ 7635, 4819 ] } ], "ref_ids": [] }, { "id": 7462, "type": "theorem", "label": "stacks-morphisms-lemma-locally-finite-type-locally-noetherian", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-locally-finite-type-locally-noetherian", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "If $f$ is locally of finite type and $\\mathcal{Y}$ is locally Noetherian,", "then $\\mathcal{X}$ is locally Noetherian." ], "refs": [], "proofs": [ { "contents": [ "Let", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "\\mathcal{X} \\ar[r] & \\mathcal{Y}", "}", "$$", "be a commutative diagram where $U$, $V$ are schemes,", "$V \\to \\mathcal{Y}$ is surjective and smooth, and", "$U \\to V \\times_\\mathcal{Y} \\mathcal{X}$ is surjective and smooth.", "Then $U \\to V$ is locally of finite type. If $\\mathcal{Y}$ is", "locally Noetherian, then $V$ is locally Noetherian. By", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}", "we see that $U$ is locally Noetherian, which means that $\\mathcal{X}$", "is locally Noetherian." ], "refs": [ "morphisms-lemma-finite-type-noetherian" ], "ref_ids": [ 5202 ] } ], "ref_ids": [] }, { "id": 7463, "type": "theorem", "label": "stacks-morphisms-lemma-check-finite-type-covering", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-check-finite-type-covering", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $W \\to \\mathcal{Y}$ be a surjective, flat, and locally of finite", "presentation where $W$ is an algebraic space. If the base change", "$W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is", "locally of finite type, then $f$ is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "Choose an algebraic space $V$ and a surjective smooth morphism", "$V \\to \\mathcal{Y}$. Choose an algebraic space $U$ and a surjective", "smooth morphism $U \\to V \\times_\\mathcal{Y} \\mathcal{X}$.", "We have to show that $U \\to V$ is locally of finite presentation.", "Now we base change everything by $W \\to \\mathcal{Y}$: Set", "$U' = W \\times_\\mathcal{Y} U$,", "$V' = W \\times_\\mathcal{Y} V$,", "$\\mathcal{X}' = W \\times_\\mathcal{Y} \\mathcal{X}$,", "and $\\mathcal{Y}' = W \\times_\\mathcal{Y} \\mathcal{Y} = W$.", "Then it is still true that $U' \\to V' \\times_{\\mathcal{Y}'} \\mathcal{X}'$", "is smooth by base change. Hence by our definition of locally finite type", "morphisms of algebraic stacks and the assumption that", "$\\mathcal{X}' \\to \\mathcal{Y}'$ is locally of finite type,", "we see that $U' \\to V'$ is locally of finite type. Then, since", "$V' \\to V$ is surjective, flat, and locally of finite presentation", "as a base change of $W \\to \\mathcal{Y}$ we see that $U \\to V$ is", "locally of finite type by", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-locally-finite-type}", "and we win." ], "refs": [ "spaces-descent-lemma-descending-property-locally-finite-type" ], "ref_ids": [ 9389 ] } ], "ref_ids": [] }, { "id": 7464, "type": "theorem", "label": "stacks-morphisms-lemma-check-finite-type-precompose", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-check-finite-type-precompose", "contents": [ "Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of", "algebraic stacks. Assume $\\mathcal{X} \\to \\mathcal{Z}$ is locally of finite", "type and that $\\mathcal{X} \\to \\mathcal{Y}$ is representable by algebraic", "spaces, surjective, flat, and locally of finite presentation.", "Then $\\mathcal{Y} \\to \\mathcal{Z}$ is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "Choose an algebraic space $W$ and a surjective smooth morphism", "$W \\to \\mathcal{Z}$. Choose an algebraic space $V$ and a surjective smooth", "morphism $V \\to W \\times_\\mathcal{Z} \\mathcal{Y}$. Set", "$U = V \\times_\\mathcal{Y} \\mathcal{X}$ which is an algebraic space.", "We know that $U \\to V$ is surjective, flat, and locally of finite presentation", "and that $U \\to W$ is locally of finite type.", "Hence the lemma reduces to the case of morphisms of algebraic spaces.", "The case of morphisms of algebraic spaces is", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-locally-finite-type-fppf-local-source}." ], "refs": [ "spaces-descent-lemma-locally-finite-type-fppf-local-source" ], "ref_ids": [ 9419 ] } ], "ref_ids": [] }, { "id": 7465, "type": "theorem", "label": "stacks-morphisms-lemma-finite-type-permanence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-finite-type-permanence", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$,", "$g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks.", "If $g \\circ f : \\mathcal{X} \\to \\mathcal{Z}$ is locally of finite type,", "then $f : \\mathcal{X} \\to \\mathcal{Y}$ is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "We can find a diagram", "$$", "\\xymatrix{", "U \\ar[r] \\ar[d] & V \\ar[r] \\ar[d] & W \\ar[d] \\\\", "\\mathcal{X} \\ar[r] & \\mathcal{Y} \\ar[r] & \\mathcal{Z}", "}", "$$", "where $U$, $V$, $W$ are schemes, the vertical arrow $W \\to \\mathcal{Z}$", "is surjective and smooth, the arrow $V \\to \\mathcal{Y} \\times_\\mathcal{Z} W$", "is surjective and smooth, and the arrow", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ is surjective and smooth.", "Then also $U \\to \\mathcal{X} \\times_\\mathcal{Z} V$ is surjective and", "smooth (as a composition of a surjective and smooth morphism with a", "base change of such). By definition we see that $U \\to W$ is locally", "of finite type. Hence $U \\to V$ is locally of finite type by", "Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type}", "which in turn means (by definition) that $\\mathcal{X} \\to \\mathcal{Y}$", "is locally of finite type." ], "refs": [ "morphisms-lemma-permanence-finite-type" ], "ref_ids": [ 5204 ] } ], "ref_ids": [] }, { "id": 7466, "type": "theorem", "label": "stacks-morphisms-lemma-point-finite-type", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-point-finite-type", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Let $x \\in |\\mathcal{X}|$. The following are equivalent:", "\\begin{enumerate}", "\\item There exists a morphism $\\Spec(k) \\to \\mathcal{X}$", "which is locally of finite type and represents $x$.", "\\item There exists a scheme $U$, a closed point $u \\in U$, and a smooth", "morphism $\\varphi : U \\to \\mathcal{X}$ such that $\\varphi(u) = x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $u \\in U$ and $U \\to \\mathcal{X}$ be as in (2). Then", "$\\Spec(\\kappa(u)) \\to U$ is of finite type, and $U \\to \\mathcal{X}$ is", "representable and locally of finite type (by", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-etale-locally-finite-presentation} and", "\\ref{spaces-morphisms-lemma-finite-presentation-finite-type}).", "Hence we see (1) holds by", "Lemma \\ref{lemma-composition-finite-type}.", "\\medskip\\noindent", "Conversely, assume $\\Spec(k) \\to \\mathcal{X}$ is locally of finite type", "and represents $x$. Let $U \\to \\mathcal{X}$ be a surjective smooth morphism", "where $U$ is a scheme. By assumption", "$U \\times_\\mathcal{X} \\Spec(k) \\to U$ is a morphism of algebraic", "spaces which is locally of finite type. Pick a finite type point $v$ of", "$U \\times_\\mathcal{X} \\Spec(k)$ (there exists at least one, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-identify-finite-type-points}).", "By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-finite-type-points-morphism}", "the image $u \\in U$ of $v$ is a finite type point of $U$.", "Hence by", "Morphisms, Lemma \\ref{morphisms-lemma-identify-finite-type-points}", "after shrinking $U$ we may assume that $u$ is a closed point of $U$, i.e.,", "(2) holds." ], "refs": [ "spaces-morphisms-lemma-etale-locally-finite-presentation", "spaces-morphisms-lemma-finite-presentation-finite-type", "stacks-morphisms-lemma-composition-finite-type", "spaces-morphisms-lemma-identify-finite-type-points", "spaces-morphisms-lemma-finite-type-points-morphism", "morphisms-lemma-identify-finite-type-points" ], "ref_ids": [ 4911, 4842, 7459, 4823, 4824, 5207 ] } ], "ref_ids": [] }, { "id": 7467, "type": "theorem", "label": "stacks-morphisms-lemma-identify-finite-type-points", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-identify-finite-type-points", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. We have", "$$", "\\mathcal{X}_{\\text{ft-pts}} =", "\\bigcup\\nolimits_{\\varphi : U \\to \\mathcal{X}\\text{ smooth}} |\\varphi|(U_0)", "$$", "where $U_0$ is the set of closed points of $U$.", "Here we may let $U$ range over all schemes smooth over $\\mathcal{X}$", "or over all affine schemes smooth over $\\mathcal{X}$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from", "Lemma \\ref{lemma-point-finite-type}." ], "refs": [ "stacks-morphisms-lemma-point-finite-type" ], "ref_ids": [ 7466 ] } ], "ref_ids": [] }, { "id": 7468, "type": "theorem", "label": "stacks-morphisms-lemma-finite-type-points-morphism", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-finite-type-points-morphism", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "If $f$ is locally of finite type, then", "$f(\\mathcal{X}_{\\text{ft-pts}}) \\subset \\mathcal{Y}_{\\text{ft-pts}}$." ], "refs": [], "proofs": [ { "contents": [ "Take $x \\in \\mathcal{X}_{\\text{ft-pts}}$. Represent $x$ by a locally", "finite type morphism $x : \\Spec(k) \\to \\mathcal{X}$. Then", "$f \\circ x$ is locally of finite type by", "Lemma \\ref{lemma-composition-finite-type}.", "Hence $f(x) \\in \\mathcal{Y}_{\\text{ft-pts}}$." ], "refs": [ "stacks-morphisms-lemma-composition-finite-type" ], "ref_ids": [ 7459 ] } ], "ref_ids": [] }, { "id": 7469, "type": "theorem", "label": "stacks-morphisms-lemma-finite-type-points-surjective-morphism", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-finite-type-points-surjective-morphism", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "If $f$ is locally of finite type and surjective, then", "$f(\\mathcal{X}_{\\text{ft-pts}}) = \\mathcal{Y}_{\\text{ft-pts}}$." ], "refs": [], "proofs": [ { "contents": [ "We have $f(\\mathcal{X}_{\\text{ft-pts}}) \\subset \\mathcal{Y}_{\\text{ft-pts}}$ by", "Lemma \\ref{lemma-finite-type-points-morphism}.", "Let $y \\in |\\mathcal{Y}|$ be a finite type point. Represent $y$ by a morphism", "$\\Spec(k) \\to \\mathcal{Y}$ which is locally of finite type.", "As $f$ is surjective the algebraic stack", "$\\mathcal{X}_k = \\Spec(k) \\times_\\mathcal{Y} \\mathcal{X}$ is nonempty,", "therefore has a finite type point $x \\in |\\mathcal{X}_k|$ by", "Lemma \\ref{lemma-identify-finite-type-points}.", "Now $\\mathcal{X}_k \\to \\mathcal{X}$ is a morphism which is locally of finite", "type as a base change of $\\Spec(k) \\to \\mathcal{Y}$", "(Lemma \\ref{lemma-base-change-finite-type}).", "Hence the image of $x$ in $\\mathcal{X}$ is a finite type point by", "Lemma \\ref{lemma-finite-type-points-morphism}", "which maps to $y$ by construction." ], "refs": [ "stacks-morphisms-lemma-finite-type-points-morphism", "stacks-morphisms-lemma-identify-finite-type-points", "stacks-morphisms-lemma-base-change-finite-type", "stacks-morphisms-lemma-finite-type-points-morphism" ], "ref_ids": [ 7468, 7467, 7460, 7468 ] } ], "ref_ids": [] }, { "id": 7470, "type": "theorem", "label": "stacks-morphisms-lemma-enough-finite-type-points", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-enough-finite-type-points", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "For any locally closed subset $T \\subset |\\mathcal{X}|$ we have", "$$", "T \\not = \\emptyset", "\\Rightarrow", "T \\cap \\mathcal{X}_{\\text{ft-pts}} \\not = \\emptyset.", "$$", "In particular, for any closed subset $T \\subset |\\mathcal{X}|$ we", "see that $T \\cap \\mathcal{X}_{\\text{ft-pts}}$ is dense in $T$." ], "refs": [], "proofs": [ { "contents": [ "Let $i : \\mathcal{Z} \\to \\mathcal{X}$ be the reduced induced substack", "structure on $T$, see", "Properties of Stacks,", "Remark \\ref{stacks-properties-remark-stack-structure-locally-closed-subset}.", "An immersion is locally of finite type, see", "Lemma \\ref{lemma-immersion-locally-finite-type}.", "Hence by", "Lemma \\ref{lemma-finite-type-points-morphism}", "we see", "$\\mathcal{Z}_{\\text{ft-pts}} \\subset \\mathcal{X}_{\\text{ft-pts}} \\cap T$.", "Finally, any nonempty affine scheme $U$ with a smooth morphism towards", "$\\mathcal{Z}$ has at least one closed point, hence $\\mathcal{Z}$ has at least", "one finite type point by", "Lemma \\ref{lemma-identify-finite-type-points}.", "The lemma follows." ], "refs": [ "stacks-properties-remark-stack-structure-locally-closed-subset", "stacks-morphisms-lemma-immersion-locally-finite-type", "stacks-morphisms-lemma-finite-type-points-morphism", "stacks-morphisms-lemma-identify-finite-type-points" ], "ref_ids": [ 8932, 7461, 7468, 7467 ] } ], "ref_ids": [] }, { "id": 7471, "type": "theorem", "label": "stacks-morphisms-lemma-point-finite-type-monomorphism", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-point-finite-type-monomorphism", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Let $x \\in |\\mathcal{X}|$. The following are equivalent:", "\\begin{enumerate}", "\\item $x$ is a finite type point,", "\\item there exists an algebraic stack $\\mathcal{Z}$", "whose underlying topological space $|\\mathcal{Z}|$ is a singleton,", "and a morphism $f : \\mathcal{Z} \\to \\mathcal{X}$ which is", "locally of finite type such that $\\{x\\} = |f|(|\\mathcal{Z}|)$, and", "\\item the residual gerbe $\\mathcal{Z}_x$ of $\\mathcal{X}$ at $x$ exists", "and the inclusion morphism $\\mathcal{Z}_x \\to \\mathcal{X}$ is locally of", "finite type.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "(All of the morphisms occurring in this paragraph are representable", "by algebraic spaces, hence the conventions and results of", "Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}", "are applicable.)", "Assume $x$ is a finite type point. Choose an affine scheme $U$,", "a closed point $u \\in U$, and a smooth morphism $\\varphi : U \\to \\mathcal{X}$", "with $\\varphi(u) = x$, see", "Lemma \\ref{lemma-identify-finite-type-points}.", "Set $u = \\Spec(\\kappa(u))$ as usual. Set $R = u \\times_\\mathcal{X} u$", "so that we obtain a groupoid in algebraic spaces", "$(u, R, s, t, c)$, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack}.", "The projection morphisms $R \\to u$ are the compositions", "$$", "R = u \\times_\\mathcal{X} u \\to", "u \\times_\\mathcal{X} U \\to", "u \\times_\\mathcal{X} X = u", "$$", "where the first arrow is of finite type (a base change of the closed", "immersion of schemes $u \\to U$) and the second arrow is smooth (a base", "change of the smooth morphism $U \\to \\mathcal{X}$). Hence", "$s, t : R \\to u$ are locally of finite type (as compositions, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-composition-finite-type}).", "Since $u$ is the spectrum of a field, it follows that", "$s, t$ are flat and locally of finite presentation (by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation}).", "We see that $\\mathcal{Z} = [u/R]$ is an algebraic stack by", "Criteria for Representability,", "Theorem \\ref{criteria-theorem-flat-groupoid-gives-algebraic-stack}.", "By", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack}", "we obtain a canonical morphism", "$$", "f : \\mathcal{Z} \\longrightarrow \\mathcal{X}", "$$", "which is fully faithful. Hence this morphism is representable by", "algebraic spaces, see", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}", "and a monomorphism, see", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-monomorphism}.", "It follows that the residual gerbe $\\mathcal{Z}_x \\subset \\mathcal{X}$", "of $\\mathcal{X}$ at $x$ exists and that $f$ factors through an", "equivalence $\\mathcal{Z} \\to \\mathcal{Z}_x$, see", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-residual-gerbe-unique}.", "By construction the diagram", "$$", "\\xymatrix{", "u \\ar[d] \\ar[r] & U \\ar[d] \\\\", "\\mathcal{Z} \\ar[r]^f & \\mathcal{X}", "}", "$$", "is commutative. By", "Criteria for Representability,", "Lemma \\ref{criteria-lemma-flat-quotient-flat-presentation}", "the left vertical arrow is surjective, flat, and locally of finite", "presentation. Consider", "$$", "\\xymatrix{", "u \\times_\\mathcal{X} U \\ar[d] \\ar[r] &", "\\mathcal{Z} \\times_\\mathcal{X} U \\ar[r] \\ar[d] & U \\ar[d] \\\\", "u \\ar[r] & \\mathcal{Z} \\ar[r]^f & \\mathcal{X}", "}", "$$", "As $u \\to \\mathcal{X}$ is locally of finite type, we see that the base change", "$u \\times_\\mathcal{X} U \\to U$ is locally of finite type. Moreover,", "$u \\times_\\mathcal{X} U \\to \\mathcal{Z} \\times_\\mathcal{X} U$ is", "surjective, flat, and locally of finite presentation as a base change of", "$u \\to \\mathcal{Z}$. Thus", "$\\{u \\times_\\mathcal{X} U \\to \\mathcal{Z} \\times_\\mathcal{X} U\\}$", "is an fppf covering of algebraic spaces, and we conclude that", "$\\mathcal{Z} \\times_\\mathcal{X} U \\to U$ is locally of finite type by", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-locally-finite-presentation-fppf-local-source}.", "By definition this means that $f$ is locally of finite type (because the", "vertical arrow $\\mathcal{Z} \\times_\\mathcal{X} U \\to \\mathcal{Z}$ is smooth", "as a base change of $U \\to \\mathcal{X}$ and surjective as $\\mathcal{Z}$ has", "only one point). Since $\\mathcal{Z} = \\mathcal{Z}_x$ we see that (3) holds.", "\\medskip\\noindent", "It is clear that (3) implies (2).", "If (2) holds then $x$ is a finite type point of $\\mathcal{X}$ by", "Lemma \\ref{lemma-finite-type-points-morphism}", "and", "Lemma \\ref{lemma-enough-finite-type-points}", "to see that $\\mathcal{Z}_{\\text{ft-pts}}$ is nonempty, i.e., the", "unique point of $\\mathcal{Z}$ is a finite type point of $\\mathcal{Z}$." ], "refs": [ "stacks-morphisms-lemma-identify-finite-type-points", "algebraic-lemma-map-space-into-stack", "spaces-morphisms-lemma-composition-finite-type", "spaces-morphisms-lemma-noetherian-finite-type-finite-presentation", "criteria-theorem-flat-groupoid-gives-algebraic-stack", "algebraic-lemma-map-space-into-stack", "algebraic-lemma-characterize-representable-by-algebraic-spaces", "stacks-properties-lemma-monomorphism", "stacks-properties-lemma-residual-gerbe-unique", "criteria-lemma-flat-quotient-flat-presentation", "spaces-descent-lemma-locally-finite-presentation-fppf-local-source", "stacks-morphisms-lemma-finite-type-points-morphism", "stacks-morphisms-lemma-enough-finite-type-points" ], "ref_ids": [ 7467, 8473, 4814, 4844, 3094, 8473, 8469, 8879, 8908, 3137, 9418, 7468, 7470 ] } ], "ref_ids": [] }, { "id": 7472, "type": "theorem", "label": "stacks-morphisms-lemma-automorphism-group-scheme", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-automorphism-group-scheme", "contents": [ "In the situation above $G_x$ is a scheme if one of the following", "holds", "\\begin{enumerate}", "\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$", "is quasi-separated", "\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$", "is locally separated,", "\\item $\\mathcal{X}$ is quasi-DM,", "\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$", "is quasi-separated,", "\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$", "is locally separated, or", "\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$", "is locally quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that (1) $\\Rightarrow$ (4), (2) $\\Rightarrow$ (5), and", "(3) $\\Rightarrow$ (6) by", "Lemma \\ref{lemma-diagonal-diagonal}.", "In case (4) we see that $G_x$ is a quasi-separated", "algebraic space and in case (5) we see that $G_x$", "is a locally separated algebraic space.", "In both cases $G_x$ is a decent algebraic space", "(Decent Spaces, Section \\ref{decent-spaces-section-reasonable-decent} and", "Lemma \\ref{decent-spaces-lemma-locally-separated-decent}).", "Then $G_x$ is separated by More on Groupoids in Spaces, Lemma", "\\ref{spaces-more-groupoids-lemma-group-scheme-over-field-separated}", "whereupon we conclude that $G_x$ is a scheme by", "More on Groupoids in Spaces, Proposition", "\\ref{spaces-more-groupoids-proposition-group-space-scheme-over-field}.", "In case (6) we see that $G_x \\to \\Spec(k)$ is locally quasi-finite", "and hence $G_x$ is a scheme by", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-locally-quasi-finite-over-field}." ], "refs": [ "stacks-morphisms-lemma-diagonal-diagonal", "decent-spaces-lemma-locally-separated-decent", "spaces-more-groupoids-lemma-group-scheme-over-field-separated", "spaces-more-groupoids-proposition-group-space-scheme-over-field", "spaces-over-fields-lemma-locally-quasi-finite-over-field" ], "ref_ids": [ 7416, 9512, 13181, 13214, 12852 ] } ], "ref_ids": [] }, { "id": 7473, "type": "theorem", "label": "stacks-morphisms-lemma-property-automorphism-groups", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-property-automorphism-groups", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$ be a point.", "Let $P$ be a property of algebraic spaces over fields which is invariant", "under ground field extensions; for example", "$P(X/k) = X \\to \\Spec(k)\\text{ is finite}$.", "The following are equivalent", "\\begin{enumerate}", "\\item for some morphism $x : \\Spec(k) \\to \\mathcal{X}$ in the", "class of $x$ the automorphism group algebraic space $G_x/k$", "has $P$, and", "\\item for any morphism $x : \\Spec(k) \\to \\mathcal{X}$ in the", "class of $x$ the automorphism group algebraic space $G_x/k$", "has $P$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7474, "type": "theorem", "label": "stacks-morphisms-lemma-iso-automorphism-groups", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-iso-automorphism-groups", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $x \\in |\\mathcal{X}|$ be a point. The following are equivalent", "\\begin{enumerate}", "\\item for some morphism $x : \\Spec(k) \\to \\mathcal{X}$ in the", "class of $x$ setting $y = f \\circ x$ the map", "$G_x \\to G_y$ of automorphism group algebraic spaces", "is an isomorphism, and", "\\item for any morphism $x : \\Spec(k) \\to \\mathcal{X}$ in the", "class of $x$ setting $y = f \\circ x$ the map", "$G_x \\to G_y$ of automorphism group algebraic spaces", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This comes down to the fact that being an isomorphism", "is fpqc local on the target, see", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-isomorphism}.", "Namely, suppose that $k'/k$ is an extension of fields and", "denote $x' : \\Spec(k') \\to \\mathcal{X}$ the composition", "and set $y' = f \\circ x'$.", "Then the morphism $G_{x'} \\to G_{y'}$ is the base change", "of $G_x \\to G_y$ by $\\Spec(k') \\to \\Spec(k)$.", "Hence $G_x \\to G_y$ is an isomorphism", "if and only if $G_{x'} \\to G_{y'}$ is an isomorphism.", "Thus we see that the property propagates through the", "equivalence class if it holds for one." ], "refs": [ "spaces-descent-lemma-descending-property-isomorphism" ], "ref_ids": [ 9395 ] } ], "ref_ids": [] }, { "id": 7475, "type": "theorem", "label": "stacks-morphisms-lemma-properties-diagonal-from-presentation", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-properties-diagonal-from-presentation", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces such that", "$s, t : R \\to U$ are flat and locally of finite presentation.", "Consider the algebraic stack $\\mathcal{X} = [U/R]$ (see above).", "\\begin{enumerate}", "\\item If $R \\to U \\times U$ is separated, then", "$\\Delta_\\mathcal{X}$ is separated.", "\\item If $U$, $R$ are separated, then $\\Delta_\\mathcal{X}$ is separated.", "\\item If $R \\to U \\times U$ is locally quasi-finite, then $\\mathcal{X}$", "is quasi-DM.", "\\item If $s, t : R \\to U$ are locally quasi-finite, then", "$\\mathcal{X}$ is quasi-DM.", "\\item If $R \\to U \\times U$ is proper, then $\\mathcal{X}$ is separated.", "\\item If $s, t : R \\to U$ are proper and $U$ is separated, then", "$\\mathcal{X}$ is separated.", "\\item Add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that the morphism $U \\to \\mathcal{X}$ is surjective, flat, and", "locally of finite presentation by", "Criteria for Representability, Lemma", "\\ref{criteria-lemma-flat-quotient-flat-presentation}.", "Hence the same is true for $U \\times U \\to \\mathcal{X} \\times \\mathcal{X}$.", "We have the cartesian diagram", "$$", "\\xymatrix{", "R = U \\times_\\mathcal{X} U \\ar[r] \\ar[d] & U \\times U \\ar[d] \\\\", "\\mathcal{X} \\ar[r] & \\mathcal{X} \\times \\mathcal{X}", "}", "$$", "(see Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-2-cartesian}).", "Thus we see that $\\Delta_\\mathcal{X}$ has one of the properties listed in", "Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}", "if and only if the morphism $R \\to U \\times U$ does, see", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}.", "This explains why (1), (3), and (5) are true.", "The condition in (2) implies $R \\to U \\times U$ is separated", "hence (2) follows from (1).", "The condition in (4) implies the condition in (3)", "hence (4) follows from (3).", "The condition in (6) implies the condition in (5)", "by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-permanence}", "hence (6) follows from (5)." ], "refs": [ "criteria-lemma-flat-quotient-flat-presentation", "spaces-groupoids-lemma-quotient-stack-2-cartesian", "stacks-properties-lemma-check-property-covering", "spaces-morphisms-lemma-universally-closed-permanence" ], "ref_ids": [ 3137, 9324, 8859, 4920 ] } ], "ref_ids": [] }, { "id": 7476, "type": "theorem", "label": "stacks-morphisms-lemma-points-presentation", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-points-presentation", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces such that", "$s, t : R \\to U$ are flat and locally of finite presentation.", "Consider the algebraic stack $\\mathcal{X} = [U/R]$ (see above).", "Then the image of $|R| \\to |U| \\times |U|$ is an equivalence relation", "and $|\\mathcal{X}|$ is the quotient of $|U|$ by this equivalence relation." ], "refs": [], "proofs": [ { "contents": [ "The induced morphism $p : U \\to \\mathcal{X}$ is surjective, flat,", "and locally of finite presentation, see", "Criteria for Representability, Lemma", "\\ref{criteria-lemma-flat-quotient-flat-presentation}.", "Hence $|U| \\to |\\mathcal{X}|$ is surjective by", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-characterize-surjective}.", "Note that $R = U \\times_\\mathcal{X} U$, see", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-2-cartesian}.", "Hence", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-points-cartesian}", "implies the map", "$$", "|R| \\longrightarrow |U| \\times_{|\\mathcal{X}|} |U|", "$$", "is surjective. Hence the image of $|R| \\to |U| \\times |U|$ is", "exactly the set of pairs $(u_1, u_2) \\in |U| \\times |U|$", "such that $u_1$ and $u_2$ have the same image in $|\\mathcal{X}|$.", "Combining these two statements we get the result of the lemma." ], "refs": [ "criteria-lemma-flat-quotient-flat-presentation", "stacks-properties-lemma-characterize-surjective", "spaces-groupoids-lemma-quotient-stack-2-cartesian", "stacks-properties-lemma-points-cartesian" ], "ref_ids": [ 3137, 8865, 9324, 8864 ] } ], "ref_ids": [] }, { "id": 7477, "type": "theorem", "label": "stacks-morphisms-lemma-slice", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-slice", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Consider a cartesian diagram", "$$", "\\xymatrix{", "U \\ar[d] & F \\ar[l]^p \\ar[d] \\\\", "\\mathcal{X} & \\Spec(k) \\ar[l]", "}", "$$", "where $U$ is an algebraic space, $k$ is a field, and $U \\to \\mathcal{X}$", "is flat and locally of finite presentation. Let", "$f_1, \\ldots, f_r \\in \\Gamma(U, \\mathcal{O}_U)$", "and $z \\in |F|$ such that $f_1, \\ldots, f_r$ map to a regular sequence", "in the local ring $\\mathcal{O}_{F, \\overline{z}}$.", "Then, after replacing $U$ by an open subspace containing $p(z)$, the morphism", "$$", "V(f_1, \\ldots, f_r) \\longrightarrow \\mathcal{X}", "$$", "is flat and locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $W$ and a surjective smooth morphism $W \\to \\mathcal{X}$.", "Choose an extension of fields $k \\subset k'$ and a morphism", "$w : \\Spec(k') \\to W$ such that $\\Spec(k') \\to W \\to \\mathcal{X}$", "is $2$-isomorphic to $\\Spec(k') \\to \\Spec(k) \\to \\mathcal{X}$.", "This is possible as $W \\to \\mathcal{X}$ is surjective.", "Consider the commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] &", "U \\times_\\mathcal{X} W \\ar[l]^-{\\text{pr}_0} \\ar[d] &", "F' \\ar[l]^-{p'} \\ar[d] \\\\", "\\mathcal{X} &", "W \\ar[l] &", "\\Spec(k') \\ar[l]", "}", "$$", "both of whose squares are cartesian. By our choice of $w$ we see that", "$F' = F \\times_{\\Spec(k)} \\Spec(k')$. Thus $F' \\to F$ is", "surjective and we can choose a point $z' \\in |F'|$ mapping to $z$.", "Since $F' \\to F$ is flat we see that", "$\\mathcal{O}_{F, \\overline{z}} \\to \\mathcal{O}_{F', \\overline{z}'}$ is", "flat, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-flat-at-point-etale-local-rings}.", "Hence $f_1, \\ldots, f_r$ map to a regular sequence in", "$\\mathcal{O}_{F', \\overline{z}'}$, see", "Algebra, Lemma \\ref{algebra-lemma-flat-increases-depth}.", "Note that $U \\times_\\mathcal{X} W \\to W$ is a morphism of algebraic spaces", "which is flat and locally of finite presentation. Hence by", "More on Morphisms of Spaces, Lemma \\ref{spaces-more-morphisms-lemma-slice}", "we see that there exists an open subspace $U'$ of $U \\times_\\mathcal{X} W$", "containing $p(z')$ such that the intersection", "$U' \\cap (V(f_1, \\ldots, f_r) \\times_\\mathcal{X} W)$ is flat and locally", "of finite presentation over $W$. Note that", "$\\text{pr}_0(U')$ is an open subspace of $U$ containing $p(z)$", "as $\\text{pr}_0$ is smooth hence open. Now we see that", "$U' \\cap (V(f_1, \\ldots, f_r) \\times_\\mathcal{X} W) \\to \\mathcal{X}$", "is flat and locally of finite presentation as the composition", "$$", "U' \\cap (V(f_1, \\ldots, f_r) \\times_\\mathcal{X} W) \\to W \\to \\mathcal{X}.", "$$", "Hence", "Properties of Stacks,", "Lemma \\ref{stacks-properties-lemma-check-property-after-precomposing}", "implies $\\text{pr}_0(U') \\cap V(f_1, \\ldots, f_r) \\to \\mathcal{X}$", "is flat and locally of finite presentation as desired." ], "refs": [ "spaces-morphisms-lemma-flat-at-point-etale-local-rings", "algebra-lemma-flat-increases-depth", "spaces-more-morphisms-lemma-slice", "stacks-properties-lemma-check-property-after-precomposing" ], "ref_ids": [ 4857, 740, 153, 8861 ] } ], "ref_ids": [] }, { "id": 7478, "type": "theorem", "label": "stacks-morphisms-lemma-quasi-finite-at-point", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-quasi-finite-at-point", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram", "$$", "\\xymatrix{", "U \\ar[d] & F \\ar[l]^p \\ar[d] \\\\", "\\mathcal{X} & \\Spec(k) \\ar[l]", "}", "$$", "where $U$ is an algebraic space, $k$ is a field, and $U \\to \\mathcal{X}$", "is locally of finite type. Let $z \\in |F|$ be such that $\\dim_z(F) = 0$.", "Then, after replacing $U$ by an open subspace containing $p(z)$, the morphism", "$$", "U \\longrightarrow \\mathcal{X}", "$$", "is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Since $f : U \\to \\mathcal{X}$ is locally of finite type there exists a", "maximal open $W(f) \\subset U$ such that the restriction", "$f|_{W(f)} : W(f) \\to \\mathcal{X}$ is locally quasi-finite, see", "Properties of Stacks, Remark", "\\ref{stacks-properties-remark-local-source-apply}", "(\\ref{stacks-properties-item-loc-quasi-finite}).", "Hence all we need to do is prove that $p(z)$ is a point of $W(f)$.", "Moreover, the remark referenced above also shows the formation of $W(f)$", "commutes with arbitrary base change by a morphism which is representable", "by algebraic spaces. Hence it suffices to show that the morphism", "$F \\to \\Spec(k)$ is locally quasi-finite at $z$. This follows", "immediately from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0}." ], "refs": [ "stacks-properties-remark-local-source-apply", "spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0" ], "ref_ids": [ 8931, 4875 ] } ], "ref_ids": [] }, { "id": 7479, "type": "theorem", "label": "stacks-morphisms-lemma-DM-residual-gerbe", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-DM-residual-gerbe", "contents": [ "Let $\\mathcal{Z}$ be a DM, locally Noetherian, reduced algebraic stack", "with $|\\mathcal{Z}|$ a singleton. Then there exists a field $k$ and", "a surjective \\'etale morphism $\\Spec(k) \\to \\mathcal{Z}$." ], "refs": [], "proofs": [ { "contents": [ "By", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-unique-point-better}", "there exists a field $k$ and a surjective, flat, locally finitely presented", "morphism $\\Spec(k) \\to \\mathcal{Z}$. Set $U = \\Spec(k)$ and", "$R = U \\times_\\mathcal{Z} U$ so we obtain a groupoid in algebraic spaces", "$(U, R, s, t, c)$, see", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids}.", "Note that by", "Algebraic Stacks, Remark \\ref{algebraic-remark-flat-fp-presentation}", "we have an equivalence", "$$", "f_{can} : [U/R] \\longrightarrow \\mathcal{Z}", "$$", "The projections $s, t : R \\to U$ are locally of finite presentation.", "As $\\mathcal{Z}$ is DM we see that the stabilizer group algebraic space", "$$", "G = U \\times_{U \\times U} R = U \\times_{U \\times U} (U \\times_\\mathcal{Z} U) =", "U \\times_{\\mathcal{Z} \\times \\mathcal{Z}, \\Delta_\\mathcal{Z}} \\mathcal{Z}", "$$", "is unramified over $U$. In particular $\\dim(G) = 0$ and by", "More on Groupoids in Spaces, Lemma", "\\ref{spaces-more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer}", "we have $\\dim(R) = 0$. This implies that $R$ is a scheme, see", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-locally-finite-type-dim-zero}.", "By", "Varieties, Lemma \\ref{varieties-lemma-algebraic-scheme-dim-0}", "we see that $R$ (and also $G$) is the disjoint union of spectra of", "Artinian local rings finite over $k$ via either $s$ or $t$. Let", "$P = \\Spec(A) \\subset R$ be the open and", "closed subscheme whose underlying point is the identity $e$ of the groupoid", "scheme $(U, R, s, t, c)$. As", "$s \\circ e = t \\circ e = \\text{id}_{\\Spec(k)}$ we see that $A$", "is an Artinian local ring whose residue field is identified with $k$", "via either $s^\\sharp : k \\to A$ or $t^\\sharp : k \\to A$.", "Note that $s, t : \\Spec(A) \\to \\Spec(k)$", "are finite (by the lemma referenced above). Since", "$G \\to \\Spec(k)$ is unramified we see that", "$$", "G \\cap P = P \\times_{U \\times U} U = \\Spec(A \\otimes_{k \\otimes k} k)", "$$", "is unramified over $k$. On the other hand $A \\otimes_{k \\otimes k} k$", "is local as a quotient of $A$ and surjects onto $k$. We conclude that", "$A \\otimes_{k \\otimes k} k = k$. It follows that $P \\to U \\times U$", "is universally injective (as $P$ has only one point with residue field $k$),", "unramified (by the computation of the fibre over the unique image point", "above), and of finite type (because $s, t$ are) hence a monomorphism (see", "\\'Etale Morphisms, Lemma", "\\ref{etale-lemma-universally-injective-unramified}).", "Thus $s|_P, t|_P : P \\to U$ define a finite flat equivalence", "relation. Thus we may apply", "Groupoids, Proposition \\ref{groupoids-proposition-finite-flat-equivalence}", "to conclude that $U/P$ exists and is a scheme $\\overline{U}$.", "Moreover, $U \\to \\overline{U}$ is finite locally free and", "$P = U \\times_{\\overline{U}} U$.", "In fact $\\overline{U} = \\Spec(k_0)$ where $k_0 \\subset k$ is the", "ring of $R$-invariant functions. As $k$ is a field it follows", "from the definition", "Groupoids, Equation (\\ref{groupoids-equation-invariants})", "that $k_0$ is a field.", "\\medskip\\noindent", "We claim that", "\\begin{equation}", "\\label{equation-etale-covering}", "\\Spec(k_0) = \\overline{U} = U/P \\to [U/R] = \\mathcal{Z}", "\\end{equation}", "is the desired surjective \\'etale morphism. It follows from", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-flat-cover-by-field}", "that this morphism is surjective. Thus it suffices to show that", "(\\ref{equation-etale-covering}) is \\'etale\\footnote{We urge the", "reader to find his/her own proof of this fact. In fact the argument", "has a lot in common with the final argument of the proof of", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}", "hence probably should be isolated into its own lemma somewhere.}.", "Instead of proving the \\'etaleness", "directly we first apply", "Bootstrap, Lemma \\ref{bootstrap-lemma-divide-subgroupoid}", "to see that there exists a groupoid scheme", "$(\\overline{U}, \\overline{R}, \\overline{s}, \\overline{t}, \\overline{c})$", "such that $(U, R, s, t, c)$ is the restriction of", "$(\\overline{U}, \\overline{R}, \\overline{s}, \\overline{t}, \\overline{c})$", "via the quotient morphism $U \\to \\overline{U}$.", "(We verified all the hypothesis of the lemma above except for the assertion", "that $j : R \\to U \\times U$ is separated and locally quasi-finite", "which follows from the fact that $R$ is a separated scheme locally quasi-finite", "over $k$.) Since $U \\to \\overline{U}$ is finite locally free", "we see that $[U/R] \\to [\\overline{U}/\\overline{R}]$ is an equivalence, see", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-restrict-equivalence}.", "\\medskip\\noindent", "Note that $s, t$ are the base changes of the morphisms", "$\\overline{s}, \\overline{t}$ by $U \\to \\overline{U}$.", "As $\\{U \\to \\overline{U}\\}$ is an fppf covering we conclude", "$\\overline{s}, \\overline{t}$ are flat, locally of finite presentation, and", "locally quasi-finite, see", "Descent, Lemmas \\ref{descent-lemma-descending-property-flat},", "\\ref{descent-lemma-descending-property-locally-finite-presentation}, and", "\\ref{descent-lemma-descending-property-quasi-finite}.", "Consider the commutative diagram", "$$", "\\xymatrix{", "U \\times_{\\overline{U}} U \\ar@{=}[r] \\ar[rd] & P \\ar[r] \\ar[d] & R \\ar[d] \\\\", "& \\overline{U} \\ar[r]^{\\overline{e}} & \\overline{R}", "}", "$$", "It is a general fact about restrictions that the outer four corners", "form a cartesian diagram. By the equality we see the inner square is", "cartesian. Since $P$ is open in $R$ we conclude that $\\overline{e}$", "is an open immersion by", "Descent, Lemma \\ref{descent-lemma-descending-property-open-immersion}.", "\\medskip\\noindent", "But of course, if $\\overline{e}$ is an open immersion and", "$\\overline{s}, \\overline{t}$ are flat and locally of finite presentation", "then the morphisms $\\overline{t}, \\overline{s}$ are \\'etale.", "For example you can see this by applying", "More on Groupoids, Lemma \\ref{more-groupoids-lemma-sheaf-differentials}", "which shows that $\\Omega_{\\overline{R}/\\overline{U}} = 0$", "implies that $\\overline{s}, \\overline{t} : \\overline{R} \\to \\overline{U}$", "is unramified (see", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-omega-zero}),", "which in turn implies that $\\overline{s}, \\overline{t}$ are \\'etale", "(see", "Morphisms, Lemma \\ref{morphisms-lemma-flat-unramified-etale}).", "Hence $\\mathcal{Z} = [\\overline{U}/\\overline{R}]$ is an \\'etale", "presentation of the algebraic stack $\\mathcal{Z}$ and we conclude that", "$\\overline{U} \\to \\mathcal{Z}$ is \\'etale by", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}." ], "refs": [ "stacks-properties-lemma-unique-point-better", "algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids", "algebraic-remark-flat-fp-presentation", "spaces-more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer", "spaces-over-fields-lemma-locally-finite-type-dim-zero", "varieties-lemma-algebraic-scheme-dim-0", "etale-lemma-universally-injective-unramified", "groupoids-proposition-finite-flat-equivalence", "stacks-properties-lemma-flat-cover-by-field", "bootstrap-theorem-final-bootstrap", "bootstrap-lemma-divide-subgroupoid", "spaces-groupoids-lemma-quotient-stack-restrict-equivalence", "descent-lemma-descending-property-flat", "descent-lemma-descending-property-locally-finite-presentation", "descent-lemma-descending-property-quasi-finite", "descent-lemma-descending-property-open-immersion", "more-groupoids-lemma-sheaf-differentials", "morphisms-lemma-unramified-omega-zero", "morphisms-lemma-flat-unramified-etale", "stacks-properties-lemma-check-property-covering" ], "ref_ids": [ 8902, 8442, 8491, 13188, 12843, 10988, 10701, 9669, 8900, 2602, 2624, 9330, 14680, 14676, 14689, 14681, 2457, 5343, 5373, 8859 ] } ], "ref_ids": [] }, { "id": 7480, "type": "theorem", "label": "stacks-morphisms-lemma-etale-at-point", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-etale-at-point", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram", "$$", "\\xymatrix{", "U \\ar[d] & F \\ar[l]^p \\ar[d] \\\\", "\\mathcal{X} & \\Spec(k) \\ar[l]", "}", "$$", "where $U$ is an algebraic space, $k$ is a field, and $U \\to \\mathcal{X}$", "is flat and locally of finite presentation. Let $z \\in |F|$ be such that", "$F \\to \\Spec(k)$ is unramified at $z$. Then, after replacing $U$ by", "an open subspace containing $p(z)$, the morphism", "$$", "U \\longrightarrow \\mathcal{X}", "$$", "is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Since $f : U \\to \\mathcal{X}$ is flat and locally of finite presentation", "there exists a maximal open $W(f) \\subset U$ such that the restriction", "$f|_{W(f)} : W(f) \\to \\mathcal{X}$ is \\'etale, see", "Properties of Stacks, Remark", "\\ref{stacks-properties-remark-local-source-apply}", "(\\ref{stacks-properties-item-etale}).", "Hence all we need to do is prove that $p(z)$ is a point of $W(f)$.", "Moreover, the remark referenced above also shows the formation of $W(f)$", "commutes with arbitrary base change by a morphism which is representable", "by algebraic spaces. Hence it suffices to show that the morphism", "$F \\to \\Spec(k)$ is \\'etale at $z$. Since it is flat and locally", "of finite presentation as a base change of $U \\to \\mathcal{X}$ and since", "$F \\to \\Spec(k)$ is unramified at $z$ by assumption, this follows", "from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-unramified-flat-lfp-etale}." ], "refs": [ "stacks-properties-remark-local-source-apply", "spaces-morphisms-lemma-unramified-flat-lfp-etale" ], "ref_ids": [ 8931, 4915 ] } ], "ref_ids": [] }, { "id": 7481, "type": "theorem", "label": "stacks-morphisms-lemma-DM", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-DM", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a DM morphism of algebraic", "stacks. Then", "\\begin{enumerate}", "\\item For every DM algebraic stack $\\mathcal{Z}$ and morphism", "$\\mathcal{Z} \\to \\mathcal{Y}$ there exists a scheme and", "a surjective \\'etale morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$.", "\\item For every algebraic space $Z$ and morphism", "$Z \\to \\mathcal{Y}$ there exists a scheme and", "a surjective \\'etale morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} Z$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). As $f$ is DM we see that the base change", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$ is DM", "by Lemma \\ref{lemma-base-change-separated}.", "Since $\\mathcal{Z}$ is DM this implies that", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ is DM", "by Lemma \\ref{lemma-separated-over-separated}. Hence there exists a", "scheme $U$ and a surjective \\'etale morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$, see", "Theorem \\ref{theorem-DM}.", "Part (2) is a special case of (1) since an algebraic space", "(when viewed as an algebraic stack) is DM by", "Lemma \\ref{lemma-trivial-implications}." ], "refs": [ "stacks-morphisms-lemma-base-change-separated", "stacks-morphisms-lemma-separated-over-separated", "stacks-morphisms-theorem-DM", "stacks-morphisms-lemma-trivial-implications" ], "ref_ids": [ 7398, 7405, 7389, 7397 ] } ], "ref_ids": [] }, { "id": 7482, "type": "theorem", "label": "stacks-morphisms-lemma-open-DM-locus", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-open-DM-locus", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. There exist open substacks", "$$", "\\mathcal{X}'' \\subset \\mathcal{X}' \\subset \\mathcal{X}", "$$", "such that $\\mathcal{X}''$ is DM, $\\mathcal{X}'$ is quasi-DM, and", "such that these are the largest open substacks with these properties." ], "refs": [], "proofs": [ { "contents": [ "All we are really saying here is that if $\\mathcal{U} \\subset \\mathcal{X}$", "and $\\mathcal{V} \\subset \\mathcal{X}$ are open substacks which are DM,", "then the open substack $\\mathcal{W} \\subset \\mathcal{X}$", "with $|\\mathcal{W}| = |\\mathcal{U}| \\cup |\\mathcal{V}|$", "is DM as well. (Similarly for quasi-DM.) Although this is a cheat, let", "us use Theorem \\ref{theorem-DM} to prove this.", "By that theorem we can choose", "schemes $U$ and $V$ and surjective \\'etale morphisms", "$U \\to \\mathcal{U}$ and $V \\to \\mathcal{V}$.", "Then of course $U \\amalg V \\to \\mathcal{W}$ is surjective and \\'etale.", "The quasi-DM case is proven by exactly the same method using", "Theorem \\ref{theorem-quasi-DM}." ], "refs": [ "stacks-morphisms-theorem-DM", "stacks-morphisms-theorem-quasi-DM" ], "ref_ids": [ 7389, 7388 ] } ], "ref_ids": [] }, { "id": 7483, "type": "theorem", "label": "stacks-morphisms-lemma-points-DM-locus", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-points-DM-locus", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$", "correspond to $x : \\Spec(k) \\to \\mathcal{X}$. Let $G_x/k$", "be the automorphism group algebraic space of $x$. Then", "\\begin{enumerate}", "\\item $x$ is in the DM locus of $\\mathcal{X}$", "if and only if $G_x \\to \\Spec(k)$ is unramified, and", "\\item $x$ is in the quasi-DM locus of $\\mathcal{X}$", "if and only if $G_x \\to \\Spec(k)$ is locally quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (2). Choose a scheme $U$ and a surjective smooth morphism", "$U \\to \\mathcal{X}$. Consider the fibre product", "$$", "\\xymatrix{", "G \\ar[r] \\ar[d] & \\mathcal{I}_\\mathcal{X} \\ar[d] \\\\", "U \\ar[r] & \\mathcal{X}", "}", "$$", "Recall that $G$ is the automorphism group algebraic space of", "$U \\to \\mathcal{X}$. By Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-open-over-which-unramified-or-lqf}", "there is a maximal open subscheme $U' \\subset U$", "such that $G_{U'} \\to U'$ is locally quasi-finite.", "Moreover, formation of $U'$ commutes with arbitrary", "base change. In particular the two inverse images of $U'$", "in $R = U \\times_\\mathcal{X} U$ are the same open subspace of $R$", "(since after all the two maps $R \\to \\mathcal{X}$ are isomorphic", "and hence have isomorphic automorphism group spaces).", "Hence $U'$ is the inverse image of an open substack", "$\\mathcal{X}' \\subset \\mathcal{X}$ by", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-substacks-presentation}", "and we have a cartesian diagram", "$$", "\\xymatrix{", "G_{U'} \\ar[r] \\ar[d] & \\mathcal{I}_{\\mathcal{X}'} \\ar[d] \\\\", "U' \\ar[r] & \\mathcal{X}'", "}", "$$", "Thus the morphism $\\mathcal{I}_{\\mathcal{X}'} \\to \\mathcal{X}'$", "is locally quasi-finite and we conclude that", "$\\mathcal{X}'$ is quasi-DM by Lemma \\ref{lemma-diagonal-diagonal}", "part (5). On the other hand, if $\\mathcal{W} \\subset \\mathcal{X}$", "is an open substack which is quasi-DM, then the inverse image", "$W \\subset U$ of $\\mathcal{W}$ must be contained in $U'$ by our", "construction of $U'$ since", "$\\mathcal{I}_\\mathcal{W} =", "\\mathcal{W} \\times_\\mathcal{X} \\mathcal{I}_\\mathcal{X}$", "is locally quasi-finite over $\\mathcal{W}$.", "Thus $\\mathcal{X}'$ is the quasi-DM locus.", "Finally, choose a field extension $K/k$ and a $2$-commutative", "diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & \\Spec(k) \\ar[d]^x \\\\", "U \\ar[r] & \\mathcal{X}", "}", "$$", "Then we find an isomorphism", "$G_x \\times_{\\Spec(k)} \\Spec(K) \\cong G \\times_U \\Spec(K)$", "of group algebraic spaces over $K$. Hence $G_x$ is locally quasi-finite", "over $k$ if and only if $\\Spec(K) \\to U$ maps into $U'$", "(use the commutation of formation of $U'$ and", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-open-over-which-unramified-or-lqf}", "applied to $\\Spec(K) \\to \\Spec(k)$ and $G_x$ to see this).", "This finishes the proof of (2). The proof of (1) is", "exactly the same." ], "refs": [ "spaces-groupoids-lemma-open-over-which-unramified-or-lqf", "stacks-properties-lemma-substacks-presentation", "stacks-morphisms-lemma-diagonal-diagonal", "spaces-groupoids-lemma-open-over-which-unramified-or-lqf" ], "ref_ids": [ 9291, 8889, 7416, 9291 ] } ], "ref_ids": [] }, { "id": 7484, "type": "theorem", "label": "stacks-morphisms-lemma-representable-by-spaces-quasi-finite", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-representable-by-spaces-quasi-finite", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Assume $f$ is representable by algebraic spaces.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is locally quasi-finite (as in Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}), and", "\\item $f$ is locally of finite type and for every morphism", "$\\Spec(k) \\to \\mathcal{Y}$ where $k$ is a field the", "space $|\\Spec(k) \\times_\\mathcal{Y} \\mathcal{X}|$ is discrete.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). In this case the morphism of algebraic spaces", "$\\mathcal{X}_k \\to \\Spec(k)$ is locally quasi-finite as a base change", "of $f$. Hence $|\\mathcal{X}_k|$ is discrete by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-locally-quasi-finite}.", "Conversely, assume (2). Pick a surjective smooth morphism", "$V \\to \\mathcal{Y}$ where $V$ is a scheme. It suffices to show that the", "morphism of algebraic spaces $V \\times_\\mathcal{Y} \\mathcal{X} \\to V$", "is locally quasi-finite, see", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}.", "The morphism $V \\times_\\mathcal{Y} \\mathcal{X} \\to V$ is locally of finite", "type by assumption. For any morphism $\\Spec(k) \\to V$ where $k$ is a", "field", "$$", "\\Spec(k) \\times_V (V \\times_\\mathcal{Y} \\mathcal{X}) =", "\\Spec(k) \\times_\\mathcal{Y} \\mathcal{X}", "$$", "has a discrete space of points by assumption. Hence we conclude that", "$V \\times_\\mathcal{Y} \\mathcal{X} \\to V$ is locally quasi-finite by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-locally-quasi-finite}." ], "refs": [ "spaces-morphisms-lemma-locally-quasi-finite", "stacks-properties-lemma-check-property-covering", "spaces-morphisms-lemma-locally-quasi-finite" ], "ref_ids": [ 4833, 8859, 4833 ] } ], "ref_ids": [] }, { "id": 7485, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-locally-quasi-finite", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-locally-quasi-finite", "contents": [ "A base change of a locally quasi-finite morphism is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "We have seen this for quasi-DM morphisms in", "Lemma \\ref{lemma-base-change-separated}", "and for locally finite type morphisms in", "Lemma \\ref{lemma-base-change-finite-type}.", "It is immediate that the condition on fibres is inherited by a base change." ], "refs": [ "stacks-morphisms-lemma-base-change-separated", "stacks-morphisms-lemma-base-change-finite-type" ], "ref_ids": [ 7398, 7460 ] } ], "ref_ids": [] }, { "id": 7486, "type": "theorem", "label": "stacks-morphisms-lemma-locally-quasi-finite-over-field", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-locally-quasi-finite-over-field", "contents": [ "Let $\\mathcal{X} \\to \\Spec(k)$ be a locally quasi-finite morphism", "where $\\mathcal{X}$ is an algebraic stack and $k$ is a field.", "Let $f : V \\to \\mathcal{X}$ be a locally quasi-finite morphism where", "$V$ is a scheme. Then $V \\to \\Spec(k)$ is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-composition-finite-type}", "we see that $V \\to \\Spec(k)$ is locally of finite type.", "Assume, to get a contradiction, that $V \\to \\Spec(k)$ is not", "locally quasi-finite. Then there exists a nontrivial specialization", "$v \\leadsto v'$ of points of $V$, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}.", "In particular $\\text{trdeg}_k(\\kappa(v)) > \\text{trdeg}_k(\\kappa(v'))$, see", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-specialization}.", "Because $|\\mathcal{X}|$ is discrete we see that $|f|(v) = |f|(v')$.", "Consider $R = V \\times_\\mathcal{X} V$. Then $R$ is an algebraic space", "and the projections $s, t : R \\to V$ are locally quasi-finite as base", "changes of $V \\to \\mathcal{X}$ (which is representable by algebraic spaces", "so this follows from the discussion in", "Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}).", "By", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-points-cartesian}", "we see that there exists an $r \\in |R|$ such that $s(r) = v$ and $t(r) = v'$.", "By", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-compare-tr-deg}", "we see that the transcendence degree of $v/k$ is equal to the", "transcendence degree of $r/k$ is equal to the transcendence degree of", "$v'/k$. This contradiction proves the lemma." ], "refs": [ "stacks-morphisms-lemma-composition-finite-type", "morphisms-lemma-quasi-finite-at-point-characterize", "morphisms-lemma-dimension-fibre-specialization", "stacks-properties-lemma-points-cartesian", "spaces-morphisms-lemma-compare-tr-deg" ], "ref_ids": [ 7459, 5226, 5283, 8864, 4867 ] } ], "ref_ids": [] }, { "id": 7487, "type": "theorem", "label": "stacks-morphisms-lemma-composition-locally-quasi-finite", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-locally-quasi-finite", "contents": [ "A composition of a locally quasi-finite morphisms is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "We have seen this for quasi-DM morphisms in", "Lemma \\ref{lemma-composition-separated}", "and for locally finite type morphisms in", "Lemma \\ref{lemma-composition-finite-type}.", "Let $\\mathcal{X} \\to \\mathcal{Y}$ and $\\mathcal{Y} \\to \\mathcal{Z}$", "be locally quasi-finite. Let $k$ be a field and let", "$\\Spec(k) \\to \\mathcal{Z}$ be a morphism.", "It suffices to show that $|\\mathcal{X}_k|$ is discrete. By", "Lemma \\ref{lemma-base-change-locally-quasi-finite}", "the morphisms $\\mathcal{X}_k \\to \\mathcal{Y}_k$", "and $\\mathcal{Y}_k \\to \\Spec(k)$ are locally quasi-finite.", "In particular we see that $\\mathcal{Y}_k$ is", "a quasi-DM algebraic stack, see", "Lemma \\ref{lemma-separated-implies-morphism-separated}.", "By", "Theorem \\ref{theorem-quasi-DM}", "we can find a scheme $V$ and a surjective, flat, locally finitely presented,", "locally quasi-finite morphism $V \\to \\mathcal{Y}_k$. By", "Lemma \\ref{lemma-locally-quasi-finite-over-field}", "we see that $V$ is locally quasi-finite over $k$, in particular", "$|V|$ is discrete. The morphism", "$V \\times_{\\mathcal{Y}_k} \\mathcal{X}_k \\to \\mathcal{X}_k$ is", "surjective, flat, and locally of finite presentation hence", "$|V \\times_{\\mathcal{Y}_k} \\mathcal{X}_k| \\to |\\mathcal{X}_k|$", "is surjective and open. Thus it suffices to show that", "$|V \\times_{\\mathcal{Y}_k} \\mathcal{X}_k|$ is discrete.", "Note that $V$ is a disjoint union of spectra of Artinian local", "$k$-algebras $A_i$ with residue fields $k_i$, see", "Varieties, Lemma \\ref{varieties-lemma-algebraic-scheme-dim-0}.", "Thus it suffices to show that each", "$$", "|\\Spec(A_i) \\times_{\\mathcal{Y}_k} \\mathcal{X}_k| =", "|\\Spec(k_i) \\times_{\\mathcal{Y}_k} \\mathcal{X}_k| =", "|\\Spec(k_i) \\times_\\mathcal{Y} \\mathcal{X}|", "$$", "is discrete, which follows from the assumption that", "$\\mathcal{X} \\to \\mathcal{Y}$ is locally quasi-finite." ], "refs": [ "stacks-morphisms-lemma-composition-separated", "stacks-morphisms-lemma-composition-finite-type", "stacks-morphisms-lemma-base-change-locally-quasi-finite", "stacks-morphisms-lemma-separated-implies-morphism-separated", "stacks-morphisms-theorem-quasi-DM", "stacks-morphisms-lemma-locally-quasi-finite-over-field", "varieties-lemma-algebraic-scheme-dim-0" ], "ref_ids": [ 7404, 7459, 7485, 7407, 7388, 7486, 10988 ] } ], "ref_ids": [] }, { "id": 7488, "type": "theorem", "label": "stacks-morphisms-lemma-characterize-quasi-DM", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-characterize-quasi-DM", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is quasi-DM,", "\\item for any morphism $V \\to \\mathcal{Y}$ with $V$ an algebraic space", "there exists a surjective, flat, locally finitely presented, locally", "quasi-finite morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ where", "$U$ is an algebraic space, and", "\\item there exist algebraic spaces $U$, $V$ and a morphism", "$V \\to \\mathcal{Y}$ which is surjective, flat, and", "locally of finite presentation, and a morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ which is surjective, flat,", "locally of finite presentation, and locally quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (3) is immediate.", "\\medskip\\noindent", "Assume (1) and let $V \\to \\mathcal{Y}$ be as in (2). Then", "$\\mathcal{X} \\times_\\mathcal{Y} V \\to V$ is quasi-DM, see", "Lemma \\ref{lemma-base-change-separated}.", "By", "Lemma \\ref{lemma-trivial-implications}", "the algebraic space $V$ is DM, hence quasi-DM. Thus", "$\\mathcal{X} \\times_\\mathcal{Y} V$ is quasi-DM by", "Lemma \\ref{lemma-separated-over-separated}.", "Hence we may apply", "Theorem \\ref{theorem-quasi-DM}", "to get the morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$", "as in (2).", "\\medskip\\noindent", "Assume (3). Let $V \\to \\mathcal{Y}$ and", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ be as in (3).", "To prove that $f$ is quasi-DM it suffices to show that", "$\\mathcal{X} \\times_\\mathcal{Y} V \\to V$ is quasi-DM, see", "Lemma \\ref{lemma-check-separated-covering}.", "By", "Lemma \\ref{lemma-properties-covering-imply-diagonal}", "we see that $\\mathcal{X} \\times_\\mathcal{Y} V$ is quasi-DM.", "Hence $\\mathcal{X} \\times_\\mathcal{Y} V \\to V$ is quasi-DM by", "Lemma \\ref{lemma-separated-implies-morphism-separated}", "and (1) holds. This finishes the proof of the lemma." ], "refs": [ "stacks-morphisms-lemma-base-change-separated", "stacks-morphisms-lemma-trivial-implications", "stacks-morphisms-lemma-separated-over-separated", "stacks-morphisms-theorem-quasi-DM", "stacks-morphisms-lemma-check-separated-covering", "stacks-morphisms-lemma-properties-covering-imply-diagonal", "stacks-morphisms-lemma-separated-implies-morphism-separated" ], "ref_ids": [ 7398, 7397, 7405, 7388, 7399, 7408, 7407 ] } ], "ref_ids": [] }, { "id": 7489, "type": "theorem", "label": "stacks-morphisms-lemma-characterize-locally-quasi-finite", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-characterize-locally-quasi-finite", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is locally quasi-finite,", "\\item $f$ is quasi-DM and for any morphism $V \\to \\mathcal{Y}$ with $V$", "an algebraic space and any locally quasi-finite morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ where $U$ is an algebraic space", "the morphism $U \\to V$ is locally quasi-finite,", "\\item for any morphism $V \\to \\mathcal{Y}$ from an algebraic space $V$", "there exists a surjective, flat, locally finitely presented, and locally", "quasi-finite morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ where", "$U$ is an algebraic space such that $U \\to V$ is locally quasi-finite,", "\\item there exists algebraic spaces $U$, $V$, a surjective, flat,", "and locally of finite presentation morphism $V \\to \\mathcal{Y}$,", "and a morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ which", "is surjective, flat, locally of finite presentation, and", "locally quasi-finite such that $U \\to V$ is locally quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Then $f$ is quasi-DM by assumption. Let", "$V \\to \\mathcal{Y}$ and $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$", "be as in (2). By", "Lemma \\ref{lemma-composition-locally-quasi-finite}", "the composition $U \\to \\mathcal{X} \\times_\\mathcal{Y} V \\to V$ is", "locally quasi-finite. Thus (1) implies (2).", "\\medskip\\noindent", "Assume (2). Let $V \\to \\mathcal{Y}$ be as in (3). By", "Lemma \\ref{lemma-characterize-quasi-DM}", "we can find an algebraic space $U$ and a surjective, flat, locally", "finitely presented, locally quasi-finite morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$. By (2) the composition", "$U \\to V$ is locally quasi-finite. Thus (2) implies (3).", "\\medskip\\noindent", "It is immediate that (3) implies (4).", "\\medskip\\noindent", "Assume (4). We will prove (1) holds, which finishes the proof. By", "Lemma \\ref{lemma-characterize-quasi-DM}", "we see that $f$ is quasi-DM. To prove that $f$ is locally of finite type", "it suffices to prove that $g : \\mathcal{X} \\times_\\mathcal{Y} V \\to V$ is", "locally of finite type, see", "Lemma \\ref{lemma-check-finite-type-covering}.", "Then it suffices to check that $g$ precomposed with", "$h : U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ is locally of finite type, see", "Lemma \\ref{lemma-check-finite-type-precompose}.", "Since $g \\circ h : U \\to V$ was assumed to be locally quasi-finite", "this holds, hence $f$ is locally of finite type.", "Finally, let $k$ be a field and let $\\Spec(k) \\to \\mathcal{Y}$", "be a morphism. Then $V \\times_\\mathcal{Y} \\Spec(k)$ is", "a nonempty algebraic space which is locally of finite presentation", "over $k$. Hence we can find a finite extension $k \\subset k'$ and", "a morphism $\\Spec(k') \\to V$ such that", "$$", "\\xymatrix{", "\\Spec(k') \\ar[r] \\ar[d] & V \\ar[d] \\\\", "\\Spec(k) \\ar[r] & \\mathcal{Y}", "}", "$$", "commutes (details omitted). Then $\\mathcal{X}_{k'} \\to \\mathcal{X}_k$", "is representable (by schemes), surjective, and finite locally free. In", "particular $|\\mathcal{X}_{k'}| \\to |\\mathcal{X}_k|$ is surjective and open.", "Thus it suffices to prove that $|\\mathcal{X}_{k'}|$ is discrete. Since", "$$", "U \\times_V \\Spec(k') =", "U \\times_{\\mathcal{X} \\times_\\mathcal{Y} V} \\mathcal{X}_{k'}", "$$", "we see that $U \\times_V \\Spec(k') \\to \\mathcal{X}_{k'}$ is", "surjective, flat, and locally of finite presentation (as a base change", "of $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$). Hence", "$|U \\times_V \\Spec(k')| \\to |\\mathcal{X}_{k'}|$ is surjective and", "open. Thus it suffices to show that $|U \\times_V \\Spec(k')|$ is", "discrete. This follows from the fact that $U \\to V$ is locally", "quasi-finite (either by our definition above or from the original definition", "for morphisms of algebraic spaces, via", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-locally-quasi-finite})." ], "refs": [ "stacks-morphisms-lemma-composition-locally-quasi-finite", "stacks-morphisms-lemma-characterize-quasi-DM", "stacks-morphisms-lemma-characterize-quasi-DM", "stacks-morphisms-lemma-check-finite-type-covering", "stacks-morphisms-lemma-check-finite-type-precompose", "spaces-morphisms-lemma-locally-quasi-finite" ], "ref_ids": [ 7487, 7488, 7488, 7463, 7464, 4833 ] } ], "ref_ids": [] }, { "id": 7490, "type": "theorem", "label": "stacks-morphisms-lemma-locally-quasi-finite-permanence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-locally-quasi-finite-permanence", "contents": [ "Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms", "of algebraic stacks. Assume that $\\mathcal{X} \\to \\mathcal{Z}$", "is locally quasi-finite and $\\mathcal{Y} \\to \\mathcal{Z}$ is quasi-DM.", "Then $\\mathcal{X} \\to \\mathcal{Y}$ is locally quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Write $\\mathcal{X} \\to \\mathcal{Y}$ as the composition", "$$", "\\mathcal{X} \\longrightarrow", "\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\longrightarrow", "\\mathcal{Y}", "$$", "The second arrow is locally quasi-finite as a base change of", "$\\mathcal{X} \\to \\mathcal{Z}$, see", "Lemma \\ref{lemma-base-change-locally-quasi-finite}.", "The first arrow is locally quasi-finite by", "Lemma \\ref{lemma-semi-diagonal}", "as $\\mathcal{Y} \\to \\mathcal{Z}$ is quasi-DM.", "Hence $\\mathcal{X} \\to \\mathcal{Y}$ is locally quasi-finite by", "Lemma \\ref{lemma-composition-locally-quasi-finite}." ], "refs": [ "stacks-morphisms-lemma-base-change-locally-quasi-finite", "stacks-morphisms-lemma-semi-diagonal", "stacks-morphisms-lemma-composition-locally-quasi-finite" ], "ref_ids": [ 7485, 7402, 7487 ] } ], "ref_ids": [] }, { "id": 7491, "type": "theorem", "label": "stacks-morphisms-lemma-composition-quasi-finite", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-quasi-finite", "contents": [ "The composition of quasi-finite morphisms is quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemmas \\ref{lemma-composition-locally-quasi-finite} and", "\\ref{lemma-composition-quasi-compact}." ], "refs": [ "stacks-morphisms-lemma-composition-locally-quasi-finite", "stacks-morphisms-lemma-composition-quasi-compact" ], "ref_ids": [ 7487, 7424 ] } ], "ref_ids": [] }, { "id": 7492, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-quasi-finite", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-quasi-finite", "contents": [ "A base change of a quasi-finite morphism is quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemmas \\ref{lemma-base-change-locally-quasi-finite} and", "\\ref{lemma-base-change-quasi-compact}." ], "refs": [ "stacks-morphisms-lemma-base-change-locally-quasi-finite", "stacks-morphisms-lemma-base-change-quasi-compact" ], "ref_ids": [ 7485, 7423 ] } ], "ref_ids": [] }, { "id": 7493, "type": "theorem", "label": "stacks-morphisms-lemma-quasi-finite-permanence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-quasi-finite-permanence", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and", "$g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks.", "If $g \\circ f$ is quasi-finite and $g$ is quasi-separated and quasi-DM", "then $f$ is quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemmas \\ref{lemma-locally-quasi-finite-permanence} and", "\\ref{lemma-quasi-compact-permanence}." ], "refs": [ "stacks-morphisms-lemma-locally-quasi-finite-permanence", "stacks-morphisms-lemma-quasi-compact-permanence" ], "ref_ids": [ 7490, 7427 ] } ], "ref_ids": [] }, { "id": 7494, "type": "theorem", "label": "stacks-morphisms-lemma-composition-flat", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-flat", "contents": [ "The composition of flat morphisms is flat." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Remark \\ref{remark-composition}", "with", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-composition-flat}." ], "refs": [ "stacks-morphisms-remark-composition", "spaces-morphisms-lemma-composition-flat" ], "ref_ids": [ 7633, 4852 ] } ], "ref_ids": [] }, { "id": 7495, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-flat", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-flat", "contents": [ "A base change of a flat morphism is flat." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Remark \\ref{remark-base-change}", "with", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-flat}." ], "refs": [ "stacks-morphisms-remark-base-change", "spaces-morphisms-lemma-base-change-flat" ], "ref_ids": [ 7634, 4853 ] } ], "ref_ids": [] }, { "id": 7496, "type": "theorem", "label": "stacks-morphisms-lemma-descent-flat", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-descent-flat", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective flat morphism of algebraic", "stacks. If the base change", "$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$", "is flat, then $f$ is flat." ], "refs": [], "proofs": [ { "contents": [ "Choose an algebraic space $W$ and a surjective smooth morphism", "$W \\to \\mathcal{Z}$. Then $W \\to \\mathcal{Z}$ is surjective and flat", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-smooth-flat})", "hence $W \\to \\mathcal{Y}$ is surjective and flat (by", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-composition-surjective}", "and", "Lemma \\ref{lemma-composition-flat}).", "Since the base change of", "$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$", "by $W \\to \\mathcal{Z}$ is a flat morphism", "(Lemma \\ref{lemma-base-change-flat})", "we may replace $\\mathcal{Z}$ by $W$.", "\\medskip\\noindent", "Choose an algebraic space $V$ and a surjective smooth morphism", "$V \\to \\mathcal{Y}$. Choose an algebraic space $U$ and a surjective", "smooth morphism $U \\to V \\times_\\mathcal{Y} \\mathcal{X}$.", "We have to show that $U \\to V$ is flat. Now we base change", "everything by $W \\to \\mathcal{Y}$: Set $U' = W \\times_\\mathcal{Y} U$,", "$V' = W \\times_\\mathcal{Y} V$,", "$\\mathcal{X}' = W \\times_\\mathcal{Y} \\mathcal{X}$,", "and $\\mathcal{Y}' = W \\times_\\mathcal{Y} \\mathcal{Y} = W$.", "Then it is still true that $U' \\to V' \\times_{\\mathcal{Y}'} \\mathcal{X}'$", "is smooth by base change. Hence by our definition of flat morphisms", "of algebraic stacks and the assumption that $\\mathcal{X}' \\to \\mathcal{Y}'$", "is flat, we see that $U' \\to V'$ is flat. Then, since", "$V' \\to V$ is surjective as a base change of $W \\to \\mathcal{Y}$ we see", "that $U \\to V$ is flat by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-module-flat} (2)", "and we win." ], "refs": [ "spaces-morphisms-lemma-smooth-flat", "stacks-properties-lemma-composition-surjective", "stacks-morphisms-lemma-composition-flat", "stacks-morphisms-lemma-base-change-flat", "spaces-morphisms-lemma-base-change-module-flat" ], "ref_ids": [ 4891, 8869, 7494, 7495, 4863 ] } ], "ref_ids": [] }, { "id": 7497, "type": "theorem", "label": "stacks-morphisms-lemma-flat-permanence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-flat-permanence", "contents": [ "Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of", "algebraic stacks. If $\\mathcal{X} \\to \\mathcal{Z}$ is flat", "and $\\mathcal{X} \\to \\mathcal{Y}$ is surjective and flat, then", "$\\mathcal{Y} \\to \\mathcal{Z}$ is flat." ], "refs": [], "proofs": [ { "contents": [ "Choose an algebraic space $W$ and a surjective smooth morphism", "$W \\to \\mathcal{Z}$. Choose an algebraic space $V$ and a surjective smooth", "morphism $V \\to W \\times_\\mathcal{Z} \\mathcal{Y}$. Choose an algebraic space", "$U$ and a surjective smooth morphism $U \\to V \\times_\\mathcal{Y} \\mathcal{X}$.", "We know that $U \\to V$ is flat and that $U \\to W$ is flat.", "Also, as $\\mathcal{X} \\to \\mathcal{Y}$ is surjective we see that", "$U \\to V$ is surjective (as a composition of surjective morphisms).", "Hence the lemma reduces to the case of morphisms of algebraic spaces.", "The case of morphisms of algebraic spaces is", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-flat-permanence}." ], "refs": [ "spaces-morphisms-lemma-flat-permanence" ], "ref_ids": [ 4865 ] } ], "ref_ids": [] }, { "id": 7498, "type": "theorem", "label": "stacks-morphisms-lemma-lift-valuation-ring-through-flat-morphism", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-lift-valuation-ring-through-flat-morphism", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a flat morphism", "of algebraic stacks. Let $\\Spec(A) \\to \\mathcal{Y}$ be a morphism", "where $A$ is a valuation ring. If the closed point of $\\Spec(A)$ maps to a", "point of $|\\mathcal{Y}|$ in the image of $|\\mathcal{X|} \\to |\\mathcal{Y}|$,", "then there exists a commutative diagram", "$$", "\\xymatrix{", "\\Spec(A') \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\", "\\Spec(A) \\ar[r] & \\mathcal{Y}", "}", "$$", "where $A \\to A'$ is an extension of valuation rings", "(More on Algebra, Definition", "\\ref{more-algebra-definition-extension-valuation-rings})." ], "refs": [ "more-algebra-definition-extension-valuation-rings" ], "proofs": [ { "contents": [ "The base change $\\mathcal{X}_A \\to \\Spec(A)$ is flat", "(Lemma \\ref{lemma-base-change-flat}) and the closed point of", "$\\Spec(A)$ is in the image of $|\\mathcal{X}_A| \\to |\\Spec(A)|$", "(Properties of Stacks, Lemma \\ref{stacks-properties-lemma-points-cartesian}).", "Thus we may assume $\\mathcal{Y} = \\Spec(A)$. Let $U \\to \\mathcal{X}$", "be a surjective smooth morphism where $U$ is a scheme.", "Then we can apply Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-lift-valuation-ring-through-flat-morphism}", "to the morphism $U \\to \\Spec(A)$ to conclude." ], "refs": [ "stacks-morphisms-lemma-base-change-flat", "stacks-properties-lemma-points-cartesian", "spaces-morphisms-lemma-lift-valuation-ring-through-flat-morphism" ], "ref_ids": [ 7495, 8864, 4933 ] } ], "ref_ids": [ 10647 ] }, { "id": 7499, "type": "theorem", "label": "stacks-morphisms-lemma-flat-at-point", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-flat-at-point", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $x \\in |\\mathcal{X}|$. Consider commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "\\mathcal{X} \\ar[r]^f & \\mathcal{Y}", "}", "}", "\\quad\\text{with points}", "\\vcenter{", "\\xymatrix{", "u \\in |U| \\ar[d] \\\\", "x \\in |\\mathcal{X}|", "}", "}", "$$", "where $U$ and $V$ are algebraic spaces, $b$ is flat, and", "$(a, h) : U \\to \\mathcal{X} \\times_\\mathcal{Y} V$", "is flat. The following are equivalent", "\\begin{enumerate}", "\\item $h$ is flat at $u$ for one diagram as above,", "\\item $h$ is flat at $u$ for every diagram as above.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose we are given a second diagram $U', V', u', a', b', h'$ as", "in the lemma. Then we can consider", "$$", "\\xymatrix{", "U \\ar[d] & U \\times_\\mathcal{X} U' \\ar[l] \\ar[d] \\ar[r] & U' \\ar[d] \\\\", "V & V \\times_\\mathcal{Y} V' \\ar[l] \\ar[r] & V'", "}", "$$", "By Properties of Stacks, Lemma \\ref{stacks-properties-lemma-points-cartesian}", "there is a point $u'' \\in |U \\times_\\mathcal{X} U'|$ mapping", "to $u$ and $u'$. If $h$ is flat at $u$, then the base change", "$U \\times_V (V \\times_\\mathcal{Y} V') \\to V \\times_\\mathcal{Y} V'$", "is flat at any point over $u$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-module-flat}.", "On the other hand, the morphism", "$$", "U \\times_\\mathcal{X} U' \\to", "U \\times_\\mathcal{X} (\\mathcal{X} \\times_\\mathcal{Y} V') =", "U \\times_\\mathcal{Y} V' =", "U \\times_V (V \\times_\\mathcal{Y} V')", "$$", "is flat as a base change of $(a', h')$, see Lemma \\ref{lemma-base-change-flat}.", "Composing and using", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-composition-module-flat}", "we conclude that $U \\times_\\mathcal{X} U' \\to V \\times_\\mathcal{Y} V'$", "is flat at $u''$. Then we can use composition by the flat map", "$V \\times_\\mathcal{Y} V' \\to V'$ to conclude that", "$U \\times_\\mathcal{X} U' \\to V'$ is flat at $u''$.", "Finally, since $U \\times_\\mathcal{X} U' \\to U'$ is flat", "at $u''$ and $u''$ maps to $u'$ we conclude that", "$U' \\to V'$ is flat at $u'$ by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-permanence}." ], "refs": [ "stacks-properties-lemma-points-cartesian", "spaces-morphisms-lemma-base-change-module-flat", "stacks-morphisms-lemma-base-change-flat", "spaces-morphisms-lemma-composition-module-flat", "spaces-morphisms-lemma-flat-permanence" ], "ref_ids": [ 8864, 4863, 7495, 4864, 4865 ] } ], "ref_ids": [] }, { "id": 7500, "type": "theorem", "label": "stacks-morphisms-lemma-composition-finite-presentation", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-finite-presentation", "contents": [ "The composition of finitely presented morphisms is of finite presentation.", "The same holds for morphisms which are locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Remark \\ref{remark-composition}", "with", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-composition-finite-presentation}." ], "refs": [ "stacks-morphisms-remark-composition", "spaces-morphisms-lemma-composition-finite-presentation" ], "ref_ids": [ 7633, 4839 ] } ], "ref_ids": [] }, { "id": 7501, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-finite-presentation", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-finite-presentation", "contents": [ "A base change of a finitely presented morphism is of finite presentation.", "The same holds for morphisms which are locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Remark \\ref{remark-base-change}", "with", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-finite-presentation}." ], "refs": [ "stacks-morphisms-remark-base-change", "spaces-morphisms-lemma-base-change-finite-presentation" ], "ref_ids": [ 7634, 4840 ] } ], "ref_ids": [] }, { "id": 7502, "type": "theorem", "label": "stacks-morphisms-lemma-finite-presentation-finite-type", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-finite-presentation-finite-type", "contents": [ "A morphism which is locally of finite presentation is locally of finite type.", "A morphism of finite presentation is of finite type." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Remark \\ref{remark-implication}", "with", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-finite-type}." ], "refs": [ "stacks-morphisms-remark-implication", "spaces-morphisms-lemma-finite-presentation-finite-type" ], "ref_ids": [ 7635, 4842 ] } ], "ref_ids": [] }, { "id": 7503, "type": "theorem", "label": "stacks-morphisms-lemma-noetherian-finite-type-finite-presentation", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-noetherian-finite-type-finite-presentation", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "\\begin{enumerate}", "\\item If $\\mathcal{Y}$ is locally Noetherian and $f$ locally of finite type", "then $f$ is locally of finite presentation.", "\\item If $\\mathcal{Y}$ is locally Noetherian and $f$ of finite type and", "quasi-separated then $f$ is of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f : \\mathcal{X} \\to \\mathcal{Y}$", "locally of finite type and $\\mathcal{Y}$ locally Noetherian.", "This means there exists a diagram as in", "Lemma \\ref{lemma-local-source-target}", "with $h$ locally of finite type and surjective vertical arrow $a$. By", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation}", "$h$ is locally of finite presentation.", "Hence $\\mathcal{X} \\to \\mathcal{Y}$", "is locally of finite presentation by definition.", "This proves (1).", "If $f$ is of finite type and quasi-separated then it is also", "quasi-compact and quasi-separated and (2) follows immediately." ], "refs": [ "stacks-morphisms-lemma-local-source-target", "spaces-morphisms-lemma-noetherian-finite-type-finite-presentation" ], "ref_ids": [ 7458, 4844 ] } ], "ref_ids": [] }, { "id": 7504, "type": "theorem", "label": "stacks-morphisms-lemma-finite-presentation-permanence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-finite-presentation-permanence", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and", "$g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks", "If $g \\circ f$ is locally of finite presentation and $g$ is locally of", "finite type, then $f$ is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Choose an algebraic space $W$ and a surjective smooth morphism", "$W \\to \\mathcal{Z}$.", "Choose an algebraic space $V$ and a surjective smooth morphism", "$V \\to \\mathcal{Y} \\times_\\mathcal{Z} W$.", "Choose an algebraic space $U$ and a surjective smooth morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$.", "The lemma follows upon applying", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-permanence}", "to the morphisms $U \\to V \\to W$." ], "refs": [ "spaces-morphisms-lemma-finite-presentation-permanence" ], "ref_ids": [ 4846 ] } ], "ref_ids": [] }, { "id": 7505, "type": "theorem", "label": "stacks-morphisms-lemma-diagonal-morphism-finite-type", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-diagonal-morphism-finite-type", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "with diagonal", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$.", "If $f$ is locally of finite type then $\\Delta$ is", "locally of finite presentation. If $f$ is", "quasi-separated and locally of finite type, then $\\Delta$ is of finite", "presentation." ], "refs": [], "proofs": [ { "contents": [ "Note that $\\Delta$ is a morphism over $\\mathcal{X}$ (via the second", "projection). If $f$ is locally of finite type, then", "$\\mathcal{X}$ is of finite presentation over $\\mathcal{X}$ and", "$\\text{pr}_2 : \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}$", "is locally of finite type by Lemma \\ref{lemma-base-change-finite-type}.", "Thus the first statement holds by", "Lemma \\ref{lemma-finite-presentation-permanence}.", "The second statement follows from the first and", "the definitions (because $f$ being quasi-separated means", "by definition that $\\Delta_f$ is quasi-compact and quasi-separated)." ], "refs": [ "stacks-morphisms-lemma-base-change-finite-type", "stacks-morphisms-lemma-finite-presentation-permanence" ], "ref_ids": [ 7460, 7504 ] } ], "ref_ids": [] }, { "id": 7506, "type": "theorem", "label": "stacks-morphisms-lemma-open-immersion-locally-finite-presentation", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-open-immersion-locally-finite-presentation", "contents": [ "An open immersion is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "In view of Properties of Stacks, Definition", "\\ref{stacks-properties-definition-immersion}", "this follows from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-open-immersion-locally-finite-presentation}." ], "refs": [ "stacks-properties-definition-immersion", "spaces-morphisms-lemma-open-immersion-locally-finite-presentation" ], "ref_ids": [ 8920, 4848 ] } ], "ref_ids": [] }, { "id": 7507, "type": "theorem", "label": "stacks-morphisms-lemma-check-property-after-fppf-base-change", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-check-property-after-fppf-base-change", "contents": [ "Let $P$ be a property of morphisms of algebraic spaces which is", "fppf local on the target and preserved by arbitrary base change.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "representable by algebraic spaces.", "Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which", "is surjective, flat, and locally of finite presentation.", "Set $\\mathcal{W} = \\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$. Then", "$$", "(f\\text{ has }P) \\Leftrightarrow", "(\\text{the projection }\\mathcal{W} \\to \\mathcal{Z}\\text{ has }P).", "$$", "For the meaning of this statement see", "Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}." ], "refs": [], "proofs": [ { "contents": [ "Choose an algebraic space $W$ and a morphism", "$W \\to \\mathcal{Z}$ which is surjective, flat, and locally of finite", "presentation. By", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-composition-surjective}", "and Lemmas \\ref{lemma-composition-flat} and", "\\ref{lemma-composition-finite-presentation}", "the composition $W \\to \\mathcal{Y}$ is also surjective, flat, and", "locally of finite presentation. Denote", "$V = W \\times_\\mathcal{Z} \\mathcal{W} = V \\times_\\mathcal{Y} \\mathcal{X}$.", "By Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}", "we see that $f$ has $\\mathcal{P}$ if and only if $V \\to W$ does", "and that $\\mathcal{W} \\to \\mathcal{Z}$ has $\\mathcal{P}$ if and only", "if $V \\to W$ does. The lemma follows." ], "refs": [ "stacks-properties-lemma-composition-surjective", "stacks-morphisms-lemma-composition-flat", "stacks-morphisms-lemma-composition-finite-presentation", "stacks-properties-lemma-check-property-covering" ], "ref_ids": [ 8869, 7494, 7500, 8859 ] } ], "ref_ids": [] }, { "id": 7508, "type": "theorem", "label": "stacks-morphisms-lemma-descent-property", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-descent-property", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces", "which is smooth local on the source-and-target and fppf local", "on the target.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective, flat, locally finitely", "presented morphism of algebraic stacks. If the base change", "$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$", "has $\\mathcal{P}$, then $f$ has $\\mathcal{P}$." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$", "has $\\mathcal{P}$. Choose an algebraic space $W$ and a surjective", "smooth morphism $W \\to \\mathcal{Z}$. Observe that", "$W \\times_\\mathcal{Z} \\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} =", "W \\times_\\mathcal{Y} \\mathcal{X}$. Thus by the very definition of", "what it means for $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$", "to have $\\mathcal{P}$ (see Definition \\ref{definition-P}", "and Lemma \\ref{lemma-local-source-target})", "we see that $W \\times_\\mathcal{Y} \\mathcal{X} \\to W$", "has $\\mathcal{P}$. On the other hand, $W \\to \\mathcal{Z}$", "is surjective, flat, and locally of finite presentation", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-smooth-flat} and", "\\ref{spaces-morphisms-lemma-smooth-locally-finite-presentation})", "hence $W \\to \\mathcal{Y}$ is surjective, flat, and locally of finite", "presentation (by", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-composition-surjective}", "and", "Lemmas \\ref{lemma-composition-flat} and", "\\ref{lemma-composition-finite-presentation}).", "Thus we may replace $\\mathcal{Z}$ by $W$.", "\\medskip\\noindent", "Choose an algebraic space $V$ and a surjective smooth morphism", "$V \\to \\mathcal{Y}$. Choose an algebraic space $U$ and a surjective", "smooth morphism $U \\to V \\times_\\mathcal{Y} \\mathcal{X}$.", "We have to show that $U \\to V$ has $\\mathcal{P}$.", "Now we base change everything by $W \\to \\mathcal{Y}$: Set", "$U' = W \\times_\\mathcal{Y} U$,", "$V' = W \\times_\\mathcal{Y} V$,", "$\\mathcal{X}' = W \\times_\\mathcal{Y} \\mathcal{X}$,", "and $\\mathcal{Y}' = W \\times_\\mathcal{Y} \\mathcal{Y} = W$.", "Then it is still true that $U' \\to V' \\times_{\\mathcal{Y}'} \\mathcal{X}'$", "is smooth by base change. Hence by Lemma \\ref{lemma-local-source-target}", "used in the definition of $\\mathcal{X}' \\to \\mathcal{Y}' = W$", "having $\\mathcal{P}$ we see that $U' \\to V'$ has $\\mathcal{P}$.", "Then, since $V' \\to V$ is surjective, flat, and locally of finite presentation", "as a base change of $W \\to \\mathcal{Y}$ we see that $U \\to V$", "has $\\mathcal{P}$ as $\\mathcal{P}$ is local in the fppf topology", "on the target." ], "refs": [ "stacks-morphisms-definition-P", "stacks-morphisms-lemma-local-source-target", "spaces-morphisms-lemma-smooth-flat", "spaces-morphisms-lemma-smooth-locally-finite-presentation", "stacks-properties-lemma-composition-surjective", "stacks-morphisms-lemma-composition-flat", "stacks-morphisms-lemma-composition-finite-presentation", "stacks-morphisms-lemma-local-source-target" ], "ref_ids": [ 7613, 7458, 4891, 4889, 8869, 7494, 7500, 7458 ] } ], "ref_ids": [] }, { "id": 7509, "type": "theorem", "label": "stacks-morphisms-lemma-descent-finite-presentation", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-descent-finite-presentation", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective, flat, locally finitely", "presented morphism of algebraic stacks. If the base change", "$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$", "is locally of finite presentation, then $f$ is locally of finite", "presentation." ], "refs": [], "proofs": [ { "contents": [ "The property", "``locally of finite presentation''", "satisfies the conditions of Lemma \\ref{lemma-descent-property}.", "Smooth local on the source-and-target we have seen in the", "introduction to this section and fppf local on the target is", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-locally-finite-presentation}." ], "refs": [ "stacks-morphisms-lemma-descent-property", "spaces-descent-lemma-descending-property-locally-finite-presentation" ], "ref_ids": [ 7508, 9390 ] } ], "ref_ids": [] }, { "id": 7510, "type": "theorem", "label": "stacks-morphisms-lemma-flat-finite-presentation-permanence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-flat-finite-presentation-permanence", "contents": [ "Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of", "algebraic stacks. If $\\mathcal{X} \\to \\mathcal{Z}$ is locally of finite", "presentation and $\\mathcal{X} \\to \\mathcal{Y}$ is surjective, flat, and", "locally of finite presentation, then $\\mathcal{Y} \\to \\mathcal{Z}$", "is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Choose an algebraic space $W$ and a surjective smooth morphism", "$W \\to \\mathcal{Z}$. Choose an algebraic space $V$ and a surjective smooth", "morphism $V \\to W \\times_\\mathcal{Z} \\mathcal{Y}$. Choose an algebraic space", "$U$ and a surjective smooth morphism $U \\to V \\times_\\mathcal{Y} \\mathcal{X}$.", "We know that $U \\to V$ is flat and locally of finite presentation", "and that $U \\to W$ is locally of finite presentation.", "Also, as $\\mathcal{X} \\to \\mathcal{Y}$ is surjective we see that", "$U \\to V$ is surjective (as a composition of surjective morphisms).", "Hence the lemma reduces to the case of morphisms of algebraic spaces.", "The case of morphisms of algebraic spaces is", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-locally-finite-presentation-fppf-local-source}." ], "refs": [ "spaces-descent-lemma-locally-finite-presentation-fppf-local-source" ], "ref_ids": [ 9418 ] } ], "ref_ids": [] }, { "id": 7511, "type": "theorem", "label": "stacks-morphisms-lemma-surjective-flat-locally-finite-presentation", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-surjective-flat-locally-finite-presentation", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "which is surjective, flat, and locally of finite presentation.", "Then for every scheme $U$ and object $y$ of $\\mathcal{Y}$ over $U$", "there exists an fppf covering $\\{U_i \\to U\\}$ and objects $x_i$", "of $\\mathcal{X}$ over $U_i$ such that $f(x_i) \\cong y|_{U_i}$ in", "$\\mathcal{Y}_{U_i}$." ], "refs": [], "proofs": [ { "contents": [ "We may think of $y$ as a morphism $U \\to \\mathcal{Y}$. By", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-base-change-surjective}", "and", "Lemmas \\ref{lemma-base-change-finite-presentation} and", "\\ref{lemma-base-change-flat}", "we see that $\\mathcal{X} \\times_\\mathcal{Y} U \\to U$ is surjective, flat,", "and locally of finite presentation. Let $V$ be a scheme and let", "$V \\to \\mathcal{X} \\times_\\mathcal{Y} U$ smooth and surjective.", "Then $V \\to \\mathcal{X} \\times_\\mathcal{Y} U$ is also surjective, flat, and", "locally of finite presentation (see", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-smooth-flat} and", "\\ref{spaces-morphisms-lemma-smooth-locally-finite-presentation}).", "Hence also $V \\to U$ is surjective, flat, and", "locally of finite presentation, see", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-composition-surjective}", "and", "Lemmas \\ref{lemma-composition-finite-presentation}, and", "\\ref{lemma-composition-flat}.", "Hence $\\{V \\to U\\}$ is the desired fppf covering and $x : V \\to \\mathcal{X}$", "is the desired object." ], "refs": [ "stacks-properties-lemma-base-change-surjective", "stacks-morphisms-lemma-base-change-finite-presentation", "stacks-morphisms-lemma-base-change-flat", "spaces-morphisms-lemma-smooth-flat", "spaces-morphisms-lemma-smooth-locally-finite-presentation", "stacks-properties-lemma-composition-surjective", "stacks-morphisms-lemma-composition-finite-presentation", "stacks-morphisms-lemma-composition-flat" ], "ref_ids": [ 8870, 7501, 7495, 4891, 4889, 8869, 7500, 7494 ] } ], "ref_ids": [] }, { "id": 7512, "type": "theorem", "label": "stacks-morphisms-lemma-surjective-family-flat-locally-finite-presentation", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-surjective-family-flat-locally-finite-presentation", "contents": [ "Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$, $j \\in J$ be a family of morphisms", "of algebraic stacks which are each flat and locally of finite presentation", "and which are jointly surjective, i.e.,", "$|\\mathcal{X}| = \\bigcup |f_j|(|\\mathcal{X}_j|)$.", "Then for every scheme $U$ and object $x$ of $\\mathcal{X}$ over $U$", "there exists an fppf covering $\\{U_i \\to U\\}_{i \\in I}$, a map", "$a : I \\to J$, and objects $x_i$ of $\\mathcal{X}_{a(i)}$ over $U_i$", "such that $f_{a(i)}(x_i) \\cong y|_{U_i}$ in $\\mathcal{X}_{U_i}$." ], "refs": [], "proofs": [ { "contents": [ "Apply", "Lemma \\ref{lemma-surjective-flat-locally-finite-presentation}", "to the morphism $\\coprod_{j \\in J} \\mathcal{X}_j \\to \\mathcal{X}$.", "(There is a slight set theoretic issue here -- due to our setup of", "things -- which we ignore.) To finish, note that a morphism", "$x_i : U_i \\to \\coprod_{j \\in J} \\mathcal{X}_j$ is given by a", "disjoint union decomposition $U_i = \\coprod U_{i, j}$ and morphisms", "$U_{i, j} \\to \\mathcal{X}_j$. Then the fppf covering $\\{U_{i, j} \\to U\\}$", "and the morphisms $U_{i, j} \\to \\mathcal{X}_j$ do the job." ], "refs": [ "stacks-morphisms-lemma-surjective-flat-locally-finite-presentation" ], "ref_ids": [ 7511 ] } ], "ref_ids": [] }, { "id": 7513, "type": "theorem", "label": "stacks-morphisms-lemma-fppf-open", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-fppf-open", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be flat and locally of finite", "presentation. Then $|f| : |\\mathcal{X}| \\to |\\mathcal{Y}|$ is open." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a smooth surjective morphism $V \\to \\mathcal{Y}$.", "Choose a scheme $U$ and a smooth surjective morphism", "$U \\to V \\times_\\mathcal{Y} \\mathcal{X}$. By assumption the morphism", "of schemes $U \\to V$ is flat and locally of finite presentation.", "Hence $U \\to V$ is open by", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}.", "By construction of the topology on $|\\mathcal{Y}|$ the map", "$|V| \\to |\\mathcal{Y}|$ is open.", "The map $|U| \\to |\\mathcal{X}|$ is surjective.", "The result follows from these facts by elementary topology." ], "refs": [ "morphisms-lemma-fppf-open" ], "ref_ids": [ 5267 ] } ], "ref_ids": [] }, { "id": 7514, "type": "theorem", "label": "stacks-morphisms-lemma-descent-quasi-compact", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-descent-quasi-compact", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective, flat, locally finitely", "presented morphism of algebraic stacks. If the base change", "$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$", "is quasi-compact, then $f$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "We have to show that given $\\mathcal{Y}' \\to \\mathcal{Y}$", "with $\\mathcal{Y}'$ quasi-compact, we have", "$\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X}$ is quasi-compact.", "Denote $\\mathcal{Z}' = \\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{Y}'$.", "Then $|\\mathcal{Z}'| \\to |\\mathcal{Y}'|$ is open, see", "Lemma \\ref{lemma-fppf-open}. Hence we can find a quasi-compact", "open substack $\\mathcal{W} \\subset \\mathcal{Z}'$ mapping onto", "$\\mathcal{Y}'$. Because", "$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$", "is quasi-compact, we know that", "$$", "\\mathcal{W} \\times_\\mathcal{Z} \\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} =", "\\mathcal{W} \\times_\\mathcal{Y} \\mathcal{X}", "$$", "is quasi-compact. And the map", "$\\mathcal{W} \\times_\\mathcal{Y} \\mathcal{X} \\to", "\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X}$", "is surjective, hence we win. Some details omitted." ], "refs": [ "stacks-morphisms-lemma-fppf-open" ], "ref_ids": [ 7513 ] } ], "ref_ids": [] }, { "id": 7515, "type": "theorem", "label": "stacks-morphisms-lemma-check-separated-on-ui-cover", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-check-separated-on-ui-cover", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$", "be composable morphisms of algebraic stacks with composition", "$h = g \\circ f : \\mathcal{X} \\to \\mathcal{Z}$.", "If $f$ is surjective, flat, locally of finite presentation,", "and universally injective and if $h$ is separated, then", "$g$ is separated." ], "refs": [], "proofs": [ { "contents": [ "Consider the diagram", "$$", "\\xymatrix{", "\\mathcal{X} \\ar[r]_\\Delta \\ar[rd] &", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\ar[r] \\ar[d] &", "\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{X} \\ar[d] \\\\", "& \\mathcal{Y} \\ar[r] & \\mathcal{Y} \\times_\\mathcal{Z} \\mathcal{Y}", "}", "$$", "The square is cartesian. We have to show the bottom horizontal arrow is proper.", "We already know that it is representable by algebraic spaces and", "locally of finite type (Lemma \\ref{lemma-properties-diagonal}).", "Since the right vertical arrow is", "surjective, flat, and locally of finite presentation", "it suffices to show the top right horizontal arrow", "is proper (Lemma \\ref{lemma-check-property-after-fppf-base-change}).", "Since $h$ is separated, the composition of the top horizontal", "arrows is proper.", "\\medskip\\noindent", "Since $f$ is universally injective $\\Delta$ is surjective", "(Lemma \\ref{lemma-universally-injective}). Since the", "composition of $\\Delta$ with the projection", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}$", "is the identity, we see that $\\Delta$ is universally closed.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-image-universally-closed-separated}", "we conclude that $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{X}$", "is separated as $\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Z} \\mathcal{X}$", "is separated. Here we use", "that implications between properties of morphisms of algebraic", "spaces can be transferred to the same implications between", "properties of morphisms of algebraic stacks representable", "by algebraic spaces; this is discussed in Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}.", "Finally, we use the same principle to conlude that", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{X}$ is proper", "from Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-image-proper-is-proper}." ], "refs": [ "stacks-morphisms-lemma-properties-diagonal", "stacks-morphisms-lemma-check-property-after-fppf-base-change", "stacks-morphisms-lemma-universally-injective", "spaces-morphisms-lemma-image-universally-closed-separated", "spaces-morphisms-lemma-image-proper-is-proper" ], "ref_ids": [ 7392, 7507, 7450, 4750, 4921 ] } ], "ref_ids": [] }, { "id": 7516, "type": "theorem", "label": "stacks-morphisms-lemma-gerbe-over-iso-classes", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-gerbe-over-iso-classes", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. If $\\mathcal{X}$ is a gerbe, then", "the sheafification of the presheaf", "$$", "(\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}, \\quad", "U \\mapsto \\Ob(\\mathcal{X}_U)/\\!\\!\\cong", "$$", "is an algebraic space and $\\mathcal{X}$ is a gerbe over it." ], "refs": [], "proofs": [ { "contents": [ "(In this proof the abuse of language introduced in", "Section \\ref{section-conventions}", "really pays off.)", "Choose a morphism $\\pi : \\mathcal{X} \\to X$ where $X$ is an", "algebraic space which turns $\\mathcal{X}$ into a gerbe over $X$.", "It suffices to prove that $X$ is the sheafification of the presheaf", "$\\mathcal{F}$ displayed in the lemma.", "It is clear that there is a map $c : \\mathcal{F} \\to X$.", "We will use", "Stacks, Lemma \\ref{stacks-lemma-when-gerbe}", "properties (2)(a) and (2)(b) to see that the map $c^\\# : \\mathcal{F}^\\# \\to X$", "is surjective and injective, hence an isomorphism, see", "Sites, Lemma \\ref{sites-lemma-mono-epi-sheaves}.", "Surjective: Let $T$ be a scheme and let $f : T \\to X$. By property (2)(a)", "there exists an fppf covering $\\{h_i : T_i \\to T\\}$ and morphisms", "$x_i : T_i \\to \\mathcal{X}$ such that $f \\circ h_i$ corresponds to", "$\\pi \\circ x_i$. Hence we see that $f|_{T_i}$ is in the image of $c$.", "Injective: Let $T$ be a scheme and let $x, y : T \\to \\mathcal{X}$", "be morphisms such that $c \\circ x = c \\circ y$.", "By (2)(b) we can find a covering $\\{T_i \\to T\\}$ and morphisms", "$x|_{T_i} \\to y|_{T_i}$ in the fibre category $\\mathcal{X}_{T_i}$.", "Hence the restrictions $x|_{T_i}, y|_{T_i}$ are equal in", "$\\mathcal{F}(T_i)$. This proves that $x, y$ give the same section", "of $\\mathcal{F}^\\#$ over $T$ as desired." ], "refs": [ "stacks-lemma-when-gerbe", "sites-lemma-mono-epi-sheaves" ], "ref_ids": [ 8975, 8517 ] } ], "ref_ids": [] }, { "id": 7517, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-gerbe", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-gerbe", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\", "\\mathcal{Y}' \\ar[r] & \\mathcal{Y}", "}", "$$", "be a fibre product of algebraic stacks.", "If $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, then", "$\\mathcal{X}'$ is a gerbe over $\\mathcal{Y}'$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions and", "Stacks, Lemma \\ref{stacks-lemma-base-change-gerbe}." ], "refs": [ "stacks-lemma-base-change-gerbe" ], "ref_ids": [ 8976 ] } ], "ref_ids": [] }, { "id": 7518, "type": "theorem", "label": "stacks-morphisms-lemma-composition-gerbe", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-gerbe", "contents": [ "Let $\\mathcal{X} \\to \\mathcal{Y}$ and $\\mathcal{Y} \\to \\mathcal{Z}$", "be morphisms of algebraic stacks. If $\\mathcal{X}$ is a gerbe over", "$\\mathcal{Y}$ and $\\mathcal{Y}$ is a gerbe over $\\mathcal{Z}$, then", "$\\mathcal{X}$ is a gerbe over $\\mathcal{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from", "Stacks, Lemma \\ref{stacks-lemma-composition-gerbe}." ], "refs": [ "stacks-lemma-composition-gerbe" ], "ref_ids": [ 8977 ] } ], "ref_ids": [] }, { "id": 7519, "type": "theorem", "label": "stacks-morphisms-lemma-gerbe-descent", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-gerbe-descent", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\", "\\mathcal{Y}' \\ar[r] & \\mathcal{Y}", "}", "$$", "be a fibre product of algebraic stacks.", "If $\\mathcal{Y}' \\to \\mathcal{Y}$ is surjective, flat, and locally", "of finite presentation and $\\mathcal{X}'$ is a gerbe over $\\mathcal{Y}'$,", "then $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Lemma \\ref{lemma-surjective-flat-locally-finite-presentation}", "and", "Stacks, Lemma \\ref{stacks-lemma-gerbe-descent}." ], "refs": [ "stacks-morphisms-lemma-surjective-flat-locally-finite-presentation", "stacks-lemma-gerbe-descent" ], "ref_ids": [ 7511, 8978 ] } ], "ref_ids": [] }, { "id": 7520, "type": "theorem", "label": "stacks-morphisms-lemma-gerbe-with-section", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-gerbe-with-section", "contents": [ "Let $\\pi : \\mathcal{X} \\to U$ be a morphism from an algebraic stack to", "an algebraic space and let $x : U \\to \\mathcal{X}$ be a section of $\\pi$.", "Set $G = \\mathit{Isom}_\\mathcal{X}(x, x)$, see", "Definition \\ref{definition-isom}.", "If $\\mathcal{X}$ is a gerbe over $U$, then", "\\begin{enumerate}", "\\item there is a canonical equivalence of stacks in groupoids", "$$", "x_{can} : [U/G] \\longrightarrow \\mathcal{X}.", "$$", "where $[U/G]$ is the quotient stack for the trivial", "action of $G$ on $U$,", "\\item $G \\to U$ is flat and locally of finite presentation, and", "\\item $U \\to \\mathcal{X}$ is surjective, flat, and locally of finite", "presentation.", "\\end{enumerate}" ], "refs": [ "stacks-morphisms-definition-isom" ], "proofs": [ { "contents": [ "Set $R = U \\times_{x, \\mathcal{X}, x} U$. The morphism $R \\to U \\times U$", "factors through the diagonal $\\Delta_U : U \\to U \\times U$ as it factors", "through $U \\times_U U = U$. Hence $R = G$ because", "\\begin{align*}", "G & = \\mathit{Isom}_\\mathcal{X}(x, x) \\\\", "& = U \\times_{x, \\mathcal{X}} \\mathcal{I}_\\mathcal{X} \\\\", "& = U \\times_{x, \\mathcal{X}}", "(\\mathcal{X}", "\\times_{\\Delta, \\mathcal{X} \\times_S \\mathcal{X}, \\Delta}", "\\mathcal{X}) \\\\", "& = (U \\times_{x, \\mathcal{X}, x} U) \\times_{U \\times U, \\Delta_U} U \\\\", "& = R \\times_{U \\times U, \\Delta_U} U \\\\", "& = R", "\\end{align*}", "for the fourth equality use", "Categories, Lemma \\ref{categories-lemma-diagonal-2}.", "Let $t, s : R \\to U$ be the projections.", "The composition law $c : R \\times_{s, U, t} R \\to R$ constructed on $R$ in", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack}", "agrees with the group law on $G$ (proof omitted). Thus", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack}", "shows we obtain a canonical fully faithful $1$-morphism", "$$", "x_{can} : [U/G] \\longrightarrow \\mathcal{X}", "$$", "of stacks in groupoids over $(\\Sch/S)_{fppf}$. To see that it is", "an equivalence it suffices to show that it is essentially surjective.", "To do this it suffices to show that any object of $\\mathcal{X}$ over", "a scheme $T$ comes fppf locally from $x$ via a morphism $T \\to U$, see", "Stacks, Lemma \\ref{stacks-lemma-characterize-essentially-surjective-when-ff}.", "However, this follows the condition that $\\pi$ turns $\\mathcal{X}$", "into a gerbe over $U$, see property (2)(a) of", "Stacks, Lemma \\ref{stacks-lemma-when-gerbe}.", "\\medskip\\noindent", "By", "Criteria for Representability, Lemma \\ref{criteria-lemma-BG-algebraic}", "we conclude that $G \\to U$ is flat and locally of finite presentation.", "Finally, $U \\to \\mathcal{X}$ is surjective, flat, and locally of finite", "presentation by", "Criteria for Representability, Lemma", "\\ref{criteria-lemma-flat-quotient-flat-presentation}." ], "refs": [ "categories-lemma-diagonal-2", "algebraic-lemma-map-space-into-stack", "algebraic-lemma-map-space-into-stack", "stacks-lemma-characterize-essentially-surjective-when-ff", "stacks-lemma-when-gerbe", "criteria-lemma-BG-algebraic", "criteria-lemma-flat-quotient-flat-presentation" ], "ref_ids": [ 12277, 8473, 8473, 8945, 8975, 3140, 3137 ] } ], "ref_ids": [ 7603 ] }, { "id": 7521, "type": "theorem", "label": "stacks-morphisms-lemma-local-structure-gerbe", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-local-structure-gerbe", "contents": [ "Let $\\pi : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, and", "\\item there exists an algebraic space $U$, a group algebraic space $G$", "flat and locally of finite presentation over $U$, and a", "surjective, flat, and locally finitely presented", "morphism $U \\to \\mathcal{Y}$ such that", "$\\mathcal{X} \\times_\\mathcal{Y} U \\cong [U/G]$ over $U$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (2). By", "Lemma \\ref{lemma-gerbe-descent}", "to prove (1) it suffices to show that $[U/G]$ is a gerbe over $U$.", "This is immediate from", "Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-group-quotient-gerbe}.", "\\medskip\\noindent", "Assume (1). Any base change of $\\pi$ is a gerbe, see", "Lemma \\ref{lemma-base-change-gerbe}.", "As a first step we choose a scheme $V$ and a surjective smooth morphism", "$V \\to \\mathcal{Y}$. Thus we may assume that $\\pi : \\mathcal{X} \\to V$", "is a gerbe over a scheme. This means that there exists an", "fppf covering $\\{V_i \\to V\\}$ such that the fibre category", "$\\mathcal{X}_{V_i}$ is nonempty, see", "Stacks, Lemma \\ref{stacks-lemma-when-gerbe} (2)(a).", "Note that $U = \\coprod V_i \\to V$ is surjective, flat, and", "locally of finite presentation. Hence we may replace $V$ by $U$ and", "assume that $\\pi : \\mathcal{X} \\to U$ is a gerbe over a scheme $U$ and", "that there exists an object $x$ of $\\mathcal{X}$ over $U$. By", "Lemma \\ref{lemma-gerbe-with-section}", "we see that $\\mathcal{X} = [U/G]$ over $U$ for some flat", "and locally finitely presented group algebraic space $G$ over $U$." ], "refs": [ "stacks-morphisms-lemma-gerbe-descent", "spaces-groupoids-lemma-group-quotient-gerbe", "stacks-morphisms-lemma-base-change-gerbe", "stacks-lemma-when-gerbe" ], "ref_ids": [ 7519, 9335, 7517, 8975 ] } ], "ref_ids": [] }, { "id": 7522, "type": "theorem", "label": "stacks-morphisms-lemma-gerbe-fppf", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-gerbe-fppf", "contents": [ "Let $\\pi : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "If $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, then $\\pi$ is surjective,", "flat, and locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "By", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-descent-surjective}", "and", "Lemmas \\ref{lemma-descent-flat} and", "\\ref{lemma-descent-finite-presentation}", "it suffices to prove to the lemma after replacing $\\pi$ by a base change", "with a surjective, flat, locally finitely presented morphism", "$\\mathcal{Y}' \\to \\mathcal{Y}$. By", "Lemma \\ref{lemma-local-structure-gerbe}", "we may assume $\\mathcal{Y} = U$ is an algebraic space and", "$\\mathcal{X} = [U/G]$ over $U$.", "Then $U \\to [U/G]$ is surjective, flat, and", "locally of finite presentation, see", "Lemma \\ref{lemma-gerbe-with-section}.", "This implies that $\\pi$ is surjective, flat, and locally", "of finite presentation by", "Properties of Stacks,", "Lemma \\ref{stacks-properties-lemma-surjective-permanence}", "and", "Lemmas \\ref{lemma-flat-permanence} and", "\\ref{lemma-flat-finite-presentation-permanence}." ], "refs": [ "stacks-properties-lemma-descent-surjective", "stacks-morphisms-lemma-descent-flat", "stacks-morphisms-lemma-descent-finite-presentation", "stacks-morphisms-lemma-local-structure-gerbe", "stacks-properties-lemma-surjective-permanence", "stacks-morphisms-lemma-flat-permanence", "stacks-morphisms-lemma-flat-finite-presentation-permanence" ], "ref_ids": [ 8871, 7496, 7509, 7521, 8872, 7497, 7510 ] } ], "ref_ids": [] }, { "id": 7523, "type": "theorem", "label": "stacks-morphisms-lemma-gerbe-isom-fppf", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-gerbe-isom-fppf", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "which makes $\\mathcal{X}$ a gerbe over $\\mathcal{Y}$. Then", "\\begin{enumerate}", "\\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$", "is flat and locally of finite presentation,", "\\item $\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "is surjective, flat, and locally of finite presentation,", "\\item given algebraic spaces $T_i$, $i = 1, 2$ and morphisms", "$x_i : T_i \\to \\mathcal{X}$, with $y_i = f \\circ x_i$ the morphism", "$$", "T_1 \\times_{x_1, \\mathcal{X}, x_2} T_2 \\longrightarrow", "T_1 \\times_{y_1, \\mathcal{Y}, y_2} T_2", "$$", "is surjective, flat, and locally of finite presentation,", "\\item given an algebraic space $T$ and morphisms", "$x_i : T \\to \\mathcal{X}$, $i = 1, 2$, with $y_i = f \\circ x_i$ the morphism", "$$", "\\mathit{Isom}_\\mathcal{X}(x_1, x_2) \\longrightarrow", "\\mathit{Isom}_\\mathcal{Y}(y_1, y_2)", "$$", "is surjective, flat, and locally of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1).", "Choose a scheme $Y$ and a surjective smooth morphism $Y \\to \\mathcal{Y}$.", "Set $\\mathcal{X}' = \\mathcal{X} \\times_\\mathcal{Y} Y$.", "By Lemma \\ref{lemma-cartesian-square-inertia} we obtain cartesian", "squares", "$$", "\\xymatrix{", "\\mathcal{I}_{\\mathcal{X}'} \\ar[r] \\ar[d] &", "\\mathcal{X}' \\ar[r] \\ar[d] & Y \\ar[d] \\\\", "\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\ar[r] &", "\\mathcal{X} \\ar[r] & \\mathcal{Y}", "}", "$$", "By Lemmas \\ref{lemma-descent-flat} and", "\\ref{lemma-descent-finite-presentation}", "it suffices to prove that $\\mathcal{I}_{\\mathcal{X}'} \\to \\mathcal{X}'$", "is flat and locally of finite presentation.", "This follows from Proposition \\ref{proposition-when-gerbe}", "(because $\\mathcal{X}'$ is a gerbe over $Y$ by", "Lemma \\ref{lemma-base-change-gerbe}).", "\\medskip\\noindent", "Proof of (2). With notation as above, note that we may assume that", "$\\mathcal{X}' = [Y/G]$ for some group algebraic space $G$ flat and", "locally of finite presentation over $Y$, see", "Lemma \\ref{lemma-local-structure-gerbe}.", "The base change of the morphism", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "over $\\mathcal{Y}$ by the morphism $Y \\to \\mathcal{Y}$", "is the morphism", "$\\Delta' : \\mathcal{X}' \\to \\mathcal{X}' \\times_Y \\mathcal{X}'$.", "Hence it suffices to show that $\\Delta'$ is", "surjective, flat, and locally of finite presentation", "(see Lemmas \\ref{lemma-descent-flat} and", "\\ref{lemma-descent-finite-presentation}).", "In other words, we have to show that", "$$", "[Y/G] \\longrightarrow [Y/G \\times_Y G]", "$$", "is surjective, flat, and locally of finite presentation.", "This is true because the base change by the surjective, flat,", "locally finitely presented morphism $Y \\to [Y/G \\times_Y G]$", "is the morphism $G \\to Y$.", "\\medskip\\noindent", "Proof of (3). Observe that the diagram", "$$", "\\xymatrix{", "T_1 \\times_{x_1, \\mathcal{X}, x_2} T_2 \\ar[d] \\ar[r] &", "T_1 \\times_{y_1, \\mathcal{Y}, y_2} T_2 \\ar[d] \\\\", "\\mathcal{X} \\ar[r] & \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}", "}", "$$", "is cartesian. Hence (3) follows from (2).", "\\medskip\\noindent", "Proof of (4). This is true because", "$$", "\\mathit{Isom}_\\mathcal{X}(x_1, x_2) =", "(T \\times_{x_1, \\mathcal{X}, x_2} T) \\times_{T \\times T, \\Delta_T} T", "$$", "hence the morphism in (4) is a base change of the morphism in (3)." ], "refs": [ "stacks-morphisms-lemma-cartesian-square-inertia", "stacks-morphisms-lemma-descent-flat", "stacks-morphisms-lemma-descent-finite-presentation", "stacks-morphisms-proposition-when-gerbe", "stacks-morphisms-lemma-base-change-gerbe", "stacks-morphisms-lemma-local-structure-gerbe", "stacks-morphisms-lemma-descent-flat", "stacks-morphisms-lemma-descent-finite-presentation" ], "ref_ids": [ 7413, 7496, 7509, 7598, 7517, 7521, 7496, 7509 ] } ], "ref_ids": [] }, { "id": 7524, "type": "theorem", "label": "stacks-morphisms-lemma-noetherian-singleton-stack-gerbe", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-noetherian-singleton-stack-gerbe", "contents": [ "Let $\\mathcal{Z}$ be a reduced, locally Noetherian algebraic stack", "such that $|\\mathcal{Z}|$ is a singleton. Then $\\mathcal{Z}$ is a gerbe", "over a reduced, locally Noetherian algebraic space $Z$ with $|Z|$ a", "singleton." ], "refs": [], "proofs": [ { "contents": [ "By", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-unique-point-better}", "there exists a surjective, flat, locally finitely presented morphism", "$\\Spec(k) \\to \\mathcal{Z}$ where $k$ is a field.", "Then $\\mathcal{I}_Z \\times_\\mathcal{Z} \\Spec(k) \\to \\Spec(k)$", "is representable by algebraic spaces and", "locally of finite type (as a base change of", "$\\mathcal{I}_\\mathcal{Z} \\to \\mathcal{Z}$, see", "Lemmas \\ref{lemma-inertia} and \\ref{lemma-base-change-finite-type}).", "Therefore it is locally of finite presentation, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation}.", "Of course it is also flat as $k$ is a field. Hence we may apply", "Lemmas \\ref{lemma-descent-flat} and", "\\ref{lemma-descent-finite-presentation}", "to see that $\\mathcal{I}_\\mathcal{Z} \\to \\mathcal{Z}$ is flat and", "locally of finite presentation. We conclude that $\\mathcal{Z}$", "is a gerbe by", "Proposition \\ref{proposition-when-gerbe}.", "Let $\\pi : \\mathcal{Z} \\to Z$ be a morphism to an algebraic space such", "that $\\mathcal{Z}$ is a gerbe over $Z$. Then $\\pi$ is surjective, flat, and", "locally of finite presentation by", "Lemma \\ref{lemma-gerbe-fppf}.", "Hence $\\Spec(k) \\to Z$ is surjective, flat, and locally of finite", "presentation as a composition, see", "Properties of Stacks,", "Lemma \\ref{stacks-properties-lemma-composition-surjective}", "and", "Lemmas \\ref{lemma-composition-flat} and", "\\ref{lemma-composition-finite-presentation}.", "Hence by", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-unique-point-better}", "we see that $|Z|$ is a singleton and that $Z$ is locally Noetherian", "and reduced." ], "refs": [ "stacks-properties-lemma-unique-point-better", "stacks-morphisms-lemma-inertia", "stacks-morphisms-lemma-base-change-finite-type", "spaces-morphisms-lemma-noetherian-finite-type-finite-presentation", "stacks-morphisms-lemma-descent-flat", "stacks-morphisms-lemma-descent-finite-presentation", "stacks-morphisms-proposition-when-gerbe", "stacks-morphisms-lemma-gerbe-fppf", "stacks-properties-lemma-composition-surjective", "stacks-morphisms-lemma-composition-flat", "stacks-morphisms-lemma-composition-finite-presentation", "stacks-properties-lemma-unique-point-better" ], "ref_ids": [ 8902, 7411, 7460, 4844, 7496, 7509, 7598, 7522, 8869, 7494, 7500, 8902 ] } ], "ref_ids": [] }, { "id": 7525, "type": "theorem", "label": "stacks-morphisms-lemma-gerbe-bijection-points", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-gerbe-bijection-points", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "If $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$ then $f$ is a", "universal homeomorphism." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-base-change-gerbe} the assumption on $f$ is", "preserved under base change. Hence it suffices to show that the map", "$|\\mathcal{X}| \\to |\\mathcal{Y}|$ is a homeomorphism of topological spaces.", "Let $k$ be a field and let $y$ be an object", "of $\\mathcal{Y}$ over $\\Spec(k)$. By", "Stacks, Lemma \\ref{stacks-lemma-when-gerbe} property (2)(a)", "there exists an fppf covering $\\{T_i \\to \\Spec(k)\\}$", "and objects $x_i$ of $\\mathcal{X}$ over $T_i$ with $f(x_i) \\cong y|_{T_i}$.", "Choose an $i$ such that $T_i \\not = \\emptyset$. Choose a", "morphism $\\Spec(K) \\to T_i$ for some field $K$.", "Then $k \\subset K$ and $x_i|_K$ is an object of $\\mathcal{X}$ lying", "over $y|_K$. Thus we see that", "$|\\mathcal{Y}| \\to |\\mathcal{X}|$. is surjective. The map", "$|\\mathcal{Y}| \\to |\\mathcal{X}|$ is also injective. Namely, if", "$x, x'$ are objects of $\\mathcal{X}$ over $\\Spec(k)$ whose images", "$f(x), f(x')$ become isomorphic (over an extension) in $\\mathcal{Y}$, then", "Stacks, Lemma \\ref{stacks-lemma-when-gerbe} property (2)(b)", "guarantees the existence of an extension of $k$ over which $x$ and $x'$", "become isomorphic (details omitted).", "Hence $|\\mathcal{X}| \\to |\\mathcal{Y}|$ is continuous and bijective", "and it suffices to show that it is also open.", "This follows from", "Lemmas \\ref{lemma-gerbe-fppf} and \\ref{lemma-fppf-open}." ], "refs": [ "stacks-morphisms-lemma-base-change-gerbe", "stacks-lemma-when-gerbe", "stacks-lemma-when-gerbe", "stacks-morphisms-lemma-gerbe-fppf", "stacks-morphisms-lemma-fppf-open" ], "ref_ids": [ 7517, 8975, 8975, 7522, 7513 ] } ], "ref_ids": [] }, { "id": 7526, "type": "theorem", "label": "stacks-morphisms-lemma-gerbe-diagonal-quasi-compact", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-gerbe-diagonal-quasi-compact", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "such that $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$.", "If $\\Delta_\\mathcal{X}$ is quasi-compact, so is $\\Delta_\\mathcal{Y}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the diagram", "$$", "\\xymatrix{", "\\mathcal{X} \\ar[r] &", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\ar[r] \\ar[d] &", "\\mathcal{X} \\times \\mathcal{X} \\ar[d] \\\\", "&", "\\mathcal{Y} \\ar[r] &", "\\mathcal{Y} \\times \\mathcal{Y}", "}", "$$", "By Proposition \\ref{proposition-when-gerbe-over} we find that", "the arrow on the top left is surjective. Since the composition", "of the top horizontal arrows is quasi-compact, we conclude", "that the top right arrow is quasi-compact by", "Lemma \\ref{lemma-surjection-from-quasi-compact}.", "The square is cartesian and the right vertical arrow is", "surjective, flat, and locally of finite presentation.", "Thus we conclude by Lemma \\ref{lemma-descent-quasi-compact}." ], "refs": [ "stacks-morphisms-proposition-when-gerbe-over", "stacks-morphisms-lemma-surjection-from-quasi-compact", "stacks-morphisms-lemma-descent-quasi-compact" ], "ref_ids": [ 7599, 7426, 7514 ] } ], "ref_ids": [] }, { "id": 7527, "type": "theorem", "label": "stacks-morphisms-lemma-gerbe-residual-gerbe-exists", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-gerbe-residual-gerbe-exists", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. If $\\mathcal{X}$ is a gerbe", "then for every $x \\in |\\mathcal{X}|$ the residual gerbe of $\\mathcal{X}$", "at $x$ exists." ], "refs": [], "proofs": [ { "contents": [ "Let $\\pi : \\mathcal{X} \\to X$ be a morphism from $\\mathcal{X}$ into", "an algebraic space $X$ which turns $\\mathcal{X}$ into a gerbe over $X$.", "Let $Z_x \\to X$ be the residual space of $X$ at $x$, see", "Decent Spaces, Definition \\ref{decent-spaces-definition-residual-space}.", "Let $\\mathcal{Z} = \\mathcal{X} \\times_X Z_x$. By", "Lemma \\ref{lemma-base-change-gerbe}", "the algebraic stack $\\mathcal{Z}$ is a gerbe over $Z_x$.", "Hence $|\\mathcal{Z}| = |Z_x|$", "(Lemma \\ref{lemma-gerbe-bijection-points})", "is a singleton. Since $\\mathcal{Z} \\to Z_x$ is locally of finite presentation", "as a base change of $\\pi$ (see", "Lemmas \\ref{lemma-gerbe-fppf} and \\ref{lemma-base-change-finite-presentation})", "we see that $\\mathcal{Z}$ is locally Noetherian, see", "Lemma \\ref{lemma-locally-finite-type-locally-noetherian}.", "Thus the residual gerbe $\\mathcal{Z}_x$ of $\\mathcal{X}$ at $x$", "exists and is equal to $\\mathcal{Z}_x = \\mathcal{Z}_{red}$ the reduction", "of the algebraic stack $\\mathcal{Z}$. Namely, we have seen above", "that $|\\mathcal{Z}_{red}|$ is a singleton mapping to $x \\in |\\mathcal{X}|$,", "it is reduced by construction, and it is locally Noetherian (as the", "reduction of a locally Noetherian algebraic stack is locally Noetherian,", "details omitted)." ], "refs": [ "decent-spaces-definition-residual-space", "stacks-morphisms-lemma-base-change-gerbe", "stacks-morphisms-lemma-gerbe-bijection-points", "stacks-morphisms-lemma-gerbe-fppf", "stacks-morphisms-lemma-base-change-finite-presentation", "stacks-morphisms-lemma-locally-finite-type-locally-noetherian" ], "ref_ids": [ 9566, 7517, 7525, 7522, 7501, 7462 ] } ], "ref_ids": [] }, { "id": 7528, "type": "theorem", "label": "stacks-morphisms-lemma-every-point-in-a-stratum", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-every-point-in-a-stratum", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack such that", "$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is quasi-compact.", "Then there exists a well-ordered index set $I$ and for every $i \\in I$", "a reduced locally closed substack $\\mathcal{U}_i \\subset \\mathcal{X}$ such that", "\\begin{enumerate}", "\\item each $\\mathcal{U}_i$ is a gerbe,", "\\item we have $|\\mathcal{X}| = \\bigcup_{i \\in I} |\\mathcal{U}_i|$,", "\\item $T_i = |\\mathcal{X}| \\setminus \\bigcup_{i' < i} |\\mathcal{U}_{i'}|$", "is closed in $|\\mathcal{X}|$ for all $i \\in I$, and", "\\item $|\\mathcal{U}_i|$ is open in $T_i$.", "\\end{enumerate}", "We can moreover arrange it so that either (a) $|\\mathcal{U}_i| \\subset T_i$", "is dense, or (b) $\\mathcal{U}_i$ is quasi-compact. In case (a), if", "we choose $\\mathcal{U}_i$ as large as possible (see proof for details), then", "the stratification is canonical." ], "refs": [], "proofs": [ { "contents": [ "Let $T \\subset |\\mathcal{X}|$ be a nonempty closed subset. We are going", "to find (resp.\\ choose) for every such $T$ a reduced locally closed substack", "$\\mathcal{U}(T) \\subset \\mathcal{X}$ with $|\\mathcal{U}(T)| \\subset T$", "open dense (resp.\\ nonempty quasi-compact). Namely, by", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-reduced-closed-substack}", "there exists a unique reduced closed substack", "$\\mathcal{X}' \\subset \\mathcal{X}$ such that $T = |\\mathcal{X}'|$.", "Note that $\\mathcal{I}_{\\mathcal{X}'} =", "\\mathcal{I}_\\mathcal{X} \\times_\\mathcal{X} \\mathcal{X}'$ by", "Lemma \\ref{lemma-monomorphism-cartesian-square-inertia}.", "Hence $\\mathcal{I}_{\\mathcal{X}'} \\to \\mathcal{X}'$ is", "quasi-compact as a base change, see", "Lemma \\ref{lemma-base-change-quasi-compact}.", "Therefore", "Proposition \\ref{proposition-open-stratum}", "implies there exists a dense maximal (see proof proposition)", "open substack $\\mathcal{U} \\subset \\mathcal{X}'$", "which is a gerbe. In case (a) we set $\\mathcal{U}(T) = \\mathcal{U}$", "(this is canonical) and in case (b) we simply choose a nonempty quasi-compact", "open $\\mathcal{U}(T) \\subset \\mathcal{U}$, see", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-space-locally-quasi-compact}", "(we can do this for all $T$", "simultaneously by the axiom of choice).", "\\medskip\\noindent", "By transfinite induction we construct for every ordinal $\\alpha$ a", "closed subset $T_\\alpha \\subset |\\mathcal{X}|$. For $\\alpha = 0$", "we set $T_0 = |\\mathcal{X}|$. Given $T_\\alpha$ set", "$$", "T_{\\alpha + 1} = T_\\alpha \\setminus |\\mathcal{U}(T_\\alpha)|.", "$$", "If $\\beta$ is a limit ordinal we set", "$$", "T_\\beta = \\bigcap\\nolimits_{\\alpha < \\beta} T_\\alpha.", "$$", "We claim that $T_\\alpha = \\emptyset$ for all $\\alpha$", "large enough. Namely, assume that $T_\\alpha \\not = \\emptyset$", "for all $\\alpha$. Then we obtain an injective map from the class", "of ordinals into the set of subsets of $|\\mathcal{X}|$ which is a", "contradiction.", "\\medskip\\noindent", "The claim implies the lemma. Namely, let", "$$", "I = \\{\\alpha \\mid \\mathcal{U}_\\alpha \\not = \\emptyset \\}.", "$$", "This is a well-ordered set by the claim. For $i = \\alpha \\in I$ we set", "$\\mathcal{U}_i = \\mathcal{U}_\\alpha$. So $\\mathcal{U}_i$ is a reduced", "locally closed substack and a gerbe, i.e., (1) holds. By construction", "$T_i = T\\alpha$ if $i = \\alpha \\in I$, hence (3) holds. Also, (4) and", "(a) or (b) hold by our choice of $\\mathcal{U}(T)$ as well. Finally, to see", "(2) let $x \\in |\\mathcal{X}|$. There exists a smallest ordinal $\\beta$", "with $x \\not \\in T_\\beta$ (because the ordinals are well-ordered).", "In this case $\\beta$ has to be a successor ordinal by the definition", "of $T_\\beta$ for limit ordinals. Hence $\\beta = \\alpha + 1$ and", "$x \\in |\\mathcal{U}(T_\\alpha)|$ and we win." ], "refs": [ "stacks-properties-lemma-reduced-closed-substack", "stacks-morphisms-lemma-monomorphism-cartesian-square-inertia", "stacks-morphisms-lemma-base-change-quasi-compact", "stacks-morphisms-proposition-open-stratum", "stacks-properties-lemma-space-locally-quasi-compact" ], "ref_ids": [ 8897, 7414, 7423, 7600, 8868 ] } ], "ref_ids": [] }, { "id": 7529, "type": "theorem", "label": "stacks-morphisms-lemma-spectral-qc-diagonal-qc", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-spectral-qc-diagonal-qc", "contents": [ "Let $\\mathcal{X}$ be a quasi-compact algebraic stack", "whose diagonal $\\Delta$ is quasi-compact.", "Then $|\\mathcal{X}|$ is a spectral topological space." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $U$ and a surjective smooth morphism", "$U \\to \\mathcal{X}$, see", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-quasi-compact-stack}.", "Then $|U| \\to |\\mathcal{X}|$ is continuous, open, and surjective, see", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-topology-points}.", "Hence the quasi-compact opens of $|\\mathcal{X}|$ form a basis", "for the topology. For $W_1, W_2 \\subset |\\mathcal{X}|$ quasi-compact open,", "we may choose a quasi-compact opens $V_1, V_2$ of $U$ mapping", "to $W_1$ and $W_2$. Since $\\Delta$ is quasi-compact,", "we see that", "$$", "V_1 \\times_\\mathcal{X} V_2 =", "(V_1 \\times V_2) \\times_{\\mathcal{X} \\times \\mathcal{X}, \\Delta} \\mathcal{X}", "$$", "is quasi-compact. Then image of $|V_1 \\times_\\mathcal{X} V_2|$ in", "$|\\mathcal{X}|$ is $W_1 \\cap W_2$ by", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-points-cartesian}.", "Thus $W_1 \\cap W_2$ is quasi-compact.", "To finish the proof, it suffices to show that $|\\mathcal{X}|$", "is sober, see Topology, Definition \\ref{topology-definition-spectral-space}.", "\\medskip\\noindent", "Let $T \\subset |\\mathcal{X}|$ be an irreducible closed subset.", "We have to show $T$ has a unique generic point.", "Let $\\mathcal{Z} \\subset \\mathcal{X}$ be the reduced induced", "closed substack corresponding to $T$, see", "Properties of Stacks, Definition", "\\ref{stacks-properties-definition-reduced-induced-stack}.", "Since $\\mathcal{Z} \\to \\mathcal{X}$ is a closed immersion,", "we see that $\\Delta_\\mathcal{Z}$ is quasi-compact:", "first show that $\\mathcal{Z} \\to \\mathcal{X} \\times \\mathcal{X}$", "is quasi-compact as the composition of $\\mathcal{Z} \\to \\mathcal{X}$", "with $\\Delta$, then write $\\mathcal{Z} \\to \\mathcal{X} \\times \\mathcal{X}$", "as the composition of $\\Delta_\\mathcal{Z}$ and", "$\\mathcal{Z} \\times \\mathcal{Z} \\to \\mathcal{X} \\times \\mathcal{X}$ and", "use Lemma \\ref{lemma-quasi-compact-permanence}", "and the fact that", "$\\mathcal{Z} \\times \\mathcal{Z} \\to \\mathcal{X} \\times \\mathcal{X}$", "is separated. Thus we reduce to the case discussed in the next", "paragraph.", "\\medskip\\noindent", "Assume $\\mathcal{X}$ is quasi-compact, $\\Delta$ is quasi-compact,", "$\\mathcal{X}$ is reduced, and $|\\mathcal{X}|$ irreducible.", "We have to show $|\\mathcal{X}|$ has a unique generic point.", "Since $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is a base change of $\\Delta$,", "we see that $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$", "is quasi-compact (Lemma \\ref{lemma-base-change-quasi-compact}).", "Thus there exists a dense open substack $\\mathcal{U} \\subset \\mathcal{X}$", "which is a gerbe by", "Proposition \\ref{proposition-open-stratum}.", "In other words, $|\\mathcal{U}| \\subset |\\mathcal{X}|$", "is open dense. Thus we may assume that $\\mathcal{X}$ is a gerbe.", "Say $\\mathcal{X} \\to X$ turns $\\mathcal{X}$ into a gerbe over the", "algebraic space $X$. Then $|\\mathcal{X}| \\cong |X|$ by", "Lemma \\ref{lemma-gerbe-bijection-points}.", "In particular, $X$ is quasi-compact.", "By Lemma \\ref{lemma-gerbe-diagonal-quasi-compact}", "we see that $X$ has quasi-compact diagonal,", "i.e., $X$ is a quasi-separated algebraic space.", "Then $|X|$ is spectral by", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-quasi-separated-spectral}", "which implies what we want is true." ], "refs": [ "stacks-properties-lemma-quasi-compact-stack", "stacks-properties-lemma-topology-points", "stacks-properties-lemma-points-cartesian", "topology-definition-spectral-space", "stacks-properties-definition-reduced-induced-stack", "stacks-morphisms-lemma-quasi-compact-permanence", "stacks-morphisms-lemma-base-change-quasi-compact", "stacks-morphisms-proposition-open-stratum", "stacks-morphisms-lemma-gerbe-bijection-points", "stacks-morphisms-lemma-gerbe-diagonal-quasi-compact", "spaces-properties-lemma-quasi-compact-quasi-separated-spectral" ], "ref_ids": [ 8873, 8867, 8864, 8370, 8922, 7427, 7423, 7600, 7525, 7526, 11853 ] } ], "ref_ids": [] }, { "id": 7530, "type": "theorem", "label": "stacks-morphisms-lemma-spectral-qcqs", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-spectral-qcqs", "contents": [ "Let $\\mathcal{X}$ be a quasi-compact and quasi-separated algebraic stack.", "Then $|\\mathcal{X}|$ is a spectral topological space." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-spectral-qc-diagonal-qc}." ], "refs": [ "stacks-morphisms-lemma-spectral-qc-diagonal-qc" ], "ref_ids": [ 7529 ] } ], "ref_ids": [] }, { "id": 7531, "type": "theorem", "label": "stacks-morphisms-lemma-sober-qs", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-sober-qs", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack whose diagonal is quasi-compact", "(for example if $\\mathcal{X}$ is quasi-separated).", "Then there is an open covering $|\\mathcal{X}| = \\bigcup U_i$", "with $U_i$ spectral. In particular $|\\mathcal{X}|$ is", "a sober topological space." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of Lemma \\ref{lemma-spectral-qc-diagonal-qc}." ], "refs": [ "stacks-morphisms-lemma-spectral-qc-diagonal-qc" ], "ref_ids": [ 7529 ] } ], "ref_ids": [] }, { "id": 7532, "type": "theorem", "label": "stacks-morphisms-lemma-every-point-residual-gerbe", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-every-point-residual-gerbe", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack such that", "$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is quasi-compact.", "Then the residual gerbe of $\\mathcal{X}$ at $x$ exists for", "every $x \\in |\\mathcal{X}|$." ], "refs": [], "proofs": [ { "contents": [ "Let $T = \\overline{\\{x\\}} \\subset |\\mathcal{X}|$ be the closure of $x$.", "By", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-reduced-closed-substack}", "there exists a reduced closed substack $\\mathcal{X}' \\subset \\mathcal{X}$", "such that $T = |\\mathcal{X}'|$. Note that", "$\\mathcal{I}_{\\mathcal{X}'} =", "\\mathcal{I}_\\mathcal{X} \\times_\\mathcal{X} \\mathcal{X}'$ by", "Lemma \\ref{lemma-monomorphism-cartesian-square-inertia}.", "Hence $\\mathcal{I}_{\\mathcal{X}'} \\to \\mathcal{X}'$ is", "quasi-compact as a base change, see", "Lemma \\ref{lemma-base-change-quasi-compact}.", "Therefore", "Proposition \\ref{proposition-open-stratum}", "implies there exists a dense open substack", "$\\mathcal{U} \\subset \\mathcal{X}'$", "which is a gerbe. Note that $x \\in |\\mathcal{U}|$ because $\\{x\\} \\subset T$", "is a dense subset too. Hence a residual gerbe", "$\\mathcal{Z}_x \\subset \\mathcal{U}$ of $\\mathcal{U}$ at $x$ exists by", "Lemma \\ref{lemma-gerbe-residual-gerbe-exists}.", "It is immediate from the definitions that $\\mathcal{Z}_x \\to \\mathcal{X}$", "is a residual gerbe of $\\mathcal{X}$ at $x$." ], "refs": [ "stacks-properties-lemma-reduced-closed-substack", "stacks-morphisms-lemma-monomorphism-cartesian-square-inertia", "stacks-morphisms-lemma-base-change-quasi-compact", "stacks-morphisms-proposition-open-stratum", "stacks-morphisms-lemma-gerbe-residual-gerbe-exists" ], "ref_ids": [ 8897, 7414, 7423, 7600, 7527 ] } ], "ref_ids": [] }, { "id": 7533, "type": "theorem", "label": "stacks-morphisms-lemma-every-point-residual-gerbe-quasi-DM", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-every-point-residual-gerbe-quasi-DM", "contents": [ "Let $\\mathcal{X}$ be a quasi-DM algebraic stack.", "Then the residual gerbe of $\\mathcal{X}$ at $x$ exists for", "every $x \\in |\\mathcal{X}|$." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective, flat, locally finite presented,", "and locally quasi-finite morphism $U \\to \\mathcal{X}$, see", "Theorem \\ref{theorem-quasi-DM}.", "Set $R = U \\times_\\mathcal{X} U$. The projections $s, t : R \\to U$", "are surjective, flat, locally of finite presentation, and", "locally quasi-finite as base changes of the morphism $U \\to \\mathcal{X}$.", "There is a canonical morphism $[U/R] \\to \\mathcal{X}$ (see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack})", "which is an equivalence because $U \\to \\mathcal{X}$ is surjective, flat,", "and locally of finite presentation, see", "Algebraic Stacks, Remark \\ref{algebraic-remark-flat-fp-presentation}.", "Thus we may assume that $\\mathcal{X} = [U/R]$ where", "$(U, R, s, t, c)$ is a groupoid in algebraic spaces such that", "$s, t : R \\to U$ are surjective, flat, locally of finite presentation,", "and locally quasi-finite. Set", "$$", "U' = \\coprod\\nolimits_{u \\in U\\text{ lying over }x} \\Spec(\\kappa(u)).", "$$", "The canonical morphism $U' \\to U$ is a monomorphism. Let", "$$", "R' = U' \\times_\\mathcal{X} U' =", "R \\times_{(U \\times U)} (U' \\times U')", "$$", "Because $U' \\to U$ is a monomorphism we see that both projections", "$s', t' : R' \\to U'$ factor as a monomorphism followed by a locally", "quasi-finite morphism. Hence, as $U'$ is a disjoint union of spectra", "of fields, using", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-mono-towards-locally-quasi-finite-over-field}", "we conclude that the morphisms $s', t' : R' \\to U'$ are locally quasi-finite.", "Again since $U'$ is a disjoint union of spectra of fields, the morphisms", "$s', t'$ are also flat. Finally, $s', t'$ locally quasi-finite", "implies $s', t'$ locally of finite type, hence $s', t'$ locally of finite", "presentation (because $U'$ is a disjoint union of spectra of fields", "in particular locally Noetherian, so that", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation}", "applies). Hence $\\mathcal{Z} = [U'/R']$ is an algebraic stack by", "Criteria for Representability, Theorem", "\\ref{criteria-theorem-flat-groupoid-gives-algebraic-stack}.", "As $R'$ is the restriction of $R$ by $U' \\to U$ we see", "$\\mathcal{Z} \\to \\mathcal{X}$ is a monomorphism by", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-restrict}", "and", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-monomorphism}.", "Since $\\mathcal{Z} \\to \\mathcal{X}$ is a monomorphism we see that", "$|\\mathcal{Z}| \\to |\\mathcal{X}|$ is injective, see", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-monomorphism-injective-points}.", "By", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-points-cartesian}", "we see that", "$$", "|U'| = |\\mathcal{Z} \\times_\\mathcal{X} U'|", "\\longrightarrow", "|\\mathcal{Z}| \\times_{|\\mathcal{X}|} |U'|", "$$", "is surjective which implies (by our choice of $U'$) that", "$|\\mathcal{Z}| \\to |\\mathcal{X}|$ has image $\\{x\\}$.", "We conclude that $|\\mathcal{Z}|$ is a singleton.", "Finally, by construction $U'$ is locally Noetherian and reduced, i.e.,", "$\\mathcal{Z}$ is reduced and locally Noetherian. This means that", "the essential image of $\\mathcal{Z} \\to \\mathcal{X}$", "is the residual gerbe of $\\mathcal{X}$ at $x$, see", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-residual-gerbe-unique}." ], "refs": [ "stacks-morphisms-theorem-quasi-DM", "algebraic-lemma-map-space-into-stack", "algebraic-remark-flat-fp-presentation", "spaces-over-fields-lemma-mono-towards-locally-quasi-finite-over-field", "spaces-morphisms-lemma-noetherian-finite-type-finite-presentation", "criteria-theorem-flat-groupoid-gives-algebraic-stack", "spaces-groupoids-lemma-quotient-stack-restrict", "stacks-properties-lemma-monomorphism", "stacks-properties-lemma-monomorphism-injective-points", "stacks-properties-lemma-points-cartesian", "stacks-properties-lemma-residual-gerbe-unique" ], "ref_ids": [ 7388, 8473, 8491, 12853, 4844, 3094, 9329, 8879, 8880, 8864, 8908 ] } ], "ref_ids": [] }, { "id": 7534, "type": "theorem", "label": "stacks-morphisms-lemma-quotient-etale", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-quotient-etale", "contents": [ "Let $Y$ be an algebraic space.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $Y$.", "Assume $U \\to Y$ is flat and locally of finite presentation", "and $R \\to U \\times_Y U$ an open immersion.", "Then $X = [U/R] = U/R$ is an algebraic space and $X \\to Y$", "is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "The quotient stack $[U/R]$ is an algebraic stacks by", "Criteria for Representability, Theorem", "\\ref{criteria-theorem-flat-groupoid-gives-algebraic-stack}.", "On the other hand, since $R \\to U \\times U$ is a monomorphism,", "it is an algebraic space (by our abuse of language and", "Algebraic Stacks, Proposition", "\\ref{algebraic-proposition-algebraic-stack-no-automorphisms})", "and of course it is equal to the algebraic space $U/R$", "(of Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}).", "Since $U \\to X$ is surjective, flat, and locally of finite presenation", "(Bootstrap, Lemma \\ref{bootstrap-lemma-covering-quotient})", "we conclude that $X \\to Y$ is flat and locally of finite presentation by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-permanence}", "and", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-flat-finitely-presented-permanence}.", "Finally, consider the cartesian diagram", "$$", "\\xymatrix{", "R \\ar[d] \\ar[r] & U \\times_Y U \\ar[d] \\\\", "X \\ar[r] & X \\times_Y X", "}", "$$", "Since the right vertical arrow is surjective, flat, and", "locally of finite presentation (small detail omitted), we", "find that $X \\to X \\times_Y X$ is an open immersion as the top horizonal arrow", "has this property by assumption (use", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}).", "Thus $X \\to Y$ is unramified by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-diagonal-unramified-morphism}.", "We conclude by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-unramified-flat-lfp-etale}." ], "refs": [ "criteria-theorem-flat-groupoid-gives-algebraic-stack", "algebraic-proposition-algebraic-stack-no-automorphisms", "bootstrap-theorem-final-bootstrap", "bootstrap-lemma-covering-quotient", "spaces-morphisms-lemma-flat-permanence", "spaces-descent-lemma-flat-finitely-presented-permanence", "stacks-properties-lemma-check-property-covering", "spaces-morphisms-lemma-diagonal-unramified-morphism", "spaces-morphisms-lemma-unramified-flat-lfp-etale" ], "ref_ids": [ 3094, 8480, 2602, 2630, 4865, 9369, 8859, 4902, 4915 ] } ], "ref_ids": [] }, { "id": 7535, "type": "theorem", "label": "stacks-morphisms-lemma-quasi-splitting-etale", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-quasi-splitting-etale", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.", "Assume $s, t$ are flat and locally of finite presentation.", "Let $P \\subset R$ be an open subspace such that", "$(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t} P})$ is a", "groupoid in algebraic spaces over $S$. Then", "$$", "[U/P] \\longrightarrow [U/R]", "$$", "is a morphism of algebraic stacks which", "is representable by algebraic spaces, surjective, and \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Since $P \\subset R$ is open, we see that $s|_P$ and $t|_P$", "are flat and locally of finite presentation.", "Thus $[U/R]$ and $[U/P]$ are algebraic stacks by", "Criteria for Representability, Theorem", "\\ref{criteria-theorem-flat-groupoid-gives-algebraic-stack}.", "To see that the morphism is representable by algebraic spaces,", "it suffices to show that $[U/P] \\to [U/R]$ is faithful on", "fibre categories, see", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}.", "This follows immediately from the fact that $P \\to R$ is a monomorphism", "and the explicit description of quotient stacks given in", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-objects}.", "Having said this, we know what it means for", "$[U/P] \\to [U/R]$ to be surjective and \\'etale by", "Algebraic Stacks, Definition", "\\ref{algebraic-definition-relative-representable-property}.", "Surjectivity follows for example from", "Criteria for Representability,", "Lemma \\ref{criteria-lemma-representable-by-spaces-surjective}", "and the description of objects of quotient stacks", "(see lemma cited above) over spectra of fields.", "It remains to prove that our morphism is \\'etale.", "\\medskip\\noindent", "To do this it suffices to show that $U \\times_{[U/R]} [U/P] \\to U$", "is \\'etale, see Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}.", "By Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-cartesian-square-of-morphism}", "the fibre product is equal to $[R/P \\times_{s, U, t} R]$", "with morphism to $U$ induced by $s : R \\to U$.", "The maps $s', t' : P \\times_{s, U, t} R \\to R$ are given by", "$s' : (p, r) \\mapsto r$ and $t' : (p, r) \\mapsto c(p, r)$. Since", "$P \\subset R$ is open we conclude that", "$(t', s') : P \\times_{s, U, t} R \\to R \\times_{s, U, s} R$", "is an open immersion.", "Thus we may apply Lemma \\ref{lemma-quotient-etale}", "to conclude." ], "refs": [ "criteria-theorem-flat-groupoid-gives-algebraic-stack", "algebraic-lemma-characterize-representable-by-algebraic-spaces", "spaces-groupoids-lemma-quotient-stack-objects", "algebraic-definition-relative-representable-property", "criteria-lemma-representable-by-spaces-surjective", "stacks-properties-lemma-check-property-covering", "spaces-groupoids-lemma-cartesian-square-of-morphism", "stacks-morphisms-lemma-quotient-etale" ], "ref_ids": [ 3094, 8469, 9328, 8483, 3110, 8859, 9322, 7534 ] } ], "ref_ids": [] }, { "id": 7536, "type": "theorem", "label": "stacks-morphisms-lemma-etale-local-quasi-DM", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-etale-local-quasi-DM", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Assume $\\mathcal{X}$ is", "quasi-DM with separated diagonal (equivalently", "$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally quasi-finite and", "separated). Let $x \\in |\\mathcal{X}|$. Then there exists a", "morphism of algebraic stacks", "$$", "\\mathcal{U} \\longrightarrow \\mathcal{X}", "$$", "with the following properties", "\\begin{enumerate}", "\\item there exists a point $u \\in |\\mathcal{U}|$ mapping to $x$,", "\\item $\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces and", "\\'etale,", "\\item $\\mathcal{U} = [U/R]$ where $(U, R, s, t, c)$ is a groupoid", "scheme with $U$, $R$ affine, and $s, t$ finite, flat, and", "locally of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "(The parenthetical statement follows from the equivalences in", "Lemma \\ref{lemma-diagonal-diagonal}).", "Choose an affine scheme $U$ and a flat, locally finitely presented,", "locally quasi-finite morphism $U \\to \\mathcal{X}$ such that $x$", "is the image of some point $u \\in U$. This is possible by", "Theorem \\ref{theorem-quasi-DM} and the assumption that $\\mathcal{X}$", "is quasi-DM. Let $(U, R, s, t, c)$ be the groupoid in algebraic spaces", "we obtain by setting $R = U \\times_\\mathcal{X} U$, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack}.", "Let $\\mathcal{X}' \\subset \\mathcal{X}$ be the open substack corresponding", "to the open image of $|U| \\to |\\mathcal{X}|$", "(Properties of Stacks, Lemmas", "\\ref{stacks-properties-lemma-topology-points} and", "\\ref{stacks-properties-lemma-open-substacks}).", "Clearly, we may replace $\\mathcal{X}$ by the open substack $\\mathcal{X}'$.", "Thus we may assume $U \\to \\mathcal{X}$ is surjective and then", "Algebraic Stacks, Remark \\ref{algebraic-remark-flat-fp-presentation}", "gives $\\mathcal{X} = [U/R]$.", "Observe that $s, t : R \\to U$ are flat, locally of finite presentation,", "and locally quasi-finite.", "Since $R = U \\times U \\times_{(\\mathcal{X} \\times \\mathcal{X})} \\mathcal{X}$", "and since the diagonal of $\\mathcal{X}$ is separated, we find that", "$R$ is separated. Hence $s, t : R \\to U$ are separated. It follows", "that $R$ is a scheme by", "Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}", "applied to $s : R \\to U$.", "\\medskip\\noindent", "Above we have verified all the assumptions of", "More on Groupoids in Spaces, Lemma", "\\ref{spaces-more-groupoids-lemma-quasi-splitting-affine-scheme}", "are satisfied for $(U, R, s, t, c)$ and $u$.", "Hence we can find an elementary \\'etale neighbourhood", "$(U', u') \\to (U, u)$ such that the restriction $R'$ of $R$ to $U'$", "is quasi-split over $u$. Note that $R' = U' \\times_\\mathcal{X} U'$", "(small detail omitted; hint: transitivity of fibre products).", "Replacing $(U, R, s, t, c)$ by $(U', R', s', t', c')$ and shrinking", "$\\mathcal{X}$ as above, we may assume that $(U, R, s, t, c)$ has", "a quasi-splitting over $u$ (the point $u$ is irrelevant from", "now on as can be seen from the footnote in", "More on Groupoids in Spaces, Definition", "\\ref{spaces-more-groupoids-definition-split-at-point}).", "Let $P \\subset R$ be a quasi-splitting of $R$ over $u$.", "Apply Lemma \\ref{lemma-quasi-splitting-etale}", "to see that", "$$", "\\mathcal{U} = [U/P] \\longrightarrow [U/R] = \\mathcal{X}", "$$", "has all the desired properties." ], "refs": [ "stacks-morphisms-lemma-diagonal-diagonal", "stacks-morphisms-theorem-quasi-DM", "algebraic-lemma-map-space-into-stack", "stacks-properties-lemma-topology-points", "stacks-properties-lemma-open-substacks", "algebraic-remark-flat-fp-presentation", "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "spaces-more-groupoids-lemma-quasi-splitting-affine-scheme", "spaces-more-groupoids-definition-split-at-point", "stacks-morphisms-lemma-quasi-splitting-etale" ], "ref_ids": [ 7416, 7388, 8473, 8867, 8890, 8491, 4983, 13213, 13216, 7535 ] } ], "ref_ids": [] }, { "id": 7537, "type": "theorem", "label": "stacks-morphisms-lemma-etale-local-quasi-DM-at-x", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-etale-local-quasi-DM-at-x", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Assume $\\mathcal{X}$ is", "quasi-DM with separated diagonal (equivalently", "$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally quasi-finite and", "separated). Let $x \\in |\\mathcal{X}|$. Assume the", "automorphism group of $\\mathcal{X}$ at $x$ is finite", "(Remark \\ref{remark-property-automorphism-groups}).", "Then there exists a morphism of algebraic stacks", "$$", "g : \\mathcal{U} \\longrightarrow \\mathcal{X}", "$$", "with the following properties", "\\begin{enumerate}", "\\item there exists a point $u \\in |\\mathcal{U}|$ mapping to $x$ and", "$g$ induces an isomorphism between automorphism groups at $u$ and $x$", "(Remark \\ref{remark-identify-automorphism-groups}),", "\\item $\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces and", "\\'etale,", "\\item $\\mathcal{U} = [U/R]$ where $(U, R, s, t, c)$ is a groupoid", "scheme with $U$, $R$ affine, and $s, t$ finite, flat, and", "locally of finite presentation.", "\\end{enumerate}" ], "refs": [ "stacks-morphisms-remark-property-automorphism-groups", "stacks-morphisms-remark-identify-automorphism-groups" ], "proofs": [ { "contents": [ "Observe that $G_x$ is a group scheme by", "Lemma \\ref{lemma-automorphism-group-scheme}.", "The first part of the proof is {\\bf exactly} the same as the first part", "of the proof of Lemma \\ref{lemma-etale-local-quasi-DM}.", "Thus we may assume $\\mathcal{X} = [U/R]$ where $(U, R, s, t, c)$", "and $u \\in U$ mapping to $x$ satisfy all the assumptions of", "More on Groupoids in Spaces, Lemma", "\\ref{spaces-more-groupoids-lemma-quasi-splitting-affine-scheme}.", "Our assumption on $G_x$ implies that $G_u$ is finite over $u$.", "Hence all the assumptions of", "More on Groupoids in Spaces, Lemma", "\\ref{spaces-more-groupoids-lemma-splitting-affine-scheme}", "are satisfied.", "Hence we can find an elementary \\'etale neighbourhood", "$(U', u') \\to (U, u)$ such that the restriction $R'$ of $R$ to $U'$", "is split over $u$. Note that $R' = U' \\times_\\mathcal{X} U'$", "(small detail omitted; hint: transitivity of fibre products).", "Replacing $(U, R, s, t, c)$ by $(U', R', s', t', c')$ and shrinking", "$\\mathcal{X}$ as above, we may assume that $(U, R, s, t, c)$ has", "a splitting over $u$. Let $P \\subset R$ be a splitting of $R$ over $u$.", "Apply Lemma \\ref{lemma-quasi-splitting-etale} to see that", "$$", "\\mathcal{U} = [U/P] \\longrightarrow [U/R] = \\mathcal{X}", "$$", "is representable by algebraic spaces and \\'etale. By construction", "$G_u$ is contained in $P$, hence this morphism defines an isomorphism", "on automorphism groups at $u$ as desired." ], "refs": [ "stacks-morphisms-lemma-automorphism-group-scheme", "stacks-morphisms-lemma-etale-local-quasi-DM", "spaces-more-groupoids-lemma-quasi-splitting-affine-scheme", "spaces-more-groupoids-lemma-splitting-affine-scheme", "stacks-morphisms-lemma-quasi-splitting-etale" ], "ref_ids": [ 7472, 7536, 13213, 13212, 7535 ] } ], "ref_ids": [ 7636, 7637 ] }, { "id": 7538, "type": "theorem", "label": "stacks-morphisms-lemma-etale-local-quasi-DM-at-x-inertia", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-etale-local-quasi-DM-at-x-inertia", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Assume $\\mathcal{X}$ is", "quasi-DM with separated diagonal (equivalently", "$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally quasi-finite and", "separated). Let $x \\in |\\mathcal{X}|$. Assume $x$ can be represented", "by a quasi-compact morphism $\\Spec(k) \\to \\mathcal{X}$.", "Then there exists a morphism of algebraic stacks", "$$", "g : \\mathcal{U} \\longrightarrow \\mathcal{X}", "$$", "with the following properties", "\\begin{enumerate}", "\\item there exists a point $u \\in |\\mathcal{U}|$ mapping to $x$ and", "$g$ induces an isomorphism between the residual gerbes at $u$ and $x$,", "\\item $\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces and", "\\'etale,", "\\item $\\mathcal{U} = [U/R]$ where $(U, R, s, t, c)$ is a groupoid", "scheme with $U$, $R$ affine, and $s, t$ finite, flat, and", "locally of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first part of the proof is {\\bf exactly} the same as the first part", "of the proof of Lemma \\ref{lemma-etale-local-quasi-DM}.", "Thus we may assume $\\mathcal{X} = [U/R]$ where $(U, R, s, t, c)$", "and $u \\in U$ mapping to $x$ satisfy all the assumptions of", "More on Groupoids in Spaces, Lemma", "\\ref{spaces-more-groupoids-lemma-quasi-splitting-affine-scheme}.", "Observe that $u = \\Spec(\\kappa(u)) \\to \\mathcal{X}$ is quasi-compact, see", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-UR-quasi-compact-above-x}.", "Consider the cartesian diagram", "$$", "\\xymatrix{", "F \\ar[d] \\ar[r] & U \\ar[d] \\\\", "u \\ar[r]^u & \\mathcal{X}", "}", "$$", "Since $U$ is an affine scheme and $F \\to U$ is quasi-compact,", "we see that $F$ is quasi-compact. Since $U \\to \\mathcal{X}$", "is locally quasi-finite, we see that $F \\to u$ is", "locally quasi-finite. Hence $F \\to u$ is quasi-finite", "and $F$ is an affine scheme whose underlying topological", "space is finite discrete (Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-locally-quasi-finite-over-field}).", "Observe that we have a monomorphism $u \\times_\\mathcal{X} u \\to F$.", "In particular the set $\\{r \\in R : s(r) = u, t(r) = u\\}$", "which is the image of $|u \\times_\\mathcal{X} u| \\to |R|$", "is finite. we conclude that all the assumptions of", "More on Groupoids in Spaces, Lemma", "\\ref{spaces-more-groupoids-lemma-strong-splitting-affine-scheme}", "hold.", "\\medskip\\noindent", "Thus we can find an elementary \\'etale neighbourhood", "$(U', u') \\to (U, u)$ such that the restriction $R'$ of $R$ to $U'$", "is strongly split over $u'$. Note that $R' = U' \\times_\\mathcal{X} U'$", "(small detail omitted; hint: transitivity of fibre products).", "Replacing $(U, R, s, t, c)$ by $(U', R', s', t', c')$ and shrinking", "$\\mathcal{X}$ as above, we may assume that $(U, R, s, t, c)$ has", "a strong splitting over $u$. Let $P \\subset R$ be a strong splitting", "of $R$ over $u$. Apply Lemma \\ref{lemma-quasi-splitting-etale} to see that", "$$", "\\mathcal{U} = [U/P] \\longrightarrow [U/R] = \\mathcal{X}", "$$", "is representable by algebraic spaces and \\'etale. Since $P \\subset R$", "is open and contains $\\{r \\in R : s(r) = u, t(r) = u\\}$ by construction", "we see that", "$u \\times_\\mathcal{U} u \\to u \\times_\\mathcal{X} u$ is an isomorphism.", "The statement on residual gerbes then follows from", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-residual-gerbe-isomorphic}", "(we observe that the residual gerbes in question exist by", "Lemma \\ref{lemma-every-point-residual-gerbe-quasi-DM})." ], "refs": [ "stacks-morphisms-lemma-etale-local-quasi-DM", "spaces-more-groupoids-lemma-quasi-splitting-affine-scheme", "stacks-properties-lemma-UR-quasi-compact-above-x", "spaces-over-fields-lemma-locally-quasi-finite-over-field", "spaces-more-groupoids-lemma-strong-splitting-affine-scheme", "stacks-morphisms-lemma-quasi-splitting-etale", "stacks-properties-lemma-residual-gerbe-isomorphic", "stacks-morphisms-lemma-every-point-residual-gerbe-quasi-DM" ], "ref_ids": [ 7536, 13213, 8912, 12852, 13211, 7535, 8910, 7533 ] } ], "ref_ids": [] }, { "id": 7539, "type": "theorem", "label": "stacks-morphisms-lemma-composition-smooth", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-smooth", "contents": [ "The composition of smooth morphisms is smooth." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Remark \\ref{remark-composition}", "with", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-composition-smooth}." ], "refs": [ "stacks-morphisms-remark-composition", "spaces-morphisms-lemma-composition-smooth" ], "ref_ids": [ 7633, 4886 ] } ], "ref_ids": [] }, { "id": 7540, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-smooth", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-smooth", "contents": [ "A base change of a smooth morphism is smooth." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Remark \\ref{remark-base-change}", "with", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-smooth}." ], "refs": [ "stacks-morphisms-remark-base-change", "spaces-morphisms-lemma-base-change-smooth" ], "ref_ids": [ 7634, 4887 ] } ], "ref_ids": [] }, { "id": 7541, "type": "theorem", "label": "stacks-morphisms-lemma-descent-smooth", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-descent-smooth", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective, flat, locally finitely", "presented morphism of algebraic stacks. If the base change", "$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$", "is smooth, then $f$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "The property ``smooth''", "satisfies the conditions of Lemma \\ref{lemma-descent-property}.", "Smooth local on the source-and-target we have seen in the", "introduction to this section and fppf local on the target is", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-smooth}." ], "refs": [ "stacks-morphisms-lemma-descent-property", "spaces-descent-lemma-descending-property-smooth" ], "ref_ids": [ 7508, 9406 ] } ], "ref_ids": [] }, { "id": 7542, "type": "theorem", "label": "stacks-morphisms-lemma-smooth-locally-finite-presentation", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-smooth-locally-finite-presentation", "contents": [ "A smooth morphism of algebraic stacks is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7543, "type": "theorem", "label": "stacks-morphisms-lemma-where-smooth", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-where-smooth", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "There is a maximal open substack $\\mathcal{U} \\subset \\mathcal{X}$", "such that $f|_\\mathcal{U} : \\mathcal{U} \\to \\mathcal{Y}$ is smooth.", "Moreover, formation of this open commutes with", "\\begin{enumerate}", "\\item precomposing by smooth morphisms,", "\\item base change by morphisms which are flat and locally of", "finite presentation,", "\\item base change by flat morphisms provided $f$ is locally of", "finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "\\mathcal{X} \\ar[r]^f & \\mathcal{Y}", "}", "$$", "where $U$ and $V$ are algebraic spaces, the vertical arrows are smooth,", "and $a : U \\to \\mathcal{X}$ surjective. There is a maximal open subspace", "$U' \\subset U$ such that $h_{U'} : U' \\to V$ is smooth, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-where-smooth}.", "Let $\\mathcal{U} \\subset \\mathcal{X}$ be the open substack", "corresponding to the image of $|U'| \\to |\\mathcal{X}|$", "(Properties of Stacks, Lemmas \\ref{stacks-properties-lemma-topology-points} and", "\\ref{stacks-properties-lemma-open-substacks}).", "By the equivalence in Lemma \\ref{lemma-local-source-target}", "we find that $f|_\\mathcal{U} : \\mathcal{U} \\to \\mathcal{Y}$ is smooth", "and that $\\mathcal{U}$ is the largest open substack with this", "property.", "\\medskip\\noindent", "Assertion (1) follows from the fact that being smooth", "is smooth local on the source (this property was used to even define", "smooth morphisms of algebraic stacks).", "Assertions (2) and (3) follow from the case of algebraic spaces, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-where-smooth}." ], "refs": [ "spaces-morphisms-lemma-where-smooth", "stacks-properties-lemma-topology-points", "stacks-properties-lemma-open-substacks", "stacks-morphisms-lemma-local-source-target", "spaces-morphisms-lemma-where-smooth" ], "ref_ids": [ 4893, 8867, 8890, 7458, 4893 ] } ], "ref_ids": [] }, { "id": 7544, "type": "theorem", "label": "stacks-morphisms-lemma-smooth-quotient-stack", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-smooth-quotient-stack", "contents": [ "Let $X \\to Y$ be a smooth morphism of algebraic spaces.", "Let $G$ be a group algebraic space over $Y$ which is flat", "and locally of finite presentation over $Y$. Let $G$ act on $X$ over $Y$.", "Then the quotient stack $[X/G]$ is smooth over $Y$." ], "refs": [], "proofs": [ { "contents": [ "The quotient $[X/G]$ is an algebraic stack by", "Criteria for Representability, Theorem", "\\ref{criteria-theorem-flat-groupoid-gives-algebraic-stack}.", "The smoothness of $[X/G]$ over $Y$ follows from the fact that smoothness", "descends under fppf coverings:", "Choose a surjective smooth morphism $U \\to [X/G]$ where $U$ is a scheme.", "Smoothness of $[X/G]$ over $Y$ is equivalent to smoothness of $U$ over $Y$.", "Observe that $U \\times_{[X/G]} X$ is smooth over $X$ and hence smooth", "over $Y$ (because compositions of smooth morphisms are smooth).", "On the other hand, $U \\times_{[X/G]} X \\to U$ is locally of", "finite presentation, flat, and surjective (because it is", "the base change of $X \\to [X/G]$ which has those properties", "for example by Criteria for Representability, Lemma", "\\ref{criteria-lemma-flat-quotient-flat-presentation}).", "Therefore we may apply Descent on Spaces,", "Lemma \\ref{spaces-descent-lemma-smooth-permanence}." ], "refs": [ "criteria-theorem-flat-groupoid-gives-algebraic-stack", "criteria-lemma-flat-quotient-flat-presentation", "spaces-descent-lemma-smooth-permanence" ], "ref_ids": [ 3094, 3137, 9371 ] } ], "ref_ids": [] }, { "id": 7545, "type": "theorem", "label": "stacks-morphisms-lemma-gerbe-smooth", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-gerbe-smooth", "contents": [ "Let $\\pi : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "If $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, then $\\pi$ is surjective", "and smooth." ], "refs": [], "proofs": [ { "contents": [ "We have seen surjectivity in Lemma \\ref{lemma-gerbe-fppf}.", "By Lemma \\ref{lemma-descent-smooth}", "it suffices to prove to the lemma after replacing $\\pi$ by a base change", "with a surjective, flat, locally finitely presented morphism", "$\\mathcal{Y}' \\to \\mathcal{Y}$. By", "Lemma \\ref{lemma-local-structure-gerbe}", "we may assume $\\mathcal{Y} = U$ is an algebraic space and", "$\\mathcal{X} = [U/G]$ over $U$ with $G \\to U$ flat and", "locally of finite presentation.", "Then we win by Lemma \\ref{lemma-smooth-quotient-stack}." ], "refs": [ "stacks-morphisms-lemma-gerbe-fppf", "stacks-morphisms-lemma-descent-smooth", "stacks-morphisms-lemma-local-structure-gerbe", "stacks-morphisms-lemma-smooth-quotient-stack" ], "ref_ids": [ 7522, 7541, 7521, 7544 ] } ], "ref_ids": [] }, { "id": 7546, "type": "theorem", "label": "stacks-morphisms-lemma-etale-smooth-local-source-target", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-etale-smooth-local-source-target", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces", "which is \\'etale-smooth local on the source-and-target.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a DM morphism of algebraic stacks.", "Consider commutative diagrams", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "\\mathcal{X} \\ar[r]^f & \\mathcal{Y}", "}", "$$", "where $U$ and $V$ are algebraic spaces, $V \\to \\mathcal{Y}$ is smooth,", "and $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ is \\'etale.", "The following are equivalent", "\\begin{enumerate}", "\\item for any diagram as above the morphism $h$ has property $\\mathcal{P}$, and", "\\item for some diagram as above with $a : U \\to \\mathcal{X}$ surjective", "the morphism $h$ has property $\\mathcal{P}$.", "\\end{enumerate}", "If $\\mathcal{X}$ and $\\mathcal{Y}$ are representable by algebraic spaces,", "then this is also equivalent to $f$ (as a morphism of algebraic spaces)", "having property $\\mathcal{P}$. If $\\mathcal{P}$ is also preserved under", "any base change, and fppf local on the base, then for morphisms $f$", "which are representable by algebraic spaces this", "is also equivalent to $f$ having property $\\mathcal{P}$ in the sense", "of", "Properties of Stacks,", "Section \\ref{stacks-properties-section-properties-morphisms}." ], "refs": [], "proofs": [ { "contents": [ "Let us prove the implication (1) $\\Rightarrow$ (2). Pick an algebraic", "space $V$ and a surjective and smooth morphism $V \\to \\mathcal{Y}$.", "As $f$ is DM there exists a scheme $U$ and a surjective \\'etale morphism", "$U \\to V \\times_\\mathcal{Y} \\mathcal{X}$, see", "Lemma \\ref{lemma-DM}. Thus we see that (2) holds.", "Note that $U \\to \\mathcal{X}$ is surjective and smooth as well, as a", "composition of the base change", "$\\mathcal{X} \\times_\\mathcal{Y} V \\to \\mathcal{X}$ and the chosen", "map $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$. Hence we obtain a", "diagram as in (1). Thus if (1) holds, then $h : U \\to V$ has property", "$\\mathcal{P}$, which means that (2) holds as $U \\to \\mathcal{X}$ is surjective.", "\\medskip\\noindent", "Conversely, assume (2) holds and let $U, V, a, b, h$ be as in (2).", "Next, let $U', V', a', b', h'$ be any diagram as in (1).", "Picture", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_h & V \\ar[d] \\\\", "\\mathcal{X} \\ar[r]^f & \\mathcal{Y}", "}", "\\quad\\quad", "\\xymatrix{", "U' \\ar[d] \\ar[r]_{h'} & V' \\ar[d] \\\\", "\\mathcal{X} \\ar[r]^f & \\mathcal{Y}", "}", "$$", "To show that (2) implies (1) we have to prove that $h'$ has $\\mathcal{P}$.", "To do this consider the commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]^h &", "U \\times_\\mathcal{X} U' \\ar[l] \\ar[d]^{(h, h')} \\ar[r] &", "U' \\ar[d]^{h'} \\\\", "V &", "V \\times_\\mathcal{Y} V' \\ar[l] \\ar[r] &", "V'", "}", "$$", "of algebraic spaces. Note that the horizontal arrows are", "smooth as base changes of the smooth morphisms", "$V \\to \\mathcal{Y}$, $V' \\to \\mathcal{Y}$, $U \\to \\mathcal{X}$, and", "$U' \\to \\mathcal{X}$. Note that the squares", "$$", "\\xymatrix{", "U \\ar[d] & U \\times_\\mathcal{X} U' \\ar[l] \\ar[d] &", "U \\times_\\mathcal{X} U' \\ar[d] \\ar[r] & U' \\ar[d] \\\\", "V \\times_\\mathcal{Y} \\mathcal{X} &", "V \\times_\\mathcal{Y} U' \\ar[l] &", "U \\times_\\mathcal{Y} V' \\ar[r] &", "\\mathcal{X} \\times_\\mathcal{Y} V'", "}", "$$", "are cartesian, hence the vertical arrows are \\'etale by our assumptions on", "$U', V', a', b', h'$ and $U, V, a, b, h$.", "Since $\\mathcal{P}$ is smooth local on the target by", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-etale-smooth-local-source-target-implies} part (2)", "we see that the base change", "$t : U \\times_\\mathcal{Y} V' \\to V \\times_\\mathcal{Y} V'$ of $h$", "has $\\mathcal{P}$. Since $\\mathcal{P}$ is \\'etale local on the source by", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-etale-smooth-local-source-target-implies} part (1)", "and $s : U \\times_\\mathcal{X} U' \\to U \\times_\\mathcal{Y} V'$ is \\'etale,", "we see the morphism $(h, h') = t \\circ s$ has $\\mathcal{P}$.", "Consider the diagram", "$$", "\\xymatrix{", "U \\times_\\mathcal{X} U' \\ar[r]_{(h, h')} \\ar[d] &", "V \\times_\\mathcal{Y} V' \\ar[d] \\\\", "U' \\ar[r]^{h'} & V'", "}", "$$", "The left vertical arrow is surjective, the right vertical arrow is smooth, and", "the induced morphism", "$$", "U \\times_\\mathcal{X} U'", "\\longrightarrow", "U' \\times_{V'} (V \\times_\\mathcal{Y} V') = V \\times_\\mathcal{Y} U'", "$$", "is \\'etale as seen above. Hence by", "Descent on Spaces, Definition", "\\ref{spaces-descent-definition-etale-smooth-local-source-target} part (3)", "we conclude that $h'$ has $\\mathcal{P}$. This finishes the proof of the", "equivalence of (1) and (2).", "\\medskip\\noindent", "If $\\mathcal{X}$ and $\\mathcal{Y}$ are representable, then", "Descent on Spaces,", "Lemma \\ref{spaces-descent-lemma-etale-smooth-local-source-target-characterize}", "applies which shows that (1) and (2) are equivalent to $f$ having", "$\\mathcal{P}$.", "\\medskip\\noindent", "Finally, suppose $f$ is representable, and $U, V, a, b, h$ are", "as in part (2) of the lemma, and that $\\mathcal{P}$ is preserved under", "arbitrary base change. We have to show that for any scheme", "$Z$ and morphism $Z \\to \\mathcal{X}$ the base change", "$Z \\times_\\mathcal{Y} \\mathcal{X} \\to Z$", "has property $\\mathcal{P}$. Consider the diagram", "$$", "\\xymatrix{", "Z \\times_\\mathcal{Y} U \\ar[d] \\ar[r] &", "Z \\times_\\mathcal{Y} V \\ar[d] \\\\", "Z \\times_\\mathcal{Y} \\mathcal{X} \\ar[r] &", "Z", "}", "$$", "Note that the top horizontal arrow is a base change of $h$ and", "hence has property $\\mathcal{P}$. The left vertical arrow is", "surjective, the induced morphism", "$$", "Z \\times_\\mathcal{Y} U \\longrightarrow", "(Z \\times_\\mathcal{Y} \\mathcal{X}) \\times_Z (Z \\times_\\mathcal{Y} V)", "$$", "is \\'etale, and the right vertical arrow is smooth. Thus", "Descent on Spaces,", "Lemma \\ref{spaces-descent-lemma-etale-smooth-local-source-target-characterize}", "kicks in and shows that $Z \\times_\\mathcal{Y} \\mathcal{X} \\to Z$", "has property $\\mathcal{P}$." ], "refs": [ "stacks-morphisms-lemma-DM", "spaces-descent-lemma-etale-smooth-local-source-target-implies", "spaces-descent-lemma-etale-smooth-local-source-target-implies", "spaces-descent-definition-etale-smooth-local-source-target", "spaces-descent-lemma-etale-smooth-local-source-target-characterize", "spaces-descent-lemma-etale-smooth-local-source-target-characterize" ], "ref_ids": [ 7481, 9430, 9430, 9443, 9431, 9431 ] } ], "ref_ids": [] }, { "id": 7547, "type": "theorem", "label": "stacks-morphisms-lemma-composition-etale", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-etale", "contents": [ "The composition of \\'etale morphisms is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Combine Remark \\ref{remark-etale-smooth-composition}", "with", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-composition-etale}." ], "refs": [ "stacks-morphisms-remark-etale-smooth-composition", "spaces-morphisms-lemma-composition-etale" ], "ref_ids": [ 7639, 4906 ] } ], "ref_ids": [] }, { "id": 7548, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-etale", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-etale", "contents": [ "A base change of an \\'etale morphism is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Remark \\ref{remark-etale-smooth-base-change}", "with", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-etale}." ], "refs": [ "stacks-morphisms-remark-etale-smooth-base-change", "spaces-morphisms-lemma-base-change-etale" ], "ref_ids": [ 7640, 4907 ] } ], "ref_ids": [] }, { "id": 7549, "type": "theorem", "label": "stacks-morphisms-lemma-open-immersion-etale", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-open-immersion-etale", "contents": [ "An open immersion is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Let $j : \\mathcal{U} \\to \\mathcal{X}$ be an open immersion of", "algebraic stacks. Since $j$ is representable, it is DM by", "Lemma \\ref{lemma-trivial-implications}. On the other hand,", "if $X \\to \\mathcal{X}$ is a smooth and surjective morphism", "where $X$ is a scheme, then $U = \\mathcal{U} \\times_\\mathcal{X} X$", "is an open subscheme of $X$. Hence $U \\to X$ is \\'etale", "(Morphisms, Lemma \\ref{morphisms-lemma-open-immersion-etale})", "and we conclude that $j$ is \\'etale from the definition." ], "refs": [ "stacks-morphisms-lemma-trivial-implications", "morphisms-lemma-open-immersion-etale" ], "ref_ids": [ 7397, 5366 ] } ], "ref_ids": [] }, { "id": 7550, "type": "theorem", "label": "stacks-morphisms-lemma-etale", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-etale", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is \\'etale,", "\\item $f$ is DM and for any morphism $V \\to \\mathcal{Y}$", "where $V$ is an algebraic space and any \\'etale morphism", "$U \\to V \\times_\\mathcal{Y} \\mathcal{X}$ where $U$ is an algebraic space,", "the morphism $U \\to V$ is \\'etale,", "\\item there exists some surjective, locally of finite presentation, and flat", "morphism $W \\to \\mathcal{Y}$ where $W$ is an algebraic space and some", "surjective \\'etale morphism $T \\to W \\times_\\mathcal{Y} \\mathcal{X}$", "where $T$ is an algebraic space such that the morphism $T \\to W$ is \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Then $f$ is DM and since being \\'etale is preserved", "by base change, we see that (2) holds.", "\\medskip\\noindent", "Assume (2). Choose a scheme $V$ and a surjective \\'etale morphism", "$V \\to \\mathcal{Y}$. Choose a scheme $U$ and a surjective \\'etale morphism", "$U \\to V \\times_\\mathcal{Y} \\mathcal{X}$ (Lemma \\ref{lemma-DM}).", "Thus we see that (3) holds.", "\\medskip\\noindent", "Assume $W \\to \\mathcal{Y}$ and $T \\to W \\times_\\mathcal{Y} \\mathcal{X}$", "are as in (3). We first check $f$ is DM. Namely, it suffices to check", "$W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is DM, see", "Lemma \\ref{lemma-check-separated-covering}.", "By Lemma \\ref{lemma-compose-after-separated}", "it suffices to check $W \\times_\\mathcal{Y} \\mathcal{X}$ is DM.", "This follows from the existence of $T \\to W \\times_\\mathcal{Y} \\mathcal{X}$", "by (the easy direction of) Theorem \\ref{theorem-DM}.", "\\medskip\\noindent", "Assume $f$ is DM and $W \\to \\mathcal{Y}$ and", "$T \\to W \\times_\\mathcal{Y} \\mathcal{X}$ are as in (3).", "Let $V$ be an algebraic space, let $V \\to \\mathcal{Y}$ be surjective smooth,", "let $U$ be an algebraic space, and let", "$U \\to V \\times_\\mathcal{Y} \\mathcal{X}$ is surjective and \\'etale", "(Lemma \\ref{lemma-DM}). We have to check that $U \\to V$ is \\'etale.", "It suffices to prove $U \\times_\\mathcal{Y} W \\to V \\times_\\mathcal{Y} W$", "is \\'etale by Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-etale}.", "We may replace $\\mathcal{X}, \\mathcal{Y}, W, T, U, V$ by", "$\\mathcal{X} \\times_\\mathcal{Y} W, W, W, T, U \\times_\\mathcal{Y} W,", "V \\times_\\mathcal{Y} W$ (small detail omitted).", "Thus we may assume that $Y = \\mathcal{Y}$ is an algebraic space, there exists", "an algebraic space $T$ and a surjective \\'etale morphism", "$T \\to \\mathcal{X}$ such that $T \\to Y$ is \\'etale, and $U$ and $V$", "are as before. In this case we know that", "$$", "U \\to V\\text{ is \\'etale}", "\\Leftrightarrow", "\\mathcal{X} \\to Y\\text{ is \\'etale}", "\\Leftrightarrow", "T \\to Y\\text{ is \\'etale}", "$$", "by the equivalence of properties (1) and (2) of", "Lemma \\ref{lemma-etale-smooth-local-source-target}", "and Definition \\ref{definition-etale}.", "This finishes the proof." ], "refs": [ "stacks-morphisms-lemma-DM", "stacks-morphisms-lemma-check-separated-covering", "stacks-morphisms-lemma-compose-after-separated", "stacks-morphisms-theorem-DM", "stacks-morphisms-lemma-DM", "spaces-descent-lemma-descending-property-etale", "stacks-morphisms-lemma-etale-smooth-local-source-target", "stacks-morphisms-definition-etale" ], "ref_ids": [ 7481, 7399, 7406, 7389, 7481, 9408, 7546, 7624 ] } ], "ref_ids": [] }, { "id": 7551, "type": "theorem", "label": "stacks-morphisms-lemma-etale-permanence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-etale-permanence", "contents": [ "Let $\\mathcal{X}, \\mathcal{Y}$ be algebraic stacks \\'etale over", "an algebraic stack $\\mathcal{Z}$. Any morphism", "$\\mathcal{X} \\to \\mathcal{Y}$ over $\\mathcal{Z}$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "The morphism $\\mathcal{X} \\to \\mathcal{Y}$ is DM by", "Lemma \\ref{lemma-compose-after-separated}.", "Let $W \\to \\mathcal{Z}$ be a surjective smooth morphism", "whose source is an algebraic space. Let", "$V \\to \\mathcal{Y} \\times_\\mathcal{Z} W$ be a surjective", "\\'etale morphism whose source is an algebraic space", "(Lemma \\ref{lemma-DM}).", "Let $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ be a surjective", "\\'etale morphism whose source is an algebraic space", "(Lemma \\ref{lemma-DM}).", "Then", "$$", "U \\longrightarrow \\mathcal{X} \\times_\\mathcal{Z} W", "$$", "is surjective \\'etale as the composition of", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$", "and the base change of $V \\to \\mathcal{Y} \\times_\\mathcal{Z} W$", "by $\\mathcal{X} \\times_\\mathcal{Z} W \\to \\mathcal{Y} \\times_\\mathcal{Z} W$.", "Hence it suffices to show that $U \\to W$ is \\'etale.", "Since $U \\to W$ and $V \\to W$ are \\'etale because", "$\\mathcal{X} \\to \\mathcal{Z}$ and $\\mathcal{Y} \\to \\mathcal{Z}$", "are \\'etale, this follows from the version of the lemma for", "algebraic spaces, namely", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-etale-permanence}." ], "refs": [ "stacks-morphisms-lemma-compose-after-separated", "stacks-morphisms-lemma-DM", "stacks-morphisms-lemma-DM", "spaces-morphisms-lemma-etale-permanence" ], "ref_ids": [ 7406, 7481, 7481, 4914 ] } ], "ref_ids": [] }, { "id": 7552, "type": "theorem", "label": "stacks-morphisms-lemma-composition-unramified", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-unramified", "contents": [ "The composition of unramified morphisms is unramified." ], "refs": [], "proofs": [ { "contents": [ "Combine Remark \\ref{remark-etale-smooth-composition}", "with", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-composition-unramified}." ], "refs": [ "stacks-morphisms-remark-etale-smooth-composition", "spaces-morphisms-lemma-composition-unramified" ], "ref_ids": [ 7639, 4896 ] } ], "ref_ids": [] }, { "id": 7553, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-unramified", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-unramified", "contents": [ "A base change of an unramified morphism is unramified." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Remark \\ref{remark-etale-smooth-base-change}", "with", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-unramified}." ], "refs": [ "stacks-morphisms-remark-etale-smooth-base-change", "spaces-morphisms-lemma-base-change-unramified" ], "ref_ids": [ 7640, 4897 ] } ], "ref_ids": [] }, { "id": 7554, "type": "theorem", "label": "stacks-morphisms-lemma-etale-unramified", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-etale-unramified", "contents": [ "An \\'etale morphism is unramified." ], "refs": [], "proofs": [ { "contents": [ "Follows from Remark \\ref{remark-etale-smooth-implication}", "and Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-etale-unramified}." ], "refs": [ "stacks-morphisms-remark-etale-smooth-implication", "spaces-morphisms-lemma-etale-unramified" ], "ref_ids": [ 7641, 4913 ] } ], "ref_ids": [] }, { "id": 7555, "type": "theorem", "label": "stacks-morphisms-lemma-immersion-unramified", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-immersion-unramified", "contents": [ "An immersion is unramified." ], "refs": [], "proofs": [ { "contents": [ "Let $j : \\mathcal{Z} \\to \\mathcal{X}$ be an immersion of", "algebraic stacks. Since $j$ is representable, it is DM by", "Lemma \\ref{lemma-trivial-implications}. On the other hand,", "if $X \\to \\mathcal{X}$ is a smooth and surjective morphism", "where $X$ is a scheme, then $Z = \\mathcal{Z} \\times_\\mathcal{X} X$", "is a locally closed subscheme of $X$. Hence $Z \\to X$ is unramified", "(Morphisms, Lemmas \\ref{morphisms-lemma-open-immersion-unramified} and", "\\ref{morphisms-lemma-closed-immersion-unramified})", "and we conclude that $j$ is unramified from the definition." ], "refs": [ "stacks-morphisms-lemma-trivial-implications", "morphisms-lemma-open-immersion-unramified", "morphisms-lemma-closed-immersion-unramified" ], "ref_ids": [ 7397, 5348, 5349 ] } ], "ref_ids": [] }, { "id": 7556, "type": "theorem", "label": "stacks-morphisms-lemma-unramified", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-unramified", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is unramified,", "\\item $f$ is DM and for any morphism $V \\to \\mathcal{Y}$", "where $V$ is an algebraic space and any \\'etale morphism", "$U \\to V \\times_\\mathcal{Y} \\mathcal{X}$ where $U$ is an algebraic space,", "the morphism $U \\to V$ is unramified,", "\\item there exists some surjective, locally of finite presentation, and flat", "morphism $W \\to \\mathcal{Y}$ where $W$ is an algebraic space and some", "surjective \\'etale morphism $T \\to W \\times_\\mathcal{Y} \\mathcal{X}$", "where $T$ is an algebraic space such that the morphism $T \\to W$ is unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Then $f$ is DM and since being unramified is preserved", "by base change, we see that (2) holds.", "\\medskip\\noindent", "Assume (2). Choose a scheme $V$ and a surjective \\'etale morphism", "$V \\to \\mathcal{Y}$. Choose a scheme $U$ and a surjective \\'etale morphism", "$U \\to V \\times_\\mathcal{Y} \\mathcal{X}$ (Lemma \\ref{lemma-DM}).", "Thus we see that (3) holds.", "\\medskip\\noindent", "Assume $W \\to \\mathcal{Y}$ and $T \\to W \\times_\\mathcal{Y} \\mathcal{X}$", "are as in (3). We first check $f$ is DM. Namely, it suffices to check", "$W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is DM, see", "Lemma \\ref{lemma-check-separated-covering}.", "By Lemma \\ref{lemma-compose-after-separated}", "it suffices to check $W \\times_\\mathcal{Y} \\mathcal{X}$ is DM.", "This follows from the existence of $T \\to W \\times_\\mathcal{Y} \\mathcal{X}$", "by (the easy direction of) Theorem \\ref{theorem-DM}.", "\\medskip\\noindent", "Assume $f$ is DM and $W \\to \\mathcal{Y}$ and", "$T \\to W \\times_\\mathcal{Y} \\mathcal{X}$ are as in (3).", "Let $V$ be an algebraic space, let $V \\to \\mathcal{Y}$ be surjective smooth,", "let $U$ be an algebraic space, and let", "$U \\to V \\times_\\mathcal{Y} \\mathcal{X}$ is surjective and \\'etale", "(Lemma \\ref{lemma-DM}). We have to check that $U \\to V$ is unramified.", "It suffices to prove $U \\times_\\mathcal{Y} W \\to V \\times_\\mathcal{Y} W$", "is unramified by Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-unramified}.", "We may replace $\\mathcal{X}, \\mathcal{Y}, W, T, U, V$ by", "$\\mathcal{X} \\times_\\mathcal{Y} W, W, W, T, U \\times_\\mathcal{Y} W,", "V \\times_\\mathcal{Y} W$ (small detail omitted).", "Thus we may assume that $Y = \\mathcal{Y}$ is an algebraic space, there exists", "an algebraic space $T$ and a surjective \\'etale morphism", "$T \\to \\mathcal{X}$ such that $T \\to Y$ is unramified, and $U$ and $V$", "are as before. In this case we know that", "$$", "U \\to V\\text{ is unramified}", "\\Leftrightarrow", "\\mathcal{X} \\to Y\\text{ is unramified}", "\\Leftrightarrow", "T \\to Y\\text{ is unramified}", "$$", "by the equivalence of properties (1) and (2) of", "Lemma \\ref{lemma-etale-smooth-local-source-target}", "and Definition \\ref{definition-unramified}.", "This finishes the proof." ], "refs": [ "stacks-morphisms-lemma-DM", "stacks-morphisms-lemma-check-separated-covering", "stacks-morphisms-lemma-compose-after-separated", "stacks-morphisms-theorem-DM", "stacks-morphisms-lemma-DM", "spaces-descent-lemma-descending-property-unramified", "stacks-morphisms-lemma-etale-smooth-local-source-target", "stacks-morphisms-definition-unramified" ], "ref_ids": [ 7481, 7399, 7406, 7389, 7481, 9407, 7546, 7625 ] } ], "ref_ids": [] }, { "id": 7557, "type": "theorem", "label": "stacks-morphisms-lemma-permanence-unramified", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-permanence-unramified", "contents": [ "Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be", "morphisms of algebraic stacks.", "If $\\mathcal{X} \\to \\mathcal{Z}$ is unramified and", "$\\mathcal{Y} \\to \\mathcal{Z}$ is DM, then", "$\\mathcal{X} \\to \\mathcal{Y}$ is unramified." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{X} \\to \\mathcal{Z}$ is unramified. By", "Lemma \\ref{lemma-compose-after-separated} the morphism", "$\\mathcal{X} \\to \\mathcal{Y}$ is DM. Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\ar[r] & W \\ar[d] \\\\", "\\mathcal{X} \\ar[r] & \\mathcal{Y} \\ar[r] & \\mathcal{Z}", "}", "$$", "with $U, V, W$ algebraic spaces,", "with $W \\to \\mathcal{Z}$ surjective smooth,", "$V \\to \\mathcal{Y} \\times_\\mathcal{Z} W$ surjective \\'etale, and", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ surjective \\'etale", "(see Lemma \\ref{lemma-DM}). Then also", "$U \\to \\mathcal{X} \\times_\\mathcal{Z} W$ is surjective and \\'etale.", "Hence we know that $U \\to W$ is unramified and we have to show that", "$U \\to V$ is unramified. This follows from", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-permanence-unramified}." ], "refs": [ "stacks-morphisms-lemma-compose-after-separated", "stacks-morphisms-lemma-DM", "spaces-morphisms-lemma-permanence-unramified" ], "ref_ids": [ 7406, 7481, 4904 ] } ], "ref_ids": [] }, { "id": 7558, "type": "theorem", "label": "stacks-morphisms-lemma-characterize-unramified", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-characterize-unramified", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is unramified, and", "\\item $f$ is locally of finite type and its diagonal is \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f$ is unramified. Then $f$ is DM hence we can choose", "algebraic spaces $U$, $V$, a smooth surjective morphism", "$V \\to \\mathcal{Y}$ and a surjective \\'etale morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ (Lemma \\ref{lemma-DM}).", "Since $f$ is unramified the induced morphism $U \\to V$ is unramified.", "Thus $U \\to V$ is locally of finite type", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-unramified-locally-finite-type})", "and we conclude that $f$ is locally of finite type. The diagonal", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "is a morphism of algebraic stacks over $\\mathcal{Y}$.", "The base change of $\\Delta$ by the surjective smooth morphism", "$V \\to \\mathcal{Y}$ is the diagonal of the base change of", "$f$, i.e., of $\\mathcal{X}_V = \\mathcal{X} \\times_\\mathcal{Y} V \\to V$.", "In other words, the diagram", "$$", "\\xymatrix{", "\\mathcal{X}_V \\ar[r] \\ar[d] & \\mathcal{X}_V \\times_V \\mathcal{X}_V \\ar[d] \\\\", "\\mathcal{X} \\ar[r] & \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}", "}", "$$", "is cartesian. Since the right vertical arrow is surjective and smooth", "it suffices to show that the top horizontal arrow is \\'etale by", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-weak-covering}.", "Consider the commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & U \\times_V U \\ar[d] \\\\", "\\mathcal{X}_V \\ar[r] & \\mathcal{X}_V \\times_V \\mathcal{X}_V", "}", "$$", "All arrows are representable by algebraic spaces,", "the vertical arrows are \\'etale, the left one is surjective, and", "the top horizontal arrow is an open immersion by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-diagonal-unramified-morphism}.", "This implies what we want: first we see that", "$U \\to \\mathcal{X}_V \\times_V \\mathcal{X}_V$ is \\'etale", "as a composition of \\'etale morphisms, and then we can use", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-after-precomposing}", "to see that $\\mathcal{X}_V \\to \\mathcal{X}_V \\times_V \\mathcal{X}_V$", "is \\'etale because being \\'etale (for morphisms of algebraic spaces)", "is local on the source in the \\'etale topology", "(Descent on Spaces, Lemma \\ref{spaces-descent-lemma-etale-etale-local-source}).", "\\medskip\\noindent", "Assume $f$ is locally of finite type and that its diagonal is \\'etale.", "Then $f$ is DM by definition (as \\'etale morphisms of algebraic spaces", "are unramified). As above this means we can choose", "algebraic spaces $U$, $V$, a smooth surjective morphism", "$V \\to \\mathcal{Y}$ and a surjective \\'etale morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ (Lemma \\ref{lemma-DM}).", "To finish the proof we have to show that $U \\to V$ is unramified.", "We already know that $U \\to V$ is locally of finite type.", "Arguing as above we find a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & U \\times_V U \\ar[d] \\\\", "\\mathcal{X}_V \\ar[r] & \\mathcal{X}_V \\times_V \\mathcal{X}_V", "}", "$$", "where all arrows are representable by algebraic spaces,", "the vertical arrows are \\'etale, and the lower horizontal", "one is \\'etale as a base change of $\\Delta$.", "It follows that $U \\to U \\times_V U$ is \\'etale", "for example by Lemma \\ref{lemma-etale-permanence}\\footnote{It is", "quite easy to deduce this directly from", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-etale-permanence}.}.", "Thus $U \\to U \\times_V U$ is an \\'etale monomorphism", "hence an open immersion (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-etale-universally-injective-open}).", "Then $U \\to V$ is unramified by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-diagonal-unramified-morphism}." ], "refs": [ "stacks-morphisms-lemma-DM", "spaces-morphisms-lemma-unramified-locally-finite-type", "stacks-properties-lemma-check-property-weak-covering", "spaces-morphisms-lemma-diagonal-unramified-morphism", "stacks-properties-lemma-check-property-after-precomposing", "spaces-descent-lemma-etale-etale-local-source", "stacks-morphisms-lemma-DM", "stacks-morphisms-lemma-etale-permanence", "spaces-morphisms-lemma-etale-permanence", "spaces-morphisms-lemma-etale-universally-injective-open", "spaces-morphisms-lemma-diagonal-unramified-morphism" ], "ref_ids": [ 7481, 4899, 8860, 4902, 8861, 9424, 7481, 7551, 4914, 4973, 4902 ] } ], "ref_ids": [] }, { "id": 7559, "type": "theorem", "label": "stacks-morphisms-lemma-characterize-etale", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-characterize-etale", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is \\'etale, and", "\\item $f$ is locally of finite presentation, flat, and unramified,", "\\item $f$ is locally of finite presentation, flat, and its diagonal", "is \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (2) and (3) follows immediately from", "Lemma \\ref{lemma-characterize-unramified}. Thus in each case", "the morphism $f$ is DM. Then we can choose Then we can choose", "algebraic spaces $U$, $V$, a smooth surjective morphism", "$V \\to \\mathcal{Y}$ and a surjective \\'etale morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ (Lemma \\ref{lemma-DM}).", "To finish the proof we have to show that", "$U \\to V$ is \\'etale if and only if it is locally of finite", "presentation, flat, and unramified.", "This follows from Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-unramified-flat-lfp-etale}", "(and the more trivial", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-etale-unramified},", "\\ref{spaces-morphisms-lemma-etale-locally-finite-presentation}, and", "\\ref{spaces-morphisms-lemma-etale-flat})." ], "refs": [ "stacks-morphisms-lemma-characterize-unramified", "stacks-morphisms-lemma-DM", "spaces-morphisms-lemma-unramified-flat-lfp-etale", "spaces-morphisms-lemma-etale-unramified", "spaces-morphisms-lemma-etale-locally-finite-presentation", "spaces-morphisms-lemma-etale-flat" ], "ref_ids": [ 7558, 7481, 4915, 4913, 4911, 4910 ] } ], "ref_ids": [] }, { "id": 7560, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-proper", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-proper", "contents": [ "A base change of a proper morphism is proper." ], "refs": [], "proofs": [ { "contents": [ "See", "Lemmas \\ref{lemma-base-change-separated},", "\\ref{lemma-base-change-finite-type}, and", "\\ref{lemma-base-change-universally-closed}." ], "refs": [ "stacks-morphisms-lemma-base-change-separated", "stacks-morphisms-lemma-base-change-finite-type", "stacks-morphisms-lemma-base-change-universally-closed" ], "ref_ids": [ 7398, 7460, 7444 ] } ], "ref_ids": [] }, { "id": 7561, "type": "theorem", "label": "stacks-morphisms-lemma-composition-proper", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-proper", "contents": [ "A composition of proper morphisms is proper." ], "refs": [], "proofs": [ { "contents": [ "See", "Lemmas \\ref{lemma-composition-separated},", "\\ref{lemma-composition-finite-type}, and", "\\ref{lemma-composition-universally-closed}." ], "refs": [ "stacks-morphisms-lemma-composition-separated", "stacks-morphisms-lemma-composition-finite-type", "stacks-morphisms-lemma-composition-universally-closed" ], "ref_ids": [ 7404, 7459, 7445 ] } ], "ref_ids": [] }, { "id": 7562, "type": "theorem", "label": "stacks-morphisms-lemma-closed-immersion-proper", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-closed-immersion-proper", "contents": [ "A closed immersion of algebraic stacks is a proper morphism of", "algebraic stacks." ], "refs": [], "proofs": [ { "contents": [ "A closed immersion is by definition representable", "(Properties of Stacks, Definition", "\\ref{stacks-properties-definition-immersion}).", "Hence this follows from the discussion in", "Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}", "and the corresponding result for morphisms of algebraic spaces, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-closed-immersion-proper}." ], "refs": [ "stacks-properties-definition-immersion", "spaces-morphisms-lemma-closed-immersion-proper" ], "ref_ids": [ 8920, 4919 ] } ], "ref_ids": [] }, { "id": 7563, "type": "theorem", "label": "stacks-morphisms-lemma-universally-closed-permanence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-universally-closed-permanence", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{X} \\ar[rr] \\ar[rd] & &", "\\mathcal{Y} \\ar[ld] \\\\", "& \\mathcal{Z} &", "}", "$$", "of algebraic stacks.", "\\begin{enumerate}", "\\item If $\\mathcal{X} \\to \\mathcal{Z}$ is universally closed and", "$\\mathcal{Y} \\to \\mathcal{Z}$ is separated,", "then the morphism $\\mathcal{X} \\to \\mathcal{Y}$ is universally closed.", "In particular, the image of $|\\mathcal{X}|$ in $|\\mathcal{Y}|$ is closed.", "\\item If $\\mathcal{X} \\to \\mathcal{Z}$ is proper and", "$\\mathcal{Y} \\to \\mathcal{Z}$ is separated, then", "the morphism $\\mathcal{X} \\to \\mathcal{Y}$ is proper.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{X} \\to \\mathcal{Z}$ is universally closed and", "$\\mathcal{Y} \\to \\mathcal{Z}$ is separated.", "We factor the morphism as", "$\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to \\mathcal{Y}$.", "The first morphism is proper (Lemma \\ref{lemma-semi-diagonal})", "hence universally closed.", "The projection $\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to \\mathcal{Y}$", "is the base change of a universally closed morphism and hence", "universally closed, see", "Lemma \\ref{lemma-base-change-universally-closed}.", "Thus $\\mathcal{X} \\to \\mathcal{Y}$ is universally closed as the composition", "of universally closed morphisms, see", "Lemma \\ref{lemma-composition-universally-closed}.", "This proves (1). To deduce (2) combine (1) with", "Lemmas \\ref{lemma-compose-after-separated},", "\\ref{lemma-quasi-compact-permanence}, and", "\\ref{lemma-finite-type-permanence}." ], "refs": [ "stacks-morphisms-lemma-semi-diagonal", "stacks-morphisms-lemma-base-change-universally-closed", "stacks-morphisms-lemma-composition-universally-closed", "stacks-morphisms-lemma-compose-after-separated", "stacks-morphisms-lemma-quasi-compact-permanence", "stacks-morphisms-lemma-finite-type-permanence" ], "ref_ids": [ 7402, 7444, 7445, 7406, 7427, 7465 ] } ], "ref_ids": [] }, { "id": 7564, "type": "theorem", "label": "stacks-morphisms-lemma-image-proper-is-proper", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-image-proper-is-proper", "contents": [ "Let $\\mathcal{Z}$ be an algebraic stack.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$", "be a morphism of algebraic stacks over $\\mathcal{Z}$.", "If $\\mathcal{X}$ is universally closed over $\\mathcal{Z}$", "and $f$ is surjective then $\\mathcal{Y}$", "is universally closed over $\\mathcal{Z}$.", "In particular, if also $\\mathcal{Y}$ is", "separated and of finite type over $\\mathcal{Z}$,", "then $\\mathcal{Y}$ is proper over $\\mathcal{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{X}$ is universally closed and $f$ surjective.", "Denote $p : \\mathcal{X} \\to \\mathcal{Z}$,", "$q : \\mathcal{Y} \\to \\mathcal{Z}$ the structure morphisms.", "Let $\\mathcal{Z}' \\to \\mathcal{Z}$ be a morphism of algebraic stacks.", "The base change $f' : \\mathcal{X}' \\to \\mathcal{Y}'$", "of $f$ by $\\mathcal{Z}' \\to \\mathcal{Z}$ is surjective", "(Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-base-change-surjective}) and the base", "change $p' : \\mathcal{X}' \\to \\mathcal{Z}'$ of $p$ is closed.", "If $T \\subset |\\mathcal{Y}'|$ is closed, then", "$(f')^{-1}(T) \\subset |\\mathcal{X}'|$ is closed, hence", "$p'((f')^{-1}(T)) = q'(T)$ is closed. So $q'$ is closed." ], "refs": [ "stacks-properties-lemma-base-change-surjective" ], "ref_ids": [ 8870 ] } ], "ref_ids": [] }, { "id": 7565, "type": "theorem", "label": "stacks-morphisms-lemma-cover-upstairs", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-cover-upstairs", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $g : \\mathcal{W} \\to \\mathcal{X}$ be a morphism of algebraic stacks", "which is surjective, flat, and locally of finite presentation.", "Then the scheme theoretic image of $f$ exists if and only if the", "scheme theoretic image of $f \\circ g$ exists and if so then these", "scheme theoretic images are the same." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{Z} \\subset \\mathcal{Y}$", "is a closed substack and $f \\circ g$ factors through $\\mathcal{Z}$.", "To prove the lemma it suffices to show", "that $f$ factors through $\\mathcal{Z}$.", "Consider a scheme $T$ and a morphism $T \\to \\mathcal{X}$", "given by an object $x$ of the fibre category of $\\mathcal{X}$ over $T$.", "We will show that $x$ is in fact in the fibre category of $\\mathcal{Z}$", "over $T$. Namely, the projection $T \\times_\\mathcal{X} \\mathcal{W} \\to T$", "is a surjective, flat, locally finitely presented morphism.", "Hence there is an fppf covering $\\{T_i \\to T\\}$ such that", "$T_i \\to T$ factors through $T \\times_\\mathcal{X} \\mathcal{W} \\to T$", "for all $i$. Then $T_i \\to \\mathcal{X}$ factors through $\\mathcal{W}$", "and hence $T_i \\to \\mathcal{Y}$ factors through $\\mathcal{Z}$.", "Thus $x|_{T_i}$ is an object of $\\mathcal{Z}$.", "Since $\\mathcal{Z}$ is a strictly full substack, we conclude", "that $x$ is an object of $\\mathcal{Z}$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7566, "type": "theorem", "label": "stacks-morphisms-lemma-scheme-theoretic-image-existence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-scheme-theoretic-image-existence", "contents": [ "Let $f : \\mathcal{Y} \\to \\mathcal{X}$ be a morphism of algebraic stacks.", "Then the scheme theoretic image of $f$ exists." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective smooth morphism $V \\to \\mathcal{Y}$.", "By Lemma \\ref{lemma-cover-upstairs} we may replace $\\mathcal{Y}$ by $V$.", "Thus it suffices to show that if $X \\to \\mathcal{X}$ is a morphism from", "a scheme to an algebraic stack, then the scheme theoretic image exists.", "Choose a scheme $U$ and a surjective smooth morphism $U \\to \\mathcal{X}$.", "Set $R = U \\times_\\mathcal{X} U$.", "We have $\\mathcal{X} = [U/R]$ by", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-stack-presentation}.", "By Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-substacks-presentation}", "the closed substacks $\\mathcal{Z}$ of $\\mathcal{X}$", "are in $1$-to-$1$ correspondence with $R$-invariant", "closed subschemes $Z \\subset U$.", "Let $Z_1 \\subset U$ be the scheme theoretic image of", "$X \\times_\\mathcal{X} U \\to U$.", "Observe that $X \\to \\mathcal{X}$ factors through $\\mathcal{Z}$", "if and only if $X \\times_\\mathcal{X} U \\to U$ factors through", "the corresponding $R$-invariant closed subscheme $Z$", "(details omitted; hint: this follows because", "$X \\times_\\mathcal{X} U \\to X$ is surjective and smooth).", "Thus we have to show that there exists a smallest $R$-invariant", "closed subscheme $Z \\subset U$ containing $Z_1$.", "\\medskip\\noindent", "Let $\\mathcal{I}_1 \\subset \\mathcal{O}_U$ be the quasi-coherent", "ideal sheaf corresponding to the closed subscheme $Z_1 \\subset U$.", "Let $Z_\\alpha$, $\\alpha \\in A$ be the set of all $R$-invariant", "closed subschemes of $U$ containing $Z_1$.", "For $\\alpha \\in A$, let $\\mathcal{I}_\\alpha \\subset \\mathcal{O}_U$", "be the quasi-coherent ideal sheaf corresponding to the closed subscheme", "$Z_\\alpha \\subset U$. The containment $Z_1 \\subset Z_\\alpha$", "means $\\mathcal{I}_\\alpha \\subset \\mathcal{I}_1$.", "The $R$-invariance of $Z_\\alpha$ means that", "$$", "s^{-1}\\mathcal{I}_\\alpha \\cdot \\mathcal{O}_R =", "t^{-1}\\mathcal{I}_\\alpha \\cdot \\mathcal{O}_R", "$$", "as (quasi-coherent) ideal sheaves on (the algebraic space) $R$.", "Consider the image", "$$", "\\mathcal{I} =", "\\Im\\left(", "\\bigoplus\\nolimits_{\\alpha \\in A} \\mathcal{I}_\\alpha \\to \\mathcal{I}_1", "\\right) =", "\\Im\\left(", "\\bigoplus\\nolimits_{\\alpha \\in A} \\mathcal{I}_\\alpha \\to \\mathcal{O}_X", "\\right)", "$$", "Since direct sums of quasi-coherent sheaves are quasi-coherent", "and since images of maps between quasi-coherent sheaves are", "quasi-coherent, we find that $\\mathcal{I}$ is quasi-coherent.", "Since pull back is exact and commutes with direct sums we find", "$$", "s^{-1}\\mathcal{I} \\cdot \\mathcal{O}_R =", "t^{-1}\\mathcal{I} \\cdot \\mathcal{O}_R", "$$", "Hence $\\mathcal{I}$ defines an $R$-invariant closed subscheme", "$Z \\subset U$ which is contained in every $Z_\\alpha$ and containes", "$Z_1$ as desired." ], "refs": [ "stacks-morphisms-lemma-cover-upstairs", "algebraic-lemma-stack-presentation", "stacks-properties-lemma-substacks-presentation" ], "ref_ids": [ 7565, 8474, 8889 ] } ], "ref_ids": [] }, { "id": 7567, "type": "theorem", "label": "stacks-morphisms-lemma-factor-factor", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-factor-factor", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{X}_1 \\ar[d] \\ar[r]_{f_1} & \\mathcal{Y}_1 \\ar[d] \\\\", "\\mathcal{X}_2 \\ar[r]^{f_2} & \\mathcal{Y}_2", "}", "$$", "be a commutative diagram of algebraic stacks.", "Let $\\mathcal{Z}_i \\subset \\mathcal{Y}_i$, $i = 1, 2$ be", "the scheme theoretic image of $f_i$. Then the morphism", "$\\mathcal{Y}_1 \\to \\mathcal{Y}_2$ induces a morphism", "$\\mathcal{Z}_1 \\to \\mathcal{Z}_2$ and a", "commutative diagram", "$$", "\\xymatrix{", "\\mathcal{X}_1 \\ar[r] \\ar[d] &", "\\mathcal{Z}_1 \\ar[d] \\ar[r] &", "\\mathcal{Y}_1 \\ar[d] \\\\", "\\mathcal{X}_2 \\ar[r] &", "\\mathcal{Z}_2 \\ar[r] &", "\\mathcal{Y}_2", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "The scheme theoretic inverse image of $\\mathcal{Z}_2$ in $\\mathcal{Y}_1$", "is a closed substack of $\\mathcal{Y}_1$ through", "which $f_1$ factors. Hence $\\mathcal{Z}_1$ is contained in this.", "This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7568, "type": "theorem", "label": "stacks-morphisms-lemma-existence-plus-flat-base-change", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-existence-plus-flat-base-change", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact", "morphism of algebraic stacks. Then formation of the scheme theoretic image", "commutes with flat base change." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{Y}' \\to \\mathcal{Y}$ be a flat morphism of algebraic stacks.", "Choose a scheme $V$ and a surjective smooth morphism $V \\to \\mathcal{Y}$.", "Choose a scheme $V'$ and a", "surjective smooth morphism $V' \\to \\mathcal{Y}' \\times_\\mathcal{Y} V$.", "We may and do assume that $V = \\coprod_{i \\in I} V_i$ is a disjoint", "union of affine schemes and that", "$V' = \\coprod_{i \\in I} \\coprod_{j \\in J_i} V_{i, j}$", "is a disjoint union of affine schemes with each $V_{i, j}$ mapping into $V_i$.", "Let", "\\begin{enumerate}", "\\item $\\mathcal{Z} \\subset \\mathcal{Y}$ be the scheme theoretic image of $f$,", "\\item $\\mathcal{Z}' \\subset \\mathcal{Y}'$ be the scheme theoretic image", "of the base change of $f$ by $\\mathcal{Y}' \\to \\mathcal{Y}$,", "\\item $Z \\subset V$ be the scheme theoretic image", "of the base change of $f$ by $V \\to \\mathcal{Y}$,", "\\item $Z' \\subset V'$ be the scheme theoretic image", "of the base change of $f$ by $V' \\to \\mathcal{Y}$.", "\\end{enumerate}", "If we can show that", "(a) $Z = V \\times_\\mathcal{Y} \\mathcal{Z}$,", "(b) $Z' = V' \\times_{\\mathcal{Y}'} \\mathcal{Z}'$, and", "(c) $Z' = V' \\times_V Z$", "then the lemma follows: the inclusion", "$\\mathcal{Z}' \\to \\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{Y}'$", "(Lemma \\ref{lemma-factor-factor})", "has to be an isomorphism because after base change by the surjective", "smooth morphism $V' \\to \\mathcal{Y}'$ it is.", "\\medskip\\noindent", "Proof of (a). Set $R = V \\times_\\mathcal{Y} V$.", "By Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-substacks-presentation}", "the rule $\\mathcal{Z} \\mapsto \\mathcal{Z} \\times_\\mathcal{Y} V$", "defines a $1$-to-$1$ correspondence between closed substacks", "of $\\mathcal{Y}$ and $R$-invariant closed subspaces of $V$.", "Moreover, $f : \\mathcal{X} \\to \\mathcal{Y}$ factors through $\\mathcal{Z}$", "if and only if the base change", "$g : \\mathcal{X} \\times_\\mathcal{Y} V \\to V$ factors through", "$\\mathcal{Z} \\times_\\mathcal{Y} V$.", "We claim: the scheme theoretic image $Z \\subset V$ of $g$", "is $R$-invariant. The claim implies (a) by what we just said.", "\\medskip\\noindent", "For each $i$ the morphism $\\mathcal{X} \\times_\\mathcal{Y} V_i \\to V_i$", "is quasi-compact and hence $\\mathcal{X} \\times_\\mathcal{Y} V_i$ is", "quasi-compact. Thus we can choose an affine scheme $W_i$ and a", "surjective smooth morphism $W_i \\to \\mathcal{X} \\times_\\mathcal{Y} V_i$.", "Observe that $W = \\coprod W_i$ is a scheme endowed with", "a smooth and surjective morphism $W \\to \\mathcal{X} \\times_\\mathcal{Y} V$", "such that the composition $W \\to V$ with $g$ is quasi-compact.", "Let $Z \\to V$ be the scheme theoretic image of $W \\to V$, see", "Morphisms, Section", "\\ref{morphisms-section-scheme-theoretic-image} and", "Morphisms of Spaces, Section", "\\ref{spaces-morphisms-section-scheme-theoretic-image}.", "It follows from Lemma \\ref{lemma-cover-upstairs}", "that $Z \\subset V$ is the scheme theoretic image of $g$.", "To show that $Z$ is $R$-invariant we claim that both", "$$", "\\text{pr}_0^{-1}(Z), \\text{pr}_1^{-1}(Z) \\subset R = V \\times_\\mathcal{Y} V", "$$", "are the scheme theoretic image of $\\mathcal{X} \\times_\\mathcal{Y} R \\to R$.", "Namely, we first use Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image}", "to see that $\\text{pr}_0^{-1}(Z)$ is the scheme theoretic image", "of the composition", "$$", "W \\times_{V, \\text{pr}_0} R = W \\times_\\mathcal{Y} V \\to", "\\mathcal{X} \\times_\\mathcal{Y} R \\to R", "$$", "Since the first arrow here is surjective and smooth we see that", "$\\text{pr}_0^{-1}(Z)$ is the scheme theoretic image of", "$\\mathcal{X} \\times_\\mathcal{Y} R \\to R$. The same argument applies", "that $\\text{pr}_1^{-1}(Z)$. Hence $Z$ is $R$-invariant.", "\\medskip\\noindent", "Statement (b) is proved in exactly the same way as one proves (a).", "\\medskip\\noindent", "Proof of (c). Let $Z_i \\subset V_i$ be the scheme theoretic image", "of $\\mathcal{X} \\times_\\mathcal{Y} V_i \\to V_i$ and let", "$Z_{i, j} \\subset V_{i, j}$ be the scheme theoretic image of", "$\\mathcal{X} \\times_\\mathcal{Y} V_{i, j} \\to V_{i, j}$.", "Clearly it suffices to show that the inverse image of $Z_i$", "in $V_{i, j}$ is $Z_{i, j}$. Above we've seen that", "$Z_i$ is the scheme theoretic image of $W_i \\to V_i$", "and by the same token $Z_{i, j}$ is the scheme theoretic", "image of $W_i \\times_{V_i} V_{i, j} \\to V_{i, j}$.", "Hence the equality follows from the case of schemes", "(Morphisms, Lemma", "\\ref{morphisms-lemma-flat-base-change-scheme-theoretic-image})", "and the fact that $V_{i, j} \\to V_i$ is flat." ], "refs": [ "stacks-morphisms-lemma-factor-factor", "stacks-properties-lemma-substacks-presentation", "stacks-morphisms-lemma-cover-upstairs", "spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image", "morphisms-lemma-flat-base-change-scheme-theoretic-image" ], "ref_ids": [ 7567, 8889, 7565, 4861, 5273 ] } ], "ref_ids": [] }, { "id": 7569, "type": "theorem", "label": "stacks-morphisms-lemma-topology-scheme-theoretic-image", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-topology-scheme-theoretic-image", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact", "morphism of algebraic stacks. Let $\\mathcal{Z} \\subset \\mathcal{Y}$", "be the scheme theoretic image of $f$. Then $|\\mathcal{Z}|$", "is the closure of the image of $|f|$." ], "refs": [], "proofs": [ { "contents": [ "Let $z \\in |\\mathcal{Z}|$ be a point.", "Choose an affine scheme $V$, a point $v \\in V$, and a smooth morphism", "$V \\to \\mathcal{Y}$ mapping $v$ to $z$.", "Then $\\mathcal{X} \\times_\\mathcal{Y} V$ is a quasi-compact algebraic stack.", "Hence we can find an affine scheme $W$ and a surjective smooth", "morphism $W \\to \\mathcal{X} \\times_\\mathcal{Y} V$.", "By Lemma \\ref{lemma-existence-plus-flat-base-change}", "the scheme theoretic image of", "$\\mathcal{X} \\times_\\mathcal{Y} V \\to V$ is", "$Z = \\mathcal{Z} \\times_\\mathcal{Y} V$.", "Hence the inverse image of $|\\mathcal{Z}|$ in $|V|$ is $|Z|$ by", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-points-cartesian}.", "By Lemma \\ref{lemma-cover-upstairs} $Z$ is", "the scheme theoretic image of $W \\to V$.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image}", "we see that the image of $|W| \\to |Z|$ is dense.", "Hence the image of $|\\mathcal{X} \\times_\\mathcal{Y} V| \\to |Z|$", "is dense. Observe that $v \\in Z$.", "Since $|V| \\to |\\mathcal{Y}|$ is open, a topology argument", "tells us that $z$ is in the closure of the image of $|f|$ as desired." ], "refs": [ "stacks-morphisms-lemma-existence-plus-flat-base-change", "stacks-properties-lemma-points-cartesian", "stacks-morphisms-lemma-cover-upstairs", "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image" ], "ref_ids": [ 7568, 8864, 7565, 4780 ] } ], "ref_ids": [] }, { "id": 7570, "type": "theorem", "label": "stacks-morphisms-lemma-scheme-theoretic-image-of-partial-section", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-scheme-theoretic-image-of-partial-section", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "which is representable by algebraic spaces and separated.", "Let $\\mathcal{V} \\subset \\mathcal{Y}$ be an open substack such that", "$\\mathcal{V} \\to \\mathcal{Y}$ is quasi-compact.", "Let $s : \\mathcal{V} \\to \\mathcal{X}$ be a morphism such that", "$f \\circ s = \\text{id}_\\mathcal{V}$.", "Let $\\mathcal{Y}'$ be the scheme theoretic image of $s$.", "Then $\\mathcal{Y}' \\to \\mathcal{Y}$ is an isomorphism over $\\mathcal{V}$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-quasi-compact-permanence}", "the morphism $s : \\mathcal{V} \\to \\mathcal{Y}$ is quasi-compact.", "Hence the construction of the scheme theoretic image $\\mathcal{Y}'$", "of $s$ commutes with flat base change by", "Lemma \\ref{lemma-existence-plus-flat-base-change}.", "Thus to prove the lemma", "we may assume $\\mathcal{Y}$ is representable by an algebraic space", "and we reduce to the case of algebraic spaces which is", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-scheme-theoretic-image-of-partial-section}." ], "refs": [ "stacks-morphisms-lemma-quasi-compact-permanence", "stacks-morphisms-lemma-existence-plus-flat-base-change" ], "ref_ids": [ 7427, 7568 ] } ], "ref_ids": [] }, { "id": 7571, "type": "theorem", "label": "stacks-morphisms-lemma-cat-dotted-arrows", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-cat-dotted-arrows", "contents": [ "In the situation of Definition \\ref{definition-fill-in-diagram}", "the category of dotted arrows is a groupoid. If $\\Delta_f$", "is separated, then it is a setoid." ], "refs": [ "stacks-morphisms-definition-fill-in-diagram" ], "proofs": [ { "contents": [ "Since $2$-arrows are invertible it is clear that the category of", "dotted arrows is a groupoid. Given a dotted arrow $(a, \\alpha, \\beta)$", "an automorphism of $(a, \\alpha, \\beta)$ is a $2$-morphism", "$\\theta : a \\to a$ satisfying two conditions. The first condition", "$\\beta = (\\text{id}_f \\star \\theta) \\circ \\beta$ signifies that", "$\\theta$ defines a morphism", "$(a, \\theta) : \\Spec(A) \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$.", "The second condition", "$\\alpha = \\alpha \\circ (\\theta \\star \\text{id}_j)$", "implies that the restriction of $(a, \\theta)$ to $\\Spec(K)$", "is the identity. Picture", "$$", "\\xymatrix{", "\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\ar[d] & &", "\\Spec(K) \\ar[d]^j \\ar[ll]_{(a \\circ j, \\text{id})} \\\\", "\\mathcal{X} & & \\Spec(A) \\ar[ll]_a \\ar[llu]_{(a, \\theta)}", "}", "$$", "In other words, if $G \\to \\Spec(A)$ is the group algebraic space", "we get by pulling back the relative inertia by $a$, then", "$\\theta$ defines a point $\\theta \\in G(A)$ whose image", "in $G(K)$ is trivial. Certainly, if the identity $e : \\Spec(A) \\to G$", "is a closed immersion, then this can happen only if", "$\\theta$ is the identity.", "Looking at Lemma \\ref{lemma-diagonal-diagonal}", "we obtain the result we want." ], "refs": [ "stacks-morphisms-lemma-diagonal-diagonal" ], "ref_ids": [ 7416 ] } ], "ref_ids": [ 7628 ] }, { "id": 7572, "type": "theorem", "label": "stacks-morphisms-lemma-cat-dotted-arrows-independent", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-cat-dotted-arrows-independent", "contents": [ "In Definition \\ref{definition-fill-in-diagram}", "assume $\\mathcal{I}_\\mathcal{Y} \\to \\mathcal{Y}$ is proper", "(for example if $\\mathcal{Y}$ is separated or if $\\mathcal{Y}$", "is separated over an algebraic space). Then the category of dotted arrows", "is independent (up to noncanonical equivalence) of the choice of $\\gamma$", "and the existence of a dotted arrow", "(for some and hence equivalently all $\\gamma$)", "is equivalent to the existence of a diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\", "\\Spec(A) \\ar[r]^-y \\ar[ru]_a & \\mathcal{Y}", "}", "$$", "with $2$-commutative triangles", "(without checking the $2$-morphisms compose correctly)." ], "refs": [ "stacks-morphisms-definition-fill-in-diagram" ], "proofs": [ { "contents": [ "Let $\\gamma, \\gamma' : y \\circ j \\longrightarrow f \\circ x$", "be two $2$-morphisms. Then $\\gamma^{-1} \\circ \\gamma'$", "is an automorphism of $y$ over $\\Spec(K)$.", "Hence if $\\mathit{Isom}_\\mathcal{Y}(y, y) \\to \\Spec(A)$", "is proper, then by the valuative criterion of properness", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-characterize-proper})", "we can find $\\delta : y \\to y$ whose restriction to", "$\\Spec(K)$ is $\\gamma^{-1} \\circ \\gamma'$.", "Then we can use $\\delta$ to define an equivalence", "between the category of dotted arrows for $\\gamma$", "to the category of dotted arrows for $\\gamma'$ by", "sending $(a, \\alpha, \\beta)$ to $(a, \\alpha, \\beta \\circ \\delta)$.", "The final statement is clear." ], "refs": [ "spaces-morphisms-lemma-characterize-proper" ], "ref_ids": [ 4938 ] } ], "ref_ids": [ 7628 ] }, { "id": 7573, "type": "theorem", "label": "stacks-morphisms-lemma-cat-dotted-arrows-base-change", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-cat-dotted-arrows-base-change", "contents": [ "Assume given a $2$-commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_-{x'} \\ar[d]_j &", "\\mathcal{X}' \\ar[d]^p \\ar[r]_q &", "\\mathcal{X} \\ar[d]^f \\\\", "\\Spec(A) \\ar[r]^-{y'} &", "\\mathcal{Y}' \\ar[r]^g &", "\\mathcal{Y}", "}", "$$", "with the right square $2$-cartesian. Choose a $2$-arrow", "$\\gamma' : y' \\circ j \\to p \\circ x'$. Set", "$x = q \\circ x'$, $y = g \\circ y'$ and let", "$\\gamma : y \\circ j \\to f \\circ x$ be the composition of", "$\\gamma'$ with the $2$-arrow implicit in the $2$-commutativity", "of the right square. Then the category of dotted arrows", "for the left square and $\\gamma'$ is equivalent to the category of dotted", "arrows for the outer rectangle and $\\gamma$." ], "refs": [], "proofs": [ { "contents": [ "This lemma, although a bit of a brain teaser, is straightforward.", "(We do not know how to prove the analogue of this lemma if instead", "of the category of dotted arrows we look at the set of isomorphism", "classes of morphisms producing two $2$-commutative", "triangles as in Lemma \\ref{lemma-cat-dotted-arrows-independent};", "in fact this analogue may very well be wrong.)", "To prove the lemma we are allowed to replace", "$\\mathcal{X}'$ by the $2$-fibre product", "$\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X}$", "as described in Categories, Lemma", "\\ref{categories-lemma-2-product-categories-over-C}.", "Then the object $x'$ becomes the triple $(y' \\circ j, x, \\gamma)$.", "Then we can go from a dotted arrow $(a, \\alpha, \\beta)$ for the", "outer rectangle to a dotted arrow $(a', \\alpha', \\beta')$", "for the left square by taking $a' = (y', a, \\beta)$ and", "$\\alpha' = (\\text{id}_{y' \\circ j}, \\alpha)$ and", "$\\beta' = \\text{id}_{y'}$. Details omitted." ], "refs": [ "stacks-morphisms-lemma-cat-dotted-arrows-independent", "categories-lemma-2-product-categories-over-C" ], "ref_ids": [ 7572, 12280 ] } ], "ref_ids": [] }, { "id": 7574, "type": "theorem", "label": "stacks-morphisms-lemma-cat-dotted-arrows-composition", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-cat-dotted-arrows-composition", "contents": [ "Assume given a $2$-commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_-x \\ar[dd]_j & \\mathcal{X} \\ar[d]^f \\\\", "& \\mathcal{Y} \\ar[d]^g \\\\", "\\Spec(A) \\ar[r]^-z & \\mathcal{Z}", "}", "$$", "Choose a $2$-arrow $\\gamma : z \\circ j \\to g \\circ f \\circ x$.", "Let $\\mathcal{C}$ be the category of dotted arrows for", "the outer rectangle and $\\gamma$. Let $\\mathcal{C}'$ be the", "category of dotted arrows for the square", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_-{f \\circ x} \\ar[d]_j & \\mathcal{Y} \\ar[d]^g \\\\", "\\Spec(A) \\ar[r]^-z & \\mathcal{Z}", "}", "$$", "and $\\gamma$. There is a canonical functor $\\mathcal{C} \\to \\mathcal{C}'$", "which turns $\\mathcal{C}$ into a category fibred in groupoids over", "$\\mathcal{C}'$ and whose fibre categories are categories of dotted arrows", "for certain squares of the form", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\", "\\Spec(A) \\ar[r]^-y & \\mathcal{Y}", "}", "$$", "and some choice of $y \\circ j \\to f \\circ x$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: If $(a, \\alpha, \\beta)$ is an object of $\\mathcal{C}$,", "then $(f \\circ a, \\text{id}_f \\star \\alpha, \\beta)$ is an object", "of $\\mathcal{C}'$. Conversely, if $(y, \\delta, \\epsilon)$ is an", "object of $\\mathcal{C}'$ and $(a, \\alpha, \\beta)$ is an object", "of the category of dotted arrows of the last displayed diagram", "with $y \\circ j \\to f \\circ x$ given by $\\delta$, then", "$(a, \\alpha, (\\text{id}_g \\star \\beta) \\circ \\epsilon)$ is an", "object of $\\mathcal{C}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7575, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-uniqueness", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-uniqueness", "contents": [ "The base change of a morphism of algebraic stacks which satisfies the", "uniqueness part of the valuative criterion by any morphism of", "algebraic stacks is a morphism of algebraic stacks which satisfies the", "uniqueness part of the valuative criterion." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-cat-dotted-arrows-base-change}", "and the definition." ], "refs": [ "stacks-morphisms-lemma-cat-dotted-arrows-base-change" ], "ref_ids": [ 7573 ] } ], "ref_ids": [] }, { "id": 7576, "type": "theorem", "label": "stacks-morphisms-lemma-composition-uniqueness", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-uniqueness", "contents": [ "The composition of morphisms of algebraic stacks which satisfy the", "uniqueness part of the valuative criterion is another", "morphism of algebraic stacks which satisfies the", "uniqueness part of the valuative criterion." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-cat-dotted-arrows-composition}", "and the definition." ], "refs": [ "stacks-morphisms-lemma-cat-dotted-arrows-composition" ], "ref_ids": [ 7574 ] } ], "ref_ids": [] }, { "id": 7577, "type": "theorem", "label": "stacks-morphisms-lemma-uniqueness-representable", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-uniqueness-representable", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "which is representable by algebraic spaces. Then the following are equivalent", "\\begin{enumerate}", "\\item $f$ satisfies the uniqueness part of the valuative criterion,", "\\item for every scheme $T$ and morphism $T \\to \\mathcal{Y}$", "the morphism $\\mathcal{X} \\times_\\mathcal{Y} T \\to T$ satisfies", "the uniqueness part of the valuative criterion as a morphism", "of algebraic spaces.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7578, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-existence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-existence", "contents": [ "The base change of a morphism of algebraic stacks which satisfies the", "existence part of the valuative criterion by any morphism of", "algebraic stacks is a morphism of algebraic stacks which satisfies the", "existence part of the valuative criterion." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-cat-dotted-arrows-base-change}", "and the definition." ], "refs": [ "stacks-morphisms-lemma-cat-dotted-arrows-base-change" ], "ref_ids": [ 7573 ] } ], "ref_ids": [] }, { "id": 7579, "type": "theorem", "label": "stacks-morphisms-lemma-composition-existence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-existence", "contents": [ "The composition of morphisms of algebraic stacks which satisfy the", "existence part of the valuative criterion is another", "morphism of algebraic stacks which satisfies the", "existence part of the valuative criterion." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-cat-dotted-arrows-composition}", "and the definition." ], "refs": [ "stacks-morphisms-lemma-cat-dotted-arrows-composition" ], "ref_ids": [ 7574 ] } ], "ref_ids": [] }, { "id": 7580, "type": "theorem", "label": "stacks-morphisms-lemma-existence-representable", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-existence-representable", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "which is representable by algebraic spaces. Then the following are equivalent", "\\begin{enumerate}", "\\item $f$ satisfies the existence part of the valuative criterion,", "\\item for every scheme $T$ and morphism $T \\to \\mathcal{Y}$", "the morphism $\\mathcal{X} \\times_\\mathcal{Y} T \\to T$ satisfies", "the existence part of the valuative criterion as a morphism", "of algebraic spaces.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7581, "type": "theorem", "label": "stacks-morphisms-lemma-closed-immersion-valuative-criteria", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-closed-immersion-valuative-criteria", "contents": [ "A closed immersion of algebraic stacks satisfies both", "the existence and uniqueness part of the valuative criterion." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: reduce to the case of a closed immersion of", "schemes by Lemmas \\ref{lemma-uniqueness-representable} and", "\\ref{lemma-existence-representable}." ], "refs": [ "stacks-morphisms-lemma-uniqueness-representable", "stacks-morphisms-lemma-existence-representable" ], "ref_ids": [ 7577, 7580 ] } ], "ref_ids": [] }, { "id": 7582, "type": "theorem", "label": "stacks-morphisms-lemma-setoids-and-diagonal", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-setoids-and-diagonal", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "If $\\Delta_f$ is quasi-separated and if for every diagram", "(\\ref{equation-diagram}) and choice of $\\gamma$ as in", "Definition \\ref{definition-fill-in-diagram}", "the category of dotted arrows", "is a setoid, then $\\Delta_f$ is separated." ], "refs": [ "stacks-morphisms-definition-fill-in-diagram" ], "proofs": [ { "contents": [ "We are going to write out a detailed proof, but we strongly urge the", "reader to find their own proof, inspired by reading the argument", "given in the proof of Lemma \\ref{lemma-cat-dotted-arrows}.", "\\medskip\\noindent", "Assume $\\Delta_f$ is quasi-separated and for every diagram", "(\\ref{equation-diagram}) and choice of $\\gamma$ as in", "Definition \\ref{definition-fill-in-diagram}", "the category of dotted arrows is a setoid.", "By Lemma \\ref{lemma-diagonal-diagonal} it suffices to show that", "$e : \\mathcal{X} \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$", "is a closed immersion. By", "Lemma \\ref{lemma-first-diagonal-separated-second-diagonal-closed}", "it in fact suffices to show that $e = \\Delta_{f, 2}$ is", "universally closed.", "Either of these lemmas tells us that $e = \\Delta_{f, 2}$ is quasi-compact", "by our assumption that $\\Delta_f$ is quasi-separated.", "\\medskip\\noindent", "In this paragraph we will show that $e$ satisfies the existence", "part of the valuative criterion. Consider a $2$-commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_x \\ar[d]_j & \\mathcal{X} \\ar[d]^e \\\\", "\\Spec(A) \\ar[r]^{(a, \\theta)} & \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}", "}", "$$", "and let $\\alpha : (a, \\theta) \\circ j \\to e \\circ x$ be any $2$-morphism", "witnessing the $2$-commutativity of the diagram (we use $\\alpha$ instead", "of the letter $\\gamma$ used in Definition \\ref{definition-fill-in-diagram}).", "Note that $f \\circ \\theta = \\text{id}$; we will use this below.", "Observe that $e \\circ x = (x, \\text{id}_x)$ and", "$(a, \\theta) \\circ j = (a \\circ j, \\theta \\star \\text{id}_j)$.", "Thus we see that $\\alpha$ is a $2$-arrow $\\alpha : a \\circ j \\to x$", "compatible with $\\theta \\star \\text{id}_j$ and $\\text{id}_x$.", "Set $y = f \\circ x$ and $\\beta = \\text{id}_{f \\circ a}$.", "Reading the arguments given in the proof of", "Lemma \\ref{lemma-cat-dotted-arrows}", "backwards, we see that $\\theta$ is an automorphism of the", "dotted arrow $(a, \\alpha, \\beta)$ with", "$$", "\\gamma : y \\circ j \\to f \\circ x", "\\quad\\text{equal to}\\quad", "\\text{id}_f \\star \\alpha : f \\circ a \\circ j \\to f \\circ x", "$$", "On the other hand, $\\text{id}_a$ is an automorphism too, hence", "we conclude $\\theta = \\text{id}_a$ from the assumption on $f$.", "Then we can take as dotted arrow for the displayed diagram above", "the morphism $a : \\Spec(A) \\to \\mathcal{X}$ with $2$-morphisms", "$(a, \\text{id}_a) \\circ j \\to (x, \\text{id}_x)$ given by $\\alpha$", "and $(a, \\theta) \\to e \\circ a$ given by $\\text{id}_a$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-base-change-existence} any base change of $e$", "satisfies the existence part of the valuative criterion.", "Since $e$ is representable by algebraic spaces, it suffices to", "show that $e$ is universally closed after a base change", "by a morphism $I \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$", "which is surjective and smooth and with $I$ an algebraic space", "(see Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}).", "This base change $e' : X' \\to I'$ is a quasi-compact", "morphism of algebraic spaces which", "satisfies the existence part of the valuative criterion", "and hence is universally closed by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-existence-universally-closed}." ], "refs": [ "stacks-morphisms-lemma-cat-dotted-arrows", "stacks-morphisms-definition-fill-in-diagram", "stacks-morphisms-lemma-diagonal-diagonal", "stacks-morphisms-lemma-first-diagonal-separated-second-diagonal-closed", "stacks-morphisms-definition-fill-in-diagram", "stacks-morphisms-lemma-cat-dotted-arrows", "stacks-morphisms-lemma-base-change-existence", "spaces-morphisms-lemma-quasi-compact-existence-universally-closed" ], "ref_ids": [ 7571, 7628, 7416, 7419, 7628, 7571, 7578, 4930 ] } ], "ref_ids": [ 7628 ] }, { "id": 7583, "type": "theorem", "label": "stacks-morphisms-lemma-helper-diagonal", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-helper-diagonal", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Consider a $2$-commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[rr]_-x \\ar[d]_j & &", "\\mathcal{X} \\ar[d]^{\\Delta_f} \\\\", "\\Spec(A) \\ar[rr]^{(a_1, a_2, \\varphi)} \\ar@{..>}[rru] & &", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$. Let", "$\\gamma : (a_1, a_2, \\varphi) \\circ j \\longrightarrow \\Delta_f \\circ x$", "be a $2$-morphism witnessing the $2$-commutativity of the diagram.", "Then", "\\begin{enumerate}", "\\item Writing $\\gamma = (\\alpha_1, \\alpha_2)$ with", "$\\alpha_i : a_i \\circ j \\to x$ we obtain two dotted arrows", "$(a_1, \\alpha_1, \\text{id})$ and $(a_2, \\alpha_2, \\varphi)$ in", "the diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\", "\\Spec(A) \\ar[r]^-{f \\circ a_1} \\ar@{..>}[ru] & \\mathcal{Y}", "}", "$$", "\\item The category of dotted arrows for the original diagram", "and $\\gamma$ is a setoid whose set of isomorphism", "classes of objects equal to the set of morphisms", "$(a_1, \\alpha_1, \\text{id}) \\to (a_2, \\alpha_2, \\varphi)$ in", "the category of dotted arrows.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $\\Delta_f$ is representable by algebraic spaces (hence the diagonal", "of $\\Delta_f$ is separated), we see that the category of dotted arrows", "in the first commutative diagram of the lemma is a setoid by", "Lemma \\ref{lemma-cat-dotted-arrows}. All the other statements", "of the lemma are consequences of $2$-diagramatic", "computations which we omit." ], "refs": [ "stacks-morphisms-lemma-cat-dotted-arrows" ], "ref_ids": [ 7571 ] } ], "ref_ids": [] }, { "id": 7584, "type": "theorem", "label": "stacks-morphisms-lemma-uniqueness-and-diagonal", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-uniqueness-and-diagonal", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Assume $f$ is quasi-separated.", "If $f$ satisfies the uniqueness part of the valuative criterion,", "then $f$ is separated." ], "refs": [], "proofs": [ { "contents": [ "The assumption on $f$ means that $\\Delta_f$ is quasi-compact", "and quasi-separated (Definition \\ref{definition-separated}).", "We have to show that $\\Delta_f$ is proper.", "Lemma \\ref{lemma-setoids-and-diagonal} says that $\\Delta_f$", "is separated. By Lemma \\ref{lemma-properties-diagonal}", "we know that $\\Delta_f$ is locally of finite type.", "To finish the proof we have to show that", "$\\Delta_f$ is universally closed. A formal argument", "(see Lemma \\ref{lemma-helper-diagonal}) shows that", "the uniqueness part of the valuative criterion implies", "that we have the existence of a dotted arrow in any solid diagram like so:", "$$", "\\xymatrix{", "\\Spec(K) \\ar[d] \\ar[r] & \\mathcal{X} \\ar[d]^{\\Delta_f} \\\\", "\\Spec(A) \\ar[r] \\ar@{..>}[ru] & \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}", "}", "$$", "Using that this property is preserved by any base change", "we conclude that any base change by $\\Delta_f$ by an algebraic", "space mapping into $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "has the existence part of the valuative criterion and", "we conclude is universally closed by the valuative criterion", "for morphisms of algebraic spaces, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-existence-universally-closed}." ], "refs": [ "stacks-morphisms-definition-separated", "stacks-morphisms-lemma-setoids-and-diagonal", "stacks-morphisms-lemma-properties-diagonal", "stacks-morphisms-lemma-helper-diagonal", "spaces-morphisms-lemma-quasi-compact-existence-universally-closed" ], "ref_ids": [ 7601, 7582, 7392, 7583, 4930 ] } ], "ref_ids": [] }, { "id": 7585, "type": "theorem", "label": "stacks-morphisms-lemma-converse-uniqueness-and-diagonal", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-converse-uniqueness-and-diagonal", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "If $f$ is separated, then $f$ satisfies the", "uniqueness part of the valuative criterion." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is separated we see that all categories of dotted arrows", "are setoids by Lemma \\ref{lemma-cat-dotted-arrows}.", "Consider a diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\", "\\Spec(A) \\ar[r]^-y \\ar@{..>}[ru] & \\mathcal{Y}", "}", "$$", "and a $2$-morphism $\\gamma : y \\circ j \\to f \\circ x$ as in", "Definition \\ref{definition-fill-in-diagram}. Consider two", "objects $(a, \\alpha, \\beta)$ and $(a', \\beta', \\alpha')$", "of the category of dotted arrows. To finish the proof we", "have to show these objects are isomorphic. The isomorphism", "$$", "f \\circ a \\xrightarrow{\\beta^{-1}} y \\xrightarrow{\\beta'} f \\circ a'", "$$", "means that $(a, a', \\beta' \\circ \\beta^{-1})$ is a morphism", "$\\Spec(A) \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$.", "On the other hand, $\\alpha$ and $\\alpha'$ define", "a $2$-arrow", "$$", "(a, a', \\beta' \\circ \\beta^{-1}) \\circ j =", "(a \\circ j, a' \\circ j,", "(\\beta' \\star \\text{id}_j) \\circ (\\beta \\star \\text{id}_j)^{-1})", "\\xrightarrow{(\\alpha, \\alpha')} (x, x, \\text{id}) = \\Delta_f \\circ x", "$$", "Here we use that both $(a, \\alpha, \\beta)$ and $(a', \\alpha', \\beta')$", "are dotted arrows with respect to $\\gamma$.", "We obtain a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[d]_j \\ar[rr]_x & & \\mathcal{X} \\ar[d]^{\\Delta_f} \\\\", "\\Spec(A) \\ar[rr]^{(a, a', \\beta' \\circ \\beta^{-1})} & &", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}", "}", "$$", "with $2$-commutativity witnessed by $(\\alpha, \\alpha')$. Now", "$\\Delta_f$ is representable by algebraic spaces", "(Lemma \\ref{lemma-properties-diagonal})", "and proper as $f$ is separated. Hence by", "Lemma \\ref{lemma-existence-representable}", "and the valuative criterion for properness for algebraic spaces", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-characterize-proper})", "we see that there exists a dotted arrow.", "Unwinding the construction, we see that this means", "$(a, \\alpha, \\beta)$ and $(a', \\alpha', \\beta')$", "are isomorphic in the category of dotted arrows as desired." ], "refs": [ "stacks-morphisms-lemma-cat-dotted-arrows", "stacks-morphisms-definition-fill-in-diagram", "stacks-morphisms-lemma-properties-diagonal", "stacks-morphisms-lemma-existence-representable", "spaces-morphisms-lemma-characterize-proper" ], "ref_ids": [ 7571, 7628, 7392, 7580, 4938 ] } ], "ref_ids": [] }, { "id": 7586, "type": "theorem", "label": "stacks-morphisms-lemma-quasi-compact-existence-universally-closed", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-quasi-compact-existence-universally-closed", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Assume", "\\begin{enumerate}", "\\item $f$ is quasi-compact, and", "\\item $f$ satisfies the existence part of the valuative criterion.", "\\end{enumerate}", "Then $f$ is universally closed." ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-base-change-quasi-compact}", "and \\ref{lemma-base-change-existence}", "properties (1) and (2) are preserved under", "any base change. By Lemma \\ref{lemma-universally-closed-local}", "we only have to show that $|T \\times_\\mathcal{Y} \\mathcal{X}| \\to |T|$", "is closed, whenever $T$ is an affine scheme mapping into $\\mathcal{Y}$.", "Hence it suffices to show: if $f : \\mathcal{X} \\to Y$ is a", "quasi-compact morphism from an algebraic stack to an affine scheme", "satisfying the existence part of the valuative criterion,", "then $|f|$ is closed. Let $T \\subset |\\mathcal{X}|$ be a closed subset.", "We have to show that $f(T)$ is closed to finish the proof.", "\\medskip\\noindent", "Let $\\mathcal{Z} \\subset \\mathcal{X}$ be the reduced induced", "algebraic stack structure on $T$ (Properties of Stacks,", "Definition \\ref{stacks-properties-definition-reduced-induced-stack}).", "Then $i : \\mathcal{Z} \\to \\mathcal{X}$ is a closed immersion", "and we have to show that the image of $|\\mathcal{Z}| \\to |Y|$", "is closed. Since closed immersions are quasi-compact", "(Lemma \\ref{lemma-closed-immersion-quasi-compact})", "and satisfies the existence part of the valuative criterion", "(Lemma \\ref{lemma-closed-immersion-valuative-criteria})", "and since compositions of quasi-compact morphisms are quasi-compact", "(Lemma \\ref{lemma-composition-quasi-compact})", "and since compositions preserve the property of satisfying", "the existence part of the valuative criterion", "(Lemma \\ref{lemma-composition-existence})", "we conclude that it suffices to show: if $f : \\mathcal{X} \\to Y$", "is a quasi-compact morphism from an algebraic stack to an affine scheme", "satisfying the existence part of the valuative criterion,", "then $|f|(|\\mathcal{X}|)$ is closed.", "\\medskip\\noindent", "Since $\\mathcal{X}$ is quasi-compact (being quasi-compact over the", "affine $Y$), we can choose an affine scheme $U$ and a surjective", "smooth morphism $U \\to \\mathcal{X}$", "(Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-quasi-compact-stack}).", "Suppose that $y \\in Y$ is in the closure of the image of $U \\to Y$", "(in other words, in the closure of the image of $|f|$).", "Then by", "Morphisms, Lemma \\ref{morphisms-lemma-reach-points-scheme-theoretic-image}", "we can find a valuation ring $A$ with fraction field $K$", "and a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & U \\ar[d] \\\\", "\\Spec(A) \\ar[r] & Y", "}", "$$", "such that the closed point of $\\Spec(A)$ maps to $y$. By assumption", "we get an extension $K'/K$ and a valuation ring $A' \\subset K'$", "dominating $A$ and the dotted arrow in the following diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] &", "\\Spec(K) \\ar[r] \\ar[d] &", "U \\ar[d] \\ar[r] &", "\\mathcal{X} \\ar[d]^f \\\\", "\\Spec(A') \\ar[r] \\ar@{..>}[rrru] &", "\\Spec(A) \\ar[r] &", "Y \\ar@{=}[r] &", "Y", "}", "$$", "Thus $y$ is in the image of $|f|$ and we win." ], "refs": [ "stacks-morphisms-lemma-base-change-quasi-compact", "stacks-morphisms-lemma-base-change-existence", "stacks-morphisms-lemma-universally-closed-local", "stacks-properties-definition-reduced-induced-stack", "stacks-morphisms-lemma-closed-immersion-quasi-compact", "stacks-morphisms-lemma-closed-immersion-valuative-criteria", "stacks-morphisms-lemma-composition-quasi-compact", "stacks-morphisms-lemma-composition-existence", "stacks-properties-lemma-quasi-compact-stack", "morphisms-lemma-reach-points-scheme-theoretic-image" ], "ref_ids": [ 7423, 7578, 7446, 8922, 7425, 7581, 7424, 7579, 8873, 5147 ] } ], "ref_ids": [] }, { "id": 7587, "type": "theorem", "label": "stacks-morphisms-lemma-converse-existence-universally-closed", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-converse-existence-universally-closed", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Assume", "\\begin{enumerate}", "\\item $f$ is quasi-separated, and", "\\item $f$ is universally closed.", "\\end{enumerate}", "Then $f$ satisfies the existence part of the valuative criterion." ], "refs": [], "proofs": [ { "contents": [ "Consider a solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\", "\\Spec(A) \\ar[r]^-y \\ar@{..>}[ru] & \\mathcal{Y}", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$", "and $\\gamma : y \\circ j \\longrightarrow f \\circ x$ as in", "Definition \\ref{definition-fill-in-diagram}. By", "Lemma \\ref{lemma-cat-dotted-arrows-base-change}", "in order to find a dotted arrow (after possibly replacing", "$K$ by an extension and $A$ by a valuation ring dominating it)", "we may replace $\\mathcal{Y}$ by $\\Spec(A)$ and $\\mathcal{X}$", "by $\\Spec(A) \\times_\\mathcal{Y} \\mathcal{X}$. Of course", "we use here that being", "quasi-separated and universally closed are preserved under base change.", "Thus we reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Consider a solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\", "\\Spec(A) \\ar@{=}[r] \\ar@{..>}[ru] & \\Spec(A)", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$ as in", "Definition \\ref{definition-fill-in-diagram}.", "By Lemma \\ref{lemma-quasi-compact-permanence} and the fact", "that $f$ is quasi-separated we have that", "the morphism $x$ is quasi-compact.", "Since $f$ is universally closed, we have in particular", "that $|f|(\\overline{\\{x\\}})$ is closed in $\\Spec(A)$.", "Since this image contains the generic point of $\\Spec(A)$", "there exists a point $x' \\in |\\mathcal{X}|$ in the closure", "of $x$ mapping to the closed point of $\\Spec(A)$.", "By Lemma \\ref{lemma-reach-points-scheme-theoretic-image}", "we can find a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[d] \\\\", "\\Spec(A') \\ar[r] & \\mathcal{X}", "}", "$$", "such that the closed point of $\\Spec(A')$ maps to $x' \\in |\\mathcal{X}|$.", "It follows that $\\Spec(A') \\to \\Spec(A)$ maps the closed point", "to the closed point, i.e., $A'$ dominates $A$ and this finishes the proof." ], "refs": [ "stacks-morphisms-definition-fill-in-diagram", "stacks-morphisms-lemma-cat-dotted-arrows-base-change", "stacks-morphisms-definition-fill-in-diagram", "stacks-morphisms-lemma-quasi-compact-permanence", "stacks-morphisms-lemma-reach-points-scheme-theoretic-image" ], "ref_ids": [ 7628, 7573, 7628, 7427, 7429 ] } ], "ref_ids": [] }, { "id": 7588, "type": "theorem", "label": "stacks-morphisms-lemma-criterion-proper", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-criterion-proper", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Assume $f$ is of finite type and quasi-separated.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $f$ is proper, and", "\\item $f$ satisfies both the uniqueness and existence parts", "of the valuative criterion.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "A proper morphism is the same thing as a separated, finite type, and", "universally closed morphism. Thus this lemma follows from Lemmas", "\\ref{lemma-uniqueness-and-diagonal},", "\\ref{lemma-converse-uniqueness-and-diagonal},", "\\ref{lemma-quasi-compact-existence-universally-closed}, and", "\\ref{lemma-converse-existence-universally-closed}." ], "refs": [ "stacks-morphisms-lemma-uniqueness-and-diagonal", "stacks-morphisms-lemma-converse-uniqueness-and-diagonal", "stacks-morphisms-lemma-quasi-compact-existence-universally-closed", "stacks-morphisms-lemma-converse-existence-universally-closed" ], "ref_ids": [ 7584, 7585, 7586, 7587 ] } ], "ref_ids": [] }, { "id": 7589, "type": "theorem", "label": "stacks-morphisms-lemma-composition-lci", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-composition-lci", "contents": [ "The composition of local complete intersection morphisms is", "a local complete intersection." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Remark \\ref{remark-composition}", "with", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-composition-lci}." ], "refs": [ "stacks-morphisms-remark-composition", "spaces-more-morphisms-lemma-composition-lci" ], "ref_ids": [ 7633, 238 ] } ], "ref_ids": [] }, { "id": 7590, "type": "theorem", "label": "stacks-morphisms-lemma-flat-base-change-lci", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-flat-base-change-lci", "contents": [ "A flat base change of a local complete intersection morphism is", "a local complete intersection morphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Argue exactly as in Remark \\ref{remark-base-change}", "(but only for flat $\\mathcal{Y}' \\to \\mathcal{Y}$) using", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-flat-base-change-lci}." ], "refs": [ "stacks-morphisms-remark-base-change", "spaces-more-morphisms-lemma-flat-base-change-lci" ], "ref_ids": [ 7634, 237 ] } ], "ref_ids": [] }, { "id": 7591, "type": "theorem", "label": "stacks-morphisms-lemma-lci-permanence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-lci-permanence", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{X} \\ar[rr]_f \\ar[rd] & & \\mathcal{Y} \\ar[ld] \\\\", "& \\mathcal{Z}", "}", "$$", "be a commutative diagram of morphisms of algebraic stacks.", "Assume $\\mathcal{Y} \\to \\mathcal{Z}$ is smooth and", "$\\mathcal{X} \\to \\mathcal{Z}$ is a local complete intersection morphism.", "Then $f : \\mathcal{X} \\to \\mathcal{Y}$ is a", "local complete intersection morphism." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $W$ and a surjective smooth morphism $W \\to \\mathcal{Z}$.", "Choose a scheme $V$ and a surjective smooth morphism", "$V \\to W \\times_\\mathcal{Z} \\mathcal{Y}$.", "Choose a scheme $U$ and a surjective smooth morphism", "$U \\to V \\times_\\mathcal{Y} \\mathcal{X}$.", "Then $U \\to W$ is a local complete intersection morphism of schemes and", "$V \\to W$ is a smooth morphism of schemes. By the result for schemes", "(More on Morphisms, Lemma \\ref{more-morphisms-lemma-lci-permanence})", "we conclude that $U \\to V$ is a local complete intersection morphism.", "By definition this means that $f$ is a local complete intersection morphism." ], "refs": [ "more-morphisms-lemma-lci-permanence" ], "ref_ids": [ 14008 ] } ], "ref_ids": [] }, { "id": 7592, "type": "theorem", "label": "stacks-morphisms-lemma-stabilizer-preserving", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-stabilizer-preserving", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "If $\\mathcal{I}_\\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$ is an isomorphism,", "then $f$ is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-second-diagonal}." ], "refs": [ "stacks-morphisms-lemma-second-diagonal" ], "ref_ids": [ 7417 ] } ], "ref_ids": [] }, { "id": 7593, "type": "theorem", "label": "stacks-morphisms-lemma-aut-iso-unramified", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-aut-iso-unramified", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be an unramified", "morphism of algebraic stacks. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{I}_\\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$", "is an isomorphism, and", "\\item $f$ induces an isomorphism between automorphism groups at $x$ and $f(x)$", "(Remark \\ref{remark-identify-automorphism-groups}) for all", "$x \\in |\\mathcal{X}|$.", "\\end{enumerate}" ], "refs": [ "stacks-morphisms-remark-identify-automorphism-groups" ], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective smooth morphism $U \\to \\mathcal{X}$.", "Denote $G \\to H$ the pullback of the morphism", "$\\mathcal{I}_\\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$", "to $U$. By Remark \\ref{remark-get-property-auts-from-diagonal} and", "Lemma \\ref{lemma-characterize-unramified} the morphism $G \\to H$ is \\'etale.", "Condition (1) is equivalent to the condition that", "$G \\to H$ is an isomorphism (this follows for example by applying", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}).", "Condition (2) is equivalent to the condition that", "for every $u \\in U$ the morphism $G_u \\to H_u$ of fibres", "is an isomorphism. Thus (1) $\\Rightarrow$ (2) is trivial.", "If (2) holds, then $G \\to H$ is a surjective, universally injective,", "\\'etale morphism of algebraic spaces. Such a morphism is an isomorphism by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-etale-universally-injective-open}." ], "refs": [ "stacks-morphisms-remark-get-property-auts-from-diagonal", "stacks-morphisms-lemma-characterize-unramified", "stacks-properties-lemma-check-property-covering", "spaces-morphisms-lemma-etale-universally-injective-open" ], "ref_ids": [ 7642, 7558, 8859, 4973 ] } ], "ref_ids": [ 7637 ] }, { "id": 7594, "type": "theorem", "label": "stacks-morphisms-lemma-stabilizer-preserving-unramified", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-stabilizer-preserving-unramified", "contents": [ "\\begin{reference}", "\\cite[Proposition 3.5]{rydh_quotients} and", "\\cite[Proposition 2.5]{alper_quotient}", "\\end{reference}", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Assume", "\\begin{enumerate}", "\\item $f$ is representable by algebraic spaces and unramified, and", "\\item $\\mathcal{I}_\\mathcal{Y} \\to \\mathcal{Y}$ is proper.", "\\end{enumerate}", "Then the set of $x \\in |\\mathcal{X}|$ such that $f$ induces an", "isomorphism between automorphism groups at $x$ and $f(x)$", "(Remark \\ref{remark-identify-automorphism-groups}) is open.", "Letting $\\mathcal{U} \\subset \\mathcal{X}$ be the corresponding open substack,", "the morphism", "$\\mathcal{I}_\\mathcal{U} \\to", "\\mathcal{U} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$", "is an isomorphism." ], "refs": [ "stacks-morphisms-remark-identify-automorphism-groups" ], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective smooth morphism $U \\to \\mathcal{X}$.", "Denote $G \\to H$ the pullback of the morphism", "$\\mathcal{I}_\\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$", "to $U$. By Remark \\ref{remark-get-property-auts-from-diagonal} and", "Lemma \\ref{lemma-characterize-unramified} the morphism", "$G \\to H$ is \\'etale. Since $f$ is representable by algebraic spaces,", "we see that $G \\to H$ is a monomorphism. Hence $G \\to H$ is an open", "immersion, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-etale-universally-injective-open}.", "By assumption $H \\to U$ is proper.", "\\medskip\\noindent", "With these preparations out of the way, we can prove the lemma as follows.", "The inverse image of the subset of $|\\mathcal{X}|$ of the lemma is", "clearly the set of $u \\in U$ such that $G_u \\to H_u$ is an isomorphism", "(since after all $G_u$ is an open sub group algebraic space of $H_u$).", "This is an open subset because the complement is", "the image of the closed subset $|H| \\setminus |G|$ and $|H| \\to |U|$", "is closed. By", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-open-substacks}", "we can consider the corresponding open substack $\\mathcal{U}$ of $\\mathcal{X}$.", "The final statement of the lemma follows from", "applying Lemma \\ref{lemma-aut-iso-unramified}", "to $\\mathcal{U} \\to \\mathcal{Y}$." ], "refs": [ "stacks-morphisms-remark-get-property-auts-from-diagonal", "stacks-morphisms-lemma-characterize-unramified", "spaces-morphisms-lemma-etale-universally-injective-open", "stacks-properties-lemma-open-substacks", "stacks-morphisms-lemma-aut-iso-unramified" ], "ref_ids": [ 7642, 7558, 4973, 8890, 7593 ] } ], "ref_ids": [ 7637 ] }, { "id": 7595, "type": "theorem", "label": "stacks-morphisms-lemma-base-change-stabilizer-preserving", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-base-change-stabilizer-preserving", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\", "\\mathcal{Y}' \\ar[r] & \\mathcal{Y}", "}", "$$", "be a cartesian diagram of algebraic stacks.", "\\begin{enumerate}", "\\item Let $x' \\in |\\mathcal{X}'|$ with image $x \\in |\\mathcal{X}|$.", "If $f$ induces an isomorphism between automorphism groups at", "$x$ and $f(x)$ (Remark \\ref{remark-identify-automorphism-groups}), then", "$f'$ induces an isomorphism between automorphism groups at $x'$ and $f(x')$.", "\\item If $\\mathcal{I}_\\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$ is an isomorphism,", "then $\\mathcal{I}_{\\mathcal{X}'} \\to", "\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{I}_{\\mathcal{Y}'}$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [ "stacks-morphisms-remark-identify-automorphism-groups" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 7637 ] }, { "id": 7596, "type": "theorem", "label": "stacks-morphisms-lemma-stabilizer-preserving-points-cartesian", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-stabilizer-preserving-points-cartesian", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\", "\\mathcal{Y}' \\ar[r]^g & \\mathcal{Y}", "}", "$$", "be a cartesian diagram of algebraic stacks. If $f$ induces an isomorphism", "between automorphism groups at points", "(Remark \\ref{remark-identify-automorphism-groups}),", "then", "$$", "\\Mor(\\Spec(k), \\mathcal{X}')", "\\longrightarrow", "\\Mor(\\Spec(k), \\mathcal{Y}') \\times \\Mor(\\Spec(k), \\mathcal{X})", "$$", "is injective on isomorphism classes for any field $k$." ], "refs": [ "stacks-morphisms-remark-identify-automorphism-groups" ], "proofs": [ { "contents": [ "We have to show that given $(y', x)$ there is at most one $x'$", "mapping to it.", "By our construction of $2$-fibre products, a morphism", "$x'$ is given by a triple $(x, y', \\alpha)$", "where $\\alpha : g \\circ y' \\to f \\circ x$ is a $2$-morphism.", "Now, suppose we have a second such triple $(x, y', \\beta)$.", "Then $\\alpha$ and $\\beta$ differ by a $k$-valued point", "$\\epsilon$ of the automorphism group algebraic space $G_{f(x)}$.", "Since $f$ induces an isomorphism $G_x \\to G_{f(x)}$ by", "assumption, this means we can lift $\\epsilon$ to a $k$-valued point", "$\\gamma$ of $G_x$. Then $(\\gamma, \\text{id}) : (x, y', \\alpha) \\to", "(x, y', \\beta)$ is an isomorphism as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 7637 ] }, { "id": 7597, "type": "theorem", "label": "stacks-morphisms-lemma-etale-iso", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-lemma-etale-iso", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Assume $f$ is separated, \\'etale, $f$ induces an isomorphism", "between automorphism groups at points", "(Remark \\ref{remark-identify-automorphism-groups})", "and for every algebraically closed field $k$ the functor", "$$", "f : \\Mor(\\Spec(k), \\mathcal{X}) \\longrightarrow \\Mor(\\Spec(k), \\mathcal{Y})", "$$", "is an equivalence. Then $f$ is an isomorphism." ], "refs": [ "stacks-morphisms-remark-identify-automorphism-groups" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-universally-injective} we see that $f$ is", "universally injective. Combining", "Lemmas \\ref{lemma-stabilizer-preserving} and", "\\ref{lemma-aut-iso-unramified}", "we see that $f$ is representable by algebraic spaces.", "Hence $f$ is an open immersion by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-etale-universally-injective-open}.", "To finish we remark that the condition in the lemma also guarantees", "that $f$ is surjective." ], "refs": [ "stacks-morphisms-lemma-universally-injective", "stacks-morphisms-lemma-stabilizer-preserving", "stacks-morphisms-lemma-aut-iso-unramified", "spaces-morphisms-lemma-etale-universally-injective-open" ], "ref_ids": [ 7450, 7592, 7593, 4973 ] } ], "ref_ids": [ 7637 ] }, { "id": 7598, "type": "theorem", "label": "stacks-morphisms-proposition-when-gerbe", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-proposition-when-gerbe", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is a gerbe, and", "\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is flat and locally of", "finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Choose a morphism $\\mathcal{X} \\to X$ into an algebraic space $X$", "which turns $\\mathcal{X}$ into a gerbe over $X$. Let $X' \\to X$ be a", "surjective, flat, locally finitely presented morphism and", "set $\\mathcal{X}' = X' \\times_X \\mathcal{X}$. Note that $\\mathcal{X}'$", "is a gerbe over $X'$ by", "Lemma \\ref{lemma-base-change-gerbe}.", "Then both squares in", "$$", "\\xymatrix{", "\\mathcal{I}_{\\mathcal{X}'} \\ar[r] \\ar[d] &", "\\mathcal{X}' \\ar[r] \\ar[d] & X' \\ar[d] \\\\", "\\mathcal{I}_\\mathcal{X} \\ar[r] &", "\\mathcal{X} \\ar[r] & X", "}", "$$", "are fibre product squares, see", "Lemma \\ref{lemma-cartesian-square-inertia}.", "Hence to prove $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is flat and", "locally of finite presentation it suffices to do so after such a base", "change by", "Lemmas \\ref{lemma-descent-flat} and", "\\ref{lemma-descent-finite-presentation}.", "Thus we can apply", "Lemma \\ref{lemma-local-structure-gerbe}", "to assume that $\\mathcal{X} = [U/G]$.", "By", "Lemma \\ref{lemma-gerbe-with-section}", "we see $G$ is flat and locally of finite presentation over $U$ and", "that $x : U \\to [U/G]$ is surjective, flat, and locally of finite", "presentation. Moreover, the pullback of $\\mathcal{I}_\\mathcal{X}$", "by $x$ is $G$ and we conclude that (2) holds by descent again, i.e., by", "Lemmas \\ref{lemma-descent-flat} and", "\\ref{lemma-descent-finite-presentation}.", "\\medskip\\noindent", "Conversely, assume (2). Choose a smooth presentation $\\mathcal{X} = [U/R]$, see", "Algebraic Stacks, Section \\ref{algebraic-section-stack-to-presentation}.", "Denote $G \\to U$ the stabilizer group algebraic space of the groupoid", "$(U, R, s, t, c, e, i)$, see", "Groupoids in Spaces, Definition", "\\ref{spaces-groupoids-definition-stabilizer-groupoid}.", "By", "Lemma \\ref{lemma-presentation-inertia}", "we see that $G \\to U$ is flat and locally of finite presentation as", "a base change of $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$, see", "Lemmas \\ref{lemma-base-change-flat} and", "\\ref{lemma-base-change-finite-presentation}.", "Consider the following action", "$$", "a : G \\times_{U, t} R \\to R, \\quad (g, r) \\mapsto c(g, r)", "$$", "of $G$ on $R$. This action is free on $T$-valued points for any", "scheme $T$ as $R$ is a groupoid. Hence $R' = R/G$ is an algebraic", "space and the quotient morphism $\\pi : R \\to R'$ is surjective,", "flat, and locally of finite presentation by", "Bootstrap, Lemma \\ref{bootstrap-lemma-quotient-free-action}.", "The projections $s, t : R \\to U$ are $G$-invariant, hence", "we obtain morphisms $s' , t' : R' \\to U$ such that $s = s' \\circ \\pi$", "and $t = t' \\circ \\pi$.", "Since $s, t : R \\to U$ are flat and locally of finite presentation", "we conclude that $s', t'$ are flat and locally of finite presentation, see", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-flat-permanence} and", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-locally-finite-presentation-fppf-local-source}.", "Consider the morphism", "$$", "j' = (t', s') : R' \\longrightarrow U \\times U.", "$$", "We claim this is a monomorphism. Namely, suppose that $T$ is a scheme", "and that $a, b : T \\to R'$ are morphisms which have the same image", "in $U \\times U$. By definition of the quotient $R' = R/G$ there", "exists an fppf covering $\\{h_j : T_j \\to T\\}$ such", "that $a \\circ h_j = \\pi \\circ a_j$ and $b \\circ h_j = \\pi \\circ b_j$", "for some morphisms $a_j, b_j : T_j \\to R$. Since $a_j, b_j$ have the same", "image in $U \\times U$ we see that $g_j = c(a_j, i(b_j))$ is a $T_j$-valued", "point of $G$ such that $c(g_j, b_j) = a_j$. In other words, $a_j$ and", "$b_j$ have the same image in $R'$ and the claim is proved.", "Since $j : R \\to U \\times U$ is a pre-equivalence relation (see", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-groupoid-pre-equivalence})", "and $R \\to R'$ is surjective (as a map of sheaves) we see that", "$j' : R' \\to U \\times U$ is an equivalence relation.", "Hence", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}", "shows that $X = U/R'$ is an algebraic space.", "Finally, we claim that the morphism", "$$", "\\mathcal{X} = [U/R] \\longrightarrow X = U/R'", "$$", "turns $\\mathcal{X}$ into a gerbe over $X$. This follows from", "Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-when-gerbe}", "as $R \\to R'$ is surjective, flat, and locally of finite presentation", "(if needed use", "Bootstrap, Lemma", "\\ref{bootstrap-lemma-surjective-flat-locally-finite-presentation}", "to see this implies the required hypothesis)." ], "refs": [ "stacks-morphisms-lemma-base-change-gerbe", "stacks-morphisms-lemma-cartesian-square-inertia", "stacks-morphisms-lemma-descent-flat", "stacks-morphisms-lemma-descent-finite-presentation", "stacks-morphisms-lemma-local-structure-gerbe", "stacks-morphisms-lemma-descent-flat", "stacks-morphisms-lemma-descent-finite-presentation", "spaces-groupoids-definition-stabilizer-groupoid", "stacks-morphisms-lemma-presentation-inertia", "stacks-morphisms-lemma-base-change-flat", "stacks-morphisms-lemma-base-change-finite-presentation", "bootstrap-lemma-quotient-free-action", "spaces-morphisms-lemma-flat-permanence", "spaces-descent-lemma-locally-finite-presentation-fppf-local-source", "spaces-groupoids-lemma-groupoid-pre-equivalence", "bootstrap-theorem-final-bootstrap", "spaces-groupoids-lemma-when-gerbe", "bootstrap-lemma-surjective-flat-locally-finite-presentation" ], "ref_ids": [ 7517, 7413, 7496, 7509, 7521, 7496, 7509, 9349, 7415, 7495, 7501, 2631, 4865, 9418, 9297, 2602, 9334, 2617 ] } ], "ref_ids": [] }, { "id": 7599, "type": "theorem", "label": "stacks-morphisms-proposition-when-gerbe-over", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-proposition-when-gerbe-over", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, and", "\\item $f : \\mathcal{X} \\to \\mathcal{Y}$ and", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "are surjective, flat, and locally of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) follows from", "Lemmas \\ref{lemma-gerbe-fppf} and \\ref{lemma-gerbe-isom-fppf}.", "\\medskip\\noindent", "Assume (2). It suffices to prove (1) for the base change of $f$", "by a surjective, flat, and locally finitely presented morphism", "$\\mathcal{Y}' \\to \\mathcal{Y}$, see", "Lemma \\ref{lemma-gerbe-descent} (note that the base change", "of the diagonal of $f$ is the diagonal of the base change). ", "Thus we may assume $\\mathcal{Y}$ is a scheme $Y$.", "In this case $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$", "is a base change of $\\Delta$ and", "we conclude that $\\mathcal{X}$ is a gerbe", "by Proposition \\ref{proposition-when-gerbe}.", "We still have to show that $\\mathcal{X}$ is a gerbe over $Y$.", "Let $\\mathcal{X} \\to X$ be the morphism of", "Lemma \\ref{lemma-gerbe-over-iso-classes}", "turning $\\mathcal{X}$ into a gerbe over the algebraic space $X$", "classifying isomorphism classes of objects of $\\mathcal{X}$.", "It is clear that $f : \\mathcal{X} \\to Y$ factors as", "$\\mathcal{X} \\to X \\to Y$. Since $f$ is surjective, flat, and", "locally of finite presentation, we conclude that $X \\to Y$ is", "surjective as a map of fppf sheaves (for example use", "Lemma \\ref{lemma-surjective-flat-locally-finite-presentation}).", "On the other hand, $X \\to Y$ is injective too: for any scheme", "$T$ and any two $T$-valued points $x_1, x_2$ of $X$ which map to", "the same point of $Y$, we can first fppf locally on $T$", "lift $x_1, x_2$ to objects $\\xi_1, \\xi_2$ of $\\mathcal{X}$ over $T$", "and second deduce that $\\xi_1$ and $\\xi_2$ are fppf locally isomorphic", "by our assumption that", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_Y \\mathcal{X}$", "is surjective, flat, and locally of finite presentation.", "Whence $x_1 = x_2$ by construction of $X$.", "Thus $X = Y$ and the proof is complete." ], "refs": [ "stacks-morphisms-lemma-gerbe-fppf", "stacks-morphisms-lemma-gerbe-isom-fppf", "stacks-morphisms-lemma-gerbe-descent", "stacks-morphisms-proposition-when-gerbe", "stacks-morphisms-lemma-gerbe-over-iso-classes", "stacks-morphisms-lemma-surjective-flat-locally-finite-presentation" ], "ref_ids": [ 7522, 7523, 7519, 7598, 7516, 7511 ] } ], "ref_ids": [] }, { "id": 7600, "type": "theorem", "label": "stacks-morphisms-proposition-open-stratum", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-proposition-open-stratum", "contents": [ "Let $\\mathcal{X}$ be a reduced algebraic stack such that", "$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is quasi-compact.", "Then there exists a dense open substack $\\mathcal{U} \\subset \\mathcal{X}$", "which is a gerbe." ], "refs": [], "proofs": [ { "contents": [ "According to", "Proposition \\ref{proposition-when-gerbe}", "it is enough to find a dense open substack $\\mathcal{U}$ such that", "$\\mathcal{I}_\\mathcal{U} \\to \\mathcal{U}$ is flat and locally of finite", "presentation. Note that", "$\\mathcal{I}_\\mathcal{U} =", "\\mathcal{I}_\\mathcal{X} \\times_\\mathcal{X} \\mathcal{U}$, see", "Lemma \\ref{lemma-cartesian-square-inertia}.", "\\medskip\\noindent", "Choose a presentation $\\mathcal{X} = [U/R]$. Let $G \\to U$ be the stabilizer", "group algebraic space of the groupoid $R$. By", "Lemma \\ref{lemma-presentation-inertia}", "we see that $G \\to U$ is the base change of", "$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ hence quasi-compact (by assumption)", "and locally of finite type (by", "Lemma \\ref{lemma-inertia}).", "Let $W \\subset U$ be the largest open (possibly empty) subscheme such that", "the restriction $G_W \\to W$ is flat and locally of finite presentation", "(we omit the proof that $W$ exists; hint: use that the properties are local).", "By", "Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-generic-flatness-reduced}", "we see that $W \\subset U$ is dense. Note that $W \\subset U$ is $R$-invariant", "by", "More on Groupoids in Spaces, Lemma", "\\ref{spaces-more-groupoids-lemma-property-G-invariant}.", "Hence $W$ corresponds to an open substack $\\mathcal{U} \\subset \\mathcal{X}$ by", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-substacks-presentation}.", "Since $|U| \\to |\\mathcal{X}|$ is open and $|W| \\subset |U|$ is dense we", "conclude that $\\mathcal{U}$ is dense in $\\mathcal{X}$.", "Finally, the morphism $\\mathcal{I}_\\mathcal{U} \\to \\mathcal{U}$", "is flat and locally of finite presentation because the base change by", "the surjective smooth morphism $W \\to \\mathcal{U}$ is the morphism", "$G_W \\to W$ which is flat and locally of finite presentation by construction.", "See", "Lemmas \\ref{lemma-descent-flat} and", "\\ref{lemma-descent-finite-presentation}." ], "refs": [ "stacks-morphisms-proposition-when-gerbe", "stacks-morphisms-lemma-cartesian-square-inertia", "stacks-morphisms-lemma-presentation-inertia", "stacks-morphisms-lemma-inertia", "spaces-morphisms-proposition-generic-flatness-reduced", "spaces-more-groupoids-lemma-property-G-invariant", "stacks-properties-lemma-substacks-presentation", "stacks-morphisms-lemma-descent-flat", "stacks-morphisms-lemma-descent-finite-presentation" ], "ref_ids": [ 7598, 7413, 7415, 7411, 4981, 13177, 8889, 7496, 7509 ] } ], "ref_ids": [] }, { "id": 7643, "type": "theorem", "label": "schemes-lemma-isomorphism-locally-ringed", "categories": [ "schemes" ], "title": "schemes-lemma-isomorphism-locally-ringed", "contents": [ "\\begin{slogan}", "An isomorphism of ringed spaces between locally ringed spaces is an", "isomorphism of locally ringed spaces.", "\\end{slogan}", "Let $X$, $Y$ be locally ringed spaces.", "If $f : X \\to Y$ is an isomorphism of", "ringed spaces, then $f$ is an isomorphism", "of locally ringed spaces." ], "refs": [], "proofs": [ { "contents": [ "This follows trivially from the corresponding fact in algebra:", "Suppose $A$, $B$ are local rings. Any isomorphism of rings", "$A \\to B$ is a local ring homomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7644, "type": "theorem", "label": "schemes-lemma-open-immersion", "categories": [ "schemes" ], "title": "schemes-lemma-open-immersion", "contents": [ "Let $f : X \\to Y$ be an open immersion of", "locally ringed spaces. Let $j : V = f(X) \\to Y$", "be the open subspace of $Y$ associated to the image of $f$.", "There is a unique isomorphism $f' : X \\cong V$ of", "locally ringed spaces such that $f = j \\circ f'$." ], "refs": [], "proofs": [ { "contents": [ "Let $f'$ be the homeomorphism between $X$ and $V$ induced by $f$. Then", "$f = j \\circ f'$ as maps of topological spaces. Since there", "is an isomorphism of sheaves", "$f^\\sharp : f^{-1}(\\mathcal{O}_Y) \\to \\mathcal{O}_X$, there is an isomorphism", "of rings", "$f^\\sharp : \\Gamma(U, f^{-1}(\\mathcal{O}_Y)) \\to \\Gamma(U, \\mathcal{O}_X)$", "for each open subset $U \\subset X$. Since", "$\\mathcal{O}_V = j^{-1}\\mathcal{O}_Y$ and $f^{-1} = f'^{-1} j^{-1}$", "(Sheaves, Lemma \\ref{sheaves-lemma-pullback-composition}) we", "see that $f^{-1}\\mathcal{O}_Y = f'^{-1}\\mathcal{O}_V$, hence", "$\\Gamma(U, f'^{-1}(\\mathcal{O}_V)) \\to \\Gamma(U, f^{-1}(\\mathcal{O}_Y))$", "is an isomorphism", "for every $U \\subset X$ open. By composing these we get an isomorphism of rings", "$$", "\\Gamma(U, f'^{-1}(\\mathcal{O}_V)) \\to \\Gamma(U, \\mathcal{O}_X)", "$$", "for each open subset $U \\subset X$, and therefore an isomorphism of sheaves", "$f^{-1}(\\mathcal{O}_V) \\to \\mathcal{O}_X$. In other words, we have an", "isomorphism $f'^{\\sharp} : f'^{-1}(\\mathcal{O}_V) \\to \\mathcal{O}_X$ and", "therefore an isomorphism of locally ringed spaces", "$(f', f'^{\\sharp}) : (X, \\mathcal{O}_X) \\to (V, \\mathcal{O}_V)$", "(use Lemma \\ref{lemma-isomorphism-locally-ringed}).", "Note that $f = j \\circ f'$ as morphisms of locally ringed spaces", "by construction.", "\\medskip\\noindent", "Suppose we have another morphism", "$f'' : (X, \\mathcal{O}_X) \\to (V, \\mathcal{O}_Y)$ such that $f = j \\circ f''$.", "At any point $x \\in X$, we have $j(f'(x)) = j(f''(x))$ from which it follows", "that $f'(x) = f''(x)$ since $j$ is the inclusion map; therefore $f'$ and $f''$", "are the same as morphisms of topological spaces. On structure sheaves,", "for each open subset $U \\subset X$ we have a commutative diagram", "$$", "\\xymatrix @R=5em{", "\\Gamma(U, f^{-1}(\\mathcal{O}_Y)) \\ar[d]_\\cong\\ar[r]^\\cong &", "\\Gamma(U, \\mathcal{O}_X) \\\\", "\\Gamma(U, f'^{-1}(\\mathcal{O}_V)) \\ar@/^/[ru]^{f'^\\sharp}", "\\ar@/_/[ru]_{f''^\\sharp} &", "}", "$$", "from which we see that $f'^\\sharp$ and $f''^\\sharp$ define", "the same morphism of sheaves." ], "refs": [ "sheaves-lemma-pullback-composition", "schemes-lemma-isomorphism-locally-ringed" ], "ref_ids": [ 14508, 7643 ] } ], "ref_ids": [] }, { "id": 7645, "type": "theorem", "label": "schemes-lemma-restrict-map-to-opens", "categories": [ "schemes" ], "title": "schemes-lemma-restrict-map-to-opens", "contents": [ "Let $f : X \\to Y$ be a morphism of locally ringed spaces.", "Let $U \\subset X$, and $V \\subset Y$ be open subsets.", "Suppose that $f(U) \\subset V$. There exists a unique", "morphism of locally ringed spaces $f|_U : U \\to V$ such", "that the following diagram is a commutative square of", "locally ringed spaces", "$$", "\\xymatrix{", "U \\ar[d]_{f|_U} \\ar[r] & X \\ar[d]^f \\\\", "V \\ar[r] & Y", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7646, "type": "theorem", "label": "schemes-lemma-closed-local-target", "categories": [ "schemes" ], "title": "schemes-lemma-closed-local-target", "contents": [ "Let $f : Z \\to X$ be a morphism of locally ringed spaces.", "In order for $f$ to be a closed immersion it suffices", "that there exists an open covering $X = \\bigcup U_i$ such", "that each $f : f^{-1}U_i \\to U_i$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7647, "type": "theorem", "label": "schemes-lemma-closed-immersion", "categories": [ "schemes" ], "title": "schemes-lemma-closed-immersion", "contents": [ "Let $f : X \\to Y$ be a closed immersion of", "locally ringed spaces. Let $\\mathcal{I}$ be the", "kernel of the map $\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$.", "Let $i : Z \\to Y$ be the closed subspace of $Y$", "associated to $\\mathcal{I}$.", "There is a unique isomorphism $f' : X \\cong Z$ of", "locally ringed spaces such that $f = i \\circ f'$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7648, "type": "theorem", "label": "schemes-lemma-characterize-closed-subspace", "categories": [ "schemes" ], "title": "schemes-lemma-characterize-closed-subspace", "contents": [ "Let $X$, $Y$ be locally ringed spaces. Let", "$\\mathcal{I} \\subset \\mathcal{O}_X$ be a sheaf of ideals locally generated", "by sections. Let $i : Z \\to X$ be the associated closed subspace.", "A morphism $f : Y \\to X$ factors through $Z$ if and only if the map", "$f^*\\mathcal{I} \\to f^*\\mathcal{O}_X = \\mathcal{O}_Y$", "is zero. If this is the case the morphism $g : Y \\to Z$", "such that $f = i \\circ g$ is unique." ], "refs": [], "proofs": [ { "contents": [ "Clearly if $f$ factors as $Y \\to Z \\to X$ then the map", "$f^*\\mathcal{I} \\to \\mathcal{O}_Y$ is zero. Conversely", "suppose that $f^*\\mathcal{I} \\to \\mathcal{O}_Y$ is zero.", "Pick any $y \\in Y$, and consider the ring map", "$f^\\sharp_y : \\mathcal{O}_{X, f(y)} \\to \\mathcal{O}_{Y, y}$.", "Since the composition", "$\\mathcal{I}_{f(y)} \\to \\mathcal{O}_{X, f(y)} \\to \\mathcal{O}_{Y, y}$", "is zero by assumption and since $f^\\sharp_y(1) = 1$", "we see that $1 \\not \\in \\mathcal{I}_{f(y)}$, i.e.,", "$\\mathcal{I}_{f(y)} \\not = \\mathcal{O}_{X, f(y)}$. We conclude that", "$f(Y) \\subset Z = \\text{Supp}(\\mathcal{O}_X/\\mathcal{I})$.", "Hence $f = i \\circ g$ where $g : Y \\to Z$ is continuous.", "Consider the map $f^\\sharp : \\mathcal{O}_X \\to f_*\\mathcal{O}_Y$.", "The assumption $f^*\\mathcal{I} \\to \\mathcal{O}_Y$ is zero implies that", "the composition $\\mathcal{I} \\to \\mathcal{O}_X \\to f_*\\mathcal{O}_Y$ is", "zero by adjointness of $f_*$ and $f^*$.", "In other words, we obtain a morphism of sheaves of rings", "$\\overline{f^\\sharp} : \\mathcal{O}_X/\\mathcal{I} \\to f_*\\mathcal{O}_Y$.", "Note that $f_*\\mathcal{O}_Y = i_*g_*\\mathcal{O}_Y$ and", "that $\\mathcal{O}_X/\\mathcal{I} = i_*\\mathcal{O}_Z$.", "By Sheaves, Lemma \\ref{sheaves-lemma-equivalence-categories-closed-structures}", "we obtain a unique morphism of sheaves of rings", "$g^\\sharp : \\mathcal{O}_Z \\to g_*\\mathcal{O}_Y$ whose pushforward", "under $i$ is $\\overline{f^\\sharp}$. We omit the verification that", "$(g, g^\\sharp)$ defines a morphism of locally ringed spaces", "and that $f = i \\circ g$ as a morphism of locally ringed spaces.", "The uniqueness of $(g, g^\\sharp)$ was pointed out above." ], "refs": [ "sheaves-lemma-equivalence-categories-closed-structures" ], "ref_ids": [ 14554 ] } ], "ref_ids": [] }, { "id": 7649, "type": "theorem", "label": "schemes-lemma-restrict-map-to-closed", "categories": [ "schemes" ], "title": "schemes-lemma-restrict-map-to-closed", "contents": [ "Let $f : X \\to Y$ be a morphism of locally ringed spaces.", "Let $\\mathcal{I} \\subset \\mathcal{O}_Y$ be a sheaf of", "ideals which is locally generated by sections.", "Let $i : Z \\to Y$ be the closed subspace associated to the", "sheaf of ideals $\\mathcal{I}$.", "Let $\\mathcal{J}$ be the image of the map", "$f^*\\mathcal{I} \\to f^*\\mathcal{O}_Y = \\mathcal{O}_X$.", "Then this ideal is locally generated by sections.", "Moreover, let $i' : Z' \\to X$ be the associated closed", "subspace of $X$. There exists a unique", "morphism of locally ringed spaces $f' : Z' \\to Z$ such", "that the following diagram is a commutative square of", "locally ringed spaces", "$$", "\\xymatrix{", "Z' \\ar[d]_{f'} \\ar[r]_{i'} & X \\ar[d]^f \\\\", "Z \\ar[r]^{i} & Y", "}", "$$", "Moreover, this diagram is a fibre square in the category of", "locally ringed spaces." ], "refs": [], "proofs": [ { "contents": [ "The ideal $\\mathcal{J}$ is locally generated by sections", "by Modules, Lemma \\ref{modules-lemma-pullback-locally-generated}.", "The rest of the lemma follows from the characterization,", "in Lemma \\ref{lemma-characterize-closed-subspace} above,", "of what it means for a morphism to factor through a closed", "subspace." ], "refs": [ "modules-lemma-pullback-locally-generated", "schemes-lemma-characterize-closed-subspace" ], "ref_ids": [ 13235, 7648 ] } ], "ref_ids": [] }, { "id": 7650, "type": "theorem", "label": "schemes-lemma-standard-open", "categories": [ "schemes" ], "title": "schemes-lemma-standard-open", "contents": [ "Let $R$ be a ring. Let $f \\in R$.", "\\begin{enumerate}", "\\item If $g\\in R$ and $D(g) \\subset D(f)$, then", "\\begin{enumerate}", "\\item $f$ is invertible in $R_g$,", "\\item $g^e = af$ for some $e \\geq 1$ and $a \\in R$,", "\\item there is a canonical ring map $R_f \\to R_g$, and", "\\item there is a canonical $R_f$-module map", "$M_f \\to M_g$ for any $R$-module $M$.", "\\end{enumerate}", "\\item Any open covering of $D(f)$ can be refined to a finite", "open covering of the form $D(f) = \\bigcup_{i = 1}^n D(g_i)$.", "\\item If $g_1, \\ldots, g_n \\in R$, then $D(f) \\subset \\bigcup D(g_i)$", "if and only if $g_1, \\ldots, g_n$ generate the unit ideal in $R_f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that $D(g) = \\Spec(R_g)$ (see", "Algebra, Lemma \\ref{algebra-lemma-standard-open}).", "Thus (a) holds because $f$", "maps to an element of $R_g$ which is not", "contained in any prime ideal, and hence invertible,", "see Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}.", "Write the inverse of $f$ in $R_g$ as $a/g^d$.", "This means $g^d - af$ is annihilated by a power of $g$, whence (b).", "For (c), the map $R_f \\to R_g$ exists by (a) from the universal property", "of localization, or we can define it by mapping $b/f^n$", "to $a^nb/g^{ne}$. The equality $M_f = M \\otimes_R R_f$", "can be used to obtain the map on modules, or", "we can define $M_f \\to M_g$ by mapping", "$x/f^n$ to $a^nx/g^{ne}$.", "\\medskip\\noindent", "Recall that $D(f)$ is quasi-compact, see", "Algebra, Lemma \\ref{algebra-lemma-qc-open}.", "Hence the second statement follows directly", "from the fact that the standard opens form", "a basis for the topology.", "\\medskip\\noindent", "The third statement follows directly from", "Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}." ], "refs": [ "algebra-lemma-standard-open", "algebra-lemma-Zariski-topology", "algebra-lemma-qc-open", "algebra-lemma-Zariski-topology" ], "ref_ids": [ 392, 389, 432, 389 ] } ], "ref_ids": [] }, { "id": 7651, "type": "theorem", "label": "schemes-lemma-spec-sheaves", "categories": [ "schemes" ], "title": "schemes-lemma-spec-sheaves", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module. Let $\\widetilde M$", "be the sheaf of $\\mathcal{O}_{\\Spec(R)}$-modules", "associated to $M$.", "\\begin{enumerate}", "\\item We have $\\Gamma(\\Spec(R), \\mathcal{O}_{\\Spec(R)}) = R$.", "\\item We have $\\Gamma(\\Spec(R), \\widetilde M) = M$ as an $R$-module.", "\\item For every $f \\in R$ we have", "$\\Gamma(D(f), \\mathcal{O}_{\\Spec(R)}) = R_f$.", "\\item For every $f\\in R$ we have $\\Gamma(D(f), \\widetilde M) = M_f$", "as an $R_f$-module.", "\\item Whenever $D(g) \\subset D(f)$ the restriction mappings", "on $\\mathcal{O}_{\\Spec(R)}$ and $\\widetilde M$", "are the maps", "$R_f \\to R_g$ and $M_f \\to M_g$ from Lemma", "\\ref{lemma-standard-open}.", "\\item Let $\\mathfrak p$ be a prime of $R$, and let $x \\in \\Spec(R)$", "be the corresponding point. We have", "$\\mathcal{O}_{\\Spec(R), x} = R_{\\mathfrak p}$.", "\\item Let $\\mathfrak p$ be a prime of $R$, and let $x \\in \\Spec(R)$", "be the corresponding point. We have $\\widetilde M_x = M_{\\mathfrak p}$", "as an $R_{\\mathfrak p}$-module.", "\\end{enumerate}", "Moreover, all these identifications are functorial in the $R$", "module $M$. In particular, the functor $M \\mapsto \\widetilde M$", "is an exact functor from the category of $R$-modules", "to the category of $\\mathcal{O}_{\\Spec(R)}$-modules." ], "refs": [ "schemes-lemma-standard-open" ], "proofs": [ { "contents": [ "Assertions (1) - (7) are clear from the discussion above.", "The exactness of the functor $M \\mapsto \\widetilde M$", "follows from the fact that the functor $M \\mapsto M_{\\mathfrak p}$", "is exact and the fact that exactness of short exact sequences", "may be checked on stalks, see", "Modules, Lemma \\ref{modules-lemma-abelian}." ], "refs": [ "modules-lemma-abelian" ], "ref_ids": [ 13221 ] } ], "ref_ids": [ 7650 ] }, { "id": 7652, "type": "theorem", "label": "schemes-lemma-morphism-into-affine-where-point-goes", "categories": [ "schemes" ], "title": "schemes-lemma-morphism-into-affine-where-point-goes", "contents": [ "Let $X$ be a locally ringed space.", "Let $Y$ be an affine scheme.", "Let $f \\in \\Mor(X, Y)$ be a morphism", "of locally ringed spaces. Given a point $x \\in X$", "consider the ring maps", "$$", "\\Gamma(Y, \\mathcal{O}_Y) \\xrightarrow{f^\\sharp}", "\\Gamma(X, \\mathcal{O}_X) \\to \\mathcal{O}_{X, x}", "$$", "Let $\\mathfrak p \\subset \\Gamma(Y, \\mathcal{O}_Y)$ denote", "the inverse image of $\\mathfrak m_x$. Let $y \\in Y$ be the", "corresponding point. Then $f(x) = y$." ], "refs": [], "proofs": [ { "contents": [ "Consider the commutative diagram", "$$", "\\xymatrix{", "\\Gamma(X, \\mathcal{O}_X) \\ar[r] &", "\\mathcal{O}_{X, x} \\\\", "\\Gamma(Y, \\mathcal{O}_Y) \\ar[r] \\ar[u] &", "\\mathcal{O}_{Y, f(x)} \\ar[u]", "}", "$$", "(see the discussion of $f$-maps below", "Sheaves, Definition \\ref{sheaves-definition-f-map}).", "Since the right vertical arrow is local", "we see that $\\mathfrak m_{f(x)}$ is the", "inverse image of $\\mathfrak m_x$. The result", "follows." ], "refs": [ "sheaves-definition-f-map" ], "ref_ids": [ 14573 ] } ], "ref_ids": [] }, { "id": 7653, "type": "theorem", "label": "schemes-lemma-f-open", "categories": [ "schemes" ], "title": "schemes-lemma-f-open", "contents": [ "Let $X$ be a locally ringed space.", "Let $f \\in \\Gamma(X, \\mathcal{O}_X)$.", "The set", "$$", "D(f) = \\{x \\in X \\mid \\text{image }f \\not\\in \\mathfrak m_x\\}", "$$", "is open. Moreover $f|_{D(f)}$ has an inverse." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Modules, Lemma \\ref{modules-lemma-s-open}, but", "we also give a direct proof.", "Suppose that $U \\subset X$ and $V \\subset X$ are", "two open subsets such that $f|_U$ has an inverse", "$g$ and $f|_V$ has an inverse $h$. Then clearly", "$g|_{U\\cap V} = h|_{U\\cap V}$. Thus it suffices", "to show that $f$ is invertible in an open neighbourhood", "of any $x \\in D(f)$. This is clear because", "$f \\not \\in \\mathfrak m_x$ implies that $f \\in \\mathcal{O}_{X, x}$", "has an inverse $g \\in \\mathcal{O}_{X, x}$ which means there", "is some open neighbourhood $x \\in U \\subset X$ so", "that $g \\in \\mathcal{O}_X(U)$ and $g\\cdot f|_U = 1$." ], "refs": [ "modules-lemma-s-open" ], "ref_ids": [ 13305 ] } ], "ref_ids": [] }, { "id": 7654, "type": "theorem", "label": "schemes-lemma-f-open-affine", "categories": [ "schemes" ], "title": "schemes-lemma-f-open-affine", "contents": [ "In Lemma \\ref{lemma-f-open} above, if $X$ is an affine scheme,", "then the open $D(f)$ agrees with the standard open $D(f)$", "defined previously (in", "Algebra, Definition \\ref{algebra-definition-spectrum-ring})." ], "refs": [ "schemes-lemma-f-open", "algebra-definition-spectrum-ring" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 7653, 1444 ] }, { "id": 7655, "type": "theorem", "label": "schemes-lemma-morphism-into-affine", "categories": [ "schemes" ], "title": "schemes-lemma-morphism-into-affine", "contents": [ "\\begin{reference}", "A reference for this fact is \\cite[II, Err 1, Prop. 1.8.1]{EGA}", "where it is attributed to J. Tate.", "\\end{reference}", "Let $X$ be a locally ringed space.", "Let $Y$ be an affine scheme.", "The map", "$$", "\\Mor(X, Y)", "\\longrightarrow", "\\Hom(\\Gamma(Y, \\mathcal{O}_Y), \\Gamma(X, \\mathcal{O}_X))", "$$", "which maps $f$ to $f^\\sharp$ (on global sections) is bijective." ], "refs": [], "proofs": [ { "contents": [ "Since $Y$ is affine we have", "$(Y, \\mathcal{O}_Y) \\cong (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$", "for some ring $R$.", "During the proof we will use facts about $Y$ and", "its structure sheaf which are direct consequences of things", "we know about the spectrum of a ring, see e.g.\\ Lemma", "\\ref{lemma-spec-sheaves}.", "\\medskip\\noindent", "Motivated by the lemmas above we construct the inverse map.", "Let $\\psi_Y : \\Gamma(Y, \\mathcal{O}_Y) \\to \\Gamma(X, \\mathcal{O}_X)$", "be a ring map. First, we define the corresponding map of", "spaces", "$$", "\\Psi : X \\longrightarrow Y", "$$", "by the rule of", "Lemma \\ref{lemma-morphism-into-affine-where-point-goes}.", "In other words, given $x \\in X$ we define $\\Psi(x)$", "to be the point of $Y$ corresponding to the prime", "in $\\Gamma(Y, \\mathcal{O}_Y)$ which is the inverse", "image of $\\mathfrak m_x$ under the composition", "$", "\\Gamma(Y, \\mathcal{O}_Y) \\xrightarrow{\\psi_Y}", "\\Gamma(X, \\mathcal{O}_X) \\to", "\\mathcal{O}_{X, x}", "$.", "\\medskip\\noindent", "We claim that the map $\\Psi : X \\to Y$ is continuous.", "The standard opens $D(g)$, for $g \\in \\Gamma(Y, \\mathcal{O}_Y)$", "are a basis for the topology of $Y$. Thus it suffices to prove", "that $\\Psi^{-1}(D(g))$ is open. By construction of $\\Psi$", "the inverse image $\\Psi^{-1}(D(g))$ is exactly the set", "$D(\\psi_Y(g)) \\subset X$ which is open by Lemma \\ref{lemma-f-open}.", "Hence $\\Psi$ is continuous.", "\\medskip\\noindent", "Next we construct a $\\Psi$-map of sheaves from", "$\\mathcal{O}_Y$ to $\\mathcal{O}_X$. By", "Sheaves, Lemma \\ref{sheaves-lemma-f-map-basis-below-structures}", "it suffices to define ring maps", "$\\psi_{D(g)} : \\Gamma(D(g), \\mathcal{O}_Y) \\to", "\\Gamma(\\Psi^{-1}(D(g)), \\mathcal{O}_X)$", "compatible with restriction maps.", "We have a canonical isomorphism", "$\\Gamma(D(g), \\mathcal{O}_Y) = \\Gamma(Y, \\mathcal{O}_Y)_g$,", "because $Y$ is an affine scheme.", "Because $\\psi_Y(g)$ is invertible on $D(\\psi_Y(g))$", "we see that there is a canonical map", "$$", "\\Gamma(Y, \\mathcal{O}_Y)_g", "\\longrightarrow", "\\Gamma(\\Psi^{-1}(D(g)), \\mathcal{O}_X)", "=", "\\Gamma(D(\\psi_Y(g)), \\mathcal{O}_X)", "$$", "extending the map $\\psi_Y$", "by the universal property of localization.", "Note that there is no choice but to take the canonical map here!", "And we take this, combined", "with the canonical identification", "$\\Gamma(D(g), \\mathcal{O}_Y) = \\Gamma(Y, \\mathcal{O}_Y)_g$, to", "be $\\psi_{D(g)}$. This is compatible with localization since the", "restriction mapping on the affine schemes are defined in terms", "of the universal properties of localization also, see", "Lemmas \\ref{lemma-spec-sheaves} and \\ref{lemma-standard-open}.", "\\medskip\\noindent", "Thus we have defined a morphism of ringed spaces", "$(\\Psi, \\psi) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "recovering $\\psi_Y$ on global sections. To see that it is", "a morphism of locally ringed spaces we have to show that", "the induced maps on local rings", "$$", "\\psi_x : \\mathcal{O}_{Y, \\Psi(x)} \\longrightarrow \\mathcal{O}_{X, x}", "$$", "are local. This follows immediately from the commutative diagram", "of the proof of Lemma \\ref{lemma-morphism-into-affine-where-point-goes}", "and the definition of $\\Psi$.", "\\medskip\\noindent", "Finally, we have to show that the constructions", "$(\\Psi, \\psi) \\mapsto \\psi_Y$ and the construction", "$\\psi_Y \\mapsto (\\Psi, \\psi)$ are inverse to each other.", "Clearly, $\\psi_Y \\mapsto (\\Psi, \\psi) \\mapsto \\psi_Y$.", "Hence the only thing to prove is that given $\\psi_Y$", "there is at most one pair $(\\Psi, \\psi)$ giving rise", "to it. The uniqueness of $\\Psi$ was shown in Lemma", "\\ref{lemma-morphism-into-affine-where-point-goes} and", "given the uniqueness of $\\Psi$ the uniqueness of the", "map $\\psi$ was pointed out during the course of the proof", "above." ], "refs": [ "schemes-lemma-spec-sheaves", "schemes-lemma-morphism-into-affine-where-point-goes", "schemes-lemma-f-open", "sheaves-lemma-f-map-basis-below-structures", "schemes-lemma-spec-sheaves", "schemes-lemma-standard-open", "schemes-lemma-morphism-into-affine-where-point-goes", "schemes-lemma-morphism-into-affine-where-point-goes" ], "ref_ids": [ 7651, 7652, 7653, 14538, 7651, 7650, 7652, 7652 ] } ], "ref_ids": [] }, { "id": 7656, "type": "theorem", "label": "schemes-lemma-category-affine-schemes", "categories": [ "schemes" ], "title": "schemes-lemma-category-affine-schemes", "contents": [ "The category of affine schemes is equivalent to the opposite of the", "category of rings. The equivalence is given by the functor that associates", "to an affine scheme the global sections of its structure sheaf." ], "refs": [], "proofs": [ { "contents": [ "This is now clear from Definition \\ref{definition-affine-scheme}", "and Lemma \\ref{lemma-morphism-into-affine}." ], "refs": [ "schemes-definition-affine-scheme", "schemes-lemma-morphism-into-affine" ], "ref_ids": [ 7741, 7655 ] } ], "ref_ids": [] }, { "id": 7657, "type": "theorem", "label": "schemes-lemma-standard-open-affine", "categories": [ "schemes" ], "title": "schemes-lemma-standard-open-affine", "contents": [ "Let $Y$ be an affine scheme.", "Let $f \\in \\Gamma(Y, \\mathcal{O}_Y)$.", "The open subspace $D(f)$ is an affine scheme." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $Y = \\Spec(R)$ and $f \\in R$.", "Consider the morphism of affine schemes", "$\\phi : U = \\Spec(R_f) \\to \\Spec(R) = Y$ induced by the ring", "map $R \\to R_f$. By Algebra, Lemma \\ref{algebra-lemma-standard-open}", "we know that it is a homeomorphism onto $D(f)$.", "On the other hand, the map $\\phi^{-1}\\mathcal{O}_Y \\to \\mathcal{O}_U$", "is an isomorphism on stalks, hence an isomorphism. Thus we see", "that $\\phi$ is an open immersion. We conclude that $D(f)$", "is isomorphic to $U$ by Lemma \\ref{lemma-open-immersion}." ], "refs": [ "algebra-lemma-standard-open", "schemes-lemma-open-immersion" ], "ref_ids": [ 392, 7644 ] } ], "ref_ids": [] }, { "id": 7658, "type": "theorem", "label": "schemes-lemma-fibre-product-affine-schemes", "categories": [ "schemes" ], "title": "schemes-lemma-fibre-product-affine-schemes", "contents": [ "The category of affine schemes has finite products, and fibre products.", "In other words, it has finite limits. Moreover, the products", "and fibre products in the category of affine schemes", "are the same as in the category of locally ringed spaces.", "In a formula, we have (in the category of locally ringed spaces)", "$$", "\\Spec(R) \\times \\Spec(S) =", "\\Spec(R \\otimes_{\\mathbf{Z}} S)", "$$", "and given ring maps $R \\to A$, $R \\to B$ we have", "$$", "\\Spec(A) \\times_{\\Spec(R)} \\Spec(B)", "=", "\\Spec(A \\otimes_R B).", "$$" ], "refs": [], "proofs": [ { "contents": [ "This is just an application of Lemma \\ref{lemma-morphism-into-affine}.", "First of all, by that lemma, the affine scheme", "$\\Spec(\\mathbf{Z})$ is the final object in the category", "of locally ringed spaces. Thus the first displayed formula", "follows from the second. To prove the second note that", "for any locally ringed space $X$ we have", "\\begin{eqnarray*}", "\\Mor(X, \\Spec(A \\otimes_R B))", "& = &", "\\Hom(A \\otimes_R B, \\mathcal{O}_X(X)) \\\\", "& = &", "\\Hom(A, \\mathcal{O}_X(X))", "\\times_{\\Hom(R, \\mathcal{O}_X(X))}", "\\Hom(B, \\mathcal{O}_X(X)) \\\\", "& = &", "\\Mor(X, \\Spec(A))", "\\times_{\\Mor(X, \\Spec(R))}", "\\Mor(X, \\Spec(B))", "\\end{eqnarray*}", "which proves the formula.", "See Categories, Section \\ref{categories-section-fibre-products} for the", "relevant definitions." ], "refs": [ "schemes-lemma-morphism-into-affine" ], "ref_ids": [ 7655 ] } ], "ref_ids": [] }, { "id": 7659, "type": "theorem", "label": "schemes-lemma-disjoint-union-affines", "categories": [ "schemes" ], "title": "schemes-lemma-disjoint-union-affines", "contents": [ "Let $X$ be a locally ringed space.", "Assume $X = U \\amalg V$ with $U$ and $V$ open and", "such that $U$, $V$ are affine schemes. Then $X$ is an affine scheme." ], "refs": [], "proofs": [ { "contents": [ "Set $R = \\Gamma(X, \\mathcal{O}_X)$.", "Note that $R = \\mathcal{O}_X(U) \\times \\mathcal{O}_X(V)$", "by the sheaf property. By Lemma \\ref{lemma-morphism-into-affine}", "there is a canonical morphism of locally ringed spaces", "$X \\to \\Spec(R)$. By Algebra, Lemma \\ref{algebra-lemma-spec-product}", "we see that as a topological space", "$\\Spec(\\mathcal{O}_X(U)) \\amalg \\Spec(\\mathcal{O}_X(V)) =", "\\Spec(R)$", "with the maps coming from the ring homomorphisms", "$R \\to \\mathcal{O}_X(U)$ and $R \\to \\mathcal{O}_X(V)$.", "This of course means that $\\Spec(R)$ is the coproduct", "in the category of locally ringed spaces as well.", "By assumption the morphism $X \\to \\Spec(R)$ induces an isomorphism", "of $\\Spec(\\mathcal{O}_X(U))$ with $U$ and similarly", "for $V$. Hence $X \\to \\Spec(R)$ is an isomorphism." ], "refs": [ "schemes-lemma-morphism-into-affine", "algebra-lemma-spec-product" ], "ref_ids": [ 7655, 404 ] } ], "ref_ids": [] }, { "id": 7660, "type": "theorem", "label": "schemes-lemma-compare-constructions", "categories": [ "schemes" ], "title": "schemes-lemma-compare-constructions", "contents": [ "Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$", "be an affine scheme. Let $M$ be an $R$-module. There exists a canonical", "isomorphism between the sheaf $\\widetilde M$ associated to the $R$-module", "$M$ (Definition \\ref{definition-structure-sheaf}) and the sheaf", "$\\mathcal{F}_M$ associated to the $R$-module $M$", "(Modules, Definition \\ref{modules-definition-sheaf-associated}).", "This isomorphism is functorial in $M$. In particular,", "the sheaves $\\widetilde M$ are quasi-coherent. Moreover, they", "are characterized by the following mapping property", "$$", "\\Hom_{\\mathcal{O}_X}(\\widetilde M, \\mathcal{F})", "=", "\\Hom_R(M, \\Gamma(X, \\mathcal{F}))", "$$", "for any sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$.", "Here a map $\\alpha : \\widetilde M \\to \\mathcal{F}$ corresponds", "to its effect on global sections." ], "refs": [ "schemes-definition-structure-sheaf", "modules-definition-sheaf-associated" ], "proofs": [ { "contents": [ "By Modules, Lemma \\ref{modules-lemma-construct-quasi-coherent-sheaves}", "we have a morphism $\\mathcal{F}_M \\to \\widetilde M$ corresponding", "to the map $M \\to \\Gamma(X, \\widetilde M) = M$. Let $x \\in X$", "correspond to the prime $\\mathfrak p \\subset R$.", "The induced map on stalks are the maps", "$\\mathcal{O}_{X, x} \\otimes_R M \\to M_{\\mathfrak p}$", "which are isomorphisms because", "$R_{\\mathfrak p} \\otimes_R M = M_{\\mathfrak p}$.", "Hence the map $\\mathcal{F}_M \\to \\widetilde M$ is an isomorphism.", "The mapping property follows from the mapping property of", "the sheaves $\\mathcal{F}_M$." ], "refs": [ "modules-lemma-construct-quasi-coherent-sheaves" ], "ref_ids": [ 13245 ] } ], "ref_ids": [ 7740, 13338 ] }, { "id": 7661, "type": "theorem", "label": "schemes-lemma-widetilde-constructions", "categories": [ "schemes" ], "title": "schemes-lemma-widetilde-constructions", "contents": [ "Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$", "be an affine scheme. There are canonical isomorphisms", "\\begin{enumerate}", "\\item", "$", "\\widetilde{M \\otimes_R N}", "\\cong", "\\widetilde M \\otimes_{\\mathcal{O}_X} \\widetilde N", "$,", "see Modules, Section \\ref{modules-section-tensor-product}.", "\\item", "$", "\\widetilde{\\text{T}^n(M)}", "\\cong", "\\text{T}^n(\\widetilde M)", "$,", "$", "\\widetilde{\\text{Sym}^n(M)}", "\\cong", "\\text{Sym}^n(\\widetilde M)", "$, and", "$", "\\widetilde{\\wedge^n(M)}", "\\cong", "\\wedge^n(\\widetilde M)", "$,", "see", "Modules, Section \\ref{modules-section-symmetric-exterior}.", "\\item if $M$ is a finitely presented $R$-module, then", "$", "\\SheafHom_{\\mathcal{O}_X}(\\widetilde M, \\widetilde N)", "\\cong", "\\widetilde{\\Hom_R(M, N)}", "$,", "see", "Modules, Section \\ref{modules-section-internal-hom}.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "Using Lemma \\ref{lemma-compare-constructions} and", "Modules, Lemma \\ref{modules-lemma-construct-quasi-coherent-sheaves}", "we see that the functor $M \\mapsto \\widetilde M$ can be viewed", "as $\\pi^*$ for a morphism $\\pi$ of ringed spaces.", "And pulling back modules commutes with tensor constructions by", "Modules, Lemmas \\ref{modules-lemma-tensor-product-pullback}", "and \\ref{modules-lemma-pullback-tensor-algebra}.", "The morphism $\\pi : (X, \\mathcal{O}_X) \\to (\\{*\\}, R)$ is", "flat for example because the stalks of $\\mathcal{O}_X$ are", "localizations of $R$ (Lemma \\ref{lemma-spec-sheaves})", "and hence flat over $R$. Thus pullback by $\\pi$ commutes", "with internal hom if the first module is finitely presented by", "Modules, Lemma \\ref{modules-lemma-pullback-internal-hom}." ], "refs": [ "schemes-lemma-compare-constructions", "modules-lemma-construct-quasi-coherent-sheaves", "modules-lemma-tensor-product-pullback", "modules-lemma-pullback-tensor-algebra", "schemes-lemma-spec-sheaves", "modules-lemma-pullback-internal-hom" ], "ref_ids": [ 7660, 13245, 13270, 13290, 7651, 13297 ] } ], "ref_ids": [] }, { "id": 7662, "type": "theorem", "label": "schemes-lemma-widetilde-pullback", "categories": [ "schemes" ], "title": "schemes-lemma-widetilde-pullback", "contents": [ "Let", "$(X, \\mathcal{O}_X) = (\\Spec(S), \\mathcal{O}_{\\Spec(S)})$,", "$(Y, \\mathcal{O}_Y) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$", "be affine schemes.", "Let $\\psi : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a", "morphism of affine schemes, corresponding to the ring map", "$\\psi^\\sharp : R \\to S$ (see Lemma \\ref{lemma-category-affine-schemes}).", "\\begin{enumerate}", "\\item We have $\\psi^* \\widetilde M = \\widetilde{S \\otimes_R M}$", "functorially in the $R$-module $M$.", "\\item We have $\\psi_* \\widetilde N = \\widetilde{N_R}$ functorially", "in the $S$-module $N$.", "\\end{enumerate}" ], "refs": [ "schemes-lemma-category-affine-schemes" ], "proofs": [ { "contents": [ "The first assertion follows from the identification in", "Lemma \\ref{lemma-compare-constructions}", "and the result of Modules, Lemma \\ref{modules-lemma-restrict-quasi-coherent}.", "The second assertion follows from the fact", "that $\\psi^{-1}(D(f)) = D(\\psi^\\sharp(f))$ and hence", "$$", "\\psi_* \\widetilde N(D(f)) = \\widetilde N(D(\\psi^\\sharp(f))) =", "N_{\\psi^\\sharp(f)} = (N_R)_f = \\widetilde{N_R}(D(f))", "$$", "as desired." ], "refs": [ "schemes-lemma-compare-constructions", "modules-lemma-restrict-quasi-coherent" ], "ref_ids": [ 7660, 13246 ] } ], "ref_ids": [ 7656 ] }, { "id": 7663, "type": "theorem", "label": "schemes-lemma-quasi-coherent-affine", "categories": [ "schemes" ], "title": "schemes-lemma-quasi-coherent-affine", "contents": [ "Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$", "be an affine scheme. Let $\\mathcal{F}$ be a", "quasi-coherent $\\mathcal{O}_X$-module. Then", "$\\mathcal{F}$ is isomorphic to the sheaf associated to", "the $R$-module $\\Gamma(X, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Since every standard open $D(f)$ is quasi-compact we see that", "$X$ is a locally quasi-compact, i.e., every point has a fundamental", "system of quasi-compact neighbourhoods, see Topology,", "Definition \\ref{topology-definition-locally-quasi-compact}.", "Hence by Modules, Lemma \\ref{modules-lemma-quasi-coherent-module}", "for every prime $\\mathfrak p \\subset R$ corresponding to $x \\in X$", "there exists an open neighbourhood $x \\in U \\subset X$ such that", "$\\mathcal{F}|_U$ is isomorphic to the quasi-coherent", "sheaf associated to some $\\mathcal{O}_X(U)$-module $M$.", "In other words, we get an open covering by $U$'s with this property.", "By Lemma \\ref{lemma-standard-open} for example we can refine this", "covering to a standard open covering.", "Thus we get a covering $\\Spec(R) = \\bigcup D(f_i)$", "and $R_{f_i}$-modules $M_i$ and isomorphisms", "$\\varphi_i : \\mathcal{F}|_{D(f_i)} \\to \\mathcal{F}_{M_i}$", "for some $R_{f_i}$-module $M_i$. On the overlaps", "we get isomorphisms", "$$", "\\xymatrix{", "\\mathcal{F}_{M_i}|_{D(f_if_j)}", "\\ar[rr]^{\\varphi_i^{-1}|_{D(f_if_j)}}", "& &", "\\mathcal{F}|_{D(f_if_j)}", "\\ar[rr]^{\\varphi_j|_{D(f_if_j)}}", "& &", "\\mathcal{F}_{M_j}|_{D(f_if_j)}.", "}", "$$", "Let us denote these $\\psi_{ij}$. It is clear that", "we have the cocycle condition", "$$", "\\psi_{jk}|_{D(f_if_jf_k)}", "\\circ", "\\psi_{ij}|_{D(f_if_jf_k)}", "=", "\\psi_{ik}|_{D(f_if_jf_k)}", "$$", "on triple overlaps.", "\\medskip\\noindent", "Recall that each of the open subspaces $D(f_i)$, $D(f_if_j)$,", "$D(f_if_jf_k)$ is an affine scheme. Hence the sheaves $\\mathcal{F}_{M_i}$", "are isomorphic to the sheaves $\\widetilde M_i$ by Lemma", "\\ref{lemma-compare-constructions} above. In particular we see that", "$\\mathcal{F}_{M_i}(D(f_if_j)) = (M_i)_{f_j}$, etc.", "Also by Lemma \\ref{lemma-compare-constructions} above we see", "that $\\psi_{ij}$ corresponds to a unique $R_{f_if_j}$-module isomorphism", "$$", "\\psi_{ij} : (M_i)_{f_j} \\longrightarrow (M_j)_{f_i}", "$$", "namely, the effect of $\\psi_{ij}$ on sections over $D(f_if_j)$.", "Moreover these then satisfy the cocycle condition that", "$$", "\\xymatrix{", "(M_i)_{f_jf_k}", "\\ar[rd]_{\\psi_{ij}}", "\\ar[rr]^{\\psi_{ik}}", "& &", "(M_k)_{f_if_j} \\\\", "&", "(M_j)_{f_if_k} \\ar[ru]_{\\psi_{jk}}", "}", "$$", "commutes (for any triple $i, j, k$).", "\\medskip\\noindent", "Now Algebra, Lemma \\ref{algebra-lemma-glue-modules}", "shows that there exist an $R$-module $M$ such that", "$M_i = M_{f_i}$ compatible with the morphisms $\\psi_{ij}$.", "Consider $\\mathcal{F}_M = \\widetilde M$. At this point it is", "a formality to show that $\\widetilde M$ is isomorphic to", "the quasi-coherent sheaf $\\mathcal{F}$ we started out with.", "Namely, the sheaves $\\mathcal{F}$ and $\\widetilde M$ give", "rise to isomorphic sets of glueing data of sheaves of $\\mathcal{O}_X$-modules", "with respect to the covering $X = \\bigcup D(f_i)$, see", "Sheaves, Section \\ref{sheaves-section-glueing-sheaves}", "and in particular Lemma \\ref{sheaves-lemma-mapping-property-glue}.", "Explicitly, in the current situation, this boils down to", "the following argument: Let us construct an $R$-module map", "$$", "M \\longrightarrow \\Gamma(X, \\mathcal{F}).", "$$", "Namely, given $m \\in M$ we get $m_i = m/1 \\in M_{f_i} = M_i$", "by construction of $M$. By construction of $M_i$ this corresponds", "to a section $s_i \\in \\mathcal{F}(U_i)$. (Namely, $\\varphi^{-1}_i(m_i)$.)", "We claim that $s_i|_{D(f_if_j)} = s_j|_{D(f_if_j)}$. This is", "true because, by construction of $M$, we have $\\psi_{ij}(m_i) = m_j$,", "and by the construction of the $\\psi_{ij}$. By the sheaf condition of", "$\\mathcal{F}$ this collection of sections gives rise to a unique", "section $s$ of $\\mathcal{F}$ over $X$. We leave it to the reader", "to show that $m \\mapsto s$ is a $R$-module map.", "By Lemma \\ref{lemma-compare-constructions} we obtain an associated", "$\\mathcal{O}_X$-module map", "$$", "\\widetilde M \\longrightarrow \\mathcal{F}.", "$$", "By construction this map reduces to the isomorphisms", "$\\varphi_i^{-1}$ on each $D(f_i)$ and hence is an isomorphism." ], "refs": [ "topology-definition-locally-quasi-compact", "modules-lemma-quasi-coherent-module", "schemes-lemma-standard-open", "schemes-lemma-compare-constructions", "schemes-lemma-compare-constructions", "algebra-lemma-glue-modules", "sheaves-lemma-mapping-property-glue", "schemes-lemma-compare-constructions" ], "ref_ids": [ 8361, 13247, 7650, 7660, 7660, 417, 14558, 7660 ] } ], "ref_ids": [] }, { "id": 7664, "type": "theorem", "label": "schemes-lemma-equivalence-quasi-coherent", "categories": [ "schemes" ], "title": "schemes-lemma-equivalence-quasi-coherent", "contents": [ "Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$", "be an affine scheme.", "The functors $M \\mapsto \\widetilde M$ and", "$\\mathcal{F} \\mapsto \\Gamma(X, \\mathcal{F})$ define quasi-inverse", "equivalences of categories", "$$", "\\xymatrix{", "\\QCoh(\\mathcal{O}_X)", "\\ar@<1ex>[r]", "&", "\\text{Mod-}R", "\\ar@<1ex>[l]", "}", "$$", "between the category of quasi-coherent $\\mathcal{O}_X$-modules", "and the category of $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "See Lemmas \\ref{lemma-compare-constructions}", "and \\ref{lemma-quasi-coherent-affine} above." ], "refs": [ "schemes-lemma-compare-constructions", "schemes-lemma-quasi-coherent-affine" ], "ref_ids": [ 7660, 7663 ] } ], "ref_ids": [] }, { "id": 7665, "type": "theorem", "label": "schemes-lemma-kernel-cokernel-quasi-coherent", "categories": [ "schemes" ], "title": "schemes-lemma-kernel-cokernel-quasi-coherent", "contents": [ "Let $X = \\Spec(R)$ be an affine scheme.", "Kernels and cokernels of maps of quasi-coherent", "$\\mathcal{O}_X$-modules are quasi-coherent." ], "refs": [], "proofs": [ { "contents": [ "This follows from the exactness of the functor $\\widetilde{\\ }$", "since by Lemma \\ref{lemma-compare-constructions} we know that any map", "$\\psi : \\widetilde{M} \\to \\widetilde{N}$ comes from", "an $R$-module map $\\varphi : M \\to N$. (So we have", "$\\Ker(\\psi) = \\widetilde{\\Ker(\\varphi)}$ and", "$\\Coker(\\psi) = \\widetilde{\\Coker(\\varphi)}$.)" ], "refs": [ "schemes-lemma-compare-constructions" ], "ref_ids": [ 7660 ] } ], "ref_ids": [] }, { "id": 7666, "type": "theorem", "label": "schemes-lemma-colimit-quasi-coherent", "categories": [ "schemes" ], "title": "schemes-lemma-colimit-quasi-coherent", "contents": [ "Let $X = \\Spec(R)$ be an affine scheme.", "The direct sum of an arbitrary collection of quasi-coherent sheaves", "on $X$ is quasi-coherent. The same holds for colimits." ], "refs": [], "proofs": [ { "contents": [ "Suppose $\\mathcal{F}_i$, $i \\in I$ is a collection of quasi-coherent", "sheaves on $X$. By Lemma \\ref{lemma-equivalence-quasi-coherent}", "above we can write $\\mathcal{F}_i = \\widetilde{M_i}$ for some $R$-module", "$M_i$. Set $M = \\bigoplus M_i$. Consider the sheaf $\\widetilde{M}$.", "For each standard open $D(f)$ we have", "$$", "\\widetilde{M}(D(f)) = M_f =", "\\left(\\bigoplus M_i\\right)_f =", "\\bigoplus M_{i, f}.", "$$", "Hence we see that the quasi-coherent $\\mathcal{O}_X$-module", "$\\widetilde{M}$ is the direct sum of the sheaves $\\mathcal{F}_i$.", "A similar argument works for general colimits." ], "refs": [ "schemes-lemma-equivalence-quasi-coherent" ], "ref_ids": [ 7664 ] } ], "ref_ids": [] }, { "id": 7667, "type": "theorem", "label": "schemes-lemma-extension-quasi-coherent", "categories": [ "schemes" ], "title": "schemes-lemma-extension-quasi-coherent", "contents": [ "Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$", "be an affine scheme. Suppose that", "$$", "0 \\to", "\\mathcal{F}_1 \\to", "\\mathcal{F}_2 \\to", "\\mathcal{F}_3 \\to", "0", "$$", "is a short exact sequence of sheaves of $\\mathcal{O}_X$-modules.", "If two out of three are quasi-coherent then so is the third." ], "refs": [], "proofs": [ { "contents": [ "This is clear in case both $\\mathcal{F}_1$ and $\\mathcal{F}_2$ are", "quasi-coherent because the functor $M \\mapsto \\widetilde M$", "is exact, see Lemma \\ref{lemma-spec-sheaves}.", "Similarly in case both $\\mathcal{F}_2$ and $\\mathcal{F}_3$ are", "quasi-coherent. Now, suppose that $\\mathcal{F}_1 = \\widetilde M_1$ and", "$\\mathcal{F}_3 = \\widetilde M_3$ are quasi-coherent.", "Set $M_2 = \\Gamma(X, \\mathcal{F}_2)$. We claim it suffices to show that", "the sequence", "$$", "0 \\to M_1 \\to M_2 \\to M_3 \\to 0", "$$", "is exact. Namely, if this is the case, then (by using the mapping", "property of Lemma \\ref{lemma-compare-constructions}) we get a commutative", "diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\widetilde M_1 \\ar[r] \\ar[d] &", "\\widetilde M_2 \\ar[r] \\ar[d] &", "\\widetilde M_3 \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{F}_1 \\ar[r] &", "\\mathcal{F}_2 \\ar[r] &", "\\mathcal{F}_3 \\ar[r] &", "0", "}", "$$", "and we win by the snake lemma.", "\\medskip\\noindent", "The ``correct'' argument here would be to show first", "that $H^1(X, \\mathcal{F}) = 0$ for any quasi-coherent sheaf $\\mathcal{F}$.", "This is actually not all that hard, but it is perhaps better to postpone", "this till later. Instead we use a small trick.", "\\medskip\\noindent", "Pick $m \\in M_3 = \\Gamma(X, \\mathcal{F}_3)$.", "Consider the following set", "$$", "I = \\{ f \\in R \\mid \\text{the element }fm\\text{ comes from }M_2\\}.", "$$", "Clearly this is an ideal. It suffices to show $1 \\in I$.", "Hence it suffices to show that for any prime $\\mathfrak p$", "there exists an $f \\in I$, $f \\not\\in \\mathfrak p$.", "Let $x \\in X$ be the point corresponding to $\\mathfrak p$.", "Because surjectivity can be checked on stalks", "there exists an open neighbourhood $U$ of $x$ such that", "$m|_U$ comes from a local section $s \\in \\mathcal{F}_2(U)$.", "In fact we may assume that $U = D(f)$ is a standard open,", "i.e., $f \\in R$, $f \\not \\in \\mathfrak p$. We will show", "that for some $N \\gg 0$ we have $f^N \\in I$, which", "will finish the proof.", "\\medskip\\noindent", "Take any point $z \\in V(f)$, say corresponding to the", "prime $\\mathfrak q \\subset R$. We can also find a $g \\in R$,", "$g \\not \\in \\mathfrak q$ such that $m|_{D(g)}$ lifts", "to some $s' \\in \\mathcal{F}_2(D(g))$.", "Consider the difference $s|_{D(fg)} - s'|_{D(fg)}$.", "This is an element $m'$ of $\\mathcal{F}_1(D(fg)) = (M_1)_{fg}$.", "For some integer $n = n(z)$ the element $f^n m'$ comes", "from some $m'_1 \\in (M_1)_g$. We see that", "$f^n s$ extends to a section $\\sigma$ of $\\mathcal{F}_2$ on $D(f) \\cup D(g)$", "because it agrees with the restriction of", "$f^n s' + m'_1$ on $D(f) \\cap D(g) = D(fg)$.", "Moreover, $\\sigma$ maps to the restriction of $f^n m$", "to $D(f) \\cup D(g)$.", "\\medskip\\noindent", "Since $V(f)$ is quasi-compact, there exists a finite list", "of elements $g_1, \\ldots, g_m \\in R$ such that", "$V(f) \\subset \\bigcup D(g_j)$, an integer $n > 0$ and sections", "$\\sigma_j \\in \\mathcal{F}_2(D(f) \\cup D(g_j))$ such that", "$\\sigma_j|_{D(f)} = f^n s$ and $\\sigma_j$ maps to the section", "$f^nm|_{D(f) \\cup D(g_j)}$ of $\\mathcal{F}_3$.", "Consider the differences", "$$", "\\sigma_j|_{D(f) \\cup D(g_jg_k)}", "-", "\\sigma_k|_{D(f) \\cup D(g_jg_k)}.", "$$", "These correspond to sections of $\\mathcal{F}_1$", "over $D(f) \\cup D(g_jg_k)$ which are zero", "on $D(f)$. In particular their images in", "$\\mathcal{F}_1(D(g_jg_k)) = (M_1)_{g_jg_k}$", "are zero in $(M_1)_{g_jg_kf}$.", "Thus some high power of $f$ kills each and every one of these.", "In other words, the elements $f^N \\sigma_j$, for some $N \\gg 0$", "satisfy the glueing condition of the sheaf property and", "give rise to a section $\\sigma $ of $\\mathcal{F}_2$", "over $\\bigcup (D(f) \\cup D(g_j)) = X$ as desired." ], "refs": [ "schemes-lemma-spec-sheaves", "schemes-lemma-compare-constructions" ], "ref_ids": [ 7651, 7660 ] } ], "ref_ids": [] }, { "id": 7668, "type": "theorem", "label": "schemes-lemma-closed-immersion-affine-case", "categories": [ "schemes" ], "title": "schemes-lemma-closed-immersion-affine-case", "contents": [ "\\begin{slogan}", "For affine schemes, closed immersions correspond to ideals.", "\\end{slogan}", "Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$", "be an affine scheme. Let $i : Z \\to X$ be any closed immersion", "of locally ringed spaces. Then there exists a unique ideal", "$I \\subset R$ such that the morphism $i : Z \\to X$ can be identified", "with the closed immersion $\\Spec(R/I) \\to \\Spec(R)$", "constructed in Example \\ref{example-closed-immersion-affines} above." ], "refs": [], "proofs": [ { "contents": [ "This is kind of silly! Namely, by Lemma \\ref{lemma-closed-immersion}", "we can identify $Z \\to X$ with the closed subspace associated to", "a sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$ as in", "Definition \\ref{definition-closed-subspace} and", "Example \\ref{example-closed-subspace}.", "By our conventions this sheaf of ideals is locally generated", "by sections as a sheaf of $\\mathcal{O}_X$-modules.", "Hence the quotient sheaf $\\mathcal{O}_X / \\mathcal{I}$", "is locally on $X$ the cokernel", "of a map $\\bigoplus_{j \\in J} \\mathcal{O}_U \\to \\mathcal{O}_U$.", "Thus by definition, $\\mathcal{O}_X / \\mathcal{I}$ is quasi-coherent.", "By our results in Section \\ref{section-quasi-coherent-affine}", "it is of the form $\\widetilde S$ for some $R$-module $S$.", "Moreover, since $\\mathcal{O}_X = \\widetilde R \\to \\widetilde S$", "is surjective we see by Lemma \\ref{lemma-extension-quasi-coherent}", "that also $\\mathcal{I}$ is quasi-coherent, say $\\mathcal{I} = \\widetilde I$.", "Of course $I \\subset R$ and $S = R/I$ and everything is clear." ], "refs": [ "schemes-lemma-closed-immersion", "schemes-definition-closed-subspace", "schemes-lemma-extension-quasi-coherent" ], "ref_ids": [ 7647, 7738, 7667 ] } ], "ref_ids": [] }, { "id": 7669, "type": "theorem", "label": "schemes-lemma-open-subspace-scheme", "categories": [ "schemes" ], "title": "schemes-lemma-open-subspace-scheme", "contents": [ "Let $X$ be a scheme. Let $j : U \\to X$ be an open immersion", "of locally ringed spaces. Then $U$ is a scheme. In particular,", "any open subspace of $X$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$. Let $u \\in U$.", "Pick an affine open neighbourhood $u \\in V \\subset X$.", "Because standard opens of $V$ form a basis of the topology", "on $V$ we see that there exists a $f\\in \\mathcal{O}_V(V)$", "such that $u \\in D(f) \\subset U$. And $D(f)$ is an affine scheme", "by Lemma \\ref{lemma-standard-open-affine}. This proves that every point", "of $U$ has an open neighbourhood which is affine." ], "refs": [ "schemes-lemma-standard-open-affine" ], "ref_ids": [ 7657 ] } ], "ref_ids": [] }, { "id": 7670, "type": "theorem", "label": "schemes-lemma-closed-subspace-scheme", "categories": [ "schemes" ], "title": "schemes-lemma-closed-subspace-scheme", "contents": [ "Let $X$ be a scheme. Let $i : Z \\to X$ be a closed immersion", "of locally ringed spaces.", "\\begin{enumerate}", "\\item The locally ringed space $Z$ is a scheme,", "\\item the kernel $\\mathcal{I}$ of the map", "$\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is a quasi-coherent", "sheaf of ideals,", "\\item for any affine open $U = \\Spec(R)$ of $X$", "the morphism $i^{-1}(U) \\to U$ can be identified with", "$\\Spec(R/I) \\to \\Spec(R)$ for some ideal $I \\subset R$, and", "\\item we have $\\mathcal{I}|_U = \\widetilde I$.", "\\end{enumerate}", "In particular, any sheaf of ideals locally generated by sections", "is a quasi-coherent sheaf of ideals (and vice versa),", "and any closed subspace of $X$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "Let $i : Z \\to X$ be a closed immersion.", "Let $z \\in Z$ be a point. Choose any affine open", "neighbourhood $i(z) \\in U \\subset X$. Say $U = \\Spec(R)$.", "By Lemma \\ref{lemma-closed-immersion-affine-case} we know", "that $i^{-1}(U) \\to U$ can be identified with the morphism", "of affine schemes $\\Spec(R/I) \\to \\Spec(R)$.", "First of all this implies that $z \\in i^{-1}(U) \\subset Z$ is an", "affine neighbourhood of $z$. Thus $Z$ is a scheme. Second", "this implies that $\\mathcal{I}|_U$ is $\\widetilde I$.", "In other words for every point $x \\in i(Z)$ there exists an", "open neighbourhood such that $\\mathcal{I}$ is quasi-coherent in", "that neighbourhood. Note that $\\mathcal{I}|_{X \\setminus i(Z)}", "\\cong \\mathcal{O}_{X \\setminus i(Z)}$. Thus the restriction", "of the sheaf of ideals is quasi-coherent on $X \\setminus i(Z)$", "also. We conclude that $\\mathcal{I}$ is quasi-coherent." ], "refs": [ "schemes-lemma-closed-immersion-affine-case" ], "ref_ids": [ 7668 ] } ], "ref_ids": [] }, { "id": 7671, "type": "theorem", "label": "schemes-lemma-immersion-when-closed", "categories": [ "schemes" ], "title": "schemes-lemma-immersion-when-closed", "contents": [ "Let $f : Y \\to X$ be an immersion of schemes. Then $f$ is a closed", "immersion if and only if $f(Y) \\subset X$ is a closed subset." ], "refs": [], "proofs": [ { "contents": [ "If $f$ is a closed immersion then $f(Y)$ is closed by definition.", "Conversely, suppose that $f(Y)$ is closed. By definition", "there exists an open subscheme $U \\subset X$ such that $f$ is the composition", "of a closed immersion $i : Y \\to U$ and the open immersion", "$j : U \\to X$. Let $\\mathcal{I} \\subset \\mathcal{O}_U$ be the", "quasi-coherent sheaf of ideals associated to the closed immersion", "$i$. Note that", "$\\mathcal{I}|_{U \\setminus i(Y)}", "= \\mathcal{O}_{U \\setminus i(Y)}", "= \\mathcal{O}_{X \\setminus i(Y)}|_{U \\setminus i(Y)}$.", "Thus we may glue (see Sheaves, Section \\ref{sheaves-section-glueing-sheaves})", "$\\mathcal{I}$ and $\\mathcal{O}_{X \\setminus i(Y)}$ to a sheaf of", "ideals $\\mathcal{J} \\subset \\mathcal{O}_X$. Since every", "point of $X$ has a neighbourhood where $\\mathcal{J}$ is", "quasi-coherent, we see that $\\mathcal{J}$ is quasi-coherent", "(in particular locally generated by sections).", "By construction $\\mathcal{O}_X/\\mathcal{J}$ is supported", "on $U$ and equal to $\\mathcal{O}_U/\\mathcal{I}$.", "Thus we see that the closed subspaces associated to $\\mathcal{I}$", "and $\\mathcal{J}$ are canonically isomorphic, see", "Example \\ref{example-closed-subspace}.", "In particular the closed subspace of $U$ associated to $\\mathcal{I}$", "is isomorphic to a closed subspace of $X$.", "Since $Y \\to U$ is identified with the closed subspace", "associated to $\\mathcal{I}$, see Lemma \\ref{lemma-closed-immersion},", "we conclude that $Y \\to U \\to X$", "is a closed immersion." ], "refs": [ "schemes-lemma-closed-immersion" ], "ref_ids": [ 7647 ] } ], "ref_ids": [] }, { "id": 7672, "type": "theorem", "label": "schemes-lemma-scheme-sober", "categories": [ "schemes" ], "title": "schemes-lemma-scheme-sober", "contents": [ "Let $X$ be a scheme.", "Any irreducible closed subset of $X$ has a unique generic point.", "In other words, $X$ is a sober topological space, see", "Topology, Definition \\ref{topology-definition-generic-point}." ], "refs": [ "topology-definition-generic-point" ], "proofs": [ { "contents": [ "Let $Z \\subset X$ be an irreducible closed subset.", "For every affine open $U \\subset X$, $U = \\Spec(R)$", "we know that $Z \\cap U = V(I)$ for a unique", "radical ideal $I \\subset R$. Note that $Z \\cap U$ is either", "empty or irreducible. In the second case (which occurs", "for at least one $U$) we see that $I = \\mathfrak p$", "is a prime ideal, which is a generic point $\\xi$ of $Z \\cap U$.", "It follows that $Z = \\overline{\\{\\xi\\}}$, in other words", "$\\xi$ is a generic point of $Z$. If $\\xi'$ was a second", "generic point, then $\\xi' \\in Z \\cap U$ and it follows", "immediately that $\\xi' = \\xi$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8354 ] }, { "id": 7673, "type": "theorem", "label": "schemes-lemma-basis-affine-opens", "categories": [ "schemes" ], "title": "schemes-lemma-basis-affine-opens", "contents": [ "Let $X$ be a scheme. The collection of affine opens", "of $X$ forms a basis for the topology on $X$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the discussion on open subschemes", "in Section \\ref{section-schemes}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7674, "type": "theorem", "label": "schemes-lemma-locally-quasi-compact", "categories": [ "schemes" ], "title": "schemes-lemma-locally-quasi-compact", "contents": [ "The underlying topological space of any scheme is", "locally quasi-compact, see", "Topology, Definition \\ref{topology-definition-locally-quasi-compact}." ], "refs": [ "topology-definition-locally-quasi-compact" ], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-basis-affine-opens} above", "and the fact that the spectrum of ring is quasi-compact, see", "Algebra, Lemma \\ref{algebra-lemma-quasi-compact}." ], "refs": [ "schemes-lemma-basis-affine-opens", "algebra-lemma-quasi-compact" ], "ref_ids": [ 7673, 395 ] } ], "ref_ids": [ 8361 ] }, { "id": 7675, "type": "theorem", "label": "schemes-lemma-standard-open-two-affines", "categories": [ "schemes" ], "title": "schemes-lemma-standard-open-two-affines", "contents": [ "Let $X$ be a scheme.", "Let $U, V$ be affine opens of $X$, and let $x \\in U \\cap V$.", "There exists an affine open neighbourhood $W$ of $x$", "such that $W$ is a standard open of both $U$ and $V$." ], "refs": [], "proofs": [ { "contents": [ "Write $U = \\Spec(A)$ and $V = \\Spec(B)$.", "Say $x$ corresponds to the prime $\\mathfrak p \\subset A$", "and the prime $\\mathfrak q \\subset B$.", "We may choose a $f \\in A$, $f \\not \\in \\mathfrak p$ such that", "$D(f) \\subset U \\cap V$. Note that any standard open of $D(f)$", "is a standard open of $\\Spec(A) = U$. Hence we may assume", "that $U \\subset V$. In other words, now we may think of $U$", "as an affine open of $V$. Next we choose a", "$g \\in B$, $g \\not \\in \\mathfrak q$ such that", "$D(g) \\subset U$. In this case we see that $D(g) = D(g_A)$", "where $g_A \\in A$ denotes the image of $g$ by the map $B \\to A$.", "Thus the lemma is proved." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7676, "type": "theorem", "label": "schemes-lemma-good-subcover", "categories": [ "schemes" ], "title": "schemes-lemma-good-subcover", "contents": [ "Let $X$ be a scheme.", "Let $X = \\bigcup_i U_i$ be an affine open covering.", "Let $V \\subset X$ be an affine open.", "There exists a standard open covering", "$V = \\bigcup_{j = 1, \\ldots, m} V_j$ (see", "Definition \\ref{definition-standard-covering})", "such that each $V_j$ is a standard open in one of the $U_i$." ], "refs": [ "schemes-definition-standard-covering" ], "proofs": [ { "contents": [ "Pick $v \\in V$. Then $v \\in U_i$ for some $i$.", "By Lemma \\ref{lemma-standard-open-two-affines} above there exists an open", "$v \\in W_v \\subset V \\cap U_i$ such that", "$W_v$ is a standard open in both $V$ and $U_i$.", "Since $V$ is quasi-compact the lemma follows." ], "refs": [ "schemes-lemma-standard-open-two-affines" ], "ref_ids": [ 7675 ] } ], "ref_ids": [ 7739 ] }, { "id": 7677, "type": "theorem", "label": "schemes-lemma-sheaf-on-affines", "categories": [ "schemes" ], "title": "schemes-lemma-sheaf-on-affines", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{B}$ be the set of affine opens of $X$.", "Let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{B}$, see", "Sheaves, Definition \\ref{sheaves-definition-presheaf-basis}. The following", "are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is the restriction of a sheaf on $X$ to $\\mathcal{B}$,", "\\item $\\mathcal{F}$ is a sheaf on $\\mathcal{B}$, and", "\\item $\\mathcal{F}(\\emptyset)$ is a singleton and", "whenever $U = V \\cup W$ with $U, V, W \\in \\mathcal{B}$ and", "$V, W \\subset U$ standard open", "(Algebra, Definition \\ref{algebra-definition-Zariski-topology})", "the map", "$$", "\\mathcal{F}(U) \\longrightarrow \\mathcal{F}(V) \\times \\mathcal{F}(W)", "$$", "is injective with image the set of pairs $(s, t)$", "such that $s|_{V \\cap W} = t|_{V \\cap W}$.", "\\end{enumerate}" ], "refs": [ "sheaves-definition-presheaf-basis", "algebra-definition-Zariski-topology" ], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is", "Sheaves, Lemma \\ref{sheaves-lemma-restrict-basis-equivalence}.", "It is clear that (2) implies (3).", "Hence it suffices to prove that (3) implies (2). By", "Sheaves, Lemma \\ref{sheaves-lemma-cofinal-systems-coverings-standard-case}", "and Lemma \\ref{lemma-standard-open} it suffices to prove the sheaf", "condition holds for standard open coverings", "(Definition \\ref{definition-standard-covering})", "of elements of $\\mathcal{B}$.", "Let $U = U_1 \\cup \\ldots \\cup U_n$ be a standard open covering", "with $U \\subset X$ affine open.", "We will prove the sheaf condition", "for this covering by induction on $n$.", "If $n = 0$, then $U$ is empty and we get the sheaf condition", "by assumption. If $n = 1$, then there is nothing to prove.", "If $n = 2$, then this is assumption (3).", "If $n > 2$, then we write $U_i = D(f_i)$ for $f_i \\in A = \\mathcal{O}_X(U)$.", "Suppose that $s_i \\in \\mathcal{F}(U_i)$ are sections such that", "$s_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}$ for all $1 \\leq i < j \\leq n$.", "Since $U = U_1 \\cup \\ldots \\cup U_n$ we have", "$1 = \\sum_{i = 1, \\ldots, n} a_i f_i$ in $A$ for some $a_i \\in A$, see", "Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}.", "Set $g = \\sum_{i = 1, \\ldots, n - 1} a_if_i$.", "Then $U = D(g) \\cup D(f_n)$.", "Observe that $D(g) = D(gf_1) \\cup \\ldots \\cup D(gf_{n - 1})$", "is a standard open covering. By induction there is a unique section", "$s' \\in \\mathcal{F}(D(g))$ which agrees with", "$s_i|_{D(gfi)}$ for $i = 1, \\ldots, n - 1$.", "We claim that $s'$ and $s_n$ have the same restriction to", "$D(gf_n)$. This is true by induction and the covering", "$D(gf_n) = D(gf_nf_1) \\cup \\ldots \\cup D(gf_nf_{n - 1})$.", "Thus there is a unique section $s \\in \\mathcal{F}(U)$", "whose restriction to $D(g)$ is $s'$ and whose restriction", "to $D(f_n)$ is $s_n$. We omit the verification that $s$", "restricts to $s_i$ on $D(f_i)$ for $i = 1, \\ldots, n - 1$", "and we omit the verification that $s$ is unique." ], "refs": [ "sheaves-lemma-restrict-basis-equivalence", "sheaves-lemma-cofinal-systems-coverings-standard-case", "schemes-lemma-standard-open", "schemes-definition-standard-covering", "algebra-lemma-Zariski-topology" ], "ref_ids": [ 14533, 14530, 7650, 7739, 389 ] } ], "ref_ids": [ 14579, 1445 ] }, { "id": 7678, "type": "theorem", "label": "schemes-lemma-scheme-finite-discrete-affine", "categories": [ "schemes" ], "title": "schemes-lemma-scheme-finite-discrete-affine", "contents": [ "Let $X$ be a scheme whose underlying topological space", "is a finite discrete set.", "Then $X$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Say $X = \\{x_1, \\ldots, x_n\\}$. Then $U_i = \\{x_i\\}$ is an open neighbourhood", "of $x_i$. By", "Lemma \\ref{lemma-basis-affine-opens}", "it is affine. Hence $X$ is a finite disjoint union of affine schemes, and", "hence is affine by", "Lemma \\ref{lemma-disjoint-union-affines}." ], "refs": [ "schemes-lemma-basis-affine-opens", "schemes-lemma-disjoint-union-affines" ], "ref_ids": [ 7673, 7659 ] } ], "ref_ids": [] }, { "id": 7679, "type": "theorem", "label": "schemes-lemma-reduced", "categories": [ "schemes" ], "title": "schemes-lemma-reduced", "contents": [ "A scheme $X$ is reduced if and only if $\\mathcal{O}_X(U)$", "is a reduced ring for all $U \\subset X$ open." ], "refs": [], "proofs": [ { "contents": [ "Assume that $X$ is reduced.", "Let $f \\in \\mathcal{O}_X(U)$ be a section such that $f^n = 0$.", "Then the image of $f$ in $\\mathcal{O}_{U, u}$ is zero for", "all $u \\in U$. Hence $f$ is zero, see", "Sheaves, Lemma \\ref{sheaves-lemma-sheaf-subset-stalks}.", "Conversely, assume that $\\mathcal{O}_X(U)$ is reduced", "for all opens $U$. Pick any nonzero element $f \\in \\mathcal{O}_{X, x}$.", "Any representative $(U, f \\in \\mathcal{O}(U))$ of $f$ is nonzero and", "hence not nilpotent. Hence $f$ is not nilpotent in $\\mathcal{O}_{X, x}$." ], "refs": [ "sheaves-lemma-sheaf-subset-stalks" ], "ref_ids": [ 14482 ] } ], "ref_ids": [] }, { "id": 7680, "type": "theorem", "label": "schemes-lemma-affine-reduced", "categories": [ "schemes" ], "title": "schemes-lemma-affine-reduced", "contents": [ "An affine scheme $\\Spec(R)$ is reduced", "if and only if $R$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "The direct implication follows immediately from", "Lemma \\ref{lemma-reduced} above.", "In the other direction it follows since any localization of", "a reduced ring is reduced, and in particular the local rings", "of a reduced ring are reduced." ], "refs": [ "schemes-lemma-reduced" ], "ref_ids": [ 7679 ] } ], "ref_ids": [] }, { "id": 7681, "type": "theorem", "label": "schemes-lemma-reduced-closed-subscheme", "categories": [ "schemes" ], "title": "schemes-lemma-reduced-closed-subscheme", "contents": [ "Let $X$ be a scheme. Let $T \\subset X$ be a closed subset.", "There exists a unique closed subscheme $Z \\subset X$ with", "the following properties: (a) the underlying topological", "space of $Z$ is equal to $T$, and (b) $Z$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the sub presheaf", "defined by the rule", "$$", "\\mathcal{I}(U) = \\{f \\in \\mathcal{O}_X(U) \\mid", "f(t) = 0\\text{ for all }t \\in T\\cap U\\}", "$$", "Here we use $f(t)$ to indicate the image of", "$f$ in the residue field $\\kappa(t)$ of $X$ at $t$.", "Because of the local nature of the condition it is", "clear that $\\mathcal{I}$ is a sheaf of ideals. Moreover,", "let $U = \\Spec(R)$ be an affine open.", "We may write $T \\cap U = V(I)$ for a unique radical", "ideal $I \\subset R$. Given a prime $\\mathfrak p \\in V(I)$", "corresponding to $t \\in T \\cap U$ and an element $f \\in R$ we have", "$f(t) = 0 \\Leftrightarrow f \\in \\mathfrak p$.", "Hence $\\mathcal{I}(U) = \\cap_{\\mathfrak p \\in V(I)} \\mathfrak p", "= I$ by Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}.", "Moreover, for any standard open $D(g) \\subset \\Spec(R) = U$", "we have $\\mathcal{I}(D(g)) = I_g$ by the same reasoning.", "Thus $\\widetilde I$ and $\\mathcal{I}|_U$ agree (as ideals)", "on a basis of opens and hence are equal. Therefore", "$\\mathcal{I}$ is a quasi-coherent sheaf of ideals.", "\\medskip\\noindent", "At this point we may define $Z$ as the closed subspace", "associated to the sheaf of ideals $\\mathcal{I}$. For every", "affine open $U = \\Spec(R)$ of $X$ we see that", "$Z \\cap U = \\Spec(R/I)$ where $I$ is a radical ideal and", "hence $Z$ is reduced (by Lemma \\ref{lemma-affine-reduced} above).", "By construction the underlying closed subset of $Z$ is $T$.", "Hence we have found a closed subscheme with properties (a) and (b).", "\\medskip\\noindent", "Let $Z' \\subset X$ be a second closed subscheme with properties (a) and (b).", "For every affine open $U = \\Spec(R)$ of $X$ we see that", "$Z' \\cap U = \\Spec(R/I')$ for some ideal $I' \\subset R$.", "By Lemma \\ref{lemma-affine-reduced} the ring $R/I'$ is", "reduced and hence $I'$ is radical. Since $V(I') = T \\cap U = V(I)$", "we deduced that $I = I'$ by", "Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}.", "Hence $Z'$ and $Z$ are defined by the same sheaf of ideals", "and hence are equal." ], "refs": [ "algebra-lemma-Zariski-topology", "schemes-lemma-affine-reduced", "schemes-lemma-affine-reduced", "algebra-lemma-Zariski-topology" ], "ref_ids": [ 389, 7680, 7680, 389 ] } ], "ref_ids": [] }, { "id": 7682, "type": "theorem", "label": "schemes-lemma-map-into-reduction", "categories": [ "schemes" ], "title": "schemes-lemma-map-into-reduction", "contents": [ "Let $X$ be a scheme.", "Let $Z \\subset X$ be a closed subscheme.", "Let $Y$ be a reduced scheme.", "A morphism $f : Y \\to X$ factors through $Z$ if and only if", "$f(Y) \\subset Z$ (set theoretically). In particular, any", "morphism $Y \\to X$ factors as $Y \\to X_{red} \\to X$." ], "refs": [], "proofs": [ { "contents": [ "Assume $f(Y) \\subset Z$ (set theoretically).", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the ideal sheaf of $Z$.", "For any affine opens $U \\subset X$, $\\Spec(B) = V \\subset Y$", "with $f(V) \\subset U$ and any $g \\in \\mathcal{I}(U)$", "the pullback $b = f^\\sharp(g) \\in \\Gamma(V, \\mathcal{O}_Y) = B$", "maps to zero in the residue field of any $y \\in V$.", "In other words $b \\in \\bigcap_{\\mathfrak p \\subset B} \\mathfrak p$.", "This implies $b = 0$ as $B$ is reduced (Lemma \\ref{lemma-reduced}, and", "Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}).", "Hence $f$ factors through", "$Z$ by Lemma \\ref{lemma-characterize-closed-subspace}." ], "refs": [ "schemes-lemma-reduced", "algebra-lemma-Zariski-topology", "schemes-lemma-characterize-closed-subspace" ], "ref_ids": [ 7679, 389, 7648 ] } ], "ref_ids": [] }, { "id": 7683, "type": "theorem", "label": "schemes-lemma-morphism-from-spec-local-ring", "categories": [ "schemes" ], "title": "schemes-lemma-morphism-from-spec-local-ring", "contents": [ "Let $X$ be a scheme. Let $R$ be a local ring.", "The construction above gives a bijective correspondence", "between morphisms $\\Spec(R) \\to X$ and pairs", "$(x, \\varphi)$ consisting of a point $x \\in X$ and", "a local homomorphism of local rings $\\varphi : \\mathcal{O}_{X, x} \\to R$." ], "refs": [], "proofs": [ { "contents": [ "Let $A$ be a ring. For any ring homomorphism $\\psi : A \\to R$", "there exists a unique prime ideal $\\mathfrak p \\subset A$", "and a factorization $A \\to A_{\\mathfrak p} \\to R$ where the", "last map is a local homomorphism of local rings. Namely,", "$\\mathfrak p = \\psi^{-1}(\\mathfrak m)$. Via", "Lemma \\ref{lemma-morphism-into-affine}", "this proves that the lemma holds if $X$ is an affine scheme.", "\\medskip\\noindent", "Let $X$ be a general scheme. Any $x \\in X$ is contained in", "an open affine $U \\subset X$. By the affine case we conclude that every pair", "$(x, \\varphi)$ occurs as the end product of the construction", "above the lemma.", "\\medskip\\noindent", "To finish the proof it suffices to show that any morphism", "$f : \\Spec(R) \\to X$ has image contained in any affine", "open containing the image $x$ of the closed", "point of $\\Spec(R)$. In fact, let $x \\in V \\subset X$", "be any open neighbourhood containing $x$. Then", "$f^{-1}(V) \\subset \\Spec(R)$ is an open containing", "the unique closed point and hence equal to $\\Spec(R)$." ], "refs": [ "schemes-lemma-morphism-into-affine" ], "ref_ids": [ 7655 ] } ], "ref_ids": [] }, { "id": 7684, "type": "theorem", "label": "schemes-lemma-specialize-points", "categories": [ "schemes" ], "title": "schemes-lemma-specialize-points", "contents": [ "Let $X$ be a scheme.", "Let $x, x' \\in X$ be points of $X$.", "Then $x' \\in X$ is a generalization of $x$ if and only if", "$x'$ is in the image of the canonical morphism", "$\\Spec(\\mathcal{O}_{X, x}) \\to X$." ], "refs": [], "proofs": [ { "contents": [ "A continuous map preserves the relation of specialization/generalization.", "Since every point of $\\Spec(\\mathcal{O}_{X, x})$ is a", "generalization of the closed point we see every point in the image", "of $\\Spec(\\mathcal{O}_{X, x}) \\to X$ is a generalization of $x$.", "Conversely, suppose that $x'$ is a generalization of $x$.", "Choose an affine open neighbourhood $U = \\Spec(R)$ of", "$x$. Then $x' \\in U$. Say $\\mathfrak p \\subset R$ and", "$\\mathfrak p' \\subset R$ are the primes corresponding", "to $x$ and $x'$. Since $x'$ is a generalization of $x$", "we see that $\\mathfrak p' \\subset \\mathfrak p$. This means", "that $\\mathfrak p'$ is in the image of the morphism", "$\\Spec(\\mathcal{O}_{X, x}) = \\Spec(R_{\\mathfrak p})", "\\to \\Spec(R) = U \\subset X$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7685, "type": "theorem", "label": "schemes-lemma-characterize-points", "categories": [ "schemes" ], "title": "schemes-lemma-characterize-points", "contents": [ "Let $X$ be a scheme. Points of $X$ correspond bijectively", "to equivalence classes of morphisms from spectra of", "fields into $X$. Moreover, each equivalence class contains", "a (unique up to unique isomorphism) smallest element", "$\\Spec(\\kappa(x)) \\to X$." ], "refs": [], "proofs": [ { "contents": [ "Follows from the discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7686, "type": "theorem", "label": "schemes-lemma-glue", "categories": [ "schemes" ], "title": "schemes-lemma-glue", "contents": [ "\\begin{slogan}", "If you have two locally ringed spaces, and a subspace of the first one", "is isomorphic to a subspace of the other, then you can glue them together", "into one big locally ringed space.", "\\end{slogan}", "Given any glueing data of locally ringed spaces there", "exists a locally ringed space $X$ and open subspaces", "$U_i \\subset X$ together with isomorphisms", "$\\varphi_i : X_i \\to U_i$ of locally ringed spaces such that", "\\begin{enumerate}", "\\item $\\varphi_i(U_{ij}) = U_i \\cap U_j$, and", "\\item $\\varphi_{ij} =", "\\varphi_j^{-1}|_{U_i \\cap U_j} \\circ \\varphi_i|_{U_{ij}}$.", "\\end{enumerate}", "The locally ringed space $X$ is characterized by the following", "mapping properties: Given a locally ringed space $Y$ we have", "\\begin{eqnarray*}", "\\Mor(X, Y) & = & \\{ (f_i)_{i\\in I} \\mid", "f_i : X_i \\to Y, \\ f_j \\circ \\varphi_{ij} = f_i|_{U_{ij}}\\} \\\\", "f & \\mapsto & (f|_{U_i} \\circ \\varphi_i)_{i \\in I} \\\\", "\\Mor(Y, X) & = &", "\\left\\{", "\\begin{matrix}", "\\text{open covering }Y = \\bigcup\\nolimits_{i \\in I} V_i\\text{ and }", "(g_i : V_i \\to X_i)_{i \\in I}", "\\text{ such that}\\\\", "g_i^{-1}(U_{ij}) = V_i \\cap V_j", "\\text{ and }", "g_j|_{V_i \\cap V_j} = \\varphi_{ij} \\circ g_i|_{V_i \\cap V_j}", "\\end{matrix}", "\\right\\} \\\\", "g & \\mapsto &", "V_i = g^{-1}(U_i), \\ g_i = \\varphi_i^{-1} \\circ g|_{V_i}", "\\end{eqnarray*}" ], "refs": [], "proofs": [ { "contents": [ "We construct $X$ in stages.", "As a set we take", "$$", "X = (\\coprod X_i) / \\sim.", "$$", "Here given $x \\in X_i$ and $x' \\in X_j$ we say", "$x \\sim x'$ if and only if $x \\in U_{ij}$, $x' \\in U_{ji}$", "and $\\varphi_{ij}(x) = x'$. This is an equivalence relation", "since if $x \\in X_i$, $x' \\in X_j$, $x'' \\in X_k$, and $x \\sim x'$ and", "$x' \\sim x''$, then $x' \\in U_{ji} \\cap U_{jk}$, hence by condition (1) of", "a glueing data also $x \\in U_{ij} \\cap U_{ik}$ and", "$x'' \\in U_{ki} \\cap U_{kj}$ and by condition (2)", "we see that $\\varphi_{ik}(x) = x''$. (Reflexivity and symmetry", "follows from our assumptions that $U_{ii} = X_i$ and", "$\\varphi_{ii} = \\text{id}_{X_i}$.)", "Denote $\\varphi_i : X_i \\to X$", "the natural maps. Denote $U_i = \\varphi_i(X_i) \\subset X$.", "Note that $\\varphi_i : X_i \\to U_i$ is a bijection.", "\\medskip\\noindent", "The topology on $X$ is defined by the rule that", "$U \\subset X$ is open if and only if $\\varphi_i^{-1}(U)$", "is open for all $i$. We leave it to the reader to verify", "that this does indeed define a topology.", "Note that in particular $U_i$ is open since $\\varphi_j^{-1}(U_i)", "= U_{ji}$ which is open in $X_j$ for all $j$.", "Moreover, for any open set $W \\subset X_i$ the image", "$\\varphi_i(W) \\subset U_i$ is open because", "$\\varphi_j^{-1}(\\varphi_i(W)) = \\varphi_{ji}^{-1}(W \\cap U_{ij})$.", "Therefore $\\varphi_i : X_i \\to U_i$ is a homeomorphism.", "\\medskip\\noindent", "To obtain a locally ringed space we have to construct the", "sheaf of rings $\\mathcal{O}_X$. We do this by glueing the", "sheaves of rings $\\mathcal{O}_{U_i} := \\varphi_{i, *} \\mathcal{O}_i$.", "Namely, in the commutative diagram", "$$", "\\xymatrix{", "U_{ij} \\ar[rr]_{\\varphi_{ij}} \\ar[rd]_{\\varphi_i|_{U_{ij}}}", "& &", "U_{ji} \\ar[ld]^{\\varphi_j|_{U_{ji}}} \\\\", "& U_i \\cap U_j &", "}", "$$", "the arrow on top is an isomorphism of ringed spaces,", "and hence we get unique isomorphisms of sheaves of rings", "$$", "\\mathcal{O}_{U_i}|_{U_i \\cap U_j}", "\\longrightarrow", "\\mathcal{O}_{U_j}|_{U_i \\cap U_j}.", "$$", "These satisfy a cocycle condition as in Sheaves,", "Section \\ref{sheaves-section-glueing-sheaves}.", "By the results of that section we obtain a sheaf of rings", "$\\mathcal{O}_X$ on $X$ such that $\\mathcal{O}_X|_{U_i}$", "is isomorphic to $\\mathcal{O}_{U_i}$ compatibly with", "the glueing maps displayed above.", "In particular $(X, \\mathcal{O}_X)$ is a locally ringed", "space since the stalks of $\\mathcal{O}_X$ are equal", "to the stalks of $\\mathcal{O}_i$ at corresponding", "points.", "\\medskip\\noindent", "The proof of the mapping properties is omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7687, "type": "theorem", "label": "schemes-lemma-glue-schemes", "categories": [ "schemes" ], "title": "schemes-lemma-glue-schemes", "contents": [ "\\begin{slogan}", "Schemes can be glued to give new schemes.", "\\end{slogan}", "In Lemma \\ref{lemma-glue} above, assume that all", "$X_i$ are schemes. Then the resulting locally ringed", "space $X$ is a scheme." ], "refs": [ "schemes-lemma-glue" ], "proofs": [ { "contents": [ "This is clear since each of the $U_i$ is a scheme", "and hence every $x \\in X$ has an affine neighbourhood." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 7686 ] }, { "id": 7688, "type": "theorem", "label": "schemes-lemma-glue-functors", "categories": [ "schemes" ], "title": "schemes-lemma-glue-functors", "contents": [ "Let $F$ be a contravariant functor on the category of schemes", "with values in the category of sets. Suppose that", "\\begin{enumerate}", "\\item $F$ satisfies the sheaf property for the Zariski topology,", "\\item there exists a set $I$ and a collection of subfunctors", "$F_i \\subset F$ such that", "\\begin{enumerate}", "\\item each $F_i$ is representable,", "\\item each $F_i \\subset F$ is representable by open immersions, and", "\\item the collection $(F_i)_{i \\in I}$ covers $F$.", "\\end{enumerate}", "\\end{enumerate}", "Then $F$ is representable." ], "refs": [], "proofs": [ { "contents": [ "Let $X_i$ be a scheme representing $F_i$ and let", "$\\xi_i \\in F_i(X_i) \\subset F(X_i)$ be the ``universal family''.", "Because $F_j \\subset F$ is representable by open immersions,", "there exists an open $U_{ij} \\subset X_i$ such that", "$T \\to X_i$ factors through $U_{ij}$ if and only if", "$\\xi_i|_T \\in F_j(T)$. In particular", "$\\xi_i|_{U_{ij}} \\in F_j(U_{ij})$ and therefore we obtain a", "canonical morphism $\\varphi_{ij} : U_{ij} \\to X_j$ such that", "$\\varphi_{ij}^*\\xi_j = \\xi_i|_{U_{ij}}$. By definition of $U_{ji}$", "this implies that $\\varphi_{ij}$ factors through $U_{ji}$.", "Since $(\\varphi_{ij} \\circ \\varphi_{ji})^*\\xi_j", "=\\varphi_{ji}^*(\\varphi_{ij}^*\\xi_j) =", "\\varphi_{ji}^*\\xi_i = \\xi_j$ we conclude that", "$\\varphi_{ij} \\circ \\varphi_{ji} = \\text{id}_{U_{ji}}$", "because the pair $(X_j, \\xi_j)$ represents $F_j$.", "In particular the maps $\\varphi_{ij} : U_{ij} \\to U_{ji}$", "are isomorphisms of schemes.", "Next we have to show that", "$\\varphi_{ij}^{-1}(U_{ji} \\cap U_{jk}) = U_{ij} \\cap U_{ik}$.", "This is true because (a) $U_{ji} \\cap U_{jk}$ is the largest", "open of $U_{ji}$ such that $\\xi_j$ restricts to an element", "of $F_k$, (b) $U_{ij} \\cap U_{ik}$ is the largest", "open of $U_{ij}$ such that $\\xi_i$ restricts to an element", "of $F_k$, and (c) $\\varphi_{ij}^*\\xi_j = \\xi_i$. Moreover,", "the cocycle condition in Section \\ref{section-glueing-schemes}", "follows because both", "$\\varphi_{jk}|_{U_{ji} \\cap U_{jk}} \\circ", "\\varphi_{ij}|_{U_{ij} \\cap U_{ik}}$ and", "$\\varphi_{ik}|_{U_{ij} \\cap U_{ik}}$ pullback $\\xi_k$", "to the element $\\xi_i$.", "Thus we may apply Lemma \\ref{lemma-glue-schemes}", "to obtain a scheme $X$ with an open", "covering $X = \\bigcup U_i$ and isomorphisms $\\varphi_i : X_i \\to U_i$", "with properties as in Lemma \\ref{lemma-glue}.", "Let $\\xi_i' = (\\varphi_i^{-1})^* \\xi_i$.", "The conditions of Lemma \\ref{lemma-glue} imply that", "$\\xi_i'|_{U_i \\cap U_j} = \\xi_j'|_{U_i \\cap U_j}$.", "Therefore, by the condition that $F$ satisfies the sheaf", "condition in the Zariski topology we see that", "there exists an element $\\xi' \\in F(X)$ such that", "$\\xi_i = \\varphi_i^*\\xi'|_{U_i}$ for all $i$.", "Since $\\varphi_i$ is an isomorphism we also get that", "$(U_i, \\xi'|_{U_i})$ represents the functor $F_i$.", "\\medskip\\noindent", "We claim that the pair $(X, \\xi')$ represents the functor $F$.", "To show this, let $T$ be a scheme and let $\\xi \\in F(T)$.", "We will construct a unique morphism $g : T \\to X$ such that", "$g^*\\xi' = \\xi$. Namely, by the condition that the subfunctors", "$F_i$ cover $F$ there exists an open covering $T = \\bigcup V_i$", "such that for each $i$ the restriction $\\xi|_{V_i} \\in F_i(V_i)$.", "Moreover, since each of the inclusions $F_i \\subset F$ are representable", "by open immersions we may assume that each $V_i \\subset T$ is maximal", "open with this property.", "Because, $(U_i, \\xi'_{U_i})$ represents the functor $F_i$ we", "get a unique morphism $g_i : V_i \\to U_i$ such that", "$g_i^*\\xi'|_{U_i} = \\xi|_{V_i}$. On the overlaps $V_i \\cap V_j$", "the morphisms $g_i$ and $g_j$ agree, for example because they both", "pull back $\\xi'|_{U_i \\cap U_j} \\in F_i(U_i \\cap U_j)$", "to the same element. Thus the morphisms $g_i$ glue to a unique morphism", "from $T \\to X$ as desired." ], "refs": [ "schemes-lemma-glue-schemes", "schemes-lemma-glue", "schemes-lemma-glue" ], "ref_ids": [ 7687, 7686, 7686 ] } ], "ref_ids": [] }, { "id": 7689, "type": "theorem", "label": "schemes-lemma-fibre-products", "categories": [ "schemes" ], "title": "schemes-lemma-fibre-products", "contents": [ "The category of schemes has a final object, products and fibre products.", "In other words, the category of schemes has finite limits, see", "Categories, Lemma \\ref{categories-lemma-finite-limits-exist}." ], "refs": [ "categories-lemma-finite-limits-exist" ], "proofs": [ { "contents": [ "Please skip this proof. It is more important to learn", "how to work with the fibre product which is explained in", "the next section.", "\\medskip\\noindent", "By Lemma \\ref{lemma-morphism-into-affine}", "the scheme $\\Spec(\\mathbf{Z})$ is a final", "object in the category of locally ringed spaces. Thus it", "suffices to prove that fibred products exist.", "\\medskip\\noindent", "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes. We have to show", "that the functor", "\\begin{eqnarray*}", "F : \\Sch^{opp} & \\longrightarrow & \\textit{Sets} \\\\", "T & \\longmapsto &", "\\Mor(T, X) \\times_{\\Mor(T, S)} \\Mor(T, Y)", "\\end{eqnarray*}", "is representable. We claim that Lemma \\ref{lemma-glue-functors}", "applies to the functor $F$. If we prove this then the lemma is proved.", "\\medskip\\noindent", "First we show that $F$ satisfies the sheaf property in the", "Zariski topology. Namely, suppose that $T$ is a scheme,", "$T = \\bigcup_{i \\in I} U_i$ is an open covering, and", "$\\xi_i \\in F(U_i)$ such that", "$\\xi_i|_{U_i \\cap U_j} = \\xi_j|_{U_i \\cap U_j}$ for", "all pairs $i, j$. By definition $\\xi_i$ corresponds to", "a pair $(a_i, b_i)$ where $a_i : U_i \\to X$ and $b_i : U_i \\to Y$", "are morphisms of schemes such that $f \\circ a_i = g \\circ b_i$.", "The glueing condition says that", "$a_i|_{U_i \\cap U_j} = a_j|_{U_i \\cap U_j}$", "and", "$b_i|_{U_i \\cap U_j} = b_j|_{U_i \\cap U_j}$.", "Thus by glueing the morphisms $a_i$ we obtain a morphism", "of locally ringed spaces (i.e., a morphism of schemes)", "$a : T \\to X$ and similarly $b : T \\to Y$ (see for example", "the mapping property of Lemma \\ref{lemma-glue}). Moreover,", "on the members of an open covering the compositions", "$f \\circ a$ and $g \\circ b$ agree. Therefore", "$f \\circ a = g \\circ b$ and the pair $(a, b)$ defines", "an element of $F(T)$ which restricts to the pairs", "$(a_i, b_i)$ on each $U_i$. The sheaf condition is verified.", "\\medskip\\noindent", "Next, we construct the family of subfunctors.", "Choose an open covering by open affines", "$S = \\bigcup\\nolimits_{i \\in I} U_i$.", "For every $i \\in I$ choose open coverings by open affines", "$f^{-1}(U_i) = \\bigcup\\nolimits_{j \\in J_i} V_j$ and", "$g^{-1}(U_i) = \\bigcup\\nolimits_{k \\in K_i} W_k$.", "Note that $X = \\bigcup_{i \\in I} \\bigcup_{j \\in J_i} V_j$", "is an open covering and similarly for $Y$.", "For any $i \\in I$ and each pair $(j, k) \\in J_i \\times K_i$", "we have a commutative diagram", "$$", "\\xymatrix{", " & W_k \\ar[d] \\ar[rd] & \\\\", "V_j \\ar[rd] \\ar[r] & U_i \\ar[rd] & Y \\ar[d] \\\\", " & X \\ar[r] & S", "}", "$$", "where all the skew arrows are open immersions. For such a", "triple we get a functor", "\\begin{eqnarray*}", "F_{i, j, k} : \\Sch^{opp} & \\longrightarrow & \\textit{Sets} \\\\", "T & \\longmapsto &", "\\Mor(T, V_j) \\times_{\\Mor(T, U_i)} \\Mor(T, W_k).", "\\end{eqnarray*}", "There is an obvious transformation of functors $F_{i, j, k} \\to F$", "(coming from the huge commutative diagram above)", "which is injective, so we may think of $F_{i, j, k}$ as a subfunctor", "of $F$.", "\\medskip\\noindent", "We check condition (2)(a) of Lemma \\ref{lemma-glue-functors}.", "This follows directly from Lemma \\ref{lemma-fibre-product-affine-schemes}.", "(Note that we use here that the fibre products in the category of", "affine schemes are also fibre products in the whole category of locally", "ringed spaces.)", "\\medskip\\noindent", "We check condition (2)(b) of Lemma \\ref{lemma-glue-functors}.", "Let $T$ be a scheme and let $\\xi \\in F(T)$. In other words,", "$\\xi = (a, b)$ where $a : T \\to X$ and $b : T \\to Y$ are", "morphisms of schemes such that $f \\circ a = g \\circ b$.", "Set $V_{i, j, k} = a^{-1}(V_j) \\cap b^{-1}(W_k)$. For any", "further morphism $h : T' \\to T$ we have", "$h^*\\xi = (a \\circ h, b \\circ h)$. Hence we see that", "$h^*\\xi \\in F_{i, j, k}(T')$ if and only if", "$a(h(T')) \\subset V_j$ and $b(h(T')) \\subset W_k$.", "In other words, if and only if $h(T') \\subset V_{i, j, k}$.", "This proves condition (2)(b).", "\\medskip\\noindent", "We check condition (2)(c) of Lemma \\ref{lemma-glue-functors}.", "Let $T$ be a scheme and let $\\xi = (a, b) \\in F(T)$ as above.", "Set $V_{i, j, k} = a^{-1}(V_j) \\cap b^{-1}(W_k)$ as above.", "Condition (2)(c) just means that $T = \\bigcup V_{i, j, k}$", "which is evident. Thus the lemma is proved and fibre products", "exist." ], "refs": [ "schemes-lemma-morphism-into-affine", "schemes-lemma-glue-functors", "schemes-lemma-glue", "schemes-lemma-glue-functors", "schemes-lemma-fibre-product-affine-schemes", "schemes-lemma-glue-functors", "schemes-lemma-glue-functors" ], "ref_ids": [ 7655, 7688, 7686, 7688, 7658, 7688, 7688 ] } ], "ref_ids": [ 12224 ] }, { "id": 7690, "type": "theorem", "label": "schemes-lemma-fibre-product-affines", "categories": [ "schemes" ], "title": "schemes-lemma-fibre-product-affines", "contents": [ "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes", "with the same target. If $X, Y, S$ are all affine then", "$X \\times_S Y$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $X = \\Spec(A)$, $Y = \\Spec(B)$", "and $S = \\Spec(R)$. By Lemma \\ref{lemma-fibre-product-affine-schemes}", "the affine scheme $\\Spec(A \\otimes_R B)$", "is the fibre product $X \\times_S Y$ in the category", "of locally ringed spaces. Hence it is a fortiori the", "fibre product in the category of schemes." ], "refs": [ "schemes-lemma-fibre-product-affine-schemes" ], "ref_ids": [ 7658 ] } ], "ref_ids": [] }, { "id": 7691, "type": "theorem", "label": "schemes-lemma-open-fibre-product", "categories": [ "schemes" ], "title": "schemes-lemma-open-fibre-product", "contents": [ "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes", "with the same target. Let $X \\times_S Y$, $p$, $q$ be the fibre product.", "Suppose that $U \\subset S$,", "$V \\subset X$, $W \\subset Y$ are open subschemes", "such that $f(V) \\subset U$ and $g(W) \\subset U$.", "Then the canonical morphism", "$V \\times_U W \\to X \\times_S Y$ is an open immersion", "which identifies $V \\times_U W$ with $p^{-1}(V) \\cap q^{-1}(W)$." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme", "Suppose $a : T \\to V$ and $b : T \\to W$ are morphisms", "such that $f \\circ a = g \\circ b$ as morphisms into $U$.", "Then they agree as morphisms into $S$.", "By the universal property of the fibre product we get", "a unique morphism $T \\to X \\times_S Y$. Of course this morphism", "has image contained in the open $p^{-1}(V) \\cap q^{-1}(W)$.", "Thus $p^{-1}(V) \\cap q^{-1}(W)$ is a fibre product of", "$V$ and $W$ over $U$. The result follows from the uniqueness", "of fibre products, see Categories, Section", "\\ref{categories-section-fibre-products}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7692, "type": "theorem", "label": "schemes-lemma-affine-covering-fibre-product", "categories": [ "schemes" ], "title": "schemes-lemma-affine-covering-fibre-product", "contents": [ "\\begin{slogan}", "Bare-hands construction of fiber products: an affine open cover of a", "fiber product of schemes can be assembled from compatible", "affine open covers of the pieces.", "\\end{slogan}", "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes", "with the same target. Let $S = \\bigcup U_i$ be any affine open", "covering of $S$. For each $i \\in I$, let", "$f^{-1}(U_i) = \\bigcup_{j \\in J_i} V_j$ be an affine open covering", "of $f^{-1}(U_i)$ and let", "$g^{-1}(U_i) = \\bigcup_{k \\in K_i} W_k$ be an affine open covering", "of $g^{-1}(U_i)$. Then", "$$", "X \\times_S Y =", "\\bigcup\\nolimits_{i \\in I}", "\\bigcup\\nolimits_{j \\in J_i, \\ k \\in K_i}", "V_j \\times_{U_i} W_k", "$$", "is an affine open covering of $X \\times_S Y$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7693, "type": "theorem", "label": "schemes-lemma-points-fibre-product", "categories": [ "schemes" ], "title": "schemes-lemma-points-fibre-product", "contents": [ "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes", "with the same target. Points $z$ of $X \\times_S Y$ are in bijective", "correspondence to quadruples", "$$", "(x, y, s, \\mathfrak p)", "$$", "where $x \\in X$, $y \\in Y$, $s \\in S$ are points with", "$f(x) = s$, $g(y) = s$ and $\\mathfrak p$ is a prime ideal", "of the ring $\\kappa(x) \\otimes_{\\kappa(s)} \\kappa(y)$.", "The residue field of $z$ corresponds to", "the residue field of the prime $\\mathfrak p$." ], "refs": [], "proofs": [ { "contents": [ "Let $z$ be a point of $X \\times_S Y$ and let us construct a", "triple as above. Recall that we may think of $z$ as a morphism", "$\\Spec(\\kappa(z)) \\to X \\times_S Y$, see", "Lemma \\ref{lemma-characterize-points}. This morphism corresponds", "to morphisms $a : \\Spec(\\kappa(z)) \\to X$", "and $b : \\Spec(\\kappa(z)) \\to Y$ such that", "$f \\circ a = g \\circ b$. By the same lemma again", "we get points $x \\in X$, $y \\in Y$ lying over the same point", "$s \\in S$ as well as field maps $\\kappa(x) \\to \\kappa(z)$,", "$\\kappa(y) \\to \\kappa(z)$ such that the compositions", "$\\kappa(s) \\to \\kappa(x) \\to \\kappa(z)$", "and", "$\\kappa(s) \\to \\kappa(y) \\to \\kappa(z)$", "are the same. In other words we get a ring map", "$\\kappa(x) \\otimes_{\\kappa(s)} \\kappa(y) \\to \\kappa(z)$.", "We let $\\mathfrak p$ be the kernel of this map.", "\\medskip\\noindent", "Conversely, given a quadruple $(x, y, s, \\mathfrak p)$ we get a", "commutative solid diagram", "$$", "\\xymatrix{", "X \\times_S Y", "\\ar@/_/[dddr] \\ar@/^/[rrrd]", "& & & \\\\", "&", "\\Spec(\\kappa(x) \\otimes_{\\kappa(s)} \\kappa(y)/\\mathfrak p)", "\\ar[r] \\ar[d] \\ar@{-->}[lu]", "&", "\\Spec(\\kappa(y)) \\ar[d] \\ar[r] &", "Y \\ar[dd] \\\\", "&", "\\Spec(\\kappa(x)) \\ar[r] \\ar[d] &", "\\Spec(\\kappa(s)) \\ar[rd] &", "\\\\", "&", "X \\ar[rr] &", "&", "S", "}", "$$", "see the discussion in Section \\ref{section-points}. Thus we get the", "dotted arrow. The corresponding point $z$ of $X \\times_S Y$ is the", "image of the generic point of", "$\\Spec(\\kappa(x) \\otimes_{\\kappa(s)} \\kappa(y)/\\mathfrak p)$.", "We omit the verification that the two constructions are inverse", "to each other." ], "refs": [ "schemes-lemma-characterize-points" ], "ref_ids": [ 7685 ] } ], "ref_ids": [] }, { "id": 7694, "type": "theorem", "label": "schemes-lemma-fibre-product-immersion", "categories": [ "schemes" ], "title": "schemes-lemma-fibre-product-immersion", "contents": [ "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes", "with the same target.", "\\begin{enumerate}", "\\item If $f : X \\to S$ is a closed immersion,", "then $X \\times_S Y \\to Y$ is a closed immersion.", "Moreover, if $X \\to S$ corresponds to the quasi-coherent", "sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_S$, then", "$X \\times_S Y \\to Y$ corresponds to the sheaf of ideals", "$\\Im(g^*\\mathcal{I} \\to \\mathcal{O}_Y)$.", "\\item If $f : X \\to S$ is an open immersion,", "then $X \\times_S Y \\to Y$ is an open immersion.", "\\item If $f : X \\to S$ is an immersion,", "then $X \\times_S Y \\to Y$ is an immersion.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume that $X \\to S$ is a closed immersion corresponding", "to the quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_S$.", "By Lemma \\ref{lemma-restrict-map-to-closed} the closed subspace", "$Z \\subset Y$ defined by the sheaf of ideals", "$\\Im(g^*\\mathcal{I} \\to \\mathcal{O}_Y)$ is the fibre product", "in the category of locally ringed spaces.", "By Lemma \\ref{lemma-closed-subspace-scheme} $Z$ is a scheme.", "Hence $Z = X \\times_S Y$ and the first statement follows.", "The second follows from Lemma \\ref{lemma-open-fibre-product}", "for example. The third is a combination of", "the first two." ], "refs": [ "schemes-lemma-restrict-map-to-closed", "schemes-lemma-closed-subspace-scheme", "schemes-lemma-open-fibre-product" ], "ref_ids": [ 7649, 7670, 7691 ] } ], "ref_ids": [] }, { "id": 7695, "type": "theorem", "label": "schemes-lemma-base-change-immersion", "categories": [ "schemes" ], "title": "schemes-lemma-base-change-immersion", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an", "immersion (resp.\\ closed immersion, resp. open immersion)", "of schemes over $S$. Then any base change of $f$ is an", "immersion (resp.\\ closed immersion, resp. open immersion)." ], "refs": [], "proofs": [ { "contents": [ "We can think of the base change of $f$ via the morphism", "$S' \\to S$ as the top left vertical arrow in the following", "commutative diagram:", "$$", "\\xymatrix{", "X_{S'} \\ar[r] \\ar[d] & X \\ar[d] \\ar@/^4ex/[dd] \\\\", "Y_{S'} \\ar[r] \\ar[d] & Y \\ar[d] \\\\", "S' \\ar[r] & S", "}", "$$", "The diagram implies $X_{S'} \\cong Y_{S'} \\times_Y X$,", "and the lemma follows from Lemma \\ref{lemma-fibre-product-immersion}." ], "refs": [ "schemes-lemma-fibre-product-immersion" ], "ref_ids": [ 7694 ] } ], "ref_ids": [] }, { "id": 7696, "type": "theorem", "label": "schemes-lemma-fibre-topological", "categories": [ "schemes" ], "title": "schemes-lemma-fibre-topological", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Consider the diagrams", "$$", "\\xymatrix{", "X_s \\ar[r] \\ar[d] &", "X \\ar[d] &", "\\Spec(\\mathcal{O}_{S, s}) \\times_S X \\ar[r] \\ar[d] &", "X \\ar[d]", "\\\\", "\\Spec(\\kappa(s)) \\ar[r] &", "S &", "\\Spec(\\mathcal{O}_{S, s}) \\ar[r] &", "S", "}", "$$", "In both cases the top horizontal arrow is a homeomorphism", "onto its image." ], "refs": [], "proofs": [ { "contents": [ "Choose an open affine $U \\subset S$ that contains $s$.", "The bottom horizontal morphisms factor through $U$, see", "Lemma \\ref{lemma-morphism-from-spec-local-ring} for example.", "Thus we may assume that $S$ is affine. If $X$ is also affine, then", "the result follows from", "Algebra, Remark \\ref{algebra-remark-fundamental-diagram}.", "In the general case the result follows by covering $X$ by open affines." ], "refs": [ "schemes-lemma-morphism-from-spec-local-ring", "algebra-remark-fundamental-diagram" ], "ref_ids": [ 7683, 1558 ] } ], "ref_ids": [] }, { "id": 7697, "type": "theorem", "label": "schemes-lemma-quasi-compact-affine", "categories": [ "schemes" ], "title": "schemes-lemma-quasi-compact-affine", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item $f : X \\to S$ is quasi-compact,", "\\item the inverse image of every affine open is quasi-compact, and", "\\item there exists some affine open covering $S = \\bigcup_{i \\in I} U_i$", "such that $f^{-1}(U_i)$ is quasi-compact for all $i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose we are given a covering $S = \\bigcup_{i \\in I} U_i$ as in (3).", "First, let $U \\subset S$ be any affine open. For any $u \\in U$", "we can find an index $i(u) \\in I$ such that $u \\in U_{i(u)}$.", "As standard opens form a basis for the topology on $U_{i(u)}$ we can find", "$W_u \\subset U \\cap U_{i(u)}$ which is standard open in $U_{i(u)}$.", "By compactness we can find finitely many points $u_1, \\ldots, u_n \\in U$", "such that $U = \\bigcup_{j = 1}^n W_{u_j}$. For each $j$ write", "$f^{-1}U_{i(u_j)} = \\bigcup_{k \\in K_j} V_{jk}$ as a finite", "union of affine opens. Since $W_{u_j} \\subset U_{i(u_j)}$ is a standard", "open we see that $f^{-1}(W_{u_j}) \\cap V_{jk}$ is a standard", "open of $V_{jk}$, see Algebra, Lemma \\ref{algebra-lemma-spec-functorial}.", "Hence $f^{-1}(W_{u_j}) \\cap V_{jk}$ is affine, and so", "$f^{-1}(W_{u_j})$ is a finite union of affines. This proves that the", "inverse image of any affine open is a finite union of affine opens.", "\\medskip\\noindent", "Next, assume that the inverse image of every affine open is a finite", "union of affine opens.", "Let $K \\subset S$ be any quasi-compact open. Since $S$ has a basis", "of the topology consisting of affine opens we see that $K$ is a finite", "union of affine opens. Hence the inverse image of $K$ is a finite", "union of affine opens. Hence $f$ is quasi-compact.", "\\medskip\\noindent", "Finally, assume that $f$ is quasi-compact. In this case the argument", "of the previous paragraph shows that the inverse image of any affine", "is a finite union of affine opens." ], "refs": [ "algebra-lemma-spec-functorial" ], "ref_ids": [ 390 ] } ], "ref_ids": [] }, { "id": 7698, "type": "theorem", "label": "schemes-lemma-quasi-compact-preserved-base-change", "categories": [ "schemes" ], "title": "schemes-lemma-quasi-compact-preserved-base-change", "contents": [ "Being quasi-compact is a property of morphisms of schemes", "over a base which is preserved under arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7699, "type": "theorem", "label": "schemes-lemma-composition-quasi-compact", "categories": [ "schemes" ], "title": "schemes-lemma-composition-quasi-compact", "contents": [ "The composition of quasi-compact morphisms is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "This follows from the definitions and", "Topology, Lemma \\ref{topology-lemma-composition-quasi-compact}." ], "refs": [ "topology-lemma-composition-quasi-compact" ], "ref_ids": [ 8228 ] } ], "ref_ids": [] }, { "id": 7700, "type": "theorem", "label": "schemes-lemma-closed-immersion-quasi-compact", "categories": [ "schemes" ], "title": "schemes-lemma-closed-immersion-quasi-compact", "contents": [ "A closed immersion is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Follows from the definitions and", "Topology, Lemma \\ref{topology-lemma-closed-in-quasi-compact}." ], "refs": [ "topology-lemma-closed-in-quasi-compact" ], "ref_ids": [ 8229 ] } ], "ref_ids": [] }, { "id": 7701, "type": "theorem", "label": "schemes-lemma-image-quasi-compact-closed", "categories": [ "schemes" ], "title": "schemes-lemma-image-quasi-compact-closed", "contents": [ "Let $f : X \\to S$ be a quasi-compact morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item $f(X) \\subset S$ is closed, and", "\\item $f(X) \\subset S$ is stable under specialization.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have (1) $\\Rightarrow$ (2) by", "Topology, Lemma \\ref{topology-lemma-open-closed-specialization}.", "Assume (2). Let $U \\subset S$ be an affine open. It suffices to prove", "that $f(X) \\cap U$ is closed. Since $U \\cap f(X)$ is stable under", "specializations in $U$, we have reduced to the case where $S$ is affine.", "Because $f$ is quasi-compact we deduce that $X = f^{-1}(S)$ is", "quasi-compact as $S$ is affine. Thus we may write", "$X = \\bigcup_{i = 1}^n U_i$ with $U_i \\subset X$ open affine.", "Say $S = \\Spec(R)$ and", "$U_i = \\Spec(A_i)$ for some $R$-algebra $A_i$.", "Then $f(X) = \\Im(\\Spec(A_1 \\times \\ldots \\times A_n)", "\\to \\Spec(R))$. Thus the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-image-stable-specialization-closed}." ], "refs": [ "topology-lemma-open-closed-specialization", "algebra-lemma-image-stable-specialization-closed" ], "ref_ids": [ 8283, 551 ] } ], "ref_ids": [] }, { "id": 7702, "type": "theorem", "label": "schemes-lemma-quasi-compact-closed", "categories": [ "schemes" ], "title": "schemes-lemma-quasi-compact-closed", "contents": [ "Let $f : X \\to S$ be a quasi-compact morphism of schemes.", "Then $f$ is closed if and only if specializations lift", "along $f$, see", "Topology, Definition \\ref{topology-definition-lift-specializations}." ], "refs": [ "topology-definition-lift-specializations" ], "proofs": [ { "contents": [ "According to", "Topology, Lemma \\ref{topology-lemma-closed-open-map-specialization}", "if $f$ is closed then specializations lift along $f$.", "Conversely, suppose that specializations lift along $f$.", "Let $Z \\subset X$ be a closed subset. We may think of $Z$", "as a scheme with the reduced induced scheme structure, see", "Definition \\ref{definition-reduced-induced-scheme}.", "Since $Z \\subset X$ is closed the restriction", "of $f$ to $Z$ is still quasi-compact. Moreover specializations lift", "along $Z \\to S$ as well,", "see Topology, Lemma \\ref{topology-lemma-lift-specialization-composition}.", "Hence it suffices to prove $f(X)$ is closed if specializations lift along $f$.", "In particular $f(X)$ is stable under specializations, see", "Topology, Lemma \\ref{topology-lemma-lift-specializations-images}.", "Thus $f(X)$ is closed by", "Lemma \\ref{lemma-image-quasi-compact-closed}." ], "refs": [ "topology-lemma-closed-open-map-specialization", "schemes-definition-reduced-induced-scheme", "topology-lemma-lift-specialization-composition", "topology-lemma-lift-specializations-images", "schemes-lemma-image-quasi-compact-closed" ], "ref_ids": [ 8287, 7745, 8285, 8286, 7701 ] } ], "ref_ids": [ 8366 ] }, { "id": 7703, "type": "theorem", "label": "schemes-lemma-specializations-lift", "categories": [ "schemes" ], "title": "schemes-lemma-specializations-lift", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item If $f$ is universally closed then specializations lift", "along any base change of $f$, see", "Topology, Definition \\ref{topology-definition-lift-specializations}.", "\\item If $f$ is quasi-compact and specializations lift", "along any base change of $f$, then $f$ is universally closed.", "\\end{enumerate}" ], "refs": [ "topology-definition-lift-specializations" ], "proofs": [ { "contents": [ "Part (1) is a direct consequence of", "Topology, Lemma \\ref{topology-lemma-closed-open-map-specialization}.", "Part (2) follows from", "Lemmas \\ref{lemma-quasi-compact-closed} and", "\\ref{lemma-quasi-compact-preserved-base-change}." ], "refs": [ "topology-lemma-closed-open-map-specialization", "schemes-lemma-quasi-compact-closed", "schemes-lemma-quasi-compact-preserved-base-change" ], "ref_ids": [ 8287, 7702, 7698 ] } ], "ref_ids": [ 8366 ] }, { "id": 7704, "type": "theorem", "label": "schemes-lemma-points-specialize", "categories": [ "schemes" ], "title": "schemes-lemma-points-specialize", "contents": [ "Let $S$ be a scheme. Let $s' \\leadsto s$ be a specialization of points of $S$.", "Then", "\\begin{enumerate}", "\\item there exists a valuation ring $A$ and a morphism", "$f : \\Spec(A) \\to S$ such that the generic point $\\eta$ of", "$\\Spec(A)$ maps to $s'$ and the special point maps to $s$, and", "\\item given a field extension $\\kappa(s') \\subset K$", "we may arrange it so that the extension", "$\\kappa(s') \\subset \\kappa(\\eta)$ induced by $f$", "is isomorphic to the given extension.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $s' \\leadsto s$ be a specialization in $S$, and let", "$\\kappa(s') \\subset K$ be an extension of fields. By", "Lemma \\ref{lemma-specialize-points}", "and the discussion following", "Lemma \\ref{lemma-characterize-points}", "this leads to ring maps $\\mathcal{O}_{S, s} \\to \\kappa(s') \\to K$.", "Let $A \\subset K$ be any valuation ring whose field of fractions is", "$K$ and which dominates the image of $\\mathcal{O}_{S, s} \\to K$, see", "Algebra, Lemma \\ref{algebra-lemma-dominate}.", "The ring map $\\mathcal{O}_{S, s} \\to A$ induces the morphism", "$f : \\Spec(A) \\to S$, see", "Lemma \\ref{lemma-morphism-from-spec-local-ring}.", "This morphism has all the desired properties by construction." ], "refs": [ "schemes-lemma-specialize-points", "schemes-lemma-characterize-points", "algebra-lemma-dominate", "schemes-lemma-morphism-from-spec-local-ring" ], "ref_ids": [ 7684, 7685, 608, 7683 ] } ], "ref_ids": [] }, { "id": 7705, "type": "theorem", "label": "schemes-lemma-lift-specializations-valuative", "categories": [ "schemes" ], "title": "schemes-lemma-lift-specializations-valuative", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item Specializations lift along any base change of $f$", "\\item The morphism $f$ satisfies the existence part of the", "valuative criterion.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) holds. Let a solid diagram as in", "Definition \\ref{definition-valuative-criterion} be given.", "In order to find the dotted arrow we may replace $X \\to S$", "by $X_{\\Spec(A)} \\to \\Spec(A)$ since after all", "the assumption is stable under base change.", "Thus we may assume $S = \\Spec(A)$.", "Let $x' \\in X$ be the image of $\\Spec(K) \\to X$, so", "that we have $\\kappa(x') \\subset K$, see", "Lemma \\ref{lemma-characterize-points}.", "By assumption there exists a specialization $x' \\leadsto x$", "in $X$ such that $x$ maps to the closed point of $S = \\Spec(A)$.", "We get a local ring map $A \\to \\mathcal{O}_{X, x}$ and a ring", "map $\\mathcal{O}_{X, x} \\to \\kappa(x')$, see", "Lemma \\ref{lemma-specialize-points} and the discussion following", "Lemma \\ref{lemma-characterize-points}. The composition", "$A \\to \\mathcal{O}_{X, x} \\to \\kappa(x') \\to K$ is the", "given injection $A \\to K$. Since $A \\to \\mathcal{O}_{X, x}$", "is local, the image of $\\mathcal{O}_{X, x} \\to K$", "dominates $A$ and hence is equal to $A$, by", "Algebra, Definition \\ref{algebra-definition-valuation-ring}.", "Thus we obtain a ring map $\\mathcal{O}_{X, x} \\to A$ and", "hence a morphism $\\Spec(A) \\to X$", "(see Lemma \\ref{lemma-morphism-from-spec-local-ring} and", "discussion following it). This proves (2).", "\\medskip\\noindent", "Conversely, assume (2) holds. It is immediate that", "the existence part of the valuative criterion holds for", "any base change $X_{S'} \\to S'$ of $f$ by considering", "the following commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X_{S'} \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] \\ar@{-->}[rru] & S' \\ar[r] & S", "}", "$$", "Namely, the more horizontal dotted arrow will lead to the", "other one by definition of the fibre product. OK, so it clearly", "suffices to show that specializations lift along $f$.", "Let $s' \\leadsto s$ be a specialization in $S$, and let", "$x' \\in X$ be a point lying over $s'$. Apply", "Lemma \\ref{lemma-points-specialize}", "to $s' \\leadsto s$ and the extension of fields", "$\\kappa(s') \\subset \\kappa(x') = K$.", "We get a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[rr] \\ar[d] & & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[rru] &", "\\Spec(\\mathcal{O}_{S, s}) \\ar[r] & S", "}", "$$", "and by condition (2) we get the dotted arrow.", "The image $x$ of the closed point of $\\Spec(A)$", "in $X$ will be a solution to our problem, i.e.,", "$x$ is a specialization of $x'$ and maps to $s$." ], "refs": [ "schemes-definition-valuative-criterion", "schemes-lemma-characterize-points", "schemes-lemma-specialize-points", "schemes-lemma-characterize-points", "algebra-definition-valuation-ring", "schemes-lemma-morphism-from-spec-local-ring", "schemes-lemma-points-specialize" ], "ref_ids": [ 7755, 7685, 7684, 7685, 1467, 7683, 7704 ] } ], "ref_ids": [] }, { "id": 7706, "type": "theorem", "label": "schemes-lemma-diagonal-affines-closed", "categories": [ "schemes" ], "title": "schemes-lemma-diagonal-affines-closed", "contents": [ "The diagonal morphism of a morphism between affines is closed." ], "refs": [], "proofs": [ { "contents": [ "The diagonal morphism associated to the morphism", "$\\Spec(S) \\to \\Spec(R)$ is the morphism on spectra", "corresponding to the ring", "map $S \\otimes_R S \\to S$, $a \\otimes b \\mapsto ab$.", "This map is clearly surjective, so $S \\cong S \\otimes_R S/J$", "for some ideal $J \\subset S \\otimes_R S$. Hence", "$\\Delta$ is a closed immersion according to", "Example \\ref{example-closed-immersion-affines}" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7707, "type": "theorem", "label": "schemes-lemma-diagonal-immersion", "categories": [ "schemes" ], "title": "schemes-lemma-diagonal-immersion", "contents": [ "\\begin{slogan}", "The diagonal morphism for relative schemes is an immersion.", "\\end{slogan}", "Let $X$ be a scheme over $S$.", "The diagonal morphism $\\Delta_{X/S}$ is an immersion." ], "refs": [], "proofs": [ { "contents": [ "Recall that if $V \\subset X$ is affine open and maps into", "$U \\subset S$ affine open, then $V \\times_U V$ is affine open", "in $X \\times_S X$, see Lemmas \\ref{lemma-fibre-product-affines}", "and \\ref{lemma-open-fibre-product}.", "Consider the open subscheme $W$ of $X \\times_S X$ which", "is the union of these affine opens $V \\times_U V$.", "By Lemma \\ref{lemma-closed-local-target} it is enough", "to show that each morphism", "$\\Delta_{X/S}^{-1}(V \\times_U V) \\to V \\times_U V$ is", "a closed immersion. Since $V = \\Delta_{X/S}^{-1}(V \\times_U V)$", "we are just checking that $\\Delta_{V/U}$ is a closed", "immersion, which is Lemma \\ref{lemma-diagonal-affines-closed}." ], "refs": [ "schemes-lemma-fibre-product-affines", "schemes-lemma-open-fibre-product", "schemes-lemma-closed-local-target", "schemes-lemma-diagonal-affines-closed" ], "ref_ids": [ 7690, 7691, 7646, 7706 ] } ], "ref_ids": [] }, { "id": 7708, "type": "theorem", "label": "schemes-lemma-where-are-they-equal", "categories": [ "schemes" ], "title": "schemes-lemma-where-are-they-equal", "contents": [ "Let $X$, $Y$ be schemes over $S$.", "Let $a, b : X \\to Y$ be morphisms of schemes over $S$.", "There exists a largest locally closed subscheme", "$Z \\subset X$ such that $a|_Z = b|_Z$. In fact $Z$ is", "the equalizer of $(a, b)$. Moreover, if $Y$ is separated", "over $S$, then $Z$ is a closed subscheme." ], "refs": [], "proofs": [ { "contents": [ "The equalizer of $(a, b)$ is for categorical reasons", "the fibre product $Z$ in the following diagram", "$$", "\\xymatrix{", "Z = Y \\times_{(Y \\times_S Y)} X \\ar[r] \\ar[d] &", " X \\ar[d]^{(a , b)} \\\\", "Y \\ar[r]^-{\\Delta_{Y/S}} & Y \\times_S Y", "}", "$$", "Thus the lemma follows from Lemmas", "\\ref{lemma-base-change-immersion}, \\ref{lemma-diagonal-immersion} and", "Definition \\ref{definition-separated}." ], "refs": [ "schemes-lemma-base-change-immersion", "schemes-lemma-diagonal-immersion", "schemes-definition-separated" ], "ref_ids": [ 7695, 7707, 7756 ] } ], "ref_ids": [] }, { "id": 7709, "type": "theorem", "label": "schemes-lemma-characterize-quasi-separated", "categories": [ "schemes" ], "title": "schemes-lemma-characterize-quasi-separated", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is quasi-separated.", "\\item For every pair of affine opens $U, V \\subset X$", "which map into a common affine open of $S$ the intersection", "$U \\cap V$ is a finite union of affine opens of $X$.", "\\item There exists an affine open covering $S = \\bigcup_{i \\in I} U_i$", "and for each $i$ an affine open covering $f^{-1}U_i = \\bigcup_{j \\in I_i} V_j$", "such that for each $i$ and each pair $j, j' \\in I_i$ the", "intersection $V_j \\cap V_{j'}$ is a finite union of affine", "opens of $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us prove that (3) implies (1).", "By Lemma \\ref{lemma-affine-covering-fibre-product}", "the covering $X \\times_S X = \\bigcup_i \\bigcup_{j, j'} V_j \\times_{U_i} V_{j'}$", "is an affine open covering of $X \\times_S X$.", "Moreover, $\\Delta_{X/S}^{-1}(V_j \\times_{U_i} V_{j'}) = V_j \\cap V_{j'}$.", "Hence the implication follows from Lemma \\ref{lemma-quasi-compact-affine}.", "\\medskip\\noindent", "The implication (1) $\\Rightarrow$ (2) follows from the fact", "that under the hypotheses of (2) the fibre product", "$U \\times_S V$ is an affine open of $X \\times_S X$.", "The implication (2) $\\Rightarrow$ (3) is trivial." ], "refs": [ "schemes-lemma-affine-covering-fibre-product", "schemes-lemma-quasi-compact-affine" ], "ref_ids": [ 7692, 7697 ] } ], "ref_ids": [] }, { "id": 7710, "type": "theorem", "label": "schemes-lemma-characterize-separated", "categories": [ "schemes" ], "title": "schemes-lemma-characterize-separated", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item If $f$ is separated then for every pair of affine", "opens $(U, V)$ of $X$ which map into a", "common affine open of $S$ we have", "\\begin{enumerate}", "\\item the intersection $U \\cap V$ is affine.", "\\item the ring map", "$\\mathcal{O}_X(U) \\otimes_{\\mathbf{Z}} \\mathcal{O}_X(V)", "\\to \\mathcal{O}_X(U \\cap V)$", "is surjective.", "\\end{enumerate}", "\\item If any pair of points $x_1, x_2 \\in X$ lying over a common", "point $s \\in S$ are contained in affine opens $x_1 \\in U$,", "$x_2 \\in V$ which map into a common affine open of $S$ such", "that (a), (b) hold, then $f$ is separated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f$ separated. Suppose $(U, V)$ is a pair as in (1).", "Let $W = \\Spec(R)$ be an affine open of $S$ containing", "both $f(U)$ and $f(V)$. Write $U = \\Spec(A)$ and", "$V = \\Spec(B)$ for $R$-algebras $A$ and $B$.", "By Lemma \\ref{lemma-open-fibre-product} we see that", "$U \\times_S V = U \\times_W V = \\Spec(A \\otimes_R B)$", "is an affine open of $X \\times_S X$. Hence, by", "Lemma \\ref{lemma-closed-subspace-scheme} we see that", "$\\Delta^{-1}(U \\times_S V) \\to U \\times_S V$", "can be identified with $\\Spec(A \\otimes_R B/J)$", "for some ideal $J \\subset A \\otimes_R B$.", "Thus $U \\cap V = \\Delta^{-1}(U \\times_S V)$ is affine.", "Assertion (1)(b) holds because", "$A \\otimes_{\\mathbf{Z}} B \\to (A \\otimes_R B)/J$ is surjective.", "\\medskip\\noindent", "Assume the hypothesis formulated in (2) holds.", "Clearly the collection of affine opens $U \\times_S V$", "for pairs $(U, V)$ as in (2) form an affine open covering", "of $X \\times_S X$ (see e.g.\\ Lemma \\ref{lemma-affine-covering-fibre-product}).", "Hence it suffices to show that each morphism", "$U \\cap V = \\Delta_{X/S}^{-1}(U \\times_S V) \\to U \\times_S V$", "is a closed immersion, see Lemma \\ref{lemma-closed-local-target}.", "By assumption (a) we have $U \\cap V = \\Spec(C)$ for some ring $C$.", "After choosing an affine open $W = \\Spec(R)$ of $S$", "into which both $U$ and $V$ map and writing $U = \\Spec(A)$,", "$V = \\Spec(B)$ we see that the assumption (b) means", "that the composition", "$$", "A \\otimes_{\\mathbf{Z}} B \\to A \\otimes_R B \\to C", "$$", "is surjective. Hence $A \\otimes_R B \\to C$ is surjective and", "we conclude that $\\Spec(C) \\to \\Spec(A \\otimes_R B)$", "is a closed immersion." ], "refs": [ "schemes-lemma-open-fibre-product", "schemes-lemma-closed-subspace-scheme", "schemes-lemma-affine-covering-fibre-product", "schemes-lemma-closed-local-target" ], "ref_ids": [ 7691, 7670, 7692, 7646 ] } ], "ref_ids": [] }, { "id": 7711, "type": "theorem", "label": "schemes-lemma-fibre-product-after-map", "categories": [ "schemes" ], "title": "schemes-lemma-fibre-product-after-map", "contents": [ "Let $f : X \\to T$ and $g : Y \\to T$ be morphisms of schemes", "with the same target. Let $h : T \\to S$ be a morphism of schemes.", "Then the induced morphism $i : X \\times_T Y \\to X \\times_S Y$ is", "an immersion. If $T \\to S$ is separated, then $i$ is a closed", "immersion. If $T \\to S$ is quasi-separated, then $i$ is a", "quasi-compact morphism." ], "refs": [], "proofs": [ { "contents": [ "By general category theory the following diagram", "$$", "\\xymatrix{", "X \\times_T Y \\ar[r] \\ar[d] & X \\times_S Y \\ar[d] \\\\", "T \\ar[r]^{\\Delta_{T/S}} \\ar[r] & T \\times_S T", "}", "$$", "is a fibre product diagram. The lemma follows", "from Lemmas \\ref{lemma-diagonal-immersion},", "\\ref{lemma-fibre-product-immersion} and", "\\ref{lemma-quasi-compact-preserved-base-change}." ], "refs": [ "schemes-lemma-diagonal-immersion", "schemes-lemma-fibre-product-immersion", "schemes-lemma-quasi-compact-preserved-base-change" ], "ref_ids": [ 7707, 7694, 7698 ] } ], "ref_ids": [] }, { "id": 7712, "type": "theorem", "label": "schemes-lemma-semi-diagonal", "categories": [ "schemes" ], "title": "schemes-lemma-semi-diagonal", "contents": [ "Let $g : X \\to Y$ be a morphism of schemes over $S$.", "The morphism $i : X \\to X \\times_S Y$ is an immersion.", "If $Y$ is separated over $S$ it is a closed immersion.", "If $Y$ is quasi-separated over $S$ it is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-fibre-product-after-map}", "applied to the morphism $X = X \\times_Y Y \\to X \\times_S Y$." ], "refs": [ "schemes-lemma-fibre-product-after-map" ], "ref_ids": [ 7711 ] } ], "ref_ids": [] }, { "id": 7713, "type": "theorem", "label": "schemes-lemma-section-immersion", "categories": [ "schemes" ], "title": "schemes-lemma-section-immersion", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $s : S \\to X$ be a section of $f$ (in a formula $f \\circ s = \\text{id}_S$).", "Then $s$ is an immersion.", "If $f$ is separated then $s$ is a closed immersion.", "If $f$ is quasi-separated, then $s$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-semi-diagonal} applied to", "$g =s$ so the morphism $i = s : S \\to S \\times_S X$." ], "refs": [ "schemes-lemma-semi-diagonal" ], "ref_ids": [ 7712 ] } ], "ref_ids": [] }, { "id": 7714, "type": "theorem", "label": "schemes-lemma-separated-permanence", "categories": [ "schemes" ], "title": "schemes-lemma-separated-permanence", "contents": [ "Permanence properties.", "\\begin{enumerate}", "\\item A composition of separated morphisms is separated.", "\\item A composition of quasi-separated morphisms is quasi-separated.", "\\item The base change of a separated morphism is separated.", "\\item The base change of a quasi-separated morphism is quasi-separated.", "\\item A (fibre) product of separated morphisms is separated.", "\\item A (fibre) product of quasi-separated morphisms is quasi-separated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X \\to Y \\to Z$ be morphisms. Assume that $X \\to Y$ and", "$Y \\to Z$ are separated. The composition", "$$", "X \\to X \\times_Y X \\to X \\times_Z X", "$$", "is closed because the first one is by assumption and the second", "one by Lemma \\ref{lemma-fibre-product-after-map}. The same argument", "works for ``quasi-separated'' (with the same references).", "\\medskip\\noindent", "Let $f : X \\to Y$ be a morphism of schemes over a base $S$.", "Let $S' \\to S$ be a morphism of schemes. Let $f' : X_{S'} \\to Y_{S'}$", "be the base change of $f$. Then the diagonal morphism", "of $f'$ is a morphism", "$$", "\\Delta_{f'} :", "X_{S'} = S' \\times_S X", "\\longrightarrow", "X_{S'} \\times_{Y_{S'}} X_{S'} = S' \\times _S (X \\times_Y X)", "$$", "which is easily seen to be the base change of $\\Delta_f$.", "Thus (3) and (4) follow from the fact that", "closed immersions and quasi-compact morphisms are preserved", "under arbitrary base change (Lemmas", "\\ref{lemma-fibre-product-immersion} and", "\\ref{lemma-quasi-compact-preserved-base-change}).", "\\medskip\\noindent", "If $f : X \\to Y$ and $g : U \\to V$ are morphisms of schemes over a base $S$,", "then $f \\times g$ is the composition of $X \\times_S U \\to X \\times_S V$", "(a base change of $g$) and $X \\times_S V \\to Y \\times_S V$ (a base change", "of $f$). Hence (5) and (6) follow from (1) -- (4)." ], "refs": [ "schemes-lemma-fibre-product-after-map", "schemes-lemma-fibre-product-immersion", "schemes-lemma-quasi-compact-preserved-base-change" ], "ref_ids": [ 7711, 7694, 7698 ] } ], "ref_ids": [] }, { "id": 7715, "type": "theorem", "label": "schemes-lemma-compose-after-separated", "categories": [ "schemes" ], "title": "schemes-lemma-compose-after-separated", "contents": [ "\\begin{slogan}", "Separated and quasi-separated morphisms satisfy cancellation.", "\\end{slogan}", "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes.", "If $g \\circ f$ is separated then so is $f$.", "If $g \\circ f$ is quasi-separated then so is $f$." ], "refs": [], "proofs": [ { "contents": [ "Assume that $g \\circ f$ is separated.", "Consider the factorization $X \\to X \\times_Y X \\to X \\times_Z X$", "of the diagonal morphism of $g \\circ f$.", "By Lemma \\ref{lemma-fibre-product-after-map}", "the last morphism is an immersion. By assumption the image", "of $X$ in $X \\times_Z X$ is closed. Hence it is also closed", "in $X \\times_Y X$. Thus we see that $X \\to X \\times_Y X$", "is a closed immersion by Lemma \\ref{lemma-immersion-when-closed}.", "\\medskip\\noindent", "Assume that $g \\circ f$ is quasi-separated.", "Let $V \\subset Y$ be an affine open which maps into an affine", "open of $Z$. Let $U_1, U_2 \\subset X$ be affine opens which", "map into $V$. Then $U_1 \\cap U_2$ is a finite union of affine", "opens because $U_1, U_2$ map into a common affine open", "of $Z$. Since we may cover $Y$ by affine opens like $V$ we", "deduce the lemma from Lemma \\ref{lemma-characterize-quasi-separated}." ], "refs": [ "schemes-lemma-fibre-product-after-map", "schemes-lemma-immersion-when-closed", "schemes-lemma-characterize-quasi-separated" ], "ref_ids": [ 7711, 7671, 7709 ] } ], "ref_ids": [] }, { "id": 7716, "type": "theorem", "label": "schemes-lemma-quasi-compact-permanence", "categories": [ "schemes" ], "title": "schemes-lemma-quasi-compact-permanence", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes.", "If $g \\circ f$ is quasi-compact and $g$ is quasi-separated", "then $f$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "This is true because $f$ equals the composition", "$(1, f) : X \\to X \\times_Z Y \\to Y$. The first map", "is quasi-compact by Lemma \\ref{lemma-section-immersion}", "because it is a section of the quasi-separated morphism $X \\times_Z Y \\to X$", "(a base change of $g$, see Lemma \\ref{lemma-separated-permanence}).", "The second map is quasi-compact as it", "is the base change of $g \\circ f$, see", "Lemma \\ref{lemma-quasi-compact-preserved-base-change}.", "And compositions of quasi-compact", "morphisms are quasi-compact, see Lemma \\ref{lemma-composition-quasi-compact}." ], "refs": [ "schemes-lemma-separated-permanence", "schemes-lemma-quasi-compact-preserved-base-change", "schemes-lemma-composition-quasi-compact" ], "ref_ids": [ 7714, 7698, 7699 ] } ], "ref_ids": [] }, { "id": 7717, "type": "theorem", "label": "schemes-lemma-affine-separated", "categories": [ "schemes" ], "title": "schemes-lemma-affine-separated", "contents": [ "An affine scheme is separated. A morphism from an affine scheme", "to another scheme is separated." ], "refs": [], "proofs": [ { "contents": [ "Let $U = \\Spec(A)$ be an affine scheme. Then $U \\to \\Spec(\\mathbf{Z})$", "has closed diagonal by Lemma \\ref{lemma-diagonal-affines-closed}.", "Thus $U$ is separated by Definition \\ref{definition-separated}.", "If $U \\to X$ is a morphism of schemes, then we can apply", "Lemma \\ref{lemma-compose-after-separated}", "to the morphisms $U \\to X \\to \\Spec(\\mathbf{Z})$ to conclude", "that $U \\to X$ is separated." ], "refs": [ "schemes-lemma-diagonal-affines-closed", "schemes-definition-separated", "schemes-lemma-compose-after-separated" ], "ref_ids": [ 7706, 7756, 7715 ] } ], "ref_ids": [] }, { "id": 7718, "type": "theorem", "label": "schemes-lemma-curiosity", "categories": [ "schemes" ], "title": "schemes-lemma-curiosity", "contents": [ "Let $f : X \\to S$ be a morphism.", "Assume $f$ is separated and $S$ is a separated scheme.", "Suppose $U \\subset X$ and $V \\subset X$ are affine.", "Then $U \\cap V$ is affine (and a closed subscheme of $U \\times V$)." ], "refs": [], "proofs": [ { "contents": [ "In this case $X$ is separated by Lemma \\ref{lemma-separated-permanence}.", "Hence $U \\cap V$ is affine by", "applying Lemma \\ref{lemma-characterize-separated} to the", "morphism $X \\to \\Spec(\\mathbf{Z})$." ], "refs": [ "schemes-lemma-separated-permanence", "schemes-lemma-characterize-separated" ], "ref_ids": [ 7714, 7710 ] } ], "ref_ids": [] }, { "id": 7719, "type": "theorem", "label": "schemes-lemma-separated-implies-valuative", "categories": [ "schemes" ], "title": "schemes-lemma-separated-implies-valuative", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "If $f$ is separated, then $f$ satisfies the uniqueness", "part of the valuative criterion." ], "refs": [], "proofs": [ { "contents": [ "Let a diagram as in Definition \\ref{definition-valuative-criterion}", "be given. Suppose there are two morphisms", "$a, b : \\Spec(A) \\to X$ fitting into the diagram.", "Let $Z \\subset \\Spec(A)$ be the equalizer of $a$ and $b$.", "By Lemma \\ref{lemma-where-are-they-equal} this is a closed", "subscheme of $\\Spec(A)$. By assumption it contains", "the generic point of $\\Spec(A)$. Since $A$ is a domain", "this implies $Z = \\Spec(A)$. Hence $a = b$ as desired." ], "refs": [ "schemes-definition-valuative-criterion", "schemes-lemma-where-are-they-equal" ], "ref_ids": [ 7755, 7708 ] } ], "ref_ids": [] }, { "id": 7720, "type": "theorem", "label": "schemes-lemma-valuative-criterion-separatedness", "categories": [ "schemes" ], "title": "schemes-lemma-valuative-criterion-separatedness", "contents": [ "\\begin{reference}", "\\cite[II Proposition 7.2.3]{EGA}", "\\end{reference}", "Let $f : X \\to S$ be a morphism.", "Assume", "\\begin{enumerate}", "\\item the morphism $f$ is quasi-separated, and", "\\item the morphism $f$ satisfies the uniqueness", "part of the valuative criterion.", "\\end{enumerate}", "Then $f$ is separated." ], "refs": [], "proofs": [ { "contents": [ "By assumption (1),", "Proposition \\ref{proposition-characterize-universally-closed}, and", "Lemmas \\ref{lemma-diagonal-immersion} and \\ref{lemma-immersion-when-closed}", "we see that it suffices to prove the morphism", "$\\Delta_{X/S} : X \\to X \\times_S X$ satisfies the existence", "part of the valuative criterion.", "Let a solid commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & X \\times_S X", "}", "$$", "be given. The lower right arrow corresponds to a", "pair of morphisms $a, b : \\Spec(A) \\to X$ over $S$.", "By (2) we see that $a = b$. Hence using $a$ as the dotted", "arrow works." ], "refs": [ "schemes-proposition-characterize-universally-closed", "schemes-lemma-diagonal-immersion", "schemes-lemma-immersion-when-closed" ], "ref_ids": [ 7733, 7707, 7671 ] } ], "ref_ids": [] }, { "id": 7721, "type": "theorem", "label": "schemes-lemma-monomorphism", "categories": [ "schemes" ], "title": "schemes-lemma-monomorphism", "contents": [ "\\begin{slogan}", "A scheme morphism is a monomorphism iff its diagonal is an isomorphism.", "\\end{slogan}", "Let $j : X \\to Y$ be a morphism of schemes.", "Then $j$ is a monomorphism if and only if the", "diagonal morphism $\\Delta_{X/Y} : X \\to X \\times_Y X$ is", "an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is true in any category with fibre products." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7722, "type": "theorem", "label": "schemes-lemma-monomorphism-separated", "categories": [ "schemes" ], "title": "schemes-lemma-monomorphism-separated", "contents": [ "A monomorphism of schemes is separated." ], "refs": [], "proofs": [ { "contents": [ "This is true because an isomorphism is a closed immersion,", "and Lemma \\ref{lemma-monomorphism} above." ], "refs": [ "schemes-lemma-monomorphism" ], "ref_ids": [ 7721 ] } ], "ref_ids": [] }, { "id": 7723, "type": "theorem", "label": "schemes-lemma-composition-monomorphism", "categories": [ "schemes" ], "title": "schemes-lemma-composition-monomorphism", "contents": [ "A composition of monomorphisms is a monomorphism." ], "refs": [], "proofs": [ { "contents": [ "True in any category." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7724, "type": "theorem", "label": "schemes-lemma-base-change-monomorphism", "categories": [ "schemes" ], "title": "schemes-lemma-base-change-monomorphism", "contents": [ "The base change of a monomorphism is a monomorphism." ], "refs": [], "proofs": [ { "contents": [ "True in any category with fibre products." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7725, "type": "theorem", "label": "schemes-lemma-injective-points", "categories": [ "schemes" ], "title": "schemes-lemma-injective-points", "contents": [ "Let $j : X \\to Y$ be a morphism of schemes.", "If $j$ is injective on points, then $j$ is separated." ], "refs": [], "proofs": [ { "contents": [ "Let $z$ be a point of $X \\times_Y X$. Then $x = \\text{pr}_1(z)$ and", "$\\text{pr}_2(z)$ are the same because $j$ maps these points to", "the same point $y$ of $Y$. Then we can choose an affine", "open neighbourhood $V \\subset Y$ of $y$ and an affine open", "neighbourhood $U \\subset X$ of $x$ with $j(U) \\subset V$.", "Then $z \\in U \\times_V U \\subset X \\times_Y X$.", "Hence $X \\times_Y X$ is the union of the affine", "opens $U \\times_V U$. Since $\\Delta_{X/Y}^{-1}(U \\times_V U) = U$", "and since $U \\to U \\times_V U$ is a closed immersion, we conclude", "that $\\Delta_{X/Y}$ is a closed immersion (see argument in the", "proof of Lemma \\ref{lemma-diagonal-immersion})." ], "refs": [ "schemes-lemma-diagonal-immersion" ], "ref_ids": [ 7707 ] } ], "ref_ids": [] }, { "id": 7726, "type": "theorem", "label": "schemes-lemma-injective-points-surjective-stalks", "categories": [ "schemes" ], "title": "schemes-lemma-injective-points-surjective-stalks", "contents": [ "Let $j : X \\to Y$ be a morphism of schemes.", "If", "\\begin{enumerate}", "\\item $j$ is injective on points, and", "\\item for any $x \\in X$ the ring map", "$j^\\sharp_x : \\mathcal{O}_{Y, j(x)} \\to \\mathcal{O}_{X, x}$", "is surjective,", "\\end{enumerate}", "then $j$ is a monomorphism." ], "refs": [], "proofs": [ { "contents": [ "Let $a, b : Z \\to X$ be two morphisms of schemes such that", "$j \\circ a = j \\circ b$.", "Then (1) implies $a = b$ as underlying", "maps of topological spaces.", "For any $z \\in Z$ we have", "$a^\\sharp_z \\circ j^\\sharp_{a(z)} = b^\\sharp_z \\circ j^\\sharp_{b(z)}$", "as maps $\\mathcal{O}_{Y, j(a(z))} \\to \\mathcal{O}_{Z, z}$.", "The surjectivity of the maps", "$j^\\sharp_x$ forces $a^\\sharp_z = b^\\sharp_z$, $\\forall z \\in Z$.", "This implies that $a^\\sharp = b^\\sharp$.", "Hence we conclude $a = b$ as morphisms of schemes", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7727, "type": "theorem", "label": "schemes-lemma-immersions-monomorphisms", "categories": [ "schemes" ], "title": "schemes-lemma-immersions-monomorphisms", "contents": [ "An immersion of schemes is a monomorphism.", "In particular, any immersion is separated." ], "refs": [], "proofs": [ { "contents": [ "We can see this by checking that the criterion of", "Lemma \\ref{lemma-injective-points-surjective-stalks} applies.", "More elegantly perhaps, we can use that", "Lemmas \\ref{lemma-restrict-map-to-opens} and", "\\ref{lemma-characterize-closed-subspace} imply that", "open and closed immersions are monomorphisms and hence", "any immersion (which is a composition of such)", "is a monomorphism." ], "refs": [ "schemes-lemma-injective-points-surjective-stalks", "schemes-lemma-restrict-map-to-opens", "schemes-lemma-characterize-closed-subspace" ], "ref_ids": [ 7726, 7645, 7648 ] } ], "ref_ids": [] }, { "id": 7728, "type": "theorem", "label": "schemes-lemma-subscheme-of-separated-scheme", "categories": [ "schemes" ], "title": "schemes-lemma-subscheme-of-separated-scheme", "contents": [ "Let $f : X \\to S$ be a separated morphism.", "Any locally closed subscheme $Z \\subset X$ is separated over $S$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-immersions-monomorphisms}", "and the fact that a composition of separated morphisms is separated", "(Lemma \\ref{lemma-separated-permanence})." ], "refs": [ "schemes-lemma-immersions-monomorphisms", "schemes-lemma-separated-permanence" ], "ref_ids": [ 7727, 7714 ] } ], "ref_ids": [] }, { "id": 7729, "type": "theorem", "label": "schemes-lemma-mono-towards-spec-field", "categories": [ "schemes" ], "title": "schemes-lemma-mono-towards-spec-field", "contents": [ "Let $k_1, \\ldots, k_n$ be fields.", "For any monomorphism of schemes", "$X \\to \\Spec(k_1 \\times \\ldots \\times k_n)$", "there exists a subset $I \\subset \\{1, \\ldots, n\\}$ such", "that $X \\cong \\Spec(\\prod_{i \\in I} k_i)$ as", "schemes over $\\Spec(k_1 \\times \\ldots \\times k_n)$.", "More generally, if $X = \\coprod_{i \\in I} \\Spec(k_i)$", "is a disjoint union of spectra of fields and $Y \\to X$ is a monomorphism,", "then there exists a subset $J \\subset I$ such that", "$Y = \\coprod_{i \\in J} \\Spec(k_i)$." ], "refs": [], "proofs": [ { "contents": [ "First reduce to the case $n = 1$ (or $\\# I = 1$)", "by taking the inverse images of the", "open and closed subschemes $\\Spec(k_i)$.", "In this case $X$ has only one point hence is affine.", "The corresponding algebra problem is this:", "If $k \\to R$ is an algebra map", "with $R \\otimes_k R \\cong R$, then $R \\cong k$ or $R = 0$.", "This holds for dimension reasons.", "See also", "Algebra, Lemma \\ref{algebra-lemma-epimorphism-over-field}" ], "refs": [ "algebra-lemma-epimorphism-over-field" ], "ref_ids": [ 954 ] } ], "ref_ids": [] }, { "id": 7730, "type": "theorem", "label": "schemes-lemma-push-forward-quasi-coherent", "categories": [ "schemes" ], "title": "schemes-lemma-push-forward-quasi-coherent", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "If $f$ is quasi-compact and quasi-separated then", "$f_*$ transforms quasi-coherent $\\mathcal{O}_X$-modules", "into quasi-coherent $\\mathcal{O}_S$-modules." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $S$ and hence we may assume that", "$S$ is affine. Because $X$ is quasi-compact we may write", "$X = \\bigcup_{i = 1}^n U_i$ with each $U_i$ open affine.", "Because $f$ is quasi-separated we may write", "$U_i \\cap U_j = \\bigcup_{k = 1}^{n_{ij}} U_{ijk}$ for some", "affine open $U_{ijk}$, see Lemma \\ref{lemma-characterize-quasi-separated}.", "Denote $f_i : U_i \\to S$ and $f_{ijk} : U_{ijk} \\to S$ the", "restrictions of $f$. For any open $V$ of $S$ and any sheaf", "$\\mathcal{F}$ on $X$ we have", "\\begin{eqnarray*}", "f_*\\mathcal{F}(V) & = & \\mathcal{F}(f^{-1}V) \\\\", "& = &", "\\Ker\\left(", "\\bigoplus\\nolimits_i \\mathcal{F}(f^{-1}V \\cap U_i)", "\\to", "\\bigoplus\\nolimits_{i, j, k} \\mathcal{F}(f^{-1}V \\cap U_{ijk})\\right) \\\\", "& = &", "\\Ker\\left(", "\\bigoplus\\nolimits_i f_{i, *}(\\mathcal{F}|_{U_i})(V)", "\\to", "\\bigoplus\\nolimits_{i, j, k} f_{ijk, *}(\\mathcal{F}|_{U_{ijk}})(V)\\right) \\\\", "& = &", "\\Ker\\left(", "\\bigoplus\\nolimits_i f_{i, *}(\\mathcal{F}|_{U_i})", "\\to", "\\bigoplus\\nolimits_{i, j, k} f_{ijk, *}(\\mathcal{F}|_{U_{ijk}})\\right)(V)", "\\end{eqnarray*}", "In other words there is an exact sequence of sheaves", "$$", "0 \\to f_*\\mathcal{F}", "\\to \\bigoplus f_{i, *}\\mathcal{F}_i", "\\to \\bigoplus f_{ijk, *}\\mathcal{F}_{ijk}", "$$", "where $\\mathcal{F}_i, \\mathcal{F}_{ijk}$ denotes the", "restriction of $\\mathcal{F}$ to the corresponding open.", "If $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module", "then $\\mathcal{F}_i$ is a quasi-coherent $\\mathcal{O}_{U_i}$-module", "and $\\mathcal{F}_{ijk}$ is a quasi-coherent $\\mathcal{O}_{U_{ijk}}$-module.", "Hence by Lemma \\ref{lemma-widetilde-pullback} we see that the second and", "third term of the exact sequence are quasi-coherent", "$\\mathcal{O}_S$-modules. Thus we conclude that", "$f_*\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_S$-module." ], "refs": [ "schemes-lemma-characterize-quasi-separated", "schemes-lemma-widetilde-pullback" ], "ref_ids": [ 7709, 7662 ] } ], "ref_ids": [] }, { "id": 7731, "type": "theorem", "label": "schemes-lemma-characterize-closed-immersions", "categories": [ "schemes" ], "title": "schemes-lemma-characterize-closed-immersions", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Suppose that", "\\begin{enumerate}", "\\item $f$ induces a homeomorphism of $X$ with a", "closed subset of $Y$, and", "\\item $f^\\sharp : \\mathcal{O}_Y \\to f_*\\mathcal{O}_X$", "is surjective.", "\\end{enumerate}", "Then $f$ is a closed immersion of schemes." ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and (2). By (1) the morphism $f$ is quasi-compact", "(see Topology, Lemma \\ref{topology-lemma-closed-in-quasi-compact}).", "Conditions (1) and (2) imply conditions (1) and (2) of", "Lemma \\ref{lemma-injective-points-surjective-stalks}.", "Hence $f : X \\to Y$ is a monomorphism. In particular, $f$", "is separated, see Lemma \\ref{lemma-monomorphism-separated}.", "Hence Lemma \\ref{lemma-push-forward-quasi-coherent}", "above applies and we conclude that $f_*\\mathcal{O}_X$ is a quasi-coherent", "$\\mathcal{O}_Y$-module. Therefore the kernel of", "$\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$ is quasi-coherent", "by Lemma \\ref{lemma-extension-quasi-coherent}. Since a quasi-coherent", "sheaf is locally generated by sections (see", "Modules, Definition \\ref{modules-definition-quasi-coherent})", "this implies that $f$ is a closed immersion, see", "Definition \\ref{definition-closed-immersion-locally-ringed-spaces}." ], "refs": [ "topology-lemma-closed-in-quasi-compact", "schemes-lemma-injective-points-surjective-stalks", "schemes-lemma-monomorphism-separated", "schemes-lemma-push-forward-quasi-coherent", "schemes-lemma-extension-quasi-coherent", "modules-definition-quasi-coherent", "schemes-definition-closed-immersion-locally-ringed-spaces" ], "ref_ids": [ 8229, 7726, 7722, 7730, 7667, 13337, 7737 ] } ], "ref_ids": [] }, { "id": 7732, "type": "theorem", "label": "schemes-lemma-composition-immersion", "categories": [ "schemes" ], "title": "schemes-lemma-composition-immersion", "contents": [ "A composition of immersions of schemes is an immersion,", "a composition of closed immersions of schemes is a closed immersion, and", "a composition of open immersions of schemes is an open immersion." ], "refs": [], "proofs": [ { "contents": [ "This is clear for the case of open immersions since an open subspace of", "an open subspace is also an open subspace.", "\\medskip\\noindent", "Suppose $a : Z \\to Y$ and $b : Y \\to X$ are closed immersions of schemes.", "We will verify that $c = b \\circ a$ is also a closed immersion.", "The assumption implies that $a$ and $b$ are homeomorphisms onto closed subsets,", "and hence also $c = b \\circ a$ is a homeomorphism onto a closed subset.", "Moreover, the map $\\mathcal{O}_X \\to c_*\\mathcal{O}_Z$ is surjective", "since it factors as the composition of the surjective maps", "$\\mathcal{O}_X \\to b_*\\mathcal{O}_Y$ and", "$b_*\\mathcal{O}_Y \\to b_*a_*\\mathcal{O}_Z$ (surjective as $b_*$ is exact,", "see Modules, Lemma \\ref{modules-lemma-i-star-exact}).", "Hence by Lemma \\ref{lemma-characterize-closed-immersions}", "above $c$ is a closed immersion.", "\\medskip\\noindent", "Finally, we come to the case of immersions.", "Suppose $a : Z \\to Y$ and $b : Y \\to X$ are immersions of schemes.", "This means there exist open subschemes", "$V \\subset Y$ and $U \\subset X$ such that", "$a(Z) \\subset V$, $b(Y) \\subset U$ and $a : Z \\to V$", "and $b : Y \\to U$ are closed immersions.", "Since the topology on $Y$ is induced from the topology on", "$U$ we can find an open $U' \\subset U$ such that", "$V = b^{-1}(U')$. Then we see that", "$Z \\to V = b^{-1}(U') \\to U'$ is a composition of", "closed immersions and hence a closed immersion.", "This proves that $Z \\to X$ is an immersion and we win." ], "refs": [ "modules-lemma-i-star-exact", "schemes-lemma-characterize-closed-immersions" ], "ref_ids": [ 13232, 7731 ] } ], "ref_ids": [] }, { "id": 7733, "type": "theorem", "label": "schemes-proposition-characterize-universally-closed", "categories": [ "schemes" ], "title": "schemes-proposition-characterize-universally-closed", "contents": [ "Let $f$ be a quasi-compact morphism of schemes.", "Then $f$ is universally closed if and only if $f$", "satisfies the existence part of the valuative criterion." ], "refs": [], "proofs": [ { "contents": [ "This is a formal consequence of", "Lemmas \\ref{lemma-specializations-lift} and", "\\ref{lemma-lift-specializations-valuative} above." ], "refs": [ "schemes-lemma-specializations-lift", "schemes-lemma-lift-specializations-valuative" ], "ref_ids": [ 7703, 7705 ] } ], "ref_ids": [] }, { "id": 7764, "type": "theorem", "label": "injectives-theorem-baer-grothendieck", "categories": [ "injectives" ], "title": "injectives-theorem-baer-grothendieck", "contents": [ "Let $\\kappa$ be the cardinality of the set of ideals in $R$, and", "let $\\alpha$ be an ordinal whose cofinality is greater than", "$\\kappa$. Then $\\mathbf{M}_\\alpha(N)$ is an injective $R$-module,", "and $N \\to \\mathbf{M}_\\alpha(N)$ is a functorial injective embedding." ], "refs": [], "proofs": [ { "contents": [ "By Baer's criterion", "Lemma \\ref{lemma-criterion-baer},", "it suffices to show that if $\\mathfrak{a} \\subset R$ is an ideal, then", "any map $f : \\mathfrak{a} \\to \\mathbf{M}_\\alpha(N)$ extends to", "$R \\to \\mathbf{M}_\\alpha(N)$. However, we know since $\\alpha$ is a limit", "ordinal that", "$$", "\\mathbf{M}_{\\alpha}(N) =", "\\colim_{\\beta < \\alpha} \\mathbf{M}_{\\beta}(N),", "$$", "so by", "Proposition \\ref{proposition-modules-are-small},", "we find that", "$$", "\\Hom_R(\\mathfrak{a}, \\mathbf{M}_{\\alpha}(N)) =", "\\colim_{\\beta < \\alpha} \\Hom_R(\\mathfrak a, \\mathbf{M}_{\\beta}(N)).", "$$", "This means in particular that there is some $\\beta' < \\alpha$", "such that $f$ factors through the submodule $\\mathbf{M}_{\\beta'}(N)$, as", "$$", "f : \\mathfrak{a} \\to \\mathbf{M}_{\\beta'}(N) \\to", "\\mathbf{M}_{\\alpha}(N).", "$$", "However, by the fundamental property of the functor $\\mathbf{M}$,", "see Lemma \\ref{lemma-construction} part (3),", "we know that the map $\\mathfrak{a} \\to \\mathbf{M}_{\\beta'}(N)$", "can be extended to", "$$", "R \\to \\mathbf{M}( \\mathbf{M}_{\\beta'}(N)) =", "\\mathbf{M}_{\\beta' + 1}(N),", "$$", "and the last object imbeds in $\\mathbf{M}_{\\alpha}(N)$ (as", "$\\beta' + 1 < \\alpha$ since $\\alpha$ is a limit ordinal).", "In particular, $f$ can be extended to $\\mathbf{M}_{\\alpha}(N)$." ], "refs": [ "injectives-lemma-criterion-baer", "injectives-proposition-modules-are-small", "injectives-lemma-construction" ], "ref_ids": [ 7771, 7804, 7772 ] } ], "ref_ids": [] }, { "id": 7765, "type": "theorem", "label": "injectives-theorem-sheaves-injectives", "categories": [ "injectives" ], "title": "injectives-theorem-sheaves-injectives", "contents": [ "The category of sheaves of abelian groups on a", "site has enough injectives. In fact there exists", "a functorial injective embedding, see", "Homology, Definition \\ref{homology-definition-functorial-injective-embedding}." ], "refs": [ "homology-definition-functorial-injective-embedding" ], "proofs": [ { "contents": [ "Let $\\mathcal{G}_i$, $i \\in I$ be a set of abelian", "sheaves such that every subsheaf of every $\\mathbf{Z}_X^\\#$", "occurs as one of the $\\mathcal{G}_i$. Apply", "Lemma \\ref{lemma-map-into-smaller} to this collection to", "get an ordinal $\\beta$. We claim that for any sheaf of abelian", "groups $\\mathcal{F}$ the map $\\mathcal{F} \\to J_\\beta(\\mathcal{F})$", "is an injection of $\\mathcal{F}$ into an injective.", "Note that by construction the assignment", "$\\mathcal{F} \\mapsto \\big(\\mathcal{F} \\to J_\\beta(\\mathcal{F})\\big)$", "is indeed functorial.", "\\medskip\\noindent", "The proof of the claim comes from the fact that by", "Lemma \\ref{lemma-characterize-injectives} it suffices to extend any", "morphism $\\gamma : \\mathcal{G} \\to J_\\beta(\\mathcal{F})$", "from a subsheaf $\\mathcal{G}$ of some $\\mathbf{Z}_X^\\#$ to all of", "$\\mathbf{Z}_X^\\#$. Then by Lemma \\ref{lemma-map-into-smaller} the", "map $\\gamma$ lifts into $J_\\alpha(\\mathcal{F})$ for some", "$\\alpha < \\beta$. Finally, we apply Lemma \\ref{lemma-map-into-next-one}", "to get the desired extension of $\\gamma$ to a morphism", "into $J_{\\alpha + 1}(\\mathcal{F}) \\to J_\\beta(\\mathcal{F})$." ], "refs": [ "injectives-lemma-map-into-smaller", "injectives-lemma-characterize-injectives", "injectives-lemma-map-into-smaller", "injectives-lemma-map-into-next-one" ], "ref_ids": [ 7777, 7778, 7777, 7776 ] } ], "ref_ids": [ 12184 ] }, { "id": 7766, "type": "theorem", "label": "injectives-theorem-sheaves-modules-injectives", "categories": [ "injectives" ], "title": "injectives-theorem-sheaves-modules-injectives", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$.", "The category of sheaves of $\\mathcal{O}$-modules on a", "site has enough injectives. In fact there exists", "a functorial injective embedding, see", "Homology, Definition \\ref{homology-definition-functorial-injective-embedding}." ], "refs": [ "homology-definition-functorial-injective-embedding" ], "proofs": [ { "contents": [ "From the discussion in this section." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12184 ] }, { "id": 7767, "type": "theorem", "label": "injectives-theorem-injective-embedding-grothendieck", "categories": [ "injectives" ], "title": "injectives-theorem-injective-embedding-grothendieck", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category.", "Then $\\mathcal{A}$ has functorial injective embeddings." ], "refs": [], "proofs": [ { "contents": [ "Please compare with the proof of", "Theorem \\ref{theorem-baer-grothendieck}.", "Choose a generator $U$ of $\\mathcal{A}$. For an object $M$ we define", "$\\mathbf{M}(M)$ by the following pushout diagram", "$$", "\\xymatrix{", "\\bigoplus_{N \\subset U}", "\\bigoplus_{\\varphi \\in \\Hom(N, M)}", "N \\ar[r] \\ar[d] & M \\ar[d] \\\\", "\\bigoplus_{N \\subset U}", "\\bigoplus_{\\varphi \\in \\Hom(N, M)}", "U \\ar[r] & \\mathbf{M}(M).", "}", "$$", "Note that $M \\to \\mathbf{M}(N)$ is a functor and that there", "exist functorial injective maps $M \\to \\mathbf{M}(M)$. By transfinite", "induction we define functors $\\mathbf{M}_\\alpha(M)$ for every", "ordinal $\\alpha$. Namely, set $\\mathbf{M}_0(M) = M$. Given", "$\\mathbf{M}_\\alpha(M)$ set", "$\\mathbf{M}_{\\alpha + 1}(M) = \\mathbf{M}(\\mathbf{M}_\\alpha(M))$.", "For a limit ordinal $\\beta$ set", "$$", "\\mathbf{M}_\\beta(M) = \\colim_{\\alpha < \\beta} \\mathbf{M}_\\alpha(M).", "$$", "Finally, pick any ordinal $\\alpha$ whose cofinality is greater than $|U|$.", "Such an ordinal exists by", "Sets, Proposition \\ref{sets-proposition-exist-ordinals-large-cofinality}.", "We claim that $M \\to \\mathbf{M}_\\alpha(M)$ is the desired functorial", "injective embedding. Namely, if $N \\subset U$ is a subobject and", "$\\varphi : N \\to \\mathbf{M}_\\alpha(M)$ is a morphism, then we see that", "$\\varphi$ factors through $\\mathbf{M}_{\\alpha'}(M)$ for some", "$\\alpha' < \\alpha$ by", "Proposition \\ref{proposition-objects-are-small}.", "By construction of $\\mathbf{M}(-)$ we see that $\\varphi$ extends to", "a morphism from $U$ into $\\mathbf{M}_{\\alpha' + 1}(M)$ and hence into", "$\\mathbf{M}_\\alpha(M)$. By", "Lemma \\ref{lemma-characterize-injective}", "we conclude that $\\mathbf{M}_\\alpha(M)$ is injective." ], "refs": [ "injectives-theorem-baer-grothendieck", "sets-proposition-exist-ordinals-large-cofinality", "injectives-proposition-objects-are-small", "injectives-lemma-characterize-injective" ], "ref_ids": [ 7764, 8802, 7807, 7787 ] } ], "ref_ids": [] }, { "id": 7768, "type": "theorem", "label": "injectives-theorem-K-injective-embedding-grothendieck", "categories": [ "injectives" ], "title": "injectives-theorem-K-injective-embedding-grothendieck", "contents": [ "\\begin{slogan}", "Existence of K-injective complexes for Grothendieck abelian categories.", "\\end{slogan}", "Let $\\mathcal{A}$ be a Grothendieck abelian category.", "For every complex $M^\\bullet$ there exists a quasi-isomorphism", "$M^\\bullet \\to I^\\bullet$ such that $M^n \\to I^n$ is injective and $I^n$", "is an injective object of $\\mathcal{A}$ for all $n$ and $I^\\bullet$", "is a K-injective complex. Moreover, the construction is functorial in", "$M^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Please compare with the proof of", "Theorem \\ref{theorem-baer-grothendieck}", "and", "Theorem \\ref{theorem-injective-embedding-grothendieck}.", "Choose a cardinal $\\kappa$ as in", "Lemmas \\ref{lemma-acyclic-quotient-complexes-bounded-size} and", "\\ref{lemma-characterize-K-injective}.", "Choose a set $(K_i^\\bullet)_{i \\in I}$", "of bounded above, acyclic complexes", "such that every bounded above acyclic complex $K^\\bullet$", "such that $|K^n| \\leq \\kappa$ is isomorphic to $K_i^\\bullet$ for some", "$i \\in I$. This is possible by", "Lemma \\ref{lemma-set-iso-classes-bounded-size}.", "Denote $\\mathbf{M}^\\bullet(-)$ the functor constructed in", "Lemma \\ref{lemma-functorial-homotopies}.", "Denote $\\mathbf{N}^\\bullet(-)$ the functor constructed in", "Lemma \\ref{lemma-functorial-injective}.", "Both of these functors come with injective transformations", "$\\text{id} \\to \\mathbf{M}$ and $\\text{id} \\to \\mathbf{N}$.", "\\medskip\\noindent", "By transfinite induction we define a sequence of functors", "$\\mathbf{T}_\\alpha(-)$ and corresponding transformations", "$\\text{id} \\to \\mathbf{T}_\\alpha$. Namely we set", "$\\mathbf{T}_0(M^\\bullet) = M^\\bullet$. If $\\mathbf{T}_\\alpha$ is", "given then we set", "$$", "\\mathbf{T}_{\\alpha + 1}(M^\\bullet) =", "\\mathbf{N}^\\bullet(\\mathbf{M}^\\bullet(\\mathbf{T}_\\alpha(M^\\bullet)))", "$$", "If $\\beta$ is a limit ordinal we set", "$$", "\\mathbf{T}_\\beta(M^\\bullet) =", "\\colim_{\\alpha < \\beta} \\mathbf{T}_\\alpha(M^\\bullet)", "$$", "The transition maps of the system are injective quasi-isomorphisms.", "By AB5 we see that the colimit is still quasi-isomorphic to $M^\\bullet$.", "We claim that $M^\\bullet \\to \\mathbf{T}_\\alpha(M^\\bullet)$", "does the job if the cofinality of $\\alpha$ is larger than", "$\\max(\\kappa, |U|)$ where $U$ is a generator of $\\mathcal{A}$.", "Namely, it suffices to check conditions (1) and (2) of", "Lemma \\ref{lemma-characterize-K-injective}.", "\\medskip\\noindent", "For (1) we use the criterion of", "Lemma \\ref{lemma-characterize-injective}.", "Suppose that $M \\subset U$ and $\\varphi : M \\to \\mathbf{T}^n_\\alpha(M^\\bullet)$", "is a morphism for some $n \\in \\mathbf{Z}$. By", "Proposition \\ref{proposition-objects-are-small}", "we see that $\\varphi$ factor through", "$\\mathbf{T}^n_{\\alpha'}(M^\\bullet)$ for some $\\alpha' < \\alpha$.", "In particular, by the construction of the functor", "$\\mathbf{N}^\\bullet(-)$ we see that $\\varphi$ factors through", "an injective object of $\\mathcal{A}$ which shows that $\\varphi$", "lifts to a morphism on $U$.", "\\medskip\\noindent", "For (2) let $w : K^\\bullet \\to \\mathbf{T}_\\alpha(M^\\bullet)$", "be a morphism of complexes where $K^\\bullet$ is a bounded above acyclic", "complex such that $|K^n| \\leq \\kappa$. Then $K^\\bullet \\cong K_i^\\bullet$", "for some $i \\in I$. Moreover, by", "Proposition \\ref{proposition-objects-are-small}", "once again we see that $w$ factor through", "$\\mathbf{T}^n_{\\alpha'}(M^\\bullet)$ for some $\\alpha' < \\alpha$.", "In particular, by the construction of the functor", "$\\mathbf{M}^\\bullet(-)$ we see that $w$ is homotopic to zero.", "This finishes the proof." ], "refs": [ "injectives-theorem-baer-grothendieck", "injectives-theorem-injective-embedding-grothendieck", "injectives-lemma-acyclic-quotient-complexes-bounded-size", "injectives-lemma-characterize-K-injective", "injectives-lemma-set-iso-classes-bounded-size", "injectives-lemma-functorial-homotopies", "injectives-lemma-functorial-injective", "injectives-lemma-characterize-K-injective", "injectives-lemma-characterize-injective", "injectives-proposition-objects-are-small", "injectives-proposition-objects-are-small" ], "ref_ids": [ 7764, 7767, 7789, 7790, 7786, 7791, 7792, 7790, 7787, 7807, 7807 ] } ], "ref_ids": [] }, { "id": 7769, "type": "theorem", "label": "injectives-theorem-gabriel-popescu", "categories": [ "injectives" ], "title": "injectives-theorem-gabriel-popescu", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category. Then there exists", "a (noncommutative) ring $R$ and functors $G : \\mathcal{A} \\to \\text{Mod}_R$", "and $F : \\text{Mod}_R \\to \\mathcal{A}$ such that", "\\begin{enumerate}", "\\item $F$ is the left adjoint to $G$,", "\\item $G$ is fully faithful, and", "\\item $F$ is exact.", "\\end{enumerate}", "Moreover, the functors are the ones constructed above." ], "refs": [], "proofs": [ { "contents": [ "We first prove $G$ is fully faithful, or equivalently that", "$F \\circ G \\to \\text{id}$ is an isomorphism, see", "Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}.", "First, given an object $A$ the map $F(G(A)) \\to A$ is surjective,", "because every map of $U \\to A$ factors through $F(G(A))$ by construction.", "On the other hand, the map $F(G(A)) \\to A$ is the adjoint of the", "map $\\text{id} : G(A) \\to G(A)$ and hence injective by", "Lemma \\ref{lemma-F-G-monos}.", "\\medskip\\noindent", "The functor $F$ is right exact as it is a left adjoint.", "Since $\\text{Mod}_R$ has enough projectives, to show that", "$F$ is exact, it is enough to show that the first left derived", "functor $L_1F$ is zero. To prove $L_1F(M) = 0$ for some $R$-module $M$", "choose an exact sequence $0 \\to K \\to P \\to M \\to 0$", "of $R$-modules with $P$ free. It suffices to show $F(K) \\to F(P)$", "is injective. Now we can write this sequence as a filtered", "colimit of sequences $0 \\to K_i \\to P_i \\to M_i \\to 0$", "with $P_i$ a finite free $R$-module: just write $P$ in this", "manner and set $K_i = K \\cap P_i$ and $M_i = \\Im(P_i \\to M)$.", "Because $F$ is a left adjoint it commutes", "with colimits and because $\\mathcal{A}$ is a Grothendieck", "abelian category, we find that $F(K) \\to F(P)$", "is injective if each $F(K_i) \\to F(P_i)$ is injective.", "Thus it suffices to check $F(K) \\to F(P)$", "is injective when $K \\subset P = R^{\\oplus n}$.", "Thus $F(K) \\to U^{\\oplus n}$ is injective by an application", "of Lemma \\ref{lemma-F-G-monos}." ], "refs": [ "categories-lemma-adjoint-fully-faithful", "injectives-lemma-F-G-monos", "injectives-lemma-F-G-monos" ], "ref_ids": [ 12248, 7801, 7801 ] } ], "ref_ids": [] }, { "id": 7770, "type": "theorem", "label": "injectives-lemma-out-of-finite", "categories": [ "injectives" ], "title": "injectives-lemma-out-of-finite", "contents": [ "Suppose that, in (\\ref{equation-compare}), $\\mathcal{C}$ is the category", "of sets and $A$ is a {\\it finite set}, then the map is a bijection." ], "refs": [], "proofs": [ { "contents": [ "Let $f : A \\to \\colim B_\\beta$.", "The range of $f$ is finite, containing say", "elements $c_1, \\ldots, c_r \\in \\colim B_\\beta$.", "These all come from some elements in $B_\\beta$ for $\\beta \\in \\alpha$", "large by definition of the colimit. Thus we can define", "$\\widetilde{f} : A \\to B_\\beta$ lifting $f$ at a finite stage.", "This proves that (\\ref{equation-compare}) is surjective.", "Next, suppose two maps $f : A \\to B_\\gamma, f' : A \\to B_{\\gamma'}$", "define the same map $A \\to \\colim B_\\beta$.", "Then each of the finitely many elements of $A$ gets sent to the same point in", "the colimit. By definition of the colimit for sets, there is", "$\\beta \\geq \\gamma, \\gamma'$ such that the finitely many elements of", "$A$ get sent to the same points in $B_\\beta$ under $f$ and $f'$.", "This proves that (\\ref{equation-compare}) is injective." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7771, "type": "theorem", "label": "injectives-lemma-criterion-baer", "categories": [ "injectives" ], "title": "injectives-lemma-criterion-baer", "contents": [ "\\begin{reference}", "\\cite[Theorem 1]{Baer}", "\\end{reference}", "Let $R$ be a ring. An $R$-module $Q$ is injective if and only if in every", "commutative diagram", "$$", "\\xymatrix{", "\\mathfrak{a} \\ar[d] \\ar[r] & Q \\\\", "R \\ar@{-->}[ru]", "}", "$$", "for $\\mathfrak{a} \\subset R$ an ideal, the dotted arrow exists." ], "refs": [], "proofs": [ { "contents": [ "This is the equivalence of (1) and (3) in", "More on Algebra, Lemma \\ref{more-algebra-lemma-characterize-injective-bis};", "please observe that the proof given there is elementary", "(and does not use $\\text{Ext}$ groups or the existence of injectives", "or projectives in the category of $R$-modules)." ], "refs": [ "more-algebra-lemma-characterize-injective-bis" ], "ref_ids": [ 10113 ] } ], "ref_ids": [] }, { "id": 7772, "type": "theorem", "label": "injectives-lemma-construction", "categories": [ "injectives" ], "title": "injectives-lemma-construction", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item The construction $M \\mapsto (M \\to \\mathbf{M}(M))$", "is functorial in $M$.", "\\item The map $M \\to \\mathbf{M}(M)$ is injective.", "\\item For any ideal $\\mathfrak{a}$ and any $R$-module map", "$\\varphi : \\mathfrak a \\to M$ there is an $R$-module map", "$\\varphi' : R \\to \\mathbf{M}(M)$ such that", "$$", "\\xymatrix{", "\\mathfrak{a} \\ar[d] \\ar[r]_\\varphi & M \\ar[d] \\\\", "R \\ar[r]^{\\varphi'} & \\mathbf{M}(M)", "}", "$$", "commutes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (2) and (3) are immediate from the construction.", "To see (1), let $\\chi : M \\to N$ be an $R$-module map. We claim there exists", "a canonical commutative diagram", "$$", "\\xymatrix{", "\\bigoplus_{\\mathfrak a}", "\\bigoplus_{\\varphi \\in \\Hom_R(\\mathfrak a, M)}", "\\mathfrak{a} \\ar[r] \\ar[d] \\ar[rrd] & M \\ar[rrd]^\\chi \\\\", "\\bigoplus_{\\mathfrak a}", "\\bigoplus_{\\varphi \\in \\Hom_R(\\mathfrak a, M)}", "R \\ar[rrd] & &", "\\bigoplus_{\\mathfrak a}", "\\bigoplus_{\\psi \\in \\Hom_R(\\mathfrak a, N)}", "\\mathfrak{a} \\ar[r] \\ar[d] & N \\\\", "& & \\bigoplus_{\\mathfrak a}", "\\bigoplus_{\\psi \\in \\Hom_R(\\mathfrak a, N)}", "R", "}", "$$", "which induces the desired map $\\mathbf{M}(M) \\to \\mathbf{M}(N)$.", "The middle east-south-east arrow maps the summand $\\mathfrak a$", "corresponding to $\\varphi$ via $\\text{id}_{\\mathfrak a}$ to the", "summand $\\mathfrak a$ corresponding to $\\psi = \\chi \\circ \\varphi$.", "Similarly for the lower east-south-east arrow. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7773, "type": "theorem", "label": "injectives-lemma-G-modules", "categories": [ "injectives" ], "title": "injectives-lemma-G-modules", "contents": [ "Let $G$ be a topological group. Let $R$ be a ring.", "The category $\\text{Mod}_{R, G}$ of $R\\text{-}G$-modules, see", "\\'Etale Cohomology, Definition", "\\ref{etale-cohomology-definition-G-module-continuous},", "has functorial injective hulls. In particular this holds", "for the category of discrete $G$-modules." ], "refs": [ "etale-cohomology-definition-G-module-continuous" ], "proofs": [ { "contents": [ "By the remark above the lemma the category $\\text{Mod}_{R[G]}$", "has functorial injective embeddings.", "Consider the forgetful functor", "$v : \\text{Mod}_{R, G} \\to \\text{Mod}_{R[G]}$.", "This functor is fully faithful, transforms injective maps into", "injective maps and has a right adjoint, namely", "$$", "u : M \\mapsto u(M) = \\{x \\in M \\mid \\text{stabilizer of }x\\text{ is open}\\}", "$$", "Since $v(M) = 0 \\Rightarrow M = 0$ we conclude by", "Homology, Lemma \\ref{homology-lemma-adjoint-functorial-injectives}." ], "refs": [ "homology-lemma-adjoint-functorial-injectives" ], "ref_ids": [ 12118 ] } ], "ref_ids": [ 6747 ] }, { "id": 7774, "type": "theorem", "label": "injectives-lemma-abelian-sheaves-space", "categories": [ "injectives" ], "title": "injectives-lemma-abelian-sheaves-space", "contents": [ "Let $X$ be a topological space.", "The category of abelian sheaves on $X$ has enough injectives.", "In fact it has functorial injective embeddings." ], "refs": [], "proofs": [ { "contents": [ "For an abelian group $A$ we denote $j : A \\to J(A)$ the functorial", "injective embedding constructed in", "More on Algebra, Section \\ref{more-algebra-section-injectives-modules}.", "Let $\\mathcal{F}$ be an abelian sheaf on $X$.", "By Sheaves, Example \\ref{sheaves-example-sheaf-product-pointwise}", "the assignment", "$$", "\\mathcal{I} : U \\mapsto", "\\mathcal{I}(U) = \\prod\\nolimits_{x\\in U} J(\\mathcal{F}_x)", "$$", "is an abelian sheaf. There is a canonical map $\\mathcal{F} \\to \\mathcal{I}$", "given by mapping $s \\in \\mathcal{F}(U)$ to $\\prod_{x \\in U} j(s_x)$", "where $s_x \\in \\mathcal{F}_x$ denotes the germ of $s$ at $x$.", "This map is injective, see", "Sheaves, Lemma \\ref{sheaves-lemma-sheaf-subset-stalks}", "for example.", "\\medskip\\noindent", "It remains to prove the following: Given a rule", "$x \\mapsto I_x$ which assigns to each point $x \\in X$ an injective", "abelian group the sheaf $\\mathcal{I} : U \\mapsto \\prod_{x \\in U} I_x$", "is injective. Note that", "$$", "\\mathcal{I} = \\prod\\nolimits_{x \\in X} i_{x, *}I_x", "$$", "is the product of the skyscraper sheaves $i_{x, *}I_x$ (see", "Sheaves, Section \\ref{sheaves-section-skyscraper-sheaves} for notation.)", "We have", "$$", "\\Mor_{\\textit{Ab}}(\\mathcal{F}_x, I_x)", "=", "\\Mor_{\\textit{Ab}(X)}(\\mathcal{F}, i_{x, *}I_x).", "$$", "see Sheaves, Lemma \\ref{sheaves-lemma-stalk-skyscraper-adjoint}. Hence it is", "clear that each $i_{x, *}I_x$ is injective. Hence the injectivity of", "$\\mathcal{I}$ follows from", "Homology, Lemma \\ref{homology-lemma-product-injectives}." ], "refs": [ "sheaves-lemma-sheaf-subset-stalks", "sheaves-lemma-stalk-skyscraper-adjoint", "homology-lemma-product-injectives" ], "ref_ids": [ 14482, 14525, 12113 ] } ], "ref_ids": [] }, { "id": 7775, "type": "theorem", "label": "injectives-lemma-sheaves-modules-space", "categories": [ "injectives" ], "title": "injectives-lemma-sheaves-modules-space", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space, see", "Sheaves, Section \\ref{sheaves-section-ringed-spaces}.", "The category of sheaves of $\\mathcal{O}_X$-modules on $X$", "has enough injectives. In fact it has functorial injective embeddings." ], "refs": [], "proofs": [ { "contents": [ "For any ring $R$ and any $R$-module $M$ we denote", "$j : M \\to J_R(M)$ the functorial", "injective embedding constructed in", "More on Algebra, Section \\ref{more-algebra-section-injectives-modules}.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules on $X$.", "By Sheaves, Examples \\ref{sheaves-example-sheaf-product-pointwise}", "and \\ref{sheaves-example-sheaf-product-pointwise-algebraic-structure}", "the assignment", "$$", "\\mathcal{I} : U \\mapsto", "\\mathcal{I}(U) = \\prod\\nolimits_{x\\in U} J_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x)", "$$", "is an abelian sheaf.", "There is a canonical map $\\mathcal{F} \\to \\mathcal{I}$", "given by mapping $s \\in \\mathcal{F}(U)$ to $\\prod_{x \\in U} j(s_x)$", "where $s_x \\in \\mathcal{F}_x$ denotes the germ of $s$ at $x$.", "This map is injective, see", "Sheaves, Lemma \\ref{sheaves-lemma-sheaf-subset-stalks}", "for example.", "\\medskip\\noindent", "It remains to prove the following: Given a rule", "$x \\mapsto I_x$ which assigns to each point $x \\in X$ an injective", "$\\mathcal{O}_{X, x}$-module", "the sheaf $\\mathcal{I} : U \\mapsto \\prod_{x \\in U} I_x$", "is injective. Note that", "$$", "\\mathcal{I} = \\prod\\nolimits_{x \\in X} i_{x, *}I_x", "$$", "is the product of the skyscraper sheaves $i_{x, *}I_x$ (see", "Sheaves, Section \\ref{sheaves-section-skyscraper-sheaves} for notation.)", "We have", "$$", "\\Hom_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x, I_x)", "=", "\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, i_{x, *}I_x).", "$$", "see Sheaves, Lemma \\ref{sheaves-lemma-stalk-skyscraper-adjoint}. Hence it is", "clear that each $i_{x, *}I_x$ is an injective $\\mathcal{O}_X$-module", "(see Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives} or argue", "directly). Hence the injectivity of $\\mathcal{I}$ follows from", "Homology, Lemma \\ref{homology-lemma-product-injectives}." ], "refs": [ "sheaves-lemma-sheaf-subset-stalks", "sheaves-lemma-stalk-skyscraper-adjoint", "homology-lemma-adjoint-preserve-injectives", "homology-lemma-product-injectives" ], "ref_ids": [ 14482, 14525, 12116, 12113 ] } ], "ref_ids": [] }, { "id": 7776, "type": "theorem", "label": "injectives-lemma-map-into-next-one", "categories": [ "injectives" ], "title": "injectives-lemma-map-into-next-one", "contents": [ "With notation as above.", "Suppose that $\\mathcal{G}_1 \\to \\mathcal{G}_2$ is an injective", "map of abelian sheaves on $\\mathcal{C}$. Let $\\alpha$ be an ordinal", "and let $\\mathcal{G}_1 \\to J_\\alpha(\\mathcal{F})$ be a morphism", "of sheaves. There exists a morphism $\\mathcal{G}_2 \\to", "J_{\\alpha + 1}(\\mathcal{F})$ such that the following diagram commutes", "$$", "\\xymatrix{", "\\mathcal{G}_1 \\ar[d] \\ar[r] & \\mathcal{G}_2 \\ar[d] \\\\", "J_{\\alpha}(\\mathcal{F}) \\ar[r] & J_{\\alpha + 1}(\\mathcal{F}) }", "$$" ], "refs": [], "proofs": [ { "contents": [ "This is because the map $i\\mathcal{G}_1 \\to i\\mathcal{G}_2$ is injective", "and hence $i\\mathcal{G}_1 \\to iJ_\\alpha(\\mathcal{F})$ extends to", "$i\\mathcal{G}_2 \\to J(iJ_\\alpha(\\mathcal{F}))$ which gives the", "desired map after applying the sheafification functor." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7777, "type": "theorem", "label": "injectives-lemma-map-into-smaller", "categories": [ "injectives" ], "title": "injectives-lemma-map-into-smaller", "contents": [ "Suppose that $\\mathcal{G}_i$, $i\\in I$ is set of abelian sheaves", "on $\\mathcal{C}$. There exists an ordinal $\\beta$ such that", "for any sheaf $\\mathcal{F}$, any $i\\in I$, and any map", "$\\varphi : \\mathcal{G}_i \\to J_\\beta(\\mathcal{F})$ there exists an", "$\\alpha < \\beta$ such that $ \\varphi $ factors through", "$J_\\alpha(\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "This reduces to the case of a single sheaf $\\mathcal{G}$", "by taking the direct sum of all the $\\mathcal{G}_i$.", "\\medskip\\noindent", "Consider the sets", "$$", "S = \\coprod\\nolimits_{U \\in \\Ob(\\mathcal{C})} \\mathcal{G}(U).", "$$", "and", "$$", "T_\\beta", "=", "\\coprod\\nolimits_{U \\in \\Ob(\\mathcal{C})} J_\\beta(\\mathcal{F})(U)", "$$", "Then $T_\\beta = \\colim_{\\alpha < \\beta} T_\\alpha$", "with injective transition maps.", "A morphism $\\mathcal{G} \\to J_\\beta(\\mathcal{F})$ factors", "through $J_\\alpha(\\mathcal{F})$ if and only if", "the associated map $S \\to T_\\beta$ factors through $T_\\alpha$.", "By", "Sets, Lemma \\ref{sets-lemma-map-from-set-lifts}", "if the cofinality of $\\beta$ is bigger than the cardinality", "of $S$, then the result of the lemma is true. Hence the lemma", "follows from the fact that there are ordinals with arbitrarily", "large cofinality, see", "Sets, Proposition \\ref{sets-proposition-exist-ordinals-large-cofinality}." ], "refs": [ "sets-lemma-map-from-set-lifts", "sets-proposition-exist-ordinals-large-cofinality" ], "ref_ids": [ 8787, 8802 ] } ], "ref_ids": [] }, { "id": 7778, "type": "theorem", "label": "injectives-lemma-characterize-injectives", "categories": [ "injectives" ], "title": "injectives-lemma-characterize-injectives", "contents": [ "Suppose $\\mathcal{J}$ is a sheaf of abelian groups with the following", "property: For all $X\\in \\Ob(\\mathcal{C})$, for any abelian subsheaf", "$\\mathcal{S} \\subset \\mathbf{Z}_X^\\#$ and any morphism", "$\\varphi : \\mathcal{S} \\to \\mathcal{J}$, there exists a morphism", "$\\mathbf{Z}_X^\\# \\to \\mathcal{J}$ extending $\\varphi$.", "Then $\\mathcal{J}$ is an injective sheaf of abelian groups." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F} \\to \\mathcal{G}$ be an injective map", "of abelian sheaves. Suppose $\\varphi : \\mathcal{F} \\to \\mathcal{J}$", "is a morphism. Arguing as in the proof of", "More on Algebra, Lemma \\ref{more-algebra-lemma-injective-abelian}", "we see that it suffices", "to prove that if $\\mathcal{F} \\not = \\mathcal{G}$, then we", "can find an abelian sheaf $\\mathcal{F}'$,", "$\\mathcal{F} \\subset \\mathcal{F}' \\subset \\mathcal{G}$", "such that (a) the inclusion $\\mathcal{F} \\subset \\mathcal{F}'$ is strict,", "and (b) $\\varphi$ can be extended to $\\mathcal{F}'$.", "To find $\\mathcal{F}'$, let $X$ be an object of $\\mathcal{C}$ such", "that the inclusion $\\mathcal{F}(X) \\subset \\mathcal{G}(X)$", "is strict. Pick $s \\in \\mathcal{G}(X)$, $s \\not \\in \\mathcal{F}(X)$.", "Let $\\psi : \\mathbf{Z}_X^\\# \\to \\mathcal{G}$ be the morphism corresponding", "to the section $s$ via (\\ref{equation-free-sheaf-on}). Set", "$\\mathcal{S} = \\psi^{-1}(\\mathcal{F})$. By assumption the morphism", "$$", "\\mathcal{S} \\xrightarrow{\\psi} \\mathcal{F} \\xrightarrow{\\varphi} \\mathcal{J}", "$$", "can be extended to a morphism $\\varphi' : \\mathbf{Z}_X^\\# \\to \\mathcal{J}$.", "Note that $\\varphi'$ annihilates the kernel of $\\psi$ (as this is true", "for $\\varphi$). Thus $\\varphi'$ gives rise to a morphism", "$\\varphi'' : \\Im(\\psi) \\to \\mathcal{J}$", "which agrees with $\\varphi$ on the intersection", "$\\mathcal{F} \\cap \\Im(\\psi)$ by construction.", "Thus $\\varphi$ and $\\varphi''$ glue to give an extension", "of $\\varphi$ to the strictly bigger subsheaf", "$\\mathcal{F}' = \\mathcal{F} + \\Im(\\psi)$." ], "refs": [ "more-algebra-lemma-injective-abelian" ], "ref_ids": [ 10110 ] } ], "ref_ids": [] }, { "id": 7779, "type": "theorem", "label": "injectives-lemma-vee-exact-sheaves", "categories": [ "injectives" ], "title": "injectives-lemma-vee-exact-sheaves", "contents": [ "The functor $\\mathcal{F} \\mapsto \\mathcal{F}^\\vee$ is exact." ], "refs": [], "proofs": [ { "contents": [ "This because $\\mathcal{J}$ is an injective abelian sheaf." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7780, "type": "theorem", "label": "injectives-lemma-ev-injective-sheaves", "categories": [ "injectives" ], "title": "injectives-lemma-ev-injective-sheaves", "contents": [ "For any $\\mathcal{O}$-module $\\mathcal{F}$ the evaluation map", "$ev : \\mathcal{F} \\to (\\mathcal{F}^\\vee)^\\vee$ is injective." ], "refs": [], "proofs": [ { "contents": [ "You can check this using the definition of $\\mathcal{J}$.", "Namely, if $s \\in \\mathcal{F}(U)$ is not zero, then let", "$j_{U!}\\mathcal{O}_U \\to \\mathcal{F}$ be the map of", "$\\mathcal{O}$-modules it corresponds to via adjunction.", "Let $\\mathcal{I}$ be the kernel of this map. There exists", "a nonzero map $\\mathcal{F} \\supset j_{U!}\\mathcal{O}_U/\\mathcal{I}", "\\to \\mathcal{J}$ which does not annihilate $s$. As $\\mathcal{J}$ is", "an injective $\\mathcal{O}$-module, this extends to a map", "$\\varphi : \\mathcal{F} \\to \\mathcal{J}$.", "Then $ev(s)(\\varphi) = \\varphi(s) \\not = 0$ which is what we had to prove." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7781, "type": "theorem", "label": "injectives-lemma-JM-injective-sheaves", "categories": [ "injectives" ], "title": "injectives-lemma-JM-injective-sheaves", "contents": [ "Let $\\mathcal{O}$ be a sheaf of rings.", "For every $\\mathcal{O}$-module $\\mathcal{F}$ the", "$\\mathcal{O}$-module $J(\\mathcal{F})$ is injective." ], "refs": [], "proofs": [ { "contents": [ "We have to show that the functor", "$\\Hom_\\mathcal{O}(\\mathcal{G}, J(\\mathcal{F}))$", "is exact. Note that", "\\begin{eqnarray*}", "\\Hom_\\mathcal{O}(\\mathcal{G}, J(\\mathcal{F}))", "& = &", "\\Hom_\\mathcal{O}(\\mathcal{G}, (F(\\mathcal{F}^\\vee))^\\vee) \\\\", "& = &", "\\Hom_\\mathcal{O}", "(\\mathcal{G}, \\SheafHom(F(\\mathcal{F}^\\vee), \\mathcal{J})) \\\\", "& = &", "\\Hom(\\mathcal{G} \\otimes_\\mathcal{O} F(\\mathcal{F}^\\vee), \\mathcal{J})", "\\end{eqnarray*}", "Thus what we want follows from the fact that $F(\\mathcal{F}^\\vee)$", "is flat and $\\mathcal{J}$ is injective." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7782, "type": "theorem", "label": "injectives-lemma-site-abelian-category", "categories": [ "injectives" ], "title": "injectives-lemma-site-abelian-category", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let", "$$", "\\text{Cov} = \\{\\{f : V \\to U\\} \\mid f\\text{ is surjective}\\}.", "$$", "Then $(\\mathcal{A}, \\text{Cov})$ is a site, see", "Sites, Definition \\ref{sites-definition-site}." ], "refs": [ "sites-definition-site" ], "proofs": [ { "contents": [ "Note that $\\Ob(\\mathcal{A})$ is a set by our conventions", "about categories. An isomorphism is a surjective morphism.", "The composition of surjective morphisms is surjective.", "And the base change of a surjective morphism in $\\mathcal{A}$", "is surjective, see", "Homology, Lemma \\ref{homology-lemma-epimorphism-universal-abelian-category}." ], "refs": [ "homology-lemma-epimorphism-universal-abelian-category" ], "ref_ids": [ 12024 ] } ], "ref_ids": [ 8652 ] }, { "id": 7783, "type": "theorem", "label": "injectives-lemma-embedding", "categories": [ "injectives" ], "title": "injectives-lemma-embedding", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $\\mathcal{C} = (\\mathcal{A}, \\text{Cov})$ be the", "site defined in", "Lemma \\ref{lemma-site-abelian-category}.", "Then $X \\mapsto h_X$ defines a fully faithful, exact functor", "$$", "\\mathcal{A} \\longrightarrow \\textit{Ab}(\\mathcal{C}).", "$$", "Moreover, the site $\\mathcal{C}$ has enough points." ], "refs": [ "injectives-lemma-site-abelian-category" ], "proofs": [ { "contents": [ "Suppose that $f : V \\to U$ is a surjective morphism of $\\mathcal{A}$.", "Let $K = \\Ker(f)$. Recall that", "$V \\times_U V = \\Ker((f, -f) : V \\oplus V \\to U)$, see", "Homology, Example \\ref{homology-example-fibre-product-pushouts}.", "In particular there exists an injection $K \\oplus K \\to V \\times_U V$.", "Let $p, q : V \\times_U V \\to V$ be the two projection morphisms.", "Note that $p - q : V \\times_U V \\to V$ is a morphism such that", "$f \\circ (p - q) = 0$. Hence $p - q$ factors through $K \\to V$.", "Let us denote this morphism by $c : V \\times_U V \\to K$.", "And since the composition $K \\oplus K \\to V \\times_U V \\to K$", "is surjective, we conclude that $c$ is surjective. It follows that", "$$", "V \\times_U V \\xrightarrow{p - q} V \\to U \\to 0", "$$", "is an exact sequence of $\\mathcal{A}$.", "Hence for an object $X$ of $\\mathcal{A}$ the sequence", "$$", "0 \\to", "\\Hom_\\mathcal{A}(U, X) \\to", "\\Hom_\\mathcal{A}(V, X) \\to", "\\Hom_\\mathcal{A}(V \\times_U V, X)", "$$", "is an exact sequence of abelian groups, see", "Homology, Lemma \\ref{homology-lemma-check-exactness}.", "This means that $h_X$ satisfies the sheaf condition", "on $\\mathcal{C}$.", "\\medskip\\noindent", "The functor is fully faithful by", "Categories, Lemma \\ref{categories-lemma-yoneda}.", "The functor is a left exact functor between abelian categories by", "Homology, Lemma \\ref{homology-lemma-check-exactness}.", "To show that it is right exact, let $X \\to Y$ be a surjective morphism", "of $\\mathcal{A}$. Let $U$ be an object of $\\mathcal{A}$, and let", "$s \\in h_Y(U) = \\Mor_\\mathcal{A}(U, Y)$ be a section of $h_Y$", "over $U$. By", "Homology, Lemma \\ref{homology-lemma-epimorphism-universal-abelian-category}", "the projection $U \\times_Y X \\to U$ is surjective.", "Hence $\\{V = U \\times_Y X \\to U\\}$ is a covering of $U$ such that", "$s|_V$ lifts to a section of $h_X$. This proves that", "$h_X \\to h_Y$ is a surjection of abelian sheaves, see", "Sites, Lemma \\ref{sites-lemma-mono-epi-sheaves}.", "\\medskip\\noindent", "The site $\\mathcal{C}$ has enough points by", "Sites, Proposition \\ref{sites-proposition-criterion-points}." ], "refs": [ "homology-lemma-check-exactness", "categories-lemma-yoneda", "homology-lemma-check-exactness", "homology-lemma-epimorphism-universal-abelian-category", "sites-lemma-mono-epi-sheaves", "sites-proposition-criterion-points" ], "ref_ids": [ 12019, 12203, 12019, 12024, 8517, 8643 ] } ], "ref_ids": [ 7782 ] }, { "id": 7784, "type": "theorem", "label": "injectives-lemma-set-of-subobjects", "categories": [ "injectives" ], "title": "injectives-lemma-set-of-subobjects", "contents": [ "Let $\\mathcal{A}$ be an abelian category with a generator $U$ and", "$X$ and object of $\\mathcal{A}$. If $\\kappa$ is the cardinality of", "$\\Mor(U, X)$ then", "\\begin{enumerate}", "\\item There does not exist a strictly increasing", "(or strictly decreasing) chain of subobjects", "of $X$ indexed by a cardinal bigger than $\\kappa$.", "\\item If $\\alpha$ is an ordinal of cofinality $> \\kappa$", "then any increasing (or decreasing) sequence of subobjects", "of $X$ indexed by $\\alpha$ is eventually constant.", "\\item The cardinality of the set of subobjects of $X$", "is $\\leq 2^\\kappa$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For (1) assume $\\kappa' > \\kappa$ is a cardinal and assume", "$X_i$, $i \\in \\kappa'$ is strictly increasing. Then take for", "each $i$ a $\\phi_i \\in \\Mor(U, X)$ such that $\\phi_i$ factors through", "$X_{i + 1}$ but not through $X_i$. Then the morphisms $\\phi_i$", "are distinct, which contradicts the definition of $\\kappa$.", "\\medskip\\noindent", "Part (2) follows from the definition of cofinality and (1).", "\\medskip\\noindent", "Proof of (3). For any subobject $Y \\subset X$", "define $S_Y \\in \\mathcal{P}(\\Mor(U, X))$ (power set) as", "$S_Y = \\{\\phi \\in \\Mor(U,X) : \\phi)\\text{ factors through }Y\\}$.", "Then $Y = Y'$ if and only if $S_Y = S_{Y'}$. Hence the cardinality", "of the set of subobjects is at most the cardinality of this power set." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7785, "type": "theorem", "label": "injectives-lemma-size-goes-down", "categories": [ "injectives" ], "title": "injectives-lemma-size-goes-down", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category.", "If $0 \\to M' \\to M \\to M'' \\to 0$ is a short exact sequence of", "$\\mathcal{A}$, then $|M'|, |M''| \\leq |M|$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7786, "type": "theorem", "label": "injectives-lemma-set-iso-classes-bounded-size", "categories": [ "injectives" ], "title": "injectives-lemma-set-iso-classes-bounded-size", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category with generator $U$.", "\\begin{enumerate}", "\\item If $|M| \\leq \\kappa$, then $M$ is the quotient of a direct", "sum of at most $\\kappa$ copies of $U$.", "\\item For every cardinal $\\kappa$ there exists a set of isomorphism classes", "of objects $M$ with $|M| \\leq \\kappa$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For (1) choose for every proper subobject $M' \\subset M$ a morphism", "$\\varphi_{M'} : U \\to M$ whose image is not contained in $M'$. Then", "$\\bigoplus_{M' \\subset M} \\varphi_{M'} : \\bigoplus_{M' \\subset M} U \\to M$", "is surjective. It is clear that (1) implies (2)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7787, "type": "theorem", "label": "injectives-lemma-characterize-injective", "categories": [ "injectives" ], "title": "injectives-lemma-characterize-injective", "contents": [ "\\begin{slogan}", "To check that an object is injective, one only needs to check that lifting", "holds for subobjects of a generator.", "\\end{slogan}", "Let $\\mathcal{A}$ be a Grothendieck abelian category with generator $U$.", "An object $I$ of $\\mathcal{A}$ is injective if and only if in every", "commutative diagram", "$$", "\\xymatrix{", "M \\ar[d] \\ar[r] & I \\\\", "U \\ar@{-->}[ru]", "}", "$$", "for $M \\subset U$ a subobject, the dotted arrow exists." ], "refs": [], "proofs": [ { "contents": [ "Please see Lemma \\ref{lemma-criterion-baer} for the case of modules.", "Choose an injection $A \\subset B$ and a morphism $\\varphi : A \\to I$.", "Consider the set $S$ of pairs $(A', \\varphi')$ consisting of", "subobjects $A \\subset A' \\subset B$ and a morphism $\\varphi' : A' \\to I$", "extending $\\varphi$. Define a partial ordering on this set in the obvious", "manner. Choose a totally ordered subset $T \\subset S$. Then", "$$", "A' = \\colim_{t \\in T} A_t \\xrightarrow{\\colim_{t \\in T} \\varphi_t} I", "$$", "is an upper bound. Hence by Zorn's lemma the set $S$ has a maximal element", "$(A', \\varphi')$. We claim that $A' = B$. If not, then choose a morphism", "$\\psi : U \\to B$ which does not factor through $A'$. Set", "$N = A' \\cap \\psi(U)$. Set $M = \\psi^{-1}(N)$. Then the map", "$$", "M \\to N \\to A' \\xrightarrow{\\varphi'} I", "$$", "can be extended to a morphism $\\chi : U \\to I$. Since", "$\\chi|_{\\Ker(\\psi)} = 0$ we see that $\\chi$ factors as", "$$", "U \\to \\Im(\\psi) \\xrightarrow{\\varphi''} I", "$$", "Since $\\varphi'$ and $\\varphi''$ agree on $N = A' \\cap \\Im(\\psi)$", "we see that combined the define a morphism $A' + \\Im(\\psi) \\to I$", "contradicting the assumed maximality of $A'$." ], "refs": [ "injectives-lemma-criterion-baer" ], "ref_ids": [ 7771 ] } ], "ref_ids": [] }, { "id": 7788, "type": "theorem", "label": "injectives-lemma-surjection-bounded-size", "categories": [ "injectives" ], "title": "injectives-lemma-surjection-bounded-size", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category with generator $U$.", "Let $c$ be the function on cardinals defined by", "$c(\\kappa) = |\\bigoplus_{\\alpha \\in \\kappa} U|$. If $\\pi : M \\to N$ is a", "surjection then there exists a subobject $M' \\subset M$ which surjects", "onto $N$ with $|N'| \\leq c(|N|)$." ], "refs": [], "proofs": [ { "contents": [ "For every proper subobject $N' \\subset N$ choose a morphism", "$\\varphi_{N'} : U \\to M$ such that $U \\to M \\to N$ does not factor", "through $N'$. Set", "$$", "N' = \\Im\\left(", "\\bigoplus\\nolimits_{N' \\subset N} \\varphi_{N'} :", "\\bigoplus\\nolimits_{N' \\subset N} U \\longrightarrow M\\right)", "$$", "Then $N'$ works." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7789, "type": "theorem", "label": "injectives-lemma-acyclic-quotient-complexes-bounded-size", "categories": [ "injectives" ], "title": "injectives-lemma-acyclic-quotient-complexes-bounded-size", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category. There exists a cardinal", "$\\kappa$ such that given any acyclic complex $M^\\bullet$ we have", "\\begin{enumerate}", "\\item if $M^\\bullet$ is nonzero, there is a nonzero subcomplex", "$N^\\bullet$ which is bounded above, acyclic, and $|N^n| \\leq \\kappa$,", "\\item there exists a surjection of complexes", "$$", "\\bigoplus\\nolimits_{i \\in I} M_i^\\bullet \\longrightarrow M^\\bullet", "$$", "where $M_i^\\bullet$ is bounded above, acyclic, and $|M_i^n| \\leq \\kappa$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a generator $U$ of $\\mathcal{A}$. Denote $c$ the function of", "Lemma \\ref{lemma-surjection-bounded-size}.", "Set $\\kappa = \\sup \\{c^n(|U|), n = 1, 2, 3, \\ldots\\}$.", "Let $n \\in \\mathbf{Z}$ and let $\\psi : U \\to M^n$ be a morphism.", "In order to prove (1) and (2) it suffices to prove there exists a subcomplex", "$N^\\bullet \\subset M^\\bullet$ which is bounded above, acyclic, and", "$|N^m| \\leq \\kappa$, such that $\\psi$ factors through $N^n$.", "To do this set $N^n = \\Im(\\psi)$, $N^{n + 1} = \\Im(U \\to M^n \\to M^{n + 1})$,", "and $N^m = 0$ for $m \\geq n + 2$.", "Suppose we have constructed $N^m \\subset M^m$ for all $m \\geq k$ such that", "\\begin{enumerate}", "\\item $\\text{d}(N^m) \\subset N^{m + 1}$, $m \\geq k$,", "\\item $\\Im(N^{m - 1} \\to N^m) = \\Ker(N^m \\to N^{m + 1})$ for", "all $m \\geq k + 1$, and", "\\item $|N^m| \\leq c^{\\max\\{n - m, 0\\}}(|U|)$.", "\\end{enumerate}", "for some $k \\leq n$. Because $M^\\bullet$ is acyclic, we see that the subobject", "$\\text{d}^{-1}(\\Ker(N^k \\to N^{k + 1})) \\subset M^{k - 1}$ surjects onto", "$\\Ker(N^k \\to N^{k + 1})$. Thus we can choose $N^{k - 1} \\subset M^{k - 1}$", "surjecting onto $\\Ker(N^k \\to N^{k + 1})$ with", "$|N^{k - 1}| \\leq c^{n - k + 1}(|U|)$ by", "Lemma \\ref{lemma-surjection-bounded-size}. The proof is finished by", "induction on $k$." ], "refs": [ "injectives-lemma-surjection-bounded-size", "injectives-lemma-surjection-bounded-size" ], "ref_ids": [ 7788, 7788 ] } ], "ref_ids": [] }, { "id": 7790, "type": "theorem", "label": "injectives-lemma-characterize-K-injective", "categories": [ "injectives" ], "title": "injectives-lemma-characterize-K-injective", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category.", "Let $\\kappa$ be a cardinal as in", "Lemma \\ref{lemma-acyclic-quotient-complexes-bounded-size}.", "Suppose that $I^\\bullet$ is a complex such that", "\\begin{enumerate}", "\\item each $I^j$ is injective, and", "\\item for every bounded above acyclic complex $M^\\bullet$", "such that $|M^n| \\leq \\kappa$", "we have $\\Hom_{K(\\mathcal{A})}(M^\\bullet, I^\\bullet) = 0$.", "\\end{enumerate}", "Then $I^\\bullet$ is an $K$-injective complex." ], "refs": [ "injectives-lemma-acyclic-quotient-complexes-bounded-size" ], "proofs": [ { "contents": [ "Let $M^\\bullet$ be an acyclic complex. We are going to construct by", "induction on the ordinal $\\alpha$ an acyclic subcomplex", "$K_\\alpha^\\bullet \\subset M^\\bullet$ as follows.", "For $\\alpha = 0$ we set $K_0^\\bullet = 0$. For $\\alpha > 0$", "we proceed as follows:", "\\begin{enumerate}", "\\item If $\\alpha = \\beta + 1$ and $K_\\beta^\\bullet = M^\\bullet$", "then we choose $K_\\alpha^\\bullet = K_\\beta^\\bullet$.", "\\item If $\\alpha = \\beta + 1$ and $K_\\beta^\\bullet \\not = M^\\bullet$", "then $M^\\bullet/K_\\beta^\\bullet$ is a nonzero acyclic complex.", "We choose a subcomplex $N_\\alpha^\\bullet \\subset M^\\bullet/K_\\beta^\\bullet$", "as in Lemma \\ref{lemma-acyclic-quotient-complexes-bounded-size}.", "Finally, we let $K_\\alpha^\\bullet \\subset M^\\bullet$", "be the inverse image of $N_\\alpha^\\bullet$.", "\\item If $\\alpha$ is a limit ordinal we set", "$K_\\beta^\\bullet = \\colim K_\\alpha^\\bullet$.", "\\end{enumerate}", "It is clear that $M^\\bullet = K_\\alpha^\\bullet$ for a suitably large", "ordinal $\\alpha$. We will prove that", "$$", "\\Hom_{K(\\mathcal{A})}(K_\\alpha^\\bullet, I^\\bullet)", "$$", "is zero by transfinite induction on $\\alpha$. It holds for $\\alpha = 0$", "since $K_0^\\bullet$ is zero. Suppose it holds for $\\beta$ and", "$\\alpha = \\beta + 1$. In case (1) of the list above the result is clear.", "In case (2) there is a short exact sequence of complexes", "$$", "0 \\to K_\\beta^\\bullet \\to K_\\alpha^\\bullet \\to N_\\alpha^\\bullet \\to 0", "$$", "Since each component of $I^\\bullet$ is injective we see that we obtain", "an exact sequence", "$$", "\\Hom_{K(\\mathcal{A})}(K_\\beta^\\bullet, I^\\bullet) \\to", "\\Hom_{K(\\mathcal{A})}(K_\\alpha^\\bullet, I^\\bullet) \\to", "\\Hom_{K(\\mathcal{A})}(N_\\alpha^\\bullet, I^\\bullet)", "$$", "By induction the term on the left is zero and by assumption on $I^\\bullet$", "the term on the right is zero. Thus the middle group is zero too.", "Finally, suppose that $\\alpha$ is a limit ordinal. Then we see that", "$$", "\\Hom^\\bullet(K_\\alpha^\\bullet, I^\\bullet) =", "\\lim_{\\beta < \\alpha} \\Hom^\\bullet(K_\\beta^\\bullet, I^\\bullet)", "$$", "with notation as in", "More on Algebra, Section \\ref{more-algebra-section-hom-complexes}.", "These complexes compute morphisms in $K(\\mathcal{A})$ by", "More on Algebra, Equation", "(\\ref{more-algebra-equation-cohomology-hom-complex}).", "Note that the transition maps in the system are surjective", "because $I^j$ is surjective for each $j$. Moreover, for a limit", "ordinal $\\alpha$ we have equality of limit and value", "(see displayed formula above). Thus we may apply", "Homology, Lemma \\ref{homology-lemma-ML-over-ordinals}", "to conclude." ], "refs": [ "injectives-lemma-acyclic-quotient-complexes-bounded-size", "homology-lemma-ML-over-ordinals" ], "ref_ids": [ 7789, 12129 ] } ], "ref_ids": [ 7789 ] }, { "id": 7791, "type": "theorem", "label": "injectives-lemma-functorial-homotopies", "categories": [ "injectives" ], "title": "injectives-lemma-functorial-homotopies", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category.", "Let $(K_i^\\bullet)_{i \\in I}$ be a set of acyclic complexes.", "There exists a functor $M^\\bullet \\mapsto \\mathbf{M}^\\bullet(M^\\bullet)$", "and a natural transformation", "$j_{M^\\bullet} : M^\\bullet \\to \\mathbf{M}^\\bullet(M^\\bullet)$", "such", "\\begin{enumerate}", "\\item $j_{M^\\bullet}$ is a (termwise) injective quasi-isomorphism, and", "\\item for every $i \\in I$ and $w : K_i^\\bullet \\to M^\\bullet$", "the morphism $j_{M^\\bullet} \\circ w$ is homotopic to zero.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For every $i \\in I$ choose a (termwise) injective map of complexes", "$K_i^\\bullet \\to L_i^\\bullet$ which is homotopic to zero with", "$L_i^\\bullet$ quasi-isomorphic to zero. For example, take $L_i^\\bullet$", "to be the cone on the identity of $K_i^\\bullet$.", "We define $\\mathbf{M}^\\bullet(M^\\bullet)$ by the following pushout diagram", "$$", "\\xymatrix{", "\\bigoplus_{i \\in I}", "\\bigoplus_{w : K_i^\\bullet \\to M^\\bullet}", "K_i^\\bullet \\ar[r] \\ar[d] & M^\\bullet \\ar[d] \\\\", "\\bigoplus_{i \\in I}", "\\bigoplus_{w : K_i^\\bullet \\to M^\\bullet}", "L_i^\\bullet \\ar[r] & \\mathbf{M}^\\bullet(M^\\bullet).", "}", "$$", "Then $M^\\bullet \\to \\mathbf{M}^\\bullet(M^\\bullet)$ is a functor. The right", "vertical arrow defines the functorial injective map $j_{M^\\bullet}$.", "The cokernel of $j_{M^\\bullet}$ is isomorphic to the direct sum of", "the cokernels of the maps $K_i^\\bullet \\to L_i^\\bullet$ hence acyclic.", "Thus $j_{M^\\bullet}$ is a quasi-isomorphism. Part (2) holds by construction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7792, "type": "theorem", "label": "injectives-lemma-functorial-injective", "categories": [ "injectives" ], "title": "injectives-lemma-functorial-injective", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category.", "There exists a functor $M^\\bullet \\mapsto \\mathbf{N}^\\bullet(M^\\bullet)$", "and a natural transformation", "$j_{M^\\bullet} : M^\\bullet \\to \\mathbf{N}^\\bullet(M^\\bullet)$", "such", "\\begin{enumerate}", "\\item $j_{M^\\bullet}$ is a (termwise) injective quasi-isomorphism, and", "\\item for every $n \\in \\mathbf{Z}$ the map $M^n \\to \\mathbf{N}^n(M^\\bullet)$", "factors through a subobject $I^n \\subset \\mathbf{N}^n(M^\\bullet)$ where $I^n$", "is an injective object of $\\mathcal{A}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a functorial injective embeddings $i_M : M \\to I(M)$, see", "Theorem \\ref{theorem-injective-embedding-grothendieck}.", "For every complex $M^\\bullet$ denote $J^\\bullet(M^\\bullet)$ the complex", "with terms $J^n(M^\\bullet) = I(M^n) \\oplus I(M^{n + 1})$ and differential", "$$", "d_{J^\\bullet(M^\\bullet)} =", "\\left(", "\\begin{matrix}", "0 & 1 \\\\", "0 & 0", "\\end{matrix}", "\\right)", "$$", "There exists a canonical injective map of complexes", "$u_{M^\\bullet} : M^\\bullet \\to J^\\bullet(M^\\bullet)$ by mapping $M^n$ to", "$I(M^n) \\oplus I(M^{n + 1})$ via the maps $i_{M^n} : M^n \\to I(M^n)$ and", "$i_{M^{n + 1}} \\circ d : M^n \\to M^{n + 1} \\to I(M^{n + 1})$. Hence a", "short exact sequence of complexes", "$$", "0 \\to M^\\bullet \\xrightarrow{u_{M^\\bullet}}", "J^\\bullet(M^\\bullet) \\xrightarrow{v_{M^\\bullet}}", "Q^\\bullet(M^\\bullet) \\to 0", "$$", "functorial in $M^\\bullet$. Set", "$$", "\\mathbf{N}^\\bullet(M^\\bullet) = C(v_{M^\\bullet})^\\bullet[-1].", "$$", "Note that", "$$", "\\mathbf{N}^n(M^\\bullet) = Q^{n - 1}(M^\\bullet) \\oplus J^n(M^\\bullet)", "$$", "with differential", "$$", "\\left(", "\\begin{matrix}", "- d^{n - 1}_{Q^\\bullet(M^\\bullet)} & - v^n_{M^\\bullet} \\\\", "0 & d^n_{J^\\bullet(M)}", "\\end{matrix}", "\\right)", "$$", "Hence we see that there is a map of complexes", "$j_{M^\\bullet} : M^\\bullet \\to \\mathbf{N}^\\bullet(M^\\bullet)$", "induced by $u$. It is injective and factors through an injective subobject", "by construction. The map $j_{M^\\bullet}$ is a quasi-isomorphism as one", "can prove by looking at the long exact sequence of cohomology associated", "to the short exact sequences of complexes above." ], "refs": [ "injectives-theorem-injective-embedding-grothendieck" ], "ref_ids": [ 7767 ] } ], "ref_ids": [] }, { "id": 7793, "type": "theorem", "label": "injectives-lemma-grothendieck-brown", "categories": [ "injectives" ], "title": "injectives-lemma-grothendieck-brown", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category.", "Let $F : \\mathcal{A}^{opp} \\to \\textit{Sets}$ be a functor.", "Then $F$ is representable if and only if $F$ commutes with colimits, i.e.,", "$$", "F(\\colim_i N_i) = \\lim F(N_i)", "$$", "for any diagram $\\mathcal{I} \\to \\mathcal{A}$, $i \\in \\mathcal{I}$." ], "refs": [], "proofs": [ { "contents": [ "If $F$ is representable, then it commutes with colimits by definition", "of colimits.", "\\medskip\\noindent", "Assume that $F$ commutes with colimits. Then $F(M \\oplus N) = F(M) \\times F(N)$", "and we can use this to define a group structure on $F(M)$. Hence we get", "$F : \\mathcal{A} \\to \\textit{Ab}$ which is additive and right exact, i.e.,", "transforms a short exact sequence $0 \\to K \\to L \\to M \\to 0$ into an exact", "sequence $F(K) \\leftarrow F(L) \\leftarrow F(M) \\leftarrow 0$ (compare with", "Homology, Section \\ref{homology-section-functors}).", "\\medskip\\noindent", "Let $U$ be a generator for $\\mathcal{A}$. Set $A = \\bigoplus_{s \\in F(U)} U$.", "Let $s_{univ} = (s)_{s \\in F(U)} \\in F(A) = \\prod_{s \\in F(U)} F(U)$. Let", "$A' \\subset A$ be the largest subobject such that $s_{univ}$ restricts to", "zero on $A'$. This exists because $\\mathcal{A}$ is a Grothendieck category", "and because $F$ commutes with colimits. Because $F$ commutes with colimits", "there exists a unique element $\\overline{s}_{univ} \\in F(A/A')$ which", "maps to $s_{univ}$ in $F(A)$. We claim that $A/A'$ represents $F$, in", "other words, the Yoneda map", "$$", "\\overline{s}_{univ} : h_{A/A'} \\longrightarrow F", "$$", "is an isomorphism. Let $M \\in \\Ob(\\mathcal{A})$ and $s \\in F(M)$. Consider", "the surjection", "$$", "c_M :", "A_M = \\bigoplus\\nolimits_{\\varphi \\in \\Hom_\\mathcal{A}(U, M)} U", "\\longrightarrow", "M.", "$$", "This gives $F(c_M)(s) = (s_\\varphi) \\in \\prod_\\varphi F(U)$.", "Consider the map", "$$", "\\psi :", "A_M = \\bigoplus\\nolimits_{\\varphi \\in \\Hom_\\mathcal{A}(U, M)} U", "\\longrightarrow", "\\bigoplus\\nolimits_{s \\in F(U)} U = A", "$$", "which maps the summand corresponding to $\\varphi$ to the summand", "corresponding to $s_\\varphi$ by the identity map on $U$. Then $s_{univ}$", "maps to $(s_\\varphi)_\\varphi$ by construction.", "in other words the right square in the diagram", "$$", "\\xymatrix{", "A' \\ar[r] &", "A \\ar@{..>}[r]_{s_{univ}} & F \\\\", "K \\ar[r] \\ar[u]^{?} & A_M \\ar[u]^\\psi \\ar[r] &", "M \\ar@{..>}[u]_s", "}", "$$", "commutes. Let $K = \\Ker(A_M \\to M)$. Since $s$ restricts to zero", "on $K$ we see that $\\psi(K) \\subset A'$ by definition of $A'$. Hence there", "is an induced morphism $M \\to A/A'$. This construction gives an inverse", "to the map $h_{A/A'}(M) \\to F(M)$ (details omitted)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7794, "type": "theorem", "label": "injectives-lemma-grothendieck-products", "categories": [ "injectives" ], "title": "injectives-lemma-grothendieck-products", "contents": [ "A Grothendieck abelian category has Ab3*." ], "refs": [], "proofs": [ { "contents": [ "Let $M_i$, $i \\in I$ be a family of objects of $\\mathcal{A}$ indexed", "by a set $I$. The functor $F = \\prod_{i \\in I} h_{M_i}$", "commutes with colimits. Hence", "Lemma \\ref{lemma-grothendieck-brown}", "applies." ], "refs": [ "injectives-lemma-grothendieck-brown" ], "ref_ids": [ 7793 ] } ], "ref_ids": [] }, { "id": 7795, "type": "theorem", "label": "injectives-lemma-derived-products", "categories": [ "injectives" ], "title": "injectives-lemma-derived-products", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category.", "Then", "\\begin{enumerate}", "\\item $D(\\mathcal{A})$ has both direct sums and products,", "\\item direct sums are obtained by taking termwise direct sums of", "any complexes,", "\\item products are obtained by taking termwise products of", "K-injective complexes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $K^\\bullet_i$, $i \\in I$ be a family of objects of $D(\\mathcal{A})$", "indexed by a set $I$. We claim that the termwise direct sum", "$\\bigoplus_{i \\in I} K^\\bullet_i$ is a direct sum in $D(\\mathcal{A})$.", "Namely, let $I^\\bullet$ be a K-injective complex. Then we have", "\\begin{align*}", "\\Hom_{D(\\mathcal{A})}(\\bigoplus\\nolimits_{i \\in I} K^\\bullet_i, I^\\bullet)", "& =", "\\Hom_{K(\\mathcal{A})}(\\bigoplus\\nolimits_{i \\in I} K^\\bullet_i, I^\\bullet) \\\\", "& =", "\\prod\\nolimits_{i \\in I} \\Hom_{K(\\mathcal{A})}(K^\\bullet_i, I^\\bullet) \\\\", "& =", "\\prod\\nolimits_{i \\in I} \\Hom_{D(\\mathcal{A})}(K^\\bullet_i, I^\\bullet)", "\\end{align*}", "as desired. This is sufficient since any complex can be represented", "by a K-injective complex by", "Theorem \\ref{theorem-K-injective-embedding-grothendieck}.", "To construct the product, choose a K-injective resolution", "$K_i^\\bullet \\to I_i^\\bullet$ for each $i$. Then we claim that", "$\\prod_{i \\in I} I_i^\\bullet$ is a product in $D(\\mathcal{A})$.", "This follows from", "Derived Categories, Lemma \\ref{derived-lemma-product-K-injective}." ], "refs": [ "injectives-theorem-K-injective-embedding-grothendieck", "derived-lemma-product-K-injective" ], "ref_ids": [ 7768, 1911 ] } ], "ref_ids": [] }, { "id": 7796, "type": "theorem", "label": "injectives-lemma-RF-commutes-with-Rlim", "categories": [ "injectives" ], "title": "injectives-lemma-RF-commutes-with-Rlim", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor of", "abelian categories. Assume", "\\begin{enumerate}", "\\item $\\mathcal{A}$ is a Grothendieck abelian category,", "\\item $\\mathcal{B}$ has exact countable products, and", "\\item $F$ commutes with countable products.", "\\end{enumerate}", "Then", "$RF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ commutes with derived limits." ], "refs": [], "proofs": [ { "contents": [ "Observe that $RF$ exists as $\\mathcal{A}$ has enough K-injectives", "(Theorem \\ref{theorem-K-injective-embedding-grothendieck}", "and", "Derived Categories, Lemma \\ref{derived-lemma-K-injective-defined}).", "The statement means that if $K = R\\lim K_n$, then", "$RF(K) = R\\lim RF(K_n)$. See", "Derived Categories, Definition \\ref{derived-definition-derived-limit}", "for notation. Since $RF$ is an exact functor of triangulated", "categories it suffices to see that $RF$ commutes with countable", "products of objects of $D(\\mathcal{A})$. In the proof of", "Lemma \\ref{lemma-derived-products}", "we have seen that products in $D(\\mathcal{A})$ are computed by", "taking products of K-injective complexes and moreover that a", "product of K-injective complexes is K-injective.", "Moreover, in Derived Categories, Lemma", "\\ref{derived-lemma-products}", "we have seen that products in $D(\\mathcal{B})$ are computed", "by taking termwise products.", "Since $RF$ is computed by applying $F$ to a K-injective", "representative and since we've assumed $F$ commutes with", "countable products, the lemma follows." ], "refs": [ "injectives-theorem-K-injective-embedding-grothendieck", "derived-lemma-K-injective-defined", "derived-definition-derived-limit", "injectives-lemma-derived-products", "derived-lemma-products" ], "ref_ids": [ 7768, 1912, 2002, 7795, 1925 ] } ], "ref_ids": [] }, { "id": 7797, "type": "theorem", "label": "injectives-lemma-K-injective-embedding-filtration", "categories": [ "injectives" ], "title": "injectives-lemma-K-injective-embedding-filtration", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category.", "Let $K^\\bullet$ be a filtered complex of $\\mathcal{A}$, see", "Homology, Definition \\ref{homology-definition-filtered-complex}.", "Then there exists a morphism $j : K^\\bullet \\to J^\\bullet$", "of filtered complexes of $\\mathcal{A}$ such that", "\\begin{enumerate}", "\\item $J^n$, $F^pJ^n$, $J^n/F^pJ^n$ and $F^pJ^n/F^{p'}J^n$ are injective", "objects of $\\mathcal{A}$,", "\\item $J^\\bullet$, $F^pJ^\\bullet$, $J^\\bullet/F^pJ^\\bullet$, and", "$F^pJ^\\bullet/F^{p'}J^\\bullet$ are K-injective complexes,", "\\item $j$ induces quasi-isomorphisms", "$K^\\bullet \\to J^\\bullet$,", "$F^pK^\\bullet \\to F^pJ^\\bullet$,", "$K^\\bullet/F^pK^\\bullet \\to J^\\bullet/F^pJ^\\bullet$, and", "$F^pK^\\bullet/F^{p'}K^\\bullet \\to F^pJ^\\bullet/F^{p'}J^\\bullet$.", "\\end{enumerate}" ], "refs": [ "homology-definition-filtered-complex" ], "proofs": [ { "contents": [ "By Theorem \\ref{theorem-K-injective-embedding-grothendieck}", "we obtain quasi-isomorphisms $i : K^\\bullet \\to I^\\bullet$ and", "$i^p : F^pK^\\bullet \\to I^{p, \\bullet}$ as well as commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "K^\\bullet \\ar[d]_i & F^pK^\\bullet \\ar[l] \\ar[d]_{i^p} \\\\", "I^\\bullet & I^{p, \\bullet} \\ar[l]_{\\alpha^p}", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "F^{p'}K^\\bullet \\ar[d]_{i^{p'}} &", "F^pK^\\bullet \\ar[l] \\ar[d]_{i^p} \\\\", "I^{p', \\bullet} &", "I^{p, \\bullet} \\ar[l]_{\\alpha^{p p'}}", "}", "}", "\\quad\\text{for }p' \\leq p", "$$", "such that $\\alpha^p \\circ \\alpha^{p' p} = \\alpha^{p'}$", "and $\\alpha^{p'p''} \\circ \\alpha^{pp'} = \\alpha^{pp''}$.", "The problem is that the maps $\\alpha^p : I^{p, \\bullet} \\to I^\\bullet$", "need not be injective. For each $p$ we choose an injection", "$t^p : I^{p, \\bullet} \\to J^{p, \\bullet}$ into an acyclic K-injective", "complex $J^{p, \\bullet}$ whose terms are injective objects of $\\mathcal{A}$", "(first map to the cone on the identity and then use the theorem).", "Choose a map of complexes $s^p : I^\\bullet \\to J^{p, \\bullet}$", "such that the following diagram commutes", "$$", "\\xymatrix{", "K^\\bullet \\ar[d]_i & F^pK^\\bullet \\ar[l] \\ar[d]_{i^p} \\\\", "I^\\bullet \\ar[rd]_{s^p} & I^{p, \\bullet} \\ar[d]^{t^p} \\\\", "& J^{p, \\bullet}", "}", "$$", "This is possible: the composition $F^pK^\\bullet \\to J^{p, \\bullet}$", "is homotopic to zero because $J^{p, \\bullet}$ is acyclic and K-injective", "(Derived Categories, Lemma \\ref{derived-lemma-K-injective}).", "Since the objects $J^{p, n - 1}$ are injective and since", "$F^pK^n \\to K^n \\to I^n$ are injective morphisms, we", "can lift the maps $F^pK^n \\to J^{p, n - 1}$ giving the homotopy", "to a map $h^n : I^n \\to J^{p, n - 1}$. Then we set $s^p$", "equal to $h \\circ \\text{d} + \\text{d} \\circ h$.", "(Warning: It will not be the case that $t^p = s^p \\circ \\alpha^p$,", "so we have to be careful not to use this below.)", "\\medskip\\noindent", "Consider", "$$", "J^\\bullet = I^\\bullet \\times \\prod\\nolimits_p J^{p, \\bullet}", "$$", "Because products in $D(\\mathcal{A})$ are given by taking", "products of K-injective complexes", "(Lemma \\ref{lemma-derived-products})", "and since $J^{p, \\bullet}$", "is isomorphic to $0$ in $D(\\mathcal{A})$ we see that", "$J^\\bullet \\to I^\\bullet$ is an isomorphism in $D(\\mathcal{A})$.", "Consider the map", "$$", "j = i \\times (s^p \\circ i)_{p \\in \\mathbf{Z}} :", "K^\\bullet", "\\longrightarrow", "I^\\bullet \\times \\prod\\nolimits_p J^{p, \\bullet} = J^\\bullet", "$$", "By our remarks above this is a quasi-isomorphism. It is also injective.", "For $p \\in \\mathbf{Z}$ we let $F^pJ^\\bullet \\subset J^\\bullet$ be", "$$", "\\Im\\left(", "\\alpha^p \\times (t^{p'} \\circ \\alpha^{pp'})_{p' \\leq p} :", "I^{p, \\bullet}", "\\to", "I^\\bullet \\times \\prod\\nolimits_{p' \\leq p} J^{p', \\bullet}", "\\right)", "\\times \\prod\\nolimits_{p' > p} J^{p', \\bullet}", "$$", "This complex is isomorphic to the complex", "$I^{p, \\bullet} \\times \\prod_{p' > p} J^{p, \\bullet}$", "as $\\alpha^{pp} = \\text{id}$ and $t^p$ is injective.", "Hence $F^pJ^\\bullet$ is quasi-isomorphic to $I^{p, \\bullet}$ (argue", "as above). We have $j(F^pK^\\bullet) \\subset F^pJ^\\bullet$ because", "of the commutativity of the diagram above. The corresponding", "map of complexes $F^pK^\\bullet \\to F^pJ^\\bullet$ is a quasi-isomorphism", "by what we just said. Finally, to see that", "$F^{p + 1}J^\\bullet \\subset F^pJ^\\bullet$", "use that $\\alpha^{p + 1p} \\circ \\alpha^{pp'} = \\alpha^{p + 1p'}$", "and the commutativity of the first displayed diagram", "in the first paragraph of the proof.", "\\medskip\\noindent", "We claim that $j : K^\\bullet \\to J^\\bullet$ is a solution to the", "problem posed by the lemma. Namely, $F^pJ^n$ is an injective object", "of $\\mathcal{A}$ because it is isomorphic to", "$I^{p, n} \\times \\prod_{p' > p} J^{p', n}$ and products of", "injectives are injective. Then the injective map $F^pJ^n \\to J^n$", "splits and hence the quotient $J^n/F^pJ^n$ is injective as well", "as a direct summand of the injective object $J^n$.", "Similarly for $F^pJ^n/F^{p'}J^n$. This in particular means", "that $0 \\to F^pJ^\\bullet \\to J^\\bullet \\to J^\\bullet/F^pJ^\\bullet \\to 0$", "is a termwise split short exact sequence of complexes, hence defines", "a distinguished triangle in $K(\\mathcal{A})$ by fiat.", "Since $J^\\bullet$ and $F^pJ^\\bullet$ are K-injective complexes", "we see that the same is true for $J^\\bullet/F^pJ^\\bullet$", "by Derived Categories, Lemma \\ref{derived-lemma-triangle-K-injective}.", "A similar argument shows that $F^pJ^\\bullet/F^{p'}J^\\bullet$", "is K-injective. By construction $j : K^\\bullet \\to J^\\bullet$", "and the induced maps $F^pK^\\bullet \\to F^pJ^\\bullet$ are", "quasi-isomorphisms. Using the long exact cohomology sequences", "of the complexes in play we find that the same holds for", "$K^\\bullet/F^pK^\\bullet \\to J^\\bullet/F^pJ^\\bullet$ and", "$F^pK^\\bullet/F^{p'}K^\\bullet \\to F^pJ^\\bullet/F^{p'}J^\\bullet$." ], "refs": [ "injectives-theorem-K-injective-embedding-grothendieck", "derived-lemma-K-injective", "injectives-lemma-derived-products", "derived-lemma-triangle-K-injective" ], "ref_ids": [ 7768, 1908, 7795, 1909 ] } ], "ref_ids": [ 12177 ] }, { "id": 7798, "type": "theorem", "label": "injectives-lemma-represent-by-filtered-complex", "categories": [ "injectives" ], "title": "injectives-lemma-represent-by-filtered-complex", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category. Suppose given an object", "$E \\in D(\\mathcal{A})$ and an inverse system $\\{E^i\\}_{i \\in \\mathbf{Z}}$", "of objects of $D(\\mathcal{A})$ over $\\mathbf{Z}$ together with", "a compatible system of maps $E^i \\to E$. Picture:", "$$", "\\ldots \\to E^{i + 1} \\to E^i \\to E^{i - 1} \\to \\ldots \\to E", "$$", "Then there exists a filtered complex $K^\\bullet$ of $\\mathcal{A}$", "(Homology, Definition \\ref{homology-definition-filtered-complex})", "such that $K^\\bullet$ represents $E$", "and $F^iK^\\bullet$ represents $E^i$ compatibly with the given maps." ], "refs": [ "homology-definition-filtered-complex" ], "proofs": [ { "contents": [ "By Theorem \\ref{theorem-K-injective-embedding-grothendieck}", "we can choose a K-injective complex $I^\\bullet$", "representing $E$ all of whose terms $I^n$ are injective", "objects of $\\mathcal{A}$.", "Choose a complex $G^{0, \\bullet}$ representing $E^0$.", "Choose a map of complexes $\\varphi^0 : G^{0, \\bullet} \\to I^\\bullet$", "representing $E^0 \\to E$.", "For $i > 0$ we inductively represent $E^i \\to E^{i - 1}$", "by a map of complexes", "$\\delta : G^{i, \\bullet} \\to G^{i - 1, \\bullet}$", "and we set $\\varphi^i = \\delta \\circ \\varphi^{i - 1}$.", "For $i < 0$ we inductively represent $E^{i + 1} \\to E^i$", "by a termwise injective map of complexes", "$\\delta : G^{i + 1, \\bullet} \\to G^{i, \\bullet}$", "(for example you can use", "Derived Categories, Lemma \\ref{derived-lemma-make-injective}).", "Claim: we can find a map of complexes", "$\\varphi^i : G^{i, \\bullet} \\to I^\\bullet$", "representing the map $E^i \\to E$ and", "fitting into the commutative diagram", "$$", "\\xymatrix{", "G^{i + 1, \\bullet} \\ar[r]_\\delta \\ar[d]_{\\varphi^{i + 1}} &", "G^{i, \\bullet} \\ar[ld]^{\\varphi^i} \\\\", "I^\\bullet", "}", "$$", "Namely, we first choose any map of complexes", "$\\varphi : G^{i, \\bullet} \\to I^\\bullet$", "representing the map", "$E^i \\to E$. Then we see that $\\varphi \\circ \\delta$", "and $\\varphi^{i + 1}$ are homotopic by some homotopy", "$h^p : G^{i + 1, p} \\to I^{p - 1}$.", "Since the terms of", "$I^\\bullet$ are injective and since $\\delta$", "is termwise injective, we can lift $h^p$ to", "$(h')^p : G^{i, p} \\to I^{p - 1}$.", "Then we set $\\varphi^i = \\varphi + h' \\circ d + d \\circ h'$", "and we get what we claimed.", "\\medskip\\noindent", "Next, we choose for every $i$ a termwise injective map of complexes", "$a^i : G^{i, \\bullet} \\to J^{i, \\bullet}$ with $J^{i, \\bullet}$", "acyclic, K-injective, with $J^{i, p}$ injective objects of $\\mathcal{A}$.", "To do this first map $G^{i, \\bullet}$ to the cone on the identity", "and then apply the theorem cited above.", "Arguing as above we can find maps of complexes", "$\\delta' : J^{i, \\bullet} \\to J^{i - 1, \\bullet}$ such that the diagrams", "$$", "\\xymatrix{", "G^{i, \\bullet} \\ar[r]_\\delta \\ar[d]_{a^i} &", "G^{i - 1, \\bullet} \\ar[d]^{a^{i - 1}} \\\\", "J^{i, \\bullet} \\ar[r]^{\\delta'} &", "J^{i - 1, \\bullet}", "}", "$$", "commute. (You could also use the functoriality of cones plus the", "functoriality in the theorem to get this.)", "Then we consider the maps", "$$", "\\xymatrix{", "G^{i + 1, \\bullet} \\times \\prod\\nolimits_{p > i + 1} J^{p, \\bullet}", "\\ar[r] \\ar[rd] &", "G^{i, \\bullet} \\times \\prod\\nolimits_{p > i} J^{p, \\bullet}", "\\ar[r] \\ar[d] &", "G^{i - 1, \\bullet} \\times \\prod\\nolimits_{p > i - 1} J^{p, \\bullet}", "\\ar[ld] \\\\", "& I^\\bullet \\times \\prod\\nolimits_p J^{p, \\bullet}", "}", "$$", "Here the arrows on $J^{p, \\bullet}$ are the obvious ones", "(identity or zero). On the factor $G^{i, \\bullet}$ we use", "$\\delta : G^{i, \\bullet} \\to G^{i - 1, \\bullet}$, the map", "$\\varphi^i : G^{i, \\bullet} \\to I^\\bullet$, the zero map", "$0 : G^{i, \\bullet} \\to J^{p, \\bullet}$ for $p > i$, the map", "$a^i : G^{i, \\bullet} \\to J^{p, \\bullet}$ for $p = i$, and", "$(\\delta')^{i - p} \\circ a^i = a^p \\circ \\delta^{i - p} :", "G^{i, \\bullet} \\to J^{p, \\bullet}$ for $p < i$.", "We omit the verification that all the arrows", "in the diagram are termwise injective. Thus we obtain a filtered", "complex. Because products in $D(\\mathcal{A})$ are given by", "taking products of K-injective complexes", "(Lemma \\ref{lemma-derived-products})", "and because $J^{p, \\bullet}$ is zero in $D(\\mathcal{A})$", "we conclude this diagram represents the given diagram in the derived", "category. This finishes the proof." ], "refs": [ "injectives-theorem-K-injective-embedding-grothendieck", "derived-lemma-make-injective", "injectives-lemma-derived-products" ], "ref_ids": [ 7768, 1797, 7795 ] } ], "ref_ids": [ 12177 ] }, { "id": 7799, "type": "theorem", "label": "injectives-lemma-represent-by-filtered-complex-bis", "categories": [ "injectives" ], "title": "injectives-lemma-represent-by-filtered-complex-bis", "contents": [ "In the situation of Lemma \\ref{lemma-represent-by-filtered-complex}", "assume we have a second inverse system $\\{(E')^i\\}_{i \\in \\mathbf{Z}}$", "and a compatible system of maps $(E')^i \\to E$.", "Then there exists a bi-filtered complex $K^\\bullet$ of $\\mathcal{A}$", "such that $K^\\bullet$ represents $E$, $F^iK^\\bullet$ represents $E^i$,", "and $(F')^iK^\\bullet$ represents $(E')^i$ compatibly with the given maps." ], "refs": [ "injectives-lemma-represent-by-filtered-complex" ], "proofs": [ { "contents": [ "Using the lemma we can first choose $K^\\bullet$ and $F$.", "Then we can choose $(K')^\\bullet$ and $F'$ which work for", "$\\{(E')^i\\}_{i \\in \\mathbf{Z}}$ and the maps $(E')^i \\to E$.", "Using Lemma \\ref{lemma-K-injective-embedding-filtration}", "we can assume $K^\\bullet$ is a K-injective complex.", "Then we can choose a map of complexes", "$(K')^\\bullet \\to K^\\bullet$ corresponding to", "the given identifications", "$(K')^\\bullet \\cong E \\cong K^\\bullet$.", "We can additionally choose a termwise injective", "map $(K')^\\bullet \\to J^\\bullet$ with", "$J^\\bullet$ acyclic and K-injective.", "(To do this first map $(K')^\\bullet$ to the cone on the identity", "and then apply Theorem \\ref{theorem-K-injective-embedding-grothendieck}.)", "Then $(K')^\\bullet \\to K^\\bullet \\times J^\\bullet$ and", "$K^\\bullet \\to K^\\bullet \\times J^\\bullet$", "are both termwise injective and quasi-isomorphisms", "(as the product represents $E$ by Lemma \\ref{lemma-derived-products}).", "Then we can simply take the images of the filtrations", "on $K^\\bullet$ and $(K')^\\bullet$ under these maps to conclude." ], "refs": [ "injectives-lemma-K-injective-embedding-filtration", "injectives-theorem-K-injective-embedding-grothendieck", "injectives-lemma-derived-products" ], "ref_ids": [ 7797, 7768, 7795 ] } ], "ref_ids": [ 7798 ] }, { "id": 7800, "type": "theorem", "label": "injectives-lemma-gabriel-popescu-left-adjoint", "categories": [ "injectives" ], "title": "injectives-lemma-gabriel-popescu-left-adjoint", "contents": [ "The functor $G$ above has a left adjoint", "$F : \\text{Mod}_R \\to \\mathcal{A}$." ], "refs": [], "proofs": [ { "contents": [ "We will give two proofs of this lemma.", "\\medskip\\noindent", "The first proof will use the adjoint functor theorem, see", "Categories, Theorem \\ref{categories-theorem-adjoint-functor}.", "Observe that that $G : \\mathcal{A} \\to \\text{Mod}_R$ is left exact and sends", "products to products. Hence $G$ commutes with limits. To check the set", "theoretical condition in the theorem, suppose that $M$ is an object of", "$\\text{Mod}_R$. Choose a suitably large cardinal $\\kappa$ and denote $E$", "a set of objects of $\\mathcal{A}$ such that every object $A$ with", "$|A| \\leq \\kappa$ is isomorphic to an element of $E$. This is possible", "by Lemma \\ref{lemma-set-iso-classes-bounded-size}. Set", "$I = \\coprod_{A \\in E} \\text{Hom}_R(M, G(A))$.", "We think of an element $i \\in I$ as a pair $(A_i, f_i)$.", "Finally, let $A$ be an arbitrary object of $\\mathcal{A}$", "and $f : M \\to G(A)$ arbitrary. We are going to think of", "elements of $\\Im(f) \\subset G(A) = \\Hom_\\mathcal{A}(U, A)$", "as maps $u : U \\to A$. Set", "$$", "A' = \\Im(\\bigoplus\\nolimits_{u \\in \\Im(f)} U \\xrightarrow{u} A)", "$$", "Since $G$ is left exact, we see that $G(A') \\subset G(A)$", "contains $\\Im(f)$ and we get $f' : M \\to G(A')$ factoring $f$.", "On the other hand, the object $A'$ is", "the quotient of a direct sum of at most $|M|$ copies of $U$.", "Hence if $\\kappa = |\\bigoplus_{|M|} U|$, then we see that $(A', f')$", "is isomorphic to an element $(A_i, f_i)$ of $E$ and we conclude that $f$", "factors as $M \\xrightarrow{f_i} G(A_i) \\to G(A)$ as desired.", "\\medskip\\noindent", "The second proof will give a construction of $F$ which will show", "that ``$F(M) = M \\otimes_R U$'' in some sense. Namely, for any", "$R$-module $M$ we can choose a resolution", "$$", "\\bigoplus\\nolimits_{j \\in J} R \\to", "\\bigoplus\\nolimits_{i \\in I} R \\to", "M \\to 0", "$$", "Then we define $F(M)$ by the corresponding exact sequence", "$$", "\\bigoplus\\nolimits_{j \\in J} U \\to", "\\bigoplus\\nolimits_{i \\in I} U \\to", "F(M) \\to 0", "$$", "This construction is independent of the choice of the resolution", "and is functorial; we omit the details.", "For any $A$ in $\\mathcal{A}$ we obtain an exact sequence", "$$", "0 \\to \\Hom_\\mathcal{A}(F(M), A) \\to", "\\prod\\nolimits_{i \\in I} G(A) \\to", "\\prod\\nolimits_{j \\in J} G(A)", "$$", "which is isomorphic to the sequence", "$$", "0 \\to \\Hom_R(M, A) \\to", "\\Hom_R(\\bigoplus\\nolimits_{i \\in I} R, G(A)) \\to", "\\Hom_R(\\bigoplus\\nolimits_{j \\in J} R, G(A))", "$$", "which shows that $F$ is the left adjoint to $G$." ], "refs": [ "categories-theorem-adjoint-functor", "injectives-lemma-set-iso-classes-bounded-size" ], "ref_ids": [ 12200, 7786 ] } ], "ref_ids": [] }, { "id": 7801, "type": "theorem", "label": "injectives-lemma-F-G-monos", "categories": [ "injectives" ], "title": "injectives-lemma-F-G-monos", "contents": [ "Let $f : M \\to G(A)$ be an injective map in $\\text{Mod}_R$.", "Then the adjoint map $f' : F(M) \\to A$ is injective too." ], "refs": [], "proofs": [ { "contents": [ "Choose a map $R^{\\oplus n} \\to M$ and consider the corresponding map", "$U^{\\oplus n} \\to F(M)$. Consider a map $v : U \\to U^{\\oplus n}$", "such that the composition $U \\to U^{\\oplus n} \\to F(M) \\to A$.", "Then this arrow $v : U \\to U^{\\oplus n}$ is an element", "$v$ of $R^{\\oplus n}$ mapping to zero in $G(A)$. Since $f$ is injective,", "we conclude that $v$ maps to zero in $M$ which means that", "$U \\to U^{\\oplus n} \\to F(M)$ is zero by construction of $F(M)$", "in the proof of Lemma \\ref{lemma-gabriel-popescu-left-adjoint}.", "Since $U$ is a generator we conclude that", "$$", "\\Ker(U^{\\oplus n} \\to F(M) \\to A) = \\Ker(U^{\\oplus n} \\to F(M))", "$$", "To finish the proof we choose a surjection $\\bigoplus_{i \\in I} R \\to M$", "and we consider the corresponding surjection", "$$", "\\pi : \\bigoplus\\nolimits_{i \\in I} U \\longrightarrow F(M)", "$$", "To prove $f'$ is injective it suffices to show that", "$\\Ker(\\pi) = \\Ker(f' \\circ \\pi)$ as subobjects of $\\bigoplus_{i \\in I} U$.", "However, now we can write $\\bigoplus_{i \\in I} U$ as the filtered colimit", "of its subobjects $\\bigoplus_{i \\in I'} U$ where $I' \\subset I$", "ranges over the finite subsets. Since filtered colimits are", "exact by AB5 for $\\mathcal{A}$, we see that", "$$", "\\Ker(\\pi) =", "\\colim_{I' \\subset I\\text{ finite}}", "\\left(\\bigoplus\\nolimits_{i \\in I'} U\\right)", "\\bigcap \\Ker(\\pi)", "$$", "and", "$$", "\\Ker(f' \\circ \\pi) =", "\\colim_{I' \\subset I\\text{ finite}}", "\\left(\\bigoplus\\nolimits_{i \\in I'} U\\right)", "\\bigcap \\Ker(f' \\circ \\pi)", "$$", "and we get equality because the same is true for each $I'$ by", "the first displayed equality above." ], "refs": [ "injectives-lemma-gabriel-popescu-left-adjoint" ], "ref_ids": [ 7800 ] } ], "ref_ids": [] }, { "id": 7802, "type": "theorem", "label": "injectives-lemma-gabriel-popescu", "categories": [ "injectives" ], "title": "injectives-lemma-gabriel-popescu", "contents": [ "\\begin{reference}", "\\cite[Corollary 4.1]{serpe}", "\\end{reference}", "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let", "$R$, $F$, $G$ be as in the Gabriel-Popescu theorem", "(Theorem \\ref{theorem-gabriel-popescu}). Then we obtain", "derived functors", "$$", "RG : D(\\mathcal{A}) \\to D(\\text{Mod}_R)", "\\quad\\text{and}\\quad", "F : D(\\text{Mod}_R) \\to D(\\mathcal{A})", "$$", "such that $F$ is left adjoint to $RG$, $RG$ is fully faithful,", "and $F \\circ RG = \\text{id}$." ], "refs": [ "injectives-theorem-gabriel-popescu" ], "proofs": [ { "contents": [ "The existence and adjointness of the functors follows from", "Theorems \\ref{theorem-gabriel-popescu} and", "\\ref{theorem-K-injective-embedding-grothendieck}", "and", "Derived Categories, Lemmas \\ref{derived-lemma-K-injective-defined},", "\\ref{derived-lemma-right-derived-exact-functor}, and", "\\ref{derived-lemma-derived-adjoint-functors}.", "The statement $F \\circ RG = \\text{id}$ follows because we can", "compute $RG$ on an object of $D(\\mathcal{A})$ by applying $G$", "to a suitable representative complex $I^\\bullet$ (for example", "a K-injective one) and then $F(G(I^\\bullet)) = I^\\bullet$", "because $F \\circ G = \\text{id}$. Fully faithfulness of $RG$", "follows from this by", "Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}." ], "refs": [ "injectives-theorem-gabriel-popescu", "injectives-theorem-K-injective-embedding-grothendieck", "derived-lemma-K-injective-defined", "derived-lemma-right-derived-exact-functor", "derived-lemma-derived-adjoint-functors", "categories-lemma-adjoint-fully-faithful" ], "ref_ids": [ 7769, 7768, 1912, 1845, 1907, 12248 ] } ], "ref_ids": [ 7769 ] }, { "id": 7803, "type": "theorem", "label": "injectives-lemma-brown", "categories": [ "injectives" ], "title": "injectives-lemma-brown", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category.", "Let $H : D(\\mathcal{A}) \\to \\textit{Ab}$ be a contravariant", "cohomological functor which transforms direct sums into products.", "Then $H$ is representable." ], "refs": [], "proofs": [ { "contents": [ "Let $R, F, G, RG$ be as in Lemma \\ref{lemma-gabriel-popescu}", "and consider the functor $H \\circ F : D(\\text{Mod}_R) \\to \\textit{Ab}$.", "Observe that since $F$ is a left adjoint it sends direct sums to", "direct sums and hence $H \\circ F$ transforms direct sums into products.", "On the other hand, the derived category $D(\\text{Mod}_R)$ is", "generated by a single compact object, namely $R$.", "By Derived Categories, Lemma \\ref{derived-lemma-brown}", "we see that $H \\circ F$ is representable, say by $L \\in D(\\text{Mod}_R)$.", "Choose a distinguished triangle", "$$", "M \\to L \\to RG(F(L)) \\to M[1]", "$$", "in $D(\\text{Mod}_R)$. Then $F(M) = 0$ because $F \\circ RG = \\text{id}$.", "Hence $H(F(M)) = 0$ hence $\\Hom(M, L) = 0$.", "It follows that $L \\to RG(F(L))$ is the inclusion of a direct summand, see", "Derived Categories, Lemma \\ref{derived-lemma-split}.", "For $A$ in $D(\\mathcal{A})$ we obtain", "\\begin{align*}", "H(A)", "& =", "H(F(RG(A)) \\\\", "& =", "\\Hom(RG(A), L) \\\\", "& \\to", "\\Hom(RG(A), RG(F(L))) \\\\", "& =", "\\Hom(F(RG(A)), F(L)) \\\\", "& =", "\\Hom(A, F(L))", "\\end{align*}", "where the arrow has a left inverse functorial in $A$. In other words, we find", "that $H$ is the direct summand of a representable functor.", "Since $D(\\mathcal{A})$ is Karoubian", "(Derived Categories, Lemma", "\\ref{derived-lemma-projectors-have-images-triangulated}) we conclude." ], "refs": [ "injectives-lemma-gabriel-popescu", "derived-lemma-brown", "derived-lemma-split", "derived-lemma-projectors-have-images-triangulated" ], "ref_ids": [ 7802, 1943, 1766, 1769 ] } ], "ref_ids": [] }, { "id": 7804, "type": "theorem", "label": "injectives-proposition-modules-are-small", "categories": [ "injectives" ], "title": "injectives-proposition-modules-are-small", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Let $\\kappa$ the cardinality of the set of submodules of $M$.", "If $\\alpha$ is an ordinal whose cofinality is bigger than $\\kappa$,", "then $M$ is $\\alpha$-small with respect to injections." ], "refs": [], "proofs": [ { "contents": [ "The proof is straightforward, but let us first think about a special case.", "If $M$ is finite, then the claim is that for any inductive system", "$\\{B_\\beta\\}$ with injections between them, parametrized by a", "limit ordinal, any map $M \\to \\colim B_\\beta$ factors through one of", "the $B_\\beta$. And this we proved in", "Lemma \\ref{lemma-out-of-finite}.", "\\medskip\\noindent", "Now we start the proof in the general case.", "We need only show that the map (\\ref{equation-compare}) is a surjection.", "Let $f : M \\to \\colim B_\\beta$ be a map.", "Consider the subobjects $\\{f^{-1}(B_\\beta)\\}$ of $M$, where $B_\\beta$", "is considered as a subobject of the colimit $B = \\bigcup_\\beta B_\\beta$.", "If one of these, say $f^{-1}(B_\\beta)$, fills $M$,", "then the map factors through $B_\\beta$.", "\\medskip\\noindent", "So suppose to the contrary that all of the $f^{-1}(B_\\beta)$ were proper", "subobjects of $M$. However, we know that", "$$", "\\bigcup f^{-1}(B_\\beta) = f^{-1}\\left(\\bigcup B_\\beta\\right) = M.", "$$", "Now there are at most $\\kappa$ different subobjects of $M$ that occur among", "the $f^{-1}(B_\\alpha)$, by hypothesis.", "Thus we can find a subset $S \\subset \\alpha$ of cardinality at most", "$\\kappa$ such that as $\\beta'$ ranges over $S$, the", "$f^{-1}(B_{\\beta'})$ range over \\emph{all} the $f^{-1}(B_\\alpha)$.", "\\medskip\\noindent", "However, $S$ has an upper bound $\\widetilde{\\alpha} < \\alpha$ as", "$\\alpha$ has cofinality bigger than $\\kappa$. In particular, all the", "$f^{-1}(B_{\\beta'})$, $\\beta' \\in S$ are contained in", "$f^{-1}(B_{\\widetilde{\\alpha}})$.", "It follows that $f^{-1}(B_{\\widetilde{\\alpha}}) = M$.", "In particular, the map $f$ factors through $B_{\\widetilde{\\alpha}}$." ], "refs": [ "injectives-lemma-out-of-finite" ], "ref_ids": [ 7770 ] } ], "ref_ids": [] }, { "id": 7805, "type": "theorem", "label": "injectives-proposition-presheaves-injectives", "categories": [ "injectives" ], "title": "injectives-proposition-presheaves-injectives", "contents": [ "For abelian presheaves on a category there is a functorial injective", "embedding." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7806, "type": "theorem", "label": "injectives-proposition-presheaves-modules", "categories": [ "injectives" ], "title": "injectives-proposition-presheaves-modules", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$.", "The category $\\textit{PMod}(\\mathcal{O})$ of presheaves of", "$\\mathcal{O}$-modules has functorial injective embeddings." ], "refs": [], "proofs": [ { "contents": [ "We could prove this along the lines of the discussion in", "Section \\ref{section-injectives-presheaves}. But instead we argue using the", "theorem above. Endow $\\mathcal{C}$ with the structure of a site by letting the", "set of coverings of an object $U$ consist of all singletons $\\{f : V \\to U\\}$", "where $f$ is an isomorphism. We omit the verification that this defines a site.", "A sheaf for this topology is the same as a presheaf (proof omitted). Hence the", "theorem applies." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7807, "type": "theorem", "label": "injectives-proposition-objects-are-small", "categories": [ "injectives" ], "title": "injectives-proposition-objects-are-small", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $M$ be an", "object of $\\mathcal{A}$. Let $\\kappa = |M|$.", "If $\\alpha$ is an ordinal whose cofinality is bigger than $\\kappa$,", "then $M$ is $\\alpha$-small with respect to injections." ], "refs": [], "proofs": [ { "contents": [ "Please compare with Proposition \\ref{proposition-modules-are-small}.", "We need only show that the map (\\ref{equation-compare}) is a surjection.", "Let $f : M \\to \\colim B_\\beta$ be a map.", "Consider the subobjects $\\{f^{-1}(B_\\beta)\\}$ of $M$, where $B_\\beta$", "is considered as a subobject of the colimit $B = \\bigcup_\\beta B_\\beta$.", "If one of these, say $f^{-1}(B_\\beta)$, fills $M$,", "then the map factors through $B_\\beta$.", "\\medskip\\noindent", "So suppose to the contrary that all of the $f^{-1}(B_\\beta)$ were proper", "subobjects of $M$. However, because $\\mathcal{A}$ has", "AB5 we have", "$$", "\\colim f^{-1}(B_\\beta) = f^{-1}\\left(\\colim B_\\beta\\right) = M.", "$$", "Now there are at most $\\kappa$ different subobjects of $M$ that occur among", "the $f^{-1}(B_\\alpha)$, by hypothesis.", "Thus we can find a subset $S \\subset \\alpha$ of cardinality at most", "$\\kappa$ such that as $\\beta'$ ranges over $S$, the", "$f^{-1}(B_{\\beta'})$ range over \\emph{all} the $f^{-1}(B_\\alpha)$.", "\\medskip\\noindent", "However, $S$ has an upper bound $\\widetilde{\\alpha} < \\alpha$ as", "$\\alpha$ has cofinality bigger than $\\kappa$. In particular, all the", "$f^{-1}(B_{\\beta'})$, $\\beta' \\in S$ are contained in", "$f^{-1}(B_{\\widetilde{\\alpha}})$.", "It follows that $f^{-1}(B_{\\widetilde{\\alpha}}) = M$.", "In particular, the map $f$ factors through $B_{\\widetilde{\\alpha}}$." ], "refs": [ "injectives-proposition-modules-are-small" ], "ref_ids": [ 7804 ] } ], "ref_ids": [] }, { "id": 7808, "type": "theorem", "label": "injectives-proposition-brown", "categories": [ "injectives" ], "title": "injectives-proposition-brown", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $\\mathcal{D}$", "be a triangulated category. Let $F : D(\\mathcal{A}) \\to \\mathcal{D}$ be an", "exact functor of triangulated categories which transforms direct sums", "into direct sums. Then $F$ has an exact right adjoint." ], "refs": [], "proofs": [ { "contents": [ "For an object $Y$ of $\\mathcal{D}$ consider the contravariant functor", "$$", "D(\\mathcal{A}) \\to \\textit{Ab},\\quad", "W \\mapsto \\Hom_\\mathcal{D}(F(W), Y)", "$$", "This is a cohomological functor as $F$ is exact and transforms direct sums", "into products as $F$ transforms direct sums into direct sums. Thus by", "Lemma \\ref{lemma-brown} we find an object $X$ of $D(\\mathcal{A})$ such that", "$\\Hom_{D(\\mathcal{A})}(W, X) = \\Hom_\\mathcal{D}(F(W), Y)$.", "The existence of the adjoint follows from", "Categories, Lemma \\ref{categories-lemma-adjoint-exists}.", "Exactness follows from", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-is-exact}." ], "refs": [ "injectives-lemma-brown", "categories-lemma-adjoint-exists", "derived-lemma-adjoint-is-exact" ], "ref_ids": [ 7803, 12246, 1792 ] } ], "ref_ids": [] }, { "id": 7822, "type": "theorem", "label": "brauer-theorem-wedderburn", "categories": [ "brauer" ], "title": "brauer-theorem-wedderburn", "contents": [ "\\begin{slogan}", "Simple finite algebras over a field are matrix algebras over a skew field.", "\\end{slogan}", "Let $A$ be a simple finite $k$-algebra. Then $A$ is a matrix algebra over", "a finite $k$-algebra $K$ which is a skew field." ], "refs": [], "proofs": [ { "contents": [ "We may choose a simple submodule $M \\subset A$ and then", "the $k$-algebra $K = \\text{End}_A(M)$ is a skew field, see", "Lemma \\ref{lemma-simple-module}.", "By", "Lemma \\ref{lemma-rieffel}", "we see that $A = \\text{End}_K(M)$. Since $K$ is a skew field and", "$M$ is finitely generated (since $\\dim_k(M) < \\infty$) we see that", "$M$ is finite free as a left $K$-module. It follows immediately that", "$A \\cong \\text{Mat}(n \\times n, K^{op})$." ], "refs": [ "brauer-lemma-simple-module", "brauer-lemma-rieffel" ], "ref_ids": [ 7827, 7826 ] } ], "ref_ids": [] }, { "id": 7823, "type": "theorem", "label": "brauer-theorem-skolem-noether", "categories": [ "brauer" ], "title": "brauer-theorem-skolem-noether", "contents": [ "Let $A$ be a finite central simple $k$-algebra. Let $B$ be a simple", "$k$-algebra. Let $f, g : B \\to A$ be two $k$-algebra homomorphisms.", "Then there exists an invertible element $x \\in A$ such that", "$f(b) = xg(b)x^{-1}$ for all $b \\in B$." ], "refs": [], "proofs": [ { "contents": [ "Choose a simple $A$-module $M$. Set $L = \\text{End}_A(M)$.", "Then $L$ is a skew field with center $k$ which acts on the left on $M$, see", "Lemmas \\ref{lemma-simple-module} and \\ref{lemma-simple-module-unique}.", "Then $M$ has two $B \\otimes_k L^{op}$-module structures defined by", "$m \\cdot_1 (b \\otimes l) = lmf(b)$ and $m \\cdot_2 (b \\otimes l) = lmg(b)$.", "The $k$-algebra $B \\otimes_k L^{op}$ is simple by", "Lemma \\ref{lemma-tensor-simple}. Since $B$ is simple, the existence of a", "$k$-algebra homomorphism $B \\to A$ implies that $B$ is finite. Thus", "$B \\otimes_k L^{op}$ is finite simple and we conclude the two", "$B \\otimes_k L^{op}$-module structures on $M$", "are isomorphic by Lemma \\ref{lemma-simple-module-unique}.", "Hence we find $\\varphi : M \\to M$ intertwining these operations.", "In particular $\\varphi$ is in the commutant of $L$ which implies that", "$\\varphi$ is multiplication by some $x \\in A$, see", "Lemma \\ref{lemma-simple-module-unique}. Working out the definitions we see", "that $x$ is a solution to our problem." ], "refs": [ "brauer-lemma-simple-module", "brauer-lemma-simple-module-unique", "brauer-lemma-tensor-simple", "brauer-lemma-simple-module-unique", "brauer-lemma-simple-module-unique" ], "ref_ids": [ 7827, 7833, 7834, 7833, 7833 ] } ], "ref_ids": [] }, { "id": 7824, "type": "theorem", "label": "brauer-theorem-centralizer", "categories": [ "brauer" ], "title": "brauer-theorem-centralizer", "contents": [ "Let $A$ be a finite central simple algebra over $k$, and let", "$B$ be a simple subalgebra of $A$. Then", "\\begin{enumerate}", "\\item the centralizer $C$ of $B$ in $A$ is simple,", "\\item $[A : k] = [B : k][C : k]$, and", "\\item the centralizer of $C$ in $A$ is $B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Throughout this proof we use the results of", "Lemma \\ref{lemma-simple-module-unique} freely.", "Choose a simple $A$-module $M$. Set $L = \\text{End}_A(M)$.", "Then $L$ is a skew field with center $k$ which acts on the left on $M$", "and $A = \\text{End}_L(M)$.", "Then $M$ is a right $B \\otimes_k L^{op}$-module and", "$C = \\text{End}_{B \\otimes_k L^{op}}(M)$.", "Since the algebra $B \\otimes_k L^{op}$ is simple by", "Lemma \\ref{lemma-tensor-simple} we see that $C$ is simple (by", "Lemma \\ref{lemma-simple-module-unique} again).", "\\medskip\\noindent", "Write $B \\otimes_k L^{op} = \\text{Mat}(m \\times m, K)$ for some", "skew field $K$ finite over $k$. Then $C = \\text{Mat}(n \\times n, K^{op})$", "if $M$ is isomorphic to a direct sum of $n$ copies of the simple", "$B \\otimes_k L^{op}$-module $K^{\\oplus m}$ (the lemma again). Thus we have", "$\\dim_k(M) = nm [K : k]$, $[B : k] [L : k] = m^2 [K : k]$,", "$[C : k] = n^2 [K : k]$, and $[A : k] [L : k] = \\dim_k(M)^2$ (by", "the lemma again). We conclude that (2) holds.", "\\medskip\\noindent", "Part (3) follows because of (2) applied to $C \\subset A$ shows", "that $[B : k] = [C' : k]$ where $C'$ is the centralizer of $C$ in $A$", "(and the obvious fact that $B \\subset C')$." ], "refs": [ "brauer-lemma-simple-module-unique", "brauer-lemma-tensor-simple", "brauer-lemma-simple-module-unique" ], "ref_ids": [ 7833, 7834, 7833 ] } ], "ref_ids": [] }, { "id": 7825, "type": "theorem", "label": "brauer-theorem-splitting", "categories": [ "brauer" ], "title": "brauer-theorem-splitting", "contents": [ "Let $A$ be a finite central simple $k$-algebra.", "Let $k \\subset k'$ be a finite field extension.", "The following are equivalent", "\\begin{enumerate}", "\\item $k'$ splits $A$, and", "\\item there exists a finite central simple algebra $B$ similar to $A$", "such that $k' \\subset B$ and $[B : k] = [k' : k]^2$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (2). It suffices to show that $B \\otimes_k k'$ is a matrix", "algebra. We know that $B \\otimes_k B^{op} \\cong \\text{End}_k(B)$.", "Since $k'$ is the centralizer of $k'$ in $B^{op}$ by", "Lemma \\ref{lemma-self-centralizing-subfield}", "we see that $B \\otimes_k k'$ is the centralizer of $k \\otimes k'$", "in $B \\otimes_k B^{op} = \\text{End}_k(B)$. Of course this centralizer", "is just $\\text{End}_{k'}(B)$ where we view $B$ as a $k'$ vector space", "via the embedding $k' \\to B$. Thus the result.", "\\medskip\\noindent", "Assume (1). This means that we have an isomorphism", "$A \\otimes_k k' \\cong \\text{End}_{k'}(V)$ for some $k'$-vector space $V$.", "Let $B$ be the commutant of $A$ in $\\text{End}_k(V)$. Note that", "$k'$ sits in $B$. By", "Lemma \\ref{lemma-when-tensor-is-equal}", "the classes of $A$ and $B$ add up to zero in $\\text{Br}(k)$.", "From the dimension formula in", "Theorem \\ref{theorem-centralizer}", "we see that", "$$", "[B : k] [A : k] =", "\\dim_k(V)^2 =", "[k' : k]^2 \\dim_{k'}(V)^2 =", "[k' : k]^2 [A : k].", "$$", "Hence $[B : k] = [k' : k]^2$. Thus we have proved the result for the", "opposite to the Brauer class of $A$. However, $k'$ splits the Brauer", "class of $A$ if and only if it splits", "the Brauer class of the opposite algebra, so we win anyway." ], "refs": [ "brauer-lemma-self-centralizing-subfield", "brauer-lemma-when-tensor-is-equal", "brauer-theorem-centralizer" ], "ref_ids": [ 7843, 7842, 7824 ] } ], "ref_ids": [] }, { "id": 7826, "type": "theorem", "label": "brauer-lemma-rieffel", "categories": [ "brauer" ], "title": "brauer-lemma-rieffel", "contents": [ "Let $A$ be a possibly noncommutative ring with $1$ which contains no", "nontrivial two-sided ideal. Let $M$ be a nonzero right ideal in $A$,", "and view $M$ as a right $A$-module. Then $A$ coincides with the", "bicommutant of $M$." ], "refs": [], "proofs": [ { "contents": [ "Let $A' = \\text{End}_A(M)$, so $M$ is a left $A'$-module.", "Set $A'' = \\text{End}_{A'}(M)$ (the bicommutant of $M$).", "We view $A''$ as an algebra so that $M$ is a right $A''$-module\\footnote{This", "means that given $a'' \\in A''$ and $m \\in M$ we have a product", "$m a'' \\in M$. In particular, the multiplication in $A''$", "is the opposite of what you'd get if you wrote elements of $A''$", "as endomorphisms acting on the left.}.", "Let $R : A \\to A''$ be the natural homomorphism such that", "$mR(a) = ma$. Then $R$ is injective, since $R(1) = \\text{id}_M$", "and $A$ contains no nontrivial two-sided ideal. We claim that $R(M)$", "is a right ideal in $A''$. Namely, $R(m)a'' = R(ma'')$ for $a'' \\in A''$", "and $m$ in $M$, because {\\it left} multiplication of $M$ by any element $n$", "of $M$ represents an element of $A'$, and so", "$(nm)a'' = n(ma'')$ for all $n$ in $M$.", "Finally, the product ideal $AM$ is a two-sided ideal, and so", "$A = AM$. Thus $R(A) = R(A)R(M)$, so that $R(A)$ is a right ideal in $A''$.", "But $R(A)$ contains the identity element of $A''$, and so $R(A) = A''$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7827, "type": "theorem", "label": "brauer-lemma-simple-module", "categories": [ "brauer" ], "title": "brauer-lemma-simple-module", "contents": [ "Let $A$ be a $k$-algebra. If $A$ is finite, then", "\\begin{enumerate}", "\\item $A$ has a simple module,", "\\item any nonzero module contains a simple submodule,", "\\item a simple module over $A$ has finite dimension over $k$, and", "\\item if $M$ is a simple $A$-module, then $\\text{End}_A(M)$ is a", "skew field.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Of course (1) follows from (2) since $A$ is a nonzero $A$-module.", "For (2), any submodule of minimal (finite) dimension as a $k$-vector", "space will be simple. There exists a finite dimensional one", "because a cyclic submodule is one. If $M$ is simple, then", "$mA \\subset M$ is a sub-module, hence we see (3). Any nonzero element", "of $\\text{End}_A(M)$ is an isomorphism, hence (4) holds." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7828, "type": "theorem", "label": "brauer-lemma-centralizer", "categories": [ "brauer" ], "title": "brauer-lemma-centralizer", "contents": [ "Let $A$, $A'$ be $k$-algebras. Let $B \\subset A$, $B' \\subset A'$ be", "subalgebras with centralizers $C$, $C'$. Then the centralizer of", "$B \\otimes_k B'$ in $A \\otimes_k A'$ is $C \\otimes_k C'$." ], "refs": [], "proofs": [ { "contents": [ "Denote $C'' \\subset A \\otimes_k A'$ the centralizer of $B \\otimes_k B'$.", "It is clear that $C \\otimes_k C' \\subset C''$. Conversely, every element", "of $C''$ commutes with $B \\otimes 1$ hence is contained in $C \\otimes_k A'$.", "Similarly $C'' \\subset A \\otimes_k C'$. Thus", "$C'' \\subset C \\otimes_k A' \\cap A \\otimes_k C' = C \\otimes_k C'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7829, "type": "theorem", "label": "brauer-lemma-center-csa", "categories": [ "brauer" ], "title": "brauer-lemma-center-csa", "contents": [ "Let $A$ be a finite simple $k$-algebra. Then the center $k'$ of $A$", "is a finite field extension of $k$." ], "refs": [], "proofs": [ { "contents": [ "Write $A = \\text{Mat}(n \\times n, K)$ for some skew field $K$ finite", "over $k$, see", "Theorem \\ref{theorem-wedderburn}.", "By", "Lemma \\ref{lemma-centralizer}", "the center of $A$ is $k \\otimes_k k'$ where $k' \\subset K$ is the", "center of $K$. Since the center of a skew field is a field, we win." ], "refs": [ "brauer-theorem-wedderburn", "brauer-lemma-centralizer" ], "ref_ids": [ 7822, 7828 ] } ], "ref_ids": [] }, { "id": 7830, "type": "theorem", "label": "brauer-lemma-generate-two-sided-sub", "categories": [ "brauer" ], "title": "brauer-lemma-generate-two-sided-sub", "contents": [ "Let $V$ be a $k$ vector space. Let $K$ be a central $k$-algebra", "which is a skew field. Let $W \\subset V \\otimes_k K$ be a two-sided", "$K$-sub vector space. Then $W$ is generated as a left $K$-vector", "space by $W \\cap (V \\otimes 1)$." ], "refs": [], "proofs": [ { "contents": [ "Let $V' \\subset V$ be the $k$-sub vector space generated by $v \\in V$", "such that $v \\otimes 1 \\in W$. Then $V' \\otimes_k K \\subset W$ and", "we have", "$$", "W/(V' \\otimes_k K) \\subset (V/V') \\otimes_k K.", "$$", "If $\\overline{v} \\in V/V'$ is a nonzero vector such that", "$\\overline{v} \\otimes 1$ is contained in $W/(V' \\otimes_k K)$,", "then we see that $v \\otimes 1 \\in W$ where $v \\in V$ lifts $\\overline{v}$.", "This contradicts our construction of $V'$. Hence we may replace", "$V$ by $V/V'$ and $W$ by $W/(V' \\otimes_k K)$ and it suffices to prove", "that $W \\cap (V \\otimes 1)$ is nonzero if $W$ is nonzero.", "\\medskip\\noindent", "To see this let $w \\in W$ be a nonzero element which can be written", "as $w = \\sum_{i = 1, \\ldots, n} v_i \\otimes k_i$ with $n$ minimal.", "We may right multiply with $k_1^{-1}$ and assume that $k_1 = 1$.", "If $n = 1$, then we win because $v_1 \\otimes 1 \\in W$.", "If $n > 1$, then we see that for any $c \\in K$", "$$", "c w - w c = \\sum\\nolimits_{i = 2, \\ldots, n} v_i \\otimes (c k_i - k_i c) \\in W", "$$", "and hence $c k_i - k_i c = 0$ by minimality of $n$.", "This implies that $k_i$ is in the center of $K$ which is $k$ by", "assumption. Hence $w = (v_1 + \\sum k_i v_i) \\otimes 1$ contradicting", "the minimality of $n$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7831, "type": "theorem", "label": "brauer-lemma-generate-two-sided-ideal", "categories": [ "brauer" ], "title": "brauer-lemma-generate-two-sided-ideal", "contents": [ "Let $A$ be a $k$-algebra. Let $K$ be a central $k$-algebra", "which is a skew field. Then any two-sided ideal $I \\subset A \\otimes_k K$", "is of the form $J \\otimes_k K$ for some two-sided ideal $J \\subset A$.", "In particular, if $A$ is simple, then so is $A \\otimes_k K$." ], "refs": [], "proofs": [ { "contents": [ "Set $J = \\{a \\in A \\mid a \\otimes 1 \\in I\\}$. This is a two-sided ideal", "of $A$. And $I = J \\otimes_k K$ by", "Lemma \\ref{lemma-generate-two-sided-sub}." ], "refs": [ "brauer-lemma-generate-two-sided-sub" ], "ref_ids": [ 7830 ] } ], "ref_ids": [] }, { "id": 7832, "type": "theorem", "label": "brauer-lemma-matrix-algebras", "categories": [ "brauer" ], "title": "brauer-lemma-matrix-algebras", "contents": [ "Let $R$ be a possibly noncommutative ring. Let $n \\geq 1$ be an integer.", "Let $R_n = \\text{Mat}(n \\times n, R)$.", "\\begin{enumerate}", "\\item The functors $M \\mapsto M^{\\oplus n}$ and", "$N \\mapsto Ne_{11}$ define quasi-inverse equivalences of categories", "$\\text{Mod}_R \\leftrightarrow \\text{Mod}_{R_n}$.", "\\item A two-sided ideal of $R_n$ is of the form $IR_n$ for some", "two-sided ideal $I$ of $R$.", "\\item The center of $R_n$ is equal to the center of $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) proves itself. If $J \\subset R_n$ is a two-sided ideal, then", "$J = \\bigoplus e_{ii}Je_{jj}$ and all of the summands $e_{ii}Je_{jj}$ are", "equal to each other and are a two-sided ideal $I$ of $R$. This proves (2).", "Part (3) is clear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7833, "type": "theorem", "label": "brauer-lemma-simple-module-unique", "categories": [ "brauer" ], "title": "brauer-lemma-simple-module-unique", "contents": [ "Let $A$ be a finite simple $k$-algebra.", "\\begin{enumerate}", "\\item There exists exactly one simple $A$-module $M$ up to isomorphism.", "\\item Any finite $A$-module is a direct sum of copies of a simple module.", "\\item Two finite $A$-modules are isomorphic if and only if they", "have the same dimension over $k$.", "\\item If $A = \\text{Mat}(n \\times n, K)$ with $K$ a finite skew field", "extension of $k$, then $M = K^{\\oplus n}$ is a simple $A$-module and", "$\\text{End}_A(M) = K^{op}$.", "\\item If $M$ is a simple $A$-module, then $L = \\text{End}_A(M)$", "is a skew field finite over $k$ acting on the left on $M$, we have", "$A = \\text{End}_L(M)$, and the centers of $A$ and $L$ agree.", "Also $[A : k] [L : k] = \\dim_k(M)^2$.", "\\item For a finite $A$-module $N$ the algebra $B = \\text{End}_A(N)$ is a", "matrix algebra over the skew field $L$ of (5). Moreover $\\text{End}_B(N) = A$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By", "Theorem \\ref{theorem-wedderburn}", "we can write $A = \\text{Mat}(n \\times n, K)$ for some finite skew", "field extension $K$ of $k$. By", "Lemma \\ref{lemma-matrix-algebras}", "the category of modules over $A$ is equivalent to the category of", "modules over $K$. Thus (1), (2), and (3) hold", "because every module over $K$ is free. Part (4) holds", "because the equivalence transforms the $K$-module $K$", "to $M = K^{\\oplus n}$. Using $M = K^{\\oplus n}$ in (5)", "we see that $L = K^{op}$. The statement about the center of $L = K^{op}$", "follows from", "Lemma \\ref{lemma-matrix-algebras}.", "The statement about $\\text{End}_L(M)$ follows from the explicit form", "of $M$. The formula of dimensions is clear.", "Part (6) follows as $N$ is isomorphic to a direct sum of", "copies of a simple module." ], "refs": [ "brauer-theorem-wedderburn", "brauer-lemma-matrix-algebras", "brauer-lemma-matrix-algebras" ], "ref_ids": [ 7822, 7832, 7832 ] } ], "ref_ids": [] }, { "id": 7834, "type": "theorem", "label": "brauer-lemma-tensor-simple", "categories": [ "brauer" ], "title": "brauer-lemma-tensor-simple", "contents": [ "Let $A$, $A'$ be two simple $k$-algebras one of which is finite and central", "over $k$. Then $A \\otimes_k A'$ is simple." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $A'$ is finite and central over $k$.", "Write $A' = \\text{Mat}(n \\times n, K')$, see", "Theorem \\ref{theorem-wedderburn}.", "Then the center of $K'$ is $k$ and we conclude that", "$A \\otimes_k K'$ is simple by", "Lemma \\ref{lemma-generate-two-sided-ideal}.", "Hence $A \\otimes_k A' = \\text{Mat}(n \\times n, A \\otimes_k K')$ is simple", "by Lemma \\ref{lemma-matrix-algebras}." ], "refs": [ "brauer-theorem-wedderburn", "brauer-lemma-generate-two-sided-ideal", "brauer-lemma-matrix-algebras" ], "ref_ids": [ 7822, 7831, 7832 ] } ], "ref_ids": [] }, { "id": 7835, "type": "theorem", "label": "brauer-lemma-tensor-central-simple", "categories": [ "brauer" ], "title": "brauer-lemma-tensor-central-simple", "contents": [ "The tensor product of finite central simple algebras over $k$ is finite,", "central, and simple." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-centralizer} and \\ref{lemma-tensor-simple}." ], "refs": [ "brauer-lemma-centralizer", "brauer-lemma-tensor-simple" ], "ref_ids": [ 7828, 7834 ] } ], "ref_ids": [] }, { "id": 7836, "type": "theorem", "label": "brauer-lemma-base-change", "categories": [ "brauer" ], "title": "brauer-lemma-base-change", "contents": [ "Let $A$ be a finite central simple algebra over $k$.", "Let $k \\subset k'$ be a field extension. Then $A' = A \\otimes_k k'$ is", "a finite central simple algebra over $k'$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-centralizer} and \\ref{lemma-tensor-simple}." ], "refs": [ "brauer-lemma-centralizer", "brauer-lemma-tensor-simple" ], "ref_ids": [ 7828, 7834 ] } ], "ref_ids": [] }, { "id": 7837, "type": "theorem", "label": "brauer-lemma-inverse", "categories": [ "brauer" ], "title": "brauer-lemma-inverse", "contents": [ "Let $A$ be a finite central simple algebra over $k$.", "Then $A \\otimes_k A^{op} \\cong \\text{Mat}(n \\times n, k)$", "where $n = [A : k]$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-tensor-central-simple} the algebra $A \\otimes_k A^{op}$", "is simple. Hence the map", "$$", "A \\otimes_k A^{op} \\longrightarrow \\text{End}_k(A),\\quad", "a \\otimes a' \\longmapsto (x \\mapsto axa')", "$$", "is injective. Since both sides of the arrow have the same dimension", "we win." ], "refs": [ "brauer-lemma-tensor-central-simple" ], "ref_ids": [ 7835 ] } ], "ref_ids": [] }, { "id": 7838, "type": "theorem", "label": "brauer-lemma-similar", "categories": [ "brauer" ], "title": "brauer-lemma-similar", "contents": [ "Similarity.", "\\begin{enumerate}", "\\item Similarity defines an equivalence relation on the set of isomorphism", "classes of finite central simple algebras over $k$.", "\\item Every similarity class contains a unique (up to isomorphism)", "finite central skew field extension of $k$.", "\\item If $A = \\text{Mat}(n \\times n, K)$ and $B = \\text{Mat}(m \\times m, K')$", "for some finite central skew fields $K$, $K'$ over $k$", "then $A$ and $B$ are similar if and only if $K \\cong K'$ as $k$-algebras.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that by Wedderburn's theorem (Theorem \\ref{theorem-wedderburn})", "we can always write a finite central simple algebra as a matrix", "algebra over a finite central skew field. Hence it suffices to prove", "the third assertion. To see this it suffices to show that if", "$A = \\text{Mat}(n \\times n, K) \\cong \\text{Mat}(m \\times m, K') = B$", "then $K \\cong K'$. To see this note that for a simple module $M$ of $A$", "we have $\\text{End}_A(M) = K^{op}$, see", "Lemma \\ref{lemma-simple-module-unique}.", "Hence $A \\cong B$ implies $K^{op} \\cong (K')^{op}$ and we win." ], "refs": [ "brauer-theorem-wedderburn", "brauer-lemma-simple-module-unique" ], "ref_ids": [ 7822, 7833 ] } ], "ref_ids": [] }, { "id": 7839, "type": "theorem", "label": "brauer-lemma-brauer-algebraically-closed", "categories": [ "brauer" ], "title": "brauer-lemma-brauer-algebraically-closed", "contents": [ "The Brauer group of an algebraically closed field is zero." ], "refs": [], "proofs": [ { "contents": [ "Let $k \\subset K$ be a finite central skew field extension.", "For any element $x \\in K$ the subring $k[x] \\subset K$ is a", "commutative finite integral $k$-sub algebra, hence a field, see", "Algebra, Lemma \\ref{algebra-lemma-integral-over-field}.", "Since $k$ is algebraically closed we conclude that", "$k[x] = k$. Since $x$ was arbitrary we conclude $k = K$." ], "refs": [ "algebra-lemma-integral-over-field" ], "ref_ids": [ 497 ] } ], "ref_ids": [] }, { "id": 7840, "type": "theorem", "label": "brauer-lemma-dimension-square", "categories": [ "brauer" ], "title": "brauer-lemma-dimension-square", "contents": [ "Let $A$ be a finite central simple algebra over a field $k$.", "Then $[A : k]$ is a square." ], "refs": [], "proofs": [ { "contents": [ "This is true because $A \\otimes_k \\overline{k}$ is a matrix", "algebra over $\\overline{k}$ by", "Lemma \\ref{lemma-brauer-algebraically-closed}." ], "refs": [ "brauer-lemma-brauer-algebraically-closed" ], "ref_ids": [ 7839 ] } ], "ref_ids": [] }, { "id": 7841, "type": "theorem", "label": "brauer-lemma-automorphism-inner", "categories": [ "brauer" ], "title": "brauer-lemma-automorphism-inner", "contents": [ "Let $A$ be a finite central simple $k$-algebra. Any automorphism of $A$ is", "inner. In particular, any automorphism of $\\text{Mat}(n \\times n, k)$", "is inner." ], "refs": [], "proofs": [ { "contents": [ "Note that $A$ is a finite central simple algebra over the center", "of $A$ which is a finite field extension of $k$, see", "Lemma \\ref{lemma-center-csa}.", "Hence the Skolem-Noether theorem (Theorem \\ref{theorem-skolem-noether})", "applies." ], "refs": [ "brauer-lemma-center-csa", "brauer-theorem-skolem-noether" ], "ref_ids": [ 7829, 7823 ] } ], "ref_ids": [] }, { "id": 7842, "type": "theorem", "label": "brauer-lemma-when-tensor-is-equal", "categories": [ "brauer" ], "title": "brauer-lemma-when-tensor-is-equal", "contents": [ "Let $A$ be a finite central simple algebra over $k$, and let", "$B$ be a simple subalgebra of $A$. If $B$ is a central", "$k$-algebra, then $A = B \\otimes_k C$ where $C$ is the (central simple)", "centralizer of $B$ in $A$." ], "refs": [], "proofs": [ { "contents": [ "We have $\\dim_k(A) = \\dim_k(B \\otimes_k C)$ by", "Theorem \\ref{theorem-centralizer}. By", "Lemma \\ref{lemma-tensor-simple}", "the tensor product is simple. Hence the natural map", "$B \\otimes_k C \\to A$ is injective hence an isomorphism." ], "refs": [ "brauer-theorem-centralizer", "brauer-lemma-tensor-simple" ], "ref_ids": [ 7824, 7834 ] } ], "ref_ids": [] }, { "id": 7843, "type": "theorem", "label": "brauer-lemma-self-centralizing-subfield", "categories": [ "brauer" ], "title": "brauer-lemma-self-centralizing-subfield", "contents": [ "Let $A$ be a finite central simple algebra over $k$.", "If $K \\subset A$ is a subfield, then the following are equivalent", "\\begin{enumerate}", "\\item $[A : k] = [K : k]^2$,", "\\item $K$ is its own centralizer, and", "\\item $K$ is a maximal commutative subring.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Theorem \\ref{theorem-centralizer}", "shows that (1) and (2) are equivalent.", "It is clear that (3) and (2) are equivalent." ], "refs": [ "brauer-theorem-centralizer" ], "ref_ids": [ 7824 ] } ], "ref_ids": [] }, { "id": 7844, "type": "theorem", "label": "brauer-lemma-maximal-subfield", "categories": [ "brauer" ], "title": "brauer-lemma-maximal-subfield", "contents": [ "\\begin{slogan}", "The dimension of a finite central skew field is the square of the dimension", "of any maximal subfield.", "\\end{slogan}", "Let $A$ be a finite central skew field over $k$.", "Then every maximal subfield $K \\subset A$ satisfies", "$[A : k] = [K : k]^2$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-self-centralizing-subfield}." ], "refs": [ "brauer-lemma-self-centralizing-subfield" ], "ref_ids": [ 7843 ] } ], "ref_ids": [] }, { "id": 7845, "type": "theorem", "label": "brauer-lemma-maximal-subfield-splits", "categories": [ "brauer" ], "title": "brauer-lemma-maximal-subfield-splits", "contents": [ "A maximal subfield of a finite central skew field $K$ over $k$ is", "a splitting field for $K$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-maximal-subfield} with", "Theorem \\ref{theorem-splitting}." ], "refs": [ "brauer-lemma-maximal-subfield", "brauer-theorem-splitting" ], "ref_ids": [ 7844, 7825 ] } ], "ref_ids": [] }, { "id": 7846, "type": "theorem", "label": "brauer-lemma-splitting-field-degree", "categories": [ "brauer" ], "title": "brauer-lemma-splitting-field-degree", "contents": [ "Consider a finite central skew field $K$ over $k$. Let $d^2 = [K : k]$.", "For any finite splitting field $k'$ for $K$ the degree $[k' : k]$ is", "divisible by $d$." ], "refs": [], "proofs": [ { "contents": [ "By Theorem \\ref{theorem-splitting} there exists a finite central", "simple algebra $B$ in the Brauer class of $K$ such that", "$[B : k] = [k' : k]^2$. By", "Lemma \\ref{lemma-similar}", "we see that $B = \\text{Mat}(n \\times n, K)$ for some $n$.", "Then $[k' : k]^2 = n^2d^2$ whence the result." ], "refs": [ "brauer-theorem-splitting", "brauer-lemma-similar" ], "ref_ids": [ 7825, 7838 ] } ], "ref_ids": [] }, { "id": 7847, "type": "theorem", "label": "brauer-lemma-finite-central-simple-algebra", "categories": [ "brauer" ], "title": "brauer-lemma-finite-central-simple-algebra", "contents": [ "Let $k$ be a field. For a $k$-algebra $A$ the following are equivalent", "\\begin{enumerate}", "\\item $A$ is finite central simple $k$-algebra,", "\\item $A$ is a finite dimensional $k$-vector space, $k$ is the center of $A$,", "and $A$ has no nontrivial two-sided ideal,", "\\item there exists $d \\geq 1$ such that", "$A \\otimes_k \\bar k \\cong \\text{Mat}(d \\times d, \\bar k)$,", "\\item there exists $d \\geq 1$ such that", "$A \\otimes_k k^{sep} \\cong \\text{Mat}(d \\times d, k^{sep})$,", "\\item there exist $d \\geq 1$ and a finite Galois extension $k \\subset k'$", "such that", "$A \\otimes_k k' \\cong \\text{Mat}(d \\times d, k')$,", "\\item there exist $n \\geq 1$ and a finite central skew field $K$", "over $k$ such that $A \\cong \\text{Mat}(n \\times n, K)$.", "\\end{enumerate}", "The integer $d$ is called the {\\it degree} of $A$." ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is a consequence of the definitions, see", "Section \\ref{section-algebras}.", "Assume (1). By", "Proposition \\ref{proposition-separable-splitting-field}", "there exists a separable splitting field $k \\subset k'$ for $A$.", "Of course, then a Galois closure of $k'/k$ is a splitting field also.", "Thus we see that (1) implies (5). It is clear that (5) $\\Rightarrow$ (4)", "$\\Rightarrow$ (3). Assume (3). Then $A \\otimes_k \\overline{k}$", "is a finite central simple $\\overline{k}$-algebra for example by", "Lemma \\ref{lemma-matrix-algebras}.", "This trivially implies that $A$ is a finite central simple $k$-algebra.", "Finally, the equivalence of (1) and (6) is Wedderburn's theorem, see", "Theorem \\ref{theorem-wedderburn}." ], "refs": [ "brauer-proposition-separable-splitting-field", "brauer-lemma-matrix-algebras", "brauer-theorem-wedderburn" ], "ref_ids": [ 7848, 7832, 7822 ] } ], "ref_ids": [] }, { "id": 7848, "type": "theorem", "label": "brauer-proposition-separable-splitting-field", "categories": [ "brauer" ], "title": "brauer-proposition-separable-splitting-field", "contents": [ "Consider a finite central skew field $K$ over $k$.", "There exists a maximal subfield $k \\subset k' \\subset K$ which", "is separable over $k$.", "In particular, every Brauer class has a finite separable", "spitting field." ], "refs": [], "proofs": [ { "contents": [ "Since every Brauer class is represented by a finite central skew", "field over $k$, we see that the second statement follows from the", "first by", "Lemma \\ref{lemma-maximal-subfield-splits}.", "\\medskip\\noindent", "To prove the first statement, suppose that we are given a separable", "subfield $k' \\subset K$. Then the centralizer $K'$ of $k'$ in $K$", "has center $k'$, and the problem reduces to finding a maximal", "subfield of $K'$ separable over $k'$. Thus it suffices to prove, if", "$k \\not = K$, that we can find an element $x \\in K$, $x \\not \\in k$", "which is separable over $k$. This statement is clear in characteristic", "zero. Hence we may assume that $k$ has characteristic $p > 0$. If the", "ground field $k$ is finite then, the result is clear as well (because", "extensions of finite fields are always separable). Thus we may assume", "that $k$ is an infinite field of positive characteristic.", "\\medskip\\noindent", "To get a contradiction assume no element of $K$ is separable over $k$.", "By the discussion in", "Fields, Section \\ref{fields-section-algebraic}", "this means the minimal polynomial of any $x \\in K$ is of the form", "$T^q - a$ where $q$ is a power of $p$ and $a \\in k$. Since it is", "clear that every element of $K$ has a minimal polynomial of degree", "$\\leq \\dim_k(K)$ we conclude that there exists a fixed $p$-power", "$q$ such that $x^q \\in k$ for all $x \\in K$.", "\\medskip\\noindent", "Consider the map", "$$", "(-)^q : K \\longrightarrow K", "$$", "and write it out in terms of a $k$-basis $\\{a_1, \\ldots, a_n\\}$ of $K$", "with $a_1 = 1$. So", "$$", "(\\sum x_i a_i)^q = \\sum f_i(x_1, \\ldots, x_n)a_i.", "$$", "Since multiplication on $A$ is $k$-bilinear we see that each $f_i$", "is a polynomial in $x_1, \\ldots, x_n$ (details omitted).", "The choice of $q$ above and the fact that $k$ is infinite shows that", "$f_i$ is identically zero for $i \\geq 2$. Hence we see that it remains", "zero on extending $k$ to its algebraic closure $\\overline{k}$. But the", "algebra $A \\otimes_k \\overline{k}$ is a matrix algebra, which implies", "there are some elements whose $q$th power is not central (e.g., $e_{11}$).", "This is the desired contradiction." ], "refs": [ "brauer-lemma-maximal-subfield-splits" ], "ref_ids": [ 7845 ] } ], "ref_ids": [] }, { "id": 7856, "type": "theorem", "label": "divisors-lemma-associated-affine-open", "categories": [ "divisors" ], "title": "divisors-lemma-associated-affine-open", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $\\Spec(A) = U \\subset X$ be an affine open, and set", "$M = \\Gamma(U, \\mathcal{F})$.", "Let $x \\in U$, and let $\\mathfrak p \\subset A$ be the corresponding prime.", "\\begin{enumerate}", "\\item If $\\mathfrak p$ is associated to $M$, then $x$ is associated", "to $\\mathcal{F}$.", "\\item If $\\mathfrak p$ is finitely generated, then the converse holds", "as well.", "\\end{enumerate}", "In particular, if $X$ is locally Noetherian, then the equivalence", "$$", "\\mathfrak p \\in \\text{Ass}(M) \\Leftrightarrow x \\in \\text{Ass}(\\mathcal{F})", "$$", "holds for all pairs $(\\mathfrak p, x)$ as above." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Algebra, Lemma \\ref{algebra-lemma-associated-primes-localize}.", "But we can also argue directly as follows.", "Suppose $\\mathfrak p$ is associated to $M$.", "Then there exists an $m \\in M$ whose annihilator is $\\mathfrak p$.", "Since localization is exact we see that", "$\\mathfrak pA_{\\mathfrak p}$ is the annihilator of", "$m/1 \\in M_{\\mathfrak p}$. Since $M_{\\mathfrak p} = \\mathcal{F}_x$", "(Schemes, Lemma \\ref{schemes-lemma-spec-sheaves})", "we conclude that $x$ is associated to $\\mathcal{F}$.", "\\medskip\\noindent", "Conversely, assume that $x$ is associated to $\\mathcal{F}$,", "and $\\mathfrak p$ is finitely generated.", "As $x$ is associated to $\\mathcal{F}$", "there exists an element $m' \\in M_{\\mathfrak p}$ whose", "annihilator is $\\mathfrak pA_{\\mathfrak p}$. Write", "$m' = m/f$ for some $f \\in A$, $f \\not \\in \\mathfrak p$.", "The annihilator $I$ of $m$ is an ideal of $A$ such that", "$IA_{\\mathfrak p} = \\mathfrak pA_{\\mathfrak p}$. Hence", "$I \\subset \\mathfrak p$, and $(\\mathfrak p/I)_{\\mathfrak p} = 0$.", "Since $\\mathfrak p$ is finitely generated,", "there exists a $g \\in A$, $g \\not \\in \\mathfrak p$ such that", "$g(\\mathfrak p/I) = 0$. Hence the annihilator of $gm$ is", "$\\mathfrak p$ and we win.", "\\medskip\\noindent", "If $X$ is locally Noetherian, then $A$ is Noetherian", "(Properties, Lemma \\ref{properties-lemma-locally-Noetherian})", "and $\\mathfrak p$ is always finitely generated." ], "refs": [ "algebra-lemma-associated-primes-localize", "schemes-lemma-spec-sheaves", "properties-lemma-locally-Noetherian" ], "ref_ids": [ 709, 7651, 2951 ] } ], "ref_ids": [] }, { "id": 7857, "type": "theorem", "label": "divisors-lemma-ass-support", "categories": [ "divisors" ], "title": "divisors-lemma-ass-support", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then $\\text{Ass}(\\mathcal{F}) \\subset \\text{Supp}(\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7858, "type": "theorem", "label": "divisors-lemma-ses-ass", "categories": [ "divisors" ], "title": "divisors-lemma-ses-ass", "contents": [ "Let $X$ be a scheme.", "Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "be a short exact sequence of quasi-coherent sheaves on $X$.", "Then", "$\\text{Ass}(\\mathcal{F}_2) \\subset", "\\text{Ass}(\\mathcal{F}_1) \\cup \\text{Ass}(\\mathcal{F}_3)$", "and", "$\\text{Ass}(\\mathcal{F}_1) \\subset \\text{Ass}(\\mathcal{F}_2)$." ], "refs": [], "proofs": [ { "contents": [ "For every point $x \\in X$ the sequence of stalks", "$0 \\to \\mathcal{F}_{1, x} \\to \\mathcal{F}_{2, x} \\to \\mathcal{F}_{3, x} \\to 0$", "is a short exact sequence of $\\mathcal{O}_{X, x}$-modules.", "Hence the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-ass}." ], "refs": [ "algebra-lemma-ass" ], "ref_ids": [ 699 ] } ], "ref_ids": [] }, { "id": 7859, "type": "theorem", "label": "divisors-lemma-finite-ass", "categories": [ "divisors" ], "title": "divisors-lemma-finite-ass", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Then $\\text{Ass}(\\mathcal{F}) \\cap U$ is finite for", "every quasi-compact open $U \\subset X$." ], "refs": [], "proofs": [ { "contents": [ "This is true because the set of associated primes of a finite module over", "a Noetherian ring is finite, see", "Algebra, Lemma \\ref{algebra-lemma-finite-ass}.", "To translate from schemes to algebra use that $U$ is a finite union of", "affine opens, each of these opens is the spectrum of a Noetherian ring", "(Properties, Lemma \\ref{properties-lemma-locally-Noetherian}),", "$\\mathcal{F}$ corresponds to a finite module over this ring", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-Noetherian}),", "and finally use", "Lemma \\ref{lemma-associated-affine-open}." ], "refs": [ "algebra-lemma-finite-ass", "properties-lemma-locally-Noetherian", "coherent-lemma-coherent-Noetherian", "divisors-lemma-associated-affine-open" ], "ref_ids": [ 701, 2951, 3308, 7856 ] } ], "ref_ids": [] }, { "id": 7860, "type": "theorem", "label": "divisors-lemma-ass-zero", "categories": [ "divisors" ], "title": "divisors-lemma-ass-zero", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a", "quasi-coherent $\\mathcal{O}_X$-module. Then", "$$", "\\mathcal{F} = 0 \\Leftrightarrow \\text{Ass}(\\mathcal{F}) = \\emptyset.", "$$" ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{F} = 0$, then $\\text{Ass}(\\mathcal{F}) = \\emptyset$", "by definition. Conversely, if $\\text{Ass}(\\mathcal{F}) = \\emptyset$,", "then $\\mathcal{F} = 0$ by", "Algebra, Lemma \\ref{algebra-lemma-ass-zero}.", "To translate from schemes to algebra, restrict to any affine and use", "Lemma \\ref{lemma-associated-affine-open}." ], "refs": [ "algebra-lemma-ass-zero", "divisors-lemma-associated-affine-open" ], "ref_ids": [ 702, 7856 ] } ], "ref_ids": [] }, { "id": 7861, "type": "theorem", "label": "divisors-lemma-restriction-injective-open-contains-ass", "categories": [ "divisors" ], "title": "divisors-lemma-restriction-injective-open-contains-ass", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. If $U \\subset X$ is open and", "$\\text{Ass}(\\mathcal{F}) \\subset U$, then", "$\\Gamma(X, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{F})$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $s \\in \\Gamma(X, \\mathcal{F})$ be a section which restricts to zero on $U$.", "Let $\\mathcal{F}' \\subset \\mathcal{F}$ be the image of the map", "$\\mathcal{O}_X \\to \\mathcal{F}$ defined by $s$. Then", "$\\text{Supp}(\\mathcal{F}') \\cap U = \\emptyset$. On the other hand,", "$\\text{Ass}(\\mathcal{F}') \\subset \\text{Ass}(\\mathcal{F})$", "by Lemma \\ref{lemma-ses-ass}. Since also", "$\\text{Ass}(\\mathcal{F}') \\subset \\text{Supp}(\\mathcal{F}')$", "(Lemma \\ref{lemma-ass-support}) we conclude", "$\\text{Ass}(\\mathcal{F}') = \\emptyset$.", "Hence $\\mathcal{F}' = 0$ by Lemma \\ref{lemma-ass-zero}." ], "refs": [ "divisors-lemma-ses-ass", "divisors-lemma-ass-support", "divisors-lemma-ass-zero" ], "ref_ids": [ 7858, 7857, 7860 ] } ], "ref_ids": [] }, { "id": 7862, "type": "theorem", "label": "divisors-lemma-minimal-support-in-ass", "categories": [ "divisors" ], "title": "divisors-lemma-minimal-support-in-ass", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $x \\in \\text{Supp}(\\mathcal{F})$ be a point in the support", "of $\\mathcal{F}$ which is not a specialization of another point of", "$\\text{Supp}(\\mathcal{F})$. Then $x \\in \\text{Ass}(\\mathcal{F})$.", "In particular, any generic point of an irreducible component of $X$", "is an associated point of $X$." ], "refs": [], "proofs": [ { "contents": [ "Since $x \\in \\text{Supp}(\\mathcal{F})$ the module $\\mathcal{F}_x$", "is not zero. Hence", "$\\text{Ass}(\\mathcal{F}_x) \\subset \\Spec(\\mathcal{O}_{X, x})$", "is nonempty by", "Algebra, Lemma \\ref{algebra-lemma-ass-zero}.", "On the other hand, by assumption", "$\\text{Supp}(\\mathcal{F}_x) = \\{\\mathfrak m_x\\}$.", "Since", "$\\text{Ass}(\\mathcal{F}_x) \\subset \\text{Supp}(\\mathcal{F}_x)$", "(Algebra, Lemma \\ref{algebra-lemma-ass-support})", "we see that $\\mathfrak m_x$ is associated to $\\mathcal{F}_x$", "and we win." ], "refs": [ "algebra-lemma-ass-zero", "algebra-lemma-ass-support" ], "ref_ids": [ 702, 698 ] } ], "ref_ids": [] }, { "id": 7863, "type": "theorem", "label": "divisors-lemma-check-injective-on-ass", "categories": [ "divisors" ], "title": "divisors-lemma-check-injective-on-ass", "contents": [ "Let $X$ be a locally Noetherian scheme. Let", "$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of", "quasi-coherent $\\mathcal{O}_X$-modules.", "Assume that for every $x \\in X$", "at least one of the following happens", "\\begin{enumerate}", "\\item $\\mathcal{F}_x \\to \\mathcal{G}_x$ is injective, or", "\\item $x \\not \\in \\text{Ass}(\\mathcal{F})$.", "\\end{enumerate}", "Then $\\varphi$ is injective." ], "refs": [], "proofs": [ { "contents": [ "The assumptions imply that $\\text{Ass}(\\Ker(\\varphi)) = \\emptyset$", "and hence $\\Ker(\\varphi) = 0$ by Lemma \\ref{lemma-ass-zero}." ], "refs": [ "divisors-lemma-ass-zero" ], "ref_ids": [ 7860 ] } ], "ref_ids": [] }, { "id": 7864, "type": "theorem", "label": "divisors-lemma-check-isomorphism-via-depth-and-ass", "categories": [ "divisors" ], "title": "divisors-lemma-check-isomorphism-via-depth-and-ass", "contents": [ "Let $X$ be a locally Noetherian scheme. Let", "$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of", "quasi-coherent $\\mathcal{O}_X$-modules. Assume $\\mathcal{F}$ is coherent", "and that for every $x \\in X$ one of the following happens", "\\begin{enumerate}", "\\item $\\mathcal{F}_x \\to \\mathcal{G}_x$ is an isomorphism, or", "\\item $\\text{depth}(\\mathcal{F}_x) \\geq 2$ and", "$x \\not \\in \\text{Ass}(\\mathcal{G})$.", "\\end{enumerate}", "Then $\\varphi$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is a translation of More on Algebra, Lemma", "\\ref{more-algebra-lemma-check-isomorphism-via-depth-and-ass}", "into the language of schemes." ], "refs": [ "more-algebra-lemma-check-isomorphism-via-depth-and-ass" ], "ref_ids": [ 9933 ] } ], "ref_ids": [] }, { "id": 7865, "type": "theorem", "label": "divisors-lemma-bourbaki", "categories": [ "divisors" ], "title": "divisors-lemma-bourbaki", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$ which is flat over $S$.", "Let $\\mathcal{G}$ be a quasi-coherent sheaf on $S$.", "Then we have", "$$", "\\text{Ass}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G})", "\\supset", "\\bigcup\\nolimits_{s \\in \\text{Ass}_S(\\mathcal{G})}", "\\text{Ass}_{X_s}(\\mathcal{F}_s)", "$$", "and equality holds if $S$ is locally Noetherian (for the notation", "$\\mathcal{F}_s$ see above)." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$ and let $s = f(x) \\in S$.", "Set $B = \\mathcal{O}_{X, x}$, $A = \\mathcal{O}_{S, s}$,", "$N = \\mathcal{F}_x$, and $M = \\mathcal{G}_s$.", "Note that the stalk of $\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}$", "at $x$ is equal to the $B$-module $M \\otimes_A N$. Hence", "$x \\in \\text{Ass}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G})$", "if and only if $\\mathfrak m_B$ is in $\\text{Ass}_B(M \\otimes_A N)$.", "Similarly $s \\in \\text{Ass}_S(\\mathcal{G})$ and", "$x \\in \\text{Ass}_{X_s}(\\mathcal{F}_s)$ if and only if", "$\\mathfrak m_A \\in \\text{Ass}_A(M)$ and", "$\\mathfrak m_B/\\mathfrak m_A B \\in", "\\text{Ass}_{B \\otimes \\kappa(\\mathfrak m_A)}(N \\otimes \\kappa(\\mathfrak m_A))$.", "Thus the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-bourbaki-fibres}." ], "refs": [ "algebra-lemma-bourbaki-fibres" ], "ref_ids": [ 719 ] } ], "ref_ids": [] }, { "id": 7866, "type": "theorem", "label": "divisors-lemma-embedded", "categories": [ "divisors" ], "title": "divisors-lemma-embedded", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Then", "\\begin{enumerate}", "\\item the generic points of irreducible components of", "$\\text{Supp}(\\mathcal{F})$ are associated points of $\\mathcal{F}$, and", "\\item an associated point of $\\mathcal{F}$ is embedded if and only", "if it is not a generic point of an irreducible component", "of $\\text{Supp}(\\mathcal{F})$.", "\\end{enumerate}", "In particular an embedded point of $X$ is an associated point of $X$", "which is not a generic point of an irreducible component of $X$." ], "refs": [], "proofs": [ { "contents": [ "Recall that in this case $Z = \\text{Supp}(\\mathcal{F})$ is closed, see", "Morphisms, Lemma \\ref{morphisms-lemma-support-finite-type}", "and that the generic points of irreducible components of $Z$ are", "associated points of $\\mathcal{F}$, see", "Lemma \\ref{lemma-minimal-support-in-ass}.", "Finally, we have $\\text{Ass}(\\mathcal{F}) \\subset Z$, by", "Lemma \\ref{lemma-ass-support}.", "These results, combined with the fact that $Z$ is a sober topological", "space and hence every point of $Z$ is a specialization of a generic", "point of $Z$, imply (1) and (2)." ], "refs": [ "morphisms-lemma-support-finite-type", "divisors-lemma-minimal-support-in-ass", "divisors-lemma-ass-support" ], "ref_ids": [ 5143, 7862, 7857 ] } ], "ref_ids": [] }, { "id": 7867, "type": "theorem", "label": "divisors-lemma-S1-no-embedded", "categories": [ "divisors" ], "title": "divisors-lemma-S1-no-embedded", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Then the following are equivalent:", "\\begin{enumerate}", "\\item $\\mathcal{F}$ has no embedded associated points, and", "\\item $\\mathcal{F}$ has property $(S_1)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is Algebra, Lemma \\ref{algebra-lemma-criterion-no-embedded-primes},", "combined with Lemma \\ref{lemma-associated-affine-open} above." ], "refs": [ "algebra-lemma-criterion-no-embedded-primes", "divisors-lemma-associated-affine-open" ], "ref_ids": [ 1309, 7856 ] } ], "ref_ids": [] }, { "id": 7868, "type": "theorem", "label": "divisors-lemma-noetherian-dim-1-CM-no-embedded-points", "categories": [ "divisors" ], "title": "divisors-lemma-noetherian-dim-1-CM-no-embedded-points", "contents": [ "Let $X$ be a locally Noetherian scheme of dimension $\\leq 1$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is Cohen-Macaulay, and", "\\item $X$ has no embedded points.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-S1-no-embedded} and the definitions." ], "refs": [ "divisors-lemma-S1-no-embedded" ], "ref_ids": [ 7867 ] } ], "ref_ids": [] }, { "id": 7869, "type": "theorem", "label": "divisors-lemma-scheme-theoretically-dense-contain-embedded-points", "categories": [ "divisors" ], "title": "divisors-lemma-scheme-theoretically-dense-contain-embedded-points", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $U \\subset X$ be an", "open subscheme. The following are equivalent", "\\begin{enumerate}", "\\item $U$ is scheme theoretically dense in $X$", "(Morphisms, Definition \\ref{morphisms-definition-scheme-theoretically-dense}),", "\\item $U$ is dense in $X$ and $U$ contains all embedded points of $X$.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-scheme-theoretically-dense" ], "proofs": [ { "contents": [ "The question is local on $X$, hence we may assume that $X = \\Spec(A)$", "where $A$ is a Noetherian ring. Then $U$ is quasi-compact", "(Properties, Lemma \\ref{properties-lemma-immersion-into-noetherian})", "hence $U = D(f_1) \\cup \\ldots \\cup D(f_n)$", "(Algebra, Lemma \\ref{algebra-lemma-qc-open}).", "In this situation $U$ is scheme theoretically dense in $X$ if and only if", "$A \\to A_{f_1} \\times \\ldots \\times A_{f_n}$ is injective, see", "Morphisms, Example \\ref{morphisms-example-scheme-theoretic-closure}.", "Condition (2) translated into algebra means that for every associated", "prime $\\mathfrak p$ of $A$ there exists an $i$ with $f_i \\not \\in \\mathfrak p$.", "\\medskip\\noindent", "Assume (1), i.e., $A \\to A_{f_1} \\times \\ldots \\times A_{f_n}$ is injective.", "If $x \\in A$ has annihilator a prime $\\mathfrak p$, then $x$ maps", "to a nonzero element of $A_{f_i}$ for some $i$ and hence", "$f_i \\not \\in \\mathfrak p$. Thus (2) holds.", "Assume (2), i.e., every associated prime $\\mathfrak p$ of $A$", "corresponds to a prime of $A_{f_i}$ for some $i$. Then", "$A \\to A_{f_1} \\times \\ldots \\times A_{f_n}$ is injective because", "$A \\to \\prod_{\\mathfrak p \\in \\text{Ass}(A)} A_\\mathfrak p$ is injective", "by Algebra, Lemma \\ref{algebra-lemma-zero-at-ass-zero}." ], "refs": [ "properties-lemma-immersion-into-noetherian", "algebra-lemma-qc-open", "algebra-lemma-zero-at-ass-zero" ], "ref_ids": [ 2952, 432, 713 ] } ], "ref_ids": [ 5540 ] }, { "id": 7870, "type": "theorem", "label": "divisors-lemma-remove-embedded-points", "categories": [ "divisors" ], "title": "divisors-lemma-remove-embedded-points", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "The set of coherent subsheaves", "$$", "\\{", "\\mathcal{K} \\subset \\mathcal{F}", "\\mid", "\\text{Supp}(\\mathcal{K})\\text{ is nowhere dense in }\\text{Supp}(\\mathcal{F})", "\\}", "$$", "has a maximal element $\\mathcal{K}$.", "Setting $\\mathcal{F}' = \\mathcal{F}/\\mathcal{K}$ we have the", "following", "\\begin{enumerate}", "\\item $\\text{Supp}(\\mathcal{F}') = \\text{Supp}(\\mathcal{F})$,", "\\item $\\mathcal{F}'$ has no embedded associated points, and", "\\item there exists a dense open $U \\subset X$ such that", "$U \\cap \\text{Supp}(\\mathcal{F})$ is dense in $\\text{Supp}(\\mathcal{F})$", "and $\\mathcal{F}'|_U \\cong \\mathcal{F}|_U$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Algebra, Lemmas \\ref{algebra-lemma-remove-embedded-primes} and", "\\ref{algebra-lemma-remove-embedded-primes-localize}.", "Note that $U$ can be taken as the complement of the closure", "of the set of embedded associated points of $\\mathcal{F}$." ], "refs": [ "algebra-lemma-remove-embedded-primes", "algebra-lemma-remove-embedded-primes-localize" ], "ref_ids": [ 736, 737 ] } ], "ref_ids": [] }, { "id": 7871, "type": "theorem", "label": "divisors-lemma-no-embedded-points-endos", "categories": [ "divisors" ], "title": "divisors-lemma-no-embedded-points-endos", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module", "without embedded associated points. Set", "$$", "\\mathcal{I}", "=", "\\Ker(\\mathcal{O}_X", "\\longrightarrow", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{F})).", "$$", "This is a coherent sheaf of ideals which defines a closed", "subscheme $Z \\subset X$ without embedded points. Moreover", "there exists a coherent sheaf $\\mathcal{G}$ on $Z$", "such that (a) $\\mathcal{F} = (Z \\to X)_*\\mathcal{G}$,", "(b) $\\mathcal{G}$ has no associated embedded points, and", "(c) $\\text{Supp}(\\mathcal{G}) = Z$ (as sets)." ], "refs": [], "proofs": [ { "contents": [ "Some of the statements we have seen in the proof of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-support-closed}.", "The others follow from", "Algebra, Lemma \\ref{algebra-lemma-no-embedded-primes-endos}." ], "refs": [ "coherent-lemma-coherent-support-closed", "algebra-lemma-no-embedded-primes-endos" ], "ref_ids": [ 3314, 738 ] } ], "ref_ids": [] }, { "id": 7872, "type": "theorem", "label": "divisors-lemma-weakly-associated-affine-open", "categories": [ "divisors" ], "title": "divisors-lemma-weakly-associated-affine-open", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $\\Spec(A) = U \\subset X$ be an affine open, and set", "$M = \\Gamma(U, \\mathcal{F})$.", "Let $x \\in U$, and let $\\mathfrak p \\subset A$ be the corresponding prime.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathfrak p$ is weakly associated to $M$, and", "\\item $x$ is weakly associated to $\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Algebra, Lemma \\ref{algebra-lemma-weakly-ass-local}." ], "refs": [ "algebra-lemma-weakly-ass-local" ], "ref_ids": [ 720 ] } ], "ref_ids": [] }, { "id": 7873, "type": "theorem", "label": "divisors-lemma-weakly-ass-support", "categories": [ "divisors" ], "title": "divisors-lemma-weakly-ass-support", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then", "$$", "\\text{Ass}(\\mathcal{F}) \\subset \\text{WeakAss}(\\mathcal{F}) \\subset", "\\text{Supp}(\\mathcal{F}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7874, "type": "theorem", "label": "divisors-lemma-ses-weakly-ass", "categories": [ "divisors" ], "title": "divisors-lemma-ses-weakly-ass", "contents": [ "Let $X$ be a scheme.", "Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "be a short exact sequence of quasi-coherent sheaves on $X$.", "Then", "$\\text{WeakAss}(\\mathcal{F}_2) \\subset", "\\text{WeakAss}(\\mathcal{F}_1) \\cup \\text{WeakAss}(\\mathcal{F}_3)$", "and", "$\\text{WeakAss}(\\mathcal{F}_1) \\subset \\text{WeakAss}(\\mathcal{F}_2)$." ], "refs": [], "proofs": [ { "contents": [ "For every point $x \\in X$ the sequence of stalks", "$0 \\to \\mathcal{F}_{1, x} \\to \\mathcal{F}_{2, x} \\to \\mathcal{F}_{3, x} \\to 0$", "is a short exact sequence of $\\mathcal{O}_{X, x}$-modules.", "Hence the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-weakly-ass}." ], "refs": [ "algebra-lemma-weakly-ass" ], "ref_ids": [ 722 ] } ], "ref_ids": [] }, { "id": 7875, "type": "theorem", "label": "divisors-lemma-weakly-ass-zero", "categories": [ "divisors" ], "title": "divisors-lemma-weakly-ass-zero", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then", "$$", "\\mathcal{F} = (0) \\Leftrightarrow \\text{WeakAss}(\\mathcal{F}) = \\emptyset", "$$" ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-weakly-associated-affine-open}", "and", "Algebra, Lemma \\ref{algebra-lemma-weakly-ass-zero}" ], "refs": [ "divisors-lemma-weakly-associated-affine-open", "algebra-lemma-weakly-ass-zero" ], "ref_ids": [ 7872, 723 ] } ], "ref_ids": [] }, { "id": 7876, "type": "theorem", "label": "divisors-lemma-restriction-injective-open-contains-weakly-ass", "categories": [ "divisors" ], "title": "divisors-lemma-restriction-injective-open-contains-weakly-ass", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. If $\\text{WeakAss}(\\mathcal{F}) \\subset U \\subset X$", "is open, then $\\Gamma(X, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{F})$", "is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $s \\in \\Gamma(X, \\mathcal{F})$ be a section which restricts to zero on $U$.", "Let $\\mathcal{F}' \\subset \\mathcal{F}$ be the image of the map", "$\\mathcal{O}_X \\to \\mathcal{F}$ defined by $s$. Then", "$\\text{Supp}(\\mathcal{F}') \\cap U = \\emptyset$. On the other hand,", "$\\text{WeakAss}(\\mathcal{F}') \\subset \\text{WeakAss}(\\mathcal{F})$", "by Lemma \\ref{lemma-ses-weakly-ass}. Since also", "$\\text{WeakAss}(\\mathcal{F}') \\subset \\text{Supp}(\\mathcal{F}')$", "(Lemma \\ref{lemma-weakly-ass-support}) we conclude", "$\\text{WeakAss}(\\mathcal{F}') = \\emptyset$.", "Hence $\\mathcal{F}' = 0$ by Lemma \\ref{lemma-weakly-ass-zero}." ], "refs": [ "divisors-lemma-ses-weakly-ass", "divisors-lemma-weakly-ass-support", "divisors-lemma-weakly-ass-zero" ], "ref_ids": [ 7874, 7873, 7875 ] } ], "ref_ids": [] }, { "id": 7877, "type": "theorem", "label": "divisors-lemma-minimal-support-in-weakly-ass", "categories": [ "divisors" ], "title": "divisors-lemma-minimal-support-in-weakly-ass", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $x \\in \\text{Supp}(\\mathcal{F})$ be a point in the support", "of $\\mathcal{F}$ which is not a specialization of another point of", "$\\text{Supp}(\\mathcal{F})$. Then", "$x \\in \\text{WeakAss}(\\mathcal{F})$.", "In particular, any generic point of an irreducible component of $X$", "is weakly associated to $\\mathcal{O}_X$." ], "refs": [], "proofs": [ { "contents": [ "Since $x \\in \\text{Supp}(\\mathcal{F})$ the module $\\mathcal{F}_x$", "is not zero. Hence", "$\\text{WeakAss}(\\mathcal{F}_x) \\subset \\Spec(\\mathcal{O}_{X, x})$", "is nonempty by", "Algebra, Lemma \\ref{algebra-lemma-weakly-ass-zero}.", "On the other hand, by assumption", "$\\text{Supp}(\\mathcal{F}_x) = \\{\\mathfrak m_x\\}$.", "Since", "$\\text{WeakAss}(\\mathcal{F}_x) \\subset \\text{Supp}(\\mathcal{F}_x)$", "(Algebra, Lemma \\ref{algebra-lemma-weakly-ass-support})", "we see that $\\mathfrak m_x$ is weakly associated to $\\mathcal{F}_x$", "and we win." ], "refs": [ "algebra-lemma-weakly-ass-zero", "algebra-lemma-weakly-ass-support" ], "ref_ids": [ 723, 724 ] } ], "ref_ids": [] }, { "id": 7878, "type": "theorem", "label": "divisors-lemma-ass-weakly-ass", "categories": [ "divisors" ], "title": "divisors-lemma-ass-weakly-ass", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "If $\\mathfrak m_x$ is a finitely generated ideal of $\\mathcal{O}_{X, x}$,", "then", "$$", "x \\in \\text{Ass}(\\mathcal{F}) \\Leftrightarrow", "x \\in \\text{WeakAss}(\\mathcal{F}).", "$$", "In particular, if $X$ is locally Noetherian, then", "$\\text{Ass}(\\mathcal{F}) = \\text{WeakAss}(\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "See", "Algebra, Lemma \\ref{algebra-lemma-ass-weakly-ass}." ], "refs": [ "algebra-lemma-ass-weakly-ass" ], "ref_ids": [ 727 ] } ], "ref_ids": [] }, { "id": 7879, "type": "theorem", "label": "divisors-lemma-weakass-pushforward", "categories": [ "divisors" ], "title": "divisors-lemma-weakass-pushforward", "contents": [ "Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $s \\in S$ be a point which is not in the image of $f$. Then", "$s$ is not weakly associated to $f_*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the base change $f' : X' \\to \\Spec(\\mathcal{O}_{S, s})$", "of $f$ by the morphism $g : \\Spec(\\mathcal{O}_{S, s}) \\to S$", "and denote $g' : X' \\to X$ the other projection.", "Then", "$$", "(f_*\\mathcal{F})_s = (g^*f_*\\mathcal{F})_s = (f'_*(g')^*\\mathcal{F})_s", "$$", "The first equality because $g$ induces an isomorphism on local", "rings at $s$ and the second by flat base change (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology}). Of course", "$s \\in \\Spec(\\mathcal{O}_{S, s})$ is not in the image of $f'$.", "Thus we may assume $S$ is the spectrum of a local ring", "$(A, \\mathfrak m)$ and $s$ corresponds to $\\mathfrak m$.", "By Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "the sheaf $f_*\\mathcal{F}$ is quasi-coherent, say corresponding", "to the $A$-module $M$. As $s$ is not in the image of $f$ we see that", "$X = \\bigcup_{a \\in \\mathfrak m} f^{-1}D(a)$ is an open covering.", "Since $X$ is quasi-compact we can find $a_1, \\ldots, a_n \\in \\mathfrak m$", "such that $X = f^{-1}D(a_1) \\cup \\ldots \\cup f^{-1}D(a_n)$. It follows", "that", "$$", "M \\to M_{a_1} \\oplus \\ldots \\oplus M_{a_r}", "$$", "is injective. Hence for any nonzero element $m$ of the stalk $M_\\mathfrak p$", "there exists an $i$ such that $a_i^n m$ is nonzero for all $n \\geq 0$.", "Thus $\\mathfrak m$ is not weakly associated to $M$." ], "refs": [ "coherent-lemma-flat-base-change-cohomology", "schemes-lemma-push-forward-quasi-coherent" ], "ref_ids": [ 3298, 7730 ] } ], "ref_ids": [] }, { "id": 7880, "type": "theorem", "label": "divisors-lemma-check-injective-on-weakass", "categories": [ "divisors" ], "title": "divisors-lemma-check-injective-on-weakass", "contents": [ "Let $X$ be a scheme. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of", "quasi-coherent $\\mathcal{O}_X$-modules. Assume that for every $x \\in X$", "at least one of the following happens", "\\begin{enumerate}", "\\item $\\mathcal{F}_x \\to \\mathcal{G}_x$ is injective, or", "\\item $x \\not \\in \\text{WeakAss}(\\mathcal{F})$.", "\\end{enumerate}", "Then $\\varphi$ is injective." ], "refs": [], "proofs": [ { "contents": [ "The assumptions imply that $\\text{WeakAss}(\\Ker(\\varphi)) = \\emptyset$", "and hence $\\Ker(\\varphi) = 0$ by Lemma \\ref{lemma-weakly-ass-zero}." ], "refs": [ "divisors-lemma-weakly-ass-zero" ], "ref_ids": [ 7875 ] } ], "ref_ids": [] }, { "id": 7881, "type": "theorem", "label": "divisors-lemma-depth-2-hartog", "categories": [ "divisors" ], "title": "divisors-lemma-depth-2-hartog", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$", "be a coherent $\\mathcal{O}_X$-module. Let $j : U \\to X$", "be an open subscheme such that for $x \\in X \\setminus U$", "we have $\\text{depth}(\\mathcal{F}_x) \\geq 2$. Then", "$$", "\\mathcal{F} \\longrightarrow j_*(\\mathcal{F}|_U)", "$$", "is an isomorphism and consequently", "$\\Gamma(X, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{F})$", "is an isomorphism too." ], "refs": [], "proofs": [ { "contents": [ "We claim Lemma \\ref{lemma-check-isomorphism-via-depth-and-ass}", "applies to the map displayed in the lemma.", "Let $x \\in X$. If $x \\in U$, then the map is an", "isomorphism on stalks as $j_*(\\mathcal{F}|_U)|_U = \\mathcal{F}|_U$.", "If $x \\in X \\setminus U$, then $x \\not \\in \\text{Ass}(j_*(\\mathcal{F}|_U))$", "(Lemmas \\ref{lemma-weakass-pushforward} and \\ref{lemma-weakly-ass-support}).", "Since we've assumed $\\text{depth}(\\mathcal{F}_x) \\geq 2$", "this finishes the proof." ], "refs": [ "divisors-lemma-check-isomorphism-via-depth-and-ass", "divisors-lemma-weakass-pushforward", "divisors-lemma-weakly-ass-support" ], "ref_ids": [ 7864, 7879, 7873 ] } ], "ref_ids": [] }, { "id": 7882, "type": "theorem", "label": "divisors-lemma-weakass-reduced", "categories": [ "divisors" ], "title": "divisors-lemma-weakass-reduced", "contents": [ "Let $X$ be a reduced scheme. Then the weakly associated points of $X$", "are exactly the generic points of the irreducible components of $X$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Algebra, Lemma \\ref{algebra-lemma-reduced-weakly-ass-minimal}." ], "refs": [ "algebra-lemma-reduced-weakly-ass-minimal" ], "ref_ids": [ 721 ] } ], "ref_ids": [] }, { "id": 7883, "type": "theorem", "label": "divisors-lemma-weakly-ass-reverse-functorial", "categories": [ "divisors" ], "title": "divisors-lemma-weakly-ass-reverse-functorial", "contents": [ "Let $f : X \\to S$ be an affine morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then we have", "$$", "\\text{WeakAss}_S(f_*\\mathcal{F}) \\subset f(\\text{WeakAss}_X(\\mathcal{F}))", "$$" ], "refs": [], "proofs": [ { "contents": [ "We may assume $X$ and $S$ affine, so $X \\to S$ comes from a ring map", "$A \\to B$. Then $\\mathcal{F} = \\widetilde M$ for some $B$-module $M$. By", "Lemma \\ref{lemma-weakly-associated-affine-open}", "the weakly associated points of $\\mathcal{F}$ correspond exactly to the", "weakly associated primes of $M$. Similarly, the weakly associated points", "of $f_*\\mathcal{F}$ correspond exactly to the weakly associated primes", "of $M$ as an $A$-module. Hence the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-weakly-ass-reverse-functorial}." ], "refs": [ "divisors-lemma-weakly-associated-affine-open", "algebra-lemma-weakly-ass-reverse-functorial" ], "ref_ids": [ 7872, 728 ] } ], "ref_ids": [] }, { "id": 7884, "type": "theorem", "label": "divisors-lemma-ass-functorial-equal", "categories": [ "divisors" ], "title": "divisors-lemma-ass-functorial-equal", "contents": [ "Let $f : X \\to S$ be an affine morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "If $X$ is locally Noetherian, then we have", "$$", "f(\\text{Ass}_X(\\mathcal{F})) =", "\\text{Ass}_S(f_*\\mathcal{F}) =", "\\text{WeakAss}_S(f_*\\mathcal{F}) =", "f(\\text{WeakAss}_X(\\mathcal{F}))", "$$" ], "refs": [], "proofs": [ { "contents": [ "We may assume $X$ and $S$ affine, so $X \\to S$ comes from a ring map", "$A \\to B$. As $X$ is locally Noetherian the ring $B$ is Noetherian, see", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian}.", "Write $\\mathcal{F} = \\widetilde M$ for some $B$-module $M$. By", "Lemma \\ref{lemma-associated-affine-open}", "the associated points of $\\mathcal{F}$ correspond exactly to the associated", "primes of $M$, and any associated prime of $M$ as an $A$-module is an", "associated points of $f_*\\mathcal{F}$.", "Hence the inclusion", "$$", "f(\\text{Ass}_X(\\mathcal{F})) \\subset \\text{Ass}_S(f_*\\mathcal{F})", "$$", "follows from", "Algebra, Lemma \\ref{algebra-lemma-ass-functorial-Noetherian}.", "We have the inclusion", "$$", "\\text{Ass}_S(f_*\\mathcal{F}) \\subset \\text{WeakAss}_S(f_*\\mathcal{F})", "$$", "by", "Lemma \\ref{lemma-weakly-ass-support}.", "We have the inclusion", "$$", "\\text{WeakAss}_S(f_*\\mathcal{F}) \\subset f(\\text{WeakAss}_X(\\mathcal{F}))", "$$", "by", "Lemma \\ref{lemma-weakly-ass-reverse-functorial}.", "The outer sets are equal by", "Lemma \\ref{lemma-ass-weakly-ass}", "hence we have equality everywhere." ], "refs": [ "properties-lemma-locally-Noetherian", "divisors-lemma-associated-affine-open", "algebra-lemma-ass-functorial-Noetherian", "divisors-lemma-weakly-ass-support", "divisors-lemma-weakly-ass-reverse-functorial", "divisors-lemma-ass-weakly-ass" ], "ref_ids": [ 2951, 7856, 707, 7873, 7883, 7878 ] } ], "ref_ids": [] }, { "id": 7885, "type": "theorem", "label": "divisors-lemma-weakly-associated-finite", "categories": [ "divisors" ], "title": "divisors-lemma-weakly-associated-finite", "contents": [ "Let $f : X \\to S$ be a finite morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then $\\text{WeakAss}(f_*\\mathcal{F}) = f(\\text{WeakAss}(\\mathcal{F}))$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $X$ and $S$ affine, so $X \\to S$ comes from a finite ring map", "$A \\to B$. Write $\\mathcal{F} = \\widetilde M$ for some $B$-module $M$. By", "Lemma \\ref{lemma-weakly-associated-affine-open}", "the weakly associated points of $\\mathcal{F}$ correspond exactly to the", "weakly associated primes of $M$. Similarly, the weakly associated points", "of $f_*\\mathcal{F}$ correspond exactly to the weakly associated primes", "of $M$ as an $A$-module. Hence the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-weakly-ass-finite-ring-map}." ], "refs": [ "divisors-lemma-weakly-associated-affine-open", "algebra-lemma-weakly-ass-finite-ring-map" ], "ref_ids": [ 7872, 729 ] } ], "ref_ids": [] }, { "id": 7886, "type": "theorem", "label": "divisors-lemma-weakly-ass-pullback", "categories": [ "divisors" ], "title": "divisors-lemma-weakly-ass-pullback", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{G}$ be a", "quasi-coherent $\\mathcal{O}_S$-module. Let $x \\in X$ with $s = f(x)$.", "If $f$ is flat at $x$, the point $x$ is a generic point of the fibre $X_s$, and", "$s \\in \\text{WeakAss}_S(\\mathcal{G})$, then", "$x \\in \\text{WeakAss}(f^*\\mathcal{G})$." ], "refs": [], "proofs": [ { "contents": [ "Let $A = \\mathcal{O}_{S, s}$, $B = \\mathcal{O}_{X, x}$, and", "$M = \\mathcal{G}_s$. Let $m \\in M$ be an element whose annihilator", "$I = \\{a \\in A \\mid am = 0\\}$ has radical $\\mathfrak m_A$. Then", "$m \\otimes 1$ has annihilator $I B$ as $A \\to B$ is", "faithfully flat. Thus it suffices to see that $\\sqrt{I B} = \\mathfrak m_B$.", "This follows from the fact that the maximal ideal of $B/\\mathfrak m_AB$", "is locally nilpotent (see", "Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring})", "and the assumption that $\\sqrt{I} = \\mathfrak m_A$.", "Some details omitted." ], "refs": [ "algebra-lemma-minimal-prime-reduced-ring" ], "ref_ids": [ 418 ] } ], "ref_ids": [] }, { "id": 7887, "type": "theorem", "label": "divisors-lemma-weakly-ass-change-fields", "categories": [ "divisors" ], "title": "divisors-lemma-weakly-ass-change-fields", "contents": [ "Let $K/k$ be a field extension. Let $X$ be a scheme over $k$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $y \\in X_K$ with image $x \\in X$. If $y$ is a weakly", "associated point of the pullback $\\mathcal{F}_K$, then $x$", "is a weakly associated point of $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "This is the translation of", "Algebra, Lemma \\ref{algebra-lemma-weakly-ass-change-fields}", "into the language of schemes." ], "refs": [ "algebra-lemma-weakly-ass-change-fields" ], "ref_ids": [ 735 ] } ], "ref_ids": [] }, { "id": 7888, "type": "theorem", "label": "divisors-lemma-depth-pushforward", "categories": [ "divisors" ], "title": "divisors-lemma-depth-pushforward", "contents": [ "Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $s \\in S$.", "\\begin{enumerate}", "\\item If $s \\not \\in f(X)$, then $s$ is not weakly associated", "to $f_*\\mathcal{F}$.", "\\item If $s \\not \\in f(X)$ and $\\mathcal{O}_{S, s}$ is Noetherian,", "then $s$ is not associated to $f_*\\mathcal{F}$.", "\\item If $s \\not \\in f(X)$, $(f_*\\mathcal{F})_s$ is a finite", "$\\mathcal{O}_{S, s}$-module, and $\\mathcal{O}_{S, s}$", "is Noetherian, then $\\text{depth}((f_*\\mathcal{F})_s) \\geq 2$.", "\\item If $\\mathcal{F}$ is flat over $S$ and $a \\in \\mathfrak m_s$", "is a nonzerodivisor, then $a$ is a nonzerodivisor on $(f_*\\mathcal{F})_s$.", "\\item If $\\mathcal{F}$ is flat over $S$ and $a, b \\in \\mathfrak m_s$", "is a regular sequence, then $a$ is a nonzerodivisor on $(f_*\\mathcal{F})_s$", "and $b$ is a nonzerodivisor on $(f_*\\mathcal{F})_s/a(f_*\\mathcal{F})_s$.", "\\item If $\\mathcal{F}$ is flat over $S$ and $(f_*\\mathcal{F})_s$", "is a finite $\\mathcal{O}_{S, s}$-module, then", "$\\text{depth}((f_*\\mathcal{F})_s) \\geq", "\\min(2, \\text{depth}(\\mathcal{O}_{S, s}))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is Lemma \\ref{lemma-weakass-pushforward}.", "Part (2) follows from (1) and Lemma \\ref{lemma-ass-weakly-ass}.", "\\medskip\\noindent", "Proof of part (3). To show the depth is $\\geq 2$ it suffices to show that", "$\\Hom_{\\mathcal{O}_{S, s}}(\\kappa(s), (f_*\\mathcal{F})_s) = 0$ and", "$\\Ext^1_{\\mathcal{O}_{S, s}}(\\kappa(s), (f_*\\mathcal{F})_s) = 0$, see", "Algebra, Lemma \\ref{algebra-lemma-depth-ext}.", "Using the exact sequence", "$0 \\to \\mathfrak m_s \\to \\mathcal{O}_{S, s} \\to \\kappa(s) \\to 0$", "it suffices to prove that the map", "$$", "\\Hom_{\\mathcal{O}_{S, s}}(\\mathcal{O}_{S, s}, (f_*\\mathcal{F})_s)", "\\to", "\\Hom_{\\mathcal{O}_{S, s}}(\\mathfrak m_s, (f_*\\mathcal{F})_s)", "$$", "is an isomorphism. By flat base change (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology})", "we may replace $S$ by", "$\\Spec(\\mathcal{O}_{S, s})$ and $X$ by $\\Spec(\\mathcal{O}_{S, s}) \\times_S X$.", "Denote $\\mathfrak m \\subset \\mathcal{O}_S$ the ideal sheaf of $s$.", "Then we see that", "$$", "\\Hom_{\\mathcal{O}_{S, s}}(\\mathfrak m_s, (f_*\\mathcal{F})_s) =", "\\Hom_{\\mathcal{O}_S}(\\mathfrak m, f_*\\mathcal{F}) =", "\\Hom_{\\mathcal{O}_X}(f^*\\mathfrak m, \\mathcal{F})", "$$", "the first equality because $S$ is local with closed point $s$", "and the second equality", "by adjunction for $f^*, f_*$ on quasi-coherent modules. However, since", "$s \\not \\in f(X)$ we see that $f^*\\mathfrak m = \\mathcal{O}_X$.", "Working backwards through the arguments we get the desired equality.", "\\medskip\\noindent", "For the proof of (4), (5), and (6) we use flat base change", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology})", "to reduce to the case where $S$ is the spectrum of", "$\\mathcal{O}_{S, s}$.", "Then a nonzerodivisor $a \\in \\mathcal{O}_{S, s}$", "deterimines a short exact sequence", "$$", "0 \\to \\mathcal{O}_S \\xrightarrow{a} \\mathcal{O}_S \\to", "\\mathcal{O}_S/a \\mathcal{O}_S \\to 0", "$$", "Since $\\mathcal{F}$ is flat over $S$, we obtain an exact sequence", "$$", "0 \\to \\mathcal{F} \\xrightarrow{a} \\mathcal{F} \\to", "\\mathcal{F}/a\\mathcal{F} \\to 0", "$$", "Pushing forward we obtain an exact sequence", "$$", "0 \\to f_*\\mathcal{F} \\xrightarrow{a} f_*\\mathcal{F} \\to", "f_*(\\mathcal{F}/a\\mathcal{F})", "$$", "This proves (4) and it shows that", "$f_*\\mathcal{F}/ af_*\\mathcal{F} \\subset f_*(\\mathcal{F}/a\\mathcal{F})$.", "If $b$ is a nonzerodivisor on", "$\\mathcal{O}_{S, s}/a\\mathcal{O}_{S, s}$, then the exact same argument shows", "$b : \\mathcal{F}/a\\mathcal{F} \\to \\mathcal{F}/a\\mathcal{F}$", "is injective. Pushing forward we conclude", "$$", "b : f_*(\\mathcal{F}/a\\mathcal{F}) \\to f_*(\\mathcal{F}/a\\mathcal{F})", "$$", "is injective and hence also", "$b : f_*\\mathcal{F}/ af_*\\mathcal{F} \\to f_*\\mathcal{F}/ af_*\\mathcal{F}$", "is injective. This proves (5). Part (6) follows from", "(4) and (5) and the definitions." ], "refs": [ "divisors-lemma-weakass-pushforward", "divisors-lemma-ass-weakly-ass", "algebra-lemma-depth-ext", "coherent-lemma-flat-base-change-cohomology", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 7879, 7878, 772, 3298, 3298 ] } ], "ref_ids": [] }, { "id": 7889, "type": "theorem", "label": "divisors-lemma-relative-assassin-affine-open", "categories": [ "divisors" ], "title": "divisors-lemma-relative-assassin-affine-open", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $U \\subset X$ and $V \\subset S$ be affine opens", "with $f(U) \\subset V$. Write $U = \\Spec(A)$, $V = \\Spec(R)$, and set", "$M = \\Gamma(U, \\mathcal{F})$.", "Let $x \\in U$, and let $\\mathfrak p \\subset A$ be the corresponding prime.", "Then", "$$", "\\mathfrak p \\in \\text{Ass}_{A/R}(M) \\Rightarrow", "x \\in \\text{Ass}_{X/S}(\\mathcal{F})", "$$", "If all fibres $X_s$ of $f$ are locally Noetherian, then", "$\\mathfrak p \\in \\text{Ass}_{A/R}(M) \\Leftrightarrow", "x \\in \\text{Ass}_{X/S}(\\mathcal{F})$", "for all pairs $(\\mathfrak p, x)$ as above." ], "refs": [], "proofs": [ { "contents": [ "The set $\\text{Ass}_{A/R}(M)$ is defined in", "Algebra, Definition \\ref{algebra-definition-relative-assassin}.", "Choose a pair $(\\mathfrak p, x)$. Let $s = f(x)$.", "Let $\\mathfrak r \\subset R$ be the prime lying under $\\mathfrak p$,", "i.e., the prime corresponding to $s$.", "Let $\\mathfrak p' \\subset A \\otimes_R \\kappa(\\mathfrak r)$", "be the prime whose inverse image is $\\mathfrak p$, i.e.,", "the prime corresponding to $x$ viewed as a point of its fibre $X_s$.", "Then $\\mathfrak p \\in \\text{Ass}_{A/R}(M)$ if and only if", "$\\mathfrak p'$ is an associated prime of", "$M \\otimes_R \\kappa(\\mathfrak r)$, see", "Algebra, Lemma \\ref{algebra-lemma-compare-relative-assassins}.", "Note that the ring $A \\otimes_R \\kappa(\\mathfrak r)$ corresponds to $U_s$", "and the module $M \\otimes_R \\kappa(\\mathfrak r)$ corresponds to the", "quasi-coherent sheaf $\\mathcal{F}_s|_{U_s}$.", "Hence $x$ is an associated point of $\\mathcal{F}_s$", "by Lemma \\ref{lemma-associated-affine-open}.", "The reverse implication holds if $\\mathfrak p'$ is finitely generated", "which is how the last sentence is seen to be true." ], "refs": [ "algebra-definition-relative-assassin", "algebra-lemma-compare-relative-assassins", "divisors-lemma-associated-affine-open" ], "ref_ids": [ 1483, 716, 7856 ] } ], "ref_ids": [] }, { "id": 7890, "type": "theorem", "label": "divisors-lemma-base-change-relative-assassin", "categories": [ "divisors" ], "title": "divisors-lemma-base-change-relative-assassin", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $g : S' \\to S$ be a morphism of schemes.", "Consider the base change diagram", "$$", "\\xymatrix{", "X' \\ar[d] \\ar[r]_{g'} & X \\ar[d] \\\\", "S' \\ar[r]^g & S", "}", "$$", "and set $\\mathcal{F}' = (g')^*\\mathcal{F}$. Let $x' \\in X'$ be a point", "with images $x \\in X$, $s' \\in S'$ and $s \\in S$.", "Assume $f$ locally of finite type.", "Then $x' \\in \\text{Ass}_{X'/S'}(\\mathcal{F}')$ if and only if", "$x \\in \\text{Ass}_{X/S}(\\mathcal{F})$ and $x'$ corresponds to", "a generic point of an irreducible component of", "$\\Spec(\\kappa(s') \\otimes_{\\kappa(s)} \\kappa(x))$." ], "refs": [], "proofs": [ { "contents": [ "Consider the morphism $X'_{s'} \\to X_s$ of fibres. As", "$X_{s'} = X_s \\times_{\\Spec(\\kappa(s))} \\Spec(\\kappa(s'))$", "this is a flat morphism. Moreover $\\mathcal{F}'_{s'}$ is the pullback", "of $\\mathcal{F}_s$ via this morphism. As $X_s$ is locally of finite", "type over the Noetherian scheme $\\Spec(\\kappa(s))$ we have that", "$X_s$ is locally Noetherian, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}.", "Thus we may apply", "Lemma \\ref{lemma-bourbaki}", "and we see that", "$$", "\\text{Ass}_{X'_{s'}}(\\mathcal{F}'_{s'}) =", "\\bigcup\\nolimits_{x \\in \\text{Ass}(\\mathcal{F}_s)} \\text{Ass}((X'_{s'})_x).", "$$", "Thus to prove the lemma it suffices to show that the associated points", "of the fibre $(X'_{s'})_x$ of the morphism $X'_{s'} \\to X_s$ over $x$", "are its generic points. Note that", "$(X'_{s'})_x = \\Spec(\\kappa(s') \\otimes_{\\kappa(s)} \\kappa(x))$", "as schemes. By", "Algebra, Lemma \\ref{algebra-lemma-tensor-fields-CM}", "the ring $\\kappa(s') \\otimes_{\\kappa(s)} \\kappa(x)$ is a Noetherian", "Cohen-Macaulay ring. Hence its associated primes are its minimal primes, see", "Algebra, Proposition \\ref{algebra-proposition-minimal-primes-associated-primes}", "(minimal primes are associated) and", "Algebra, Lemma \\ref{algebra-lemma-criterion-no-embedded-primes}", "(no embedded primes)." ], "refs": [ "morphisms-lemma-finite-type-noetherian", "divisors-lemma-bourbaki", "algebra-lemma-tensor-fields-CM", "algebra-proposition-minimal-primes-associated-primes", "algebra-lemma-criterion-no-embedded-primes" ], "ref_ids": [ 5202, 7865, 1387, 1412, 1309 ] } ], "ref_ids": [] }, { "id": 7891, "type": "theorem", "label": "divisors-lemma-relative-weak-assassin-assassin-finite-type", "categories": [ "divisors" ], "title": "divisors-lemma-relative-weak-assassin-assassin-finite-type", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then $\\text{WeakAss}_{X/S}(\\mathcal{F}) = \\text{Ass}_{X/S}(\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "This is true because the fibres of $f$ are locally Noetherian schemes,", "and associated and weakly associated points agree on locally Noetherian", "schemes, see", "Lemma \\ref{lemma-ass-weakly-ass}." ], "refs": [ "divisors-lemma-ass-weakly-ass" ], "ref_ids": [ 7878 ] } ], "ref_ids": [] }, { "id": 7892, "type": "theorem", "label": "divisors-lemma-relative-weak-assassin-finite", "categories": [ "divisors" ], "title": "divisors-lemma-relative-weak-assassin-finite", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $i : Z \\to X$ be a finite morphism.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_Z$-module.", "Then $\\text{WeakAss}_{X/S}(i_*\\mathcal{F}) =", "i(\\text{WeakAss}_{Z/S}(\\mathcal{F}))$." ], "refs": [], "proofs": [ { "contents": [ "Let $i_s : Z_s \\to X_s$ be the induced morphism between fibres.", "Then $(i_*\\mathcal{F})_s = i_{s, *}(\\mathcal{F}_s)$ by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-affine-base-change}", "and the fact that $i$ is affine. Hence", "we may apply Lemma \\ref{lemma-weakly-associated-finite} to conclude." ], "refs": [ "coherent-lemma-affine-base-change", "divisors-lemma-weakly-associated-finite" ], "ref_ids": [ 3297, 7885 ] } ], "ref_ids": [] }, { "id": 7893, "type": "theorem", "label": "divisors-lemma-base-change-fitting-ideal", "categories": [ "divisors" ], "title": "divisors-lemma-base-change-fitting-ideal", "contents": [ "Let $f : T \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_S$-module.", "Then", "$f^{-1}\\text{Fit}_i(\\mathcal{F}) \\cdot \\mathcal{O}_T =", "\\text{Fit}_i(f^*\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from More on Algebra, Lemma", "\\ref{more-algebra-lemma-fitting-ideal-basics} part (3)." ], "refs": [ "more-algebra-lemma-fitting-ideal-basics" ], "ref_ids": [ 9834 ] } ], "ref_ids": [] }, { "id": 7894, "type": "theorem", "label": "divisors-lemma-fitting-ideal-of-finitely-presented", "categories": [ "divisors" ], "title": "divisors-lemma-fitting-ideal-of-finitely-presented", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_S$-module.", "Then $\\text{Fit}_r(\\mathcal{F})$ is a quasi-coherent ideal of finite type." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from More on Algebra, Lemma", "\\ref{more-algebra-lemma-fitting-ideal-basics} part (4)." ], "refs": [ "more-algebra-lemma-fitting-ideal-basics" ], "ref_ids": [ 9834 ] } ], "ref_ids": [] }, { "id": 7895, "type": "theorem", "label": "divisors-lemma-on-subscheme-cut-out-by-Fit-0", "categories": [ "divisors" ], "title": "divisors-lemma-on-subscheme-cut-out-by-Fit-0", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_S$-module.", "Let $Z_0 \\subset S$ be the closed subscheme cut out by", "$\\text{Fit}_0(\\mathcal{F})$.", "Let $Z \\subset S$ be the scheme theoretic support of $\\mathcal{F}$.", "Then", "\\begin{enumerate}", "\\item $Z \\subset Z_0 \\subset S$ as closed subschemes,", "\\item $Z = Z_0 = \\text{Supp}(\\mathcal{F})$ as closed subsets,", "\\item there exists a finite type, quasi-coherent $\\mathcal{O}_{Z_0}$-module", "$\\mathcal{G}_0$ with", "$$", "(Z_0 \\to X)_*\\mathcal{G}_0 = \\mathcal{F}.", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that $Z$ is locally cut out by the annihilator of $\\mathcal{F}$, see", "Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-support}", "(which uses Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-support}", "to define $Z$). Hence we see that $Z \\subset Z_0$ scheme theoretically", "by More on Algebra, Lemma", "\\ref{more-algebra-lemma-fitting-ideal-basics} part (6).", "On the other hand we have $Z = \\text{Supp}(\\mathcal{F})$", "set theoretically by", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-support}", "and we have $Z_0 = Z$ set theoretically by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-fitting-ideal-basics} part (7).", "Finally, to get $\\mathcal{G}_0$ as in part (3) we can either use", "that we have $\\mathcal{G}$ on $Z$ as in", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-support}", "and set $\\mathcal{G}_0 = (Z \\to Z_0)_*\\mathcal{G}$", "or we can use Morphisms, Lemma \\ref{morphisms-lemma-i-star-equivalence}", "and the fact that $\\text{Fit}_0(\\mathcal{F})$ annihilates", "$\\mathcal{F}$ by More on Algebra, Lemma", "\\ref{more-algebra-lemma-fitting-ideal-basics} part (6)." ], "refs": [ "morphisms-definition-scheme-theoretic-support", "morphisms-lemma-scheme-theoretic-support", "more-algebra-lemma-fitting-ideal-basics", "morphisms-lemma-scheme-theoretic-support", "more-algebra-lemma-fitting-ideal-basics", "morphisms-lemma-scheme-theoretic-support", "morphisms-lemma-i-star-equivalence", "more-algebra-lemma-fitting-ideal-basics" ], "ref_ids": [ 5538, 5144, 9834, 5144, 9834, 5144, 5136, 9834 ] } ], "ref_ids": [] }, { "id": 7896, "type": "theorem", "label": "divisors-lemma-fitting-ideal-generate-locally", "categories": [ "divisors" ], "title": "divisors-lemma-fitting-ideal-generate-locally", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type, quasi-coherent", "$\\mathcal{O}_S$-module. Let $s \\in S$. Then $\\mathcal{F}$ can be", "generated by $r$ elements in a neighbourhood of $s$ if and only", "if $\\text{Fit}_r(\\mathcal{F})_s = \\mathcal{O}_{S, s}$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "More on Algebra, Lemma \\ref{more-algebra-lemma-fitting-ideal-generate-locally}." ], "refs": [ "more-algebra-lemma-fitting-ideal-generate-locally" ], "ref_ids": [ 9835 ] } ], "ref_ids": [] }, { "id": 7897, "type": "theorem", "label": "divisors-lemma-fitting-ideal-finite-locally-free", "categories": [ "divisors" ], "title": "divisors-lemma-fitting-ideal-finite-locally-free", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type, quasi-coherent", "$\\mathcal{O}_S$-module. Let $r \\geq 0$. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is finite locally free of rank $r$", "\\item $\\text{Fit}_{r - 1}(\\mathcal{F}) = 0$ and", "$\\text{Fit}_r(\\mathcal{F}) = \\mathcal{O}_S$, and", "\\item $\\text{Fit}_k(\\mathcal{F}) = 0$ for $k < r$ and", "$\\text{Fit}_k(\\mathcal{F}) = \\mathcal{O}_S$ for $k \\geq r$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-fitting-ideal-finite-locally-free}." ], "refs": [ "more-algebra-lemma-fitting-ideal-finite-locally-free" ], "ref_ids": [ 9836 ] } ], "ref_ids": [] }, { "id": 7898, "type": "theorem", "label": "divisors-lemma-locally-free-rank-r-pullback", "categories": [ "divisors" ], "title": "divisors-lemma-locally-free-rank-r-pullback", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type, quasi-coherent", "$\\mathcal{O}_S$-module. The closed subschemes", "$$", "S = Z_{-1} \\supset Z_0 \\supset Z_1 \\supset Z_2 \\ldots", "$$", "defined by the Fitting ideals of $\\mathcal{F}$ have the following", "properties", "\\begin{enumerate}", "\\item The intersection $\\bigcap Z_r$ is empty.", "\\item The functor $(\\Sch/S)^{opp} \\to \\textit{Sets}$ defined by the rule", "$$", "T \\longmapsto", "\\left\\{", "\\begin{matrix}", "\\{*\\} & \\text{if }\\mathcal{F}_T\\text{ is locally generated by }", "\\leq r\\text{ sections} \\\\", "\\emptyset & \\text{otherwise}", "\\end{matrix}", "\\right.", "$$", "is representable by the open subscheme $S \\setminus Z_r$.", "\\item The functor $F_r : (\\Sch/S)^{opp} \\to \\textit{Sets}$ defined by the rule", "$$", "T \\longmapsto", "\\left\\{", "\\begin{matrix}", "\\{*\\} & \\text{if }\\mathcal{F}_T\\text{ locally free rank }r\\\\", "\\emptyset & \\text{otherwise}", "\\end{matrix}", "\\right.", "$$", "is representable by the locally closed subscheme $Z_{r - 1} \\setminus Z_r$", "of $S$.", "\\end{enumerate}", "If $\\mathcal{F}$ is of finite presentation, then", "$Z_r \\to S$, $S \\setminus Z_r \\to S$, and $Z_{r - 1} \\setminus Z_r \\to S$", "are of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Part (1) is true because over every affine open $U$ there is an integer $n$", "such that $\\text{Fit}_n(\\mathcal{F})|_U = \\mathcal{O}_U$. Namely, we can", "take $n$ to be the number of generators of $\\mathcal{F}$ over $U$, see", "More on Algebra, Section \\ref{more-algebra-section-fitting-ideals}.", "\\medskip\\noindent", "For any morphism $g : T \\to S$ we see from", "Lemmas \\ref{lemma-base-change-fitting-ideal} and", "\\ref{lemma-fitting-ideal-generate-locally}", "that $\\mathcal{F}_T$ is locally generated by $\\leq r$ sections if and only if", "$\\text{Fit}_r(\\mathcal{F}) \\cdot \\mathcal{O}_T = \\mathcal{O}_T$.", "This proves (2).", "\\medskip\\noindent", "For any morphism $g : T \\to S$ we see from", "Lemmas \\ref{lemma-base-change-fitting-ideal} and", "\\ref{lemma-fitting-ideal-finite-locally-free}", "that $\\mathcal{F}_T$ is free of rank $r$ if and only if", "$\\text{Fit}_r(\\mathcal{F}) \\cdot \\mathcal{O}_T = \\mathcal{O}_T$ and", "$\\text{Fit}_{r - 1}(\\mathcal{F}) \\cdot \\mathcal{O}_T = 0$.", "This proves (3).", "\\medskip\\noindent", "Part (4) follows from the fact that if", "$\\mathcal{F}$ is of finite presentation, then each of the morphisms", "$Z_r \\to S$ is of finite presentation as $\\text{Fit}_r(\\mathcal{F})$", "is of finite type (Lemma \\ref{lemma-fitting-ideal-of-finitely-presented} and", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-finite-presentation}).", "This implies that $Z_{r - 1} \\setminus Z_r$ is a retrocompact open in $Z_r$", "(Properties, Lemma \\ref{properties-lemma-quasi-coherent-finite-type-ideals})", "and hence the morphism $Z_{r - 1} \\setminus Z_r \\to Z_r$", "is of finite presentation as well." ], "refs": [ "divisors-lemma-base-change-fitting-ideal", "divisors-lemma-fitting-ideal-generate-locally", "divisors-lemma-base-change-fitting-ideal", "divisors-lemma-fitting-ideal-finite-locally-free", "divisors-lemma-fitting-ideal-of-finitely-presented", "morphisms-lemma-closed-immersion-finite-presentation", "properties-lemma-quasi-coherent-finite-type-ideals" ], "ref_ids": [ 7893, 7896, 7893, 7897, 7894, 5243, 3033 ] } ], "ref_ids": [] }, { "id": 7899, "type": "theorem", "label": "divisors-lemma-finite-presentation-module", "categories": [ "divisors" ], "title": "divisors-lemma-finite-presentation-module", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ be an $\\mathcal{O}_S$-module", "of finite presentation. Let $S = Z_{-1} \\supset Z_0 \\supset Z_1 \\supset \\ldots$", "be as in Lemma \\ref{lemma-locally-free-rank-r-pullback}.", "Set $S_r = Z_{r - 1} \\setminus Z_r$.", "Then $S' = \\coprod_{r \\geq 0} S_r$ represents the functor", "$$", "F_{flat} : \\Sch/S \\longrightarrow \\textit{Sets},\\quad\\quad", "T \\longmapsto", "\\left\\{", "\\begin{matrix}", "\\{*\\} & \\text{if }\\mathcal{F}_T\\text{ flat over }T\\\\", "\\emptyset & \\text{otherwise}", "\\end{matrix}", "\\right.", "$$", "Moreover, $\\mathcal{F}|_{S_r}$ is locally free of rank $r$ and the", "morphisms $S_r \\to S$ and $S' \\to S$ are of finite presentation." ], "refs": [ "divisors-lemma-locally-free-rank-r-pullback" ], "proofs": [ { "contents": [ "Suppose that $g : T \\to S$ is a morphism of schemes such that the pullback", "$\\mathcal{F}_T = g^*\\mathcal{F}$ is flat. Then $\\mathcal{F}_T$ is a flat", "$\\mathcal{O}_T$-module of finite presentation. Hence", "$\\mathcal{F}_T$ is finite locally free, see", "Properties, Lemma \\ref{properties-lemma-finite-locally-free}.", "Thus $T = \\coprod_{r \\geq 0} T_r$, where $\\mathcal{F}_T|_{T_r}$ is locally", "free of rank $r$. This implies that", "$$", "F_{flat} = \\coprod\\nolimits_{r \\geq 0} F_r", "$$", "in the category of Zariski sheaves on $\\Sch/S$ where $F_r$ is as in", "Lemma \\ref{lemma-locally-free-rank-r-pullback}. It follows", "that $F_{flat}$ is represented by", "$\\coprod_{r \\geq 0} (Z_{r - 1} \\setminus Z_r)$ where", "$Z_r$ is as in", "Lemma \\ref{lemma-locally-free-rank-r-pullback}.", "The other statements also follow from the lemma." ], "refs": [ "properties-lemma-finite-locally-free", "divisors-lemma-locally-free-rank-r-pullback", "divisors-lemma-locally-free-rank-r-pullback" ], "ref_ids": [ 3014, 7898, 7898 ] } ], "ref_ids": [ 7898 ] }, { "id": 7900, "type": "theorem", "label": "divisors-lemma-base-change-and-fitting-ideal-omega", "categories": [ "divisors" ], "title": "divisors-lemma-base-change-and-fitting-ideal-omega", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $X = Z_{-1} \\supset Z_0 \\supset Z_1 \\supset \\ldots$", "be the closed subschemes defined by the fitting ideals", "of $\\Omega_{X/S}$. Then the formation of $Z_i$ commutes", "with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\Omega_{X/S}$ is a finite type quasi-coherent", "$\\mathcal{O}_X$-module", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-type-differentials})", "hence the fitting ideals are defined. If $f' : X' \\to S'$", "is the base change of $f$ by $g : S' \\to S$, then", "$\\Omega_{X'/S'} = (g')^*\\Omega_{X/S}$ where $g' : X' \\to X$", "is the projection", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-differentials}).", "Hence $(g')^{-1}\\text{Fit}_i(\\Omega_{X/S}) \\cdot \\mathcal{O}_{X'} =", "\\text{Fit}_i(\\Omega_{X'/S'})$. This means that", "$$", "Z'_i = (g')^{-1}(Z_i) = Z_i \\times_X X'", "$$", "scheme theoretically and this is the meaning of the statement of", "the lemma." ], "refs": [ "morphisms-lemma-finite-type-differentials", "morphisms-lemma-base-change-differentials" ], "ref_ids": [ 5316, 5314 ] } ], "ref_ids": [] }, { "id": 7901, "type": "theorem", "label": "divisors-lemma-zero-fitting-ideal-omega-unramified", "categories": [ "divisors" ], "title": "divisors-lemma-zero-fitting-ideal-omega-unramified", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "The closed subscheme $Z \\subset X$ cut out by the $0$th fitting ideal of", "$\\Omega_{X/S}$ is exactly the set of points where $f$ is not unramified." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-on-subscheme-cut-out-by-Fit-0} the complement of $Z$", "is exactly the locus where $\\Omega_{X/S}$ is zero. This is exactly", "the set of points where $f$ is unramified by", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-omega-zero}." ], "refs": [ "divisors-lemma-on-subscheme-cut-out-by-Fit-0", "morphisms-lemma-unramified-omega-zero" ], "ref_ids": [ 7895, 5343 ] } ], "ref_ids": [] }, { "id": 7902, "type": "theorem", "label": "divisors-lemma-d-fitting-ideal-omega-smooth", "categories": [ "divisors" ], "title": "divisors-lemma-d-fitting-ideal-omega-smooth", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $d \\geq 0$ be an integer.", "Assume", "\\begin{enumerate}", "\\item $f$ is flat,", "\\item $f$ is locally of finite presentation, and", "\\item every nonempty fibre of $f$ is equidimensional of dimension $d$.", "\\end{enumerate}", "Let $Z \\subset X$ be the closed subscheme cut out by the $d$th fitting", "ideal of $\\Omega_{X/S}$. Then $Z$ is exactly the set of points", "where $f$ is not smooth." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-locally-free-rank-r-pullback} the complement of $Z$", "is exactly the locus where $\\Omega_{X/S}$ can be generated by at most", "$d$ elements. Hence the lemma follows from", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-at-point}." ], "refs": [ "divisors-lemma-locally-free-rank-r-pullback", "morphisms-lemma-smooth-at-point" ], "ref_ids": [ 7898, 5335 ] } ], "ref_ids": [] }, { "id": 7903, "type": "theorem", "label": "divisors-lemma-torsion-sections", "categories": [ "divisors" ], "title": "divisors-lemma-torsion-sections", "contents": [ "Let $X$ be an integral scheme with generic point $\\eta$. Let $\\mathcal{F}$", "be a quasi-coherent $\\mathcal{O}_X$-module. Let $U \\subset X$ be nonempty", "open and $s \\in \\mathcal{F}(U)$. The following are equivalent", "\\begin{enumerate}", "\\item for some $x \\in U$ the image of $s$ in $\\mathcal{F}_x$ is torsion,", "\\item for all $x \\in U$ the image of $s$ in $\\mathcal{F}_x$ is torsion,", "\\item the image of $s$ in $\\mathcal{F}_\\eta$ is zero,", "\\item the image of $s$ in $j_*\\mathcal{F}_\\eta$ is zero, where $j : \\eta \\to X$", "is the inclusion morphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7904, "type": "theorem", "label": "divisors-lemma-check-torsion-on-affines", "categories": [ "divisors" ], "title": "divisors-lemma-check-torsion-on-affines", "contents": [ "Let $X$ be an integral scheme. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is torsion free,", "\\item for $U \\subset X$ affine open $\\mathcal{F}(U)$", "is a torsion free $\\mathcal{O}(U)$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7905, "type": "theorem", "label": "divisors-lemma-torsion", "categories": [ "divisors" ], "title": "divisors-lemma-torsion", "contents": [ "Let $X$ be an integral scheme. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. The torsion sections of $\\mathcal{F}$ form", "a quasi-coherent $\\mathcal{O}_X$-submodule", "$\\mathcal{F}_{tors} \\subset \\mathcal{F}$.", "The quotient module $\\mathcal{F}/\\mathcal{F}_{tors}$ is torsion free." ], "refs": [], "proofs": [ { "contents": [ "Omitted. See More on Algebra, Lemma \\ref{more-algebra-lemma-torsion}", "for the algebraic analogue." ], "refs": [ "more-algebra-lemma-torsion" ], "ref_ids": [ 9912 ] } ], "ref_ids": [] }, { "id": 7906, "type": "theorem", "label": "divisors-lemma-flat-torsion-free", "categories": [ "divisors" ], "title": "divisors-lemma-flat-torsion-free", "contents": [ "Let $X$ be an integral scheme. Any flat quasi-coherent $\\mathcal{O}_X$-module", "is torsion free." ], "refs": [], "proofs": [ { "contents": [ "Omitted. See More on Algebra, Lemma \\ref{more-algebra-lemma-flat-torsion-free}." ], "refs": [ "more-algebra-lemma-flat-torsion-free" ], "ref_ids": [ 9919 ] } ], "ref_ids": [] }, { "id": 7907, "type": "theorem", "label": "divisors-lemma-flat-pullback-torsion", "categories": [ "divisors" ], "title": "divisors-lemma-flat-pullback-torsion", "contents": [ "Let $f : X \\to Y$ be a flat morphism of integral schemes.", "Let $\\mathcal{G}$ be a torsion free quasi-coherent $\\mathcal{O}_Y$-module.", "Then $f^*\\mathcal{G}$ is a torsion free $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "Omitted. See", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-pullback-reflexive}", "for the algebraic analogue." ], "refs": [ "more-algebra-lemma-flat-pullback-reflexive" ], "ref_ids": [ 9928 ] } ], "ref_ids": [] }, { "id": 7908, "type": "theorem", "label": "divisors-lemma-flat-over-integral-integral-fibre", "categories": [ "divisors" ], "title": "divisors-lemma-flat-over-integral-integral-fibre", "contents": [ "Let $f : X \\to Y$ be a flat morphism of schemes. If $Y$ is integral", "and the generic fibre of $f$ is integral, then $X$ is integral." ], "refs": [], "proofs": [ { "contents": [ "The algebraic analogue is this: let $A$ be a domain with fraction", "field $K$ and let $B$ be a flat $A$-algebra such that $B \\otimes_A K$", "is a domain. Then $B$ is a domain. This is true because $B$ is", "torsion free by More on Algebra, Lemma", "\\ref{more-algebra-lemma-flat-torsion-free}", "and hence $B \\subset B \\otimes_A K$." ], "refs": [ "more-algebra-lemma-flat-torsion-free" ], "ref_ids": [ 9919 ] } ], "ref_ids": [] }, { "id": 7909, "type": "theorem", "label": "divisors-lemma-check-torsion", "categories": [ "divisors" ], "title": "divisors-lemma-check-torsion", "contents": [ "Let $X$ be an integral scheme. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is torsion free if and only if", "$\\mathcal{F}_x$ is a torsion free $\\mathcal{O}_{X, x}$-module for all $x \\in X$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. See More on Algebra, Lemma", "\\ref{more-algebra-lemma-check-torsion}." ], "refs": [ "more-algebra-lemma-check-torsion" ], "ref_ids": [ 9916 ] } ], "ref_ids": [] }, { "id": 7910, "type": "theorem", "label": "divisors-lemma-extension-torsion-free", "categories": [ "divisors" ], "title": "divisors-lemma-extension-torsion-free", "contents": [ "Let $X$ be an integral scheme. Let", "$0 \\to \\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{F}'' \\to 0$", "be a short exact sequence of quasi-coherent $\\mathcal{O}_X$-modules.", "If $\\mathcal{F}$ and $\\mathcal{F}''$ are torsion free, then $\\mathcal{F}'$", "is torsion free." ], "refs": [], "proofs": [ { "contents": [ "Omitted. See", "More on Algebra, Lemma \\ref{more-algebra-lemma-extension-torsion-free}", "for the algebraic analogue." ], "refs": [ "more-algebra-lemma-extension-torsion-free" ], "ref_ids": [ 9915 ] } ], "ref_ids": [] }, { "id": 7911, "type": "theorem", "label": "divisors-lemma-torsion-free-finite-noetherian-domain", "categories": [ "divisors" ], "title": "divisors-lemma-torsion-free-finite-noetherian-domain", "contents": [ "Let $X$ be a locally Noetherian integral scheme with generic point $\\eta$.", "Let $\\mathcal{F}$ be a nonzero coherent $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is torsion free,", "\\item $\\eta$ is the only associated prime of $\\mathcal{F}$,", "\\item $\\eta$ is in the support of $\\mathcal{F}$ and $\\mathcal{F}$", "has property $(S_1)$, and", "\\item $\\eta$ is in the support of $\\mathcal{F}$ and $\\mathcal{F}$", "has no embedded associated prime.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a translation of More on Algebra, Lemma", "\\ref{more-algebra-lemma-torsion-free-finite-noetherian-domain}", "into the language of schemes. We omit the translation." ], "refs": [ "more-algebra-lemma-torsion-free-finite-noetherian-domain" ], "ref_ids": [ 9918 ] } ], "ref_ids": [] }, { "id": 7912, "type": "theorem", "label": "divisors-lemma-torsion-free-over-regular-dim-1", "categories": [ "divisors" ], "title": "divisors-lemma-torsion-free-over-regular-dim-1", "contents": [ "Let $X$ be an integral regular scheme of dimension $\\leq 1$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is torsion free,", "\\item $\\mathcal{F}$ is finite locally free.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that a finite locally free module is torsion free.", "For the converse, we will show that if $\\mathcal{F}$ is", "torsion free, then $\\mathcal{F}_x$ is a free $\\mathcal{O}_{X, x}$-module", "for all $x \\in X$. This is enough by", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}", "and the fact that $\\mathcal{F}$ is coherent.", "If $\\dim(\\mathcal{O}_{X, x}) = 0$, then", "$\\mathcal{O}_{X, x}$ is a field and the statement is clear.", "If $\\dim(\\mathcal{O}_{X, x}) = 1$, then $\\mathcal{O}_{X, x}$", "is a discrete valuation ring", "(Algebra, Lemma \\ref{algebra-lemma-characterize-dvr})", "and $\\mathcal{F}_x$ is torsion free.", "Hence $\\mathcal{F}_x$ is free by More on Algebra, Lemma", "\\ref{more-algebra-lemma-dedekind-torsion-free-flat}." ], "refs": [ "algebra-lemma-finite-projective", "algebra-lemma-characterize-dvr", "more-algebra-lemma-dedekind-torsion-free-flat" ], "ref_ids": [ 795, 1023, 9921 ] } ], "ref_ids": [] }, { "id": 7913, "type": "theorem", "label": "divisors-lemma-hom-into-torsion-free", "categories": [ "divisors" ], "title": "divisors-lemma-hom-into-torsion-free", "contents": [ "Let $X$ be an integral scheme. Let $\\mathcal{F}$, $\\mathcal{G}$ be", "quasi-coherent $\\mathcal{O}_X$-modules.", "If $\\mathcal{G}$ is torsion free and $\\mathcal{F}$ is of finite presentation,", "then $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is torsion free." ], "refs": [], "proofs": [ { "contents": [ "The statement makes sense because", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$", "is quasi-coherent by Schemes, Section \\ref{schemes-section-quasi-coherent}.", "To see the statement is true, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-hom-into-torsion-free}.", "Some details omitted." ], "refs": [ "more-algebra-lemma-hom-into-torsion-free" ], "ref_ids": [ 9922 ] } ], "ref_ids": [] }, { "id": 7914, "type": "theorem", "label": "divisors-lemma-isom-depth-2-torsion-free", "categories": [ "divisors" ], "title": "divisors-lemma-isom-depth-2-torsion-free", "contents": [ "Let $X$ be an integral locally Noetherian scheme. Let", "$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of", "quasi-coherent $\\mathcal{O}_X$-modules. Assume $\\mathcal{F}$ is coherent,", "$\\mathcal{G}$ is torsion free, and that for every $x \\in X$ one of the", "following happens", "\\begin{enumerate}", "\\item $\\mathcal{F}_x \\to \\mathcal{G}_x$ is an isomorphism, or", "\\item $\\text{depth}(\\mathcal{F}_x) \\geq 2$.", "\\end{enumerate}", "Then $\\varphi$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is a translation of More on Algebra, Lemma", "\\ref{more-algebra-lemma-isom-depth-2-torsion-free}", "into the language of schemes." ], "refs": [ "more-algebra-lemma-isom-depth-2-torsion-free" ], "ref_ids": [ 9934 ] } ], "ref_ids": [] }, { "id": 7915, "type": "theorem", "label": "divisors-lemma-check-reflexive-on-affines", "categories": [ "divisors" ], "title": "divisors-lemma-check-reflexive-on-affines", "contents": [ "Let $X$ be an integral locally Noetherian scheme. Let $\\mathcal{F}$ be a", "coherent $\\mathcal{O}_X$-module. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is reflexive,", "\\item for $U \\subset X$ affine open $\\mathcal{F}(U)$", "is a reflexive $\\mathcal{O}(U)$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7916, "type": "theorem", "label": "divisors-lemma-reflexive-torsion-free", "categories": [ "divisors" ], "title": "divisors-lemma-reflexive-torsion-free", "contents": [ "Let $X$ be an integral locally Noetherian scheme. Let $\\mathcal{F}$", "be a coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is reflexive, then $\\mathcal{F}$ is torsion free.", "\\item The map $j : \\mathcal{F} \\longrightarrow \\mathcal{F}^{**}$", "is injective if and only if $\\mathcal{F}$ is torsion free", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. See More on Algebra, Lemma", "\\ref{more-algebra-lemma-reflexive-torsion-free}." ], "refs": [ "more-algebra-lemma-reflexive-torsion-free" ], "ref_ids": [ 9923 ] } ], "ref_ids": [] }, { "id": 7917, "type": "theorem", "label": "divisors-lemma-check-reflexive", "categories": [ "divisors" ], "title": "divisors-lemma-check-reflexive", "contents": [ "Let $X$ be an integral locally Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is reflexive,", "\\item $\\mathcal{F}_x$ is a reflexive $\\mathcal{O}_{X, x}$-module", "for all $x \\in X$,", "\\item $\\mathcal{F}_x$ is a reflexive $\\mathcal{O}_{X, x}$-module", "for all closed points $x \\in X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Modules, Lemma \\ref{modules-lemma-stalk-internal-hom} we see that", "(1) and (2) are equivalent. Since every point of $X$ specializes to", "a closed point", "(Properties, Lemma \\ref{properties-lemma-locally-Noetherian-closed-point})", "we see that (2) and (3) are equivalent." ], "refs": [ "modules-lemma-stalk-internal-hom", "properties-lemma-locally-Noetherian-closed-point" ], "ref_ids": [ 13296, 2958 ] } ], "ref_ids": [] }, { "id": 7918, "type": "theorem", "label": "divisors-lemma-flat-pullback-reflexive", "categories": [ "divisors" ], "title": "divisors-lemma-flat-pullback-reflexive", "contents": [ "Let $f : X \\to Y$ be a flat morphism of integral locally Noetherian schemes.", "Let $\\mathcal{G}$ be a coherent reflexive $\\mathcal{O}_Y$-module.", "Then $f^*\\mathcal{G}$ is a coherent reflexive $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "Omitted. See", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-pullback-torsion}", "for the algebraic analogue." ], "refs": [ "more-algebra-lemma-flat-pullback-torsion" ], "ref_ids": [ 9914 ] } ], "ref_ids": [] }, { "id": 7919, "type": "theorem", "label": "divisors-lemma-sequence-reflexive", "categories": [ "divisors" ], "title": "divisors-lemma-sequence-reflexive", "contents": [ "Let $X$ be an integral locally Noetherian scheme.", "Let $0 \\to \\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{F}''$", "an exact sequence of coherent $\\mathcal{O}_X$-modules.", "If $\\mathcal{F}'$ is reflexive and $\\mathcal{F}''$ is torsion free,", "then $\\mathcal{F}$ is reflexive." ], "refs": [], "proofs": [ { "contents": [ "Omitted. See More on Algebra, Lemma \\ref{more-algebra-lemma-sequence-reflexive}." ], "refs": [ "more-algebra-lemma-sequence-reflexive" ], "ref_ids": [ 9926 ] } ], "ref_ids": [] }, { "id": 7920, "type": "theorem", "label": "divisors-lemma-dual-reflexive", "categories": [ "divisors" ], "title": "divisors-lemma-dual-reflexive", "contents": [ "Let $X$ be an integral locally Noetherian scheme.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be", "coherent $\\mathcal{O}_X$-modules.", "If $\\mathcal{G}$ is reflexive,", "then $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is reflexive." ], "refs": [], "proofs": [ { "contents": [ "The statement makes sense because", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$", "is coherent by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-tensor-hom-coherent}.", "To see the statement is true, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-dual-reflexive}.", "Some details omitted." ], "refs": [ "coherent-lemma-tensor-hom-coherent", "more-algebra-lemma-dual-reflexive" ], "ref_ids": [ 3311, 9929 ] } ], "ref_ids": [] }, { "id": 7921, "type": "theorem", "label": "divisors-lemma-reflexive-depth-2", "categories": [ "divisors" ], "title": "divisors-lemma-reflexive-depth-2", "contents": [ "Let $X$ be an integral locally Noetherian scheme. Let $\\mathcal{F}$", "be a coherent $\\mathcal{O}_X$-module. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is reflexive,", "\\item for each $x \\in X$ one of the following happens", "\\begin{enumerate}", "\\item $\\mathcal{F}_x$ is a reflexive $\\mathcal{O}_{X, x}$-module, or", "\\item $\\text{depth}(\\mathcal{F}_x) \\geq 2$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. See More on Algebra, Lemma \\ref{more-algebra-lemma-reflexive-depth-2}." ], "refs": [ "more-algebra-lemma-reflexive-depth-2" ], "ref_ids": [ 9935 ] } ], "ref_ids": [] }, { "id": 7922, "type": "theorem", "label": "divisors-lemma-reflexive-S2", "categories": [ "divisors" ], "title": "divisors-lemma-reflexive-S2", "contents": [ "Let $X$ be an integral locally Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent reflexive $\\mathcal{O}_X$-module.", "Let $x \\in X$.", "\\begin{enumerate}", "\\item If $\\text{depth}(\\mathcal{O}_{X, x}) \\geq 2$, then", "$\\text{depth}(\\mathcal{F}_x) \\geq 2$.", "\\item If $X$ is $(S_2)$, then $\\mathcal{F}$ is $(S_2)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. See More on Algebra, Lemma \\ref{more-algebra-lemma-reflexive-S2}." ], "refs": [ "more-algebra-lemma-reflexive-S2" ], "ref_ids": [ 9936 ] } ], "ref_ids": [] }, { "id": 7923, "type": "theorem", "label": "divisors-lemma-reflexive-S2-extend", "categories": [ "divisors" ], "title": "divisors-lemma-reflexive-S2-extend", "contents": [ "Let $X$ be an integral locally Noetherian scheme. Let $j : U \\to X$", "be an open subscheme with complement $Z$. Assume $\\mathcal{O}_{X, z}$", "has depth $\\geq 2$ for all $z \\in Z$. Then $j^*$ and $j_*$ define", "an equivalence of categories between the category of coherent reflexive", "$\\mathcal{O}_X$-modules and the category of coherent reflexive", "$\\mathcal{O}_U$-modules." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a coherent reflexive $\\mathcal{O}_X$-module. For $z \\in Z$", "the stalk $\\mathcal{F}_z$ has depth $\\geq 2$ by Lemma \\ref{lemma-reflexive-S2}.", "Thus $\\mathcal{F} \\to j_*j^*\\mathcal{F}$ is an isomorphism by", "Lemma \\ref{lemma-depth-2-hartog}. Conversely, let $\\mathcal{G}$", "be a coherent reflexive $\\mathcal{O}_U$-module. It suffices to show that", "$j_*\\mathcal{G}$ is a coherent reflexive $\\mathcal{O}_X$-module.", "To prove this we may assume $X$ is affine. By Properties, Lemma", "\\ref{properties-lemma-lift-finite-presentation}", "there exists a coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "with $\\mathcal{G} = j^*\\mathcal{F}$. After replacing $\\mathcal{F}$", "by its reflexive hull, we may assume $\\mathcal{F}$ is reflexive", "(see discussion above and in particular Lemma \\ref{lemma-dual-reflexive}).", "By the above $j_*\\mathcal{G} = j_*j^*\\mathcal{F} = \\mathcal{F}$", "as desired." ], "refs": [ "divisors-lemma-reflexive-S2", "divisors-lemma-depth-2-hartog", "properties-lemma-lift-finite-presentation", "divisors-lemma-dual-reflexive" ], "ref_ids": [ 7922, 7881, 3022, 7920 ] } ], "ref_ids": [] }, { "id": 7924, "type": "theorem", "label": "divisors-lemma-reflexive-over-normal", "categories": [ "divisors" ], "title": "divisors-lemma-reflexive-over-normal", "contents": [ "Let $X$ be an integral locally Noetherian normal scheme.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is reflexive,", "\\item $\\mathcal{F}$ is torsion free and has property $(S_2)$, and", "\\item there exists an open subscheme $j : U \\to X$ such that", "\\begin{enumerate}", "\\item every irreducible component of $X \\setminus U$", "has codimension $\\geq 2$ in $X$,", "\\item $j^*\\mathcal{F}$ is finite locally free, and", "\\item $\\mathcal{F} = j_*j^*\\mathcal{F}$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Using Lemma \\ref{lemma-check-reflexive-on-affines}", "the equivalence of (1) and (2) follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-reflexive-over-normal}.", "Let $U \\subset X$ be as in (3). By", "Properties, Lemma \\ref{properties-lemma-criterion-normal}", "we see that $\\text{depth}(\\mathcal{O}_{X, x}) \\geq 2$", "for $x \\not \\in U$. Since a finite locally free module is reflexive,", "we conclude (3) implies (1) by Lemma \\ref{lemma-reflexive-S2-extend}.", "\\medskip\\noindent", "Assume (1). Let $U \\subset X$ be the maximal open subscheme such", "that $j^*\\mathcal{F} = \\mathcal{F}|_U$", "is finite locally free. So (3)(b) holds. Let $x \\in X$ be a point.", "If $\\mathcal{F}_x$ is a free $\\mathcal{O}_{X, x}$-module, then", "$x \\in U$, see", "Modules, Lemma \\ref{modules-lemma-finite-presentation-stalk-free}.", "If $\\dim(\\mathcal{O}_{X, x}) \\leq 1$, then $\\mathcal{O}_{X, x}$", "is either a field or a discrete valuation ring", "(Properties, Lemma \\ref{properties-lemma-criterion-normal})", "and hence $\\mathcal{F}_x$ is free (More on Algebra, Lemma", "\\ref{more-algebra-lemma-dedekind-torsion-free-flat}).", "Thus $x \\not \\in U \\Rightarrow \\dim(\\mathcal{O}_{X, x}) \\geq 2$.", "Then Properties, Lemma \\ref{properties-lemma-codimension-local-ring}", "shows (3)(a) holds. By the already used", "Properties, Lemma \\ref{properties-lemma-criterion-normal}", "we also see that $\\text{depth}(\\mathcal{O}_{X, x}) \\geq 2$", "for $x \\not \\in U$ and hence (3)(c) follows from", "Lemma \\ref{lemma-reflexive-S2-extend}." ], "refs": [ "divisors-lemma-check-reflexive-on-affines", "more-algebra-lemma-reflexive-over-normal", "properties-lemma-criterion-normal", "divisors-lemma-reflexive-S2-extend", "modules-lemma-finite-presentation-stalk-free", "properties-lemma-criterion-normal", "more-algebra-lemma-dedekind-torsion-free-flat", "properties-lemma-codimension-local-ring", "properties-lemma-criterion-normal", "divisors-lemma-reflexive-S2-extend" ], "ref_ids": [ 7915, 9937, 2989, 7923, 13253, 2989, 9921, 2979, 2989, 7923 ] } ], "ref_ids": [] }, { "id": 7925, "type": "theorem", "label": "divisors-lemma-describe-reflexive-hull", "categories": [ "divisors" ], "title": "divisors-lemma-describe-reflexive-hull", "contents": [ "Let $X$ be an integral locally Noetherian normal scheme with", "generic point $\\eta$. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent", "$\\mathcal{O}_X$-modules. Let $T : \\mathcal{G}_\\eta \\to \\mathcal{F}_\\eta$", "be a linear map. Then $T$ extends to a map", "$\\mathcal{G} \\to \\mathcal{F}^{**}$ of $\\mathcal{O}_X$-modules", "if and only if", "\\begin{itemize}", "\\item[(*)] for every $x \\in X$ with $\\dim(\\mathcal{O}_{X, x}) = 1$", "we have", "$$", "T\\left(\\Im(\\mathcal{G}_x \\to \\mathcal{G}_\\eta)\\right) \\subset", "\\Im(\\mathcal{F}_x \\to \\mathcal{F}_\\eta).", "$$", "\\end{itemize}" ], "refs": [], "proofs": [ { "contents": [ "Because $\\mathcal{F}^{**}$ is torsion free and", "$\\mathcal{F}_\\eta = \\mathcal{F}^{**}_\\eta$ an extension, if it exists,", "is unique. Thus it suffices to prove the lemma over the members of an", "open covering of $X$, i.e., we may assume $X$ is affine. In this case", "we are asking the following algebra question: Let $R$ be a Noetherian", "normal domain with fraction field $K$, let $M$, $N$ be finite $R$-modules,", "let $T : M \\otimes_R K \\to N \\otimes_R K$ be a $K$-linear map. When", "does $T$ extend to a map $N \\to M^{**}$? By More on Algebra, Lemma", "\\ref{more-algebra-lemma-describe-reflexive-hull}", "this happens if and only if $N_\\mathfrak p$ maps into", "$(M/M_{tors})_\\mathfrak p$ for every height $1$ prime $\\mathfrak p$ of $R$.", "This is exactly condition $(*)$ of the lemma." ], "refs": [ "more-algebra-lemma-describe-reflexive-hull" ], "ref_ids": [ 9938 ] } ], "ref_ids": [] }, { "id": 7926, "type": "theorem", "label": "divisors-lemma-reflexive-over-regular-dim-2", "categories": [ "divisors" ], "title": "divisors-lemma-reflexive-over-regular-dim-2", "contents": [ "Let $X$ be a regular scheme of dimension $\\leq 2$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is reflexive,", "\\item $\\mathcal{F}$ is finite locally free.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that a finite locally free module is reflexive.", "For the converse, we will show that if $\\mathcal{F}$ is", "reflexive, then $\\mathcal{F}_x$ is a free $\\mathcal{O}_{X, x}$-module", "for all $x \\in X$. This is enough by", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}", "and the fact that $\\mathcal{F}$ is coherent.", "If $\\dim(\\mathcal{O}_{X, x}) = 0$, then", "$\\mathcal{O}_{X, x}$ is a field and the statement is clear.", "If $\\dim(\\mathcal{O}_{X, x}) = 1$, then $\\mathcal{O}_{X, x}$", "is a discrete valuation ring", "(Algebra, Lemma \\ref{algebra-lemma-characterize-dvr})", "and $\\mathcal{F}_x$ is torsion free.", "Hence $\\mathcal{F}_x$ is free by More on Algebra, Lemma", "\\ref{more-algebra-lemma-dedekind-torsion-free-flat}.", "If $\\dim(\\mathcal{O}_{X, x}) = 2$, then $\\mathcal{O}_{X, x}$", "is a regular local ring of dimension $2$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-reflexive-over-normal}", "we see that $\\mathcal{F}_x$ has depth $\\geq 2$.", "Hence $\\mathcal{F}$ is free by", "Algebra, Lemma \\ref{algebra-lemma-regular-mcm-free}." ], "refs": [ "algebra-lemma-finite-projective", "algebra-lemma-characterize-dvr", "more-algebra-lemma-dedekind-torsion-free-flat", "more-algebra-lemma-reflexive-over-normal", "algebra-lemma-regular-mcm-free" ], "ref_ids": [ 795, 1023, 9921, 9937, 944 ] } ], "ref_ids": [] }, { "id": 7927, "type": "theorem", "label": "divisors-lemma-characterize-effective-Cartier-divisor", "categories": [ "divisors" ], "title": "divisors-lemma-characterize-effective-Cartier-divisor", "contents": [ "Let $S$ be a scheme.", "Let $D \\subset S$ be a closed subscheme.", "The following are equivalent:", "\\begin{enumerate}", "\\item The subscheme $D$ is an effective Cartier divisor on $S$.", "\\item For every $x \\in D$ there exists an affine open neighbourhood", "$\\Spec(A) = U \\subset S$ of $x$ such that", "$U \\cap D = \\Spec(A/(f))$ with $f \\in A$ a nonzerodivisor.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). For every $x \\in D$ there exists an affine open neighbourhood", "$\\Spec(A) = U \\subset S$ of $x$ such that", "$\\mathcal{I}_D|_U \\cong \\mathcal{O}_U$. In other words, there exists", "a section $f \\in \\Gamma(U, \\mathcal{I}_D)$ which freely generates the", "restriction $\\mathcal{I}_D|_U$. Hence $f \\in A$, and the multiplication", "map $f : A \\to A$ is injective. Also, since $\\mathcal{I}_D$ is", "quasi-coherent we see that $D \\cap U = \\Spec(A/(f))$.", "\\medskip\\noindent", "Assume (2). Let $x \\in D$. By assumption there exists an affine open", "neighbourhood $\\Spec(A) = U \\subset S$ of $x$ such that", "$U \\cap D = \\Spec(A/(f))$ with $f \\in A$ a nonzerodivisor.", "Then $\\mathcal{I}_D|_U \\cong \\mathcal{O}_U$ since it is equal to", "$\\widetilde{(f)} \\cong \\widetilde{A} \\cong \\mathcal{O}_U$.", "Of course $\\mathcal{I}_D$ restricted to the open subscheme", "$S \\setminus D$ is isomorphic to $\\mathcal{O}_{S \\setminus D}$.", "Hence $\\mathcal{I}_D$ is an invertible $\\mathcal{O}_S$-module." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7928, "type": "theorem", "label": "divisors-lemma-complement-locally-principal-closed-subscheme", "categories": [ "divisors" ], "title": "divisors-lemma-complement-locally-principal-closed-subscheme", "contents": [ "Let $S$ be a scheme. Let $Z \\subset S$ be a locally principal closed", "subscheme. Let $U = S \\setminus Z$. Then $U \\to S$ is an affine morphism." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $S$, see", "Morphisms, Lemmas \\ref{morphisms-lemma-characterize-affine}.", "Thus we may assume $S = \\Spec(A)$ and $Z = V(f)$ for some $f \\in A$.", "In this case $U = D(f) = \\Spec(A_f)$ is affine hence $U \\to S$ is affine." ], "refs": [ "morphisms-lemma-characterize-affine" ], "ref_ids": [ 5172 ] } ], "ref_ids": [] }, { "id": 7929, "type": "theorem", "label": "divisors-lemma-complement-effective-Cartier-divisor", "categories": [ "divisors" ], "title": "divisors-lemma-complement-effective-Cartier-divisor", "contents": [ "Let $S$ be a scheme. Let $D \\subset S$ be an effective Cartier divisor.", "Let $U = S \\setminus D$. Then $U \\to S$ is an affine morphism and $U$", "is scheme theoretically dense in $S$." ], "refs": [], "proofs": [ { "contents": [ "Affineness is Lemma \\ref{lemma-complement-locally-principal-closed-subscheme}.", "The density question is local on $S$, see", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-scheme-theoretically-dense}.", "Thus we may assume $S = \\Spec(A)$ and $D$ corresponding to the", "nonzerodivisor $f \\in A$, see", "Lemma \\ref{lemma-characterize-effective-Cartier-divisor}.", "Thus $A \\subset A_f$ which implies that $U \\subset S$ is", "scheme theoretically dense, see", "Morphisms, Example \\ref{morphisms-example-scheme-theoretic-closure}." ], "refs": [ "divisors-lemma-complement-locally-principal-closed-subscheme", "morphisms-lemma-characterize-scheme-theoretically-dense", "divisors-lemma-characterize-effective-Cartier-divisor" ], "ref_ids": [ 7928, 5152, 7927 ] } ], "ref_ids": [] }, { "id": 7930, "type": "theorem", "label": "divisors-lemma-effective-Cartier-makes-dimension-drop", "categories": [ "divisors" ], "title": "divisors-lemma-effective-Cartier-makes-dimension-drop", "contents": [ "Let $S$ be a scheme.", "Let $D \\subset S$ be an effective Cartier divisor.", "Let $s \\in D$.", "If $\\dim_s(S) < \\infty$, then $\\dim_s(D) < \\dim_s(S)$." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\dim_s(S) < \\infty$.", "Let $U = \\Spec(A) \\subset S$ be an affine open neighbourhood", "of $s$ such that $\\dim(U) = \\dim_s(S)$ and such that $D = V(f)$", "for some nonzerodivisor $f \\in A$ (see", "Lemma \\ref{lemma-characterize-effective-Cartier-divisor}).", "Recall that $\\dim(U)$ is the Krull dimension of the ring $A$", "and that $\\dim(U \\cap D)$ is the Krull dimension of the ring $A/(f)$.", "Then $f$ is not contained in any minimal prime of $A$.", "Hence any maximal chain of primes in $A/(f)$, viewed as a chain", "of primes in $A$, can be extended by adding a minimal prime." ], "refs": [ "divisors-lemma-characterize-effective-Cartier-divisor" ], "ref_ids": [ 7927 ] } ], "ref_ids": [] }, { "id": 7931, "type": "theorem", "label": "divisors-lemma-sum-effective-Cartier-divisors", "categories": [ "divisors" ], "title": "divisors-lemma-sum-effective-Cartier-divisors", "contents": [ "The sum of two effective Cartier divisors is an effective", "Cartier divisor." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Locally $f_1, f_2 \\in A$ are nonzerodivisors, then also", "$f_1f_2 \\in A$ is a nonzerodivisor." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7932, "type": "theorem", "label": "divisors-lemma-difference-effective-Cartier-divisors", "categories": [ "divisors" ], "title": "divisors-lemma-difference-effective-Cartier-divisors", "contents": [ "Let $X$ be a scheme.", "Let $D, D'$ be two effective Cartier divisors on $X$.", "If $D \\subset D'$ (as closed subschemes of $X$), then", "there exists an effective Cartier divisor $D''$ such", "that $D' = D + D''$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7933, "type": "theorem", "label": "divisors-lemma-sum-closed-subschemes-effective-Cartier", "categories": [ "divisors" ], "title": "divisors-lemma-sum-closed-subschemes-effective-Cartier", "contents": [ "Let $X$ be a scheme. Let $Z, Y$ be two closed subschemes of $X$", "with ideal sheaves $\\mathcal{I}$ and $\\mathcal{J}$. If $\\mathcal{I}\\mathcal{J}$", "defines an effective Cartier divisor $D \\subset X$, then $Z$ and $Y$", "are effective Cartier divisors and $D = Z + Y$." ], "refs": [], "proofs": [ { "contents": [ "Applying Lemma \\ref{lemma-characterize-effective-Cartier-divisor} we obtain", "the following algebra situation: $A$ is a ring, $I, J \\subset A$", "ideals and $f \\in A$ a nonzerodivisor such that $IJ = (f)$.", "Thus the result follows from", "Algebra, Lemma \\ref{algebra-lemma-product-ideals-principal}." ], "refs": [ "divisors-lemma-characterize-effective-Cartier-divisor", "algebra-lemma-product-ideals-principal" ], "ref_ids": [ 7927, 1040 ] } ], "ref_ids": [] }, { "id": 7934, "type": "theorem", "label": "divisors-lemma-sum-effective-Cartier-divisors-union", "categories": [ "divisors" ], "title": "divisors-lemma-sum-effective-Cartier-divisors-union", "contents": [ "Let $X$ be a scheme. Let $D, D' \\subset X$ be effective Cartier divisors", "such that the scheme theoretic intersection $D \\cap D'$ is an effective", "Cartier divisor on $D'$. Then $D + D'$ is the scheme theoretic", "union of $D$ and $D'$." ], "refs": [], "proofs": [ { "contents": [ "See Morphisms, Definition", "\\ref{morphisms-definition-scheme-theoretic-intersection-union}", "for the definition of scheme theoretic intersection and union.", "To prove the lemma working locally", "(using Lemma \\ref{lemma-characterize-effective-Cartier-divisor})", "we obtain the following algebra problem: Given a ring $A$", "and nonzerodivisors $f_1, f_2 \\in A$ such that $f_1$ maps", "to a nonzerodivisor in $A/f_2A$, show that $f_1A \\cap f_2A = f_1f_2A$.", "We omit the straightforward argument." ], "refs": [ "morphisms-definition-scheme-theoretic-intersection-union", "divisors-lemma-characterize-effective-Cartier-divisor" ], "ref_ids": [ 5537, 7927 ] } ], "ref_ids": [] }, { "id": 7935, "type": "theorem", "label": "divisors-lemma-pullback-locally-principal", "categories": [ "divisors" ], "title": "divisors-lemma-pullback-locally-principal", "contents": [ "Let $f : S' \\to S$ be a morphism of schemes. Let $Z \\subset S$", "be a locally principal closed subscheme. Then the inverse image", "$f^{-1}(Z)$ is a locally principal closed subscheme of $S'$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7936, "type": "theorem", "label": "divisors-lemma-pullback-effective-Cartier-defined", "categories": [ "divisors" ], "title": "divisors-lemma-pullback-effective-Cartier-defined", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $D \\subset Y$ be an effective Cartier divisor.", "The pullback of $D$ by $f$ is defined in each of the following cases:", "\\begin{enumerate}", "\\item $f(x) \\not \\in D$ for any weakly associated point $x$ of $X$,", "\\item $X$, $Y$ integral and $f$ dominant,", "\\item $X$ reduced and $f(\\xi) \\not \\in D$ for any generic point $\\xi$ of any", "irreducible component of $X$,", "\\item $X$ is locally Noetherian and $f(x) \\not \\in D$ for any associated point", "$x$ of $X$,", "\\item $X$ is locally Noetherian, has no embedded points, and", "$f(\\xi) \\not \\in D$ for any generic point $\\xi$ of an irreducible component of", "$X$,", "\\item $f$ is flat, and", "\\item add more here as needed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$, and hence we reduce to the case", "where $X = \\Spec(A)$, $Y = \\Spec(R)$, $f$ is", "given by $\\varphi : R \\to A$ and", "$D = \\Spec(R/(t))$ where $t \\in R$ is a nonzerodivisor.", "The goal in each case is to show that $\\varphi(t) \\in A$", "is a nonzerodivisor.", "\\medskip\\noindent", "In case (1) this follows from", "Algebra, Lemma \\ref{algebra-lemma-weakly-ass-zero-divisors}.", "Case (4) is a special case of (1) by Lemma \\ref{lemma-ass-weakly-ass}.", "Case (5) follows from (4) and the definitions.", "Case (3) is a special case of (1) by", "Lemma \\ref{lemma-weakass-reduced}.", "Case (2) is a special case of (3).", "If $R \\to A$ is flat, then $t : R \\to R$ being injective", "shows that $t : A \\to A$ is injective. This proves (6)." ], "refs": [ "algebra-lemma-weakly-ass-zero-divisors", "divisors-lemma-ass-weakly-ass", "divisors-lemma-weakass-reduced" ], "ref_ids": [ 725, 7878, 7882 ] } ], "ref_ids": [] }, { "id": 7937, "type": "theorem", "label": "divisors-lemma-pullback-effective-Cartier-divisors-additive", "categories": [ "divisors" ], "title": "divisors-lemma-pullback-effective-Cartier-divisors-additive", "contents": [ "Let $f : S' \\to S$ be a morphism of schemes.", "Let $D_1$, $D_2$ be effective Cartier divisors on $S$.", "If the pullbacks of $D_1$ and $D_2$ are defined then the", "pullback of $D = D_1 + D_2$ is defined and", "$f^*D = f^*D_1 + f^*D_2$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7938, "type": "theorem", "label": "divisors-lemma-conormal-effective-Cartier-divisor", "categories": [ "divisors" ], "title": "divisors-lemma-conormal-effective-Cartier-divisor", "contents": [ "Let $S$ be a scheme and let $D \\subset S$ be an effective Cartier divisor.", "Then the conormal sheaf is $\\mathcal{C}_{D/S} = \\mathcal{I}_D|D =", "\\mathcal{O}_S(-D)|_D$ and the normal sheaf is", "$\\mathcal{N}_{D/S} = \\mathcal{O}_S(D)|_D$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Morphisms, Lemma \\ref{morphisms-lemma-affine-conormal}." ], "refs": [ "morphisms-lemma-affine-conormal" ], "ref_ids": [ 5303 ] } ], "ref_ids": [] }, { "id": 7939, "type": "theorem", "label": "divisors-lemma-ses-add-divisor", "categories": [ "divisors" ], "title": "divisors-lemma-ses-add-divisor", "contents": [ "Let $X$ be a scheme. Let $D, C \\subset X$ be", "effective Cartier divisors with $C \\subset D$ and let $D' = D + C$.", "Then there is a short exact sequence", "$$", "0 \\to \\mathcal{O}_X(-D)|_C \\to \\mathcal{O}_{D'} \\to \\mathcal{O}_D \\to 0", "$$", "of $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "In the statement of the lemma and in the proof we use the equivalence of", "Morphisms, Lemma \\ref{morphisms-lemma-i-star-equivalence} to think of", "quasi-coherent modules on closed subschemes of $X$", "as quasi-coherent modules on $X$. Let $\\mathcal{I}$ be the ideal", "sheaf of $D$ in $D'$. Then there is a short exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_{D'} \\to \\mathcal{O}_D \\to 0", "$$", "because $D \\to D'$ is a closed immersion. There is a", "canonical surjection", "$\\mathcal{I} \\to \\mathcal{I}/\\mathcal{I}^2 = \\mathcal{C}_{D/D'}$.", "We have $\\mathcal{C}_{D/X} = \\mathcal{O}_X(-D)|_D$ by", "Lemma \\ref{lemma-conormal-effective-Cartier-divisor}", "and there is a canonical surjective map", "$$", "\\mathcal{C}_{D/X} \\longrightarrow \\mathcal{C}_{D/D'}", "$$", "see Morphisms, Lemmas \\ref{morphisms-lemma-conormal-functorial} and", "\\ref{morphisms-lemma-conormal-functorial-flat}.", "Thus it suffices to show: (a) $\\mathcal{I}^2 = 0$ and (b)", "$\\mathcal{I}$ is an invertible $\\mathcal{O}_C$-module.", "Both (a) and (b) can be checked locally, hence we may assume", "$X = \\Spec(A)$, $D = \\Spec(A/fA)$ and $C = \\Spec(A/gA)$ where", "$f, g \\in A$ are nonzerodivisors", "(Lemma \\ref{lemma-characterize-effective-Cartier-divisor}).", "Since $C \\subset D$ we see", "that $f \\in gA$. Then $I = fA/fgA$ has square zero and is invertible", "as an $A/gA$-module as desired." ], "refs": [ "morphisms-lemma-i-star-equivalence", "divisors-lemma-conormal-effective-Cartier-divisor", "morphisms-lemma-conormal-functorial", "morphisms-lemma-conormal-functorial-flat", "divisors-lemma-characterize-effective-Cartier-divisor" ], "ref_ids": [ 5136, 7938, 5304, 5305, 7927 ] } ], "ref_ids": [] }, { "id": 7940, "type": "theorem", "label": "divisors-lemma-invertible-sheaf-sum-effective-Cartier-divisors", "categories": [ "divisors" ], "title": "divisors-lemma-invertible-sheaf-sum-effective-Cartier-divisors", "contents": [ "Let $S$ be a scheme.", "Let $D_1$, $D_2$ be effective Cartier divisors on $S$.", "Let $D = D_1 + D_2$.", "Then there is a unique isomorphism", "$$", "\\mathcal{O}_S(D_1) \\otimes_{\\mathcal{O}_S} \\mathcal{O}_S(D_2)", "\\longrightarrow", "\\mathcal{O}_S(D)", "$$", "which maps $1_{D_1} \\otimes 1_{D_2}$ to $1_D$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7941, "type": "theorem", "label": "divisors-lemma-pullback-effective-Cartier-divisors", "categories": [ "divisors" ], "title": "divisors-lemma-pullback-effective-Cartier-divisors", "contents": [ "Let $f : S' \\to S$ be a morphism of schemes.", "Let $D$ be a effective Cartier divisors on $S$.", "If the pullback of $D$ is defined then", "$f^*\\mathcal{O}_S(D) = \\mathcal{O}_{S'}(f^*D)$", "and the canonical section $1_D$ pulls back to", "the canonical section $1_{f^*D}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7942, "type": "theorem", "label": "divisors-lemma-regular-section-structure-sheaf", "categories": [ "divisors" ], "title": "divisors-lemma-regular-section-structure-sheaf", "contents": [ "Let $X$ be a locally ringed space. Let $f \\in \\Gamma(X, \\mathcal{O}_X)$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is a regular section, and", "\\item for any $x \\in X$ the image $f \\in \\mathcal{O}_{X, x}$", "is a nonzerodivisor.", "\\end{enumerate}", "If $X$ is a scheme these are also equivalent to", "\\begin{enumerate}", "\\item[(3)] for any affine open $\\Spec(A) = U \\subset X$", "the image $f \\in A$ is a nonzerodivisor,", "\\item[(4)] there exists an affine open covering", "$X = \\bigcup \\Spec(A_i)$ such that", "the image of $f$ in $A_i$ is a nonzerodivisor for all $i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7943, "type": "theorem", "label": "divisors-lemma-zero-scheme", "categories": [ "divisors" ], "title": "divisors-lemma-zero-scheme", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{L}$ be an invertible sheaf.", "Let $s \\in \\Gamma(X, \\mathcal{L})$.", "\\begin{enumerate}", "\\item Consider closed immersions $i : Z \\to X$ such that", "$i^*s \\in \\Gamma(Z, i^*\\mathcal{L})$ is zero", "ordered by inclusion. The zero scheme $Z(s)$ is the", "maximal element of this ordered set.", "\\item For any morphism of schemes $f : Y \\to X$ we have", "$f^*s = 0$ in $\\Gamma(Y, f^*\\mathcal{L})$ if and only if", "$f$ factors through $Z(s)$.", "\\item The zero scheme $Z(s)$ is a locally principal closed subscheme.", "\\item The zero scheme $Z(s)$ is an effective Cartier divisor", "if and only if $s$ is a regular section of $\\mathcal{L}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7944, "type": "theorem", "label": "divisors-lemma-characterize-OD", "categories": [ "divisors" ], "title": "divisors-lemma-characterize-OD", "contents": [ "\\begin{slogan}", "Effective Cartier divisors on a scheme are the same as invertible sheaves", "with fixed regular global section.", "\\end{slogan}", "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item If $D \\subset X$ is an effective Cartier divisor, then", "the canonical section $1_D$ of $\\mathcal{O}_X(D)$ is regular.", "\\item Conversely, if $s$ is a regular section of the invertible", "sheaf $\\mathcal{L}$, then there exists a unique effective", "Cartier divisor $D = Z(s) \\subset X$ and a unique isomorphism", "$\\mathcal{O}_X(D) \\to \\mathcal{L}$ which maps $1_D$ to $s$.", "\\end{enumerate}", "The constructions", "$D \\mapsto (\\mathcal{O}_X(D), 1_D)$ and $(\\mathcal{L}, s) \\mapsto Z(s)$", "give mutually inverse maps", "$$", "\\left\\{", "\\begin{matrix}", "\\text{effective Cartier divisors on }X", "\\end{matrix}", "\\right\\}", "\\leftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{isomorphism classes of pairs }(\\mathcal{L}, s)\\\\", "\\text{consisting of an invertible }", "\\mathcal{O}_X\\text{-module}\\\\", "\\mathcal{L}\\text{ and a regular global section }s", "\\end{matrix}", "\\right\\}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7945, "type": "theorem", "label": "divisors-lemma-regular-section-associated-points", "categories": [ "divisors" ], "title": "divisors-lemma-regular-section-associated-points", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{L}$ be an invertible", "$\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$. Then $s$", "is a regular section if and only if $s$ does not vanish in the associated", "points of $X$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: reduce to the affine case and $\\mathcal{L}$ trivial", "and then use Lemma \\ref{lemma-regular-section-structure-sheaf} and", "Algebra, Lemma \\ref{algebra-lemma-ass-zero-divisors}." ], "refs": [ "algebra-lemma-ass-zero-divisors" ], "ref_ids": [ 704 ] } ], "ref_ids": [] }, { "id": 7946, "type": "theorem", "label": "divisors-lemma-effective-Cartier-in-points", "categories": [ "divisors" ], "title": "divisors-lemma-effective-Cartier-in-points", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a closed subscheme", "corresponding to the quasi-coherent ideal sheaf", "$\\mathcal{I} \\subset \\mathcal{O}_X$.", "\\begin{enumerate}", "\\item If for every $x \\in D$ the ideal", "$\\mathcal{I}_x \\subset \\mathcal{O}_{X, x}$", "can be generated by one element, then $D$ is locally principal.", "\\item If for every $x \\in D$ the ideal", "$\\mathcal{I}_x \\subset \\mathcal{O}_{X, x}$", "can be generated by a single nonzerodivisor, then $D$ is an", "effective Cartier divisor.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\Spec(A)$ be an affine neighbourhood of a point $x \\in D$.", "Let $\\mathfrak p \\subset A$ be the prime corresponding to $x$.", "Let $I \\subset A$ be the ideal defining the trace of $D$ on", "$\\Spec(A)$. Since $A$ is Noetherian (as $X$ is Noetherian)", "the ideal $I$ is generated by finitely many elements, say", "$I = (f_1, \\ldots, f_r)$. Under the assumption of (1) we have", "$I_\\mathfrak p = (f)$ for some $f \\in A_\\mathfrak p$.", "Then $f_i = g_i f$ for some $g_i \\in A_\\mathfrak p$.", "Write $g_i = a_i/h_i$ and $f = f'/h$ for some", "$a_i, h_i, f', h \\in A$, $h_i, h \\not \\in \\mathfrak p$.", "Then $I_{h_1 \\ldots h_r h} \\subset A_{h_1 \\ldots h_r h}$ is", "principal, because it is generated by $f'$. This proves (1).", "For (2) we may assume $I = (f)$. The assumption implies", "that the image of $f$ in $A_\\mathfrak p$ is a nonzerodivisor.", "Then $f$ is a nonzerodivisor on a neighbourhood of $x$ by", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-neighbourhood}.", "This proves (2)." ], "refs": [ "algebra-lemma-regular-sequence-in-neighbourhood" ], "ref_ids": [ 741 ] } ], "ref_ids": [] }, { "id": 7947, "type": "theorem", "label": "divisors-lemma-effective-Cartier-codimension-1", "categories": [ "divisors" ], "title": "divisors-lemma-effective-Cartier-codimension-1", "contents": [ "Let $X$ be a locally Noetherian scheme.", "\\begin{enumerate}", "\\item Let $D \\subset X$ be a locally principal closed subscheme.", "Let $\\xi \\in D$ be a generic point of an irreducible component of $D$.", "Then $\\dim(\\mathcal{O}_{X, \\xi}) \\leq 1$.", "\\item Let $D \\subset X$ be an effective Cartier divisor.", "Let $\\xi \\in D$ be a generic point of an irreducible component of $D$.", "Then $\\dim(\\mathcal{O}_{X, \\xi}) = 1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). By assumption we may assume $X = \\Spec(A)$ and", "$D = \\Spec(A/(f))$ where $A$ is a Noetherian ring and $f \\in A$.", "Let $\\xi$ correspond to the prime ideal $\\mathfrak p \\subset A$.", "The assumption that $\\xi$ is a generic point of an irreducible", "component of $D$ signifies $\\mathfrak p$ is minimal over $(f)$.", "Thus $\\dim(A_\\mathfrak p) \\leq 1$ by", "Algebra, Lemma \\ref{algebra-lemma-minimal-over-1}.", "\\medskip\\noindent", "Proof of (2). By part (1) we see that $\\dim(\\mathcal{O}_{X, \\xi}) \\leq 1$.", "On the other hand, the local equation $f$ is a nonzerodivisor in", "$A_\\mathfrak p$ by Lemma \\ref{lemma-characterize-effective-Cartier-divisor}", "which implies the dimension is at least $1$ (because there must be a", "prime in $A_\\mathfrak p$ not containing $f$ by the elementary", "Algebra, Lemma \\ref{algebra-lemma-Zariski-topology})." ], "refs": [ "algebra-lemma-minimal-over-1", "divisors-lemma-characterize-effective-Cartier-divisor", "algebra-lemma-Zariski-topology" ], "ref_ids": [ 683, 7927, 389 ] } ], "ref_ids": [] }, { "id": 7948, "type": "theorem", "label": "divisors-lemma-integral-effective-Cartier-divisor-dvr", "categories": [ "divisors" ], "title": "divisors-lemma-integral-effective-Cartier-divisor-dvr", "contents": [ "Let $X$ be a Noetherian scheme. Let $D \\subset X$ be an", "integral closed subscheme which is also an", "effective Cartier divisor. Then the local ring of $X$", "at the generic point of $D$ is a discrete valuation ring." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-characterize-effective-Cartier-divisor}", "we may assume $X = \\Spec(A)$ and $D = \\Spec(A/(f))$", "where $A$ is a Noetherian ring and $f \\in A$ is a nonzerodivisor.", "The assumption that $D$ is integral signifies that $(f)$ is prime.", "Hence the local ring of $X$ at the generic point is $A_{(f)}$", "which is a Noetherian local ring whose maximal ideal is generated by", "a nonzerodivisor. Thus it is a discrete valuation ring by", "Algebra, Lemma \\ref{algebra-lemma-characterize-dvr}." ], "refs": [ "divisors-lemma-characterize-effective-Cartier-divisor", "algebra-lemma-characterize-dvr" ], "ref_ids": [ 7927, 1023 ] } ], "ref_ids": [] }, { "id": 7949, "type": "theorem", "label": "divisors-lemma-effective-Cartier-divisor-Sk", "categories": [ "divisors" ], "title": "divisors-lemma-effective-Cartier-divisor-Sk", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be an", "effective Cartier divisor. If $X$ is $(S_k)$, then $D$ is $(S_{k - 1})$." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in D$. Then $\\mathcal{O}_{D, x} = \\mathcal{O}_{X, x}/(f)$ where", "$f \\in \\mathcal{O}_{X, x}$ is a nonzerodivisor. By assumption we have", "$\\text{depth}(\\mathcal{O}_{X, x}) \\geq \\min(\\dim(\\mathcal{O}_{X, x}), k)$.", "By Algebra, Lemma \\ref{algebra-lemma-depth-drops-by-one} we have", "$\\text{depth}(\\mathcal{O}_{D, x}) = \\text{depth}(\\mathcal{O}_{X, x}) - 1$", "and by Algebra, Lemma \\ref{algebra-lemma-one-equation}", "$\\dim(\\mathcal{O}_{D, x}) = \\dim(\\mathcal{O}_{X, x}) - 1$.", "It follows that", "$\\text{depth}(\\mathcal{O}_{D, x}) \\geq \\min(\\dim(\\mathcal{O}_{D, x}), k - 1)$", "as desired." ], "refs": [ "algebra-lemma-depth-drops-by-one" ], "ref_ids": [ 774 ] } ], "ref_ids": [] }, { "id": 7950, "type": "theorem", "label": "divisors-lemma-normal-effective-Cartier-divisor-S1", "categories": [ "divisors" ], "title": "divisors-lemma-normal-effective-Cartier-divisor-S1", "contents": [ "Let $X$ be a locally Noetherian normal scheme. Let $D \\subset X$ be an", "effective Cartier divisor. Then $D$ is $(S_1)$." ], "refs": [], "proofs": [ { "contents": [ "By Properties, Lemma \\ref{properties-lemma-criterion-normal}", "we see that $X$ is $(S_2)$. Thus we conclude by", "Lemma \\ref{lemma-effective-Cartier-divisor-Sk}." ], "refs": [ "properties-lemma-criterion-normal", "divisors-lemma-effective-Cartier-divisor-Sk" ], "ref_ids": [ 2989, 7949 ] } ], "ref_ids": [] }, { "id": 7951, "type": "theorem", "label": "divisors-lemma-weil-divisor-is-cartier-UFD", "categories": [ "divisors" ], "title": "divisors-lemma-weil-divisor-is-cartier-UFD", "contents": [ "Let $X$ be a Noetherian scheme. Let $D \\subset X$ be a integral", "closed subscheme. Assume that", "\\begin{enumerate}", "\\item $D$ has codimension $1$ in $X$, and", "\\item $\\mathcal{O}_{X, x}$ is a UFD for all $x \\in D$.", "\\end{enumerate}", "Then $D$ is an effective Cartier divisor." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in D$ and set $A = \\mathcal{O}_{X, x}$. Let $\\mathfrak p \\subset A$", "correspond to the generic point of $D$. Then $A_\\mathfrak p$ has dimension", "$1$ by assumption (1). Thus $\\mathfrak p$ is a prime ideal of height $1$.", "Since $A$ is a UFD this implies that $\\mathfrak p = (f)$ for some $f \\in A$.", "Of course $f$ is a nonzerodivisor and we conclude by", "Lemma \\ref{lemma-effective-Cartier-in-points}." ], "refs": [ "divisors-lemma-effective-Cartier-in-points" ], "ref_ids": [ 7946 ] } ], "ref_ids": [] }, { "id": 7952, "type": "theorem", "label": "divisors-lemma-codim-1-part", "categories": [ "divisors" ], "title": "divisors-lemma-codim-1-part", "contents": [ "Let $X$ be a Noetherian scheme. Let $Z \\subset X$ be a closed subscheme.", "Assume there exist integral effective Cartier divisors $D_i \\subset X$", "and a closed subset $Z' \\subset X$ of codimension $\\geq 2$ such that", "$Z \\subset Z' \\cup \\bigcup D_i$ set-theoretically.", "Then there exists an effective Cartier divisor of the form", "$$", "D = \\sum a_i D_i \\subset Z", "$$", "such that $D \\to Z$ is an isomorphism away from codimension $2$ in $X$.", "The existence of the $D_i$ is guaranteed if $\\mathcal{O}_{X, x}$", "is a UFD for all $x \\in Z$ or if $X$ is regular." ], "refs": [], "proofs": [ { "contents": [ "Let $\\xi_i \\in D_i$ be the generic point and let", "$\\mathcal{O}_i = \\mathcal{O}_{X, \\xi_i}$ be the local ring", "which is a discrete valuation ring by", "Lemma \\ref{lemma-integral-effective-Cartier-divisor-dvr}.", "Let $a_i \\geq 0$ be the minimal valuation of an element of", "$\\mathcal{I}_{Z, \\xi_i} \\subset \\mathcal{O}_i$.", "We claim that the effective Cartier divisor $D = \\sum a_i D_i$ works.", "\\medskip\\noindent", "Namely, suppose that $x \\in X$. Let $A = \\mathcal{O}_{X, x}$.", "Let $f_i \\in A$ be a local equation for $D_i$;", "we only consider those $i$ such that $x \\in D_i$. Then $f_i$ is", "a prime element of $A$ and $\\mathcal{O}_i = A_{(f_i)}$. Let", "$I = \\mathcal{I}_{Z, x} \\subset A$. By our choice of $a_i$ we have", "$I A_{(f_i)} = f_i^{a_i}A_{(f_i)}$. It follows that", "$I \\subset (\\prod f_i^{a_i})$ because the $f_i$ are prime elements of $A$.", "This proves that $\\mathcal{I}_Z \\subset \\mathcal{I}_D$, i.e., that", "$D \\subset Z$. Moreover, we also see that $D$ and $Z$ agree at the $\\xi_i$,", "which proves the final assertion.", "\\medskip\\noindent", "To see the final statements we argue as follows. A regular local", "ring is a UFD (More on Algebra, Lemma", "\\ref{more-algebra-lemma-regular-local-UFD}) hence it suffices", "to argue in the UFD case. In that case, let", "$D_i$ be the irreducible components of $Z$", "which have codimension $1$ in $X$.", "By Lemma \\ref{lemma-weil-divisor-is-cartier-UFD} each $D_i$", "is an effective Cartier divisor." ], "refs": [ "divisors-lemma-integral-effective-Cartier-divisor-dvr", "more-algebra-lemma-regular-local-UFD", "divisors-lemma-weil-divisor-is-cartier-UFD" ], "ref_ids": [ 7948, 10544, 7951 ] } ], "ref_ids": [] }, { "id": 7953, "type": "theorem", "label": "divisors-lemma-codimension-1-is-effective-Cartier", "categories": [ "divisors" ], "title": "divisors-lemma-codimension-1-is-effective-Cartier", "contents": [ "Let $Z \\subset X$ be a closed subscheme of a Noetherian scheme. Assume", "\\begin{enumerate}", "\\item $Z$ has no embedded points,", "\\item every irreducible component of $Z$ has codimension $1$ in $X$,", "\\item every local ring $\\mathcal{O}_{X, x}$, $x \\in Z$ is", "a UFD or $X$ is regular.", "\\end{enumerate}", "Then $Z$ is an effective Cartier divisor." ], "refs": [], "proofs": [ { "contents": [ "Let $D = \\sum a_i D_i$ be as in Lemma \\ref{lemma-codim-1-part}", "where $D_i \\subset Z$ are the irreducible components of $Z$.", "If $D \\to Z$ is not an isomorphism, then $\\mathcal{O}_Z \\to \\mathcal{O}_D$", "has a nonzero kernel sitting in codimension $\\geq 2$. This", "would mean that $Z$ has embedded points, which is forbidden", "by assumption (1). Hence $D \\cong Z$ as desired." ], "refs": [ "divisors-lemma-codim-1-part" ], "ref_ids": [ 7952 ] } ], "ref_ids": [] }, { "id": 7954, "type": "theorem", "label": "divisors-lemma-UFD-one-equation-CM", "categories": [ "divisors" ], "title": "divisors-lemma-UFD-one-equation-CM", "contents": [ "Let $R$ be a Noetherian UFD. Let $I \\subset R$ be an ideal", "such that $R/I$ has no embedded primes and such that", "every minimal prime over $I$ has height $1$.", "Then $I = (f)$ for some $f \\in R$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-codimension-1-is-effective-Cartier}", "the ideal sheaf $\\tilde I$ is invertible on $\\Spec(R)$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-UFD-Pic-trivial}", "it is generated by a single element." ], "refs": [ "divisors-lemma-codimension-1-is-effective-Cartier", "more-algebra-lemma-UFD-Pic-trivial" ], "ref_ids": [ 7953, 10534 ] } ], "ref_ids": [] }, { "id": 7955, "type": "theorem", "label": "divisors-lemma-effective-Cartier-divisor-is-a-sum", "categories": [ "divisors" ], "title": "divisors-lemma-effective-Cartier-divisor-is-a-sum", "contents": [ "Let $X$ be a Noetherian scheme. Let $D \\subset X$ be an effective", "Cartier divisor. Assume that there exist integral effective Cartier", "divisors $D_i \\subset X$ such that $D \\subset \\bigcup D_i$", "set theoretically. Then $D = \\sum a_i D_i$ for some $a_i \\geq 0$.", "The existence of the $D_i$ is guaranteed if $\\mathcal{O}_{X, x}$", "is a UFD for all $x \\in D$ or if $X$ is regular." ], "refs": [], "proofs": [ { "contents": [ "Choose $a_i$ as in Lemma \\ref{lemma-codim-1-part} and set $D' = \\sum a_i D_i$.", "Then $D' \\to D$ is an inclusion of effective Cartier divisors which", "is an isomorphism away from codimension $2$ on $X$. Pick $x \\in X$.", "Set $A = \\mathcal{O}_{X, x}$ and let $f, f' \\in A$ be the nonzerodivisor", "generating the ideal of $D, D'$ in $A$. Then $f = gf'$ for some $g \\in A$.", "Moreover, for every prime $\\mathfrak p$ of height $\\leq 1$ of $A$ we see", "that $g$ maps to a unit of $A_\\mathfrak p$. This implies that $g$ is", "a unit because the minimal primes over $(g)$ have height $1$", "(Algebra, Lemma \\ref{algebra-lemma-minimal-over-1})." ], "refs": [ "divisors-lemma-codim-1-part", "algebra-lemma-minimal-over-1" ], "ref_ids": [ 7952, 683 ] } ], "ref_ids": [] }, { "id": 7956, "type": "theorem", "label": "divisors-lemma-quasi-projective-Noetherian-pic-effective-Cartier", "categories": [ "divisors" ], "title": "divisors-lemma-quasi-projective-Noetherian-pic-effective-Cartier", "contents": [ "\\begin{slogan}", "On a projective scheme, every line bundle has a regular meromorphic section.", "\\end{slogan}", "Let $X$ be a Noetherian scheme which has an ample invertible sheaf.", "Then every invertible $\\mathcal{O}_X$-module is isomorphic to", "$$", "\\mathcal{O}_X(D - D') =", "\\mathcal{O}_X(D) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(D')^{\\otimes -1}", "$$", "for some effective Cartier divisors $D, D'$ in $X$. Moreover, given a", "finite subset $E \\subset X$ we may choose $D, D'$ such that", "$E \\cap D = \\emptyset$ and $E \\cap D' = \\emptyset$. If", "$X$ is quasi-affine, then we may choose $D' = \\emptyset$." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1, \\ldots, x_n$ be the associated points of $X$", "(Lemma \\ref{lemma-finite-ass}).", "\\medskip\\noindent", "If $X$ is quasi-affine and $\\mathcal{N}$ is any invertible", "$\\mathcal{O}_X$-module, then we can pick a section $t$ of", "$\\mathcal{N}$ which does not vanish at any of the points", "of $E \\cup \\{x_1, \\ldots, x_n\\}$, see Properties, Lemma", "\\ref{properties-lemma-quasi-affine-invertible-nonvanishing-section}.", "Then $t$ is a regular section of $\\mathcal{N}$ by", "Lemma \\ref{lemma-regular-section-associated-points}.", "Hence $\\mathcal{N} \\cong \\mathcal{O}_X(D)$ where", "$D = Z(t)$ is the effective Cartier divisor corresponding to $t$, see", "Lemma \\ref{lemma-characterize-OD}. Since $E \\cap D = \\emptyset$", "by construction we are done in this case.", "\\medskip\\noindent", "Returning to the general case, let $\\mathcal{L}$ be an ample invertible sheaf", "on $X$. There exists an $n > 0$ and a section", "$s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ such that $X_s$", "is affine and such that $E \\cup \\{x_1, \\ldots, x_n\\} \\subset X_s$", "(Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-principal-affine}).", "\\medskip\\noindent", "Let $\\mathcal{N}$ be an arbitrary invertible $\\mathcal{O}_X$-module.", "By the quasi-affine case, we can find a section", "$t \\in \\mathcal{N}(X_s)$ which does not vanish at any point", "of $E \\cup \\{x_1, \\ldots, x_n\\}$.", "By Properties, Lemma \\ref{properties-lemma-invert-s-sections}", "we see that for some $e \\geq 0$ the section $s^e|_{X_s} t$ extends to", "a global section $\\tau$ of $\\mathcal{L}^{\\otimes e} \\otimes \\mathcal{N}$.", "Thus both $\\mathcal{L}^{\\otimes e} \\otimes \\mathcal{N}$ and", "$\\mathcal{L}^{\\otimes e}$ are invertible sheaves which have global sections", "which do not vanish at any point of $E \\cup \\{x_1, \\ldots, x_n\\}$.", "Thus these are regular sections by", "Lemma \\ref{lemma-regular-section-associated-points}.", "Hence $\\mathcal{L}^{\\otimes e} \\otimes \\mathcal{N} \\cong \\mathcal{O}_X(D)$", "and $\\mathcal{L}^{\\otimes e} \\cong \\mathcal{O}_X(D')$ for some", "effective Cartier divisors $D$ and $D'$, see Lemma \\ref{lemma-characterize-OD}.", "By construction $E \\cap D = \\emptyset$ and $E \\cap D' = \\emptyset$", "and the proof is complete." ], "refs": [ "divisors-lemma-finite-ass", "divisors-lemma-characterize-OD", "properties-lemma-ample-finite-set-in-principal-affine", "properties-lemma-invert-s-sections", "divisors-lemma-characterize-OD" ], "ref_ids": [ 7859, 7944, 3063, 3005, 7944 ] } ], "ref_ids": [] }, { "id": 7957, "type": "theorem", "label": "divisors-lemma-wedge-product-ses", "categories": [ "divisors" ], "title": "divisors-lemma-wedge-product-ses", "contents": [ "Let $X$ be an integral regular scheme of dimension $2$.", "Let $i : D \\to X$ be the immersion of an effective Cartier divisor.", "Let $\\mathcal{F} \\to \\mathcal{F}' \\to i_*\\mathcal{G} \\to 0$", "be an exact sequence of coherent $\\mathcal{O}_X$-modules.", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{F}, \\mathcal{F}'$ are locally free of rank $r$ on a nonempty", "open of $X$,", "\\item $D$ is an integral scheme,", "\\item $\\mathcal{G}$ is a finite locally free $\\mathcal{O}_D$-module", "of rank $s$.", "\\end{enumerate}", "Then $\\mathcal{L} = (\\wedge^r\\mathcal{F})^{**}$ and", "$\\mathcal{L}' = (\\wedge^r \\mathcal{F}')^{**}$", "are invertible $\\mathcal{O}_X$-modules and", "$\\mathcal{L}' \\cong \\mathcal{L}(k D)$ for some", "$k \\in \\{0, \\ldots, \\min(s, r)\\}$." ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from Lemma \\ref{lemma-reflexive-over-regular-dim-2}", "as assumption (1) implies that $\\mathcal{L}$ and $\\mathcal{L}'$", "have rank $1$. Taking $\\wedge^r$ and double duals are functors, hence", "we obtain a canonical map $\\sigma : \\mathcal{L} \\to \\mathcal{L}'$", "which is an isomorphism over the nonempty open of (1), hence", "nonzero. To finish the proof, it suffices to see that", "$\\sigma$ viewed as a global section of", "$\\mathcal{L}' \\otimes \\mathcal{L}^{\\otimes -1}$ does not", "vanish at any codimension point of $X$, except at the generic", "point of $D$ and there with vanishing order at most $\\min(s, r)$.", "\\medskip\\noindent", "Translated into algebra, we arrive at the following problem:", "Let $(A, \\mathfrak m, \\kappa)$ be a discrete valuation ring", "with fraction field $K$. Let $M \\to M' \\to N \\to 0$ be an exact sequence", "of finite $A$-modules with $\\dim_K(M \\otimes K) = \\dim_K(M' \\otimes K) = r$", "and with $N \\cong \\kappa^{\\oplus s}$. Show that the induced map", "$L = \\wedge^r(M)^{**} \\to L' = \\wedge^r(M')^{**}$ vanishes to", "order at most $\\min(s, r)$. We will use the structure theorem for", "modules over $A$, see", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-generalized-valuation-ring-modules} or", "\\ref{more-algebra-lemma-modules-PID}.", "Dividing out a finite $A$-module by a torsion submodule does not", "change the double dual.", "Thus we may replace $M$ by $M/M_{tors}$ and $M'$ by", "$M'/\\Im(M_{tors} \\to M')$ and assume that $M$ is torsion free.", "Then $M \\to M'$ is injective and $M'_{tors} \\to N$ is injective.", "Hence we may replace $M'$ by $M'/M'_{tors}$ and $N$ by $N/M'_{tors}$.", "Thus we reduce to the case where $M$ and $M'$ are free of rank $r$", "and $N \\cong \\kappa^{\\oplus s}$. In this case $\\sigma$", "is the determinant of $M \\to M'$ and vanishes to order $s$", "for example by Algebra, Lemma \\ref{algebra-lemma-order-vanishing-determinant}." ], "refs": [ "divisors-lemma-reflexive-over-regular-dim-2", "more-algebra-lemma-generalized-valuation-ring-modules", "more-algebra-lemma-modules-PID", "algebra-lemma-order-vanishing-determinant" ], "ref_ids": [ 7926, 10556, 10561, 1046 ] } ], "ref_ids": [] }, { "id": 7958, "type": "theorem", "label": "divisors-lemma-affine-punctured-spec", "categories": [ "divisors" ], "title": "divisors-lemma-affine-punctured-spec", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "The punctured spectrum $U = \\Spec(A) \\setminus \\{\\mathfrak m\\}$", "of $A$ is affine if and only if $\\dim(A) \\leq 1$." ], "refs": [], "proofs": [ { "contents": [ "If $\\dim(A) = 0$, then $U$ is empty hence affine (equal to the spectrum of", "the $0$ ring). If $\\dim(A) = 1$, then we can choose an element", "$f \\in \\mathfrak m$ not contained in any of the finite number of minimal", "primes of $A$", "(Algebra, Lemmas \\ref{algebra-lemma-Noetherian-irreducible-components} and", "\\ref{algebra-lemma-silly}). Then $U = \\Spec(A_f)$", "is affine.", "\\medskip\\noindent", "The converse is more interesting. We will give a somewhat nonstandard proof", "and discuss the standard argument in a remark below.", "Assume $U = \\Spec(B)$ is affine. Since affineness and dimension are not", "affecting by going to the reduction we may replace $A$ by the quotient by", "its ideal of nilpotent elements and assume $A$ is reduced.", "Set $Q = B/A$ viewed as an $A$-module.", "The support of $Q$ is $\\{\\mathfrak m\\}$ as $A_\\mathfrak p = B_\\mathfrak p$", "for all nonmaximal primes $\\mathfrak p$ of $A$.", "We may assume $\\dim(A) \\geq 1$, hence as above we can pick", "$f \\in \\mathfrak m$ not contained in any of the minimal ideals of $A$.", "Since $A$ is reduced this implies that $f$ is a nonzerodivisor.", "In particular $\\dim(A/fA) = \\dim(A) - 1$, see", "Algebra, Lemma \\ref{algebra-lemma-one-equation}.", "Applying the snake lemma to multiplication by $f$ on the short", "exact sequence $0 \\to A \\to B \\to Q \\to 0$ we obtain", "$$", "0 \\to Q[f] \\to A/fA \\to B/fB \\to Q/fQ \\to 0", "$$", "where $Q[f] = \\Ker(f : Q \\to Q)$.", "This implies that $Q[f]$ is a finite $A$-module. Since the support of", "$Q[f]$ is $\\{\\mathfrak m\\}$ we see $l = \\text{length}_A(Q[f]) < \\infty$", "(Algebra, Lemma \\ref{algebra-lemma-support-point}).", "Set $l_n = \\text{length}_A(Q[f^n])$. The exact sequence", "$$", "0 \\to Q[f^n] \\to Q[f^{n + 1}] \\xrightarrow{f^n} Q[f]", "$$", "shows inductively that $l_n < \\infty$ and that $l_n \\leq l_{n + 1}$.", "Considering the exact sequence", "$$", "0 \\to Q[f] \\to Q[f^{n + 1}] \\xrightarrow{f} Q[f^n] \\to Q/fQ", "$$", "and we see that the image of $Q[f^n]$ in $Q/fQ$ has length", "$l_n - l_{n + 1} + l \\leq l$. Since $Q = \\bigcup Q[f^n]$ we", "find that the length of $Q/fQ$ is at most $l$, i.e., bounded.", "Thus $Q/fQ$ is a finite $A$-module. Hence $A/fA \\to B/fB$ is a", "finite ring map, in particular induces a closed map on spectra", "(Algebra, Lemmas \\ref{algebra-lemma-integral-going-up} and", "\\ref{algebra-lemma-going-up-closed}).", "On the other hand $\\Spec(B/fB)$ is the punctured spectrum of $\\Spec(A/fA)$.", "This is a contradiction unless $\\Spec(B/fB) = \\emptyset$ which", "means that $\\dim(A/fA) = 0$ as desired." ], "refs": [ "algebra-lemma-Noetherian-irreducible-components", "algebra-lemma-silly", "algebra-lemma-support-point", "algebra-lemma-integral-going-up", "algebra-lemma-going-up-closed" ], "ref_ids": [ 453, 378, 693, 500, 552 ] } ], "ref_ids": [] }, { "id": 7959, "type": "theorem", "label": "divisors-lemma-complement-affine-open-immersion", "categories": [ "divisors" ], "title": "divisors-lemma-complement-affine-open-immersion", "contents": [ "\\begin{reference}", "\\cite[EGA IV, Corollaire 21.12.7]{EGA4}", "\\end{reference}", "Let $X$ be a locally Noetherian scheme. Let $U \\subset X$ be an open subscheme", "such that the inclusion morphism $U \\to X$ is affine.", "For every generic point $\\xi$ of an irreducible component of", "$X \\setminus U$ the local ring $\\mathcal{O}_{X, \\xi}$", "has dimension $\\leq 1$. If $U$ is dense or if $\\xi$ is in the closure", "of $U$, then $\\dim(\\mathcal{O}_{X, \\xi}) = 1$." ], "refs": [], "proofs": [ { "contents": [ "Since $\\xi$ is a generic point of $X \\setminus U$, we see that", "$$", "U_\\xi = U \\times_X \\Spec(\\mathcal{O}_{X, \\xi}) \\subset", "\\Spec(\\mathcal{O}_{X, \\xi})", "$$", "is the punctured spectrum of $\\mathcal{O}_{X, \\xi}$ (hint: use", "Schemes, Lemma \\ref{schemes-lemma-specialize-points}).", "As $U \\to X$ is affine, we see that $U_\\xi \\to \\Spec(\\mathcal{O}_{X, \\xi})$", "is affine (Morphisms, Lemma \\ref{morphisms-lemma-base-change-affine})", "and we conclude that $U_\\xi$ is affine.", "Hence $\\dim(\\mathcal{O}_{X, \\xi}) \\leq 1$ by", "Lemma \\ref{lemma-affine-punctured-spec}.", "If $\\xi \\in \\overline{U}$, then there is a specialization", "$\\eta \\to \\xi$ where $\\eta \\in U$ (just take $\\eta$ a generic", "point of an irreducible component of $\\overline{U}$ which", "contains $\\xi$; since $\\overline{U}$ is locally Noetherian,", "hence locally has finitely many irreducible components, we see that", "$\\eta \\in U$). Then $\\eta \\in \\Spec(\\mathcal{O}_{X, \\xi})$ and", "we see that the dimension cannot be $0$." ], "refs": [ "schemes-lemma-specialize-points", "morphisms-lemma-base-change-affine", "divisors-lemma-affine-punctured-spec" ], "ref_ids": [ 7684, 5176, 7958 ] } ], "ref_ids": [] }, { "id": 7960, "type": "theorem", "label": "divisors-lemma-complement-affine-open", "categories": [ "divisors" ], "title": "divisors-lemma-complement-affine-open", "contents": [ "Let $X$ be a separated locally Noetherian scheme. Let $U \\subset X$ be an", "affine open. For every generic point $\\xi$ of an irreducible component of", "$X \\setminus U$ the local ring $\\mathcal{O}_{X, \\xi}$", "has dimension $\\leq 1$. If $U$ is dense or if $\\xi$ is in the closure", "of $U$, then $\\dim(\\mathcal{O}_{X, \\xi}) = 1$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-complement-affine-open-immersion}", "because the morphism $U \\to X$ is affine by", "Morphisms, Lemma \\ref{morphisms-lemma-affine-permanence}." ], "refs": [ "divisors-lemma-complement-affine-open-immersion", "morphisms-lemma-affine-permanence" ], "ref_ids": [ 7959, 5179 ] } ], "ref_ids": [] }, { "id": 7961, "type": "theorem", "label": "divisors-lemma-complement-open-affine-effective-cartier-divisor", "categories": [ "divisors" ], "title": "divisors-lemma-complement-open-affine-effective-cartier-divisor", "contents": [ "Let $X$ be a Noetherian separated scheme. Let $U \\subset X$ be", "a dense affine open. If $\\mathcal{O}_{X, x}$ is a UFD for all", "$x \\in X \\setminus U$, then there exists an effective Cartier", "divisor $D \\subset X$ with $U = X \\setminus D$." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is Noetherian, the complement $X \\setminus U$ has finitely", "many irreducible components $D_1, \\ldots, D_r$", "(Properties, Lemma \\ref{properties-lemma-Noetherian-irreducible-components}", "applied to the reduced induced subscheme structure on $X \\setminus U$).", "Each $D_i \\subset X$ has codimension $1$ by", "Lemma \\ref{lemma-complement-affine-open}", "(and Properties, Lemma \\ref{properties-lemma-codimension-local-ring}).", "Thus $D_i$ is an effective Cartier divisor by", "Lemma \\ref{lemma-weil-divisor-is-cartier-UFD}.", "Hence we can take $D = D_1 + \\ldots + D_r$." ], "refs": [ "properties-lemma-Noetherian-irreducible-components", "divisors-lemma-complement-affine-open", "properties-lemma-codimension-local-ring", "divisors-lemma-weil-divisor-is-cartier-UFD" ], "ref_ids": [ 2956, 7960, 2979, 7951 ] } ], "ref_ids": [] }, { "id": 7962, "type": "theorem", "label": "divisors-lemma-complement-open-affine-effective-cartier-divisor-bis", "categories": [ "divisors" ], "title": "divisors-lemma-complement-open-affine-effective-cartier-divisor-bis", "contents": [ "Let $X$ be a Noetherian scheme with affine diagonal. Let $U \\subset X$ be", "a dense affine open. If $\\mathcal{O}_{X, x}$ is a UFD for all", "$x \\in X \\setminus U$, then there exists an effective Cartier", "divisor $D \\subset X$ with $U = X \\setminus D$." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is Noetherian, the complement $X \\setminus U$ has finitely", "many irreducible components $D_1, \\ldots, D_r$", "(Properties, Lemma \\ref{properties-lemma-Noetherian-irreducible-components}", "applied to the reduced induced subscheme structure on $X \\setminus U$).", "We view $D_i$ as a reduced closed subscheme of $X$.", "Let $X = \\bigcup_{j \\in J} X_j$ be an affine open covering of $X$. For all", "$j$ in $J$, set $U_j = U \\cap X_j$. Since $X$ has affine diagonal,", "the scheme", "$$", "U_j = X \\times_{(X \\times X)} (U \\times X_j)", "$$", "is affine. Therefore, as $X_j$ is separated, it follows from", "Lemma \\ref{lemma-complement-open-affine-effective-cartier-divisor}", "and its proof that for all $j \\in J$ and $1 \\leq i \\leq r$ the", "intersection $D_i \\cap X_j$ is either empty or an", "effective Cartier divisor in $X_j$.", "Thus $D_i \\subset X$ is an effective Cartier divisor (as this is", "a local property). Hence we can take $D = D_1 + \\ldots + D_r$." ], "refs": [ "properties-lemma-Noetherian-irreducible-components", "divisors-lemma-complement-open-affine-effective-cartier-divisor" ], "ref_ids": [ 2956, 7961 ] } ], "ref_ids": [] }, { "id": 7963, "type": "theorem", "label": "divisors-lemma-finite-trivialize-invertible-upstairs", "categories": [ "divisors" ], "title": "divisors-lemma-finite-trivialize-invertible-upstairs", "contents": [ "Let $\\pi : X \\to Y$ be a finite morphism of schemes.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $y \\in Y$. There exists an open neighbourhood", "$V \\subset Y$ of $y$ such that $\\mathcal{L}|_{\\pi^{-1}(V)}$ is trivial." ], "refs": [], "proofs": [ { "contents": [ "Clearly we may assume $Y$ and hence $X$ affine. Since $\\pi$ is finite the", "fibre $\\pi^{-1}(\\{y\\})$ over $y$ is finite.", "Since $X$ is affine, we can pick $s \\in \\Gamma(X, \\mathcal{L})$", "not vanishing in any point of $\\pi^{-1}(\\{y\\})$. This follows", "from Properties, Lemma", "\\ref{properties-lemma-quasi-affine-invertible-nonvanishing-section}", "but we also give a direct argument. Namely, we can", "pick a finite set $E \\subset X$ of closed points such that", "every $x \\in \\pi^{-1}(\\{y\\})$ specializes to some point of $E$.", "For $x \\in E$ denote $i_x : x \\to X$ the closed immersion.", "Then", "$\\mathcal{L} \\to \\bigoplus_{x \\in E} i_{x, *}i_x^*\\mathcal{L}$", "is a surjective map of quasi-coherent $\\mathcal{O}_X$-modules,", "and hence the map", "$$", "\\Gamma(X, \\mathcal{L}) \\to", "\\bigoplus\\nolimits_{x \\in E} \\mathcal{L}_x/\\mathfrak m_x\\mathcal{L}_x", "$$", "is surjective (as taking global sections is an exact functor on the", "category of quasi-coherent $\\mathcal{O}_X$-modules, see", "Schemes, Lemma \\ref{schemes-lemma-equivalence-quasi-coherent}).", "Thus we can find an $s \\in \\Gamma(X, \\mathcal{L})$", "not vanishing at any point specializing to a point of $E$.", "Then $X_s \\subset X$ is an open neighbourhood of $\\pi^{-1}(\\{y\\})$.", "Since $\\pi$ is finite, hence closed, we conclude that there is an", "open neighbourhood $V \\subset Y$ of $y$ whose inverse image", "is contained in $X_s$ as desired." ], "refs": [ "schemes-lemma-equivalence-quasi-coherent" ], "ref_ids": [ 7664 ] } ], "ref_ids": [] }, { "id": 7964, "type": "theorem", "label": "divisors-lemma-norm-invertible", "categories": [ "divisors" ], "title": "divisors-lemma-norm-invertible", "contents": [ "Let $\\pi : X \\to Y$ be a finite morphism of schemes. If there exists", "a norm of degree $d$ for $\\pi$, then there exists a homomorphism of", "abelian groups", "$$", "\\text{Norm}_\\pi : \\Pic(X) \\to \\Pic(Y)", "$$", "such that $\\text{Norm}_\\pi(\\pi^*\\mathcal{N}) \\cong \\mathcal{N}^{\\otimes d}$", "for all invertible $\\mathcal{O}_Y$-modules $\\mathcal{N}$." ], "refs": [], "proofs": [ { "contents": [ "We will use the correspondence between isomorphism classes of", "invertible $\\mathcal{O}_X$-modules and elements of", "$H^1(X, \\mathcal{O}_X^*)$ given in", "Cohomology, Lemma \\ref{cohomology-lemma-h1-invertible}", "without further mention. We explain how to take the norm of an invertible", "$\\mathcal{O}_X$-module $\\mathcal{L}$. Namely, by", "Lemma \\ref{lemma-finite-trivialize-invertible-upstairs}", "there exists an open covering $Y = \\bigcup V_j$ such that", "$\\mathcal{L}|_{\\pi^{-1}V_j}$ is trivial. Choose a generating section", "$s_j \\in \\mathcal{L}(\\pi^{-1}V_j)$ for each $j$.", "On the overlaps $\\pi^{-1}V_j \\cap \\pi^{-1}V_{j'}$ we can write", "$$", "s_j = u_{jj'} s_{j'}", "$$", "for a unique $u_{jj'} \\in \\mathcal{O}^*_X(\\pi^{-1}V_j \\cap \\pi^{-1}V_{j'})$.", "Thus we can consider the elements", "$$", "v_{jj'} = \\text{Norm}_\\pi(u_{jj'}) \\in \\mathcal{O}_Y^*(V_j \\cap V_{j'})", "$$", "These elements satisfy the cocycle condition (because the", "$u_{jj'}$ do and $\\text{Norm}_\\pi$ is multiplicative) and", "therefore define an invertible $\\mathcal{O}_Y$-module.", "We omit the verification that: this is well defined,", "additive on Picard groups, and satisfies the property", "$\\text{Norm}_\\pi(\\pi^*\\mathcal{N}) \\cong \\mathcal{N}^{\\otimes d}$", "for all invertible $\\mathcal{O}_Y$-modules $\\mathcal{N}$." ], "refs": [ "cohomology-lemma-h1-invertible", "divisors-lemma-finite-trivialize-invertible-upstairs" ], "ref_ids": [ 2036, 7963 ] } ], "ref_ids": [] }, { "id": 7965, "type": "theorem", "label": "divisors-lemma-norm-map-invertible", "categories": [ "divisors" ], "title": "divisors-lemma-norm-map-invertible", "contents": [ "Let $\\pi : X \\to Y$ be a finite morphism of schemes. Assume there exists", "a norm of degree $d$ for $\\pi$. For any $\\mathcal{O}_X$-linear map", "$\\varphi : \\mathcal{L} \\to \\mathcal{L}'$", "of invertible $\\mathcal{O}_X$-modules there is an $\\mathcal{O}_Y$-linear", "map", "$$", "\\text{Norm}_\\pi(\\varphi) :", "\\text{Norm}_\\pi(\\mathcal{L})", "\\longrightarrow", "\\text{Norm}_\\pi(\\mathcal{L}')", "$$", "with $\\text{Norm}_\\pi(\\mathcal{L})$, $\\text{Norm}_\\pi(\\mathcal{L}')$", "as in Lemma \\ref{lemma-norm-invertible}. Moreover, for", "$y \\in Y$ the following are equivalent", "\\begin{enumerate}", "\\item $\\varphi$ is zero at a point of $x \\in X$ with $\\pi(x) = y$, and", "\\item $\\text{Norm}_\\pi(\\varphi)$ is zero at $y$.", "\\end{enumerate}" ], "refs": [ "divisors-lemma-norm-invertible" ], "proofs": [ { "contents": [ "We choose an open covering $Y = \\bigcup V_j$ such that", "$\\mathcal{L}$ and $\\mathcal{L}'$ are trivial over the opens $\\pi^{-1}V_j$.", "This is possible by", "Lemma \\ref{lemma-finite-trivialize-invertible-upstairs}.", "Choose generating sections", "$s_j$ and $s'_j$ of $\\mathcal{L}$ and $\\mathcal{L}'$", "over the opens $\\pi^{-1}V_j$. Then $\\varphi(s_j) = f_js'_j$", "for some $f_j \\in \\mathcal{O}_X(\\pi^{-1}V_j)$.", "Define $\\text{Norm}_\\pi(\\varphi)$ to be multiplication", "by $\\text{Norm}_\\pi(f_j)$ on $V_j$. An simple", "calculation involving the cocycles used to construct", "$\\text{Norm}_\\pi(\\mathcal{L})$, $\\text{Norm}_\\pi(\\mathcal{L}')$", "in the proof of Lemma \\ref{lemma-norm-invertible}", "shows that this defines", "a map as stated in the lemma. The final statement follows", "from condition (2) in the definition of a norm map of degree $d$.", "Some details omitted." ], "refs": [ "divisors-lemma-finite-trivialize-invertible-upstairs", "divisors-lemma-norm-invertible" ], "ref_ids": [ 7963, 7964 ] } ], "ref_ids": [ 7964 ] }, { "id": 7966, "type": "theorem", "label": "divisors-lemma-norm-ample", "categories": [ "divisors" ], "title": "divisors-lemma-norm-ample", "contents": [ "Let $\\pi : X \\to Y$ be a finite morphism of schemes. Assume $X$ has", "an ample invertible sheaf and there exists a norm of degree $d$", "for $\\pi$. Then $Y$ has an ample invertible sheaf." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{L}$ be the ample invertible sheaf on $X$ given to us", "by assumption. We will prove that $\\mathcal{N} = \\text{Norm}_\\pi(\\mathcal{L})$", "is ample on $Y$.", "\\medskip\\noindent", "Since $X$ is quasi-compact (Properties, Definition", "\\ref{properties-definition-ample}) and $X \\to Y$ surjective", "(by the existence of $\\text{Norm}_\\pi$)", "we see that $Y$ is quasi-compact.", "Let $y \\in Y$ be a point. To finish the proof", "we will show that there exists a section $t$ of some positive tensor", "power of $\\mathcal{N}$ which does not vanish at $y$ such that $Y_t$", "is affine. To do this, choose an affine open neighbourhood $V \\subset Y$", "of $y$. Choose $n \\gg 0$ and a section", "$s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$", "such that", "$$", "\\pi^{-1}(\\{y\\}) \\subset X_s \\subset \\pi^{-1}V", "$$", "by", "Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-principal-affine}.", "Then $t = \\text{Norm}_\\pi(s)$ is a section of $\\mathcal{N}^{\\otimes n}$", "which does not vanish at $x$ and with $Y_t \\subset V$, see", "Lemma \\ref{lemma-norm-map-invertible}. Then $Y_t$", "is affine by Properties, Lemma \\ref{properties-lemma-affine-cap-s-open}." ], "refs": [ "properties-definition-ample", "properties-lemma-ample-finite-set-in-principal-affine", "divisors-lemma-norm-map-invertible", "properties-lemma-affine-cap-s-open" ], "ref_ids": [ 3088, 3063, 7965, 3042 ] } ], "ref_ids": [] }, { "id": 7967, "type": "theorem", "label": "divisors-lemma-norm-quasi-affine", "categories": [ "divisors" ], "title": "divisors-lemma-norm-quasi-affine", "contents": [ "Let $\\pi : X \\to Y$ be a finite morphism of schemes. Assume $X$ is quasi-affine", "and there exists a norm of degree $d$ for $\\pi$. Then $Y$ is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "By Properties, Lemma \\ref{properties-lemma-quasi-affine-O-ample}", "we see that $\\mathcal{O}_X$ is an ample invertible sheaf on $X$.", "The proof of Lemma \\ref{lemma-norm-ample} shows that", "$\\text{Norm}_\\pi(\\mathcal{O}_X) = \\mathcal{O}_Y$", "is an ample invertible $\\mathcal{O}_Y$-module. Hence", "Properties, Lemma \\ref{properties-lemma-quasi-affine-O-ample}", "shows that $Y$ is quasi-affine." ], "refs": [ "properties-lemma-quasi-affine-O-ample", "divisors-lemma-norm-ample", "properties-lemma-quasi-affine-O-ample" ], "ref_ids": [ 3053, 7966, 3053 ] } ], "ref_ids": [] }, { "id": 7968, "type": "theorem", "label": "divisors-lemma-finite-locally-free-has-norm", "categories": [ "divisors" ], "title": "divisors-lemma-finite-locally-free-has-norm", "contents": [ "Let $\\pi : X \\to Y$ be a finite locally free morphism of degree $d \\geq 1$.", "Then there exists a canonical norm of degree $d$ whose formation commutes", "with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset Y$ be an affine open such that $(\\pi_*\\mathcal{O}_X)|_V$", "is finite free of rank $d$. Choosing a basis we obtain an isomorphism", "$$", "\\mathcal{O}_V^{\\oplus d} \\cong (\\pi_*\\mathcal{O}_X)|_V", "$$", "For every $f \\in \\pi_*\\mathcal{O}_X(V) = \\mathcal{O}_X(\\pi^{-1}(V))$", "multiplication by $f$ defines a $\\mathcal{O}_V$-linear endomorphism", "$m_f$ of the displayed free vector bundle. Thus we get a $d \\times d$", "matrix $M_f \\in \\text{Mat}(d \\times d, \\mathcal{O}_Y(V))$ and we can set", "$$", "\\text{Norm}_\\pi(f) = \\det(M_f)", "$$", "Since the determinant of a matrix is independent of the choice of", "the basis chosen we see that this is well defined which also means", "that this construction will glue to a global map as desired.", "Compatibility with base change is straightforward from the construction.", "\\medskip\\noindent", "Property (1) follows from the fact that the determinant of a", "$d \\times d$ diagonal matrix with entries $g, g, \\ldots, g$ is $g^d$.", "To see property (2) we may base change and assume that $Y$ is the", "spectrum of a field $k$. Then $X = \\Spec(A)$ with $A$ a $k$-algebra", "with $\\dim_k(A) = d$. If there exists an $x \\in X$ such that", "$f \\in A$ vanishes at $x$, then there exists a map $A \\to \\kappa$", "into a field such that $f$ maps to zero in $\\kappa$. Then", "$f : A \\to A$ cannot be surjective, hence $\\det(f : A \\to A) = 0$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7969, "type": "theorem", "label": "divisors-lemma-norm-in-normal-case", "categories": [ "divisors" ], "title": "divisors-lemma-norm-in-normal-case", "contents": [ "Let $\\pi : X \\to Y$ be a finite surjective morphism with $X$ and $Y$", "integral and $Y$ normal. Then there exists a norm of degree", "$[R(X) : R(Y)]$ for $\\pi$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\Spec(B) \\subset Y$ be an affine open subset and let", "$\\Spec(A) \\subset X$ be its inverse image. Then $A$ and $B$", "are domains. Let $K$ be the fraction", "field of $A$ and $L$ the fraction field of $B$. Picture:", "$$", "\\xymatrix{", "L \\ar[r] & K \\\\", "B \\ar[u] \\ar[r] & A \\ar[u]", "}", "$$", "Since $K/L$ is a finite extension, there is a norm map", "$\\text{Norm}_{K/L} : K^* \\to L^*$ of degree $d = [K : L]$; this is given by", "mapping $f \\in K$ to $\\det_L(f : K \\to K)$ as in the proof", "of Lemma \\ref{lemma-finite-locally-free-has-norm}.", "Observe that the characteristic polynomial of $f : K \\to K$", "is a power of the minimal polynomial of $f$ over $L$;", "in particular $\\text{Norm}_{K/L}(f)$ is a power of the constant", "coefficient of the minimal polynomial of $f$ over $L$. Hence by", "Algebra, Lemma \\ref{algebra-lemma-minimal-polynomial-normal-domain}", "$\\text{Norm}_{K/L}$ maps $A$ into $B$.", "This determines a compatible system of maps", "on sections over affines and hence a global norm map", "$\\text{Norm}_\\pi$ of degree $d$.", "\\medskip\\noindent", "Property (1) is immediate from the construction.", "To see property (2) let $f \\in A$ be contained in the", "prime ideal $\\mathfrak p \\subset A$. Let", "$f^m + b_1 f^{m - 1} + \\ldots + b_m$ be the minimal", "polynomial of $f$ over $L$. By", "Algebra, Lemma \\ref{algebra-lemma-minimal-polynomial-normal-domain}", "we have $b_i \\in B$. Hence $b_0 \\in B \\cap \\mathfrak p$.", "Since $\\text{Norm}_{K/L}(f) = b_0^{d/m}$ (see above)", "we conclude that the norm vanishes in the image point of $\\mathfrak p$." ], "refs": [ "divisors-lemma-finite-locally-free-has-norm", "algebra-lemma-minimal-polynomial-normal-domain", "algebra-lemma-minimal-polynomial-normal-domain" ], "ref_ids": [ 7968, 521, 521 ] } ], "ref_ids": [] }, { "id": 7970, "type": "theorem", "label": "divisors-lemma-Frobenius-gives-norm-for-reduction", "categories": [ "divisors" ], "title": "divisors-lemma-Frobenius-gives-norm-for-reduction", "contents": [ "Let $X$ be a Noetherian scheme. Let $p$ be a prime number such that", "$p\\mathcal{O}_X = 0$. Then for some $e > 0$ there exists a norm", "of degree $p^e$ for $X_{red} \\to X$ where $X_{red}$ is the reduction", "of $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $A$ be a Noetherian ring with $pA = 0$. Let $I \\subset A$ be the", "ideal of nilpotent elements. Then $I^n = 0$ for some $n$ (Algebra,", "Lemma \\ref{algebra-lemma-Noetherian-power}).", "Pick $e$ such that $p^e \\geq n$. Then", "$$", "A/I \\longrightarrow A,\\quad", "f \\bmod I \\longmapsto f^{p^e}", "$$", "is well defined. This produces a norm of degree $p^e$ for", "$\\Spec(A/I) \\to \\Spec(A)$. Now if $X$ is obtained by glueing some", "affine schemes $\\Spec(A_i)$ then for some $e \\gg 0$ these maps", "glue to a norm map for $X_{red} \\to X$. Details omitted." ], "refs": [ "algebra-lemma-Noetherian-power" ], "ref_ids": [ 460 ] } ], "ref_ids": [] }, { "id": 7971, "type": "theorem", "label": "divisors-lemma-push-down-quasi-affine", "categories": [ "divisors" ], "title": "divisors-lemma-push-down-quasi-affine", "contents": [ "Let $\\pi : X \\to Y$ be a finite surjective morphism of schemes.", "Assume that $X$ is quasi-affine. If either", "\\begin{enumerate}", "\\item $\\pi$ is finite locally free, or", "\\item $Y$ is an integral normal scheme", "\\end{enumerate}", "then $Y$ is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "Case (1) follows from a combination of Lemmas", "\\ref{lemma-finite-locally-free-has-norm} and \\ref{lemma-norm-quasi-affine}.", "In case (2) we first replace $X$ by an irreducible component of $X$", "which dominates $Y$ (viewed as a reduced closed subscheme of $X$).", "Then we can apply Lemma \\ref{lemma-norm-in-normal-case}." ], "refs": [ "divisors-lemma-finite-locally-free-has-norm", "divisors-lemma-norm-quasi-affine", "divisors-lemma-norm-in-normal-case" ], "ref_ids": [ 7968, 7967, 7969 ] } ], "ref_ids": [] }, { "id": 7972, "type": "theorem", "label": "divisors-lemma-relative-Cartier", "categories": [ "divisors" ], "title": "divisors-lemma-relative-Cartier", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $D \\subset X$ be a closed subscheme.", "Assume", "\\begin{enumerate}", "\\item $D$ is an effective Cartier divisor, and", "\\item $D \\to S$ is a flat morphism.", "\\end{enumerate}", "Then for every morphism of schemes $g : S' \\to S$ the pullback", "$(g')^{-1}D$ is an effective Cartier divisor on $X' = S' \\times_S X$", "where $g' : X' \\to X$ is the projection." ], "refs": [], "proofs": [ { "contents": [ "Using", "Lemma \\ref{lemma-characterize-effective-Cartier-divisor}", "we translate this as follows into algebra. Let $A \\to B$ be a ring", "map and $h \\in B$. Assume $h$ is a nonzerodivisor and that $B/hB$ is flat", "over $A$. Then", "$$", "0 \\to B \\xrightarrow{h} B \\to B/hB \\to 0", "$$", "is a short exact sequence of $A$-modules with $B/hB$ flat over $A$. By", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}", "this sequence remains exact on tensoring over $A$ with any module, in", "particular with any $A$-algebra $A'$." ], "refs": [ "divisors-lemma-characterize-effective-Cartier-divisor", "algebra-lemma-flat-tor-zero" ], "ref_ids": [ 7927, 532 ] } ], "ref_ids": [] }, { "id": 7973, "type": "theorem", "label": "divisors-lemma-sum-relative-effective-Cartier-divisor", "categories": [ "divisors" ], "title": "divisors-lemma-sum-relative-effective-Cartier-divisor", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. If $D_1, D_2 \\subset X$", "are relative effective Cartier divisor on $X/S$ then so", "is $D_1 + D_2$ (Definition \\ref{definition-sum-effective-Cartier-divisors})." ], "refs": [ "divisors-definition-sum-effective-Cartier-divisors" ], "proofs": [ { "contents": [ "This translates into the following algebra fact:", "Let $A \\to B$ be a ring map and $h_1, h_2 \\in B$.", "Assume the $h_i$ are nonzerodivisors and that $B/h_iB$ is flat over $A$.", "Then $h_1h_2$ is a nonzerodivisor and $B/h_1h_2B$ is flat over $A$.", "The reason is that we have a short exact sequence", "$$", "0 \\to B/h_1B \\to B/h_1h_2B \\to B/h_2B \\to 0", "$$", "where the first arrow is given by multiplication by $h_2$. Since", "the outer two are flat modules over $A$, so is the middle one, see", "Algebra, Lemma \\ref{algebra-lemma-flat-ses}." ], "refs": [ "algebra-lemma-flat-ses" ], "ref_ids": [ 533 ] } ], "ref_ids": [ 8090 ] }, { "id": 7974, "type": "theorem", "label": "divisors-lemma-difference-relative-effective-Cartier-divisor", "categories": [ "divisors" ], "title": "divisors-lemma-difference-relative-effective-Cartier-divisor", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. If $D_1, D_2 \\subset X$", "are relative effective Cartier divisor on $X/S$ and $D_1 \\subset D_2$", "as closed subschemes, then the effective Cartier divisor $D$", "such that $D_2 = D_1 + D$", "(Lemma \\ref{lemma-difference-effective-Cartier-divisors}) is", "a relative effective Cartier divisor on $X/S$." ], "refs": [ "divisors-lemma-difference-effective-Cartier-divisors" ], "proofs": [ { "contents": [ "This translates into the following algebra fact:", "Let $A \\to B$ be a ring map and $h_1, h_2 \\in B$.", "Assume the $h_i$ are nonzerodivisors, that $B/h_iB$ is flat over $A$, and", "that $(h_2) \\subset (h_1)$. Then we can write $h_2 = h h_1$", "where $h \\in B$ is a nonzerodivisor. We get a short exact sequence", "$$", "0 \\to B/hB \\to B/h_2B \\to B/h_1B \\to 0", "$$", "where the first arrow is given by multiplication by $h_1$. Since", "the right two are flat modules over $A$, so is the middle one, see", "Algebra, Lemma \\ref{algebra-lemma-flat-ses}." ], "refs": [ "algebra-lemma-flat-ses" ], "ref_ids": [ 533 ] } ], "ref_ids": [ 7932 ] }, { "id": 7975, "type": "theorem", "label": "divisors-lemma-flat-at-x", "categories": [ "divisors" ], "title": "divisors-lemma-flat-at-x", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $D \\subset X$ be a relative effective Cartier divisor on $X/S$.", "If $x \\in D$ and $\\mathcal{O}_{X, x}$ is Noetherian, then $f$ is flat at $x$." ], "refs": [], "proofs": [ { "contents": [ "Set $A = \\mathcal{O}_{S, f(x)}$ and $B = \\mathcal{O}_{X, x}$.", "Let $h \\in B$ be an element which generates the ideal of $D$.", "Then $h$ is a nonzerodivisor in $B$ such that $B/hB$ is a flat", "local $A$-algebra. Let $I \\subset A$ be a finitely generated ideal.", "Consider the commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "B \\ar[r]_h &", "B \\ar[r] &", "B/hB \\ar[r] & 0 \\\\", "0 \\ar[r] &", "B \\otimes_A I \\ar[r]^h \\ar[u] &", "B \\otimes_A I \\ar[r] \\ar[u] &", "B/hB \\otimes_A I \\ar[r] \\ar[u] & 0", "}", "$$", "The lower sequence is short exact as $B/hB$ is flat over $A$, see", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}.", "The right vertical arrow is injective as $B/hB$ is flat over $A$, see", "Algebra, Lemma \\ref{algebra-lemma-flat}.", "Hence multiplication by $h$ is surjective on the kernel $K$ of", "the middle vertical arrow. By Nakayama's lemma, see", "Algebra, Lemma \\ref{algebra-lemma-NAK}", "we conclude that $K= 0$. Hence $B$ is flat over $A$, see", "Algebra, Lemma \\ref{algebra-lemma-flat}." ], "refs": [ "algebra-lemma-flat-tor-zero", "algebra-lemma-flat", "algebra-lemma-NAK", "algebra-lemma-flat" ], "ref_ids": [ 532, 525, 401, 525 ] } ], "ref_ids": [] }, { "id": 7976, "type": "theorem", "label": "divisors-lemma-flat-relative-Cartier-divisor", "categories": [ "divisors" ], "title": "divisors-lemma-flat-relative-Cartier-divisor", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $D \\subset X$ be a relative effective Cartier divisor.", "If $f$ is locally of finite presentation, then there exists", "an open subscheme $U \\subset X$ such that $D \\subset U$ and", "such that $f|_U : U \\to S$ is flat." ], "refs": [], "proofs": [ { "contents": [ "Pick $x \\in D$. It suffices to find an open neighbourhood $U \\subset X$", "of $x$ such that $f|_U$ is flat. Hence the lemma reduces to the case", "that $X = \\Spec(B)$ and $S = \\Spec(A)$ are affine", "and that $D$ is given by a nonzerodivisor $h \\in B$. By assumption", "$B$ is a finitely presented $A$-algebra and $B/hB$ is a flat", "$A$-algebra. We are going to use absolute Noetherian approximation.", "\\medskip\\noindent", "Write $B = A[x_1, \\ldots, x_n]/(g_1, \\ldots, g_m)$. Assume", "$h$ is the image of $h' \\in A[x_1, \\ldots, x_n]$. Choose a finite type", "$\\mathbf{Z}$-subalgebra $A_0 \\subset A$ such that all the coefficients", "of the polynomials $h', g_1, \\ldots, g_m$ are in $A_0$. Then we can set", "$B_0 = A_0[x_1, \\ldots, x_n]/(g_1, \\ldots, g_m)$ and $h_0$ the image", "of $h'$ in $B_0$. Then $B = B_0 \\otimes_{A_0} A$ and", "$B/hB = B_0/h_0B_0 \\otimes_{A_0} A$. By Algebra, Lemma", "\\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "we may, after enlarging $A_0$, assume that $B_0/h_0B_0$ is flat", "over $A_0$. Let $K_0 = \\Ker(h_0 : B_0 \\to B_0)$.", "As $B_0$ is of finite type over $\\mathbf{Z}$ we see that $K_0$ is", "a finitely generated ideal. Let $A_1 \\subset A$ be a finite type", "$\\mathbf{Z}$-subalgebra containing $A_0$ and denote $B_1$, $h_1$, $K_1$", "the corresponding objects over $A_1$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-H1-regular}", "the map $K_0 \\otimes_{A_0} A_1 \\to K_1$ is surjective. On the other hand,", "the kernel of $h : B \\to B$ is zero by assumption. Hence every element", "of $K_0$ maps to zero in $K_1$ for sufficiently large subrings", "$A_1 \\subset A$. Since $K_0$ is finitely generated, we conclude that", "$K_1 = 0$ for a suitable choice of $A_1$.", "\\medskip\\noindent", "Set $f_1 : X_1 \\to S_1$ equal to $\\Spec$ of the", "ring map $A_1 \\to B_1$. Set $D_1 = \\Spec(B_1/h_1B_1)$.", "Since $B = B_1 \\otimes_{A_1} A$, i.e., $X = X_1 \\times_{S_1} S$,", "it now suffices to prove the lemma for $X_1 \\to S_1$ and the relative", "effective Cartier divisor $D_1$, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-module-flat}.", "Hence we have reduced to the case where $A$ is a Noetherian ring.", "In this case we know that the ring map $A \\to B$ is flat at every", "prime $\\mathfrak q$ of $V(h)$ by", "Lemma \\ref{lemma-flat-at-x}.", "Combined with the fact that the flat locus is open in this case, see", "Algebra, Theorem \\ref{algebra-theorem-openness-flatness}", "we win." ], "refs": [ "algebra-lemma-flat-finite-presentation-limit-flat", "more-algebra-lemma-base-change-H1-regular", "morphisms-lemma-base-change-module-flat", "divisors-lemma-flat-at-x", "algebra-theorem-openness-flatness" ], "ref_ids": [ 1389, 9991, 5264, 7975, 326 ] } ], "ref_ids": [] }, { "id": 7977, "type": "theorem", "label": "divisors-lemma-michael-artin", "categories": [ "divisors" ], "title": "divisors-lemma-michael-artin", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $D \\subset X$ be a relative effective Cartier divisor on $X/S$.", "If $f$ is flat at all points of $X \\setminus D$, then $f$ is flat." ], "refs": [], "proofs": [ { "contents": [ "This translates into the following algebra fact:", "Let $A \\to B$ be a ring map and $h \\in B$.", "Assume $h$ is a nonzerodivisor, that $B/hB$ is flat over $A$, and", "that the localization $B_h$ is flat over $A$. Then $B$ is flat over $A$.", "The reason is that we have a short exact sequence", "$$", "0 \\to B \\to B_h \\to \\colim_n (1/h^n)B/B \\to 0", "$$", "and that the second and third terms are flat over $A$, which implies", "that $B$ is flat over $A$ (see", "Algebra, Lemma \\ref{algebra-lemma-flat-ses}). Note that a filtered", "colimit of flat modules is flat (see", "Algebra, Lemma \\ref{algebra-lemma-colimit-flat})", "and that by induction on $n$ each $(1/h^n)B/B \\cong B/h^nB$ is flat over", "$A$ since it fits into the short exact sequence", "$$", "0 \\to B/h^{n - 1}B \\xrightarrow{h} B/h^nB \\to B/hB \\to 0", "$$", "Some details omitted." ], "refs": [ "algebra-lemma-flat-ses", "algebra-lemma-colimit-flat" ], "ref_ids": [ 533, 523 ] } ], "ref_ids": [] }, { "id": 7978, "type": "theorem", "label": "divisors-lemma-fibre-Cartier", "categories": [ "divisors" ], "title": "divisors-lemma-fibre-Cartier", "contents": [ "Let $\\varphi : X \\to S$ be a flat morphism which is locally of finite", "presentation. Let $Z \\subset X$ be a closed subscheme.", "Let $x \\in Z$ with image $s \\in S$.", "\\begin{enumerate}", "\\item If $Z_s \\subset X_s$ is a Cartier divisor in a neighbourhood of $x$,", "then there exists an open $U \\subset X$ and a", "relative effective Cartier divisor $D \\subset U$ such that", "$Z \\cap U \\subset D$ and $Z_s \\cap U = D_s$.", "\\item If $Z_s \\subset X_s$ is a Cartier divisor in a neighbourhood of $x$,", "the morphism $Z \\to X$ is of finite presentation, and $Z \\to S$ is flat at", "$x$, then we can choose $U$ and $D$ such that $Z \\cap U = D$.", "\\item If $Z_s \\subset X_s$ is a Cartier divisor in a neighbourhood of $x$", "and $Z$ is a locally principal closed subscheme of $X$ in a neighbourhood", "of $x$, then we can choose $U$ and $D$ such that $Z \\cap U = D$.", "\\end{enumerate}", "In particular, if $Z \\to S$ is locally of finite presentation and flat and", "all fibres $Z_s \\subset X_s$ are effective Cartier divisors, then", "$Z$ is a relative effective Cartier divisor. Similarly, if $Z$", "is a locally principal closed subscheme of $X$ such that all fibres", "$Z_s \\subset X_s$ are effective Cartier divisors, then", "$Z$ is a relative effective Cartier divisor." ], "refs": [], "proofs": [ { "contents": [ "Choose affine open neighbourhoods $\\Spec(A)$ of $s$ and", "$\\Spec(B)$ of $x$ such that", "$\\varphi(\\Spec(B)) \\subset \\Spec(A)$.", "Let $\\mathfrak p \\subset A$ be the prime ideal corresponding to $s$.", "Let $\\mathfrak q \\subset B$ be the prime ideal corresponding to $x$.", "Let $I \\subset B$ be the ideal corresponding to $Z$.", "By the initial assumption of the lemma we know that", "$A \\to B$ is flat and of finite presentation.", "The assumption in (1) means that, after shrinking $\\Spec(B)$, we may", "assume $I(B \\otimes_A \\kappa(\\mathfrak p))$ is generated by a single", "element which is a nonzerodivisor in $B \\otimes_A \\kappa(\\mathfrak p)$.", "Say $f \\in I$ maps to this generator. We claim that after inverting", "an element $g \\in B$, $g \\not \\in \\mathfrak q$ the closed subscheme", "$D = V(f) \\subset \\Spec(B_g)$ is a relative effective Cartier", "divisor.", "\\medskip\\noindent", "By", "Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "we can find a flat finite type ring map $A_0 \\to B_0$ of Noetherian", "rings, an element $f_0 \\in B_0$, a ring map $A_0 \\to A$ and an isomorphism", "$A \\otimes_{A_0} B_0 \\cong B$. If $\\mathfrak p_0 = A_0 \\cap \\mathfrak p$", "then we see that", "$$", "B \\otimes_A \\kappa(\\mathfrak p) =", "\\left(B_0 \\otimes_{A_0} \\kappa(\\mathfrak p_0)\\right)", "\\otimes_{\\kappa(\\mathfrak p_0))} \\kappa(\\mathfrak p)", "$$", "hence $f_0$ is a nonzerodivisor in $B_0 \\otimes_{A_0} \\kappa(\\mathfrak p_0)$.", "By", "Algebra, Lemma \\ref{algebra-lemma-grothendieck}", "we see that $f_0$ is a nonzerodivisor in $(B_0)_{\\mathfrak q_0}$", "where $\\mathfrak q_0 = B_0 \\cap \\mathfrak q$ and", "that $(B_0/f_0B_0)_{\\mathfrak q_0}$ is flat over $A_0$. Hence by", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-neighbourhood}", "and", "Algebra, Theorem \\ref{algebra-theorem-openness-flatness}", "there exists a $g_0 \\in B_0$, $g_0 \\not \\in \\mathfrak q_0$ such", "that $f_0$ is a nonzerodivisor in $(B_0)_{g_0}$ and such that", "$(B_0/f_0B_0)_{g_0}$ is flat over $A_0$. Hence we see that", "$D_0 = V(f_0) \\subset \\Spec((B_0)_{g_0})$ is a relative effective", "Cartier divisor. Since we know that this property is preserved under", "base change, see", "Lemma \\ref{lemma-relative-Cartier},", "we obtain the claim mentioned above with $g$ equal to the image of $g_0$", "in $B$.", "\\medskip\\noindent", "At this point we have proved (1). To see (2) consider the closed", "immersion $Z \\to D$. The surjective ring map", "$u : \\mathcal{O}_{D, x} \\to \\mathcal{O}_{Z, x}$", "is a map of flat local $\\mathcal{O}_{S, s}$-algebras which", "are essentially of finite presentation, and which becomes an", "isomorphisms after dividing by $\\mathfrak m_s$. Hence it is", "an isomorphism, see", "Algebra, Lemma \\ref{algebra-lemma-mod-injective-general}.", "It follows that $Z \\to D$ is an isomorphism in a neighbourhood", "of $x$, see", "Algebra, Lemma \\ref{algebra-lemma-local-isomorphism}.", "To see (3), after possibly shrinking $U$ we may assume that", "the ideal of $D$ is generated by a single nonzerodivisor $f$", "and the ideal of $Z$ is generated by an element $g$. Then", "$f = gh$. But $g|_{U_s}$ and $f|_{U_s}$ cut out the same", "effective Cartier divisor in a neighbourhood of $x$. Hence", "$h|_{X_s}$ is a unit in $\\mathcal{O}_{X_s, x}$, hence $h$ is", "a unit in $\\mathcal{O}_{X, x}$ hence $h$ is a unit in an", "open neighbourhood of $x$. I.e., $Z \\cap U = D$ after shrinking $U$.", "\\medskip\\noindent", "The final statements of the lemma follow immediately from", "parts (2) and (3), combined with the fact that $Z \\to S$", "is locally of finite presentation if and only if $Z \\to X$ is", "of finite presentation, see", "Morphisms, Lemmas \\ref{morphisms-lemma-composition-finite-presentation} and", "\\ref{morphisms-lemma-finite-presentation-permanence}." ], "refs": [ "algebra-lemma-flat-finite-presentation-limit-flat", "algebra-lemma-grothendieck", "algebra-lemma-regular-sequence-in-neighbourhood", "algebra-theorem-openness-flatness", "divisors-lemma-relative-Cartier", "algebra-lemma-mod-injective-general", "algebra-lemma-local-isomorphism", "morphisms-lemma-composition-finite-presentation", "morphisms-lemma-finite-presentation-permanence" ], "ref_ids": [ 1389, 884, 741, 326, 7972, 1110, 1084, 5239, 5247 ] } ], "ref_ids": [] }, { "id": 7979, "type": "theorem", "label": "divisors-lemma-affine-conormal-sheaf", "categories": [ "divisors" ], "title": "divisors-lemma-affine-conormal-sheaf", "contents": [ "Let $i : Z \\to X$ be an immersion. The conormal algebra", "of $i$ has the following properties:", "\\begin{enumerate}", "\\item Let $U \\subset X$ be any open such that $i(Z)$ is", "a closed subset of $U$. Let $\\mathcal{I} \\subset \\mathcal{O}_U$", "be the sheaf of ideals corresponding to the closed subscheme", "$i(Z) \\subset U$. Then", "$$", "\\mathcal{C}_{Z/X, *} =", "i^*\\left(\\bigoplus\\nolimits_{n \\geq 0} \\mathcal{I}^n\\right) =", "i^{-1}\\left(", "\\bigoplus\\nolimits_{n \\geq 0} \\mathcal{I}^n/\\mathcal{I}^{n + 1}", "\\right)", "$$", "\\item", "For any affine open $\\Spec(R) = U \\subset X$", "such that $Z \\cap U = \\Spec(R/I)$ there is a", "canonical isomorphism", "$\\Gamma(Z \\cap U, \\mathcal{C}_{Z/X, *}) = \\bigoplus_{n \\geq 0} I^n/I^{n + 1}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Mostly clear from the definitions. Note that given a ring $R$ and", "an ideal $I$ of $R$ we have $I^n/I^{n + 1} = I^n \\otimes_R R/I$.", "Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7980, "type": "theorem", "label": "divisors-lemma-conormal-algebra-functorial", "categories": [ "divisors" ], "title": "divisors-lemma-conormal-algebra-functorial", "contents": [ "Let", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\", "Z' \\ar[r]^{i'} & X'", "}", "$$", "be a commutative diagram in the category of schemes.", "Assume $i$, $i'$ immersions. There is a canonical map", "of graded $\\mathcal{O}_Z$-algebras", "$$", "f^*\\mathcal{C}_{Z'/X', *}", "\\longrightarrow", "\\mathcal{C}_{Z/X, *}", "$$", "characterized by the following property: For every pair of affine opens", "$(\\Spec(R) = U \\subset X, \\Spec(R') = U' \\subset X')$ with", "$f(U) \\subset U'$ such that", "$Z \\cap U = \\Spec(R/I)$ and $Z' \\cap U' = \\Spec(R'/I')$", "the induced map", "$$", "\\Gamma(Z' \\cap U', \\mathcal{C}_{Z'/X', *}) =", "\\bigoplus\\nolimits (I')^n/(I')^{n + 1}", "\\longrightarrow", "\\bigoplus\\nolimits_{n \\geq 0} I^n/I^{n + 1} =", "\\Gamma(Z \\cap U, \\mathcal{C}_{Z/X, *})", "$$", "is the one induced by the ring map $f^\\sharp : R' \\to R$ which", "has the property $f^\\sharp(I') \\subset I$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\partial Z' = \\overline{Z'} \\setminus Z'$ and", "$\\partial Z = \\overline{Z} \\setminus Z$. These are closed subsets of $X'$ and", "of $X$. Replacing $X'$ by $X' \\setminus \\partial Z'$ and $X$ by", "$X \\setminus \\big(g^{-1}(\\partial Z') \\cup \\partial Z\\big)$ we", "see that we may assume that $i$ and $i'$ are closed immersions.", "\\medskip\\noindent", "The fact that $g \\circ i$ factors through $i'$ implies that", "$g^*\\mathcal{I}'$ maps into $\\mathcal{I}$ under the canonical", "map $g^*\\mathcal{I}' \\to \\mathcal{O}_X$, see", "Schemes, Lemmas", "\\ref{schemes-lemma-characterize-closed-subspace} and", "\\ref{schemes-lemma-restrict-map-to-closed}.", "Hence we get an induced map of quasi-coherent sheaves", "$g^*((\\mathcal{I}')^n/(\\mathcal{I}')^{n + 1}) \\to", "\\mathcal{I}^n/\\mathcal{I}^{n + 1}$.", "Pulling back by $i$ gives", "$i^*g^*((\\mathcal{I}')^n/(\\mathcal{I}')^{n + 1}) \\to", "i^*(\\mathcal{I}^n/\\mathcal{I}^{n + 1})$.", "Note that", "$i^*(\\mathcal{I}^n/\\mathcal{I}^{n + 1}) = \\mathcal{C}_{Z/X, n}$.", "On the other hand,", "$i^*g^*((\\mathcal{I}')^n/(\\mathcal{I}')^{n + 1}) =", "f^*(i')^*((\\mathcal{I}')^n/(\\mathcal{I}')^{n + 1}) =", "f^*\\mathcal{C}_{Z'/X', n}$.", "This gives the desired map.", "\\medskip\\noindent", "Checking that the map is locally described as the given map", "$(I')^n/(I')^{n + 1} \\to I^n/I^{n + 1}$ is a matter of unwinding the", "definitions and is omitted. Another observation is that given any", "$x \\in i(Z)$ there do exist affine open neighbourhoods $U$, $U'$", "with $f(U) \\subset U'$ and $Z \\cap U$ as well as $U' \\cap Z'$", "closed such that $x \\in U$. Proof omitted. Hence the requirement", "of the lemma indeed characterizes the map (and could have been used", "to define it)." ], "refs": [ "schemes-lemma-characterize-closed-subspace", "schemes-lemma-restrict-map-to-closed" ], "ref_ids": [ 7648, 7649 ] } ], "ref_ids": [] }, { "id": 7981, "type": "theorem", "label": "divisors-lemma-conormal-algebra-functorial-flat", "categories": [ "divisors" ], "title": "divisors-lemma-conormal-algebra-functorial-flat", "contents": [ "Let", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\", "Z' \\ar[r]^{i'} & X'", "}", "$$", "be a fibre product diagram in the category of schemes with", "$i$, $i'$ immersions. Then the canonical map", "$f^*\\mathcal{C}_{Z'/X', *} \\to \\mathcal{C}_{Z/X, *}$ of", "Lemma \\ref{lemma-conormal-algebra-functorial}", "is surjective. If $g$ is flat, then it is an isomorphism." ], "refs": [ "divisors-lemma-conormal-algebra-functorial" ], "proofs": [ { "contents": [ "Let $R' \\to R$ be a ring map, and $I' \\subset R'$ an ideal.", "Set $I = I'R$. Then $(I')^n/(I')^{n + 1} \\otimes_{R'} R \\to I^n/I^{n + 1}$", "is surjective. If $R' \\to R$ is flat, then $I^n = (I')^n \\otimes_{R'} R$", "and we see the map is an isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 7980 ] }, { "id": 7982, "type": "theorem", "label": "divisors-lemma-types-regular-sequences-implications", "categories": [ "divisors" ], "title": "divisors-lemma-types-regular-sequences-implications", "contents": [ "Let $X$ be a ringed space.", "Let $f_1, \\ldots, f_r \\in \\Gamma(X, \\mathcal{O}_X)$.", "We have the following implications", "$f_1, \\ldots, f_r$ is a regular sequence $\\Rightarrow$", "$f_1, \\ldots, f_r$ is a Koszul-regular sequence $\\Rightarrow$", "$f_1, \\ldots, f_r$ is an $H_1$-regular sequence $\\Rightarrow$", "$f_1, \\ldots, f_r$ is a quasi-regular sequence." ], "refs": [], "proofs": [ { "contents": [ "Since we may check exactness at stalks, a", "sequence $f_1, \\ldots, f_r$ is a regular sequence if and only", "if the maps", "$$", "f_i :", "\\mathcal{O}_{X, x}/(f_1, \\ldots, f_{i - 1})", "\\longrightarrow", "\\mathcal{O}_{X, x}/(f_1, \\ldots, f_{i - 1})", "$$", "are injective for all $x \\in X$. In other words, the image of the sequence", "$f_1, \\ldots, f_r$ in the ring $\\mathcal{O}_{X, x}$ is a", "regular sequence for all $x \\in X$. The other types of regularity", "can be checked stalkwise as well (details omitted).", "Hence the implications follow from", "More on Algebra, Lemmas", "\\ref{more-algebra-lemma-regular-koszul-regular},", "\\ref{more-algebra-lemma-koszul-regular-H1-regular}, and", "\\ref{more-algebra-lemma-H1-regular-quasi-regular}." ], "refs": [ "more-algebra-lemma-regular-koszul-regular", "more-algebra-lemma-koszul-regular-H1-regular", "more-algebra-lemma-H1-regular-quasi-regular" ], "ref_ids": [ 9973, 9974, 9977 ] } ], "ref_ids": [] }, { "id": 7983, "type": "theorem", "label": "divisors-lemma-regular-quasi-regular-scheme", "categories": [ "divisors" ], "title": "divisors-lemma-regular-quasi-regular-scheme", "contents": [ "Let $X$ be a ringed space. Let $\\mathcal{J}$ be a sheaf of ideals.", "We have the following implications:", "$\\mathcal{J}$ is regular $\\Rightarrow$", "$\\mathcal{J}$ is Koszul-regular $\\Rightarrow$", "$\\mathcal{J}$ is $H_1$-regular $\\Rightarrow$", "$\\mathcal{J}$ is quasi-regular." ], "refs": [], "proofs": [ { "contents": [ "The lemma immediately reduces to", "Lemma \\ref{lemma-types-regular-sequences-implications}." ], "refs": [ "divisors-lemma-types-regular-sequences-implications" ], "ref_ids": [ 7982 ] } ], "ref_ids": [] }, { "id": 7984, "type": "theorem", "label": "divisors-lemma-quasi-regular-ideal", "categories": [ "divisors" ], "title": "divisors-lemma-quasi-regular-ideal", "contents": [ "Let $X$ be a locally ringed space. Let $\\mathcal{J} \\subset \\mathcal{O}_X$", "be a sheaf of ideals. Then $\\mathcal{J}$ is quasi-regular if and", "only if the following conditions are satisfied:", "\\begin{enumerate}", "\\item $\\mathcal{J}$ is an $\\mathcal{O}_X$-module of finite type,", "\\item $\\mathcal{J}/\\mathcal{J}^2$ is a finite locally free", "$\\mathcal{O}_X/\\mathcal{J}$-module, and", "\\item the canonical maps", "$$", "\\text{Sym}^n_{\\mathcal{O}_X/\\mathcal{J}}(\\mathcal{J}/\\mathcal{J}^2)", "\\longrightarrow", "\\mathcal{J}^n/\\mathcal{J}^{n + 1}", "$$", "are isomorphisms for all $n \\geq 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that if $U \\subset X$ is an open such that", "$\\mathcal{J}|_U$ is generated by a quasi-regular sequence", "$f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ then $\\mathcal{J}|_U$", "is of finite type, $\\mathcal{J}|_U/\\mathcal{J}^2|_U$ is free", "with basis $f_1, \\ldots, f_r$, and the maps in (3) are isomorphisms", "because they are coordinate free formulation of the degree $n$", "part of (\\ref{equation-map-quasi-regular}). Hence it is clear that", "being quasi-regular implies conditions (1), (2), and (3).", "\\medskip\\noindent", "Conversely, suppose that (1), (2), and (3) hold. Pick a point", "$x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$. Then there exists", "a neighbourhood $U \\subset X$ of $x$ such that", "$\\mathcal{J}|_U/\\mathcal{J}^2|_U$", "is free of rank $r$ over $\\mathcal{O}_U/\\mathcal{J}|_U$.", "After possibly shrinking $U$ we may assume there exist", "$f_1, \\ldots, f_r \\in \\mathcal{J}(U)$ which map to a basis", "of $\\mathcal{J}|_U/\\mathcal{J}^2|_U$ as an", "$\\mathcal{O}_U/\\mathcal{J}|_U$-module.", "In particular we see that the images of $f_1, \\ldots, f_r$ in", "$\\mathcal{J}_x/\\mathcal{J}^2_x$ generate. Hence by Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK})", "we see that $f_1, \\ldots, f_r$ generate the stalk $\\mathcal{J}_x$.", "Hence, since $\\mathcal{J}$ is of finite type, by", "Modules, Lemma \\ref{modules-lemma-finite-type-surjective-on-stalk}", "after shrinking $U$ we may assume that $f_1, \\ldots, f_r$ generate", "$\\mathcal{J}$. Finally, from (3) and the isomorphism", "$\\mathcal{J}|_U/\\mathcal{J}^2|_U = \\bigoplus \\mathcal{O}_U/\\mathcal{J}|_U f_i$", "it is clear that $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$", "is a quasi-regular sequence." ], "refs": [ "algebra-lemma-NAK", "modules-lemma-finite-type-surjective-on-stalk" ], "ref_ids": [ 401, 13238 ] } ], "ref_ids": [] }, { "id": 7985, "type": "theorem", "label": "divisors-lemma-generate-regular-ideal", "categories": [ "divisors" ], "title": "divisors-lemma-generate-regular-ideal", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a locally ringed space.", "Let $\\mathcal{J} \\subset \\mathcal{O}_X$ be a sheaf of ideals.", "Let $x \\in X$ and $f_1, \\ldots, f_r \\in \\mathcal{J}_x$ whose images", "give a basis for the $\\kappa(x)$-vector space", "$\\mathcal{J}_x/\\mathfrak m_x\\mathcal{J}_x$.", "\\begin{enumerate}", "\\item If $\\mathcal{J}$ is quasi-regular, then there exists an open", "neighbourhood such that $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$", "form a quasi-regular sequence generating $\\mathcal{J}|_U$.", "\\item If $\\mathcal{J}$ is $H_1$-regular, then there exists an open", "neighbourhood such that $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$", "form an $H_1$-regular sequence generating $\\mathcal{J}|_U$.", "\\item If $\\mathcal{J}$ is Koszul-regular, then there exists an open", "neighbourhood such that $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$", "form an Koszul-regular sequence generating $\\mathcal{J}|_U$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First assume that $\\mathcal{J}$ is quasi-regular. We may choose an", "open neighbourhood $U \\subset X$ of $x$ and a quasi-regular sequence", "$g_1, \\ldots, g_s \\in \\mathcal{O}_X(U)$ which generates $\\mathcal{J}|_U$.", "Note that this implies that $\\mathcal{J}/\\mathcal{J}^2$ is free of", "rank $s$ over $\\mathcal{O}_U/\\mathcal{J}|_U$ (see", "Lemma \\ref{lemma-quasi-regular-ideal}", "and its proof) and hence $r = s$.", "We may shrink $U$ and assume $f_1, \\ldots, f_r \\in \\mathcal{J}(U)$.", "Thus we may write", "$$", "f_i = \\sum a_{ij} g_j", "$$", "for some $a_{ij} \\in \\mathcal{O}_X(U)$. By assumption the matrix", "$A = (a_{ij})$ maps to an invertible matrix over $\\kappa(x)$.", "Hence, after shrinking $U$ once more, we may assume that $(a_{ij})$", "is invertible. Thus we see that $f_1, \\ldots, f_r$ give a basis", "for $(\\mathcal{J}/\\mathcal{J}^2)|_U$ which proves that $f_1, \\ldots, f_r$", "is a quasi-regular sequence over $U$.", "\\medskip\\noindent", "Note that in order to prove (2) and (3) we may, because the assumptions", "of (2) and (3) are stronger than the assumption in (1), already assume that", "$f_1, \\ldots, f_r \\in \\mathcal{J}(U)$ and $f_i = \\sum a_{ij}g_j$", "with $(a_{ij})$ invertible as above, where now $g_1, \\ldots, g_r$", "is a $H_1$-regular or Koszul-regular sequence. Since the Koszul complex", "on $f_1, \\ldots, f_r$ is isomorphic to the Koszul complex on", "$g_1, \\ldots, g_r$ via the matrix $(a_{ij})$ (see", "More on Algebra, Lemma \\ref{more-algebra-lemma-change-basis})", "we conclude that $f_1, \\ldots, f_r$ is $H_1$-regular or Koszul-regular", "as desired." ], "refs": [ "divisors-lemma-quasi-regular-ideal", "more-algebra-lemma-change-basis" ], "ref_ids": [ 7984, 9958 ] } ], "ref_ids": [] }, { "id": 7986, "type": "theorem", "label": "divisors-lemma-regular-ideal-sheaf-quasi-coherent", "categories": [ "divisors" ], "title": "divisors-lemma-regular-ideal-sheaf-quasi-coherent", "contents": [ "Any regular, Koszul-regular, $H_1$-regular, or quasi-regular sheaf", "of ideals on a scheme is a finite type quasi-coherent sheaf of ideals." ], "refs": [], "proofs": [ { "contents": [ "This follows as such a sheaf of ideals is locally generated by", "finitely many sections. And any sheaf of ideals locally generated", "by sections on a scheme is quasi-coherent, see", "Schemes, Lemma \\ref{schemes-lemma-closed-subspace-scheme}." ], "refs": [ "schemes-lemma-closed-subspace-scheme" ], "ref_ids": [ 7670 ] } ], "ref_ids": [] }, { "id": 7987, "type": "theorem", "label": "divisors-lemma-regular-ideal-sheaf-scheme", "categories": [ "divisors" ], "title": "divisors-lemma-regular-ideal-sheaf-scheme", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{J}$ be a sheaf of ideals.", "Then $\\mathcal{J}$ is regular", "(resp.\\ Koszul-regular, $H_1$-regular, quasi-regular) if and only if", "for every $x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists", "an affine open neighbourhood $x \\in U \\subset X$, $U = \\Spec(A)$", "such that $\\mathcal{J}|_U = \\widetilde{I}$ and such that $I$", "is generated by a regular (resp.\\ Koszul-regular, $H_1$-regular,", "quasi-regular) sequence $f_1, \\ldots, f_r \\in A$." ], "refs": [], "proofs": [ { "contents": [ "By assumption we can find an open neighbourhood $U$ of $x$ over which", "$\\mathcal{J}$ is generated by a", "regular (resp.\\ Koszul-regular, $H_1$-regular, quasi-regular)", "sequence $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$. After shrinking", "$U$ we may assume that $U$ is affine, say $U = \\Spec(A)$.", "Since $\\mathcal{J}$ is quasi-coherent by", "Lemma \\ref{lemma-regular-ideal-sheaf-quasi-coherent}", "we see that $\\mathcal{J}|_U = \\widetilde{I}$ for some ideal $I \\subset A$.", "Now we can use the fact that", "$$", "\\widetilde{\\ } : \\text{Mod}_A \\longrightarrow \\QCoh(\\mathcal{O}_U)", "$$", "is an equivalence of categories which preserves exactness. For example", "the fact that the functions $f_i$ generate $\\mathcal{J}$ means that", "the $f_i$, seen as elements of $A$ generate $I$. The fact that", "(\\ref{equation-map-regular}) is injective", "(resp.\\ (\\ref{equation-koszul}) is exact, (\\ref{equation-koszul}) is exact", "in degree $1$, (\\ref{equation-map-quasi-regular}) is an isomorphism)", "implies the corresponding property of the map", "$A/(f_1, \\ldots, f_{i - 1}) \\to A/(f_1, \\ldots, f_{i - 1})$", "(resp.\\ the complex $K_\\bullet(A, f_1, \\ldots, f_r)$, the", "map $A/I[T_1, \\ldots, T_r] \\to \\bigoplus I^n/I^{n + 1}$).", "Thus $f_1, \\ldots, f_r \\in A$ is a regular", "(resp.\\ Koszul-regular, $H_1$-regular, quasi-regular)", "sequence of the ring $A$." ], "refs": [ "divisors-lemma-regular-ideal-sheaf-quasi-coherent" ], "ref_ids": [ 7986 ] } ], "ref_ids": [] }, { "id": 7988, "type": "theorem", "label": "divisors-lemma-Noetherian-scheme-regular-ideal", "categories": [ "divisors" ], "title": "divisors-lemma-Noetherian-scheme-regular-ideal", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{J} \\subset \\mathcal{O}_X$", "be a quasi-coherent sheaf of ideals. Let $x$ be a point of the support of", "$\\mathcal{O}_X/\\mathcal{J}$. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{J}_x$ is generated by a regular sequence in", "$\\mathcal{O}_{X, x}$,", "\\item $\\mathcal{J}_x$ is generated by a Koszul-regular sequence in", "$\\mathcal{O}_{X, x}$,", "\\item $\\mathcal{J}_x$ is generated by an $H_1$-regular sequence in", "$\\mathcal{O}_{X, x}$,", "\\item $\\mathcal{J}_x$ is generated by a quasi-regular sequence in", "$\\mathcal{O}_{X, x}$,", "\\item there exists an affine neighbourhood $U = \\Spec(A)$ of $x$ such", "that $\\mathcal{J}|_U = \\widetilde{I}$ and $I$ is generated by a", "regular sequence in $A$, and", "\\item there exists an affine neighbourhood $U = \\Spec(A)$ of $x$ such", "that $\\mathcal{J}|_U = \\widetilde{I}$ and $I$ is generated by a", "Koszul-regular sequence in $A$, and", "\\item there exists an affine neighbourhood $U = \\Spec(A)$ of $x$ such", "that $\\mathcal{J}|_U = \\widetilde{I}$ and $I$ is generated by an", "$H_1$-regular sequence in $A$, and", "\\item there exists an affine neighbourhood $U = \\Spec(A)$ of $x$ such", "that $\\mathcal{J}|_U = \\widetilde{I}$ and $I$ is generated by a", "quasi-regular sequence in $A$,", "\\item there exists a neighbourhood $U$ of $x$ such that $\\mathcal{J}|_U$", "is regular, and", "\\item there exists a neighbourhood $U$ of $x$ such that $\\mathcal{J}|_U$", "is Koszul-regular, and", "\\item there exists a neighbourhood $U$ of $x$ such that $\\mathcal{J}|_U$", "is $H_1$-regular, and", "\\item there exists a neighbourhood $U$ of $x$ such that $\\mathcal{J}|_U$", "is quasi-regular.", "\\end{enumerate}", "In particular, on a locally Noetherian scheme the notions of", "regular, Koszul-regular, $H_1$-regular, or quasi-regular ideal sheaf all agree." ], "refs": [], "proofs": [ { "contents": [ "It follows from", "Lemma \\ref{lemma-regular-ideal-sheaf-scheme}", "that (5) $\\Leftrightarrow$ (9), (6) $\\Leftrightarrow$ (10),", "(7) $\\Leftrightarrow$ (11), and (8) $\\Leftrightarrow$ (12).", "It is clear that (5) $\\Rightarrow$ (1), (6) $\\Rightarrow$ (2),", "(7) $\\Rightarrow$ (3), and (8) $\\Rightarrow$ (4).", "We have (1) $\\Rightarrow$ (5) by", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-neighbourhood}.", "We have (9) $\\Rightarrow$ (10) $\\Rightarrow$ (11) $\\Rightarrow$ (12) by", "Lemma \\ref{lemma-regular-quasi-regular-scheme}.", "Finally, (4) $\\Rightarrow$ (1) by", "Algebra, Lemma \\ref{algebra-lemma-quasi-regular-regular}.", "Now all 12 statements are equivalent." ], "refs": [ "divisors-lemma-regular-ideal-sheaf-scheme", "algebra-lemma-regular-sequence-in-neighbourhood", "divisors-lemma-regular-quasi-regular-scheme", "algebra-lemma-quasi-regular-regular" ], "ref_ids": [ 7987, 741, 7983, 750 ] } ], "ref_ids": [] }, { "id": 7989, "type": "theorem", "label": "divisors-lemma-regular-quasi-regular-immersion", "categories": [ "divisors" ], "title": "divisors-lemma-regular-quasi-regular-immersion", "contents": [ "Let $i : Z \\to X$ be an immersion of schemes.", "We have the following implications:", "$i$ is regular $\\Rightarrow$", "$i$ is Koszul-regular $\\Rightarrow$", "$i$ is $H_1$-regular $\\Rightarrow$", "$i$ is quasi-regular." ], "refs": [], "proofs": [ { "contents": [ "The lemma immediately reduces to", "Lemma \\ref{lemma-regular-quasi-regular-scheme}." ], "refs": [ "divisors-lemma-regular-quasi-regular-scheme" ], "ref_ids": [ 7983 ] } ], "ref_ids": [] }, { "id": 7990, "type": "theorem", "label": "divisors-lemma-regular-immersion-noetherian", "categories": [ "divisors" ], "title": "divisors-lemma-regular-immersion-noetherian", "contents": [ "Let $i : Z \\to X$ be an immersion of schemes.", "Assume $X$ is locally Noetherian. Then", "$i$ is regular $\\Leftrightarrow$", "$i$ is Koszul-regular $\\Leftrightarrow$", "$i$ is $H_1$-regular $\\Leftrightarrow$", "$i$ is quasi-regular." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Lemma \\ref{lemma-regular-quasi-regular-immersion}", "and", "Lemma \\ref{lemma-Noetherian-scheme-regular-ideal}." ], "refs": [ "divisors-lemma-regular-quasi-regular-immersion", "divisors-lemma-Noetherian-scheme-regular-ideal" ], "ref_ids": [ 7989, 7988 ] } ], "ref_ids": [] }, { "id": 7991, "type": "theorem", "label": "divisors-lemma-flat-base-change-regular-immersion", "categories": [ "divisors" ], "title": "divisors-lemma-flat-base-change-regular-immersion", "contents": [ "Let $i : Z \\to X$ be a regular (resp.\\ Koszul-regular,", "$H_1$-regular, quasi-regular) immersion. Let $X' \\to X$ be a flat", "morphism. Then the base change $i' : Z \\times_X X' \\to X'$", "is a regular (resp.\\ Koszul-regular,", "$H_1$-regular, quasi-regular) immersion." ], "refs": [], "proofs": [ { "contents": [ "Via", "Lemma \\ref{lemma-regular-ideal-sheaf-scheme}", "this translates into the algebraic statements in", "Algebra, Lemmas \\ref{algebra-lemma-flat-increases-depth} and", "\\ref{algebra-lemma-flat-base-change-quasi-regular}", "and", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-koszul-regular-flat-base-change}." ], "refs": [ "divisors-lemma-regular-ideal-sheaf-scheme", "algebra-lemma-flat-increases-depth", "algebra-lemma-flat-base-change-quasi-regular", "more-algebra-lemma-koszul-regular-flat-base-change" ], "ref_ids": [ 7987, 740, 747, 9976 ] } ], "ref_ids": [] }, { "id": 7992, "type": "theorem", "label": "divisors-lemma-quasi-regular-immersion", "categories": [ "divisors" ], "title": "divisors-lemma-quasi-regular-immersion", "contents": [ "Let $i : Z \\to X$ be an immersion of schemes. Then $i$ is a quasi-regular", "immersion if and only if the following conditions are satisfied", "\\begin{enumerate}", "\\item $i$ is locally of finite presentation,", "\\item the conormal sheaf $\\mathcal{C}_{Z/X}$ is finite locally free, and", "\\item the map (\\ref{equation-conormal-algebra-quotient}) is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "An open immersion is locally of finite presentation. Hence we may", "replace $X$ by an open subscheme $U \\subset X$ such that $i$ identifies", "$Z$ with a closed subscheme of $U$, i.e., we may assume that $i$", "is a closed immersion. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the", "corresponding quasi-coherent sheaf of ideals. Recall, see", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-finite-presentation}", "that $\\mathcal{I}$ is of finite type if and only if $i$ is locally", "of finite presentation. Hence the equivalence follows from", "Lemma \\ref{lemma-quasi-regular-ideal}", "and unwinding the definitions." ], "refs": [ "morphisms-lemma-closed-immersion-finite-presentation", "divisors-lemma-quasi-regular-ideal" ], "ref_ids": [ 5243, 7984 ] } ], "ref_ids": [] }, { "id": 7993, "type": "theorem", "label": "divisors-lemma-transitivity-conormal-quasi-regular", "categories": [ "divisors" ], "title": "divisors-lemma-transitivity-conormal-quasi-regular", "contents": [ "Let $Z \\to Y \\to X$ be immersions of schemes. Assume that", "$Z \\to Y$ is $H_1$-regular. Then the canonical sequence of", "Morphisms, Lemma \\ref{morphisms-lemma-transitivity-conormal}", "$$", "0 \\to i^*\\mathcal{C}_{Y/X} \\to", "\\mathcal{C}_{Z/X} \\to", "\\mathcal{C}_{Z/Y} \\to 0", "$$", "is exact and locally split." ], "refs": [ "morphisms-lemma-transitivity-conormal" ], "proofs": [ { "contents": [ "Since $\\mathcal{C}_{Z/Y}$ is finite locally free (see", "Lemma \\ref{lemma-quasi-regular-immersion}", "and", "Lemma \\ref{lemma-regular-quasi-regular-scheme})", "it suffices to prove that the sequence is exact. By what was proven in", "Morphisms, Lemma \\ref{morphisms-lemma-transitivity-conormal}", "it suffices to show that the first map is injective.", "Working affine locally this reduces to the following question:", "Suppose that we have a ring $A$ and ideals $I \\subset J \\subset A$.", "Assume that $J/I \\subset A/I$ is generated by an $H_1$-regular sequence.", "Does this imply that $I/I^2 \\otimes_A A/J \\to J/J^2$ is injective?", "Note that $I/I^2 \\otimes_A A/J = I/IJ$. Hence we are trying to prove", "that $I \\cap J^2 = IJ$. This is the result of", "More on Algebra, Lemma \\ref{more-algebra-lemma-conormal-sequence-H1-regular}." ], "refs": [ "divisors-lemma-quasi-regular-immersion", "divisors-lemma-regular-quasi-regular-scheme", "morphisms-lemma-transitivity-conormal", "more-algebra-lemma-conormal-sequence-H1-regular" ], "ref_ids": [ 7992, 7983, 5306, 9980 ] } ], "ref_ids": [ 5306 ] }, { "id": 7994, "type": "theorem", "label": "divisors-lemma-composition-regular-immersion", "categories": [ "divisors" ], "title": "divisors-lemma-composition-regular-immersion", "contents": [ "Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of schemes.", "\\begin{enumerate}", "\\item If $i$ and $j$ are regular immersions, so is $j \\circ i$.", "\\item If $i$ and $j$ are Koszul-regular immersions, so is $j \\circ i$.", "\\item If $i$ and $j$ are $H_1$-regular immersions, so is $j \\circ i$.", "\\item If $i$ is an $H_1$-regular immersion and $j$ is a quasi-regular", "immersion, then $j \\circ i$ is a quasi-regular immersion.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The algebraic version of (1) is", "Algebra, Lemma \\ref{algebra-lemma-join-regular-sequences}.", "The algebraic version of (2) is", "More on Algebra, Lemma \\ref{more-algebra-lemma-join-koszul-regular-sequences}.", "The algebraic version of (3) is", "More on Algebra, Lemma \\ref{more-algebra-lemma-join-H1-regular-sequences}.", "The algebraic version of (4) is", "More on Algebra, Lemma \\ref{more-algebra-lemma-join-quasi-regular-H1-regular}." ], "refs": [ "algebra-lemma-join-regular-sequences", "more-algebra-lemma-join-koszul-regular-sequences", "more-algebra-lemma-join-H1-regular-sequences", "more-algebra-lemma-join-quasi-regular-H1-regular" ], "ref_ids": [ 742, 9984, 9982, 9981 ] } ], "ref_ids": [] }, { "id": 7995, "type": "theorem", "label": "divisors-lemma-permanence-regular-immersion", "categories": [ "divisors" ], "title": "divisors-lemma-permanence-regular-immersion", "contents": [ "Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of schemes. Assume", "that the sequence", "$$", "0 \\to i^*\\mathcal{C}_{Y/X} \\to", "\\mathcal{C}_{Z/X} \\to", "\\mathcal{C}_{Z/Y} \\to 0", "$$", "of", "Morphisms, Lemma \\ref{morphisms-lemma-transitivity-conormal}", "is exact and locally split.", "\\begin{enumerate}", "\\item If $j \\circ i$ is a quasi-regular immersion, so is $i$.", "\\item If $j \\circ i$ is a $H_1$-regular immersion, so is $i$.", "\\item If both $j$ and $j \\circ i$ are Koszul-regular immersions, so is $i$.", "\\end{enumerate}" ], "refs": [ "morphisms-lemma-transitivity-conormal" ], "proofs": [ { "contents": [ "After shrinking $Y$ and $X$ we may assume that $i$ and $j$ are closed", "immersions. Denote $\\mathcal{I} \\subset \\mathcal{O}_X$ the ideal sheaf", "of $Y$ and $\\mathcal{J} \\subset \\mathcal{O}_X$ the ideal sheaf of $Z$.", "The conormal sequence is $0 \\to \\mathcal{I}/\\mathcal{I}\\mathcal{J}", "\\to \\mathcal{J}/\\mathcal{J}^2 \\to", "\\mathcal{J}/(\\mathcal{I} + \\mathcal{J}^2) \\to 0$.", "Let $z \\in Z$ and set $y = i(z)$, $x = j(y) = j(i(z))$.", "Choose $f_1, \\ldots, f_n \\in \\mathcal{I}_x$ which map to a basis of", "$\\mathcal{I}_x/\\mathfrak m_z\\mathcal{I}_x$. Extend this to", "$f_1, \\ldots, f_n, g_1, \\ldots, g_m \\in \\mathcal{J}_x$", "which map to a basis of $\\mathcal{J}_x/\\mathfrak m_z\\mathcal{J}_x$.", "This is possible as we have assumed that the sequence of conormal", "sheaves is split in a neighbourhood of $z$, hence", "$\\mathcal{I}_x/\\mathfrak m_x\\mathcal{I}_x \\to", "\\mathcal{J}_x/\\mathfrak m_x\\mathcal{J}_x$ is injective.", "\\medskip\\noindent", "Proof of (1). By", "Lemma \\ref{lemma-generate-regular-ideal}", "we can find an affine open neighbourhood $U$ of $x$ such that", "$f_1, \\ldots, f_n, g_1, \\ldots, g_m$ forms a quasi-regular sequence", "generating $\\mathcal{J}$. Hence by", "Algebra, Lemma \\ref{algebra-lemma-truncate-quasi-regular}", "we see that $g_1, \\ldots, g_m$ induces a quasi-regular sequence on", "$Y \\cap U$ cutting out $Z$.", "\\medskip\\noindent", "Proof of (2). Exactly the same as the proof of (1) except using", "More on Algebra, Lemma \\ref{more-algebra-lemma-truncate-H1-regular}.", "\\medskip\\noindent", "Proof of (3). By", "Lemma \\ref{lemma-generate-regular-ideal}", "(applied twice)", "we can find an affine open neighbourhood $U$ of $x$ such that", "$f_1, \\ldots, f_n$ forms a Koszul-regular sequence generating", "$\\mathcal{I}$ and $f_1, \\ldots, f_n, g_1, \\ldots, g_m$ forms a", "Koszul-regular sequence generating $\\mathcal{J}$. Hence by", "More on Algebra, Lemma \\ref{more-algebra-lemma-truncate-koszul-regular}", "we see that $g_1, \\ldots, g_m$ induces a Koszul-regular sequence on", "$Y \\cap U$ cutting out $Z$." ], "refs": [ "divisors-lemma-generate-regular-ideal", "algebra-lemma-truncate-quasi-regular", "more-algebra-lemma-truncate-H1-regular", "divisors-lemma-generate-regular-ideal", "more-algebra-lemma-truncate-koszul-regular" ], "ref_ids": [ 7985, 749, 9983, 7985, 9985 ] } ], "ref_ids": [ 5306 ] }, { "id": 7996, "type": "theorem", "label": "divisors-lemma-extra-permanence-regular-immersion-noetherian", "categories": [ "divisors" ], "title": "divisors-lemma-extra-permanence-regular-immersion-noetherian", "contents": [ "Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of schemes.", "Pick $z \\in Z$ and denote $y \\in Y$, $x \\in X$ the corresponding points.", "Assume $X$ is locally Noetherian.", "The following are equivalent", "\\begin{enumerate}", "\\item $i$ is a regular immersion in a neighbourhood of $z$ and $j$", "is a regular immersion in a neighbourhood of $y$,", "\\item $i$ and $j \\circ i$ are regular immersions in a neighbourhood of $z$,", "\\item $j \\circ i$ is a regular immersion in a neighbourhood of $z$ and the", "conormal sequence", "$$", "0 \\to i^*\\mathcal{C}_{Y/X} \\to", "\\mathcal{C}_{Z/X} \\to", "\\mathcal{C}_{Z/Y} \\to 0", "$$", "is split exact in a neighbourhood of $z$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $X$ (and hence $Y$) is locally Noetherian all 4 types of regular", "immersions agree, and moreover we may check whether a morphism is a", "regular immersion on the level of local rings, see", "Lemma \\ref{lemma-Noetherian-scheme-regular-ideal}.", "The implication (1) $\\Rightarrow$ (2) is", "Lemma \\ref{lemma-composition-regular-immersion}.", "The implication (2) $\\Rightarrow$ (3) is", "Lemma \\ref{lemma-transitivity-conormal-quasi-regular}.", "Thus it suffices to prove that (3) implies (1).", "\\medskip\\noindent", "Assume (3). Set $A = \\mathcal{O}_{X, x}$. Denote $I \\subset A$ the kernel", "of the surjective map $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{Y, y}$ and", "denote $J \\subset A$ the kernel", "of the surjective map $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{Z, z}$.", "Note that any minimal sequence of elements generating $J$ in $A$", "is a quasi-regular hence regular sequence, see", "Lemma \\ref{lemma-generate-regular-ideal}.", "By assumption the conormal sequence", "$$", "0 \\to I/IJ \\to J/J^2 \\to J/(I + J^2) \\to 0", "$$", "is split exact as a sequence of $A/J$-modules. Hence we can pick", "a minimal system of generators $f_1, \\ldots, f_n, g_1, \\ldots, g_m$", "of $J$ with $f_1, \\ldots, f_n \\in I$ a minimal system of generators of $I$.", "As pointed out above $f_1, \\ldots, f_n, g_1, \\ldots, g_m$ is a regular", "sequence in $A$. It follows directly from the definition of a regular", "sequence that $f_1, \\ldots, f_n$ is a regular sequence in $A$ and", "$\\overline{g}_1, \\ldots, \\overline{g}_m$ is a regular sequence in", "$A/I$. Thus $j$ is a regular immersion at $y$ and $i$ is a regular", "immersion at $z$." ], "refs": [ "divisors-lemma-Noetherian-scheme-regular-ideal", "divisors-lemma-composition-regular-immersion", "divisors-lemma-transitivity-conormal-quasi-regular", "divisors-lemma-generate-regular-ideal" ], "ref_ids": [ 7988, 7994, 7993, 7985 ] } ], "ref_ids": [] }, { "id": 7997, "type": "theorem", "label": "divisors-lemma-koszul-regular-smooth-locally-regular", "categories": [ "divisors" ], "title": "divisors-lemma-koszul-regular-smooth-locally-regular", "contents": [ "Let $i : Z \\to X$ be a Koszul regular closed immersion.", "Then there exists a surjective smooth morphism $X' \\to X$ such", "that the base change $i' : Z \\times_X X' \\to X'$ of $i$ is", "a regular immersion." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $X$ is affine and the ideal of $Z$ generated by", "a Koszul-regular sequence by replacing $X$ by the members of a suitable", "affine open covering (affine opens as in", "Lemma \\ref{lemma-regular-ideal-sheaf-scheme}).", "The affine case is", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-Koszul-regular-flat-locally-regular}." ], "refs": [ "divisors-lemma-regular-ideal-sheaf-scheme", "more-algebra-lemma-Koszul-regular-flat-locally-regular" ], "ref_ids": [ 7987, 9988 ] } ], "ref_ids": [] }, { "id": 7998, "type": "theorem", "label": "divisors-lemma-immersion-regular-regular-immersion", "categories": [ "divisors" ], "title": "divisors-lemma-immersion-regular-regular-immersion", "contents": [ "Let $i : Z \\to X$ be an immersion. If $Z$ and $X$ are", "regular schemes, then $i$ is a regular immersion." ], "refs": [], "proofs": [ { "contents": [ "Let $z \\in Z$. By Lemma \\ref{lemma-Noetherian-scheme-regular-ideal}", "it suffices to show that the kernel of", "$\\mathcal{O}_{X, z} \\to \\mathcal{O}_{Z, z}$", "is generated by a regular sequence. This follows from", "Algebra, Lemmas \\ref{algebra-lemma-regular-quotient-regular} and", "\\ref{algebra-lemma-regular-ring-CM}." ], "refs": [ "divisors-lemma-Noetherian-scheme-regular-ideal", "algebra-lemma-regular-quotient-regular", "algebra-lemma-regular-ring-CM" ], "ref_ids": [ 7988, 942, 941 ] } ], "ref_ids": [] }, { "id": 7999, "type": "theorem", "label": "divisors-lemma-relative-regular-immersion", "categories": [ "divisors" ], "title": "divisors-lemma-relative-regular-immersion", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $i : Z \\subset X$ be an immersion.", "Assume", "\\begin{enumerate}", "\\item $i$ is an $H_1$-regular (resp.\\ quasi-regular) immersion, and", "\\item $Z \\to S$ is a flat morphism.", "\\end{enumerate}", "Then for every morphism of schemes $g : S' \\to S$ the base change", "$Z' = S' \\times_S Z \\to X' = S' \\times_S X$", "is an $H_1$-regular (resp.\\ quasi-regular) immersion." ], "refs": [], "proofs": [ { "contents": [ "Unwinding the definitions and using", "Lemma \\ref{lemma-regular-ideal-sheaf-scheme}", "this translates into More on Algebra, Lemma", "\\ref{more-algebra-lemma-relative-regular-immersion-algebra}." ], "refs": [ "divisors-lemma-regular-ideal-sheaf-scheme", "more-algebra-lemma-relative-regular-immersion-algebra" ], "ref_ids": [ 7987, 9992 ] } ], "ref_ids": [] }, { "id": 8000, "type": "theorem", "label": "divisors-lemma-quasi-regular-immersion-flat-at-x", "categories": [ "divisors" ], "title": "divisors-lemma-quasi-regular-immersion-flat-at-x", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $Z \\to X$ be a relative quasi-regular immersion.", "If $x \\in Z$ and $\\mathcal{O}_{X, x}$ is Noetherian, then $f$ is flat at $x$." ], "refs": [], "proofs": [ { "contents": [ "Let $f_1, \\ldots, f_r \\in \\mathcal{O}_{X, x}$ be a quasi-regular", "sequence cutting out the ideal of $Z$ at $x$. By", "Algebra, Lemma \\ref{algebra-lemma-quasi-regular-regular}", "we know that $f_1, \\ldots, f_r$ is a regular sequence.", "Hence $f_r$ is a nonzerodivisor on", "$\\mathcal{O}_{X, x}/(f_1, \\ldots, f_{r - 1})$ such that the", "quotient is a flat $\\mathcal{O}_{S, f(x)}$-module.", "By", "Lemma \\ref{lemma-flat-at-x}", "we conclude that $\\mathcal{O}_{X, x}/(f_1, \\ldots, f_{r - 1})$", "is a flat $\\mathcal{O}_{S, f(x)}$-module.", "Continuing by induction we find that $\\mathcal{O}_{X, x}$", "is a flat $\\mathcal{O}_{S, s}$-module." ], "refs": [ "algebra-lemma-quasi-regular-regular", "divisors-lemma-flat-at-x" ], "ref_ids": [ 750, 7975 ] } ], "ref_ids": [] }, { "id": 8001, "type": "theorem", "label": "divisors-lemma-relative-regular-immersion-flat-in-neighbourhood", "categories": [ "divisors" ], "title": "divisors-lemma-relative-regular-immersion-flat-in-neighbourhood", "contents": [ "Let $X \\to S$ be a morphism of schemes.", "Let $Z \\to X$ be an immersion.", "Assume", "\\begin{enumerate}", "\\item $X \\to S$ is flat and locally of finite presentation,", "\\item $Z \\to X$ is a relative quasi-regular immersion.", "\\end{enumerate}", "Then $Z \\to X$ is a regular immersion and", "the same remains true after any base change." ], "refs": [], "proofs": [ { "contents": [ "Pick $x \\in Z$ with image $s \\in S$. To prove this it suffices to", "find an affine neighbourhood of $x$ contained in $U$ such that the", "result holds on that affine open. Hence we may assume that $X$ is affine", "and there exist a quasi-regular sequence", "$f_1, \\ldots, f_r \\in \\Gamma(X, \\mathcal{O}_X)$", "such that $Z = V(f_1, \\ldots, f_r)$. By", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-relative-regular-immersion-algebra}", "the sequence $f_1|_{X_s}, \\ldots, f_r|_{X_s}$ is a", "quasi-regular sequence in $\\Gamma(X_s, \\mathcal{O}_{X_s})$.", "Since $X_s$ is Noetherian, this implies, possibly after shrinking", "$X$ a bit, that $f_1|_{X_s}, \\ldots, f_r|_{X_s}$ is a regular", "sequence, see", "Algebra, Lemmas \\ref{algebra-lemma-quasi-regular-regular} and", "\\ref{algebra-lemma-regular-sequence-in-neighbourhood}.", "By", "Lemma \\ref{lemma-fibre-Cartier}", "it follows that $Z_1 = V(f_1) \\subset X$ is a relative effective", "Cartier divisor, again after possibly shrinking $X$ a bit.", "Applying the same lemma again, but now to $Z_2 = V(f_1, f_2) \\subset Z_1$", "we see that $Z_2 \\subset Z_1$ is a relative effective Cartier divisor.", "And so on until on reaches $Z = Z_n = V(f_1, \\ldots, f_n)$.", "Since being a relative effective Cartier divisor is preserved under", "arbitrary base change, see", "Lemma \\ref{lemma-relative-Cartier},", "we also see that the final statement of the lemma holds." ], "refs": [ "more-algebra-lemma-relative-regular-immersion-algebra", "algebra-lemma-quasi-regular-regular", "algebra-lemma-regular-sequence-in-neighbourhood", "divisors-lemma-fibre-Cartier", "divisors-lemma-relative-Cartier" ], "ref_ids": [ 9992, 750, 741, 7978, 7972 ] } ], "ref_ids": [] }, { "id": 8002, "type": "theorem", "label": "divisors-lemma-flat-relative-H1-regular", "categories": [ "divisors" ], "title": "divisors-lemma-flat-relative-H1-regular", "contents": [ "Let $X \\to S$ be a morphism of schemes.", "Let $Z \\to X$ be a relative $H_1$-regular immersion.", "Assume $X \\to S$ is locally of finite presentation. Then", "\\begin{enumerate}", "\\item there exists an open subscheme $U \\subset X$ such that", "$Z \\subset U$ and such that $U \\to S$ is flat, and", "\\item $Z \\to X$ is a regular immersion and the same remains", "true after any base change.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Pick $x \\in Z$. To prove (1) suffices to find an open neighbourhood", "$U \\subset X$ of $x$ such that $U \\to S$ is flat. Hence the lemma reduces", "to the case that $X = \\Spec(B)$ and $S = \\Spec(A)$ are affine", "and that $Z$ is given by an $H_1$-regular sequence $f_1, \\ldots, f_r \\in B$.", "By assumption $B$ is a finitely presented $A$-algebra and", "$B/(f_1, \\ldots, f_r)B$ is a flat $A$-algebra. We are going to use", "absolute Noetherian approximation.", "\\medskip\\noindent", "Write $B = A[x_1, \\ldots, x_n]/(g_1, \\ldots, g_m)$. Assume", "$f_i$ is the image of $f_i' \\in A[x_1, \\ldots, x_n]$. Choose a finite type", "$\\mathbf{Z}$-subalgebra $A_0 \\subset A$ such that all the coefficients", "of the polynomials $f_1', \\ldots, f_r', g_1, \\ldots, g_m$ are in $A_0$.", "We set $B_0 = A_0[x_1, \\ldots, x_n]/(g_1, \\ldots, g_m)$ and we denote", "$f_{i, 0}$ the image of $f_i'$ in $B_0$. Then $B = B_0 \\otimes_{A_0} A$", "and", "$$", "B/(f_1, \\ldots, f_r) =", "B_0/(f_{0, 1}, \\ldots, f_{0, r}) \\otimes_{A_0} A.", "$$", "By", "Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "we may, after enlarging $A_0$, assume that", "$B_0/(f_{0, 1}, \\ldots, f_{0, r})$ is flat over $A_0$.", "It may not be the case at this point that the Koszul cohomology group", "$H_1(K_\\bullet(B_0, f_{0, 1}, \\ldots, f_{0, r}))$ is zero.", "On the other hand, as $B_0$ is Noetherian, it is a finitely", "generated $B_0$-module. Let", "$\\xi_1, \\ldots, \\xi_n \\in H_1(K_\\bullet(B_0, f_{0, 1}, \\ldots, f_{0, r}))$", "be generators. Let $A_0 \\subset A_1 \\subset A$ be a larger finite type", "$\\mathbf{Z}$-subalgebra of $A$. Denote $f_{1, i}$ the image", "of $f_{0, i}$ in $B_1 = B_0 \\otimes_{A_0} A_1$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-H1-regular}", "the map", "$$", "H_1(K_\\bullet(B_0, f_{0, 1}, \\ldots, f_{0, r})) \\otimes_{A_0} A_1", "\\longrightarrow", "H_1(K_\\bullet(B_1, f_{1, 1}, \\ldots, f_{1, r}))", "$$", "is surjective. Furthermore, it is clear that the colimit (over all", "choices of $A_1$ as above) of the", "complexes $K_\\bullet(B_1, f_{1, 1}, \\ldots, f_{1, r})$ is the complex", "$K_\\bullet(B, f_1, \\ldots, f_r)$ which is acyclic in degree $1$. Hence", "$$", "\\colim_{A_0 \\subset A_1 \\subset A}", "H_1(K_\\bullet(B_1, f_{1, 1}, \\ldots, f_{1, r}))", "= 0", "$$", "by", "Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}.", "Thus we can find a choice of $A_1$ such that $\\xi_1, \\ldots, \\xi_n$", "all map to zero in $H_1(K_\\bullet(B_1, f_{1, 1}, \\ldots, f_{1, r}))$.", "In other words, the Koszul cohomology group", "$H_1(K_\\bullet(B_1, f_{1, 1}, \\ldots, f_{1, r}))$", "is zero.", "\\medskip\\noindent", "Consider the morphism of affine schemes", "$X_1 \\to S_1$ equal to $\\Spec$ of the", "ring map $A_1 \\to B_1$ and", "$Z_1 = \\Spec(B_1/(f_{1, 1}, \\ldots, f_{1, r}))$.", "Since $B = B_1 \\otimes_{A_1} A$, i.e., $X = X_1 \\times_{S_1} S$,", "and similarly $Z = Z_1 \\times_S S_1$,", "it now suffices to prove (1) for $X_1 \\to S_1$ and the relative", "$H_1$-regular immersion $Z_1 \\to X_1$, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-module-flat}.", "Hence we have reduced to the case where $X \\to S$ is a finite type", "morphism of Noetherian schemes.", "In this case we know that $X \\to S$ is flat at every", "point of $Z$ by", "Lemma \\ref{lemma-quasi-regular-immersion-flat-at-x}.", "Combined with the fact that the flat locus is open in this case, see", "Algebra, Theorem \\ref{algebra-theorem-openness-flatness}", "we see that (1) holds. Part (2) then follows from an application of", "Lemma \\ref{lemma-relative-regular-immersion-flat-in-neighbourhood}." ], "refs": [ "algebra-lemma-flat-finite-presentation-limit-flat", "more-algebra-lemma-base-change-H1-regular", "algebra-lemma-directed-colimit-exact", "morphisms-lemma-base-change-module-flat", "divisors-lemma-quasi-regular-immersion-flat-at-x", "algebra-theorem-openness-flatness", "divisors-lemma-relative-regular-immersion-flat-in-neighbourhood" ], "ref_ids": [ 1389, 9991, 343, 5264, 8000, 326, 8001 ] } ], "ref_ids": [] }, { "id": 8003, "type": "theorem", "label": "divisors-lemma-fibre-quasi-regular", "categories": [ "divisors" ], "title": "divisors-lemma-fibre-quasi-regular", "contents": [ "Let $\\varphi : X \\to S$ be a flat morphism which is locally of finite", "presentation. Let $T \\subset X$ be a closed subscheme.", "Let $x \\in T$ with image $s \\in S$.", "\\begin{enumerate}", "\\item If $T_s \\subset X_s$ is a quasi-regular immersion", "in a neighbourhood of $x$, then there exists an open", "$U \\subset X$ and a relative quasi-regular immersion", "$Z \\subset U$ such that $Z_s = T_s \\cap U_s$ and $T \\cap U \\subset Z$.", "\\item If $T_s \\subset X_s$ is a quasi-regular immersion", "in a neighbourhood of $x$, the morphism $T \\to X$ is of finite", "presentation, and $T \\to S$ is flat at $x$, then we can choose $U$ and", "$Z$ as in (1) such that $T \\cap U = Z$.", "\\item If $T_s \\subset X_s$ is a quasi-regular immersion in a neighbourhood", "of $x$, and $T$ is cut out by $c$ equations in a neighbourhood of $x$,", "where $c = \\dim_x(X_s) - \\dim_x(T_s)$, then we can choose $U$ and $Z$ as in (1)", "such that $T \\cap U = Z$.", "\\end{enumerate}", "In each case $Z \\to U$ is a regular immersion by", "Lemma \\ref{lemma-relative-regular-immersion-flat-in-neighbourhood}.", "In particular, if $T \\to S$ is locally of finite presentation and flat and", "all fibres $T_s \\subset X_s$ are quasi-regular immersions, then", "$T \\to X$ is a relative quasi-regular immersion." ], "refs": [ "divisors-lemma-relative-regular-immersion-flat-in-neighbourhood" ], "proofs": [ { "contents": [ "Choose affine open neighbourhoods $\\Spec(A)$ of $s$ and", "$\\Spec(B)$ of $x$ such that", "$\\varphi(\\Spec(B)) \\subset \\Spec(A)$.", "Let $\\mathfrak p \\subset A$ be the prime ideal corresponding to $s$.", "Let $\\mathfrak q \\subset B$ be the prime ideal corresponding to $x$.", "Let $I \\subset B$ be the ideal corresponding to $T$.", "By the initial assumption of the lemma we know that", "$A \\to B$ is flat and of finite presentation.", "The assumption in (1) means that, after shrinking $\\Spec(B)$, we may", "assume $I(B \\otimes_A \\kappa(\\mathfrak p))$ is generated by a", "quasi-regular sequence of elements. After possibly localizing $B$", "at some $g \\in B$, $g \\not \\in \\mathfrak q$ we may assume there", "exist $f_1, \\ldots, f_r \\in I$ which map to a quasi-regular", "sequence in $B \\otimes_A \\kappa(\\mathfrak p)$ which generates", "$I(B \\otimes_A \\kappa(\\mathfrak p))$. By", "Algebra, Lemmas \\ref{algebra-lemma-quasi-regular-regular} and", "\\ref{algebra-lemma-regular-sequence-in-neighbourhood}", "we may assume after another localization that", "$f_1, \\ldots, f_r \\in I$ form a regular", "sequence in $B \\otimes_A \\kappa(\\mathfrak p)$. By", "Lemma \\ref{lemma-fibre-Cartier}", "it follows that $Z_1 = V(f_1) \\subset \\Spec(B)$", "is a relative effective Cartier divisor, again after possibly", "localizing $B$. Applying the same lemma again, but now to", "$Z_2 = V(f_1, f_2) \\subset Z_1$ we see that $Z_2 \\subset Z_1$", "is a relative effective Cartier divisor. And so on until one", "reaches $Z = Z_n = V(f_1, \\ldots, f_n)$. Then", "$Z \\to \\Spec(B)$ is a regular immersion and $Z$ is", "flat over $S$, in particular $Z \\to \\Spec(B)$ is", "a relative quasi-regular immersion over $\\Spec(A)$.", "This proves (1).", "\\medskip\\noindent", "To see (2) consider the closed immersion $Z \\to D$. The surjective", "ring map $u : \\mathcal{O}_{D, x} \\to \\mathcal{O}_{Z, x}$", "is a map of flat local $\\mathcal{O}_{S, s}$-algebras which", "are essentially of finite presentation, and which becomes an", "isomorphisms after dividing by $\\mathfrak m_s$. Hence it is", "an isomorphism, see", "Algebra, Lemma \\ref{algebra-lemma-mod-injective-general}.", "It follows that $Z \\to D$ is an isomorphism in a neighbourhood", "of $x$, see", "Algebra, Lemma \\ref{algebra-lemma-local-isomorphism}.", "\\medskip\\noindent", "To see (3), after possibly shrinking $U$ we may assume that", "the ideal of $Z$ is generated by a regular sequence $f_1, \\ldots, f_r$", "(see our construction of $Z$ above)", "and the ideal of $T$ is generated by $g_1, \\ldots, g_c$.", "We claim that $c = r$. Namely,", "\\begin{align*}", "\\dim_x(X_s) & = \\dim(\\mathcal{O}_{X_s, x}) +", "\\text{trdeg}_{\\kappa(s)}(\\kappa(x)), \\\\", "\\dim_x(T_s) & = \\dim(\\mathcal{O}_{T_s, x}) +", "\\text{trdeg}_{\\kappa(s)}(\\kappa(x)), \\\\", "\\dim(\\mathcal{O}_{X_s, x}) & = \\dim(\\mathcal{O}_{T_s, x}) + r", "\\end{align*}", "the first two equalities by", "Algebra, Lemma \\ref{algebra-lemma-dimension-at-a-point-finite-type-field}", "and the second by $r$ times applying", "Algebra, Lemma \\ref{algebra-lemma-one-equation}.", "As $T \\subset Z$ we see that", "$f_i = \\sum b_{ij} g_j$. But the ideals of $Z$ and $T$ cut out the same", "quasi-regular closed subscheme of $X_s$ in a neighbourhood of $x$. Hence", "the matrix $(b_{ij}) \\bmod \\mathfrak m_x$ is invertible (some details", "omitted). Hence $(b_{ij})$ is invertible in an", "open neighbourhood of $x$. In other words,", "$T \\cap U = Z$ after shrinking $U$.", "\\medskip\\noindent", "The final statements of the lemma follow immediately from", "part (2), combined with the fact that $Z \\to S$", "is locally of finite presentation if and only if $Z \\to X$ is", "of finite presentation, see", "Morphisms, Lemmas \\ref{morphisms-lemma-composition-finite-presentation} and", "\\ref{morphisms-lemma-finite-presentation-permanence}." ], "refs": [ "algebra-lemma-quasi-regular-regular", "algebra-lemma-regular-sequence-in-neighbourhood", "divisors-lemma-fibre-Cartier", "algebra-lemma-mod-injective-general", "algebra-lemma-local-isomorphism", "algebra-lemma-dimension-at-a-point-finite-type-field", "morphisms-lemma-composition-finite-presentation", "morphisms-lemma-finite-presentation-permanence" ], "ref_ids": [ 750, 741, 7978, 1110, 1084, 1007, 5239, 5247 ] } ], "ref_ids": [ 8001 ] }, { "id": 8004, "type": "theorem", "label": "divisors-lemma-section-smooth-regular-immersion", "categories": [ "divisors" ], "title": "divisors-lemma-section-smooth-regular-immersion", "contents": [ "Let $f : X \\to S$ be a smooth morphism of schemes.", "Let $\\sigma : S \\to X$ be a section of $f$.", "Then $\\sigma$ is a regular immersion." ], "refs": [], "proofs": [ { "contents": [ "By", "Schemes, Lemma \\ref{schemes-lemma-semi-diagonal}", "the morphism $\\sigma$ is an immersion.", "After replacing $X$ by an open neighbourhood of $\\sigma(S)$", "we may assume that $\\sigma$ is a closed immersion.", "Let $T = \\sigma(S)$ be the corresponding closed subscheme of $X$.", "Since $T \\to S$ is an isomorphism it is flat and of finite presentation.", "Also a smooth morphism is flat and locally of finite presentation, see", "Morphisms, Lemmas \\ref{morphisms-lemma-smooth-flat} and", "\\ref{morphisms-lemma-smooth-locally-finite-presentation}.", "Thus, according to", "Lemma \\ref{lemma-fibre-quasi-regular},", "it suffices to show that $T_s \\subset X_s$ is a quasi-regular closed", "subscheme. This follows immediately from", "Morphisms, Lemma \\ref{morphisms-lemma-section-smooth-morphism}", "but we can also see it directly as follows.", "Let $k$ be a field and let $A$ be a smooth $k$-algebra.", "Let $\\mathfrak m \\subset A$ be a maximal ideal whose residue field is $k$.", "Then $\\mathfrak m$ is generated by a quasi-regular sequence, possibly", "after replacing $A$ by $A_g$ for some $g \\in A$, $g \\not \\in \\mathfrak m$.", "In", "Algebra, Lemma \\ref{algebra-lemma-characterize-smooth-over-field}", "we proved that $A_{\\mathfrak m}$ is a regular local ring,", "hence $\\mathfrak mA_{\\mathfrak m}$ is generated by a regular sequence.", "This does indeed imply that $\\mathfrak m$ is generated by a", "regular sequence (after replacing $A$ by $A_g$ for some $g \\in A$,", "$g \\not \\in \\mathfrak m$), see", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-neighbourhood}." ], "refs": [ "schemes-lemma-semi-diagonal", "morphisms-lemma-smooth-flat", "morphisms-lemma-smooth-locally-finite-presentation", "divisors-lemma-fibre-quasi-regular", "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-regular-sequence-in-neighbourhood" ], "ref_ids": [ 7712, 5331, 5330, 8003, 1223, 741 ] } ], "ref_ids": [] }, { "id": 8005, "type": "theorem", "label": "divisors-lemma-lift-regular-immersion-to-smooth", "categories": [ "divisors" ], "title": "divisors-lemma-lift-regular-immersion-to-smooth", "contents": [ "Let", "$$", "\\xymatrix{", "Y \\ar[rd]_j \\ar[rr]_i & & X \\ar[ld] \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes.", "Assume $X \\to S$ smooth, and $i$, $j$ immersions.", "If $j$ is a regular (resp.\\ Koszul-regular, $H_1$-regular, quasi-regular)", "immersion, then so is $i$." ], "refs": [], "proofs": [ { "contents": [ "We can write $i$ as the composition", "$$", "Y \\to Y \\times_S X \\to X", "$$", "By", "Lemma \\ref{lemma-section-smooth-regular-immersion}", "the first arrow is a regular immersion.", "The second arrow is a flat base change of $Y \\to S$, hence is a", "regular (resp.\\ Koszul-regular, $H_1$-regular, quasi-regular) immersion, see", "Lemma \\ref{lemma-flat-base-change-regular-immersion}.", "We conclude by an application of", "Lemma \\ref{lemma-composition-regular-immersion}." ], "refs": [ "divisors-lemma-flat-base-change-regular-immersion", "divisors-lemma-composition-regular-immersion" ], "ref_ids": [ 7991, 7994 ] } ], "ref_ids": [] }, { "id": 8006, "type": "theorem", "label": "divisors-lemma-immersion-lci-into-smooth-regular-immersion", "categories": [ "divisors" ], "title": "divisors-lemma-immersion-lci-into-smooth-regular-immersion", "contents": [ "Let", "$$", "\\xymatrix{", "Y \\ar[rd] \\ar[rr]_i & & X \\ar[ld] \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes.", "Assume that $Y \\to S$ is syntomic, $X \\to S$ smooth, and", "$i$ an immersion. Then $i$ is a regular immersion." ], "refs": [], "proofs": [ { "contents": [ "After replacing $X$ by an open neighbourhood of $i(Y)$", "we may assume that $i$ is a closed immersion.", "Let $T = i(Y)$ be the corresponding closed subscheme of $X$. Since", "$T \\cong Y$ the morphism $T \\to S$ is flat and of finite presentation", "(Morphisms, Lemmas", "\\ref{morphisms-lemma-syntomic-locally-finite-presentation} and", "\\ref{morphisms-lemma-syntomic-flat}).", "Also a smooth morphism is flat and locally of finite presentation", "(Morphisms, Lemmas", "\\ref{morphisms-lemma-smooth-flat} and", "\\ref{morphisms-lemma-smooth-locally-finite-presentation}).", "Thus, according to", "Lemma \\ref{lemma-fibre-quasi-regular},", "it suffices to show that $T_s \\subset X_s$ is a quasi-regular closed", "subscheme. As $X_s$ is locally of finite type over a field, it is Noetherian", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}).", "Thus we can check that $T_s \\subset X_s$ is a quasi-regular immersion", "at points, see", "Lemma \\ref{lemma-Noetherian-scheme-regular-ideal}.", "Take $t \\in T_s$. By", "Morphisms, Lemma \\ref{morphisms-lemma-local-complete-intersection}", "the local ring $\\mathcal{O}_{T_s, t}$ is a local complete intersection", "over $\\kappa(s)$.", "The local ring $\\mathcal{O}_{X_s, t}$ is regular, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-smooth-over-field}.", "By", "Algebra, Lemma \\ref{algebra-lemma-lci-local}", "we see that the kernel of the surjection", "$\\mathcal{O}_{X_s, t} \\to \\mathcal{O}_{T_s, t}$ is generated by a regular", "sequence, which is what we had to show." ], "refs": [ "morphisms-lemma-syntomic-locally-finite-presentation", "morphisms-lemma-syntomic-flat", "morphisms-lemma-smooth-flat", "morphisms-lemma-smooth-locally-finite-presentation", "divisors-lemma-fibre-quasi-regular", "morphisms-lemma-finite-type-noetherian", "divisors-lemma-Noetherian-scheme-regular-ideal", "morphisms-lemma-local-complete-intersection", "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-lci-local" ], "ref_ids": [ 5293, 5294, 5331, 5330, 8003, 5202, 7988, 5296, 1223, 1169 ] } ], "ref_ids": [] }, { "id": 8007, "type": "theorem", "label": "divisors-lemma-immersion-smooth-into-smooth-regular-immersion", "categories": [ "divisors" ], "title": "divisors-lemma-immersion-smooth-into-smooth-regular-immersion", "contents": [ "Let", "$$", "\\xymatrix{", "Y \\ar[rd] \\ar[rr]_i & & X \\ar[ld] \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes.", "Assume that $Y \\to S$ is smooth, $X \\to S$ smooth, and", "$i$ an immersion. Then $i$ is a regular immersion." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-immersion-lci-into-smooth-regular-immersion}", "because a smooth morphism is syntomic, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-syntomic}." ], "refs": [ "divisors-lemma-immersion-lci-into-smooth-regular-immersion", "morphisms-lemma-smooth-syntomic" ], "ref_ids": [ 8006, 5329 ] } ], "ref_ids": [] }, { "id": 8008, "type": "theorem", "label": "divisors-lemma-push-regular-immersion-thru-smooth", "categories": [ "divisors" ], "title": "divisors-lemma-push-regular-immersion-thru-smooth", "contents": [ "Let", "$$", "\\xymatrix{", "Y \\ar[rd]_j \\ar[rr]_i & & X \\ar[ld] \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes.", "Assume $X \\to S$ smooth, and $i$, $j$ immersions.", "If $i$ is a Koszul-regular (resp.\\ $H_1$-regular, quasi-regular)", "immersion, then so is $j$." ], "refs": [], "proofs": [ { "contents": [ "Let $y \\in Y$ be any point. Set $x = i(y)$ and set $s = j(y)$.", "It suffices to prove the result after replacing $X, S$ by open", "neighbourhoods $U, V$ of $x, s$ and $Y$ by an open neighbourhood", "of $y$ in $i^{-1}(U) \\cap j^{-1}(V)$. Hence we may assume that", "$Y$, $X$ and $S$ are affine. In this case we can choose a closed", "immersion $h : X \\to \\mathbf{A}^n_S$ over $S$ for some $n$. Note that", "$h$ is a regular immersion by", "Lemma \\ref{lemma-immersion-smooth-into-smooth-regular-immersion}.", "Hence $h \\circ i$ is a Koszul-regular (resp.\\ $H_1$-regular, quasi-regular)", "immersion, see", "Lemmas \\ref{lemma-composition-regular-immersion} and", "\\ref{lemma-regular-quasi-regular-immersion}.", "In this way we reduce to the case $X = \\mathbf{A}^n_S$ and $S$ affine.", "\\medskip\\noindent", "After replacing $S$ by an affine open $V$ and replacing $Y$ by", "$j^{-1}(V)$ we may assume that $i$ is a closed immersion and $S$", "affine. Write $S = \\Spec(A)$. Then $j : Y \\to S$ defines an", "isomorphism of $Y$ to the closed subscheme $\\Spec(A/I)$ for", "some ideal $I \\subset A$. The map", "$i : Y = \\Spec(A/I) \\to", "\\mathbf{A}^n_S = \\Spec(A[x_1, \\ldots, x_n])$", "corresponds to an $A$-algebra homomorphism", "$i^\\sharp : A[x_1, \\ldots, x_n] \\to A/I$.", "Choose $a_i \\in A$ which map to $i^\\sharp(x_i)$ in $A/I$.", "Observe that the ideal of the closed immersion $i$ is", "$$", "J = (x_1 - a_1, \\ldots, x_n - a_n) + IA[x_1, \\ldots, x_n].", "$$", "Set $K = (x_1 - a_1, \\ldots, x_n - a_n)$. We claim the sequence", "$$", "0 \\to K/KJ \\to J/J^2 \\to J/(K + J^2) \\to 0", "$$", "is split exact. To see this note that $K/K^2$ is free with basis", "$x_i - a_i$ over the ring $A[x_1, \\ldots, x_n]/K \\cong A$.", "Hence $K/KJ$ is free with the same basis over the ring", "$A[x_1, \\ldots, x_n]/J \\cong A/I$. On the other hand, taking derivatives", "gives a map", "$$", "\\text{d}_{A[x_1, \\ldots, x_n]/A} :", "J/J^2", "\\longrightarrow", "\\Omega_{A[x_1, \\ldots, x_n]/A} \\otimes_{A[x_1, \\ldots, x_n]}", "A[x_1, \\ldots, x_n]/J", "$$", "which maps the generators $x_i - a_i$ to the basis elements $\\text{d}x_i$", "of the free module on the right. The claim follows. Moreover, note that", "$x_1 - a_1, \\ldots, x_n - a_n$ is a regular sequence in", "$A[x_1, \\ldots, x_n]$ with quotient ring", "$A[x_1, \\ldots, x_n]/(x_1 - a_1, \\ldots, x_n - a_n) \\cong A$.", "Thus we have a factorization", "$$", "Y \\to V(x_1 - a_1, \\ldots, x_n - a_n) \\to \\mathbf{A}^n_S", "$$", "of our closed immersion $i$ where the composition is", "Koszul-regular (resp.\\ $H_1$-regular, quasi-regular),", "the second arrow is a regular immersion, and the associated conormal", "sequence is split. Now the result follows from", "Lemma \\ref{lemma-permanence-regular-immersion}." ], "refs": [ "divisors-lemma-immersion-smooth-into-smooth-regular-immersion", "divisors-lemma-composition-regular-immersion", "divisors-lemma-regular-quasi-regular-immersion", "divisors-lemma-permanence-regular-immersion" ], "ref_ids": [ 8007, 7994, 7989, 7995 ] } ], "ref_ids": [] }, { "id": 8009, "type": "theorem", "label": "divisors-lemma-pullback-meromorphic-sections-defined", "categories": [ "divisors" ], "title": "divisors-lemma-pullback-meromorphic-sections-defined", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "In each of the following cases pullbacks of meromorphic", "functions are defined.", "\\begin{enumerate}", "\\item every weakly associated point of $X$ maps to", "a generic point of an irreducible component of $Y$,", "\\item $X$, $Y$ are integral and $f$ is dominant,", "\\item $X$ is integral and the generic point of $X$ maps", "to a generic point of an irreducible component of $Y$,", "\\item $X$ is reduced and every generic point of every irreducible", "component of $X$ maps to the generic point of an irreducible component", "of $Y$,", "\\item $X$ is locally Noetherian, and any associated point of", "$X$ maps to a generic point of an irreducible component of $Y$,", "\\item $X$ is locally Noetherian, has no embedded points and", "any generic point of an irreducible component of", "$X$ maps to the generic point of an irreducible component of $Y$, and", "\\item $f$ is flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$ and $Y$. Hence we reduce to the case where", "$X = \\Spec(A)$, $Y = \\Spec(R)$ and $f$ is given by a ring map", "$\\varphi : R \\to A$.", "By the characterization of regular sections of the structure sheaf", "in Lemma \\ref{lemma-regular-section-structure-sheaf} we have to", "show that $R \\to A$ maps nonzerodivisors to nonzerodivisors.", "Let $t \\in R$ be a nonzerodivisor.", "\\medskip\\noindent", "If $R \\to A$ is flat, then $t : R \\to R$ being injective", "shows that $t : A \\to A$ is injective. This proves (7).", "\\medskip\\noindent", "In the other cases we note that $t$ is not contained in any of", "the minimal primes of $R$ (because every element of a minimal", "prime in a ring is a zerodivisor).", "Hence in case (1) we see that $\\varphi(t)$ is not contained", "in any weakly associated prime of $A$. Thus this case follows from", "Algebra, Lemma \\ref{algebra-lemma-weakly-ass-zero-divisors}.", "Case (5) is a special case of (1) by Lemma \\ref{lemma-ass-weakly-ass}.", "Case (6) follows from (5) and the definitions.", "Case (4) is a special case of (1) by", "Lemma \\ref{lemma-weakass-reduced}.", "Cases (2) and (3) are special cases of (4)." ], "refs": [ "algebra-lemma-weakly-ass-zero-divisors", "divisors-lemma-ass-weakly-ass", "divisors-lemma-weakass-reduced" ], "ref_ids": [ 725, 7878, 7882 ] } ], "ref_ids": [] }, { "id": 8010, "type": "theorem", "label": "divisors-lemma-meromorphic-weakass-finite", "categories": [ "divisors" ], "title": "divisors-lemma-meromorphic-weakass-finite", "contents": [ "Let $X$ be a scheme such that", "\\begin{enumerate}", "\\item[(a)] every weakly associated point of $X$ is a generic point of an", "irreducible component of $X$, and", "\\item[(b)] any quasi-compact open has a finite number of irreducible components.", "\\end{enumerate}", "Let $X^0$ be the set of generic points of irreducible components of $X$.", "Then we have", "$$", "\\mathcal{K}_X =", "\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta} =", "\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta}", "$$", "where $j_\\eta : \\Spec(\\mathcal{O}_{X, \\eta}) \\to X$ is the canonical map", "of Schemes, Section \\ref{schemes-section-points}. Moreover", "\\begin{enumerate}", "\\item $\\mathcal{K}_X$ is a quasi-coherent sheaf of", "$\\mathcal{O}_X$-algebras,", "\\item for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the sheaf", "$$", "\\mathcal{K}_X(\\mathcal{F}) =", "\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta =", "\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta", "$$", "of meromorphic sections of $\\mathcal{F}$", "is quasi-coherent,", "\\item $\\mathcal{S}_x \\subset \\mathcal{O}_{X, x}$", "is the set of nonzerodivisors for any $x \\in X$,", "\\item $\\mathcal{K}_{X, x}$ is the total quotient ring of $\\mathcal{O}_{X, x}$", "for any $x \\in X$,", "\\item $\\mathcal{K}_X(U)$ equals the total quotient ring of $\\mathcal{O}_X(U)$", "for any affine open $U \\subset X$,", "\\item the ring of rational functions of $X$", "(Morphisms, Definition \\ref{morphisms-definition-rational-function})", "is the ring of meromorphic", "functions on $X$, in a formula: $R(X) = \\Gamma(X, \\mathcal{K}_X)$.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-rational-function" ], "proofs": [ { "contents": [ "Observe that a locally finite direct sum of sheaves of modules", "is equal to the product since you can check this on stalks for", "example. Then since $\\mathcal{K}_X(\\mathcal{F}) =", "\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{K}_X$", "we see that (2) follows from the other statements.", "Also, observe that part (6) follows from the initial", "statement of the lemma and Morphisms, Lemma", "\\ref{morphisms-lemma-integral-scheme-rational-functions}", "when $X^0$ is finite; the general case of (6) follows from this", "by glueing (argument omitted).", "\\medskip\\noindent", "Let $j : Y = \\coprod\\nolimits_{\\eta \\in X^0} \\Spec(\\mathcal{O}_{X, \\eta}) \\to X$", "be the product of the morphisms $j_\\eta$. We have to show that", "$\\mathcal{K}_X = j_*\\mathcal{O}_Y$.", "First note that $\\mathcal{K}_Y = \\mathcal{O}_Y$ as $Y$ is a disjoint", "union of spectra of local rings of dimension $0$: in a local", "ring of dimension zero any nonzerodivisor is a unit.", "Next, note that pullbacks of meromorphic", "functions are defined for $j$ by", "Lemma \\ref{lemma-pullback-meromorphic-sections-defined}.", "This gives a map", "$$", "\\mathcal{K}_X \\longrightarrow j_*\\mathcal{O}_Y.", "$$", "Let $\\Spec(A) = U \\subset X$ be an affine open. Then $A$ is a ring", "with finitely many minimal primes $\\mathfrak q_1, \\ldots, \\mathfrak q_t$", "and every weakly associated prime of $A$ is one of the $\\mathfrak q_i$.", "We obtain $Q(A) = \\prod A_{\\mathfrak q_i}$", "by Algebra, Lemmas \\ref{algebra-lemma-total-ring-fractions-no-embedded-points}", "and \\ref{algebra-lemma-weakly-ass-zero-divisors}.", "In other words, already the value of the presheaf", "$U \\mapsto \\mathcal{S}(U)^{-1}\\mathcal{O}_X(U)$ agrees with", "$j_*\\mathcal{O}_Y(U)$ on our affine open $U$. Hence the displayed", "map is an isomorphism which proves the first displayed equality in", "the statement of the lemma.", "\\medskip\\noindent", "Finally, we prove (1), (3), (4), and (5).", "Part (5) we saw during the course of the proof that", "$\\mathcal{K}_X = j_*\\mathcal{O}_Y$.", "The morphism $j$ is quasi-compact by our assumption", "that the set of irreducible components of $X$ is locally finite.", "Hence $j$ is quasi-compact and quasi-separated (as $Y$ is separated).", "By Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "$j_*\\mathcal{O}_Y$ is quasi-coherent. This proves (1).", "Let $x \\in X$. We may choose an affine open neighbourhood", "$U = \\Spec(A)$ of $x$ all of whose irreducible components", "pass through $x$. Then $A \\subset A_\\mathfrak p$ because every", "weakly associated prime of $A$ is contained in $\\mathfrak p$", "hence elements of $A \\setminus \\mathfrak p$ are nonzerodivisors", "by Algebra, Lemma \\ref{algebra-lemma-weakly-ass-zero-divisors}.", "It follows easily that any nonzerodivisor of $A_\\mathfrak p$", "is the image of a nonzerodivisor on a (possibly smaller)", "affine open neighbourhood of $x$. This proves (3).", "Part (4) follows from part (3) by computing stalks." ], "refs": [ "morphisms-lemma-integral-scheme-rational-functions", "divisors-lemma-pullback-meromorphic-sections-defined", "algebra-lemma-total-ring-fractions-no-embedded-points", "algebra-lemma-weakly-ass-zero-divisors", "schemes-lemma-push-forward-quasi-coherent", "algebra-lemma-weakly-ass-zero-divisors" ], "ref_ids": [ 5478, 8009, 421, 725, 7730, 725 ] } ], "ref_ids": [ 5580 ] }, { "id": 8011, "type": "theorem", "label": "divisors-lemma-meromorphic-sections-pullback", "categories": [ "divisors" ], "title": "divisors-lemma-meromorphic-sections-pullback", "contents": [ "Let $f : X \\to Y$ be a morphism of locally ringed spaces.", "Assume that pullbacks of meromorphic functions are defined", "for $f$ (see", "Definition \\ref{definition-pullback-meromorphic-sections}).", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_Y$-modules.", "There is a canonical pullback map", "$f^* : \\Gamma(Y, \\mathcal{K}_Y(\\mathcal{F})) \\to", "\\Gamma(X, \\mathcal{K}_X(f^*\\mathcal{F}))$", "for meromorphic sections of $\\mathcal{F}$.", "\\item Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "A regular meromorphic section $s$ of $\\mathcal{L}$ pulls back", "to a regular meromorphic section $f^*s$ of $f^*\\mathcal{L}$.", "\\end{enumerate}" ], "refs": [ "divisors-definition-pullback-meromorphic-sections" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8103 ] }, { "id": 8012, "type": "theorem", "label": "divisors-lemma-regular-meromorphic-ideal-denominators", "categories": [ "divisors" ], "title": "divisors-lemma-regular-meromorphic-ideal-denominators", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s$ be a regular meromorphic section of $\\mathcal{L}$.", "Let us denote $\\mathcal{I} \\subset \\mathcal{O}_X$ the", "sheaf of ideals defined by the rule", "$$", "\\mathcal{I}(V)", "=", "\\{f \\in \\mathcal{O}_X(V) \\mid fs \\in \\mathcal{L}(V)\\}.", "$$", "The formula makes sense since", "$\\mathcal{L}(V) \\subset \\mathcal{K}_X(\\mathcal{L})(V)$.", "Then $\\mathcal{I}$ is a quasi-coherent sheaf of ideals and", "we have injective maps", "$$", "1 : \\mathcal{I} \\longrightarrow \\mathcal{O}_X,", "\\quad", "s : \\mathcal{I} \\longrightarrow \\mathcal{L}", "$$", "whose cokernels are supported on closed nowhere dense subsets of $X$." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$.", "Hence we may assume that $X = \\Spec(A)$,", "and $\\mathcal{L} = \\mathcal{O}_X$. After shrinking further", "we may assume that $s = a/b$ with $a, b \\in A$ {\\it both}", "nonzerodivisors in $A$. Set $I = \\{x \\in A \\mid x(a/b) \\in A\\}$.", "\\medskip\\noindent", "To show that $\\mathcal{I}$ is quasi-coherent we have to show", "that $I_f = \\{x \\in A_f \\mid x(a/b) \\in A_f\\}$ for every", "$f \\in A$. If $c/f^n \\in A_f$, $(c/f^n)(a/b) \\in A_f$, then we see", "that $f^mc(a/b) \\in A$ for some $m$, hence $c/f^n \\in I_f$.", "Conversely it is easy to see that $I_f$ is contained in", "$\\{x \\in A_f \\mid x(a/b) \\in A_f\\}$. This proves quasi-coherence.", "\\medskip\\noindent", "Let us prove the final statement. It is clear that $(b) \\subset I$.", "Hence $V(I) \\subset V(b)$ is a nowhere dense subset as $b$ is", "a nonzerodivisor. Thus the cokernel of $1$ is supported in a nowhere", "dense closed set. The same argument works for the cokernel", "of $s$ since $s(b) = (a) \\subset sI \\subset A$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8013, "type": "theorem", "label": "divisors-lemma-meromorphic-section-restricts-to-zero", "categories": [ "divisors" ], "title": "divisors-lemma-meromorphic-section-restricts-to-zero", "contents": [ "Let $X$ be a quasi-compact scheme. Let $h \\in \\Gamma(X, \\mathcal{O}_X)$ and", "$f \\in \\Gamma(X, \\mathcal{K}_X)$ such that $f$ restricts", "to zero on $X_h$. Then $h^n f = 0$ for some $n \\gg 0$." ], "refs": [], "proofs": [ { "contents": [ "We can find a covering of $X$ by affine opens $U$ such that $f|_U = s^{-1}a$", "with $a \\in \\mathcal{O}_X(U)$ and $s \\in \\mathcal{S}(U)$. Since $X$ is", "quasi-compact we can cover it by finitely many affine opens of this form.", "Thus it suffices to prove the lemma when $X = \\Spec(A)$ and $f = s^{-1}a$.", "Note that $s \\in A$ is a nonzerodivisor hence it suffices to prove", "the result when $f = a$. The condition $f|_{X_h} = 0$ implies that", "$a$ maps to zero in $A_h = \\mathcal{O}_X(X_h)$ as", "$\\mathcal{O}_X \\subset \\mathcal{K}_X$. Thus $h^na = 0$ for some $n > 0$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8014, "type": "theorem", "label": "divisors-lemma-locally-Noetherian-K", "categories": [ "divisors" ], "title": "divisors-lemma-locally-Noetherian-K", "contents": [ "Let $X$ be a locally Noetherian scheme.", "\\begin{enumerate}", "\\item For any $x \\in X$ we have $\\mathcal{S}_x \\subset \\mathcal{O}_{X, x}$", "is the set of nonzerodivisors, and hence $\\mathcal{K}_{X, x}$", "is the total quotient ring of $\\mathcal{O}_{X, x}$.", "\\item For any affine open $U \\subset X$ the ring", "$\\mathcal{K}_X(U)$ equals the total quotient ring of $\\mathcal{O}_X(U)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove this lemma we may assume $X$ is the spectrum of a Noetherian", "ring $A$. Say $x \\in X$ corresponds to $\\mathfrak p \\subset A$.", "\\medskip\\noindent", "Proof of (1). It is clear that $\\mathcal{S}_x$ is contained", "in the set of nonzerodivisors of $\\mathcal{O}_{X, x} = A_\\mathfrak p$.", "For the converse, let $f, g \\in A$, $g \\not \\in \\mathfrak p$ and", "assume $f/g$ is a nonzerodivisor in $A_{\\mathfrak p}$. Let", "$I = \\{a \\in A \\mid af = 0\\}$. Then we see that $I_{\\mathfrak p} = 0$ by", "exactness of localization. Since $A$ is Noetherian we see that $I$", "is finitely generated and hence that $g'I = 0$ for some $g' \\in A$,", "$g' \\not \\in \\mathfrak p$. Hence $f$ is a nonzerodivisor", "in $A_{g'}$, i.e., in a Zariski open neighbourhood of $\\mathfrak p$.", "Thus $f/g$ is an element of $\\mathcal{S}_x$.", "\\medskip\\noindent", "Proof of (2). Let $f \\in \\Gamma(X, \\mathcal{K}_X)$ be a meromorphic function.", "Set $I = \\{a \\in A \\mid af \\in A\\}$. Fix a prime $\\mathfrak p \\subset A$", "corresponding to the point $x \\in X$. By (1) we can write the image of $f$", "in the stalk at $\\mathfrak p$ as $a/b$, $a, b \\in A_{\\mathfrak p}$ with", "$b \\in A_{\\mathfrak p}$ not a zerodivisor. Write $b = c/d$ with", "$c, d \\in A$, $d \\not \\in \\mathfrak p$. Then $ad - cf$ is a section of", "$\\mathcal{K}_X$ which vanishes in an open neighbourhood of $x$. Say it", "vanishes on $D(e)$ with $e \\in A$, $e \\not \\in \\mathfrak p$. Then", "$e^n(ad - cf) = 0$ for some $n \\gg 0$ by", "Lemma \\ref{lemma-meromorphic-section-restricts-to-zero}.", "Thus $e^nc \\in I$ and $e^nc$ maps to a nonzerodivisor in", "$A_{\\mathfrak p}$. Let", "$\\text{Ass}(A) = \\{\\mathfrak q_1, \\ldots, \\mathfrak q_t\\}$ be the", "associated primes of $A$. By looking at $IA_{\\mathfrak q_i}$ and", "using Algebra, Lemma \\ref{algebra-lemma-associated-primes-localize}", "the above says that", "$I \\not \\subset \\mathfrak q_i$ for each $i$. By", "Algebra, Lemma \\ref{algebra-lemma-silly}", "there exists an element $x \\in I$, $x \\not \\in \\bigcup \\mathfrak q_i$.", "By Algebra, Lemma \\ref{algebra-lemma-ass-zero-divisors}", "we see that $x$ is not a zerodivisor on $A$.", "Hence $f = (xf)/x$ is an element of the total ring of fractions of $A$.", "This proves (2)." ], "refs": [ "algebra-lemma-associated-primes-localize", "algebra-lemma-silly", "algebra-lemma-ass-zero-divisors" ], "ref_ids": [ 709, 378, 704 ] } ], "ref_ids": [] }, { "id": 8015, "type": "theorem", "label": "divisors-lemma-quasi-coherent-K", "categories": [ "divisors" ], "title": "divisors-lemma-quasi-coherent-K", "contents": [ "Let $X$ be a locally Noetherian scheme having no embedded points.", "Let $X^0$ be the set of generic points of irreducible components of $X$.", "Then we have", "$$", "\\mathcal{K}_X =", "\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta} =", "\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta}", "$$", "where $j_\\eta : \\Spec(\\mathcal{O}_{X, \\eta}) \\to X$ is the canonical map", "of Schemes, Section \\ref{schemes-section-points}. Moreover", "\\begin{enumerate}", "\\item $\\mathcal{K}_X$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras,", "\\item for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the sheaf", "$$", "\\mathcal{K}_X(\\mathcal{F}) =", "\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta =", "\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta", "$$", "of meromorphic sections of $\\mathcal{F}$ is quasi-coherent, and", "\\item the ring of rational functions of $X$ is the ring of meromorphic", "functions on $X$, in a formula: $R(X) = \\Gamma(X, \\mathcal{K}_X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma is a special case of", "Lemma \\ref{lemma-meromorphic-weakass-finite}", "because in the locally Noetherian case", "weakly associated points are the same thing", "as associated points by Lemma \\ref{lemma-ass-weakly-ass}." ], "refs": [ "divisors-lemma-meromorphic-weakass-finite", "divisors-lemma-ass-weakly-ass" ], "ref_ids": [ 8010, 7878 ] } ], "ref_ids": [] }, { "id": 8016, "type": "theorem", "label": "divisors-lemma-regular-meromorphic-section-exists-noetherian", "categories": [ "divisors" ], "title": "divisors-lemma-regular-meromorphic-section-exists-noetherian", "contents": [ "Let $X$ be a locally Noetherian scheme having no embedded points.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Then $\\mathcal{L}$ has a regular meromorphic section." ], "refs": [], "proofs": [ { "contents": [ "For each generic point $\\eta$ of $X$ pick a generator", "$s_\\eta$ of the free rank $1$ module $\\mathcal{L}_\\eta$", "over the artinian local ring $\\mathcal{O}_{X, \\eta}$.", "It follows immediately from the description of", "$\\mathcal{K}_X$ and $\\mathcal{K}_X(\\mathcal{L})$ in", "Lemma \\ref{lemma-quasi-coherent-K} that $s = \\prod s_\\eta$", "is a regular meromorphic section of $\\mathcal{L}$." ], "refs": [ "divisors-lemma-quasi-coherent-K" ], "ref_ids": [ 8015 ] } ], "ref_ids": [] }, { "id": 8017, "type": "theorem", "label": "divisors-lemma-make-maps-regular-section", "categories": [ "divisors" ], "title": "divisors-lemma-make-maps-regular-section", "contents": [ "Suppose given", "\\begin{enumerate}", "\\item $X$ a locally Noetherian scheme,", "\\item $\\mathcal{L}$ an invertible $\\mathcal{O}_X$-module,", "\\item $s$ a regular meromorphic section of $\\mathcal{L}$, and", "\\item $\\mathcal{F}$ coherent on $X$", "without embedded associated points and $\\text{Supp}(\\mathcal{F}) = X$.", "\\end{enumerate}", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the ideal of", "denominators of $s$. Let $T \\subset X$ be the union", "of the supports of $\\mathcal{O}_X/\\mathcal{I}$ and", "$\\mathcal{L}/s(\\mathcal{I})$ which is a nowhere dense closed", "subset $T \\subset X$ according to", "Lemma \\ref{lemma-regular-meromorphic-ideal-denominators}.", "Then there are canonical injective maps", "$$", "1 : \\mathcal{I}\\mathcal{F} \\to \\mathcal{F}, \\quad", "s : \\mathcal{I}\\mathcal{F} \\to \\mathcal{F} \\otimes_{\\mathcal{O}_X}\\mathcal{L}", "$$", "whose cokernels are supported on $T$." ], "refs": [ "divisors-lemma-regular-meromorphic-ideal-denominators" ], "proofs": [ { "contents": [ "Reduce to the affine case with $\\mathcal{L} \\cong \\mathcal{O}_X$,", "and $s = a/b$ with $a, b \\in A$ both nonzerodivisors.", "Proof of reduction step omitted.", "Write $\\mathcal{F} = \\widetilde{M}$.", "Let $I = \\{x \\in A \\mid x(a/b) \\in A\\}$", "so that $\\mathcal{I} = \\widetilde{I}$ (see", "proof of Lemma \\ref{lemma-regular-meromorphic-ideal-denominators}).", "Note that $T = V(I) \\cup V((a/b)I)$.", "For any $A$-module $M$ consider the map $1 : IM \\to M$; this is the", "map that gives rise to the map $1$ of the lemma.", "Consider on the other hand the map", "$\\sigma : IM \\to M_b, x \\mapsto ax/b$.", "Since $b$ is not a zerodivisor in $A$, and since", "$M$ has support $\\Spec(A)$ and no embedded primes we", "see that $b$ is a nonzerodivisor on $M$ also. Hence $M \\subset M_b$.", "By definition of $I$ we have $\\sigma(IM) \\subset M$ as submodules", "of $M_b$. Hence we get an $A$-module map $s : IM \\to M$ (namely the", "unique map such that $s(z)/1 = \\sigma(z)$ in $M_b$ for all $z \\in IM$).", "It is injective because $a$ is a nonzerodivisor also (on both $A$ and $M$).", "It is clear that $M/IM$ is annihilated by $I$ and that", "$M/s(IM)$ is annihilated by $(a/b)I$. Thus the lemma follows." ], "refs": [ "divisors-lemma-regular-meromorphic-ideal-denominators" ], "ref_ids": [ 8012 ] } ], "ref_ids": [ 8012 ] }, { "id": 8018, "type": "theorem", "label": "divisors-lemma-reduced-finite-irreducible", "categories": [ "divisors" ], "title": "divisors-lemma-reduced-finite-irreducible", "contents": [ "Let $X$ be a reduced scheme such that any quasi-compact open", "has a finite number of irreducible components. Let $X^0$ be the set", "of generic points of irreducible components of $X$. Then we have", "$$", "\\mathcal{K}_X =", "\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\kappa(\\eta) =", "\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\kappa(\\eta)", "$$", "where $j_\\eta : \\Spec(\\kappa(\\eta)) \\to X$ is the canonical map", "of Schemes, Section \\ref{schemes-section-points}. Moreover", "\\begin{enumerate}", "\\item $\\mathcal{K}_X$ is a quasi-coherent sheaf of", "$\\mathcal{O}_X$-algebras,", "\\item for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the sheaf", "$$", "\\mathcal{K}_X(\\mathcal{F}) =", "\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta =", "\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta", "$$", "of meromorphic sections of $\\mathcal{F}$", "is quasi-coherent,", "\\item $\\mathcal{S}_x \\subset \\mathcal{O}_{X, x}$", "is the set of nonzerodivisors for any $x \\in X$,", "\\item $\\mathcal{K}_{X, x}$ is the total quotient ring of $\\mathcal{O}_{X, x}$", "for any $x \\in X$,", "\\item $\\mathcal{K}_X(U)$ equals the total quotient ring of $\\mathcal{O}_X(U)$", "for any affine open $U \\subset X$,", "\\item the ring of rational functions of $X$ is the ring of meromorphic", "functions on $X$, in a formula: $R(X) = \\Gamma(X, \\mathcal{K}_X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma is a special case of", "Lemma \\ref{lemma-meromorphic-weakass-finite}", "because on a reduced scheme the weakly associated", "points are the generic points by", "Lemma \\ref{lemma-weakass-reduced}." ], "refs": [ "divisors-lemma-meromorphic-weakass-finite", "divisors-lemma-weakass-reduced" ], "ref_ids": [ 8010, 7882 ] } ], "ref_ids": [] }, { "id": 8019, "type": "theorem", "label": "divisors-lemma-reduced-normalization", "categories": [ "divisors" ], "title": "divisors-lemma-reduced-normalization", "contents": [ "Let $X$ be a scheme.", "Assume $X$ is reduced and any quasi-compact open $U \\subset X$", "has a finite number of irreducible components.", "Then the normalization morphism $\\nu : X^\\nu \\to X$ is the", "morphism", "$$", "\\underline{\\Spec}_X(\\mathcal{O}') \\longrightarrow X", "$$", "where $\\mathcal{O}' \\subset \\mathcal{K}_X$ is the integral", "closure of $\\mathcal{O}_X$ in the sheaf of meromorphic functions." ], "refs": [], "proofs": [ { "contents": [ "Compare the definition of the normalization morphism", "$\\nu : X^\\nu \\to X$ (see", "Morphisms, Definition \\ref{morphisms-definition-normalization})", "with the description of $\\mathcal{K}_X$ in", "Lemma \\ref{lemma-reduced-finite-irreducible} above." ], "refs": [ "morphisms-definition-normalization", "divisors-lemma-reduced-finite-irreducible" ], "ref_ids": [ 5592, 8018 ] } ], "ref_ids": [] }, { "id": 8020, "type": "theorem", "label": "divisors-lemma-meromorphic-functions-integral-scheme", "categories": [ "divisors" ], "title": "divisors-lemma-meromorphic-functions-integral-scheme", "contents": [ "Let $X$ be an integral scheme with generic point $\\eta$. We have", "\\begin{enumerate}", "\\item the sheaf of meromorphic functions is", "isomorphic to the constant sheaf with value the", "function field (see", "Morphisms, Definition \\ref{morphisms-definition-function-field})", "of $X$.", "\\item for any quasi-coherent sheaf $\\mathcal{F}$ on $X$ the", "sheaf $\\mathcal{K}_X(\\mathcal{F})$ is isomorphic to the", "constant sheaf with value $\\mathcal{F}_\\eta$.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-function-field" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 5582 ] }, { "id": 8021, "type": "theorem", "label": "divisors-lemma-regular-meromorphic-section-exists", "categories": [ "divisors" ], "title": "divisors-lemma-regular-meromorphic-section-exists", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "In each of the following cases $\\mathcal{L}$ has a regular meromorphic", "section:", "\\begin{enumerate}", "\\item $X$ is integral,", "\\item $X$ is reduced and any quasi-compact open has a finite", "number of irreducible components,", "\\item $X$ is locally Noetherian and has no embedded points.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In case (1) let $\\eta \\in X$ be the generic point. We have seen in", "Lemma \\ref{lemma-meromorphic-functions-integral-scheme}", "that $\\mathcal{K}_X$, resp.\\ $\\mathcal{K}_X(\\mathcal{L})$", "is the constant sheaf with value", "$\\kappa(\\eta)$, resp.\\ $\\mathcal{L}_\\eta$.", "Since $\\dim_{\\kappa(\\eta)} \\mathcal{L}_\\eta = 1$", "we can pick a nonzero element $s \\in \\mathcal{L}_\\eta$.", "Clearly $s$ is a regular meromorphic section of $\\mathcal{L}$.", "In case (2) pick $s_\\eta \\in \\mathcal{L}_\\eta$ nonzero", "for all generic points $\\eta$ of $X$; this is possible", "as $\\mathcal{L}_\\eta$ is a $1$-dimensional vector space", "over $\\kappa(\\eta)$.", "It follows immediately from the description of", "$\\mathcal{K}_X$ and $\\mathcal{K}_X(\\mathcal{L})$", "in Lemma \\ref{lemma-reduced-finite-irreducible}", "that $s = \\prod s_\\eta$ is a regular meromorphic section of $\\mathcal{L}$.", "Case (3) is Lemma \\ref{lemma-regular-meromorphic-section-exists-noetherian}." ], "refs": [ "divisors-lemma-meromorphic-functions-integral-scheme", "divisors-lemma-reduced-finite-irreducible" ], "ref_ids": [ 8020, 8018 ] } ], "ref_ids": [] }, { "id": 8022, "type": "theorem", "label": "divisors-lemma-components-locally-finite", "categories": [ "divisors" ], "title": "divisors-lemma-components-locally-finite", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $Z \\subset X$ be a closed", "subscheme. The collection of irreducible components of $Z$", "is locally finite in $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be a quasi-compact open subscheme. Then $U$ is a Noetherian", "scheme, and hence has a Noetherian underlying topological space", "(Properties, Lemma \\ref{properties-lemma-Noetherian-topology}).", "Hence every subspace is Noetherian and", "has finitely many irreducible components", "(see Topology, Lemma \\ref{topology-lemma-Noetherian})." ], "refs": [ "properties-lemma-Noetherian-topology", "topology-lemma-Noetherian" ], "ref_ids": [ 2954, 8220 ] } ], "ref_ids": [] }, { "id": 8023, "type": "theorem", "label": "divisors-lemma-divisor-locally-finite", "categories": [ "divisors" ], "title": "divisors-lemma-divisor-locally-finite", "contents": [ "Let $X$ be a locally Noetherian integral scheme. Let $f \\in R(X)^*$.", "Then the collections", "$$", "\\{Z \\subset X \\mid Z\\text{ a prime divisor with generic point }\\xi", "\\text{ and }f\\text{ not in }\\mathcal{O}_{X, \\xi}\\}", "$$", "and", "$$", "\\{Z \\subset X \\mid Z \\text{ a prime divisor and }\\text{ord}_Z(f) \\not = 0\\}", "$$", "are locally finite in $X$." ], "refs": [], "proofs": [ { "contents": [ "There exists a nonempty open subscheme $U \\subset X$ such that $f$", "corresponds to a section of $\\Gamma(U, \\mathcal{O}_X^*)$. Hence the", "prime divisors which can occur in the sets of the lemma are all", "irreducible components of $X \\setminus U$.", "Hence Lemma \\ref{lemma-components-locally-finite} gives the desired result." ], "refs": [ "divisors-lemma-components-locally-finite" ], "ref_ids": [ 8022 ] } ], "ref_ids": [] }, { "id": 8024, "type": "theorem", "label": "divisors-lemma-div-additive", "categories": [ "divisors" ], "title": "divisors-lemma-div-additive", "contents": [ "Let $X$ be a locally Noetherian integral scheme. Let $f, g \\in R(X)^*$. Then", "$$", "\\text{div}_X(fg) = \\text{div}_X(f) + \\text{div}_X(g)", "$$", "as Weil divisors on $X$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the additivity of the $\\text{ord}$ functions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8025, "type": "theorem", "label": "divisors-lemma-divisor-meromorphic-locally-finite", "categories": [ "divisors" ], "title": "divisors-lemma-divisor-meromorphic-locally-finite", "contents": [ "Let $X$ be a locally Noetherian integral scheme. Let $\\mathcal{L}$ be an", "invertible $\\mathcal{O}_X$-module. Let $s \\in \\mathcal{K}_X(\\mathcal{L})$ be a", "regular (i.e., nonzero) meromorphic section of $\\mathcal{L}$. Then the sets", "$$", "\\{Z \\subset X \\mid Z \\text{ a prime divisor with generic point }\\xi", "\\text{ and }s\\text{ not in }\\mathcal{L}_\\xi\\}", "$$", "and", "$$", "\\{Z \\subset X \\mid Z \\text{ is a prime divisor and }", "\\text{ord}_{Z, \\mathcal{L}}(s) \\not = 0\\}", "$$", "are locally finite in $X$." ], "refs": [], "proofs": [ { "contents": [ "There exists a nonempty open subscheme $U \\subset X$ such that $s$", "corresponds to a section of $\\Gamma(U, \\mathcal{L})$ which generates", "$\\mathcal{L}$ over $U$. Hence the prime divisors which can occur", "in the sets of the lemma are all irreducible components of $X \\setminus U$.", "Hence Lemma \\ref{lemma-components-locally-finite}.", "gives the desired result." ], "refs": [ "divisors-lemma-components-locally-finite" ], "ref_ids": [ 8022 ] } ], "ref_ids": [] }, { "id": 8026, "type": "theorem", "label": "divisors-lemma-divisor-meromorphic-well-defined", "categories": [ "divisors" ], "title": "divisors-lemma-divisor-meromorphic-well-defined", "contents": [ "Let $X$ be a locally Noetherian integral scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s, s' \\in \\mathcal{K}_X(\\mathcal{L})$ be nonzero", "meromorphic sections of $\\mathcal{L}$. Then $f = s/s'$", "is an element of $R(X)^*$ and we have", "$$", "\\sum \\text{ord}_{Z, \\mathcal{L}}(s)[Z]", "=", "\\sum \\text{ord}_{Z, \\mathcal{L}}(s')[Z]", "+", "\\text{div}(f)", "$$", "as Weil divisors." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions.", "Note that Lemma \\ref{lemma-divisor-meromorphic-locally-finite}", "guarantees that the sums are indeed Weil divisors." ], "refs": [ "divisors-lemma-divisor-meromorphic-locally-finite" ], "ref_ids": [ 8025 ] } ], "ref_ids": [] }, { "id": 8027, "type": "theorem", "label": "divisors-lemma-c1-additive", "categories": [ "divisors" ], "title": "divisors-lemma-c1-additive", "contents": [ "Let $X$ be a locally Noetherian integral scheme.", "Let $\\mathcal{L}$, $\\mathcal{N}$ be invertible $\\mathcal{O}_X$-modules.", "Let $s$, resp.\\ $t$ be a nonzero meromorphic section", "of $\\mathcal{L}$, resp.\\ $\\mathcal{N}$. Then $st$ is a nonzero", "meromorphic section of $\\mathcal{L} \\otimes \\mathcal{N}$, and", "$$", "\\text{div}_{\\mathcal{L} \\otimes \\mathcal{N}}(st)", "=", "\\text{div}_\\mathcal{L}(s) + \\text{div}_\\mathcal{N}(t)", "$$", "in $\\text{Div}(X)$. In particular, the Weil divisor class of", "$\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}$ is the sum", "of the Weil divisor classes of $\\mathcal{L}$ and $\\mathcal{N}$." ], "refs": [], "proofs": [ { "contents": [ "Let $s$, resp.\\ $t$ be a nonzero meromorphic section", "of $\\mathcal{L}$, resp.\\ $\\mathcal{N}$. Then $st$ is a nonzero", "meromorphic section of $\\mathcal{L} \\otimes \\mathcal{N}$.", "Let $Z \\subset X$ be a prime divisor. Let $\\xi \\in Z$ be its generic", "point. Choose generators $s_\\xi \\in \\mathcal{L}_\\xi$, and", "$t_\\xi \\in \\mathcal{N}_\\xi$. Then $s_\\xi t_\\xi$ is a generator", "for $(\\mathcal{L} \\otimes \\mathcal{N})_\\xi$.", "So $st/(s_\\xi t_\\xi) = (s/s_\\xi)(t/t_\\xi)$.", "Hence we see that", "$$", "\\text{div}_{\\mathcal{L} \\otimes \\mathcal{N}, Z}(st)", "=", "\\text{div}_{\\mathcal{L}, Z}(s) + \\text{div}_{\\mathcal{N}, Z}(t)", "$$", "by the additivity of the $\\text{ord}_Z$ function." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8028, "type": "theorem", "label": "divisors-lemma-normal-c1-injective", "categories": [ "divisors" ], "title": "divisors-lemma-normal-c1-injective", "contents": [ "Let $X$ be a locally Noetherian integral scheme. If $X$ is normal,", "then the map (\\ref{equation-c1}) $\\Pic(X) \\to \\text{Cl}(X)$", "is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module whose", "associated Weil divisor class is trivial. Let $s$ be a regular", "meromorphic section of $\\mathcal{L}$. The assumption means that", "$\\text{div}_\\mathcal{L}(s) = \\text{div}(f)$ for some", "$f \\in R(X)^*$. Then we see that $t = f^{-1}s$ is a regular", "meromorphic section of $\\mathcal{L}$ with", "$\\text{div}_\\mathcal{L}(t) = 0$, see", "Lemma \\ref{lemma-divisor-meromorphic-well-defined}.", "We will show that $t$ defines a trivialization of $\\mathcal{L}$", "which finishes the proof of the lemma.", "In order to prove this we may work locally on $X$.", "Hence we may assume that $X = \\Spec(A)$ is affine", "and that $\\mathcal{L}$ is trivial. Then $A$ is a Noetherian normal", "domain and $t$ is an element of its fraction field", "such that $\\text{ord}_{A_\\mathfrak p}(t) = 0$", "for all height $1$ primes $\\mathfrak p$ of $A$.", "Our goal is to show that $t$ is a unit of $A$.", "Since $A_\\mathfrak p$ is a discrete valuation ring for height", "one primes of $A$ (Algebra, Lemma \\ref{algebra-lemma-criterion-normal}), the", "condition signifies that $t \\in A_\\mathfrak p^*$ for all primes $\\mathfrak p$", "of height $1$. This implies $t \\in A$ and $t^{-1} \\in A$ by", "Algebra, Lemma", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}", "and the proof is complete." ], "refs": [ "divisors-lemma-divisor-meromorphic-well-defined", "algebra-lemma-criterion-normal", "algebra-lemma-normal-domain-intersection-localizations-height-1" ], "ref_ids": [ 8026, 1311, 1313 ] } ], "ref_ids": [] }, { "id": 8029, "type": "theorem", "label": "divisors-lemma-local-rings-UFD-c1-bijective", "categories": [ "divisors" ], "title": "divisors-lemma-local-rings-UFD-c1-bijective", "contents": [ "Let $X$ be a locally Noetherian integral scheme. Consider the map", "(\\ref{equation-c1}) $\\Pic(X) \\to \\text{Cl}(X)$.", "The following are equivalent", "\\begin{enumerate}", "\\item the local rings of $X$ are UFDs, and", "\\item $X$ is normal and $\\Pic(X) \\to \\text{Cl}(X)$", "is surjective.", "\\end{enumerate}", "In this case $\\Pic(X) \\to \\text{Cl}(X)$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "If (1) holds, then $X$ is normal by", "Algebra, Lemma \\ref{algebra-lemma-UFD-normal}.", "Hence the map (\\ref{equation-c1}) is injective by", "Lemma \\ref{lemma-normal-c1-injective}. Moreover,", "every prime divisor $D \\subset X$ is an effective", "Cartier divisor by Lemma \\ref{lemma-weil-divisor-is-cartier-UFD}.", "In this case the canonical section $1_D$ of $\\mathcal{O}_X(D)$", "(Definition \\ref{definition-invertible-sheaf-effective-Cartier-divisor})", "vanishes exactly along $D$ and we see that the class of $D$ is the", "image of $\\mathcal{O}_X(D)$ under the map (\\ref{equation-c1}).", "Thus the map is surjective as well.", "\\medskip\\noindent", "Assume (2) holds. Pick a prime divisor $D \\subset X$.", "Since (\\ref{equation-c1}) is surjective there exists an invertible", "sheaf $\\mathcal{L}$, a regular meromorphic section $s$, and $f \\in R(X)^*$", "such that $\\text{div}_\\mathcal{L}(s) + \\text{div}(f) = [D]$.", "In other words, $\\text{div}_\\mathcal{L}(fs) = [D]$.", "Let $x \\in X$ and let $A = \\mathcal{O}_{X, x}$. Thus $A$ is", "a Noetherian local normal domain with fraction field $K = R(X)$.", "Every height $1$ prime of $A$ corresponds to a prime divisor on $X$", "and every invertible $\\mathcal{O}_X$-module restricts to the", "trivial invertible module on $\\Spec(A)$. It follows that for every", "height $1$ prime $\\mathfrak p \\subset A$ there exists an element $f \\in K$", "such that $\\text{ord}_{A_\\mathfrak p}(f) = 1$ and", "$\\text{ord}_{A_{\\mathfrak p'}}(f) = 0$ for every other", "height one prime $\\mathfrak p'$. Then $f \\in A$ by Algebra, Lemma", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}.", "Arguing in the same fashion we see that every element $g \\in \\mathfrak p$", "is of the form $g = af$ for some $a \\in A$. Thus we see that every", "height one prime ideal of $A$ is principal and $A$ is a UFD", "by Algebra, Lemma \\ref{algebra-lemma-characterize-UFD-height-1}." ], "refs": [ "algebra-lemma-UFD-normal", "divisors-lemma-normal-c1-injective", "divisors-lemma-weil-divisor-is-cartier-UFD", "divisors-definition-invertible-sheaf-effective-Cartier-divisor", "algebra-lemma-normal-domain-intersection-localizations-height-1", "algebra-lemma-characterize-UFD-height-1" ], "ref_ids": [ 1037, 8028, 7951, 8092, 1313, 1032 ] } ], "ref_ids": [] }, { "id": 8030, "type": "theorem", "label": "divisors-lemma-in-image-pullback", "categories": [ "divisors" ], "title": "divisors-lemma-in-image-pullback", "contents": [ "Let $\\varphi : X \\to Y$ be a morphism of schemes. Let $\\mathcal{L}$", "be an invertible $\\mathcal{O}_X$-module. Assume that", "\\begin{enumerate}", "\\item $X$ is locally Noetherian,", "\\item $Y$ is locally Noetherian, integral, and normal,", "\\item $\\varphi$ is flat with integral (hence nonempty) fibres,", "\\item $\\varphi$ is either quasi-compact or locally of finite type,", "\\item $\\mathcal{L}$ is trivial when restricted to the generic fibre of", "$\\varphi$.", "\\end{enumerate}", "Then $\\mathcal{L} \\cong \\varphi^*\\mathcal{N}$ for some invertible", "$\\mathcal{O}_Y$-module $\\mathcal{N}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\xi \\in Y$ be the generic point. Let $X_\\xi$ be the scheme theoretic", "fibre of $\\varphi$ over $\\xi$. Denote $\\mathcal{L}_\\xi$ the pullback of", "$\\mathcal{L}$ to $X_\\xi$. Assumption (5) means that $\\mathcal{L}_\\xi$", "is trivial. Choose a trivializing section", "$s \\in \\Gamma(X_\\xi, \\mathcal{L}_\\xi)$. Observe that $X$ is integral by", "Lemma \\ref{lemma-flat-over-integral-integral-fibre}.", "Hence we can think of $s$ as a regular meromorphic section of $\\mathcal{L}$.", "Pullbacks of meromorphic functions are defined for", "$\\varphi$ by Lemma \\ref{lemma-pullback-meromorphic-sections-defined}.", "Let $\\mathcal{N} \\subset \\mathcal{K}_Y$ be the $\\mathcal{O}_Y$-module", "whose sections over an open $V \\subset Y$ are those meromorphic functions", "$g \\in \\mathcal{K}_Y(V)$ such that", "$\\varphi^*(g)s \\in \\mathcal{L}(\\varphi^{-1}V)$.", "A priori $\\varphi^*(g)s$ is a section of $\\mathcal{K}_X(\\mathcal{L})$ over", "$\\varphi^{-1}V$. We claim that $\\mathcal{N}$ is an invertible", "$\\mathcal{O}_Y$-module and that the map", "$$", "\\varphi^*\\mathcal{N} \\longrightarrow \\mathcal{L},\\quad", "g \\longmapsto gs", "$$", "is an isomorphism.", "\\medskip\\noindent", "We first prove the claim in the following situation:", "$X$ and $Y$ are affine and $\\mathcal{L}$ trivial. Say $Y = \\Spec(R)$,", "$X = \\Spec(A)$ and $s$ given by the element $s \\in A \\otimes_R K$", "where $K$ is the fraction field of $R$. We can write $s = a/r$", "for some nonzero $r \\in R$ and $a \\in A$. Since $s$ generates $\\mathcal{L}$", "on the generic fibre we see that there exists an $s' \\in A \\otimes_R K$", "such that $ss' = 1$. Thus we see that $s = r'/a'$ for some", "nonzero $r' \\in R$ and $a' \\in A$. Let", "$\\mathfrak p_1, \\ldots, \\mathfrak p_n \\subset R$", "be the minimal primes over $rr'$. Each $R_{\\mathfrak p_i}$", "is a discrete valuation ring", "(Algebra, Lemmas \\ref{algebra-lemma-minimal-over-1} and", "\\ref{algebra-lemma-criterion-normal}). By assumption", "$\\mathfrak q_i = \\mathfrak p_i A$ is a prime. Hence", "$\\mathfrak q_i A_{\\mathfrak q_i}$ is generated by a single element", "and we find that $A_{\\mathfrak q_i}$ is a discrete valuation ring as", "well (Algebra, Lemma \\ref{algebra-lemma-characterize-dvr}). Of course", "$R_{\\mathfrak p_i} \\to A_{\\mathfrak q_i}$ has ramification index $1$.", "Let $e_i, e'_i \\geq 0$ be the valuation of $a, a'$ in $A_{\\mathfrak q_i}$.", "Then $e_i + e'_i$ is the valuation of $rr'$ in $R_{\\mathfrak p_i}$. Note that", "$$", "\\mathfrak p_1^{(e_1 + e'_1)} \\cap \\ldots \\cap \\mathfrak p_i^{(e_n + e'_n)} =", "(rr')", "$$", "in $R$ by Algebra, Lemma", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}.", "Set", "$$", "I = \\mathfrak p_1^{(e_1)} \\cap \\ldots \\cap \\mathfrak p_i^{(e_n)}", "\\quad\\text{and}\\quad", "I' = \\mathfrak p_1^{(e'_1)} \\cap \\ldots \\cap \\mathfrak p_i^{(e'_n)}", "$$", "so that $II' \\subset (rr')$. Observe that", "$$", "IA =", "(\\mathfrak p_1^{(e_1)} \\cap \\ldots \\cap \\mathfrak p_i^{(e_n)})A =", "(\\mathfrak p_1A)^{(e_1)} \\cap \\ldots \\cap (\\mathfrak p_i A)^{(e_n)}", "$$", "by Algebra, Lemmas \\ref{algebra-lemma-symbolic-power-flat-extension} and", "\\ref{algebra-lemma-flat-intersect-ideals}. Similarly for $I'A$. Hence", "$a \\in IA$ and $a' \\in I'A$.", "We conclude that $IA \\otimes_A I'A \\to rr'A$ is surjective.", "By faithful flatness of $R \\to A$ we find that", "$I \\otimes_R I' \\to (rr')$ is surjective as well.", "It follows that $II' = (rr')$ and $I$ and $I'$ are finite locally", "free of rank $1$, see", "Algebra, Lemma \\ref{algebra-lemma-product-ideals-principal}.", "Thus Zariski locally on $R$ we can write $I = (g)$ and $I' = (g')$", "with $gg' = rr'$. Then $a = ug$ and $a' = u'g'$ for some $u, u' \\in A$.", "We conclude that $u, u'$ are units. Thus Zariski locally on $R$", "we have $s = ug/r$ and the claim follows in this case.", "\\medskip\\noindent", "Let $y \\in Y$ be a point.", "Pick $x \\in X$ mapping to $y$. We may apply the result of the previous", "paragraph to $\\Spec(\\mathcal{O}_{X, x}) \\to \\Spec(\\mathcal{O}_{Y, y})$.", "We conclude there exists an element $g \\in R(Y)^*$ well defined up to", "multiplication by an element of $\\mathcal{O}_{Y, y}^*$ such that", "$\\varphi^*(g)s$ generates $\\mathcal{L}_x$. Hence $\\varphi^*(g)s$", "generates $\\mathcal{L}$ in a neighbourhood $U$ of $x$.", "Suppose $x'$ is a second point lying over $y$ and $g' \\in R(Y)^*$ is", "such that $\\varphi^*(g')s$ generates $\\mathcal{L}$ in an open neighbourhood", "$U'$ of $x'$. Then we can choose a point", "$x''$ in $U \\cap U' \\cap \\varphi^{-1}(\\{y\\})$", "because the fibre is irreducible. By the uniqueness for", "the ring map $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x''}$", "we find that $g$ and $g'$ differ (multiplicatively)", "by an element in $\\mathcal{O}_{Y, y}^*$. Hence we see that $\\varphi^*(g)s$", "is a generator for $\\mathcal{L}$ on an open neighbourhood", "of $\\varphi^{-1}(y)$. Let $Z \\subset X$ be the set of points", "$z \\in X$ such that $\\varphi^*(g)s$ does not generate $\\mathcal{L}_z$.", "The arguments above show that $Z$ is closed and that $Z = \\varphi^{-1}(T)$", "for some subset $T \\subset Y$ with $y \\not \\in T$. If we can show that", "$T$ is closed, then $g$ will be a generator for $\\mathcal{N}$ as an", "$\\mathcal{O}_Y$-module in the open neighbourhood $Y \\setminus T$ of $y$", "thereby finishing the proof (some details omitted).", "\\medskip\\noindent", "If $\\varphi$ is quasi-compact, then $T$ is closed by", "Morphisms, Lemma \\ref{morphisms-lemma-fpqc-quotient-topology}.", "If $\\varphi$ is locally of finite type, then $\\varphi$ is open", "by Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}.", "Then $Y \\setminus T$ is open as the image of the open $X \\setminus Z$." ], "refs": [ "divisors-lemma-flat-over-integral-integral-fibre", "divisors-lemma-pullback-meromorphic-sections-defined", "algebra-lemma-minimal-over-1", "algebra-lemma-criterion-normal", "algebra-lemma-characterize-dvr", "algebra-lemma-normal-domain-intersection-localizations-height-1", "algebra-lemma-symbolic-power-flat-extension", "algebra-lemma-flat-intersect-ideals", "algebra-lemma-product-ideals-principal", "morphisms-lemma-fpqc-quotient-topology", "morphisms-lemma-fppf-open" ], "ref_ids": [ 7908, 8009, 683, 1311, 1023, 1313, 715, 522, 1040, 5269, 5267 ] } ], "ref_ids": [] }, { "id": 8031, "type": "theorem", "label": "divisors-lemma-closure-effective-cartier-divisor", "categories": [ "divisors" ], "title": "divisors-lemma-closure-effective-cartier-divisor", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $U \\subset X$ be", "an open and let $D \\subset U$ be an effective Cartier divisor.", "If $\\mathcal{O}_{X, x}$ is a UFD for all $x \\in X \\setminus U$,", "then there exists an effective Cartier divisor $D' \\subset X$", "with $D = U \\cap D'$." ], "refs": [], "proofs": [ { "contents": [ "Let $D' \\subset X$ be the scheme theoretic image of the morphism $D \\to X$.", "Since $X$ is locally Noetherian the morphism $D \\to X$ is quasi-compact, see", "Properties, Lemma \\ref{properties-lemma-immersion-into-noetherian}.", "Hence the formation of $D'$ commutes with passing to opens in $X$ by", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}.", "Thus we may assume $X = \\Spec(A)$ is affine.", "Let $I \\subset A$ be the ideal corresponding to $D'$.", "Let $\\mathfrak p \\subset A$ be a prime ideal corresponding to a point", "of $X \\setminus U$.", "To finish the proof it is enough to show that $I_\\mathfrak p$ is generated", "by one element, see Lemma \\ref{lemma-effective-Cartier-in-points}.", "Thus we may replace $X$ by $\\Spec(A_\\mathfrak p)$, see", "Morphisms, Lemma \\ref{morphisms-lemma-flat-base-change-scheme-theoretic-image}.", "In other words, we may assume that $X$ is the spectrum of a local", "UFD $A$. Then all local rings of $A$ are UFD's. It follows that", "$D = \\sum a_i D_i$ with $D_i \\subset U$ an integral effective Cartier divisor,", "see Lemma \\ref{lemma-effective-Cartier-divisor-is-a-sum}.", "The generic points $\\xi_i$ of $D_i$ correspond to prime ideals", "$\\mathfrak p_i \\subset A$ of height $1$, see", "Lemma \\ref{lemma-effective-Cartier-codimension-1}.", "Then $\\mathfrak p_i = (f_i)$ for some prime element $f_i \\in A$", "and we conclude that $D'$ is cut out by $\\prod f_i^{a_i}$ as desired." ], "refs": [ "properties-lemma-immersion-into-noetherian", "morphisms-lemma-quasi-compact-scheme-theoretic-image", "divisors-lemma-effective-Cartier-in-points", "morphisms-lemma-flat-base-change-scheme-theoretic-image", "divisors-lemma-effective-Cartier-divisor-is-a-sum", "divisors-lemma-effective-Cartier-codimension-1" ], "ref_ids": [ 2952, 5146, 7946, 5273, 7955, 7947 ] } ], "ref_ids": [] }, { "id": 8032, "type": "theorem", "label": "divisors-lemma-extend-invertible-module", "categories": [ "divisors" ], "title": "divisors-lemma-extend-invertible-module", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $U \\subset X$ be", "an open and let $\\mathcal{L}$ be an invertible $\\mathcal{O}_U$-module.", "If $\\mathcal{O}_{X, x}$ is a UFD for all $x \\in X \\setminus U$,", "then there exists an invertible $\\mathcal{O}_X$-module $\\mathcal{L}'$", "with $\\mathcal{L} \\cong \\mathcal{L}'|_U$." ], "refs": [], "proofs": [ { "contents": [ "Choose $x \\in X$, $x \\not \\in U$. We will show there exists an", "affine open neighbourhood $W \\subset X$, such that $\\mathcal{L}|_{W \\cap U}$", "extends to an invertible sheaf on $W$. This implies by glueing of", "sheaves (Sheaves, Section \\ref{sheaves-section-glueing-sheaves})", "that we can extend $\\mathcal{L}$ to the strictly bigger open $U \\cup W$.", "Let $W = \\Spec(A)$ be an affine open neighbourhood.", "Since $U \\cap W$ is quasi-affine, we see that we can write", "$\\mathcal{L}|_{W \\cap U}$ as", "$\\mathcal{O}(D_1) \\otimes \\mathcal{O}(D_2)^{\\otimes -1}$ for some", "effective Cartier divisors $D_1, D_2 \\subset W \\cap U$, see", "Lemma \\ref{lemma-quasi-projective-Noetherian-pic-effective-Cartier}.", "Then $D_1$ and $D_2$ extend to effective Cartier divisors of", "$W$ by Lemma \\ref{lemma-closure-effective-cartier-divisor}", "which gives us the extension of the invertible sheaf.", "\\medskip\\noindent", "If $X$ is Noetherian (which is the case most used in practice), the above", "combined with Noetherian induction finishes the proof. In the general case", "we argue as follows. First, because every local ring of a point outside", "of $U$ is a domain and $X$ is locally Noetherian, we see that the closure", "of $U$ in $X$ is open. Thus we may assume that $U \\subset X$ is dense", "and schematically dense.", "Now we consider the set $T$ of triples $(U', \\mathcal{L}', \\alpha)$", "where $U \\subset U' \\subset X$ is an open subscheme, $\\mathcal{L}'$", "is an invertible $\\mathcal{O}_{U'}$-module, and", "$\\alpha : \\mathcal{L}'|_U \\to \\mathcal{L}$ is an isomorphism.", "We endow $T$ with a partial ordering $\\leq$ defined by the rule", "$(U', \\mathcal{L}', \\alpha) \\leq (U'', \\mathcal{L}'', \\alpha')$", "if and only if $U' \\subset U''$ and there exists an isomorphism", "$\\beta : \\mathcal{L}''|_{U'} \\to \\mathcal{L}'$ compatible with", "$\\alpha$ and $\\alpha'$. Observe that $\\beta$ is unique (if it exists)", "because $U \\subset X$ is dense. The first part of the proof shows that", "for any element $t = (U', \\mathcal{L}', \\alpha)$ of $T$ with $U' \\not = X$", "there exists a $t' \\in T$ with $t' > t$. Hence to finish the proof it", "suffices to show that Zorn's lemma applies. Thus consider a", "totally ordered subset $I \\subset T$. If $i \\in I$ corresponds to", "the triple $(U_i, \\mathcal{L}_i, \\alpha_i)$, then we can construct", "an invertible module $\\mathcal{L}'$ on $U' = \\bigcup U_i$ as follows.", "For $W \\subset U'$ open and quasi-compact we see that", "$W \\subset U_i$ for some $i$ and we set", "$$", "\\mathcal{L}'(W) = \\mathcal{L}_i(W)", "$$", "For the transition maps we use the $\\beta$'s (which are unique and hence", "compose correctly). This defines an invertible $\\mathcal{O}$-module", "$\\mathcal{L}'$ on the basis of quasi-compact opens of $U'$ which is sufficient", "to define an invertible module (Sheaves, Section \\ref{sheaves-section-bases}).", "We omit the details." ], "refs": [ "divisors-lemma-quasi-projective-Noetherian-pic-effective-Cartier", "divisors-lemma-closure-effective-cartier-divisor" ], "ref_ids": [ 7956, 8031 ] } ], "ref_ids": [] }, { "id": 8033, "type": "theorem", "label": "divisors-lemma-open-subscheme-UFD", "categories": [ "divisors" ], "title": "divisors-lemma-open-subscheme-UFD", "contents": [ "Let $R$ be a UFD. The Picard groups of the following are", "trivial.", "\\begin{enumerate}", "\\item $\\Spec(R)$ and any open subscheme of it.", "\\item $\\mathbf{A}^n_R = \\Spec(R[x_1, \\ldots, x_n])$ and any open subscheme", "of it.", "\\end{enumerate}", "In particular, the Picard group of any open subscheme of affine", "$n$-space $\\mathbf{A}^n_k$ over a field $k$ is trivial." ], "refs": [], "proofs": [ { "contents": [ "Since $R$ is a UFD so is any localization of it and any polynomial", "ring over it (Algebra, Lemma \\ref{algebra-lemma-polynomial-ring-UFD}).", "Thus if $U \\subset \\mathbf{A}^n_R$ is open, then the map", "$\\Pic(\\mathbf{A}^n_R) \\to \\Pic(U)$ is surjective", "by Lemma \\ref{lemma-extend-invertible-module}.", "The vanishing of $\\Pic(\\mathbf{A}^n_R)$ is equivalent to", "the vanishing of the picard group of the UFD $R[x_1, \\ldots, x_n]$", "which is proved in", "More on Algebra, Lemma \\ref{more-algebra-lemma-UFD-Pic-trivial}." ], "refs": [ "algebra-lemma-polynomial-ring-UFD", "divisors-lemma-extend-invertible-module", "more-algebra-lemma-UFD-Pic-trivial" ], "ref_ids": [ 1036, 8032, 10534 ] } ], "ref_ids": [] }, { "id": 8034, "type": "theorem", "label": "divisors-lemma-Pic-projective-space-UFD", "categories": [ "divisors" ], "title": "divisors-lemma-Pic-projective-space-UFD", "contents": [ "Let $R$ be a UFD. The Picard group of $\\mathbf{P}^n_R$", "is $\\mathbf{Z}$. More precisely, there is an isomorphism", "$$", "\\mathbf{Z} \\longrightarrow \\Pic(\\mathbf{P}^n_R),\\quad", "m \\longmapsto \\mathcal{O}_{\\mathbf{P}^n_R}(m)", "$$", "In particular, the Picard group of $\\mathbf{P}^n_k$ of projective", "space over a field $k$ is $\\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the local rings of $X = \\mathbf{P}^n_R$ are", "UFDs because $X$ is covered by affine pieces isomorphic", "to $\\mathbf{A}^n_R$ and $R[x_1, \\ldots, x_n]$ is a UFD", "(Algebra, Lemma \\ref{algebra-lemma-polynomial-ring-UFD}).", "Hence $X$ is an integral Noetherian scheme all of whose", "local rings are UFDs and we see that $\\Pic(X) = \\text{Cl}(X)$", "by Lemma \\ref{lemma-local-rings-UFD-c1-bijective}.", "\\medskip\\noindent", "The displayed map is a group homomorphism by", "Constructions, Lemma \\ref{constructions-lemma-apply-modules}.", "The map is injective because $H^0$ of", "$\\mathcal{O}_X$ and $\\mathcal{O}_X(m)$ are non-isomorphic $R$-modules", "if $m > 0$, see Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cohomology-projective-space-over-ring}.", "Let $\\mathcal{L}$ be an invertible module on $X$.", "Consider the open $U = D_+(T_0) \\cong \\mathbf{A}^n_R$.", "The complement $H = X \\setminus U$ is a prime divisor because it is", "isomorphic to $\\text{Proj}(R[T_1, \\ldots, T_n])$ which is", "integral by the discussion in the previous paragraph.", "In fact $H$ is the zero scheme of the regular global", "section $T_0$ of $\\mathcal{O}_X(1)$", "hence $\\mathcal{O}_X(1)$ maps to the class of $H$ in $\\text{Cl}(X)$.", "By Lemma \\ref{lemma-open-subscheme-UFD} we see that", "$\\mathcal{L}|_U \\cong \\mathcal{O}_U$.", "Let $s \\in \\mathcal{L}(U)$ be a trivializing section.", "Then we can think of $s$ as a regular meromorphic section", "of $\\mathcal{L}$ and we see that necessarily", "$\\text{div}_\\mathcal{L}(s) = m[H]$ for some $m \\in \\mathbf{Z}$", "as $H$ is the only prime divisor of $X$ not meeting $U$.", "In other words, we see that $\\mathcal{L}$ and", "$\\mathcal{O}_X(m)$ map to the same element of $\\text{Cl}(X)$", "and hence $\\mathcal{L} \\cong \\mathcal{O}_X(m)$", "as desired." ], "refs": [ "algebra-lemma-polynomial-ring-UFD", "divisors-lemma-local-rings-UFD-c1-bijective", "constructions-lemma-apply-modules", "coherent-lemma-cohomology-projective-space-over-ring", "divisors-lemma-open-subscheme-UFD" ], "ref_ids": [ 1036, 8029, 12603, 3304, 8033 ] } ], "ref_ids": [] }, { "id": 8035, "type": "theorem", "label": "divisors-lemma-reflexive-normal", "categories": [ "divisors" ], "title": "divisors-lemma-reflexive-normal", "contents": [ "Let $X$ be an integral locally Noetherian normal scheme.", "For $\\mathcal{F}$ and $\\mathcal{G}$ coherent reflexive", "$\\mathcal{O}_X$-modules the map", "$$", "(\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X)", "\\otimes_{\\mathcal{O}_X} \\mathcal{G})^{**} \\to", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})", "$$", "is an isomorphism. The rule $\\mathcal{F}, \\mathcal{G} \\mapsto", "(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G})^{**}$", "defines an abelian group law on the set of isomorphism classes of rank $1$", "coherent reflexive $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "Although not strictly necessary, we recommend reading", "Remark \\ref{remark-tensor} before proceeding with the proof.", "Choose an open subscheme $j : U \\to X$ such that", "every irreducible component of $X \\setminus U$ has codimension $\\geq 2$", "in $X$ and such that $j^*\\mathcal{F}$ and $j^*\\mathcal{G}$ are finite", "locally free, see Lemma \\ref{lemma-reflexive-over-normal}.", "The map", "$$", "\\SheafHom_{\\mathcal{O}_U}(j^*\\mathcal{F}, \\mathcal{O}_U)", "\\otimes_{\\mathcal{O}_U} j^*\\mathcal{G} \\to", "\\SheafHom_{\\mathcal{O}_U}(j^*\\mathcal{F}, j^*\\mathcal{G})", "$$", "is an isomorphism, because we may check it locally and it is", "clear when the modules are finite free. Observe that $j^*$", "applied to the displayed arrow of the lemma gives the arrow", "we've just shown is an isomorphism (small detail omitted).", "Since $j^*$ defines an equivalence between coherent reflexive modules on $U$", "and coherent reflexive modules on $X$", "(by Lemma \\ref{lemma-reflexive-S2-extend} and Serre's criterion", "Properties, Lemma \\ref{properties-lemma-criterion-normal}),", "we conclude that the arrow of the lemma is an isomorphism too.", "If $\\mathcal{F}$ has rank $1$, then $j^*\\mathcal{F}$", "is an invertible $\\mathcal{O}_U$-module and the reflexive module", "$\\mathcal{F}^\\vee = \\SheafHom(\\mathcal{F}, \\mathcal{O}_X)$", "restricts to its inverse. It follows in the same manner as before that", "$(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{F}^\\vee)^{**} = \\mathcal{O}_X$.", "In this way we see that we have inverses for the group law", "given in the statement of the lemma." ], "refs": [ "divisors-remark-tensor", "divisors-lemma-reflexive-over-normal", "divisors-lemma-reflexive-S2-extend", "properties-lemma-criterion-normal" ], "ref_ids": [ 8117, 7924, 7923, 2989 ] } ], "ref_ids": [] }, { "id": 8036, "type": "theorem", "label": "divisors-lemma-normal-class-group", "categories": [ "divisors" ], "title": "divisors-lemma-normal-class-group", "contents": [ "Let $X$ be an integral locally Noetherian normal scheme.", "The group of rank $1$ coherent reflexive $\\mathcal{O}_X$-modules", "is isomorphic to the Weil divisor class group $\\text{Cl}(X)$ of $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a rank $1$ coherent reflexive $\\mathcal{O}_X$-module.", "Choose an open $U \\subset X$ such that", "every irreducible component of $X \\setminus U$ has codimension $\\geq 2$", "in $X$ and such that $\\mathcal{F}|_U$ is invertible, see", "Lemma \\ref{lemma-reflexive-over-normal}.", "Observe that $\\text{Cl}(U) = \\text{Cl}(X)$", "as the Weil divisor class group of $X$ only depends on", "its field of rational functions and the points of", "codimension $1$ and their local rings.", "Thus we can define the Weil divisor class of $\\mathcal{F}$", "to be the Weil divisor class of $\\mathcal{F}|_U$", "in $\\text{Cl}(U)$. We omit the verification that this", "is independent of the choice of $U$.", "\\medskip\\noindent", "Denote $\\text{Cl}'(X)$ the set of isomorphism classes of", "rank $1$ coherent reflexive $\\mathcal{O}_X$-modules. The", "construction above gives a group homorphism", "$$", "\\text{Cl}'(X) \\longrightarrow \\text{Cl}(X)", "$$", "because for any pair $\\mathcal{F}, \\mathcal{G}$ of elements", "of $\\text{Cl}'(X)$ we can choose a $U$ which works for both", "and the assignment (\\ref{equation-c1}) sending an invertible", "module to its Weil divisor class is a homorphism.", "If $\\mathcal{F}$ is in the kernel of this map, then we find that", "$\\mathcal{F}|_U$ is trivial (Lemma \\ref{lemma-normal-c1-injective})", "and hence $\\mathcal{F}$ is trivial too by", "Lemma \\ref{lemma-reflexive-S2-extend} and Serre's criterion", "Properties, Lemma \\ref{properties-lemma-criterion-normal}.", "To finish the proof it suffices to check the map is surjective.", "\\medskip\\noindent", "Let $D = \\sum n_Z Z$ be a Weil divisor on $X$.", "We claim that there is an open $U \\subset X$ such that", "every irreducible component of $X \\setminus U$ has codimension $\\geq 2$", "in $X$ and such that $Z|_U$ is an effective Cartier divisor", "for $n_Z \\not = 0$. To prove the claim we may assume $X$ is affine.", "Then we may assume $D = n_1 Z_1 + \\ldots + n_r Z_r$ is a finite sum", "with $Z_1, \\ldots, Z_r$ pairwise distinct. After throwing out", "$Z_i \\cap Z_j$ for $i \\not = j$ we may assume $Z_1, \\ldots, Z_r$", "are pairwise disjoint. This reduces us to the case of a single", "prime divisor $Z$ on $X$. As $X$ is $(R_1)$ by", "Properties, Lemma \\ref{properties-lemma-criterion-normal}", "the local ring", "$\\mathcal{O}_{X, \\xi}$ at the generic point $\\xi$ of $Z$ is a discrete", "valuation ring. Let $f \\in \\mathcal{O}_{X, \\xi}$ be a uniformizer.", "Let $V \\subset X$ be an open neighbourhood of $\\xi$ such that", "$f$ is the image of an element $f \\in \\mathcal{O}_X(V)$.", "After shrinking $V$ we may assume that $Z \\cap V = V(f)$", "scheme theoretically, since this is true in the local ring", "at $\\xi$. In this case taking", "$$", "U = X \\setminus (Z \\setminus V) = (X \\setminus Z) \\cup V", "$$", "gives the desired open, thereby proving the claim.", "\\medskip\\noindent", "In order to show that the divisor class of $D$ is in the image,", "we may write $D = \\sum_{n_Z < 0} n_Z Z - \\sum_{n_Z > 0} (-n_Z) Z$.", "By additivity of the map constructed above, we", "may and do assume $n_Z \\leq 0$ for all prime divisors $Z$", "(this step may be avoided if the reader so desires).", "Let $U \\subset X$ be as in the claim above. If $U$ is quasi-compact,", "then we write $D|_U = -n_1 Z_1 - \\ldots - n_r Z_r$ for", "pairwise distinct prime divisors $Z_i$ and $n_i > 0$ and", "we consider the invertible $\\mathcal{O}_U$-module", "$$", "\\mathcal{L} =", "\\mathcal{I}_1^{n_1} \\ldots \\mathcal{I}_r^{n_r} \\subset \\mathcal{O}_U", "$$", "where $\\mathcal{I}_i$ is the ideal sheaf of $Z_i$.", "This is invertible by our choice of $U$ and", "Lemma \\ref{lemma-sum-effective-Cartier-divisors}.", "Also $\\text{div}_\\mathcal{L}(1) = D|_U$.", "Since $\\mathcal{L} = \\mathcal{F}|_U$ for some rank $1$ coherent", "reflexive $\\mathcal{O}_X$-module $\\mathcal{F}$ by", "Lemma \\ref{lemma-reflexive-S2-extend} we find that $D$ is", "in the image of our map.", "\\medskip\\noindent", "If $U$ is not quasi-compact, then we define", "$\\mathcal{L} \\subset \\mathcal{O}_U$ locally by the displayed formula", "above. The reader shows that the construction glues and", "finishes the proof exactly as before. Details omitted." ], "refs": [ "divisors-lemma-reflexive-over-normal", "divisors-lemma-normal-c1-injective", "divisors-lemma-reflexive-S2-extend", "properties-lemma-criterion-normal", "properties-lemma-criterion-normal", "divisors-lemma-sum-effective-Cartier-divisors", "divisors-lemma-reflexive-S2-extend" ], "ref_ids": [ 7924, 8028, 7923, 2989, 2989, 7931, 7923 ] } ], "ref_ids": [] }, { "id": 8037, "type": "theorem", "label": "divisors-lemma-structure-sheaf-Xs", "categories": [ "divisors" ], "title": "divisors-lemma-structure-sheaf-Xs", "contents": [ "Let $X$ be an integral locally Noetherian normal scheme.", "Let $\\mathcal{F}$ be a rank 1 coherent reflexive $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{F})$. Let", "$$", "U = \\{x \\in X \\mid s : \\mathcal{O}_{X, x} \\to \\mathcal{F}_x", "\\text{ is an isomorphism}\\}", "$$", "Then $j : U \\to X$ is an open subscheme of $X$ and", "$$", "j_*\\mathcal{O}_U =", "\\colim (\\mathcal{O}_X \\xrightarrow{s} \\mathcal{F}", "\\xrightarrow{s} \\mathcal{F}^{[2]}", "\\xrightarrow{s} \\mathcal{F}^{[3]}", "\\xrightarrow{s} \\ldots)", "$$", "where $\\mathcal{F}^{[1]} = \\mathcal{F}$ and", "inductively $\\mathcal{F}^{[n + 1]} =", "(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{F}^{[n]})^{**}$." ], "refs": [], "proofs": [ { "contents": [ "The set $U$ is open by Modules, Lemmas", "\\ref{modules-lemma-finite-type-surjective-on-stalk} and", "\\ref{modules-lemma-finite-type-to-coherent-injective-on-stalk}.", "Observe that $j$ is quasi-compact by", "Properties, Lemma \\ref{properties-lemma-immersion-into-noetherian}.", "To prove the final statement it suffices to show for every", "quasi-compact open $W \\subset X$ there is an isomorphism", "$$", "\\colim \\Gamma(W, \\mathcal{F}^{[n]})", "\\longrightarrow", "\\Gamma(U \\cap W, \\mathcal{O}_U)", "$$", "of $\\mathcal{O}_X(W)$-modules compatible with restriction maps.", "We will omit the verification of compatibilities.", "After replacing $X$ by $W$ and rewriting the above in", "terms of homs, we see that it suffices to construct an isomorphism", "$$", "\\colim \\Hom_{\\mathcal{O}_X}(\\mathcal{O}_X, \\mathcal{F}^{[n]})", "\\longrightarrow", "\\Hom_{\\mathcal{O}_U}(\\mathcal{O}_U, \\mathcal{O}_U)", "$$", "Choose an open $V \\subset X$ such that every irreducible component of", "$X \\setminus V$ has codimension $\\geq 2$ in $X$ and such that", "$\\mathcal{F}|_V$ is invertible, see Lemma \\ref{lemma-reflexive-over-normal}.", "Then restriction defines an equivalence of categories", "between rank $1$ coherent reflexive modules on $X$ and $V$", "and between rank $1$ coherent reflexive modules on $U$ and $V \\cap U$.", "See Lemma \\ref{lemma-reflexive-S2-extend} and Serre's criterion", "Properties, Lemma \\ref{properties-lemma-criterion-normal}.", "Thus it suffices to construct an isomorphism", "$$", "\\colim \\Gamma(V, (\\mathcal{F}|_V)^{\\otimes n}) \\longrightarrow", "\\Gamma(V \\cap U, \\mathcal{O}_U)", "$$", "Since $\\mathcal{F}|_V$ is invertible and since $U \\cap V$ is", "equal to the set of points where $s|_V$ generates this invertible module,", "this is a special case of", "Properties, Lemma \\ref{properties-lemma-invert-s-sections}", "(there is an explicit formula for the map as well)." ], "refs": [ "modules-lemma-finite-type-surjective-on-stalk", "modules-lemma-finite-type-to-coherent-injective-on-stalk", "properties-lemma-immersion-into-noetherian", "divisors-lemma-reflexive-over-normal", "divisors-lemma-reflexive-S2-extend", "properties-lemma-criterion-normal", "properties-lemma-invert-s-sections" ], "ref_ids": [ 13238, 13257, 2952, 7924, 7923, 2989, 3005 ] } ], "ref_ids": [] }, { "id": 8038, "type": "theorem", "label": "divisors-lemma-Xs-codim-complement", "categories": [ "divisors" ], "title": "divisors-lemma-Xs-codim-complement", "contents": [ "Assumptions and notation as in Lemma \\ref{lemma-structure-sheaf-Xs}.", "If $s$ is nonzero, then every irreducible component of $X \\setminus U$", "has codimension $1$ in $X$." ], "refs": [ "divisors-lemma-structure-sheaf-Xs" ], "proofs": [ { "contents": [ "Let $\\xi \\in X$ be a generic point of an irreducible component $Z$ of", "$X \\setminus U$. After replacing $X$ by an open neighbourhood of", "$\\xi$ we may assume that $Z = X \\setminus U$ is irreducible. Since", "$s : \\mathcal{O}_U \\to \\mathcal{F}|_U$ is an isomorphism, if", "the codimension of $Z$ in $X$ is $\\geq 2$, then", "$s : \\mathcal{O}_X \\to \\mathcal{F}$ is an isomorphism by", "Lemma \\ref{lemma-reflexive-S2-extend} and Serre's criterion", "Properties, Lemma \\ref{properties-lemma-criterion-normal}.", "This would mean that $Z = \\emptyset$, a contradiction." ], "refs": [ "divisors-lemma-reflexive-S2-extend", "properties-lemma-criterion-normal" ], "ref_ids": [ 7923, 2989 ] } ], "ref_ids": [ 8037 ] }, { "id": 8039, "type": "theorem", "label": "divisors-lemma-affine-Xs", "categories": [ "divisors" ], "title": "divisors-lemma-affine-Xs", "contents": [ "Assumptions and notation as in Lemma \\ref{lemma-structure-sheaf-Xs}.", "The following are equivalent", "\\begin{enumerate}", "\\item the inclusion morphism $j : U \\to X$ is affine, and", "\\item for every $x \\in X \\setminus U$ there is an $n > 0$", "such that $s^n \\in \\mathfrak m_x \\mathcal{F}^{[n]}_x$.", "\\end{enumerate}" ], "refs": [ "divisors-lemma-structure-sheaf-Xs" ], "proofs": [ { "contents": [ "Assume (1). Then for $x \\in X \\setminus U$ the inverse image $U_x$ of $U$", "under the canonical morphism $f_x : \\Spec(\\mathcal{O}_{X, x}) \\to X$ is affine", "and does not contain $x$. Thus $\\mathfrak m_x \\Gamma(U_x, \\mathcal{O}_{U_x})$", "is the unit ideal. In particular, we see that we can write", "$$", "1 = \\sum f_i g_i", "$$", "with $f_i \\in \\mathfrak m_x$ and $g_i \\in \\Gamma(U_x, \\mathcal{O}_{U_x})$.", "By Lemma \\ref{lemma-structure-sheaf-Xs} we have", "$\\Gamma(U_x, \\mathcal{O}_{U_x}) = \\colim \\mathcal{F}^{[n]}_x$", "with transition maps given by multiplication by $s$.", "Hence for some $n > 0$ we have", "$$", "s^n = \\sum f_i t_i", "$$", "for some $t_i = s^ng_i \\in \\mathcal{F}^{[n]}_x$. Thus (2) holds.", "\\medskip\\noindent", "Conversely, assume that (2) holds. To prove $j$ is affine is local on $X$,", "see Morphisms, Lemma \\ref{morphisms-lemma-characterize-affine}.", "Thus we may and do assume that $X$ is affine. Our goal is to", "show that $U$ is affine.", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-affine-if-quasi-affine}", "it suffices to show that $H^p(U, \\mathcal{O}_U) = 0$ for $p > 0$.", "Since $H^p(U, \\mathcal{O}_U) = H^0(X, R^pj_*\\mathcal{O}_U)$", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images-application})", "and since $R^pj_*\\mathcal{O}_U$ is quasi-coherent", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images})", "it is enough to show the stalk $(R^pj_*\\mathcal{O}_U)_x$", "at a point $x \\in X$ is zero. Consider the base change diagram", "$$", "\\xymatrix{", "U_x \\ar[d]_{j_x} \\ar[r] & U \\ar[d]^j \\\\", "\\Spec(\\mathcal{O}_{X, x}) \\ar[r] & X", "}", "$$", "By Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology} we have", "$(R^pj_*\\mathcal{O}_U)_x = R^pj_{x, *}\\mathcal{O}_{U_x}$.", "Hence we may assume $X$ is local with closed point $x$", "and we have to show $U$ is affine (because this is equivalent to", "the desired vanishing by the reference given above).", "In particular $d = \\dim(X)$ is finite", "(Algebra, Proposition \\ref{algebra-proposition-dimension}).", "If $x \\in U$, then $U = X$ and the result is clear.", "If $d = 0$ and $x \\not \\in U$, then $U = \\emptyset$", "and the result is clear. Now assume $d > 0$ and $x \\not \\in U$.", "Since $j_*\\mathcal{O}_U = \\colim \\mathcal{F}^{[n]}$", "our assumption means that we can write", "$$", "1 = \\sum f_i g_i", "$$", "for some $n > 0$, $f_i \\in \\mathfrak m_x$, and $g_i \\in \\mathcal{O}(U)$.", "By induction on $d$ we know that $D(f_i) \\cap U$ is affine", "for all $i$: going through the whole argument just given with", "$X$ replaced by $D(f_i)$ we end up with Noetherian local rings", "whose dimension is strictly smaller than $d$. Hence $U$", "is affine by Properties, Lemma \\ref{properties-lemma-characterize-affine}", "as desired." ], "refs": [ "divisors-lemma-structure-sheaf-Xs", "morphisms-lemma-characterize-affine", "coherent-lemma-affine-if-quasi-affine", "coherent-lemma-quasi-coherence-higher-direct-images-application", "coherent-lemma-quasi-coherence-higher-direct-images", "coherent-lemma-flat-base-change-cohomology", "algebra-proposition-dimension", "properties-lemma-characterize-affine" ], "ref_ids": [ 8037, 5172, 3353, 3296, 3295, 3298, 1411, 3055 ] } ], "ref_ids": [ 8037 ] }, { "id": 8040, "type": "theorem", "label": "divisors-lemma-relative-proj-quasi-compact", "categories": [ "divisors" ], "title": "divisors-lemma-relative-proj-quasi-compact", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded", "$\\mathcal{O}_S$-algebra. Let", "$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative", "Proj of $\\mathcal{A}$. If one of the following holds", "\\begin{enumerate}", "\\item $\\mathcal{A}$ is of finite type as a sheaf of", "$\\mathcal{A}_0$-algebras,", "\\item $\\mathcal{A}$ is generated by $\\mathcal{A}_1$ as an", "$\\mathcal{A}_0$-algebra and $\\mathcal{A}_1$ is a finite type", "$\\mathcal{A}_0$-module,", "\\item there exists a finite type quasi-coherent $\\mathcal{A}_0$-submodule", "$\\mathcal{F} \\subset \\mathcal{A}_{+}$ such that", "$\\mathcal{A}_{+}/\\mathcal{F}\\mathcal{A}$ is a locally nilpotent", "sheaf of ideals of $\\mathcal{A}/\\mathcal{F}\\mathcal{A}$,", "\\end{enumerate}", "then $p$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "The question is local on the base, see", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-affine}.", "Thus we may assume $S$ is affine.", "Say $S = \\Spec(R)$ and $\\mathcal{A}$ corresponds to the", "graded $R$-algebra $A$. Then $X = \\text{Proj}(A)$, see", "Constructions, Section \\ref{constructions-section-relative-proj-via-glueing}.", "In case (1) we may after possibly localizing more", "assume that $A$ is generated by homogeneous elements", "$f_1, \\ldots, f_n \\in A_{+}$ over $A_0$. Then", "$A_{+} = (f_1, \\ldots, f_n)$ by", "Algebra, Lemma \\ref{algebra-lemma-S-plus-generated}.", "In case (3) we see that $\\mathcal{F} = \\widetilde{M}$", "for some finite type $A_0$-module $M \\subset A_{+}$. Say", "$M = \\sum A_0f_i$. Say $f_i = \\sum f_{i, j}$ is the decomposition", "into homogeneous pieces. The condition in (3) signifies that", "$A_{+} \\subset \\sqrt{(f_{i, j})}$. Thus in both cases we conclude that", "$\\text{Proj}(A)$ is quasi-compact by", "Constructions, Lemma \\ref{constructions-lemma-proj-quasi-compact}.", "Finally, (2) follows from (1)." ], "refs": [ "schemes-lemma-quasi-compact-affine", "algebra-lemma-S-plus-generated", "constructions-lemma-proj-quasi-compact" ], "ref_ids": [ 7697, 667, 12598 ] } ], "ref_ids": [] }, { "id": 8041, "type": "theorem", "label": "divisors-lemma-relative-proj-finite-type", "categories": [ "divisors" ], "title": "divisors-lemma-relative-proj-finite-type", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded", "$\\mathcal{O}_S$-algebra. Let", "$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative", "Proj of $\\mathcal{A}$. If $\\mathcal{A}$ is of finite type as a sheaf of", "$\\mathcal{O}_S$-algebras, then $p$ is of finite type and $\\mathcal{O}_X(d)$", "is a finite type $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "The assumption implies that $p$ is quasi-compact, see", "Lemma \\ref{lemma-relative-proj-quasi-compact}. Hence it suffices", "to show that $p$ is locally of finite type.", "Thus the question is local on the base and target, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-finite-type-characterize}.", "Say $S = \\Spec(R)$ and $\\mathcal{A}$ corresponds to the", "graded $R$-algebra $A$. After further localizing on $S$ we may", "assume that $A$ is a finite type $R$-algebra. The scheme $X$ is constructed", "out of glueing the spectra of the rings $A_{(f)}$ for $f \\in A_{+}$", "homogeneous. Each of these is of finite type over $R$ by", "Algebra, Lemma \\ref{algebra-lemma-dehomogenize-finite-type} part (1).", "Thus $\\text{Proj}(A)$ is of finite type over $R$.", "To see the statement on $\\mathcal{O}_X(d)$ use part (2) of", "Algebra, Lemma \\ref{algebra-lemma-dehomogenize-finite-type}." ], "refs": [ "divisors-lemma-relative-proj-quasi-compact", "morphisms-lemma-locally-finite-type-characterize", "algebra-lemma-dehomogenize-finite-type", "algebra-lemma-dehomogenize-finite-type" ], "ref_ids": [ 8040, 5198, 665, 665 ] } ], "ref_ids": [] }, { "id": 8042, "type": "theorem", "label": "divisors-lemma-relative-proj-universally-closed", "categories": [ "divisors" ], "title": "divisors-lemma-relative-proj-universally-closed", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded", "$\\mathcal{O}_S$-algebra. Let", "$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative", "Proj of $\\mathcal{A}$. If $\\mathcal{O}_S \\to \\mathcal{A}_0$", "is an integral algebra map\\footnote{In other words, the integral", "closure of $\\mathcal{O}_S$ in $\\mathcal{A}_0$, see", "Morphisms, Definition \\ref{morphisms-definition-integral-closure}, equals", "$\\mathcal{A}_0$.} and $\\mathcal{A}$ is of finite type as an", "$\\mathcal{A}_0$-algebra, then $p$ is universally closed." ], "refs": [ "morphisms-definition-integral-closure" ], "proofs": [ { "contents": [ "The question is local on the base. Thus we may assume that $X = \\Spec(R)$", "is affine. Let $\\mathcal{A}$ be the quasi-coherent $\\mathcal{O}_X$-algebra", "associated to the graded $R$-algebra $A$. The assumption is that $R \\to A_0$", "is integral and $A$ is of finite type over $A_0$.", "Write $X \\to \\Spec(R)$ as the composition $X \\to \\Spec(A_0) \\to \\Spec(R)$.", "Since $R \\to A_0$ is an integral ring map, we see that", "$\\Spec(A_0) \\to \\Spec(R)$ is universally closed, see", "Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed}.", "The quasi-compact (see", "Constructions, Lemma \\ref{constructions-lemma-proj-quasi-compact}) morphism", "$$", "X = \\text{Proj}(A) \\to \\Spec(A_0)", "$$", "satisfies the existence part of the valuative criterion by", "Constructions, Lemma \\ref{constructions-lemma-proj-valuative-criterion}", "and hence it is universally closed by", "Schemes, Proposition \\ref{schemes-proposition-characterize-universally-closed}.", "Thus $X \\to \\Spec(R)$ is universally closed as a composition of", "universally closed morphisms." ], "refs": [ "morphisms-lemma-integral-universally-closed", "constructions-lemma-proj-quasi-compact", "constructions-lemma-proj-valuative-criterion", "schemes-proposition-characterize-universally-closed" ], "ref_ids": [ 5441, 12598, 12600, 7733 ] } ], "ref_ids": [ 5590 ] }, { "id": 8043, "type": "theorem", "label": "divisors-lemma-relative-proj-proper", "categories": [ "divisors" ], "title": "divisors-lemma-relative-proj-proper", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded", "$\\mathcal{O}_S$-algebra. Let", "$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative", "Proj of $\\mathcal{A}$. The following conditions are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{A}_0$ is a finite type $\\mathcal{O}_S$-module", "and $\\mathcal{A}$ is of finite type as an $\\mathcal{A}_0$-algebra,", "\\item $\\mathcal{A}_0$ is a finite type $\\mathcal{O}_S$-module", "and $\\mathcal{A}$ is of finite type as an $\\mathcal{O}_S$-algebra", "\\end{enumerate}", "If these conditions hold, then $p$ is locally projective and in", "particular proper." ], "refs": [], "proofs": [ { "contents": [ "Assume that $\\mathcal{A}_0$ is a finite type $\\mathcal{O}_S$-module.", "Choose an affine open $U = \\Spec(R) \\subset X$ such that $\\mathcal{A}$", "corresponds to a graded $R$-algebra $A$ with $A_0$ a finite $R$-module.", "Condition (1) means that (after possibly localizing further on $S$)", "that $A$ is a finite type $A_0$-algebra and condition (2) means that", "(after possibly localizing further on $S$) that $A$ is a finite type", "$R$-algebra. Thus these conditions imply each other by", "Algebra, Lemma \\ref{algebra-lemma-compose-finite-type}.", "\\medskip\\noindent", "A locally projective morphism is proper, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-projective-proper}.", "Thus we may now assume that $S = \\Spec(R)$ and $X = \\text{Proj}(A)$", "and that $A_0$ is finite over $R$ and $A$ of finite type over $R$.", "We will show that $X = \\text{Proj}(A) \\to \\Spec(R)$ is projective.", "We urge the reader to prove this for themselves, by directly constructing", "a closed immersion of $X$ into a projective space over $R$, instead", "of reading the argument we give below.", "\\medskip\\noindent", "By Lemma \\ref{lemma-relative-proj-finite-type}", "we see that $X$ is of finite type over $\\Spec(R)$.", "Constructions, Lemma \\ref{constructions-lemma-ample-on-proj}", "tells us that $\\mathcal{O}_X(d)$ is ample on $X$ for some $d \\geq 1$", "(see Properties, Section \\ref{properties-section-ample}).", "Hence $X \\to \\Spec(R)$ is quasi-projective (by", "Morphisms, Definition \\ref{morphisms-definition-quasi-projective}).", "By Morphisms, Lemma \\ref{morphisms-lemma-quasi-projective-open-projective}", "we conclude that $X$ is isomorphic to an open subscheme of a scheme", "projective over $\\Spec(R)$. Therefore, to finish the proof, it suffices", "to show that $X \\to \\Spec(R)$ is universally closed (use", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}).", "This follows from Lemma \\ref{lemma-relative-proj-universally-closed}." ], "refs": [ "algebra-lemma-compose-finite-type", "morphisms-lemma-locally-projective-proper", "divisors-lemma-relative-proj-finite-type", "constructions-lemma-ample-on-proj", "morphisms-definition-quasi-projective", "morphisms-lemma-quasi-projective-open-projective", "morphisms-lemma-image-proper-scheme-closed", "divisors-lemma-relative-proj-universally-closed" ], "ref_ids": [ 333, 5422, 8041, 12606, 5570, 5429, 5411, 8042 ] } ], "ref_ids": [] }, { "id": 8044, "type": "theorem", "label": "divisors-lemma-relative-proj-projective", "categories": [ "divisors" ], "title": "divisors-lemma-relative-proj-projective", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded", "$\\mathcal{O}_S$-algebra. Let", "$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative", "Proj of $\\mathcal{A}$. If $\\mathcal{A}$ is generated by", "$\\mathcal{A}_1$ over $\\mathcal{A}_0$ and $\\mathcal{A}_1$", "is a finite type $\\mathcal{O}_S$-module, then $p$ is projective." ], "refs": [], "proofs": [ { "contents": [ "Namely, the morphism associated to the graded $\\mathcal{O}_S$-algebra map", "$$", "\\text{Sym}_{\\mathcal{O}_X}^*(\\mathcal{A}_1)", "\\longrightarrow", "\\mathcal{A}", "$$", "is a closed immersion $X \\to \\mathbf{P}(\\mathcal{A}_1)$, see", "Constructions, Lemma", "\\ref{constructions-lemma-surjective-generated-degree-1-map-relative-proj}." ], "refs": [ "constructions-lemma-surjective-generated-degree-1-map-relative-proj" ], "ref_ids": [ 12648 ] } ], "ref_ids": [] }, { "id": 8045, "type": "theorem", "label": "divisors-lemma-relative-proj-flat", "categories": [ "divisors" ], "title": "divisors-lemma-relative-proj-flat", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded", "$\\mathcal{O}_S$-algebra. Let", "$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative", "Proj of $\\mathcal{A}$. If $\\mathcal{A}_d$ is a flat $\\mathcal{O}_S$-module", "for $d \\gg 0$, then $p$ is flat and $\\mathcal{O}_X(d)$ is", "flat over $S$." ], "refs": [], "proofs": [ { "contents": [ "Affine locally flatness of $X$ over $S$ reduces to the following statement:", "Let $R$ be a ring, let $A$ be a graded $R$-algebra with", "$A_d$ flat over $R$ for $d \\gg 0$, let $f \\in A_d$", "for some $d > 0$, then $A_{(f)}$ is flat over $R$.", "Since $A_{(f)} = \\colim A_{nd}$ where the transition maps", "are given by multiplication by $f$, this follows from", "Algebra, Lemma \\ref{algebra-lemma-colimit-flat}.", "Argue similarly to get flatness of $\\mathcal{O}_X(d)$ over $S$." ], "refs": [ "algebra-lemma-colimit-flat" ], "ref_ids": [ 523 ] } ], "ref_ids": [] }, { "id": 8046, "type": "theorem", "label": "divisors-lemma-relative-proj-finite-presentation", "categories": [ "divisors" ], "title": "divisors-lemma-relative-proj-finite-presentation", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded", "$\\mathcal{O}_S$-algebra. Let", "$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative", "Proj of $\\mathcal{A}$. If $\\mathcal{A}$ is a finitely presented", "$\\mathcal{O}_S$-algebra, then $p$ is of finite presentation", "and $\\mathcal{O}_X(d)$ is an $\\mathcal{O}_X$-module of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Affine locally this reduces to the following statement:", "Let $R$ be a ring and let $A$ be a finitely presented graded $R$-algebra.", "Then $\\text{Proj}(A) \\to \\Spec(R)$ is of finite presentation", "and $\\mathcal{O}_{\\text{Proj}(A)}(d)$ is a", "$\\mathcal{O}_{\\text{Proj}(A)}$-module of finite presentation.", "The finite presentation condition implies we can choose", "a presentation", "$$", "A = R[X_1, \\ldots, X_n]/(F_1, \\ldots, F_m)", "$$", "where $R[X_1, \\ldots, X_n]$ is a polynomial ring graded by giving", "weights $d_i$ to $X_i$ and $F_1, \\ldots, F_m$ are homogeneous polynomials", "of degree $e_j$. Let $R_0 \\subset R$ be the subring generated by", "the coefficients of the polynomials $F_1, \\ldots, F_m$.", "Then we set $A_0 = R_0[X_1, \\ldots, X_n]/(F_1, \\ldots, F_m)$.", "By construction $A = A_0 \\otimes_{R_0} R$.", "Thus by", "Constructions, Lemma \\ref{constructions-lemma-base-change-map-proj}", "it suffices to prove the result for $X_0 = \\text{Proj}(A_0)$ over $R_0$.", "By Lemma \\ref{lemma-relative-proj-finite-type}", "we know $X_0$ is of finite type over $R_0$ and", "$\\mathcal{O}_{X_0}(d)$ is a quasi-coherent $\\mathcal{O}_{X_0}$-module", "of finite type.", "Since $R_0$ is Noetherian (as a finitely generated $\\mathbf{Z}$-algebra)", "we see that $X_0$ is of finite presentation over $R_0$", "(Morphisms, Lemma", "\\ref{morphisms-lemma-noetherian-finite-type-finite-presentation})", "and $\\mathcal{O}_{X_0}(d)$ is of finite presentation by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-Noetherian}.", "This finishes the proof." ], "refs": [ "constructions-lemma-base-change-map-proj", "divisors-lemma-relative-proj-finite-type", "morphisms-lemma-noetherian-finite-type-finite-presentation", "coherent-lemma-coherent-Noetherian" ], "ref_ids": [ 12613, 8041, 5245, 3308 ] } ], "ref_ids": [] }, { "id": 8047, "type": "theorem", "label": "divisors-lemma-closed-subscheme-proj", "categories": [ "divisors" ], "title": "divisors-lemma-closed-subscheme-proj", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded", "$\\mathcal{O}_S$-algebra. Let", "$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative", "Proj of $\\mathcal{A}$. Let $i : Z \\to X$ be a closed subscheme. Denote", "$\\mathcal{I} \\subset \\mathcal{A}$ the kernel of the canonical map", "$$", "\\mathcal{A}", "\\longrightarrow", "\\bigoplus\\nolimits_{d \\geq 0} p_*\\left((i_*\\mathcal{O}_Z)(d)\\right)", "$$", "If $p$ is quasi-compact, then there is an isomorphism", "$Z = \\underline{\\text{Proj}}_S(\\mathcal{A}/\\mathcal{I})$." ], "refs": [], "proofs": [ { "contents": [ "The morphism $p$ is separated by", "Constructions, Lemma \\ref{constructions-lemma-relative-proj-separated}.", "As $p$ is quasi-compact, $p_*$ transforms quasi-coherent modules", "into quasi-coherent modules, see", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}.", "Hence $\\mathcal{I}$ is a quasi-coherent $\\mathcal{O}_S$-module.", "In particular, $\\mathcal{B} = \\mathcal{A}/\\mathcal{I}$ is a", "quasi-coherent graded $\\mathcal{O}_S$-algebra. The functoriality", "morphism $Z' = \\underline{\\text{Proj}}_S(\\mathcal{B}) \\to", "\\underline{\\text{Proj}}_S(\\mathcal{A})$ is everywhere defined and", "a closed immersion, see Constructions, Lemma", "\\ref{constructions-lemma-surjective-graded-rings-map-relative-proj}.", "Hence it suffices to prove $Z = Z'$ as closed subschemes of $X$.", "\\medskip\\noindent", "Having said this, the question is local on the base and we may assume", "that $S = \\Spec(R)$ and that $X = \\text{Proj}(A)$ for some", "graded $R$-algebra $A$. Assume $\\mathcal{I} = \\widetilde{I}$", "for $I \\subset A$ a graded ideal. By", "Constructions, Lemma \\ref{constructions-lemma-proj-quasi-compact}", "there exist $f_0, \\ldots, f_n \\in A_{+}$ such that", "$A_{+} \\subset \\sqrt{(f_0, \\ldots, f_n)}$ in other words", "$X = \\bigcup D_{+}(f_i)$. Therefore, it suffices to check that", "$Z \\cap D_{+}(f_i) = Z' \\cap D_{+}(f_i)$ for each $i$.", "By renumbering we may assume $i = 0$.", "Say $Z \\cap D_{+}(f_0)$, resp.\\ $Z' \\cap D_{+}(f_0)$", "is cut out by the ideal $J$, resp.\\ $J'$ of $A_{(f_0)}$.", "\\medskip\\noindent", "The inclusion $J' \\subset J$.", "Let $d$ be the least common multiple of $\\deg(f_0), \\ldots, \\deg(f_n)$.", "Note that each of the twists $\\mathcal{O}_X(nd)$ is invertible, trivialized", "by $f_i^{nd/\\deg(f_i)}$ over $D_{+}(f_i)$, and that for any quasi-coherent", "module $\\mathcal{F}$ on $X$ the multiplication maps", "$\\mathcal{O}_X(nd) \\otimes_{\\mathcal{O}_X} \\mathcal{F}(m)", "\\to \\mathcal{F}(nd + m)$ are isomorphisms, see", "Constructions, Lemma \\ref{constructions-lemma-when-invertible}.", "Observe that $J'$ is the ideal generated by the elements $g/f_0^e$ where", "$g \\in I$ is homogeneous of degree $e\\deg(f_0)$ (see proof of", "Constructions, Lemma", "\\ref{constructions-lemma-surjective-graded-rings-map-proj}).", "Of course, by replacing $g$ by $f_0^lg$ for suitable $l$", "we may always assume that $d | e$. Then, since $g$ vanishes as a section of", "$\\mathcal{O}_X(e\\deg(f_0))$ restricted to $Z$ we see that", "$g/f_0^d$ is an element of $J$. Thus $J' \\subset J$.", "\\medskip\\noindent", "Conversely, suppose that $g/f_0^e \\in J$. Again we may assume $d | e$.", "Pick $i \\in \\{1, \\ldots, n\\}$. Then $Z \\cap D_{+}(f_i)$ is", "cut out by some ideal $J_i \\subset A_{(f_i)}$. Moreover,", "$$", "J \\cdot A_{(f_0f_i)} = J_i \\cdot A_{(f_0f_i)}", "$$", "The right hand side is the localization of $J_i$ with respect to", "$f_0^{\\deg(f_i)}/f_i^{\\deg(f_0)}$. It follows that", "$$", "f_0^{e_i}g/f_i^{(e_i + e)\\deg(f_0)/\\deg(f_i)} \\in J_i", "$$", "for some $e_i \\gg 0$ sufficiently divisible. This proves that", "$f_0^{\\max(e_i)}g$ is an element of $I$, because its restriction to each", "affine open $D_{+}(f_i)$ vanishes on the closed subscheme", "$Z \\cap D_{+}(f_i)$. Hence $g \\in J'$ and we conclude $J \\subset J'$", "as desired." ], "refs": [ "constructions-lemma-relative-proj-separated", "schemes-lemma-push-forward-quasi-coherent", "constructions-lemma-surjective-graded-rings-map-relative-proj", "constructions-lemma-proj-quasi-compact", "constructions-lemma-when-invertible", "constructions-lemma-surjective-graded-rings-map-proj" ], "ref_ids": [ 12640, 7730, 12646, 12598, 12602, 12610 ] } ], "ref_ids": [] }, { "id": 8048, "type": "theorem", "label": "divisors-lemma-equation-codim-1-in-projective-space", "categories": [ "divisors" ], "title": "divisors-lemma-equation-codim-1-in-projective-space", "contents": [ "Let $R$ be a UFD. Let $Z \\subset \\mathbf{P}^n_R$ be a closed subscheme", "which has no embedded points such that every irreducible component", "of $Z$ has codimension $1$ in $\\mathbf{P}^n_R$.", "Then the ideal $I(Z) \\subset R[T_0, \\ldots, T_n]$ corresponding", "to $Z$ is principal." ], "refs": [], "proofs": [ { "contents": [ "Observe that the local rings of $X = \\mathbf{P}^n_R$ are", "UFDs because $X$ is covered by affine pieces isomorphic", "to $\\mathbf{A}^n_R$ and $R[x_1, \\ldots, x_n]$ is a UFD", "(Algebra, Lemma \\ref{algebra-lemma-polynomial-ring-UFD}).", "Thus $Z$ is an effective Cartier divisor by", "Lemma \\ref{lemma-codimension-1-is-effective-Cartier}.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent", "sheaf of ideals corresponding to $Z$.", "Choose an isomorphism $\\mathcal{O}(m) \\to \\mathcal{I}$", "for some $m \\in \\mathbf{Z}$, see", "Lemma \\ref{lemma-Pic-projective-space-UFD}.", "Then the composition", "$$", "\\mathcal{O}_X(m) \\to \\mathcal{I} \\to \\mathcal{O}_X", "$$", "is nonzero. We conclude that $m \\leq 0$ and that the corresponding", "section of $\\mathcal{O}_X(m)^{\\otimes -1} = \\mathcal{O}_X(-m)$", "is given by some $F \\in R[T_0, \\ldots, T_n]$ of degree $-m$, see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cohomology-projective-space-over-ring}.", "Thus on the $i$th standard open $U_i = D_+(T_i)$ the", "closed subscheme $Z \\cap U_i$ is cut out by the ideal", "$$", "(F(T_0/T_i, \\ldots, T_n/T_i)) \\subset R[T_0/T_i, \\ldots, T_n/T_i]", "$$", "Thus the homogeneous elements of the graded ideal", "$I(Z) = \\Ker(R[T_0, \\ldots, T_n] \\to \\bigoplus \\Gamma(\\mathcal{O}_Z(m)))$", "is the set of homogeneous polynomials $G$ such that", "$$", "G(T_0/T_i, \\ldots, T_n/T_i) \\in (F(T_0/T_i, \\ldots, T_n/T_i))", "$$", "for $i = 0, \\ldots, n$. Clearing denominators, we see there exist", "$e_i \\geq 0$ such that", "$$", "T_i^{e_i}G \\in (F)", "$$", "for $i = 0, \\ldots, n$. As $R$ is a UFD, so is $R[T_0, \\ldots, T_n]$.", "Then $F | T_0^{e_0}G$ and $F | T_1^{e_1}G$ implies $F | G$ as", "$T_0^{e_0}$ and $T_1^{e_1}$ have no factor in common. Thus $I(Z) = (F)$." ], "refs": [ "algebra-lemma-polynomial-ring-UFD", "divisors-lemma-codimension-1-is-effective-Cartier", "divisors-lemma-Pic-projective-space-UFD", "coherent-lemma-cohomology-projective-space-over-ring" ], "ref_ids": [ 1036, 7953, 8034, 3304 ] } ], "ref_ids": [] }, { "id": 8049, "type": "theorem", "label": "divisors-lemma-closed-subscheme-proj-finite", "categories": [ "divisors" ], "title": "divisors-lemma-closed-subscheme-proj-finite", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra. Let", "$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative", "Proj of $\\mathcal{A}$. Let $i : Z \\to X$ be a closed subscheme.", "If $p$ is quasi-compact and $i$ of finite presentation, then there exists", "a $d > 0$ and a quasi-coherent finite type $\\mathcal{O}_S$-submodule", "$\\mathcal{F} \\subset \\mathcal{A}_d$ such that", "$Z = \\underline{\\text{Proj}}_S(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-closed-subscheme-proj} we know there exists a", "quasi-coherent graded sheaf of ideals $\\mathcal{I} \\subset \\mathcal{A}$", "such that $Z = \\underline{\\text{Proj}}(\\mathcal{A}/\\mathcal{I})$.", "Since $S$ is quasi-compact we can choose a finite affine open covering", "$S = U_1 \\cup \\ldots \\cup U_n$. Say $U_i = \\Spec(R_i)$. Let", "$\\mathcal{A}|_{U_i}$ correspond to the graded $R_i$-algebra $A_i$ and", "$\\mathcal{I}|_{U_i}$ to the graded ideal $I_i \\subset A_i$. Note that", "$p^{-1}(U_i) = \\text{Proj}(A_i)$ as schemes over $R_i$.", "Since $p$ is quasi-compact we can choose finitely many homogeneous", "elements $f_{i, j} \\in A_{i, +}$ such that $p^{-1}(U_i) = D_{+}(f_{i, j})$.", "The condition on $Z \\to X$ means that the ideal sheaf of $Z$ in", "$\\mathcal{O}_X$ is of finite type, see", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-finite-presentation}.", "Hence we can find finitely many homogeneous elements", "$h_{i, j, k} \\in I_i \\cap A_{i, +}$ such that the ideal of", "$Z \\cap D_{+}(f_{i, j})$ is generated by the elements", "$h_{i, j, k}/f_{i, j}^{e_{i, j, k}}$. Choose $d > 0$ to be a common multiple", "of all the integers $\\deg(f_{i, j})$ and $\\deg(h_{i, j, k})$.", "By Properties, Lemma \\ref{properties-lemma-quasi-coherent-colimit-finite-type}", "there exists a finite type quasi-coherent $\\mathcal{F} \\subset \\mathcal{I}_d$", "such that all the local sections", "$$", "h_{i, j, k}f_{i, j}^{(d - \\deg(h_{i, j, k}))/\\deg(f_{i, j})}", "$$", "are sections of $\\mathcal{F}$. By construction $\\mathcal{F}$ is a solution." ], "refs": [ "divisors-lemma-closed-subscheme-proj", "morphisms-lemma-closed-immersion-finite-presentation", "properties-lemma-quasi-coherent-colimit-finite-type" ], "ref_ids": [ 8047, 5243, 3020 ] } ], "ref_ids": [] }, { "id": 8050, "type": "theorem", "label": "divisors-lemma-closed-subscheme-proj-finite-type", "categories": [ "divisors" ], "title": "divisors-lemma-closed-subscheme-proj-finite-type", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra.", "Let $p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative", "Proj of $\\mathcal{A}$. Let $i : Z \\to X$ be a closed subscheme.", "Let $U \\subset X$ be an open. Assume that", "\\begin{enumerate}", "\\item $p$ is quasi-compact,", "\\item $i$ of finite presentation,", "\\item $U \\cap p(i(Z)) = \\emptyset$,", "\\item $U$ is quasi-compact,", "\\item $\\mathcal{A}_n$ is a finite type $\\mathcal{O}_S$-module for all $n$.", "\\end{enumerate}", "Then there exists a $d > 0$ and a quasi-coherent finite type", "$\\mathcal{O}_S$-submodule $\\mathcal{F} \\subset \\mathcal{A}_d$ with (a)", "$Z = \\underline{\\text{Proj}}_S(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$", "and (b) the support of $\\mathcal{A}_d/\\mathcal{F}$ is disjoint from $U$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\subset \\mathcal{A}$ be the sheaf of quasi-coherent", "graded ideals constructed in Lemma \\ref{lemma-closed-subscheme-proj}.", "Let $U_i$, $R_i$, $A_i$, $I_i$, $f_{i, j}$, $h_{i, j, k}$, and $d$", "be as constructed in the proof of", "Lemma \\ref{lemma-closed-subscheme-proj-finite}.", "Since $U \\cap p(i(Z)) = \\emptyset$ we see that", "$\\mathcal{I}_d|_U = \\mathcal{A}_d|_U$ (by our construction of", "$\\mathcal{I}$ as a kernel). Since $U$ is quasi-compact we", "can choose a finite affine open covering $U = W_1 \\cup \\ldots \\cup W_m$.", "Since $\\mathcal{A}_d$ is of finite type we can find finitely many sections", "$g_{t, s} \\in \\mathcal{A}_d(W_t)$ which generate", "$\\mathcal{A}_d|_{W_t} = \\mathcal{I}_d|_{W_t}$", "as an $\\mathcal{O}_{W_t}$-module. To finish the proof, note that by", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-colimit-finite-type}", "there exists a finite type $\\mathcal{F} \\subset \\mathcal{I}_d$", "such that all the local sections", "$$", "h_{i, j, k}f_{i, j}^{(d - \\deg(h_{i, j, k}))/\\deg(f_{i, j})}", "\\quad\\text{and}\\quad", "g_{t, s}", "$$", "are sections of $\\mathcal{F}$. By construction $\\mathcal{F}$ is a solution." ], "refs": [ "divisors-lemma-closed-subscheme-proj", "divisors-lemma-closed-subscheme-proj-finite", "properties-lemma-quasi-coherent-colimit-finite-type" ], "ref_ids": [ 8047, 8049, 3020 ] } ], "ref_ids": [] }, { "id": 8051, "type": "theorem", "label": "divisors-lemma-conormal-sheaf-section-projective-bundle", "categories": [ "divisors" ], "title": "divisors-lemma-conormal-sheaf-section-projective-bundle", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{E}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. There is a bijection", "$$", "\\left\\{", "\\begin{matrix}", "\\text{sections }\\sigma\\text{ of the } \\\\", "\\text{morphism } \\mathbf{P}(\\mathcal{E}) \\to X", "\\end{matrix}", "\\right\\}", "\\leftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{surjections }\\mathcal{E} \\to \\mathcal{L}\\text{ where} \\\\", "\\mathcal{L}\\text{ is an invertible }\\mathcal{O}_X\\text{-module}", "\\end{matrix}", "\\right\\}", "$$", "In this case $\\sigma$ is a closed immersion and there is a canonical", "isomorphism", "$$", "\\Ker(\\mathcal{E} \\to \\mathcal{L})", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes -1}", "\\longrightarrow", "\\mathcal{C}_{\\sigma(X)/\\mathbf{P}(\\mathcal{E})}", "$$", "Both the bijection and isomorphism are compatible with base change." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\pi : \\mathbf{P}(\\mathcal{E}) \\to X$ is the relative proj of the", "symmetric algebra on $\\mathcal{E}$, see", "Constructions, Definition \\ref{constructions-definition-projective-bundle}.", "Hence the descriptions of sections $\\sigma$ follows immediately from", "the description of the functor of points of $\\mathbf{P}(\\mathcal{E})$", "in Constructions, Lemma \\ref{constructions-lemma-apply-relative}.", "Since $\\pi$ is separated, any section is a closed immersion", "(Constructions, Lemma \\ref{constructions-lemma-relative-proj-separated} and", "Schemes, Lemma \\ref{schemes-lemma-section-immersion}).", "Let $U \\subset X$ be an affine open and $k \\in \\mathcal{E}(U)$ and", "$s \\in \\mathcal{E}(U)$ be local sections such that $k$ maps to", "zero in $\\mathcal{L}$ and $s$ maps to a generator $\\overline{s}$", "of $\\mathcal{L}$.", "Then $f = k/s$ is a section of $\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}$", "defined in an open neighbourhood $D_+(s)$ of $s(U)$ in $\\pi^{-1}(U)$.", "Moreover, since $k$ maps to zero in $\\mathcal{L}$ we see that", "$f$ is a section of the ideal sheaf of $s(U)$ in $\\pi^{-1}(U)$.", "Thus we can take the image $\\overline{f}$ of $f$ in", "$\\mathcal{C}_{\\sigma(X)/\\mathbf{P}(\\mathcal{E})}(U)$.", "We claim (1) that the image $\\overline{f}$ depends only on the", "sections $k$ and $\\overline{s}$ and not on the choice of $s$", "and (2) that we get an isomorphism over $U$ in this manner (see below).", "However, once (1) and (2) are established, we see that", "the construction is compatible with base change by $U' \\to U$", "where $U'$ is affine, which proves that these local maps glue", "and are compatible with arbitrary base change.", "\\medskip\\noindent", "To prove (1) and (2) we make explicit what is going on.", "Namely, say $U = \\Spec(A)$ and say $\\mathcal{E} \\to \\mathcal{L}$", "corresponds to the map of $A$-modules $M \\to N$. Then", "$k \\in K = \\Ker(M \\to N)$ and $s \\in M$ maps to a generator $\\overline{s}$", "of $N$. Hence $M = K \\oplus A s$. Thus", "$$", "\\text{Sym}(M) = \\text{Sym}(K)[s]", "$$", "Consider the identification $\\text{Sym}(K) \\to \\text{Sym}(M)_{(s)}$", "via the rule $g \\mapsto g/s^n$ for $g \\in \\text{Sym}^n(K)$.", "This gives an isomorphism $D_+(s) = \\Spec(\\text{Sym}(K))$ such", "that $\\sigma$ corresponds to the ring map $\\text{Sym}(K) \\to A$", "mapping $K$ to zero. Via this isomorphism we see that the quasi-coherent", "module corresponding to $K$ is identified with", "$\\mathcal{C}_{\\sigma(U)/D_+(s)}$ proving (2).", "Finally, suppose that $s' = k' + s$ for some $k' \\in K$.", "Then", "$$", "k/s' = (k/s) (s/s') = (k/s) (s'/s)^{-1} = (k/s) (1 + k'/s)^{-1}", "$$", "in an open neighbourhood of $\\sigma(U)$ in $D_+(s)$. Thus we see that", "$s'/s$ restricts to $1$ on $\\sigma(U)$ and we see that $k/s'$ maps to", "the same element of the conormal sheaf as does $k/s$ thereby proving (1)." ], "refs": [ "constructions-definition-projective-bundle", "constructions-lemma-apply-relative", "constructions-lemma-relative-proj-separated" ], "ref_ids": [ 12666, 12642, 12640 ] } ], "ref_ids": [] }, { "id": 8052, "type": "theorem", "label": "divisors-lemma-blowing-up-affine", "categories": [ "divisors" ], "title": "divisors-lemma-blowing-up-affine", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a", "quasi-coherent sheaf of ideals. Let $U = \\Spec(A)$ be an affine open", "subscheme of $X$ and let $I \\subset A$ be the ideal corresponding to", "$\\mathcal{I}|_U$. If $b : X' \\to X$ is the blowup of $X$ in $\\mathcal{I}$,", "then there is a canonical isomorphism", "$$", "b^{-1}(U) = \\text{Proj}(\\bigoplus\\nolimits_{d \\geq 0} I^d)", "$$", "of $b^{-1}(U)$ with the homogeneous spectrum of the Rees algebra", "of $I$ in $A$. Moreover, $b^{-1}(U)$ has an affine open covering by", "spectra of the affine blowup algebras $A[\\frac{I}{a}]$." ], "refs": [], "proofs": [ { "contents": [ "The first statement is clear from the construction of the relative Proj via", "glueing, see Constructions, Section", "\\ref{constructions-section-relative-proj-via-glueing}.", "For $a \\in I$ denote $a^{(1)}$ the element $a$ seen as an element of", "degree $1$ in the Rees algebra $\\bigoplus_{n \\geq 0} I^n$.", "Since these elements generate the Rees algebra over $A$ we see that", "$\\text{Proj}(\\bigoplus_{d \\geq 0} I^d)$ is covered by the affine opens", "$D_{+}(a^{(1)})$. The affine scheme $D_{+}(a^{(1)})$ is the spectrum of", "the affine blowup algebra $A' = A[\\frac{I}{a}]$, see", "Algebra, Definition \\ref{algebra-definition-blow-up}.", "This finishes the proof." ], "refs": [ "algebra-definition-blow-up" ], "ref_ids": [ 1488 ] } ], "ref_ids": [] }, { "id": 8053, "type": "theorem", "label": "divisors-lemma-flat-base-change-blowing-up", "categories": [ "divisors" ], "title": "divisors-lemma-flat-base-change-blowing-up", "contents": [ "\\begin{slogan}", "Blowing up commutes with flat base change.", "\\end{slogan}", "Let $X_1 \\to X_2$ be a flat morphism of schemes. Let $Z_2 \\subset X_2$ be a", "closed subscheme. Let $Z_1$ be the inverse image of $Z_2$ in $X_1$.", "Let $X'_i$ be the blowup of $Z_i$ in $X_i$. Then there exists a cartesian", "diagram", "$$", "\\xymatrix{", "X_1' \\ar[r] \\ar[d] & X_2' \\ar[d] \\\\", "X_1 \\ar[r] & X_2", "}", "$$", "of schemes." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I}_2$ be the ideal sheaf of $Z_2$ in $X_2$.", "Denote $g : X_1 \\to X_2$ the given morphism. Then the ideal sheaf", "$\\mathcal{I}_1$ of $Z_1$ is the image of", "$g^*\\mathcal{I}_2 \\to \\mathcal{O}_{X_1}$", "(by definition of the inverse image, see", "Schemes, Definition \\ref{schemes-definition-inverse-image-closed-subscheme}).", "By Constructions, Lemma \\ref{constructions-lemma-relative-proj-base-change}", "we see that $X_1 \\times_{X_2} X_2'$ is the relative Proj of", "$\\bigoplus_{n \\geq 0} g^*\\mathcal{I}_2^n$. Because $g$ is flat the map", "$g^*\\mathcal{I}_2^n \\to \\mathcal{O}_{X_1}$ is injective with image", "$\\mathcal{I}_1^n$. Thus we see that $X_1 \\times_{X_2} X_2' = X_1'$." ], "refs": [ "schemes-definition-inverse-image-closed-subscheme", "constructions-lemma-relative-proj-base-change" ], "ref_ids": [ 7749, 12641 ] } ], "ref_ids": [] }, { "id": 8054, "type": "theorem", "label": "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "categories": [ "divisors" ], "title": "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "contents": [ "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme.", "The blowing up $b : X' \\to X$ of $Z$ in $X$", "has the following properties:", "\\begin{enumerate}", "\\item $b|_{b^{-1}(X \\setminus Z)} : b^{-1}(X \\setminus Z) \\to X \\setminus Z$", "is an isomorphism,", "\\item the exceptional divisor $E = b^{-1}(Z)$ is an effective Cartier divisor", "on $X'$,", "\\item there is a canonical isomorphism", "$\\mathcal{O}_{X'}(-1) = \\mathcal{O}_{X'}(E)$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As blowing up commutes with restrictions to open subschemes", "(Lemma \\ref{lemma-flat-base-change-blowing-up}) the first statement", "just means that $X' = X$ if $Z = \\emptyset$. In this case we are blowing", "up in the ideal sheaf $\\mathcal{I} = \\mathcal{O}_X$ and the result follows from", "Constructions, Example \\ref{constructions-example-trivial-proj}.", "\\medskip\\noindent", "The second statement is local on $X$, hence we may assume $X$ affine.", "Say $X = \\Spec(A)$ and $Z = \\Spec(A/I)$. By Lemma \\ref{lemma-blowing-up-affine}", "we see that $X'$ is covered by the spectra of the affine blowup algebras", "$A' = A[\\frac{I}{a}]$. Then $IA' = aA'$ and $a$ maps to a nonzerodivisor", "in $A'$ according to Algebra, Lemma \\ref{algebra-lemma-affine-blowup}.", "This proves the lemma as the inverse image of $Z$ in $\\Spec(A')$", "corresponds to $\\Spec(A'/IA') \\subset \\Spec(A')$.", "\\medskip\\noindent", "Consider the canonical map", "$\\psi_{univ, 1} : b^*\\mathcal{I} \\to \\mathcal{O}_{X'}(1)$, see", "discussion following Constructions, Definition", "\\ref{constructions-definition-relative-proj}.", "We claim that this factors through an isomorphism", "$\\mathcal{I}_E \\to \\mathcal{O}_{X'}(1)$ (which proves the final assertion).", "Namely, on the affine open corresponding to the blowup algebra", "$A' = A[\\frac{I}{a}]$ mentioned above $\\psi_{univ, 1}$ corresponds to", "the $A'$-module map", "$$", "I \\otimes_A A'", "\\longrightarrow", "\\left(\\Big(\\bigoplus\\nolimits_{d \\geq 0} I^d\\Big)_{a^{(1)}}\\right)_1", "$$", "where $a^{(1)}$ is as in Algebra, Definition \\ref{algebra-definition-blow-up}.", "We omit the verification that this is the map", "$I \\otimes_A A' \\to IA' = aA'$." ], "refs": [ "divisors-lemma-flat-base-change-blowing-up", "divisors-lemma-blowing-up-affine", "algebra-lemma-affine-blowup", "constructions-definition-relative-proj", "algebra-definition-blow-up" ], "ref_ids": [ 8053, 8052, 752, 12665, 1488 ] } ], "ref_ids": [] }, { "id": 8055, "type": "theorem", "label": "divisors-lemma-universal-property-blowing-up", "categories": [ "divisors" ], "title": "divisors-lemma-universal-property-blowing-up", "contents": [ "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme.", "Let $\\mathcal{C}$ be the full subcategory of $(\\Sch/X)$ consisting", "of $Y \\to X$ such that the inverse image of $Z$ is an effective", "Cartier divisor on $Y$. Then the blowing up $b : X' \\to X$ of $Z$ in $X$", "is a final object of $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "We see that $b : X' \\to X$ is an object of $\\mathcal{C}$ according to", "Lemma \\ref{lemma-blowing-up-gives-effective-Cartier-divisor}.", "Let $f : Y \\to X$ be an object of $\\mathcal{C}$. We have to show there exists", "a unique morphism $Y \\to X'$ over $X$. Let $D = f^{-1}(Z)$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the ideal sheaf of $Z$", "and let $\\mathcal{I}_D$ be the ideal sheaf of $D$. Then", "$f^*\\mathcal{I} \\to \\mathcal{I}_D$ is a surjection", "to an invertible $\\mathcal{O}_Y$-module. This extends to a map", "$\\psi : \\bigoplus f^*\\mathcal{I}^d \\to \\bigoplus \\mathcal{I}_D^d$", "of graded $\\mathcal{O}_Y$-algebras. (We observe that", "$\\mathcal{I}_D^d = \\mathcal{I}_D^{\\otimes d}$ as $D$ is an", "effective Cartier divisor.) By the material in", "Constructions, Section \\ref{constructions-section-relative-proj}", "the triple $(1, f : Y \\to X, \\psi)$ defines a morphism $Y \\to X'$ over $X$.", "The restriction", "$$", "Y \\setminus D \\longrightarrow X' \\setminus b^{-1}(Z) = X \\setminus Z", "$$", "is unique. The open $Y \\setminus D$ is scheme theoretically dense in $Y$", "according to Lemma \\ref{lemma-complement-effective-Cartier-divisor}.", "Thus the morphism $Y \\to X'$ is unique by", "Morphisms, Lemma \\ref{morphisms-lemma-equality-of-morphisms}", "(also $b$ is separated by Constructions, Lemma", "\\ref{constructions-lemma-relative-proj-separated})." ], "refs": [ "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "divisors-lemma-complement-effective-Cartier-divisor", "morphisms-lemma-equality-of-morphisms", "constructions-lemma-relative-proj-separated" ], "ref_ids": [ 8054, 7929, 5157, 12640 ] } ], "ref_ids": [] }, { "id": 8056, "type": "theorem", "label": "divisors-lemma-characterize-affine-blowup", "categories": [ "divisors" ], "title": "divisors-lemma-characterize-affine-blowup", "contents": [ "Let $b : X' \\to X$ be the blowing up of the scheme $X$ along a closed", "subscheme $Z$. Let $U = \\Spec(A)$ be an affine open of $X$ and let", "$I \\subset A$ be the ideal corresponding to $Z \\cap U$.", "Let $a \\in I$ and let $x' \\in X'$ be a point mapping to a point of $U$.", "Then $x'$ is a point of the affine open $U' = \\Spec(A[\\frac{I}{a}])$", "if and only if the image of $a$ in $\\mathcal{O}_{X', x'}$ cuts", "out the exceptional divisor." ], "refs": [], "proofs": [ { "contents": [ "Since the exceptional divisor over $U'$ is cut out by the image of", "$a$ in $A' = A[\\frac{I}{a}]$ one direction is clear. Conversely, assume", "that the image of $a$ in $\\mathcal{O}_{X', x'}$ cuts out $E$.", "Since every element of $I$ maps to an element of the ideal", "defining $E$ over $b^{-1}(U)$ we see that elements of $I$ become", "divisible by $a$ in $\\mathcal{O}_{X', x'}$. Thus for $f \\in I^n$", "we can write $f = \\psi(f) a^n$ for some $\\psi(f) \\in \\mathcal{O}_{X', x'}$.", "Observe that since $a$ maps to a nonzerodivisor of $\\mathcal{O}_{X', x'}$", "the element $\\psi(f)$ is uniquely characterized by this. Then we", "define", "$$", "A' \\longrightarrow \\mathcal{O}_{X', x'},\\quad", "f/a^n \\longmapsto \\psi(f)", "$$", "Here we use the description of blowup algebras given following", "Algebra, Definition \\ref{definition-blow-up}. The uniqueness mentioned", "above shows that this is an $A$-algebra homomorphism.", "This gives a morphism $\\Spec(\\mathcal{O}_{X', x\"}) \\to \\Spec(A') = U'$.", "By the universal property of blowing up", "(Lemma \\ref{lemma-universal-property-blowing-up})", "this is a morphism over", "$X'$, which of course implies that $x' \\in U'$." ], "refs": [ "divisors-definition-blow-up", "divisors-lemma-universal-property-blowing-up" ], "ref_ids": [ 8112, 8055 ] } ], "ref_ids": [] }, { "id": 8057, "type": "theorem", "label": "divisors-lemma-blow-up-effective-Cartier-divisor", "categories": [ "divisors" ], "title": "divisors-lemma-blow-up-effective-Cartier-divisor", "contents": [ "Let $X$ be a scheme. Let $Z \\subset X$ be an effective Cartier divisor.", "The blowup of $X$ in $Z$ is the identity morphism of $X$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the universal property of blowups", "(Lemma \\ref{lemma-universal-property-blowing-up})." ], "refs": [ "divisors-lemma-universal-property-blowing-up" ], "ref_ids": [ 8055 ] } ], "ref_ids": [] }, { "id": 8058, "type": "theorem", "label": "divisors-lemma-blow-up-reduced-scheme", "categories": [ "divisors" ], "title": "divisors-lemma-blow-up-reduced-scheme", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a", "quasi-coherent sheaf of ideals. If $X$ is reduced, then the", "blowup $X'$ of $X$ in $\\mathcal{I}$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-blowing-up-affine}", "with Algebra, Lemma \\ref{algebra-lemma-blowup-reduced}." ], "refs": [ "divisors-lemma-blowing-up-affine", "algebra-lemma-blowup-reduced" ], "ref_ids": [ 8052, 757 ] } ], "ref_ids": [] }, { "id": 8059, "type": "theorem", "label": "divisors-lemma-blow-up-integral-scheme", "categories": [ "divisors" ], "title": "divisors-lemma-blow-up-integral-scheme", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a", "nonzero quasi-coherent sheaf of ideals. If $X$ is integral, then the", "blowup $X'$ of $X$ in $\\mathcal{I}$ is integral." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-blowing-up-affine}", "with Algebra, Lemma \\ref{algebra-lemma-blowup-domain}." ], "refs": [ "divisors-lemma-blowing-up-affine", "algebra-lemma-blowup-domain" ], "ref_ids": [ 8052, 758 ] } ], "ref_ids": [] }, { "id": 8060, "type": "theorem", "label": "divisors-lemma-blow-up-and-irreducible-components", "categories": [ "divisors" ], "title": "divisors-lemma-blow-up-and-irreducible-components", "contents": [ "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme.", "Let $b : X' \\to X$ be the blowing up of $X$ along $Z$. Then", "$b$ induces an bijective map from the set of generic points", "of irreducible components of $X'$ to the set of generic points of", "irreducible components of $X$ which are not in $Z$." ], "refs": [], "proofs": [ { "contents": [ "The exceptional divisor $E \\subset X'$ is an effective Cartier divisor", "(Lemma \\ref{lemma-blowing-up-gives-effective-Cartier-divisor})", "hence is nowhere dense in $X'$", "(Lemma \\ref{lemma-complement-effective-Cartier-divisor}).", "On the other hand, $X' \\setminus E \\to X \\setminus Z$ is an", "isomorphism. The lemma follows." ], "refs": [ "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "divisors-lemma-complement-effective-Cartier-divisor" ], "ref_ids": [ 8054, 7929 ] } ], "ref_ids": [] }, { "id": 8061, "type": "theorem", "label": "divisors-lemma-blow-up-pullback-effective-Cartier", "categories": [ "divisors" ], "title": "divisors-lemma-blow-up-pullback-effective-Cartier", "contents": [ "Let $X$ be a scheme. Let $b : X' \\to X$ be a blowup of $X$ in a closed", "subscheme. The pullback $b^{-1}D$ is defined", "for all effective Cartier divisors $D \\subset X$", "and pullbacks of meromorphic functions are defined for $b$", "(Definitions", "\\ref{definition-pullback-effective-Cartier-divisor} and", "\\ref{definition-pullback-meromorphic-sections})." ], "refs": [ "divisors-definition-pullback-effective-Cartier-divisor", "divisors-definition-pullback-meromorphic-sections" ], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-blowing-up-affine} and", "\\ref{lemma-characterize-effective-Cartier-divisor}", "this reduces to the following algebra fact:", "Let $A$ be a ring, $I \\subset A$ an ideal, $a \\in I$, and $x \\in A$", "a nonzerodivisor. Then the image of $x$ in $A[\\frac{I}{a}]$ is a", "nonzerodivisor. Namely, suppose that $x (y/a^n) = 0$ in $A[\\frac{I}{a}]$.", "Then $a^mxy = 0$ in $A$ for some $m$. Hence $a^my = 0$ as $x$ is a", "nonzerodivisor. Whence $y/a^n$ is zero in $A[\\frac{I}{a}]$ as desired." ], "refs": [ "divisors-lemma-blowing-up-affine", "divisors-lemma-characterize-effective-Cartier-divisor" ], "ref_ids": [ 8052, 7927 ] } ], "ref_ids": [ 8091, 8103 ] }, { "id": 8062, "type": "theorem", "label": "divisors-lemma-blowing-up-two-ideals", "categories": [ "divisors" ], "title": "divisors-lemma-blowing-up-two-ideals", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{I}, \\mathcal{J} \\subset \\mathcal{O}_X$", "be quasi-coherent sheaves of ideals. Let $b : X' \\to X$", "be the blowing up of $X$ in $\\mathcal{I}$. Let $b' : X'' \\to X'$ be the", "blowing up of $X'$ in $b^{-1}\\mathcal{J} \\mathcal{O}_{X'}$. Then $X'' \\to X$", "is canonically isomorphic to the blowing up of $X$ in $\\mathcal{I}\\mathcal{J}$." ], "refs": [], "proofs": [ { "contents": [ "Let $E \\subset X'$ be the exceptional divisor of $b$ which is an effective", "Cartier divisor by", "Lemma \\ref{lemma-blowing-up-gives-effective-Cartier-divisor}.", "Then $(b')^{-1}E$ is an effective Cartier divisor on $X''$ by", "Lemma \\ref{lemma-blow-up-pullback-effective-Cartier}.", "Let $E' \\subset X''$ be the exceptional divisor of $b'$ (also an effective", "Cartier divisor). Consider the effective Cartier divisor", "$E'' = E' + (b')^{-1}E$. By construction the ideal of $E''$ is", "$(b \\circ b')^{-1}\\mathcal{I} (b \\circ b')^{-1}\\mathcal{J} \\mathcal{O}_{X''}$.", "Hence according to Lemma \\ref{lemma-universal-property-blowing-up}", "there is a canonical morphism from $X''$ to the blowup $c : Y \\to X$", "of $X$ in $\\mathcal{I}\\mathcal{J}$. Conversely, as $\\mathcal{I}\\mathcal{J}$", "pulls back to an invertible ideal we see that", "$c^{-1}\\mathcal{I}\\mathcal{O}_Y$ defines", "an effective Cartier divisor, see", "Lemma \\ref{lemma-sum-closed-subschemes-effective-Cartier}.", "Thus a morphism $c' : Y \\to X'$ over $X$ by", "Lemma \\ref{lemma-universal-property-blowing-up}.", "Then $(c')^{-1}b^{-1}\\mathcal{J}\\mathcal{O}_Y = c^{-1}\\mathcal{J}\\mathcal{O}_Y$", "which also defines an effective Cartier divisor. Thus a morphism", "$c'' : Y \\to X''$ over $X'$. We omit the verification that this", "morphism is inverse to the morphism $X'' \\to Y$ constructed earlier." ], "refs": [ "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "divisors-lemma-blow-up-pullback-effective-Cartier", "divisors-lemma-universal-property-blowing-up", "divisors-lemma-sum-closed-subschemes-effective-Cartier", "divisors-lemma-universal-property-blowing-up" ], "ref_ids": [ 8054, 8061, 8055, 7933, 8055 ] } ], "ref_ids": [] }, { "id": 8063, "type": "theorem", "label": "divisors-lemma-blowing-up-projective", "categories": [ "divisors" ], "title": "divisors-lemma-blowing-up-projective", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a", "quasi-coherent sheaf of ideals. Let $b : X' \\to X$ be the blowing up of $X$", "in the ideal sheaf $\\mathcal{I}$. If $\\mathcal{I}$ is of finite type, then", "\\begin{enumerate}", "\\item $b : X' \\to X$ is a projective morphism, and", "\\item $\\mathcal{O}_{X'}(1)$ is a $b$-relatively ample invertible sheaf.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The surjection of graded $\\mathcal{O}_X$-algebras", "$$", "\\text{Sym}_{\\mathcal{O}_X}^*(\\mathcal{I})", "\\longrightarrow", "\\bigoplus\\nolimits_{d \\geq 0} \\mathcal{I}^d", "$$", "defines via Constructions, Lemma", "\\ref{constructions-lemma-surjective-generated-degree-1-map-relative-proj}", "a closed immersion", "$$", "X' = \\underline{\\text{Proj}}_X (\\bigoplus\\nolimits_{d \\geq 0} \\mathcal{I}^d)", "\\longrightarrow", "\\mathbf{P}(\\mathcal{I}).", "$$", "Hence $b$ is projective, see", "Morphisms, Definition \\ref{morphisms-definition-projective}.", "The second statement follows for example from the characterization", "of relatively ample invertible sheaves in", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-relatively-ample}.", "Some details omitted." ], "refs": [ "constructions-lemma-surjective-generated-degree-1-map-relative-proj", "morphisms-definition-projective", "morphisms-lemma-characterize-relatively-ample" ], "ref_ids": [ 12648, 5572, 5380 ] } ], "ref_ids": [] }, { "id": 8064, "type": "theorem", "label": "divisors-lemma-composition-finite-type-blowups", "categories": [ "divisors" ], "title": "divisors-lemma-composition-finite-type-blowups", "contents": [ "\\begin{slogan}", "Composition of blowing ups is a blowing up", "\\end{slogan}", "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $Z \\subset X$ be a closed subscheme of finite presentation.", "Let $b : X' \\to X$ be the blowing up with center $Z$. Let $Z' \\subset X'$ be", "a closed subscheme of finite presentation.", "Let $X'' \\to X'$ be the blowing up with center $Z'$.", "There exists a closed subscheme $Y \\subset X$ of finite presentation,", "such that", "\\begin{enumerate}", "\\item $Y = Z \\cup b(Z')$ set theoretically, and", "\\item the composition $X'' \\to X$ is isomorphic to the blowing up", "of $X$ in $Y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The condition that $Z \\to X$ is of finite presentation means that", "$Z$ is cut out by a finite type quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_X$, see", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-finite-presentation}.", "Write $\\mathcal{A} = \\bigoplus_{n \\geq 0} \\mathcal{I}^n$ so that", "$X' = \\underline{\\text{Proj}}(\\mathcal{A})$.", "Note that $X \\setminus Z$ is a quasi-compact open of $X$ by", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-finite-type-ideals}.", "Since $b^{-1}(X \\setminus Z) \\to X \\setminus Z$ is an isomorphism", "(Lemma \\ref{lemma-blowing-up-gives-effective-Cartier-divisor}) the same", "result shows that", "$b^{-1}(X \\setminus Z) \\setminus Z'$ is quasi-compact open in $X'$.", "Hence $U = X \\setminus (Z \\cup b(Z'))$ is quasi-compact open in $X$.", "By Lemma \\ref{lemma-closed-subscheme-proj-finite-type}", "there exist a $d > 0$ and a finite type", "$\\mathcal{O}_X$-submodule $\\mathcal{F} \\subset \\mathcal{I}^d$ such", "that $Z' = \\underline{\\text{Proj}}(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$", "and such that the support of $\\mathcal{I}^d/\\mathcal{F}$ is contained", "in $X \\setminus U$.", "\\medskip\\noindent", "Since $\\mathcal{F} \\subset \\mathcal{I}^d$ is an $\\mathcal{O}_X$-submodule", "we may think of $\\mathcal{F} \\subset \\mathcal{I}^d \\subset \\mathcal{O}_X$", "as a finite type quasi-coherent sheaf of ideals on $X$. Let's denote this", "$\\mathcal{J} \\subset \\mathcal{O}_X$ to prevent confusion. Since", "$\\mathcal{I}^d / \\mathcal{J}$ and $\\mathcal{O}/\\mathcal{I}^d$", "are supported on $X \\setminus U$ we see that $V(\\mathcal{J})$ is contained", "in $X \\setminus U$. Conversely, as $\\mathcal{J} \\subset \\mathcal{I}^d$", "we see that $Z \\subset V(\\mathcal{J})$. Over", "$X \\setminus Z \\cong X' \\setminus b^{-1}(Z)$ the sheaf of ideals", "$\\mathcal{J}$ cuts out $Z'$ (see displayed formula below). Hence", "$V(\\mathcal{J})$ equals $Z \\cup b(Z')$. It follows that also", "$V(\\mathcal{I}\\mathcal{J}) = Z \\cup b(Z')$ set theoretically. Moreover,", "$\\mathcal{I}\\mathcal{J}$ is an ideal of finite type as a product of two such.", "We claim that $X'' \\to X$ is isomorphic to the blowing up of $X$ in", "$\\mathcal{I}\\mathcal{J}$ which finishes the proof of the lemma by setting", "$Y = V(\\mathcal{I}\\mathcal{J})$.", "\\medskip\\noindent", "First, recall that the blowup of $X$ in $\\mathcal{I}\\mathcal{J}$", "is the same as the blowup of $X'$ in $b^{-1}\\mathcal{J} \\mathcal{O}_{X'}$,", "see Lemma \\ref{lemma-blowing-up-two-ideals}.", "Hence it suffices to show that the blowup of $X'$ in", "$b^{-1}\\mathcal{J} \\mathcal{O}_{X'}$ agrees with the blowup of $X'$", "in $Z'$. We will show that", "$$", "b^{-1}\\mathcal{J} \\mathcal{O}_{X'} = \\mathcal{I}_E^d \\mathcal{I}_{Z'}", "$$", "as ideal sheaves on $X''$. This will prove what we want as", "$\\mathcal{I}_E^d$ cuts out the effective Cartier divisor $dE$", "and we can use Lemmas \\ref{lemma-blow-up-effective-Cartier-divisor} and", "\\ref{lemma-blowing-up-two-ideals}.", "\\medskip\\noindent", "To see the displayed equality of the ideals we may work locally.", "With notation $A$, $I$, $a \\in I$ as in Lemma \\ref{lemma-blowing-up-affine}", "we see that $\\mathcal{F}$ corresponds to an $R$-submodule $M \\subset I^d$", "mapping isomorphically to an ideal $J \\subset R$. The condition", "$Z' = \\underline{\\text{Proj}}(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$", "means that $Z' \\cap \\Spec(A[\\frac{I}{a}])$ is cut out by the ideal", "generated by the elements $m/a^d$, $m \\in M$. Say the element $m \\in M$", "corresponds to the function $f \\in J$. Then in the affine blowup algebra", "$A' = A[\\frac{I}{a}]$ we see that $f = (a^dm)/a^d = a^d (m/a^d)$.", "Thus the equality holds." ], "refs": [ "morphisms-lemma-closed-immersion-finite-presentation", "properties-lemma-quasi-coherent-finite-type-ideals", "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "divisors-lemma-closed-subscheme-proj-finite-type", "divisors-lemma-blowing-up-two-ideals", "divisors-lemma-blow-up-effective-Cartier-divisor", "divisors-lemma-blowing-up-two-ideals", "divisors-lemma-blowing-up-affine" ], "ref_ids": [ 5243, 3033, 8054, 8050, 8062, 8057, 8062, 8052 ] } ], "ref_ids": [] }, { "id": 8065, "type": "theorem", "label": "divisors-lemma-strict-transform", "categories": [ "divisors" ], "title": "divisors-lemma-strict-transform", "contents": [ "In the situation of Definition \\ref{definition-strict-transform}.", "\\begin{enumerate}", "\\item The strict transform $X'$ of $X$ is the blowup of $X$ in the closed", "subscheme $f^{-1}Z$ of $X$.", "\\item For a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the", "strict transform $\\mathcal{F}'$ is canonically isomorphic to", "the pushforward along $X' \\to X \\times_S S'$ of the strict transform of", "$\\mathcal{F}$ relative to the blowing up $X' \\to X$.", "\\end{enumerate}" ], "refs": [ "divisors-definition-strict-transform" ], "proofs": [ { "contents": [ "Let $X'' \\to X$ be the blowup of $X$ in $f^{-1}Z$. By the universal", "property of blowing up (Lemma \\ref{lemma-universal-property-blowing-up})", "there exists a commutative diagram", "$$", "\\xymatrix{", "X'' \\ar[r] \\ar[d] & X \\ar[d] \\\\", "S' \\ar[r] & S", "}", "$$", "whence a morphism $X'' \\to X \\times_S S'$. Thus the first assertion", "is that this morphism is a closed immersion with image $X'$.", "The question is local on $X$. Thus we may assume $X$", "and $S$ are affine. Say that $S = \\Spec(A)$, $X = \\Spec(B)$, and $Z$", "is cut out by the ideal $I \\subset A$. Set $J = IB$. The map", "$B \\otimes_A \\bigoplus_{n \\geq 0} I^n \\to \\bigoplus_{n \\geq 0} J^n$", "defines a closed immersion $X'' \\to X \\times_S S'$, see", "Constructions, Lemmas", "\\ref{constructions-lemma-base-change-map-proj} and", "\\ref{constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj}.", "We omit the verification that this morphism is the same as the", "one constructed above from the universal property.", "Pick $a \\in I$ corresponding to the affine open", "$\\Spec(A[\\frac{I}{a}]) \\subset S'$, see Lemma \\ref{lemma-blowing-up-affine}.", "The inverse image of $\\Spec(A[\\frac{I}{a}])$ in the strict transform", "$X'$ of $X$ is the spectrum of", "$$", "B' = (B \\otimes_A A[\\textstyle{\\frac{I}{a}}])/a\\text{-power-torsion}", "$$", "see Properties, Lemma", "\\ref{properties-lemma-sections-supported-on-closed-subset}.", "On the other hand, letting $b \\in J$ be the image of $a$ we see that", "$\\Spec(B[\\frac{J}{b}])$ is the inverse image of $\\Spec(A[\\frac{I}{a}])$", "in $X''$. By Algebra, Lemma \\ref{algebra-lemma-blowup-base-change}", "the open $\\Spec(B[\\frac{J}{b}])$ maps isomorphically to the open subscheme", "$\\text{pr}_{S'}^{-1}(\\Spec(A[\\frac{I}{a}]))$ of $X'$.", "Thus $X'' \\to X'$ is an isomorphism.", "\\medskip\\noindent", "In the notation above, let $\\mathcal{F}$ correspond to the $B$-module $N$.", "The strict transform of $\\mathcal{F}$ corresponds to the", "$B \\otimes_A A[\\frac{I}{a}]$-module", "$$", "N' = (N \\otimes_A A[\\textstyle{\\frac{I}{a}}])/a\\text{-power-torsion}", "$$", "see Properties, Lemma", "\\ref{properties-lemma-sections-supported-on-closed-subset}.", "The strict transform of $\\mathcal{F}$ relative to the blowup of", "$X$ in $f^{-1}Z$ corresponds to the $B[\\frac{J}{b}]$-module", "$N \\otimes_B B[\\frac{J}{b}]/b\\text{-power-torsion}$. In exactly the same", "way as above one proves that these two modules are isomorphic.", "Details omitted." ], "refs": [ "divisors-lemma-universal-property-blowing-up", "constructions-lemma-base-change-map-proj", "constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj", "divisors-lemma-blowing-up-affine", "properties-lemma-sections-supported-on-closed-subset", "algebra-lemma-blowup-base-change", "properties-lemma-sections-supported-on-closed-subset" ], "ref_ids": [ 8055, 12613, 12612, 8052, 3036, 753, 3036 ] } ], "ref_ids": [ 8113 ] }, { "id": 8066, "type": "theorem", "label": "divisors-lemma-strict-transform-flat", "categories": [ "divisors" ], "title": "divisors-lemma-strict-transform-flat", "contents": [ "In the situation of Definition \\ref{definition-strict-transform}.", "\\begin{enumerate}", "\\item If $X$ is flat over $S$ at all points lying over $Z$, then", "the strict transform of $X$ is equal to the base change $X \\times_S S'$.", "\\item Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "If $\\mathcal{F}$ is flat over $S$ at all points lying over $Z$, then", "the strict transform $\\mathcal{F}'$ of $\\mathcal{F}$ is equal to the", "pullback $\\text{pr}_X^*\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [ "divisors-definition-strict-transform" ], "proofs": [ { "contents": [ "We will prove part (2) as it implies part (1) by the definition of the", "strict transform of a scheme over $S$. The question is local on $X$.", "Thus we may assume that $S = \\Spec(A)$, $X = \\Spec(B)$, and that", "$\\mathcal{F}$ corresponds to the $B$-module $N$. Then $\\mathcal{F}'$", "over the open $\\Spec(B \\otimes_A A[\\frac{I}{a}])$ of $X \\times_S S'$", "corresponds to the module", "$$", "N' = (N \\otimes_A A[\\textstyle{\\frac{I}{a}}])/a\\text{-power-torsion}", "$$", "see Properties, Lemma", "\\ref{properties-lemma-sections-supported-on-closed-subset}.", "Thus we have to show that the $a$-power-torsion of", "$N \\otimes_A A[\\frac{I}{a}]$ is zero. Let $y \\in N \\otimes_A A[\\frac{I}{a}]$", "with $a^n y = 0$. If $\\mathfrak q \\subset B$", "is a prime and $a \\not \\in \\mathfrak q$, then $y$ maps to", "zero in $(N \\otimes_A A[\\frac{I}{a}])_\\mathfrak q$. on the other hand,", "if $a \\in \\mathfrak q$, then $N_\\mathfrak q$ is a flat $A$-module", "and we see that", "$N_\\mathfrak q \\otimes_A A[\\frac{I}{a}]", "=(N \\otimes_A A[\\frac{I}{a}])_\\mathfrak q$", "has no $a$-power torsion (as $A[\\frac{I}{a}]$ doesn't).", "Hence $y$ maps to zero in this localization as well. We conclude that", "$y$ is zero by", "Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}." ], "refs": [ "properties-lemma-sections-supported-on-closed-subset", "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 3036, 410 ] } ], "ref_ids": [ 8113 ] }, { "id": 8067, "type": "theorem", "label": "divisors-lemma-strict-transform-affine", "categories": [ "divisors" ], "title": "divisors-lemma-strict-transform-affine", "contents": [ "Let $S$ be a scheme. Let $Z \\subset S$ be a closed subscheme.", "Let $b : S' \\to S$ be the blowing up of $Z$ in $S$. Let", "$g : X \\to Y$ be an affine morphism of schemes over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $g' : X \\times_S S' \\to Y \\times_S S'$ be the base change", "of $g$. Let $\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$", "relative to $b$. Then $g'_*\\mathcal{F}'$ is the strict transform", "of $g_*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $g'_*\\text{pr}_X^*\\mathcal{F} = \\text{pr}_Y^*g_*\\mathcal{F}$", "by Cohomology of Schemes, Lemma \\ref{coherent-lemma-affine-base-change}.", "Let $\\mathcal{K} \\subset \\text{pr}_X^*\\mathcal{F}$ be the subsheaf", "of sections supported in the inverse image of $Z$ in $X \\times_S S'$.", "By Properties, Lemma", "\\ref{properties-lemma-push-sections-supported-on-closed-subset}", "the pushforward $g'_*\\mathcal{K}$ is the subsheaf of sections of", "$\\text{pr}_Y^*g_*\\mathcal{F}$ supported in the inverse", "image of $Z$ in $Y \\times_S S'$. As $g'$ is affine", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-affine})", "we see that $g'_*$ is exact, hence we conclude." ], "refs": [ "coherent-lemma-affine-base-change", "properties-lemma-push-sections-supported-on-closed-subset", "morphisms-lemma-base-change-affine" ], "ref_ids": [ 3297, 3037, 5176 ] } ], "ref_ids": [] }, { "id": 8068, "type": "theorem", "label": "divisors-lemma-strict-transform-different-centers", "categories": [ "divisors" ], "title": "divisors-lemma-strict-transform-different-centers", "contents": [ "Let $S$ be a scheme. Let $Z \\subset S$ be a closed subscheme.", "Let $D \\subset S$ be an effective Cartier divisor.", "Let $Z' \\subset S$ be the closed subscheme cut out by the product", "of the ideal sheaves of $Z$ and $D$.", "Let $S' \\to S$ be the blowup of $S$ in $Z$.", "\\begin{enumerate}", "\\item The blowup of $S$ in $Z'$ is isomorphic to $S' \\to S$.", "\\item Let $f : X \\to S$ be a morphism of schemes and let $\\mathcal{F}$", "be a quasi-coherent $\\mathcal{O}_X$-module. If $\\mathcal{F}$ has", "no nonzero local sections supported in $f^{-1}D$, then the", "strict transform of $\\mathcal{F}$ relative to the blowing up", "in $Z$ agrees with the strict transform of $\\mathcal{F}$ relative", "to the blowing up of $S$ in $Z'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first statement follows on combining", "Lemmas \\ref{lemma-blowing-up-two-ideals} and", "\\ref{lemma-blow-up-effective-Cartier-divisor}.", "Using Lemma \\ref{lemma-blowing-up-affine} the second statement", "translates into the", "following algebra problem. Let $A$ be a ring, $I \\subset A$ an ideal,", "$x \\in A$ a nonzerodivisor, and $a \\in I$. Let $M$ be an $A$-module", "whose $x$-torsion is zero. To show: the $a$-power torsion in", "$M \\otimes_A A[\\frac{I}{a}]$ is equal to the $xa$-power torsion.", "The reason for this is that the kernel and cokernel of the map", "$A \\to A[\\frac{I}{a}]$ is $a$-power torsion, so this map becomes an", "isomorphism after inverting $a$. Hence the kernel", "and cokernel of $M \\to M \\otimes_A A[\\frac{I}{a}]$ are $a$-power", "torsion too. This implies the result." ], "refs": [ "divisors-lemma-blowing-up-two-ideals", "divisors-lemma-blow-up-effective-Cartier-divisor", "divisors-lemma-blowing-up-affine" ], "ref_ids": [ 8062, 8057, 8052 ] } ], "ref_ids": [] }, { "id": 8069, "type": "theorem", "label": "divisors-lemma-strict-transform-composition-blowups", "categories": [ "divisors" ], "title": "divisors-lemma-strict-transform-composition-blowups", "contents": [ "Let $S$ be a scheme. Let $Z \\subset S$ be a closed subscheme.", "Let $b : S' \\to S$ be the blowing up with center $Z$. Let $Z' \\subset S'$ be", "a closed subscheme. Let $S'' \\to S'$ be the blowing up with center $Z'$.", "Let $Y \\subset S$ be a closed subscheme such that", "$Y = Z \\cup b(Z')$ set theoretically and the composition $S'' \\to S$", "is isomorphic to the blowing up of $S$ in $Y$.", "In this situation, given any scheme $X$ over $S$ and", "$\\mathcal{F} \\in \\QCoh(\\mathcal{O}_X)$ we have", "\\begin{enumerate}", "\\item the strict transform of $\\mathcal{F}$ with respect to the blowing", "up of $S$ in $Y$ is equal to the strict transform with respect to the", "blowup $S'' \\to S'$ in $Z'$ of the strict transform of $\\mathcal{F}$", "with respect to the blowup $S' \\to S$ of $S$ in $Z$, and", "\\item the strict transform of $X$ with respect to the blowing", "up of $S$ in $Y$ is equal to the strict transform with respect to the", "blowup $S'' \\to S'$ in $Z'$ of the strict transform of $X$", "with respect to the blowup $S' \\to S$ of $S$ in $Z$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$ with respect", "to the blowup $S' \\to S$ of $S$ in $Z$.", "Let $\\mathcal{F}''$ be the strict transform of $\\mathcal{F}'$ with respect", "to the blowup $S'' \\to S'$ of $S'$ in $Z'$.", "Let $\\mathcal{G}$ be the strict transform of $\\mathcal{F}$ with respect", "to the blowup $S'' \\to S$ of $S$ in $Y$.", "We also label the morphisms", "$$", "\\xymatrix{", "X \\times_S S'' \\ar[r]_q \\ar[d]^{f''} &", "X \\times_S S' \\ar[r]_p \\ar[d]^{f'} &", "X \\ar[d]^f \\\\", "S'' \\ar[r] & S' \\ar[r] & S", "}", "$$", "By definition there is a surjection $p^*\\mathcal{F} \\to \\mathcal{F}'$", "and a surjection $q^*\\mathcal{F}' \\to \\mathcal{F}''$ which combine", "by right exactness of $q^*$ to a surjection", "$(p \\circ q)^*\\mathcal{F} \\to \\mathcal{F}''$. Also we have the surjection", "$(p \\circ q)^*\\mathcal{F} \\to \\mathcal{G}$. Thus it suffices to prove", "that these two surjections have the same kernel.", "\\medskip\\noindent", "The kernel of the surjection $p^*\\mathcal{F} \\to \\mathcal{F}'$", "is supported on $(f \\circ p)^{-1}Z$, so this map is an isomorphism at", "points in the complement. Hence the kernel of", "$q^*p^*\\mathcal{F} \\to q^*\\mathcal{F}'$", "is supported on $(f \\circ p \\circ q)^{-1}Z$. The kernel of", "$q^*\\mathcal{F}' \\to \\mathcal{F}''$ is supported on $(f' \\circ q)^{-1}Z'$.", "Combined we see that the kernel of", "$(p \\circ q)^*\\mathcal{F} \\to \\mathcal{F}''$ is supported on", "$(f \\circ p \\circ q)^{-1}Z \\cup (f' \\circ q)^{-1}Z' =", "(f \\circ p \\circ q)^{-1}Y$.", "By construction of $\\mathcal{G}$ we see that we obtain a factorization", "$(p \\circ q)^*\\mathcal{F} \\to \\mathcal{F}'' \\to \\mathcal{G}$.", "To finish the proof it suffices to show that $\\mathcal{F}''$ has no", "nonzero (local) sections supported on", "$(f \\circ p \\circ q)^{-1}(Y) =", "(f \\circ p \\circ q)^{-1}Z \\cup (f' \\circ q)^{-1}Z'$.", "This follows from Lemma \\ref{lemma-strict-transform-different-centers}", "applied to $\\mathcal{F}'$ on $X \\times_S S'$ over $S'$, the closed", "subscheme $Z'$ and the effective Cartier divisor $b^{-1}Z$." ], "refs": [ "divisors-lemma-strict-transform-different-centers" ], "ref_ids": [ 8068 ] } ], "ref_ids": [] }, { "id": 8070, "type": "theorem", "label": "divisors-lemma-strict-transform-universally-injective", "categories": [ "divisors" ], "title": "divisors-lemma-strict-transform-universally-injective", "contents": [ "In the situation of Definition \\ref{definition-strict-transform}.", "Suppose that", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0", "$$", "is an exact sequence of quasi-coherent sheaves on $X$ which remains", "exact after any base change $T \\to S$. Then the strict transforms of", "$\\mathcal{F}_i'$ relative to any blowup $S' \\to S$", "form a short exact sequence", "$0 \\to \\mathcal{F}'_1 \\to \\mathcal{F}'_2 \\to \\mathcal{F}'_3 \\to 0$ too." ], "refs": [ "divisors-definition-strict-transform" ], "proofs": [ { "contents": [ "We may localize on $S$ and $X$ and assume both are affine.", "Then we may push $\\mathcal{F}_i$ to $S$, see", "Lemma \\ref{lemma-strict-transform-affine}.", "We may assume that our blowup is the morphism $1 : S \\to S$", "associated to an effective Cartier divisor $D \\subset S$.", "Then the translation into algebra is the following: Suppose that $A$", "is a ring and $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ is a universally", "exact sequence of $A$-modules. Let $a\\in A$. Then the sequence", "$$", "0 \\to", "M_1/a\\text{-power torsion} \\to", "M_2/a\\text{-power torsion} \\to", "M_3/a\\text{-power torsion} \\to 0", "$$", "is exact too. Namely, surjectivity of the last map and injectivity of", "the first map are immediate. The problem is exactness in the middle.", "Suppose that $x \\in M_2$ maps to zero in $M_3/a\\text{-power torsion}$.", "Then $y = a^n x \\in M_1$ for some $n$. Then $y$ maps to zero in", "$M_2/a^nM_2$. Since $M_1 \\to M_2$ is universally injective we see that", "$y$ maps to zero in $M_1/a^nM_1$. Thus $y = a^n z$ for some $z \\in M_1$.", "Thus $a^n(x - y) = 0$. Hence $y$ maps to the class of $x$ in", "$M_2/a\\text{-power torsion}$ as desired." ], "refs": [ "divisors-lemma-strict-transform-affine" ], "ref_ids": [ 8067 ] } ], "ref_ids": [ 8113 ] }, { "id": 8071, "type": "theorem", "label": "divisors-lemma-composition-admissible-blowups", "categories": [ "divisors" ], "title": "divisors-lemma-composition-admissible-blowups", "contents": [ "\\begin{slogan}", "Admissible blowups are stable under composition.", "\\end{slogan}", "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $U \\subset X$ be a quasi-compact open subscheme.", "Let $b : X' \\to X$ be a $U$-admissible blowup.", "Let $X'' \\to X'$ be a $U$-admissible blowup.", "Then the composition $X'' \\to X$ is a $U$-admissible blowup." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the more precise", "Lemma \\ref{lemma-composition-finite-type-blowups}." ], "refs": [ "divisors-lemma-composition-finite-type-blowups" ], "ref_ids": [ 8064 ] } ], "ref_ids": [] }, { "id": 8072, "type": "theorem", "label": "divisors-lemma-extend-admissible-blowups", "categories": [ "divisors" ], "title": "divisors-lemma-extend-admissible-blowups", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $U, V \\subset X$ be quasi-compact open subschemes.", "Let $b : V' \\to V$ be a $U \\cap V$-admissible blowup.", "Then there exists a $U$-admissible blowup $X' \\to X$", "whose restriction to $V$ is $V'$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\subset \\mathcal{O}_V$ be the finite type", "quasi-coherent sheaf of ideals such that $V(\\mathcal{I})$ is", "disjoint from $U \\cap V$ and such that $V'$ is isomorphic to the", "blowup of $V$ in $\\mathcal{I}$. Let", "$\\mathcal{I}' \\subset \\mathcal{O}_{U \\cup V}$ be the quasi-coherent", "sheaf of ideals whose restriction to $U$ is $\\mathcal{O}_U$ and", "whose restriction to $V$ is $\\mathcal{I}$ (see Sheaves, Section", "\\ref{sheaves-section-glueing-sheaves}).", "By Properties, Lemma \\ref{properties-lemma-extend}", "there exists a finite type quasi-coherent sheaf of ideals", "$\\mathcal{J} \\subset \\mathcal{O}_X$ whose restriction to $U \\cup V$ is", "$\\mathcal{I}'$. The lemma follows." ], "refs": [ "properties-lemma-extend" ], "ref_ids": [ 3019 ] } ], "ref_ids": [] }, { "id": 8073, "type": "theorem", "label": "divisors-lemma-dominate-admissible-blowups", "categories": [ "divisors" ], "title": "divisors-lemma-dominate-admissible-blowups", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $U \\subset X$ be a quasi-compact open subscheme.", "Let $b_i : X_i \\to X$, $i = 1, \\ldots, n$ be $U$-admissible blowups.", "There exists a $U$-admissible blowup $b : X' \\to X$ such that", "(a) $b$ factors as $X' \\to X_i \\to X$ for $i = 1, \\ldots, n$ and", "(b) each of the morphisms $X' \\to X_i$ is a $U$-admissible blowup." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I}_i \\subset \\mathcal{O}_X$ be the finite type", "quasi-coherent sheaf of ideals such that $V(\\mathcal{I}_i)$ is", "disjoint from $U$ and such that $X_i$ is isomorphic to the", "blowup of $X$ in $\\mathcal{I}_i$. Set", "$\\mathcal{I} = \\mathcal{I}_1 \\cdot \\ldots \\cdot \\mathcal{I}_n$", "and let $X'$ be the blowup of $X$ in $\\mathcal{I}$. Then", "$X' \\to X$ factors through $b_i$ by Lemma \\ref{lemma-blowing-up-two-ideals}." ], "refs": [ "divisors-lemma-blowing-up-two-ideals" ], "ref_ids": [ 8062 ] } ], "ref_ids": [] }, { "id": 8074, "type": "theorem", "label": "divisors-lemma-separate-disjoint-opens-by-blowing-up", "categories": [ "divisors" ], "title": "divisors-lemma-separate-disjoint-opens-by-blowing-up", "contents": [ "\\begin{slogan}", "Separate irreducible components by blowing up.", "\\end{slogan}", "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $U, V$ be quasi-compact disjoint open subschemes of $X$.", "Then there exist a $U \\cup V$-admissible blowup $b : X' \\to X$", "such that $X'$ is a disjoint union of open subschemes", "$X' = X'_1 \\amalg X'_2$ with $b^{-1}(U) \\subset X'_1$ and", "$b^{-1}(V) \\subset X'_2$." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite type quasi-coherent sheaf of ideals $\\mathcal{I}$,", "resp.\\ $\\mathcal{J}$ such that $X \\setminus U = V(\\mathcal{I})$,", "resp.\\ $X \\setminus V = V(\\mathcal{J})$, see", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-finite-type-ideals}.", "Then $V(\\mathcal{I}\\mathcal{J}) = X$ set theoretically, hence", "$\\mathcal{I}\\mathcal{J}$ is a locally nilpotent sheaf of ideals.", "Since $\\mathcal{I}$ and $\\mathcal{J}$ are of finite type and $X$", "is quasi-compact there exists an $n > 0$ such that", "$\\mathcal{I}^n \\mathcal{J}^n = 0$. We may and do replace $\\mathcal{I}$", "by $\\mathcal{I}^n$ and $\\mathcal{J}$ by $\\mathcal{J}^n$. Whence", "$\\mathcal{I} \\mathcal{J} = 0$. Let $b : X' \\to X$ be the blowing", "up in $\\mathcal{I} + \\mathcal{J}$. This is $U \\cup V$-admissible", "as $V(\\mathcal{I} + \\mathcal{J}) = X \\setminus U \\cup V$. We will show that", "$X'$ is a disjoint union of open subschemes $X' = X'_1 \\amalg X'_2$", "such that $b^{-1}\\mathcal{I}|_{X'_2} = 0$ and $b^{-1}\\mathcal{J}|_{X'_1} = 0$", "which will prove the lemma.", "\\medskip\\noindent", "We will use the description of the blowing up in", "Lemma \\ref{lemma-blowing-up-affine}. Suppose that $U = \\Spec(A) \\subset X$", "is an affine open such that $\\mathcal{I}|_U$, resp.\\ $\\mathcal{J}|_U$", "corresponds to the finitely generated ideal $I \\subset A$, resp.\\ $J \\subset A$.", "Then", "$$", "b^{-1}(U) = \\text{Proj}(A \\oplus (I + J) \\oplus (I + J)^2 \\oplus \\ldots)", "$$", "This is covered by the affine open subsets $A[\\frac{I + J}{x}]$", "and $A[\\frac{I + J}{y}]$ with $x \\in I$ and $y \\in J$. Since $x \\in I$ is a", "nonzerodivisor in $A[\\frac{I + J}{x}]$ and $IJ = 0$ we see that", "$J A[\\frac{I + J}{x}] = 0$. Since $y \\in J$ is a nonzerodivisor", "in $A[\\frac{I + J}{y}]$ and $IJ = 0$ we see that", "$I A[\\frac{I + J}{y}] = 0$. Moreover,", "$$", "\\Spec(A[\\textstyle{\\frac{I + J}{x}}]) \\cap", "\\Spec(A[\\textstyle{\\frac{I + J}{y}}]) =", "\\Spec(A[\\textstyle{\\frac{I + J}{xy}}]) = \\emptyset", "$$", "because $xy$ is both a nonzerodivisor and zero. Thus $b^{-1}(U)$", "is the disjoint union of the open subscheme $U_1$ defined as the union", "of the standard opens $\\Spec(A[\\frac{I + J}{x}])$ for $x \\in I$ and the open", "subscheme $U_2$ which is the union of the affine opens", "$\\Spec(A[\\frac{I + J}{y}])$ for $y \\in J$. We have seen that", "$b^{-1}\\mathcal{I}\\mathcal{O}_{X'}$ restricts to zero on $U_2$", "and $b^{-1}\\mathcal{I}\\mathcal{O}_{X'}$ restricts to zero on $U_1$.", "We omit the verification that these open subschemes glue to global", "open subschemes $X'_1$ and $X'_2$." ], "refs": [ "properties-lemma-quasi-coherent-finite-type-ideals", "divisors-lemma-blowing-up-affine" ], "ref_ids": [ 3033, 8052 ] } ], "ref_ids": [] }, { "id": 8075, "type": "theorem", "label": "divisors-lemma-blowing-up-denominators", "categories": [ "divisors" ], "title": "divisors-lemma-blowing-up-denominators", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s$ be a regular meromorphic section of $\\mathcal{L}$.", "Let $U \\subset X$ be the maximal open subscheme such that", "$s$ corresponds to a section of $\\mathcal{L}$ over $U$.", "The blowup $b : X' \\to X$ in the ideal of denominators", "of $s$ is $U$-admissible. There exists an effective Cartier divisor", "$D \\subset X'$ and an isomorphism", "$$", "b^*\\mathcal{L} = \\mathcal{O}_{X'}(D - E),", "$$", "where $E \\subset X'$ is the exceptional divisor such that the", "meromorphic section $b^*s$ corresponds, via the isomorphism,", "to the meromorphic section $1_D \\otimes (1_E)^{-1}$." ], "refs": [], "proofs": [ { "contents": [ "From the definition of the ideal of denominators in", "Definition", "\\ref{definition-regular-meromorphic-ideal-denominators}", "we immediately see that $b$ is a $U$-admissible blowup.", "For the notation $1_D$, $1_E$, and $\\mathcal{O}_{X'}(D - E)$", "please see Definition", "\\ref{definition-invertible-sheaf-effective-Cartier-divisor}.", "The pullback $b^*s$ is defined by", "Lemmas \\ref{lemma-blow-up-pullback-effective-Cartier} and", "\\ref{lemma-meromorphic-sections-pullback}.", "Thus the statement of the lemma makes sense.", "We can reinterpret the final assertion as saying", "that $b^*s$ is a global regular section of", "$b^*\\mathcal{L}(E)$ whose zero scheme is $D$.", "This uniquely defines $D$ hence", "to prove the lemma we may work affine locally on $X$ and $X'$.", "Assume $X = \\Spec(A)$ is affine and", "$\\mathcal{L} = \\mathcal{O}_X$. Then $s$ is a regular meromorphic", "function and shrinking further we may assume", "$s = a'/a$ with $a', a \\in A$ nonzerodivisors.", "Then the ideal of denominators of $s$ corresponds", "to the ideal $I = \\{x \\in A \\mid xa' \\in aA\\}$.", "Recall that $X'$ is covered by spectra of affine blowup", "algebras $A' = A[\\frac{I}{x}]$ with $x \\in I$", "(Lemma \\ref{lemma-blowing-up-affine}). Fix $x \\in I$ and", "write $xa' = a a''$ for some $a'' \\in A$.", "The divisor $E \\subset X'$ is cut out by $x \\in A'$ over", "the spectrum of $A'$ and hence $1/x$ is a ", "generator of $\\mathcal{O}_{X'}(E)$ over $\\Spec(A')$.", "Finally, in the total quotient ring", "of $A'$ we have $a'/a = a''/x$. Hence $b^*s = a'/a$ restricts", "to a regular section of $\\mathcal{O}_{X'}(E)$ which is", "over $\\Spec(A')$ given by $a''/x$. This finishes the proof.", "(The divisor $D \\cap \\Spec(A')$ is cut out by the image of", "$a''$ in $A'$.)" ], "refs": [ "divisors-definition-regular-meromorphic-ideal-denominators", "divisors-definition-invertible-sheaf-effective-Cartier-divisor", "divisors-lemma-blow-up-pullback-effective-Cartier", "divisors-lemma-meromorphic-sections-pullback", "divisors-lemma-blowing-up-affine" ], "ref_ids": [ 8105, 8092, 8061, 8011, 8052 ] } ], "ref_ids": [] }, { "id": 8076, "type": "theorem", "label": "divisors-lemma-strict-transform-blowup-fitting-ideal", "categories": [ "divisors" ], "title": "divisors-lemma-strict-transform-blowup-fitting-ideal", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type", "quasi-coherent $\\mathcal{O}_S$-module. Let $Z_k \\subset S$ be the closed", "subscheme cut out by $\\text{Fit}_k(\\mathcal{F})$, see", "Section \\ref{section-fitting-ideals}.", "Let $S' \\to S$ be the blowup of $S$ in $Z_k$ and let", "$\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$.", "Then $\\mathcal{F}'$ can locally be generated by $\\leq k$", "sections." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\mathcal{F}'$ can locally be generated by $\\leq k$", "sections if and only if $\\text{Fit}_k(\\mathcal{F}') = \\mathcal{O}_{S'}$, see", "Lemma \\ref{lemma-fitting-ideal-generate-locally}.", "Hence this lemma is a translation of", "More on Algebra, Lemma \\ref{more-algebra-lemma-blowup-fitting-ideal}." ], "refs": [ "divisors-lemma-fitting-ideal-generate-locally", "more-algebra-lemma-blowup-fitting-ideal" ], "ref_ids": [ 7896, 9949 ] } ], "ref_ids": [] }, { "id": 8077, "type": "theorem", "label": "divisors-lemma-strict-transform-blowup-fitting-ideal-locally-free", "categories": [ "divisors" ], "title": "divisors-lemma-strict-transform-blowup-fitting-ideal-locally-free", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type", "quasi-coherent $\\mathcal{O}_S$-module. Let $Z_k \\subset S$ be the closed", "subscheme cut out by $\\text{Fit}_k(\\mathcal{F})$, see", "Section \\ref{section-fitting-ideals}.", "Assume that $\\mathcal{F}$ is locally free of rank $k$ on $S \\setminus Z_k$.", "Let $S' \\to S$ be the blowup of $S$ in $Z_k$ and let", "$\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$.", "Then $\\mathcal{F}'$ is locally free of rank $k$." ], "refs": [], "proofs": [ { "contents": [ "Translation of More on Algebra, Lemma", "\\ref{more-algebra-lemma-blowup-fitting-ideal-locally-free}." ], "refs": [ "more-algebra-lemma-blowup-fitting-ideal-locally-free" ], "ref_ids": [ 9950 ] } ], "ref_ids": [] }, { "id": 8078, "type": "theorem", "label": "divisors-lemma-blowup-fitting-ideal", "categories": [ "divisors" ], "title": "divisors-lemma-blowup-fitting-ideal", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{F}$ be a finitely presented", "$\\mathcal{O}_X$-module. Let $U \\subset X$ be a scheme theoretically", "dense open such that $\\mathcal{F}|_U$ is finite locally free of", "constant rank $r$. Then", "\\begin{enumerate}", "\\item the blowup $b : X' \\to X$ of $X$ in the $r$th Fitting", "ideal of $\\mathcal{F}$ is $U$-admissible,", "\\item the strict transform $\\mathcal{F}'$ of $\\mathcal{F}$", "with respect to $b$ is locally free of rank $r$,", "\\item the kernel $\\mathcal{K}$ of the surjection", "$b^*\\mathcal{F} \\to \\mathcal{F}'$ is", "finitely presented and $\\mathcal{K}|_U = 0$,", "\\item $b^*\\mathcal{F}$ and $\\mathcal{K}$ are perfect", "$\\mathcal{O}_{X'}$-modules of tor dimension $\\leq 1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The ideal $\\text{Fit}_r(\\mathcal{F})$ is of finite type", "by Lemma \\ref{lemma-fitting-ideal-of-finitely-presented}", "and its restriction to $U$ is equal to $\\mathcal{O}_U$ by", "Lemma \\ref{lemma-fitting-ideal-finite-locally-free}.", "Hence $b : X' \\to X$ is $U$-admissible, see", "Definition \\ref{definition-admissible-blowup}.", "\\medskip\\noindent", "By Lemma \\ref{lemma-fitting-ideal-finite-locally-free}", "the restriction of $\\text{Fit}_{r - 1}(\\mathcal{F})$", "to $U$ is zero, and since $U$ is scheme theoretically dense", "we conclude that $\\text{Fit}_{r - 1}(\\mathcal{F}) = 0$", "on all of $X$. Thus it follows from", "Lemma \\ref{lemma-fitting-ideal-finite-locally-free}", "that $\\mathcal{F}$ is locally free of rank $r$", "on the complement of subscheme cut out by the $r$th", "Fitting ideal of $\\mathcal{F}$ (this complement may", "be bigger than $U$ which is why we had to do this step", "in the argument). Hence by", "Lemma \\ref{lemma-strict-transform-blowup-fitting-ideal-locally-free}", "the strict transform", "$$", "b^*\\mathcal{F} \\longrightarrow \\mathcal{F}'", "$$", "is locally free of rank $r$. The kernel $\\mathcal{K}$", "of this map is supported on the exceptional divisor", "of the blowup $b$ and hence $\\mathcal{K}|_U = 0$.", "Finally, since $\\mathcal{F}'$ is finite locally free", "and since the displayed arrow is surjective, we can", "locally on $X'$ write $b^*\\mathcal{F}$ as the", "direct sum of $\\mathcal{K}$ and $\\mathcal{F}'$.", "Since $b^*\\mathcal{F}'$ is finitely presented", "(Modules, Lemma \\ref{modules-lemma-pullback-finite-presentation})", "the same is true for $\\mathcal{K}$.", "\\medskip\\noindent", "The statement on tor dimension follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-fitting-ideals-and-pd1}." ], "refs": [ "divisors-lemma-fitting-ideal-of-finitely-presented", "divisors-lemma-fitting-ideal-finite-locally-free", "divisors-definition-admissible-blowup", "divisors-lemma-fitting-ideal-finite-locally-free", "divisors-lemma-fitting-ideal-finite-locally-free", "divisors-lemma-strict-transform-blowup-fitting-ideal-locally-free", "modules-lemma-pullback-finite-presentation", "more-algebra-lemma-fitting-ideals-and-pd1" ], "ref_ids": [ 7894, 7897, 8114, 7897, 7897, 8077, 13250, 9838 ] } ], "ref_ids": [] }, { "id": 8079, "type": "theorem", "label": "divisors-lemma-filter-after-modification", "categories": [ "divisors" ], "title": "divisors-lemma-filter-after-modification", "contents": [ "Let $X$ be an integral scheme. Let $\\mathcal{E}$ be a finite locally free", "$\\mathcal{O}_X$-module. There exists a modification $f : X' \\to X$", "such that $f^*\\mathcal{E}$ has a filtration whose successive quotients", "are invertible $\\mathcal{O}_{X'}$-modules." ], "refs": [], "proofs": [ { "contents": [ "We prove this by induction on the rank $r$ of $\\mathcal{E}$.", "If $r = 1$ or $r = 0$ the lemma is obvious. Assume $r > 1$.", "Let $P = \\mathbf{P}(\\mathcal{E})$ with structure morphism $\\pi : P \\to X$,", "see Constructions, Section \\ref{constructions-section-projective-bundle}.", "Then $\\pi$ is proper (Lemma \\ref{lemma-relative-proj-proper}).", "There is a canonical surjection", "$$", "\\pi^*\\mathcal{E} \\to \\mathcal{O}_P(1)", "$$", "whose kernel is finite locally free of rank $r - 1$.", "Choose a nonempty open subscheme $U \\subset X$ such that", "$\\mathcal{E}|_U \\cong \\mathcal{O}_U^{\\oplus r}$.", "Then $P_U = \\pi^{-1}(U)$ is isomorphic to $\\mathbf{P}^{r - 1}_U$.", "In particular, there exists a section $s : U \\to P_U$ of $\\pi$.", "Let $X' \\subset P$ be the scheme theoretic image of the", "morphism $U \\to P_U \\to P$. Then $X'$ is integral", "(Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-image-reduced}),", "the morphism $f = \\pi|_{X'} : X' \\to X$ is proper (Morphisms, Lemmas", "\\ref{morphisms-lemma-closed-immersion-proper} and", "\\ref{morphisms-lemma-composition-proper}), and", "$f^{-1}(U) \\to U$ is an isomorphism. Hence $f$ is a modification", "(Morphisms, Definition \\ref{morphisms-definition-modification}).", "By construction the pullback $f^*\\mathcal{E}$ has a two step", "filtration whose quotient is invertible because it is equal to", "$\\mathcal{O}_P(1)|_{X'}$ and whose sub $\\mathcal{E}'$ is locally free of rank", "$r - 1$. By induction we can find a modification $g : X'' \\to X'$", "such that $g^*\\mathcal{E}'$ has a filtration as in the statement of", "the lemma. Thus $f \\circ g : X'' \\to X$ is the required modification." ], "refs": [ "divisors-lemma-relative-proj-proper", "morphisms-lemma-scheme-theoretic-image-reduced", "morphisms-lemma-closed-immersion-proper", "morphisms-lemma-composition-proper", "morphisms-definition-modification" ], "ref_ids": [ 8043, 5149, 5410, 5408, 5588 ] } ], "ref_ids": [] }, { "id": 8080, "type": "theorem", "label": "divisors-lemma-extend-rational-map-after-modification", "categories": [ "divisors" ], "title": "divisors-lemma-extend-rational-map-after-modification", "contents": [ "Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$.", "Assume $X$ is Noetherian and $Y$ is proper over $S$.", "Given an $S$-rational map $f : U \\to Y$ from $X$ to $Y$", "there exists a morphism $p : X' \\to X$ and an", "$S$-morphism $f' : X' \\to Y$ such that", "\\begin{enumerate}", "\\item $p$ is proper and $p^{-1}(U) \\to U$ is an isomorphism,", "\\item $f'|_{p^{-1}(U)}$ is equal to $f \\circ p|_{p^{-1}(U)}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $j : U \\to X$ the inclusion morphism. Let $X' \\subset Y \\times_S X$", "be the scheme theoretic image of $(f, j) : U \\to Y \\times_S X$", "(Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-image}).", "The projection $g : Y \\times_S X \\to X$ is proper", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-proper}).", "The composition $p : X' \\to X$ of $X' \\to Y \\times_S X$ and $g$ is proper", "(Morphisms, Lemmas \\ref{morphisms-lemma-closed-immersion-proper} and", "\\ref{morphisms-lemma-composition-proper}).", "Since $g$ is separated and $U \\subset X$ is retrocompact (as $X$ is Noetherian)", "we conclude that $p^{-1}(U) \\to U$ is an isomorphism by", "Morphisms, Lemma", "\\ref{morphisms-lemma-scheme-theoretic-image-of-partial-section}.", "On the other hand, the composition $f' : X' \\to Y$ of $X' \\to Y \\times_S X$", "and the projection $Y \\times_S X \\to Y$ agrees with $f$ on $p^{-1}(U)$." ], "refs": [ "morphisms-definition-scheme-theoretic-image", "morphisms-lemma-base-change-proper", "morphisms-lemma-closed-immersion-proper", "morphisms-lemma-composition-proper" ], "ref_ids": [ 5539, 5409, 5410, 5408 ] } ], "ref_ids": [] }, { "id": 8081, "type": "theorem", "label": "divisors-proposition-push-down-ample", "categories": [ "divisors" ], "title": "divisors-proposition-push-down-ample", "contents": [ "Let $\\pi : X \\to Y$ be a finite surjective morphism of schemes.", "Assume that $X$ has an ample invertible $\\mathcal{O}_X$-module. If", "\\begin{enumerate}", "\\item $\\pi$ is finite locally free, or", "\\item $Y$ is an integral normal scheme, or", "\\item $Y$ is Noetherian, $p\\mathcal{O}_Y = 0$, and $X = Y_{red}$,", "\\end{enumerate}", "then $Y$ has an ample invertible $\\mathcal{O}_Y$-module." ], "refs": [], "proofs": [ { "contents": [ "Case (1) follows from a combination of", "Lemmas \\ref{lemma-finite-locally-free-has-norm} and \\ref{lemma-norm-ample}.", "Case (3) follows from a combination of", "Lemmas \\ref{lemma-Frobenius-gives-norm-for-reduction} and", "\\ref{lemma-norm-ample}.", "In case (2) we first replace $X$ by an irreducible component of $X$", "which dominates $Y$ (viewed as a reduced closed subscheme of $X$).", "Then we can apply Lemma \\ref{lemma-norm-in-normal-case}." ], "refs": [ "divisors-lemma-finite-locally-free-has-norm", "divisors-lemma-norm-ample", "divisors-lemma-Frobenius-gives-norm-for-reduction", "divisors-lemma-norm-ample", "divisors-lemma-norm-in-normal-case" ], "ref_ids": [ 7968, 7966, 7970, 7966, 7969 ] } ], "ref_ids": [] }, { "id": 8124, "type": "theorem", "label": "spaces-theorem-presentation", "categories": [ "spaces" ], "title": "spaces-theorem-presentation", "contents": [ "Let $S$ be a scheme. Let $U$ be a scheme over $S$.", "Let $j = (s, t) : R \\to U \\times_S U$", "be an \\'etale equivalence relation on $U$ over $S$.", "Then the quotient $U/R$ is an algebraic space,", "and $U \\to U/R$ is \\'etale and surjective, in other words", "$(U, R, U \\to U/R)$ is a presentation of $U/R$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-when-it-works-it-works}", "it suffices to prove that $U/R$ is an algebraic space.", "Let $U' \\to U$ be a surjective, \\'etale morphism.", "Then $\\{U' \\to U\\}$ is in particular an fppf covering.", "Let $R'$ be the restriction of $R$ to $U'$, see", "Groupoids, Definition \\ref{groupoids-definition-restrict-relation}.", "According to", "Groupoids, Lemma \\ref{groupoids-lemma-quotient-groupoid-restrict}", "we see that $U/R \\cong U'/R'$.", "By Lemma \\ref{lemma-pullback-etale-equivalence-relation} $R'$ is an", "\\'etale equivalence relation on $U'$. Thus we may replace $U$ by $U'$.", "\\medskip\\noindent", "We apply the previous remark to $U' = \\coprod U_i$, where", "$U = \\bigcup U_i$ is an affine open covering of $S$. Hence we", "may and do assume that $U = \\coprod U_i$ where", "each $U_i$ is an affine scheme.", "\\medskip\\noindent", "Consider the restriction $R_i$ of $R$ to $U_i$.", "By Lemma \\ref{lemma-pullback-etale-equivalence-relation}", "this is an \\'etale equivalence relation.", "Set $F_i = U_i/R_i$ and $F = U/R$.", "It is clear that $\\coprod F_i \\to F$ is surjective.", "By Lemma \\ref{lemma-finding-opens} each $F_i \\to F$", "is representable, and an open immersion.", "By Lemma \\ref{lemma-presentation-quasi-compact}", "applied to $(U_i, R_i)$ we see that $F_i$ is an algebraic space.", "Then by Lemma \\ref{lemma-when-it-works-it-works} we see that", "$U_i \\to F_i$ is \\'etale and surjective.", "From Lemma \\ref{lemma-coproduct-algebraic-spaces}", "it follows that $\\coprod F_i$ is an algebraic space.", "Finally, we have verified all", "hypotheses of Lemma \\ref{lemma-glueing-algebraic-spaces}", "and it follows that $F = U/R$ is an algebraic space." ], "refs": [ "spaces-lemma-when-it-works-it-works", "groupoids-definition-restrict-relation", "groupoids-lemma-quotient-groupoid-restrict", "spaces-lemma-pullback-etale-equivalence-relation", "spaces-lemma-pullback-etale-equivalence-relation", "spaces-lemma-finding-opens", "spaces-lemma-presentation-quasi-compact", "spaces-lemma-when-it-works-it-works", "spaces-lemma-coproduct-algebraic-spaces", "spaces-lemma-glueing-algebraic-spaces" ], "ref_ids": [ 8152, 9671, 9651, 8150, 8150, 8151, 8153, 8152, 8147, 8148 ] } ], "ref_ids": [] }, { "id": 8125, "type": "theorem", "label": "spaces-lemma-morphism-schemes-gives-representable-transformation", "categories": [ "spaces" ], "title": "spaces-lemma-morphism-schemes-gives-representable-transformation", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$ and let", "$X$, $Y$ be objects of $(\\Sch/S)_{fppf}$.", "Let $f : X \\to Y$ be a morphism of schemes.", "Then", "$$", "h_f : h_X \\longrightarrow h_Y", "$$", "is a representable transformation of functors." ], "refs": [], "proofs": [ { "contents": [ "This is formal and relies only on the fact that", "the category $(\\Sch/S)_{fppf}$ has fibre products." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8126, "type": "theorem", "label": "spaces-lemma-composition-representable-transformations", "categories": [ "spaces" ], "title": "spaces-lemma-composition-representable-transformations", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$, $b : G \\to H$ be representable transformations of functors.", "Then", "$$", "b \\circ a : F \\longrightarrow H", "$$", "is a representable transformation of functors." ], "refs": [], "proofs": [ { "contents": [ "This is entirely formal and works in any category." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8127, "type": "theorem", "label": "spaces-lemma-base-change-representable-transformations", "categories": [ "spaces" ], "title": "spaces-lemma-base-change-representable-transformations", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$ be a representable transformation of functors.", "Let $b : H \\to G$ be any transformation of functors.", "Consider the fibre product diagram", "$$", "\\xymatrix{", "H \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\", "H \\ar[r]^b & G", "}", "$$", "Then the base change $a'$ is a representable transformation of functors." ], "refs": [], "proofs": [ { "contents": [ "This is entirely formal and works in any category." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8128, "type": "theorem", "label": "spaces-lemma-product-representable-transformations", "categories": [ "spaces" ], "title": "spaces-lemma-product-representable-transformations", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F_i, G_i : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$, $i = 1, 2$.", "Let $a_i : F_i \\to G_i$, $i = 1, 2$", "be representable transformations of functors.", "Then", "$$", "a_1 \\times a_2 : F_1 \\times F_2 \\longrightarrow G_1 \\times G_2", "$$", "is a representable transformation of functors." ], "refs": [], "proofs": [ { "contents": [ "Write $a_1 \\times a_2$ as the composition", "$F_1 \\times F_2 \\to G_1 \\times F_2 \\to G_1 \\times G_2$.", "The first arrow is the base change of $a_1$ by the map", "$G_1 \\times F_2 \\to G_1$, and the second arrow", "is the base change of $a_2$ by the map", "$G_1 \\times G_2 \\to G_2$. Hence this lemma is a formal", "consequence of Lemmas \\ref{lemma-composition-representable-transformations}", "and \\ref{lemma-base-change-representable-transformations}." ], "refs": [ "spaces-lemma-composition-representable-transformations", "spaces-lemma-base-change-representable-transformations" ], "ref_ids": [ 8126, 8127 ] } ], "ref_ids": [] }, { "id": 8129, "type": "theorem", "label": "spaces-lemma-representable-transformation-to-sheaf", "categories": [ "spaces" ], "title": "spaces-lemma-representable-transformation-to-sheaf", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$ be a representable transformation of functors.", "If $G$ is a sheaf, then so is $F$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{\\varphi_i : T_i \\to T\\}$ be a covering of the site", "$(\\Sch/S)_{fppf}$.", "Let $s_i \\in F(T_i)$ which satisfy the sheaf condition.", "Then $\\sigma_i = a(s_i) \\in G(T_i)$ satisfy the sheaf condition", "also. Hence there exists a unique $\\sigma \\in G(T)$ such", "that $\\sigma_i = \\sigma|_{T_i}$. By assumption", "$F' = h_T \\times_{\\sigma, G, a} F$ is a representable presheaf", "and hence (see remarks in Section \\ref{section-general}) a sheaf.", "Note that $(\\varphi_i, s_i) \\in F'(T_i)$ satisfy the", "sheaf condition also, and hence come from some unique", "$(\\text{id}_T, s) \\in F'(T)$. Clearly $s$ is the section of", "$F$ we are looking for." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8130, "type": "theorem", "label": "spaces-lemma-representable-transformation-diagonal", "categories": [ "spaces" ], "title": "spaces-lemma-representable-transformation-diagonal", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$ be a representable transformation of functors.", "Then $\\Delta_{F/G} : F \\to F \\times_G F$ is representable." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\in \\Ob((\\Sch/S)_{fppf})$. Let", "$\\xi = (\\xi_1, \\xi_2) \\in (F \\times_G F)(U)$.", "Set $\\xi' = a(\\xi_1) = a(\\xi_2) \\in G(U)$.", "By assumption there exist a scheme $V$ and a morphism $V \\to U$", "representing the fibre product $h_U \\times_{\\xi', G} F$.", "In particular, the elements $\\xi_1, \\xi_2$ give morphisms", "$f_1, f_2 : U \\to V$ over $U$. Because $V$ represents the", "fibre product $h_U \\times_{\\xi', G} F$ and because", "$\\xi' = a \\circ \\xi_1 = a \\circ \\xi_2$", "we see that if $g : U' \\to U$ is a morphism then", "$$", "g^*\\xi_1 = g^*\\xi_2", "\\Leftrightarrow", "f_1 \\circ g = f_2 \\circ g.", "$$", "In other words, we see that $h_U \\times_{\\xi, F \\times_G F} F$", "is represented by $V \\times_{\\Delta, V \\times V, (f_1, f_2)} U$", "which is a scheme." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8131, "type": "theorem", "label": "spaces-lemma-morphism-schemes-gives-representable-transformation-property", "categories": [ "spaces" ], "title": "spaces-lemma-morphism-schemes-gives-representable-transformation-property", "contents": [ "Let $S$, $X$, $Y$ be objects of $\\Sch_{fppf}$.", "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\mathcal{P}$ be as in", "Definition \\ref{definition-relative-representable-property}.", "Then $h_X \\longrightarrow h_Y$ has property $\\mathcal{P}$ if", "and only if $f$ has property $\\mathcal{P}$." ], "refs": [ "spaces-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Note that the lemma makes sense by", "Lemma \\ref{lemma-morphism-schemes-gives-representable-transformation}.", "Proof omitted." ], "refs": [ "spaces-lemma-morphism-schemes-gives-representable-transformation" ], "ref_ids": [ 8125 ] } ], "ref_ids": [ 8173 ] }, { "id": 8132, "type": "theorem", "label": "spaces-lemma-composition-representable-transformations-property", "categories": [ "spaces" ], "title": "spaces-lemma-composition-representable-transformations-property", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $\\mathcal{P}$ be a property as in", "Definition \\ref{definition-relative-representable-property}", "which is stable under composition.", "Let $a : F \\to G$, $b : G \\to H$ be representable transformations of functors.", "If $a$ and $b$ have property $\\mathcal{P}$ so does", "$b \\circ a : F \\longrightarrow H$." ], "refs": [ "spaces-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Note that the lemma makes sense by", "Lemma \\ref{lemma-composition-representable-transformations}.", "Proof omitted." ], "refs": [ "spaces-lemma-composition-representable-transformations" ], "ref_ids": [ 8126 ] } ], "ref_ids": [ 8173 ] }, { "id": 8133, "type": "theorem", "label": "spaces-lemma-base-change-representable-transformations-property", "categories": [ "spaces" ], "title": "spaces-lemma-base-change-representable-transformations-property", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $\\mathcal{P}$ be a property as in", "Definition \\ref{definition-relative-representable-property}.", "Let $a : F \\to G$ be a representable transformations of functors.", "Let $b : H \\to G$ be any transformation of functors.", "Consider the fibre product diagram", "$$", "\\xymatrix{", "H \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\", "H \\ar[r]^b & G", "}", "$$", "If $a$ has property $\\mathcal{P}$ then also the base change $a'$", "has property $\\mathcal{P}$." ], "refs": [ "spaces-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Note that the lemma makes sense by", "Lemma \\ref{lemma-base-change-representable-transformations}.", "Proof omitted." ], "refs": [ "spaces-lemma-base-change-representable-transformations" ], "ref_ids": [ 8127 ] } ], "ref_ids": [ 8173 ] }, { "id": 8134, "type": "theorem", "label": "spaces-lemma-descent-representable-transformations-property", "categories": [ "spaces" ], "title": "spaces-lemma-descent-representable-transformations-property", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $\\mathcal{P}$ be a property as in", "Definition \\ref{definition-relative-representable-property}.", "Let $a : F \\to G$ be a representable transformations of functors.", "Let $b : H \\to G$ be any transformation of functors.", "Consider the fibre product diagram", "$$", "\\xymatrix{", "H \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\", "H \\ar[r]^b & G", "}", "$$", "Assume that $b$ induces a surjective map of fppf sheaves $H^\\# \\to G^\\#$.", "In this case, if $a'$ has property $\\mathcal{P}$, then also $a$", "has property $\\mathcal{P}$." ], "refs": [ "spaces-definition-relative-representable-property" ], "proofs": [ { "contents": [ "First we remark that by", "Lemma \\ref{lemma-base-change-representable-transformations}", "the transformation $a'$ is representable.", "Let $U \\in \\Ob((\\Sch/S)_{fppf})$, and let", "$\\xi \\in G(U)$. By assumption there exists an fppf covering", "$\\{U_i \\to U\\}_{i \\in I}$ and elements $\\xi_i \\in H(U_i)$ mapping", "to $\\xi|_U$ via $b$. From general category theory it follows that for", "each $i$ we have a fibre product diagram", "$$", "\\xymatrix{", "U_i \\times_{\\xi_i, H, a'} (H \\times_{b, G, a} F) \\ar[r] \\ar[d] &", "U \\times_{\\xi, G, a} F \\ar[d] \\\\", "U_i \\ar[r] & U", "}", "$$", "By assumption the left vertical arrow is a morphism of schemes which", "has property $\\mathcal{P}$. Since $\\mathcal{P}$ is local in the fppf", "topology this implies that also the right vertical arrow has property", "$\\mathcal{P}$ as desired." ], "refs": [ "spaces-lemma-base-change-representable-transformations" ], "ref_ids": [ 8127 ] } ], "ref_ids": [ 8173 ] }, { "id": 8135, "type": "theorem", "label": "spaces-lemma-product-representable-transformations-property", "categories": [ "spaces" ], "title": "spaces-lemma-product-representable-transformations-property", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F_i, G_i : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$,", "$i = 1, 2$.", "Let $a_i : F_i \\to G_i$, $i = 1, 2$ be representable transformations", "of functors.", "Let $\\mathcal{P}$ be a property as in", "Definition \\ref{definition-relative-representable-property}", "which is stable under composition.", "If $a_1$ and $a_2$ have property $\\mathcal{P}$ so does", "$a_1 \\times a_2 : F_1 \\times F_2 \\longrightarrow G_1 \\times G_2$." ], "refs": [ "spaces-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Note that the lemma makes sense by", "Lemma \\ref{lemma-product-representable-transformations}.", "Proof omitted." ], "refs": [ "spaces-lemma-product-representable-transformations" ], "ref_ids": [ 8128 ] } ], "ref_ids": [ 8173 ] }, { "id": 8136, "type": "theorem", "label": "spaces-lemma-representable-transformations-property-implication", "categories": [ "spaces" ], "title": "spaces-lemma-representable-transformations-property-implication", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Let $a : F \\to G$ be a representable transformation of functors.", "Let $\\mathcal{P}$, $\\mathcal{P}'$ be properties as in", "Definition \\ref{definition-relative-representable-property}.", "Suppose that for any morphism of schemes $f : X \\to Y$", "we have $\\mathcal{P}(f) \\Rightarrow \\mathcal{P}'(f)$.", "If $a$ has property $\\mathcal{P}$ then", "$a$ has property $\\mathcal{P}'$." ], "refs": [ "spaces-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8173 ] }, { "id": 8137, "type": "theorem", "label": "spaces-lemma-surjective-flat-locally-finite-presentation", "categories": [ "spaces" ], "title": "spaces-lemma-surjective-flat-locally-finite-presentation", "contents": [ "Let $S$ be a scheme.", "Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be sheaves.", "Let $a : F \\to G$ be representable, flat,", "locally of finite presentation, and surjective.", "Then $a : F \\to G$ is surjective as a map of sheaves." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme over $S$ and let $g : T \\to G$ be a $T$-valued point of", "$G$. By assumption $T' = F \\times_G T$ is (representable by) a scheme and", "the morphism $T' \\to T$ is a flat, locally of finite presentation, and", "surjective. Hence $\\{T' \\to T\\}$ is an fppf covering such", "that $g|_{T'} \\in G(T')$ comes from an element of $F(T')$, namely", "the map $T' \\to F$. This proves the map is surjective as", "a map of sheaves, see", "Sites, Definition \\ref{sites-definition-sheaves-injective-surjective}." ], "refs": [ "sites-definition-sheaves-injective-surjective" ], "ref_ids": [ 8660 ] } ], "ref_ids": [] }, { "id": 8138, "type": "theorem", "label": "spaces-lemma-representable-diagonal", "categories": [ "spaces" ], "title": "spaces-lemma-representable-diagonal", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F$ be a presheaf of sets on $(\\Sch/S)_{fppf}$.", "The following are equivalent:", "\\begin{enumerate}", "\\item the diagonal $F \\to F \\times F$ is representable,", "\\item for $U \\in \\Ob((\\Sch/S)_{fppf})$ and any $a \\in F(U)$", "the map $a : h_U \\to F$ is representable,", "\\item for every pair $U, V \\in \\Ob((\\Sch/S)_{fppf})$", "and any $a \\in F(U)$, $b \\in F(V)$ the fibre product", "$h_U \\times_{a, F, b} h_V$ is representable.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is completely formal, see", "Categories, Lemma \\ref{categories-lemma-representable-diagonal}.", "It depends only on the fact that the category $(\\Sch/S)_{fppf}$", "has products of pairs of objects and fibre products, see", "Topologies, Lemma \\ref{topologies-lemma-fibre-products-fppf}." ], "refs": [ "categories-lemma-representable-diagonal", "topologies-lemma-fibre-products-fppf" ], "ref_ids": [ 12208, 12476 ] } ], "ref_ids": [] }, { "id": 8139, "type": "theorem", "label": "spaces-lemma-transformation-diagonal-properties", "categories": [ "spaces" ], "title": "spaces-lemma-transformation-diagonal-properties", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F$ be a presheaf of sets on $(\\Sch/S)_{fppf}$.", "Let $\\mathcal{P}$ be a property as in", "Definition \\ref{definition-relative-representable-property}.", "If for every $U, V \\in \\Ob((\\Sch/S)_{fppf})$ and $a \\in F(U)$,", "$b \\in F(V)$ we have", "\\begin{enumerate}", "\\item $h_U \\times_{a, F, b} h_V$ is representable, say by the scheme $W$, and", "\\item the morphism $W \\to U \\times_S V$ corresponding to", "$h_U \\times_{a, F, b} h_V \\to h_U \\times h_V$ has property $\\mathcal{P}$,", "\\end{enumerate}", "then $\\Delta : F \\to F \\times F$ is representable and has", "property $\\mathcal{P}$." ], "refs": [ "spaces-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Observe that $\\Delta$ is representable by", "Lemma \\ref{lemma-representable-diagonal}.", "We can formulate condition (2) as saying that", "the transformation $h_U \\times_{a, F, b} h_V \\to h_{U \\times_S V}$", "has property $\\mathcal{P}$, see Lemma", "\\ref{lemma-morphism-schemes-gives-representable-transformation-property}.", "Consider $T \\in \\Ob((\\Sch/S)_{fppf})$ and $(a, b) \\in (F \\times F)(T)$.", "Observe that we have the commutative diagram", "$$", "\\xymatrix{", "F \\times_{\\Delta, F \\times F, (a, b)} h_T \\ar[d] \\ar[r] &", "h_T \\ar[d]^{\\Delta_{T/S}} \\\\", "h_T \\times_{a, F, b} h_T \\ar[r] \\ar[d] &", "h_{T \\times_S T} \\ar[d]^{(a, b)} \\\\", "F \\ar[r]^\\Delta & F \\times F", "}", "$$", "both of whose squares are cartesian. In this way we see that", "the morphism $F \\times_{F \\times F} h_T \\to h_T$ is the base", "change of a morphism having property $\\mathcal{P}$ by", "$\\Delta_{T/S}$. Since $\\mathcal{P}$ is preserved under base", "change this finishes the proof." ], "refs": [ "spaces-lemma-representable-diagonal", "spaces-lemma-morphism-schemes-gives-representable-transformation-property" ], "ref_ids": [ 8138, 8131 ] } ], "ref_ids": [ 8173 ] }, { "id": 8140, "type": "theorem", "label": "spaces-lemma-scheme-is-space", "categories": [ "spaces" ], "title": "spaces-lemma-scheme-is-space", "contents": [ "A scheme is an algebraic space. More precisely,", "given a scheme $T \\in \\Ob((\\Sch/S)_{fppf})$", "the representable functor $h_T$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "The functor $h_T$ is a sheaf by our remarks in Section \\ref{section-general}.", "The diagonal $h_T \\to h_T \\times h_T = h_{T \\times T}$ is", "representable because $(\\Sch/S)_{fppf}$ has fibre products.", "The identity map $h_T \\to h_T$ is surjective \\'etale." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8141, "type": "theorem", "label": "spaces-lemma-product-spaces", "categories": [ "spaces" ], "title": "spaces-lemma-product-spaces", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G$ be algebraic spaces over $S$.", "Then $F \\times G$ is an algebraic space, and is a product", "in the category of algebraic spaces over $S$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $H = F \\times G$ is a sheaf.", "The diagonal of $H$ is simply the product of the", "diagonals of $F$ and $G$. Hence it is representable by", "Lemma \\ref{lemma-product-representable-transformations}.", "Finally, if $U \\to F$ and $V \\to G$ are surjective", "\\'etale morphisms, with $U, V \\in \\Ob((\\Sch/S)_{fppf})$,", "then $U \\times V \\to F \\times G$ is surjective \\'etale", "by Lemma \\ref{lemma-product-representable-transformations-property}." ], "refs": [ "spaces-lemma-product-representable-transformations", "spaces-lemma-product-representable-transformations-property" ], "ref_ids": [ 8128, 8135 ] } ], "ref_ids": [] }, { "id": 8142, "type": "theorem", "label": "spaces-lemma-fibre-product-spaces-over-sheaf-with-representable-diagonal", "categories": [ "spaces" ], "title": "spaces-lemma-fibre-product-spaces-over-sheaf-with-representable-diagonal", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $H$ be a sheaf on $(\\Sch/S)_{fppf}$ whose diagonal", "is representable. Let $F, G$ be algebraic spaces over $S$.", "Let $F \\to H$, $G \\to H$ be maps of sheaves.", "Then $F \\times_H G$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "We check the 3 conditions of", "Definition \\ref{definition-algebraic-space}.", "A fibre product of sheaves is a sheaf, hence $F \\times_H G$ is a sheaf.", "The diagonal of $F \\times_H G$ is the left vertical arrow in", "$$", "\\xymatrix{", "F \\times_H G \\ar[r] \\ar[d]_\\Delta &", "F \\times G \\ar[d]^{\\Delta_F \\times \\Delta_G} \\\\", "(F \\times F) \\times_{(H \\times H)} (G \\times G) \\ar[r] &", "(F \\times F) \\times (G \\times G)", "}", "$$", "which is cartesian. Hence $\\Delta$ is representable as the base change", "of the morphism on the right which is representable, see", "Lemmas \\ref{lemma-product-representable-transformations} and", "\\ref{lemma-base-change-representable-transformations}.", "Finally, let $U, V \\in \\Ob((\\Sch/S)_{fppf})$", "and $a : U \\to F$, $b : V \\to G$ be surjective and \\'etale.", "As $\\Delta_H$ is representable, we see that $U \\times_H V$ is a scheme.", "The morphism", "$$", "U \\times_H V \\longrightarrow F \\times_H G", "$$", "is surjective and \\'etale as a composition of the base changes", "$U \\times_H V \\to U \\times_H G$ and $U \\times_H G \\to F \\times_H G$", "of the \\'etale surjective morphisms $U \\to F$ and $V \\to G$, see", "Lemmas \\ref{lemma-composition-representable-transformations} and", "\\ref{lemma-base-change-representable-transformations}.", "This proves the last condition of", "Definition \\ref{definition-algebraic-space}", "holds and we conclude that $F \\times_H G$ is an algebraic space." ], "refs": [ "spaces-definition-algebraic-space", "spaces-lemma-product-representable-transformations", "spaces-lemma-base-change-representable-transformations", "spaces-lemma-composition-representable-transformations", "spaces-lemma-base-change-representable-transformations", "spaces-definition-algebraic-space" ], "ref_ids": [ 8174, 8128, 8127, 8126, 8127, 8174 ] } ], "ref_ids": [] }, { "id": 8143, "type": "theorem", "label": "spaces-lemma-fibre-product-spaces", "categories": [ "spaces" ], "title": "spaces-lemma-fibre-product-spaces", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F \\to H$, $G \\to H$ be morphisms of algebraic spaces over $S$.", "Then $F \\times_H G$ is an algebraic space, and is a fibre product", "in the category of algebraic spaces over $S$." ], "refs": [], "proofs": [ { "contents": [ "It follows from the stronger", "Lemma \\ref{lemma-fibre-product-spaces-over-sheaf-with-representable-diagonal}", "that $F \\times_H G$ is an algebraic space.", "It is clear that $F \\times_H G$", "is a fibre product in the category of algebraic spaces over $S$", "since that is a full subcategory of the category", "of (pre)sheaves of sets on $(\\Sch/S)_{fppf}$." ], "refs": [ "spaces-lemma-fibre-product-spaces-over-sheaf-with-representable-diagonal" ], "ref_ids": [ 8142 ] } ], "ref_ids": [] }, { "id": 8144, "type": "theorem", "label": "spaces-lemma-coproduct-sheaves-open-and-closed", "categories": [ "spaces" ], "title": "spaces-lemma-coproduct-sheaves-open-and-closed", "contents": [ "Let $S \\in \\Ob(\\Sch_{fppf})$. Let $F$ and $G$ be sheaves on", "$(\\Sch/S)_{fppf}^{opp}$ and denote $F \\amalg G$ the coproduct", "in the category of sheaves. The map $F \\to F \\amalg G$ is representable by", "open and closed immersions." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $\\xi \\in (F \\amalg G)(\\xi)$. Recall the", "coproduct in the category of sheaves is the sheafification of", "the coproduct presheaf (Sites, Lemma \\ref{sites-lemma-colimit-sheaves}).", "Thus there exists an fppf covering $\\{g_i : U_i \\to U\\}_{i \\in I}$", "and a disjoint union decomposition $I = I' \\amalg I''$ such that", "$U_i \\to U \\to F \\amalg G$ factors through $F$, resp.\\ $G$", "if and only if $i \\in I'$, resp.\\ $i \\in I''$. Since $F$ and", "$G$ have empty intersection in $F \\amalg G$ we conclude that", "$U_i \\times_U U_j$ is empty if $i \\in I'$ and $j \\in I''$.", "Hence $U' = \\bigcup_{i \\in I'} g_i(U_i)$ and", "$U'' = \\bigcup_{i \\in I''} g_i(U_i)$ are disjoint open", "(Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}) subschemes of $U$", "with $U = U' \\amalg U''$.", "We omit the verification that $U' = U \\times_{F \\amalg G} F$." ], "refs": [ "sites-lemma-colimit-sheaves", "morphisms-lemma-fppf-open" ], "ref_ids": [ 8514, 5267 ] } ], "ref_ids": [] }, { "id": 8145, "type": "theorem", "label": "spaces-lemma-representable-sheaf-coproduct-sheaves", "categories": [ "spaces" ], "title": "spaces-lemma-representable-sheaf-coproduct-sheaves", "contents": [ "Let $S \\in \\Ob(\\Sch_{fppf})$.", "Let $U \\in \\Ob((\\Sch/S)_{fppf})$.", "Given a set $I$ and sheaves $F_i$ on $\\Ob((\\Sch/S)_{fppf})$,", "if $U \\cong \\coprod_{i\\in I} F_i$", "as sheaves, then each $F_i$ is representable by an open and closed", "subscheme $U_i$ and $U \\cong \\coprod U_i$ as schemes." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-coproduct-sheaves-open-and-closed}", "the map $F_i \\to U$ is representable by open and closed immersions.", "Hence $F_i$ is representable by an open and closed subscheme $U_i$ of $U$.", "We have $U = \\coprod U_i$ because we have $U \\cong \\coprod F_i$", "as sheaves and we can test the equality on points." ], "refs": [ "spaces-lemma-coproduct-sheaves-open-and-closed" ], "ref_ids": [ 8144 ] } ], "ref_ids": [] }, { "id": 8146, "type": "theorem", "label": "spaces-lemma-algebraic-space-coproduct-sheaves", "categories": [ "spaces" ], "title": "spaces-lemma-algebraic-space-coproduct-sheaves", "contents": [ "Let $S \\in \\Ob(\\Sch_{fppf})$.", "Let $F$ be an algebraic space over $S$.", "Given a set $I$ and sheaves $F_i$ on", "$\\Ob((\\Sch/S)_{fppf})$,", "if $F \\cong \\coprod_{i\\in I} F_i$ as sheaves,", "then each $F_i$ is an algebraic space over $S$." ], "refs": [], "proofs": [ { "contents": [ "The representability of $F \\to F \\times F$ implies that each diagonal morphism", "$F_i \\to F_i \\times F_i$ is representable (immediate from the definitions", "and the fact that $F \\times_{(F \\times F)} (F_i \\times F_i) = F_i$).", "Choose a scheme $U$ in $(\\Sch/S)_{fppf}$ and a surjective", "\\'etale morphism $U \\to F$ (this exist by hypothesis).", "The base change $U \\times_F F_i \\to F_i$ is surjective and \\'etale", "by Lemma \\ref{lemma-base-change-representable-transformations-property}.", "On the other hand, $U \\times_F F_i$ is a scheme by", "Lemma \\ref{lemma-coproduct-sheaves-open-and-closed}.", "Thus we have verified all the conditions in", "Definition \\ref{definition-algebraic-space}", "and $F_i$ is an algebraic space." ], "refs": [ "spaces-lemma-base-change-representable-transformations-property", "spaces-lemma-coproduct-sheaves-open-and-closed", "spaces-definition-algebraic-space" ], "ref_ids": [ 8133, 8144, 8174 ] } ], "ref_ids": [] }, { "id": 8147, "type": "theorem", "label": "spaces-lemma-coproduct-algebraic-spaces", "categories": [ "spaces" ], "title": "spaces-lemma-coproduct-algebraic-spaces", "contents": [ "Let $S \\in \\Ob(\\Sch_{fppf})$.", "Suppose given a set $I$ and algebraic spaces $F_i$, $i \\in I$.", "Then $F = \\coprod_{i \\in I} F_i$ is an algebraic space", "provided $I$, and the $F_i$ are not too ``large'': for example if we", "can choose surjective \\'etale morphisms $U_i \\to F_i$ such that", "$\\coprod_{i \\in I} U_i$ is isomorphic to an object of", "$(\\Sch/S)_{fppf}$, then $F$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "By construction $F$ is a sheaf. We omit the verification that the", "diagonal morphism of $F$ is representable. Finally, if $U$ is an", "object of $(\\Sch/S)_{fppf}$ isomorphic to $\\coprod_{i \\in I} U_i$", "then it is straightforward to verify that the resulting map", "$U \\to \\coprod F_i$ is surjective and \\'etale." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8148, "type": "theorem", "label": "spaces-lemma-glueing-algebraic-spaces", "categories": [ "spaces" ], "title": "spaces-lemma-glueing-algebraic-spaces", "contents": [ "Let $S \\in \\Ob(\\Sch_{fppf})$.", "Let $F$ be a presheaf of sets on $(\\Sch/S)_{fppf}$.", "Assume", "\\begin{enumerate}", "\\item $F$ is a sheaf,", "\\item there exists an index set $I$", "and subfunctors $F_i \\subset F$ such that", "\\begin{enumerate}", "\\item each $F_i$ is an algebraic space,", "\\item each $F_i \\to F$ is representable,", "\\item each $F_i \\to F$ is an open immersion (see", "Definition \\ref{definition-relative-representable-property}),", "\\item the map $\\coprod F_i \\to F$ is surjective as a map of sheaves, and", "\\item $\\coprod F_i$ is an algebraic space (set theoretic condition, see", "Lemma \\ref{lemma-coproduct-algebraic-spaces}).", "\\end{enumerate}", "\\end{enumerate}", "Then $F$ is an algebraic space." ], "refs": [ "spaces-definition-relative-representable-property", "spaces-lemma-coproduct-algebraic-spaces" ], "proofs": [ { "contents": [ "Let $T$ be an object of $(\\Sch/S)_{fppf}$. Let $T \\to F$ be a morphism.", "By assumption (2)(b) and (2)(c) the fibre product $F_i \\times_F T$", "is representable by an open subscheme $V_i \\subset T$. It follows that", "$(\\coprod F_i) \\times_F T$ is represented by the scheme $\\coprod V_i$ over $T$.", "By assumption (2)(d) there exists an fppf covering $\\{T_j \\to T\\}_{j \\in J}$", "such that $T_j \\to T \\to F$ factors through $F_i$, $i = i(j)$.", "Hence $T_j \\to T$ factors through the open subscheme $V_{i(j)} \\subset T$.", "Since $\\{T_j \\to T\\}$ is jointly surjective, it follows that", "$T = \\bigcup V_i$ is an open covering. In particular, the transformation", "of functors $\\coprod F_i \\to F$ is representable", "and surjective in the sense of", "Definition \\ref{definition-relative-representable-property}", "(see Remark \\ref{remark-warning} for a discussion).", "\\medskip\\noindent", "Next, let $T' \\to F$ be a second morphism from an object in $(\\Sch/S)_{fppf}$.", "Write as above $T' = \\bigcup V'_i$ with $V'_i = T' \\times_F F_i$.", "To show that the diagonal $F \\to F \\times F$ is representable", "we have to show that $G = T \\times_F T'$ is representable, see", "Lemma \\ref{lemma-representable-diagonal}.", "Consider the subfunctors $G_i = G \\times_F F_i$.", "Note that $G_i = V_i \\times_{F_i} V'_i$, and hence is representable", "as $F_i$ is an algebraic space.", "By the above the $G_i$ form a Zariski covering of $G$.", "Hence by Schemes, Lemma \\ref{schemes-lemma-glue-functors}", "we see $G$ is representable.", "\\medskip\\noindent", "Choose a scheme $U \\in \\Ob((\\Sch/S)_{fppf})$ and a surjective", "\\'etale morphism $U \\to \\coprod F_i$ (this exists by hypothesis).", "We may write $U = \\coprod U_i$ with $U_i$ the inverse image of $F_i$,", "see Lemma \\ref{lemma-representable-sheaf-coproduct-sheaves}.", "We claim that $U \\to F$ is surjective and \\'etale. Surjectivity follows", "as $\\coprod F_i \\to F$ is surjective (see first paragraph of the proof)", "by applying", "Lemma \\ref{lemma-composition-representable-transformations-property}.", "Consider the fibre product $U \\times_F T$ where $T \\to F$ is as", "above. We have to show that $U \\times_F T \\to T$ is \\'etale.", "Since $U \\times_F T = \\coprod U_i \\times_F T$ it suffices to show", "each $U_i \\times_F T \\to T$ is \\'etale. Since", "$U_i \\times_F T = U_i \\times_{F_i} V_i$ this follows from the", "fact that $U_i \\to F_i$ is \\'etale and $V_i \\to T$ is an open immersion", "(and Morphisms, Lemmas \\ref{morphisms-lemma-open-immersion-etale}", "and \\ref{morphisms-lemma-composition-etale})." ], "refs": [ "spaces-definition-relative-representable-property", "spaces-remark-warning", "spaces-lemma-representable-diagonal", "schemes-lemma-glue-functors", "spaces-lemma-representable-sheaf-coproduct-sheaves", "spaces-lemma-composition-representable-transformations-property", "morphisms-lemma-open-immersion-etale", "morphisms-lemma-composition-etale" ], "ref_ids": [ 8173, 8187, 8138, 7688, 8145, 8132, 5366, 5360 ] } ], "ref_ids": [ 8173, 8147 ] }, { "id": 8149, "type": "theorem", "label": "spaces-lemma-space-presentation", "categories": [ "spaces" ], "title": "spaces-lemma-space-presentation", "contents": [ "Let $F$ be an algebraic space over $S$. Let $f : U \\to F$ be a", "surjective \\'etale morphism from a scheme to $F$. Set $R = U \\times_F U$.", "Then", "\\begin{enumerate}", "\\item $j : R \\to U \\times_S U$ defines an equivalence relation on", "$U$ over $S$ (see", "Groupoids, Definition \\ref{groupoids-definition-equivalence-relation}).", "\\item the morphisms $s, t : R \\to U$ are \\'etale, and", "\\item the diagram", "$$", "\\xymatrix{", "R \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "U \\ar[r] &", "F", "}", "$$", "is a coequalizer diagram in $\\Sh((\\Sch/S)_{fppf})$.", "\\end{enumerate}" ], "refs": [ "groupoids-definition-equivalence-relation" ], "proofs": [ { "contents": [ "Let $T/S$ be an object of $(\\Sch/S)_{fppf}$.", "Then $R(T) = \\{(a, b) \\in U(T) \\times U(T) \\mid f \\circ a = f \\circ b\\}$", "which is clearly defines an equivalence relation on $U(T)$.", "The morphisms $s, t : R \\to U$ are \\'etale because the morphism", "$U \\to F$ is \\'etale.", "\\medskip\\noindent", "To prove (3) we first show that", "$U \\to F$ is a surjection of sheaves, see", "Sites, Definition \\ref{sites-definition-sheaves-injective-surjective}.", "Let $\\xi \\in F(T)$ with $T$ as above. Let $V = T \\times_{\\xi, F, f}U$.", "By assumption $V$ is a scheme and $V \\to T$ is surjective \\'etale.", "Hence $\\{V \\to T\\}$ is a covering for the fppf topology.", "Since $\\xi|_V$ factors through $U$ by construction we", "conclude $U \\to F$ is surjective. Surjectivity implies that", "$F$ is the coequalizer of the diagram by", "Sites, Lemma \\ref{sites-lemma-coequalizer-surjection}." ], "refs": [ "sites-definition-sheaves-injective-surjective", "sites-lemma-coequalizer-surjection" ], "ref_ids": [ 8660, 8518 ] } ], "ref_ids": [ 9670 ] }, { "id": 8150, "type": "theorem", "label": "spaces-lemma-pullback-etale-equivalence-relation", "categories": [ "spaces" ], "title": "spaces-lemma-pullback-etale-equivalence-relation", "contents": [ "Let $S$ be a scheme. Let $U$ be a scheme over $S$.", "Let $j = (s, t) : R \\to U \\times_S U$", "be an \\'etale equivalence relation on $U$ over $S$.", "Let $U' \\to U$ be an \\'etale morphism.", "Let $R'$ be the restriction of $R$ to $U'$, see", "Groupoids, Definition \\ref{groupoids-definition-restrict-relation}.", "Then $j' : R' \\to U' \\times_S U'$ is an \\'etale equivalence", "relation also." ], "refs": [ "groupoids-definition-restrict-relation" ], "proofs": [ { "contents": [ "It is clear from the description of $s', t'$ in", "Groupoids, Lemma \\ref{groupoids-lemma-restrict-groupoid}", "that $s' , t' : R' \\to U'$ are \\'etale", "as compositions of base changes of \\'etale morphisms", "(see Morphisms, Lemma \\ref{morphisms-lemma-base-change-etale}", "and \\ref{morphisms-lemma-composition-etale})." ], "refs": [ "groupoids-lemma-restrict-groupoid", "morphisms-lemma-base-change-etale", "morphisms-lemma-composition-etale" ], "ref_ids": [ 9642, 5361, 5360 ] } ], "ref_ids": [ 9671 ] }, { "id": 8151, "type": "theorem", "label": "spaces-lemma-finding-opens", "categories": [ "spaces" ], "title": "spaces-lemma-finding-opens", "contents": [ "Let $S$ be a scheme.", "Let $U$ be a scheme over $S$.", "Let $j = (s, t) : R \\to U \\times_S U$ be a pre-relation.", "Let $g : U' \\to U$ be a morphism.", "Assume", "\\begin{enumerate}", "\\item $j$ is an equivalence relation,", "\\item $s, t : R \\to U$ are surjective, flat and", "locally of finite presentation,", "\\item $g$ is flat and locally of finite presentation.", "\\end{enumerate}", "Let $R' = R|_{U'}$ be the restriction of $R$ to $U'$. Then", "$U'/R' \\to U/R$ is representable, and is an open immersion." ], "refs": [], "proofs": [ { "contents": [ "By Groupoids, Lemma \\ref{groupoids-lemma-restrict-relation}", "the morphism $j' = (t', s') : R' \\to U' \\times_S U'$", "defines an equivalence relation. Since $g$ is flat and locally of", "finite presentation we see that $g$ is universally open as well", "(Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}).", "For the same reason $s, t$ are universally open as well.", "Let $W^1 = g(U') \\subset U$, and let $W = t(s^{-1}(W^1))$.", "Then $W^1$ and $W$ are open in $U$. Moreover, as $j$ is an", "equivalence relation we have $t(s^{-1}(W)) = W$ (see", "Groupoids, Lemma \\ref{groupoids-lemma-constructing-invariant-opens}", "for example).", "\\medskip\\noindent", "By", "Groupoids,", "Lemma \\ref{groupoids-lemma-quotient-pre-equivalence-relation-restrict}", "the map of sheaves $F' = U'/R' \\to F = U/R$ is injective.", "Let $a : T \\to F$ be a morphism from a scheme into $U/R$.", "We have to show that $T \\times_F F'$ is representable", "by an open subscheme of $T$.", "\\medskip\\noindent", "The morphism $a$ is given by the following data:", "an fppf covering $\\{\\varphi_j : T_j \\to T\\}_{j \\in J}$ of $T$ and", "morphisms $a_j : T_j \\to U$ such that the maps", "$$", "a_j \\times a_{j'} :", "T_j \\times_T T_{j'}", "\\longrightarrow", "U \\times_S U", "$$", "factor through $j : R \\to U \\times_S U$ via some (unique) maps", "$r_{jj'} : T_j \\times_T T_{j'} \\to R$. The system", "$(a_j)$ corresponds to $a$ in the sense that the diagrams", "$$", "\\xymatrix{", "T_j \\ar[r]_{a_j} \\ar[d] & U \\ar[d] \\\\", "T \\ar[r]^a & F", "}", "$$", "commute.", "\\medskip\\noindent", "Consider the open subsets $W_j = a_j^{-1}(W) \\subset T_j$.", "Since $t(s^{-1}(W)) = W$ we see that", "$$", "W_j \\times_T T_{j'} =", "r_{jj'}^{-1}(t^{-1}(W)) = r_{jj'}^{-1}(s^{-1}(W)) =", "T_j \\times_T W_{j'}.", "$$", "By", "Descent, Lemma \\ref{descent-lemma-open-fpqc-covering}", "this means there exists an open", "$W_T \\subset T$ such that $\\varphi_j^{-1}(W_T) = W_j$ for all $j \\in J$.", "We claim that $W_T \\to T$ represents $T \\times_F F' \\to T$.", "\\medskip\\noindent", "First, let us show that $W_T \\to T \\to F$ is an element of", "$F'(W_T)$. Since $\\{W_j \\to W_T\\}_{j \\in J}$ is an", "fppf covering of $W_T$, it is enough to show that", "each $W_j \\to U \\to F$ is an element of $F'(W_j)$ (as $F'$ is a sheaf", "for the fppf topology). Consider the commutative diagram", "$$", "\\xymatrix{", "W'_j \\ar[rr] \\ar[dd] \\ar[rd] & & U' \\ar[d]^g \\\\", "& s^{-1}(W^1) \\ar[r]_s \\ar[d]^t & W^1 \\ar[d] \\\\", "W_j \\ar[r]^{a_j|_{W_j}} & W \\ar[r] & F", "}", "$$", "where $W'_j = W_j \\times_W s^{-1}(W^1) \\times_{W^1} U'$.", "Since $t$ and $g$ are surjective, flat and locally of finite", "presentation, so is $W'_j \\to W_j$. Hence the restriction of", "the element $W_j \\to U \\to F$ to $W'_j$ is an element of $F'$", "as desired.", "\\medskip\\noindent", "Suppose that $f : T' \\to T$ is a morphism of schemes", "such that $a|_{T'} \\in F'(T')$. We have to show that", "$f$ factors through the open $W_T$. Since", "$\\{T' \\times_T T_j \\to T'\\}$ is an fppf covering of $T'$", "it is enough to show each $T' \\times_T T_j \\to T$", "factors through $W_T$. Hence we may assume $f$ factors", "as $\\varphi_j \\circ f_j : T' \\to T_j \\to T$ for some $j$.", "In this case the condition $a|_{T'} \\in F'(T')$ means that there exists", "some fppf covering $\\{\\psi_i : T'_i \\to T'\\}_{i \\in I}$ and some", "morphisms $b_i : T'_i \\to U'$ such that", "$$", "\\xymatrix{", "T'_i \\ar[r]_{b_i} \\ar[d]_{f_j \\circ \\psi_i} & U' \\ar[r]_g & U \\ar[d] \\\\", "T_j \\ar[r]^{a_j} & U \\ar[r] & F", "}", "$$", "is commutative. This commutativity means that there exists a", "morphism $r'_i : T'_i \\to R$ such that", "$t \\circ r'_i = a_j \\circ f_j \\circ \\psi_i$, and", "$s \\circ r'_i = g \\circ b_i$. This implies that", "$\\Im(f_j \\circ \\psi_i) \\subset W_j$ and we win." ], "refs": [ "groupoids-lemma-restrict-relation", "morphisms-lemma-fppf-open", "groupoids-lemma-constructing-invariant-opens", "groupoids-lemma-quotient-pre-equivalence-relation-restrict", "descent-lemma-open-fpqc-covering" ], "ref_ids": [ 9578, 5267, 9645, 9650, 14637 ] } ], "ref_ids": [] }, { "id": 8152, "type": "theorem", "label": "spaces-lemma-when-it-works-it-works", "categories": [ "spaces" ], "title": "spaces-lemma-when-it-works-it-works", "contents": [ "Let $S$ be a scheme. Let $U$ be a scheme over $S$.", "Let $j = (s, t) : R \\to U \\times_S U$", "be an \\'etale equivalence relation on $U$ over $S$.", "If the quotient $U/R$ is an algebraic space, then", "$U \\to U/R$ is \\'etale and surjective. Hence", "$(U, R, U \\to U/R)$ is a presentation of the algebraic", "space $U/R$." ], "refs": [], "proofs": [ { "contents": [ "Denote $c : U \\to U/R$ the morphism in question.", "Let $T$ be a scheme and let $a : T \\to U/R$ be a morphism.", "We have to show that the morphism (of schemes)", "$\\pi : T \\times_{a, U/R, c} U \\to T$ is \\'etale and surjective.", "The morphism $a$ corresponds to an fppf covering", "$\\{\\varphi_i : T_i \\to T\\}$ and morphisms $a_i : T_i \\to U$ such", "that", "$a_i \\times a_{i'} : T_i \\times_T T_{i'} \\to U \\times_S U$", "factors through $R$, and such that $c \\circ a_i = a \\circ \\varphi_i$.", "Hence", "$$", "T_i \\times_{\\varphi_i, T} T \\times_{a, U/R, c} U =", "T_i \\times_{c \\circ a_i, U/R, c} U =", "T_i \\times_{a_i, U} U \\times_{c, U/R, c} U = T_i \\times_{a_i, U, t} R.", "$$", "Since $t$ is \\'etale and surjective we conclude that", "the base change of $\\pi$ to $T_i$ is surjective and \\'etale.", "Since the property of being surjective and \\'etale is local", "on the base in the fpqc topology (see", "Remark \\ref{remark-list-properties-fpqc-local-base})", "we win." ], "refs": [ "spaces-remark-list-properties-fpqc-local-base" ], "ref_ids": [ 8186 ] } ], "ref_ids": [] }, { "id": 8153, "type": "theorem", "label": "spaces-lemma-presentation-quasi-compact", "categories": [ "spaces" ], "title": "spaces-lemma-presentation-quasi-compact", "contents": [ "Let $S$ be a scheme.", "Let $U$ be a scheme over $S$.", "Let $j = (s, t) : R \\to U \\times_S U$", "be an \\'etale equivalence relation on $U$ over $S$.", "Assume that $U$ is affine. Then the quotient $F = U/R$", "is an algebraic space, and $U \\to F$ is \\'etale and surjective." ], "refs": [], "proofs": [ { "contents": [ "Since $j : R \\to U \\times_S U$ is a monomorphism we see that $j$ is separated", "(see Schemes, Lemma \\ref{schemes-lemma-monomorphism-separated}).", "Since $U$ is affine we see that $U \\times_S U$", "(which comes equipped with a monomorphism into the affine scheme", "$U \\times U$) is separated. Hence we see that $R$ is separated.", "In particular the morphisms $s, t$ are separated as well as \\'etale.", "\\medskip\\noindent", "Since the composition $R \\to U \\times_S U \\to U$ is", "locally of finite type we conclude that", "$j$ is locally of finite type (see", "Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type}).", "As $j$ is also a monomorphism it has finite fibres and", "we see that $j$ is locally quasi-finite by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-fibre}.", "Altogether we see that $j$ is separated and locally quasi-finite.", "\\medskip\\noindent", "Our first step is to show that the quotient map", "$c : U \\to F$ is representable.", "Consider a scheme $T$ and a morphism $a : T \\to F$.", "We have to show that the sheaf $G = T \\times_{a, F, c} U$", "is representable.", "As seen in the proofs of Lemmas \\ref{lemma-finding-opens} and", "\\ref{lemma-when-it-works-it-works} there exists an fppf covering", "$\\{\\varphi_i : T_i \\to T\\}_{i \\in I}$ and morphisms $a_i : T_i \\to U$", "such that $a_i \\times a_{i'} : T_i \\times_T T_{i'} \\to U \\times_S U$", "factors through $R$, and such that $c \\circ a_i = a \\circ \\varphi_i$.", "As in the proof of Lemma \\ref{lemma-when-it-works-it-works} we see that", "\\begin{eqnarray*}", "T_i \\times_{\\varphi_i, T} G & = &", "T_i \\times_{\\varphi_i, T} T \\times_{a, U/R, c} U \\\\", "& = & T_i \\times_{c \\circ a_i, U/R, c} U \\\\", "& = & T_i \\times_{a_i, U} U \\times_{c, U/R, c} U \\\\", "& = & T_i \\times_{a_i, U, t} R", "\\end{eqnarray*}", "Since $t$ is separated and \\'etale, and in particular", "separated and locally quasi-finite (by Morphisms, Lemmas", "\\ref{morphisms-lemma-unramified-quasi-finite} and", "\\ref{morphisms-lemma-flat-unramified-etale})", "we see that the restriction", "of $G$ to each $T_i$ is representable by a morphism of schemes", "$X_i \\to T_i$ which is separated and locally quasi-finite. By", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves}", "we obtain a descent datum $(X_i, \\varphi_{ii'})$ relative", "to the fppf-covering $\\{T_i \\to T\\}$. Since each", "$X_i \\to T_i$ is separated and locally quasi-finite we see by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend}", "that this descent datum is effective.", "Hence by", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves} (2)", "we conclude that $G$ is representable as desired.", "\\medskip\\noindent", "The second step of the proof is to show that $U \\to F$ is surjective and", "\\'etale. This is clear from the above since in the first step above we", "saw that $G = T \\times_{a, F, c} U$ is a scheme over $T$ which base changes", "to schemes $X_i \\to T_i$ which are surjective and \\'etale. Thus $G \\to T$", "is surjective and \\'etale (see", "Remark \\ref{remark-list-properties-fpqc-local-base}).", "Alternatively one can reread the proof of", "Lemma \\ref{lemma-when-it-works-it-works} in the current", "situation.", "\\medskip\\noindent", "The third and final step is to show that the diagonal map $F \\to F \\times F$", "is representable. We first observe that the diagram", "$$", "\\xymatrix{", "R \\ar[r] \\ar[d]_j & F \\ar[d]^\\Delta \\\\", "U \\times_S U \\ar[r] & F \\times F", "}", "$$", "is a fibre product square. By", "Lemma \\ref{lemma-product-representable-transformations} the morphism", "$U \\times_S U \\to F \\times F$ is representable (note that", "$h_U \\times h_U = h_{U \\times_S U}$). Moreover, by", "Lemma \\ref{lemma-product-representable-transformations-property}", "the morphism $U \\times_S U \\to F \\times F$ is surjective", "and \\'etale (note also that \\'etale and surjective occur in the lists of", "Remarks \\ref{remark-list-properties-fpqc-local-base}", "and \\ref{remark-list-properties-stable-composition}).", "It follows either from", "Lemma \\ref{lemma-base-change-representable-transformations}", "and the diagram above, or by writing $R \\to F$ as $R \\to U \\to F$ and", "Lemmas", "\\ref{lemma-morphism-schemes-gives-representable-transformation} and", "\\ref{lemma-composition-representable-transformations} that", "$R \\to F$ is representable as well. Let $T$ be a scheme and let", "$a : T \\to F \\times F$ be a morphism. We have to show that", "$G = T \\times_{a, F \\times F, \\Delta} F$ is representable.", "By what was said above the morphism (of schemes)", "$$", "T' = (U \\times_S U) \\times_{F \\times F, a} T \\longrightarrow T", "$$", "is surjective and \\'etale. Hence $\\{T' \\to T\\}$ is an \\'etale", "covering of $T$. Note also that", "$$", "T' \\times_T G = T' \\times_{U \\times_S U, j} R", "$$", "as can be seen contemplating the following cube", "$$", "\\xymatrix{", "& R \\ar[rr] \\ar[dd] & & F \\ar[dd] \\\\", "T' \\times_T G \\ar[rr] \\ar[dd] \\ar[ru] & & G \\ar[dd] \\ar[ru] & \\\\", "& U \\times_S U \\ar'[r][rr] & & F \\times F \\\\", "T' \\ar[rr] \\ar[ru] & & T \\ar[ru]", "}", "$$", "Hence we see that the restriction of $G$ to $T'$ is representable", "by a scheme $X$, and moreover that the morphism $X \\to T'$ is", "a base change of the morphism $j$. Hence $X \\to T'$ is", "separated and locally quasi-finite (see second paragraph of the proof).", "By Descent, Lemma \\ref{descent-lemma-descent-data-sheaves}", "we obtain a descent datum $(X, \\varphi)$ relative", "to the fppf-covering $\\{T' \\to T\\}$. Since", "$X \\to T$ is separated and locally quasi-finite we see by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend}", "that this descent datum is effective.", "Hence by", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves} (2)", "we conclude that $G$ is representable as desired." ], "refs": [ "schemes-lemma-monomorphism-separated", "morphisms-lemma-permanence-finite-type", "morphisms-lemma-finite-fibre", "spaces-lemma-finding-opens", "spaces-lemma-when-it-works-it-works", "spaces-lemma-when-it-works-it-works", "morphisms-lemma-unramified-quasi-finite", "morphisms-lemma-flat-unramified-etale", "descent-lemma-descent-data-sheaves", "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "descent-lemma-descent-data-sheaves", "spaces-remark-list-properties-fpqc-local-base", "spaces-lemma-when-it-works-it-works", "spaces-lemma-product-representable-transformations", "spaces-lemma-product-representable-transformations-property", "spaces-remark-list-properties-fpqc-local-base", "spaces-remark-list-properties-stable-composition", "spaces-lemma-base-change-representable-transformations", "spaces-lemma-morphism-schemes-gives-representable-transformation", "spaces-lemma-composition-representable-transformations", "descent-lemma-descent-data-sheaves", "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "descent-lemma-descent-data-sheaves" ], "ref_ids": [ 7722, 5204, 5227, 8151, 8152, 8152, 5351, 5373, 14751, 13949, 14751, 8186, 8152, 8128, 8135, 8186, 8185, 8127, 8125, 8126, 14751, 13949, 14751 ] } ], "ref_ids": [] }, { "id": 8154, "type": "theorem", "label": "spaces-lemma-etale-locally-representable-gives-space", "categories": [ "spaces" ], "title": "spaces-lemma-etale-locally-representable-gives-space", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F$ be a sheaf on $(\\Sch/S)_{fppf}$", "such that there exists $U \\in \\Ob((\\Sch/S)_{fppf})$ and a map", "$U \\to F$ which is representable, surjective, and \\'etale.", "Then $F$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "Set $R = U \\times_F U$. This is a scheme as $U \\to F$ is assumed representable.", "The projections $s, t : R \\to U$ are \\'etale as $U \\to F$ is assumed \\'etale.", "The map $j = (t, s) : R \\to U \\times_S U$ is a monomorphism and an equivalence", "relation as $R = U \\times_F U$. By Theorem \\ref{theorem-presentation}", "the quotient sheaf $F' = U/R$ is an algebraic space and $U \\to F'$", "is surjective and \\'etale. Again since $R = U \\times_F U$ we obtain", "a canonical factorization $U \\to F' \\to F$ and $F' \\to F$ is an injective", "map of sheaves. On the other hand, $U \\to F$ is surjective as a map", "of sheaves by Lemma \\ref{lemma-surjective-flat-locally-finite-presentation}.", "Thus $F' \\to F$ is also surjective and we conclude $F' = F$ is an", "algebraic space." ], "refs": [ "spaces-theorem-presentation", "spaces-lemma-surjective-flat-locally-finite-presentation" ], "ref_ids": [ 8124, 8137 ] } ], "ref_ids": [] }, { "id": 8155, "type": "theorem", "label": "spaces-lemma-etale-locally-representable-by-space-gives-space", "categories": [ "spaces" ], "title": "spaces-lemma-etale-locally-representable-by-space-gives-space", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $G$ be an algebraic", "space over $S$, let $F$ be a sheaf on $(\\Sch/S)_{fppf}$, and let", "$G \\to F$ be a representable transformation of functors which is", "surjective and \\'etale. Then $F$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "Pick a scheme $U$ and a surjective \\'etale morphism $U \\to G$.", "Since $G$ is an algebraic space $U \\to G$ is representable.", "Hence the composition $U \\to G \\to F$ is representable,", "surjective, and \\'etale. See Lemmas", "\\ref{lemma-composition-representable-transformations} and", "\\ref{lemma-composition-representable-transformations-property}.", "Thus $F$ is an algebraic space by", "Lemma \\ref{lemma-etale-locally-representable-gives-space}." ], "refs": [ "spaces-lemma-composition-representable-transformations", "spaces-lemma-composition-representable-transformations-property", "spaces-lemma-etale-locally-representable-gives-space" ], "ref_ids": [ 8126, 8132, 8154 ] } ], "ref_ids": [] }, { "id": 8156, "type": "theorem", "label": "spaces-lemma-representable-over-space", "categories": [ "spaces" ], "title": "spaces-lemma-representable-over-space", "contents": [ "\\begin{slogan}", "A functor that admits a representable morphism to an algebraic space is", "an algebraic space.", "\\end{slogan}", "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F$ be an algebraic space over $S$.", "Let $G \\to F$ be a representable transformation of functors.", "Then $G$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-representable-transformation-to-sheaf}", "we see that $G$ is a sheaf. The diagram", "$$", "\\xymatrix{", "G \\times_F G \\ar[r] \\ar[d] & F \\ar[d]^{\\Delta_F} \\\\", "G \\times G \\ar[r] & F \\times F", "}", "$$", "is cartesian. Hence we see that $G \\times_F G \\to G \\times G$", "is representable by", "Lemma \\ref{lemma-base-change-representable-transformations}.", "By", "Lemma \\ref{lemma-representable-transformation-diagonal}", "we see that $G \\to G \\times_F G$ is representable.", "Hence $\\Delta_G : G \\to G \\times G$ is representable as a composition", "of representable transformations, see", "Lemma \\ref{lemma-composition-representable-transformations}.", "Finally, let $U$ be an object of $(\\Sch/S)_{fppf}$", "and let $U \\to F$ be surjective and \\'etale. By assumption", "$U \\times_F G$ is representable by a scheme $U'$. By", "Lemma \\ref{lemma-base-change-representable-transformations-property}", "the morphism $U' \\to G$ is surjective and \\'etale. This verifies", "the final condition of Definition \\ref{definition-algebraic-space} and we win." ], "refs": [ "spaces-lemma-representable-transformation-to-sheaf", "spaces-lemma-base-change-representable-transformations", "spaces-lemma-representable-transformation-diagonal", "spaces-lemma-composition-representable-transformations", "spaces-lemma-base-change-representable-transformations-property", "spaces-definition-algebraic-space" ], "ref_ids": [ 8129, 8127, 8130, 8126, 8133, 8174 ] } ], "ref_ids": [] }, { "id": 8157, "type": "theorem", "label": "spaces-lemma-representable-morphisms-spaces-property", "categories": [ "spaces" ], "title": "spaces-lemma-representable-morphisms-spaces-property", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F$, $G$ be algebraic spaces over $S$.", "Let $G \\to F$ be a representable morphism.", "Let $U \\in \\Ob((\\Sch/S)_{fppf})$, and $q : U \\to F$", "surjective and \\'etale. Set $V = G \\times_F U$.", "Finally, let $\\mathcal{P}$ be a property of morphisms", "of schemes as in Definition \\ref{definition-relative-representable-property}.", "Then $G \\to F$ has property $\\mathcal{P}$ if and only if", "$V \\to U$ has property $\\mathcal{P}$." ], "refs": [ "spaces-definition-relative-representable-property" ], "proofs": [ { "contents": [ "(This lemma follows from", "Lemmas \\ref{lemma-base-change-representable-transformations-property} and", "\\ref{lemma-descent-representable-transformations-property},", "but we give a direct proof here also.)", "It is clear from the definitions that if $G \\to F$ has property", "$\\mathcal{P}$, then $V \\to U$ has property $\\mathcal{P}$.", "Conversely, assume $V \\to U$ has property $\\mathcal{P}$.", "Let $T \\to F$ be a morphism from a scheme to $F$.", "Let $T' = T \\times_F G$ which is a scheme since $G \\to F$ is", "representable. We have to show that $T' \\to T$ has property $\\mathcal{P}$.", "Consider the commutative diagram of schemes", "$$", "\\xymatrix{", "V \\ar[d] & T \\times_F V \\ar[d] \\ar[l] \\ar[r] &", "T \\times_F G \\ar[d] \\ar@{=}[r] & T' \\\\", "U & T \\times_F U \\ar[l] \\ar[r] & T", "}", "$$", "where both squares are fibre product squares. Hence we conclude", "the middle arrow has property $\\mathcal{P}$ as a base change", "of $V \\to U$. Finally, $\\{T \\times_F U \\to T\\}$ is a fppf covering", "as it is surjective \\'etale, and hence we conclude that", "$T' \\to T$ has property $\\mathcal{P}$ as it is local on the", "base in the fppf topology." ], "refs": [ "spaces-lemma-base-change-representable-transformations-property", "spaces-lemma-descent-representable-transformations-property" ], "ref_ids": [ 8133, 8134 ] } ], "ref_ids": [ 8173 ] }, { "id": 8158, "type": "theorem", "label": "spaces-lemma-morphism-sheaves-with-P-effective-descent-etale", "categories": [ "spaces" ], "title": "spaces-lemma-morphism-sheaves-with-P-effective-descent-etale", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $G \\to F$ be a transformation of presheaves on $(\\Sch/S)_{fppf}$.", "Let $\\mathcal{P}$ be a property of morphisms of schemes.", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{P}$ is preserved under any base change, fppf local on the", "base, and morphisms of type $\\mathcal{P}$ satisfy descent for fppf coverings,", "see Descent, Definition \\ref{descent-definition-descending-types-morphisms},", "\\item $G$ is a sheaf,", "\\item $F$ is an algebraic space,", "\\item there exists a $U \\in \\Ob((\\Sch/S)_{fppf})$", "and a surjective \\'etale morphism $U \\to F$ such that", "$V = G \\times_F U$ is representable, and", "\\item $V \\to U$ has $\\mathcal{P}$.", "\\end{enumerate}", "Then $G$ is an algebraic space, $G \\to F$ is representable and has property", "$\\mathcal{P}$." ], "refs": [ "descent-definition-descending-types-morphisms" ], "proofs": [ { "contents": [ "Let $R = U \\times_F U$, and denote $t, s : R \\to U$ the projection morphisms", "as usual. Let $T$ be a scheme and let $T \\to F$ be a morphism. Then", "$U \\times_F T \\to T$ is surjective \\'etale, hence $\\{U \\times_F T \\to T\\}$ is", "a covering for the \\'etale topology. Consider", "$$", "W = G \\times_F (U \\times_F T) = V \\times_F T = V \\times_U (U \\times_F T).", "$$", "It is a scheme since $F$ is an algebraic space. The morphism", "$W \\to U \\times_F T$ has property $\\mathcal{P}$ since it is a", "base change of $V \\to U$. There is an isomorphism", "\\begin{align*}", "W \\times_T (U \\times_F T) & =", "(G \\times_F (U \\times_F T)) \\times_T (U \\times_F T) \\\\", "& = (U \\times_F T) \\times_T (G \\times_F (U \\times_F T)) \\\\", "& = (U \\times_F T) \\times_T W", "\\end{align*}", "over $(U \\times_F T) \\times_T (U \\times_F T)$. The middle equality maps", "$((g, (u_1, t)), (u_2, t))$ to $((u_1, t), (g, (u_2, t)))$.", "This defines a descent datum for $W/U \\times_F T/T$, see", "Descent, Definition \\ref{descent-definition-descent-datum}.", "This follows from", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves}.", "Namely we have a sheaf $G \\times_F T$, whose", "base change to $U \\times_F T$ is represented by $W$ and the isomorphism", "above is the one from the proof of", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves}.", "By assumption on $\\mathcal{P}$ the descent datum above is representable.", "Hence by the last statement of", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves}", "we see that $G \\times_F T$ is representable. This proves that", "$G \\to F$ is a representable transformation of functors.", "\\medskip\\noindent", "As $G \\to F$ is representable, we see that $G$ is an algebraic space by", "Lemma \\ref{lemma-representable-over-space}. The fact that $G \\to F$ has", "property $\\mathcal{P}$ now follows from", "Lemma \\ref{lemma-representable-morphisms-spaces-property}." ], "refs": [ "descent-definition-descent-datum", "descent-lemma-descent-data-sheaves", "descent-lemma-descent-data-sheaves", "descent-lemma-descent-data-sheaves", "spaces-lemma-representable-over-space", "spaces-lemma-representable-morphisms-spaces-property" ], "ref_ids": [ 14776, 14751, 14751, 14751, 8156, 8157 ] } ], "ref_ids": [ 14782 ] }, { "id": 8159, "type": "theorem", "label": "spaces-lemma-lift-morphism-presentations", "categories": [ "spaces" ], "title": "spaces-lemma-lift-morphism-presentations", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F, G$ be algebraic spaces over $S$.", "Let $a : F \\to G$ be a morphism.", "Given any $V \\in \\Ob((\\Sch/S)_{fppf})$", "and a surjective \\'etale morphism $q : V \\to G$ there exists", "a $U \\in \\Ob((\\Sch/S)_{fppf})$", "and a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_p \\ar[r]_\\alpha &", "V \\ar[d]^q \\\\", "F \\ar[r]^a & G", "}", "$$", "with $p$ surjective and \\'etale." ], "refs": [], "proofs": [ { "contents": [ "First choose $W \\in \\Ob((\\Sch/S)_{fppf})$", "with surjective \\'etale morphism $W \\to F$.", "Next, put $U = W \\times_G V$. Since $G$ is an algebraic space", "we see that $U$ is isomorphic to an object of $(\\Sch/S)_{fppf}$.", "As $q$ is surjective \\'etale, we see that $U \\to W$ is surjective", "\\'etale (see", "Lemma \\ref{lemma-base-change-representable-transformations-property}).", "Thus $U \\to F$ is surjective \\'etale as a composition of surjective", "\\'etale morphisms (see", "Lemma \\ref{lemma-composition-representable-transformations-property})." ], "refs": [ "spaces-lemma-base-change-representable-transformations-property", "spaces-lemma-composition-representable-transformations-property" ], "ref_ids": [ 8133, 8132 ] } ], "ref_ids": [] }, { "id": 8160, "type": "theorem", "label": "spaces-lemma-composition-immersions", "categories": [ "spaces" ], "title": "spaces-lemma-composition-immersions", "contents": [ "Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme.", "A composition of (closed, resp.\\ open) immersions of", "algebraic spaces over $S$ is a (closed, resp.\\ open)", "immersion of algebraic spaces over $S$." ], "refs": [], "proofs": [ { "contents": [ "See Lemma \\ref{lemma-composition-representable-transformations-property} and", "Remarks \\ref{remark-list-properties-fpqc-local-base} (see very last line of", "that remark) and \\ref{remark-list-properties-stable-composition}." ], "refs": [ "spaces-lemma-composition-representable-transformations-property", "spaces-remark-list-properties-fpqc-local-base", "spaces-remark-list-properties-stable-composition" ], "ref_ids": [ 8132, 8186, 8185 ] } ], "ref_ids": [] }, { "id": 8161, "type": "theorem", "label": "spaces-lemma-base-change-immersions", "categories": [ "spaces" ], "title": "spaces-lemma-base-change-immersions", "contents": [ "Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme.", "A base change of a (closed, resp.\\ open) immersion", "of algebraic spaces over $S$ is a (closed, resp.\\ open)", "immersion of algebraic spaces over $S$." ], "refs": [], "proofs": [ { "contents": [ "See Lemma \\ref{lemma-base-change-representable-transformations-property} and", "Remark \\ref{remark-list-properties-fpqc-local-base} (see very last line of", "that remark)." ], "refs": [ "spaces-lemma-base-change-representable-transformations-property", "spaces-remark-list-properties-fpqc-local-base" ], "ref_ids": [ 8133, 8186 ] } ], "ref_ids": [] }, { "id": 8162, "type": "theorem", "label": "spaces-lemma-sub-subspaces", "categories": [ "spaces" ], "title": "spaces-lemma-sub-subspaces", "contents": [ "Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme.", "Let $F$ be an algebraic space over $S$. Let $F_1$, $F_2$ be", "locally closed subspaces of $F$. If $F_1 \\subset F_2$ as subfunctors", "of $F$, then $F_1$ is a locally closed subspace of $F_2$.", "Similarly for closed and open subspaces." ], "refs": [], "proofs": [ { "contents": [ "Let $T \\to F_2$ be a morphism with $T$ a scheme.", "Since $F_2 \\to F$ is a monomorphism, we see that", "$T \\times_{F_2} F_1 = T \\times_F F_1$. The lemma follows", "formally from this." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8163, "type": "theorem", "label": "spaces-lemma-properties-diagonal", "categories": [ "spaces" ], "title": "spaces-lemma-properties-diagonal", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F$ be an algebraic space over $S$.", "Let $\\Delta : F \\to F \\times F$ be the diagonal morphism.", "Then", "\\begin{enumerate}", "\\item $\\Delta$ is locally of finite type,", "\\item $\\Delta$ is a monomorphism,", "\\item $\\Delta$ is separated, and", "\\item $\\Delta$ is locally quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $F = U/R$ be a presentation of $F$.", "As in the proof of Lemma \\ref{lemma-presentation-quasi-compact} the diagram", "$$", "\\xymatrix{", "R \\ar[r] \\ar[d]_j & F \\ar[d]^\\Delta \\\\", "U \\times_S U \\ar[r] & F \\times F", "}", "$$", "is cartesian. Hence according to", "Lemma \\ref{lemma-representable-morphisms-spaces-property}", "it suffices to show that $j$ has the properties listed in the lemma.", "(Note that each of the properties (1) -- (4) occur in the lists", "of Remarks \\ref{remark-list-properties-stable-base-change}", "and \\ref{remark-list-properties-fpqc-local-base}.)", "Since $j$ is an equivalence relation it is a monomorphism.", "Hence it is separated by", "Schemes, Lemma \\ref{schemes-lemma-monomorphism-separated}.", "As $R$ is an \\'etale equivalence relation we see that", "$s, t : R \\to U$ are \\'etale. Hence $s, t$ are locally of finite", "type. Then it follows from", "Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type} that", "$j$ is locally of finite type. Finally, as it is a monomorphism", "its fibres are finite. Thus we conclude that it is locally quasi-finite by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-fibre}." ], "refs": [ "spaces-lemma-presentation-quasi-compact", "spaces-lemma-representable-morphisms-spaces-property", "spaces-remark-list-properties-stable-base-change", "spaces-remark-list-properties-fpqc-local-base", "schemes-lemma-monomorphism-separated", "morphisms-lemma-permanence-finite-type", "morphisms-lemma-finite-fibre" ], "ref_ids": [ 8153, 8157, 8184, 8186, 7722, 5204, 5227 ] } ], "ref_ids": [] }, { "id": 8164, "type": "theorem", "label": "spaces-lemma-quotient", "categories": [ "spaces" ], "title": "spaces-lemma-quotient", "contents": [ "Let $U \\to S$ be a morphism of $\\Sch_{fppf}$.", "Let $G$ be an abstract group. Let $G \\to \\text{Aut}_S(U)$", "be a group homomorphism. Assume", "\\begin{itemize}", "\\item[(*)] if $u \\in U$ is a point, and $g(u) = u$", "for some non-identity element $g \\in G$, then $g$", "induces a nontrivial automorphism of $\\kappa(u)$.", "\\end{itemize}", "Then", "$$", "j :", "R = \\coprod\\nolimits_{g \\in G} U", "\\longrightarrow", "U \\times_S U,", "\\quad", "(g, x) \\longmapsto (g(x), x)", "$$", "is an \\'etale equivalence relation and hence", "$$", "F = U/R", "$$", "is an algebraic space by Theorem \\ref{theorem-presentation}." ], "refs": [ "spaces-theorem-presentation" ], "proofs": [ { "contents": [ "In the statement of the lemma the symbol $\\text{Aut}_S(U)$ denotes", "the group of automorphisms of $U$ over $S$.", "Assume $(*)$ holds. Let us show that", "$$", "j :", "R = \\coprod\\nolimits_{g \\in G} U", "\\longrightarrow", "U \\times_S U,", "\\quad", "(g, x) \\longmapsto (g(x), x)", "$$", "is a monomorphism. This signifies that if $T$ is a nonempty", "scheme, and $h : T \\to U$ is a $T$-valued point such that", "$g \\circ h = g' \\circ h$ then $g = g'$. Suppose", "$T \\not = \\emptyset$, $h : T \\to U$ and $g \\circ h = g' \\circ h$.", "Let $t \\in T$. Consider the composition", "$\\Spec(\\kappa(t)) \\to \\Spec(\\kappa(h(t))) \\to U$.", "Then we conclude that $g^{-1} \\circ g'$ fixes $u = h(t)$ and", "acts as the identity on its residue field. Hence $g = g'$ by $(*)$.", "\\medskip\\noindent", "Thus if $(*)$ holds we see that $j$ is a relation (see", "Groupoids, Definition \\ref{groupoids-definition-equivalence-relation}).", "Moreover, it is an equivalence relation since on $T$-valued points", "for a connected scheme $T$ we see that", "$R(T) = G \\times U(T) \\to U(T) \\times U(T)$ (recall that we always", "work over $S$). Moreover, the morphisms $s, t : R \\to U$ are \\'etale", "since $R$ is a disjoint product of copies of $U$.", "This proves that $j : R \\to U \\times_S U$ is an \\'etale equivalence relation." ], "refs": [ "groupoids-definition-equivalence-relation" ], "ref_ids": [ 9670 ] } ], "ref_ids": [ 8124 ] }, { "id": 8165, "type": "theorem", "label": "spaces-lemma-quotient-finite-separated", "categories": [ "spaces" ], "title": "spaces-lemma-quotient-finite-separated", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-quotient}.", "Assume $G$ is finite. Then", "\\begin{enumerate}", "\\item if $U \\to S$ is quasi-separated, then $U/G$ is quasi-separated", "over $S$, and", "\\item if $U \\to S$ is separated, then $U/G$ is separated over $S$.", "\\end{enumerate}" ], "refs": [ "spaces-lemma-quotient" ], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-properties-diagonal}", "we saw that it suffices to prove the", "corresponding properties for the morphism $j : R \\to U \\times_S U$.", "If $U \\to S$ is quasi-separated, then for every affine open $V \\subset U$", "which maps into an affine of $S$", "the opens $g(V) \\cap V$ are quasi-compact. It follows that $j$ is", "quasi-compact.", "If $U \\to S$ is separated, the diagonal $\\Delta_{U/S}$ is a closed", "immersion. Hence $j : R \\to U \\times_S U$ is a finite coproduct", "of closed immersions with disjoint images. Hence $j$ is a closed immersion." ], "refs": [ "spaces-lemma-properties-diagonal" ], "ref_ids": [ 8163 ] } ], "ref_ids": [ 8164 ] }, { "id": 8166, "type": "theorem", "label": "spaces-lemma-quotient-field-map", "categories": [ "spaces" ], "title": "spaces-lemma-quotient-field-map", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-quotient}.", "If $\\Spec(k) \\to U/G$ is a morphism, then there exist", "\\begin{enumerate}", "\\item a finite Galois extension $k'/k$,", "\\item a finite subgroup $H \\subset G$,", "\\item an isomorphism $H \\to \\text{Gal}(k'/k)$, and", "\\item an $H$-equivariant morphism $\\Spec(k') \\to U$.", "\\end{enumerate}", "Conversely, such data determine a morphism $\\Spec(k) \\to U/G$." ], "refs": [ "spaces-lemma-quotient" ], "proofs": [ { "contents": [ "Consider the fibre product $V = \\Spec(k) \\times_{U/G} U$.", "Here is a diagram", "$$", "\\xymatrix{", "V \\ar[r] \\ar[d] & U \\ar[d] \\\\", "\\Spec(k) \\ar[r] & U/G", "}", "$$", "Then $V$ is a nonempty scheme \\'etale over $\\Spec(k)$ and hence is a", "disjoint union $V = \\coprod_{i \\in I} \\Spec(k_i)$", "of spectra of fields $k_i$ finite separable over $k$", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}).", "We have", "\\begin{align*}", "V \\times_{\\Spec(k)} V", "& =", "(\\Spec(k) \\times_{U/G} U) \\times_{\\Spec(k)}(\\Spec(k) \\times_{U/G} U) \\\\", "& = ", "\\Spec(k) \\times_{U/G} U \\times_{U/G} U \\\\", "& =", "\\Spec(k) \\times_{U/G} U \\times G \\\\", "& =", "V \\times G", "\\end{align*}", "The action of $G$ on $U$ induces an action of $a : G \\times V \\to V$.", "The displayed equality means that", "$G \\times V \\to V \\times_{\\Spec(k)} V$, $(g, v) \\mapsto (a(g, v), v)$", "is an isomorphism. In particular we see that for every $i$ we have", "an isomorphism $H_i \\times \\Spec(k_i) \\to \\Spec(k_i \\otimes_k k_i)$", "where $H_i \\subset G$ is the subgroup of elements fixing $i \\in I$.", "Thus $H_i$ is finite and is the Galois group of $k_i/k$.", "We omit the converse construction." ], "refs": [ "morphisms-lemma-etale-over-field" ], "ref_ids": [ 5364 ] } ], "ref_ids": [ 8164 ] }, { "id": 8167, "type": "theorem", "label": "spaces-lemma-change-big-site", "categories": [ "spaces" ], "title": "spaces-lemma-change-big-site", "contents": [ "Suppose given big sites $\\Sch_{fppf}$ and $\\Sch'_{fppf}$.", "Assume that $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$,", "see Topologies, Section \\ref{topologies-section-change-alpha}.", "Let $S$ be an object of $\\Sch_{fppf}$. Let", "\\begin{align*}", "g : \\Sh((\\Sch/S)_{fppf})", "\\longrightarrow", "\\Sh((\\Sch'/S)_{fppf}), \\\\", "f : \\Sh((\\Sch'/S)_{fppf})", "\\longrightarrow", "\\Sh((\\Sch/S)_{fppf})", "\\end{align*}", "be the morphisms of topoi of", "Topologies, Lemma \\ref{topologies-lemma-change-alpha}.", "Let $F$ be a sheaf of sets on $(\\Sch/S)_{fppf}$. Then", "\\begin{enumerate}", "\\item if $F$ is representable by a scheme", "$X \\in \\Ob((\\Sch/S)_{fppf})$ over $S$,", "then $f^{-1}F$ is representable too, in fact it is representable by the", "same scheme $X$, now viewed as an object of $(\\Sch'/S)_{fppf}$, and", "\\item if $F$ is an algebraic space over $S$, then $f^{-1}F$ is an algebraic", "space over $S$ also.", "\\end{enumerate}" ], "refs": [ "topologies-lemma-change-alpha" ], "proofs": [ { "contents": [ "Let $X \\in \\Ob((\\Sch/S)_{fppf})$. Let us write $h_X$ for the", "representable sheaf on $(\\Sch/S)_{fppf}$ associated to $X$, and", "$h'_X$ for the representable sheaf on $(\\Sch'/S)_{fppf}$ associated to", "$X$. By the description of $f^{-1}$ in", "Topologies, Section \\ref{topologies-section-change-alpha}", "we see that $f^{-1}h_X = h'_X$. This proves (1).", "\\medskip\\noindent", "Next, suppose that $F$", "is an algebraic space over $S$. By Lemma \\ref{lemma-space-presentation}", "this means that $F = h_U/h_R$ for some \\'etale equivalence relation", "$R \\to U \\times_S U$ in $(\\Sch/S)_{fppf}$. Since $f^{-1}$ is an", "exact functor we conclude that $f^{-1}F = h'_U/h'_R$. Hence", "$f^{-1}F$ is an algebraic space over $S$ by Theorem \\ref{theorem-presentation}." ], "refs": [ "spaces-lemma-space-presentation", "spaces-theorem-presentation" ], "ref_ids": [ 8149, 8124 ] } ], "ref_ids": [ 12516 ] }, { "id": 8168, "type": "theorem", "label": "spaces-lemma-fully-faithful", "categories": [ "spaces" ], "title": "spaces-lemma-fully-faithful", "contents": [ "Suppose $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$.", "Let $S$ be an object of $\\Sch_{fppf}$. Denote", "$\\textit{Spaces}/S$ the category of algebraic spaces over $S$", "defined using $\\Sch_{fppf}$. Similarly, denote", "$\\textit{Spaces}'/S$ the category of algebraic spaces over $S$", "defined using $\\Sch'_{fppf}$. The construction of", "Lemma \\ref{lemma-change-big-site}", "defines a fully faithful functor", "$$", "\\textit{Spaces}/S \\longrightarrow \\textit{Spaces}'/S", "$$", "whose essential image consists of those $X' \\in \\Ob(\\textit{Spaces}'/S)$", "such that there exist $U, R \\in \\Ob((\\Sch/S)_{fppf})$\\footnote{Requiring the", "existence of $R$ is necessary because of our choice of the function $Bound$ in", "Sets, Equation (\\ref{sets-equation-bound}). The size of the fibre product", "$U \\times_{X'} U$ can grow faster than $Bound$ in terms of the size of $U$. We", "can illustrate this by setting $S = \\Spec(A)$, $U = \\Spec(A[x_i, i \\in I])$ and", "$R = \\coprod_{(\\lambda_i) \\in A^I} \\Spec(A[x_i, y_i]/(x_i - \\lambda_i y_i))$.", "In this case the size of $R$ grows like $\\kappa^\\kappa$ where $\\kappa$ is the", "size of $U$.} and morphisms", "$$", "U \\longrightarrow X'", "\\quad\\text{and}\\quad", "R \\longrightarrow U \\times_{X'} U", "$$", "in $\\Sh((\\Sch'/S)_{fppf})$ which are surjective as maps of sheaves", "(for example if the displayed morphisms are surjective and \\'etale)." ], "refs": [ "spaces-lemma-change-big-site" ], "proofs": [ { "contents": [ "In Sites, Lemma \\ref{sites-lemma-bigger-site} we have seen that the functor", "$f^{-1} : \\Sh((\\Sch/S)_{fppf}) \\to \\Sh((\\Sch'/S)_{fppf})$", "is fully faithful (see discussion in", "Topologies, Section \\ref{topologies-section-change-alpha}).", "Hence we see that the displayed functor of the lemma is fully faithful.", "\\medskip\\noindent", "Suppose that $X' \\in \\Ob(\\textit{Spaces}'/S)$ such that there exists", "$U \\in \\Ob((\\Sch/S)_{fppf})$ and a map $U \\to X'$ in", "$\\Sh((\\Sch'/S)_{fppf})$ which is surjective as a map of sheaves.", "Let $U' \\to X'$ be a surjective \\'etale morphism with", "$U' \\in \\Ob((\\Sch'/S)_{fppf})$. Let $\\kappa = \\text{size}(U)$, see", "Sets, Section \\ref{sets-section-categories-schemes}.", "Then $U$ has an affine open covering $U = \\bigcup_{i \\in I} U_i$", "with $|I| \\leq \\kappa$. Observe that $U' \\times_{X'} U \\to U$ is \\'etale", "and surjective. For each $i$ we can pick a quasi-compact", "open $U'_i \\subset U'$ such that $U'_i \\times_{X'} U_i \\to U_i$", "is surjective (because the scheme $U' \\times_{X'} U_i$ is the", "union of the Zariski opens $W \\times_{X'} U_i$ for $W \\subset U'$", "affine and because $U' \\times_{X'} U_i \\to U_i$ is \\'etale hence open).", "Then $\\coprod_{i \\in I} U'_i \\to X$ is surjective \\'etale", "because of our assumption that $U \\to X$ and hence $\\coprod U_i \\to X$", "is a surjection of sheaves (details omitted).", "Because $U'_i \\times_{X'} U \\to U'_i$ is a surjection of sheaves", "and because $U'_i$ is quasi-compact,", "we can find a quasi-compact open $W_i \\subset U'_i \\times_{X'} U$", "such that $W_i \\to U'_i$ is surjective as a map of sheaves", "(details omitted). Then $W_i \\to U$ is \\'etale and we conclude", "that $\\text{size}(W_i) \\leq \\text{size}(U)$, see", "Sets, Lemma \\ref{sets-lemma-bound-finite-type}. By", "Sets, Lemma \\ref{sets-lemma-bound-by-covering} we conclude", "that $\\text{size}(U'_i) \\leq \\text{size}(U)$.", "Hence $\\coprod_{i \\in I} U'_i$ is isomorphic to an object of", "$(\\Sch/S)_{fppf}$ by Sets, Lemma \\ref{sets-lemma-bound-size}.", "\\medskip\\noindent", "Now let $X'$, $U \\to X'$ and $R \\to U \\times_{X'} U$ be as in the", "statement of the lemma. In the previous paragraph we have seen that", "we can find $U' \\in \\Ob((\\Sch/S)_{fppf})$ and a surjective \\'etale morphism", "$U' \\to X'$ in $\\Sh((\\Sch'/S)_{fppf})$. Then", "$U' \\times_{X'} U \\to U'$ is a surjection of sheaves, i.e., we can find an fppf", "covering $\\{U'_i \\to U'\\}$ such that $U'_i \\to U'$ factors through", "$U' \\times_{X'} U \\to U'$.", "By Sets, Lemma \\ref{sets-lemma-bound-fppf-covering}", "we can find", "$\\tilde U \\to U'$", "which is surjective, flat, and locally of finite presentation,", "with $\\text{size}(\\tilde U) \\leq \\text{size}(U')$,", "such that $\\tilde U \\to U'$ factors through", "$U' \\times_{X'} U \\to U'$. Then we consider", "$$", "\\xymatrix{", "U' \\times_{X'} U' \\ar[d] &", "\\tilde U \\times_{X'} \\tilde U \\ar[l] \\ar[d] \\ar[r] &", "U \\times_{X'} U \\ar[d] \\\\", "U' \\times_S U' & \\tilde U \\times_S \\tilde U \\ar[l] \\ar[r] & U \\times_S U", "}", "$$", "The squares are cartesian. We know the objects of the bottom row", "are represented by objects of $(\\Sch/S)_{fppf}$. By the result of the", "argument of the previous paragraph, the same is true for $U \\times_{X'} U$", "(as we have the surjection of sheaves $R \\to U \\times_{X'} U$", "by assumption). Since $(\\Sch/S)_{fppf}$ is closed under fibre", "products (by construction), we see that $\\tilde U \\times_{X'} \\tilde U$", "is represented by an object of $(\\Sch/S)_{fppf}$. Finally, the", "map $\\tilde U \\times_{X'} \\tilde U \\to U' \\times_{X'} U'$ is", "a surjection of fppf sheaves as $\\tilde U \\to U'$ is so.", "Thus we can once more apply the result of the previous paragraph", "to conclude that $R' = U' \\times_{X'} U'$ is represented by an object", "of $(\\Sch/S)_{fppf}$. At this point", "Lemma \\ref{lemma-space-presentation} and", "Theorem \\ref{theorem-presentation} imply that $X = h_{U'}/h_{R'}$", "is an object of $\\textit{Spaces}/S$ such that $f^{-1}X \\cong X'$", "as desired." ], "refs": [ "sites-lemma-bigger-site", "sets-lemma-bound-finite-type", "sets-lemma-bound-by-covering", "sets-lemma-bound-size", "sets-lemma-bound-fppf-covering", "spaces-lemma-space-presentation", "spaces-theorem-presentation" ], "ref_ids": [ 8548, 8793, 8796, 8791, 8797, 8149, 8124 ] } ], "ref_ids": [ 8167 ] }, { "id": 8169, "type": "theorem", "label": "spaces-lemma-change-base-scheme", "categories": [ "spaces" ], "title": "spaces-lemma-change-base-scheme", "contents": [ "Suppose given a big site $\\Sch_{fppf}$.", "Let $g : S \\to S'$ be morphism of $\\Sch_{fppf}$.", "Let $j : (\\Sch/S)_{fppf} \\to (\\Sch/S')_{fppf}$ be", "the corresponding localization functor.", "Let $F$ be a sheaf of sets on $(\\Sch/S)_{fppf}$.", "Then", "\\begin{enumerate}", "\\item for a scheme $T'$ over $S'$ we have", "$j_!F(T'/S') =", "\\coprod\\nolimits_{\\varphi : T' \\to S} F(T' \\xrightarrow{\\varphi} S),$", "\\item if $F$ is representable by a scheme", "$X \\in \\Ob((\\Sch/S)_{fppf})$,", "then $j_!F$ is representable by $j(X)$ which is", "$X$ viewed as a scheme over $S'$, and", "\\item if $F$ is an algebraic space over $S$, then $j_!F$ is an algebraic", "space over $S'$, and if $F = U/R$ is a presentation, then", "$j_!F = j(U)/j(R)$ is a presentation.", "\\end{enumerate}", "Let $F'$ be a sheaf of sets on $(\\Sch/S')_{fppf}$. Then", "\\begin{enumerate}", "\\item[(4)] for a scheme $T$ over $S$ we have $j^{-1}F'(T/S) = F'(T/S')$,", "\\item[(5)] if $F'$ is representable by a scheme", "$X' \\in \\Ob((\\Sch/S')_{fppf})$, then", "$j^{-1}F'$ is representable, namely by $X'_S = S \\times_{S'} X'$, and", "\\item[(6)] if $F'$ is an algebraic space, then", "$j^{-1}F'$ is an algebraic space, and if $F' = U'/R'$ is a presentation,", "then $j^{-1}F' = U'_S/R'_S$ is a presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The functors $j_!$, $j_*$ and $j^{-1}$ are defined in", "Sites, Lemma \\ref{sites-lemma-relocalize}", "where it is also shown that $j = j_{S/S'}$ is the localization", "of $(\\Sch/S')_{fppf}$ at the object $S/S'$. Hence", "all of the material on localization functors is available for $j$.", "The formula in (1) is", "Sites, Lemma \\ref{sites-lemma-describe-j-shriek-good-site}.", "By definition $j_!$ is the left adjoint to restriction $j^{-1}$,", "hence $j_!$ is right exact. By", "Sites, Lemma \\ref{sites-lemma-j-shriek-commutes-equalizers-fibre-products}", "it also commutes with fibre products and equalizers.", "By", "Sites, Lemma \\ref{sites-lemma-describe-j-shriek-representable}", "we see that $j_!h_X = h_{j(X)}$ hence (2) holds.", "If $F$ is an algebraic space over $S$, then we can write $F = U/R$", "(Lemma \\ref{lemma-space-presentation})", "and we get", "$$", "j_!F = j(U)/j(R)", "$$", "because $j_!$ being right exact commutes with coequalizers, and moreover", "$j(R) = j(U) \\times_{j_!F} j(U)$ as $j_!$ commutes with fibre products.", "Since the morphisms $j(s), j(t) : j(R) \\to j(U)$ are simply the morphisms", "$s, t : R \\to U$ (but viewed as morphisms of schemes over $S'$), they", "are still \\'etale. Thus $(j(U), j(R), s, t)$ is an \\'etale equivalence relation.", "Hence by", "Theorem \\ref{theorem-presentation}", "we conclude that $j_!F$ is an algebraic space.", "\\medskip\\noindent", "Proof of (4), (5), and (6). The description of $j^{-1}$ is in", "Sites, Section \\ref{sites-section-localize}.", "The restriction of the representable sheaf associated to $X'/S'$", "is the representable sheaf associated to", "$X'_S = S \\times_{S'} Y'$ by", "Sites, Lemma \\ref{sites-lemma-localize-given-products}.", "The restriction functor $j^{-1}$ is exact, hence $j^{-1}F' = U'_S/R'_S$.", "Again by exactness the sheaf $R'_S$ is still an equivalence relation on", "$U'_S$. Finally the two maps $R'_S \\to U'_S$ are \\'etale as base changes", "of the \\'etale morphisms $R' \\to U'$. Hence $j^{-1}F' = U'_S/R'_S$ is", "an algebraic space by", "Theorem \\ref{theorem-presentation}", "and we win." ], "refs": [ "sites-lemma-relocalize", "sites-lemma-describe-j-shriek-good-site", "sites-lemma-j-shriek-commutes-equalizers-fibre-products", "sites-lemma-describe-j-shriek-representable", "spaces-lemma-space-presentation", "spaces-theorem-presentation", "sites-lemma-localize-given-products", "spaces-theorem-presentation" ], "ref_ids": [ 8559, 8566, 8556, 8554, 8149, 8124, 8567, 8124 ] } ], "ref_ids": [] }, { "id": 8170, "type": "theorem", "label": "spaces-lemma-category-of-spaces-over-smaller-base-scheme", "categories": [ "spaces" ], "title": "spaces-lemma-category-of-spaces-over-smaller-base-scheme", "contents": [ "Let $\\Sch_{fppf}$ be a big fppf site.", "Let $S \\to S'$ be a morphism of this site.", "The construction above give an equivalence of", "categories", "$$", "\\left\\{", "\\begin{matrix}", "\\text{category of algebraic}\\\\", "\\text{spaces over }S", "\\end{matrix}", "\\right\\}", "\\leftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{category of pairs }(F', F' \\to S)\\text{ consisting}\\\\", "\\text{of an algebraic space }F'\\text{ over }S'\\text{ and a}\\\\", "\\text{morphism }F' \\to S\\text{ of algebraic spaces over }S'", "\\end{matrix}", "\\right\\}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $F$ be an algebraic space over $S$. The functor from left to right", "assigns the pair $(j_!F, j_!F \\to S)$ ot $F$", "which is an object of the right hand side by", "Lemma \\ref{lemma-change-base-scheme}.", "Since this defines an equivalence of categories of sheaves by", "Sites, Lemma \\ref{sites-lemma-essential-image-j-shriek}", "to finish the proof it suffices to show:", "if $F$ is a sheaf and $j_!F$ is an algebraic space, then $F$", "is an algebraic space. To do this, write", "$j_!F = U'/R'$ as in", "Lemma \\ref{lemma-space-presentation}", "with $U', R' \\in \\Ob((\\Sch/S')_{fppf})$.", "Then the compositions $U' \\to j_!F \\to S$ and $R' \\to j_!F \\to S$", "are morphisms of schemes over $S'$. Denote $U, R$ the corresponding", "objects of $(\\Sch/S)_{fppf}$. The two morphisms", "$R' \\to U'$ are morphisms over $S$ and hence correspond to", "morphisms $R \\to U$. Since these are simply the same", "morphisms (but viewed over $S$) we see that we get an \\'etale", "equivalence relation over $S$. As $j_!$ defines an equivalence of", "categories of sheaves (see reference above) we see that", "$F = U/R$ and by", "Theorem \\ref{theorem-presentation}", "we see that $F$ is an algebraic space." ], "refs": [ "spaces-lemma-change-base-scheme", "sites-lemma-essential-image-j-shriek", "spaces-lemma-space-presentation", "spaces-theorem-presentation" ], "ref_ids": [ 8169, 8555, 8149, 8124 ] } ], "ref_ids": [] }, { "id": 8171, "type": "theorem", "label": "spaces-lemma-rephrase", "categories": [ "spaces" ], "title": "spaces-lemma-rephrase", "contents": [ "Let $\\Sch_{fppf}$ be a big fppf site.", "Let $S \\to S'$ be a morphism of this site.", "Let $F'$ be a sheaf on $(\\Sch/S')_{fppf}$.", "The following are equivalent:", "\\begin{enumerate}", "\\item The restriction $F'|_{(\\Sch/S)_{fppf}}$", "is an algebraic space over $S$, and", "\\item the sheaf $h_S \\times F'$ is an algebraic space over $S'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The restriction and the product match under", "the equivalence of categories of", "Sites, Lemma \\ref{sites-lemma-essential-image-j-shriek}", "so that", "Lemma \\ref{lemma-category-of-spaces-over-smaller-base-scheme}", "above gives the result." ], "refs": [ "sites-lemma-essential-image-j-shriek", "spaces-lemma-category-of-spaces-over-smaller-base-scheme" ], "ref_ids": [ 8555, 8170 ] } ], "ref_ids": [] }, { "id": 8172, "type": "theorem", "label": "spaces-lemma-viewed-as-properties", "categories": [ "spaces" ], "title": "spaces-lemma-viewed-as-properties", "contents": [ "Let $\\Sch_{fppf}$ be a big fppf site.", "Let $S \\to S'$ be a morphism of this site.", "Let $F$ be an algebraic space over $S$.", "Let $T$ be a scheme over $S$ and let $f : T \\to F$ be", "a morphism over $S$.", "Let $f' : T' \\to F'$ be the morphism over $S'$ we get from", "$f$ by applying the equivalence of categories described in", "Lemma \\ref{lemma-category-of-spaces-over-smaller-base-scheme}.", "For any property $\\mathcal{P}$ as in", "Definition \\ref{definition-relative-representable-property}", "we have $\\mathcal{P}(f') \\Leftrightarrow \\mathcal{P}(f)$." ], "refs": [ "spaces-lemma-category-of-spaces-over-smaller-base-scheme", "spaces-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Suppose that $U$ is a scheme over $S$, and $U \\to F$ is a surjective \\'etale", "morphism. Denote $U'$ the scheme $U$ viewed as a scheme over $S'$. In", "Lemma \\ref{lemma-change-base-scheme}", "we have seen that $U' \\to F'$ is surjective \\'etale. Since", "$$", "j(T \\times_{f, F} U) = T' \\times_{f', F'} U'", "$$", "the morphism of schemes $T \\times_{f, F} U \\to U$", "is identified with the morphism of schemes", "$T' \\times_{f', F'} U' \\to U'$.", "It is the same morphism, just viewed over different base schemes.", "Hence the lemma follows from", "Lemma \\ref{lemma-representable-morphisms-spaces-property}." ], "refs": [ "spaces-lemma-change-base-scheme", "spaces-lemma-representable-morphisms-spaces-property" ], "ref_ids": [ 8169, 8157 ] } ], "ref_ids": [ 8170, 8173 ] }, { "id": 8188, "type": "theorem", "label": "topology-theorem-tychonov", "categories": [ "topology" ], "title": "topology-theorem-tychonov", "contents": [ "A product of quasi-compact spaces is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Let $I$ be a set and for $i \\in I$ let $X_i$ be a quasi-compact topological", "space. Set $X = \\prod X_i$. Let $\\mathcal{B}$ be the set of subsets of $X$", "of the form $U_i \\times \\prod_{i' \\in I, i' \\not = i} X_{i'}$ where", "$U_i \\subset X_i$ is open. By construction this family is a subbasis", "for the topology on $X$. By Lemma \\ref{lemma-subbase-theorem} it", "suffices to show that any covering $X = \\bigcup_{j \\in J} B_j$", "by elements $B_j$ of $\\mathcal{B}$ has a finite refinement.", "We can decompose $J = \\coprod J_i$ so that if $j \\in J_i$, then", "$B_j = U_j \\times \\prod_{i' \\not = i} X_{i'}$ with $U_j \\subset X_i$", "open. If $X_i = \\bigcup_{j \\in J_i} U_j$, then there is a finite", "refinement and we conclude that $X = \\bigcup_{j \\in J} B_j$", "has a finite refinement. If this is not the case, then for every $i$", "we can choose an point $x_i \\in X_i$ which is not in", "$\\bigcup_{j \\in J_i} U_j$. But then the point $x = (x_i)_{i \\in I}$", "is an element of $X$ not contained in $\\bigcup_{j \\in J} B_j$, a", "contradiction." ], "refs": [ "topology-lemma-subbase-theorem" ], "ref_ids": [ 8241 ] } ], "ref_ids": [] }, { "id": 8189, "type": "theorem", "label": "topology-theorem-characterize-proper", "categories": [ "topology" ], "title": "topology-theorem-characterize-proper", "contents": [ "\\begin{reference}", "In \\cite[I, p. 75, Theorem 1]{Bourbaki} you can find:", "(2) $\\Leftrightarrow$ (4).", "In \\cite[I, p. 77, Proposition 6]{Bourbaki} you can find:", "(2) $\\Rightarrow$ (1).", "\\end{reference}", "Let $f: X\\to Y$ be a continuous map between", "topological spaces. The following conditions are equivalent:", "\\begin{enumerate}", "\\item The map $f$ is quasi-proper and closed.", "\\item The map $f$ is proper.", "\\item The map $f$ is universally closed.", "\\item The map $f$ is closed and $f^{-1}(y)$ is quasi-compact for any", "$y\\in Y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "(See also the remark below.)", "If the map $f$ satisfies (1), it automatically satisfies (4) because", "any single point is quasi-compact.", "\\medskip\\noindent", "Assume map $f$ satisfies (4).", "We will prove it is universally closed, i.e., (3) holds.", "Let $g : Z \\to Y$ be a continuous map of topological spaces", "and consider the diagram", "$$", "\\xymatrix{", "Z \\times_Y X \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Z \\ar[r]^g & Y", "}", "$$", "During the proof we will use that $Z \\times_Y X \\to Z \\times X$", "is a homeomorphism onto its image, i.e., that we may identify", "$Z \\times_Y X$ with the corresponding subset of $Z \\times X$ with", "the induced topology.", "The image of $f' : Z \\times_Y X \\to Z$ is", "$\\Im(f') = \\{z : g(z) \\in f(X)\\}$.", "Because $f(X)$ is closed, we see that", "$\\Im(f')$ is a closed subspace of $Z$.", "Consider a closed subset $P \\subset Z \\times_Y X$.", "Let $z \\in Z$, $z \\not \\in f'(P)$.", "If $z \\not \\in \\Im(f')$, then $Z \\setminus \\Im(f')$", "is an open neighbourhood which avoids $f'(P)$.", "If $z$ is in $\\Im(f')$", "then $(f')^{-1}\\{z\\} = \\{z\\} \\times f^{-1}\\{g(z)\\}$", "and $f^{-1}\\{g(z)\\}$", "is quasi-compact by assumption. Because $P$ is a closed", "subset of $Z \\times_Y X$, we have a closed $P'$ of $Z \\times X$ such", "that $P = P' \\cap Z \\times_Y X$.", "Since $(f')^{-1}\\{z\\}$ is a subset of $P^c = P'^c \\cup (Z \\times_Y X)^c$,", "and since $(f')^{-1}\\{z\\}$ is disjoint from $(Z \\times_Y X)^c$", "we see that $(f')^{-1}\\{z\\}$ is contained in $P'^c$.", "We may apply the Tube Lemma \\ref{lemma-tube} to", "$(f')^{-1}\\{z\\} = \\{z\\} \\times f^{-1}\\{g(z)\\}", "\\subset (P')^c \\subset Z \\times X$.", "This gives $V \\times U$ containing", "$(f')^{-1}\\{z\\}$ where $U$ and $V$ are open sets in $X$ and $Z$", "respectively and $V \\times U$ has empty intersection with $P'$.", "Then the set $V \\cap g^{-1}(Y-f(U^c))$ is open in $Z$ since $f$", "is closed, contains $z$, and has empty intersection with the image of $P$.", "Thus $f'(P)$ is closed. In other words, the map $f$ is universally closed.", "\\medskip\\noindent", "The implication (3) $\\Rightarrow$ (2) is trivial.", "Namely, given any topological space $Z$ consider the projection", "morphism $g : Z \\times Y \\to Y$. Then it is easy to see", "that $f'$ is the map $Z \\times X \\to Z \\times Y$, in other", "words that $(Z \\times Y) \\times_Y X = Z \\times X$. (This identification", "is a purely categorical property having nothing to do with", "topological spaces per se.)", "\\medskip\\noindent", "Assume $f$ satisfies (2). We will prove it satisfies (1).", "Note that $f$ is closed as $f$ can be identified with the map", "$\\{pt\\} \\times X \\to \\{pt\\} \\times Y$ which is assumed closed.", "Choose any quasi-compact subset $K \\subset Y$.", "Let $Z$ be any topological space.", "Because $Z \\times X \\to Z \\times Y$ is closed", "we see the map $Z \\times f^{-1}(K) \\to Z \\times K$", "is closed (if $T$ is closed in $Z \\times f^{-1}(K)$, write", "$T = Z \\times f^{-1}(K) \\cap T'$ for some closed", "$T' \\subset Z \\times X$). Because $K$ is quasi-compact,", "$K \\times Z\\to Z$ is closed by Lemma \\ref{lemma-characterize-quasi-compact}.", "Hence the composition $Z \\times f^{-1}(K)\\to Z \\times K \\to Z$", "is closed and therefore $f^{-1}(K)$ must be quasi-compact", "by Lemma \\ref{lemma-characterize-quasi-compact} again." ], "refs": [ "topology-lemma-tube", "topology-lemma-characterize-quasi-compact", "topology-lemma-characterize-quasi-compact" ], "ref_ids": [ 8272, 8273, 8273 ] } ], "ref_ids": [] }, { "id": 8190, "type": "theorem", "label": "topology-lemma-Hausdorff", "categories": [ "topology" ], "title": "topology-lemma-Hausdorff", "contents": [ "Let $X$ be a topological space. The following are equivalent:", "\\begin{enumerate}", "\\item $X$ is Hausdorff,", "\\item the diagonal $\\Delta(X) \\subset X \\times X$ is closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8191, "type": "theorem", "label": "topology-lemma-graph-closed", "categories": [ "topology" ], "title": "topology-lemma-graph-closed", "contents": [ "\\begin{slogan}", "Graphs of maps to Hausdorff spaces are closed.", "\\end{slogan}", "Let $f : X \\to Y$ be a continuous map of topological spaces.", "If $Y$ is Hausdorff, then the graph of $f$ is closed in $X \\times Y$." ], "refs": [], "proofs": [ { "contents": [ "The graph is the inverse image of the diagonal under the map", "$X \\times Y \\to Y \\times Y$. Thus the lemma follows from", "Lemma \\ref{lemma-Hausdorff}." ], "refs": [ "topology-lemma-Hausdorff" ], "ref_ids": [ 8190 ] } ], "ref_ids": [] }, { "id": 8192, "type": "theorem", "label": "topology-lemma-section-closed", "categories": [ "topology" ], "title": "topology-lemma-section-closed", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Let $s : Y \\to X$ be a continuous map such that $f \\circ s = \\text{id}_Y$.", "If $X$ is Hausdorff, then $s(Y)$ is closed." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-Hausdorff} as", "$s(Y) = \\{x \\in X \\mid x = s(f(x))\\}$." ], "refs": [ "topology-lemma-Hausdorff" ], "ref_ids": [ 8190 ] } ], "ref_ids": [] }, { "id": 8193, "type": "theorem", "label": "topology-lemma-fibre-product-closed", "categories": [ "topology" ], "title": "topology-lemma-fibre-product-closed", "contents": [ "Let $X \\to Z$ and $Y \\to Z$ be continuous maps of topological spaces.", "If $Z$ is Hausdorff, then $X \\times_Z Y$ is closed in $X \\times Y$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-Hausdorff} as", "$X \\times_Z Y$ is the inverse image of $\\Delta(Z)$", "under $X \\times Y \\to Z \\times Z$." ], "refs": [ "topology-lemma-Hausdorff" ], "ref_ids": [ 8190 ] } ], "ref_ids": [] }, { "id": 8194, "type": "theorem", "label": "topology-lemma-separated", "categories": [ "topology" ], "title": "topology-lemma-separated", "contents": [ "Let $f : X \\to Y$ be continuous map of topological spaces.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is separated,", "\\item $\\Delta(X) \\subset X \\times_Y X$ is a closed subset,", "\\item given distinct points $x, x' \\in X$ mapping to the same point of", "$Y$, there exist disjoint open neighbourhoods of $x$ and $x'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8195, "type": "theorem", "label": "topology-lemma-from-hausdorff", "categories": [ "topology" ], "title": "topology-lemma-from-hausdorff", "contents": [ "Let $f : X \\to Y$ be continuous map of topological spaces.", "If $X$ is Hausdorff, then $f$ is separated." ], "refs": [], "proofs": [ { "contents": [ "Clear from Lemma \\ref{lemma-separated}." ], "refs": [ "topology-lemma-separated" ], "ref_ids": [ 8194 ] } ], "ref_ids": [] }, { "id": 8196, "type": "theorem", "label": "topology-lemma-base-change-separated", "categories": [ "topology" ], "title": "topology-lemma-base-change-separated", "contents": [ "Let $f : X \\to Y$ and $Z \\to Y$ be continuous maps of topological spaces.", "If $f$ is separated, then $f' : Z \\times_Y X \\to Z$ is separated." ], "refs": [], "proofs": [ { "contents": [ "Follows from characterization (3) of Lemma \\ref{lemma-separated}." ], "refs": [ "topology-lemma-separated" ], "ref_ids": [ 8194 ] } ], "ref_ids": [] }, { "id": 8197, "type": "theorem", "label": "topology-lemma-make-base", "categories": [ "topology" ], "title": "topology-lemma-make-base", "contents": [ "Let $X$ be a set and let $\\mathcal{B}$ be a collection of subsets.", "Assume that $X = \\bigcup_{B \\in \\mathcal{B}} B$ and that given", "$x \\in B_1 \\cap B_2$ with $B_1, B_2 \\in \\mathcal{B}$ there is a", "$B_3 \\in \\mathcal{B}$ with $x \\in B_3 \\subset B_1 \\cap B_2$.", "Then there is a unique topology on $X$ such that $\\mathcal{B}$", "is a basis for this topology." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8198, "type": "theorem", "label": "topology-lemma-refine-covering-basis", "categories": [ "topology" ], "title": "topology-lemma-refine-covering-basis", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{B}$ be a basis for the topology on $X$.", "Let $\\mathcal{U} : U = \\bigcup_i U_i$ be an open covering of", "$U \\subset X$. There exists an open covering $U = \\bigcup V_j$", "which is a refinement of $\\mathcal{U}$ such that each", "$V_j$ is an element of the basis $\\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8199, "type": "theorem", "label": "topology-lemma-subbase", "categories": [ "topology" ], "title": "topology-lemma-subbase", "contents": [ "Let $X$ be a set. Given any collection $\\mathcal{B}$ of subsets of $X$", "there is a unique topology on $X$ such that $\\mathcal{B}$ is a subbase", "for this topology." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8200, "type": "theorem", "label": "topology-lemma-create-map-from-subcollection", "categories": [ "topology" ], "title": "topology-lemma-create-map-from-subcollection", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{B}$ be a collection", "of opens of $X$. Assume $X = \\bigcup_{U \\in \\mathcal{B}} U$ and", "for $U, V \\in \\mathcal{B}$ we have", "$U \\cap V = \\bigcup_{W \\in \\mathcal{B}, W \\subset U \\cap V} W$.", "Then there is a continuous map $f : X \\to Y$ of topological spaces", "such that", "\\begin{enumerate}", "\\item for $U \\in \\mathcal{B}$ the image $f(U)$ is open,", "\\item for $U \\in \\mathcal{B}$ we have $f^{-1}(f(U)) = U$, and", "\\item the opens $f(U)$, $U \\in \\mathcal{B}$", "form a basis for the topology on $Y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Define an equivalence relation $\\sim$ on points of $X$", "by the rule", "$$", "x \\sim y \\Leftrightarrow", "(\\forall U \\in \\mathcal{B} : x \\in U \\Leftrightarrow y \\in U)", "$$", "Let $Y$ be the set of equivalence classes and $f : X \\to Y$", "the natural map. Part (2) holds by construction.", "The assumptions on $\\mathcal{B}$ exactly", "mirror the assumptions in Lemma \\ref{lemma-make-base}", "on the set of subsets $f(U)$, $U \\in \\mathcal{B}$.", "Hence there is a unique topology on $Y$ such that (3) holds.", "Then (1) is clear as well." ], "refs": [ "topology-lemma-make-base" ], "ref_ids": [ 8197 ] } ], "ref_ids": [] }, { "id": 8201, "type": "theorem", "label": "topology-lemma-induced", "categories": [ "topology" ], "title": "topology-lemma-induced", "contents": [ "Let $X$ be a topological space. Let $Y$ be a set and let", "$f : Y \\to X$ be an injective map of sets. The induced", "topology on $Y$ is the topology characterized by", "each of the following statements:", "\\begin{enumerate}", "\\item it is the weakest topology on $Y$ such that $f$ is continuous,", "\\item the open subsets of $Y$ are $f^{-1}(U)$ for $U \\subset X$ open,", "\\item the closed subsets of $Y$ are the sets $f^{-1}(Z)$ for $Z \\subset X$", "closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8202, "type": "theorem", "label": "topology-lemma-quotient", "categories": [ "topology" ], "title": "topology-lemma-quotient", "contents": [ "Let $X$ be a topological space. Let $Y$ be a set and let $f : X \\to Y$", "be a surjective map of sets. The quotient topology on $Y$ is the", "topology characterized by each of the following statements:", "\\begin{enumerate}", "\\item it is the strongest topology on $Y$ such that $f$ is continuous,", "\\item a subset $V$ of $Y$ is open if and only if $f^{-1}(V)$ is open,", "\\item a subset $Z$ of $Y$ is closed if and only if $f^{-1}(Z)$ is closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8203, "type": "theorem", "label": "topology-lemma-open-morphism-quotient-topology", "categories": [ "topology" ], "title": "topology-lemma-open-morphism-quotient-topology", "contents": [ "Let $f : X \\to Y$ be surjective, open, continuous map of topological spaces.", "Let $T \\subset Y$ be a subset. Then", "\\begin{enumerate}", "\\item $f^{-1}(\\overline{T}) = \\overline{f^{-1}(T)}$,", "\\item $T \\subset Y$ is closed if and only if $f^{-1}(T)$ is closed,", "\\item $T \\subset Y$ is open if and only if $f^{-1}(T)$ is open, and", "\\item $T \\subset Y$ is locally closed if and only if $f^{-1}(T)$", "is locally closed.", "\\end{enumerate}", "In particular we see that $f$ is submersive." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $\\overline{f^{-1}(T)} \\subset f^{-1}(\\overline{T})$.", "If $x \\in X$, and $x \\not \\in \\overline{f^{-1}(T)}$, then there", "exists an open neighbourhood $x \\in U \\subset X$ with", "$U \\cap f^{-1}(T) = \\emptyset$. Since $f$ is open we see that", "$f(U)$ is an open neighbourhood of $f(x)$ not meeting $T$.", "Hence $x \\not \\in f^{-1}(\\overline{T})$. This proves (1).", "Part (2) is an easy consequence of (1).", "Part (3) is obvious from the fact that $f$ is open and surjective.", "For (4), if $f^{-1}(T)$ is locally closed, then", "$f^{-1}(T) \\subset \\overline{f^{-1}(T)} = f^{-1}(\\overline{T})$", "is open, and hence by (3) applied to the map", "$f^{-1}(\\overline{T}) \\to \\overline{T}$ we see that", "$T$ is open in $\\overline{T}$, i.e., $T$ is locally closed." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8204, "type": "theorem", "label": "topology-lemma-closed-morphism-quotient-topology", "categories": [ "topology" ], "title": "topology-lemma-closed-morphism-quotient-topology", "contents": [ "Let $f : X \\to Y$ be surjective, closed, continuous map of topological spaces.", "Let $T \\subset Y$ be a subset. Then", "\\begin{enumerate}", "\\item $\\overline{T} = f(\\overline{f^{-1}(T)})$,", "\\item $T \\subset Y$ is closed if and only if $f^{-1}(T)$ is closed,", "\\item $T \\subset Y$ is open if and only if $f^{-1}(T)$ is open, and", "\\item $T \\subset Y$ is locally closed if and only if", "$f^{-1}(T)$ is locally closed.", "\\end{enumerate}", "In particular we see that $f$ is submersive." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $\\overline{f^{-1}(T)} \\subset f^{-1}(\\overline{T})$.", "Then $T \\subset f(\\overline{f^{-1}(T)}) \\subset \\overline{T}$", "is a closed subset, hence we get (1). Part (2) is obvious from", "the fact that $f$ is closed and surjective.", "Part (3) follows from (2) applied to the complement of $T$.", "For (4), if $f^{-1}(T)$ is locally closed, then", "$f^{-1}(T) \\subset \\overline{f^{-1}(T)}$ is open.", "Since the map $\\overline{f^{-1}(T)} \\to \\overline{T}$ is surjective", "by (1) we can apply part (3) to the map $\\overline{f^{-1}(T)} \\to \\overline{T}$", "induced by $f$ to conclude that $T$ is open in", "$\\overline{T}$, i.e., $T$ is locally closed." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8205, "type": "theorem", "label": "topology-lemma-image-connected-space", "categories": [ "topology" ], "title": "topology-lemma-image-connected-space", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "If $E \\subset X$ is a connected subset, then $f(E) \\subset Y$", "is connected as well." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8206, "type": "theorem", "label": "topology-lemma-connected-components", "categories": [ "topology" ], "title": "topology-lemma-connected-components", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item If $T \\subset X$ is connected, then so is its closure.", "\\item Any connected component of $X$ is closed (but not necessarily open).", "\\item Every connected subset of $X$ is contained in a unique connected", "component of $X$.", "\\item Every point of $X$ is contained in a unique connected component, in other", "words, $X$ is the union of its connected components.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\overline{T}$ be the closure of the connected subset $T$.", "Suppose $\\overline{T} = T_1 \\amalg T_2$ with $T_i \\subset \\overline{T}$", "open and closed. Then $T = (T\\cap T_1) \\amalg (T \\cap T_2)$. Hence", "$T$ equals one of the two, say $T = T_1 \\cap T$. Thus clearly", "$\\overline{T} \\subset T_1$ as desired.", "\\medskip\\noindent", "Pick a point $x\\in X$. Consider the set $A$ of connected subsets", "$x \\in T_\\alpha \\subset X$. Note that $A$ is nonempty since", "$\\{x\\} \\in A$. There is a partial ordering on $A$ coming from", "inclusion: $\\alpha \\leq \\alpha' \\Leftrightarrow T_\\alpha \\subset T_{\\alpha'}$.", "Choose a maximal totally ordered subset $A' \\subset A$, and let", "$T = \\bigcup_{\\alpha \\in A'} T_\\alpha$. We claim that $T$ is", "connected. Namely, suppose that $T = T_1 \\amalg T_2$ is a disjoint", "union of two open and closed subsets of $T$.", "For each $\\alpha \\in A'$ we have either $T_\\alpha \\subset T_1$", "or $T_\\alpha \\subset T_2$, by connectedness of $T_\\alpha$.", "Suppose that for some $\\alpha_0 \\in A'$ we have", "$T_{\\alpha_0} \\not\\subset T_1$ (say, if not we're done anyway).", "Then, since $A'$ is totally ordered we see immediately that", "$T_\\alpha \\subset T_2$ for all $\\alpha \\in A'$. Hence $T = T_2$.", "\\medskip\\noindent", "To get an example where connected components are not open, just take", "an infinite product $\\prod_{n \\in \\mathbf{N}} \\{0, 1\\}$", "with the product topology. Its connected components are singletons,", "which are not open." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8207, "type": "theorem", "label": "topology-lemma-connected-fibres-quotient-topology-connected-components", "categories": [ "topology" ], "title": "topology-lemma-connected-fibres-quotient-topology-connected-components", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Assume that", "\\begin{enumerate}", "\\item all fibres of $f$ are connected, and", "\\item a set $T \\subset Y$ is closed if and only if $f^{-1}(T)$ is closed.", "\\end{enumerate}", "Then $f$ induces a bijection between the sets of connected", "components of $X$ and $Y$." ], "refs": [], "proofs": [ { "contents": [ "Let $T \\subset Y$ be a connected component.", "Note that $T$ is closed, see Lemma \\ref{lemma-connected-components}.", "The lemma follows if we show that $f^{-1}(T)$ is connected", "because any connected subset of $X$ maps into a connected component", "of $Y$ by Lemma \\ref{lemma-image-connected-space}.", "Suppose that $f^{-1}(T) = Z_1 \\amalg Z_2$", "with $Z_1$, $Z_2$ closed. For any $t \\in T$ we see that", "$f^{-1}(\\{t\\}) = Z_1 \\cap f^{-1}(\\{t\\}) \\amalg Z_2 \\cap f^{-1}(\\{t\\})$.", "By (1) we see $f^{-1}(\\{t\\})$ is connected we conclude that", "either $f^{-1}(\\{t\\}) \\subset Z_1$ or $f^{-1}(\\{t\\}) \\subset Z_2$.", "In other words $T = T_1 \\amalg T_2$ with $f^{-1}(T_i) = Z_i$.", "By (2) we conclude that $T_i$ is closed in $Y$.", "Hence either $T_1 = \\emptyset$ or $T_2 = \\emptyset$ as desired." ], "refs": [ "topology-lemma-connected-components", "topology-lemma-image-connected-space" ], "ref_ids": [ 8206, 8205 ] } ], "ref_ids": [] }, { "id": 8208, "type": "theorem", "label": "topology-lemma-connected-fibres-connected-components", "categories": [ "topology" ], "title": "topology-lemma-connected-fibres-connected-components", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Assume that", "(a) $f$ is open,", "(b) all fibres of $f$ are connected.", "Then $f$ induces a bijection between the sets of connected", "components of $X$ and $Y$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-connected-fibres-quotient-topology-connected-components}." ], "refs": [ "topology-lemma-connected-fibres-quotient-topology-connected-components" ], "ref_ids": [ 8207 ] } ], "ref_ids": [] }, { "id": 8209, "type": "theorem", "label": "topology-lemma-finite-fibre-connected-components", "categories": [ "topology" ], "title": "topology-lemma-finite-fibre-connected-components", "contents": [ "Let $f : X \\to Y$ be a continuous map of nonempty topological spaces. Assume", "that", "(a) $Y$ is connected,", "(b) $f$ is open and closed, and", "(c) there is a point $y\\in Y$ such that the fiber $f^{-1}(y)$ is a finite set.", "Then $X$ has at most $|f^{-1}(y)|$ connected components. Hence any connected ", "component $T$ of $X$ is open and closed, and $f(T)$ is a nonempty open and ", "closed subset of $Y$, which is therefore equal to $Y$." ], "refs": [], "proofs": [ { "contents": [ "If the topological space $X$ has at least $N$ connected components for some", "$N \\in \\mathbf{N}$, we find by induction a decomposition", "$X = X_1 \\amalg \\ldots \\amalg X_N$ of $X$ as a disjoint union of $N$ nonempty", "open and closed subsets $X_1, \\ldots , X_N$ of $X$. As $f$ is open and closed,", "each $f(X_i)$ is a nonempty open and closed subset of $Y$ and is hence equal to", "$Y$. In particular the intersection $X_i \\cap f^{-1}(y)$ is nonempty for each", "$1 \\leq i \\leq N$. Hence $f^{-1}(y)$ has at least $N$ elements." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8210, "type": "theorem", "label": "topology-lemma-space-connected-components", "categories": [ "topology" ], "title": "topology-lemma-space-connected-components", "contents": [ "Let $X$ be a topological space. Let $\\pi_0(X)$ be the set of connected", "components of $X$. Let $X \\to \\pi_0(X)$ be the map which sends", "$x \\in X$ to the connected component of $X$ passing through $x$.", "Endow $\\pi_0(X)$ with the quotient topology. Then $\\pi_0(X)$ is a", "totally disconnected space and any continuous map $X \\to Y$", "from $X$ to a totally disconnected space $Y$ factors through $\\pi_0(X)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma", "\\ref{lemma-connected-fibres-quotient-topology-connected-components}", "the connected components of $\\pi_0(X)$ are the singletons.", "We omit the proof of the second statement." ], "refs": [ "topology-lemma-connected-fibres-quotient-topology-connected-components" ], "ref_ids": [ 8207 ] } ], "ref_ids": [] }, { "id": 8211, "type": "theorem", "label": "topology-lemma-locally-connected", "categories": [ "topology" ], "title": "topology-lemma-locally-connected", "contents": [ "Let $X$ be a topological space. If $X$ is locally connected, then", "\\begin{enumerate}", "\\item any open subset of $X$ is locally connected, and", "\\item the connected components of $X$ are open.", "\\end{enumerate}", "So also the connected components of open subsets of $X$ are open.", "In particular, every point has a fundamental system of open connected", "neighbourhoods." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8212, "type": "theorem", "label": "topology-lemma-image-irreducible-space", "categories": [ "topology" ], "title": "topology-lemma-image-irreducible-space", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "If $E \\subset X$ is an irreducible subset, then $f(E) \\subset Y$", "is irreducible as well." ], "refs": [], "proofs": [ { "contents": [ "Clearly we may assume $E = X$ (i.e., $X$ irreducible)", "and $f(E) = Y$ (i.e., $f$ surjective). First, $Y \\not = \\emptyset$", "since $X \\not = \\emptyset$. Next, assume $Y = Y_1 \\cup Y_2$ with", "$Y_1$, $Y_2$ closed. Then $X = X_1 \\cup X_2$ with $X_i = f^{-1}(Y_i)$", "closed in $X$. By assumption on $X$, we must have", "$X = X_1$ or $X = X_2$, hence $Y = Y_1$ or $Y = Y_2$", "since $f$ is surjective." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8213, "type": "theorem", "label": "topology-lemma-irreducible", "categories": [ "topology" ], "title": "topology-lemma-irreducible", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item If $T \\subset X$ is irreducible so is its closure in $X$.", "\\item Any irreducible component of $X$ is closed.", "\\item Any irreducible subset of $X$ is contained in an", "irreducible component of $X$.", "\\item Every point of $X$ is contained in some irreducible component", "of $X$, in other words, $X$ is the union of its irreducible components.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\overline{T}$ be the closure of the irreducible subset $T$.", "If $\\overline{T} = Z_1 \\cup Z_2$ with $Z_i \\subset \\overline{T}$", "closed, then $T = (T\\cap Z_1) \\cup (T \\cap Z_2)$ and hence", "$T$ equals one of the two, say $T = Z_1 \\cap T$. Thus clearly", "$\\overline{T} \\subset Z_1$. This proves (1). Part (2) follows", "immediately from (1) and the definition of irreducible components.", "\\medskip\\noindent", "Let $T \\subset X$ be irreducible. Consider the set $A$ of irreducible subsets", "$T \\subset T_\\alpha \\subset X$. Note that $A$ is nonempty since", "$T \\in A$. There is a partial ordering on $A$ coming from", "inclusion: $\\alpha \\leq \\alpha' \\Leftrightarrow T_\\alpha \\subset T_{\\alpha'}$.", "Choose a maximal totally ordered subset $A' \\subset A$, and let", "$T' = \\bigcup_{\\alpha \\in A'} T_\\alpha$. We claim that $T'$ is", "irreducible. Namely, suppose that $T' = Z_1 \\cup Z_2$ is a union", "of two closed subsets of $T'$. For each $\\alpha \\in A'$ we have", "either $T_\\alpha \\subset Z_1$ or $T_\\alpha \\subset Z_2$, by irreducibility", "of $T_\\alpha$. Suppose that for some $\\alpha_0 \\in A'$ we have", "$T_{\\alpha_0} \\not\\subset Z_1$ (say, if not we're done anyway).", "Then, since $A'$ is totally ordered we see immediately that", "$T_\\alpha \\subset Z_2$ for all $\\alpha \\in A'$. Hence $T' = Z_2$.", "This proves (3). Part (4) is an immediate consequence of (3)", "as a singleton space is irreducible." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8214, "type": "theorem", "label": "topology-lemma-pick-irreducible-components", "categories": [ "topology" ], "title": "topology-lemma-pick-irreducible-components", "contents": [ "Let $X$ be a topological space and suppose $X = \\bigcup_{i = 1, \\ldots, n} X_i$", "where each $X_i$ is an irreducible closed subset of $X$ and no $X_i$", "is contained in the union of the other members. Then each $X_i$ is an", "irreducible component of $X$ and each irreducible component of $X$", "is one of the $X_i$." ], "refs": [], "proofs": [ { "contents": [ "Let $Y$ be an irreducible component of $X$. Write", "$Y = \\bigcup_{i = 1, \\ldots, n} (Y \\cap X_i)$", "and note that each $Y \\cap X_i$ is closed in $Y$ since $X_i$ is closed in $X$.", "By irreducibility of $Y$ we see that $Y$ is equal to one of the $Y \\cap X_i$,", "i.e., $Y \\subset X_i$. By maximality of irreducible components we", "get $Y = X_i$.", "\\medskip\\noindent", "Conversely, take one of the $X_i$ and expand it to an irreducible component", "$Y$, which we have already shown is one of the $X_j$.", "So $X_i \\subset X_j$ and since the original union does not have", "redundant members, $X_i = X_j$, which is an irreducible component." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8215, "type": "theorem", "label": "topology-lemma-sober-subspace", "categories": [ "topology" ], "title": "topology-lemma-sober-subspace", "contents": [ "Let $X$ be a topological space and let $Y\\subset X$.", "\\begin{enumerate}", "\\item If $X$ is Kolmogorov then so is $Y$.", "\\item Suppose $Y$ is locally closed in $X$. If $X$ is quasi-sober then", "so is $Y$.", "\\item Suppose $Y$ is locally closed in $X$. If $X$ is sober then so is $Y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Suppose $X$ is Kolmogorov. Let $x,y\\in Y$ with $x\\neq y$. Then", "$\\overline{\\overline{\\{x\\}}\\cap Y}=\\overline{\\{x\\}}\\neq\\overline{\\{y\\}}=", "\\overline{\\overline{\\{y\\}}\\cap Y}$. Hence", "$\\overline{\\{x\\}}\\cap Y\\neq\\overline{\\{y\\}}\\cap Y$. This shows that $Y$ is", "Kolmogorov.", "\\medskip\\noindent", "Proof of (2). Suppose $X$ is quasi-sober. It suffices to consider the", "cases $Y$ is closed and $Y$ is open. First, suppose $Y$ is closed. Let", "$Z$ be an irreducible closed subset of $Y$. Then $Z$ is an irreducible closed", "subset of $X$. Hence there exists $x \\in Z$ with $\\overline{\\{x\\}} = Z$. It", "follows $\\overline{\\{x\\}} \\cap Y = Z$. This shows $Y$ is quasi-sober. Second,", "suppose $Y$ is open. Let $Z$ be an irreducible closed subset of $Y$. Then", "$\\overline{Z}$ is an irreducible closed subset of $X$. Hence there", "exists $x \\in \\overline{Z}$ with $\\overline{\\{x\\}}=\\overline{Z}$. If", "$x\\notin Y$ we get the contradiction", "$Z=Z\\cap Y\\subset\\overline{Z}\\cap Y=\\overline{\\{x\\}}\\cap Y=\\emptyset$.", "Therefore $x\\in Y$. It follows $Z=\\overline{Z}\\cap Y=\\overline{\\{x\\}}\\cap Y$.", "This shows $Y$ is quasi-sober.", "\\medskip\\noindent", "Proof of (3). Immediately from (1) and (2)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8216, "type": "theorem", "label": "topology-lemma-sober-local", "categories": [ "topology" ], "title": "topology-lemma-sober-local", "contents": [ "Let $X$ be a topological space and let $(X_i)_{i\\in I}$ be a covering of $X$.", "\\begin{enumerate}", "\\item Suppose $X_i$ is locally closed in $X$ for every $i\\in I$. Then, $X$ is", "Kolmogorov if and only if $X_i$ is Kolmogorov for every $i\\in I$.", "\\item Suppose $X_i$ is open in $X$ for every $i\\in I$. Then, $X$ is", "quasi-sober if and only if $X_i$ is quasi-sober for every $i\\in I$.", "\\item Suppose $X_i$ is open in $X$ for every $i\\in I$. Then, $X$ is sober if", "and only if $X_i$ is sober for every $i\\in I$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). If $X$ is Kolmogorov then so is $X_i$ for every $i\\in I$ by", "Lemma \\ref{lemma-sober-subspace}. Suppose $X_i$ is Kolmogorov for every", "$i\\in I$. Let $x,y\\in X$ with $\\overline{\\{x\\}}=\\overline{\\{y\\}}$. There exists", "$i\\in I$ with $x\\in X_i$. There exists an open subset $U\\subset X$ such that", "$X_i$ is a closed subset of $U$. If $y\\notin U$ we get the contradiction", "$x\\in\\overline{\\{x\\}}\\cap U=\\overline{\\{y\\}}\\cap U=\\emptyset$. Hence $y\\in U$.", "It follows $y\\in\\overline{\\{y\\}}\\cap U=\\overline{\\{x\\}}\\cap U\\subset X_i$.", "This shows $y\\in X_i$. It follows", "$\\overline{\\{x\\}}\\cap X_i=\\overline{\\{y\\}}\\cap X_i$. Since $X_i$ is", "Kolmogorov we get $x=y$. This shows $X$ is Kolmogorov.", "\\medskip\\noindent", "Proof of (2). If $X$ is quasi-sober then so is $X_i$ for every $i\\in I$ by", "Lemma \\ref{lemma-sober-subspace}. Suppose $X_i$ is quasi-sober for every", "$i\\in I$. Let $Y$ be an irreducible closed subset of $X$. As $Y\\neq\\emptyset$", "there exists $i\\in I$ with $X_i\\cap Y\\neq\\emptyset$. As $X_i$ is open in $X$ it", "follows $X_i\\cap Y$ is non-empty and open in $Y$, hence irreducible", "and dense in $Y$. Thus $X_i\\cap Y$ is an irreducible closed subset of $X_i$. As", "$X_i$ is quasi-sober there exists $x\\in X_i\\cap Y$ with", "$X_i\\cap Y=\\overline{\\{x\\}}\\cap X_i\\subset\\overline{\\{x\\}}$. Since", "$X_i\\cap Y$ is dense in $Y$ and $Y$ is closed in $X$ it follows", "$Y=\\overline{X_i\\cap Y}\\cap Y\\subset\\overline{X_i\\cap Y}\\subset", "\\overline{\\{x\\}}\\subset Y$. Therefore", "$Y=\\overline{\\{x\\}}$. This shows $X$ is quasi-sober.", "\\medskip\\noindent", "Proof of (3). Immediately from (1) and (2)." ], "refs": [ "topology-lemma-sober-subspace", "topology-lemma-sober-subspace" ], "ref_ids": [ 8215, 8215 ] } ], "ref_ids": [] }, { "id": 8217, "type": "theorem", "label": "topology-lemma-irreducible-on-top", "categories": [ "topology" ], "title": "topology-lemma-irreducible-on-top", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Assume that", "(a) $Y$ is irreducible,", "(b) $f$ is open, and", "(c) there exists a dense collection of points $y \\in Y$ such", "that $f^{-1}(y)$ is irreducible.", "Then $X$ is irreducible." ], "refs": [], "proofs": [ { "contents": [ "Suppose $X = Z_1 \\cup Z_2$ with $Z_i$ closed.", "Consider the open sets $U_1 = Z_1 \\setminus Z_2 = X \\setminus Z_2$ and", "$U_2 = Z_2 \\setminus Z_1 = X \\setminus Z_1$. To get a contradiction", "assume that $U_1$ and $U_2$ are both nonempty. By (b) we see that $f(U_i)$", "is open. By (a) we have $Y$ irreducible and hence", "$f(U_1) \\cap f(U_2) \\not = \\emptyset$. By (c) there is a point $y$ which", "corresponds to a point of this intersection such that the fibre", "$X_y = f^{-1}(y)$ is irreducible. Then $X_y \\cap U_1$ and", "$X_y \\cap U_2$ are nonempty disjoint open subsets of $X_y$ which is", "a contradiction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8218, "type": "theorem", "label": "topology-lemma-irreducible-fibres-irreducible-components", "categories": [ "topology" ], "title": "topology-lemma-irreducible-fibres-irreducible-components", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Assume that (a) $f$ is open, and", "(b) for every $y \\in Y$ the fibre $f^{-1}(y)$ is irreducible.", "Then $f$ induces a bijection between irreducible components." ], "refs": [], "proofs": [ { "contents": [ "We point out that assumption (b) implies that $f$ is surjective (see", "Definition \\ref{definition-irreducible-components}).", "Let $T \\subset Y$ be an irreducible component.", "Note that $T$ is closed, see Lemma \\ref{lemma-irreducible}.", "The lemma follows if we show that $f^{-1}(T)$ is irreducible", "because any irreducible subset of $X$ maps into an irreducible component", "of $Y$ by Lemma \\ref{lemma-image-irreducible-space}.", "Note that $f^{-1}(T) \\to T$ satisfies the assumptions", "of Lemma \\ref{lemma-irreducible-on-top}. Hence we win." ], "refs": [ "topology-definition-irreducible-components", "topology-lemma-irreducible", "topology-lemma-image-irreducible-space", "topology-lemma-irreducible-on-top" ], "ref_ids": [ 8353, 8213, 8212, 8217 ] } ], "ref_ids": [] }, { "id": 8219, "type": "theorem", "label": "topology-lemma-make-sober", "categories": [ "topology" ], "title": "topology-lemma-make-sober", "contents": [ "Let $X$ be a topological space. There is a canonical continuous map", "$$", "c : X \\longrightarrow X'", "$$", "from $X$ to a sober topological space $X'$ which is universal", "among continuous maps from $X$ to sober topological spaces.", "Moreover, the assignment $U' \\mapsto c^{-1}(U')$ is a bijection", "between opens of $X'$ and $X$ which commutes with finite intersections", "and arbitrary unions.", "The image $c(X)$ is a Kolmogorov topological space and the", "map $c : X \\to c(X)$ is universal for maps of $X$ into Kolmogorov spaces." ], "refs": [], "proofs": [ { "contents": [ "Let $X'$ be the set of irreducible closed subsets of $X$ and let", "$$", "c : X \\to X', \\quad x \\mapsto \\overline{\\{x\\}}.", "$$", "For $U \\subset X$ open, let $U' \\subset X'$ denote the set", "of irreducible closed subsets of $X$ which meet $U$.", "Then $c^{-1}(U') = U$. In particular, if $U_1 \\not = U_2$ are open in", "$X$, then $U'_1 \\not = U_2'$. Hence $c$ induces", "a bijection between the subsets of $X'$ of the form $U'$ and the", "opens of $X$.", "\\medskip\\noindent", "Let $U_1, U_2$ be open in $X$. Suppose that $Z \\in U'_1$ and", "$Z \\in U'_2$. Then $Z \\cap U_1$ and $Z \\cap U_2$ are nonempty", "open subsets of the irreducible space $Z$ and hence $Z \\cap U_1 \\cap U_2$", "is nonempty. Thus $(U_1 \\cap U_2)' = U'_1 \\cap U'_2$.", "The rule $U \\mapsto U'$ is also compatible with arbitrary unions", "(details omitted). Thus it is clear that the collection of", "$U'$ form a topology on $X'$ and that we have a bijection as", "stated in the lemma.", "\\medskip\\noindent", "Next we show that $X'$ is sober. Let $T \\subset X'$ be an irreducible", "closed subset. Let $U \\subset X$ be the open such that $X' \\setminus T = U'$.", "Then $Z = X \\setminus U$ is irreducible because of the properties", "of the bijection of the lemma. We claim that $Z \\in T$ is the unique generic", "point. Namely, any open of the form $V' \\subset X'$", "which does not contain $Z$ must come from an open $V \\subset X$", "which misses $Z$, i.e., is contained in $U$.", "\\medskip\\noindent", "Finally, we check the universal property. Let $f : X \\to Y$ be a continuous", "map to a sober topological space. Then we let $f' : X' \\to Y$ be the map", "which sends the irreducible closed $Z \\subset X$ to the unique generic", "point of $\\overline{f(Z)}$. It follows immediately that", "$f' \\circ c = f$ as maps of sets, and the properties of $c$ imply that", "$f'$ is continuous. We omit the verification that the continuous", "map $f'$ is unique. We also omit the proof of the statements on", "Kolmogorov spaces." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8220, "type": "theorem", "label": "topology-lemma-Noetherian", "categories": [ "topology" ], "title": "topology-lemma-Noetherian", "contents": [ "Let $X$ be a Noetherian topological space.", "\\begin{enumerate}", "\\item Any subset of $X$ with the induced topology is Noetherian.", "\\item The space $X$ has finitely many irreducible components.", "\\item Each irreducible component of $X$ contains a nonempty open of $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $T \\subset X$ be a subset of $X$.", "Let $T_1 \\supset T_2 \\supset \\ldots$", "be a descending chain of closed subsets of $T$.", "Write $T_i = T \\cap Z_i$ with $Z_i \\subset X$ closed.", "Consider the descending chain of closed subsets", "$Z_1 \\supset Z_1\\cap Z_2 \\supset Z_1 \\cap Z_2 \\cap Z_3 \\ldots$", "This stabilizes by assumption and hence the original sequence", "of $T_i$ stabilizes. Thus $T$ is Noetherian.", "\\medskip\\noindent", "Let $A$ be the set of closed subsets of $X$ which do not", "have finitely many irreducible components. Assume that", "$A$ is not empty to arrive at a contradiction.", "The set $A$ is partially ordered by inclusion: $\\alpha \\leq \\alpha'", "\\Leftrightarrow Z_{\\alpha} \\subset Z_{\\alpha'}$.", "By the descending chain condition we may find a", "smallest element of $A$, say $Z$. As $Z$ is not a finite", "union of irreducible components, it is not irreducible.", "Hence we can write $Z = Z' \\cup Z''$ and both are strictly smaller", "closed subsets. By construction $Z' = \\bigcup Z'_i$ and", "$Z'' = \\bigcup Z''_j$ are finite unions of their irreducible", "components. Hence $Z = \\bigcup Z'_i \\cup \\bigcup Z''_j$ is", "a finite union of irreducible closed subsets.", "After removing redundant members of this expression,", "this will be the decomposition of $Z$ into its irreducible", "components (Lemma \\ref{lemma-pick-irreducible-components}), a contradiction.", "\\medskip\\noindent", "Let $Z \\subset X$ be an irreducible component of $X$.", "Let $Z_1, \\ldots, Z_n$ be the other irreducible components", "of $X$. Consider $U = Z \\setminus (Z_1\\cup\\ldots\\cup Z_n)$.", "This is not empty since otherwise the irreducible space", "$Z$ would be contained in one of the other $Z_i$.", "Because $X = Z \\cup Z_1 \\cup \\ldots Z_n$ (see Lemma \\ref{lemma-irreducible}),", "also $U = X \\setminus (Z_1\\cup\\ldots\\cup Z_n)$", "and hence open in $X$. Thus $Z$ contains a nonempty", "open of $X$." ], "refs": [ "topology-lemma-pick-irreducible-components", "topology-lemma-irreducible" ], "ref_ids": [ 8214, 8213 ] } ], "ref_ids": [] }, { "id": 8221, "type": "theorem", "label": "topology-lemma-image-Noetherian", "categories": [ "topology" ], "title": "topology-lemma-image-Noetherian", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "\\begin{enumerate}", "\\item If $X$ is Noetherian, then $f(X)$ is Noetherian.", "\\item If $X$ is locally Noetherian and $f$ open, then $f(X)$ is", "locally Noetherian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In case (1), suppose that $Z_1 \\supset Z_2 \\supset Z_3 \\supset \\ldots$", "is a descending chain of closed subsets of $f(X)$ (as usual with the induced", "topology as a subset of $Y$). Then", "$f^{-1}(Z_1) \\supset f^{-1}(Z_2) \\supset f^{-1}(Z_3) \\supset \\ldots$ is", "a descending chain of closed subsets of $X$. Hence this chain stabilizes.", "Since $f(f^{-1}(Z_i)) = Z_i$ we conclude that", "$Z_1 \\supset Z_2 \\supset Z_3 \\supset \\ldots$", "stabilizes also. In case (2), let $y \\in f(X)$. Choose $x \\in X$ with", "$f(x) = y$. By assumption there exists a neighbourhood $E \\subset X$ of", "$x$ which is Noetherian. Then $f(E) \\subset f(X)$ is a neighbourhood", "which is Noetherian by part (1)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8222, "type": "theorem", "label": "topology-lemma-finite-union-Noetherian", "categories": [ "topology" ], "title": "topology-lemma-finite-union-Noetherian", "contents": [ "Let $X$ be a topological space.", "Let $X_i \\subset X$, $i = 1, \\ldots, n$ be a finite collection of subsets.", "If each $X_i$ is Noetherian (with the induced topology), then", "$\\bigcup_{i = 1, \\ldots, n} X_i$ is Noetherian (with the induced topology)." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8223, "type": "theorem", "label": "topology-lemma-locally-Noetherian-locally-connected", "categories": [ "topology" ], "title": "topology-lemma-locally-Noetherian-locally-connected", "contents": [ "Let $X$ be a locally Noetherian topological space.", "Then $X$ is locally connected." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$. Let $E$ be a neighbourhood of $x$.", "We have to find a connected neighbourhood of $x$ contained", "in $E$. By assumption there exists a neighbourhood $E'$ of $x$", "which is Noetherian. Then $E \\cap E'$ is Noetherian, see", "Lemma \\ref{lemma-Noetherian}.", "Let $E \\cap E' = Y_1 \\cup \\ldots \\cup Y_n$ be the decomposition", "into irreducible components, see", "Lemma \\ref{lemma-Noetherian}.", "Let $E'' = \\bigcup_{x \\in Y_i} Y_i$. This is a connected", "subset of $E \\cap E'$ containing $x$. It contains the open", "$E \\cap E' \\setminus (\\bigcup_{x \\not \\in Y_i} Y_i)$ of $E \\cap E'$", "and hence it is a neighbourhood of $x$ in $X$. This proves the lemma." ], "refs": [ "topology-lemma-Noetherian", "topology-lemma-Noetherian" ], "ref_ids": [ 8220, 8220 ] } ], "ref_ids": [] }, { "id": 8224, "type": "theorem", "label": "topology-lemma-dimension-supremum-local-dimensions", "categories": [ "topology" ], "title": "topology-lemma-dimension-supremum-local-dimensions", "contents": [ "Let $X$ be a topological space. Then $\\dim(X) = \\sup \\dim_x(X)$", "where the supremum runs over the points $x$ of $X$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $\\dim(X) \\geq \\dim_x(X)$ for all $x \\in X$ (see", "discussion following Definition \\ref{definition-Krull}).", "Thus an inequality in one direction. For the converse, let $n \\geq 0$", "and suppose that $\\dim(X) \\geq n$. Then we can find a chain of irreducible", "closed subsets $Z_0 \\subset Z_1 \\subset \\ldots \\subset Z_n \\subset X$.", "Pick $x \\in Z_0$. Then we see that every open neighbourhood $U$ of $x$", "has a chain of irreducible closed subsets", "$Z_0 \\cap U \\subset Z_1 \\cap U \\subset \\ldots Z_n \\cap U \\subset U$.", "In this way we see that $\\dim_x(X) \\geq n$ which proves the other", "inequality." ], "refs": [ "topology-definition-Krull" ], "ref_ids": [ 8356 ] } ], "ref_ids": [] }, { "id": 8225, "type": "theorem", "label": "topology-lemma-codimension-at-generic-point", "categories": [ "topology" ], "title": "topology-lemma-codimension-at-generic-point", "contents": [ "Let $X$ be a topological space.", "Let $Y \\subset X$ be an irreducible closed subset.", "Let $U \\subset X$ be an open subset such that $Y \\cap U$ is nonempty.", "Then", "$$", "\\text{codim}(Y, X) = \\text{codim}(Y \\cap U, U)", "$$" ], "refs": [], "proofs": [ { "contents": [ "The rule $T \\mapsto \\overline{T}$ defines a bijective", "inclusion preserving map between the closed irreducible subsets", "of $U$ and the closed irreducible subsets of $X$ which meet $U$.", "Using this the lemma easily follows. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8226, "type": "theorem", "label": "topology-lemma-catenary", "categories": [ "topology" ], "title": "topology-lemma-catenary", "contents": [ "Let $X$ be a topological space.", "The following are equivalent:", "\\begin{enumerate}", "\\item $X$ is catenary,", "\\item $X$ has an open covering by catenary spaces.", "\\end{enumerate}", "Moreover, in this case any locally closed subspace of $X$ is catenary." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $X$ is catenary and that $U \\subset X$ is an open", "subset. The rule $T \\mapsto \\overline{T}$ defines a bijective", "inclusion preserving map between the closed irreducible subsets", "of $U$ and the closed irreducible subsets of $X$ which meet $U$.", "Using this the lemma easily follows. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8227, "type": "theorem", "label": "topology-lemma-catenary-in-codimension", "categories": [ "topology" ], "title": "topology-lemma-catenary-in-codimension", "contents": [ "Let $X$ be a topological space. The following are equivalent:", "\\begin{enumerate}", "\\item $X$ is catenary, and", "\\item for every pair of irreducible closed subsets $Y \\subset Y'$ we have", "$\\text{codim}(Y, Y') < \\infty$ and for every triple", "$Y \\subset Y' \\subset Y''$ of irreducible closed subsets we have", "$$", "\\text{codim}(Y, Y'') = \\text{codim}(Y, Y') + \\text{codim}(Y', Y'').", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8228, "type": "theorem", "label": "topology-lemma-composition-quasi-compact", "categories": [ "topology" ], "title": "topology-lemma-composition-quasi-compact", "contents": [ "A composition of quasi-compact maps is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8229, "type": "theorem", "label": "topology-lemma-closed-in-quasi-compact", "categories": [ "topology" ], "title": "topology-lemma-closed-in-quasi-compact", "contents": [ "A closed subset of a quasi-compact topological space is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Let $E \\subset X$ be a closed subset of the quasi-compact space $X$.", "Let $E = \\bigcup V_j$ be an open covering. Choose $U_j \\subset X$", "open such that $V_j = E \\cap U_j$. Then $X = (X \\setminus E) \\cup \\bigcup U_j$", "is an open covering of $X$. Hence", "$X = (X \\setminus E) \\cup U_{j_1} \\cup \\ldots \\cup U_{j_n}$ for some", "$n$ and indices $j_i$. Thus $E = V_{j_1} \\cup \\ldots \\cup V_{j_n}$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8230, "type": "theorem", "label": "topology-lemma-quasi-compact-in-Hausdorff", "categories": [ "topology" ], "title": "topology-lemma-quasi-compact-in-Hausdorff", "contents": [ "Let $X$ be a Hausdorff topological space.", "\\begin{enumerate}", "\\item If $E \\subset X$ is quasi-compact, then it is closed.", "\\item If $E_1, E_2 \\subset X$ are disjoint quasi-compact subsets", "then there exists opens $E_i \\subset U_i$ with $U_1 \\cap U_2 = \\emptyset$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $x \\in X$, $x \\not \\in E$.", "For every $e \\in E$ we can find disjoint opens $V_e$ and $U_e$", "with $e \\in V_e$ and $x \\in U_e$. Since $E \\subset \\bigcup V_e$", "we can find finitely many $e_1, \\ldots, e_n$ such that", "$E \\subset V_{e_1} \\cup \\ldots \\cup V_{e_n}$. Then", "$U = U_{e_1} \\cap \\ldots \\cap U_{e_n}$ is an open neighbourhood", "of $x$ which avoids $V_{e_1} \\cup \\ldots \\cup V_{e_n}$. In particular", "it avoids $E$. Thus $E$ is closed.", "\\medskip\\noindent", "Proof of (2). In the proof of (1) we have seen that given $x \\in E_1$", "we can find an open neighbourhood $x \\in U_x$ and an open", "$E_2 \\subset V_x$ such that $U_x \\cap V_x = \\emptyset$. Because", "$E_1$ is quasi-compact we can find a finite number $x_i \\in E_1$", "such that $E_1 \\subset U = U_{x_1} \\cup \\ldots \\cup U_{x_n}$.", "We take $V = V_{x_1} \\cap \\ldots \\cap V_{x_n}$ to finish the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8231, "type": "theorem", "label": "topology-lemma-closed-in-compact", "categories": [ "topology" ], "title": "topology-lemma-closed-in-compact", "contents": [ "Let $X$ be a quasi-compact Hausdorff space. Let $E \\subset X$.", "The following are equivalent: (a) $E$ is closed in $X$, (b)", "$E$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "The implication (a) $\\Rightarrow$ (b) is", "Lemma \\ref{lemma-closed-in-quasi-compact}.", "The implication (b) $\\Rightarrow$ (a) is", "Lemma \\ref{lemma-quasi-compact-in-Hausdorff}." ], "refs": [ "topology-lemma-closed-in-quasi-compact", "topology-lemma-quasi-compact-in-Hausdorff" ], "ref_ids": [ 8229, 8230 ] } ], "ref_ids": [] }, { "id": 8232, "type": "theorem", "label": "topology-lemma-intersection-closed-in-quasi-compact", "categories": [ "topology" ], "title": "topology-lemma-intersection-closed-in-quasi-compact", "contents": [ "Let $X$ be a quasi-compact topological space.", "If $\\{Z_\\alpha\\}_{\\alpha \\in A}$ is a collection of closed subsets", "such that the intersection of each finite subcollection", "is nonempty, then $\\bigcap_{\\alpha \\in A} Z_\\alpha$ is nonempty." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8233, "type": "theorem", "label": "topology-lemma-image-quasi-compact", "categories": [ "topology" ], "title": "topology-lemma-image-quasi-compact", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "\\begin{enumerate}", "\\item If $X$ is quasi-compact, then $f(X)$ is quasi-compact.", "\\item If $f$ is quasi-compact, then $f(X)$ is retrocompact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $f(X) = \\bigcup V_i$ is an open covering, then $X = \\bigcup f^{-1}(V_i)$", "is an open covering. Hence if $X$ is quasi-compact then", "$X = f^{-1}(V_{i_1}) \\cup \\ldots \\cup f^{-1}(V_{i_n})$ for some", "$i_1, \\ldots, i_n \\in I$ and hence", "$f(X) = V_{i_1} \\cup \\ldots \\cup V_{i_n}$. This proves (1).", "Assume $f$ is quasi-compact, and let $V \\subset Y$ be quasi-compact open.", "Then $f^{-1}(V)$ is quasi-compact, hence by (1) we see that", "$f(f^{-1}(V)) = f(X) \\cap V$ is quasi-compact. Hence $f(X)$", "is retrocompact." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8234, "type": "theorem", "label": "topology-lemma-quasi-compact-closed-point", "categories": [ "topology" ], "title": "topology-lemma-quasi-compact-closed-point", "contents": [ "Let $X$ be a topological space. Assume that", "\\begin{enumerate}", "\\item $X$ is nonempty,", "\\item $X$ is quasi-compact, and", "\\item $X$ is Kolmogorov.", "\\end{enumerate}", "Then $X$ has a closed point." ], "refs": [], "proofs": [ { "contents": [ "Consider the set", "$$", "\\mathcal{T} =", "\\{Z \\subset X \\mid Z = \\overline{\\{x\\}} \\text{ for some }x \\in X\\}", "$$", "of all closures of singletons in $X$. It is nonempty since $X$ is", "nonempty. Make $\\mathcal{T}$ into a", "partially ordered set using the relation of inclusion.", "Suppose $Z_\\alpha$, $\\alpha \\in A$ is a totally ordered subset of $\\mathcal{T}$.", "By Lemma \\ref{lemma-intersection-closed-in-quasi-compact} we see", "that $\\bigcap_{\\alpha \\in A} Z_\\alpha \\not = \\emptyset$. Hence there exists", "some $x \\in \\bigcap_{\\alpha \\in A} Z_\\alpha$ and we see that", "$Z = \\overline{\\{x\\}}\\in \\mathcal{T}$ is a lower bound for", "the family. By Zorn's lemma there exists a minimal element", "$Z \\in \\mathcal{T}$. As $X$ is Kolmogorov we conclude that", "$Z = \\{x\\}$ for some $x$ and $x \\in X$ is a closed point." ], "refs": [ "topology-lemma-intersection-closed-in-quasi-compact" ], "ref_ids": [ 8232 ] } ], "ref_ids": [] }, { "id": 8235, "type": "theorem", "label": "topology-lemma-closed-points-quasi-compact", "categories": [ "topology" ], "title": "topology-lemma-closed-points-quasi-compact", "contents": [ "Let $X$ be a quasi-compact Kolmogorov space. Then the set $X_0$ of", "closed points of $X$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Let $X_0 = \\bigcup U_{i, 0}$ be an open covering.", "Write $U_{i, 0} = X_0 \\cap U_i$ for some open $U_i \\subset X$.", "Consider the complement $Z$ of $\\bigcup U_i$. This is a closed subset of", "$X$, hence quasi-compact (Lemma \\ref{lemma-closed-in-quasi-compact})", "and Kolmogorov. By Lemma \\ref{lemma-quasi-compact-closed-point}", "if $Z$ is nonempty it would have a closed", "point which contradicts the fact that $X_0 \\subset \\bigcup U_i$.", "Hence $Z = \\emptyset$ and $X = \\bigcup U_i$. Since $X$ is quasi-compact", "this covering has a finite subcover and we conclude." ], "refs": [ "topology-lemma-closed-in-quasi-compact", "topology-lemma-quasi-compact-closed-point" ], "ref_ids": [ 8229, 8234 ] } ], "ref_ids": [] }, { "id": 8236, "type": "theorem", "label": "topology-lemma-connected-component-intersection", "categories": [ "topology" ], "title": "topology-lemma-connected-component-intersection", "contents": [ "Let $X$ be a topological space.", "Assume", "\\begin{enumerate}", "\\item $X$ is quasi-compact,", "\\item $X$ has a basis for the topology consisting of quasi-compact opens, and", "\\item the intersection of two quasi-compact opens is quasi-compact.", "\\end{enumerate}", "For any $x \\in X$ the connected component of $X$ containing", "$x$ is the intersection of all open and closed subsets", "of $X$ containing $x$." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be the connected component containing $x$.", "Let $S = \\bigcap_{\\alpha \\in A} Z_\\alpha$ be the intersection of all", "open and closed subsets $Z_\\alpha$ of $X$ containing $x$.", "Note that $S$ is closed in $X$.", "Note that any finite intersection of $Z_\\alpha$'s is a $Z_\\alpha$.", "Because $T$ is connected and $x \\in T$ we have $T \\subset S$.", "It suffices to show that $S$ is connected.", "If not, then there exists a disjoint union decomposition", "$S = B \\amalg C$ with $B$ and $C$ open and closed in $S$.", "In particular, $B$ and $C$ are closed in $X$, and so quasi-compact by", "Lemma \\ref{lemma-closed-in-quasi-compact} and assumption (1).", "By assumption (2) there exist quasi-compact opens", "$U, V \\subset X$ with $B = S \\cap U$ and $C = S \\cap V$ (details omitted).", "Then $U \\cap V \\cap S = \\emptyset$.", "Hence $\\bigcap_\\alpha U \\cap V \\cap Z_\\alpha = \\emptyset$.", "By assumption (3) the intersection $U \\cap V$ is quasi-compact.", "By Lemma \\ref{lemma-intersection-closed-in-quasi-compact}", "for some $\\alpha' \\in A$ we have $U \\cap V \\cap Z_{\\alpha'} = \\emptyset$.", "Since $X \\setminus (U \\cup V)$ is disjoint from $S$", "and closed in $X$ hence quasi-compact, we can use the same lemma", "to see that $Z_{\\alpha''} \\subset U \\cup V$ for some $\\alpha'' \\in A$.", "Then $Z_\\alpha = Z_{\\alpha'} \\cap Z_{\\alpha''}$ is contained", "in $U \\cup V$ and disjoint from $U \\cap V$.", "Hence $Z_\\alpha = U \\cap Z_\\alpha \\amalg V \\cap Z_\\alpha$", "is a decomposition into two open pieces,", "hence $U \\cap Z_\\alpha$ and $V \\cap Z_\\alpha$ are open and closed in $X$.", "Thus, if $x \\in B$ say, then we see that $S \\subset U \\cap Z_\\alpha$", "and we conclude that $C = \\emptyset$." ], "refs": [ "topology-lemma-closed-in-quasi-compact", "topology-lemma-intersection-closed-in-quasi-compact" ], "ref_ids": [ 8229, 8232 ] } ], "ref_ids": [] }, { "id": 8237, "type": "theorem", "label": "topology-lemma-connected-component-intersection-compact-Hausdorff", "categories": [ "topology" ], "title": "topology-lemma-connected-component-intersection-compact-Hausdorff", "contents": [ "Let $X$ be a topological space. Assume $X$ is quasi-compact and Hausdorff.", "For any $x \\in X$ the connected component of $X$ containing", "$x$ is the intersection of all open and closed subsets", "of $X$ containing $x$." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be the connected component containing $x$.", "Let $S = \\bigcap_{\\alpha \\in A} Z_\\alpha$ be the intersection of all", "open and closed subsets $Z_\\alpha$ of $X$ containing $x$.", "Note that $S$ is closed in $X$.", "Note that any finite intersection of $Z_\\alpha$'s is a $Z_\\alpha$.", "Because $T$ is connected and $x \\in T$ we have $T \\subset S$.", "It suffices to show that $S$ is connected.", "If not, then there exists a disjoint union decomposition", "$S = B \\amalg C$ with $B$ and $C$ open and closed in $S$.", "In particular, $B$ and $C$ are closed in $X$, and so quasi-compact by", "Lemma \\ref{lemma-closed-in-quasi-compact}.", "By Lemma \\ref{lemma-quasi-compact-in-Hausdorff}", "there exist disjoint opens $U, V \\subset X$ with $B \\subset U$ and", "$C \\subset V$. Then $X \\setminus U \\cup V$ is closed in $X$", "hence quasi-compact (Lemma \\ref{lemma-closed-in-quasi-compact}).", "It follows that $(X \\setminus U \\cup V) \\cap Z_\\alpha = \\emptyset$", "for some $\\alpha$ by Lemma \\ref{lemma-intersection-closed-in-quasi-compact}.", "In other words, $Z_\\alpha \\subset U \\cup V$. Thus", "$Z_\\alpha = Z_\\alpha \\cap V \\amalg Z_\\alpha \\cap U$", "is a decomposition into two open pieces,", "hence $U \\cap Z_\\alpha$ and $V \\cap Z_\\alpha$ are open and closed in $X$.", "Thus, if $x \\in B$ say, then we see that $S \\subset U \\cap Z_\\alpha$", "and we conclude that $C = \\emptyset$." ], "refs": [ "topology-lemma-closed-in-quasi-compact", "topology-lemma-quasi-compact-in-Hausdorff", "topology-lemma-closed-in-quasi-compact", "topology-lemma-intersection-closed-in-quasi-compact" ], "ref_ids": [ 8229, 8230, 8229, 8232 ] } ], "ref_ids": [] }, { "id": 8238, "type": "theorem", "label": "topology-lemma-closed-union-connected-components", "categories": [ "topology" ], "title": "topology-lemma-closed-union-connected-components", "contents": [ "Let $X$ be a topological space.", "Assume", "\\begin{enumerate}", "\\item $X$ is quasi-compact,", "\\item $X$ has a basis for the topology consisting of quasi-compact opens, and", "\\item the intersection of two quasi-compact opens is quasi-compact.", "\\end{enumerate}", "For a subset $T \\subset X$ the following are equivalent:", "\\begin{enumerate}", "\\item[(a)] $T$ is an intersection of open and closed subsets of $X$, and", "\\item[(b)] $T$ is closed in $X$ and is a union of connected components of $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (a) implies (b).", "Assume (b). Let $x \\in X$, $x \\not \\in T$. Let $x \\in C \\subset X$", "be the connected component of $X$ containing $x$. By", "Lemma \\ref{lemma-connected-component-intersection}", "we see that $C = \\bigcap V_\\alpha$ is the intersection of all open and", "closed subsets $V_\\alpha$ of $X$ which contain $C$.", "In particular, any pairwise intersection $V_\\alpha \\cap V_\\beta$", "occurs as a $V_\\alpha$.", "As $T$ is a union of connected components", "of $X$ we see that $C \\cap T = \\emptyset$. Hence", "$T \\cap \\bigcap V_\\alpha = \\emptyset$. Since $T$ is quasi-compact as a", "closed subset of a quasi-compact space (see", "Lemma \\ref{lemma-closed-in-quasi-compact})", "we deduce that $T \\cap V_\\alpha = \\emptyset$ for some $\\alpha$, see", "Lemma \\ref{lemma-intersection-closed-in-quasi-compact}.", "For this $\\alpha$ we see that $U_\\alpha = X \\setminus V_\\alpha$", "is an open and closed subset of $X$ which contains $T$ and not $x$.", "The lemma follows." ], "refs": [ "topology-lemma-connected-component-intersection", "topology-lemma-closed-in-quasi-compact", "topology-lemma-intersection-closed-in-quasi-compact" ], "ref_ids": [ 8236, 8229, 8232 ] } ], "ref_ids": [] }, { "id": 8239, "type": "theorem", "label": "topology-lemma-Noetherian-quasi-compact", "categories": [ "topology" ], "title": "topology-lemma-Noetherian-quasi-compact", "contents": [ "Let $X$ be a Noetherian topological space.", "\\begin{enumerate}", "\\item The space $X$ is quasi-compact.", "\\item Any subset of $X$ is retrocompact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose $X = \\bigcup U_i$ is an open covering of $X$ indexed", "by the set $I$ which does not have a refinement by a finite", "open covering. Choose $i_1, i_2, \\ldots $ elements of $I$ inductively", "in the following way: Choose $i_{n + 1}$ such that $U_{i_{n + 1}}$", "is not contained in $U_{i_1} \\cup \\ldots \\cup U_{i_n}$. Thus we see that", "$X \\supset (X \\setminus U_{i_1}) \\supset", "(X \\setminus U_{i_1} \\cup U_{i_2}) \\supset \\ldots$ is a strictly", "decreasing infinite sequence of closed subsets. This contradicts", "the fact that $X$ is Noetherian. This proves the first assertion.", "The second assertion is now clear since every subset of $X$ is Noetherian by", "Lemma \\ref{lemma-Noetherian}." ], "refs": [ "topology-lemma-Noetherian" ], "ref_ids": [ 8220 ] } ], "ref_ids": [] }, { "id": 8240, "type": "theorem", "label": "topology-lemma-quasi-compact-locally-Noetherian-Noetherian", "categories": [ "topology" ], "title": "topology-lemma-quasi-compact-locally-Noetherian-Noetherian", "contents": [ "A quasi-compact locally Noetherian space is Noetherian." ], "refs": [], "proofs": [ { "contents": [ "The conditions imply immediately that $X$ has a finite covering by", "Noetherian subsets, and hence is Noetherian by", "Lemma \\ref{lemma-finite-union-Noetherian}." ], "refs": [ "topology-lemma-finite-union-Noetherian" ], "ref_ids": [ 8222 ] } ], "ref_ids": [] }, { "id": 8241, "type": "theorem", "label": "topology-lemma-subbase-theorem", "categories": [ "topology" ], "title": "topology-lemma-subbase-theorem", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{B}$ be a subbase for $X$.", "If every covering of $X$ by elements of $\\mathcal{B}$ has a finite", "refinement, then $X$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Assume there is an open covering of $X$ which does not have a finite", "refinement. Using Zorn's lemma we can choose a maximal open covering", "$X = \\bigcup_{i \\in I} U_i$ which does not have a finite refinement", "(details omitted).", "In other words, if $U \\subset X$ is any open which does not occur as", "one of the $U_i$, then the covering $X = U \\cup \\bigcup_{i \\in I} U_i$", "does have a finite refinement. Let $I' \\subset I$ be the set of indices", "such that $U_i \\in \\mathcal{B}$. Then $\\bigcup_{i \\in I'} U_i \\not = X$,", "since otherwise we would get a finite refinement covering $X$ by our", "assumption on $\\mathcal{B}$. Pick $x \\in X$,", "$x \\not \\in \\bigcup_{i \\in I'} U_i$. Pick $i \\in I$ with $x \\in U_i$.", "Pick $V_1, \\ldots, V_n \\in \\mathcal{B}$ such that", "$x \\in V_1 \\cap \\ldots \\cap V_n \\subset U_i$. This is", "possible as $\\mathcal{B}$ is a subbasis for $X$. Note that", "$V_j$ does not occur as a $U_i$. By maximality of the chosen", "covering we see that for each $j$ there exist", "$i_{j, 1}, \\ldots, i_{j, n_j} \\in I$ such that", "$X = V_j \\cup U_{i_{j, 1}} \\cup \\ldots \\cup U_{i_{j, n_j}}$.", "Since $V_1 \\cap \\ldots \\cap V_n \\subset U_i$ we conclude that", "$X = U_i \\cup \\bigcup U_{i_{j, l}}$ a contradiction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8242, "type": "theorem", "label": "topology-lemma-locally-quasi-compact-Hausdorff", "categories": [ "topology" ], "title": "topology-lemma-locally-quasi-compact-Hausdorff", "contents": [ "A Hausdorff space is locally quasi-compact if and only if every point", "has a quasi-compact neighbourhood." ], "refs": [], "proofs": [ { "contents": [ "Let $X$ be a Hausdorff space. Let $x \\in X$ and let $x \\in E \\subset X$", "be a quasi-compact neighbourhood. Then $E$ is closed by", "Lemma \\ref{lemma-quasi-compact-in-Hausdorff}.", "Suppose that $x \\in U \\subset X$ is an open neighbourhood of $x$.", "Then $Z = E \\setminus U$ is a closed subset of $E$ not containing $x$.", "Hence we can find a pair of disjoint open subsets $W, V \\subset E$", "of $E$ such that $x \\in V$ and $Z \\subset W$, see", "Lemma \\ref{lemma-quasi-compact-in-Hausdorff}.", "It follows that $\\overline{V} \\subset E$ is a closed neighbourhood", "of $x$ contained in $E \\cap U$. Also $\\overline{V}$ is quasi-compact", "as a closed subset of $E$ (Lemma \\ref{lemma-closed-in-quasi-compact}).", "In this way we obtain a fundamental system of quasi-compact neighbourhoods", "of $x$." ], "refs": [ "topology-lemma-quasi-compact-in-Hausdorff", "topology-lemma-quasi-compact-in-Hausdorff", "topology-lemma-closed-in-quasi-compact" ], "ref_ids": [ 8230, 8230, 8229 ] } ], "ref_ids": [] }, { "id": 8243, "type": "theorem", "label": "topology-lemma-baire-category-locally-compact", "categories": [ "topology" ], "title": "topology-lemma-baire-category-locally-compact", "contents": [ "Let $X$ be a locally quasi-compact Hausdorff space.", "Let $U_n \\subset X$, $n \\geq 1$ be dense open subsets. Then", "$\\bigcap_{n \\geq 1} U_n$ is dense in $X$." ], "refs": [], "proofs": [ { "contents": [ "After replacing $U_n$ by $\\bigcap_{i = 1, \\ldots, n} U_i$", "we may assume that $U_1 \\supset U_2 \\supset \\ldots$.", "Let $x \\in X$. We will show that $x$ is in the closure of", "$\\bigcap_{n \\geq 1} U_n$. Thus let $E$ be a neighbourhood of $x$.", "To show that $E \\cap \\bigcap_{n \\geq 1} U_n$ is nonempty we", "may replace $E$ by a smaller neighbourhood. After replacing", "$E$ by a smaller neighbourhood, we may assume that $E$ is quasi-compact.", "\\medskip\\noindent", "Set $x_0 = x$ and $E_0 = E$. Below, we will inductively choose", "a point $x_i \\in E_{i - 1} \\cap U_i$ and a quasi-compact", "neighbourhood $E_i$ of $x_i$ with $E_i \\subset E_{i - 1} \\cap U_i$.", "Because $X$ is Hausdorff, the subsets $E_i \\subset X$ are closed", "(Lemma \\ref{lemma-quasi-compact-in-Hausdorff}).", "Since the $E_i$ are also nonempty we conclude that", "$\\bigcap_{i \\geq 1} E_i$ is nonempty", "(Lemma \\ref{lemma-intersection-closed-in-quasi-compact}).", "Since $\\bigcap_{i \\geq 1} E_i \\subset E \\cap \\bigcap_{n \\geq 1} U_n$", "this proves the lemma.", "\\medskip\\noindent", "The base case $i = 0$ we have done above. Induction step.", "Since $E_{i - 1}$ is a neighbourhood of $x_{i - 1}$ we can", "find an open $x_{i - 1} \\in W \\subset E_{i - 1}$.", "Since $U_i$ is dense in $X$", "we see that $W \\cap U_i$ is nonempty.", "Pick any $x_i \\in W \\cap U_i$.", "By definition of locally quasi-compact spaces we can", "find a quasi-compact neighbourhood $E_i$ of $x_i$", "contained in $W \\cap U_i$. Then $E_i \\subset E_{i - 1} \\cap U_i$", "as desired." ], "refs": [ "topology-lemma-quasi-compact-in-Hausdorff", "topology-lemma-intersection-closed-in-quasi-compact" ], "ref_ids": [ 8230, 8232 ] } ], "ref_ids": [] }, { "id": 8244, "type": "theorem", "label": "topology-lemma-relatively-compact-refinement", "categories": [ "topology" ], "title": "topology-lemma-relatively-compact-refinement", "contents": [ "Let $X$ be a Hausdorff and quasi-compact space.", "Let $X = \\bigcup_{i \\in I} U_i$ be an open covering.", "Then there exists an open covering $X = \\bigcup_{i \\in I} V_i$", "such that $\\overline{V_i} \\subset U_i$ for all $i$." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$. Choose an $i(x) \\in I$ such that $x \\in U_{i(x)}$.", "Since $X \\setminus U_{i(x)}$ and $\\{x\\}$ are disjoint closed", "subsets of $X$, by Lemmas \\ref{lemma-closed-in-quasi-compact} and", "\\ref{lemma-quasi-compact-in-Hausdorff}", "there exists an open neighbourhood $U_x$ of $x$", "whose closure is disjoint from $X \\setminus U_{i(x)}$.", "Thus $\\overline{U_x} \\subset U_{i(x)}$. Since $X$ is quasi-compact,", "there is a finite list of points $x_1, \\ldots, x_m$ such that", "$X = U_{x_1} \\cup \\ldots \\cup U_{x_m}$. Setting", "$V_i = \\bigcup_{i = i(x_j)} U_{x_j}$ the proof is finished." ], "refs": [ "topology-lemma-closed-in-quasi-compact", "topology-lemma-quasi-compact-in-Hausdorff" ], "ref_ids": [ 8229, 8230 ] } ], "ref_ids": [] }, { "id": 8245, "type": "theorem", "label": "topology-lemma-refine-covering", "categories": [ "topology" ], "title": "topology-lemma-refine-covering", "contents": [ "Let $X$ be a Hausdorff and quasi-compact space.", "Let $X = \\bigcup_{i \\in I} U_i$ be an open covering.", "Suppose given an integer $p \\geq 0$ and for every $(p + 1)$-tuple", "$i_0, \\ldots, i_p$ of $I$ an open covering", "$U_{i_0} \\cap \\ldots \\cap U_{i_p} = \\bigcup W_{i_0 \\ldots i_p, k}$.", "Then there exists an open covering $X = \\bigcup_{j \\in J} V_j$", "and a map $\\alpha : J \\to I$ such that $\\overline{V_j} \\subset U_{\\alpha(j)}$", "and such that each $V_{j_0} \\cap \\ldots \\cap V_{j_p}$", "is contained in $W_{\\alpha(j_0) \\ldots \\alpha(j_p), k}$", "for some $k$." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is quasi-compact, there is a reduction to the", "case where $I$ is finite (details omitted).", "We prove the result for $I$ finite by induction on $p$.", "The base case $p = 0$ is immediate by taking a covering as in", "Lemma \\ref{lemma-relatively-compact-refinement}", "refining the open covering $X = \\bigcup W_{i_0, k}$.", "\\medskip\\noindent", "Induction step. Assume the lemma proven for $p - 1$.", "For all $p$-tuples $i'_0, \\ldots, i'_{p - 1}$ of $I$ let", "$U_{i'_0} \\cap \\ldots \\cap U_{i'_{p - 1}} =", "\\bigcup W_{i'_0 \\ldots i'_{p - 1}, k}$", "be a common refinement of the coverings", "$U_{i_0} \\cap \\ldots \\cap U_{i_p} = \\bigcup W_{i_0 \\ldots i_p, k}$", "for those $(p + 1)$-tuples such that", "$\\{i'_0, \\ldots, i'_{p - 1}\\} = \\{i_0, \\ldots, i_p\\}$ (equality of sets).", "(There are finitely many of these as $I$ is finite.)", "By induction there exists a solution for these opens, say", "$X = \\bigcup V_j$ and $\\alpha : J \\to I$.", "At this point the covering $X = \\bigcup_{j \\in J} V_j$", "and $\\alpha$ satisfy $\\overline{V_j} \\subset U_{\\alpha(j)}$", "and each $V_{j_0} \\cap \\ldots \\cap V_{j_p}$", "is contained in $W_{\\alpha(j_0) \\ldots \\alpha(j_p), k}$", "for some $k$ if there is a repetition in $\\alpha(j_0), \\ldots, \\alpha(j_p)$.", "Of course, we may and do assume that $J$ is finite.", "\\medskip\\noindent", "Fix $i_0, \\ldots, i_p \\in I$ pairwise distinct. Consider $(p + 1)$-tuples", "$j_0, \\ldots, j_p \\in J$ with $i_0 = \\alpha(j_0), \\ldots, i_p = \\alpha(j_p)$", "such that $V_{j_0} \\cap \\ldots \\cap V_{j_p}$", "is {\\bf not} contained in $W_{\\alpha(j_0) \\ldots \\alpha(j_p), k}$ for any $k$.", "Let $N$ be the number of such $(p + 1)$-tuples. We will show how to decrease", "$N$. Since", "$$", "\\overline{V_{j_0}} \\cap \\ldots \\cap \\overline{V_{j_p}} \\subset", "U_{i_0} \\cap \\ldots \\cap U_{i_p} = \\bigcup W_{i_0 \\ldots i_p, k}", "$$", "we find a finite set $K = \\{k_1, \\ldots, k_t\\}$ such that the LHS", "is contained in $\\bigcup_{k \\in K} W_{i_0 \\ldots i_p, k}$.", "Then we consider the open covering", "$$", "V_{j_0} =", "(V_{j_0} \\setminus (\\overline{V_{j_1}} \\cap \\ldots \\cap \\overline{V_{j_p}}))", "\\cup (\\bigcup\\nolimits_{k \\in K} V_{j_0} \\cap W_{i_0 \\ldots i_p, k})", "$$", "The first open on the RHS intersects $V_{j_1} \\cap \\ldots \\cap V_{j_p}$", "in the empty set and the other opens $V_{j_0, k}$ of the RHS", "satisfy $V_{j_0, k} \\cap V_{j_1} \\ldots \\cap V_{j_p} \\subset", "W_{\\alpha(j_0) \\ldots \\alpha(j_p), k}$.", "Set $J' = J \\amalg K$. For $j \\in J$ set $V'_j = V_j$ if $j \\not = j_0$", "and set $V'_{j_0} =", "V_{j_0} \\setminus (\\overline{V_{j_1}} \\cap \\ldots \\cap \\overline{V_{j_p}})$.", "For $k \\in K$ set $V'_k = V_{j_0, k}$. Finally, the map $\\alpha' : J' \\to I$", "is given by $\\alpha$ on $J$ and maps every element of $K$ to $i_0$.", "A simple check shows that $N$ has decreased by one under this replacement.", "Repeating this procedure $N$ times we arrive at the situation where", "$N = 0$.", "\\medskip\\noindent", "To finish the proof we argue by induction on the number $M$ of $(p + 1)$-tuples", "$i_0, \\ldots, i_p \\in I$ with pairwise distinct entries for which there exists", "a $(p + 1)$-tuple $j_0, \\ldots, j_p \\in J$ with", "$i_0 = \\alpha(j_0), \\ldots, i_p = \\alpha(j_p)$ such that", "$V_{j_0} \\cap \\ldots \\cap V_{j_p}$ is {\\bf not} contained in", "$W_{\\alpha(j_0) \\ldots \\alpha(j_p), k}$ for any $k$. To do this, we", "claim that the operation performed in the previous paragraph does not", "increase $M$. This follows formally from the fact that the map", "$\\alpha' : J' \\to I$ factors through a map $\\beta : J' \\to J$", "such that $V'_{j'} \\subset V_{\\beta(j')}$." ], "refs": [ "topology-lemma-relatively-compact-refinement" ], "ref_ids": [ 8244 ] } ], "ref_ids": [] }, { "id": 8246, "type": "theorem", "label": "topology-lemma-lift-covering-of-a-closed", "categories": [ "topology" ], "title": "topology-lemma-lift-covering-of-a-closed", "contents": [ "Let $X$ be a Hausdorff and locally quasi-compact space.", "Let $Z \\subset X$ be a quasi-compact (hence closed) subset.", "Suppose given an integer $p \\geq 0$, a set $I$, for every $i \\in I$", "an open $U_i \\subset X$, and for every $(p + 1)$-tuple", "$i_0, \\ldots, i_p$ of $I$ an open", "$W_{i_0 \\ldots i_p} \\subset U_{i_0} \\cap \\ldots \\cap U_{i_p}$", "such that", "\\begin{enumerate}", "\\item $Z \\subset \\bigcup U_i$, and", "\\item for every $i_0, \\ldots, i_p$ we have", "$W_{i_0 \\ldots i_p} \\cap Z = U_{i_0} \\cap \\ldots \\cap U_{i_p} \\cap Z$.", "\\end{enumerate}", "Then there exist opens $V_i$ of $X$ such that", "we have $Z \\subset \\bigcup V_i$,", "for all $i$ we have $\\overline{V_i} \\subset U_i$, and", "we have $V_{i_0} \\cap \\ldots \\cap V_{i_p} \\subset W_{i_0 \\ldots i_p}$", "for all $(p + 1)$-tuples $i_0, \\ldots, i_p$." ], "refs": [], "proofs": [ { "contents": [ "Since $Z$ is quasi-compact, there is a reduction to the", "case where $I$ is finite (details omitted).", "Because $X$ is locally quasi-compact and $Z$ is quasi-compact,", "we can find a neighbourhood $Z \\subset E$ which is quasi-compact,", "i.e., $E$ is quasi-compact and contains an open neighbourhood", "of $Z$ in $X$. If we prove the result after replacing $X$ by $E$,", "then the result follows. Hence we may assume $X$ is quasi-compact.", "\\medskip\\noindent", "We prove the result in case $I$ is finite and $X$ is quasi-compact", "by induction on $p$. The base case is $p = 0$. In this case we have", "$X = (X \\setminus Z) \\cup \\bigcup W_i$. By", "Lemma \\ref{lemma-relatively-compact-refinement}", "we can find a covering $X = V \\cup \\bigcup V_i$ by", "opens $V_i \\subset W_i$ and $V \\subset X \\setminus Z$", "with $\\overline{V_i} \\subset W_i$ for all $i$. Then we see that", "we obtain a solution of the problem posed by the lemma.", "\\medskip\\noindent", "Induction step. Assume the lemma proven for $p - 1$.", "Set $W_{j_0 \\ldots j_{p - 1}}$ equal to the intersection of", "all $W_{i_0 \\ldots i_p}$ with", "$\\{j_0, \\ldots, j_{p - 1}\\} = \\{i_0, \\ldots, i_p\\}$ (equality of sets).", "By induction there exists a solution for these opens, say", "$V_i \\subset U_i$.", "It follows from our choice of $W_{j_0 \\ldots j_{p - 1}}$ that we have", "$V_{i_0} \\cap \\ldots \\cap V_{i_p} \\subset W_{i_0 \\ldots i_p}$", "for all $(p + 1)$-tuples $i_0, \\ldots, i_p$ where $i_a = i_b$ for", "some $0 \\leq a < b \\leq p$.", "Thus we only need to modify our choice of", "$V_i$ if $V_{i_0} \\cap \\ldots \\cap V_{i_p} \\not \\subset W_{i_0 \\ldots i_p}$", "for some $(p + 1)$-tuple $i_0, \\ldots, i_p$ with pairwise distinct elements.", "In this case we have", "$$", "T =", "\\overline{V_{i_0} \\cap \\ldots \\cap V_{i_p} \\setminus W_{i_0 \\ldots i_p}}", "\\subset ", "\\overline{V_{i_0}} \\cap \\ldots \\cap \\overline{V_{i_p}} \\setminus", "W_{i_0 \\ldots i_p}", "$$", "is a closed subset of $X$ contained in $U_{i_0} \\cap \\ldots \\cap U_{i_p}$", "not meeting $Z$. Hence we can replace $V_{i_0}$ by $V_{i_0} \\setminus T$", "to ``fix'' the problem. After repeating this finitely many times for each", "of the problem tuples, the lemma is proven." ], "refs": [ "topology-lemma-relatively-compact-refinement" ], "ref_ids": [ 8244 ] } ], "ref_ids": [] }, { "id": 8247, "type": "theorem", "label": "topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset", "categories": [ "topology" ], "title": "topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset", "contents": [ "Let $X$ be a topological space. Let $Z \\subset X$ be a quasi-compact subset", "such that any two points of $Z$ have disjoint open neighbourhoods in $X$.", "Suppose given an integer $p \\geq 0$, a set $I$, for every $i \\in I$", "an open $U_i \\subset X$, and for every $(p + 1)$-tuple", "$i_0, \\ldots, i_p$ of $I$ an open", "$W_{i_0 \\ldots i_p} \\subset U_{i_0} \\cap \\ldots \\cap U_{i_p}$", "such that", "\\begin{enumerate}", "\\item $Z \\subset \\bigcup U_i$, and", "\\item for every $i_0, \\ldots, i_p$ we have", "$W_{i_0 \\ldots i_p} \\cap Z = U_{i_0} \\cap \\ldots \\cap U_{i_p} \\cap Z$.", "\\end{enumerate}", "Then there exist opens $V_i$ of $X$ such that", "\\begin{enumerate}", "\\item $Z \\subset \\bigcup V_i$,", "\\item $V_i \\subset U_i$ for all $i$,", "\\item $\\overline{V_i} \\cap Z \\subset U_i$ for all $i$, and", "\\item $V_{i_0} \\cap \\ldots \\cap V_{i_p} \\subset W_{i_0 \\ldots i_p}$", "for all $(p + 1)$-tuples $i_0, \\ldots, i_p$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $Z$ is quasi-compact, there is a reduction to the", "case where $I$ is finite (details omitted).", "We prove the result in case $I$ is finite by induction on $p$.", "\\medskip\\noindent", "The base case is $p = 0$.", "For $z \\in Z \\cap U_i$ and $z' \\in Z \\setminus U_i$ there exist", "disjoint opens $z \\in V_{z, z'}$ and $z' \\in W_{z, z'}$ of $X$.", "Since $Z \\setminus U_i$ is quasi-compact", "(Lemma \\ref{lemma-closed-in-quasi-compact}),", "we can choose a finite nunber $z'_1, \\ldots, z'_r$ such that", "$Z \\setminus U_i \\subset W_{z, z'_1} \\cup \\ldots \\cup W_{z, z'_r}$.", "Then we see that", "$V_z = V_{z, z'_1} \\cap \\ldots \\cap V_{z, z'_r} \\cap U_i$", "is an open neighbourhood of $z$ contained in $U_i$", "with the property that $\\overline{V_z} \\cap Z \\subset U_i$.", "Since $z$ and $i$ where arbitrary and since $Z$ is quasi-compact", "we can find a finite list $z_1, i_1, \\ldots, z_t, i_t$", "and opens $V_{z_j} \\subset U_{i_j}$ with", "$\\overline{V_{z_j}} \\cap Z \\subset U_{i_j}$", "and $Z \\subset \\bigcup V_{z_j}$.", "Then we can set $V_i = W_i \\cap (\\bigcup_{j : i = i_j} V_{z_j})$", "to solve the problem for $p = 0$.", "\\medskip\\noindent", "Induction step. Assume the lemma proven for $p - 1$.", "Set $W_{j_0 \\ldots j_{p - 1}}$ equal to the intersection of", "all $W_{i_0 \\ldots i_p}$ with", "$\\{j_0, \\ldots, j_{p - 1}\\} = \\{i_0, \\ldots, i_p\\}$ (equality of sets).", "By induction there exists a solution for these opens, say", "$V_i \\subset U_i$.", "It follows from our choice of $W_{j_0 \\ldots j_{p - 1}}$ that we have", "$V_{i_0} \\cap \\ldots \\cap V_{i_p} \\subset W_{i_0 \\ldots i_p}$", "for all $(p + 1)$-tuples $i_0, \\ldots, i_p$ where $i_a = i_b$ for", "some $0 \\leq a < b \\leq p$.", "Thus we only need to modify our choice of", "$V_i$ if $V_{i_0} \\cap \\ldots \\cap V_{i_p} \\not \\subset W_{i_0 \\ldots i_p}$", "for some $(p + 1)$-tuple $i_0, \\ldots, i_p$ with pairwise distinct elements.", "In this case we have", "$$", "T =", "\\overline{V_{i_0} \\cap \\ldots \\cap V_{i_p} \\setminus W_{i_0 \\ldots i_p}}", "\\subset", "\\overline{V_{i_0}} \\cap \\ldots \\cap \\overline{V_{i_p}} \\setminus", "W_{i_0 \\ldots i_p}", "$$", "is a closed subset of $X$ not meeting $Z$ by our property (3) of the", "opens $V_i$. Hence we can replace $V_{i_0}$ by $V_{i_0} \\setminus T$", "to ``fix'' the problem. After repeating this finitely many times for each", "of the problem tuples, the lemma is proven." ], "refs": [ "topology-lemma-closed-in-quasi-compact" ], "ref_ids": [ 8229 ] } ], "ref_ids": [] }, { "id": 8248, "type": "theorem", "label": "topology-lemma-limits", "categories": [ "topology" ], "title": "topology-lemma-limits", "contents": [ "The category of topological spaces has limits and the forgetful functor", "to sets commutes with them." ], "refs": [], "proofs": [ { "contents": [ "This follows from the discussion above and", "Categories, Lemma \\ref{categories-lemma-limits-products-equalizers}.", "It follows from the description above that the forgetful functor", "commutes with limits. Another way to see this is to use", "Categories, Lemma \\ref{categories-lemma-adjoint-exact} and use that", "the forgetful functor has a left adjoint, namely the functor which", "assigns to a set the corresponding discrete topological space." ], "refs": [ "categories-lemma-limits-products-equalizers", "categories-lemma-adjoint-exact" ], "ref_ids": [ 12213, 12249 ] } ], "ref_ids": [] }, { "id": 8249, "type": "theorem", "label": "topology-lemma-describe-limits", "categories": [ "topology" ], "title": "topology-lemma-describe-limits", "contents": [ "Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram", "of topological spaces over $\\mathcal{I}$. Let $X = \\lim X_i$ be the limit", "with projection maps $f_i : X \\to X_i$.", "\\begin{enumerate}", "\\item Any open of $X$ is of the form $\\bigcup_{j \\in J} f_j^{-1}(U_j)$", "for some subset $J \\subset I$ and opens $U_j \\subset X_j$.", "\\item Any quasi-compact open of $X$ is of the form", "$f_i^{-1}(U_i)$ for some $i$ and some $U_i \\subset X_i$ open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The construction of the limit given above shows that $X \\subset \\prod X_i$", "with the induced topology. A basis for the topology of $\\prod X_i$ are", "the opens $\\prod U_i$ where $U_i \\subset X_i$ is open and $U_i = X_i$", "for almost all $i$. Say $i_1, \\ldots, i_n \\in \\Ob(\\mathcal{I})$ are the", "objects such that $U_{i_j} \\not = X_{i_j}$. Then", "$$", "X \\cap \\prod U_i = f_{i_1}^{-1}(U_{i_1}) \\cap \\ldots \\cap f_{i_n}^{-1}(U_{i_n})", "$$", "For a general limit of topological spaces these form a basis for the", "topology on $X$. However, if $\\mathcal{I}$ is cofiltered as in the statement", "of the lemma, then we can pick a $j \\in \\Ob(\\mathcal{I})$ and morphisms", "$j \\to i_l$, $l = 1, \\ldots, n$. Let", "$$", "U_j =", "(X_j \\to X_{i_1})^{-1}(U_{i_1}) \\cap \\ldots \\cap", "(X_j \\to X_{i_n})^{-1}(U_{i_n})", "$$", "Then it is clear that $X \\cap \\prod U_i = f_j^{-1}(U_j)$. Thus for any open", "$W$ of $X$ there is a set $A$ and a map $\\alpha : A \\to \\Ob(\\mathcal{I})$ and", "opens $U_a \\subset X_{\\alpha(a)}$ such that", "$W = \\bigcup f_{\\alpha(a)}^{-1}(U_a)$. Set $J = \\Im(\\alpha)$ and", "for $j \\in J$ set $U_j = \\bigcup_{\\alpha(a) = j} U_a$ to see that", "$W = \\bigcup_{j \\in J} f_j^{-1}(U_j)$.", "This proves (1).", "\\medskip\\noindent", "To see (2) suppose that $\\bigcup_{j \\in J} f_j^{-1}(U_j)$ is quasi-compact.", "Then it is equal to", "$f_{j_1}^{-1}(U_{j_1}) \\cup \\ldots \\cup f_{j_m}^{-1}(U_{j_m})$", "for some $j_1, \\ldots, j_m \\in J$. Since $\\mathcal{I}$ is cofiltered,", "we can pick a $i \\in \\Ob(\\mathcal{I})$ and morphisms", "$i \\to j_l$, $l = 1, \\ldots, m$. Let", "$$", "U_i =", "(X_i \\to X_{j_1})^{-1}(U_{j_1}) \\cup \\ldots \\cup", "(X_i \\to X_{j_m})^{-1}(U_{j_m})", "$$", "Then our open equals $f_i^{-1}(U_i)$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8250, "type": "theorem", "label": "topology-lemma-characterize-limit", "categories": [ "topology" ], "title": "topology-lemma-characterize-limit", "contents": [ "Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram", "of topological spaces over $\\mathcal{I}$. Let $X$ be a topological", "space such that", "\\begin{enumerate}", "\\item $X = \\lim X_i$ as a set (denote $f_i$ the projection maps),", "\\item the sets $f_i^{-1}(U_i)$ for $i \\in \\Ob(\\mathcal{I})$ and", "$U_i \\subset X_i$ open form a basis for the topology of $X$.", "\\end{enumerate}", "Then $X$ is the limit of the $X_i$ as a topological space." ], "refs": [], "proofs": [ { "contents": [ "Follows from the description of the limit topology", "in Lemma \\ref{lemma-describe-limits}." ], "refs": [ "topology-lemma-describe-limits" ], "ref_ids": [ 8249 ] } ], "ref_ids": [] }, { "id": 8251, "type": "theorem", "label": "topology-lemma-inverse-limit-quasi-compact", "categories": [ "topology" ], "title": "topology-lemma-inverse-limit-quasi-compact", "contents": [ "Let $\\mathcal{I}$ be a category and let $i \\mapsto X_i$", "be a diagram over $\\mathcal{I}$ in the category of topological", "spaces. If each $X_i$ is quasi-compact and Hausdorff, then", "$\\lim X_i$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\lim X_i$ is a subspace of $\\prod X_i$. By", "Theorem \\ref{theorem-tychonov} this product is quasi-compact. Hence it", "suffices to show that $\\lim X_i$ is a closed subspace of $\\prod X_i$", "(Lemma \\ref{lemma-closed-in-quasi-compact}).", "If $\\varphi : j \\to k$ is a morphism of $\\mathcal{I}$, then", "let $\\Gamma_\\varphi \\subset X_j \\times X_k$ denote the graph", "of the corresponding continuous map $X_j \\to X_k$. By", "Lemma \\ref{lemma-graph-closed} this graph is closed.", "It is clear that $\\lim X_i$ is the intersection of the", "closed subsets", "$$", "\\Gamma_\\varphi \\times \\prod\\nolimits_{l \\not = j, k} X_l", "\\subset \\prod X_i", "$$", "Thus the result follows." ], "refs": [ "topology-theorem-tychonov", "topology-lemma-closed-in-quasi-compact", "topology-lemma-graph-closed" ], "ref_ids": [ 8188, 8229, 8191 ] } ], "ref_ids": [] }, { "id": 8252, "type": "theorem", "label": "topology-lemma-nonempty-limit", "categories": [ "topology" ], "title": "topology-lemma-nonempty-limit", "contents": [ "Let $\\mathcal{I}$ be a cofiltered category and let $i \\mapsto X_i$", "be a diagram over $\\mathcal{I}$ in the category of topological", "spaces. If each $X_i$ is quasi-compact, Hausdorff, and nonempty, then", "$\\lim X_i$ is nonempty." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-inverse-limit-quasi-compact}", "we have seen that $X = \\lim X_i$ is the intersection of the", "closed subsets", "$$", "Z_\\varphi = \\Gamma_\\varphi \\times \\prod\\nolimits_{l \\not = j, k} X_l", "$$", "inside the quasi-compact space $\\prod X_i$ where $\\varphi : j \\to k$", "is a morphism of $\\mathcal{I}$ and $\\Gamma_\\varphi \\subset X_j \\times X_k$", "is the graph of the corresponding morphism $X_j \\to X_k$. Hence by", "Lemma \\ref{lemma-intersection-closed-in-quasi-compact}", "it suffices to show any finite intersection of these subsets is nonempty.", "Assume $\\varphi_t : j_t \\to k_t$, $t = 1, \\ldots, n$ is a finite collection", "of morphisms of $\\mathcal{I}$. Since $\\mathcal{I}$ is cofiltered, we can", "pick an object $j$ and a morphism $\\psi_t : j \\to j_t$ for each $t$.", "For each pair $t, t'$ such that either (a) $j_t = j_{t'}$, or", "(b) $j_t = k_{t'}$, or (c) $k_t = k_{t'}$ we obtain two morphisms", "$j \\to l$ with $l = j_t$ in case (a), (b) or $l = k_t$ in case (c).", "Because $\\mathcal{I}$ is cofiltered and since there are finitely", "many pairs $(t, t')$ we may choose a map $j' \\to j$ which equalizes", "these two morphisms for all such pairs $(t, t')$. Pick an element", "$x \\in X_{j'}$ and for each $t$ let $x_{j_t}$, resp.\\ $x_{k_t}$", "be the image of $x$ under the morphism $X_{j'} \\to X_j \\to X_{j_t}$,", "resp.\\ $X_{j'} \\to X_j \\to X_{j_t} \\to X_{k_t}$.", "For any index $l \\in \\Ob(\\mathcal{I})$ which is not equal to", "$j_t$ or $k_t$ for some $t$ we pick an arbitrary element $x_l \\in X_l$", "(using the axiom of choice). Then $(x_i)_{i \\in \\Ob(\\mathcal{I})}$", "is in the intersection ", "$$", "Z_{\\varphi_1} \\cap \\ldots \\cap Z_{\\varphi_n}", "$$", "by construction and the proof is complete." ], "refs": [ "topology-lemma-inverse-limit-quasi-compact", "topology-lemma-intersection-closed-in-quasi-compact" ], "ref_ids": [ 8251, 8232 ] } ], "ref_ids": [] }, { "id": 8253, "type": "theorem", "label": "topology-lemma-constructible", "categories": [ "topology" ], "title": "topology-lemma-constructible", "contents": [ "The collection of constructible sets is closed under", "finite intersections, finite unions and complements." ], "refs": [], "proofs": [ { "contents": [ "Note that if $U_1$, $U_2$ are open and retrocompact in $X$", "then so is $U_1 \\cup U_2$ because the union of two quasi-compact", "subsets of $X$ is quasi-compact. It is also true that", "$U_1 \\cap U_2$ is retrocompact. Namely, suppose $U \\subset X$", "is quasi-compact open, then $U_2 \\cap U$ is quasi-compact because", "$U_2$ is retrocompact in $X$, and then we conclude", "$U_1 \\cap (U_2 \\cap U)$ is quasi-compact because $U_1$ is", "retrocompact in $X$. From this it is formal to show that", "the complement of a constructible set is constructible,", "that finite unions of constructibles are constructible, and", "that finite intersections of constructibles are constructible." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8254, "type": "theorem", "label": "topology-lemma-inverse-images-constructibles", "categories": [ "topology" ], "title": "topology-lemma-inverse-images-constructibles", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "If the inverse image of every retrocompact open subset of $Y$", "is retrocompact in $X$, then inverse images of constructible", "sets are constructible." ], "refs": [], "proofs": [ { "contents": [ "This is true because $f^{-1}(U \\cap V^c) = f^{-1}(U) \\cap f^{-1}(V)^c$,", "combined with the definition of constructible sets." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8255, "type": "theorem", "label": "topology-lemma-open-immersion-constructible-inverse-image", "categories": [ "topology" ], "title": "topology-lemma-open-immersion-constructible-inverse-image", "contents": [ "Let $U \\subset X$ be open. For a constructible set", "$E \\subset X$ the intersection $E \\cap U$ is constructible", "in $U$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $V \\subset X$ is retrocompact open in $X$.", "It suffices to show that $V \\cap U$ is retrocompact in $U$", "by Lemma \\ref{lemma-inverse-images-constructibles}. To show this", "let $W \\subset U$ be open and quasi-compact. Then $W$", "is open and quasi-compact in $X$. Hence $V \\cap W = V \\cap U \\cap W$", "is quasi-compact as $V$ is retrocompact in $X$." ], "refs": [ "topology-lemma-inverse-images-constructibles" ], "ref_ids": [ 8254 ] } ], "ref_ids": [] }, { "id": 8256, "type": "theorem", "label": "topology-lemma-quasi-compact-open-immersion-constructible-image", "categories": [ "topology" ], "title": "topology-lemma-quasi-compact-open-immersion-constructible-image", "contents": [ "Let $U \\subset X$ be a retrocompact open. Let $E \\subset U$.", "If $E$ is constructible in $U$, then $E$ is constructible in $X$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $V, W \\subset U$ are retrocompact open in $U$.", "Then $V, W$ are retrocompact open in $X$", "(Lemma \\ref{lemma-composition-quasi-compact}).", "Hence $V \\cap (U \\setminus W) = V \\cap (X \\setminus W)$", "is constructible in $X$. We conclude since every constructible subset of $U$", "is a finite union of subsets of the form $V \\cap (U \\setminus W)$." ], "refs": [ "topology-lemma-composition-quasi-compact" ], "ref_ids": [ 8228 ] } ], "ref_ids": [] }, { "id": 8257, "type": "theorem", "label": "topology-lemma-collate-constructible", "categories": [ "topology" ], "title": "topology-lemma-collate-constructible", "contents": [ "Let $X$ be a topological space. Let $E \\subset X$ be a subset.", "Let $X = V_1 \\cup \\ldots \\cup V_m$ be a finite covering by", "retrocompact opens.", "Then $E$ is constructible in $X$ if and only if $E \\cap V_j$", "is constructible in $V_j$ for each $j = 1, \\ldots, m$." ], "refs": [], "proofs": [ { "contents": [ "If $E$ is constructible in $X$, then by", "Lemma \\ref{lemma-open-immersion-constructible-inverse-image}", "we see that $E \\cap V_j$ is constructible in $V_j$ for all $j$.", "Conversely, suppose that $E \\cap V_j$", "is constructible in $V_j$ for each $j = 1, \\ldots, m$.", "Then $E = \\bigcup E \\cap V_j$ is a finite union of", "constructible sets by", "Lemma \\ref{lemma-quasi-compact-open-immersion-constructible-image}", "and hence constructible." ], "refs": [ "topology-lemma-open-immersion-constructible-inverse-image", "topology-lemma-quasi-compact-open-immersion-constructible-image" ], "ref_ids": [ 8255, 8256 ] } ], "ref_ids": [] }, { "id": 8258, "type": "theorem", "label": "topology-lemma-intersect-constructible-with-closed", "categories": [ "topology" ], "title": "topology-lemma-intersect-constructible-with-closed", "contents": [ "Let $X$ be a topological space. Let $Z \\subset X$ be a closed", "subset such that $X \\setminus Z$ is quasi-compact.", "Then for a constructible set $E \\subset X$ the intersection", "$E \\cap Z$ is constructible in $Z$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $V \\subset X$ is retrocompact open in $X$.", "It suffices to show that $V \\cap Z$ is retrocompact in $Z$", "by Lemma \\ref{lemma-inverse-images-constructibles}. To show this", "let $W \\subset Z$ be open and quasi-compact. The subset", "$W' = W \\cup (X \\setminus Z)$ is quasi-compact, open, and $W = Z \\cap W'$.", "Hence $V \\cap Z \\cap W = V \\cap Z \\cap W'$", "is a closed subset of the quasi-compact open $V \\cap W'$", "as $V$ is retrocompact in $X$. Thus $V \\cap Z \\cap W$ is quasi-compact", "by Lemma \\ref{lemma-closed-in-quasi-compact}." ], "refs": [ "topology-lemma-inverse-images-constructibles", "topology-lemma-closed-in-quasi-compact" ], "ref_ids": [ 8254, 8229 ] } ], "ref_ids": [] }, { "id": 8259, "type": "theorem", "label": "topology-lemma-intersect-constructible-with-retrocompact", "categories": [ "topology" ], "title": "topology-lemma-intersect-constructible-with-retrocompact", "contents": [ "Let $X$ be a topological space. Let $T \\subset X$ be a subset. Suppose", "\\begin{enumerate}", "\\item $T$ is retrocompact in $X$,", "\\item quasi-compact opens form a basis for the topology on $X$.", "\\end{enumerate}", "Then for a constructible set $E \\subset X$ the intersection $E \\cap T$ is", "constructible in $T$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $V \\subset X$ is retrocompact open in $X$.", "It suffices to show that $V \\cap T$ is retrocompact in $T$", "by Lemma \\ref{lemma-inverse-images-constructibles}. To show this", "let $W \\subset T$ be open and quasi-compact. By assumption (2)", "we can find a quasi-compact open $W' \\subset X$", "such that $W = T \\cap W'$ (details omitted).", "Hence $V \\cap T \\cap W = V \\cap T \\cap W'$", "is the intersection of $T$ with the quasi-compact open $V \\cap W'$", "as $V$ is retrocompact in $X$. Thus $V \\cap T \\cap W$ is quasi-compact." ], "refs": [ "topology-lemma-inverse-images-constructibles" ], "ref_ids": [ 8254 ] } ], "ref_ids": [] }, { "id": 8260, "type": "theorem", "label": "topology-lemma-closed-constructible-image", "categories": [ "topology" ], "title": "topology-lemma-closed-constructible-image", "contents": [ "Let $Z \\subset X$ be a closed subset whose complement is retrocompact open.", "Let $E \\subset Z$. If $E$ is constructible in $Z$, then $E$ is constructible", "in $X$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $V \\subset Z$ is retrocompact open in $Z$. Consider the open", "subset $\\tilde V = V \\cup (X \\setminus Z)$ of $X$. Let $W \\subset X$ be", "quasi-compact open. Then", "$$", "W \\cap \\tilde V =", "\\left(V \\cap W\\right) \\cup \\left((X \\setminus Z) \\cap W\\right).", "$$", "The first part is quasi-compact as $V \\cap W = V \\cap (Z \\cap W)$ and", "$(Z \\cap W)$ is quasi-compact open in $Z$", "(Lemma \\ref{lemma-closed-in-quasi-compact}) and $V$ is retrocompact in $Z$.", "The second part is quasi-compact as $(X \\setminus Z)$ is retrocompact in $X$.", "In this way we see that $\\tilde V$ is retrocompact in $X$.", "Thus if $V_1, V_2 \\subset Z$ are retrocompact open, then", "$$", "V_1 \\cap (Z \\setminus V_2) = \\tilde V_1 \\cap (X \\setminus \\tilde V_2)", "$$", "is constructible in $X$. We conclude since every constructible subset of $Z$", "is a finite union of subsets of the form $V_1 \\cap (Z \\setminus V_2)$." ], "refs": [ "topology-lemma-closed-in-quasi-compact" ], "ref_ids": [ 8229 ] } ], "ref_ids": [] }, { "id": 8261, "type": "theorem", "label": "topology-lemma-constructible-is-retrocompact", "categories": [ "topology" ], "title": "topology-lemma-constructible-is-retrocompact", "contents": [ "Let $X$ be a topological space. Every constructible", "subset of $X$ is retrocompact." ], "refs": [], "proofs": [ { "contents": [ "Let $E = \\bigcup_{i = 1, \\ldots, n} U_i \\cap V_i^c$ with $U_i, V_i$", "retrocompact open in $X$. Let $W \\subset X$ be quasi-compact open.", "Then $E \\cap W = \\bigcup_{i = 1, \\ldots, n} U_i \\cap V_i^c \\cap W$.", "Thus it suffices to show that $U \\cap V^c \\cap W$ is quasi-compact", "if $U, V$ are retrocompact open and $W$ is quasi-compact", "open. This is true because $U \\cap V^c \\cap W$ is a closed", "subset of the quasi-compact $U \\cap W$ so", "Lemma \\ref{lemma-closed-in-quasi-compact}", "applies." ], "refs": [ "topology-lemma-closed-in-quasi-compact" ], "ref_ids": [ 8229 ] } ], "ref_ids": [] }, { "id": 8262, "type": "theorem", "label": "topology-lemma-intersect-constructible-with-constructible", "categories": [ "topology" ], "title": "topology-lemma-intersect-constructible-with-constructible", "contents": [ "Let $X$ be a topological space. Assume", "$X$ has a basis consisting of quasi-compact opens.", "For $E, E'$ constructible in $X$, the intersection", "$E \\cap E'$ is constructible in $E$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-intersect-constructible-with-retrocompact} and", "\\ref{lemma-constructible-is-retrocompact}." ], "refs": [ "topology-lemma-intersect-constructible-with-retrocompact", "topology-lemma-constructible-is-retrocompact" ], "ref_ids": [ 8259, 8261 ] } ], "ref_ids": [] }, { "id": 8263, "type": "theorem", "label": "topology-lemma-constructible-in-constructible", "categories": [ "topology" ], "title": "topology-lemma-constructible-in-constructible", "contents": [ "Let $X$ be a topological space. Assume", "$X$ has a basis consisting of quasi-compact opens.", "Let $E$ be constructible in $X$ and $F \\subset E$ constructible in $E$.", "Then $F$ is constructible in $X$." ], "refs": [], "proofs": [ { "contents": [ "Observe that any retrocompact subset $T$ of $X$ has a basis for the induced", "topology consisting of quasi-compact opens. In particular this holds", "for any constructible subset", "(Lemma \\ref{lemma-constructible-is-retrocompact}).", "Write $E = E_1 \\cup \\ldots \\cup E_n$ with $E_i = U_i \\cap V_i^c$", "where $U_i, V_i \\subset X$ are retrocompact open.", "Note that $E_i = E \\cap E_i$ is constructible in $E$ by", "Lemma \\ref{lemma-intersect-constructible-with-constructible}.", "Hence $F \\cap E_i$ is constructible in $E_i$ by", "Lemma \\ref{lemma-intersect-constructible-with-constructible}.", "Thus it suffices to prove the lemma in case $E = U \\cap V^c$", "where $U, V \\subset X$ are retrocompact open.", "In this case the inclusion $E \\subset X$ is a composition", "$$", "E = U \\cap V^c \\to U \\to X", "$$", "Then we can apply Lemma \\ref{lemma-closed-constructible-image}", "to the first inclusion and", "Lemma \\ref{lemma-quasi-compact-open-immersion-constructible-image}", "to the second." ], "refs": [ "topology-lemma-constructible-is-retrocompact", "topology-lemma-intersect-constructible-with-constructible", "topology-lemma-intersect-constructible-with-constructible", "topology-lemma-closed-constructible-image", "topology-lemma-quasi-compact-open-immersion-constructible-image" ], "ref_ids": [ 8261, 8262, 8262, 8260, 8256 ] } ], "ref_ids": [] }, { "id": 8264, "type": "theorem", "label": "topology-lemma-locally-closed-constructible-image", "categories": [ "topology" ], "title": "topology-lemma-locally-closed-constructible-image", "contents": [ "Let $X$ be a quasi-compact topological space having a basis consisting of", "quasi-compact opens such that the intersection of any two", "quasi-compact opens is quasi-compact.", "Let $T \\subset X$ be a locally closed subset", "such that $T$ is quasi-compact and $T^c$ is retrocompact in $X$.", "Then $T$ is constructible in $X$." ], "refs": [], "proofs": [ { "contents": [ "Note that $T$ is quasi-compact and open in $\\overline{T}$.", "Using our basis of quasi-compact opens we can write", "$T = U \\cap \\overline{T}$ where $U$ is quasi-compact open in $X$.", "Then $U \\setminus T = U \\cap T^c$ is retrocompact in $U$ as $T^c$", "is retrocompact in $X$. Hence the inclusion $T \\subset X$ can be written", "as the composition of the inclusion $T \\subset U$ of a closed", "subset with retrocompact complement and the inclusion $U \\subset X$", "which is retrocompact by our assumption on intersections of quasi-compact", "opens. Thus the lemma is a consequence of", "Lemmas \\ref{lemma-quasi-compact-open-immersion-constructible-image} and", "\\ref{lemma-closed-constructible-image}." ], "refs": [ "topology-lemma-quasi-compact-open-immersion-constructible-image", "topology-lemma-closed-constructible-image" ], "ref_ids": [ 8256, 8260 ] } ], "ref_ids": [] }, { "id": 8265, "type": "theorem", "label": "topology-lemma-collate-constructible-from-constructible", "categories": [ "topology" ], "title": "topology-lemma-collate-constructible-from-constructible", "contents": [ "Let $X$ be a topological space which has a basis for the topology", "consisting of quasi-compact opens. Let $E \\subset X$ be a subset.", "Let $X = E_1 \\cup \\ldots \\cup E_m$ be a finite covering by constructible", "subsets. Then $E$ is constructible in $X$ if and only if $E \\cap E_j$", "is constructible in $E_j$ for each $j = 1, \\ldots, m$." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemmas \\ref{lemma-intersect-constructible-with-constructible} and", "\\ref{lemma-constructible-in-constructible}." ], "refs": [ "topology-lemma-intersect-constructible-with-constructible", "topology-lemma-constructible-in-constructible" ], "ref_ids": [ 8262, 8263 ] } ], "ref_ids": [] }, { "id": 8266, "type": "theorem", "label": "topology-lemma-generic-point-in-constructible", "categories": [ "topology" ], "title": "topology-lemma-generic-point-in-constructible", "contents": [ "Let $X$ be a topological space. Suppose that", "$Z \\subset X$ is irreducible. Let $E \\subset X$", "be a finite union of locally closed subsets (e.g.\\ $E$", "is constructible). The following are equivalent", "\\begin{enumerate}", "\\item The intersection $E \\cap Z$ contains an open", "dense subset of $Z$.", "\\item The intersection $E \\cap Z$ is dense in $Z$.", "\\end{enumerate}", "If $Z$ has a generic point $\\xi$, then this is", "also equivalent to", "\\begin{enumerate}", "\\item[(3)] We have $\\xi \\in E$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $E = \\bigcup U_i \\cap Z_i$ as the finite union of", "intersections of open sets $U_i$ and closed sets $Z_i$.", "Suppose that $E \\cap Z$ is dense in $Z$. Note that", "the closure of $E \\cap Z$ is the union of the closures", "of the intersections $U_i \\cap Z_i \\cap Z$. As $Z$ is irreducible we", "conclude that the closure of $U_i \\cap Z_i \\cap Z$ is $Z$ for some $i$.", "Fix such an $i$. It follows that $Z \\subset Z_i$ since otherwise", "the closed subset $Z \\cap Z_i$ of $Z$ would not be dense in $Z$.", "Then $U_i \\cap Z_i \\cap Z = U_i \\cap Z$ is an open nonempty subset of $Z$.", "Because $Z$ is irreducible, it is open dense. Hence $E \\cap Z$", "contains an open dense subset of $Z$.", "The converse is obvious.", "\\medskip\\noindent", "Suppose that $\\xi \\in Z$ is a generic point. Of course if", "(1) $\\Leftrightarrow$ (2) holds, then $\\xi \\in E$. Conversely,", "if $\\xi \\in E$, then $\\xi \\in U_i \\cap Z_i$ for some $i = i_0$.", "Clearly this implies $Z \\subset Z_{i_0}$ and hence", "$U_{i_0} \\cap Z_{i_0} \\cap Z = U_{i_0} \\cap Z$ is an open", "not empty subset of $Z$. We conclude as before." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8267, "type": "theorem", "label": "topology-lemma-constructible-Noetherian-space", "categories": [ "topology" ], "title": "topology-lemma-constructible-Noetherian-space", "contents": [ "Let $X$ be a Noetherian topological space.", "The constructible sets in $X$ are precisely the finite unions", "of locally closed subsets of $X$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from", "Lemma \\ref{lemma-Noetherian-quasi-compact}." ], "refs": [ "topology-lemma-Noetherian-quasi-compact" ], "ref_ids": [ 8239 ] } ], "ref_ids": [] }, { "id": 8268, "type": "theorem", "label": "topology-lemma-constructible-map-Noetherian", "categories": [ "topology" ], "title": "topology-lemma-constructible-map-Noetherian", "contents": [ "Let $f : X \\to Y$ be a continuous map of Noetherian topological spaces.", "If $E \\subset Y$ is constructible in $Y$, then $f^{-1}(E)$ is constructible", "in $X$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Lemma \\ref{lemma-constructible-Noetherian-space}", "and the definition of a continuous map." ], "refs": [ "topology-lemma-constructible-Noetherian-space" ], "ref_ids": [ 8267 ] } ], "ref_ids": [] }, { "id": 8269, "type": "theorem", "label": "topology-lemma-characterize-constructible-Noetherian", "categories": [ "topology" ], "title": "topology-lemma-characterize-constructible-Noetherian", "contents": [ "Let $X$ be a Noetherian topological space.", "Let $E \\subset X$ be a subset.", "The following are equivalent:", "\\begin{enumerate}", "\\item $E$ is constructible in $X$, and", "\\item for every irreducible closed $Z \\subset X$ the intersection", "$E \\cap Z$ either contains a nonempty open of $Z$ or is not dense in $Z$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $E$ is constructible and $Z \\subset X$ irreducible closed.", "Then $E \\cap Z$ is constructible in $Z$ by", "Lemma \\ref{lemma-constructible-map-Noetherian}.", "Hence $E \\cap Z$ is a finite union of nonempty locally closed subsets", "$T_i$ of $Z$. Clearly if none of the $T_i$ is open in $Z$, then", "$E \\cap Z$ is not dense in $Z$. In this way we see that (1) implies (2).", "\\medskip\\noindent", "Conversely, assume (2) holds. Consider the set $\\mathcal{S}$ of closed", "subsets $Y$ of $X$ such that $E \\cap Y$ is not constructible in $Y$.", "If $\\mathcal{S} \\not = \\emptyset$, then it has a smallest element $Y$", "as $X$ is Noetherian.", "Let $Y = Y_1 \\cup \\ldots \\cup Y_r$ be the decomposition of $Y$ into its", "irreducible components, see", "Lemma \\ref{lemma-Noetherian}.", "If $r > 1$, then each $Y_i \\cap E$ is constructible in $Y_i$ and hence", "a finite union of locally closed subsets of $Y_i$. Thus $E \\cap Y$", "is a finite union of locally closed subsets of $Y$ too and we conclude", "that $E \\cap Y$ is constructible in $Y$ by", "Lemma \\ref{lemma-constructible-Noetherian-space}.", "This is a contradiction and so $r = 1$. If $r = 1$, then $Y$ is", "irreducible, and by assumption (2) we see that $E \\cap Y$ either", "(a) contains an open $V$ of $Y$ or (b) is not dense in $Y$.", "In case (a) we see, by minimality of $Y$, that $E \\cap (Y \\setminus V)$", "is a finite union of locally closed subsets of $Y \\setminus V$. Thus", "$E \\cap Y$ is a finite union of locally closed subsets of $Y$ and is", "constructible by", "Lemma \\ref{lemma-constructible-Noetherian-space}.", "This is a contradiction and so we must be in case (b).", "In case (b) we see that $E \\cap Y = E \\cap Y'$ for some proper closed", "subset $Y' \\subset Y$. By minimality of $Y$ we see that", "$E \\cap Y'$ is a finite union of locally closed subsets of $Y'$ and", "we see that $E \\cap Y' = E \\cap Y$ is a finite union of locally closed", "subsets of $Y$ and is constructible by", "Lemma \\ref{lemma-constructible-Noetherian-space}.", "This contradiction finishes the proof of the lemma." ], "refs": [ "topology-lemma-constructible-map-Noetherian", "topology-lemma-Noetherian", "topology-lemma-constructible-Noetherian-space", "topology-lemma-constructible-Noetherian-space", "topology-lemma-constructible-Noetherian-space" ], "ref_ids": [ 8268, 8220, 8267, 8267, 8267 ] } ], "ref_ids": [] }, { "id": 8270, "type": "theorem", "label": "topology-lemma-constructible-neighbourhood-Noetherian", "categories": [ "topology" ], "title": "topology-lemma-constructible-neighbourhood-Noetherian", "contents": [ "Let $X$ be a Noetherian topological space.", "Let $x \\in X$.", "Let $E \\subset X$ be constructible in $X$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $E$ is a neighbourhood of $x$, and", "\\item for every irreducible closed subset $Y$ of $X$ which contains", "$x$ the intersection $E \\cap Y$ is dense in $Y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2). Assume (2).", "Consider the set $\\mathcal{S}$ of closed subsets $Y$ of $X$ containing $x$", "such that $E \\cap Y$ is not a neighbourhood of $x$ in $Y$.", "If $\\mathcal{S} \\not = \\emptyset$, then it has a minimal element $Y$", "as $X$ is Noetherian. Suppose $Y = Y_1 \\cup Y_2$ with two smaller nonempty", "closed subsets $Y_1$, $Y_2$. If $x \\in Y_i$ for $i = 1, 2$, then $Y_i \\cap E$", "is a neighbourhood of $x$ in $Y_i$ and we conclude $Y \\cap E$ is a", "neighbourhood of $x$ in $Y$ which is a contradiction. If $x \\in Y_1$ but", "$x \\not\\in Y_2$ (say), then $Y_1 \\cap E$ is a neighbourhood of $x$ in", "$Y_1$ and hence also in $Y$, which is a contradiction as well.", "We conclude that $Y$ is irreducible closed. By assumption (2) we see that", "$E \\cap Y$ is dense in $Y$. Thus $E \\cap Y$ contains an open $V$ of $Y$, see", "Lemma \\ref{lemma-characterize-constructible-Noetherian}.", "If $x \\in V$ then $E \\cap Y$ is a neighbourhood of $x$ in $Y$ which", "is a contradiction. If $x \\not \\in V$, then $Y' = Y \\setminus V$ is a", "proper closed subset of $Y$ containing $x$. By minimality of $Y$", "we see that $E \\cap Y'$ contains an open neighbourhood $V' \\subset Y'$", "of $x$ in $Y'$. But then $V' \\cup V$ is an open neighbourhood of $x$", "in $Y$ contained in $E$, a contradiction.", "This contradiction finishes the proof of the lemma." ], "refs": [ "topology-lemma-characterize-constructible-Noetherian" ], "ref_ids": [ 8269 ] } ], "ref_ids": [] }, { "id": 8271, "type": "theorem", "label": "topology-lemma-characterize-open-Noetherian", "categories": [ "topology" ], "title": "topology-lemma-characterize-open-Noetherian", "contents": [ "Let $X$ be a Noetherian topological space.", "Let $E \\subset X$ be a subset.", "The following are equivalent:", "\\begin{enumerate}", "\\item $E$ is open in $X$, and", "\\item for every irreducible closed subset $Y$ of $X$", "the intersection $E \\cap Y$ is either empty or", "contains a nonempty open of $Y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows formally from", "Lemmas \\ref{lemma-characterize-constructible-Noetherian} and", "\\ref{lemma-constructible-neighbourhood-Noetherian}." ], "refs": [ "topology-lemma-characterize-constructible-Noetherian", "topology-lemma-constructible-neighbourhood-Noetherian" ], "ref_ids": [ 8269, 8270 ] } ], "ref_ids": [] }, { "id": 8272, "type": "theorem", "label": "topology-lemma-tube", "categories": [ "topology" ], "title": "topology-lemma-tube", "contents": [ "Let $X$ and $Y$ be topological spaces.", "Let $A \\subset X$ and $B \\subset Y$ be quasi-compact subsets.", "Let $A \\times B \\subset W \\subset X \\times Y$ with $W$", "open in $X \\times Y$. Then there exists opens $A \\subset U \\subset X$", "and $B \\subset V \\subset Y$ such that $U \\times V \\subset W$." ], "refs": [], "proofs": [ { "contents": [ "For every $a \\in A$ and $b \\in B$ there exist opens", "$U_{(a, b)}$ of $X$ and $V_{(a, b)}$ of $Y$ such that", "$(a, b) \\in U_{(a, b)} \\times V_{(a, b)} \\subset W$.", "Fix $b$ and we see there exist a finite number $a_1, \\ldots, a_n$", "such that $A \\subset U_{(a_1, b)} \\cup \\ldots \\cup U_{(a_n, b)}$.", "Hence", "$$", "A \\times \\{b\\} \\subset", "(U_{(a_1, b)} \\cup \\ldots \\cup U_{(a_n, b)}) \\times", "(V_{(a_1, b)} \\cap \\ldots \\cap V_{(a_n, b)}) \\subset W.", "$$", "Thus for every $b \\in B$ there exists opens $U_b \\subset X$ and", "$V_b \\subset Y$ such that $A \\times \\{b\\} \\subset U_b \\times V_b \\subset W$.", "As above there exist a finite number $b_1, \\ldots, b_m$ such", "that $B \\subset V_{b_1} \\cup \\ldots \\cup V_{b_m}$.", "Then we win because", "$A \\times B \\subset", "(U_{b_1} \\cap \\ldots \\cap U_{b_m}) \\times", "(V_{b_1} \\cup \\ldots \\cup V_{b_m})$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8273, "type": "theorem", "label": "topology-lemma-characterize-quasi-compact", "categories": [ "topology" ], "title": "topology-lemma-characterize-quasi-compact", "contents": [ "\\begin{reference}", "Combination of", "\\cite[I, p. 75, Lemme 1]{Bourbaki} and", "\\cite[I, p. 76, Corrolaire 1]{Bourbaki}.", "\\end{reference}", "A topological space $X$ is quasi-compact if and only if the", "projection map $Z \\times X \\to Z$ is closed for", "any topological space $Z$." ], "refs": [], "proofs": [ { "contents": [ "(See also remark below.)", "If $X$ is not quasi-compact, there exists an open covering", "$X = \\bigcup_{i \\in I} U_i$ such that no finite", "number of $U_i$ cover $X$.", "Let $Z$ be the subset of the power set $\\mathcal{P}(I)$ of $I$", "consisting of $I$ and all nonempty finite subsets of $I$.", "Define a topology on $Z$ with as a basis for the topology", "the following sets:", "\\begin{enumerate}", "\\item All subsets of $Z\\setminus\\{I\\}$.", "\\item For every finite subset $K$ of $I$ the set", "$U_K := \\{J\\subset I \\mid J \\in Z, \\ K\\subset J \\})$.", "\\end{enumerate}", "It is left to the reader to verify this is the basis for a topology.", "Consider the subset of $Z \\times X$ defined by the formula", "$$", "M = \\{(J, x) \\mid J \\in Z, \\ x \\in \\bigcap\\nolimits_{i \\in J} U_i^c)\\}", "$$", "If $(J, x) \\not \\in M$, then $x \\in U_i$ for some $i \\in J$.", "Hence $U_{\\{i\\}} \\times U_i \\subset Z \\times X$ is an open", "subset containing $(J, x)$ and not intersecting $M$. Hence", "$M$ is closed. The projection of $M$ to $Z$ is $Z-\\{I\\}$", "which is not closed. Hence $Z \\times X \\to Z$ is not closed.", "\\medskip\\noindent", "Assume $X$ is quasi-compact. Let $Z$ be a topological space.", "Let $M \\subset Z \\times X$ be closed. Let $z \\in Z$ be a point", "which is not in $\\text{pr}_1(M)$. By the Tube Lemma \\ref{lemma-tube}", "there exists an open $U \\subset Z$ such that $U \\times X$ is", "contained in the complement of $M$. Hence $\\text{pr}_1(M)$ is closed." ], "refs": [ "topology-lemma-tube" ], "ref_ids": [ 8272 ] } ], "ref_ids": [] }, { "id": 8274, "type": "theorem", "label": "topology-lemma-closed-map", "categories": [ "topology" ], "title": "topology-lemma-closed-map", "contents": [ "\\begin{slogan}", "A map from a compact space to a Hausdorff space is a proper.", "\\end{slogan}", "Let $f : X \\to Y$ be a continuous map of topological spaces.", "If $X$ is quasi-compact and $Y$ is Hausdorff, then $f$ is proper." ], "refs": [], "proofs": [ { "contents": [ "Since every point of $Y$ is closed, we see from", "Lemma \\ref{lemma-closed-in-quasi-compact}", "that the closed subset $f^{-1}(y)$ of $X$ is quasi-compact for all $y \\in Y$.", "Thus, by Theorem \\ref{theorem-characterize-proper}", "it suffices to show that $f$ is closed.", "If $E \\subset X$ is closed, then it is quasi-compact", "(Lemma \\ref{lemma-closed-in-quasi-compact}),", "hence $f(E) \\subset Y$ is quasi-compact", "(Lemma \\ref{lemma-image-quasi-compact}),", "hence $f(E)$ is closed in $Y$", "(Lemma \\ref{lemma-quasi-compact-in-Hausdorff})." ], "refs": [ "topology-lemma-closed-in-quasi-compact", "topology-theorem-characterize-proper", "topology-lemma-closed-in-quasi-compact", "topology-lemma-image-quasi-compact", "topology-lemma-quasi-compact-in-Hausdorff" ], "ref_ids": [ 8229, 8189, 8229, 8233, 8230 ] } ], "ref_ids": [] }, { "id": 8275, "type": "theorem", "label": "topology-lemma-bijective-map", "categories": [ "topology" ], "title": "topology-lemma-bijective-map", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "If $f$ is bijective, $X$ is quasi-compact, and $Y$ is Hausdorff,", "then $f$ is a homeomorphism." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from Lemma \\ref{lemma-closed-map}", "which tells us that $f$ is closed, i.e., $f^{-1}$ is", "continuous." ], "refs": [ "topology-lemma-closed-map" ], "ref_ids": [ 8274 ] } ], "ref_ids": [] }, { "id": 8276, "type": "theorem", "label": "topology-lemma-jacobson-check-irreducible-closed", "categories": [ "topology" ], "title": "topology-lemma-jacobson-check-irreducible-closed", "contents": [ "Let $X$ be a topological space. Let $X_0$ be the set", "of closed points of $X$.", "Suppose that for every point $x\\in X$", "the intersection $X_0 \\cap \\overline{\\{x\\}}$ is dense in $\\overline{\\{x\\}}$.", "Then $X$ is Jacobson." ], "refs": [], "proofs": [ { "contents": [ "Let $Z$ be closed subset of $X$", "and $U$ be and open subset of $X$", "such that $U\\cap Z$ is nonempty.", "Then for $x\\in U\\cap Z$ we have that $\\overline{\\{x\\}}\\cap U$ is a nonempty", "subset of $Z\\cap U$,", "and by hypothesis it contains a point closed in $X$ as required." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8277, "type": "theorem", "label": "topology-lemma-non-jacobson-Noetherian-characterize", "categories": [ "topology" ], "title": "topology-lemma-non-jacobson-Noetherian-characterize", "contents": [ "Let $X$ be a Kolmogorov topological space with a basis of quasi-compact", "open sets.", "If $X$ is not Jacobson, then there exists a non-closed point", "$x \\in X$ such that $\\{x\\}$ is locally closed." ], "refs": [], "proofs": [ { "contents": [ "As $X$ is not Jacobson there exists a closed set $Z$ and an open set $U$", "in $X$ such that $Z \\cap U$ is nonempty and does not contain points closed", "in $X$. As $X$ has a basis of quasi-compact open sets we may replace $U$", "by an open quasi-compact neighborhood of a point in $Z\\cap U$ and so we may", "assume that $U$ is quasi-compact open. By", "Lemma \\ref{lemma-quasi-compact-closed-point}, there exists a point", "$x \\in Z \\cap U$ closed in $Z \\cap U$,", "and so $\\{x\\}$ is locally closed but not closed in $X$." ], "refs": [ "topology-lemma-quasi-compact-closed-point" ], "ref_ids": [ 8234 ] } ], "ref_ids": [] }, { "id": 8278, "type": "theorem", "label": "topology-lemma-jacobson-local", "categories": [ "topology" ], "title": "topology-lemma-jacobson-local", "contents": [ "Let $X$ be a topological space.", "Let $X = \\bigcup U_i$ be an open covering.", "Then $X$ is Jacobson if and only if each $U_i$ is Jacobson.", "Moreover, in this case $X_0 = \\bigcup U_{i, 0}$." ], "refs": [], "proofs": [ { "contents": [ "Let $X$ be a topological space.", "Let $X_0$ be the set of closed points of $X$.", "Let $U_{i, 0}$ be the set of closed points of", "$U_i$. Then $X_0 \\cap U_i \\subset U_{i, 0}$", "but equality may not hold in general.", "\\medskip\\noindent", "First, assume that each $U_i$ is Jacobson.", "We claim that in this case $X_0 \\cap U_i = U_{i, 0}$.", "Namely, suppose that $x \\in U_{i, 0}$, i.e., $x$ is closed in", "$U_i$. Let $\\overline{\\{x\\}}$ be the closure", "in $X$. Consider $\\overline{\\{x\\}} \\cap U_j$.", "If $x \\not \\in U_j$, then $\\overline{\\{x\\}} \\cap U_j = \\emptyset$.", "If $x \\in U_j$, then $U_i \\cap U_j \\subset U_j$", "is an open subset of $U_j$ containing $x$.", "Let $T' = U_j \\setminus U_i \\cap U_j$ and", "$T = \\{x\\} \\amalg T'$. Then $T$, $T'$", "are closed subsets of $U_j$ and $T$ contains", "$x$. As $U_j$ is Jacobson we see that the closed points of", "$U_j$ are dense in $T$. Because $T = \\{x\\} \\amalg T'$", "this can only be the case if $x$ is closed in $U_j$.", "Hence $\\overline{\\{x\\}} \\cap U_j = \\{x\\}$. We conclude", "that $\\overline{\\{x\\}} = \\{ x \\}$ as desired.", "\\medskip\\noindent", "Let $Z \\subset X$ be a closed subset (still", "assuming each $U_i$ is Jacobson).", "Since now we know that $X_0 \\cap Z \\cap U_i", "= U_{i, 0} \\cap Z$ are dense in $Z \\cap U_i$", "it follows immediately that $X_0 \\cap Z$ is", "dense in $Z$.", "\\medskip\\noindent", "Conversely, assume that $X$ is Jacobson.", "Let $Z \\subset U_i$ be closed. Then", "$X_0 \\cap \\overline{Z}$ is dense in $\\overline{Z}$.", "Hence also $X_0 \\cap Z$ is dense in $Z$, because", "$\\overline{Z} \\setminus Z$ is closed. As $X_0 \\cap U_i", "\\subset U_{i, 0}$ we see that", "$U_{i, 0} \\cap Z$ is dense in $Z$.", "Thus $U_i$ is Jacobson as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8279, "type": "theorem", "label": "topology-lemma-jacobson-inherited", "categories": [ "topology" ], "title": "topology-lemma-jacobson-inherited", "contents": [ "Let $X$ be Jacobson. The following types of subsets $T \\subset X$", "are Jacobson:", "\\begin{enumerate}", "\\item Open subspaces.", "\\item Closed subspaces.", "\\item Locally closed subspaces.", "\\item Unions of locally closed subspaces.", "\\item Constructible sets.", "\\item Any subset $T \\subset X$ which locally on $X$", "is a union of locally closed subsets.", "\\end{enumerate}", "In each of these cases closed points of $T$ are", "closed in $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $X_0$ be the set of closed points of $X$. For any subset", "$T \\subset X$ we let $(*)$ denote the property:", "\\begin{itemize}", "\\item[(*)] Every nonempty locally closed subset of $T$ has a point", "closed in $X$.", "\\end{itemize}", "Note that always $X_0 \\cap T \\subset T_0$. Hence property $(*)$", "implies that $T$ is Jacobson. In addition it clearly implies", "that every closed point of $T$ is closed in $X$.", "\\medskip\\noindent", "Suppose that $T=\\bigcup_i T_i$ with $T_i$ locally closed in $X$.", "Take $A\\subset T$ a locally closed nonempty subset in $T$,", "then there exists a $T_i$ such that $A\\cap T_i$ is nonempty, it is", "locally closed in $T_i$ and so in $X$.", "As $X$ is Jacobson $A$ has a point closed in $X$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8280, "type": "theorem", "label": "topology-lemma-finite-jacobson", "categories": [ "topology" ], "title": "topology-lemma-finite-jacobson", "contents": [ "A finite Jacobson space is discrete." ], "refs": [], "proofs": [ { "contents": [ "If $X$ is finite Jacobson, $X_0 \\subset X$ the subset of closed points,", "then, on the one hand, $\\overline{X_0} = X$. On the other hand, $X$,", "and hence $X_0$ is finite, so", "$X_0 =\\{x_1, \\ldots, x_n\\} = \\bigcup_{i = 1, \\ldots, n} \\{x_i\\}$", "is a finite union of closed sets, hence closed, so", "$X = \\overline{X_0} = X_0$. Every point is closed, and by", "finiteness, every point is open." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8281, "type": "theorem", "label": "topology-lemma-jacobson-equivalent-locally-closed", "categories": [ "topology" ], "title": "topology-lemma-jacobson-equivalent-locally-closed", "contents": [ "\\begin{slogan}", "For Jacobson spaces, closed points see everything about the topology.", "\\end{slogan}", "Suppose $X$ is a Jacobson topological space.", "Let $X_0$ be the set of closed points of $X$.", "There is a bijective, inclusion preserving correspondence", "$$", "\\{\\text{finite unions loc.\\ closed subsets of } X\\}", "\\leftrightarrow", "\\{\\text{finite unions loc.\\ closed subsets of } X_0\\}", "$$", "given by $E \\mapsto E \\cap X_0$. This correspondence preserves", "the subsets of locally closed, of open and of closed subsets." ], "refs": [], "proofs": [ { "contents": [ "We just prove that the correspondence $E \\mapsto E \\cap X_0$ is injective.", "Indeed if $E\\neq E'$ then without loss of generality $E\\setminus E'$ is", "nonempty, and it is a finite union of locally closed sets (details omitted).", "As $X$ is Jacobson, we see that", "$(E \\setminus E') \\cap X_0 = E \\cap X_0 \\setminus E' \\cap X_0$ is not empty." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8282, "type": "theorem", "label": "topology-lemma-jacobson-equivalent-constructible", "categories": [ "topology" ], "title": "topology-lemma-jacobson-equivalent-constructible", "contents": [ "Suppose $X$ is a Jacobson topological space.", "Let $X_0$ be the set of closed points of $X$.", "There is a bijective, inclusion preserving correspondence", "$$", "\\{\\text{constructible subsets of } X\\}", "\\leftrightarrow", "\\{\\text{constructible subsets of } X_0\\}", "$$", "given by $E \\mapsto E \\cap X_0$. This correspondence preserves", "the subset of retrocompact open subsets, as well as complements", "of these." ], "refs": [], "proofs": [ { "contents": [ "From Lemma \\ref{lemma-jacobson-equivalent-locally-closed} above,", "we just have to see that if $U$ is open in $X$ then $U\\cap X_0$ is", "retrocompact in $X_0$ if and only if $U$ is retrocompact in $X$.", "This follows if we prove that for $U$ open in $X$ then $U\\cap X_0$ is", "quasi-compact if and only if $U$ is quasi-compact.", "From Lemma \\ref{lemma-jacobson-inherited} it follows that we may replace", "$X$ by $U$ and assume that $U = X$.", "Finally notice that any collection of opens $\\mathcal{U}$ of $X$ cover", "$X$ if and only if they cover $X_0$, using the Jacobson property", "of $X$ in the closed $X\\setminus \\bigcup \\mathcal{U}$ to find a point", "in $X_0$ if it were nonempty." ], "refs": [ "topology-lemma-jacobson-equivalent-locally-closed", "topology-lemma-jacobson-inherited" ], "ref_ids": [ 8281, 8279 ] } ], "ref_ids": [] }, { "id": 8283, "type": "theorem", "label": "topology-lemma-open-closed-specialization", "categories": [ "topology" ], "title": "topology-lemma-open-closed-specialization", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item Any closed subset of $X$ is stable under specialization.", "\\item Any open subset of $X$ is stable under generalization.", "\\item A subset $T \\subset X$ is stable under specialization", "if and only if", "the complement $T^c$ is stable under generalization.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8284, "type": "theorem", "label": "topology-lemma-stable-specialization", "categories": [ "topology" ], "title": "topology-lemma-stable-specialization", "contents": [ "Let $T \\subset X$ be a subset of a topological space $X$.", "The following are equivalent", "\\begin{enumerate}", "\\item $T$ is stable under specialization, and", "\\item $T$ is a (directed) union of closed subsets of $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8285, "type": "theorem", "label": "topology-lemma-lift-specialization-composition", "categories": [ "topology" ], "title": "topology-lemma-lift-specialization-composition", "contents": [ "Suppose $f : X \\to Y$ and $g : Y \\to Z$ are continuous maps", "of topological spaces. If specializations lift along both $f$ and $g$", "then specializations lift along $g \\circ f$. Similarly for", "``generalizations lift along''." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8286, "type": "theorem", "label": "topology-lemma-lift-specializations-images", "categories": [ "topology" ], "title": "topology-lemma-lift-specializations-images", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "\\begin{enumerate}", "\\item If specializations lift along $f$, and if $T \\subset X$", "is stable under specialization, then $f(T) \\subset Y$ is", "stable under specialization.", "\\item If generalizations lift along $f$, and if $T \\subset X$", "is stable under generalization, then $f(T) \\subset Y$ is", "stable under generalization.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8287, "type": "theorem", "label": "topology-lemma-closed-open-map-specialization", "categories": [ "topology" ], "title": "topology-lemma-closed-open-map-specialization", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "\\begin{enumerate}", "\\item If $f$ is closed then specializations lift along $f$.", "\\item If $f$ is open, $X$ is a Noetherian topological space,", "each irreducible closed subset of $X$ has a generic point,", "and $Y$ is Kolmogorov then generalizations lift along $f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f$ is closed. Let $y' \\leadsto y$ in $Y$ and any $x'\\in X$ with", "$f(x') = y'$ be given. Consider the closed subset $T = \\overline{\\{x'\\}}$", "of $X$. Then $f(T) \\subset Y$ is a closed subset, and $y' \\in f(T)$.", "Hence also $y \\in f(T)$. Hence $y = f(x)$ with $x \\in T$, i.e.,", "$x' \\leadsto x$.", "\\medskip\\noindent", "Assume $f$ is open, $X$ Noetherian, every irreducible closed subset of $X$", "has a generic point, and $Y$ is Kolmogorov.", "Let $y' \\leadsto y$ in $Y$ and any $x \\in X$ with", "$f(x) = y$ be given. Consider $T = f^{-1}(\\{y'\\}) \\subset X$.", "Take an open neighbourhood $x \\in U \\subset X$ of $x$.", "Then $f(U) \\subset Y$ is open and $y \\in f(U)$. Hence also $y' \\in f(U)$.", "In other words, $T \\cap U \\not = \\emptyset$. This proves that", "$x \\in \\overline{T}$. Since $X$ is Noetherian, $T$ is Noetherian", "(Lemma \\ref{lemma-Noetherian}).", "Hence it has a decomposition $T = T_1 \\cup \\ldots \\cup T_n$ into irreducible", "components. Then correspondingly", "$\\overline{T} = \\overline{T_1} \\cup \\ldots \\cup \\overline{T_n}$.", "By the above $x \\in \\overline{T_i}$ for some $i$. By assumption", "there exists a generic point $x' \\in \\overline{T_i}$, and", "we see that $x' \\leadsto x$. As $x' \\in \\overline{T}$ we see that", "$f(x') \\in \\overline{\\{y'\\}}$. Note that", "$f(\\overline{T_i}) = f(\\overline{\\{x'\\}}) \\subset \\overline{\\{f(x')\\}}$.", "If $f(x') \\not = y'$, then since $Y$ is Kolmogorov $f(x')$ is not a generic", "point of the irreducible closed subset $\\overline{\\{y'\\}}$ and the inclusion", "$\\overline{\\{f(x')\\}} \\subset \\overline{\\{y'\\}}$", "is strict, i.e., $y' \\not \\in f(\\overline{T_i})$.", "This contradicts the fact that $f(T_i) = \\{y'\\}$.", "Hence $f(x') = y'$ and we win." ], "refs": [ "topology-lemma-Noetherian" ], "ref_ids": [ 8220 ] } ], "ref_ids": [] }, { "id": 8288, "type": "theorem", "label": "topology-lemma-quotient-kolmogorov", "categories": [ "topology" ], "title": "topology-lemma-quotient-kolmogorov", "contents": [ "Suppose that $s, t : R \\to U$ and $\\pi : U \\to X$ are continuous maps", "of topological spaces such that", "\\begin{enumerate}", "\\item $\\pi$ is open,", "\\item $U$ is sober,", "\\item $s, t$ have finite fibres,", "\\item generalizations lift along $s, t$,", "\\item $(t, s)(R) \\subset U \\times U$ is an equivalence relation on $U$ and", "$X$ is the quotient of $U$ by this equivalence relation (as a set).", "\\end{enumerate}", "Then $X$ is Kolmogorov." ], "refs": [], "proofs": [ { "contents": [ "Properties (3) and (5) imply that a point $x$ corresponds to an", "finite equivalence class $\\{u_1, \\ldots, u_n\\} \\subset U$ of the equivalence", "relation. Suppose that $x' \\in X$ is a second point corresponding to", "the equivalence class $\\{u'_1, \\ldots, u'_m\\} \\subset U$.", "Suppose that $u_i \\leadsto u'_j$ for some $i, j$. Then for any", "$r' \\in R$ with $s(r') = u'_j$ by (4) we can find $r \\leadsto r'$", "with $s(r) = u_i$. Hence $t(r) \\leadsto t(r')$. Since", "$\\{u'_1, \\ldots, u'_m\\} = t(s^{-1}(\\{u'_j\\}))$ we conclude that", "every element of $\\{u'_1, \\ldots, u'_m\\}$ is the specialization of", "an element of $\\{u_1, \\ldots, u_n\\}$.", "Thus $\\overline{\\{u_1\\}} \\cup \\ldots \\cup \\overline{\\{u_n\\}}$ is", "a union of equivalence classes, hence of the form $\\pi^{-1}(Z)$", "for some subset $Z \\subset X$. By (1) we see that $Z$ is closed in $X$", "and in fact $Z = \\overline{\\{x\\}}$ because $\\pi(\\overline{\\{u_i\\}})", "\\subset \\overline{\\{x\\}}$ for each $i$. In other words, $x \\leadsto x'$", "if and only if some lift of $x$ in $U$ specializes to some lift of", "$x'$ in $U$, if and only if every lift of $x'$ in $U$ is a specialization", "of some lift of $x$ in $U$.", "\\medskip\\noindent", "Suppose that both $x \\leadsto x'$ and $x' \\leadsto x$. Say $x$ corresponds", "to $\\{u_1, \\ldots, u_n\\}$ and $x'$ corresponds to $\\{u'_1, \\ldots, u'_m\\}$", "as above. Then, by the results of the preceding paragraph, we can find a", "sequence", "$$", "\\ldots \\leadsto u'_{j_3} \\leadsto u_{i_3} \\leadsto u'_{j_2} \\leadsto", "u_{i_2} \\leadsto u'_{j_1} \\leadsto u_{i_1}", "$$", "which must repeat, hence by (2) we conclude that", "$\\{u_1, \\ldots, u_n\\} = \\{u'_1, \\ldots, u'_m\\}$, i.e., $x = x'$.", "Thus $X$ is Kolmogorov." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8289, "type": "theorem", "label": "topology-lemma-dimension-specializations-lift", "categories": [ "topology" ], "title": "topology-lemma-dimension-specializations-lift", "contents": [ "Let $f : X \\to Y$ be a morphism of topological spaces.", "Suppose that $Y$ is a sober topological space, and $f$ is surjective.", "If either specializations or generalizations lift along $f$, then", "$\\dim(X) \\geq \\dim(Y)$." ], "refs": [], "proofs": [ { "contents": [ "Assume specializations lift along $f$.", "Let $Z_0 \\subset Z_1 \\subset \\ldots Z_e \\subset Y$ be a chain", "of irreducible closed subsets of $X$. Let $\\xi_e \\in X$ be a point", "mapping to the generic point of $Z_e$. By assumption there", "exists a specialization", "$\\xi_e \\leadsto \\xi_{e - 1}$ in $X$ such that $\\xi_{e - 1}$ maps to", "the generic point of $Z_{e - 1}$. Continuing in this manner we find", "a sequence of specializations", "$$", "\\xi_e \\leadsto \\xi_{e - 1} \\leadsto \\ldots \\leadsto \\xi_0", "$$", "with $\\xi_i$ mapping to the generic point of $Z_i$.", "This clearly implies the sequence of irreducible closed", "subsets", "$$", "\\overline{\\{\\xi_0\\}} \\subset", "\\overline{\\{\\xi_1\\}} \\subset \\ldots", "\\overline{\\{\\xi_e\\}}", "$$", "is a chain of length $e$ in $X$.", "The case when generalizations lift along $f$ is similar." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8290, "type": "theorem", "label": "topology-lemma-characterize-closed-Noetherian", "categories": [ "topology" ], "title": "topology-lemma-characterize-closed-Noetherian", "contents": [ "Let $X$ be a Noetherian sober topological space.", "Let $E \\subset X$ be a subset of $X$.", "\\begin{enumerate}", "\\item If $E$ is constructible and stable under specialization, then", "$E$ is closed.", "\\item If $E$ is constructible and stable under generalization, then", "$E$ is open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $E$ be constructible and stable under generalization.", "Let $Y \\subset X$ be an irreducible closed subset with generic point", "$\\xi \\in Y$. If $E \\cap Y$ is nonempty, then it contains $\\xi$ (by", "stability under generalization) and hence is dense in $Y$, hence it", "contains a nonempty open of $Y$, see", "Lemma \\ref{lemma-characterize-constructible-Noetherian}.", "Thus $E$ is open by", "Lemma \\ref{lemma-characterize-open-Noetherian}.", "This proves (2). To prove (1) apply (2) to the complement of $E$ in $X$." ], "refs": [ "topology-lemma-characterize-constructible-Noetherian", "topology-lemma-characterize-open-Noetherian" ], "ref_ids": [ 8269, 8271 ] } ], "ref_ids": [] }, { "id": 8291, "type": "theorem", "label": "topology-lemma-dimension-function-catenary", "categories": [ "topology" ], "title": "topology-lemma-dimension-function-catenary", "contents": [ "Let $X$ be a topological space. If $X$ is sober and has a dimension", "function, then $X$ is catenary. Moreover, for any $x \\leadsto y$", "we have", "$$", "\\delta(x) - \\delta(y) =", "\\text{codim}\\left(\\overline{\\{y\\}}, \\ \\overline{\\{x\\}}\\right).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Suppose $Y \\subset Y' \\subset X$ are irreducible closed subsets.", "Let $\\xi \\in Y$, $\\xi' \\in Y'$ be their generic points.", "Then we see immediately from the definitions that", "$\\text{codim}(Y, Y') \\leq \\delta(\\xi) - \\delta(\\xi') < \\infty$.", "In fact the first inequality is an equality. Namely, suppose", "$$", "Y = Y_0 \\subset Y_1 \\subset \\ldots \\subset Y_e = Y'", "$$", "is any maximal chain of irreducible closed subsets. Let", "$\\xi_i \\in Y_i$ denote the generic point. Then we see that", "$\\xi_i \\leadsto \\xi_{i + 1}$ is an immediate specialization.", "Hence we see that $e = \\delta(\\xi) - \\delta(\\xi')$ as desired.", "This also proves the last statement of the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8292, "type": "theorem", "label": "topology-lemma-dimension-function-unique", "categories": [ "topology" ], "title": "topology-lemma-dimension-function-unique", "contents": [ "Let $X$ be a topological space.", "Let $\\delta$, $\\delta'$ be two dimension functions on $X$.", "If $X$ is locally Noetherian and sober then $\\delta - \\delta'$ is", "locally constant on $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$ be a point. We will show that $\\delta - \\delta'$ is", "constant in a neighbourhood of $x$.", "We may replace $X$ by an open neighbourhood", "of $x$ in $X$ which is Noetherian. Hence we may assume $X$ is", "Noetherian and sober.", "Let $Z_1, \\ldots, Z_r$ be the irreducible", "components of $X$ passing through $x$. (There are finitely many as", "$X$ is Noetherian, see Lemma \\ref{lemma-Noetherian}.)", "Let $\\xi_i \\in Z_i$ be the generic point.", "Note $Z_1 \\cup \\ldots \\cup Z_r$ is a neighbourhood of $x$ in $X$", "(not necessarily closed). We claim that $\\delta - \\delta'$ is", "constant on $Z_1 \\cup \\ldots \\cup Z_r$. Namely, if $y \\in Z_i$,", "then", "$$", "\\delta(x) - \\delta(y) = \\delta(x) - \\delta(\\xi_i) + \\delta(\\xi_i) - \\delta(y)", "= - \\text{codim}(\\overline{\\{x\\}}, Z_i)", "+ \\text{codim}(\\overline{\\{y\\}}, Z_i)", "$$", "by Lemma \\ref{lemma-dimension-function-catenary}.", "Similarly for $\\delta'$. Whence the result." ], "refs": [ "topology-lemma-Noetherian", "topology-lemma-dimension-function-catenary" ], "ref_ids": [ 8220, 8291 ] } ], "ref_ids": [] }, { "id": 8293, "type": "theorem", "label": "topology-lemma-locally-dimension-function", "categories": [ "topology" ], "title": "topology-lemma-locally-dimension-function", "contents": [ "Let $X$ be locally Noetherian, sober and catenary.", "Then any point has an open neighbourhood", "$U \\subset X$ which has a dimension function." ], "refs": [], "proofs": [ { "contents": [ "We will use repeatedly", "that an open subspace of a catenary space is catenary, see", "Lemma \\ref{lemma-catenary} and that a Noetherian topological space", "has finitely many irreducible components, see Lemma \\ref{lemma-Noetherian}.", "In the proof of Lemma \\ref{lemma-dimension-function-unique} we saw how to", "construct such a function. Namely, we first replace $X$ by a Noetherian", "open neighbourhood of $x$. Next, we let $Z_1, \\ldots, Z_r \\subset X$", "be the irreducible components of $X$. Let", "$$", "Z_i \\cap Z_j = \\bigcup Z_{ijk}", "$$", "be the decomposition into irreducible components. We replace", "$X$ by", "$$", "X \\setminus \\left(", "\\bigcup\\nolimits_{x \\not \\in Z_i} Z_i", "\\cup", "\\bigcup\\nolimits_{x \\not \\in Z_{ijk}} Z_{ijk}", "\\right)", "$$", "so that we may assume $x \\in Z_i$ for all $i$ and", "$x \\in Z_{ijk}$ for all $i, j, k$. For $y \\in X$ choose any", "$i$ such that $y \\in Z_i$ and set", "$$", "\\delta(y) = - \\text{codim}(\\overline{\\{x\\}}, Z_i)", "+ \\text{codim}(\\overline{\\{y\\}}, Z_i).", "$$", "We claim this is a dimension function. First we show that it", "is well defined, i.e., independent of the choice of $i$.", "Namely, suppose that $y \\in Z_{ijk}$ for some $i, j, k$.", "Then we have (using Lemma \\ref{lemma-catenary-in-codimension})", "\\begin{align*}", "\\delta(y) & =", "- \\text{codim}(\\overline{\\{x\\}}, Z_i)", "+ \\text{codim}(\\overline{\\{y\\}}, Z_i) \\\\", "& =", "- \\text{codim}(\\overline{\\{x\\}}, Z_{ijk})", "- \\text{codim}(Z_{ijk}, Z_i)", "+ \\text{codim}(\\overline{\\{y\\}}, Z_{ijk})", "+ \\text{codim}(Z_{ijk}, Z_i) \\\\", "& =", "- \\text{codim}(\\overline{\\{x\\}}, Z_{ijk})", "+ \\text{codim}(\\overline{\\{y\\}}, Z_{ijk})", "\\end{align*}", "which is symmetric in $i$ and $j$.", "We omit the proof that it is a dimension function." ], "refs": [ "topology-lemma-catenary", "topology-lemma-Noetherian", "topology-lemma-dimension-function-unique", "topology-lemma-catenary-in-codimension" ], "ref_ids": [ 8226, 8220, 8292, 8227 ] } ], "ref_ids": [] }, { "id": 8294, "type": "theorem", "label": "topology-lemma-nowhere-dense", "categories": [ "topology" ], "title": "topology-lemma-nowhere-dense", "contents": [ "Let $X$ be a topological space. The union of a finite number of nowhere", "dense sets is a nowhere dense set." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8295, "type": "theorem", "label": "topology-lemma-image-nowhere-dense-open", "categories": [ "topology" ], "title": "topology-lemma-image-nowhere-dense-open", "contents": [ "Let $X$ be a topological space.", "Let $U \\subset X$ be an open.", "Let $T \\subset U$ be a subset.", "If $T$ is nowhere dense in $U$, then $T$ is nowhere dense in $X$." ], "refs": [], "proofs": [ { "contents": [ "Assume $T$ is nowhere dense in $U$.", "Suppose that $x \\in X$ is an interior point of the closure", "$\\overline{T}$ of $T$ in $X$. Say $x \\in V \\subset \\overline{T}$", "with $V \\subset X$ open in $X$. Note that $\\overline{T} \\cap U$ is", "the closure of $T$ in $U$. Hence the interior of $\\overline{T} \\cap U$", "being empty implies $V \\cap U = \\emptyset$. Thus $x$ cannot be in the", "closure of $U$, a fortiori cannot be in the closure of $T$, a contradiction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8296, "type": "theorem", "label": "topology-lemma-nowhere-dense-local", "categories": [ "topology" ], "title": "topology-lemma-nowhere-dense-local", "contents": [ "Let $X$ be a topological space.", "Let $X = \\bigcup U_i$ be an open covering.", "Let $T \\subset X$ be a subset.", "If $T \\cap U_i$ is nowhere dense in $U_i$ for all $i$,", "then $T$ is nowhere dense in $X$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. (Hint: closure commutes with intersecting with opens.)" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8297, "type": "theorem", "label": "topology-lemma-closed-image-nowhere-dense", "categories": [ "topology" ], "title": "topology-lemma-closed-image-nowhere-dense", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Let $T \\subset X$ be a subset.", "If $f$ is a homeomorphism of $X$ onto a closed subset of $Y$", "and $T$ is nowhere dense in $X$, then also $f(T)$ is nowhere dense in $Y$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8298, "type": "theorem", "label": "topology-lemma-open-inverse-image-closed-nowhere-dense", "categories": [ "topology" ], "title": "topology-lemma-open-inverse-image-closed-nowhere-dense", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Let $T \\subset Y$ be a subset.", "If $f$ is open and $T$ is a closed nowhere dense subset of $Y$,", "then also $f^{-1}(T)$ is a closed nowhere dense subset of $X$.", "If $f$ is surjective and open, then", "$T$ is closed nowhere dense if and only", "if $f^{-1}(T)$ is closed nowhere dense." ], "refs": [], "proofs": [ { "contents": [ "Omitted. (Hint: In the first case the interior of $f^{-1}(T)$", "maps into the interior of $T$, and in the second case the interior of", "$f^{-1}(T)$ maps onto the interior of $T$.)" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8299, "type": "theorem", "label": "topology-lemma-profinite", "categories": [ "topology" ], "title": "topology-lemma-profinite", "contents": [ "Let $X$ be a topological space.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is a profinite space, and", "\\item $X$ is Hausdorff, quasi-compact, and totally disconnected.", "\\end{enumerate}", "If this is true, then $X$ is a cofiltered limit of finite discrete", "spaces." ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Choose a diagram $i \\mapsto X_i$ of finite discrete spaces", "such that $X = \\lim X_i$. As each $X_i$ is Hausdorff and quasi-compact we find", "that $X$ is quasi-compact by Lemma \\ref{lemma-inverse-limit-quasi-compact}.", "If $x, x' \\in X$ are distinct points, then $x$ and $x'$ map to", "distinct points in some $X_i$. Hence $x$ and $x'$ have disjoint", "open neighbourhoods, i.e., $X$ is Hausdorff. In exactly the same way", "we see that $X$ is totally disconnected.", "\\medskip\\noindent", "Assume (2). Let $\\mathcal{I}$ be the set of finite disjoint union", "decompositions $X = \\coprod_{i \\in I} U_i$ with $U_i$ nonempty open", "(and closed) for all $i \\in I$.", "For each $I \\in \\mathcal{I}$ there is a continuous map", "$X \\to I$ sending a point of $U_i$ to $i$. We define a partial", "ordering: $I \\leq I'$ for $I, I' \\in \\mathcal{I}$ if and only", "if the covering corresponding to $I'$ refines the covering corresponding", "to $I$. In this case we obtain a canonical map $I' \\to I$. In other", "words we obtain an inverse system of finite discrete spaces over $\\mathcal{I}$.", "The maps $X \\to I$ fit together and we obtain a continuous map", "$$", "X \\longrightarrow \\lim_{I \\in \\mathcal{I}} I", "$$", "We claim this map is a homeomorphism, which finishes the proof.", "(The final assertion follows too as the partially ordered set", "$\\mathcal{I}$ is directed: given two disjoint union decompositions", "of $X$ we can find a third refining both.)", "Namely, the map is injective as $X$ is totally disconnected", "and hence $\\{x\\}$ is the intersection of all open and closed subsets", "of $X$ containing $x$", "(Lemma \\ref{lemma-connected-component-intersection-compact-Hausdorff})", "and the map is surjective by", "Lemma \\ref{lemma-intersection-closed-in-quasi-compact}.", "By Lemma \\ref{lemma-bijective-map} the map is a homeomorphism." ], "refs": [ "topology-lemma-inverse-limit-quasi-compact", "topology-lemma-connected-component-intersection-compact-Hausdorff", "topology-lemma-intersection-closed-in-quasi-compact", "topology-lemma-bijective-map" ], "ref_ids": [ 8251, 8237, 8232, 8275 ] } ], "ref_ids": [] }, { "id": 8300, "type": "theorem", "label": "topology-lemma-directed-inverse-limit-profinite", "categories": [ "topology" ], "title": "topology-lemma-directed-inverse-limit-profinite", "contents": [ "A limit of profinite spaces is profinite." ], "refs": [], "proofs": [ { "contents": [ "Let $i \\mapsto X_i$ be a diagram of profinite spaces", "over the index category $\\mathcal{I}$.", "Let us use the characterization of profinite spaces in", "Lemma \\ref{lemma-profinite}. In particular each $X_i$ is", "Hausdorff, quasi-compact, and totally disconnected.", "By Lemma \\ref{lemma-limits} the limit $X = \\lim X_i$ exists.", "By Lemma \\ref{lemma-inverse-limit-quasi-compact}", "the limit $X$ is quasi-compact. Let $x, x' \\in X$ be distinct points.", "Then there exists an $i$ such that $x$ and $x'$", "have distinct images $x_i$ and $x'_i$ in $X_i$", "under the projection $X \\to X_i$. Then $x_i$ and $x'_i$", "have disjoint open neighbourhoods in $X_i$. Taking the", "inverse images of these opens we conclude that $X$ is Hausdorff.", "Similarly, $x_i$ and $x'_i$ are in distinct connected components", "of $X_i$ whence necessarily $x$ and $x'$ must be in distinct", "connected components of $X$. Hence $X$ is totally disconnected.", "This finishes the proof." ], "refs": [ "topology-lemma-profinite", "topology-lemma-limits", "topology-lemma-inverse-limit-quasi-compact" ], "ref_ids": [ 8299, 8248, 8251 ] } ], "ref_ids": [] }, { "id": 8301, "type": "theorem", "label": "topology-lemma-profinite-refine-open-covering", "categories": [ "topology" ], "title": "topology-lemma-profinite-refine-open-covering", "contents": [ "Let $X$ be a profinite space. Every open covering of $X$ has a refinement", "by a finite covering $X = \\coprod U_i$ with $U_i$ open and closed." ], "refs": [], "proofs": [ { "contents": [ "Write $X = \\lim X_i$ as a limit of an inverse system of finite discrete", "spaces over a directed set $I$ (Lemma \\ref{lemma-profinite}).", "Denote $f_i : X \\to X_i$ the projection.", "For every point $x = (x_i) \\in X$ a fundamental system of open neighbourhoods", "is the collection $f_i^{-1}(\\{x_i\\})$. Thus, as $X$ is quasi-compact, we may", "assume we have an open covering", "$$", "X = f_{i_1}^{-1}(\\{x_{i_1}\\}) \\cup \\ldots \\cup f_{i_n}^{-1}(\\{x_{i_n}\\})", "$$", "Choose $i \\in I$ with $i \\geq i_j$ for $j = 1, \\ldots, n$ (this is possible", "as $I$ is a directed set). Then we see", "that the covering", "$$", "X = \\coprod\\nolimits_{t \\in X_i} f_i^{-1}(\\{t\\})", "$$", "refines the given covering and is of the desired form." ], "refs": [ "topology-lemma-profinite" ], "ref_ids": [ 8299 ] } ], "ref_ids": [] }, { "id": 8302, "type": "theorem", "label": "topology-lemma-pi0-profinite", "categories": [ "topology" ], "title": "topology-lemma-pi0-profinite", "contents": [ "Let $X$ be a topological space. If $X$ is quasi-compact", "and every connected component of $X$ is the intersection", "of the open and closed subsets containing it, then $\\pi_0(X)$", "is a profinite space." ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-profinite} to prove this.", "Since $\\pi_0(X)$ is the image of a quasi-compact space it is", "quasi-compact (Lemma \\ref{lemma-image-quasi-compact}).", "It is totally disconnected by construction", "(Lemma \\ref{lemma-space-connected-components}).", "Let $C, D \\subset X$ be distinct connected components of $X$.", "Write $C = \\bigcap U_\\alpha$ as the intersection of the open and", "closed subsets of $X$ containing $C$. Any finite intersection", "of $U_\\alpha$'s is another. Since $\\bigcap U_\\alpha \\cap D = \\emptyset$", "we conclude that $U_\\alpha \\cap D = \\emptyset$ for some $\\alpha$", "(use Lemmas \\ref{lemma-connected-components},", "\\ref{lemma-closed-in-quasi-compact} and", "\\ref{lemma-intersection-closed-in-quasi-compact})", "Since $U_\\alpha$ is open and closed, it is the union of the", "connected components it contains, i.e., $U_\\alpha$ is the inverse", "image of some open and closed subset $V_\\alpha \\subset \\pi_0(X)$.", "This proves that the points corresponding to $C$ and $D$", "are contained in disjoint open subsets, i.e., $\\pi_0(X)$ is", "Hausdorff." ], "refs": [ "topology-lemma-profinite", "topology-lemma-image-quasi-compact", "topology-lemma-space-connected-components", "topology-lemma-connected-components", "topology-lemma-closed-in-quasi-compact", "topology-lemma-intersection-closed-in-quasi-compact" ], "ref_ids": [ 8299, 8233, 8210, 8206, 8229, 8232 ] } ], "ref_ids": [] }, { "id": 8303, "type": "theorem", "label": "topology-lemma-constructible-hausdorff-quasi-compact", "categories": [ "topology" ], "title": "topology-lemma-constructible-hausdorff-quasi-compact", "contents": [ "Let $X$ be a spectral space. The constructible topology is", "Hausdorff, totally disconnected, and quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Let $x, y \\in X$ with $x \\not = y$. Since $X$ is sober, there", "is an open subset $U$ containing exactly one of the two points $x, y$.", "Say $x \\in U$. We may replace $U$ by a quasi-compact open", "neighbourhood of $x$ contained in $U$. Then $U$ and $U^c$ are open and", "closed in the constructible topology. Hence $X$ is Hausdorff in the", "constructible topology because $x \\in U$ and $y \\in U^c$ are", "disjoint opens in the constructible topology. The existence of $U$", "also implies $x$ and $y$ are in distinct connected components in the", "constructible topology, whence $X$ is totally disconnected in the", "constructible topology.", "\\medskip\\noindent", "Let $\\mathcal{B}$ be the collection of subsets $B \\subset X$", "with $B$ either quasi-compact open or closed with quasi-compact", "complement. If $B \\in \\mathcal{B}$ then $B^c \\in \\mathcal{B}$.", "It suffices to show every covering $X = \\bigcup_{i \\in I} B_i$", "with $B_i \\in \\mathcal{B}$ has a finite refinement, see", "Lemma \\ref{lemma-subbase-theorem}.", "Taking complements we see that we have to show that any family", "$\\{B_i\\}_{i \\in I}$ of elements of $\\mathcal{B}$", "such that $B_{i_1} \\cap \\ldots \\cap B_{i_n} \\not = \\emptyset$", "for all $n$ and all $i_1, \\ldots, i_n \\in I$", "has a common point of intersection. We may and do assume", "$B_i \\not = B_{i'}$ for $i \\not = i'$.", "\\medskip\\noindent", "To get a contradiction assume $\\{B_i\\}_{i \\in I}$ is a family", "of elements of $\\mathcal{B}$ having the finite intersection property", "but empty intersection. An application of Zorn's lemma shows that we", "may assume our family is maximal (details omitted).", "Let $I' \\subset I$ be those indices such that", "$B_i$ is closed and set $Z = \\bigcap_{i \\in I'} B_i$.", "This is a closed subset of $X$ which is nonempty by Lemma", "\\ref{lemma-intersection-closed-in-quasi-compact}.", "If $Z$ is reducible, then we can write $Z = Z' \\cup Z''$", "as a union of two closed subsets, neither equal to $Z$. This means in", "particular that we can find a quasi-compact open $U' \\subset X$ meeting", "$Z'$ but not $Z''$. Similarly, we can find a quasi-compact open", "$U'' \\subset X$ meeting $Z''$ but not $Z'$. Set $B' = X \\setminus U'$ and", "$B'' = X \\setminus U''$. Note that $Z'' \\subset B'$ and $Z' \\subset B''$.", "If there exist a finite number of indices $i_1, \\ldots, i_n \\in I$ such", "that $B' \\cap B_{i_1} \\cap \\ldots \\cap B_{i_n} = \\emptyset$", "as well as a finite number of indices $j_1, \\ldots, j_m \\in I$ such that", "$B'' \\cap B_{j_1} \\cap \\ldots \\cap B_{j_m} = \\emptyset$", "then we find that", "$Z \\cap B_{i_1} \\cap \\ldots \\cap B_{i_n} \\cap B_{j_1} \\cap \\ldots \\cap B_{j_m}", "= \\emptyset$.", "However, the set", "$B_{i_1} \\cap \\ldots \\cap B_{i_n} \\cap B_{j_1} \\cap \\ldots \\cap B_{j_m}$", "is quasi-compact hence we would find a finite number of", "indices $i'_1, \\ldots, i'_l \\in I'$ with", "$B_{i_1} \\cap \\ldots \\cap B_{i_n} \\cap B_{j_1} \\cap \\ldots \\cap", "B_{j_m} \\cap B_{i'_1} \\cap \\ldots \\cap B_{i'_l} = \\emptyset$, a contradiction.", "Thus we see that we may add either $B'$ or $B''$ to the given family", "contradicting maximality. We conclude that $Z$ is irreducible. However,", "this leads to a contradiction as well, as now every nonempty (by the", "same argument as above) open $Z \\cap B_i$ for $i \\in I \\setminus I'$", "contains the unique generic point of $Z$. This contradiction proves the lemma." ], "refs": [ "topology-lemma-subbase-theorem", "topology-lemma-intersection-closed-in-quasi-compact" ], "ref_ids": [ 8241, 8232 ] } ], "ref_ids": [] }, { "id": 8304, "type": "theorem", "label": "topology-lemma-fibres-spectral-map-quasi-compact", "categories": [ "topology" ], "title": "topology-lemma-fibres-spectral-map-quasi-compact", "contents": [ "Let $f : X \\to Y$ be a spectral map of spectral spaces. Then", "\\begin{enumerate}", "\\item $f$ is continuous in the constructible topology,", "\\item the fibres of $f$ are quasi-compact, and", "\\item the image is closed in the constructible topology.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X'$ and $Y'$ denote $X$ and $Y$ endowed with the constructible topology", "which are quasi-compact Hausdorff spaces by", "Lemma \\ref{lemma-constructible-hausdorff-quasi-compact}.", "Part (1) says $X' \\to Y'$ is continuous and follows immediately from the", "definitions. Part (3) follows as $f(X')$ is a quasi-compact subset of the", "Hausdorff space $Y'$, see Lemma \\ref{lemma-quasi-compact-in-Hausdorff}.", "We have a commutative diagram", "$$", "\\xymatrix{", "X' \\ar[r] \\ar[d] & X \\ar[d] \\\\", "Y' \\ar[r] & Y", "}", "$$", "of continuous maps of topological spaces. Since $Y'$ is Hausdorff", "we see that the fibres $X'_y$ are closed in $X'$. As $X'$ is quasi-compact", "we see that $X'_y$ is quasi-compact", "(Lemma \\ref{lemma-closed-in-quasi-compact}).", "As $X'_y \\to X_y$ is a surjective continuous map we conclude", "that $X_y$ is quasi-compact (Lemma \\ref{lemma-image-quasi-compact})." ], "refs": [ "topology-lemma-constructible-hausdorff-quasi-compact", "topology-lemma-quasi-compact-in-Hausdorff", "topology-lemma-closed-in-quasi-compact", "topology-lemma-image-quasi-compact" ], "ref_ids": [ 8303, 8230, 8229, 8233 ] } ], "ref_ids": [] }, { "id": 8305, "type": "theorem", "label": "topology-lemma-spectral-if-continuous-wrt-constructible-top", "categories": [ "topology" ], "title": "topology-lemma-spectral-if-continuous-wrt-constructible-top", "contents": [ "Let $X$ and $Y$ be spectral spaces. Let $f : X \\to Y$ be a continuous map.", "Then $f$ is spectral if and only if $f$ is continuous in the constructible", "topology." ], "refs": [], "proofs": [ { "contents": [ "The only if part of this is", "Lemma \\ref{lemma-fibres-spectral-map-quasi-compact}.", "Assume $f$ is continuous in the constructible topology.", "Let $V \\subset Y$ be quasi-compact open.", "Then $V$ is open and closed in the constructible topology.", "Hence $f^{-1}(V)$ is open and closed in the constructible topology.", "Hence $f^{-1}(V)$ is quasi-compact in the constructible topology", "as $X$ is quasi-compact in the constructible topology by", "Lemma \\ref{lemma-constructible-hausdorff-quasi-compact}.", "Since the identity $f^{-1}(V) \\to f^{-1}(V)$ is surjective", "and continuous from the constructible topology to the usual", "topology, we conclude that $f^{-1}(V)$ is quasi-compact", "in the topology of $X$ by Lemma \\ref{lemma-image-quasi-compact}.", "This finishes the proof." ], "refs": [ "topology-lemma-fibres-spectral-map-quasi-compact", "topology-lemma-constructible-hausdorff-quasi-compact", "topology-lemma-image-quasi-compact" ], "ref_ids": [ 8304, 8303, 8233 ] } ], "ref_ids": [] }, { "id": 8306, "type": "theorem", "label": "topology-lemma-spectral-sub", "categories": [ "topology" ], "title": "topology-lemma-spectral-sub", "contents": [ "Let $X$ be a spectral space. Let $E \\subset X$ be closed in the constructible", "topology (for example constructible or closed). Then $E$ with the induced", "topology is a spectral space." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset E$ be a closed irreducible subset. Let $\\eta$ be the generic", "point of the closure $\\overline{Z}$ of $Z$ in $X$. To prove that $E$", "is sober, we show that $\\eta \\in E$. If not, then since $E$ is closed", "in the constructible topology, there exists a constructible subset", "$F \\subset X$ such that $\\eta \\in F$ and $F \\cap E = \\emptyset$.", "By Lemma \\ref{lemma-generic-point-in-constructible} this implies", "$F \\cap \\overline{Z}$ contains a nonempty open subset of $\\overline{Z}$.", "But this is impossible as $\\overline{Z}$ is the closure of $Z$ and", "$Z \\cap F = \\emptyset$.", "\\medskip\\noindent", "Since $E$ is closed in the constructible topology, it is quasi-compact", "in the constructible topology", "(Lemmas \\ref{lemma-closed-in-quasi-compact} and", "\\ref{lemma-constructible-hausdorff-quasi-compact}). Hence a fortiori it is", "quasi-compact in the topology coming from $X$. If $U \\subset X$", "is a quasi-compact open, then $E \\cap U$ is closed in the constructible", "topology, hence quasi-compact (as seen above). It follows that the", "quasi-compact open subsets of $E$ are the intersections $E \\cap U$", "with $U$ quasi-compact open in $X$. These form a basis for the topology.", "Finally, given two $U, U' \\subset X$ quasi-compact opens, the intersection", "$(E \\cap U) \\cap (E \\cap U') = E \\cap (U \\cap U')$ and $U \\cap U'$", "is quasi-compact as $X$ is spectral. This finishes the proof." ], "refs": [ "topology-lemma-generic-point-in-constructible", "topology-lemma-closed-in-quasi-compact", "topology-lemma-constructible-hausdorff-quasi-compact" ], "ref_ids": [ 8266, 8229, 8303 ] } ], "ref_ids": [] }, { "id": 8307, "type": "theorem", "label": "topology-lemma-constructible-stable-specialization-closed", "categories": [ "topology" ], "title": "topology-lemma-constructible-stable-specialization-closed", "contents": [ "Let $X$ be a spectral space. Let $E \\subset X$ be a subset closed", "in the constructible topology (for example constructible).", "\\begin{enumerate}", "\\item If $x \\in \\overline{E}$, then $x$ is the specialization of a point of", "$E$.", "\\item If $E$ is stable under specialization, then $E$ is closed.", "\\item If $E' \\subset X$ is open in the constructible topology", "(for example constructible) and stable under generalization, then $E'$ is open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $x \\in \\overline{E}$. Let $\\{U_i\\}$ be the set of", "quasi-compact open neighbourhoods of $x$. A finite intersection of the", "$U_i$ is another one. The intersection $U_i \\cap E$ is nonempty for all", "$i$. Since the subsets $U_i \\cap E$ are closed in the constructible topology", "we see that $\\bigcap (U_i \\cap E)$ is nonempty by", "Lemma \\ref{lemma-constructible-hausdorff-quasi-compact} and", "Lemma \\ref{lemma-intersection-closed-in-quasi-compact}.", "Since $X$ is a sober space and $\\{U_i\\}$ is a", "fundamental system of open neighbourhoods of $x$, we see that", "$\\bigcap U_i$ is the set of generalizations of $x$. Thus", "$x$ is a specialization of a point of $E$.", "\\medskip\\noindent", "Part (2) is immediate from (1).", "\\medskip\\noindent", "Proof of (3). Assume $E'$ is as in (3). The complement of $E'$ is closed", "in the constructible topology (Lemma \\ref{lemma-constructible})", "and closed under specialization", "(Lemma \\ref{lemma-open-closed-specialization}).", "Hence the complement is closed by (2), i.e., $E'$ is open." ], "refs": [ "topology-lemma-constructible-hausdorff-quasi-compact", "topology-lemma-intersection-closed-in-quasi-compact", "topology-lemma-constructible", "topology-lemma-open-closed-specialization" ], "ref_ids": [ 8303, 8232, 8253, 8283 ] } ], "ref_ids": [] }, { "id": 8308, "type": "theorem", "label": "topology-lemma-two-points", "categories": [ "topology" ], "title": "topology-lemma-two-points", "contents": [ "Let $X$ be a spectral space. Let $x, y \\in X$. Then either there exists", "a third point specializing to both $x$ and $y$, or there exist disjoint", "open neighbourhoods containing $x$ and $y$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{U_i\\}$ be the set of quasi-compact open neighbourhoods of $x$.", "A finite intersection of the $U_i$ is another one.", "Let $\\{V_j\\}$ be the set of quasi-compact open neighbourhoods of $y$.", "A finite intersection of the $V_j$ is another one.", "If $U_i \\cap V_j$ is empty for some $i, j$ we are done.", "If not, then the intersection $U_i \\cap V_j$ is nonempty", "for all $i$ and $j$. The sets $U_i \\cap V_j$ are closed in the constructible", "topology on $X$. By", "Lemma \\ref{lemma-constructible-hausdorff-quasi-compact}", "we see that $\\bigcap (U_i \\cap V_j)$ is nonempty", "(Lemma \\ref{lemma-intersection-closed-in-quasi-compact}).", "Since $X$ is a sober space and $\\{U_i\\}$ is a", "fundamental system of open neighbourhoods of $x$, we see that", "$\\bigcap U_i$ is the set of generalizations of $x$.", "Similarly, $\\bigcap V_j$ is the set of generalizations of $y$.", "Thus any element of $\\bigcap (U_i \\cap V_j)$ specializes to both", "$x$ and $y$." ], "refs": [ "topology-lemma-constructible-hausdorff-quasi-compact", "topology-lemma-intersection-closed-in-quasi-compact" ], "ref_ids": [ 8303, 8232 ] } ], "ref_ids": [] }, { "id": 8309, "type": "theorem", "label": "topology-lemma-characterize-profinite-spectral", "categories": [ "topology" ], "title": "topology-lemma-characterize-profinite-spectral", "contents": [ "Let $X$ be a spectral space. The following are equivalent:", "\\begin{enumerate}", "\\item $X$ is profinite,", "\\item $X$ is Hausdorff,", "\\item $X$ is totally disconnected,", "\\item every quasi-compact open is closed,", "\\item there are no nontrivial specializations between points,", "\\item every point of $X$ is closed,", "\\item every point of $X$ is the generic point of an irreducible component", "of $X$,", "\\item the constructible topology equals the given topology on $X$, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Lemma \\ref{lemma-profinite} shows the implication (1) $\\Rightarrow$ (3).", "Irreducible components are closed, so if $X$ is totally disconnected, then", "every point is closed. So (3) implies (6). The equivalence of (6) and (5)", "is immediate, and (6) $\\Leftrightarrow$ (7) holds because $X$ is sober.", "Assume (5). Then all constructible subsets of $X$ are closed", "(Lemma \\ref{lemma-constructible-stable-specialization-closed}), in", "particular all quasi-compact opens are closed. So (5) implies (4).", "Since $X$ is sober, for any two points there is a quasi-compact open", "containing exactly one of them, hence (4) implies (2).", "Parts (4) and (8) are equivalent by the definition of the", "constructible topology.", "It remains to prove (2) implies (1). Suppose $X$ is Hausdorff. Every", "quasi-compact open is also closed", "(Lemma \\ref{lemma-quasi-compact-in-Hausdorff}). This implies $X$", "is totally disconnected. Hence it is profinite, by", "Lemma \\ref{lemma-profinite}." ], "refs": [ "topology-lemma-profinite", "topology-lemma-constructible-stable-specialization-closed", "topology-lemma-quasi-compact-in-Hausdorff", "topology-lemma-profinite" ], "ref_ids": [ 8299, 8307, 8230, 8299 ] } ], "ref_ids": [] }, { "id": 8310, "type": "theorem", "label": "topology-lemma-spectral-pi0", "categories": [ "topology" ], "title": "topology-lemma-spectral-pi0", "contents": [ "If $X$ is a spectral space, then $\\pi_0(X)$ is a profinite space." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-connected-component-intersection} and", "\\ref{lemma-pi0-profinite}." ], "refs": [ "topology-lemma-connected-component-intersection", "topology-lemma-pi0-profinite" ], "ref_ids": [ 8236, 8302 ] } ], "ref_ids": [] }, { "id": 8311, "type": "theorem", "label": "topology-lemma-product-spectral-spaces", "categories": [ "topology" ], "title": "topology-lemma-product-spectral-spaces", "contents": [ "The product of two spectral spaces is spectral." ], "refs": [], "proofs": [ { "contents": [ "Let $X$, $Y$ be spectral spaces. Denote $p : X \\times Y \\to X$ and", "$q : X \\times Y \\to Y$ the projections. Let $Z \\subset X \\times Y$ be a", "closed irreducible subset. Then $p(Z) \\subset X$ is irreducible", "and $q(Z) \\subset Y$ is irreducible. Let $x \\in X$ be the generic", "point of the closure of $p(X)$ and let $y \\in Y$ be the generic", "point of the closure of $q(Y)$. If $(x, y) \\not \\in Z$, then", "there exist opens $x \\in U \\subset X$, $y \\in V \\subset Y$ such", "that $Z \\cap U \\times V = \\emptyset$. Hence $Z$ is contained", "in $(X \\setminus U) \\times Y \\cup X \\times (Y \\setminus V)$.", "Since $Z$ is irreducible, we see that either", "$Z \\subset (X \\setminus U) \\times Y$ or $Z \\subset X \\times (Y \\setminus V)$.", "In the first case $p(Z) \\subset (X \\setminus U)$ and in the", "second case $q(Z) \\subset (Y \\setminus V)$. Both cases are absurd", "as $x$ is in the closure of $p(Z)$ and $y$ is in the closure of", "$q(Z)$. Thus we conclude that $(x, y) \\in Z$, which means that", "$(x, y)$ is the generic point for $Z$.", "\\medskip\\noindent", "A basis of the topology of $X \\times Y$ are the opens of the form", "$U \\times V$ with $U \\subset X$ and $V \\subset Y$ quasi-compact open", "(here we use that $X$ and $Y$ are spectral). Then $U \\times V$ is", "quasi-compact as the product of quasi-compact spaces is quasi-compact.", "Moreover, any quasi-compact open of $X \\times Y$ is a finite union", "of such quasi-compact rectangles $U \\times V$. It follows that", "the intersection of two such is again quasi-compact", "(since $X$ and $Y$ are spectral). This concludes the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8312, "type": "theorem", "label": "topology-lemma-spectral-bijective", "categories": [ "topology" ], "title": "topology-lemma-spectral-bijective", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces. If", "\\begin{enumerate}", "\\item $X$ and $Y$ are spectral,", "\\item $f$ is spectral and bijective, and", "\\item generalizations (resp.\\ specializations) lift along $f$.", "\\end{enumerate}", "Then $f$ is a homeomorphism." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is spectral it defines a continuous map between $X$ and $Y$ in", "the constructible topology. By", "Lemmas \\ref{lemma-constructible-hausdorff-quasi-compact} and", "\\ref{lemma-bijective-map}", "it follows that $X \\to Y$ is a homeomorphism in the constructible", "topology. Let $U \\subset X$ be quasi-compact open.", "Then $f(U)$ is constructible in $Y$. Let $y \\in Y$ specialize", "to a point in $f(U)$. By the last assumption we see that $f^{-1}(y)$", "specializes to a point of $U$. Hence $f^{-1}(y) \\in U$. Thus $y \\in f(U)$.", "It follows that $f(U)$ is open, see", "Lemma \\ref{lemma-constructible-stable-specialization-closed}.", "Whence $f$ is a homeomorphism.", "To prove the lemma in case specializations lift along $f$", "one shows instead that $f(Z)$ is closed if $X \\setminus Z$ is a", "quasi-compact open of $X$." ], "refs": [ "topology-lemma-constructible-hausdorff-quasi-compact", "topology-lemma-bijective-map", "topology-lemma-constructible-stable-specialization-closed" ], "ref_ids": [ 8303, 8275, 8307 ] } ], "ref_ids": [] }, { "id": 8313, "type": "theorem", "label": "topology-lemma-directed-inverse-limit-finite-sober-spectral-spaces", "categories": [ "topology" ], "title": "topology-lemma-directed-inverse-limit-finite-sober-spectral-spaces", "contents": [ "The inverse limit of a directed inverse system of finite sober", "topological spaces is a spectral topological space." ], "refs": [], "proofs": [ { "contents": [ "Let $I$ be a directed set. Let $X_i$ be an inverse", "system of finite sober spaces over $I$. Let $X = \\lim X_i$ which exists", "by Lemma \\ref{lemma-limits}. As a set $X = \\lim X_i$. Denote", "$p_i : X \\to X_i$ the projection.", "Because $I$ is directed we may apply Lemma \\ref{lemma-describe-limits}.", "A basis for the topology is given by the opens", "$p_i^{-1}(U_i)$ for $U_i \\subset X_i$ open. Since an open covering of", "$p_i^{-1}(U_i)$ is in particular an open covering in the profinite", "topology, we conclude that $p_i^{-1}(U_i)$ is quasi-compact.", "Given $U_i \\subset X_i$ and $U_j \\subset X_j$, then", "$p_i^{-1}(U_i) \\cap p_j^{-1}(U_j) = p_k^{-1}(U_k)$", "for some $k \\geq i, j$ and open $U_k \\subset X_k$. Finally, if $Z \\subset X$", "is irreducible and closed, then $p_i(Z) \\subset X_i$ is irreducible", "and therefore has a unique generic point $\\xi_i$ (because $X_i$", "is a finite sober topological space). Then $\\xi = \\lim \\xi_i$ is a", "generic point of $Z$ (it is a point of $Z$ as $Z$ is closed).", "This finishes the proof." ], "refs": [ "topology-lemma-limits", "topology-lemma-describe-limits" ], "ref_ids": [ 8248, 8249 ] } ], "ref_ids": [] }, { "id": 8314, "type": "theorem", "label": "topology-lemma-spectral-closed-in-product-two-point-space", "categories": [ "topology" ], "title": "topology-lemma-spectral-closed-in-product-two-point-space", "contents": [ "Let $W$ be the topological space with two points, one closed,", "the other not. A topological space is spectral if and only if", "it is homeomorphic to a subspace of a product of", "copies of $W$ which is closed in the constructible topology." ], "refs": [], "proofs": [ { "contents": [ "Write $W = \\{0, 1\\}$ where $0$ is a specialization of $1$ but not vice versa.", "Let $I$ be a set. The space $\\prod_{i \\in I} W$ is spectral by", "Lemma \\ref{lemma-directed-inverse-limit-finite-sober-spectral-spaces}.", "Thus we see that a subspace of $\\prod_{i \\in I} W$ closed in the", "constructible topology is", "a spectral space by Lemma \\ref{lemma-spectral-sub}.", "\\medskip\\noindent", "For the converse, let $X$ be a spectral space. Let $U \\subset X$ be a", "quasi-compact open. Consider the continuous map", "$$", "f_U : X \\longrightarrow W", "$$", "which maps every point in $U$ to $1$ and every point in $X \\setminus U$ to $0$.", "Taking the product of these maps we obtain a continuous map", "$$", "f = \\prod f_U : X \\longrightarrow \\prod\\nolimits_U W", "$$", "By construction the map $f : X \\to Y$ is spectral. By", "Lemma \\ref{lemma-fibres-spectral-map-quasi-compact}", "the image of $f$ is closed in the constructible topology.", "If $x', x \\in X$ are distinct, then since $X$ is sober either $x'$", "is not a specialization of $x$ or conversely. In either case (as the", "quasi-compact opens form a basis for the topology of $X$) there", "exists a quasi-compact open $U \\subset X$ such that $f_U(x') \\not = f_U(x)$.", "Thus $f$ is injective. Let $Y = f(X)$ endowed with the induced topology.", "Let $y' \\leadsto y$ be a specialization in $Y$ and say", "$f(x') = y'$ and $f(x) = y$. Arguing as above we see that", "$x' \\leadsto x$, since otherwise there is a $U$ such that", "$x \\in U$ and $x' \\not \\in U$, which would imply", "$f_U(x') \\not \\leadsto f_U(x)$.", "We conclude that $f : X \\to Y$ is a homeomorphism by", "Lemma \\ref{lemma-spectral-bijective}." ], "refs": [ "topology-lemma-directed-inverse-limit-finite-sober-spectral-spaces", "topology-lemma-spectral-sub", "topology-lemma-fibres-spectral-map-quasi-compact", "topology-lemma-spectral-bijective" ], "ref_ids": [ 8313, 8306, 8304, 8312 ] } ], "ref_ids": [] }, { "id": 8315, "type": "theorem", "label": "topology-lemma-spectral-inverse-limit-finite-sober-spaces", "categories": [ "topology" ], "title": "topology-lemma-spectral-inverse-limit-finite-sober-spaces", "contents": [ "A topological space is spectral if and only if it is a directed", "inverse limit of finite sober topological spaces." ], "refs": [], "proofs": [ { "contents": [ "One direction is given by", "Lemma \\ref{lemma-directed-inverse-limit-finite-sober-spectral-spaces}.", "For the converse, assume $X$ is spectral. Then we", "may assume $X \\subset \\prod_{i \\in I} W$ is a subset closed", "in the constructible topology where $W = \\{0, 1\\}$ as in", "Lemma \\ref{lemma-spectral-closed-in-product-two-point-space}.", "We can write", "$$", "\\prod\\nolimits_{i \\in I} W =", "\\lim_{J \\subset I\\text{ finite }} \\prod\\nolimits_{j \\in J} W", "$$", "as a cofiltered limit.", "For each $J$, let $X_J \\subset \\prod_{j \\in J} W$ be the image of $X$.", "Then we see that $X = \\lim X_J$ as sets because $X$ is closed in", "the product with the constructible topology (detail omitted).", "A formal argument (omitted) on limits shows that $X = \\lim X_J$ as topological", "spaces." ], "refs": [ "topology-lemma-directed-inverse-limit-finite-sober-spectral-spaces", "topology-lemma-spectral-closed-in-product-two-point-space" ], "ref_ids": [ 8313, 8314 ] } ], "ref_ids": [] }, { "id": 8316, "type": "theorem", "label": "topology-lemma-Noetherian-goes-to-spectral", "categories": [ "topology" ], "title": "topology-lemma-Noetherian-goes-to-spectral", "contents": [ "Let $X$ be a topological space and let $c : X \\to X'$ be the universal", "map from $X$ to a sober topological space, see", "Lemma \\ref{lemma-make-sober}.", "\\begin{enumerate}", "\\item If $X$ is quasi-compact, so is $X'$.", "\\item If $X$ is quasi-compact, has a basis of quasi-compact opens,", "and the intersection of two quasi-compact opens is quasi-compact, then ", "$X'$ is spectral.", "\\item If $X$ is Noetherian, then $X'$ is a Noetherian spectral space.", "\\end{enumerate}" ], "refs": [ "topology-lemma-make-sober" ], "proofs": [ { "contents": [ "Let $U \\subset X$ be open and let $U' \\subset X'$ be the corresponding", "open, i.e., the open such that $c^{-1}(U') = U$.", "Then $U$ is quasi-compact if and only if $U'$ is quasi-compact,", "as pulling back by $c$ is a bijection between the opens of", "$X$ and $X'$ which commutes with unions. This in particular proves (1).", "\\medskip\\noindent", "Proof of (2). It follows from the above that $X'$ has a basis of", "quasi-compact opens. Since $c^{-1}$ also", "commutes with intersections of pairs of opens, we see that", "the intersection of two quasi-compact opens $X'$ is quasi-compact.", "Finally, $X'$ is quasi-compact by (1) and sober by construction.", "Hence $X'$ is spectral.", "\\medskip\\noindent", "Proof of (3). It is immediate that $X'$ is Noetherian as this is defined", "in terms of the acc for open subsets which holds for $X$. We have", "already seen in (2) that $X'$ is spectral." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8219 ] }, { "id": 8317, "type": "theorem", "label": "topology-lemma-inverse-limit-spectral-spaces-quasi-compact", "categories": [ "topology" ], "title": "topology-lemma-inverse-limit-spectral-spaces-quasi-compact", "contents": [ "Let $\\mathcal{I}$ be a category. Let $i \\mapsto X_i$ be a diagram", "of spectral spaces such that for $a : j \\to i$ in $\\mathcal{I}$", "the corresponding map $f_a : X_j \\to X_i$ is spectral.", "\\begin{enumerate}", "\\item Given subsets $Z_i \\subset X_i$ closed in the constructible", "topology with $f_a(Z_j) \\subset Z_i$ for all $a : j \\to i$ in $\\mathcal{I}$,", "then $\\lim Z_i$ is quasi-compact.", "\\item The space $X = \\lim X_i$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The limit $Z = \\lim Z_i$ exists by Lemma \\ref{lemma-limits}.", "Denote $X'_i$ the space $X_i$ endowed with the constructible topology", "and $Z'_i$ the corresponding subspace of $X'_i$.", "Let $a : j \\to i$ in $\\mathcal{I}$ be a morphism. As $f_a$ is spectral", "it defines a continuous map $f_a : X'_j \\to X'_i$. Thus", "$f_a|_{Z_j} : Z'_j \\to Z'_i$ is a continuous map of quasi-compact", "Hausdorff spaces (by", "Lemmas \\ref{lemma-constructible-hausdorff-quasi-compact} and", "\\ref{lemma-closed-in-quasi-compact}).", "Thus $Z' = \\lim Z_i$ is quasi-compact by", "Lemma \\ref{lemma-inverse-limit-quasi-compact}.", "The maps $Z'_i \\to Z_i$ are continuous, hence $Z' \\to Z$", "is continuous and a bijection on underlying sets. Hence $Z$", "is quasi-compact as the image of the surjective continuous map $Z' \\to Z$", "(Lemma \\ref{lemma-image-quasi-compact})." ], "refs": [ "topology-lemma-limits", "topology-lemma-constructible-hausdorff-quasi-compact", "topology-lemma-closed-in-quasi-compact", "topology-lemma-inverse-limit-quasi-compact", "topology-lemma-image-quasi-compact" ], "ref_ids": [ 8248, 8303, 8229, 8251, 8233 ] } ], "ref_ids": [] }, { "id": 8318, "type": "theorem", "label": "topology-lemma-inverse-limit-spectral-spaces-nonempty", "categories": [ "topology" ], "title": "topology-lemma-inverse-limit-spectral-spaces-nonempty", "contents": [ "Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram", "of spectral spaces such that for $a : j \\to i$ in $\\mathcal{I}$", "the corresponding map $f_a : X_j \\to X_i$ is spectral.", "\\begin{enumerate}", "\\item Given nonempty subsets $Z_i \\subset X_i$ closed in the constructible", "topology with $f_a(Z_j) \\subset Z_i$ for all $a : j \\to i$ in $\\mathcal{I}$,", "then $\\lim Z_i$ is nonempty.", "\\item If each $X_i$ is nonempty, then $X = \\lim X_i$ is nonempty.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $X'_i$ the space $X_i$ endowed with the constructible topology", "and $Z'_i$ the corresponding subspace of $X'_i$.", "Let $a : j \\to i$ in $\\mathcal{I}$ be a morphism. As $f_a$ is spectral", "it defines a continuous map $f_a : X'_j \\to X'_i$. Thus", "$f_a|_{Z_j} : Z'_j \\to Z'_i$ is a continuous map of quasi-compact", "Hausdorff spaces (by", "Lemmas \\ref{lemma-constructible-hausdorff-quasi-compact} and", "\\ref{lemma-closed-in-quasi-compact}). By", "Lemma \\ref{lemma-nonempty-limit} the space $\\lim Z'_i$ is nonempty.", "Since $\\lim Z'_i = \\lim Z_i$ as sets we conclude." ], "refs": [ "topology-lemma-constructible-hausdorff-quasi-compact", "topology-lemma-closed-in-quasi-compact", "topology-lemma-nonempty-limit" ], "ref_ids": [ 8303, 8229, 8252 ] } ], "ref_ids": [] }, { "id": 8319, "type": "theorem", "label": "topology-lemma-inverse-limit-spectral-spaces-equal", "categories": [ "topology" ], "title": "topology-lemma-inverse-limit-spectral-spaces-equal", "contents": [ "Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram", "of spectral spaces such that for $a : j \\to i$ in $\\mathcal{I}$", "the corresponding map $f_a : X_j \\to X_i$ is spectral. Let $X = \\lim X_i$", "with projections $p_i : X \\to X_i$. Let $i \\in \\Ob(\\mathcal{I})$ and let", "$E, F \\subset X_i$ be subsets with $E$ closed in the constructible topology", "and $F$ open in the constructible topology.", "Then $p_i^{-1}(E) \\subset p_i^{-1}(F)$ if and only if there is a morphism", "$a : j \\to i$ in $\\mathcal{I}$ such that $f_a^{-1}(E) \\subset f_a^{-1}(F)$." ], "refs": [], "proofs": [ { "contents": [ "Observe that", "$$", "p_i^{-1}(E) \\setminus p_i^{-1}(F) =", "\\lim_{a : j \\to i} f_a^{-1}(E) \\setminus f_a^{-1}(F)", "$$", "Since $f_a$ is a spectral map, it is continuous in the constructible topology", "hence the set $f_a^{-1}(E) \\setminus f_a^{-1}(F)$ is closed in the", "constructible topology. Hence", "Lemma \\ref{lemma-inverse-limit-spectral-spaces-nonempty}", "applies to show that the LHS is nonempty if and only if each of the spaces", "of the RHS is nonempty." ], "refs": [ "topology-lemma-inverse-limit-spectral-spaces-nonempty" ], "ref_ids": [ 8318 ] } ], "ref_ids": [] }, { "id": 8320, "type": "theorem", "label": "topology-lemma-inverse-limit-spectral-spaces-constructible", "categories": [ "topology" ], "title": "topology-lemma-inverse-limit-spectral-spaces-constructible", "contents": [ "Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram", "of spectral spaces such that for $a : j \\to i$ in $\\mathcal{I}$", "the corresponding map $f_a : X_j \\to X_i$ is spectral. Let $X = \\lim X_i$", "with projections $p_i : X \\to X_i$. Let $E \\subset X$ be a constructible", "subset. Then there exists an $i \\in \\Ob(\\mathcal{I})$ and a constructible", "subset $E_i \\subset X_i$ such that $p_i^{-1}(E_i) = E$. If $E$ is open,", "resp.\\ closed, we may choose $E_i$ open, resp.\\ closed." ], "refs": [], "proofs": [ { "contents": [ "Assume $E$ is a quasi-compact open of $X$. By", "Lemma \\ref{lemma-describe-limits} we can write $E = p_i^{-1}(U_i)$", "for some $i$ and some open $U_i \\subset X_i$.", "Write $U_i = \\bigcup U_{i, \\alpha}$ as a union of quasi-compact opens.", "As $E$ is quasi-compact we can find $\\alpha_1, \\ldots, \\alpha_n$", "such that $E = p_i^{-1}(U_{i, \\alpha_1} \\cup \\ldots \\cup U_{i, \\alpha_n})$.", "Hence $E_i = U_{i, \\alpha_1} \\cup \\ldots \\cup U_{i, \\alpha_n}$ works.", "\\medskip\\noindent", "Assume $E$ is a constructible closed subset. Then $E^c$ is quasi-compact", "open. So $E^c = p_i^{-1}(F_i)$ for some $i$ and", "quasi-compact open $F_i \\subset X_i$ by the result of the previous paragraph.", "Then $E = p_i^{-1}(F_i^c)$ as desired.", "\\medskip\\noindent", "If $E$ is general we can write $E = \\bigcup_{l = 1, \\ldots, n} U_l \\cap Z_l$", "with $U_l$ constructible open and $Z_l$ constructible closed. By the result", "of the previous paragraphs we may write $U_l = p_{i_l}^{-1}(U_{l, i_l})$", "and $Z_l = p_{j_l}^{-1}(Z_{l, j_l})$", "with $U_{l, i_l} \\subset X_{i_l}$ constructible open and", "$Z_{l, j_l} \\subset X_{j_l}$ constructible closed. As $\\mathcal{I}$ is", "cofiltered we may choose an object $k$ of $\\mathcal{I}$ and morphism", "$a_l : k \\to i_l$ and $b_l : k \\to j_l$. Then taking", "$E_k = \\bigcup_{l = 1, \\ldots, n}", "f_{a_l}^{-1}(U_{l, i_l}) \\cap f_{b_l}^{-1}(Z_{l, j_l})$", "we obtain a constructible subset of $X_k$ whose inverse image in $X$ is $E$." ], "refs": [ "topology-lemma-describe-limits" ], "ref_ids": [ 8249 ] } ], "ref_ids": [] }, { "id": 8321, "type": "theorem", "label": "topology-lemma-directed-inverse-limit-spectral-spaces", "categories": [ "topology" ], "title": "topology-lemma-directed-inverse-limit-spectral-spaces", "contents": [ "Let $\\mathcal{I}$ be a cofiltered index category.", "Let $i \\mapsto X_i$ be a diagram of spectral spaces such", "that for $a : j \\to i$ in $\\mathcal{I}$ the corresponding map", "$f_a : X_j \\to X_i$ is spectral. Then", "the inverse limit $X = \\lim X_i$ is a spectral topological space", "and the projection maps $p_i : X \\to X_i$ are spectral." ], "refs": [], "proofs": [ { "contents": [ "The limit $X = \\lim X_i$ exists (Lemma \\ref{lemma-limits})", "and is quasi-compact by", "Lemma \\ref{lemma-inverse-limit-spectral-spaces-quasi-compact}.", "\\medskip\\noindent", "Denote $p_i : X \\to X_i$ the projection.", "Because $\\mathcal{I}$ is cofiltered we can apply", "Lemma \\ref{lemma-describe-limits}.", "Hence a basis for the topology on $X$ is given by the opens", "$p_i^{-1}(U_i)$ for $U_i \\subset X_i$ open. Since a basis for", "the topology of $X_i$ is given by the quasi-compact open, we conclude", "that a basis for the topology on $X$ is given by $p_i^{-1}(U_i)$", "with $U_i \\subset X_i$ quasi-compact open. A formal argument", "shows that", "$$", "p_i^{-1}(U_i) = \\lim_{a : j \\to i} f_a^{-1}(U_i)", "$$", "as topological spaces. Since each $f_a$ is spectral the sets", "$f_a^{-1}(U_i)$ are closed in the constructible topology of $X_j$", "and hence $p_i^{-1}(U_i)$ is quasi-compact", "by Lemma \\ref{lemma-inverse-limit-spectral-spaces-quasi-compact}.", "Thus $X$ has a basis for the topology consisting of quasi-compact opens.", "\\medskip\\noindent", "Any quasi-compact open $U$ of $X$ is of the form $U = p_i^{-1}(U_i)$", "for some $i$ and some quasi-compact open $U_i \\subset X_i$", "(see Lemma \\ref{lemma-inverse-limit-spectral-spaces-constructible}).", "Given $U_i \\subset X_i$ and $U_j \\subset X_j$ quasi-compact open, then", "$p_i^{-1}(U_i) \\cap p_j^{-1}(U_j) = p_k^{-1}(U_k)$ for some $k$", "and quasi-compact open $U_k \\subset X_k$. Namely, choose $k$", "and morphisms $k \\to i$ and $k \\to j$ and let $U_k$ be the intersection of the", "pullbacks of $U_i$ and $U_j$ to $X_k$. Thus we see that the intersection", "of two quasi-compact opens of $X$ is quasi-compact open.", "\\medskip\\noindent", "Finally, let $Z \\subset X$ be irreducible and closed. Then $p_i(Z) \\subset X_i$", "is irreducible and therefore $Z_i = \\overline{p_i(Z)}$ has a unique generic", "point $\\xi_i$ (because $X_i$ is a spectral space). Then $f_a(\\xi_j) = \\xi_i$", "for $a : j \\to i$ in $\\mathcal{I}$ because $\\overline{f_a(Z_j)} = Z_i$.", "Hence $\\xi = \\lim \\xi_i$ is a point of $X$. Claim: $\\xi \\in Z$. Namely,", "if not we can find a quasi-compact open containing $\\xi$ disjoint", "from $Z$. This would be of the form $p_i^{-1}(U_i)$ for some $i$ and", "quasi-compact open $U_i \\subset X_i$. Then $\\xi_i \\in U_i$ but", "$p_i(Z) \\cap U_i = \\emptyset$ which contradicts $\\xi_i \\in \\overline{p_i(Z)}$.", "So $\\xi \\in Z$ and hence $\\overline{\\{\\xi\\}} \\subset Z$. Conversely,", "every $z \\in Z$ is in the closure of $\\xi$. Namely, given a quasi-compact", "open neighbourhood $U$ of $z$ we write $U = p_i^{-1}(U_i)$ for some $i$", "and quasi-compact open $U_i \\subset X_i$. We see that $p_i(z) \\in U_i$", "hence $\\xi_i \\in U_i$ hence $\\xi \\in U$. Thus $\\xi$ is a generic point", "of $Z$. We omit the proof that $\\xi$ is the unique generic point of $Z$", "(hint: show that a second generic point has to be equal to $\\xi$", "by showing that it has to map to $\\xi_i$ in $X_i$ since", "by spectrality of $X_i$ the irreducible $Z_i$ has a unique generic point).", "This finishes the proof." ], "refs": [ "topology-lemma-limits", "topology-lemma-inverse-limit-spectral-spaces-quasi-compact", "topology-lemma-describe-limits", "topology-lemma-inverse-limit-spectral-spaces-quasi-compact", "topology-lemma-inverse-limit-spectral-spaces-constructible" ], "ref_ids": [ 8248, 8317, 8249, 8317, 8320 ] } ], "ref_ids": [] }, { "id": 8322, "type": "theorem", "label": "topology-lemma-descend-opens", "categories": [ "topology" ], "title": "topology-lemma-descend-opens", "contents": [ "Let $\\mathcal{I}$ be a cofiltered index category.", "Let $i \\mapsto X_i$ be a diagram of spectral spaces such", "that for $a : j \\to i$ in $\\mathcal{I}$ the corresponding map", "$f_a : X_j \\to X_i$ is spectral. Set $X = \\lim X_i$ and denote", "$p_i : X \\to X_i$ the projection.", "\\begin{enumerate}", "\\item Given any quasi-compact open $U \\subset X$", "there exists an $i \\in \\Ob(\\mathcal{I})$ and a quasi-compact open", "$U_i \\subset X_i$ such that $p_i^{-1}(U_i) = U$.", "\\item Given $U_i \\subset X_i$ and $U_j \\subset X_j$", "quasi-compact opens such that $p_i^{-1}(U_i) \\subset p_j^{-1}(U_j)$", "there exist $k \\in \\Ob(\\mathcal{I})$ and morphisms", "$a : k \\to i$ and $b : k \\to j$ such that $f_a^{-1}(U_i) \\subset f_b^{-1}(U_j)$.", "\\item If $U_i, U_{1, i}, \\ldots, U_{n, i} \\subset X_i$ are quasi-compact", "opens and", "$p_i^{-1}(U_i) = p_i^{-1}(U_{1, i}) \\cup \\ldots \\cup p_i^{-1}(U_{n, i})$", "then", "$f_a^{-1}(U_i) = f_a^{-1}(U_{1, i}) \\cup \\ldots \\cup f_a^{-1}(U_{n, i})$", "for some morphism $a : j \\to i$ in $\\mathcal{I}$.", "\\item Same statement as in (3) but for intersections.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is a special case of", "Lemma \\ref{lemma-inverse-limit-spectral-spaces-constructible}.", "Part (2) is a special case of", "Lemma \\ref{lemma-inverse-limit-spectral-spaces-equal}", "as quasi-compact opens are both open and closed in the constructible", "topology. Parts (3) and (4) follow formally from (1) and (2)", "and the fact that taking inverse images of subsets commutes with", "taking unions and intersections." ], "refs": [ "topology-lemma-inverse-limit-spectral-spaces-constructible", "topology-lemma-inverse-limit-spectral-spaces-equal" ], "ref_ids": [ 8320, 8319 ] } ], "ref_ids": [] }, { "id": 8323, "type": "theorem", "label": "topology-lemma-make-spectral-space", "categories": [ "topology" ], "title": "topology-lemma-make-spectral-space", "contents": [ "Let $W$ be a subset of a spectral space $X$. The following are equivalent:", "\\begin{enumerate}", "\\item $W$ is an intersection of constructible sets and", "closed under generalizations,", "\\item $W$ is quasi-compact and closed under generalizations,", "\\item there exists a quasi-compact subset $E \\subset X$ such that", "$W$ is the set of points specializing to $E$,", "\\item $W$ is an intersection of quasi-compact open subsets,", "\\item", "\\label{item-intersection-quasi-compact-open}", "there exists a nonempty set $I$ and quasi-compact opens", "$U_i \\subset X$, $i \\in I$", "such that $W = \\bigcap U_i$ and for all $i, j \\in I$ there exists a", "$k \\in I$ with $U_k \\subset U_i \\cap U_j$.", "\\end{enumerate}", "In this case we have (a) $W$ is a spectral space, (b) $W = \\lim U_i$", "as topological spaces, and (c) for any open $U$ containing $W$", "there exists an $i$ with $U_i \\subset U$." ], "refs": [], "proofs": [ { "contents": [ "Let $W \\subset X$ satisfy (1). Then $W$ is closed in the constructible", "topology, hence quasi-compact in the constructible topology (by", "Lemmas \\ref{lemma-constructible-hausdorff-quasi-compact} and", "\\ref{lemma-closed-in-quasi-compact}), hence quasi-compact in the topology", "of $X$ (because opens in $X$ are open in the constructible topology). Thus", "(2) holds.", "\\medskip\\noindent", "It is clear that (2) implies (3) by taking $E = W$.", "\\medskip\\noindent", "Let $X$ be a spectral space and let $E \\subset W$ be as in (3).", "Since every point of $W$ specializes to a point of $E$ we see that", "an open of $W$ which contains $E$ is equal to $W$. Hence since $E$", "is quasi-compact, so is $W$.", "If $x \\in X$, $x \\not \\in W$, then $Z = \\overline{\\{x\\}}$ is", "disjoint from $W$. Since $W$ is quasi-compact we can find a", "quasi-compact open $U$ with $W \\subset U$ and $U \\cap Z = \\emptyset$.", "We conclude that (4) holds.", "\\medskip\\noindent", "If $W = \\bigcap_{j \\in J} U_j$ then setting $I$ equal to the set of", "finite subsets of $J$ and $U_i = U_{j_1} \\cap \\ldots \\cap U_{j_r}$", "for $i = \\{j_1, \\ldots, j_r\\}$ shows that (4) implies (5). It is immediate", "that (5) implies (1).", "\\medskip\\noindent", "Let $I$ and $U_i$ be as in (5).", "Since $W = \\bigcap U_i$ we have $W = \\lim U_i$ by the universal property", "of limits. Then $W$ is a spectral space by", "Lemma \\ref{lemma-directed-inverse-limit-spectral-spaces}.", "Let $U \\subset X$ be an open neighbourhood of $W$.", "Then $E_i = U_i \\cap (X \\setminus U)$ is a family of constructible", "subsets of the spectral space $Z = X \\setminus U$", "with empty intersection. Using that the spectral topology on $Z$", "is quasi-compact (Lemma \\ref{lemma-constructible-hausdorff-quasi-compact})", "we conclude from", "Lemma \\ref{lemma-intersection-closed-in-quasi-compact}", "that $E_i = \\emptyset$ for some $i$." ], "refs": [ "topology-lemma-constructible-hausdorff-quasi-compact", "topology-lemma-closed-in-quasi-compact", "topology-lemma-directed-inverse-limit-spectral-spaces", "topology-lemma-constructible-hausdorff-quasi-compact", "topology-lemma-intersection-closed-in-quasi-compact" ], "ref_ids": [ 8303, 8229, 8321, 8303, 8232 ] } ], "ref_ids": [] }, { "id": 8324, "type": "theorem", "label": "topology-lemma-make-spectral-space-minus", "categories": [ "topology" ], "title": "topology-lemma-make-spectral-space-minus", "contents": [ "Let $X$ be a spectral space. Let $E \\subset X$ be a constructible subset.", "Let $W \\subset X$ be the set of points of $X$ which specialize", "to a point of $E$. Then $W \\setminus E$ is a spectral space.", "If $W = \\bigcap U_i$ with $U_i$ as in", "Lemma \\ref{lemma-make-spectral-space}", "(\\ref{item-intersection-quasi-compact-open})", "then $W \\setminus E = \\lim (U_i \\setminus E)$." ], "refs": [ "topology-lemma-make-spectral-space" ], "proofs": [ { "contents": [ "Since $E$ is constructible, it is quasi-compact and hence", "Lemma \\ref{lemma-make-spectral-space} applies to $W$.", "If $E$ is constructible, then $E$ is constructible in $U_i$", "for all $i \\in I$. Hence $U_i \\setminus E$", "is spectral by Lemma \\ref{lemma-spectral-sub}.", "Since $W \\setminus E = \\bigcap (U_i \\setminus E)$ we have", "$W \\setminus E = \\lim U_i \\setminus E$ by the universal property of limits.", "Then $W \\setminus E$ is a spectral space by", "Lemma \\ref{lemma-directed-inverse-limit-spectral-spaces}." ], "refs": [ "topology-lemma-make-spectral-space", "topology-lemma-spectral-sub", "topology-lemma-directed-inverse-limit-spectral-spaces" ], "ref_ids": [ 8323, 8306, 8321 ] } ], "ref_ids": [ 8323 ] }, { "id": 8325, "type": "theorem", "label": "topology-lemma-dense-image", "categories": [ "topology" ], "title": "topology-lemma-dense-image", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces. Assume that", "$f(X)$ is dense in $Y$ and that $Y$ is Hausdorff. Then the cardinality", "of $Y$ is at most the cardinality of $P(P(X))$ where $P$ is the power", "set operation." ], "refs": [], "proofs": [ { "contents": [ "Let $S = f(X) \\subset Y$. Let $\\mathcal{D}$ be the set of all", "closed domains of $Y$, i.e., subsets $D \\subset Y$ which", "equal the closure of its interior. Note that the closure of an", "open subset of $Y$ is a closed domain. For $y \\in Y$ consider the set", "$$", "I_y = \\{T \\subset S \\mid \\text{ there exists }", "D \\in \\mathcal{D}\\text{ with }T = S \\cap D\\text{ and }y \\in D\\}.", "$$", "Since $S$ is dense in $Y$ for every closed domain $D$ we see", "that $S \\cap D$ is dense in $D$. Hence, if", "$D \\cap S = D' \\cap S$ for $D, D' \\in \\mathcal{D}$, then", "$D = D'$. Thus $I_y = I_{y'}$ implies that $y = y'$ because the", "Hausdorff condition assures us that we can find a closed domain", "containing $y$ but not $y'$. The result follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8326, "type": "theorem", "label": "topology-lemma-one-point-compactification", "categories": [ "topology" ], "title": "topology-lemma-one-point-compactification", "contents": [ "Let $X$ be a Hausdorff, locally quasi-compact space.", "There exists a map $X \\to X^*$ which identifies $X$ as an open", "subspace of a quasi-compact Hausdorff space $X^*$ such that", "$X^* \\setminus X$ is a singleton (one point compactification).", "In particular, the map $X \\to \\beta(X)$ identifies $X$", "with an open subspace of $\\beta(X)$." ], "refs": [], "proofs": [ { "contents": [ "Set $X^* = X \\amalg \\{\\infty\\}$. We declare a subset $V$ of $X^*$ to be", "open if either $V \\subset X$ is open in $X$, or $\\infty \\in V$ and", "$U = V \\cap X$ is an open of $X$ such that $X \\setminus U$ is quasi-compact.", "We omit the verification that this defines a topology. It is clear", "that $X \\to X^*$ identifies $X$ with an open subspace of $X$.", "\\medskip\\noindent", "Since $X$ is locally quasi-compact, every point $x \\in X$ has a", "quasi-compact neighbourhood $x \\in E \\subset X$. Then $E$", "is closed (Lemma \\ref{lemma-closed-in-quasi-compact}) and", "$V = (X \\setminus E) \\amalg \\{\\infty\\}$ is an open neighbourhood", "of $\\infty$ disjoint from the interior of $E$. Thus $X^*$ is Hausdorff.", "\\medskip\\noindent", "Let $X^* = \\bigcup V_i$ be an open covering. Then for some $i$, say $i_0$,", "we have $\\infty \\in V_{i_0}$. By construction $Z = X^* \\setminus V_{i_0}$", "is quasi-compact. Hence the covering", "$Z \\subset \\bigcup_{i \\not = i_0} Z \\cap V_i$ has a finite refinement which", "implies that the given covering of $X^*$ has a finite refinement.", "Thus $X^*$ is quasi-compact.", "\\medskip\\noindent", "The map $X \\to X^*$ factors as $X \\to \\beta(X) \\to X^*$ by the universal", "property of the Stone-{\\v C}ech compactification. Let", "$\\varphi : \\beta(X) \\to X^*$ be this factorization.", "Then $X \\to \\varphi^{-1}(X)$ is a section to", "$\\varphi^{-1}(X) \\to X$ hence has closed image", "(Lemma \\ref{lemma-section-closed}).", "Since the image of $X \\to \\beta(X)$ is dense we conclude that", "$X = \\varphi^{-1}(X)$." ], "refs": [ "topology-lemma-closed-in-quasi-compact" ], "ref_ids": [ 8229 ] } ], "ref_ids": [] }, { "id": 8327, "type": "theorem", "label": "topology-lemma-image-open-technical", "categories": [ "topology" ], "title": "topology-lemma-image-open-technical", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Assume $f$ is surjective and $f(E) \\not = Y$ for all proper", "closed subsets $E \\subset X$. Then for $U \\subset X$ open the subset", "$f(U)$ is contained in the closure of $Y \\setminus f(X \\setminus U)$." ], "refs": [], "proofs": [ { "contents": [ "Pick $y \\in f(U)$ and let $V \\subset Y$ be any open neighbourhood of $y$.", "We will show that $V$ intersects $Y \\setminus f(X \\setminus U)$.", "Note that $W = U \\cap f^{-1}(V)$ is a nonempty open subset of $X$, hence", "$f(X \\setminus W) \\not = Y$. Take $y' \\in Y$, $y' \\not \\in f(X \\setminus W)$.", "It is elementary to show that $y' \\in V$ and", "$y' \\in Y \\setminus f(X \\setminus U)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8328, "type": "theorem", "label": "topology-lemma-intersection-empty", "categories": [ "topology" ], "title": "topology-lemma-intersection-empty", "contents": [ "Let $X$ be an extremally disconnected space.", "If $U, V \\subset X$ are disjoint open subsets, then", "$\\overline{U}$ and $\\overline{V}$ are disjoint too." ], "refs": [], "proofs": [ { "contents": [ "By assumption $\\overline{U}$ is open, hence $V \\cap \\overline{U}$", "is open and disjoint from $U$, hence empty because $\\overline{U}$", "is the intersection of all the closed subsets of $X$ containing $U$.", "This means the open $\\overline{V} \\cap \\overline{U}$ avoids $V$", "hence is empty by the same argument." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8329, "type": "theorem", "label": "topology-lemma-isomorphism", "categories": [ "topology" ], "title": "topology-lemma-isomorphism", "contents": [ "Let $f : X \\to Y$ be a continuous map of Hausdorff quasi-compact", "topological spaces. If $Y$ is extremally disconnected, $f$ is surjective,", "and $f(Z) \\not = Y$ for every proper closed subset $Z$ of $X$, then", "$f$ is a homeomorphism." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-bijective-map} it suffices to show that $f$ is injective.", "Suppose that $x, x' \\in X$ are distinct points with $y = f(x) = f(x')$.", "Choose disjoint open neighbourhoods $U, U' \\subset X$ of $x, x'$.", "Observe that $f$ is closed (Lemma \\ref{lemma-closed-map}) hence", "$T = f(X \\setminus U)$ and $T' = f(X \\setminus U')$ are closed in $Y$.", "Since $X$ is the union of $X \\setminus U$ and $X \\setminus U'$ we see that", "$Y = T \\cup T'$. By Lemma \\ref{lemma-image-open-technical} we", "see that $y$ is contained in the closure of $Y \\setminus T$ and", "the closure of $Y \\setminus T'$. On the other hand, by", "Lemma \\ref{lemma-intersection-empty},", "this intersection is empty. In this way we obtain the desired contradiction." ], "refs": [ "topology-lemma-bijective-map", "topology-lemma-closed-map", "topology-lemma-image-open-technical", "topology-lemma-intersection-empty" ], "ref_ids": [ 8275, 8274, 8327, 8328 ] } ], "ref_ids": [] }, { "id": 8330, "type": "theorem", "label": "topology-lemma-find-compact-subset", "categories": [ "topology" ], "title": "topology-lemma-find-compact-subset", "contents": [ "Let $f : X \\to Y$ be a continuous surjective map of Hausdorff quasi-compact", "topological spaces. There exists a quasi-compact subset $E \\subset X$", "such that $f(E) = Y$ but $f(E') \\not = Y$ for all proper closed subsets", "$E' \\subset E$." ], "refs": [], "proofs": [ { "contents": [ "We will use without further mention that the quasi-compact subsets", "of $X$ are exactly the closed subsets", "(Lemma \\ref{lemma-closed-in-compact}).", "Consider the collection $\\mathcal{E}$ of all quasi-compact subsets", "$E \\subset X$ with $f(E) = Y$ ordered by inclusion. We will use", "Zorn's lemma to show that", "$\\mathcal{E}$ has a minimal element. To do this it suffices to show", "that given a totally ordered family $E_\\lambda$ of elements of $\\mathcal{E}$", "the intersection $\\bigcap E_\\lambda$ is an element of $\\mathcal{E}$.", "It is quasi-compact as it is closed.", "For every $y \\in Y$ the sets $E_\\lambda \\cap f^{-1}(\\{y\\})$", "are nonempty and closed, hence the intersection", "$\\bigcap E_\\lambda \\cap f^{-1}(\\{y\\}) = \\bigcap (E_\\lambda \\cap f^{-1}(\\{y\\}))$", "is nonempty by", "Lemma \\ref{lemma-intersection-closed-in-quasi-compact}.", "This finishes the proof." ], "refs": [ "topology-lemma-closed-in-compact", "topology-lemma-intersection-closed-in-quasi-compact" ], "ref_ids": [ 8231, 8232 ] } ], "ref_ids": [] }, { "id": 8331, "type": "theorem", "label": "topology-lemma-rainwater", "categories": [ "topology" ], "title": "topology-lemma-rainwater", "contents": [ "Let $f : X \\to X$ be a surjective continuous selfmap of a Hausdorff", "topological space. If $f$ is not $\\text{id}_X$, then there exists a", "proper closed subset $E \\subset X$ such that $X = E \\cup f(E)$." ], "refs": [], "proofs": [ { "contents": [ "Pick $p \\in X$ with $f(p) \\not = p$. Choose disjoint open neighbourhoods", "$p \\in U$, $f(p) \\in V$ and set $E = X \\setminus U \\cap f^{-1}(V)$.", "Then $p \\not \\in E$ hence $E$ is a proper closed subset.", "If $x \\in X$, then either $x \\in E$, or if not, then $x \\in U \\cap f^{-1}(V)$", "and writing $x = f(y)$ (possible as $f$ is surjective) we find", "$y \\in V \\subset E$ and $x \\in f(E)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8332, "type": "theorem", "label": "topology-lemma-existence-projective-cover", "categories": [ "topology" ], "title": "topology-lemma-existence-projective-cover", "contents": [ "\\begin{slogan}", "Every quasi-compact Hausdorff space has a canonical", "extremally disconnected cover", "\\end{slogan}", "Let $X$ be a quasi-compact Hausdorff space.", "There exists a continuous surjection $X' \\to X$ with $X'$", "quasi-compact, Hausdorff, and extremally disconnected.", "If we require that every proper closed subset of $X'$ does not", "map onto $X$, then $X'$ is unique up to isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Let $Y = X$ but endowed with the discrete topology. Let $X' = \\beta(Y)$.", "The continuous map $Y \\to X$ factors as $Y \\to X' \\to X$. This", "proves the first statement of the lemma by", "Example \\ref{example-stone-Cech-discrete}.", "\\medskip\\noindent", "By Lemma \\ref{lemma-find-compact-subset} we can find a quasi-compact subset", "$E \\subset X'$ surjecting onto $X$ such that no proper closed subset of $E$", "surjects onto $X$.", "Because $X'$ is extremally disconnected there exists a continuous map", "$f : X' \\to E$ over $X$", "(Proposition \\ref{proposition-projective-in-category-hausdorff-qc}).", "Composing $f$ with the map $E \\to X'$ gives a continuous selfmap", "$f|_E : E \\to E$. Observe that $f|_E$ has to be surjective as otherwise", "the image would be a proper closed subset surjecting onto $X$.", "Hence $f|_E$ has to be $\\text{id}_E$ as otherwise", "Lemma \\ref{lemma-rainwater} shows that $E$ isn't minimal.", "Thus the $\\text{id}_E$ factors through the extremally disconnected", "space $X'$. A formal, categorical argument (using the characterization of", "Proposition \\ref{proposition-projective-in-category-hausdorff-qc})", "shows that $E$ is extremally disconnected.", "\\medskip\\noindent", "To prove uniqueness, suppose we have a second $X'' \\to X$", "minimal cover. By the lifting property proven in", "Proposition \\ref{proposition-projective-in-category-hausdorff-qc}", "we can find a continuous map $g : X' \\to X''$ over $X$.", "Observe that $g$ is a closed map (Lemma \\ref{lemma-closed-map}).", "Hence $g(X') \\subset X''$ is a closed subset surjecting onto $X$", "and we conclude $g(X') = X''$ by minimality of $X''$.", "On the other hand, if $E \\subset X'$ is a proper closed subset,", "then $g(E) \\not = X''$ as $E$ does not map onto $X$ by minimality", "of $X'$. By Lemma \\ref{lemma-isomorphism} we see that $g$ is an isomorphism." ], "refs": [ "topology-lemma-find-compact-subset", "topology-proposition-projective-in-category-hausdorff-qc", "topology-lemma-rainwater", "topology-proposition-projective-in-category-hausdorff-qc", "topology-proposition-projective-in-category-hausdorff-qc", "topology-lemma-closed-map", "topology-lemma-isomorphism" ], "ref_ids": [ 8330, 8345, 8331, 8345, 8345, 8274, 8329 ] } ], "ref_ids": [] }, { "id": 8333, "type": "theorem", "label": "topology-lemma-topology-quasi-separated-scheme", "categories": [ "topology" ], "title": "topology-lemma-topology-quasi-separated-scheme", "contents": [ "Let $X$ be a topological space which", "\\begin{enumerate}", "\\item has a basis of the topology consisting of quasi-compact opens, and", "\\item has the property that the intersection of any two quasi-compact", "opens is quasi-compact.", "\\end{enumerate}", "Then", "\\begin{enumerate}", "\\item $X$ is locally quasi-compact,", "\\item a quasi-compact open $U \\subset X$ is retrocompact,", "\\item any quasi-compact open $U \\subset X$ has a cofinal system of open", "coverings $\\mathcal{U} : U = \\bigcup_{j\\in J} U_j$ with $J$ finite", "and all $U_j$ and $U_j \\cap U_{j'}$ quasi-compact,", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8334, "type": "theorem", "label": "topology-lemma-partition-refined-by-stratification", "categories": [ "topology" ], "title": "topology-lemma-partition-refined-by-stratification", "contents": [ "Let $X$ be a topological space. Let $X = \\coprod X_i$ be a finite partition", "of $X$. Then there exists a finite stratification of $X$ refining it." ], "refs": [], "proofs": [ { "contents": [ "Let $T_i = \\overline{X_i}$ and $\\Delta_i = T_i \\setminus X_i$.", "Let $S$ be the set of all intersections of $T_i$ and $\\Delta_i$.", "(For example $T_1 \\cap T_2 \\cap \\Delta_4$ is an element of $S$.)", "Then $S = \\{Z_s\\}$ is a finite collection of closed subsets of $X$ such that", "$Z_s \\cap Z_{s'} \\in S$ for all $s, s' \\in S$. Define a partial ordering", "on $S$ by inclusion. Then set $Y_s = Z_s \\setminus \\bigcup_{s' < s} Z_{s'}$", "to get the desired stratification." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8335, "type": "theorem", "label": "topology-lemma-constructible-partition-refined-by-stratification", "categories": [ "topology" ], "title": "topology-lemma-constructible-partition-refined-by-stratification", "contents": [ "Let $X$ be a topological space. Suppose $X = T_1 \\cup \\ldots \\cup T_n$", "is written as a union of constructible subsets. There exists a finite", "stratification $X = \\coprod X_i$ with each $X_i$ constructible", "such that each $T_k$ is a union of strata." ], "refs": [], "proofs": [ { "contents": [ "By definition of constructible subsets, we can write each $T_i$ as a", "finite union of $U \\cap V^c$ with $U, V \\subset X$ retrocompact open.", "Hence we may assume that $T_i = U_i \\cap V_i^c$", "with $U_i, V_i \\subset X$ retrocompact open. Let $S$ be the", "finite set of closed subsets of $X$ consisting of", "$\\emptyset, X, U_i^c, V_i^c$ and finite intersections of these.", "Write $S = \\{Z_s\\}$. If $s \\in S$, then $Z_s$ is constructible", "(Lemma \\ref{lemma-constructible}).", "Moreover, $Z_s \\cap Z_{s'} \\in S$ for all $s, s' \\in S$.", "Define a partial ordering on $S$ by inclusion. Then set", "$Y_s = Z_s \\setminus \\bigcup_{s' < s} Z_{s'}$", "to get the desired stratification." ], "refs": [ "topology-lemma-constructible" ], "ref_ids": [ 8253 ] } ], "ref_ids": [] }, { "id": 8336, "type": "theorem", "label": "topology-lemma-noetherian-partition-refined-by-stratification", "categories": [ "topology" ], "title": "topology-lemma-noetherian-partition-refined-by-stratification", "contents": [ "Let $X$ be a Noetherian topological space. Any finite partition", "of $X$ can be refined by a finite good stratification." ], "refs": [], "proofs": [ { "contents": [ "Let $X = \\coprod X_i$ be a finite partition of $X$.", "Let $Z$ be an irreducible component of $X$. Since $X = \\bigcup \\overline{X_i}$", "with finite index set, there is an $i$ such that", "$Z \\subset \\overline{X_i}$. Since $X_i$ is locally closed this", "implies that $Z \\cap X_i$ contains an open of $Z$. Thus", "$Z \\cap X_i$ contains an open $U$ of $X$ (Lemma \\ref{lemma-Noetherian}).", "Write $X_i = U \\amalg X_i^1 \\amalg X_i^2$ with", "$X_i^1 = (X_i \\setminus U) \\cap \\overline{U}$ and", "$X_i^2 = (X_i \\setminus U) \\cap \\overline{U}^c$.", "For $i' \\not = i$ we set", "$X_{i'}^1 = X_{i'} \\cap \\overline{U}$ and", "$X_{i'}^2 = X_{i'} \\cap \\overline{U}^c$.", "Then", "$$", "X \\setminus U = \\coprod X^k_l", "$$", "is a partition such that $\\overline{U} \\setminus U = \\bigcup X_l^1$.", "Note that $X \\setminus U$ is closed and strictly smaller than $X$.", "By Noetherian induction we can refine this partition", "by a finite good stratification", "$X \\setminus U = \\coprod_{\\alpha \\in A} T_\\alpha$.", "Then $X = U \\amalg \\coprod_{\\alpha \\in A} T_\\alpha$ is a finite", "good stratification of $X$ refining the partition we started with." ], "refs": [ "topology-lemma-Noetherian" ], "ref_ids": [ 8220 ] } ], "ref_ids": [] }, { "id": 8337, "type": "theorem", "label": "topology-lemma-colimits", "categories": [ "topology" ], "title": "topology-lemma-colimits", "contents": [ "The category of topological spaces has colimits and the forgetful functor", "to sets commutes with them." ], "refs": [], "proofs": [ { "contents": [ "This follows from the discussion above and", "Categories, Lemma \\ref{categories-lemma-colimits-coproducts-coequalizers}.", "Another proof of existence of colimits is sketched in", "Categories, Remark \\ref{categories-remark-how-to-use-it}.", "It follows from the above that the forgetful functor", "commutes with colimits. Another way to see this is to use", "Categories, Lemma \\ref{categories-lemma-adjoint-exact} and use that", "the forgetful functor has a right adjoint, namely the functor which", "assigns to a set the corresponding chaotic (or indiscrete) topological space." ], "refs": [ "categories-lemma-colimits-coproducts-coequalizers", "categories-remark-how-to-use-it", "categories-lemma-adjoint-exact" ], "ref_ids": [ 12214, 12422, 12249 ] } ], "ref_ids": [] }, { "id": 8338, "type": "theorem", "label": "topology-lemma-topological-group-limits", "categories": [ "topology" ], "title": "topology-lemma-topological-group-limits", "contents": [ "The category of topological groups has limits and limits commute", "with the forgetful functors to (a) the category of topological spaces and", "(b) the category of groups." ], "refs": [], "proofs": [ { "contents": [ "It is enough to prove the existence and commutation for products and", "equalizers, see", "Categories, Lemma \\ref{categories-lemma-limits-products-equalizers}.", "Let $G_i$, $i \\in I$ be a collection of topological groups.", "Take the usual product $G = \\prod G_i$ with the product topology.", "Since $G \\times G = \\prod (G_i \\times G_i)$ as a topological space", "(because products commutes with products in any category), we", "see that multiplication on $G$ is continuous. Similarly for", "the inverse map. Let $a, b : G \\to H$ be two homomorphisms of", "topological groups. Then as the equalizer we can simply take", "the equalizer of $a$ and $b$ as maps of topological spaces,", "which is the same thing as the equalizer as maps of groups", "endowed with the induced topology." ], "refs": [ "categories-lemma-limits-products-equalizers" ], "ref_ids": [ 12213 ] } ], "ref_ids": [] }, { "id": 8339, "type": "theorem", "label": "topology-lemma-profinite-group", "categories": [ "topology" ], "title": "topology-lemma-profinite-group", "contents": [ "Let $G$ be a topological group. The following are equivalent", "\\begin{enumerate}", "\\item $G$ as a topological space is profinite,", "\\item $G$ is a limit of a diagram of finite discrete topological groups,", "\\item $G$ is a cofiltered limit of finite discrete topological groups.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have the corresponding result for topological spaces, see", "Lemma \\ref{lemma-profinite}. Combined with", "Lemma \\ref{lemma-topological-group-limits}", "we see that it suffices to prove that (1) implies (3).", "\\medskip\\noindent", "We first prove that every neighbourhood $E$ of the neutral element $e$ contains", "an open subgroup. Namely, since $G$ is the cofiltered limit of finite", "discrete topological spaces (Lemma \\ref{lemma-profinite}),", "we can choose a continuous map $f : G \\to T$ to a finite discrete", "space $T$ such that $f^{-1}(f(\\{e\\})) \\subset E$.", "Consider", "$$", "H = \\{g \\in G \\mid f(gg') = f(g')\\text{ for all }g' \\in G\\}", "$$", "This is a subgroup of $G$ and contained in $E$. Thus it suffices to", "show that $H$ is open. Pick $t \\in T$ and set $W = f^{-1}(\\{t\\})$.", "Observe that $W \\subset G$ is open and closed, in particular quasi-compact.", "For each $w \\in W$ there exist open neighbourhoods $e \\in U_w \\subset G$", "and $w \\in U'_w \\subset W$ such that $U_wU'_w \\subset W$.", "By quasi-compactness we can find $w_1, \\ldots, w_n$ such that", "$W = \\bigcup U'_{w_i}$. Then", "$U_t = U_{w_1} \\cap \\ldots \\cap U_{w_n}$ is an open neighbourhood", "of $e$ such that $f(gw) = t$ for all $w \\in W$.", "Since $T$ is finite we see that $\\bigcap_{t \\in T} U_t \\subset H$", "is an open neighbourhood of $e$. Since $H \\subset G$ is a subgroup", "it follows that $H$ is open.", "\\medskip\\noindent", "Suppose that $H \\subset G$ is an open subgroup. Since $G$ is quasi-compact", "we see that the index of $H$ in $G$ is finite. Say", "$G = Hg_1 \\cup \\ldots \\cup Hg_n$. Then", "$N = \\bigcap_{i = 1, \\ldots, n} g_iHg_i^{-1}$ is", "an open normal subgroup contained in $H$. Since $N$ also has finite", "index we see that $G \\to G/N$ is a surjection to a finite discrete", "topological group.", "\\medskip\\noindent", "Consider the map", "$$", "G \\longrightarrow \\lim_{N \\subset G\\text{ open and normal}} G/N", "$$", "We claim that this map is an isomorphism of topological groups.", "This finishes the proof of the lemma as the limit on the right", "is cofiltered (the intersection of two open normal subgroups is", "open and normal). The map is continuous as each $G \\to G/N$ is continuous.", "The map is injective as $G$ is Hausdorff and every neighbourhood", "of $e$ contains an $N$ by the arguments above.", "The map is surjective by", "Lemma \\ref{lemma-intersection-closed-in-quasi-compact}.", "By Lemma \\ref{lemma-bijective-map} the map is a homeomorphism." ], "refs": [ "topology-lemma-profinite", "topology-lemma-topological-group-limits", "topology-lemma-profinite", "topology-lemma-intersection-closed-in-quasi-compact", "topology-lemma-bijective-map" ], "ref_ids": [ 8299, 8338, 8299, 8232, 8275 ] } ], "ref_ids": [] }, { "id": 8340, "type": "theorem", "label": "topology-lemma-topological-group-colimits", "categories": [ "topology" ], "title": "topology-lemma-topological-group-colimits", "contents": [ "The category of topological groups has colimits and colimits commute", "with the forgetful functor to the category of groups." ], "refs": [], "proofs": [ { "contents": [ "We will use the argument of", "Categories, Remark \\ref{categories-remark-how-to-use-it} to prove", "existence of colimits. Namely, suppose", "that $\\mathcal{I} \\to \\textit{Top}$, $i \\mapsto G_i$ is a functor", "into the category $\\textit{TopGroup}$ of topological groups.", "Then we can consider", "$$", "F : \\textit{TopGroup} \\longrightarrow \\textit{Sets},\\quad", "H \\longmapsto \\lim_\\mathcal{I} \\Mor_{\\textit{TopGroup}}(G_i, H)", "$$", "This functor commutes with limits. Moreover, given any topological group", "$H$ and an element $(\\varphi_i : G_i \\to H)$ of $F(H)$, there is", "a subgroup $H' \\subset H$ of cardinality at most $|\\coprod G_i|$", "(coproduct in the category of groups, i.e., the free product on the $G_i$)", "such that the morphisms $\\varphi_i$ map into $H'$. Namely, we can", "take the induced topology on the subgroup generated by the images", "of the $\\varphi_i$. Thus it is clear that the hypotheses of", "Categories, Lemma \\ref{categories-lemma-a-version-of-brown}", "are satisfied and we find a topological group $G$", "representing the functor $F$, which precisely means that $G$ is", "the colimit of the diagram $i \\mapsto G_i$.", "\\medskip\\noindent", "To see the statement on commutation with the forgetful functor to", "groups we will use", "Categories, Lemma \\ref{categories-lemma-adjoint-exact}.", "Indeed, the forgetful functor has a right adjoint, namely the functor which", "assigns to a group the corresponding chaotic (or indiscrete) topological group." ], "refs": [ "categories-remark-how-to-use-it", "categories-lemma-a-version-of-brown", "categories-lemma-adjoint-exact" ], "ref_ids": [ 12422, 12253, 12249 ] } ], "ref_ids": [] }, { "id": 8341, "type": "theorem", "label": "topology-lemma-topological-ring-limits", "categories": [ "topology" ], "title": "topology-lemma-topological-ring-limits", "contents": [ "The category of topological rings has limits and limits commute", "with the forgetful functors to (a) the category of topological spaces and", "(b) the category of rings." ], "refs": [], "proofs": [ { "contents": [ "It is enough to prove the existence and commutation for products and", "equalizers, see", "Categories, Lemma \\ref{categories-lemma-limits-products-equalizers}.", "Let $R_i$, $i \\in I$ be a collection of topological rings.", "Take the usual product $R = \\prod R_i$ with the product topology.", "Since $R \\times R = \\prod (R_i \\times R_i)$ as a topological space", "(because products commutes with products in any category), we", "see that addition and multiplication on $R$ are continuous. Let", "$a, b : R \\to R'$ be two homomorphisms of topological rings.", "Then as the equalizer we can simply take the equalizer of $a$ and $b$", "as maps of topological spaces, which is the same thing as the equalizer", "as maps of rings endowed with the induced topology." ], "refs": [ "categories-lemma-limits-products-equalizers" ], "ref_ids": [ 12213 ] } ], "ref_ids": [] }, { "id": 8342, "type": "theorem", "label": "topology-lemma-topological-ring-colimits", "categories": [ "topology" ], "title": "topology-lemma-topological-ring-colimits", "contents": [ "The category of topological rings has colimits and colimits commute", "with the forgetful functor to the category of rings." ], "refs": [], "proofs": [ { "contents": [ "The exact same argument as used in the proof of", "Lemma \\ref{lemma-topological-group-colimits} shows existence of colimits.", "To see the statement on commutation with the forgetful functor to", "rings we will use Categories, Lemma \\ref{categories-lemma-adjoint-exact}.", "Indeed, the forgetful functor has a right adjoint, namely the functor which", "assigns to a ring the corresponding chaotic (or indiscrete) topological ring." ], "refs": [ "topology-lemma-topological-group-colimits", "categories-lemma-adjoint-exact" ], "ref_ids": [ 8340, 12249 ] } ], "ref_ids": [] }, { "id": 8343, "type": "theorem", "label": "topology-lemma-topological-module-limits", "categories": [ "topology" ], "title": "topology-lemma-topological-module-limits", "contents": [ "Let $R$ be a topological ring. The category of topological modules over $R$", "has limits and limits commute with the forgetful functors to", "(a) the category of topological spaces and", "(b) the category of $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "It is enough to prove the existence and commutation for products and", "equalizers, see", "Categories, Lemma \\ref{categories-lemma-limits-products-equalizers}.", "Let $M_i$, $i \\in I$ be a collection of topological modules over $R$.", "Take the usual product $M = \\prod M_i$ with the product topology.", "Since $M \\times M = \\prod (M_i \\times M_i)$ as a topological space", "(because products commutes with products in any category), we", "see that addition on $M$ is continuous. Similarly for multiplication", "$R \\times M \\to M$.", "Let $a, b : M \\to M'$ be two homomorphisms of topological modules over $R$.", "Then as the equalizer we can simply take the equalizer of $a$ and $b$", "as maps of topological spaces, which is the same thing as the equalizer", "as maps of modules endowed with the induced topology." ], "refs": [ "categories-lemma-limits-products-equalizers" ], "ref_ids": [ 12213 ] } ], "ref_ids": [] }, { "id": 8344, "type": "theorem", "label": "topology-lemma-topological-module-colimits", "categories": [ "topology" ], "title": "topology-lemma-topological-module-colimits", "contents": [ "Let $R$ be a topological ring. The category of topological modules over $R$", "has colimits and colimits commute with the forgetful functor to the category", "of modules over $R$." ], "refs": [], "proofs": [ { "contents": [ "The exact same argument as used in the proof of", "Lemma \\ref{lemma-topological-group-colimits} shows existence of colimits.", "To see the statement on commutation with the forgetful functor to", "$R$-modules we will use Categories, Lemma \\ref{categories-lemma-adjoint-exact}.", "Indeed, the forgetful functor has a right adjoint, namely the functor which", "assigns to a module the corresponding chaotic (or indiscrete) topological", "module." ], "refs": [ "topology-lemma-topological-group-colimits", "categories-lemma-adjoint-exact" ], "ref_ids": [ 8340, 12249 ] } ], "ref_ids": [] }, { "id": 8345, "type": "theorem", "label": "topology-proposition-projective-in-category-hausdorff-qc", "categories": [ "topology" ], "title": "topology-proposition-projective-in-category-hausdorff-qc", "contents": [ "Let $X$ be a Hausdorff, quasi-compact topological space.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is extremally disconnected,", "\\item for any surjective continuous map $f : Y \\to X$ with $Y$ Hausdorff", "quasi-compact there exists a continuous section, and", "\\item for any solid commutative diagram", "$$", "\\xymatrix{", "& Y \\ar[d] \\\\", "X \\ar@{..>}[ru] \\ar[r] & Z", "}", "$$", "of continuous maps of quasi-compact Hausdorff spaces with $Y \\to Z$", "surjective, there is a dotted arrow", "in the category of topological spaces making the diagram commute.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (3) implies (2). On the other hand, if (2) holds", "and $X \\to Z$ and $Y \\to Z$ are as in (3), then (2) assures there", "is a section to the projection $X \\times_Z Y \\to X$ which implies", "a suitable dotted arrow exists (details omitted). Thus (3)", "is equivalent to (2).", "\\medskip\\noindent", "Assume $X$ is extremally disconnected and let $f : Y \\to X$ be as in (2).", "By Lemma \\ref{lemma-find-compact-subset} there exists a quasi-compact subset", "$E \\subset Y$ such that $f(E) = X$ but $f(E') \\not = X$ for all", "proper closed subsets $E' \\subset E$. By", "Lemma \\ref{lemma-isomorphism} we find that $f|_E : E \\to X$", "is a homeomorphism, the inverse of which gives the desired section.", "\\medskip\\noindent", "Assume (2). Let $U \\subset X$ be open with complement $Z$.", "Consider the continuous surjection $f : \\overline{U} \\amalg Z \\to X$.", "Let $\\sigma$ be a section. Then $\\overline{U} = \\sigma^{-1}(\\overline{U})$", "is open. Thus $X$ is extremally disconnected." ], "refs": [ "topology-lemma-find-compact-subset", "topology-lemma-isomorphism" ], "ref_ids": [ 8330, 8329 ] } ], "ref_ids": [] }, { "id": 8387, "type": "theorem", "label": "hypercovering-theorem-cohomology-hypercoverings", "categories": [ "hypercovering" ], "title": "hypercovering-theorem-cohomology-hypercoverings", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X$ be an object of $\\mathcal{C}$. Let $i \\geq 0$.", "The functors", "\\begin{eqnarray*}", "\\textit{Ab}(\\mathcal{C}) & \\longrightarrow & \\textit{Ab} \\\\", "\\mathcal{F} & \\longmapsto & H^i(X, \\mathcal{F}) \\\\", "\\mathcal{F} & \\longmapsto & \\check{H}^i_{\\text{HC}}(X, \\mathcal{F})", "\\end{eqnarray*}", "are canonically isomorphic." ], "refs": [], "proofs": [ { "contents": [ "[Proof using spectral sequences.]", "Suppose that $\\xi \\in H^p(X, \\mathcal{F})$ for some $p \\geq 0$.", "Let us show that $\\xi$ is in the image of the map", "$\\check{H}^p(X, \\mathcal{F}) \\to H^p(X, \\mathcal{F})$ of", "Lemma \\ref{lemma-cech-spectral-sequence}", "for some hypercovering $K$ of $X$.", "\\medskip\\noindent", "This is true if $p = 0$ by Lemma \\ref{lemma-h0-cech}.", "If $p = 1$, choose a {\\v C}ech hypercovering $K$ of $X$ as in", "Example \\ref{example-cech} starting with a covering", "$K_0 = \\{U_i \\to X\\}$ in the site $\\mathcal{C}$ such that", "$\\xi|_{U_i} = 0$, see", "Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-kill-cohomology-class-on-covering}.", "It follows immediately from the spectral sequence", "in Lemma \\ref{lemma-cech-spectral-sequence} that $\\xi$ comes", "from an element of $\\check{H}^1(K, \\mathcal{F})$ in this case.", "In general, choose any hypercovering $K$ of $X$ such", "that $\\xi$ maps to zero in $\\underline{H}^p(\\mathcal{F})(K_0)$", "(using Example \\ref{example-cech} and", "Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-kill-cohomology-class-on-covering}", "again).", "By the spectral sequence of Lemma \\ref{lemma-cech-spectral-sequence}", "the obstruction for $\\xi$ to come from an element of", "$\\check{H}^p(K, \\mathcal{F})$ is a sequence of elements", "$\\xi_1, \\ldots, \\xi_{p - 1}$ with", "$\\xi_q \\in \\check{H}^{p - q}(K, \\underline{H}^q(\\mathcal{F}))$", "(more precisely the images of the $\\xi_q$ in certain subquotients", "of these groups).", "\\medskip\\noindent", "We can inductively replace the hypercovering $K$ by refinements", "such that the obstructions $\\xi_1, \\ldots, \\xi_{p - 1}$ restrict to zero", "(and not just the images", "in the subquotients -- so no subtlety here). Indeed, suppose we have", "already managed to reach the situation where", "$\\xi_{q + 1}, \\ldots, \\xi_{p - 1}$ are zero.", "Note that $\\xi_q \\in \\check{H}^{p - q}(K, \\underline{H}^q(\\mathcal{F}))$", "is the class of some element", "$$", "\\tilde \\xi_q \\in", "\\underline{H}^q(\\mathcal{F})(K_{p - q}) =", "\\prod H^q(U_i, \\mathcal{F})", "$$", "if $K_{p - q} = \\{U_i \\to X\\}_{i \\in I}$. Let $\\xi_{q, i}$", "be the component of $\\tilde \\xi_q$ in $H^q(U_i, \\mathcal{F})$.", "As $q \\geq 1$ we can use", "Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-kill-cohomology-class-on-covering}", "yet again to choose coverings $\\{U_{i, j} \\to U_i\\}$", "of the site such that each restriction $\\xi_{q, i}|_{U_{i, j}} = 0$.", "Consider the object $Z = \\{U_{i, j} \\to X\\}$ of the category", "$\\text{SR}(\\mathcal{C}, X)$ and its obvious morphism", "$u : Z \\to K_{p - q}$. It is clear that $u$ is a covering, see", "Definition \\ref{definition-covering-SR}. By", "Lemma \\ref{lemma-covering} there", "exists a morphism $L \\to K$ of hypercoverings of $X$ such that", "$L_{p - q} \\to K_{p - q}$ factors through $u$. Then clearly the", "image of $\\xi_q$ in $\\underline{H}^q(\\mathcal{F})(L_{p - q})$.", "is zero. Since the spectral sequence of", "Lemma \\ref{lemma-cech-spectral-sequence}", "is functorial this means that after replacing $K$ by $L$ we reach the", "situation where $\\xi_q, \\ldots, \\xi_{p - 1}$ are all zero.", "Continuing like this we end up with a hypercovering where they are all", "zero and hence $\\xi$ is in the image of the map", "$\\check{H}^p(X, \\mathcal{F}) \\to H^p(X, \\mathcal{F})$.", "\\medskip\\noindent", "Suppose that $K$ is a hypercovering of $X$, that", "$\\xi \\in \\check{H}^p(K, \\mathcal{F})$ and that the image of", "$\\xi$ under the map", "$\\check{H}^p(X, \\mathcal{F}) \\to H^p(X, \\mathcal{F})$ of", "Lemma \\ref{lemma-cech-spectral-sequence}", "is zero. To finish the proof of the theorem we have to show that", "there exists a morphism of hypercoverings $L \\to K$ such that", "$\\xi$ restricts to zero in $\\check{H}^p(L, \\mathcal{F})$.", "By the spectral sequence of Lemma \\ref{lemma-cech-spectral-sequence}", "the vanishing of the image of $\\xi$ in $H^p(X, \\mathcal{F})$", "means that there exist elements $\\xi_1, \\ldots, \\xi_{p - 2}$", "with $\\xi_q \\in \\check{H}^{p - 1 - q}(K, \\underline{H}^q(\\mathcal{F}))$", "(more precisely the images of these in certain subquotients)", "such that the images $d_{q + 1}^{p - 1 - q, q}\\xi_q$ (in the spectral", "sequence) add up to $\\xi$. Hence by exactly the same mechanism as above", "we can find a morphism of hypercoverings $L \\to K$ such that", "the restrictions of the elements $\\xi_q$, $q = 1, \\ldots, p - 2$", "in $\\check{H}^{p - 1 - q}(L, \\underline{H}^q(\\mathcal{F}))$ are zero.", "Then it follows that $\\xi$ is zero since the morphism $L \\to K$", "induces a morphism of spectral sequences according to", "Lemma \\ref{lemma-cech-spectral-sequence}." ], "refs": [ "hypercovering-lemma-cech-spectral-sequence", "hypercovering-lemma-h0-cech", "sites-cohomology-lemma-kill-cohomology-class-on-covering", "hypercovering-lemma-cech-spectral-sequence", "sites-cohomology-lemma-kill-cohomology-class-on-covering", "hypercovering-lemma-cech-spectral-sequence", "sites-cohomology-lemma-kill-cohomology-class-on-covering", "hypercovering-definition-covering-SR", "hypercovering-lemma-covering", "hypercovering-lemma-cech-spectral-sequence", "hypercovering-lemma-cech-spectral-sequence", "hypercovering-lemma-cech-spectral-sequence", "hypercovering-lemma-cech-spectral-sequence" ], "ref_ids": [ 8398, 8396, 4188, 8398, 4188, 8398, 4188, 8423, 8405, 8398, 8398, 8398, 8398 ] } ], "ref_ids": [] }, { "id": 8388, "type": "theorem", "label": "hypercovering-lemma-coprod-prod-SR", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-coprod-prod-SR", "contents": [ "Let $\\mathcal{C}$ be a category.", "\\begin{enumerate}", "\\item the category $\\text{SR}(\\mathcal{C})$ has coproducts", "and $F$ commutes with them,", "\\item the functor $F : \\text{SR}(\\mathcal{C}) \\to \\textit{PSh}(\\mathcal{C})$", "commutes with limits,", "\\item if $\\mathcal{C}$ has fibre products, then $\\text{SR}(\\mathcal{C})$", "has fibre products,", "\\item if $\\mathcal{C}$ has products of pairs, then", "$\\text{SR}(\\mathcal{C})$ has products of pairs,", "\\item if $\\mathcal{C}$ has equalizers, so does $\\text{SR}(\\mathcal{C})$, and", "\\item if $\\mathcal{C}$ has a final object, so does $\\text{SR}(\\mathcal{C})$.", "\\end{enumerate}", "Let $X \\in \\Ob(\\mathcal{C})$.", "\\begin{enumerate}", "\\item the category $\\text{SR}(\\mathcal{C}, X)$ has coproducts", "and $F$ commutes with them,", "\\item if $\\mathcal{C}$ has fibre products, then $\\text{SR}(\\mathcal{C}, X)$", "has finite limits and", "$F : \\text{SR}(\\mathcal{C}, X) \\to \\textit{PSh}(\\mathcal{C})/h_X$", "commutes with them.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of the results on $\\text{SR}(\\mathcal{C})$.", "Proof of (1). The coproduct of $\\{U_i\\}_{i \\in I}$ and $\\{V_j\\}_{j \\in J}$ is", "$\\{U_i\\}_{i \\in I} \\amalg \\{V_j\\}_{j \\in J}$, in other words, the family", "of objects whose index set is $I \\amalg J$ and for an element", "$k \\in I \\amalg J$ gives $U_i$ if $k = i \\in I$ and gives $V_j$ if", "$k = j \\in J$. Similarly for coproducts", "of families of objects. It is clear that $F$ commutes with these.", "\\medskip\\noindent", "Proof of (2). For $U$ in $\\Ob(\\mathcal{C})$ consider the object $\\{U\\}$ of", "$\\text{SR}(\\mathcal{C})$. It is clear that", "$\\Mor_{\\text{SR}(\\mathcal{C})}(\\{U\\}, K)) = F(K)(U)$", "for $K \\in \\Ob(\\text{SR}(\\mathcal{C}))$. Since limits of presheaves", "are computed at the level of sections", "(Sites, Section \\ref{sites-section-limits-colimits-PSh})", "we conclude that $F$ commutes with limits.", "\\medskip\\noindent", "Proof of (3). Suppose given a morphism", "$(\\alpha, f_i) : \\{U_i\\}_{i \\in I} \\to \\{V_j\\}_{j \\in J}$", "and a morphism", "$(\\beta, g_k) : \\{W_k\\}_{k \\in K} \\to \\{V_j\\}_{j \\in J}$.", "The fibred product of these morphisms is given by", "$$", "\\{ U_i \\times_{f_i, V_j, g_k} W_k\\}_{(i, j, k) \\in I \\times J \\times K", "\\text{ such that } j = \\alpha(i) = \\beta(k)}", "$$", "The fibre products exist if $\\mathcal{C}$ has fibre products.", "\\medskip\\noindent", "Proof of (4). The product of $\\{U_i\\}_{i \\in I}$ and $\\{V_j\\}_{j \\in J}$ is", "$\\{U_i \\times V_j\\}_{i \\in I, j \\in J}$. The products exist if", "$\\mathcal{C}$ has products.", "\\medskip\\noindent", "Proof of (5). The equalizer of two maps", "$(\\alpha, f_i), (\\alpha', f'_i) : \\{U_i\\}_{i \\in I} \\to \\{V_j\\}_{j \\in J}$", "is", "$$", "\\{", "\\text{Eq}(f_i, f'_i : U_i \\to V_{\\alpha(i)})", "\\}_{i \\in I,\\ \\alpha(i) = \\alpha'(i)}", "$$", "The equalizers exist if $\\mathcal{C}$ has equalizers.", "\\medskip\\noindent", "Proof of (6). If $X$ is a final object of $\\mathcal{C}$, then", "$\\{X\\}$ is a final object of $\\text{SR}(\\mathcal{C})$.", "\\medskip\\noindent", "Proof of the statements about $\\text{SR}(\\mathcal{C}, X)$.", "These follow from the results above applied to the category", "$\\mathcal{C}/X$ using that", "$\\text{SR}(\\mathcal{C}/X) = \\text{SR}(\\mathcal{C}, X)$ and that", "$\\textit{PSh}(\\mathcal{C}/X) = \\textit{PSh}(\\mathcal{C})/h_X$", "(Sites, Lemma \\ref{sites-lemma-essential-image-j-shriek} applied", "to $\\mathcal{C}$ endowed with the chaotic topology). However", "we also argue directly as follows.", "It is clear that the coproduct of", "$\\{U_i \\to X\\}_{i \\in I}$ and $\\{V_j \\to X\\}_{j \\in J}$", "is $\\{U_i \\to X\\}_{i \\in I} \\amalg \\{V_j \\to X\\}_{j \\in J}$", "and similarly for coproducts of", "families of families of morphisms with target $X$.", "The object $\\{X \\to X\\}$ is a final", "object of $\\text{SR}(\\mathcal{C}, X)$.", "Suppose given a morphism", "$(\\alpha, f_i) : \\{U_i \\to X\\}_{i \\in I} \\to \\{V_j \\to X\\}_{j \\in J}$", "and a morphism", "$(\\beta, g_k) : \\{W_k \\to X\\}_{k \\in K} \\to \\{V_j \\to X\\}_{j \\in J}$.", "The fibred product of these morphisms is given by", "$$", "\\{ U_i \\times_{f_i, V_j, g_k} W_k \\to X \\}_{(i, j, k) \\in I \\times J \\times K", "\\text{ such that } j = \\alpha(i) = \\beta(k)}", "$$", "The fibre products exist by the assumption that", "$\\mathcal{C}$ has fibre products.", "Thus $\\text{SR}(\\mathcal{C}, X)$ has finite limits,", "see Categories, Lemma \\ref{categories-lemma-finite-limits-exist}.", "We omit verifying the statements on the functor $F$ in this case." ], "refs": [ "sites-lemma-essential-image-j-shriek", "categories-lemma-finite-limits-exist" ], "ref_ids": [ 8555, 12224 ] } ], "ref_ids": [] }, { "id": 8389, "type": "theorem", "label": "hypercovering-lemma-covering-permanence", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-covering-permanence", "contents": [ "Let $\\mathcal{C}$ be a site.", "\\begin{enumerate}", "\\item A composition of coverings in $\\text{SR}(\\mathcal{C})$", "is a covering.", "\\item If $K \\to L$ is a covering in $\\text{SR}(\\mathcal{C})$", "and $L' \\to L$ is a morphism, then $L' \\times_L K$ exists", "and $L' \\times_L K \\to L'$ is a covering.", "\\item If $\\mathcal{C}$ has products of pairs, and", "$A \\to B$ and $K \\to L$ are coverings in $\\text{SR}(\\mathcal{C})$,", "then $A \\times K \\to B \\times L$ is a covering.", "\\end{enumerate}", "Let $X \\in \\Ob(\\mathcal{C})$. Then (1) and (2) holds for", "$\\text{SR}(\\mathcal{C}, X)$ and (3) holds if $\\mathcal{C}$", "has fibre products." ], "refs": [], "proofs": [ { "contents": [ "Part (1) is immediate from the axioms of a site.", "Part (2) follows by the construction of fibre products", "in $\\text{SR}(\\mathcal{C})$ in the proof of", "Lemma \\ref{lemma-coprod-prod-SR}", "and the requirement that the morphisms in a covering", "of $\\mathcal{C}$ are representable.", "Part (3) follows by thinking of $A \\times K \\to B \\times L$", "as the composition $A \\times K \\to B \\times K \\to B \\times L$", "and hence a composition of basechanges of coverings.", "The final statement follows because $\\text{SR}(\\mathcal{C}, X) =", "\\text{SR}(\\mathcal{C}/X)$." ], "refs": [ "hypercovering-lemma-coprod-prod-SR" ], "ref_ids": [ 8388 ] } ], "ref_ids": [] }, { "id": 8390, "type": "theorem", "label": "hypercovering-lemma-hypercoverings-set", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-hypercoverings-set", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X \\in \\Ob(\\mathcal{C})$ be an object of $\\mathcal{C}$.", "The collection of all hypercoverings of $X$ forms a set." ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathcal{C}$ is a site, the set of all coverings of", "$X$ forms a set. Thus we see that the collection", "of possible $K_0$ forms a set. Suppose we have shown that", "the collection of all possible $K_0, \\ldots, K_n$ form", "a set. Then it is enough to show that given", "$K_0, \\ldots, K_n$ the collection of all possible", "$K_{n + 1}$ forms a set. And this is clearly true since", "we have to choose $K_{n + 1}$ among all possible coverings", "of $(\\text{cosk}_n \\text{sk}_n K)_{n + 1}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8391, "type": "theorem", "label": "hypercovering-lemma-hypercovering-F", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-hypercovering-F", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X \\in \\Ob(\\mathcal{C})$ be an object of $\\mathcal{C}$.", "Let $K$ be a hypercovering of $X$.", "Consider the simplicial object $F(K)$ of $\\textit{PSh}(\\mathcal{C})$,", "endowed with its augmentation to the constant simplicial presheaf $h_X$.", "\\begin{enumerate}", "\\item The morphism of presheaves $F(K)_0 \\to h_X$ becomes", "a surjection after sheafification.", "\\item The morphism", "$$", "(d^1_0, d^1_1) :", "F(K)_1", "\\longrightarrow", "F(K)_0 \\times_{h_X} F(K)_0", "$$", "becomes a surjection after sheafification.", "\\item For every $n \\geq 1$ the morphism", "$$", "F(K)_{n + 1} \\longrightarrow (\\text{cosk}_n \\text{sk}_n F(K))_{n + 1}", "$$", "turns into a surjection after sheafification.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use the fact that if", "$\\{U_i \\to U\\}_{i \\in I}$ is a covering of the site", "$\\mathcal{C}$, then the morphism", "$$", "\\amalg_{i \\in I} h_{U_i} \\to h_U", "$$", "becomes surjective after sheafification, see", "Sites, Lemma \\ref{sites-lemma-covering-surjective-after-sheafification}.", "Thus the first assertion follows immediately.", "\\medskip\\noindent", "For the second assertion, note that according to", "Simplicial, Example \\ref{simplicial-example-cosk0}", "the simplicial object $\\text{cosk}_0 \\text{sk}_0 K$", "has terms $K_0 \\times \\ldots \\times K_0$. Thus", "according to the definition of a hypercovering we", "see that $(d^1_0, d^1_1) : K_1 \\to K_0 \\times K_0$ is a", "covering. Hence (2) follows from the claim above", "and the fact that $F$ transforms products into fibred", "products over $h_X$.", "\\medskip\\noindent", "For the third, we claim that", "$\\text{cosk}_n \\text{sk}_n F(K) =", "F(\\text{cosk}_n \\text{sk}_n K)$ for $n \\geq 1$.", "To prove this, denote temporarily $F'$ the functor", "$\\text{SR}(\\mathcal{C}, X) \\to \\textit{PSh}(\\mathcal{C})/h_X$.", "By Lemma \\ref{lemma-coprod-prod-SR} the functor", "$F'$ commutes with finite limits.", "By our description of the $\\text{cosk}_n$ functor in", "Simplicial, Section \\ref{simplicial-section-skeleton}", "we see that $\\text{cosk}_n \\text{sk}_n F'(K) =", "F'(\\text{cosk}_n \\text{sk}_n K)$.", "Recall that the category used in the description of", "$(\\text{cosk}_n U)_m$ in", "Simplicial, Lemma \\ref{simplicial-lemma-existence-cosk}", "is the category $(\\Delta/[m])^{opp}_{\\leq n}$. It is an", "amusing exercise to show that $(\\Delta/[m])_{\\leq n}$ is", "a connected category (see", "Categories, Definition \\ref{categories-definition-category-connected})", "as soon as $n \\geq 1$. Hence,", "Categories, Lemma \\ref{categories-lemma-connected-limit-over-X}", "shows that $\\text{cosk}_n \\text{sk}_n F'(K) =", "\\text{cosk}_n \\text{sk}_n F(K)$. Whence the claim.", "Property (2) follows from this, because now we see that", "the morphism in (2) is the result of applying the", "functor $F$ to a covering as in Definition \\ref{definition-covering-SR},", "and the result follows from the first fact mentioned", "in this proof." ], "refs": [ "sites-lemma-covering-surjective-after-sheafification", "hypercovering-lemma-coprod-prod-SR", "simplicial-lemma-existence-cosk", "categories-definition-category-connected", "categories-lemma-connected-limit-over-X", "hypercovering-definition-covering-SR" ], "ref_ids": [ 8519, 8388, 14835, 12360, 12215, 8423 ] } ], "ref_ids": [] }, { "id": 8392, "type": "theorem", "label": "hypercovering-lemma-compare-cosk0", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-compare-cosk0", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{F} \\to \\mathcal{G}$ be a morphism", "of presheaves of sets. Denote $K$ the simplicial", "object of $\\textit{PSh}(\\mathcal{C})$ whose $n$th", "term is the $(n + 1)$st fibre product of $\\mathcal{F}$", "over $\\mathcal{G}$, see", "Simplicial, Example \\ref{simplicial-example-fibre-products-simplicial-object}.", "Then, if $\\mathcal{F} \\to \\mathcal{G}$ is surjective after", "sheafification, we have", "$$", "H_i(K) =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & i > 0\\\\", "\\mathbf{Z}_\\mathcal{G}^\\# & \\text{if} & i = 0", "\\end{matrix}", "\\right.", "$$", "The isomorphism in degree $0$ is given by the", "morphism $H_0(K) \\to \\mathbf{Z}_\\mathcal{G}^\\#$", "coming from the map $(\\mathbf{Z}_K^\\#)_0 =", "\\mathbf{Z}_\\mathcal{F}^\\# \\to \\mathbf{Z}_\\mathcal{G}^\\#$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{G}' \\subset \\mathcal{G}$ be the image of", "the morphism $\\mathcal{F} \\to \\mathcal{G}$.", "Let $U \\in \\Ob(\\mathcal{C})$. Set", "$A = \\mathcal{F}(U)$ and $B = \\mathcal{G}'(U)$.", "Then the simplicial set $K(U)$ is equal to the simplicial", "set with $n$-simplices given by", "$$", "A \\times_B A \\times_B \\ldots \\times_B A\\ (n + 1 \\text{ factors)}.", "$$", "By Simplicial, Lemma \\ref{simplicial-lemma-cosk-minus-one-equivalence}", "the morphism $K(U) \\to B$ is a trivial Kan fibration.", "Thus it is a homotopy equivalence", "(Simplicial, Lemma \\ref{simplicial-lemma-cosk-minus-one-equivalence}).", "Hence applying the functor ``free abelian group on'' to this", "we deduce that", "$$", "\\mathbf{Z}_K(U) \\longrightarrow \\mathbf{Z}_B", "$$", "is a homotopy equivalence. Note that $s(\\mathbf{Z}_B)$ is", "the complex", "$$", "\\ldots \\to", "\\bigoplus\\nolimits_{b \\in B}\\mathbf{Z} \\xrightarrow{0}", "\\bigoplus\\nolimits_{b \\in B}\\mathbf{Z} \\xrightarrow{1}", "\\bigoplus\\nolimits_{b \\in B}\\mathbf{Z} \\xrightarrow{0}", "\\bigoplus\\nolimits_{b \\in B}\\mathbf{Z} \\to 0", "$$", "see Simplicial, Lemma \\ref{simplicial-lemma-homology-eilenberg-maclane}.", "Thus we see that", "$H_i(s(\\mathbf{Z}_K(U))) = 0$ for $i > 0$, and", "$H_0(s(\\mathbf{Z}_K(U))) = \\bigoplus_{b \\in B}\\mathbf{Z}", "= \\bigoplus_{s \\in \\mathcal{G}'(U)} \\mathbf{Z}$.", "These identifications are compatible with restriction", "maps.", "\\medskip\\noindent", "We conclude that $H_i(s(\\mathbf{Z}_K)) = 0$ for $i > 0$ and", "$H_0(s(\\mathbf{Z}_K)) = \\mathbf{Z}_{\\mathcal{G}'}$, where here", "we compute homology groups in $\\textit{PAb}(\\mathcal{C})$. Since", "sheafification is an exact functor we deduce the result", "of the lemma. Namely, the exactness implies", "that $H_0(s(\\mathbf{Z}_K))^\\# = H_0(s(\\mathbf{Z}_K^\\#))$,", "and similarly for other indices." ], "refs": [ "simplicial-lemma-cosk-minus-one-equivalence", "simplicial-lemma-cosk-minus-one-equivalence", "simplicial-lemma-homology-eilenberg-maclane" ], "ref_ids": [ 14903, 14903, 14863 ] } ], "ref_ids": [] }, { "id": 8393, "type": "theorem", "label": "hypercovering-lemma-acyclicity", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-acyclicity", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $f : L \\to K$ be a morphism of", "simplicial objects of $\\textit{PSh}(\\mathcal{C})$.", "Let $n \\geq 0$ be an integer.", "Assume that", "\\begin{enumerate}", "\\item For $i < n$ the morphism $L_i \\to K_i$ is an isomorphism.", "\\item The morphism $L_n \\to K_n$ is surjective after sheafification.", "\\item The canonical map $L \\to \\text{cosk}_n \\text{sk}_n L$ is an isomorphism.", "\\item The canonical map $K \\to \\text{cosk}_n \\text{sk}_n K$ is an isomorphism.", "\\end{enumerate}", "Then $H_i(f) : H_i(L) \\to H_i(K)$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This proof is exactly the same as the proof of", "Lemma \\ref{lemma-compare-cosk0} above. Namely,", "we first let $K_n' \\subset K_n$ be the sub presheaf", "which is the image of the map $L_n \\to K_n$. Assumption", "(2) means that the sheafification of $K_n'$ is equal to", "the sheafification of $K_n$. Moreover, since $L_i = K_i$", "for all $i < n$ we see that get an $n$-truncated", "simplicial presheaf $U$ by taking", "$U_0 = L_0 = K_0, \\ldots, U_{n - 1} = L_{n - 1} = K_{n - 1}, U_n = K'_n$.", "Denote $K' = \\text{cosk}_n U$, a simplicial presheaf.", "Because we can construct $K'_m$ as a finite limit, and", "since sheafification is exact, we see that", "$(K'_m)^\\# = K_m$. In other words, $(K')^\\# = K^\\#$.", "We conclude, by exactness of sheafification once more,", "that $H_i(K) = H_i(K')$. Thus it suffices to prove the lemma", "for the morphism $L \\to K'$, in other words, we may", "assume that $L_n \\to K_n$ is a surjective morphism", "of {\\it presheaves}!", "\\medskip\\noindent", "In this case, for any object $U$ of $\\mathcal{C}$ we", "see that the morphism of simplicial sets", "$$", "L(U) \\longrightarrow K(U)", "$$", "satisfies all the assumptions of", "Simplicial, Lemma \\ref{simplicial-lemma-section}.", "Hence it is a trivial Kan fibration. In particular it is", "a homotopy equivalence", "(Simplicial, Lemma \\ref{simplicial-lemma-trivial-kan-homotopy}).", "Thus", "$$", "\\mathbf{Z}_L(U) \\longrightarrow \\mathbf{Z}_K(U)", "$$", "is a homotopy equivalence too. This for all $U$.", "The result follows." ], "refs": [ "hypercovering-lemma-compare-cosk0", "simplicial-lemma-trivial-kan-homotopy" ], "ref_ids": [ 8392, 14892 ] } ], "ref_ids": [] }, { "id": 8394, "type": "theorem", "label": "hypercovering-lemma-acyclic-hypercover-sheaves", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-acyclic-hypercover-sheaves", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $K$ be a simplicial presheaf.", "Let $\\mathcal{G}$ be a presheaf.", "Let $K \\to \\mathcal{G}$ be an augmentation of $K$", "towards $\\mathcal{G}$. Assume that", "\\begin{enumerate}", "\\item The morphism of presheaves $K_0 \\to \\mathcal{G}$ becomes", "a surjection after sheafification.", "\\item The morphism", "$$", "(d^1_0, d^1_1) :", "K_1", "\\longrightarrow", "K_0 \\times_\\mathcal{G} K_0", "$$", "becomes a surjection after sheafification.", "\\item For every $n \\geq 1$ the morphism", "$$", "K_{n + 1} \\longrightarrow (\\text{cosk}_n \\text{sk}_n K)_{n + 1}", "$$", "turns into a surjection after sheafification.", "\\end{enumerate}", "Then $H_i(K) = 0$ for $i > 0$ and", "$H_0(K) = \\mathbf{Z}_\\mathcal{G}^\\#$." ], "refs": [], "proofs": [ { "contents": [ "Denote $K^n = \\text{cosk}_n \\text{sk}_n K$ for $n \\geq 1$.", "Define $K^0$ as the simplicial object with terms", "$(K^0)_n$ equal to the $(n + 1)$-fold fibred product", "$K_0 \\times_\\mathcal{G} \\ldots \\times_\\mathcal{G} K_0$,", "see Simplicial,", "Example \\ref{simplicial-example-fibre-products-simplicial-object}.", "We have morphisms", "$$", "K \\longrightarrow \\ldots \\to K^n \\to K^{n - 1} \\to \\ldots \\to K^1 \\to K^0.", "$$", "The morphisms $K \\to K^i$, $K^j \\to K^i$ for $j \\geq i \\geq 1$ come", "from the universal properties of the $\\text{cosk}_n$ functors.", "The morphism $K^1 \\to K^0$ is the canonical morphism", "from", "Simplicial, Remark \\ref{simplicial-remark-augmentation}.", "We also recall that $K^0 \\to \\text{cosk}_1 \\text{sk}_1 K^0$", "is an isomorphism, see", "Simplicial, Lemma \\ref{simplicial-lemma-cosk-minus-one}.", "\\medskip\\noindent", "By Lemma \\ref{lemma-compare-cosk0} we see that", "$H_i(K^0) = 0$ for $i > 0$ and $H_0(K^0) = \\mathbf{Z}_\\mathcal{G}^\\#$.", "\\medskip\\noindent", "Pick $n \\geq 1$. Consider the morphism $K^n \\to K^{n - 1}$.", "It is an isomorphism on terms of degree $< n$.", "Note that $K^n \\to \\text{cosk}_n \\text{sk}_n K^n$ and", "$K^{n - 1} \\to \\text{cosk}_n \\text{sk}_n K^{n - 1}$", "are isomorphisms. Note that $(K^n)_n = K_n$ and", "that $(K^{n - 1})_n = (\\text{cosk}_{n - 1} \\text{sk}_{n - 1} K)_n$.", "Hence by assumption, we have that $(K^n)_n \\to (K^{n - 1})_n$", "is a morphism of presheaves which becomes surjective after", "sheafification. By Lemma \\ref{lemma-acyclicity} we conclude that", "$H_i(K^n) = H_i(K^{n - 1})$.", "Combined with the above this proves the lemma." ], "refs": [ "simplicial-remark-augmentation", "simplicial-lemma-cosk-minus-one", "hypercovering-lemma-compare-cosk0", "hypercovering-lemma-acyclicity" ], "ref_ids": [ 14938, 14846, 8392, 8393 ] } ], "ref_ids": [] }, { "id": 8395, "type": "theorem", "label": "hypercovering-lemma-hypercovering-acyclic", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-hypercovering-acyclic", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X$ be an object of $\\mathcal{C}$.", "Let $K$ be a hypercovering of $X$.", "The homology of the simplicial presheaf $F(K)$ is", "$0$ in degrees $> 0$ and equal to $\\mathbf{Z}_X^\\#$", "in degree $0$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-acyclic-hypercover-sheaves}", "and \\ref{lemma-hypercovering-F}." ], "refs": [ "hypercovering-lemma-acyclic-hypercover-sheaves", "hypercovering-lemma-hypercovering-F" ], "ref_ids": [ 8394, 8391 ] } ], "ref_ids": [] }, { "id": 8396, "type": "theorem", "label": "hypercovering-lemma-h0-cech", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-h0-cech", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X$ be an object of $\\mathcal{C}$.", "Let $K$ be a hypercovering of $X$.", "Let $\\mathcal{F}$ be a sheaf of abelian groups on $\\mathcal{C}$.", "Then $\\check{H}^0(K, \\mathcal{F}) = \\mathcal{F}(X)$." ], "refs": [], "proofs": [ { "contents": [ "We have", "$$", "\\check{H}^0(K, \\mathcal{F})", "=", "\\Ker(\\mathcal{F}(K_0) \\longrightarrow \\mathcal{F}(K_1))", "$$", "Write $K_0 = \\{U_i \\to X\\}$. It is a covering in the site", "$\\mathcal{C}$. As well, we have that $K_1 \\to K_0 \\times K_0$", "is a covering in $\\text{SR}(\\mathcal{C}, X)$. Hence we may", "write $K_1 = \\amalg_{i_0, i_1 \\in I} \\{V_{i_0i_1j} \\to X\\}$", "so that the morphism $K_1 \\to K_0 \\times K_0$ is given", "by coverings $\\{V_{i_0i_1j} \\to U_{i_0} \\times_X U_{i_1}\\}$", "of the site $\\mathcal{C}$. Thus we can further identify", "$$", "\\check{H}^0(K, \\mathcal{F})", "=", "\\Ker(", "\\prod\\nolimits_i \\mathcal{F}(U_i)", "\\longrightarrow", "\\prod\\nolimits_{i_0i_1 j} \\mathcal{F}(V_{i_0i_1j})", ")", "$$", "with obvious map. The sheaf property of $\\mathcal{F}$", "implies that $\\check{H}^0(K, \\mathcal{F}) = H^0(X, \\mathcal{F})$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8397, "type": "theorem", "label": "hypercovering-lemma-injective-trivial-cech", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-injective-trivial-cech", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X$ be an object of $\\mathcal{C}$.", "Let $K$ be a hypercovering of $X$.", "Let $\\mathcal{I}$ be an injective sheaf of abelian groups on $\\mathcal{C}$.", "Then", "$$", "\\check{H}^p(K, \\mathcal{I}) =", "\\left\\{", "\\begin{matrix}", "\\mathcal{I}(X) & \\text{if} & p = 0 \\\\", "0 & \\text{if} & p > 0", "\\end{matrix}", "\\right.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Observe that for any object $Z = \\{U_i \\to X\\}$ of", "$\\text{SR}(\\mathcal{C}, X)$ and any abelian sheaf", "$\\mathcal{F}$ on $\\mathcal{C}$ we have", "\\begin{eqnarray*}", "\\mathcal{F}(Z)", "& = &", "\\prod \\mathcal{F}(U_i) \\\\", "& = &", "\\prod \\Mor_{\\textit{PSh}(\\mathcal{C})}(h_{U_i}, \\mathcal{F})\\\\", "& = &", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(F(Z), \\mathcal{F})\\\\", "& = &", "\\Mor_{\\textit{PAb}(\\mathcal{C})}(\\mathbf{Z}_{F(Z)}, \\mathcal{F}) \\\\", "& = &", "\\Mor_{\\textit{Ab}(\\mathcal{C})}(\\mathbf{Z}_{F(Z)}^\\#, \\mathcal{F})", "\\end{eqnarray*}", "Thus we see, for any simplicial object $K$ of", "$\\text{SR}(\\mathcal{C}, X)$ that we have", "\\begin{equation}", "\\label{equation-identify-cech}", "s(\\mathcal{F}(K))", "=", "\\Hom_{\\textit{Ab}(\\mathcal{C})}(s(\\mathbf{Z}_{F(K)}^\\#), \\mathcal{F})", "\\end{equation}", "see Definition \\ref{definition-homology} for notation.", "The complex of sheaves $s(\\mathbf{Z}_{F(K)}^\\#)$ is quasi-isomorphic", "to $\\mathbf{Z}_X^\\#$ if $K$ is a hypercovering, see", "Lemma \\ref{lemma-hypercovering-acyclic}. We conclude", "that if $\\mathcal{I}$ is an injective abelian sheaf, and", "$K$ a hypercovering, then the complex $s(\\mathcal{I}(K))$", "is acyclic except possibly in degree $0$.", "In other words, we have", "$$", "\\check{H}^i(K, \\mathcal{I}) = 0", "$$", "for $i > 0$. Combined with Lemma \\ref{lemma-h0-cech} the lemma is proved." ], "refs": [ "hypercovering-definition-homology", "hypercovering-lemma-hypercovering-acyclic", "hypercovering-lemma-h0-cech" ], "ref_ids": [ 8425, 8395, 8396 ] } ], "ref_ids": [] }, { "id": 8398, "type": "theorem", "label": "hypercovering-lemma-cech-spectral-sequence", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-cech-spectral-sequence", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X$ be an object of $\\mathcal{C}$.", "Let $K$ be a hypercovering of $X$.", "Let $\\mathcal{F}$ be a sheaf of abelian groups on $\\mathcal{C}$.", "There is a map", "$$", "s(\\mathcal{F}(K))", "\\longrightarrow", "R\\Gamma(X, \\mathcal{F})", "$$", "in $D^{+}(\\textit{Ab})$ functorial in $\\mathcal{F}$, which induces", "natural transformations", "$$", "\\check{H}^i(K, -) \\longrightarrow H^i(X, -)", "$$", "as functors $\\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}$. Moreover,", "there is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with", "$$", "E_2^{p, q} = \\check{H}^p(K, \\underline{H}^q(\\mathcal{F}))", "$$", "converging to $H^{p + q}(X, \\mathcal{F})$.", "This spectral sequence is functorial in $\\mathcal{F}$ and", "in the hypercovering $K$." ], "refs": [], "proofs": [ { "contents": [ "We could prove this by the same method as employed in the corresponding", "lemma in the chapter on cohomology. Instead let us prove this by a", "double complex argument.", "\\medskip\\noindent", "Choose an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$", "in the category of abelian sheaves on $\\mathcal{C}$. Consider the", "double complex $A^{\\bullet, \\bullet}$ with terms", "$$", "A^{p, q} = \\mathcal{I}^q(K_p)", "$$", "where the differential $d_1^{p, q} : A^{p, q} \\to A^{p + 1, q}$ is the one", "coming from the differential on the complex $s(\\mathcal{I}^q(K))$", "associated to the cosimplicial abelian group $\\mathcal{I}^p(K)$", "and the differential $d_2^{p, q} : A^{p, q} \\to A^{p, q + 1}$ is the one", "coming from the differential $\\mathcal{I}^q \\to \\mathcal{I}^{q + 1}$.", "Denote $\\text{Tot}(A^{\\bullet, \\bullet})$ the total complex associated to", "the double complex $A^{\\bullet, \\bullet}$, see", "Homology, Section \\ref{homology-section-double-complexes}.", "We will use the two spectral", "sequences $({}'E_r, {}'d_r)$ and $({}''E_r, {}''d_r)$", "associated to this double complex, see", "Homology, Section \\ref{homology-section-double-complex}.", "\\medskip\\noindent", "By Lemma \\ref{lemma-injective-trivial-cech}", "the complexes $s(\\mathcal{I}^q(K))$ are acyclic in", "positive degrees and have $H^0$ equal to $\\mathcal{I}^q(X)$.", "Hence by", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}", "the natural map", "$$", "\\mathcal{I}^\\bullet(X) \\longrightarrow \\text{Tot}(A^{\\bullet, \\bullet})", "$$", "is a quasi-isomorphism of complexes of abelian groups. In particular", "we conclude that $H^n(\\text{Tot}(A^{\\bullet, \\bullet})) = H^n(X, \\mathcal{F})$.", "\\medskip\\noindent", "The map $s(\\mathcal{F}(K)) \\longrightarrow R\\Gamma(X, \\mathcal{F})$ of", "the lemma is the composition of the map", "$s(\\mathcal{F}(K)) \\to \\text{Tot}(A^{\\bullet, \\bullet})$", "followed by the inverse", "of the displayed quasi-isomorphism above. This works because", "$\\mathcal{I}^\\bullet(X)$ is a representative of $R\\Gamma(X, \\mathcal{F})$.", "\\medskip\\noindent", "Consider the spectral sequence $({}'E_r, {}'d_r)_{r \\geq 0}$. By", "Homology, Lemma \\ref{homology-lemma-ss-double-complex}", "we see that", "$$", "{}'E_2^{p, q} = H^p_I(H^q_{II}(A^{\\bullet, \\bullet}))", "$$", "In other words, we first take cohomology with respect to", "$d_2$ which gives the groups", "${}'E_1^{p, q} = \\underline{H}^q(\\mathcal{F})(K_p)$.", "Hence it is indeed the case (by the description of the differential", "${}'d_1$) that", "${}'E_2^{p, q} = \\check{H}^p(K, \\underline{H}^q(\\mathcal{F}))$.", "By the above and Homology, Lemma \\ref{homology-lemma-first-quadrant-ss}", "we see that this converges to $H^n(X, \\mathcal{F})$ as desired.", "\\medskip\\noindent", "We omit the proof of the statements regarding the functoriality of", "the above constructions in the abelian sheaf $\\mathcal{F}$ and the", "hypercovering $K$." ], "refs": [ "hypercovering-lemma-injective-trivial-cech", "homology-lemma-double-complex-gives-resolution", "homology-lemma-ss-double-complex", "homology-lemma-first-quadrant-ss" ], "ref_ids": [ 8397, 12106, 12104, 12105 ] } ], "ref_ids": [] }, { "id": 8399, "type": "theorem", "label": "hypercovering-lemma-h0-cech-variant", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-h0-cech-variant", "contents": [ "Let $\\mathcal{C}$ be a site with equalizers and fibre products.", "Let $\\mathcal{G}$ be a presheaf on $\\mathcal{C}$.", "Let $K$ be a hypercovering of $\\mathcal{G}$.", "Let $\\mathcal{F}$ be a sheaf of abelian groups on $\\mathcal{C}$.", "Then $\\check{H}^0(K, \\mathcal{F}) = H^0(\\mathcal{G}, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the definition of $H^0(\\mathcal{G}, \\mathcal{F})$", "and the fact that", "$$", "\\xymatrix{", "F(K_1) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "F(K_0) \\ar[r] & \\mathcal{G}", "}", "$$", "becomes an coequalizer diagram after sheafification." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8400, "type": "theorem", "label": "hypercovering-lemma-injective-trivial-cech-variant", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-injective-trivial-cech-variant", "contents": [ "Let $\\mathcal{C}$ be a site with equalizers and fibre products.", "Let $\\mathcal{G}$ be a presheaf on $\\mathcal{C}$.", "Let $K$ be a hypercovering of $\\mathcal{G}$.", "Let $\\mathcal{I}$ be an injective sheaf of abelian groups on $\\mathcal{C}$.", "Then", "$$", "\\check{H}^p(K, \\mathcal{I}) =", "\\left\\{", "\\begin{matrix}", "H^0(\\mathcal{G}, \\mathcal{I}) & \\text{if} & p = 0 \\\\", "0 & \\text{if} & p > 0", "\\end{matrix}", "\\right.", "$$" ], "refs": [], "proofs": [ { "contents": [ "By (\\ref{equation-identify-cech}) we have", "$$", "s(\\mathcal{F}(K))", "=", "\\Hom_{\\textit{Ab}(\\mathcal{C})}(s(\\mathbf{Z}_{F(K)}^\\#), \\mathcal{F})", "$$", "The complex $s(\\mathbf{Z}_{F(K)}^\\#)$ is quasi-isomorphic", "to $\\mathbf{Z}_\\mathcal{G}^\\#$, see", "Lemma \\ref{lemma-acyclic-hypercover-sheaves}. We conclude", "that if $\\mathcal{I}$ is an injective abelian sheaf, then", "the complex $s(\\mathcal{I}(K))$ is acyclic except possibly in degree $0$.", "In other words, we have $\\check{H}^i(K, \\mathcal{I}) = 0$", "for $i > 0$. Combined with Lemma \\ref{lemma-h0-cech-variant}", "the lemma is proved." ], "refs": [ "hypercovering-lemma-acyclic-hypercover-sheaves", "hypercovering-lemma-h0-cech-variant" ], "ref_ids": [ 8394, 8399 ] } ], "ref_ids": [] }, { "id": 8401, "type": "theorem", "label": "hypercovering-lemma-cech-spectral-sequence-variant", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-cech-spectral-sequence-variant", "contents": [ "Let $\\mathcal{C}$ be a site with equalizers and fibre products.", "Let $\\mathcal{G}$ be a presheaf on $\\mathcal{C}$.", "Let $K$ be a hypercovering of $\\mathcal{G}$.", "Let $\\mathcal{F}$ be a sheaf of abelian groups on $\\mathcal{C}$.", "There is a map", "$$", "s(\\mathcal{F}(K)) \\longrightarrow R\\Gamma(\\mathcal{G}, \\mathcal{F})", "$$", "in $D^{+}(\\textit{Ab})$ functorial in $\\mathcal{F}$, which induces", "a natural transformation", "$$", "\\check{H}^i(K, -) \\longrightarrow H^i(\\mathcal{G}, -)", "$$", "of functors $\\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}$. Moreover,", "there is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with", "$$", "E_2^{p, q} = \\check{H}^p(K, \\underline{H}^q(\\mathcal{F}))", "$$", "converging to $H^{p + q}(\\mathcal{G}, \\mathcal{F})$.", "This spectral sequence is functorial in $\\mathcal{F}$ and", "in the hypercovering $K$." ], "refs": [], "proofs": [ { "contents": [ "Choose an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$", "in the category of abelian sheaves on $\\mathcal{C}$. Consider the", "double complex $A^{\\bullet, \\bullet}$ with terms", "$$", "A^{p, q} = \\mathcal{I}^q(K_p)", "$$", "where the differential $d_1^{p, q} : A^{p, q} \\to A^{p + 1, q}$", "is the one coming from the differential $\\mathcal{I}^p \\to \\mathcal{I}^{p + 1}$", "and the differential $d_2^{p, q} : A^{p, q} \\to A^{p, q + 1}$ is the", "one coming from the differential on the complex", "$s(\\mathcal{I}^p(K))$ associated to the cosimplicial abelian group", "$\\mathcal{I}^p(K)$ as explained above.", "We will use the two spectral", "sequences $({}'E_r, {}'d_r)$ and $({}''E_r, {}''d_r)$", "associated to this double complex, see", "Homology, Section \\ref{homology-section-double-complex}.", "\\medskip\\noindent", "By Lemma \\ref{lemma-injective-trivial-cech-variant} the complexes", "$s(\\mathcal{I}^p(K))$ are acyclic in positive degrees and have", "$H^0$ equal to $H^0(\\mathcal{G}, \\mathcal{I}^p)$. Hence by", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}", "and its proof the spectral sequence $({}'E_r, {}'d_r)$ degenerates,", "and the natural map", "$$", "H^0(\\mathcal{G}, \\mathcal{I}^\\bullet) \\longrightarrow", "\\text{Tot}(A^{\\bullet, \\bullet})", "$$", "is a quasi-isomorphism of complexes of abelian groups. The map", "$s(\\mathcal{F}(K)) \\longrightarrow R\\Gamma(\\mathcal{G}, \\mathcal{F})$", "of the lemma is the composition of the natural map", "$s(\\mathcal{F}(K)) \\to \\text{Tot}(A^{\\bullet, \\bullet})$", "followed by the inverse of the displayed quasi-isomorphism above.", "This works because $H^0(\\mathcal{G}, \\mathcal{I}^\\bullet)$", "is a representative of $R\\Gamma(\\mathcal{G}, \\mathcal{F})$.", "\\medskip\\noindent", "Consider the spectral sequence $({}''E_r, {}''d_r)_{r \\geq 0}$. By", "Homology, Lemma \\ref{homology-lemma-ss-double-complex}", "we see that", "$$", "{}''E_2^{p, q} = H^p_{II}(H^q_I(A^{\\bullet, \\bullet}))", "$$", "In other words, we first take cohomology with respect to", "$d_1$ which gives the groups", "${}''E_1^{p, q} = \\underline{H}^p(\\mathcal{F})(K_q)$.", "Hence it is indeed the case (by the description of the differential", "${}''d_1$) that", "${}''E_2^{p, q} = \\check{H}^p(K, \\underline{H}^q(\\mathcal{F}))$.", "Since this spectral sequence converges to the cohomology of", "$\\text{Tot}(A^{\\bullet, \\bullet})$ the proof is finished." ], "refs": [ "hypercovering-lemma-injective-trivial-cech-variant", "homology-lemma-double-complex-gives-resolution", "homology-lemma-ss-double-complex" ], "ref_ids": [ 8400, 12106, 12104 ] } ], "ref_ids": [] }, { "id": 8402, "type": "theorem", "label": "hypercovering-lemma-cech-spectral-sequence-verdier", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-cech-spectral-sequence-verdier", "contents": [ "Let $\\mathcal{C}$ be a site with equalizers and fibre products.", "Let $K$ be a hypercovering.", "Let $\\mathcal{F}$ be an abelian sheaf. There is a", "spectral sequence $(E_r, d_r)_{r \\geq 0}$ with", "$$", "E_2^{p, q} = \\check{H}^p(K, \\underline{H}^q(\\mathcal{F}))", "$$", "converging to the global cohomology groups $H^{p + q}(\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-cech-spectral-sequence-variant}." ], "refs": [ "hypercovering-lemma-cech-spectral-sequence-variant" ], "ref_ids": [ 8401 ] } ], "ref_ids": [] }, { "id": 8403, "type": "theorem", "label": "hypercovering-lemma-funny-gamma", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-funny-gamma", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X$ be an object of $\\mathcal{C}$.", "Let $K, L, M$ be simplicial objects of $\\text{SR}(\\mathcal{C}, X)$.", "Let $a : K \\to L$, $b : M \\to L$ be morphisms.", "Assume", "\\begin{enumerate}", "\\item $K$ is a hypercovering of $X$,", "\\item the morphism $M_0 \\to L_0$ is a covering, and", "\\item for all $n \\geq 0$ in the diagram", "$$", "\\xymatrix{", "M_{n + 1} \\ar[dd] \\ar[rr] \\ar[rd]^\\gamma &", "&", "(\\text{cosk}_n \\text{sk}_n M)_{n + 1} \\ar[dd] \\\\", "&", "L_{n + 1}", "\\times_{(\\text{cosk}_n \\text{sk}_n L)_{n + 1}}", "(\\text{cosk}_n \\text{sk}_n M)_{n + 1}", "\\ar[ld] \\ar[ru]", "& \\\\", "L_{n + 1} \\ar[rr] & & (\\text{cosk}_n \\text{sk}_n L)_{n + 1}", "}", "$$", "the arrow $\\gamma$ is a covering.", "\\end{enumerate}", "Then the fibre product $K \\times_L M$ is a hypercovering of $X$." ], "refs": [], "proofs": [ { "contents": [ "The morphism $(K \\times_L M)_0 = K_0 \\times_{L_0} M_0 \\to K_0$", "is a base change of a covering by (2), hence a covering, see", "Lemma \\ref{lemma-covering-permanence}. And $K_0 \\to \\{X \\to X\\}$", "is a covering by (1). Thus $(K \\times_L M)_0 \\to \\{X \\to X\\}$", "is a covering by Lemma \\ref{lemma-covering-permanence}. Hence", "$K \\times_L M$ satisfies the first condition of Definition", "\\ref{definition-hypercovering}.", "\\medskip\\noindent", "We still have to check that", "$$", "K_{n + 1} \\times_{L_{n + 1}} M_{n + 1} = (K \\times_L M)_{n + 1}", "\\longrightarrow", "(\\text{cosk}_n \\text{sk}_n (K \\times_L M))_{n + 1}", "$$", "is a covering for all $n \\geq 0$. We abbreviate as follows:", "$A = (\\text{cosk}_n \\text{sk}_n K)_{n + 1}$,", "$B = (\\text{cosk}_n \\text{sk}_n L)_{n + 1}$, and", "$C = (\\text{cosk}_n \\text{sk}_n M)_{n + 1}$.", "The functor $\\text{cosk}_n \\text{sk}_n$ commutes with fibre products,", "see Simplicial, Lemma \\ref{simplicial-lemma-cosk-fibre-product}.", "Thus the right hand side above is equal to $A \\times_B C$.", "Consider the following commutative diagram", "$$", "\\xymatrix{", "K_{n + 1} \\times_{L_{n + 1}} M_{n + 1} \\ar[r] \\ar[d] &", "M_{n + 1} \\ar[d] \\ar[rd]_\\gamma \\ar[rrd] &", "& \\\\", "K_{n + 1} \\ar[r] \\ar[rd] &", "L_{n + 1} \\ar[rrd] &", "L_{n + 1} \\times_B C \\ar[l] \\ar[r] &", "C \\ar[d] \\\\", "&", "A \\ar[rr] &", "&", "B", "}", "$$", "This diagram shows that", "$$", "K_{n + 1} \\times_{L_{n + 1}} M_{n + 1}", "=", "(K_{n + 1} \\times_B C)", "\\times_{(L_{n + 1} \\times_B C), \\gamma}", "M_{n + 1}", "$$", "Now, $K_{n + 1} \\times_B C \\to A \\times_B C$", "is a base change of the covering $K_{n + 1} \\to A$", "via the morphism $A \\times_B C \\to A$, hence is a", "covering. By assumption (3) the morphism $\\gamma$ is a covering.", "Hence the morphism", "$$", "(K_{n + 1} \\times_B C)", "\\times_{(L_{n + 1} \\times_B C), \\gamma}", "M_{n + 1}", "\\longrightarrow", "K_{n + 1} \\times_B C", "$$", "is a covering as a base change of a covering.", "The lemma follows as a composition of coverings", "is a covering." ], "refs": [ "hypercovering-lemma-covering-permanence", "hypercovering-lemma-covering-permanence", "hypercovering-definition-hypercovering", "simplicial-lemma-cosk-fibre-product" ], "ref_ids": [ 8389, 8389, 8424, 14842 ] } ], "ref_ids": [] }, { "id": 8404, "type": "theorem", "label": "hypercovering-lemma-product-hypercoverings", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-product-hypercoverings", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X$ be an object of $\\mathcal{C}$.", "If $K, L$ are hypercoverings of $X$, then", "$K \\times L$ is a hypercovering of $X$." ], "refs": [], "proofs": [ { "contents": [ "You can either verify this directly, or use", "Lemma \\ref{lemma-funny-gamma} above and check that $L \\to \\{X \\to X\\}$", "has property (3)." ], "refs": [ "hypercovering-lemma-funny-gamma" ], "ref_ids": [ 8403 ] } ], "ref_ids": [] }, { "id": 8405, "type": "theorem", "label": "hypercovering-lemma-covering", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-covering", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X$ be an object of $\\mathcal{C}$.", "Let $K$ be a hypercovering of $X$.", "Let $k \\geq 0$ be an integer.", "Let $u : Z \\to K_k$ be a covering", "in $\\text{SR}(\\mathcal{C}, X)$.", "Then there exists a morphism of hypercoverings", "$f: L \\to K$ such that $L_k \\to K_k$", "factors through $u$." ], "refs": [], "proofs": [ { "contents": [ "Denote $Y = K_k$. Let $C[k]$ be the cosimplicial set defined in", "Simplicial, Example \\ref{simplicial-example-simplex-cosimplicial-set}.", "We will use the description of $\\Hom(C[k], Y)$ and $\\Hom(C[k], Z)$", "given in", "Simplicial, Lemma \\ref{simplicial-lemma-morphism-into-product}.", "There is a canonical morphism", "$K \\to \\Hom(C[k], Y)$ corresponding to $\\text{id} : K_k = Y \\to Y$.", "Consider the morphism $\\Hom(C[k], Z) \\to \\Hom(C[k], Y)$", "which on degree $n$ terms is the morphism", "$$", "\\prod\\nolimits_{\\alpha : [k] \\to [n]} Z", "\\longrightarrow", "\\prod\\nolimits_{\\alpha : [k] \\to [n]} Y", "$$", "using the given morphism $Z \\to Y$ on each factor. Set", "$$", "L = K \\times_{\\Hom(C[k], Y)} \\Hom(C[k], Z).", "$$", "The morphism $L_k \\to K_k$ sits in to a commutative diagram", "$$", "\\xymatrix{", "L_k \\ar[r] \\ar[d] &", "\\prod_{\\alpha : [k] \\to [k]} Z \\ar[r]^-{\\text{pr}_{\\text{id}_{[k]}}} \\ar[d] &", "Z \\ar[d] \\\\", "K_k \\ar[r] &", "\\prod_{\\alpha : [k] \\to [k]} Y \\ar[r]^-{\\text{pr}_{\\text{id}_{[k]}}} &", "Y", "}", "$$", "Since the composition of the two bottom arrows is the identity", "we conclude that we have the desired factorization.", "\\medskip\\noindent", "We still have to show that $L$ is a hypercovering of $X$.", "To see this we will use Lemma \\ref{lemma-funny-gamma}.", "Condition (1) is satisfied by assumption.", "For (2), the morphism", "$$", "\\Hom(C[k], Z)_0 \\to \\Hom(C[k], Y)_0", "$$", "is a covering because it is isomorphic to $Z \\to Y$ as", "there is only one morphism $[k] \\to [0]$.", "\\medskip\\noindent", "Let us consider condition (3) for $n = 0$. Then, since", "$(\\text{cosk}_0 T)_1 = T \\times T$", "(Simplicial, Example \\ref{simplicial-example-cosk0})", "and since $\\Hom(C[k], Z)_1 = \\prod_{\\alpha : [k] \\to [1]} Z$", "we obtain the diagram", "$$", "\\xymatrix{", "\\prod\\nolimits_{\\alpha : [k] \\to [1]} Z \\ar[r] \\ar[d] &", "Z \\times Z \\ar[d] \\\\", "\\prod\\nolimits_{\\alpha : [k] \\to [1]} Y \\ar[r] &", "Y \\times Y", "}", "$$", "with horizontal arrows corresponding to the projection onto the factors", "corresponding to the two nonsurjective $\\alpha$. Thus the arrow $\\gamma$", "is the morphism", "$$", "\\prod\\nolimits_{\\alpha : [k] \\to [1]} Z", "\\longrightarrow", "\\prod\\nolimits_{\\alpha : [k] \\to [1]\\text{ not onto}} Z", "\\times", "\\prod\\nolimits_{\\alpha : [k] \\to [1]\\text{ onto}} Y", "$$", "which is a product of coverings and hence a covering by", "Lemma \\ref{lemma-covering-permanence}.", "\\medskip\\noindent", "Let us consider condition (3) for $n > 0$. We claim there is an", "injective map $\\tau : S' \\to S$ of finite sets, such that for any", "object $T$ of $\\text{SR}(\\mathcal{C}, X)$ the morphism", "\\begin{equation}", "\\label{equation-map}", "\\Hom(C[k], T)_{n + 1} \\to", "(\\text{cosk}_n \\text{sk}_n \\Hom(C[k], T))_{n + 1}", "\\end{equation}", "is isomorphic to the projection $\\prod_{s \\in S} T \\to \\prod_{s' \\in S'} T$", "functorially in $T$. If this is true, then we see, arguing as in the previous", "paragraph, that the arrow $\\gamma$ is the morphism", "$$", "\\prod\\nolimits_{s \\in S} Z", "\\longrightarrow", "\\prod\\nolimits_{s \\in S'} Z", "\\times", "\\prod\\nolimits_{s \\not\\in \\tau(S')} Y", "$$", "which is a product of coverings and hence a covering by", "Lemma \\ref{lemma-covering-permanence}. By construction, we have", "$\\Hom(C[k], T)_{n + 1} = \\prod_{\\alpha : [k] \\to [n + 1]} T$", "(see Simplicial, Lemma \\ref{simplicial-lemma-morphism-into-product}).", "Correspondingly we take $S = \\text{Map}([k], [n + 1])$.", "On the other hand, Simplicial, Lemma \\ref{simplicial-lemma-formula-limit},", "provides a description of points of", "$(\\text{cosk}_n \\text{sk}_n \\Hom(C[k], T))_{n + 1}$", "as sequences $(f_0, \\ldots, f_{n + 1})$ of points of $\\Hom(C[k], T)_n$", "satisfying $d^n_{j - 1} f_i = d^n_i f_j$ for $0 \\leq i < j \\leq n + 1$.", "We can write $f_i = (f_{i, \\alpha})$ with $f_{i, \\alpha}$ a point of $T$", "and $\\alpha \\in \\text{Map}([k], [n])$. The conditions translate into", "$$", "f_{i, \\delta^n_{j - 1} \\circ \\beta} = f_{j, \\delta_i^n \\circ \\beta}", "$$", "for any $0 \\leq i < j \\leq n + 1$ and $\\beta : [k] \\to [n - 1]$. Thus we", "see that", "$$", "S' = \\{0, \\ldots, n + 1\\} \\times \\text{Map}([k], [n]) / \\sim", "$$", "where the equivalence relation is generated by the equivalences", "$$", "(i, \\delta^n_{j - 1} \\circ \\beta) \\sim (j, \\delta_i^n \\circ \\beta)", "$$", "for $0 \\leq i < j \\leq n + 1$ and $\\beta : [k] \\to [n - 1]$.", "A computation (omitted) shows that the morphism (\\ref{equation-map})", "corresponds to the map $S' \\to S$ which sends $(i, \\alpha)$ to", "$\\delta^{n + 1}_i \\circ \\alpha \\in S$. (It may be a comfort to the", "reader to see that this map is well defined by part (1) of", "Simplicial, Lemma \\ref{simplicial-lemma-relations-face-degeneracy}.)", "To finish the proof it suffices to show that if", "$\\alpha, \\alpha' : [k] \\to [n]$ and $0 \\leq i < j \\leq n + 1$", "are such that", "$$", "\\delta^{n + 1}_i \\circ \\alpha = \\delta^{n + 1}_j \\circ \\alpha'", "$$", "then we have $\\alpha = \\delta^n_{j - 1} \\circ \\beta$", "and $\\alpha' = \\delta_i^n \\circ \\beta$ for some $\\beta : [k] \\to [n - 1]$.", "This is easy to see and omitted." ], "refs": [ "simplicial-lemma-morphism-into-product", "hypercovering-lemma-funny-gamma", "hypercovering-lemma-covering-permanence", "hypercovering-lemma-covering-permanence", "simplicial-lemma-morphism-into-product", "simplicial-lemma-formula-limit", "simplicial-lemma-relations-face-degeneracy" ], "ref_ids": [ 14822, 8403, 8389, 8389, 14822, 14838, 14806 ] } ], "ref_ids": [] }, { "id": 8406, "type": "theorem", "label": "hypercovering-lemma-covering-sheaf", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-covering-sheaf", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X$ be an object of $\\mathcal{C}$.", "Let $K$ be a hypercovering of $X$.", "Let $n \\geq 0$ be an integer.", "Let $u : \\mathcal{F} \\to F(K_n)$ be a morphism", "of presheaves which becomes surjective", "on sheafification.", "Then there exists a morphism of hypercoverings", "$f: L \\to K$ such that $F(f_n) : F(L_n) \\to F(K_n)$", "factors through $u$." ], "refs": [], "proofs": [ { "contents": [ "Write $K_n = \\{U_i \\to X\\}_{i \\in I}$.", "Thus the map $u$ is a morphism of presheaves of sets", "$u : \\mathcal{F} \\to \\amalg h_{u_i}$.", "The assumption on $u$ means that for every", "$i \\in I$ there exists a covering $\\{U_{ij} \\to U_i\\}_{j \\in I_i}$", "of the site $\\mathcal{C}$ and a morphism of presheaves", "$t_{ij} : h_{U_{ij}} \\to \\mathcal{F}$ such that", "$u \\circ t_{ij}$ is the map $h_{U_{ij}} \\to h_{U_i}$", "coming from the morphism $U_{ij} \\to U_i$.", "Set $J = \\amalg_{i \\in I} I_i$, and let", "$\\alpha : J \\to I$ be the obvious map.", "For $j \\in J$ denote $V_j = U_{\\alpha(j)j}$. Set", "$Z = \\{V_j \\to X\\}_{j \\in J}$.", "Finally, consider the morphism", "$u' : Z \\to K_n$ given by $\\alpha : J \\to I$", "and the morphisms $V_j = U_{\\alpha(j)j} \\to U_{\\alpha(j)}$", "above. Clearly, this is a covering in the", "category $\\text{SR}(\\mathcal{C}, X)$, and by", "construction $F(u') : F(Z) \\to F(K_n)$ factors through $u$.", "Thus the result follows from Lemma \\ref{lemma-covering} above." ], "refs": [ "hypercovering-lemma-covering" ], "ref_ids": [ 8405 ] } ], "ref_ids": [] }, { "id": 8407, "type": "theorem", "label": "hypercovering-lemma-one-more-simplex", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-one-more-simplex", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X$ be an object of $\\mathcal{C}$.", "Let $K$ be a hypercovering of $X$.", "Let $U \\subset V$ be simplicial sets, with $U_n, V_n$", "finite nonempty for all $n$.", "Assume that $U$ has finitely many nondegenerate simplices.", "Suppose $n \\geq 0$ and $x \\in V_n$,", "$x \\not \\in U_n$ are such that", "\\begin{enumerate}", "\\item $V_i = U_i$ for $i < n$,", "\\item $V_n = U_n \\cup \\{x\\}$,", "\\item any $z \\in V_j$, $z \\not \\in U_j$ for $j > n$", "is degenerate.", "\\end{enumerate}", "Then the morphism", "$$", "\\Hom(V, K)_0", "\\longrightarrow", "\\Hom(U, K)_0", "$$", "of $\\text{SR}(\\mathcal{C}, X)$ is a covering." ], "refs": [], "proofs": [ { "contents": [ "If $n = 0$, then it follows easily that $V = U \\amalg \\Delta[0]$", "(see below). In this case $\\Hom(V, K)_0 =", "\\Hom(U, K)_0 \\times K_0$. The result, in this case, then follows", "from Lemma \\ref{lemma-covering-permanence}.", "\\medskip\\noindent", "Let $a : \\Delta[n] \\to V$ be the morphism associated to $x$", "as in Simplicial, Lemma \\ref{simplicial-lemma-simplex-map}.", "Let us write $\\partial \\Delta[n] = i_{(n-1)!} \\text{sk}_{n - 1} \\Delta[n]$", "for the $(n - 1)$-skeleton of $\\Delta[n]$.", "Let $b : \\partial \\Delta[n] \\to U$ be the restriction", "of $a$ to the $(n - 1)$ skeleton of $\\Delta[n]$. By", "Simplicial, Lemma \\ref{simplicial-lemma-glue-simplex}", "we have $V = U \\amalg_{\\partial \\Delta[n]} \\Delta[n]$. By", "Simplicial, Lemma", "\\ref{simplicial-lemma-hom-from-coprod}", "we get that", "$$", "\\xymatrix{", "\\Hom(V, K)_0 \\ar[r] \\ar[d] &", "\\Hom(U, K)_0 \\ar[d] \\\\", "\\Hom(\\Delta[n], K)_0 \\ar[r] &", "\\Hom(\\partial \\Delta[n], K)_0", "}", "$$", "is a fibre product square. Thus it suffices to show that", "the bottom horizontal arrow is a covering. By", "Simplicial, Lemma \\ref{simplicial-lemma-cosk-shriek}", "this arrow is identified with", "$$", "K_n \\to (\\text{cosk}_{n - 1} \\text{sk}_{n - 1} K)_n", "$$", "and hence is a covering by definition of a hypercovering." ], "refs": [ "hypercovering-lemma-covering-permanence", "simplicial-lemma-simplex-map", "simplicial-lemma-glue-simplex", "simplicial-lemma-hom-from-coprod", "simplicial-lemma-cosk-shriek" ], "ref_ids": [ 8389, 14817, 14852, 14826, 14856 ] } ], "ref_ids": [] }, { "id": 8408, "type": "theorem", "label": "hypercovering-lemma-add-simplices", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-add-simplices", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X$ be an object of $\\mathcal{C}$.", "Let $K$ be a hypercovering of $X$.", "Let $U \\subset V$ be simplicial sets, with $U_n, V_n$", "finite nonempty for all $n$.", "Assume that $U$ and $V$ have finitely many nondegenerate simplices.", "Then the morphism", "$$", "\\Hom(V, K)_0", "\\longrightarrow", "\\Hom(U, K)_0", "$$", "of $\\text{SR}(\\mathcal{C}, X)$ is a covering." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-one-more-simplex}", "above, it suffices to prove a simple lemma", "about inclusions of simplicial sets $U \\subset V$ as in the", "lemma. And this is exactly the result of", "Simplicial, Lemma \\ref{simplicial-lemma-add-simplices}." ], "refs": [ "hypercovering-lemma-one-more-simplex", "simplicial-lemma-add-simplices" ], "ref_ids": [ 8407, 14853 ] } ], "ref_ids": [] }, { "id": 8409, "type": "theorem", "label": "hypercovering-lemma-degeneracy-maps-coverings", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-degeneracy-maps-coverings", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of", "$\\mathcal{C}$. Let $K$ be a hypercovering of $X$. Then", "\\begin{enumerate}", "\\item $K_n$ is a covering of $X$ for each $n \\geq 0$,", "\\item $d^n_i : K_n \\to K_{n - 1}$ is a covering for all $n \\geq 1$", "and $0 \\leq i \\leq n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that $K_0$ is a covering of $X$ by", "Definition \\ref{definition-hypercovering}", "and that this is equivalent to saying that", "$K_0 \\to \\{X \\to X\\}$ is a covering in the sense", "of Definition \\ref{definition-covering-SR}.", "Hence (1) follows from (2) because it will prove that", "the composition", "$K_n \\to K_{n - 1} \\to \\ldots \\to K_0 \\to \\{X \\to X\\}$", "is a covering by Lemma \\ref{lemma-covering-permanence}.", "\\medskip\\noindent", "Proof of (2). Observe that", "$\\Mor(\\Delta[n], K)_0 = K_n$ by", "Simplicial, Lemma \\ref{simplicial-lemma-exists-hom-from-simplicial-set-finite}.", "Therefore (2) follows from Lemma \\ref{lemma-add-simplices}", "applied to the $n + 1$ different inclusions $\\Delta[n - 1] \\to \\Delta[n]$." ], "refs": [ "hypercovering-definition-hypercovering", "hypercovering-definition-covering-SR", "hypercovering-lemma-covering-permanence", "simplicial-lemma-exists-hom-from-simplicial-set-finite", "hypercovering-lemma-add-simplices" ], "ref_ids": [ 8424, 8423, 8389, 14825, 8408 ] } ], "ref_ids": [] }, { "id": 8410, "type": "theorem", "label": "hypercovering-lemma-hom-hypercovering", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-hom-hypercovering", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X$ be an object of $\\mathcal{C}$.", "Let $L$ be a simplicial object of $\\text{SR}(\\mathcal{C}, X)$.", "Let $n \\geq 0$. Consider the commutative diagram", "\\begin{equation}", "\\label{equation-diagram}", "\\xymatrix{", "\\Hom(\\Delta[1], L)_{n + 1} \\ar[r] \\ar[d] &", "(\\text{cosk}_n \\text{sk}_n \\Hom(\\Delta[1], L))_{n + 1} \\ar[d] \\\\", "(L \\times L)_{n + 1} \\ar[r] &", "(\\text{cosk}_n \\text{sk}_n (L \\times L))_{n + 1}", "}", "\\end{equation}", "coming from the morphism defined above.", "We can identify the terms in this diagram as follows,", "where", "$\\partial \\Delta[n + 1] = i_{n!}\\text{sk}_n \\Delta[n + 1]$", "is the $n$-skeleton of the $(n + 1)$-simplex:", "\\begin{eqnarray*}", "\\Hom(\\Delta[1], L)_{n + 1}", "& = &", "\\Hom(\\Delta[1] \\times \\Delta[n + 1], L)_0 \\\\", "(\\text{cosk}_n \\text{sk}_n \\Hom(\\Delta[1], L))_{n + 1}", "& = &", "\\Hom(\\Delta[1] \\times \\partial \\Delta[n + 1], L)_0 \\\\", "(L \\times L)_{n + 1}", "& = &", "\\Hom(", "(\\Delta[n + 1] \\amalg \\Delta[n + 1], L)_0 \\\\", "(\\text{cosk}_n \\text{sk}_n (L \\times L))_{n + 1}", "& = &", "\\Hom(", "\\partial \\Delta[n + 1]", "\\amalg", "\\partial \\Delta[n + 1], L)_0", "\\end{eqnarray*}", "and the morphism between these objects of $\\text{SR}(\\mathcal{C}, X)$", "come from the commutative diagram of simplicial sets", "\\begin{equation}", "\\label{equation-dual-diagram}", "\\xymatrix{", "\\Delta[1] \\times \\Delta[n + 1] &", "\\Delta[1] \\times \\partial\\Delta[n + 1] \\ar[l] \\\\", "\\Delta[n + 1] \\amalg \\Delta[n + 1] \\ar[u] &", "\\partial\\Delta[n + 1] \\amalg \\partial\\Delta[n + 1]", "\\ar[l] \\ar[u]", "}", "\\end{equation}", "Moreover the fibre product of the bottom arrow and the", "right arrow in (\\ref{equation-diagram}) is equal to", "$$", "\\Hom(U, L)_0", "$$", "where $U \\subset \\Delta[1] \\times \\Delta[n + 1]$", "is the smallest simplicial subset such that both", "$\\Delta[n + 1] \\amalg \\Delta[n + 1]$ and", "$\\Delta[1] \\times \\partial\\Delta[n + 1]$ map into it." ], "refs": [], "proofs": [ { "contents": [ "The first and third equalities are", "Simplicial, Lemma \\ref{simplicial-lemma-exists-hom-from-simplicial-set-finite}.", "The second and fourth follow from the cited lemma combined with", "Simplicial, Lemma \\ref{simplicial-lemma-cosk-shriek}.", "The last assertion follows from the fact that", "$U$ is the push-out of the bottom and right arrow of the", "diagram (\\ref{equation-dual-diagram}), via", "Simplicial, Lemma \\ref{simplicial-lemma-hom-from-coprod}.", "To see that $U$ is equal to this push-out it suffices", "to see that the intersection of", "$\\Delta[n + 1] \\amalg \\Delta[n + 1]$ and", "$\\Delta[1] \\times \\partial\\Delta[n + 1]$", "in $\\Delta[1] \\times \\Delta[n + 1]$ is equal to", "$\\partial\\Delta[n + 1] \\amalg \\partial\\Delta[n + 1]$.", "This we leave to the reader." ], "refs": [ "simplicial-lemma-exists-hom-from-simplicial-set-finite", "simplicial-lemma-cosk-shriek", "simplicial-lemma-hom-from-coprod" ], "ref_ids": [ 14825, 14856, 14826 ] } ], "ref_ids": [] }, { "id": 8411, "type": "theorem", "label": "hypercovering-lemma-homotopy", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-homotopy", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $X$ be an object of $\\mathcal{C}$.", "Let $K, L$ be hypercoverings of $X$.", "Let $a, b : K \\to L$ be morphisms of hypercoverings.", "There exists a morphism of hypercoverings", "$c : K' \\to K$ such that $a \\circ c$ is homotopic", "to $b \\circ c$." ], "refs": [], "proofs": [ { "contents": [ "Consider the following commutative diagram", "$$", "\\xymatrix{", "K' \\ar@{=}[r]^-{def} \\ar[rd]_c &", "K \\times_{(L \\times L)} \\Hom(\\Delta[1], L)", "\\ar[r] \\ar[d] & \\Hom(\\Delta[1], L) \\ar[d] \\\\", "& K \\ar[r]^{(a, b)} & L \\times L", "}", "$$", "By the functorial property of $\\Hom(\\Delta[1], L)$", "the composition of the horizontal morphisms", "corresponds to a morphism $K' \\times \\Delta[1] \\to L$ which", "defines a homotopy between $c \\circ a$ and $c \\circ b$.", "Thus if we can show that $K'$ is a", "hypercovering of $X$, then we obtain the lemma.", "To see this we will apply Lemma \\ref{lemma-funny-gamma}", "to the pair of morphisms $K \\to L \\times L$", "and $\\Hom(\\Delta[1], L) \\to L \\times L$.", "Condition (1) of Lemma \\ref{lemma-funny-gamma} is satisfied.", "Condition (2) of Lemma \\ref{lemma-funny-gamma} is true because", "$\\Hom(\\Delta[1], L)_0 = L_1$, and the morphism", "$(d^1_0, d^1_1) : L_1 \\to L_0 \\times L_0$ is a", "covering of $\\text{SR}(\\mathcal{C}, X)$ by our", "assumption that $L$ is a hypercovering.", "To prove condition (3) of Lemma \\ref{lemma-funny-gamma}", "we use Lemma \\ref{lemma-hom-hypercovering} above. According", "to this lemma the morphism $\\gamma$ of condition (3) of Lemma", "\\ref{lemma-funny-gamma} is the morphism", "$$", "\\Hom(\\Delta[1] \\times \\Delta[n + 1], L)_0", "\\longrightarrow", "\\Hom(U, L)_0", "$$", "where $U \\subset \\Delta[1] \\times \\Delta[n + 1]$.", "According to Lemma \\ref{lemma-add-simplices}", "this is a covering and hence the claim has been proven." ], "refs": [ "hypercovering-lemma-funny-gamma", "hypercovering-lemma-funny-gamma", "hypercovering-lemma-funny-gamma", "hypercovering-lemma-funny-gamma", "hypercovering-lemma-hom-hypercovering", "hypercovering-lemma-funny-gamma", "hypercovering-lemma-add-simplices" ], "ref_ids": [ 8403, 8403, 8403, 8403, 8410, 8403, 8408 ] } ], "ref_ids": [] }, { "id": 8412, "type": "theorem", "label": "hypercovering-lemma-basis-hypercovering", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-basis-hypercovering", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{B}$ be a basis for the topology of $X$.", "There exists a hypercovering $(I, \\{U_i\\})$ of $X$", "such that each $U_i$ is an element of $\\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "Let $n \\geq 0$.", "Let us say that an {\\it $n$-truncated hypercovering} of $X$ is", "given by an $n$-truncated simplicial set $I$ and for each", "$i \\in I_a$, $0 \\leq a \\leq n$ an open $U_i$ of $X$ such that", "the conditions defining a hypercovering hold whenever they make sense.", "In other words we require the inclusion relations and covering", "conditions only when all simplices that occur in them", "are $a$-simplices with $a \\leq n$. The lemma follows if we can prove", "that given a $n$-truncated hypercovering $(I, \\{U_i\\})$ with", "all $U_i \\in \\mathcal{B}$ we can extend it to an $(n + 1)$-truncated", "hypercovering without adding any $a$-simplices for $a \\leq n$.", "This we do as follows. First we consider the $(n + 1)$-truncated", "simplicial set $I'$ defined by", "$I' = \\text{sk}_{n + 1}(\\text{cosk}_n I)$.", "Recall that", "$$", "I'_{n + 1} =", "\\left\\{", "\\begin{matrix}", "(i_0, \\ldots, i_{n + 1}) \\in (I_n)^{n + 2} \\text{ such that}\\\\", "d^n_{b - 1}(i_a) = d^n_a(i_b) \\text{ for all }0\\leq a < b\\leq n + 1", "\\end{matrix}", "\\right\\}", "$$", "If $i' \\in I'_{n + 1}$ is degenerate, say $i' = s^n_a(i)$ then we set", "$U_{i'} = U_i$ (this is forced on us anyway by the second condition).", "We also set $J_{i'} = \\{i'\\}$ in this case.", "If $i' \\in I'_{n + 1}$ is nondegenerate, say", "$i' = (i_0, \\ldots, i_{n + 1})$, then we choose a set", "$J_{i'}$ and an open covering", "\\begin{equation}", "\\label{equation-choose-covering}", "U_{i_0} \\cap \\ldots \\cap U_{i_{n + 1}} =", "\\bigcup\\nolimits_{i \\in J_{i'}} U_i,", "\\end{equation}", "with $U_i \\in \\mathcal{B}$ for $i \\in J_{i'}$.", "Set", "$$", "I_{n + 1} = \\coprod\\nolimits_{i' \\in I'_{n + 1}} J_{i'}", "$$", "There is a canonical map $\\pi : I_{n + 1} \\to I'_{n + 1}$ which is", "a bijection over the set of degenerate simplices in $I'_{n + 1}$ by", "construction.", "For $i \\in I_{n + 1}$ we define $d^{n + 1}_a(i) = d^{n + 1}_a(\\pi(i))$.", "For $i \\in I_n$ we define $s^n_a(i) \\in I_{n + 1}$ as the unique", "simplex lying over the degenerate simplex $s^n_a(i) \\in I'_{n + 1}$.", "We omit the verification that this defines an $(n + 1)$-truncated", "hypercovering of $X$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8413, "type": "theorem", "label": "hypercovering-lemma-quasi-separated-quasi-compact-hypercovering", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-quasi-separated-quasi-compact-hypercovering", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{B}$ be a basis for the topology of $X$.", "Assume that", "\\begin{enumerate}", "\\item $X$ is quasi-compact,", "\\item each $U \\in \\mathcal{B}$ is quasi-compact open, and", "\\item the intersection of any two quasi-compact opens in", "$X$ is quasi-compact.", "\\end{enumerate}", "Then there exists a hypercovering $(I, \\{U_i\\})$ of $X$ with the", "following properties", "\\begin{enumerate}", "\\item each $U_i$ is an element of the basis $\\mathcal{B}$,", "\\item each of the $I_n$ is a finite set, and in particular", "\\item each of the coverings (\\ref{equation-covering-X}),", "(\\ref{equation-covering-two}), and (\\ref{equation-covering-general})", "is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows directly from the construction in the proof of", "Lemma \\ref{lemma-basis-hypercovering} if we choose finite coverings", "by elements of $\\mathcal{B}$ in (\\ref{equation-choose-covering}).", "Details omitted." ], "refs": [ "hypercovering-lemma-basis-hypercovering" ], "ref_ids": [ 8412 ] } ], "ref_ids": [] }, { "id": 8414, "type": "theorem", "label": "hypercovering-lemma-split", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-split", "contents": [ "Let $\\mathcal{C}$ be a site. Let $K$ be an $r$-truncated simplicial object", "of $\\text{SR}(\\mathcal{C})$. The following are equivalent", "\\begin{enumerate}", "\\item $K$ is split (Simplicial, Definition \\ref{simplicial-definition-split}),", "\\item $f_{\\varphi, i} : U_{n, i} \\to U_{m, \\alpha(\\varphi)(i)}$", "is an isomorphism for $r \\geq n \\geq 0$,", "$\\varphi : [m] \\to [n]$ surjective, $i \\in I_n$, and", "\\item $f_{\\sigma^n_j, i} : U_{n, i} \\to U_{n + 1, \\alpha(\\sigma^n_j)(i)}$", "is an isomorphism for $0 \\leq j \\leq n < r$, $i \\in I_n$.", "\\end{enumerate}", "The same holds for simplicial objects if in (2) and (3)", "we set $r = \\infty$." ], "refs": [ "simplicial-definition-split" ], "proofs": [ { "contents": [ "The splitting of a simplicial set is unique and is given by", "the nondegenerate indices $N(I_n)$ in each degree $n$, see", "Simplicial, Lemma \\ref{simplicial-lemma-splitting-simplicial-sets}.", "The coproduct of two objects $\\{U_i\\}_{i \\in I}$ and $\\{U_j\\}_{j \\in J}$", "of $\\text{SR}(\\mathcal{C})$ is given by $\\{U_l\\}_{l \\in I \\amalg J}$", "with obvious notation. Hence a splitting of $K$ must be given by", "$N(K_n) = \\{U_i\\}_{i \\in N(I_n)}$. The equivalence of (1) and (2)", "now follows by unwinding the definitions. The equivalence of (2)", "and (3) follows from the fact that any surjection", "$\\varphi : [m] \\to [n]$ is a composition of morphisms", "$\\sigma^k_j$ with $k = n, n + 1, \\ldots, m - 1$." ], "refs": [ "simplicial-lemma-splitting-simplicial-sets" ], "ref_ids": [ 14827 ] } ], "ref_ids": [ 14924 ] }, { "id": 8415, "type": "theorem", "label": "hypercovering-lemma-hypercovering-object", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-hypercovering-object", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products.", "Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset.", "Assume", "\\begin{enumerate}", "\\item any object $U$ of $\\mathcal{C}$ has a covering", "$\\{U_j \\to U\\}_{j \\in J}$ with $U_j \\in \\mathcal{B}$, and", "\\item if $\\{U_j \\to U\\}_{j \\in J}$ is a covering", "with $U_j \\in \\mathcal{B}$ and $\\{U' \\to U\\}$ is a morphism with", "$U' \\in \\mathcal{B}$, then $\\{U_j \\to U\\}_{j \\in J} \\amalg \\{U' \\to U\\}$", "is a covering.", "\\end{enumerate}", "Then for any $X$ in $\\mathcal{C}$ there is a hypercovering $K$", "of $X$ such that $K_n = \\{U_{n, i}\\}_{i \\in I_n}$", "with $U_{n, i} \\in \\mathcal{B}$ for all $i \\in I_n$." ], "refs": [], "proofs": [ { "contents": [ "A warmup for this proof is the proof of", "Lemma \\ref{lemma-basis-hypercovering} and", "we encourage the reader to read that proof first.", "\\medskip\\noindent", "First we replace $\\mathcal{C}$ by the site $\\mathcal{C}/X$.", "After doing so we may assume that $X$ is the final object", "of $\\mathcal{C}$ and that $\\mathcal{C}$ has all finite limits", "(Categories, Lemma \\ref{categories-lemma-finite-limits-exist}).", "\\medskip\\noindent", "Let $n \\geq 0$. Let us say that an", "{\\it $n$-truncated $\\mathcal{B}$-hypercovering of $X$}", "is given by an $n$-truncated simplicial object $K$", "of $\\text{SR}(\\mathcal{C})$", "such that for $i \\in I_a$, $0 \\leq a \\leq n$", "we have $U_{a, i} \\in \\mathcal{B}$ and such that", "$K_0$ is a covering of $X$ and", "$K_{a + 1} \\to (\\text{cosk}_a \\text{sk}_a K)_{a + 1}$", "for $a = 0, \\ldots, n - 1$", "is a covering as in Definition \\ref{definition-covering-SR}.", "\\medskip\\noindent", "Since $X$ has a covering $\\{U_{0, i} \\to X\\}_{i \\in I_0}$", "with $U_i \\in \\mathcal{B}$ by assumption, we get a $0$-truncated", "$\\mathcal{B}$-hypercovering of $X$. Observe that any $0$-truncated", "$\\mathcal{B}$-hypercovering of $X$ is split, see", "Lemma \\ref{lemma-split}.", "\\medskip\\noindent", "The lemma follows if we can prove for $n \\geq 0$ that given a", "split $n$-truncated $\\mathcal{B}$-hypercovering $K$ of $X$", "we can extend it to a", "split $(n + 1)$-truncated $\\mathcal{B}$-hypercovering of $X$.", "\\medskip\\noindent", "Construction of the extension. Consider the $(n + 1)$-truncated simplicial", "object $K' = \\text{sk}_{n + 1}(\\text{cosk}_n K)$ of $\\text{SR}(\\mathcal{C})$.", "Write", "$$", "K'_{n + 1} = \\{U'_{n + 1, i}\\}_{i \\in I'_{n + 1}}", "$$", "Since $K = \\text{sk}_n K'$ we have $K_a = K'_a$ for $0 \\leq a \\leq n$.", "For every $i' \\in I'_{n + 1}$ we choose a covering", "\\begin{equation}", "\\label{equation-choose-covering-B}", "\\{g_{n + 1, j} : U_{n + 1, j} \\to U'_{n + 1, i'}\\}_{j \\in J_{i'}}", "\\end{equation}", "with $U_{n + 1, j} \\in \\mathcal{B}$ for $j \\in J_{i'}$.", "This is possible by our assumption on $\\mathcal{B}$ in the lemma.", "For $0 \\leq m \\leq n$ denote $N_m \\subset I_m$ the subset of", "nondegenerate indices. We set", "$$", "I_{n + 1} =", "\\coprod\\nolimits_{\\varphi : [n + 1] \\to [m]\\text{ surjective, }0\\leq m \\leq n}", "N_m \\amalg", "\\coprod\\nolimits_{i' \\in I'_{n + 1}} J_{i'}", "$$", "For $j \\in I_{n + 1}$ we set", "$$", "U_{n + 1, j} =", "\\left\\{", "\\begin{matrix}", "U_{m, i} & \\text{if} &", "j = (\\varphi, i) & \\text{where} & \\varphi : [n + 1] \\to [m], i \\in N_m \\\\", "U_{n + 1, j} & \\text{if} &", "j \\in J_{i'} & \\text{where} & i' \\in I'_{n + 1}", "\\end{matrix}", "\\right.", "$$", "with obvious notation. We set $K_{n + 1} = \\{U_{n + 1, j}\\}_{j \\in I_{n + 1}}$.", "By construction $U_{n + 1, j}$ is an element", "of $\\mathcal{B}$ for all $j \\in I_{n + 1}$. Let us define compatible", "maps", "$$", "I_{n + 1} \\to I'_{n + 1}", "\\quad\\text{and}\\quad", "K_{n + 1} \\to K'_{n + 1}", "$$", "Namely, the first map is given by", "$(\\varphi, i) \\mapsto \\alpha'(\\varphi)(i)$ and", "$(j \\in J_{i'}) \\mapsto i'$.", "For the second map we use the morphisms", "$$", "f'_{\\varphi, i} : U_{m, i} \\to U'_{n + 1, \\alpha'(\\varphi)(i)}", "\\quad\\text{and}\\quad", "g_{n + 1, j} : U_{n + 1, j} \\to U'_{n + 1, i'}", "$$", "We claim the morphism", "$$", "K_{n + 1} \\to K'_{n + 1} =", "(\\text{cosk}_n \\text{sk}_n K')_{n + 1} =", "(\\text{cosk}_n K)_{n + 1}", "$$", "is a covering as in Definition \\ref{definition-covering-SR}.", "Namely, if $i' \\in I'_{n + 1}$, then either $i'$ is nondegenerate", "and the inverse image of $i'$ in $I_{n + 1}$ is equal to $J_{i'}$", "and we get a covering of $U'_{n + 1, i'}$ by our choice", "(\\ref{equation-choose-covering-B}), or $i'$ is degenerate and", "the inverse image of $i'$ in $I_{n + 1}$ is", "$J_{i'} \\amalg \\{(\\varphi, i)\\}$ for a unique pair $(\\varphi, i)$", "and we get a covering by our choice (\\ref{equation-choose-covering-B})", "and assumption (2) of the lemma.", "\\medskip\\noindent", "To finish the proof we have to define the morphisms", "$K(\\varphi) : K_{n + 1} \\to K_m$ corresponding to morphisms", "$\\varphi : [m] \\to [n + 1]$, $0 \\leq m \\leq n$ and the morphisms", "$K(\\varphi) : K_m \\to K_{n + 1}$ corresponding to morphisms", "$\\varphi : [n + 1] \\to [m]$, $0 \\leq m \\leq n$", "satisfying suitable composition relations.", "For the first kind we use the composition", "$$", "K_{n + 1} \\to K'_{n + 1} \\xrightarrow{K'(\\varphi)} K'_m = K_m", "$$", "to define $K(\\varphi) : K_{n + 1} \\to K_m$.", "For the second kind, suppose given $\\varphi : [n + 1] \\to [m]$,", "$0 \\leq m \\leq n$. We define the corresponding morphism", "$K(\\varphi) : K_m \\to K_{n + 1}$ as follows:", "\\begin{enumerate}", "\\item for $i \\in I_m$ there is a unique surjective map", "$\\psi : [m] \\to [m_0]$ and a unique $i_0 \\in I_{m_0}$ nondegenerate", "such that $\\alpha(\\psi)(i_0) = i$\\footnote{For example, if $i$ is", "nondegenerate, then $m = m_0$ and $\\psi = \\text{id}_{[m]}$.},", "\\item we set $\\varphi_0 = \\psi_0 \\circ \\varphi : [n + 1] \\to [m_0]$", "and we map", "$i \\in I_m$ to $(\\varphi_0, i_0) \\in I_{n + 1}$, in other words,", "$\\alpha(\\varphi)(i) = (\\varphi_0, i_0)$, and", "\\item the morphism", "$f_{\\varphi, i} : U_{m, i} \\to U_{n + 1, \\alpha(\\varphi)(i)} = U_{m_0, i_0}$", "is the inverse of the isomorphism $f_{\\psi, i_0} : U_{m_0, i_0} \\to U_{m, i}$", "(see Lemma \\ref{lemma-split}).", "\\end{enumerate}", "We omit the straightforward but cumbersome verification that this defines", "a split $(n + 1)$-truncated $\\mathcal{B}$-hypercovering of $X$", "extending the given $n$-truncated one. In fact, everything is clear", "from the above, except for the verification that the morphisms", "$K(\\varphi)$ compose correctly for all $\\varphi : [a] \\to [b]$", "with $0 \\leq a, b \\leq n + 1$." ], "refs": [ "hypercovering-lemma-basis-hypercovering", "categories-lemma-finite-limits-exist", "hypercovering-definition-covering-SR", "hypercovering-lemma-split", "hypercovering-definition-covering-SR", "hypercovering-lemma-split" ], "ref_ids": [ 8412, 12224, 8423, 8414, 8423, 8414 ] } ], "ref_ids": [] }, { "id": 8416, "type": "theorem", "label": "hypercovering-lemma-hypercovering-site", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-hypercovering-site", "contents": [ "Let $\\mathcal{C}$ be a site with equalizers and fibre products.", "Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume", "that any object of $\\mathcal{C}$ has a covering", "whose members are elements of $\\mathcal{B}$.", "Then there is a hypercovering $K$ such that", "$K_n = \\{U_i\\}_{i \\in I_n}$ with $U_i \\in \\mathcal{B}$", "for all $i \\in I_n$." ], "refs": [], "proofs": [ { "contents": [ "This proof is almost the same as the proof of", "Lemma \\ref{lemma-hypercovering-object}. We will", "only explain the differences.", "\\medskip\\noindent", "Let $n \\geq 1$. Let us say that an", "{\\it $n$-truncated $\\mathcal{B}$-hypercovering}", "is given by an $n$-truncated simplicial", "object $K$ of $\\text{SR}(\\mathcal{C})$", "such that for $i \\in I_a$, $0 \\leq a \\leq n$", "we have $U_{a, i} \\in \\mathcal{B}$ and such that", "\\begin{enumerate}", "\\item $F(K_0)^\\# \\to *$ is surjective,", "\\item $F(K_1)^\\# \\to F(K_0)^\\# \\times F(K_0)^\\#$ is surjective,", "\\item $F(K_{a + 1})^\\# \\to F((\\text{cosk}_a \\text{sk}_a K)_{a + 1})^\\#$", "for $a = 1, \\ldots, n - 1$ is surjective.", "\\end{enumerate}", "We first explicitly construct a split $1$-truncated $\\mathcal{B}$-hypercovering.", "\\medskip\\noindent", "Take $I_0 = \\mathcal{B}$ and $K_0 = \\{U\\}_{U \\in \\mathcal{B}}$.", "Then (1) holds by our assumption on $\\mathcal{B}$. Set", "$$", "\\Omega =", "\\{(U, V, W, a, b) \\mid U, V, W \\in \\mathcal{B}, a : U \\to V, b : U \\to W\\}", "$$", "Then we set $I_1 = I_0 \\amalg \\Omega$. For $i \\in I_1$ we set", "$U_{1, i} = U_{0, i}$ if $i \\in I_0$ and $U_{1, i} = U$", "if $i = (U, V, W, a, b) \\in \\Omega$. The map", "$K(\\sigma^0_0) : K_0 \\to K_1$ corresponds to the", "inclusion $\\alpha(\\sigma^0_0) : I_0 \\to I_1$", "and the identity $f_{\\sigma^0_0, i} : U_{0, i} \\to U_{1, i}$", "on objects. The maps $K(\\delta^1_0), K(\\delta^1_1) : K_1 \\to K_0$", "correspond to the two maps $I_1 \\to I_0$ which are the", "identity on $I_0 \\subset I_1$ and map $(U, V, W, a, b) \\in \\Omega \\subset I_1$", "to $V$, resp.\\ $W$. The corresponding morphisms", "$f_{\\delta^1_0, i}, f_{\\delta^1_1, i} : U_{1, i} \\to U_{0, i}$ are", "the identity if $i \\in I_0$ and $a, b$ in case $i = (U, V, W, a, b) \\in \\Omega$.", "The reason that (2) holds is that any section of", "$F(K_0)^\\# \\times F(K_0)^\\#$ over an object $U$ of $\\mathcal{C}$", "comes, after replacing $U$ by the members of a covering,", "from a map $U \\to F(K_0) \\times F(K_0)$.", "This in turn means we have $V, W \\in \\mathcal{B}$", "and two morphisms $U \\to V$ and $U \\to W$. Further replacing", "$U$ by the members of a covering we may assume $U \\in \\mathcal{B}$", "as desired.", "\\medskip\\noindent", "The lemma follows if we can prove that given a split", "$n$-truncated $\\mathcal{B}$-hypercovering $K$ for $n \\geq 1$", "we can extend it to a split $(n + 1)$-truncated $\\mathcal{B}$-hypercovering.", "Here the argument proceeds exactly as in the proof of", "Lemma \\ref{lemma-hypercovering-object}.", "We omit the precise details, except for the following comments.", "First, we do not need assumption (2) in the proof of the current", "lemma as we do not need the morphism", "$K_{n + 1} \\to (\\text{cosk}_n K)_{n + 1}$ to be covering;", "we only need it to induce a surjection on associated sheaves of sets", "which follows from", "Sites, Lemma \\ref{sites-lemma-covering-surjective-after-sheafification}.", "Second, the assumption that $\\mathcal{C}$ has fibre products and equalizers", "guarantees that $\\text{SR}(\\mathcal{C})$ has fibre products", "and equalizers and $F$ commutes with these", "(Lemma \\ref{lemma-coprod-prod-SR}). This suffices", "assure us the coskeleton functors used exist (see", "Simplicial, Remark \\ref{simplicial-remark-existence-cosk} and", "Categories, Lemma \\ref{categories-lemma-fibre-products-equalizers-exist})." ], "refs": [ "hypercovering-lemma-hypercovering-object", "hypercovering-lemma-hypercovering-object", "sites-lemma-covering-surjective-after-sheafification", "hypercovering-lemma-coprod-prod-SR", "simplicial-remark-existence-cosk", "categories-lemma-fibre-products-equalizers-exist" ], "ref_ids": [ 8415, 8415, 8519, 8388, 14937, 12222 ] } ], "ref_ids": [] }, { "id": 8417, "type": "theorem", "label": "hypercovering-lemma-hypercovering-morphism-sites", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-hypercovering-morphism-sites", "contents": [ "Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites", "given by the functor $u : \\mathcal{D} \\to \\mathcal{C}$.", "Assume $\\mathcal{D}$ and $\\mathcal{C}$ have equalizers and", "fibre products and $u$ commutes with them.", "If a simplicial object $K$ of $\\text{SR}(\\mathcal{D})$", "is a hypercovering, then $u(K)$ is a hypercovering." ], "refs": [], "proofs": [ { "contents": [ "If we write $K_n = \\{U_{n, i}\\}_{i \\in I_n}$ as in the introduction", "to this section, then $u(K)$ is the object of $\\text{SR}(\\mathcal{C})$", "given by $u(K_n) = \\{u(U_i)\\}_{i \\in I_n}$.", "By Sites, Lemma \\ref{sites-lemma-pullback-representable-sheaf}", "we have $f^{-1}h_U^\\# = h_{u(U)}^\\#$ for $U \\in \\Ob(\\mathcal{D})$.", "This means that $f^{-1}F(K_n)^\\# = F(u(K_n))^\\#$ for all $n$.", "Let us check the conditions (1), (2), (3) for $u(K)$ to be a", "hypercovering from Definition \\ref{definition-hypercovering-variant}.", "Since $f^{-1}$ is an exact functor, we find that", "$$", "F(u(K_0))^\\# = f^{-1}F(K_0)^\\# \\to f^{-1}* = *", "$$", "is surjective as a pullback of a surjective map and we get (1).", "Similarly,", "$$", "F(u(K_1))^\\# = f^{-1}F(K_1)^\\# \\to", "f^{-1} (F(K_0) \\times F(K_0))^\\# = F(u(K_0))^\\# \\times F(u(K_0))^\\#", "$$", "is surjective as a pullback and we get (2). For condition (3),", "in order to conclude by the same method it suffices if", "$$", "F((\\text{cosk}_n \\text{sk}_n u(K))_{n + 1})^\\# =", "f^{-1}F((\\text{cosk}_n \\text{sk}_n K)_{n + 1})^\\#", "$$", "The above shows that $f^{-1}F(-) = F(u(-))$. Thus it suffices to show", "that $u$ commutes with the limits used in defining", "$(\\text{cosk}_n \\text{sk}_n K)_{n + 1}$ for $n \\geq 1$.", "By Simplicial, Remark \\ref{simplicial-remark-existence-cosk}", "these limits are finite connected limits and $u$ commutes with these", "by assumption." ], "refs": [ "sites-lemma-pullback-representable-sheaf", "hypercovering-definition-hypercovering-variant", "simplicial-remark-existence-cosk" ], "ref_ids": [ 8524, 8426, 14937 ] } ], "ref_ids": [] }, { "id": 8418, "type": "theorem", "label": "hypercovering-lemma-hypercovering-continuous-functor", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-hypercovering-continuous-functor", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let", "$u : \\mathcal{D} \\to \\mathcal{C}$ be a continuous functor.", "Assume $\\mathcal{D}$ and $\\mathcal{C}$ have fibre products", "and $u$ commutes with them. Let $Y \\in \\mathcal{D}$ and", "$K \\in \\text{SR}(\\mathcal{D}, Y)$ a hypercovering of $Y$.", "Then $u(K)$ is a hypercovering of $u(Y)$." ], "refs": [], "proofs": [ { "contents": [ "This is easier than the proof of Lemma \\ref{lemma-hypercovering-morphism-sites}", "because the notion of being a hypercovering of an object is stronger, see", "Definitions \\ref{definition-hypercovering} and \\ref{definition-covering-SR}.", "Namely, $u$ sends coverings to coverings by the definition of", "a morphism of sites. It suffices to check $u$ commutes with the", "limits used in defining", "$(\\text{cosk}_n \\text{sk}_n K)_{n + 1}$ for $n \\geq 1$.", "This is clear because the induced functor", "$\\mathcal{D}/Y \\to \\mathcal{C}/X$ commutes with all finite limits", "(and source and target have all finite limits by", "Categories, Lemma \\ref{categories-lemma-finite-limits-exist})." ], "refs": [ "hypercovering-lemma-hypercovering-morphism-sites", "hypercovering-definition-hypercovering", "hypercovering-definition-covering-SR", "categories-lemma-finite-limits-exist" ], "ref_ids": [ 8417, 8424, 8423, 12224 ] } ], "ref_ids": [] }, { "id": 8419, "type": "theorem", "label": "hypercovering-lemma-w-contractible", "categories": [ "hypercovering" ], "title": "hypercovering-lemma-w-contractible", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$", "be a subset. Assume", "\\begin{enumerate}", "\\item $\\mathcal{C}$ has fibre products,", "\\item for all $X \\in \\Ob(\\mathcal{C})$ there exists a finite covering", "$\\{U_i \\to X\\}_{i \\in I}$ with $U_i \\in \\mathcal{B}$,", "\\item if $\\{U_i \\to X\\}_{i \\in I}$ is a finite covering with", "$U_i \\in \\mathcal{B}$ and $U \\to X$ is a morphism with $U \\in \\mathcal{B}$,", "then $\\{U_i \\to X\\}_{i \\in I} \\amalg \\{U \\to X\\}$ is a covering.", "\\end{enumerate}", "Then for every $X$ there exists a hypercovering $K$ of $X$", "such that each $K_n = \\{U_{n, i} \\to X\\}_{i \\in I_n}$ with", "$I_n$ finite and $U_{n, i} \\in \\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "This lemma is the analogue of", "Lemma \\ref{lemma-quasi-separated-quasi-compact-hypercovering}", "for sites. To prove the lemma we follow exactly the proof of", "Lemma \\ref{lemma-hypercovering-object}", "paying attention to the following two points", "\\begin{enumerate}", "\\item[(a)] We choose our initial covering $\\{U_{0, i} \\to X\\}_{i \\in I_0}$", "with $U_{0, i} \\in \\mathcal{B}$ such that the index set $I_0$ is finite, and", "\\item[(b)] in choosing the coverings", "(\\ref{equation-choose-covering-B})", "we choose $J_{i'}$ finite.", "\\end{enumerate}", "The reader sees easily that with these modifications we end up", "with finite index sets $I_n$ for all $n$." ], "refs": [ "hypercovering-lemma-quasi-separated-quasi-compact-hypercovering", "hypercovering-lemma-hypercovering-object" ], "ref_ids": [ 8413, 8415 ] } ], "ref_ids": [] }, { "id": 8420, "type": "theorem", "label": "hypercovering-proposition-cohomology-hypercoverings", "categories": [ "hypercovering" ], "title": "hypercovering-proposition-cohomology-hypercoverings", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products and products of pairs.", "Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}$.", "Let $i \\geq 0$. Then", "\\begin{enumerate}", "\\item for every $\\xi \\in H^i(\\mathcal{F})$ there exists a hypercovering", "$K$ such that $\\xi$ is in the image of the canonical map", "$\\check{H}^i(K, \\mathcal{F}) \\to H^i(\\mathcal{F})$, and", "\\item if $K, L$ are hypercoverings and $\\xi_K \\in \\check{H}^i(K, \\mathcal{F})$,", "$\\xi_L \\in \\check{H}^i(L, \\mathcal{F})$ are elements mapping", "to the same element of $H^i(\\mathcal{F})$, then there exists", "a hypercovering $M$ and morphisms $M \\to K$ and $M \\to L$ such", "that $\\xi_K$ and $\\xi_L$ map to the same element of", "$\\check{H}^i(M, \\mathcal{F})$.", "\\end{enumerate}", "In other words, modulo set theoretical issues, the cohomology", "groups of $\\mathcal{F}$ on $\\mathcal{C}$ are the colimit of", "the {\\v C}ech cohomology groups of $\\mathcal{F}$ over all hypercoverings." ], "refs": [], "proofs": [ { "contents": [ "This result is a trivial consequence of", "Theorem \\ref{theorem-cohomology-hypercoverings}.", "Namely, we can artificially replace $\\mathcal{C}$ with a slightly", "bigger site $\\mathcal{C}'$ such that", "(I) $\\mathcal{C}'$ has a final object $X$ and (II)", "hypercoverings in $\\mathcal{C}$ are more or less the", "same thing as hypercoverings of $X$ in $\\mathcal{C}'$.", "But due to the nature of things, there is quite a bit of", "bookkeeping to do.", "\\medskip\\noindent", "Let us call a family of morphisms $\\{U_i \\to U\\}$ in $\\mathcal{C}$", "with fixed target a {\\it weak covering} if the sheafification of the", "map $\\coprod_{i \\in I} h_{U_i} \\to h_U$ becomes surjective.", "We construct a new site $\\mathcal{C}'$ as follows", "\\begin{enumerate}", "\\item as a category set $\\Ob(\\mathcal{C}') = \\Ob(\\mathcal{C}) \\amalg \\{X\\}$", "and add a unique morphism to $X$ from every object of $\\mathcal{C}'$,", "\\item $\\mathcal{C}'$ has fibre products as fibre products and products", "of pairs exist in $\\mathcal{C}$,", "\\item coverings of $\\mathcal{C}'$ are weak coverings of $\\mathcal{C}$", "together with those $\\{U_i \\to X\\}_{i \\in I}$ such that either $U_i = X$", "for some $i$, or $U_i \\not = X$ for all $i$ and the map", "$\\coprod h_{U_i} \\to *$ of presheaves on $\\mathcal{C}$ becomes", "surjective after sheafification on $\\mathcal{C}$,", "\\item we apply Sets, Lemma \\ref{sets-lemma-coverings-site}", "to restrict the coverings to obtain our site $\\mathcal{C}'$.", "\\end{enumerate}", "Then $\\Sh(\\mathcal{C}') = \\Sh(\\mathcal{C})$ because the inclusion", "functor $\\mathcal{C} \\to \\mathcal{C}'$ is a special cocontinuous functor", "(see Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}).", "We omit the straightforward verifications.", "\\medskip\\noindent", "Choose a covering $\\{U_i \\to X\\}$ of $\\mathcal{C}'$ such that $U_i$ is an", "object of $\\mathcal{C}$ for all $i$ (possible because", "$\\mathcal{C} \\to \\mathcal{C}'$ is special cocontinuous).", "Then $K_0 = \\{U_i \\to X\\}$ is a covering in the", "site $\\mathcal{C}'$ constructed above. We view $K_0$ as an object of", "$\\text{SR}(\\mathcal{C}', X)$ and we set $K_{init} = \\text{cosk}_0(K_0)$.", "Then $K_{init}$ is a hypercovering of $X$, see", "Example \\ref{example-cech}. Note that every $K_{init, n}$ has the shape", "$\\{W_j \\to X\\}$ with $W_j \\in \\Ob(\\mathcal{C})$.", "\\medskip\\noindent", "Proof of (1). Choose $\\xi \\in H^i(\\mathcal{F}) = H^i(X, \\mathcal{F}')$", "where $\\mathcal{F}'$ is the abelian sheaf on $\\mathcal{C}'$ corresponding", "to $\\mathcal{F}$ on $\\mathcal{C}$. By", "Theorem \\ref{theorem-cohomology-hypercoverings}", "there exists a morphism of hypercoverings $K' \\to K_{init}$", "of $X$ in $\\mathcal{C}'$ such that $\\xi$ comes from an element", "of $\\check{H}^i(K', \\mathcal{F})$.", "Write $K'_n = \\{U_{n, j} \\to X\\}$. Now since $K'_n$ maps to", "$K_{init, n}$ we see that $U_{n, j}$ is an object of $\\mathcal{C}$.", "Hence we can define a simplicial object $K$ of $\\text{SR}(\\mathcal{C})$", "by setting $K_n = \\{U_{n, j}\\}$. Since coverings in", "$\\mathcal{C}'$ consisting of families of morphisms of $\\mathcal{C}$", "are weak coverings, we see that $K$ is a hypercovering in the sense", "of Definition \\ref{definition-hypercovering-variant}.", "Finally, since $\\mathcal{F}'$ is the unique sheaf on $\\mathcal{C}'$", "whose restriction to $\\mathcal{C}$ is equal to $\\mathcal{F}$", "we see that the {\\v C}ech complexes $s(\\mathcal{F}(K))$", "and $s(\\mathcal{F}'(K'))$ are identical and (1) follows.", "(Compatibility with map into cohomology groups omitted.)", "\\medskip\\noindent", "Proof of (2). Let $K$ and $L$ be hypercoverings in $\\mathcal{C}$.", "Let $K'$ and $L'$ be the simplicial objects of $\\text{SR}(\\mathcal{C}', X)$", "gotten from $K$ and $L$ by the functor", "$\\text{SR}(\\mathcal{C}) \\to \\text{SR}(\\mathcal{C}', X)$,", "$\\{U_i\\} \\mapsto \\{U_i \\to X\\}$. As before we have equality of", "{\\v C}ech complexes and hence we obtain $\\xi_{K'}$ and", "$\\xi_{L'}$ mapping to the same cohomology class of $\\mathcal{F}'$", "over $\\mathcal{C}'$. After possibly enlarging our choice", "of coverings in $\\mathcal{C}'$ (due to a set theoretical issue)", "we may assume that $K'$ and $L'$ are hypercoverings of $X$ in", "$\\mathcal{C}'$; this is true by our definition of hypercoverings in", "Definition \\ref{definition-hypercovering-variant} and", "the fact that weak coverings in $\\mathcal{C}$ give coverings in", "$\\mathcal{C}'$. By", "Theorem \\ref{theorem-cohomology-hypercoverings}", "there exists a hypercovering $M'$ of $X$ in $\\mathcal{C}'$", "and morphisms $M' \\to K'$, $M' \\to L'$, and $M' \\to K_{init}$", "such that $\\xi_{K'}$ and $\\xi_{L'}$ restrict to the same element of", "$\\check{H}^i(M', \\mathcal{F})$. Unwinding this statement as above", "we find that (2) is true." ], "refs": [ "hypercovering-theorem-cohomology-hypercoverings", "sets-lemma-coverings-site", "sites-definition-special-cocontinuous-functor", "hypercovering-theorem-cohomology-hypercoverings", "hypercovering-definition-hypercovering-variant", "hypercovering-definition-hypercovering-variant", "hypercovering-theorem-cohomology-hypercoverings" ], "ref_ids": [ 8387, 8800, 8672, 8387, 8426, 8426, 8387 ] } ], "ref_ids": [] }, { "id": 8435, "type": "theorem", "label": "algebraic-theorem-smooth-groupoid-gives-algebraic-stack", "categories": [ "algebraic" ], "title": "algebraic-theorem-smooth-groupoid-gives-algebraic-stack", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces over $S$.", "Then the quotient stack $[U/R]$ is an algebraic stack over $S$." ], "refs": [], "proofs": [ { "contents": [ "We check the three conditions of", "Definition \\ref{definition-algebraic-stack}.", "By construction we have that $[U/R]$ is a stack in groupoids", "which is the first condition.", "\\medskip\\noindent", "The second condition follows from the stronger", "Lemma \\ref{lemma-diagonal-quotient-stack}.", "\\medskip\\noindent", "Finally, we have to show there exists a scheme $W$ over $S$", "and a surjective smooth $1$-morphism", "$(\\Sch/W)_{fppf} \\longrightarrow \\mathcal{X}$.", "First choose $W \\in \\Ob((\\Sch/S)_{fppf})$ and a", "surjective \\'etale morphism $W \\to U$. Note that this", "gives a surjective \\'etale morphism $\\mathcal{S}_W \\to \\mathcal{S}_U$", "of categories fibred in sets, see", "Lemma", "\\ref{lemma-map-presheaves-representable-by-spaces-transformation-property}.", "Of course then $\\mathcal{S}_W \\to \\mathcal{S}_U$ is also surjective and", "smooth, see", "Lemma \\ref{lemma-representable-transformations-property-implication}.", "Hence $\\mathcal{S}_W \\to \\mathcal{S}_U \\to [U/R]$ is surjective", "and smooth by a combination of", "Lemmas \\ref{lemma-smooth-quotient-smooth-presentation} and", "\\ref{lemma-composition-representable-transformations-property}." ], "refs": [ "algebraic-definition-algebraic-stack", "algebraic-lemma-diagonal-quotient-stack", "algebraic-lemma-map-presheaves-representable-by-spaces-transformation-property", "algebraic-lemma-representable-transformations-property-implication", "algebraic-lemma-smooth-quotient-smooth-presentation", "algebraic-lemma-composition-representable-transformations-property" ], "ref_ids": [ 8484, 8475, 8453, 8459, 8476, 8455 ] } ], "ref_ids": [] }, { "id": 8436, "type": "theorem", "label": "algebraic-lemma-morphism-schemes-gives-representable-transformation", "categories": [ "algebraic" ], "title": "algebraic-lemma-morphism-schemes-gives-representable-transformation", "contents": [ "Let $S$, $X$, $Y$ be objects of $\\Sch_{fppf}$.", "Let $f : X \\to Y$ be a morphism of schemes.", "Then the $1$-morphism induced by $f$", "$$", "(\\Sch/X)_{fppf} \\longrightarrow (\\Sch/Y)_{fppf}", "$$", "is a representable $1$-morphism." ], "refs": [], "proofs": [ { "contents": [ "This is formal and relies only on the fact that", "the category $(\\Sch/S)_{fppf}$ has fibre products." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8437, "type": "theorem", "label": "algebraic-lemma-representable-morphism-equivalent", "categories": [ "algebraic" ], "title": "algebraic-lemma-representable-morphism-equivalent", "contents": [ "Let $S$ be an object of $\\Sch_{fppf}$.", "Consider a $2$-commutative diagram", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\", "\\mathcal{Y}' \\ar[r] & \\mathcal{Y}", "}", "$$", "of $1$-morphisms of categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$.", "Assume the horizontal arrows are equivalences.", "Then $f$ is representable if and only if $f'$ is representable." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8438, "type": "theorem", "label": "algebraic-lemma-composition-representable-transformations", "categories": [ "algebraic" ], "title": "algebraic-lemma-composition-representable-transformations", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$", "be categories fibred in groupoids over $(\\Sch/S)_{fppf}$", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$", "be representable $1$-morphisms. Then", "$$", "g \\circ f : \\mathcal{X} \\longrightarrow \\mathcal{Z}", "$$", "is a representable $1$-morphism." ], "refs": [], "proofs": [ { "contents": [ "This is entirely formal and works in any category." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8439, "type": "theorem", "label": "algebraic-lemma-base-change-representable-transformations", "categories": [ "algebraic" ], "title": "algebraic-lemma-base-change-representable-transformations", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$", "be categories fibred in groupoids over $(\\Sch/S)_{fppf}$", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a representable $1$-morphism.", "Let $g : \\mathcal{Z} \\to \\mathcal{Y}$ be any $1$-morphism.", "Consider the fibre product diagram", "$$", "\\xymatrix{", "\\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X} \\ar[r]_-{g'} \\ar[d]_{f'} &", "\\mathcal{X} \\ar[d]^f \\\\", "\\mathcal{Z} \\ar[r]^g & \\mathcal{Y}", "}", "$$", "Then the base change $f'$ is a representable $1$-morphism." ], "refs": [], "proofs": [ { "contents": [ "This is entirely formal and works in any category." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8440, "type": "theorem", "label": "algebraic-lemma-product-representable-transformations", "categories": [ "algebraic" ], "title": "algebraic-lemma-product-representable-transformations", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}_i, \\mathcal{Y}_i$ be categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$, $i = 1, 2$.", "Let $f_i : \\mathcal{X}_i \\to \\mathcal{Y}_i$, $i = 1, 2$", "be representable $1$-morphisms.", "Then", "$$", "f_1 \\times f_2 :", "\\mathcal{X}_1 \\times \\mathcal{X}_2", "\\longrightarrow", "\\mathcal{Y}_1 \\times \\mathcal{Y}_2", "$$", "is a representable $1$-morphism." ], "refs": [], "proofs": [ { "contents": [ "Write $f_1 \\times f_2$ as the composition", "$\\mathcal{X}_1 \\times \\mathcal{X}_2 \\to", "\\mathcal{Y}_1 \\times \\mathcal{X}_2 \\to", "\\mathcal{Y}_1 \\times \\mathcal{Y}_2$.", "The first arrow is the base change of $f_1$ by the map", "$\\mathcal{Y}_1 \\times \\mathcal{X}_2 \\to \\mathcal{Y}_1$, and the second arrow", "is the base change of $f_2$ by the map", "$\\mathcal{Y}_1 \\times \\mathcal{Y}_2 \\to \\mathcal{Y}_2$.", "Hence this lemma is a formal", "consequence of Lemmas \\ref{lemma-composition-representable-transformations}", "and \\ref{lemma-base-change-representable-transformations}." ], "refs": [ "algebraic-lemma-composition-representable-transformations", "algebraic-lemma-base-change-representable-transformations" ], "ref_ids": [ 8438, 8439 ] } ], "ref_ids": [] }, { "id": 8441, "type": "theorem", "label": "algebraic-lemma-characterize-representable-by-space", "categories": [ "algebraic" ], "title": "algebraic-lemma-characterize-representable-by-space", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$", "be a category fibred in groupoids.", "Then $\\mathcal{X}$ is representable by an algebraic space over $S$", "if and only if the following conditions are satisfied:", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is fibred in setoids\\footnote{This means that", "it is fibred in groupoids and objects in the fibre categories", "have no nontrivial automorphisms, see Categories,", "Definition \\ref{categories-definition-category-fibred-sets}.}, and", "\\item the presheaf $U \\mapsto \\Ob(\\mathcal{X}_U)/\\!\\!\\cong$ is", "an algebraic space.", "\\end{enumerate}" ], "refs": [ "categories-definition-category-fibred-sets" ], "proofs": [ { "contents": [ "Omitted, but see Categories,", "Lemma \\ref{categories-lemma-characterize-representable-fibred-category}." ], "refs": [ "categories-lemma-characterize-representable-fibred-category" ], "ref_ids": [ 12316 ] } ], "ref_ids": [ 12397 ] }, { "id": 8442, "type": "theorem", "label": "algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids", "categories": [ "algebraic" ], "title": "algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism", "of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "The following are necessary and sufficient conditions for", "$f$ to be representable by algebraic spaces:", "\\begin{enumerate}", "\\item for each scheme $U/S$ the", "functor $f_U : \\mathcal{X}_U \\longrightarrow \\mathcal{Y}_U$", "between fibre categories is faithful, and", "\\item for each $U$ and each $y \\in \\Ob(\\mathcal{Y}_U)$ the presheaf", "$$", "(h : V \\to U)", "\\longmapsto", "\\{(x, \\phi) \\mid x \\in \\Ob(\\mathcal{X}_V), \\phi : h^*y \\to f(x)\\}/\\cong", "$$", "is an algebraic space over $U$.", "\\end{enumerate}", "Here we have made a choice of pullbacks for $\\mathcal{Y}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the description of fibre categories of the $2$-fibre products", "$(\\Sch/U)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}$ in", "Categories, Lemma \\ref{categories-lemma-identify-fibre-product}", "combined with", "Lemma \\ref{lemma-characterize-representable-by-space}." ], "refs": [ "categories-lemma-identify-fibre-product", "algebraic-lemma-characterize-representable-by-space" ], "ref_ids": [ 12319, 8441 ] } ], "ref_ids": [] }, { "id": 8443, "type": "theorem", "label": "algebraic-lemma-representable-by-spaces-morphism-equivalent", "categories": [ "algebraic" ], "title": "algebraic-lemma-representable-by-spaces-morphism-equivalent", "contents": [ "Let $S$ be an object of $\\Sch_{fppf}$.", "Consider a $2$-commutative diagram", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\", "\\mathcal{Y}' \\ar[r] & \\mathcal{Y}", "}", "$$", "of $1$-morphisms of categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$.", "Assume the horizontal arrows are equivalences.", "Then $f$ is representable by algebraic spaces", "if and only if $f'$ is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8444, "type": "theorem", "label": "algebraic-lemma-morphism-spaces-gives-representable-by-spaces", "categories": [ "algebraic" ], "title": "algebraic-lemma-morphism-spaces-gives-representable-by-spaces", "contents": [ "Let $S$ be an object of $\\Sch_{fppf}$.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$", "be a $1$-morphism of categories fibred in groupoids over $S$.", "If $\\mathcal{X}$ and $\\mathcal{Y}$ are representable by", "algebraic spaces over $S$, then the $1$-morphism $f$", "is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Omitted. This relies only on the fact that", "the category of algebraic spaces over $S$ has fibre products,", "see Spaces, Lemma \\ref{spaces-lemma-fibre-product-spaces}." ], "refs": [ "spaces-lemma-fibre-product-spaces" ], "ref_ids": [ 8143 ] } ], "ref_ids": [] }, { "id": 8445, "type": "theorem", "label": "algebraic-lemma-map-presheaves-representable-by-algebraic-spaces", "categories": [ "algebraic" ], "title": "algebraic-lemma-map-presheaves-representable-by-algebraic-spaces", "contents": [ "Let $S$ be an object of $\\Sch_{fppf}$.", "Let $a : F \\to G$ be a map of presheaves of sets on $(\\Sch/S)_{fppf}$.", "Denote $a' : \\mathcal{S}_F \\to \\mathcal{S}_G$ the associated", "map of categories fibred in sets.", "Then $a$ is representable by algebraic spaces (see", "Bootstrap,", "Definition \\ref{bootstrap-definition-morphism-representable-by-spaces})", "if and only if $a'$ is representable by algebraic spaces." ], "refs": [ "bootstrap-definition-morphism-representable-by-spaces" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 2637 ] }, { "id": 8446, "type": "theorem", "label": "algebraic-lemma-map-fibred-setoids-representable-algebraic-spaces", "categories": [ "algebraic" ], "title": "algebraic-lemma-map-fibred-setoids-representable-algebraic-spaces", "contents": [ "Let $S$ be an object of $\\Sch_{fppf}$.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of", "categories fibred in setoids over $(\\Sch/S)_{fppf}$.", "Let $F$, resp.\\ $G$ be the presheaf which to $T$ associates", "the set of isomorphism classes of objects of", "$\\mathcal{X}_T$, resp.\\ $\\mathcal{Y}_T$.", "Let $a : F \\to G$ be the map of presheaves corresponding to $f$.", "Then $a$ is representable by algebraic spaces (see", "Bootstrap,", "Definition \\ref{bootstrap-definition-morphism-representable-by-spaces})", "if and only if $f$ is representable by algebraic spaces." ], "refs": [ "bootstrap-definition-morphism-representable-by-spaces" ], "proofs": [ { "contents": [ "Omitted. Hint: Combine", "Lemmas \\ref{lemma-representable-by-spaces-morphism-equivalent}", "and \\ref{lemma-map-presheaves-representable-by-algebraic-spaces}." ], "refs": [ "algebraic-lemma-representable-by-spaces-morphism-equivalent", "algebraic-lemma-map-presheaves-representable-by-algebraic-spaces" ], "ref_ids": [ 8443, 8445 ] } ], "ref_ids": [ 2637 ] }, { "id": 8447, "type": "theorem", "label": "algebraic-lemma-base-change-representable-by-spaces", "categories": [ "algebraic" ], "title": "algebraic-lemma-base-change-representable-by-spaces", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$", "be categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism", "representable by algebraic spaces.", "Let $g : \\mathcal{Z} \\to \\mathcal{Y}$ be any $1$-morphism.", "Consider the fibre product diagram", "$$", "\\xymatrix{", "\\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X} \\ar[r]_-{g'} \\ar[d]_{f'} &", "\\mathcal{X} \\ar[d]^f \\\\", "\\mathcal{Z} \\ar[r]^g & \\mathcal{Y}", "}", "$$", "Then the base change $f'$ is a $1$-morphism representable by", "algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8448, "type": "theorem", "label": "algebraic-lemma-base-change-by-space-representable-by-space", "categories": [ "algebraic" ], "title": "algebraic-lemma-base-change-by-space-representable-by-space", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$", "be categories fibred in groupoids over $(\\Sch/S)_{fppf}$", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$,", "$g : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms.", "Assume", "\\begin{enumerate}", "\\item $f$ is representable by algebraic spaces, and", "\\item $\\mathcal{Z}$ is representable by an algebraic space over $S$.", "\\end{enumerate}", "Then the $2$-fibre product", "$\\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X}$", "is representable by an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of", "Bootstrap, Lemma \\ref{bootstrap-lemma-representable-by-spaces-over-space}.", "First note that", "$\\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X}$", "is fibred in setoids over $(\\Sch/S)_{fppf}$.", "Hence it is equivalent to $\\mathcal{S}_F$ for some presheaf", "$F$ on $(\\Sch/S)_{fppf}$, see", "Categories, Lemma \\ref{categories-lemma-setoid-fibres}.", "Moreover, let $G$ be an algebraic space which represents", "$\\mathcal{Z}$. The $1$-morphism", "$\\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X} \\to \\mathcal{Z}$", "is representable by algebraic spaces by", "Lemma \\ref{lemma-base-change-representable-by-spaces}.", "And $\\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X} \\to \\mathcal{Z}$", "corresponds to a morphism $F \\to G$ by", "Categories, Lemma \\ref{categories-lemma-2-category-fibred-setoids}.", "Then $F \\to G$ is representable by algebraic spaces by", "Lemma \\ref{lemma-map-fibred-setoids-representable-algebraic-spaces}.", "Hence", "Bootstrap, Lemma \\ref{bootstrap-lemma-representable-by-spaces-over-space}", "implies that $F$ is an algebraic space as desired." ], "refs": [ "bootstrap-lemma-representable-by-spaces-over-space", "categories-lemma-setoid-fibres", "algebraic-lemma-base-change-representable-by-spaces", "categories-lemma-2-category-fibred-setoids", "algebraic-lemma-map-fibred-setoids-representable-algebraic-spaces", "bootstrap-lemma-representable-by-spaces-over-space" ], "ref_ids": [ 2607, 12312, 8447, 12313, 8446, 2607 ] } ], "ref_ids": [] }, { "id": 8449, "type": "theorem", "label": "algebraic-lemma-composition-representable-by-spaces", "categories": [ "algebraic" ], "title": "algebraic-lemma-composition-representable-by-spaces", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$", "be categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "If $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$", "are $1$-morphisms representable by algebraic spaces, then", "$$", "g \\circ f : \\mathcal{X} \\longrightarrow \\mathcal{Z}", "$$", "is a $1$-morphism representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-base-change-by-space-representable-by-space}.", "Details omitted." ], "refs": [ "algebraic-lemma-base-change-by-space-representable-by-space" ], "ref_ids": [ 8448 ] } ], "ref_ids": [] }, { "id": 8450, "type": "theorem", "label": "algebraic-lemma-product-representable-by-spaces", "categories": [ "algebraic" ], "title": "algebraic-lemma-product-representable-by-spaces", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}_i, \\mathcal{Y}_i$ be categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$, $i = 1, 2$.", "Let $f_i : \\mathcal{X}_i \\to \\mathcal{Y}_i$, $i = 1, 2$", "be $1$-morphisms representable by algebraic spaces.", "Then", "$$", "f_1 \\times f_2 :", "\\mathcal{X}_1 \\times \\mathcal{X}_2", "\\longrightarrow", "\\mathcal{Y}_1 \\times \\mathcal{Y}_2", "$$", "is a $1$-morphism representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "Write $f_1 \\times f_2$ as the composition", "$\\mathcal{X}_1 \\times \\mathcal{X}_2 \\to", "\\mathcal{Y}_1 \\times \\mathcal{X}_2 \\to", "\\mathcal{Y}_1 \\times \\mathcal{Y}_2$.", "The first arrow is the base change of $f_1$ by the map", "$\\mathcal{Y}_1 \\times \\mathcal{X}_2 \\to \\mathcal{Y}_1$, and the second arrow", "is the base change of $f_2$ by the map", "$\\mathcal{Y}_1 \\times \\mathcal{Y}_2 \\to \\mathcal{Y}_2$.", "Hence this lemma is a formal", "consequence of Lemmas \\ref{lemma-composition-representable-by-spaces}", "and \\ref{lemma-base-change-representable-by-spaces}." ], "refs": [ "algebraic-lemma-composition-representable-by-spaces", "algebraic-lemma-base-change-representable-by-spaces" ], "ref_ids": [ 8449, 8447 ] } ], "ref_ids": [] }, { "id": 8451, "type": "theorem", "label": "algebraic-lemma-get-a-stack", "categories": [ "algebraic" ], "title": "algebraic-lemma-get-a-stack", "contents": [ "\\begin{reference}", "Lemma in an email of Matthew Emerton dated June 15, 2016", "\\end{reference}", "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X} \\to \\mathcal{Z}$ and $\\mathcal{Y} \\to \\mathcal{Z}$", "be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "If $\\mathcal{X} \\to \\mathcal{Z}$ is representable by algebraic spaces", "and $\\mathcal{Y}$ is a stack in groupoids, then", "$\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y}$ is a stack in groupoids." ], "refs": [], "proofs": [ { "contents": [ "The property of a morphism being representable by algebraic spaces", "is preserved under base-change", "(Lemma \\ref{lemma-base-change-by-space-representable-by-space}),", "and so, passing to the base-change", "$\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y}$ over $\\mathcal{Y}$,", "we may reduce to the case of a morphism of categories", "fibred in groupoids $\\mathcal{X} \\to \\mathcal{Y}$", "which is representable by algebraic spaces, and", "whose target is a stack in groupoids; our goal is then to prove", "that $\\mathcal{X}$ is also a stack in groupoids.", "This follows from Stacks, Lemma", "\\ref{stacks-lemma-relative-sheaf-over-stack-is-stack}", "whose assumptions are satisfied as a result of", "Lemma \\ref{lemma-criterion-map-representable-spaces-fibred-in-groupoids}." ], "refs": [ "algebraic-lemma-base-change-by-space-representable-by-space", "stacks-lemma-relative-sheaf-over-stack-is-stack", "algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids" ], "ref_ids": [ 8448, 8958, 8442 ] } ], "ref_ids": [] }, { "id": 8452, "type": "theorem", "label": "algebraic-lemma-property-morphism-equivalent", "categories": [ "algebraic" ], "title": "algebraic-lemma-property-morphism-equivalent", "contents": [ "Let $S$ be an object of $\\Sch_{fppf}$.", "Let $\\mathcal{P}$ be as in", "Definition \\ref{definition-relative-representable-property}.", "Consider a $2$-commutative diagram", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\", "\\mathcal{Y}' \\ar[r] & \\mathcal{Y}", "}", "$$", "of $1$-morphisms of categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$.", "Assume the horizontal arrows are equivalences and $f$ (or equivalently $f'$)", "is representably by algebraic spaces.", "Then $f$ has $\\mathcal{P}$ if and only if $f'$ has $\\mathcal{P}$." ], "refs": [ "algebraic-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Note that this makes sense by", "Lemma \\ref{lemma-representable-by-spaces-morphism-equivalent}.", "Proof omitted." ], "refs": [ "algebraic-lemma-representable-by-spaces-morphism-equivalent" ], "ref_ids": [ 8443 ] } ], "ref_ids": [ 8483 ] }, { "id": 8453, "type": "theorem", "label": "algebraic-lemma-map-presheaves-representable-by-spaces-transformation-property", "categories": [ "algebraic" ], "title": "algebraic-lemma-map-presheaves-representable-by-spaces-transformation-property", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $a : F \\to G$ be a map of presheaves on $(\\Sch/S)_{fppf}$.", "Let $\\mathcal{P}$ be as in", "Definition \\ref{definition-relative-representable-property}.", "Assume $a$ is representable by algebraic spaces.", "Then $a : F \\to G$ has property $\\mathcal{P}$ (see", "Bootstrap, Definition \\ref{bootstrap-definition-property-transformation})", "if and only if the corresponding morphism", "$\\mathcal{S}_F \\to \\mathcal{S}_G$ of categories fibred in groupoids", "has property $\\mathcal{P}$." ], "refs": [ "algebraic-definition-relative-representable-property", "bootstrap-definition-property-transformation" ], "proofs": [ { "contents": [ "Note that the lemma makes sense by", "Lemma \\ref{lemma-map-presheaves-representable-by-algebraic-spaces}.", "Proof omitted." ], "refs": [ "algebraic-lemma-map-presheaves-representable-by-algebraic-spaces" ], "ref_ids": [ 8445 ] } ], "ref_ids": [ 8483, 2638 ] }, { "id": 8454, "type": "theorem", "label": "algebraic-lemma-map-fibred-setoids-property", "categories": [ "algebraic" ], "title": "algebraic-lemma-map-fibred-setoids-property", "contents": [ "Let $S$ be an object of $\\Sch_{fppf}$. Let $\\mathcal{P}$ be as in", "Definition \\ref{definition-relative-representable-property}.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of", "categories fibred in setoids over $(\\Sch/S)_{fppf}$.", "Let $F$, resp.\\ $G$ be the presheaf which to $T$ associates", "the set of isomorphism classes of objects of", "$\\mathcal{X}_T$, resp.\\ $\\mathcal{Y}_T$.", "Let $a : F \\to G$ be the map of presheaves corresponding to $f$.", "Then $a$ has $\\mathcal{P}$ if and only if $f$ has $\\mathcal{P}$." ], "refs": [ "algebraic-definition-relative-representable-property" ], "proofs": [ { "contents": [ "The lemma makes sense by", "Lemma \\ref{lemma-map-fibred-setoids-representable-algebraic-spaces}.", "The lemma follows on combining", "Lemmas \\ref{lemma-property-morphism-equivalent}", "and \\ref{lemma-map-presheaves-representable-by-spaces-transformation-property}." ], "refs": [ "algebraic-lemma-map-fibred-setoids-representable-algebraic-spaces", "algebraic-lemma-property-morphism-equivalent", "algebraic-lemma-map-presheaves-representable-by-spaces-transformation-property" ], "ref_ids": [ 8446, 8452, 8453 ] } ], "ref_ids": [ 8483 ] }, { "id": 8455, "type": "theorem", "label": "algebraic-lemma-composition-representable-transformations-property", "categories": [ "algebraic" ], "title": "algebraic-lemma-composition-representable-transformations-property", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}$, $\\mathcal{Y}$, $\\mathcal{Z}$ be categories fibred", "in groupoids over $(\\Sch/S)_{fppf}$.", "Let $\\mathcal{P}$ be a property as in", "Definition \\ref{definition-relative-representable-property}", "which is stable under composition.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$,", "$g : \\mathcal{Y} \\to \\mathcal{Z}$ be $1$-morphisms which", "are representable by algebraic spaces.", "If $f$ and $g$ have property $\\mathcal{P}$ so does", "$g \\circ f : \\mathcal{X} \\to \\mathcal{Z}$." ], "refs": [ "algebraic-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Note that the lemma makes sense by", "Lemma \\ref{lemma-composition-representable-by-spaces}.", "Proof omitted." ], "refs": [ "algebraic-lemma-composition-representable-by-spaces" ], "ref_ids": [ 8449 ] } ], "ref_ids": [ 8483 ] }, { "id": 8456, "type": "theorem", "label": "algebraic-lemma-base-change-representable-transformations-property", "categories": [ "algebraic" ], "title": "algebraic-lemma-base-change-representable-transformations-property", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$", "be categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "Let $\\mathcal{P}$ be a property as in", "Definition \\ref{definition-relative-representable-property}.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism", "representable by algebraic spaces.", "Let $g : \\mathcal{Z} \\to \\mathcal{Y}$ be any $1$-morphism.", "Consider the $2$-fibre product diagram", "$$", "\\xymatrix{", "\\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X} \\ar[r]_-{g'} \\ar[d]_{f'} &", "\\mathcal{X} \\ar[d]^f \\\\", "\\mathcal{Z} \\ar[r]^g & \\mathcal{Y}", "}", "$$", "If $f$ has $\\mathcal{P}$, then the base change $f'$", "has $\\mathcal{P}$." ], "refs": [ "algebraic-definition-relative-representable-property" ], "proofs": [ { "contents": [ "The lemma makes sense by", "Lemma \\ref{lemma-base-change-representable-by-spaces}.", "Proof omitted." ], "refs": [ "algebraic-lemma-base-change-representable-by-spaces" ], "ref_ids": [ 8447 ] } ], "ref_ids": [ 8483 ] }, { "id": 8457, "type": "theorem", "label": "algebraic-lemma-descent-representable-transformations-property", "categories": [ "algebraic" ], "title": "algebraic-lemma-descent-representable-transformations-property", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$", "be categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "Let $\\mathcal{P}$ be a property as in", "Definition \\ref{definition-relative-representable-property}.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism", "representable by algebraic spaces.", "Let $g : \\mathcal{Z} \\to \\mathcal{Y}$ be any $1$-morphism.", "Consider the fibre product diagram", "$$", "\\xymatrix{", "\\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X} \\ar[r]_-{g'} \\ar[d]_{f'} &", "\\mathcal{X} \\ar[d]^f \\\\", "\\mathcal{Z} \\ar[r]^g & \\mathcal{Y}", "}", "$$", "Assume that for every scheme $U$ and object $x$ of $\\mathcal{Y}_U$,", "there exists an fppf covering $\\{U_i \\to U\\}$ such that $x|_{U_i}$", "is in the essential image of the functor", "$g : \\mathcal{Z}_{U_i} \\to \\mathcal{Y}_{U_i}$.", "In this case, if $f'$ has $\\mathcal{P}$, then $f$ has $\\mathcal{P}$." ], "refs": [ "algebraic-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Proof omitted. Hint: Compare with the proof of", "Spaces,", "Lemma \\ref{spaces-lemma-descent-representable-transformations-property}." ], "refs": [ "spaces-lemma-descent-representable-transformations-property" ], "ref_ids": [ 8134 ] } ], "ref_ids": [ 8483 ] }, { "id": 8458, "type": "theorem", "label": "algebraic-lemma-product-representable-transformations-property", "categories": [ "algebraic" ], "title": "algebraic-lemma-product-representable-transformations-property", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{P}$ be a property as in", "Definition \\ref{definition-relative-representable-property}", "which is stable under composition.", "Let $\\mathcal{X}_i, \\mathcal{Y}_i$ be categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$, $i = 1, 2$.", "Let $f_i : \\mathcal{X}_i \\to \\mathcal{Y}_i$, $i = 1, 2$", "be $1$-morphisms representable by algebraic spaces.", "If $f_1$ and $f_2$ have property $\\mathcal{P}$ so does", "$", "f_1 \\times f_2 :", "\\mathcal{X}_1 \\times \\mathcal{X}_2", "\\to", "\\mathcal{Y}_1 \\times \\mathcal{Y}_2", "$." ], "refs": [ "algebraic-definition-relative-representable-property" ], "proofs": [ { "contents": [ "The lemma makes sense by", "Lemma \\ref{lemma-product-representable-by-spaces}.", "Proof omitted." ], "refs": [ "algebraic-lemma-product-representable-by-spaces" ], "ref_ids": [ 8450 ] } ], "ref_ids": [ 8483 ] }, { "id": 8459, "type": "theorem", "label": "algebraic-lemma-representable-transformations-property-implication", "categories": [ "algebraic" ], "title": "algebraic-lemma-representable-transformations-property-implication", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids", "over $(\\Sch/S)_{fppf}$.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism representable", "by algebraic spaces.", "Let $\\mathcal{P}$, $\\mathcal{P}'$ be properties as in", "Definition \\ref{definition-relative-representable-property}.", "Suppose that for any morphism of algebraic spaces $a : F \\to G$", "we have $\\mathcal{P}(a) \\Rightarrow \\mathcal{P}'(a)$.", "If $f$ has property $\\mathcal{P}$ then", "$f$ has property $\\mathcal{P}'$." ], "refs": [ "algebraic-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8483 ] }, { "id": 8460, "type": "theorem", "label": "algebraic-lemma-open-fibred-category-is-full", "categories": [ "algebraic" ], "title": "algebraic-lemma-open-fibred-category-is-full", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $j : \\mathcal X \\to \\mathcal Y$ be a $1$-morphism of", "categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "Assume $j$ is representable by algebraic spaces and a monomorphism", "(see", "Definition \\ref{definition-relative-representable-property}", "and", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-monomorphism}).", "Then $j$ is fully faithful on fibre categories." ], "refs": [ "algebraic-definition-relative-representable-property", "spaces-descent-lemma-descending-property-monomorphism" ], "proofs": [ { "contents": [ "We have seen in", "Lemma \\ref{lemma-criterion-map-representable-spaces-fibred-in-groupoids}", "that $j$ is faithful on fibre categories. Consider a scheme $U$,", "two objects $u, v$ of $\\mathcal{X}_U$, and an isomorphism", "$t : j(u) \\to j(v)$ in $\\mathcal{Y}_U$. We have to construct an", "isomorphism in $\\mathcal{X}_U$ between $u$ and $v$.", "By the $2$-Yoneda lemma (see Section \\ref{section-2-yoneda})", "we think of $u$, $v$ as $1$-morphisms", "$u, v : (\\Sch/U)_{fppf} \\to \\mathcal{X}$", "and we consider the $2$-fibre product", "$$", "(\\Sch/U)_{fppf} \\times_{j \\circ v, \\mathcal{Y}} \\mathcal{X}.", "$$", "By assumption this is representable by an algebraic space", "$F_{j \\circ v}$, over $U$ and the morphism", "$F_{j \\circ v} \\to U$ is a monomorphism.", "But since $(1_U, v, 1_{j(v)})$ gives a $1$-morphism of", "$(\\Sch/U)_{fppf}$ into the displayed $2$-fibre product,", "we see that $F_{j \\circ v} = U$ (here we use", "that if $V \\to U$ is a monomorphism of algebraic spaces which has a", "section, then $V = U$). Therefore the $1$-morphism projecting to", "the first coordinate", "$$", "(\\Sch/U)_{fppf} \\times_{j \\circ v, \\mathcal{Y}} \\mathcal{X}", "\\to (\\Sch/U)_{fppf}", "$$", "is an equivalence of fibre categories.", "Since $(1_U, u, t)$ and $(1_U, v, 1_{j(v)})$ give two", "objects in $((\\Sch/U)_{fppf} \\times_{j \\circ v, \\mathcal{Y}}", "\\mathcal{X})_U$ which have the same first coordinate, there must", "be a $2$-morphism between them in the $2$-fibre product.", "This is by definition a morphism $\\tilde t : u \\to v$ such that", "$j(\\tilde t) = t$." ], "refs": [ "algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids" ], "ref_ids": [ 8442 ] } ], "ref_ids": [ 8483, 9410 ] }, { "id": 8461, "type": "theorem", "label": "algebraic-lemma-representable-diagonal", "categories": [ "algebraic" ], "title": "algebraic-lemma-representable-diagonal", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}$ be a category fibred in groupoids over", "$(\\Sch/S)_{fppf}$. The following are equivalent:", "\\begin{enumerate}", "\\item the diagonal $\\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$", "is representable by algebraic spaces,", "\\item for every scheme $U$ over $S$, and any", "$x, y \\in \\Ob(\\mathcal{X}_U)$ the sheaf", "$\\mathit{Isom}(x, y)$ is an algebraic space over $U$,", "\\item for every scheme $U$ over $S$, and any $x \\in \\Ob(\\mathcal{X}_U)$", "the associated $1$-morphism $x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$", "is representable by algebraic spaces,", "\\item for every pair of schemes $T_1, T_2$ over $S$, and any", "$x_i \\in \\Ob(\\mathcal{X}_{T_i})$, $i = 1, 2$ the $2$-fibre product", "$(\\Sch/T_1)_{fppf} \\times_{x_1, \\mathcal{X}, x_2}", "(\\Sch/T_2)_{fppf}$", "is representable by an algebraic space,", "\\item for every representable category fibred in groupoids $\\mathcal{U}$", "over $(\\Sch/S)_{fppf}$ every $1$-morphism", "$\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces,", "\\item for every pair $\\mathcal{T}_1, \\mathcal{T}_2$ of representable", "categories fibred in groupoids over $(\\Sch/S)_{fppf}$ and any", "$1$-morphisms $x_i : \\mathcal{T}_i \\to \\mathcal{X}$, $i = 1, 2$ the", "$2$-fibre product $\\mathcal{T}_1 \\times_{x_1, \\mathcal{X}, x_2} \\mathcal{T}_2$", "is representable by an algebraic space,", "\\item for every category fibred in groupoids $\\mathcal{U}$", "over $(\\Sch/S)_{fppf}$ which is", "representable by an algebraic space every $1$-morphism", "$\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces,", "\\item for every pair $\\mathcal{T}_1, \\mathcal{T}_2$ of categories fibred", "in groupoids over $(\\Sch/S)_{fppf}$ which are representable", "by algebraic spaces, and any $1$-morphisms", "$x_i : \\mathcal{T}_i \\to \\mathcal{X}$ the", "$2$-fibre product $\\mathcal{T}_1 \\times_{x_1, \\mathcal{X}, x_2} \\mathcal{T}_2$", "is representable by an algebraic space.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows from", "Stacks, Lemma \\ref{stacks-lemma-isom-as-2-fibre-product}", "and the definitions.", "Let us prove the equivalence of (1) and (3).", "Write $\\mathcal{C} = (\\Sch/S)_{fppf}$ for the base category.", "We will use some of the observations of the proof of the similar", "Categories, Lemma \\ref{categories-lemma-representable-diagonal-groupoids}.", "We will use the symbol $\\cong$ to mean ``equivalence of categories fibred", "in groupoids over $\\mathcal{C} = (\\Sch/S)_{fppf}$''.", "Assume (1). Suppose given $U$ and $x$ as in (3). For any scheme $V$", "and $y \\in \\Ob(\\mathcal{X}_V)$ we see (compare reference above) that", "$$", "\\mathcal{C}/U", "\\times_{x, \\mathcal{X}, y}", "\\mathcal{C}/V", "\\cong", "(\\mathcal{C}/U \\times_S V)", "\\times_{(x, y), \\mathcal{X} \\times \\mathcal{X}, \\Delta}", "\\mathcal{X}", "$$", "which is representable by an algebraic space by assumption. Conversely,", "assume (3). Consider any scheme $U$ over $S$ and a pair $(x, x')$", "of objects of $\\mathcal{X}$ over $U$. We have to show that", "$\\mathcal{X} \\times_{\\Delta, \\mathcal{X} \\times \\mathcal{X}, (x, x')} U$", "is representable by an algebraic space. This is clear because", "(compare reference above)", "$$", "\\mathcal{X}", "\\times_{\\Delta, \\mathcal{X} \\times \\mathcal{X}, (x, x')}", "\\mathcal{C}/U", "\\cong", "(\\mathcal{C}/U \\times_{x, \\mathcal{X}, x'} \\mathcal{C}/U)", "\\times_{\\mathcal{C}/U \\times_S U, \\Delta}", "\\mathcal{C}/U", "$$", "and the right hand side is representable by an algebraic space by assumption", "and the fact that the category of algebraic spaces over $S$ has fibre products", "and contains $U$ and $S$.", "\\medskip\\noindent", "The equivalences", "(3) $\\Leftrightarrow$ (4),", "(5) $\\Leftrightarrow$ (6),", "and", "(7) $\\Leftrightarrow$ (8)", "are formal. The equivalences", "(3) $\\Leftrightarrow$ (5) and", "(4) $\\Leftrightarrow$ (6)", "follow from", "Lemma \\ref{lemma-representable-by-spaces-morphism-equivalent}.", "Assume (3), and let $\\mathcal{U} \\to \\mathcal{X}$ be as in (7).", "To prove (7) we have to show that for every scheme $V$ and $1$-morphism", "$y : (\\Sch/V)_{fppf} \\to \\mathcal{X}$ the $2$-fibre product", "$(\\Sch/V)_{fppf} \\times_{y, \\mathcal{X}} \\mathcal{U}$", "is representable by an algebraic space. Property (3) tells us", "that $y$ is representable by algebraic spaces hence", "Lemma \\ref{lemma-base-change-by-space-representable-by-space}", "implies what we want. Finally, (7) directly implies (3)." ], "refs": [ "stacks-lemma-isom-as-2-fibre-product", "categories-lemma-representable-diagonal-groupoids", "algebraic-lemma-representable-by-spaces-morphism-equivalent", "algebraic-lemma-base-change-by-space-representable-by-space" ], "ref_ids": [ 8936, 12323, 8443, 8448 ] } ], "ref_ids": [] }, { "id": 8462, "type": "theorem", "label": "algebraic-lemma-equivalent", "categories": [ "algebraic" ], "title": "algebraic-lemma-equivalent", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}$, $\\mathcal{Y}$ be categories over $(\\Sch/S)_{fppf}$.", "Assume $\\mathcal{X}$, $\\mathcal{Y}$ are equivalent as categories over", "$(\\Sch/S)_{fppf}$. Then $\\mathcal{X}$ is an algebraic stack if and", "only if $\\mathcal{Y}$ is an algebraic stack. Similarly, $\\mathcal{X}$", "is a Deligne-Mumford stack if and only if $\\mathcal{Y}$ is a Deligne-Mumford", "stack." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{X}$ is an algebraic stack (resp.\\ a Deligne-Mumford stack). By", "Stacks, Lemma \\ref{stacks-lemma-stack-in-groupoids-equivalent}", "this implies that $\\mathcal{Y}$ is a stack in groupoids over", "$\\Sch_{fppf}$. Choose an equivalence $f : \\mathcal{X} \\to \\mathcal{Y}$", "over $\\Sch_{fppf}$. This gives a $2$-commutative diagram", "$$", "\\xymatrix{", "\\mathcal{X} \\ar[r]_f \\ar[d]_{\\Delta_\\mathcal{X}} &", "\\mathcal{Y} \\ar[d]^{\\Delta_\\mathcal{Y}} \\\\", "\\mathcal{X} \\times \\mathcal{X} \\ar[r]^{f \\times f} &", "\\mathcal{Y} \\times \\mathcal{Y}", "}", "$$", "whose horizontal arrows are equivalences. This implies that", "$\\Delta_\\mathcal{Y}$ is representable by algebraic spaces according to", "Lemma \\ref{lemma-representable-by-spaces-morphism-equivalent}.", "Finally, let $U$ be a scheme over $S$, and let", "$x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$ be a $1$-morphism which", "is surjective and smooth (resp.\\ \\'etale). Considering the diagram", "$$", "\\xymatrix{", "(\\Sch/U)_{fppf} \\ar[r]_{\\text{id}} \\ar[d]_x &", "(\\Sch/U)_{fppf} \\ar[d]^{f \\circ x} \\\\", "\\mathcal{X} \\ar[r]^f &", "\\mathcal{Y}", "}", "$$", "and applying", "Lemma \\ref{lemma-property-morphism-equivalent}", "we conclude that $f \\circ x$ is surjective and smooth (resp.\\ \\'etale)", "as desired." ], "refs": [ "stacks-lemma-stack-in-groupoids-equivalent", "algebraic-lemma-representable-by-spaces-morphism-equivalent", "algebraic-lemma-property-morphism-equivalent" ], "ref_ids": [ 8948, 8443, 8452 ] } ], "ref_ids": [] }, { "id": 8463, "type": "theorem", "label": "algebraic-lemma-representable-algebraic", "categories": [ "algebraic" ], "title": "algebraic-lemma-representable-algebraic", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "\\begin{enumerate}", "\\item A category fibred in groupoids", "$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$", "which is representable by an algebraic space is a Deligne-Mumford stack.", "\\item If $F$ is an algebraic space over $S$, then the associated", "category fibred in groupoids", "$p : \\mathcal{S}_F \\to (\\Sch/S)_{fppf}$", "is a Deligne-Mumford stack.", "\\item If $X \\in \\Ob((\\Sch/S)_{fppf})$, then", "$(\\Sch/X)_{fppf} \\to (\\Sch/S)_{fppf}$ is", "a Deligne-Mumford stack.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (2) implies (3).", "Parts (1) and (2) are equivalent by Lemma \\ref{lemma-equivalent}.", "Hence it suffices to prove (2).", "First, we note that $\\mathcal{S}_F$ is stack in sets since", "$F$ is a sheaf (Stacks, Lemma", "\\ref{stacks-lemma-stack-in-setoids-characterize}).", "A fortiori it is a stack in groupoids. Second the diagonal", "morphism $\\mathcal{S}_F \\to \\mathcal{S}_F \\times \\mathcal{S}_F$", "is the same as the morphism $\\mathcal{S}_F \\to \\mathcal{S}_{F \\times F}$", "which comes from the diagonal of $F$. Hence this is representable", "by algebraic spaces according to", "Lemma \\ref{lemma-morphism-spaces-gives-representable-by-spaces}.", "Actually it is even representable (by schemes), as the diagonal of", "an algebraic space is representable, but we do not need this.", "Let $U$ be a scheme and let $h_U \\to F$ be a surjective \\'etale morphism.", "We may think of this as a surjective \\'etale morphism of algebraic spaces.", "Hence by", "Lemma", "\\ref{lemma-map-presheaves-representable-by-spaces-transformation-property}", "the corresponding $1$-morphism $(\\Sch/U)_{fppf} \\to \\mathcal{S}_F$", "is surjective and \\'etale." ], "refs": [ "algebraic-lemma-equivalent", "stacks-lemma-stack-in-setoids-characterize", "algebraic-lemma-morphism-spaces-gives-representable-by-spaces", "algebraic-lemma-map-presheaves-representable-by-spaces-transformation-property" ], "ref_ids": [ 8462, 8951, 8444, 8453 ] } ], "ref_ids": [] }, { "id": 8464, "type": "theorem", "label": "algebraic-lemma-algebraic-stack-no-automorphisms", "categories": [ "algebraic" ], "title": "algebraic-lemma-algebraic-stack-no-automorphisms", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}$ be an algebraic stack over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is a Deligne-Mumford stack and is a stack in setoids,", "\\item $\\mathcal{X}$ is a Deligne-Mumford stack such that the", "canonical $1$-morphism $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$", "is an equivalence, and", "\\item $\\mathcal{X}$ is representable by an algebraic space.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows from", "Stacks, Lemma \\ref{stacks-lemma-characterize-stack-in-setoids}.", "The implication (3) $\\Rightarrow$ (1) follows from", "Lemma \\ref{lemma-representable-algebraic}.", "Finally, assume (1). By", "Stacks, Lemma \\ref{stacks-lemma-stack-in-setoids-characterize}", "there exists a sheaf $F$ on $(\\Sch/S)_{fppf}$", "and an equivalence $j : \\mathcal{X} \\to \\mathcal{S}_F$. By", "Lemma \\ref{lemma-map-presheaves-representable-by-algebraic-spaces}", "the fact that $\\Delta_\\mathcal{X}$ is representable by algebraic", "spaces, means that $\\Delta_F : F \\to F \\times F$", "is representable by algebraic spaces.", "Let $U$ be a scheme, and let $x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$", "be a surjective \\'etale morphism. The composition", "$j \\circ x : (\\Sch/U)_{fppf} \\to \\mathcal{S}_F$", "corresponds to a morphism $h_U \\to F$ of sheaves. By", "Bootstrap, Lemma \\ref{bootstrap-lemma-representable-diagonal}", "this morphism is representable by algebraic spaces.", "Hence by", "Lemma \\ref{lemma-map-fibred-setoids-property}", "we conclude that $h_U \\to F$ is surjective and \\'etale.", "Finally, we apply", "Bootstrap, Theorem \\ref{bootstrap-theorem-bootstrap}", "to see that $F$ is an algebraic space." ], "refs": [ "stacks-lemma-characterize-stack-in-setoids", "algebraic-lemma-representable-algebraic", "stacks-lemma-stack-in-setoids-characterize", "algebraic-lemma-map-presheaves-representable-by-algebraic-spaces", "bootstrap-lemma-representable-diagonal", "algebraic-lemma-map-fibred-setoids-property", "bootstrap-theorem-bootstrap" ], "ref_ids": [ 8960, 8463, 8951, 8445, 2618, 8454, 2601 ] } ], "ref_ids": [] }, { "id": 8465, "type": "theorem", "label": "algebraic-lemma-product-spaces", "categories": [ "algebraic" ], "title": "algebraic-lemma-product-spaces", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}$, $\\mathcal{Y}$ be algebraic stacks over $S$.", "Then $\\mathcal{X} \\times_{(\\Sch/S)_{fppf}} \\mathcal{Y}$", "is an algebraic stack, and is a product in the $2$-category of", "algebraic stacks over $S$." ], "refs": [], "proofs": [ { "contents": [ "An object of $\\mathcal{X} \\times_{(\\Sch/S)_{fppf}} \\mathcal{Y}$", "over $T$ is just a pair $(x, y)$ where $x$ is an object of $\\mathcal{X}_T$", "and $y$ is an object of $\\mathcal{Y}_T$. Hence it is immediate from", "the definitions that", "$\\mathcal{X} \\times_{(\\Sch/S)_{fppf}} \\mathcal{Y}$ is a", "stack in groupoids. If $(x, y)$ and $(x', y')$ are", "two objects of $\\mathcal{X} \\times_{(\\Sch/S)_{fppf}} \\mathcal{Y}$", "over $T$, then", "$$", "\\mathit{Isom}((x, y), (x', y')) =", "\\mathit{Isom}(x, x') \\times \\mathit{Isom}(y, y').", "$$", "Hence it follows from the equivalences in", "Lemma \\ref{lemma-representable-diagonal}", "and the fact that the category of algebraic spaces has products", "that the diagonal of $\\mathcal{X} \\times_{(\\Sch/S)_{fppf}} \\mathcal{Y}$", "is representable by algebraic spaces.", "Finally, suppose that $U, V \\in \\Ob((\\Sch/S)_{fppf})$,", "and let $x, y$ be surjective smooth morphisms", "$x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$,", "$y : (\\Sch/V)_{fppf} \\to \\mathcal{Y}$.", "Note that", "$$", "(\\Sch/U \\times_S V)_{fppf} =", "(\\Sch/U)_{fppf}", "\\times_{(\\Sch/S)_{fppf}} (\\Sch/V)_{fppf}.", "$$", "The object $(\\text{pr}_U^*x, \\text{pr}_V^*y)$ of", "$\\mathcal{X} \\times_{(\\Sch/S)_{fppf}} \\mathcal{Y}$ over", "$(\\Sch/U \\times_S V)_{fppf}$ thus defines a $1$-morphism", "$$", "(\\Sch/U \\times_S V)_{fppf}", "\\longrightarrow", "\\mathcal{X} \\times_{(\\Sch/S)_{fppf}} \\mathcal{Y}", "$$", "which is the composition of base changes of $x$ and $y$, hence", "is surjective and smooth, see", "Lemmas \\ref{lemma-base-change-representable-transformations-property} and", "\\ref{lemma-composition-representable-transformations-property}.", "We conclude that $\\mathcal{X} \\times_{(\\Sch/S)_{fppf}} \\mathcal{Y}$", "is indeed an algebraic stack. We omit the verification that it", "really is a product." ], "refs": [ "algebraic-lemma-representable-diagonal", "algebraic-lemma-base-change-representable-transformations-property", "algebraic-lemma-composition-representable-transformations-property" ], "ref_ids": [ 8461, 8456, 8455 ] } ], "ref_ids": [] }, { "id": 8466, "type": "theorem", "label": "algebraic-lemma-2-fibre-product-general", "categories": [ "algebraic" ], "title": "algebraic-lemma-2-fibre-product-general", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{Z}$ be a stack in groupoids over $(\\Sch/S)_{fppf}$", "whose diagonal is representable by algebraic spaces.", "Let $\\mathcal{X}$, $\\mathcal{Y}$ be algebraic stacks over $S$.", "Let $f : \\mathcal{X} \\to \\mathcal{Z}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$", "be $1$-morphisms of stacks in groupoids. Then the $2$-fibre product", "$\\mathcal{X} \\times_{f, \\mathcal{Z}, g} \\mathcal{Y}$ is an algebraic stack." ], "refs": [], "proofs": [ { "contents": [ "We have to check conditions (1), (2), and (3) of", "Definition \\ref{definition-algebraic-stack}.", "The first condition follows from", "Stacks, Lemma \\ref{stacks-lemma-2-product-stacks-in-groupoids}.", "\\medskip\\noindent", "The second condition we have to check is that the $\\mathit{Isom}$-sheaves", "are representable by algebraic spaces. To do this, suppose that", "$T$ is a scheme over $S$, and $u, v$ are objects of", "$(\\mathcal{X} \\times_{f, \\mathcal{Z}, g} \\mathcal{Y})_T$.", "By our construction of $2$-fibre products (which goes all the way", "back to", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C})", "we may write $u = (x, y, \\alpha)$ and $v = (x', y', \\alpha')$.", "Here $\\alpha : f(x) \\to g(y)$ and similarly for $\\alpha'$.", "Then it is clear that", "$$", "\\xymatrix{", "\\mathit{Isom}(u, v) \\ar[d] \\ar[rr] & &", "\\mathit{Isom}(y, y') \\ar[d]^{\\phi \\mapsto g(\\phi) \\circ \\alpha} \\\\", "\\mathit{Isom}(x, x') \\ar[rr]^-{\\psi \\mapsto \\alpha' \\circ f(\\psi)} & &", "\\mathit{Isom}(f(x), g(y'))", "}", "$$", "is a cartesian diagram of sheaves on $(\\Sch/T)_{fppf}$.", "Since by assumption the sheaves", "$\\mathit{Isom}(y, y')$, $\\mathit{Isom}(x, x')$, $\\mathit{Isom}(f(x), g(y'))$", "are algebraic spaces (see", "Lemma \\ref{lemma-representable-diagonal})", "we see that $\\mathit{Isom}(u, v)$", "is an algebraic space.", "\\medskip\\noindent", "Let $U, V \\in \\Ob((\\Sch/S)_{fppf})$,", "and let $x, y$ be surjective smooth morphisms", "$x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$,", "$y : (\\Sch/V)_{fppf} \\to \\mathcal{Y}$.", "Consider the morphism", "$$", "(\\Sch/U)_{fppf}", "\\times_{f \\circ x, \\mathcal{Z}, g \\circ y}", "(\\Sch/V)_{fppf}", "\\longrightarrow", "\\mathcal{X} \\times_{f, \\mathcal{Z}, g} \\mathcal{Y}.", "$$", "As the diagonal of $\\mathcal{Z}$ is representable by algebraic spaces", "the source of this arrow is representable by an algebraic space $F$, see", "Lemma \\ref{lemma-representable-diagonal}.", "Moreover, the morphism is the composition", "of base changes of $x$ and $y$, hence surjective and smooth, see", "Lemmas \\ref{lemma-base-change-representable-transformations-property} and", "\\ref{lemma-composition-representable-transformations-property}.", "Choosing a scheme $W$ and a surjective \\'etale morphism $W \\to F$", "we see that the composition of the displayed $1$-morphism", "with the corresponding $1$-morphism", "$$", "(\\Sch/W)_{fppf}", "\\longrightarrow", "(\\Sch/U)_{fppf}", "\\times_{f \\circ x, \\mathcal{Z}, g \\circ y}", "(\\Sch/V)_{fppf}", "$$", "is surjective and smooth which proves the last condition." ], "refs": [ "algebraic-definition-algebraic-stack", "stacks-lemma-2-product-stacks-in-groupoids", "categories-lemma-2-product-categories-over-C", "algebraic-lemma-representable-diagonal", "algebraic-lemma-representable-diagonal", "algebraic-lemma-base-change-representable-transformations-property", "algebraic-lemma-composition-representable-transformations-property" ], "ref_ids": [ 8484, 8949, 12280, 8461, 8461, 8456, 8455 ] } ], "ref_ids": [] }, { "id": 8467, "type": "theorem", "label": "algebraic-lemma-2-fibre-product", "categories": [ "algebraic" ], "title": "algebraic-lemma-2-fibre-product", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ be algebraic stacks over $S$.", "Let $f : \\mathcal{X} \\to \\mathcal{Z}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$", "be $1$-morphisms of algebraic stacks. Then the $2$-fibre product", "$\\mathcal{X} \\times_{f, \\mathcal{Z}, g} \\mathcal{Y}$ is an algebraic stack.", "It is also the $2$-fibre product in the $2$-category of algebraic stacks", "over $(\\Sch/S)_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "The fact that $\\mathcal{X} \\times_{f, \\mathcal{Z}, g} \\mathcal{Y}$ is an", "algebraic stack follows from the stronger", "Lemma \\ref{lemma-2-fibre-product-general}.", "The fact that $\\mathcal{X} \\times_{f, \\mathcal{Z}, g} \\mathcal{Y}$", "is a $2$-fibre product in the $2$-category of algebraic stacks over $S$", "follows formally from the fact that the $2$-category of algebraic stacks", "over $S$ is a full sub $2$-category of the $2$-category of stacks in", "groupoids over $(\\Sch/S)_{fppf}$." ], "refs": [ "algebraic-lemma-2-fibre-product-general" ], "ref_ids": [ 8466 ] } ], "ref_ids": [] }, { "id": 8468, "type": "theorem", "label": "algebraic-lemma-lift-morphism-presentations", "categories": [ "algebraic" ], "title": "algebraic-lemma-lift-morphism-presentations", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of algebraic", "stacks over $S$.", "Let $V \\in \\Ob((\\Sch/S)_{fppf})$.", "Let $y : (\\Sch/V)_{fppf} \\to \\mathcal{Y}$ be surjective and smooth.", "Then there exists an object $U \\in \\Ob((\\Sch/S)_{fppf})$", "and a $2$-commutative diagram", "$$", "\\xymatrix{", "(\\Sch/U)_{fppf} \\ar[r]_a \\ar[d]_x &", "(\\Sch/V)_{fppf} \\ar[d]^y \\\\", "\\mathcal{X} \\ar[r]^f & \\mathcal{Y}", "}", "$$", "with $x$ surjective and smooth." ], "refs": [], "proofs": [ { "contents": [ "First choose $W \\in \\Ob((\\Sch/S)_{fppf})$ and a surjective", "smooth $1$-morphism $z : (\\Sch/W)_{fppf} \\to \\mathcal{X}$.", "As $\\mathcal{Y}$ is an algebraic stack we may choose an equivalence", "$$", "j :", "\\mathcal{S}_F", "\\longrightarrow", "(\\Sch/W)_{fppf}", "\\times_{f \\circ z, \\mathcal{Y}, y}", "(\\Sch/V)_{fppf}", "$$", "where $F$ is an algebraic space. By", "Lemma \\ref{lemma-base-change-representable-transformations-property}", "the morphism", "$\\mathcal{S}_F \\to (\\Sch/W)_{fppf}$ is surjective and smooth", "as a base change of $y$. Hence by", "Lemma \\ref{lemma-composition-representable-transformations-property}", "we see that $\\mathcal{S}_F \\to \\mathcal{X}$ is surjective and smooth.", "Choose an object $U \\in \\Ob((\\Sch/S)_{fppf})$", "and a surjective \\'etale morphism $U \\to F$. Then applying", "Lemma \\ref{lemma-composition-representable-transformations-property}", "once more we obtain the desired properties." ], "refs": [ "algebraic-lemma-base-change-representable-transformations-property", "algebraic-lemma-composition-representable-transformations-property", "algebraic-lemma-composition-representable-transformations-property" ], "ref_ids": [ 8456, 8455, 8455 ] } ], "ref_ids": [] }, { "id": 8469, "type": "theorem", "label": "algebraic-lemma-characterize-representable-by-algebraic-spaces", "categories": [ "algebraic" ], "title": "algebraic-lemma-characterize-representable-by-algebraic-spaces", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of algebraic", "stacks over $S$. The following are equivalent:", "\\begin{enumerate}", "\\item for $U \\in \\Ob((\\Sch/S)_{fppf})$", "the functor $f : \\mathcal{X}_U \\to \\mathcal{Y}_U$ is faithful,", "\\item the functor $f$ is faithful, and", "\\item $f$ is representable by algebraic spaces.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1) and (2) are equivalent by general properties of $1$-morphisms", "of categories fibred in groupoids, see", "Categories, Lemma \\ref{categories-lemma-equivalence-fibred-categories}.", "We see that (3) implies (2) by", "Lemma \\ref{lemma-criterion-map-representable-spaces-fibred-in-groupoids}.", "Finally, assume (2).", "Let $U$ be a scheme. Let $y \\in \\Ob(\\mathcal{Y}_U)$.", "We have to prove that", "$$", "\\mathcal{W} = (\\Sch/U)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}", "$$", "is representable by an algebraic space over $U$. Since", "$(\\Sch/U)_{fppf}$ is an algebraic stack we see from", "Lemma \\ref{lemma-2-fibre-product}", "that $\\mathcal{W}$ is an algebraic stack.", "On the other hand the explicit description of objects of $\\mathcal{W}$", "as triples $(V, x, \\alpha : y(V) \\to f(x))$ and the fact that $f$ is", "faithful, shows that the fibre categories of $\\mathcal{W}$ are setoids. Hence", "Proposition \\ref{proposition-algebraic-stack-no-automorphisms}", "guarantees that $\\mathcal{W}$ is representable by an algebraic space." ], "refs": [ "categories-lemma-equivalence-fibred-categories", "algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids", "algebraic-lemma-2-fibre-product", "algebraic-proposition-algebraic-stack-no-automorphisms" ], "ref_ids": [ 12297, 8442, 8467, 8480 ] } ], "ref_ids": [] }, { "id": 8470, "type": "theorem", "label": "algebraic-lemma-smooth-surjective-morphism-implies-algebraic", "categories": [ "algebraic" ], "title": "algebraic-lemma-smooth-surjective-morphism-implies-algebraic", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $u : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of", "stacks in groupoids over $(\\Sch/S)_{fppf}$. If", "\\begin{enumerate}", "\\item $\\mathcal{U}$ is representable by an algebraic space, and", "\\item $u$ is representable by algebraic spaces, surjective and smooth,", "\\end{enumerate}", "then $\\mathcal X$ is an algebraic stack over $S$." ], "refs": [], "proofs": [ { "contents": [ "We have to show that $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$", "is representable by algebraic spaces, see", "Definition \\ref{definition-algebraic-stack}.", "Given two schemes $T_1$, $T_2$ over $S$ denote", "$\\mathcal{T}_i = (\\Sch/T_i)_{fppf}$ the associated representable", "fibre categories. Suppose given $1$-morphisms", "$f_i : \\mathcal{T}_i \\to \\mathcal{X}$.", "According to", "Lemma \\ref{lemma-representable-diagonal}", "it suffices to prove that the $2$-fibered", "product $\\mathcal{T}_1 \\times_\\mathcal{X} \\mathcal{T}_2$", "is representable by an algebraic space. By", "Stacks, Lemma", "\\ref{stacks-lemma-2-fibre-product-stacks-in-setoids-over-stack-in-groupoids}", "this is in any case a stack in setoids. Thus", "$\\mathcal{T}_1 \\times_\\mathcal{X} \\mathcal{T}_2$ corresponds", "to some sheaf $F$ on $(\\Sch/S)_{fppf}$, see", "Stacks, Lemma \\ref{stacks-lemma-stack-in-setoids-characterize}.", "Let $U$ be the algebraic space which represents $\\mathcal{U}$.", "By assumption", "$$", "\\mathcal{T}_i' = \\mathcal{U} \\times_{u, \\mathcal{X}, f_i} \\mathcal{T}_i", "$$", "is representable by an algebraic space $T'_i$ over $S$. Hence", "$\\mathcal{T}_1' \\times_\\mathcal{U} \\mathcal{T}_2'$ is representable", "by the algebraic space $T'_1 \\times_U T'_2$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "&", "\\mathcal{T}_1 \\times_{\\mathcal X} \\mathcal{T}_2 \\ar[rr]\\ar'[d][dd] & &", "\\mathcal{T}_1 \\ar[dd] \\\\", "\\mathcal{T}_1' \\times_\\mathcal{U} \\mathcal{T}_2' \\ar[ur]\\ar[rr]\\ar[dd] & &", "\\mathcal{T}_1' \\ar[ur]\\ar[dd] \\\\", "&", "\\mathcal{T}_2 \\ar'[r][rr] & &", "\\mathcal X \\\\", "\\mathcal{T}_2' \\ar[rr]\\ar[ur] & &", "\\mathcal{U} \\ar[ur] }", "$$", "In this diagram the bottom square, the right square, the back square, and", "the front square are $2$-fibre products. A formal argument then shows", "that $\\mathcal{T}_1' \\times_\\mathcal{U} \\mathcal{T}_2' \\to", "\\mathcal{T}_1 \\times_{\\mathcal X} \\mathcal{T}_2$", "is the ``base change'' of $\\mathcal{U} \\to \\mathcal{X}$, more precisely", "the diagram", "$$", "\\xymatrix{", "\\mathcal{T}_1' \\times_\\mathcal{U} \\mathcal{T}_2' \\ar[d] \\ar[r] &", "\\mathcal{U} \\ar[d] \\\\", "\\mathcal{T}_1 \\times_{\\mathcal X} \\mathcal{T}_2 \\ar[r] &", "\\mathcal{X}", "}", "$$", "is a $2$-fibre square.", "Hence $T'_1 \\times_U T'_2 \\to F$ is representable by algebraic spaces,", "smooth, and surjective, see", "Lemmas \\ref{lemma-map-fibred-setoids-representable-algebraic-spaces},", "\\ref{lemma-base-change-representable-by-spaces},", "\\ref{lemma-map-fibred-setoids-property}, and", "\\ref{lemma-base-change-representable-transformations-property}.", "Therefore $F$ is an algebraic space by", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}", "and we win." ], "refs": [ "algebraic-definition-algebraic-stack", "algebraic-lemma-representable-diagonal", "stacks-lemma-2-fibre-product-stacks-in-setoids-over-stack-in-groupoids", "stacks-lemma-stack-in-setoids-characterize", "algebraic-lemma-map-fibred-setoids-representable-algebraic-spaces", "algebraic-lemma-base-change-representable-by-spaces", "algebraic-lemma-map-fibred-setoids-property", "algebraic-lemma-base-change-representable-transformations-property", "bootstrap-theorem-final-bootstrap" ], "ref_ids": [ 8484, 8461, 8955, 8951, 8446, 8447, 8454, 8456, 2602 ] } ], "ref_ids": [] }, { "id": 8471, "type": "theorem", "label": "algebraic-lemma-representable-morphism-to-algebraic", "categories": [ "algebraic" ], "title": "algebraic-lemma-representable-morphism-to-algebraic", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of stacks in groupoids", "over $(\\Sch/S)_{fppf}$. Assume that", "\\begin{enumerate}", "\\item $\\mathcal{X} \\to \\mathcal{Y}$ is representable by algebraic spaces, and", "\\item $\\mathcal{Y}$ is an algebraic stack over $S$.", "\\end{enumerate}", "Then $\\mathcal{X}$ is an algebraic stack over $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{V} \\to \\mathcal{Y}$ be a surjective smooth $1$-morphism", "from a representable stack in groupoids to $\\mathcal{Y}$. This exists by", "Definition \\ref{definition-algebraic-stack}.", "Then the $2$-fibre product", "$\\mathcal{U} = \\mathcal{V} \\times_{\\mathcal Y} \\mathcal X$", "is representable by an algebraic space by", "Lemma \\ref{lemma-base-change-by-space-representable-by-space}.", "The $1$-morphism $\\mathcal{U} \\to \\mathcal X$ is representable by algebraic", "spaces, smooth, and surjective, see", "Lemmas \\ref{lemma-base-change-representable-by-spaces} and", "\\ref{lemma-base-change-representable-transformations-property}.", "By", "Lemma \\ref{lemma-smooth-surjective-morphism-implies-algebraic}", "we conclude that $\\mathcal{X}$ is an algebraic stack." ], "refs": [ "algebraic-definition-algebraic-stack", "algebraic-lemma-base-change-by-space-representable-by-space", "algebraic-lemma-base-change-representable-by-spaces", "algebraic-lemma-base-change-representable-transformations-property", "algebraic-lemma-smooth-surjective-morphism-implies-algebraic" ], "ref_ids": [ 8484, 8448, 8447, 8456, 8470 ] } ], "ref_ids": [] }, { "id": 8472, "type": "theorem", "label": "algebraic-lemma-open-fibred-category-is-algebraic", "categories": [ "algebraic" ], "title": "algebraic-lemma-open-fibred-category-is-algebraic", "contents": [ "\\begin{reference}", "Removing the hypothesis that $j$ is a monomorphism was observed", "in an email from Matthew Emerton dates June 15, 2016", "\\end{reference}", "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $j : \\mathcal X \\to \\mathcal Y$ be a $1$-morphism of", "categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "Assume $j$ is representable by algebraic spaces.", "Then, if $\\mathcal{Y}$ is a stack in groupoids", "(resp.\\ an algebraic stack), so is $\\mathcal{X}$." ], "refs": [], "proofs": [ { "contents": [ "The statement on algebraic stacks will follow from the statement on", "stacks in groupoids by Lemma \\ref{lemma-representable-morphism-to-algebraic}.", "If $j$ is representable by algebraic spaces, then $j$ is", "faithful on fibre categories and for each $U$ and each", "$y \\in \\Ob(\\mathcal{Y}_U)$ the presheaf", "$$", "(h : V \\to U)", "\\longmapsto", "\\{(x, \\phi) \\mid x \\in \\Ob(\\mathcal{X}_V), \\phi : h^*y \\to f(x)\\}/\\cong", "$$", "is an algebraic space over $U$. See", "Lemma \\ref{lemma-criterion-map-representable-spaces-fibred-in-groupoids}.", "In particular this presheaf is a sheaf and the conclusion follows", "from Stacks, Lemma \\ref{stacks-lemma-relative-sheaf-over-stack-is-stack}." ], "refs": [ "algebraic-lemma-representable-morphism-to-algebraic", "algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids", "stacks-lemma-relative-sheaf-over-stack-is-stack" ], "ref_ids": [ 8471, 8442, 8958 ] } ], "ref_ids": [] }, { "id": 8473, "type": "theorem", "label": "algebraic-lemma-map-space-into-stack", "categories": [ "algebraic" ], "title": "algebraic-lemma-map-space-into-stack", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}$ be an algebraic stack over $S$.", "Let $\\mathcal{U}$ be an algebraic stack over $S$ which", "is representable by an algebraic space.", "Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a 1-morphism. Then", "\\begin{enumerate}", "\\item the $2$-fibre product", "$\\mathcal{R} = \\mathcal{U} \\times_{f, \\mathcal{X}, f} \\mathcal{U}$", "is representable by an algebraic space,", "\\item there is a canonical equivalence", "$$", "\\mathcal{U} \\times_{f, \\mathcal{X}, f} \\mathcal{U}", "\\times_{f, \\mathcal{X}, f} \\mathcal{U} =", "\\mathcal{R} \\times_{\\text{pr}_1, \\mathcal{U}, \\text{pr}_0} \\mathcal{R},", "$$", "\\item the projection $\\text{pr}_{02}$ induces via (2) a $1$-morphism", "$$", "\\text{pr}_{02} :", "\\mathcal{R} \\times_{\\text{pr}_1, \\mathcal{U}, \\text{pr}_0} \\mathcal{R}", "\\longrightarrow", "\\mathcal{R}", "$$", "\\item let $U$, $R$ be the algebraic spaces representing", "$\\mathcal{U}, \\mathcal{R}$ and $t, s : R \\to U$ and", "$c : R \\times_{s, U, t} R \\to R$ are the morphisms corresponding", "to the $1$-morphisms", "$\\text{pr}_0, \\text{pr}_1 : \\mathcal{R} \\to \\mathcal{U}$", "and", "$\\text{pr}_{02} :", "\\mathcal{R} \\times_{\\text{pr}_1, \\mathcal{U}, \\text{pr}_0} \\mathcal{R} \\to", "\\mathcal{R}$ above, then the quintuple $(U, R, s, t, c)$ is a groupoid in", "algebraic spaces over $S$,", "\\item the morphism $f$ induces a canonical $1$-morphism", "$f_{can} : [U/R] \\to \\mathcal{X}$", "of stacks in groupoids over $(\\Sch/S)_{fppf}$, and", "\\item the $1$-morphism $f_{can} : [U/R] \\to \\mathcal{X}$ is fully faithful.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). By definition $\\Delta_\\mathcal{X}$ is representable", "by algebraic spaces so", "Lemma \\ref{lemma-representable-diagonal}", "applies to show that $\\mathcal{U} \\to \\mathcal{X}$ is representable", "by algebraic spaces. Hence the result follows from", "Lemma \\ref{lemma-base-change-by-space-representable-by-space}.", "\\medskip\\noindent", "Let $T$ be a scheme over $S$. By construction of the $2$-fibre product (see", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C})", "we see that the objects of the fibre category $\\mathcal{R}_T$", "are triples $(a, b, \\alpha)$ where $a, b \\in \\Ob(\\mathcal{U}_T)$", "and $\\alpha : f(a) \\to f(b)$", "is a morphism in the fibre category $\\mathcal{X}_T$.", "\\medskip\\noindent", "Proof of (2). The equivalence comes from repeatedly applying", "Categories, Lemmas \\ref{categories-lemma-associativity-2-fibre-product} and", "\\ref{categories-lemma-2-fibre-product-erase-factor}.", "Let us identify", "$\\mathcal{U} \\times_\\mathcal{X} \\mathcal{U} \\times_\\mathcal{X} \\mathcal{U}$", "with", "$(\\mathcal{U} \\times_\\mathcal{X} \\mathcal{U})", "\\times_\\mathcal{X} \\mathcal{U}$.", "If $T$ is a scheme over $S$, then on fibre categories over $T$", "this equivalence maps the object", "$((a, b, \\alpha), c, \\beta)$ on the left hand side", "to the object $((a, b, \\alpha), (b, c, \\beta))$ of the right hand side.", "\\medskip\\noindent", "Proof of (3). The $1$-morphism $\\text{pr}_{02}$ is constructed in the proof of", "Categories, Lemma \\ref{categories-lemma-triple-2-fibre-product-pr02}.", "In terms of the description of objects of the fibre category", "above we see that $((a, b, \\alpha), (b, c, \\beta))$", "maps to $(a, c, \\beta \\circ \\alpha)$.", "\\medskip\\noindent", "Unfortunately, this is {\\it not compatible} with our conventions on", "groupoids where we always have $j = (t, s) : R \\to U$, and we ``think''", "of a $T$-valued point $r$ of $R$ as a morphism $r : s(r) \\to t(r)$.", "However, this does not affect the proof of (4), since the opposite of", "a groupoid is a groupoid. But in the proof of (5) it is responsible", "for the inverses in the displayed formula below.", "\\medskip\\noindent", "Proof of (4). Recall that the sheaf $U$ is isomorphic to the sheaf", "$T \\mapsto \\Ob(\\mathcal{U}_T)/\\!\\cong$, and", "similarly for $R$, see", "Lemma \\ref{lemma-characterize-representable-by-space}.", "It follows from", "Categories,", "Lemma \\ref{categories-lemma-category-fibred-setoids-presheaves-products}", "that this description is compatible with $2$-fibre products", "so we get a similar matching of", "$\\mathcal{R} \\times_{\\text{pr}_1, \\mathcal{U}, \\text{pr}_0} \\mathcal{R}$", "and $R \\times_{s, U, t} R$.", "The morphisms $t, s : R \\to U$ and $c : R \\times_{s, U, t} R \\to R$", "we get from the general equality (\\ref{equation-morphisms-spaces}).", "Explicitly these maps are the transformations of functors that come", "from letting $\\text{pr}_0$, $\\text{pr}_0$, $\\text{pr}_{02}$", "act on isomorphism classes of objects of fibre categories.", "Hence to show that we obtain a groupoid in algebraic", "spaces it suffices to show that for every scheme $T$ over $S$", "the structure", "$$", "(\\Ob(\\mathcal{U}_T)/\\!\\cong,", "\\Ob(\\mathcal{R}_T)/\\!\\cong,", "\\text{pr}_1, \\text{pr}_0, \\text{pr}_{02})", "$$", "is a groupoid which is clear from our description of objects of", "$\\mathcal{R}_T$ above.", "\\medskip\\noindent", "Proof of (5). We will eventually apply", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-2-coequalizer}", "to obtain the functor $[U/R] \\to \\mathcal{X}$.", "Consider the $1$-morphism $f : \\mathcal{U} \\to \\mathcal{X}$.", "We have a $2$-arrow $\\tau : f \\circ \\text{pr}_1 \\to f \\circ \\text{pr}_0$", "by definition of $\\mathcal{R}$ as the $2$-fibre product.", "Namely, on an object $(a, b, \\alpha)$ of $\\mathcal{R}$ over $T$ it is", "the map $\\alpha^{-1} : b \\to a$. We claim that", "$$", "\\tau \\circ \\text{id}_{\\text{pr}_{02}} =", "(\\tau \\star \\text{id}_{\\text{pr}_0})", "\\circ", "(\\tau \\star \\text{id}_{\\text{pr}_1}).", "$$", "This identity says that given an object", "$((a, b, \\alpha), (b, c, \\beta))$ of", "$\\mathcal{R} \\times_{\\text{pr}_1, \\mathcal{U}, \\text{pr}_0} \\mathcal{R}$", "over $T$, then the composition of", "$$", "\\xymatrix{", "c \\ar[r]^{\\beta^{-1}} & b \\ar[r]^{\\alpha^{-1}} & a", "}", "$$", "is the same as the arrow $(\\beta \\circ \\alpha)^{-1} : a \\to c$. This is", "clearly true, hence the claim holds. In this way we see that all the", "assumption of", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-2-coequalizer}", "are satisfied for the structure", "$(\\mathcal{U}, \\mathcal{R}, \\text{pr}_0, \\text{pr}_1, \\text{pr}_{02})$", "and the $1$-morphism $f$ and the $2$-morphism $\\tau$.", "Except, to apply the lemma we need to prove this holds", "for the structure $(\\mathcal{S}_U, \\mathcal{S}_R, s, t, c)$", "with suitable morphisms.", "\\medskip\\noindent", "Now there should be some general abstract nonsense", "argument which transfer these data between the two, but it seems to", "be quite long. Instead, we use the following trick.", "Pick a quasi-inverse $j^{-1} : \\mathcal{S}_U \\to \\mathcal{U}$", "of the canonical equivalence $j : \\mathcal{U} \\to \\mathcal{S}_U$ which comes", "from $U(T) = \\Ob(\\mathcal{U}_T)/\\!\\!\\cong$.", "This just means that for every scheme $T/S$ and every", "object $a \\in \\mathcal{U}_T$ we have picked out a particular", "element of its isomorphism class, namely $j^{-1}(j(a))$.", "Using $j^{-1}$ we may therefore see $\\mathcal{S}_U$", "as a subcategory of $\\mathcal{U}$. Having chosen this subcategory", "we can consider those objects $(a, b, \\alpha)$ of $\\mathcal{R}_T$", "such that $a, b$ are objects of $(\\mathcal{S}_U)_T$, i.e., such", "that $j^{-1}(j(a)) = a$ and $j^{-1}(j(b)) = b$. Then it is clear that", "this forms a subcategory of $\\mathcal{R}$ which maps isomorphically", "to $\\mathcal{S}_R$ via the canonical equivalence", "$\\mathcal{R} \\to \\mathcal{S}_R$. Moreover, this is clearly compatible", "with forming the $2$-fibre product", "$\\mathcal{R} \\times_{\\text{pr}_1, \\mathcal{U}, \\text{pr}_0} \\mathcal{R}$.", "Hence we see that we may simply restrict", "$f$ to $\\mathcal{S}_U$ and restrict $\\tau$ to a transformation", "between functors $\\mathcal{S}_R \\to \\mathcal{X}$. Hence it is clear that", "the displayed equality of", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-2-coequalizer}", "holds since it holds even as an equality of transformations of functors", "$\\mathcal{R} \\times_{\\text{pr}_1, \\mathcal{U}, \\text{pr}_0} \\mathcal{R}", "\\to \\mathcal{X}$ before restricting to the subcategory", "$\\mathcal{S}_{R \\times_{s, U, t} R}$.", "\\medskip\\noindent", "This proves that", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-2-coequalizer}", "applies and we get our desired morphism of stacks", "$f_{can} : [U/R] \\to \\mathcal{X}$. We briefly spell out how", "$f_{can}$ is defined in this special case.", "On an object $a$ of $\\mathcal{S}_U$ over $T$", "we have $f_{can}(a) = f(a)$, where we think of", "$\\mathcal{S}_U \\subset \\mathcal{U}$ by the chosen embedding above.", "If $a, b$ are objects of $\\mathcal{S}_U$ over $T$, then a morphism", "$\\varphi : a \\to b$ in $[U/R]$ is by definition an object of the", "form $\\varphi = (b, a, \\alpha)$ of $\\mathcal{R}$ over $T$. (Note", "switch.) And the rule in the proof of", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-2-coequalizer}", "is that", "\\begin{equation}", "\\label{equation-on-morphisms}", "f_{can}(\\varphi) = \\Big(f(a) \\xrightarrow{\\alpha^{-1}} f(b)\\Big).", "\\end{equation}", "Proof of (6). Both $[U/R]$ and $\\mathcal{X}$ are stacks.", "Hence given a scheme $T/S$ and objects $a, b$ of $[U/R]$", "over $T$ we obtain a transformation of fppf sheaves", "$$", "\\mathit{Isom}(a, b) \\longrightarrow \\mathit{Isom}(f_{can}(a), f_{can}(b))", "$$", "on $(\\Sch/T)_{fppf}$. We have to show that this is an", "isomorphism. We may work fppf locally on $T$, hence we may assume that", "$a, b$ come from morphisms $a, b : T \\to U$. By the embedding", "$\\mathcal{S}_U \\subset \\mathcal{U}$ above we may also think of $a, b$ as", "objects of $\\mathcal{U}$ over $T$. In", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-morphisms}", "we have seen that the left hand sheaf is represented by the algebraic space", "$$", "R \\times_{(t, s), U \\times_S U, (b, a)} T", "$$", "over $T$. On the other hand, the right hand side is by", "Stacks, Lemma \\ref{stacks-lemma-isom-as-2-fibre-product}", "equal to the sheaf associated to the following stack in setoids:", "$$", "\\mathcal{X}", "\\times_{\\mathcal{X} \\times \\mathcal{X}, (f \\circ b, f \\circ a)} T =", "\\mathcal{X}", "\\times_{\\mathcal{X} \\times \\mathcal{X}, (f, f)}", "(\\mathcal{U} \\times \\mathcal{U})", "\\times_{\\mathcal{U} \\times \\mathcal{U}, (b, a)} T =", "\\mathcal{R}", "\\times_{(\\text{pr}_0, \\text{pr}_1), \\mathcal{U} \\times \\mathcal{U}, (b, a)} T", "$$", "which is representable by the fibre product displayed above.", "At this point we have shown that the two $\\mathit{Isom}$-sheaves", "are isomorphic. Our $1$-morphism $f_{can} : [U/R] \\to \\mathcal{X}$ induces", "this isomorphism on $\\mathit{Isom}$-sheaves by", "Equation (\\ref{equation-on-morphisms})." ], "refs": [ "algebraic-lemma-representable-diagonal", "algebraic-lemma-base-change-by-space-representable-by-space", "categories-lemma-2-product-categories-over-C", "categories-lemma-associativity-2-fibre-product", "categories-lemma-2-fibre-product-erase-factor", "categories-lemma-triple-2-fibre-product-pr02", "algebraic-lemma-characterize-representable-by-space", "categories-lemma-category-fibred-setoids-presheaves-products", "spaces-groupoids-lemma-quotient-stack-2-coequalizer", "spaces-groupoids-lemma-quotient-stack-2-coequalizer", "spaces-groupoids-lemma-quotient-stack-2-coequalizer", "spaces-groupoids-lemma-quotient-stack-2-coequalizer", "spaces-groupoids-lemma-quotient-stack-2-coequalizer", "spaces-groupoids-lemma-quotient-stack-morphisms", "stacks-lemma-isom-as-2-fibre-product" ], "ref_ids": [ 8461, 8448, 12280, 12273, 12275, 12274, 8441, 12315, 9327, 9327, 9327, 9327, 9327, 9323, 8936 ] } ], "ref_ids": [] }, { "id": 8474, "type": "theorem", "label": "algebraic-lemma-stack-presentation", "categories": [ "algebraic" ], "title": "algebraic-lemma-stack-presentation", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}$ be an algebraic stack over $S$.", "Let $U$ be an algebraic space over $S$.", "Let $f : \\mathcal{S}_U \\to \\mathcal{X}$ be a surjective smooth morphism.", "Let $(U, R, s, t, c)$ be the groupoid in algebraic spaces", "and $f_{can} : [U/R] \\to \\mathcal{X}$ be the result of applying", "Lemma \\ref{lemma-map-space-into-stack}", "to $U$ and $f$. Then", "\\begin{enumerate}", "\\item the morphisms $s$, $t$ are smooth, and", "\\item the $1$-morphism $f_{can} : [U/R] \\to \\mathcal{X}$", "is an equivalence.", "\\end{enumerate}" ], "refs": [ "algebraic-lemma-map-space-into-stack" ], "proofs": [ { "contents": [ "The morphisms $s, t$ are smooth by", "Lemmas \\ref{lemma-property-morphism-equivalent} and", "\\ref{lemma-map-presheaves-representable-by-spaces-transformation-property}.", "As the $1$-morphism $f$ is smooth and", "surjective it is clear that given any scheme $T$ and any object", "$a \\in \\Ob(\\mathcal{X}_T)$ there exists a smooth and surjective", "morphism $T' \\to T$ such that $a|_T'$ comes from an object of", "$[U/R]_{T'}$. Since $f_{can} : [U/R] \\to \\mathcal{X}$", "is fully faithful, we deduce that", "$[U/R] \\to \\mathcal{X}$ is essentially surjective as", "descent data on objects are effective on both sides, see", "Stacks, Lemma \\ref{stacks-lemma-characterize-essentially-surjective-when-ff}." ], "refs": [ "algebraic-lemma-property-morphism-equivalent", "algebraic-lemma-map-presheaves-representable-by-spaces-transformation-property", "stacks-lemma-characterize-essentially-surjective-when-ff" ], "ref_ids": [ 8452, 8453, 8945 ] } ], "ref_ids": [ 8473 ] }, { "id": 8475, "type": "theorem", "label": "algebraic-lemma-diagonal-quotient-stack", "categories": [ "algebraic" ], "title": "algebraic-lemma-diagonal-quotient-stack", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.", "Then the diagonal of $[U/R]$ is representable by algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that the $\\mathit{Isom}$-sheaves are algebraic", "spaces, see", "Lemma \\ref{lemma-representable-diagonal}.", "This follows from", "Bootstrap, Lemma \\ref{bootstrap-lemma-quotient-stack-isom}." ], "refs": [ "algebraic-lemma-representable-diagonal", "bootstrap-lemma-quotient-stack-isom" ], "ref_ids": [ 8461, 2629 ] } ], "ref_ids": [] }, { "id": 8476, "type": "theorem", "label": "algebraic-lemma-smooth-quotient-smooth-presentation", "categories": [ "algebraic" ], "title": "algebraic-lemma-smooth-quotient-smooth-presentation", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces over $S$.", "Then the morphism $\\mathcal{S}_U \\to [U/R]$ is smooth and surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme and let $x : (\\Sch/T)_{fppf} \\to [U/R]$", "be a $1$-morphism. We have to show that the projection", "$$", "\\mathcal{S}_U \\times_{[U/R]} (\\Sch/T)_{fppf}", "\\longrightarrow", "(\\Sch/T)_{fppf}", "$$", "is surjective and smooth. We already know that the left hand side", "is representable by an algebraic space $F$, see", "Lemmas \\ref{lemma-diagonal-quotient-stack} and", "\\ref{lemma-representable-diagonal}.", "Hence we have to show the corresponding morphism $F \\to T$ of", "algebraic spaces is surjective and smooth.", "Since we are working with properties of morphisms of algebraic", "spaces which are local on the target in the fppf topology we", "may check this fppf locally on $T$. By construction, there exists", "an fppf covering $\\{T_i \\to T\\}$ of $T$ such that", "$x|_{(\\Sch/T_i)_{fppf}}$ comes from a morphism", "$x_i : T_i \\to U$. (Note that $F \\times_T T_i$ represents the", "$2$-fibre product $\\mathcal{S}_U \\times_{[U/R]} (\\Sch/T_i)_{fppf}$", "so everything is compatible with the base change via $T_i \\to T$.)", "Hence we may assume that $x$ comes from $x : T \\to U$.", "In this case we see that", "$$", "\\mathcal{S}_U \\times_{[U/R]} (\\Sch/T)_{fppf}", "=", "(\\mathcal{S}_U \\times_{[U/R]} \\mathcal{S}_U)", "\\times_{\\mathcal{S}_U} (\\Sch/T)_{fppf}", "=", "\\mathcal{S}_R \\times_{\\mathcal{S}_U} (\\Sch/T)_{fppf}", "$$", "The first equality by", "Categories, Lemma \\ref{categories-lemma-2-fibre-product-erase-factor}", "and the second equality by", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-2-cartesian}.", "Clearly the last $2$-fibre product is represented by the algebraic", "space $F = R \\times_{s, U, x} T$ and the projection", "$R \\times_{s, U, x} T \\to T$ is smooth as the base change of", "the smooth morphism of algebraic spaces $s : R \\to U$.", "It is also surjective as $s$ has a section (namely the identity", "$e : U \\to R$ of the groupoid).", "This proves the lemma." ], "refs": [ "algebraic-lemma-diagonal-quotient-stack", "algebraic-lemma-representable-diagonal", "categories-lemma-2-fibre-product-erase-factor", "spaces-groupoids-lemma-quotient-stack-2-cartesian" ], "ref_ids": [ 8475, 8461, 12275, 9324 ] } ], "ref_ids": [] }, { "id": 8477, "type": "theorem", "label": "algebraic-lemma-change-big-site", "categories": [ "algebraic" ], "title": "algebraic-lemma-change-big-site", "contents": [ "Suppose given big sites $\\Sch_{fppf}$ and $\\Sch'_{fppf}$.", "Assume that $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$,", "see Topologies, Section \\ref{topologies-section-change-alpha}.", "Let $S$ be an object of $\\Sch_{fppf}$.", "Let $f : (\\Sch'/S)_{fppf} \\to (\\Sch/S)_{fppf}$ the morphism", "of sites corresponding to the inclusion functor", "$u : (\\Sch/S)_{fppf} \\to (\\Sch'/S)_{fppf}$.", "Let $\\mathcal{X}$ be a stack in groupoids over $(\\Sch/S)_{fppf}$.", "\\begin{enumerate}", "\\item if $\\mathcal{X}$ is representable by some", "$X \\in \\Ob((\\Sch/S)_{fppf})$, then", "$f^{-1}\\mathcal{X}$ is representable too, in fact it is representable by the", "same scheme $X$, now viewed as an object of $(\\Sch'/S)_{fppf}$,", "\\item if $\\mathcal{X}$ is representable by", "$F \\in \\Sh((\\Sch/S)_{fppf})$ which is", "an algebraic space, then $f^{-1}\\mathcal{X}$ is representable", "by the algebraic space $f^{-1}F$,", "\\item if $\\mathcal{X}$ is an algebraic stack, then $f^{-1}\\mathcal{X}$", "is an algebraic stack, and", "\\item if $\\mathcal{X}$ is a Deligne-Mumford stack, then $f^{-1}\\mathcal{X}$", "is a Deligne-Mumford stack too.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us prove (3). By", "Lemma \\ref{lemma-stack-presentation}", "we may write $\\mathcal{X} = [U/R]$ for some smooth", "groupoid in algebraic spaces $(U, R, s, t, c)$. By", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-change-big-site}", "we see that $f^{-1}[U/R] = [f^{-1}U/f^{-1}R]$.", "Of course $(f^{-1}U, f^{-1}R, f^{-1}s, f^{-1}t, f^{-1}c)$", "is a smooth groupoid in algebraic spaces too. Hence (3) is proved.", "\\medskip\\noindent", "Now the other cases (1), (2), (4) each mean that $\\mathcal{X}$ has", "a presentation $[U/R]$ of a particular kind, and hence translate into the", "same kind of presentation for $f^{-1}\\mathcal{X} = [f^{-1}U/f^{-1}R]$.", "Whence the lemma is proved." ], "refs": [ "algebraic-lemma-stack-presentation", "spaces-groupoids-lemma-quotient-stack-change-big-site" ], "ref_ids": [ 8474, 9336 ] } ], "ref_ids": [] }, { "id": 8478, "type": "theorem", "label": "algebraic-lemma-fully-faithful", "categories": [ "algebraic" ], "title": "algebraic-lemma-fully-faithful", "contents": [ "Suppose $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$.", "Let $S$ be an object of $\\Sch_{fppf}$. Denote", "$\\textit{Algebraic-Stacks}/S$ the $2$-category of algebraic stacks over $S$", "defined using $\\Sch_{fppf}$. Similarly, denote", "$\\textit{Algebraic-Stacks}'/S$ the $2$-category of algebraic stacks over $S$", "defined using $\\Sch'_{fppf}$. The rule", "$\\mathcal{X} \\mapsto f^{-1}\\mathcal{X}$ of", "Lemma \\ref{lemma-change-big-site}", "defines a functor of $2$-categories", "$$", "\\textit{Algebraic-Stacks}/S \\longrightarrow \\textit{Algebraic-Stacks}'/S", "$$", "which defines equivalences of morphism categories", "$$", "\\Mor_{\\textit{Algebraic-Stacks}/S}(\\mathcal{X}, \\mathcal{Y})", "\\longrightarrow", "\\Mor_{\\textit{Algebraic-Stacks}'/S}(f^{-1}\\mathcal{X}, f^{-1}\\mathcal{Y})", "$$", "for every objects $\\mathcal{X}, \\mathcal{Y}$ of", "$\\textit{Algebraic-Stacks}/S$. An object", "$\\mathcal{X}'$ of $\\textit{Algebraic-Stacks}'/S$", "is equivalence to $f^{-1}\\mathcal{X}$ for some", "$\\mathcal{X}$ in $\\textit{Algebraic-Stacks}/S$", "if and only if it has a presentation $\\mathcal{X} = [U'/R']$", "with $U', R'$ isomorphic to $f^{-1}U$, $f^{-1}R$ for some", "$U, R \\in \\textit{Spaces}/S$." ], "refs": [ "algebraic-lemma-change-big-site" ], "proofs": [ { "contents": [ "The statement on morphism categories is a consequence of the more general", "Stacks, Lemma \\ref{stacks-lemma-bigger-site}.", "The characterization of the ``essential image'' follows from the description", "of $f^{-1}$ in the proof of", "Lemma \\ref{lemma-change-big-site}." ], "refs": [ "stacks-lemma-bigger-site", "algebraic-lemma-change-big-site" ], "ref_ids": [ 8989, 8477 ] } ], "ref_ids": [ 8477 ] }, { "id": 8479, "type": "theorem", "label": "algebraic-lemma-category-of-spaces-over-smaller-base-scheme", "categories": [ "algebraic" ], "title": "algebraic-lemma-category-of-spaces-over-smaller-base-scheme", "contents": [ "Let $\\Sch_{fppf}$ be a big fppf site.", "Let $S \\to S'$ be a morphism of this site.", "The constructions A and B of", "Stacks, Section \\ref{stacks-section-localize}", "above give isomorphisms of $2$-categories", "$$", "\\left\\{", "\\begin{matrix}", "2\\text{-category of algebraic}\\\\", "\\text{stacks }\\mathcal{X}\\text{ over }S", "\\end{matrix}", "\\right\\}", "\\leftrightarrow", "\\left\\{", "\\begin{matrix}", "2\\text{-category of pairs }(\\mathcal{X}', f)\\text{ consisting of an}\\\\", "\\text{algebraic stack }\\mathcal{X}'\\text{ over }S'\\text{ and a morphism}\\\\", "f : \\mathcal{X}' \\to (\\Sch/S)_{fppf}\\text{ of algebraic stacks over }S'", "\\end{matrix}", "\\right\\}", "$$" ], "refs": [], "proofs": [ { "contents": [ "The statement makes sense as the functor", "$j : (\\Sch/S)_{fppf} \\to (\\Sch/S')_{fppf}$", "is the localization functor associated to the object $S/S'$", "of $(\\Sch/S')_{fppf}$. By", "Stacks, Lemma \\ref{stacks-lemma-localize-stacks}", "the only thing to show is that the constructions A and B", "preserve the subcategories of algebraic stacks.", "For example, if $\\mathcal{X} = [U/R]$ then construction A", "applied to $\\mathcal{X}$ just produces", "$\\mathcal{X}' = \\mathcal{X}$. Conversely, if $\\mathcal{X}' = [U'/R']$", "the morphism $p$ induces morphisms of algebraic spaces", "$U' \\to S$ and $R' \\to S$, and then $\\mathcal{X} = [U'/R']$", "but now viewed as a stack over $S$. Hence the lemma is clear." ], "refs": [ "stacks-lemma-localize-stacks" ], "ref_ids": [ 8991 ] } ], "ref_ids": [] }, { "id": 8480, "type": "theorem", "label": "algebraic-proposition-algebraic-stack-no-automorphisms", "categories": [ "algebraic" ], "title": "algebraic-proposition-algebraic-stack-no-automorphisms", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}$ be an algebraic stack over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is a stack in setoids,", "\\item the canonical $1$-morphism $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$", "is an equivalence, and", "\\item $\\mathcal{X}$ is representable by an algebraic space.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows from", "Stacks, Lemma \\ref{stacks-lemma-characterize-stack-in-setoids}.", "The implication (3) $\\Rightarrow$ (1) follows from", "Lemma \\ref{lemma-algebraic-stack-no-automorphisms}.", "Finally, assume (1). By", "Stacks, Lemma \\ref{stacks-lemma-stack-in-setoids-characterize}", "there exists an equivalence $j : \\mathcal{X} \\to \\mathcal{S}_F$", "where $F$ is a sheaf on $(\\Sch/S)_{fppf}$. By", "Lemma \\ref{lemma-map-presheaves-representable-by-algebraic-spaces}", "the fact that $\\Delta_\\mathcal{X}$ is representable by algebraic", "spaces, means that $\\Delta_F : F \\to F \\times F$", "is representable by algebraic spaces.", "Let $U$ be a scheme and let $x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$", "be a surjective smooth morphism. The composition", "$j \\circ x : (\\Sch/U)_{fppf} \\to \\mathcal{S}_F$", "corresponds to a morphism $h_U \\to F$ of sheaves. By", "Bootstrap, Lemma \\ref{bootstrap-lemma-representable-diagonal}", "this morphism is representable by algebraic spaces.", "Hence by", "Lemma \\ref{lemma-map-fibred-setoids-property}", "we conclude that $h_U \\to F$ is surjective and smooth.", "In particular it is surjective, flat and locally of finite presentation", "(by", "Lemma \\ref{lemma-representable-transformations-property-implication}", "and the fact that a smooth morphism of algebraic spaces is flat and", "locally of finite presentation, see", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-smooth-locally-finite-presentation} and", "\\ref{spaces-morphisms-lemma-smooth-flat}).", "Finally, we apply", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}", "to see that $F$ is an algebraic space." ], "refs": [ "stacks-lemma-characterize-stack-in-setoids", "algebraic-lemma-algebraic-stack-no-automorphisms", "stacks-lemma-stack-in-setoids-characterize", "algebraic-lemma-map-presheaves-representable-by-algebraic-spaces", "bootstrap-lemma-representable-diagonal", "algebraic-lemma-map-fibred-setoids-property", "algebraic-lemma-representable-transformations-property-implication", "spaces-morphisms-lemma-smooth-locally-finite-presentation", "spaces-morphisms-lemma-smooth-flat", "bootstrap-theorem-final-bootstrap" ], "ref_ids": [ 8960, 8464, 8951, 8445, 2618, 8454, 8459, 4889, 4891, 2602 ] } ], "ref_ids": [] }, { "id": 8492, "type": "theorem", "label": "sites-theorem-plus", "categories": [ "sites" ], "title": "sites-theorem-plus", "contents": [ "With $\\mathcal{F}$ as above", "\\begin{enumerate}", "\\item", "\\label{item-sep}", "The presheaf $\\mathcal{F}^+$ is separated.", "\\item", "\\label{item-sheaf}", "If $\\mathcal{F}$ is separated, then $\\mathcal{F}^+$ is a sheaf", "and the map of presheaves $\\mathcal{F} \\to \\mathcal{F}^+$ is injective.", "\\item", "\\label{item-plus-iso}", "If $\\mathcal{F}$ is a sheaf, then $\\mathcal{F} \\to \\mathcal{F}^+$", "is an isomorphism.", "\\item", "\\label{item-plusplus}", "The presheaf $\\mathcal{F}^{++}$ is always a sheaf.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (\\ref{item-sep}).", "Suppose that $s, s' \\in \\mathcal{F}^+(U)$ and suppose that", "there exists some covering $\\{U_i \\to U\\}$ such that", "$s|_{U_i} = s'|_{U_i}$ for all $i$. We now have three coverings", "of $U$: the covering $\\{U_i \\to U\\}$ above, a covering $\\mathcal{U}$", "for $s$ as in Lemma \\ref{lemma-plus-surjective},", "and a similar covering $\\mathcal{U}'$ for $s'$. By Lemma", "\\ref{lemma-common-refinement}, we can find a common refinement,", "say $\\{W_j \\to U\\}$. This means we have $s_j, s'_j \\in \\mathcal{F}(W_j)$", "such that $s|_{W_j} = \\theta(s_j)$, similarly for $s'|_{W_j}$, and", "such that $\\theta(s_j) = \\theta(s'_j)$. This last equality means", "that there exists some covering $\\{W_{jk} \\to W_j\\}$ such that", "$s_j|_{W_{jk}} = s'_j|_{W_{jk}}$. Then since $\\{W_{jk} \\to U\\}$", "is a covering we see that $s, s'$ map to the same element of", "$H^0(\\{W_{jk} \\to U\\}, \\mathcal{F})$ as desired.", "\\medskip\\noindent", "Proof of (\\ref{item-sheaf}). It is clear that $\\mathcal{F} \\to", "\\mathcal{F}^+$ is injective because all the maps", "$\\mathcal{F}(U) \\to H^0(\\mathcal{U}, \\mathcal{F})$", "are injective. It is also clear that, if $\\mathcal{U} \\to", "\\mathcal{U}'$ is a refinement, then $H^0(\\mathcal{U}', \\mathcal{F})", "\\to H^0(\\mathcal{U}, \\mathcal{F})$ is injective. Now,", "suppose that $\\{U_i \\to U\\}$ is a covering, and let", "$(s_i)$ be a family of elements of $\\mathcal{F}^+(U_i)$", "satisfying the sheaf condition", "$s_i|_{U_i \\times_U U_j} = s_j|_{U_i \\times_U U_j}$", "for all $i, j \\in I$. Choose coverings (as in", "Lemma \\ref{lemma-plus-surjective}) $\\{U_{ij} \\to U_i\\}$", "such that $s_i|_{U_{ij}}$ is the image of the (unique)", "element $s_{ij} \\in \\mathcal{F}(U_{ij})$. The sheaf condition", "implies that $s_{ij}$ and $s_{i'j'}$ agree over", "$U_{ij} \\times_U U_{i'j'}$ because it maps to", "$U_i \\times_U U_{i'}$ and we have the equality there.", "Hence $(s_{ij}) \\in H^0(\\{U_{ij} \\to U\\}, \\mathcal{F})$", "gives rise to an element $s \\in \\mathcal{F}^+(U)$. We leave", "it to the reader to verify that $s|_{U_i} = s_i$.", "\\medskip\\noindent", "Proof of (\\ref{item-plus-iso}). This is immediate from the definitions", "because the sheaf property says exactly that every map", "$\\mathcal{F} \\to H^0(\\mathcal{U}, \\mathcal{F})$ is bijective", "(for every covering $\\mathcal{U}$ of $U$).", "\\medskip\\noindent", "Statement (\\ref{item-plusplus}) is now obvious." ], "refs": [ "sites-lemma-plus-surjective", "sites-lemma-common-refinement", "sites-lemma-plus-surjective" ], "ref_ids": [ 8513, 8511, 8513 ] } ], "ref_ids": [] }, { "id": 8493, "type": "theorem", "label": "sites-theorem-L-topology", "categories": [ "sites" ], "title": "sites-theorem-L-topology", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $J$ be a topology on $\\mathcal{C}$.", "Let $\\mathcal{F}$ be a presheaf of sets.", "\\begin{enumerate}", "\\item The presheaf $L\\mathcal{F}$ is separated.", "\\item If $\\mathcal{F}$ is separated, then $L\\mathcal{F}$ is a sheaf", "and the map of presheaves $\\mathcal{F} \\to L\\mathcal{F}$ is injective.", "\\item If $\\mathcal{F}$ is a sheaf, then $\\mathcal{F} \\to L\\mathcal{F}$", "is an isomorphism.", "\\item The presheaf $LL\\mathcal{F}$ is always a sheaf.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (3) is trivial from the definition of $L$ and", "the definition of a sheaf (Definition \\ref{definition-sheaf-sets-topology}).", "Part (4) follows formally from the others.", "\\medskip\\noindent", "We sketch the proof of (1). Suppose $S$ is a covering sieve", "of the object $U$. Suppose that $\\varphi_i \\in L\\mathcal{F}(U)$,", "$i = 1, 2$ map to the same element in", "$\\Mor_{\\textit{PSh}(\\mathcal{C})}(S, L\\mathcal{F})$.", "We may find a single covering sieve $S'$ on $U$ such", "that both $\\varphi_i$ are represented by elements", "$\\varphi_i \\in \\Mor_{\\textit{PSh}(\\mathcal{C})}(S', \\mathcal{F})$.", "We may assume that $S' = S$ by replacing both $S$ and $S'$ by", "$S' \\cap S$ which is also a covering sieve, see Lemma \\ref{lemma-sieves-set}.", "Suppose $V\\in \\Ob(\\mathcal{C})$, and", "$\\alpha : V \\to U$ in $S(V)$.", "Then we have $S \\times_U V = h_V$,", "see Lemma \\ref{lemma-pullback-sieve-section}. Thus the restrictions", "of $\\varphi_i$ via $V \\to U$ correspond to sections $s_{i, V, \\alpha}$", "of $\\mathcal{F}$ over $V$. The assumption is that there exist", "a covering sieve $S_{V, \\alpha}$ of $V$ such that", "$s_{i, V, \\alpha}$ restrict to the same element of", "$\\Mor_{\\textit{PSh}(\\mathcal{C})}(S_{V, \\alpha}, \\mathcal{F})$.", "Consider the sieve $S''$ on $U$ defined by the rule", "\\begin{eqnarray}", "\\label{equation-S-prime-prime}", "(f : T \\to U) \\in S''(T)", "& \\Leftrightarrow &", "\\exists\\ V , \\ \\alpha : V \\to U, \\ \\alpha \\in S(V), \\nonumber \\\\", "& &", "\\exists\\ g : T \\to V, \\ g \\in S_{V, \\alpha}(T), \\\\", "& &", "f = \\alpha \\circ g \\nonumber", "\\end{eqnarray}", "By axiom (2) of a topology we see that $S''$ is a covering", "sieve on $U$. By construction we see that $\\varphi_1$", "and $\\varphi_2$ restrict to the same element of", "$\\Mor_{\\textit{PSh}(\\mathcal{C})}(S'', L\\mathcal{F})$", "as desired.", "\\medskip\\noindent", "We sketch the proof of (2). Assume that $\\mathcal{F}$ is a", "separated presheaf of sets on $\\mathcal{C}$ with respect to", "the topology $J$.", "Let $S$ be a covering sieve of the object $U$ of $\\mathcal{C}$.", "Suppose that $\\varphi \\in \\Mor_\\mathcal{C}(S, L\\mathcal{F})$.", "We have to find an element $s \\in L\\mathcal{F}(U)$ restricting", "to $\\varphi$. Suppose $V\\in \\Ob(\\mathcal{C})$, and", "$\\alpha : V \\to U$ in $S(V)$. The value $\\varphi(\\alpha)", "\\in L\\mathcal{F}(V)$ is given by a covering sieve", "$S_{V, \\alpha}$ of $V$ and a morphism of presheaves", "$\\varphi_{V, \\alpha} : S_{V, \\alpha} \\to \\mathcal{F}$.", "As in the proof above, define a covering sieve $S''$ on $U$ by", "Equation (\\ref{equation-S-prime-prime}). We define", "$$", "\\varphi'' : S'' \\longrightarrow \\mathcal{F}", "$$", "by the following simple rule: For every $f : T \\to U$,", "$f \\in S''(T)$ choose $V, \\alpha, g$ as in", "Equation (\\ref{equation-S-prime-prime}). Then set", "$$", "\\varphi''(f) = \\varphi_{V, \\alpha}(g).", "$$", "We claim this is independent of the", "choice of $V, \\alpha, g$.", "Consider a second such choice$ V', \\alpha', g'$.", "The restrictions of $\\varphi_{V, \\alpha}$ and", "$\\varphi_{V', \\alpha'}$ to the intersection", "of the following covering sieves on $T$", "$$", "(S_{V, \\alpha} \\times_{V, g} T) \\cap (S_{V', \\alpha'} \\times_{V', g'} T)", "$$", "agree. Namely, these restrictions both correspond to the", "restriction of $\\varphi$ to $T$ (via $f$) and the desired", "equality follows because $\\mathcal{F}$ is separated.", "Denote the common restriction $\\psi$.", "The independence of choice follows because", "$\\varphi_{V, \\alpha}(g) = \\psi(\\text{id}_T) =", "\\varphi_{V', \\alpha'}(g')$. OK, so now $\\varphi''$", "gives an element $s \\in L\\mathcal{F}(U)$. We leave it to", "the reader to check that $s$ restricts to $\\varphi$." ], "refs": [ "sites-definition-sheaf-sets-topology", "sites-lemma-sieves-set" ], "ref_ids": [ 8695, 8630 ] } ], "ref_ids": [] }, { "id": 8494, "type": "theorem", "label": "sites-theorem-topology-and-topos", "categories": [ "sites" ], "title": "sites-theorem-topology-and-topos", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $J$, $J'$ be topologies on $\\mathcal{C}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $J = J'$,", "\\item sheaves for the topology $J$ are the same as", "sheaves for the topology $J'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is a tautology that if $J = J'$ then the notions of sheaves", "are the same. Conversely, Lemma \\ref{lemma-sieve-sheafification}", "characterizes covering sieves in terms of the sheafification", "functor. But the sheafification functor", "$\\textit{PSh}(\\mathcal{C}) \\to \\Sh(\\mathcal{C}, J)$", "is the right adjoint of the inclusion functor", "$\\Sh(\\mathcal{C}, J) \\to \\textit{PSh}(\\mathcal{C})$.", "Hence if the subcategories", "$\\Sh(\\mathcal{C}, J)$ and", "$\\Sh(\\mathcal{C}, J')$ are the same, then the sheafification", "functors are the same and hence the collections of covering", "sieves are the same." ], "refs": [ "sites-lemma-sieve-sheafification" ], "ref_ids": [ 8637 ] } ], "ref_ids": [] }, { "id": 8495, "type": "theorem", "label": "sites-lemma-mono-epi", "categories": [ "sites" ], "title": "sites-lemma-mono-epi", "contents": [ "The injective (resp.\\ surjective) maps defined above", "are exactly the monomorphisms (resp.\\ epimorphisms) of", "$\\textit{PSh}(\\mathcal{C})$. A map is an isomorphism", "if and only if it is both injective and surjective." ], "refs": [], "proofs": [ { "contents": [ "We shall show that $\\varphi : \\mathcal{F} \\to", "\\mathcal{G}$ is injective if and only if it is a monomorphism", "of $\\textit{PSh}(\\mathcal{C})$. Indeed, the ``only if''", "direction is straightforward, so let us show the ``if''", "direction. Assume that $\\varphi$ is a monomorphism. Let", "$U \\in \\Ob(\\mathcal{C})$; we need to show that $\\varphi_U$ is", "injective. So let $a, b \\in \\mathcal{F}(U)$ be such that", "$\\varphi_U (a) = \\varphi_U (b)$; we need to check that $a = b$.", "Under the isomorphism", "(\\ref{equation-map-representable-into-presheaf}), the elements", "$a$ and $b$ of $\\mathcal{F}(U)$ correspond to two natural", "transformations", "$a', b' \\in \\Mor_{\\textit{PSh}(\\mathcal{C})}(h_U, \\mathcal{F})$.", "Similarly, under the analogous isomorphism", "$\\Mor_{\\textit{PSh}(\\mathcal{C})}(h_U, \\mathcal{G})", "= \\mathcal{G}(U)$,", "the two equal elements $\\varphi_U (a)$ and $\\varphi_U (b)$ of", "$\\mathcal{G}(U)$ correspond to the two natural transformations", "$\\varphi \\circ a', \\varphi \\circ b'", "\\in \\Mor_{\\textit{PSh}(\\mathcal{C})}(h_U, \\mathcal{G})$,", "which therefore must also be equal. So", "$\\varphi \\circ a' = \\varphi \\circ b'$, and thus $a' = b'$", "(since $\\varphi$ is monic), whence $a = b$. This finishes (1).", "\\medskip\\noindent", "We shall show that $\\varphi : \\mathcal{F} \\to", "\\mathcal{G}$ is surjective if and only if it is an epimorphism", "of $\\textit{PSh}(\\mathcal{C})$. Indeed, the ``only if''", "direction is straightforward, so let us show the ``if''", "direction. Assume that $\\varphi$ is an epimorphism.", "\\medskip\\noindent", "For any two morphisms $f : A \\to B$ and $g : A \\to C$ in the", "category $\\textit{Sets}$, we let $\\text{inl}_{f,g}$ and", "$\\text{inr}_{f,g}$ denote the two canonical maps from", "$B$ and $C$ to $B \\coprod_A C$. (Here, the pushout is", "evaluated in $\\textit{Sets}$.)", "\\medskip\\noindent", "Now, we define a presheaf $\\mathcal{H}$ of sets on $\\mathcal{C}$", "by setting $\\mathcal{H}(U)", "= \\mathcal{G}(U) \\coprod_{\\mathcal{F}(U)} \\mathcal{G}(U)$ (where", "the pushout is evaluated in $\\textit{Sets}$ and induced by", "the map $\\varphi_U : \\mathcal{F}(U) \\to \\mathcal{G}(U)$) for", "every $U \\in \\Ob(\\mathcal{C})$; its action on morphisms is", "defined in the obvious way (by the functoriality of pushout).", "Then, there are two natural", "transformations $i_1 : \\mathcal{G} \\to \\mathcal{H}$ and", "$i_2 : \\mathcal{G} \\to \\mathcal{H}$ whose components at an object", "$U \\in \\Ob(\\mathcal{C})$ are given by the maps", "$\\text{inl}_{\\varphi_U, \\varphi_U}$ and", "$\\text{inr}_{\\varphi_U, \\varphi_U}$, respectively. The", "definition of a pushout shows that $i_1 \\circ \\varphi", "= i_2 \\circ \\varphi$, whence $i_1 = i_2$ (since $\\varphi$ is an", "epimorphism). Thus, for every $U \\in \\Ob(\\mathcal{C})$, we have", "$\\text{inl}_{\\varphi_U, \\varphi_U}", "= \\text{inr}_{\\varphi_U, \\varphi_U}$. Thus, $\\varphi_U$", "must be surjective (since a simple combinatorial argument shows", "that if $f : A \\to B$ is a morphism in $\\textit{Sets}$, then", "$\\text{inl}_{f,f} = \\text{inr}_{f,f}$ if and", "only if $f$ is surjective). In other words, $\\varphi$ is", "surjective, and (2) is proven.", "\\medskip\\noindent", "We shall show that $\\varphi : \\mathcal{F} \\to", "\\mathcal{G}$ is both injective and surjective if and only if it", "is an isomorphism of $\\textit{PSh}(\\mathcal{C})$. This time,", "the ``if'' direction is straightforward. To prove the ``only if''", "direction, it suffices to observe that if $\\varphi$ is both", "injective and surjective, then $\\varphi_U$ is an invertible map", "for every $U \\in \\Ob(\\mathcal{C})$, and the inverses of these", "maps for all $U$ can be combined to a natural transformation", "$\\mathcal{G} \\to \\mathcal{F}$ which is an inverse to $\\varphi$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8496, "type": "theorem", "label": "sites-lemma-image", "categories": [ "sites" ], "title": "sites-lemma-image", "contents": [ "Let $\\mathcal{C}$ be a category.", "Suppose that $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a", "morphism of presheaves of sets on $\\mathcal{C}$.", "There exists a unique subpresheaf $\\mathcal{G}' \\subset \\mathcal{G}$", "such that $\\varphi$ factors as", "$\\mathcal{F} \\to \\mathcal{G}' \\to \\mathcal{G}$", "and such that the first map is surjective." ], "refs": [], "proofs": [ { "contents": [ "To prove existence, just set", "$\\mathcal{G}'(U) = \\varphi_U \\left(\\mathcal{F}(U)\\right)$", "for every $U \\in \\Ob(C)$ (and inherit the action on morphisms", "from $\\mathcal{G}$), and prove that this defines a", "subpresheaf of $\\mathcal{G}$ and that $\\varphi$ factors as", "$\\mathcal{F} \\to \\mathcal{G}' \\to \\mathcal{G}$ with the", "first map being surjective. Uniqueness is straightforward." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8497, "type": "theorem", "label": "sites-lemma-almost-directed", "categories": [ "sites" ], "title": "sites-lemma-almost-directed", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor between categories.", "Suppose that $\\mathcal{C}$ has fibre products and equalizers, and that", "$u$ commutes with them. Then the categories $(\\mathcal{I}_V)^{opp}$", "satisfy the hypotheses of", "Categories, Lemma \\ref{categories-lemma-split-into-directed}." ], "refs": [ "categories-lemma-split-into-directed" ], "proofs": [ { "contents": [ "There are two conditions to check.", "\\medskip\\noindent", "First, suppose we are given three objects", "$\\phi : V \\to u(U)$, $\\phi' : V \\to u(U')$, and $\\phi'' : V \\to u(U'')$", "and morphisms $a : U' \\to U$, $b : U'' \\to U$ such that", "$u(a) \\circ \\phi' = \\phi$ and $u(b) \\circ \\phi'' = \\phi$.", "We have to show there exists another object $\\phi''' : V \\to u(U''')$", "and morphisms $c : U''' \\to U'$ and $d : U''' \\to U''$ such that", "$u(c) \\circ \\phi''' = \\phi'$, $u(d) \\circ \\phi''' = \\phi''$ and", "$a \\circ c = b \\circ d$. We take $U''' = U' \\times_U U''$", "with $c$ and $d$ the projection morphisms. This works as $u$ commutes", "with fibre products; we omit the verification.", "\\medskip\\noindent", "Second, suppose we are given two objects", "$\\phi : V \\to u(U)$ and $\\phi' : V \\to u(U')$", "and morphisms $a, b : (U, \\phi) \\to (U', \\phi')$.", "We have to find a morphism $c : (U'', \\phi'') \\to (U, \\phi)$", "which equalizes $a$ and $b$. Let $c : U'' \\to U$ be the equalizer of", "$a$ and $b$ in the category $\\mathcal{C}$. As $u$ commutes", "with equalizers and since $u(a) \\circ \\phi = u(b) \\circ \\phi = \\phi'$", "we obtain a morphism $\\phi'' : V \\to u(U'')$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12234 ] }, { "id": 8498, "type": "theorem", "label": "sites-lemma-directed", "categories": [ "sites" ], "title": "sites-lemma-directed", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor between categories.", "Assume", "\\begin{enumerate}", "\\item the category $\\mathcal{C}$ has a final object $X$ and", "$u(X)$ is a final object of $\\mathcal{D}$ , and", "\\item the category $\\mathcal{C}$ has fibre products and", "$u$ commutes with them.", "\\end{enumerate}", "Then the index categories $(\\mathcal{I}^u_V)^{opp}$ are filtered (see", "Categories, Definition \\ref{categories-definition-directed})." ], "refs": [ "categories-definition-directed" ], "proofs": [ { "contents": [ "The assumptions imply that the assumptions of", "Lemma \\ref{lemma-almost-directed}", "are satisfied (see the discussion in", "Categories, Section \\ref{categories-section-finite-limits}).", "By", "Categories, Lemma \\ref{categories-lemma-split-into-directed}", "we see that $\\mathcal{I}_V$ is a (possibly empty) disjoint union of", "directed categories.", "Hence it suffices to show that $\\mathcal{I}_V$ is connected.", "\\medskip\\noindent", "First, we show that $\\mathcal{I}_V$ is nonempty.", "Namely, let $X$ be the final object of $\\mathcal{C}$,", "which exists by assumption.", "Let $V \\to u(X)$ be the morphism coming from the fact", "that $u(X)$ is final in $\\mathcal{D}$ by assumption.", "This gives an object of $\\mathcal{I}_V$.", "\\medskip\\noindent", "Second, we show that $\\mathcal{I}_V$ is connected.", "Let $\\phi_1 : V \\to u(U_1)$ and $\\phi_2 : V \\to u(U_2)$ be", "in $\\Ob(\\mathcal{I}_V)$. By assumption $U_1\\times U_2$", "exists and $u(U_1\\times U_2) = u(U_1)\\times u(U_2)$.", "Consider the morphism $\\phi : V \\to u(U_1\\times U_2)$", "corresponding to $(\\phi_1, \\phi_2)$ by the universal property", "of products. Clearly the object $\\phi : V \\to u(U_1\\times U_2)$", "maps to both $\\phi_1 : V \\to u(U_1)$ and $\\phi_2 : V \\to u(U_2)$." ], "refs": [ "sites-lemma-almost-directed", "categories-lemma-split-into-directed" ], "ref_ids": [ 8497, 12234 ] } ], "ref_ids": [ 12363 ] }, { "id": 8499, "type": "theorem", "label": "sites-lemma-recover", "categories": [ "sites" ], "title": "sites-lemma-recover", "contents": [ "There is a canonical map", "$\\mathcal{F}(U) \\to u_p\\mathcal{F}(u(U))$,", "which is compatible with restriction maps", "(on $\\mathcal{F}$ and on $u_p\\mathcal{F}$)." ], "refs": [], "proofs": [ { "contents": [ "This is just the map $c(\\text{id}_{u(U)})$ introduced above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8500, "type": "theorem", "label": "sites-lemma-adjoints-u", "categories": [ "sites" ], "title": "sites-lemma-adjoints-u", "contents": [ "The functor $u_p$ is a left adjoint to the functor $u^p$.", "In other words the formula", "$$", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(\\mathcal{F}, u^p\\mathcal{G})", "=", "\\Mor_{\\textit{PSh}(\\mathcal{D})}(u_p\\mathcal{F}, \\mathcal{G})", "$$", "holds bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{G}$ be a presheaf on $\\mathcal{D}$ and let", "$\\mathcal{F}$ be a presheaf on $\\mathcal{C}$.", "We will show that the displayed formula holds", "by constructing maps either way. We will leave", "it to the reader to verify they are each others inverse.", "\\medskip\\noindent", "Given a map $\\alpha : u_p \\mathcal{F} \\to \\mathcal{G}$", "we get $u^p\\alpha : u^p u_p \\mathcal{F} \\to u^p \\mathcal{G}$.", "Lemma \\ref{lemma-recover} says that there is a", "map $\\mathcal{F} \\to u^p u_p \\mathcal{F}$. The composition", "of the two gives the desired map. (The good thing about this construction", "is that it is clearly functorial in everything in sight.)", "\\medskip\\noindent", "Conversely, given a map $\\beta : \\mathcal{F} \\to u^p\\mathcal{G}$", "we get a map $u_p\\beta : u_p\\mathcal{F} \\to u_p u^p\\mathcal{G}$.", "We claim that the functor $u^p\\mathcal{G}_Y$ on $\\mathcal{I}_Y$", "has a canonical map to the constant functor with value $\\mathcal{G}(Y)$.", "Namely, for every object $(X, \\phi)$ of $\\mathcal{I}_Y$,", "the value of $u^p\\mathcal{G}_Y$ on this object is $\\mathcal{G}(u(X))$", "which maps to $\\mathcal{G}(Y)$ by $\\mathcal{G}(\\phi) = \\phi^* $.", "This is a transformation of functors because $\\mathcal{G}$ is a functor", "itself. This leads to a map $u_p u^p \\mathcal{G}(Y) \\to \\mathcal{G}(Y)$.", "Another trivial verification shows that this is functorial in $Y$", "leading to a map of presheaves $u_p u^p \\mathcal{G} \\to \\mathcal{G}$.", "The composition $u_p\\mathcal{F} \\to u_p u^p\\mathcal{G} \\to", "\\mathcal{G}$ is the desired map." ], "refs": [ "sites-lemma-recover" ], "ref_ids": [ 8499 ] } ], "ref_ids": [] }, { "id": 8501, "type": "theorem", "label": "sites-lemma-pullback-representable-presheaf", "categories": [ "sites" ], "title": "sites-lemma-pullback-representable-presheaf", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor between categories.", "For any object $U$ of $\\mathcal{C}$ we have $u_ph_U = h_{u(U)}$." ], "refs": [], "proofs": [ { "contents": [ "By adjointness of $u_p$ and $u^p$ we have", "$$", "\\Mor_{\\textit{PSh}(\\mathcal{D})}(u_ph_U, \\mathcal{G})", "=", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(h_U, u^p\\mathcal{G})", "=", "u^p\\mathcal{G}(U) =", "\\mathcal{G}(u(U))", "$$", "and hence by Yoneda's lemma we see that $u_ph_U = h_{u(U)}$ as", "presheaves." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8502, "type": "theorem", "label": "sites-lemma-tautological-combinatorial", "categories": [ "sites" ], "title": "sites-lemma-tautological-combinatorial", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{U} = \\{\\varphi_i : U_i \\to U\\}_{i\\in I}$, and", "$\\mathcal{V} = \\{\\psi_j : V_j \\to U\\}_{j\\in J}$ be two families of morphisms", "with the same fixed target.", "\\begin{enumerate}", "\\item If $\\mathcal{U}$ and $\\mathcal{V}$ are combinatorially equivalent", "then they are tautologically equivalent.", "\\item If $\\mathcal{U}$ and $\\mathcal{V}$ are tautologically equivalent", "then $\\mathcal{U}$ is a refinement of $\\mathcal{V}$ and", "$\\mathcal{V}$ is a refinement of $\\mathcal{U}$.", "\\item The relation ``being combinatorially equivalent'' is an", "equivalence relation on all families of morphisms with fixed target.", "\\item The relation ``being tautologically equivalent'' is an", "equivalence relation on all families of morphisms with fixed target.", "\\item The relation ``$\\mathcal{U}$ refines $\\mathcal{V}$ and", "$\\mathcal{V}$ refines $\\mathcal{U}$'' is an equivalence relation on", "all families of morphisms with fixed target.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8503, "type": "theorem", "label": "sites-lemma-tautological-same-sheaf", "categories": [ "sites" ], "title": "sites-lemma-tautological-same-sheaf", "contents": [ "Let $\\mathcal{C}$ be a category. Let", "$\\mathcal{U} = \\{\\varphi_i : U_i \\to U\\}_{i\\in I}$, and", "$\\mathcal{V} = \\{\\psi_j : V_j \\to U\\}_{j\\in J}$ be two families of morphisms", "with the same fixed target. Assume that the fibre products", "$U_i \\times_U U_{i'}$ and $V_j \\times_U V_{j'}$ exist.", "If $\\mathcal{U}$ and $\\mathcal{V}$ are", "tautologically equivalent, then for any presheaf $\\mathcal{F}$ on", "$\\mathcal{C}$ the sheaf condition for $\\mathcal{F}$ with respect to", "$\\mathcal{U}$ is equivalent to the sheaf condition for $\\mathcal{F}$", "with respect to $\\mathcal{V}$." ], "refs": [], "proofs": [ { "contents": [ "First, note that if $\\varphi : A \\to B$ is an isomorphism in the", "category $\\mathcal{C}$, then $\\varphi^* : \\mathcal{F}(B) \\to \\mathcal{F}(A)$", "is an isomorphism. Let $\\beta : J \\to I$ be a map and let", "$\\chi_j : V_j \\to U_{\\beta(j)}$ be isomorphisms over $U$ which", "are assumed to exist by hypothesis. Let us show that the sheaf", "condition for $\\mathcal{V}$ implies the sheaf condition for $\\mathcal{U}$.", "Suppose given sections $s_i \\in \\mathcal{F}(U_i)$ such that", "$$", "s_i|_{U_i \\times_U U_{i'}} = s_{i'}|_{U_i \\times_U U_{i'}}", "$$", "in $\\mathcal{F}(U_i \\times_U U_{i'})$ for all pairs $(i, i') \\in I \\times I$.", "Then we can define $s_j = \\chi_j^*s_{\\beta(j)}$. For any pair", "$(j, j') \\in J \\times J$ the morphism", "$\\chi_j \\times_{\\text{id}_U} \\chi_{j'} : V_j \\times_U V_{j'} \\to", "U_{\\beta(j)} \\times_U U_{\\beta(j')}$ is an isomorphism as well.", "Hence by transport of structure we see that", "$$", "s_j|_{V_j \\times_U V_{j'}} = s_{j'}|_{V_j \\times_U V_{j'}}", "$$", "as well. The sheaf condition w.r.t.\\ $\\mathcal{V}$ implies there", "exists a unique $s$ such that $s|_{V_j} = s_j$ for all $j \\in J$.", "By the first remark of the proof this implies that $s|_{U_i} = s_i$", "for all $i \\in \\Im(\\beta)$ as well. Suppose that $i \\in I$,", "$i \\not \\in \\Im(\\beta)$. For such an $i$ we have isomorphisms", "$U_i \\to V_{\\alpha(i)} \\to U_{\\beta(\\alpha(i))}$ over $U$. This gives a", "morphism $U_i \\to U_i \\times_U U_{\\beta(\\alpha(i))}$ which is a", "section of the projection. Because $s_i$ and $s_{\\beta(\\alpha(i))}$", "restrict to the same element on the fibre product we conclude that", "$s_{\\beta(\\alpha(i))}$ pulls back to $s_i$ via $U_i \\to U_{\\beta(\\alpha(i))}$.", "Thus we see that also $s_i = s|_{U_i}$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8504, "type": "theorem", "label": "sites-lemma-compare-separated-presheaf-condition", "categories": [ "sites" ], "title": "sites-lemma-compare-separated-presheaf-condition", "contents": [ "Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I} \\to", "\\mathcal{V} = \\{V_j \\to U\\}_{j \\in J}$ be a morphism of families of maps", "with fixed target of $\\mathcal{C}$ given by $\\text{id} : U \\to U$,", "$\\alpha : J \\to I$ and $f_j : V_j \\to U_{\\alpha(j)}$. Let $\\mathcal{F}$", "be a presheaf on $\\mathcal{C}$. If", "$\\mathcal{F}(U) \\to \\prod_{j \\in J} \\mathcal{F}(V_j)$ is", "injective then", "$\\mathcal{F}(U) \\to \\prod_{i \\in I} \\mathcal{F}(U_i)$ is", "injective." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8505, "type": "theorem", "label": "sites-lemma-compare-sheaf-condition", "categories": [ "sites" ], "title": "sites-lemma-compare-sheaf-condition", "contents": [ "Let $\\mathcal{C}$ be a category. Let $\\mathcal{V} = \\{V_j \\to U\\}_{j \\in J} \\to", "\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a morphism of families of maps", "with fixed target of $\\mathcal{C}$ given by $\\text{id} : U \\to U$,", "$\\alpha : J \\to I$ and $f_j : V_j \\to U_{\\alpha(j)}$. Let $\\mathcal{F}$", "be a presheaf on $\\mathcal{C}$. If", "\\begin{enumerate}", "\\item the fibre products $U_i \\times_U U_{i'}$, $U_i \\times_U V_j$,", "$V_j \\times_U V_{j'}$ exist,", "\\item $\\mathcal{F}$ satisfies the sheaf condition with respect to", "$\\mathcal{V}$, and", "\\item for every $i \\in I$ the map", "$\\mathcal{F}(U_i) \\to \\prod_{j \\in J} \\mathcal{F}(V_j \\times_U U_i)$", "is injective.", "\\end{enumerate}", "Then $\\mathcal{F}$ satisfies the sheaf condition with respect to $\\mathcal{U}$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-compare-separated-presheaf-condition} the map", "$\\mathcal{F}(U) \\to \\prod \\mathcal{F}(U_i)$ is injective.", "Suppose given", "$s_i \\in \\mathcal{F}(U_i)$ such that $s_i|_{U_i \\times_U U_{i'}}", "= s_{i'}|_{U_i \\times_U U_{i'}}$ for all $i, i' \\in I$.", "Set $s_j = f_j^*(s_{\\alpha(j)}) \\in \\mathcal{F}(V_j)$.", "Since the morphisms $f_j$ are morphisms over $U$ we obtain", "induced morphisms $f_{jj'} : V_j \\times_U V_{j'} \\to", "U_{\\alpha(i)} \\times_U U_{\\alpha(i')}$ compatible with the", "$f_j, f_{j'}$ via the projection maps. It follows that", "$$", "s_j|_{V_j \\times_U V_{j'}}", "= f_{jj'}^*(s_{\\alpha(j)}|_{U_{\\alpha(j)} \\times_U U_{\\alpha(j')}})", "= f_{jj'}^*(s_{\\alpha(j')}|_{U_{\\alpha(j)} \\times_U U_{\\alpha(j')}})", "= s_{j'}|_{V_j \\times_U V_{j'}}", "$$", "for all $j, j' \\in J$. Hence, by the sheaf condition", "for $\\mathcal{F}$ with respect to $\\mathcal{V}$, we get a section", "$s \\in \\mathcal{F}(U)$ which restricts to $s_j$ on each $V_j$.", "We are done if we show $s$ restricts to $s_i$ on $U_i$", "for any $i \\in I$. Since $\\mathcal{F}$ satisfies (3) it", "suffices to show that $s$ and $s_i$ restrict to the", "same element over $U_i \\times_U V_j$ for all $j \\in J$.", "To see this we use", "$$", "s|_{U_i \\times_U V_j} = s_j|_{U_i \\times_U V_j} =", "(\\text{id} \\times f_j)^*s_{\\alpha(j)}|_{U_i \\times_U U_{\\alpha(j)}} =", "(\\text{id} \\times f_j)^*s_i|_{U_i \\times_U U_{\\alpha(j)}} =", "s_i|_{U_i \\times_U V_j}", "$$", "as desired." ], "refs": [ "sites-lemma-compare-separated-presheaf-condition" ], "ref_ids": [ 8504 ] } ], "ref_ids": [] }, { "id": 8506, "type": "theorem", "label": "sites-lemma-refine-same-topology", "categories": [ "sites" ], "title": "sites-lemma-refine-same-topology", "contents": [ "Let $\\mathcal{C}$ be a category. Let $\\text{Cov}_i$, $i = 1, 2$", "be two sets of families of morphisms with fixed target which", "each define the structure of a site on $\\mathcal{C}$.", "\\begin{enumerate}", "\\item If every $\\mathcal{U} \\in \\text{Cov}_1$ is tautologically", "equivalent to some $\\mathcal{V} \\in \\text{Cov}_2$, then", "$\\Sh(\\mathcal{C}, \\text{Cov}_2) \\subset", "\\Sh(\\mathcal{C}, \\text{Cov}_1)$.", "If also, every $\\mathcal{U} \\in \\text{Cov}_2$ is tautologically", "equivalent to some $\\mathcal{V} \\in \\text{Cov}_1$ then", "the category of sheaves are equal.", "\\item Suppose", "that for each $\\mathcal{U} \\in \\text{Cov}_1$ there exists a", "$\\mathcal{V} \\in \\text{Cov}_2$ such that $\\mathcal{V}$ refines", "$\\mathcal{U}$. In this case", "$\\Sh(\\mathcal{C}, \\text{Cov}_2) \\subset", "\\Sh(\\mathcal{C}, \\text{Cov}_1)$.", "If also for every $\\mathcal{U} \\in \\text{Cov}_2$", "there exists a $\\mathcal{V} \\in \\text{Cov}_1$ such that $\\mathcal{V}$", "refines $\\mathcal{U}$, then the categories of sheaves", "are equal.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows directly from Lemma \\ref{lemma-tautological-same-sheaf}", "and the definitions.", "\\medskip\\noindent", "Proof of (2). Let $\\mathcal{F}$ be a sheaf of sets for the site", "$(\\mathcal{C}, \\text{Cov}_2)$. Let $\\mathcal{U} \\in \\text{Cov}_1$,", "say $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$. By assumption we may choose a", "refinement $\\mathcal{V} \\in \\text{Cov}_2$ of $\\mathcal{U}$, say", "$\\mathcal{V} = \\{V_j \\to U\\}_{j \\in J}$ and refinement given", "by $\\alpha : J \\to I$ and $f_j : V_j \\to U_{\\alpha(j)}$.", "Observe that $\\mathcal{F}$ satisfies the sheaf condition for", "$\\mathcal{V}$ and for the coverings $\\{V_j \\times_U U_i \\to U_i\\}_{j \\in J}$", "as these are in $\\text{Cov}_2$. Hence $\\mathcal{F}$ satisfies", "the sheaf condition for $\\mathcal{U}$ by", "Lemma \\ref{lemma-compare-sheaf-condition}." ], "refs": [ "sites-lemma-tautological-same-sheaf", "sites-lemma-compare-sheaf-condition" ], "ref_ids": [ 8503, 8505 ] } ], "ref_ids": [] }, { "id": 8507, "type": "theorem", "label": "sites-lemma-choice-set-coverings-immaterial", "categories": [ "sites" ], "title": "sites-lemma-choice-set-coverings-immaterial", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\text{Cov}(\\mathcal{C})$ be a proper class of coverings", "satisfying conditions (1), (2) and (3) of Definition \\ref{definition-site}.", "Let $\\text{Cov}_1, \\text{Cov}_2 \\subset \\text{Cov}(\\mathcal{C})$", "be two subsets of $\\text{Cov}(\\mathcal{C})$ which endow", "$\\mathcal{C}$ with the structure of a site. If", "every covering $\\mathcal{U} \\in \\text{Cov}(\\mathcal{C})$", "is combinatorially equivalent to a covering in", "$\\text{Cov}_1$ and combinatorially equivalent to a", "covering in $\\text{Cov}_2$, then", "$\\Sh(\\mathcal{C}, \\text{Cov}_1) =", "\\Sh(\\mathcal{C}, \\text{Cov}_2)$." ], "refs": [ "sites-definition-site" ], "proofs": [ { "contents": [ "This is clear from Lemmas \\ref{lemma-refine-same-topology}", "and \\ref{lemma-tautological-combinatorial} above as the hypothesis", "implies that every covering", "$\\mathcal{U} \\in \\text{Cov}_1 \\subset \\text{Cov}(\\mathcal{C})$", "is combinatorially equivalent to an element of $\\text{Cov}_2$,", "and similarly with the roles of $\\text{Cov}_1$ and $\\text{Cov}_2$", "reversed." ], "refs": [ "sites-lemma-refine-same-topology", "sites-lemma-tautological-combinatorial" ], "ref_ids": [ 8506, 8502 ] } ], "ref_ids": [ 8652 ] }, { "id": 8508, "type": "theorem", "label": "sites-lemma-limit-sheaf", "categories": [ "sites" ], "title": "sites-lemma-limit-sheaf", "contents": [ "Let $\\mathcal{F} : \\mathcal{I} \\to \\Sh(\\mathcal{C})$", "be a diagram. Then $\\lim_\\mathcal{I} \\mathcal{F}$ exists", "and is equal to the limit in the category of presheaves." ], "refs": [], "proofs": [ { "contents": [ "Let $\\lim_i \\mathcal{F}_i$ be the limit as a presheaf.", "We will show that this is a sheaf and then it will trivially follow", "that it is a limit in the category of sheaves. To prove the sheaf", "property, let $\\mathcal{V} = \\{V_j \\to V\\}_{j\\in J}$ be a covering.", "Let $(s_j)_{j\\in J}$ be an element of $H^0(\\mathcal{V}, \\lim_i \\mathcal{F}_i)$.", "Using the projection maps we get elements $(s_{j, i})_{j\\in J}$", "in $H^0(\\mathcal{V}, \\mathcal{F}_i)$. By the sheaf property for", "$\\mathcal{F}_i$ we see that there is a unique $s_i \\in \\mathcal{F}_i(V)$", "such that $s_{j, i} = s_i|_{V_j}$. Let $\\phi : i \\to i'$ be a morphism", "of the index category. We would like to show that", "$\\mathcal{F}(\\phi) : \\mathcal{F}_i \\to \\mathcal{F}_{i'}$", "maps $s_i$ to $s_{i'}$. We know this is true for the sections", "$s_{i, j}$ and $s_{i', j}$ for all $j$ and hence by the sheaf property", "for $\\mathcal{F}_{i'}$ this is true. At this point we have an", "element $s = (s_i)_{i \\in \\Ob(\\mathcal{I})}$ of", "$(\\lim_i \\mathcal{F}_i)(V)$. We leave it to the reader to see", "this element has the required property that $s_j = s|_{V_j}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8509, "type": "theorem", "label": "sites-lemma-plus-presheaf", "categories": [ "sites" ], "title": "sites-lemma-plus-presheaf", "contents": [ "The constructions above define a presheaf", "$\\mathcal{F}^+$ together with a canonical", "map of presheaves $\\mathcal{F} \\to \\mathcal{F}^+$." ], "refs": [], "proofs": [ { "contents": [ "All we have to do is to show that given morphisms", "$W \\to V \\to U$ the composition $\\mathcal{F}^+(U)", "\\to \\mathcal{F}^+(V) \\to \\mathcal{F}^+(W)$", "equals the map $\\mathcal{F}^+(U) \\to \\mathcal{F}^+(W)$.", "This can be shown directly by verifying that, given", "a covering $\\{U_i \\to U\\}$ and", "$s = (s_i) \\in H^0(\\{U_i \\to U\\}, \\mathcal{F})$,", "we have canonically", "$W \\times_U U_i \\cong W \\times_V (V \\times_U U_i)$,", "and", "$s_i|_{W \\times_U U_i}$", "corresponds to", "$(s_i|_{V \\times_U U_i})|_{W \\times_V (V \\times_U U_i)}$", "via this isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8510, "type": "theorem", "label": "sites-lemma-plus-functorial", "categories": [ "sites" ], "title": "sites-lemma-plus-functorial", "contents": [ "The association $\\mathcal{F} \\mapsto", "(\\mathcal{F} \\to \\mathcal{F}^+)$", "is a functor." ], "refs": [], "proofs": [ { "contents": [ "Instead of proving this we state exactly what needs to be proven.", "Let $\\mathcal{F} \\to \\mathcal{G}$ be a map of presheaves. Prove", "the commutativity of:", "$$", "\\xymatrix{", "\\mathcal{F} \\ar[r] \\ar[d]", "&", "\\mathcal{F}^{+} \\ar[d]", "\\\\", "\\mathcal{G} \\ar[r]", "&", "\\mathcal{G}^{+}", "}", "$$" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8511, "type": "theorem", "label": "sites-lemma-common-refinement", "categories": [ "sites" ], "title": "sites-lemma-common-refinement", "contents": [ "Given a pair of coverings $\\{U_i \\to U\\}$", "and $\\{V_j \\to U\\}$ of a given object $U$ of the site", "$\\mathcal{C}$, there exists a covering which is a", "common refinement." ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathcal{C}$ is a site we have that for every $i$ the", "family $\\{V_j \\times_U U_i \\to U_i\\}_j$ is a covering.", "And, then another axiom implies that $\\{V_j \\times_U U_i \\to U\\}_{i, j}$", "is a covering of $U$. Clearly this covering refines both given", "coverings." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8512, "type": "theorem", "label": "sites-lemma-independent-refinement", "categories": [ "sites" ], "title": "sites-lemma-independent-refinement", "contents": [ "Any two morphisms $f, g: \\mathcal{U} \\to \\mathcal{V}$ of coverings", "inducing the same morphism $U \\to V$ induce the same", "map $H^0(\\mathcal{V}, \\mathcal{F}) \\to H^0(\\mathcal{U}, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{U} = \\{U_i \\to U\\}_{i\\in I}$ and", "$\\mathcal{V} = \\{V_j \\to V\\}_{j\\in J}$.", "The morphism $f$ consists of a map $U\\to V$, a map $\\alpha : I \\to J$ and", "maps $f_i : U_i \\to V_{\\alpha(i)}$.", "Likewise, $g$~determines a map $\\beta : I \\to J$ and maps", "$g_i : U_i \\to V_{\\beta(i)}$.", "As $f$ and $g$ induce the same map $U\\to V$, the diagram", "$$", "\\xymatrix{", "&", "V_{\\alpha(i)} \\ar[dr]", "\\\\", "U_i \\ar[ur]^{f_i} \\ar[dr]_{g_i}", "&", "&", "V", "\\\\", "&", "V_{\\beta(i)} \\ar[ur]", "}", "$$", "is commutative for every $i\\in I$. Hence $f$ and $g$ factor through", "the fibre product", "$$", "\\xymatrix{", "&", "V_{\\alpha(i)}", "\\\\", "U_i \\ar[r]^-\\varphi \\ar[ur]^{f_i} \\ar[dr]_{g_i}", "&", "V_{\\alpha(i)} \\times_V V_{\\beta(i)} \\ar[u]_{\\text{pr}_1} \\ar[d]^{\\text{pr}_2}", "\\\\", "&", "V_{\\beta(i)}.", "}", "$$", "Now let $s = (s_j)_j \\in H^0(\\mathcal{V}, \\mathcal{F})$.", "Then for all $i\\in I$:", "$$", "(f^*s)_i", "=", "f_i^*(s_{\\alpha(i)})", "=", "\\varphi^*\\text{pr}_1^*(s_{\\alpha(i)})", "=", "\\varphi^*\\text{pr}_2^*(s_{\\beta(i)})", "=", "g_i^*(s_{\\beta(i)})", "=", "(g^*s)_i,", "$$", "where the middle equality is given by the definition", "of $H^0(\\mathcal{V}, \\mathcal{F})$.", "This shows that the maps", "$H^0(\\mathcal{V}, \\mathcal{F}) \\to H^0(\\mathcal{U}, \\mathcal{F})$", "induced by $f$ and $g$ are equal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8513, "type": "theorem", "label": "sites-lemma-plus-surjective", "categories": [ "sites" ], "title": "sites-lemma-plus-surjective", "contents": [ "The map $\\theta : \\mathcal{F} \\to \\mathcal{F}^+$ has the following", "property: For every object $U$ of $\\mathcal{C}$ and every section", "$s \\in \\mathcal{F}^+(U)$ there exists a covering $\\{U_i \\to U\\}$", "such that $s|_{U_i}$ is in the image of $\\theta : \\mathcal{F}(U_i)", "\\to \\mathcal{F}^{+}(U_i)$." ], "refs": [], "proofs": [ { "contents": [ "Namely, let $\\{U_i \\to U\\}$ be a covering such that $s$ arises", "from the element $(s_i) \\in H^0(\\{U_i \\to U\\}, \\mathcal{F})$.", "According to Lemma \\ref{lemma-independent-refinement} we may", "consider the covering $\\{U_i \\to U_i\\}$ and the (obvious) morphism", "of coverings $\\{U_i \\to U_i\\} \\to \\{U_i \\to U\\}$ to compute the", "pullback of $s$ to an element of $\\mathcal{F}^+(U_i)$. And indeed,", "using this covering we get exactly $\\theta(s_i)$ for the restriction", "of $s$ to $U_i$." ], "refs": [ "sites-lemma-independent-refinement" ], "ref_ids": [ 8512 ] } ], "ref_ids": [] }, { "id": 8514, "type": "theorem", "label": "sites-lemma-colimit-sheaves", "categories": [ "sites" ], "title": "sites-lemma-colimit-sheaves", "contents": [ "Let $\\mathcal{F} : \\mathcal{I} \\to \\Sh(\\mathcal{C})$", "be a diagram. Then $\\colim_\\mathcal{I} \\mathcal{F}$ exists", "and is the sheafification of the colimit in the category of presheaves." ], "refs": [], "proofs": [ { "contents": [ "Since the sheafification functor is a left adjoint it commutes", "with all colimits, see Categories,", "Lemma \\ref{categories-lemma-adjoint-exact}.", "Hence, since $\\textit{PSh}(\\mathcal{C})$ has colimits, we deduce", "that $\\Sh(\\mathcal{C})$ has colimits (which are the", "sheafifications of the colimits in presheaves)." ], "refs": [ "categories-lemma-adjoint-exact" ], "ref_ids": [ 12249 ] } ], "ref_ids": [] }, { "id": 8515, "type": "theorem", "label": "sites-lemma-sheafification-exact", "categories": [ "sites" ], "title": "sites-lemma-sheafification-exact", "contents": [ "The functor $\\textit{PSh}(\\mathcal{C}) \\to \\Sh(\\mathcal{C})$,", "$\\mathcal{F} \\mapsto \\mathcal{F}^\\#$ is exact." ], "refs": [], "proofs": [ { "contents": [ "Since it is a left adjoint it is right exact, see", "Categories, Lemma \\ref{categories-lemma-exact-adjoint}.", "On the other hand, by Lemmas \\ref{lemma-common-refinement}", "and Lemma \\ref{lemma-independent-refinement} the colimits", "in the construction of $\\mathcal{F}^+$ are really over the", "directed set $\\Ob(\\mathcal{J}_U)$ where", "$\\mathcal{U} \\geq \\mathcal{U}'$ if and only if", "$\\mathcal{U}$ is a refinement of $\\mathcal{U}'$. Hence by", "Categories, Lemma \\ref{categories-lemma-directed-commutes}", "we see that $\\mathcal{F} \\to \\mathcal{F}^+$ commutes", "with finite limits (as a functor from presheaves to", "presheaves). Then we conclude using", "Lemma \\ref{lemma-limit-sheaf}." ], "refs": [ "categories-lemma-exact-adjoint", "sites-lemma-common-refinement", "sites-lemma-independent-refinement", "categories-lemma-directed-commutes", "sites-lemma-limit-sheaf" ], "ref_ids": [ 12250, 8511, 8512, 12228, 8508 ] } ], "ref_ids": [] }, { "id": 8516, "type": "theorem", "label": "sites-lemma-sections-sheafification", "categories": [ "sites" ], "title": "sites-lemma-sections-sheafification", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{C}$.", "Denote $\\theta^2 : \\mathcal{F} \\to \\mathcal{F}^\\#$ the canonical", "map of $\\mathcal{F}$ into its sheafification.", "Let $U$ be an object of $\\mathcal{C}$.", "Let $s \\in \\mathcal{F}^\\#(U)$. There exists", "a covering $\\{U_i \\to U\\}$ and sections", "$s_i \\in \\mathcal{F}(U_i)$ such that", "\\begin{enumerate}", "\\item $s|_{U_i} = \\theta^2(s_i)$, and", "\\item for every $i, j$ there exists a covering", "$\\{U_{ijk} \\to U_i \\times_U U_j\\}$ of $\\mathcal{C}$ such that", "the pullback of $s_i$ and $s_j$ to each $U_{ijk}$ agree.", "\\end{enumerate}", "Conversely, given any covering $\\{U_i \\to U\\}$, elements", "$s_i \\in \\mathcal{F}(U_i)$ such that (2) holds, then there", "exists a unique section $s \\in \\mathcal{F}^\\#(U)$ such", "that (1) holds." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8517, "type": "theorem", "label": "sites-lemma-mono-epi-sheaves", "categories": [ "sites" ], "title": "sites-lemma-mono-epi-sheaves", "contents": [ "The injective (resp.\\ surjective) maps defined above", "are exactly the monomorphisms (resp.\\ epimorphisms) of", "the category $\\Sh(\\mathcal{C})$. A map of sheaves", "is an isomorphism if and only if it is both injective", "and surjective." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8518, "type": "theorem", "label": "sites-lemma-coequalizer-surjection", "categories": [ "sites" ], "title": "sites-lemma-coequalizer-surjection", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F} \\to \\mathcal{G}$", "be a surjection of sheaves of sets. Then the diagram", "$$", "\\xymatrix{", "\\mathcal{F} \\times_\\mathcal{G} \\mathcal{F}", "\\ar@<1ex>[r] \\ar@<-1ex>[r]", "&", "\\mathcal{F} \\ar[r]", "&", "\\mathcal{G}}", "$$", "represents $\\mathcal{G}$ as a coequalizer." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{H}$ be a sheaf of sets and let", "$\\varphi : \\mathcal{F} \\to \\mathcal{H}$ be a map of sheaves equalizing", "the two maps $\\mathcal{F} \\times_\\mathcal{G} \\mathcal{F} \\to \\mathcal{F}$.", "Let $\\mathcal{G}' \\subset \\mathcal{G}$ be the presheaf image of", "the map $\\mathcal{F} \\to \\mathcal{G}$. As the product", "$\\mathcal{F} \\times_\\mathcal{G} \\mathcal{F}$ may be computed in the", "category of presheaves we see that it is equal to the presheaf product", "$\\mathcal{F} \\times_{\\mathcal{G}'} \\mathcal{F}$. Hence $\\varphi$", "induces a unique map of presheaves $\\psi' : \\mathcal{G}' \\to \\mathcal{H}$.", "Since $\\mathcal{G}$ is the sheafification of $\\mathcal{G}'$ by", "Lemma \\ref{lemma-mono-epi-sheaves}", "we conclude that $\\psi'$ extends uniquely to a map of sheaves", "$\\psi : \\mathcal{G} \\to \\mathcal{H}$. We omit the verification that", "$\\varphi$ is equal to the composition of $\\psi$ and the given map." ], "refs": [ "sites-lemma-mono-epi-sheaves" ], "ref_ids": [ 8517 ] } ], "ref_ids": [] }, { "id": 8519, "type": "theorem", "label": "sites-lemma-covering-surjective-after-sheafification", "categories": [ "sites" ], "title": "sites-lemma-covering-surjective-after-sheafification", "contents": [ "\\begin{slogan}", "Coverings become surjective after sheafification.", "\\end{slogan}", "Let $\\mathcal{C}$ be a site. If", "$\\{U_i \\to U\\}_{i \\in I}$ is a covering of the site", "$\\mathcal{C}$, then the morphism of presheaves of sets", "$$", "\\coprod\\nolimits_{i \\in I} h_{U_i} \\to h_U", "$$", "becomes surjective after sheafification." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-mono-epi-sheaves} above we have to show that", "$\\coprod\\nolimits_{i \\in I} h_{U_i}^\\# \\to h_U^\\#$", "is an epimorphism. Let $\\mathcal{F}$ be a sheaf of sets.", "A morphism $h_U^\\# \\to \\mathcal{F}$", "corresponds to a section $s \\in \\mathcal{F}(U)$.", "Hence the injectivity of $\\Mor(h_U^\\#, \\mathcal{F})", "\\to \\prod_i \\Mor(h_{U_i}^\\#, \\mathcal{F})$ follows", "directly from the sheaf property of $\\mathcal{F}$." ], "refs": [ "sites-lemma-mono-epi-sheaves" ], "ref_ids": [ 8517 ] } ], "ref_ids": [] }, { "id": 8520, "type": "theorem", "label": "sites-lemma-sheaf-coequalizer-representable", "categories": [ "sites" ], "title": "sites-lemma-sheaf-coequalizer-representable", "contents": [ "Let $\\mathcal{C}$ be a site. Let $E \\subset \\Ob(\\mathcal{C})$ be a", "subset such that every object of $\\mathcal{C}$ has a covering by", "elements of $E$. Let $\\mathcal{F}$ be a sheaf of sets. There exists a", "diagram of sheaves of sets", "$$", "\\xymatrix{", "\\mathcal{F}_1 \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\mathcal{F}_0 \\ar[r] &", "\\mathcal{F}", "}", "$$", "which represents $\\mathcal{F}$ as a coequalizer,", "such that $\\mathcal{F}_i$, $i = 0, 1$ are coproducts", "of sheaves of the form $h_U^\\#$ with $U \\in E$." ], "refs": [], "proofs": [ { "contents": [ "First we show there is an epimorphism $\\mathcal{F}_0 \\to \\mathcal{F}$", "of the desired type. Namely, just take", "$$", "\\mathcal{F}_0 =", "\\coprod\\nolimits_{U \\in E, s \\in \\mathcal{F}(U)}", "(h_U)^\\# \\longrightarrow \\mathcal{F}", "$$", "Here the arrow restricted to the component corresponding to $(U, s)$ maps", "the element $\\text{id}_U \\in h_U^\\#(U)$ to the section $s \\in \\mathcal{F}(U)$.", "This is an epimorphism according to Lemma \\ref{lemma-mono-epi-sheaves} and", "our condition on $E$. To construct $\\mathcal{F}_1$ first set", "$\\mathcal{G} = \\mathcal{F}_0 \\times_\\mathcal{F} \\mathcal{F}_0$ and", "then construct an epimorphism $\\mathcal{F}_1 \\to \\mathcal{G}$", "as above. See Lemma \\ref{lemma-coequalizer-surjection}." ], "refs": [ "sites-lemma-mono-epi-sheaves", "sites-lemma-coequalizer-surjection" ], "ref_ids": [ 8517, 8518 ] } ], "ref_ids": [] }, { "id": 8521, "type": "theorem", "label": "sites-lemma-pushforward-sheaf", "categories": [ "sites" ], "title": "sites-lemma-pushforward-sheaf", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor.", "If $\\mathcal{F}$ is a sheaf on $\\mathcal{D}$ then", "$u^p\\mathcal{F}$ is a sheaf as well." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{V_i \\to V\\}$ be a covering.", "By assumption $\\{u(V_i) \\to u(V)\\}$ is a covering", "in $\\mathcal{D}$ and $u(V_i \\times_V V_j) =", "u(V_i)\\times_{u(V)}u(V_j)$. Hence the sheaf condition for", "$u^p\\mathcal{F}$ and the covering $\\{V_i \\to V\\}$", "is precisely the same as the sheaf condition for $\\mathcal{F}$", "and the covering $\\{u(V_i) \\to u(V)\\}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8522, "type": "theorem", "label": "sites-lemma-adjoint-sheaves", "categories": [ "sites" ], "title": "sites-lemma-adjoint-sheaves", "contents": [ "In the situation of Lemma \\ref{lemma-pushforward-sheaf}.", "The functor $u_s : \\mathcal{G} \\mapsto (u_p \\mathcal{G})^\\#$", "is a left adjoint to $u^s$." ], "refs": [ "sites-lemma-pushforward-sheaf" ], "proofs": [ { "contents": [ "Follows directly from Lemma \\ref{lemma-adjoints-u} and", "Proposition \\ref{proposition-sheafification-adjoint}." ], "refs": [ "sites-lemma-adjoints-u", "sites-proposition-sheafification-adjoint" ], "ref_ids": [ 8500, 8640 ] } ], "ref_ids": [ 8521 ] }, { "id": 8523, "type": "theorem", "label": "sites-lemma-technical-up", "categories": [ "sites" ], "title": "sites-lemma-technical-up", "contents": [ "In the situation of Lemma \\ref{lemma-pushforward-sheaf}.", "For any presheaf $\\mathcal{G}$ on $\\mathcal{C}$", "we have $(u_p\\mathcal{G})^\\# = (u_p(\\mathcal{G}^\\#))^\\#$." ], "refs": [ "sites-lemma-pushforward-sheaf" ], "proofs": [ { "contents": [ "For any sheaf $\\mathcal{F}$ on $\\mathcal{D}$ we have", "\\begin{eqnarray*}", "\\Mor_{\\Sh(\\mathcal{D})}(u_s(\\mathcal{G}^\\#), \\mathcal{F})", "& = &", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{G}^\\#, u^s\\mathcal{F}) \\\\", "& = &", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(\\mathcal{G}^\\#, u^p\\mathcal{F}) \\\\", "& = &", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(\\mathcal{G}, u^p\\mathcal{F}) \\\\", "& = &", "\\Mor_{\\textit{PSh}(\\mathcal{D})}(u_p\\mathcal{G}, \\mathcal{F}) \\\\", "& = &", "\\Mor_{\\Sh(\\mathcal{D})}((u_p\\mathcal{G})^\\#, \\mathcal{F})", "\\end{eqnarray*}", "and the result follows from the Yoneda lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8521 ] }, { "id": 8524, "type": "theorem", "label": "sites-lemma-pullback-representable-sheaf", "categories": [ "sites" ], "title": "sites-lemma-pullback-representable-sheaf", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor", "between sites.", "For any object $U$ of $\\mathcal{C}$ we have $u_sh_U^\\# = h_{u(U)}^\\#$." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemmas \\ref{lemma-pullback-representable-presheaf}", "and \\ref{lemma-technical-up}." ], "refs": [ "sites-lemma-pullback-representable-presheaf", "sites-lemma-technical-up" ], "ref_ids": [ 8501, 8523 ] } ], "ref_ids": [] }, { "id": 8525, "type": "theorem", "label": "sites-lemma-composition-morphisms-sites", "categories": [ "sites" ], "title": "sites-lemma-composition-morphisms-sites", "contents": [ "Let $\\mathcal{C}_i$, $i = 1, 2, 3$ be sites. Let", "$u : \\mathcal{C}_2 \\to \\mathcal{C}_1$ and", "$v : \\mathcal{C}_3 \\to \\mathcal{C}_2$ be continuous functors", "which induce morphisms of sites. Then the functor", "$u \\circ v : \\mathcal{C}_3 \\to \\mathcal{C}_1$ is continuous and", "defines a morphism of sites $\\mathcal{C}_1 \\to \\mathcal{C}_3$." ], "refs": [], "proofs": [ { "contents": [ "It is immediate from the definitions that $u \\circ v$ is a continuous functor.", "In addition, we clearly have $(u \\circ v)^p = v^p \\circ u^p$, and hence", "$(u \\circ v)^s = v^s \\circ u^s$. Hence functors $(u \\circ v)_s$ and", "$u_s \\circ v_s$ are both left adjoints of $(u \\circ v)^s$. Therefore", "$(u \\circ v)_s \\cong u_s \\circ v_s$ and we conclude that $(u \\circ v)_s$", "is exact as a composition of exact functors." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8526, "type": "theorem", "label": "sites-lemma-directed-morphism", "categories": [ "sites" ], "title": "sites-lemma-directed-morphism", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let", "$u : \\mathcal{C} \\to \\mathcal{D}$ be continuous.", "Assume all the categories $(\\mathcal{I}_V^u)^{opp}$ of", "Section \\ref{section-functoriality-PSh}", "are filtered. Then $u$ defines a morphism of sites $\\mathcal{D} \\to", "\\mathcal{C}$, in other words $u_s$ is exact." ], "refs": [], "proofs": [ { "contents": [ "Since $u_s$ is the left adjoint of $u^s$ we see that $u_s$ is right", "exact, see Categories, Lemma \\ref{categories-lemma-exact-adjoint}.", "Hence it suffices to show that $u_s$ is left exact. In other words", "we have to show that $u_s$ commutes with finite limits.", "Because the categories $\\mathcal{I}_Y^{opp}$ are filtered", "we see that $u_p$ commutes with finite limits, see", "Categories, Lemma \\ref{categories-lemma-directed-commutes}", "(this also uses the description of limits in $\\textit{PSh}$,", "see Section \\ref{section-limits-colimits-PSh}).", "And since sheafification commutes with finite limits as well", "(Lemma \\ref{lemma-sheafification-exact}) we conclude because", "$u_s = \\# \\circ u_p$." ], "refs": [ "categories-lemma-exact-adjoint", "categories-lemma-directed-commutes", "sites-lemma-sheafification-exact" ], "ref_ids": [ 12250, 12228, 8515 ] } ], "ref_ids": [] }, { "id": 8527, "type": "theorem", "label": "sites-lemma-morphism-of-sites-covering", "categories": [ "sites" ], "title": "sites-lemma-morphism-of-sites-covering", "contents": [ "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites given by the", "functor $u : \\mathcal{C} \\to \\mathcal{D}$. Given any object $V$ of", "$\\mathcal{D}$ there exists a covering $\\{V_j \\to V\\}$ such that for every", "$j$ there exists a morphism $V_j \\to u(U_j)$ for some object $U_j$", "of $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Since $f^{-1} = u_s$ is exact we have $f^{-1}* = *$ where $*$ denotes the", "final object of the category of sheaves", "(Example \\ref{example-singleton-sheaf}).", "Since $f^{-1}* = u_s*$ is the sheafification of $u_p*$ we see", "there exists a covering $\\{V_j \\to V\\}$ such that $(u_p*)(V_j)$", "is nonempty. Since $(u_p*)(V_j)$ is a colimit over the category", "$\\mathcal{I}^u_{V_j}$ whose objects are morphisms $V_j \\to u(U)$", "the lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8528, "type": "theorem", "label": "sites-lemma-morphism-sites-topoi", "categories": [ "sites" ], "title": "sites-lemma-morphism-sites-topoi", "contents": [ "Given a morphism of sites $f : \\mathcal{D} \\to \\mathcal{C}$", "corresponding to the functor $u : \\mathcal{C} \\to \\mathcal{D}$", "the pair of functors $(f^{-1} = u_s, f_* = u^s)$ is a morphism of topoi." ], "refs": [], "proofs": [ { "contents": [ "This is obvious from Definition \\ref{definition-morphism-sites}." ], "refs": [ "sites-definition-morphism-sites" ], "ref_ids": [ 8665 ] } ], "ref_ids": [] }, { "id": 8529, "type": "theorem", "label": "sites-lemma-conclude-quasi-compact", "categories": [ "sites" ], "title": "sites-lemma-conclude-quasi-compact", "contents": [ "Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$.", "Consider the following conditions", "\\begin{enumerate}", "\\item $U$ is quasi-compact,", "\\item for every covering $\\{U_i \\to U\\}_{i \\in I}$ in $\\mathcal{C}$", "there exists a finite covering $\\{V_j \\to U\\}_{j = 1, \\ldots, m}$", "of $\\mathcal{C}$ refining $\\mathcal{U}$, and", "\\item for every covering $\\{U_i \\to U\\}_{i \\in I}$ in $\\mathcal{C}$", "there exists a finite subset $I' \\subset I$ such that", "$\\{U_i \\to U\\}_{i \\in I'}$ is a covering in $\\mathcal{C}$.", "\\end{enumerate}", "Then we always have (3) $\\Rightarrow$ (2) $\\Rightarrow$ (1)", "but the reverse implications do not hold in general." ], "refs": [], "proofs": [ { "contents": [ "The implications are immediate from the definitions.", "Let $X = [0, 1] \\subset \\mathbf{R}$", "as a topological space (with the usual $\\epsilon$-$\\delta$ topology).", "Let $\\mathcal{C}$ be the category of open subspaces of $X$ with", "inclusions as morphisms and usual open coverings (compare with", "Example \\ref{example-site-topological}). However, then we change the notion", "of covering in $\\mathcal{C}$ to exclude all finite coverings, except", "for the coverings of the form $\\{U \\to U\\}$. It is easy to see that this", "will be a site as in Definition \\ref{definition-site}.", "In this site the object $X = U = [0, 1]$ is quasi-compact in the sense of", "Definition \\ref{definition-quasi-compact} but $U$ does not satisfy (2).", "We leave it to the reader to make an example where (2) holds but not (3)." ], "refs": [ "sites-definition-site", "sites-definition-quasi-compact" ], "ref_ids": [ 8652, 8668 ] } ], "ref_ids": [] }, { "id": 8530, "type": "theorem", "label": "sites-lemma-quasi-compact", "categories": [ "sites" ], "title": "sites-lemma-quasi-compact", "contents": [ "Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $U$ is quasi-compact, and", "\\item for every surjection of sheaves", "$\\coprod_{i \\in I} \\mathcal{F}_i \\to h_U^\\#$", "there is a finite subset $J \\subset I$ such that", "$\\coprod_{i \\in J} \\mathcal{F}_i \\to h_U^\\#$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and let $\\coprod_{i \\in I} \\mathcal{F}_i \\to h_U^\\#$", "be a surjection. Then $\\text{id}_U$ is a section of", "$h_U^\\#$ over $U$. Hence there exists a covering", "$\\{U_a \\to U\\}_{a \\in A}$ and for each $a \\in A$", "a section $s_a$ of $\\coprod_{i \\in I} \\mathcal{F}_i$", "over $U_a$ mapping to $\\text{id}_U$. By the construction of coproducts as", "sheafification of coproducts of presheaves", "(Lemma \\ref{lemma-colimit-sheaves}), for each $a$", "there exists a covering $\\{U_{ab} \\to U_a\\}_{b \\in B_a}$ and", "for all $b \\in B_a$ an $\\iota(b) \\in I$ and a section", "$s_{b}$ of $\\mathcal{F}_{\\iota(b)}$ over $U_{ab}$", "mapping to $\\text{id}_U|_{U_{ab}}$. Thus after replacing", "the covering $\\{U_a \\to U\\}_{a \\in A}$ by", "$\\{U_{ab} \\to U\\}_{a \\in A, b \\in B_a}$", "we may assume we have a map $\\iota : A \\to I$", "and for each $a \\in A$ a section $s_a$ of $\\mathcal{F}_{\\iota(a)}$", "over $U_a$ mapping to $\\text{id}_U$.", "Since $U$ is quasi-compact, there is a covering", "$\\{V_c \\to U\\}_{c \\in C}$, a map $\\alpha : C \\to A$", "with finite image, and $V_c \\to U_{\\alpha(c)}$ over $U$.", "Then we see that $J = \\Im(\\iota \\circ \\alpha) \\subset I$ works", "because $\\coprod_{c \\in C} h_{V_c}^\\# \\to h_U^\\#$ is surjective", "(Lemma \\ref{lemma-covering-surjective-after-sheafification})", "and factors through $\\coprod_{i \\in J} \\mathcal{F}_i \\to h_U^\\#$.", "(Here we use that the composition", "$h_{V_c}^\\# \\to h_{U_{\\alpha(c)}}", "\\xrightarrow{s_{\\alpha(c)}} \\mathcal{F}_{\\iota(\\alpha(c))} \\to h_U^\\#$", "is the map $h_{V_c}^\\# \\to h_U^\\#$ coming from the morphism", "$V_c \\to U$ because $s_{\\alpha(c)}$ maps to $\\text{id}_U|_{U_{\\alpha(c)}}$.)", "\\medskip\\noindent", "Assume (2). Let $\\{U_i \\to U\\}_{i \\in I}$ be a covering.", "By Lemma \\ref{lemma-covering-surjective-after-sheafification}", "we see that $\\coprod_{i \\in I} h_{U_i}^\\# \\to h_U^\\#$ is surjective.", "Thus we find a finite subset $J \\subset I$ such that", "$\\coprod_{j \\in J} h_{U_j}^\\# \\to h_U^\\#$ is surjective.", "Then arguing as above we find a covering", "$\\{V_c \\to U\\}_{c \\in C}$ of $U$ in $\\mathcal{C}$", "and a map $\\iota : C \\to J$", "such that $\\text{id}_U$ lifts to a section", "of $s_c$ of $h_{U_{\\iota(c)}}^\\#$ over $V_c$.", "Refining the covering even further we may assume", "$s_c \\in h_{U_{\\iota(c)}}(V_c)$ mapping to $\\text{id}_U$.", "Then $s_c : V_c \\to U_{\\iota(c)}$ is a morphism over $U$", "and we conclude." ], "refs": [ "sites-lemma-colimit-sheaves", "sites-lemma-covering-surjective-after-sheafification", "sites-lemma-covering-surjective-after-sheafification" ], "ref_ids": [ 8514, 8519, 8519 ] } ], "ref_ids": [] }, { "id": 8531, "type": "theorem", "label": "sites-lemma-directed-colimits-sections", "categories": [ "sites" ], "title": "sites-lemma-directed-colimits-sections", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$\\mathcal{I} \\to \\Sh(\\mathcal{C})$, $i \\mapsto \\mathcal{F}_i$", "be a filtered diagram of sheaves of sets.", "Let $U \\in \\Ob(\\mathcal{C})$.", "Consider the canonical map", "$$", "\\Psi :", "\\colim_i \\mathcal{F}_i(U)", "\\longrightarrow", "\\left(\\colim_i \\mathcal{F}_i\\right)(U)", "$$", "With the terminology introduced above:", "\\begin{enumerate}", "\\item If all the transition maps are injective then", "$\\Psi$ is injective for any $U$.", "\\item If $U$ is quasi-compact, then $\\Psi$ is injective.", "\\item If $U$ is quasi-compact and all the transition maps are injective", "then $\\Psi$ is an isomorphism.", "\\item If $U$ has a cofinal system of coverings", "$\\{U_j \\to U\\}_{j \\in J}$ with", "$J$ finite and $U_j \\times_U U_{j'}$ quasi-compact", "for all $j, j' \\in J$, then $\\Psi$ is bijective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume all the transition maps are injective. In this case the presheaf", "$\\mathcal{F}' : V \\mapsto \\colim_i \\mathcal{F}_i(V)$ is", "separated (see Definition \\ref{definition-separated}).", "By Lemma \\ref{lemma-colimit-sheaves}", "we have", "$(\\mathcal{F}')^\\# = \\colim_i \\mathcal{F}_i$.", "By Theorem \\ref{theorem-plus}", "we see that $\\mathcal{F}' \\to (\\mathcal{F}')^\\#$ is injective.", "This proves (1).", "\\medskip\\noindent", "Assume $U$ is quasi-compact. Suppose that $s \\in \\mathcal{F}_i(U)$ and", "$s' \\in \\mathcal{F}_{i'}(U)$ give rise to elements on", "the left hand side which have the same image under $\\Psi$.", "This means we can choose a covering $\\{U_a \\to U\\}_{a \\in A}$", "and for each $a \\in A$ an index $i_a \\in I$, $i_a \\geq i$, $i_a \\geq i'$", "such that $\\varphi_{ii_a}(s) = \\varphi_{i'i_a}(s')$.", "Because $U$ is quasi-compact we can choose a covering", "$\\{V_b \\to U\\}_{b \\in B}$, a map $\\alpha : B \\to A$ with finite image,", "and morphisms $V_b \\to U_{\\alpha(b)}$ over $U$.", "Pick $i''\\in I$ to be $\\geq$ than all of the $i_{\\alpha(b)}$", "which is possible because the image of $\\alpha$ is finite.", "We conclude that $\\varphi_{ii''}(s)$ and $\\varphi_{i'i''}(s)$", "agree on $V_b$ for all $b \\in B$ and hence that", "$\\varphi_{ii''}(s) = \\varphi_{i'i''}(s)$. This proves (2).", "\\medskip\\noindent", "Assume $U$ is quasi-compact and all transition maps injective.", "Let $s$ be an element of the target of $\\Psi$. There exists a covering", "$\\{U_a \\to U\\}_{a \\in A}$ and for each $a \\in A$ an index $i_a \\in I$", "and a section $s_a \\in \\mathcal{F}_{i_a}(U_a)$", "such that $s|_{U_a}$ comes from $s_a$ for all $a \\in A$.", "Because $U$ is quasi-compact we can choose a covering", "$\\{V_b \\to U\\}_{b \\in B}$, a map $\\alpha : B \\to A$ with finite image,", "and morphisms $V_b \\to U_{\\alpha(b)}$ over $U$.", "Pick $i \\in I$ to be $\\geq$ than all of the $i_{\\alpha(b)}$", "which is possible because the image of $\\alpha$ is finite.", "By (1) the sections", "$s_b = \\varphi_{i_{\\alpha(b)} i}(s_{\\alpha(b)})|_{V_b}$", "agree over $V_b \\times_U V_{b'}$.", "Hence they glue to a section", "$s' \\in \\mathcal{F}_i(U)$ which maps to $s$ under $\\Psi$.", "This proves (3).", "\\medskip\\noindent", "Assume the hypothesis of (4).", "Let $s$ be an element of the target of $\\Psi$.", "By assumption there exists a finite covering", "$\\{U_j \\to U\\}_{j = 1, \\ldots, m} U_j$, with $U_j \\times_U U_{j'}$", "quasi-compact for all $j, j' \\in J$ and", "for each $j$ an index $i_j \\in I$ and $s_j \\in \\mathcal{F}_{i_j}(U_j)$", "such that $s|_{U_j}$ is the image of $s_j$ for all $j$.", "Since $U_j \\times_U U_{j'}$ is quasi-compact we can apply (2)", "and we see that there exists an $i_{jj'} \\in I$,", "$i_{jj'} \\geq i_j$, $i_{jj'} \\geq i_{j'}$ such that", "$\\varphi_{i_ji_{jj'}}(s_j)$ and $\\varphi_{i_{j'}i_{jj'}}(s_{j'})$", "agree over $U_j \\times_U U_{j'}$. Choose an index $i \\in I$", "wich is bigger or equal than all the $i_{jj'}$. Then we see that", "the sections $\\varphi_{i_ji}(s_j)$ of $\\mathcal{F}_i$ glue", "to a section of $\\mathcal{F}_i$ over $U$. This section is mapped", "to the element $s$ as desired." ], "refs": [ "sites-definition-separated", "sites-lemma-colimit-sheaves", "sites-theorem-plus" ], "ref_ids": [ 8658, 8514, 8492 ] } ], "ref_ids": [] }, { "id": 8532, "type": "theorem", "label": "sites-lemma-colimit-sites", "categories": [ "sites" ], "title": "sites-lemma-colimit-sites", "contents": [ "In Situation \\ref{situation-inverse-limit-sites} we can construct", "a site $(\\mathcal{C}, \\text{Cov}(\\mathcal{C}))$ as follows", "\\begin{enumerate}", "\\item as a category $\\mathcal{C} = \\colim \\mathcal{C}_i$, and", "\\item $\\text{Cov}(\\mathcal{C})$ is the union of the images", "of $\\text{Cov}(\\mathcal{C}_i)$ by $u_i : \\mathcal{C}_i \\to \\mathcal{C}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Our definition of composition of morphisms of sites implies that", "$u_b \\circ u_a = u_c$ whenever $c = a \\circ b$ in $\\mathcal{I}$.", "The formula $\\mathcal{C} = \\colim \\mathcal{C}_i$ means that", "$\\Ob(\\mathcal{C}) = \\colim \\Ob(\\mathcal{C}_i)$ and", "$\\text{Arrows}(\\mathcal{C}) = \\colim \\text{Arrows}(\\mathcal{C}_i)$.", "Then source, target, and composition are inherited from the", "source, target, and composition on $\\text{Arrows}(\\mathcal{C}_i)$.", "In this way we obtain a category. ", "Denote $u_i : \\mathcal{C}_i \\to \\mathcal{C}$ the obvious functor.", "Remark that given any finite diagram in $\\mathcal{C}$", "there exists an $i$ such that this diagram is", "the image of a diagram in $\\mathcal{C}_i$.", "\\medskip\\noindent", "Let $\\{U^t \\to U\\}$ be a covering of $\\mathcal{C}$. We first prove that if", "$V \\to U$ is a morphism of $\\mathcal{C}$, then $U^t \\times_U V$ exists.", "By our remark above and our definition of coverings, we can find an", "$i$, a covering $\\{U_i^t \\to U_i\\}$ of $\\mathcal{C}_i$ and a morphism", "$V_i \\to U_i$ whose image by $u_i$ is the given data. We claim that", "$U^t \\times_U V$ is the image of $U^t_i \\times_{U_i} V_i$ by $u_i$.", "Namely, for every $a : j \\to i$ in $\\mathcal{I}$ the functor $u_a$", "is continuous, hence", "$u_a(U^t_i \\times_{U_i} V_i) = u_a(U^t_i) \\times_{u_a(U_i)} u_a(V_i)$.", "In particular we can replace $i$ by $j$, if we so desire.", "Thus, if $W$ is another object of $\\mathcal{C}$, then we may", "assume $W = u_i(W_i)$ and we see that", "\\begin{align*}", "& \\Mor_\\mathcal{C}(W, u_i(U^t_i \\times_{U_i} V_i)) \\\\", "& =", "\\colim_{a : j \\to i}", "\\Mor_{\\mathcal{C}_j}(u_a(W_i), u_a(U^t_i \\times_{U_i} V_i)) \\\\", "& =", "\\colim_{a : j \\to i}", "\\Mor_{\\mathcal{C}_j}(u_a(W_i), u_a(U^t_i))", "\\times_{\\Mor_{\\mathcal{C}_j}(u_a(W_i), u_a(U_i))}", "\\Mor_{\\mathcal{C}_j}(u_a(W_i), u_a(V_i)) \\\\", "& =", "\\Mor_\\mathcal{C}(W, U^t)", "\\times_{\\Mor_\\mathcal{C}(W, U)}", "\\Mor_\\mathcal{C}(W, V)", "\\end{align*}", "as filtered colimits commute with finite limits", "(Categories, Lemma \\ref{categories-lemma-directed-commutes}).", "It also follows that", "$\\{U^t \\times_U V \\to V\\}$ is a covering in $\\mathcal{C}$.", "In this way we see that axiom (3) of Definition \\ref{definition-site} holds.", "\\medskip\\noindent", "To verify axiom (2) of Definition \\ref{definition-site}", "let $\\{U^t \\to U\\}_{t \\in T}$ be a covering of $\\mathcal{C}$", "and for each $t$ let $\\{U^{ts} \\to U^t\\}$ be a covering of", "$\\mathcal{C}$. Then we can find an $i$ and a covering", "$\\{U^t_i \\to U_i\\}_{t \\in T}$ of $\\mathcal{C}_i$ whose image by $u_i$ is", "$\\{U^t \\to U\\}$. Since $T$ is {\\bf finite} we may choose an $a : j \\to i$", "in $\\mathcal{I}$ and coverings $\\{U^{ts}_j \\to u_a(U^t_i)\\}$ of", "$\\mathcal{C}_j$ whose image by $u_j$ gives $\\{U^{ts} \\to U^t\\}$.", "Then we conclude that $\\{U^{ts} \\to U\\}$ is a covering of $\\mathcal{C}$", "by an application of axiom (2) to the site $\\mathcal{C}_j$.", "\\medskip\\noindent", "We omit the proof of axiom (1) of Definition \\ref{definition-site}." ], "refs": [ "categories-lemma-directed-commutes", "sites-definition-site", "sites-definition-site", "sites-definition-site" ], "ref_ids": [ 12228, 8652, 8652, 8652 ] } ], "ref_ids": [] }, { "id": 8533, "type": "theorem", "label": "sites-lemma-compute-pullback-to-limit", "categories": [ "sites" ], "title": "sites-lemma-compute-pullback-to-limit", "contents": [ "In Situation \\ref{situation-inverse-limit-sites} let", "$u_i : \\mathcal{C}_i \\to \\mathcal{C}$ be as constructed in", "Lemma \\ref{lemma-colimit-sites}. Then $u_i$ defines a morphism", "of sites $f_i : \\mathcal{C} \\to \\mathcal{C}_i$. For", "$U_i \\in \\Ob(\\mathcal{C}_i)$ and sheaf $\\mathcal{F}$ on $\\mathcal{C}_i$ we have", "\\begin{equation}", "\\label{equation-compute-pullback-to-limit}", "f_i^{-1}\\mathcal{F}(u_i(U_i)) =", "\\colim_{a : j \\to i} f_a^{-1}\\mathcal{F}(u_a(U_i))", "\\end{equation}" ], "refs": [ "sites-lemma-colimit-sites" ], "proofs": [ { "contents": [ "It is immediate from the arguments in the proof of", "Lemma \\ref{lemma-colimit-sites} that the functors $u_i$ are continuous.", "To finish the proof we have to show that $f_i^{-1} := u_{i, s}$", "is an exact functor $\\Sh(\\mathcal{C}_i) \\to \\Sh(\\mathcal{C})$.", "In fact it suffices to show that $f_i^{-1}$ is left exact, because", "it is right exact as a left adjoint", "(Categories, Lemma \\ref{categories-lemma-exact-adjoint}).", "We first prove (\\ref{equation-compute-pullback-to-limit})", "and then we deduce exactness.", "\\medskip\\noindent", "For an arbitrary object $V$ of $\\mathcal{C}$ we can pick a $a : j \\to i$", "and an object $V_j \\in \\Ob(\\mathcal{C})$ with $V = u_j(V_j)$. Then we", "can set", "$$", "\\mathcal{G}(V) = \\colim_{b : k \\to j} f_{a \\circ b}^{-1}\\mathcal{F}(u_b(V_j))", "$$", "The value $\\mathcal{G}(V)$ of the colimit is independent of the choice", "of $b : j \\to i$ and of the object $V_j$ with $u_j(V_j) = V$; we omit", "the verification. Moreover, if $\\alpha : V \\to V'$ is a morphism of", "$\\mathcal{C}$, then we can choose $b : j \\to i$ and a morphism", "$\\alpha_j : V_j \\to V'_j$ with $u_j(\\alpha_j) = \\alpha$. This induces", "a map $\\mathcal{G}(V') \\to \\mathcal{G}(V)$ by using the restrictions", "along the morphisms $u_b(\\alpha_j) : u_b(V_j) \\to u_b(V'_j)$. A check", "shows that $\\mathcal{G}$ is a presheaf (omitted).", "In fact, $\\mathcal{G}$ satisfies the sheaf condition. Namely,", "any covering $\\mathcal{U} = \\{U^t \\to U\\}$ in $\\mathcal{C}$", "comes from a finite level. Say $\\mathcal{U}_j = \\{U^t_j \\to U_j\\}$", "is mapped to $\\mathcal{U}$ by $u_j$ for some $a : j \\to i$ in $\\mathcal{I}$.", "Then we have", "$$", "H^0(\\mathcal{U}, \\mathcal{G}) =", "\\colim_{b : k \\to j} H^0(u_b(\\mathcal{U}_j), f_{b \\circ a}^{-1}\\mathcal{F}) =", "\\colim_{b : k \\to j} f_{b \\circ a}^{-1}\\mathcal{F}(u_b(U_j)) =", "\\mathcal{G}(U)", "$$", "as desired. The first equality holds because filtered colimits commute", "with finite limits", "(Categories, Lemma \\ref{categories-lemma-directed-commutes}).", "By construction $\\mathcal{G}(U)$ is given by the right hand side of", "(\\ref{equation-compute-pullback-to-limit}).", "Hence (\\ref{equation-compute-pullback-to-limit}) is true if we can", "show that $\\mathcal{G}$ is equal to $f_i^{-1}\\mathcal{F}$.", "\\medskip\\noindent", "In this paragraph we check that $\\mathcal{G}$ is canonically isomorphic to", "$f_i^{-1}\\mathcal{F}$. We strongly encourage the reader to skip this paragraph.", "To check this we have to show there is a bijection", "$\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{G}, \\mathcal{H}) =", "\\Mor_{\\Sh(\\mathcal{C}_i)}(\\mathcal{F}, f_{i, *}\\mathcal{H})$", "functorial in the sheaf $\\mathcal{H}$ on $\\mathcal{C}$", "where $f_{i, *} = u_i^p$. A map", "$\\mathcal{G} \\to \\mathcal{H}$ is the same thing as a compatible", "system of maps", "$$", "\\varphi_{a, b, V_j} :", "f_{a \\circ b}^{-1}\\mathcal{F}(u_b(V_j))", "\\longrightarrow", "\\mathcal{H}(u_j(V_j))", "$$", "for all $a : j \\to i$, $b : k \\to j$ and $V_j \\in \\Ob(\\mathcal{C}_j)$.", "The compatibilities force the maps $\\varphi_{a, b, V_j}$ to be equal", "to $\\varphi_{a \\circ b, \\text{id}, u_b(V_j)}$. Given $a : j \\to i$, the", "family of maps $\\varphi_{a, \\text{id}, V_j}$ corresponds to a map", "of sheaves $\\varphi_a : f_a^{-1}\\mathcal{F} \\to f_{j, *}\\mathcal{H}$.", "The compatibilities between the $\\varphi_{a, \\text{id}, u_a(V_i)}$", "and the $\\varphi_{\\text{id}, \\text{id}, V_i}$ implies that $\\varphi_a$", "is the adjoint of the map $\\varphi_{id}$ via", "$$", "\\Mor_{\\Sh(\\mathcal{C}_j)}(f_a^{-1}\\mathcal{F}, f_{j, *}\\mathcal{H}) =", "\\Mor_{\\Sh(\\mathcal{C}_i)}(\\mathcal{F}, f_{a, *}f_{j, *}\\mathcal{H}) =", "\\Mor_{\\Sh(\\mathcal{C}_i)}(\\mathcal{F}, f_{i, *}\\mathcal{H})", "$$", "Thus finally we see that the whole system of maps $\\varphi_{a, b, V_j}$", "is determined by the map $\\varphi_{id} : \\mathcal{F} \\to f_{i, *}\\mathcal{H}$.", "Conversely, given such a map $\\psi : \\mathcal{F} \\to f_{i, *}\\mathcal{H}$", "we can read the argument just given backwards to construct the family", "of maps $\\varphi_{a, b, V_j}$. This finishes the proof that", "$\\mathcal{G} = f_i^{-1}\\mathcal{F}$.", "\\medskip\\noindent", "Assume (\\ref{equation-compute-pullback-to-limit}) holds. Then the functor", "$\\mathcal{F} \\mapsto f_i^{-1}\\mathcal{F}(U)$ commutes with finite limits", "because finite limits of sheaves are computed in the category of presheaves", "(Lemma \\ref{lemma-limit-sheaf}), the functors $f_a^{-1}$ commutes with finite", "limits, and filtered colimits commute with finite limits. To see that", "$\\mathcal{F} \\mapsto f_i^{-1}\\mathcal{F}(V)$ commutes with finite limits", "for a general object $V$ of $\\mathcal{C}$, we can use the same argument using", "the formula for $f_i^{-1}\\mathcal{F}(V) = \\mathcal{G}(V)$ given above.", "Thus $f_i^{-1}$ is left exact and the proof of the lemma is complete." ], "refs": [ "sites-lemma-colimit-sites", "categories-lemma-exact-adjoint", "categories-lemma-directed-commutes", "sites-lemma-limit-sheaf" ], "ref_ids": [ 8532, 12250, 12228, 8508 ] } ], "ref_ids": [ 8532 ] }, { "id": 8534, "type": "theorem", "label": "sites-lemma-colimit", "categories": [ "sites" ], "title": "sites-lemma-colimit", "contents": [ "In Situation \\ref{situation-inverse-limit-sites} assume given", "\\begin{enumerate}", "\\item a sheaf $\\mathcal{F}_i$ on $\\mathcal{C}_i$ for all", "$i \\in \\Ob(\\mathcal{I})$,", "\\item for $a : j \\to i$ a map", "$\\varphi_a : f_a^{-1}\\mathcal{F}_i \\to \\mathcal{F}_j$", "of sheaves on $\\mathcal{C}_j$", "\\end{enumerate}", "such that $\\varphi_c = \\varphi_b \\circ f_b^{-1}\\varphi_a$", "whenever $c = a \\circ b$. Set $\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$", "on the site $\\mathcal{C}$ of Lemma \\ref{lemma-colimit-sites}.", "Let $i \\in \\Ob(\\mathcal{I})$ and $X_i \\in \\text{Ob}(\\mathcal{C}_i)$. Then", "$$", "\\colim_{a : j \\to i} \\mathcal{F}_j(u_a(X_i)) = \\mathcal{F}(u_i(X_i))", "$$" ], "refs": [ "sites-lemma-colimit-sites" ], "proofs": [ { "contents": [ "A formal argument shows that", "$$", "\\colim_{a : j \\to i} \\mathcal{F}_i(u_a(X_i)) =", "\\colim_{a : j \\to i} \\colim_{b : k \\to j}", "f_b^{-1}\\mathcal{F}_j(u_{a \\circ b}(X_i))", "$$", "By (\\ref{equation-compute-pullback-to-limit})", "we see that the inner colimit is equal to", "$f_j^{-1}\\mathcal{F}_j(u_i(X_i))$ hence we conclude by", "Lemma \\ref{lemma-directed-colimits-sections}." ], "refs": [ "sites-lemma-directed-colimits-sections" ], "ref_ids": [ 8531 ] } ], "ref_ids": [ 8532 ] }, { "id": 8535, "type": "theorem", "label": "sites-lemma-colimit-push-pull", "categories": [ "sites" ], "title": "sites-lemma-colimit-push-pull", "contents": [ "In Situation \\ref{situation-inverse-limit-sites} assume", "we have a sheaf $\\mathcal{F}$ on $\\mathcal{C}$. Then", "$$", "\\mathcal{F} = \\colim f_i^{-1}f_{i, *}\\mathcal{F}", "$$", "where the transition maps are $f_j^{-1}\\varphi_a$", "for $a : j \\to i$ where", "$\\varphi_a : f_a^{-1}f_{i, *}\\mathcal{F} \\to f_{j, *}\\mathcal{F}$", "is a canonical map", "satisfying a cocycle condition as in Lemma \\ref{lemma-colimit}." ], "refs": [ "sites-lemma-colimit" ], "proofs": [ { "contents": [ "For the morphism", "$$", "\\varphi_a : f_a^{-1}f_{i, *}\\mathcal{F} \\to f_{j, *}\\mathcal{F}", "$$", "we choose the adjoint to the identity map", "$$", "f_{i, *}\\mathcal{F} \\to f_{a, *}f_{j, *}\\mathcal{F}", "$$", "Hence $\\varphi_a$ is the counit for the adjunction given by", "$(f_a^{-1}, f_{a, *})$. We must prove that for all", "$a : j \\to i$ and $b : k \\to i$ with composition $c = a \\circ b$", "we have", "$\\varphi_c = \\varphi_b \\circ f_b^{-1}\\varphi_a$.", "This follows from Categories, Lemma \\ref{categories-lemma-compose-counits}.", "Lastly, we must prove that the map given by adjunction", "$$", "\\colim_{i \\in I} f_i^{-1}f_{i, *}\\mathcal{F}", "\\longrightarrow", "\\mathcal{F}", "$$", "is an isomorphism. For an object $U$ of $\\mathcal{C}$", "we need to show the map", "$$", "(\\colim_{i \\in I} f_i^{-1}\\mathcal{F}_i)(U) \\to \\mathcal{F}(U)", "$$", "is bijective. Choose an $i$ and an object $U_i$ of $\\mathcal{C}_i$", "with $u_i(U_i) = U$. Then the left hand side is equal to", "$$", "(\\colim_{i \\in I} f_i^{-1}\\mathcal{F}_i)(U) =", "\\colim_{a : j \\to i} f_{j, *}\\mathcal{F}(u_a(U_i))", "$$", "by Lemma \\ref{lemma-colimit}. Since $u_j(u_a(U_i)) = U$", "we have $f_{j, *}\\mathcal{F}(u_a(U_i)) = \\mathcal{F}(U)$", "for all $a : j \\to i$ by definition. Hence the value of the colimit is", "$\\mathcal{F}(U)$ and the proof is complete." ], "refs": [ "categories-lemma-compose-counits", "sites-lemma-colimit" ], "ref_ids": [ 12252, 8534 ] } ], "ref_ids": [ 8534 ] }, { "id": 8536, "type": "theorem", "label": "sites-lemma-recover-pu", "categories": [ "sites" ], "title": "sites-lemma-recover-pu", "contents": [ "There is a canonical map", "${}_pu\\mathcal{F}(u(U)) \\to \\mathcal{F}(U)$,", "which is compatible with restriction maps." ], "refs": [], "proofs": [ { "contents": [ "This is just the projection map $c(\\text{id}_{u(U)})$ above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8537, "type": "theorem", "label": "sites-lemma-adjoints-pu", "categories": [ "sites" ], "title": "sites-lemma-adjoints-pu", "contents": [ "The functor ${}_pu$ is a right adjoint to the functor $u^p$.", "In other words the formula", "$$", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(u^p\\mathcal{G}, \\mathcal{F})", "=", "\\Mor_{\\textit{PSh}(\\mathcal{D})}(\\mathcal{G}, {}_pu\\mathcal{F})", "$$", "holds bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "This is proved in exactly the same way as the proof", "of Lemma \\ref{lemma-adjoints-u}. We note that the map", "$u^p{}_pu \\mathcal{F} \\to \\mathcal{F}$ from", "Lemma \\ref{lemma-recover-pu} is the map that", "is used to go from the right to the left.", "\\medskip\\noindent", "Alternately, think", "of a presheaf of sets $\\mathcal{F}$ on $\\mathcal{C}$ as a presheaf", "$\\mathcal{F}'$ on $\\mathcal{C}^{opp}$ with values in $\\textit{Sets}^{opp}$,", "and similarly on $\\mathcal{D}$.", "Check that $({}_pu \\mathcal{F})' = u_p(\\mathcal{F}')$,", "and that $(u^p\\mathcal{G})' = u^p(\\mathcal{G}')$.", "By Remark \\ref{remark-functoriality-presheaves-values}", "we have the adjointness of $u_p$ and $u^p$ for", "presheaves with values in $\\textit{Sets}^{opp}$.", "The result then follows formally from this." ], "refs": [ "sites-lemma-adjoints-u", "sites-lemma-recover-pu", "sites-remark-functoriality-presheaves-values" ], "ref_ids": [ 8500, 8536, 8702 ] } ], "ref_ids": [] }, { "id": 8538, "type": "theorem", "label": "sites-lemma-adjoint-functors", "categories": [ "sites" ], "title": "sites-lemma-adjoint-functors", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ and $v : \\mathcal{D} \\to \\mathcal{C}$", "be functors of categories. Assume that $v$ is right adjoint to $u$.", "Then we have", "\\begin{enumerate}", "\\item $u^ph_V = h_{v(V)}$ for any $V$ in $\\mathcal{D}$,", "\\item the category $\\mathcal{I}^v_U$ has an initial object,", "\\item the category ${}_V^u\\mathcal{I}$ has a final object,", "\\item ${}_pu = v^p$, and", "\\item $u^p = v_p$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $V$ be an object of $\\mathcal{D}$. We have", "$u^ph_V = h_{v(V)}$ because", "$u^ph_V(U) = \\Mor_\\mathcal{D}(u(U), V) = \\Mor_\\mathcal{C}(U, v(V))$", "by assumption.", "\\medskip\\noindent", "Proof of (2). Let $U$ be an object of $\\mathcal{C}$. Let", "$\\eta : U \\to v(u(U))$ be the map adjoint to the map", "$\\text{id} : u(U) \\to u(U)$. Then we claim $(u(U), \\eta)$", "is an initial object of $\\mathcal{I}_U^v$. Namely, given", "an object $(V, \\phi : U \\to v(V))$ of $\\mathcal{I}_U^v$", "the morphism $\\phi$ is adjoint to a map $\\psi : u(U) \\to V$", "which then defines a morphism $(u(U), \\eta) \\to (V, \\phi)$.", "\\medskip\\noindent", "Proof of (3). Let $V$ be an object of $\\mathcal{D}$. Let", "$\\xi : u(v(V)) \\to V$ be the map adjoint to the map", "$\\text{id} : v(V) \\to v(V)$. Then we claim $(v(V), \\xi)$", "is a final object of ${}_V^u\\mathcal{I}$. Namely, given", "an object $(U, \\psi : u(U) \\to V)$ of ${}_V^u\\mathcal{I}$", "the morphism $\\psi$ is adjoint to a map $\\phi : U \\to v(V)$", "which then defines a morphism $(U, \\psi) \\to (v(V), \\xi)$.", "\\medskip\\noindent", "Hence for any presheaf $\\mathcal{F}$ on $\\mathcal{C}$ we have", "\\begin{eqnarray*}", "v^p\\mathcal{F}(V)", "& = &", "\\mathcal{F}(v(V)) \\\\", "& = &", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(h_{v(V)}, \\mathcal{F}) \\\\", "& = &", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(u^ph_V, \\mathcal{F}) \\\\", "& = &", "\\Mor_{\\textit{PSh}(\\mathcal{D})}(h_V, {}_pu\\mathcal{F}) \\\\", "& = &", "{}_pu\\mathcal{F}(V)", "\\end{eqnarray*}", "which proves part (4). Part (5) follows by the uniqueness of adjoint functors." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8539, "type": "theorem", "label": "sites-lemma-continuous-with-continuous-left-adjoint", "categories": [ "sites" ], "title": "sites-lemma-continuous-with-continuous-left-adjoint", "contents": [ "A continuous functor of sites which has a continuous left adjoint", "defines a morphism of sites." ], "refs": [], "proofs": [ { "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor of sites.", "Let $w : \\mathcal{D} \\to \\mathcal{C}$ be a continuous left adjoint.", "Then $u_p = w^p$ by Lemma \\ref{lemma-adjoint-functors}.", "Hence $u_s = w^s$ has a left adjoint, namely $w_s$", "(Lemma \\ref{lemma-adjoint-sheaves}). Thus $u_s$ has both a right and a", "left adjoint, whence is exact", "(Categories, Lemma \\ref{categories-lemma-exact-adjoint})." ], "refs": [ "sites-lemma-adjoint-functors", "sites-lemma-adjoint-sheaves", "categories-lemma-exact-adjoint" ], "ref_ids": [ 8538, 8522, 12250 ] } ], "ref_ids": [] }, { "id": 8540, "type": "theorem", "label": "sites-lemma-pu-sheaf", "categories": [ "sites" ], "title": "sites-lemma-pu-sheaf", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be cocontinuous.", "Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.", "Then ${}_pu\\mathcal{F}$ is a sheaf on $\\mathcal{D}$,", "which we will denote ${}_su\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{V_j \\to V\\}_{j \\in J}$ be a covering of the site $\\mathcal{D}$.", "We have to show that", "$$", "\\xymatrix{", "&\\ \\phantom{}_pu\\mathcal{F}(V) \\ar[r] &", "\\prod {}_pu\\mathcal{F}(V_j) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\prod {}_pu\\mathcal{F}(V_j \\times_V V_{j'}) &", "}", "$$", "is an equalizer diagram. Since ${}_pu$ is right adjoint to $u^p$", "we have", "$$", "{}_pu\\mathcal{F}(V) =", "\\Mor_{\\textit{PSh}(\\mathcal{D})}(h_V, {}_pu\\mathcal{F}) =", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(u^ph_V, \\mathcal{F}) =", "\\Mor_{\\Sh(\\mathcal{C})}((u^ph_V)^\\#, \\mathcal{F})", "$$", "Hence it suffices to show that", "\\begin{equation}", "\\label{equation-coequalizer}", "\\xymatrix{", "\\coprod u^p h_{V_j \\times_V V_{j'}} \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\coprod u^p h_{V_j} \\ar[r] &", "u^p h_V", "}", "\\end{equation}", "becomes a coequalizer diagram after sheafification. (Recall that a coproduct", "in the category of sheaves is the sheafification of the coproduct in the", "category of presheaves, see Lemma \\ref{lemma-colimit-sheaves}.)", "\\medskip\\noindent", "We first show that the second arrow of (\\ref{equation-coequalizer})", "becomes surjective after sheafification.", "To do this we use Lemma \\ref{lemma-mono-epi-sheaves}. Thus it suffices to", "show a section $s$ of $u^ph_V$ over $U$ lifts", "to a section of $\\coprod u^p h_{V_j}$ on the members of a covering of $U$.", "Note that $s$ is a morphism $s : u(U) \\to V$. Then", "$\\{V_j \\times_{V, s} u(U) \\to u(U)\\}$ is a covering of $\\mathcal{D}$.", "Hence, as $u$ is cocontinuous, there is a covering $\\{U_i \\to U\\}$", "such that $\\{u(U_i) \\to u(U)\\}$ refines $\\{V_j \\times_{V, s} u(U) \\to u(U)\\}$.", "This means that each restriction $s|_{U_i} : u(U_i) \\to V$ factors", "through a morphism $s_i : u(U_i) \\to V_j$ for some $j$, i.e., $s|_{U_i}$", "is in the image of $u^ph_{V_j}(U_i) \\to u^ph_V(U_i)$ as desired.", "\\medskip\\noindent", "Let $s, s' \\in (\\coprod u^ph_{V_j})^\\#(U)$ map to the same element", "of $(u^ph_V)^\\#(U)$. To finish the proof of the lemma we show that", "after replacing $U$ by the members of a covering that $s, s'$ are", "the image of the same section of $\\coprod u^p h_{V_j \\times_V V_{j'}}$", "by the two maps of (\\ref{equation-coequalizer}). We may first replace $U$", "by the members of a covering and assume that $s \\in u^ph_{V_j}(U)$", "and $s' \\in u^ph_{V_{j'}}(U)$. A second such replacement guarantees", "that $s$ and $s'$ have the same image in $u^ph_V(U)$ instead of in", "the sheafification. Hence $s : u(U) \\to V_j$ and $s' : u(U) \\to V_{j'}$", "are morphisms of $\\mathcal{D}$ such that", "$$", "\\xymatrix{", "u(U) \\ar[r]_{s'} \\ar[d]_s & V_{j'} \\ar[d] \\\\", "V_j \\ar[r] & V", "}", "$$", "is commutative. Thus we obtain $t = (s, s') : u(U) \\to V_j \\times_V V_{j'}$,", "i.e., a section $t \\in u^ph_{V_j \\times_V V_{j'}}(U)$", "which maps to $s, s'$ as desired." ], "refs": [ "sites-lemma-colimit-sheaves", "sites-lemma-mono-epi-sheaves" ], "ref_ids": [ 8514, 8517 ] } ], "ref_ids": [] }, { "id": 8541, "type": "theorem", "label": "sites-lemma-exact-cocontinuous", "categories": [ "sites" ], "title": "sites-lemma-exact-cocontinuous", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be cocontinuous.", "The functor", "$\\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$,", "$\\mathcal{G} \\mapsto (u^p\\mathcal{G})^\\#$", "is a left adjoint to the functor ${}_su$ introduced", "in Lemma \\ref{lemma-pu-sheaf} above. Moreover, it", "is exact." ], "refs": [ "sites-lemma-pu-sheaf" ], "proofs": [ { "contents": [ "Let us prove the adjointness property as follows", "\\begin{eqnarray*}", "\\Mor_{\\Sh(\\mathcal{C})}", "((u^p\\mathcal{G})^\\#, \\mathcal{F})", "& = &", "\\Mor_{\\textit{PSh}(\\mathcal{C})}", "(u^p\\mathcal{G}, \\mathcal{F}) \\\\", "& = &", "\\Mor_{\\textit{PSh}(\\mathcal{D})}", "(\\mathcal{G}, {}_pu\\mathcal{F}) \\\\", "& = &", "\\Mor_{\\Sh(\\mathcal{D})}", "(\\mathcal{G}, {}_su\\mathcal{F}).", "\\end{eqnarray*}", "Thus it is a left adjoint and hence right exact,", "see Categories, Lemma \\ref{categories-lemma-exact-adjoint}.", "We have seen that sheafification is left exact,", "see Lemma \\ref{lemma-sheafification-exact}.", "Moreover, the inclusion", "$i : \\Sh(\\mathcal{D}) \\to \\textit{PSh}(\\mathcal{D})$", "is left exact by Lemma \\ref{lemma-limit-sheaf}. Finally, the functor", "$u^p$ is left exact because it is a right adjoint", "(namely to $u_p$). Thus the functor is the composition", "${}^\\# \\circ u^p \\circ i$ of left exact functors,", "hence left exact." ], "refs": [ "categories-lemma-exact-adjoint", "sites-lemma-sheafification-exact", "sites-lemma-limit-sheaf" ], "ref_ids": [ 12250, 8515, 8508 ] } ], "ref_ids": [ 8540 ] }, { "id": 8542, "type": "theorem", "label": "sites-lemma-technical-pu", "categories": [ "sites" ], "title": "sites-lemma-technical-pu", "contents": [ "In the situation of Lemma \\ref{lemma-exact-cocontinuous}.", "For any presheaf $\\mathcal{G}$ on $\\mathcal{D}$", "we have $(u^p\\mathcal{G})^\\# = (u^p(\\mathcal{G}^\\#))^\\#$." ], "refs": [ "sites-lemma-exact-cocontinuous" ], "proofs": [ { "contents": [ "For any sheaf $\\mathcal{F}$ on $\\mathcal{C}$ we have", "\\begin{eqnarray*}", "\\Mor_{\\Sh(\\mathcal{C})}((u^p(\\mathcal{G}^\\#))^\\#, \\mathcal{F})", "& = &", "\\Mor_{\\Sh(\\mathcal{D})}(\\mathcal{G}^\\#, {}_su\\mathcal{F}) \\\\", "& = &", "\\Mor_{\\Sh(\\mathcal{D})}(\\mathcal{G}^\\#, {}_pu\\mathcal{F}) \\\\", "& = &", "\\Mor_{\\textit{PSh}(\\mathcal{D})}(\\mathcal{G}, {}_pu\\mathcal{F}) \\\\", "& = &", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(u^p\\mathcal{G}, \\mathcal{F}) \\\\", "& = &", "\\Mor_{\\Sh(\\mathcal{C})}((u^p\\mathcal{G})^\\#, \\mathcal{F})", "\\end{eqnarray*}", "and the result follows from the Yoneda lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8541 ] }, { "id": 8543, "type": "theorem", "label": "sites-lemma-cocontinuous-morphism-topoi", "categories": [ "sites" ], "title": "sites-lemma-cocontinuous-morphism-topoi", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be cocontinuous.", "The functors $g_* = {}_su$ and $g^{-1} = (u^p\\ )^\\#$", "define a morphism of topoi", "$g$ from $\\Sh(\\mathcal{C})$ to $\\Sh(\\mathcal{D})$." ], "refs": [], "proofs": [ { "contents": [ "This is exactly the content of Lemma \\ref{lemma-exact-cocontinuous}." ], "refs": [ "sites-lemma-exact-cocontinuous" ], "ref_ids": [ 8541 ] } ], "ref_ids": [] }, { "id": 8544, "type": "theorem", "label": "sites-lemma-composition-cocontinuous", "categories": [ "sites" ], "title": "sites-lemma-composition-cocontinuous", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$, and $v : \\mathcal{D} \\to \\mathcal{E}$", "be cocontinuous functors. Then $v \\circ u$ is cocontinuous and we", "have $h = g \\circ f$", "where $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$,", "resp.\\ $g : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{E})$,", "resp.\\ $h : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{E})$ is the", "morphism of topoi associated to $u$, resp.\\ $v$, resp.\\ $v \\circ u$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\in \\Ob(\\mathcal{C})$.", "Let $\\{E_i \\to v(u(U))\\}$ be a covering of $U$ in $\\mathcal{E}$.", "By assumption there exists a covering $\\{D_j \\to u(U)\\}$ in $\\mathcal{D}$", "such that $\\{v(D_j) \\to v(u(U))\\}$ refines $\\{E_i \\to v(u(U))\\}$. Also", "by assumption there exists a covering $\\{C_l \\to U\\}$ in $\\mathcal{C}$", "such that $\\{u(C_l) \\to u(U)\\}$ refines $\\{D_j \\to u(U)\\}$. Then it is", "true that $\\{v(u(C_l)) \\to v(u(U))\\}$ refines the covering", "$\\{E_i \\to v(u(U))\\}$. This proves that $v \\circ u$ is cocontinuous.", "To prove the last assertion it suffices to show that", "${}_sv \\circ {}_su = {}_s(v \\circ u)$. It suffices to prove that", "${}_pv \\circ {}_pu = {}_p(v \\circ u)$, see Lemma \\ref{lemma-pu-sheaf}.", "Since ${}_pu$, resp.\\ ${}_pv$, resp.\\ ${}_p(v \\circ u)$ is right adjoint to", "$u^p$, resp.\\ $v^p$, resp.\\ $(v \\circ u)^p$ it suffices to prove that", "$u^p \\circ v^p = (v \\circ u)^p$. And this is direct from the definitions." ], "refs": [ "sites-lemma-pu-sheaf" ], "ref_ids": [ 8540 ] } ], "ref_ids": [] }, { "id": 8545, "type": "theorem", "label": "sites-lemma-when-shriek", "categories": [ "sites" ], "title": "sites-lemma-when-shriek", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "Assume that", "\\begin{enumerate}", "\\item[(a)] $u$ is cocontinuous, and", "\\item[(b)] $u$ is continuous.", "\\end{enumerate}", "Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "be the associated morphism of topoi. Then", "\\begin{enumerate}", "\\item sheafification in the formula $g^{-1} = (u^p\\ )^\\#$ is", "unnecessary, in other words $g^{-1}(\\mathcal{G})(U) = \\mathcal{G}(u(U))$,", "\\item $g^{-1}$ has a left adjoint $g_{!} = (u_p\\ )^\\#$, and", "\\item $g^{-1}$ commutes with arbitrary limits and colimits.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-pushforward-sheaf} for any sheaf $\\mathcal{G}$", "on $\\mathcal{D}$ the presheaf $u^p\\mathcal{G}$ is a sheaf on $\\mathcal{C}$.", "And then we see the adjointness by the following string of", "equalities", "\\begin{eqnarray*}", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F}, g^{-1}\\mathcal{G})", "& = &", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(\\mathcal{F}, u^p\\mathcal{G})", "\\\\", "& = &", "\\Mor_{\\textit{PSh}(\\mathcal{D})}(u_p\\mathcal{F}, \\mathcal{G})", "\\\\", "& = &", "\\Mor_{\\Sh(\\mathcal{D})}(g_{!}\\mathcal{F}, \\mathcal{G})", "\\end{eqnarray*}", "The statement on limits and colimits follows from the", "discussion in Categories, Section \\ref{categories-section-adjoint}." ], "refs": [ "sites-lemma-pushforward-sheaf" ], "ref_ids": [ 8521 ] } ], "ref_ids": [] }, { "id": 8546, "type": "theorem", "label": "sites-lemma-preserve-equalizers", "categories": [ "sites" ], "title": "sites-lemma-preserve-equalizers", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "Assume that", "\\begin{enumerate}", "\\item[(a)] $u$ is cocontinuous,", "\\item[(b)] $u$ is continuous, and", "\\item[(c)] fibre products and equalizers exist in $\\mathcal{C}$ and", "$u$ commutes with them.", "\\end{enumerate}", "In this case the functor $g_!$ above commutes with fibre products and", "equalizers (and more generally with finite connected limits)." ], "refs": [], "proofs": [ { "contents": [ "Assume (a), (b), and (c).", "We have $g_! = (u_p\\ )^\\#$. Recall (Lemma \\ref{lemma-limit-sheaf}) that", "limits of sheaves are equal to the corresponding limits as presheaves.", "And sheafification commutes with finite limits", "(Lemma \\ref{lemma-sheafification-exact}). Thus it", "suffices to show that $u_p$ commutes with fibre products and equalizers.", "To do this it suffices that colimits over the categories", "$(\\mathcal{I}_V^u)^{opp}$ of", "Section \\ref{section-functoriality-PSh}", "commute with fibre products and equalizers. This follows", "from", "Lemma \\ref{lemma-almost-directed}", "and", "Categories, Lemma \\ref{categories-lemma-almost-directed-commutes-equalizers}." ], "refs": [ "sites-lemma-limit-sheaf", "sites-lemma-sheafification-exact", "sites-lemma-almost-directed", "categories-lemma-almost-directed-commutes-equalizers" ], "ref_ids": [ 8508, 8515, 8497, 12235 ] } ], "ref_ids": [] }, { "id": 8547, "type": "theorem", "label": "sites-lemma-back-and-forth", "categories": [ "sites" ], "title": "sites-lemma-back-and-forth", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "Assume that", "\\begin{enumerate}", "\\item[(a)] $u$ is cocontinuous,", "\\item[(b)] $u$ is continuous, and", "\\item[(c)] $u$ is fully faithful.", "\\end{enumerate}", "For $g_!, g^{-1}, g_*$ as above", "the canonical maps $\\mathcal{F} \\to g^{-1}g_!\\mathcal{F}$", "and $g^{-1}g_*\\mathcal{F} \\to \\mathcal{F}$ are isomorphisms", "for all sheaves $\\mathcal{F}$ on $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Let $X$ be an object of $\\mathcal{C}$.", "In Lemmas \\ref{lemma-pu-sheaf} and \\ref{lemma-when-shriek} we have seen that", "sheafification is not necessary for the functors", "$g^{-1} = (u^p\\ )^\\#$ and $g_{*} = ({}_pu\\ )^\\#$.", "We may compute", "$(g^{-1}g_{*}\\mathcal{F})(X) = g_{*}\\mathcal{F}(u(X))", "= \\lim \\mathcal{F}(Y)$. Here the limit", "is over the category of pairs $(Y, u(Y) \\to u(X))$", "where the morphisms $u(Y) \\to u(X)$ are not required to be", "of the form $u(\\alpha)$ with $\\alpha$ a morphism of $\\mathcal{C}$.", "By assumption (c) we see that they automatically come from", "morphisms of $\\mathcal{C}$ and we deduce that the limit is the", "value on $(X, u(\\text{id}_X))$, i.e., $\\mathcal{F}(X)$.", "This proves that $g^{-1}g_{*}\\mathcal{F} = \\mathcal{F}$.", "\\medskip\\noindent", "On the other hand, $(g^{-1}g_{!}\\mathcal{F})(X) =", "g_{!}\\mathcal{F}(u(X)) = (u_p\\mathcal{F})^\\#(u(X))$, and", "$u_p\\mathcal{F}(u(X)) = \\colim \\mathcal{F}(Y)$.", "Here the colimit is over the category of pairs $(Y, u(X) \\to u(Y))$", "where the morphisms $u(X) \\to u(Y)$ are not required to be", "of the form $u(\\alpha)$ with $\\alpha$ a morphism of $\\mathcal{C}$.", "By assumption (c) we see that they automatically come", "from morphisms of $\\mathcal{C}$ and we deduce that the colimit is the", "value on $(X, u(\\text{id}_X))$, i.e., $\\mathcal{F}(X)$. Thus for every", "$X \\in \\Ob(\\mathcal{C})$ we have", "$u_p\\mathcal{F}(u(X)) = \\mathcal{F}(X)$.", "Since $u$ is cocontinuous and continuous any covering of $u(X)$ in", "$\\mathcal{D}$ can be refined by a covering (!) $\\{u(X_i) \\to u(X)\\}$", "of $\\mathcal{D}$ where $\\{X_i \\to X\\}$ is a covering in $\\mathcal{C}$.", "This implies that $(u_p\\mathcal{F})^+(u(X)) = \\mathcal{F}(X)$ also,", "since in the colimit defining the value of $(u_p\\mathcal{F})^+$", "on $u(X)$ we may restrict to the cofinal system of coverings", "$\\{u(X_i) \\to u(X)\\}$ as above. Hence we see that", "$(u_p\\mathcal{F})^+(u(X)) = \\mathcal{F}(X)$ for all objects $X$", "of $\\mathcal{C}$ as well. Repeating this argument one more time", "gives the equality $(u_p\\mathcal{F})^\\#(u(X)) = \\mathcal{F}(X)$", "for all objects $X$ of $\\mathcal{C}$. This produces the desired", "equality $g^{-1}g_!\\mathcal{F} = \\mathcal{F}$." ], "refs": [ "sites-lemma-pu-sheaf", "sites-lemma-when-shriek" ], "ref_ids": [ 8540, 8545 ] } ], "ref_ids": [] }, { "id": 8548, "type": "theorem", "label": "sites-lemma-bigger-site", "categories": [ "sites" ], "title": "sites-lemma-bigger-site", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "Assume that", "\\begin{enumerate}", "\\item[(a)] $u$ is cocontinuous,", "\\item[(b)] $u$ is continuous,", "\\item[(c)] $u$ is fully faithful,", "\\item[(d)] fibre products exist in $\\mathcal{C}$ and $u$ commutes with them,", "and", "\\item[(e)] there exist final objects", "$e_\\mathcal{C} \\in \\Ob(\\mathcal{C})$,", "$e_\\mathcal{D} \\in \\Ob(\\mathcal{D})$ such that", "$u(e_\\mathcal{C}) = e_\\mathcal{D}$.", "\\end{enumerate}", "Let $g_!, g^{-1}, g_*$ be as above. Then, $u$ defines a morphism of sites", "$f : \\mathcal{D} \\to \\mathcal{C}$ with $f_* = g^{-1}$, $f^{-1} = g_!$.", "The composition", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C}) \\ar[r]^g &", "\\Sh(\\mathcal{D}) \\ar[r]^f &", "\\Sh(\\mathcal{C})", "}", "$$", "is isomorphic to the identity morphism of the topos", "$\\Sh(\\mathcal{C})$. Moreover, the functor $f^{-1}$ is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "By assumption the functor $u$ satisfies the hypotheses of", "Proposition \\ref{proposition-get-morphism}. Hence $u$ defines", "a morphism of sites and hence a morphism of topoi $f$ as in", "Lemma \\ref{lemma-morphism-sites-topoi}. The formulas", "$f_* = g^{-1}$ and $f^{-1} = g_!$ are clear from the lemma cited and", "Lemma \\ref{lemma-when-shriek}.", "We have", "$f_* \\circ g_* = g^{-1} \\circ g_* \\cong \\text{id}$, and", "$g^{-1} \\circ f^{-1} = g^{-1} \\circ g_! \\cong \\text{id}$", "by Lemma \\ref{lemma-back-and-forth}.", "\\medskip\\noindent", "We still have to show that $f^{-1}$ is fully faithful.", "Let $\\mathcal{F}, \\mathcal{G} \\in \\Ob(\\Sh(\\mathcal{C}))$.", "We have to show that the map", "$$", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F}, \\mathcal{G})", "\\longrightarrow", "\\Mor_{\\Sh(\\mathcal{D})}(f^{-1}\\mathcal{F}, f^{-1}\\mathcal{G})", "$$", "is bijective. But the right hand side is equal to", "\\begin{align*}", "\\Mor_{\\Sh(\\mathcal{D})}(f^{-1}\\mathcal{F}, f^{-1}\\mathcal{G})", "& =", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F}, f_*f^{-1}\\mathcal{G}) \\\\", "& =", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F}, g^{-1}f^{-1}\\mathcal{G}) \\\\", "& =", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F}, \\mathcal{G})", "\\end{align*}", "(the first equality by adjunction) which proves what we want." ], "refs": [ "sites-proposition-get-morphism", "sites-lemma-morphism-sites-topoi", "sites-lemma-when-shriek", "sites-lemma-back-and-forth" ], "ref_ids": [ 8641, 8528, 8545, 8547 ] } ], "ref_ids": [] }, { "id": 8549, "type": "theorem", "label": "sites-lemma-have-functor-other-way", "categories": [ "sites" ], "title": "sites-lemma-have-functor-other-way", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let", "$u : \\mathcal{C} \\to \\mathcal{D}$, and $v : \\mathcal{D} \\to \\mathcal{C}$", "be functors. Assume that $u$ is cocontinuous,", "and that $v$ is a right adjoint to $u$.", "Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be", "the morphism of topoi associated to $u$, see", "Lemma \\ref{lemma-cocontinuous-morphism-topoi}.", "Then $g_*\\mathcal{F}$ is equal to the presheaf", "$v^p\\mathcal{F}$, in other words, $(g_*\\mathcal{F})(V) = \\mathcal{F}(v(V))$." ], "refs": [ "sites-lemma-cocontinuous-morphism-topoi" ], "proofs": [ { "contents": [ "We have $u^ph_V = h_{v(V)}$ by Lemma \\ref{lemma-adjoint-functors}.", "By Lemma \\ref{lemma-technical-pu} this implies that", "$g^{-1}(h_V^\\#) = (u^ph_V^\\#)^\\# = (u^ph_V)^\\# = h_{v(V)}^\\#$.", "Hence for any sheaf $\\mathcal{F}$ on $\\mathcal{C}$ we have", "\\begin{eqnarray*}", "(g_*\\mathcal{F})(V)", "& = &", "\\Mor_{\\Sh(\\mathcal{D})}(h_V^\\#, g_*\\mathcal{F}) \\\\", "& = &", "\\Mor_{\\Sh(\\mathcal{C})}(g^{-1}(h_V^\\#), \\mathcal{F}) \\\\", "& = &", "\\Mor_{\\Sh(\\mathcal{C})}(h_{v(V)}^\\#, \\mathcal{F}) \\\\", "& = &", "\\mathcal{F}(v(V))", "\\end{eqnarray*}", "which proves the lemma." ], "refs": [ "sites-lemma-adjoint-functors", "sites-lemma-technical-pu" ], "ref_ids": [ 8538, 8542 ] } ], "ref_ids": [ 8543 ] }, { "id": 8550, "type": "theorem", "label": "sites-lemma-have-functor-other-way-morphism", "categories": [ "sites" ], "title": "sites-lemma-have-functor-other-way-morphism", "contents": [ "In the situation of Lemma \\ref{lemma-have-functor-other-way}.", "We have $g_* = v^s = v^p$ and $g^{-1} = v_s = (v_p\\ )^\\#$.", "If $v$ is continuous then $v$ defines a morphism of sites $f$", "from $\\mathcal{C}$ to $\\mathcal{D}$ whose associated morphism", "of topoi is equal to the morphism $g$ associated to the cocontinuous", "functor $u$. In other words, a continuous functor which has a", "cocontinuous left adjoint defines a morphism of sites." ], "refs": [ "sites-lemma-have-functor-other-way" ], "proofs": [ { "contents": [ "Clear from the discussion above the lemma and", "Definitions \\ref{definition-morphism-sites} and", "Lemma \\ref{lemma-morphism-sites-topoi}." ], "refs": [ "sites-definition-morphism-sites", "sites-lemma-morphism-sites-topoi" ], "ref_ids": [ 8665, 8528 ] } ], "ref_ids": [ 8549 ] }, { "id": 8551, "type": "theorem", "label": "sites-lemma-have-left-adjoint", "categories": [ "sites" ], "title": "sites-lemma-have-left-adjoint", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let", "$g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be", "the morphism of topoi associated to a continuous and cocontinuous functor", "$u : \\mathcal{C} \\to \\mathcal{D}$, see", "Lemmas \\ref{lemma-cocontinuous-morphism-topoi} and", "\\ref{lemma-when-shriek}.", "\\begin{enumerate}", "\\item If $w : \\mathcal{D} \\to \\mathcal{C}$ is a left adjoint to $u$, then", "\\begin{enumerate}", "\\item $g_!\\mathcal{F}$ is the sheaf associated to the presheaf", "$w^p\\mathcal{F}$, and", "\\item $g_!$ is exact.", "\\end{enumerate}", "\\item if $w$ is a continuous left adjoint, then $g_!$", "has a left adjoint.", "\\item If $w$ is a cocontinuous left adjoint, then $g_! = h^{-1}$ and", "$g^{-1} = h_*$ where $h : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ is", "the morphism of topoi associated to $w$.", "\\end{enumerate}" ], "refs": [ "sites-lemma-cocontinuous-morphism-topoi", "sites-lemma-when-shriek" ], "proofs": [ { "contents": [ "Recall that $g_!\\mathcal{F}$ is the sheafification of $u_p\\mathcal{F}$.", "Hence (1)(a) follows from the fact that $u_p = w^p$ by", "Lemma \\ref{lemma-adjoint-functors}.", "\\medskip\\noindent", "To see (1)(b) note that $g_!$ commutes with all colimits as $g_!$", "is a left adjoint (Categories, Lemma \\ref{categories-lemma-adjoint-exact}).", "Let $i \\mapsto \\mathcal{F}_i$ be a finite diagram in $\\Sh(\\mathcal{C})$.", "Then $\\lim \\mathcal{F}_i$ is computed in the category of presheaves", "(Lemma \\ref{lemma-limit-sheaf}). Since $w^p$ is a right adjoint", "(Lemma \\ref{lemma-adjoints-u})", "we see that $w^p \\lim \\mathcal{F}_i = \\lim w^p\\mathcal{F}_i$. Since", "sheafification is exact", "(Lemma \\ref{lemma-sheafification-exact})", "we conclude by (1)(a).", "\\medskip\\noindent", "Assume $w$ is continuous. Then $g_! = (w^p\\ )^\\# = w^s$ but sheafification", "isn't necessary and one has the left adjoint $w_s$, see", "Lemmas \\ref{lemma-pushforward-sheaf} and \\ref{lemma-adjoint-sheaves}.", "\\medskip\\noindent", "Assume $w$ is cocontinuous. The equality $g_! = h^{-1}$ follows from (1)(a)", "and the definitions. The equality $g^{-1} = h_*$ follows from the equality", "$g_! = h^{-1}$ and uniqueness of adjoint functor. Alternatively one can deduce", "it from Lemma \\ref{lemma-have-functor-other-way}." ], "refs": [ "sites-lemma-adjoint-functors", "categories-lemma-adjoint-exact", "sites-lemma-limit-sheaf", "sites-lemma-adjoints-u", "sites-lemma-sheafification-exact", "sites-lemma-pushforward-sheaf", "sites-lemma-adjoint-sheaves", "sites-lemma-have-functor-other-way" ], "ref_ids": [ 8538, 12249, 8508, 8500, 8515, 8521, 8522, 8549 ] } ], "ref_ids": [ 8543, 8545 ] }, { "id": 8552, "type": "theorem", "label": "sites-lemma-existence-lower-shriek", "categories": [ "sites" ], "title": "sites-lemma-existence-lower-shriek", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be two sites.", "Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi.", "Let $E \\subset \\Ob(\\mathcal{D})$ be a subset such that", "\\begin{enumerate}", "\\item for $V \\in E$ there exists a sheaf $\\mathcal{G}$", "on $\\mathcal{C}$ such that $f^{-1}\\mathcal{F}(V) = ", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{G}, \\mathcal{F})$ functorially", "for $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$,", "\\item every object of $\\mathcal{D}$ has a covering by objects of $E$.", "\\end{enumerate}", "Then $f^{-1}$ has a left adjoint $f_!$." ], "refs": [], "proofs": [ { "contents": [ "By the Yoneda lemma (Categories, Lemma \\ref{categories-lemma-yoneda})", "the sheaf $\\mathcal{G}_V$ corresponding to $V \\in E$", "is defined up to unique isomorphism by the formula", "$f^{-1}\\mathcal{F}(V) = \\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{G}_V, \\mathcal{F})$.", "Recall that", "$f^{-1}\\mathcal{F}(V) = \\Mor_{\\Sh(\\mathcal{D})}(h_V^\\#, f^{-1}\\mathcal{F})$.", "Denote $i_V : h_V^\\# \\to f^{-1}\\mathcal{G}_V$ the map corresponding to", "$\\text{id}$ in $\\Mor(\\mathcal{G}_V, \\mathcal{G}_V)$.", "Functoriality in (1) implies that the bijection is given by", "$$", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{G}_V, \\mathcal{F}) \\to", "\\Mor_{\\Sh(\\mathcal{D})}(h_V^\\#, f^{-1}\\mathcal{F}),\\quad", "\\varphi \\mapsto f^{-1}\\varphi \\circ i_V", "$$", "For any $V_1, V_2 \\in E$ there is a canonical map", "$$", "\\Mor_{\\Sh(\\mathcal{D})}(h^\\#_{V_2}, h^\\#_{V_1})", "\\to", "\\Hom_{\\Sh(\\mathcal{C})}(\\mathcal{G}_{V_2}, \\mathcal{G}_{V_1}),\\quad", "\\varphi \\mapsto f_!(\\varphi)", "$$", "which is characterized by", "$f^{-1}(f_!(\\varphi)) \\circ i_{V_2} = i_{V_1} \\circ \\varphi$.", "Note that $\\varphi \\mapsto f_!(\\varphi)$ is", "compatible with composition; this can be seen directly", "from the characterization. Hence $h_V^\\# \\mapsto \\mathcal{G}_V$", "and $\\varphi \\mapsto f_!\\varphi$ is a functor from", "the full subcategory of $\\Sh(\\mathcal{D})$ whose objects are the $h_V^\\#$.", "\\medskip\\noindent", "Let $J$ be a set and let $J \\to E$, $j \\mapsto V_j$ be a map.", "Then we have a functorial bijection", "$$", "\\Mor_{\\Sh(\\mathcal{C})}(\\coprod \\mathcal{G}_{V_j}, \\mathcal{F})", "\\longrightarrow", "\\Mor_{\\Sh(\\mathcal{D})}(\\coprod h_{V_j}^\\#, f^{-1}\\mathcal{F})", "$$", "using the product of the bijections above. Hence we can extend the", "functor $f_!$ to the full subcategory of $\\Sh(\\mathcal{D})$ whose", "objects are coproducts of $h_V^\\#$ with $V \\in E$.", "\\medskip\\noindent", "Given an arbitrary sheaf $\\mathcal{H}$ on $\\mathcal{D}$ we choose a", "coequalizer diagram", "$$", "\\xymatrix{", "\\mathcal{H}_1 \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\mathcal{H}_0 \\ar[r] &", "\\mathcal{H}", "}", "$$", "where $\\mathcal{H}_i = \\coprod h_{V_{i, j}}^\\#$", "is a coproduct with $V_{i, j} \\in E$.", "This is possible by assumption (2), see", "Lemma \\ref{lemma-sheaf-coequalizer-representable}", "(for those worried about set theoretical issues, note that", "the construction given in", "Lemma \\ref{lemma-sheaf-coequalizer-representable} is canonical).", "Define $f_!(\\mathcal{H})$ to be the sheaf on $\\mathcal{C}$", "which makes", "$$", "\\xymatrix{", "f_!\\mathcal{H}_1 \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "f_!\\mathcal{H}_0 \\ar[r] &", "f_!\\mathcal{H}", "}", "$$", "a coequalizer diagram. Then", "\\begin{align*}", "\\Mor(f_!\\mathcal{H}, \\mathcal{F})", "& =", "\\text{Equalizer}(", "\\xymatrix{", "\\Mor(f_!\\mathcal{H}_0, \\mathcal{F}) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\Mor(f_!\\mathcal{H}_1, \\mathcal{F})", "}", ") \\\\", "& =", "\\text{Equalizer}(", "\\xymatrix{", "\\Mor(\\mathcal{H}_0, f^{-1}\\mathcal{F}) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\Mor(\\mathcal{H}_1, f^{-1}\\mathcal{F})", "}", ") \\\\", "& =", "\\Hom(\\mathcal{H}, f^{-1}\\mathcal{F})", "\\end{align*}", "Hence we see that we can extend $f_!$ to the whole category of sheaves", "on $\\mathcal{D}$." ], "refs": [ "categories-lemma-yoneda", "sites-lemma-sheaf-coequalizer-representable", "sites-lemma-sheaf-coequalizer-representable" ], "ref_ids": [ 12203, 8520, 8520 ] } ], "ref_ids": [] }, { "id": 8553, "type": "theorem", "label": "sites-lemma-describe-j-shriek", "categories": [ "sites" ], "title": "sites-lemma-describe-j-shriek", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $U \\in \\Ob(\\mathcal{C})$.", "Let $\\mathcal{G}$ be a presheaf on $\\mathcal{C}/U$.", "Then $j_{U!}(\\mathcal{G}^\\#)$ is the sheaf associated to the presheaf", "$$", "V", "\\longmapsto", "\\coprod\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{G}(V \\xrightarrow{\\varphi} U)", "$$", "with obvious restriction mappings." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-when-shriek} we have", "$j_{U!}(\\mathcal{G}^\\#) = ((j_U)_p\\mathcal{G}^\\#)^\\#$.", "By Lemma \\ref{lemma-technical-up} this is equal to $((j_U)_p\\mathcal{G})^\\#$.", "Hence it suffices to prove that $(j_U)_p$ is given by", "the formula above for any presheaf $\\mathcal{G}$ on $\\mathcal{C}/U$.", "OK, and by the definition in Section \\ref{section-functoriality-PSh} we have", "$$", "(j_U)_p\\mathcal{G}(V)", "=", "\\colim_{(W/U, V \\to W)} \\mathcal{G}(W)", "$$", "Now it is clear that the category of pairs $(W/U, V \\to W)$", "has an object $O_\\varphi = (\\varphi : V \\to U, \\text{id} : V \\to V)$ for every", "$\\varphi : V \\to U$, and moreover for any object there is a unique", "morphism from one of the $O_\\varphi$ into it. The result follows." ], "refs": [ "sites-lemma-when-shriek", "sites-lemma-technical-up" ], "ref_ids": [ 8545, 8523 ] } ], "ref_ids": [] }, { "id": 8554, "type": "theorem", "label": "sites-lemma-describe-j-shriek-representable", "categories": [ "sites" ], "title": "sites-lemma-describe-j-shriek-representable", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $U \\in \\Ob(\\mathcal{C})$.", "Let $X/U$ be an object of $\\mathcal{C}/U$.", "Then we have $j_{U!}(h_{X/U}^\\#) = h_X^\\#$." ], "refs": [], "proofs": [ { "contents": [ "Denote $p : X \\to U$ the structure morphism of $X$.", "By Lemma \\ref{lemma-describe-j-shriek} we see $j_{U!}(h_{X/U}^\\#)$", "is the sheaf associated to the presheaf", "$$", "V", "\\longmapsto", "\\coprod\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\{\\psi : V \\to X \\mid p \\circ \\psi = \\varphi\\}", "$$", "This is clearly the same thing as $\\Mor_\\mathcal{C}(V, X)$.", "Hence the lemma follows." ], "refs": [ "sites-lemma-describe-j-shriek" ], "ref_ids": [ 8553 ] } ], "ref_ids": [] }, { "id": 8555, "type": "theorem", "label": "sites-lemma-essential-image-j-shriek", "categories": [ "sites" ], "title": "sites-lemma-essential-image-j-shriek", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $U \\in \\Ob(\\mathcal{C})$.", "The functor $j_{U!}$ gives an equivalence of categories", "$$", "\\Sh(\\mathcal{C}/U)", "\\longrightarrow", "\\Sh(\\mathcal{C})/h_U^\\#", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let us denote objects of $\\mathcal{C}/U$ as pairs $(X, a)$", "where $X$ is an object of $\\mathcal{C}$ and $a : X \\to U$ is", "a morphism of $\\mathcal{C}$. Similarly, objects of", "$\\Sh(\\mathcal{C})/h_U^\\#$ are pairs $(\\mathcal{F}, \\varphi)$.", "The functor $\\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})/h_U^\\#$", "sends $\\mathcal{G}$ to the pair $(j_{U!}\\mathcal{G}, \\gamma)$", "where $\\gamma$ is the composition of", "$j_{U!}\\mathcal{G} \\to j_{U!}*$ with the identification", "$j_{U!}* = h_U^\\#$.", "\\medskip\\noindent", "Let us construct a functor from", "$\\Sh(\\mathcal{C})/h_U^\\#$ to $\\Sh(\\mathcal{C}/U)$.", "Suppose that $(\\mathcal{F}, \\varphi)$ is given.", "For an object $(X, a)$ of $\\mathcal{C}/U$", "we consider the set $\\mathcal{F}_\\varphi(X, a)$", "of elements $s \\in \\mathcal{F}(X)$ which under $\\varphi$ map to the image", "of $a \\in \\Mor_\\mathcal{C}(X, U) = h_U(X)$ in", "$h_U^\\#(X)$. It is easy to see that", "$(X, a) \\mapsto \\mathcal{F}_\\varphi(X, a)$ is", "a sheaf on $\\mathcal{C}/U$. Clearly, the rule", "$(\\mathcal{F}, \\varphi) \\mapsto \\mathcal{F}_\\varphi$", "defines a functor $\\Sh(\\mathcal{C})/h_U^\\# \\to \\Sh(\\mathcal{C}/U)$.", "\\medskip\\noindent", "Consider also the functor", "$\\textit{PSh}(\\mathcal{C})/h_U \\to \\textit{PSh}(\\mathcal{C}/U)$,", "$(\\mathcal{F}, \\varphi) \\mapsto \\mathcal{F}_\\varphi$", "where $\\mathcal{F}_\\varphi(X, a)$ is defined as the set of elements", "of $\\mathcal{F}(X)$ mapping to $a \\in h_U(X)$.", "We claim that the diagram", "$$", "\\xymatrix{", "\\textit{PSh}(\\mathcal{C})/h_U \\ar[r] \\ar[d] &", "\\textit{PSh}(\\mathcal{C}/U) \\ar[d] \\\\", "\\Sh(\\mathcal{C})/h_U^\\# \\ar[r] &", "\\Sh(\\mathcal{C}/U)", "}", "$$", "commutes, where the vertical arrows are given by sheafification.", "To see this\\footnote{An alternative is to describe", "$\\mathcal{F}_\\varphi$ by the cartesian diagram", "$$", "\\vcenter{", "\\xymatrix{", "\\mathcal{F}_\\varphi \\ar[r] \\ar[d] & {*} \\ar[d] \\\\", "\\mathcal{F}|_{\\mathcal{C}/U} \\ar[r] & h_U|_{\\mathcal{C}/U}", "}", "}", "\\quad\\text{for presheaves and}\\quad", "\\vcenter{", "\\xymatrix{", "\\mathcal{F}_\\varphi \\ar[r] \\ar[d] & {*} \\ar[d] \\\\", "\\mathcal{F}|_{\\mathcal{C}/U} \\ar[r] & h_U^\\#|_{\\mathcal{C}/U}", "}", "}", "$$", "for sheaves and use that restriction to $\\mathcal{C}/U$ commutes", "with sheafification.}, it", "suffices to prove that the construction commutes with", "the functor $\\mathcal{F} \\mapsto \\mathcal{F}^+$ of", "Lemmas \\ref{lemma-plus-presheaf} and \\ref{lemma-plus-functorial}", "and Theorem \\ref{theorem-plus}.", "Commutation with $\\mathcal{F} \\mapsto \\mathcal{F}^+$ follows from the fact", "that given $(X, a)$ the categories of coverings of $(X, a)$ in", "$\\mathcal{C}/U$ and coverings of $X$ in $\\mathcal{C}$", "are canonically identified.", "\\medskip\\noindent", "Next, let $\\textit{PSh}(\\mathcal{C}/U) \\to \\textit{PSh}(\\mathcal{C})/h_U$", "send $\\mathcal{G}$ to the pair $(j_{U!}^{PSh}\\mathcal{G}, \\gamma)$", "where $j_{U!}^{PSh}\\mathcal{G}$ the presheaf defined by the formula", "in Lemma \\ref{lemma-describe-j-shriek} and $\\gamma$ is the composition of", "$j_{U!}^{PSh}\\mathcal{G} \\to j_{U!}*$ with the identification", "$j_{U!}^{PSh}* = h_U$ (obvious from the formula).", "Then it is immediately clear that the diagram", "$$", "\\xymatrix{", "\\textit{PSh}(\\mathcal{C}/U) \\ar[r] \\ar[d] &", "\\textit{PSh}(\\mathcal{C})/h_U \\ar[d] \\\\", "\\Sh(\\mathcal{C}/U) \\ar[r] &", "\\Sh(\\mathcal{C})/h_U^\\#", "}", "$$", "commutes, where the vertical arrows are sheafification.", "Putting everything together it suffices to show there are", "functorial isomorphisms $(j_{U!}^{PSh}\\mathcal{G})_\\gamma = \\mathcal{G}$", "for $\\mathcal{G}$ in $\\textit{PSh}(\\mathcal{C}/U)$", "and $j_{U!}^{PSh}\\mathcal{F}_\\varphi = \\mathcal{F}$", "for $(\\mathcal{F}, \\varphi)$ in $\\textit{PSh}(\\mathcal{C})/h_U$.", "The value of the presheaf $(j_{U!}^{PSh}\\mathcal{G})_\\gamma$", "on $(X, a)$ is the fibre of the map", "$$", "\\coprod\\nolimits_{a' : X \\to U} \\mathcal{G}(X, a') \\to \\Mor_\\mathcal{C}(X, U)", "$$", "over $a$ which is $\\mathcal{G}(X, a)$. This proves the first equality.", "The value of the presheaf $j_{U!}^{PSh}\\mathcal{F}_\\varphi$ is", "on $X$ is", "$$", "\\coprod\\nolimits_{a : X \\to U} \\mathcal{F}_\\varphi(X, a) =", "\\mathcal{F}(X)", "$$", "because given a set map $S \\to S'$ the set $S$ is the disjoint", "union of its fibres." ], "refs": [ "sites-lemma-plus-presheaf", "sites-lemma-plus-functorial", "sites-theorem-plus", "sites-lemma-describe-j-shriek" ], "ref_ids": [ 8509, 8510, 8492, 8553 ] } ], "ref_ids": [] }, { "id": 8556, "type": "theorem", "label": "sites-lemma-j-shriek-commutes-equalizers-fibre-products", "categories": [ "sites" ], "title": "sites-lemma-j-shriek-commutes-equalizers-fibre-products", "contents": [ "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$.", "The functor $j_{U!}$ commutes with fibre products and equalizers (and", "more generally finite connected limits). In particular, if", "$\\mathcal{F} \\subset \\mathcal{F}'$ in $\\Sh(\\mathcal{C}/U)$, then", "$j_{U!}\\mathcal{F} \\subset j_{U!}\\mathcal{F}'$." ], "refs": [], "proofs": [ { "contents": [ "Via Lemma \\ref{lemma-essential-image-j-shriek}", "and the fact that an equivalence of categories commutes", "with all limits, this reduces to the fact that the functor", "$\\Sh(\\mathcal{C})/h_U^\\# \\rightarrow \\Sh(\\mathcal{C})$", "commutes with fibre products and equalizers. Alternatively, one can", "prove this directly using the description of $j_{U!}$ in", "Lemma \\ref{lemma-describe-j-shriek}", "using that sheafification is exact. (Also, in case $\\mathcal{C}$ has", "fibre products and equalizers, the result follows from", "Lemma \\ref{lemma-preserve-equalizers}.)" ], "refs": [ "sites-lemma-essential-image-j-shriek", "sites-lemma-describe-j-shriek", "sites-lemma-preserve-equalizers" ], "ref_ids": [ 8555, 8553, 8546 ] } ], "ref_ids": [] }, { "id": 8557, "type": "theorem", "label": "sites-lemma-j-shriek-reflects-injections-surjections", "categories": [ "sites" ], "title": "sites-lemma-j-shriek-reflects-injections-surjections", "contents": [ "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$.", "The functor $j_{U!}$ reflects injections and surjections." ], "refs": [], "proofs": [ { "contents": [ "We have to show $j_{U!}$ reflects monomorphisms and epimorphisms, see", "Lemma \\ref{lemma-mono-epi-sheaves}.", "Via Lemma \\ref{lemma-essential-image-j-shriek}", "this reduces to the fact that the functor", "$\\Sh(\\mathcal{C})/h_U^\\# \\to \\Sh(\\mathcal{C})$", "reflects monomorphisms and epimorphisms." ], "refs": [ "sites-lemma-mono-epi-sheaves", "sites-lemma-essential-image-j-shriek" ], "ref_ids": [ 8517, 8555 ] } ], "ref_ids": [] }, { "id": 8558, "type": "theorem", "label": "sites-lemma-compute-j-shriek-restrict", "categories": [ "sites" ], "title": "sites-lemma-compute-j-shriek-restrict", "contents": [ "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$.", "For any sheaf $\\mathcal{F}$ on $\\mathcal{C}$ we have", "$j_{U!}j_U^{-1}\\mathcal{F} = \\mathcal{F} \\times h_U^\\#$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the description of $j_{U!}$ in", "Lemma \\ref{lemma-describe-j-shriek}." ], "refs": [ "sites-lemma-describe-j-shriek" ], "ref_ids": [ 8553 ] } ], "ref_ids": [] }, { "id": 8559, "type": "theorem", "label": "sites-lemma-relocalize", "categories": [ "sites" ], "title": "sites-lemma-relocalize", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $f : V \\to U$ be a morphism of $\\mathcal{C}$.", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{C}/V \\ar[rd]_{j_V} \\ar[rr]_j & &", "\\mathcal{C}/U \\ar[ld]^{j_U} \\\\", "& \\mathcal{C} &", "}", "$$", "of continuous and cocontinuous functors.", "The functor $j : \\mathcal{C}/V \\to \\mathcal{C}/U$,", "$(a : W \\to V) \\mapsto (f \\circ a : W \\to U)$", "is identified with the functor", "$j_{V/U} : (\\mathcal{C}/U)/(V/U) \\to \\mathcal{C}/U$", "via the identification $(\\mathcal{C}/U)/(V/U) = \\mathcal{C}/V$.", "Moreover we have $j_{V!} = j_{U!} \\circ j_!$,", "$j_V^{-1} = j^{-1} \\circ j_U^{-1}$, and", "$j_{V*} = j_{U*} \\circ j_*$." ], "refs": [], "proofs": [ { "contents": [ "The commutativity of the diagram is immediate.", "The agreement of $j$ with $j_{V/U}$ follows from the definitions. By", "Lemma \\ref{lemma-composition-cocontinuous}", "we see that the following diagram of morphisms of topoi", "\\begin{equation}", "\\label{equation-relocalize}", "\\vcenter{", "\\xymatrix{", "\\Sh(\\mathcal{C}/V) \\ar[rd]_{j_V} \\ar[rr]_j & &", "\\Sh(\\mathcal{C}/U) \\ar[ld]^{j_U} \\\\", "& \\Sh(\\mathcal{C}) &", "}", "}", "\\end{equation}", "is commutative. This proves that", "$j_V^{-1} = j^{-1} \\circ j_U^{-1}$ and $j_{V*} = j_{U*} \\circ j_*$.", "The equality $j_{V!} = j_{U!} \\circ j_!$", "follows formally from adjointness properties." ], "refs": [ "sites-lemma-composition-cocontinuous" ], "ref_ids": [ 8544 ] } ], "ref_ids": [] }, { "id": 8560, "type": "theorem", "label": "sites-lemma-relocalize-explicit", "categories": [ "sites" ], "title": "sites-lemma-relocalize-explicit", "contents": [ "Notation $\\mathcal{C}$, $f : V \\to U$, $j_U$, $j_V$, and $j$ as in", "Lemma \\ref{lemma-relocalize}. Via the identifications", "$\\Sh(\\mathcal{C}/V) = \\Sh(\\mathcal{C})/h_V^\\#$", "and", "$\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$", "of", "Lemma \\ref{lemma-essential-image-j-shriek}", "we have", "\\begin{enumerate}", "\\item the functor $j^{-1}$ has the following description", "$$", "j^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} h_U^\\#)", "=", "(\\mathcal{H} \\times_{\\varphi, h_U^\\#, f} h_V^\\# \\to h_V^\\#).", "$$", "\\item the functor $j_!$ has the following description", "$$", "j_!(\\mathcal{H} \\xrightarrow{\\varphi} h_V^\\#) =", "(\\mathcal{H} \\xrightarrow{h_f \\circ \\varphi} h_U^\\#)", "$$", "\\end{enumerate}" ], "refs": [ "sites-lemma-relocalize", "sites-lemma-essential-image-j-shriek" ], "proofs": [ { "contents": [ "Proof of (2). Recall that the identification", "$\\Sh(\\mathcal{C}/V) \\to \\Sh(\\mathcal{C})/h_V^\\#$", "sends $\\mathcal{G}$ to $j_{V!}\\mathcal{G} \\to j_{V!}(*) = h_V^\\#$", "and similarly for", "$\\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})/h_U^\\#$.", "Thus $j_!\\mathcal{G}$ is mapped to", "$j_{U!}(j_!\\mathcal{G}) \\to j_{U!}(*) = h_U^\\#$", "and (2) follows because $j_{U!}j_! = j_{V!}$", "by Lemma \\ref{lemma-relocalize}.", "\\medskip\\noindent", "The reader can now prove (1) by using that $j^{-1}$ is the", "right adjoint to $j_!$ and using that the rule in (1)", "is the right adjoint to the rule in (2). Here is a direct proof.", "Suppose that $\\varphi : \\mathcal{H} \\to h_U^\\#$ is an object of", "$\\Sh(\\mathcal{C})/h_U^\\#$. By the proof of", "Lemma \\ref{lemma-essential-image-j-shriek}", "this corresponds to the sheaf", "$\\mathcal{H}_\\varphi$ on $\\mathcal{C}/U$ defined by the rule", "$$", "(a : W \\to U)", "\\longmapsto", "\\{ s \\in \\mathcal{H}(W) \\mid \\varphi(s) = a\\}", "$$", "on $\\mathcal{C}/U$. The pullback $j^{-1}\\mathcal{H}_\\varphi$ to", "$\\mathcal{C}/V$ is given by the rule", "$$", "(a : W \\to V)", "\\longmapsto", "\\{ s \\in \\mathcal{H}(W) \\mid \\varphi(s) = f \\circ a\\}", "$$", "by the description of $j^{-1} = j_{U/V}^{-1}$ as the restriction", "of $\\mathcal{H}_\\varphi$ to $\\mathcal{C}/V$.", "On the other hand, applying the rule to the object", "$$", "\\xymatrix{", "\\mathcal{H}' = \\mathcal{H} \\times_{\\varphi, h_U^\\#, f} h_V^\\#", "\\ar[rr]^-{\\varphi'} & & h_V^\\#", "}", "$$", "of $\\Sh(\\mathcal{C})/h_V^\\#$", "we get $\\mathcal{H}'_{\\varphi'}$", "given by", "\\begin{align*}", "(a : W \\to V)", "\\longmapsto", "& \\{ s' \\in \\mathcal{H}'(W) \\mid \\varphi'(s') = a\\} \\\\", "= &", "\\{ (s, a') \\in \\mathcal{H}(W) \\times h_V^\\#(W) \\mid", "a' = a \\text{ and } \\varphi(s) = f \\circ a'\\}", "\\end{align*}", "which is exactly the same rule as the one describing", "$j^{-1}\\mathcal{H}_\\varphi$ above." ], "refs": [ "sites-lemma-relocalize", "sites-lemma-essential-image-j-shriek" ], "ref_ids": [ 8559, 8555 ] } ], "ref_ids": [ 8559, 8555 ] }, { "id": 8561, "type": "theorem", "label": "sites-lemma-glue-maps", "categories": [ "sites" ], "title": "sites-lemma-glue-maps", "contents": [ "\\begin{slogan}", "Maps of sheaves glue.", "\\end{slogan}", "Let $\\mathcal{C}$ be a site.", "Let $\\{U_i \\to U\\}$ be a covering of $\\mathcal{C}$.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be sheaves on $\\mathcal{C}$.", "Given a collection", "$$", "\\varphi_i :", "\\mathcal{F}|_{\\mathcal{C}/U_i}", "\\longrightarrow", "\\mathcal{G}|_{\\mathcal{C}/U_i}", "$$", "of maps of sheaves such that for all $i, j \\in I$ the maps", "$\\varphi_i, \\varphi_j$ restrict to the same map", "$\\varphi_{ij} : \\mathcal{F}|_{\\mathcal{C}/U_i \\times_U U_j} \\to", "\\mathcal{G}|_{\\mathcal{C}/U_i \\times_U U_j}$", "then there exists a unique map of sheaves", "$$", "\\varphi :", "\\mathcal{F}|_{\\mathcal{C}/U}", "\\longrightarrow", "\\mathcal{G}|_{\\mathcal{C}/U}", "$$", "whose restriction to each $\\mathcal{C}/U_i$ agrees with $\\varphi_i$." ], "refs": [], "proofs": [ { "contents": [ "The restrictions used in the lemma are those of", "Lemma \\ref{lemma-relocalize}.", "Let $V/U$ be an object of $\\mathcal{C}/U$.", "Set $V_i = U_i \\times_U V$ and denote $\\mathcal{V} = \\{V_i \\to V\\}$.", "Observe that $(U_i \\times_U U_j) \\times_U V = V_i \\times_V V_j$.", "Then we have", "$\\mathcal{F}|_{\\mathcal{C}/U_i}(V_i/U_i) = \\mathcal{F}(V_i)$", "and", "$\\mathcal{F}|_{\\mathcal{C}/U_i \\times_U U_j}(V_i \\times_V V_j/U_i \\times_U U_j)", "= \\mathcal{F}(V_i \\times_V V_j)$", "and similarly for $\\mathcal{G}$.", "Thus we can define $\\varphi$ on sections over $V$ ", "as the dotted arrows in the diagram", "$$", "\\xymatrix{", "\\mathcal{F}(V) \\ar@{=}[r] &", "H^0(\\mathcal{V}, \\mathcal{F}) \\ar@{..>}[d] \\ar[r] &", "\\prod \\mathcal{F}(V_i)", "\\ar[d]_{\\prod \\varphi_i}", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\prod \\mathcal{F}(V_i \\times_V V_j) \\ar[d]_{\\prod \\varphi_{ij}} \\\\", "\\mathcal{G}(V) \\ar@{=}[r] &", "H^0(\\mathcal{V}, \\mathcal{G}) \\ar[r] &", "\\prod \\mathcal{G}(V_i)", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\prod \\mathcal{G}(V_i \\times_V V_j)", "}", "$$", "The equality signs come from the sheaf condition; see", "Section \\ref{section-sheafification} for the notation", "$H^0(\\mathcal{V}, -)$.", "We omit the verification that these maps are compatible", "with the restriction maps." ], "refs": [ "sites-lemma-relocalize" ], "ref_ids": [ 8559 ] } ], "ref_ids": [] }, { "id": 8562, "type": "theorem", "label": "sites-lemma-internal-hom-sheaf", "categories": [ "sites" ], "title": "sites-lemma-internal-hom-sheaf", "contents": [ "\\begin{slogan}", "The category of sheaves on a site is cartesian closed", "\\end{slogan}", "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$, $\\mathcal{G}$ and", "$\\mathcal{H}$ be sheaves on $\\mathcal{C}$. There is a canonical bijection", "$$", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F}\\times\\mathcal{G},\\mathcal{H}) =", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F},\\SheafHom(\\mathcal{G},\\mathcal{H}))", "$$", "which is functorial in all three entries." ], "refs": [], "proofs": [ { "contents": [ "The lemma says that the functors $-\\times\\mathcal{G}$ and", "$\\SheafHom(\\mathcal{G},-)$ are adjoint to each other. ", "To show this, we use the notion of unit and counit, see", "Categories, Section \\ref{categories-section-adjoint}.", "The unit", "$$", "\\eta_\\mathcal{F} :", "\\mathcal{F}", "\\longrightarrow", "\\SheafHom(\\mathcal{G},\\mathcal{F}\\times\\mathcal{G})", "$$", "sends $s \\in \\mathcal{F}(U)$ to the map", "$\\mathcal{G}|_{\\mathcal{C}/U} \\to", "\\mathcal{F}|_{\\mathcal{C}/U}\\times\\mathcal{G}|_{\\mathcal{C}/U}$", "which over $V/U$ is given by", "$$", "\\mathcal{G}(V) \\longrightarrow \\mathcal{F}(V)\\times \\mathcal{G}(V), \\quad", "t \\longmapsto (s|_{V},t).", "$$", "The counit", "$$", "\\epsilon_{\\mathcal{H}} :", "\\SheafHom(\\mathcal{G}, \\mathcal{H}) \\times \\mathcal{G}", "\\longrightarrow", "\\mathcal{H}", "$$", "is the evaluation map. It is given by the rule", "$$", "\\Mor_{\\Sh(\\mathcal{C}/U)}(", "\\mathcal{G}|_{\\mathcal{C}/U}, \\mathcal{H}|_{\\mathcal{C}/U})", "\\times \\mathcal{G}(U)", "\\longrightarrow", "\\mathcal{H}(U),\\quad", "(\\varphi, s) \\longmapsto \\varphi(s).", "$$", "Then for each $\\varphi : \\mathcal{F} \\times \\mathcal{G} \\to \\mathcal{H}$,", "the corresponding morphism", "$\\mathcal{F} \\to \\SheafHom(\\mathcal{G},\\mathcal{H})$", "is given by mapping each section $s \\in \\mathcal{F}(U)$", "to the morphism of sheaves on $\\mathcal{C}/U$ which on", "sections over $V/U$ is given by", "$$", "\\mathcal{G}(V) \\longrightarrow \\mathcal{H}(V),\\quad", "t \\longmapsto \\varphi(s|_V, t).", "$$", "Conversely, for each", "$\\psi : \\mathcal{F} \\to \\SheafHom(\\mathcal{G}, \\mathcal{H})$,", "the corresponding morphism", "$\\mathcal{F} \\times \\mathcal{G} \\to \\mathcal{H}$ is given by", "$$", "\\mathcal{F}(U) \\times \\mathcal{G}(U) \\longrightarrow \\mathcal{H}(U),\\quad", "(s, t) \\longmapsto \\psi(s)(t)", "$$", "on sections over an object $U$. We omit the details of the proof showing", "that these constructions are mutually inverse." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8563, "type": "theorem", "label": "sites-lemma-hom-sheaf-hU", "categories": [ "sites" ], "title": "sites-lemma-hom-sheaf-hU", "contents": [ "Let $\\mathcal{C}$ be a site and $U \\in \\Ob(\\mathcal{C})$.", "Then $\\SheafHom(h_U^\\#, \\mathcal{F}) = j_*(\\mathcal{F}|_{\\mathcal{C}/U})$", "for $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "This can be shown by directly constructing an isomorphism", "of sheaves. Instead we argue as follows.", "Let $\\mathcal{G}$ be a sheaf on $\\mathcal{C}$.", "Then", "\\begin{align*}", "\\Mor(\\mathcal{G}, j_*(\\mathcal{F}|_{\\mathcal{C}/U}))", "& =", "\\Mor(\\mathcal{G}|_{\\mathcal{C}/U}, \\mathcal{F}|_{\\mathcal{C}/U}) \\\\", "& =", "\\Mor(j_!(\\mathcal{G}|_{\\mathcal{C}/U}), \\mathcal{F}) \\\\", "& =", "\\Mor(\\mathcal{G} \\times h_U^\\#, \\mathcal{F}) \\\\", "& =", "\\Mor(\\mathcal{G}, \\SheafHom(h_U^\\#, \\mathcal{F}))", "\\end{align*}", "and we conclude by the Yoneda lemma. Here we used", "Lemmas \\ref{lemma-internal-hom-sheaf} and", "\\ref{lemma-compute-j-shriek-restrict}." ], "refs": [ "sites-lemma-internal-hom-sheaf", "sites-lemma-compute-j-shriek-restrict" ], "ref_ids": [ 8562, 8558 ] } ], "ref_ids": [] }, { "id": 8564, "type": "theorem", "label": "sites-lemma-glue-sheaves", "categories": [ "sites" ], "title": "sites-lemma-glue-sheaves", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$.", "Given any glueing data $(\\mathcal{F}_i, \\varphi_{ij})$", "for sheaves of sets with respect to the covering $\\{U_i \\to U\\}_{i \\in I}$", "there exists a sheaf of sets $\\mathcal{F}$ on $\\mathcal{C}/U$", "together with isomorphisms", "$$", "\\varphi_i : \\mathcal{F}|_{\\mathcal{C}/U_i} \\to \\mathcal{F}_i", "$$", "such that the diagrams", "$$", "\\xymatrix{", "\\mathcal{F}|_{\\mathcal{C}/U_i \\times_U U_j}", "\\ar[d]_{\\text{id}} \\ar[r]_{\\varphi_i} &", "\\mathcal{F}_i|_{\\mathcal{C}/U_i \\times_U U_j} \\ar[d]^{\\varphi_{ij}} \\\\", "\\mathcal{F}|_{\\mathcal{C}/U_i \\times_U U_j} \\ar[r]^{\\varphi_j} &", "\\mathcal{F}_j|_{\\mathcal{C}/U_i \\times_U U_j}", "}", "$$", "are commutative." ], "refs": [], "proofs": [ { "contents": [ "Let us describe how to construct the sheaf $\\mathcal{F}$ on", "$\\mathcal{C}/U$. Let $a : V \\to U$ be an object of $\\mathcal{C}/U$.", "Then", "$$", "\\mathcal{F}(V/U) = \\{", "(s_i)_{i \\in I} \\in \\prod_{i \\in I} \\mathcal{F}_i(U_i \\times_U V/U_i)", "\\mid", "\\varphi_{ij}(s_i|_{U_i \\times_U U_j \\times_U V})", "=", "s_j|_{U_i \\times_U U_j \\times_U V}", "\\}", "$$", "We omit the construction of the restriction mappings.", "We omit the verification that this is a sheaf.", "We omit the construction of the isomorphisms $\\varphi_i$,", "and we omit proving the commutativity of the diagrams", "of the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8565, "type": "theorem", "label": "sites-lemma-mapping-property-glue", "categories": [ "sites" ], "title": "sites-lemma-mapping-property-glue", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$.", "The category $\\Sh(\\mathcal{C}/U)$ is equivalent", "to the category of glueing data via the functor that associates", "to $\\mathcal{F}$ on $\\mathcal{C}/U$ the canonical glueing data." ], "refs": [], "proofs": [ { "contents": [ "In", "Lemma \\ref{lemma-glue-maps}", "we saw that the functor is fully faithful, and in", "Lemma \\ref{lemma-glue-sheaves}", "we proved that it is essentially surjective (by explicitly constructing", "a quasi-inverse functor)." ], "refs": [ "sites-lemma-glue-maps", "sites-lemma-glue-sheaves" ], "ref_ids": [ 8561, 8564 ] } ], "ref_ids": [] }, { "id": 8566, "type": "theorem", "label": "sites-lemma-describe-j-shriek-good-site", "categories": [ "sites" ], "title": "sites-lemma-describe-j-shriek-good-site", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $U \\in \\Ob(\\mathcal{C})$.", "If the topology on $\\mathcal{C}$ is subcanonical, see", "Definition \\ref{definition-weaker-than-canonical},", "and if $\\mathcal{G}$ is a sheaf on $\\mathcal{C}/U$, then", "$$", "j_{U!}(\\mathcal{G})(V)", "=", "\\coprod\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{G}(V \\xrightarrow{\\varphi} U),", "$$", "in other words sheafification is not necessary in", "Lemma \\ref{lemma-describe-j-shriek}." ], "refs": [ "sites-definition-weaker-than-canonical", "sites-lemma-describe-j-shriek" ], "proofs": [ { "contents": [ "Let $\\mathcal{V} = \\{V_i \\to V\\}_{i \\in I}$ be a covering of $V$", "in the site $\\mathcal{C}$.", "We are going to check the sheaf condition for the presheaf $\\mathcal{H}$", "of Lemma \\ref{lemma-describe-j-shriek} directly.", "Let $(s_i, \\varphi_i)_{i \\in I} \\in \\prod_i \\mathcal{H}(V_i)$,", "This means $\\varphi_i : V_i \\to U$ is a morphism in $\\mathcal{C}$, and", "$s_i \\in \\mathcal{G}(V_i \\xrightarrow{\\varphi_i} U)$.", "The restriction of the pair $(s_i, \\varphi_i)$ to", "$V_i \\times_V V_j$ is the pair", "$(s_i|_{V_i \\times_V V_j/U}, \\text{pr}_1 \\circ \\varphi_i)$, and", "likewise the restriction of the pair $(s_j, \\varphi_j)$ to", "$V_i \\times_V V_j$ is the pair", "$(s_j|_{V_i \\times_V V_j/U}, \\text{pr}_2 \\circ \\varphi_j)$.", "Hence, if the family $(s_i, \\varphi_i)$ lies in", "$\\check{H}^0(\\mathcal{V}, \\mathcal{H})$, then we see that", "$\\text{pr}_1 \\circ \\varphi_i = \\text{pr}_2 \\circ \\varphi_j$.", "The condition that the topology on $\\mathcal{C}$ is weaker than the canonical", "topology then implies that there exists a unique morphism", "$\\varphi : V \\to U$ such that $\\varphi_i$ is the composition", "of $V_i \\to V$ with $\\varphi$. At this point the sheaf condition for", "$\\mathcal{G}$ guarantees that the sections $s_i$ glue to a unique", "section $s \\in \\mathcal{G}(V \\xrightarrow{\\varphi} U)$.", "Hence $(s, \\varphi) \\in \\mathcal{H}(V)$ as desired." ], "refs": [ "sites-lemma-describe-j-shriek" ], "ref_ids": [ 8553 ] } ], "ref_ids": [ 8662, 8553 ] }, { "id": 8567, "type": "theorem", "label": "sites-lemma-localize-given-products", "categories": [ "sites" ], "title": "sites-lemma-localize-given-products", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $U \\in \\Ob(\\mathcal{C})$.", "Assume $\\mathcal{C}$ has products of pairs of objects.", "Then", "\\begin{enumerate}", "\\item the functor $j_U$ has a continuous right adjoint,", "namely the functor $v(X) = X \\times U / U$,", "\\item the functor $v$ defines a morphism of sites", "$\\mathcal{C}/U \\to \\mathcal{C}$ whose associated morphism of topoi equals", "$j_U : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})$, and", "\\item we have $j_{U*}\\mathcal{F}(X) = \\mathcal{F}(X \\times U/U)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The functor $v$ being right adjoint to $j_U$ means that given $Y/U$ and $X$", "we have", "$$", "\\Mor_\\mathcal{C}(Y, X)", "=", "\\Mor_{\\mathcal{C}/U}(Y/U, X \\times U/U)", "$$", "which is clear. To check that $v$ is continuous let $\\{X_i \\to X\\}$ be", "a covering of $\\mathcal{C}$. By the third axiom of a site", "(Definition \\ref{definition-site})", "we see that", "$$", "\\{X_i \\times_X (X \\times U) \\to X \\times_X (X \\times U)\\}", "=", "\\{X_i \\times U \\to X \\times U\\}", "$$", "is a covering of $\\mathcal{C}$ also. Hence $v$ is continuous. The other", "statements of the lemma follow from Lemmas \\ref{lemma-have-functor-other-way}", "and \\ref{lemma-have-functor-other-way-morphism}." ], "refs": [ "sites-definition-site", "sites-lemma-have-functor-other-way", "sites-lemma-have-functor-other-way-morphism" ], "ref_ids": [ 8652, 8549, 8550 ] } ], "ref_ids": [] }, { "id": 8568, "type": "theorem", "label": "sites-lemma-relocalize-given-fibre-products", "categories": [ "sites" ], "title": "sites-lemma-relocalize-given-fibre-products", "contents": [ "Let $\\mathcal{C}$ be a site. Let $U \\to V$ be a morphism of $\\mathcal{C}$.", "Assume $\\mathcal{C}$ has fibre products. Let $j$ be as in", "Lemma \\ref{lemma-relocalize}. Then", "\\begin{enumerate}", "\\item the functor $j : \\mathcal{C}/U \\to \\mathcal{C}/V$", "has a continuous right adjoint, namely the functor", "$v : (X/V) \\mapsto (X \\times_V U/U)$,", "\\item the functor $v$ defines a morphism of sites", "$\\mathcal{C}/U \\to \\mathcal{C}/V$ whose associated morphism of topoi equals", "$j$, and", "\\item we have $j_*\\mathcal{F}(X/V) = \\mathcal{F}(X \\times_V U/U)$.", "\\end{enumerate}" ], "refs": [ "sites-lemma-relocalize" ], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-localize-given-products} since $j$ may be viewed", "as a localization functor by Lemma \\ref{lemma-relocalize}." ], "refs": [ "sites-lemma-localize-given-products", "sites-lemma-relocalize" ], "ref_ids": [ 8567, 8559 ] } ], "ref_ids": [ 8559 ] }, { "id": 8569, "type": "theorem", "label": "sites-lemma-restrict-back", "categories": [ "sites" ], "title": "sites-lemma-restrict-back", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $U \\in \\Ob(\\mathcal{C})$.", "Assume that every $X$ in $\\mathcal{C}$ has at most", "one morphism to $U$. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}/U$.", "The canonical maps $\\mathcal{F} \\to j_U^{-1}j_{U!}\\mathcal{F}$", "and $j_U^{-1}j_{U*}\\mathcal{F} \\to \\mathcal{F}$ are", "isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-back-and-forth}", "because the assumption on $U$ is equivalent to the fully faithfulness", "of the localization functor $\\mathcal{C}/U \\to \\mathcal{C}$." ], "refs": [ "sites-lemma-back-and-forth" ], "ref_ids": [ 8547 ] } ], "ref_ids": [] }, { "id": 8570, "type": "theorem", "label": "sites-lemma-localize-cartesian-square", "categories": [ "sites" ], "title": "sites-lemma-localize-cartesian-square", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$$", "\\xymatrix{", "U' \\ar[d] \\ar[r] & U \\ar[d] \\\\", "V' \\ar[r] & V", "}", "$$", "be a commutative diagram of $\\mathcal{C}$. The", "morphisms of Lemma \\ref{lemma-relocalize}", "produce commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "\\mathcal{C}/U' \\ar[d]_{j_{U'/V'}} \\ar[r]_{j_{U'/U}} &", "\\mathcal{C}/U \\ar[d]^{j_{U/V}} \\\\", "\\mathcal{C}/V' \\ar[r]^{j_{V'/V}} & \\mathcal{C}/V", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "\\Sh(\\mathcal{C}/U') \\ar[d]_{j_{U'/V'}} \\ar[r]_{j_{U'/U}} &", "\\Sh(\\mathcal{C}/U) \\ar[d]^{j_{U/V}} \\\\", "\\Sh(\\mathcal{C}/V') \\ar[r]^{j_{V'/V}} &", "\\Sh(\\mathcal{C}/V)", "}", "}", "$$", "of continuous and cocontinuous functors and of topoi.", "Moreover, if the initial diagram of $\\mathcal{C}$ is cartesian,", "then we have", "$j_{V'/V}^{-1} \\circ j_{U/V, *} = j_{U'/V', *} \\circ j_{U'/U}^{-1}$." ], "refs": [ "sites-lemma-relocalize" ], "proofs": [ { "contents": [ "The commutativity of the left square in the first statement of the lemma", "is immediate from the definitions. It implies the commutativity", "of the diagram of topoi by Lemma \\ref{lemma-composition-cocontinuous}.", "Assume the diagram is cartesian.", "By the uniqueness of adjoint functors, to show", "$j_{V'/V}^{-1} \\circ j_{U/V, *} = j_{U'/V', *} \\circ j_{U'/U}^{-1}$", "is equivalent to showing", "$j_{U/V}^{-1} \\circ j_{V'/V!} = j_{U'/U!} \\circ j_{U'/V'}^{-1}$.", "Via the identifications of Lemma \\ref{lemma-essential-image-j-shriek}", "we may think of our diagram of topoi as", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C})/h_{U'}^\\# \\ar[d] \\ar[r] &", "\\Sh(\\mathcal{C})/h_U^\\# \\ar[d] \\\\", "\\Sh(\\mathcal{C})/h_{V'}^\\# \\ar[r] &", "\\Sh(\\mathcal{C})/h_V^\\#", "}", "$$", "and we know how to interpret the functors $j^{-1}$ and $j_!$", "by Lemma \\ref{lemma-relocalize-explicit}. Thus we have to show", "given $\\mathcal{F} \\to h_{V'}^\\#$ that", "$$", "\\mathcal{F} \\times_{h_{V'}^\\#} h_{U'}^\\# =", "\\mathcal{F} \\times_{h_V^\\#} h_U^\\#", "$$", "as sheaves with map to $h_U^\\#$.", "This is true because $h_{U'} = h_{V'} \\times_{h_V} h_U$", "and hence also", "$$", "h_{U'}^\\# = h_{V'}^\\# \\times_{h_V^\\#} h_U^\\#", "$$", "as sheafification is exact." ], "refs": [ "sites-lemma-composition-cocontinuous", "sites-lemma-essential-image-j-shriek", "sites-lemma-relocalize-explicit" ], "ref_ids": [ 8544, 8555, 8560 ] } ], "ref_ids": [ 8559 ] }, { "id": 8571, "type": "theorem", "label": "sites-lemma-localize-morphism", "categories": [ "sites" ], "title": "sites-lemma-localize-morphism", "contents": [ "Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites", "corresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.", "Let $V \\in \\Ob(\\mathcal{D})$ and set $U = u(V)$.", "Then the functor $u' : \\mathcal{D}/V \\to \\mathcal{C}/U$,", "$V'/V \\mapsto u(V')/U$ determines a morphism of sites", "$f' : \\mathcal{C}/U \\to \\mathcal{D}/V$.", "The morphism $f'$ fits into a commutative diagram of topoi", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C}/U) \\ar[r]_{j_U} \\ar[d]_{f'} &", "\\Sh(\\mathcal{C}) \\ar[d]^f \\\\", "\\Sh(\\mathcal{D}/V) \\ar[r]^{j_V} &", "\\Sh(\\mathcal{D}).", "}", "$$", "Using the identifications", "$\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$ and", "$\\Sh(\\mathcal{D}/V) = \\Sh(\\mathcal{D})/h_V^\\#$ of", "Lemma \\ref{lemma-essential-image-j-shriek}", "the functor $(f')^{-1}$ is described by the rule", "$$", "(f')^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} h_V^\\#)", "=", "(f^{-1}\\mathcal{H} \\xrightarrow{f^{-1}\\varphi} h_U^\\#).", "$$", "Finally, we have $f'_*j_U^{-1} = j_V^{-1}f_*$." ], "refs": [ "sites-lemma-essential-image-j-shriek" ], "proofs": [ { "contents": [ "It is clear that $u'$ is continuous, and hence we get functors", "$f'_* = (u')^s = (u')^p$ (see", "Sections \\ref{section-functoriality-PSh}", "and \\ref{section-continuous-functors})", "and an adjoint $(f')^{-1} = (u')_s = ((u')_p\\ )^\\#$. The assertion", "$f'_*j_U^{-1} = j_V^{-1}f_*$ follows as", "$$", "(j_V^{-1}f_*\\mathcal{F})(V'/V)", "= f_*\\mathcal{F}(V') = \\mathcal{F}(u(V'))", "= (j_U^{-1}\\mathcal{F})(u(V')/U)", "= (f'_*j_U^{-1}\\mathcal{F})(V'/V)", "$$", "which holds even for presheaves. What isn't clear a priori is", "that $(f')^{-1}$ is exact, that the diagram commutes, and that", "the description of $(f')^{-1}$ holds.", "\\medskip\\noindent", "Let $\\mathcal{H}$ be a sheaf on $\\mathcal{D}/V$.", "Let us compute $j_{U!}(f')^{-1}\\mathcal{H}$. We have", "\\begin{eqnarray*}", "j_{U!}(f')^{-1}\\mathcal{H}", "& =", "((j_U)_p(u'_p\\mathcal{H})^\\#)^\\# \\\\", "& =", "((j_U)_pu'_p\\mathcal{H})^\\# \\\\", "& =", "(u_p(j_V)_p\\mathcal{H})^\\# \\\\", "& =", "f^{-1}j_{V!}\\mathcal{H}", "\\end{eqnarray*}", "The first equality by unwinding the definitions.", "The second equality by Lemma \\ref{lemma-technical-up}.", "The third equality because $u \\circ j_V = j_U \\circ u'$.", "The fourth equality by Lemma \\ref{lemma-technical-up} again.", "All of the equalities above are isomorphisms of functors, and", "hence we may interpret this as saying that the following", "diagram of categories and functors is commutative", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C}/U) \\ar[r]_{j_{U!}} &", "\\Sh(\\mathcal{C})/h_U^\\# \\ar[r] &", "\\Sh(\\mathcal{C}) \\\\", "\\Sh(\\mathcal{D}/V) \\ar[r]^{j_{V!}} \\ar[u]^{(f')^{-1}} &", "\\Sh(\\mathcal{D})/h_V^\\# \\ar[r] \\ar[u]^{f^{-1}} &", "\\Sh(\\mathcal{D}) \\ar[u]^{f^{-1}}", "}", "$$", "The middle arrow makes sense as $f^{-1}h_V^\\# = (h_{u(V)})^\\# = h_U^\\#$, see", "Lemma \\ref{lemma-pullback-representable-sheaf}.", "In particular this proves the description of $(f')^{-1}$ given", "in the statement of the lemma.", "Since by", "Lemma \\ref{lemma-essential-image-j-shriek}", "the left horizontal arrows are equivalences", "and since $f^{-1}$ is exact by assumption we conclude that", "$(f')^{-1} = u'_s$ is exact. Namely, because it is a left adjoint", "it is already right exact", "(Categories, Lemma \\ref{categories-lemma-adjoint-exact}).", "Hence we only need to show that", "it transforms a final object into a final object and commutes", "with fibre products", "(Categories, Lemma \\ref{categories-lemma-characterize-left-exact}).", "Both are clear for the induced functor", "$f^{-1} : \\Sh(\\mathcal{D})/h_V^\\# \\to \\Sh(\\mathcal{C})/h_U^\\#$.", "This proves that $f'$ is a morphism of sites.", "\\medskip\\noindent", "We still have to verify that $(f')^{-1}j_V^{-1} = j_U^{-1}f^{-1}$.", "To see this use the formula above and the description", "in Lemma \\ref{lemma-compute-j-shriek-restrict}. Namely,", "combined these give, for any sheaf $\\mathcal{G}$ on $\\mathcal{D}$, that", "$$", "j_{U!}(f')^{-1}j_V^{-1}\\mathcal{G}", "=", "f^{-1}j_{V!}j_V^{-1}\\mathcal{G}", "=", "f^{-1}(\\mathcal{G} \\times h_V^\\#)", "=", "f^{-1}\\mathcal{G} \\times h_U^\\#", "=", "j_{U!}j_U^{-1}f^{-1}\\mathcal{G}.", "$$", "Since the functor $j_{U!}$ induces an equivalence", "$\\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})/h_U^\\#$", "we conclude." ], "refs": [ "sites-lemma-technical-up", "sites-lemma-technical-up", "sites-lemma-pullback-representable-sheaf", "sites-lemma-essential-image-j-shriek", "categories-lemma-adjoint-exact", "categories-lemma-characterize-left-exact", "sites-lemma-compute-j-shriek-restrict" ], "ref_ids": [ 8523, 8523, 8524, 8555, 12249, 12245, 8558 ] } ], "ref_ids": [ 8555 ] }, { "id": 8572, "type": "theorem", "label": "sites-lemma-localize-morphism-strong", "categories": [ "sites" ], "title": "sites-lemma-localize-morphism-strong", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{D} \\to \\mathcal{C}$ be a functor.", "Let $V \\in \\Ob(\\mathcal{D})$. Set $U = u(V)$.", "Assume that", "\\begin{enumerate}", "\\item $\\mathcal{C}$ and $\\mathcal{D}$ have", "all finite limits,", "\\item $u$ is continuous, and", "\\item $u$ commutes with finite limits.", "\\end{enumerate}", "There exists a commutative diagram of morphisms of sites", "$$", "\\xymatrix{", "\\mathcal{C}/U \\ar[r]_{j_U} \\ar[d]_{f'} & \\mathcal{C} \\ar[d]^f \\\\", "\\mathcal{D}/V \\ar[r]^{j_V} & \\mathcal{D}", "}", "$$", "where the right vertical arrow corresponds to $u$,", "the left vertical arrow corresponds to the", "functor $u' : \\mathcal{D}/V \\to \\mathcal{C}/U$, $V'/V \\mapsto u(V')/u(V)$", "and the horizontal arrows correspond to the functors", "$\\mathcal{C} \\to \\mathcal{C}/U$, $X \\mapsto X \\times U$", "and $\\mathcal{D} \\to \\mathcal{D}/V$, $Y \\mapsto Y \\times V$", "as in Lemma \\ref{lemma-localize-given-products}.", "Moreover, the associated diagram of morphisms of topoi is", "equal to the diagram of", "Lemma \\ref{lemma-localize-morphism}.", "In particular we have $f'_*j_U^{-1} = j_V^{-1}f_*$." ], "refs": [ "sites-lemma-localize-given-products", "sites-lemma-localize-morphism" ], "proofs": [ { "contents": [ "Note that $u$ satisfies the assumptions of", "Proposition \\ref{proposition-get-morphism} and hence induces", "a morphism of sites $f : \\mathcal{C} \\to \\mathcal{D}$ by that proposition.", "It is clear that $u$ induces a functor $u'$ as indicated.", "It is clear that this functor also satisfies the assumptions of", "Proposition \\ref{proposition-get-morphism}.", "Hence we get a morphism of sites $f' : \\mathcal{C}/U \\to \\mathcal{D}/V$.", "The diagram commutes by our definition of composition of morphisms of", "sites (see Definition \\ref{definition-composition-morphisms-sites})", "and because", "$$", "u(Y \\times V) = u(Y) \\times u(V) = u(Y) \\times U", "$$", "which shows that the diagram of categories and functors opposite to", "the diagram of the lemma commutes." ], "refs": [ "sites-proposition-get-morphism", "sites-proposition-get-morphism", "sites-definition-composition-morphisms-sites" ], "ref_ids": [ 8641, 8641, 8666 ] } ], "ref_ids": [ 8567, 8571 ] }, { "id": 8573, "type": "theorem", "label": "sites-lemma-relocalize-morphism", "categories": [ "sites" ], "title": "sites-lemma-relocalize-morphism", "contents": [ "Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites", "corresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.", "Let $V \\in \\Ob(\\mathcal{D})$, $U \\in \\Ob(\\mathcal{C})$", "and $c : U \\to u(V)$ a morphism of $\\mathcal{C}$.", "There exists a commutative diagram of topoi", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C}/U) \\ar[r]_{j_U} \\ar[d]_{f_c} &", "\\Sh(\\mathcal{C}) \\ar[d]^f \\\\", "\\Sh(\\mathcal{D}/V) \\ar[r]^{j_V} &", "\\Sh(\\mathcal{D}).", "}", "$$", "We have $f_c = f' \\circ j_{U/u(V)}$ where", "$f' : \\Sh(\\mathcal{C}/u(V)) \\to \\Sh(\\mathcal{D}/V)$", "is as in", "Lemma \\ref{lemma-localize-morphism}", "and", "$j_{U/u(V)} : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C}/u(V))$", "is as in", "Lemma \\ref{lemma-relocalize}.", "Using the identifications", "$\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$ and", "$\\Sh(\\mathcal{D}/V) = \\Sh(\\mathcal{D})/h_V^\\#$ of", "Lemma \\ref{lemma-essential-image-j-shriek}", "the functor $(f_c)^{-1}$ is described by the rule", "$$", "(f_c)^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} h_V^\\#)", "=", "(f^{-1}\\mathcal{H} \\times_{f^{-1}\\varphi, h_{u(V)}^\\#, c} h_U^\\#", "\\rightarrow h_U^\\#).", "$$", "Finally, given any morphisms $b : V' \\to V$, $a : U' \\to U$ and", "$c' : U' \\to u(V')$ such that", "$$", "\\xymatrix{", "U' \\ar[r]_-{c'} \\ar[d]_a & u(V') \\ar[d]^{u(b)} \\\\", "U \\ar[r]^-c & u(V)", "}", "$$", "commutes, then the diagram", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C}/U') \\ar[r]_{j_{U'/U}} \\ar[d]_{f_{c'}} &", "\\Sh(\\mathcal{C}/U) \\ar[d]^{f_c} \\\\", "\\Sh(\\mathcal{D}/V') \\ar[r]^{j_{V'/V}} &", "\\Sh(\\mathcal{D}/V).", "}", "$$", "commutes." ], "refs": [ "sites-lemma-localize-morphism", "sites-lemma-relocalize", "sites-lemma-essential-image-j-shriek" ], "proofs": [ { "contents": [ "This lemma proves itself, and is more a collection of things we know", "at this stage of the development of theory. For example the commutativity", "of the first square follows from the commutativity of", "Diagram (\\ref{equation-relocalize})", "and the commutativity of the diagram in", "Lemma \\ref{lemma-localize-morphism}.", "The description of $f_c^{-1}$ follows on combining", "Lemma \\ref{lemma-relocalize-explicit}", "with", "Lemma \\ref{lemma-localize-morphism}.", "The commutativity of the last square then follows from the", "equality", "$$", "f^{-1}\\mathcal{H} \\times_{h_{u(V)}^\\#, c} h_U^\\# \\times_{h_U^\\#} h_{U'}^\\#", "=", "f^{-1}(\\mathcal{H} \\times_{h_V^\\#} h_{V'}^\\#)", "\\times_{h_{u(V'), c'}^\\#} h_{U'}^\\#", "$$", "which is formal using that $f^{-1}h_V^\\# = h_{u(V)}^\\#$ and", "$f^{-1}h_{V'}^\\# = h_{u(V')}^\\#$, see", "Lemma \\ref{lemma-pullback-representable-sheaf}." ], "refs": [ "sites-lemma-localize-morphism", "sites-lemma-relocalize-explicit", "sites-lemma-localize-morphism", "sites-lemma-pullback-representable-sheaf" ], "ref_ids": [ 8571, 8560, 8571, 8524 ] } ], "ref_ids": [ 8571, 8559, 8555 ] }, { "id": 8574, "type": "theorem", "label": "sites-lemma-localize-cocontinuous", "categories": [ "sites" ], "title": "sites-lemma-localize-cocontinuous", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a cocontinuous functor.", "Let $U$ be an object of $\\mathcal{C}$, and set $V = u(U)$.", "We have a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{C}/U \\ar[r]_{j_U} \\ar[d]_{u'} & \\mathcal{C} \\ar[d]^u \\\\", "\\mathcal{D}/V \\ar[r]^-{j_V} & \\mathcal{D}", "}", "$$", "where the left vertical arrow is", "$u' : \\mathcal{C}/U \\to \\mathcal{D}/V$, $U'/U \\mapsto V'/V$.", "Then $u'$ is cocontinuous also and we get a commutative diagram of topoi", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C}/U) \\ar[r]_{j_U} \\ar[d]_{f'} &", "\\Sh(\\mathcal{C}) \\ar[d]^f \\\\", "\\Sh(\\mathcal{D}/V) \\ar[r]^-{j_V} &", "\\Sh(\\mathcal{D})", "}", "$$", "where $f$ (resp.\\ $f'$) corresponds to $u$ (resp.\\ $u'$)." ], "refs": [], "proofs": [ { "contents": [ "The commutativity of the first diagram is clear.", "It implies the commutativity of the second diagram provided we", "show that $u'$ is cocontinuous.", "\\medskip\\noindent", "Let $U'/U$ be an object of $\\mathcal{C}/U$.", "Let $\\{V_j/V \\to u(U')/V\\}_{j \\in J}$ be a covering of $u(U')/V$", "in $\\mathcal{D}/V$. Since $u$ is cocontinuous there exists a", "covering $\\{U_i' \\to U'\\}_{i \\in I}$ such that the family", "$\\{u(U_i') \\to u(U')\\}$ refines the covering", "$\\{V_j \\to u(U')\\}$ in $\\mathcal{D}$. In other words, there exists", "a map of index sets $\\alpha : I \\to J$ and morphisms", "$\\phi_i : u(U_i') \\to V_{\\alpha(i)}$ over $U'$.", "Think of $U_i'$ as an object over", "$U$ via the composition $U'_i \\to U' \\to U$. Then", "$\\{U'_i/U \\to U'/U\\}$ is a covering of $\\mathcal{C}/U$ such that", "$\\{u(U_i')/V \\to u(U')/V\\}$ refines $\\{V_j/V \\to u(U')/V\\}$", "(use the same $\\alpha$ and the same maps $\\phi_i$). Hence", "$u' : \\mathcal{C}/U \\to \\mathcal{D}/V$ is cocontinuous." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8575, "type": "theorem", "label": "sites-lemma-localize-cocontinuous-downstairs", "categories": [ "sites" ], "title": "sites-lemma-localize-cocontinuous-downstairs", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a cocontinuous functor.", "Let $V$ be an object of $\\mathcal{D}$. Let", "${}^u_V\\mathcal{I}$ be the category introduced in", "Section \\ref{section-more-functoriality-PSh}.", "We have a commutative diagram", "$$", "\\vcenter{", "\\xymatrix{", "\\,_V^u\\mathcal{I} \\ar[r]_j \\ar[d]_{u'} &", "\\mathcal{C} \\ar[d]^u \\\\", "\\mathcal{D}/V \\ar[r]^-{j_V} &", "\\mathcal{D}", "}", "}", "\\quad\\text{where}\\quad", "\\begin{matrix}", "j : (U, \\psi) \\mapsto U \\\\", "u' : (U, \\psi) \\mapsto (\\psi : u(U) \\to V)", "\\end{matrix}", "$$", "Declare a family of morphisms $\\{(U_i, \\psi_i) \\to (U, \\psi)\\}$", "of ${}^u_V\\mathcal{I}$ to be a covering if and only if", "$\\{U_i \\to U\\}$ is a covering in $\\mathcal{C}$.", "Then", "\\begin{enumerate}", "\\item ${}^u_V\\mathcal{I}$ is a site,", "\\item $j$ is continuous and cocontinuous,", "\\item $u'$ is cocontinuous,", "\\item we get a commutative diagram of topoi", "$$", "\\xymatrix{", "\\Sh({}^u_V\\mathcal{I}) \\ar[r]_j \\ar[d]_{f'} &", "\\Sh(\\mathcal{C}) \\ar[d]^f \\\\", "\\Sh(\\mathcal{D}/V) \\ar[r]^-{j_V} &", "\\Sh(\\mathcal{D})", "}", "$$", "where $f$ (resp.\\ $f'$) corresponds to $u$ (resp.\\ $u'$), and", "\\item we have $f'_*j^{-1} = j_V^{-1}f_*$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1), (2), (3), and (4) are straightforward consequences of the", "definitions and the fact that the functor $j$ commutes with fibre products.", "We omit the details. To see (5) recall that $f_*$ is given by", "${}_su = {}_pu$. Hence the value of $j_V^{-1}f_*\\mathcal{F}$ on", "$V'/V$ is the value of ${}_pu\\mathcal{F}$ on $V'$ which is the", "limit of the values of $\\mathcal{F}$ on the category", "${}^u_{V'}\\mathcal{I}$. Clearly, there is an equivalence", "of categories", "$$", "{}^u_{V'}\\mathcal{I} \\to {}^{u'}_{V'/V}\\mathcal{I}", "$$", "Since the value of $f'_*j^{-1}\\mathcal{F}$ on $V'/V$ is", "given by the limit of the values of $j^{-1}\\mathcal{F}$", "on the category ${}^{u'}_{V'/V}\\mathcal{I}$ and since", "the values of $j^{-1}\\mathcal{F}$ on objects of", "${}^u_V\\mathcal{I}$ are just the values of $\\mathcal{F}$", "(by Lemma \\ref{lemma-when-shriek} as $j$ is continuous and", "cocontinuous)", "we see that (5) is true." ], "refs": [ "sites-lemma-when-shriek" ], "ref_ids": [ 8545 ] } ], "ref_ids": [] }, { "id": 8576, "type": "theorem", "label": "sites-lemma-special-square-cocontinuous", "categories": [ "sites" ], "title": "sites-lemma-special-square-cocontinuous", "contents": [ "Assume given sites $\\mathcal{C}', \\mathcal{C}, \\mathcal{D}', \\mathcal{D}$", "and functors", "$$", "\\xymatrix{", "\\mathcal{C}' \\ar[r]_{v'} \\ar[d]_{u'} &", "\\mathcal{C} \\ar[d]^u \\\\", "\\mathcal{D}' \\ar[r]^v &", "\\mathcal{D}", "}", "$$", "Assume", "\\begin{enumerate}", "\\item $u$, $u'$, $v$, and $v'$ are cocontinuous giving rise to morphisms of", "topoi $f$, $f'$, $g$, and $g'$ by Lemma \\ref{lemma-cocontinuous-morphism-topoi},", "\\item $v \\circ u' = u \\circ v'$,", "\\item $v$ and $v'$ are continuous as well as cocontinuous, and", "\\item for any object $V'$ of $\\mathcal{D}'$ the functor", "${}^{u'}_{V'}\\mathcal{I} \\to {}^{\\ \\ \\ u}_{v(V')}\\mathcal{I}$", "given by $v$ is cofinal.", "\\end{enumerate}", "Then $f'_* \\circ (g')^{-1} = g^{-1} \\circ f_*$ and", "$g'_! \\circ (f')^{-1} = f^{-1} \\circ g_!$." ], "refs": [ "sites-lemma-cocontinuous-morphism-topoi" ], "proofs": [ { "contents": [ "The categories ${}^{u'}_{V'}\\mathcal{I}$ and", "${}^{\\ \\ \\ u}_{v(V')}\\mathcal{I}$ are defined in", "Section \\ref{section-more-functoriality-PSh}.", "The functor in condition (4) sends the object", "$\\psi : u'(U') \\to V'$ of ${}^{u'}_{V'}\\mathcal{I}$ to the object", "$v(\\psi) : uv'(U') = vu'(U') \\to v(V')$ of ${}^{\\ \\ \\ u}_{v(V')}\\mathcal{I}$.", "Recall that $g^{-1}$ is given by $v^p$ (Lemma \\ref{lemma-when-shriek}) and", "$f_*$ is given by ${}_su = {}_pu$. Hence the value of", "$g^{-1}f_*\\mathcal{F}$ on $V'$ is the value of ${}_pu\\mathcal{F}$", "on $v(V')$ which is the limit", "$$", "\\lim_{u(U) \\to v(V') \\in \\Ob({}^{\\ \\ \\ u}_{v(V')}\\mathcal{I}^{opp})}", "\\mathcal{F}(U)", "$$", "By the same reasoning, the value of $f'_*(g')^{-1}\\mathcal{F}$", "on $V'$ is given by the limit", "$$", "\\lim_{u'(U') \\to V' \\in \\Ob({}^{u'}_{V'}\\mathcal{I}^{opp})} \\mathcal{F}(v'(U'))", "$$", "Thus assumption (4) and Categories, Lemma \\ref{categories-lemma-initial}", "show that these agree and the first equality of the", "lemma is proved. The second equality follows from the first by uniqueness of", "adjoints." ], "refs": [ "sites-lemma-when-shriek", "categories-lemma-initial" ], "ref_ids": [ 8545, 12218 ] } ], "ref_ids": [ 8543 ] }, { "id": 8577, "type": "theorem", "label": "sites-lemma-special-square-continuous", "categories": [ "sites" ], "title": "sites-lemma-special-square-continuous", "contents": [ "Assume given sites $\\mathcal{C}', \\mathcal{C}, \\mathcal{D}', \\mathcal{D}$", "and functors", "$$", "\\xymatrix{", "\\mathcal{C}' \\ar[r]_{v'} &", "\\mathcal{C} \\\\", "\\mathcal{D}' \\ar[r]^v \\ar[u]^{u'} &", "\\mathcal{D} \\ar[u]_u", "}", "$$", "With notation as in Sections \\ref{section-morphism-sites}", "and \\ref{section-cocontinuous-morphism-topoi} assume", "\\begin{enumerate}", "\\item $u$ and $u'$ are continuous giving rise to morphisms of", "sites $f$ and $f'$,", "\\item $v$ and $v'$ are cocontinuous giving rise to morphisms", "of topoi $g$ and $g'$,", "\\item $u \\circ v = v' \\circ u'$, and", "\\item $v$ and $v'$ are continuous as well as cocontinuous.", "\\end{enumerate}", "Then\\footnote{In this generality", "we don't know $f \\circ g'$ is equal to $g \\circ f'$", "as morphisms of topoi (there is a canonical $2$-arrow from", "the first to the second which may not be an isomorphism).}", "$f'_* \\circ (g')^{-1} = g^{-1} \\circ f_*$ and", "$g'_! \\circ (f')^{-1} = f^{-1} \\circ g_!$." ], "refs": [], "proofs": [ { "contents": [ "Namely, we have", "$$", "f'_*(g')^{-1}\\mathcal{F} = (u')^p((v')^p\\mathcal{F})^\\# =", "(u')^p(v')^p\\mathcal{F}", "$$", "The first equality by definition and the", "second by Lemma \\ref{lemma-when-shriek}. We have", "$$", "g^{-1}f_*\\mathcal{F} = (v^pu^p\\mathcal{F})^\\# =", "((u')^p(v')^p\\mathcal{F})^\\# = (u')^p(v')^p\\mathcal{F}", "$$", "The first equality by definition, the second because", "$u \\circ v = v' \\circ u'$, the third because we already saw that", "$(u')^p(v')^p\\mathcal{F}$ is a sheaf. This proves", "$f'_* \\circ (g')^{-1} = g^{-1} \\circ f_*$ and the equality", "$g'_! \\circ (f')^{-1} = f^{-1} \\circ g_!$ follows by", "uniqueness of left adjoints." ], "refs": [ "sites-lemma-when-shriek" ], "ref_ids": [ 8545 ] } ], "ref_ids": [] }, { "id": 8578, "type": "theorem", "label": "sites-lemma-equivalence", "categories": [ "sites" ], "title": "sites-lemma-equivalence", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "Assume that", "\\begin{enumerate}", "\\item $u$ is cocontinuous,", "\\item $u$ is continuous,", "\\item given $a, b : U' \\to U$ in $\\mathcal{C}$ such that", "$u(a) = u(b)$, then there exists a covering $\\{f_i : U'_i \\to U'\\}$", "in $\\mathcal{C}$ such that $a \\circ f_i = b \\circ f_i$,", "\\item given $U', U \\in \\Ob(\\mathcal{C})$ and", "a morphism $c : u(U') \\to u(U)$ in $\\mathcal{D}$ there exists", "a covering $\\{f_i : U_i' \\to U'\\}$ in $\\mathcal{C}$", "and morphisms $c_i : U_i' \\to U$ such that $u(c_i) = c \\circ u(f_i)$, and", "\\item given $V \\in \\Ob(\\mathcal{D})$ there exists a covering", "of $V$ in $\\mathcal{D}$ of the form $\\{u(U_i) \\to V\\}_{i \\in I}$.", "\\end{enumerate}", "Then the morphism of topoi", "$$", "g : \\Sh(\\mathcal{C}) \\longrightarrow \\Sh(\\mathcal{D})", "$$", "associated to the cocontinuous functor $u$ by", "Lemma \\ref{lemma-cocontinuous-morphism-topoi}", "is an equivalence." ], "refs": [ "sites-lemma-cocontinuous-morphism-topoi" ], "proofs": [ { "contents": [ "Assume $u$ satisfies properties (1) -- (5). We will show that", "the adjunction mappings", "$$", "\\mathcal{G} \\longrightarrow g_*g^{-1}\\mathcal{G}", "\\quad\\text{and}\\quad", "g^{-1}g_*\\mathcal{F} \\longrightarrow \\mathcal{F}", "$$", "are isomorphisms.", "\\medskip\\noindent", "Note that Lemma \\ref{lemma-when-shriek} applies and we have", "$g^{-1}\\mathcal{G}(U) = \\mathcal{G}(u(U))$ for any sheaf $\\mathcal{G}$", "on $\\mathcal{D}$. Next, let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$,", "and let $V$ be an object of $\\mathcal{D}$. By definition we have", "$g_*\\mathcal{F}(V) = \\lim_{u(U) \\to V} \\mathcal{F}(U)$.", "Hence", "$$", "g^{-1}g_*\\mathcal{F}(U) = \\lim_{U', u(U') \\to u(U)} \\mathcal{F}(U')", "$$", "where the morphisms $\\psi : u(U') \\to u(U)$ need not be of the form", "$u(\\alpha)$. The category of such pairs $(U', \\psi)$ has a final", "object, namely $(U, \\text{id})$, which gives rise to the map from", "the limit into $\\mathcal{F}(U)$. Let $(s_{(U', \\psi)})$ be an element", "of the limit. We want to show that $s_{(U', \\psi)}$ is uniquely determined", "by the value $s_{(U, \\text{id})} \\in \\mathcal{F}(U)$. By property (4) given", "any $(U', \\psi)$ there exists a covering $\\{U'_i \\to U'\\}$ such that the", "compositions $u(U'_i) \\to u(U') \\to u(U)$ are of the form $u(c_i)$", "for some $c_i : U'_i \\to U$ in $\\mathcal{C}$. Hence", "$$", "s_{(U', \\psi)}|_{U'_i} = c_i^*(s_{(U, \\text{id})}).", "$$", "Since $\\mathcal{F}$ is a sheaf it follows that indeed $s_{(U', \\psi)}$", "is determined by $s_{(U, \\text{id})}$. This proves uniqueness.", "For existence, assume given any", "$s \\in \\mathcal{F}(U)$, $\\psi : u(U') \\to u(U)$, $\\{f_i : U_i' \\to U'\\}$", "and $c_i : U_i' \\to U$ such that $\\psi \\circ u(f_i) = u(c_i)$ as above.", "We claim there exists a (unique) element", "$s_{(U', \\psi)} \\in \\mathcal{F}(U')$ such that", "$$", "s_{(U', \\psi)}|_{U'_i} = c_i^*(s).", "$$", "Namely, a priori it is not clear the elements", "$c_i^*(s)|_{U_i' \\times_{U'} U_j'}$", "and $c_j^*(s)|_{U_i' \\times_{U'} U_j'}$ agree, since", "the diagram", "$$", "\\xymatrix{", "U_i' \\times_{U'} U_j' \\ar[r]_-{\\text{pr}_2} \\ar[d]_{\\text{pr}_1} &", "U_j' \\ar[d]^{c_j} \\\\", "U_i' \\ar[r]^{c_i} & U}", "$$", "need not commute. But condition (3) of the lemma guarantees that there", "exist coverings", "$\\{f_{ijk} : U'_{ijk} \\to U_i' \\times_{U'} U_j'\\}_{k \\in K_{ij}}$ such that", "$c_i \\circ \\text{pr}_1 \\circ f_{ijk} = c_j \\circ \\text{pr}_2 \\circ f_{ijk}$.", "Hence", "$$", "f_{ijk}^* \\left(c_i^*s|_{U_i' \\times_{U'} U_j'}\\right)", "=", "f_{ijk}^* \\left(c_j^*s|_{U_i' \\times_{U'} U_j'}\\right)", "$$", "Hence $c_i^*(s)|_{U_i' \\times_{U'} U_j'} = c_j^*(s)|_{U_i' \\times_{U'} U_j'}$", "by the sheaf condition for $\\mathcal{F}$ and hence the existence of", "$s_{(U', \\psi)}$ also by the sheaf condition for $\\mathcal{F}$. The uniqueness", "guarantees that the collection $(s_{(U', \\psi)})$ so obtained is an element", "of the limit with $s_{(U, \\psi)} = s$. This proves", "that $g^{-1}g_*\\mathcal{F} \\to \\mathcal{F}$ is an isomorphism.", "\\medskip\\noindent", "Let $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$. Let $V$ be an", "object of $\\mathcal{D}$. Then we see that", "$$", "g_*g^{-1}\\mathcal{G}(V) =", "\\lim_{U, \\psi : u(U) \\to V} \\mathcal{G}(u(U))", "$$", "By the preceding paragraph we see that the value of the sheaf", "$g_*g^{-1}\\mathcal{G}$ on an object $V$ of the form $V = u(U)$", "is equal to $\\mathcal{G}(u(U))$. (Formally, this holds because", "we have $g^{-1}g_*g^{-1} \\cong g^{-1}$, and the description", "of $g^{-1}$ given at the beginning of the proof; informally just by", "comparing limits here and above.)", "Hence the adjunction mapping $\\mathcal{G} \\to g_*g^{-1}\\mathcal{G}$ has", "the property that it is a bijection on sections over any object of the", "form $u(U)$. Since by axiom (5) there", "exists a covering of $V$ by objects of the form $u(U)$ we see", "easily that the adjunction map is an isomorphism." ], "refs": [ "sites-lemma-when-shriek" ], "ref_ids": [ 8545 ] } ], "ref_ids": [ 8543 ] }, { "id": 8579, "type": "theorem", "label": "sites-lemma-localize-special-cocontinuous", "categories": [ "sites" ], "title": "sites-lemma-localize-special-cocontinuous", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a special cocontinuous functor.", "For every object $U$ of $\\mathcal{C}$ we have a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{C}/U \\ar[r]_{j_U} \\ar[d] & \\mathcal{C} \\ar[d]^u \\\\", "\\mathcal{D}/u(U) \\ar[r]^-{j_{u(U)}} & \\mathcal{D}", "}", "$$", "as in Lemma \\ref{lemma-localize-cocontinuous}.", "The left vertical arrow is a special cocontinuous functor.", "Hence in the commutative diagram of topoi", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C}/U) \\ar[r]_{j_U} \\ar[d] &", "\\Sh(\\mathcal{C}) \\ar[d]^u \\\\", "\\Sh(\\mathcal{D}/u(U)) \\ar[r]^-{j_{u(U)}} &", "\\Sh(\\mathcal{D})", "}", "$$", "the vertical arrows are equivalences." ], "refs": [ "sites-lemma-localize-cocontinuous" ], "proofs": [ { "contents": [ "We have seen the existence and commutativity of the diagrams in", "Lemma \\ref{lemma-localize-cocontinuous}. We have to check", "hypotheses (1) -- (5) of Lemma \\ref{lemma-equivalence} for the", "induced functor $u : \\mathcal{C}/U \\to \\mathcal{D}/u(U)$.", "This is completely mechanical.", "\\medskip\\noindent", "Property (1). This is Lemma \\ref{lemma-localize-cocontinuous}.", "\\medskip\\noindent", "Property (2). Let $\\{U_i'/U \\to U'/U\\}_{i \\in I}$ be a covering", "of $U'/U$ in $\\mathcal{C}/U$. Because $u$ is continuous we see that", "$\\{u(U_i')/u(U) \\to u(U')/u(U)\\}_{i \\in I}$ is a covering", "of $u(U')/u(U)$ in $\\mathcal{D}/u(U)$. Hence (2) holds", "for $u : \\mathcal{C}/U \\to \\mathcal{D}/u(U)$.", "\\medskip\\noindent", "Property (3). Let $a, b : U''/U \\to U'/U$ in $\\mathcal{C}/U$", "be morphisms such that $u(a) = u(b)$ in $\\mathcal{D}/u(U)$.", "Because $u$ satisfies (3) we see there exists a covering", "$\\{f_i : U''_i \\to U''\\}$ in $\\mathcal{C}$ such that", "$a \\circ f_i = b \\circ f_i$. This gives a covering", "$\\{f_i : U''_i/U \\to U''/U\\}$ in $\\mathcal{C}/U$ such that", "$a \\circ f_i = b \\circ f_i$. Hence (3) holds", "for $u : \\mathcal{C}/U \\to \\mathcal{D}/u(U)$.", "\\medskip\\noindent", "Property (4). Let $U''/U, U'/U \\in \\Ob(\\mathcal{C}/U)$ and", "a morphism $c : u(U'')/u(U) \\to u(U')/u(U)$ in $\\mathcal{D}/u(U)$", "be given. Because $u$ satisfies property (4) there exists", "a covering $\\{f_i : U_i'' \\to U''\\}$ in $\\mathcal{C}$", "and morphisms $c_i : U_i'' \\to U'$ such that $u(c_i) = c \\circ u(f_i)$.", "We think of $U_i''$ as an object over $U$ via the composition", "$U_i'' \\to U'' \\to U$.", "It may not be true that $c_i$ is a morphism over $U$!", "But since $u(c_i)$ is a morphism over $u(U)$ we may apply", "property (3) for $u$ and find coverings $\\{f_{ik} : U''_{ik} \\to U''_i\\}$", "such that $c_{ik} = c_i \\circ f_{ik} : U''_{ik} \\to U'$ are morphisms over $U$.", "Hence $\\{f_i \\circ f_{ik} : U''_{ik}/U \\to U''/U\\}$ is a covering", "in $\\mathcal{C}/U$ such that $u(c_{ik}) = c \\circ u(f_{ik})$.", "Hence (4) holds", "for $u : \\mathcal{C}/U \\to \\mathcal{D}/u(U)$.", "\\medskip\\noindent", "Property (5). Let $h : V \\to u(U)$ be an object of $\\mathcal{D}/u(U)$.", "Because $u$ satisfies property (5) there exists a covering", "$\\{c_i : u(U_i) \\to V\\}$ in $\\mathcal{D}$. By property (4) we can find", "coverings $\\{f_{ij} : U_{ij} \\to U_i\\}$ and morphisms", "$c_{ij} : U_{ij} \\to U$ such that $u(c_{ij}) = h \\circ c_i \\circ u(f_{ij})$.", "Hence $\\{u(U_{ij})/u(U) \\to V/u(U)\\}$ is a covering in", "$\\mathcal{D}/u(U)$ of the desired shape and we conclude that", "(5) holds for $u : \\mathcal{C}/U \\to \\mathcal{D}/u(U)$." ], "refs": [ "sites-lemma-localize-cocontinuous", "sites-lemma-equivalence", "sites-lemma-localize-cocontinuous" ], "ref_ids": [ 8574, 8578, 8574 ] } ], "ref_ids": [ 8574 ] }, { "id": 8580, "type": "theorem", "label": "sites-lemma-special-equivalence", "categories": [ "sites" ], "title": "sites-lemma-special-equivalence", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$\\mathcal{C}' \\subset \\Sh(\\mathcal{C})$", "be a full subcategory (with a set of objects) such that", "\\begin{enumerate}", "\\item $h_U^\\# \\in \\Ob(\\mathcal{C}')$ for all", "$U \\in \\Ob(\\mathcal{C})$, and", "\\item $\\mathcal{C}'$ is preserved under fibre products in", "$\\Sh(\\mathcal{C})$.", "\\end{enumerate}", "Declare a covering of $\\mathcal{C}'$ to be any family", "$\\{\\mathcal{F}_i \\to \\mathcal{F}\\}_{i \\in I}$ of maps such that", "$\\coprod_{i \\in I} \\mathcal{F}_i \\to \\mathcal{F}$ is a surjective", "map of sheaves. Then", "\\begin{enumerate}", "\\item $\\mathcal{C}'$ is a site (after", "choosing a set of coverings, see Sets, Lemma \\ref{sets-lemma-coverings-site}),", "\\item representable presheaves on $\\mathcal{C}'$ are sheaves", "(i.e., the topology on $\\mathcal{C}'$ is subcanonical, see", "Definition \\ref{definition-weaker-than-canonical}),", "\\item the functor $v : \\mathcal{C} \\to \\mathcal{C}'$,", "$U \\mapsto h_U^\\#$ is a special cocontinuous functor, hence induces an", "equivalence $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$,", "\\item for any $\\mathcal{F} \\in \\Ob(\\mathcal{C}')$ we have", "$g^{-1}h_\\mathcal{F} = \\mathcal{F}$, and", "\\item for any $U \\in \\Ob(\\mathcal{C})$ we have", "$g_*h_U^\\# = h_{v(U)} = h_{h_U^\\#}$.", "\\end{enumerate}" ], "refs": [ "sets-lemma-coverings-site", "sites-definition-weaker-than-canonical" ], "proofs": [ { "contents": [ "Warning: Some of the statements above may look be a bit confusing at first;", "this is because objects of $\\mathcal{C}'$ can also be viewed as sheaves on", "$\\mathcal{C}$! We omit the proof that the coverings of $\\mathcal{C}'$ as", "described in the lemma satisfy the conditions of", "Definition \\ref{definition-site}.", "\\medskip\\noindent", "Suppose that $\\{\\mathcal{F}_i \\to \\mathcal{F}\\}$ is a surjective", "family of morphisms of sheaves. Let $\\mathcal{G}$ be another sheaf.", "Part (2) of the lemma says that the equalizer of", "$$", "\\xymatrix{", "\\Mor_{\\Sh(\\mathcal{C})}(", "\\coprod_{i \\in I} \\mathcal{F}_i, \\mathcal{G})", "\\ar@<1ex>[r] \\ar@<-1ex>[r]", "&", "\\Mor_{\\Sh(\\mathcal{C})}(", "\\coprod_{(i_0, i_1) \\in I \\times I}", "\\mathcal{F}_{i_0} \\times_\\mathcal{F} \\mathcal{F}_{i_1}, \\mathcal{G})", "}", "$$", "is $\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F}, \\mathcal{G}).$", "This is clear (for example use Lemma \\ref{lemma-coequalizer-surjection}).", "\\medskip\\noindent", "To prove (3) we", "have to check conditions (1) -- (5) of Lemma \\ref{lemma-equivalence}.", "The fact that $v$ is cocontinuous is", "equivalent to the description of surjective maps of sheaves in", "Lemma \\ref{lemma-mono-epi-sheaves}.", "The functor $v$ is continuous because", "$U \\mapsto h_U^\\#$ commutes with fibre products,", "and transforms coverings into coverings (see", "Lemma \\ref{lemma-sheafification-exact}, and", "Lemma \\ref{lemma-covering-surjective-after-sheafification}).", "Properties (3), (4) of Lemma \\ref{lemma-equivalence}", "are statements about morphisms $f : h_{U'}^\\# \\to h_U^\\#$.", "Such a morphism is the same thing as an element of $h_U^\\#(U')$.", "Hence (3) and (4) are immediate from the construction of the sheafification.", "Property (5) of Lemma \\ref{lemma-equivalence} is", "Lemma \\ref{lemma-sheaf-coequalizer-representable}.", "Denote $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$ the", "equivalence of topoi associated with $v$ by Lemma \\ref{lemma-equivalence}.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be as in part (4) of the lemma.", "For any $U \\in \\Ob(\\mathcal{C})$ we have", "$$", "g^{-1}h_\\mathcal{F}(U) = h_\\mathcal{F}(v(U))", "= \\Mor_{\\Sh(\\mathcal{C})}(h_U^\\#, \\mathcal{F})", "= \\mathcal{F}(U)", "$$", "The first equality by", "Lemma \\ref{lemma-when-shriek}.", "Thus part (4) holds.", "\\medskip\\noindent", "Let $\\mathcal{F} \\in \\Ob(\\mathcal{C}')$.", "Let $U \\in \\Ob(\\mathcal{C})$.", "Then", "\\begin{align*}", "g_*h_U^\\#(\\mathcal{F})", "& =", "\\Mor_{\\Sh(\\mathcal{C}')}(h_\\mathcal{F}, g_*h_U^\\#) \\\\", "& =", "\\Mor_{\\Sh(\\mathcal{C})}(g^{-1}h_\\mathcal{F}, h_U^\\#) \\\\", "& =", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F}, h_U^\\#) \\\\", "& =", "\\Mor_{\\mathcal{C}'}(\\mathcal{F}, h_U^\\#)", "\\end{align*}", "as desired (where the third equality was shown above)." ], "refs": [ "sites-definition-site", "sites-lemma-coequalizer-surjection", "sites-lemma-equivalence", "sites-lemma-mono-epi-sheaves", "sites-lemma-sheafification-exact", "sites-lemma-covering-surjective-after-sheafification", "sites-lemma-equivalence", "sites-lemma-equivalence", "sites-lemma-sheaf-coequalizer-representable", "sites-lemma-equivalence", "sites-lemma-when-shriek" ], "ref_ids": [ 8652, 8518, 8578, 8517, 8515, 8519, 8578, 8578, 8520, 8578, 8545 ] } ], "ref_ids": [ 8800, 8662 ] }, { "id": 8581, "type": "theorem", "label": "sites-lemma-topos-good-site", "categories": [ "sites" ], "title": "sites-lemma-topos-good-site", "contents": [ "Let $\\Sh(\\mathcal{C})$ be a topos. Let $\\{\\mathcal{F}_i\\}_{i \\in I}$", "be a set of sheaves on $\\mathcal{C}$. There exists an equivalence of topoi", "$g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$ induced by a special", "cocontinuous functor $u : \\mathcal{C} \\to \\mathcal{C}'$ of sites", "such that", "\\begin{enumerate}", "\\item $\\mathcal{C}'$ has a subcanonical topology,", "\\item a family $\\{V_j \\to V\\}$ of morphisms of $\\mathcal{C}'$", "is (combinatorially equivalent to) a covering of $\\mathcal{C}'$", "if and only if $\\coprod h_{V_j} \\to h_V$ is surjective,", "\\item $\\mathcal{C}'$ has fibre products and a final object", "(i.e., $\\mathcal{C}'$ has all finite limits),", "\\item every subsheaf of a representable sheaf on $\\mathcal{C}'$", "is representable, and", "\\item each $g_*\\mathcal{F}_i$ is a representable sheaf.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the full subcategory", "$\\mathcal{C}_1 \\subset \\Sh(\\mathcal{C})$ consisting of all", "$h_U^\\#$ for all $U \\in \\Ob(\\mathcal{C})$, the given sheaves", "$\\mathcal{F}_i$ and the final sheaf $*$ (see", "Example \\ref{example-singleton-sheaf}). We are going to inductively", "define full subcategories", "$$", "\\mathcal{C}_1 \\subset \\mathcal{C}_2 \\subset \\mathcal{C}_2 \\subset \\ldots", "\\subset \\Sh(\\mathcal{C})", "$$", "Namely, given $\\mathcal{C}_n$ let $\\mathcal{C}_{n + 1}$ be the full", "subcategory consisting of all fibre products and subsheaves of objects", "of $\\mathcal{C}_n$. (Note that $\\mathcal{C}_{n + 1}$ has a set", "of objects.) Set", "$\\mathcal{C}' = \\bigcup_{n \\geq 1} \\mathcal{C}_n$.", "A covering in $\\mathcal{C}'$ is any family", "$\\{\\mathcal{G}_j \\to \\mathcal{G}\\}_{j \\in J}$ of morphisms of objects", "of $\\mathcal{C}'$ such that", "$\\coprod \\mathcal{G}_j \\to \\mathcal{G}$ is surjective", "as a map of sheaves on $\\mathcal{C}$.", "The functor $v : \\mathcal{C} \\to \\mathcal{C'}$ is given by", "$U \\mapsto h_U^\\#$. Apply", "Lemma \\ref{lemma-special-equivalence}." ], "refs": [ "sites-lemma-special-equivalence" ], "ref_ids": [ 8580 ] } ], "ref_ids": [] }, { "id": 8582, "type": "theorem", "label": "sites-lemma-morphism-topoi-comes-from-morphism-sites", "categories": [ "sites" ], "title": "sites-lemma-morphism-topoi-comes-from-morphism-sites", "contents": [ "\\begin{reference}", "This statement is closely related to", "\\cite[Proposition 4.9.4. Expos\\'e IV]{SGA4}.", "In order to get the whole result, one should also use", "\\cite[Remarque 4.7.4, Expos\\'e IV]{SGA4}.", "\\end{reference}", "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be a", "morphism of topoi.", "Then there exists a site $\\mathcal{C}'$ and a diagram of functors", "$$", "\\xymatrix{", "\\mathcal{C} \\ar[r]_v & \\mathcal{C}' & \\mathcal{D} \\ar[l]^u", "}", "$$", "such that", "\\begin{enumerate}", "\\item the functor $v$ is a special cocontinuous functor,", "\\item the functor $u$ commutes with fibre products, is", "continuous and defines a morphism of sites", "$\\mathcal{C}' \\to \\mathcal{D}$, and", "\\item the morphism of topoi $f$ agrees with the composition", "of morphisms of topoi", "$$", "\\Sh(\\mathcal{C}) \\longrightarrow", "\\Sh(\\mathcal{C}') \\longrightarrow", "\\Sh(\\mathcal{D})", "$$", "where the first arrow comes from $v$ via Lemma \\ref{lemma-equivalence}", "and the second arrow from $u$ via Lemma \\ref{lemma-morphism-sites-topoi}.", "\\end{enumerate}" ], "refs": [ "sites-lemma-equivalence", "sites-lemma-morphism-sites-topoi" ], "proofs": [ { "contents": [ "Consider the full subcategory", "$\\mathcal{C}_1 \\subset \\Sh(\\mathcal{C})$ consisting of all", "$h_U^\\#$ and all $f^{-1}h_V^\\#$ for all", "$U \\in \\Ob(\\mathcal{C})$ and all $V \\in \\Ob(\\mathcal{D})$.", "Let $\\mathcal{C}_{n + 1}$ be a full subcategory consisting of all", "fibre products of objects of $\\mathcal{C}_n$. Set", "$\\mathcal{C}' = \\bigcup_{n \\geq 1} \\mathcal{C}_n$.", "A covering in $\\mathcal{C}'$ is any family", "$\\{\\mathcal{F}_i \\to \\mathcal{F}\\}_{i \\in I}$ such that", "$\\coprod_{i \\in I} \\mathcal{F}_i \\to \\mathcal{F}$ is surjective", "as a map of sheaves on $\\mathcal{C}$.", "The functor $v : \\mathcal{C} \\to \\mathcal{C'}$ is given by", "$U \\mapsto h_U^\\#$.", "The functor $u : \\mathcal{D} \\to \\mathcal{C'}$ is given by", "$V \\mapsto f^{-1}h_V^\\#$.", "\\medskip\\noindent", "Part (1) follows from Lemma \\ref{lemma-special-equivalence}.", "\\medskip\\noindent", "Proof of (2) and (3) of the lemma. The functor $u$ commutes with fibre", "products as both $V \\mapsto h_V^\\#$ and $f^{-1}$ do. Moreover,", "since $f^{-1}$ is exact and commutes with arbitrary colimits we see", "that it transforms a covering into a surjective family of morphisms of", "sheaves. Hence $u$ is continuous. To see that it defines a morphism of", "sites we still have to see that $u_s$ is exact. In order to do this", "we will show that $g^{-1} \\circ u_s = f^{-1}$. Namely, then since $g^{-1}$", "is an equivalence and $f^{-1}$ is exact we will conclude.", "Because $g^{-1}$ is adjoint to $g_*$, and $u_s$ is adjoint to", "$u^s$, and $f^{-1}$ is adjoint to $f_*$ it also suffices to prove that", "$u^s \\circ g_* = f_*$.", "Let $U$ be an object of $\\mathcal{C}$ and let", "$V$ be an object of $\\mathcal{D}$. Then", "\\begin{align*}", "(u^sg_*h_U^\\#)(V)", "& =", "g_*h_U^\\#(f^{-1}h_V^\\#) \\\\", "& =", "\\Mor_{\\Sh(\\mathcal{C})}(f^{-1}h_V^\\#, h_U^\\#) \\\\", "& =", "\\Mor_{\\Sh(\\mathcal{D})}(h_V^\\#, f_*h_U^\\#) \\\\", "& = f_*h_U^\\#(V)", "\\end{align*}", "The first equality because $u^s = u^p$. The second equality", "by Lemma \\ref{lemma-special-equivalence} (5). The third equality", "by adjointness of $f_*$ and $f^{-1}$ and the final equality by", "properties of sheafification and the Yoneda lemma.", "We omit the verification that these identities are functorial in $U$", "and $V$. Hence we see that", "we have $u^s \\circ g_* = f_*$ for sheaves of the form $h_U^\\#$.", "This implies that $u^s \\circ g_* = f_*$ and we win (some details omitted)." ], "refs": [ "sites-lemma-special-equivalence", "sites-lemma-special-equivalence" ], "ref_ids": [ 8580, 8580 ] } ], "ref_ids": [ 8578, 8528 ] }, { "id": 8583, "type": "theorem", "label": "sites-lemma-localize-topos", "categories": [ "sites" ], "title": "sites-lemma-localize-topos", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.", "Then the category $\\Sh(\\mathcal{C})/\\mathcal{F}$", "is a topos. There is a canonical morphism of topoi", "$$", "j_\\mathcal{F} :", "\\Sh(\\mathcal{C})/\\mathcal{F}", "\\longrightarrow", "\\Sh(\\mathcal{C})", "$$", "which is a localization as in", "Section \\ref{section-localize}", "such that", "\\begin{enumerate}", "\\item the functor $j_\\mathcal{F}^{-1}$ is the functor", "$\\mathcal{H} \\mapsto \\mathcal{H} \\times \\mathcal{F}/\\mathcal{F}$, and", "\\item the functor $j_{\\mathcal{F}!}$ is the forgetful", "functor $\\mathcal{G}/\\mathcal{F} \\mapsto \\mathcal{G}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Apply", "Lemma \\ref{lemma-topos-good-site}.", "This means we may assume $\\mathcal{C}$ is a site", "with subcanonical topology, and $\\mathcal{F} = h_U = h_U^\\#$", "for some $U \\in \\Ob(\\mathcal{C})$.", "Hence the material of", "Section \\ref{section-localize}", "applies. In particular, there is an equivalence", "$\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$", "such that the composition", "$$", "\\Sh(\\mathcal{C}/U)", "\\to", "\\Sh(\\mathcal{C})/h_U^\\#", "\\to \\Sh(\\mathcal{C})", "$$", "is equal to $j_{U!}$, see", "Lemma \\ref{lemma-essential-image-j-shriek}.", "Denote", "$a : \\Sh(\\mathcal{C})/h_U^\\# \\to \\Sh(\\mathcal{C}/U)$", "the inverse functor, so $j_{\\mathcal{F}!} = j_{U!} \\circ a$,", "$j_\\mathcal{F}^{-1} = a^{-1} \\circ j_U^{-1}$, and", "$j_{\\mathcal{F}, *} = j_{U, *} \\circ a$. The description of", "$j_{\\mathcal{F}!}$ follows from the above. The description of", "$j_\\mathcal{F}^{-1}$ follows from", "Lemma \\ref{lemma-compute-j-shriek-restrict}." ], "refs": [ "sites-lemma-topos-good-site", "sites-lemma-essential-image-j-shriek", "sites-lemma-compute-j-shriek-restrict" ], "ref_ids": [ 8581, 8555, 8558 ] } ], "ref_ids": [] }, { "id": 8584, "type": "theorem", "label": "sites-lemma-localize-pushforward", "categories": [ "sites" ], "title": "sites-lemma-localize-pushforward", "contents": [ "In the situation of Lemma \\ref{lemma-localize-topos}, the functor", "$j_{\\mathcal{F}, *}$ is the one associates to", "$\\varphi : \\mathcal{G} \\to \\mathcal{F}$ the sheaf", "$$", "U", "\\longmapsto", "\\{\\alpha : \\mathcal{F}|_U \\to \\mathcal{G}|_U", "\\text{ such that } \\alpha \\text{ is a right inverse to }\\varphi|_U \\}.", "$$" ], "refs": [ "sites-lemma-localize-topos" ], "proofs": [ { "contents": [ "For any $\\varphi : \\mathcal{G} \\to \\mathcal{F}$, let us use", "the notation $\\mathcal{G}/\\mathcal{F}$ to denote the corresponding", "object of $\\Sh(\\mathcal{C})/\\mathcal{F}$. We have", "$$", "(j_{\\mathcal{F}, *}(\\mathcal{G}/\\mathcal{F}))(U) =", "\\Mor_{\\Sh(\\mathcal{C})}(h_U^\\#, j_{\\mathcal{F}, *}(\\mathcal{G}/\\mathcal{F})) =", "\\Mor_{\\Sh(\\mathcal{C})/\\mathcal{F}}(j_{\\mathcal{F}}^{-1}h_U^{\\#},", "(\\mathcal{G}/\\mathcal{F})).", "$$", "By Lemma \\ref{lemma-localize-topos} this set is the fiber over", "the element $h_U^\\# \\times \\mathcal{F} \\to \\mathcal{F}$ under the", "map of sets", "$$", "\\Mor_{\\Sh(\\mathcal{C})}(h_U^\\# \\times \\mathcal{F}, \\mathcal{G})", "\\xrightarrow{\\varphi \\circ}", "\\Mor_{\\Sh(\\mathcal{C})}(h_U^\\# \\times \\mathcal{F}, \\mathcal{F}).", "$$", "By the adjunction in Lemma \\ref{lemma-internal-hom-sheaf}, we have", "\\begin{align*}", "\\Mor_{\\Sh(\\mathcal{C})}(h_U^{\\#}\\times\\mathcal{F}, \\mathcal{G})", "& =", "\\Mor_{\\Sh(\\mathcal{C})}(h_U^{\\#},\\SheafHom(\\mathcal{F}, \\mathcal{G})) \\\\", "& =", "\\Mor_{\\Sh(\\mathcal{C}/U)}(\\mathcal{F}|_{\\mathcal{C}/U},", "\\mathcal{G}|_{\\mathcal{C}/U}), \\\\", "\\Mor_{\\Sh(\\mathcal{C})}(h_U^{\\#} \\times \\mathcal{F}, \\mathcal{F})", "& =", "\\Mor_{\\Sh(\\mathcal{C})}(h_U^{\\#},\\SheafHom(\\mathcal{F},\\mathcal{F})) \\\\", "& =", "\\Mor_{\\Sh(\\mathcal{C}/U)}(\\mathcal{F}|_{\\mathcal{C}/U},", "\\mathcal{F}|_{\\mathcal{C}/U}),", "\\end{align*}", "and under the adjunction, the map $\\varphi\\circ$ becomes the map", "$$", "\\Mor_{\\Sh(\\mathcal{C}/U)}(\\mathcal{F}|_{\\mathcal{C}/U},", "\\mathcal{G}|_{\\mathcal{C}/U})", "\\longrightarrow", "\\Mor_{\\Sh(\\mathcal{C}/U)}(\\mathcal{F}|_{\\mathcal{C}/U},", "\\mathcal{F}|_{\\mathcal{C}/U}),\\quad", "\\psi \\longmapsto \\varphi|_{\\mathcal{C}/U} \\circ \\psi,", "$$", "the element $h_U^\\# \\times \\mathcal{F} \\to \\mathcal{F}$", "becomes $\\text{id}_{\\mathcal{F}|_{\\mathcal{C}/U}}$.", "Therefore $(j_{\\mathcal{F}, *}\\mathcal{G}/\\mathcal{F})(U)$", "is isomorphic to the fiber of", "$\\text{id}_{\\mathcal{F}|_{\\mathcal{C}/U}}$ under the map", "$$", "\\Mor_{\\Sh(\\mathcal{C}/U)}(\\mathcal{F}|_{\\mathcal{C}/U},", "\\mathcal{G}|_{\\mathcal{C}/U})", "\\xrightarrow{\\varphi|_{\\mathcal{C}/U}\\circ}", "\\Mor_{\\Sh(\\mathcal{C}/U)}(\\mathcal{F}|_{\\mathcal{C}/U},", "\\mathcal{F}|_{\\mathcal{C}/U}),", "$$", "which is $\\{\\alpha : \\mathcal{F}|_U \\to \\mathcal{G}|_U", "\\text{ such that } \\alpha \\text{ is a right inverse to }\\varphi|_U \\}$", "as desired." ], "refs": [ "sites-lemma-localize-topos", "sites-lemma-internal-hom-sheaf" ], "ref_ids": [ 8583, 8562 ] } ], "ref_ids": [ 8583 ] }, { "id": 8585, "type": "theorem", "label": "sites-lemma-localize-topos-site", "categories": [ "sites" ], "title": "sites-lemma-localize-topos-site", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.", "Let $\\mathcal{C}/\\mathcal{F}$ be the category of pairs $(U, s)$ where", "$U \\in \\Ob(\\mathcal{C})$ and $s \\in \\mathcal{F}(U)$. Let a covering in", "$\\mathcal{C}/\\mathcal{F}$ be a family $\\{(U_i, s_i) \\to (U, s)\\}$", "such that $\\{U_i \\to U\\}$ is a covering of $\\mathcal{C}$.", "Then $j : \\mathcal{C}/\\mathcal{F} \\to \\mathcal{C}$ is a continuous", "and cocontinuous functor of sites which induces a morphism of topoi", "$j : \\Sh(\\mathcal{C}/\\mathcal{F}) \\to \\Sh(\\mathcal{C})$. In fact, there", "is an equivalence $\\Sh(\\mathcal{C}/\\mathcal{F}) =", "\\Sh(\\mathcal{C})/\\mathcal{F}$ which turns $j$ into $j_\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "We omit the verification that $\\mathcal{C}/\\mathcal{F}$ is a site and", "that $j$ is continuous and cocontinuous. By", "Lemma \\ref{lemma-when-shriek} there exists a morphism of topoi", "$j$ as indicated, with $j^{-1}\\mathcal{G}(U, s) = \\mathcal{G}(U)$,", "and there is a left adjoint $j_!$ to $j^{-1}$. A morphism", "$\\varphi : * \\to j^{-1}\\mathcal{G}$ on $\\mathcal{C}/\\mathcal{F}$", "is the same thing as a rule which assigns to every pair $(U, s)$ a", "section $\\varphi(s) \\in \\mathcal{G}(U)$ compatible with restriction maps.", "Hence this is the same thing as a morphism", "$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ over $\\mathcal{C}$.", "We conclude that $j_!* = \\mathcal{F}$. In particular, for every", "$\\mathcal{H} \\in \\Sh(\\mathcal{C}/\\mathcal{F})$ there is a canonical map", "$$", "j_!\\mathcal{H} \\to j_!* = \\mathcal{F}", "$$", "i.e., we obtain a functor", "$j'_! : \\Sh(\\mathcal{C}/\\mathcal{F}) \\to \\Sh(\\mathcal{C})/\\mathcal{F}$.", "An inverse to this functor is the rule which assigns to an object", "$\\varphi : \\mathcal{G} \\to \\mathcal{F}$ of $\\Sh(\\mathcal{C})/\\mathcal{F}$ the", "sheaf", "$$", "a(\\mathcal{G}/\\mathcal{F}) :", "(U, s) \\longmapsto \\{t \\in \\mathcal{G}(U) \\mid \\varphi(t) = s\\}", "$$", "We omit the verification that $a(\\mathcal{G}/\\mathcal{F})$ is a sheaf", "and that $a$ is inverse to $j'_!$." ], "refs": [ "sites-lemma-when-shriek" ], "ref_ids": [ 8545 ] } ], "ref_ids": [] }, { "id": 8586, "type": "theorem", "label": "sites-lemma-localize-compare", "categories": [ "sites" ], "title": "sites-lemma-localize-compare", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F} = h_U^\\#$ for some object $U$", "of $\\mathcal{C}$. Then $j_\\mathcal{F} : \\Sh(\\mathcal{C})/\\mathcal{F}", "\\to \\Sh(\\mathcal{C})$ constructed in", "Lemma \\ref{lemma-localize-topos}", "agrees with the morphism of topoi", "$j_U : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})$", "constructed in", "Section \\ref{section-localize}", "via the identification", "$\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$", "of", "Lemma \\ref{lemma-essential-image-j-shriek}." ], "refs": [ "sites-lemma-localize-topos", "sites-lemma-essential-image-j-shriek" ], "proofs": [ { "contents": [ "We have seen in", "Lemma \\ref{lemma-essential-image-j-shriek}", "that the composition", "$\\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})/h_U^\\#", "\\to \\Sh(\\mathcal{C})$", "is $j_{U!}$. The functor", "$\\Sh(\\mathcal{C})/h_U^\\# \\to \\Sh(\\mathcal{C})$", "is $j_{\\mathcal{F}!}$ by", "Lemma \\ref{lemma-localize-topos}.", "Hence $j_{\\mathcal{F}!} = j_{U!}$ via the identification.", "So $j_\\mathcal{F}^{-1} = j_U^{-1}$ (by adjointness) and so", "$j_{\\mathcal{F}, *} = j_{U, *}$ (by adjointness again)." ], "refs": [ "sites-lemma-essential-image-j-shriek", "sites-lemma-localize-topos" ], "ref_ids": [ 8555, 8583 ] } ], "ref_ids": [ 8583, 8555 ] }, { "id": 8587, "type": "theorem", "label": "sites-lemma-relocalize-topos", "categories": [ "sites" ], "title": "sites-lemma-relocalize-topos", "contents": [ "Let $\\mathcal{C}$ be a site.", "If $s : \\mathcal{G} \\to \\mathcal{F}$ is a morphism of sheaves", "on $\\mathcal{C}$ then there exists a natural commutative diagram of", "morphisms of topoi", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C})/\\mathcal{G} \\ar[rd]_{j_\\mathcal{G}} \\ar[rr]_j & &", "\\Sh(\\mathcal{C})/\\mathcal{F} \\ar[ld]^{j_\\mathcal{F}} \\\\", "& \\Sh(\\mathcal{C}) &", "}", "$$", "where $j = j_{\\mathcal{G}/\\mathcal{F}}$ is the localization of the", "topos $\\Sh(\\mathcal{C})/\\mathcal{F}$ at the object", "$\\mathcal{G}/\\mathcal{F}$. In particular we have", "$$", "j^{-1}(\\mathcal{H} \\to \\mathcal{F}) =", "(\\mathcal{H} \\times_\\mathcal{F} \\mathcal{G} \\to \\mathcal{G})", "$$", "and", "$$", "j_!(\\mathcal{E} \\xrightarrow{e} \\mathcal{F}) =", "(\\mathcal{E} \\xrightarrow{s \\circ e} \\mathcal{G}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "The description of $j^{-1}$ and $j_!$ comes from the description", "of those functors in", "Lemma \\ref{lemma-localize-topos}.", "The equality of functors", "$j_{\\mathcal{G}!} = j_{\\mathcal{F}!} \\circ j_!$ is clear", "from the description of these functors (as forgetful functors).", "By adjointness we also obtain the equalities", "$j_\\mathcal{G}^{-1} = j^{-1} \\circ j_\\mathcal{F}^{-1}$, and", "$j_{\\mathcal{G}, *} = j_{\\mathcal{F}, *} \\circ j_*$." ], "refs": [ "sites-lemma-localize-topos" ], "ref_ids": [ 8583 ] } ], "ref_ids": [] }, { "id": 8588, "type": "theorem", "label": "sites-lemma-relocalize-compare", "categories": [ "sites" ], "title": "sites-lemma-relocalize-compare", "contents": [ "Assume $\\mathcal{C}$ and $s : \\mathcal{G} \\to \\mathcal{F}$ are as in", "Lemma \\ref{lemma-relocalize-topos}.", "If $\\mathcal{G} = h_V^\\#$ and $\\mathcal{F} = h_U^\\#$ and", "$s : \\mathcal{G} \\to \\mathcal{F}$ comes from a morphism $V \\to U$", "of $\\mathcal{C}$ then the diagram in", "Lemma \\ref{lemma-relocalize-topos}", "is identified with", "diagram (\\ref{equation-relocalize})", "via the identifications", "$\\Sh(\\mathcal{C}/V) = \\Sh(\\mathcal{C})/h_V^\\#$", "and", "$\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$", "of", "Lemma \\ref{lemma-essential-image-j-shriek}." ], "refs": [ "sites-lemma-relocalize-topos", "sites-lemma-relocalize-topos", "sites-lemma-essential-image-j-shriek" ], "proofs": [ { "contents": [ "This is true because the descriptions of $j^{-1}$ agree.", "See", "Lemma \\ref{lemma-relocalize-explicit}", "and", "Lemma \\ref{lemma-relocalize-topos}." ], "refs": [ "sites-lemma-relocalize-explicit", "sites-lemma-relocalize-topos" ], "ref_ids": [ 8560, 8587 ] } ], "ref_ids": [ 8587, 8587, 8555 ] }, { "id": 8589, "type": "theorem", "label": "sites-lemma-localize-morphism-topoi", "categories": [ "sites" ], "title": "sites-lemma-localize-morphism-topoi", "contents": [ "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "be a morphism of topoi. Let $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$.", "Set $\\mathcal{F} = f^{-1}\\mathcal{G}$. Then there exists", "a commutative diagram of topoi", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C})/\\mathcal{F} \\ar[r]_{j_\\mathcal{F}} \\ar[d]_{f'} &", "\\Sh(\\mathcal{C}) \\ar[d]^f \\\\", "\\Sh(\\mathcal{D})/\\mathcal{G} \\ar[r]^{j_\\mathcal{G}} &", "\\Sh(\\mathcal{D}).", "}", "$$", "The morphism $f'$ is characterized by the property that", "$$", "(f')^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} \\mathcal{G})", "=", "(f^{-1}\\mathcal{H} \\xrightarrow{f^{-1}\\varphi} \\mathcal{F})", "$$", "and we have $f'_*j_\\mathcal{F}^{-1} = j_\\mathcal{G}^{-1}f_*$." ], "refs": [], "proofs": [ { "contents": [ "Since the statement is about topoi and does not refer to the underlying", "sites we may change sites at will. Hence by the discussion in", "Remark \\ref{remark-morphism-topoi-comes-from-morphism-sites}", "we may assume that $f$ is given by a continuous functor", "$u : \\mathcal{D} \\to \\mathcal{C}$ satisfying the assumptions of", "Proposition \\ref{proposition-get-morphism}", "between sites having all finite limits and subcanonical topologies,", "and such that $\\mathcal{G} = h_V$ for some object $V$ of", "$\\mathcal{D}$. Then $\\mathcal{F} = f^{-1}h_V = h_{u(V)}$ by", "Lemma \\ref{lemma-pullback-representable-sheaf}.", "By", "Lemma \\ref{lemma-localize-morphism}", "we obtain a commutative diagram of morphisms of topoi", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C}/U) \\ar[r]_{j_U} \\ar[d]_{f'} &", "\\Sh(\\mathcal{C}) \\ar[d]^f \\\\", "\\Sh(\\mathcal{D}/V) \\ar[r]^{j_V} &", "\\Sh(\\mathcal{D}),", "}", "$$", "and we have $f'_*j_U^{-1} = j_V^{-1}f_*$. By", "Lemma \\ref{lemma-localize-compare}", "we may identify $j_\\mathcal{F}$ and $j_U$", "and $j_\\mathcal{G}$ and $j_V$. The description of $(f')^{-1}$", "is given in", "Lemma \\ref{lemma-localize-morphism}." ], "refs": [ "sites-remark-morphism-topoi-comes-from-morphism-sites", "sites-proposition-get-morphism", "sites-lemma-pullback-representable-sheaf", "sites-lemma-localize-morphism", "sites-lemma-localize-compare", "sites-lemma-localize-morphism" ], "ref_ids": [ 8715, 8641, 8524, 8571, 8586, 8571 ] } ], "ref_ids": [] }, { "id": 8590, "type": "theorem", "label": "sites-lemma-localize-morphism-compare", "categories": [ "sites" ], "title": "sites-lemma-localize-morphism-compare", "contents": [ "Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites given", "by the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.", "Let $V$ be an object of $\\mathcal{D}$. Set $U = u(V)$.", "Set $\\mathcal{G} = h_V^\\#$, and", "$\\mathcal{F} = h_U^\\# = f^{-1}h_V^\\#$ (see", "Lemma \\ref{lemma-pullback-representable-sheaf}).", "Then the diagram of morphisms of topoi of", "Lemma \\ref{lemma-localize-morphism-topoi}", "agrees with the diagram of morphisms of topoi of", "Lemma \\ref{lemma-localize-morphism}", "via the identifications $j_\\mathcal{F}= j_U$", "and $j_\\mathcal{G} = j_V$ of", "Lemma \\ref{lemma-localize-compare}." ], "refs": [ "sites-lemma-pullback-representable-sheaf", "sites-lemma-localize-morphism-topoi", "sites-lemma-localize-morphism", "sites-lemma-localize-compare" ], "proofs": [ { "contents": [ "This is not a complete triviality as the choice of morphism of sites", "giving rise to $f$ made in the proof of", "Lemma \\ref{lemma-localize-morphism-topoi}", "may be different from the morphisms of sites given to us in the lemma.", "But in both cases the functor $(f')^{-1}$ is described by the same", "rule. Hence they agree and the associated morphism of topoi is the same.", "Some details omitted." ], "refs": [ "sites-lemma-localize-morphism-topoi" ], "ref_ids": [ 8589 ] } ], "ref_ids": [ 8524, 8589, 8571, 8586 ] }, { "id": 8591, "type": "theorem", "label": "sites-lemma-relocalize-morphism-topoi", "categories": [ "sites" ], "title": "sites-lemma-relocalize-morphism-topoi", "contents": [ "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "be a morphism of topoi.", "Let $\\mathcal{G} \\in \\Sh(\\mathcal{D})$,", "$\\mathcal{F} \\in \\Sh(\\mathcal{C})$", "and $s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ a morphism of sheaves.", "There exists a commutative diagram of topoi", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C})/\\mathcal{F} \\ar[r]_{j_\\mathcal{F}} \\ar[d]_{f_s} &", "\\Sh(\\mathcal{C}) \\ar[d]^f \\\\", "\\Sh(\\mathcal{D})/\\mathcal{G} \\ar[r]^{j_\\mathcal{G}} &", "\\Sh(\\mathcal{D}).", "}", "$$", "We have $f_s = f' \\circ j_{\\mathcal{F}/f^{-1}\\mathcal{G}}$ where", "$f' :", "\\Sh(\\mathcal{C})/f^{-1}\\mathcal{G}", "\\to", "\\Sh(\\mathcal{D})/\\mathcal{F}$", "is as in", "Lemma \\ref{lemma-localize-morphism-topoi}", "and", "$j_{\\mathcal{F}/f^{-1}\\mathcal{G}} :", "\\Sh(\\mathcal{C})/\\mathcal{F}", "\\to", "\\Sh(\\mathcal{C})/f^{-1}\\mathcal{G}$", "is as in", "Lemma \\ref{lemma-relocalize-topos}.", "The functor $(f_s)^{-1}$ is described by the rule", "$$", "(f_s)^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} \\mathcal{G})", "=", "(f^{-1}\\mathcal{H} \\times_{f^{-1}\\varphi, f^{-1}\\mathcal{G}, s} \\mathcal{F}", "\\rightarrow \\mathcal{F}).", "$$", "Finally, given any morphisms $b : \\mathcal{G}' \\to \\mathcal{G}$,", "$a : \\mathcal{F}' \\to \\mathcal{F}$ and", "$s' : \\mathcal{F}' \\to f^{-1}\\mathcal{G}'$ such that", "$$", "\\xymatrix{", "\\mathcal{F}' \\ar[r]_-{s'} \\ar[d]_a & f^{-1}\\mathcal{G}' \\ar[d]^{f^{-1}b} \\\\", "\\mathcal{F} \\ar[r]^-s & f^{-1}\\mathcal{G}", "}", "$$", "commutes, then the diagram", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C})/\\mathcal{F}'", "\\ar[r]_{j_{\\mathcal{F}'/\\mathcal{F}}} \\ar[d]_{f_{s'}} &", "\\Sh(\\mathcal{C})/\\mathcal{F} \\ar[d]^{f_s} \\\\", "\\Sh(\\mathcal{D})/\\mathcal{G}' \\ar[r]^{j_{\\mathcal{G}'/\\mathcal{G}}} &", "\\Sh(\\mathcal{D})/\\mathcal{G}.", "}", "$$", "commutes." ], "refs": [ "sites-lemma-localize-morphism-topoi", "sites-lemma-relocalize-topos" ], "proofs": [ { "contents": [ "The commutativity of the first square follows from the commutativity of", "the diagram in", "Lemma \\ref{lemma-relocalize-topos}", "and the commutativity of the diagram in", "Lemma \\ref{lemma-localize-morphism-topoi}.", "The description of $f_s^{-1}$ follows on combining the descriptions", "of $(f')^{-1}$ in", "Lemma \\ref{lemma-localize-morphism-topoi}", "with the description of", "$(j_{\\mathcal{F}/f^{-1}\\mathcal{G}})^{-1}$ in", "Lemma \\ref{lemma-relocalize-topos}.", "The commutativity of the last square then follows from the", "equality", "$$", "f^{-1}\\mathcal{H} \\times_{f^{-1}\\mathcal{G}, s} \\mathcal{F}", "\\times_\\mathcal{F} \\mathcal{F}'", "=", "f^{-1}(\\mathcal{H} \\times_\\mathcal{G} \\mathcal{G}')", "\\times_{f^{-1}\\mathcal{G}', s'} \\mathcal{F}'", "$$", "which is formal." ], "refs": [ "sites-lemma-relocalize-topos", "sites-lemma-localize-morphism-topoi", "sites-lemma-localize-morphism-topoi", "sites-lemma-relocalize-topos" ], "ref_ids": [ 8587, 8589, 8589, 8587 ] } ], "ref_ids": [ 8589, 8587 ] }, { "id": 8592, "type": "theorem", "label": "sites-lemma-relocalize-morphism-compare", "categories": [ "sites" ], "title": "sites-lemma-relocalize-morphism-compare", "contents": [ "Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites given", "by the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.", "Let $V$ be an object of $\\mathcal{D}$. Let $c : U \\to u(V)$ be a morphism.", "Set $\\mathcal{G} = h_V^\\#$ and $\\mathcal{F} = h_U^\\# = f^{-1}h_V^\\#$.", "Let $s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ be the map induced by $c$.", "Then the diagram of morphisms of topoi of", "Lemma \\ref{lemma-relocalize-morphism}", "agrees with the diagram of morphisms of topoi of", "Lemma \\ref{lemma-relocalize-morphism-topoi}", "via the identifications $j_\\mathcal{F} = j_U$", "and $j_\\mathcal{G} = j_V$ of", "Lemma \\ref{lemma-localize-compare}." ], "refs": [ "sites-lemma-relocalize-morphism", "sites-lemma-relocalize-morphism-topoi", "sites-lemma-localize-compare" ], "proofs": [ { "contents": [ "This follows on combining", "Lemmas \\ref{lemma-relocalize-compare} and", "\\ref{lemma-localize-morphism-compare}." ], "refs": [ "sites-lemma-relocalize-compare", "sites-lemma-localize-morphism-compare" ], "ref_ids": [ 8588, 8590 ] } ], "ref_ids": [ 8573, 8591, 8586 ] }, { "id": 8593, "type": "theorem", "label": "sites-lemma-points-recover", "categories": [ "sites" ], "title": "sites-lemma-points-recover", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $p = u : \\mathcal{C} \\to \\textit{Sets}$ be a functor.", "There are functorial isomorphisms", "$(h_U)_p = u(U)$ for $U \\in \\Ob(\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "An element of $(h_U)_p$ is given by a triple $(V, y, f)$,", "where $V \\in \\Ob(\\mathcal{C})$, $y\\in u(V)$ and", "$f \\in h_U(V) = \\Mor_\\mathcal{C}(V, U)$.", "Two such $(V, y, f)$, $(V', y', f')$ determine the same object if", "there exists a morphism $\\phi : V \\to V'$ such that", "$u(\\phi)(y) = y'$ and $f' \\circ \\phi = f$, and in general you have", "to take chains of identities like this to get the correct equivalence", "relation. In any case, every $(V, y, f)$ is equivalent to", "the element $(U, u(f)(y), \\text{id}_U)$. If $\\phi$ exists as above,", "then the triples $(V, y, f)$, $(V', y', f')$ determine the same triple", "$(U, u(f)(y), \\text{id}_U) = (U, u(f')(y'), \\text{id}_U)$.", "This proves that the map", "$u(U) \\to (h_U)_p$, $x \\mapsto \\text{class of }(U, x, \\text{id}_U)$", "is bijective." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8594, "type": "theorem", "label": "sites-lemma-adjoint-point-push-stalk", "categories": [ "sites" ], "title": "sites-lemma-adjoint-point-push-stalk", "contents": [ "For any functor $u : \\mathcal{C} \\to \\textit{Sets}$.", "The functor $u^p$ is a right adjoint to the stalk functor", "on presheaves." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a presheaf on $\\mathcal{C}$.", "Let $E$ be a set. A morphism $\\mathcal{F} \\to u^pE$", "is given by a compatible system of maps", "$\\mathcal{F}(U) \\to \\text{Map}(u(U), E)$, i.e.,", "a compatible system of maps $\\mathcal{F}(U) \\times u(U) \\to E$.", "And by definition of $\\mathcal{F}_p$ a map $\\mathcal{F}_p \\to E$", "is given by a rule associating with each triple $(U, x, \\sigma)$", "an element in $E$ such that equivalent triples map to the same element, see", "discussion surrounding", "Equation (\\ref{equation-stalk}).", "This also means a compatible system of maps $\\mathcal{F}(U) \\times u(U) \\to E$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8595, "type": "theorem", "label": "sites-lemma-point-pushforward-sheaf", "categories": [ "sites" ], "title": "sites-lemma-point-pushforward-sheaf", "contents": [ "Let $\\mathcal{C}$ be a site. Let $p = u : \\mathcal{C} \\to \\textit{Sets}$", "be a functor. Suppose that for every covering $\\{U_i \\to U\\}$ of $\\mathcal{C}$", "\\begin{enumerate}", "\\item the map $\\coprod u(U_i) \\to u(U)$ is surjective, and", "\\item the maps", "$u(U_i \\times_U U_j) \\to u(U_i) \\times_{u(U)} u(U_j)$ are surjective.", "\\end{enumerate}", "Then we have", "\\begin{enumerate}", "\\item the presheaf $u^pE$ is a sheaf for all sets $E$, denote it $u^sE$,", "\\item the stalk functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$", "and the functor $u^s: \\textit{Sets} \\to \\Sh(\\mathcal{C})$ are", "adjoint, and", "\\item we have $\\mathcal{F}_p = \\mathcal{F}^\\#_p$", "for every presheaf of sets $\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first assertion is immediate from the definition of a sheaf, assumptions", "(1) and (2), and the definition of $u^pE$. The second is a restatement of the", "adjointness of $u^p$ and the stalk functor", "(Lemma \\ref{lemma-adjoint-point-push-stalk}) restricted to sheaves.", "The third assertion follows as, for any set $E$, we have", "$$", "\\text{Map}(\\mathcal{F}_p, E) =", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(\\mathcal{F}, u^pE) =", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F}^\\#, u^sE) =", "\\text{Map}(\\mathcal{F}^\\#_p, E)", "$$", "by the adjointness property of sheafification." ], "refs": [ "sites-lemma-adjoint-point-push-stalk" ], "ref_ids": [ 8594 ] } ], "ref_ids": [] }, { "id": 8596, "type": "theorem", "label": "sites-lemma-point-site-topos", "categories": [ "sites" ], "title": "sites-lemma-point-site-topos", "contents": [ "Let $\\mathcal{C}$ be a site.", "\\begin{enumerate}", "\\item Let $p$ be a point of the site $\\mathcal{C}$.", "Then the pair of functors $(p_*, p^{-1})$ introduced", "above define a morphism of topoi", "$\\Sh(pt) \\to \\Sh(\\mathcal{C})$.", "\\item Let $p = (p_*, p^{-1})$", "be a point of the topos $\\Sh(\\mathcal{C})$.", "Then the functor $u : U \\mapsto p^{-1}(h_U^\\#)$ gives", "rise to a point $p'$ of the site $\\mathcal{C}$", "whose associated morphism of topoi $(p'_*, (p')^{-1})$", "is equal to $p$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). By the above the functors $p_*$ and $p^{-1}$ are adjoint.", "The functor $p^{-1}$ is required to be exact by", "Definition \\ref{definition-point}.", "Hence the conditions imposed in", "Definition \\ref{definition-topos}", "are all satisfied and we see that (1) holds.", "\\medskip\\noindent", "Proof of (2). Let $\\{U_i \\to U\\}$ be a covering of $\\mathcal{C}$.", "Then $\\coprod (h_{U_i})^\\# \\to h_U^\\#$ is surjective, see", "Lemma \\ref{lemma-covering-surjective-after-sheafification}.", "Since $p^{-1}$ is exact (by definition of a morphism of topoi) we conclude", "that $\\coprod u(U_i) \\to u(U)$ is surjective.", "This proves part (1) of", "Definition \\ref{definition-point}.", "Sheafification is exact, see", "Lemma \\ref{lemma-sheafification-exact}.", "Hence if $U \\times_V W$ exists in $\\mathcal{C}$, then", "$$", "h_{U \\times_V W}^\\# = h_U^\\# \\times_{h_V^\\#} h_W^\\#", "$$", "and we see that $u(U \\times_V W) = u(U) \\times_{u(V)} u(W)$ since $p^{-1}$", "is exact. This proves part (2) of", "Definition \\ref{definition-point}.", "Let $p' = u$, and let $\\mathcal{F}_{p'}$ be the stalk functor", "defined by Equation (\\ref{equation-stalk}) using $u$. There is", "a canonical comparison map", "$c : \\mathcal{F}_{p'} \\to \\mathcal{F}_p = p^{-1}\\mathcal{F}$.", "Namely, given a triple $(U, x, \\sigma)$ representing an element $\\xi$", "of $\\mathcal{F}_{p'}$ we think of $\\sigma$ as a map", "$\\sigma : h_U^\\# \\to \\mathcal{F}$ and we can set", "$c(\\xi) = p^{-1}(\\sigma)(x)$ since $x \\in u(U) = p^{-1}(h_U^\\#)$. By", "Lemma \\ref{lemma-points-recover}", "we see that $(h_U)_{p'} = u(U)$. Since conditions (1) and (2) of", "Definition \\ref{definition-point}", "hold for $p'$ we also have $(h_U^\\#)_{p'} = (h_U)_{p'}$ by", "Lemma \\ref{lemma-point-pushforward-sheaf}.", "Hence we have", "$$", "(h_U^\\#)_{p'} = (h_U)_{p'} = u(U) = p^{-1}(h_U^\\#)", "$$", "We claim this bijection equals the comparison map", "$c : (h_U^\\#)_{p'} \\to p^{-1}(h_U^\\#)$ (verification omitted).", "Any sheaf on $\\mathcal{C}$ is a coequalizer of maps", "of coproducts of sheaves of the form $h_U^\\#$, see", "Lemma \\ref{lemma-sheaf-coequalizer-representable}.", "The stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_{p'}$ and", "the functor $p^{-1}$ commute with arbitrary colimits (as they", "are both left adjoints).", "We conclude $c$ is an isomorphism for every sheaf $\\mathcal{F}$.", "Thus the stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_{p'}$", "is isomorphic to $p^{-1}$ and we in particular see that it is exact.", "This proves condition (3) of", "Definition \\ref{definition-point}", "holds and $p'$ is a point. The final assertion has already been shown", "above, since we saw that $p^{-1} = (p')^{-1}$." ], "refs": [ "sites-definition-point", "sites-definition-topos", "sites-lemma-covering-surjective-after-sheafification", "sites-definition-point", "sites-lemma-sheafification-exact", "sites-definition-point", "sites-lemma-points-recover", "sites-definition-point", "sites-lemma-point-pushforward-sheaf", "sites-lemma-sheaf-coequalizer-representable", "sites-definition-point" ], "ref_ids": [ 8675, 8667, 8519, 8675, 8515, 8675, 8593, 8675, 8595, 8520, 8675 ] } ], "ref_ids": [] }, { "id": 8597, "type": "theorem", "label": "sites-lemma-site-point-morphism", "categories": [ "sites" ], "title": "sites-lemma-site-point-morphism", "contents": [ "Let $\\mathcal{C}$ be a site. Let $p$ be a point of $\\mathcal{C}$ given by", "$u : \\mathcal{C} \\to \\textit{Sets}$. Let $S_0$ be an infinite set such that", "$u(U) \\subset S_0$ for all $U \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{S}$", "be the site constructed out of the powerset $S = \\mathcal{P}(S_0)$ in", "Remark \\ref{remark-pt-topos}.", "Then", "\\begin{enumerate}", "\\item there is an equivalence", "$i : \\Sh(pt) \\to \\Sh(\\mathcal{S})$,", "\\item the functor $u : \\mathcal{C} \\to \\mathcal{S}$ induces a morphism of", "sites $f : \\mathcal{S} \\to \\mathcal{C}$, and", "\\item the composition", "$$", "\\Sh(pt) \\to", "\\Sh(\\mathcal{S}) \\to", "\\Sh(\\mathcal{C})", "$$", "is the morphism of topoi $(p_*, p^{-1})$ of", "Lemma \\ref{lemma-point-site-topos}.", "\\end{enumerate}" ], "refs": [ "sites-remark-pt-topos", "sites-lemma-point-site-topos" ], "proofs": [ { "contents": [ "Part (1) we saw in", "Remark \\ref{remark-pt-topos}.", "Moreover, recall that the equivalence associates to the set $E$", "the sheaf $i_*E$ on $\\mathcal{S}$ defined by the rule", "$V \\mapsto \\Mor_{\\textit{Sets}}(V, E)$.", "Part (2) is clear from the definition of a point of $\\mathcal{C}$", "(Definition \\ref{definition-point})", "and the definition of a morphism of sites", "(Definition \\ref{definition-morphism-sites}).", "Finally, consider $f_*i_*E$. By construction we have", "$$", "f_*i_*E(U) = i_*E(u(U)) = \\Mor_{\\textit{Sets}}(u(U), E)", "$$", "which is equal to $p_*E(U)$, see", "Equation (\\ref{equation-skyscraper}).", "This proves (3)." ], "refs": [ "sites-remark-pt-topos", "sites-definition-point", "sites-definition-morphism-sites" ], "ref_ids": [ 8709, 8675, 8665 ] } ], "ref_ids": [ 8709, 8596 ] }, { "id": 8598, "type": "theorem", "label": "sites-lemma-stalk-skyscraper", "categories": [ "sites" ], "title": "sites-lemma-stalk-skyscraper", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$p : \\Sh(pt) \\to \\Sh(\\mathcal{C})$ be a point of", "the topos associated to $\\mathcal{C}$.", "For any set $E$ there are canonical maps", "$$", "E \\longrightarrow (p_*E)_p \\longrightarrow E", "$$", "whose composition is $\\text{id}_E$." ], "refs": [], "proofs": [ { "contents": [ "There is always an adjunction map $(p_*E)_p = p^{-1}p_*E \\to E$.", "This map is an isomorphism when $E = \\{*\\}$ because $p_*$", "and $p^{-1}$ are both left exact, hence transform the final", "object into the final object. Hence given $e \\in E$ we can consider", "the map $i_e : \\{*\\} \\to E$ which gives", "$$", "\\xymatrix{", "p^{-1}p_*\\{*\\} \\ar[rr]_{p^{-1}p_*i_e} \\ar[d]_{\\cong} & & p^{-1}p_*E \\ar[d] \\\\", "\\{*\\} \\ar[rr]^{i_e} & & E", "}", "$$", "whence the map $E \\to (p_*E)_p = p^{-1}p_*E$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8599, "type": "theorem", "label": "sites-lemma-skyscraper-functor-exact", "categories": [ "sites" ], "title": "sites-lemma-skyscraper-functor-exact", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$p : \\Sh(pt) \\to \\Sh(\\mathcal{C})$ be a point of", "the topos associated to $\\mathcal{C}$.", "The functor $p_* : \\textit{Sets} \\to \\Sh(\\mathcal{C})$", "has the following properties: It commutes with arbitrary limits,", "it is left exact, it is faithful, it transforms surjections into surjections,", "it commutes with coequalizers, it reflects injections, it reflects", "surjections, and it reflects isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "Because $p_*$ is a right adjoint it commutes with arbitrary limits and", "it is left exact. The fact that $p^{-1}p_*E \\to E$ is a canonically", "split surjection implies that $p_*$ is faithful, reflects injections,", "reflects surjections, and reflects isomorphisms. By", "Lemma \\ref{lemma-point-site-topos}", "we may assume that $p$ comes from a point $u : \\mathcal{C} \\to \\textit{Sets}$", "of the underlying site $\\mathcal{C}$. In this case the sheaf $p_*E$ is given by", "$$", "p_*E(U) = \\Mor_{\\textit{Sets}}(u(U), E)", "$$", "see Equation (\\ref{equation-skyscraper}) and", "Definition \\ref{definition-pushforward-point}.", "It follows immediately from this formula that $p_*$ transforms", "surjections into surjections and coequalizers into coequalizers." ], "refs": [ "sites-lemma-point-site-topos", "sites-definition-pushforward-point" ], "ref_ids": [ 8596, 8676 ] } ], "ref_ids": [] }, { "id": 8600, "type": "theorem", "label": "sites-lemma-neighbourhoods-cofiltered", "categories": [ "sites" ], "title": "sites-lemma-neighbourhoods-cofiltered", "contents": [ "Let $\\mathcal{C}$ be a site. Let $p = u : \\mathcal{C} \\to \\textit{Sets}$", "be a functor. If the category of neighbourhoods of $p$ is", "cofiltered, then the stalk functor (\\ref{equation-stalk})", "is left exact." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\to \\Sh(\\mathcal{C})$,", "$i \\mapsto \\mathcal{F}_i$ be a finite diagram of sheaves.", "We have to show that the stalk of the limit of this", "system agrees with the limit of the stalks.", "Let $\\mathcal{F}$ be the limit of the system as a {\\it presheaf}.", "According to Lemma \\ref{lemma-limit-sheaf} this is a sheaf and", "it is the limit in the category of sheaves.", "Hence we have to show that", "$\\mathcal{F}_p = \\lim_\\mathcal{I} \\mathcal{F}_{i, p}$.", "Recall also that $\\mathcal{F}$ has a simple description, see", "Section \\ref{section-limits-colimits-PSh}. Thus we have to show that", "$$", "\\lim_i \\colim_{\\{(U, x)\\}^{opp}} \\mathcal{F}_i(U)", "=", "\\colim_{\\{(U, x)\\}^{opp}} \\lim_i \\mathcal{F}_i(U).", "$$", "This holds, by Categories, Lemma \\ref{categories-lemma-directed-commutes},", "because the opposite of the category of neighbourhoods is filtered", "by assumption." ], "refs": [ "sites-lemma-limit-sheaf", "categories-lemma-directed-commutes" ], "ref_ids": [ 8508, 12228 ] } ], "ref_ids": [] }, { "id": 8601, "type": "theorem", "label": "sites-lemma-neighbourhoods-directed", "categories": [ "sites" ], "title": "sites-lemma-neighbourhoods-directed", "contents": [ "Let $\\mathcal{C}$ be a site. Assume that $\\mathcal{C}$ has", "a final object $X$ and fibred products.", "Let $p = u : \\mathcal{C} \\to \\textit{Sets}$ be a functor such that", "\\begin{enumerate}", "\\item $u(X)$ is a singleton set, and", "\\item for every pair of morphisms $U \\to W$ and $V \\to W$ with", "the same target the map", "$u(U \\times_W V) \\to u(U) \\times_{u(W)} u(V)$ is bijective.", "\\end{enumerate}", "Then the the category of neighbourhoods of $p$ is cofiltered", "and consequently the stalk functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$,", "$\\mathcal{F} \\to \\mathcal{F}_p$ commutes with finite limits." ], "refs": [], "proofs": [ { "contents": [ "Please note the analogy with Lemma \\ref{lemma-directed}.", "The assumptions on $\\mathcal{C}$ imply that $\\mathcal{C}$ has finite limits.", "See Categories, Lemma \\ref{categories-lemma-finite-limits-exist}.", "Assumption (1) implies that the category of neighbourhoods", "is nonempty. Suppose $(U, x)$ and $(V, y)$ are neighbourhoods.", "Then", "$u(U \\times V) = u(U \\times_X V) =", "u(U) \\times_{u(X)} u(V) = u(U) \\times u(V)$ by (2).", "Hence there exists a neighbourhood $(U \\times_X V, z)$ mapping", "to both $(U, x)$ and $(V, y)$.", "Let $a, b : (V, y) \\to (U, x)$ be two morphisms", "in the category of neighbourhoods. Let $W$ be the equalizer of", "$a, b : V \\to U$. As in the proof of", "Categories, Lemma \\ref{categories-lemma-finite-limits-exist}", "we may write $W$ in terms of fibre products:", "$$", "W = (V \\times_{a, U, b} V) \\times_{(pr_1, pr_2), V \\times V, \\Delta} V", "$$", "The bijectivity in (2) guarantees there exists an element $z \\in u(W)$", "which maps to $((y, y), y)$.", "Then $(W, z) \\to (V, y)$ equalizes $a, b$ as desired.", "The ``consequently'' clause is Lemma \\ref{lemma-neighbourhoods-cofiltered}." ], "refs": [ "sites-lemma-directed", "categories-lemma-finite-limits-exist", "categories-lemma-finite-limits-exist", "sites-lemma-neighbourhoods-cofiltered" ], "ref_ids": [ 8498, 12224, 12224, 8600 ] } ], "ref_ids": [] }, { "id": 8602, "type": "theorem", "label": "sites-lemma-point-functor", "categories": [ "sites" ], "title": "sites-lemma-point-functor", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Let", "$v : \\mathcal{D} \\to \\textit{Sets}$ be a functor and set", "$w = v \\circ u$. Denote $q$, resp., $p$ the stalk functor", "(\\ref{equation-stalk}) associated to $v$, resp.\\ $w$.", "Then $(u_p\\mathcal{F})_q = \\mathcal{F}_p$ functorially in the", "presheaf $\\mathcal{F}$ on $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "This is a simple categorical fact. We have", "\\begin{align*}", "(u_p\\mathcal{F})_q", "& =", "\\colim_{(V, y)} \\colim_{U, \\phi : V \\to u(U)} \\mathcal{F}(U) \\\\", "& = \\colim_{(V, y, U, \\phi : V \\to u(U))} \\mathcal{F}(U) \\\\", "& = \\colim_{(U, x)} \\mathcal{F}(U) \\\\", "& = \\mathcal{F}_p", "\\end{align*}", "The first equality holds by the definition of $u_p$ and the", "definition of the stalk functor. Observe that $y \\in v(V)$.", "In the second equality we simply combine colimits.", "To see the third equality we apply", "Categories, Lemma \\ref{categories-lemma-colimit-constant-connected-fibers}", "to the functor $F$ of diagram categories defined by the rule", "$$", "F((V, y, U, \\phi : V \\to u(U))) = (U, v(\\phi)(y)).", "$$", "This makes sense because $w(U) = v(u(U))$. Let us check the hypotheses of", "Categories, Lemma \\ref{categories-lemma-colimit-constant-connected-fibers}.", "Observe that $F$ has a right inverse, namely", "$(U, x) \\mapsto (u(U), x, U, \\text{id} : u(U) \\to u(U))$.", "Again this makes sense because $x \\in w(U) = v(u(U))$. On the other hand,", "there is always a morphism", "$$", "(V, y, U, \\phi : V \\to u(U))", "\\longrightarrow", "(u(U), v(\\phi)(y), U, \\text{id} : u(U) \\to u(U))", "$$", "in the fibre category over $(U, x)$ which shows the fibre categories", "are connected. The fourth and final equality is clear." ], "refs": [ "categories-lemma-colimit-constant-connected-fibers", "categories-lemma-colimit-constant-connected-fibers" ], "ref_ids": [ 12219, 12219 ] } ], "ref_ids": [] }, { "id": 8603, "type": "theorem", "label": "sites-lemma-point-morphism-sites", "categories": [ "sites" ], "title": "sites-lemma-point-morphism-sites", "contents": [ "\\begin{slogan}", "A map of sites defines a map on points, and pullback respects stalks.", "\\end{slogan}", "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites", "given by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$.", "Let $q$ be a point of $\\mathcal{D}$ given by the functor", "$v : \\mathcal{D} \\to \\textit{Sets}$, see", "Definition \\ref{definition-point}.", "Then the functor $v \\circ u : \\mathcal{C} \\to \\textit{Sets}$", "defines a point $p$ of $\\mathcal{C}$ and moreover there is", "a canonical identification", "$$", "(f^{-1}\\mathcal{F})_q = \\mathcal{F}_p", "$$", "for any sheaf $\\mathcal{F}$ on $\\mathcal{C}$." ], "refs": [ "sites-definition-point" ], "proofs": [ { "contents": [ "[First proof Lemma \\ref{lemma-point-morphism-sites}]", "Note that since $u$ is continuous and since $v$ defines a point,", "it is immediate that $v \\circ u$ satisfies conditions (1) and (2) of", "Definition \\ref{definition-point}. Let us prove the displayed equality.", "Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$. Then", "$$", "(f^{-1}\\mathcal{F})_q = (u_s\\mathcal{F})_q =", "(u_p \\mathcal{F})_q = \\mathcal{F}_p", "$$", "The first equality since $f^{-1} = u_s$, the second equality", "by Lemma \\ref{lemma-point-pushforward-sheaf}, and the third", "by Lemma \\ref{lemma-point-functor}.", "Hence now we see that $p$ also satisfies condition (3) of", "Definition \\ref{definition-point}", "because it is a composition of exact functors. This finishes the proof." ], "refs": [ "sites-lemma-point-morphism-sites", "sites-definition-point", "sites-lemma-point-pushforward-sheaf", "sites-lemma-point-functor", "sites-definition-point" ], "ref_ids": [ 8603, 8675, 8595, 8602, 8675 ] } ], "ref_ids": [ 8675 ] }, { "id": 8604, "type": "theorem", "label": "sites-lemma-point-morphism-topoi", "categories": [ "sites" ], "title": "sites-lemma-point-morphism-topoi", "contents": [ "Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$", "be a morphism of topoi. Let $q : \\Sh(pt) \\to \\Sh(\\mathcal{D})$", "be a point. Then $p = f \\circ q$ is a point of the topos", "$\\Sh(\\mathcal{C})$ and we have", "a canonical identification", "$$", "(f^{-1}\\mathcal{F})_q = \\mathcal{F}_p", "$$", "for any sheaf $\\mathcal{F}$ on $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions and the fact that we can", "compose morphisms of topoi." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8605, "type": "theorem", "label": "sites-lemma-point-localize", "categories": [ "sites" ], "title": "sites-lemma-point-localize", "contents": [ "Let $\\mathcal{C}$ be a site. Let $p$ be a point of $\\mathcal{C}$ given by", "$u : \\mathcal{C} \\to \\textit{Sets}$. Let $U$ be an object of $\\mathcal{C}$", "and let $x \\in u(U)$. The functor", "$$", "v : \\mathcal{C}/U \\longrightarrow \\textit{Sets}, \\quad", "(\\varphi : V \\to U) \\longmapsto \\{y \\in u(V) \\mid u(\\varphi)(y) = x\\}", "$$", "defines a point $q$ of the site $\\mathcal{C}/U$ such that the diagram", "$$", "\\xymatrix{", "& \\Sh(pt) \\ar[d]^p \\ar[ld]_q \\\\", "\\Sh(\\mathcal{C}/U) \\ar[r]^{j_U} &", "\\Sh(\\mathcal{C})", "}", "$$", "commutes. In other words", "$\\mathcal{F}_p = (j_U^{-1}\\mathcal{F})_q$ for any", "sheaf on $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Choose $S$ and $\\mathcal{S}$ as in", "Lemma \\ref{lemma-site-point-morphism}.", "We may identify $\\Sh(pt) = \\Sh(\\mathcal{S})$", "as in that lemma, and we may write", "$p = f : \\Sh(\\mathcal{S}) \\to \\Sh(\\mathcal{C})$", "for the morphism of topoi induced by $u$.", "By", "Lemma \\ref{lemma-localize-morphism}", "we get a commutative diagram of topoi", "$$", "\\xymatrix{", "\\Sh(\\mathcal{S}/u(U)) \\ar[r]_-{j_{u(U)}} \\ar[d]_{p'} &", "\\Sh(\\mathcal{S}) \\ar[d]^p \\\\", "\\Sh(\\mathcal{C}/U) \\ar[r]^{j_U} &", "\\Sh(\\mathcal{C}),", "}", "$$", "where $p'$ is given by the functor $u' : \\mathcal{C}/U \\to \\mathcal{S}/u(U)$,", "$V/U \\mapsto u(V)/u(U)$.", "Consider the functor $j_x : \\mathcal{S} \\cong \\mathcal{S}/x$ obtained", "by assigning to a set $E$ the set $E$ endowed with the constant map", "$E \\to u(U)$ with value $x$.", "Then $j_x$ is a fully faithful cocontinuous functor which has a", "continuous right adjoint", "$v_x : (\\psi : E \\to u(U)) \\mapsto \\psi^{-1}(\\{x\\})$.", "Note that $j_{u(U)} \\circ j_x = \\text{id}_\\mathcal{S}$, and", "$v_x \\circ u' = v$.", "These observations imply that we have the following commutative", "diagram of topoi", "$$", "\\xymatrix{", "\\Sh(\\mathcal{S}) \\ar[rd]^a \\ar[rdd]_q", "\\ar `r[rrr] `d[dd]^p [rrdd] & & & \\\\", "& \\Sh(\\mathcal{S}/u(U)) \\ar[r]_-{j_{u(U)}} \\ar[d]^{p'} &", "\\Sh(\\mathcal{S}) \\ar[d]^p & \\\\", "& \\Sh(\\mathcal{C}/U) \\ar[r]^{j_U} &", "\\Sh(\\mathcal{C}) &", "}", "$$", "Namely:", "\\begin{enumerate}", "\\item The morphism", "$a : \\Sh(\\mathcal{S}) \\to \\Sh(\\mathcal{S}/u(U))$", "is the morphism of topoi associated to the cocontinuous functor", "$j_x$, which equals the morphism associated to the continuous", "functor $v_x$, see", "Lemma \\ref{lemma-cocontinuous-morphism-topoi}", "and", "Section \\ref{section-cocontinuous-adjoint}.", "\\item The composition $p \\circ j_{u(U)} \\circ a = p$ since", "$j_{u(U)} \\circ j_x = \\text{id}_\\mathcal{S}$.", "\\item The composition $p' \\circ a$ gives a morphism of topoi.", "Moreover, it is the morphism of topoi associated to the continuous", "functor $v_x \\circ u' = v$. Hence $v$ does indeed define a point $q$ of", "$\\mathcal{C}/U$ which fits into the diagram above by construction.", "\\end{enumerate}", "This ends the proof of the lemma." ], "refs": [ "sites-lemma-site-point-morphism", "sites-lemma-localize-morphism", "sites-lemma-cocontinuous-morphism-topoi" ], "ref_ids": [ 8597, 8571, 8543 ] } ], "ref_ids": [] }, { "id": 8606, "type": "theorem", "label": "sites-lemma-points-above-point", "categories": [ "sites" ], "title": "sites-lemma-points-above-point", "contents": [ "Let $\\mathcal{C}$, $p$, $u$, $U$ be as in", "Lemma \\ref{lemma-point-localize}.", "The construction of", "Lemma \\ref{lemma-point-localize}", "gives a one to one correspondence between points $q$", "of $\\mathcal{C}/U$ lying over $p$ and elements $x$ of $u(U)$." ], "refs": [ "sites-lemma-point-localize", "sites-lemma-point-localize" ], "proofs": [ { "contents": [ "Let $q$ be a point of $\\mathcal{C}/U$ given by the functor", "$v : \\mathcal{C}/U \\to \\textit{Sets}$ such that $j_U \\circ q = p$", "as morphisms of topoi. Recall that $u(V) = p^{-1}(h_V^\\#)$ for any", "object $V$ of $\\mathcal{C}$, see", "Lemma \\ref{lemma-point-site-topos}.", "Similarly $v(V/U) = q^{-1}(h_{V/U}^\\#)$ for any object", "$V/U$ of $\\mathcal{C}/U$.", "Consider the following two diagrams", "$$", "\\vcenter{", "\\xymatrix{", "\\Mor_{\\mathcal{C}/U}(W/U, V/U) \\ar[r] \\ar[d] &", "\\Mor_\\mathcal{C}(W, V) \\ar[d] \\\\", "\\Mor_{\\mathcal{C}/U}(W/U, U/U) \\ar[r] &", "\\Mor_\\mathcal{C}(W, U)", "}", "}", "\\quad", "\\vcenter{", "\\xymatrix{", "h_{V/U}^\\# \\ar[r] \\ar[d] & j_U^{-1}(h_V^\\#) \\ar[d] \\\\", "h_{U/U}^\\# \\ar[r] & j_U^{-1}(h_U^\\#)", "}", "}", "$$", "The right hand diagram is the sheafification of the diagram of", "presheaves on $\\mathcal{C}/U$ which maps $W/U$ to the left hand", "diagram of sets. (There is a small technical point to make here, namely, that", "we have $(j_U^{-1}h_V)^\\# = j_U^{-1}(h_V^\\#)$ and similarly for $h_U$, see", "Lemma \\ref{lemma-technical-pu}.)", "Note that the left hand diagram of sets is cartesian.", "Since sheafification is exact", "(Lemma \\ref{lemma-sheafification-exact})", "we conclude that the right hand diagram is cartesian.", "\\medskip\\noindent", "Apply the exact functor $q^{-1}$ to the right", "hand diagram to get a cartesian diagram", "$$", "\\xymatrix{", "v(V/U) \\ar[r] \\ar[d] & u(V) \\ar[d] \\\\", "v(U/U) \\ar[r] & u(U)", "}", "$$", "of sets. Here we have used that", "$q^{-1} \\circ j^{-1} = p^{-1}$. Since $U/U$ is a final object of", "$\\mathcal{C}/U$ we see that $v(U/U)$ is a singleton. Hence the", "image of $v(U/U)$ in $u(U)$ is an element $x$, and the top horizontal", "map gives a bijection", "$v(V/U) \\to \\{y \\in u(V) \\mid y \\mapsto x\\text{ in }u(U)\\}$", "as desired." ], "refs": [ "sites-lemma-point-site-topos", "sites-lemma-technical-pu", "sites-lemma-sheafification-exact" ], "ref_ids": [ 8596, 8542, 8515 ] } ], "ref_ids": [ 8605, 8605 ] }, { "id": 8607, "type": "theorem", "label": "sites-lemma-stalk-j-shriek", "categories": [ "sites" ], "title": "sites-lemma-stalk-j-shriek", "contents": [ "Let $\\mathcal{C}$ be a site. Let $p$ be a point of $\\mathcal{C}$ given by", "$u : \\mathcal{C} \\to \\textit{Sets}$. Let $U$ be an object of $\\mathcal{C}$.", "For any sheaf $\\mathcal{G}$ on $\\mathcal{C}/U$ we have", "$$", "(j_{U!}\\mathcal{G})_p =", "\\coprod\\nolimits_q \\mathcal{G}_q", "$$", "where the coproduct is over the points $q$ of $\\mathcal{C}/U$", "associated to elements $x \\in u(U)$ as in", "Lemma \\ref{lemma-point-localize}." ], "refs": [ "sites-lemma-point-localize" ], "proofs": [ { "contents": [ "We use the description of $j_{U!}\\mathcal{G}$ as the sheaf associated", "to the presheaf", "$V \\mapsto", "\\coprod\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{G}(V/_\\varphi U)$", "of", "Lemma \\ref{lemma-describe-j-shriek}.", "Also, the stalk of $j_{U!}\\mathcal{G}$ at $p$ is equal to the", "stalk of this presheaf, see", "Lemma \\ref{lemma-point-pushforward-sheaf}.", "Hence we see that", "$$", "(j_{U!}\\mathcal{G})_p =", "\\colim_{(V, y)} \\coprod\\nolimits_{\\varphi : V \\to U} \\mathcal{G}(V/_\\varphi U)", "$$", "To each element $(V, y, \\varphi, s)$ of this colimit, we can assign", "$x = u(\\varphi)(y) \\in u(U)$. Hence we obtain", "$$", "(j_{U!}\\mathcal{G})_p =", "\\coprod\\nolimits_{x \\in u(U)}", "\\colim_{(\\varphi : V \\to U, y), \\ u(\\varphi)(y) = x} \\mathcal{G}(V/_\\varphi U).", "$$", "This is equal to the expression of the lemma by our", "construction of the points $q$." ], "refs": [ "sites-lemma-describe-j-shriek", "sites-lemma-point-pushforward-sheaf" ], "ref_ids": [ 8553, 8595 ] } ], "ref_ids": [ 8605 ] }, { "id": 8608, "type": "theorem", "label": "sites-lemma-maps-u-points", "categories": [ "sites" ], "title": "sites-lemma-maps-u-points", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $u, u' : \\mathcal{C} \\to \\textit{Sets}$ be two", "functors, and let $t : u' \\to u$ be a transformation of functors.", "Then we obtain a canonical transformation of stalk", "functors $t_{stalk} : \\mathcal{F}_{p'} \\to \\mathcal{F}_p$", "which agrees with $t$ via the identifications of", "Lemma \\ref{lemma-points-recover}." ], "refs": [ "sites-lemma-points-recover" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8593 ] }, { "id": 8609, "type": "theorem", "label": "sites-lemma-exactness-stalks", "categories": [ "sites" ], "title": "sites-lemma-exactness-stalks", "contents": [ "Let $\\mathcal{C}$ be a site and let $\\{p_i\\}_{i\\in I}$ be a conservative", "family of points. Then", "\\begin{enumerate}", "\\item Given any map of sheaves $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "we have $\\forall i, \\varphi_{p_i}$ injective implies $\\varphi$ injective.", "\\item Given any map of sheaves $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "we have $\\forall i, \\varphi_{p_i}$ surjective implies $\\varphi$ surjective.", "\\item Given any pair of maps of sheaves", "$\\varphi_1, \\varphi_2 : \\mathcal{F} \\to \\mathcal{G}$", "we have $\\forall i, \\varphi_{1, p_i} = \\varphi_{2, p_i}$", "implies $\\varphi_1 = \\varphi_2$.", "\\item Given a finite diagram $\\mathcal{G} : \\mathcal{J}", "\\to \\Sh(\\mathcal{C})$, a sheaf $\\mathcal{F}$ and morphisms", "$q_j : \\mathcal{F} \\to \\mathcal{G}_j$ then $(\\mathcal{F}, q_j)$", "is a limit of the diagram if and only if for each $i$ the stalk", "$(\\mathcal{F}_{p_i}, (q_j)_{p_i})$ is one.", "\\item Given a finite diagram $\\mathcal{F} : \\mathcal{J}", "\\to \\Sh(\\mathcal{C})$, a sheaf $\\mathcal{G}$ and morphisms", "$e_j : \\mathcal{F}_j \\to \\mathcal{G}$ then $(\\mathcal{G}, e_j)$", "is a colimit of the diagram if and only if for each $i$ the stalk", "$(\\mathcal{G}_{p_i}, (e_j)_{p_i})$ is one.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use over and over again that all the stalk functors commute", "with any finite limits and colimits and hence with products, fibred", "products, etc. We will also use that injective maps are the monomorphisms", "and the surjective maps are the epimorphisms.", "A map of sheaves $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "is injective if and only if", "$\\mathcal{F} \\to \\mathcal{F} \\times_\\mathcal{G}\\mathcal{F}$", "is an isomorphism. Hence (1).", "Similarly, $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "is surjective if and only if", "$\\mathcal{G} \\amalg_\\mathcal{F} \\mathcal{G} \\to \\mathcal{G}$", "is an isomorphism. Hence (2).", "The maps $a, b : \\mathcal{F} \\to \\mathcal{G}$", "are equal if and only if $\\mathcal{F} \\times_{a, \\mathcal{G}, b}\\mathcal{F}", "\\to \\mathcal{F} \\times \\mathcal{F}$ is an isomorphism. Hence (3).", "The assertions (4) and (5) follow immediately from the definitions", "and the remarks at the start of this proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8610, "type": "theorem", "label": "sites-lemma-enough", "categories": [ "sites" ], "title": "sites-lemma-enough", "contents": [ "Let $\\mathcal{C}$ be a site and let $\\{(p_i, u_i)\\}_{i\\in I}$ be a", "family of points. The family is conservative if and only if for every", "sheaf $\\mathcal{F}$ and every $U\\in \\Ob(\\mathcal{C})$ and every", "pair of distinct sections $s, s' \\in \\mathcal{F}(U)$, $s \\not = s'$ there", "exists an $i$ and $x\\in u_i(U)$ such that the triples", "$(U, x, s)$ and $(U, x, s')$ define distinct elements of", "$\\mathcal{F}_{p_i}$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that the family is conservative and that $\\mathcal{F}$, $U$, and", "$s, s'$ are as in the lemma. The sections $s$, $s'$ define maps", "$a, a' : (h_U)^\\# \\to \\mathcal{F}$ which are distinct. Hence, by Lemma", "\\ref{lemma-exactness-stalks} there is an $i$ such that $a_{p_i}", "\\not = a'_{p_i}$. Recall that $(h_U)^\\#_{p_i} = u_i(U)$, by", "Lemmas \\ref{lemma-points-recover} and", "\\ref{lemma-point-pushforward-sheaf}.", "Hence there exists an $x \\in u_i(U)$ such that $a_{p_i}(x)", "\\not = a'_{p_i}(x)$ in $\\mathcal{F}_{p_i}$.", "Unwinding the definitions you see that $(U, x, s)$", "and $(U, x, s')$ are as in the statement of the lemma.", "\\medskip\\noindent", "To prove the converse, assume the condition on the existence of", "points of the lemma. Let $\\phi : \\mathcal{F} \\to \\mathcal{G}$", "be a map of sheaves which is an isomorphism at all the stalks.", "We have to show that $\\phi$ is both", "injective and surjective, see Lemma \\ref{lemma-mono-epi-sheaves}.", "Injectivity is an immediate consequence of the assumption.", "To show surjectivity we have to show that", "$\\mathcal{G} \\amalg_\\mathcal{F} \\mathcal{G} \\to \\mathcal{G}$", "is an isomorphism", "(Categories, Lemma \\ref{categories-lemma-characterize-mono-epi}).", "Since this map is clearly surjective, it suffices to check injectivity", "which follows as $\\mathcal{G} \\amalg_\\mathcal{F} \\mathcal{G} \\to \\mathcal{G}$", "is injective on all stalks by assumption." ], "refs": [ "sites-lemma-exactness-stalks", "sites-lemma-points-recover", "sites-lemma-point-pushforward-sheaf", "sites-lemma-mono-epi-sheaves", "categories-lemma-characterize-mono-epi" ], "ref_ids": [ 8609, 8593, 8595, 8517, 12209 ] } ], "ref_ids": [] }, { "id": 8611, "type": "theorem", "label": "sites-lemma-localize-enough", "categories": [ "sites" ], "title": "sites-lemma-localize-enough", "contents": [ "Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$.", "let $\\{(p_i, u_i)\\}_{i\\in I}$ be a family of points of $\\mathcal{C}$.", "For $x \\in u_i(U)$ let $q_{i, x}$ be the point of $\\mathcal{C}/U$", "constructed in", "Lemma \\ref{lemma-point-localize}.", "If $\\{p_i\\}$ is a conservative family of points, then", "$\\{q_{i, x}\\}_{i \\in I, x \\in u_i(U)}$ is a conservative", "family of points of $\\mathcal{C}/U$.", "In particular, if $\\mathcal{C}$ has enough points, then so", "does every localization $\\mathcal{C}/U$." ], "refs": [ "sites-lemma-point-localize" ], "proofs": [ { "contents": [ "We know that $j_{U!}$ induces an equivalence", "$j_{U!} : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})/h_U^\\#$, see", "Lemma \\ref{lemma-essential-image-j-shriek}.", "Moreover, we know that", "$(j_{U!}\\mathcal{G})_{p_i} = \\coprod_x \\mathcal{G}_{q_{i, x}}$, see", "Lemma \\ref{lemma-stalk-j-shriek}.", "Hence the result follows formally." ], "refs": [ "sites-lemma-essential-image-j-shriek", "sites-lemma-stalk-j-shriek" ], "ref_ids": [ 8555, 8607 ] } ], "ref_ids": [ 8605 ] }, { "id": 8612, "type": "theorem", "label": "sites-lemma-enough-points-local", "categories": [ "sites" ], "title": "sites-lemma-enough-points-local", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\{U_i\\}_{i \\in I}$ be a family of", "objects of $\\mathcal{C}$. Assume", "\\begin{enumerate}", "\\item $\\coprod h_{U_i}^\\# \\to *$ is a surjective map of sheaves, and", "\\item each localization $\\mathcal{C}/U_i$ has enough points.", "\\end{enumerate}", "Then $\\mathcal{C}$ has enough points." ], "refs": [], "proofs": [ { "contents": [ "For each $i \\in I$ let $\\{p_j\\}_{j \\in J_i}$ be a conservative", "family of points of $\\mathcal{C}/U_i$. For $j \\in J_i$ denote", "$q_j : \\Sh(pt) \\to \\Sh(\\mathcal{C})$ the composition", "of $p_j$ with the localization morphism", "$\\Sh(\\mathcal{C}/U_i) \\to \\Sh(\\mathcal{C})$.", "Then $q_j$ is a point, see", "Lemma \\ref{lemma-point-morphism-topoi}.", "We claim that the family of points $\\{q_j\\}_{j \\in \\coprod J_i}$", "is conservative.", "Namely, let $\\mathcal{F} \\to \\mathcal{G}$ be a map of sheaves", "on $\\mathcal{C}$ such that $\\mathcal{F}_{q_j} \\to \\mathcal{G}_{q_j}$", "is an isomorphism for all $j \\in \\coprod J_i$.", "Let $W$ be an object of $\\mathcal{C}$.", "By assumption (1) there exists a covering $\\{W_a \\to W\\}$ and", "morphisms $W_a \\to U_{i(a)}$.", "Since $(\\mathcal{F}|_{\\mathcal{C}/U_{i(a)}})_{p_j} = \\mathcal{F}_{q_j}$", "and $(\\mathcal{G}|_{\\mathcal{C}/U_{i(a)}})_{p_j} = \\mathcal{G}_{q_j}$ by", "Lemma \\ref{lemma-point-morphism-topoi}", "we see that", "$\\mathcal{F}|_{U_{i(a)}} \\to \\mathcal{G}|_{U_{i(a)}}$ is an isomorphism", "since the family of points $\\{p_j\\}_{j \\in J_{i(a)}}$ is conservative.", "Hence $\\mathcal{F}(W_a) \\to \\mathcal{G}(W_a)$ is bijective for each $a$.", "Similarly $\\mathcal{F}(W_a \\times_W W_b) \\to \\mathcal{G}(W_a \\times_W W_b)$", "is bijective for each $a, b$.", "By the sheaf condition this shows that", "$\\mathcal{F}(W) \\to \\mathcal{G}(W)$ is bijective, i.e.,", "$\\mathcal{F} \\to \\mathcal{G}$ is an isomorphism." ], "refs": [ "sites-lemma-point-morphism-topoi", "sites-lemma-point-morphism-topoi" ], "ref_ids": [ 8604, 8604 ] } ], "ref_ids": [] }, { "id": 8613, "type": "theorem", "label": "sites-lemma-check-morphism-sites", "categories": [ "sites" ], "title": "sites-lemma-check-morphism-sites", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor of sites.", "Let $\\{(q_i, v_i)\\}_{i\\in I}$ be a conservative family of points of", "$\\mathcal{D}$. If each functor $u_i = v_i \\circ u$ defines", "a point of $\\mathcal{C}$,", "then $u$ defines a morphism of sites $f : \\mathcal{D} \\to \\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Denote $p_i$ the stalk functor (\\ref{equation-stalk}) on", "$\\textit{PSh}(\\mathcal{C})$ corresponding to the functor $u_i$. We have", "$$", "(f^{-1}\\mathcal{F})_{q_i} =", "(u_s\\mathcal{F})_{q_i} =", "(u_p\\mathcal{F})_{q_i} =", "\\mathcal{F}_{p_i}", "$$", "The first equality since $f^{-1} = u_s$, the second equality", "by Lemma \\ref{lemma-point-pushforward-sheaf}, and the third", "by Lemma \\ref{lemma-point-functor}.", "Hence if $p_i$ is a point, then pulling back by $f$", "and then taking stalks at $q_i$ is an exact functor.", "Since the family of points $\\{q_i\\}$ is conservative, this", "implies that $f^{-1}$ is an exact functor and we see", "that $f$ is a morphism of sites by Definition \\ref{definition-morphism-sites}." ], "refs": [ "sites-lemma-point-pushforward-sheaf", "sites-lemma-point-functor", "sites-definition-morphism-sites" ], "ref_ids": [ 8595, 8602, 8665 ] } ], "ref_ids": [] }, { "id": 8614, "type": "theorem", "label": "sites-lemma-refine", "categories": [ "sites" ], "title": "sites-lemma-refine", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $(J, \\geq, V_j, g_{jj'})$ be a system as above with associated", "pair of functors $(u', p')$.", "Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.", "Let $s, s' \\in \\mathcal{F}_{p'}$ be distinct elements.", "Let $\\{W_k \\to W\\}$ be a finite covering of $\\mathcal{C}$.", "Let $f \\in u'(W)$.", "There exists a refinement $(I, \\geq, U_i, f_{ii'})$", "of $(J, \\geq, V_j, g_{jj'})$ such that $s, s'$ map", "to distinct elements of $\\mathcal{F}_p$ and that", "the image of $f$ in $u(W)$ is in the image of one of", "the $u(W_k)$." ], "refs": [], "proofs": [ { "contents": [ "There exists a $j_0 \\in J$ such that $f$ is defined by $f' : V_{j_0} \\to W$.", "For $j \\geq j_0$ we set $V_{j, k} = V_j \\times_{f'\\circ f_{j j_0}, W} W_k$.", "Then $\\{V_{j, k} \\to V_j\\}$ is a finite covering in the site", "$\\mathcal{C}$. Hence", "$\\mathcal{F}(V_j) \\subset \\prod_k \\mathcal{F}(V_{j, k})$.", "By Categories, Lemma \\ref{categories-lemma-directed-commutes}", "once again we see that", "$$", "\\mathcal{F}_{p'} =", "\\colim_j \\mathcal{F}(V_j)", "\\longrightarrow", "\\prod\\nolimits_k \\colim_j \\mathcal{F}(V_{j, k})", "$$", "is injective. Hence there exists a $k$ such that $s$ and $s'$", "have distinct image in $\\colim_j \\mathcal{F}(V_{j, k})$.", "Let $J_0 = \\{j \\in J, j \\geq j_0\\}$ and $I = J \\amalg J_0$.", "We order $I$ so that no element of the second summand", "is smaller than any element of the first, but otherwise", "using the ordering on $J$. If $j \\in I$ is in the first", "summand then we use $V_j$ and if $j \\in I$ is in the second", "summand then we use $V_{j, k}$. We omit the definition", "of the transition maps of the inverse system. By the above", "it follows that $s, s'$ have distinct image in $\\mathcal{F}_p$.", "Moreover, the restriction of $f'$ to $V_{j, k}$ factors", "through $W_k$ by construction." ], "refs": [ "categories-lemma-directed-commutes" ], "ref_ids": [ 12228 ] } ], "ref_ids": [] }, { "id": 8615, "type": "theorem", "label": "sites-lemma-refine-all-at-once", "categories": [ "sites" ], "title": "sites-lemma-refine-all-at-once", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $(J, \\geq, V_j, g_{jj'})$ be a system as above with associated", "pair of functors $(u', p')$.", "Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.", "Let $s, s' \\in \\mathcal{F}_{p'}$ be distinct elements.", "There exists a refinement $(I, \\geq, U_i, f_{ii'})$", "of $(J, \\geq, V_j, g_{jj'})$ such that $s, s'$ map", "to distinct elements of $\\mathcal{F}_p$ and such that", "for every finite covering $\\{W_k \\to W\\}$ of the site", "$\\mathcal{C}$, and any $f \\in u'(W)$ the image of $f$ in $u(W)$", "is in the image of one of the $u(W_k)$." ], "refs": [], "proofs": [ { "contents": [ "Let $E$ be the set of pairs $(\\{W_k \\to W\\}, f\\in u'(W))$.", "Consider pairs $(E' \\subset E, (I, \\geq, U_i, f_{ii'}))$", "such that", "\\begin{enumerate}", "\\item $(I, \\geq, U_i, g_{ii'})$ is a refinement of $(J, \\geq, V_j, g_{jj'})$,", "\\item $s, s'$ map to distinct elements of $\\mathcal{F}_p$, and", "\\item for every pair $(\\{W_k \\to W\\}, f\\in u'(W)) \\in E'$ we have that", "the image of $f$ in $u(W)$ is in the image of one of the $u(W_k)$.", "\\end{enumerate}", "We order such pairs by inclusion in the first factor and", "by refinement in the second. Denote $\\mathcal{S}$ the class", "of all pairs $(E' \\subset E, (I, \\geq, U_i, f_{ii'}))$ as above.", "We claim that the hypothesis of Zorn's lemma holds for $\\mathcal{S}$. Namely,", "suppose that $(E'_a, (I_a, \\geq, U_i, f_{ii'}))_{a \\in A}$", "is a totally ordered subset of $\\mathcal{S}$. Then we can define", "$E' = \\bigcup_{a \\in A} E'_a$ and we can set $I = \\bigcup_{a \\in A} I_a$.", "We claim that the corresponding pair", "$(E' , (I, \\geq, U_i, f_{ii'}))$ is an element of $\\mathcal{S}$.", "Conditions (1) and (3) are clear. For condition (2) you note", "that", "$$", "u = \\colim_{a \\in A} u_a", "\\text{ and correspondingly }", "\\mathcal{F}_p = \\colim_{a \\in A} \\mathcal{F}_{p_a}", "$$", "The distinctness of the images of $s, s'$ in this stalk follows", "from the description of a directed colimit of sets, see", "Categories, Section \\ref{categories-section-directed-colimits}.", "We will simply write", "$(E', (I, \\ldots)) = \\bigcup_{a \\in A}(E'_a, (I_a, \\ldots))$", "in this situation.", "\\medskip\\noindent", "OK, so Zorn's Lemma would apply if $\\mathcal{S}$ was a set,", "and this would, combined with Lemma \\ref{lemma-refine} above easily prove", "the lemma. It doesn't since $\\mathcal{S}$ is a class. In order", "to circumvent this we choose a well ordering on $E$.", "For $e \\in E$ set $E'_e = \\{e' \\in E \\mid e' \\leq e\\}$.", "By transfinite induction we construct pairs", "$(E'_e, (I_e, \\ldots)) \\in \\mathcal{S}$ such that", "$e_1 \\leq e_2 \\Rightarrow (E'_{e_1}, (I_{e_1}, \\ldots))", "\\leq (E'_{e_2}, (I_{e_2}, \\ldots))$.", "Let $e \\in E$, say $e = (\\{W_k \\to W\\}, f\\in u'(W))$.", "If $e$ has a predecessor $e - 1$, then we let", "$(I_e, \\ldots)$ be a refinement of $(I_{e - 1}, \\ldots)$", "as in Lemma \\ref{lemma-refine} with respect to the system", "$e = (\\{W_k \\to W\\}, f\\in u'(W))$.", "If $e$ does not have a predecessor, then we let", "$(I_e, \\ldots)$ be a refinement of $\\bigcup_{e' < e} (I_{e'}, \\ldots)$", "with respect to the system", "$e = (\\{W_k \\to W\\}, f\\in u'(W))$.", "Finally, the union $\\bigcup_{e \\in E} I_e$ will be a solution to", "the problem posed in the lemma." ], "refs": [ "sites-lemma-refine", "sites-lemma-refine" ], "ref_ids": [ 8614, 8614 ] } ], "ref_ids": [] }, { "id": 8616, "type": "theorem", "label": "sites-lemma-criterion-points", "categories": [ "sites" ], "title": "sites-lemma-criterion-points", "contents": [ "Let $\\mathcal{C}$ be a site. Let $I$ be a set and for", "$i \\in I$ let $U_i$ be an object of $\\mathcal{C}$ such that", "\\begin{enumerate}", "\\item $\\coprod h_{U_i}$ surjects onto", "the final object of $\\Sh(\\mathcal{C})$, and", "\\item $\\mathcal{C}/U_i$ satisfies the hypotheses of", "Proposition \\ref{proposition-criterion-points}.", "\\end{enumerate}", "Then $\\mathcal{C}$ has enough points." ], "refs": [ "sites-proposition-criterion-points" ], "proofs": [ { "contents": [ "By assumption (2) and the proposition $\\mathcal{C}/U_i$ has enough points.", "The points of $\\mathcal{C}/U_i$ give points of $\\mathcal{C}$", "via the procedure of Lemma \\ref{lemma-point-morphism-sites}.", "Thus it suffices to show: if $\\phi : \\mathcal{F} \\to \\mathcal{G}$", "is a map of sheaves on $\\mathcal{C}$ such that $\\phi|_{\\mathcal{C}/U_i}$", "is an isomorphism for all $i$, then $\\phi$ is an isomorphism.", "By assumption (1) for every object $W$ of $\\mathcal{C}$", "there is a covering $\\{W_j \\to W\\}_{j \\in J}$", "such that for $j \\in J$ there is an $i \\in I$ and a morphism", "$f_j : W_j \\to U_i$. Then the maps", "$\\mathcal{F}(W_j) \\to \\mathcal{G}(W_j)$", "are bijective and similarly for", "$\\mathcal{F}(W_j \\times_W W_{j'}) \\to \\mathcal{G}(W_j \\times_W W_{j'})$.", "The sheaf condition tells us that $\\mathcal{F}(W) \\to \\mathcal{G}(W)$", "is bijective as desired." ], "refs": [ "sites-lemma-point-morphism-sites" ], "ref_ids": [ 8603 ] } ], "ref_ids": [ 8643 ] }, { "id": 8617, "type": "theorem", "label": "sites-lemma-w-contractible", "categories": [ "sites" ], "title": "sites-lemma-w-contractible", "contents": [ "Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$.", "The following conditions are equivalent", "\\begin{enumerate}", "\\item For every covering $\\{U_i \\to U\\}$ there exists a map of", "sheaves $h_U^\\# \\to \\coprod h_{U_i}^\\#$ right inverse to the sheafification", "of $\\coprod h_{U_i} \\to h_U$.", "\\item For every surjection of sheaves of sets $\\mathcal{F} \\to \\mathcal{G}$", "the map $\\mathcal{F}(U) \\to \\mathcal{G}(U)$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and let $\\mathcal{F} \\to \\mathcal{G}$ be a surjective map of", "sheaves of sets. For $s \\in \\mathcal{G}(U)$ there exists a covering", "$\\{U_i \\to U\\}$ and $t_i \\in \\mathcal{F}(U_i)$ mapping to", "$s|_{U_i}$, see Definition \\ref{definition-sheaves-injective-surjective}.", "Think of $t_i$ as a map $t_i : h_{U_i}^\\# \\to \\mathcal{F}$ via", "(\\ref{equation-map-representable-into-sheaf}).", "Then precomposing $\\coprod t_i : \\coprod h_{U_i}^\\# \\to \\mathcal{F}$", "with the map $h_U^\\# \\to \\coprod h_{U_i}^\\#$ we get from (1)", "we obtain a section $t \\in \\mathcal{F}(U)$ mapping to $s$.", "Thus (2) holds.", "\\medskip\\noindent", "Assume (2) holds. Let $\\{U_i \\to U\\}$ be a covering.", "Then $\\coprod h_{U_i}^\\# \\to h_U^\\#$ is surjective", "(Lemma \\ref{lemma-covering-surjective-after-sheafification}).", "Hence by (2) there exists a section $s$ of $\\coprod h_{U_i}^\\#$ mapping", "to the section $\\text{id}_U$ of $h_U^\\#$. This section corresponds to a map", "$h_U^\\# \\to \\coprod h_{U_i}^\\#$ which is right inverse to the sheafification", "of $\\coprod h_{U_i} \\to h_U$ which proves (1)." ], "refs": [ "sites-definition-sheaves-injective-surjective", "sites-lemma-covering-surjective-after-sheafification" ], "ref_ids": [ 8660, 8519 ] } ], "ref_ids": [] }, { "id": 8618, "type": "theorem", "label": "sites-lemma-exactness-properties", "categories": [ "sites" ], "title": "sites-lemma-exactness-properties", "contents": [ "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be", "a morphism of topoi. Consider the following properties (on sheaves", "of sets):", "\\begin{enumerate}", "\\item $f_*$ is faithful,", "\\item $f_*$ is fully faithful,", "\\item $f^{-1}f_*\\mathcal{F} \\to \\mathcal{F}$ is surjective for", "all $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$,", "\\item $f_*$ transforms surjections into surjections,", "\\item $f_*$ commutes with coequalizers,", "\\item $f_*$ commutes with pushouts,", "\\item $f^{-1}f_*\\mathcal{F} \\to \\mathcal{F}$ is an isomorphism for", "all $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$,", "\\item $f_*$ reflects injections,", "\\item $f_*$ reflects surjections,", "\\item $f_*$ reflects bijections, and", "\\item for any surjection $\\mathcal{F} \\to f^{-1}\\mathcal{G}$ there", "exists a surjection $\\mathcal{G}' \\to \\mathcal{G}$ such that", "$f^{-1}\\mathcal{G}' \\to f^{-1}\\mathcal{G}$ factors through", "$\\mathcal{F} \\to f^{-1}\\mathcal{G}$.", "\\end{enumerate}", "Then we have the following implications", "\\begin{enumerate}", "\\item[(a)] (2) $\\Rightarrow$ (1),", "\\item[(b)] (3) $\\Rightarrow$ (1),", "\\item[(c)] (7) $\\Rightarrow$ (1), (2), (3), (8), (9), (10).", "\\item[(d)] (3) $\\Leftrightarrow$ (9),", "\\item[(e)] (6) $\\Rightarrow$ (4) and (5) $\\Rightarrow$ (4),", "\\item[(f)] (4) $\\Leftrightarrow$ (11),", "\\item[(g)] (9) $\\Rightarrow$ (8), (10), and", "\\item[(h)] (2) $\\Leftrightarrow$ (7).", "\\end{enumerate}", "Picture", "$$", "\\xymatrix{", "(6) \\ar@{=>}[rd] & & & & & (9) \\ar@{=>}[r] \\ar@{=>}[rd] & (8) \\\\", "& (4) \\ar@{<=>}[r] & (11) &", "(2) \\ar@{<=>}[r] &", "(7) \\ar@{=>}[ru] \\ar@{=>}[rd] & & (10) \\\\", "(5) \\ar@{=>}[ur] & & & & & (3) \\ar@{=>}[r] & (1)", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Proof of (a): This is immediate from the definitions.", "\\medskip\\noindent", "Proof of (b). Suppose that $a, b : \\mathcal{F} \\to \\mathcal{F}'$ are", "maps of sheaves on $\\mathcal{C}$. If $f_*a = f_*b$, then", "$f^{-1}f_*a = f^{-1}f_*b$. Consider the commutative diagram", "$$", "\\xymatrix{", "\\mathcal{F} \\ar@<-1ex>[r] \\ar@<1ex>[r] & \\mathcal{F}' \\\\", "f^{-1}f_*\\mathcal{F} \\ar@<-1ex>[r] \\ar@<1ex>[r] \\ar[u] &", "f^{-1}f_*\\mathcal{F}' \\ar[u]", "}", "$$", "If the bottom two arrows are equal and the vertical arrows are surjective", "then the top two arrows are equal. Hence (b) follows.", "\\medskip\\noindent", "Proof of (c). Suppose that $a : \\mathcal{F} \\to \\mathcal{F}'$ is a", "map of sheaves on $\\mathcal{C}$. Consider the commutative diagram", "$$", "\\xymatrix{", "\\mathcal{F} \\ar[r] & \\mathcal{F}' \\\\", "f^{-1}f_*\\mathcal{F} \\ar[r] \\ar[u] &", "f^{-1}f_*\\mathcal{F}' \\ar[u]", "}", "$$", "If (7) holds, then the vertical arrows are isomorphisms.", "Hence if $f_*a$ is injective (resp.\\ surjective, resp.\\ bijective)", "then the bottom arrow is injective (resp.\\ surjective, resp.\\ bijective) and", "hence the top arrow is injective (resp.\\ surjective, resp.\\ bijective).", "Thus we see that (7) implies (8), (9), (10). It is clear that (7) implies (3).", "The implications (7) $\\Rightarrow$ (2), (1) follow from (a) and (h) which", "we will see below.", "\\medskip\\noindent", "Proof of (d). Assume (3). Suppose that $a : \\mathcal{F} \\to \\mathcal{F}'$", "is a map of sheaves on $\\mathcal{C}$ such that $f_*a$ is surjective.", "As $f^{-1}$ is exact this implies that", "$f^{-1}f_*a : f^{-1}f_*\\mathcal{F} \\to f^{-1}f_*\\mathcal{F}'$", "is surjective. Combined with (3) this implies that $a$ is surjective.", "This means that (9) holds.", "Assume (9). Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.", "We have to show that the map $f^{-1}f_*\\mathcal{F} \\to \\mathcal{F}$ is", "surjective. It suffices to show that", "$f_*f^{-1}f_*\\mathcal{F} \\to f_*\\mathcal{F}$ is surjective.", "And this is true because there is a canonical map", "$f_*\\mathcal{F} \\to f_*f^{-1}f_*\\mathcal{F}$ which is a one-sided inverse.", "\\medskip\\noindent", "Proof of (e). We use", "Categories, Lemma \\ref{categories-lemma-characterize-mono-epi}", "without further mention.", "If $\\mathcal{F} \\to \\mathcal{F}'$ is surjective then", "$\\mathcal{F}' \\amalg_\\mathcal{F} \\mathcal{F}' \\to \\mathcal{F}'$", "is an isomorphism. Hence (6) implies that", "$$", "f_*\\mathcal{F}' \\amalg_{f_*\\mathcal{F}} f_*\\mathcal{F}' =", "f_*(\\mathcal{F}' \\amalg_\\mathcal{F} \\mathcal{F}')", "\\longrightarrow", "f_*\\mathcal{F}'", "$$", "is an isomorphism also. And this in turn implies that", "$f_*\\mathcal{F} \\to f_*\\mathcal{F}'$ is surjective.", "Hence we see that (6) implies (4). If $\\mathcal{F} \\to \\mathcal{F}'$", "is surjective then $\\mathcal{F}'$ is the coequalizer of the two", "projections $\\mathcal{F} \\times_{\\mathcal{F}'} \\mathcal{F} \\to \\mathcal{F}$", "by Lemma \\ref{lemma-coequalizer-surjection}.", "Hence if (5) holds, then $f_*\\mathcal{F}'$ is the coequalizer of", "the two projections", "$$", "f_*(\\mathcal{F} \\times_{\\mathcal{F}'} \\mathcal{F}) =", "f_*\\mathcal{F} \\times_{f_*\\mathcal{F}'} f_*\\mathcal{F}", "\\longrightarrow", "f_*\\mathcal{F}", "$$", "which clearly means that $f_*\\mathcal{F} \\to f_*\\mathcal{F}'$ is", "surjective. Hence (5) implies (4) as well.", "\\medskip\\noindent", "Proof of (f). Assume (4). Let $\\mathcal{F} \\to f^{-1}\\mathcal{G}$ be", "a surjective map of sheaves on $\\mathcal{C}$. By (4) we see that", "$f_*\\mathcal{F} \\to f_*f^{-1}\\mathcal{G}$ is surjective. Let", "$\\mathcal{G}'$ be the fibre product", "$$", "\\xymatrix{", "f_*\\mathcal{F} \\ar[r] & f_*f^{-1}\\mathcal{G} \\\\", "\\mathcal{G}' \\ar[u] \\ar[r] & \\mathcal{G} \\ar[u]", "}", "$$", "so that $\\mathcal{G}' \\to \\mathcal{G}$ is surjective also. Consider", "the commutative diagram", "$$", "\\xymatrix{", "\\mathcal{F} \\ar[r] & f^{-1}\\mathcal{G} \\\\", "f^{-1}f_*\\mathcal{F} \\ar[r] \\ar[u] & f^{-1}f_*f^{-1}\\mathcal{G} \\ar[u] \\\\", "f^{-1}\\mathcal{G}' \\ar[u] \\ar[r] & f^{-1}\\mathcal{G} \\ar[u]", "}", "$$", "and we see the required result. Conversely, assume (11).", "Let $a : \\mathcal{F} \\to \\mathcal{F}'$ be surjective map of sheaves", "on $\\mathcal{C}$. Consider the fibre product diagram", "$$", "\\xymatrix{", "\\mathcal{F} \\ar[r] & \\mathcal{F}' \\\\", "\\mathcal{F}'' \\ar[u] \\ar[r] & f^{-1}f_*\\mathcal{F}' \\ar[u]", "}", "$$", "Because the lower horizontal arrow is surjective and by (11) we can", "find a surjection $\\gamma : \\mathcal{G}' \\to f_*\\mathcal{F}'$ such", "that $f^{-1}\\gamma$ factors through $\\mathcal{F}'' \\to f^{-1}f_*\\mathcal{F}'$:", "$$", "\\xymatrix{", "& \\mathcal{F} \\ar[r] & \\mathcal{F}' \\\\", "f^{-1}\\mathcal{G}' \\ar[r] & \\mathcal{F}'' \\ar[u] \\ar[r] &", "f^{-1}f_*\\mathcal{F}' \\ar[u]", "}", "$$", "Pushing this down using $f_*$ we get a commutative diagram", "$$", "\\xymatrix{", "& f_*\\mathcal{F} \\ar[r] & f_*\\mathcal{F}' \\\\", "f_*f^{-1}\\mathcal{G}' \\ar[r] & f_*\\mathcal{F}'' \\ar[u] \\ar[r] &", "f_*f^{-1}f_*\\mathcal{F}' \\ar[u] \\\\", "\\mathcal{G}' \\ar[u] \\ar[rr] & & f_*\\mathcal{F}' \\ar[u]", "}", "$$", "which proves that (4) holds.", "\\medskip\\noindent", "Proof of (g). Assume (9). We use", "Categories, Lemma \\ref{categories-lemma-characterize-mono-epi}", "without further mention. Let $a : \\mathcal{F} \\to \\mathcal{F}'$ be a map", "of sheaves on $\\mathcal{C}$ such that $f_*a$ is injective. This means that", "$f_*\\mathcal{F} \\to f_*\\mathcal{F} \\times_{f_*\\mathcal{F}'} f_*\\mathcal{F} =", "f_*(\\mathcal{F} \\times_{\\mathcal{F}'} \\mathcal{F})$ is an isomorphism.", "Thus by (9) we see that", "$\\mathcal{F} \\to \\mathcal{F} \\times_{\\mathcal{F}'} \\mathcal{F}$ is", "surjective, i.e., an isomorphism. Thus $a$ is injective, i.e., (8) holds.", "Since (10) is trivially equivalent to (8) $+$ (9) we are done with (g).", "\\medskip\\noindent", "Proof of (h). This is", "Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}." ], "refs": [ "categories-lemma-characterize-mono-epi", "sites-lemma-coequalizer-surjection", "categories-lemma-characterize-mono-epi", "categories-lemma-adjoint-fully-faithful" ], "ref_ids": [ 12209, 8518, 12209, 12248 ] } ], "ref_ids": [] }, { "id": 8619, "type": "theorem", "label": "sites-lemma-weaker", "categories": [ "sites" ], "title": "sites-lemma-weaker", "contents": [ "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites associated to", "the continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$.", "Assume that for any object $U$ of $\\mathcal{C}$ and any covering", "$\\{V_j \\to u(U)\\}$ in $\\mathcal{D}$ there exists a covering $\\{U_i \\to U\\}$", "in $\\mathcal{C}$ such that the map of sheaves", "$$", "\\coprod h_{u(U_i)}^\\# \\to h_{u(U)}^\\#", "$$", "factors through the map of sheaves", "$$", "\\coprod h_{V_j}^\\# \\to h_{u(U)}^\\#.", "$$", "Then $f_*$ transforms surjective maps of sheaves into surjective maps of", "sheaves." ], "refs": [], "proofs": [ { "contents": [ "Let $a : \\mathcal{F} \\to \\mathcal{G}$ be a surjective map of sheaves on", "$\\mathcal{D}$. Let $U$ be an object of $\\mathcal{C}$ and let", "$s \\in f_*\\mathcal{G}(U) = \\mathcal{G}(u(U))$. By assumption there exists", "a covering $\\{V_j \\to u(U)\\}$ and sections $s_j \\in \\mathcal{F}(V_j)$", "with $a(s_j) = s|_{V_j}$. Now we may think of the sections $s$,", "$s_j$ and $a$ as giving a commutative diagram of maps of sheaves", "$$", "\\xymatrix{", "\\coprod h_{V_j}^\\# \\ar[r]_-{\\coprod s_j} \\ar[d] & \\mathcal{F} \\ar[d]^a \\\\", "h_{u(U)}^\\# \\ar[r]^s & \\mathcal{G}", "}", "$$", "By assumption there exists a covering $\\{U_i \\to U\\}$ such that we can", "enlarge the commutative diagram above as follows", "$$", "\\xymatrix{", "& \\coprod h_{V_j}^\\# \\ar[r]_-{\\coprod s_j} \\ar[d] & \\mathcal{F} \\ar[d]^a \\\\", "\\coprod h_{u(U_i)}^\\# \\ar[r] \\ar[ur] &", "h_{u(U)}^\\# \\ar[r]^s & \\mathcal{G}", "}", "$$", "Because $\\mathcal{F}$ is a sheaf the map from the left lower corner to", "the right upper corner corresponds to a family of sections", "$s_i \\in \\mathcal{F}(u(U_i))$, i.e., sections $s_i \\in f_*\\mathcal{F}(U_i)$.", "The commutativity of the diagram implies that $a(s_i)$ is equal to the", "restriction of $s$ to $U_i$. In other words we have shown that $f_*a$ is a", "surjective map of sheaves." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8620, "type": "theorem", "label": "sites-lemma-cover-from-below", "categories": [ "sites" ], "title": "sites-lemma-cover-from-below", "contents": [ "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites given", "by the functor $u : \\mathcal{C} \\to \\mathcal{D}$.", "Assume that for every object $V$ of $\\mathcal{D}$ there exist objects", "$U_i$ of $\\mathcal{C}$ and morphisms $u(U_i) \\to V$ such that", "$\\{u(U_i) \\to V\\}$ is a covering of $\\mathcal{D}$. In this case the functor", "$f_* : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ reflects", "injections and surjections." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha : \\mathcal{F} \\to \\mathcal{G}$ be maps of sheaves", "on $\\mathcal{D}$. By assumption for every object $V$ of $\\mathcal{D}$", "we get $\\mathcal{F}(V) \\subset \\prod \\mathcal{F}(u(U_i)) =", "\\prod f_*\\mathcal{F}(U_i)$ by the sheaf condition", "for some $U_i \\in \\Ob(\\mathcal{C})$ and similarly for $\\mathcal{G}$.", "Hence it is clear that if $f_*\\alpha$ is injective, then", "$\\alpha$ is injective. In other words $f_*$ reflects injections.", "\\medskip\\noindent", "Suppose that $f_*\\alpha$ is surjective. Then for $V, U_i, u(U_i) \\to V$", "as above and a section $s \\in \\mathcal{G}(V)$, there exist coverings", "$\\{U_{ij} \\to U_i\\}$ such that $s|_{u(U_{ij})}$ is in the image", "of $\\mathcal{F}(u(U_{ij}))$. Since $\\{u(U_{ij}) \\to V\\}$ is a covering", "(as $u$ is continuous and by the axioms of a site) we conclude that", "$s$ is locally in the image. Thus $\\alpha$ is surjective. In other", "words $f_*$ reflects surjections." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8621, "type": "theorem", "label": "sites-lemma-characterize-empty", "categories": [ "sites" ], "title": "sites-lemma-characterize-empty", "contents": [ "Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $U$ is sheaf theoretically empty,", "\\item $\\mathcal{F}(U)$ is a singleton for each sheaf $\\mathcal{F}$,", "\\item $\\emptyset^\\#(U)$ is a singleton,", "\\item $\\emptyset^\\#(U)$ is nonempty, and", "\\item the empty family is a covering of $U$ in $\\mathcal{C}$.", "\\end{enumerate}", "Moreover, if $U$ is sheaf theoretically empty, then for any morphism", "$U' \\to U$ of $\\mathcal{C}$ the object $U'$ is sheaf theoretically empty." ], "refs": [], "proofs": [ { "contents": [ "For any sheaf $\\mathcal{F}$ we have", "$\\mathcal{F}(U) = \\Mor_{\\Sh(\\mathcal{C})}(h_U^\\#, \\mathcal{F})$.", "Hence, we see that (1) and (2) are equivalent.", "It is clear that (2) implies (3) implies (4).", "If every covering of $U$ is given by a nonempty family,", "then $\\emptyset^+(U)$ is empty by definition of the plus construction.", "Note that $\\emptyset^+ = \\emptyset^\\#$ as $\\emptyset$ is a separated", "presheaf, see", "Theorem \\ref{theorem-plus}.", "Thus we see that (4) implies (5). If (5) holds, then", "$\\mathcal{F}(U)$ is a singleton for every sheaf $\\mathcal{F}$", "by the sheaf condition for $\\mathcal{F}$, see", "Remark \\ref{remark-sheaf-condition-empty-covering}.", "Thus (5) implies (2) and (1) -- (5) are equivalent. The final", "assertion of the lemma follows from Axiom (3) of", "Definition \\ref{definition-site}", "applied the empty covering of $U$." ], "refs": [ "sites-theorem-plus", "sites-remark-sheaf-condition-empty-covering", "sites-definition-site" ], "ref_ids": [ 8492, 8704, 8652 ] } ], "ref_ids": [] }, { "id": 8622, "type": "theorem", "label": "sites-lemma-almost-cocontinuous-sheafification", "categories": [ "sites" ], "title": "sites-lemma-almost-cocontinuous-sheafification", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "Assume that $u$ is continuous and almost cocontinuous.", "Let $\\mathcal{G}$ be a presheaf on $\\mathcal{D}$ such that $\\mathcal{G}(V)$ is", "a singleton whenever $V$ is sheaf theoretically empty.", "Then $(u^p\\mathcal{G})^\\# = u^p(\\mathcal{G}^\\#)$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\in \\Ob(\\mathcal{C})$. We have to show that", "$(u^p\\mathcal{G})^\\#(U) = u^p(\\mathcal{G}^\\#)(U)$.", "It suffices to show that", "$(u^p\\mathcal{G})^+(U) = u^p(\\mathcal{G}^+)(U)$", "since $\\mathcal{G}^+$ is another presheaf for which the", "assumption of the lemma holds. We have", "$$", "u^p(\\mathcal{G}^+)(U) =", "\\mathcal{G}^+(u(U)) =", "\\colim_\\mathcal{V} \\check H^0(\\mathcal{V}, \\mathcal{G})", "$$", "where the colimit is over the coverings $\\mathcal{V}$ of $u(U)$ in", "$\\mathcal{D}$. On the other hand, we see that", "$$", "u^p(\\mathcal{G})^+(U) =", "\\colim_\\mathcal{U} \\check H^0(u(\\mathcal{U}), \\mathcal{G})", "$$", "where the colimit is over the category of coverings", "$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ of $U$ in $\\mathcal{C}$ and", "$u(\\mathcal{U}) = \\{u(U_i) \\to u(U)\\}_{i \\in I}$. The condition", "that $u$ is continuous means that each $u(\\mathcal{U})$ is a covering.", "Write $I = I_1 \\amalg I_2$, where", "$$", "I_2 = \\{i \\in I \\mid u(U_i)\\text{ is sheaf theoretically empty}\\}", "$$", "Then $u(\\mathcal{U})' = \\{u(U_i) \\to u(U)\\}_{i \\in I_1}$ is still a covering of", "because each of the other pieces can be covered by the empty family", "and hence can be dropped by Axiom (2) of", "Definition \\ref{definition-site}.", "Moreover,", "$\\check H^0(u(\\mathcal{U}), \\mathcal{G}) =", "\\check H^0(u(\\mathcal{U})', \\mathcal{G})$", "by our assumption on $\\mathcal{G}$. Finally, the condition that $u$ is", "almost cocontinuous implies that for every covering $\\mathcal{V}$", "of $u(U)$ there exists a covering $\\mathcal{U}$ of $U$ such that", "$u(\\mathcal{U})'$ refines $\\mathcal{V}$. It follows that the two colimits", "displayed above have the same value as desired." ], "refs": [ "sites-definition-site" ], "ref_ids": [ 8652 ] } ], "ref_ids": [] }, { "id": 8623, "type": "theorem", "label": "sites-lemma-continuous-almost-cocontinuous", "categories": [ "sites" ], "title": "sites-lemma-continuous-almost-cocontinuous", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "Assume that $u$ is continuous and almost cocontinuous.", "Then $u^s = u^p : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$", "commutes with pushouts and coequalizers (and more generally", "finite connected colimits)." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I}$ be a finite connected index category.", "Let $\\mathcal{I} \\to \\Sh(\\mathcal{D})$,", "$i \\mapsto \\mathcal{G}_i$ by a diagram. We know that the colimit of", "this diagram is the sheafification of the colimit in the category of", "presheaves, see", "Lemma \\ref{lemma-colimit-sheaves}.", "Denote $\\colim^{Psh}$ the colimit in the category", "of presheaves. Since $\\mathcal{I}$ is finite and connected", "we see that $\\colim^{Psh}_i \\mathcal{G}_i$", "is a presheaf satisfying the assumptions of", "Lemma \\ref{lemma-almost-cocontinuous-sheafification}", "(because a finite connected colimit of singleton sets is a", "singleton). Hence that lemma gives", "\\begin{align*}", "u^s(\\colim_i \\mathcal{G}_i) & =", "u^s((\\colim^{Psh}_i \\mathcal{G}_i)^\\#) \\\\", "& = (u^p(\\colim^{Psh}_i \\mathcal{G}_i))^\\# \\\\", "& = (\\colim^{PSh}_i u^p(\\mathcal{G}_i))^\\# \\\\", "& = \\colim_i u^s(\\mathcal{G}_i)", "\\end{align*}", "as desired." ], "refs": [ "sites-lemma-colimit-sheaves", "sites-lemma-almost-cocontinuous-sheafification" ], "ref_ids": [ 8514, 8622 ] } ], "ref_ids": [] }, { "id": 8624, "type": "theorem", "label": "sites-lemma-morphism-of-sites-almost-cocontinuous", "categories": [ "sites" ], "title": "sites-lemma-morphism-of-sites-almost-cocontinuous", "contents": [ "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites", "associated to the continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$.", "If $u$ is almost cocontinuous then $f_*$ commutes with", "pushouts and coequalizers (and more generally finite connected colimits)." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-continuous-almost-cocontinuous}." ], "refs": [ "sites-lemma-continuous-almost-cocontinuous" ], "ref_ids": [ 8623 ] } ], "ref_ids": [] }, { "id": 8625, "type": "theorem", "label": "sites-lemma-open-subtopos", "categories": [ "sites" ], "title": "sites-lemma-open-subtopos", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a sheaf on", "$\\mathcal{C}$. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a subobject of the final object of", "$\\Sh(\\mathcal{C})$, and", "\\item the topos $\\Sh(\\mathcal{C})/\\mathcal{F}$ is a subtopos of", "$\\Sh(\\mathcal{C})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have seen in Lemma \\ref{lemma-localize-topos} that", "$\\Sh(\\mathcal{C})/\\mathcal{F}$ is a topos. In fact, we recall the", "proof. First we apply Lemma \\ref{lemma-topos-good-site}", "to see that we may assume $\\mathcal{C}$ is a site with a subcanonical", "topology, fibre products, a final object $X$, and an object $U$ with", "$\\mathcal{F} = h_U$. The proof of", "Lemma \\ref{lemma-localize-topos}", "shows that the morphism of topoi", "$j_\\mathcal{F} : \\Sh(\\mathcal{C})/\\mathcal{F} \\to \\Sh(\\mathcal{C})$", "is equal (modulo certain identifications) to the localization morphism", "$j_U : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})$.", "\\medskip\\noindent", "Assume (2). This means that $j_U^{-1}j_{U, *}\\mathcal{G} \\to \\mathcal{G}$", "is an isomorphism for all sheaves $\\mathcal{G}$ on $\\mathcal{C}/U$.", "For any object $Z/U$ of $\\mathcal{C}/U$ we have", "$$", "(j_{U, *}h_{Z/U})(U) = \\Mor_{\\mathcal{C}/U}(U \\times_X U/U, Z/U)", "$$", "by Lemma \\ref{lemma-localize-given-products}.", "Setting $\\mathcal{G} = h_{Z/U}$ in the equality above we obtain", "$$", "\\Mor_{\\mathcal{C}/U}(U \\times_X U/U, Z/U) = \\Mor_{\\mathcal{C}/U}(U, Z/U)", "$$", "for all $Z/U$. By Yoneda's lemma", "(Categories, Lemma \\ref{categories-lemma-yoneda})", "this implies $U \\times_X U = U$. By", "Categories, Lemma \\ref{categories-lemma-characterize-mono-epi}", "$U \\to X$ is a monomorphism, in other words (1) holds.", "\\medskip\\noindent", "Assume (1). Then $j_U^{-1} j_{U, *} = \\text{id}$ by", "Lemma \\ref{lemma-restrict-back}." ], "refs": [ "sites-lemma-localize-topos", "sites-lemma-topos-good-site", "sites-lemma-localize-topos", "sites-lemma-localize-given-products", "categories-lemma-yoneda", "categories-lemma-characterize-mono-epi", "sites-lemma-restrict-back" ], "ref_ids": [ 8583, 8581, 8583, 8567, 12203, 12209, 8569 ] } ], "ref_ids": [] }, { "id": 8626, "type": "theorem", "label": "sites-lemma-closed-subtopos", "categories": [ "sites" ], "title": "sites-lemma-closed-subtopos", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a subsheaf of the final", "object $*$ of $\\Sh(\\mathcal{C})$. The full subcategory of sheaves", "$\\mathcal{G}$ such that $\\mathcal{F} \\times \\mathcal{G} \\to \\mathcal{F}$", "is an isomorphism is a subtopos of $\\Sh(\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "We apply Lemma \\ref{lemma-topos-good-site} to see that we may assume", "$\\mathcal{C}$ is a site with the properties listed in that lemma.", "In particular $\\mathcal{C}$ has a final object $X$ (so that", "$* = h_X$) and an object $U$ with $\\mathcal{F} = h_U$.", "\\medskip\\noindent", "Let $\\mathcal{D} = \\mathcal{C}$ as a category but a covering", "is a family $\\{V_j \\to V\\}$ of morphisms such that", "$\\{V_i \\to V\\} \\cup \\{U \\times_X V \\to V\\}$ is a covering.", "By our choice of $\\mathcal{C}$ this means exactly that", "$$", "h_{U \\times_X V} \\amalg \\coprod h_{V_i} \\longrightarrow h_V", "$$", "is surjective. We claim that $\\mathcal{D}$ is a site, i.e., the coverings", "satisfy the conditions (1), (2), (3) of Definition \\ref{definition-site}.", "Condition (1) holds. For condition (2) suppose that", "$\\{V_i \\to V\\}$ and $\\{V_{ij} \\to V_i\\}$ are coverings of $\\mathcal{D}$.", "Then the composition", "$$", "\\coprod \\left(", "h_{U \\times_X V_i} \\amalg \\coprod h_{V_{ij}}", "\\right) \\longrightarrow", "h_{U \\times_X V} \\amalg \\coprod h_{V_i} \\longrightarrow h_V", "$$", "is surjective. Since each of the morphisms $U \\times_X V_i \\to V$", "factors through $U \\times_X V$ we see that", "$$", "h_{U \\times_X V} \\amalg \\coprod h_{V_{ij}} \\longrightarrow h_V", "$$", "is surjective, i.e., $\\{V_{ij} \\to V\\}$ is a covering of $V$ in", "$\\mathcal{D}$. Condition (3) follows similarly as a base change of", "a surjective map of sheaves is surjective.", "\\medskip\\noindent", "Note that the (identity) functor $u : \\mathcal{C} \\to \\mathcal{D}$ is", "continuous and commutes with fibre products and final objects. Hence", "we obtain a morphism $f : \\mathcal{D} \\to \\mathcal{C}$ of sites", "(Proposition \\ref{proposition-get-morphism}).", "Observe that $f_*$ is the identity functor on underlying", "presheaves, hence fully faithful. To finish the proof we have to", "show that the essential image of $f_*$ is the full subcategory", "$E \\subset \\Sh(\\mathcal{C})$ singled out in the lemma. To do this, note", "that $\\mathcal{G} \\in \\Ob(\\Sh(\\mathcal{C}))$ is in $E$ if and only if", "$\\mathcal{G}(U \\times_X V)$ is a singleton for all objects", "$V$ of $\\mathcal{C}$. Thus such a sheaf satisfies the", "sheaf property for all coverings of $\\mathcal{D}$ (argument omitted).", "Conversely, if $\\mathcal{G}$ satisfies the sheaf property", "for all coverings of $\\mathcal{D}$, then $\\mathcal{G}(U \\times_X V)$", "is a singleton, as in $\\mathcal{D}$ the object $U \\times_X V$ is", "covered by the empty covering." ], "refs": [ "sites-lemma-topos-good-site", "sites-definition-site", "sites-proposition-get-morphism" ], "ref_ids": [ 8581, 8652, 8641 ] } ], "ref_ids": [] }, { "id": 8627, "type": "theorem", "label": "sites-lemma-closed-immersion", "categories": [ "sites" ], "title": "sites-lemma-closed-immersion", "contents": [ "Let $i : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a closed immersion of", "topoi. Then $i_*$ is fully faithful, transforms surjections into surjections,", "commutes with coequalizers, commutes with pushouts, reflects injections,", "reflects surjections, and reflects bijections." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a subsheaf of the final object $*$ of $\\Sh(\\mathcal{C})$", "and let $E \\subset \\Sh(\\mathcal{C})$ be the full subcategory consisting", "of those $\\mathcal{G}$ such that", "$\\mathcal{F} \\times \\mathcal{G} \\to \\mathcal{F}$ is an isomorphism.", "By Lemma \\ref{lemma-closed-subtopos}", "the functor $i_*$ is isomorphic to the", "inclusion functor $\\iota : E \\to \\Sh(\\mathcal{C})$.", "\\medskip\\noindent", "Let $j_{\\mathcal{F}} : \\Sh(\\mathcal{C})/\\mathcal{F} \\to \\Sh(\\mathcal{C})$", "be the localization functor (Lemma \\ref{lemma-localize-topos}).", "Note that $E$ can also be described as", "the collection of sheaves $\\mathcal{G}$ such that", "$j_\\mathcal{F}^{-1}\\mathcal{G} = *$.", "\\medskip\\noindent", "Let $a, b : \\mathcal{G}_1 \\to \\mathcal{G}_2$ be two morphism of $E$.", "To prove $\\iota$ commutes with coequalizers it suffices to show that", "the coequalizer of $a$, $b$ in $\\Sh(\\mathcal{C})$ lies in $E$.", "This is clear because", "the coequalizer of two morphisms $* \\to *$ is $*$ and because", "$j_\\mathcal{F}^{-1}$ is exact. Similarly for pushouts.", "\\medskip\\noindent", "Thus $i_*$ satisfies properties (5), (6), and (7) of", "Lemma \\ref{lemma-exactness-properties} and hence", "the morphism $i$ satisfies all properties mentioned in that", "lemma, in particular the ones mentioned in this lemma." ], "refs": [ "sites-lemma-closed-subtopos", "sites-lemma-localize-topos", "sites-lemma-exactness-properties" ], "ref_ids": [ 8626, 8583, 8618 ] } ], "ref_ids": [] }, { "id": 8628, "type": "theorem", "label": "sites-lemma-push-pull-good-case", "categories": [ "sites" ], "title": "sites-lemma-push-pull-good-case", "contents": [ "Suppose the functor $u : \\mathcal{C} \\to \\mathcal{D}$ satisfies", "the hypotheses of Proposition \\ref{proposition-get-morphism},", "and hence gives rise to a morphism of sites", "$f : \\mathcal{D} \\to \\mathcal{C}$. In this case", "the pullback functor $f^{-1}$ (resp.\\ $u_p$) and the pushforward", "functor $f_*$ (resp. $u^p$) extend to an adjoint pair of functors on", "the categories of sheaves (resp.\\ presheaves) of algebraic structures.", "Moreover, these functors commute with taking", "the underlying sheaf (resp.\\ presheaf) of sets." ], "refs": [ "sites-proposition-get-morphism" ], "proofs": [ { "contents": [ "We have defined $f_* = u^p$ above.", "In the course of the proof of Proposition \\ref{proposition-get-morphism}", "we saw that all the colimits used to define $u_p$ are", "filtered under the assumptions of the proposition.", "Hence we conclude from the definition of a type of", "algebraic structure that we may define $u_p$ by exactly", "the same colimits as a functor on presheaves of algebraic structures.", "Adjointness of $u_p$ and $u^p$ is proved in exactly the", "same way as the proof of Lemma \\ref{lemma-adjoints-u}.", "The discussion of sheafification of presheaves of", "algebraic structures above then implies that we may define", "$f^{-1}(\\mathcal{F}) = (u_p\\mathcal{F})^\\#$." ], "refs": [ "sites-proposition-get-morphism", "sites-lemma-adjoints-u" ], "ref_ids": [ 8641, 8500 ] } ], "ref_ids": [ 8641 ] }, { "id": 8629, "type": "theorem", "label": "sites-lemma-compute-global-sections", "categories": [ "sites" ], "title": "sites-lemma-compute-global-sections", "contents": [ "Let $\\mathcal{C}$ be a site. Let $a, b : V \\to U$ be objects of $\\mathcal{C}$", "such that", "$$", "\\xymatrix{", "h_V^\\# \\ar@<1ex>[r] \\ar@<-1ex>[r] & h_U^\\# \\ar[r] & {*}", "}", "$$", "is a coequalizer in $\\Sh(\\mathcal{C})$. Then", "$\\Gamma(\\mathcal{C}, \\mathcal{F})$ is the equalizer of", "$a^*, b^* : \\mathcal{F}(U) \\to \\mathcal{F}(V)$." ], "refs": [], "proofs": [ { "contents": [ "Since $\\Mor_{\\Sh(\\mathcal{C})}(h_U^\\#, \\mathcal{F}) = \\mathcal{F}(U)$", "this is clear from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8630, "type": "theorem", "label": "sites-lemma-sieves-set", "categories": [ "sites" ], "title": "sites-lemma-sieves-set", "contents": [ "Let $\\mathcal{C}$ be a category. Let $U \\in \\Ob(\\mathcal{C})$.", "\\begin{enumerate}", "\\item The collection of sieves on $U$ is a set.", "\\item Inclusion defines a partial ordering on this set.", "\\item Unions and intersections of sieves are sieves.", "\\item", "\\label{item-sieve-generated}", "Given a family of morphisms $\\{U_i \\to U\\}_{i\\in I}$", "of $\\mathcal{C}$ with target $U$", "there exists a unique smallest sieve $S$ on $U$ such that", "each $U_i \\to U$ belongs to $S(U_i)$.", "\\item The sieve $S = h_U$ is the maximal sieve.", "\\item The empty subpresheaf is the minimal sieve.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By our definition of subpresheaf, the collection of", "all subpresheaves of a presheaf $\\mathcal{F}$ is a subset of", "$\\prod_{U \\in \\Ob(\\mathcal{C})} \\mathcal{P}(\\mathcal{F}(U))$.", "And this is a set. (Here $\\mathcal{P}(A)$ denotes", "the powerset of $A$.) Hence the collection of sieves on $U$", "is a set.", "\\medskip\\noindent", "The partial ordering is defined by: $S \\leq S'$ if and only if", "$S(T) \\subset S'(T)$ for all $T \\to U$. Notation: $S \\subset S'$.", "\\medskip\\noindent", "Given a collection of sieves $S_i$, $i \\in I$ on $U$ we can", "define $\\bigcup S_i$ as the sieve with values", "$(\\bigcup S_i)(T) = \\bigcup S_i(T)$ for all", "$T \\in \\Ob(\\mathcal{C})$.", "We define the intersection $\\bigcap S_i$ in the same way.", "\\medskip\\noindent", "Given $\\{U_i \\to U\\}_{i\\in I}$ as in the statement, consider", "the morphisms of presheaves $h_{U_i} \\to h_U$. We simply", "define $S$ as the union of the images (Definition \\ref{definition-image})", "of these maps of presheaves.", "\\medskip\\noindent", "The last two statements of the lemma are obvious." ], "refs": [ "sites-definition-image" ], "ref_ids": [ 8650 ] } ], "ref_ids": [] }, { "id": 8631, "type": "theorem", "label": "sites-lemma-pullback-sieve-section", "categories": [ "sites" ], "title": "sites-lemma-pullback-sieve-section", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $U \\in \\Ob(\\mathcal{C})$.", "Let $S$ be a sieve on $U$.", "If $f : V \\to U$ is in $S$, then", "$S \\times_U V = h_V$ is maximal." ], "refs": [], "proofs": [ { "contents": [ "Trivial from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8632, "type": "theorem", "label": "sites-lemma-topology-basic", "categories": [ "sites" ], "title": "sites-lemma-topology-basic", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $J$ be a topology on $\\mathcal{C}$.", "Let $U \\in \\Ob(\\mathcal{C})$.", "\\begin{enumerate}", "\\item Finite intersections of elements of $J(U)$ are in $J(U)$.", "\\item If $S \\in J(U)$ and $S' \\supset S$, then $S' \\in J(U)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $S, S' \\in J(U)$. Consider $S'' = S \\cap S'$. For every", "$V \\to U$ in $S(U)$ we have", "$$", "S' \\times_U V = S'' \\times_U V", "$$", "simply because $V \\to U$ already is in $S$. Hence by the second", "axiom of the definition we see that $S'' \\in J(U)$.", "\\medskip\\noindent", "Let $S \\in J(U)$ and $S' \\supset S$. For every", "$V \\to U$ in $S(U)$ we have $S' \\times_U V = h_V$ by", "Lemma \\ref{lemma-pullback-sieve-section}. Thus", "$S' \\times_U V \\in J(V)$ by the third axiom. Hence", "$S' \\in J(U)$ by the second axiom." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8633, "type": "theorem", "label": "sites-lemma-play-with-topologies", "categories": [ "sites" ], "title": "sites-lemma-play-with-topologies", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\{J_i\\}_{i\\in I}$ be a set of topologies.", "\\begin{enumerate}", "\\item The rule $J(U) = \\bigcap J_i(U)$ defines", "a topology on $\\mathcal{C}$.", "\\item There is a coarsest topology finer than", "all of the topologies $J_i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first part is direct from the definitions.", "The second follows by taking the intersection", "of all topologies finer than all of the $J_i$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8634, "type": "theorem", "label": "sites-lemma-topology-presheaves-sheaves", "categories": [ "sites" ], "title": "sites-lemma-topology-presheaves-sheaves", "contents": [ "Let $\\mathcal{C}$ be a category. Let $\\{ \\mathcal{F}_i \\}_{i\\in I}$ be a", "collection of presheaves of sets on $\\mathcal{C}$. For each", "$U \\in \\Ob(\\mathcal{C})$ denote", "$J(U)$ the set of sieves $S$ with the following property:", "For every morphism $V \\to U$, the maps", "$$", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(h_V, \\mathcal{F}_i)", "\\longrightarrow", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(S \\times_U V, \\mathcal{F}_i)", "$$", "are bijective for all $i \\in I$. Then $J$ defines a", "topology on $\\mathcal{C}$. This topology is the finest", "topology in which all of the $\\mathcal{F}_i$ are sheaves." ], "refs": [], "proofs": [ { "contents": [ "If we show that $J$ is a topology, then the last statement of", "the lemma immediately follows. The first and third axioms of", "a topology are immediately verified. Thus, assume that", "we have an object $U$, and sieves $S, S'$ of $U$", "such that $S \\in J(U)$, and for all $V \\to U$ in $S(V)$", "we have $S' \\times_U V \\in J(V)$. We have to show that", "$S' \\in J(U)$. In other words, we have to show that for", "any $f : W \\to U$, the maps", "$$", "\\mathcal{F}_i(W) =", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(h_W, \\mathcal{F}_i)", "\\longrightarrow", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(S' \\times_U W, \\mathcal{F}_i)", "$$", "are bijective for all $i \\in I$. Pick an element", "$i \\in I$ and pick an element", "$\\varphi \\in", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(S' \\times_U W, \\mathcal{F}_i)$.", "We will construct a section $s \\in \\mathcal{F}_i(W)$", "mapping to $\\varphi$.", "\\medskip\\noindent", "Suppose $\\alpha : V \\to W$ is an element of $S \\times_U W$.", "According to the definition of pullbacks we see that", "the composition $f \\circ\\alpha : V \\to W \\to U$ is in $S$. Hence", "$S' \\times_U V$ is in $J(W)$ by assumption on the pair", "of sieves $S, S'$. Now we have a commutative diagram", "of presheaves", "$$", "\\xymatrix{", "S' \\times_U V \\ar[r] \\ar[d] & h_V \\ar[d] \\\\", "S' \\times_U W \\ar[r] & h_W", "}", "$$", "The restriction of $\\varphi$ to $S' \\times_U V$", "corresponds to an element $s_{V, \\alpha} \\in \\mathcal{F}_i(V)$.", "This we see from the definition of $J$,", "and because $S' \\times_U V$ is in $J(W)$.", "We leave it to the reader to check", "that the rule $(V, \\alpha) \\mapsto s_{V, \\alpha}$ defines", "an element", "$\\psi \\in", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(S \\times_U W, \\mathcal{F}_i)$.", "Since $S \\in J(U)$ we see immediately from the definition of $J$", "that $\\psi$ corresponds to an element $s$ of $\\mathcal{F}_i(W)$.", "\\medskip\\noindent", "We leave it to the reader to verify that the construction", "$\\varphi \\mapsto s$ is inverse to the natural map displayed above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8635, "type": "theorem", "label": "sites-lemma-site-gives-topology", "categories": [ "sites" ], "title": "sites-lemma-site-gives-topology", "contents": [ "Let $\\mathcal{C}$ be a site with coverings $\\text{Cov}(\\mathcal{C})$.", "For every object $U$ of $\\mathcal{C}$, let $J(U)$ denote", "the set of sieves $S$ on $U$ with the following property:", "there exists a covering", "$\\{f_i : U_i \\to U\\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$", "so that the sieve $S'$ generated by the $f_i$ (see Definition", "\\ref{definition-sieve-generated}) is contained in $S$.", "\\begin{enumerate}", "\\item This $J$ is a topology on $\\mathcal{C}$.", "\\item A presheaf $\\mathcal{F}$ is a sheaf for this topology", "(see Definition \\ref{definition-sheaf-sets-topology})", "if and only if it is a sheaf on the site (see", "Definition \\ref{definition-sheaf-sets}).", "\\end{enumerate}" ], "refs": [ "sites-definition-sieve-generated", "sites-definition-sheaf-sets-topology", "sites-definition-sheaf-sets" ], "proofs": [ { "contents": [ "To prove the first assertion we just note that axioms", "(1), (2) and (3) of the definition of a site", "(Definition \\ref{definition-site})", "directly imply the axioms", "(3), (2) and (1) of the definition of a topology", "(Definition \\ref{definition-topology}). As an example we", "prove $J$ has property (2). Namely, let $U$ be an object", "of $\\mathcal{C}$, let $S, S'$ be sieves on $U$ such that", "$S \\in J(U)$, and such that for every $V \\to U$ in $S(V)$", "we have $S' \\times_U V \\in J(V)$. By definition of $J(U)$", "we can find a covering $\\{f_i : U_i \\to U\\}$ of the site", "such that $S$ the image of $h_{U_i} \\to h_U$ is contained", "in $S$. Since each $S'\\times_U U_i$ is in $J(U_i)$ we", "see that there are coverings $\\{U_{ij} \\to U_i\\}$ of the", "site such that $h_{U_{ij}} \\to h_{U_i}$ is contained", "in $S' \\times_U U_i$. By definition of the base change", "this means that $h_{U_{ij}} \\to h_U$ is contained", "in the subpresheaf $S' \\subset h_U$. By axiom (2) for", "sites we see that $\\{U_{ij} \\to U\\}$ is a covering of", "$U$ and we conclude that $S' \\in J(U)$ by definition of $J$.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be a presheaf. Suppose that $\\mathcal{F}$", "is a sheaf in the topology $J$. We will show that $\\mathcal{F}$", "is a sheaf on the site as well. Let $\\{f_i : U_i \\to U\\}_{i\\in I}$", "be a covering of the site. Let $s_i \\in \\mathcal{F}(U_i)$ be a", "family of sections such that", "$s_i|_{U_i \\times_U U_j} = s_j|_{U_i \\times_U U_j}$ for all", "$i, j$. We have to show that there exists a unique section", "$s \\in \\mathcal{F}(U)$ restricting back to the $s_i$ on the $U_i$.", "Let $S \\subset h_U$ be the sieve generated by the $f_i$.", "Note that $S \\in J(U)$ by definition. In stead of constructing", "$s$, by the sheaf condition in the topology, it suffices", "to construct an element", "$$", "\\varphi \\in \\Mor_{\\textit{PSh}(\\mathcal{C})}(S, \\mathcal{F}).", "$$", "Take $\\alpha \\in S(T)$ for some object $T \\in \\mathcal{U}$.", "This means exactly that $\\alpha : T \\to U$ is a morphism", "which factors through $f_i$ for some $i\\in I$ (and maybe more than $1$).", "Pick such an index $i$ and a factorization $\\alpha = f_i \\circ \\alpha_i$.", "Define $\\varphi(\\alpha) = \\alpha_i^* s_i$. If $i'$, $\\alpha", "= f_i \\circ \\alpha_{i'}'$ is a second choice, then", "$\\alpha_i^* s_i = (\\alpha_{i'}')^* s_{i'}$ exactly because of our", "condition $s_i|_{U_i \\times_U U_j} = s_j|_{U_i \\times_U U_j}$ for all", "$i, j$. Thus $\\varphi(\\alpha)$ is well defined. We leave it to the reader", "to verify that $\\varphi$, which in turn determines $s$ is correct", "in the sense that $s$ restricts back to $s_i$.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be a presheaf. Suppose that $\\mathcal{F}$", "is a sheaf on the site $(\\mathcal{C}, \\text{Cov}(\\mathcal{C}))$.", "We will show that $\\mathcal{F}$ is a sheaf for the topology $J$", "as well. Let $U$ be an object of $\\mathcal{C}$. Let", "$S$ be a covering sieve on $U$ with respect to the topology $J$.", "Let", "$$", "\\varphi \\in \\Mor_{\\textit{PSh}(\\mathcal{C})}(S, \\mathcal{F}).", "$$", "We have to show there is a unique element in", "$\\mathcal{F}(U) = \\Mor_{\\textit{PSh}(\\mathcal{C})}(h_U, \\mathcal{F})$", "which restricts back to $\\varphi$. By definition there exists", "a covering $\\{f_i : U_i \\to U\\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$", "such that $f_i : U_i \\in U$ belongs to $S(U_i)$. Hence", "we can set $s_i = \\varphi(f_i) \\in \\mathcal{F}(U_i)$.", "Then it is a pleasant exercise to see that", "$s_i|_{U_i \\times_U U_j} = s_j|_{U_i \\times_U U_j}$ for all", "$i, j$. Thus we obtain the desired section $s$ by the sheaf", "condition for $\\mathcal{F}$ on the site", "$(\\mathcal{C}, \\text{Cov}(\\mathcal{C}))$.", "Details left to the reader." ], "refs": [ "sites-definition-site", "sites-definition-topology" ], "ref_ids": [ 8652, 8693 ] } ], "ref_ids": [ 8691, 8695, 8653 ] }, { "id": 8636, "type": "theorem", "label": "sites-lemma-L-presheaf", "categories": [ "sites" ], "title": "sites-lemma-L-presheaf", "contents": [ "In the situation above.", "\\begin{enumerate}", "\\item The assignment $U \\mapsto L\\mathcal{F}(U)$ combined with the", "restriction mappings defined above is a presheaf.", "\\item The maps $\\ell$ glue to give a morphism of presheaves", "$\\ell : \\mathcal{F} \\to L\\mathcal{F}$.", "\\item The rule $\\mathcal{F} \\mapsto (\\mathcal{F} \\xrightarrow{\\ell}", "L\\mathcal{F})$ is a functor.", "\\item If $\\mathcal{F}$ is a subpresheaf of $\\mathcal{G}$, then", "$L\\mathcal{F}$ is a subpresheaf of $L\\mathcal{G}$.", "\\item The map $\\ell : \\mathcal{F} \\to L\\mathcal{F}$ has the", "following property: For every section $s \\in L\\mathcal{F}(U)$", "there exists a covering sieve $S$ on $U$ and an element", "$\\varphi \\in \\Mor_{\\textit{PSh}(\\mathcal{C})}(S, \\mathcal{F})$", "such that $\\ell(\\varphi)$ equals the restriction of", "$s$ to $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8637, "type": "theorem", "label": "sites-lemma-sieve-sheafification", "categories": [ "sites" ], "title": "sites-lemma-sieve-sheafification", "contents": [ "Let $\\mathcal{C}$ be a category endowed with a topology $J$.", "Let $U$ be an object of $\\mathcal{C}$.", "Let $S$ be a sieve on $U$. The following are equivalent", "\\begin{enumerate}", "\\item The sieve $S$ is a covering sieve.", "\\item The sheafification $S^\\# \\to h_U^\\#$", "of the map $S \\to h_U$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First we make a couple of general remarks.", "We will use that $S^\\# = LLS$, and $h_U^\\# = LLh_U$.", "In particular, by Lemma \\ref{lemma-L-presheaf}, we see that", "$S^\\# \\to h_U^\\#$ is injective. Note that", "$\\text{id}_U \\in h_U(U)$. Hence it gives rise to", "sections of $Lh_U$ and $h_U^\\# = LLh_U$ over $U$ which", "we will also denote $\\text{id}_U$.", "\\medskip\\noindent", "Suppose $S$ is a covering sieve. It clearly suffices to", "find a morphism $h_U \\to S^\\#$ such that the composition", "$h_U \\to h_U^\\#$ is the canonical map. To find such a map", "it suffices to find a section $s \\in S^\\#(U)$ wich restricts", "to $\\text{id}_U$. But since $S$ is a", "covering sieve, the element", "$\\text{id}_S \\in \\Mor_{\\textit{PSh}(\\mathcal{C})}(S, S)$", "gives rise to a section of $LS$ over $U$ which restricts to", "$\\text{id}_U$ in $Lh_U$. Hence we win.", "\\medskip\\noindent", "Suppose that $S^\\# \\to h_U^\\#$ is an isomorphism.", "Let $1 \\in S^\\#(U)$ be the element corresponding to", "$\\text{id}_U$ in $h_U^\\#(U)$. Because $S^\\# = LLS$", "there exists a covering sieve $S'$ on $U$ such that", "$1$ comes from a", "$$", "\\varphi \\in \\Mor_{\\textit{PSh}(\\mathcal{C})}(S', LS).", "$$", "This in turn means that for every $\\alpha : V \\to U$,", "$\\alpha\\in S'(V)$ there exists a covering sieve $S_{V, \\alpha}$", "on $V$ such that $\\varphi(\\text{id}_V)$ corresponds to", "a morphism of presheaves $S_{V, \\alpha} \\to S$. In other words", "$S_{V, \\alpha}$ is contained in $S \\times_U V$. By the second", "axiom of a topology we see that $S$ is a covering sieve." ], "refs": [ "sites-lemma-L-presheaf" ], "ref_ids": [ 8636 ] } ], "ref_ids": [] }, { "id": 8638, "type": "theorem", "label": "sites-lemma-finer-topology", "categories": [ "sites" ], "title": "sites-lemma-finer-topology", "contents": [ "Assumption and notation as in Theorem \\ref{theorem-topology-and-topos}.", "Then $J \\subset J'$ if and only if every sheaf for the", "topology $J'$ is a sheaf for the topology $J$." ], "refs": [ "sites-theorem-topology-and-topos" ], "proofs": [ { "contents": [ "One direction is clear. For the other direction suppose that", "$\\Sh(\\mathcal{C}, J') \\subset \\Sh(\\mathcal{C}, J)$.", "By formal nonsense this implies", "that if $\\mathcal{F}$ is a presheaf of sets,", "and $\\mathcal{F} \\to \\mathcal{F}^\\#$,", "resp.\\ $\\mathcal{F} \\to \\mathcal{F}^{\\#, \\prime}$", "is the sheafification wrt $J$, resp.\\ $J'$ then there", "is a canonical map $\\mathcal{F}^\\# \\to \\mathcal{F}^{\\#, \\prime}$", "such that", "$\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{F}^{\\#, \\prime}$", "equals the canonical map $\\mathcal{F} \\to \\mathcal{F}^{\\#, \\prime}$.", "Of course, $\\mathcal{F}^\\# \\to \\mathcal{F}^{\\#, \\prime}$", "identifies the second sheaf as the sheafification of the first", "with respect to the topology $J'$.", "Apply this to the map $S \\to h_U$ of", "Lemma \\ref{lemma-sieve-sheafification}. We get a commutative", "diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar[d] &", "S^\\# \\ar[r] \\ar[d] &", "S^{\\#, \\prime} \\ar[d] \\\\", "h_U \\ar[r] &", "h_U^\\# \\ar[r] &", "h_U^{\\#, \\prime}", "}", "$$", "And clearly, if $S$ is a covering sieve for the topology $J$", "then the middle vertical map is an isomorphism (by the lemma)", "and we conclude that the right vertical map is an isomorphism as", "it is the sheafification of the one in the middle wrt $J'$.", "By the lemma again we conclude that $S$ is a covering sieve", "for $J'$ as well." ], "refs": [ "sites-lemma-sieve-sheafification" ], "ref_ids": [ 8637 ] } ], "ref_ids": [ 8494 ] }, { "id": 8639, "type": "theorem", "label": "sites-proposition-sheaves-on-group", "categories": [ "sites" ], "title": "sites-proposition-sheaves-on-group", "contents": [ "The functors $\\mathcal{F} \\mapsto \\mathcal{F}({}_GG)$", "and $S \\mapsto \\mathcal{F}_S$ define quasi-inverse", "equivalences between $\\Sh(\\mathcal{T}_G)$", "and $G\\textit{-Sets}$." ], "refs": [], "proofs": [ { "contents": [ "We have already seen that composing the functors one way around", "is isomorphic to the identity functor.", "In the other direction, for any sheaf $\\mathcal{H}$ there is a natural", "map of sheaves", "$$", "can :", "\\mathcal{H}", "\\longrightarrow", "\\mathcal{F}_{\\mathcal{H}({}_GG)}.", "$$", "Namely, for any object $U$ of $\\mathcal{T}_G$ we let $can_U$", "be the map", "$$", "\\begin{matrix}", "\\mathcal{H}(U)", "&", "\\longrightarrow", "&", "\\mathcal{F}_{\\mathcal{H}({}_GG)}(U)", "=", "\\Mor_G(U, \\mathcal{H}({}_GG))", "\\\\", "s", "&", "\\longmapsto", "&", "(u \\mapsto \\alpha_u^*s).", "\\end{matrix}", "$$", "Here $\\alpha_u : {}_GG \\to U$ is the map", "$\\alpha_u(g) = gu$ and $\\alpha_u^* : \\mathcal{H}(U)", "\\to \\mathcal{H}({}_GG)$ is the pullback map. A trivial", "but confusing verification shows that this is indeed a map", "of presheaves. We have to show that $can$ is an isomorphism.", "We do this by showing $can_U$ is an isomorphism for all", "$U \\in \\Ob(\\mathcal{T}_G)$. We leave the (important but", "easy) case that $U = {}_GG$ to the reader.", "A general object $U$ of $\\mathcal{T}_G$ is a disjoint union of", "$G$-orbits: $U = \\coprod_{i\\in I} O_i$. The family of maps", "$\\{O_i \\to U\\}_{i \\in I}$ is tautologically equivalent", "to a covering in $\\mathcal{T}_G$ (by the properties of $\\mathcal{T}_G$", "listed at the beginning of this section). Hence by Lemma", "\\ref{lemma-tautological-same-sheaf} the sheaf $\\mathcal{H}$", "satisfies the sheaf property with respect to", "$\\{O_i \\to U\\}_{i \\in I}$. The sheaf property for this covering", "implies $\\mathcal{H}(U) = \\prod_i \\mathcal{H}(O_i)$.", "Hence it suffices to show that $can_U$ is an", "isomorphism when $U$ consists of a single $G$-orbit. Let $u \\in U$", "and let $H \\subset G$ be its stabilizer. Clearly,", "$\\Mor_G(U, \\mathcal{H}({}_GG)) = \\mathcal{H}({}_GG)^H$", "equals the subset of $H$-invariant elements. On the other hand", "consider the covering $\\{{}_GG \\to U\\}$ given by $g \\mapsto", "gu$ (again it is just combinatorially equivalent to some covering", "of $\\mathcal{T}_G$, and again this doesn't matter).", "Note that the fibre product $({}_GG)\\times_U ({}_GG)$", "is equal to $\\{(g, gh), g\\in G, h\\in H\\} \\cong \\coprod_{h\\in H} {}_GG$.", "Hence the sheaf property for this covering reads as", "$$", "\\xymatrix{", "\\mathcal{H}(U) \\ar[r]", "&", "\\mathcal{H}({}_GG)", "\\ar@<1ex>[r]^-{\\text{pr}_0^*} \\ar@<-1ex>[r]_-{\\text{pr}_1^*}", "&", "\\prod_{h \\in H}", "\\mathcal{H}({}_GG).", "}", "$$", "The two maps $\\text{pr}_i^*$ into the factor", "$\\mathcal{H}({}_GG)$ differ by multiplication by $h$.", "Now the result follows from this and the fact that $can$", "is an isomorphism for $U = {}_GG$." ], "refs": [ "sites-lemma-tautological-same-sheaf" ], "ref_ids": [ 8503 ] } ], "ref_ids": [] }, { "id": 8640, "type": "theorem", "label": "sites-proposition-sheafification-adjoint", "categories": [ "sites" ], "title": "sites-proposition-sheafification-adjoint", "contents": [ "The canonical map $\\mathcal{F} \\to \\mathcal{F}^\\#$ has the", "following universal property: For any map $\\mathcal{F} \\to \\mathcal{G}$,", "where $\\mathcal{G}$ is a sheaf of sets, there is a unique map", "$\\mathcal{F}^\\# \\to \\mathcal{G}$ such that $\\mathcal{F} \\to \\mathcal{F}^\\#", "\\to \\mathcal{G}$ equals the given map." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-plus-functorial} we get a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{F} \\ar[r] \\ar[d]", "&", "\\mathcal{F}^{+} \\ar[r] \\ar[d]", "&", "\\mathcal{F}^{++} \\ar[d]", "\\\\", "\\mathcal{G} \\ar[r]", "&", "\\mathcal{G}^{+} \\ar[r]", "&", "\\mathcal{G}^{++}", "}", "$$", "and by Theorem \\ref{theorem-plus} the lower horizontal maps", "are isomorphisms. The uniqueness follows from Lemma", "\\ref{lemma-plus-surjective} which says that every section of", "$\\mathcal{F}^\\#$ locally comes from sections of $\\mathcal{F}$." ], "refs": [ "sites-lemma-plus-functorial", "sites-theorem-plus", "sites-lemma-plus-surjective" ], "ref_ids": [ 8510, 8492, 8513 ] } ], "ref_ids": [] }, { "id": 8641, "type": "theorem", "label": "sites-proposition-get-morphism", "categories": [ "sites" ], "title": "sites-proposition-get-morphism", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let", "$u : \\mathcal{C} \\to \\mathcal{D}$ be continuous.", "Assume furthermore the following:", "\\begin{enumerate}", "\\item the category $\\mathcal{C}$ has a final object $X$ and", "$u(X)$ is a final object of $\\mathcal{D}$ , and", "\\item the category $\\mathcal{C}$ has fibre products and", "$u$ commutes with them.", "\\end{enumerate}", "Then $u$ defines a morphism of sites $\\mathcal{D} \\to", "\\mathcal{C}$, in other words $u_s$ is exact." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-directed} and", "\\ref{lemma-directed-morphism}." ], "refs": [ "sites-lemma-directed", "sites-lemma-directed-morphism" ], "ref_ids": [ 8498, 8526 ] } ], "ref_ids": [] }, { "id": 8642, "type": "theorem", "label": "sites-proposition-point-limits", "categories": [ "sites" ], "title": "sites-proposition-point-limits", "contents": [ "Let $\\mathcal{C}$ be a site. Assume that finite limits exist", "in $\\mathcal{C}$. (I.e., $\\mathcal{C}$ has fibre products, and a", "final object.) A point $p$ of such a site $\\mathcal{C}$", "is given by a functor $u : \\mathcal{C} \\to \\textit{Sets}$ such that", "\\begin{enumerate}", "\\item $u$ commutes with finite limits, and", "\\item if $\\{U_i \\to U\\}$ is a covering, then", "$\\coprod_i u(U_i) \\to u(U)$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose first that $p$ is a point (Definition \\ref{definition-point})", "given by a functor $u$. Condition (2) is satisfied directly from", "the definition of a point. By Lemma \\ref{lemma-points-recover}", "we have $(h_U)_p = u(U)$. By Lemma \\ref{lemma-point-pushforward-sheaf}", "we have $(h_U^\\#)_p = (h_U)_p$. Thus we see that $u$", "is equal to the composition of functors", "$$", "\\mathcal{C} \\xrightarrow{h}", "\\textit{PSh}(\\mathcal{C}) \\xrightarrow{{}^\\#}", "\\Sh(\\mathcal{C}) \\xrightarrow{()_p}", "\\textit{Sets}", "$$", "Each of these functors is left exact, and hence we see $u$ satisfies (1).", "\\medskip\\noindent", "Conversely, suppose that $u$ satisfies (1) and (2).", "In this case we immediately see that $u$ satisfies the first two", "conditions of Definition \\ref{definition-point}. And its", "stalk functor is exact, because it is a left adjoint by", "Lemma \\ref{lemma-point-pushforward-sheaf} and it commutes", "with finite limits by Lemma \\ref{lemma-neighbourhoods-directed}." ], "refs": [ "sites-definition-point", "sites-lemma-points-recover", "sites-lemma-point-pushforward-sheaf", "sites-definition-point", "sites-lemma-point-pushforward-sheaf", "sites-lemma-neighbourhoods-directed" ], "ref_ids": [ 8675, 8593, 8595, 8675, 8595, 8601 ] } ], "ref_ids": [] }, { "id": 8643, "type": "theorem", "label": "sites-proposition-criterion-points", "categories": [ "sites" ], "title": "sites-proposition-criterion-points", "contents": [ "\\begin{reference}", "\\cite[Expos\\'e VI, Appendix by Deligne, Proposition 9.0]{SGA4}", "\\end{reference}", "Let $\\mathcal{C}$ be a site. Assume that", "\\begin{enumerate}", "\\item finite limits exist in $\\mathcal{C}$, and", "\\item every covering $\\{U_i \\to U\\}_{i \\in I}$", "has a refinement by a finite covering of $\\mathcal{C}$.", "\\end{enumerate}", "Then $\\mathcal{C}$ has enough points." ], "refs": [], "proofs": [ { "contents": [ "We have to show that given any sheaf", "$\\mathcal{F}$ on $\\mathcal{C}$, any $U \\in \\Ob(\\mathcal{C})$,", "and any distinct sections $s, s' \\in \\mathcal{F}(U)$, there exists", "a point $p$ such that $s, s'$ have distinct image in", "$\\mathcal{F}_p$. See Lemma \\ref{lemma-enough}.", "Consider the system $(J, \\geq, V_j, g_{jj'})$", "with $J = \\{1\\}$, $V_1 = U$, $g_{11} = \\text{id}_U$.", "Apply Lemma \\ref{lemma-refine-all-at-once}.", "By the result of that lemma we get a system", "$(I, \\geq, U_i, f_{ii'})$ refining our system such", "that $s_p \\not = s'_p$ and such that moreover for every", "finite covering $\\{W_k \\to W\\}$ of the site $\\mathcal{C}$ the map", "$\\coprod_k u(W_k) \\to u(W)$ is surjective.", "Since every covering of $\\mathcal{C}$ can be refined by", "a finite covering we conclude that", "$\\coprod_k u(W_k) \\to u(W)$ is surjective for {\\it any}", "covering $\\{W_k \\to W\\}$ of the site $\\mathcal{C}$.", "This implies that $u = p$ is a point, see", "Proposition \\ref{proposition-point-limits} (and the discussion", "at the beginning of this section which guarantees that $u$", "commutes with finite limits)." ], "refs": [ "sites-lemma-enough", "sites-lemma-refine-all-at-once", "sites-proposition-point-limits" ], "ref_ids": [ 8610, 8615, 8642 ] } ], "ref_ids": [] }, { "id": 8644, "type": "theorem", "label": "sites-proposition-functoriality-algebraic-structures-topoi", "categories": [ "sites" ], "title": "sites-proposition-functoriality-algebraic-structures-topoi", "contents": [ "\\begin{slogan}", "Morphisms of topoi preserve algebraic structure.", "\\end{slogan}", "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $f = (f^{-1}, f_*)$ be a morphism of topoi", "from $\\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$.", "The method introduced above gives rise to an adjoint", "pair of functors $(f^{-1}, f_*)$ on sheaves of algebraic structures", "compatible with taking the underlying sheaves of sets", "for the following types of algebraic structures:", "\\begin{enumerate}", "\\item pointed sets,", "\\item abelian groups,", "\\item groups,", "\\item monoids,", "\\item rings,", "\\item modules over a fixed ring, and", "\\item lie algebras over a fixed field.", "\\end{enumerate}", "Moreover, in each of these cases the results above labeled ($\\alpha$),", "($\\beta$), ($\\gamma$), ($\\delta$), ($\\epsilon$), and ($\\zeta$) hold." ], "refs": [], "proofs": [ { "contents": [ "The final statement of the proposition holds simply since each of the listed", "categories, endowed with the obvious forgetful functor, is indeed a type of", "algebraic structure in the sense explained at the beginning of this section.", "See Sheaves, Lemma \\ref{sheaves-lemma-list-algebraic-structures}.", "\\medskip\\noindent", "Proof of (2). We think of a sheaf of abelian groups as", "a quadruple $(\\mathcal{F}, +, 0, i)$ as explained in the discussion preceding", "the proposition.", "If $(\\mathcal{F}, +, 0, i)$ lives on $\\mathcal{C}$, then its pullback", "is defined as $(f^{-1}\\mathcal{F}, f^{-1}+, f^{-1}0, f^{-1}i)$.", "If $(\\mathcal{G}, +, 0, i)$ lives on $\\mathcal{D}$, then its pushforward", "is defined as $(f_*\\mathcal{G}, f_*+, f_*0, f_*i)$. This works because", "$f_*(\\mathcal{G} \\times \\mathcal{G}) = f_*\\mathcal{G} \\times f_*\\mathcal{G}$.", "Adjointness follows from adjointness of the set based functors,", "since", "$$", "\\Mor_{\\textit{Ab}(\\mathcal{C})}", "((\\mathcal{F}_1, +, 0, i), (\\mathcal{F}_2, +, 0, i))", "=", "\\left\\{", "\\begin{matrix}", "\\varphi \\in \\Mor_{\\Sh(\\mathcal{C})}", "(\\mathcal{F}_1, \\mathcal{F}_2) \\\\", "\\varphi \\text{ is compatible with }+, 0, i", "\\end{matrix}", "\\right\\}", "$$", "Details left to the reader.", "\\medskip\\noindent", "This method also works for sheaves of rings by thinking", "of a sheaf of rings (with unit) as a sixtuple", "$(\\mathcal{O}, + , 0, i, \\cdot, 1)$ satisfying a list", "of axioms that you can find in any elementary", "algebra book.", "\\medskip\\noindent", "A sheaf of pointed sets is a pair $(\\mathcal{F}, p)$, where", "$\\mathcal{F}$ is a sheaf of sets, and $p : * \\to \\mathcal{F}$", "is a map of sheaves of sets.", "\\medskip\\noindent", "A sheaf of groups is given by a quadruple $(\\mathcal{F}, \\cdot, 1, i)$", "with suitable axioms.", "\\medskip\\noindent", "A sheaf of monoids is given by a pair $(\\mathcal{F}, \\cdot)$", "with suitable axiom.", "\\medskip\\noindent", "Let $R$ be a ring. An sheaf of $R$-modules is given by", "a quintuple $(\\mathcal{F}, +, 0, i, \\{\\lambda_r\\}_{r \\in R})$,", "where the quadruple $(\\mathcal{F}, +, 0, i)$ is a sheaf of", "abelian groups as above, and $\\lambda_r : \\mathcal{F} \\to \\mathcal{F}$", "is a family of morphisms of sheaves of sets", "such that", "$\\lambda_r \\circ 0 = 0$,", "$\\lambda_r \\circ + = + \\circ (\\lambda_r, \\lambda_r)$,", "$\\lambda_{r + r'} =", "+ \\circ \\lambda_r \\times \\lambda_{r'} \\circ (\\text{id}, \\text{id})$,", "$\\lambda_{rr'} = \\lambda_r \\circ \\lambda_{r'}$,", "$\\lambda_1 = \\text{id}$, $\\lambda_0 = 0 \\circ (\\mathcal{F} \\to *)$." ], "refs": [ "sheaves-lemma-list-algebraic-structures" ], "ref_ids": [ 14487 ] } ], "ref_ids": [] }, { "id": 8645, "type": "theorem", "label": "sites-proposition-sheafification-adjoint-topology", "categories": [ "sites" ], "title": "sites-proposition-sheafification-adjoint-topology", "contents": [ "Let $\\mathcal{C}$ be a category endowed with a topology.", "Let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{C}$.", "The canonical map $\\mathcal{F} \\to \\mathcal{F}^\\#$ has the", "following universal property: For any map", "$\\mathcal{F} \\to \\mathcal{G}$,", "where $\\mathcal{G}$ is a sheaf of sets, there is a unique map", "$\\mathcal{F}^\\# \\to \\mathcal{G}$ such that $\\mathcal{F} \\to \\mathcal{F}^\\#", "\\to \\mathcal{G}$ equals the given map." ], "refs": [], "proofs": [ { "contents": [ "Same as the proof of Proposition \\ref{proposition-sheafification-adjoint}." ], "refs": [ "sites-proposition-sheafification-adjoint" ], "ref_ids": [ 8640 ] } ], "ref_ids": [] }, { "id": 8724, "type": "theorem", "label": "examples-defos-lemma-finite-projective-modules-RS", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-finite-projective-modules-RS", "contents": [ "Example \\ref{example-finite-projective-modules}", "satisfies the Rim-Schlessinger condition (RS).", "In particular, $\\Deformationcategory_V$ is a deformation category", "for any finite dimensional vector space $V$ over $k$." ], "refs": [], "proofs": [ { "contents": [ "Let $A_1 \\to A$ and $A_2 \\to A$ be morphisms of $\\mathcal{C}_\\Lambda$.", "Assume $A_2 \\to A$ is surjective. According to", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-RS-2-categorical}", "it suffices to show that the functor", "$\\mathcal{F}(A_1 \\times_A A_2) \\to", "\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$", "is an equivalence of categories.", "\\medskip\\noindent", "Thus we have to show that the category of finite projective modules", "over $A_1 \\times_A A_2$ is equivalent to the fibre product", "of the categories of finite projective modules over $A_1$ and $A_2$", "over the category of finite projective modules over $A$.", "This is a special case of More on Algebra, Lemma", "\\ref{more-algebra-lemma-finitely-presented-module-over-fibre-product}.", "We recall that the inverse functor sends the triple", "$(M_1, M_2, \\varphi)$ where", "$M_1$ is a finite projective $A_1$-module,", "$M_2$ is a finite projective $A_2$-module, and", "$\\varphi : M_1 \\otimes_{A_1} A \\to M_2 \\otimes_{A_2} A$", "is an isomorphism of $A$-module, to the finite projective", "$A_1 \\times_A A_2$-module $M_1 \\times_\\varphi M_2$." ], "refs": [ "formal-defos-lemma-RS-2-categorical", "more-algebra-lemma-finitely-presented-module-over-fibre-product" ], "ref_ids": [ 3468, 9825 ] } ], "ref_ids": [] }, { "id": 8725, "type": "theorem", "label": "examples-defos-lemma-finite-projective-modules-TI", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-finite-projective-modules-TI", "contents": [ "In Example \\ref{example-finite-projective-modules}", "let $V$ be a finite dimensional $k$-vector space. Then", "$$", "T\\Deformationcategory_V = (0)", "\\quad\\text{and}\\quad", "\\text{Inf}(\\Deformationcategory_V) = \\text{End}_k(V)", "$$", "are finite dimensional." ], "refs": [], "proofs": [ { "contents": [ "With $\\mathcal{F}$ as in Example \\ref{example-finite-projective-modules}", "set $x_0 = (k, V) \\in \\Ob(\\mathcal{F}(k))$.", "Recall that $T\\Deformationcategory_V = T_{x_0}\\mathcal{F}$", "is the set of isomorphism", "classes of pairs $(x, \\alpha)$ consisting of an object $x$ of $\\mathcal{F}", "$ over the dual numbers $k[\\epsilon]$ and a morphism", "$\\alpha : x \\to x_0$ of $\\mathcal{F}$ lying over $k[\\epsilon] \\to k$.", "\\medskip\\noindent", "Up to isomorphism, there is a unique pair $(M, \\alpha)$ consisting of a", "finite projective module $M$ over $k[\\epsilon]$", "and $k[\\epsilon]$-linear map $\\alpha : M \\to V$", "which induces an isomorphism $M \\otimes_{k[\\epsilon]} k \\to V$.", "For example, if $V = k^{\\oplus n}$, then we take", "$M = k[\\epsilon]^{\\oplus n}$ with the obvious map $\\alpha$.", "\\medskip\\noindent", "Similarly, $\\text{Inf}(\\Deformationcategory_V) = \\text{Inf}_{x_0}(\\mathcal{F})$", "is the set of automorphisms", "of the trivial deformation $x'_0$ of $x_0$ over $k[\\epsilon]$.", "See Formal Deformation Theory, Definition", "\\ref{formal-defos-definition-infinitesimal-auts} for details.", "\\medskip\\noindent", "Given $(M, \\alpha)$ as in the second paragraph, we see that an element of", "$\\text{Inf}_{x_0}(\\mathcal{F})$ is an automorphism $\\gamma : M \\to M$ with", "$\\gamma \\bmod \\epsilon = \\text{id}$. Then we can write", "$\\gamma = \\text{id}_M + \\epsilon \\psi$ where", "$\\psi : M/\\epsilon M \\to M/\\epsilon M$ is $k$-linear.", "Using $\\alpha$ we can think of $\\psi$ as an element of", "$\\text{End}_k(V)$ and this finishes the proof." ], "refs": [ "formal-defos-definition-infinitesimal-auts" ], "ref_ids": [ 3533 ] } ], "ref_ids": [] }, { "id": 8726, "type": "theorem", "label": "examples-defos-lemma-representations-RS", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-representations-RS", "contents": [ "Example \\ref{example-representations}", "satisfies the Rim-Schlessinger condition (RS).", "In particular, $\\Deformationcategory_{V, \\rho_0}$ is a deformation category", "for any finite dimensional representation", "$\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$." ], "refs": [], "proofs": [ { "contents": [ "Let $A_1 \\to A$ and $A_2 \\to A$ be morphisms of $\\mathcal{C}_\\Lambda$.", "Assume $A_2 \\to A$ is surjective. According to", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-RS-2-categorical}", "it suffices to show that the functor", "$\\mathcal{F}(A_1 \\times_A A_2) \\to", "\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$", "is an equivalence of categories.", "\\medskip\\noindent", "Consider an object", "$$", "((A_1, M_1, \\rho_1), (A_2, M_2, \\rho_2), (\\text{id}_A, \\varphi))", "$$", "of the category $\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$.", "Then, as seen in the proof of Lemma \\ref{lemma-finite-projective-modules-RS},", "we can consider the finite projective", "$A_1 \\times_A A_2$-module $M_1 \\times_\\varphi M_2$.", "Since $\\varphi$ is compatible with the given actions we obtain", "$$", "\\rho_1 \\times \\rho_2 : \\Gamma \\longrightarrow", "\\text{GL}_{A_1 \\times_A A_2}(M_1 \\times_\\varphi M_2)", "$$", "Then $(M_1 \\times_\\varphi M_2, \\rho_1 \\times \\rho_2)$", "is an object of $\\mathcal{F}(A_1 \\times_A A_2)$.", "This construction determines a quasi-inverse to our functor." ], "refs": [ "formal-defos-lemma-RS-2-categorical", "examples-defos-lemma-finite-projective-modules-RS" ], "ref_ids": [ 3468, 8724 ] } ], "ref_ids": [] }, { "id": 8727, "type": "theorem", "label": "examples-defos-lemma-representations-TI", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-representations-TI", "contents": [ "In Example \\ref{example-representations} let ", "$\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$", "be a finite dimensional representation. Then", "$$", "T\\Deformationcategory_{V, \\rho_0} = \\Ext^1_{k[\\Gamma]}(V, V) =", "H^1(\\Gamma, \\text{End}_k(V))", "\\quad\\text{and}\\quad", "\\text{Inf}(\\Deformationcategory_{V, \\rho_0}) = H^0(\\Gamma, \\text{End}_k(V))", "$$", "Thus $\\text{Inf}(\\Deformationcategory_{V, \\rho_0})$", "is always finite dimensional", "and $T\\Deformationcategory_{V, \\rho_0}$ is finite dimensional", "if $\\Gamma$ is finitely generated." ], "refs": [], "proofs": [ { "contents": [ "We first deal with the infinitesimal automorphisms.", "Let $M = V \\otimes_k k[\\epsilon]$ with induced action", "$\\rho_0' : \\Gamma \\to \\text{GL}_n(M)$.", "Then an infinitesimal automorphism, i.e., an element of", "$\\text{Inf}(\\Deformationcategory_{V, \\rho_0})$,", "is given by an automorphism", "$\\gamma = \\text{id} + \\epsilon \\psi : M \\to M$", "as in the proof of Lemma \\ref{lemma-finite-projective-modules-TI},", "where moreover $\\psi$ has to commute", "with the action of $\\Gamma$ (given by $\\rho_0$).", "Thus we see that", "$$", "\\text{Inf}(\\Deformationcategory_{V, \\rho_0}) = H^0(\\Gamma, \\text{End}_k(V))", "$$", "as predicted in the lemma.", "\\medskip\\noindent", "Next, let $(k[\\epsilon], M, \\rho)$ be an object of $\\mathcal{F}$", "over $k[\\epsilon]$ and let $\\alpha : M \\to V$ be a $\\Gamma$-equivariant map", "inducing an isomorphism $M/\\epsilon M \\to V$.", "Since $M$ is free as a $k[\\epsilon]$-module we obtain", "an extension of $\\Gamma$-modules", "$$", "0 \\to V \\to M \\xrightarrow{\\alpha} V \\to 0", "$$", "We omit the detailed construction of the map on the left.", "Conversely, if we have an extension of $\\Gamma$-modules as", "above, then we can use this to make a $k[\\epsilon]$-module", "structure on $M$ and get an object of $\\mathcal{F}(k[\\epsilon])$", "together with a map $\\alpha$ as above.", "It follows that", "$$", "T\\Deformationcategory_{V, \\rho_0} = \\Ext^1_{k[\\Gamma]}(V, V)", "$$", "as predicted in the lemma. This is equal to", "$H^1(\\Gamma, \\text{End}_k(V))$ by", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-ext-modules-hom}.", "\\medskip\\noindent", "The statement on dimensions follows from", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-finite-dim-group-cohomology}." ], "refs": [ "examples-defos-lemma-finite-projective-modules-TI", "etale-cohomology-lemma-ext-modules-hom", "etale-cohomology-lemma-finite-dim-group-cohomology" ], "ref_ids": [ 8725, 6486, 6487 ] } ], "ref_ids": [] }, { "id": 8728, "type": "theorem", "label": "examples-defos-lemma-representations-hull", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-representations-hull", "contents": [ "In Example \\ref{example-representations} assume $\\Gamma$ finitely generated.", "Let $\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$ be a finite dimensional representation.", "Assume $\\Lambda$ is a complete local ring with residue field $k$", "(the classical case). Then the functor", "$$", "F : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad", "A \\longmapsto \\Ob(\\Deformationcategory_{V, \\rho_0}(A))/\\cong", "$$", "of isomorphism classes of objects has a hull. If", "$H^0(\\Gamma, \\text{End}_k(V)) = k$, then $F$ is", "prorepresentable." ], "refs": [], "proofs": [ { "contents": [ "The existence of a hull follows from Lemmas \\ref{lemma-representations-RS} and", "\\ref{lemma-representations-TI} and", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-RS-implies-S1-S2}", "and Remark \\ref{formal-defos-remark-compose-minimal-into-iso-classes}.", "\\medskip\\noindent", "Assume $H^0(\\Gamma, \\text{End}_k(V)) = k$. To see that $F$", "is prorepresentable it suffices to show that $F$ is a", "deformation functor, see Formal Deformation Theory, Theorem", "\\ref{formal-defos-theorem-Schlessinger-prorepresentability}.", "In other words, we have to show $F$ satisfies (RS).", "For this we can use the criterion of Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-RS-associated-functor}.", "The required surjectivity of automorphism groups will follow if we", "show that", "$$", "A \\cdot \\text{id}_M =", "\\text{End}_{A[\\Gamma]}(M)", "$$", "for any object $(A, M, \\rho)$ of $\\mathcal{F}$ such that", "$M \\otimes_A k$ is isomorphic to $V$ as a representation of $\\Gamma$.", "Since the left hand side is contained in the right hand side,", "it suffices to show", "$\\text{length}_A \\text{End}_{A[\\Gamma]}(M) \\leq \\text{length}_A A$.", "Choose pairwise distinct ideals", "$(0) = I_n \\subset \\ldots \\subset I_1 \\subset A$", "with $n = \\text{length}(A)$. By correspondingly filtering", "$M$, we see that it suffices to prove $\\Hom_{A[\\Gamma]}(M, I_tM/I_{t + 1}M)$", "has length $1$. Since $I_tM/I_{t + 1}M \\cong M \\otimes_A k$", "and since any $A[\\Gamma]$-module map $M \\to M \\otimes_A k$ factors", "uniquely through the quotient map $M \\to M \\otimes_A k$", "to give an element of", "$$", "\\text{End}_{A[\\Gamma]}(M \\otimes_A k) = \\text{End}_{k[\\Gamma]}(V) = k", "$$", "we conclude." ], "refs": [ "examples-defos-lemma-representations-RS", "examples-defos-lemma-representations-TI", "formal-defos-lemma-RS-implies-S1-S2", "formal-defos-remark-compose-minimal-into-iso-classes", "formal-defos-theorem-Schlessinger-prorepresentability", "formal-defos-lemma-RS-associated-functor" ], "ref_ids": [ 8726, 8727, 3469, 3567, 3411, 3470 ] } ], "ref_ids": [] }, { "id": 8729, "type": "theorem", "label": "examples-defos-lemma-continuous-representations-RS", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-continuous-representations-RS", "contents": [ "Example \\ref{example-continuous-representations}", "satisfies the Rim-Schlessinger condition (RS).", "In particular, $\\Deformationcategory_{V, \\rho_0}$ is a deformation category", "for any finite dimensional continuous representation", "$\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Lemma \\ref{lemma-representations-RS}." ], "refs": [ "examples-defos-lemma-representations-RS" ], "ref_ids": [ 8726 ] } ], "ref_ids": [] }, { "id": 8730, "type": "theorem", "label": "examples-defos-lemma-continuous-representations-TI", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-continuous-representations-TI", "contents": [ "In Example \\ref{example-continuous-representations} let", "$\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$ be a finite dimensional", "continuous representation. Then", "$$", "T\\Deformationcategory_{V, \\rho_0} = H^1(\\Gamma, \\text{End}_k(V))", "\\quad\\text{and}\\quad", "\\text{Inf}(\\Deformationcategory_{V, \\rho_0}) = H^0(\\Gamma, \\text{End}_k(V))", "$$", "Thus $\\text{Inf}(\\Deformationcategory_{V, \\rho_0})$", "is always finite dimensional", "and $T\\Deformationcategory_{V, \\rho_0}$ is finite dimensional", "if $\\Gamma$ is topologically finitely generated." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Lemma \\ref{lemma-representations-TI}." ], "refs": [ "examples-defos-lemma-representations-TI" ], "ref_ids": [ 8727 ] } ], "ref_ids": [] }, { "id": 8731, "type": "theorem", "label": "examples-defos-lemma-continuous-representations-hull", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-continuous-representations-hull", "contents": [ "In Example \\ref{example-continuous-representations} assume $\\Gamma$", "is topologically finitely generated.", "Let $\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$ be a finite dimensional representation.", "Assume $\\Lambda$ is a complete local ring with residue field $k$", "(the classical case). Then the functor", "$$", "F : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad", "A \\longmapsto \\Ob(\\Deformationcategory_{V, \\rho_0}(A))/\\cong", "$$", "of isomorphism classes of objects has a hull. If", "$H^0(\\Gamma, \\text{End}_k(V)) = k$, then $F$ is", "prorepresentable." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Lemma \\ref{lemma-representations-hull}." ], "refs": [ "examples-defos-lemma-representations-hull" ], "ref_ids": [ 8728 ] } ], "ref_ids": [] }, { "id": 8732, "type": "theorem", "label": "examples-defos-lemma-graded-algebras-RS", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-graded-algebras-RS", "contents": [ "Example \\ref{example-graded-algebras}", "satisfies the Rim-Schlessinger condition (RS).", "In particular, $\\Deformationcategory_P$ is a deformation category", "for any graded $k$-algebra $P$." ], "refs": [], "proofs": [ { "contents": [ "Let $A_1 \\to A$ and $A_2 \\to A$ be morphisms of $\\mathcal{C}_\\Lambda$.", "Assume $A_2 \\to A$ is surjective. According to", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-RS-2-categorical}", "it suffices to show that the functor", "$\\mathcal{F}(A_1 \\times_A A_2) \\to", "\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$", "is an equivalence of categories.", "\\medskip\\noindent", "Consider an object", "$$", "((A_1, P_1), (A_2, P_2), (\\text{id}_A, \\varphi))", "$$", "of the category $\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$.", "Then we consider $P_1 \\times_\\varphi P_2$. Since", "$\\varphi : P_1 \\otimes_{A_1} A \\to P_2 \\otimes_{A_2} A$", "is an isomorphism of graded algebras, we see that the graded pieces", "of $P_1 \\times_\\varphi P_2$ are finite projective $A_1 \\times_A A_2$-modules,", "see proof of Lemma \\ref{lemma-finite-projective-modules-RS}.", "Thus $P_1 \\times_\\varphi P_2$ is an object of $\\mathcal{F}(A_1 \\times_A A_2)$.", "This construction determines a quasi-inverse to our functor", "and the proof is complete." ], "refs": [ "formal-defos-lemma-RS-2-categorical", "examples-defos-lemma-finite-projective-modules-RS" ], "ref_ids": [ 3468, 8724 ] } ], "ref_ids": [] }, { "id": 8733, "type": "theorem", "label": "examples-defos-lemma-graded-algebras-TI", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-graded-algebras-TI", "contents": [ "In Example \\ref{example-graded-algebras} let $P$ be a graded $k$-algebra.", "Then", "$$", "T\\Deformationcategory_P", "\\quad\\text{and}\\quad", "\\text{Inf}(\\Deformationcategory_P) = \\text{Der}_k(P, P)", "$$", "are finite dimensional if $P$ is finitely generated over $k$." ], "refs": [], "proofs": [ { "contents": [ "We first deal with the infinitesimal automorphisms.", "Let $Q = P \\otimes_k k[\\epsilon]$.", "Then an element of $\\text{Inf}(\\Deformationcategory_P)$", "is given by an automorphism", "$\\gamma = \\text{id} + \\epsilon \\delta : Q \\to Q$", "as above where now $\\delta : P \\to P$.", "The fact that $\\gamma$ is graded implies that", "$\\delta$ is homogeneous of degree $0$.", "The fact that $\\gamma$ is $k$-linear implies that", "$\\delta$ is $k$-linear.", "The fact that $\\gamma$ is multiplicative implies that", "$\\delta$ is a $k$-derivation.", "Conversely, given a $k$-derivation $\\delta : P \\to P$", "homogeneous of degree $0$, we obtain an automorphism", "$\\gamma = \\text{id} + \\epsilon \\delta$ as above.", "Thus we see that", "$$", "\\text{Inf}(\\Deformationcategory_P) = \\text{Der}_k(P, P)", "$$", "as predicted in the lemma.", "Clearly, if $P$ is generated in degrees $P_i$,", "$0 \\leq i \\leq N$, then $\\delta$ is determined by", "the linear maps $\\delta_i : P_i \\to P_i$ for", "$0 \\leq i \\leq N$ and we see that", "$$", "\\dim_k \\text{Der}_k(P, P) < \\infty", "$$", "as desired.", "\\medskip\\noindent", "To finish the proof of the lemma we show that there is a finite", "dimensional deformation space. To do this we", "choose a presentation", "$$", "k[X_1, \\ldots, X_n]/(F_1, \\ldots, F_m) \\longrightarrow P", "$$", "of graded $k$-algebras where $\\deg(X_i) = d_i$ and", "$F_j$ is homogeneous of degree $e_j$.", "Let $Q$ be any graded $k[\\epsilon]$-algebra", "finite free in each degree which comes with an isomorphsm", "$\\alpha : Q/\\epsilon Q \\to P$ so that $(Q, \\alpha)$ defines", "an element of $T\\Deformationcategory_P$.", "Choose a homogeneous element $q_i \\in Q$ of degree $d_i$", "mapping to the image of $X_i$ in $P$.", "Then we obtain", "$$", "k[\\epsilon][X_1, \\ldots, X_n] \\longrightarrow Q,\\quad", "X_i \\longmapsto q_i", "$$", "and since $P = Q/\\epsilon Q$ this map is surjective by Nakayama's lemma.", "A small diagram chase shows we can choose homogeneous elements", "$F_{\\epsilon, j} \\in k[\\epsilon][X_1, \\ldots, X_n]$ of degree $e_j$", "mapping to zero in $Q$ and mapping to $F_j$ in $k[X_1, \\ldots, X_n]$.", "Then", "$$", "k[\\epsilon][X_1, \\ldots, X_n]/(F_{\\epsilon, 1}, \\ldots, F_{\\epsilon, m})", "\\longrightarrow Q", "$$", "is a presentation of $Q$ by flatness of $Q$ over $k[\\epsilon]$.", "Write", "$$", "F_{\\epsilon, j} = F_j + \\epsilon G_j", "$$", "There is some ambiguity in the vector $(G_1, \\ldots, G_m)$.", "First, using different choices of $F_{\\epsilon, j}$", "we can modify $G_j$ by an arbitrary element of degree $e_j$", "in the kernel of $k[X_1, \\ldots, X_n] \\to P$.", "Hence, instead of $(G_1, \\ldots, G_m)$, we remember the", "element", "$$", "(g_1, \\ldots, g_m) \\in P_{e_1} \\oplus \\ldots \\oplus P_{e_m}", "$$", "where $g_j$ is the image of $G_j$ in $P_{e_j}$.", "Moreover, if we change our choice of $q_i$ into $q_i + \\epsilon p_i$", "with $p_i$ of degree $d_i$ then a computation (omitted) shows", "that $g_j$ changes into", "$$", "g_j^{new} = g_j - \\sum\\nolimits_{i = 1}^n p_i \\partial F_j / \\partial X_i", "$$", "We conclude that the isomorphism class of $Q$ is determined by the", "image of the vector $(G_1, \\ldots, G_m)$ in the $k$-vector space", "$$", "W = \\Coker(P_{d_1} \\oplus \\ldots \\oplus P_{d_n}", "\\xrightarrow{(\\frac{\\partial F_j}{\\partial X_i})}", "P_{e_1} \\oplus \\ldots \\oplus P_{e_m})", "$$", "In this way we see that we obtain an injection", "$$", "T\\Deformationcategory_P \\longrightarrow W", "$$", "Since $W$ visibly has finite dimension, we conclude that the lemma is true." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8734, "type": "theorem", "label": "examples-defos-lemma-graded-algebras-hull", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-graded-algebras-hull", "contents": [ "In Example \\ref{example-graded-algebras} assume $P$ is a finitely generated", "graded $k$-algebra. Assume $\\Lambda$ is a complete local ring", "with residue field $k$", "(the classical case). Then the functor", "$$", "F : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad", "A \\longmapsto \\Ob(\\Deformationcategory_P(A))/\\cong", "$$", "of isomorphism classes of objects has a hull." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from Lemmas \\ref{lemma-graded-algebras-RS} and", "\\ref{lemma-graded-algebras-TI} and", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-RS-implies-S1-S2}", "and Remark \\ref{formal-defos-remark-compose-minimal-into-iso-classes}." ], "refs": [ "examples-defos-lemma-graded-algebras-RS", "examples-defos-lemma-graded-algebras-TI", "formal-defos-lemma-RS-implies-S1-S2", "formal-defos-remark-compose-minimal-into-iso-classes" ], "ref_ids": [ 8732, 8733, 3469, 3567 ] } ], "ref_ids": [] }, { "id": 8735, "type": "theorem", "label": "examples-defos-lemma-rings-RS", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-rings-RS", "contents": [ "Example \\ref{example-rings}", "satisfies the Rim-Schlessinger condition (RS).", "In particular, $\\Deformationcategory_P$ is a deformation category", "for any $k$-algebra $P$." ], "refs": [], "proofs": [ { "contents": [ "Let $A_1 \\to A$ and $A_2 \\to A$ be morphisms of $\\mathcal{C}_\\Lambda$.", "Assume $A_2 \\to A$ is surjective. According to", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-RS-2-categorical}", "it suffices to show that the functor", "$\\mathcal{F}(A_1 \\times_A A_2) \\to", "\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$", "is an equivalence of categories.", "This is a special case of More on Algebra, Lemma", "\\ref{more-algebra-lemma-properties-algebras-over-fibre-product}." ], "refs": [ "formal-defos-lemma-RS-2-categorical", "more-algebra-lemma-properties-algebras-over-fibre-product" ], "ref_ids": [ 3468, 9831 ] } ], "ref_ids": [] }, { "id": 8736, "type": "theorem", "label": "examples-defos-lemma-rings-TI", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-rings-TI", "contents": [ "In Example \\ref{example-rings} let $P$ be a $k$-algebra. Then", "$$", "T\\Deformationcategory_P = \\text{Ext}^1_P(\\NL_{P/k}, P)", "\\quad\\text{and}\\quad", "\\text{Inf}(\\Deformationcategory_P) = \\text{Der}_k(P, P)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\text{Inf}(\\Deformationcategory_P)$ is the set of", "automorphisms of the trivial deformation", "$P[\\epsilon] = P \\otimes_k k[\\epsilon]$ of $P$ to $k[\\epsilon]$", "equal to the identity modulo $\\epsilon$.", "By Deformation Theory, Lemma \\ref{defos-lemma-huge-diagram}", "this is equal to $\\Hom_P(\\Omega_{P/k}, P)$ which in turn is", "equal to $\\text{Der}_k(P, P)$ by", "Algebra, Lemma \\ref{algebra-lemma-universal-omega}.", "\\medskip\\noindent", "Recall that $T\\Deformationcategory_P$ is the set of isomorphism classes", "of flat deformations $Q$ of $P$ to $k[\\epsilon]$, more precisely,", "the set of isomorphism classes of $\\Deformationcategory_P(k[\\epsilon])$.", "Recall that a $k[\\epsilon]$-algebra $Q$ with $Q/\\epsilon Q = P$", "is flat over $k[\\epsilon]$ if and only if", "$$", "0 \\to P \\xrightarrow{\\epsilon} Q \\to P \\to 0", "$$", "is exact. This is proven in More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-deform} and more generally in", "Deformation Theory, Lemma \\ref{defos-lemma-deform-module}.", "Thus we may apply", "Deformation Theory, Lemma \\ref{defos-lemma-choices}", "to see that the set of isomorphism classes of such", "deformations is equal to $\\text{Ext}^1_P(\\NL_{P/k}, P)$." ], "refs": [ "defos-lemma-huge-diagram", "algebra-lemma-universal-omega", "more-morphisms-lemma-deform", "defos-lemma-deform-module", "defos-lemma-choices" ], "ref_ids": [ 13369, 1129, 13723, 13379, 13371 ] } ], "ref_ids": [] }, { "id": 8737, "type": "theorem", "label": "examples-defos-lemma-smooth", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-smooth", "contents": [ "In Example \\ref{example-rings} let $P$ be a smooth $k$-algebra. Then", "$T\\Deformationcategory_P = (0)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-rings-TI} we have to show", "$\\text{Ext}^1_P(\\NL_{P/k}, P) = (0)$.", "Since $k \\to P$ is smooth $\\NL_{P/k}$ is quasi-isomorphic to the", "complex consisting of a finite projective", "$P$-module placed in degree $0$." ], "refs": [ "examples-defos-lemma-rings-TI" ], "ref_ids": [ 8736 ] } ], "ref_ids": [] }, { "id": 8738, "type": "theorem", "label": "examples-defos-lemma-finite-type-rings-TI", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-finite-type-rings-TI", "contents": [ "In Lemma \\ref{lemma-rings-TI} if $P$ is a finite type $k$-algebra, then", "\\begin{enumerate}", "\\item $\\text{Inf}(\\Deformationcategory_P)$ is finite dimensional if and only if", "$\\dim(P) = 0$, and", "\\item $T\\Deformationcategory_P$ is finite dimensional if", "$\\Spec(P) \\to \\Spec(k)$ is smooth except at a finite number of points.", "\\end{enumerate}" ], "refs": [ "examples-defos-lemma-rings-TI" ], "proofs": [ { "contents": [ "Proof of (1). We view $\\text{Der}_k(P, P)$ as a $P$-module.", "If it has finite dimension over $k$, then it has finite length", "as a $P$-module, hence it is supported in finitely many", "closed points of $\\Spec(P)$", "(Algebra, Lemma \\ref{algebra-lemma-simple-pieces}).", "Since $\\text{Der}_k(P, P) = \\Hom_P(\\Omega_{P/k}, P)$", "we see that", "$\\text{Der}_k(P, P)_\\mathfrak p = \\text{Der}_k(P_\\mathfrak p, P_\\mathfrak p)$", "for any prime $\\mathfrak p \\subset P$", "(this uses Algebra, Lemmas", "\\ref{algebra-lemma-differentials-localize},", "\\ref{algebra-lemma-differentials-finitely-presented}, and", "\\ref{algebra-lemma-hom-from-finitely-presented}).", "Let $\\mathfrak p$ be a minimal prime ideal of $P$", "corresponding to an irreducible component of dimension $d > 0$.", "Then $P_\\mathfrak p$ is an Artinian local ring", "essentially of finite type over $k$ with residue field", "and $\\Omega_{P_\\mathfrak p/k}$ is nonzero for example by", "Algebra, Lemma \\ref{algebra-lemma-characterize-smooth-over-field}.", "Any nonzero finite module over an Artinian local ring", "has both a sub and a quotient module isomorphic to the residue field.", "Thus we find that", "$\\text{Der}_k(P_\\mathfrak p, P_\\mathfrak p) =", "\\Hom_{P_\\mathfrak p}(\\Omega_{P_\\mathfrak p/k}, P_\\mathfrak p)$", "is nonzero too. Combining all of the above we find that (1) is true.", "\\medskip\\noindent", "Proof of (2). For a prime $\\mathfrak p$ of $P$ we will use that", "$\\NL_{P_\\mathfrak p/k} = (\\NL_{P/k})_\\mathfrak p$", "(Algebra, Lemma \\ref{algebra-lemma-localize-NL})", "and we will", "use that", "$\\text{Ext}_P^1(\\NL_{P/k}, P)_\\mathfrak p =", "\\text{Ext}_{P_\\mathfrak p}^1(\\NL_{P_\\mathfrak p/k}, P_\\mathfrak p)$", "(More on Algebra, Lemma", "\\ref{more-algebra-lemma-pseudo-coherence-and-base-change-ext}).", "Given a prime $\\mathfrak p \\subset P$", "then $k \\to P$ is smooth at $\\mathfrak p$ if and only if", "$(\\NL_{P/k})_\\mathfrak p$ is quasi-isomorphic", "to a finite projective module placed in degree $0$ (this follows", "immediately from the definition of a smooth ring map but it also", "follows from the stronger Algebra, Lemma \\ref{algebra-lemma-smooth-at-point}).", "\\medskip\\noindent", "Assume that $P$ is smooth over $k$ at all but finitely many primes.", "Then these ``bad'' primes are maximal ideals", "$\\mathfrak m_1, \\ldots, \\mathfrak m_n \\subset P$ by", "Algebra, Lemma \\ref{algebra-lemma-finite-type-algebra-finite-nr-primes}", "and the fact that the ``bad'' primes form a closed subset of $\\Spec(P)$.", "For $\\mathfrak p \\not \\in \\{\\mathfrak m_1, \\ldots, \\mathfrak m_n\\}$", "we have $\\text{Ext}^1_P(\\NL_{P/k}, P)_\\mathfrak p = 0$ by the results above.", "Thus $\\text{Ext}^1_P(\\NL_{P/k}, P)$ is a finite $P$-module", "whose support is contained in $\\{\\mathfrak m_1, \\ldots, \\mathfrak m_r\\}$.", "By Algebra, Proposition", "\\ref{algebra-proposition-minimal-primes-associated-primes}", "for example, we find that the dimension over $k$ of", "$\\text{Ext}^1_P(\\NL_{P/k}, P)$ is a finite integer combination", "of $\\dim_k \\kappa(\\mathfrak m_i)$ and hence finite by", "the Hilbert Nullstellensatz", "(Algebra, Theorem \\ref{algebra-theorem-nullstellensatz})." ], "refs": [ "algebra-lemma-simple-pieces", "algebra-lemma-differentials-localize", "algebra-lemma-differentials-finitely-presented", "algebra-lemma-hom-from-finitely-presented", "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-localize-NL", "more-algebra-lemma-pseudo-coherence-and-base-change-ext", "algebra-lemma-smooth-at-point", "algebra-lemma-finite-type-algebra-finite-nr-primes", "algebra-proposition-minimal-primes-associated-primes", "algebra-theorem-nullstellensatz" ], "ref_ids": [ 638, 1134, 1141, 353, 1223, 1161, 10165, 1196, 689, 1412, 316 ] } ], "ref_ids": [ 8736 ] }, { "id": 8739, "type": "theorem", "label": "examples-defos-lemma-rings-hull", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-rings-hull", "contents": [ "In Example \\ref{example-rings} assume $P$ is a finite type", "$k$-algebra such that $\\Spec(P) \\to \\Spec(k)$ is smooth except", "at a finite number of points.", "Assume $\\Lambda$ is a complete local ring with residue field $k$", "(the classical case). Then the functor", "$$", "F : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad", "A \\longmapsto \\Ob(\\Deformationcategory_P(A))/\\cong", "$$", "of isomorphism classes of objects has a hull." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from Lemmas \\ref{lemma-rings-RS} and", "\\ref{lemma-finite-type-rings-TI} and", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-RS-implies-S1-S2}", "and Remark \\ref{formal-defos-remark-compose-minimal-into-iso-classes}." ], "refs": [ "examples-defos-lemma-rings-RS", "examples-defos-lemma-finite-type-rings-TI", "formal-defos-lemma-RS-implies-S1-S2", "formal-defos-remark-compose-minimal-into-iso-classes" ], "ref_ids": [ 8735, 8738, 3469, 3567 ] } ], "ref_ids": [] }, { "id": 8740, "type": "theorem", "label": "examples-defos-lemma-localization", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-localization", "contents": [ "In Example \\ref{example-rings} let $P$ be a $k$-algebra.", "Let $S \\subset P$ be a multiplicative subset. There is a natural functor", "$$", "\\Deformationcategory_P \\longrightarrow \\Deformationcategory_{S^{-1}P}", "$$", "of deformation categories." ], "refs": [], "proofs": [ { "contents": [ "Given a deformation of $P$ we can take the localization", "of it to get a deformation of the localization; this is", "clear and we encourage the reader to skip the proof. More precisely,", "let $(A, Q) \\to (k, P)$ be a morphism in $\\mathcal{F}$, i.e.,", "an object of $\\Deformationcategory_P$. Let $S_Q \\subset Q$ be the", "inverse image of $S$. Then", "Hence $(A, S_Q^{-1}Q) \\to (k, S^{-1}P)$", "is the desired object of $\\Deformationcategory_{S^{-1}P}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8741, "type": "theorem", "label": "examples-defos-lemma-henselization", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-henselization", "contents": [ "In Example \\ref{example-rings} let $P$ be a $k$-algebra.", "Let $J \\subset P$ be an ideal.", "Denote $(P^h, J^h)$ the henselization of the pair $(P, J)$.", "There is a natural functor", "$$", "\\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^h}", "$$", "of deformation categories." ], "refs": [], "proofs": [ { "contents": [ "Given a deformation of $P$ we can take the henselization", "of it to get a deformation of the henselization; this is", "clear and we encourage the reader to skip the proof. More precisely,", "let $(A, Q) \\to (k, P)$ be a morphism in $\\mathcal{F}$, i.e.,", "an object of $\\Deformationcategory_P$. Denote $J_Q \\subset Q$ the inverse", "image of $J$ in $Q$. Let $(Q^h, J_Q^h)$ be the henselization of", "the pair $(Q, J_Q)$. Recall that $Q \\to Q^h$ is flat", "(More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-flat})", "and hence $Q^h$ is flat over $A$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-integral}", "we see that the map $Q^h \\to P^h$ induces an isomorphism", "$Q^h \\otimes_A k = Q^h \\otimes_Q P = P^h$.", "Hence $(A, Q^h) \\to (k, P^h)$ is the desired object of", "$\\Deformationcategory_{P^h}$." ], "refs": [ "more-algebra-lemma-henselization-flat", "more-algebra-lemma-henselization-integral" ], "ref_ids": [ 9872, 9877 ] } ], "ref_ids": [] }, { "id": 8742, "type": "theorem", "label": "examples-defos-lemma-strict-henselization", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-strict-henselization", "contents": [ "In Example \\ref{example-rings} let $P$ be a $k$-algebra.", "Assume $P$ is a local ring and let $P^{sh}$ be a strict henselization of $P$.", "There is a natural functor", "$$", "\\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^{sh}}", "$$", "of deformation categories." ], "refs": [], "proofs": [ { "contents": [ "Given a deformation of $P$ we can take the strict henselization", "of it to get a deformation of the strict henselization; this is", "clear and we encourage the reader to skip the proof. More precisely,", "let $(A, Q) \\to (k, P)$ be a morphism in $\\mathcal{F}$, i.e.,", "an object of $\\Deformationcategory_P$. Since the kernel of the surjection", "$Q \\to P$ is nilpotent, we find that $Q$ is a local ring with the", "same residue field as $P$. Let $Q^{sh}$ be the strict henselization", "of $Q$. Recall that $Q \\to Q^{sh}$ is flat", "(More on Algebra, Lemma \\ref{more-algebra-lemma-dumb-properties-henselization})", "and hence $Q^{sh}$ is flat over $A$.", "By Algebra, Lemma \\ref{algebra-lemma-quotient-strict-henselization}", "we see that the map $Q^{sh} \\to P^{sh}$ induces an isomorphism", "$Q^{sh} \\otimes_A k = Q^{sh} \\otimes_Q P = P^{sh}$.", "Hence $(A, Q^{sh}) \\to (k, P^{sh})$ is the desired object of", "$\\Deformationcategory_{P^{sh}}$." ], "refs": [ "more-algebra-lemma-dumb-properties-henselization", "algebra-lemma-quotient-strict-henselization" ], "ref_ids": [ 10055, 1307 ] } ], "ref_ids": [] }, { "id": 8743, "type": "theorem", "label": "examples-defos-lemma-completion", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-completion", "contents": [ "In Example \\ref{example-rings} let $P$ be a $k$-algebra.", "Assume $P$ Noetherian and let $J \\subset P$ be an ideal.", "Denote $P^\\wedge$ the $J$-adic completion.", "There is a natural functor", "$$", "\\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^\\wedge}", "$$", "of deformation categories." ], "refs": [], "proofs": [ { "contents": [ "Given a deformation of $P$ we can take the completion", "of it to get a deformation of the completion; this is", "clear and we encourage the reader to skip the proof. More precisely,", "let $(A, Q) \\to (k, P)$ be a morphism in $\\mathcal{F}$, i.e.,", "an object of $\\Deformationcategory_P$. Observe that $Q$ is a Noetherian", "ring: the kernel of the surjective ring map $Q \\to P$ is", "nilpotent and finitely generated and $P$ is Noetherian; apply", "Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian}.", "Denote $J_Q \\subset Q$ the inverse", "image of $J$ in $Q$. Let $Q^\\wedge$ be the $J_Q$-adic completion of $Q$.", "Recall that $Q \\to Q^\\wedge$ is flat", "(Algebra, Lemma \\ref{algebra-lemma-completion-flat})", "and hence $Q^\\wedge$ is flat over $A$.", "The induced map $Q^\\wedge \\to P^\\wedge$ induces an isomorphism", "$Q^\\wedge \\otimes_A k = Q^\\wedge \\otimes_Q P = P^\\wedge$ by", "Algebra, Lemma \\ref{algebra-lemma-completion-tensor} for example.", "Hence $(A, Q^\\wedge) \\to (k, P^\\wedge)$", "is the desired object of $\\Deformationcategory_{P^\\wedge}$." ], "refs": [ "algebra-lemma-completion-Noetherian", "algebra-lemma-completion-flat", "algebra-lemma-completion-tensor" ], "ref_ids": [ 873, 870, 869 ] } ], "ref_ids": [] }, { "id": 8744, "type": "theorem", "label": "examples-defos-lemma-power-series-rings-TI", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-power-series-rings-TI", "contents": [ "In Lemma \\ref{lemma-rings-TI} if $P = k[[x_1, \\ldots, x_n]]/(f)$", "for some nonzero $f \\in (x_1, \\ldots, x_n)^2$, then", "\\begin{enumerate}", "\\item $\\text{Inf}(\\Deformationcategory_P)$ is finite dimensional", "if and only if $n = 1$, and", "\\item $T\\Deformationcategory_P$ is finite dimensional if", "$$", "\\sqrt{(f, \\partial f/\\partial x_1, \\ldots, \\partial f/\\partial x_n)} =", "(x_1, \\ldots, x_n)", "$$", "\\end{enumerate}" ], "refs": [ "examples-defos-lemma-rings-TI" ], "proofs": [ { "contents": [ "Proof of (1). Consider the derivations $\\partial/\\partial x_i$ of", "$k[[x_1, \\ldots, x_n]]$ over $k$. Write $f_i = \\partial f/\\partial x_i$.", "The derivation", "$$", "\\theta = \\sum h_i \\partial/\\partial x_i", "$$", "of $k[[x_1, \\ldots, x_n]]$", "induces a derivation of $P = k[[x_1, \\ldots, x_n]]/(f)$", "if and only if", "$\\sum h_i f_i \\in (f)$. Moreover, the induced derivation of $P$", "is zero if and only if $h_i \\in (f)$ for $i = 1, \\ldots, n$.", "Thus we find", "$$", "\\Ker((f_1, \\ldots, f_n) : P^{\\oplus n} \\longrightarrow P) \\subset", "\\text{Der}_k(P, P)", "$$", "The left hand side is a finite dimensional $k$-vector space only if", "$n = 1$; we omit the proof. We also leave it to the reader to see", "that the right hand side has finite dimension if $n = 1$.", "This proves (1).", "\\medskip\\noindent", "Proof of (2). Let $Q$ be a flat deformation of $P$ over $k[\\epsilon]$", "as in the proof of Lemma \\ref{lemma-rings-TI}. Choose lifts $q_i \\in Q$", "of the image of $x_i$ in $P$. Then $Q$ is a complete local ring", "with maximal ideal generated by $q_1, \\ldots, q_n$ and $\\epsilon$", "(small argument omitted). Thus we get a surjection", "$$", "k[\\epsilon][[x_1, \\ldots, x_n]] \\longrightarrow Q,\\quad", "x_i \\longmapsto q_i", "$$", "Choose an element of the form", "$f + \\epsilon g \\in k[\\epsilon][[x_1, \\ldots, x_n]]$", "mapping to zero in $Q$. Observe that $g$ is well defined modulo $(f)$.", "Since $Q$ is flat over $k[\\epsilon]$ we get", "$$", "Q = k[\\epsilon][[x_1, \\ldots, x_n]]/(f + \\epsilon g)", "$$", "Finally, if we changing the choice of $q_i$ amounts to", "changing the coordinates $x_i$ into $x_i + \\epsilon h_i$", "for some $h_i \\in k[[x_1, \\ldots, x_n]]$. Then", "$f + \\epsilon g$ changes into $f + \\epsilon (g + \\sum h_i f_i)$", "where $f_i = \\partial f/\\partial x_i$. Thus we see that the", "isomorphism class of the deformation $Q$ is determined", "by an element of", "$$", "k[[x_1, \\ldots, x_n]]/", "(f, \\partial f/\\partial x_1, \\ldots, \\partial f/\\partial x_n)", "$$", "This has finite dimension over $k$ if and only if", "its support is the closed point of $k[[x_1, \\ldots, x_n]]$", "if and only if", "$\\sqrt{(f, \\partial f/\\partial x_1, \\ldots, \\partial f/\\partial x_n)} =", "(x_1, \\ldots, x_n)$." ], "refs": [ "examples-defos-lemma-rings-TI" ], "ref_ids": [ 8736 ] } ], "ref_ids": [ 8736 ] }, { "id": 8745, "type": "theorem", "label": "examples-defos-lemma-schemes-RS", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-schemes-RS", "contents": [ "Example \\ref{example-schemes}", "satisfies the Rim-Schlessinger condition (RS).", "In particular, $\\Deformationcategory_X$ is a deformation category", "for any scheme $X$ over $k$." ], "refs": [], "proofs": [ { "contents": [ "Let $A_1 \\to A$ and $A_2 \\to A$ be morphisms of $\\mathcal{C}_\\Lambda$.", "Assume $A_2 \\to A$ is surjective. According to", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-RS-2-categorical}", "it suffices to show that the functor", "$\\mathcal{F}(A_1 \\times_A A_2) \\to", "\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$", "is an equivalence of categories.", "Observe that", "$$", "\\xymatrix{", "\\Spec(A) \\ar[r] \\ar[d] & \\Spec(A_2) \\ar[d] \\\\", "\\Spec(A_1) \\ar[r] &", "\\Spec(A_1 \\times_A A_2)", "}", "$$", "is a pushout diagram as in More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-pushout-along-thickening}.", "Thus the lemma is a special case of More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat}." ], "refs": [ "formal-defos-lemma-RS-2-categorical", "more-morphisms-lemma-pushout-along-thickening", "more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat" ], "ref_ids": [ 3468, 13762, 13765 ] } ], "ref_ids": [] }, { "id": 8746, "type": "theorem", "label": "examples-defos-lemma-schemes-TI", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-schemes-TI", "contents": [ "In Example \\ref{example-schemes} let $X$ be a scheme over $k$. Then", "$$", "\\text{Inf}(\\Deformationcategory_X) =", "\\text{Ext}^0_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X) =", "\\Hom_{\\mathcal{O}_X}(\\Omega_{X/k}, \\mathcal{O}_X) =", "\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)", "$$", "and", "$$", "T\\Deformationcategory_X =", "\\text{Ext}^1_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\text{Inf}(\\Deformationcategory_X)$ is the set of", "automorphisms of the trivial deformation", "$X' = X \\times_{\\Spec(k)} \\Spec(k[\\epsilon])$ of $X$ to $k[\\epsilon]$", "equal to the identity modulo $\\epsilon$.", "By Deformation Theory, Lemma \\ref{defos-lemma-deform}", "this is equal to $\\text{Ext}^0_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X)$.", "The equality $\\text{Ext}^0_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X) =", "\\Hom_{\\mathcal{O}_X}(\\Omega_{X/k}, \\mathcal{O}_X)$ follows from", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-netherlander-quasi-coherent}.", "The equality", "$\\Hom_{\\mathcal{O}_X}(\\Omega_{X/k}, \\mathcal{O}_X) =", "\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)$", "follows from Morphisms, Lemma", "\\ref{morphisms-lemma-universal-derivation-universal}.", "\\medskip\\noindent", "Recall that $T_{x_0}\\Deformationcategory_X$ is the set of isomorphism classes", "of flat deformations $X'$ of $X$ to $k[\\epsilon]$, more precisely,", "the set of isomorphism classes of $\\Deformationcategory_X(k[\\epsilon])$.", "Thus the second statement of the lemma follows from", "Deformation Theory, Lemma \\ref{defos-lemma-deform}." ], "refs": [ "defos-lemma-deform", "more-morphisms-lemma-netherlander-quasi-coherent", "morphisms-lemma-universal-derivation-universal", "defos-lemma-deform" ], "ref_ids": [ 13391, 13746, 5307, 13391 ] } ], "ref_ids": [] }, { "id": 8747, "type": "theorem", "label": "examples-defos-lemma-proper-schemes-TI", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-proper-schemes-TI", "contents": [ "In Lemma \\ref{lemma-schemes-TI} if $X$ is proper over $k$, then", "$\\text{Inf}(\\Deformationcategory_X)$ and $T\\Deformationcategory_X$ are", "finite dimensional." ], "refs": [ "examples-defos-lemma-schemes-TI" ], "proofs": [ { "contents": [ "By the lemma we have to show", "$\\Ext^1_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X)$ and", "$\\Ext^0_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X)$ are finite", "dimensional. By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-netherlander-fp}", "and the fact that $X$ is Noetherian, we see that", "$\\NL_{X/k}$ has coherent cohomology sheaves zero except", "in degrees $0$ and $-1$.", "By Derived Categories of Schemes, Lemma \\ref{perfect-lemma-ext-finite}", "the displayed $\\Ext$-groups are finite $k$-vector spaces", "and the proof is complete." ], "refs": [ "more-morphisms-lemma-netherlander-fp", "perfect-lemma-ext-finite" ], "ref_ids": [ 13747, 6988 ] } ], "ref_ids": [ 8746 ] }, { "id": 8748, "type": "theorem", "label": "examples-defos-lemma-schemes-hull", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-schemes-hull", "contents": [ "In Example \\ref{example-schemes} assume $X$ is a proper $k$-scheme.", "Assume $\\Lambda$ is a complete local ring with residue field $k$", "(the classical case). Then the functor", "$$", "F : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad", "A \\longmapsto \\Ob(\\Deformationcategory_X(A))/\\cong", "$$", "of isomorphism classes of objects has a hull. If", "$\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = 0$, then", "$F$ is prorepresentable." ], "refs": [], "proofs": [ { "contents": [ "The existence of a hull follows immediately from", "Lemmas \\ref{lemma-schemes-RS} and \\ref{lemma-proper-schemes-TI} and", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-RS-implies-S1-S2}", "and Remark \\ref{formal-defos-remark-compose-minimal-into-iso-classes}.", "\\medskip\\noindent", "Assume $\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = 0$. Then", "$\\Deformationcategory_X$ and $F$ are equivalent by", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-infdef-trivial}.", "Hence $F$ is a deformation functor (because $\\Deformationcategory_X$ is a", "deformation category) with finite tangent space and we can apply", "Formal Deformation Theory, Theorem", "\\ref{formal-defos-theorem-Schlessinger-prorepresentability}." ], "refs": [ "examples-defos-lemma-schemes-RS", "examples-defos-lemma-proper-schemes-TI", "formal-defos-lemma-RS-implies-S1-S2", "formal-defos-remark-compose-minimal-into-iso-classes", "formal-defos-lemma-infdef-trivial", "formal-defos-theorem-Schlessinger-prorepresentability" ], "ref_ids": [ 8745, 8747, 3469, 3567, 3481, 3411 ] } ], "ref_ids": [] }, { "id": 8749, "type": "theorem", "label": "examples-defos-lemma-open", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-open", "contents": [ "In Example \\ref{example-schemes} let $X$ be a scheme over $k$.", "Let $U \\subset X$ be an open subscheme.", "There is a natural functor", "$$", "\\Deformationcategory_X \\longrightarrow \\Deformationcategory_U", "$$", "of deformation categories." ], "refs": [], "proofs": [ { "contents": [ "Given a deformation of $X$ we can take the corresponding open", "of it to get a deformation of $U$. We omit the details." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8750, "type": "theorem", "label": "examples-defos-lemma-affine", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-affine", "contents": [ "In Example \\ref{example-schemes} let $X = \\Spec(P)$ be an", "affine scheme over $k$. With $\\Deformationcategory_P$ as in", "Example \\ref{example-rings} there is a natural equivalence", "$$", "\\Deformationcategory_X \\longrightarrow \\Deformationcategory_P", "$$", "of deformation categories." ], "refs": [], "proofs": [ { "contents": [ "The functor sends $(A, Y)$ to $\\Gamma(Y, \\mathcal{O}_Y)$.", "This works because", "any deformation of $X$ is affine by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-thickening-affine-scheme}." ], "refs": [ "more-morphisms-lemma-thickening-affine-scheme" ], "ref_ids": [ 13678 ] } ], "ref_ids": [] }, { "id": 8751, "type": "theorem", "label": "examples-defos-lemma-local-ring", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-local-ring", "contents": [ "In Example \\ref{example-schemes} let $X$ be a scheme over $k$", "Let $p \\in X$ be a point. With $\\Deformationcategory_{\\mathcal{O}_{X, p}}$", "as in Example \\ref{example-rings} there is a natural functor", "$$", "\\Deformationcategory_X", "\\longrightarrow", "\\Deformationcategory_{\\mathcal{O}_{X, p}}", "$$", "of deformation categories." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine open $U = \\Spec(P) \\subset X$ containing $p$.", "Then $\\mathcal{O}_{X, p}$ is a localization of $P$. We combine", "the functors from", "Lemmas \\ref{lemma-open}, \\ref{lemma-affine}, and \\ref{lemma-localization}." ], "refs": [ "examples-defos-lemma-open", "examples-defos-lemma-affine", "examples-defos-lemma-localization" ], "ref_ids": [ 8749, 8750, 8740 ] } ], "ref_ids": [] }, { "id": 8752, "type": "theorem", "label": "examples-defos-lemma-glueing", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-glueing", "contents": [ "In Situation \\ref{situation-glueing}", "there is an equivalence", "$$", "\\Deformationcategory_X =", "\\Deformationcategory_{P_1}", "\\times_{\\Deformationcategory_{P_{12}}}", "\\Deformationcategory_{P_2}", "$$", "of deformation categories, see Examples \\ref{example-schemes} and", "\\ref{example-rings}." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that the functors of Lemma \\ref{lemma-open}", "define an equivalence", "$$", "\\Deformationcategory_X \\longrightarrow", "\\Deformationcategory_{U_1}", "\\times_{\\Deformationcategory_{U_{12}}}", "\\Deformationcategory_{U_2}", "$$", "because then we can apply Lemma \\ref{lemma-affine} to translate into rings.", "To do this we construct a quasi-inverse. Denote", "$F_i : \\Deformationcategory_{U_i} \\to \\Deformationcategory_{U_{12}}$", "the functor of Lemma \\ref{lemma-open}.", "An object of the RHS is given by an $A$ in $\\mathcal{C}_\\Lambda$,", "objects $(A, V_1) \\to (k, U_1)$ and $(A, V_2) \\to (k, U_2)$, and", "a morphism", "$$", "g : F_1(A, V_1) \\to F_2(A, V_2)", "$$", "Now $F_i(A, V_i) = (A, V_{i, 3 - i})$ where $V_{i, 3 - i} \\subset V_i$", "is the open subscheme whose base change to $k$ is $U_{12} \\subset U_i$.", "The morphism $g$ defines an isomorphism", "$V_{1, 2} \\to V_{2, 1}$ of schemes over $A$ compatible", "with $\\text{id} : U_{12} \\to U_{12}$ over $k$.", "Thus $(\\{1, 2\\}, V_i, V_{i, 3 - i}, g, g^{-1})$ is a glueing", "data as in Schemes, Section \\ref{schemes-section-glueing-schemes}.", "Let $Y$ be the glueing, see Schemes, Lemma \\ref{schemes-lemma-glue}.", "Then $Y$ is a scheme over $A$ and the", "compatibilities mentioned above show that", "there is a canonical isomorphism", "$Y \\times_{\\Spec(A)} \\Spec(k) = X$.", "Thus $(A, Y) \\to (k, X)$ is an object of $\\Deformationcategory_X$.", "We omit the verification that this construction is a functor", "and is quasi-inverse to the given one." ], "refs": [ "examples-defos-lemma-open", "examples-defos-lemma-affine", "examples-defos-lemma-open", "schemes-lemma-glue" ], "ref_ids": [ 8749, 8750, 8749, 7686 ] } ], "ref_ids": [] }, { "id": 8753, "type": "theorem", "label": "examples-defos-lemma-schemes-morphisms-RS", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-schemes-morphisms-RS", "contents": [ "Example \\ref{example-schemes-morphisms}", "satisfies the Rim-Schlessinger condition (RS).", "In particular, $\\Deformationcategory_{X \\to Y}$ is a deformation category", "for any morphism of schemes $X \\to Y$ over $k$." ], "refs": [], "proofs": [ { "contents": [ "Let $A_1 \\to A$ and $A_2 \\to A$ be morphisms of $\\mathcal{C}_\\Lambda$.", "Assume $A_2 \\to A$ is surjective. According to", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-RS-2-categorical}", "it suffices to show that the functor", "$\\mathcal{F}(A_1 \\times_A A_2) \\to", "\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$", "is an equivalence of categories.", "Observe that", "$$", "\\xymatrix{", "\\Spec(A) \\ar[r] \\ar[d] & \\Spec(A_2) \\ar[d] \\\\", "\\Spec(A_1) \\ar[r] &", "\\Spec(A_1 \\times_A A_2)", "}", "$$", "is a pushout diagram as in More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-pushout-along-thickening}.", "Thus the lemma follows immediately from", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat}", "as this describes the category of schemes flat over $A_1 \\times_A A_2$", "as the fibre product of the category of schemes flat over $A_1$", "with the category of schemes flat over $A_2$ over the category of", "schemes flat over $A$." ], "refs": [ "formal-defos-lemma-RS-2-categorical", "more-morphisms-lemma-pushout-along-thickening", "more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat" ], "ref_ids": [ 3468, 13762, 13765 ] } ], "ref_ids": [] }, { "id": 8754, "type": "theorem", "label": "examples-defos-lemma-schemes-morphisms-TI", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-schemes-morphisms-TI", "contents": [ "In Example \\ref{example-schemes} let $f : X \\to Y$ be a morphism of schemes", "over $k$. There is a canonical exact sequence of $k$-vector spaces", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\text{Inf}(\\Deformationcategory_{X \\to Y}) \\ar[r] &", "\\text{Inf}(\\Deformationcategory_X \\times \\Deformationcategory_Y) \\ar[r] &", "\\text{Der}_k(\\mathcal{O}_Y, f_*\\mathcal{O}_X) \\ar[lld] \\\\", "& T\\Deformationcategory_{X \\to Y} \\ar[r] &", "T(\\Deformationcategory_X \\times \\Deformationcategory_Y) \\ar[r] &", "\\text{Ext}^1_{\\mathcal{O}_X}(Lf^*\\NL_{Y/k}, \\mathcal{O}_X)", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "The obvious map of deformation categories", "$\\Deformationcategory_{X \\to Y} \\to", "\\Deformationcategory_X \\times \\Deformationcategory_Y$", "gives two of the arrows in the exact sequence of the lemma.", "Recall that $\\text{Inf}(\\Deformationcategory_{X \\to Y})$", "is the set of automorphisms of the trivial deformation", "$$", "f' : X' = X \\times_{\\Spec(k)} \\Spec(k[\\epsilon])", "\\xrightarrow{f \\times \\text{id}}", "Y' = Y \\times_{\\Spec(k)} \\Spec(k[\\epsilon])", "$$", "of $X \\to Y$ to $k[\\epsilon]$ equal to the identity modulo $\\epsilon$.", "This is clearly the same thing as pairs", "$(\\alpha, \\beta) \\in", "\\text{Inf}(\\Deformationcategory_X \\times \\Deformationcategory_Y)$", "of infinitesimal automorphisms of $X$ and $Y$ compatible with $f'$, i.e.,", "such that $f' \\circ \\alpha = \\beta \\circ f'$.", "By Deformation Theory, Lemma \\ref{defos-lemma-huge-diagram-ringed-spaces}", "for an arbitrary pair $(\\alpha, \\beta)$ the difference between", "the morphism $f' : X' \\to Y'$ and the morphism", "$\\beta^{-1} \\circ f' \\circ \\alpha : X' \\to Y'$ defines an elment", "in", "$$", "\\text{Der}_k(\\mathcal{O}_Y, f_*\\mathcal{O}_X) =", "\\Hom_{\\mathcal{O}_Y}(\\Omega_{Y/k}, f_*\\mathcal{O}_X)", "$$", "Equality by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-netherlander-quasi-coherent}.", "This defines the last top horizontal arrow and shows exactness", "in the first two places. For the map", "$$", "\\text{Der}_k(\\mathcal{O}_Y, f_*\\mathcal{O}_X)", "\\to", "T\\Deformationcategory_{X \\to Y}", "$$", "we interpret elements of the source as morphisms", "$f_\\epsilon : X' \\to Y'$ over $\\Spec(k[\\epsilon])$", "equal to $f$ modulo $\\epsilon$", "using Deformation Theory, Lemma \\ref{defos-lemma-huge-diagram-ringed-spaces}.", "We send $f_\\epsilon$ to the isomorphism class of", "$(f_\\epsilon : X' \\to Y')$ in $T\\Deformationcategory_{X \\to Y}$.", "Note that $(f_\\epsilon : X' \\to Y')$ is isomorphic to the", "trivial deformation $(f' : X' \\to Y')$ exactly when", "$f_\\epsilon = \\beta^{-1} \\circ f \\circ \\alpha$ for some", "pair $(\\alpha, \\beta)$ which implies exactness in the third spot.", "Clearly, if some first order deformation", "$(f_\\epsilon : X_\\epsilon \\to Y_\\epsilon)$", "maps to zero in $T(\\Deformationcategory_X \\times \\Deformationcategory_Y)$,", "then we can choose isomorphisms $X' \\to X_\\epsilon$ and $Y' \\to Y_\\epsilon$", "and we conclude we are in the image of the south-west arrow.", "Therefore we have exactness at the fourth spot.", "Finally, given two first order deformations $X_\\epsilon$, $Y_\\epsilon$", "of $X$, $Y$ there is an obstruction in", "$$", "ob(X_\\epsilon, Y_\\epsilon) \\in", "\\text{Ext}^1_{\\mathcal{O}_X}(Lf^*\\NL_{Y/k}, \\mathcal{O}_X)", "$$", "which vanishes if and only if $f : X \\to Y$ lifts to", "$X_\\epsilon \\to Y_\\epsilon$, see", "Deformation Theory, Lemma \\ref{defos-lemma-huge-diagram-ringed-spaces}.", "This finishes the proof." ], "refs": [ "defos-lemma-huge-diagram-ringed-spaces", "more-morphisms-lemma-netherlander-quasi-coherent", "defos-lemma-huge-diagram-ringed-spaces", "defos-lemma-huge-diagram-ringed-spaces" ], "ref_ids": [ 13387, 13746, 13387, 13387 ] } ], "ref_ids": [] }, { "id": 8755, "type": "theorem", "label": "examples-defos-lemma-proper-schemes-morphisms-TI", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-proper-schemes-morphisms-TI", "contents": [ "In Lemma \\ref{lemma-schemes-morphisms-TI} if $X$ and $Y$ are both", "proper over $k$, then", "$\\text{Inf}(\\Deformationcategory_{X \\to Y})$ and", "$T\\Deformationcategory_{X \\to Y}$ are finite dimensional." ], "refs": [ "examples-defos-lemma-schemes-morphisms-TI" ], "proofs": [ { "contents": [ "Omitted. Hint: argue as in Lemma \\ref{lemma-proper-schemes-TI}", "and use the exact sequence of the lemma." ], "refs": [ "examples-defos-lemma-proper-schemes-TI" ], "ref_ids": [ 8747 ] } ], "ref_ids": [ 8754 ] }, { "id": 8756, "type": "theorem", "label": "examples-defos-lemma-schemes-morphisms-hull", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-schemes-morphisms-hull", "contents": [ "In Example \\ref{example-schemes-morphisms} assume $X \\to Y$", "is a morphism of proper $k$-schemes.", "Assume $\\Lambda$ is a complete local ring with residue field $k$", "(the classical case). Then the functor", "$$", "F : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad", "A \\longmapsto \\Ob(\\Deformationcategory_{X \\to Y}(A))/\\cong", "$$", "of isomorphism classes of objects has a hull. If", "$\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) =", "\\text{Der}_k(\\mathcal{O}_Y, \\mathcal{O}_Y) = 0$, then", "$F$ is prorepresentable." ], "refs": [], "proofs": [ { "contents": [ "The existence of a hull follows immediately from", "Lemmas \\ref{lemma-schemes-morphisms-RS} and", "\\ref{lemma-proper-schemes-morphisms-TI} and", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-RS-implies-S1-S2}", "and Remark \\ref{formal-defos-remark-compose-minimal-into-iso-classes}.", "\\medskip\\noindent", "Assume $\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) =", "\\text{Der}_k(\\mathcal{O}_Y, \\mathcal{O}_Y) = 0$. Then", "the exact sequence of Lemma \\ref{lemma-schemes-morphisms-TI}", "combined with Lemma \\ref{lemma-schemes-TI}", "shows that $\\text{Inf}(\\Deformationcategory_{X \\to Y}) = 0$.", "Then $\\Deformationcategory_{X \\to Y}$ and $F$ are equivalent by", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-infdef-trivial}.", "Hence $F$ is a deformation functor (because", "$\\Deformationcategory_{X \\to Y}$ is a", "deformation category) with finite tangent space and we can apply", "Formal Deformation Theory, Theorem", "\\ref{formal-defos-theorem-Schlessinger-prorepresentability}." ], "refs": [ "examples-defos-lemma-schemes-morphisms-RS", "examples-defos-lemma-proper-schemes-morphisms-TI", "formal-defos-lemma-RS-implies-S1-S2", "formal-defos-remark-compose-minimal-into-iso-classes", "examples-defos-lemma-schemes-morphisms-TI", "examples-defos-lemma-schemes-TI", "formal-defos-lemma-infdef-trivial", "formal-defos-theorem-Schlessinger-prorepresentability" ], "ref_ids": [ 8753, 8755, 3469, 3567, 8754, 8746, 3481, 3411 ] } ], "ref_ids": [] }, { "id": 8757, "type": "theorem", "label": "examples-defos-lemma-schemes-morphisms-smooth-to-base", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-schemes-morphisms-smooth-to-base", "contents": [ "\\begin{reference}", "This is discussed in \\cite[Section 5.3]{Ravi-Murphys-Law} and", "\\cite[Theorem 3.3]{Ran-deformations}.", "\\end{reference}", "In Example \\ref{example-schemes} let $f : X \\to Y$ be a morphism of schemes", "over $k$. If $f_*\\mathcal{O}_X = \\mathcal{O}_Y$ and $R^1f_*\\mathcal{O}_X = 0$,", "then the morphism of deformation categories", "$$", "\\Deformationcategory_{X \\to Y} \\to \\Deformationcategory_X", "$$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "We construct a quasi-inverse to the forgetful functor of the lemma.", "Namely, suppose that $(A, U)$ is an object of $\\Deformationcategory_X$.", "The given map $X \\to U$ is a finite order thickening and we can use", "it to identify the underlying topological spaces of $U$ and $X$, see", "More on Morphisms, Section \\ref{more-morphisms-section-thickenings}.", "Thus we may and do think of $\\mathcal{O}_U$ as a sheaf of", "$A$-algebras on $X$; moreover the fact that $U \\to \\Spec(A)$ is", "flat, means that $\\mathcal{O}_U$ is flat as a sheaf of $A$-modules.", "In particular, we have a filtration", "$$", "0 = \\mathfrak m_A^n\\mathcal{O}_U \\subset", "\\mathfrak m_A^{n - 1}\\mathcal{O}_U \\subset \\ldots \\subset", "\\mathfrak m_A^2\\mathcal{O}_U \\subset", "\\mathfrak m_A\\mathcal{O}_U \\subset \\mathcal{O}_U", "$$", "with subquotients equal to", "$\\mathcal{O}_X \\otimes_k \\mathfrak m_A^i/\\mathfrak m_A^{i + 1}$", "by flatness, see More on Morphisms, Lemma \\ref{more-morphisms-lemma-deform}", "or the more general Deformation Theory, Lemma \\ref{defos-lemma-deform-module}.", "Set", "$$", "\\mathcal{O}_V = f_*\\mathcal{O}_U", "$$", "viewed as sheaf of $A$-algebras on $Y$. Since", "$R^1f_*\\mathcal{O}_X = 0$ we find by the description above that", "$R^1f_*(\\mathfrak m_A^i\\mathcal{O}_U/\\mathfrak m_A^{i + 1}\\mathcal{O}_U) = 0$", "for all $i$. This implies that the sequences", "$$", "0 \\to", "(f_*\\mathcal{O}_X) \\otimes_k \\mathfrak m_A^i/\\mathfrak m_A^{i + 1} \\to", "f_*(\\mathcal{O}_U/\\mathfrak m_A^{i + 1}\\mathcal{O}_U) \\to", "f_*(\\mathcal{O}_U/\\mathfrak m_A^i\\mathcal{O}_U) \\to 0", "$$", "are exact for all $i$. Reading the references given above backwards", "(and using induction) we find that $\\mathcal{O}_V$ is a flat", "sheaf of $A$-algebras with", "$\\mathcal{O}_V/\\mathfrak m_A\\mathcal{O}_V = \\mathcal{O}_Y$.", "Using More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-first-order-thickening}", "we find that $(Y, \\mathcal{O}_V)$ is a scheme, call it $V$.", "The equality $\\mathcal{O}_V = f_*\\mathcal{O}_U$ defines a", "morphism of ringed spaces $U \\to V$ which is easily seen to be", "a morphism of schemes. This finishes the proof by the", "flatness already established." ], "refs": [ "more-morphisms-lemma-deform", "defos-lemma-deform-module", "more-morphisms-lemma-first-order-thickening" ], "ref_ids": [ 13723, 13379, 13677 ] } ], "ref_ids": [] }, { "id": 8758, "type": "theorem", "label": "examples-defos-lemma-spaces-RS", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-spaces-RS", "contents": [ "Example \\ref{example-spaces}", "satisfies the Rim-Schlessinger condition (RS).", "In particular, $\\Deformationcategory_X$ is a deformation category", "for any algebraic space $X$ over $k$." ], "refs": [], "proofs": [ { "contents": [ "Let $A_1 \\to A$ and $A_2 \\to A$ be morphisms of $\\mathcal{C}_\\Lambda$.", "Assume $A_2 \\to A$ is surjective. According to", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-RS-2-categorical}", "it suffices to show that the functor", "$\\mathcal{F}(A_1 \\times_A A_2) \\to", "\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$", "is an equivalence of categories.", "Observe that", "$$", "\\xymatrix{", "\\Spec(A) \\ar[r] \\ar[d] & \\Spec(A_2) \\ar[d] \\\\", "\\Spec(A_1) \\ar[r] &", "\\Spec(A_1 \\times_A A_2)", "}", "$$", "is a pushout diagram as in Pushouts of Spaces, Lemma", "\\ref{spaces-pushouts-lemma-pushout-along-thickening}.", "Thus the lemma is a special case of Pushouts of Spaces, Lemma", "\\ref{spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat}." ], "refs": [ "formal-defos-lemma-RS-2-categorical", "spaces-pushouts-lemma-pushout-along-thickening", "spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat" ], "ref_ids": [ 3468, 10859, 10864 ] } ], "ref_ids": [] }, { "id": 8759, "type": "theorem", "label": "examples-defos-lemma-spaces-TI", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-spaces-TI", "contents": [ "In Example \\ref{example-spaces} let $X$ be an algebraic space over $k$. Then", "$$", "\\text{Inf}(\\Deformationcategory_X) =", "\\text{Ext}^0_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X) =", "\\Hom_{\\mathcal{O}_X}(\\Omega_{X/k}, \\mathcal{O}_X) =", "\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)", "$$", "and", "$$", "T\\Deformationcategory_X =", "\\text{Ext}^1_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\text{Inf}(\\Deformationcategory_X)$ is the set of", "automorphisms of the trivial deformation", "$X' = X \\times_{\\Spec(k)} \\Spec(k[\\epsilon])$ of $X$ to $k[\\epsilon]$", "equal to the identity modulo $\\epsilon$.", "By Deformation Theory, Lemma \\ref{defos-lemma-deform-spaces}", "this is equal to $\\text{Ext}^0_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X)$.", "The equality $\\text{Ext}^0_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X) =", "\\Hom_{\\mathcal{O}_X}(\\Omega_{X/k}, \\mathcal{O}_X)$ follows from", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-netherlander-quasi-coherent}.", "The equality", "$\\Hom_{\\mathcal{O}_X}(\\Omega_{X/k}, \\mathcal{O}_X) =", "\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)$", "follows from More on Morphisms of Spaces, Definition", "\\ref{spaces-more-morphisms-definition-sheaf-differentials} and", "Modules on Sites, Definition", "\\ref{sites-modules-definition-module-differentials}.", "\\medskip\\noindent", "Recall that $T_{x_0}\\Deformationcategory_X$ is the set of isomorphism classes", "of flat deformations $X'$ of $X$ to $k[\\epsilon]$, more precisely,", "the set of isomorphism classes of $\\Deformationcategory_X(k[\\epsilon])$.", "Thus the second statement of the lemma follows from", "Deformation Theory, Lemma \\ref{defos-lemma-deform-spaces}." ], "refs": [ "defos-lemma-deform-spaces", "spaces-more-morphisms-lemma-netherlander-quasi-coherent", "spaces-more-morphisms-definition-sheaf-differentials", "sites-modules-definition-module-differentials", "defos-lemma-deform-spaces" ], "ref_ids": [ 13412, 124, 282, 14296, 13412 ] } ], "ref_ids": [] }, { "id": 8760, "type": "theorem", "label": "examples-defos-lemma-proper-spaces-TI", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-proper-spaces-TI", "contents": [ "In Lemma \\ref{lemma-spaces-TI} if $X$ is proper over $k$, then", "$\\text{Inf}(\\Deformationcategory_X)$ and $T\\Deformationcategory_X$ are", "finite dimensional." ], "refs": [ "examples-defos-lemma-spaces-TI" ], "proofs": [ { "contents": [ "By the lemma we have to show", "$\\Ext^1_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X)$ and", "$\\Ext^0_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X)$ are finite", "dimensional. By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-netherlander-fp}", "and the fact that $X$ is Noetherian, we see that", "$\\NL_{X/k}$ has coherent cohomology sheaves zero except", "in degrees $0$ and $-1$.", "By Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-ext-finite}", "the displayed $\\Ext$-groups are finite $k$-vector spaces", "and the proof is complete." ], "refs": [ "spaces-more-morphisms-lemma-netherlander-fp", "spaces-perfect-lemma-ext-finite" ], "ref_ids": [ 125, 2668 ] } ], "ref_ids": [ 8759 ] }, { "id": 8761, "type": "theorem", "label": "examples-defos-lemma-spaces-hull", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-spaces-hull", "contents": [ "In Example \\ref{example-spaces} assume $X$ is a proper algebraic space over $k$.", "Assume $\\Lambda$ is a complete local ring with residue field $k$", "(the classical case). Then the functor", "$$", "F : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad", "A \\longmapsto \\Ob(\\Deformationcategory_X(A))/\\cong", "$$", "of isomorphism classes of objects has a hull. If", "$\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = 0$, then", "$F$ is prorepresentable." ], "refs": [], "proofs": [ { "contents": [ "The existence of a hull follows immediately from", "Lemmas \\ref{lemma-spaces-RS} and \\ref{lemma-proper-spaces-TI} and", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-RS-implies-S1-S2}", "and Remark \\ref{formal-defos-remark-compose-minimal-into-iso-classes}.", "\\medskip\\noindent", "Assume $\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = 0$. Then", "$\\Deformationcategory_X$ and $F$ are equivalent by", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-infdef-trivial}.", "Hence $F$ is a deformation functor (because $\\Deformationcategory_X$ is a", "deformation category) with finite tangent space and we can apply", "Formal Deformation Theory, Theorem", "\\ref{formal-defos-theorem-Schlessinger-prorepresentability}." ], "refs": [ "examples-defos-lemma-spaces-RS", "examples-defos-lemma-proper-spaces-TI", "formal-defos-lemma-RS-implies-S1-S2", "formal-defos-remark-compose-minimal-into-iso-classes", "formal-defos-lemma-infdef-trivial", "formal-defos-theorem-Schlessinger-prorepresentability" ], "ref_ids": [ 8758, 8760, 3469, 3567, 3481, 3411 ] } ], "ref_ids": [] }, { "id": 8762, "type": "theorem", "label": "examples-defos-lemma-lift-equivalence-module-derived", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-lift-equivalence-module-derived", "contents": [ "Let $A' \\to A$ be a surjection of rings with nilpotent kernel.", "Let $A' \\to P'$ be a flat ring map.", "Set $P = P' \\otimes_{A'} A$.", "Let $M$ be an $A$-flat $P$-module.", "Then the following are equivalent", "\\begin{enumerate}", "\\item there is an $A'$-flat $P'$-module $M'$ with", "$M' \\otimes_{P'} P = M$, and", "\\item there is an object $K' \\in D^-(P')$ with", "$K' \\otimes_{P'}^\\mathbf{L} P = M$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $M'$ is as in (1). Then", "$$", "M = M' \\otimes_P P' = M' \\otimes_{A'} A =", "M' \\otimes_A^\\mathbf{L} A' = M' \\otimes_{P'}^\\mathbf{L} P", "$$", "The first two equalities are clear, the third holds because", "$M'$ is flat over $A'$, and the fourth holds by", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-comparison}.", "Thus (2) holds. Conversely, suppose $K'$ is as in (2).", "We may and do assume $M$ is nonzero.", "Let $t$ be the largest integer such that $H^t(K')$ is nonzero", "(exists because $M$ is nonzero).", "Then $H^t(K') \\otimes_{P'} P = H^t(K' \\otimes_{P'}^\\mathbf{L} P)$", "is zero if $t > 0$. Since the kernel of $P' \\to P$ is nilpotent", "this implies $H^t(K') = 0$ by Nakayama's lemma a contradiction.", "Hence $t = 0$ (the case $t < 0$ is absurd as well).", "Then $M' = H^0(K')$ is a $P'$-module such that $M = M' \\otimes_{P'} P$", "and the spectral sequence for Tor gives an injective map", "$$", "\\text{Tor}_1^{P'}(M', P) \\to H^{-1}(M' \\otimes_{P'}^\\mathbf{L} P) = 0", "$$", "By the reference on derived base change above", "$0 = \\text{Tor}_1^{P'}(M', P) = \\text{Tor}_1^{A'}(M', A)$.", "We conclude that $M'$ is $A'$-flat by", "Algebra, Lemma \\ref{algebra-lemma-what-does-it-mean}." ], "refs": [ "more-algebra-lemma-base-change-comparison", "algebra-lemma-what-does-it-mean" ], "ref_ids": [ 10139, 890 ] } ], "ref_ids": [] }, { "id": 8763, "type": "theorem", "label": "examples-defos-lemma-lift-equivalence-module", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-lift-equivalence-module", "contents": [ "Consider a commutative diagram of Noetherian rings", "$$", "\\xymatrix{", "A' \\ar[d] \\ar[r] &", "P' \\ar[d] \\ar[r] &", "Q' \\ar[d] \\\\", "A \\ar[r] &", "P \\ar[r] &", "Q", "}", "$$", "with cartesian squares, with flat horizontal arrows, and with", "surjective vertial arrows whose kernels are nilpotent.", "Let $J' \\subset P'$ be an ideal such that $P'/J' = Q'/J'Q'$.", "Let $M$ be an $A$-flat $P$-module.", "Assume for all $g \\in J'$ there exists an $A'$-flat $(P')_g$-module", "lifting $M_g$. Then the following are equivalent", "\\begin{enumerate}", "\\item $M$ has an $A'$-flat lift to a $P'$-module, and", "\\item $M \\otimes_P Q$ has an $A'$-flat lift to a $Q'$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $I = \\Ker(A' \\to A)$. By induction on the integer $n > 1$", "such that $I^n = 0$ we reduce to the case where $I$ is an ideal", "of square zero; details omitted.", "We translate the condition of liftability of", "$M$ into the problem of finding an object of $D^-(P')$ as in", "Lemma \\ref{lemma-lift-equivalence-module-derived}.", "The obstruction to doing this is the element", "$$", "\\omega(M) \\in \\text{Ext}^2_P(M, M \\otimes_P^\\mathbf{L} IP) =", "\\text{Ext}^2_P(M, M \\otimes_P IP)", "$$", "constructed in", "Deformation Theory, Lemma \\ref{defos-lemma-canonical-class-algebra}.", "The equality in the displayed formula holds as", "$M \\otimes_P^\\mathbf{L} IP = M \\otimes_P IP$", "since $M$ and $P$ are $A$-flat\\footnote{Choose a resolution", "$F_\\bullet \\to I$ by free $A$-modules. Since $A \\to P$ is flat,", "$P \\otimes_A F_\\bullet$ is a free resolution of $IP$.", "Hence $M \\otimes_P^\\mathbf{L} IP$ is represented by", "$M \\otimes_P P \\otimes_A F_\\bullet = M \\otimes_A F_\\bullet$.", "This only has cohomology in degree $0$ as $M$ is $A$-flat.}.", "The obstruction for lifting $M \\otimes_P Q$ is similarly", "the element", "$$", "\\omega(M \\otimes_P Q) \\in", "\\text{Ext}^2_Q(M \\otimes_P Q, (M \\otimes_P Q) \\otimes_Q IQ)", "$$", "which is the image of $\\omega(M)$ by the functoriality", "of the construction $\\omega(-)$ of", "Deformation Theory, Lemma \\ref{defos-lemma-canonical-class-algebra}.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom}", "we have", "$$", "\\text{Ext}^2_Q(M \\otimes_P Q, (M \\otimes_P Q) \\otimes_Q IQ) =", "\\text{Ext}^2_P(M, M \\otimes_P IP) \\otimes_P Q", "$$", "here we use that $P$ is Noetherian and $M$ finite.", "Our assumption on $P' \\to Q'$ guarantees that for an $P$-module $E$", "the map $E \\to E \\otimes_P Q$ is bijective on $J'$-power torsion, see", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-neighbourhood-equivalence}.", "Thus we conclude that it suffices to show $\\omega(M)$", "is $J'$-power torsion. In other words, it suffices to show that", "$\\omega(M)$ dies in", "$$", "\\text{Ext}^2_P(M, M \\otimes_P IP)_g =", "\\text{Ext}^2_{P_g}(M_g, M_g \\otimes_{P_g} IP_g)", "$$", "for all $g \\in J'$. Howeover, by the compatibility of formation of $\\omega(M)$", "with base change again, we conclude that this is true as $M_g$", "is assumed to have a lift (of course you have to use the whole", "string of equivalences again)." ], "refs": [ "examples-defos-lemma-lift-equivalence-module-derived", "defos-lemma-canonical-class-algebra", "defos-lemma-canonical-class-algebra", "more-algebra-lemma-base-change-RHom", "more-algebra-lemma-neighbourhood-equivalence" ], "ref_ids": [ 8762, 13415, 13415, 10418, 10341 ] } ], "ref_ids": [] }, { "id": 8764, "type": "theorem", "label": "examples-defos-lemma-lift-equivalence", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-lift-equivalence", "contents": [ "Let $A' \\to A$ be a surjective map of Noetherian rings with nilpotent kernel.", "Let $A \\to B$ be a finite type flat ring map.", "Let $\\mathfrak b \\subset B$ be an ideal such that", "$\\Spec(B) \\to \\Spec(A)$ is syntomic on the complement of $V(\\mathfrak b)$.", "Then $B$ has a flat lift to $A'$ if and only if the $\\mathfrak b$-adic", "completion $B^\\wedge$ has a flat lift to $A'$." ], "refs": [], "proofs": [ { "contents": [ "Choose an $A$-algebra surjection $P = A[x_1, \\ldots, x_n] \\to B$.", "Let $\\mathfrak p \\subset P$ be the inverse image of $\\mathfrak b$.", "Set $P' = A'[x_1, \\ldots, x_n]$ and denote $\\mathfrak p' \\subset P'$", "the inverse image of $\\mathfrak p$. (Of course $\\mathfrak p$", "and $\\mathfrak p'$ do not designate prime ideals here.)", "We will denote $P^\\wedge$ and $(P')^\\wedge$ the respective completions.", "\\medskip\\noindent", "Suppose $A' \\to B'$ is a flat lift of $A \\to B$, in other words,", "$A' \\to B'$ is flat and there is an $A$-algebra isomorphism", "$B = B' \\otimes_{A'} A$. Then we can choose an $A'$-algebra map", "$P' \\to B'$ lifting the given surjection $P \\to B$.", "By Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "we find that $B'$ is a quotient of $P'$. In particular, we find", "that we can endow $B'$ with an $A'$-flat $P'$-module structure", "lifting $B$ as an $A$-flat $P$-module.", "Conversely, if we can lift $B$ to a $P'$-module $M'$ flat over $A'$,", "then $M'$ is a cyclic module $M' \\cong P'/J'$ (using Nakayama again)", "and setting $B' = P'/J'$ we find a flat lift of $B$ as an algebra.", "\\medskip\\noindent", "Set $C = B^\\wedge$ and $\\mathfrak c = \\mathfrak bC$.", "Suppose that $A' \\to C'$ is a flat lift of $A \\to C$.", "Then $C'$ is complete with respect to the inverse image $\\mathfrak c'$", "of $\\mathfrak c$", "(Algebra, Lemma \\ref{algebra-lemma-complete-modulo-nilpotent}).", "We choose an $A'$-algebra map $P' \\to C'$ lifting", "the $A$-algebra map $P \\to C$. These maps pass through", "completions to give surjections $P^\\wedge \\to C$ and $(P')^\\wedge \\to C'$", "(for the second again using Nakayama's lemma).", "In particular, we find that we can endow $C'$ with an $A'$-flat", "$(P')^\\wedge$-module structure lifting $C$ as an $A$-flat $P^\\wedge$-module.", "Conversely, if we can lift $C$ to a $(P')^\\wedge$-module $N'$ flat over $A'$,", "then $N'$ is a cyclic module $N' \\cong (P')^\\wedge/\\tilde J$", "(using Nakayama again) and setting $C' = (P')^\\wedge/\\tilde J$", "we find a flat lift of $C$ as an algebra.", "\\medskip\\noindent", "Observe that $P' \\to (P')^\\wedge$ is a flat ring map which", "induces an isomorphism $P'/\\mathfrak p' = (P')^\\wedge/\\mathfrak p'(P')^\\wedge$.", "We conclude that our lemma is a consequence of", "Lemma \\ref{lemma-lift-equivalence-module} provided we can", "show that $B_g$ lifts to an $A'$-flat $P'_g$-module for", "$g \\in \\mathfrak p'$. However, the ring map $A \\to B_g$ is syntomic", "and hence lifts to an $A'$-flat algebra $B'$ by", "Smoothing Ring Maps, Proposition \\ref{smoothing-proposition-lift-smooth}.", "Since $A' \\to P'_g$ is smooth, we can lift $P_g \\to B_g$", "to a surjective map $P'_g \\to B'$ as before and we get what we want." ], "refs": [ "algebra-lemma-NAK", "algebra-lemma-complete-modulo-nilpotent", "examples-defos-lemma-lift-equivalence-module", "smoothing-proposition-lift-smooth" ], "ref_ids": [ 401, 878, 8763, 5645 ] } ], "ref_ids": [] }, { "id": 8765, "type": "theorem", "label": "examples-defos-lemma-first-order-completion", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-first-order-completion", "contents": [ "Let $k$ be a field. Let $B$ be a finite type $k$-algebra.", "Let $J \\subset B$ be an ideal such that", "$\\Spec(B) \\to \\Spec(k)$ is smooth on the complement of $V(J)$.", "Let $N$ be a finite $B$-module.", "Then there is a canonical bijection", "$$", "\\text{Exal}_k(B, N) \\to \\text{Exal}_k(B^\\wedge, N^\\wedge)", "$$", "Here $B^\\wedge$ and $N^\\wedge$ are the $J$-adic completions." ], "refs": [], "proofs": [ { "contents": [ "The map is given by completion: given $0 \\to N \\to C \\to B \\to 0$", "in $\\text{Exal}_k(B, N)$ we send it to the completion $C^\\wedge$", "of $C$ with respect to the inverse image of $J$. Compare with", "the proof of Lemma \\ref{lemma-completion}.", "\\medskip\\noindent", "Since $k \\to B$ is of finite presentation the complex", "$\\NL_{B/k}$ can be represented by a complex", "$N^{-1} \\to N^0$ where $N^i$ is a finite $B$-module, see", "Algebra, Section \\ref{algebra-section-netherlander} and", "in particular", "Algebra, Lemma \\ref{algebra-lemma-NL-homotopy}.", "As $B$ is Noetherian, this means that $\\NL_{B/k}$", "is pseudo-coherent. For $g \\in J$ the $k$-algebra $B_g$", "is smooth and hence $(\\NL_{B/k})_g = \\NL_{B_g/k}$", "is quasi-isomorphic to a finite projective $B$-module sitting in degree $0$.", "Thus $\\text{Ext}^i_B(\\NL_{B/k}, N)_g = 0$ for $i \\geq 1$", "and any $B$-module $N$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-ext-annihilated-into}", "we conclude that", "$$", "\\text{Ext}^1_B(\\NL_{B/k}, N) \\longrightarrow", "\\lim_n \\text{Ext}^1_B(\\NL_{B/k}, N/J^n N)", "$$", "is an isomorphism for any finite $B$-module $N$.", "\\medskip\\noindent", "Injectivity of the map.", "Suppose that $0 \\to N \\to C \\to B \\to 0$ is in $\\text{Exal}_k(B, N)$", "and maps to zero in $\\text{Exal}_k(B^\\wedge, N^\\wedge)$.", "Choose a splitting $C^\\wedge = B^\\wedge \\oplus N^\\wedge$.", "Then the induced map $C \\to C^\\wedge \\to N^\\wedge$", "gives maps $C \\to N/J^nN$ for all $n$.", "Hence we see that our element is in the kernel of the maps", "$$", "\\text{Ext}^1_B(\\NL_{B/k}, N) \\to", "\\text{Ext}^1_B(\\NL_{B/k}, N/J^n N)", "$$", "for all $n$. By the previous paragraph we conclude that", "our element is zero.", "\\medskip\\noindent", "Surjectivity of the map. Let $0 \\to N^\\wedge \\to C' \\to B^\\wedge \\to 0$", "be an element of $\\text{Exal}_k(B^\\wedge, N^\\wedge)$.", "Pulling back by $B \\to B^\\wedge$ we get an element", "$0 \\to N^\\wedge \\to C'' \\to B \\to 0$ in", "$\\text{Exal}_k(B, N^\\wedge)$.", "we have", "$$", "\\text{Ext}^1_B(\\NL_{B/k}, N^\\wedge) =", "\\text{Ext}^1_B(\\NL_{B/k}, N) \\otimes_B B^\\wedge =", "\\text{Ext}^1_B(\\NL_{B/k}, N)", "$$", "The first equality as $N^\\wedge = N \\otimes_B B^\\wedge$", "(Algebra, Lemma \\ref{algebra-lemma-completion-tensor})", "and", "More on Algebra, Lemma \\ref{more-algebra-lemma-pseudo-coherence-and-ext}.", "The second equality because $\\text{Ext}^1_B(\\NL_{B/k}, N)$", "is $J$-power torsion (see above), $B \\to B^\\wedge$ is flat and induces", "an isomorphism $B/J \\to B^\\wedge/JB^\\wedge$, and", "More on Algebra, Lemma \\ref{more-algebra-lemma-neighbourhood-equivalence}.", "Thus we can find a $C \\in \\text{Exal}_k(B, N)$ mapping to $C''$ in", "$\\text{Exal}_k(B, N^\\wedge)$.", "Thus", "$$", "0 \\to N^\\wedge \\to C' \\to B^\\wedge \\to 0", "\\quad\\text{and}\\quad", "0 \\to N^\\wedge \\to C^\\wedge \\to B^\\wedge \\to 0", "$$", "are two elements of $\\text{Exal}_k(B^\\wedge, N^\\wedge)$", "mapping to the same element of $\\text{Exal}_k(B, N^\\wedge)$.", "Taking the difference we get an element", "$0 \\to N^\\wedge \\to C' \\to B^\\wedge \\to 0$ of", "$\\text{Exal}_k(B^\\wedge, N^\\wedge)$", "whose image in $\\text{Exal}_k(B, N^\\wedge)$ is zero.", "This means there exists", "$$", "\\xymatrix{", "0 \\ar[r] &", "N^\\wedge \\ar[r] &", "C' \\ar[r] &", "B^\\wedge \\ar[r] & 0 \\\\", "& & B \\ar[u]^\\sigma \\ar[ru]", "}", "$$", "Let $J' \\subset C'$ be the inverse image of $JB^\\wedge \\subset B^\\wedge$.", "To finish the proof it suffices to note that", "$\\sigma$ is continuous for the $J$-adic topology on $B$", "and the $J'$-adic topology on $C'$ and that $C'$ is $J'$-adically complete by", "Algebra, Lemma \\ref{algebra-lemma-complete-modulo-nilpotent}", "(here we also use that $C'$ is Noetherian; small detail omitted).", "Namely, this means that $\\sigma$ factors through the", "completion $B^\\wedge$ and $C' = 0$ in $\\text{Exal}_k(B^\\wedge, N^\\wedge)$." ], "refs": [ "examples-defos-lemma-completion", "algebra-lemma-NL-homotopy", "more-algebra-lemma-ext-annihilated-into", "algebra-lemma-completion-tensor", "more-algebra-lemma-pseudo-coherence-and-ext", "more-algebra-lemma-neighbourhood-equivalence", "algebra-lemma-complete-modulo-nilpotent" ], "ref_ids": [ 8743, 1151, 10434, 869, 10164, 10341, 878 ] } ], "ref_ids": [] }, { "id": 8766, "type": "theorem", "label": "examples-defos-lemma-smooth-completion", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-smooth-completion", "contents": [ "In Example \\ref{example-rings} let $P$ be a $k$-algebra.", "Let $J \\subset P$ be an ideal.", "Denote $P^\\wedge$ the $J$-adic completion. If", "\\begin{enumerate}", "\\item $k \\to P$ is of finite type, and", "\\item $\\Spec(P) \\to \\Spec(k)$ is smooth on the complement of $V(J)$.", "\\end{enumerate}", "then the functor between deformation categories of", "Lemma \\ref{lemma-completion}", "$$", "\\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^\\wedge}", "$$", "is smooth and induces an isomorphism on tangent spaces." ], "refs": [ "examples-defos-lemma-completion" ], "proofs": [ { "contents": [ "We know that $\\Deformationcategory_P$ and $\\Deformationcategory_{P^\\wedge}$", "are deformation categories by Lemma \\ref{lemma-rings-RS}.", "Thus it suffices to check", "our functor identifies tangent spaces and a correspondence", "between liftability, see", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-easy-check-smooth}.", "The property on liftability is proven in", "Lemma \\ref{lemma-lift-equivalence}", "and the isomorphism on tangent spaces is the special case of", "Lemma \\ref{lemma-first-order-completion} where $N = B$." ], "refs": [ "examples-defos-lemma-rings-RS", "formal-defos-lemma-easy-check-smooth", "examples-defos-lemma-lift-equivalence", "examples-defos-lemma-first-order-completion" ], "ref_ids": [ 8735, 3484, 8764, 8765 ] } ], "ref_ids": [ 8743 ] }, { "id": 8767, "type": "theorem", "label": "examples-defos-lemma-lift-equivalence-localization", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-lift-equivalence-localization", "contents": [ "Let $A' \\to A$ be a surjective map of Noetherian rings with nilpotent kernel.", "Let $A \\to B$ be a finite type flat ring map.", "Let $S \\subset B$ be a multiplicative subset such that", "if $\\Spec(B) \\to \\Spec(A)$ is not syntomic at $\\mathfrak q$,", "then $S \\cap \\mathfrak q = \\emptyset$.", "Then $B$ has a flat lift to $A'$ if and only if", "$S^{-1}B$ has a flat lift to $A'$." ], "refs": [], "proofs": [ { "contents": [ "This proof is the same as the proof of", "Lemma \\ref{lemma-lift-equivalence} but easier. We suggest the", "reader to skip the proof.", "Choose an $A$-algebra surjection $P = A[x_1, \\ldots, x_n] \\to B$.", "Let $S_P \\subset P$ be the inverse image of $S$.", "Set $P' = A'[x_1, \\ldots, x_n]$ and denote $S_{P'} \\subset P'$", "the inverse image of $S_P$.", "\\medskip\\noindent", "Suppose $A' \\to B'$ is a flat lift of $A \\to B$, in other words,", "$A' \\to B'$ is flat and there is an $A$-algebra isomorphism", "$B = B' \\otimes_{A'} A$. Then we can choose an $A'$-algebra map", "$P' \\to B'$ lifting the given surjection $P \\to B$.", "By Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "we find that $B'$ is a quotient of $P'$. In particular, we find", "that we can endow $B'$ with an $A'$-flat $P'$-module structure", "lifting $B$ as an $A$-flat $P$-module.", "Conversely, if we can lift $B$ to a $P'$-module $M'$ flat over $A'$,", "then $M'$ is a cyclic module $M' \\cong P'/J'$ (using Nakayama again)", "and setting $B' = P'/J'$ we find a flat lift of $B$ as an algebra.", "\\medskip\\noindent", "Set $C = S^{-1}B$. Suppose that $A' \\to C'$ is a flat lift of $A \\to C$.", "Elements of $C'$ which map to invertible elements of $C$ are invertible.", "We choose an $A'$-algebra map $P' \\to C'$ lifting", "the $A$-algebra map $P \\to C$. By the remark above", "these maps pass through localizations to give surjections", "$S_P^{-1}P \\to C$ and $S_{P'}^{-1}P' \\to C'$", "(for the second use Nakayama's lemma).", "In particular, we find that we can endow $C'$ with an $A'$-flat", "$S_{P'}^{-1}P'$-module structure lifting $C$ as an $A$-flat", "$S_P^{-1}P$-module. Conversely, if we can lift $C$ to a", "$S_{P'}^{-1}P'$-module $N'$ flat over $A'$, then $N'$", "is a cyclic module $N' \\cong S_{P'}^{-1}P'/\\tilde J$", "(using Nakayama again) and setting $C' = S_{P'}^{-1}P'/\\tilde J$", "we find a flat lift of $C$ as an algebra.", "\\medskip\\noindent", "The syntomic locus of a morphism of schemes is open by definition.", "Let $J_B \\subset B$ be an ideal cutting out the set of points", "in $\\Spec(B)$ where $\\Spec(B) \\to \\Spec(A)$ is not syntomic.", "Denote $J_P \\subset P$ and $J_{P'} \\subset P'$ the corresponding", "ideals. Observe that $P' \\to S_{P'}^{-1}P'$ is a flat ring map which", "induces an isomorphism $P'/J_{P'} = S_{P'}^{-1}P'/J_{P'}S_{P'}^{-1}P'$", "by our assumption on $S$ in the lemma, namely, the assumption", "in the lemma is exactly that $B/J_B = S^{-1}(B/J_B)$.", "We conclude that our lemma is a consequence of", "Lemma \\ref{lemma-lift-equivalence-module} provided we can", "show that $B_g$ lifts to an $A'$-flat $P'_g$-module for", "$g \\in J_B$. However, the ring map $A \\to B_g$ is syntomic", "and hence lifts to an $A'$-flat algebra $B'$ by", "Smoothing Ring Maps, Proposition \\ref{smoothing-proposition-lift-smooth}.", "Since $A' \\to P'_g$ is smooth, we can lift $P_g \\to B_g$", "to a surjective map $P'_g \\to B'$ as before and we get what we want." ], "refs": [ "examples-defos-lemma-lift-equivalence", "algebra-lemma-NAK", "examples-defos-lemma-lift-equivalence-module", "smoothing-proposition-lift-smooth" ], "ref_ids": [ 8764, 401, 8763, 5645 ] } ], "ref_ids": [] }, { "id": 8768, "type": "theorem", "label": "examples-defos-lemma-first-order-localization", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-first-order-localization", "contents": [ "Let $k$ be a field. Let $B$ be a finite type $k$-algebra.", "Let $S \\subset B$ be a multiplicative subset ideal such that", "if $\\Spec(B) \\to \\Spec(k)$ is not smooth at $\\mathfrak q$", "then $S \\cap \\mathfrak q = \\emptyset$.", "Let $N$ be a finite $B$-module.", "Then there is a canonical bijection", "$$", "\\text{Exal}_k(B, N) \\to \\text{Exal}_k(S^{-1}B, S^{-1}N)", "$$" ], "refs": [], "proofs": [ { "contents": [ "This proof is the same as the proof of", "Lemma \\ref{lemma-first-order-completion} but easier. We suggest the", "reader to skip the proof.", "The map is given by localization: given $0 \\to N \\to C \\to B \\to 0$", "in $\\text{Exal}_k(B, N)$ we send it to the localization $S_C^{-1}C$", "of $C$ with respect to the inverse image $S_C \\subset C$ of $S$.", "Compare with the proof of Lemma \\ref{lemma-localization}.", "\\medskip\\noindent", "The smooth locus of a morphism of schemes is open by definition.", "Let $J \\subset B$ be an ideal cutting out the set of points", "in $\\Spec(B)$ where $\\Spec(B) \\to \\Spec(A)$ is not smooth.", "Since $k \\to B$ is of finite presentation the complex", "$\\NL_{B/k}$ can be represented by a complex", "$N^{-1} \\to N^0$ where $N^i$ is a finite $B$-module, see", "Algebra, Section \\ref{algebra-section-netherlander} and", "in particular", "Algebra, Lemma \\ref{algebra-lemma-NL-homotopy}.", "As $B$ is Noetherian, this means that $\\NL_{B/k}$", "is pseudo-coherent. For $g \\in J$ the $k$-algebra $B_g$", "is smooth and hence $(\\NL_{B/k})_g = \\NL_{B_g/k}$", "is quasi-isomorphic to a finite projective $B$-module sitting in degree $0$.", "Thus $\\text{Ext}^i_B(\\NL_{B/k}, N)_g = 0$ for $i \\geq 1$", "and any $B$-module $N$. Finally, we have", "$$", "\\text{Ext}^1_{S^{-1}B}(\\NL_{S^{-1}B/k}, S^{-1}N) =", "\\text{Ext}^1_B(\\NL_{B/k}, N) \\otimes_B S^{-1}B =", "\\text{Ext}^1_B(\\NL_{B/k}, N)", "$$", "The first equality by", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom}", "and Algebra, Lemma \\ref{algebra-lemma-localize-NL}.", "The second because $\\text{Ext}^1_B(\\NL_{B/k}, N)$ is $J$-power", "torsion and elements of $S$ act invertibly on $J$-power torsion modules.", "This concludes the proof by the description of $\\text{Exal}_A(B, N)$", "as $\\text{Ext}^1_B(\\NL_{B/A}, N)$ given just above", "Lemma \\ref{lemma-first-order-completion}." ], "refs": [ "examples-defos-lemma-first-order-completion", "examples-defos-lemma-localization", "algebra-lemma-NL-homotopy", "more-algebra-lemma-base-change-RHom", "algebra-lemma-localize-NL", "examples-defos-lemma-first-order-completion" ], "ref_ids": [ 8765, 8740, 1151, 10418, 1161, 8765 ] } ], "ref_ids": [] }, { "id": 8769, "type": "theorem", "label": "examples-defos-lemma-smooth-localization", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-smooth-localization", "contents": [ "In Example \\ref{example-rings} let $P$ be a $k$-algebra.", "Let $S \\subset P$ be a multiplicative subset. If", "\\begin{enumerate}", "\\item $k \\to P$ is of finite type, and", "\\item $\\Spec(P) \\to \\Spec(k)$ is smooth at all points of", "$V(g)$ for all $g \\in S$.", "\\end{enumerate}", "then the functor between deformation categories of", "Lemma \\ref{lemma-localization}", "$$", "\\Deformationcategory_P \\longrightarrow \\Deformationcategory_{S^{-1}P}", "$$", "is smooth and induces an isomorphism on tangent spaces." ], "refs": [ "examples-defos-lemma-localization" ], "proofs": [ { "contents": [ "We know that $\\Deformationcategory_P$ and $\\Deformationcategory_{S^{-1}P}$", "are deformation categories by Lemma \\ref{lemma-rings-RS}.", "Thus it suffices to check", "our functor identifies tangent spaces and a correspondence", "between liftability, see", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-easy-check-smooth}.", "The property on liftability is proven in", "Lemma \\ref{lemma-lift-equivalence-localization}", "and the isomorphism on tangent spaces is the special case of", "Lemma \\ref{lemma-first-order-localization} where $N = B$." ], "refs": [ "examples-defos-lemma-rings-RS", "formal-defos-lemma-easy-check-smooth", "examples-defos-lemma-lift-equivalence-localization", "examples-defos-lemma-first-order-localization" ], "ref_ids": [ 8735, 3484, 8767, 8768 ] } ], "ref_ids": [ 8740 ] }, { "id": 8770, "type": "theorem", "label": "examples-defos-lemma-lift-equivalence-henselization", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-lift-equivalence-henselization", "contents": [ "Let $A' \\to A$ be a surjective map of Noetherian rings with nilpotent kernel.", "Let $A \\to B$ be a finite type flat ring map.", "Let $\\mathfrak b \\subset B$ be an ideal such that", "$\\Spec(B) \\to \\Spec(A)$ is syntomic on the complement of $V(\\mathfrak b)$.", "Let $(B^h, \\mathfrak b^h)$ be the henselization of the pair $(B, \\mathfrak b)$.", "Then $B$ has a flat lift to $A'$ if and only if $B^h$ has a flat lift to $A'$." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "This proof is a cheat. Namely, if $B$ has a flat lift $B'$, then", "taking the henselization $(B')^h$ we obtain a flat lift of $B^h$", "(compare with the proof of Lemma \\ref{lemma-henselization}).", "Conversely, suppose that $C'$ is an $A'$-flat lift of $(B')^h$.", "Then let $\\mathfrak c' \\subset C'$ be the inverse image of the", "ideal $\\mathfrak b^h$. Then the completion $(C')^\\wedge$ of", "$C'$ with respect to $\\mathfrak c'$ is a lift of $B^\\wedge$ (details omitted).", "Hence we see that $B$ has a flat lift by", "Lemma \\ref{lemma-lift-equivalence}." ], "refs": [ "examples-defos-lemma-henselization", "examples-defos-lemma-lift-equivalence" ], "ref_ids": [ 8741, 8764 ] } ], "ref_ids": [] }, { "id": 8771, "type": "theorem", "label": "examples-defos-lemma-first-order-henselization", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-first-order-henselization", "contents": [ "Let $k$ be a field. Let $B$ be a finite type $k$-algebra.", "Let $J \\subset B$ be an ideal such that", "$\\Spec(B) \\to \\Spec(k)$ is smooth on the complement of $V(J)$.", "Let $N$ be a finite $B$-module.", "Then there is a canonical bijection", "$$", "\\text{Exal}_k(B, N) \\to \\text{Exal}_k(B^h, N^h)", "$$", "Here $(B^h, J^h)$ is the henselization of $(B, J)$", "and $N^h = N \\otimes_B B^h$." ], "refs": [], "proofs": [ { "contents": [ "This proof is the same as the proof of", "Lemma \\ref{lemma-first-order-completion} but easier. We suggest the", "reader to skip the proof.", "The map is given by henselization: given $0 \\to N \\to C \\to B \\to 0$", "in $\\text{Exal}_k(B, N)$ we send it to the", "henselization $C^h$", "of $C$ with respect to the inverse image $J_C \\subset C$ of $J$.", "Compare with the proof of Lemma \\ref{lemma-henselization}.", "\\medskip\\noindent", "Since $k \\to B$ is of finite presentation the complex", "$\\NL_{B/k}$ can be represented by a complex", "$N^{-1} \\to N^0$ where $N^i$ is a finite $B$-module, see", "Algebra, Section \\ref{algebra-section-netherlander} and", "in particular", "Algebra, Lemma \\ref{algebra-lemma-NL-homotopy}.", "As $B$ is Noetherian, this means that $\\NL_{B/k}$", "is pseudo-coherent. For $g \\in J$ the $k$-algebra $B_g$", "is smooth and hence $(\\NL_{B/k})_g = \\NL_{B_g/k}$", "is quasi-isomorphic to a finite projective $B$-module sitting in degree $0$.", "Thus $\\text{Ext}^i_B(\\NL_{B/k}, N)_g = 0$ for $i \\geq 1$", "and any $B$-module $N$. Finally, we have", "\\begin{align*}", "\\text{Ext}^1_{B^h}(\\NL_{B^h/k}, N^h)", "& =", "\\text{Ext}^1_{B^h}(\\NL_{B/k} \\otimes_B B^h, N \\otimes_B B^h) \\\\", "& =", "\\text{Ext}^1_B(\\NL_{B/k}, N) \\otimes_B B^h \\\\", "& =", "\\text{Ext}^1_B(\\NL_{B/k}, N)", "\\end{align*}", "The first equality by", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-NL}", "(or rather its analogue for henselizations of pairs).", "The second by", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom}.", "The third because $\\text{Ext}^1_B(\\NL_{B/k}, N)$ is $J$-power", "torsion, the map $B \\to B^h$ is flat and induces an isomorphism", "$B/J \\to B^h/JB^h$ (More on Algebra, Lemma", "\\ref{more-algebra-lemma-henselization-flat}), and", "More on Algebra, Lemma \\ref{more-algebra-lemma-neighbourhood-equivalence}.", "This concludes the proof by the description of $\\text{Exal}_A(B, N)$", "as $\\text{Ext}^1_B(\\NL_{B/A}, N)$ given just above", "Lemma \\ref{lemma-first-order-completion}." ], "refs": [ "examples-defos-lemma-first-order-completion", "examples-defos-lemma-henselization", "algebra-lemma-NL-homotopy", "more-algebra-lemma-henselization-NL", "more-algebra-lemma-base-change-RHom", "more-algebra-lemma-henselization-flat", "more-algebra-lemma-neighbourhood-equivalence", "examples-defos-lemma-first-order-completion" ], "ref_ids": [ 8765, 8741, 1151, 10005, 10418, 9872, 10341, 8765 ] } ], "ref_ids": [] }, { "id": 8772, "type": "theorem", "label": "examples-defos-lemma-smooth-henselization", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-smooth-henselization", "contents": [ "In Example \\ref{example-rings} let $P$ be a $k$-algebra.", "Let $J \\subset P$ be an ideal.", "Denote $(P^h, J^h)$ the henselization of the pair $(P, J)$. If", "\\begin{enumerate}", "\\item $k \\to P$ is of finite type, and", "\\item $\\Spec(P) \\to \\Spec(k)$ is smooth on the complement of $V(J)$,", "\\end{enumerate}", "then the functor between deformation categories of", "Lemma \\ref{lemma-henselization}", "$$", "\\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^h}", "$$", "is smooth and induces an isomorphism on tangent spaces." ], "refs": [ "examples-defos-lemma-henselization" ], "proofs": [ { "contents": [ "We know that $\\Deformationcategory_P$ and $\\Deformationcategory_{P^h}$", "are deformation categories by Lemma \\ref{lemma-rings-RS}.", "Thus it suffices to check", "our functor identifies tangent spaces and a correspondence", "between liftability, see", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-easy-check-smooth}.", "The property on liftability is proven in", "Lemma \\ref{lemma-lift-equivalence-henselization}", "and the isomorphism on tangent spaces is the special case of", "Lemma \\ref{lemma-first-order-henselization} where $N = B$." ], "refs": [ "examples-defos-lemma-rings-RS", "formal-defos-lemma-easy-check-smooth", "examples-defos-lemma-lift-equivalence-henselization", "examples-defos-lemma-first-order-henselization" ], "ref_ids": [ 8735, 3484, 8770, 8771 ] } ], "ref_ids": [ 8741 ] }, { "id": 8773, "type": "theorem", "label": "examples-defos-lemma-isolated", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-isolated", "contents": [ "In Example \\ref{example-rings} let $P$ be a $k$-algebra.", "Assume that $k \\to P$ is of finite type and that $\\Spec(P) \\to \\Spec(k)$", "is smooth except at the maximal ideals", "$\\mathfrak m_1, \\ldots, \\mathfrak m_n$ of $P$.", "Let $P_{\\mathfrak m_i}$, $P_{\\mathfrak m_i}^h$, $P_{\\mathfrak m_i}^\\wedge$", "be the local ring, henselization, completion.", "Then the maps of deformation categories", "$$", "\\Deformationcategory_P \\to", "\\prod \\Deformationcategory_{P_{\\mathfrak m_i}} \\to", "\\prod \\Deformationcategory_{P_{\\mathfrak m_i}^h} \\to", "\\prod \\Deformationcategory_{P_{\\mathfrak m_i}^\\wedge}", "$$", "are smooth and induce isomorphisms on their finite dimensional", "tangent spaces." ], "refs": [], "proofs": [ { "contents": [ "The tangent space is finite dimensional by", "Lemma \\ref{lemma-finite-type-rings-TI}.", "The functors between the categories are constructed", "in Lemmas \\ref{lemma-localization}, \\ref{lemma-henselization}, and", "\\ref{lemma-completion} (we omit some verifications of the form:", "the completion of the henselization is the completion).", "\\medskip\\noindent", "Set $J = \\mathfrak m_1 \\cap \\ldots \\cap \\mathfrak m_n$ and apply", "Lemma \\ref{lemma-smooth-completion} to get that", "$\\Deformationcategory_P \\to \\Deformationcategory_{P^\\wedge}$", "is smooth and induces an isomorphism on tangent spaces", "where $P^\\wedge$ is the $J$-adic completion of $P$.", "However, since $P^\\wedge = \\prod P_{\\mathfrak m_i}^\\wedge$", "we see that the map $\\Deformationcategory_P \\to", "\\prod \\Deformationcategory_{P_{\\mathfrak m_i}^\\wedge}$", "is smooth and induces an isomorphism on tangent spaces.", "\\medskip\\noindent", "Let $(P^h, J^h)$ be the henselization of the pair $(P, J)$.", "Then $P^h = \\prod P_{\\mathfrak m_i}^h$ (look at idempotents", "and use More on Algebra, Lemma", "\\ref{more-algebra-lemma-characterize-henselian-pair}).", "Hence we can apply Lemma \\ref{lemma-smooth-henselization}", "to conclude as in the case of completion.", "\\medskip\\noindent", "To get the final case it suffices to show that", "$\\Deformationcategory_{P_{\\mathfrak m_i}} \\to", "\\Deformationcategory_{P_{\\mathfrak m_i}^\\wedge}$", "is smooth and induce isomorphisms on tangent spaces for each $i$ separately.", "To do this, we may replace $P$ by a principal localization", "whose only singular point is a maximal ideal $\\mathfrak m$", "(corresponding to $\\mathfrak m_i$ in the original $P$).", "Then we can apply", "Lemma \\ref{lemma-smooth-localization}", "with multiplicative subset $S = P \\setminus \\mathfrak m$ to conclude.", "Minor details omitted." ], "refs": [ "examples-defos-lemma-finite-type-rings-TI", "examples-defos-lemma-localization", "examples-defos-lemma-henselization", "examples-defos-lemma-completion", "examples-defos-lemma-smooth-completion", "more-algebra-lemma-characterize-henselian-pair", "examples-defos-lemma-smooth-henselization", "examples-defos-lemma-smooth-localization" ], "ref_ids": [ 8738, 8740, 8741, 8743, 8766, 9861, 8772, 8769 ] } ], "ref_ids": [] }, { "id": 8774, "type": "theorem", "label": "examples-defos-lemma-lci-unobstructed", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-lci-unobstructed", "contents": [ "In Example \\ref{example-rings} let $P$ be a local complete", "intersection over $k$ (Algebra, Definition \\ref{algebra-definition-lci-field}).", "Then $\\Deformationcategory_P$ is unobstructed." ], "refs": [ "algebra-definition-lci-field" ], "proofs": [ { "contents": [ "Let $(A, Q) \\to (k, P)$ be an object of $\\Deformationcategory_P$.", "Then we see that $A \\to Q$ is a syntomic ring map by", "Algebra, Definition \\ref{algebra-definition-lci}.", "Hence for any surjection $A' \\to A$ in $\\mathcal{C}_\\Lambda$", "we see that there is a morphism $(A', Q') \\to (A, Q)$", "lifting $A' \\to A$ by", "Smoothing Ring Maps, Proposition \\ref{smoothing-proposition-lift-smooth}.", "This proves the lemma." ], "refs": [ "algebra-definition-lci", "smoothing-proposition-lift-smooth" ], "ref_ids": [ 1532, 5645 ] } ], "ref_ids": [ 1530 ] }, { "id": 8775, "type": "theorem", "label": "examples-defos-lemma-glueing-smooth", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-glueing-smooth", "contents": [ "In Situation \\ref{situation-glueing} if $U_{12} \\to \\Spec(k)$ is smooth,", "then the morphism", "$$", "\\Deformationcategory_X", "\\longrightarrow", "\\Deformationcategory_{U_1} \\times \\Deformationcategory_{U_2} =", "\\Deformationcategory_{P_1} \\times \\Deformationcategory_{P_2}", "$$", "is smooth. If in addition", "$U_1$ is a local complete intersection over $k$, then", "$$", "\\Deformationcategory_X", "\\longrightarrow", "\\Deformationcategory_{U_2} = \\Deformationcategory_{P_2}", "$$", "is smooth." ], "refs": [], "proofs": [ { "contents": [ "The equality signs hold by Lemma \\ref{lemma-affine}.", "Let us think of $\\mathcal{C}_\\Lambda$ as a deformation", "category over $\\mathcal{C}_\\Lambda$ as in", "Formal Deformation Theory, Section \\ref{formal-defos-section-smooth}.", "Then", "$$", "\\Deformationcategory_{P_1} \\times \\Deformationcategory_{P_2} =", "\\Deformationcategory_{P_1}", "\\times_{\\mathcal{C}_\\Lambda}", "\\Deformationcategory_{P_2},", "$$", "see Formal Deformation Theory, Remarks", "\\ref{formal-defos-remarks-cofibered-groupoids}", "(\\ref{formal-defos-item-product}).", "Using", "Lemma \\ref{lemma-glueing}", "the first statement is that the functor", "$$", "\\Deformationcategory_{P_1}", "\\times_{\\Deformationcategory_{P_{12}}}", "\\Deformationcategory_{P_2}", "\\longrightarrow", "\\Deformationcategory_{P_1}", "\\times_{\\mathcal{C}_\\Lambda}", "\\Deformationcategory_{P_2}", "$$", "is smooth. This follows from Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-map-fibre-products-smooth} as long as", "we can show that $T\\Deformationcategory_{P_{12}} = (0)$.", "This vanishing follows from Lemma \\ref{lemma-smooth}", "as $P_{12}$ is smooth over $k$.", "For the second statement it suffices to show that", "$\\Deformationcategory_{P_1} \\to \\mathcal{C}_\\Lambda$", "is smooth, see Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-smooth-properties}.", "In other words, we have to show $\\Deformationcategory_{P_1}$", "is unobstructed, which is Lemma \\ref{lemma-lci-unobstructed}." ], "refs": [ "examples-defos-lemma-affine", "formal-defos-remarks-cofibered-groupoids", "examples-defos-lemma-glueing", "formal-defos-lemma-map-fibre-products-smooth", "examples-defos-lemma-smooth", "formal-defos-lemma-smooth-properties", "examples-defos-lemma-lci-unobstructed" ], "ref_ids": [ 8750, 3585, 8752, 3483, 8737, 3433, 8774 ] } ], "ref_ids": [] }, { "id": 8776, "type": "theorem", "label": "examples-defos-lemma-curve-isolated", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-curve-isolated", "contents": [ "In Example \\ref{example-schemes} let $X$ be a scheme over $k$. Assume", "\\begin{enumerate}", "\\item $X$ is separated, finite type over $k$ and $\\dim(X) \\leq 1$,", "\\item $X \\to \\Spec(k)$ is smooth except at the closed", "points $p_1, \\ldots, p_n \\in X$.", "\\end{enumerate}", "Let $\\mathcal{O}_{X, p_1}$, $\\mathcal{O}_{X, p_1}^h$,", "$\\mathcal{O}_{X, p_1}^\\wedge$ be the local ring, henselization, completion.", "Consider the maps of deformation categories", "$$", "\\Deformationcategory_X", "\\longrightarrow", "\\prod \\Deformationcategory_{\\mathcal{O}_{X, p_i}}", "\\longrightarrow", "\\prod \\Deformationcategory_{\\mathcal{O}_{X, p_i}^h}", "\\longrightarrow", "\\prod \\Deformationcategory_{\\mathcal{O}_{X, p_i}^\\wedge}", "$$", "The first arrow is smooth and the second and third arrows", "are smooth and induce isomorphisms on tangent spaces." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine open $U_2 \\subset X$ containing", "$p_1, \\ldots, p_n$ and the generic point of every irreducible", "component of $X$. This is possible by", "Varieties, Lemma \\ref{varieties-lemma-dim-1-quasi-projective}", "and Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-affine}.", "Then $X \\setminus U_2$ is finite and we can choose an affine open", "$U_1 \\subset X \\setminus \\{p_1, \\ldots, p_n\\}$ such that", "$X = U_1 \\cup U_2$. Set $U_{12} = U_1 \\cap U_2$.", "Then $U_1$ and $U_{12}$ are smooth affine schemes over $k$.", "We conclude that", "$$", "\\Deformationcategory_X \\longrightarrow \\Deformationcategory_{U_2}", "$$", "is smooth by Lemma \\ref{lemma-glueing-smooth}.", "Applying Lemmas \\ref{lemma-affine} and \\ref{lemma-isolated} we win." ], "refs": [ "varieties-lemma-dim-1-quasi-projective", "properties-lemma-ample-finite-set-in-affine", "examples-defos-lemma-glueing-smooth", "examples-defos-lemma-affine", "examples-defos-lemma-isolated" ], "ref_ids": [ 11098, 3062, 8775, 8750, 8773 ] } ], "ref_ids": [] }, { "id": 8777, "type": "theorem", "label": "examples-defos-lemma-curve-isolated-lci", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-curve-isolated-lci", "contents": [ "In Example \\ref{example-schemes} let $X$ be a scheme over $k$. Assume", "\\begin{enumerate}", "\\item $X$ is separated, finite type over $k$ and $\\dim(X) \\leq 1$,", "\\item $X$ is a local complete intersection over $k$, and", "\\item $X \\to \\Spec(k)$ is smooth except at finitely many points.", "\\end{enumerate}", "Then $\\Deformationcategory_X$ is unobstructed." ], "refs": [], "proofs": [ { "contents": [ "Let $p_1, \\ldots, p_n \\in X$ be the points where $X \\to \\Spec(k)$", "isn't smooth. Choose an affine open $U_2 \\subset X$ containing", "$p_1, \\ldots, p_n$ and the generic point of every irreducible", "component of $X$. This is possible by", "Varieties, Lemma \\ref{varieties-lemma-dim-1-quasi-projective}", "and Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-affine}.", "Then $X \\setminus U_2$ is finite and we can choose an affine open", "$U_1 \\subset X \\setminus \\{p_1, \\ldots, p_n\\}$ such that", "$X = U_1 \\cup U_2$. Set $U_{12} = U_1 \\cap U_2$.", "Then $U_1$ and $U_{12}$ are smooth affine schemes over $k$.", "We conclude that", "$$", "\\Deformationcategory_X \\longrightarrow \\Deformationcategory_{U_2}", "$$", "is smooth by Lemma \\ref{lemma-glueing-smooth}.", "Applying Lemmas \\ref{lemma-affine} and \\ref{lemma-lci-unobstructed} we win." ], "refs": [ "varieties-lemma-dim-1-quasi-projective", "properties-lemma-ample-finite-set-in-affine", "examples-defos-lemma-glueing-smooth", "examples-defos-lemma-affine", "examples-defos-lemma-lci-unobstructed" ], "ref_ids": [ 11098, 3062, 8775, 8750, 8774 ] } ], "ref_ids": [] }, { "id": 8778, "type": "theorem", "label": "examples-defos-lemma-criterion-smoothing", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-criterion-smoothing", "contents": [ "Let $k$ be a field. Set $S = \\Spec(k[[t]])$ and", "$S_n = \\Spec(k[t]/(t^n))$. Let $Y \\to S$ be a proper, flat morphism", "of schemes whose special fibre $X$ is Cohen-Macaulay and", "equidimensional of dimension $d$. Denote $X_n = Y \\times_S S_n$.", "If for some $n \\geq 1$ the $d$the Fitting ideal of $\\Omega_{X_n/S_n}$", "contains $t^{n - 1}$, then the generic fibre of $Y \\to S$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-flat-finite-presentation-CM-open}", "we see that $Y \\to S$ is a Cohen-Macaulay morphism.", "By Morphisms, Lemma", "\\ref{morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension}", "we see that $Y \\to S$ has relative dimension $d$.", "By Divisors, Lemma \\ref{divisors-lemma-d-fitting-ideal-omega-smooth}", "the $d$th Fitting ideal $\\mathcal{I} \\subset \\mathcal{O}_Y$", "of $\\Omega_{Y/S}$ cuts out the singular locus of the morphism $Y \\to S$.", "In other words, $V(\\mathcal{I}) \\subset Y$ is the closed subset", "of points where $Y \\to S$ is not smooth.", "By Divisors, Lemma \\ref{divisors-lemma-base-change-and-fitting-ideal-omega}", "formation of this Fitting ideal commutes with base change.", "By assumption we see that $t^{n - 1}$ is a section of", "$\\mathcal{I} + t^n\\mathcal{O}_Y$. Thus for every", "$x \\in X = V(t) \\subset Y$ we conclude that", "$t^{n - 1} \\in \\mathcal{I}_x$ where $\\mathcal{I}_x$ is the stalk at $x$.", "This implies that $V(\\mathcal{I}) \\subset V(t)$ in an", "open neighbourhood of $X$ in $Y$. Since $Y \\to S$", "is proper, this implies $V(\\mathcal{I}) \\subset V(t)$", "as desired." ], "refs": [ "more-morphisms-lemma-flat-finite-presentation-CM-open", "morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension", "divisors-lemma-d-fitting-ideal-omega-smooth", "divisors-lemma-base-change-and-fitting-ideal-omega" ], "ref_ids": [ 13789, 5286, 7902, 7900 ] } ], "ref_ids": [] }, { "id": 8779, "type": "theorem", "label": "examples-defos-lemma-jouanolou-type-thing", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-jouanolou-type-thing", "contents": [ "Let $k$ be a field. Let $1 \\leq c \\leq n$ be integers.", "Let $f_1, \\ldots, f_c \\in k[x_1, \\ldots x_n]$ be elements.", "Let $a_{ij}$, $0 \\leq i \\leq n$, $1 \\leq j \\leq c$ be", "variables. Consider", "$$", "g_j = f_j + a_{0j} + a_{1j}x_1 + \\ldots + a_{nj}x_n \\in", "k[a_{ij}][x_1, \\ldots, x_n]", "$$", "Denote $Y \\subset \\mathbf{A}^{n + c(n + 1)}_k$", "the closed subscheme cut out by $g_1, \\ldots, g_c$.", "Denote $\\pi : Y \\to \\mathbf{A}^{c(n + 1)}_k$ the projection", "onto the affine space with variables $a_{ij}$.", "Then there is a nonempty Zariski open ", "of $\\mathbf{A}^{c(n + 1)}_k$ over which $\\pi$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "Recall that the set of points where $\\pi$ is smooth is open.", "Thus the complement, i.e., the singular locus, is closed.", "By Chevalley's theorem (in the form of", "Morphisms, Lemma \\ref{morphisms-lemma-chevalley})", "the image of the singular locus is constructible.", "Hence if the generic point of $\\mathbf{A}^{c(n + 1)}_k$", "is not in the image of the singular locus, then", "the lemma follows (by Topology, Lemma", "\\ref{topology-lemma-generic-point-in-constructible} for example).", "Thus we have to show there is no point", "$y \\in Y$ where $\\pi$ is not smooth mapping to", "the generic point of $\\mathbf{A}^{c(n + 1)}_k$.", "Consider the matrix of partial derivatives", "$$", "(\\frac{\\partial g_j}{\\partial x_i}) =", "(\\frac{\\partial f_j}{\\partial x_i} + a_{ij})", "$$", "The image of this matrix in $\\kappa(y)$ must have rank $< c$", "since otherwise $\\pi$ would be smooth at $y$, see discussion in", "Smoothing Ring Maps, Section \\ref{smoothing-section-singular-ideal}.", "Thus we can find $\\lambda_1, \\ldots, \\lambda_c \\in \\kappa(y)$", "not all zero such that the vector $(\\lambda_1, \\ldots, \\lambda_c)$", "is in the kernel of this matrix.", "After renumbering we may assume $\\lambda_1 \\not = 0$.", "Dividing by $\\lambda_1$ we may assume our vector has", "the form $(1, \\lambda_2, \\ldots, \\lambda_c)$.", "Then we obtain", "$$", "a_{i1} = -", "\\frac{\\partial f_j}{\\partial x_1} -", "\\sum\\nolimits_{j = 2, \\ldots, c} \\lambda_j(\\frac{\\partial f_j}{\\partial x_i} + a_{ij})", "$$", "in $\\kappa(y)$ for $i = 1, \\ldots, n$. Moreover, since $y \\in Y$ we also", "have", "$$", "a_{0j} = -f_j - a_{1j}x_1 - \\ldots - a_{nj}x_n", "$$", "in $\\kappa(y)$. This means that the subfield of $\\kappa(y)$", "generated by $a_{ij}$ is contained in the subfield of $\\kappa(y)$", "generated by the images of $x_1, \\ldots, x_n, \\lambda_2, \\ldots, \\lambda_c$,", "and $a_{ij}$ except for $a_{i1}$ and $a_{0j}$.", "We count and we see that the transcendence degree of this is", "at most $c(n + 1) - 1$. Hence $y$ cannot map to the generic point", "as desired." ], "refs": [ "morphisms-lemma-chevalley", "topology-lemma-generic-point-in-constructible" ], "ref_ids": [ 5250, 8266 ] } ], "ref_ids": [] }, { "id": 8780, "type": "theorem", "label": "examples-defos-lemma-smoothing-affine-lci", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-smoothing-affine-lci", "contents": [ "Let $k$ be a field. Let $A$ be a global complete interesection", "over $k$. There exists a flat finite type ring map", "$k[[t]] \\to B$ with $B/tB \\cong A$ such that", "$B[1/t]$ is smooth over $k((t))$." ], "refs": [], "proofs": [ { "contents": [ "Write $A = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ as in", "Algebra, Definition \\ref{algebra-definition-lci-field}.", "We are going to choose", "$a_{ij} \\in (t) \\subset k[[t]]$ and set", "$$", "g_j = f_j + a_{0j} + a_{1j}x_1 + \\ldots + a_{nj}x_n \\in", "k[[t]][x_1, \\ldots, x_n]", "$$", "After doing this we take", "$B = k[[t]][x_1, \\ldots, x_n]/(g_1, \\ldots, g_c)$.", "We claim that $k[[t]] \\to B$ is flat at every prime ideal", "lying over $(t)$. Namely, the elements $f_1, \\ldots, f_c$", "form a regular sequence in the local ring at any prime ideal", "$\\mathfrak p$ of $k[x_1, \\ldots, x_n]$ containing $f_1, \\ldots, f_c$", "(Algebra, Lemma \\ref{algebra-lemma-lci}). Thus $g_1, \\ldots, g_c$", "is locally a lift of a regular sequence and we can apply", "Algebra, Lemma \\ref{algebra-lemma-grothendieck-regular-sequence}.", "Flatness at primes lying over $(0) \\subset k[[t]]$ is automatic", "because $k((t)) = k[[t]]_{(0)}$ is a field. Thus $B$ is flat", "over $k[[t]]$.", "\\medskip\\noindent", "All that remains is to show that for suitable choices", "of $a_{ij}$ the generic fibre $B_{(0)}$ is smooth over", "$k((t))$. For this we have to show that we can choose", "our $a_{ij}$ so that the induced morphism", "$$", "(a_{ij}) : \\Spec(k[[t]]) \\longrightarrow \\mathbf{A}^{c(n + 1)}_k", "$$", "maps into the nonempty Zariski open of", "Lemma \\ref{lemma-jouanolou-type-thing}.", "This is clear because there is no nonzero polynomial in the", "$a_{ij}$ which vanishes on $(t)^{\\oplus c(n + 1)}$.", "(We leave this as an exercise to the reader.)" ], "refs": [ "algebra-definition-lci-field", "algebra-lemma-lci", "algebra-lemma-grothendieck-regular-sequence", "examples-defos-lemma-jouanolou-type-thing" ], "ref_ids": [ 1530, 1167, 885, 8779 ] } ], "ref_ids": [] }, { "id": 8781, "type": "theorem", "label": "examples-defos-lemma-smoothing-artinian-lci", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-smoothing-artinian-lci", "contents": [ "Let $k$ be a field. Let $A$ be a finite dimensional $k$-algebra", "which is a local complete intersection over $k$. Then there is", "a finite flat $k[[t]]$-algebra $B$ with $B/tB \\cong A$", "and $B[1/t]$ \\'etale over $k((t))$." ], "refs": [], "proofs": [ { "contents": [ "Since $A$ is Artinian", "(Algebra, Lemma \\ref{algebra-lemma-finite-dimensional-algebra}),", "we can write $A$ as a product of local Artinian rings", "(Algebra, Lemma \\ref{algebra-lemma-artinian-finite-length}).", "Thus it suffices to prove the lemma if $A$ is local", "(this uses that being a local complete intersection is", "preserved under taking principal localizations, see", "Algebra, Lemma \\ref{algebra-lemma-localize-lci}).", "In this case $A$ is a global complete intersection.", "Consider the algebra $B$ constructed in", "Lemma \\ref{lemma-smoothing-affine-lci}.", "Then $k[[t]] \\to B$ is quasi-finite at the unique prime of $B$", "lying over $(t)$ (Algebra, Definition \\ref{algebra-definition-quasi-finite}).", "Observe that $k[[t]]$ is a henselian local ring", "(Algebra, Lemma \\ref{algebra-lemma-complete-henselian}).", "Thus $B = B' \\times C$ where $B'$ is finite over $k[[t]]$", "and $C$ has no prime lying over $(t)$, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-henselian}.", "Then $B'$ is the ring we are looking for", "(recall that \\'etale is the same thing as", "smooth of relative dimension $0$)." ], "refs": [ "algebra-lemma-finite-dimensional-algebra", "algebra-lemma-artinian-finite-length", "algebra-lemma-localize-lci", "examples-defos-lemma-smoothing-affine-lci", "algebra-definition-quasi-finite", "algebra-lemma-complete-henselian", "algebra-lemma-characterize-henselian" ], "ref_ids": [ 642, 646, 1165, 8780, 1522, 1282, 1276 ] } ], "ref_ids": [] }, { "id": 8782, "type": "theorem", "label": "examples-defos-lemma-smoothing-at-lci-point", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-smoothing-at-lci-point", "contents": [ "Let $k$ be a field. Let $A$ be a $k$-algebra. Assume", "\\begin{enumerate}", "\\item $A$ is a local ring essentially of finite type over $k$,", "\\item $A$ is a complete intersection over $k$", "(Algebra, Definition \\ref{algebra-definition-lci-local-ring}).", "\\end{enumerate}", "Set $d = \\dim(A) + \\text{trdeg}_k(\\kappa)$ where $\\kappa$", "is the residue field of $A$. Then there exists an integer $n$", "and a flat, essentially of finite type ring map", "$k[[t]] \\to B$ with $B/tB \\cong A$ such that $t^n$ is in the", "$d$th Fitting ideal of $\\Omega_{B/k[[t]]}$." ], "refs": [ "algebra-definition-lci-local-ring" ], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-lci-local} we can write $A$ as the", "localization at a prime $\\mathfrak p$ of a global complete intersection $P$", "over $k$. Observe that $\\dim(P) = d$ by", "Algebra, Lemma \\ref{algebra-lemma-dimension-at-a-point-finite-type-field}.", "By Lemma \\ref{lemma-smoothing-affine-lci} we can find a", "flat, finite type ring map $k[[t]] \\to Q$ such that $P \\cong Q/tQ$ and", "such that $k((t)) \\to Q[1/t]$ is smooth. It follows from the construction", "of $Q$ in the lemma that $k[[t]] \\to Q$ is a relative global", "complete intersection of relative dimension $d$; alternatively,", "Algebra, Lemma \\ref{algebra-lemma-syntomic} tells us that $Q$ or a", "suitable principal localization of $Q$ is such a global complete intersection.", "Hence by Divisors, Lemma \\ref{divisors-lemma-d-fitting-ideal-omega-smooth}", "the $d$th Fitting ideal $I \\subset Q$ of $\\Omega_{Q/k[[t]]}$", "cuts out the singular locus of $\\Spec(Q) \\to \\Spec(k[[t]])$.", "Thus $t^n \\in I$ for some $n$.", "Let $\\mathfrak q \\subset Q$", "be the inverse image of $\\mathfrak p$. Set $B = Q_\\mathfrak q$.", "The lemma is proved." ], "refs": [ "algebra-lemma-lci-local", "algebra-lemma-dimension-at-a-point-finite-type-field", "examples-defos-lemma-smoothing-affine-lci", "algebra-lemma-syntomic", "divisors-lemma-d-fitting-ideal-omega-smooth" ], "ref_ids": [ 1169, 1007, 8780, 1185, 7902 ] } ], "ref_ids": [ 1531 ] }, { "id": 8783, "type": "theorem", "label": "examples-defos-lemma-smoothing-proper-curve-isolated-lci", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-smoothing-proper-curve-isolated-lci", "contents": [ "Let $X$ be a scheme over a field $k$. Assume", "\\begin{enumerate}", "\\item $X$ is proper over $k$,", "\\item $X$ is a local complete intersection over $k$,", "\\item $X$ has dimension $\\leq 1$, and", "\\item $X \\to \\Spec(k)$ is smooth except at finitely many points.", "\\end{enumerate}", "Then there exists a flat projective morphism $Y \\to \\Spec(k[[t]])$", "whose generic fibre is smooth and whose special fibre is", "isomorphic to $X$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $X$ is Cohen-Macaulay, see", "Algebra, Lemma \\ref{algebra-lemma-lci-CM}.", "Thus $X = X' \\amalg X''$ with $\\dim(X') = 0$", "and $X''$ equidimensional of dimension $1$, see Morphisms, Lemma", "\\ref{morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension}.", "Since $X'$ is finite over $k$ (Varieties, Lemma", "\\ref{varieties-lemma-algebraic-scheme-dim-0})", "we can find $Y' \\to \\Spec(k[[t]])$ with special", "fibre $X'$ and generic fibre smooth by", "Lemma \\ref{lemma-smoothing-artinian-lci}.", "Thus it suffices to prove the lemma for $X''$.", "After replacing $X$ by $X''$ we have $X$ is", "Cohen-Macaulay and equidimensional of dimension $1$.", "\\medskip\\noindent", "We are going to use deformation theory for the situation $\\Lambda = k \\to k$.", "Let $p_1, \\ldots, p_r \\in X$ be the closed singular points of $X$, i.e.,", "the points where $X \\to \\Spec(k)$ isn't smooth. For each $i$ we pick", "an integer $n_i$ and a flat, essentially of finite type ring map", "$$", "k[[t]] \\longrightarrow B_i", "$$", "with $B_i/tB_i \\cong \\mathcal{O}_{X, p_i}$ such that", "$t^{n_i}$ is in the $1$st Fitting ideal of $\\Omega_{B_i/k[[t]]}$.", "This is possible by Lemma \\ref{lemma-smoothing-at-lci-point}.", "Observe that the system $(B_i/t^nB_i)$ defines a formal object of", "$\\Deformationcategory_{\\mathcal{O}_{X, p_i}}$ over $k[[t]]$.", "By Lemma \\ref{lemma-curve-isolated} the map", "$$", "\\Deformationcategory_X", "\\longrightarrow", "\\prod\\nolimits_{i = 1, \\ldots, r} \\Deformationcategory_{\\mathcal{O}_{X, p_i}}", "$$", "is a smooth map between deformation categories. Hence by", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-smooth-morphism-essentially-surjective}", "there exists a formal object $(X_n)$ in $\\Deformationcategory_X$", "mapping to the formal object $\\prod_i (B_i/t^n)$ by the arrow above.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-formal-algebraic-space-proper-reldim-1}", "there exists a projective scheme $Y$ over $k[[t]]$ and compatible", "isomorphisms $Y \\times_{\\Spec(k[[t]])} \\Spec(k[t]/(t^n)) \\cong X_n$.", "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-check-flatness-on-infinitesimal-nbhds}", "we see that $Y \\to \\Spec(k[[t]])$ is flat.", "Since $X$ is Cohen-Macaulay and equidimensional of dimension $1$", "we may apply Lemma \\ref{lemma-criterion-smoothing}", "to check $Y$ has smooth generic fibre\\footnote{Warning: in general it is", "{\\bf not} true that the local ring of $Y$ at the point", "$p_i$ is isomorphic to $B_i$. We only know that this is true after", "dividing by $t^n$ on both sides!}.", "Choose $n$ strictly larger than the maximum of the integers $n_i$ found above.", "It we can show $t^{n - 1}$ is in the first Fitting ideal of", "$\\Omega_{X_n/S_n}$ with $S_n = \\Spec(k[t]/(t^n))$, then the proof is done.", "To do this it suffices to prove this is true in each of", "the local rings of $X_n$ at closed points $p$.", "However, if $p$ corresponds to a smooth point for $X \\to \\Spec(k)$,", "then $\\Omega_{X_n/S_n, p}$ is free of rank $1$ and the first Fitting", "ideal is equal to the local ring. If $p = p_i$ for some $i$, then", "$$", "\\Omega_{X_n/S_n, p_i} =", "\\Omega_{(B_i/t^nB_i)/(k[t]/(t^n))} =", "\\Omega_{B_i/k[[t]]}/t^n\\Omega_{B_i/k[[t]]}", "$$", "Since taking Fitting ideals commutes with base change", "(with already used this but in this algebraic setting", "it follows from More on Algebra, Lemma", "\\ref{more-algebra-lemma-fitting-ideal-basics}),", "and since $n - 1 \\geq n_i$ we see that $t^{n - 1}$ is", "in the Fitting ideal of this module over $B_i/t^nB_i$ as desired." ], "refs": [ "algebra-lemma-lci-CM", "morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension", "varieties-lemma-algebraic-scheme-dim-0", "examples-defos-lemma-smoothing-artinian-lci", "examples-defos-lemma-smoothing-at-lci-point", "examples-defos-lemma-curve-isolated", "formal-defos-lemma-smooth-morphism-essentially-surjective", "spaces-more-morphisms-lemma-formal-algebraic-space-proper-reldim-1", "more-morphisms-lemma-check-flatness-on-infinitesimal-nbhds", "examples-defos-lemma-criterion-smoothing", "more-algebra-lemma-fitting-ideal-basics" ], "ref_ids": [ 1166, 5286, 10988, 8781, 8782, 8776, 3434, 212, 13744, 8778, 9834 ] } ], "ref_ids": [] }, { "id": 8784, "type": "theorem", "label": "examples-defos-lemma-smoothing-curve-isolated-lci", "categories": [ "examples-defos" ], "title": "examples-defos-lemma-smoothing-curve-isolated-lci", "contents": [ "Let $k$ be a field and let $X$ be a scheme over $k$. Assume", "\\begin{enumerate}", "\\item $X$ is separated, finite type over $k$ and $\\dim(X) \\leq 1$,", "\\item $X$ is a local complete intersection over $k$, and", "\\item $X \\to \\Spec(k)$ is smooth except at finitely many points.", "\\end{enumerate}", "Then there exists a flat, separated, finite type morphism $Y \\to \\Spec(k[[t]])$", "whose generic fibre is smooth and whose special fibre is", "isomorphic to $X$." ], "refs": [], "proofs": [ { "contents": [ "If $X$ is reduced, then we can choose an embedding", "$X \\subset \\overline{X}$ as in", "Varieties, Lemma \\ref{varieties-lemma-reduced-dim-1-projective-completion}.", "Writing $X = \\overline{X} \\setminus \\{x_1, \\ldots, x_n\\}$", "we see that $\\mathcal{O}_{\\overline{X}, x_i}$ is a discrete", "valuation ring and hence in particular a local complete intersection", "(Algebra, Definition \\ref{algebra-definition-lci-local-ring}).", "Thus $\\overline{X}$ is a local complete intersection", "over $k$ because this holds over the open $X$ and", "at the points $x_i$ by Algebra, Lemma \\ref{algebra-lemma-lci-local}.", "Thus we may apply Lemma \\ref{lemma-smoothing-proper-curve-isolated-lci}", "to find a projective flat morphism $\\overline{Y} \\to \\Spec(k[[t]])$", "whose generic fibre is smooth and whose special fibre", "is $\\overline{X}$. Then we remove $x_1, \\ldots, x_n$", "from $\\overline{Y}$ to obtain $Y$.", "\\medskip\\noindent", "In the general case, write $X = X' \\amalg X''$ where", "with $\\dim(X') = 0$ and $X''$ equidimensional of dimension $1$.", "Then $X''$ is reduced and the first paragraph applies to it.", "On the other hand, $X'$ can be dealt with", "as in the proof of Lemma \\ref{lemma-smoothing-proper-curve-isolated-lci}.", "Some details omitted." ], "refs": [ "varieties-lemma-reduced-dim-1-projective-completion", "algebra-definition-lci-local-ring", "algebra-lemma-lci-local", "examples-defos-lemma-smoothing-proper-curve-isolated-lci", "examples-defos-lemma-smoothing-proper-curve-isolated-lci" ], "ref_ids": [ 11101, 1531, 1169, 8783, 8783 ] } ], "ref_ids": [] }, { "id": 8785, "type": "theorem", "label": "sets-theorem-reflection-principle", "categories": [ "sets" ], "title": "sets-theorem-reflection-principle", "contents": [ "Suppose given $\\phi_1(x_1, \\ldots, x_n), \\ldots, \\phi_m(x_1, \\ldots, x_n)$", "a {\\bf finite} collection of", "formulas of set theory. Let $M_0$ be a set.", "There exists a set $M$ such that", "$M_0 \\subset M$ and", "$\\forall x_1, \\ldots, x_n \\in M$, we have", "$$", "\\forall i = 1, \\ldots, m, \\ \\phi_i^{M}(x_1, \\ldots, x_n)", "\\Leftrightarrow", "\\forall i = 1, \\ldots, m, \\ \\phi_i(x_1, \\ldots, x_n).", "$$", "In fact we may take $M = V_\\alpha$ for some limit ordinal $\\alpha$." ], "refs": [], "proofs": [ { "contents": [ "See \\cite[Theorem 12.14]{Jech} or \\cite[Theorem 7.4]{Kunen}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8786, "type": "theorem", "label": "sets-lemma-axiom-regularity", "categories": [ "sets" ], "title": "sets-lemma-axiom-regularity", "contents": [ "Every set is an element of $V_\\alpha$ for some ordinal $\\alpha$." ], "refs": [], "proofs": [ { "contents": [ "See \\cite[Lemma 6.3]{Jech}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8787, "type": "theorem", "label": "sets-lemma-map-from-set-lifts", "categories": [ "sets" ], "title": "sets-lemma-map-from-set-lifts", "contents": [ "Suppose that $T = \\colim_{\\alpha < \\beta} T_\\alpha$", "is a colimit of sets indexed by ordinals less than a given ordinal $\\beta$.", "Suppose that $\\varphi : S \\to T$ is a map of sets.", "Then $\\varphi$ lifts to a map into $T_\\alpha$ for some $\\alpha < \\beta$", "provided that $\\beta$ is not a limit of ordinals indexed by $S$,", "in other words, if $\\beta$ is an ordinal with $\\text{cf}(\\beta) > |S|$." ], "refs": [], "proofs": [ { "contents": [ "For each element $s \\in S$ pick a $\\alpha_s < \\beta$ and an element", "$t_s \\in T_{\\alpha_s}$ which maps to $\\varphi(s)$ in $T$.", "By assumption $\\alpha = \\sup_{s \\in S} \\alpha_s$ is strictly smaller", "than $\\beta$. Hence the map $\\varphi_\\alpha : S \\to T_\\alpha$", "which assigns to $s$ the image of $t_s$ in $T_\\alpha$ is a solution." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8788, "type": "theorem", "label": "sets-lemma-bounded-size", "categories": [ "sets" ], "title": "sets-lemma-bounded-size", "contents": [ "For every cardinal $\\kappa$, there exists a set $A$ such", "that every element of $A$ is a scheme and such that for every", "scheme $S$ with $\\text{size}(S) \\leq \\kappa$, there is", "an element $X \\in A$ such that $X \\cong S$ (isomorphism", "of schemes)." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: think about how any scheme is isomorphic to a scheme", "obtained by glueing affines." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8789, "type": "theorem", "label": "sets-lemma-construct-category", "categories": [ "sets" ], "title": "sets-lemma-construct-category", "contents": [ "With notations $\\text{size}$, $Bound$ and $\\Sch_\\alpha$ as above.", "Let $S_0$ be a set of schemes. There exists a limit ordinal", "$\\alpha$ with the following properties:", "\\begin{enumerate}", "\\item", "\\label{item-inclusion}", "We have $S_0 \\subset V_\\alpha$; in other words,", "$S_0 \\subset \\Ob(\\Sch_\\alpha)$.", "\\item", "\\label{item-bounded}", "For any $S \\in \\Ob(\\Sch_\\alpha)$ and any", "scheme $T$ with $\\text{size}(T) \\leq Bound(\\text{size}(S))$,", "there exists a scheme $S' \\in \\Ob(\\Sch_\\alpha)$", "such that $T \\cong S'$.", "\\item", "\\label{item-limit}", "For any countable\\footnote{Both the set of objects and", "the morphism sets are countable. In fact you can prove the lemma with", "$\\aleph_0$ replaced by any cardinal whatsoever in (3) and (4).} diagram", "category $\\mathcal{I}$ and", "any functor $F : \\mathcal{I} \\to \\Sch_\\alpha$, the limit", "$\\lim_\\mathcal{I} F$ exists in $\\Sch_\\alpha$ if and", "only if it exists in $\\Sch$ and moreover, in this case,", "the natural morphism between them is an isomorphism.", "\\item", "\\label{item-colimit}", "For any countable diagram category $\\mathcal{I}$ and", "any functor $F : \\mathcal{I} \\to \\Sch_\\alpha$, the colimit", "$\\colim_\\mathcal{I} F$ exists in $\\Sch_\\alpha$ if and", "only if it exists in $\\Sch$ and moreover, in this case,", "the natural morphism between them is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We define, by transfinite induction, a function $f$ which associates", "to every ordinal an ordinal as follows. Let $f(0) = 0$.", "Given $f(\\alpha)$, we define $f(\\alpha + 1)$ to be the least", "ordinal $\\beta$ such that the following hold:", "\\begin{enumerate}", "\\item We have $\\alpha + 1 \\leq \\beta$ and $f(\\alpha) \\leq \\beta$.", "\\item For any $S \\in \\Ob(\\Sch_{f(\\alpha)})$ and any", "scheme $T$ with $\\text{size}(T) \\leq Bound(\\text{size}(S))$,", "there exists a scheme $S' \\in \\Ob(\\Sch_\\beta)$", "such that $T \\cong S'$.", "\\item For any countable diagram category $\\mathcal{I}$ and", "any functor $F : \\mathcal{I} \\to \\Sch_{f(\\alpha)}$, if", "the limit $\\lim_\\mathcal{I} F$ or the colimit", "$\\colim_\\mathcal{I} F$ exists in $\\Sch$,", "then it is isomorphic to a scheme in $\\Sch_\\beta$.", "\\end{enumerate}", "To see $\\beta$ exists, we argue as follows. Since", "$\\Ob(\\Sch_{f(\\alpha)})$ is a set, we see that", "$\\kappa =", "\\sup_{S \\in \\Ob(\\Sch_{f(\\alpha)})} Bound(\\text{size}(S))$", "exists and is a cardinal.", "Let $A$ be a set of schemes obtained starting with $\\kappa$", "as in Lemma \\ref{lemma-bounded-size}.", "There is a set $CountCat$ of countable", "categories such that any countable category is isomorphic to", "an element of $CountCat$. Hence in (3) above we may assume", "that $\\mathcal{I}$ is an element in $CountCat$. This means that", "the pairs $(\\mathcal{I}, F)$ in (3) range over a set.", "Thus, there exists a set $B$ whose elements are schemes", "such that for every $(\\mathcal{I}, F)$ as in (3), if the", "limit or colimit exists, then it is isomorphic to an element in $B$.", "Hence, if we pick any $\\beta$ such that $A \\cup B \\subset V_\\beta$", "and $\\beta > \\max\\{\\alpha + 1, f(\\alpha)\\}$, then (1)--(3) hold.", "Since every nonempty collection of ordinals has a least element,", "we see that $f(\\alpha + 1)$ is well defined. Finally, if $\\alpha$", "is a limit ordinal, then we set", "$f(\\alpha) = \\sup_{\\alpha' < \\alpha} f(\\alpha')$.", "\\medskip\\noindent", "Pick $\\beta_0$ such that $S_0 \\subset V_{\\beta_0}$.", "By construction $f(\\beta) \\geq \\beta$ and we see that", "also $S_0 \\subset V_{f(\\beta_0)}$. Moreover, as $f$ is", "nondecreasing, we see $S_0 \\subset V_{f(\\beta)}$ is true for any", "$\\beta \\geq \\beta_0$.", "Next, choose any ordinal $\\beta_1 > \\beta_0$ with cofinality", "$\\text{cf}(\\beta_1) > \\omega = \\aleph_0$. This is possible", "since the cofinality of ordinals gets arbitrarily large, see", "Proposition \\ref{proposition-exist-ordinals-large-cofinality}.", "We claim that", "$\\alpha = f(\\beta_1)$ is a solution to the problem posed in the lemma.", "\\medskip\\noindent", "The first property of the lemma holds by our choice", "of $\\beta_1 > \\beta_0$ above.", "\\medskip\\noindent", "Since $\\beta_1$ is a limit ordinal (as its cofinality is infinite),", "we get $f(\\beta_1) = \\sup_{\\beta < \\beta_1} f(\\beta)$.", "Hence $\\{f(\\beta) \\mid \\beta < \\beta_1\\} \\subset f(\\beta_1)$ is a", "cofinal subset. Hence we see that", "$$", "V_\\alpha = V_{f(\\beta_1)} = \\bigcup\\nolimits_{\\beta < \\beta_1} V_{f(\\beta)}.", "$$", "Now, let $S \\in \\Ob(\\Sch_\\alpha)$. We define", "$\\beta(S)$ to be the least ordinal $\\beta$ such that", "$S \\in \\Ob(\\Sch_{f(\\beta)})$. By the above we see", "that always $\\beta(S) < \\beta_1$. Since", "$\\Ob(\\Sch_{f(\\beta + 1)}) \\subset", "\\Ob(\\Sch_\\alpha)$, we", "see by construction of $f$ above that the second property of the lemma", "is satisfied.", "\\medskip\\noindent", "Suppose that $\\{S_1, S_2, \\ldots\\} \\subset \\Ob(\\Sch_\\alpha)$", "is a countable collection. Consider the function", "$\\omega \\to \\beta_1$, $n \\mapsto \\beta(S_n)$. Since the cofinality", "of $\\beta_1$ is $> \\omega$, the image of this function cannot be a", "cofinal subset. Hence there exists a $\\beta < \\beta_1$ such", "that $\\{S_1, S_2, \\ldots\\} \\subset \\Ob(\\Sch_{f(\\beta)})$.", "It follows that any functor $F : \\mathcal{I} \\to \\Sch_\\alpha$", "factors through one of the subcategories $\\Sch_{f(\\beta)}$.", "Thus, if there exists a scheme $X$ that is the colimit or limit", "of the diagram $F$, then, by construction of $f$, we see", "$X$ is isomorphic to an object", "of $\\Sch_{f(\\beta + 1)}$ which is a subcategory of", "$\\Sch_\\alpha$. This proves the last two assertions of", "the lemma." ], "refs": [ "sets-lemma-bounded-size", "sets-proposition-exist-ordinals-large-cofinality" ], "ref_ids": [ 8788, 8802 ] } ], "ref_ids": [] }, { "id": 8790, "type": "theorem", "label": "sets-lemma-bound-affine", "categories": [ "sets" ], "title": "sets-lemma-bound-affine", "contents": [ "Let $S$ be an affine scheme.", "Let $R = \\Gamma(S, \\mathcal{O}_S)$.", "Then the size of $S$ is equal to $\\max\\{ \\aleph_0, |R|\\}$." ], "refs": [], "proofs": [ { "contents": [ "There are at most $\\max\\{|R|, \\aleph_0\\}$ affine opens of", "$\\Spec(R)$. This is clear since any affine open", "$U \\subset \\Spec(R)$ is a finite union of principal", "opens $D(f_1) \\cup \\ldots \\cup D(f_n)$ and hence the number", "of affine opens is at most $\\sup_n |R|^n = \\max\\{|R|, \\aleph_0\\}$,", "see \\cite[Ch. I, 10.13]{Kunen}. On the other hand, we see that", "$\\Gamma(U, \\mathcal{O}) \\subset R_{f_1} \\times \\ldots \\times R_{f_n}$", "and hence $|\\Gamma(U, \\mathcal{O})| \\leq", "\\max\\{\\aleph_0, |R_{f_1}|, \\ldots, |R_{f_n}|\\}$. Thus", "it suffices to prove that $|R_f| \\leq \\max\\{\\aleph_0, |R|\\}$", "which is omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8791, "type": "theorem", "label": "sets-lemma-bound-size", "categories": [ "sets" ], "title": "sets-lemma-bound-size", "contents": [ "Let $S$ be a scheme. Let $S = \\bigcup_{i \\in I} S_i$ be", "an open covering. Then", "$\\text{size}(S) \\leq \\max\\{|I|, \\sup_i\\{\\text{size}(S_i)\\}\\}$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset S$ be any affine open. Since $U$ is quasi-compact", "there exist finitely many elements $i_1, \\ldots, i_n \\in I$", "and affine opens $U_i \\subset U \\cap S_i$ such that", "$U = U_1 \\cup U_2 \\cup \\ldots \\cup U_n$. Thus", "$$", "|\\Gamma(U, \\mathcal{O}_U)|", "\\leq", "|\\Gamma(U_1, \\mathcal{O})|", "\\otimes", "\\ldots", "\\otimes", "|\\Gamma(U_n, \\mathcal{O})|", "\\leq \\sup\\nolimits_i\\{\\text{size}(S_i)\\}", "$$", "Moreover, it shows that the set of affine opens of $S$ has", "cardinality less than or equal to the cardinality of the set", "$$", "\\coprod_{n \\in \\omega}", "\\coprod_{i_1, \\ldots, i_n \\in I}", "\\{\\text{affine opens of }S_{i_1}\\}", "\\times", "\\ldots", "\\times", "\\{\\text{affine opens of }S_{i_n}\\}.", "$$", "Each of the sets inside the disjoint union has cardinality at most", "$\\sup_i\\{\\text{size}(S_i)\\}$. The index set has cardinality at most", "$\\max\\{|I|, \\aleph_0\\}$, see \\cite[Ch. I, 10.13]{Kunen}.", "Hence by \\cite[Lemma 5.8]{Jech} the cardinality", "of the coproduct is at most $\\max\\{\\aleph_0, |I|\\}", "\\otimes \\sup_i\\{\\text{size}(S_i)\\}$. The lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8792, "type": "theorem", "label": "sets-lemma-bound-size-fibre-product", "categories": [ "sets" ], "title": "sets-lemma-bound-size-fibre-product", "contents": [ "Let $f : X \\to S$, $g : Y \\to S$ be morphisms of schemes.", "Then we have", "$\\text{size}(X \\times_S Y) \\leq \\max\\{\\text{size}(X), \\text{size}(Y)\\}$." ], "refs": [], "proofs": [ { "contents": [ "Let $S = \\bigcup_{k \\in K} S_k$ be an affine open covering.", "Let $X = \\bigcup_{i \\in I} U_i$, $Y = \\bigcup_{j \\in J} V_j$", "be affine open coverings with $I$, $J$ of cardinality", "$\\leq \\text{size}(X), \\text{size}(Y)$.", "For each $i \\in I$ there exists a finite set $K_i$ of $k \\in K$", "such that $f(U_i) \\subset \\bigcup_{k \\in K_i} S_k$.", "For each $j \\in J$ there exists a finite set $K_j$ of $k \\in K$", "such that $g(V_j) \\subset \\bigcup_{k \\in K_j} S_k$.", "Hence $f(X), g(Y)$ are contained in", "$S' = \\bigcup_{k \\in K'} S_k$ with", "$K' = \\bigcup_{i \\in I} K_i \\cup \\bigcup_{j \\in J} K_j$.", "Note that the cardinality of $K'$", "is at most $\\max\\{\\aleph_0, |I|, |J|\\}$. Applying", "Lemma \\ref{lemma-bound-size}", "we see that it suffices to prove that", "$\\text{size}(f^{-1}(S_k) \\times_{S_k} g^{-1}(S_k))", "\\leq \\max\\{\\text{size}(X), \\text{size}(Y))\\}$ for $k \\in K'$.", "In other words, we may assume that $S$ is affine.", "\\medskip\\noindent", "Assume $S$ affine.", "Let $X = \\bigcup_{i \\in I} U_i$, $Y = \\bigcup_{j \\in J} V_j$", "be affine open coverings with $I$, $J$ of cardinality", "$\\leq \\text{size}(X), \\text{size}(Y)$.", "Again by", "Lemma \\ref{lemma-bound-size}", "it suffices to prove the lemma for the products", "$U_i \\times_S V_j$. By", "Lemma \\ref{lemma-bound-affine}", "we see that it suffices to show that", "$$", "|A \\otimes_C B| \\leq \\max\\{\\aleph_0, |A|, |B|\\}.", "$$", "We omit the proof of this inequality." ], "refs": [ "sets-lemma-bound-size", "sets-lemma-bound-size", "sets-lemma-bound-affine" ], "ref_ids": [ 8791, 8791, 8790 ] } ], "ref_ids": [] }, { "id": 8793, "type": "theorem", "label": "sets-lemma-bound-finite-type", "categories": [ "sets" ], "title": "sets-lemma-bound-finite-type", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to S$ be locally of finite type with $X$ quasi-compact.", "Then $\\text{size}(X) \\leq \\text{size}(S)$." ], "refs": [], "proofs": [ { "contents": [ "We can find a finite affine open covering $X = \\bigcup_{i = 1, \\ldots n} U_i$", "such that each $U_i$ maps into an affine open $S_i$ of $S$. Thus by", "Lemma \\ref{lemma-bound-size}", "we reduce to the case where both $S$ and $X$ are affine. In this case by", "Lemma \\ref{lemma-bound-affine}", "we see that it suffices to show", "$$", "|A[x_1, \\ldots, x_n]| \\leq \\max\\{\\aleph_0, |A|\\}.", "$$", "We omit the proof of this inequality." ], "refs": [ "sets-lemma-bound-size", "sets-lemma-bound-affine" ], "ref_ids": [ 8791, 8790 ] } ], "ref_ids": [] }, { "id": 8794, "type": "theorem", "label": "sets-lemma-bound-monomorphism", "categories": [ "sets" ], "title": "sets-lemma-bound-monomorphism", "contents": [ "Let $f : X \\to Y$ be a monomorphism of schemes.", "If at least one of the following properties", "holds, then $\\text{size}(X) \\leq \\text{size}(Y)$:", "\\begin{enumerate}", "\\item $f$ is quasi-compact,", "\\item $f$ is locally of finite presentation,", "\\item add more here as needed.", "\\end{enumerate}", "But the bound does not hold for monomorphisms", "which are locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "Let $Y = \\bigcup_{j \\in J} V_j$ be an affine open covering of $Y$", "with $|J| \\leq \\text{size}(Y)$. By Lemma \\ref{lemma-bound-size}", "it suffices to bound the size of the inverse image of $V_j$ in $X$.", "Hence we reduce to the case that $Y$ is affine, say $Y = \\Spec(B)$.", "For any affine open $\\Spec(A) \\subset X$ we have", "$|A| \\leq \\max(|B|, \\aleph_0) = \\text{size}(Y)$, see remark above", "and Lemma \\ref{lemma-bound-affine}. Thus it suffices to show", "that $X$ has at most $\\text{size}(Y)$ affine opens. This is clear", "if $X$ is quasi-compact, whence case (1) holds.", "In case (2) the number of isomorphism classes of $B$-algebras $A$", "that can occur is bounded by $\\text{size}(B)$, because each", "$A$ is of finite type over $B$, hence isomorphic to an algebra", "$B[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$", "for some $n, m$, and $f_j \\in B[x_1, \\ldots, x_n]$. However, as", "$X \\to Y$ is a monomorphism, there is a unique morphism", "$\\Spec(A) \\to X$ over $Y = \\Spec(B)$ if there is one,", "hence the number of affine", "opens of $X$ is bounded by the number of these isomorphism classes.", "\\medskip\\noindent", "To prove the final statement of the lemma consider the ring", "$B = \\prod_{n \\in \\mathbf{N}} \\mathbf{F}_2$ and set $Y = \\Spec(B)$.", "For every ultrafilter $\\mathcal{U}$ on $\\mathbf{N}$ we obtain a maximal", "ideal $\\mathfrak m_\\mathcal{U}$ with residue field $\\mathbf{F}_2$;", "the map $B \\to \\mathbf{F}_2$ sends the element $(x_n)$ to", "$\\lim_\\mathcal{U} x_n$. Details omitted.", "The morphism of schemes $X = \\coprod_\\mathcal{U} \\Spec(\\mathbf{F}_2) \\to Y$", "is a monomorphism as all the points are distinct. However the cardinality", "of the set of affine open subschemes of $X$ is equal to the cardinality", "of the set of ultrafilters on $\\mathbf{N}$ which is", "$2^{2^{\\aleph_0}}$. We conclude as $|B| = 2^{\\aleph_0} < 2^{2^{\\aleph_0}}$." ], "refs": [ "sets-lemma-bound-size", "sets-lemma-bound-affine" ], "ref_ids": [ 8791, 8790 ] } ], "ref_ids": [] }, { "id": 8795, "type": "theorem", "label": "sets-lemma-what-is-in-it", "categories": [ "sets" ], "title": "sets-lemma-what-is-in-it", "contents": [ "Let $\\alpha$ be an ordinal as in Lemma \\ref{lemma-construct-category} above.", "The category $\\Sch_\\alpha$ satisfies the following", "properties:", "\\begin{enumerate}", "\\item If $X, Y, S \\in \\Ob(\\Sch_\\alpha)$, then", "for any morphisms $f : X \\to S$, $g : Y \\to S$ the fibre", "product $X \\times_S Y$ in $\\Sch_\\alpha$ exists", "and is a fibre product in the category of schemes.", "\\item Given any at most countable collection $S_1, S_2, \\ldots$", "of elements of $\\Ob(\\Sch_\\alpha)$, the coproduct", "$\\coprod_i S_i$ exists in $\\Ob(\\Sch_\\alpha)$ and", "is a coproduct in the category of schemes.", "\\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and", "any open immersion $U \\to S$, there exists a", "$V \\in \\Ob(\\Sch_\\alpha)$ with $V \\cong U$.", "\\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and", "any closed immersion $T \\to S$, there exists a", "$S' \\in \\Ob(\\Sch_\\alpha)$ with $S' \\cong T$.", "\\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and", "any finite type morphism $T \\to S$, there exists a", "$S' \\in \\Ob(\\Sch_\\alpha)$ with $S' \\cong T$.", "\\item Suppose $S$ is a scheme which has an open covering", "$S = \\bigcup_{i \\in I} S_i$ such that there exists", "a $T \\in \\Ob(\\Sch_\\alpha)$ with", "(a) $\\text{size}(S_i) \\leq \\text{size}(T)^{\\aleph_0}$ for all", "$i \\in I$, and (b) $|I| \\leq \\text{size}(T)^{\\aleph_0}$.", "Then $S$ is isomorphic to an object of $\\Sch_\\alpha$.", "\\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and", "any morphism $f : T \\to S$ locally of finite type such", "that $T$ can be covered by at most", "$\\text{size}(S)^{\\aleph_0}$ open affines, there exists a", "$S' \\in \\Ob(\\Sch_\\alpha)$ with $S' \\cong T$.", "For example this holds if $T$ can be covered by at most", "$|\\mathbf{R}| = 2^{\\aleph_0} = \\aleph_0^{\\aleph_0}$ open affines.", "\\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and", "any monomorphism $T \\to S$ which is either locally of finite presentation", "or quasi-compact, there exists a", "$S' \\in \\Ob(\\Sch_\\alpha)$ with $S' \\cong T$.", "\\item Suppose that $T \\in \\Ob(\\Sch_\\alpha)$ is", "affine. Write $R = \\Gamma(T, \\mathcal{O}_T)$.", "Then any of the following schemes is isomorphic to a scheme", "in $\\Sch_\\alpha$:", "\\begin{enumerate}", "\\item For any ideal $I \\subset R$ with completion", "$R^* = \\lim_n R/I^n$, the scheme $\\Spec(R^*)$.", "\\item For any finite type $R$-algebra $R'$, the", "scheme $\\Spec(R')$.", "\\item For any localization $S^{-1}R$, the scheme $\\Spec(S^{-1}R)$.", "\\item For any prime $\\mathfrak p \\subset R$, the scheme", "$\\Spec(\\overline{\\kappa(\\mathfrak p)})$.", "\\item For any subring $R' \\subset R$, the scheme", "$\\Spec(R')$.", "\\item Any scheme of finite type over a ring of cardinality at most", "$|R|^{\\aleph_0}$.", "\\item And so on.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [ "sets-lemma-construct-category" ], "proofs": [ { "contents": [ "Statements (1) and (2) follow directly from the definitions.", "Statement (3) follows as the size of an open subscheme $U$ of $S$ is", "clearly smaller than or equal to the size of $S$.", "Statement (4) follows from (5).", "Statement (5) follows from (7).", "Statement (6) follows as the size of $S$ is", "$\\leq \\max\\{|I|, \\sup_i \\text{size}(S_i)\\} \\leq \\text{size}(T)^{\\aleph_0}$", "by Lemma \\ref{lemma-bound-size}. Statement (7) follows from (6).", "Namely, for any affine open $V \\subset T$ we have", "$\\text{size}(V) \\leq \\text{size}(S)$ by", "Lemma \\ref{lemma-bound-finite-type}.", "Thus, we see that (6) applies in the situation of (7).", "Part (8) follows from", "Lemma \\ref{lemma-bound-monomorphism}.", "\\medskip\\noindent", "Statement (9) is translated, via Lemma \\ref{lemma-bound-affine},", "into an upper bound on the cardinality of the rings", "$R^*$, $S^{-1}R$, $\\overline{\\kappa(\\mathfrak p)}$, $R'$, etc.", "Perhaps the most interesting one is the ring $R^*$. As a", "set, it is the image of a surjective map $R^{\\mathbf{N}} \\to R^*$.", "Since $|R^{\\mathbf{N}}| = |R|^{\\aleph_0}$, we see that", "it works by our choice of $Bound(\\kappa)$ being at least $\\kappa^{\\aleph_0}$.", "Phew! (The cardinality of the algebraic closure of a field", "is the same as the cardinality of the field, or it is $\\aleph_0$.)" ], "refs": [ "sets-lemma-bound-size", "sets-lemma-bound-finite-type", "sets-lemma-bound-monomorphism", "sets-lemma-bound-affine" ], "ref_ids": [ 8791, 8793, 8794, 8790 ] } ], "ref_ids": [ 8789 ] }, { "id": 8796, "type": "theorem", "label": "sets-lemma-bound-by-covering", "categories": [ "sets" ], "title": "sets-lemma-bound-by-covering", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume there exists an", "fpqc covering $\\{g_j : Y_j \\to Y\\}_{j \\in J}$ such that $g_j$ factors", "through $f$. Then $\\text{size}(Y) \\leq \\text{size}(X)$." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset Y$ be an affine open. By definition there exist", "$n \\geq 0$ and $a : \\{1, \\ldots, n\\} \\to J$ and affine opens", "$V_i \\subset Y_{a(i)}$ such that", "$V = g_{a(1)}(V_1) \\cup \\ldots \\cup g_{a(n)}(V_n)$.", "Denote $h_j : Y_j \\to X$ a morphism such that $f \\circ h_j = g_j$.", "Then $h_{a(1)}(V_1) \\cup \\ldots \\cup h_{a(n)}(V_n)$ is", "a quasi-compact subset of $f^{-1}(V)$. Hence we can find a", "quasi-compact open $W \\subset f^{-1}(V)$ which contains", "$h_{a(i)}(V_i)$ for $i = 1, \\ldots, n$.", "In particular $V = f(W)$.", "\\medskip\\noindent", "On the one hand this shows that the cardinality of the set of", "affine opens of $Y$ is at most the cardinality of the set $S$ of", "quasi-compact opens of $X$. Since every quasi-compact open of", "$X$ is a finite union of affines, we see that the cardinality", "of this set is at most $\\sup |S|^n = \\max(\\aleph_0, |S|)$.", "On the other hand, we have", "$\\mathcal{O}_Y(V) \\subset \\prod_{i = 1, \\ldots, n} \\mathcal{O}_{Y_{a(i)}}(V_i)$", "because $\\{V_i \\to V\\}$ is an fpqc covering. Hence", "$\\mathcal{O}_Y(V) \\subset \\mathcal{O}_X(W)$ because $V_i \\to V$", "factors through $W$. Again since $W$ has a finite covering by", "affine opens of $X$ we conclude that $|\\mathcal{O}_Y(V)|$", "is bounded by the size of $X$. The lemma now follows", "from the definition of the size of a scheme." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8797, "type": "theorem", "label": "sets-lemma-bound-fppf-covering", "categories": [ "sets" ], "title": "sets-lemma-bound-fppf-covering", "contents": [ "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fppf covering of a scheme.", "There exists an fppf covering $\\{W_j \\to X\\}_{j \\in J}$", "which is a refinement of $\\{X_i \\to X\\}_{i \\in I}$ such that", "$\\text{size}(\\coprod W_j) \\leq \\text{size}(X)$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine open covering $X = \\bigcup_{a \\in A} U_a$ with", "$|A| \\leq \\text{size}(X)$. For each $a$ we can choose", "a finite subset $I_a \\subset I$ and for $i \\in I_a$ a quasi-compact open", "$W_{a, i} \\subset X_i$ such that $U_a = \\bigcup_{i \\in I_a} f_i(W_{a, i})$.", "Then $\\text{size}(W_{a, i}) \\leq \\text{size}(X)$ by", "Lemma \\ref{lemma-bound-finite-type}.", "We conclude that", "$\\text{size}(\\coprod_a \\coprod_{i \\in I_a} W_{i, a}) \\leq \\text{size}(X)$", "by Lemma \\ref{lemma-bound-size}." ], "refs": [ "sets-lemma-bound-finite-type", "sets-lemma-bound-size" ], "ref_ids": [ 8793, 8791 ] } ], "ref_ids": [] }, { "id": 8798, "type": "theorem", "label": "sets-lemma-sets-with-group-action", "categories": [ "sets" ], "title": "sets-lemma-sets-with-group-action", "contents": [ "With notations $G$, $G\\textit{-Sets}_\\alpha$, $\\text{size}$,", "and $Bound$ as above. Let $S_0$ be a set of $G$-sets.", "There exists a limit ordinal $\\alpha$ with the following properties:", "\\begin{enumerate}", "\\item We have $S_0 \\cup \\{{}_GG\\} \\subset \\Ob(G\\textit{-Sets}_\\alpha)$.", "\\item For any $S \\in \\Ob(G\\textit{-Sets}_\\alpha)$ and any", "$G$-set $T$ with $\\text{size}(T) \\leq Bound(\\text{size}(S))$,", "there exists a $S' \\in \\Ob(G\\textit{-Sets}_\\alpha)$", "that is isomorphic to $T$.", "\\item For any countable diagram category $\\mathcal{I}$ and", "any functor $F : \\mathcal{I} \\to G\\textit{-Sets}_\\alpha$, the", "limit $\\lim_\\mathcal{I} F$ and colimit", "$\\colim_\\mathcal{I} F$ exist in $G\\textit{-Sets}_\\alpha$", "and are the same as in $G\\textit{-Sets}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Similar to but easier than the proof of", "Lemma \\ref{lemma-construct-category} above." ], "refs": [ "sets-lemma-construct-category" ], "ref_ids": [ 8789 ] } ], "ref_ids": [] }, { "id": 8799, "type": "theorem", "label": "sets-lemma-what-is-in-it-G-sets", "categories": [ "sets" ], "title": "sets-lemma-what-is-in-it-G-sets", "contents": [ "Let $\\alpha$ be an ordinal as in Lemma \\ref{lemma-sets-with-group-action}", "above. The category $G\\textit{-Sets}_\\alpha$ satisfies the following", "properties:", "\\begin{enumerate}", "\\item The $G$-set ${}_GG$ is an object of $G\\textit{-Sets}_\\alpha$.", "\\item (Co)Products, fibre products, and pushouts", "exist in $G\\textit{-Sets}_\\alpha$", "and are the same as their counterparts in $G\\textit{-Sets}$.", "\\item Given an object $U$ of $G\\textit{-Sets}_\\alpha$,", "any $G$-stable subset $O \\subset U$ is isomorphic to an object", "of $G\\textit{-Sets}_\\alpha$.", "\\end{enumerate}" ], "refs": [ "sets-lemma-sets-with-group-action" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8798 ] }, { "id": 8800, "type": "theorem", "label": "sets-lemma-coverings-site", "categories": [ "sets" ], "title": "sets-lemma-coverings-site", "contents": [ "With notations as above.", "Let $\\text{Cov}_0 \\subset \\text{Cov}(\\mathcal{C})$", "be a set contained in $\\text{Cov}(\\mathcal{C})$.", "There exist a cardinal $\\kappa$ and a limit ordinal $\\alpha$", "with the following properties:", "\\begin{enumerate}", "\\item We have $\\text{Cov}_0 \\subset \\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha}$.", "\\item The set of coverings", "$\\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha}$ satisfies", "(1), (2), and (3) of Sites, Definition \\ref{sites-definition-site} (see above).", "In other words $(\\mathcal{C}, \\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha})$", "is a site.", "\\item Every covering in $\\text{Cov}(\\mathcal{C})$", "is combinatorially equivalent", "to a covering in $\\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha}$.", "\\end{enumerate}" ], "refs": [ "sites-definition-site" ], "proofs": [ { "contents": [ "To prove this, we first consider the set $\\mathcal{S}$ of all", "sets of morphisms of $\\mathcal{C}$ with fixed target.", "In other words, an element of $\\mathcal{S}$ is a subset $T$", "of $\\text{Arrows}(\\mathcal{C})$ such that all", "elements of $T$ have the same target. Given a family", "$\\mathcal{U} = \\{\\varphi_i : U_i \\to U\\}_{i\\in I}$ of morphisms with fixed", "target, we define", "$Supp(\\mathcal{U}) = \\{ \\varphi \\in \\text{Arrows}(\\mathcal{C})", "\\mid \\exists i\\in I, \\varphi = \\varphi_i\\}$.", "Note that two families $\\mathcal{U} = \\{\\varphi_i : U_i \\to U\\}_{i\\in I}$", "and $\\mathcal{V} = \\{V_j \\to V\\}_{j \\in J}$ are combinatorially", "equivalent if and only if $Supp(\\mathcal{U}) = Supp(\\mathcal{V})$.", "Next, we define", "$\\mathcal{S}_\\tau \\subset \\mathcal{S}$ to be the subset", "$\\mathcal{S}_\\tau = \\{ T \\in \\mathcal{S} \\mid", "\\exists\\ \\mathcal{U} \\in \\text{Cov}(\\mathcal{C}) \\ T = Supp(\\mathcal{U})\\}$.", "For every element $T \\in \\mathcal{S}_\\tau$, set", "$\\beta(T)$ to equal the least ordinal $\\beta$ such that", "there exists a $\\mathcal{U} \\in \\text{Cov}(\\mathcal{C})_\\beta$", "such that $T = \\text{Supp}(\\mathcal{U})$. Finally, set", "$\\beta_0 = \\sup_{T \\in S_\\tau} \\beta(T)$.", "At this point it follows that every $\\mathcal{U} \\in \\text{Cov}(\\mathcal{C})$", "is combinatorially equivalent to some element", "of $\\text{Cov}(\\mathcal{C})_{\\beta_0}$.", "\\medskip\\noindent", "Let $\\kappa$ be the maximum of $\\aleph_0$,", "the cardinality $|\\text{Arrows}(\\mathcal{C})|$,", "$$", "\\sup\\nolimits_{\\{U_i \\to U\\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})_{\\beta_0}}", "|I|,", "\\quad\\text{and}\\quad", "\\sup\\nolimits_{\\{U_i \\to U\\}_{i\\in I} \\in \\text{Cov}_0} |I|.", "$$", "Since $\\kappa$ is an infinite cardinal, we have", "$\\kappa \\otimes \\kappa = \\kappa$. Note that obviously", "$\\text{Cov}(\\mathcal{C})_{\\beta_0} =", "\\text{Cov}(\\mathcal{C})_{\\kappa, \\beta_0}$.", "\\medskip\\noindent", "We define, by transfinite induction, a function $f$ which associates", "to every ordinal an ordinal as follows. Let $f(0) = 0$.", "Given $f(\\alpha)$, we define $f(\\alpha + 1)$ to be the least", "ordinal $\\beta$ such that the following hold:", "\\begin{enumerate}", "\\item We have $\\alpha + 1 \\leq \\beta$ and $f(\\alpha) \\leq \\beta$.", "\\item If $\\{U_i \\to U\\}_{i\\in I}", "\\in \\text{Cov}(\\mathcal{C})_{\\kappa, f(\\alpha)}$", "and for each $i$ we have", "$\\{W_{ij} \\to U_i\\}_{j\\in J_i}", "\\in \\text{Cov}(\\mathcal{C})_{\\kappa, f(\\alpha)}$,", "then", "$\\{W_{ij} \\to U\\}_{i \\in I, j\\in J_i}", "\\in \\text{Cov}(\\mathcal{C})_{\\kappa, \\beta}$.", "\\item If $\\{U_i \\to U\\}_{i\\in I}", "\\in \\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha}$", "and $W \\to U$ is a morphism of $\\mathcal{C}$, then", "$\\{U_i \\times_U W \\to W \\}_{i\\in I}", "\\in \\text{Cov}(\\mathcal{C})_{\\kappa, \\beta}$.", "\\end{enumerate}", "To see $\\beta$ exists we note that clearly the collection of all", "coverings $\\{W_{ij} \\to U\\}$ and $\\{U_i \\times_U W \\to W \\}$ that occur in", "(2) and (3) form a set. Hence there is some ordinal $\\beta$ such that", "$V_\\beta$ contains all of these coverings. Moreover, the index set", "of the covering $\\{W_{ij} \\to U\\}$ has cardinality", "$\\sum_{i \\in I} |J_i| \\leq \\kappa \\otimes \\kappa = \\kappa$, and", "hence these coverings are contained in", "$\\text{Cov}(\\mathcal{C})_{\\kappa, \\beta}$.", "Since every nonempty collection of ordinals has a least element", "we see that $f(\\alpha + 1)$ is well defined. Finally, if $\\alpha$", "is a limit ordinal, then we set", "$f(\\alpha) = \\sup_{\\alpha' < \\alpha} f(\\alpha')$.", "\\medskip\\noindent", "Pick an ordinal $\\beta_1$ such that", "$\\text{Arrows}(\\mathcal{C}) \\subset V_{\\beta_1}$,", "$\\text{Cov}_0 \\subset V_{\\beta_0}$,", "and $\\beta_1 \\geq \\beta_0$.", "By construction $f(\\beta_1) \\geq \\beta_1$ and we see that", "the same properties hold for $V_{f(\\beta_1)}$. Moreover, as $f$ is", "nondecreasing this remains true for any $\\beta \\geq \\beta_1$.", "Next, choose any ordinal $\\beta_2 > \\beta_1$ with", "cofinality $\\text{cf}(\\beta_2) > \\kappa$. This is possible", "since the cofinality of ordinals gets arbitrarily large, see", "Proposition \\ref{proposition-exist-ordinals-large-cofinality}.", "We claim that the pair $\\kappa$,", "$\\alpha = f(\\beta_2)$ is a solution to the problem posed in the lemma.", "\\medskip\\noindent", "The first and third property of the lemma holds by our choices", "of $\\kappa$, $\\beta_2 > \\beta_1 > \\beta_0$ above.", "\\medskip\\noindent", "Since $\\beta_2$ is a limit ordinal (as its cofinality is infinite)", "we get $f(\\beta_2) = \\sup_{\\beta < \\beta_2} f(\\beta)$.", "Hence $\\{f(\\beta) \\mid \\beta < \\beta_2\\} \\subset f(\\beta_2)$ is a", "cofinal subset. Hence we see that", "$$", "V_\\alpha = V_{f(\\beta_2)} = \\bigcup\\nolimits_{\\beta < \\beta_2} V_{f(\\beta)}.", "$$", "Now, let $\\mathcal{U} \\in \\text{Cov}_{\\kappa, \\alpha}$.", "We define $\\beta(\\mathcal{U})$ to be the least ordinal $\\beta$ such that", "$\\mathcal{U} \\in \\text{Cov}_{\\kappa, f(\\beta)}$. By the above we see", "that always $\\beta(\\mathcal{U}) < \\beta_2$.", "\\medskip\\noindent", "We have to show properties (1), (2), and (3) defining a site", "hold for the pair $(\\mathcal{C}, \\text{Cov}_{\\kappa, \\alpha})$.", "The first holds because by our choice of $\\beta_2$", "all arrows of $\\mathcal{C}$ are contained in $V_{f(\\beta_2)}$.", "For the third, we use that given a covering", "$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}", "\\in \\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha}$", "we have $\\beta(\\mathcal{U}) < \\beta_2$ and hence", "any base change of $\\mathcal{U}$ is by construction of", "$f$ contained in $\\text{Cov}(\\mathcal{C})_{\\kappa, f(\\beta + 1)}$", "and hence in $\\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha}$.", "\\medskip\\noindent", "Finally, for the second condition, suppose that $\\{U_i \\to U\\}_{i\\in I}", "\\in \\text{Cov}(\\mathcal{C})_{\\kappa, f(\\alpha)}$", "and for each $i$ we have", "$\\mathcal{W}_i = \\{W_{ij} \\to U_i\\}_{j\\in J_i}", "\\in \\text{Cov}(\\mathcal{C})_{\\kappa, f(\\alpha)}$.", "Consider the function", "$I \\to \\beta_2$, $i \\mapsto \\beta(\\mathcal{W}_i)$. Since the cofinality", "of $\\beta_2$ is $> \\kappa \\geq |I|$ the image of this function cannot be a", "cofinal subset. Hence there exists a $\\beta < \\beta_1$ such", "that $\\mathcal{W}_i \\in \\text{Cov}_{\\kappa, f(\\beta)}$ for all $i \\in I$.", "It follows that the covering $\\{W_{ij} \\to U\\}_{i\\in I, j \\in J_i}$", "is an element of $\\text{Cov}(\\mathcal{C})_{\\kappa, f(\\beta + 1)}", "\\subset \\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha}$ as desired." ], "refs": [ "sets-proposition-exist-ordinals-large-cofinality" ], "ref_ids": [ 8802 ] } ], "ref_ids": [ 8652 ] }, { "id": 8801, "type": "theorem", "label": "sets-lemma-abelian-injectives", "categories": [ "sets" ], "title": "sets-lemma-abelian-injectives", "contents": [ "Suppose given a big category $\\mathcal{A}$ (see", "Categories, Remark \\ref{categories-remark-big-categories}).", "Assume $\\mathcal{A}$ is abelian and has enough injectives.", "See Homology, Definitions \\ref{homology-definition-abelian-category}", "and \\ref{homology-definition-enough-injectives}.", "Then for any given set of objects $\\{A_s\\}_{s\\in S}$", "of $\\mathcal{A}$, there is an abelian subcategory", "$\\mathcal{A}' \\subset \\mathcal{A}$", "with the following properties:", "\\begin{enumerate}", "\\item $\\Ob(\\mathcal{A}')$ is a set,", "\\item $\\Ob(\\mathcal{A}')$ contains $A_s$ for each $s \\in S$,", "\\item $\\mathcal{A}'$ has enough injectives, and", "\\item an object of $\\mathcal{A}'$ is injective if and only if it", "is an injective object of $\\mathcal{A}$.", "\\end{enumerate}" ], "refs": [ "categories-remark-big-categories", "homology-definition-abelian-category", "homology-definition-enough-injectives" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12410, 12137, 12183 ] }, { "id": 8802, "type": "theorem", "label": "sets-proposition-exist-ordinals-large-cofinality", "categories": [ "sets" ], "title": "sets-proposition-exist-ordinals-large-cofinality", "contents": [ "Let $\\kappa$ be a cardinal. Then there exists an ordinal", "whose cofinality is bigger than $\\kappa$." ], "refs": [], "proofs": [ { "contents": [ "If $\\kappa$ is finite, then $\\omega = \\text{cf}(\\omega)$ works.", "Let us thus assume that $\\kappa$ is infinite.", "Consider the smallest ordinal $\\alpha$ whose cardinality is strictly greater", "than $\\kappa$. We claim that $\\text{cf}(\\alpha) > \\kappa$.", "Note that $\\alpha$ is a limit ordinal, since if $\\alpha = \\beta + 1$, then", "$|\\alpha| = |\\beta|$ (because $\\alpha$ and $\\beta$ are infinite) and", "this contradicts the minimality of $\\alpha$. (Of course $\\alpha$ is also", "a cardinal, but we do not need this.) To get a contradiction", "suppose $S \\subset \\alpha$ is a cofinal", "subset with $|S| \\leq \\kappa$. For $\\beta \\in S$, i.e., $\\beta < \\alpha$,", "we have $|\\beta| \\leq \\kappa$ by minimality of $\\alpha$. As $\\alpha$ is", "a limit ordinal and $S$ cofinal in $\\alpha$ we obtain", "$\\alpha = \\bigcup_{\\beta \\in S} \\beta$. Hence", "$|\\alpha| \\leq |S| \\otimes \\kappa \\leq \\kappa \\otimes \\kappa \\leq \\kappa$", "which is a contradiction with our choice of $\\alpha$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8806, "type": "theorem", "label": "more-etale-lemma-section-support-in-locally-closed-pre", "categories": [ "more-etale" ], "title": "more-etale-lemma-section-support-in-locally-closed-pre", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$.", "Let $\\varphi : U' \\to U$ be a morphism of $X_\\etale$. Let $Z' \\subset U'$ be a", "closed subscheme such that $Z' \\to U' \\to U$ is a closed immersion", "with image $Z \\subset U$. Then there is a canonical bijection", "$$", "\\{s \\in \\mathcal{F}(U) \\mid \\text{Supp}(s) \\subset Z\\} =", "\\{s' \\in \\mathcal{F}(U') \\mid \\text{Supp}(s') \\subset Z'\\}", "$$", "which is given by restriction if $\\varphi^{-1}(Z) = Z'$." ], "refs": [], "proofs": [ { "contents": [ "Consider the closed subscheme $Z'' = \\varphi^{-1}(Z)$ of $U'$.", "Then $Z' \\subset Z''$ is closed because $Z'$ is closed in $U'$.", "On the other hand, $Z' \\to Z''$ is an \\'etale morphism", "(as a morphism between schemes \\'etale over $Z$) and hence", "open. Thus $Z'' = Z' \\amalg T$ for some closed subset $T$.", "The open covering $U' = (U' \\setminus T) \\cup (U' \\setminus Z')$", "shows that", "$$", "\\{s' \\in \\mathcal{F}(U') \\mid \\text{Supp}(s') \\subset Z'\\} =", "\\{s' \\in \\mathcal{F}(U' \\setminus T) \\mid \\text{Supp}(s') \\subset Z'\\}", "$$", "and the \\'etale covering $\\{U' \\setminus T \\to U, U \\setminus Z \\to U\\}$", "shows that", "$$", "\\{s \\in \\mathcal{F}(U) \\mid \\text{Supp}(s) \\subset Z\\} =", "\\{s' \\in \\mathcal{F}(U' \\setminus T) \\mid \\text{Supp}(s') \\subset Z'\\}", "$$", "This finishes the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8807, "type": "theorem", "label": "more-etale-lemma-section-support-in-locally-closed", "categories": [ "more-etale" ], "title": "more-etale-lemma-section-support-in-locally-closed", "contents": [ "Let $X$ be a scheme. Let $Z \\subset X$ be a locally closed subscheme.", "Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Given", "$U, U' \\subset X$ open containing $Z$ as a closed subscheme,", "there is a canonical bijection", "$$", "\\{s \\in \\mathcal{F}(U) \\mid \\text{Supp}(s) \\subset Z\\} =", "\\{s \\in \\mathcal{F}(U') \\mid \\text{Supp}(s) \\subset Z\\}", "$$", "which is given by restriction if $U' \\subset U$." ], "refs": [], "proofs": [ { "contents": [ "Since $Z$ is a closed subscheme of $U \\cap U'$, it suffices to", "prove the lemma when $U' \\subset U$. Then it is a special case", "of Lemma \\ref{lemma-section-support-in-locally-closed-pre}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8808, "type": "theorem", "label": "more-etale-lemma-f-shriek-separated", "categories": [ "more-etale" ], "title": "more-etale-lemma-f-shriek-separated", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes which is locally of finite type.", "Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. The rule", "$$", "Y_\\etale \\longrightarrow \\textit{Ab},\\quad", "V \\longmapsto \\{s \\in f_*\\mathcal{F}(V) = \\mathcal{F}(X_V) \\mid", "\\text{Supp}(s) \\subset X_V \\text{ is proper over }V\\}", "$$", "is an abelian subsheaf of $f_*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Recall that the support of a section is closed", "(\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-support-section-closed})", "hence the material in", "Cohomology of Schemes, Section \\ref{coherent-section-proper-over-base}", "applies. By the lemma above and", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-union-closed-proper-over-base}", "we find that our subset of $f_*\\mathcal{F}(V)$ is a subgroup.", "By Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-base-change-closed-proper-over-base}", "we see that our rule defines a sub presheaf.", "Finally, suppose that we have $s \\in f_*\\mathcal{F}(V)$", "and an \\'etale covering $\\{V_i \\to V\\}$ such that", "$s|_{V_i}$ has support proper over $V_i$.", "Observe that the support of $s|_{V_i}$ is the inverse", "image of the support of $s|_V$ (use the characterization", "of the support in terms of stalks and", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-stalk-pullback}).", "Whence the support of $s$ is proper over $V$ by", "Descent, Lemma \\ref{descent-lemma-descending-property-proper-over-base}.", "This proves that our rule satisfies the sheaf condition." ], "refs": [ "coherent-lemma-union-closed-proper-over-base", "coherent-lemma-base-change-closed-proper-over-base", "etale-cohomology-lemma-stalk-pullback", "descent-lemma-descending-property-proper-over-base" ], "ref_ids": [ 3390, 3388, 6436, 14703 ] } ], "ref_ids": [] }, { "id": 8809, "type": "theorem", "label": "more-etale-lemma-separated-etale-shriek", "categories": [ "more-etale" ], "title": "more-etale-lemma-separated-etale-shriek", "contents": [ "Let $j : U \\to X$ be a separated \\'etale morphism. Let $\\mathcal{F}$", "be an abelian sheaf on $U_\\etale$. The image of the injective map", "$j_!\\mathcal{F} \\to j_*\\mathcal{F}$ of", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-shriek-into-star-separated-etale}", "is the subsheaf of Lemma \\ref{lemma-f-shriek-separated}." ], "refs": [ "etale-cohomology-lemma-shriek-into-star-separated-etale", "more-etale-lemma-f-shriek-separated" ], "proofs": [ { "contents": [ "The construction of $j_!\\mathcal{F} \\to j_*\\mathcal{F}$ in the proof of", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-shriek-into-star-separated-etale}", "is via the construction of a map", "$j_{p!}\\mathcal{F} \\to j_*\\mathcal{F}$ of presheaves", "whose image is clearly contained in the subsheaf of", "Lemma \\ref{lemma-f-shriek-separated}.", "Hence since $j_!\\mathcal{F}$ is the sheafification of", "$j_{p!}\\mathcal{F}$ we conclude the image of", "$j_!\\mathcal{F} \\to j_*\\mathcal{F}$ is contained in", "this subsheaf. Conversely, let $s \\in j_*\\mathcal{F}(V)$", "have support $Z$ proper over $V$. Then $Z \\to V$ is", "finite with closed image $Z' \\subset V$, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-characterize-finite}.", "The restriction of $s$ to $V \\setminus Z'$ is zero and the zero section is", "contained in the image of $j_!\\mathcal{F} \\to j_*\\mathcal{F}$.", "On the other hand, if $v \\in Z'$, then we can find", "an \\'etale neighbourhood", "$(V', v') \\to (V, v)$ such that we have a decomposition", "$U_{V'} = W \\amalg U'_1 \\amalg \\ldots \\amalg U'_n$", "into open and closed subschemes with $U'_i \\to V'$ an isomorphism", "and with $T_{V'} \\subset U'_1 \\amalg \\ldots \\amalg U'_n$, see", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-etale-etale-local-technical}.", "Inverting the isomorphisms $U'_i \\to V'$", "we obtain $n$ morphisms $\\varphi'_i : V' \\to U$", "and sections $s'_i$ over $V'$ by pulling back $s$.", "Then the section $\\sum (\\varphi'_i, s'_i)$ of", "$j_{p!}\\mathcal{F}$ over $V'$, see formula for $j_{p!}\\mathcal{F}(V')$", "in proof of \\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-shriek-into-star-separated-etale},", "maps to the restriction of $s$ to $V'$ by construction.", "We conclude that $s$ is \\'etale locally in the image", "of $j_!\\mathcal{F} \\to j_*\\mathcal{F}$ and the proof is complete." ], "refs": [ "etale-cohomology-lemma-shriek-into-star-separated-etale", "more-etale-lemma-f-shriek-separated", "more-morphisms-lemma-characterize-finite", "etale-lemma-etale-etale-local-technical", "etale-cohomology-lemma-shriek-into-star-separated-etale" ], "ref_ids": [ 6524, 8808, 13903, 10713, 6524 ] } ], "ref_ids": [ 6524, 8808 ] }, { "id": 8810, "type": "theorem", "label": "more-etale-lemma-proper-f-shriek", "categories": [ "more-etale" ], "title": "more-etale-lemma-proper-f-shriek", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes.", "Then $f_! = f_*$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the construction of $f_!$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8811, "type": "theorem", "label": "more-etale-lemma-compactify-f-shriek-separated", "categories": [ "more-etale" ], "title": "more-etale-lemma-compactify-f-shriek-separated", "contents": [ "Let $Y$ be a scheme. Let $j : X \\to \\overline{X}$ be an open", "immersion of schemes over $Y$ with $\\overline{X}$ proper over $Y$.", "Denote $f : X \\to Y$ and $\\overline{f} : \\overline{X} \\to Y$", "the structure morphisms. For $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$", "there is a canonical isomorphism (see proof)", "$$", "f_!\\mathcal{F} \\longrightarrow \\overline{f}_!j_!\\mathcal{F}", "$$", "As we have $\\overline{f}_! = \\overline{f}_*$ by", "Lemma \\ref{lemma-proper-f-shriek} we obtain", "$\\overline{f}_* \\circ j_! = f_!$ as functors", "$\\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$." ], "refs": [ "more-etale-lemma-proper-f-shriek" ], "proofs": [ { "contents": [ "We have $(j_!\\mathcal{F})|_X = \\mathcal{F}$, see", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-jshriek-open}.", "Thus the displayed arrow is the injective map", "$f_!(\\mathcal{G}|_X) \\to \\overline{f}_!\\mathcal{G}$", "of Remark \\ref{remark-covariance-f-shriek-separated}", "for $\\mathcal{G} = j_!\\mathcal{F}$. The explicit nature", "of this map implies that it now suffices to show: if $V \\in Y_\\etale$ and", "$s \\in \\overline{f}_!\\mathcal{G}(V) = \\overline{f}_*\\mathcal{G}(V) =", "\\mathcal{G}(\\overline{X}_V)$", "is a section, then the support of $s$ is contained in the open", "$X_V \\subset \\overline{X}_V$. This is immediate from the fact", "that the stalks of $\\mathcal{G}$ are zero at geometric", "points of $\\overline{X} \\setminus X$." ], "refs": [ "etale-cohomology-lemma-jshriek-open", "more-etale-remark-covariance-f-shriek-separated" ], "ref_ids": [ 6522, 8849 ] } ], "ref_ids": [ 8810 ] }, { "id": 8812, "type": "theorem", "label": "more-etale-lemma-proper-compact-support", "categories": [ "more-etale" ], "title": "more-etale-lemma-proper-compact-support", "contents": [ "Let $X$ be a proper scheme over a field $k$. Then", "$H^0_c(X, \\mathcal{F}) = H^0(X, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the construction of $H^0_c$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8813, "type": "theorem", "label": "more-etale-lemma-compactify-compact-support", "categories": [ "more-etale" ], "title": "more-etale-lemma-compactify-compact-support", "contents": [ "Let $k$ be a field. Let $j : X \\to \\overline{X}$ be an open", "immersion of schemes over $k$ with $\\overline{X}$ proper over $k$.", "For $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$", "there is a canonical isomorphism (see proof)", "$$", "H^0_c(X, \\mathcal{F}) \\longrightarrow", "H^0_c(\\overline{X}, j_!\\mathcal{F}) =", "H^0(\\overline{X}, j_!\\mathcal{F})", "$$", "where we have the equality on the right by", "Lemma \\ref{lemma-proper-compact-support}." ], "refs": [ "more-etale-lemma-proper-compact-support" ], "proofs": [ { "contents": [ "We have $(j_!\\mathcal{F})|_X = \\mathcal{F}$, see", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-jshriek-open}.", "Thus the displayed arrow is the injective map", "$H^0_c(X, \\mathcal{G}|_X) \\to H^0_c(\\overline{X}, \\mathcal{G})$", "of Remark \\ref{remark-covariance-compact-support}", "for $\\mathcal{G} = j_!\\mathcal{F}$. The explicit nature", "of this map implies that it now suffices to show: if", "$s \\in H^0(\\overline{X}, \\mathcal{G})$ is a section, then the support of", "$s$ is contained in the open $X$. This is immediate from the fact", "that the stalks of $\\mathcal{G}$ are zero at geometric", "points of $\\overline{X} \\setminus X$." ], "refs": [ "etale-cohomology-lemma-jshriek-open", "more-etale-remark-covariance-compact-support" ], "ref_ids": [ 6522, 8850 ] } ], "ref_ids": [ 8812 ] }, { "id": 8814, "type": "theorem", "label": "more-etale-lemma-stalk-f-shriek-separated", "categories": [ "more-etale" ], "title": "more-etale-lemma-stalk-f-shriek-separated", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes which is separated and", "locally of finite type. Let $\\mathcal{F}$ be an abelian sheaf on", "$X_\\etale$. Then there is a canonical isomorphism", "$$", "(f_!\\mathcal{F})_{\\overline{y}}", "\\longrightarrow", "H^0_c(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}})", "$$", "for any geometric point $\\overline{y} : \\Spec(k) \\to Y$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $(f_*\\mathcal{F})_{\\overline{y}} = \\colim f_*\\mathcal{F}(V)$", "where the colimit is over the \\'etale neighbourhoods $(V, \\overline{v})$", "of $\\overline{y}$. If $s \\in f_*\\mathcal{F}(V) = \\mathcal{F}(X_V)$,", "then we can pullback $s$ to a section of $\\mathcal{F}$ over", "$(X_V)_{\\overline{v}} = X_{\\overline{y}}$. Thus we obtain a canonical map", "$$", "c_{\\overline{y}} :", "(f_*\\mathcal{F})_{\\overline{y}}", "\\longrightarrow ", "H^0(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}})", "$$", "We claim that this map induces a bijection between the subgroups", "$(f_!\\mathcal{F})_{\\overline{y}}$ and", "$H^0_c(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}})$.", "The claim implies the lemma, but is a little bit more precise", "in that it describes the identification of the lemma as given", "by pullbacks of sections of $\\mathcal{F}$ to the geometric fibre of $f$.", "\\medskip\\noindent", "Observe that any element", "$s \\in (f_!\\mathcal{F})_{\\overline{y}} \\subset (f_*\\mathcal{F})_{\\overline{y}}$", "is mapped by $c_{\\overline{y}}$ to an element of", "$H^0_c(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}}) \\subset", "H^0(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}})$.", "This is true because taking the support of a section", "commutes with pullback and because properness is preserved by", "base change. This at least produces the map in the statement of the lemma.", "To prove that it is an isomorphism we may work Zariski", "locally on $Y$ and hence we may and do assume $Y$ is affine.", "\\medskip\\noindent", "An observation that we will use below", "is that given an open subscheme $X' \\subset X$", "and if $f' = f|_{X'}$, then we obtain a commutative diagram", "$$", "\\xymatrix{", "(f'_!(\\mathcal{F}|_{X'}))_{\\overline{y}} \\ar[r] \\ar[d] &", "H^0_c(X'_{\\overline{y}}, \\mathcal{F}|_{X'_{\\overline{y}}}) \\ar[d] \\\\", "(f_!\\mathcal{F})_{\\overline{y}} \\ar[r] &", "H^0_c(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}})", "}", "$$", "where the horizontal arrows are the maps constructed above and", "the vertical arrows are given in", "Remarks \\ref{remark-covariance-f-shriek-separated} and", "\\ref{remark-covariance-compact-support}.", "The reason is that given an \\'etale neighbourhood $(V, \\overline{v})$", "of $\\overline{y}$ and a section $s \\in f_*\\mathcal{F}(V) = \\mathcal{F}(X_V)$", "whose support $Z$ happens to be contained in $X'_V$ and is proper over $V$,", "so that $s$ gives rise to an element of both", "$(f'_!(\\mathcal{F}|_{X'}))_{\\overline{y}}$ and", "$(f_!\\mathcal{F})_{\\overline{y}}$ which correspond via", "the vertical arrow of the diagram, then these elements are mapped via the", "horizontal arrows to the pullback $s|_{X_{\\overline{y}}}$ of $s$ to", "$X_{\\overline{y}}$ whose support $Z_{\\overline{y}}$ is contained in", "$X'_{\\overline{y}}$ and hence this restriction gives rise to", "a compatible pair of elements of", "$H^0_c(X'_{\\overline{y}}, \\mathcal{F}|_{X'_{\\overline{y}}})$", "and", "$H^0_c(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}})$.", "\\medskip\\noindent", "Suppose $s \\in (f_!\\mathcal{F})_{\\overline{y}}$ maps to zero in", "$H^0_c(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}})$.", "Say $s$ corresponds to $s \\in f_*\\mathcal{F}(V) = \\mathcal{F}(X_V)$", "with support $Z$ proper over $V$. We may assume that $V$ is affine", "and hence $Z$ is quasi-compact. Then we may choose a quasi-compact open", "$X' \\subset X$ containing the image of $Z$. Then $Z$ is contained in", "$X'_V$ and hence $s$ is the image of an element", "$s' \\in f'_!(\\mathcal{F}|_{X'})(V)$ where $f' = f|_{X'}$ as in", "the previous paragraph. Then $s'$ maps to zero in", "$H^0_c(X'_{\\overline{y}}, \\mathcal{F}|_{X'_{\\overline{y}}})$.", "Hence in order to prove injectivity, we may replace $X$ by", "$X'$, i.e., we may assume $X$ is quasi-compact. We will prove", "this case below.", "\\medskip\\noindent", "Suppose that", "$t \\in H^0_c(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}})$.", "Then the support of $t$ is contained in a quasi-compact", "open subscheme $W \\subset X_{\\overline{y}}$.", "Hence we can find a quasi-compact open subscheme", "$X' \\subset X$ such that $X'_{\\overline{y}}$ contains $W$.", "Then it is clear that $t$ is contained in the image", "of the injective map", "$H^0_c(X'_{\\overline{y}}, \\mathcal{F}|_{X'_{\\overline{y}}}) \\to", "H^0_c(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}})$.", "Hence in order to show surjectivity, we may replace $X$", "by $X'$, i.e., we may assume $X$ is quasi-compact.", "We will prove this case below.", "\\medskip\\noindent", "In this last paragraph of the proof we prove the lemma in case", "$X$ is quasi-compact and $Y$ is affine. By More on Flatness, Theorem", "\\ref{flat-theorem-nagata} there exists a compactification", "$j : X \\to \\overline{X}$ over $Y$. Set $\\mathcal{G} = j_!\\mathcal{F}$", "so that $\\mathcal{F} = \\mathcal{G}|_X$ by", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-jshriek-open}.", "By the disussion above we get a commutative diagram", "$$", "\\xymatrix{", "(f_!\\mathcal{F})_{\\overline{y}} \\ar[r] \\ar[d] &", "H^0_c(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}}) \\ar[d] \\\\", "(\\overline{f}_!\\mathcal{G})_{\\overline{y}} \\ar[r] &", "H^0_c(\\overline{X}_{\\overline{y}}, \\mathcal{G}|_{\\overline{X}_{\\overline{y}}})", "}", "$$", "By Lemmas \\ref{lemma-compactify-f-shriek-separated} and", "\\ref{lemma-compactify-compact-support} the vertical maps", "are isomorphisms. This reduces us to the case of the proper", "morphism $\\overline{X} \\to Y$. For a proper morphism our map", "is an isomorphism by", "Lemmas \\ref{lemma-proper-f-shriek} and \\ref{lemma-proper-compact-support}", "and proper base change for pushforwards, see", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-proper-pushforward-stalk}." ], "refs": [ "more-etale-remark-covariance-f-shriek-separated", "more-etale-remark-covariance-compact-support", "flat-theorem-nagata", "etale-cohomology-lemma-jshriek-open", "more-etale-lemma-compactify-f-shriek-separated", "more-etale-lemma-compactify-compact-support", "more-etale-lemma-proper-f-shriek", "more-etale-lemma-proper-compact-support", "etale-cohomology-lemma-proper-pushforward-stalk" ], "ref_ids": [ 8849, 8850, 5976, 6522, 8811, 8813, 8810, 8812, 6619 ] } ], "ref_ids": [] }, { "id": 8815, "type": "theorem", "label": "more-etale-lemma-base-change-f-shriek-separated", "categories": [ "more-etale" ], "title": "more-etale-lemma-base-change-f-shriek-separated", "contents": [ "Consider a cartesian square", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "of schemes with $f$ separated and locally of finite type.", "For any abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have", "$f'_!(g')^{-1}\\mathcal{F} = g^{-1}f_!\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "In great generality there is a pullback map", "$g^{-1}f_*\\mathcal{F} \\to f'_*(g')^{-1}\\mathcal{F}$, see", "Sites, Section \\ref{sites-section-pullback}.", "We claim that this map sends $g^{-1}f_!\\mathcal{F}$", "into the subsheaf $f'_!(g')^{-1}\\mathcal{F}$", "and induces the isomorphism in the lemma.", "\\medskip\\noindent", "Choose a geometric point $\\overline{y}': \\Spec(k) \\to Y'$ and denote", "$\\overline{y} = g \\circ \\overline{y}'$ the image in $Y$. There is a", "commutative diagram", "$$", "\\xymatrix{", "(f_*\\mathcal{F})_{\\overline{y}} \\ar[r] \\ar[d] &", "H^0(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}}) \\ar[d] \\\\", "(f'_*(g')^{-1}\\mathcal{F})_{\\overline{y}'} \\ar[r] &", "H^0(X'_{\\overline{y}'}, (g')^{-1}\\mathcal{F}|_{X'_{\\overline{y}'}})", "}", "$$", "where the horizontal maps were used in the proof of", "Lemma \\ref{lemma-stalk-f-shriek-separated}", "and the vertical maps are the pullback maps above.", "The diagram commutes because each of the four maps", "in question is given by pulling back local sections along", "a morphism of schemes and the underlying diagram of morphisms", "of schemes commutes. Since the diagram in the statement of the lemma", "is cartesian we have $X'_{\\overline{y}'} = X_{\\overline{y}}$.", "Hence by Lemma \\ref{lemma-stalk-f-shriek-separated}", "and its proof we obtain a commutative diagram", "$$", "\\xymatrix{", "(f_*\\mathcal{F})_{\\overline{y}} \\ar[rrr] \\ar[ddd] & & &", "H^0(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}}) \\ar[ddd] \\\\", "& (f_!\\mathcal{F})_{\\overline{y}} \\ar[r] \\ar@{..>}[d] \\ar[lu] &", "H^0_c(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}}) \\ar[d] \\ar[ru] \\\\", "& (f'_!(g')^{-1}\\mathcal{F})_{\\overline{y}'} \\ar[r] \\ar[ld] &", "H^0_c(X'_{\\overline{y}'}, (g')^{-1}\\mathcal{F}|_{X'_{\\overline{y}'}}) \\ar[rd]\\\\", "(f'_*(g')^{-1}\\mathcal{F})_{\\overline{y}'} \\ar[rrr] & & &", "H^0(X'_{\\overline{y}'}, (g')^{-1}\\mathcal{F}|_{X'_{\\overline{y}'}})", "}", "$$", "where the horizontal arrows of the inner square are isomorphisms", "and the two right vertical arrows are equalities. Also, the", "se, sw, ne, nw arrows are injective. It follows that there is a unique", "bijective dotted arrow fitting into the diagram. We conclude that", "$g^{-1}f_!\\mathcal{F} \\subset g^{-1}f_*\\mathcal{F} \\to f'_*(g')^{-1}\\mathcal{F}$", "is mapped into the subsheaf", "$f'_!(g')^{-1}\\mathcal{F} \\subset f'_*(g')^{-1}\\mathcal{F}$", "because this is true on stalks, see", "\\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-exactness-stalks}.", "The same theorem then implies that the induced map is an isomorphism", "and the proof is complete." ], "refs": [ "more-etale-lemma-stalk-f-shriek-separated", "more-etale-lemma-stalk-f-shriek-separated", "etale-cohomology-theorem-exactness-stalks" ], "ref_ids": [ 8814, 8814, 6376 ] } ], "ref_ids": [] }, { "id": 8816, "type": "theorem", "label": "more-etale-lemma-f-shriek-composition", "categories": [ "more-etale" ], "title": "more-etale-lemma-f-shriek-composition", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be composable morphisms of schemes which", "are separated and locally of finite type. Let $\\mathcal{F}$ be an abelian", "sheaf on $X_\\etale$. Then $g_!f_!\\mathcal{F} = (g \\circ f)_!\\mathcal{F}$", "as subsheaves of $(g \\circ f)_*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "We strongly urge the reader to prove this for themselves.", "Let $W \\in Z_\\etale$ and", "$s \\in (g \\circ f)_*\\mathcal{F}(W) = \\mathcal{F}(X_W)$.", "Denote $T \\subset X_W$ the support of $s$; this is a closed", "subset. Observe that $s$ is a section of $(g \\circ f)_!\\mathcal{F}$", "if and only if $T$ is proper over $W$. We have", "$f_!\\mathcal{F} \\subset f_*\\mathcal{F}$ and hence", "$g_!f_!\\mathcal{F} \\subset g_!f_*\\mathcal{F} \\subset g_*f_*\\mathcal{F}$.", "On the other hand, $s$ is a section of $g_!f_!\\mathcal{F}$ if and only", "if (a) $T$ is proper over $Y_W$ and (b) the support $T'$ of $s$", "viewed as section of $f_!\\mathcal{F}$ is proper over $W$.", "If (a) holds, then the image of $T$ in $Y_W$ is closed and since", "$f_!\\mathcal{F} \\subset f_*\\mathcal{F}$ we see that", "$T' \\subset Y_W$ is the image of $T$ (details omitted; look at stalks).", "\\medskip\\noindent", "The conclusion is that we have to show a closed subset $T \\subset X_W$", "is proper over $W$ if and only if $T$ is proper over $Y_W$", "and the image of $T$ in $Y_W$ is proper over $W$. Let us endow $T$", "with the reduced induced closed subscheme structure.", "If $T$ is proper over $W$, then $T \\to Y_W$ is proper by", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}", "and the image of $T$ in $Y_W$ is proper over $W$ by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-functoriality-closed-proper-over-base}.", "Conversely, if $T$ is proper over $Y_W$", "and the image of $T$ in $Y_W$ is proper over $W$,", "then the morphism $T \\to W$ is proper as a composition", "of proper morphisms (here we endow the closed image of $T$", "in $Y_W$ with its reduced induced scheme structure to turn the", "question into one about morphisms of schemes), see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-proper}." ], "refs": [ "morphisms-lemma-image-proper-scheme-closed", "coherent-lemma-functoriality-closed-proper-over-base", "morphisms-lemma-composition-proper" ], "ref_ids": [ 5411, 3389, 5408 ] } ], "ref_ids": [] }, { "id": 8817, "type": "theorem", "label": "more-etale-lemma-colim-f-shriek-separated", "categories": [ "more-etale" ], "title": "more-etale-lemma-colim-f-shriek-separated", "contents": [ "Let $f : X \\to Y$ be morphism of schemes which is separated and", "locally of finite type. Let $X = \\bigcup_{i \\in I} X_i$ be an", "open covering such that for all $i, j \\in I$ there exists a $k$", "with $X_i \\cup X_j \\subset X_k$. Denote $f_i : X_i \\to Y$", "the restriction of $f$. Then", "$$", "f_!\\mathcal{F} = \\colim_{i \\in I} f_{i, !}(\\mathcal{F}|_{X_i})", "$$", "functorially in $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$", "where the transition maps are the ones constructed in", "Remark \\ref{remark-covariance-f-shriek-separated}." ], "refs": [ "more-etale-remark-covariance-f-shriek-separated" ], "proofs": [ { "contents": [ "It suffices to show that the canonical map from", "right to left is a bijection when evaluated on a quasi-compact", "object $V$ of $Y_\\etale$.", "Observe that the colimit on the right hand side is directed", "and has injective transition maps.", "Thus we can use", "Sites, Lemma \\ref{sites-lemma-directed-colimits-sections}", "to evaluate the colimit. Hence, the statement comes down", "to the observation that a closed subset $Z \\subset X_V$ proper over $V$", "is quasi-compact and hence is contained in $X_{i, V}$ for some $i$." ], "refs": [ "sites-lemma-directed-colimits-sections" ], "ref_ids": [ 8531 ] } ], "ref_ids": [ 8849 ] }, { "id": 8818, "type": "theorem", "label": "more-etale-lemma-f-shriek-separated-direct-sums", "categories": [ "more-etale" ], "title": "more-etale-lemma-f-shriek-separated-direct-sums", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes which is separated and", "locally of finite type. Then functor $f_!$ commutes with direct sums." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F} = \\bigoplus \\mathcal{F}_i$. To show that the map", "$\\bigoplus f_!\\mathcal{F}_i \\to f_!\\mathcal{F}$ is an isomorphism,", "it suffices to show that these sheaves have the same sections over", "a quasi-compact object $V$ of $Y_\\etale$. Replacing $Y$ by $V$", "it suffices to show", "$H^0(Y, f_!\\mathcal{F}) \\subset H^0(X, \\mathcal{F})$", "is equal to", "$\\bigoplus H^0(Y, f_!\\mathcal{F}_i)", "\\subset \\bigoplus H^0(X, \\mathcal{F}_i)", "\\subset H^0(X, \\bigoplus \\mathcal{F}_i)$.", "In this case, by writing $X$ as the union of its quasi-compact opens", "and using Lemma \\ref{lemma-colim-f-shriek-separated}", "we reduce to the case where $X$ is quasi-compact as well.", "Then $H^0(X, \\mathcal{F}) = \\bigoplus H^0(X, \\mathcal{F}_i)$", "by \\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-colimit}.", "Looking at supports of sections the reader easily concludes." ], "refs": [ "more-etale-lemma-colim-f-shriek-separated", "etale-cohomology-theorem-colimit" ], "ref_ids": [ 8817, 6384 ] } ], "ref_ids": [] }, { "id": 8819, "type": "theorem", "label": "more-etale-lemma-lqf-f-shriek-separated-colimits", "categories": [ "more-etale" ], "title": "more-etale-lemma-lqf-f-shriek-separated-colimits", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes which is separated and", "locally quasi-finite. Then", "\\begin{enumerate}", "\\item for $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$ and a geometric", "point $\\overline{y} : \\Spec(k) \\to Y$ we have", "$$", "(f_!\\mathcal{F})_{\\overline{y}} =", "\\bigoplus\\nolimits_{f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}}", "$$", "functorially in $\\mathcal{F}$, and", "\\item the functor $f_!$ is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The functor $f_!$ is left exact by construction. Right exactness may", "be checked on stalks", "(\\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-exactness-stalks}).", "Thus it suffices to prove part (1).", "\\medskip\\noindent", "Let $\\overline{y} : \\Spec(k) \\to Y$ be a geometric point.", "The scheme $X_{\\overline{y}}$ has a discrete underlying", "topological space", "(Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-fibres})", "and all the residue fields at the points are equal to $k$", "(as finite extensions of $k$). Hence", "$\\{\\overline{x} : \\Spec(k) \\to X : f(\\overline{x}) = \\overline{y}\\}$", "is equal to the set of points of $X_{\\overline{y}}$.", "Thus the computation of the stalk follows from the more general", "Lemma \\ref{lemma-stalk-f-shriek-separated}." ], "refs": [ "etale-cohomology-theorem-exactness-stalks", "morphisms-lemma-locally-quasi-finite-fibres", "more-etale-lemma-stalk-f-shriek-separated" ], "ref_ids": [ 6376, 5228, 8814 ] } ], "ref_ids": [] }, { "id": 8820, "type": "theorem", "label": "more-etale-lemma-finite-support-f-shriek-separated", "categories": [ "more-etale" ], "title": "more-etale-lemma-finite-support-f-shriek-separated", "contents": [ "Let $f : X \\to Y$ be a separated and locally quasi-finite morphism", "of schemes. Functorially in $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$", "there is a canonical isomorphism(!)", "$$", "f_{p!}\\mathcal{F} \\longrightarrow f_!\\mathcal{F}", "$$", "of abelian presheaves which identifies the sheaf", "$f_!\\mathcal{F}$ of Definition \\ref{definition-f-shriek-separated}", "with the presheaf $f_{p!}\\mathcal{F}$ constructed above." ], "refs": [ "more-etale-definition-f-shriek-separated" ], "proofs": [ { "contents": [ "Let $V$ be an object of $Y_\\etale$. If $Z \\subset X_V$ is locally closed", "and finite over $V$, then, since $f$ is separated, we see that", "the morphism $Z \\to X_V$ is a closed immersion. Moreover, if", "$Z_i$, $i = 1, \\ldots, n$ are closed subschemes of $X_V$ finite", "over $V$, then $Z_1 \\cup \\ldots \\cup Z_n$ (scheme theoretic union)", "is a closed subscheme finite over $V$. Hence in this case the colimit", "(\\ref{equation-colimit-definition}) defining $f_{p!}\\mathcal{F}(V)$", "is directed and we find that $f_{!p}\\mathcal{F}(V)$ is simply equal", "to the set of sections of $\\mathcal{F}(X_V)$ whose support is finite over $V$.", "Since any closed subset of $X_V$ which is proper over $V$ is", "actually finite over $V$ (as $f$ is locally quasi-finite)", "we conclude that this is equal to $f_!\\mathcal{F}(V)$", "by its very definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8846 ] }, { "id": 8821, "type": "theorem", "label": "more-etale-lemma-finite-support-stalk", "categories": [ "more-etale" ], "title": "more-etale-lemma-finite-support-stalk", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes which is locally quasi-finite.", "Let $\\overline{y} : \\Spec(k) \\to Y$ be a geometric point.", "Functorially in $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$ we have", "$$", "(f_{p!}\\mathcal{F})_{\\overline{y}} =", "\\bigoplus\\nolimits_{f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Recall that the stalk at $\\overline{y}$ of a presheaf is defined by the", "usual colimit over \\'etale neighbourhoods $(V, \\overline{v})$", "of $\\overline{y}$, see \\'Etale Cohomology, Definition", "\\ref{etale-cohomology-definition-stalk}. Accordingly", "suppose $s = \\sum_{i = 1, \\ldots, n} (Z_i, s_i)$ as in", "(\\ref{equation-formal-sum}) is an element of $f_{p!}\\mathcal{F}(V)$", "where $(V, \\overline{v})$ is an \\'etale neighbourhood of $\\overline{y}$.", "Then since", "$$", "X_{\\overline{y}} = (X_V)_{\\overline{v}} \\supset Z_{i, \\overline{v}}", "$$", "and since $s_i$ is a section of $\\mathcal{F}$ on an open neighbourhood", "of $Z_i$ in $X_V$ we can send $s$ to", "$$", "\\sum\\nolimits_{i = 1, \\ldots, n} ", "\\sum\\nolimits_{\\overline{x} \\in Z_{i, \\overline{v}}}", "\\left(\\text{class of }s_i\\text{ in }\\mathcal{F}_{\\overline{x}}\\right)", "\\quad\\in\\quad", "\\bigoplus\\nolimits_{f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}}", "$$", "We omit the verification that this is compatible with restriction", "maps and that the relations (\\ref{item-sum}) $(Z, s) + (Z, s') - (Z, s + s')$", "and (\\ref{item-sub}) $(Z, s) - (Z', s)$ if $Z \\subset Z'$ are sent to zero.", "Thus we obtain a map", "$$", "(f_{p!}\\mathcal{F})_{\\overline{y}}", "\\longrightarrow", "\\bigoplus\\nolimits_{f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}}", "$$", "\\medskip\\noindent", "Let us prove this arrow is surjective. For this it suffices to pick", "an $\\overline{x}$ with $f(\\overline{x}) = \\overline{y}$ and prove that", "an element $s$ in the summand $\\mathcal{F}_{\\overline{x}}$ is in the", "image. Let $s$ correspond to the element $s \\in \\mathcal{F}(U)$", "where $(U, \\overline{u})$ is an \\'etale neighbourhood of $\\overline{x}$.", "Since $f$ is locally quasi-finite, the morphism $U \\to Y$", "is locally quasi-finite too. By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points-var}", "we can find an \\'etale neighbourhood $(V, \\overline{v})$ of", "$\\overline{y}$, an open subscheme", "$$", "W \\subset U \\times_Y V,", "$$", "and a geometric point $\\overline{w}$ mapping to $\\overline{u}$ and", "$\\overline{v}$ such that $W \\to V$ is finite and $\\overline{w}$ is the", "only geometric point of $W$ mapping to $\\overline{v}$. (We omit the translation", "between the language of geometric points we are currently using and the", "language of points and residue field extensions used in the", "statement of the lemma.) Observe that $W \\to X_V = X \\times_Y V$", "is \\'etale. Choose an affine open neighbourhood $W' \\subset X_V$", "of the image $\\overline{w}'$ of $\\overline{w}$. Since $\\overline{w}$", "is the only point of $W$ over $\\overline{v}$ and since $W \\to V$", "is closed, after replacing $V$ by an open neighbourhood of $\\overline{v}$,", "we may assume $W \\to X_V$ maps into $W'$. Then $W \\to W'$ is finite and", "\\'etale and there is a unique geometric point $\\overline{w}$ of $W$", "lying over $\\overline{w}'$. It follows that $W \\to W'$ is an open immersion", "over an open neighbourhood of $\\overline{w}'$ in $W'$, see", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-finite-etale-one-point}.", "Shrinking $V$ and $W'$ we may assume $W \\to W'$ is an isomorphism.", "Thus $s$ may be viewed as a section $s'$ of $\\mathcal{F}$ over", "the open subscheme $W' \\subset X_V$ which is finite over $V$.", "Hence by definition $(W', s')$ defines an element of $j_{p!}\\mathcal{F}(V)$", "which maps to $s$ as desired.", "\\medskip\\noindent", "Let us prove the arrow is injective. To do this, let", "$s = \\sum_{i = 1, \\ldots, n} (Z_i, s_i)$ as in (\\ref{equation-formal-sum})", "be an element of $f_{p!}\\mathcal{F}(V)$ where $(V, \\overline{v})$ is an", "\\'etale neighbourhood of $\\overline{y}$. Assume $s$ maps to zero", "under the map constructed above. First, after replacing", "$(V, \\overline{v})$ by an \\'etale neighbourhood of itself,", "we may assume there exist decompositions", "$Z_i = Z_{i, 1} \\amalg \\ldots \\amalg Z_{i, m_i}$ into open and closed", "subschemes such that each $Z_{i, j}$ has exactly one geometric point", "over $\\overline{v}$. Say under the obvious direct sum decomposition", "$$", "H_{Z_i}(\\mathcal{F}) = \\bigoplus H_{Z_{i, j}}(\\mathcal{F})", "$$", "the element $s_i$ corresponds to $\\sum s_{i, j}$. We may use relations", "(\\ref{item-sum}) and (\\ref{item-sub}) to replace $s$ by", "$\\sum_{i = 1, \\ldots, n} \\sum_{j = 1, \\ldots, m_i} (Z_{i, j}, s_{i, j})$.", "In other words, we may assume $Z_i$ has a unique geometric point", "lying over $\\overline{v}$. Let $\\overline{x}_1, \\ldots, \\overline{x}_m$", "be the geometric points of $X$ over $\\overline{y}$ corresponding to", "the geometric points of our $Z_i$ over $\\overline{v}$; note that for", "one $j \\in \\{1, \\ldots, m\\}$ there may be multiple indices $i$ for which", "$\\overline{x}_j$ corresponds to a point of $Z_i$.", "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points-var}", "applied to both $X_V \\to V$", "after replacing $(V, \\overline{v})$ by an \\'etale neighbourhood of itself", "we may assume there exist open subschemes", "$$", "W_j \\subset X \\times_Y V,\\quad j = 1, \\ldots, m", "$$", "and a geometric point $\\overline{w}_j$ of $W_j$ mapping to $\\overline{x}_j$ and", "$\\overline{v}$ such that $W_j \\to V$ is finite and $\\overline{w}_j$ is the", "only geometric point of $W_j$ mapping to $\\overline{v}$.", "After shrinking $V$ we may assume $Z_i \\subset W_j$ for some $j$", "and we have the map $H_{Z_i}(\\mathcal{F}) \\to H_{W_j}(\\mathcal{F})$.", "Thus by the relation (\\ref{item-sub})", "we see that our element is equivalent to an element of the form", "$$", "\\sum\\nolimits_{j = 1, \\ldots, m} (W_j, t_j)", "$$", "for some $t_j \\in H_{W_j}(\\mathcal{F})$. Clearly, this element is mapped", "simply to the class of $t_j$ in the summand $\\mathcal{F}_{\\overline{x}_j}$.", "Since $s$ maps to zero, we find that $t_j$ maps to zero in", "$\\mathcal{F}_{\\overline{x}_j}$. This implies that $t_j$ restricts", "to zero on an open neighbourhood of $\\overline{w}_j$ in $W_j$, see", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-zero-over-image}.", "Shrinking $V$ once more we obtain $t_j = 0$ for all $j$ as desired." ], "refs": [ "etale-cohomology-definition-stalk", "more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points-var", "etale-lemma-finite-etale-one-point", "more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points-var", "etale-cohomology-lemma-zero-over-image" ], "ref_ids": [ 6736, 13894, 10707, 13894, 6429 ] } ], "ref_ids": [] }, { "id": 8822, "type": "theorem", "label": "more-etale-lemma-finite-support-etale-shriek", "categories": [ "more-etale" ], "title": "more-etale-lemma-finite-support-etale-shriek", "contents": [ "Let $f = j : U \\to X$ be an \\'etale of schemes. Denote $j_{p!}$", "the construction of \\'Etale Cohomology, Equation", "(\\ref{etale-cohomology-equation-j-p-shriek})", "and denote $f_{p!}$ the construction above. Functorially in", "$\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ there is a canonical map", "$$", "j_{p!}\\mathcal{F} \\longrightarrow f_{p!}\\mathcal{F}", "$$", "of abelian presheaves which identifies the sheaf", "$j_!\\mathcal{F} = (j_{p!}\\mathcal{F})^\\#$ of \\'Etale Cohomology,", "Definition \\ref{etale-cohomology-definition-extension-zero}", "with $(f_{p!}\\mathcal{F})^\\#$." ], "refs": [ "etale-cohomology-definition-extension-zero" ], "proofs": [ { "contents": [ "Please read the proof of \\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-shriek-into-star-separated-etale}", "before reading the proof of this lemma.", "Let $V$ be an object of $X_\\etale$. Recall that", "$$", "j_{p!}\\mathcal{F}(V) =", "\\bigoplus\\nolimits_{\\varphi : V \\to U} \\mathcal{F}(V \\xrightarrow{\\varphi} U)", "$$", "Given $\\varphi$ we obtain an open subscheme", "$Z_\\varphi \\subset U_V = U \\times_X V$, namely,", "the image of the graph of $\\varphi$. Via $\\varphi$", "we obtain an isomorphism $V \\to Z_\\varphi$ over $U$", "and we can think of an element", "$$", "s_\\varphi \\in \\mathcal{F}(V \\xrightarrow{\\varphi} U) =", "\\mathcal{F}(Z_\\varphi) = H_{Z_\\varphi}(\\mathcal{F})", "$$", "as a section of $\\mathcal{F}$ over $Z_{\\varphi}$. Since", "$Z_\\varphi \\subset U_V$ is open, we actually have", "$H_{Z_\\varphi}(\\mathcal{F}) = \\mathcal{F}(Z_\\varphi)$", "and we can think of $s_\\varphi$ as an element of $H_{Z_\\varphi}(\\mathcal{F})$.", "Having said this, our map $j_{p!}\\mathcal{F} \\to f_{p!}\\mathcal{F}$", "is defined by the rule", "$$", "\\sum\\nolimits_{i = 1, \\ldots, n} s_{\\varphi_i}", "\\longmapsto", "\\sum\\nolimits_{i = 1, \\ldots, n} (Z_{\\varphi_i}, s_{\\varphi_i})", "$$", "with right hand side a sum as in (\\ref{equation-formal-sum}).", "We omit the verification that this is compatible with restriction", "mappings and functorial in $\\mathcal{F}$.", "\\medskip\\noindent", "To finish the proof, we claim that given a geometric point", "$\\overline{y} : \\Spec(k) \\to Y$ there is a commutative diagram", "$$", "\\xymatrix{", "(j_{p!}\\mathcal{F})_{\\overline{y}} \\ar[r] \\ar[d] &", "\\bigoplus_{j(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}}", "\\ar@{=}[d] \\\\", "(f_{p!}\\mathcal{F})_{\\overline{y}} \\ar[r] &", "\\bigoplus_{f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}}", "}", "$$", "where the top horizontal arrow is constructed in the proof of", "\\'Etale Cohomology, Proposition", "\\ref{etale-cohomology-proposition-describe-jshriek},", "the bottom horizontal arrow is constructed in the proof of", "Lemma \\ref{lemma-finite-support-stalk},", "the right vertical arrow is the obvious equality, and", "the left veritical arrow is the map defined in the previous", "paragraph on stalks. The claim follows in a straightforward manner", "from the explicit description of all of the arrows involved", "here and in the references given.", "Since the horizontal arrows are isomorphisms", "we conclude so is the left vertical arrow. Hence we find that", "our map induces an isomorphism on sheafifications by", "\\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-exactness-stalks}." ], "refs": [ "etale-cohomology-lemma-shriek-into-star-separated-etale", "etale-cohomology-proposition-describe-jshriek", "more-etale-lemma-finite-support-stalk", "etale-cohomology-theorem-exactness-stalks" ], "ref_ids": [ 6524, 6705, 8821, 6376 ] } ], "ref_ids": [ 6755 ] }, { "id": 8823, "type": "theorem", "label": "more-etale-lemma-lqf-f-shriek-stalk", "categories": [ "more-etale" ], "title": "more-etale-lemma-lqf-f-shriek-stalk", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes. Then", "\\begin{enumerate}", "\\item for $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$ and a geometric", "point $\\overline{y} : \\Spec(k) \\to Y$ we have", "$$", "(f_!\\mathcal{F})_{\\overline{y}} =", "\\bigoplus\\nolimits_{f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}}", "$$", "functorially in $\\mathcal{F}$, and", "\\item the functor $f_! : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$", "is exact and commutes with direct sums.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The formula for the stalks is immediate (and in fact equivalent) to", "Lemma \\ref{lemma-finite-support-stalk}.", "The exactness of the functor follows immediately from this", "and the fact that exactness may be checked on stalks, see", "\\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-exactness-stalks}." ], "refs": [ "more-etale-lemma-finite-support-stalk", "etale-cohomology-theorem-exactness-stalks" ], "ref_ids": [ 8821, 6376 ] } ], "ref_ids": [] }, { "id": 8824, "type": "theorem", "label": "more-etale-lemma-lqf-colimit-f-shriek", "categories": [ "more-etale" ], "title": "more-etale-lemma-lqf-colimit-f-shriek", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes.", "Let $X = \\bigcup_{i \\in I} X_i$ be an open covering. Then there", "exists an exact complex", "$$", "\\ldots \\to", "\\bigoplus\\nolimits_{i_0, i_1, i_2} f_{i_0i_1i_2, !}", "\\mathcal{F}|_{X_{i_0i_1i_2}} \\to", "\\bigoplus\\nolimits_{i_0, i_1} f_{i_0i_1, !} \\mathcal{F}|_{X_{i_0i_1}} \\to", "\\bigoplus\\nolimits_{i_0} f_{i_0, !} \\mathcal{F}|_{X_{i_0}}", "\\to f_!\\mathcal{F} \\to 0", "$$", "functorial in $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$, see", "proof for details." ], "refs": [], "proofs": [ { "contents": [ "Here as usual we set $X_{i_0 \\ldots i_p} = X_{i_0} \\cap \\ldots \\cap X_{i_p}$", "and we denote $f_{i_0 \\ldots i_p}$ the restriction of $f$ to", "$X_{i_0 \\ldots i_p}$. The maps in the complex are the maps", "constructed in Remark \\ref{remark-covariance-lqf-f-shriek}", "with sign rules as in the {\\v C}ech complex.", "Exactness follows easily from the description of stalks in", "Lemma \\ref{lemma-lqf-f-shriek-stalk}. Details omitted." ], "refs": [ "more-etale-remark-covariance-lqf-f-shriek", "more-etale-lemma-lqf-f-shriek-stalk" ], "ref_ids": [ 8852, 8823 ] } ], "ref_ids": [] }, { "id": 8825, "type": "theorem", "label": "more-etale-lemma-lqf-base-change-f-shriek", "categories": [ "more-etale" ], "title": "more-etale-lemma-lqf-base-change-f-shriek", "contents": [ "Consider a cartesian square", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "of schemes with $f$ locally quasi-finite. For any abelian sheaf $\\mathcal{F}$", "on $X_\\etale$ we have $f'_!(g')^{-1}\\mathcal{F} = g^{-1}f_!\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "With conventions as in Remark \\ref{remark-construct-map-presheaves-downstairs}", "we will explicitly construct a map", "$$", "c : f_{p!}\\mathcal{F} \\longrightarrow g_*f'_{p!}(g')^{-1}\\mathcal{F}", "$$", "of abelian presheaves on $Y_\\etale$. By the discussion in", "Remark \\ref{remark-construct-map-presheaves-downstairs}", "this will determine a canonical map", "$g^{-1}f_!\\mathcal{F} \\to f'_!(g')^{-1}\\mathcal{F}$. Finally, we", "will show this map induces isomorphisms on stalks and conclude by", "\\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-exactness-stalks}.", "\\medskip\\noindent", "Construction of the map $c$. Let $V \\in Y_\\etale$ and consider a section", "$s = \\sum_{i = 1, \\ldots, n} (Z_i, s_i)$ as in", "(\\ref{equation-formal-sum}) defining an element of $f_{p!}\\mathcal{F}(V)$.", "The value of $g_*f'_{p!}(g')^{-1}\\mathcal{F}$ at $V$ is", "$f'_{p!}(g')^{-1}\\mathcal{F}(V')$ where $V' = V \\times_Y Y'$.", "Denote $Z'_i \\subset X'_{V'}$ the base change of $Z_i$ to $V'$.", "By (\\ref{item-pullback}) there is a pullback map", "$H_{Z_i}(\\mathcal{F}) \\to H_{Z'_i}((g')^{-1}\\mathcal{F})$.", "Denoting $s'_i \\in H_{Z'_i}((g')^{-1}\\mathcal{F})$ the image of $s_i$", "under pullback, we set $c(s) = \\sum_{i = 1, \\ldots, n} (Z'_i, s'_i)$ as in", "(\\ref{equation-formal-sum}) defining an element of", "$f'_{p!}(g')^{-1}\\mathcal{F}(V')$. We omit the verification", "that this construction is compatible the relations", "(\\ref{item-sum}) and (\\ref{item-sub}) and compatible", "with restriction mappings. The construction is clearly", "functorial in $\\mathcal{F}$.", "\\medskip\\noindent", "Let $\\overline{y}' : \\Spec(k) \\to Y'$ be a geometric point with image", "$\\overline{y} = g \\circ \\overline{y}'$ in $Y$. Observe that", "$X'_{\\overline{y}'} = X_{\\overline{y}}$ by transitivity of", "fibre products. Hence $g'$ produces a bijection", "$\\{f'(\\overline{x}') = \\overline{y}'\\} \\to \\{f(\\overline{x}) = \\overline{y}\\}$", "and if $\\overline{x}'$ maps to $\\overline{x}$, then", "$((g')^{-1}\\mathcal{F})_{\\overline{x}'} = \\mathcal{F}_{\\overline{x}}$", "by \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-stalk-pullback}.", "Now we claim that the diagram", "$$", "\\xymatrix{", "(g^{-1}f_!\\mathcal{F})_{\\overline{y}'} \\ar@{=}[r] \\ar[d] &", "(f_!\\mathcal{F})_{\\overline{y}} \\ar[r] \\ar[ld] &", "\\bigoplus\\nolimits_{f(\\overline{x}) = \\overline{y}}", "\\mathcal{F}_{\\overline{x}} \\ar[d]", "\\\\", "(f'_!(g')^{-1}\\mathcal{F})_{\\overline{y}'} \\ar[rr] & &", "\\bigoplus\\nolimits_{f'(\\overline{x}') = \\overline{y}'}", "(g')^{-1}\\mathcal{F}_{\\overline{x}'}", "}", "$$", "commutes where the horizontal arrows are given in the proof of", "Lemma \\ref{lemma-finite-support-stalk} and where the right vertical", "arrow is an equality by what we just said above. The southwest arrow is", "described in Remark \\ref{remark-construct-map-presheaves-downstairs}", "as the pullback map, i.e.,", "simply given by our construction $c$ above. Then the simple", "description of the image of a sum $\\sum (Z_i, z_i)$ in the", "stalk at $\\overline{x}$ given in the proof of", "Lemma \\ref{lemma-finite-support-stalk} immediately shows the", "diagram commutes. This finishes the proof of the lemma." ], "refs": [ "more-etale-remark-construct-map-presheaves-downstairs", "more-etale-remark-construct-map-presheaves-downstairs", "etale-cohomology-theorem-exactness-stalks", "etale-cohomology-lemma-stalk-pullback", "more-etale-lemma-finite-support-stalk", "more-etale-remark-construct-map-presheaves-downstairs", "more-etale-lemma-finite-support-stalk" ], "ref_ids": [ 8854, 8854, 6376, 6436, 8821, 8854, 8821 ] } ], "ref_ids": [] }, { "id": 8826, "type": "theorem", "label": "more-etale-lemma-lqf-separated-shriek-composition", "categories": [ "more-etale" ], "title": "more-etale-lemma-lqf-separated-shriek-composition", "contents": [ "Let $f' : X \\to Y'$ and $g : Y' \\to Y$ be composable morphisms of schemes", "with $f'$ and $f = g \\circ f'$ locally quasi-finite and $g$ separated and", "locally of finite type. Then there is a canonical isomorphism of functors", "$g_! \\circ f'_! = f_!$. This isomorphism is compatible with", "\\begin{enumerate}", "\\item[(a)] covariance with respect to open embeddings as in", "Remarks \\ref{remark-covariance-f-shriek-separated} and", "\\ref{remark-covariance-lqf-f-shriek},", "\\item[(b)] the base change isomorphisms of", "Lemmas \\ref{lemma-lqf-base-change-f-shriek}", "and \\ref{lemma-base-change-f-shriek-separated}, and", "\\item[(c)] equal to the isomorphism of Lemma \\ref{lemma-f-shriek-composition}", "via the identifications of Lemma \\ref{lemma-finite-support-f-shriek-separated}", "in case $f'$ is separated.", "\\end{enumerate}" ], "refs": [ "more-etale-remark-covariance-f-shriek-separated", "more-etale-remark-covariance-lqf-f-shriek", "more-etale-lemma-lqf-base-change-f-shriek", "more-etale-lemma-base-change-f-shriek-separated", "more-etale-lemma-f-shriek-composition", "more-etale-lemma-finite-support-f-shriek-separated" ], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. With conventions as in", "Remark \\ref{remark-construct-map-presheaves-downstairs} we will explicitly", "construct a map", "$$", "c : f_{p!}\\mathcal{F} \\longrightarrow g_*f'_{p!}\\mathcal{F}", "$$", "of abelian presheaves on $Y_\\etale$. By the discussion in", "Remark \\ref{remark-construct-map-presheaves-downstairs}", "this will determine a canonical map", "$c^\\# : f_!\\mathcal{F} \\to g_*f'_!\\mathcal{F}$.", "We will show that $c^\\#$ has image contained in the subsheaf", "$g_!f'_!\\mathcal{F}$, thereby obtaining a map", "$c' : f_!\\mathcal{F} \\to g_!f'_!\\mathcal{F}$. Next, we will prove", "(a), (b), and (c) that. Finally, part (b)", "will allow us to show that $c'$ is an isomorphism.", "\\medskip\\noindent", "Construction of the map $c$. Let $V \\in Y_\\etale$ and", "let $s = \\sum (Z_i, s_i)$ be a sum as in (\\ref{equation-formal-sum})", "defining an element of $f_{p!}\\mathcal{F}(V)$.", "Recall that $Z_i \\subset X_V = X \\times_Y V$", "is a locally closed subscheme finite over $V$.", "Setting $V' = Y' \\times_Y V$ we get $X_{V'} = X \\times_{Y'} V' = X_V$.", "Hence $Z_i \\subset X_{V'}$ is locally closed and", "$Z_i$ is finite over $V'$ because $g$ is separated", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-permanence}).", "Hence we may set $c(s) = \\sum (Z_i, s_i)$ but now viewed", "as an element of $f'_{p!}\\mathcal{F}(V') = (g_*f'_{p!}\\mathcal{F})(V)$.", "The construction is clearly compatible with relations", "(\\ref{item-sum}) and (\\ref{item-sub})", "and compatible with restriction mappings and hence we obtain the map $c$.", "\\medskip\\noindent", "Observe that in the discussion above our section $c(s) = \\sum (Z_i, s_i)$ of", "$f'_!\\mathcal{F}$ over $V'$ restricts to zero on", "$V' \\setminus \\Im(\\coprod Z_i \\to V')$. Since $\\Im(\\coprod Z_i \\to V')$", "is proper over $V$ (for example by Morphisms, Lemma", "\\ref{morphisms-lemma-scheme-theoretic-image-is-proper})", "we conclude that $c(s)$ defines a section of", "$g_!f'_!\\mathcal{F} \\subset g_*f'_!\\mathcal{F}$ over $V$.", "Since every local section of $f_!\\mathcal{F}$ locally comes from a", "local section of $f_{p!}\\mathcal{F}$ we conclude that the image", "of $c^\\#$ is contained in $g_!f'_!\\mathcal{F}$.", "Thus we obtain an induced map $c' : f_!\\mathcal{F} \\to g_!f'_!\\mathcal{F}$", "factoring $c^\\#$ as predicted in the first paragraph of the proof.", "\\medskip\\noindent", "Proof of (a). Let $Y'_1 \\subset Y'$ be an open subscheme", "and set $X_1 = (f')^{-1}(W')$. We obtain a diagram", "$$", "\\xymatrix{", "X_1 \\ar[d]_{f'_1} \\ar[r]_a \\ar@/_2em/[dd]_{f_1} &", "X \\ar[d]^{f'} \\ar@/^2em/[dd]^f \\\\", "Y'_1 \\ar[d]_{g_1} \\ar[r]_{b'} &", "Y' \\ar[d]^g \\\\", "Y \\ar@{=}[r] &", "Y", "}", "$$", "where the horizontal arrows are open immersions. Then our claim is that", "the diagram", "$$", "\\xymatrix{", "f_{1, !}\\mathcal{F}|_{X_1} \\ar[r]_{c'_1} \\ar[dd] &", "g_{1, !}f'_{1, !}\\mathcal{F}|_{X_1} \\ar@{=}[d] \\\\", "& g_{1, !}(f'_!\\mathcal{F})|_{Y'_1} \\ar[d] \\\\", "f_!\\mathcal{F} \\ar[r]^{c'} &", "g_!f'_!\\mathcal{F} \\ar[r] & g_*f'_!\\mathcal{F}", "}", "$$", "commutes where the left vertical arrow is", "Remark \\ref{remark-covariance-lqf-f-shriek} and", "the right vertical arrow is Remark \\ref{remark-covariance-f-shriek-separated}.", "The equality sign in the diagram comes about because $f'_1$", "is the restriction of $f'$ to $Y'_1$ and our construction", "of $f'_!$ is local on the base.", "Finally, to prove the commutativity we choose an object $V$ of", "$Y_\\etale$ and a formal sum $s_1 = \\sum (Z_{1, i}, s_{1, i})$ as in", "(\\ref{equation-formal-sum}) defining an element of", "$f_{1, p!}\\mathcal{F}|_{X_1}(V)$. Recall this means", "$Z_{1, i} \\subset X_1 \\times_Y V$ is locally closed finite over $V$", "and $s_{1, i} \\in H_{Z_{1, i}}(\\mathcal{F})$.", "Then we chase this section", "across the maps involved, but we only need to show we", "end up with the same element of", "$g_*f'_!\\mathcal{F}(V) = f'_!\\mathcal{F}(Y' \\times_Y V)$.", "Going around both sides of the diagram the reader immediately", "sees we end up with the element $\\sum (Z_{1, i}, s_{1, i})$", "where now $Z_{1, i}$ is viewed as a locally closed subscheme", "of $X \\times_{Y'} (Y' \\times_Y V) = X \\times_Y V$ finite over", "$Y' \\times_Y V$.", "\\medskip\\noindent", "Proof of (b). Let $b : Y_1 \\to Y$ be a morphism of schemes. Let us form the", "commutative diagram", "$$", "\\xymatrix{", "X_1 \\ar[d]_{f'_1} \\ar[r]_a \\ar@/_2em/[dd]_{f_1} &", "X \\ar[d]^{f'} \\ar@/^2em/[dd]^f \\\\", "Y'_1 \\ar[d]_{g_1} \\ar[r]_{b'} &", "Y' \\ar[d]^g \\\\", "Y_1 \\ar[r]^b &", "Y", "}", "$$", "with cartesian squares. We claim that our construction is compatible", "with the base change maps of Lemmas \\ref{lemma-lqf-base-change-f-shriek}", "and \\ref{lemma-base-change-f-shriek-separated}, i.e.,", "that the top rectangle of the diagram", "$$", "\\xymatrix{", "b^{-1}f_!\\mathcal{F} \\ar[rr] \\ar[d]_{b^{-1}c'} & &", "f_{1, !}a^{-1}\\mathcal{F} \\ar[d]^{c_1'} \\\\", "b^{-1}g_!f'_!\\mathcal{F} \\ar[r] \\ar[d] &", "g_{1, !}(b')^{-1}f'_!\\mathcal{F} \\ar[r] \\ar[d] &", "g_{1, !}f'_{1, !}a^{-1}\\mathcal{F} \\ar[d] \\\\", "b^{-1}g_*f'_!\\mathcal{F} \\ar[r] &", "g_{1, *}(b')^{-1}f'_!\\mathcal{F} \\ar[r] &", "g_{1, *}f'_{1, !}a^{-1}\\mathcal{F}", "}", "$$", "commutes. The verification of this is completely routine and we", "urge the reader to skip it. Since the arrows going from the middle", "row down to the bottom row are injective, it suffices to show that", "the outer diagram commutes.", "To show this it suffices to take a local section of", "$b^{-1}f_!\\mathcal{F}$ and show we end up with the same local", "section of $g_{1, *}f'_{1, !}a^{-1}\\mathcal{F}$", "going around either way. However, in fact it suffices to check", "this for local sections which are of the the pullback by $b$ of", "a section $s = \\sum (Z_i, s_i)$ of $f_{p!}\\mathcal{F}(V)$", "as above (since such pullbacks generate the abelian sheaf", "$b^{-1}f_!\\mathcal{F}$). Denote $V_1$, $V'_1$, and $Z_{1, i}$", "the base change of $V$, $V' = Y' \\times_Y V$, $Z_i$ by $Y_1 \\to Y$.", "Recall that $Z_i$ is a locally closed subscheme of $X_V = X_{V'}$", "and hence $Z_{1, i}$ is a locally closed subscheme", "of $(X_1)_{V_1} = (X_1)_{V'_1}$. Then $b^{-1}c'$ sends the pullback", "of $s$ to the pullback of the local section $c(s) \\sum (Z_i, s_i)$ viewed", "as an element of $f'_{p!}\\mathcal{F}(V') = (g_*f'_{p!}\\mathcal{F})(V)$.", "The composition of the bottom two base change maps", "simply maps this to $\\sum (Z_{i, 1}, s_{1, i})$ viewed as an", "element of $f'_{1, p!}a^{-1}\\mathcal{F}(V'_1) =", "g_{1, *}f'_{1, p!}a^{-1}\\mathcal{F}(V_1)$.", "On the other hand, the base change map at the top of the diagram", "sends the pullback of $s$ to $\\sum (Z_{1, i}, s_{1, i})$ viewed", "as an element of $f_{1, !}a^{-1}\\mathcal{F}(V_1)$.", "Then finally $c'_1$ by its very construction does indeed", "map this to $\\sum (Z_{i, 1}, s_{1, i})$ viewed as an", "element of $f'_{1, p!}a^{-1}\\mathcal{F}(V'_1) =", "g_{1, *}f'_{1, p!}a^{-1}\\mathcal{F}(V_1)$ and the commutativity", "has been verified.", "\\medskip\\noindent", "Proof of (c). This follows from comparing the definitions", "for both maps; we omit the details.", "\\medskip\\noindent", "To finish the proof it suffices to show that the pullback of", "$c'$ via any geometric point $\\overline{y} : \\Spec(k) \\to Y$", "is an isomorphism. Namely, pulling back by $\\overline{y}$", "is the same thing as taking stalks and $\\overline{y}$", "(\\'Etale Cohomology, Remark \\ref{etale-cohomology-remark-stalk-pullback})", "and hence we can invoke", "\\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-exactness-stalks}.", "By the compatibility (b) just shown, we ", "conclude that we may assume $Y$ is the spectrum of $k$", "and we have to show that $c'$ is an isomorphism.", "To do this it suffices to show that the induced map", "$$", "\\bigoplus\\nolimits_{x \\in X} \\mathcal{F}_x = H^0(Y, f_!\\mathcal{F})", "\\longrightarrow", "H^0(Y, g_!f'_!\\mathcal{F}) = H^0_c(Y', f'_!\\mathcal{F})", "$$", "is an isomorphism. The equalities hold by", "Lemmas \\ref{lemma-lqf-f-shriek-stalk} and", "\\ref{lemma-stalk-f-shriek-separated}.", "Recall that $X$ is a disjoint union of", "spectra of Artinian local rings with residue field $k$, see", "Varieties, Lemma \\ref{varieties-lemma-algebraic-scheme-dim-0}.", "Since the left and right hand side commute with direct", "sums (details omitted) we may assume that $\\mathcal{F}$ is a skyscraper", "sheaf $x_*A$ supported at some $x \\in X$.", "Then $f'_!\\mathcal{F}$ is the skyscraper sheaf at the", "image $y'$ of $x$ in $Y$ by Lemma \\ref{lemma-lqf-f-shriek-stalk}.", "In this case it is obvious that our construction", "produces the identity map $A \\to H^0_c(Y', y'_*A) = A$", "as desired." ], "refs": [ "more-etale-remark-construct-map-presheaves-downstairs", "more-etale-remark-construct-map-presheaves-downstairs", "morphisms-lemma-finite-permanence", "morphisms-lemma-scheme-theoretic-image-is-proper", "more-etale-remark-covariance-lqf-f-shriek", "more-etale-remark-covariance-f-shriek-separated", "more-etale-lemma-lqf-base-change-f-shriek", "more-etale-lemma-base-change-f-shriek-separated", "etale-cohomology-remark-stalk-pullback", "etale-cohomology-theorem-exactness-stalks", "more-etale-lemma-lqf-f-shriek-stalk", "more-etale-lemma-stalk-f-shriek-separated", "varieties-lemma-algebraic-scheme-dim-0", "more-etale-lemma-lqf-f-shriek-stalk" ], "ref_ids": [ 8854, 8854, 5448, 5414, 8852, 8849, 8825, 8815, 6788, 6376, 8823, 8814, 10988, 8823 ] } ], "ref_ids": [ 8849, 8852, 8825, 8815, 8816, 8820 ] }, { "id": 8827, "type": "theorem", "label": "more-etale-lemma-lqf-shriek-composition", "categories": [ "more-etale" ], "title": "more-etale-lemma-lqf-shriek-composition", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be composable locally quasi-finite", "morphisms of schemes. Then there is a canonical isomorphism of functors", "$$", "(g \\circ f)_! \\longrightarrow g_! \\circ f_!", "$$", "These isomorphisms satisfy the following properties:", "\\begin{enumerate}", "\\item If $f$ and $g$ are separated, then the isomorphism agrees", "with Lemma \\ref{lemma-f-shriek-composition}.", "\\item If $g$ is separated, then the isomorphism agrees with", "Lemma \\ref{lemma-lqf-separated-shriek-composition}.", "\\item For a geometric point $\\overline{z} : \\Spec(k) \\to Z$ the diagram", "$$", "\\xymatrix{", "((g \\circ f)_!\\mathcal{F})_{\\overline{z}} \\ar[d] \\ar[rr] & &", "\\bigoplus\\nolimits_{g(f(\\overline{x})) = \\overline{z}}", "\\mathcal{F}_{\\overline{x}} \\ar@{=}[d] \\\\", "(g_!f_!\\mathcal{F})_{\\overline{z}} \\ar[r] &", "\\bigoplus\\nolimits_{g(\\overline{y}) = \\overline{z}}", "(f_!\\mathcal{F})_{\\overline{y}} \\ar[r] &", "\\bigoplus\\nolimits_{g(f(\\overline{x})) = \\overline{z}}", "\\mathcal{F}_{\\overline{x}}", "}", "$$", "is commutative where the horizontal arrows are given by", "Lemma \\ref{lemma-lqf-f-shriek-stalk}.", "\\item Let $h : Z \\to T$ be a third locally quasi-finite", "morphism of schemes. Then the diagram", "$$", "\\xymatrix{", "(h \\circ g \\circ f)_! \\ar[r] \\ar[d] &", "(h \\circ g)_! \\circ f_! \\ar[d] \\\\", "h_! \\circ (g \\circ f)_! \\ar[r] &", "h_! \\circ g_! \\circ f_!", "}", "$$", "commutes.", "\\item Suppose that we have a diagram of schemes", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_c & X \\ar[d]^f \\\\", "Y' \\ar[d]_{g'} \\ar[r]_b & Y \\ar[d]^g \\\\", "Z' \\ar[r]^a & Z", "}", "$$", "with both squares cartesian and $f$ and $g$", "locally quasi-finite. Then the diagram", "$$", "\\xymatrix{", "a^{-1} \\circ (g \\circ f)_! \\ar[d] \\ar[rr] & &", "(g' \\circ f')_! \\circ c^{-1} \\ar[d] \\\\", "a^{-1} \\circ g_! \\circ f_! \\ar[r] &", "g'_! \\circ b^{-1} \\circ f_! \\ar[r] &", "g'_! \\circ f'_! \\circ c^{-1}", "}", "$$", "commutes where the horizontal arrows are those of", "Lemma \\ref{lemma-lqf-base-change-f-shriek}.", "\\end{enumerate}" ], "refs": [ "more-etale-lemma-f-shriek-composition", "more-etale-lemma-lqf-separated-shriek-composition", "more-etale-lemma-lqf-f-shriek-stalk", "more-etale-lemma-lqf-base-change-f-shriek" ], "proofs": [ { "contents": [ "If $f$ and $g$ are separated, then this is a special case of", "Lemma \\ref{lemma-f-shriek-composition}.", "If $g$ is separated, then this is a special case of", "Lemma \\ref{lemma-lqf-separated-shriek-composition}", "which moreover agrees with the case where $f$ and $g$ are separated.", "\\medskip\\noindent", "Construction in the general case. Choose an open covering $Y = \\bigcup Y_i$", "such that the restriction $g_i : Y_i \\to Z$ of $g$ is separated.", "Set $X_i = f^{-1}(Y_i)$ and denote $f_i : X_i \\to Y_i$ the restriction", "of $f$. Also denote $h = g \\circ f$ and $h_i : X_i \\to Z$ the restriction", "of $h$. Consider the following diagram", "$$", "\\xymatrix{", "\\bigoplus\\nolimits_{i_0, i_1}", "h_{i_0i_1, !}\\mathcal{F}|_{X_{i_0i_1}} \\ar[r] \\ar[d] &", "\\bigoplus\\nolimits_{i_0} h_{i_0, !}\\mathcal{F}|_{X_{i_0}} \\ar[r] \\ar[d] &", "h_!\\mathcal{F} \\ar[r] \\ar@{..>}[dd] &", "0 \\\\", "\\bigoplus\\nolimits_{i_0, i_1}", "g_{i_0i_1, !} f_{i_0i_1, !}\\mathcal{F}|_{X_{i_0i_1}} \\ar[r] \\ar[d] &", "\\bigoplus\\nolimits_{i_0}", "g_{i_0, !} f_{i_0, !}\\mathcal{F}|_{X_{i_0}} \\ar[d] \\\\", "\\bigoplus\\nolimits_{i_0, i_1}", "g_{i_0i_1, !} (f_!\\mathcal{F})|_{Y_{i_0i_1}} \\ar[r] &", "\\bigoplus\\nolimits_{i_0}", "g_{i_0, !} (f_!\\mathcal{F})|_{Y_{i_0}} \\ar[r] &", "g_!f_!\\mathcal{F} \\ar[r] &", "0", "}", "$$", "By Lemma \\ref{lemma-lqf-colimit-f-shriek} the top and bottom row", "in the diagram are exact. By Lemma \\ref{lemma-lqf-separated-shriek-composition}", "the top left square commutes. The vertical arrows in the", "lower left square come about because", "$(f_!\\mathcal{F})|_{Y_{i_0i_1}} = f_{i_0i_1, !}\\mathcal{F}|_{X_{i_0i_1}}$ and", "$(f_!\\mathcal{F})|_{Y_{i_0}} = f_{i_0, !}\\mathcal{F}|_{X_{i_0}}$", "as the construction of $f_!$ is local on the base. Moreover, these", "equalities are (of course) compatible with the identifications", "$((f_!\\mathcal{F})|_{Y_{i_0}})|_{Y_{i_0i_1}} =", "(f_!\\mathcal{F})|_{Y_{i_0i_1}}$ and", "$(f_{i_0, !}\\mathcal{F}|_{X_{i_0}})|_{Y_{i_0i_1}} =", "f_{i_0i_1, !}\\mathcal{F}|_{X_{i_0i_1}}$", "which are used (together with the covariance for open embeddings", "for $Y_{i_0i_1} \\subset Y_{i_0}$)", "to define the horizontal maps of the lower left square.", "Thus this square commutes as well.", "In this way we conclude there is a unique", "dotted arrow as indicated in the diagram and", "moreover this arrow is an isomorphism.", "\\medskip\\noindent", "Proof of properties (1) -- (5). Fix the open covering $Y = \\bigcup Y_i$.", "Observe that if $Y \\to Z$ happens to be separated, then we get a dotted", "arrow fitting into the huge diagram above by using the map of", "Lemma \\ref{lemma-lqf-separated-shriek-composition}", "(by the very properties of that lemma).", "This proves (2) and hence also (1) by the compatibility of the", "maps of Lemma \\ref{lemma-lqf-separated-shriek-composition}", "and Lemma \\ref{lemma-f-shriek-composition}.", "Next, for any scheme $Z'$ over $Z$, we obtain the compatibility in (5)", "for the map $(g' \\circ f')_! \\to g'_! \\circ f'_!$", "constructed using the open covering $Y' = \\bigcup b^{-1}(Y_i)$.", "This is clear from the corresponding compatibility of the maps", "constructed in Lemma \\ref{lemma-lqf-separated-shriek-composition}.", "In particular, we can consider a geometric point", "$\\overline{z} : \\Spec(k) \\to Z$. Since", "$X_{\\overline{z}} \\to Y_{\\overline{z}} \\to \\Spec(k)$", "are separated maps, we find that the base change of", "$(g \\circ f)_!\\mathcal{F} \\to g_! f_! \\mathcal{F}$", "by $\\overline{z}$ is equal to the map of", "Lemma \\ref{lemma-f-shriek-composition}.", "The reader then immediately sees that we obtain property (3).", "Of course, property (3) guarantees that our transformation of functors", "$(g \\circ f)_! \\to g_! \\circ f_!$ constructed using the open covering", "$Y = \\bigcup Y_i$ doesn't depend on the choice of this open covering.", "Finally, property (4) follows by looking at what happens on stalks", "using the already proven property (3)." ], "refs": [ "more-etale-lemma-f-shriek-composition", "more-etale-lemma-lqf-separated-shriek-composition", "more-etale-lemma-lqf-colimit-f-shriek", "more-etale-lemma-lqf-separated-shriek-composition", "more-etale-lemma-lqf-separated-shriek-composition", "more-etale-lemma-lqf-separated-shriek-composition", "more-etale-lemma-f-shriek-composition", "more-etale-lemma-lqf-separated-shriek-composition", "more-etale-lemma-f-shriek-composition" ], "ref_ids": [ 8816, 8826, 8824, 8826, 8826, 8826, 8816, 8826, 8816 ] } ], "ref_ids": [ 8816, 8826, 8823, 8825 ] }, { "id": 8828, "type": "theorem", "label": "more-etale-lemma-lqf-f-upper-shriek", "categories": [ "more-etale" ], "title": "more-etale-lemma-lqf-f-upper-shriek", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes. The functor", "$f_! : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$", "has a right adjoint", "$f^! : \\textit{Ab}(Y_\\etale) \\to \\textit{Ab}(X_\\etale)$.", "Moreover, we have $f^!(\\overline{y}_*A) =", "\\prod_{f(\\overline{x}) = \\overline{y}} \\overline{x}_*A$." ], "refs": [], "proofs": [ { "contents": [ "Let $E \\subset \\Ob(\\textit{Ab}(Y_\\etale))$ be the class consisting of", "products of skyscraper sheaves. We claim that", "\\begin{enumerate}", "\\item every $\\mathcal{G}$ in $\\textit{Ab}(Y_\\etale)$ is a subsheaf", "of an element of $E$, and", "\\item for every $\\mathcal{G} \\in E$ there exists an object", "$\\mathcal{H}$ of $\\textit{Ab}(X_\\etale)$ such that", "$\\Hom(f_!\\mathcal{F}, \\mathcal{G}) = \\Hom(\\mathcal{F}, \\mathcal{H})$", "functorially in $\\mathcal{F}$.", "\\end{enumerate}", "Once the claim has been verified, the dual of", "Homology, Lemma \\ref{homology-lemma-partially-defined-adjoint}", "produces the adjoint functor $f^!$.", "\\medskip\\noindent", "Part (1) is true because we can map $\\mathcal{G}$ to the sheaf", "$\\prod \\overline{y}_*\\mathcal{G}_{\\overline{y}}$ where the", "product is over all geometric points of $Y$. This is an injection by", "\\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-exactness-stalks}.", "(This is the first step in the Godement resolution when", "done in the setting of abelian sheaves on topological spaces.)", "\\medskip\\noindent", "Part (2) and the final statement of the lemma can be seen as follows.", "Suppose that", "$\\mathcal{G} = \\prod \\overline{y}_*A_{\\overline{y}}$", "for some abelian groups $A_{\\overline{y}}$. Then", "$$", "\\Hom(f_!\\mathcal{F}, \\mathcal{G}) =", "\\prod \\Hom(f_!\\mathcal{F}, \\overline{y}_*A_{\\overline{y}})", "$$", "Thus it suffices to find abelian sheaves $\\mathcal{H}_{\\overline{y}}$", "on $X_\\etale$ representing the functors", "$\\mathcal{F} \\mapsto \\Hom(f_!\\mathcal{F}, \\overline{y}_*A_{\\overline{y}})$", "and to take $\\mathcal{H} = \\prod \\mathcal{H}_{\\overline{y}}$.", "This reduces us to the case $\\mathcal{H} = \\overline{y}_*A$", "for some fixed geometric point $\\overline{y} : \\Spec(k) \\to Y$", "and some fixed abelian group $A$. We claim that in this case", "$\\mathcal{H} = \\prod_{f(\\overline{x}) = \\overline{y}} \\overline{x}_*A$ works.", "This will finish the proof of the lemma.", "Namely, we have", "$$", "\\Hom(f_!\\mathcal{F}, \\overline{y}_*A) =", "\\Hom_{\\textit{Ab}}((f_!\\mathcal{F})_{\\overline{y}}, A) =", "\\Hom_{\\textit{Ab}}(\\bigoplus\\nolimits_{f(\\overline{x}) = \\overline{y}}", "\\mathcal{F}_{\\overline{x}}, A)", "$$", "by the description of stalks in", "Lemma \\ref{lemma-lqf-f-shriek-stalk}", "on the one hand and on the other hand we have", "$$", "\\Hom(\\mathcal{F}, \\mathcal{H}) =", "\\prod\\nolimits_{f(\\overline{x}) = \\overline{y}}", "\\Hom(\\mathcal{F}, \\overline{x}_*A) =", "\\prod\\nolimits_{f(\\overline{x}) = \\overline{y}}", "\\Hom_{\\textit{Ab}}(\\mathcal{F}_{\\overline{x}}, A)", "$$", "We leave it to the reader to identify these as functors of $\\mathcal{F}$." ], "refs": [ "homology-lemma-partially-defined-adjoint", "etale-cohomology-theorem-exactness-stalks", "more-etale-lemma-lqf-f-shriek-stalk" ], "ref_ids": [ 12119, 6376, 8823 ] } ], "ref_ids": [] }, { "id": 8829, "type": "theorem", "label": "more-etale-lemma-etale-upper-shriek", "categories": [ "more-etale" ], "title": "more-etale-lemma-etale-upper-shriek", "contents": [ "Let $j : U \\to X$ be an \\'etale morphism. Then $j^! = j^{-1}$." ], "refs": [], "proofs": [ { "contents": [ "This is true because $j_!$ as defined in", "Section \\ref{section-finite-support}", "agrees with $j_!$ as defined in \\'Etale Cohomology, Section", "\\ref{etale-cohomology-section-extension-by-zero}, see", "Lemma \\ref{lemma-finite-support-etale-shriek}. Finally, in", "\\'Etale Cohomology, Section \\ref{etale-cohomology-section-extension-by-zero}", "the functor $j_!$ is defined", "as the left adjoint of $j^{-1}$ and hence we conclude by", "uniqueness of adjoint functors." ], "refs": [ "more-etale-lemma-finite-support-etale-shriek" ], "ref_ids": [ 8822 ] } ], "ref_ids": [] }, { "id": 8830, "type": "theorem", "label": "more-etale-lemma-upper-shriek-restriction", "categories": [ "more-etale" ], "title": "more-etale-lemma-upper-shriek-restriction", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be separated and locally quasi-finite", "morphisms. There is a canonical isomorphism $(g \\circ f)^! \\to f^! \\circ g^!$.", "Given a third locally quasi-finite morphism $h : Z \\to T$", "the diagram", "$$", "\\xymatrix{", "(h \\circ g \\circ f)^! \\ar[r] \\ar[d] &", "f^! \\circ (h \\circ g)^! \\ar[d] \\\\", "(g \\circ f)^! \\circ h^! \\ar[r] & f^! \\circ g^! \\circ h^!", "}", "$$", "commutes." ], "refs": [], "proofs": [ { "contents": [ "By uniqueness of adjoint functors, this immediately translates", "into the corresponding (dual) statement for the functors $f_!$.", "See Lemma \\ref{lemma-lqf-shriek-composition}." ], "refs": [ "more-etale-lemma-lqf-shriek-composition" ], "ref_ids": [ 8827 ] } ], "ref_ids": [] }, { "id": 8831, "type": "theorem", "label": "more-etale-lemma-upper-shriek-restriction-etale", "categories": [ "more-etale" ], "title": "more-etale-lemma-upper-shriek-restriction-etale", "contents": [ "Let $j : U \\to X$ and $j' : V \\to U$ be \\'etale morphisms.", "The isomorphism $(j \\circ j')^{-1} = (j')^{-1} \\circ j^{-1}$", "and the isomorphism $(j \\circ j')^! = (j')^! \\circ j^!$ of", "Lemma \\ref{lemma-upper-shriek-restriction}", "agree via the isomorphism of Lemma \\ref{lemma-etale-upper-shriek}." ], "refs": [ "more-etale-lemma-upper-shriek-restriction", "more-etale-lemma-etale-upper-shriek" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8830, 8829 ] }, { "id": 8832, "type": "theorem", "label": "more-etale-lemma-lqf-base-change-upper-shriek", "categories": [ "more-etale" ], "title": "more-etale-lemma-lqf-base-change-upper-shriek", "contents": [ "Consider a cartesian square", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "of schemes with $f$ locally quasi-finite. For any abelian sheaf $\\mathcal{F}$", "on $Y'_\\etale$ we have $(g')_*(f')^!\\mathcal{F} = f^!g_*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "By uniqueness of adjoint functors, this follows from", "the corresponding (dual) statement for the functors $f_!$.", "See Lemma \\ref{lemma-lqf-base-change-f-shriek}." ], "refs": [ "more-etale-lemma-lqf-base-change-f-shriek" ], "ref_ids": [ 8825 ] } ], "ref_ids": [] }, { "id": 8833, "type": "theorem", "label": "more-etale-lemma-lqf-shriek-derived", "categories": [ "more-etale" ], "title": "more-etale-lemma-lqf-shriek-derived", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes. The functors", "$f_!$ and $f^!$ of Definition \\ref{definition-f-shriek-lqf} and", "Lemma \\ref{lemma-lqf-f-upper-shriek}", "induce adjoint functors $f_! : D(X_\\etale) \\to D(Y_\\etale)$", "and $Rf^! : D(Y_\\etale) \\to D(X_\\etale)$ on derived categories." ], "refs": [ "more-etale-definition-f-shriek-lqf", "more-etale-lemma-lqf-f-upper-shriek" ], "proofs": [ { "contents": [ "This follows immediately from", "Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors},", "the fact that $f_!$ is exact (Lemma \\ref{lemma-lqf-f-shriek-stalk})", "and hence $Lf_! = f_!$", "and the fact that we have enough K-injective complexes of abelian sheaves", "on $Y_\\etale$ so that $Rf^!$ is defined." ], "refs": [ "derived-lemma-derived-adjoint-functors", "more-etale-lemma-lqf-f-shriek-stalk" ], "ref_ids": [ 1907, 8823 ] } ], "ref_ids": [ 8848, 8828 ] }, { "id": 8834, "type": "theorem", "label": "more-etale-lemma-shriek-proper-and-open", "categories": [ "more-etale" ], "title": "more-etale-lemma-shriek-proper-and-open", "contents": [ "Consider a commutative diagram of schemes", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "with $f$ and $f'$ proper and $g$ and $g'$ separated and locally quasi-finite.", "For a torsion ring $\\Lambda$ and $K$ in $D(X'_\\etale, \\Lambda)$", "there is a canonical isomorphism $g_!Rf'_*K \\to Rf_*(g'_!K)$", "in $D(Y_\\etale, \\Lambda)$." ], "refs": [], "proofs": [ { "contents": [ "Represent $K$ by a K-injective complex $\\mathcal{J}^\\bullet$ of", "sheaves of $\\Lambda$-modules on $X'_\\etale$. Choose a quasi-isomorphism", "$g'_!\\mathcal{J}^\\bullet \\to \\mathcal{I}^\\bullet$ to a", "K-injective complex $\\mathcal{I}^\\bullet$ of", "sheaves of $\\Lambda$-modules on $X_\\etale$.", "Then we can consider the map", "$$", "g_!f'_*\\mathcal{J}^\\bullet =", "g_!f'_!\\mathcal{J}^\\bullet =", "f_!g'_!\\mathcal{J}^\\bullet =", "f_*g'_!\\mathcal{J}^\\bullet \\to", "f_*\\mathcal{I}^\\bullet", "$$", "where the first and third equality come from", "Lemma \\ref{lemma-proper-f-shriek}", "and the second equality comes from", "Lemma \\ref{lemma-f-shriek-composition} which tells us that both", "$g_! \\circ f'_!$ and $f_! \\circ g'_!$ are equal to", "$(g \\circ f')_! = (f \\circ g')_!$ as subsheaves of", "$(g \\circ f')_* = (f \\circ g')_*$.", "To finish the proof it suffices to show that", "$f_*g'_!\\mathcal{J}^\\bullet \\to f_*\\mathcal{I}^\\bullet$", "is a quasi-isomorphism.", "\\medskip\\noindent", "The question is local on $Y$. Hence we may assume that the dimension", "of fibres of $f$ is bounded, see Morphisms, Lemma", "\\ref{morphisms-lemma-morphism-finite-type-bounded-dimension}.", "Then we see that $Rf_*$ has finite cohomological dimension, see", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-cohomological-dimension-proper}.", "Hence by Derived Categories, Lemma \\ref{derived-lemma-unbounded-right-derived},", "if we show that $R^qf_*(g'_!\\mathcal{J}) = 0$ for $q > 0$", "and any injective sheaf of $\\Lambda$-modules $\\mathcal{J}$", "on $X'_\\etale$, then the result follows.", "\\medskip\\noindent", "The stalk of $R^qf_*(g'_!\\mathcal{J})$ at a geometric point", "$\\overline{y}$ is equal to", "$H^q(X_{\\overline{y}}, g'_!\\mathcal{J}|_{X_{\\overline{y}}})$ by", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-proper-base-change-stalk}.", "Since formation of $g'_!$ commutes with base change", "this is equal to", "$$", "H^q(X_{\\overline{y}}, g'_{\\overline{y}, !}(\\mathcal{J}|_{X'_{\\overline{y}}}))", "$$", "Since $Y' \\to Y$ is locally quasi-finite, we see that", "$X'_{\\overline{y}}$ is a disjoint union of the fibres", "$X'_{\\overline{y}'}$ at geometric points $\\overline{y}'$ of $Y'$", "lying over $\\overline{y}$. Hence we get", "$$", "H^q(X_{\\overline{y}},", "\\bigoplus g'_{\\overline{y}', !}(\\mathcal{J}|_{X'_{\\overline{y}'}}))", "$$", "for example by Lemma \\ref{lemma-colim-f-shriek-separated} (but it", "is also obvious from the definition of $g'_{\\overline{y}, !}$", "in Section \\ref{section-compact-support}).", "Since taking \\'etale cohomology over $X_{\\overline{y}}$", "commutes with direct sums", "(\\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-colimit})", "we conclude it suffices to show that", "$$", "H^q(X_{\\overline{y}}, g'_{\\overline{y}', !}(\\mathcal{J}|_{X'_{\\overline{y}'}}))", "$$", "is zero. Observe that", "$g_{\\overline{y}'} : X'_{\\overline{y}'} \\to X_{\\overline{y}}$", "is a morphism between proper scheme over $\\overline{y}$ and hence is", "proper itself. As it is locally quasi-finite as well we conclude that", "$g_{\\overline{y}'}$ is finite. Thus we see that", "$g'_{\\overline{y}', !} = g'_{\\overline{y}', *} = Rg'_{\\overline{y}', *}$.", "By Leray we conlude that we have to show", "$$", "H^q(X'_{\\overline{y}'}, \\mathcal{J}|_{X'_{\\overline{y}'}}))", "$$", "is zero. This of course follows from proper base change", "(in the form we quoted it before) as the higher direct images", "of $\\mathcal{J}$ under $f'$ are zero." ], "refs": [ "more-etale-lemma-proper-f-shriek", "more-etale-lemma-f-shriek-composition", "morphisms-lemma-morphism-finite-type-bounded-dimension", "etale-cohomology-lemma-cohomological-dimension-proper", "derived-lemma-unbounded-right-derived", "etale-cohomology-lemma-proper-base-change-stalk", "more-etale-lemma-colim-f-shriek-separated", "etale-cohomology-theorem-colimit" ], "ref_ids": [ 8810, 8816, 5281, 6629, 1917, 6627, 8817, 6384 ] } ], "ref_ids": [] }, { "id": 8835, "type": "theorem", "label": "more-etale-lemma-shriek-proper-and-open-compose", "categories": [ "more-etale" ], "title": "more-etale-lemma-shriek-proper-and-open-compose", "contents": [ "Consider a commutative diagram of schemes", "$$", "\\xymatrix{", "X' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]_l \\ar[d]_{g'} & Y \\ar[d]^g \\\\", "Z' \\ar[r]^m & Z", "}", "$$", "with $f$, $f'$, $g$ and $g'$ proper and", "$k$, $l$, and $m$ separated and locally quasi-finite.", "Then the isomorphisms of Lemma \\ref{lemma-shriek-proper-and-open}", "for the two squares compose to give the isomorphism", "for the outer rectangle (see proof for a precise statement)." ], "refs": [ "more-etale-lemma-shriek-proper-and-open" ], "proofs": [ { "contents": [ "The statement means that if we write", "$R(g \\circ f)_* = Rg_* \\circ Rf_*$ and", "$R(g' \\circ f')_* = Rg'_* \\circ Rf'_*$,", "then the isomorphism", "$m_! \\circ Rg'_* \\circ Rf'_* \\to Rg_* \\circ Rf_* \\circ k_!$", "of the outer rectangle is equal to the composition", "$$", "m_! \\circ Rg'_* \\circ Rf'_* \\to", "Rg_* \\circ l_! \\circ Rf'_* \\to", "Rg_* \\circ Rf_* \\circ k_!", "$$", "of the two maps of the squares in the diagram. To prove this choose", "a K-injective complex $\\mathcal{J}^\\bullet$ of $\\Lambda$-modules", "on $X'_\\etale$ and a quasi-isomorphism", "$k_!\\mathcal{J}^\\bullet \\to \\mathcal{I}^\\bullet$", "to a K-injective complex $\\mathcal{I}^\\bullet$ of $\\Lambda$-modules", "on $X_\\etale$. The proof of Lemma \\ref{lemma-shriek-proper-and-open}", "shows that the canonical map", "$$", "a : l_!f'_*\\mathcal{J}^\\bullet \\to f_*\\mathcal{I}^\\bullet", "$$", "is a quasi-isomorphism and this quasi-isomorphism produces the", "second arrow on applying $Rg_*$. By", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-K-injective-flat}", "the complex $f_*\\mathcal{I}^\\bullet$, resp.\\ $f'_*\\mathcal{J}^\\bullet$", "is a K-injective complex of $\\Lambda$-modules on ", "$Y_\\etale$, resp.\\ $Y'_\\etale$.", "(Using this is cheating and could be avoided.)", "In particular, the same reasoning gives that the canonical map", "$$", "b : m_!g'_*f'_*\\mathcal{J}^\\bullet \\to g_*f_*\\mathcal{I}^\\bullet", "$$", "is a quasi-isomorphism and this quasi-isomorphism represents", "the first arrow. Finally, the proof of Lemma \\ref{lemma-shriek-proper-and-open}", "show that $g_*l_!f'_!\\mathcal{J}^\\bullet$ represents", "$Rg_*(l_!f'_*\\mathcal{J}^\\bullet)$ because $f'_*\\mathcal{J}^\\bullet$", "is K-injective. Hence $Rg_*(a) = g_*(a)$ and the composition", "$g_*(a) \\circ b$ is the arrow of Lemma \\ref{lemma-shriek-proper-and-open}", "for the rectangle." ], "refs": [ "more-etale-lemma-shriek-proper-and-open", "sites-cohomology-lemma-K-injective-flat", "more-etale-lemma-shriek-proper-and-open", "more-etale-lemma-shriek-proper-and-open" ], "ref_ids": [ 8834, 4261, 8834, 8834 ] } ], "ref_ids": [ 8834 ] }, { "id": 8836, "type": "theorem", "label": "more-etale-lemma-shriek-proper-and-open-compose-horizontal", "categories": [ "more-etale" ], "title": "more-etale-lemma-shriek-proper-and-open-compose-horizontal", "contents": [ "Consider a commutative diagram of schemes", "$$", "\\xymatrix{", "X'' \\ar[r]_{g'} \\ar[d]_{f''} &", "X' \\ar[r]_g \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "Y'' \\ar[r]^{h'} &", "Y' \\ar[r]^h &", "Y", "}", "$$", "with $f$, $f'$, and $f''$ proper and", "$g$, $g'$, $h$, and $h'$ separated and locally quasi-finite.", "Then the isomorphisms of Lemma \\ref{lemma-shriek-proper-and-open}", "for the two squares compose to give the isomorphism", "for the outer rectangle (see proof for a precise statement)." ], "refs": [ "more-etale-lemma-shriek-proper-and-open" ], "proofs": [ { "contents": [ "The statement means that if we write", "$(h \\circ h')_! = h_! \\circ h'_!$ and", "$(g \\circ g')_! = g_! \\circ g'_!$", "using the equalities of Lemma \\ref{lemma-f-shriek-composition},", "then the isomorphism", "$h_! \\circ h'_! \\circ Rf''_* \\to Rf_* \\circ g_! \\circ g'_!$", "of the outer rectangle is equal to the composition", "$$", "h_! \\circ h'_! \\circ Rf''_* \\to", "h_! \\circ Rf'_* \\circ g'_! \\to", "Rf_* \\circ g_! \\circ g'_!", "$$", "of the two maps of the squares in the diagram. To prove this choose", "a K-injective complex $\\mathcal{I}^\\bullet$ of $\\Lambda$-modules", "on $X''_\\etale$ and a quasi-isomorphism", "$g'_!\\mathcal{I}^\\bullet \\to \\mathcal{J}^\\bullet$", "to a K-injective complex $\\mathcal{J}^\\bullet$ of $\\Lambda$-modules", "on $X'_\\etale$. Next, choose a quasi-isomorphism", "$g_!\\mathcal{J}^\\bullet \\to \\mathcal{K}^\\bullet$", "to a K-injective complex $\\mathcal{K}^\\bullet$ of $\\Lambda$-modules", "on $X_\\etale$.", "The proof of Lemma \\ref{lemma-shriek-proper-and-open} shows that the canonical", "maps", "$$", "h'_!f''_*\\mathcal{I}^\\bullet \\to f'_*\\mathcal{J}^\\bullet", "\\quad\\text{and}\\quad", "h_!f'_*\\mathcal{J}^\\bullet \\to f_*\\mathcal{K}^\\bullet", "$$", "are quasi-isomorphisms and these quasi-isomorphisms define the", "first and second arrow above. Since $g_!$ is an exact functor", "(Lemma \\ref{lemma-lqf-f-shriek-separated-colimits})", "we find that $g_!g'_!\\mathcal{I}^\\bullet \\to \\mathcal{K}^\\bullet$", "is a quasi-ismorphism and hence the canonical map", "$$", "h_!h'_!f''_*\\mathcal{I}^\\bullet \\to f_*\\mathcal{K}^\\bullet", "$$", "is a quasi-isomorphism and represents the map for the outer", "rectangle in the derived category. Clearly this map is the", "composition of the other two and the proof is complete." ], "refs": [ "more-etale-lemma-f-shriek-composition", "more-etale-lemma-shriek-proper-and-open", "more-etale-lemma-lqf-f-shriek-separated-colimits" ], "ref_ids": [ 8816, 8834, 8819 ] } ], "ref_ids": [ 8834 ] }, { "id": 8837, "type": "theorem", "label": "more-etale-lemma-shriek-proper-and-open-base-change", "categories": [ "more-etale" ], "title": "more-etale-lemma-shriek-proper-and-open-base-change", "contents": [ "Let $b : Y_1 \\to Y$ be a morphism of schemes. Consider a commutative diagram", "of schemes", "$$", "\\vcenter{", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "}", "\\quad\\text{and let}\\quad", "\\vcenter{", "\\xymatrix{", "X'_1 \\ar[r]_{g'_1} \\ar[d]_{f'_1} & X_1 \\ar[d]^{f_1} \\\\", "Y'_1 \\ar[r]^{g_1} & Y_1", "}", "}", "$$", "be the base change by $b$. Assume $f$ and $f'$ proper and", "$g$ and $g'$ separated and locally quasi-finite.", "For a torsion ring $\\Lambda$ and $K$ in $D(X'_\\etale, \\Lambda)$", "there is commutative diagram", "$$", "\\xymatrix{", "b^{-1}g_!Rf'_*K \\ar[d] \\ar[r] &", "g_{1, !}(b')^{-1}Rf'_*K \\ar[r] &", "g_{1, !}Rf'_{1, *}(a')^{-1}K \\ar[d] \\\\", "b^{-1}Rf_*g'_!K \\ar[r] &", "Rf_{1, *}a^{-1}g'_!K \\ar[r] &", "Rf_{1, *}g'_{1, !}(a')^{-1}K", "}", "$$", "in $D(Y_{1, \\etale}, \\Lambda)$ where $a : X_1 \\to X$, $a' : X'_1 \\to X'$,", "$b' : Y'_1 \\to Y'$ are the projections, the vertical maps are the arrows", "of Lemma \\ref{lemma-shriek-proper-and-open}", "and the horizontal arrows are the base change map", "(from \\'Etale Cohomology, Section", "\\ref{etale-cohomology-section-base-change-preliminaries})", "and the base change map of Lemma \\ref{lemma-base-change-f-shriek-separated}." ], "refs": [ "more-etale-lemma-shriek-proper-and-open", "more-etale-lemma-base-change-f-shriek-separated" ], "proofs": [ { "contents": [ "Represent $K$ by a K-injective complex $\\mathcal{J}^\\bullet$ of", "sheaves of $\\Lambda$-modules on $X'_\\etale$. Choose a quasi-isomorphism", "$g'_!\\mathcal{J}^\\bullet \\to \\mathcal{I}^\\bullet$ to a", "K-injective complex $\\mathcal{I}^\\bullet$ of", "sheaves of $\\Lambda$-modules on $X_\\etale$.", "The proof of Lemma \\ref{lemma-shriek-proper-and-open}", "constructs $g_!Rf'_*K \\to Rf_*g'_!K$ as", "$$", "g_!f'_*\\mathcal{J}^\\bullet =", "g_!f'_!\\mathcal{J}^\\bullet =", "f_!g'_!\\mathcal{J}^\\bullet =", "f_*g'_!\\mathcal{J}^\\bullet \\to", "f_*\\mathcal{I}^\\bullet", "$$", "Choose a quasi-isomorphism", "$(a')^{-1}\\mathcal{J}^\\bullet \\to \\mathcal{J}_1^\\bullet$", "to a K-injective complex $\\mathcal{J}_1^\\bullet$ of", "sheaves of $\\Lambda$-modules on $X'_{1, \\etale}$.", "Then we can pick a diagram of complexes", "$$", "\\xymatrix{", "g'_{1, !}\\mathcal{J}_1^\\bullet \\ar[rr] & &", "\\mathcal{I}_1^\\bullet \\\\", "g'_{1, !}(a')^{-1}\\mathcal{J}^\\bullet \\ar[u] \\ar@{=}[r] &", "a^{-1}g'_!\\mathcal{J}^\\bullet \\ar[r] &", "a^{-1}\\mathcal{I}^\\bullet \\ar[u]", "}", "$$", "commuting up to homotopy where all arrows are quasi-isomorphisms, the", "equality comes from Lemma \\ref{lemma-proper-f-shriek},", "and $\\mathcal{I}_1^\\bullet$ is a K-injective complex of sheaves of", "$\\Lambda$-modules on $X_{1, \\etale}$. The map", "$g_{1, !}Rf'_{1, *}(a')^{-1}K \\to Rf_{1, *}g'_{1, !}(a')^{-1}K$ is given by", "$$", "g_{1, !}f'_{1, *}\\mathcal{J}_1^\\bullet =", "g_{1, !}f'_{1, !}\\mathcal{J}_1^\\bullet =", "f_{1, !}g'_{1, !}\\mathcal{J}_1^\\bullet =", "f_{1, *}g'_{1, !}\\mathcal{J}_1^\\bullet \\to", "f_{1, *}\\mathcal{I}_1^\\bullet", "$$", "The identifications across the $3$ equal signs in both arrows", "are compatible with pullback maps, i.e., the diagram", "$$", "\\xymatrix{", "b^{-1}g_!f'_*\\mathcal{J}^\\bullet \\ar@{=}[d] \\ar[r] &", "g_{1, !}(b')^{-1}f'_*\\mathcal{J}^\\bullet \\ar[r] &", "g_{1, !}f'_{1, *}(a')^{-1}\\mathcal{J}^\\bullet \\ar@{=}[d] \\\\", "b^{-1}f_*g'_!\\mathcal{J}^\\bullet \\ar[r] &", "f_{1, *}a^{-1}g'_!\\mathcal{J}^\\bullet \\ar[r] &", "f_{1, *}g'_{1, !}(a')^{-1}\\mathcal{J}^\\bullet", "}", "$$", "of complexes of abelian sheaves commutes. To show this it is enough to", "show the diagram commutes with $g_!, g_{1, !}, g'_!, g'_{1, !}$", "replaced by $g_*, g_{1, *}, g'_*, g'_{1, *}$ (because the shriek", "functors are defined as subfunctors of the $*$ functors and the", "base change maps are defined in a manner compatible with this, see", "proof of Lemma \\ref{lemma-base-change-f-shriek-separated}).", "For this new diagram the commutativity follows from the compatibility", "of pullback maps with horizontal and vertical stacking of diagrams, see", "Sites, Remarks \\ref{sites-remark-compose-base-change} and", "\\ref{sites-remark-compose-base-change-horizontal}", "so that going around the diagram in either direction is the pullback", "map for the base change of $f \\circ g' = g \\circ f'$ by $b$.", "Since of course", "$$", "\\xymatrix{", "g_{1, !}f'_{1, *}(a')^{-1}\\mathcal{J}^\\bullet \\ar@{=}[d] \\ar[r] &", "g_{1, !}f'_{1, *}\\mathcal{J}_1^\\bullet \\ar@{=}[d] \\\\", "f_{1, *}g'_{1, !}(a')^{-1}\\mathcal{J}^\\bullet \\ar[r] &", "f_{1, *}g'_{1, !}\\mathcal{J}_1^\\bullet ", "}", "$$", "commutes, to finish the proof it suffices to show that", "$$", "\\xymatrix{", "b^{-1}f_*g'_!\\mathcal{J}^\\bullet \\ar[r] \\ar[d] &", "f_{1, *}a^{-1}g'_!\\mathcal{J}^\\bullet \\ar[r] \\ar[d] &", "f_{1, *}g'_{1, !}(a')^{-1}\\mathcal{J}^\\bullet \\ar[r] &", "f_{1, *}g'_{1, !}\\mathcal{J}_1^\\bullet \\ar[d] \\\\", "b^{-1}f_*\\mathcal{I}^\\bullet \\ar[r] &", "f_{1, *}a^{-1}\\mathcal{I}^\\bullet \\ar[rr] & &", "f_{1, *}\\mathcal{I}_1^\\bullet", "}", "$$", "commutes in the derived category, which holds by our choice of", "maps earlier." ], "refs": [ "more-etale-lemma-shriek-proper-and-open", "more-etale-lemma-proper-f-shriek", "more-etale-lemma-base-change-f-shriek-separated", "sites-remark-compose-base-change", "sites-remark-compose-base-change-horizontal" ], "ref_ids": [ 8834, 8810, 8815, 8720, 8721 ] } ], "ref_ids": [ 8834, 8815 ] }, { "id": 8838, "type": "theorem", "label": "more-etale-lemma-shriek-lqf-and-proper", "categories": [ "more-etale" ], "title": "more-etale-lemma-shriek-lqf-and-proper", "contents": [ "Consider a commutative diagram of schemes", "$$", "\\xymatrix{", "X \\ar[r]_f \\ar[rd]_g & Y \\ar[d]^h \\\\", "& Z", "}", "$$", "with $f$ and $g$ locally quasi-finite and $h$ proper. For any torsion ring", "$\\Lambda$ and $K$ in $D(X_\\etale, \\Lambda)$ there is a canonical", "isomorphism $g_!K \\to Rh_*(f_!K)$ in $D(Z_\\etale, \\Lambda)$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-shriek-proper-and-open}", "if $f$ and $g$ are separated. We urge the reader to skip the proof", "in the general case", "as we'll mainly use the case where $f$ and $g$ are separated.", "\\medskip\\noindent", "Represent $K$ by a complex $\\mathcal{K}^\\bullet$ of sheaves of", "$\\Lambda$-modules on $X_\\etale$. Choose a quasi-isomorphism", "$f_!\\mathcal{K}^\\bullet \\to \\mathcal{I}^\\bullet$ into a K-injective", "complex $\\mathcal{I}^\\bullet$ of sheaves of $\\Lambda$-modules on $Y_\\etale$.", "Consider the map", "$$", "g_!\\mathcal{K}^\\bullet =", "h_!f_!\\mathcal{K}^\\bullet =", "h_*f_!\\mathcal{K}^\\bullet", "\\longrightarrow", "h_*\\mathcal{I}^\\bullet", "$$", "where the equalities are", "Lemmas \\ref{lemma-lqf-separated-shriek-composition}", "and \\ref{lemma-proper-f-shriek}. This map of complexes determines the map", "$g_!K \\to Rh_*(f_!K)$ of the statement of the lemma.", "To check the map is an isomorphism we may work locally on $Z$.", "Hence we may assume that the dimension of fibres of $h$ is bounded,", "see Morphisms, Lemma", "\\ref{morphisms-lemma-morphism-finite-type-bounded-dimension}.", "Then we see that $Rh_*$ has finite cohomological dimension, see", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-cohomological-dimension-proper}.", "Hence by Derived Categories, Lemma \\ref{derived-lemma-unbounded-right-derived},", "if we show that $R^qh_*(f_!\\mathcal{F}) = 0$ for $q > 0$", "and any sheaf $\\mathcal{F}$ of $\\Lambda$-modules on $X_\\etale$, then", "$h_*f_!\\mathcal{K}^\\bullet \\to h_*\\mathcal{I}^\\bullet$ is a quasi-isomorphism.", "\\medskip\\noindent", "Observe that $\\mathcal{G} = f_!\\mathcal{F}$ is a sheaf of $\\Lambda$-modules", "on $Y$ whose stalks are nonzero only at points $y \\in Y$ such that", "$\\kappa(y)/\\kappa(h(y))$ is a finite extension. This follows from the", "description of stalks of $f_!\\mathcal{F}$ in", "Lemma \\ref{lemma-lqf-f-shriek-stalk}", "and the fact that both $f$ and $g$ are locally quasi-finite.", "Hence by the proper base change theorem (\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-proper-base-change-stalk})", "it suffices to show that $H^q(Y_{\\overline{z}}, \\mathcal{H}) = 0$", "where $\\mathcal{H}$ is a sheaf on the proper scheme $Y_{\\overline{z}}$", "over $\\kappa(\\overline{z})$ whose support is contained in the set", "of closed points. Thus the required vanishing by \\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-supported-in-closed-points}." ], "refs": [ "more-etale-lemma-shriek-proper-and-open", "more-etale-lemma-lqf-separated-shriek-composition", "more-etale-lemma-proper-f-shriek", "morphisms-lemma-morphism-finite-type-bounded-dimension", "etale-cohomology-lemma-cohomological-dimension-proper", "derived-lemma-unbounded-right-derived", "more-etale-lemma-lqf-f-shriek-stalk", "etale-cohomology-lemma-proper-base-change-stalk", "etale-cohomology-lemma-supported-in-closed-points" ], "ref_ids": [ 8834, 8826, 8810, 5281, 6629, 1917, 8823, 6627, 6645 ] } ], "ref_ids": [] }, { "id": 8839, "type": "theorem", "label": "more-etale-lemma-shriek-well-defined", "categories": [ "more-etale" ], "title": "more-etale-lemma-shriek-well-defined", "contents": [ "Let $f : X \\to Y$ be a finite type separated morphism of quasi-compact", "and quasi-separated schemes. The functor $Rf_!$ is, up to canonical", "isomorphism, independent of the choice of the compactification." ], "refs": [], "proofs": [ { "contents": [ "Consider the category of compactifications of $X$ over $Y$, which is", "cofiltered according to More on Flatness, Theorem \\ref{flat-theorem-nagata} and", "Lemmas \\ref{flat-lemma-compactifications-cofiltered} and", "\\ref{flat-lemma-compactifyable}.", "To every choice of a compactification", "$$", "j : X \\to \\overline{X},\\quad \\overline{f} : \\overline{X} \\to Y", "$$", "the construction above associates the functor $R\\overline{f}_* \\circ j_! :", "D(X_\\etale, \\Lambda) \\to D(Y_\\etale, \\Lambda)$.", "Let's be a little more explicit. Given a complex $\\mathcal{K}^\\bullet$", "of sheaves of $\\Lambda$-modules on $X_\\etale$, we choose a quasi-isomorphism", "$j_!\\mathcal{K}^\\bullet \\to \\mathcal{I}^\\bullet$ into a K-injective", "complex of sheaves of $\\Lambda$-modules on $\\overline{X}_\\etale$.", "Then our functor sends $\\mathcal{K}^\\bullet$ to", "$\\overline{f}_*\\mathcal{I}^\\bullet$.", "\\medskip\\noindent", "Suppose given a morphism $g : \\overline{X}_1 \\to \\overline{X}_2$", "between compactifications $j_i : X \\to \\overline{X}_i$ over $Y$.", "Then we get an isomorphism", "$$", "R\\overline{f}_{2, *} \\circ j_{2, !} =", "R\\overline{f}_{2, *} \\circ Rg_* \\circ j_{1, !} =", "R\\overline{f}_{1, *} \\circ j_{1, !}", "$$", "using Lemma \\ref{lemma-shriek-lqf-and-proper} in the first equality.", "\\medskip\\noindent", "To finish the proof, since the category of compactifications of $X$ over $Y$", "is cofiltered, it suffices to show compositions of morphisms of", "compactifications of $X$ over $Y$ are turned into compositions of", "isomorphisms of functors\\footnote{Namely, if $\\alpha, \\beta : F \\to G$", "are morphisms of functors and $\\gamma : G \\to H$ is an isomorphism", "of functors such that $\\gamma \\circ \\alpha = \\gamma \\circ \\beta$, then", "we conclude $\\alpha = \\beta$.}. To do this, suppose that", "$j_3 : X \\to \\overline{X}_3$", "is a third compactification and that $h : \\overline{X}_2 \\to \\overline{X}_3$", "is a morphism of compactifications. Then we have to show that the", "composition", "$$", "R\\overline{f}_{3, *} \\circ j_{3, !} =", "R\\overline{f}_{3, *} \\circ Rh_* \\circ j_{2, !} =", "R\\overline{f}_{2, *} \\circ j_{2, !} =", "R\\overline{f}_{2, *} \\circ Rg_* \\circ j_{1, !} =", "R\\overline{f}_{1, *} \\circ j_{1, !}", "$$", "is equal to the isomorphism of functors constructed using simply", "$j_3$, $g \\circ h$, and $j_1$. A calculation shows that it suffices to", "prove that the composition of the maps", "$$", "j_{3, !} \\to Rh_* \\circ j_{2, !} \\to Rh_* \\circ Rg_* \\circ j_{1, !}", "$$", "of Lemma \\ref{lemma-shriek-lqf-and-proper} agrees with the corresponding", "map $j_{3, !} \\to R(h \\circ g)_* \\circ j_{1, !}$", "via the identification $R(h \\circ g)_* = Rh_* \\circ Rg_*$.", "Since the map of Lemma \\ref{lemma-shriek-lqf-and-proper}", "is a special case of the map of Lemma \\ref{lemma-shriek-proper-and-open}", "(as $j_1$ and $j_2$ are separated) this follows immediately from", "Lemma \\ref{lemma-shriek-proper-and-open-compose}." ], "refs": [ "flat-theorem-nagata", "flat-lemma-compactifications-cofiltered", "flat-lemma-compactifyable", "more-etale-lemma-shriek-lqf-and-proper", "more-etale-lemma-shriek-lqf-and-proper", "more-etale-lemma-shriek-lqf-and-proper", "more-etale-lemma-shriek-proper-and-open", "more-etale-lemma-shriek-proper-and-open-compose" ], "ref_ids": [ 5976, 6128, 6129, 8838, 8838, 8838, 8834, 8835 ] } ], "ref_ids": [] }, { "id": 8840, "type": "theorem", "label": "more-etale-lemma-shriek-composition", "categories": [ "more-etale" ], "title": "more-etale-lemma-shriek-composition", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be separated morphisms of finite type", "of quasi-compact and quasi-separated schemes. Then there is a canonical", "isomorphism $Rg_! \\circ Rf_! \\to R(g \\circ f)_!$." ], "refs": [], "proofs": [ { "contents": [ "Choose a compactification $i : Y \\to \\overline{Y}$ of $Y$ over $Z$.", "Choose a compactification $X \\to \\overline{X}$ of $X$ over", "$\\overline{Y}$. This uses More on Flatness, Theorem \\ref{flat-theorem-nagata}", "and Lemma \\ref{flat-lemma-compactifyable} twice.", "Let $U$ be the inverse image of $Y$ in $\\overline{X}$", "so that we get the commutative diagram", "$$", "\\xymatrix{", "X \\ar[r]_j \\ar[d]_f &", "U \\ar[dl]^{f'} \\ar[r]_{j'} &", "\\overline{X} \\ar[dl]^{\\overline{f}} \\\\", "Y \\ar[r]_i \\ar[d]_g &", "\\overline{Y} \\ar[dl]^{\\overline{g}} \\\\", "Z", "}", "$$", "Then we have", "\\begin{align*}", "R(g \\circ f)_!", "& =", "R(\\overline{g} \\circ \\overline{f})_* \\circ (j' \\circ j)_! \\\\", "& =", "R\\overline{g}_* \\circ R\\overline{f}_* \\circ j'_! \\circ j_! \\\\", "& =", "R\\overline{g}_* \\circ i_! \\circ Rf'_* \\circ j_! \\\\", "& =", "Rg_! \\circ Rf_!", "\\end{align*}", "The first equality is the definition of $R(g \\circ f)_!$.", "The second equality uses the identifications", "$R(\\overline{g} \\circ \\overline{f})_* = R\\overline{g}_* \\circ R\\overline{f}_*$", "and $(j' \\circ j)_! = j'_! \\circ j_!$ of Lemma \\ref{lemma-f-shriek-composition}.", "The identification $i_! \\circ Rf'_* \\to R\\overline{f}_* \\circ j_!$", "used in the third equality is Lemma \\ref{lemma-shriek-proper-and-open}.", "The final fourth equality is the definition of $Rg_!$ and $Rf_!$.", "To finish the proof we show that", "this isomorphism is independent of choices made.", "\\medskip\\noindent", "Suppose we have two diagrams", "$$", "\\vcenter{", "\\xymatrix{", "X \\ar[r]_{j_1} \\ar[d] &", "U_1 \\ar[dl]^{f_1} \\ar[r]_{j'_1} &", "\\overline{X}_1 \\ar[dl]^{\\overline{f}_1} \\\\", "Y \\ar[r]_{i_1} \\ar[d] &", "\\overline{Y}_1 \\ar[dl]^{\\overline{g}_1} \\\\", "Z", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "X \\ar[r]_{j_2} \\ar[d] &", "U_2 \\ar[dl]^{f_2} \\ar[r]_{j'_2} &", "\\overline{X}_2 \\ar[dl]^{\\overline{f}_2} \\\\", "Y \\ar[r]_{i_2} \\ar[d] &", "\\overline{Y}_2 \\ar[dl]^{\\overline{g}_2} \\\\", "Z", "}", "}", "$$", "We can first choose a compactification $i : Y \\to \\overline{Y}$", "of $Y$ over $Z$ which dominates both $\\overline{Y}_1$ and $\\overline{Y}_2$,", "see More on Flatness, Lemma \\ref{flat-lemma-compactifications-cofiltered}.", "By More on Flatness, Lemma \\ref{flat-lemma-right-multiplicative-system} and", "Categories, Lemmas \\ref{categories-lemma-morphisms-right-localization} and", "\\ref{categories-lemma-equality-morphisms-right-localization}", "we can choose a compactification $X \\to \\overline{X}$ of", "$X$ over $\\overline{Y}$ with morphisms $\\overline{X} \\to \\overline{X}_1$", "and $\\overline{X} \\to \\overline{X}_2$ and such that the composition", "$\\overline{X} \\to \\overline{Y} \\to \\overline{Y}_1$ is equal to", "the composition $\\overline{X} \\to \\overline{X}_1 \\to \\overline{Y}_1$", "and such that the composition", "$\\overline{X} \\to \\overline{Y} \\to \\overline{Y}_2$ is equal to", "the composition $\\overline{X} \\to \\overline{X}_2 \\to \\overline{Y}_2$.", "Thus we see that it suffices to compare the maps", "determined by our diagrams when we have a commutative diagram", "as follows", "$$", "\\xymatrix{", "X \\ar[rr]_{j_1} \\ar@{=}[d] & &", "U_1 \\ar[d]^{h'} \\ar[ddll] \\ar[rr]_{j'_1} & &", "\\overline{X}_1 \\ar[d]^h \\ar[ddll] \\\\", "X \\ar'[r][rr]^-{j_2} \\ar[d] & &", "U_2 \\ar'[dl][ddll] \\ar'[r][rr]^-{j'_2} & &", "\\overline{X}_2 \\ar[ddll] \\\\", "Y \\ar[rr]^{i_1} \\ar@{=}[d] & & \\overline{Y}_1 \\ar[d]^k \\\\", "Y \\ar[rr]^{i_2} \\ar[d] & & \\overline{Y}_2 \\ar[dll] \\\\", "Z", "}", "$$", "Each of the squares", "$$", "\\xymatrix{", "X \\ar[r]_{j_1} \\ar[d]_{\\text{id}} \\ar@{}[dr]|A &", "U_1 \\ar[d]^{h'} \\\\", "X \\ar[r]^{j_2} &", "U_2", "}", "\\quad", "\\xymatrix{", "U_2 \\ar[r]_{j_2'} \\ar[d]_{f_2} \\ar@{}[dr]|B &", "\\overline{X}_2 \\ar[d]^{\\overline{f}_2} \\\\", "Y \\ar[r]^{i_2} &", "\\overline{Y}_2", "}", "\\quad", "\\xymatrix{", "U_1 \\ar[r]_{j_1'} \\ar[d]_{f_1} \\ar@{}[dr]|C &", "\\overline{X}_1 \\ar[d]^{\\overline{f}_1} \\\\", "Y \\ar[r]^{i_1} &", "\\overline{Y}_1", "}", "\\quad", "\\xymatrix{", "Y \\ar[r]_{i_1} \\ar[d]_{\\text{id}} \\ar@{}[dr]|D &", "\\overline{Y}_1 \\ar[d]^k \\\\", "Y \\ar[r]^{i_2} &", "\\overline{Y}_2", "}", "\\quad", "\\xymatrix{", "X \\ar[r]_{j_1' \\circ j_1} \\ar[d]_{\\text{id}} \\ar@{}[dr]|E &", "\\overline{X}_1 \\ar[d]^h \\\\", "X \\ar[r]^{j_2} &", "\\overline{X}_2", "}", "$$", "gives rise to an isomorphism as follows", "\\begin{align*}", "\\gamma_A & :", "j_{2, !} \\to Rh'_* \\circ j_{1, !} \\\\", "\\gamma_B & :", "i_{2, !} \\circ Rf_{2, *} \\to R\\overline{f}_{2, *} \\circ j'_{2, !} \\\\", "\\gamma_C & :", "i_{1, !} \\circ Rf_{1, *} \\to R\\overline{f}_{1, *} \\circ j'_{1, !} \\\\", "\\gamma_D & :", "i_{2, !} \\to Rk_* \\circ i_{1, !} \\\\", "\\gamma_E & :", "j_{2, !} \\to Rh_* \\circ (j'_1 \\circ j_1)_!", "\\end{align*}", "by applying the map from Lemma \\ref{lemma-shriek-proper-and-open}", "(which is the same as the map in Lemma \\ref{lemma-shriek-lqf-and-proper}", "in case the left vertical arrow is the identity). Let us write", "\\begin{align*}", "F_1 & = Rf_{1, *} \\circ j_{1, !} \\\\", "F_2 & = Rf_{2, *} \\circ j_{2, !} \\\\", "G_1 & = R\\overline{g}_{1, *} \\circ i_{1, !} \\\\", "G_2 & = R\\overline{g}_{2, *} \\circ i_{2, !} \\\\", "C_1 & = R(\\overline{g}_1 \\circ \\overline{f}_1)_* \\circ (j'_1 \\circ j_1)_! \\\\", "C_2 & = R(\\overline{g}_2 \\circ \\overline{f}_2)_* \\circ (j'_2 \\circ j_2)_!", "\\end{align*}", "The construction given in the first paragraph of the proof", "and in Lemma \\ref{lemma-shriek-well-defined} uses", "\\begin{enumerate}", "\\item $\\gamma_C$ for the map $G_1 \\circ F_1 \\to C_1$,", "\\item $\\gamma_B$ for the map $G_2 \\circ F_2 \\to C_2 $,", "\\item $\\gamma_A$ for the map $F_2 \\to F_1$,", "\\item $\\gamma_D$ for the map $G_2 \\to G_1$, and", "\\item $\\gamma_E$ for the map $C_2 \\to C_1$.", "\\end{enumerate}", "This implies that we have to show that the diagram", "$$", "\\xymatrix{", "C_2 \\ar[rr]_{\\gamma_E} & &", "C_1 \\\\", "G_2 \\circ F_2 \\ar[rr]^{\\gamma_D \\circ \\gamma_A} \\ar[u]^{\\gamma_B} & &", "G_1 \\circ F_1 \\ar[u]_{\\gamma_C}", "}", "$$", "is commutative. We will use", "Lemmas \\ref{lemma-shriek-proper-and-open-compose} and", "\\ref{lemma-shriek-proper-and-open-compose-horizontal}", "and with (abuse of) notation as in", "Remark \\ref{remark-going-around} (in particular", "dropping $\\star$ products with identity transformations", "from the notation).", "We can write $\\gamma_E = \\gamma_F \\circ \\gamma_A$ where", "$$", "\\xymatrix{", "U_1 \\ar[r]_{j'_1} \\ar[d]_{h'} \\ar@{}[rd]|F &", "\\overline{X}_1 \\ar[d]^h \\\\", "U_2 \\ar[r]^{j'_2} &", "\\overline{X}_2", "}", "$$", "Thus we see that", "$$", "\\gamma_E \\circ \\gamma_B = \\gamma_F \\circ \\gamma_A \\circ \\gamma_B", "= \\gamma_F \\circ \\gamma_B \\circ \\gamma_A", "$$", "the last equality because the two squares $A$ and $B$ only", "intersect in one point (similar to the last argument in", "Remark \\ref{remark-going-around}). Thus it suffices to prove that", "$\\gamma_C \\circ \\gamma_D = \\gamma_F \\circ \\gamma_B$.", "Since both of these are equal to the map for the square", "$$", "\\xymatrix{", "U_1 \\ar[r] \\ar[d] & \\overline{X}_1 \\ar[d] \\\\", "Y \\ar[r] & \\overline{Y}_2", "}", "$$", "we conclude." ], "refs": [ "flat-theorem-nagata", "flat-lemma-compactifyable", "more-etale-lemma-f-shriek-composition", "more-etale-lemma-shriek-proper-and-open", "flat-lemma-compactifications-cofiltered", "flat-lemma-right-multiplicative-system", "categories-lemma-morphisms-right-localization", "categories-lemma-equality-morphisms-right-localization", "more-etale-lemma-shriek-proper-and-open", "more-etale-lemma-shriek-lqf-and-proper", "more-etale-lemma-shriek-well-defined", "more-etale-lemma-shriek-proper-and-open-compose", "more-etale-lemma-shriek-proper-and-open-compose-horizontal", "more-etale-remark-going-around", "more-etale-remark-going-around" ], "ref_ids": [ 5976, 6129, 8816, 8834, 6128, 6130, 12262, 12263, 8834, 8838, 8839, 8835, 8836, 8856, 8856 ] } ], "ref_ids": [] }, { "id": 8841, "type": "theorem", "label": "more-etale-lemma-pseudo-functor", "categories": [ "more-etale" ], "title": "more-etale-lemma-pseudo-functor", "contents": [ "Let $f : X \\to Y$, $g : Y \\to Z$, $h : Z \\to T$ be separated morphisms of", "finite type of quasi-compact and quasi-separated schemes. Then", "the diagram", "$$", "\\xymatrix{", "Rh_! \\circ Rg_! \\circ Rf_! \\ar[r]_{\\gamma_C} \\ar[d]^{\\gamma_A} &", "R(h \\circ g)_! \\circ Rf_! \\ar[d]_{\\gamma_{A + B}} \\\\", "Rh_! \\circ R(g \\circ f)_! \\ar[r]^{\\gamma_{B + C}} &", "R(h \\circ g \\circ f)_!", "}", "$$", "of isomorphisms of Lemma \\ref{lemma-shriek-composition} commutes", "(for the meaning of the $\\gamma$'s see proof)." ], "refs": [ "more-etale-lemma-shriek-composition" ], "proofs": [ { "contents": [ "To do this we choose a compactification $\\overline{Z}$", "of $Z$ over $T$, then a compactification $\\overline{Y}$ of $Y$ over", "$\\overline{Z}$, and then a compactification $\\overline{X}$ of", "$X$ over $\\overline{Y}$. This uses", "More on Flatness, Theorem \\ref{flat-theorem-nagata} and", "Lemma \\ref{flat-lemma-compactifyable}.", "Let $W \\subset \\overline{Y}$ be the inverse image of $Z$ under", "$\\overline{Y} \\to \\overline{Z}$ and let $U \\subset V \\subset \\overline{X}$", "be the inverse images of $Y \\subset W$ under $\\overline{X} \\to \\overline{Y}$.", "This produces the following diagram", "$$", "\\xymatrix{", "X \\ar[d]_f \\ar[r] & U \\ar[r] \\ar[d] \\ar@{}[dr]|A &", "V \\ar[d] \\ar[r] \\ar@{}[rd]|B & \\overline{X} \\ar[d] \\\\", "Y \\ar[d]_g \\ar[r] & Y \\ar[r] \\ar[d] & W \\ar[r] \\ar[d] \\ar@{}[rd]|C &", "\\overline{Y} \\ar[d] \\\\", "Z \\ar[d]_h \\ar[r] & Z \\ar[d] \\ar[r] & Z \\ar[d] \\ar[r] & \\overline{Z} \\ar[d] \\\\", "T \\ar[r] & T \\ar[r] & T \\ar[r] & T", "}", "$$", "Without introducing tons of notation but arguing exactly", "as in the proof of Lemma \\ref{lemma-shriek-composition}", "we see that the maps in the first displayed diagram use the", "maps of Lemma \\ref{lemma-shriek-proper-and-open} for the rectangles", "$A + B$, $B + C$, $A$, and $C$ as indicated in the diagram in", "the statement of the lemma. Since by", "Lemmas \\ref{lemma-shriek-proper-and-open-compose} and", "\\ref{lemma-shriek-proper-and-open-compose-horizontal}", "we have $\\gamma_{A + B} = \\gamma_B \\circ \\gamma_A$ and", "$\\gamma_{B + C} = \\gamma_B \\circ \\gamma_C$ we conclude", "that the desired equality holds provided", "$\\gamma_A \\circ \\gamma_C = \\gamma_C \\circ \\gamma_A$.", "This is true because the two squares $A$ and $C$ only", "intersect in one point (similar to the last argument in", "Remark \\ref{remark-going-around})." ], "refs": [ "flat-theorem-nagata", "flat-lemma-compactifyable", "more-etale-lemma-shriek-composition", "more-etale-lemma-shriek-proper-and-open", "more-etale-lemma-shriek-proper-and-open-compose", "more-etale-lemma-shriek-proper-and-open-compose-horizontal", "more-etale-remark-going-around" ], "ref_ids": [ 5976, 6129, 8840, 8834, 8835, 8836, 8856 ] } ], "ref_ids": [ 8840 ] }, { "id": 8842, "type": "theorem", "label": "more-etale-lemma-base-change-shriek", "categories": [ "more-etale" ], "title": "more-etale-lemma-base-change-shriek", "contents": [ "Consider a cartesian square", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "of quasi-compact and quasi-separated schemes with $f$ separated and", "of finite type. Then there is a canonical isomorphism", "$$", "g^{-1} \\circ Rf_! \\to Rf'_! \\circ (g')^{-1}", "$$", "Moreover, these isomorphisms are compatible with the isomorphisms", "of Lemma \\ref{lemma-shriek-composition}." ], "refs": [ "more-etale-lemma-shriek-composition" ], "proofs": [ { "contents": [ "Choose a compactification $j : X \\to \\overline{X}$ over $Y$", "and denote $\\overline{f} : \\overline{X} \\to Y$ the structure morphism.", "Let $j' : X' \\to \\overline{X}'$ and $\\overline{f}' : \\overline{X}' \\to Y'$", "denote the base changes of $j$ and $\\overline{f}$.", "Since $Rf_! = R\\overline{f} \\circ j_!$ and $Rf'_! = R\\overline{f}' \\circ j'_!$", "the isomorphism can be constructed via", "$$", "g^{-1} \\circ R\\overline{f} \\circ j_! \\to", "R\\overline{f}' \\circ (\\overline{g}')^{-1} \\circ j_! \\to", "R\\overline{f}' \\circ j'_! \\circ (g')^{-1} ", "$$", "where the first arrow is the isomorphism given to us by the", "proper base change theorem (\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-proper-base-change-mod-n})", "and the second arrow is the isomorphism of", "Lemma \\ref{lemma-base-change-f-shriek-separated}.", "\\medskip\\noindent", "To finish the proof we have to show two things: first we have to show", "that the isomorphism of functors so obtained does not depend on", "the choice of the compactification and second we have to show that", "if we vertically stack two base change diagrams as in the lemma, then", "these base change isomorphisms are compatible with the isomorphisms", "of Lemma \\ref{lemma-shriek-composition}.", "A straightforward argument which we omit shows that both follow", "if we can show that the isomorphisms", "\\begin{enumerate}", "\\item $Rg_* \\circ Rf_* = R(g \\circ f)_*$ for $f : X \\to Y$ and $g : Y \\to Z$", "proper,", "\\item $g_! \\circ f_! = (g \\circ f)_!$ for $f : X \\to Y$ and $g : Y \\to Z$", "separated and quasi-finite, and", "\\item $g_! \\circ Rf'_* = Rf_* \\circ g'_!$ for $f : X \\to Y$ and", "$f' : X' \\to Y'$ proper and $g : Y' \\to Y$ and $g' : X' \\to X$", "separated and quasi-finite with $f \\circ g' = g \\circ f'$", "\\end{enumerate}", "are compatible with base change. This holds for (1) by", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-compose-base-change},", "for (2) by Remark \\ref{remark-f-shriek-base-change-composition}, and", "(3) by Lemma \\ref{lemma-shriek-proper-and-open-base-change}." ], "refs": [ "etale-cohomology-lemma-proper-base-change-mod-n", "more-etale-lemma-base-change-f-shriek-separated", "more-etale-lemma-shriek-composition", "sites-cohomology-remark-compose-base-change", "more-etale-remark-f-shriek-base-change-composition", "more-etale-lemma-shriek-proper-and-open-base-change" ], "ref_ids": [ 6630, 8815, 8840, 4425, 8851, 8837 ] } ], "ref_ids": [ 8840 ] }, { "id": 8843, "type": "theorem", "label": "more-etale-lemma-derived-lower-shriek-commute-direct-sums", "categories": [ "more-etale" ], "title": "more-etale-lemma-derived-lower-shriek-commute-direct-sums", "contents": [ "Let $f : X \\to Y$ be a finite type separated morphism of quasi-compact", "and quasi-separated schemes. The functor $Rf_!$ commutes with direct sums." ], "refs": [], "proofs": [ { "contents": [ "By construction it suffices to prove this when $f$ is an open immersion", "and when $f$ is a proper morphism. For any open immersion $j : U \\to X$", "of schemes, the functor $j_! : D(U_\\etale) \\to D(X_\\etale)$ is a left", "adjoint to pullback $j^{-1} : D(X_\\etale) \\to D(U_\\etale)$", "and hence commutes with direct sums, see Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-adjoint-lower-shriek-restrict}.", "In the proper case we have $Rf_! = Rf_*$ and we get the result from", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-proper-mod-n-direct-sums}", "and the fact that our coefficient ring $\\Lambda$ is a torsion ring." ], "refs": [ "sites-cohomology-lemma-adjoint-lower-shriek-restrict", "etale-cohomology-lemma-proper-mod-n-direct-sums" ], "ref_ids": [ 4260, 6641 ] } ], "ref_ids": [] }, { "id": 8844, "type": "theorem", "label": "more-etale-lemma-derived-lower-shriek-bounded", "categories": [ "more-etale" ], "title": "more-etale-lemma-derived-lower-shriek-bounded", "contents": [ "Let $f : X \\to Y$ be a finite type separated morphism of quasi-compact", "and quasi-separated schemes. The functor $Rf_!$ is bounded in the", "following sense: There exists an integer $N$ such that", "for $E \\in D(X_\\etale, \\Lambda)$ we have", "\\begin{enumerate}", "\\item $H^i(Rf_!(\\tau_{\\leq a}E) \\to H^i(Rf_!(E))$ is an isomorphism", "for $i \\leq a$,", "\\item $H^i(Rf_!(E)) \\to H^i(Rf_!(\\tau_{\\geq b - N + 1}E))$ is an isomorphism", "for $i \\geq b$,", "\\item if $H^i(E) = 0$ for $i \\not \\in [a, b]$ for some", "$-\\infty \\leq a \\leq b \\leq \\infty$, then $H^i(Rf_!(E)) = 0$", "for $i \\not \\in [a, b + N - 1]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By construction it suffices to prove this when $f$ is an open immersion", "and when $f$ is a proper morphism. For any open immersion $j : U \\to X$", "of schemes, the functor $j_! : D(U_\\etale) \\to D(X_\\etale)$ is exact", "and hence the statement holds with $N = 0$ in this case.", "If $f$ is proper then $Rf_! = Rf_*$, i.e., it is a right derived", "functor. Hence the bound on the left by", "Derived Categories, Lemma \\ref{derived-lemma-negative-vanishing}.", "Moreover in this case $f_* : \\textit{Mod}(X_\\etale, \\Lambda)", "\\to \\textit{Mod}(Y_\\etale, \\Lambda)$ has bounded cohomological dimension by", "Morphisms, Lemma \\ref{morphisms-lemma-morphism-finite-type-bounded-dimension}", "and \\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-cohomological-dimension-proper}.", "Thus we conclude by", "Derived Categories, Lemma \\ref{derived-lemma-unbounded-right-derived}." ], "refs": [ "derived-lemma-negative-vanishing", "morphisms-lemma-morphism-finite-type-bounded-dimension", "etale-cohomology-lemma-cohomological-dimension-proper", "derived-lemma-unbounded-right-derived" ], "ref_ids": [ 1839, 5281, 6629, 1917 ] } ], "ref_ids": [] }, { "id": 8845, "type": "theorem", "label": "more-etale-lemma-upper-shriek-derived", "categories": [ "more-etale" ], "title": "more-etale-lemma-upper-shriek-derived", "contents": [ "Let $f : X \\to Y$ be a finite type separated morphism of quasi-compact", "and quasi-separated schemes. The functor", "$Rf_! : D(X_\\etale, \\Lambda) \\to D(Y_\\etale, \\Lambda)$", "has a right adjoint", "$Rf^! : D(Y_\\etale, \\Lambda) \\to D(X_\\etale, \\Lambda)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Injectives, Proposition \\ref{injectives-proposition-brown}", "and Lemma \\ref{lemma-derived-lower-shriek-commute-direct-sums} above." ], "refs": [ "injectives-proposition-brown", "more-etale-lemma-derived-lower-shriek-commute-direct-sums" ], "ref_ids": [ 7808, 8843 ] } ], "ref_ids": [] }, { "id": 8857, "type": "theorem", "label": "stacks-properties-lemma-check-representable-covering", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-check-representable-covering", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $W$ be an algebraic space and let $W \\to \\mathcal{Y}$ be surjective,", "locally of finite presentation, and flat. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is representable by algebraic spaces, and", "\\item $W \\times_\\mathcal{Y} \\mathcal{X}$ is an algebraic space.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) is", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-base-change-by-space-representable-by-space}.", "Conversely, let $W \\to \\mathcal{Y}$ be as in (2). To prove (1) it", "suffices to show that $f$ is faithful on fibre categories, see", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}.", "Assumption (2) implies in particular that", "$W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is faithful.", "Hence the faithfulness of $f$ follows from", "Stacks, Lemma \\ref{stacks-lemma-faithful-descent}." ], "refs": [ "algebraic-lemma-base-change-by-space-representable-by-space", "algebraic-lemma-characterize-representable-by-algebraic-spaces", "stacks-lemma-faithful-descent" ], "ref_ids": [ 8448, 8469, 8956 ] } ], "ref_ids": [] }, { "id": 8858, "type": "theorem", "label": "stacks-properties-lemma-property-spaces-too", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-property-spaces-too", "contents": [ "Let $P$ be a property of morphisms of algebraic spaces as above.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "representable by algebraic spaces. The following are equivalent:", "\\begin{enumerate}", "\\item $f$ has $P$,", "\\item for every algebraic space $Z$ and morphism $Z \\to \\mathcal{Y}$", "the morphism $Z \\times_\\mathcal{Y} \\mathcal{X} \\to Z$ has $P$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is immediate. Assume (1).", "Let $Z \\to \\mathcal{Y}$ be as in (2). Choose a scheme $U$ and a", "surjective \\'etale morphism $U \\to Z$. By assumption the morphism", "$U \\times_\\mathcal{Y} \\mathcal{X} \\to U$ has $P$. But the diagram", "$$", "\\xymatrix{", "U \\times_\\mathcal{Y} \\mathcal{X} \\ar[d] \\ar[r] &", "Z \\times_\\mathcal{Y} \\mathcal{X} \\ar[d] \\\\", "U \\ar[r] & Z", "}", "$$", "is cartesian, hence the right vertical arrow has $P$ as", "$\\{U \\to Z\\}$ is an fppf covering." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8859, "type": "theorem", "label": "stacks-properties-lemma-check-property-covering", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-check-property-covering", "contents": [ "Let $P$ be a property of morphisms of algebraic spaces as above.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "representable by algebraic spaces.", "Let $W$ be an algebraic space and let $W \\to \\mathcal{Y}$ be surjective,", "locally of finite presentation, and flat.", "Set $V = W \\times_\\mathcal{Y} \\mathcal{X}$. Then", "$$", "(f\\text{ has }P) \\Leftrightarrow (\\text{the projection }V \\to W\\text{ has }P).", "$$" ], "refs": [], "proofs": [ { "contents": [ "The implication from left to right follows from", "Lemma \\ref{lemma-property-spaces-too}.", "Assume $V \\to W$ has $P$. Let $T$ be a scheme, and let", "$T \\to \\mathcal{Y}$ be a morphism. Consider the commutative diagram", "$$", "\\xymatrix{", "T \\times_\\mathcal{Y} \\mathcal{X} \\ar[d] &", "T \\times_\\mathcal{Y} W \\ar[d] \\ar[l] \\ar[r] &", "W \\ar[d] \\\\", "T & T \\times_\\mathcal{Y} V \\ar[l] \\ar[r] & V", "}", "$$", "of algebraic spaces. The squares are cartesian.", "The bottom left morphism is a surjective, flat morphism which is locally of", "finite presentation, hence $\\{T \\times_\\mathcal{Y} V \\to T\\}$ is an", "fppf covering. Hence the fact that the right vertical arrow has property", "$P$ implies that the left vertical arrow has property $P$." ], "refs": [ "stacks-properties-lemma-property-spaces-too" ], "ref_ids": [ 8858 ] } ], "ref_ids": [] }, { "id": 8860, "type": "theorem", "label": "stacks-properties-lemma-check-property-weak-covering", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-check-property-weak-covering", "contents": [ "Let $P$ be a property of morphisms of algebraic spaces as above.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "representable by algebraic spaces.", "Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which", "is representable by algebraic spaces, surjective, flat, and", "locally of finite presentation.", "Set $\\mathcal{W} = \\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$. Then", "$$", "(f\\text{ has }P) \\Leftrightarrow", "(\\text{the projection }\\mathcal{W} \\to \\mathcal{Z}\\text{ has }P).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose an algebraic space $W$ and a morphism", "$W \\to \\mathcal{Z}$ which is surjective, flat, and locally of finite", "presentation. By the discussion above the composition", "$W \\to \\mathcal{Y}$ is also surjective, flat, and locally of finite", "presentation. Denote", "$V = W \\times_\\mathcal{Z} \\mathcal{W} = V \\times_\\mathcal{Y} \\mathcal{X}$.", "By", "Lemma \\ref{lemma-check-property-covering}", "we see that $f$ has $\\mathcal{P}$ if and only if $V \\to W$ does", "and that $\\mathcal{W} \\to \\mathcal{Z}$ has $\\mathcal{P}$ if and only", "if $V \\to W$ does. The lemma follows." ], "refs": [ "stacks-properties-lemma-check-property-covering" ], "ref_ids": [ 8859 ] } ], "ref_ids": [] }, { "id": 8861, "type": "theorem", "label": "stacks-properties-lemma-check-property-after-precomposing", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-check-property-after-precomposing", "contents": [ "Let $P$ be a property of morphisms of algebraic spaces as above.", "Let $\\tau \\in \\{\\etale, smooth, syntomic, fppf\\}$.", "Let $\\mathcal{X} \\to \\mathcal{Y}$ and $\\mathcal{Y} \\to \\mathcal{Z}$", "be morphisms of algebraic stacks representable by algebraic spaces.", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{X} \\to \\mathcal{Y}$ is surjective and", "\\'etale, smooth, syntomic, or flat and locally of finite presentation,", "\\item the composition has $P$, and", "\\item $P$ is local on the source in the $\\tau$ topology.", "\\end{enumerate}", "Then $\\mathcal{Y} \\to \\mathcal{Z}$ has property $P$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z$ be a scheme and let $Z \\to \\mathcal{Z}$ be a morphism.", "Set $X = \\mathcal{X} \\times_\\mathcal{Z} Z$,", "$Y = \\mathcal{Y} \\times_\\mathcal{Z} Z$. By (1) $\\{X \\to Y\\}$", "is a $\\tau$ covering of algebraic spaces and by (2) $X \\to Z$ has property", "$P$. By (3) this implies that $Y \\to Z$ has property $P$", "and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8862, "type": "theorem", "label": "stacks-properties-lemma-representable-in-terms-presentations", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-representable-in-terms-presentations", "contents": [ "Let $g : \\mathcal{X}' \\to \\mathcal{X}$ be a morphism of algebraic stacks", "which is representable by algebraic spaces. Let $[U/R] \\to \\mathcal{X}$", "be a presentation. Set $U' = U \\times_\\mathcal{X} \\mathcal{X}'$,", "and $R' = R \\times_\\mathcal{X} \\mathcal{X}'$.", "Then there exists a groupoid in algebraic spaces of the form", "$(U', R', s', t', c')$, a presentation $[U'/R'] \\to \\mathcal{X}'$,", "and the diagram", "$$", "\\xymatrix{", "[U'/R'] \\ar[d]_{[\\text{pr}]} \\ar[r] & \\mathcal{X}' \\ar[d]^g \\\\", "[U/R] \\ar[r] & \\mathcal{X}", "}", "$$", "is $2$-commutative where the morphism $[\\text{pr}]$ comes from a", "morphism of groupoids", "$\\text{pr} : (U', R', s', t', c') \\to (U, R, s, t, c)$." ], "refs": [], "proofs": [ { "contents": [ "Since $U \\to \\mathcal{Y}$ is surjective and smooth, see", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-smooth-quotient-smooth-presentation}", "the base change $U' \\to \\mathcal{X}'$ is also surjective and smooth.", "Hence, by", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-stack-presentation}", "it suffices to show that $R' = U' \\times_{\\mathcal{X}'} U'$ in order to", "get a smooth groupoid $(U', R', s', t', c')$ and a presentation", "$[U'/R'] \\to \\mathcal{X}'$.", "Using that $R = V \\times_\\mathcal{Y} V$ (see", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-2-cartesian})", "this follows from", "$$", "R' =", "U \\times_\\mathcal{X} U \\times_\\mathcal{X} \\mathcal{X}' =", "(U \\times_\\mathcal{X} \\mathcal{X}')", "\\times_{\\mathcal{X}'}", "(U \\times_\\mathcal{X} \\mathcal{X}')", "$$", "see", "Categories, Lemmas \\ref{categories-lemma-associativity-2-fibre-product} and", "\\ref{categories-lemma-2-fibre-product-erase-factor}.", "Clearly the projection morphisms $U' \\to U$ and $R' \\to R$ give the", "desired morphism of groupoids", "$\\text{pr} : (U', R', s', t', c') \\to (U, R, s, t, c)$.", "Hence the morphism $[\\text{pr}]$ of quotient stacks by", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-functorial}.", "\\medskip\\noindent", "We still have to show that the diagram $2$-commutes.", "It is clear that the diagram", "$$", "\\xymatrix{", "U' \\ar[d]_{\\text{pr}_U} \\ar[r]_{f'} & \\mathcal{X}' \\ar[d]^g \\\\", "U \\ar[r]^f & \\mathcal{X}", "}", "$$", "$2$-commutes where $\\text{pr}_U : U' \\to U$ is the projection.", "There is a canonical $2$-arrow", "$\\tau : f \\circ t \\to f \\circ s$ in $\\Mor(R, \\mathcal{X})$", "coming from $R = U \\times_\\mathcal{X} U$, $t = \\text{pr}_0$, and", "$s = \\text{pr}_1$. Using the isomorphism", "$R' \\to U' \\times_{\\mathcal{X}'} U'$ we get similarly an isomorphism", "$\\tau' : f' \\circ t' \\to f' \\circ s'$. Note that", "$g \\circ f' \\circ t' = f \\circ t \\circ \\text{pr}_R$ and", "$g \\circ f' \\circ s' = f \\circ s \\circ \\text{pr}_R$, where", "$\\text{pr}_R : R' \\to R$ is the projection. Thus it makes sense to ask", "if", "\\begin{equation}", "\\label{equation-verify}", "\\tau \\star \\text{id}_{\\text{pr}_R} = \\text{id}_g \\star \\tau'.", "\\end{equation}", "Now we make two claims: (1) if Equation (\\ref{equation-verify}) holds,", "then the diagram $2$-commutes, and (2) Equation (\\ref{equation-verify}) holds.", "We omit the proof of both claims. Hints: part (1) follows from the", "construction of $f = f_{can}$ and $f' = f'_{can}$ in", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack}.", "Part (2) follows by carefuly working through the definitions." ], "refs": [ "algebraic-lemma-smooth-quotient-smooth-presentation", "algebraic-lemma-stack-presentation", "spaces-groupoids-lemma-quotient-stack-2-cartesian", "categories-lemma-associativity-2-fibre-product", "categories-lemma-2-fibre-product-erase-factor", "spaces-groupoids-lemma-quotient-stack-functorial", "algebraic-lemma-map-space-into-stack" ], "ref_ids": [ 8476, 8474, 9324, 12273, 12275, 9321, 8473 ] } ], "ref_ids": [] }, { "id": 8863, "type": "theorem", "label": "stacks-properties-lemma-equivalence", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-equivalence", "contents": [ "The notion above does indeed define an equivalence relation on", "morphisms from spectra of fields into the algebraic stack $\\mathcal{X}$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that the relation is reflexive and symmetric.", "Hence we have to prove that it is transitive. This comes down", "to the following: Given a diagram", "$$", "\\xymatrix{", "\\Spec(\\Omega) \\ar[r]_b \\ar[d]_a &", "\\Spec(L) \\ar[d]^q & \\Spec(\\Omega') \\ar[l]^{b'} \\ar[d]^{a'} \\\\", "\\Spec(K) \\ar[r]^p &", "\\mathcal{X} &", "\\Spec(K') \\ar[l]_{p'}", "}", "$$", "with both squares $2$-commutative we have to show that $p$ is equivalent to", "$p'$. By the $2$-Yoneda lemma (see", "Algebraic Stacks, Section \\ref{algebraic-section-2-yoneda})", "the morphisms $p$, $p'$, and $q$ are given by objects", "$x$, $x'$, and $y$ in the fibre categories of $\\mathcal{X}$ over", "$\\Spec(K)$, $\\Spec(K')$, and $\\Spec(L)$. The", "$2$-commutativity of the squares means that there are isomorphisms", "$\\alpha : a^*x \\to b^*y$ and $\\alpha' : (a')^*x' \\to (b')^*y$", "in the fibre categories", "of $\\mathcal{X}$ over $\\Spec(\\Omega)$ and $\\Spec(\\Omega')$.", "Choose any field $\\Omega''$ and embeddings", "$\\Omega \\to \\Omega''$ and $\\Omega' \\to \\Omega''$ agreeing on $L$.", "Then we can extend the diagram above to", "$$", "\\xymatrix{", "& \\Spec(\\Omega'') \\ar[ld]_c \\ar[d]^{q'} \\ar[rd]^{c'} \\\\", "\\Spec(\\Omega) \\ar[r]_b \\ar[d]_a &", "\\Spec(L) \\ar[d]^q & \\Spec(\\Omega') \\ar[l]^{b'} \\ar[d]^{a'} \\\\", "\\Spec(K) \\ar[r]^p &", "\\mathcal{X} &", "\\Spec(K') \\ar[l]_{p'}", "}", "$$", "with commutative triangles and", "$$", "(q')^*(\\alpha')^{-1} \\circ (q')^*\\alpha :", "(a \\circ c)^*x", "\\longrightarrow", "(a' \\circ c')^*x'", "$$", "is an isomorphism in the fibre category over $\\Spec(\\Omega'')$.", "Hence $p$ is equivalent to $p'$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8864, "type": "theorem", "label": "stacks-properties-lemma-points-cartesian", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-points-cartesian", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\ar[r] \\ar[d] &", "\\mathcal{X} \\ar[d] \\\\", "\\mathcal{Z} \\ar[r] & \\mathcal{Y}", "}", "$$", "be a fibre product of algebraic stacks. Then the map of sets", "of points", "$$", "|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}|", "\\longrightarrow", "|\\mathcal{Z}| \\times_{|\\mathcal{Y}|} |\\mathcal{X}|", "$$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Namely, suppose given fields $K$, $L$ and morphisms", "$\\Spec(K) \\to \\mathcal{X}$, $\\Spec(L) \\to \\mathcal{Z}$,", "then the assumption that they agree as elements of $|\\mathcal{Y}|$ means that", "there is a common extension $K \\subset M$ and $L \\subset M$", "such that", "$\\Spec(M) \\to \\Spec(K) \\to \\mathcal{X} \\to \\mathcal{Y}$ and", "$\\Spec(M) \\to \\Spec(L) \\to \\mathcal{Z} \\to \\mathcal{Y}$", "are $2$-isomorphic. And this is exactly the condition which says you get a", "morphism $\\Spec(M) \\to \\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8865, "type": "theorem", "label": "stacks-properties-lemma-characterize-surjective", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-characterize-surjective", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "which is representable by algebraic spaces. The following are equivalent:", "\\begin{enumerate}", "\\item $|f| : |\\mathcal{X}| \\to |\\mathcal{Y}|$ is surjective, and", "\\item $f$ is surjective", "(in the sense of Section \\ref{section-properties-morphisms}).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Let $T \\to \\mathcal{Y}$ be a morphism whose source is a scheme.", "To prove (2) we have to show that the morphism of algebraic spaces", "$T \\times_\\mathcal{Y} \\mathcal{X} \\to T$ is surjective. By", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-surjective}", "this means we have to show that", "$|T \\times_\\mathcal{Y} \\mathcal{X}| \\to |T|$ is surjective.", "Applying", "Lemma \\ref{lemma-points-cartesian}", "we see that this follows from (1).", "\\medskip\\noindent", "Conversely, assume (2). Let $y : \\Spec(K) \\to \\mathcal{Y}$ be a", "morphism from the spectrum of a field into $\\mathcal{Y}$. By assumption the", "morphism", "$\\Spec(K) \\times_{y, \\mathcal{Y}} \\mathcal{X} \\to \\Spec(K)$", "of algebraic spaces is surjective. By", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-surjective}", "this means there exists a field extension", "$K \\subset K'$ and a morphism", "$\\Spec(K') \\to \\Spec(K) \\times_{y, \\mathcal{Y}} \\mathcal{X}$", "such that the left square of the diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] &", "\\Spec(K) \\times_{y, \\mathcal{Y}} \\mathcal{X} \\ar[d] \\ar[r] &", "\\mathcal{X} \\ar[d]", "\\\\", "\\Spec(K) \\ar@{=}[r] &", "\\Spec(K) \\ar[r]^-y &", "\\mathcal{Y}", "}", "$$", "is commutative. This shows that $|X| \\to |\\mathcal{Y}|$ is surjective." ], "refs": [ "spaces-morphisms-definition-surjective", "stacks-properties-lemma-points-cartesian", "spaces-morphisms-definition-surjective" ], "ref_ids": [ 4985, 8864, 4985 ] } ], "ref_ids": [] }, { "id": 8866, "type": "theorem", "label": "stacks-properties-lemma-points-presentation", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-points-presentation", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Let $\\mathcal{X} = [U/R]$ be a presentation of $\\mathcal{X}$, see", "Algebraic Stacks, Definition \\ref{algebraic-definition-presentation}.", "Then the image of $|R| \\to |U| \\times |U|$ is an equivalence relation", "and $|\\mathcal{X}|$ is the quotient of $|U|$ by this equivalence relation." ], "refs": [ "algebraic-definition-presentation" ], "proofs": [ { "contents": [ "The assumption means that we have a smooth groupoid $(U, R, s, t, c)$", "in algebraic spaces, and an equivalence $f : [U/R] \\to \\mathcal{X}$.", "We may assume $\\mathcal{X} = [U/R]$.", "The induced morphism $p : U \\to \\mathcal{X}$ is smooth and surjective, see", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-smooth-quotient-smooth-presentation}.", "Hence $|U| \\to |\\mathcal{X}|$ is surjective by", "Lemma \\ref{lemma-characterize-surjective}.", "Note that $R = U \\times_\\mathcal{X} U$, see", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-2-cartesian}.", "Hence", "Lemma \\ref{lemma-points-cartesian}", "implies the map", "$$", "|R| \\longrightarrow |U| \\times_{|\\mathcal{X}|} |U|", "$$", "is surjective. Hence the image of $|R| \\to |U| \\times |U|$ is", "exactly the set of pairs $(u_1, u_2) \\in |U| \\times |U|$", "such that $u_1$ and $u_2$ have the same image in $|\\mathcal{X}|$.", "Combining these two statements we get the result of the lemma." ], "refs": [ "algebraic-lemma-smooth-quotient-smooth-presentation", "stacks-properties-lemma-characterize-surjective", "spaces-groupoids-lemma-quotient-stack-2-cartesian", "stacks-properties-lemma-points-cartesian" ], "ref_ids": [ 8476, 8865, 9324, 8864 ] } ], "ref_ids": [ 8488 ] }, { "id": 8867, "type": "theorem", "label": "stacks-properties-lemma-topology-points", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-topology-points", "contents": [ "There exists a unique topology on the sets of points", "of algebraic stacks with the following properties:", "\\begin{enumerate}", "\\item for every morphism of algebraic stacks $\\mathcal{X} \\to \\mathcal{Y}$", "the map $|\\mathcal{X}| \\to |\\mathcal{Y}|$ is continuous, and", "\\item for every morphism $U \\to \\mathcal{X}$ which is flat and locally", "of finite presentation with $U$ an algebraic space", "the map of topological spaces $|U| \\to |\\mathcal{X}|$ is continuous and open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a morphism $p : U \\to \\mathcal{X}$ which is", "surjective, flat, and locally of finite presentation", "with $U$ an algebraic space. Such exist by the definition of an algebraic", "stack, as a smooth morphism is flat and locally of finite presentation", "(see", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-smooth-locally-finite-presentation} and", "\\ref{spaces-morphisms-lemma-smooth-flat}).", "We define a topology on $|\\mathcal{X}|$ by the rule:", "$W \\subset |\\mathcal{X}|$ is open if and only if $|p|^{-1}(W)$ is open", "in $|U|$. To show that this is independent of the choice of $p$, let", "$p' : U' \\to \\mathcal{X}$ be another morphism which is surjective, flat,", "locally of finite presentation from an algebraic space to", "$\\mathcal{X}$. Set $U'' = U \\times_\\mathcal{X} U'$", "so that we have a $2$-commutative diagram", "$$", "\\xymatrix{", "U'' \\ar[r] \\ar[d] & U' \\ar[d] \\\\", "U \\ar[r] & \\mathcal{X}", "}", "$$", "As $U \\to \\mathcal{X}$ and $U' \\to \\mathcal{X}$ are surjective, flat,", "locally of finite presentation we see that $U'' \\to U'$ and $U'' \\to U$", "are surjective, flat and locally of finite presentation, see", "Lemma \\ref{lemma-property-spaces-too}.", "Hence the maps $|U''| \\to |U'|$ and $|U''| \\to |U|$ are continuous, open", "and surjective, see", "Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-surjective} and", "Lemma \\ref{spaces-morphisms-lemma-fppf-open}.", "This clearly implies that our definition is independent of the choice", "of $p : U \\to \\mathcal{X}$.", "\\medskip\\noindent", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "By", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-lift-morphism-presentations}", "we can find a $2$-commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_x \\ar[r]_a & V \\ar[d]^y \\\\", "\\mathcal{X} \\ar[r]^f & \\mathcal{Y}", "}", "$$", "with surjective smooth vertical arrows.", "Consider the associated commutative diagram", "$$", "\\xymatrix{", "|U| \\ar[d]_{|x|} \\ar[r]_{|a|} & |V| \\ar[d]^{|y|} \\\\", "|\\mathcal{X}| \\ar[r]^{|f|} & |\\mathcal{Y}|", "}", "$$", "of sets. If $W \\subset |\\mathcal{Y}|$ is open, then by the definition", "above this means exactly that $|y|^{-1}(W)$ is open in $|V|$. Since", "$|a|$ is continuous we conclude that", "$|a|^{-1}|y|^{-1}(W) = |x|^{-1}|f|^{-1}(W)$ is open in $|W|$ which means", "by definition that $|f|^{-1}(W)$ is open in $|\\mathcal{X}|$.", "Thus $|f|$ is continuous.", "\\medskip\\noindent", "Finally, we have to show that if $U$ is an algebraic space, and", "$U \\to \\mathcal{X}$ is flat and locally of finite presentation, then", "$|U| \\to |\\mathcal{X}|$ is open. Let $V \\to \\mathcal{X}$ be surjective,", "flat, and locally of finite presentation with $V$ an algebraic space.", "Consider the commutative diagram", "$$", "\\xymatrix{", "|U \\times_\\mathcal{X} V| \\ar[r]_e \\ar[rd]_f &", "|U| \\times_{|\\mathcal{X}|} |V| \\ar[d]_c \\ar[r]_d &", "|V| \\ar[d]^b \\\\", "& |U| \\ar[r]^a & |\\mathcal{X}|", "}", "$$", "Now the morphism $U \\times_\\mathcal{X} V \\to U$ is surjective, i.e,", "$f : |U \\times_\\mathcal{X} V| \\to |U|$ is surjective.", "The left top horizontal arrow is surjective, see", "Lemma \\ref{lemma-points-cartesian}.", "The morphism $U \\times_\\mathcal{X} V \\to V$ is flat and locally of finite", "presentation, hence $d \\circ e : |U \\times_\\mathcal{X} V| \\to |V|$ is open,", "see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-fppf-open}.", "Pick $W \\subset |U|$ open. The properties above imply that", "$b^{-1}(a(W)) = (d \\circ e)(f^{-1}(W))$ is open, which by construction means", "that $a(W)$ is open as desired." ], "refs": [ "spaces-morphisms-lemma-smooth-locally-finite-presentation", "spaces-morphisms-lemma-smooth-flat", "stacks-properties-lemma-property-spaces-too", "spaces-morphisms-definition-surjective", "spaces-morphisms-lemma-fppf-open", "algebraic-lemma-lift-morphism-presentations", "stacks-properties-lemma-points-cartesian", "spaces-morphisms-lemma-fppf-open" ], "ref_ids": [ 4889, 4891, 8858, 4985, 4855, 8468, 8864, 4855 ] } ], "ref_ids": [] }, { "id": 8868, "type": "theorem", "label": "stacks-properties-lemma-space-locally-quasi-compact", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-space-locally-quasi-compact", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Every point of $|\\mathcal{X}|$ has a fundamental system of", "quasi-compact open neighbourhoods.", "In particular $|\\mathcal{X}|$ is locally quasi-compact in the sense of", "Topology, Definition \\ref{topology-definition-locally-quasi-compact}." ], "refs": [ "topology-definition-locally-quasi-compact" ], "proofs": [ { "contents": [ "This follows formally from the fact that there exists a scheme", "$U$ and a surjective, open, continuous map $U \\to |\\mathcal{X}|$", "of topological spaces. Namely, if $U \\to \\mathcal{X}$ is surjective and", "smooth, then", "Lemma \\ref{lemma-topology-points}", "guarantees that $|U| \\to |\\mathcal{X}|$ is continuous, surjective, and open." ], "refs": [ "stacks-properties-lemma-topology-points" ], "ref_ids": [ 8867 ] } ], "ref_ids": [ 8361 ] }, { "id": 8869, "type": "theorem", "label": "stacks-properties-lemma-composition-surjective", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-composition-surjective", "contents": [ "The composition of surjective morphisms is surjective." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8870, "type": "theorem", "label": "stacks-properties-lemma-base-change-surjective", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-base-change-surjective", "contents": [ "The base change of a surjective morphism is surjective." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use", "Lemma \\ref{lemma-points-cartesian}." ], "refs": [ "stacks-properties-lemma-points-cartesian" ], "ref_ids": [ 8864 ] } ], "ref_ids": [] }, { "id": 8871, "type": "theorem", "label": "stacks-properties-lemma-descent-surjective", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-descent-surjective", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $\\mathcal{Y}' \\to \\mathcal{Y}$ be a surjective morphism of algebraic", "stacks. If the base change $f' : \\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X}", "\\to \\mathcal{Y}'$ of $f$ is surjective, then $f$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Immediate from", "Lemma \\ref{lemma-points-cartesian}." ], "refs": [ "stacks-properties-lemma-points-cartesian" ], "ref_ids": [ 8864 ] } ], "ref_ids": [] }, { "id": 8872, "type": "theorem", "label": "stacks-properties-lemma-surjective-permanence", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-surjective-permanence", "contents": [ "Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of", "algebraic stacks. If $\\mathcal{X} \\to \\mathcal{Z}$ is surjective", "so is $\\mathcal{Y} \\to \\mathcal{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Immediate." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8873, "type": "theorem", "label": "stacks-properties-lemma-quasi-compact-stack", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-quasi-compact-stack", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "The following are equivalent:", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is quasi-compact,", "\\item there exists a surjective smooth morphism $U \\to \\mathcal{X}$", "with $U$ a quasi-compact scheme,", "\\item there exists a surjective smooth morphism $U \\to \\mathcal{X}$", "with $U$ a quasi-compact algebraic space, and", "\\item there exists a surjective morphism $\\mathcal{U} \\to \\mathcal{X}$", "of algebraic stacks such that $\\mathcal{U}$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-characterize-surjective}.", "Suppose $\\mathcal{U}$ and $\\mathcal{U} \\to \\mathcal{X}$ are as in (4).", "Then since $|\\mathcal{U}| \\to |\\mathcal{X}|$ is surjective and", "continuous we conclude that $|\\mathcal{X}|$ is quasi-compact.", "Thus (4) implies (1). The implications (2) $\\Rightarrow$ (3) $\\Rightarrow$ (4)", "are immediate. Assume (1), i.e., $\\mathcal{X}$ is quasi-compact, i.e., that", "$|\\mathcal{X}|$ is quasi-compact. Choose a scheme $U$ and a surjective", "smooth morphism $U \\to \\mathcal{X}$. Then since $|U| \\to |\\mathcal{X}|$", "is open we see that there exists a quasi-compact open $U' \\subset U$", "such that $|U'| \\to |X|$ is surjective (and still smooth).", "Hence (2) holds." ], "refs": [ "stacks-properties-lemma-characterize-surjective" ], "ref_ids": [ 8865 ] } ], "ref_ids": [] }, { "id": 8874, "type": "theorem", "label": "stacks-properties-lemma-finite-disjoint-quasi-compact", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-finite-disjoint-quasi-compact", "contents": [ "A finite disjoint union of quasi-compact algebraic stacks is", "a quasi-compact algebraic stack." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the corresponding topological fact." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8875, "type": "theorem", "label": "stacks-properties-lemma-type-property", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-type-property", "contents": [ "Let $\\mathcal{P}$ be a property of schemes which is local in the smooth", "topology, see", "Descent, Definition \\ref{descent-definition-property-local}.", "Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent", "\\begin{enumerate}", "\\item for some scheme $U$ and some surjective smooth morphism", "$U \\to \\mathcal{X}$ the scheme $U$ has property $\\mathcal{P}$,", "\\item for every scheme $U$ and every smooth morphism $U \\to \\mathcal{X}$", "the scheme $U$ has property $\\mathcal{P}$,", "\\item for some algebraic space $U$ and some surjective smooth morphism", "$U \\to \\mathcal{X}$ the algebraic space $U$ has property $\\mathcal{P}$, and", "\\item for every algebraic space $U$ and every smooth morphism", "$U \\to \\mathcal{X}$ the algebraic space $U$ has property $\\mathcal{P}$.", "\\end{enumerate}", "If $\\mathcal{X}$ is a scheme this is equivalent to $\\mathcal{P}(U)$.", "If $\\mathcal{X}$ is an algebraic space this is equivalent to", "$X$ having property $\\mathcal{P}$." ], "refs": [ "descent-definition-property-local" ], "proofs": [ { "contents": [ "Let $U \\to \\mathcal{X}$ surjective and smooth with $U$ an algebraic space.", "Let $V \\to \\mathcal{X}$ be a smooth morphism with $V$ an algebraic space.", "Choose schemes $U'$ and $V'$ and surjective \\'etale morphisms", "$U' \\to U$ and $V' \\to V$. Finally, choose a scheme $W$ and a", "surjective \\'etale morphism $W \\to V' \\times_\\mathcal{X} U'$.", "Then $W \\to V'$ and $W \\to U'$ are smooth morphisms of schemes", "as compositions of \\'etale and smooth morphisms of algebraic spaces, see", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-etale-smooth} and", "\\ref{spaces-morphisms-lemma-composition-smooth}.", "Moreover, $W \\to V'$ is surjective as $U' \\to \\mathcal{X}$ is surjective.", "Hence, we have", "$$", "\\mathcal{P}(U) \\Leftrightarrow", "\\mathcal{P}(U') \\Rightarrow", "\\mathcal{P}(W) \\Rightarrow", "\\mathcal{P}(V') \\Leftrightarrow \\mathcal{P}(V)", "$$", "where the equivalences are by definition of property $\\mathcal{P}$ for", "algebraic spaces, and the two implications come from", "Descent, Definition \\ref{descent-definition-property-local}.", "This proves (3) $\\Rightarrow$ (4).", "\\medskip\\noindent", "The implications (2) $\\Rightarrow$ (1), (1) $\\Rightarrow$ (3),", "and (4) $\\Rightarrow$ (2) are immediate." ], "refs": [ "spaces-morphisms-lemma-etale-smooth", "spaces-morphisms-lemma-composition-smooth", "descent-definition-property-local" ], "ref_ids": [ 4909, 4886, 14768 ] } ], "ref_ids": [ 14768 ] }, { "id": 8876, "type": "theorem", "label": "stacks-properties-lemma-local-source-target-at-point", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-local-source-target-at-point", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Let $x \\in |\\mathcal{X}|$ be a point of $\\mathcal{X}$.", "Let $\\mathcal{P}$ be a property of germs of schemes which is smooth local, see", "Descent, Definition \\ref{descent-definition-local-at-point}.", "The following are equivalent", "\\begin{enumerate}", "\\item for any smooth morphism $U \\to \\mathcal{X}$ with $U$ a scheme", "and $u \\in U$ with $a(u) = x$ we have $\\mathcal{P}(U, u)$,", "\\item for some smooth morphism $U \\to \\mathcal{X}$ with $U$ a scheme", "and some $u \\in U$ with $a(u) = x$ we have $\\mathcal{P}(U, u)$,", "\\item for any smooth morphism $U \\to \\mathcal{X}$ with $U$ an algebraic space", "and $u \\in |U|$ with $a(u) = x$ the algebraic space $U$ has property", "$\\mathcal{P}$ at $u$, and", "\\item for some smooth morphism $U \\to \\mathcal{X}$ with $U$ a an", "algebraic space and some $u \\in |U|$ with $a(u) = x$ the algebraic space", "$U$ has property $\\mathcal{P}$ at $u$.", "\\end{enumerate}", "If $\\mathcal{X}$ is representable, then this is equivalent to", "$\\mathcal{P}(\\mathcal{X}, x)$. If $\\mathcal{X}$ is an algebraic space then", "this is equivalent to $\\mathcal{X}$ having property $\\mathcal{P}$ at $x$." ], "refs": [ "descent-definition-local-at-point" ], "proofs": [ { "contents": [ "Let $a : U \\to \\mathcal{X}$ and $u \\in |U|$ as in (3). Let", "$b : V \\to \\mathcal{X}$ be another smooth morphism with $V$ an algebraic", "space and $v \\in |V|$ with $b(v) = x$ also.", "Choose a scheme $U'$, an \\'etale morphism $U' \\to U$ and $u' \\in U'$", "mapping to $u$. Choose a scheme $V'$, an \\'etale morphism $V' \\to V$", "and $v' \\in V'$ mapping to $v$. By", "Lemma \\ref{lemma-points-cartesian}", "there exists a point $\\overline{w} \\in |V' \\times_\\mathcal{X} U'|$", "mapping to $u'$ and $v'$. Choose a scheme $W$ and a surjective \\'etale", "morphism $W \\to V' \\times_\\mathcal{X} U'$. We may choose a", "$w \\in |W|$ mapping to $\\overline{w}$ (see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-characterize-surjective}).", "Then $W \\to V'$ and $W \\to U'$ are smooth morphisms of schemes", "as compositions of \\'etale and smooth morphisms of algebraic spaces, see", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-etale-smooth} and", "\\ref{spaces-morphisms-lemma-composition-smooth}.", "Hence", "$$", "\\mathcal{P}(U, u)", "\\Leftrightarrow", "\\mathcal{P}(U', u')", "\\Leftrightarrow", "\\mathcal{P}(W, w)", "\\Leftrightarrow", "\\mathcal{P}(V', v')", "\\Leftrightarrow", "\\mathcal{P}(V, v)", "$$", "The outer two equivalences by", "Properties of Spaces,", "Definition \\ref{spaces-properties-definition-property-at-point}", "and the other two by what it means to be a smooth local property", "of germs of schemes. This proves (4) $\\Rightarrow$ (3).", "\\medskip\\noindent", "The implications (1) $\\Rightarrow$ (2), (2) $\\Rightarrow$ (4),", "and (3) $\\Rightarrow$ (1) are immediate." ], "refs": [ "stacks-properties-lemma-points-cartesian", "spaces-properties-lemma-characterize-surjective", "spaces-morphisms-lemma-etale-smooth", "spaces-morphisms-lemma-composition-smooth", "spaces-properties-definition-property-at-point" ], "ref_ids": [ 8864, 11820, 4909, 4886, 11927 ] } ], "ref_ids": [ 14771 ] }, { "id": 8877, "type": "theorem", "label": "stacks-properties-lemma-base-change-monomorphism", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-base-change-monomorphism", "contents": [ "Let $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a monomorphism.", "Then $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}$", "is a monomorphism." ], "refs": [], "proofs": [ { "contents": [ "This follows from the general discussion in", "Section \\ref{section-properties-morphisms}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8878, "type": "theorem", "label": "stacks-properties-lemma-composition-monomorphism", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-composition-monomorphism", "contents": [ "Compositions of monomorphisms of algebraic stacks are monomorphisms." ], "refs": [], "proofs": [ { "contents": [ "This follows from the general discussion in", "Section \\ref{section-properties-morphisms}", "and", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-composition-monomorphism}." ], "refs": [ "spaces-morphisms-lemma-composition-monomorphism" ], "ref_ids": [ 4753 ] } ], "ref_ids": [] }, { "id": 8879, "type": "theorem", "label": "stacks-properties-lemma-monomorphism", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-monomorphism", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is a monomorphism,", "\\item $f$ is fully faithful,", "\\item the diagonal", "$\\Delta_f : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "is an equivalence, and", "\\item there exists an algebraic space $W$ and a surjective, flat morphism", "$W \\to \\mathcal{Y}$ which is locally of finite presentation such that", "$V = \\mathcal{X} \\times_\\mathcal{Y} W$ is an algebraic space, and the", "morphism $V \\to W$ is a monomorphism of algebraic spaces.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (4) follows from the general discussion in", "Section \\ref{section-properties-morphisms}", "and in particular", "Lemmas \\ref{lemma-check-representable-covering} and", "\\ref{lemma-check-property-covering}.", "\\medskip\\noindent", "The equivalence of (2) and (3) is", "Categories, Lemma \\ref{categories-lemma-fully-faithful-diagonal-equivalence}.", "\\medskip\\noindent", "Assume the equivalent conditions (2) and (3). Then $f$ is representable", "by algebraic spaces according to", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}.", "Moreover, the $2$-Yoneda lemma combined with the fully faithfulness", "implies that for every scheme $T$ the functor", "$$", "\\Mor(T, \\mathcal{X})", "\\longrightarrow", "\\Mor(T, \\mathcal{Y})", "$$", "is fully faithful. Hence given a morphism $y : T \\to \\mathcal{Y}$ there exists", "up to unique $2$-isomorphism at most one morphism $x : T \\to \\mathcal{X}$", "such that $y \\cong f \\circ x$. In particular, given a morphism of schemes", "$h : T' \\to T$ there exists at most one lift", "$\\tilde h : T' \\to T \\times_\\mathcal{Y} \\mathcal{X}$ of $h$.", "Thus $T \\times_\\mathcal{Y} \\mathcal{X} \\to T$ is a monomorphism of", "algebraic spaces, which proves that (1) holds.", "\\medskip\\noindent", "Finally, assume that (1) holds. Then for any scheme $T$ and morphism", "$y : T \\to \\mathcal{Y}$ the fibre product $T \\times_\\mathcal{Y} \\mathcal{X}$", "is an algebraic space, and $T \\times_\\mathcal{Y} \\mathcal{X} \\to T$", "is a monomorphism. Hence there exists up to unique isomorphism", "exactly one pair $(x, \\alpha)$ where $x : T \\to \\mathcal{X}$ is a morphism", "and $\\alpha : f \\circ x \\to y$ is a $2$-morphism. Applying", "the $2$-Yoneda lemma this says exactly that $f$ is fully faithful, i.e.,", "that (2) holds." ], "refs": [ "stacks-properties-lemma-check-representable-covering", "stacks-properties-lemma-check-property-covering", "categories-lemma-fully-faithful-diagonal-equivalence", "algebraic-lemma-characterize-representable-by-algebraic-spaces" ], "ref_ids": [ 8857, 8859, 12298, 8469 ] } ], "ref_ids": [] }, { "id": 8880, "type": "theorem", "label": "stacks-properties-lemma-monomorphism-injective-points", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-monomorphism-injective-points", "contents": [ "\\begin{slogan}", "Monomorphisms of stacks are injective on points.", "\\end{slogan}", "A monomorphism of algebraic stacks induces an injective map of", "sets of points." ], "refs": [], "proofs": [ { "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a monomorphism of algebraic stacks.", "Suppose that $x_i : \\Spec(K_i) \\to \\mathcal{X}$ be morphisms such that", "$f \\circ x_1$ and $f \\circ x_2$ define the same element of $|\\mathcal{Y}|$.", "Applying the definition we find a common extension $\\Omega$ with corresponding", "morphisms $c_i : \\Spec(\\Omega) \\to \\Spec(K_i)$ and a", "$2$-isomorphism $\\beta : f \\circ x_1 \\circ c_1 \\to f \\circ x_1 \\circ c_2$.", "As $f$ is fully faithful, see", "Lemma \\ref{lemma-monomorphism},", "we can lift $\\beta$ to an isomorphism", "$\\alpha : f \\circ x_1 \\circ c_1 \\to f \\circ x_1 \\circ c_2$.", "Hence $x_1$ and $x_2$ define the same point of $|\\mathcal{X}|$", "as desired." ], "refs": [ "stacks-properties-lemma-monomorphism" ], "ref_ids": [ 8879 ] } ], "ref_ids": [] }, { "id": 8881, "type": "theorem", "label": "stacks-properties-lemma-monomorphism-diagonal", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-monomorphism-diagonal", "contents": [ "Let $\\mathcal{X} \\to \\mathcal{X}' \\to \\mathcal{Y}$ be morphisms", "of algebraic stacks. If $\\mathcal{X} \\to \\mathcal{X}'$ is a monomorphism", "then the canonical diagram", "$$", "\\xymatrix{", "\\mathcal{X} \\ar[r] \\ar[d] &", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\ar[d] \\\\", "\\mathcal{X}' \\ar[r] &", "\\mathcal{X}' \\times_\\mathcal{Y} \\mathcal{X}'", "}", "$$", "is a fibre product square." ], "refs": [], "proofs": [ { "contents": [ "We have $\\mathcal{X} = \\mathcal{X} \\times_{\\mathcal{X}'} \\mathcal{X}$", "by Lemma \\ref{lemma-monomorphism}. Thus the result by applying", "Categories, Lemma \\ref{categories-lemma-fibre-product-after-map}." ], "refs": [ "stacks-properties-lemma-monomorphism", "categories-lemma-fibre-product-after-map" ], "ref_ids": [ 8879, 12278 ] } ], "ref_ids": [] }, { "id": 8882, "type": "theorem", "label": "stacks-properties-lemma-base-change-immersion", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-base-change-immersion", "contents": [ "Let $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a", "(closed, resp.\\ open) immersion.", "Then $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}$", "is a (closed, resp.\\ open) immersion." ], "refs": [], "proofs": [ { "contents": [ "This follows from the general discussion in", "Section \\ref{section-properties-morphisms}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8883, "type": "theorem", "label": "stacks-properties-lemma-composition-immersion", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-composition-immersion", "contents": [ "Compositions of immersions of algebraic stacks are immersions.", "Similarly for closed immersions and open immersions." ], "refs": [], "proofs": [ { "contents": [ "This follows from the general discussion in", "Section \\ref{section-properties-morphisms}", "and", "Spaces, Lemma \\ref{spaces-lemma-composition-immersions}." ], "refs": [ "spaces-lemma-composition-immersions" ], "ref_ids": [ 8160 ] } ], "ref_ids": [] }, { "id": 8884, "type": "theorem", "label": "stacks-properties-lemma-check-immersion-covering", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-check-immersion-covering", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "let $W$ be an algebraic space and let $W \\to \\mathcal{Y}$ be a surjective,", "flat morphism which is locally of finite presentation. The following", "are equivalent:", "\\begin{enumerate}", "\\item $f$ is an (open, resp.\\ closed) immersion, and", "\\item $V = W \\times_\\mathcal{Y} \\mathcal{X}$ is an algebraic space, and", "$V \\to W$ is an (open, resp.\\ closed) immersion.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from the general discussion in", "Section \\ref{section-properties-morphisms}", "and in particular", "Lemmas \\ref{lemma-check-representable-covering} and", "\\ref{lemma-check-property-covering}." ], "refs": [ "stacks-properties-lemma-check-representable-covering", "stacks-properties-lemma-check-property-covering" ], "ref_ids": [ 8857, 8859 ] } ], "ref_ids": [] }, { "id": 8885, "type": "theorem", "label": "stacks-properties-lemma-immersion-monomorphism", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-immersion-monomorphism", "contents": [ "An immersion is a monomorphism." ], "refs": [], "proofs": [ { "contents": [ "See", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-immersions-monomorphisms}." ], "refs": [ "spaces-morphisms-lemma-immersions-monomorphisms" ], "ref_ids": [ 4756 ] } ], "ref_ids": [] }, { "id": 8886, "type": "theorem", "label": "stacks-properties-lemma-immersion-into-presentation", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-immersion-into-presentation", "contents": [ "Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces.", "Let $i : \\mathcal{Z} \\to [U/R]$ be an immersion.", "Then there exists an $R$-invariant locally closed subspace", "$Z \\subset U$ and a presentation $[Z/R_Z] \\to \\mathcal{Z}$", "where $R_Z$ is the restriction of $R$ to $Z$ such that", "$$", "\\xymatrix{", "[Z/R_Z] \\ar[dr] \\ar[rr] & & \\mathcal{Z} \\ar[ld]^i \\\\", "& [U/R]", "}", "$$", "is $2$-commutative. If $i$ is a closed (resp.\\ open) immersion", "then $Z$ is a closed (resp.\\ open) subspace of $U$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-representable-in-terms-presentations}", "we get a commutative diagram", "$$", "\\xymatrix{", "[U'/R'] \\ar[dr] \\ar[rr] & & \\mathcal{Z} \\ar[ld] \\\\", "& [U/R]", "}", "$$", "where $U' = \\mathcal{Z} \\times_{[U/R]} U$ and", "$R' = \\mathcal{Z} \\times_{[U/R]} R$.", "Since $\\mathcal{Z} \\to [U/R]$ is an immersion we see that", "$U' \\to U$ is an immersion of algebraic spaces. Let $Z \\subset U$", "be the locally closed subspace such that $U' \\to U$ factors through", "$Z$ and induces an isomorphism $U' \\to Z$.", "It is clear from the construction of $R'$ that", "$R' = U' \\times_{U, t} R = R \\times_{s, U} U'$.", "This implies that $Z \\cong U'$ is $R$-invariant and that the image of", "$R' \\to R$ identifies $R'$ with the restriction", "$R_Z = s^{-1}(Z) = t^{-1}(Z)$ of $R$ to $Z$. Hence the lemma holds." ], "refs": [ "stacks-properties-lemma-representable-in-terms-presentations" ], "ref_ids": [ 8862 ] } ], "ref_ids": [] }, { "id": 8887, "type": "theorem", "label": "stacks-properties-lemma-immersion-presentation", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-immersion-presentation", "contents": [ "Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces.", "Let $\\mathcal{X} = [U/R]$ be the associated algebraic stack, see", "Algebraic Stacks,", "Theorem \\ref{algebraic-theorem-smooth-groupoid-gives-algebraic-stack}.", "Let $Z \\subset U$ be an $R$-invariant locally closed subspace. Then", "$$", "[Z/R_Z] \\longrightarrow [U/R]", "$$", "is an immersion of algebraic stacks, where $R_Z$ is the restriction", "of $R$ to $Z$. If $Z \\subset U$ is open (resp.\\ closed) then the morphism", "is an open (resp.\\ closed) immersion of algebraic stacks." ], "refs": [ "algebraic-theorem-smooth-groupoid-gives-algebraic-stack" ], "proofs": [ { "contents": [ "Recall that by", "Groupoids in Spaces,", "Definition \\ref{spaces-groupoids-definition-invariant-open}", "(see also discussion following the definition)", "we have $R_Z = s^{-1}(Z) = t^{-1}(Z)$ as locally closed subspaces", "of $R$. Hence the two morphisms $R_Z \\to Z$ are smooth as base changes", "of $s$ and $t$. Hence", "$(Z, R_Z, s|_{R_Z}, t|_{R_Z}, c|_{R_Z \\times_{s, Z, t} R_Z})$ is", "a smooth groupoid in algebraic spaces, and we see that", "$[Z/R_Z]$ is an algebraic stack, see", "Algebraic Stacks,", "Theorem \\ref{algebraic-theorem-smooth-groupoid-gives-algebraic-stack}.", "The assumptions of", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-criterion-fibre-product}", "are all satisfied and it follows that we have a $2$-fibre square", "$$", "\\xymatrix{", "Z \\ar[d] \\ar[r] & [Z/R_Z] \\ar[d] \\\\", "U \\ar[r] & [U/R]", "}", "$$", "It follows from this and", "Lemma \\ref{lemma-check-representable-covering}", "that $[Z/R_Z] \\to [U/R]$ is representable by algebraic spaces,", "whereupon it follows from", "Lemma \\ref{lemma-check-property-covering}", "that the right vertical arrow is an immersion (resp.\\ closed immersion,", "resp.\\ open immersion) if and only if the left vertical arrow is." ], "refs": [ "spaces-groupoids-definition-invariant-open", "algebraic-theorem-smooth-groupoid-gives-algebraic-stack", "spaces-groupoids-lemma-criterion-fibre-product", "stacks-properties-lemma-check-representable-covering", "stacks-properties-lemma-check-property-covering" ], "ref_ids": [ 9351, 8435, 9331, 8857, 8859 ] } ], "ref_ids": [ 8435 ] }, { "id": 8888, "type": "theorem", "label": "stacks-properties-lemma-substack-image", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-substack-image", "contents": [ "For any immersion $i : \\mathcal{Z} \\to \\mathcal{X}$ there exists a", "unique locally closed substack $\\mathcal{X}' \\subset \\mathcal{X}$", "such that $i$ factors as the composition of", "an equivalence $i' : \\mathcal{Z} \\to \\mathcal{X}'$", "followed by the inclusion morphism $\\mathcal{X}' \\to \\mathcal{X}$.", "If $i$ is a closed (resp.\\ open) immersion, then $\\mathcal{X}'$", "is a closed (resp.\\ open) substack of $\\mathcal{X}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8889, "type": "theorem", "label": "stacks-properties-lemma-substacks-presentation", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-substacks-presentation", "contents": [ "Let $[U/R] \\to \\mathcal{X}$ be a presentation of an algebraic stack.", "There is a canonical bijection", "$$", "\\text{locally closed substacks }\\mathcal{Z}\\text{ of }\\mathcal{X}", "\\longrightarrow", "R\\text{-invariant locally closed subspaces }Z\\text{ of }U", "$$", "which sends $\\mathcal{Z}$ to $U \\times_\\mathcal{X} \\mathcal{Z}$.", "Moreover, a morphism of algebraic stacks $f : \\mathcal{Y} \\to \\mathcal{X}$", "factors through $\\mathcal{Z}$ if and only if", "$\\mathcal{Y} \\times_\\mathcal{X} U \\to U$ factors through $Z$.", "Similarly for closed substacks and open substacks." ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-immersion-into-presentation} and", "\\ref{lemma-immersion-presentation}", "we find that the map is a bijection.", "If $\\mathcal{Y} \\to \\mathcal{X}$ factors through $\\mathcal{Z}$", "then of course the base change $\\mathcal{Y} \\times_\\mathcal{X} U \\to U$", "factors through $Z$. Converse, suppose that $\\mathcal{Y} \\to \\mathcal{X}$", "is a morphism such that $\\mathcal{Y} \\times_\\mathcal{X} U \\to U$", "factors through $Z$. We will show that for every scheme $T$ and", "morphism $T \\to \\mathcal{Y}$,", "given by an object $y$ of the fibre category of $\\mathcal{Y}$ over $T$,", "the object $y$ is in fact in the fibre category of $\\mathcal{Z}$ over $T$.", "Namely, the fibre product $T \\times_\\mathcal{X} U$ is an algebraic", "space and $T \\times_\\mathcal{X} U \\to T$ is a surjective smooth morphism.", "Hence there is an fppf covering $\\{T_i \\to T\\}$ such that", "$T_i \\to T$ factors through $T \\times_\\mathcal{X} U \\to T$ for all $i$.", "Then $T_i \\to \\mathcal{X}$ factors through $\\mathcal{Y} \\times_\\mathcal{X} U$", "and hence through $Z \\subset U$. Thus $y|_{T_i}$ is an object of", "$\\mathcal{Z}$ (as $Z$ is the fibre product of $U$ with", "$\\mathcal{Z}$ over $\\mathcal{X}$).", "Since $\\mathcal{Z}$ is a strictly full substack, we conclude", "that $y$ is an object of $\\mathcal{Z}$ as desired." ], "refs": [ "stacks-properties-lemma-immersion-into-presentation", "stacks-properties-lemma-immersion-presentation" ], "ref_ids": [ 8886, 8887 ] } ], "ref_ids": [] }, { "id": 8890, "type": "theorem", "label": "stacks-properties-lemma-open-substacks", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-open-substacks", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. The rule", "$\\mathcal{U} \\mapsto |\\mathcal{U}|$ defines an inclusion preserving", "bijection between open substacks of $\\mathcal{X}$ and open subsets", "of $|\\mathcal{X}|$." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $[U/R] \\to \\mathcal{X}$, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-stack-presentation}.", "By", "Lemma \\ref{lemma-substacks-presentation}", "we see that open substacks correspond to $R$-invariant open subschemes", "of $U$. On the other hand", "Lemmas \\ref{lemma-points-presentation} and \\ref{lemma-topology-points}", "guarantee these correspond bijectively to open subsets of $|\\mathcal{X}|$." ], "refs": [ "algebraic-lemma-stack-presentation", "stacks-properties-lemma-substacks-presentation", "stacks-properties-lemma-points-presentation", "stacks-properties-lemma-topology-points" ], "ref_ids": [ 8474, 8889, 8866, 8867 ] } ], "ref_ids": [] }, { "id": 8891, "type": "theorem", "label": "stacks-properties-lemma-open-image-substack", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-open-image-substack", "contents": [ "Let $\\mathcal X$ be an algebraic stack. Let $U$ be an algebraic space and", "$U \\to \\mathcal X$ a surjective smooth morphism. For an open immersion", "$V \\hookrightarrow U$, there exists an algebraic stack $\\mathcal Y$, an", "open immersion $\\mathcal Y \\to \\mathcal X$, and a surjective smooth", "morphism $V \\to \\mathcal Y$." ], "refs": [], "proofs": [ { "contents": [ "We define a category fibred in groupoids $\\mathcal Y$ by letting the fiber", "category $\\mathcal{Y}_T$ over an object $T$ of $(\\Sch/S)_{fppf}$ be", "the full subcategory of $\\mathcal{X}_T$ consisting of all", "$y \\in \\Ob(\\mathcal{X}_T)$ such that the projection morphism", "$V \\times_{\\mathcal X, y} T \\to T$ surjective. Now for any morphism", "$x : T \\to \\mathcal X$, the $2$-fibred product", "$T \\times_{x, \\mathcal X} \\mathcal Y$ has fiber category over $T'$ consisting", "of triples $(f : T' \\to T, y \\in \\mathcal{X}_{T'}, f^*x \\simeq y)$ such that", "$V \\times_{\\mathcal X, y} T' \\to T'$ is surjective.", "Note that $T \\times_{x, \\mathcal X} \\mathcal Y$ is fibered in setoids since", "$\\mathcal Y \\to \\mathcal X$ is faithful (see", "Stacks, Lemma \\ref{stacks-lemma-2-fibre-product-gives-stack-in-setoids}).", "Now the isomorphism $f^*x \\simeq y$ gives the diagram", "$$", "\\xymatrix{", "V \\times_{\\mathcal X, y} T' \\ar[d] \\ar[r] &", "V \\times_{\\mathcal X, x} T \\ar[r] \\ar[d] &", "V \\ar[d] \\\\", "T' \\ar[r]^f &", "T \\ar[r]^x &", "\\mathcal X", "}", "$$", "where both squares are cartesian. The morphism", "$V \\times_{\\mathcal X, x} T \\to T$ is smooth by base change, and hence open.", "Let $T_0 \\subset T$ be its image. From the cartesian squares we deduce that", "$V \\times_{\\mathcal X, y} T' \\to T'$ is surjective if and only if $f$ lands", "in $T_0$. Therefore $T \\times_{x, \\mathcal X} \\mathcal Y$ is representable by", "$T_0$, so the inclusion $\\mathcal Y \\to \\mathcal X$ is an open immersion.", "By", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-open-fibred-category-is-algebraic}", "we conclude that $\\mathcal{Y}$ is an algebraic stack.", "Lastly if we denote the morphism $V \\to \\mathcal X$ by $g$, we have", "$V \\times_{\\mathcal X} V \\to V$ is surjective (the diagonal gives a", "section). Hence $g$ is in the image of $\\mathcal{Y}_V \\to \\mathcal{X}_V$, i.e.,", "we obtain a morphism $g' : V \\to \\mathcal{Y}$ fitting into the commutative", "diagram", "$$", "\\xymatrix{", "V \\ar[r] \\ar[d]^{g'} & U \\ar[d] \\\\", "\\mathcal{Y} \\ar[r] & \\mathcal{X}", "}", "$$", "Since $V \\times_{g, \\mathcal X} \\mathcal Y \\to V$ is a", "monomorphism, it is in fact an isomorphism since $(1, g')$ defines a section.", "Therefore $g' : V \\to \\mathcal Y$ is a smooth morphism, as it is the", "base change of the smooth morphism $g : V \\to \\mathcal{X}$.", "It is surjective by our construction of $\\mathcal{Y}$ which finishes", "the proof of the lemma." ], "refs": [ "stacks-lemma-2-fibre-product-gives-stack-in-setoids", "algebraic-lemma-open-fibred-category-is-algebraic" ], "ref_ids": [ 8954, 8472 ] } ], "ref_ids": [] }, { "id": 8892, "type": "theorem", "label": "stacks-properties-lemma-union-open-substacks", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-union-open-substacks", "contents": [ "Let $\\mathcal X$ be an algebraic stack and $\\mathcal{X}_i \\subset \\mathcal X$", "a collection of open substacks indexed by $i \\in I$. Then there exists an", "open substack, which we denote", "$\\bigcup_{i\\in I} \\mathcal{X}_i \\subset \\mathcal X$, such that", "the $\\mathcal{X}_i$ are open substacks covering it." ], "refs": [], "proofs": [ { "contents": [ "We define a fibred subcategory", "$\\mathcal{X}' = \\bigcup_{i \\in I} \\mathcal{X}_i$", "by letting the fiber category over an object $T$ of $(\\Sch/S)_{fppf}$", "be the full subcategory of $\\mathcal{X}_T$ consisting of all", "$x \\in \\Ob(\\mathcal{X}_T)$ such that the morphism", "$\\coprod_{i \\in I} (\\mathcal{X}_i \\times_{\\mathcal X} T) \\to T$", "is surjective. Let $x_i \\in \\Ob((\\mathcal{X}_i)_T)$.", "Then $(x_i, 1)$ gives a section of", "$\\mathcal{X}_i \\times_{\\mathcal X} T \\to T$, so we have an isomorphism. Thus", "$\\mathcal{X}_i \\subset \\mathcal{X}'$ is a full subcategory.", "Now let $x \\in \\Ob(\\mathcal{X}_T)$. Then", "$\\mathcal{X}_i \\times_{\\mathcal X} T$ is representable", "by an open subscheme $T_i \\subset T$. The $2$-fibred product", "$\\mathcal{X}' \\times_{\\mathcal X} T$ has fiber over $T'$ consisting", "of $(y \\in \\mathcal{X}_{T'}, f : T' \\to T, f^*x \\simeq y)$ such that", "$\\coprod (\\mathcal{X}_i \\times_{\\mathcal X, y} T') \\to T'$ is surjective.", "The isomorphism $f^*x \\simeq y$ induces an isomorphism", "$\\mathcal{X}_i \\times_{\\mathcal X, y} T' \\simeq T_i \\times_T T'$.", "Then the $T_i \\times_T T'$ cover $T'$ if and only if $f$ lands in", "$\\bigcup T_i$. Therefore we have a diagram", "$$", "\\xymatrix{", "T_i \\ar[r] \\ar[d] &", "\\bigcup T_i \\ar[r] \\ar[d] &", "T \\ar[d] \\\\", "\\mathcal{X}_i \\ar[r] &", "\\mathcal{X}' \\ar[r] &", "\\mathcal{X}", "}", "$$", "with both squares cartesian. By", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-open-fibred-category-is-algebraic}", "we conclude that $\\mathcal{X'} \\subset \\mathcal{X}$ is algebraic and an", "open substack. It is also clear from the cartesian squares above that the", "morphism $\\coprod_{i \\in I} \\mathcal{X}_i \\to \\mathcal{X}'$ which", "finishes the proof of the lemma." ], "refs": [ "algebraic-lemma-open-fibred-category-is-algebraic" ], "ref_ids": [ 8472 ] } ], "ref_ids": [] }, { "id": 8893, "type": "theorem", "label": "stacks-properties-lemma-quasi-compact-finite-subcover", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-quasi-compact-finite-subcover", "contents": [ "Let $\\mathcal X$ be an algebraic stack and $\\mathcal X' \\subset \\mathcal X$", "a quasi-compact open substack. Suppose that we have a collection of open", "substacks $\\mathcal{X}_i \\subset \\mathcal X$ indexed by $i \\in I$ such", "that $\\mathcal{X}' \\subset \\bigcup_{i \\in I} \\mathcal{X}_i$,", "where we define the union as in Lemma \\ref{lemma-union-open-substacks}.", "Then there exists a finite subset $I' \\subset I$ such that", "$\\mathcal{X}' \\subset \\bigcup_{i \\in I'} \\mathcal{X}_i$." ], "refs": [ "stacks-properties-lemma-union-open-substacks" ], "proofs": [ { "contents": [ "Since $\\mathcal X$ is algebraic, there exists a scheme $U$ with a surjective", "smooth morphism $U \\to \\mathcal X$. Let $U_i \\subset U$ be the open subscheme", "representing $\\mathcal{X}_i \\times_{\\mathcal X} U$ and $U' \\subset U$ the", "open subscheme representing $\\mathcal{X}' \\times_{\\mathcal X} U$. By", "hypothesis, $U'\\subset \\bigcup_{i\\in I} U_i$. From the proof of", "Lemma \\ref{lemma-quasi-compact-stack},", "there is a quasi-compact open $V \\subset U'$ such that $V \\to \\mathcal{X}'$", "is a surjective smooth morphism. Therefore there exists a finite subset", "$I' \\subset I$ such that $V \\subset \\bigcup_{i \\in I'}", "U_i$. We claim that $\\mathcal{X}' \\subset \\bigcup_{i \\in I'} \\mathcal{X}_i$.", "Take $x \\in \\Ob(\\mathcal{X}'_T)$ for", "$T \\in \\Ob((\\Sch/S)_{fppf})$.", "Since $\\mathcal{X}' \\to \\mathcal{X}$ is a monomorphism, we have cartesian", "squares", "$$", "\\xymatrix{", "V \\times_\\mathcal{X} T \\ar[r] \\ar[d] &", "T \\ar[d]^x \\ar@{=}[r] &", "T \\ar[d]^x \\\\", "V \\ar[r] &", "\\mathcal{X}' \\ar[r] &", "\\mathcal X", "}", "$$", "By base change, $V \\times_{\\mathcal X} T \\to T$ is surjective. Therefore", "$\\bigcup_{i \\in I'} U_i \\times_{\\mathcal X} T \\to T$ is also surjective.", "Let $T_i \\subset T$ be the open subscheme representing", "$\\mathcal{X}_i \\times_{\\mathcal X} T$.", "By a formal argument, we have a Cartesian square", "$$", "\\xymatrix{", "U_i \\times_{\\mathcal{X}_i} T_i \\ar[r] \\ar[d] &", "U \\times_{\\mathcal X} T \\ar[d] \\\\", "T_i \\ar[r] & T", "}", "$$", "where the vertical arrows are surjective by base change. Since", "$U_i \\times_{\\mathcal{X}_i} T_i \\simeq U_i \\times_{\\mathcal X} T$,", "we find that $\\bigcup_{i \\in I'} T_i = T$. Hence", "$x$ is an object of $(\\bigcup_{i\\in I'} \\mathcal{X}_i)_T$ by", "definition of the union. Observe that the inclusion", "$\\mathcal{X}' \\subset \\bigcup_{i \\in I'} \\mathcal{X}_i$", "is automatically an open substack." ], "refs": [ "stacks-properties-lemma-quasi-compact-stack" ], "ref_ids": [ 8873 ] } ], "ref_ids": [ 8892 ] }, { "id": 8894, "type": "theorem", "label": "stacks-properties-lemma-zariski-open-cover-stack-is-space", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-zariski-open-cover-stack-is-space", "contents": [ "Let $\\mathcal X$ be an algebraic stack.", "Let $\\mathcal{X}_i$, $i \\in I$ be a set of open substacks of $\\mathcal{X}$.", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{X} = \\bigcup_{i \\in I} \\mathcal{X}_i$, and", "\\item each $\\mathcal{X}_i$ is an algebraic space.", "\\end{enumerate}", "Then $\\mathcal{X}$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "Apply", "Stacks, Lemma \\ref{stacks-lemma-stack-in-setoids-descent}", "to the morphism $\\coprod_{i \\in I} \\mathcal{X}_i \\to \\mathcal{X}$", "and the morphism $\\text{id} : \\mathcal{X} \\to \\mathcal{X}$ to", "see that $\\mathcal{X}$ is a stack in setoids.", "Hence $\\mathcal{X}$ is an algebraic space, see", "Algebraic Stacks,", "Proposition \\ref{algebraic-proposition-algebraic-stack-no-automorphisms}." ], "refs": [ "stacks-lemma-stack-in-setoids-descent", "algebraic-proposition-algebraic-stack-no-automorphisms" ], "ref_ids": [ 8957, 8480 ] } ], "ref_ids": [] }, { "id": 8895, "type": "theorem", "label": "stacks-properties-lemma-zariski-open-cover-stack-is-scheme", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-zariski-open-cover-stack-is-scheme", "contents": [ "Let $\\mathcal X$ be an algebraic stack.", "Let $\\mathcal{X}_i$, $i \\in I$ be a set of open substacks of $\\mathcal{X}$.", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{X} = \\bigcup_{i \\in I} \\mathcal{X}_i$, and", "\\item each $\\mathcal{X}_i$ is a scheme", "\\end{enumerate}", "Then $\\mathcal{X}$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-zariski-open-cover-stack-is-space}", "we see that $\\mathcal{X}$ is an algebraic space. Since any algebraic", "space has a largest open subspace which is a scheme, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme}", "we see that $\\mathcal{X}$ is a scheme." ], "refs": [ "stacks-properties-lemma-zariski-open-cover-stack-is-space", "spaces-properties-lemma-subscheme" ], "ref_ids": [ 8894, 11848 ] } ], "ref_ids": [] }, { "id": 8896, "type": "theorem", "label": "stacks-properties-lemma-local-source", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-local-source", "contents": [ "Let $\\mathcal{P}, \\mathcal{Q}, \\mathcal{R}$ be properties of morphisms", "of algebraic spaces. Assume", "\\begin{enumerate}", "\\item $\\mathcal{P}, \\mathcal{Q}, \\mathcal{R}$ are fppf local on the target", "and stable under arbitrary base change,", "\\item $\\text{smooth} \\Rightarrow \\mathcal{R}$,", "\\item for any morphism $f : X \\to Y$ which has $\\mathcal{Q}$ there exists a", "largest open subspace $W(\\mathcal{P}, f) \\subset X$ such that", "$f|_{W(\\mathcal{P}, f)}$ has $\\mathcal{P}$, and", "\\item for any morphism $f : X \\to Y$ which has $\\mathcal{Q}$,", "and any morphism $Y' \\to Y$ which has $\\mathcal{R}$ we have", "$Y' \\times_Y W(\\mathcal{P}, f) = W(\\mathcal{P}, f')$, where", "$f' : X_{Y'} \\to Y'$ is the base change of $f$.", "\\end{enumerate}", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "representable by algebraic spaces. Assume $f$ has $\\mathcal{Q}$. Then", "\\begin{enumerate}", "\\item[(A)] there exists a largest open substack", "$\\mathcal{X}' \\subset \\mathcal{X}$ such that $f|_{\\mathcal{X}'}$ has", "$\\mathcal{P}$, and", "\\item[(B)] if $\\mathcal{Z} \\to \\mathcal{Y}$ is a morphism of algebraic", "stacks representable by algebraic spaces which has $\\mathcal{R}$", "then $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}'$ is the largest open", "substack of $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$ over which", "the base change $\\text{id}_\\mathcal{Z} \\times f$ has property $\\mathcal{P}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective smooth morphism $V \\to \\mathcal{Y}$.", "Set $U = V \\times_\\mathcal{Y} \\mathcal{X}$ and let $f' : U \\to V$ be the", "base change of $f$. The morphism of algebraic spaces $f' : U \\to V$ has", "property $\\mathcal{Q}$. Thus we obtain the open $W(\\mathcal{P}, f') \\subset U$", "by assumption (3). Note that", "$U \\times_\\mathcal{X} U =", "(V \\times_\\mathcal{Y} V) \\times_\\mathcal{Y} \\mathcal{X}$", "hence the morphism $f'' : U \\times_\\mathcal{X} U \\to V \\times_\\mathcal{Y} V$", "is the base change of $f$ via either projection", "$V \\times_\\mathcal{Y} V \\to V$. By our choice of $V$ these projections", "are smooth, hence have property $\\mathcal{R}$ by (2). Thus by (4) we see", "that the inverse images of $W(\\mathcal{P}, f')$ under the two projections", "$\\text{pr}_i : U \\times_\\mathcal{X} U \\to U$ agree. In other words,", "$W(\\mathcal{P}, f')$ is an $R$-invariant subspace of $U$ (where", "$R = U \\times_\\mathcal{X} U$). Let $\\mathcal{X}'$ be the open substack of", "$\\mathcal{X}$ corresponding to $W(\\mathcal{P}, f)$ via", "Lemma \\ref{lemma-immersion-into-presentation}.", "By construction $W(\\mathcal{P}, f') = \\mathcal{X}' \\times_\\mathcal{Y} V$", "hence $f|_{\\mathcal{X}'}$ has property $\\mathcal{P}$ by", "Lemma \\ref{lemma-check-property-covering}.", "Also, $\\mathcal{X}'$ is the largest open substack", "such that $f|_{\\mathcal{X}'}$ has $\\mathcal{P}$ as the same maximality holds", "for $W(\\mathcal{P}, f)$. This proves (A).", "\\medskip\\noindent", "Finally, if $\\mathcal{Z} \\to \\mathcal{Y}$ is a", "morphism of algebraic stacks representable by algebraic spaces which has", "$\\mathcal{R}$ then we set $T = V \\times_\\mathcal{Y} \\mathcal{Z}$ and", "we see that $T \\to V$ is a morphism of algebraic spaces having", "property $\\mathcal{R}$. Set $f'_T : T \\times_V U \\to T$ the base change", "of $f'$. By (4) again we see that $W(\\mathcal{P}, f'_T)$ is the", "inverse image of $W(\\mathcal{P}, f)$ in $T \\times_V U$. This implies", "(B); some details omitted." ], "refs": [ "stacks-properties-lemma-immersion-into-presentation", "stacks-properties-lemma-check-property-covering" ], "ref_ids": [ 8886, 8859 ] } ], "ref_ids": [] }, { "id": 8897, "type": "theorem", "label": "stacks-properties-lemma-reduced-closed-substack", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-reduced-closed-substack", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Let $T \\subset |\\mathcal{X}|$ be a closed subset.", "There exists a unique closed substack $\\mathcal{Z} \\subset \\mathcal{X}$", "with the following properties:", "(a) we have $|\\mathcal{Z}| = T$, and (b) $\\mathcal{Z}$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to \\mathcal{X}$ be a surjective smooth morphism, where $U$ is an", "algebraic space. Set $R = U \\times_\\mathcal{X} U$, so that there is a", "presentation $[U/R] \\to \\mathcal{X}$, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-stack-presentation}.", "As usual we denote $s, t : R \\to U$ the two smooth projection morphisms.", "By Lemma \\ref{lemma-points-presentation}", "we see that $T$ corresponds to a closed subset $T' \\subset |U|$ such", "that $|s|^{-1}(T') = |t|^{-1}(T')$.", "Let $Z \\subset U$ be the reduced induced algebraic space", "structure on $T'$, see", "Properties of Spaces,", "Definition \\ref{spaces-properties-definition-reduced-induced-space}.", "The fibre products", "$Z \\times_{U, t} R$ and $R \\times_{s, U} Z$ are closed subspaces", "of $R$", "(Spaces, Lemma \\ref{spaces-lemma-base-change-immersions}).", "The projections $Z \\times_{U, t} R \\to Z$ and", "$R \\times_{s, U} Z \\to Z$ are smooth by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-smooth}.", "Thus as $Z$ is reduced, it follows that", "$Z \\times_{U, t} R$ and $R \\times_{s, U} Z$ are reduced, see", "Remark \\ref{remark-list-properties-local-smooth-topology}.", "Since", "$$", "|Z \\times_{U, t} R| = |t|^{-1}(T') = |s|^{-1}(T') = R \\times_{s, U} Z", "$$", "we conclude from the uniqueness in", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-reduced-closed-subspace}", "that $Z \\times_{U, t} R = R \\times_{s, U} Z$.", "Hence $Z$ is an $R$-invariant closed subspace of $U$.", "By the correspondence of", "Lemma \\ref{lemma-substacks-presentation}", "we obtain a closed substack $\\mathcal{Z} \\subset \\mathcal{X}$", "with $Z = \\mathcal{Z} \\times_\\mathcal{X} U$. Then", "$[Z/R_Z] \\to \\mathcal{Z}$ is a presentation", "(Lemma \\ref{lemma-immersion-into-presentation}).", "Then $|\\mathcal{Z}| = |Z|/|R_Z| = |T'|/\\sim$ is the given", "closed subset $T$. We omit the proof of unicity." ], "refs": [ "algebraic-lemma-stack-presentation", "stacks-properties-lemma-points-presentation", "spaces-properties-definition-reduced-induced-space", "spaces-lemma-base-change-immersions", "spaces-morphisms-lemma-base-change-smooth", "stacks-properties-remark-list-properties-local-smooth-topology", "spaces-properties-lemma-reduced-closed-subspace", "stacks-properties-lemma-substacks-presentation", "stacks-properties-lemma-immersion-into-presentation" ], "ref_ids": [ 8474, 8866, 11932, 8161, 4887, 8929, 11846, 8889, 8886 ] } ], "ref_ids": [] }, { "id": 8898, "type": "theorem", "label": "stacks-properties-lemma-reduced-stack-determined-by-points", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-reduced-stack-determined-by-points", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "If $\\mathcal{X}' \\subset \\mathcal{X}$", "is a closed substack, $\\mathcal{X}$ is reduced and", "$|\\mathcal{X}'| = |\\mathcal{X}|$, then $\\mathcal{X}' = \\mathcal{X}$." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $[U/R] \\to \\mathcal{X}$ with $U$ a scheme.", "As $\\mathcal{X}$ is reduced, we see that $U$ is reduced (by definition", "of reduced algebraic stacks). By", "Lemma \\ref{lemma-substacks-presentation}", "$\\mathcal{X}'$ corresponds to an $R$-invariant closed subscheme $Z \\subset U$.", "But now $|Z| \\subset |U|$ is the inverse image of $|\\mathcal{X}'|$, and", "hence $|Z| = |U|$. Hence $Z$ is a closed subscheme of $U$ whose underlying", "sets of points agree. By", "Schemes, Lemma \\ref{schemes-lemma-map-into-reduction}", "the map $\\text{id}_U : U \\to U$ factors through $Z \\to U$, and hence", "$Z = U$, i.e., $\\mathcal{X}' = \\mathcal{X}$." ], "refs": [ "stacks-properties-lemma-substacks-presentation", "schemes-lemma-map-into-reduction" ], "ref_ids": [ 8889, 7682 ] } ], "ref_ids": [] }, { "id": 8899, "type": "theorem", "label": "stacks-properties-lemma-map-into-reduction", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-map-into-reduction", "contents": [ "Let $\\mathcal{X}$, $\\mathcal{Y}$ be algebraic stacks.", "Let $\\mathcal{Z} \\subset \\mathcal{X}$ be a closed substack", "Assume $\\mathcal{Y}$ is reduced.", "A morphism $f : \\mathcal{Y} \\to \\mathcal{X}$ factors through", "$\\mathcal{Z}$ if and only if", "$f(|\\mathcal{Y}|) \\subset |\\mathcal{Z}|$." ], "refs": [], "proofs": [ { "contents": [ "Assume $f(|\\mathcal{Y}|) \\subset |\\mathcal{Z}|$. Consider", "$\\mathcal{Y} \\times_\\mathcal{X} \\mathcal{Z} \\to \\mathcal{Y}$.", "There is an equivalence", "$\\mathcal{Y} \\times_\\mathcal{X} \\mathcal{Z} \\to \\mathcal{Y}'$", "where $\\mathcal{Y}'$ is a closed substack of $\\mathcal{Y}$, see", "Lemmas \\ref{lemma-base-change-immersion} and", "\\ref{lemma-substack-image}.", "Using", "Lemmas \\ref{lemma-points-cartesian},", "\\ref{lemma-monomorphism-injective-points}, and", "\\ref{lemma-immersion-monomorphism}", "we see that $|\\mathcal{Y}'| = |\\mathcal{Y}|$. Hence we have", "reduced the lemma to", "Lemma \\ref{lemma-reduced-stack-determined-by-points}." ], "refs": [ "stacks-properties-lemma-base-change-immersion", "stacks-properties-lemma-substack-image", "stacks-properties-lemma-points-cartesian", "stacks-properties-lemma-monomorphism-injective-points", "stacks-properties-lemma-immersion-monomorphism", "stacks-properties-lemma-reduced-stack-determined-by-points" ], "ref_ids": [ 8882, 8888, 8864, 8880, 8885, 8898 ] } ], "ref_ids": [] }, { "id": 8900, "type": "theorem", "label": "stacks-properties-lemma-flat-cover-by-field", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-flat-cover-by-field", "contents": [ "Let $\\mathcal{Z}$ be an algebraic stack. Let $k$ be a field and let", "$\\Spec(k) \\to \\mathcal{Z}$ be surjective and flat. Then any", "morphism $\\Spec(k') \\to \\mathcal{Z}$ where $k'$ is a field is", "surjective and flat." ], "refs": [], "proofs": [ { "contents": [ "Consider the fibre square", "$$", "\\xymatrix{", "T \\ar[d] \\ar[r] & \\Spec(k) \\ar[d] \\\\", "\\Spec(k') \\ar[r] & \\mathcal{Z}", "}", "$$", "Note that $T \\to \\Spec(k')$ is flat and surjective hence $T$", "is not empty. On the other hand $T \\to \\Spec(k)$ is flat as", "$k$ is a field. Hence $T \\to \\mathcal{Z}$ is flat and surjective.", "It follows from", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-permanence}", "(via the discussion in", "Section \\ref{section-properties-morphisms})", "that $\\Spec(k') \\to \\mathcal{Z}$ is flat. It is clear that it", "is surjective as by assumption $|\\mathcal{Z}|$ is a singleton." ], "refs": [ "spaces-morphisms-lemma-flat-permanence" ], "ref_ids": [ 4865 ] } ], "ref_ids": [] }, { "id": 8901, "type": "theorem", "label": "stacks-properties-lemma-unique-point", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-unique-point", "contents": [ "Let $\\mathcal{Z}$ be an algebraic stack. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{Z}$ is reduced and $|\\mathcal{Z}|$ is a singleton,", "\\item there exists a surjective flat morphism $\\Spec(k) \\to \\mathcal{Z}$", "where $k$ is a field, and", "\\item there exists a locally of finite type, surjective, flat morphism", "$\\Spec(k) \\to \\mathcal{Z}$ where $k$ is a field.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Let $W$ be a scheme and", "let $W \\to \\mathcal{Z}$ be a surjective smooth morphism. Then $W$ is", "a reduced scheme. Let $\\eta \\in W$ be a generic point of an irreducible", "component of $W$. Since $W$ is reduced we have", "$\\mathcal{O}_{W, \\eta} = \\kappa(\\eta)$. It follows that the canonical", "morphism $\\eta = \\Spec(\\kappa(\\eta)) \\to W$ is flat. We see that the", "composition $\\eta \\to \\mathcal{Z}$ is flat (see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-composition-flat}).", "It is also surjective as $|\\mathcal{Z}|$ is a singleton. In other words", "(2) holds.", "\\medskip\\noindent", "Assume (2). Let $W$ be a scheme and", "let $W \\to \\mathcal{Z}$ be a surjective smooth morphism. Choose a field", "$k$ and a surjective flat morphism $\\Spec(k) \\to \\mathcal{Z}$.", "Then $W \\times_\\mathcal{Z} \\Spec(k)$ is an algebraic space smooth", "over $k$, hence regular (see", "Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-smooth-regular})", "and in particular reduced. Since $W \\times_\\mathcal{Z} \\Spec(k) \\to W$", "is surjective and flat we conclude that $W$ is reduced", "(Descent on Spaces, Lemma \\ref{spaces-descent-lemma-descend-reduced}).", "In other words (1) holds.", "\\medskip\\noindent", "It is clear that (3) implies (2). Finally, assume (2). Pick a nonempty", "affine scheme $W$ and a smooth morphism $W \\to \\mathcal{Z}$. Pick a closed", "point $w \\in W$ and set $k = \\kappa(w)$. The composition", "$$", "\\Spec(k) \\xrightarrow{w} W \\longrightarrow \\mathcal{Z}", "$$", "is locally of finite type by", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-composition-finite-type} and", "\\ref{spaces-morphisms-lemma-smooth-locally-finite-type}.", "It is also flat and surjective by", "Lemma \\ref{lemma-flat-cover-by-field}.", "Hence (3) holds." ], "refs": [ "spaces-morphisms-lemma-composition-flat", "spaces-over-fields-lemma-smooth-regular", "spaces-descent-lemma-descend-reduced", "spaces-morphisms-lemma-composition-finite-type", "spaces-morphisms-lemma-smooth-locally-finite-type", "stacks-properties-lemma-flat-cover-by-field" ], "ref_ids": [ 4852, 12872, 9374, 4814, 4890, 8900 ] } ], "ref_ids": [] }, { "id": 8902, "type": "theorem", "label": "stacks-properties-lemma-unique-point-better", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-unique-point-better", "contents": [ "Let $\\mathcal{Z}$ be an algebraic stack. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{Z}$ is reduced, locally Noetherian, and $|\\mathcal{Z}|$", "is a singleton, and", "\\item there exists a locally finitely presented, surjective, flat morphism", "$\\Spec(k) \\to \\mathcal{Z}$ where $k$ is a field.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (2) holds. By", "Lemma \\ref{lemma-unique-point}", "we see that $\\mathcal{Z}$ is reduced and $|\\mathcal{Z}|$ is a singleton.", "Let $W$ be a scheme and let $W \\to \\mathcal{Z}$ be a surjective smooth", "morphism. Choose a field $k$ and a locally finitely presented, surjective,", "flat morphism $\\Spec(k) \\to \\mathcal{Z}$.", "Then $W \\times_\\mathcal{Z} \\Spec(k)$ is an algebraic space", "smooth over $k$, hence locally Noetherian (see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}).", "Since $W \\times_\\mathcal{Z} \\Spec(k) \\to W$", "is flat, surjective, and locally of finite presentation, we see", "that $\\{W \\times_\\mathcal{Z} \\Spec(k) \\to W\\}$ is an fppf covering", "and we conclude that $W$ is locally Noetherian", "(Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descend-locally-Noetherian}).", "In other words (1) holds.", "\\medskip\\noindent", "Assume (1). Pick a nonempty affine scheme $W$ and a smooth morphism", "$W \\to \\mathcal{Z}$. Pick a closed point $w \\in W$ and set", "$k = \\kappa(w)$. Because $W$ is locally Noetherian the morphism", "$w : \\Spec(k) \\to W$ is of finite presentation, see", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-finite-presentation}.", "Hence the composition", "$$", "\\Spec(k) \\xrightarrow{w} W \\longrightarrow \\mathcal{Z}", "$$", "is locally of finite presentation by", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-composition-finite-presentation} and", "\\ref{spaces-morphisms-lemma-smooth-locally-finite-presentation}.", "It is also flat and surjective by", "Lemma \\ref{lemma-flat-cover-by-field}.", "Hence (2) holds." ], "refs": [ "stacks-properties-lemma-unique-point", "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "spaces-descent-lemma-descend-locally-Noetherian", "morphisms-lemma-closed-immersion-finite-presentation", "spaces-morphisms-lemma-composition-finite-presentation", "spaces-morphisms-lemma-smooth-locally-finite-presentation", "stacks-properties-lemma-flat-cover-by-field" ], "ref_ids": [ 8901, 4817, 9375, 5243, 4839, 4889, 8900 ] } ], "ref_ids": [] }, { "id": 8903, "type": "theorem", "label": "stacks-properties-lemma-monomorphism-into-point", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-monomorphism-into-point", "contents": [ "Let $\\mathcal{Z}' \\to \\mathcal{Z}$ be a monomorphism of algebraic stacks.", "Assume there exists a field $k$ and a locally finitely presented, surjective,", "flat morphism $\\Spec(k) \\to \\mathcal{Z}$. Then either $\\mathcal{Z}'$", "is empty or $\\mathcal{Z}' \\to \\mathcal{Z}$ is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $\\mathcal{Z}'$ is nonempty. In this case the", "fibre product $T = \\mathcal{Z}' \\times_\\mathcal{Z} \\Spec(k)$", "is nonempty, see", "Lemma \\ref{lemma-points-cartesian}.", "Now $T$ is an algebraic space and the projection $T \\to \\Spec(k)$", "is a monomorphism. Hence $T = \\Spec(k)$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-monomorphism-toward-field}.", "We conclude that $\\Spec(k) \\to \\mathcal{Z}$ factors through", "$\\mathcal{Z}'$. Suppose the morphism $z : \\Spec(k) \\to \\mathcal{Z}$", "is given by the object $\\xi$ over $\\Spec(k)$. We have just seen that", "$\\xi$ is isomorphic to an object $\\xi'$ of $\\mathcal{Z}'$ over", "$\\Spec(k)$. Since $z$", "is surjective, flat, and locally of finite presentation we see that", "every object of $\\mathcal{Z}$ over any scheme is fppf locally isomorphic", "to a pullback of $\\xi$, hence also to a pullback of $\\xi'$. By descent of", "objects for stacks in groupoids this implies that", "$\\mathcal{Z}' \\to \\mathcal{Z}$ is essentially surjective (as well as", "fully faithful, see", "Lemma \\ref{lemma-monomorphism}).", "Hence we win." ], "refs": [ "stacks-properties-lemma-points-cartesian", "spaces-morphisms-lemma-monomorphism-toward-field", "stacks-properties-lemma-monomorphism" ], "ref_ids": [ 8864, 4757, 8879 ] } ], "ref_ids": [] }, { "id": 8904, "type": "theorem", "label": "stacks-properties-lemma-improve-unique-point", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-improve-unique-point", "contents": [ "Let $\\mathcal{Z}$ be an algebraic stack. Assume $\\mathcal{Z}$ satisfies", "the equivalent conditions of", "Lemma \\ref{lemma-unique-point}.", "Then there exists a unique strictly full subcategory", "$\\mathcal{Z}' \\subset \\mathcal{Z}$ such that", "$\\mathcal{Z}'$ is an algebraic stack which satisfies the equivalent", "conditions of", "Lemma \\ref{lemma-unique-point-better}.", "The inclusion morphism $\\mathcal{Z}' \\to \\mathcal{Z}$ is a monomorphism", "of algebraic stacks." ], "refs": [ "stacks-properties-lemma-unique-point", "stacks-properties-lemma-unique-point-better" ], "proofs": [ { "contents": [ "The last part is immediate from the first part and", "Lemma \\ref{lemma-monomorphism}.", "Pick a field $k$ and a morphism $\\Spec(k) \\to \\mathcal{Z}$", "which is surjective, flat, and locally of finite type.", "Set $U = \\Spec(k)$ and $R = U \\times_\\mathcal{Z} U$.", "The projections $s, t : R \\to U$ are locally of finite type.", "Since $U$ is the spectrum of a field, it follows that", "$s, t$ are flat and locally of finite presentation (by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation}).", "We see that $\\mathcal{Z}' = [U/R]$ is an algebraic stack by", "Criteria for Representability,", "Theorem \\ref{criteria-theorem-flat-groupoid-gives-algebraic-stack}.", "By", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack}", "we obtain a canonical morphism", "$$", "f : \\mathcal{Z}' \\longrightarrow \\mathcal{Z}", "$$", "which is fully faithful. Hence this morphism is representable by", "algebraic spaces, see", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}", "and a monomorphism, see", "Lemma \\ref{lemma-monomorphism}.", "By", "Criteria for Representability,", "Lemma \\ref{criteria-lemma-flat-quotient-flat-presentation}", "the morphism $U \\to \\mathcal{Z}'$ is surjective, flat, and locally of finite", "presentation. Hence $\\mathcal{Z}'$ is an algebraic stack which satisfies", "the equivalent conditions of", "Lemma \\ref{lemma-unique-point-better}.", "By", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-equivalent}", "we may replace $\\mathcal{Z}'$ by its essential image in $\\mathcal{Z}$.", "Hence we have proved all the assertions of the lemma except for the", "uniqueness of $\\mathcal{Z}' \\subset \\mathcal{Z}$. Suppose that", "$\\mathcal{Z}'' \\subset \\mathcal{Z}$ is a second such algebraic stack.", "Then the projections", "$$", "\\mathcal{Z}'", "\\longleftarrow", "\\mathcal{Z}' \\times_\\mathcal{Z} \\mathcal{Z}''", "\\longrightarrow", "\\mathcal{Z}''", "$$", "are monomorphisms. The algebraic stack in the middle is nonempty by", "Lemma \\ref{lemma-points-cartesian}.", "Hence the two projections are isomorphisms by", "Lemma \\ref{lemma-monomorphism-into-point}", "and we win." ], "refs": [ "stacks-properties-lemma-monomorphism", "spaces-morphisms-lemma-noetherian-finite-type-finite-presentation", "criteria-theorem-flat-groupoid-gives-algebraic-stack", "algebraic-lemma-map-space-into-stack", "algebraic-lemma-characterize-representable-by-algebraic-spaces", "stacks-properties-lemma-monomorphism", "criteria-lemma-flat-quotient-flat-presentation", "stacks-properties-lemma-unique-point-better", "algebraic-lemma-equivalent", "stacks-properties-lemma-points-cartesian", "stacks-properties-lemma-monomorphism-into-point" ], "ref_ids": [ 8879, 4844, 3094, 8473, 8469, 8879, 3137, 8902, 8462, 8864, 8903 ] } ], "ref_ids": [ 8901, 8902 ] }, { "id": 8905, "type": "theorem", "label": "stacks-properties-lemma-residual-gerbe", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-residual-gerbe", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$ be a point.", "The following are equivalent", "\\begin{enumerate}", "\\item there exists an algebraic stack $\\mathcal{Z}$ and a monomorphism", "$\\mathcal{Z} \\to \\mathcal{X}$ such that $|\\mathcal{Z}|$ is a singleton", "and such that the image of $|\\mathcal{Z}|$ in $|\\mathcal{X}|$ is $x$,", "\\item there exists a reduced algebraic stack $\\mathcal{Z}$ and a monomorphism", "$\\mathcal{Z} \\to \\mathcal{X}$ such that $|\\mathcal{Z}|$ is a singleton", "and such that the image of $|\\mathcal{Z}|$ in $|\\mathcal{X}|$ is $x$,", "\\item there exists an algebraic stack $\\mathcal{Z}$, a monomorphism", "$f : \\mathcal{Z} \\to \\mathcal{X}$, and a surjective flat morphism", "$z : \\Spec(k) \\to \\mathcal{Z}$ where $k$ is a field such that", "$x = f(z)$.", "\\end{enumerate}", "Moreover, if these conditions hold, then there exists a unique", "strictly full subcategory $\\mathcal{Z}_x \\subset \\mathcal{X}$", "such that $\\mathcal{Z}_x$ is a reduced, locally Noetherian algebraic", "stack and $|\\mathcal{Z}_x|$ is a singleton which maps to $x$", "via the map $|\\mathcal{Z}_x| \\to |\\mathcal{X}|$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{Z} \\to \\mathcal{X}$ is as in (1), then", "$\\mathcal{Z}_{red} \\to \\mathcal{X}$ is as in (2). (See", "Section \\ref{section-reduced}", "for the notion of the reduction of an algebraic stack.)", "Hence (1) implies (2).", "It is immediate that (2) implies (1).", "The equivalence of (2) and (3) is immediate from", "Lemma \\ref{lemma-unique-point}.", "\\medskip\\noindent", "At this point we've seen the equivalence of (1) -- (3).", "Pick a monomorphism $f : \\mathcal{Z} \\to \\mathcal{X}$ as in (2).", "Note that this implies that $f$ is fully faithful, see", "Lemma \\ref{lemma-monomorphism}.", "Denote $\\mathcal{Z}' \\subset \\mathcal{X}$ the essential image of the functor", "$f$. Then $f : \\mathcal{Z} \\to \\mathcal{Z}'$ is an equivalence and hence", "$\\mathcal{Z}'$ is an algebraic stack, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-equivalent}.", "Apply", "Lemma \\ref{lemma-improve-unique-point}", "to get a strictly full subcategory $\\mathcal{Z}_x \\subset \\mathcal{Z}'$", "as in the statement of the lemma.", "This proves all the statements of the lemma except for uniqueness.", "\\medskip\\noindent", "In order to prove the uniqueness suppose that", "$\\mathcal{Z}_x \\subset \\mathcal{X}$", "and", "$\\mathcal{Z}'_x \\subset \\mathcal{X}$", "are two strictly full subcategories as in the statement of the lemma.", "Then the projections", "$$", "\\mathcal{Z}'_x", "\\longleftarrow", "\\mathcal{Z}'_x \\times_\\mathcal{X} \\mathcal{Z}_x", "\\longrightarrow", "\\mathcal{Z}_x", "$$", "are monomorphisms. The algebraic stack in the middle is nonempty by", "Lemma \\ref{lemma-points-cartesian}.", "Hence the two projections are isomorphisms by", "Lemma \\ref{lemma-monomorphism-into-point}", "and we win." ], "refs": [ "stacks-properties-lemma-unique-point", "stacks-properties-lemma-monomorphism", "algebraic-lemma-equivalent", "stacks-properties-lemma-improve-unique-point", "stacks-properties-lemma-points-cartesian", "stacks-properties-lemma-monomorphism-into-point" ], "ref_ids": [ 8901, 8879, 8462, 8904, 8864, 8903 ] } ], "ref_ids": [] }, { "id": 8906, "type": "theorem", "label": "stacks-properties-lemma-residual-gerbe-regular", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-residual-gerbe-regular", "contents": [ "A reduced, locally Noetherian algebraic stack $\\mathcal{Z}$ such that", "$|\\mathcal{Z}|$ is a singleton is regular." ], "refs": [], "proofs": [ { "contents": [ "Let $W \\to \\mathcal{Z}$ be a surjective smooth morphism where $W$ is a scheme.", "Let $k$ be a field and let $\\Spec(k) \\to \\mathcal{Z}$ be surjective,", "flat, and locally of finite presentation (see", "Lemma \\ref{lemma-unique-point-better}).", "The algebraic space $T = W \\times_\\mathcal{Z} \\Spec(k)$ is", "smooth over $k$ in particular regular, see", "Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-smooth-regular}.", "Since $T \\to W$ is locally of finite presentation, flat, and surjective it", "follows that $W$ is regular, see", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-descend-regular}.", "By definition this means that $\\mathcal{Z}$ is regular." ], "refs": [ "stacks-properties-lemma-unique-point-better", "spaces-over-fields-lemma-smooth-regular", "spaces-descent-lemma-descend-regular" ], "ref_ids": [ 8902, 12872, 9376 ] } ], "ref_ids": [] }, { "id": 8907, "type": "theorem", "label": "stacks-properties-lemma-residual-gerbe-points", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-residual-gerbe-points", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$.", "Assume that the residual gerbe $\\mathcal{Z}_x$ of $\\mathcal{X}$ exists.", "Let $f : \\Spec(K) \\to \\mathcal{X}$ be a morphism where $K$ is a field", "in the equivalence class of $x$. Then $f$ factors through the inclusion", "morphism $\\mathcal{Z}_x \\to \\mathcal{X}$." ], "refs": [], "proofs": [ { "contents": [ "Choose a field $k$ and a surjective flat locally finite presentation", "morphism $\\Spec(k) \\to \\mathcal{Z}_x$. Set", "$T = \\Spec(K) \\times_\\mathcal{X} \\mathcal{Z}_x$. By", "Lemma \\ref{lemma-points-cartesian}", "we see that $T$ is nonempty. As $\\mathcal{Z}_x \\to \\mathcal{X}$", "is a monomorphism we see that $T \\to \\Spec(K)$ is a monomorphism.", "Hence by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-monomorphism-toward-field}", "we see that $T = \\Spec(K)$ which proves the lemma." ], "refs": [ "stacks-properties-lemma-points-cartesian", "spaces-morphisms-lemma-monomorphism-toward-field" ], "ref_ids": [ 8864, 4757 ] } ], "ref_ids": [] }, { "id": 8908, "type": "theorem", "label": "stacks-properties-lemma-residual-gerbe-unique", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-residual-gerbe-unique", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$.", "Let $\\mathcal{Z}$ be an algebraic stack satisfying the equivalent conditions of", "Lemma \\ref{lemma-unique-point-better}", "and let $\\mathcal{Z} \\to \\mathcal{X}$ be a monomorphism such that the image", "of $|\\mathcal{Z}| \\to |\\mathcal{X}|$ is $x$. Then the residual gerbe", "$\\mathcal{Z}_x$ of $\\mathcal{X}$ at $x$ exists and", "$\\mathcal{Z} \\to \\mathcal{X}$ factors as", "$\\mathcal{Z} \\to \\mathcal{Z}_x \\to \\mathcal{X}$ where the first arrow", "is an equivalence." ], "refs": [ "stacks-properties-lemma-unique-point-better" ], "proofs": [ { "contents": [ "Let $\\mathcal{Z}_x \\subset \\mathcal{X}$ be the full subcategory corresponding", "to the essential image of the functor $\\mathcal{Z} \\to \\mathcal{X}$.", "Then $\\mathcal{Z} \\to \\mathcal{Z}_x$ is an equivalence, hence", "$\\mathcal{Z}_x$ is an algebraic stack, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-equivalent}.", "Since $\\mathcal{Z}_x$ inherits all the properties of $\\mathcal{Z}$ from", "this equivalence it is clear from the uniqueness in", "Lemma \\ref{lemma-residual-gerbe}", "that $\\mathcal{Z}_x$ is the residual gerbe of $\\mathcal{X}$ at $x$." ], "refs": [ "algebraic-lemma-equivalent", "stacks-properties-lemma-residual-gerbe" ], "ref_ids": [ 8462, 8905 ] } ], "ref_ids": [ 8902 ] }, { "id": 8909, "type": "theorem", "label": "stacks-properties-lemma-residual-gerbe-functorial", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-residual-gerbe-functorial", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $x \\in |\\mathcal{X}|$ with image $y \\in |\\mathcal{Y}|$.", "If the residual gerbes $\\mathcal{Z}_x \\subset \\mathcal{X}$", "and $\\mathcal{Z}_y \\subset \\mathcal{Y}$ of $x$ and $y$ exist,", "then $f$ induces a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{X} \\ar[d]_f & \\mathcal{Z}_x \\ar[l] \\ar[d] \\\\", "\\mathcal{Y} & \\mathcal{Z}_y \\ar[l]", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose a field $k$ and a surjective, flat, locally finitely presented", "morphism $\\Spec(k) \\to \\mathcal{Z}_x$. The morphism", "$\\Spec(k) \\to \\mathcal{Y}$ factors through $\\mathcal{Z}_y$ by", "Lemma \\ref{lemma-residual-gerbe-points}.", "Thus $\\mathcal{Z}_x \\times_\\mathcal{Y} \\mathcal{Z}_y$", "is a nonempty substack of $\\mathcal{Z}_x$", "hence equal to $\\mathcal{Z}_x$ by Lemma \\ref{lemma-monomorphism-into-point}." ], "refs": [ "stacks-properties-lemma-residual-gerbe-points", "stacks-properties-lemma-monomorphism-into-point" ], "ref_ids": [ 8907, 8903 ] } ], "ref_ids": [] }, { "id": 8910, "type": "theorem", "label": "stacks-properties-lemma-residual-gerbe-isomorphic", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-residual-gerbe-isomorphic", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $x \\in |\\mathcal{X}|$ with image $y \\in |\\mathcal{Y}|$.", "Assume the residual gerbes $\\mathcal{Z}_x \\subset \\mathcal{X}$", "and $\\mathcal{Z}_y \\subset \\mathcal{Y}$ of $x$ and $y$ exist", "and that there exists a morphism $\\Spec(k) \\to \\mathcal{X}$", "in the equivalence class of $x$ such that", "$$", "\\Spec(k) \\times_\\mathcal{X} \\Spec(k)", "\\longrightarrow", "\\Spec(k) \\times_\\mathcal{Y} \\Spec(k)", "$$", "is an isomorphism. Then $\\mathcal{Z}_x \\to \\mathcal{Z}_y$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Let $k'/k$ be an extension of fields. Then", "$$", "\\Spec(k') \\times_\\mathcal{X} \\Spec(k')", "\\longrightarrow", "\\Spec(k') \\times_\\mathcal{Y} \\Spec(k')", "$$", "is the base change of the morphism in the lemma by the", "faithfully flat morphism $\\Spec(k' \\otimes k') \\to \\Spec(k \\otimes k)$.", "Thus the property described in the lemma is independent of the", "choice of the morphism $\\Spec(k) \\to \\mathcal{X}$ in the", "equivalence class of $x$. Thus we may assume that", "$\\Spec(k) \\to \\mathcal{Z}_x$ is surjective, flat, and", "locally of finite presentation. In this situation we have", "$$", "\\mathcal{Z}_x = [\\Spec(k)/R]", "$$", "with $R = \\Spec(k) \\times_\\mathcal{X} \\Spec(k)$. See", "proof of Lemma \\ref{lemma-improve-unique-point}.", "Since also $R = \\Spec(k) \\times_\\mathcal{Y} \\Spec(k)$", "we conclude that the morphism $\\mathcal{Z}_x \\to \\mathcal{Z}_y$", "of Lemma \\ref{lemma-residual-gerbe-functorial}", "is fully faithful by", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack}.", "We conclude for example by Lemma \\ref{lemma-residual-gerbe-unique}." ], "refs": [ "stacks-properties-lemma-improve-unique-point", "stacks-properties-lemma-residual-gerbe-functorial", "algebraic-lemma-map-space-into-stack", "stacks-properties-lemma-residual-gerbe-unique" ], "ref_ids": [ 8904, 8909, 8473, 8908 ] } ], "ref_ids": [] }, { "id": 8911, "type": "theorem", "label": "stacks-properties-lemma-dimension-at-point-well-defined", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-dimension-at-point-well-defined", "contents": [ "Let $\\mathcal{X}$ be a locally Noetherian algebraic stack over a scheme $S$.", "Let $x \\in |\\mathcal{X}|$ be a point of $\\mathcal{X}$.", "Let $[U/R] \\to \\mathcal{X}$ be a presentation", "(Algebraic Stacks, Definition \\ref{algebraic-definition-presentation})", "where $U$ is a scheme. Let $u \\in U$ be a point that maps to $x$.", "Let $e : U \\to R$ be the ``identity'' map and let $s : R \\to U$ be the", "``source'' map, which is a smooth morphism of algebraic spaces. Let $R_u$", "be the fiber of $s : R \\to U$ over $u$. The element", "$$", "\\dim_x(\\mathcal{X}) = \\dim_u(U) - \\dim_{e(u)}(R_u) \\in \\mathbf{Z} \\cup \\infty", "$$", "is independent of the choice of presentation and the point $u$ over $x$." ], "refs": [ "algebraic-definition-presentation" ], "proofs": [ { "contents": [ "Since $R \\to U$ is smooth, the scheme $R_u$ is smooth over $\\kappa(u)$", "and hence has finite dimension. On the other hand, the scheme $U$", "is locally Noetherian, but this does not guarantee that", "$\\dim_u(U)$ is finite. Thus the difference is an element of", "$\\mathbf{Z} \\cup \\{\\infty\\}$.", "\\medskip\\noindent", "Let $[U'/R'] \\to \\mathcal{X}$ and $u' \\in U'$ be a second presentation", "where $U'$ is a scheme and $u'$ maps to $x$. Consider the algebraic", "space $P = U \\times_\\mathcal{X} U'$. By", "Lemma \\ref{lemma-points-cartesian} there exists a $p \\in |P|$ mapping to", "$u$ and $u'$. Since $P \\to U$ and $P \\to U'$ are smooth we see that", "$\\dim_p(P) = \\dim_u(U) + \\dim_p(P_u)$ and", "$\\dim_p(P) = \\dim_{u'}(U') + \\dim_p(P_{u'})$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-smoothness-dimension-spaces}.", "Note that", "$$", "R'_{u'} = \\Spec(\\kappa(u')) \\times_\\mathcal{X} U'", "\\quad\\text{and}\\quad", "P_u = \\Spec(\\kappa(u)) \\times_\\mathcal{X} U'", "$$", "Let us represent $p \\in |P|$ by a morphism $\\Spec(\\Omega) \\to P$.", "Since $p$ maps to both $u$ and $u'$ it induces a $2$-morphism", "between the compositions", "$\\Spec(\\Omega) \\to \\Spec(\\kappa(u')) \\to \\mathcal{X}$ and", "$\\Spec(\\Omega) \\to \\Spec(\\kappa(u)) \\to \\mathcal{X}$", "which in turn defines an isomorphism", "$$", "\\Spec(\\Omega) \\times_{\\Spec(\\kappa(u'))} R'_{u'}", "\\cong", "\\Spec(\\Omega) \\times_{\\Spec(\\kappa(u))} P_u", "$$", "as algebraic spaces over $\\Spec(\\Omega)$ mapping the $\\Omega$-rational", "point $(1, e'(u'))$ to $(1, p)$ (some details omitted). We conclude that", "$$", "\\dim_{e'(u')}(R'_{u'}) = \\dim_p(P_u)", "$$", "by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}.", "By symmetry we have", "$\\dim_{e(u)}(R_u) = \\dim_p(P_{u'})$.", "Putting everything together we obtain the independence of choices." ], "refs": [ "stacks-properties-lemma-points-cartesian", "spaces-morphisms-lemma-smoothness-dimension-spaces", "spaces-morphisms-lemma-dimension-fibre-after-base-change" ], "ref_ids": [ 8864, 4894, 4872 ] } ], "ref_ids": [ 8488 ] }, { "id": 8912, "type": "theorem", "label": "stacks-properties-lemma-UR-quasi-compact-above-x", "categories": [ "stacks-properties" ], "title": "stacks-properties-lemma-UR-quasi-compact-above-x", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$", "be a point. The following are equivalent", "\\begin{enumerate}", "\\item some morphism $\\Spec(k) \\to \\mathcal{X}$ in the equivalence", "class of $x$ is quasi-compact, and", "\\item any morphism $\\Spec(k) \\to \\mathcal{X}$ in the equivalence", "class of $x$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\Spec(k) \\to \\mathcal{X}$ be in the equivalence", "class of $x$. Let $k'/k$ be a field extension.", "Then we have to show that $\\Spec(k) \\to \\mathcal{X}$ is", "quasi-compact if and only if $\\Spec(k') \\to \\mathcal{X}$", "is quasi-compact. This follows from", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-surjection-from-quasi-compact}", "and the principle of Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-representable-transformations-property-implication}." ], "refs": [ "spaces-morphisms-lemma-surjection-from-quasi-compact", "algebraic-lemma-representable-transformations-property-implication" ], "ref_ids": [ 4740, 8459 ] } ], "ref_ids": [] }, { "id": 8934, "type": "theorem", "label": "stacks-lemma-painful", "categories": [ "stacks" ], "title": "stacks-lemma-painful", "contents": [ "This actually does give a presheaf." ], "refs": [], "proofs": [ { "contents": [ "Let $g : V'/U \\to V/U$ be as above and similarly", "$g' : V''/U \\to V'/U$ be morphisms in $\\mathcal{C}/U$.", "So $f' = f \\circ g$ and $f'' = f' \\circ g' = f \\circ g \\circ g'$.", "Let $\\phi \\in \\Mor_{\\mathcal{S}_V}(f^\\ast x, f^\\ast y)$.", "Then we have", "\\begin{eqnarray*}", "& &", "(\\alpha_{g \\circ g', f})_y^{-1} \\circ", "(g \\circ g')^\\ast \\phi \\circ", "(\\alpha_{g \\circ g', f})_x", "\\\\", "& = &", "(\\alpha_{g \\circ g', f})_y^{-1} \\circ", "(\\alpha_{g', g})_{f^*y}^{-1} \\circ", "(g')^*g^\\ast \\phi \\circ", "(\\alpha_{g', g})_{f^*x} \\circ", "(\\alpha_{g \\circ g', f})_x", "\\\\", "& = &", "(\\alpha_{g', f'})_y^{-1} \\circ", "(g')^*(\\alpha_{g, f})_y^{-1} \\circ", "(g')^* g^\\ast \\phi \\circ", "(g')^*(\\alpha_{g, f})_x", "\\circ", "(\\alpha_{g', f'})_x", "\\\\", "& = &", "(\\alpha_{g', f'})_y^{-1} \\circ", "(g')^*\\Big(", "(\\alpha_{g, f})_y^{-1} \\circ", "g^\\ast \\phi \\circ", "(\\alpha_{g, f})_x", "\\Big) \\circ", "(\\alpha_{g', f'})_x", "\\end{eqnarray*}", "which is what we want, namely $\\phi|_{V''} = (\\phi|_{V'})|_{V''}$.", "The first equality holds because", "$\\alpha_{g', g}$ is a transformation of functors, and hence", "$$", "\\xymatrix{", "(g \\circ g')^*f^*x", "\\ar[rr]_{(g \\circ g')^\\ast \\phi}", "\\ar[d]_{(\\alpha_{g', g})_{f^*x}} & &", "(g \\circ g')^*f^*y", "\\ar[d]^{(\\alpha_{g', g})_{f^*y}} \\\\", "(g')^*g^*f^*x", "\\ar[rr]^{(g')^*g^\\ast \\phi} & &", "(g')^*g^*f^*y", "}", "$$", "commutes. The second equality holds because of property (d) of", "a pseudo functor since $f' = f \\circ g$ (see", "Categories, Definition \\ref{categories-definition-functor-into-2-category}).", "The last equality follows from the fact that $(g')^*$ is a functor." ], "refs": [ "categories-definition-functor-into-2-category" ], "ref_ids": [ 12381 ] } ], "ref_ids": [] }, { "id": 8935, "type": "theorem", "label": "stacks-lemma-presheaf-mor-map-fibred-categories", "categories": [ "stacks" ], "title": "stacks-lemma-presheaf-mor-map-fibred-categories", "contents": [ "Let $F : \\mathcal{S}_1 \\to \\mathcal{S}_2$ be a $1$-morphism of fibred", "categories over the category $\\mathcal{C}$. Let $U \\in \\Ob(\\mathcal{C})$", "and $x, y\\in \\Ob((\\mathcal{S}_1)_U)$. Then $F$ defines a canonical", "morphism of presheaves", "$$", "\\mathit{Mor}_{\\mathcal{S}_1}(x, y)", "\\longrightarrow", "\\mathit{Mor}_{\\mathcal{S}_2}(F(x), F(y))", "$$", "on $\\mathcal{C}/U$." ], "refs": [], "proofs": [ { "contents": [ "By", "Categories, Definition \\ref{categories-definition-fibred-categories-over-C}", "the functor $F$ maps strongly cartesian morphisms to strongly cartesian", "morphisms. Hence if $f : V \\to U$ is a morphism in $\\mathcal{C}$, then", "there are canonical isomorphisms $\\alpha_V : f^*F(x) \\to F(f^*x)$,", "$\\beta_V : f^*F(y) \\to F(f^*y)$ such that $f^*F(x) \\to F(f^*x) \\to F(x)$", "is the canonical morphism $f^*F(x) \\to F(x)$, and similarly for $\\beta_V$.", "Thus we may define", "$$", "\\xymatrix{", "\\mathit{Mor}_{\\mathcal{S}_1}(x, y)(f : V \\to U) \\ar@{=}[r] &", "\\Mor_{\\mathcal{S}_{1, V}}(f^\\ast x, f^\\ast y) \\ar[d] \\\\", "\\mathit{Mor}_{\\mathcal{S}_2}(F(x), F(y))(f : V \\to U) \\ar@{=}[r] &", "\\Mor_{\\mathcal{S}_{2, V}}(f^\\ast F(x), f^\\ast F(y))", "}", "$$", "by $\\phi \\mapsto \\beta_V^{-1} \\circ F(\\phi) \\circ \\alpha_V$.", "We omit the verification that this is compatible with the restriction", "mappings." ], "refs": [ "categories-definition-fibred-categories-over-C" ], "ref_ids": [ 12390 ] } ], "ref_ids": [] }, { "id": 8936, "type": "theorem", "label": "stacks-lemma-isom-as-2-fibre-product", "categories": [ "stacks" ], "title": "stacks-lemma-isom-as-2-fibre-product", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category,", "see Categories, Section \\ref{categories-section-fibred-categories}.", "Let $U \\in \\Ob(\\mathcal{C})$ and let $x, y \\in \\Ob(\\mathcal{S}_U)$.", "Denote $x, y : \\mathcal{C}/U \\to \\mathcal{S}$ also the corresponding", "$1$-morphisms, see", "Categories, Lemma \\ref{categories-lemma-yoneda-2category}.", "Then", "\\begin{enumerate}", "\\item the $2$-fibre product", "$\\mathcal{S} \\times_{\\mathcal{S} \\times \\mathcal{S}, (x, y)} \\mathcal{C}/U$", "is fibred in setoids over $\\mathcal{C}/U$, and", "\\item $\\mathit{Isom}(x, y)$ is the presheaf of sets corresponding", "to this category fibred in setoids, see", "Categories, Lemma \\ref{categories-lemma-2-category-fibred-setoids}.", "\\end{enumerate}" ], "refs": [ "categories-lemma-yoneda-2category", "categories-lemma-2-category-fibred-setoids" ], "proofs": [ { "contents": [ "Omitted. Hint: Objects of the $2$-fibre product are", "$(a : V \\to U, z, a : V \\to U, (\\alpha, \\beta))$ where", "$\\alpha : z \\to a^*x$ and $\\beta : z \\to a^*y$ are isomorphisms", "in $\\mathcal{S}_V$. Thus the relationship with $\\mathit{Isom}(x, y)$", "comes by assigning to such an object the isomorphism", "$\\beta \\circ \\alpha^{-1}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12318, 12313 ] }, { "id": 8937, "type": "theorem", "label": "stacks-lemma-pullback", "categories": [ "stacks" ], "title": "stacks-lemma-pullback", "contents": [ "(Pullback of descent data.)", "Let $\\mathcal{C}$ be a category.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category.", "Make a choice pullbacks as in Categories,", "Definition \\ref{categories-definition-pullback-functor-fibred-category}.", "Let $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$, and", "$\\mathcal{V} = \\{V_j \\to V\\}_{j \\in J}$", "be a families of morphisms of $\\mathcal{C}$ with fixed target.", "Assume all the fibre products", "$U_i \\times_U U_{i'}$, $U_i \\times_U U_{i'} \\times_U U_{i''}$,", "$V_j \\times_V V_{j'}$, and $V_j \\times_V V_{j'} \\times_V V_{j''}$ exist.", "Let $\\alpha : I \\to J$, $h : U \\to V$ and", "$g_i : U_i \\to V_{\\alpha(i)}$ be a morphism of families", "of maps with fixed target, see", "Sites, Definition \\ref{sites-definition-morphism-coverings}.", "\\begin{enumerate}", "\\item Let $(Y_j, \\varphi_{jj'})$ be a descent datum relative to the", "family $\\{V_j \\to V\\}$. The system", "$$", "\\left(", "g_i^*Y_{\\alpha(i)},", "(g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')}", "\\right)", "$$", "is a descent datum relative to $\\mathcal{U}$.", "\\item This construction defines a functor between descent data relative", "to $\\mathcal{V}$ and descent data relative to $\\mathcal{U}$.", "\\item Given a second $\\alpha' : I \\to J$, $h' : U \\to V$ and", "$g'_i : U_i \\to V_{\\alpha'(i)}$ morphism of families", "of maps with fixed target, then if $h = h'$ the two resulting functors", "between descent data are canonically isomorphic.", "\\end{enumerate}" ], "refs": [ "categories-definition-pullback-functor-fibred-category", "sites-definition-morphism-coverings" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12389, 8656 ] }, { "id": 8938, "type": "theorem", "label": "stacks-lemma-trivial-cocycle", "categories": [ "stacks" ], "title": "stacks-lemma-trivial-cocycle", "contents": [ "In the situation of", "Definition \\ref{definition-effective-descent-datum} part (2) the maps", "$can_{ij} : \\text{pr}_0^*f_i^*X \\to \\text{pr}_1^*f_j^*X$ are equal to", "$(\\alpha_{\\text{pr}_1, f_j})_X \\circ (\\alpha_{\\text{pr}_0, f_i})_X^{-1}$", "where $\\alpha_{\\cdot, \\cdot}$ is as in", "Categories, Lemma \\ref{categories-lemma-fibred}", "and where we", "use the equality $f_i \\circ \\text{pr}_0 = f_j \\circ \\text{pr}_1$", "as maps $U_i \\times_U U_j \\to U$." ], "refs": [ "stacks-definition-effective-descent-datum", "categories-lemma-fibred" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8995, 12285 ] }, { "id": 8939, "type": "theorem", "label": "stacks-lemma-compare-descent-condition", "categories": [ "stacks" ], "title": "stacks-lemma-compare-descent-condition", "contents": [ "Let $\\mathcal{C}$ be a category. Let", "$\\mathcal{V} = \\{V_j \\to U\\}_{j \\in J} \\to", "\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$", "be a morphism of families of maps with fixed target of", "$\\mathcal{C}$ given by $\\text{id} : U \\to U$,", "$\\alpha : J \\to I$ and $f_j : V_j \\to U_{\\alpha(j)}$. Let", "$p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category. If", "\\begin{enumerate}", "\\item for $0 \\leq p \\leq 3$ and $0 \\leq q \\leq 3$ with $p + q \\geq 2$", "and $i_1, \\ldots, i_p \\in I$ and $j_1, \\ldots, j_q \\in J$", "the fibre products $U_{i_1} \\times_U \\ldots \\times_U U_{i_p} \\times_U", "V_{j_1} \\times_U \\ldots \\times_U V_{j_q}$ exist,", "\\item the functor $\\mathcal{S}_U \\to DD(\\mathcal{V})$", "is an equivalence,", "\\item for every $i \\in I$ the functor", "$\\mathcal{S}_{U_i} \\to DD(\\mathcal{V}_i)$", "is fully faithful, and", "\\item for every $i, i' \\in I$ the functor", "$\\mathcal{S}_{U_i \\times_U U_{i'}} \\to DD(\\mathcal{V}_{ii'})$", "is faithful.", "\\end{enumerate}", "Here $\\mathcal{V}_i = \\{U_i \\times_U V_j \\to U_i\\}_{j \\in J}$ and", "$\\mathcal{V}_{ii'} =", "\\{U_i \\times_U U_{i'} \\times_U V_j \\to U_i \\times_U U_{i'}\\}_{j \\in J}$.", "Then $\\mathcal{S}_U \\to DD(\\mathcal{U})$ is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Condition (1) guarantees we have enough fibre products so that", "the statement makes sense.", "We will show that the functor $\\mathcal{S}_U \\to DD(\\mathcal{U})$ is", "essentially surjective. Suppose given a descent datum", "$(X_i, \\varphi_{ii'})$ relative to $\\mathcal{U}$.", "By Lemma \\ref{lemma-pullback} we can pull this back to a descent datum", "$(X_j, \\varphi_{jj'})$ for $\\mathcal{V}$.", "By assumption (2) this descent datum is effective, hence we", "get an object $X$ of $\\mathcal{S}_U$ such that the", "pullback of the trivial descent datum $(X, \\text{id}_X)$", "by the morphism $\\mathcal{V} \\to \\{U \\to U\\}$ is isomorphic to", "$(X_j, \\varphi_{jj'})$. Next, observe that we have a diagram", "$$", "\\xymatrix{", "\\mathcal{V}_i \\ar[r] \\ar[d] &", "\\mathcal{V} \\ar[r] &", "\\mathcal{U} \\ar[d] \\\\", "\\{U_i \\to U_i\\} \\ar[rr] \\ar[rru] & &", "\\{U \\to U\\}", "}", "$$", "of morphisms of families of maps with fixed target of $\\mathcal{C}$.", "This diagram does not commute, but by Lemma \\ref{lemma-pullback}", "the pullback functors on descent data one gets are canonically", "isomorphic. Hence $(X, \\text{id}_X)$ and $(X_i, \\text{id}_{X_i})$", "pull back to isomorphic objects in $DD(\\mathcal{V}_i)$.", "Hence by assumption (3) we obtain an isomorphism", "$(U_i \\to U)^*X \\to X_i$ in the category $\\mathcal{S}_{U_i}$.", "We omit the verification that these arrows are compatible", "with the morphisms $\\varphi_{ii'}$; hint: use the faithfulness", "of the functors in condition (4).", "We also omit the verification that the functor", "$\\mathcal{S}_U \\to DD(\\mathcal{U})$ is", "fully faithful." ], "refs": [ "stacks-lemma-pullback", "stacks-lemma-pullback" ], "ref_ids": [ 8937, 8937 ] } ], "ref_ids": [] }, { "id": 8940, "type": "theorem", "label": "stacks-lemma-stack-equivalences", "categories": [ "stacks" ], "title": "stacks-lemma-stack-equivalences", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category", "over $\\mathcal{C}$. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{S}$ is a stack over $\\mathcal{C}$, and", "\\item for any covering $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$", "of the site $\\mathcal{C}$ the functor", "$$", "\\mathcal{S}_U \\longrightarrow DD(\\mathcal{U})", "$$", "which associates to an", "object its canonical descent datum is an equivalence.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8941, "type": "theorem", "label": "stacks-lemma-substack", "categories": [ "stacks" ], "title": "stacks-lemma-substack", "contents": [ "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a stack over the site $\\mathcal{C}$.", "Let $\\mathcal{S}'$ be a subcategory of $\\mathcal{S}$.", "Assume", "\\begin{enumerate}", "\\item if $\\varphi : y \\to x$ is a strongly cartesian", "morphism of $\\mathcal{S}$ and", "$x$ is an object of $\\mathcal{S}'$, then $y$ is isomorphic to an", "object of $\\mathcal{S}'$,", "\\item $\\mathcal{S}'$ is a full subcategory of $\\mathcal{S}$, and", "\\item if $\\{f_i : U_i \\to U\\}$ is a covering of $\\mathcal{C}$,", "and $x$ an object of $\\mathcal{S}$ over $U$ such that $f_i^*x$", "is isomorphic to an object of $\\mathcal{S}'$ for each $i$,", "then $x$ is isomorphic to an object of $\\mathcal{S}'$.", "\\end{enumerate}", "Then $\\mathcal{S}' \\to \\mathcal{C}$ is a stack." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints:", "The first condition guarantees that $\\mathcal{S}'$ is a fibred category.", "The second condition guarantees that the $\\mathit{Isom}$-presheaves", "of $\\mathcal{S}'$ are sheaves (as they are identical to their counter parts", "in $\\mathcal{S}$). The third condition guarantees that the descent condition", "holds in $\\mathcal{S}'$ as we can first descend in $\\mathcal{S}$ and", "then (3) implies the resulting object is isomorphic to an object of", "$\\mathcal{S}'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8942, "type": "theorem", "label": "stacks-lemma-stack-equivalent", "categories": [ "stacks" ], "title": "stacks-lemma-stack-equivalent", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{S}_1$, $\\mathcal{S}_2$ be categories over $\\mathcal{C}$.", "Suppose that $\\mathcal{S}_1$ and $\\mathcal{S}_2$ are equivalent", "as categories over $\\mathcal{C}$.", "Then $\\mathcal{S}_1$ is a stack over $\\mathcal{C}$ if and only if", "$\\mathcal{S}_2$ is a stack over $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Let $F : \\mathcal{S}_1 \\to \\mathcal{S}_2$,", "$G : \\mathcal{S}_2 \\to \\mathcal{S}_1$ be functors over $\\mathcal{C}$, and let", "$i : F \\circ G \\to \\text{id}_{\\mathcal{S}_2}$,", "$j : G \\circ F \\to \\text{id}_{\\mathcal{S}_1}$ be isomorphisms of", "functors over $\\mathcal{C}$. By", "Categories, Lemma \\ref{categories-lemma-fibred-equivalent}", "we see that $\\mathcal{S}_1$ is fibred if and only if $\\mathcal{S}_2$", "is fibred over $\\mathcal{C}$. Hence we may assume that both", "$\\mathcal{S}_1$ and $\\mathcal{S}_2$ are fibred. Moreover, the proof of", "Categories, Lemma \\ref{categories-lemma-fibred-equivalent}", "shows that $F$ and $G$ map strongly cartesian morphisms to strongly", "cartesian morphisms, i.e., $F$ and $G$ are $1$-morphisms of fibred", "categories over $\\mathcal{C}$. This means that given", "$U \\in \\Ob(\\mathcal{C})$, and $x, y \\in \\mathcal{S}_{1, U}$ then", "the presheaves", "$$", "\\mathit{Mor}_{\\mathcal{S}_1}(x, y),", "\\mathit{Mor}_{\\mathcal{S}_1}(F(x), F(y)) :", "(\\mathcal{C}/U)^{opp} \\longrightarrow \\textit{Sets}.", "$$", "are identified, see", "Lemma \\ref{lemma-presheaf-mor-map-fibred-categories}. Hence", "the first is a sheaf if and only if the second is a sheaf.", "Finally, we have to show that if every descent datum in $\\mathcal{S}_1$", "is effective, then so is every descent datum in $\\mathcal{S}_2$.", "To do this, let $(X_i, \\varphi_{ii'})$ be a descent datum", "in $\\mathcal{S}_2$ relative the covering $\\{U_i \\to U\\}$ of the site", "$\\mathcal{C}$. Then $(G(X_i), G(\\varphi_{ii'}))$ is a descent datum", "in $\\mathcal{S}_1$ relative the covering $\\{U_i \\to U\\}$.", "Let $X$ be an object of $\\mathcal{S}_{1, U}$ such that the", "descent datum $(f_i^*X, can)$ is isomorphic to", "$(G(X_i), G(\\varphi_{ii'}))$. Then $F(X)$ is an object of $\\mathcal{S}_{2, U}$", "such that the descent datum $(f_i^*F(X), can)$ is isomorphic to", "$(F(G(X_i)), F(G(\\varphi_{ii'})))$ which in turn is isomorphic to", "the original descent datum $(X_i, \\varphi_{ii'})$ using $i$." ], "refs": [ "categories-lemma-fibred-equivalent", "categories-lemma-fibred-equivalent", "stacks-lemma-presheaf-mor-map-fibred-categories" ], "ref_ids": [ 12286, 12286, 8935 ] } ], "ref_ids": [] }, { "id": 8943, "type": "theorem", "label": "stacks-lemma-2-product-stacks", "categories": [ "stacks" ], "title": "stacks-lemma-2-product-stacks", "contents": [ "Let $\\mathcal{C}$ be a site.", "The $(2, 1)$-category of stacks over $\\mathcal{C}$", "has 2-fibre products, and they are described as in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}." ], "refs": [ "categories-lemma-2-product-categories-over-C" ], "proofs": [ { "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{S}$ and", "$g : \\mathcal{Y} \\to \\mathcal{S}$ be", "$1$-morphisms of stacks over $\\mathcal{C}$", "as defined above. The category", "$\\mathcal{X} \\times_\\mathcal{S} \\mathcal{Y}$", "described in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C} is a", "fibred category according to", "Categories, Lemma \\ref{categories-lemma-2-product-fibred-categories-over-C}.", "(This is where we use that $f$ and $g$ preserve strongly cartesian", "morphisms.) It remains to show that the morphism presheaves are sheaves", "and that descent relative to coverings of $\\mathcal{C}$ is effective.", "\\medskip\\noindent", "Recall that an object of $\\mathcal{X} \\times_\\mathcal{S} \\mathcal{Y}$", "is given by a quadruple $(U, x, y, \\phi)$.", "It lies over the object", "$U$ of $\\mathcal{C}$. Next, let $(U, x', y', \\phi')$ be second", "object lying over $U$.", "Recall that $\\phi : f(x) \\to g(y)$, and $\\phi' : f(x') \\to g(y')$", "are isomorphisms in the category $\\mathcal{S}_U$. Let us", "use these isomorphisms to identify $z = f(x) = g(y)$ and", "$z' = f(x') = g(y')$. With this identifications", "it is clear that", "$$", "\\mathit{Mor}((U, x, y, \\phi), (U, x', y', \\phi'))", "=", "\\mathit{Mor}(x, x')", "\\times_{\\mathit{Mor}(z, z')}", "\\mathit{Mor}(y, y')", "$$", "as presheaves. However, as the fibred product in the category of", "presheaves preserves sheaves (Sites, Lemma \\ref{sites-lemma-limit-sheaf})", "we see that this is a sheaf.", "\\medskip\\noindent", "Let $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ be a covering of the site", "$\\mathcal{C}$. Let $(X_i, \\chi_{ij})$ be a descent datum", "in $\\mathcal{X} \\times_\\mathcal{S} \\mathcal{Y}$ relative to $\\mathcal{U}$.", "Write $X_i = (U_i, x_i, y_i, \\phi_i)$ as above. Write", "$\\chi_{ij} = (\\varphi_{ij}, \\psi_{ij})$ as in the definition of", "the category $\\mathcal{X} \\times_\\mathcal{S} \\mathcal{Y}$ (see", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}).", "It is clear that $(x_i, \\varphi_{ij})$ is a descent datum in", "$\\mathcal{X}$ and that $(y_i, \\psi_{ij})$ is a descent datum in", "$\\mathcal{Y}$. Since $\\mathcal{X}$ and $\\mathcal{Y}$ are stacks these", "descent data are effective. Thus we get", "$x \\in \\Ob(\\mathcal{X}_U)$, and $y \\in \\Ob(\\mathcal{Y}_U)$", "with $x_i = x|_{U_i}$, and $y_i = y|_{U_i}$ compatibly with descent data.", "Set $z = f(x)$ and $z' = g(y)$ which are both objects of $\\mathcal{S}_U$.", "The morphisms $\\phi_i$ are elements of", "$\\mathit{Isom}(z, z')(U_i)$ with the property that", "$\\phi_i|_{U_i \\times_U U_j} = \\phi_j|_{U_i \\times_U U_j}$.", "Hence by the sheaf property of $\\mathit{Isom}(z, z')$", "we obtain an isomorphism $\\phi : z = f(x) \\to z' = g(y)$.", "We omit the verification that the canonical descent datum associated to", "the object $(U, x, y, \\phi)$ of", "$(\\mathcal{X} \\times_\\mathcal{S} \\mathcal{Y})_U$ is isomorphic", "to the descent datum we started with." ], "refs": [ "categories-lemma-2-product-categories-over-C", "categories-lemma-2-product-fibred-categories-over-C", "sites-lemma-limit-sheaf", "categories-lemma-2-product-categories-over-C" ], "ref_ids": [ 12280, 12287, 8508, 12280 ] } ], "ref_ids": [ 12280 ] }, { "id": 8944, "type": "theorem", "label": "stacks-lemma-characterize-ff", "categories": [ "stacks" ], "title": "stacks-lemma-characterize-ff", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{S}_1$, $\\mathcal{S}_2$ be stacks over $\\mathcal{C}$.", "Let $F : \\mathcal{S}_1 \\to \\mathcal{S}_2$ be a $1$-morphism.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $F$ is fully faithful,", "\\item for every $U \\in \\Ob(\\mathcal{C})$ and for every", "$x, y \\in \\Ob(\\mathcal{S}_{1, U})$ the map", "$$", "F :", "\\mathit{Mor}_{\\mathcal{S}_1}(x, y)", "\\longrightarrow", "\\mathit{Mor}_{\\mathcal{S}_2}(F(x), F(y))", "$$", "is an isomorphism of sheaves on $\\mathcal{C}/U$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). For $U, x, y$ as in (2) the displayed map $F$ evaluates to the map", "$F : \\Mor_{\\mathcal{S}_{1, V}}(x|_V, y|_V) \\to", "\\Mor_{\\mathcal{S}_{2, V}}(F(x|_V), F(y|_V))$", "on an object $V$ of $\\mathcal{C}$ lying over $U$.", "Now, since $F$ is fully faithful, the corresponding map", "$\\Mor_{\\mathcal{S}_1}(x|_V, y|_V) \\to \\Mor_{\\mathcal{S}_2}(F(x|_V), F(y|_V))$", "is a bijection. Morphisms in the fibre category $\\mathcal{S}_{1, V}$ are", "exactly those morphisms between $x|_V$ and $y|_V$ in $\\mathcal{S}_1$ lying", "over $\\text{id}_V$. Similarly, morphisms in the fibre category", "$\\mathcal{S}_{2, V}$ are exactly those morphisms between $F(x|_V)$ and", "$F(y|_V)$ in $\\mathcal{S}_2$ lying over $\\text{id}_V$. Thus we find that $F$", "induces a bijection between these also. Hence (2) holds.", "\\medskip\\noindent", "Assume (2). Suppose given objects $U$, $V$ of $\\mathcal{C}$ and", "$x \\in \\Ob(\\mathcal{S}_{1, U})$ and", "$y \\in \\Ob(\\mathcal{S}_{1, V})$. To show that $F$ is fully faithful,", "it suffices to prove it induces a bijection on", "morphisms lying over a fixed $f : U \\to V$. Choose a strongly Cartesian", "$f^*y \\to y$ in $\\mathcal{S}_1$ lying above $f$. This results in a", "bijection between the set of morphisms $x \\to y$ in $\\mathcal{S}_1$ lying", "over $f$ and $\\Mor_{\\mathcal{S}_{1, U}}(x, f^*y)$. Since $F$ preserves", "strongly Cartesian morphisms as a $1$-morphism in the $2$-category", "of stacks over $\\mathcal{C}$, we also get a bijection", "between the set of morphisms $F(x) \\to F(y)$ in $\\mathcal{S}_2$ lying", "over $f$ and $\\Mor_{\\mathcal{S}_{2, U}}(F(x), F(f^*y))$.", "Since $F$ induces a bijection", "$\\Mor_{\\mathcal{S}_{1, U}}(x, f^*y) \\to", "\\Mor_{\\mathcal{S}_{2, U}}(F(x), F(f^*y))$", "we conclude (1) holds." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8945, "type": "theorem", "label": "stacks-lemma-characterize-essentially-surjective-when-ff", "categories": [ "stacks" ], "title": "stacks-lemma-characterize-essentially-surjective-when-ff", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{S}_1$, $\\mathcal{S}_2$ be stacks over $\\mathcal{C}$.", "Let $F : \\mathcal{S}_1 \\to \\mathcal{S}_2$ be a $1$-morphism which is", "fully faithful. Then the following are equivalent", "\\begin{enumerate}", "\\item $F$ is an equivalence,", "\\item for every $U \\in \\Ob(\\mathcal{C})$ and for every", "$x \\in \\Ob(\\mathcal{S}_{2, U})$ there exists a covering", "$\\{f_i : U_i \\to U\\}$ such that $f_i^*x$ is in the essential image", "of the functor $F : \\mathcal{S}_{1, U_i} \\to \\mathcal{S}_{2, U_i}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) is immediate.", "To see that (2) implies (1) we have to show that every", "$x$ as in (2) is in the essential image of the functor $F$.", "To do this choose a covering as in (2),", "$x_i \\in \\Ob(\\mathcal{S}_{1, U_i})$, and", "isomorphisms $\\varphi_i : F(x_i) \\to f_i^*x$. Then we get a descent", "datum for $\\mathcal{S}_1$ relative to $\\{f_i : U_i \\to U\\}$", "by taking", "$$", "\\varphi_{ij} :", "x_i|_{U_i \\times_U U_j}", "\\longrightarrow", "x_j|_{U_i \\times_U U_j}", "$$", "the arrow such that $F(\\varphi_{ij}) = \\varphi_j^{-1} \\circ \\varphi_i$.", "This descent datum is effective by the axioms of a stack, and hence", "we obtain an object $x_1$ of $\\mathcal{S}_1$ over $U$. We omit the", "verification that $F(x_1)$ is isomorphic to $x$ over $U$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8946, "type": "theorem", "label": "stacks-lemma-stack-in-groupoids-stack", "categories": [ "stacks" ], "title": "stacks-lemma-stack-in-groupoids-stack", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{S}$ is a stack in groupoids over $\\mathcal{C}$,", "\\item $\\mathcal{S}$ is a stack over $\\mathcal{C}$ and all", "fibre categories are groupoids, and", "\\item $\\mathcal{S}$ is fibred in groupoids over $\\mathcal{C}$", "and is a stack over $\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted, but see Categories, Lemma \\ref{categories-lemma-fibred-groupoids}." ], "refs": [ "categories-lemma-fibred-groupoids" ], "ref_ids": [ 12294 ] } ], "ref_ids": [] }, { "id": 8947, "type": "theorem", "label": "stacks-lemma-stack-gives-stack-groupoids", "categories": [ "stacks" ], "title": "stacks-lemma-stack-gives-stack-groupoids", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a stack.", "Let $p' : \\mathcal{S}' \\to \\mathcal{C}$", "be the category fibred in groupoids associated to $\\mathcal{S}$", "constructed in", "Categories, Lemma \\ref{categories-lemma-fibred-gives-fibred-groupoids}.", "Then $p' : \\mathcal{S}' \\to \\mathcal{C}$ is a stack in groupoids." ], "refs": [ "categories-lemma-fibred-gives-fibred-groupoids" ], "proofs": [ { "contents": [ "Recall that the morphisms in $\\mathcal{S}'$ are exactly the", "strongly cartesian morphisms of $\\mathcal{S}$, and that any isomorphism of", "$\\mathcal{S}$ is such a morphism. Hence descent data in $\\mathcal{S}'$", "are exactly the same thing as descent data in $\\mathcal{S}$. Now apply", "Lemma \\ref{lemma-stack-equivalences}. Some details omitted." ], "refs": [ "stacks-lemma-stack-equivalences" ], "ref_ids": [ 8940 ] } ], "ref_ids": [ 12295 ] }, { "id": 8948, "type": "theorem", "label": "stacks-lemma-stack-in-groupoids-equivalent", "categories": [ "stacks" ], "title": "stacks-lemma-stack-in-groupoids-equivalent", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{S}_1$, $\\mathcal{S}_2$ be categories over $\\mathcal{C}$.", "Suppose that $\\mathcal{S}_1$ and $\\mathcal{S}_2$ are equivalent", "as categories over $\\mathcal{C}$.", "Then $\\mathcal{S}_1$ is a stack in groupoids over $\\mathcal{C}$ if and only if", "$\\mathcal{S}_2$ is a stack in groupoids over $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Follows by combining", "Lemmas \\ref{lemma-stack-in-groupoids-stack} and \\ref{lemma-stack-equivalent}." ], "refs": [ "stacks-lemma-stack-in-groupoids-stack", "stacks-lemma-stack-equivalent" ], "ref_ids": [ 8946, 8942 ] } ], "ref_ids": [] }, { "id": 8949, "type": "theorem", "label": "stacks-lemma-2-product-stacks-in-groupoids", "categories": [ "stacks" ], "title": "stacks-lemma-2-product-stacks-in-groupoids", "contents": [ "Let $\\mathcal{C}$ be a category.", "The $2$-category of stacks in groupoids over $\\mathcal{C}$", "has 2-fibre products, and they are described as in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}." ], "refs": [ "categories-lemma-2-product-categories-over-C" ], "proofs": [ { "contents": [ "This is clear from", "Categories, Lemma \\ref{categories-lemma-2-product-fibred-categories}", "and Lemmas \\ref{lemma-stack-in-groupoids-stack}", "and \\ref{lemma-2-product-stacks}." ], "refs": [ "categories-lemma-2-product-fibred-categories", "stacks-lemma-stack-in-groupoids-stack", "stacks-lemma-2-product-stacks" ], "ref_ids": [ 12296, 8946, 8943 ] } ], "ref_ids": [ 12280 ] }, { "id": 8950, "type": "theorem", "label": "stacks-lemma-when-stack-in-sets", "categories": [ "stacks" ], "title": "stacks-lemma-when-stack-in-sets", "contents": [ "Let $\\mathcal{C}$ be a site. Under the equivalence", "$$", "\\left\\{", "\\begin{matrix}", "\\text{the category of presheaves}\\\\", "\\text{of sets over }\\mathcal{C}", "\\end{matrix}", "\\right\\}", "\\leftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{the category of categories}\\\\", "\\text{fibred in sets over }\\mathcal{C}", "\\end{matrix}", "\\right\\}", "$$", "of", "Categories, Lemma \\ref{categories-lemma-2-category-fibred-sets}", "the stacks in sets correspond precisely to the sheaves." ], "refs": [ "categories-lemma-2-category-fibred-sets" ], "proofs": [ { "contents": [ "Omitted. Hint: Show that effectivity of descent corresponds exactly to", "the sheaf condition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12310 ] }, { "id": 8951, "type": "theorem", "label": "stacks-lemma-stack-in-setoids-characterize", "categories": [ "stacks" ], "title": "stacks-lemma-stack-in-setoids-characterize", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{S}$ be a category fibred in setoids over $\\mathcal{C}$.", "Then $\\mathcal{S}$ is a stack in setoids if and only if the unique", "equivalent category $\\mathcal{S}'$ fibred in sets (see", "Categories, Lemma \\ref{categories-lemma-setoid-fibres})", "is a stack in sets. In other words, if and only if the presheaf", "$$", "U \\longmapsto \\Ob(\\mathcal{S}_U)/\\!\\!\\cong", "$$", "is a sheaf." ], "refs": [ "categories-lemma-setoid-fibres" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12312 ] }, { "id": 8952, "type": "theorem", "label": "stacks-lemma-stack-in-setoids-equivalent", "categories": [ "stacks" ], "title": "stacks-lemma-stack-in-setoids-equivalent", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{S}_1$, $\\mathcal{S}_2$ be categories over $\\mathcal{C}$.", "Suppose that $\\mathcal{S}_1$ and $\\mathcal{S}_2$ are equivalent", "as categories over $\\mathcal{C}$.", "Then $\\mathcal{S}_1$ is a stack in setoids over $\\mathcal{C}$ if and only if", "$\\mathcal{S}_2$ is a stack in setoids over $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "By", "Categories, Lemma \\ref{categories-lemma-setoid-fibres}", "we see that a category $\\mathcal{S}$ over $\\mathcal{C}$ is fibred in setoids", "over $\\mathcal{C}$ if and only if it is equivalent over $\\mathcal{C}$", "to a category fibred in sets.", "Hence we see that $\\mathcal{S}_1$ is fibred in setoids over $\\mathcal{C}$", "if and only if $\\mathcal{S}_2$ is fibred in setoids over $\\mathcal{C}$.", "Hence now the lemma follows from", "Lemma \\ref{lemma-stack-in-setoids-characterize}." ], "refs": [ "categories-lemma-setoid-fibres", "stacks-lemma-stack-in-setoids-characterize" ], "ref_ids": [ 12312, 8951 ] } ], "ref_ids": [] }, { "id": 8953, "type": "theorem", "label": "stacks-lemma-2-product-stacks-in-setoids", "categories": [ "stacks" ], "title": "stacks-lemma-2-product-stacks-in-setoids", "contents": [ "Let $\\mathcal{C}$ be a site.", "The $2$-category of stacks in setoids over $\\mathcal{C}$", "has 2-fibre products, and they are described as in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}." ], "refs": [ "categories-lemma-2-product-categories-over-C" ], "proofs": [ { "contents": [ "This is clear from", "Categories, Lemmas \\ref{categories-lemma-2-product-fibred-categories} and", "\\ref{categories-lemma-2-product-categories-fibred-setoids}", "and", "Lemmas \\ref{lemma-stack-in-groupoids-stack} and", "\\ref{lemma-2-product-stacks}." ], "refs": [ "categories-lemma-2-product-fibred-categories", "categories-lemma-2-product-categories-fibred-setoids", "stacks-lemma-stack-in-groupoids-stack", "stacks-lemma-2-product-stacks" ], "ref_ids": [ 12296, 12311, 8946, 8943 ] } ], "ref_ids": [ 12280 ] }, { "id": 8954, "type": "theorem", "label": "stacks-lemma-2-fibre-product-gives-stack-in-setoids", "categories": [ "stacks" ], "title": "stacks-lemma-2-fibre-product-gives-stack-in-setoids", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{S}, \\mathcal{T}$ be stacks in groupoids over $\\mathcal{C}$", "and let $\\mathcal{R}$ be a stack in setoids over $\\mathcal{C}$.", "Let $f : \\mathcal{T} \\to \\mathcal{S}$ and $g : \\mathcal{R} \\to \\mathcal{S}$", "be $1$-morphisms. If $f$ is faithful, then the $2$-fibre product", "$$", "\\mathcal{T} \\times_{f, \\mathcal{S}, g} \\mathcal{R}", "$$", "is a stack in setoids over $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the explicit description of the $2$-fibre product in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}." ], "refs": [ "categories-lemma-2-product-categories-over-C" ], "ref_ids": [ 12280 ] } ], "ref_ids": [] }, { "id": 8955, "type": "theorem", "label": "stacks-lemma-2-fibre-product-stacks-in-setoids-over-stack-in-groupoids", "categories": [ "stacks" ], "title": "stacks-lemma-2-fibre-product-stacks-in-setoids-over-stack-in-groupoids", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{S}$ be a stack in groupoids over $\\mathcal{C}$ and", "let $\\mathcal{S}_i$, $i = 1, 2$ be stacks in setoids over $\\mathcal{C}$.", "Let $f_i : \\mathcal{S}_i \\to \\mathcal{S}$ be $1$-morphisms.", "Then the $2$-fibre product", "$$", "\\mathcal{S}_1 \\times_{f_1, \\mathcal{S}, f_2} \\mathcal{S}_2", "$$", "is a stack in setoids over $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-2-fibre-product-gives-stack-in-setoids}", "as $f_2$ is faithful." ], "refs": [ "stacks-lemma-2-fibre-product-gives-stack-in-setoids" ], "ref_ids": [ 8954 ] } ], "ref_ids": [] }, { "id": 8956, "type": "theorem", "label": "stacks-lemma-faithful-descent", "categories": [ "stacks" ], "title": "stacks-lemma-faithful-descent", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$$", "\\xymatrix{", "\\mathcal{T}_2 \\ar[r] \\ar[d]_{G'} & \\mathcal{T}_1 \\ar[d]^G \\\\", "\\mathcal{S}_2 \\ar[r]^F & \\mathcal{S}_1", "}", "$$", "be a $2$-cartesian diagram of stacks in groupoids over $\\mathcal{C}$.", "Assume", "\\begin{enumerate}", "\\item for every $U \\in \\Ob(\\mathcal{C})$ and", "$x \\in \\Ob((\\mathcal{S}_1)_U)$ there exists a covering", "$\\{U_i \\to U\\}$ such that $x|_{U_i}$ is in the essential", "image of $F : (\\mathcal{S}_2)_{U_i} \\to (\\mathcal{S}_1)_{U_i}$, and", "\\item $G'$ is faithful,", "\\end{enumerate}", "then $G$ is faithful." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $\\mathcal{T}_2$ is the category", "$\\mathcal{S}_2 \\times_{\\mathcal{S}_1} \\mathcal{T}_1$", "described in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.", "By", "Categories, Lemma \\ref{categories-lemma-equivalence-fibred-categories}", "the faithfulness of $G, G'$ can be checked on fibre categories.", "Suppose that $y, y'$ are objects of $\\mathcal{T}_1$ over the object $U$", "of $\\mathcal{C}$. Let $\\alpha, \\beta : y \\to y'$ be morphisms of", "$(\\mathcal{T}_1)_U$ such that $G(\\alpha) = G(\\beta)$. Our object is to", "show that $\\alpha = \\beta$. Considering instead", "$\\gamma = \\alpha^{-1} \\circ \\beta$ we see that $G(\\gamma) = \\text{id}_{G(y)}$", "and we have to show that $\\gamma = \\text{id}_y$. By assumption we", "can find a covering $\\{U_i \\to U\\}$ such that $G(y)|_{U_i}$ is in the", "essential image of $F :(\\mathcal{S}_2)_{U_i} \\to (\\mathcal{S}_1)_{U_i}$.", "Since it suffices to show that $\\gamma|_{U_i} = \\text{id}$ for each $i$,", "we may therefore assume that we have", "$f : F(x) \\to G(y)$ for some object $x$ of $\\mathcal{S}_2$ over $U$", "and morphisms $f$ of $(\\mathcal{S}_1)_U$. In this case we get", "a morphism", "$$", "(1, \\gamma) : (U, x, y, f) \\longrightarrow (U, x, y, f)", "$$", "in the fibre category of $\\mathcal{S}_2 \\times_{\\mathcal{S}_1} \\mathcal{T}_1$", "over $U$ whose image under $G'$ in $\\mathcal{S}_1$ is $\\text{id}_x$.", "As $G'$ is faithful we conclude that $\\gamma = \\text{id}_y$ and we win." ], "refs": [ "categories-lemma-2-product-categories-over-C", "categories-lemma-equivalence-fibred-categories" ], "ref_ids": [ 12280, 12297 ] } ], "ref_ids": [] }, { "id": 8957, "type": "theorem", "label": "stacks-lemma-stack-in-setoids-descent", "categories": [ "stacks" ], "title": "stacks-lemma-stack-in-setoids-descent", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let", "$$", "\\xymatrix{", "\\mathcal{T}_2 \\ar[r] \\ar[d] & \\mathcal{T}_1 \\ar[d]^G \\\\", "\\mathcal{S}_2 \\ar[r]^F & \\mathcal{S}_1", "}", "$$", "be a $2$-cartesian diagram of stacks in groupoids over $\\mathcal{C}$.", "If", "\\begin{enumerate}", "\\item $F : \\mathcal{S}_2 \\to \\mathcal{S}_1$ is fully faithful,", "\\item for every $U \\in \\Ob(\\mathcal{C})$ and", "$x \\in \\Ob((\\mathcal{S}_1)_U)$ there exists a covering", "$\\{U_i \\to U\\}$ such that $x|_{U_i}$ is in the essential", "image of $F : (\\mathcal{S}_2)_{U_i} \\to (\\mathcal{S}_1)_{U_i}$, and", "\\item $\\mathcal{T}_2$ is a stack in setoids.", "\\end{enumerate}", "then $\\mathcal{T}_1$ is a stack in setoids." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $\\mathcal{T}_2$ is the category", "$\\mathcal{S}_2 \\times_{\\mathcal{S}_1} \\mathcal{T}_1$", "described in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.", "Pick $U \\in \\Ob(\\mathcal{C})$ and", "$y \\in \\Ob((\\mathcal{T}_1)_U)$.", "We have to show that the sheaf $\\mathit{Aut}(y)$ on $\\mathcal{C}/U$", "is trivial. To to this we may replace $U$ by the members of", "a covering of $U$. Hence by assumption (2) we may assume that", "there exists an object $x \\in \\Ob((\\mathcal{S}_2)_U)$", "and an isomorphism $f : F(x) \\to G(y)$.", "Then $y' = (U, x, y, f)$ is an object of $\\mathcal{T}_2$ over $U$", "which is mapped to $y$ under the projection $\\mathcal{T}_2 \\to \\mathcal{T}_1$.", "Because $F$ is fully faithful by (1) the map", "$\\mathit{Aut}(y') \\to \\mathit{Aut}(y)$ is surjective, use the explicit", "description of morphisms in $\\mathcal{T}_2$ in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.", "Since by (3) the sheaf $\\mathit{Aut}(y')$ is trivial", "we get the result of the lemma." ], "refs": [ "categories-lemma-2-product-categories-over-C", "categories-lemma-2-product-categories-over-C" ], "ref_ids": [ 12280, 12280 ] } ], "ref_ids": [] }, { "id": 8958, "type": "theorem", "label": "stacks-lemma-relative-sheaf-over-stack-is-stack", "categories": [ "stacks" ], "title": "stacks-lemma-relative-sheaf-over-stack-is-stack", "contents": [ "Let $\\mathcal{C}$ be a site. Let $F : \\mathcal{S} \\to \\mathcal{T}$", "be a $1$-morphism of categories fibred in groupoids over $\\mathcal{C}$.", "Assume that", "\\begin{enumerate}", "\\item $\\mathcal{T}$ is a stack in groupoids over $\\mathcal{C}$,", "\\item for every $U \\in \\Ob(\\mathcal{C})$ the functor", "$\\mathcal{S}_U \\to \\mathcal{T}_U$ of fibre categories is faithful,", "\\item for each $U$ and each $y \\in \\Ob(\\mathcal{T}_U)$ the presheaf", "$$", "(h : V \\to U)", "\\longmapsto", "\\{(x, f) \\mid x \\in \\Ob(\\mathcal{S}_V), f : F(x) \\to f^*y\\text{ over }V\\}/\\cong", "$$", "is a sheaf on $\\mathcal{C}/U$.", "\\end{enumerate}", "Then $\\mathcal{S}$ is a stack in groupoids over $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "We have to prove descent for morphisms and descent for objects.", "\\medskip\\noindent", "Descent for morphisms. Let $\\{U_i \\to U\\}$ be a covering of $\\mathcal{C}$.", "Let $x, x'$ be objects of $\\mathcal{S}$ over $U$. For each $i$", "let $\\alpha_i : x|_{U_i} \\to x'|_{U_i}$ be a morphism over $U_i$", "such that $\\alpha_i$ and $\\alpha_j$ restrict to the same morphism", "$x|_{U_i \\times_U U_j} \\to x'|_{U_i \\times_U U_j}$.", "Because $\\mathcal{T}$ is a stack in groupoids, there is a morphism", "$\\beta : F(x) \\to F(x')$ over $U$ whose restriction to $U_i$ is $F(\\alpha_i)$.", "Then we can think of $\\xi = (x, \\beta)$ and $\\xi' = (x', \\text{id}_{F(x')})$", "as sections of the presheaf associated to $y = F(x')$ over $U$", "in assumption (3). On the other hand, the restrictions of", "$\\xi$ and $\\xi'$ to $U_i$ are $(x|_{U_i}, F(\\alpha_i))$", "and $(x'|_{U_i}, \\text{id}_{F(x'|_{U_i})})$.", "These are isomorphic to each other by the morphism $\\alpha_i$.", "Thus $\\xi$ and $\\xi'$ are isomorphic by assumption (3). This means there is a", "morphism $\\alpha : x \\to x'$ over $U$ with $F(\\alpha) = \\beta$.", "Since $F$ is faithful on fibre categories we obtain", "$\\alpha|_{U_i} = \\alpha_i$.", "\\medskip\\noindent", "Descent of objects. Let $\\{U_i \\to U\\}$ be a covering of $\\mathcal{C}$.", "Let $(x_i, \\varphi_{ij})$ be a descent datum for $\\mathcal{S}$ with", "respect to the given covering. Because $\\mathcal{T}$ is a stack in groupoids,", "there is an object $y$ in $\\mathcal{T}_U$ and isomorphisms", "$\\beta_i : F(x_i) \\to y|_{U_i}$ such that", "$F(\\varphi_{ij}) = \\beta_j|_{U_i \\times_U U_j} \\circ", "(\\beta_i|_{U_i \\times_U U_j})^{-1}$.", "Then $(x_i, \\beta_i)$ are sections of the presheaf associated to", "$y$ over $U$ defined in assumption (3).", "Moreover, $\\varphi_{ij}$ defines an isomorphism", "from the pair $(x_i, \\beta_i)|_{U_i \\times_U U_j}$ to ", "the pair $(x_j, \\beta_j)|_{U_i \\times_U U_j}$.", "Hence by assumption (3) there exists a pair $(x, \\beta)$ over $U$", "whose restriction to $U_i$ is isomorphic to $(x_i, \\beta_i)$.", "This means there are morphisms $\\alpha_i : x_i \\to x|_{U_i}$", "with $\\beta_i = \\beta|_{U_i} \\circ F(\\alpha_i)$.", "Since $F$ is faithful on fibre categories a calculation shows", "that $\\varphi_{ij} = \\alpha_j|_{U_i \\times_U U_j} \\circ", "(\\alpha_i|_{U_i \\times_U U_j})^{-1}$. This finishes the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8959, "type": "theorem", "label": "stacks-lemma-inertia", "categories": [ "stacks" ], "title": "stacks-lemma-inertia", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$p : \\mathcal{S} \\to \\mathcal{C}$ and", "$p' : \\mathcal{S}' \\to \\mathcal{C}$", "be stacks over the site $\\mathcal{C}$.", "Let $F : \\mathcal{S} \\to \\mathcal{S}'$ be a $1$-morphism of", "stacks over $\\mathcal{C}$.", "\\begin{enumerate}", "\\item The inertia $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ and", "$\\mathcal{I}_\\mathcal{S}$ are stacks over $\\mathcal{C}$.", "\\item If $\\mathcal{S}, \\mathcal{S}'$ are stacks in groupoids over", "$\\mathcal{C}$, then so are $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ and", "$\\mathcal{I}_\\mathcal{S}$.", "\\item If $\\mathcal{S}, \\mathcal{S}'$ are stacks in setoids over $\\mathcal{C}$,", "then so are $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ and", "$\\mathcal{I}_\\mathcal{S}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first three assertions follow from", "Lemmas \\ref{lemma-2-product-stacks},", "\\ref{lemma-2-product-stacks-in-groupoids}, and", "\\ref{lemma-2-product-stacks-in-setoids}", "and the equivalence in", "Categories, Lemma \\ref{categories-lemma-inertia-fibred-category} part (1)." ], "refs": [ "stacks-lemma-2-product-stacks", "stacks-lemma-2-product-stacks-in-groupoids", "stacks-lemma-2-product-stacks-in-setoids", "categories-lemma-inertia-fibred-category" ], "ref_ids": [ 8943, 8949, 8953, 12292 ] } ], "ref_ids": [] }, { "id": 8960, "type": "theorem", "label": "stacks-lemma-characterize-stack-in-setoids", "categories": [ "stacks" ], "title": "stacks-lemma-characterize-stack-in-setoids", "contents": [ "Let $\\mathcal{C}$ be a site.", "If $\\mathcal{S}$ is a stack in groupoids, then the", "canonical $1$-morphism $\\mathcal{I}_\\mathcal{S} \\to \\mathcal{S}$", "is an equivalence if and only if $\\mathcal{S}$ is a stack in setoids." ], "refs": [], "proofs": [ { "contents": [ "Follows directly from", "Categories, Lemma \\ref{categories-lemma-characterize-fibred-setoids-inertia}." ], "refs": [ "categories-lemma-characterize-fibred-setoids-inertia" ], "ref_ids": [ 12314 ] } ], "ref_ids": [] }, { "id": 8961, "type": "theorem", "label": "stacks-lemma-stackify", "categories": [ "stacks" ], "title": "stacks-lemma-stackify", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category over $\\mathcal{C}$.", "There exists a stack $p' : \\mathcal{S}' \\to \\mathcal{C}$ and a", "$1$-morphism $G : \\mathcal{S} \\to \\mathcal{S}'$", "of fibred categories over $\\mathcal{C}$ (see", "Categories, Definition \\ref{categories-definition-fibred-categories-over-C})", "such that", "\\begin{enumerate}", "\\item for every $U \\in \\Ob(\\mathcal{C})$, and any", "$x, y \\in \\Ob(\\mathcal{S}_U)$ the map", "$$", "\\mathit{Mor}(x, y) \\longrightarrow \\mathit{Mor}(G(x), G(y))", "$$", "induced by $G$ identifies the right hand side with the sheafification", "of the left hand side, and", "\\item for every $U \\in \\Ob(\\mathcal{C})$, and any", "$x' \\in \\Ob(\\mathcal{S}'_U)$ there exists a covering", "$\\{U_i \\to U\\}_{i \\in I}$ such that for every $i \\in I$ the", "object $x'|_{U_i}$ is in the essential image of the", "functor $G : \\mathcal{S}_{U_i} \\to \\mathcal{S}'_{U_i}$.", "\\end{enumerate}", "Moreover the stack $\\mathcal{S}'$ is determined up to unique", "$2$-isomorphism by these conditions." ], "refs": [ "categories-definition-fibred-categories-over-C" ], "proofs": [ { "contents": [ "[Proof by naive method]", "In this proof method we proceed in stages:", "\\medskip\\noindent", "First, given $x$ lying over $U$ and any object $y$ of", "$\\mathcal{S}$, we say that two morphisms", "$a, b : x \\to y$ of $\\mathcal{S}$", "lying over the same arrow of $\\mathcal{C}$", "are {\\it locally equal}", "if there exists a covering $\\{f_i : U_i \\to U\\}$ of $\\mathcal{C}$", "such that the compositions", "$$", "f_i^*x \\to x \\xrightarrow{a} y,", "\\quad", "f_i^*x \\to x \\xrightarrow{b} y", "$$", "are equal. This gives an equivalence relation $\\sim$", "on arrows of $\\mathcal{S}$. If $b \\sim b'$ then", "$a \\circ b \\circ c \\sim a \\circ b' \\circ c$ (verification omitted).", "Hence we can quotient out by this equivalence relation to", "obtain a new category $\\mathcal{S}^1$ over $\\mathcal{C}$", "together with a morphism $G^1 : \\mathcal{S} \\to \\mathcal{S}^1$.", "\\medskip\\noindent", "One checks that $G^1$ preserves strongly cartesian morphisms", "and that $\\mathcal{S}^1$ is a fibred category over $\\mathcal{C}$.", "Checks omitted. Thus we reduce to the case where locally equal", "morphisms are equal.", "\\medskip\\noindent", "Next, we add morphisms as follows. Given", "$x$ lying over $U$ and any object $y$ of lying over $V$", "a {\\it locally defined morphism from $x$ to $y$} is given by", "\\begin{enumerate}", "\\item a morphism $f : U \\to V$,", "\\item a covering $\\{f_i : U_i \\to U\\}$ of $U$, and", "\\item morphisms $a_i : f_i^*x \\to y$ with $p(a_i) = f \\circ f_i$", "\\end{enumerate}", "with the property that the compositions", "$$", "(f_i \\times f_j)^*x \\to f_i^*x \\xrightarrow{a_i} y,", "\\quad", "(f_i \\times f_j)^*x \\to f_j^*x \\xrightarrow{a_j} y", "$$", "are equal. Note that a usual morphism $a : x \\to y$ gives a locally", "defined morphism $(p(a) : U \\to V, \\{\\text{id}_U\\}, a)$.", "We say two locally defined morphisms", "$(f, \\{f_i : U_i \\to U\\}, a_i)$ and $(g, \\{g_j : U'_j \\to U\\}, b_j)$", "are {\\it equal} if $f = g$ and the compositions", "$$", "(f_i \\times g_j)^*x \\to f_i^*x \\xrightarrow{a_i} y,", "\\quad", "(f_i \\times g_j)^*x \\to g_j^*x \\xrightarrow{b_j} y", "$$", "are equal (this is the right condition since we are in the", "situation where locally equal morphisms are equal).", "To compose locally defined morphisms", "$(f, \\{f_i : U_i \\to U\\}, a_i)$ from $x$ to $y$ and", "$(g, \\{g_j : V_j \\to V\\}, b_j)$ from $y$ to $z$ lying over $W$,", "just take $g \\circ f : U \\to W$, the covering", "$\\{U_i \\times_V V_j \\to U\\}$, and as maps the compositions", "$$", "x|_{U_i \\times_V V_j}", "\\xrightarrow{\\text{pr}_0^*a_i}", "y|_{V_j}", "\\xrightarrow{b_j}", "z", "$$", "We omit the verification that this is a locally defined morphism.", "\\medskip\\noindent", "One checks that $\\mathcal{S}^2$ with the same objects as", "$\\mathcal{S}$ and with locally defined morphisms as morphisms", "is a category over $\\mathcal{C}$, that there is a functor", "$G^2 : \\mathcal{S} \\to \\mathcal{S}^2$ over $\\mathcal{C}$,", "that this functor preserves strongly cartesian objects,", "and that $\\mathcal{S}^2$ is a fibred category over $\\mathcal{C}$.", "Checks omitted. This reduces one to the case where the", "morphism presheaves of $\\mathcal{S}$ are all sheaves, by", "checking that the effect of using locally defined morphisms", "is to take the sheafification of the (separated) morphisms", "presheaves.", "\\medskip\\noindent", "Finally, in the case where the morphism presheaves are all sheaves", "we have to add objects in order to make sure descent conditions are", "effective in the end result. The simplest way to do this is to", "consider the category $\\mathcal{S}'$ whose objects are", "pairs $(\\mathcal{U}, \\xi)$ where", "$\\mathcal{U} = \\{U_i \\to U\\}$ is a covering of $\\mathcal{C}$ and", "$\\xi = (X_i, \\varphi_{ii'})$ is a descent datum relative $\\mathcal{U}$.", "Suppose given two such data", "$(\\mathcal{U}, \\xi) = (\\{f_i : U_i \\to U\\}, x_i, \\varphi_{ii'})$ and", "$(\\mathcal{V}, \\eta) = (\\{g_j : V_j \\to V\\}, y_j, \\psi_{jj'})$.", "We define", "$$", "\\Mor_{\\mathcal{S}'}((\\mathcal{U}, \\xi), (\\mathcal{V}, \\eta))", "$$", "as the set of $(f, a_{ij})$, where $f : U \\to V$ and", "$$", "a_{ij} :", "x_i|_{U_i \\times_V V_j}", "\\longrightarrow", "y_j", "$$", "are morphisms of $\\mathcal{S}$ lying over $U_i \\times_V V_j \\to V_j$.", "These have to satisfy the following condition: for any", "$i, i' \\in I$ and $j, j' \\in J$ set", "$W = (U_i \\times_U U_{i'}) \\times_V (V_j \\times_V V_{j'})$. Then", "$$", "\\xymatrix{", "x_i|_W \\ar[r]_{a_{ij}|_W} \\ar[d]_{\\varphi_{ii'}|_W} &", "y_j|_W \\ar[d]^{\\psi_{jj'}|_W} \\\\", "x_{i'}|_W \\ar[r]^{a_{i'j'}|_W} &", "y_{j'}|_W", "}", "$$", "commutes. At this point you have to verify the following things:", "\\begin{enumerate}", "\\item there is a well defined composition on morphisms as above,", "\\item this turns $\\mathcal{S}'$ into a category over $\\mathcal{C}$,", "\\item there is a functor $G : \\mathcal{S} \\to \\mathcal{S}'$ over $\\mathcal{C}$,", "\\item for $x, y$ objects of $\\mathcal{S}$ we have", "$\\Mor_\\mathcal{S}(x, y) = \\Mor_{\\mathcal{S}'}(G(x), G(y))$,", "\\item any object of $\\mathcal{S}'$ locally comes from an object of", "$\\mathcal{S}$, i.e., part (2) of the lemma holds,", "\\item $G$ preserves strongly cartesian morphisms,", "\\item $\\mathcal{S}'$ is a fibred category over $\\mathcal{C}$, and", "\\item $\\mathcal{S}'$ is a stack over $\\mathcal{C}$.", "\\end{enumerate}", "This is all not hard but there is a lot of it. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12390 ] }, { "id": 8962, "type": "theorem", "label": "stacks-lemma-stackify-universal-property", "categories": [ "stacks" ], "title": "stacks-lemma-stackify-universal-property", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category over $\\mathcal{C}$.", "Let $p' : \\mathcal{S}' \\to \\mathcal{C}$ and $G : \\mathcal{S} \\to \\mathcal{S}'$", "the stack and $1$-morphism constructed in Lemma \\ref{lemma-stackify}.", "This construction has the following universal property: Given a stack", "$q : \\mathcal{X} \\to \\mathcal{C}$ and a $1$-morphism", "$F : \\mathcal{S} \\to \\mathcal{X}$ of fibred categories over $\\mathcal{C}$", "there exists a $1$-morphism $H : \\mathcal{S}' \\to \\mathcal{X}$", "such that the diagram", "$$", "\\xymatrix{", "\\mathcal{S} \\ar[rr]_F \\ar[rd]_G & & \\mathcal{X} \\\\", "& \\mathcal{S}' \\ar[ru]_H", "}", "$$", "is $2$-commutative." ], "refs": [ "stacks-lemma-stackify" ], "proofs": [ { "contents": [ "Omitted. Hint: Suppose that $x' \\in \\Ob(\\mathcal{S}'_U)$.", "By the result of Lemma \\ref{lemma-stackify}", "there exists a covering $\\{U_i \\to U\\}_{i \\in I}$", "such that $x'|_{U_i} = G(x_i)$ for some $x_i \\in \\Ob(\\mathcal{S}_{U_i})$.", "Moreover, there exist coverings $\\{U_{ijk} \\to U_i \\times_U U_j\\}$", "and isomorphisms $\\alpha_{ijk} : x_i|_{U_{ijk}} \\to x_j|_{U_{ijk}}$", "with $G(\\alpha_{ijk}) = \\text{id}_{x'|_{U_{ijk}}}$. Set $y_i = F(x_i)$.", "Then you can check that", "$$", "F(\\alpha_{ijk}) : y_i|_{U_{ijk}} \\to y_j|_{U_{ijk}}", "$$", "agree on overlaps and therefore (as $\\mathcal{X}$ is a stack) define", "a morphism $\\beta_{ij} : y_i|_{U_i \\times_U U_j} \\to y_j|_{U_i \\times_U U_j}$.", "Next, you check that the $\\beta_{ij}$ define a descent datum. Since", "$\\mathcal{X}$ is a stack these descent data are effective and we find", "an object $y$ of $\\mathcal{X}_U$ agreeing with $G(x_i)$ over $U_i$.", "The hint is to set $H(x') = y$." ], "refs": [ "stacks-lemma-stackify" ], "ref_ids": [ 8961 ] } ], "ref_ids": [ 8961 ] }, { "id": 8963, "type": "theorem", "label": "stacks-lemma-stackify-universal-property-more", "categories": [ "stacks" ], "title": "stacks-lemma-stackify-universal-property-more", "contents": [ "Notation and assumptions as in", "Lemma \\ref{lemma-stackify-universal-property}.", "There is a canonical equivalence of categories", "$$", "\\Mor_{\\textit{Fib}/\\mathcal{C}}(\\mathcal{S}, \\mathcal{X})", "=", "\\Mor_{\\textit{Stacks}/\\mathcal{C}}(\\mathcal{S}', \\mathcal{X})", "$$", "given by the constructions in the proof of the aforementioned lemma." ], "refs": [ "stacks-lemma-stackify-universal-property" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8962 ] }, { "id": 8964, "type": "theorem", "label": "stacks-lemma-stackification-fibre-product-fibred-categories", "categories": [ "stacks" ], "title": "stacks-lemma-stackification-fibre-product-fibred-categories", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Z} \\to \\mathcal{Y}$", "be morphisms of fibred categories over $\\mathcal{C}$.", "In this case the stackification of the $2$-fibre product is the $2$-fibre", "product of the stackifications." ], "refs": [], "proofs": [ { "contents": [ "Let us denote $\\mathcal{X}', \\mathcal{Y}', \\mathcal{Z}'$ the stackifications", "and $\\mathcal{W}$ the stackification of", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$. By construction of $2$-fibre", "products there is a canonical $1$-morphism", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to", "\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{Z}'$.", "As the second $2$-fibre product is a stack (see", "Lemma \\ref{lemma-2-product-stacks})", "this $1$-morphism induces a $1$-morphism", "$h : \\mathcal{W} \\to \\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{Z}'$", "by the universal property of stackification, see", "Lemma \\ref{lemma-stackify-universal-property}.", "Now $h$ is a morphism of stacks, and we may check that it is an", "equivalence using", "Lemmas \\ref{lemma-characterize-ff} and", "\\ref{lemma-characterize-essentially-surjective-when-ff}.", "\\medskip\\noindent", "Thus we first prove that $h$ induces isomorphisms of $\\mathit{Mor}$-sheaves.", "Let $\\xi, \\xi'$ be objects of $\\mathcal{W}$ over", "$U \\in \\Ob(\\mathcal{C})$. We want to show that", "$$", "h : \\mathit{Mor}(\\xi, \\xi') \\longrightarrow \\mathit{Mor}(h(\\xi), h(\\xi'))", "$$", "is an isomorphism. To do this we may work locally on $U$ (see", "Sites, Section \\ref{sites-section-glueing-sheaves}).", "Hence by construction of $\\mathcal{W}$ (see", "Lemma \\ref{lemma-stackify})", "we may assume that $\\xi, \\xi'$", "actually come from objects $(x, z, \\alpha)$ and $(x', z', \\alpha')$", "of $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ over $U$.", "By the same lemma once more we see that in this case", "$\\mathit{Mor}(\\xi, \\xi')$ is the sheafification of", "$$", "V/U \\longmapsto", "\\Mor_{\\mathcal{X}_V}(x|_V, x'|_V)", "\\times_{\\Mor_{\\mathcal{Y}_V}(f(x)|_V, f(x')|_V)}", "\\Mor_{\\mathcal{Z}_V}(z|_V, z'|_V)", "$$", "and that $\\mathit{Mor}(h(\\xi), h(\\xi'))$ is equal to the fibre product", "$$", "\\mathit{Mor}(i(x), i(x'))", "\\times_{\\mathit{Mor}(j(f(x)), j(f(x'))}", "\\mathit{Mor}(k(z), k(z'))", "$$", "where $i : \\mathcal{X} \\to \\mathcal{X}'$,", "$j : \\mathcal{Y} \\to \\mathcal{Y}'$, and", "$k : \\mathcal{Z} \\to \\mathcal{Z}'$ are the canonical functors.", "Thus the first displayed map of this paragraph is an isomorphism as", "sheafification is exact (and hence the sheafification of a fibre product", "of presheaves is the fibre product of the sheafifications).", "\\medskip\\noindent", "Finally, we have to check that any object of", "$\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{Z}'$", "over $U$ is locally on $U$ in the essential image of $h$.", "Write such an object as a triple $(x', z', \\alpha)$.", "Then $x'$ locally comes from an object of $\\mathcal{X}$,", "$z'$ locally comes from an object of $\\mathcal{Z}$, and", "having made suitable replacements for $x'$, $z'$ the morphism", "$\\alpha$ of $\\mathcal{Y}'_U$ locally comes from a morphism of", "$\\mathcal{Y}$. In other words, we have shown that any object of", "$\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{Z}'$", "over $U$ is locally on $U$ in the essential image of", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to", "\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{Z}'$, hence", "a fortiori it is locally in the essential image of $h$." ], "refs": [ "stacks-lemma-2-product-stacks", "stacks-lemma-stackify-universal-property", "stacks-lemma-characterize-ff", "stacks-lemma-characterize-essentially-surjective-when-ff", "stacks-lemma-stackify" ], "ref_ids": [ 8943, 8962, 8944, 8945, 8961 ] } ], "ref_ids": [] }, { "id": 8965, "type": "theorem", "label": "stacks-lemma-stackification-inertia", "categories": [ "stacks" ], "title": "stacks-lemma-stackification-inertia", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{X}$ be a fibred category over $\\mathcal{C}$.", "The stackification of the inertia fibred category $\\mathcal{I}_\\mathcal{X}$", "is inertia of the stackification of $\\mathcal{X}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that stackification is compatible", "with $2$-fibre products by", "Lemma \\ref{lemma-stackification-fibre-product-fibred-categories}", "and the fact that there is a formula for the inertia in terms of", "$2$-fibre products of categories over $\\mathcal{C}$, see", "Categories, Lemma \\ref{categories-lemma-inertia-fibred-category}." ], "refs": [ "stacks-lemma-stackification-fibre-product-fibred-categories", "categories-lemma-inertia-fibred-category" ], "ref_ids": [ 8964, 12292 ] } ], "ref_ids": [] }, { "id": 8966, "type": "theorem", "label": "stacks-lemma-stackify-groupoids", "categories": [ "stacks" ], "title": "stacks-lemma-stackify-groupoids", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category", "fibred in groupoids over $\\mathcal{C}$.", "There exists a stack in groupoids", "$p' : \\mathcal{S}' \\to \\mathcal{C}$ and a", "$1$-morphism $G : \\mathcal{S} \\to \\mathcal{S}'$", "of categories fibred in groupoids over $\\mathcal{C}$ (see", "Categories, Definition", "\\ref{categories-definition-categories-fibred-in-groupoids-over-C})", "such that", "\\begin{enumerate}", "\\item for every $U \\in \\Ob(\\mathcal{C})$, and any", "$x, y \\in \\Ob(\\mathcal{S}_U)$ the map", "$$", "\\mathit{Mor}(x, y) \\longrightarrow \\mathit{Mor}(G(x), G(y))", "$$", "induced by $G$ identifies the right hand side with the sheafification", "of the left hand side, and", "\\item for every $U \\in \\Ob(\\mathcal{C})$, and any", "$x' \\in \\Ob(\\mathcal{S}'_U)$ there exists a covering", "$\\{U_i \\to U\\}_{i \\in I}$ such that for every $i \\in I$ the", "object $x'|_{U_i}$ is in the essential image of the", "functor $G : \\mathcal{S}_{U_i} \\to \\mathcal{S}'_{U_i}$.", "\\end{enumerate}", "Moreover the stack in groupoids $\\mathcal{S}'$ is determined up to unique", "$2$-isomorphism by these conditions." ], "refs": [ "categories-definition-categories-fibred-in-groupoids-over-C" ], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-stackify}. The result will be a", "stack in groupoids by applying Lemma \\ref{lemma-stack-in-groupoids-stack}." ], "refs": [ "stacks-lemma-stackify", "stacks-lemma-stack-in-groupoids-stack" ], "ref_ids": [ 8961, 8946 ] } ], "ref_ids": [ 12393 ] }, { "id": 8967, "type": "theorem", "label": "stacks-lemma-stackify-groupoids-universal-property", "categories": [ "stacks" ], "title": "stacks-lemma-stackify-groupoids-universal-property", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids", "over $\\mathcal{C}$. Let $p' : \\mathcal{S}' \\to \\mathcal{C}$ and", "$G : \\mathcal{S} \\to \\mathcal{S}'$", "the stack in groupoids and $1$-morphism constructed in", "Lemma \\ref{lemma-stackify-groupoids}.", "This construction has the following universal property: Given a stack", "in groupoids $q : \\mathcal{X} \\to \\mathcal{C}$ and a $1$-morphism", "$F : \\mathcal{S} \\to \\mathcal{X}$ of categories over $\\mathcal{C}$", "there exists a $1$-morphism $H : \\mathcal{S}' \\to \\mathcal{X}$", "such that the diagram", "$$", "\\xymatrix{", "\\mathcal{S} \\ar[rr]_F \\ar[rd]_G & & \\mathcal{X} \\\\", "& \\mathcal{S}' \\ar[ru]_H", "}", "$$", "is $2$-commutative." ], "refs": [ "stacks-lemma-stackify-groupoids" ], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-stackify-universal-property}." ], "refs": [ "stacks-lemma-stackify-universal-property" ], "ref_ids": [ 8962 ] } ], "ref_ids": [ 8966 ] }, { "id": 8968, "type": "theorem", "label": "stacks-lemma-stackification-fibre-product-categories-fibred-in-groupoids", "categories": [ "stacks" ], "title": "stacks-lemma-stackification-fibre-product-categories-fibred-in-groupoids", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Z} \\to \\mathcal{Y}$", "be morphisms of categories fibred in groupoids over $\\mathcal{C}$.", "In this case the stackification of the $2$-fibre product is the $2$-fibre", "product of the stackifications." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-stackification-fibre-product-fibred-categories}." ], "refs": [ "stacks-lemma-stackification-fibre-product-fibred-categories" ], "ref_ids": [ 8964 ] } ], "ref_ids": [] }, { "id": 8969, "type": "theorem", "label": "stacks-lemma-topology-inherited", "categories": [ "stacks" ], "title": "stacks-lemma-topology-inherited", "contents": [ "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$", "be a fibred category. Let $\\text{Cov}(\\mathcal{S})$", "be the set of families $\\{x_i \\to x\\}_{i \\in I}$ of morphisms in $\\mathcal{S}$", "with fixed target such that (a) each $x_i \\to x$ is strongly cartesian,", "and (b) $\\{p(x_i) \\to p(x)\\}_{i \\in I}$ is a covering of $\\mathcal{C}$.", "Then $(\\mathcal{S}, \\text{Cov}(\\mathcal{S}))$ is a site." ], "refs": [], "proofs": [ { "contents": [ "We have to check the three conditions of", "Sites, Definition \\ref{sites-definition-site}.", "\\begin{enumerate}", "\\item If $x \\to y$ is an isomorphism of $\\mathcal{S}$, then", "it is strongly cartesian by", "Categories, Lemma \\ref{categories-lemma-composition-cartesian}", "and $p(x) \\to p(y)$ is an isomorphism of $\\mathcal{C}$. Thus", "$\\{p(x) \\to p(y)\\}$ is a covering of $\\mathcal{C}$ whence", "$\\{x \\to y\\} \\in \\text{Cov}(\\mathcal{S})$.", "\\item If $\\{x_i \\to x\\}_{i\\in I} \\in \\text{Cov}(\\mathcal{S})$ and for each", "$i$ we have $\\{y_{ij} \\to x_i\\}_{j\\in J_i} \\in \\text{Cov}(\\mathcal{S})$, then", "each composition $p(y_{ij}) \\to p(x)$ is strongly cartesian by", "Categories, Lemma \\ref{categories-lemma-composition-cartesian}", "and $\\{p(y_{ij}) \\to p(x)\\}_{i \\in I, j\\in J_i} \\in \\text{Cov}(\\mathcal{C})$.", "Hence also $\\{y_{ij} \\to x\\}_{i \\in I, j\\in J_i} \\in \\text{Cov}(\\mathcal{S})$.", "\\item Suppose $\\{x_i \\to x\\}_{i\\in I}\\in \\text{Cov}(\\mathcal{S})$", "and $y \\to x$ is a morphism of $\\mathcal{S}$. As $\\{p(x_i) \\to p(x)\\}$", "is a covering of $\\mathcal{C}$ we see that $p(x_i) \\times_{p(x)} p(y)$", "exists. Hence", "Categories, Lemma \\ref{categories-lemma-fibred-category-representable-goes-up}", "implies that $x_i \\times_x y$ exists, that $p(x_i \\times_x y) =", "p(x_i) \\times_{p(x)} p(y)$, and that $x_i \\times_x y \\to y$ is", "strongly cartesian. Since also", "$\\{p(x_i) \\times_{p(x)} p(y) \\to p(y) \\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$", "we conclude that", "$\\{x_i \\times_x y \\to y \\}_{i\\in I} \\in \\text{Cov}(\\mathcal{S})$", "\\end{enumerate}", "This finishes the proof." ], "refs": [ "sites-definition-site", "categories-lemma-composition-cartesian", "categories-lemma-composition-cartesian", "categories-lemma-fibred-category-representable-goes-up" ], "ref_ids": [ 8652, 12282, 12282, 12290 ] } ], "ref_ids": [] }, { "id": 8970, "type": "theorem", "label": "stacks-lemma-topology-inherited-functorial", "categories": [ "stacks" ], "title": "stacks-lemma-topology-inherited-functorial", "contents": [ "Let $\\mathcal{C}$ be a site. Let $F : \\mathcal{X} \\to \\mathcal{Y}$", "be a $1$-morphism of fibred categories over $\\mathcal{C}$.", "Then $F$ is a continuous and cocontinuous functor between the structure", "of sites inherited from $\\mathcal{C}$. Hence $F$ induces a morphism of topoi", "$f : \\Sh(\\mathcal{X}) \\to \\Sh(\\mathcal{Y})$ with", "$f_* = {}_sF = {}_pF$ and $f^{-1} = F^s = F^p$. In particular", "$f^{-1}(\\mathcal{G})(x) = \\mathcal{G}(F(x))$", "for a sheaf $\\mathcal{G}$ on $\\mathcal{Y}$ and object $x$ of $\\mathcal{X}$." ], "refs": [], "proofs": [ { "contents": [ "We first prove that $F$ is continuous.", "Let $\\{x_i \\to x\\}_{i \\in I}$ be a covering of $\\mathcal{X}$. By", "Categories, Definition \\ref{categories-definition-fibred-categories-over-C}", "the functor $F$ transforms strongly cartesian morphisms into strongly", "cartesian morphisms, hence $\\{F(x_i) \\to F(x)\\}_{i \\in I}$ is a covering", "of $\\mathcal{Y}$. This proves part (1) of", "Sites, Definition \\ref{sites-definition-continuous}.", "Moreover, let $x' \\to x$ be a morphism of $\\mathcal{X}$. By", "Categories, Lemma \\ref{categories-lemma-fibred-category-representable-goes-up}", "the fibre product $x_i \\times_x x'$ exists and $x_i \\times_x x' \\to x'$", "is strongly cartesian. Hence $F(x_i \\times_x x') \\to F(x')$ is strongly", "cartesian. By", "Categories, Lemma \\ref{categories-lemma-fibred-category-representable-goes-up}", "applied to $\\mathcal{Y}$ this means that", "$F(x_i \\times_x x') = F(x_i) \\times_{F(x)} F(x')$.", "This proves part (2) of", "Sites, Definition \\ref{sites-definition-continuous}", "and we conclude that $F$ is continuous.", "\\medskip\\noindent", "Next we prove that $F$ is cocontinuous.", "Let $x \\in \\Ob(\\mathcal{X})$ and let $\\{y_i \\to F(x)\\}_{i \\in I}$", "be a covering in $\\mathcal{Y}$. Denote $\\{U_i \\to U\\}_{i \\in I}$ the", "corresponding covering of $\\mathcal{C}$. For each $i$ choose a strongly", "cartesian morphism $x_i \\to x$ in $\\mathcal{X}$ lying over $U_i \\to U$.", "Then $F(x_i) \\to F(x)$ and $y_i \\to F(x)$ are both a strongly cartesian", "morphisms in $\\mathcal{Y}$ lying over $U_i \\to U$. Hence there exists", "a unique isomorphism $F(x_i) \\to y_i$ in $\\mathcal{Y}_{U_i}$ compatible", "with the maps to $F(x)$. Thus $\\{x_i \\to x\\}_{i \\in I}$ is a covering of", "$\\mathcal{X}$ such that $\\{F(x_i) \\to F(x)\\}_{i \\in I}$ is isomorphic to", "$\\{y_i \\to F(x)\\}_{i \\in I}$. Hence $F$ is cocontinuous, see", "Sites, Definition \\ref{sites-definition-cocontinuous}.", "\\medskip\\noindent", "The final assertion follows from the first two, see", "Sites, Lemmas", "\\ref{sites-lemma-cocontinuous-morphism-topoi},", "\\ref{sites-lemma-pu-sheaf}, and", "\\ref{sites-lemma-when-shriek}." ], "refs": [ "categories-definition-fibred-categories-over-C", "sites-definition-continuous", "categories-lemma-fibred-category-representable-goes-up", "categories-lemma-fibred-category-representable-goes-up", "sites-definition-continuous", "sites-definition-cocontinuous", "sites-lemma-cocontinuous-morphism-topoi", "sites-lemma-pu-sheaf", "sites-lemma-when-shriek" ], "ref_ids": [ 12390, 8664, 12290, 12290, 8664, 8670, 8543, 8540, 8545 ] } ], "ref_ids": [] }, { "id": 8971, "type": "theorem", "label": "stacks-lemma-localizing", "categories": [ "stacks" ], "title": "stacks-lemma-localizing", "contents": [ "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{X} \\to \\mathcal{C}$", "be a category fibred in groupoids. Let $x \\in \\Ob(\\mathcal{X})$", "lying over $U = p(x)$. The functor $p$ induces an equivalence of sites", "$\\mathcal{X}/x \\to \\mathcal{C}/U$ where $\\mathcal{X}$ is endowed with", "the topology inherited from $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Here $\\mathcal{C}/U$ is the localization of the site", "$\\mathcal{C}$ at the object $U$ and similarly for $\\mathcal{X}/x$.", "It follows from", "Categories, Definition \\ref{categories-definition-fibred-groupoids}", "that the rule $x'/x \\mapsto p(x')/p(x)$ defines an equivalence of", "categories $\\mathcal{X}/x \\to \\mathcal{C}/U$. Whereupon it follows from", "Definition \\ref{definition-topology-inherited}", "that coverings of $x'$ in $\\mathcal{X}/x$ are in bijective correspondence", "with coverings of $p(x')$ in $\\mathcal{C}/U$." ], "refs": [ "categories-definition-fibred-groupoids", "stacks-definition-topology-inherited" ], "ref_ids": [ 12392, 9002 ] } ], "ref_ids": [] }, { "id": 8972, "type": "theorem", "label": "stacks-lemma-stack-in-groupoids-over-stack-in-groupoids", "categories": [ "stacks" ], "title": "stacks-lemma-stack-in-groupoids-over-stack-in-groupoids", "contents": [ "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{X} \\to \\mathcal{C}$", "and $q : \\mathcal{Y} \\to \\mathcal{C}$", "be stacks in groupoids. Let $F : \\mathcal{X} \\to \\mathcal{Y}$", "be a $1$-morphism of categories over $\\mathcal{C}$. If $F$ turns", "$\\mathcal{X}$ into a category fibred in groupoids over $\\mathcal{Y}$,", "then $\\mathcal{X}$ is a stack in groupoids over $\\mathcal{Y}$ (with", "topology inherited from $\\mathcal{C}$)." ], "refs": [], "proofs": [ { "contents": [ "Let us prove descent for objects. Let $\\{y_i \\to y\\}$ be a covering", "of $\\mathcal{Y}$. Let $(x_i, \\varphi_{ij})$ be a descent datum in $\\mathcal{X}$", "with respect to this covering. Then $(x_i, \\varphi_{ij})$ is also a descent", "datum with respect to the covering $\\{q(y_i) \\to q(y)\\}$ of", "$\\mathcal{C}$. As $\\mathcal{X}$ is a stack in groupoids we obtain an", "object $x$ over $q(y)$ and isomorphisms $\\psi_i : x|_{q(y_i)} \\to x_i$", "over $q(y_i)$ compatible with the $\\varphi_{ij}$, i.e., such that", "$$", "\\varphi_{ij} = \\psi_j|_{q(y_i) \\times_{q(y)} q(y_j)}", "\\circ \\psi_i^{-1}|_{q(y_i) \\times_{q(y)} q(y_j)}.", "$$", "Consider the sheaf $\\mathit{I} = \\mathit{Isom}_\\mathcal{Y}(F(x), y)$", "on $\\mathcal{C}/p(x)$. Note that $s_i = F(\\psi_i) \\in \\mathit{I}(q(x_i))$", "because $F(x_i) = y_i$. Because $F(\\varphi_{ij}) = \\text{id}$ (as we started", "with a descent datum over $\\{y_i \\to y\\}$) the displayed formula shows", "that $s_i|_{q(y_i) \\times_{q(y)} q(y_j)} = s_j|_{q(y_i) \\times_{q(y)} q(y_j)}$.", "Hence the local sections $s_i$ glue to $s : F(x) \\to y$. As $F$ is fibred in", "groupoids we see that $x$ is isomorphic to an object $x'$ with $F(x') = y$.", "We omit the verification that $x'$ in the fibre category of $\\mathcal{X}$", "over $y$ is a solution to the problem of descent posed by the descent datum", "$(x_i, \\varphi_{ij})$. We also omit the proof of the sheaf property", "of the $\\mathit{Isom}$-presheaves of $\\mathcal{X}/\\mathcal{Y}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 8973, "type": "theorem", "label": "stacks-lemma-stack-over-stack", "categories": [ "stacks" ], "title": "stacks-lemma-stack-over-stack", "contents": [ "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{X} \\to \\mathcal{C}$", "be a stack. Endow $\\mathcal{X}$ with the topology inherited from", "$\\mathcal{C}$ and let $q : \\mathcal{Y} \\to \\mathcal{X}$ be a stack.", "Then $\\mathcal{Y}$ is a stack over $\\mathcal{C}$.", "If $p$ and $q$ define stacks in groupoids, then", "$\\mathcal{Y}$ is a stack in groupoids over $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "We check the three conditions in Definition \\ref{definition-stack}", "to prove that $\\mathcal{Y}$ is a stack over $\\mathcal{C}$.", "By Categories, Lemma \\ref{categories-lemma-fibred-over-fibred}", "we find that $\\mathcal{Y}$ is a fibred category over $\\mathcal{C}$.", "Thus condition (1) holds.", "\\medskip\\noindent", "Let $U$ be an object of $\\mathcal{C}$ and let $y_1, y_2$ be objects", "of $\\mathcal{Y}$ over $U$. Denote $x_i = q(y_i)$ in $\\mathcal{X}$.", "Consider the map of presheaves", "$$", "q : \\mathit{Mor}_{\\mathcal{Y}/\\mathcal{C}}(y_1, y_2)", "\\longrightarrow", "\\mathit{Mor}_{\\mathcal{X}/\\mathcal{C}}(x_1, x_2)", "$$", "on $\\mathcal{C}/U$, see Lemma \\ref{lemma-presheaf-mor-map-fibred-categories}.", "Let $\\{U_i \\to U\\}$ be a covering and let $\\varphi_i$ be a section", "of the presheaf on the left over $U_i$ such that $\\varphi_i$ and", "$\\varphi_j$ restrict to the same section over $U_i \\times_U U_j$.", "We have to find a morphism $\\varphi : x_1 \\to x_2$ restricting to $\\varphi_i$.", "Note that $q(\\varphi_i) = \\psi|_{U_i}$ for some morphism", "$\\psi : x_1 \\to x_2$ over $U$ because the second presheaf is a sheaf", "(by assumption). Let $y_{12} \\to y_2$ be the stronly $\\mathcal{X}$-cartesian", "morphism of $\\mathcal{Y}$ lying over $\\psi$. Then $\\varphi_i$ corresponds", "to a morphism $\\varphi'_i : y_1|_{U_i} \\to y_{12}|_{U_i}$ over $x_1|_{U_i}$.", "In other words, $\\varphi'_i$ now define local sections of the presheaf", "$$", "\\mathit{Mor}_{\\mathcal{Y}/\\mathcal{X}}(y_1, y_{12})", "$$", "over the members of the covering $\\{x_1|_{U_i} \\to x_1\\}$. By assumption these", "glue to a unique morphism $y_1 \\to y_{12}$ which composed with the given", "morphism $y_{12} \\to y_2$ produces the desired morphism $y_1 \\to y_2$.", "\\medskip\\noindent", "Finally, we show that descent data are effective. Let $\\{f_i : U_i \\to U\\}$", "be a covering of $\\mathcal{C}$ and let $(y_i, \\varphi_{ij})$ be a descent", "datum relative to this covering (Definition \\ref{definition-descent-data}).", "Setting $x_i = q(y_i)$ and $\\psi_{ij} = q(\\varphi_{ij})$", "we obtain a descent datum $(x_i, \\psi_{ij})$ for the covering in $\\mathcal{X}$.", "By assumption on $\\mathcal{X}$ we may assume $x_i = x|_{U_i}$", "and the $\\psi_{ij}$ equal to the canonical descent datum", "(Definition \\ref{definition-effective-descent-datum}).", "In this case $\\{x|_{U_i} \\to x\\}$ is a covering and we can view", "$(y_i, \\varphi_{ij})$ as a descent datum relative to this covering.", "By our assumption that $\\mathcal{Y}$ is a stack over $\\mathcal{C}$", "we see that it is effective which finishes the proof of condition (3).", "\\medskip\\noindent", "The final assertion follows because $\\mathcal{Y}$ is a stack over", "$\\mathcal{C}$ and is fibred in groupoids by", "Categories, Lemma", "\\ref{categories-lemma-fibred-in-groupoids-over-fibred-in-groupoids}." ], "refs": [ "stacks-definition-stack", "categories-lemma-fibred-over-fibred", "stacks-lemma-presheaf-mor-map-fibred-categories", "stacks-definition-descent-data", "stacks-definition-effective-descent-datum", "categories-lemma-fibred-in-groupoids-over-fibred-in-groupoids" ], "ref_ids": [ 8996, 12289, 8935, 8993, 8995, 12302 ] } ], "ref_ids": [] }, { "id": 8974, "type": "theorem", "label": "stacks-lemma-gerbe-equivalent", "categories": [ "stacks" ], "title": "stacks-lemma-gerbe-equivalent", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{S}_1$, $\\mathcal{S}_2$ be categories over $\\mathcal{C}$.", "Suppose that $\\mathcal{S}_1$ and $\\mathcal{S}_2$ are equivalent", "as categories over $\\mathcal{C}$.", "Then $\\mathcal{S}_1$ is a gerbe over $\\mathcal{C}$ if and only if", "$\\mathcal{S}_2$ is a gerbe over $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{S}_1$ is a gerbe over $\\mathcal{C}$. By", "Lemma \\ref{lemma-stack-in-groupoids-equivalent}", "we see $\\mathcal{S}_2$ is a stack in groupoids over $\\mathcal{C}$.", "Let $F : \\mathcal{S}_1 \\to \\mathcal{S}_2$,", "$G : \\mathcal{S}_2 \\to \\mathcal{S}_1$ be equivalences of categories over", "$\\mathcal{C}$. Given $U \\in \\Ob(\\mathcal{C})$ we see that there exists", "a covering $\\{U_i \\to U\\}$ such that $(\\mathcal{S}_1)_{U_i}$ is", "nonempty. Applying $F$ we see that $(\\mathcal{S}_2)_{U_i}$ is nonempty.", "Given $U \\in \\Ob(\\mathcal{C})$ and", "$x, y \\in \\Ob((\\mathcal{S}_2)_U)$ there exists", "a covering $\\{U_i \\to U\\}$ in $\\mathcal{C}$ such that", "$G(x)|_{U_i} \\cong G(y)|_{U_i}$ in $(\\mathcal{S}_1)_{U_i}$. By", "Categories, Lemma \\ref{categories-lemma-equivalence-fibred-categories}", "this implies $x|_{U_i} \\cong y|_{U_i}$ in $(\\mathcal{S}_2)_{U_i}$." ], "refs": [ "stacks-lemma-stack-in-groupoids-equivalent", "categories-lemma-equivalence-fibred-categories" ], "ref_ids": [ 8948, 12297 ] } ], "ref_ids": [] }, { "id": 8975, "type": "theorem", "label": "stacks-lemma-when-gerbe", "categories": [ "stacks" ], "title": "stacks-lemma-when-gerbe", "contents": [ "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{X} \\to \\mathcal{C}$", "and $q : \\mathcal{Y} \\to \\mathcal{C}$ be stacks in groupoids.", "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "over $\\mathcal{C}$. The following are equivalent", "\\begin{enumerate}", "\\item For some (equivalently any) factorization $F = F' \\circ a$ where", "$a : \\mathcal{X} \\to \\mathcal{X}'$ is an equivalence of categories over", "$\\mathcal{C}$ and $F'$ is fibred in groupoids, the map", "$F' : \\mathcal{X}' \\to \\mathcal{Y}$ is a gerbe (with the topology", "on $\\mathcal{Y}$ inherited from $\\mathcal{C}$).", "\\item The following two conditions are satisfied", "\\begin{enumerate}", "\\item for $y \\in \\Ob(\\mathcal{Y})$ lying over", "$U \\in \\Ob(\\mathcal{C})$ there exists a covering", "$\\{U_i \\to U\\}$ in $\\mathcal{C}$ and objects $x_i$ of $\\mathcal{X}$", "over $U_i$ such that $F(x_i) \\cong y|_{U_i}$ in $\\mathcal{Y}_{U_i}$, and", "\\item for $U \\in \\Ob(\\mathcal{C})$,", "$x, x' \\in \\Ob(\\mathcal{X}_U)$, and $b : F(x) \\to F(x')$ in", "$\\mathcal{Y}_U$ there exists", "a covering $\\{U_i \\to U\\}$ in $\\mathcal{C}$ and morphisms", "$a_i : x|_{U_i} \\to x'|_{U_i}$ in $\\mathcal{X}_{U_i}$ with", "$F(a_i) = b|_{U_i}$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By", "Categories, Lemma", "\\ref{categories-lemma-ameliorate-morphism-categories-fibred-groupoids}", "there exists a factorization $F = F' \\circ a$ where", "$a : \\mathcal{X} \\to \\mathcal{X}'$ is an equivalence of categories over", "$\\mathcal{C}$ and $F'$ is fibred in groupoids. By", "Categories, Lemma \\ref{categories-lemma-amelioration-unique}", "given any two such factorizations $F = F' \\circ a = F'' \\circ b$", "we have that $\\mathcal{X}'$ is equivalent to $\\mathcal{X}''$ as", "categories over $\\mathcal{Y}$. Hence", "Lemma \\ref{lemma-gerbe-equivalent}", "guarantees that the condition (1) is independent of the choice of the", "factorization. Moreover, this means that we may assume", "$\\mathcal{X}' = \\mathcal{X} \\times_{F, \\mathcal{Y}, \\text{id}} \\mathcal{Y}$", "as in the proof of", "Categories, Lemma", "\\ref{categories-lemma-ameliorate-morphism-categories-fibred-groupoids}", "\\medskip\\noindent", "Let us prove that (a) and (b) imply that $\\mathcal{X}' \\to \\mathcal{Y}$", "is a gerbe. First of all, by", "Lemma \\ref{lemma-stack-in-groupoids-over-stack-in-groupoids}", "we see that $\\mathcal{X}' \\to \\mathcal{Y}$ is a stack in groupoids.", "Next, let $y$ be an object of $\\mathcal{Y}$ lying over", "$U \\in \\Ob(\\mathcal{C})$. By (a) we can find a covering", "$\\{U_i \\to U\\}$ in $\\mathcal{C}$ and objects $x_i$ of $\\mathcal{X}$", "over $U_i$ and isomorphisms $f_i : F(x_i) \\to y|_{U_i}$ in", "$\\mathcal{Y}_{U_i}$. Then $(U_i, x_i, y|_{U_i}, f_i)$ are objects", "of $\\mathcal{X}'_{U_i}$, i.e., the second condition of", "Definition \\ref{definition-gerbe}", "holds. Finally, let $(U, x, y, f)$ and $(U, x', y, f')$ be objects", "of $\\mathcal{X}'$ lying over the same object $y \\in \\Ob(\\mathcal{Y})$.", "Set $b = (f')^{-1} \\circ f$. By condition (b) we can find a covering", "$\\{U_i \\to U\\}$ and isomorphisms $a_i : x|_{U_i} \\to x'|_{U_i}$", "in $\\mathcal{X}_{U_i}$ with $F(a_i) = b|_{U_i}$. Then", "$$", "(a_i, \\text{id}) : (U, x, y, f)|_{U_i} \\to (U, x', y, f')|_{U_i}", "$$", "is a morphism in $\\mathcal{X}'_{U_i}$ as desired. This proves that", "(2) implies (1).", "\\medskip\\noindent", "To prove that (1) implies (2) one reads the arguments", "in the preceding paragraph backwards. Details omitted." ], "refs": [ "categories-lemma-ameliorate-morphism-categories-fibred-groupoids", "categories-lemma-amelioration-unique", "stacks-lemma-gerbe-equivalent", "categories-lemma-ameliorate-morphism-categories-fibred-groupoids", "stacks-lemma-stack-in-groupoids-over-stack-in-groupoids", "stacks-definition-gerbe" ], "ref_ids": [ 12304, 12305, 8974, 12304, 8972, 9003 ] } ], "ref_ids": [] }, { "id": 8976, "type": "theorem", "label": "stacks-lemma-base-change-gerbe", "categories": [ "stacks" ], "title": "stacks-lemma-base-change-gerbe", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[r]_{G'} \\ar[d]_{F'} & \\mathcal{X} \\ar[d]^F \\\\", "\\mathcal{Y}' \\ar[r]^G & \\mathcal{Y}", "}", "$$", "be a $2$-fibre product of stacks in groupoids over $\\mathcal{C}$.", "If $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, then", "$\\mathcal{X}'$ is a gerbe over $\\mathcal{Y}'$." ], "refs": [], "proofs": [ { "contents": [ "By the uniqueness property of a $2$-fibre product may assume that", "$\\mathcal{X}' = \\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X}$", "as in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.", "Let us prove properties (2)(a) and (2)(b) of", "Lemma \\ref{lemma-when-gerbe}", "for $\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Y}'$.", "\\medskip\\noindent", "Let $y'$ be an object of $\\mathcal{Y}'$ lying over the object $U$", "of $\\mathcal{C}$. By assumption there exists", "a covering $\\{U_i \\to U\\}$ of $U$ and objects", "$x_i \\in \\mathcal{X}_{U_i}$ with isomorphisms", "$\\alpha_i : G(y')|_{U_i} \\to F(x_i)$.", "Then $(U_i, y'|_{U_i}, x_i, \\alpha_i)$ is an object of", "$\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X}$ over $U_i$", "whose image in $\\mathcal{Y}'$ is $y'|_{U_i}$. Thus (2)(a) holds.", "\\medskip\\noindent", "Let $U \\in \\Ob(\\mathcal{C})$, let $x'_1, x'_2$ be objects of", "$\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X}$ over $U$, and let", "$b' : F'(x'_1) \\to F'(x'_2)$ be a morphism in $\\mathcal{Y}'_U$.", "Write $x'_i = (U, y'_i, x_i, \\alpha_i)$. Note that $F'(x'_i) = x_i$", "and $G'(x'_i) = y'_i$. By assumption there exists", "a covering $\\{U_i \\to U\\}$ in $\\mathcal{C}$ and morphisms", "$a_i : x_1|_{U_i} \\to x_2|_{U_i}$ in $\\mathcal{X}_{U_i}$ with", "$F(a_i) = G(b')|_{U_i}$. Then $(b'|_{U_i}, a_i)$ is a morphism", "$x'_1|_{U_i} \\to x'_2|_{U_i}$ as required in (2)(b)." ], "refs": [ "categories-lemma-2-product-categories-over-C", "stacks-lemma-when-gerbe" ], "ref_ids": [ 12280, 8975 ] } ], "ref_ids": [] }, { "id": 8977, "type": "theorem", "label": "stacks-lemma-composition-gerbe", "categories": [ "stacks" ], "title": "stacks-lemma-composition-gerbe", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$F : \\mathcal{X} \\to \\mathcal{Y}$ and $G : \\mathcal{Y} \\to \\mathcal{Z}$", "be $1$-morphisms of stacks in groupoids over $\\mathcal{C}$.", "If $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$ and", "$\\mathcal{Y}$ is a gerbe over $\\mathcal{Z}$, then", "$\\mathcal{X}$ is a gerbe over $\\mathcal{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Let us prove properties (2)(a) and (2)(b) of", "Lemma \\ref{lemma-when-gerbe}", "for $\\mathcal{X} \\to \\mathcal{Z}$.", "\\medskip\\noindent", "Let $z$ be an object of $\\mathcal{Z}$ lying over the object $U$", "of $\\mathcal{C}$. By assumption on $G$ there exists", "a covering $\\{U_i \\to U\\}$ of $U$ and objects", "$y_i \\in \\mathcal{Y}_{U_i}$ such that $G(y_i) \\cong z|_{U_i}$.", "By assumption on $F$ there exist coverings $\\{U_{ij} \\to U_i\\}$", "and objects $x_{ij} \\in \\mathcal{X}_{U_{ij}}$ such that", "$F(x_{ij}) \\cong y_i|_{U_{ij}}$.", "Then $\\{U_{ij} \\to U\\}$ is a covering of $\\mathcal{C}$", "and $(G \\circ F)(x_{ij}) \\cong z|_{U_{ij}}$. Thus (2)(a) holds.", "\\medskip\\noindent", "Let $U \\in \\Ob(\\mathcal{C})$, let $x_1, x_2$ be objects of", "$\\mathcal{X}$ over $U$, and let", "$c : (G \\circ F)(x_1) \\to (G \\circ F)(x_2)$ be a morphism in", "$\\mathcal{Z}_U$. By assumption on $G$ there exists", "a covering $\\{U_i \\to U\\}$ of $U$ and morphisms", "$b_i : F(x_1)|_{U_i} \\to F(x_2)|_{U_i}$ in $\\mathcal{Y}_{U_i}$", "such that $G(b_i) = c|_{U_i}$.", "By assumption on $F$ there exist coverings $\\{U_{ij} \\to U_i\\}$", "and morphisms $a_{ij} : x_1|_{U_{ij}} \\to x_2|_{U_{ij}}$ in", "$\\mathcal{X}_{U_{ij}}$ such that $F(a_{ij}) = b_i|_{U_{ij}}$.", "Then $\\{U_{ij} \\to U\\}$ is a covering of $\\mathcal{C}$", "and $(G \\circ F)(a_{ij}) = c|_{U_{ij}}$ as required in (2)(b)." ], "refs": [ "stacks-lemma-when-gerbe" ], "ref_ids": [ 8975 ] } ], "ref_ids": [] }, { "id": 8978, "type": "theorem", "label": "stacks-lemma-gerbe-descent", "categories": [ "stacks" ], "title": "stacks-lemma-gerbe-descent", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[r]_{G'} \\ar[d]_{F'} & \\mathcal{X} \\ar[d]^F \\\\", "\\mathcal{Y}' \\ar[r]^G & \\mathcal{Y}", "}", "$$", "be a $2$-cartesian diagram of stacks in groupoids over $\\mathcal{C}$.", "If for every $U \\in \\Ob(\\mathcal{C})$ and", "$x \\in \\Ob(\\mathcal{Y}_U)$ there exists a covering", "$\\{U_i \\to U\\}$ such that $x|_{U_i}$ is in the essential", "image of $G : \\mathcal{Y}'_{U_i} \\to \\mathcal{Y}_{U_i}$ and", "$\\mathcal{X}'$ is a gerbe over $\\mathcal{Y}'$, then", "$\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$." ], "refs": [], "proofs": [ { "contents": [ "By the uniqueness property of a $2$-fibre product may assume that", "$\\mathcal{X}' = \\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X}$", "as in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.", "Let us prove properties (2)(a) and (2)(b) of", "Lemma \\ref{lemma-when-gerbe}", "for $\\mathcal{X} \\to \\mathcal{Y}$.", "\\medskip\\noindent", "Let $y$ be an object of $\\mathcal{Y}$ lying over the object $U$", "of $\\mathcal{C}$. By assumption there exists", "a covering $\\{U_i \\to U\\}$ of $U$ and objects", "$y'_i \\in \\mathcal{Y}'_{U_i}$ with $G(y'_i) \\cong y|_{U_i}$.", "By (2)(a) for $\\mathcal{X}' \\to \\mathcal{Y}'$ there exist", "coverings $\\{U_{ij} \\to U_i\\}$ and objects $x'_{ij}$ of", "$\\mathcal{X}'$ over $U_{ij}$ with $F'(x'_{ij})$ isomorphic", "to the restriction of $y'_i$ to $U_{ij}$. Then", "$\\{U_{ij} \\to U\\}$ is a covering of $\\mathcal{C}$ and", "$G'(x'_{ij})$ are objects of $\\mathcal{X}$ over $U_{ij}$", "whose images in $\\mathcal{Y}$ are isomorphic to the restrictions", "$y|_{U_{ij}}$. This proves (2)(a) for $\\mathcal{X} \\to \\mathcal{Y}$.", "\\medskip\\noindent", "Let $U \\in \\Ob(\\mathcal{C})$, let $x_1, x_2$ be objects of", "$\\mathcal{X}$ over $U$, and let $b : F(x_1) \\to F(x_2)$ be a morphism", "in $\\mathcal{Y}_U$. By assumption we may choose a covering", "$\\{U_i \\to U\\}$ and objects $y'_i$ of $\\mathcal{Y}'$", "over $U_i$ such that there exist isomorphisms", "$\\alpha_i : G(y'_i) \\to F(x_1)|_{U_i}$.", "Then we get objects", "$$", "x'_{1i} = (U_i, y'_i, x_1|_{U_i}, \\alpha_i)", "\\quad\\text{and}\\quad", "x'_{2i} = (U_i, y'_i, x_2|_{U_i}, b|_{U_i} \\circ \\alpha_i)", "$$", "of $\\mathcal{X}'$ over $U_i$. The identity morphism on $y'_i$ is a", "morphism $F'(x'_{1i}) \\to F'(x'_{2i})$. By (2)(b) for", "$\\mathcal{X}' \\to \\mathcal{Y}'$ there exist coverings", "$\\{U_{ij} \\to U_i\\}$ and morphisms", "$a'_{ij} : x'_{1i}|_{U_{ij}} \\to x'_{2i}|_{U_{ij}}$", "such that $F'(a'_{ij}) = \\text{id}_{y'_i}|_{U_{ij}}$. Unwinding the definition", "of morphisms in $\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X}$", "we see that $G'(a'_{ij}) : x_1|_{U_{ij}} \\to x_2|_{U_{ij}}$ are", "the morphisms we're looking for, i.e., (2)(b) holds for", "$\\mathcal{X} \\to \\mathcal{Y}$." ], "refs": [ "categories-lemma-2-product-categories-over-C", "stacks-lemma-when-gerbe" ], "ref_ids": [ 12280, 8975 ] } ], "ref_ids": [] }, { "id": 8979, "type": "theorem", "label": "stacks-lemma-gerbe-abelian-auts", "categories": [ "stacks" ], "title": "stacks-lemma-gerbe-abelian-auts", "contents": [ "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a gerbe over a site $\\mathcal{C}$.", "Assume that for all $U \\in \\Ob(\\mathcal{C})$ and $x \\in \\Ob(\\mathcal{S}_U)$", "the sheaf of groups $\\mathit{Aut}(x) = \\mathit{Isom}(x, x)$ on $\\mathcal{C}/U$", "is abelian. Then there exist", "\\begin{enumerate}", "\\item a sheaf $\\mathcal{G}$ of abelian groups on $\\mathcal{C}$,", "\\item for every $U \\in \\Ob(\\mathcal{C})$ and every $x \\in \\Ob(\\mathcal{S}_U)$", "an isomorphism $\\mathcal{G}|_U \\to \\mathit{Aut}(x)$", "\\end{enumerate}", "such that for every $U$ and every morphism $\\varphi : x \\to y$", "in $\\mathcal{S}_U$ the diagram", "$$", "\\xymatrix{", "\\mathcal{G}|_U \\ar[d] \\ar@{=}[rr] & & \\mathcal{G}|_U \\ar[d] \\\\", "\\mathit{Aut}(x)", "\\ar[rr]^{\\alpha \\mapsto \\varphi \\circ \\alpha \\circ \\varphi^{-1}} & &", "\\mathit{Aut}(y)", "}", "$$", "is commutative." ], "refs": [], "proofs": [ { "contents": [ "Let $x, y$ be two objects of $\\mathcal{S}$ with $U = p(x) = p(y)$.", "\\medskip\\noindent", "If there is a morphism $\\varphi : x \\to y$ over $U$, then it is an", "isomorphism and then we indeed get an isomorphism", "$\\mathit{Aut}(x) \\to \\mathit{Aut}(y)$ sending", "$\\alpha$ to $\\varphi \\circ \\alpha \\circ \\varphi^{-1}$.", "Moreover, since we are assuming $\\mathit{Aut}(x)$ is", "commutative, this isomorphism is independent of the choice", "of $\\varphi$ by a simple computation: namely, if $\\psi$ is", "a second such map, then", "$$", "\\varphi \\circ \\alpha \\circ \\varphi^{-1} =", "\\psi \\circ \\psi^{-1} \\circ \\varphi \\circ \\alpha \\circ \\varphi^{-1} =", "\\psi \\circ \\alpha \\circ \\psi^{-1} \\circ \\varphi \\circ \\varphi^{-1} =", "\\psi \\circ \\alpha \\circ \\psi^{-1}", "$$", "The upshot is a canonical isomorphism of sheaves", "$\\mathit{Aut}(x) \\to \\mathit{Aut}(y)$. Furthermore, if there", "is a third object $z$ and a morphism $y \\to z$ (and hence", "also a morphism $x \\to z$), then the", "canonical isomorphisms $\\mathit{Aut}(x) \\to \\mathit{Aut}(y)$,", "$\\mathit{Aut}(y) \\to \\mathit{Aut}(z)$, and", "$\\mathit{Aut}(x) \\to \\mathit{Aut}(z)$ are compatible in the sense", "that", "$$", "\\xymatrix{", "\\mathit{Aut}(x) \\ar[rd] \\ar[rr] & & \\mathit{Aut}(z) \\\\", "& \\mathit{Aut}(y) \\ar[ru]", "}", "$$", "commutes.", "\\medskip\\noindent", "If there is no morphism from $x$ to $y$ over $U$, then we can", "choose a covering $\\{U_i \\to U\\}$ such that there exist morphisms", "$x|_{U_i} \\to y|_{U_i}$. This gives canonical isomorphisms", "$$", "\\mathit{Aut}(x)|_{U_i} \\longrightarrow \\mathit{Aut}(y)|_{U_i}", "$$", "which agree over $U_i \\times_U U_j$ (by canonicity). By glueing of", "sheaves (Sites, Lemma \\ref{sites-lemma-glue-maps}) we get a unique", "isomorphism $\\mathit{Aut}(x) \\to \\mathit{Aut}(y)$ whose restriction", "to any $U_i$ is the canonical isomorphism of the previous paragraph.", "Similarly to the above these canonical isomorphisms satisfy", "a compatibility if we have a third object over $U$.", "\\medskip\\noindent", "What if the fibre category of $\\mathcal{S}$ over $U$ is empty?", "Well, in this case we can find a covering $\\{U_i \\to U\\}$", "and objects $x_i$ of $\\mathcal{S}$ over $U_i$. Then we set", "$\\mathcal{G}_i = \\mathit{Aut}(x_i)$. By the above we obtain", "canonical isomorphisms", "$$", "\\varphi_{ij} :", "\\mathcal{G}_i|_{U_i \\times_U U_j}", "\\longrightarrow", "\\mathcal{G}_j|_{U_i \\times_U U_j}", "$$", "whose restrictions to $U_i \\times_U U_j \\times_U U_k$ satisfy", "the cocycle condition explained in Sites, Section", "\\ref{sites-section-glueing-sheaves}.", "By Sites, Lemma \\ref{sites-lemma-glue-sheaves}", "we obtain a sheaf $\\mathcal{G}$ over $U$ whose", "restriction to $U_i$ gives $\\mathcal{G}_i$", "in a manner compatible with the glueing maps $\\varphi_{ij}$.", "\\medskip\\noindent", "If $\\mathcal{C}$ has a final object $U$, then this finishes the proof", "as we can take $\\mathcal{G}$ equal to the sheaf we just constructed.", "In the general case we need to verify that the sheaves $\\mathcal{G}$", "constructed over varying $U$ are compatible in a canonical manner.", "This is omitted." ], "refs": [ "sites-lemma-glue-maps", "sites-lemma-glue-sheaves" ], "ref_ids": [ 8561, 8564 ] } ], "ref_ids": [] }, { "id": 8980, "type": "theorem", "label": "stacks-lemma-fibred-category-pushforward", "categories": [ "stacks" ], "title": "stacks-lemma-fibred-category-pushforward", "contents": [ "In the situation above, if $\\mathcal{S}$ is a fibred category over", "$\\mathcal{D}$ then $u^p\\mathcal{S}$ is a fibred category over $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Please take a look at the discussion surrounding", "Categories, Definitions \\ref{categories-definition-cartesian-over-C} and", "\\ref{categories-definition-fibred-category}", "before reading this proof.", "Let $(a, \\beta) : (U, y) \\to (U', y')$ be a morphism of $u^p\\mathcal{S}$.", "We claim that $(a, \\beta)$ is strongly cartesian if and only if", "$\\beta$ is strongly cartesian. First, assume $\\beta$ is strongly cartesian.", "Consider any second morphism", "$(a_1, \\beta_1) : (U_1, y_1) \\to (U', y')$ of $u^p\\mathcal{S}$.", "Then", "\\begin{align*}", "& \\Mor_{u^p\\mathcal{S}}((U_1, y_1), (U, y)) \\\\", "& =", "\\Mor_\\mathcal{C}(U_1, U)", "\\times_{\\Mor_\\mathcal{D}(u(U_1), u(U))}", "\\Mor_\\mathcal{S}(y_1, y) \\\\", "& =", "\\Mor_\\mathcal{C}(U_1, U)", "\\times_{\\Mor_\\mathcal{D}(u(U_1), u(U))}", "\\Mor_\\mathcal{S}(y_1, y')", "\\times_{\\Mor_\\mathcal{D}(u(U_1), u(U'))}", "\\Mor_\\mathcal{D}(u(U_1), u(U)) \\\\", "& =", "\\Mor_\\mathcal{S}(y_1, y')", "\\times_{\\Mor_\\mathcal{D}(u(U_1), u(U'))}", "\\Mor_\\mathcal{C}(U_1, U) \\\\", "& =", "\\Mor_{u^p\\mathcal{S}}((U_1, y_1), (U', y'))", "\\times_{\\Mor_\\mathcal{C}(U_1, U')}", "\\Mor_\\mathcal{C}(U_1, U)", "\\end{align*}", "the second equality as $\\beta$ is strongly cartesian. Hence we see that", "indeed $(a, \\beta)$ is strongly cartesian. Conversely, suppose that", "$(a, \\beta)$ is strongly cartesian. Choose a strongly cartesian morphism", "$\\beta' : y'' \\to y'$ in $\\mathcal{S}$ with $p(\\beta') = u(a)$.", "Then bot $(a, \\beta) : (U, y) \\to (U, y')$ and", "$(a, \\beta') : (U, y'') \\to (U, y)$ are strongly cartesian and", "lift $a$. Hence, by the uniqueness of strongly cartesian morphisms", "(see discussion in", "Categories, Section \\ref{categories-section-fibred-categories})", "there exists an isomorphism $\\iota : y \\to y''$ in $\\mathcal{S}_{u(U)}$", "such that $\\beta = \\beta' \\circ \\iota$, which implies that", "$\\beta$ is strongly cartesian in $\\mathcal{S}$ by", "Categories, Lemma \\ref{categories-lemma-composition-cartesian}.", "\\medskip\\noindent", "Finally, we have to show", "that given $(U', y')$ and $U \\to U'$ we can find a strongly", "cartesian morphism $(U, y) \\to (U', y')$ in $u^p\\mathcal{S}$", "lifting the morphism $U \\to U'$. This follows from the above as by assumption", "we can find a strongly cartesian morphism $y \\to y'$ lifting the", "morphism $u(U) \\to u(U')$." ], "refs": [ "categories-definition-cartesian-over-C", "categories-definition-fibred-category", "categories-lemma-composition-cartesian" ], "ref_ids": [ 12387, 12388, 12282 ] } ], "ref_ids": [] }, { "id": 8981, "type": "theorem", "label": "stacks-lemma-stack-pushforward", "categories": [ "stacks" ], "title": "stacks-lemma-stack-pushforward", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor of sites.", "Let $p : \\mathcal{S} \\to \\mathcal{D}$ be a stack over $\\mathcal{D}$.", "Then $u^p\\mathcal{S}$ is a stack over $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "We have seen in", "Lemma \\ref{lemma-fibred-category-pushforward}", "that $u^p\\mathcal{S}$ is a fibred category over $\\mathcal{C}$.", "Moreover, in the proof of that lemma we have seen that a morphism", "$(a, \\beta)$ of $u^p\\mathcal{S}$ is strongly cartesian if and", "only $\\beta$ is strongly cartesian in $\\mathcal{S}$. Hence,", "given a morphism $a : U \\to U'$ of $\\mathcal{C}$, not only", "do we have the equalities", "$(u^p\\mathcal{S})_U = \\mathcal{S}_U$ and", "$(u^p\\mathcal{S})_{U'} = \\mathcal{S}_{U'}$, but via these", "equalities the pullback functors agree; in a formula", "$a^*(U', y') = (U, u(a)^*y')$.", "\\medskip\\noindent", "Having said this, let $\\mathcal{U} = \\{U_i \\to U\\}$ be a covering", "of $\\mathcal{C}$. As $u$ is continuous we see that", "$\\mathcal{V} = \\{u(U_i) \\to u(U)\\}$ is a covering of $\\mathcal{D}$,", "and that $u(U_i \\times_U U_j) = u(U_i) \\times_{u(U)} u(U_j)$ and", "similarly for the triple fibre products $U_i \\times_U U_j \\times_U U_k$.", "As we have the identifications of fibre categories and pullbacks", "we see that descend data relative to $\\mathcal{U}$ are identical", "to descend data relative to $\\mathcal{V}$. Since by assumption we have", "effective descent in $\\mathcal{S}$ we conclude the same holds for", "$u^p\\mathcal{S}$." ], "refs": [ "stacks-lemma-fibred-category-pushforward" ], "ref_ids": [ 8980 ] } ], "ref_ids": [] }, { "id": 8982, "type": "theorem", "label": "stacks-lemma-stack-in-groupoids-pushforward", "categories": [ "stacks" ], "title": "stacks-lemma-stack-in-groupoids-pushforward", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor of sites.", "Let $p : \\mathcal{S} \\to \\mathcal{D}$ be a stack in groupoids", "over $\\mathcal{D}$. Then $u^p\\mathcal{S}$ is a stack in groupoids", "over $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from", "Lemma \\ref{lemma-stack-pushforward}", "and the fact that all fibre categories are groupoids." ], "refs": [ "stacks-lemma-stack-pushforward" ], "ref_ids": [ 8981 ] } ], "ref_ids": [] }, { "id": 8983, "type": "theorem", "label": "stacks-lemma-right-multiplicative-system", "categories": [ "stacks" ], "title": "stacks-lemma-right-multiplicative-system", "contents": [ "In the situation above assume", "\\begin{enumerate}", "\\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category,", "\\item $\\mathcal{C}$ has nonempty finite limits, and", "\\item $u : \\mathcal{C} \\to \\mathcal{D}$ commutes with nonempty finite limits.", "\\end{enumerate}", "Consider the set $R \\subset \\text{Arrows}(u_{pp}\\mathcal{S})$ of morphisms", "of the form", "$$", "(a, \\text{id}_V, \\alpha) :", "(U', \\phi' : V \\to u(U'), x')", "\\longrightarrow", "(U, \\phi : V \\to u(U), x)", "$$", "with $\\alpha$ strongly cartesian. Then $R$ is a right multiplicative system." ], "refs": [], "proofs": [ { "contents": [ "According to", "Categories, Definition \\ref{categories-definition-multiplicative-system}", "we have to check RMS1, RMS2, RMS3.", "Condition RMS1 holds as a composition of strongly cartesian morphisms is", "strongly cartesian, see", "Categories, Lemma \\ref{categories-lemma-composition-cartesian}.", "\\medskip\\noindent", "To check RMS2 suppose we have a morphism", "$$", "(a, b, \\alpha) :", "(U_1, \\phi_1 : V_1 \\to u(U_1), x_1)", "\\longrightarrow", "(U, \\phi : V \\to u(U), x)", "$$", "of $u_{pp}\\mathcal{S}$ and a morphism", "$$", "(c, \\text{id}_V, \\gamma) :", "(U', \\phi' : V \\to u(U'), x')", "\\longrightarrow", "(U, \\phi : V \\to u(U), x)", "$$", "with $\\gamma$ strongly cartesian from $R$. In this situation set", "$U'_1 = U_1 \\times_U U'$, and denote $a' : U'_1 \\to U'$ and", "$c' : U'_1 \\to U_1$ the projections.", "As $u(U'_1) = u(U_1) \\times_{u(U)} u(U')$", "we see that $\\phi'_1 = (\\phi_1, \\phi') : V_1 \\to u(U'_1)$ is", "a morphism in $\\mathcal{D}$. Let $\\gamma_1 : x_1' \\to x_1$", "be a strongly cartesian morphism of $\\mathcal{S}$ with", "$p(\\gamma_1) = \\phi'_1$ (which exists because $\\mathcal{S}$ is a", "fibred category over $\\mathcal{C}$). Then as $\\gamma : x' \\to x$ is", "strongly cartesian there exists a unique morphism", "$\\alpha' : x'_1 \\to x'$ with $p(\\alpha') = a'$.", "At this point we see that", "$$", "(a', b, \\alpha') :", "(U_1, \\phi_1 : V_1 \\to u(U'_1), x'_1)", "\\longrightarrow", "(U, \\phi : V \\to u(U'), x')", "$$", "is a morphism and that", "$$", "(c', \\text{id}_{V_1}, \\gamma_1) :", "(U'_1, \\phi'_1 : V_1 \\to u(U'_1), x'_1)", "\\longrightarrow", "(U_1, \\phi : V_1 \\to u(U_1), x_1)", "$$", "is an element of $R$ which form a solution of the existence problem", "posed by RMS2.", "\\medskip\\noindent", "Finally, suppose that", "$$", "(a, b, \\alpha), (a', b', \\alpha') :", "(U_1, \\phi_1 : V_1 \\to u(U_1), x_1)", "\\longrightarrow", "(U, \\phi : V \\to u(U), x)", "$$", "are two morphisms of $u_{pp}\\mathcal{S}$ and suppose that", "$$", "(c, \\text{id}_V, \\gamma) :", "(U, \\phi : V \\to u(U), x)", "\\longrightarrow", "(U', \\phi : V \\to u(U'), x')", "$$", "is an element of $R$ which equalizes the morphisms", "$(a, b, \\alpha)$ and $(a', b', \\alpha')$. This implies in particular", "that $b = b'$. Let $d : U_2 \\to U_1$ be the equalizer of $a, a'$ which", "exists (see", "Categories, Lemma \\ref{categories-lemma-almost-finite-limits-exist}).", "Moreover, $u(d) : u(U_2) \\to u(U_1)$ is the equalizer of $u(a), u(a')$", "hence (as $b = b'$) there is a morphism $\\phi_2 : V_1 \\to u(U_2)$ such that", "$\\phi_1 = u(d) \\circ \\phi_1$. Let $\\delta : x_2 \\to x_1$ be a strongly", "cartesian morphism of $\\mathcal{S}$ with $p(\\delta) = u(d)$.", "Now we claim that $\\alpha \\circ \\delta = \\alpha' \\circ \\delta$.", "This is true because", "$\\gamma$ is strongly cartesian,", "$\\gamma \\circ \\alpha \\circ \\delta = \\gamma \\circ \\alpha' \\circ \\delta$, and", "$p(\\alpha \\circ \\delta) = p(\\alpha' \\circ \\delta)$.", "Hence the arrow", "$$", "(d, \\text{id}_{V_1}, \\delta) :", "(U_2, \\phi_2 : V_1 \\to u(U_2), x_2)", "\\longrightarrow", "(U_1, \\phi_1 : V_1 \\to u(U_1), x_1)", "$$", "is an element of $R$ and equalizes $(a, b, \\alpha)$ and $(a', b', \\alpha')$.", "Hence $R$ satisfies RMS3 as well." ], "refs": [ "categories-definition-multiplicative-system", "categories-lemma-composition-cartesian", "categories-lemma-almost-finite-limits-exist" ], "ref_ids": [ 12373, 12282, 12223 ] } ], "ref_ids": [] }, { "id": 8984, "type": "theorem", "label": "stacks-lemma-fibred-category-pullback", "categories": [ "stacks" ], "title": "stacks-lemma-fibred-category-pullback", "contents": [ "With notation and assumptions as in", "Lemma \\ref{lemma-right-multiplicative-system}.", "Set $u_p\\mathcal{S} = R^{-1}u_{pp}\\mathcal{S}$, see", "Categories, Section \\ref{categories-section-localization}.", "Then $u_p\\mathcal{S}$ is a fibred category over $\\mathcal{D}$." ], "refs": [ "stacks-lemma-right-multiplicative-system" ], "proofs": [ { "contents": [ "We use the description of $u_p\\mathcal{S}$ given just above", "Categories, Lemma \\ref{categories-lemma-right-localization}.", "Note that the functor $p_{pp} : u_{pp}\\mathcal{S} \\to \\mathcal{D}$", "transforms every element of $R$ to an identity morphism.", "Hence by", "Categories, Lemma \\ref{categories-lemma-properties-right-localization}", "we obtain a canonical functor $p_p : u_p\\mathcal{S} \\to \\mathcal{D}$", "extending the given functor. This is how we think of", "$u_p\\mathcal{S}$ as a category over $\\mathcal{D}$.", "\\medskip\\noindent", "First we want to characterize the $\\mathcal{D}$-strongly cartesian morphisms", "in $u_p\\mathcal{S}$.", "A morphism $f : X \\to Y$ of $u_p\\mathcal{S}$ is the equivalence class of", "a pair $(f' : X' \\to Y, r : X' \\to X)$ with $r \\in R$.", "In fact, in $u_p\\mathcal{S}$ we have $f = (f', 1) \\circ (r, 1)^{-1}$", "with obvious notation.", "Note that an isomorphism is always strongly cartesian, as are", "compositions of strongly cartesian morphisms, see", "Categories, Lemma \\ref{categories-lemma-composition-cartesian}.", "Hence $f$ is strongly cartesian if and only if $(f', 1)$ is so.", "Thus the following claim completely characterizes strongly cartesian", "morphisms. Claim: A morphism", "$$", "(a, b, \\alpha) :", "X_1 = (U_1, \\phi_1 : V_1 \\to u(U_1), x_1)", "\\longrightarrow", "(U_2, \\phi_2 : V_2 \\to u(U_2), x_2) = X_2", "$$", "of $u_{pp}\\mathcal{S}$ has image $f = ((a, b, \\alpha), 1)$", "strongly cartesian in $u_p\\mathcal{S}$ if and only if $\\alpha$", "is a strongly cartesian morphism of $\\mathcal{S}$.", "\\medskip\\noindent", "Assume $\\alpha$ strongly cartesian.", "Let $X = (U, \\phi : V \\to u(U), x)$ be another object, and let", "$f_2 : X \\to X_2$ be a morphism of $u_p\\mathcal{S}$", "such that $p_p(f_2) = b \\circ b_1$ for some $b_1 : U \\to U_1$.", "To show that $f$ is strongly cartesian we have to show", "that there exists a unique morphism $f_1 : X \\to X_1$ in $u_p\\mathcal{S}$", "such that $p_p(f_1) = b_1$ and $f_2 = f \\circ f_1$ in $u_p\\mathcal{S}$.", "Write $f_2 = (f'_2 : X' \\to X_2, r : X' \\to X)$. Again we can write", "$f_2 = (f'_2, 1) \\circ (r, 1)^{-1}$ in $u_p\\mathcal{S}$. Since $(r, 1)$", "is an isomorphism whose image in $\\mathcal{D}$ is an identity we", "see that finding a morphism $f_1 : X \\to X_1$ with the required", "properties is the same thing as finding a morphism $f'_1 : X' \\to X_1$", "in $u_p\\mathcal{S}$ with $p(f'_1) = b_1$ and $f'_2 = f \\circ f'_1$.", "Hence we may assume that $f_2$ is of the form", "$f_2 = ((a_2, b_2, \\alpha_2), 1)$ with $b_2 = b \\circ b_1$. Here is", "a picture", "$$", "\\xymatrix{", "& & (U_1, V_1 \\to u(U_1), x_1) \\ar[d]^{(a, b, \\alpha)} \\\\", "(U, V \\to u(U), x) \\ar[rr]^{(a_2, b_2, \\alpha_2)} & &", "(U_2, V_2 \\to u(U_2), x_2)", "}", "$$", "Now it is clear how to construct the morphism $f_1$.", "Namely, set $U' = U \\times_{U_2} U_1$ with projections", "$c : U' \\to U$ and $a_1 : U' \\to U_1$.", "Pick a strongly cartesian morphism $\\gamma : x' \\to x$", "lifting the morphism $c$. Since $b_2 = b \\circ b_1$,", "and since $u(U') = u(U) \\times_{u(U_2)} u(U_1)$ we see that", "$\\phi' = (\\phi, \\phi_1 \\circ b_1) : V \\to u(U')$. Since", "$\\alpha$ is strongly cartesian, and", "$a \\circ a_1 = a_2 \\circ c = p(\\alpha_2 \\circ \\gamma)$", "there exists a morphism", "$\\alpha_1 : x' \\to x_1$ lifting $a_1$ such that", "$\\alpha \\circ \\alpha_1 = \\alpha_2 \\circ \\gamma$.", "Set $X' = (U', \\phi' : V \\to u(U'), x')$.", "Thus we see that", "$$", "f_1 =", "((a_1, b_1, \\alpha_1) : X' \\to X_1, (c, \\text{id}_V, \\gamma) : X' \\to X) :", "X \\longrightarrow X_1", "$$", "works, in fact the diagram", "$$", "\\xymatrix{", "(U', \\phi' : V \\to u(U'), x')", "\\ar[d]_{(c, \\text{id}_V, \\gamma)}", "\\ar[rr]_{(a_1, b_1, \\alpha_1)}", "& & (U_1, V_1 \\to u(U_1), x_1) \\ar[d]^{(a, b, \\alpha)} \\\\", "(U, V \\to u(U), x) \\ar[rr]^{(a_2, b_2, \\alpha_2)} & &", "(U_2, V_2 \\to u(U_2), x_2)", "}", "$$", "is commutative by construction. This proves existence.", "\\medskip\\noindent", "Next we prove uniqueness, still in the special case", "$f = ((a, b, \\alpha), 1)$ and $f_2 = ((a_2, b_2, \\alpha_2), 1)$.", "We strongly advise the reader to skip this part.", "Suppose that $g_1, g'_1 : X \\to X_1$ are two morphisms of", "$u_p\\mathcal{S}$ such that $p_p(g_1) = p_p(g'_1) = b_1$ and", "$f_2 = f \\circ g_1 = f \\circ g'_1$. Our goal is to show that", "$g_1 = g'_1$. By", "Categories, Lemma \\ref{categories-lemma-morphisms-right-localization}", "we may represent $g_1$ and $g'_1$ as the equivalence classes of", "$(f_1 : X' \\to X_1, r : X' \\to X)$ and", "$(f'_1 : X' \\to X_1, r : X' \\to X)$ for some $r \\in R$. By", "Categories, Lemma \\ref{categories-lemma-equality-morphisms-right-localization}", "we see that $f_2 = f \\circ g_1 = f \\circ g'_1$ means that there", "exists a morphism $r' : X'' \\to X'$ in $u_{pp}\\mathcal{S}$ such that", "$r' \\circ r \\in R$ and", "$$", "(a, b, \\alpha) \\circ f_1 \\circ r' =", "(a, b, \\alpha) \\circ f'_1 \\circ r' =", "(a_2, b_2, \\alpha_2) \\circ r'", "$$", "in $u_{pp}\\mathcal{S}$. Note that", "now $g_1$ is represented by $(f_1 \\circ r', r \\circ r')$ and", "similarly for $g'_1$. Hence we may assume that", "$$", "(a, b, \\alpha) \\circ f_1 =", "(a, b, \\alpha) \\circ f'_1 =", "(a_2, b_2, \\alpha_2).", "$$", "Write $r = (c, \\text{id}_V, \\gamma) : (U', \\phi' : V \\to u(U'), x')$,", "$f_1 = (a_1, b_1, \\alpha_1)$, and $f'_1 = (a'_1, b_1, \\alpha'_1)$.", "Here we have used the condition that $p_p(g_1) = p_p(g'_1)$.", "The equalities above are now equivalent to", "$a \\circ a_1 = a \\circ a'_1 = a_2 \\circ c$ and", "$\\alpha \\circ \\alpha_1 = \\alpha \\circ \\alpha'_1 = \\alpha_2 \\circ \\gamma$.", "It need not be the case that $a_1 = a'_1$ in this situation.", "Thus we have to precompose by one more morphism from $R$.", "Namely, let $U'' = \\text{Eq}(a_1, a'_1)$ be the equalizer of", "$a_1$ and $a'_1$ which is a subobject of $U'$. Denote", "$c' : U'' \\to U'$ the canonical monomorphism.", "Because of the relations among the morphisms", "above we see that $V \\to u(U')$ maps into", "$u(U'') = u(\\text{Eq}(a_1, a'_1)) = \\text{Eq}(u(a_1), u(a'_1))$.", "Hence we get a new object $(U'', \\phi'' : V \\to u(U''), x'')$, where", "$\\gamma' : x'' \\to x'$ is a strongly cartesian morphism lifting $\\gamma$.", "Then we see that we may precompose $f_1$ and $f'_1$ with the element", "$(c', \\text{id}_V, \\gamma')$ of $R$. After doing this, i.e., replacing", "$(U', \\phi' : V \\to u(U'), x')$ with", "$(U'', \\phi'' : V \\to u(U''), x'')$, we get back to the previous", "situation where in addition we now have that $a_1 = a'_1$.", "In this case it follows formally from the fact that $\\alpha$", "is strongly cartesian (!) that $\\alpha_1 = \\alpha'_1$.", "This shows that $g_1 = g'_1$ as desired.", "\\medskip\\noindent", "We omit the proof of the fact that for any strongly cartesian morphism", "of $u_p\\mathcal{S}$ of the form $((a, b, \\alpha), 1)$ the morphism", "$\\alpha$ is strongly cartesian in $\\mathcal{S}$.", "(We do not need the characterization of strongly cartesian morphisms", "in the rest of the proof, although we do use it later in this section.)", "\\medskip\\noindent", "Let $(U, \\phi : V \\to u(U), x)$ be an object of $u_p\\mathcal{S}$.", "Let $b : V' \\to V$ be a morphism of $\\mathcal{D}$. Then the morphism", "$$", "(\\text{id}_U, b, \\text{id}_x) :", "(U, \\phi \\circ b : V' \\to u(U), x)", "\\longrightarrow", "(U, \\phi : V \\to u(U), x)", "$$", "is strongly cartesian by the result of the preceding paragraphs and we win." ], "refs": [ "categories-lemma-right-localization", "categories-lemma-properties-right-localization", "categories-lemma-composition-cartesian", "categories-lemma-morphisms-right-localization", "categories-lemma-equality-morphisms-right-localization" ], "ref_ids": [ 12261, 12264, 12282, 12262, 12263 ] } ], "ref_ids": [ 8983 ] }, { "id": 8985, "type": "theorem", "label": "stacks-lemma-fibred-groupoids-category-pullback", "categories": [ "stacks" ], "title": "stacks-lemma-fibred-groupoids-category-pullback", "contents": [ "With notation and assumptions as in", "Lemma \\ref{lemma-fibred-category-pullback}.", "If $\\mathcal{S}$ is fibred in groupoids, then $u_p\\mathcal{S}$ is fibred", "in groupoids." ], "refs": [ "stacks-lemma-fibred-category-pullback" ], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-fibred-category-pullback}", "we know that $u_p\\mathcal{S}$ is a fibred category.", "Let $f : X \\to Y$ be a morphism of $u_p\\mathcal{S}$ with", "$p_p(f) = \\text{id}_V$. We are done if we can show that $f$ is invertible, see", "Categories, Lemma \\ref{categories-lemma-fibred-groupoids}.", "Write $f$ as the equivalence class of", "a pair $((a, b, \\alpha), r)$ with $r \\in R$. Then $p_p(r) = \\text{id}_V$,", "hence $p_{pp}((a, b, \\alpha)) = \\text{id}_V$. Hence $b = \\text{id}_V$.", "But any morphism of $\\mathcal{S}$ is strongly cartesian, see", "Categories, Lemma \\ref{categories-lemma-fibred-groupoids}", "hence we see that $(a, b, \\alpha) \\in R$ is invertible", "in $u_p\\mathcal{S}$ as desired." ], "refs": [ "stacks-lemma-fibred-category-pullback", "categories-lemma-fibred-groupoids", "categories-lemma-fibred-groupoids" ], "ref_ids": [ 8984, 12294, 12294 ] } ], "ref_ids": [ 8984 ] }, { "id": 8986, "type": "theorem", "label": "stacks-lemma-adjointness-pullback-pushforward", "categories": [ "stacks" ], "title": "stacks-lemma-adjointness-pullback-pushforward", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ and $q : \\mathcal{T} \\to \\mathcal{D}$", "be categories over $\\mathcal{C}$ and $\\mathcal{D}$. Assume that", "\\begin{enumerate}", "\\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category,", "\\item $q : \\mathcal{T} \\to \\mathcal{D}$ is a fibred category,", "\\item $\\mathcal{C}$ has nonempty finite limits, and", "\\item $u : \\mathcal{C} \\to \\mathcal{D}$ commutes with nonempty finite limits.", "\\end{enumerate}", "Then we have a canonical equivalence of categories", "$$", "\\Mor_{\\textit{Fib}/\\mathcal{C}}(\\mathcal{S}, u^p\\mathcal{T})", "=", "\\Mor_{\\textit{Fib}/\\mathcal{D}}(u_p\\mathcal{S}, \\mathcal{T})", "$$", "of morphism categories." ], "refs": [], "proofs": [ { "contents": [ "In this proof we use the notation $x/U$ to denote an object", "$x$ of $\\mathcal{S}$ which lies over $U$ in $\\mathcal{C}$.", "Similarly $y/V$ denotes an object $y$ of $\\mathcal{T}$", "which lies over $V$ in $\\mathcal{D}$. In the same vein", "$\\alpha/a : x/U \\to x'/U'$ denotes the morphism", "$\\alpha : x \\to x'$ with image $a : U \\to U'$ in $\\mathcal{C}$.", "\\medskip\\noindent", "Let $G : u_p\\mathcal{S} \\to \\mathcal{T}$ be a $1$-morphism of fibred", "categories over $\\mathcal{D}$. Denote $G' : u_{pp}\\mathcal{S} \\to \\mathcal{T}$", "the composition of $G$ with the canonical (localization) functor", "$u_{pp}\\mathcal{S} \\to u_p\\mathcal{S}$. Then consider the functor", "$H : \\mathcal{S} \\to u^p\\mathcal{T}$ given by", "$$", "H(x/U) = (U, G'(U, \\text{id}_{u(U)} : u(U) \\to u(U), x))", "$$", "on objects and by", "$$", "H((\\alpha, a) : x/U \\to x'/U') = G'(a, u(a), \\alpha)", "$$", "on morphisms. Since $G$ transforms strongly cartesian morphisms", "into strongly cartesian morphisms, we see that if $\\alpha$ is strongly", "cartesian, then $H(\\alpha)$ is strongly cartesian.", "Namely, we've seen in the proof of", "Lemma \\ref{lemma-fibred-category-pullback}", "that in this case the map $(a, u(a), \\alpha)$ becomes", "strongly cartesian in $u_p\\mathcal{S}$. Clearly this construction is", "functorial in $G$ and we obtain a functor", "$$", "A :", "\\Mor_{\\textit{Fib}/\\mathcal{D}}(u_p\\mathcal{S}, \\mathcal{T})", "\\longrightarrow", "\\Mor_{\\textit{Fib}/\\mathcal{C}}(\\mathcal{S}, u^p\\mathcal{T})", "$$", "\\medskip\\noindent", "Conversely, let $H : \\mathcal{S} \\to u^p\\mathcal{T}$ be a $1$-morphism", "of fibred categories. Recall that an object of $u^p\\mathcal{T}$ is", "a pair $(U, y)$ with $y \\in \\Ob(\\mathcal{T}_{u(U)})$. We denote", "$\\text{pr} : u^p\\mathcal{T} \\to \\mathcal{T}$ the functor $(U, y) \\mapsto y$.", "In this case we define a functor", "$G' : u_{pp}\\mathcal{S} \\to \\mathcal{T}$ by the rules", "$$", "G'(U, \\phi : V \\to u(U), x) = \\phi^*\\text{pr}(H(x))", "$$", "on objects and we let", "$$", "G'((a, b, \\alpha) : (U, \\phi : V \\to u(U), x)", "\\to (U', \\phi' : V' \\to u(U'), x'))", "=", "\\beta", "$$", "be the unique morphism", "$\\beta : \\phi^*\\text{pr}(H(x)) \\to (\\phi')^*\\text{pr}(H(x'))$", "such that $q(\\beta) = b$ and the diagram", "$$", "\\xymatrix{", "\\phi^*\\text{pr}(H(x)) \\ar[d] \\ar[r]_-{\\beta} &", "(\\phi')^*\\text{pr}(H(x')) \\ar[d] \\\\", "\\text{pr}(H(x)) \\ar[r]^{\\text{pr}(H(a, \\alpha))} & \\text{pr}(H(x'))", "}", "$$", "Such a morphism exists and is unique because $\\mathcal{T}$ is a fibred", "category.", "\\medskip\\noindent", "We check that $G'(r)$ is an isomorphism if $r \\in R$.", "Namely, if", "$$", "(a, \\text{id}_V, \\alpha) :", "(U', \\phi' : V \\to u(U'), x')", "\\longrightarrow", "(U, \\phi : V \\to u(U), x)", "$$", "with $\\alpha$ strongly cartesian is an element of the right multiplicative", "system $R$ of", "Lemma \\ref{lemma-right-multiplicative-system}", "then $H(\\alpha)$ is strongly cartesian, and", "$\\text{pr}(H(\\alpha))$ is strongly cartesian, see proof of", "Lemma \\ref{lemma-fibred-category-pushforward}.", "Hence in this case the morphism $\\beta$ has $q(\\beta) = \\text{id}_V$", "and is strongly cartesian. Hence $\\beta$ is an isomorphism by", "Categories, Lemma \\ref{categories-lemma-composition-cartesian}.", "Thus by", "Categories, Lemma \\ref{categories-lemma-properties-right-localization}", "we obtain a canonical extension $G : u_p\\mathcal{S} \\to \\mathcal{T}$.", "\\medskip\\noindent", "Next, let us prove that $G$ transforms strongly cartesian morphisms", "into strongly cartesian morphisms.", "Suppose that $f : X \\to Y$ is a strongly cartesian. By the characterization", "of strongly cartesian morphisms in $u_p\\mathcal{S}$ we can write $f$ as", "$((a, b, \\alpha) : X' \\to Y, r : X' \\to Y)$ where $r \\in R$ and", "$\\alpha$ strongly cartesian in $\\mathcal{S}$. By the above it suffices", "to show that $G'(a, b \\alpha)$ is strongly cartesian. As before", "the condition that $\\alpha$ is strongly cartesian implies that", "$\\text{pr}(H(a, \\alpha)) : \\text{pr}(H(x)) \\to \\text{pr}(H(x'))$", "is strongly cartesian in $\\mathcal{T}$. Since in the commutative", "square above now all arrows except possibly $\\beta$ is strongly cartesian", "it follows that also $\\beta$ is strongly cartesian as desired.", "Clearly the construction $H \\mapsto G$ is functorial in $H$ and we", "obtain a functor", "$$", "B :", "\\Mor_{\\textit{Fib}/\\mathcal{C}}(\\mathcal{S}, u^p\\mathcal{T})", "\\longrightarrow", "\\Mor_{\\textit{Fib}/\\mathcal{D}}(u_p\\mathcal{S}, \\mathcal{T})", "$$", "To finish the proof of the lemma we have to show that the functors", "$A$ and $B$ are mutually quasi-inverse. We omit the verifications." ], "refs": [ "stacks-lemma-fibred-category-pullback", "stacks-lemma-right-multiplicative-system", "stacks-lemma-fibred-category-pushforward", "categories-lemma-composition-cartesian", "categories-lemma-properties-right-localization" ], "ref_ids": [ 8984, 8983, 8980, 12282, 12264 ] } ], "ref_ids": [] }, { "id": 8987, "type": "theorem", "label": "stacks-lemma-adjointness-pullback-pushforward-stacks", "categories": [ "stacks" ], "title": "stacks-lemma-adjointness-pullback-pushforward-stacks", "contents": [ "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites", "given by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$", "satisfying the hypotheses and conclusions of", "Sites, Proposition \\ref{sites-proposition-get-morphism}.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ and", "$q : \\mathcal{T} \\to \\mathcal{D}$ be stacks.", "Then we have a canonical equivalence of categories", "$$", "\\Mor_{\\textit{Stacks}/\\mathcal{C}}(\\mathcal{S}, f_*\\mathcal{T})", "=", "\\Mor_{\\textit{Stacks}/\\mathcal{D}}(f^{-1}\\mathcal{S}, \\mathcal{T})", "$$", "of morphism categories." ], "refs": [ "sites-proposition-get-morphism" ], "proofs": [ { "contents": [ "For $i = 1, 2$ an $i$-morphism of stacks is the same thing as a", "$i$-morphism of fibred categories, see", "Definition \\ref{definition-stacks-over-C}.", "By", "Lemma \\ref{lemma-adjointness-pullback-pushforward}", "we have already", "$$", "\\Mor_{\\textit{Fib}/\\mathcal{C}}(\\mathcal{S}, u^p\\mathcal{T})", "=", "\\Mor_{\\textit{Fib}/\\mathcal{D}}(u_p\\mathcal{S}, \\mathcal{T})", "$$", "Hence the result follows from", "Lemma \\ref{lemma-stackify-universal-property-more}", "as $u^p\\mathcal{T} = f_*\\mathcal{T}$ and $f^{-1}\\mathcal{S}$ is the", "stackification of $u_p\\mathcal{S}$." ], "refs": [ "stacks-definition-stacks-over-C", "stacks-lemma-adjointness-pullback-pushforward", "stacks-lemma-stackify-universal-property-more" ], "ref_ids": [ 8997, 8986, 8963 ] } ], "ref_ids": [ 8641 ] }, { "id": 8988, "type": "theorem", "label": "stacks-lemma-technical-up", "categories": [ "stacks" ], "title": "stacks-lemma-technical-up", "contents": [ "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites", "given by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$", "satisfying the hypotheses and conclusions of", "Sites, Proposition \\ref{sites-proposition-get-morphism}.", "Let $\\mathcal{S} \\to \\mathcal{C}$ be a fibred category, and", "let $\\mathcal{S} \\to \\mathcal{S}'$ be the stackification of $\\mathcal{S}$.", "Then $f^{-1}\\mathcal{S}'$ is the stackification of", "$u_p\\mathcal{S}$." ], "refs": [ "sites-proposition-get-morphism" ], "proofs": [ { "contents": [ "Omitted. Hint:", "This is the analogue of Sites, Lemma \\ref{sites-lemma-technical-up}." ], "refs": [ "sites-lemma-technical-up" ], "ref_ids": [ 8523 ] } ], "ref_ids": [ 8641 ] }, { "id": 8989, "type": "theorem", "label": "stacks-lemma-bigger-site", "categories": [ "stacks" ], "title": "stacks-lemma-bigger-site", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor satisfying the", "assumptions of", "Sites, Lemma \\ref{sites-lemma-bigger-site}.", "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be the corresponding", "morphism of sites. Then", "\\begin{enumerate}", "\\item for every stack $p : \\mathcal{S} \\to \\mathcal{C}$ the", "canonical functor $\\mathcal{S} \\to f_*f^{-1}\\mathcal{S}$ is", "an equivalence of stacks,", "\\item given stacks $\\mathcal{S}, \\mathcal{S}'$ over $\\mathcal{C}$", "the construction $f^{-1}$", "induces an equivalence", "$$", "\\Mor_{\\textit{Stacks}/\\mathcal{C}}(\\mathcal{S}, \\mathcal{S}')", "\\longrightarrow", "\\Mor_{\\textit{Stacks}/\\mathcal{D}}(f^{-1}\\mathcal{S}, f^{-1}\\mathcal{S}')", "$$", "of morphism categories.", "\\end{enumerate}" ], "refs": [ "sites-lemma-bigger-site" ], "proofs": [ { "contents": [ "Note that by", "Lemma \\ref{lemma-adjointness-pullback-pushforward-stacks}", "we have an equivalence of categories", "$$", "\\Mor_{\\textit{Stacks}/\\mathcal{D}}(f^{-1}\\mathcal{S}, f^{-1}\\mathcal{S}')", "=", "\\Mor_{\\textit{Stacks}/\\mathcal{C}}(\\mathcal{S}, f_*f^{-1}\\mathcal{S}')", "$$", "Hence (2) follows from (1).", "\\medskip\\noindent", "To prove (1) we are going to use", "Lemma \\ref{lemma-characterize-essentially-surjective-when-ff}.", "This lemma tells us that we have to show that", "$can : \\mathcal{S} \\to f_*f^{-1}\\mathcal{S}$ is", "fully faithful and that all objects of $f_*f^{-1}\\mathcal{S}$", "are locally in the essential image.", "\\medskip\\noindent", "We quickly describe the functor $can$, see proof of", "Lemma \\ref{lemma-adjointness-pullback-pushforward}.", "To do this we introduce the functor", "$c'' : \\mathcal{S} \\to u_{pp}\\mathcal{S}$ defined by", "$c''(x/U) = (U, \\text{id} : u(U) \\to u(U), x)$, and", "$c''(\\alpha/a) = (a, u(a), \\alpha)$. We set", "$c' : \\mathcal{S} \\to u_p\\mathcal{S}$ equal to the composition", "of $c''$ and the canonical functor $u_{pp}\\mathcal{S} \\to u_p\\mathcal{S}$.", "We set $c : \\mathcal{S} \\to f^{-1}\\mathcal{S}$ equal to the composition", "of $c'$ and the canonical functor $u_p\\mathcal{S} \\to f^{-1}\\mathcal{S}$.", "Then $can : \\mathcal{S} \\to f_*f^{-1}\\mathcal{S}$ is the functor", "which to $x/U$ associates the pair $(U, c(x))$ and to", "$\\alpha/a$ the morphism $(a, c(\\alpha))$.", "\\medskip\\noindent", "Fully faithfulness. To prove this we are going to use", "Lemma \\ref{lemma-characterize-ff}.", "Let $U \\in \\Ob(\\mathcal{C})$.", "Let $x, y \\in \\mathcal{S}_U$.", "First off, as $u$ is fully faithful, we have", "$$", "\\Mor_{(f_*f^{-1}\\mathcal{S})_U}(can(x), can(y))", "=", "\\Mor_{(f^{-1}\\mathcal{S})_{u(U)}}(c(x), c(y))", "$$", "directly from the definition of $f_*$. Similar holds after", "pulling back to any $U'/U$. Because $f^{-1}\\mathcal{S}$ is the", "stackification of $u_p\\mathcal{S}$, and since $u$ is continuous", "and cocontinuous the presheaf", "$$", "U'/U \\longmapsto", "\\Mor_{(f^{-1}\\mathcal{S})_{u(U')}}(c(x|_{U'}), c(y|_{U'}))", "$$", "is the sheafification of the presheaf", "$$", "U'/U \\longmapsto", "\\Mor_{(u_p\\mathcal{S})_{u(U')}}(c'(x|_{U'}), c'(y|_{U'}))", "$$", "Hence to finish the proof of fully faithfulness it suffices", "to show that for any $U$ and $x, y$ the map", "$$", "\\Mor_{\\mathcal{S}_U}(x, y)", "\\longrightarrow", "\\Mor_{(u_p\\mathcal{S})_U}(c'(x), c'(y))", "$$", "is bijective. A morphism $f : x \\to y$ in $u_p\\mathcal{S}$ over $u(U)$", "is given by an equivalence class of diagrams", "$$", "\\xymatrix{", "(U', \\phi : u(U) \\to u(U'), x')", "\\ar[d]_{(c, \\text{id}_{u(U)}, \\gamma)}", "\\ar[r]_{(a, b, \\alpha)} &", "(U, \\text{id} : u(U) \\to u(U), y)", "\\\\", "(U, \\text{id} : u(U) \\to u(U), x)", "}", "$$", "with $\\gamma$ strongly cartesian and $b = \\text{id}_{u(U)}$.", "But since $u$ is fully faithful we can write $\\phi = u(c')$ for some", "morphism $c' : U \\to U'$", "and then we see that $a \\circ c' = \\text{id}_U$ and", "$c \\circ c' = \\text{id}_{U'}$. Because $\\gamma$ is strongly cartesian", "we can find a morphism $\\gamma' : x \\to x'$ lifting $c'$ such that", "$\\gamma \\circ \\gamma' = \\text{id}_x$. By definition of the equivalence", "classes defining morphisms in $u_p\\mathcal{S}$ it follows that", "the morphism", "$$", "\\xymatrix{", "(U, \\text{id} : u(U) \\to u(U), x)", "\\ar[rr]_{(\\text{id}, \\text{id}, \\alpha \\circ \\gamma')} & &", "(U, \\text{id} : u(U) \\to u(U), y)", "}", "$$", "of $u_{pp}\\mathcal{S}$ induces the morphism $f$ in $u_p\\mathcal{S}$.", "This proves that the map is surjective. We omit the proof that it is", "injective.", "\\medskip\\noindent", "Finally, we have to show that any object of $f_*f^{-1}\\mathcal{S}$", "locally comes from an object of $\\mathcal{S}$. This is clear from the", "constructions (details omitted)." ], "refs": [ "stacks-lemma-adjointness-pullback-pushforward-stacks", "stacks-lemma-characterize-essentially-surjective-when-ff", "stacks-lemma-adjointness-pullback-pushforward", "stacks-lemma-characterize-ff" ], "ref_ids": [ 8987, 8945, 8986, 8944 ] } ], "ref_ids": [ 8548 ] }, { "id": 8990, "type": "theorem", "label": "stacks-lemma-when-localization-stack", "categories": [ "stacks" ], "title": "stacks-lemma-when-localization-stack", "contents": [ "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$.", "Then $j_U : \\mathcal{C}/U \\to \\mathcal{C}$ is a stack over $\\mathcal{C}$", "if and only if $h_U$ is a sheaf." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemma \\ref{lemma-stack-in-setoids-characterize}", "with", "Categories,", "Example \\ref{categories-example-fibred-category-from-functor-of-points}." ], "refs": [ "stacks-lemma-stack-in-setoids-characterize" ], "ref_ids": [ 8951 ] } ], "ref_ids": [] }, { "id": 8991, "type": "theorem", "label": "stacks-lemma-localize-stacks", "categories": [ "stacks" ], "title": "stacks-lemma-localize-stacks", "contents": [ "Assume that $\\mathcal{C}$ is a site,", "and $U$ is an object of $\\mathcal{C}$ whose associated representable", "presheaf is a sheaf. Constructions A and B above define", "mutually inverse (!) functors of $2$-categories", "$$", "\\left\\{", "\\begin{matrix}", "2\\text{-category of}\\\\", "\\text{stacks over }\\mathcal{C}/U", "\\end{matrix}", "\\right\\}", "\\leftrightarrow", "\\left\\{", "\\begin{matrix}", "2\\text{-category of pairs }(\\mathcal{T}, p)", "\\text{ consisting} \\\\", "\\text{of a stack }\\mathcal{T}\\text{ over }\\mathcal{C}\\text{ and a morphism} \\\\", "p : \\mathcal{T} \\to \\mathcal{C}/U\\text{ of stacks over }\\mathcal{C}", "\\end{matrix}", "\\right\\}", "$$" ], "refs": [], "proofs": [ { "contents": [ "This is clear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9010, "type": "theorem", "label": "spaces-simplicial-lemma-simplicial-site", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-simplicial-site", "contents": [ "Let $X$ be a simplicial space. Then $X_{Zar}$", "as defined above is a site." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9011, "type": "theorem", "label": "spaces-simplicial-lemma-describe-sheaves-simplicial-site", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-describe-sheaves-simplicial-site", "contents": [ "Let $X$ be a simplicial space. There is an equivalence of", "categories between", "\\begin{enumerate}", "\\item $\\Sh(X_{Zar})$, and", "\\item category of systems $(\\mathcal{F}_n, \\mathcal{F}(\\varphi))$", "described above.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9012, "type": "theorem", "label": "spaces-simplicial-lemma-simplicial-space-site-functorial", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-simplicial-space-site-functorial", "contents": [ "Let $f : Y \\to X$ be a morphism of simplicial spaces.", "Then the functor $u : X_{Zar} \\to Y_{Zar}$", "which associates to the open $U \\subset X_n$ the open", "$f_n^{-1}(U) \\subset Y_n$ defines a morphism of sites", "$f_{Zar} : Y_{Zar} \\to X_{Zar}$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $u$ is a continuous functor. Hence we obtain functors", "$f_{Zar, *} = u^s$ and $f_{Zar}^{-1} = u_s$, see", "Sites, Section \\ref{sites-section-morphism-sites}.", "To see that we obtain a morphism of sites we have to show", "that $u_s$ is exact. We will use", "Sites, Lemma \\ref{sites-lemma-directed-morphism} to see this.", "Let $V \\subset Y_n$ be an open subset. The category", "$\\mathcal{I}_V^u$ (see Sites, Section \\ref{sites-section-functoriality-PSh})", "consists of pairs $(U, \\varphi)$ where", "$\\varphi : [m] \\to [n]$ and $U \\subset X_m$ open such that", "$Y(\\varphi)(V) \\subset f_m^{-1}(U)$. Moreover, a morphism", "$(U, \\varphi) \\to (U', \\varphi')$ is given by a", "$\\psi : [m'] \\to [m]$ such that $X(\\psi)(U) \\subset U'$", "and $\\varphi \\circ \\psi = \\varphi'$.", "It is our task to show that $\\mathcal{I}_V^u$ is cofiltered.", "\\medskip\\noindent", "We verify the conditions of", "Categories, Definition \\ref{categories-definition-codirected}.", "Condition (1) holds because $(X_n, \\text{id}_{[n]})$ is an object.", "Let $(U, \\varphi)$ be an object. The condition", "$Y(\\varphi)(V) \\subset f_m^{-1}(U)$ is equivalent to", "$V \\subset f_n^{-1}(X(\\varphi)^{-1}(U))$. Hence we obtain a morphism", "$(X(\\varphi)^{-1}(U), \\text{id}_{[n]}) \\to (U, \\varphi)$ given", "by setting $\\psi = \\varphi$. Moreover, given a pair of objects", "of the form $(U, \\text{id}_{[n]})$ and $(U', \\text{id}_{[n]})$", "we see there exists an object, namely $(U \\cap U', \\text{id}_{[n]})$,", "which maps to both of them. Thus condition (2) holds.", "To verify condition (3) suppose given two morphisms", "$a, a': (U, \\varphi) \\to (U', \\varphi')$ given by $\\psi, \\psi' : [m'] \\to [m]$.", "Then precomposing with the morphism", "$(X(\\varphi)^{-1}(U), \\text{id}_{[n]}) \\to (U, \\varphi)$ given", "by $\\varphi$ equalizes $a, a'$ because", "$\\varphi \\circ \\psi = \\varphi' = \\varphi \\circ \\psi'$.", "This finishes the proof." ], "refs": [ "sites-lemma-directed-morphism", "categories-definition-codirected" ], "ref_ids": [ 8526, 12364 ] } ], "ref_ids": [] }, { "id": 9013, "type": "theorem", "label": "spaces-simplicial-lemma-describe-functoriality", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-describe-functoriality", "contents": [ "Let $f : Y \\to X$ be a morphism of simplicial spaces. In terms of the", "description of sheaves in", "Lemma \\ref{lemma-describe-sheaves-simplicial-site} the", "morphism $f_{Zar}$ of Lemma \\ref{lemma-simplicial-space-site-functorial}", "can be described as follows.", "\\begin{enumerate}", "\\item If $\\mathcal{G}$ is a sheaf on $Y$, then", "$(f_{Zar, *}\\mathcal{G})_n = f_{n, *}\\mathcal{G}_n$.", "\\item If $\\mathcal{F}$ is a sheaf on $X$, then", "$(f_{Zar}^{-1}\\mathcal{F})_n = f_n^{-1}\\mathcal{F}_n$.", "\\end{enumerate}" ], "refs": [ "spaces-simplicial-lemma-describe-sheaves-simplicial-site", "spaces-simplicial-lemma-simplicial-space-site-functorial" ], "proofs": [ { "contents": [ "The first part is immediate from the definitions. For the second part, note", "that in the proof of", "Lemma \\ref{lemma-simplicial-space-site-functorial}", "we have shown that for a $V \\subset Y_n$ open the category", "$(\\mathcal{I}_V^u)^{opp}$ contains as a cofinal subcategory", "the category of opens $U \\subset X_n$ with $f_n^{-1}(U) \\supset V$", "and morphisms given by inclusions. Hence we see that the restriction", "of $u_p\\mathcal{F}$ to opens of $Y_n$ is the presheaf", "$f_{n, p}\\mathcal{F}_n$ as defined in", "Sheaves, Lemma \\ref{sheaves-lemma-pullback-presheaves}.", "Since $f_{Zar}^{-1}\\mathcal{F} = u_s\\mathcal{F}$ is the sheafification", "of $u_p\\mathcal{F}$ and since sheafification uses only coverings and", "since coverings in $Y_{Zar}$ use only inclusions between opens on the", "same $Y_n$, the result follows from the fact that $f_n^{-1}\\mathcal{F}_n$", "is (correspondingly) the sheafification of $f_{n, p}\\mathcal{F}_n$, see", "Sheaves, Section \\ref{sheaves-section-presheaves-functorial}." ], "refs": [ "spaces-simplicial-lemma-simplicial-space-site-functorial", "sheaves-lemma-pullback-presheaves" ], "ref_ids": [ 9012, 14505 ] } ], "ref_ids": [ 9011, 9012 ] }, { "id": 9014, "type": "theorem", "label": "spaces-simplicial-lemma-restriction-to-components", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-restriction-to-components", "contents": [ "Let $X$ be a simplicial space. The functor", "$X_{n, Zar} \\to X_{Zar}$, $U \\mapsto U$ is continuous", "and cocontinuous. The associated morphism of", "topoi $g_n : \\Sh(X_n) \\to \\Sh(X_{Zar})$ satisfies", "\\begin{enumerate}", "\\item $g_n^{-1}$ associates to the sheaf $\\mathcal{F}$ on $X$", "the sheaf $\\mathcal{F}_n$ on $X_n$,", "\\item $g_n^{-1} : \\Sh(X_{Zar}) \\to \\Sh(X_n)$ has a left adjoint $g^{Sh}_{n!}$,", "\\item $g^{Sh}_{n!}$ commutes with finite connected limits,", "\\item $g_n^{-1} : \\textit{Ab}(X_{Zar}) \\to \\textit{Ab}(X_n)$", "has a left adjoint $g_{n!}$, and", "\\item $g_{n!}$ is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Besides the properties of our functor mentioned in the statement,", "the category $X_{n, Zar}$ has fibre products and equalizers", "and the functor commutes with them (beware that $X_{Zar}$ does not", "have all fibre products). Hence the lemma follows from the discussion in", "Sites, Sections \\ref{sites-section-cocontinuous-functors} and", "\\ref{sites-section-cocontinuous-morphism-topoi}", "and", "Modules on Sites, Section \\ref{sites-modules-section-exactness-lower-shriek}.", "More precisely,", "Sites, Lemmas \\ref{sites-lemma-cocontinuous-morphism-topoi},", "\\ref{sites-lemma-when-shriek}, and", "\\ref{sites-lemma-preserve-equalizers}", "and", "Modules on Sites, Lemmas", "\\ref{sites-modules-lemma-g-shriek-adjoint} and", "\\ref{sites-modules-lemma-exactness-lower-shriek}." ], "refs": [ "sites-lemma-cocontinuous-morphism-topoi", "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers", "sites-modules-lemma-g-shriek-adjoint", "sites-modules-lemma-exactness-lower-shriek" ], "ref_ids": [ 8543, 8545, 8546, 14164, 14165 ] } ], "ref_ids": [] }, { "id": 9015, "type": "theorem", "label": "spaces-simplicial-lemma-restriction-injective-to-component", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-restriction-injective-to-component", "contents": [ "Let $X$ be a simplicial space. If $\\mathcal{I}$ is an injective abelian", "sheaf on $X_{Zar}$, then $\\mathcal{I}_n$ is an injective abelian sheaf", "on $X_n$." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}", "and", "Lemma \\ref{lemma-restriction-to-components}." ], "refs": [ "homology-lemma-adjoint-preserve-injectives", "spaces-simplicial-lemma-restriction-to-components" ], "ref_ids": [ 12116, 9014 ] } ], "ref_ids": [] }, { "id": 9016, "type": "theorem", "label": "spaces-simplicial-lemma-restriction-to-components-functorial", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-restriction-to-components-functorial", "contents": [ "Let $f : Y \\to X$ be a morphism of simplicial spaces. Then", "$$", "\\xymatrix{", "\\Sh(Y_n) \\ar[d] \\ar[r]_{f_n} & \\Sh(X_n) \\ar[d] \\\\", "\\Sh(Y_{Zar}) \\ar[r]^{f_{Zar}} & \\Sh(X_{Zar})", "}", "$$", "is a commutative diagram of topoi." ], "refs": [], "proofs": [ { "contents": [ "Direct from the description of pullback functors in", "Lemmas \\ref{lemma-describe-functoriality} and", "\\ref{lemma-restriction-to-components}." ], "refs": [ "spaces-simplicial-lemma-describe-functoriality", "spaces-simplicial-lemma-restriction-to-components" ], "ref_ids": [ 9013, 9014 ] } ], "ref_ids": [] }, { "id": 9017, "type": "theorem", "label": "spaces-simplicial-lemma-augmentation", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-augmentation", "contents": [ "Let $Y$ be a simplicial space and let $a : Y \\to X$ be an augmentation", "(Simplicial, Definition \\ref{simplicial-definition-augmentation}).", "Let $a_n : Y_n \\to X$ be the corresponding morphisms of topological spaces.", "There is a canonical morphism of topoi", "$$", "a : \\Sh(Y_{Zar}) \\to \\Sh(X)", "$$", "with the following properties:", "\\begin{enumerate}", "\\item $a^{-1}\\mathcal{F}$ is the sheaf restricting to $a_n^{-1}\\mathcal{F}$", "on $Y_n$,", "\\item $a_m \\circ Y(\\varphi) = a_n$ for all $\\varphi : [m] \\to [n]$,", "\\item $a \\circ g_n = a_n$ as morphisms of topoi with", "$g_n$ as in Lemma \\ref{lemma-restriction-to-components},", "\\item $a_*\\mathcal{G}$ for $\\mathcal{G} \\in \\Sh(Y_{Zar})$", "is the equalizer of the two maps", "$a_{0, *}\\mathcal{G}_0 \\to a_{1, *}\\mathcal{G}_1$.", "\\end{enumerate}" ], "refs": [ "simplicial-definition-augmentation", "spaces-simplicial-lemma-restriction-to-components" ], "proofs": [ { "contents": [ "Part (2) holds for augmentations of simplicial objects in any category.", "Thus $Y(\\varphi)^{-1} a_m^{-1} \\mathcal{F} = a_n^{-1}\\mathcal{F}$", "which defines an $Y(\\varphi)$-map from $a_m^{-1}\\mathcal{F}$", "to $a_n^{-1}\\mathcal{F}$.", "Thus we can use (1) as the definition of $a^{-1}\\mathcal{F}$ (using", "Lemma \\ref{lemma-describe-sheaves-simplicial-site}) and", "(4) as the definition of $a_*$. If this defines a morphism of topoi", "then part (3) follows because we'll have $g_n^{-1} \\circ a^{-1} = a_n^{-1}$", "by construction. To check $a$ is a morphism of topoi we have to show", "that $a^{-1}$ is left adjoint to $a_*$ and we have to show that", "$a^{-1}$ is exact. The last fact is immediate from the exactness of", "the functors $a_n^{-1}$.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be an object of $\\Sh(X)$ and let $\\mathcal{G}$", "be an object of $\\Sh(Y_{Zar})$. Given", "$\\beta : a^{-1}\\mathcal{F} \\to \\mathcal{G}$ we can look at the", "components $\\beta_n : a_n^{-1}\\mathcal{F} \\to \\mathcal{G}_n$.", "These maps are adjoint to maps", "$\\beta_n : \\mathcal{F} \\to a_{n, *}\\mathcal{G}_n$.", "Compatibility with the simplicial structure shows that", "$\\beta_0$ maps into $a_*\\mathcal{G}$.", "Conversely, suppose given a map $\\alpha : \\mathcal{F} \\to a_*\\mathcal{G}$.", "For any $n$ choose a $\\varphi : [0] \\to [n]$. Then we can look at", "the composition", "$$", "\\mathcal{F} \\xrightarrow{\\alpha} a_*\\mathcal{G}", "\\to a_{0, *}\\mathcal{G}_0 \\xrightarrow{\\mathcal{G}(\\varphi)}", "a_{n, *}\\mathcal{G}_n", "$$", "These are adjoint to maps $a_n^{-1}\\mathcal{F} \\to \\mathcal{G}_n$", "which define a morphism of sheaves $a^{-1}\\mathcal{F} \\to \\mathcal{G}$.", "We omit the proof that the constructions given above define", "mutually inverse bijections", "$$", "\\Mor_{\\Sh(Y_{Zar})}(a^{-1}\\mathcal{F}, \\mathcal{G}) =", "\\Mor_{\\Sh(X)}(\\mathcal{F}, a_*\\mathcal{G})", "$$", "This finishes the proof. An interesting observation is here that", "this morphism of topoi does not correspond to any obvious geometric", "functor between the sites defining the topoi." ], "refs": [ "spaces-simplicial-lemma-describe-sheaves-simplicial-site" ], "ref_ids": [ 9011 ] } ], "ref_ids": [ 14925, 9014 ] }, { "id": 9018, "type": "theorem", "label": "spaces-simplicial-lemma-simplicial-resolution-Z", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-simplicial-resolution-Z", "contents": [ "Let $X$ be a simplicial topological space. The complex of", "abelian presheaves on $X_{Zar}$", "$$", "\\ldots \\to \\mathbf{Z}_{X_2} \\to \\mathbf{Z}_{X_1} \\to \\mathbf{Z}_{X_0}", "$$", "with boundary $\\sum (-1)^i d^n_i$ is a resolution", "of the constant presheaf $\\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X_m$ be an object of $X_{Zar}$. Then the value of", "the complex above on $U$ is the complex of abelian groups", "$$", "\\ldots \\to", "\\mathbf{Z}[\\Mor_\\Delta([2], [m])] \\to", "\\mathbf{Z}[\\Mor_\\Delta([1], [m])] \\to", "\\mathbf{Z}[\\Mor_\\Delta([0], [m])]", "$$", "In other words, this is the complex associated to the", "free abelian group on the simplicial set $\\Delta[m]$, see", "Simplicial, Example \\ref{simplicial-example-simplex-simplicial-set}.", "Since $\\Delta[m]$ is homotopy equivalent to $\\Delta[0]$, see", "Simplicial, Example \\ref{simplicial-example-simplex-contractible},", "and since ``taking free abelian groups'' is a functor,", "we see that the complex above is homotopy equivalent to", "the free abelian group on $\\Delta[0]$", "(Simplicial, Remark \\ref{simplicial-remark-homotopy-better} and", "Lemma \\ref{simplicial-lemma-homotopy-equivalence-s-N}).", "This complex is acyclic in positive degrees", "and equal to $\\mathbf{Z}$ in degree $0$." ], "refs": [ "simplicial-remark-homotopy-better", "simplicial-lemma-homotopy-equivalence-s-N" ], "ref_ids": [ 14941, 14877 ] } ], "ref_ids": [] }, { "id": 9019, "type": "theorem", "label": "spaces-simplicial-lemma-simplicial-sheaf-cohomology", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-simplicial-sheaf-cohomology", "contents": [ "Let $X$ be a simplicial topological space. Let $\\mathcal{F}$ be an abelian", "sheaf on $X$. There is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with", "$$", "E_1^{p, q} = H^q(X_p, \\mathcal{F}_p)", "$$", "converging to $H^{p + q}(X_{Zar}, \\mathcal{F})$.", "This spectral sequence is functorial in $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ be an injective resolution.", "Consider the double complex with terms", "$$", "A^{p, q} = \\mathcal{I}^q(X_p)", "$$", "and first differential given by the alternating sum along the maps", "$d^{p + 1}_i$-maps $\\mathcal{I}_p^q \\to \\mathcal{I}_{p + 1}^q$, see", "Lemma \\ref{lemma-describe-sheaves-simplicial-site}. Note that", "$$", "A^{p, q} = \\Gamma(X_p, \\mathcal{I}_p^q) =", "\\Mor_{\\textit{PSh}}(h_{X_p}, \\mathcal{I}^q) =", "\\Mor_{\\textit{PAb}}(\\mathbf{Z}_{X_p}, \\mathcal{I}^q)", "$$", "Hence it follows from Lemma \\ref{lemma-simplicial-resolution-Z} and", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-injective-abelian-sheaf-injective-presheaf}", "that the rows of the double complex are exact in positive degrees and", "evaluate to $\\Gamma(X_{Zar}, \\mathcal{I}^q)$ in degree $0$.", "On the other hand, since restriction is exact", "(Lemma \\ref{lemma-restriction-to-components})", "the map", "$$", "\\mathcal{F}_p \\to \\mathcal{I}_p^\\bullet", "$$", "is a resolution. The sheaves $\\mathcal{I}_p^q$ are injective", "abelian sheaves on $X_p$", "(Lemma \\ref{lemma-restriction-injective-to-component}).", "Hence the cohomology of the columns computes the groups", "$H^q(X_p, \\mathcal{F}_p)$. We conclude by applying", "Homology, Lemmas \\ref{homology-lemma-first-quadrant-ss} and", "\\ref{homology-lemma-double-complex-gives-resolution}." ], "refs": [ "spaces-simplicial-lemma-describe-sheaves-simplicial-site", "spaces-simplicial-lemma-simplicial-resolution-Z", "sites-cohomology-lemma-injective-abelian-sheaf-injective-presheaf", "spaces-simplicial-lemma-restriction-to-components", "spaces-simplicial-lemma-restriction-injective-to-component", "homology-lemma-first-quadrant-ss", "homology-lemma-double-complex-gives-resolution" ], "ref_ids": [ 9011, 9018, 4197, 9014, 9015, 12105, 12106 ] } ], "ref_ids": [] }, { "id": 9020, "type": "theorem", "label": "spaces-simplicial-lemma-augmentation-pushforward-higher-direct-image", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-augmentation-pushforward-higher-direct-image", "contents": [ "Let $X$ be a simplicial space and let $a : X \\to Y$", "be an augmentation. Let $\\mathcal{F}$ be an abelian sheaf", "on $X_{Zar}$. Then $R^na_*\\mathcal{F}$ is the sheaf associated", "to the presheaf", "$$", "V \\longmapsto H^n((X \\times_Y V)_{Zar}, \\mathcal{F}|_{(X \\times_Y V)_{Zar}})", "$$" ], "refs": [], "proofs": [ { "contents": [ "This is the analogue of", "Cohomology, Lemma \\ref{cohomology-lemma-describe-higher-direct-images} or of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-higher-direct-images}", "and we strongly encourge the reader to skip the proof.", "Choosing an injective resolution of $\\mathcal{F}$ on", "$X_{Zar}$ and using the definitions we see that it suffices to show:", "(1) the restriction of an injective abelian", "sheaf on $X_{Zar}$ to $(X \\times_Y V)_{Zar}$ is an injective abelian sheaf and", "(2) $a_*\\mathcal{F}$ is equal to the rule", "$$", "V \\longmapsto H^0((X \\times_Y V)_{Zar}, \\mathcal{F}|_{(X \\times_Y V)_{Zar}})", "$$", "Part (2) follows from the following facts", "\\begin{enumerate}", "\\item[(2a)] $a_*\\mathcal{F}$ is the equalizer of the two maps", "$a_{0, *}\\mathcal{F}_0 \\to a_{1, *}\\mathcal{F}_1$", "by Lemma \\ref{lemma-augmentation},", "\\item[(2b)] $a_{0, *}\\mathcal{F}_0(V) =", "H^0(a_0^{-1}(V), \\mathcal{F}_0)$ and", "$a_{1, *}\\mathcal{F}_1(V) = H^0(a_1^{-1}(V), \\mathcal{F}_1)$,", "\\item[(2c)] $X_0 \\times_Y V = a_0^{-1}(V)$ and $X_1 \\times_Y V = a_1^{-1}(V)$,", "\\item[(2d)] $H^0((X \\times_Y V)_{Zar}, \\mathcal{F}|_{(X \\times_Y V)_{Zar}})$", "is the equalizer of the two maps", "$H^0(X_0 \\times_Y V, \\mathcal{F}_0) \\to H^0(X_1 \\times_Y V, \\mathcal{F}_1)$", "for example by Lemma \\ref{lemma-simplicial-sheaf-cohomology}.", "\\end{enumerate}", "Part (1) follows after one defines an exact left adjoint", "$j_! : \\textit{Ab}((X \\times_Y V)_{Zar}) \\to \\textit{Ab}(X_{Zar})$", "(extension by zero) to restriction", "$\\textit{Ab}(X_{Zar}) \\to \\textit{Ab}((X \\times_Y V)_{Zar})$", "and using Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}." ], "refs": [ "cohomology-lemma-describe-higher-direct-images", "sites-cohomology-lemma-higher-direct-images", "spaces-simplicial-lemma-augmentation", "spaces-simplicial-lemma-simplicial-sheaf-cohomology", "homology-lemma-adjoint-preserve-injectives" ], "ref_ids": [ 2039, 4189, 9017, 9019, 12116 ] } ], "ref_ids": [] }, { "id": 9021, "type": "theorem", "label": "spaces-simplicial-lemma-constant-simplicial-space", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-constant-simplicial-space", "contents": [ "Let $X$ be a topological space. Let $X_\\bullet$ be the constant", "simplicial topological space with value $X$. The functor", "$$", "X_{\\bullet, Zar} \\longrightarrow X_{Zar},\\quad", "U \\longmapsto U", "$$", "is continuous and cocontinuous and defines a morphism of", "topoi $g : \\Sh(X_{\\bullet, Zar}) \\to \\Sh(X)$ as well as a left adjoint", "$g_!$ to $g^{-1}$. We have", "\\begin{enumerate}", "\\item $g^{-1}$ associates to a sheaf on $X$ the constant cosimplicial", "sheaf on $X$,", "\\item $g_!$ associates to a sheaf $\\mathcal{F}$ on $X_{\\bullet, Zar}$ the", "sheaf $\\mathcal{F}_0$, and", "\\item $g_*$ associates to a sheaf $\\mathcal{F}$ on $X_{\\bullet, Zar}$ the", "equalizer of the two maps $\\mathcal{F}_0 \\to \\mathcal{F}_1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The statements about the functor are straightforward to verify.", "The existence of $g$ and $g_!$ follow from", "Sites, Lemmas \\ref{sites-lemma-cocontinuous-morphism-topoi} and", "\\ref{sites-lemma-when-shriek}. The description of", "$g^{-1}$ is immediate from Sites, Lemma \\ref{sites-lemma-when-shriek}.", "The description of $g_*$ and $g_!$ follows as the functors given are", "right and left adjoint to $g^{-1}$." ], "refs": [ "sites-lemma-cocontinuous-morphism-topoi", "sites-lemma-when-shriek", "sites-lemma-when-shriek" ], "ref_ids": [ 8543, 8545, 8545 ] } ], "ref_ids": [] }, { "id": 9022, "type": "theorem", "label": "spaces-simplicial-lemma-simplicial-site-site", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-simplicial-site-site", "contents": [ "Let $\\mathcal{C}$ be a simplicial object in the category of sites.", "With notation as above we construct a site $\\mathcal{C}_{total}$ as follows.", "\\begin{enumerate}", "\\item An object of $\\mathcal{C}_{total}$ is an object $U$ of", "$\\mathcal{C}_n$ for some $n$,", "\\item a morphism $(\\varphi, f) : U \\to V$ of $\\mathcal{C}_{total}$", "is given by a map $\\varphi : [m] \\to [n]$ with", "$U \\in \\Ob(\\mathcal{C}_n)$, $V \\in \\Ob(\\mathcal{C}_m)$", "and a morphism $f : U \\to u_\\varphi(V)$ of $\\mathcal{C}_n$, and", "\\item a covering $\\{(\\text{id}, f_i) : U_i \\to U\\}$ in $\\mathcal{C}_{total}$", "is given by an $n$ and a covering $\\{f_i : U_i \\to U\\}$", "of $\\mathcal{C}_n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Composition of $(\\varphi, f) : U \\to V$ with $(\\psi, g) : V \\to W$", "is given by $(\\varphi \\circ \\psi, u_\\varphi(g) \\circ f)$.", "This uses that $u_\\varphi \\circ u_\\psi = u_{\\varphi \\circ \\psi}$.", "\\medskip\\noindent", "Let $\\{(\\text{id}, f_i) : U_i \\to U\\}$ be a covering as in (3)", "and let $(\\varphi, g) : W \\to U$ be a morphism with", "$W \\in \\Ob(\\mathcal{C}_m)$. We claim that", "$$", "W \\times_{(\\varphi, g), U, (\\text{id}, f_i)} U_i =", "W \\times_{g, u_\\varphi(U), u_\\varphi(f_i)} u_\\varphi(U_i)", "$$", "in the category $\\mathcal{C}_{total}$. This makes sense as by our", "definition of morphisms of sites, the required fibre products", "in $\\mathcal{C}_m$ exist since $u_\\varphi$ transforms coverings into", "coverings. The same reasoning implies the claim (details omitted).", "Thus we see that the collection of coverings is stable under base", "change. The other axioms of a site are immediate." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9023, "type": "theorem", "label": "spaces-simplicial-lemma-simplicial-cocontinuous-site", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-simplicial-cocontinuous-site", "contents": [ "Let $\\mathcal{C}$ be a simplicial object in the category whose objects are", "sites and whose morphisms are cocontinuous functors. With notation as above,", "assume the functors $u_\\varphi : \\mathcal{C}_n \\to \\mathcal{C}_m$", "have property $P$ of Sites, Remark \\ref{sites-remark-cartesian-cocontinuous}.", "Then we can construct a site $\\mathcal{C}_{total}$ as follows.", "\\begin{enumerate}", "\\item An object of $\\mathcal{C}_{total}$ is an object $U$ of", "$\\mathcal{C}_n$ for some $n$,", "\\item a morphism $(\\varphi, f) : U \\to V$ of $\\mathcal{C}_{total}$", "is given by a map $\\varphi : [m] \\to [n]$ with", "$U \\in \\Ob(\\mathcal{C}_n)$, $V \\in \\Ob(\\mathcal{C}_m)$", "and a morphism $f : u_\\varphi(U) \\to V$ of $\\mathcal{C}_m$, and", "\\item a covering $\\{(\\text{id}, f_i) : U_i \\to U\\}$ in $\\mathcal{C}_{total}$", "is given by an $n$ and a covering $\\{f_i : U_i \\to U\\}$", "of $\\mathcal{C}_n$.", "\\end{enumerate}" ], "refs": [ "sites-remark-cartesian-cocontinuous" ], "proofs": [ { "contents": [ "Composition of $(\\varphi, f) : U \\to V$ with $(\\psi, g) : V \\to W$", "is given by $(\\varphi \\circ \\psi, g \\circ u_\\psi(f))$.", "This uses that $u_\\psi \\circ u_\\varphi = u_{\\varphi \\circ \\psi}$.", "\\medskip\\noindent", "Let $\\{(\\text{id}, f_i) : U_i \\to U\\}$ be a covering as in (3)", "and let $(\\varphi, g) : W \\to U$ be a morphism with", "$W \\in \\Ob(\\mathcal{C}_m)$. We claim that", "$$", "W \\times_{(\\varphi, g), U, (\\text{id}, f_i)} U_i =", "W \\times_{g, U, f_i} U_i", "$$", "in the category $\\mathcal{C}_{total}$ where the right hand side", "is the object of $\\mathcal{C}_m$ defined in", "Sites, Remark \\ref{sites-remark-cartesian-cocontinuous}", "which exists by property $P$. Compatibility of this type of fibre product", "with compositions of functors implies the claim (details omitted).", "Since the family $\\{W \\times_{g, U, f_i} U_i \\to W\\}$ is a", "covering of $\\mathcal{C}_m$ by property $P$ we see that", "the collection of coverings is stable under base", "change. The other axioms of a site are immediate." ], "refs": [ "sites-remark-cartesian-cocontinuous" ], "ref_ids": [ 8712 ] } ], "ref_ids": [ 8712 ] }, { "id": 9024, "type": "theorem", "label": "spaces-simplicial-lemma-describe-sheaves-simplicial-site-site", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-describe-sheaves-simplicial-site-site", "contents": [ "In Situation \\ref{situation-simplicial-site} there is an equivalence of", "categories between", "\\begin{enumerate}", "\\item $\\Sh(\\mathcal{C}_{total})$, and", "\\item the category of systems $(\\mathcal{F}_n, \\mathcal{F}(\\varphi))$", "described above.", "\\end{enumerate}", "In particular, the topos $\\Sh(\\mathcal{C}_{total})$ only depends on", "the topoi $\\Sh(\\mathcal{C}_n)$ and the morphisms of topoi $f_\\varphi$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9025, "type": "theorem", "label": "spaces-simplicial-lemma-restriction-to-components-site", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-restriction-to-components-site", "contents": [ "In Situation \\ref{situation-simplicial-site} the functor", "$\\mathcal{C}_n \\to \\mathcal{C}_{total}$, $U \\mapsto U$ is continuous", "and cocontinuous. The associated morphism of", "topoi $g_n : \\Sh(\\mathcal{C}_n) \\to \\Sh(\\mathcal{C}_{total})$ satisfies", "\\begin{enumerate}", "\\item $g_n^{-1}$ associates to the sheaf $\\mathcal{F}$ on $\\mathcal{C}_{total}$", "the sheaf $\\mathcal{F}_n$ on $\\mathcal{C}_n$,", "\\item $g_n^{-1} : \\Sh(\\mathcal{C}_{total}) \\to \\Sh(\\mathcal{C}_n)$", "has a left adjoint $g^{Sh}_{n!}$,", "\\item for $\\mathcal{G}$ in $\\Sh(\\mathcal{C}_n)$ the restriction of", "$g_{n!}^{Sh}\\mathcal{G}$ to $\\mathcal{C}_m$ is", "$\\coprod\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^{-1}\\mathcal{G}$,", "\\item $g_{n!}^{Sh}$ commutes with finite connected limits,", "\\item $g_n^{-1} : \\textit{Ab}(\\mathcal{C}_{total}) \\to", "\\textit{Ab}(\\mathcal{C}_n)$ has a left adjoint $g_{n!}$,", "\\item for $\\mathcal{G}$ in $\\textit{Ab}(\\mathcal{C}_n)$ the restriction of", "$g_{n!}\\mathcal{G}$ to $\\mathcal{C}_m$ is", "$\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^{-1}\\mathcal{G}$, and", "\\item $g_{n!}$ is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Case A. If $\\{U_i \\to U\\}_{i \\in I}$ is a covering in $\\mathcal{C}_n$", "then the image $\\{U_i \\to U\\}_{i \\in I}$ is a covering in $\\mathcal{C}_{total}$", "by definition (Lemma \\ref{lemma-simplicial-site-site}). For a morphism", "$V \\to U$ of $\\mathcal{C}_n$, the fibre product", "$V \\times_U U_i$ in $\\mathcal{C}_n$ is also", "the fibre product in $\\mathcal{C}_{total}$ (by the claim in the", "proof of Lemma \\ref{lemma-simplicial-site-site}).", "Therefore our functor is continuous. On the other hand, our functor", "defines a bijection between coverings of $U$ in $\\mathcal{C}_n$", "and coverings of $U$ in $\\mathcal{C}_{total}$. Therefore it is", "certainly the case that our functor is cocontinuous.", "\\medskip\\noindent", "Case B. If $\\{U_i \\to U\\}_{i \\in I}$ is a covering in $\\mathcal{C}_n$", "then the image $\\{U_i \\to U\\}_{i \\in I}$ is a covering in $\\mathcal{C}_{total}$", "by definition (Lemma \\ref{lemma-simplicial-cocontinuous-site}). For a morphism", "$V \\to U$ of $\\mathcal{C}_n$, the fibre product", "$V \\times_U U_i$ in $\\mathcal{C}_n$ is also", "the fibre product in $\\mathcal{C}_{total}$ (by the claim in the", "proof of Lemma \\ref{lemma-simplicial-cocontinuous-site}).", "Therefore our functor is continuous. On the other hand, our functor", "defines a bijection between coverings of $U$ in $\\mathcal{C}_n$", "and coverings of $U$ in $\\mathcal{C}_{total}$. Therefore it is", "certainly the case that our functor is cocontinuous.", "\\medskip\\noindent", "At this point part (1) and the existence of $g^{Sh}_{n!}$ and $g_{n!}$", "in cases A and B follows from", "Sites, Lemmas \\ref{sites-lemma-cocontinuous-morphism-topoi} and", "\\ref{sites-lemma-when-shriek}", "and", "Modules on Sites, Lemma \\ref{sites-modules-lemma-g-shriek-adjoint}.", "\\medskip\\noindent", "Proof of (3). Let $\\mathcal{G}$ be a sheaf on $\\mathcal{C}_n$.", "Consider the sheaf $\\mathcal{H}$ on $\\mathcal{C}_{total}$", "whose degree $m$ part is the sheaf", "$$", "\\mathcal{H}_m = \\coprod\\nolimits_{\\varphi : [n] \\to [m]}", "f_\\varphi^{-1}\\mathcal{G}", "$$", "given in part (3) of the statement of the lemma.", "Given a map $\\psi : [m] \\to [m']$ the map", "$\\mathcal{H}(\\psi) : f_\\psi^{-1}\\mathcal{H}_m \\to \\mathcal{H}_{m'}$", "is given on components by the identifications", "$$", "f_\\psi^{-1} f_\\varphi^{-1} \\mathcal{G} \\to", "f_{\\psi \\circ \\varphi}^{-1}\\mathcal{G}", "$$", "Observe that given a map $\\alpha : \\mathcal{H} \\to \\mathcal{F}$", "of sheaves on $\\mathcal{C}_{total}$ we obtain a map", "$\\mathcal{G} \\to \\mathcal{F}_n$", "corresponding to the restriction of $\\alpha_n$ to the component", "$\\mathcal{G}$ in $\\mathcal{H}_n$. Conversely, given a map", "$\\beta : \\mathcal{G} \\to \\mathcal{F}_n$ of sheaves on $\\mathcal{C}_n$", "we can define", "$\\alpha : \\mathcal{H} \\to \\mathcal{F}$ by letting $\\alpha_m$", "be the map which on components", "$$", "f_\\varphi^{-1}\\mathcal{G} \\to \\mathcal{F}_m", "$$", "uses the maps adjoint to $\\mathcal{F}(\\varphi) \\circ f_\\varphi^{-1}\\beta$.", "We omit the arguments showing these two constructions give", "mutually inverse maps", "$$", "\\Mor_{\\Sh(\\mathcal{C}_n)}(\\mathcal{G}, \\mathcal{F}_n) =", "\\Mor_{\\Sh(\\mathcal{C}_{total})}(\\mathcal{H}, \\mathcal{F})", "$$", "Thus $\\mathcal{H} = g^{Sh}_{n!}\\mathcal{G}$ as desired.", "\\medskip\\noindent", "Proof of (4). If $\\mathcal{G}$ is an abelian sheaf on $\\mathcal{C}_n$,", "then we proceed in exactly the same ammner as above, except that", "we define $\\mathcal{H}$ is the abelian sheaf on $\\mathcal{C}_{total}$", "whose degree $m$ part is the sheaf", "$$", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^{-1}\\mathcal{G}", "$$", "with transition maps defined exactly as above. The bijection", "$$", "\\Mor_{\\textit{Ab}(\\mathcal{C}_n)}(\\mathcal{G}, \\mathcal{F}_n) =", "\\Mor_{\\textit{Ab}(\\mathcal{C}_{total})}(\\mathcal{H}, \\mathcal{F})", "$$", "is proved exactly as above.", "Thus $\\mathcal{H} = g_{n!}\\mathcal{G}$ as desired.", "\\medskip\\noindent", "The exactness properties of $g^{Sh}_{n!}$ and $g_{n!}$ follow", "from formulas given for these functors." ], "refs": [ "spaces-simplicial-lemma-simplicial-site-site", "spaces-simplicial-lemma-simplicial-site-site", "spaces-simplicial-lemma-simplicial-cocontinuous-site", "spaces-simplicial-lemma-simplicial-cocontinuous-site", "sites-lemma-cocontinuous-morphism-topoi", "sites-lemma-when-shriek", "sites-modules-lemma-g-shriek-adjoint" ], "ref_ids": [ 9022, 9022, 9023, 9023, 8543, 8545, 14164 ] } ], "ref_ids": [] }, { "id": 9026, "type": "theorem", "label": "spaces-simplicial-lemma-restriction-injective-to-component-site", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-restriction-injective-to-component-site", "contents": [ "\\begin{slogan}", "An injective abelian sheaf on a simplicial site is injective on each component", "\\end{slogan}", "In Situation \\ref{situation-simplicial-site}.", "If $\\mathcal{I}$ is injective in $\\textit{Ab}(\\mathcal{C}_{total})$,", "then $\\mathcal{I}_n$ is injective in $\\textit{Ab}(\\mathcal{C}_n)$.", "If $\\mathcal{I}^\\bullet$ is a K-injective complex in", "$\\textit{Ab}(\\mathcal{C}_{total})$,", "then $\\mathcal{I}_n^\\bullet$ is K-injective in $\\textit{Ab}(\\mathcal{C}_n)$." ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}", "and", "Lemma \\ref{lemma-restriction-to-components-site}.", "The second statement from", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}", "and", "Lemma \\ref{lemma-restriction-to-components-site}." ], "refs": [ "homology-lemma-adjoint-preserve-injectives", "spaces-simplicial-lemma-restriction-to-components-site", "derived-lemma-adjoint-preserve-K-injectives", "spaces-simplicial-lemma-restriction-to-components-site" ], "ref_ids": [ 12116, 9025, 1915, 9025 ] } ], "ref_ids": [] }, { "id": 9027, "type": "theorem", "label": "spaces-simplicial-lemma-augmentation-site", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-augmentation-site", "contents": [ "In Situation \\ref{situation-simplicial-site} let $a_0$ be an", "augmentation towards a site $\\mathcal{D}$ as in", "Remark \\ref{remark-augmentation-site}. Then $a_0$ induces", "\\begin{enumerate}", "\\item a morphism of topoi $a_n : \\Sh(\\mathcal{C}_n) \\to \\Sh(\\mathcal{D})$", "for all $n \\geq 0$,", "\\item a morphism of topoi $a : \\Sh(\\mathcal{C}_{total}) \\to \\Sh(\\mathcal{D})$", "\\end{enumerate}", "such that", "\\begin{enumerate}", "\\item for all $\\varphi : [m] \\to [n]$ we have $a_m \\circ f_\\varphi = a_n$,", "\\item if $g_n : \\Sh(\\mathcal{C}_n) \\to \\Sh(\\mathcal{C}_{total})$", "is as in Lemma \\ref{lemma-restriction-to-components-site}, then", "$a \\circ g_n = a_n$, and", "\\item $a_*\\mathcal{F}$ for $\\mathcal{F} \\in \\Sh(\\mathcal{C}_{total})$", "is the equalizer of the two maps", "$a_{0, *}\\mathcal{F}_0 \\to a_{1, *}\\mathcal{F}_1$.", "\\end{enumerate}" ], "refs": [ "spaces-simplicial-remark-augmentation-site", "spaces-simplicial-lemma-restriction-to-components-site" ], "proofs": [ { "contents": [ "Case A. Let $u_n : \\mathcal{D} \\to \\mathcal{C}_n$ be the common", "value of the functors $u_\\varphi \\circ u_0$ for $\\varphi : [0] \\to [n]$.", "Then $u_n$ corresponds to a morphism of sites", "$a_n : \\mathcal{C}_n \\to \\mathcal{D}$, see", "Sites, Lemma \\ref{sites-lemma-composition-morphisms-sites}.", "The same lemma shows that for all $\\varphi : [m] \\to [n]$ we have", "$a_m \\circ f_\\varphi = a_n$.", "\\medskip\\noindent", "Case B. Let $u_n : \\mathcal{C}_n \\to \\mathcal{D}$ be the common", "value of the functors $u_0 \\circ u_\\varphi$ for $\\varphi : [0] \\to [n]$.", "Then $u_n$ is cocontinuous and hence defines a morphism of topoi", "$a_n : \\Sh(\\mathcal{C}_n) \\to \\Sh(\\mathcal{D)}$, see", "Sites, Lemma \\ref{sites-lemma-composition-cocontinuous}.", "The same lemma shows that for all $\\varphi : [m] \\to [n]$ we have", "$a_m \\circ f_\\varphi = a_n$.", "\\medskip\\noindent", "Consider the functor $a^{-1} : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C}_{total})$", "which to a sheaf of sets $\\mathcal{G}$ associates the sheaf", "$\\mathcal{F} = a^{-1}\\mathcal{G}$ whose components are $a_n^{-1}\\mathcal{G}$", "and whose transition maps $\\mathcal{F}(\\varphi)$ are the identifications", "$$", "f_\\varphi^{-1}\\mathcal{F}_m =", "f_\\varphi^{-1} a_m^{-1}\\mathcal{G} =", "a_n^{-1}\\mathcal{G} =", "\\mathcal{F}_n", "$$", "for $\\varphi : [m] \\to [n]$, see the description of", "$\\Sh(\\mathcal{C}_{total})$ in", "Lemma \\ref{lemma-describe-sheaves-simplicial-site-site}.", "Since the functors $a_n^{-1}$ are exact, $a^{-1}$ is an exact functor.", "Finally, for $a_* : \\Sh(\\mathcal{C}_{total}) \\to \\Sh(\\mathcal{D})$", "we take the functor which to a sheaf $\\mathcal{F}$ on $\\Sh(\\mathcal{D})$", "associates", "$$", "\\xymatrix{", "a_*\\mathcal{F} \\ar@{=}[r] &", "\\text{Equalizer}(a_{0, *}\\mathcal{F}_0", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "a_{1, *}\\mathcal{F}_1)", "}", "$$", "Here the two maps come from the two maps $\\varphi : [0] \\to [1]$", "via", "$$", "a_{0, *}\\mathcal{F}_0 \\to", "a_{0, *}f_{\\varphi, *} f_\\varphi^{-1}\\mathcal{F}_0", "\\xrightarrow{\\mathcal{F}(\\varphi)}", "a_{0, *}f_{\\varphi, *} \\mathcal{F}_0 = a_{1, *}\\mathcal{F}_1", "$$", "where the first arrow comes from $1 \\to f_{\\varphi, *} f_\\varphi^{-1}$.", "Let $\\mathcal{G}_\\bullet$ denote the constant simplicial sheaf", "with value $\\mathcal{G}$ and let $a_{\\bullet, *}\\mathcal{F}$", "denote the simplicial sheaf having $a_{n, *}\\mathcal{F}_n$ in degree $n$.", "By the usual adjuntion for the morphisms of topoi $a_n$ we see that", "a map $a^{-1}\\mathcal{G} \\to \\mathcal{F}$", "is the same thing as a map", "$$", "\\mathcal{G}_\\bullet \\longrightarrow a_{\\bullet, *}\\mathcal{F}", "$$", "of simplicial sheaves.", "By Simplicial, Lemma \\ref{simplicial-lemma-augmentation-howto}", "this is the same thing as a map $\\mathcal{G} \\to a_*\\mathcal{F}$.", "Thus $a^{-1}$ and $a_*$ are adjoint functors and we obtain", "our morphism of topoi $a$\\footnote{In case B the morphism $a$", "corresponds to the cocontinuous functor", "$\\mathcal{C}_{total} \\to \\mathcal{D}$ sending", "$U$ in $\\mathcal{C}_n$ to $u_n(U)$.}. The equalities", "$a \\circ g_n = f_n$ follow immediately from the definitions." ], "refs": [ "sites-lemma-composition-morphisms-sites", "sites-lemma-composition-cocontinuous", "spaces-simplicial-lemma-describe-sheaves-simplicial-site-site", "simplicial-lemma-augmentation-howto" ], "ref_ids": [ 8525, 8544, 9024, 14845 ] } ], "ref_ids": [ 9152, 9025 ] }, { "id": 9028, "type": "theorem", "label": "spaces-simplicial-lemma-morphism-simplicial-sites", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-morphism-simplicial-sites", "contents": [ "Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi$ and", "$\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi$ be as in", "Situation \\ref{situation-simplicial-site}.", "Let $h$ be a morphism between simplicial sites as in", "Remark \\ref{remark-morphism-simplicial-sites}.", "Then we obtain a morphism of topoi", "$$", "h_{total} : \\Sh(\\mathcal{C}_{total}) \\to \\Sh(\\mathcal{C}'_{total})", "$$", "and commutative diagrams", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C}_n) \\ar[d]_{g_n} \\ar[r]_{h_n} &", "\\Sh(\\mathcal{C}'_n) \\ar[d]^{g'_n} \\\\", "\\Sh(\\mathcal{C}_{total}) \\ar[r]^{h_{total}} &", "\\Sh(\\mathcal{C}'_{total})", "}", "$$", "Moreover, we have $(g'_n)^{-1} \\circ h_{total, *} = h_{n, *} \\circ g_n^{-1}$." ], "refs": [ "spaces-simplicial-remark-morphism-simplicial-sites" ], "proofs": [ { "contents": [ "Case A. Say $h_n$ corresponds to the continuous functor", "$v_n : \\mathcal{C}'_n \\to \\mathcal{C}_n$. Then we can define", "a functor $v_{total} : \\mathcal{C}'_{total} \\to \\mathcal{C}_{total}$", "by using $v_n$ in degree $n$. This is clearly a continuous functor", "(see definition of coverings in Lemma \\ref{lemma-simplicial-site-site}).", "Let", "$h_{total}^{-1} = v_{total, s} :", "\\Sh(\\mathcal{C}'_{total}) \\to \\Sh(\\mathcal{C}_{total})$ and", "$h_{total, *} = v_{total}^s = v_{total}^p :", "\\Sh(\\mathcal{C}_{total}) \\to \\Sh(\\mathcal{C}'_{total})$", "be the adjoint pair of functors constructed and studied in", "Sites, Sections \\ref{sites-section-continuous-functors} and", "\\ref{sites-section-morphism-sites}.", "To see that $h_{total}$ is a morphism of topoi", "we still have to verify that $h_{total}^{-1}$ is exact.", "We first observe that", "$(g'_n)^{-1} \\circ h_{total, *} = h_{n, *} \\circ g_n^{-1}$;", "this is immediate by computing sections over an object $U$", "of $\\mathcal{C}'_n$. Thus, if we think of a sheaf $\\mathcal{F}$", "on $\\mathcal{C}_{total}$ as a system $(\\mathcal{F}_n, \\mathcal{F}(\\varphi))$", "as in Lemma \\ref{lemma-describe-sheaves-simplicial-site-site}, then", "$h_{total, *}\\mathcal{F}$ corresponds to", "the system $(h_{n, *}\\mathcal{F}_n, h_{n, *}\\mathcal{F}(\\varphi))$.", "Clearly, the functor", "$(\\mathcal{F}'_n, \\mathcal{F}'(\\varphi)) \\to", "(h_n^{-1}\\mathcal{F}'_n, h_n^{-1}\\mathcal{F}'(\\varphi))$", "is its left adjoint. By uniqueness of adjoints, we conclude that", "$h_{total}^{-1}$ is given by this rule on systems. In particular,", "$h_{total}^{-1}$ is exact (by the description of sheaves on", "$\\mathcal{C}_{total}$ given in the lemma and the exactness of", "the functors $h_n^{-1}$) and we have our morphism of topoi.", "Finally, we obtain $g_n^{-1} \\circ h_{total}^{-1} =", "h_n^{-1} \\circ (g'_n)^{-1}$ as well, which proves that the", "displayed diagram of the lemma commutes.", "\\medskip\\noindent", "Case B. Here we have a functor", "$v_{total} : \\mathcal{C}_{total} \\to \\mathcal{C}'_{total}$", "by using $v_n$ in degree $n$. This is clearly a cocontinuous functor", "(see definition of coverings in Lemma \\ref{lemma-simplicial-cocontinuous-site}).", "Let $h_{total}$ be the morphism of topoi associated to $v_{total}$.", "The commutativity of the displayed diagram of the lemma follows", "immediately from Sites, Lemma \\ref{sites-lemma-composition-cocontinuous}.", "Taking left adjoints the final equality of the lemma becomes", "$$", "h_{total}^{-1} \\circ (g'_n)^{Sh}_! = g^{Sh}_{n!} \\circ h_n^{-1}", "$$", "This follows immediately from the explicit description of the functors", "$(g'_n)^{Sh}_!$ and $g^{Sh}_{n!}$ in", "Lemma \\ref{lemma-restriction-to-components-site},", "the fact that $h_n^{-1} \\circ (f'_\\varphi)^{-1} =", "f_\\varphi^{-1} \\circ h_m^{-1}$ for $\\varphi : [m] \\to [n]$, and", "the fact that we already know $h_{total}^{-1}$ commutes", "with restrictions to the degree $n$ parts of the simplicial sites." ], "refs": [ "spaces-simplicial-lemma-simplicial-site-site", "spaces-simplicial-lemma-describe-sheaves-simplicial-site-site", "spaces-simplicial-lemma-simplicial-cocontinuous-site", "sites-lemma-composition-cocontinuous", "spaces-simplicial-lemma-restriction-to-components-site" ], "ref_ids": [ 9022, 9024, 9023, 8544, 9025 ] } ], "ref_ids": [ 9153 ] }, { "id": 9029, "type": "theorem", "label": "spaces-simplicial-lemma-direct-image-morphism-simplicial-sites", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-direct-image-morphism-simplicial-sites", "contents": [ "With notation and hypotheses as in Lemma \\ref{lemma-morphism-simplicial-sites}.", "For $K \\in D(\\mathcal{C}_{total})$ we have", "$(g'_n)^{-1}Rh_{total, *}K = Rh_{n, *}g_n^{-1}K$." ], "refs": [ "spaces-simplicial-lemma-morphism-simplicial-sites" ], "proofs": [ { "contents": [ "Let $\\mathcal{I}^\\bullet$ be a K-injective complex on $\\mathcal{C}_{total}$", "representing $K$. Then $g_n^{-1}K$ is represented by", "$g_n^{-1}\\mathcal{I}^\\bullet = \\mathcal{I}_n^\\bullet$", "which is K-injective by", "Lemma \\ref{lemma-restriction-injective-to-component-site}.", "We have $(g'_n)^{-1}h_{total, *}\\mathcal{I}^\\bullet =", "h_{n, *}g_n^{-1}\\mathcal{I}_n^\\bullet$ by", "Lemma \\ref{lemma-morphism-simplicial-sites}", "which gives the desired equality." ], "refs": [ "spaces-simplicial-lemma-restriction-injective-to-component-site", "spaces-simplicial-lemma-morphism-simplicial-sites" ], "ref_ids": [ 9026, 9028 ] } ], "ref_ids": [ 9028 ] }, { "id": 9030, "type": "theorem", "label": "spaces-simplicial-lemma-morphism-augmentation-simplicial-sites", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-morphism-augmentation-simplicial-sites", "contents": [ "Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi, \\mathcal{D}, a_0$,", "$\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi, \\mathcal{D}', a'_0$, and", "$h_n$, $n \\geq -1$ be as in", "Remark \\ref{remark-morphism-augmentation-simplicial-sites}.", "Then we obtain a commutative diagram", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C}_{total}) \\ar[d]_a \\ar[r]_{h_{total}} &", "\\Sh(\\mathcal{C}'_{total}) \\ar[d]^{a'} \\\\", "\\Sh(\\mathcal{D}) \\ar[r]^{h_{-1}} &", "\\Sh(\\mathcal{D}')", "}", "$$" ], "refs": [ "spaces-simplicial-remark-morphism-augmentation-simplicial-sites" ], "proofs": [ { "contents": [ "The morphism $h$ is defined in Lemma \\ref{lemma-morphism-simplicial-sites}.", "The morphisms $a$ and $a'$ are defined in Lemma \\ref{lemma-augmentation-site}.", "Thus the only thing is to prove the commutativity of the diagram.", "To do this, we prove that", "$a^{-1} \\circ h_{-1}^{-1} = h_{total}^{-1} \\circ (a')^{-1}$.", "By the commutative diagrams of", "Lemma \\ref{lemma-morphism-simplicial-sites}", "and the description of $\\Sh(\\mathcal{C}_{total})$", "and $\\Sh(\\mathcal{C}'_{total})$ in terms of components", "in Lemma \\ref{lemma-describe-sheaves-simplicial-site-site},", "it suffices to show that", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C}_n) \\ar[d]_{a_n} \\ar[r]_{h_n} &", "\\Sh(\\mathcal{C}'_n) \\ar[d]^{a'_n} \\\\", "\\Sh(\\mathcal{D}) \\ar[r]^{h_{-1}} &", "\\Sh(\\mathcal{D}')", "}", "$$", "commutes for all $n$. This follows from the case for $n = 0$", "(which is an assumption in", "Remark \\ref{remark-morphism-augmentation-simplicial-sites})", "and for $n > 0$ we pick $\\varphi : [0] \\to [n]$", "and then the required commutativity follows from the case $n = 0$", "and the relations $a_n = a_0 \\circ f_\\varphi$", "and $a'_n = a'_0 \\circ f'_\\varphi$", "as well as the commutation relations", "$f'_\\varphi \\circ h_n = h_0 \\circ f_\\varphi$." ], "refs": [ "spaces-simplicial-lemma-morphism-simplicial-sites", "spaces-simplicial-lemma-augmentation-site", "spaces-simplicial-lemma-morphism-simplicial-sites", "spaces-simplicial-lemma-describe-sheaves-simplicial-site-site", "spaces-simplicial-remark-morphism-augmentation-simplicial-sites" ], "ref_ids": [ 9028, 9027, 9028, 9024, 9154 ] } ], "ref_ids": [ 9154 ] }, { "id": 9031, "type": "theorem", "label": "spaces-simplicial-lemma-restriction-module-to-components-site", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-restriction-module-to-components-site", "contents": [ "In Situation \\ref{situation-simplicial-site}. Let $\\mathcal{O}$", "be a sheaf of rings on $\\mathcal{C}_{total}$.", "There is a canonical morphism of ringed topoi", "$g_n : (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n) \\to", "(\\Sh(\\mathcal{C}_{total}), \\mathcal{O})$", "agreeing with the morphism $g_n$ of", "Lemma \\ref{lemma-restriction-to-components-site} on underlying topoi.", "The functor", "$g_n^* : \\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_n)$", "has a left adjoint $g_{n!}$.", "For $\\mathcal{G}$ in $\\textit{Mod}(\\mathcal{O}_n)$-modules the", "restriction of $g_{n!}\\mathcal{G}$ to $\\mathcal{C}_m$ is", "$$", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^*\\mathcal{G}", "$$", "where $f_\\varphi : (\\Sh(\\mathcal{C}_m), \\mathcal{O}_m) \\to", "(\\Sh(\\mathcal{C}_n), \\mathcal{O}_n)$ is the morphism of ringed topoi", "agreeing with the previously defined $f_\\varphi$ on topoi and", "using the map", "$\\mathcal{O}(\\varphi) : f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$", "on sheaves of rings." ], "refs": [ "spaces-simplicial-lemma-restriction-to-components-site" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-restriction-to-components-site} we have", "$g_n^{-1}\\mathcal{O} = \\mathcal{O}_n$ and hence we obtain our", "morphism of ringed topoi. By Modules on Sites, Lemma", "\\ref{sites-modules-lemma-lower-shriek-modules}", "we obtain the adjoint $g_{n!}$. To prove the formula for $g_{n!}$", "we first define a sheaf of $\\mathcal{O}$-modules $\\mathcal{H}$", "on $\\mathcal{C}_{total}$ with degree $m$ component", "the $\\mathcal{O}_m$-module", "$$", "\\mathcal{H}_m =", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^*\\mathcal{G}", "$$", "Given a map $\\psi : [m] \\to [m']$ the map", "$\\mathcal{H}(\\psi) : f_\\psi^{-1}\\mathcal{H}_m \\to \\mathcal{H}_{m'}$", "is given on components by", "$$", "f_\\psi^{-1} f_\\varphi^*\\mathcal{G} \\to", "f_\\psi^* f_\\varphi^*\\mathcal{G} \\to", "f_{\\psi \\circ \\varphi}^*\\mathcal{G}", "$$", "Since this map $f_\\psi^{-1}\\mathcal{H}_m \\to \\mathcal{H}_{m'}$ is", "$\\mathcal{O}(\\psi) : f_\\psi^{-1}\\mathcal{O}_m \\to \\mathcal{O}_{m'}$-semi-linear,", "this indeed does define an $\\mathcal{O}$-module", "(use Lemma \\ref{lemma-describe-sheaves-simplicial-site-site}).", "Then one proves directly that", "$$", "\\Mor_{\\mathcal{O}_n}(\\mathcal{G}, \\mathcal{F}_n) =", "\\Mor_{\\mathcal{O}}(\\mathcal{H}, \\mathcal{F})", "$$", "proceeding as in the proof of Lemma \\ref{lemma-restriction-to-components-site}.", "Thus $\\mathcal{H} = g_{n!}\\mathcal{G}$ as desired." ], "refs": [ "spaces-simplicial-lemma-restriction-to-components-site", "sites-modules-lemma-lower-shriek-modules", "spaces-simplicial-lemma-describe-sheaves-simplicial-site-site", "spaces-simplicial-lemma-restriction-to-components-site" ], "ref_ids": [ 9025, 14262, 9024, 9025 ] } ], "ref_ids": [ 9025 ] }, { "id": 9032, "type": "theorem", "label": "spaces-simplicial-lemma-restriction-injective-to-component-limp", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-restriction-injective-to-component-limp", "contents": [ "In Situation \\ref{situation-simplicial-site}.", "Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$.", "If $\\mathcal{I}$ is injective in $\\textit{Mod}(\\mathcal{O})$, then", "$\\mathcal{I}_n$ is a totally acyclic sheaf on $\\mathcal{C}_n$." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-pullback-injective-limp}", "applied to the inclusion functor $\\mathcal{C}_n \\to \\mathcal{C}_{total}$", "and its properties proven in Lemma \\ref{lemma-restriction-to-components-site}." ], "refs": [ "sites-cohomology-lemma-pullback-injective-limp", "spaces-simplicial-lemma-restriction-to-components-site" ], "ref_ids": [ 4338, 9025 ] } ], "ref_ids": [] }, { "id": 9033, "type": "theorem", "label": "spaces-simplicial-lemma-exactness-g-shriek-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-exactness-g-shriek-modules", "contents": [ "With assumptions as in", "Lemma \\ref{lemma-restriction-module-to-components-site} the functor", "$g_{n!} : \\textit{Mod}(\\mathcal{O}_n) \\to \\textit{Mod}(\\mathcal{O})$", "is exact if the maps $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$", "are flat for all $\\varphi : [n] \\to [m]$." ], "refs": [ "spaces-simplicial-lemma-restriction-module-to-components-site" ], "proofs": [ { "contents": [ "Recall that $g_{n!}\\mathcal{G}$ is the $\\mathcal{O}$-module", "whose degree $m$ part is the $\\mathcal{O}_m$-module", "$$", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^*\\mathcal{G}", "$$", "Here the morphism of ringed topoi", "$f_\\varphi : (\\Sh(\\mathcal{C}_m), \\mathcal{O}_m) \\to", "(\\Sh(\\mathcal{C}_n), \\mathcal{O}_n)$ uses the map", "$f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$ of the", "statement of the lemma. If these maps are flat, then", "$f_\\varphi^*$ is exact", "(Modules on Sites, Lemma \\ref{sites-modules-lemma-flat-pullback-exact}).", "By definition of the site $\\mathcal{C}_{total}$ we see that these", "functors have the desired exactness properties and we conclude." ], "refs": [ "sites-modules-lemma-flat-pullback-exact" ], "ref_ids": [ 14223 ] } ], "ref_ids": [ 9031 ] }, { "id": 9034, "type": "theorem", "label": "spaces-simplicial-lemma-restriction-injective-to-component-site-module", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-restriction-injective-to-component-site-module", "contents": [ "In Situation \\ref{situation-simplicial-site}.", "Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$", "such that $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$", "is flat for all $\\varphi : [n] \\to [m]$.", "If $\\mathcal{I}$ is injective in $\\textit{Mod}(\\mathcal{O})$, then", "$\\mathcal{I}_n$ is injective in $\\textit{Mod}(\\mathcal{O}_n)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}", "and", "Lemma \\ref{lemma-exactness-g-shriek-modules}." ], "refs": [ "homology-lemma-adjoint-preserve-injectives", "spaces-simplicial-lemma-exactness-g-shriek-modules" ], "ref_ids": [ 12116, 9033 ] } ], "ref_ids": [] }, { "id": 9035, "type": "theorem", "label": "spaces-simplicial-lemma-morphism-simplicial-sites-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-morphism-simplicial-sites-modules", "contents": [ "Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi$ and", "$\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi$ be as in", "Situation \\ref{situation-simplicial-site}.", "Let $\\mathcal{O}$ and $\\mathcal{O}'$", "be a sheaf of rings on $\\mathcal{C}_{total}$ and $\\mathcal{C}'_{total}$.", "Let $(h, h^\\sharp)$ be a morphism between simplicial sites as in", "Remark \\ref{remark-morphism-simplicial-sites-modules}.", "Then we obtain a morphism of ringed topoi", "$$", "h_{total} :", "(\\Sh(\\mathcal{C}_{total}, \\mathcal{O})", "\\to", "(\\Sh(\\mathcal{C}'_{total}), \\mathcal{O}')", "$$", "and commutative diagrams", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}_n), \\mathcal{O}_n) \\ar[d]_{g_n} \\ar[r]_{h_n} &", "(\\Sh(\\mathcal{C}'_n), \\mathcal{O}'_n) \\ar[d]^{g'_n} \\\\", "(\\Sh(\\mathcal{C}_{total}), \\mathcal{O}) \\ar[r]^{h_{total}} &", "(\\Sh(\\mathcal{C}'_{total}), \\mathcal{O}')", "}", "$$", "of ringed topoi where $g_n$ and $g'_n$ are as in", "Lemma \\ref{lemma-restriction-module-to-components-site}.", "Moreover, we have", "$(g'_n)^* \\circ h_{total, *} = h_{n, *} \\circ g_n^*$", "as functor $\\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}'_n)$." ], "refs": [ "spaces-simplicial-remark-morphism-simplicial-sites-modules", "spaces-simplicial-lemma-restriction-module-to-components-site" ], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-morphism-simplicial-sites} and", "\\ref{lemma-restriction-module-to-components-site}", "by keeping track of the sheaves of rings.", "A small point is that in order to define $h_n$ as a morphism", "of ringed topoi we set", "$h_n^\\sharp = g_n^{-1}h^\\sharp :", "g_n^{-1}h_{total}^{-1}\\mathcal{O}' \\to g_n^{-1}\\mathcal{O}$", "which makes sense because", "$g_n^{-1}h_{total}^{-1}\\mathcal{O}' = h_n^{-1}(g'_n)^{-1}\\mathcal{O}' =", "h_n^{-1}\\mathcal{O}'_n$ and $g_n^{-1}\\mathcal{O} = \\mathcal{O}_n$.", "Note that $g_n^*\\mathcal{F} = g_n^{-1}\\mathcal{F}$", "for a sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$", "and similarly for $g'_n$ and this helps explain why", "$(g'_n)^* \\circ h_{total, *} = h_{n, *} \\circ g_n^*$", "follows from the corresponding statement of", "Lemma \\ref{lemma-morphism-simplicial-sites}." ], "refs": [ "spaces-simplicial-lemma-morphism-simplicial-sites", "spaces-simplicial-lemma-restriction-module-to-components-site", "spaces-simplicial-lemma-morphism-simplicial-sites" ], "ref_ids": [ 9028, 9031, 9028 ] } ], "ref_ids": [ 9155, 9031 ] }, { "id": 9036, "type": "theorem", "label": "spaces-simplicial-lemma-direct-image-morphism-simplicial-sites-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-direct-image-morphism-simplicial-sites-modules", "contents": [ "With notation and hypotheses as in", "Lemma \\ref{lemma-morphism-simplicial-sites-modules}.", "For $K \\in D(\\mathcal{O})$ we have", "$(g'_n)^*Rh_{total, *}K = Rh_{n, *}g_n^*K$." ], "refs": [ "spaces-simplicial-lemma-morphism-simplicial-sites-modules" ], "proofs": [ { "contents": [ "Recall that $g_n^* = g_n^{-1}$ because $g_n^{-1}\\mathcal{O} = \\mathcal{O}_n$", "by the construction in Lemma \\ref{lemma-restriction-module-to-components-site}.", "In particular $g_n^*$ is exact and $Lg_n^*$ is given by applying $g_n^*$", "to any representative complex of modules. Similarly for $g'_n$.", "There is a canonical base change map", "$(g'_n)^*Rh_{total, *}K \\to Rh_{n, *}g_n^*K$, see", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change}.", "By Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-modules-abelian-unbounded}", "the image of this in $D(\\mathcal{C}'_n)$ is the map", "$(g'_n)^{-1}Rh_{total, *}K_{ab} \\to Rh_{n, *}g_n^{-1}K_{ab}$", "where $K_{ab}$ is the image of $K$ in $D(\\mathcal{C}_{total})$.", "This we proved to be an isomorphism in", "Lemma \\ref{lemma-direct-image-morphism-simplicial-sites}", "and the result follows." ], "refs": [ "spaces-simplicial-lemma-restriction-module-to-components-site", "sites-cohomology-remark-base-change", "sites-cohomology-lemma-modules-abelian-unbounded", "spaces-simplicial-lemma-direct-image-morphism-simplicial-sites" ], "ref_ids": [ 9031, 4424, 4259, 9029 ] } ], "ref_ids": [ 9035 ] }, { "id": 9037, "type": "theorem", "label": "spaces-simplicial-lemma-simplicial-resolution-Z-site", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-simplicial-resolution-Z-site", "contents": [ "In Situation \\ref{situation-simplicial-site} and with notation as above", "there is a complex", "$$", "\\ldots \\to g_{2!}\\mathbf{Z} \\to g_{1!}\\mathbf{Z} \\to g_{0!}\\mathbf{Z}", "$$", "of abelian sheaves on $\\mathcal{C}_{total}$ which forms a resolution of", "the constant sheaf with value $\\mathbf{Z}$ on $\\mathcal{C}_{total}$." ], "refs": [], "proofs": [ { "contents": [ "We will use the description of the functors $g_{n!}$ in", "Lemma \\ref{lemma-restriction-to-components-site} without further mention.", "As maps of the complex we take $\\sum (-1)^i d^n_i$ where", "$d^n_i : g_{n!}\\mathbf{Z} \\to g_{n - 1!}\\mathbf{Z}$ is the", "adjoint to the map $\\mathbf{Z} \\to", "\\bigoplus_{[n - 1] \\to [n]} \\mathbf{Z} = g_n^{-1}g_{n - 1!}\\mathbf{Z}$", "corresponding to the factor labeled with $\\delta^n_i : [n - 1] \\to [n]$.", "Then $g_m^{-1}$ applied to the complex gives the complex", "$$", "\\ldots \\to", "\\bigoplus\\nolimits_{\\alpha \\in \\Mor_\\Delta([2], [m])]} \\mathbf{Z} \\to", "\\bigoplus\\nolimits_{\\alpha \\in \\Mor_\\Delta([1], [m])]} \\mathbf{Z} \\to", "\\bigoplus\\nolimits_{\\alpha \\in \\Mor_\\Delta([0], [m])]} \\mathbf{Z}", "$$", "on $\\mathcal{C}_m$.", "In other words, this is the complex associated to the", "free abelian sheaf on the simplicial set $\\Delta[m]$, see", "Simplicial, Example \\ref{simplicial-example-simplex-simplicial-set}.", "Since $\\Delta[m]$ is homotopy equivalent to $\\Delta[0]$, see", "Simplicial, Example \\ref{simplicial-example-simplex-contractible},", "and since ``taking free abelian sheaf on'' is a functor,", "we see that the complex above is homotopy equivalent to", "the free abelian sheaf on $\\Delta[0]$", "(Simplicial, Remark \\ref{simplicial-remark-homotopy-better} and", "Lemma \\ref{simplicial-lemma-homotopy-equivalence-s-N}).", "This complex is acyclic in positive degrees", "and equal to $\\mathbf{Z}$ in degree $0$." ], "refs": [ "spaces-simplicial-lemma-restriction-to-components-site", "simplicial-remark-homotopy-better", "simplicial-lemma-homotopy-equivalence-s-N" ], "ref_ids": [ 9025, 14941, 14877 ] } ], "ref_ids": [] }, { "id": 9038, "type": "theorem", "label": "spaces-simplicial-lemma-cech-complex", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-cech-complex", "contents": [ "In Situation \\ref{situation-simplicial-site}. Let $\\mathcal{F}$ be an abelian", "sheaf on $\\mathcal{C}_{total}$ there is a canonical complex", "$$", "0 \\to \\Gamma(\\mathcal{C}_{total}, \\mathcal{F}) \\to", "\\Gamma(\\mathcal{C}_0, \\mathcal{F}_0) \\to", "\\Gamma(\\mathcal{C}_1, \\mathcal{F}_1) \\to", "\\Gamma(\\mathcal{C}_2, \\mathcal{F}_2) \\to \\ldots", "$$", "which is exact in degrees $-1, 0$ and exact everywhere", "if $\\mathcal{F}$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Observe that", "$\\Hom(\\mathbf{Z}, \\mathcal{F}) = \\Gamma(\\mathcal{C}_{total}, \\mathcal{F})$", "and", "$\\Hom(g_{n!}\\mathbf{Z}, \\mathcal{F}) = \\Gamma(\\mathcal{C}_n, \\mathcal{F}_n)$.", "Hence this lemma is an immediate consequence of", "Lemma \\ref{lemma-simplicial-resolution-Z-site}", "and the fact that $\\Hom(-, \\mathcal{F})$ is exact if", "$\\mathcal{F}$ is injective." ], "refs": [ "spaces-simplicial-lemma-simplicial-resolution-Z-site" ], "ref_ids": [ 9037 ] } ], "ref_ids": [] }, { "id": 9039, "type": "theorem", "label": "spaces-simplicial-lemma-simplicial-sheaf-cohomology-site", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-simplicial-sheaf-cohomology-site", "contents": [ "In Situation \\ref{situation-simplicial-site}. For $K$ in", "$D^+(\\mathcal{C}_{total})$ there is a spectral sequence", "$(E_r, d_r)_{r \\geq 0}$ with", "$$", "E_1^{p, q} = H^q(\\mathcal{C}_p, K_p),\\quad", "d_1^{p, q} : E_1^{p, q} \\to E_1^{p + 1, q}", "$$", "converging to $H^{p + q}(\\mathcal{C}_{total}, K)$.", "This spectral sequence is functorial in $K$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I}^\\bullet$ be a bounded below complex of injectives", "representing $K$. Consider the double complex with terms", "$$", "A^{p, q} = \\Gamma(\\mathcal{C}_p, \\mathcal{I}^q_p)", "$$", "where the horizontal arrows come from Lemma \\ref{lemma-cech-complex}", "and the vertical arrows from the differentials of the", "complex $\\mathcal{I}^\\bullet$. The rows of the double complex are exact", "in positive degrees and evaluate to", "$\\Gamma(\\mathcal{C}_{total}, \\mathcal{I}^q)$ in degree $0$.", "On the other hand, since restriction to $\\mathcal{C}_p$ is exact", "(Lemma \\ref{lemma-restriction-to-components-site})", "the complex $\\mathcal{I}_p^\\bullet$ represents $K_p$ in", "$D(\\mathcal{C}_p)$. The sheaves $\\mathcal{I}_p^q$ are injective", "abelian sheaves on $\\mathcal{C}_p$", "(Lemma \\ref{lemma-restriction-injective-to-component-site}).", "Hence the cohomology of the columns computes the groups", "$H^q(\\mathcal{C}_p, K_p)$. We conclude by applying", "Homology, Lemmas \\ref{homology-lemma-first-quadrant-ss} and", "\\ref{homology-lemma-double-complex-gives-resolution}." ], "refs": [ "spaces-simplicial-lemma-cech-complex", "spaces-simplicial-lemma-restriction-to-components-site", "spaces-simplicial-lemma-restriction-injective-to-component-site", "homology-lemma-first-quadrant-ss", "homology-lemma-double-complex-gives-resolution" ], "ref_ids": [ 9038, 9025, 9026, 12105, 12106 ] } ], "ref_ids": [] }, { "id": 9040, "type": "theorem", "label": "spaces-simplicial-lemma-sanity-check", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-sanity-check", "contents": [ "Let $\\mathcal{C}$ be as in Situation \\ref{situation-simplicial-site}.", "Let $U \\in \\Ob(\\mathcal{C}_n)$. Let", "$\\mathcal{F} \\in \\textit{Ab}(\\mathcal{C}_{total})$.", "Then $H^p(U, \\mathcal{F}) = H^p(U, g_n^{-1}\\mathcal{F})$", "where on the left hand side $U$ is viewed as an object of $\\mathcal{C}_{total}$." ], "refs": [], "proofs": [ { "contents": [ "Observe that ``$U$ viewed as object of $\\mathcal{C}_{total}$''", "is explained by the construction of $\\mathcal{C}_{total}$ in", "Lemma \\ref{lemma-simplicial-site-site} in case (A) and", "Lemma \\ref{lemma-simplicial-cocontinuous-site} in case (B).", "The equality then follows from", "Lemma \\ref{lemma-restriction-injective-to-component-site}", "and the definition of cohomology." ], "refs": [ "spaces-simplicial-lemma-simplicial-site-site", "spaces-simplicial-lemma-simplicial-cocontinuous-site", "spaces-simplicial-lemma-restriction-injective-to-component-site" ], "ref_ids": [ 9022, 9023, 9026 ] } ], "ref_ids": [] }, { "id": 9041, "type": "theorem", "label": "spaces-simplicial-lemma-simplicial-resolution-augmentation", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-simplicial-resolution-augmentation", "contents": [ "In Situation \\ref{situation-simplicial-site} let", "$a_0$ be an augmentation towards a site $\\mathcal{D}$", "as in Remark \\ref{remark-augmentation-site}.", "For any abelian sheaf $\\mathcal{G}$ on $\\mathcal{D}$ ", "there is an exact complex", "$$", "\\ldots \\to", "g_{2!}(a_2^{-1}\\mathcal{G}) \\to", "g_{1!}(a_1^{-1}\\mathcal{G}) \\to", "g_{0!}(a_0^{-1}\\mathcal{G}) \\to", "a^{-1}\\mathcal{G} \\to 0", "$$", "of abelian sheaves on $\\mathcal{C}_{total}$." ], "refs": [ "spaces-simplicial-remark-augmentation-site" ], "proofs": [ { "contents": [ "We encourage the reader to read the proof of", "Lemma \\ref{lemma-simplicial-resolution-Z-site} first.", "We will use Lemma \\ref{lemma-augmentation-site} and", "the description of the functors $g_{n!}$ in", "Lemma \\ref{lemma-restriction-to-components-site} without further mention.", "In particular $g_{n!}(a_n^{-1}\\mathcal{G})$ is the", "sheaf on $\\mathcal{C}_{total}$ whose restriction to $\\mathcal{C}_m$", "is the sheaf", "$$", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^{-1}a_n^{-1}\\mathcal{G} =", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} a_m^{-1}\\mathcal{G}", "$$", "As maps of the complex we take $\\sum (-1)^i d^n_i$ where", "$d^n_i : g_{n!}(a_n^{-1}\\mathcal{G}) \\to g_{n - 1!}(a_{n - 1}^{-1}\\mathcal{G})$", "is the adjoint to the map", "$a_n^{-1}\\mathcal{G} \\to \\bigoplus_{[n - 1] \\to [n]} a_n^{-1}\\mathcal{G} =", "g_n^{-1}g_{n - 1!}(a_{n - 1}^{-1}\\mathcal{G})$", "corresponding to the factor labeled with $\\delta^n_i : [n - 1] \\to [n]$.", "The map $g_{0!}(a_0^{-1}\\mathcal{G}) \\to a^{-1}\\mathcal{G}$ is adjoint", "to the identity map of $a_0^{-1}\\mathcal{G}$.", "Then $g_m^{-1}$ applied to the chain complex in degrees", "$\\ldots, 2, 1, 0$ gives the complex", "$$", "\\ldots \\to", "\\bigoplus\\nolimits_{\\alpha \\in \\Mor_\\Delta([2], [m])]} a_m^{-1}\\mathcal{G} \\to", "\\bigoplus\\nolimits_{\\alpha \\in \\Mor_\\Delta([1], [m])]} a_m^{-1}\\mathcal{G} \\to", "\\bigoplus\\nolimits_{\\alpha \\in \\Mor_\\Delta([0], [m])]} a_m^{-1}\\mathcal{G}", "$$", "on $\\mathcal{C}_m$. This is equal to $a_m^{-1}\\mathcal{G}$", "tensored over the constant sheaf $\\mathbf{Z}$ with the complex", "$$", "\\ldots \\to", "\\bigoplus\\nolimits_{\\alpha \\in \\Mor_\\Delta([2], [m])]} \\mathbf{Z} \\to", "\\bigoplus\\nolimits_{\\alpha \\in \\Mor_\\Delta([1], [m])]} \\mathbf{Z} \\to", "\\bigoplus\\nolimits_{\\alpha \\in \\Mor_\\Delta([0], [m])]} \\mathbf{Z}", "$$", "discussed in the proof of Lemma \\ref{lemma-simplicial-resolution-Z-site}.", "There we have seen that this complex is homotopy equivalent to", "$\\mathbf{Z}$ placed in degree $0$ which finishes the proof." ], "refs": [ "spaces-simplicial-lemma-simplicial-resolution-Z-site", "spaces-simplicial-lemma-augmentation-site", "spaces-simplicial-lemma-restriction-to-components-site", "spaces-simplicial-lemma-simplicial-resolution-Z-site" ], "ref_ids": [ 9037, 9027, 9025, 9037 ] } ], "ref_ids": [ 9152 ] }, { "id": 9042, "type": "theorem", "label": "spaces-simplicial-lemma-augmentation-cech-complex", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-augmentation-cech-complex", "contents": [ "In Situation \\ref{situation-simplicial-site} let", "$a_0$ be an augmentation towards a site $\\mathcal{D}$", "as in Remark \\ref{remark-augmentation-site}.", "For an abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}_{total}$", "there is a canonical complex", "$$", "0 \\to a_*\\mathcal{F} \\to a_{0, *}\\mathcal{F}_0 \\to a_{1, *}\\mathcal{F}_1 \\to", "a_{2, *}\\mathcal{F}_2 \\to \\ldots", "$$", "on $\\mathcal{D}$ which is exact in degrees $-1, 0$ and", "exact everywhere if $\\mathcal{F}$ is injective." ], "refs": [ "spaces-simplicial-remark-augmentation-site" ], "proofs": [ { "contents": [ "To construct the complex, by the Yoneda lemma, it suffices for any", "abelian sheaf $\\mathcal{G}$ on $\\mathcal{D}$ to construct a complex", "$$", "0 \\to \\Hom(\\mathcal{G}, a_*\\mathcal{F}) \\to", "\\Hom(\\mathcal{G}, a_{0, *}\\mathcal{F}_0) \\to", "\\Hom(\\mathcal{G}, a_{1, *}\\mathcal{F}_1) \\to \\ldots", "$$", "functorially in $\\mathcal{G}$. To do this apply $\\Hom(-, \\mathcal{F})$", "to the exact complex of Lemma \\ref{lemma-simplicial-resolution-augmentation}", "and use adjointness of pullback and pushforward.", "The exactness properties in degrees $-1, 0$ follow from", "the construction as $\\Hom(-, \\mathcal{F})$ is left exact.", "If $\\mathcal{F}$ is an injective abelian sheaf, then the", "complex is exact because $\\Hom(-, \\mathcal{F})$ is exact." ], "refs": [ "spaces-simplicial-lemma-simplicial-resolution-augmentation" ], "ref_ids": [ 9041 ] } ], "ref_ids": [ 9152 ] }, { "id": 9043, "type": "theorem", "label": "spaces-simplicial-lemma-augmentation-spectral-sequence", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-augmentation-spectral-sequence", "contents": [ "In Situation \\ref{situation-simplicial-site} let", "$a_0$ be an augmentation towards a site $\\mathcal{D}$", "as in Remark \\ref{remark-augmentation-site}.", "For any $K$ in $D^+(\\mathcal{C}_{total})$ there is a spectral", "sequence ", "$(E_r, d_r)_{r \\geq 0}$ with", "$$", "E_1^{p, q} = R^qa_{p, *} K_p,\\quad", "d_1^{p, q} : E_1^{p, q} \\to E_1^{p + 1, q}", "$$", "converging to $R^{p + q}a_*K$. This spectral sequence is functorial in $K$." ], "refs": [ "spaces-simplicial-remark-augmentation-site" ], "proofs": [ { "contents": [ "Let $\\mathcal{I}^\\bullet$ be a bounded below complex of injectives", "representing $K$. Consider the double complex with terms", "$$", "A^{p, q} = a_{p, *}\\mathcal{I}^q_p", "$$", "where the horizontal arrows come from", "Lemma \\ref{lemma-augmentation-cech-complex}", "and the vertical arrows from the differentials of the", "complex $\\mathcal{I}^\\bullet$. The rows of the double complex are exact", "in positive degrees and evaluate to $a_*\\mathcal{I}^q$ in degree $0$.", "On the other hand, since restriction to $\\mathcal{C}_p$ is exact", "(Lemma \\ref{lemma-restriction-to-components-site})", "the complex $\\mathcal{I}_p^\\bullet$ represents $K_p$ in", "$D(\\mathcal{C}_p)$. The sheaves $\\mathcal{I}_p^q$ are injective", "abelian sheaves on $\\mathcal{C}_p$", "(Lemma \\ref{lemma-restriction-injective-to-component-site}).", "Hence the cohomology of the columns computes $R^qa_{p, *}K_p$.", "We conclude by applying", "Homology, Lemmas \\ref{homology-lemma-first-quadrant-ss} and", "\\ref{homology-lemma-double-complex-gives-resolution}." ], "refs": [ "spaces-simplicial-lemma-augmentation-cech-complex", "spaces-simplicial-lemma-restriction-to-components-site", "spaces-simplicial-lemma-restriction-injective-to-component-site", "homology-lemma-first-quadrant-ss", "homology-lemma-double-complex-gives-resolution" ], "ref_ids": [ 9042, 9025, 9026, 12105, 12106 ] } ], "ref_ids": [ 9152 ] }, { "id": 9044, "type": "theorem", "label": "spaces-simplicial-lemma-simplicial-resolution-ringed", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-simplicial-resolution-ringed", "contents": [ "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$", "be a sheaf of rings on $\\mathcal{C}_{total}$. There is a complex", "$$", "\\ldots \\to g_{2!}\\mathcal{O}_2 \\to g_{1!}\\mathcal{O}_1 \\to g_{0!}\\mathcal{O}_0", "$$", "of $\\mathcal{O}$-modules which forms a resolution of", "$\\mathcal{O}$.", "Here $g_{n!}$ is as in Lemma \\ref{lemma-restriction-module-to-components-site}." ], "refs": [ "spaces-simplicial-lemma-restriction-module-to-components-site" ], "proofs": [ { "contents": [ "We will use the description of $g_{n!}$ given in", "Lemma \\ref{lemma-restriction-to-components-site}.", "As maps of the complex we take $\\sum (-1)^i d^n_i$ where", "$d^n_i : g_{n!}\\mathcal{O}_n \\to g_{n - 1!}\\mathcal{O}_{n - 1}$", "is the adjoint to the map", "$\\mathcal{O}_n \\to \\bigoplus_{[n - 1] \\to [n]} \\mathcal{O}_n =", "g_n^*g_{n - 1!}\\mathcal{O}_{n - 1}$", "corresponding to the factor labeled with $\\delta^n_i : [n - 1] \\to [n]$.", "Then $g_m^{-1}$ applied to the complex gives the complex", "$$", "\\ldots \\to", "\\bigoplus\\nolimits_{\\alpha \\in \\Mor_\\Delta([2], [m])]} \\mathcal{O}_m \\to", "\\bigoplus\\nolimits_{\\alpha \\in \\Mor_\\Delta([1], [m])]} \\mathcal{O}_m \\to", "\\bigoplus\\nolimits_{\\alpha \\in \\Mor_\\Delta([0], [m])]} \\mathcal{O}_m", "$$", "on $\\mathcal{C}_m$.", "In other words, this is the complex associated to the", "free $\\mathcal{O}_m$-module on the simplicial set $\\Delta[m]$, see", "Simplicial, Example \\ref{simplicial-example-simplex-simplicial-set}.", "Since $\\Delta[m]$ is homotopy equivalent to $\\Delta[0]$, see", "Simplicial, Example \\ref{simplicial-example-simplex-contractible},", "and since ``taking free abelian sheaf on'' is a functor,", "we see that the complex above is homotopy equivalent to", "the free abelian sheaf on $\\Delta[0]$", "(Simplicial, Remark \\ref{simplicial-remark-homotopy-better} and", "Lemma \\ref{simplicial-lemma-homotopy-equivalence-s-N}).", "This complex is acyclic in positive degrees", "and equal to $\\mathcal{O}_m$ in degree $0$." ], "refs": [ "spaces-simplicial-lemma-restriction-to-components-site", "simplicial-remark-homotopy-better", "simplicial-lemma-homotopy-equivalence-s-N" ], "ref_ids": [ 9025, 14941, 14877 ] } ], "ref_ids": [ 9031 ] }, { "id": 9045, "type": "theorem", "label": "spaces-simplicial-lemma-cech-complex-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-cech-complex-modules", "contents": [ "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$", "be a sheaf of rings. Let $\\mathcal{F}$ be a", "sheaf of $\\mathcal{O}$-modules. There is a canonical complex", "$$", "0 \\to \\Gamma(\\mathcal{C}_{total}, \\mathcal{F}) \\to", "\\Gamma(\\mathcal{C}_0, \\mathcal{F}_0) \\to", "\\Gamma(\\mathcal{C}_1, \\mathcal{F}_1) \\to", "\\Gamma(\\mathcal{C}_2, \\mathcal{F}_2) \\to \\ldots", "$$", "which is exact in degrees $-1, 0$ and exact everywhere", "if $\\mathcal{F}$ is an injective $\\mathcal{O}$-module." ], "refs": [], "proofs": [ { "contents": [ "Observe that", "$\\Hom(\\mathcal{O}, \\mathcal{F}) = \\Gamma(\\mathcal{C}_{total}, \\mathcal{F})$", "and", "$\\Hom(g_{n!}\\mathcal{O}_n, \\mathcal{F}) = \\Gamma(\\mathcal{C}_n, \\mathcal{F}_n)$.", "Hence this lemma is an immediate consequence of", "Lemma \\ref{lemma-simplicial-resolution-ringed}", "and the fact that $\\Hom(-, \\mathcal{F})$ is exact if", "$\\mathcal{F}$ is injective." ], "refs": [ "spaces-simplicial-lemma-simplicial-resolution-ringed" ], "ref_ids": [ 9044 ] } ], "ref_ids": [] }, { "id": 9046, "type": "theorem", "label": "spaces-simplicial-lemma-simplicial-module-cohomology-site", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-simplicial-module-cohomology-site", "contents": [ "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$", "be a sheaf of rings. For $K$ in $D^+(\\mathcal{O})$", "there is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with", "$$", "E_1^{p, q} = H^q(\\mathcal{C}_p, K_p),\\quad", "d_1^{p, q} : E_1^{p, q} \\to E_1^{p + 1, q}", "$$", "converging to $H^{p + q}(\\mathcal{C}_{total}, K)$.", "This spectral sequence is functorial in $K$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I}^\\bullet$ be a bounded below complex of injective", "$\\mathcal{O}$-modules representing $K$. Consider the double complex with terms", "$$", "A^{p, q} = \\Gamma(\\mathcal{C}_p, \\mathcal{I}^q_p)", "$$", "where the horizontal arrows come from", "Lemma \\ref{lemma-cech-complex-modules}", "and the vertical arrows from the differentials of the", "complex $\\mathcal{I}^\\bullet$. Observe that", "$\\Gamma(\\mathcal{D}, -) =", "\\Hom_{\\mathcal{O}_\\mathcal{D}}(\\mathcal{O}_\\mathcal{D}, -)$", "on $\\textit{Mod}(\\mathcal{O}_\\mathcal{D})$. Hence the lemma", "says rows of the double complex are exact", "in positive degrees and evaluate to", "$\\Gamma(\\mathcal{C}_{total}, \\mathcal{I}^q)$ in degree $0$.", "Thus the total complex associated to the double complex", "computes $R\\Gamma(\\mathcal{C}_{total}, K)$ by", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}.", "On the other hand, since restriction to $\\mathcal{C}_p$ is exact", "(Lemma \\ref{lemma-restriction-to-components-site})", "the complex $\\mathcal{I}_p^\\bullet$ represents $K_p$ in", "$D(\\mathcal{C}_p)$. The sheaves $\\mathcal{I}_p^q$", "are totally acyclic on $\\mathcal{C}_p$", "(Lemma \\ref{lemma-restriction-injective-to-component-limp}).", "Hence the cohomology of the columns computes the groups", "$H^q(\\mathcal{C}_p, K_p)$ by Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "and", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-limp-acyclic}.", "We conclude by applying", "Homology, Lemma \\ref{homology-lemma-first-quadrant-ss}." ], "refs": [ "spaces-simplicial-lemma-cech-complex-modules", "homology-lemma-double-complex-gives-resolution", "spaces-simplicial-lemma-restriction-to-components-site", "spaces-simplicial-lemma-restriction-injective-to-component-limp", "derived-lemma-leray-acyclicity", "sites-cohomology-lemma-limp-acyclic", "homology-lemma-first-quadrant-ss" ], "ref_ids": [ 9045, 12106, 9025, 9032, 1844, 4219, 12105 ] } ], "ref_ids": [] }, { "id": 9047, "type": "theorem", "label": "spaces-simplicial-lemma-sanity-check-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-sanity-check-modules", "contents": [ "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$", "be a sheaf of rings. Let $U \\in \\Ob(\\mathcal{C}_n)$. Let", "$\\mathcal{F} \\in \\textit{Mod}(\\mathcal{O})$.", "Then $H^p(U, \\mathcal{F}) = H^p(U, g_n^*\\mathcal{F})$", "where on the left hand side $U$ is viewed as an object of", "$\\mathcal{C}_{total}$." ], "refs": [], "proofs": [ { "contents": [ "Observe that ``$U$ viewed as object of $\\mathcal{C}_{total}$''", "is explained by the construction of $\\mathcal{C}_{total}$ in", "Lemma \\ref{lemma-simplicial-site-site} in case (A) and", "Lemma \\ref{lemma-simplicial-cocontinuous-site} in case (B).", "In both cases the functor $\\mathcal{C}_n \\to \\mathcal{C}$", "is continuous and cocontinuous, see", "Lemma \\ref{lemma-restriction-to-components-site}, and", "$g_n^{-1}\\mathcal{O} = \\mathcal{O}_n$ by definition.", "Hence the result is a special case of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-pullback-same-cohomology}." ], "refs": [ "spaces-simplicial-lemma-simplicial-site-site", "spaces-simplicial-lemma-simplicial-cocontinuous-site", "spaces-simplicial-lemma-restriction-to-components-site", "sites-cohomology-lemma-pullback-same-cohomology" ], "ref_ids": [ 9022, 9023, 9025, 4339 ] } ], "ref_ids": [] }, { "id": 9048, "type": "theorem", "label": "spaces-simplicial-lemma-flat-augmentation-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-flat-augmentation-modules", "contents": [ "With notation as above. The morphism", "$a : (\\Sh(\\mathcal{C}_{total}), \\mathcal{O}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "is flat if and only if", "$a_n : (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "is flat for $n \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Since $g_n : (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n) \\to", "(\\Sh(\\mathcal{C}_{total}), \\mathcal{O})$ is flat, we see", "that if $a$ is flat, then $a_n = a \\circ g_n$ is flat as", "a composition. Conversely, suppose that $a_n$ is flat for all $n$.", "We have to check that $\\mathcal{O}$ is flat as a sheaf of", "$a^{-1}\\mathcal{O}_\\mathcal{D}$-modules. Let $\\mathcal{F} \\to \\mathcal{G}$", "be an injective map of $a^{-1}\\mathcal{O}_\\mathcal{D}$-modules.", "We have to show that", "$$", "\\mathcal{F} \\otimes_{a^{-1}\\mathcal{O}_\\mathcal{D}} \\mathcal{O}", "\\to", "\\mathcal{G} \\otimes_{a^{-1}\\mathcal{O}_\\mathcal{D}} \\mathcal{O}", "$$", "is injective. We can check this on $\\mathcal{C}_n$, i.e., after", "applying $g_n^{-1}$. Since $g_n^* = g_n^{-1}$ because", "$g_n^{-1}\\mathcal{O} = \\mathcal{O}_n$ we obtain", "$$", "g_n^{-1}\\mathcal{F} \\otimes_{g_n^{-1}a^{-1}\\mathcal{O}_\\mathcal{D}}", "\\mathcal{O}_n", "\\to", "g_n^{-1}\\mathcal{G} \\otimes_{g_n^{-1}a^{-1}\\mathcal{O}_\\mathcal{D}}", "\\mathcal{O}_n", "$$", "which is injective because", "$g_n^{-1}a^{-1}\\mathcal{O}_\\mathcal{D} = a_n^{-1}\\mathcal{O}_\\mathcal{D}$", "and we assume $a_n$ was flat." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9049, "type": "theorem", "label": "spaces-simplicial-lemma-simplicial-resolution-augmentation-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-simplicial-resolution-augmentation-modules", "contents": [ "With notation as above. For a $\\mathcal{O}_\\mathcal{D}$-module $\\mathcal{G}$", "there is an exact complex", "$$", "\\ldots \\to", "g_{2!}(a_2^*\\mathcal{G}) \\to", "g_{1!}(a_1^*\\mathcal{G}) \\to", "g_{0!}(a_0^*\\mathcal{G}) \\to", "a^*\\mathcal{G} \\to 0", "$$", "of sheaves of $\\mathcal{O}$-modules on $\\mathcal{C}_{total}$.", "Here $g_{n!}$ is as in Lemma \\ref{lemma-restriction-module-to-components-site}." ], "refs": [ "spaces-simplicial-lemma-restriction-module-to-components-site" ], "proofs": [ { "contents": [ "Observe that $a^*\\mathcal{G}$ is the $\\mathcal{O}$-module on", "$\\mathcal{C}_{total}$ whose restriction to $\\mathcal{C}_m$", "is the $\\mathcal{O}_m$-module $a_m^*\\mathcal{G}$.", "The description of the functors $g_{n!}$ on modules", "in Lemma \\ref{lemma-restriction-module-to-components-site}", "shows that $g_{n!}(a_n^*\\mathcal{G})$ is the", "$\\mathcal{O}$-module on $\\mathcal{C}_{total}$", "whose restriction to $\\mathcal{C}_m$ is the $\\mathcal{O}_m$-module", "$$", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^*a_n^*\\mathcal{G} =", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} a_m^*\\mathcal{G}", "$$", "The rest of the proof is exactly the same as the proof of", "Lemma \\ref{lemma-simplicial-resolution-augmentation},", "replacing $a_m^{-1}\\mathcal{G}$ by $a_m^*\\mathcal{G}$." ], "refs": [ "spaces-simplicial-lemma-restriction-module-to-components-site", "spaces-simplicial-lemma-simplicial-resolution-augmentation" ], "ref_ids": [ 9031, 9041 ] } ], "ref_ids": [ 9031 ] }, { "id": 9050, "type": "theorem", "label": "spaces-simplicial-lemma-augmentation-cech-complex-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-augmentation-cech-complex-modules", "contents": [ "With notation as above.", "For an $\\mathcal{O}$-module $\\mathcal{F}$ on $\\mathcal{C}_{total}$", "there is a canonical complex", "$$", "0 \\to a_*\\mathcal{F} \\to a_{0, *}\\mathcal{F}_0 \\to a_{1, *}\\mathcal{F}_1 \\to", "a_{2, *}\\mathcal{F}_2 \\to \\ldots", "$$", "of $\\mathcal{O}_\\mathcal{D}$-modules which is exact in degrees $-1, 0$.", "If $\\mathcal{F}$ is an injective $\\mathcal{O}$-module, then the complex", "is exact in all degrees and remains exact on applying the functor", "$\\Hom_{\\mathcal{O}_\\mathcal{D}}(\\mathcal{G}, -)$ for any", "$\\mathcal{O}_\\mathcal{D}$-module $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "To construct the complex, by the Yoneda lemma, it suffices for any", "$\\mathcal{O}_\\mathcal{D}$-modules $\\mathcal{G}$ on $\\mathcal{D}$", "to construct a complex", "$$", "0 \\to \\Hom_{\\mathcal{O}_\\mathcal{D}}(\\mathcal{G}, a_*\\mathcal{F}) \\to", "\\Hom_{\\mathcal{O}_\\mathcal{D}}(\\mathcal{G}, a_{0, *}\\mathcal{F}_0) \\to", "\\Hom_{\\mathcal{O}_\\mathcal{D}}(\\mathcal{G}, a_{1, *}\\mathcal{F}_1) \\to \\ldots", "$$", "functorially in $\\mathcal{G}$. To do this apply", "$\\Hom_\\mathcal{O}(-, \\mathcal{F})$", "to the exact complex of", "Lemma \\ref{lemma-simplicial-resolution-augmentation-modules}", "and use adjointness of pullback and pushforward.", "The exactness properties in degrees $-1, 0$ follow from", "the construction as $\\Hom_\\mathcal{O}(-, \\mathcal{F})$ is left exact.", "If $\\mathcal{F}$ is an injective $\\mathcal{O}$-module, then the", "complex is exact because $\\Hom_\\mathcal{O}(-, \\mathcal{F})$ is exact." ], "refs": [ "spaces-simplicial-lemma-simplicial-resolution-augmentation-modules" ], "ref_ids": [ 9049 ] } ], "ref_ids": [] }, { "id": 9051, "type": "theorem", "label": "spaces-simplicial-lemma-augmentation-spectral-sequence-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-augmentation-spectral-sequence-modules", "contents": [ "With notation as above for any $K$ in $D^+(\\mathcal{O})$ there is a spectral", "sequence $(E_r, d_r)_{r \\geq 0}$ in $\\textit{Mod}(\\mathcal{O}_\\mathcal{D})$", "with", "$$", "E_1^{p, q} = R^qa_{p, *} K_p", "$$", "converging to $R^{p + q}a_*K$. This spectral sequence is functorial in $K$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I}^\\bullet$ be a bounded below complex of injective", "$\\mathcal{O}$-modules representing $K$. Consider the double complex with terms", "$$", "A^{p, q} = a_{p, *}\\mathcal{I}^q_p", "$$", "where the horizontal arrows come from", "Lemma \\ref{lemma-augmentation-cech-complex-modules}", "and the vertical arrows from the differentials of the", "complex $\\mathcal{I}^\\bullet$. The lemma", "says rows of the double complex are exact", "in positive degrees and evaluate to", "$a_*\\mathcal{I}^q$ in degree $0$.", "Thus the total complex associated to the double complex", "computes $Ra_*K$ by", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}.", "On the other hand, since restriction to $\\mathcal{C}_p$ is exact", "(Lemma \\ref{lemma-restriction-to-components-site})", "the complex $\\mathcal{I}_p^\\bullet$ represents $K_p$ in", "$D(\\mathcal{C}_p)$. The sheaves $\\mathcal{I}_p^q$", "are totally acyclic on $\\mathcal{C}_p$", "(Lemma \\ref{lemma-restriction-injective-to-component-limp}).", "Hence the cohomology of the columns are the sheaves", "$R^qa_{p, *}K_p$ by Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "and", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-limp-acyclic}.", "We conclude by applying", "Homology, Lemma \\ref{homology-lemma-first-quadrant-ss}." ], "refs": [ "spaces-simplicial-lemma-augmentation-cech-complex-modules", "homology-lemma-double-complex-gives-resolution", "spaces-simplicial-lemma-restriction-to-components-site", "spaces-simplicial-lemma-restriction-injective-to-component-limp", "derived-lemma-leray-acyclicity", "sites-cohomology-lemma-limp-acyclic", "homology-lemma-first-quadrant-ss" ], "ref_ids": [ 9050, 12106, 9025, 9032, 1844, 4219, 12105 ] } ], "ref_ids": [] }, { "id": 9052, "type": "theorem", "label": "spaces-simplicial-lemma-check-cartesian-module", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-check-cartesian-module", "contents": [ "In Situation \\ref{situation-simplicial-site}.", "\\begin{enumerate}", "\\item A sheaf $\\mathcal{F}$ of sets or abelian groups is cartesian", "if and only if the maps", "$(f_{\\delta^n_j})^{-1}\\mathcal{F}_{n - 1} \\to \\mathcal{F}_n$", "are isomorphisms.", "\\item An object $K$ of $D(\\mathcal{C}_{total})$ is cartesian", "if and only if the maps", "$(f_{\\delta^n_j})^{-1}K_{n - 1} \\to K_n$", "are isomorphisms.", "\\item If $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}_{total}$", "a sheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules is cartesian", "if and only if the maps", "$(f_{\\delta^n_j})^*\\mathcal{F}_{n - 1} \\to \\mathcal{F}_n$", "are isomorphisms.", "\\item If $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}_{total}$", "an object $K$ of $D(\\mathcal{O})$ is cartesian", "if and only if the maps", "$L(f_{\\delta^n_j})^*K_{n - 1} \\to K_n$", "are isomorphisms.", "\\item Add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In each case the key is that the pullback functors", "compose to pullback functor; for part (4) see", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-derived-pullback-composition}.", "We show how the argument works in case (1) and omit the proof", "in the other cases.", "The category $\\Delta$ is generated by the morphisms", "the morphisms $\\delta^n_j$ and $\\sigma^n_j$, see", "Simplicial, Lemma \\ref{simplicial-lemma-face-degeneracy}.", "Hence we only need to check the maps", "$(f_{\\delta^n_j})^{-1}\\mathcal{F}_{n - 1} \\to \\mathcal{F}_n$", "and $(f_{\\sigma^n_j})^{-1}\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n$ are", "isomorphisms, see", "Simplicial, Lemma \\ref{simplicial-lemma-characterize-simplicial-object}", "for notation. Since $\\sigma^n_j \\circ \\delta_j^{n + 1} = \\text{id}_{[n]}$", "the composition", "$$", "\\mathcal{F}_n =", "(f_{\\sigma^n_j})^{-1}", "(f_{\\delta_j^{n + 1}})^{-1}", "\\mathcal{F}_n \\to", "(f_{\\sigma^n_j})^{-1}", "\\mathcal{F}_{n + 1} \\to", "\\mathcal{F}_n", "$$", "is the identity. Thus the result for $\\delta^{n + 1}_j$ implies the result", "for $\\sigma^n_j$." ], "refs": [ "sites-cohomology-lemma-derived-pullback-composition", "simplicial-lemma-face-degeneracy", "simplicial-lemma-characterize-simplicial-object" ], "ref_ids": [ 4243, 14805, 14808 ] } ], "ref_ids": [] }, { "id": 9053, "type": "theorem", "label": "spaces-simplicial-lemma-augmentation-cartesian-module", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-augmentation-cartesian-module", "contents": [ "In Situation \\ref{situation-simplicial-site} let", "$a_0$ be an augmentation towards a site $\\mathcal{D}$ as in", "Remark \\ref{remark-augmentation-site}.", "\\begin{enumerate}", "\\item The pullback $a^{-1}\\mathcal{G}$ of a sheaf of sets or abelian groups", "on $\\mathcal{D}$ is cartesian.", "\\item The pullback $a^{-1}K$ of an object $K$ of $D(\\mathcal{D})$", "is cartesian.", "\\end{enumerate}", "Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$ and", "$\\mathcal{O}_\\mathcal{D}$ a sheaf of rings on $\\mathcal{D}$", "and $a^\\sharp : \\mathcal{O}_\\mathcal{D} \\to a_*\\mathcal{O}$ a", "morphism as in", "Section \\ref{section-cohomology-augmentation-ringed-simplicial-sites}.", "\\begin{enumerate}", "\\item[(3)] The pullback $a^*\\mathcal{F}$ of a sheaf of", "$\\mathcal{O}_\\mathcal{D}$-modules is cartesian.", "\\item[(4)] The derived pullback $La^*K$ of an object", "$K$ of $D(\\mathcal{O}_\\mathcal{D})$ is cartesian.", "\\end{enumerate}" ], "refs": [ "spaces-simplicial-remark-augmentation-site" ], "proofs": [ { "contents": [ "This follows immediately from the identities", "$a_m \\circ f_\\varphi = a_n$ for all $\\varphi : [m] \\to [n]$.", "See Lemma \\ref{lemma-augmentation-site} and the discussion in", "Section \\ref{section-cohomology-augmentation-ringed-simplicial-sites}." ], "refs": [ "spaces-simplicial-lemma-augmentation-site" ], "ref_ids": [ 9027 ] } ], "ref_ids": [ 9152 ] }, { "id": 9054, "type": "theorem", "label": "spaces-simplicial-lemma-characterize-cartesian", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-characterize-cartesian", "contents": [ "In Situation \\ref{situation-simplicial-site}.", "The category of cartesian sheaves of sets (resp.\\ abelian groups)", "is equivalent to the category of pairs $(\\mathcal{F}, \\alpha)$", "where $\\mathcal{F}$ is a sheaf of sets (resp.\\ abelian groups)", "on $\\mathcal{C}_0$ and", "$$", "\\alpha :", "(f_{\\delta_1^1})^{-1}\\mathcal{F}", "\\longrightarrow (f_{\\delta_0^1})^{-1}\\mathcal{F}", "$$", "is an isomorphism of sheaves of sets (resp.\\ abelian groups)", "on $\\mathcal{C}_1$ such that", "$(f_{\\delta^2_1})^{-1}\\alpha =", "(f_{\\delta^2_0})^{-1}\\alpha \\circ (f_{\\delta^2_2})^{-1}\\alpha$", "as maps of sheaves on $\\mathcal{C}_2$." ], "refs": [], "proofs": [ { "contents": [ "We abbreviate", "$d^n_j = f_{\\delta^n_j} : \\Sh(\\mathcal{C}_n) \\to \\Sh(\\mathcal{C}_{n - 1})$.", "The condition on $\\alpha$ in the statement of the lemma makes sense because", "$$", "d^1_1 \\circ d^2_2 = d^1_1 \\circ d^2_1, \\quad", "d^1_1 \\circ d^2_0 = d^1_0 \\circ d^2_2, \\quad", "d^1_0 \\circ d^2_0 = d^1_0 \\circ d^2_1", "$$", "as morphisms of topoi $\\Sh(\\mathcal{C}_2) \\to \\Sh(\\mathcal{C}_0)$, see", "Simplicial, Remark \\ref{simplicial-remark-relations}. Hence we", "can picture these maps as follows", "$$", "\\xymatrix{", "& (d^2_0)^{-1}(d^1_1)^{-1}\\mathcal{F} \\ar[r]_-{(d^2_0)^{-1}\\alpha} &", "(d^2_0)^{-1}(d^1_0)^{-1}\\mathcal{F} \\ar@{=}[rd] & \\\\", "(d^2_2)^{-1}(d^1_0)^{-1}\\mathcal{F} \\ar@{=}[ru] & & &", "(d^2_1)^{-1}(d^1_0)^{-1}\\mathcal{F} \\\\", "& (d^2_2)^{-1}(d^1_1)^{-1}\\mathcal{F} \\ar[lu]^{(d^2_2)^{-1}\\alpha} \\ar@{=}[r] &", "(d^2_1)^{-1}(d^1_1)^{-1}\\mathcal{F} \\ar[ru]_{(d^2_1)^{-1}\\alpha}", "}", "$$", "and the condition signifies the diagram is commutative. It is clear that", "given a cartesian sheaf $\\mathcal{G}$ of sets (resp.\\ abelian groups)", "on $\\mathcal{C}_{total}$", "we can set $\\mathcal{F} = \\mathcal{G}_0$ and $\\alpha$ equal to the composition", "$$", "(d_1^1)^{-1}\\mathcal{G}_0 \\to \\mathcal{G}_1", "\\leftarrow (d_1^0)^{-1}\\mathcal{G}_0", "$$", "where the arrows are invertible as $\\mathcal{G}$ is cartesian.", "To prove this functor", "is an equivalence we construct a quasi-inverse. The construction of", "the quasi-inverse is analogous to the construction discussed in", "Descent, Section \\ref{descent-section-descent-modules} from which we borrow", "the notation $\\tau^n_i : [0] \\to [n]$, $0 \\mapsto i$ and", "$\\tau^n_{ij} : [1] \\to [n]$, $0 \\mapsto i$, $1 \\mapsto j$.", "Namely, given a pair $(\\mathcal{F}, \\alpha)$", "as in the lemma we set $\\mathcal{G}_n = (f_{\\tau^n_n})^{-1}\\mathcal{F}$.", "Given $\\varphi : [n] \\to [m]$ we define", "$\\mathcal{G}(\\varphi) : (f_\\varphi)^{-1}\\mathcal{G}_n \\to \\mathcal{G}_m$", "using", "$$", "\\xymatrix{", "(f_\\varphi)^{-1}\\mathcal{G}_n \\ar@{=}[r] &", "(f_\\varphi)^{-1}(f_{\\tau^n_n})^{-1}\\mathcal{F} \\ar@{=}[r] &", "(f_{\\tau^m_{\\varphi(n)}})^{-1}\\mathcal{F} \\ar@{=}[r] &", "(f_{\\tau^m_{\\varphi(n)m}})^{-1}(d^1_1)^{-1}\\mathcal{F}", "\\ar[d]^{(f_{\\tau^m_{\\varphi(n)m}})^{-1}\\alpha} \\\\", "&", "\\mathcal{G}_m \\ar@{=}[r] &", "(f_{\\tau^m_m})^{-1}\\mathcal{F} \\ar@{=}[r] &", "(f_{\\tau^m_{\\varphi(n)m}})^{-1}(d^1_0)^{-1}\\mathcal{F}", "}", "$$", "We omit the verification that the commutativity of the displayed diagram", "above implies the maps compose correctly and hence give rise to a", "sheaf on $\\mathcal{C}_{total}$, see", "Lemma \\ref{lemma-describe-sheaves-simplicial-site-site}.", "We also omit the verification", "that the two functors are quasi-inverse to each other." ], "refs": [ "simplicial-remark-relations", "spaces-simplicial-lemma-describe-sheaves-simplicial-site-site" ], "ref_ids": [ 14932, 9024 ] } ], "ref_ids": [] }, { "id": 9055, "type": "theorem", "label": "spaces-simplicial-lemma-characterize-cartesian-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-characterize-cartesian-modules", "contents": [ "In Situation \\ref{situation-simplicial-site}", "let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$.", "The category of cartesian $\\mathcal{O}$-modules", "is equivalent to the category of pairs $(\\mathcal{F}, \\alpha)$", "where $\\mathcal{F}$ is a $\\mathcal{O}_0$-module", "and", "$$", "\\alpha :", "(f_{\\delta_1^1})^*\\mathcal{F}", "\\longrightarrow (f_{\\delta_0^1})^*\\mathcal{F}", "$$", "is an isomorphism of $\\mathcal{O}_1$-modules such that", "$(f_{\\delta^2_1})^*\\alpha =", "(f_{\\delta^2_0})^*\\alpha \\circ (f_{\\delta^2_2})^*\\alpha$", "as $\\mathcal{O}_2$-module maps." ], "refs": [], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Lemma \\ref{lemma-characterize-cartesian}", "with pullback of sheaves of abelian groups replaced", "by pullback of modules." ], "refs": [ "spaces-simplicial-lemma-characterize-cartesian" ], "ref_ids": [ 9054 ] } ], "ref_ids": [] }, { "id": 9056, "type": "theorem", "label": "spaces-simplicial-lemma-Serre-subcat-cartesian-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-Serre-subcat-cartesian-modules", "contents": [ "In Situation \\ref{situation-simplicial-site}.", "\\begin{enumerate}", "\\item The full subcategory of cartesian abelian sheaves forms a", "weak Serre subcategory of $\\textit{Ab}(\\mathcal{C}_{total})$.", "Colimits of systems of cartesian abelian sheaves are cartesian.", "\\item Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$", "such that the morphisms", "$$", "f_{\\delta^n_j} : (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n)", "\\to (\\Sh(\\mathcal{C}_{n - 1}), \\mathcal{O}_{n - 1})", "$$", "are flat. The full subcategory of cartesian $\\mathcal{O}$-modules forms a", "weak Serre subcategory of $\\textit{Mod}(\\mathcal{O})$.", "Colimits of systems of cartesian $\\mathcal{O}$-modules are cartesian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To see we obtain a weak Serre subcategory in (1)", "we check the conditions listed in", "Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}.", "First, if $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a map", "between cartesian abelian sheaves, then", "$\\Ker(\\varphi)$ and $\\Coker(\\varphi)$ are cartesian too", "because the restriction functors", "$\\Sh(\\mathcal{C}_{total}) \\to \\Sh(\\mathcal{C}_n)$", "and the functors $f_\\varphi^{-1}$ are exact.", "Similarly, if", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{H} \\to \\mathcal{G} \\to 0", "$$", "is a short exact sequence of abelian sheaves on $\\mathcal{C}_{total}$", "with $\\mathcal{F}$ and $\\mathcal{G}$ cartesian, then it follows that", "$\\mathcal{H}$ is cartesian from the 5-lemma. To see the property of", "colimits, use that colimits commute with pullback as pullback is a", "left adjoint. In the case of modules", "we argue in the same manner, using the exactness of flat pullback", "(Modules on Sites, Lemma \\ref{sites-modules-lemma-flat-pullback-exact})", "and the fact that it suffices to check the condition", "for $f_{\\delta^n_j}$, see Lemma \\ref{lemma-check-cartesian-module}." ], "refs": [ "homology-lemma-characterize-weak-serre-subcategory", "sites-modules-lemma-flat-pullback-exact", "spaces-simplicial-lemma-check-cartesian-module" ], "ref_ids": [ 12046, 14223, 9052 ] } ], "ref_ids": [] }, { "id": 9057, "type": "theorem", "label": "spaces-simplicial-lemma-derived-cartesian-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-derived-cartesian-modules", "contents": [ "In Situation \\ref{situation-simplicial-site}.", "\\begin{enumerate}", "\\item An object $K$ of $D(\\mathcal{C}_{total})$ is cartesian if and only", "if $H^q(K)$ is a cartesian abelian sheaf for all $q$.", "\\item Let $\\mathcal{O}$ be a sheaf", "of rings on $\\mathcal{C}_{total}$ such that the morphisms", "$f_{\\delta^n_j} : (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n)", "\\to (\\Sh(\\mathcal{C}_{n - 1}), \\mathcal{O}_{n - 1})$ are flat.", "Then an object $K$ of $D(\\mathcal{O})$ is cartesian if and only", "if $H^q(K)$ is a cartesian $\\mathcal{O}$-module for all $q$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is true because the pullback functors $(f_\\varphi)^{-1}$", "are exact. Part (2) follows from the characterization in", "Lemma \\ref{lemma-check-cartesian-module}", "and the fact that $L(f_{\\delta^n_j})^* = (f_{\\delta^n_j})^*$", "by flatness." ], "refs": [ "spaces-simplicial-lemma-check-cartesian-module" ], "ref_ids": [ 9052 ] } ], "ref_ids": [] }, { "id": 9058, "type": "theorem", "label": "spaces-simplicial-lemma-derived-cartesian-shriek", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-derived-cartesian-shriek", "contents": [ "In Situation \\ref{situation-simplicial-site}.", "\\begin{enumerate}", "\\item An object $K$ of $D(\\mathcal{C}_{total})$ is cartesian if and only", "the canonical map", "$$", "g_{n!}K_n \\longrightarrow", "g_{n!}\\mathbf{Z} \\otimes^\\mathbf{L}_\\mathbf{Z} K", "$$", "is an isomorphism for all $n$.", "\\item Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$", "such that the morphisms $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$", "are flat for all $\\varphi : [n] \\to [m]$. Then an object $K$ of", "$D(\\mathcal{O})$ is cartesian if and only if the canonical map", "$$", "g_{n!}K_n \\longrightarrow", "g_{n!}\\mathcal{O}_n \\otimes^\\mathbf{L}_\\mathcal{O} K", "$$", "is an isomorphism for all $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Since $g_{n!}$ is exact, it induces a functor", "on derived categories adjoint to $g_n^{-1}$.", "The map is the adjoint of the map", "$K_n \\to (g_n^{-1}g_{n!}\\mathbf{Z}) \\otimes^\\mathbf{L}_\\mathbf{Z} K_n$", "corresponding to $\\mathbf{Z} \\to g_n^{-1}g_{n!}\\mathbf{Z}$", "which in turn is adjoint to", "$\\text{id} : g_{n!}\\mathbf{Z} \\to g_{n!}\\mathbf{Z}$.", "Using the description of $g_{n!}$", "given in Lemma \\ref{lemma-restriction-to-components-site}", "we see that the restriction to $\\mathcal{C}_m$ of this map", "is", "$$", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^{-1}K_n", "\\longrightarrow", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} K_m", "$$", "Thus the statement is clear.", "\\medskip\\noindent", "Proof of (2). Since $g_{n!}$ is exact", "(Lemma \\ref{lemma-exactness-g-shriek-modules}), it induces a functor", "on derived categories adjoint to $g_n^*$ (also exact).", "The map is the adjoint of the map", "$K_n \\to (g_n^*g_{n!}\\mathcal{O}_n) \\otimes^\\mathbf{L}_{\\mathcal{O}_n} K_n$", "corresponding to $\\mathcal{O}_n \\to g_n^*g_{n!}\\mathcal{O}_n$", "which in turn is adjoint to", "$\\text{id} : g_{n!}\\mathcal{O}_n \\to g_{n!}\\mathcal{O}_n$.", "Using the description of $g_{n!}$", "given in Lemma \\ref{lemma-restriction-module-to-components-site}", "we see that the restriction to $\\mathcal{C}_m$ of this map", "is", "$$", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^*K_n", "\\longrightarrow", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]}", "f_\\varphi^*\\mathcal{O}_n \\otimes_{\\mathcal{O}_m} K_m =", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} K_m", "$$", "Thus the statement is clear." ], "refs": [ "spaces-simplicial-lemma-restriction-to-components-site", "spaces-simplicial-lemma-exactness-g-shriek-modules", "spaces-simplicial-lemma-restriction-module-to-components-site" ], "ref_ids": [ 9025, 9033, 9031 ] } ], "ref_ids": [] }, { "id": 9059, "type": "theorem", "label": "spaces-simplicial-lemma-quasi-coherent-sheaf", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-quasi-coherent-sheaf", "contents": [ "In Situation \\ref{situation-simplicial-site}", "let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.", "Then $\\mathcal{F}$ is quasi-coherent in the sense of", "Modules on Sites, Definition \\ref{sites-modules-definition-site-local}", "if and only if $\\mathcal{F}$ is cartesian", "and $\\mathcal{F}_n$ is a quasi-coherent $\\mathcal{O}_n$-module for all $n$." ], "refs": [ "sites-modules-definition-site-local" ], "proofs": [ { "contents": [ "Assume $\\mathcal{F}$ is quasi-coherent. Since pullbacks of", "quasi-coherent modules are quasi-coherent", "(Modules on Sites, Lemma \\ref{sites-modules-lemma-local-pullback})", "we see that $\\mathcal{F}_n$ is a quasi-coherent $\\mathcal{O}_n$-module", "for all $n$. To show that $\\mathcal{F}$ is cartesian, let $U$", "be an object of $\\mathcal{C}_n$ for some $n$. Let us view $U$", "as an object of $\\mathcal{C}_{total}$. Because $\\mathcal{F}$", "is quasi-coherent there exists a covering $\\{U_i \\to U\\}$", "and for each $i$ a presentation", "$$", "\\bigoplus\\nolimits_{j \\in J_i} \\mathcal{O}_{\\mathcal{C}_{total}/U_i} \\to", "\\bigoplus\\nolimits_{k \\in K_i} \\mathcal{O}_{\\mathcal{C}_{total}/U_i} \\to", "\\mathcal{F}|_{\\mathcal{C}_{total}/U_i} \\to 0", "$$", "Observe that $\\{U_i \\to U\\}$ is a covering of $\\mathcal{C}_n$ by", "the construction of the site $\\mathcal{C}_{total}$.", "Next, let $V$ be an object of $\\mathcal{C}_m$ for some $m$ and let", "$V \\to U$ be a morphism of $\\mathcal{C}_{total}$ lying over", "$\\varphi : [n] \\to [m]$. The fibre products $V_i = V \\times_U U_i$", "exist and we get an induced covering $\\{V_i \\to V\\}$ in $\\mathcal{C}_m$.", "Restricting the presentation above to the sites", "$\\mathcal{C}_n/U_i$ and $\\mathcal{C}_m/V_i$ we obtain", "presentations", "$$", "\\bigoplus\\nolimits_{j \\in J_i} \\mathcal{O}_{\\mathcal{C}_m/U_i} \\to", "\\bigoplus\\nolimits_{k \\in K_i} \\mathcal{O}_{\\mathcal{C}_m/U_i} \\to", "\\mathcal{F}_n|_{\\mathcal{C}_n/U_i} \\to 0", "$$", "and", "$$", "\\bigoplus\\nolimits_{j \\in J_i} \\mathcal{O}_{\\mathcal{C}_m/V_i} \\to", "\\bigoplus\\nolimits_{k \\in K_i} \\mathcal{O}_{\\mathcal{C}_m/V_i} \\to", "\\mathcal{F}_m|_{\\mathcal{C}_m/V_i} \\to 0", "$$", "These presentations are compatible with the map", "$\\mathcal{F}(\\varphi) : f_\\varphi^*\\mathcal{F}_n \\to \\mathcal{F}_m$", "(as this map is defined using the restriction maps of $\\mathcal{F}$", "along morphisms of $\\mathcal{C}_{total}$ lying over $\\varphi$).", "We conclude that $\\mathcal{F}(\\varphi)|_{\\mathcal{C}_m/V_i}$", "is an isomorphism. As $\\{V_i \\to V\\}$ is a covering we conclude", "$\\mathcal{F}(\\varphi)|_{\\mathcal{C}_m/V}$ is an isomorphism.", "Since $V$ and $U$ were arbitrary this proves that $\\mathcal{F}$ is cartesian.", "(In case A use Sites, Lemma \\ref{sites-lemma-morphism-of-sites-covering}.)", "\\medskip\\noindent", "Conversely, assume $\\mathcal{F}_n$ is quasi-coherent", "for all $n$ and that $\\mathcal{F}$ is cartesian.", "Then for any $n$ and object $U$ of $\\mathcal{C}_n$ we", "can choose a covering $\\{U_i \\to U\\}$ of $\\mathcal{C}_n$", "and for each $i$ a presentation", "$$", "\\bigoplus\\nolimits_{j \\in J_i} \\mathcal{O}_{\\mathcal{C}_m/U_i} \\to", "\\bigoplus\\nolimits_{k \\in K_i} \\mathcal{O}_{\\mathcal{C}_m/U_i} \\to", "\\mathcal{F}_n|_{\\mathcal{C}_n/U_i} \\to 0", "$$", "Pulling back to $\\mathcal{C}_{total}/U_i$ we obtain complexes", "$$", "\\bigoplus\\nolimits_{j \\in J_i} \\mathcal{O}_{\\mathcal{C}_{total}/U_i} \\to", "\\bigoplus\\nolimits_{k \\in K_i} \\mathcal{O}_{\\mathcal{C}_{total}/U_i} \\to", "\\mathcal{F}|_{\\mathcal{C}_{total}/U_i} \\to 0", "$$", "of modules on $\\mathcal{C}_{total}/U_i$. Then the property that", "$\\mathcal{F}$ is cartesian implies that this is exact.", "We omit the details." ], "refs": [ "sites-modules-lemma-local-pullback", "sites-lemma-morphism-of-sites-covering" ], "ref_ids": [ 14186, 8527 ] } ], "ref_ids": [ 14289 ] }, { "id": 9060, "type": "theorem", "label": "spaces-simplicial-lemma-cartesian-objects-derived", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-cartesian-objects-derived", "contents": [ "In Situation \\ref{situation-simplicial-site}.", "If $K \\in D(\\mathcal{C}_{total})$ is an object,", "then $(K_n, K(\\varphi))$ is a simplicial system of the derived category.", "If $K$ is cartesian, so is the system." ], "refs": [], "proofs": [ { "contents": [ "This is obvious." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9061, "type": "theorem", "label": "spaces-simplicial-lemma-abelian-postnikov", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-abelian-postnikov", "contents": [ "In Situation \\ref{situation-simplicial-site}. Let $K$ be", "an object of $D(\\mathcal{C}_{total})$. Set", "$$", "X_n = (g_{n!}\\mathbf{Z})", "\\otimes^\\mathbf{L}_\\mathbf{Z} K", "\\quad\\text{and}\\quad", "Y_n =", "(g_{n!}\\mathbf{Z} \\to \\ldots \\to g_{0!}\\mathbf{Z})[-n]", "\\otimes^\\mathbf{L}_\\mathbf{Z} K", "$$", "as objects of $D(\\mathcal{C}_{total})$ where the maps are", "as in Lemma \\ref{lemma-simplicial-resolution-Z-site}.", "With the evident canonical maps $Y_n \\to X_n$ and", "$Y_0 \\to Y_1[1] \\to Y_2[2] \\to \\ldots$ we have", "\\begin{enumerate}", "\\item the distinguished triangles $Y_n \\to X_n \\to Y_{n - 1} \\to Y_n[1]$", "define a Postnikov system", "(Derived Categories, Definition \\ref{derived-definition-postnikov-system})", "for $\\ldots \\to X_2 \\to X_1 \\to X_0$,", "\\item $K = \\text{hocolim} Y_n[n]$ in $D(\\mathcal{C}_{total})$.", "\\end{enumerate}" ], "refs": [ "spaces-simplicial-lemma-simplicial-resolution-Z-site", "derived-definition-postnikov-system" ], "proofs": [ { "contents": [ "First, if $K = \\mathbf{Z}$, then this is the construction of", "Derived Categories, Example \\ref{derived-example-key-postnikov}", "applied to the complex", "$$", "\\ldots \\to", "g_{2!}\\mathbf{Z} \\to", "g_{1!}\\mathbf{Z} \\to", "g_{0!}\\mathbf{Z}", "$$", "in $\\textit{Ab}(\\mathcal{C}_{total})$ combined with the fact that", "this complex represents $K = \\mathbf{Z}$ in $D(\\mathcal{C}_{total})$", "by Lemma \\ref{lemma-simplicial-resolution-Z-site}.", "The general case follows from this, the fact that the exact functor", "$- \\otimes^\\mathbf{L}_\\mathbf{Z} K$ sends Postnikov systems to", "Postnikov systems, and", "that $- \\otimes^\\mathbf{L}_\\mathbf{Z} K$ commutes with homotopy colimits." ], "refs": [ "spaces-simplicial-lemma-simplicial-resolution-Z-site" ], "ref_ids": [ 9037 ] } ], "ref_ids": [ 9037, 2008 ] }, { "id": 9062, "type": "theorem", "label": "spaces-simplicial-lemma-nullity-cartesian-objects-derived", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-nullity-cartesian-objects-derived", "contents": [ "In Situation \\ref{situation-simplicial-site}.", "If $K, K' \\in D(\\mathcal{C}_{total})$.", "Assume", "\\begin{enumerate}", "\\item $K$ is cartesian,", "\\item $\\Hom(K_i[i], K'_i) = 0$ for $i > 0$, and", "\\item $\\Hom(K_i[i + 1], K'_i) = 0$ for $i \\geq 0$.", "\\end{enumerate}", "Then any map $K \\to K'$ which induces the zero map $K_0 \\to K'_0$ is zero." ], "refs": [], "proofs": [ { "contents": [ "Consider the objects $X_n$ and the Postnikov system $Y_n$", "associated to $K$ in Lemma \\ref{lemma-abelian-postnikov}.", "As $K = \\text{hocolim} Y_n[n]$ the map $K \\to K'$ induces", "a compatible family of morphisms $Y_n[n] \\to K'$.", "By (1) and Lemma \\ref{lemma-derived-cartesian-shriek} we have", "$X_n = g_{n!}K_n$. Since $Y_0 = X_0$ we find that", "$K_0 \\to K'_0$ being zero implies $Y_0 \\to K'$ is zero.", "Suppose we've shown that the map $Y_n[n] \\to K'$ is zero", "for some $n \\geq 0$. From the distinguished triangle", "$$", "Y_n[n] \\to Y_{n + 1}[n + 1] \\to X_{n + 1}[n + 1] \\to Y_n[n + 1]", "$$", "we get an exact sequence", "$$", "\\Hom(X_{n + 1}[n + 1], K') \\to", "\\Hom(Y_{n + 1}[n + 1], K') \\to", "\\Hom(Y_n[n], K')", "$$", "As $X_{n + 1}[n + 1] = g_{n + 1!}K_{n + 1}[n + 1]$ the first group is equal to", "$$", "\\Hom(K_{n + 1}[n + 1], K'_{n + 1})", "$$", "which is zero by assumption (2). By induction we conclude all the maps", "$Y_n[n] \\to K'$ are zero. Consider the defining distinguished triangle", "$$", "\\bigoplus Y_n[n] \\to", "\\bigoplus Y_n[n] \\to", "K \\to", "(\\bigoplus Y_n[n])[1]", "$$", "for the homotopy colimit. Arguing as above, we find that it suffices", "to show that", "$$", "\\Hom((\\bigoplus Y_n[n])[1], K') = \\prod \\Hom(Y_n[n + 1], K')", "$$", "is zero for all $n \\geq 0$. To see this, arguing as above,", "it suffices to show that", "$$", "\\Hom(K_n[n + 1], K'_n) = 0", "$$", "for all $n \\geq 0$ which follows from condition (3)." ], "refs": [ "spaces-simplicial-lemma-abelian-postnikov", "spaces-simplicial-lemma-derived-cartesian-shriek" ], "ref_ids": [ 9061, 9058 ] } ], "ref_ids": [] }, { "id": 9063, "type": "theorem", "label": "spaces-simplicial-lemma-hom-cartesian-objects-derived", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hom-cartesian-objects-derived", "contents": [ "In Situation \\ref{situation-simplicial-site}.", "If $K, K' \\in D(\\mathcal{C}_{total})$.", "Assume", "\\begin{enumerate}", "\\item $K$ is cartesian,", "\\item $\\Hom(K_i[i - 1], K'_i) = 0$ for $i > 1$.", "\\end{enumerate}", "Then any map $\\{K_n \\to K'_n\\}$ between the associated simplicial systems ", "of $K$ and $K'$ comes from a map $K \\to K'$ in $D(\\mathcal{C}_{total})$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{K_n \\to K'_n\\}_{n \\geq 0}$", "be a morphism of simplicial systems of the derived category.", "Consider the objects $X_n$ and Postnikov system $Y_n$", "associated to $K$ of Lemma \\ref{lemma-abelian-postnikov}.", "By (1) and Lemma \\ref{lemma-derived-cartesian-shriek} we have", "$X_n = g_{n!}K_n$. In particular, the map $K_0 \\to K'_0$", "induces a morphism $X_0 \\to K'$. Since $\\{K_n \\to K'_n\\}$", "is a morphism of systems, a computation (omitted) shows that", "the composition", "$$", "X_1 \\to X_0 \\to K'", "$$", "is zero. As $Y_0 = X_0$ and as $Y_1$ fits into a distinguished", "triangle", "$$", "Y_1 \\to X_1 \\to Y_0 \\to Y_1[1]", "$$", "we conclude that there exists a morphism $Y_1[1] \\to K'$ whose", "composition with $X_0 = Y_0 \\to Y_1[1]$ is the morphism $X_0 \\to K'$", "given above. Suppose given a map $Y_n[n] \\to K'$ for $n \\geq 1$.", "From the distinguished triangle", "$$", "X_{n + 1}[n] \\to Y_n[n] \\to Y_{n + 1}[n + 1] \\to X_{n + 1}[n + 1]", "$$", "we get an exact sequence", "$$", "\\Hom(Y_{n + 1}[n + 1], K') \\to", "\\Hom(Y_n[n], K') \\to", "\\Hom(X_{n + 1}[n], K')", "$$", "As $X_{n + 1}[n] = g_{n + 1!}K_{n + 1}[n]$ the last group is equal to", "$$", "\\Hom(K_{n + 1}[n], K'_{n + 1})", "$$", "which is zero by assumption (2). By induction we get a system of", "maps $Y_n[n] \\to K'$ compatible with transition maps and reducing", "to the given map on $Y_0$. This produces a map", "$$", "\\gamma :", "K = \\text{hocolim} Y_n[n]", "\\longrightarrow", "K'", "$$", "This map in any case has the property that the diagram", "$$", "\\xymatrix{", "X_0 \\ar[rd] \\ar[r] &", "K \\ar[d]^\\gamma \\\\", "& K'", "}", "$$", "is commutative. Restricting to", "$\\mathcal{C}_0$ we deduce that the map $\\gamma_0 : K_0 \\to K'_0$", "is the same as the first map $K_0 \\to K'_0$ of the morphism", "of simplicial systems. Since $K$ is cartesian, this easily gives that", "$\\{\\gamma_n\\}$ is the map of simplicial systems we started out with." ], "refs": [ "spaces-simplicial-lemma-abelian-postnikov", "spaces-simplicial-lemma-derived-cartesian-shriek" ], "ref_ids": [ 9061, 9058 ] } ], "ref_ids": [] }, { "id": 9064, "type": "theorem", "label": "spaces-simplicial-lemma-cartesian-object-derived-from-simplicial", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-cartesian-object-derived-from-simplicial", "contents": [ "In Situation \\ref{situation-simplicial-site}. Let", "$(K_n, K_\\varphi)$ be a simplicial system of the derived category.", "Assume", "\\begin{enumerate}", "\\item $(K_n, K_\\varphi)$ is cartesian,", "\\item $\\Hom(K_i[t], K_i) = 0$ for $i \\geq 0$ and $t > 0$.", "\\end{enumerate}", "Then there exists a cartesian object $K$ of $D(\\mathcal{C}_{total})$", "whose associated simplicial system is isomorphic to $(K_n, K_\\varphi)$." ], "refs": [], "proofs": [ { "contents": [ "Set $X_n = g_{n!}K_n$ in $D(\\mathcal{C}_{total})$. For each $n \\geq 1$", "we have", "$$", "\\Hom(X_n, X_{n - 1}) =", "\\Hom(K_n, g_n^{-1}g_{n - 1!}K_{n - 1}) =", "\\bigoplus\\nolimits_{\\varphi : [n - 1] \\to [n]}", "\\Hom(K_n, f_\\varphi^{-1}K_{n - 1})", "$$", "Thus we get a map $X_n \\to X_{n - 1}$ corresponding to the", "alternating sum of the maps", "$K_\\varphi^{-1} : K_n \\to f_\\varphi^{-1}K_{n - 1}$", "where $\\varphi$ runs over $\\delta^n_0, \\ldots, \\delta^n_n$.", "We can do this because $K_\\varphi$ is invertible by assumption (1).", "Please observe the similarity with the definition of the maps", "in the proof of Lemma \\ref{lemma-simplicial-resolution-Z-site}.", "We obtain a complex", "$$", "\\ldots \\to X_2 \\to X_1 \\to X_0", "$$", "in $D(\\mathcal{C}_{total})$. We omit the computation which shows", "that the compositions are zero. By", "Derived Categories, Lemma \\ref{derived-lemma-existence-postnikov-system}", "if we have", "$$", "\\Hom(X_i[i - j - 2], X_j) = 0\\text{ for }i > j + 2", "$$", "then we can extend this complex to a Postnikov system.", "The group is equal to", "$$", "\\Hom(K_i[i - j - 2], g_i^{-1}g_{j!}K_j)", "$$", "Again using that $(K_n, K_\\varphi)$ is cartesian we see that", "$g_i^{-1}g_{j!}K_j$ is isomorphic to a finite direct sum of copies of", "$K_i$. Hence the group vanishes by assumption (2).", "Let the Postnikov system be given by $Y_0 = X_0$ and distinguished", "sequences $Y_n \\to X_n \\to Y_{n - 1} \\to Y_n[1]$ for $n \\geq 1$.", "We set", "$$", "K = \\text{hocolim} Y_n[n]", "$$", "To finish the proof we have to show that $g_m^{-1}K$ is isomorphic", "to $K_m$ for all $m$ compatible with the maps $K_\\varphi$. Observe that", "$$", "g_m^{-1} K = \\text{hocolim} g_m^{-1}Y_n[n]", "$$", "and that $g_m^{-1}Y_n[n]$ is a Postnikov system for $g_m^{-1}X_n$.", "Consider the isomorphisms", "$$", "g_m^{-1}X_n =", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^{-1}K_n", "\\xrightarrow{\\bigoplus K_\\varphi}", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} K_m", "$$", "These maps define an isomorphism of complexes", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "g_m^{-1}X_2 \\ar[r] \\ar[d] &", "g_m^{-1}X_1 \\ar[r] \\ar[d] &", "g_m^{-1}X_0 \\ar[d] \\\\", "\\ldots \\ar[r] &", "\\bigoplus\\nolimits_{\\varphi : [2] \\to [m]} K_m \\ar[r] &", "\\bigoplus\\nolimits_{\\varphi : [1] \\to [m]} K_m \\ar[r] &", "\\bigoplus\\nolimits_{\\varphi : [0] \\to [m]} K_m", "}", "$$", "in $D(\\mathcal{C}_m)$ where the arrows in the bottom row are as", "in the proof of Lemma \\ref{lemma-simplicial-resolution-Z-site}.", "The squares commute by our choice of the arrows of the complex", "$\\ldots \\to X_2 \\to X_1 \\to X_0$; we omit the computation.", "The bottom row complex has a postnikov tower given by", "$$", "Y'_{m, n} =", "\\left(\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} \\mathbf{Z} \\to", "\\ldots \\to", "\\bigoplus\\nolimits_{\\varphi : [0] \\to [m]} \\mathbf{Z}\\right)[-n]", "\\otimes^\\mathbf{L}_\\mathbf{Z} K_m", "$$", "and $\\text{hocolim} Y'_{m, n} = K_m$", "(please compare with the proof of Lemma \\ref{lemma-abelian-postnikov}", "and Derived Categories, Example \\ref{derived-example-key-postnikov}).", "Applying the second part of", "Derived Categories, Lemma \\ref{derived-lemma-existence-postnikov-system}", "the vertical maps in the big diagram extend to an isomorphism", "of Postnikov systems provided we have", "$$", "\\Hom(g_m^{-1}X_i[i - j - 1], \\bigoplus\\nolimits_{\\varphi : [j] \\to [m]} K_m)", "= 0\\text{ for }i > j + 1", "$$", "The is true if $\\Hom(K_m[i - j - 1], K_m) = 0$ for $i > j + 1$", "which holds by assumption (2). Choose an isomorphism given", "by $\\gamma_{m, n} : g_m^{-1}Y_n \\to Y'_{m, n}$ of Postnikov systems", "in $D(\\mathcal{C}_m)$. By uniqueness of homotopy colimits,", "we can find an isomorphism", "$$", "g_m^{-1} K = \\text{hocolim} g_m^{-1}Y_n[n]", "\\xrightarrow{\\gamma_m}", "\\text{hocolim} Y'_{m, n} = K_m", "$$", "compatible with $\\gamma_{m, n}$.", "\\medskip\\noindent", "We still have to prove that the maps $\\gamma_m$ fit into commutative diagrams", "$$", "\\xymatrix{", "f_\\varphi^{-1}g_m^{-1}K \\ar[d]_{f_\\varphi^{-1}\\gamma_m} \\ar[r]_{K(\\varphi)} &", "g_n^{-1}K \\ar[d]^{\\gamma_n} \\\\", "f_\\varphi^{-1}K_m \\ar[r]^{K_\\varphi} &", "K_n", "}", "$$", "for every $\\varphi : [m] \\to [n]$. Consider the diagram", "$$", "\\xymatrix{", "f_\\varphi^{-1}(\\bigoplus_{\\psi : [0] \\to [m]} f_\\psi^{-1}K_0)", "\\ar@{=}[r] \\ar[d]_{f_\\varphi^{-1}(\\bigoplus K_\\psi)} &", "f_\\varphi^{-1}g_m^{-1}X_0 \\ar[d] \\ar[r]_{X_0(\\varphi)} &", "g_n^{-1}X_0 \\ar[d] &", "\\bigoplus_{\\chi : [0] \\to [n]} f_\\chi^{-1}K_0", "\\ar@{=}[l] \\ar[d]^{\\bigoplus K_\\chi} \\\\", "f_\\varphi^{-1}(\\bigoplus_{\\psi : [0] \\to [m]} K_m) \\ar@{=}[d] &", "f_\\varphi^{-1}g_m^{-1}K \\ar[d]_{f_\\varphi^{-1}\\gamma_m} \\ar[r]_{K(\\varphi)} &", "g_n^{-1}K \\ar[d]^{\\gamma_n} &", "\\bigoplus_{\\chi : [0] \\to [n]} K_n \\ar@{=}[d] \\\\", "f_\\varphi^{-1}Y'_{0, m} \\ar[r] &", "f_\\varphi^{-1}K_m \\ar[r]^{K_\\varphi} &", "K_n &", "Y'_{0, n} \\ar[l]", "}", "$$", "The top middle square is commutative as $X_0 \\to K$ is a morphism", "of simplicial objects. The left, resp.\\ the right rectangles are", "commutative as $\\gamma_m$, resp.\\ $\\gamma_n$ is compatible with", "$\\gamma_{0, m}$, resp.\\ $\\gamma_{0, n}$ which are the arrows", "$\\bigoplus K_\\psi$ and $\\bigoplus K_\\chi$ in the diagram.", "Going around the outer rectangle of the diagram", "is commutative as $(K_n, K_\\varphi)$ is a simplical system", "and the map $X_0(\\varphi)$ is given by the obvious identifications", "$f_\\varphi^{-1}f_\\psi^{-1}K_0 = f_{\\varphi \\circ \\psi}^{-1}K_0$.", "Note that the arrow $\\bigoplus_\\psi K_m \\to Y'_{0, m} \\to K_m$", "induces an isomorphism on any of the direct summands", "(because of our explicit construction of the Postnikov", "systems $Y'_{i, j}$ above).", "Hence, if we take a direct summand of", "the upper left and corner, then this maps isomorphically to", "$f_\\varphi^{-1}g_m^{-1}K$ as $\\gamma_m$ is an isomorphism.", "Working out what the above says,", "but looking only at this direct summand we conclude the lower", "middle square commutes as we well. This concludes the proof." ], "refs": [ "spaces-simplicial-lemma-simplicial-resolution-Z-site", "derived-lemma-existence-postnikov-system", "spaces-simplicial-lemma-simplicial-resolution-Z-site", "spaces-simplicial-lemma-abelian-postnikov", "derived-lemma-existence-postnikov-system" ], "ref_ids": [ 9037, 1952, 9037, 9061, 1952 ] } ], "ref_ids": [] }, { "id": 9065, "type": "theorem", "label": "spaces-simplicial-lemma-cartesian-objects-derived-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-cartesian-objects-derived-modules", "contents": [ "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be a", "sheaf of rings on $\\mathcal{C}_{total}$.", "If $K \\in D(\\mathcal{O})$ is an object, then $(K_n, K(\\varphi))$", "is a simplicial system of the derived category of modules.", "If $K$ is cartesian, so is the system." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9066, "type": "theorem", "label": "spaces-simplicial-lemma-modules-postnikov", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-modules-postnikov", "contents": [ "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$", "be a sheaf of rings on $\\mathcal{C}_{total}$. Let $K$ be", "an object of $D(\\mathcal{C}_{total})$. Set", "$$", "X_n = (g_{n!}\\mathcal{O}_n)", "\\otimes^\\mathbf{L}_\\mathcal{O} K", "\\quad\\text{and}\\quad", "Y_n =", "(g_{n!}\\mathcal{O}_n \\to \\ldots \\to g_{0!}\\mathcal{O}_0)[-n]", "\\otimes^\\mathbf{L}_\\mathcal{O} K", "$$", "as objects of $D(\\mathcal{O})$ where the maps are", "as in Lemma \\ref{lemma-simplicial-resolution-Z-site}.", "With the evident canonical maps $Y_n \\to X_n$ and", "$Y_0 \\to Y_1[1] \\to Y_2[2] \\to \\ldots$ we have", "\\begin{enumerate}", "\\item the distinguished triangles $Y_n \\to X_n \\to Y_{n - 1} \\to Y_n[1]$", "define a Postnikov system", "(Derived Categories, Definition \\ref{derived-definition-postnikov-system})", "for $\\ldots \\to X_2 \\to X_1 \\to X_0$,", "\\item $K = \\text{hocolim} Y_n[n]$ in $D(\\mathcal{O})$.", "\\end{enumerate}" ], "refs": [ "spaces-simplicial-lemma-simplicial-resolution-Z-site", "derived-definition-postnikov-system" ], "proofs": [ { "contents": [ "First, if $K = \\mathcal{O}$, then this is the construction of", "Derived Categories, Example \\ref{derived-example-key-postnikov}", "applied to the complex", "$$", "\\ldots \\to", "g_{2!}\\mathcal{O}_2 \\to", "g_{1!}\\mathcal{O}_1 \\to", "g_{0!}\\mathcal{O}_0", "$$", "in $\\textit{Ab}(\\mathcal{C}_{total})$ combined with the fact that", "this complex represents $K = \\mathcal{O}$ in $D(\\mathcal{C}_{total})$", "by Lemma \\ref{lemma-simplicial-resolution-ringed}.", "The general case follows from this, the fact that the exact functor", "$- \\otimes^\\mathbf{L}_\\mathcal{O} K$ sends Postnikov systems to", "Postnikov systems, and", "that $- \\otimes^\\mathbf{L}_\\mathcal{O} K$ commutes with homotopy colimits." ], "refs": [ "spaces-simplicial-lemma-simplicial-resolution-ringed" ], "ref_ids": [ 9044 ] } ], "ref_ids": [ 9037, 2008 ] }, { "id": 9067, "type": "theorem", "label": "spaces-simplicial-lemma-nullity-cartesian-modules-derived", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-nullity-cartesian-modules-derived", "contents": [ "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be", "a sheaf of rings on $\\mathcal{C}_{total}$.", "If $K, K' \\in D(\\mathcal{O})$.", "Assume", "\\begin{enumerate}", "\\item $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$ is flat for", "$\\varphi : [m] \\to [n]$,", "\\item $K$ is cartesian,", "\\item $\\Hom(K_i[i], K'_i) = 0$ for $i > 0$, and", "\\item $\\Hom(K_i[i + 1], K'_i) = 0$ for $i \\geq 0$.", "\\end{enumerate}", "Then any map $K \\to K'$ which induces the zero map $K_0 \\to K'_0$ is zero." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Lemma \\ref{lemma-nullity-cartesian-objects-derived} except using", "Lemma \\ref{lemma-modules-postnikov} instead of", "Lemma \\ref{lemma-abelian-postnikov}." ], "refs": [ "spaces-simplicial-lemma-nullity-cartesian-objects-derived", "spaces-simplicial-lemma-modules-postnikov", "spaces-simplicial-lemma-abelian-postnikov" ], "ref_ids": [ 9062, 9066, 9061 ] } ], "ref_ids": [] }, { "id": 9068, "type": "theorem", "label": "spaces-simplicial-lemma-hom-cartesian-modules-derived", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hom-cartesian-modules-derived", "contents": [ "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be", "a sheaf of rings on $\\mathcal{C}_{total}$.", "If $K, K' \\in D(\\mathcal{O})$.", "Assume", "\\begin{enumerate}", "\\item $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$ is flat for", "$\\varphi : [m] \\to [n]$,", "\\item $K$ is cartesian,", "\\item $\\Hom(K_i[i - 1], K'_i) = 0$ for $i > 1$.", "\\end{enumerate}", "Then any map $\\{K_n \\to K'_n\\}$ between the associated simplicial systems ", "of $K$ and $K'$ comes from a map $K \\to K'$ in $D(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Lemma \\ref{lemma-hom-cartesian-objects-derived} except using", "Lemma \\ref{lemma-modules-postnikov} instead of", "Lemma \\ref{lemma-abelian-postnikov}." ], "refs": [ "spaces-simplicial-lemma-hom-cartesian-objects-derived", "spaces-simplicial-lemma-modules-postnikov", "spaces-simplicial-lemma-abelian-postnikov" ], "ref_ids": [ 9063, 9066, 9061 ] } ], "ref_ids": [] }, { "id": 9069, "type": "theorem", "label": "spaces-simplicial-lemma-cartesian-module-derived-from-simplicial", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-cartesian-module-derived-from-simplicial", "contents": [ "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be", "a sheaf of rings on $\\mathcal{C}_{total}$. Let", "$(K_n, K_\\varphi)$ be a simplicial system of the derived category", "of modules. Assume", "\\begin{enumerate}", "\\item $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$ is flat for", "$\\varphi : [m] \\to [n]$,", "\\item $(K_n, K_\\varphi)$ is cartesian,", "\\item $\\Hom(K_i[t], K_i) = 0$ for $i \\geq 0$ and $t > 0$.", "\\end{enumerate}", "Then there exists a cartesian object $K$ of $D(\\mathcal{O})$", "whose associated simplicial system is isomorphic to $(K_n, K_\\varphi)$." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Lemma \\ref{lemma-cartesian-object-derived-from-simplicial}", "with the following changes", "\\begin{enumerate}", "\\item use $g_n^* = Lg_n^*$ everywhere instead of $g_n^{-1}$,", "\\item use $f_\\varphi^* = Lf_\\varphi^*$ everywhere instead of $f_\\varphi^{-1}$,", "\\item refer to Lemma \\ref{lemma-simplicial-resolution-ringed}", "instead of Lemma \\ref{lemma-simplicial-resolution-Z-site},", "\\item in the construction of $Y'_{m, n}$ use", "$\\mathcal{O}_m$ instead of $\\mathbf{Z}$,", "\\item compare with the proof of Lemma \\ref{lemma-modules-postnikov}", "rather than the proof of Lemma \\ref{lemma-abelian-postnikov}.", "\\end{enumerate}", "This ends the proof." ], "refs": [ "spaces-simplicial-lemma-cartesian-object-derived-from-simplicial", "spaces-simplicial-lemma-simplicial-resolution-ringed", "spaces-simplicial-lemma-simplicial-resolution-Z-site", "spaces-simplicial-lemma-modules-postnikov", "spaces-simplicial-lemma-abelian-postnikov" ], "ref_ids": [ 9064, 9044, 9037, 9066, 9061 ] } ], "ref_ids": [] }, { "id": 9070, "type": "theorem", "label": "spaces-simplicial-lemma-has-P", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-has-P", "contents": [ "Let $\\mathcal{C}$ be a site.", "\\begin{enumerate}", "\\item For $K$ in $\\text{SR}(\\mathcal{C})$ the functor", "$j : \\mathcal{C}/K \\to \\mathcal{C}$ is continuous,", "cocontinuous, and has property P of", "Sites, Remark \\ref{sites-remark-cartesian-cocontinuous}.", "\\item For $f : K \\to L$ in $\\text{SR}(\\mathcal{C})$", "the functor $v : \\mathcal{C}/K \\to \\mathcal{C}/L$ (see above)", "is continuous, cocontinuous, and has property P of", "Sites, Remark \\ref{sites-remark-cartesian-cocontinuous}.", "\\end{enumerate}" ], "refs": [ "sites-remark-cartesian-cocontinuous", "sites-remark-cartesian-cocontinuous" ], "proofs": [ { "contents": [ "Proof of (2). In the notation of the discussion preceding the lemma,", "the localization functors $\\mathcal{C}/U_i \\to \\mathcal{C}/V_{\\alpha(i)}$", "are continuous and cocontinuous by", "Sites, Section \\ref{sites-section-localize}", "and satisfy $P$ by", "Sites, Remark \\ref{sites-remark-localization-cartesian-cocontinuous}.", "It is formal to deduce $v$ is continuous and cocontinuous and has $P$.", "We omit the details. We also omit the proof of (1)." ], "refs": [ "sites-remark-localization-cartesian-cocontinuous" ], "ref_ids": [ 8714 ] } ], "ref_ids": [ 8712, 8712 ] }, { "id": 9071, "type": "theorem", "label": "spaces-simplicial-lemma-push-pull-localization", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-push-pull-localization", "contents": [ "Let $\\mathcal{C}$ be a site and $K$ in $\\text{SR}(\\mathcal{C})$.", "For $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$ we have", "$$", "j_*j^{-1}\\mathcal{F} = \\SheafHom(F(K)^\\#, \\mathcal{F})", "$$", "where $F$ is as in", "Hypercoverings, Definition \\ref{hypercovering-definition-SR-F}." ], "refs": [ "hypercovering-definition-SR-F" ], "proofs": [ { "contents": [ "Say $K = \\{U_i\\}_{i \\in I}$.", "Using the description of the functors $j^{-1}$ and $j_*$", "given above we see that", "$$", "j_*j^{-1}\\mathcal{F} =", "\\prod\\nolimits_{i \\in I} j_{i, *}(\\mathcal{F}|_{\\mathcal{C}/U_i}) =", "\\prod\\nolimits_{i \\in I} \\SheafHom(h_{U_i}^\\#, \\mathcal{F})", "$$", "The second equality by Sites, Lemma \\ref{sites-lemma-hom-sheaf-hU}.", "Since $F(K) = \\coprod h_{U_i}$ in $\\textit{PSh}(\\mathcal{C}$,", "we have $F(K)^\\# = \\coprod h_{U_i}^\\#$ in $\\Sh(\\mathcal{C})$", "and since $\\SheafHom(-, \\mathcal{F})$ turns coproducts into", "products (immediate from the construction in", "Sites, Section \\ref{sites-section-glueing-sheaves}), we conclude." ], "refs": [ "sites-lemma-hom-sheaf-hU" ], "ref_ids": [ 8563 ] } ], "ref_ids": [ 8422 ] }, { "id": 9072, "type": "theorem", "label": "spaces-simplicial-lemma-localize-compare", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-localize-compare", "contents": [ "Let $\\mathcal{C}$ be a site.", "\\begin{enumerate}", "\\item For $K$ in $\\text{SR}(\\mathcal{C})$ the functor $j_!$", "gives an equivalence $\\Sh(\\mathcal{C}/K) \\to \\Sh(\\mathcal{C})/F(K)^\\#$", "where $F$ is as in", "Hypercoverings, Definition \\ref{hypercovering-definition-SR-F}.", "\\item The functor $j^{-1} : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}/K)$", "corresponds via the identification of (1) with", "$\\mathcal{F} \\mapsto (\\mathcal{F} \\times F(K)^\\# \\to F(K)^\\#)$.", "\\item For $f : K \\to L$ in $\\text{SR}(\\mathcal{C})$ the functor", "$f^{-1}$ corresponds via the identifications of (1) to the functor", "$\\Sh(\\mathcal{C})/F(L)^\\# \\to \\Sh(\\mathcal{C})/F(K)^\\#$,", "$(\\mathcal{G} \\to F(L)^\\#) \\mapsto", "(\\mathcal{G} \\times_{F(L)^\\#} F(K)^\\# \\to F(K)^\\#)$.", "\\end{enumerate}" ], "refs": [ "hypercovering-definition-SR-F" ], "proofs": [ { "contents": [ "Observe that if $K = \\{U_i\\}_{i \\in I}$ then the category", "$\\Sh(\\mathcal{C}/K)$ decomposes as the product of the categories", "$\\Sh(\\mathcal{C}/U_i)$. Observe that", "$F(K)^\\# = \\coprod_{i \\in I} h_{U_i}^\\#$ (coproduct in sheaves).", "Hence $\\Sh(\\mathcal{C})/F(K)^\\#$ is the product of the", "categories $\\Sh(\\mathcal{C})/h_{U_i}^\\#$.", "Thus (1) and (2) follow from the corresponding", "statements for each $i$, see", "Sites, Lemmas \\ref{sites-lemma-essential-image-j-shriek} and", "\\ref{sites-lemma-compute-j-shriek-restrict}.", "Similarly, if $L = \\{V_j\\}_{j \\in J}$ and $f$ is given", "by $\\alpha : I \\to J$ and $f_i : U_i \\to V_{\\alpha(i)}$,", "then we can apply", "Sites, Lemma \\ref{sites-lemma-relocalize-explicit}", "to each of the re-localization morphisms", "$\\mathcal{C}/U_i \\to \\mathcal{C}/V_{\\alpha(i)}$", "to get (3)." ], "refs": [ "sites-lemma-essential-image-j-shriek", "sites-lemma-compute-j-shriek-restrict", "sites-lemma-relocalize-explicit" ], "ref_ids": [ 8555, 8558, 8560 ] } ], "ref_ids": [ 8422 ] }, { "id": 9073, "type": "theorem", "label": "spaces-simplicial-lemma-localize-injective", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-localize-injective", "contents": [ "Let $\\mathcal{C}$ be a site. For $K$ in $\\text{SR}(\\mathcal{C})$", "the functor $j^{-1}$ sends injective abelian sheaves to injective", "abelian sheaves. Similarly, the functor $j^{-1}$ sends K-injective", "complexes of abelian sheaves to K-injective complexes of", "abelian sheaves." ], "refs": [], "proofs": [ { "contents": [ "The first statement is the natural generalization of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-cohomology-of-open}", "to semi-representable objects.", "In fact, it follows from this lemma", "by the product decomposition of $\\Sh(\\mathcal{C}/K)$", "and the description of the functor $j^{-1}$ given above.", "The second statement is the natural generalization of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-restrict-K-injective-to-open}", "and follows from it by the product decomposition of the topos.", "\\medskip\\noindent", "Alternative: since $j$ induces a localization of topoi by", "Lemma \\ref{lemma-localize-compare} part (1)", "it also follows immediately from", "Cohomology on Sites, Lemmas \\ref{sites-cohomology-lemma-cohomology-of-open}", "and \\ref{sites-cohomology-lemma-restrict-K-injective-to-open}", "by enlarging the site; compare with the proof of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-cohomology-on-sheaf-sets}", "in the case of injective sheaves." ], "refs": [ "sites-cohomology-lemma-cohomology-of-open", "sites-cohomology-lemma-restrict-K-injective-to-open", "spaces-simplicial-lemma-localize-compare", "sites-cohomology-lemma-cohomology-of-open", "sites-cohomology-lemma-restrict-K-injective-to-open", "sites-cohomology-lemma-cohomology-on-sheaf-sets" ], "ref_ids": [ 4186, 4253, 9072, 4186, 4253, 4215 ] } ], "ref_ids": [] }, { "id": 9074, "type": "theorem", "label": "spaces-simplicial-lemma-augmentation-simplicial-semi-representable", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-augmentation-simplicial-semi-representable", "contents": [ "Let $\\mathcal{C}$ be a site. Let $K$ be a simplicial object of", "$\\text{SR}(\\mathcal{C})$. The localization functor", "$j_0 : \\mathcal{C}/K_0 \\to \\mathcal{C}$ defines an augmentation", "$a_0 : \\Sh(\\mathcal{C}/K_0) \\to \\Sh(\\mathcal{C})$, as in case (B) of", "Remark \\ref{remark-augmentation-site}.", "The corresponding morphisms of topoi", "$$", "a_n : \\Sh(\\mathcal{C}/K_n) \\longrightarrow \\Sh(\\mathcal{C}),\\quad", "a : \\Sh((\\mathcal{C}/K)_{total}) \\longrightarrow \\Sh(\\mathcal{C})", "$$", "of Lemma \\ref{lemma-augmentation-site}", "are equal to the morphisms of topoi associated to the", "continuous and cocontinuous localization functors", "$j_n : \\mathcal{C}/K_n \\to \\mathcal{C}$ and", "$j_{total} : (\\mathcal{C}/K)_{total} \\to \\mathcal{C}$." ], "refs": [ "spaces-simplicial-remark-augmentation-site", "spaces-simplicial-lemma-augmentation-site" ], "proofs": [ { "contents": [ "This is immediate from working through the definitions.", "See in particular the footnote in the proof of", "Lemma \\ref{lemma-augmentation-site}", "for the relationship between $a$ and $j_{total}$." ], "refs": [ "spaces-simplicial-lemma-augmentation-site" ], "ref_ids": [ 9027 ] } ], "ref_ids": [ 9152, 9027 ] }, { "id": 9075, "type": "theorem", "label": "spaces-simplicial-lemma-comparison", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-comparison", "contents": [ "With assumption and notation as in", "Lemma \\ref{lemma-augmentation-simplicial-semi-representable}", "we have the following properties:", "\\begin{enumerate}", "\\item there is a functor", "$a^{Sh}_! : \\Sh((\\mathcal{C}/K)_{total}) \\to \\Sh(\\mathcal{C})$", "left adjoint to $a^{-1} : \\Sh(\\mathcal{C}) \\to \\Sh((\\mathcal{C}/K)_{total})$,", "\\item there is a functor", "$a_! : \\textit{Ab}((\\mathcal{C}/K)_{total}) \\to \\textit{Ab}(\\mathcal{C})$", "left adjoint to", "$a^{-1} : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}((\\mathcal{C}/K)_{total})$,", "\\item the functor $a^{-1}$ associates to", "$\\mathcal{F}$ in $\\Sh(\\mathcal{C})$ the sheaf on $(\\mathcal{C}/K)_{total}$", "wich in degree $n$ is equal to $a_n^{-1}\\mathcal{F}$,", "\\item the functor $a_*$ associates to $\\mathcal{G}$ in", "$\\textit{Ab}((\\mathcal{C}/K)_{total})$ the equalizer of the two maps", "$j_{0, *}\\mathcal{G}_0 \\to j_{1, *}\\mathcal{G}_1$,", "\\end{enumerate}" ], "refs": [ "spaces-simplicial-lemma-augmentation-simplicial-semi-representable" ], "proofs": [ { "contents": [ "Parts (3) and (4) hold for any augmentation of a", "simplicial site, see Lemma \\ref{lemma-augmentation-site}.", "Parts (1) and (2) follow as $j_{total}$ is continuous and cocontinuous.", "The functor $a^{Sh}_!$ is constructed in", "Sites, Lemma \\ref{sites-lemma-when-shriek}", "and the functor $a_!$ is constructed in", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-g-shriek-adjoint}." ], "refs": [ "spaces-simplicial-lemma-augmentation-site", "sites-lemma-when-shriek", "sites-modules-lemma-g-shriek-adjoint" ], "ref_ids": [ 9027, 8545, 14164 ] } ], "ref_ids": [ 9074 ] }, { "id": 9076, "type": "theorem", "label": "spaces-simplicial-lemma-sanity-check-simplicial-semi-representable", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-sanity-check-simplicial-semi-representable", "contents": [ "Let $\\mathcal{C}$ be a site. Let $K$ be a simplicial object of", "$\\text{SR}(\\mathcal{C})$. Let $U/U_{n, i}$ be an object of", "$\\mathcal{C}/K_n$. Let", "$\\mathcal{F} \\in \\textit{Ab}((\\mathcal{C}/K)_{total})$.", "Then", "$$", "H^p(U, \\mathcal{F}) = H^p(U, \\mathcal{F}_{n, i})", "$$", "where", "\\begin{enumerate}", "\\item on the left hand side $U$ is viewed as an object of", "$\\mathcal{C}_{total}$, and", "\\item on the right hand side $\\mathcal{F}_{n, i}$ is the $i$th", "component of the sheaf $\\mathcal{F}_n$ on $\\mathcal{C}/K_n$", "in the decomposition $\\Sh(\\mathcal{C}/K_n) = \\prod \\Sh(\\mathcal{C}/U_{n, i})$", "of Section \\ref{section-semi-representable}.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from Lemma \\ref{lemma-sanity-check}", "and the product decompositions of Section \\ref{section-semi-representable}." ], "refs": [ "spaces-simplicial-lemma-sanity-check" ], "ref_ids": [ 9040 ] } ], "ref_ids": [] }, { "id": 9077, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-descent-sheaves", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-descent-sheaves", "contents": [ "Let $\\mathcal{C}$ be a site with equalizers and fibre products.", "Let $K$ be a hypercovering. Then", "\\begin{enumerate}", "\\item $a^{-1} : \\Sh(\\mathcal{C}) \\to \\Sh((\\mathcal{C}/K)_{total})$", "is fully faithful with essential image the cartesian sheaves of sets,", "\\item $a^{-1} : \\textit{Ab}(\\mathcal{C}) \\to", "\\textit{Ab}((\\mathcal{C}/K)_{total})$", "is fully faithful with essential image the cartesian sheaves", "of abelian groups.", "\\end{enumerate}", "In both cases $a_*$ provides the quasi-inverse functor." ], "refs": [], "proofs": [ { "contents": [ "The case of abelian sheaves follows immediately from the case", "of sheaves of sets as the functor $a^{-1}$ commutes with products.", "In the rest of the proof we work with sheaves of sets.", "Observe that $a^{-1}\\mathcal{F}$ is cartesian for", "$\\mathcal{F}$ in $\\Sh(\\mathcal{C})$ by", "Lemma \\ref{lemma-augmentation-cartesian-module}.", "It suffices to show that the adjunction map", "$\\mathcal{F} \\to a_*a^{-1}\\mathcal{F}$", "is an isomorphism $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$", "and that for a cartesian sheaf", "$\\mathcal{G}$ on $(\\mathcal{C}/K)_{total}$", "the adjunction map", "$a^{-1}a_*\\mathcal{G} \\to \\mathcal{G}$ is an isomorphism.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.", "Recall that $a_*a^{-1}\\mathcal{F}$ is the equalizer", "of the two maps $a_{0, *}a_0^{-1}\\mathcal{F} \\to a_{1, *}a_1^{-1}\\mathcal{F}$,", "see Lemma \\ref{lemma-comparison}.", "By Lemma \\ref{lemma-push-pull-localization}", "$$", "a_{0, *}a_0^{-1}\\mathcal{F} = \\SheafHom(F(K_0)^\\#, \\mathcal{F})", "\\quad\\text{and}\\quad", "a_{1, *}a_1^{-1}\\mathcal{F} = \\SheafHom(F(K_1)^\\#, \\mathcal{F})", "$$", "On the other hand, we know that", "$$", "\\xymatrix{", "F(K_1)^\\# \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "F(K_0)^\\# \\ar[r] & \\text{final object }*\\text{ of }\\Sh(\\mathcal{C})", "}", "$$", "is a coequalizer diagram in sheaves of sets by definition of", "a hypercovering. Thus it suffices to prove", "that $\\SheafHom(-, \\mathcal{F})$ transforms coequalizers", "into equalizers which is immediate from the construction", "in Sites, Section \\ref{sites-section-glueing-sheaves}.", "\\medskip\\noindent", "Let $\\mathcal{G}$ be a cartesian sheaf on $(\\mathcal{C}/K)_{total}$.", "We will show that $\\mathcal{G} = a^{-1}\\mathcal{F}$ for some sheaf", "$\\mathcal{F}$ on $\\mathcal{C}$. This will finish the proof because", "then $a^{-1}a_*\\mathcal{G} = a^{-1}a_*a^{-1}\\mathcal{F} =", "a^{-1}\\mathcal{F} = \\mathcal{G}$ by the result of the previous paragraph.", "Set $\\mathcal{K}_n = F(K_n)^\\#$ for $n \\geq 0$. Then we have maps of sheaves", "$$", "\\xymatrix{", "\\mathcal{K}_2", "\\ar@<1ex>[r]", "\\ar@<0ex>[r]", "\\ar@<-1ex>[r]", "&", "\\mathcal{K}_1", "\\ar@<0.5ex>[r]", "\\ar@<-0.5ex>[r]", "&", "\\mathcal{K}_0", "}", "$$", "coming from the fact that $K$ is a simplicial semi-representable object.", "The fact that $K$ is a hypercovering means that", "$$", "\\mathcal{K}_1 \\to \\mathcal{K}_0 \\times \\mathcal{K}_0", "\\quad\\text{and}\\quad", "\\mathcal{K}_2 \\to", "\\left(\\text{cosq}_1(", "\\xymatrix{", "\\mathcal{K}_1", "\\ar@<0.5ex>[r]", "\\ar@<-0.5ex>[r]", "&", "\\mathcal{K}_0 \\ar[l]", "})\\right)_2", "$$", "are surjective maps of sheaves. Using the description of cartesian sheaves on", "$(\\mathcal{C}/K)_{total}$ given in Lemma \\ref{lemma-characterize-cartesian}", "and using the description of $\\Sh(\\mathcal{C}/K_n)$ in", "Lemma \\ref{lemma-localize-compare}", "we find that our problem can be entirely formulated\\footnote{Even though it", "does not matter what the precise formulation is, we spell it out:", "the problem is to show that given an object", "$\\mathcal{G}_0/\\mathcal{K}_0$ of $\\Sh(\\mathcal{C})/\\mathcal{K}_0$", "and an isomorphism", "$$", "\\alpha :", "\\mathcal{G}_0 \\times_{\\mathcal{K}_0, \\mathcal{K}(\\delta^1_1)} \\mathcal{K}_1 \\to", "\\mathcal{G}_0 \\times_{\\mathcal{K}_0, \\mathcal{K}(\\delta^1_0)} \\mathcal{K}_1", "$$", "over $\\mathcal{K}_1$ satisfying a cocycle condtion in", "$\\Sh(\\mathcal{C})/\\mathcal{K}_2$, there exists", "$\\mathcal{F}$ in $\\Sh(\\mathcal{C})$ and an isomorphism", "$\\mathcal{F} \\times \\mathcal{K}_0 \\to \\mathcal{G}_0$ over $\\mathcal{K}_0$", "compatible with $\\alpha$.} in terms of", "\\begin{enumerate}", "\\item the topos $\\Sh(\\mathcal{C})$, and", "\\item the simplicial object $\\mathcal{K}$ in $\\Sh(\\mathcal{C})$", "whose terms are $\\mathcal{K}_n$.", "\\end{enumerate}", "Thus, after replacing $\\mathcal{C}$ by a different site $\\mathcal{C}'$", "as in Sites, Lemma \\ref{sites-lemma-topos-good-site}, we may assume", "$\\mathcal{C}$ has all finite limits,", "the topology on $\\mathcal{C}$ is subcanonical,", "a family $\\{V_j \\to V\\}$ of morphisms of $\\mathcal{C}$", "is a covering if and only if $\\coprod h_{V_j} \\to V$ is surjective, and", "there exists a simplicial object $U$ of $\\mathcal{C}$", "such that $\\mathcal{K}_n = h_{U_n}$ as simplicial sheaves.", "Working backwards through the equivalences we may assume", "$K_n = \\{U_n\\}$ for all $n$.", "\\medskip\\noindent", "Let $X$ be the final object of $\\mathcal{C}$.", "Then $\\{U_0 \\to X\\}$ is a covering,", "$\\{U_1 \\to U_0 \\times U_0\\}$ is a covering, and", "$\\{U_2 \\to (\\text{cosq}_1 \\text{sk}_1 U)_2\\}$ is a covering.", "Let us use $d^n_i : U_n \\to U_{n - 1}$ and", "$s^n_j : U_n \\to U_{n + 1}$ the morphisms corresponding", "to $\\delta^n_i$ and $\\sigma^n_j$ as in", "Simplicial, Definition \\ref{simplicial-definition-face-degeneracy}.", "By abuse of notation, given a morphism", "$c : V \\to W$ of $\\mathcal{C}$ we denote the morphism of topoi", "$c : \\Sh(\\mathcal{C}/V) \\to \\Sh(\\mathcal{C}/W)$ by the same letter.", "Now $\\mathcal{G}$ is given by a sheaf $\\mathcal{G}_0$", "on $\\mathcal{C}/U_0$ and an isomorphism", "$\\alpha : (d^1_1)^{-1}\\mathcal{G}_0 \\to (d^1_0)^{-1}\\mathcal{G}_0$", "satisfying the cocycle condition on $\\mathcal{C}/U_2$", "formulated in Lemma \\ref{lemma-characterize-cartesian}.", "Since $\\{U_2 \\to (\\text{cosq}_1 \\text{sk}_1 U)_2\\}$", "is a covering, the corresponding pullback functor", "on sheaves is faithful (small detail omitted).", "Hence we may replace $U$ by $\\text{cosk}_1 \\text{sk}_1 U$, because", "this replaces $U_2$ by $(\\text{cosq}_1 \\text{sk}_1 U)_2$ and leaves", "$U_1$ and $U_0$ unchanged. Then", "$$", "(d^2_0, d^2_1, d^2_2) : U_2 \\to U_1 \\times U_1 \\times U_1", "$$", "is a monomorphism whose its image on $T$-valued points is", "described in Simplicial, Lemma \\ref{simplicial-lemma-work-out}.", "In particular, there is a morphism $c$ fitting into a commutative diagram", "$$", "\\xymatrix{", "U_1 \\times_{(d^1_1, d^1_0), U_0 \\times U_0, (d^1_1, d^1_0)} U_1", "\\ar[d] \\ar[rr]_c & & U_2 \\ar[d] \\\\", "U_1 \\times U_1", "\\ar[rr]^{(\\text{pr}_1, \\text{pr}_2, s^0_0 \\circ d^1_1 \\circ \\text{pr}_1)} & &", "U_1 \\times U_1 \\times U_1", "}", "$$", "as going around the other way defines a point of $U_2$.", "Pulling back the cocycle condition for $\\alpha$ on $U_2$", "translates into the condition that the pullbacks of $\\alpha$", "via the projections to", "$U_1 \\times_{(d^1_1, d^1_0), U_0 \\times U_0, (d^1_1, d^1_0)} U_1$", "are the same as the pullback of $\\alpha$ via", "$s^0_0 \\circ d^1_1 \\circ \\text{pr}_1$ is the identity map", "(namely, the pullback of $\\alpha$ by $s^0_0$ is the identity).", "By Sites, Lemma \\ref{sites-lemma-glue-maps}", "this means that $\\alpha$ comes from an isomorphism", "$$", "\\alpha' : \\text{pr}_1^{-1}\\mathcal{G}_0 \\to \\text{pr}_2^{-1}\\mathcal{G}_0", "$$", "of sheaves on $\\mathcal{C}/U_0 \\times U_0$.", "Then finally, the morphism $U_2 \\to U_0 \\times U_0 \\times U_0$", "is surjective on associated sheaves as is easily seen using the", "surjectivity of $U_1 \\to U_0 \\times U_0$", "and the description of $U_2$ given above. Therefore $\\alpha'$", "satisfies the cocycle condition on $U_0 \\times U_0 \\times U_0$.", "The proof is finished by an application of", "Sites, Lemma \\ref{sites-lemma-mapping-property-glue}", "to the covering $\\{U_0 \\to X\\}$." ], "refs": [ "spaces-simplicial-lemma-augmentation-cartesian-module", "spaces-simplicial-lemma-comparison", "spaces-simplicial-lemma-push-pull-localization", "spaces-simplicial-lemma-characterize-cartesian", "spaces-simplicial-lemma-localize-compare", "sites-lemma-topos-good-site", "simplicial-definition-face-degeneracy", "spaces-simplicial-lemma-characterize-cartesian", "simplicial-lemma-work-out", "sites-lemma-glue-maps", "sites-lemma-mapping-property-glue" ], "ref_ids": [ 9053, 9075, 9071, 9054, 9072, 8581, 14910, 9054, 14839, 8561, 8565 ] } ], "ref_ids": [] }, { "id": 9078, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-cech-complex", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-cech-complex", "contents": [ "Let $\\mathcal{C}$ be a site with equalizers and fibre products.", "Let $K$ be a hypercovering. The {\\v C}ech complex", "of Lemma \\ref{lemma-augmentation-cech-complex} associated to", "$a^{-1}\\mathcal{F}$", "$$", "a_{0, *}a_0^{-1}\\mathcal{F} \\to a_{1, *}a_1^{-1}\\mathcal{F} \\to", "a_{2, *}a_2^{-1}\\mathcal{F} \\to \\ldots", "$$", "is equal to the complex $\\SheafHom(s(\\mathbf{Z}_{F(K)}^\\#), \\mathcal{F})$.", "Here $s(\\mathbf{Z}_{F(K)}^\\#)$ is as in", "Hypercoverings, Definition \\ref{hypercovering-definition-homology}." ], "refs": [ "spaces-simplicial-lemma-augmentation-cech-complex", "hypercovering-definition-homology" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-push-pull-localization} we have", "$$", "a_{n, *}a_n^{-1}\\mathcal{F} = \\SheafHom'(F(K_n)^\\#, \\mathcal{F})", "$$", "where $\\SheafHom'$ is as in Sites, Section \\ref{sites-section-glueing-sheaves}.", "The boundary maps in the complex of", "Lemma \\ref{lemma-augmentation-cech-complex}", "come from the simplicial structure.", "Thus the equality of complexes comes ", "from the canonical identifications", "$\\SheafHom'(\\mathcal{G}, \\mathcal{F}) =", "\\SheafHom(\\mathbf{Z}_\\mathcal{G}, \\mathcal{F})$ for", "$\\mathcal{G}$ in $\\Sh(\\mathcal{C})$." ], "refs": [ "spaces-simplicial-lemma-push-pull-localization", "spaces-simplicial-lemma-augmentation-cech-complex" ], "ref_ids": [ 9071, 9042 ] } ], "ref_ids": [ 9042, 8425 ] }, { "id": 9079, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-descent-bounded-abelian", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-descent-bounded-abelian", "contents": [ "Let $\\mathcal{C}$ be a site with equalizers and fibre products.", "Let $K$ be a hypercovering. For", "$E \\in D(\\mathcal{C})$ the map", "$$", "E \\longrightarrow Ra_*a^{-1}E", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "First, let $\\mathcal{I}$ be an injective abelian sheaf on $\\mathcal{C}$.", "Then the spectral sequence of", "Lemma \\ref{lemma-augmentation-spectral-sequence}", "for the sheaf $a^{-1}\\mathcal{I}$ degenerates as", "$(a^{-1}\\mathcal{I})_p = a_p^{-1}\\mathcal{I}$", "is injective by Lemma \\ref{lemma-localize-injective}.", "Thus the complex", "$$", "a_{0, *}a_0^{-1}\\mathcal{I} \\to", "a_{1, *}a_1^{-1}\\mathcal{I} \\to", "a_{2, *}a_2^{-1}\\mathcal{I} \\to\\ldots", "$$", "computes $Ra_*a^{-1}\\mathcal{I}$. By", "Lemma \\ref{lemma-hypercovering-cech-complex}", "this is equal to the complex", "$\\SheafHom(s(\\mathbf{Z}_{F(K)}^\\#), \\mathcal{I})$.", "Because $K$ is a hypercovering, we see that", "$s(\\mathbf{Z}_{F(K)}^\\#)$ is exact in degrees $> 0$ by", "Hypercoverings, Lemma \\ref{hypercovering-lemma-acyclic-hypercover-sheaves}", "applied to the simplicial presheaf $F(K)$.", "Since $\\mathcal{I}$ is injective, the functor $\\SheafHom(-, \\mathcal{I})$", "is exact and we conclude that", "$\\SheafHom(s(\\mathbf{Z}_{F(K)}^\\#), \\mathcal{I})$", "is exact in positive degrees. We conclude that", "$R^pa_*a^{-1}\\mathcal{I} = 0$ for $p > 0$.", "On the other hand, we have $\\mathcal{I} = a_*a^{-1}\\mathcal{I}$", "by Lemma \\ref{lemma-hypercovering-descent-sheaves}.", "\\medskip\\noindent", "Bounded case. Let $E \\in D^+(\\mathcal{C})$.", "Choose a bounded below complex $\\mathcal{I}^\\bullet$ of injectives", "representing $E$. By the result of the first paragraph and", "Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "$Ra_*a^{-1}\\mathcal{I}^\\bullet$", "is computed by the complex", "$a_*a^{-1}\\mathcal{I}^\\bullet = \\mathcal{I}^\\bullet$", "and we conclude the lemma is true in this case.", "\\medskip\\noindent", "Unbounded case. We urge the reader to skip this, since the argument", "is the same as above, except that we use explicit representation", "by double complexes to get around convergence issues.", "Let $E \\in D(\\mathcal{C})$.", "To show the map $E \\to Ra_*a^{-1}E$ is an isomorphism,", "it suffices to show for every object $U$ of $\\mathcal{C}$ that", "$$", "R\\Gamma(U, E) = R\\Gamma(U, Ra_*a^{-1}E)", "$$", "We will compute both sides and show the map $E \\to Ra_*a^{-1}E$", "induces an isomorphism. Choose a K-injective", "complex $\\mathcal{I}^\\bullet$ representing $E$. Choose a quasi-isomorphism", "$a^{-1}\\mathcal{I}^\\bullet \\to \\mathcal{J}^\\bullet$", "for some K-injective complex $\\mathcal{J}^\\bullet$ on", "$(\\mathcal{C}/K)_{total}$. We have", "$$", "R\\Gamma(U, E) = R\\Hom(\\mathbf{Z}_U^\\#, E)", "$$", "and", "$$", "R\\Gamma(U, Ra_*a^{-1}E) = R\\Hom(\\mathbf{Z}_U^\\#, Ra_*a^{-1}E) =", "R\\Hom(a^{-1}\\mathbf{Z}_U^\\#, a^{-1}E)", "$$", "By Lemma \\ref{lemma-simplicial-resolution-augmentation}", "we have a quasi-isomorphism", "$$", "\\Big(\\ldots \\to", "g_{2!}(a_2^{-1}\\mathbf{Z}_U^\\#) \\to", "g_{1!}(a_1^{-1}\\mathbf{Z}_U^\\#) \\to", "g_{0!}(a_0^{-1}\\mathbf{Z}_U^\\#)\\Big)", "\\longrightarrow", "a^{-1}\\mathbf{Z}_U^\\#", "$$", "Hence $R\\Hom(a^{-1}\\mathbf{Z}_U^\\#, a^{-1}E)$ is equal to", "$$", "R\\Gamma((\\mathcal{C}/K)_{total},", "R\\SheafHom(", "\\ldots \\to", "g_{2!}(a_2^{-1}\\mathbf{Z}_U^\\#) \\to", "g_{1!}(a_1^{-1}\\mathbf{Z}_U^\\#) \\to", "g_{0!}(a_0^{-1}\\mathbf{Z}_U^\\#),", "\\mathcal{J}^\\bullet))", "$$", "By the construction in Cohomology on Sites, Section", "\\ref{sites-cohomology-section-internal-hom}", "and since $\\mathcal{J}^\\bullet$ is K-injective, we see that", "this is represented by the complex of abelian groups with terms", "$$", "\\prod\\nolimits_{p + q = n}", "\\Hom(g_{p!}(a_p^{-1}\\mathbf{Z}_U^\\#), \\mathcal{J}^q) =", "\\prod\\nolimits_{p + q = n}", "\\Hom(a_p^{-1}\\mathbf{Z}_U^\\#, g_p^{-1}\\mathcal{J}^q)", "$$", "See Cohomology on Sites, Lemmas", "\\ref{sites-cohomology-lemma-RHom-into-K-injective} and", "\\ref{sites-cohomology-lemma-section-RHom-over-U} for more information.", "Thus we find that $R\\Gamma(U, Ra_*a^{-1}E)$ is computed by", "the product total complex $\\text{Tot}_\\pi(B^{\\bullet, \\bullet})$", "with $B^{p, q} = \\Hom(a_p^{-1}\\mathbf{Z}_U^\\#, g_p^{-1}\\mathcal{J}^q)$.", "For the other side we argue similarly. First we note that", "$$", "s(\\mathbf{Z}_{F(K)}^\\#) \\longrightarrow \\mathbf{Z}", "$$", "is a quasi-isomorphism of complexes on $\\mathcal{C}$", "by Hypercoverings, Lemma \\ref{hypercovering-lemma-acyclic-hypercover-sheaves}.", "Since $\\mathbf{Z}_U^\\#$ is a flat sheaf of $\\mathbf{Z}$-modules", "we see that", "$$", "s(\\mathbf{Z}_{F(K)}^\\#) \\otimes_\\mathbf{Z} \\mathbf{Z}_U^\\#", "\\longrightarrow", "\\mathbf{Z}_U^\\#", "$$", "is a quasi-isomorphism. Therefore", "$R\\Hom(\\mathbf{Z}_U^\\#, E)$ is equal to", "$$", "R\\Gamma(\\mathcal{C}, R\\SheafHom(", "s(\\mathbf{Z}_{F(K)}^\\#) \\otimes_\\mathbf{Z} \\mathbf{Z}_U^\\#,", "\\mathcal{I}^\\bullet))", "$$", "By the construction of $R\\SheafHom$ and since $\\mathcal{I}^\\bullet$", "is K-injective, this is represented by the complex of abelian groups", "with terms", "$$", "\\prod\\nolimits_{p + q = n}", "\\Hom(\\mathbf{Z}^\\#_{K_p} \\otimes_\\mathbf{Z} \\mathbf{Z}_U^\\#, \\mathcal{I}^q)", "=", "\\prod\\nolimits_{p + q = n}", "\\Hom(a_p^{-1}\\mathbf{Z}_U^\\#, a_p^{-1}\\mathcal{I}^q)", "$$", "The equality of terms follows from the fact that", "$\\mathbf{Z}^\\#_{K_p} \\otimes_\\mathbf{Z} \\mathbf{Z}_U^\\# =", "a_{p!}a_p^{-1}\\mathbf{Z}_U^\\#$ by Modules on Sites,", "Remark \\ref{sites-modules-remark-j-shriek-tensor}.", "Thus we find that $R\\Gamma(U, E)$ is computed by", "the product total complex $\\text{Tot}_\\pi(A^{\\bullet, \\bullet})$", "with $A^{p, q} = \\Hom(a_p^{-1}\\mathbf{Z}_U^\\#, a_p^{-1}\\mathcal{I}^q)$.", "\\medskip\\noindent", "Since $\\mathcal{I}^\\bullet$ is K-injective we see that", "$a_p^{-1}\\mathcal{I}^\\bullet$ is K-injective, see", "Lemma \\ref{lemma-localize-injective}.", "Since $\\mathcal{J}^\\bullet$ is K-injective we see that", "$g_p^{-1}\\mathcal{J}^\\bullet$ is K-injective, see", "Lemma \\ref{lemma-restriction-injective-to-component-site}.", "Both represent the object $a_p^{-1}E$.", "Hence for every $p \\geq 0$ the map of complexes", "$$", "A^{p, \\bullet} = \\Hom(a_p^{-1}\\mathbf{Z}_U^\\#, a_p^{-1}\\mathcal{I}^\\bullet)", "\\longrightarrow", "\\Hom(a_p^{-1}\\mathbf{Z}_U^\\#, g_p^{-1}\\mathcal{J}^\\bullet) = B^{p, \\bullet}", "$$", "induced by $g_p^{-1}$ applied to the given map", "$a^{-1}\\mathcal{I}^\\bullet \\to \\mathcal{J}^\\bullet$", "is a quasi-isomorphisms as these complexes both compute", "$$", "R\\Hom(a_p^{-1}\\mathbf{Z}_U^\\#, a_p^{-1}E)", "$$", "By More on Algebra, Lemma \\ref{more-algebra-lemma-prod-qis-gives-qis}", "we conclude that the right vertical arrow in the commutative diagram", "$$", "\\xymatrix{", "R\\Gamma(U, E) \\ar[r] \\ar[d] &", "\\text{Tot}_\\pi(A^{\\bullet, \\bullet}) \\ar[d] \\\\", "R\\Gamma(U, Ra_*a^{-1}E) \\ar[r] &", "\\text{Tot}_\\pi(B^{\\bullet, \\bullet})", "}", "$$", "is a quasi-isomorphism. Since we saw above that the horizontal arrows", "are quasi-isomorphisms, so is the left vertical arrow." ], "refs": [ "spaces-simplicial-lemma-augmentation-spectral-sequence", "spaces-simplicial-lemma-localize-injective", "spaces-simplicial-lemma-hypercovering-cech-complex", "hypercovering-lemma-acyclic-hypercover-sheaves", "spaces-simplicial-lemma-hypercovering-descent-sheaves", "derived-lemma-leray-acyclicity", "spaces-simplicial-lemma-simplicial-resolution-augmentation", "sites-cohomology-lemma-RHom-into-K-injective", "hypercovering-lemma-acyclic-hypercover-sheaves", "sites-modules-remark-j-shriek-tensor", "spaces-simplicial-lemma-localize-injective", "spaces-simplicial-lemma-restriction-injective-to-component-site", "more-algebra-lemma-prod-qis-gives-qis" ], "ref_ids": [ 9043, 9073, 9078, 8394, 9077, 1844, 9041, 4324, 8394, 14309, 9073, 9026, 10438 ] } ], "ref_ids": [] }, { "id": 9080, "type": "theorem", "label": "spaces-simplicial-lemma-compare-cohomology-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-compare-cohomology-hypercovering", "contents": [ "Let $\\mathcal{C}$ be a site with equalizers and fibre products.", "Let $K$ be a hypercovering.", "Then we have a canonical isomorphism", "$$", "R\\Gamma(\\mathcal{C}, E) =", "R\\Gamma((\\mathcal{C}/K)_{total}, a^{-1}E)", "$$", "for $E \\in D(\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-hypercovering-descent-bounded-abelian}", "because $R\\Gamma((\\mathcal{C}/K)_{total}, -) =", "R\\Gamma(\\mathcal{C}, -) \\circ Ra_*$ by", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-before-Leray}." ], "refs": [ "spaces-simplicial-lemma-hypercovering-descent-bounded-abelian", "sites-cohomology-remark-before-Leray" ], "ref_ids": [ 9079, 4423 ] } ], "ref_ids": [] }, { "id": 9081, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-equivalence-bounded", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-equivalence-bounded", "contents": [ "Let $\\mathcal{C}$ be a site with equalizers and fibre products.", "Let $K$ be a hypercovering.", "Let $\\mathcal{A} \\subset \\textit{Ab}((\\mathcal{C}/K)_{total})$", "denote the weak Serre subcategory of cartesian abelian sheaves.", "Then the functor $a^{-1}$ defines an equivalence", "$$", "D^+(\\mathcal{C}) \\longrightarrow D_\\mathcal{A}^+((\\mathcal{C}/K)_{total})", "$$", "with quasi-inverse $Ra_*$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\mathcal{A}$ is a weak Serre subcategory by", "Lemma \\ref{lemma-Serre-subcat-cartesian-modules}.", "The equivalence is a", "formal consequence of the results obtained so far. Use", "Lemmas \\ref{lemma-hypercovering-descent-sheaves} and", "\\ref{lemma-hypercovering-descent-bounded-abelian} and", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-equivalence-bounded}" ], "refs": [ "spaces-simplicial-lemma-Serre-subcat-cartesian-modules", "spaces-simplicial-lemma-hypercovering-descent-sheaves", "spaces-simplicial-lemma-hypercovering-descent-bounded-abelian", "sites-cohomology-lemma-equivalence-bounded" ], "ref_ids": [ 9056, 9077, 9079, 4290 ] } ], "ref_ids": [] }, { "id": 9082, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-descent-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-descent-modules", "contents": [ "Let $\\mathcal{C}$ be a site with equalizers and fibre products.", "Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.", "Let $K$ be a hypercovering. With notation as above", "$$", "a^* : \\textit{Mod}(\\mathcal{O}_\\mathcal{C}) \\to \\textit{Mod}(\\mathcal{O})", "$$", "is fully faithful with essential image the cartesian $\\mathcal{O}$-modules.", "The functor $a_*$ provides the quasi-inverse." ], "refs": [], "proofs": [ { "contents": [ "Since $a^{-1}\\mathcal{O}_\\mathcal{C} = \\mathcal{O}$ we have", "$a^* = a^{-1}$. Hence the lemma follows", "immediately from Lemma \\ref{lemma-hypercovering-descent-sheaves}." ], "refs": [ "spaces-simplicial-lemma-hypercovering-descent-sheaves" ], "ref_ids": [ 9077 ] } ], "ref_ids": [] }, { "id": 9083, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-descent-bounded-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-descent-bounded-modules", "contents": [ "Let $\\mathcal{C}$ be a site with equalizers and fibre products.", "Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.", "Let $K$ be a hypercovering. For", "$E \\in D(\\mathcal{O}_\\mathcal{C})$ the map", "$$", "E \\longrightarrow Ra_*La^*E", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Since $a^{-1}\\mathcal{O}_\\mathcal{C} = \\mathcal{O}$ we have", "$La^* = a^* = a^{-1}$. Moreover $Ra_*$ agrees with", "$Ra_*$ on abelian sheaves, see", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-modules-abelian-unbounded}.", "Hence the lemma follows", "immediately from Lemma \\ref{lemma-hypercovering-descent-bounded-abelian}." ], "refs": [ "sites-cohomology-lemma-modules-abelian-unbounded", "spaces-simplicial-lemma-hypercovering-descent-bounded-abelian" ], "ref_ids": [ 4259, 9079 ] } ], "ref_ids": [] }, { "id": 9084, "type": "theorem", "label": "spaces-simplicial-lemma-compare-cohomology-hypercovering-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-compare-cohomology-hypercovering-modules", "contents": [ "Let $\\mathcal{C}$ be a site with equalizers and fibre products.", "Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.", "Let $K$ be a hypercovering.", "Then we have a canonical isomorphism", "$$", "R\\Gamma(\\mathcal{C}, E) =", "R\\Gamma((\\mathcal{C}/K)_{total}, La^*E)", "$$", "for $E \\in D(\\mathcal{O}_\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-hypercovering-descent-bounded-modules}", "because $R\\Gamma((\\mathcal{C}/K)_{total}, -) =", "R\\Gamma(\\mathcal{C}, -) \\circ Ra_*$ by", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-before-Leray}", "or by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-Leray-unbounded}." ], "refs": [ "spaces-simplicial-lemma-hypercovering-descent-bounded-modules", "sites-cohomology-remark-before-Leray", "sites-cohomology-lemma-Leray-unbounded" ], "ref_ids": [ 9083, 4423, 4257 ] } ], "ref_ids": [] }, { "id": 9085, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-equivalence-bounded-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-equivalence-bounded-modules", "contents": [ "Let $\\mathcal{C}$ be a site with equalizers and fibre products.", "Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.", "Let $K$ be a hypercovering.", "Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$", "denote the weak Serre subcategory of cartesian $\\mathcal{O}$-modules.", "Then the functor $La^*$ defines an equivalence", "$$", "D^+(\\mathcal{O}_\\mathcal{C}) \\longrightarrow D_\\mathcal{A}^+(\\mathcal{O})", "$$", "with quasi-inverse $Ra_*$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\mathcal{A}$ is a weak Serre subcategory by", "Lemma \\ref{lemma-Serre-subcat-cartesian-modules}", "(the required hypotheses hold by the discussion in", "Remark \\ref{remark-augmentation-ringed}).", "The equivalence is a", "formal consequence of the results obtained so far. Use", "Lemmas \\ref{lemma-hypercovering-descent-modules} and", "\\ref{lemma-hypercovering-descent-bounded-modules} and", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-equivalence-bounded}." ], "refs": [ "spaces-simplicial-lemma-Serre-subcat-cartesian-modules", "spaces-simplicial-remark-augmentation-ringed", "spaces-simplicial-lemma-hypercovering-descent-modules", "spaces-simplicial-lemma-hypercovering-descent-bounded-modules", "sites-cohomology-lemma-equivalence-bounded" ], "ref_ids": [ 9056, 9161, 9082, 9083, 4290 ] } ], "ref_ids": [] }, { "id": 9086, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-X-descent-sheaves", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-X-descent-sheaves", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.", "Let $K$ be a hypercovering of $X$. Then", "\\begin{enumerate}", "\\item $a^{-1} : \\Sh(\\mathcal{C}/X) \\to \\Sh((\\mathcal{C}/K)_{total})$", "is fully faithful with essential image the cartesian sheaves of sets,", "\\item $a^{-1} : \\textit{Ab}(\\mathcal{C}/X) \\to", "\\textit{Ab}((\\mathcal{C}/K)_{total})$", "is fully faithful with essential image the cartesian sheaves", "of abelian groups.", "\\end{enumerate}", "In both cases $a_*$ provides the quasi-inverse functor." ], "refs": [], "proofs": [ { "contents": [ "Via Remarks \\ref{remark-semi-representable-over-object} and", "\\ref{remark-augmentation-over-object} and the discussion in", "the introduction to this section", "this follows from Lemma \\ref{lemma-hypercovering-descent-sheaves}." ], "refs": [ "spaces-simplicial-remark-semi-representable-over-object", "spaces-simplicial-remark-augmentation-over-object", "spaces-simplicial-lemma-hypercovering-descent-sheaves" ], "ref_ids": [ 9157, 9160, 9077 ] } ], "ref_ids": [] }, { "id": 9087, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-X-descent-bounded-abelian", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-X-descent-bounded-abelian", "contents": [ "Let $\\mathcal{C}$ be a site with fibre product and $X \\in \\Ob(\\mathcal{C})$.", "Let $K$ be a hypercovering of $X$. For", "$E \\in D(\\mathcal{C}/X)$ the map", "$$", "E \\longrightarrow Ra_*a^{-1}E", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Via Remarks \\ref{remark-semi-representable-over-object} and", "\\ref{remark-augmentation-over-object} and the discussion in", "the introduction to this section", "this follows from Lemma \\ref{lemma-hypercovering-descent-bounded-abelian}." ], "refs": [ "spaces-simplicial-remark-semi-representable-over-object", "spaces-simplicial-remark-augmentation-over-object", "spaces-simplicial-lemma-hypercovering-descent-bounded-abelian" ], "ref_ids": [ 9157, 9160, 9079 ] } ], "ref_ids": [] }, { "id": 9088, "type": "theorem", "label": "spaces-simplicial-lemma-compare-cohomology-hypercovering-X", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-compare-cohomology-hypercovering-X", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.", "Let $K$ be a hypercovering of $X$.", "Then we have a canonical isomorphism", "$$", "R\\Gamma(X, E) = R\\Gamma((\\mathcal{C}/K)_{total}, a^{-1}E)", "$$", "for $E \\in D(\\mathcal{C}/X)$." ], "refs": [], "proofs": [ { "contents": [ "Via Remarks \\ref{remark-semi-representable-over-object} and", "\\ref{remark-augmentation-over-object}", "this follows from Lemma \\ref{lemma-compare-cohomology-hypercovering}." ], "refs": [ "spaces-simplicial-remark-semi-representable-over-object", "spaces-simplicial-remark-augmentation-over-object", "spaces-simplicial-lemma-compare-cohomology-hypercovering" ], "ref_ids": [ 9157, 9160, 9080 ] } ], "ref_ids": [] }, { "id": 9089, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-X-equivalence-bounded", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-X-equivalence-bounded", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.", "Let $K$ be a hypercovering of $X$.", "Let $\\mathcal{A} \\subset \\textit{Ab}((\\mathcal{C}/K)_{total})$", "denote the weak Serre subcategory of cartesian abelian sheaves.", "Then the functor $a^{-1}$ defines an equivalence", "$$", "D^+(\\mathcal{C}/X) \\longrightarrow D_\\mathcal{A}^+((\\mathcal{C}/K)_{total})", "$$", "with quasi-inverse $Ra_*$." ], "refs": [], "proofs": [ { "contents": [ "Via Remarks \\ref{remark-semi-representable-over-object} and", "\\ref{remark-augmentation-over-object}", "this follows from Lemma \\ref{lemma-hypercovering-equivalence-bounded}." ], "refs": [ "spaces-simplicial-remark-semi-representable-over-object", "spaces-simplicial-remark-augmentation-over-object", "spaces-simplicial-lemma-hypercovering-equivalence-bounded" ], "ref_ids": [ 9157, 9160, 9081 ] } ], "ref_ids": [] }, { "id": 9090, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-X-descent-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-X-descent-modules", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.", "Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.", "Let $K$ be a hypercovering of $X$. With notation as above", "$$", "a^* : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O})", "$$", "is fully faithful with essential image the cartesian $\\mathcal{O}$-modules.", "The functor $a_*$ provides the quasi-inverse." ], "refs": [], "proofs": [ { "contents": [ "Via Remarks \\ref{remark-semi-representable-ringed-over-object} and", "\\ref{remark-augmentation-ringed-over-object} and the discussion in", "the introduction to this section", "this follows from Lemma \\ref{lemma-hypercovering-descent-modules}." ], "refs": [ "spaces-simplicial-remark-semi-representable-ringed-over-object", "spaces-simplicial-remark-augmentation-ringed-over-object", "spaces-simplicial-lemma-hypercovering-descent-modules" ], "ref_ids": [ 9159, 9162, 9082 ] } ], "ref_ids": [] }, { "id": 9091, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-X-descent-bounded-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-X-descent-bounded-modules", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.", "Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.", "Let $K$ be a hypercovering of $X$. For", "$E \\in D(\\mathcal{O}_X)$ the map", "$$", "E \\longrightarrow Ra_*La^*E", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Via Remarks \\ref{remark-semi-representable-ringed-over-object} and", "\\ref{remark-augmentation-ringed-over-object} and the discussion in", "the introduction to this section", "this follows from Lemma \\ref{lemma-hypercovering-descent-bounded-modules}." ], "refs": [ "spaces-simplicial-remark-semi-representable-ringed-over-object", "spaces-simplicial-remark-augmentation-ringed-over-object", "spaces-simplicial-lemma-hypercovering-descent-bounded-modules" ], "ref_ids": [ 9159, 9162, 9083 ] } ], "ref_ids": [] }, { "id": 9092, "type": "theorem", "label": "spaces-simplicial-lemma-compare-cohomology-hypercovering-X-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-compare-cohomology-hypercovering-X-modules", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.", "Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.", "Let $K$ be a hypercovering of $X$.", "Then we have a canonical isomorphism", "$$", "R\\Gamma(X, E) = R\\Gamma((\\mathcal{C}/K)_{total}, La^*E)", "$$", "for $E \\in D(\\mathcal{O}_\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "Via Remarks \\ref{remark-semi-representable-ringed-over-object} and", "\\ref{remark-augmentation-ringed-over-object} and the discussion in", "the introduction to this section", "this follows from Lemma \\ref{lemma-compare-cohomology-hypercovering-modules}." ], "refs": [ "spaces-simplicial-remark-semi-representable-ringed-over-object", "spaces-simplicial-remark-augmentation-ringed-over-object", "spaces-simplicial-lemma-compare-cohomology-hypercovering-modules" ], "ref_ids": [ 9159, 9162, 9084 ] } ], "ref_ids": [] }, { "id": 9093, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-X-equivalence-bounded-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-X-equivalence-bounded-modules", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.", "Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.", "Let $K$ be a hypercovering of $X$.", "Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$", "denote the weak Serre subcategory of cartesian $\\mathcal{O}$-modules.", "Then the functor $La^*$ defines an equivalence", "$$", "D^+(\\mathcal{O}_X) \\longrightarrow D_\\mathcal{A}^+(\\mathcal{O})", "$$", "with quasi-inverse $Ra_*$." ], "refs": [], "proofs": [ { "contents": [ "Via Remarks \\ref{remark-semi-representable-ringed-over-object} and", "\\ref{remark-augmentation-ringed-over-object} and the discussion in", "the introduction to this section", "this follows from Lemma \\ref{lemma-hypercovering-equivalence-bounded-modules}." ], "refs": [ "spaces-simplicial-remark-semi-representable-ringed-over-object", "spaces-simplicial-remark-augmentation-ringed-over-object", "spaces-simplicial-lemma-hypercovering-equivalence-bounded-modules" ], "ref_ids": [ 9159, 9162, 9085 ] } ], "ref_ids": [] }, { "id": 9094, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-X-simple-descent-sheaves", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-X-simple-descent-sheaves", "contents": [ "Let $\\mathcal{C}$ be a site with fibre product and $X \\in \\Ob(\\mathcal{C})$.", "Let $a : U \\to X$ be a hypercovering of $X$ in $\\mathcal{C}$ as defined above.", "Then", "\\begin{enumerate}", "\\item $a^{-1} : \\Sh(\\mathcal{C}/X) \\to \\Sh((\\mathcal{C}/U)_{total})$", "is fully faithful with essential image the cartesian sheaves of sets,", "\\item $a^{-1} : \\textit{Ab}(\\mathcal{C}/X) \\to", "\\textit{Ab}((\\mathcal{C}/U)_{total})$", "is fully faithful with essential image the cartesian sheaves", "of abelian groups.", "\\end{enumerate}", "In both cases $a_*$ provides the quasi-inverse functor." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-hypercovering-X-descent-sheaves}." ], "refs": [ "spaces-simplicial-lemma-hypercovering-X-descent-sheaves" ], "ref_ids": [ 9086 ] } ], "ref_ids": [] }, { "id": 9095, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-X-simple-descent-bounded-abelian", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-X-simple-descent-bounded-abelian", "contents": [ "Let $\\mathcal{C}$ be a site with fibre product and $X \\in \\Ob(\\mathcal{C})$.", "Let $a : U \\to X$ be a hypercovering of $X$ in $\\mathcal{C}$ as defined above.", "For $E \\in D(\\mathcal{C}/X)$ the map", "$$", "E \\longrightarrow Ra_*a^{-1}E", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-hypercovering-X-descent-bounded-abelian}." ], "refs": [ "spaces-simplicial-lemma-hypercovering-X-descent-bounded-abelian" ], "ref_ids": [ 9087 ] } ], "ref_ids": [] }, { "id": 9096, "type": "theorem", "label": "spaces-simplicial-lemma-compare-cohomology-hypercovering-X-simple", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-compare-cohomology-hypercovering-X-simple", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.", "Let $a : U \\to X$ be a hypercovering of $X$ in $\\mathcal{C}$ as defined above.", "Then we have a canonical isomorphism", "$$", "R\\Gamma(X, E) = R\\Gamma((\\mathcal{C}/U)_{total}, a^{-1}E)", "$$", "for $E \\in D(\\mathcal{C}/X)$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-compare-cohomology-hypercovering-X}." ], "refs": [ "spaces-simplicial-lemma-compare-cohomology-hypercovering-X" ], "ref_ids": [ 9088 ] } ], "ref_ids": [] }, { "id": 9097, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-X-simple-equivalence-bounded", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-X-simple-equivalence-bounded", "contents": [ "Let $\\mathcal{C}$ be a site with fibre product and $X \\in \\Ob(\\mathcal{C})$.", "Let $a : U \\to X$ be a hypercovering of $X$ in $\\mathcal{C}$ as defined above.", "Let $\\mathcal{A} \\subset \\textit{Ab}((\\mathcal{C}/U)_{total})$", "denote the weak Serre subcategory of cartesian abelian sheaves.", "Then the functor $a^{-1}$ defines an equivalence", "$$", "D^+(\\mathcal{C}/X) \\longrightarrow D_\\mathcal{A}^+((\\mathcal{C}/U)_{total})", "$$", "with quasi-inverse $Ra_*$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-hypercovering-X-equivalence-bounded}" ], "refs": [ "spaces-simplicial-lemma-hypercovering-X-equivalence-bounded" ], "ref_ids": [ 9089 ] } ], "ref_ids": [] }, { "id": 9098, "type": "theorem", "label": "spaces-simplicial-lemma-sr-when-fibre-products", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-sr-when-fibre-products", "contents": [ "Let $U$ be a simplicial object of a site $\\mathcal{C}$", "with fibre products.", "\\begin{enumerate}", "\\item $\\mathcal{C}/U$ has the structure of a simplicial object", "in the category whose objects are sites and", "whose morphisms are morphisms of sites,", "\\item the construction of Lemma \\ref{lemma-simplicial-site-site}", "applied to the structure in (1)", "reproduces the site $(\\mathcal{C}/U)_{total}$ above,", "\\item if $a : U \\to X$ is an augmentation, then", "$a_0 : \\mathcal{C}/U_0 \\to \\mathcal{C}/X$ is an augmentation", "as in Remark \\ref{remark-augmentation-site} part (A) and gives the", "same morphism of topoi", "$a : \\Sh((\\mathcal{C}/U)_{total}) \\to \\Sh(\\mathcal{C}/X)$", "as the one above.", "\\end{enumerate}" ], "refs": [ "spaces-simplicial-lemma-simplicial-site-site", "spaces-simplicial-remark-augmentation-site" ], "proofs": [ { "contents": [ "Given a morphism of objects $V \\to W$ of $\\mathcal{C}$ the localization", "morphism $j : \\mathcal{C}/V \\to \\mathcal{C}/W$ is a left adjoint to", "the base change functor $\\mathcal{C}/W \\to \\mathcal{C}/V$.", "The base change functor is continuous and induces the same morphism of", "topoi as $j$. See", "Sites, Lemma \\ref{sites-lemma-relocalize-given-fibre-products}.", "This proves (1).", "\\medskip\\noindent", "Part (2) holds because a morphism $V/U_n \\to W/U_m$", "of the category constructed", "in Lemma \\ref{lemma-simplicial-site-site}", "is a morphism $V \\to W \\times_{U_m, f_\\varphi} U_n$ over $U_n$", "which is the same thing as a morphism $f : V \\to W$", "over the morphism $f_\\varphi : U_n \\to U_m$, i.e.,", "the same thing as a morphism in the category $(\\mathcal{C}/U)_{total}$", "defined above. Equality of sets of coverings is", "immediate from the definition.", "\\medskip\\noindent", "We omit the proof of (3)." ], "refs": [ "sites-lemma-relocalize-given-fibre-products", "spaces-simplicial-lemma-simplicial-site-site" ], "ref_ids": [ 8568, 9022 ] } ], "ref_ids": [ 9022, 9152 ] }, { "id": 9099, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-X-simple-descent-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-X-simple-descent-modules", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.", "Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.", "Let $U$ be a hypercovering of $X$ in $\\mathcal{C}$. With notation as above", "$$", "a^* : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O})", "$$", "is fully faithful with essential image the cartesian $\\mathcal{O}$-modules.", "The functor $a_*$ provides the quasi-inverse." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-hypercovering-X-descent-modules}." ], "refs": [ "spaces-simplicial-lemma-hypercovering-X-descent-modules" ], "ref_ids": [ 9090 ] } ], "ref_ids": [] }, { "id": 9100, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-X-simple-descent-bounded-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-X-simple-descent-bounded-modules", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.", "Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.", "Let $U$ be a hypercovering of $X$ in $\\mathcal{C}$. For", "$E \\in D(\\mathcal{O}_X)$ the map", "$$", "E \\longrightarrow Ra_*La^*E", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-hypercovering-X-descent-bounded-modules}." ], "refs": [ "spaces-simplicial-lemma-hypercovering-X-descent-bounded-modules" ], "ref_ids": [ 9091 ] } ], "ref_ids": [] }, { "id": 9101, "type": "theorem", "label": "spaces-simplicial-lemma-compare-cohomology-hypercovering-X-simple-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-compare-cohomology-hypercovering-X-simple-modules", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.", "Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.", "Let $U$ be a hypercovering of $X$ in $\\mathcal{C}$.", "Then we have a canonical isomorphism", "$$", "R\\Gamma(X, E) = R\\Gamma((\\mathcal{C}/U)_{total}, La^*E)", "$$", "for $E \\in D(\\mathcal{O}_\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-compare-cohomology-hypercovering-X-modules}." ], "refs": [ "spaces-simplicial-lemma-compare-cohomology-hypercovering-X-modules" ], "ref_ids": [ 9092 ] } ], "ref_ids": [] }, { "id": 9102, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-X-simple-equivalence-bounded-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-X-simple-equivalence-bounded-modules", "contents": [ "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.", "Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.", "Let $U$ be a hypercovering of $X$ in $\\mathcal{C}$.", "Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$", "denote the weak Serre subcategory of cartesian $\\mathcal{O}$-modules.", "Then the functor $La^*$ defines an equivalence", "$$", "D^+(\\mathcal{O}_X) \\longrightarrow D_\\mathcal{A}^+(\\mathcal{O})", "$$", "with quasi-inverse $Ra_*$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-hypercovering-X-equivalence-bounded-modules}." ], "refs": [ "spaces-simplicial-lemma-hypercovering-X-equivalence-bounded-modules" ], "ref_ids": [ 9093 ] } ], "ref_ids": [] }, { "id": 9103, "type": "theorem", "label": "spaces-simplicial-lemma-hypercovering-equivalence-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-hypercovering-equivalence-modules", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O}_\\mathcal{C})$ be a ringed site.", "Assume given weak Serre subcategories", "$\\mathcal{A}_U \\subset \\textit{Mod}(\\mathcal{O}_U)$", "satisfying conditions (\\ref{item-restriction}),", "(\\ref{item-local}), and (\\ref{item-bounded-dimension}) above.", "Assume $\\mathcal{C}$ has equalizers and fibre products and", "let $K$ be a hypercovering.", "Let $((\\mathcal{C}/K)_{total}, \\mathcal{O})$ be as in", "Remark \\ref{remark-augmentation-ringed}.", "Let $\\mathcal{A}_{total} \\subset \\textit{Mod}(\\mathcal{O})$", "denote the weak Serre subcategory of cartesian $\\mathcal{O}$-modules", "$\\mathcal{F}$ whose restriction $\\mathcal{F}_n$ is in", "$\\mathcal{A}_{K_n}$ for all $n$ (as defined above).", "Then the functor $La^*$ defines an equivalence", "$$", "D_\\mathcal{A}(\\mathcal{O}_\\mathcal{C})", "\\longrightarrow", "D_{\\mathcal{A}_{total}}(\\mathcal{O})", "$$", "with quasi-inverse $Ra_*$." ], "refs": [ "spaces-simplicial-remark-augmentation-ringed" ], "proofs": [ { "contents": [ "The cartesian $\\mathcal{O}$-modules form a weak Serre subcategory by", "Lemma \\ref{lemma-Serre-subcat-cartesian-modules}", "(the required hypotheses hold by the discussion in", "Remark \\ref{remark-augmentation-ringed}).", "Since the restriction functor", "$g_n^* : \\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_n)$", "are exact, it follows that $\\mathcal{A}_{total}$ is a weak Serre", "subcategory.", "\\medskip\\noindent", "Let us show that $a^* : \\mathcal{A} \\to \\mathcal{A}_{total}$", "is an equivalence of categories with inverse given by $La_*$.", "We already know that $La_*a^*\\mathcal{F} = \\mathcal{F}$ by the", "bounded version", "(Lemma \\ref{lemma-hypercovering-equivalence-bounded-modules}).", "It is clear that $a^*\\mathcal{F}$ is in $\\mathcal{A}_{total}$", "for $\\mathcal{F}$ in $\\mathcal{A}$. Conversely, assume that", "$\\mathcal{G} \\in \\mathcal{A}_{total}$. Because $\\mathcal{G}$", "is cartesian we see that $\\mathcal{G} = a^*\\mathcal{F}$", "for some $\\mathcal{O}_\\mathcal{C}$-module $\\mathcal{F}$ by", "Lemma \\ref{lemma-hypercovering-descent-modules}.", "We want to show that $\\mathcal{F}$ is in $\\mathcal{A}$.", "Take $U \\in \\Ob(\\mathcal{C})$. We have to show that the", "restriction of $\\mathcal{F}$ to $\\mathcal{C}/U$ is in $\\mathcal{A}_U$.", "As usual, write $K_0 = \\{U_{0, i}\\}_{i \\in I_0}$.", "Since $K$ is a hypercovering, the map $\\coprod_{i \\in I_0} h_{U_{0, i}} \\to *$", "becomes surjective after sheafification. This implies there is", "a covering $\\{U_j \\to U\\}_{j \\in J}$ and a map $\\tau : J \\to I_0$", "and for each $j \\in J$ a morphism $\\varphi_j : U_j \\to U_{0, \\tau(j)}$.", "Since $\\mathcal{G}_0 = a_0^*\\mathcal{F}$ we find", "that the restriction of $\\mathcal{F}$ to $\\mathcal{C}/U_j$", "is equal to the restriction of the $\\tau(j)$th component of", "$\\mathcal{G}_0$ to $\\mathcal{C}/U_j$ via the morphism", "$\\varphi_j : U_j \\to U_{0, \\tau(i)}$. Hence by", "(\\ref{item-restriction}) we find that $\\mathcal{F}|_{\\mathcal{C}/U_j}$", "is in $\\mathcal{A}_{U_j}$ and in turn by", "(\\ref{item-local}) we find that $\\mathcal{F}|_{\\mathcal{C}/U}$", "is in $\\mathcal{A}_U$.", "\\medskip\\noindent", "In particular the statement of the lemma makes sense.", "The lemma now follows from Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-equivalence-unbounded-one}.", "Assumption (1) is clear (see Remark \\ref{remark-augmentation-ringed}).", "Assumptions (2) and (3) we proved in the preceding paragraph.", "Assumption (4) is immediate from (\\ref{item-bounded-dimension}).", "For assumption (5) let $\\mathcal{B}_{total}$ be the set of", "objects $U/U_{n, i}$ of the site $(\\mathcal{C}/K)_{total}$", "such that $U \\in \\mathcal{B}$ where $\\mathcal{B}$ is as in", "(\\ref{item-bounded-dimension}). Here we use the description of", "the site $(\\mathcal{C}/K)_{total}$ given in", "Section \\ref{section-simplicial-semi-representable}.", "Moreover, we set $\\text{Cov}_{U/U_{n, i}}$ equal to $\\text{Cov}_U$", "and $d_{U/U_{n, i}}$ equal $d_U$ where $\\text{Cov}_U$ and $d_U$", "are given to us by (\\ref{item-bounded-dimension}).", "Then we claim that condition (5) holds with these choices.", "This follows immediately from", "Lemma \\ref{lemma-sanity-check-simplicial-semi-representable}", "and the fact that $\\mathcal{F} \\in \\mathcal{A}_{total}$", "implies $\\mathcal{F}_n \\in \\mathcal{A}_{K_n}$ and hence", "$\\mathcal{F}_{n, i} \\in \\mathcal{A}_{U_{n, i}}$.", "(The reader who worries about the difference between", "cohomology of abelian sheaves versus cohomology", "of sheaves of modules may consult Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-cohomology-modules-abelian-agree}.)" ], "refs": [ "spaces-simplicial-lemma-Serre-subcat-cartesian-modules", "spaces-simplicial-remark-augmentation-ringed", "spaces-simplicial-lemma-hypercovering-equivalence-bounded-modules", "spaces-simplicial-lemma-hypercovering-descent-modules", "sites-cohomology-lemma-equivalence-unbounded-one", "spaces-simplicial-remark-augmentation-ringed", "spaces-simplicial-lemma-sanity-check-simplicial-semi-representable", "sites-cohomology-lemma-cohomology-modules-abelian-agree" ], "ref_ids": [ 9056, 9161, 9085, 9082, 4291, 9161, 9076, 4210 ] } ], "ref_ids": [ 9161 ] }, { "id": 9104, "type": "theorem", "label": "spaces-simplicial-lemma-prepare-bbd-glueing", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-prepare-bbd-glueing", "contents": [ "In Situation \\ref{situation-locally-given}.", "Assume negative self-exts of $E_U$ in $D(\\mathcal{O}_{u(U)})$ are zero.", "Let $L$ be a simplicial object of $\\text{SR}(\\mathcal{B})$.", "Consider the simplicial object $K = u(L)$ of $\\text{SR}(\\mathcal{C})$", "and let $((\\mathcal{C}/K)_{total}, \\mathcal{O})$ be as in", "Remark \\ref{remark-augmentation-ringed}.", "There exists a cartesian object $E$ of $D(\\mathcal{O})$", "such that writing $L_n = \\{U_{n, i}\\}_{i \\in I_n}$", "the restriction of $E$ to $D(\\mathcal{O}_{\\mathcal{C}/u(U_{n, i})})$", "is $E_{U_{n, i}}$ compatibly (see proof for details).", "Moreover, $E$ is unique up to unique isomorphism." ], "refs": [ "spaces-simplicial-remark-augmentation-ringed" ], "proofs": [ { "contents": [ "Recall that", "$\\Sh(\\mathcal{C}/K_n) = \\prod_{i \\in I_n} \\Sh(\\mathcal{C}/u(U_{n, i}))$", "and similarly for the categories of modules. This product decomposition", "is also inherited by the derived categories of sheaves of modules.", "Moreover, this product decomposition is compatible with", "the morphisms in the simplicial semi-representable object $K$.", "See Section \\ref{section-semi-representable}.", "Hence we can set $E_n = \\prod_{i \\in I_n} E_{U_{n, i}}$", "(``formal'' product) in $D(\\mathcal{O}_n)$.", "Taking (formal) products of the maps $\\rho_a$ of", "Situation \\ref{situation-locally-given}", "we obtain isomorphisms $E_\\varphi : f_\\varphi^*E_n \\to E_m$.", "The assumption about compostions of the maps $\\rho_a$", "immediately implies that $(E_n, E_\\varphi)$", "defines a simplicial system of the derived category of modules", "as in Definition \\ref{definition-cartesian-derived-modules}.", "The vanishing of negative exts assumed in the lemma implies that", "$\\Hom(E_n[t], E_n) = 0$ for $n \\geq 0$ and $t > 0$.", "Thus by", "Lemma \\ref{lemma-cartesian-module-derived-from-simplicial}", "we obtain $E$.", "Uniqueness up to unique isomorphism follows from", "Lemmas \\ref{lemma-nullity-cartesian-modules-derived} and", "\\ref{lemma-hom-cartesian-modules-derived}." ], "refs": [ "spaces-simplicial-definition-cartesian-derived-modules", "spaces-simplicial-lemma-cartesian-module-derived-from-simplicial", "spaces-simplicial-lemma-nullity-cartesian-modules-derived", "spaces-simplicial-lemma-hom-cartesian-modules-derived" ], "ref_ids": [ 9149, 9069, 9067, 9068 ] } ], "ref_ids": [ 9161 ] }, { "id": 9105, "type": "theorem", "label": "spaces-simplicial-lemma-bbd-glueing", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-bbd-glueing", "contents": [ "In Situation \\ref{situation-locally-given}. Assume", "\\begin{enumerate}", "\\item $\\mathcal{C}$ has equalizers and fibre products,", "\\item there is a morphism of sites $f : \\mathcal{C} \\to \\mathcal{D}$", "given by a continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$", "such that", "\\begin{enumerate}", "\\item $\\mathcal{D}$ has equalizers and fibre products and $u$", "commutes with them,", "\\item $\\mathcal{B}$ is a full subcategory of $\\mathcal{D}$", "and $u : \\mathcal{B} \\to \\mathcal{C}$ is the restriction of $u$,", "\\item every object of $\\mathcal{D}$ has a covering whose members", "are objects of $\\mathcal{B}$,", "\\end{enumerate}", "\\item all negative self-exts of $E_U$ in $D(\\mathcal{O}_{u(U)})$ are zero, and", "\\item there exists a $t \\in \\mathbf{Z}$ such that $H^i(E_U) = 0$ for $i < t$", "and $U \\in \\Ob(\\mathcal{B})$.", "\\end{enumerate}", "Then there exists a solution unique up to unique isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By Hypercoverings, Lemma \\ref{hypercovering-lemma-hypercovering-site}", "there exists a hypercovering $L$ for the site $\\mathcal{D}$ such that", "$L_n = \\{U_{n, i}\\}_{i \\in I_n}$ with $U_{i, n} \\in \\Ob(\\mathcal{B})$.", "Set $K = u(L)$. Apply Lemma \\ref{lemma-prepare-bbd-glueing}", "to get a cartesian object $E$ of $D(\\mathcal{O})$ on the site", "$(\\mathcal{C}/K)_{total}$ restricting to $E_{U_{n, i}}$ on", "$\\mathcal{C}/u(U_{n, i})$ compatibly.", "The assumption on $t$ implies that $E \\in D^+(\\mathcal{O})$.", "By Hypercoverings, Lemma \\ref{hypercovering-lemma-hypercovering-morphism-sites}", "we see that $K$ is a hypercovering too.", "By Lemma \\ref{lemma-hypercovering-equivalence-bounded-modules}", "we find that $E = a^*F$ for some $F$ in $D^+(\\mathcal{O}_\\mathcal{C})$.", "\\medskip\\noindent", "To prove that $F$ is a solution we will use the construction of", "$L_0$ and $L_1$ given in the proof of", "Hypercoverings, Lemma \\ref{hypercovering-lemma-hypercovering-site}.", "(This is a bit inelegant but there does not seem to be a completely", "straightforward way around it.)", "\\medskip\\noindent", "Namely, we have $I_0 = \\Ob(\\mathcal{B})$ and so", "$L_0 = \\{U\\}_{U \\in \\Ob(\\mathcal{B})}$.", "Hence the isomorphism $a^*F \\to E$ restricted to the components", "$\\mathcal{C}/u(U)$ of $\\mathcal{C}/K_0$ defines isomorphisms", "$\\rho_U : F|_{\\mathcal{C}/u(U)} \\to E_U$ for $U \\in \\Ob(\\mathcal{B})$", "by our choice of $E$.", "\\medskip\\noindent", "To prove that $\\rho_U$ satisfy the requirement of compatibility", "with the maps $\\rho_a$ of Situation \\ref{situation-locally-given}", "we use that $I_1$ contains the set", "$$", "\\Omega =", "\\{(U, V, W, a, b) \\mid U, V, W \\in \\mathcal{B}, a : U \\to V, b : U \\to W\\}", "$$", "and that for $i = (U, V, W, a, b)$ in $\\Omega$ we have", "$U_{1, i} = U$. Moreover, the component maps $f_{\\delta^1_0, i}$ and", "$f_{\\delta^1_1, i}$ of the two morphisms $K_1 \\to K_0$ are the morphisms", "$$", "a : U \\to V \\quad\\text{and}\\quad b : U \\to V", "$$", "Hence the compatibility mentioned in", "Lemma \\ref{lemma-prepare-bbd-glueing} gives that", "$$", "\\rho_a \\circ \\rho_V|_{\\mathcal{C}/u(U)} = \\rho_U", "\\quad\\text{and}\\quad", "\\rho_b \\circ \\rho_W|_{\\mathcal{C}/u(U)} = \\rho_U", "$$", "Taking $i = (U, V, U, a, \\text{id}_U) \\in \\Omega$ for example, we find", "that we have the desired compatibility. The uniqueness of $F$ follows", "from the uniqueness of $E$ in the previous lemma (small detail omitted)." ], "refs": [ "hypercovering-lemma-hypercovering-site", "spaces-simplicial-lemma-prepare-bbd-glueing", "hypercovering-lemma-hypercovering-morphism-sites", "spaces-simplicial-lemma-hypercovering-equivalence-bounded-modules", "hypercovering-lemma-hypercovering-site", "spaces-simplicial-lemma-prepare-bbd-glueing" ], "ref_ids": [ 8416, 9104, 8417, 9085, 8416, 9104 ] } ], "ref_ids": [] }, { "id": 9106, "type": "theorem", "label": "spaces-simplicial-lemma-bbd-unbounded-glueing", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-bbd-unbounded-glueing", "contents": [ "In Situation \\ref{situation-locally-given}. Assume", "\\begin{enumerate}", "\\item $\\mathcal{C}$ has equalizers and fibre products,", "\\item there is a morphism of sites $f : \\mathcal{C} \\to \\mathcal{D}$", "given by a continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$", "such that", "\\begin{enumerate}", "\\item $\\mathcal{D}$ has equalizers and fibre products and $u$", "commutes with them,", "\\item $\\mathcal{B}$ is a full subcategory of $\\mathcal{D}$", "and $u : \\mathcal{B} \\to \\mathcal{C}$ is the restriction of $u$,", "\\item every object of $\\mathcal{D}$ has a covering whose members", "are objects of $\\mathcal{B}$,", "\\end{enumerate}", "\\item all negative self-exts of $E_U$ in $D(\\mathcal{O}_{u(U)})$ are zero, and", "\\item there exist weak Serre subcategories", "$\\mathcal{A}_U \\subset \\textit{Mod}(\\mathcal{O}_U)$ for all", "$U \\in \\Ob(\\mathcal{C})$ satisfying conditions (\\ref{item-restriction}),", "(\\ref{item-local}), and (\\ref{item-bounded-dimension}),", "\\item $E_U \\in D_{\\mathcal{A}_U}(\\mathcal{O}_U)$.", "\\end{enumerate}", "Then there exists a solution unique up to unique isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The proof is {\\bf exactly} the same as the proof of", "Lemma \\ref{lemma-bbd-glueing}. The only change is that", "$E$ is an object of $D_{\\mathcal{A}_{total}}(\\mathcal{O})$", "and hence we use Lemma \\ref{lemma-hypercovering-equivalence-modules}", "to obtain $F$ with $E = a^*F$", "instead of Lemma \\ref{lemma-hypercovering-equivalence-bounded-modules}." ], "refs": [ "spaces-simplicial-lemma-bbd-glueing", "spaces-simplicial-lemma-hypercovering-equivalence-modules", "spaces-simplicial-lemma-hypercovering-equivalence-bounded-modules" ], "ref_ids": [ 9105, 9103, 9085 ] } ], "ref_ids": [] }, { "id": 9107, "type": "theorem", "label": "spaces-simplicial-lemma-compare-simplicial-objects", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-compare-simplicial-objects", "contents": [ "Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$", "be an augmentation. There is a commutative diagram", "$$", "\\xymatrix{", "\\Sh((\\textit{LC}_{qc}/U)_{total}) \\ar[r]_-h \\ar[d]_{a_{qc}} &", "\\Sh(U_{Zar}) \\ar[d]^a \\\\", "\\Sh(\\textit{LC}_{qc}/X) \\ar[r]^-{h_{-1}} &", "\\Sh(X)", "}", "$$", "where the left vertical arrow is defined in", "Section \\ref{section-hypercovering}", "and the right vertical arrow is defined in", "Lemma \\ref{lemma-augmentation}." ], "refs": [ "spaces-simplicial-lemma-augmentation" ], "proofs": [ { "contents": [ "Write $\\Sh(X) = \\Sh(X_{Zar})$. Observe that both", "$(\\textit{LC}_{qc}/U)_{total}$ and $U_{Zar}$ fall", "into case A of Situation \\ref{situation-simplicial-site}.", "This is immediate from the construction of", "$U_{Zar}$ in Section \\ref{section-simplicial-top}", "and it follows from Lemma \\ref{lemma-sr-when-fibre-products}", "for $(\\textit{LC}_{qc}/U)_{total}$.", "Next, consider the functors", "$U_{n, Zar} \\to \\textit{LC}_{qc}/U_n$, $U \\mapsto U/U_n$", "and", "$X_{Zar} \\to \\textit{LC}_{qc}/X$, $U \\mapsto U/X$.", "We have seen that these define morphisms of sites", "in Cohomology on Sites, Section \\ref{sites-cohomology-section-cohomology-LC}.", "Thus we obtain a morphism of simplicial sites compatible with", "augmentations as in Remark \\ref{remark-morphism-augmentation-simplicial-sites}", "and we may apply", "Lemma \\ref{lemma-morphism-augmentation-simplicial-sites} to conclude." ], "refs": [ "spaces-simplicial-lemma-sr-when-fibre-products", "spaces-simplicial-remark-morphism-augmentation-simplicial-sites", "spaces-simplicial-lemma-morphism-augmentation-simplicial-sites" ], "ref_ids": [ 9098, 9154, 9030 ] } ], "ref_ids": [ 9017 ] }, { "id": 9108, "type": "theorem", "label": "spaces-simplicial-lemma-descent-sheaves-for-proper-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-descent-sheaves-for-proper-hypercovering", "contents": [ "Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$", "be an augmentation. If $a : U \\to X$ gives a proper hypercovering of $X$,", "then", "$$", "a^{-1} : \\Sh(X) \\to \\Sh(U_{Zar})", "\\quad\\text{and}\\quad", "a^{-1} : \\textit{Ab}(X) \\to \\textit{Ab}(U_{Zar})", "$$", "are fully faithful with essential image the cartesian sheaves and", "quasi-inverse given by $a_*$. Here $a : \\Sh(U_{Zar}) \\to \\Sh(X)$ is as in", "Lemma \\ref{lemma-augmentation}." ], "refs": [ "spaces-simplicial-lemma-augmentation" ], "proofs": [ { "contents": [ "We will prove the statement for sheaves of sets. It will be an", "almost formal consequence of results already established.", "Consider the diagram of Lemma \\ref{lemma-compare-simplicial-objects}.", "By Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-describe-pullback-pi}", "the functor $(h_{-1})^{-1}$ is fully faithful with quasi-inverse $h_{-1, *}$.", "The same holds true for the components $h_n$ of $h$.", "By the description of the functors $h^{-1}$ and $h_*$ of", "Lemma \\ref{lemma-morphism-simplicial-sites}", "we conclude that $h^{-1}$ is fully faithful with quasi-inverse $h_*$.", "Observe that $U$ is a hypercovering of $X$ in $\\textit{LC}_{qc}$", "(as defined in Section \\ref{section-hypercovering}) by", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-proper-surjective-is-qc-covering}.", "By Lemma \\ref{lemma-hypercovering-X-simple-descent-sheaves}", "we see that $a_{qc}^{-1}$ is fully faithful with quasi-inverse $a_{qc, *}$", "and with essential image the cartesian sheaves on", "$(\\textit{LC}_{qc}/U)_{total}$.", "A formal argument (chasing around the diagram) now shows that", "$a^{-1}$ is fully faithful.", "\\medskip\\noindent", "Finally, suppose that $\\mathcal{G}$ is a cartesian sheaf on $U_{Zar}$.", "Then $h^{-1}\\mathcal{G}$ is a cartesian sheaf on $\\textit{LC}_{qc}/U$.", "Hence $h^{-1}\\mathcal{G} = a_{qc}^{-1}\\mathcal{H}$ for some sheaf", "$\\mathcal{H}$ on $\\textit{LC}_{qc}/X$.", "We compute", "\\begin{align*}", "(h_{-1})^{-1}(a_*\\mathcal{G})", "& =", "(h_{-1})^{-1}", "\\text{Eq}(", "\\xymatrix{", "a_{0, *}\\mathcal{G}_0", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "a_{1, *}\\mathcal{G}_1", "}", ") \\\\", "& =", "\\text{Eq}(", "\\xymatrix{", "(h_{-1})^{-1}a_{0, *}\\mathcal{G}_0", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "(h_{-1})^{-1}a_{1, *}\\mathcal{G}_1", "}", ") \\\\", "& =", "\\text{Eq}(", "\\xymatrix{", "a_{qc, 0, *}h_0^{-1}\\mathcal{G}_0", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "a_{qc, 1, *}h_1^{-1}\\mathcal{G}_1", "}", ") \\\\", "& =", "\\text{Eq}(", "\\xymatrix{", "a_{qc, 0, *}a_{qc, 0}^{-1}\\mathcal{H}", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "a_{qc, 1, *}a_{qc, 1}^{-1}\\mathcal{H}", "}", ") \\\\", "& =", "a_{qc, *}a_{qc}^{-1}\\mathcal{H} \\\\", "& =", "\\mathcal{H}", "\\end{align*}", "Here the first equality follows from Lemma \\ref{lemma-augmentation},", "the second equality follows as $(h_{-1})^{-1}$ is an exact functor,", "the third equality follows from", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-push-pull-LC}", "(here we use that $a_0 : U_0 \\to X$ and $a_1: U_1 \\to X$ are proper),", "the fourth follows from $a_{qc}^{-1}\\mathcal{H} = h^{-1}\\mathcal{G}$,", "the fifth from Lemma \\ref{lemma-augmentation-site}, and the", "sixth we've seen above. Since $a_{qc}^{-1}\\mathcal{H} = h^{-1}\\mathcal{G}$", "we deduce that $h^{-1}\\mathcal{G} \\cong h^{-1}a^{-1}a_*\\mathcal{G}$", "which ends the proof by fully faithfulness of $h^{-1}$." ], "refs": [ "spaces-simplicial-lemma-compare-simplicial-objects", "sites-cohomology-lemma-describe-pullback-pi", "spaces-simplicial-lemma-morphism-simplicial-sites", "sites-cohomology-lemma-proper-surjective-is-qc-covering", "spaces-simplicial-lemma-hypercovering-X-simple-descent-sheaves", "spaces-simplicial-lemma-augmentation", "sites-cohomology-lemma-push-pull-LC", "spaces-simplicial-lemma-augmentation-site" ], "ref_ids": [ 9107, 4308, 9028, 4307, 9094, 9017, 4310, 9027 ] } ], "ref_ids": [ 9017 ] }, { "id": 9109, "type": "theorem", "label": "spaces-simplicial-lemma-cohomological-descent-for-proper-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-cohomological-descent-for-proper-hypercovering", "contents": [ "Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$", "be an augmentation. If $a : U \\to X$ gives a proper hypercovering of $X$,", "then for $K \\in D^+(X)$", "$$", "K \\to Ra_*(a^{-1}K)", "$$", "is an isomorphism where $a : \\Sh(U_{Zar}) \\to \\Sh(X)$ is as in", "Lemma \\ref{lemma-augmentation}." ], "refs": [ "spaces-simplicial-lemma-augmentation" ], "proofs": [ { "contents": [ "Consider the diagram of Lemma \\ref{lemma-compare-simplicial-objects}.", "Observe that $Rh_{n, *}h_n^{-1}$ is the identity functor", "on $D^+(U_n)$ by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-cohomological-descent-LC}.", "Hence $Rh_*h^{-1}$ is the identity functor on", "$D^+(U_{Zar})$ by", "Lemma \\ref{lemma-direct-image-morphism-simplicial-sites}.", "We have", "\\begin{align*}", "Ra_*(a^{-1}K)", "& =", "Ra_*Rh_*h^{-1}a^{-1}K \\\\", "& =", "Rh_{-1, *}Ra_{qc, *}a_{qc}^{-1}(h_{-1})^{-1}K \\\\", "& =", "Rh_{-1, *}(h_{-1})^{-1}K \\\\", "& =", "K", "\\end{align*}", "The first equality by the discussion above, the second equality", "because of the commutativity of the diagram in", "Lemma \\ref{lemma-compare-simplicial-objects}, the third equality by", "Lemma \\ref{lemma-hypercovering-X-simple-descent-bounded-abelian}", "($U$ is a hypercovering of $X$ in $\\textit{LC}_{qc}$ by", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-proper-surjective-is-qc-covering}),", "and the last equality by the already used Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-cohomological-descent-LC}." ], "refs": [ "spaces-simplicial-lemma-compare-simplicial-objects", "sites-cohomology-lemma-cohomological-descent-LC", "spaces-simplicial-lemma-direct-image-morphism-simplicial-sites", "spaces-simplicial-lemma-compare-simplicial-objects", "spaces-simplicial-lemma-hypercovering-X-simple-descent-bounded-abelian", "sites-cohomology-lemma-proper-surjective-is-qc-covering", "sites-cohomology-lemma-cohomological-descent-LC" ], "ref_ids": [ 9107, 4313, 9029, 9107, 9095, 4307, 4313 ] } ], "ref_ids": [ 9017 ] }, { "id": 9110, "type": "theorem", "label": "spaces-simplicial-lemma-compute-via-proper-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-compute-via-proper-hypercovering", "contents": [ "Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$", "be an augmentation. If $U$ is a proper hypercovering of $X$, then", "$$", "R\\Gamma(X, K) = R\\Gamma(U_{Zar}, a^{-1}K)", "$$", "for $K \\in D^+(X)$ where $a : \\Sh(U_{Zar}) \\to \\Sh(X)$", "is as in Lemma \\ref{lemma-augmentation}." ], "refs": [ "spaces-simplicial-lemma-augmentation" ], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-cohomological-descent-for-proper-hypercovering}", "because $R\\Gamma(U_{Zar}, -) = R\\Gamma(X, -) \\circ Ra_*$ by", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-before-Leray}." ], "refs": [ "spaces-simplicial-lemma-cohomological-descent-for-proper-hypercovering", "sites-cohomology-remark-before-Leray" ], "ref_ids": [ 9109, 4423 ] } ], "ref_ids": [ 9017 ] }, { "id": 9111, "type": "theorem", "label": "spaces-simplicial-lemma-proper-hypercovering-equivalence-bounded", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-proper-hypercovering-equivalence-bounded", "contents": [ "Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$", "be an augmentation.", "Let $\\mathcal{A} \\subset \\textit{Ab}(U_{Zar})$", "denote the weak Serre subcategory of cartesian abelian sheaves.", "If $U$ is a proper hypercovering of $X$, then", "the functor $a^{-1}$ defines an equivalence", "$$", "D^+(X) \\longrightarrow D_\\mathcal{A}^+(U_{Zar})", "$$", "with quasi-inverse $Ra_*$ where $a : \\Sh(U_{Zar}) \\to \\Sh(X)$", "is as in Lemma \\ref{lemma-augmentation}." ], "refs": [ "spaces-simplicial-lemma-augmentation" ], "proofs": [ { "contents": [ "Observe that $\\mathcal{A}$ is a weak Serre subcategory by", "Lemma \\ref{lemma-Serre-subcat-cartesian-modules}.", "The equivalence is a", "formal consequence of the results obtained so far. Use", "Lemmas \\ref{lemma-descent-sheaves-for-proper-hypercovering} and", "\\ref{lemma-cohomological-descent-for-proper-hypercovering} and", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-equivalence-bounded}." ], "refs": [ "spaces-simplicial-lemma-Serre-subcat-cartesian-modules", "spaces-simplicial-lemma-descent-sheaves-for-proper-hypercovering", "spaces-simplicial-lemma-cohomological-descent-for-proper-hypercovering", "sites-cohomology-lemma-equivalence-bounded" ], "ref_ids": [ 9056, 9108, 9109, 4290 ] } ], "ref_ids": [ 9017 ] }, { "id": 9112, "type": "theorem", "label": "spaces-simplicial-lemma-spectral-sequence-proper-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-spectral-sequence-proper-hypercovering", "contents": [ "Let $U$ be a simplicial object of $\\textit{LC}$ and let", "$a : U \\to X$ be an augmentation. Let $\\mathcal{F}$ be an abelian sheaf", "on $X$. Let $\\mathcal{F}_n$ be the pullback to $U_n$.", "If $U$ is a proper hypercovering of $X$, then", "there exists a canonical spectral sequence", "$$", "E_1^{p, q} = H^q(U_p, \\mathcal{F}_p)", "$$", "converging to $H^{p + q}(X, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of Lemmas \\ref{lemma-compute-via-proper-hypercovering}", "and \\ref{lemma-simplicial-sheaf-cohomology}." ], "refs": [ "spaces-simplicial-lemma-compute-via-proper-hypercovering", "spaces-simplicial-lemma-simplicial-sheaf-cohomology" ], "ref_ids": [ 9110, 9019 ] } ], "ref_ids": [] }, { "id": 9113, "type": "theorem", "label": "spaces-simplicial-lemma-characterize-cartesian-schemes", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-characterize-cartesian-schemes", "contents": [ "Let $X$ be a simplicial scheme. The category of simplicial schemes cartesian", "over $X$ is equivalent to the category of pairs $(V, \\varphi)$", "where $V$ is a scheme over $X_0$ and", "$$", "\\varphi :", "V \\times_{X_0, d^1_1} X_1", "\\longrightarrow", "X_1 \\times_{d^1_0, X_0} V", "$$", "is an isomorphism over $X_1$ such that", "$(s_0^0)^*\\varphi = \\text{id}_V$ and such that", "$$", "(d^2_1)^*\\varphi = (d^2_0)^*\\varphi \\circ (d^2_2)^*\\varphi", "$$", "as morphisms of schemes over $X_2$." ], "refs": [], "proofs": [ { "contents": [ "The statement of the displayed equality makes sense because", "$d^1_1 \\circ d^2_2 = d^1_1 \\circ d^2_1$,", "$d^1_1 \\circ d^2_0 = d^1_0 \\circ d^2_2$, and", "$d^1_0 \\circ d^2_0 = d^1_0 \\circ d^2_1$ as morphisms $X_2 \\to X_0$, see", "Simplicial, Remark \\ref{simplicial-remark-relations} hence we", "can picture these maps as follows", "$$", "\\xymatrix{", "&", "X_2 \\times_{d^1_1 \\circ d^2_0, X_0} V", "\\ar[r]_-{(d^2_0)^*\\varphi} &", "X_2 \\times_{d^1_0 \\circ d^2_0, X_0} V", "\\ar@{=}[rd] & \\\\", "X_2 \\times_{d^1_0 \\circ d^2_2, X_0} V", "\\ar@{=}[ru] & & &", "X_2 \\times_{d^1_0 \\circ d^2_1, X_0} V \\\\", "&", "X_2 \\times_{d^1_1 \\circ d^2_2, X_0} V", "\\ar[lu]^{(d^2_2)^*\\varphi} \\ar@{=}[r] &", "X_2 \\times_{d^1_1 \\circ d^2_1, X_0} V", "\\ar[ru]_{(d^2_1)^*\\varphi}", "}", "$$", "and the condition signifies the diagram is commutative. It is clear that", "given a simplicial scheme $Y$ cartesian over $X$ we can", "set $V = Y_0$ and $\\varphi$ equal to the composition", "$$", "V \\times_{X_0, d^1_1} X_1 =", "Y_0 \\times_{X_0, d^1_1} X_1 = Y_1 =", "X_1 \\times_{X_0, d^1_0} Y_0 =", "X_1 \\times_{X_0, d^1_0} V", "$$", "of identifications given by the cartesian structure. To prove this functor", "is an equivalence we construct a quasi-inverse. The construction of", "the quasi-inverse is analogous to the construction discussed in", "Descent, Section \\ref{descent-section-descent-modules} from which we borrow", "the notation $\\tau^n_i : [0] \\to [n]$, $0 \\mapsto i$ and", "$\\tau^n_{ij} : [1] \\to [n]$, $0 \\mapsto i$, $1 \\mapsto j$.", "Namely, given a pair $(V, \\varphi)$", "as in the lemma we set $Y_n = X_n \\times_{X(\\tau^n_n), X_0} V$.", "Then given $\\beta : [n] \\to [m]$ we define", "$V(\\beta) : Y_m \\to Y_n$ as the pullback by $X(\\tau^m_{\\beta(n)m})$", "of the map $\\varphi$ postcomposed by the projection", "$X_m \\times_{X(\\beta), X_n} Y_n \\to Y_n$. This makes sense because", "$$", "X_m \\times_{X(\\tau^m_{\\beta(n)m}), X_1} X_1 \\times_{d^1_1, X_0} V", "=", "X_m \\times_{X(\\tau^m_m), X_0} V = Y_m", "$$", "and", "$$", "X_m \\times_{X(\\tau^m_{\\beta(n)m}), X_1} X_1 \\times_{d^1_0, X_0} V =", "X_m \\times_{X(\\tau^m_{\\beta(n)}), X_0} V =", "X_m \\times_{X(\\beta), X_n} Y_n.", "$$", "We omit the verification that the commutativity", "of the displayed diagram", "above implies the maps compose correctly. We also omit the verification", "that the two functors are quasi-inverse to each other." ], "refs": [ "simplicial-remark-relations" ], "ref_ids": [ 14932 ] } ], "ref_ids": [] }, { "id": 9114, "type": "theorem", "label": "spaces-simplicial-lemma-cartesian-over", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-cartesian-over", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\pi : Y \\to (X/S)_\\bullet$ be a cartesian morphism", "of simplicial schemes.", "Set $V = Y_0$ considered as a scheme over $X$.", "The morphisms $d^1_0, d^1_1 : Y_1 \\to Y_0$ and the morphism", "$\\pi_1 : Y_1 \\to X \\times_S X$ induce isomorphisms", "$$", "\\xymatrix{", "V \\times_S X & &", "Y_1 \\ar[ll]_-{(d^1_1, \\text{pr}_1 \\circ \\pi_1)}", "\\ar[rr]^-{(\\text{pr}_0 \\circ \\pi_1, d^1_0)} & &", "X \\times_S V.", "}", "$$", "Denote $\\varphi : V \\times_S X \\to X \\times_S V$ the", "resulting isomorphism.", "Then the pair $(V, \\varphi)$ is a descent datum relative", "to $X \\to S$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of (part of)", "Lemma \\ref{lemma-characterize-cartesian-schemes}", "as the displayed equation of that lemma is", "equivalent to the cocycle condition of", "Descent, Definition \\ref{descent-definition-descent-datum}." ], "refs": [ "spaces-simplicial-lemma-characterize-cartesian-schemes", "descent-definition-descent-datum" ], "ref_ids": [ 9113, 14776 ] } ], "ref_ids": [] }, { "id": 9115, "type": "theorem", "label": "spaces-simplicial-lemma-cartesian-equivalent-descent-datum", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-cartesian-equivalent-descent-datum", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. The construction", "$$", "\\begin{matrix}", "\\text{category of cartesian } \\\\", "\\text{schemes over } (X/S)_\\bullet", "\\end{matrix}", "\\longrightarrow", "\\begin{matrix}", "\\text{ category of descent data} \\\\", "\\text{ relative to } X/S", "\\end{matrix}", "$$", "of Lemma \\ref{lemma-cartesian-over}", "is an equivalence of categories." ], "refs": [ "spaces-simplicial-lemma-cartesian-over" ], "proofs": [ { "contents": [ "The functor from left to right is given in", "Lemma \\ref{lemma-cartesian-over}.", "Hence this is a special case of", "Lemma \\ref{lemma-characterize-cartesian-schemes}." ], "refs": [ "spaces-simplicial-lemma-cartesian-over", "spaces-simplicial-lemma-characterize-cartesian-schemes" ], "ref_ids": [ 9114, 9113 ] } ], "ref_ids": [ 9114 ] }, { "id": 9116, "type": "theorem", "label": "spaces-simplicial-lemma-pullback-cartesian-module", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-pullback-cartesian-module", "contents": [ "Let $f : V \\to U$ be a morphism of simplicial schemes. Given a", "quasi-coherent module $\\mathcal{F}$ on $U_{Zar}$ the pullback", "$f^*\\mathcal{F}$ is a quasi-coherent module on $V_{Zar}$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\mathcal{F}$ is cartesian with $\\mathcal{F}_n$", "quasi-coherent, see Lemma \\ref{lemma-quasi-coherent-sheaf}.", "By Lemma \\ref{lemma-describe-functoriality} we see that", "$(f^*\\mathcal{F})_n = f_n^*\\mathcal{F}_n$ (some details omitted).", "Hence $(f^*\\mathcal{F})_n$ is quasi-coherent.", "The same fact and the cartesian property for $\\mathcal{F}$", "imply the cartesian property for $f^*\\mathcal{F}$.", "Thus $\\mathcal{F}$ is quasi-coherent by", "Lemma \\ref{lemma-quasi-coherent-sheaf} again." ], "refs": [ "spaces-simplicial-lemma-quasi-coherent-sheaf", "spaces-simplicial-lemma-describe-functoriality", "spaces-simplicial-lemma-quasi-coherent-sheaf" ], "ref_ids": [ 9059, 9013, 9059 ] } ], "ref_ids": [] }, { "id": 9117, "type": "theorem", "label": "spaces-simplicial-lemma-pushforward-cartesian-module", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-pushforward-cartesian-module", "contents": [ "Let $f : V \\to U$ be a cartesian morphism of simplicial schemes.", "Assume the morphisms $d^n_j : U_n \\to U_{n - 1}$ are", "flat and the morphisms $V_n \\to U_n$ are quasi-compact and quasi-separated.", "For a quasi-coherent module $\\mathcal{G}$ on $V_{Zar}$", "the pushforward $f_*\\mathcal{G}$ is a quasi-coherent module on $U_{Zar}$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{F} = f_* \\mathcal{G}$, then", "$\\mathcal{F}_n = f_{n , *}\\mathcal{G}_n$ by", "Lemma \\ref{lemma-describe-functoriality}.", "The maps $\\mathcal{F}(\\varphi)$ are defined using the base change maps, see", "Cohomology, Section \\ref{cohomology-section-base-change-map}.", "The sheaves $\\mathcal{F}_n$ are quasi-coherent by", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "and the fact that $\\mathcal{G}_n$ is quasi-coherent", "by Lemma \\ref{lemma-quasi-coherent-sheaf}.", "The base change maps along the degeneracies", "$d^n_j$ are isomorphisms by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology}", "and the fact that $\\mathcal{G}$ is cartesian", "by Lemma \\ref{lemma-quasi-coherent-sheaf}.", "Hence $\\mathcal{F}$ is cartesian by", "Lemma \\ref{lemma-check-cartesian-module}.", "Thus $\\mathcal{F}$ is quasi-coherent by", "Lemma \\ref{lemma-quasi-coherent-sheaf}." ], "refs": [ "spaces-simplicial-lemma-describe-functoriality", "schemes-lemma-push-forward-quasi-coherent", "spaces-simplicial-lemma-quasi-coherent-sheaf", "coherent-lemma-flat-base-change-cohomology", "spaces-simplicial-lemma-quasi-coherent-sheaf", "spaces-simplicial-lemma-check-cartesian-module", "spaces-simplicial-lemma-quasi-coherent-sheaf" ], "ref_ids": [ 9013, 7730, 9059, 3298, 9059, 9052, 9059 ] } ], "ref_ids": [] }, { "id": 9118, "type": "theorem", "label": "spaces-simplicial-lemma-adjoint-functors-cartesian-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-adjoint-functors-cartesian-modules", "contents": [ "Let $f : V \\to U$ be a cartesian morphism of", "simplicial schemes. Assume the morphisms $d^n_j : U_n \\to U_{n - 1}$ are", "flat and the morphisms $V_n \\to U_n$ are quasi-compact and quasi-separated.", "Then $f^*$ and $f_*$ form an adjoint pair of functors", "between the categories of quasi-coherent modules on $U_{Zar}$ and $V_{Zar}$." ], "refs": [], "proofs": [ { "contents": [ "We have seen in Lemmas \\ref{lemma-pullback-cartesian-module} and", "\\ref{lemma-pushforward-cartesian-module}", "that the statement makes sense. The adjointness property follows", "immediately from the fact that each $f_n^*$ is adjoint to $f_{n, *}$." ], "refs": [ "spaces-simplicial-lemma-pullback-cartesian-module", "spaces-simplicial-lemma-pushforward-cartesian-module" ], "ref_ids": [ 9116, 9117 ] } ], "ref_ids": [] }, { "id": 9119, "type": "theorem", "label": "spaces-simplicial-lemma-cartesian-modules-with-section", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-cartesian-modules-with-section", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which has a", "section\\footnote{In fact, it would be enough to assume that $f$", "has fpqc locally on $S$ a section, since we have descent of", "quasi-coherent modules by Descent,", "Section \\ref{descent-section-fpqc-descent-quasi-coherent}.}.", "Let $(X/S)_\\bullet$ be the simplicial", "scheme associated to $X \\to S$, see", "Definition \\ref{definition-fibre-products-simplicial-scheme}.", "Then pullback defines an equivalence between the category of", "quasi-coherent $\\mathcal{O}_S$-modules and the category of", "quasi-coherent modules on $((X/S)_\\bullet)_{Zar}$." ], "refs": [ "spaces-simplicial-definition-fibre-products-simplicial-scheme" ], "proofs": [ { "contents": [ "Let $\\sigma : S \\to X$ be a section of $f$. Let $(\\mathcal{F}, \\alpha)$", "be a pair as in Lemma \\ref{lemma-characterize-cartesian-modules}.", "Set $\\mathcal{G} = \\sigma^*\\mathcal{F}$. Consider the diagram", "$$", "\\xymatrix{", "X \\ar[r]_-{(\\sigma \\circ f, 1)} \\ar[d]_f &", "X \\times_S X \\ar[d]^{\\text{pr}_0} \\ar[r]_-{\\text{pr}_1} & X \\\\", "S \\ar[r]^\\sigma & X", "}", "$$", "Note that $\\text{pr}_0 = d^1_1$ and $\\text{pr}_1 = d^1_0$. Hence we", "see that $(\\sigma \\circ f, 1)^*\\alpha$ defines an isomorphism", "$$", "f^*\\mathcal{G} = (\\sigma \\circ f, 1)^*\\text{pr}_0^*\\mathcal{F}", "\\longrightarrow", "(\\sigma \\circ f, 1)^*\\text{pr}_1^*\\mathcal{F} = \\mathcal{F}", "$$", "We omit the verification that this isomorphism is compatible", "with $\\alpha$ and the canonical isomorphism", "$\\text{pr}_0^*f^*\\mathcal{G} \\to \\text{pr}_1^*f^*\\mathcal{G}$." ], "refs": [ "spaces-simplicial-lemma-characterize-cartesian-modules" ], "ref_ids": [ 9055 ] } ], "ref_ids": [ 9151 ] }, { "id": 9120, "type": "theorem", "label": "spaces-simplicial-lemma-groupoid-simplicial", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-groupoid-simplicial", "contents": [ "Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$.", "There exists a simplicial scheme $X$ over $S$", "with the following properties", "\\begin{enumerate}", "\\item $X_0 = U$, $X_1 = R$, $X_2 = R \\times_{s, U, t} R$,", "\\item $s_0^0 = e : X_0 \\to X_1$,", "\\item $d^1_0 = s : X_1 \\to X_0$, $d^1_1 = t : X_1 \\to X_0$,", "\\item $s_0^1 = (e \\circ t, 1) : X_1 \\to X_2$,", "$s_1^1 = (1, e \\circ t) : X_1 \\to X_2$,", "\\item $d^2_0 = \\text{pr}_1 : X_2 \\to X_1$,", "$d^2_1 = c : X_2 \\to X_1$,", "$d^2_2 = \\text{pr}_0$, and", "\\item $X = \\text{cosk}_2 \\text{sk}_2 X$.", "\\end{enumerate}", "For all $n$ we have $X_n = R \\times_{s, U, t} \\ldots \\times_{s, U, t} R$", "with $n$ factors. The map $d^n_j : X_n \\to X_{n - 1}$ is given on", "functors of points by", "$$", "(r_1, \\ldots, r_n) \\longmapsto (r_1, \\ldots, c(r_j, r_{j + 1}), \\ldots, r_n)", "$$", "for $1 \\leq j \\leq n - 1$ whereas", "$d^n_0(r_1, \\ldots, r_n) = (r_2, \\ldots, r_n)$ and", "$d^n_n(r_1, \\ldots, r_n) = (r_1, \\ldots, r_{n - 1})$." ], "refs": [], "proofs": [ { "contents": [ "We only have to verify that the rules prescribed in (1), (2), (3), (4), (5)", "define a $2$-truncated simplicial scheme $U'$ over $S$, since then (6)", "allows us to set $X = \\text{cosk}_2 U'$, see", "Simplicial, Lemma \\ref{simplicial-lemma-existence-cosk}.", "Using the functor of points approach, all we have to verify is that", "if $(\\text{Ob}, \\text{Arrows}, s, t, c, e, i)$ is a groupoid, then", "$$", "\\xymatrix{", "\\text{Arrows} \\times_{s, \\text{Ob}, t} \\text{Arrows}", "\\ar@<8ex>[d]^{\\text{pr}_0}", "\\ar@<0ex>[d]_c", "\\ar@<-8ex>[d]_{\\text{pr}_1}", "\\\\", "\\text{Arrows}", "\\ar@<4ex>[d]^t", "\\ar@<-4ex>[d]_s", "\\ar@<4ex>[u]^{1, e}", "\\ar@<-4ex>[u]_{e, 1}", "\\\\", "\\text{Ob}", "\\ar@<0ex>[u]_e", "}", "$$", "is a $2$-truncated simplicial set. We omit the details.", "\\medskip\\noindent", "Finally, the description of $X_n$ for $n > 2$ follows by induction from", "the description of $X_0$, $X_1$, $X_2$, and", "Simplicial, Remark \\ref{simplicial-remark-inductive-coskeleton} and", "Lemma \\ref{simplicial-lemma-work-out}. Alternately, one shows that", "$\\text{cosk}_2$ applied to the $2$-truncated simplicial set displayed above", "gives a simplicial set whose $n$th term equals", "$\\text{Arrows} \\times_{s, \\text{Ob}, t} \\ldots \\times_{s, \\text{Ob}, t}", "\\text{Arrows}$ with $n$ factors and degeneracy maps as given in the lemma.", "Some details omitted." ], "refs": [ "simplicial-lemma-existence-cosk", "simplicial-remark-inductive-coskeleton", "simplicial-lemma-work-out" ], "ref_ids": [ 14835, 14936, 14839 ] } ], "ref_ids": [] }, { "id": 9121, "type": "theorem", "label": "spaces-simplicial-lemma-quasi-coherent-groupoid-simplicial", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-quasi-coherent-groupoid-simplicial", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. Let $X$ be the simplicial scheme over $S$ constructed", "in Lemma \\ref{lemma-groupoid-simplicial}.", "Then the category of quasi-coherent modules on $(U, R, s, t, c)$", "is equivalent to the category of quasi-coherent modules on $X_{Zar}$." ], "refs": [ "spaces-simplicial-lemma-groupoid-simplicial" ], "proofs": [ { "contents": [ "This is clear from Lemmas", "\\ref{lemma-quasi-coherent-sheaf} and", "\\ref{lemma-characterize-cartesian-modules}", "and Groupoids, Definition \\ref{groupoids-definition-groupoid-module}." ], "refs": [ "spaces-simplicial-lemma-quasi-coherent-sheaf", "spaces-simplicial-lemma-characterize-cartesian-modules", "groupoids-definition-groupoid-module" ], "ref_ids": [ 9059, 9055, 9682 ] } ], "ref_ids": [ 9120 ] }, { "id": 9122, "type": "theorem", "label": "spaces-simplicial-lemma-quasi-coherent-groupoid-R-cartesian", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-quasi-coherent-groupoid-R-cartesian", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$.", "Let $X$ be the simplicial scheme over $S$ constructed", "in Lemma \\ref{lemma-groupoid-simplicial}.", "Let $(R/U)_\\bullet$ be the simplicial", "scheme associated to $s : R \\to U$, see", "Definition \\ref{definition-fibre-products-simplicial-scheme}.", "There exists a cartesian morphism $t_\\bullet : (R/U)_\\bullet \\to X$", "of simplicial schemes with low degree morphisms given by", "$$", "\\xymatrix{", "R \\times_{s, U, s} R \\times_{s, U, s} R", "\\ar@<3ex>[r]_-{\\text{pr}_{12}}", "\\ar@<0ex>[r]_-{\\text{pr}_{02}}", "\\ar@<-3ex>[r]_-{\\text{pr}_{01}}", "\\ar[dd]_{(r_0, r_1, r_2) \\mapsto (r_0 \\circ r_1^{-1}, r_1 \\circ r_2^{-1})} &", "R \\times_{s, U, s} R", "\\ar@<1ex>[r]_-{\\text{pr}_1} \\ar@<-2ex>[r]_-{\\text{pr}_0}", "\\ar[dd]_{(r_0, r_1) \\mapsto r_0 \\circ r_1^{-1}} &", "R \\ar[dd]^t", "\\\\", "\\\\", "R \\times_{s, U, t} R", "\\ar@<3ex>[r]_{\\text{pr}_1}", "\\ar@<0ex>[r]_c", "\\ar@<-3ex>[r]_{\\text{pr}_0} &", "R \\ar@<1ex>[r]_s \\ar@<-2ex>[r]_t &", "U", "}", "$$" ], "refs": [ "spaces-simplicial-lemma-groupoid-simplicial", "spaces-simplicial-definition-fibre-products-simplicial-scheme" ], "proofs": [ { "contents": [ "For arbitrary $n$ we define $(R/U)_\\bullet \\to X_n$ by the rule", "$$", "(r_0, \\ldots, r_n)", "\\longrightarrow", "(r_0 \\circ r_1^{-1}, \\ldots, r_{n - 1} \\circ r_n^{-1})", "$$", "Compatibility with degeneracy maps is clear from the description of the", "degeneracies in Lemma \\ref{lemma-groupoid-simplicial}.", "We omit the verification that the maps respect the morphisms $s^n_j$.", "Groupoids, Lemma \\ref{groupoids-lemma-diagram-pull}", "(with the roles of $s$ and $t$ reversed)", "shows that the two right squares are cartesian. In exactly the same manner", "one shows all the other squares are cartesian too. Hence", "the morphism is cartesian." ], "refs": [ "spaces-simplicial-lemma-groupoid-simplicial", "groupoids-lemma-diagram-pull" ], "ref_ids": [ 9120, 9623 ] } ], "ref_ids": [ 9120, 9151 ] }, { "id": 9123, "type": "theorem", "label": "spaces-simplicial-lemma-equivalence-relation", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-equivalence-relation", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\pi : Y \\to (X/S)_\\bullet$ be a cartesian morphism of simplicial", "schemes, see Definitions \\ref{definition-cartesian-morphism} and", "\\ref{definition-fibre-products-simplicial-scheme}.", "Then the morphism", "$$", "j = (d^1_1, d^1_0) : Y_1 \\to Y_0 \\times_S Y_0", "$$", "defines an equivalence relation on $Y_0$ over $S$,", "see Groupoids, Definition \\ref{groupoids-definition-equivalence-relation}." ], "refs": [ "spaces-simplicial-definition-cartesian-morphism", "spaces-simplicial-definition-fibre-products-simplicial-scheme", "groupoids-definition-equivalence-relation" ], "proofs": [ { "contents": [ "Note that $j$ is a monomorphism. Namely the", "composition $Y_1 \\to Y_0 \\times_S Y_0 \\to Y_0 \\times_S X$", "is an isomorphism as $\\pi$ is cartesian.", "\\medskip\\noindent", "Consider the morphism", "$$", "(d^2_2, d^2_0) : Y_2 \\to Y_1 \\times_{d^1_0, Y_0, d^1_1} Y_1.", "$$", "This works because $d_0 \\circ d_2 = d_1 \\circ d_0$,", "see Simplicial, Remark \\ref{simplicial-remark-relations}.", "Also, it is a morphism over $(X/S)_2$. It is an isomorphism", "because $Y \\to (X/S)_\\bullet$ is cartesian. Note for example that the", "right hand side is isomorphic to", "$Y_0 \\times_{\\pi_0, X, \\text{pr}_1} (X \\times_S X \\times_S X) =", "X \\times_S Y_0 \\times_S X$", "because $\\pi$ is cartesian. Details omitted.", "\\medskip\\noindent", "As in Groupoids, Definition \\ref{groupoids-definition-equivalence-relation}", "we denote $t = \\text{pr}_0 \\circ j = d^1_1$ and", "$s = \\text{pr}_1 \\circ j = d^1_0$.", "The isomorphism above, combined with the morphism", "$d^2_1 : Y_2 \\to Y_1$ give us a composition morphism", "$$", "c : Y_1 \\times_{s, Y_0, t} Y_1 \\longrightarrow Y_1", "$$", "over $Y_0 \\times_S Y_0$. This immediately implies", "that for any scheme $T/S$ the relation", "$Y_1(T) \\subset Y_0(T) \\times Y_0(T)$ is transitive.", "\\medskip\\noindent", "Reflexivity follows from the fact that the", "restriction of the morphism $j$ to the diagonal", "$\\Delta : X \\to X \\times_S X$ is an isomorphism", "(again use the cartesian property of $\\pi$).", "\\medskip\\noindent", "To see symmetry we consider the morphism", "$$", "(d^2_2, d^2_1) : Y_2 \\to Y_1 \\times_{d^1_1, Y_0, d^1_1} Y_1.", "$$", "This works because $d_1 \\circ d_2 = d_1 \\circ d_1$,", "see Simplicial, Remark \\ref{simplicial-remark-relations}.", "It is an isomorphism", "because $Y \\to (X/S)_\\bullet$ is cartesian.", "Note for example that the", "right hand side is isomorphic to", "$Y_0 \\times_{\\pi_0, X, \\text{pr}_0} (X \\times_S X \\times_S X) =", "Y_0 \\times_S X \\times_S X$", "because $\\pi$ is cartesian. Details omitted.", "\\medskip\\noindent", "Let $T/S$ be a scheme. Let $a \\sim b$ for $a, b \\in Y_0(T)$", "be synonymous with $(a, b) \\in Y_1(T)$.", "The isomorphism $(d^2_2, d^2_1)$ above", "implies that if $a \\sim b$ and $a \\sim c$, then $b \\sim c$.", "Combined with reflexivity this shows that $\\sim$ is", "an equivalence relation." ], "refs": [ "simplicial-remark-relations", "groupoids-definition-equivalence-relation", "simplicial-remark-relations" ], "ref_ids": [ 14932, 9670, 14932 ] } ], "ref_ids": [ 9150, 9151, 9670 ] }, { "id": 9124, "type": "theorem", "label": "spaces-simplicial-lemma-equivalence-classes-points", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-equivalence-classes-points", "contents": [ "Let $X \\to S$ be a morphism of schemes. Suppose $Y \\to (X/S)_\\bullet$", "is a cartesian morphism of simplicial schemes. For $y \\in Y_0$ a point define", "$$", "T_y = \\{y' \\in Y_0 \\mid \\exists\\ y_1 \\in Y_1:", "d^1_1(y_1) = y, d^1_0(y_1) = y'\\}", "$$", "as a subset of $Y_0$. Then $y \\in T_y$ and", "$T_y \\cap T_{y'} \\not = \\emptyset \\Rightarrow T_y = T_{y'}$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-equivalence-relation} and", "Groupoids, Lemma", "\\ref{groupoids-lemma-pre-equivalence-equivalence-relation-points}." ], "refs": [ "spaces-simplicial-lemma-equivalence-relation", "groupoids-lemma-pre-equivalence-equivalence-relation-points" ], "ref_ids": [ 9123, 9579 ] } ], "ref_ids": [] }, { "id": 9125, "type": "theorem", "label": "spaces-simplicial-lemma-quasi-compact", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-quasi-compact", "contents": [ "Let $X \\to S$ be a morphism of schemes.", "Suppose $Y \\to (X/S)_\\bullet$ is a cartesian morphism of simplicial schemes.", "Let $y \\in Y_0$ be a point. If $X \\to S$ is quasi-compact, then", "$$", "T_y = \\{y' \\in Y_0 \\mid \\exists\\ y_1 \\in Y_1:", "d^1_1(y_1) = y, d^1_0(y_1) = y'\\}", "$$", "is a quasi-compact subset of $Y_0$." ], "refs": [], "proofs": [ { "contents": [ "Let $F_y$ be the scheme theoretic fibre of $d^1_1 : Y_1 \\to Y_0$", "at $y$. Then we see that $T_y$ is the image of the morphism", "$$", "\\xymatrix{", "F_y \\ar[r] \\ar[d] &", "Y_1 \\ar[r]^{d^1_0} \\ar[d]^{d^1_1} &", "Y_0 \\\\", "y \\ar[r] &", "Y_0 &", "}", "$$", "Note that $F_y$ is quasi-compact. This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9126, "type": "theorem", "label": "spaces-simplicial-lemma-descent-disjoint-union-Artinian-along-fields", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-descent-disjoint-union-Artinian-along-fields", "contents": [ "Let $X \\to S$ be a quasi-compact flat surjective morphism.", "Let $(V, \\varphi)$ be a descent datum relative", "to $X \\to S$. If $V$ is a disjoint union of", "spectra of Artinian rings, then $(V, \\varphi)$ is effective." ], "refs": [], "proofs": [ { "contents": [ "Let $Y \\to (X/S)_\\bullet$ be the cartesian morphism of simplicial", "schemes corresponding to $(V, \\varphi)$ by", "Lemma \\ref{lemma-cartesian-equivalent-descent-datum}.", "Observe that $Y_0 = V$.", "Write $V = \\coprod_{i \\in I} \\Spec(A_i)$ with each $A_i$ local", "Artinian. Moreover, let $v_i \\in V$ be the unique closed point of", "$\\Spec(A_i)$ for all $i \\in I$. Write $i \\sim j$ if and only if", "$v_i \\in T_{v_j}$ with notation as in", "Lemma \\ref{lemma-equivalence-classes-points} above.", "By Lemmas \\ref{lemma-equivalence-classes-points} and \\ref{lemma-quasi-compact}", "this is an equivalence relation with finite equivalence", "classes. Let $\\overline{I} = I/\\sim$. Then we can write", "$V = \\coprod_{\\overline{i} \\in \\overline{I}} V_{\\overline{i}}$", "with", "$V_{\\overline{i}} = \\coprod_{i \\in \\overline{i}} \\Spec(A_i)$.", "By construction we see that", "$\\varphi : V \\times_S X \\to X \\times_S V$ maps", "the open and closed subspaces $V_{\\overline{i}} \\times_S X$", "into the open and closed subspaces $X \\times_S V_{\\overline{i}}$.", "In other words, we get descent data", "$(V_{\\overline{i}}, \\varphi_{\\overline{i}})$, and", "$(V, \\varphi)$ is the coproduct of them in the category of", "descent data.", "Since each of the $V_{\\overline{i}}$ is a finite union of", "spectra of Artinian local rings the morphism $V_{\\overline{i}} \\to X$", "is affine, see Morphisms, Lemma \\ref{morphisms-lemma-Artinian-affine}.", "Since $\\{X \\to S\\}$ is an fpqc covering we see that all", "the descent data $(V_{\\overline{i}}, \\varphi_{\\overline{i}})$ are effective", "by Descent, Lemma \\ref{descent-lemma-affine}." ], "refs": [ "spaces-simplicial-lemma-cartesian-equivalent-descent-datum", "spaces-simplicial-lemma-equivalence-classes-points", "spaces-simplicial-lemma-equivalence-classes-points", "spaces-simplicial-lemma-quasi-compact", "morphisms-lemma-Artinian-affine", "descent-lemma-affine" ], "ref_ids": [ 9115, 9124, 9124, 9125, 5181, 14748 ] } ], "ref_ids": [] }, { "id": 9127, "type": "theorem", "label": "spaces-simplicial-lemma-compare-simplicial-objects-fppf-etale", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-compare-simplicial-objects-fppf-etale", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$.", "Let $a : U \\to X$ be an augmentation. There is a commutative diagram", "$$", "\\xymatrix{", "\\Sh((\\textit{Spaces}/U)_{fppf, total}) \\ar[r]_-h \\ar[d]_{a_{fppf}} &", "\\Sh(U_\\etale) \\ar[d]^a \\\\", "\\Sh((\\textit{Spaces}/X)_{fppf}) \\ar[r]^-{h_{-1}} &", "\\Sh(X_\\etale)", "}", "$$", "where the left vertical arrow is defined in", "Section \\ref{section-hypercovering}", "and the right vertical arrow is defined in", "Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "The notation $(\\textit{Spaces}/U)_{fppf, total}$ indicates that", "we are using the construction of", "Section \\ref{section-hypercovering}", "for the site $(\\textit{Spaces}/S)_{fppf}$ and the", "simplicial object $U$ of this site\\footnote{We could also", "use the \\'etale topology and this would be denoted", "$(\\textit{Spaces}/U)_{\\etale, total}$.}.", "We will use the sites $X_{spaces, \\etale}$ and $U_{spaces, \\etale}$", "for the topoi on the right hand side; this is permissible", "see discussion in Section \\ref{section-simplicial-algebraic-spaces}.", "\\medskip\\noindent", "Observe that both $(\\textit{Spaces}/U)_{fppf, total}$ and", "$U_{spaces, \\etale}$", "fall into case A of Situation \\ref{situation-simplicial-site}.", "This is immediate from the construction of", "$U_\\etale$ in Section \\ref{section-simplicial-algebraic-spaces}", "and it follows from Lemma \\ref{lemma-sr-when-fibre-products}", "for $(\\textit{Spaces}/U)_{fppf, total}$.", "Next, consider the functors", "$U_{n, spaces, \\etale} \\to (\\textit{Spaces}/U_n)_{fppf}$, $U \\mapsto U/U_n$", "and", "$X_{spaces, \\etale} \\to (\\textit{Spaces}/X)_{fppf}$, $U \\mapsto U/X$.", "We have seen that these define morphisms of sites in", "More on Cohomology of Spaces, Section", "\\ref{spaces-more-cohomology-section-fppf-etale}", "where these were denoted $a_{U_n} = \\epsilon_{U_n} \\circ \\pi_{u_n}$", "and $a_X = \\epsilon_X \\circ \\pi_X$.", "Thus we obtain a morphism of simplicial sites compatible with", "augmentations as in Remark \\ref{remark-morphism-augmentation-simplicial-sites}", "and we may apply", "Lemma \\ref{lemma-morphism-augmentation-simplicial-sites} to conclude." ], "refs": [ "spaces-simplicial-lemma-sr-when-fibre-products", "spaces-simplicial-remark-morphism-augmentation-simplicial-sites", "spaces-simplicial-lemma-morphism-augmentation-simplicial-sites" ], "ref_ids": [ 9098, 9154, 9030 ] } ], "ref_ids": [] }, { "id": 9128, "type": "theorem", "label": "spaces-simplicial-lemma-descent-sheaves-for-fppf-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-descent-sheaves-for-fppf-hypercovering", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$,", "then", "$$", "a^{-1} : \\Sh(X_\\etale) \\to \\Sh(U_\\etale)", "\\quad\\text{and}\\quad", "a^{-1} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(U_\\etale)", "$$", "are fully faithful with essential image the cartesian sheaves and", "quasi-inverse given by $a_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$", "is as in Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "We will prove the statement for sheaves of sets. It will be an", "almost formal consequence of results already established.", "Consider the diagram of", "Lemma \\ref{lemma-compare-simplicial-objects-fppf-etale}.", "In the proof of this lemma we have seen that", "$h_{-1}$ is the morphism $a_X$ of", "More on Cohomology of Spaces, Section", "\\ref{spaces-more-cohomology-section-fppf-etale}.", "Thus it follows from", "More on Cohomology of Spaces, Lemma", "\\ref{spaces-more-cohomology-lemma-comparison-fppf-etale}", "that $(h_{-1})^{-1}$ is fully faithful with quasi-inverse $h_{-1, *}$.", "The same holds true for the components $h_n$ of $h$.", "By the description of the functors $h^{-1}$ and $h_*$ of", "Lemma \\ref{lemma-morphism-simplicial-sites}", "we conclude that $h^{-1}$ is fully faithful with quasi-inverse $h_*$.", "Observe that $U$ is a hypercovering of $X$ in $(\\textit{Spaces}/S)_{fppf}$", "as defined in Section \\ref{section-hypercovering}.", "By Lemma \\ref{lemma-hypercovering-X-simple-descent-sheaves}", "we see that $a_{fppf}^{-1}$ is fully faithful with quasi-inverse", "$a_{fppf, *}$ and with essential image the cartesian sheaves", "on $(\\textit{Spaces}/U)_{fppf, total}$.", "A formal argument (chasing around the diagram) now shows that", "$a^{-1}$ is fully faithful.", "\\medskip\\noindent", "Finally, suppose that $\\mathcal{G}$ is a cartesian sheaf on $U_\\etale$.", "Then $h^{-1}\\mathcal{G}$ is a cartesian sheaf on", "$(\\textit{Spaces}/U)_{fppf, total}$. Hence", "$h^{-1}\\mathcal{G} = a_{fppf}^{-1}\\mathcal{H}$ for some sheaf", "$\\mathcal{H}$ on $(\\textit{Spaces}/X)_{fppf}$.", "In particular we find that", "$h_0^{-1}\\mathcal{G}_0 = (a_{0, big, fppf})^{-1}\\mathcal{H}$.", "Recalling that $h_0 = a_{U_0}$ and that $U_0 \\to X$ is", "flat, locally of finite presentation, and surjective, we", "find from More on Cohomology of Spaces,", "Lemma \\ref{spaces-more-cohomology-lemma-descent-sheaf-fppf-etale}", "that there exists a sheaf $\\mathcal{F}$ on $X_\\etale$ and isomorphism", "$\\mathcal{H} = (h_{-1})^{-1}\\mathcal{F}$.", "Since $a_{fppf}^{-1}\\mathcal{H} = h^{-1}\\mathcal{G}$", "we deduce that $h^{-1}\\mathcal{G} \\cong h^{-1}a^{-1}\\mathcal{F}$.", "By fully faithfulness of $h^{-1}$ we conclude that", "$a^{-1}\\mathcal{F} \\cong \\mathcal{G}$.", "\\medskip\\noindent", "Fix an isomorphism $\\theta : a^{-1}\\mathcal{F} \\to \\mathcal{G}$.", "To finish the proof we have to show $\\mathcal{G} = a^{-1}a_*\\mathcal{G}$", "(in order to show that the quasi-inverse is given by $a_*$; everything", "else has been proven above).", "Because $a^{-1}$ is fully faithful we have $\\text{id} \\cong a_*a^{-1}$ by", "Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}.", "Thus $\\mathcal{F} \\cong a_*a^{-1}\\mathcal{F}$ and", "$a_*\\theta : a_*a^{-1}\\mathcal{F} \\to a_*\\mathcal{G}$", "combine to an isomorphism $\\mathcal{F} \\to a_*\\mathcal{G}$.", "Pulling back by $a$ and precomposing by $\\theta^{-1}$", "we find the desired isomorphism." ], "refs": [ "spaces-simplicial-lemma-compare-simplicial-objects-fppf-etale", "spaces-more-cohomology-lemma-comparison-fppf-etale", "spaces-simplicial-lemma-morphism-simplicial-sites", "spaces-simplicial-lemma-hypercovering-X-simple-descent-sheaves", "spaces-more-cohomology-lemma-descent-sheaf-fppf-etale", "categories-lemma-adjoint-fully-faithful" ], "ref_ids": [ 9127, 3262, 9028, 9094, 3268, 12248 ] } ], "ref_ids": [] }, { "id": 9129, "type": "theorem", "label": "spaces-simplicial-lemma-cohomological-descent-for-fppf-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-cohomological-descent-for-fppf-hypercovering", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$,", "then for $K \\in D^+(X_\\etale)$", "$$", "K \\to Ra_*(a^{-1}K)", "$$", "is an isomorphism. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$", "is as in Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "Consider the diagram of Lemma \\ref{lemma-compare-simplicial-objects-fppf-etale}.", "Observe that $Rh_{n, *}h_n^{-1}$ is the identity functor", "on $D^+(U_{n, \\etale})$ by", "More on Cohomology of Spaces, Lemma", "\\ref{spaces-more-cohomology-lemma-cohomological-descent-etale-fppf}.", "Hence $Rh_*h^{-1}$ is the identity functor on", "$D^+(U_\\etale)$ by", "Lemma \\ref{lemma-direct-image-morphism-simplicial-sites}.", "We have", "\\begin{align*}", "Ra_*(a^{-1}K)", "& =", "Ra_*Rh_*h^{-1}a^{-1}K \\\\", "& =", "Rh_{-1, *}Ra_{fppf, *}a_{fppf}^{-1}(h_{-1})^{-1}K \\\\", "& =", "Rh_{-1, *}(h_{-1})^{-1}K \\\\", "& =", "K", "\\end{align*}", "The first equality by the discussion above, the second equality", "because of the commutativity of the diagram in", "Lemma \\ref{lemma-compare-simplicial-objects}, the third equality by", "Lemma \\ref{lemma-hypercovering-X-simple-descent-bounded-abelian}", "as $U$ is a hypercovering of $X$ in $(\\textit{Spaces}/S)_{fppf}$,", "and the last equality by the already used", "More on Cohomology of Spaces, Lemma", "\\ref{spaces-more-cohomology-lemma-cohomological-descent-etale-fppf}." ], "refs": [ "spaces-simplicial-lemma-compare-simplicial-objects-fppf-etale", "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf", "spaces-simplicial-lemma-direct-image-morphism-simplicial-sites", "spaces-simplicial-lemma-compare-simplicial-objects", "spaces-simplicial-lemma-hypercovering-X-simple-descent-bounded-abelian", "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf" ], "ref_ids": [ 9127, 3263, 9029, 9107, 9095, 3263 ] } ], "ref_ids": [] }, { "id": 9130, "type": "theorem", "label": "spaces-simplicial-lemma-compute-via-fppf-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-compute-via-fppf-hypercovering", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$, then", "$$", "R\\Gamma(X_\\etale, K) = R\\Gamma(U_\\etale, a^{-1}K)", "$$", "for $K \\in D^+(X_\\etale)$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$", "is as in Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-cohomological-descent-for-fppf-hypercovering}", "because $R\\Gamma(U_\\etale, -) = R\\Gamma(X_\\etale, -) \\circ Ra_*$ by", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-before-Leray}." ], "refs": [ "spaces-simplicial-lemma-cohomological-descent-for-fppf-hypercovering", "sites-cohomology-remark-before-Leray" ], "ref_ids": [ 9129, 4423 ] } ], "ref_ids": [] }, { "id": 9131, "type": "theorem", "label": "spaces-simplicial-lemma-fppf-hypercovering-equivalence-bounded", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-fppf-hypercovering-equivalence-bounded", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation.", "Let $\\mathcal{A} \\subset \\textit{Ab}(U_\\etale)$", "denote the weak Serre subcategory of cartesian abelian sheaves.", "If $U$ is an fppf hypercovering of $X$, then", "the functor $a^{-1}$ defines an equivalence", "$$", "D^+(X_\\etale) \\longrightarrow D_\\mathcal{A}^+(U_\\etale)", "$$", "with quasi-inverse $Ra_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$", "is as in Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\mathcal{A}$ is a weak Serre subcategory by", "Lemma \\ref{lemma-Serre-subcat-cartesian-modules}.", "The equivalence is a", "formal consequence of the results obtained so far. Use", "Lemmas \\ref{lemma-descent-sheaves-for-fppf-hypercovering} and", "\\ref{lemma-cohomological-descent-for-fppf-hypercovering} and", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-equivalence-bounded}." ], "refs": [ "spaces-simplicial-lemma-Serre-subcat-cartesian-modules", "spaces-simplicial-lemma-descent-sheaves-for-fppf-hypercovering", "spaces-simplicial-lemma-cohomological-descent-for-fppf-hypercovering", "sites-cohomology-lemma-equivalence-bounded" ], "ref_ids": [ 9056, 9128, 9129, 4290 ] } ], "ref_ids": [] }, { "id": 9132, "type": "theorem", "label": "spaces-simplicial-lemma-spectral-sequence-fppf-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-spectral-sequence-fppf-hypercovering", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation. Let $\\mathcal{F}$ be an abelian sheaf", "on $X_\\etale$. Let $\\mathcal{F}_n$ be the pullback to $U_{n, \\etale}$.", "If $U$ is an fppf hypercovering of $X$, then", "there exists a canonical spectral sequence", "$$", "E_1^{p, q} = H^q_\\etale(U_p, \\mathcal{F}_p)", "$$", "converging to $H^{p + q}_\\etale(X, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of Lemmas \\ref{lemma-compute-via-fppf-hypercovering}", "and \\ref{lemma-simplicial-sheaf-cohomology-site}." ], "refs": [ "spaces-simplicial-lemma-compute-via-fppf-hypercovering", "spaces-simplicial-lemma-simplicial-sheaf-cohomology-site" ], "ref_ids": [ 9130, 9039 ] } ], "ref_ids": [] }, { "id": 9133, "type": "theorem", "label": "spaces-simplicial-lemma-compare-simplicial-objects-fppf-etale-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-compare-simplicial-objects-fppf-etale-modules", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$.", "Let $a : U \\to X$ be an augmentation. There is a commutative diagram", "$$", "\\xymatrix{", "(\\Sh((\\textit{Spaces}/U)_{fppf, total}), \\mathcal{O}_{big, total})", "\\ar[r]_-h \\ar[d]_{a_{fppf}} &", "(\\Sh(U_\\etale), \\mathcal{O}_U) \\ar[d]^a \\\\", "(\\Sh((\\textit{Spaces}/X)_{fppf}), \\mathcal{O}_{big}) \\ar[r]^-{h_{-1}} &", "(\\Sh(X_\\etale), \\mathcal{O}_X)", "}", "$$", "of ringed topoi where the left vertical arrow is defined in", "Section \\ref{section-hypercovering-modules}", "and the right vertical arrow is defined in", "Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "For the underlying diagram of topoi we refer to the discussion in", "the proof of Lemma \\ref{lemma-compare-simplicial-objects-fppf-etale}.", "The sheaf $\\mathcal{O}_U$ is the structure sheaf of the", "simplicial algebraic space $U$ as defined in", "Section \\ref{section-simplicial-algebraic-spaces}.", "The sheaf $\\mathcal{O}_X$ is the usual structure sheaf of the algebraic", "space $X$. The sheaves of rings $\\mathcal{O}_{big, total}$ and", "$\\mathcal{O}_{big}$ come from the structure sheaf on", "$(\\textit{Spaces}/S)_{fppf}$ in the manner explained in", "Section \\ref{section-hypercovering-modules}", "which also constructs $a_{fppf}$ as a morphism of ringed topoi.", "The component morphisms $h_n = a_{U_n}$ and $h_{-1} = a_X$", "are morphisms of ringed topoi by", "More on Cohomology of Spaces, Section", "\\ref{spaces-more-cohomology-section-fppf-etale-modules}.", "Finally, since the continuous functor", "$u : U_{spaces, \\etale} \\to (\\textit{Spaces}/U)_{fppf, total}$", "used to define $h$\\footnote{This happened in the proof of", "Lemma \\ref{lemma-compare-simplicial-objects-fppf-etale}", "via an application of", "Lemma \\ref{lemma-morphism-augmentation-simplicial-sites}.}", "is given by $V/U_n \\mapsto V/U_n$", "we see that $h_*\\mathcal{O}_{big, total} = \\mathcal{O}_U$", "which is how we endow $h$ with the structure of a morphism", "of ringed simplicial sites as in", "Remark \\ref{remark-morphism-simplicial-sites-modules}.", "Then we obtain $h$ as a morphism of ringed topoi", "by Lemma \\ref{lemma-morphism-simplicial-sites-modules}.", "Please observe that the morphisms $h_n$ indeed agree", "with the morphisms $a_{U_n}$ described above.", "We omit the verification", "that the diagram is commutative (as a diagram of", "ringed topoi -- we already know it is commutative", "as a diagram of topoi)." ], "refs": [ "spaces-simplicial-lemma-compare-simplicial-objects-fppf-etale", "spaces-simplicial-lemma-compare-simplicial-objects-fppf-etale", "spaces-simplicial-lemma-morphism-augmentation-simplicial-sites", "spaces-simplicial-remark-morphism-simplicial-sites-modules", "spaces-simplicial-lemma-morphism-simplicial-sites-modules" ], "ref_ids": [ 9127, 9127, 9030, 9155, 9035 ] } ], "ref_ids": [] }, { "id": 9134, "type": "theorem", "label": "spaces-simplicial-lemma-descent-qcoh-for-fppf-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-descent-qcoh-for-fppf-hypercovering", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$,", "then", "$$", "a^* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_U)", "$$", "is an equivalence fully faithful with quasi-inverse given by $a_*$.", "Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$", "is as in Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "Consider the diagram of", "Lemma \\ref{lemma-compare-simplicial-objects-fppf-etale-modules}.", "In the proof of this lemma we have seen that", "$h_{-1}$ is the morphism $a_X$ of", "More on Cohomology of Spaces, Section", "\\ref{spaces-more-cohomology-section-fppf-etale-modules}.", "Thus it follows from", "More on Cohomology of Spaces, Lemma", "\\ref{spaces-more-cohomology-lemma-review-quasi-coherent}", "that", "$$", "(h_{-1})^* :", "\\QCoh(\\mathcal{O}_X)", "\\longrightarrow", "\\QCoh(\\mathcal{O}_{big})", "$$", "is an equivalence with quasi-inverse $h_{-1, *}$.", "The same holds true for the components $h_n$ of $h$.", "Recall that $\\QCoh(\\mathcal{O}_U)$ and $\\QCoh(\\mathcal{O}_{big, total})$", "consist of cartesian modules whose components are quasi-coherent, see", "Lemma \\ref{lemma-quasi-coherent-sheaf}.", "Since the functors $h^*$ and $h_*$ of", "Lemma \\ref{lemma-morphism-simplicial-sites-modules}", "agree with the functors $h_n^*$ and $h_{n, *}$ on components", "we conclude that", "$$", "h^* :", "\\QCoh(\\mathcal{O}_U)", "\\longrightarrow ", "\\QCoh(\\mathcal{O}_{big, total})", "$$", "is an equivalence with quasi-inverse $h_*$.", "Observe that $U$ is a hypercovering of $X$ in $(\\textit{Spaces}/S)_{fppf}$", "as defined in Section \\ref{section-hypercovering}.", "By Lemma \\ref{lemma-hypercovering-X-simple-descent-modules}", "we see that $a_{fppf}^*$ is fully faithful with quasi-inverse", "$a_{fppf, *}$ and with essential image the cartesian sheaves", "of $\\mathcal{O}_{fppf, total}$-modules.", "Thus, by the description of $\\QCoh(\\mathcal{O}_{big})$ and", "$\\QCoh(\\mathcal{O}_{big, total})$ of Lemma \\ref{lemma-quasi-coherent-sheaf},", "we get an equivalence", "$$", "a_{fppf}^* :", "\\QCoh(\\mathcal{O}_{big})", "\\longrightarrow", "\\QCoh(\\mathcal{O}_{big, total})", "$$", "with quasi-inverse given by $a_{fppf, *}$.", "A formal argument (chasing around the diagram) now shows that", "$a^*$ is fully faithful on $\\QCoh(\\mathcal{O}_X)$ and has", "image contained in $\\QCoh(\\mathcal{O}_U)$.", "\\medskip\\noindent", "Finally, suppose that $\\mathcal{G}$ is in $\\QCoh(\\mathcal{O}_U)$.", "Then $h^*\\mathcal{G}$ is in $\\QCoh(\\mathcal{O}_{big, total})$.", "Hence $h^*\\mathcal{G} = a_{fppf}^*\\mathcal{H}$ with", "$\\mathcal{H} = a_{fppf, *}h^*\\mathcal{G}$", "in $\\QCoh(\\mathcal{O}_{big})$ (see above).", "In turn we see that $\\mathcal{H} = (h_{-1})^*\\mathcal{F}$", "with $\\mathcal{F} = h_{-1, *}\\mathcal{H}$ in $\\QCoh(\\mathcal{O}_X)$.", "Going around the diagram we deduce that", "$h^*\\mathcal{G} \\cong h^*a^*\\mathcal{F}$.", "By fully faithfulness of $h^*$ we conclude that", "$a^*\\mathcal{F} \\cong \\mathcal{G}$.", "Since $\\mathcal{F} = h_{-1, *}a_{fppf, *}h^*\\mathcal{G} =", "a_*h_*h^*\\mathcal{G} = a_*\\mathcal{G}$ we also obtain", "the statement that the quasi-inverse is given by $a_*$." ], "refs": [ "spaces-simplicial-lemma-compare-simplicial-objects-fppf-etale-modules", "spaces-more-cohomology-lemma-review-quasi-coherent", "spaces-simplicial-lemma-quasi-coherent-sheaf", "spaces-simplicial-lemma-morphism-simplicial-sites-modules", "spaces-simplicial-lemma-hypercovering-X-simple-descent-modules", "spaces-simplicial-lemma-quasi-coherent-sheaf" ], "ref_ids": [ 9133, 3269, 9059, 9035, 9099, 9059 ] } ], "ref_ids": [] }, { "id": 9135, "type": "theorem", "label": "spaces-simplicial-lemma-cohomological-descent-qcoh-for-fppf-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-cohomological-descent-qcoh-for-fppf-hypercovering", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$,", "then for $\\mathcal{F}$ a quasi-coherent $\\mathcal{O}_X$-module", "the map", "$$", "\\mathcal{F} \\to Ra_*(a^*\\mathcal{F})", "$$", "is an isomorphism. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$", "is as in Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "Consider the diagram of Lemma \\ref{lemma-compare-simplicial-objects-fppf-etale}.", "Let $\\mathcal{F}_n = a_n^*\\mathcal{F}$ be the $n$th component of", "$a^*\\mathcal{F}$. This is a quasi-coherent $\\mathcal{O}_{U_n}$-module.", "Then $\\mathcal{F}_n = Rh_{n, *}h_n^*\\mathcal{F}_n$", "by More on Cohomology of Spaces, Lemma", "\\ref{spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules}.", "Hence $a^*\\mathcal{F} = Rh_*h^*a^*\\mathcal{F}$ by", "Lemma \\ref{lemma-direct-image-morphism-simplicial-sites-modules}.", "We have", "\\begin{align*}", "Ra_*(a^*\\mathcal{F})", "& =", "Ra_*Rh_*h^*a^*\\mathcal{F} \\\\", "& =", "Rh_{-1, *}Ra_{fppf, *}a_{fppf}^*(h_{-1})^*\\mathcal{F} \\\\", "& =", "Rh_{-1, *}(h_{-1})^*\\mathcal{F} \\\\", "& =", "\\mathcal{F}", "\\end{align*}", "The first equality by the discussion above, the second equality", "because of the commutativity of the diagram in", "Lemma \\ref{lemma-compare-simplicial-objects}, the third equality by", "Lemma \\ref{lemma-hypercovering-X-simple-descent-bounded-modules}", "as $U$ is a hypercovering of $X$ in $(\\textit{Spaces}/S)_{fppf}$", "and $La_{fppf}^* = a_{fppf}^*$ as $a_{fppf}$ is flat", "(namely $a_{fppf}^{-1}\\mathcal{O}_{big} = \\mathcal{O}_{big, total}$,", "see Remark \\ref{remark-augmentation-ringed}), and", "the last equality by the already used", "More on Cohomology of Spaces, Lemma", "\\ref{spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules}." ], "refs": [ "spaces-simplicial-lemma-compare-simplicial-objects-fppf-etale", "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules", "spaces-simplicial-lemma-direct-image-morphism-simplicial-sites-modules", "spaces-simplicial-lemma-compare-simplicial-objects", "spaces-simplicial-lemma-hypercovering-X-simple-descent-bounded-modules", "spaces-simplicial-remark-augmentation-ringed", "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules" ], "ref_ids": [ 9127, 3270, 9036, 9107, 9100, 9161, 3270 ] } ], "ref_ids": [] }, { "id": 9136, "type": "theorem", "label": "spaces-simplicial-lemma-coh-descent-qcoh-unbounded-for-fppf-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-coh-descent-qcoh-unbounded-for-fppf-hypercovering", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation. Assume $a : U \\to X$ is an fppf hypercovering of $X$.", "Then $\\QCoh(\\mathcal{O}_U)$ is a weak Serre subcategory of", "$\\textit{Mod}(\\mathcal{O}_U)$ and", "$$", "a^* : D_\\QCoh(\\mathcal{O}_X) \\longrightarrow D_\\QCoh(\\mathcal{O}_U)", "$$", "is an equivalence of categories with quasi-inverse given by", "$Ra_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$", "is as in Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "First observe that the maps $a_n : U_n \\to X$ and $d^n_i : U_n \\to U_{n - 1}$", "are flat, locally of finite presentation, and surjective by", "Hypercoverings, Remark \\ref{hypercovering-remark-P-covering}.", "\\medskip\\noindent", "Recall that an $\\mathcal{O}_U$-module $\\mathcal{F}$ is quasi-coherent if and", "only if it is cartesian and $\\mathcal{F}_n$ is quasi-coherent for all $n$.", "See Lemma \\ref{lemma-quasi-coherent-sheaf}.", "By Lemma \\ref{lemma-Serre-subcat-cartesian-modules}", "(and flatness of the maps $d^n_i : U_n \\to U_{n - 1}$ shown above)", "the cartesian modules for a weak Serre subcategory of", "$\\textit{Mod}(\\mathcal{O}_U)$. On the other hand", "$\\QCoh(\\mathcal{O}_{U_n}) \\subset \\textit{Mod}(\\mathcal{O}_{U_n})$", "is a weak Serre subcategory for each $n$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-properties-quasi-coherent}).", "Combined we see that", "$\\QCoh(\\mathcal{O}_U) \\subset \\textit{Mod}(\\mathcal{O}_U)$", "is a weak Serre subcategory.", "\\medskip\\noindent", "To finish the proof we check the conditions (1) -- (5) of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-equivalence-unbounded-one} one by one.", "\\medskip\\noindent", "Ad (1). This holds since $a_n$ flat (seen above) implies $a$ is flat", "by Lemma \\ref{lemma-flat-augmentation-modules}.", "\\medskip\\noindent", "Ad (2). This is the content of", "Lemma \\ref{lemma-descent-qcoh-for-fppf-hypercovering}.", "\\medskip\\noindent", "Ad (3). This is the content of", "Lemma \\ref{lemma-cohomological-descent-qcoh-for-fppf-hypercovering}.", "\\medskip\\noindent", "Ad (4). Recall that we can use either the site $U_\\etale$ or", "$U_{spaces, \\etale}$ to define the small \\'etale topos", "$\\Sh(U_\\etale)$, see Section \\ref{section-simplicial-algebraic-spaces}.", "The assumption of", "Cohomology on Sites, Situation \\ref{sites-cohomology-situation-olsson-laszlo}", "holds for the triple", "$(U_{spaces, \\etale}, \\mathcal{O}_U, \\QCoh(\\mathcal{O}_U))$", "and by the same reasoning for the triple", "$(U_\\etale, \\mathcal{O}_U, \\QCoh(\\mathcal{O}_U))$.", "Namely, take", "$$", "\\mathcal{B} \\subset \\Ob(U_\\etale) \\subset \\Ob(U_{spaces, \\etale})", "$$", "to be the set of affine objects. For $V/U_n \\in \\mathcal{B}$", "take $d_{V/U_n} = 0$ and take $\\text{Cov}_{V/U_n}$ to be the set of", "\\'etale coverings $\\{V_i \\to V\\}$ with $V_i$ affine.", "Then we get the desired vanishing because for", "$\\mathcal{F} \\in \\QCoh(\\mathcal{O}_U)$", "and any $V/U_n \\in \\mathcal{B}$ we have", "$$", "H^p(V/U_n, \\mathcal{F}) = H^p(V, \\mathcal{F}_n)", "$$", "by Lemma \\ref{lemma-sanity-check-modules}. Here on the", "right hand side we have the cohomology of the quasi-coherent", "sheaf $\\mathcal{F}_n$ on $U_n$ over the affine obect $V$", "of $U_{n, \\etale}$. This vanishes for $p > 0$ by the discussion in", "Cohomology of Spaces, Section", "\\ref{spaces-cohomology-section-higher-direct-image} and", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "\\medskip\\noindent", "Ad (5). Follows by taking $\\mathcal{B} \\subset \\Ob(X_{spaces, \\etale})$", "the set of affine objects and the references given above." ], "refs": [ "hypercovering-remark-P-covering", "spaces-simplicial-lemma-quasi-coherent-sheaf", "spaces-simplicial-lemma-Serre-subcat-cartesian-modules", "spaces-properties-lemma-properties-quasi-coherent", "sites-cohomology-lemma-equivalence-unbounded-one", "spaces-simplicial-lemma-flat-augmentation-modules", "spaces-simplicial-lemma-descent-qcoh-for-fppf-hypercovering", "spaces-simplicial-lemma-cohomological-descent-qcoh-for-fppf-hypercovering", "spaces-simplicial-lemma-sanity-check-modules", "coherent-lemma-quasi-coherent-affine-cohomology-zero" ], "ref_ids": [ 8428, 9059, 9056, 11912, 4291, 9048, 9134, 9135, 9047, 3282 ] } ], "ref_ids": [] }, { "id": 9137, "type": "theorem", "label": "spaces-simplicial-lemma-compute-via-fppf-hypercovering-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-compute-via-fppf-hypercovering-modules", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$, then", "$$", "R\\Gamma(X_\\etale, K) = R\\Gamma(U_\\etale, a^*K)", "$$", "for $K \\in D_\\QCoh(\\mathcal{O}_X)$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$", "is as in Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-coh-descent-qcoh-unbounded-for-fppf-hypercovering}", "because $R\\Gamma(U_\\etale, -) = R\\Gamma(X_\\etale, -) \\circ Ra_*$ by", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-before-Leray}." ], "refs": [ "spaces-simplicial-lemma-coh-descent-qcoh-unbounded-for-fppf-hypercovering", "sites-cohomology-remark-before-Leray" ], "ref_ids": [ 9136, 4423 ] } ], "ref_ids": [] }, { "id": 9138, "type": "theorem", "label": "spaces-simplicial-lemma-spectral-sequence-fppf-hypercovering-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-spectral-sequence-fppf-hypercovering-modules", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation. Let $\\mathcal{F}$ be quasi-coherent", "$\\mathcal{O}_X$-module. Let $\\mathcal{F}_n$ be the pullback to", "$U_{n, \\etale}$. If $U$ is an fppf hypercovering of $X$, then", "there exists a canonical spectral sequence", "$$", "E_1^{p, q} = H^q_\\etale(U_p, \\mathcal{F}_p)", "$$", "converging to $H^{p + q}_\\etale(X, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Lemmas \\ref{lemma-compute-via-fppf-hypercovering-modules}", "and \\ref{lemma-simplicial-module-cohomology-site}." ], "refs": [ "spaces-simplicial-lemma-compute-via-fppf-hypercovering-modules", "spaces-simplicial-lemma-simplicial-module-cohomology-site" ], "ref_ids": [ 9137, 9046 ] } ], "ref_ids": [] }, { "id": 9139, "type": "theorem", "label": "spaces-simplicial-lemma-fppf-neg-ext-zero-hom", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-fppf-neg-ext-zero-hom", "contents": [ "Let $X$ be an algebraic space over a scheme $S$.", "Let $K, E \\in D_\\QCoh(\\mathcal{O}_X)$.", "Let $a : U \\to X$ be an fppf hypercovering.", "Assume that for all $n \\geq 0$ we have", "$$", "\\Ext_{\\mathcal{O}_{U_n}}^i(La_n^*K, La_n^*E) = 0", "\\text{ for } i < 0", "$$", "Then we have", "\\begin{enumerate}", "\\item $\\Ext_{\\mathcal{O}_X}^i(K, E) = 0$ for $i < 0$, and", "\\item there is an exact sequence", "$$", "0", "\\to", "\\Hom_{\\mathcal{O}_X}(K, E)", "\\to", "\\Hom_{\\mathcal{O}_{U_0}}(La_0^*K, La_0^*E)", "\\to", "\\Hom_{\\mathcal{O}_{U_1}}(La_1^*K, La_1^*E)", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $K_n = La_n^*K$ and $E_n = La_n^*E$. Then these are the", "simplicial systems of the derived category of modules", "(Definition \\ref{definition-cartesian-derived-modules})", "associated to $La^*K$ and $La^*E$", "(Lemma \\ref{lemma-cartesian-objects-derived-modules})", "where $a : U_\\etale \\to X_\\etale$ is as in", "Section \\ref{section-simplicial-algebraic-spaces}.", "Let us prove (2) first. By", "Lemma \\ref{lemma-coh-descent-qcoh-unbounded-for-fppf-hypercovering}", "we have", "$$", "\\Hom_{\\mathcal{O}_X}(K, E) =", "\\Hom_{\\mathcal{O}_U}(La^*K, La^*E)", "$$", "Thus the sequence looks like this:", "$$", "0", "\\to", "\\Hom_{\\mathcal{O}_U}(La^*K, La^*E)", "\\to", "\\Hom_{\\mathcal{O}_{U_0}}(K_0, E_0)", "\\to", "\\Hom_{\\mathcal{O}_{U_1}}(K_1, E_1)", "$$", "The first arrow is injective by", "Lemma \\ref{lemma-nullity-cartesian-modules-derived}.", "The image of this arrow is the kernel of the second", "by Lemma \\ref{lemma-hom-cartesian-modules-derived}.", "This finishes the proof of (2).", "Part (1) follows by applying part (2) with", "$K[i]$ and $E$ for $i > 0$." ], "refs": [ "spaces-simplicial-definition-cartesian-derived-modules", "spaces-simplicial-lemma-cartesian-objects-derived-modules", "spaces-simplicial-lemma-coh-descent-qcoh-unbounded-for-fppf-hypercovering", "spaces-simplicial-lemma-nullity-cartesian-modules-derived", "spaces-simplicial-lemma-hom-cartesian-modules-derived" ], "ref_ids": [ 9149, 9065, 9136, 9067, 9068 ] } ], "ref_ids": [] }, { "id": 9140, "type": "theorem", "label": "spaces-simplicial-lemma-fppf-glue-neg-ext-zero", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-fppf-glue-neg-ext-zero", "contents": [ "Let $X$ be an algebraic space over a scheme $S$.", "Let $a : U \\to X$ be an fppf hypercovering.", "Suppose given $K_0 \\in D_\\QCoh(U_0)$ and an isomorphism", "$$", "\\alpha :", "L(f_{\\delta_1^1})^*K_0", "\\longrightarrow", "L(f_{\\delta_0^1})^*K_0", "$$", "satisfying the cocycle condition on $U_1$. Set", "$\\tau^n_i : [0] \\to [n]$, $0 \\mapsto i$ and", "set $K_n = Lf_{\\tau^n_n}^*K_0$.", "Assume $\\Ext^i_{\\mathcal{O}_{U_n}}(K_n, K_n) = 0$ for $i < 0$.", "Then there exists an object $K \\in D_\\QCoh(\\mathcal{O}_X)$", "and an isomorphism $La_0^*K \\to K$ compatible with $\\alpha$." ], "refs": [], "proofs": [ { "contents": [ "We claim that the objects $K_n$ form the members of a ", "simplicial system of the derived category of modules", "(Definition \\ref{definition-cartesian-derived-modules})", "of the ringed simplicial site $U_\\etale$ of", "Section \\ref{section-simplicial-algebraic-spaces}.", "The construction is analogous to the construction discussed in", "Descent, Section \\ref{descent-section-descent-modules} from which we borrow", "the notation $\\tau^n_i : [0] \\to [n]$, $0 \\mapsto i$ and", "$\\tau^n_{ij} : [1] \\to [n]$, $0 \\mapsto i$, $1 \\mapsto j$.", "Given $\\varphi : [n] \\to [m]$ we define", "$K_\\varphi : L(f_\\varphi)^*K_n \\to K_m$", "using", "$$", "\\xymatrix{", "L(f_\\varphi)^*K_n \\ar@{=}[r] &", "L(f_\\varphi)^* L(f_{\\tau^n_n})^*K_0 \\ar@{=}[r] &", "L(f_{\\tau^m_{\\varphi(n)}})^*K_0 \\ar@{=}[r] &", "L(f_{\\tau^m_{\\varphi(n)m}})^* L(f_{\\delta^1_1})^*K_0", "\\ar[d]_{L(f_{\\tau^m_{\\varphi(n)m}})^*\\alpha} \\\\", "&", "K_m \\ar@{=}[r] &", "L(f_{\\tau^m_m})^*K_0 \\ar@{=}[r] &", "L(f_{\\tau^m_{\\varphi(n)m}})^* L(f_{\\delta^1_0})^*K_0", "}", "$$", "We omit the verification that the cocycle condition", "implies the maps compose correctly (in their respective", "derived categories) and hence give rise to a", "simplicial systems of the derived category of modules\\footnote{This", "verification is the same as that done in the proof", "of Lemma \\ref{lemma-characterize-cartesian}", "as well as in the chapter on descent referenced", "above. We should probably write this as a general lemma about", "fibred and cofibred categories over $\\Delta$.}.", "Once this is verified, we obtain an object", "$K' \\in D_\\QCoh(\\mathcal{O}_{U_\\etale})$", "such that $(K_n, K_\\varphi)$ is the system deduced from $K'$, see", "Lemma \\ref{lemma-cartesian-module-derived-from-simplicial}.", "Finally, we apply", "Lemma \\ref{lemma-coh-descent-qcoh-unbounded-for-fppf-hypercovering}", "to see that $K' = La^*K$ for some $K \\in D_\\QCoh(\\mathcal{O}_X)$", "as desired." ], "refs": [ "spaces-simplicial-definition-cartesian-derived-modules", "spaces-simplicial-lemma-characterize-cartesian", "spaces-simplicial-lemma-cartesian-module-derived-from-simplicial", "spaces-simplicial-lemma-coh-descent-qcoh-unbounded-for-fppf-hypercovering" ], "ref_ids": [ 9149, 9054, 9069, 9136 ] } ], "ref_ids": [] }, { "id": 9141, "type": "theorem", "label": "spaces-simplicial-lemma-compare-simplicial-objects-ph-etale", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-compare-simplicial-objects-ph-etale", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$.", "Let $a : U \\to X$ be an augmentation. There is a commutative diagram", "$$", "\\xymatrix{", "\\Sh((\\textit{Spaces}/U)_{ph, total}) \\ar[r]_-h \\ar[d]_{a_{ph}} &", "\\Sh(U_\\etale) \\ar[d]^a \\\\", "\\Sh((\\textit{Spaces}/X)_{ph}) \\ar[r]^-{h_{-1}} &", "\\Sh(X_\\etale)", "}", "$$", "where the left vertical arrow is defined in", "Section \\ref{section-hypercovering}", "and the right vertical arrow is defined in", "Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "The notation $(\\textit{Spaces}/U)_{ph, total}$ indicates that", "we are using the construction of", "Section \\ref{section-hypercovering}", "for the site $(\\textit{Spaces}/S)_{ph}$ and the", "simplicial object $U$ of this site\\footnote{To distinguish from", "$(\\textit{Spaces}/U)_{fppf, total}$ defined using the fppf", "topology in Section \\ref{section-fppf-hypercovering}.}.", "We will use the sites $X_{spaces, \\etale}$ and $U_{spaces, \\etale}$", "for the topoi on the right hand side; this is permissible", "see discussion in Section \\ref{section-simplicial-algebraic-spaces}.", "\\medskip\\noindent", "Observe that both $(\\textit{Spaces}/U)_{ph, total}$ and", "$U_{spaces, \\etale}$", "fall into case A of Situation \\ref{situation-simplicial-site}.", "This is immediate from the construction of", "$U_\\etale$ in Section \\ref{section-simplicial-algebraic-spaces}", "and it follows from Lemma \\ref{lemma-sr-when-fibre-products}", "for $(\\textit{Spaces}/U)_{ph, total}$.", "Next, consider the functors", "$U_{n, spaces, \\etale} \\to (\\textit{Spaces}/U_n)_{ph}$, $U \\mapsto U/U_n$", "and", "$X_{spaces, \\etale} \\to (\\textit{Spaces}/X)_{ph}$, $U \\mapsto U/X$.", "We have seen that these define morphisms of sites in", "More on Cohomology of Spaces, Section", "\\ref{spaces-more-cohomology-section-ph-etale}", "where these were denoted $a_{U_n} = \\epsilon_{U_n} \\circ \\pi_{u_n}$", "and $a_X = \\epsilon_X \\circ \\pi_X$.", "Thus we obtain a morphism of simplicial sites compatible with", "augmentations as in Remark \\ref{remark-morphism-augmentation-simplicial-sites}", "and we may apply", "Lemma \\ref{lemma-morphism-augmentation-simplicial-sites} to conclude." ], "refs": [ "spaces-simplicial-lemma-sr-when-fibre-products", "spaces-simplicial-remark-morphism-augmentation-simplicial-sites", "spaces-simplicial-lemma-morphism-augmentation-simplicial-sites" ], "ref_ids": [ 9098, 9154, 9030 ] } ], "ref_ids": [] }, { "id": 9142, "type": "theorem", "label": "spaces-simplicial-lemma-descent-sheaves-for-ph-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-descent-sheaves-for-ph-hypercovering", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation. If $a : U \\to X$ is a proper hypercovering of $X$,", "then", "$$", "a^{-1} : \\Sh(X_\\etale) \\to \\Sh(U_\\etale)", "\\quad\\text{and}\\quad", "a^{-1} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(U_\\etale)", "$$", "are fully faithful with essential image the cartesian sheaves and", "quasi-inverse given by $a_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$", "is as in Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "We will prove the statement for sheaves of sets. It will be an", "almost formal consequence of results already established.", "Consider the diagram of", "Lemma \\ref{lemma-compare-simplicial-objects-ph-etale}.", "In the proof of this lemma we have seen that", "$h_{-1}$ is the morphism $a_X$ of", "More on Cohomology of Spaces, Section", "\\ref{spaces-more-cohomology-section-ph-etale}.", "Thus it follows from", "More on Cohomology of Spaces, Lemma", "\\ref{spaces-more-cohomology-lemma-comparison-ph-etale}", "that $(h_{-1})^{-1}$ is fully faithful with quasi-inverse $h_{-1, *}$.", "The same holds true for the components $h_n$ of $h$.", "By the description of the functors $h^{-1}$ and $h_*$ of", "Lemma \\ref{lemma-morphism-simplicial-sites}", "we conclude that $h^{-1}$ is fully faithful with quasi-inverse $h_*$.", "Observe that $U$ is a hypercovering of $X$ in $(\\textit{Spaces}/S)_{ph}$", "as defined in Section \\ref{section-hypercovering} since a surjective", "proper morphism gives a ph covering by Topologies on Spaces, Lemma", "\\ref{spaces-topologies-lemma-surjective-proper-ph}.", "By Lemma \\ref{lemma-hypercovering-X-simple-descent-sheaves}", "we see that $a_{ph}^{-1}$ is fully faithful with quasi-inverse", "$a_{ph, *}$ and with essential image the cartesian sheaves", "on $(\\textit{Spaces}/U)_{ph, total}$.", "A formal argument (chasing around the diagram) now shows that", "$a^{-1}$ is fully faithful.", "\\medskip\\noindent", "Finally, suppose that $\\mathcal{G}$ is a cartesian sheaf on $U_\\etale$.", "Then $h^{-1}\\mathcal{G}$ is a cartesian sheaf on", "$(\\textit{Spaces}/U)_{ph, total}$.", "Hence $h^{-1}\\mathcal{G} = a_{ph}^{-1}\\mathcal{H}$ for some sheaf", "$\\mathcal{H}$ on $(\\textit{Spaces}/X)_{ph}$.", "We compute using somewhat pedantic notation", "\\begin{align*}", "(h_{-1})^{-1}(a_*\\mathcal{G})", "& =", "(h_{-1})^{-1}", "\\text{Eq}(", "\\xymatrix{", "a_{0, small, *}\\mathcal{G}_0", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "a_{1, small, *}\\mathcal{G}_1", "}", ") \\\\", "& =", "\\text{Eq}(", "\\xymatrix{", "(h_{-1})^{-1}a_{0, small, *}\\mathcal{G}_0", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "(h_{-1})^{-1}a_{1, small, *}\\mathcal{G}_1", "}", ") \\\\", "& =", "\\text{Eq}(", "\\xymatrix{", "a_{0, big, ph, *}h_0^{-1}\\mathcal{G}_0", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "a_{1, big, ph, *}h_1^{-1}\\mathcal{G}_1", "}", ") \\\\", "& =", "\\text{Eq}(", "\\xymatrix{", "a_{0, big, ph, *}(a_{0, big, ph})^{-1}\\mathcal{H}", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "a_{1, big, ph, *}(a_{1, big, ph})^{-1}\\mathcal{H}", "}", ") \\\\", "& =", "a_{ph, *}a_{ph}^{-1}\\mathcal{H} \\\\", "& =", "\\mathcal{H}", "\\end{align*}", "Here the first equality follows from Lemma \\ref{lemma-augmentation-site},", "the second equality follows as $(h_{-1})^{-1}$ is an exact functor,", "the third equality follows from", "More on Cohomology of Spaces, Lemma", "\\ref{spaces-more-cohomology-lemma-proper-push-pull-ph-etale}", "(here we use that $a_0 : U_0 \\to X$ and $a_1: U_1 \\to X$ are proper),", "the fourth follows from $a_{ph}^{-1}\\mathcal{H} = h^{-1}\\mathcal{G}$,", "the fifth from Lemma \\ref{lemma-augmentation-site}, and the", "sixth we've seen above. Since $a_{ph}^{-1}\\mathcal{H} = h^{-1}\\mathcal{G}$", "we deduce that $h^{-1}\\mathcal{G} \\cong h^{-1}a^{-1}a_*\\mathcal{G}$", "which ends the proof by fully faithfulness of $h^{-1}$." ], "refs": [ "spaces-simplicial-lemma-compare-simplicial-objects-ph-etale", "spaces-more-cohomology-lemma-comparison-ph-etale", "spaces-simplicial-lemma-morphism-simplicial-sites", "spaces-topologies-lemma-surjective-proper-ph", "spaces-simplicial-lemma-hypercovering-X-simple-descent-sheaves", "spaces-simplicial-lemma-augmentation-site", "spaces-more-cohomology-lemma-proper-push-pull-ph-etale", "spaces-simplicial-lemma-augmentation-site" ], "ref_ids": [ 9141, 3273, 9028, 3671, 9094, 9027, 3277, 9027 ] } ], "ref_ids": [] }, { "id": 9143, "type": "theorem", "label": "spaces-simplicial-lemma-cohomological-descent-for-ph-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-cohomological-descent-for-ph-hypercovering", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation. If $a : U \\to X$ is a proper hypercovering of $X$,", "then for $K \\in D^+(X_\\etale)$", "$$", "K \\to Ra_*(a^{-1}K)", "$$", "is an isomorphism. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$", "is as in Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "Consider the diagram of Lemma \\ref{lemma-compare-simplicial-objects-ph-etale}.", "Observe that $Rh_{n, *}h_n^{-1}$ is the identity functor", "on $D^+(U_{n, \\etale})$ by", "More on Cohomology of Spaces, Lemma", "\\ref{spaces-more-cohomology-lemma-cohomological-descent-etale-ph}.", "Hence $Rh_*h^{-1}$ is the identity functor on", "$D^+(U_\\etale)$ by", "Lemma \\ref{lemma-direct-image-morphism-simplicial-sites}.", "We have", "\\begin{align*}", "Ra_*(a^{-1}K)", "& =", "Ra_*Rh_*h^{-1}a^{-1}K \\\\", "& =", "Rh_{-1, *}Ra_{ph, *}a_{ph}^{-1}(h_{-1})^{-1}K \\\\", "& =", "Rh_{-1, *}(h_{-1})^{-1}K \\\\", "& =", "K", "\\end{align*}", "The first equality by the discussion above, the second equality", "because of the commutativity of the diagram in", "Lemma \\ref{lemma-compare-simplicial-objects}, the third equality by", "Lemma \\ref{lemma-hypercovering-X-simple-descent-bounded-abelian}", "as $U$ is a hypercovering of $X$ in $(\\textit{Spaces}/S)_{ph}$", "by Topologies on Spaces, Lemma", "\\ref{spaces-topologies-lemma-surjective-proper-ph},", "and the last equality by the already used", "More on Cohomology of Spaces, Lemma", "\\ref{spaces-more-cohomology-lemma-cohomological-descent-etale-ph}." ], "refs": [ "spaces-simplicial-lemma-compare-simplicial-objects-ph-etale", "spaces-more-cohomology-lemma-cohomological-descent-etale-ph", "spaces-simplicial-lemma-direct-image-morphism-simplicial-sites", "spaces-simplicial-lemma-compare-simplicial-objects", "spaces-simplicial-lemma-hypercovering-X-simple-descent-bounded-abelian", "spaces-topologies-lemma-surjective-proper-ph", "spaces-more-cohomology-lemma-cohomological-descent-etale-ph" ], "ref_ids": [ 9141, 3274, 9029, 9107, 9095, 3671, 3274 ] } ], "ref_ids": [] }, { "id": 9144, "type": "theorem", "label": "spaces-simplicial-lemma-compute-via-ph-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-compute-via-ph-hypercovering", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation. If $a : U \\to X$ is a proper hypercovering of $X$, then", "$$", "R\\Gamma(X_\\etale, K) = R\\Gamma(U_\\etale, a^{-1}K)", "$$", "for $K \\in D^+(X_\\etale)$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$", "is as in Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-cohomological-descent-for-ph-hypercovering}", "because $R\\Gamma(U_\\etale, -) = R\\Gamma(X_\\etale, -) \\circ Ra_*$ by", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-before-Leray}." ], "refs": [ "spaces-simplicial-lemma-cohomological-descent-for-ph-hypercovering", "sites-cohomology-remark-before-Leray" ], "ref_ids": [ 9143, 4423 ] } ], "ref_ids": [] }, { "id": 9145, "type": "theorem", "label": "spaces-simplicial-lemma-ph-hypercovering-equivalence-bounded", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-ph-hypercovering-equivalence-bounded", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation.", "Let $\\mathcal{A} \\subset \\textit{Ab}(U_\\etale)$", "denote the weak Serre subcategory of cartesian abelian sheaves.", "If $U$ is a proper hypercovering of $X$, then", "the functor $a^{-1}$ defines an equivalence", "$$", "D^+(X_\\etale) \\longrightarrow D_\\mathcal{A}^+(U_\\etale)", "$$", "with quasi-inverse $Ra_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$", "is as in Section \\ref{section-simplicial-algebraic-spaces}." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\mathcal{A}$ is a weak Serre subcategory by", "Lemma \\ref{lemma-Serre-subcat-cartesian-modules}.", "The equivalence is a", "formal consequence of the results obtained so far. Use", "Lemmas \\ref{lemma-descent-sheaves-for-ph-hypercovering} and", "\\ref{lemma-cohomological-descent-for-ph-hypercovering} and", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-equivalence-bounded}." ], "refs": [ "spaces-simplicial-lemma-Serre-subcat-cartesian-modules", "spaces-simplicial-lemma-descent-sheaves-for-ph-hypercovering", "spaces-simplicial-lemma-cohomological-descent-for-ph-hypercovering", "sites-cohomology-lemma-equivalence-bounded" ], "ref_ids": [ 9056, 9142, 9143, 4290 ] } ], "ref_ids": [] }, { "id": 9146, "type": "theorem", "label": "spaces-simplicial-lemma-spectral-sequence-ph-hypercovering", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-lemma-spectral-sequence-ph-hypercovering", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$", "be an augmentation. Let $\\mathcal{F}$ be an abelian sheaf", "on $X_\\etale$. Let $\\mathcal{F}_n$ be the pullback to $U_{n, \\etale}$.", "If $U$ is a ph hypercovering of $X$, then", "there exists a canonical spectral sequence", "$$", "E_1^{p, q} = H^q_\\etale(U_p, \\mathcal{F}_p)", "$$", "converging to $H^{p + q}_\\etale(X, \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of Lemmas \\ref{lemma-compute-via-ph-hypercovering}", "and \\ref{lemma-simplicial-sheaf-cohomology-site}." ], "refs": [ "spaces-simplicial-lemma-compute-via-ph-hypercovering", "spaces-simplicial-lemma-simplicial-sheaf-cohomology-site" ], "ref_ids": [ 9144, 9039 ] } ], "ref_ids": [] }, { "id": 9164, "type": "theorem", "label": "examples-stacks-lemma-quasi-coherent-strongly-cartesian", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-quasi-coherent-strongly-cartesian", "contents": [ "A morphism $(f, \\varphi) : (Y, \\mathcal{G}) \\to (X, \\mathcal{F})$", "of $\\QCohstack$ is strongly cartesian if and only if the", "map $\\varphi$ induces an isomorphism $f^*\\mathcal{F} \\to \\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Let $(X, \\mathcal{F}) \\in \\Ob(\\QCohstack)$.", "Let $f : Y \\to X$ be a morphism of $(\\Sch/S)_{fppf}$.", "Note that there is a canonical $f$-map $c : \\mathcal{F} \\to f^*\\mathcal{F}$", "and hence we get a morphism", "$(f, c) : (Y, f^*\\mathcal{F}) \\to (X, \\mathcal{F})$.", "We claim that $(f, c)$ is strongly cartesian.", "Namely, for any object $(Z, \\mathcal{H})$ of $\\QCohstack$ we have", "\\begin{align*}", "\\Mor_{\\QCohstack}((Z, \\mathcal{H}), (Y, f^*\\mathcal{F}))", "& =", "\\coprod\\nolimits_{g \\in \\Mor_S(Z, Y)}", "\\Mor_{\\QCoh(\\mathcal{O}_Z)}(g^*f^*\\mathcal{F}, \\mathcal{H}) \\\\", "& =", "\\coprod\\nolimits_{g \\in \\Mor_S(Z, Y)}", "\\Mor_{\\QCoh(\\mathcal{O}_Z)}((f \\circ g)^*\\mathcal{F}, \\mathcal{H}) \\\\", "& =", "\\Mor_{\\QCohstack}((Z, \\mathcal{H}), (X, \\mathcal{F}))", "\\times_{\\Mor_S(Z, X)} \\Mor_S(Z, Y)", "\\end{align*}", "where we have used Equation (\\ref{equation-morphisms-qcoh}) twice.", "This proves that the condition of", "Categories, Definition \\ref{categories-definition-cartesian-over-C}", "holds for $(f, c)$, and hence our claim is true. Now by", "Categories, Lemma \\ref{categories-lemma-composition-cartesian}", "we see that isomorphisms are strongly cartesian and", "compositions of strongly cartesian morphisms are strongly cartesian", "which proves the ``if'' part of the lemma. For the converse, note", "that given $(X, \\mathcal{F})$ and $f : Y \\to X$, if there exists a", "strongly cartesian morphism lifting $f$ with target $(X, \\mathcal{F})$", "then it has to be isomorphic to $(f, c)$ (see discussion following", "Categories, Definition \\ref{categories-definition-cartesian-over-C}).", "Hence the \"only if\" part of the lemma holds." ], "refs": [ "categories-definition-cartesian-over-C", "categories-lemma-composition-cartesian", "categories-definition-cartesian-over-C" ], "ref_ids": [ 12387, 12282, 12387 ] } ], "ref_ids": [] }, { "id": 9165, "type": "theorem", "label": "examples-stacks-lemma-stack-of-quasi-coherent-sheaves", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-stack-of-quasi-coherent-sheaves", "contents": [ "The functor $p : \\QCohstack \\to (\\Sch/S)_{fppf}$", "satisfies conditions (1), (2) and (3) of", "Stacks, Definition \\ref{stacks-definition-stack}." ], "refs": [ "stacks-definition-stack" ], "proofs": [ { "contents": [ "It is clear from", "Lemma \\ref{lemma-quasi-coherent-strongly-cartesian}", "that $\\QCohstack$ is a fibred category over $(\\Sch/S)_{fppf}$.", "Given covering $\\mathcal{U} = \\{X_i \\to X\\}_{i \\in I}$ of", "$(\\Sch/S)_{fppf}$ the functor", "$$", "\\QCoh(\\mathcal{O}_X) \\longrightarrow DD(\\mathcal{U})", "$$", "is fully faithful and essentially surjective, see", "Descent, Proposition \\ref{descent-proposition-fpqc-descent-quasi-coherent}.", "Hence", "Stacks, Lemma \\ref{stacks-lemma-stack-equivalences}", "applies to show that $\\QCohstack$ satisfies all the", "axioms of a stack." ], "refs": [ "examples-stacks-lemma-quasi-coherent-strongly-cartesian", "descent-proposition-fpqc-descent-quasi-coherent", "stacks-lemma-stack-equivalences" ], "ref_ids": [ 9164, 14753, 8940 ] } ], "ref_ids": [ 8996 ] }, { "id": 9166, "type": "theorem", "label": "examples-stacks-lemma-stack-of-finite-type-quasi-coherent-sheaves", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-stack-of-finite-type-quasi-coherent-sheaves", "contents": [ "The functor $p_{fg} : \\QCohstack_{fg} \\to (\\Sch/S)_{fppf}$", "satisfies conditions (1), (2) and (3) of", "Stacks, Definition \\ref{stacks-definition-stack}." ], "refs": [ "stacks-definition-stack" ], "proofs": [ { "contents": [ "We will verify assumptions (1), (2), (3) of", "Stacks, Lemma \\ref{stacks-lemma-substack}", "to prove this. By", "Lemma \\ref{lemma-quasi-coherent-strongly-cartesian}", "a morphism $(Y, \\mathcal{G}) \\to (X, \\mathcal{F})$ is", "strongly cartesian if and only if it induces an isomorphism", "$f^*\\mathcal{F} \\to \\mathcal{G}$. By", "Modules, Lemma \\ref{modules-lemma-pullback-finite-type}", "the pullback of a finite type $\\mathcal{O}_X$-module is of finite", "type. Hence assumption (1) of", "Stacks, Lemma \\ref{stacks-lemma-substack}", "holds. Assumption (2) holds trivially.", "Finally, to prove assumption (3) we have to show:", "If $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module", "and $\\{f_i : X_i \\to X\\}$ is an fppf covering such that each", "$f_i^*\\mathcal{F}$ is of finite type, then $\\mathcal{F}$ is of", "finite type. Considering the restriction of $\\mathcal{F}$ to", "an affine open of $X$ this reduces to the following algebra statement:", "Suppose that $R \\to S$ is a finitely presented, faithfully flat ring map", "and $M$ an $R$-module. If $M \\otimes_R S$ is a finitely generated", "$S$-module, then $M$ is a finitely generated $R$-module.", "A stronger form of the algebra fact can be found in", "Algebra, Lemma \\ref{algebra-lemma-descend-properties-modules}." ], "refs": [ "stacks-lemma-substack", "examples-stacks-lemma-quasi-coherent-strongly-cartesian", "modules-lemma-pullback-finite-type", "stacks-lemma-substack", "algebra-lemma-descend-properties-modules" ], "ref_ids": [ 8941, 9164, 13236, 8941, 819 ] } ], "ref_ids": [ 8996 ] }, { "id": 9167, "type": "theorem", "label": "examples-stacks-lemma-finite-type", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-finite-type", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "\\begin{enumerate}", "\\item The category of finite type $\\mathcal{O}_X$-modules has a", "set of isomorphism classes.", "\\item The category of finite type quasi-coherent", "$\\mathcal{O}_X$-modules has a set of isomorphism classes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) follows from part (1) as the category in (2) is a full subcategory", "of the category in (1). Consider any open covering", "$\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$. Denote $j_i : U_i \\to X$", "the inclusion maps. Consider any map $r : I \\to \\mathbf{N}$.", "If $\\mathcal{F}$ is an $\\mathcal{O}_X$-module whose restriction to", "$U_i$ is generated by at most $r(i)$ sections from $\\mathcal{F}(U_i)$,", "then $\\mathcal{F}$ is a quotient of the sheaf", "$$", "\\mathcal{H}_{\\mathcal{U}, r} =", "\\bigoplus\\nolimits_{i \\in I} j_{i, !}\\mathcal{O}_{U_i}^{\\oplus r(i)}", "$$", "By definition, if $\\mathcal{F}$ is of finite type, then there exists", "some open covering with $\\mathcal{U}$ whose index set is $I = X$", "such that this condition is true. Hence it suffices to show that", "there is a set of possible choices for $\\mathcal{U}$ (obvious),", "a set of possible choices for $r : I \\to \\mathbf{N}$ (obvious), and", "a set of possible quotient modules of $\\mathcal{H}_{\\mathcal{U}, r}$", "for each $\\mathcal{U}$ and $r$. In other words, it suffices to show", "that given an $\\mathcal{O}_X$-module $\\mathcal{H}$ there is at most", "a set of isomorphism classes of quotients.", "This last assertion becomes obvious", "by thinking of the kernels of a quotient map", "$\\mathcal{H} \\to \\mathcal{F}$", "as being parametrized by a subset of the power set of", "$\\prod_{U \\subset X\\text{ open}} \\mathcal{H}(U)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9168, "type": "theorem", "label": "examples-stacks-lemma-stack-fg-quasi-coherent", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-stack-fg-quasi-coherent", "contents": [ "There exists a subcategory", "$\\QCohstack_{fg, small} \\subset \\QCohstack_{fg}$", "with the following properties:", "\\begin{enumerate}", "\\item the inclusion functor", "$\\QCohstack_{fg, small} \\to \\QCohstack_{fg}$ is", "fully faithful and essentially surjective, and", "\\item the functor", "$p_{fg, small} : \\QCohstack_{fg, small} \\to (\\Sch/S)_{fppf}$", "turns $\\QCohstack_{fg, small}$ into a stack over $(\\Sch/S)_{fppf}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have seen in", "Lemmas \\ref{lemma-stack-of-finite-type-quasi-coherent-sheaves} and", "\\ref{lemma-finite-type}", "that $p_{fg} : \\QCohstack_{fg} \\to (\\Sch/S)_{fppf}$", "satisfies (1), (2) and (3) of", "Stacks, Definition \\ref{stacks-definition-stack}", "as well as the additional condition (4) of", "Stacks, Remark \\ref{stacks-remark-stack-make-small}.", "Hence we obtain $\\QCohstack_{fg, small}$ from the discussion", "in that remark." ], "refs": [ "examples-stacks-lemma-stack-of-finite-type-quasi-coherent-sheaves", "examples-stacks-lemma-finite-type", "stacks-definition-stack", "stacks-remark-stack-make-small" ], "ref_ids": [ 9166, 9167, 8996, 9008 ] } ], "ref_ids": [] }, { "id": 9169, "type": "theorem", "label": "examples-stacks-lemma-finite-etale-stack", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-finite-etale-stack", "contents": [ "The functor", "$$", "p : \\textit{F\\'Et} \\longrightarrow (\\Sch/S)_{fppf}", "$$", "defines a stack in groupoids over $(\\Sch/S)_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "Fppf descent for finite \\'etale morphisms follows from", "Descent, Lemmas \\ref{descent-lemma-affine},", "\\ref{descent-lemma-descending-property-finite}, and", "\\ref{descent-lemma-descending-property-etale}.", "Details omitted." ], "refs": [ "descent-lemma-affine", "descent-lemma-descending-property-finite", "descent-lemma-descending-property-etale" ], "ref_ids": [ 14748, 14688, 14694 ] } ], "ref_ids": [] }, { "id": 9170, "type": "theorem", "label": "examples-stacks-lemma-spaces-strongly-cartesian", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-spaces-strongly-cartesian", "contents": [ "A morphism $(f, g) : X/U \\to Y/V$", "of $\\Spacesstack$ is strongly cartesian if and only if the", "map $f$ induces an isomorphism $X \\to U \\times_{g, V} Y$." ], "refs": [], "proofs": [ { "contents": [ "Let $Y/V \\in \\Ob(\\Spacesstack)$.", "Let $g : U \\to V$ be a morphism of $(\\Sch/S)_{fppf}$.", "Note that the projection $p : U \\times_{g, V} Y \\to Y$", "gives rise a morphism", "$(p, g) : U \\times_{g, V} Y/U \\to Y/V$ of $\\Spacesstack$.", "We claim that $(p, g)$ is strongly cartesian.", "Namely, for any object $Z/W$ of $\\Spacesstack$ we have", "\\begin{align*}", "\\Mor_{\\Spacesstack}(Z/W, U \\times_{g, V} Y/U)", "& =", "\\coprod\\nolimits_{h \\in \\Mor_S(W, U)}", "\\Mor_{\\textit{Spaces}/W}(Z, W \\times_{h, U} U \\times_{g, V} Y) \\\\", "& =", "\\coprod\\nolimits_{h \\in \\Mor_S(W, U)}", "\\Mor_{\\textit{Spaces}/W}(Z, W \\times_{g \\circ h, V} Y) \\\\", "& =", "\\Mor_{\\Spacesstack}(Z/W, Y/V)", "\\times_{\\Mor_S(W, V)} \\Mor_S(W, U)", "\\end{align*}", "where we have used Equation (\\ref{equation-morphisms-spaces}) twice.", "This proves that the condition of", "Categories, Definition \\ref{categories-definition-cartesian-over-C}", "holds for $(p, g)$, and hence our claim is true. Now by", "Categories, Lemma \\ref{categories-lemma-composition-cartesian}", "we see that isomorphisms are strongly cartesian and", "compositions of strongly cartesian morphisms are strongly cartesian", "which proves the ``if'' part of the lemma. For the converse, note", "that given $Y/V$ and $g : U \\to V$, if there exists a", "strongly cartesian morphism lifting $g$ with target $Y/V$", "then it has to be isomorphic to $(p, g)$ (see discussion following", "Categories, Definition \\ref{categories-definition-cartesian-over-C}).", "Hence the \"only if\" part of the lemma holds." ], "refs": [ "categories-definition-cartesian-over-C", "categories-lemma-composition-cartesian", "categories-definition-cartesian-over-C" ], "ref_ids": [ 12387, 12282, 12387 ] } ], "ref_ids": [] }, { "id": 9171, "type": "theorem", "label": "examples-stacks-lemma-pre-stack-of-spaces", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-pre-stack-of-spaces", "contents": [ "The functor $p : \\Spacesstack \\to (\\Sch/S)_{fppf}$", "satisfies conditions (1) and (2) of", "Stacks, Definition \\ref{stacks-definition-stack}." ], "refs": [ "stacks-definition-stack" ], "proofs": [ { "contents": [ "It is follows from", "Lemma \\ref{lemma-spaces-strongly-cartesian}", "that $\\Spacesstack$ is a fibred category over $(\\Sch/S)_{fppf}$", "which proves (1).", "Suppose that $\\{U_i \\to U\\}_{i \\in I}$ is a covering of", "$(\\Sch/S)_{fppf}$. Suppose that $X, Y$ are algebraic spaces over", "$U$. Finally, suppose that $\\varphi_i : X_{U_i} \\to Y_{U_i}$ are morphisms", "of $\\textit{Spaces}/U_i$ such that $\\varphi_i$ and $\\varphi_j$ restrict", "to the same morphisms $X_{U_i \\times_U U_j} \\to Y_{U_i \\times_U U_j}$", "of algebraic spaces over $U_i \\times_U U_j$.", "To prove (2) we have to show that there exists a unique morphism", "$\\varphi : X \\to Y$ over $U$ whose base change to $U_i$ is", "equal to $\\varphi_i$. As a morphism from $X$ to $Y$ is the same thing", "as a map of sheaves this follows directly from", "Sites, Lemma \\ref{sites-lemma-glue-maps}." ], "refs": [ "examples-stacks-lemma-spaces-strongly-cartesian", "sites-lemma-glue-maps" ], "ref_ids": [ 9170, 8561 ] } ], "ref_ids": [ 8996 ] }, { "id": 9172, "type": "theorem", "label": "examples-stacks-lemma-stack-of-finite-type-spaces", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-stack-of-finite-type-spaces", "contents": [ "The functor", "$p_{ft} : \\Spacesstack_{ft} \\to (\\Sch/S)_{fppf}$", "satisfies the conditions (1), (2) and (3) of", "Stacks, Definition \\ref{stacks-definition-stack}." ], "refs": [ "stacks-definition-stack" ], "proofs": [ { "contents": [ "We are going to write this out in ridiculous detail (which may make", "it hard to see what is going on).", "\\medskip\\noindent", "We have seen in", "Lemma \\ref{lemma-spaces-strongly-cartesian}", "that a morphism $(f, g) : X/U \\to Y/V$ of $\\Spacesstack$ is", "strongly cartesian if the induced morphism $f : X \\to U \\times_V Y$", "is an isomorphism. Note that if $Y \\to V$ is of finite type", "then also $U \\times_V Y \\to U$ is of finite type, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-base-change-finite-type}.", "So if $(f, g) : X/U \\to Y/V$ of $\\Spacesstack$ is", "strongly cartesian in $\\Spacesstack$ and $Y/V$ is an object", "of $\\Spacesstack_{ft}$ then automatically also $X/U$ is an", "object of $\\Spacesstack_{ft}$, and of course $(f, g)$ is", "also strongly cartesian in $\\Spacesstack_{ft}$. In this way", "we conclude that $\\Spacesstack_{ft}$ is a fibred category over", "$(\\Sch/S)_{fppf}$. This proves (1).", "\\medskip\\noindent", "The argument above also shows that the inclusion", "functor $\\Spacesstack_{ft} \\to \\Spacesstack$ transforms", "strongly cartesian morphisms into strongly cartesian morphisms.", "In other words $\\Spacesstack_{ft} \\to \\Spacesstack$ is", "a $1$-morphism of fibred categories over $(\\Sch/S)_{fppf}$.", "\\medskip\\noindent", "Let $U \\in \\Ob((\\Sch/S)_{fppf})$.", "Let $X, Y$ be algebraic spaces of finite type over $U$. By", "Stacks, Lemma \\ref{stacks-lemma-presheaf-mor-map-fibred-categories}", "we obtain a map of presheaves", "$$", "\\mathit{Mor}_{\\Spacesstack_{ft}}(X, Y)", "\\longrightarrow", "\\mathit{Mor}_{\\Spacesstack}(X, Y)", "$$", "which is an isomorphism as $\\Spacesstack_{ft}$ is a full subcategory of", "$\\Spacesstack$. Hence the left hand side is a sheaf, because in", "Lemma \\ref{lemma-pre-stack-of-spaces}", "we showed the right hand side is a sheaf. This proves (2).", "\\medskip\\noindent", "To prove condition (3) of", "Stacks, Definition \\ref{stacks-definition-stack}", "we have to show the following: Given", "\\begin{enumerate}", "\\item a covering $\\{U_i \\to U\\}_{i \\in I}$ of $(\\Sch/S)_{fppf}$,", "\\item for each $i \\in I$ an algebraic space $X_i$ of finite type over $U_i$,", "and", "\\item for each $i, j \\in I$ an isomorphism", "$\\varphi_{ij} : X_i \\times_U U_j \\to U_i \\times_U X_j$ of algebraic spaces", "over $U_i \\times_U U_j$ satisfying the cocycle condition over", "$U_i \\times_U U_j \\times_U U_k$,", "\\end{enumerate}", "there exists an algebraic space $X$ of finite type over $U$ and isomorphisms", "$X_{U_i} \\cong X_i$ over $U_i$ recovering the isomorphisms $\\varphi_{ij}$.", "This follows from", "Bootstrap, Lemma \\ref{bootstrap-lemma-descend-algebraic-space} part (2). By", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-locally-finite-presentation}", "we see that $X \\to U$ is of finite type which concludes the proof." ], "refs": [ "examples-stacks-lemma-spaces-strongly-cartesian", "spaces-morphisms-lemma-base-change-finite-type", "stacks-lemma-presheaf-mor-map-fibred-categories", "examples-stacks-lemma-pre-stack-of-spaces", "stacks-definition-stack", "bootstrap-lemma-descend-algebraic-space", "spaces-descent-lemma-descending-property-locally-finite-presentation" ], "ref_ids": [ 9170, 4815, 8935, 9171, 8996, 2627, 9390 ] } ], "ref_ids": [ 8996 ] }, { "id": 9173, "type": "theorem", "label": "examples-stacks-lemma-stack-ft-spaces", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-stack-ft-spaces", "contents": [ "There exists a subcategory", "$\\Spacesstack_{ft, small} \\subset \\Spacesstack_{ft}$", "with the following properties:", "\\begin{enumerate}", "\\item the inclusion functor", "$\\Spacesstack_{ft, small} \\to \\Spacesstack_{ft}$ is", "fully faithful and essentially surjective, and", "\\item the functor", "$p_{ft, small} : \\Spacesstack_{ft, small} \\to (\\Sch/S)_{fppf}$", "turns $\\Spacesstack_{ft, small}$ into a stack over", "$(\\Sch/S)_{fppf}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have seen in", "Lemmas \\ref{lemma-stack-of-finite-type-spaces}", "that $p_{ft} : \\Spacesstack_{ft} \\to (\\Sch/S)_{fppf}$", "satisfies (1), (2) and (3) of", "Stacks, Definition \\ref{stacks-definition-stack}.", "The additional condition (4) of", "Stacks, Remark \\ref{stacks-remark-stack-make-small}", "holds because every algebraic space $X$ over $S$ is of the", "form $U/R$ for $U, R \\in \\Ob((\\Sch/S)_{fppf})$, see", "Spaces, Lemma \\ref{spaces-lemma-space-presentation}.", "Thus there is only a set worth of isomorphism classes of objects.", "Hence we obtain $\\Spacesstack_{ft, small}$ from the discussion", "in that remark." ], "refs": [ "examples-stacks-lemma-stack-of-finite-type-spaces", "stacks-definition-stack", "stacks-remark-stack-make-small", "spaces-lemma-space-presentation" ], "ref_ids": [ 9172, 8996, 9008, 8149 ] } ], "ref_ids": [] }, { "id": 9174, "type": "theorem", "label": "examples-stacks-lemma-torsors-sheaf-stack-in-groupoids", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-torsors-sheaf-stack-in-groupoids", "contents": [ "Up to a replacement as in", "Stacks, Remark \\ref{stacks-remark-stack-make-small}", "the functor", "$$", "p : \\mathcal{G}\\textit{-Torsors} \\longrightarrow (\\Sch/S)_{fppf}", "$$", "defines a stack in groupoids over $(\\Sch/S)_{fppf}$." ], "refs": [ "stacks-remark-stack-make-small" ], "proofs": [ { "contents": [ "The most difficult part of the proof is to show that", "we have descent for objects.", "Let $\\{U_i \\to U\\}_{i \\in I}$ be a covering of $(\\Sch/S)_{fppf}$.", "Suppose that for each $i$ we are given a $\\mathcal{G}|_{U_i}$-torsor", "$\\mathcal{F}_i$, and for each $i, j \\in I$ an isomorphism", "$\\varphi_{ij} :", "\\mathcal{F}_i|_{U_i \\times_U U_j} \\to \\mathcal{F}_j|_{U_i \\times_U U_j}$", "of $\\mathcal{G}|_{U_i \\times_U U_j}$-torsors", "satisfying a suitable cocycle condition on $U_i \\times_U U_j \\times_U U_k$.", "Then by", "Sites, Section \\ref{sites-section-glueing-sheaves}", "we obtain a sheaf $\\mathcal{F}$ on $(\\Sch/U)_{fppf}$", "whose restriction to each $U_i$ recovers $\\mathcal{F}_i$ as well", "as recovering the descent data. By the equivalence of categories in", "Sites, Lemma \\ref{sites-lemma-mapping-property-glue}", "the action maps $\\mathcal{G}|_{U_i} \\times \\mathcal{F}_i \\to \\mathcal{F}_i$", "glue to give a map $a : \\mathcal{G}|_U \\times \\mathcal{F} \\to \\mathcal{F}$.", "Now we have to show that $a$ is an action and that $\\mathcal{F}$ becomes", "a $\\mathcal{G}|_U$-torsor. Both properties may be checked locally, and", "hence follow from the corresponding properties of the actions", "$\\mathcal{G}|_{U_i} \\times \\mathcal{F}_i \\to \\mathcal{F}_i$.", "This proves that descent for objects holds in", "$\\mathcal{G}\\textit{-Torsors}$.", "Some details omitted." ], "refs": [ "sites-lemma-mapping-property-glue" ], "ref_ids": [ 8565 ] } ], "ref_ids": [ 9008 ] }, { "id": 9175, "type": "theorem", "label": "examples-stacks-lemma-variant-torsors-sheaf-stack-in-groupoids", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-variant-torsors-sheaf-stack-in-groupoids", "contents": [ "Up to a replacement as in", "Stacks, Remark \\ref{stacks-remark-stack-make-small}", "the functor", "$$", "p :", "\\mathcal{G}/\\mathcal{B}\\textit{-Torsors}", "\\longrightarrow", "(\\Sch/S)_{fppf}", "$$", "defines a stack in groupoids over $(\\Sch/S)_{fppf}$." ], "refs": [ "stacks-remark-stack-make-small" ], "proofs": [ { "contents": [ "This proof is a repeat of the proof of", "Lemma \\ref{lemma-torsors-sheaf-stack-in-groupoids}.", "The reader is encouraged to read that proof first since", "the notation is less cumbersome.", "The most difficult part of the proof is to show that", "we have descent for objects. Let $\\{U_i \\to U\\}_{i \\in I}$", "be a covering of $(\\Sch/S)_{fppf}$.", "Suppose that for each $i$ we are given a pair $(b_i, \\mathcal{F}_i)$", "consisting of a morphism $b_i : U_i \\to \\mathcal{B}$ and a", "$U_i \\times_{b_i, \\mathcal{B}} \\mathcal{G}$-torsor", "$\\mathcal{F}_i$, and for each $i, j \\in I$", "we have $b_i|_{U_i \\times_U U_j} = b_j|_{U_i \\times_U U_j}$ and", "we are given an isomorphism", "$\\varphi_{ij} :", "\\mathcal{F}_i|_{U_i \\times_U U_j} \\to \\mathcal{F}_j|_{U_i \\times_U U_j}$", "of $(U_i \\times_U U_j) \\times_\\mathcal{B} \\mathcal{G}$-torsors", "satisfying a suitable cocycle condition on $U_i \\times_U U_j \\times_U U_k$.", "Then by", "Sites, Section \\ref{sites-section-glueing-sheaves}", "we obtain a sheaf $\\mathcal{F}$ on $(\\Sch/U)_{fppf}$", "whose restriction to each $U_i$ recovers $\\mathcal{F}_i$ as well", "as recovering the descent data. By the sheaf axiom for $\\mathcal{B}$", "the morphisms $b_i$ come from a unique morphism $b : U \\to \\mathcal{B}$.", "By the equivalence of categories in", "Sites, Lemma \\ref{sites-lemma-mapping-property-glue}", "the action maps", "$(U_i \\times_{b_i, \\mathcal{B}} \\mathcal{G}) \\times_{U_i} \\mathcal{F}_i", "\\to \\mathcal{F}_i$", "glue to give a map", "$(U \\times_{b, \\mathcal{B}} \\mathcal{G}) \\times \\mathcal{F} \\to \\mathcal{F}$.", "Now we have to show that this is an action and that $\\mathcal{F}$ becomes", "a $U \\times_{b, \\mathcal{B}} \\mathcal{G}$-torsor.", "Both properties may be checked locally, and", "hence follow from the corresponding properties of the actions", "on the $\\mathcal{F}_i$.", "This proves that descent for objects holds in", "$\\mathcal{G}/\\mathcal{B}\\textit{-Torsors}$.", "Some details omitted." ], "refs": [ "examples-stacks-lemma-torsors-sheaf-stack-in-groupoids", "sites-lemma-mapping-property-glue" ], "ref_ids": [ 9174, 8565 ] } ], "ref_ids": [ 9008 ] }, { "id": 9176, "type": "theorem", "label": "examples-stacks-lemma-torsors-stack-in-groupoids", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-torsors-stack-in-groupoids", "contents": [ "Up to a replacement as in", "Stacks, Remark \\ref{stacks-remark-stack-make-small}", "the functor", "$$", "p : G\\textit{-Torsors} \\longrightarrow (\\Sch/S)_{fppf}", "$$", "defines a stack in groupoids over $(\\Sch/S)_{fppf}$." ], "refs": [ "stacks-remark-stack-make-small" ], "proofs": [ { "contents": [ "The most difficult part of the proof is to show that we have descent for", "objects, which is", "Bootstrap, Lemma \\ref{bootstrap-lemma-descent-torsor}.", "We omit the proof of axioms (1) and (2) of", "Stacks, Definition \\ref{stacks-definition-stack-in-groupoids}." ], "refs": [ "bootstrap-lemma-descent-torsor", "stacks-definition-stack-in-groupoids" ], "ref_ids": [ 2632, 8998 ] } ], "ref_ids": [ 9008 ] }, { "id": 9177, "type": "theorem", "label": "examples-stacks-lemma-compare-torsors", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-compare-torsors", "contents": [ "Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic", "space over $B$. Denote $\\mathcal{G}$, resp.\\ $\\mathcal{B}$ the algebraic", "space $G$, resp.\\ $B$ seen as a sheaf on $(\\Sch/S)_{fppf}$.", "The functor", "$$", "G\\textit{-Torsors} \\longrightarrow \\mathcal{G}/\\mathcal{B}\\textit{-Torsors}", "$$", "which associates to a triple $(U, b, X)$ the triple", "$(U, b, \\mathcal{X})$ where $\\mathcal{X}$ is $X$ viewed as a sheaf", "is an equivalence of stacks in groupoids over $(\\Sch/S)_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "We will use the result of", "Stacks, Lemma \\ref{stacks-lemma-characterize-essentially-surjective-when-ff}", "to prove this. The functor is fully faithful since the category of", "algebraic spaces over $S$ is a full subcategory of the category of", "sheaves on $(\\Sch/S)_{fppf}$.", "Moreover, all objects (on both sides) are locally trivial torsors", "so condition (2) of the lemma referenced above holds.", "Hence the functor is an equivalence." ], "refs": [ "stacks-lemma-characterize-essentially-surjective-when-ff" ], "ref_ids": [ 8945 ] } ], "ref_ids": [] }, { "id": 9178, "type": "theorem", "label": "examples-stacks-lemma-group-quotient-stack-in-groupoids", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-group-quotient-stack-in-groupoids", "contents": [ "Up to a replacement as in", "Stacks, Remark \\ref{stacks-remark-stack-make-small}", "the functor", "$$", "p : [[X/G]] \\longrightarrow (\\Sch/S)_{fppf}", "$$", "defines a stack in groupoids over $(\\Sch/S)_{fppf}$." ], "refs": [ "stacks-remark-stack-make-small" ], "proofs": [ { "contents": [ "The most difficult part of the proof is to show that we have descent for", "objects. Suppose that $\\{U_i \\to U\\}_{i \\in I}$ is a covering in", "$(\\Sch/S)_{fppf}$. Let", "$\\xi_i = (U_i, b_i, P_i, \\varphi_i)$ be objects of $[[X/G]]$ over $U_i$,", "and let $\\varphi_{ij} : \\text{pr}_0^*\\xi_i \\to \\text{pr}_1^*\\xi_j$", "be a descent datum. This in particular implies that we get a descent", "datum on the triples $(U_i, b_i, P_i)$ for the stack in groupoids", "$G\\textit{-Torsors}$ by applying the functor", "$[[X/G]] \\to G\\textit{-Torsors}$ above. We have seen that", "$G\\textit{-Torsors}$ is a stack in groupoids", "(Lemma \\ref{lemma-torsors-stack-in-groupoids}).", "Hence we may assume that $b_i = b|_{U_i}$ for some morphism $b : U \\to B$, and", "that $P_i = U_i \\times_U P$ for some fppf $G_U = U \\times_{b, B} G$-torsor", "$P$ over $U$. The morphisms $\\varphi_i$ are compatible", "with the canonical descent datum on the restrictions $U_i \\times_U P$", "and hence define a morphism $\\varphi : P \\to X$. (For example you", "can use", "Sites, Lemma \\ref{sites-lemma-mapping-property-glue}", "or you can use", "Descent on Spaces,", "Lemma \\ref{spaces-descent-lemma-fpqc-universal-effective-epimorphisms}", "to get $\\varphi$.)", "This proves descent for objects.", "We omit the proof of axioms (1) and (2) of", "Stacks, Definition \\ref{stacks-definition-stack-in-groupoids}." ], "refs": [ "examples-stacks-lemma-torsors-stack-in-groupoids", "sites-lemma-mapping-property-glue", "spaces-descent-lemma-fpqc-universal-effective-epimorphisms", "stacks-definition-stack-in-groupoids" ], "ref_ids": [ 9176, 8565, 9367, 8998 ] } ], "ref_ids": [ 9008 ] }, { "id": 9179, "type": "theorem", "label": "examples-stacks-lemma-classifying-stacks", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-classifying-stacks", "contents": [ "\\begin{slogan}", "The classifying stack of a group scheme or group algebraic space.", "\\end{slogan}", "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $G$ be a group algebraic space over $B$. Then the stacks", "in groupoids", "$$", "[B/G],\\quad", "[[B/G]],\\quad", "G\\textit{-Torsors},\\quad", "\\mathcal{G}/\\mathcal{B}\\textit{-Torsors}", "$$", "are all canonically equivalent.", "If $G \\to B$ is flat and locally", "of finite presentation, then these are also equivalent to", "$G\\textit{-Principal}$." ], "refs": [], "proofs": [ { "contents": [ "The equivalence", "$G\\textit{-Torsors} \\to \\mathcal{G}/\\mathcal{B}\\textit{-Torsors}$", "is given in Lemma \\ref{lemma-compare-torsors}.", "The equivalence $[B/G] \\to [[B/G]]$ is given in", "Proposition \\ref{proposition-equal-quotient-stacks}.", "Unwinding the definition of $[[B/G]]$ given in", "Section \\ref{section-group-quotient-stacks}", "we see that $[[B//G]] = G\\textit{-Torsors}$.", "\\medskip\\noindent", "Finally, assume $G \\to B$ is flat and locally of finite presentation.", "To show that the natural functor", "$G\\textit{-Torsors} \\to G\\textit{-Principal}$ is an equivalence", "it suffices to show that for a scheme $U$ over $B$", "a principal homogeneous $G_U$-space $X \\to U$", "is fppf locally trivial. By our definition of principal homogeneous spaces", "(Groupoids in Spaces,", "Definition \\ref{spaces-groupoids-definition-principal-homogeneous-space})", "there exists an fpqc covering $\\{U_i \\to U\\}$ such that", "$U_i \\times_U X \\cong G \\times_B U_i$ as algebraic spaces over $U_i$.", "This implies that $X \\to U$ is surjective, flat, and locally of finite", "presentation, see", "Descent on Spaces, Lemmas", "\\ref{spaces-descent-lemma-descending-property-surjective},", "\\ref{spaces-descent-lemma-descending-property-flat}, and", "\\ref{spaces-descent-lemma-descending-property-locally-finite-presentation}.", "Choose a scheme $W$ and a surjective \\'etale morphism $W \\to X$.", "Then it follows from what we just said that $\\{W \\to U\\}$ is an fppf covering", "such that $X_W \\to W$ has a section. Hence $X$ is an fppf $G_U$-torsor." ], "refs": [ "examples-stacks-lemma-compare-torsors", "examples-stacks-proposition-equal-quotient-stacks", "spaces-groupoids-definition-principal-homogeneous-space", "spaces-descent-lemma-descending-property-surjective", "spaces-descent-lemma-descending-property-flat", "spaces-descent-lemma-descending-property-locally-finite-presentation" ], "ref_ids": [ 9177, 9183, 9345, 9386, 9393, 9390 ] } ], "ref_ids": [] }, { "id": 9180, "type": "theorem", "label": "examples-stacks-lemma-picard-stack", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-picard-stack", "contents": [ "Up to a replacement as in", "Stacks, Remark \\ref{stacks-remark-stack-make-small}", "the functor", "$$", "\\Picardstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf}", "$$", "defines a stack in groupoids over $(\\Sch/S)_{fppf}$." ], "refs": [ "stacks-remark-stack-make-small" ], "proofs": [ { "contents": [ "As usual, the hardest part is to show descent for objects.", "To see this let $\\{U_i \\to U\\}$ be a covering of $(\\Sch/S)_{fppf}$.", "Let $\\xi_i = (U_i, b_i, \\mathcal{L}_i)$ be an object of", "$\\Picardstack_{X/B}$ lying over $U_i$, and let", "$\\varphi_{ij} : \\text{pr}_0^*\\xi_i \\to \\text{pr}_1^*\\xi_j$", "be a descent datum. This implies in particular that the morphisms", "$b_i$ are the restrictions of a morphism $b : U \\to B$.", "Write $X_U = U \\times_{b, B} X$ and", "$X_i = U_i \\times_{b_i, B} X =", "U_i \\times_U U \\times_{b, B} X = U_i \\times_U X_U$.", "Observe that $\\mathcal{L}_i$ is an invertible $\\mathcal{O}_{X_i}$-module.", "Note that $\\{X_i \\to X_U\\}$ forms an fppf covering as well.", "Moreover, the descent datum $\\varphi_{ij}$ translates into a", "descent datum on the invertible sheaves $\\mathcal{L}_i$ relative", "to the fppf covering $\\{X_i \\to X_U\\}$.", "Hence by", "Descent on Spaces,", "Proposition \\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}", "we obtain a unique invertible sheaf $\\mathcal{L}$ on $X_U$", "which recovers $\\mathcal{L}_i$ and the descent data over $X_i$.", "The triple $(U, b, \\mathcal{L})$ is therefore the object of", "$\\Picardstack_{X/B}$ over $U$ we were looking for.", "Details omitted." ], "refs": [ "spaces-descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 9437 ] } ], "ref_ids": [ 9008 ] }, { "id": 9181, "type": "theorem", "label": "examples-stacks-lemma-hilbert-d-stack", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-hilbert-d-stack", "contents": [ "The category $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ endowed with", "the functor $p$ above defines a stack in groupoids over", "$(\\Sch/S)_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "As usual, the hardest part is to show descent for objects.", "To see this let $\\{U_i \\to U\\}$ be a covering of $(\\Sch/S)_{fppf}$.", "Let $\\xi_i = (U_i, Z_i, y_i, x_i, \\alpha_i)$ be an object of", "$\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ lying over $U_i$, and let", "$\\varphi_{ij} : \\text{pr}_0^*\\xi_i \\to \\text{pr}_1^*\\xi_j$", "be a descent datum. First, observe that $\\varphi_{ij}$", "induces a descent datum $(Z_i/U_i, \\varphi_{ij})$ which is effective by", "Descent, Lemma \\ref{descent-lemma-affine}", "This produces a scheme $Z/U$ which is finite locally free of degree $d$ by", "Descent, Lemma \\ref{descent-lemma-descending-property-finite-locally-free}.", "From now on we identify $Z_i$ with $Z \\times_U U_i$.", "Next, the objects $y_i$ in the fibre categories $\\mathcal{Y}_{U_i}$", "descend to an object $y$ in $\\mathcal{Y}_U$ because $\\mathcal{Y}$ is a", "stack in groupoids. Similarly the objects $x_i$ in the fibre categories", "$\\mathcal{X}_{Z_i}$ descend to an object $x$ in $\\mathcal{X}_Z$ because", "$\\mathcal{X}$ is a stack in groupoids. Finally, the given", "isomorphisms", "$$", "\\alpha_i :", "(y|_Z)_{Z_i} = y_i|_{Z_i}", "\\longrightarrow", "F(x_i) = F(x|_{Z_i})", "$$", "glue to a morphism $\\alpha : y|_Z \\to F(x)$ as the $\\mathcal{Y}$", "is a stack and hence $\\mathit{Isom}_\\mathcal{Y}(y|_Z, F(x))$ is", "a sheaf. Details omitted." ], "refs": [ "descent-lemma-affine", "descent-lemma-descending-property-finite-locally-free" ], "ref_ids": [ 14748, 14695 ] } ], "ref_ids": [] }, { "id": 9182, "type": "theorem", "label": "examples-stacks-lemma-faithful-hilbert", "categories": [ "examples-stacks" ], "title": "examples-stacks-lemma-faithful-hilbert", "contents": [ "The $1$-morphism", "$\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y}) \\to \\mathcal{H}_d(\\mathcal{X})$", "is faithful." ], "refs": [], "proofs": [ { "contents": [ "To check that", "$\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y}) \\to \\mathcal{H}_d(\\mathcal{X})$", "is faithful it suffices to prove that it is faithful on fibre categories.", "Suppose that $\\xi = (U, Z, y, x, \\alpha)$ and $\\xi' = (U, Z', y', x', \\alpha')$", "are two objects of $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ over the", "scheme $U$. Let $(g, b, a), (g', b', a') : \\xi \\to \\xi'$ be two morphisms", "in the fibre category of $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ over $U$.", "The image of these morphisms in $\\mathcal{H}_d(\\mathcal{X})$ agree", "if and only if $g = g'$ and $a = a'$. Then the commutative diagram", "$$", "\\xymatrix{", "y|_Z \\ar[rr]_\\alpha \\ar[d]_{b|_Z, \\ b'|_Z} & &", "F(x) \\ar[d]^{F(a) = F(a')} \\\\", "y'|_Z \\ar[rr]^-{\\alpha'} & &", "F(g^*x') = F((g')^*x') \\\\", "}", "$$", "implies that $b|_Z = b'|_Z$. Since $Z \\to U$ is finite locally free of degree", "$d$ we see $\\{Z \\to U\\}$ is an fppf covering, hence $b = b'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9183, "type": "theorem", "label": "examples-stacks-proposition-equal-quotient-stacks", "categories": [ "examples-stacks" ], "title": "examples-stacks-proposition-equal-quotient-stacks", "contents": [ "In", "Situation \\ref{situation-quotient-stack}", "there exists a canonical equivalence", "$$", "[X/G] \\longrightarrow [[X/G]]", "$$", "of stacks in groupoids over $(\\Sch/S)_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "We write this out in detail, to make sure that all the definitions", "work out in exactly the correct manner.", "Recall that $[X/G]$ is the quotient stack", "associated to the groupoid in algebraic spaces", "$(X, G \\times_B X, s, t, c)$, see", "Groupoids in Spaces,", "Definition \\ref{spaces-groupoids-definition-quotient-stack}.", "This means that $[X/G]$ is the stackification of the", "category fibred in groupoids $[X/_{\\!p}G]$ associated to the functor", "$$", "(\\Sch/S)_{fppf} \\longrightarrow \\textit{Groupoids},", "\\quad", "U \\longmapsto (X(U), G(U) \\times_{B(U)} X(U), s, t, c)", "$$", "where $s(g, x) = x$, $t(g, x) = a(g, x)$, and", "$c((g, x), (g', x')) = (m(g, g'), x')$. By the construction of", "Categories, Example \\ref{categories-example-functor-groupoids}", "an object of $[X/_{\\!p}G]$ is a pair $(U, x)$ with $x \\in X(U)$", "and a morphism $(f, g) : (U, x) \\to (U', x')$ of $[X/_{\\!p}G]$", "is given by a morphism of schemes $f : U \\to U'$ and an element", "$g \\in G(U)$ such that $a(g, x) = x' \\circ f$.", "Hence we can define a $1$-morphism of stacks in groupoids", "$$", "F_p : [X/_{\\!p}G] \\longrightarrow [[X/G]]", "$$", "by the following rules: On objects we set", "$$", "F_p(U, x) =", "(U, \\pi \\circ x, G \\times_{B, \\pi \\circ x} U, a \\circ (\\text{id}_G \\times x))", "$$", "This makes sense because the diagram", "$$", "\\xymatrix{", "G \\times_{B, \\pi \\circ x} U \\ar[d] \\ar[r]_{\\text{id}_G \\times x} &", "G \\times_{B, \\pi} X \\ar[r]_-a &", "X \\ar[d]^\\pi \\\\", "U \\ar[rr]^{\\pi \\circ x} & & B", "}", "$$", "commutes, and the two horizontal arrows are $G$-equivariant if we think", "of the fibre products as trivial $G$-torsors over $U$, resp.\\ $X$.", "On morphisms $(f, g) : (U, x) \\to (U', x')$ we set $F_p(f, g) = (f, R_g)$", "where $R_g$ denotes right translation by $g$. More precisely, the", "morphism of $F_p(f, g) : F_p(U, x) \\to F_p(U', x')$ is given by the cartesian", "diagram", "$$", "\\xymatrix{", "G \\times_{B, \\pi \\circ x} U \\ar[d] \\ar[r]_{R_{g^{-1}}} &", "G \\times_{B, \\pi \\circ x'} U' \\ar[d] \\\\", "U \\ar[r]^f & U'", "}", "$$", "where $R_{g^{-1}}$ on $T$-valued points is given by", "$$", "R_{g^{-1}}(g', u) = (m(g', i(g)), f(u))", "$$", "To see that this works we have to verify that", "$$", "a \\circ (\\text{id}_G \\times x)", "=", "a \\circ (\\text{id}_G \\times x') \\circ R_{g^{-1}}", "$$", "which is true because the right hand side applied to the $T$-valued point", "$(g', u)$ gives the desired equality", "\\begin{align*}", "a((\\text{id}_G \\times x')(m(g', i(g)), f(u)))", "& =", "a(m(g', i(g)), x'(f(u))) \\\\", "& =", "a(g', a(i(g), x'(f(u)))) \\\\", "& =", "a(g', x(u))", "\\end{align*}", "because $a(g, x) = x' \\circ f$ and hence $a(i(g), x' \\circ f) = x$.", "\\medskip\\noindent", "By the universal property of stackification from", "Stacks, Lemma \\ref{stacks-lemma-stackify-groupoids-universal-property}", "we obtain a canonical extension $F : [X/G] \\to [[X/G]]$ of the $1$-morphism", "$F_p$ above. We first prove that $F$ is fully faithful.", "To do this, since both source and target are stacks in groupoids,", "it suffices to prove that the $\\mathit{Isom}$-sheaves are identified", "under $F$. Pick a scheme $U$ and objects $\\xi, \\xi'$ of", "$[X/G]$ over $U$. We want to show that", "$$", "F :", "\\mathit{Isom}_{[X/G]}(\\xi, \\xi')", "\\longrightarrow", "\\mathit{Isom}_{[[X/G]]}(F(\\xi), F(\\xi'))", "$$", "is an isomorphism of sheaves. To do this it suffices to work locally", "on $U$, and hence we may assume that $\\xi, \\xi'$ come from objects", "$(U, x)$, $(U, x')$ of $[X/_{\\!p}G]$ over $U$; this follows directly", "from the construction of the stackification, and it is also worked", "out in detail in", "Groupoids in Spaces,", "Section \\ref{spaces-groupoids-section-explicit-quotient-stacks}.", "Either by directly using the description of morphisms in", "$[X/_{\\!p}G]$ above, or using", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-quotient-stack-morphisms}", "we see that in this case", "$$", "\\mathit{Isom}_{[X/G]}(\\xi, \\xi') =", "U \\times_{(x, x'), X \\times_S X, (s, t)} (G \\times_B X)", "$$", "A $T$-valued point of this fibre product corresponds to a pair", "$(u, g)$ with $u \\in U(T)$, and $g \\in G(T)$ such that", "$a(g, x \\circ u) = x' \\circ u$. (Note that this implies", "$\\pi \\circ x \\circ u = \\pi \\circ x' \\circ u$.)", "On the other hand, a $T$-valued", "point of $\\mathit{Isom}_{[[X/G]]}(F(\\xi), F(\\xi'))$ by definition", "corresponds to a morphism $u : T \\to U$ such that", "$\\pi \\circ x \\circ u = \\pi \\circ x' \\circ u : T \\to B$ and an isomorphism", "$$", "R :", "G \\times_{B, \\pi \\circ x \\circ u} T", "\\longrightarrow", "G \\times_{B, \\pi \\circ x' \\circ u} T", "$$", "of trivial $G_T$-torsors compatible with the given maps to $X$.", "Since the torsors are trivial we see that $R = R_{g^{-1}}$", "(right multiplication) by some $g \\in G(T)$. Compatibility with the maps", "$a \\circ (1_G, x \\circ u), a \\circ (1_G, x' \\circ u) : G \\times_B T \\to X$", "is equivalent to the condition that $a(g, x \\circ u) = x' \\circ u$.", "Hence we obtain the desired equality of $\\mathit{Isom}$-sheaves.", "\\medskip\\noindent", "Now that we know that $F$ is fully faithful we see that", "Stacks, Lemma \\ref{stacks-lemma-characterize-essentially-surjective-when-ff}", "applies. Thus to show that $F$ is an equivalence it suffices", "to show that objects of $[[X/G]]$ are fppf locally in the essential image", "of $F$. This is clear as fppf torsors are fppf locally trivial, and hence", "we win." ], "refs": [ "spaces-groupoids-definition-quotient-stack", "stacks-lemma-stackify-groupoids-universal-property", "spaces-groupoids-lemma-quotient-stack-morphisms", "stacks-lemma-characterize-essentially-surjective-when-ff" ], "ref_ids": [ 9354, 8967, 9323, 8945 ] } ], "ref_ids": [] }, { "id": 9190, "type": "theorem", "label": "models-theorem-semistable-reduction", "categories": [ "models" ], "title": "models-theorem-semistable-reduction", "contents": [ "\\begin{reference}", "\\cite[Corollary 2.7]{DM}", "\\end{reference}", "Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a", "smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$.", "Then there exists an extension of discrete valuation rings", "$R \\subset R'$ which induces a finite separable extension of", "fraction fields $K \\subset K'$ such that $C_{K'}$ has semistable reduction.", "More precisely, we have the following", "\\begin{enumerate}", "\\item If the genus of $C$ is zero, then there exists a degree $2$", "separable extension $K'/K$ such that $C_{K'} \\cong \\mathbf{P}^1_{K'}$", "and hence $C_{K'}$ is isomorphic to the generic fibre of the", "smooth projective scheme $\\mathbf{P}^1_{R'}$ over the integral closure", "$R'$ of $R$ in $K'$.", "\\item If the genus of $C$ is one, then there exists a finite separable", "extension $K'/K$ such that $C_{K'}$ has semistable reduction", "over $R'_\\mathfrak m$ for every maximal ideal $\\mathfrak m$", "of the integral closure $R'$ of $R$ in $K'$. Moreover, the special", "fibre of the (unique) minimal model of $C_{K'}$ over $R'_\\mathfrak m$", "is either a smooth genus one curve or a cycle of rational curves.", "\\item If the genus $g$ of $C$ is greater than one, then there exists a", "finite separable extension $K'/K$ of degree at most", "$B_g$ (\\ref{equation-bound}) such that $C_{K'}$ has semistable reduction", "over $R'_\\mathfrak m$ for every maximal ideal $\\mathfrak m$", "of the integral closure $R'$ of $R$ in $K'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For the case of genus zero, see", "Section \\ref{section-semistable-reduction-genus-zero}.", "For the case of genus one, see", "Section \\ref{section-semistable-reduction-genus-one}.", "For the case of genus greater than one, see", "Section \\ref{section-semistable-reduction-genus-at-least-two}.", "To see that we have a bound on the degree $[K' : K]$", "you can use the bound on the degree of the extension needed", "to make all $\\ell$ or $\\ell'$ torsion visible proved in", "Algebraic Curves, Lemma \\ref{curves-lemma-torsion-picard-becomes-visible}.", "(The reason for using $\\ell$ and $\\ell'$ is that we need to", "avoid the characteristic of the residue field $k$.)" ], "refs": [ "curves-lemma-torsion-picard-becomes-visible" ], "ref_ids": [ 6297 ] } ], "ref_ids": [] }, { "id": 9191, "type": "theorem", "label": "models-lemma-recurring", "categories": [ "models" ], "title": "models-lemma-recurring", "contents": [ "\\begin{reference}", "\\cite[Theorem I]{Taussky}", "\\end{reference}", "Let $A = (a_{ij})$ be a complex $n \\times n$ matrix.", "\\begin{enumerate}", "\\item If $|a_{ii}| > \\sum_{j \\not = i} |a_{ij}|$ for each $i$, then", "$\\det(A)$ is nonzero.", "\\item If there exists a real vector $m = (m_1, \\ldots, m_n)$", "with $m_i > 0$ such that $|a_{ii} m_i| > \\sum_{j \\not = i} |a_{ij}m_j|$", "for each $i$, then $\\det(A)$ is nonzero.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $A$ is as in (1) and $\\det(A) = 0$, then there is a nonzero vector", "$z$ with $Az = 0$. Choose $r$ with $|z_r|$ maximal. Then", "$$", "|a_{rr} z_r| = |\\sum\\nolimits_{k \\not = r} a_{rk}z_k| \\leq", "\\sum\\nolimits_{k \\not = r} |a_{rk}||z_k| \\leq", "|z_r| \\sum\\nolimits_{k \\not = r} |a_{rk}| < |a_{rr}||z_r|", "$$", "which is a contradiction. To prove (2) apply (1) to the matrix", "$(a_{ij}m_j)$ whose determinant is $m_1 \\ldots m_n \\det(A)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9192, "type": "theorem", "label": "models-lemma-recurring-real", "categories": [ "models" ], "title": "models-lemma-recurring-real", "contents": [ "Let $A = (a_{ij})$ be a real $n \\times n$ matrix with", "$a_{ij} \\geq 0$ for $i \\not = j$. Let $m = (m_1, \\ldots, m_n)$ be a real", "vector with $m_i > 0$. For $I \\subset \\{1, \\ldots, n\\}$ let", "$x_I \\in \\mathbf{R}^n$", "be the vector whose $i$th coordinate is $m_i$ if $i \\in I$", "and $0$ otherwise. If", "\\begin{equation}", "\\label{equation-ineq}", "-a_{ii}m_i \\geq \\sum\\nolimits_{j \\not = i} a_{ij}m_j", "\\end{equation}", "for each $i$, then $\\Ker(A)$ is the vector space", "spanned by the vectors $x_I$ such that", "\\begin{enumerate}", "\\item $a_{ij} = 0$ for $i \\in I$, $j \\not \\in I$, and", "\\item equality holds in (\\ref{equation-ineq}) for $i \\in I$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "After replacing $a_{ij}$ by $a_{ij}m_j$ we may assume $m_i = 1$ for all $i$.", "If $I \\subset \\{1, \\ldots, n\\}$ such that (1) and (2) are true,", "then a simple computation shows that $x_I$ is in the kernel of $A$.", "Conversely, let $x = (x_1, \\ldots, x_n) \\in \\mathbf{R}^n$ be a", "nonzero vector in the kernel of $A$. We will show by induction", "on the number of nonzero coordinates of $x$ that $x$ is in the", "span of the vectors $x_I$ satisfying (1) and (2). Let", "$I \\subset \\{1, \\ldots, n\\}$ be the set of indices $r$ with $|x_r|$ maximal.", "For $r \\in I$ we have", "$$", "|a_{rr} x_r| = |\\sum\\nolimits_{k \\not = r} a_{rk}x_k| \\leq", "\\sum\\nolimits_{k \\not = r} a_{rk}|x_k| \\leq", "|x_r| \\sum\\nolimits_{k \\not = r} a_{rk} \\leq |a_{rr}||x_r|", "$$", "Thus equality holds everywhere. In particular, we see that", "$a_{rk} = 0$ if $r \\in I$, $k \\not \\in I$ and equality holds", "in (\\ref{equation-ineq}) for $r \\in I$. Then we see that we", "can substract a suitable multiple of $x_I$ from $x$ to decrease", "the number of nonzero coordinates." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9193, "type": "theorem", "label": "models-lemma-recurring-symmetric-real", "categories": [ "models" ], "title": "models-lemma-recurring-symmetric-real", "contents": [ "Let $A = (a_{ij})$ be a symmetric real $n \\times n$ matrix with", "$a_{ij} \\geq 0$ for $i \\not = j$.", "Let $m = (m_1, \\ldots, m_n)$ be a real vector with $m_i > 0$.", "Assume", "\\begin{enumerate}", "\\item $Am = 0$,", "\\item there is no proper nonempty subset $I \\subset \\{1, \\ldots, n\\}$", "such that $a_{ij} = 0$ for $i \\in I$ and $j \\not \\in I$.", "\\end{enumerate}", "Then $x^t A x \\leq 0$ with equality if and only if $x = qm$", "for some $q \\in \\mathbf{R}$." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "After replacing $a_{ij}$ by $a_{ij}m_im_j$ we may assume $m_i = 1$", "for all $i$. Condition (1) means $-a_{ii} = \\sum_{j \\not = i} a_{ij}$", "for all $i$. Recall that $x^tAx = \\sum_{i, j} x_ia_{ij}x_j$.", "Then", "\\begin{align*}", "\\sum\\nolimits_{i \\not = j} -a_{ij}(x_j - x_i)^2 & =", "\\sum\\nolimits_{i \\not = j} -a_{ij}x_j^2 + 2a_{ij}x_ix_i - a_{ij}x_i^2 \\\\", "& =", "\\sum\\nolimits_j a_{jj} x_j^2 +", "\\sum\\nolimits_{i \\not = j} 2a_{ij}x_ix_i +", "\\sum\\nolimits_j a_{jj} x_i^2 \\\\", "& = 2x^tAx", "\\end{align*}", "This is clearly $\\leq 0$. If equality holds, then let $I$ be the set", "of indices $i$ with $x_i \\not = x_1$. Then $a_{ij} = 0$ for $i \\in I$", "and $j \\not \\in I$. Thus $I = \\{1, \\ldots, n\\}$ by condition (2) and", "$x$ is a multiple of $m = (1, \\ldots, 1)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9194, "type": "theorem", "label": "models-lemma-orthogonal-direct-sum", "categories": [ "models" ], "title": "models-lemma-orthogonal-direct-sum", "contents": [ "Let $L$ be a finite free $\\mathbf{Z}$-module endowed", "with an integral symmetric bilinear positive definite", "form $\\langle\\ ,\\ \\rangle : L \\times L \\to \\mathbf{Z}$.", "Let $A \\subset L$ be a submodule with $L/A$ torsion free. Set", "$B = \\{b \\in L \\mid \\langle a, b\\rangle = 0,\\ \\forall a \\in A\\}$.", "Then we have injective maps", "$$", "A^\\#/A \\leftarrow L/(A \\oplus B) \\rightarrow B^\\#/B", "$$", "whose cokernels are quotients of $L^\\#/L$. Here", "$A^\\# = \\{a' \\in A \\otimes \\mathbf{Q} \\mid", "\\langle a, a'\\rangle \\in \\mathbf{Z},\\ \\forall a \\in A\\}$", "and similarly for $B$ and $L$." ], "refs": [], "proofs": [ { "contents": [ "Observe that", "$L \\otimes \\mathbf{Q} = A \\otimes \\mathbf{Q} \\oplus B \\otimes \\mathbf{Q}$", "because the form is nondegenerate on $A$ (by positivity).", "We denote $\\pi_B : L \\otimes \\mathbf{Q} \\to B \\otimes \\mathbf{Q}$", "the projection. Observe that $\\pi_B(x) \\in B^\\#$ for $x \\in L$", "because the form is integral. This gives an exact sequence", "$$", "0 \\to A \\to L \\xrightarrow{\\pi_B} B^\\# \\to Q \\to 0", "$$", "where $Q$ is the cokernel of $L \\to B^\\#$. Observe that $Q$", "is a quotient of $L^\\#/L$ as the map $L^\\# \\to B^\\#$ is surjective", "since it is the $\\mathbf{Z}$-linear dual to $B \\to L$ which is split", "as a map of $\\mathbf{Z}$-modules.", "Dividing by $A \\oplus B$ we get a short exact sequence", "$$", "0 \\to L/(A \\oplus B) \\to B^\\#/B \\to Q \\to 0", "$$", "This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9195, "type": "theorem", "label": "models-lemma-coker", "categories": [ "models" ], "title": "models-lemma-coker", "contents": [ "Let $L_0$, $L_1$ be a finite free $\\mathbf{Z}$-modules endowed", "with integral symmetric bilinear positive definite", "forms $\\langle\\ ,\\ \\rangle : L_i \\times L_i \\to \\mathbf{Z}$.", "Let $\\text{d} : L_0 \\to L_1$ and $\\text{d}^* : L_1 \\to L_0$", "be adjoint. If $\\langle\\ ,\\ \\rangle$ on $L_0$ is unimodular, then", "there is an isomorphism", "$$", "\\Phi :", "\\Coker(\\text{d}^*\\text{d})_{torsion}", "\\longrightarrow", "\\Im(\\text{d})^\\#/\\Im(\\text{d})", "$$", "with notation as in Lemma \\ref{lemma-orthogonal-direct-sum}." ], "refs": [ "models-lemma-orthogonal-direct-sum" ], "proofs": [ { "contents": [ "Let $x \\in L_0$ be an element representing a torsion", "class in $\\Coker(\\text{d}^*\\text{d})$.", "Then for some $a > 0$ we can write $ax = \\text{d}^*\\text{d}(y)$.", "For any $z \\in \\Im(\\text{d})$, say $z = \\text{d}(y')$, we have", "$$", "\\langle (1/a)\\text{d}(y), z \\rangle =", "\\langle (1/a)\\text{d}(y), \\text{d}(y') \\rangle =", "\\langle x, y' \\rangle \\in \\mathbf{Z}", "$$", "Hence $(1/a)\\text{d}(y) \\in \\Im(\\text{d})^\\#$. We define", "$\\Phi(x) = (1/a)\\text{d}(y) \\bmod \\Im(\\text{d})$.", "We omit the proof that $\\Phi$ is well defined, additive, and injective.", "\\medskip\\noindent", "To prove $\\Phi$ is surjective, let $z \\in \\Im(\\text{d})^\\#$.", "Then $z$ defines a linear map $L_0 \\to \\mathbf{Z}$", "by the rule $x \\mapsto \\langle z, \\text{d}(x)\\rangle$.", "Since the pairing on $L_0$ is unimodular by assumption", "we can find an $x' \\in L_0$ with", "$\\langle x', x \\rangle = \\langle z, \\text{d}(x)\\rangle$", "for all $x \\in L_0$. In particular, we see", "that $x'$ pairs to zero with $\\Ker(\\text{d})$.", "Since $\\Im(\\text{d}^*\\text{d}) \\otimes \\mathbf{Q}$", "is the orthogonal complement of $\\Ker(\\text{d}) \\otimes \\mathbf{Q}$", "this means that $x'$ defines a torsion class in", "$\\Coker(\\text{d}^*\\text{d})$. We claim that $\\Phi(x') = z$.", "Namely, write $a x' = \\text{d}^*\\text{d}(y)$", "for some $y \\in L_0$ and $a > 0$.", "For any $x \\in L_0$ we get", "$$", "\\langle z, \\text{d}(x)\\rangle =", "\\langle x', x \\rangle =", "\\langle (1/a)\\text{d}^*\\text{d}(y), x \\rangle =", "\\langle (1/a)\\text{d}(y),\\text{d}(x) \\rangle", "$$", "Hence $z = \\Phi(x')$ and the proof is complete." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9194 ] }, { "id": 9196, "type": "theorem", "label": "models-lemma-recurring-symmetric-integer", "categories": [ "models" ], "title": "models-lemma-recurring-symmetric-integer", "contents": [ "Let $A = (a_{ij})$ be a symmetric $n \\times n$ integer matrix with", "$a_{ij} \\geq 0$ for $i \\not = j$. Let $m = (m_1, \\ldots, m_n)$ be an", "integer vector with $m_i > 0$. Assume", "\\begin{enumerate}", "\\item $Am = 0$,", "\\item there is no proper nonempty subset $I \\subset \\{1, \\ldots, n\\}$", "such that $a_{ij} = 0$ for $i \\in I$ and $j \\not \\in I$.", "\\end{enumerate}", "Let $e$ be the number of pairs $(i, j)$ with $i < j$ and $a_{ij} > 0$.", "Then for $\\ell$ a prime number coprime with all $a_{ij}$ and $m_i$", "we have", "$$", "\\dim_{\\mathbf{F}_\\ell}(\\Coker(A)[\\ell]) \\leq 1 - n + e", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-recurring-symmetric-real} the rank of $A$ is $n - 1$.", "The composition", "$$", "\\mathbf{Z}^{\\oplus n} \\xrightarrow{\\text{diag}(m_1, \\ldots, m_n)}", "\\mathbf{Z}^{\\oplus n} \\xrightarrow{(a_{ij})}", "\\mathbf{Z}^{\\oplus n} \\xrightarrow{\\text{diag}(m_1, \\ldots, m_n)}", "\\mathbf{Z}^{\\oplus n}", "$$", "has matrix $a_{ij}m_im_j$. Since the cokernel of the first and last", "maps are torsion of order prime to $\\ell$ by our restriction on $\\ell$", "we see that it suffices to prove the lemma for the matrix", "with entries $a_{ij}m_im_j$. Thus we may assume $m = (1, \\ldots, 1)$.", "\\medskip\\noindent", "Assume $m = (1, \\ldots, 1)$. Set $V = \\{1, \\ldots, n\\}$ and", "$E = \\{(i, j) \\mid i < j\\text{ and }a_{ij} > 0\\}$. For", "$e = (i, j) \\in E$ set $a_e = a_{ij}$. Define maps", "$s, t : E \\to V$ by setting $s(i, j) = i$ and $t(i, j) = j$.", "Set", "$\\mathbf{Z}(V) = \\bigoplus_{i \\in V} \\mathbf{Z}i$ and", "$\\mathbf{Z}(E) = \\bigoplus_{e \\in E} \\mathbf{Z}e$.", "We define symmetric positive definite integer valued pairings", "on $\\mathbf{Z}(V)$ and $\\mathbf{Z}(E)$ by setting", "$$", "\\langle i, i \\rangle = 1\\text{ for }i \\in V, \\quad", "\\langle e, e \\rangle = a_e\\text{ for }e \\in E", "$$", "and all other pairings zero. Consider the maps", "$$", "\\text{d} : \\mathbf{Z}(V) \\to \\mathbf{Z}(E), \\quad", "i \\longmapsto", "\\sum\\nolimits_{e \\in E,\\ s(e) = i} e - \\sum\\nolimits_{e \\in E,\\ t(e) = i} e", "$$", "and", "$$", "\\text{d}^*(e) = a_e(s(e) - t(e))", "$$", "A computation shows that", "$$", "\\langle d(x), y\\rangle = \\langle x, \\text{d}^*(y) \\rangle", "$$", "in other words, $\\text{d}$ and $\\text{d}^*$ are adjoint. Next we compute", "\\begin{align*}", "\\text{d}^*\\text{d}(i)", "& = ", "\\text{d}^*(", "\\sum\\nolimits_{e \\in E,\\ s(e) = i} e - \\sum\\nolimits_{e \\in E,\\ t(e) = i} e) \\\\", "& =", "\\sum\\nolimits_{e \\in E,\\ s(e) = i} a_e(s(e) - t(e)) -", "\\sum\\nolimits_{e \\in E,\\ t(e) = i} a_e(s(e) - t(e))", "\\end{align*}", "The coefficient of $i$ in $\\text{d}^*\\text{d}(i)$ is", "$$", "\\sum\\nolimits_{e \\in E,\\ s(e) = i} a_e +", "\\sum\\nolimits_{e \\in E,\\ t(e) = i} a_e = - a_{ii}", "$$", "because $\\sum_j a_{ij} = 0$ and the coefficient of", "$j \\not = i$ in $\\text{d}^*\\text{d}(i)$ is $-a_{ij}$.", "Hence $\\Coker(A) = \\Coker(\\text{d}^*\\text{d})$.", "\\medskip\\noindent", "Consider the inclusion", "$$", "\\Im(\\text{d}) \\oplus \\Ker(\\text{d}^*) \\subset \\mathbf{Z}(E)", "$$", "The left hand side is an orthogonal direct sum. Clearly", "$\\mathbf{Z}(E)/\\Ker(\\text{d}^*)$ is torsion free.", "We claim $\\mathbf{Z}(E)/\\Im(\\text{d})$ is torsion free as well.", "Namely, say $x = \\sum x_e e \\in \\mathbf{Z}(E)$ and $a > 1$ are such", "that $ax = \\text{d}y$ for some $y = \\sum y_i i \\in \\mathbf{Z}(V)$.", "Then $a x_e = y_{s(e)} - y_{t(e)}$. By property (2) we conclude", "that all $y_i$ have the same congruence class modulo $a$.", "Hence we can write $y = a y' + (y_1, y_1, \\ldots, y_1)$.", "Since $\\text{d}(y_1, y_1, \\ldots, y_1) = 0$ we conclude", "that $x = \\text{d}(y')$ which is what we had to show.", "\\medskip\\noindent", "Hence we may apply Lemma \\ref{lemma-orthogonal-direct-sum}", "to get injective maps", "$$", "\\Im(\\text{d})^\\#/\\Im(\\text{d}) \\leftarrow", "\\mathbf{Z}(E)/(\\Im(\\text{d}) \\oplus \\Ker(\\text{d}^*)) \\rightarrow", "\\Ker(\\text{d}^*)^\\#/\\Ker(\\text{d}^*)", "$$", "whose cokernels are annihilated by the product of the $a_e$", "(which is prime to $\\ell$). Since $\\Ker(\\text{d}^*)$ is", "a lattice of rank $1 - n + e$ we see that the proof is complete", "if we prove that there exists an isomorphism", "$$", "\\Phi : M_{torsion} \\longrightarrow \\Im(\\text{d})^\\#/\\Im(\\text{d})", "$$", "This is proved in Lemma \\ref{lemma-coker}." ], "refs": [ "models-lemma-recurring-symmetric-real", "models-lemma-orthogonal-direct-sum", "models-lemma-coker" ], "ref_ids": [ 9193, 9194, 9195 ] } ], "ref_ids": [] }, { "id": 9197, "type": "theorem", "label": "models-lemma-genus", "categories": [ "models" ], "title": "models-lemma-genus", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type. Then the expression", "$$", "g = 1 + \\sum m_i(w_i(g_i - 1) - \\frac{1}{2} a_{ii})", "$$", "is an integer." ], "refs": [], "proofs": [ { "contents": [ "To prove $g$ is an integer we have to show that $\\sum a_{ii}m_i$ is even.", "This we can see by computing modulo $2$ as follows", "\\begin{align*}", "\\sum\\nolimits_i a_{ii} m_i ", "& \\equiv", "\\sum\\nolimits_{i,\\ m_i\\text{ odd}} a_{ii}m_i \\\\", "& \\equiv", "\\sum\\nolimits_{i,\\ m_i\\text{ odd}} \\sum\\nolimits_{j \\not = i} a_{ij}m_j \\\\", "& \\equiv", "\\sum\\nolimits_{i,\\ m_i\\text{ odd}}", "\\sum\\nolimits_{j \\not = i,\\ m_j\\text{ odd}} a_{ij}m_j \\\\", "& \\equiv", "\\sum\\nolimits_{i < j,\\ m_i\\text{ and }m_j\\text{ odd}} a_{ij}(m_i + m_j) \\\\", "& \\equiv", "0", "\\end{align*}", "where we have used that $a_{ij} = a_{ji}$ and that $\\sum_j a_{ij}m_j = 0$", "for all $i$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9198, "type": "theorem", "label": "models-lemma-irreducible", "categories": [ "models" ], "title": "models-lemma-irreducible", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$.", "If $n = 1$, then $a_{11} = 0$ and $g = 1 + m_1w_1(g_1 - 1)$.", "Moreover, we can classify all such numerical types as follows", "\\begin{enumerate}", "\\item If $g < 0$, then $g_1 = 0$ and there are finitely many possible", "numerical types of genus $g$ with $n = 1$ corresponding to factorizations", "$m_1w_1 = 1 - g$.", "\\item If $g = 0$, then $m_1 = 1$, $w_1 = 1$, $g_1 = 0$", "as in Lemma \\ref{lemma-genus-zero}.", "\\item If $g = 1$, then we conclude $g_1 = 1$ but $m_1, w_1$ can be arbitrary", "positive integers; this is case", "(\\ref{item-one}) of Lemma \\ref{lemma-genus-one}.", "\\item If $g > 1$, then $g_1 > 1$ and there are finitely many possible", "numerical types of genus $g$ with $n = 1$ corresponding to", "factorizations $m_1w_1(g_1 - 1) = g - 1$.", "\\end{enumerate}" ], "refs": [ "models-lemma-genus-zero", "models-lemma-genus-one" ], "proofs": [ { "contents": [ "The lemma proves itself." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9226, 9227 ] }, { "id": 9199, "type": "theorem", "label": "models-lemma-diagonal-negative", "categories": [ "models" ], "title": "models-lemma-diagonal-negative", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$.", "If $n > 1$, then $a_{ii} < 0$ for all $i$." ], "refs": [], "proofs": [ { "contents": [ "Lemma \\ref{lemma-recurring-symmetric-real} applies to the matrix $A$." ], "refs": [ "models-lemma-recurring-symmetric-real" ], "ref_ids": [ 9193 ] } ], "ref_ids": [] }, { "id": 9200, "type": "theorem", "label": "models-lemma-minus-one", "categories": [ "models" ], "title": "models-lemma-minus-one", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$.", "Assume $n > 1$. If $i$ is such that the contribution", "$m_i(w_i(g_i - 1) - \\frac{1}{2} a_{ii})$", "to the genus $g$ is $< 0$, then $g_i = 0$ and $a_{ii} = -w_i$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-diagonal-negative} and", "$w_i > 0$, $g_i \\geq 0$, and $w_i | a_{ii}$." ], "refs": [ "models-lemma-diagonal-negative" ], "ref_ids": [ 9199 ] } ], "ref_ids": [] }, { "id": 9201, "type": "theorem", "label": "models-lemma-contract", "categories": [ "models" ], "title": "models-lemma-contract", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$.", "Assume $n$ is a $(-1)$-index. Then there is a numerical", "type $T'$ given by $n', m'_i, a'_{ij}, w'_i, g'_i$ with", "\\begin{enumerate}", "\\item $n' = n - 1$,", "\\item $m'_i = m_i$,", "\\item $a'_{ij} = a_{ij} - a_{in}a_{jn}/a_{nn}$,", "\\item $w'_i = w_i/2$ if $a_{in}/w_n$ even and $a_{in}/w_i$ odd", "and $w'_i = w_i$ else,", "\\item $g'_i =", "\\frac{w_i}{w'_i}(g_i - 1) + 1 + \\frac{a_{in}^2 - w_na_{in}}{2w'_iw_n}$.", "\\end{enumerate}", "Moreover, we have $g = g'$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $n > 1$ for example by Lemma \\ref{lemma-irreducible}", "and hence $n' \\geq 1$. We check conditions (1) -- (5) of", "Definition \\ref{definition-type} for $n', m'_i, a'_{ij}, w'_i, g'_i$.", "\\medskip\\noindent", "Condition (1) is immediate.", "\\medskip\\noindent", "Condition (2). Symmetry of $A' = (a'_{ij})$ is immediate", "and since $a_{nn} < 0$ by Lemma \\ref{lemma-diagonal-negative}", "we see that $a'_{ij} \\geq a_{ij} \\geq 0$ if $i \\not = j$.", "\\medskip\\noindent", "Condition (3). Suppose that $I \\subset \\{1, \\ldots, n - 1\\}$ such that", "$a'_{ii'} = 0$ for $i \\in I$ and $i' \\in \\{1, \\ldots, n - 1\\} \\setminus I$.", "Then we see that for each $i \\in I$ and $i' \\in I'$ we have", "$a_{in}a_{i'n} = 0$. Thus either $a_{in} = 0$ for all $i \\in I$ and", "$I \\subset \\{1, \\ldots, n\\}$ is a contradiction for property (3) for $T$,", "or $a_{i'n} = 0$ for all $i' \\in \\{1, \\ldots, n - 1\\} \\setminus I$", "and $I \\cup \\{n\\} \\subset \\{1, \\ldots, n\\}$ is a contradiction for", "property (3) of $T$. Hence (3) holds for $T'$.", "\\medskip\\noindent", "Condition (4). We compute", "$$", "\\sum\\nolimits_{j = 1}^{n - 1} a'_{ij}m_j =", "\\sum\\nolimits_{j = 1}^{n - 1}", "(a_{ij}m_j - \\frac{a_{in}a_{jn}m_j}{a_{nn}}) =", "- a_{in}m_n - \\frac{a_{in}}{a_{nn}}(-a_{nn}m_n) = 0", "$$", "as desired.", "\\medskip\\noindent", "Condition (5). We have to show that $w'_i$ divides $a_{in}a_{jn}/a_{nn}$.", "This is clear because $a_{nn} = -w_n$ and $w_n | a_{jn}$ and $w_i | a_{in}$.", "\\medskip\\noindent", "To show that $g = g'$ we first write", "\\begin{align*}", "g", "& =", "1 + \\sum\\nolimits_{i = 1}^n m_i(w_i(g_i - 1) - \\frac{1}{2}a_{ii}) \\\\", "& =", "1 + \\sum\\nolimits_{i = 1}^{n - 1} m_i(w_i(g_i - 1) - \\frac{1}{2}a_{ii})", "-\\frac{1}{2}m_nw_n \\\\", "& =", "1 + \\sum\\nolimits_{i = 1}^{n - 1}", "m_i(w_i(g_i - 1) - \\frac{1}{2}a_{ii} - \\frac{1}{2}a_{in})", "\\end{align*}", "Comparing with the expression for $g'$ we see that it suffices if", "$$", "w'_i(g'_i - 1) - \\frac{1}{2}a'_{ii} =", "w_i(g_i - 1) - \\frac{1}{2}a_{in} - \\frac{1}{2}a_{ii}", "$$", "for $i \\leq n - 1$. In other words, we have", "$$", "g'_i = \\frac{2w_i(g_i - 1) - a_{in} - a_{ii} + a'_{ii} + 2w'_i}{2w'_i} =", "\\frac{w_i}{w'_i}(g_i - 1) + 1 + \\frac{a_{in}^2 - w_na_{in}}{2w'_iw_n}", "$$", "It is elementary to check that this is an integer $\\geq 0$", "if we choose $w'_i$ as in (4)." ], "refs": [ "models-lemma-irreducible", "models-definition-type", "models-lemma-diagonal-negative" ], "ref_ids": [ 9198, 9269, 9199 ] } ], "ref_ids": [] }, { "id": 9202, "type": "theorem", "label": "models-lemma-top-genus", "categories": [ "models" ], "title": "models-lemma-top-genus", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type.", "Let $e$ be the number of pairs $(i, j)$ with $i < j$ and $a_{ij} > 0$.", "Then the expression $g_{top} = 1 - n + e$ is $\\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "If not, then $e < n - 1$ which means there exists an $i$ such that", "$a_{ij} = 0$ for all $j \\not = i$. This contradicts assumption", "(3) of Definition \\ref{definition-type}." ], "refs": [ "models-definition-type" ], "ref_ids": [ 9269 ] } ], "ref_ids": [] }, { "id": 9203, "type": "theorem", "label": "models-lemma-non-irreducible-minimal-type-genus-at-least-one", "categories": [ "models" ], "title": "models-lemma-non-irreducible-minimal-type-genus-at-least-one", "contents": [ "If $n, m_i, a_{ij}, w_i, g_i$ is a minimal numerical type", "with $n > 1$, then $g \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "This is true because $g = 1 + \\sum \\Phi_i$ with", "$\\Phi_i = m_i(w_i(g_i - 1) - \\frac{1}{2} a_{ii})$ nonnegative", "by Lemma \\ref{lemma-minus-one} and the definition of minimal types." ], "refs": [ "models-lemma-minus-one" ], "ref_ids": [ 9200 ] } ], "ref_ids": [] }, { "id": 9204, "type": "theorem", "label": "models-lemma-genus-nonnegative", "categories": [ "models" ], "title": "models-lemma-genus-nonnegative", "contents": [ "If $n, m_i, a_{ij}, w_i, g_i$ is a minimal numerical type", "with $n > 1$, then $g \\geq g_{top}$." ], "refs": [], "proofs": [ { "contents": [ "The reader who is only interested in the case of numerical types", "associated to proper regular models can skip this proof as we will", "reprove this in the geometric situation later.", "We can write", "$$", "g_{top} = 1 - n + \\frac{1}{2}\\sum\\nolimits_{a_{ij} > 0} 1 =", "1 + \\sum\\nolimits_i (-1 +", "\\frac{1}{2}\\sum\\nolimits_{j \\not = i,\\ a_{ij} > 0} 1) ", "$$", "On the other hand, we have", "\\begin{align*}", "g & =", "1 + \\sum m_i(w_i(g_i - 1) - \\frac{1}{2} a_{ii}) \\\\", "& =", "1 + \\sum m_iw_ig_i - \\sum m_iw_i +", "\\frac{1}{2} \\sum\\nolimits_{i \\not = j} a_{ij}m_j \\\\", "& =", "1 + \\sum\\nolimits_i", "m_iw_i(-1 + g_i + \\frac{1}{2} \\sum\\nolimits_{j \\not = i} \\frac{a_{ij}}{w_i})", "\\end{align*}", "The first equality is the definition, the second equality uses that", "$\\sum a_{ij}m_j = 0$, and the last equality uses that", "uses $a_{ij} = a_{ji}$ and switching order of", "summation. Comparing with the formula for $g_{top}$ we conclude", "that the lemma holds if", "$$", "\\Psi_i =", "m_iw_i(-1 + g_i + \\frac{1}{2} \\sum\\nolimits_{j \\not = i} \\frac{a_{ij}}{w_i})", "- (-1 + \\frac{1}{2}\\sum\\nolimits_{j \\not = i,\\ a_{ij} > 0} 1)", "$$", "is $\\geq 0$ for each $i$. However, this may not be the case.", "Let us analyze for which indices we can have $\\Psi_i < 0$.", "First, observe that", "$$", "(-1 + g_i + \\frac{1}{2}\\sum\\nolimits_{j \\not = i} \\frac{a_{ij}}{w_i}) \\geq", "(-1 + \\frac{1}{2}\\sum\\nolimits_{j \\not = i,\\ a_{ij} > 0} 1)", "$$", "because $a_{ij}/w_i$ is a nonnegative integer. Since $m_iw_i$ is", "a positive integer we conclude that $\\Psi_i \\geq 0$ as soon as", "either $m_iw_i = 1$ or the left hand side of the inequality is $\\geq 0$", "which happens if $g_i > 0$, or $a_{ij} > 0$ for at least two indices $j$, or", "if there is a $j$ with $a_{ij} > w_i$. Thus", "$$", "P = \\{i : \\Psi_i < 0\\}", "$$", "is the set of indices $i$ such that $m_iw_i > 1$, $g_i = 0$,", "$a_{ij} > 0$ for a unique $j$, and $a_{ij} = w_i$ for this $j$.", "Moreover", "$$", "i \\in P \\Rightarrow \\Psi_i = \\frac{1}{2}(-m_iw_i + 1)", "$$", "The strategy of proof is to show that given $i \\in P$ we can borrow a bit", "from $\\Psi_j$ where $j$ is the neighbour of $i$, i.e., $a_{ij} > 0$.", "However, this won't quite work because $j$ may be an index with $\\Psi_j = 0$.", "\\medskip\\noindent", "Consider the set", "$$", "Z = \\{j : g_j = 0\\text{ and }", "j\\text{ has exactly two neighbours }i, k\\text{ with }", "a_{ij} = w_j = a_{jk}\\}", "$$", "For $j \\in Z$ we have $\\Psi_j = 0$. We will consider sequences", "$M = (i, j_1, \\ldots, j_s)$ where $s \\geq 0$,", "$i \\in P$, $j_1, \\ldots, j_s \\in Z$, and", "$a_{ij_1} > 0, a_{j_1j_2} > 0, \\ldots, a_{j_{s - 1}j_s} > 0$.", "If our numerical type consists of two indices which are in $P$", "or more generally if our numerical type consists of", "two indices which are in $P$ and all other indices in $Z$, then", "$g_{top} = 0$ and we win by", "Lemma \\ref{lemma-non-irreducible-minimal-type-genus-at-least-one}.", "We may and do discard these cases.", "\\medskip\\noindent", "Let $M = (i, j_1, \\ldots, j_s)$ be a maximal sequence and let", "$k$ be the second neighbour of $j_s$. (If $s = 0$, then $k$", "is the unique neighbour of $i$.) By maximality $k \\not \\in Z$", "and by what we just said $k \\not \\in P$. Observe that", "$w_i = a_{ij_1} = w_{j_1} = a_{j_1j_2} = \\ldots = w_{j_s} =", "a_{j_sk}$. Looking at the definition", "of a numerical type we see that", "\\begin{align*}", "m_ia_{ii} + m_{j_1}w_i & = 0,\\\\", "m_iw_i + m_{j_1}a_{j_1j_1} + m_{j_2}w_i & = 0,\\\\", "\\ldots & \\ldots \\\\", "m_{j_{s - 1}}w_i + m_{j_s}a_{j_sj_s} + m_kw_i & = 0", "\\end{align*}", "The first equality implies $m_{j_1} \\geq 2m_i$ because the", "numerical type is minimal. Then the second equality implies", "$m_{j_2} \\geq 3m_i$, and so on. In any case, we conclude that", "$m_k \\geq 2m_i$ (including when $s = 0$).", "\\medskip\\noindent", "Let $k$ be an index such that we have a $t > 0$ and pairwise distinct", "maximal sequences $M_1, \\ldots, M_t$ as above, with", "$M_b = (i_b, j_{b, 1}, \\ldots, j_{b, s_b})$", "such that $k$ is a neighbour of $j_{b, s_b}$ for $b = 1, \\ldots, t$.", "We will show that $\\Phi_j + \\sum_{b = 1, \\ldots, t} \\Phi_{i_b} \\geq 0$.", "This will finish the proof of the lemma by what we said above.", "Let $M$ be the union of the indices occurring in $M_b$, $b = 1, \\ldots, t$.", "We write", "$$", "\\Psi_k =", "-\\sum\\nolimits_{b = 1, \\ldots, t} \\Psi_{i_b} + \\Psi_k'", "$$", "where", "\\begin{align*}", "\\Psi_k' & =", "m_kw_k\\left(-1 + g_k +", "\\frac{1}{2} \\sum\\nolimits_{b = 1, \\ldots t}", "(\\frac{a_{kj_{b, s_b}}}{w_k} - \\frac{m_{i_b}w_{i_b}}{m_kw_k}) +", "\\frac{1}{2} \\sum\\nolimits_{l \\not = k,\\ l \\not \\in M}", "\\frac{a_{kl}}{w_k}", "\\right) \\\\", "&", "-\\left(", "-1 + \\frac{1}{2}\\sum\\nolimits_{l \\not = k,\\ l \\not \\in M,\\ a_{kl} > 0} 1", "\\right)", "\\end{align*}", "Assume $\\Psi_k' < 0$ to get a contradiction.", "If the set $\\{l : l \\not = k,\\ l \\not \\in M,\\ a_{kl} > 0\\}$ is empty,", "then $\\{1, \\ldots, n\\} = M \\cup \\{k\\}$ and $g_{top} = 0$", "because $e = n - 1$ in this case and the result holds by", "Lemma \\ref{lemma-non-irreducible-minimal-type-genus-at-least-one}.", "Thus we may assume there is at least one such $l$ which contributes", "$(1/2)a_{kl}/w_k \\geq 1/2$ to the sum inside the first brackets.", "For each $b = 1, \\ldots, t$ we have", "$$", "\\frac{a_{kj_{b, s_b}}}{w_k} - \\frac{m_{i_b}w_{i_b}}{m_kw_k} =", "\\frac{w_{i_b}}{w_k}(1 - \\frac{m_{i_b}}{m_k})", "$$", "This expression is $\\geq \\frac{1}{2}$ because $m_k \\geq 2m_{i_b}$", "by the previous paragraph and is $\\geq 1$ if $w_k < w_{i_b}$.", "It follows that $\\Psi_k' < 0$ implies $g_k = 0$.", "If $t \\geq 2$ or $t = 1$ and $w_k < w_{i_1}$, then $\\Psi_k' \\geq 0$", "(here we use the existence of an $l$ as shown above) which", "is a contradiction too.", "Thus $t = 1$ and $w_k = w_{i_1}$. If there at least two nonzero terms", "in the sum over $l$ or if there is one such $k$ and $a_{kl} > w_k$, then", "$\\Psi_k' \\geq 0$ as well. The final possibility is that $t = 1$ and", "there is one $l$ with $a_{kl} = w_k$. This is disallowed as this would", "mean $k \\in Z$ contradicting the maximality of $M_1$." ], "refs": [ "models-lemma-non-irreducible-minimal-type-genus-at-least-one", "models-lemma-non-irreducible-minimal-type-genus-at-least-one" ], "ref_ids": [ 9203, 9203 ] } ], "ref_ids": [] }, { "id": 9205, "type": "theorem", "label": "models-lemma-minus-two", "categories": [ "models" ], "title": "models-lemma-minus-two", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$.", "Assume $n > 1$. If $i$ is such that the contribution", "$m_i(w_i(g_i - 1) - \\frac{1}{2} a_{ii})$", "to the genus $g$ is $0$, then $g_i = 0$ and $a_{ii} = -2w_i$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-diagonal-negative} and", "$w_i > 0$, $g_i \\geq 0$, and $w_i | a_{ii}$." ], "refs": [ "models-lemma-diagonal-negative" ], "ref_ids": [ 9199 ] } ], "ref_ids": [] }, { "id": 9206, "type": "theorem", "label": "models-lemma-picard-rank-1", "categories": [ "models" ], "title": "models-lemma-picard-rank-1", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$.", "The Picard group of $T$ is a finitely generated abelian group of rank $1$." ], "refs": [], "proofs": [ { "contents": [ "If $n = 1$, then $A = (a_{ij})$ is the zero matrix and", "the result is clear. For $n > 1$ the matrix $A$ has rank", "$n - 1$ by either Lemma \\ref{lemma-recurring-real} or", "Lemma \\ref{lemma-recurring-symmetric-real}.", "Of course the rank is not affected by scaling the rows", "by $1/w_i$. This proves the lemma." ], "refs": [ "models-lemma-recurring-real", "models-lemma-recurring-symmetric-real" ], "ref_ids": [ 9192, 9193 ] } ], "ref_ids": [] }, { "id": 9207, "type": "theorem", "label": "models-lemma-picard-T-and-A", "categories": [ "models" ], "title": "models-lemma-picard-T-and-A", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$.", "Then $\\Pic(T) \\subset \\Coker(A)$ where $A = (a_{ij})$." ], "refs": [], "proofs": [ { "contents": [ "Since $\\Pic(T)$ is the cokernel of $(a_{ij}/w_i)$", "we see that there is a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathbf{Z}^{\\oplus n} \\ar[rr]_A & &", "\\mathbf{Z}^{\\oplus n} \\ar[rr] & &", "\\Coker(A) \\ar[r] & 0 \\\\", "0 \\ar[r] &", "\\mathbf{Z}^{\\oplus n} \\ar[rr]^{(a_{ij}/w_i)} \\ar[u]_{\\text{id}} & &", "\\mathbf{Z}^{\\oplus n} \\ar[rr] \\ar[u]_{\\text{diag}(w_1, \\ldots, w_n)} & &", "\\Pic(T) \\ar[r] \\ar[u] & 0", "}", "$$", "with exact rows. By the snake lemma we conclude that", "$\\Pic(T) \\subset \\Coker(A)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9208, "type": "theorem", "label": "models-lemma-contract-picard-group", "categories": [ "models" ], "title": "models-lemma-contract-picard-group", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$.", "Assume $n$ is a $(-1)$-index. Let $T'$ be the numerical", "type constructed in Lemma \\ref{lemma-contract}. There exists an", "injective map", "$$", "\\Pic(T) \\to \\Pic(T')", "$$", "whose cokernel is an elementary abelian $2$-group." ], "refs": [ "models-lemma-contract" ], "proofs": [ { "contents": [ "Recall that $n' = n - 1$. Let $e_i$, resp., $e'_i$ be the $i$th", "basis vector of $\\mathbf{Z}^{\\oplus n}$, resp.\\ $\\mathbf{Z}^{\\oplus n - 1}$.", "First we denote", "$$", "q : \\mathbf{Z}^{\\oplus n} \\to \\mathbf{Z}^{\\oplus n - 1},", "\\quad e_n \\mapsto 0\\text{ and }e_i \\mapsto e'_i\\text{ for }i \\leq n - 1", "$$", "and we set", "$$", "p : \\mathbf{Z}^{\\oplus n} \\to \\mathbf{Z}^{\\oplus n - 1},\\quad", "e_n \\mapsto \\sum\\nolimits_{j = 1}^{n - 1} \\frac{a_{nj}}{w'_j} e'_j", "\\text{ and }", "e_i \\mapsto \\frac{w_i}{w'_i} e'_i\\text{ for }i \\leq n - 1", "$$", "A computation (which we omit) shows there is a commutative diagram", "$$", "\\xymatrix{", "\\mathbf{Z}^{\\oplus n} \\ar[rr]_{(a_{ij}/w_i)} \\ar[d]_q & &", "\\mathbf{Z}^{\\oplus n} \\ar[d]^p \\\\", "\\mathbf{Z}^{\\oplus n'} \\ar[rr]^{(a'_{ij}/w'_i)} & &", "\\mathbf{Z}^{\\oplus n'}", "}", "$$", "Since the cokernel of the top arrow is", "$\\Pic(T)$ and the cokernel of the bottom arrow", "is $\\Pic(T')$, we obtain the desired homomorphism", "of Picard groups. Since $\\frac{w_i}{w'_i} \\in \\{1, 2\\}$", "we see that the cokernel of $\\Pic(T) \\to \\Pic(T')$", "is annihilated by $2$ (because $2e'_i$ is in the image of $p$", "for all $i \\leq n - 1$).", "Finally, we show $\\Pic(T) \\to \\Pic(T')$ is injective.", "Let $L = (l_1, \\ldots, l_n)$ be a representative", "of an element of $\\Pic(T)$ mapping to zero in $\\Pic(T')$.", "Since $q$ is surjective, a diagram chase shows that we can assume", "$L$ is in the kernel of $p$. This means that", "$l_na_{ni}/w'_i + l_iw_i/w'_i = 0$, i.e., $l_i = - a_{ni}/w_i l_n$.", "Thus $L$ is the image of $-l_ne_n$ under the map $(a_{ij}/w_j)$", "and the lemma is proved." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9201 ] }, { "id": 9209, "type": "theorem", "label": "models-lemma-picard-group-genus-nonpositive", "categories": [ "models" ], "title": "models-lemma-picard-group-genus-nonpositive", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$.", "If the genus $g$ of $T$ is $\\leq 0$, then $\\Pic(T) = \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "By induction on $n$. If $n = 1$, then the assertion is clear.", "If $n > 1$, then $T$ is not minimal by", "Lemma \\ref{lemma-non-irreducible-minimal-type-genus-at-least-one}.", "After replacing $T$ by an equivalent type", "we may assume $n$ is a $(-1)$-index.", "By Lemma \\ref{lemma-contract-picard-group}", "we find $\\Pic(T) \\subset \\Pic(T')$.", "By Lemma \\ref{lemma-contract} we see that the genus", "of $T'$ is equal to the genus of $T$ and we conclude by", "induction." ], "refs": [ "models-lemma-non-irreducible-minimal-type-genus-at-least-one", "models-lemma-contract-picard-group", "models-lemma-contract" ], "ref_ids": [ 9203, 9208, 9201 ] } ], "ref_ids": [] }, { "id": 9210, "type": "theorem", "label": "models-lemma-two-by-two", "categories": [ "models" ], "title": "models-lemma-two-by-two", "contents": [ "Classification of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] & \\bullet", "}", "$$", "If $n > 2$, then given a pair $i, j$ of $(-2)$-indices with $a_{ij} > 0$,", "then up to ordering we have the $m$'s, $a$'s, $w$'s", "\\begin{enumerate}", "\\item", "\\label{item-A2}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w \\\\", "w & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w", "\\end{matrix}", "\\right)", "$$", "with $w$ arbitrary and $2m_1 \\geq m_2$ and $2m_2 \\geq m_1$, or", "\\item", "\\label{item-B2}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & 2w \\\\", "2w & -4w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "2w", "\\end{matrix}", "\\right)", "$$", "with $w$ arbitrary and $m_1 \\geq m_2$ and $2m_2 \\geq m_1$, or", "\\item", "\\label{item-G2}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & 3w \\\\", "3w & -6w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "3w", "\\end{matrix}", "\\right)", "$$", "with $w$ arbitrary and $2m_1 \\geq 3m_2$ and $2m_2 \\geq m_1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9211, "type": "theorem", "label": "models-lemma-three-by-three", "categories": [ "models" ], "title": "models-lemma-three-by-three", "contents": [ "Classification of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet", "}", "$$", "If $n > 3$, then given a triple $i, j, k$ of $(-2)$-indices", "with at least two $a_{ij}, a_{ik}, a_{jk}$ nonzero, then up", "to ordering we have the $m$'s, $a$'s, $w$'s", "\\begin{enumerate}", "\\item", "\\label{item-A3}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 \\\\", "w & -2w & w \\\\", "0 & w & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2$, or", "\\item", "\\label{item-C3}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 \\\\", "w & -2w & 2w \\\\", "0 & 2w & -4w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "2w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + 2m_3$, $2m_3 \\geq m_2$, or", "\\item", "\\label{item-B3}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-4w & 2w & 0 \\\\", "2w & -4w & 2w \\\\", "0 & 2w & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "2w \\\\", "2w \\\\", "w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $m_3 \\geq m_2$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9212, "type": "theorem", "label": "models-lemma-four-by-four", "categories": [ "models" ], "title": "models-lemma-four-by-four", "contents": [ "Classification of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet", "}", "$$", "If $n > 4$, then given four $(-2)$-indices $i, j, k, l$", "with $a_{ij}, a_{jk}, a_{kl}$ nonzero, then up", "to ordering we have the $m$'s, $a$'s, $w$'s", "\\begin{enumerate}", "\\item", "\\label{item-A4}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3 \\\\", "m_4", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & 0 \\\\", "w & -2w & w & 0 \\\\", "0 & w & -2w & w \\\\", "0 & 0 & w & -2w ", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$,", "and $2m_4 \\geq m_3$, or", "\\item", "\\label{item-C4}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3 \\\\", "m_4", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & 0 \\\\", "w & -2w & w & 0 \\\\", "0 & w & -2w & 2w \\\\", "0 & 0 & 2w & -4w ", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "2w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + 2m_4$,", "and $2m_4 \\geq m_3$, or", "\\item", "\\label{item-B4}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3 \\\\", "m_4", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-4w & 2w & 0 & 0 \\\\", "2w & -4w & 2w & 0 \\\\", "0 & 2w & -4w & 2w \\\\", "0 & 0 & 2w & -2w ", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "2w \\\\", "2w \\\\", "2w \\\\", "w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$,", "and $m_4 \\geq m_3$, or", "\\item", "\\label{item-F4}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3 \\\\", "m_4", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & 0 \\\\", "w & -2w & 2w & 0 \\\\", "0 & 2w & -4w & 2w \\\\", "0 & 0 & 2w & -4w ", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "2w \\\\", "2w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + 2m_3$, $2m_3 \\geq m_2 + m_4$,", "and $2m_4 \\geq m_3$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9213, "type": "theorem", "label": "models-lemma-D4", "categories": [ "models" ], "title": "models-lemma-D4", "contents": [ "Classification of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\\\", "& \\bullet", "}", "$$", "If $n > 4$, then given four $(-2)$-indices $i, j, k, l$", "with $a_{ij}, a_{ik}, a_{il}$ nonzero, then up", "to ordering we have the $m$'s, $a$'s, $w$'s", "\\begin{enumerate}", "\\item", "\\label{item-D4}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3 \\\\", "m_4", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & w & w \\\\", "w & -2w & 0 & 0 \\\\", "w & 0 & -2w & 0 \\\\", "w & 0 & 0 & -2w ", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2 + m_3 + m_4$, $2m_2 \\geq m_1$, $2m_3 \\geq m_1$,", "$2m_4 \\geq m_1$. Observe that this implies $m_1 \\geq \\max(m_2, m_3, m_4)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9214, "type": "theorem", "label": "models-lemma-five-by-five", "categories": [ "models" ], "title": "models-lemma-five-by-five", "contents": [ "Classification of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet", "}", "$$", "If $n > 5$, then given five $(-2)$-indices $h, i, j, k, l$", "with $a_{hi}, a_{ij}, a_{jk}, a_{kl}$ nonzero, then up", "to ordering we have the $m$'s, $a$'s, $w$'s", "\\begin{enumerate}", "\\item", "\\label{item-A5}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3 \\\\", "m_4 \\\\", "m_5", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & 0 & 0 \\\\", "w & -2w & w & 0 & 0 \\\\", "0 & w & -2w & w & 0 \\\\", "0 & 0 & w & -2w & w \\\\", "0 & 0 & 0 & w & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$,", "$2m_4 \\geq m_3 + m_5$, and $2m_5 \\geq m_4$, or", "\\item", "\\label{item-C5}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3 \\\\", "m_4 \\\\", "m_5", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & 0 & 0 \\\\", "w & -2w & w & 0 & 0 \\\\", "0 & w & -2w & w & 0 \\\\", "0 & 0 & w & -2w & 2w \\\\", "0 & 0 & 0 & 2w & -4w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "2w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + 2m_4$,", "$2m_4 \\geq m_3 + m_5$, and $2m_5 \\geq m_4$, or", "\\item", "\\label{item-B5}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3 \\\\", "m_4 \\\\", "m_5", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-4w & 2w & 0 & 0 & 0 \\\\", "2w & -4w & 2w & 0 & 0 \\\\", "0 & 2w & -4w & 2w & 0 \\\\", "0 & 0 & 2w & -4w & 2w \\\\", "0 & 0 & 0 & 2w & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "2w \\\\", "2w \\\\", "2w \\\\", "2w \\\\", "w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$,", "$2m_4 \\geq m_3 + m_5$, and $m_4 \\geq m_3$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9215, "type": "theorem", "label": "models-lemma-fourfold", "categories": [ "models" ], "title": "models-lemma-fourfold", "contents": [ "Nonexistence of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[ld] \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\\\", "\\bullet & \\bullet", "}", "$$", "If $n > 5$, there do {\\bf not} exist five $(-2)$-indices", "$h$, $i$, $j$, $k$ with $a_{hi} > 0$, $a_{hj} > 0$, $a_{hk} > 0$, and", "$a_{hl} > 0$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9216, "type": "theorem", "label": "models-lemma-D5", "categories": [ "models" ], "title": "models-lemma-D5", "contents": [ "Classification of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\\\", "& & \\bullet", "}", "$$", "If $n > 5$, then given five $(-2)$-indices $h, i, j, k, l$", "with $a_{hi}, a_{ij}, a_{jk}, a_{jl}$ nonzero, then up", "to ordering we have the $m$'s, $a$'s, $w$'s", "\\begin{enumerate}", "\\item", "\\label{item-D5}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3 \\\\", "m_4 \\\\", "m_5", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & 0 & 0 \\\\", "w & -2w & w & 0 & 0 \\\\", "0 & w & -2w & w & w \\\\", "0 & 0 & w & -2w & 0 \\\\", "0 & 0 & w & 0 & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4 + m_5$,", "$2m_4 \\geq m_3$, and $2m_5 \\geq m_3$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9217, "type": "theorem", "label": "models-lemma-long", "categories": [ "models" ], "title": "models-lemma-long", "contents": [ "Classification of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{..}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet", "}", "$$", "Let $t > 5$ and $n > t$. Then given $t$ distinct $(-2)$-indices", "$i_1, \\ldots, i_t$ such that $a_{i_ji_{j + 1}}$ is nonzero for", "$j = 1, \\ldots, t - 1$, then up to reversing the order of these indices", "we have the $a$'s and $w$'s", "\\begin{enumerate}", "\\item", "\\label{item-An}", "are given by $w_{i_1} = w_{i_2} = \\ldots = w_{i_t} = w$,", "$a_{i_ji_{j + 1}} = w$, and $a_{i_ji_k} = 0$ if $k > j + 1$, or", "\\item", "\\label{item-Cn}", "are given by $w_{i_1} = w_{i_2} = \\ldots = w_{i_{t - 1}} = w$,", "$w_{j_t} = 2w$, $a_{i_ji_{j + 1}} = w$ for $j < t - 1$,", "$a_{i_{t - 1}i_t} = 2w$, and $a_{i_ji_k} = 0$ if $k > j + 1$, or", "\\item", "\\label{item-Bn}", "are given by $w_{i_1} = w_{i_2} = \\ldots = w_{i_{t - 1}} = 2w$,", "$w_{j_t} = w$, $a_{i_ji_{j + 1}} = 2w$, and", "$a_{i_{t - 1}i_t} = 2w$, and $a_{i_ji_k} = 0$ if $k > j + 1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9218, "type": "theorem", "label": "models-lemma-Dn", "categories": [ "models" ], "title": "models-lemma-Dn", "contents": [ "Classification of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{..}[r] & \\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\\\", "& & & \\bullet", "}", "$$", "Let $t > 4$ and $n > t + 1$. Then given $t + 1$ distinct", "$(-2)$-indices $i_1, \\ldots, i_{t + 1}$ such that $a_{i_ji_{j + 1}}$", "is nonzero for $j = 1, \\ldots, t - 1$ and $a_{i_{t - 1}i_{t + 1}}$", "is nonzero, then we have the $a$'s and $w$'s", "\\begin{enumerate}", "\\item", "\\label{item-Dn}", "are given by $w_{i_1} = w_{i_2} = \\ldots = w_{i_{t + 1}} = w$,", "$a_{i_ji_{j + 1}} = w$ for $j = 1, \\ldots, t - 1$,", "$a_{i_{t - 1}i_{t + 1}} = w$ and $a_{i_ji_k} = 0$ for other", "pairs $(j, k)$ with $j > k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9219, "type": "theorem", "label": "models-lemma-E6", "categories": [ "models" ], "title": "models-lemma-E6", "contents": [ "Classification of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] &", "\\bullet \\ar@{-}[r] & \\bullet \\\\", "& & \\bullet", "}", "$$", "Let $n > 6$. Then given $6$ distinct $(-2)$-indices $i_1, \\ldots, i_6$", "such that $a_{12}, a_{23}, a_{34}, a_{45}, a_{36}$ are nonzero, then", "we have the $m$'s, $a$'s, and $w$'s", "\\begin{enumerate}", "\\item", "\\label{item-E6}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3 \\\\", "m_4 \\\\", "m_5 \\\\", "m_6", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & 0 & 0 & 0 \\\\", "w & -2w & w & 0 & 0 & 0 \\\\", "0 & w & -2w & w & 0 & w \\\\", "0 & 0 & w & -2w & w & 0 \\\\", "0 & 0 & 0 & w & -2w & 0 \\\\", "0 & 0 & w & 0 & 0 & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4 + m_6$,", "$2m_4 \\geq m_3 + m_5$, $2m_5 \\geq m_3$, and $2m_6 \\geq m_3$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9220, "type": "theorem", "label": "models-lemma-double-triple", "categories": [ "models" ], "title": "models-lemma-double-triple", "contents": [ "Nonexistence of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{..}[r] \\ar@{-}[d] &", "\\bullet \\ar@{-}[d] \\ar@{-}[r] & \\bullet \\\\", "& \\bullet & \\bullet", "}", "$$", "Assume $t \\geq 4$ and $n > t + 2$.", "There do {\\bf not} exist $t + 2$ distinct", "$(-2)$-indices $i_0, \\ldots, i_{t + 1}$ such that", "$a_{i_ji_{j + 1}} > 0$ for $j = 1, \\ldots, t - 1$", "and $a_{i_0i_2} > 0$ and $a_{i_{t - 1}i_{t + 1}} > 0$." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9221, "type": "theorem", "label": "models-lemma-E6-completed", "categories": [ "models" ], "title": "models-lemma-E6-completed", "contents": [ "Nonexistence of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] &", "\\bullet \\ar@{-}[r] & \\bullet \\\\", "& & \\bullet \\ar@{-}[d] \\\\", "& & \\bullet", "}", "$$", "Assume $n > 7$. There do {\\bf not} exist $7$ distinct", "$(-2)$-indices $f, g, h, i, j, k, l$", "such that $a_{fg}, a_{gh}, a_{ij}, a_{jh}, a_{kl}, a_{lh}$ are nonzero." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9222, "type": "theorem", "label": "models-lemma-E7", "categories": [ "models" ], "title": "models-lemma-E7", "contents": [ "Classification of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\ar@{-}[r] & \\bullet \\\\", "& & & \\bullet", "}", "$$", "Let $n > 7$. Then given $7$ distinct $(-2)$-indices $i_1, \\ldots, i_7$", "such that $a_{12}, a_{23}, a_{34}, a_{45}, a_{56}, a_{47}$ are nonzero,", "then we have the $m$'s, $a$'s, and $w$'s", "\\begin{enumerate}", "\\item", "\\label{item-E7}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3 \\\\", "m_4 \\\\", "m_5 \\\\", "m_6 \\\\", "m_7", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & 0 & 0 & 0 & 0 \\\\", "w & -2w & w & 0 & 0 & 0 & 0 \\\\", "0 & w & -2w & w & 0 & 0 & 0 \\\\", "0 & 0 & w & -2w & w & 0 & w \\\\", "0 & 0 & 0 & w & -2w & w & 0 \\\\", "0 & 0 & 0 & 0 & w & -2w & 0 \\\\", "0 & 0 & 0 & w & 0 & 0 & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$,", "$2m_4 \\geq m_3 + m_5 + m_7$, $2m_5 \\geq m_4 + m_6$, $2m_6 \\geq m_5$,", "and $2m_7 \\geq m_4$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9223, "type": "theorem", "label": "models-lemma-E8", "categories": [ "models" ], "title": "models-lemma-E8", "contents": [ "Classification of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] &", "\\bullet \\ar@{-}[r] & \\bullet \\\\", "& & & & \\bullet", "}", "$$", "Let $n > 8$. Then given $8$ distinct $(-2)$-indices $i_1, \\ldots, i_8$", "such that $a_{12}, a_{23}, a_{34}, a_{45}, a_{56}, a_{65}, a_{57}$", "are nonzero, then we have the $m$'s, $a$'s, and $w$'s", "\\begin{enumerate}", "\\item", "\\label{item-E8}", "are given by", "$$", "\\left(", "\\begin{matrix}", "m_1 \\\\", "m_2 \\\\", "m_3 \\\\", "m_4 \\\\", "m_5 \\\\", "m_6 \\\\", "m_7 \\\\", "m_8", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\\\", "w & -2w & w & 0 & 0 & 0 & 0 & 0 \\\\", "0 & w & -2w & w & 0 & 0 & 0 & 0 \\\\", "0 & 0 & w & -2w & w & 0 & 0 & 0 \\\\", "0 & 0 & 0 & w & -2w & w & 0 & w \\\\", "0 & 0 & 0 & 0 & w & -2w & w & 0 \\\\", "0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\\\", "0 & 0 & 0 & 0 & w & 0 & 0 & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right)", "$$", "with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$,", "$2m_4 \\geq m_3 + m_5$, $2m_5 \\geq m_4 + m_6 + m_8$, $2m_6 \\geq m_5 + m_7$,", "$2m_7 \\geq m_6$, and $2m_8 \\geq m_5$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9224, "type": "theorem", "label": "models-lemma-E7-completed", "categories": [ "models" ], "title": "models-lemma-E7-completed", "contents": [ "Nonexistence of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] &", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\\\", "& & & \\bullet", "}", "$$", "Assume $n > 8$. There do {\\bf not} exist $8$ distinct", "$(-2)$-indices $e, f, g, h, i, j, k, l$", "such that $a_{ef}, a_{fg}, a_{gh}, a_{hi}, a_{ij}, a_{jk}, a_{lh}$", "are nonzero." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9225, "type": "theorem", "label": "models-lemma-E8-completed", "categories": [ "models" ], "title": "models-lemma-E8-completed", "contents": [ "Nonexistence of proper subgraphs of the form", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] &", "\\bullet \\ar@{-}[r] & \\bullet \\\\", "& & & & & \\bullet", "}", "$$", "Assume $n > 9$. There do {\\bf not} exist $9$ distinct", "$(-2)$-indices $d, e, f, g, h, i, j, k, l$", "such that $a_{de}, a_{ef}, a_{fg}, a_{gh}, a_{hi}, a_{ij}, a_{jk}, a_{lh}$", "are nonzero." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9226, "type": "theorem", "label": "models-lemma-genus-zero", "categories": [ "models" ], "title": "models-lemma-genus-zero", "contents": [ "The only minimal numerical type of genus zero is", "$n = 1$, $m_1 = 1$, $a_{11} = 0$, $w_1 = 1$, $g_1 = 0$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemmas \\ref{lemma-non-irreducible-minimal-type-genus-at-least-one}", "and \\ref{lemma-irreducible}." ], "refs": [ "models-lemma-non-irreducible-minimal-type-genus-at-least-one", "models-lemma-irreducible" ], "ref_ids": [ 9203, 9198 ] } ], "ref_ids": [] }, { "id": 9227, "type": "theorem", "label": "models-lemma-genus-one", "categories": [ "models" ], "title": "models-lemma-genus-one", "contents": [ "The minimal numerical types of genus one are up to equivalence", "\\begin{enumerate}", "\\item", "\\label{item-one}", "$n = 1$, $a_{11} = 0$, $g_1 = 1$, $m_1, w_1 \\geq 1$ arbitrary,", "\\item", "\\label{item-two-cycle}", "$n = 2$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & 2w \\\\", "2w & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-up4}", "$n = 2$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "2m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & 4w \\\\", "4w & -8w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "4w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-three-cycle}", "$n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & w \\\\", "w & -2w & w \\\\", "w & w & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-equal-up3}", "$n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "2m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 \\\\", "w & -2w & 3w \\\\", "0 & 3w & -6w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "3w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-equal-down3}", "$n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "2m \\\\", "3m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-6w & 3w & 0 \\\\", "3w & -6w & 3w \\\\", "0 & 3w & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "3w \\\\", "3w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-up-up}", "$n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "2m \\\\", "2m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & 2w & 0 \\\\", "2w & -4w & 4w \\\\", "0 & 4w & -8w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "2w \\\\", "4w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-up-down}", "$n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & 2w & 0 \\\\", "2w & -4w & 2w \\\\", "0 & 2w & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "2w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-down-up}", "$n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "2m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-4w & 2w & 0 \\\\", "2w & -2w & 2w \\\\", "0 & 2w & -4w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "2w \\\\", "w \\\\", "2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-four-cycle}", "$n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "m \\\\", "m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & w \\\\", "w & -2w & w & 0 \\\\", "0 & w & -2w & w \\\\", "w & 0 & w & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-up-equal-up}", "$n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "2m \\\\", "2m \\\\", "2m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & 2w & 0 & 0 \\\\", "2w & -4w & 2w & 0 \\\\", "0 & 2w & -4w & 4w \\\\", "0 & 0 & 4w & -8w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "2w \\\\", "2w \\\\", "4w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-up-equal-down}", "$n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "m \\\\", "m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & 2w & 0 & 0 \\\\", "2w & -4w & 2w & 0 \\\\", "0 & 2w & -4w & 2w \\\\", "0 & 0 & 2w & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "2w \\\\", "2w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-down-equal-up}", "$n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "2m \\\\", "2m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-4w & 2w & 0 & 0 \\\\", "2w & -2w & w & 0 \\\\", "0 & w & -2w & 2w \\\\", "0 & 0 & 2w & -4w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "2w \\\\", "w \\\\", "w \\\\", "2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-triple-with-up}", "$n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "2m \\\\", "m \\\\", "m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & w & 2w \\\\", "w & -2w & 0 & 0 \\\\", "w & 0 & -2w & 0 \\\\", "2w & 0 & 0 & -4w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-triple-with-down}", "$n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "2m \\\\", "m \\\\", "m \\\\", "2m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-4w & 2w & 2w & 2w \\\\", "2w & -4w & 0 & 0 \\\\", "2w & 0 & -4w & 0 \\\\", "2w & 0 & 0 & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "2w \\\\", "2w \\\\", "2w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-five-cycle}", "$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "m \\\\", "m \\\\", "m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & 0 & w \\\\", "w & -2w & w & 0 & 0 \\\\", "0 & w & -2w & w & 0 \\\\", "0 & 0 & w & -2w & w \\\\", "w & 0 & 0 & w & -2w \\\\", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-equal-equal-up-equal}", "$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "2m \\\\", "3m \\\\", "2m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & 0 & 0 \\\\", "w & -2w & w & 0 & 0 \\\\", "0 & w & -2w & 2w & 0 \\\\", "0 & 0 & 2w & -4w & 2w \\\\", "0 & 0 & 0 & 2w & -4w \\\\", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "2w \\\\", "2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-equal-equal-down-equal}", "$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "2m \\\\", "3m \\\\", "4m \\\\", "2m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-4w & 2w & 0 & 0 & 0 \\\\", "2w & -4w & 2w & 0 & 0 \\\\", "0 & 2w & -4w & 2w & 0 \\\\", "0 & 0 & 2w & -2w & w \\\\", "0 & 0 & 0 & w & -2w \\\\", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "2w \\\\", "2w \\\\", "2w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-up-equal-equal-up}", "$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "2m \\\\", "2m \\\\", "2m \\\\", "2m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & 2w & 0 & 0 & 0 \\\\", "2w & -4w & 2w & 0 & 0 \\\\", "0 & 2w & -4w & 2w & 0 \\\\", "0 & 0 & 2w & -4w & 4w \\\\", "0 & 0 & 0 & 4w & -8w \\\\", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "2w \\\\", "2w \\\\", "2w \\\\", "4w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-up-equal-equal-down}", "$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "m \\\\", "m \\\\", "m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & 2w & 0 & 0 & 0 \\\\", "2w & -4w & 2w & 0 & 0 \\\\", "0 & 2w & -4w & 2w & 0 \\\\", "0 & 0 & 2w & -4w & 2w \\\\", "0 & 0 & 0 & 2w & -2w \\\\", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "2w \\\\", "2w \\\\", "2w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-down-equal-equal-up}", "$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "2m \\\\", "2m \\\\", "2m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-4w & 2w & 0 & 0 & 0 \\\\", "2w & -2w & w & 0 & 0 \\\\", "0 & w & -2w & w & 0 \\\\", "0 & 0 & w & -2w & 2w \\\\", "0 & 0 & 0 & 2w & -4w \\\\", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "2w \\\\", "w \\\\", "w \\\\", "w \\\\", "2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-quadruple}", "$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "2m \\\\", "m \\\\", "m \\\\", "m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & w & w & w \\\\", "w & -2w & 0 & 0 & 0 \\\\", "w & 0 & -2w & 0 & 0 \\\\", "w & 0 & 0 & -2w & 0 \\\\", "w & 0 & 0 & 0 & -2w \\\\", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-triple-extended-up}", "$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "2m \\\\", "2m \\\\", "m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-4w & 2w & 0 & 0 & 0 \\\\", "2w & -2w & w & 0 & 0 \\\\", "0 & w & -2w & w & w \\\\", "0 & 0 & w & -2w & 0 \\\\", "0 & 0 & w & 0 & -2w \\\\", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "2w \\\\", "w \\\\", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-triple-extended-down}", "$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "2m \\\\", "2m \\\\", "2m \\\\", "m \\\\", "m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & 2w & 0 & 0 & 0 \\\\", "2w & -4w & 2w & 0 & 0 \\\\", "0 & 2w & -4w & 2w & 2w \\\\", "0 & 0 & 2w & -4w & 0 \\\\", "0 & 0 & 2w & 0 & -4w \\\\", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "2w \\\\", "2w \\\\", "2w \\\\", "2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-n-cycle}", "$n \\geq 6$ and we have an $n$-cycle generalizing (\\ref{item-five-cycle}):", "\\begin{enumerate}", "\\item $m_1 = \\ldots = m_n = m$,", "\\item $a_{12} = \\ldots = a_{(n - 1) n} = w$, $a_{1n} = w$,", "and for other $i < j$ we have $a_{ij} = 0$,", "\\item $w_1 = \\ldots = w_n = w$", "\\end{enumerate}", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-up-chain-equal-up}", "$n \\geq 6$ and we have a chain generalizing (\\ref{item-up-equal-equal-up}):", "\\begin{enumerate}", "\\item $m_1 = \\ldots = m_{n - 1} = 2m$, $m_n = m$,", "\\item $a_{12} = \\ldots = a_{(n - 2) (n - 1)} = 2w$, $a_{(n - 1) n} = 4w$,", "and for other $i < j$ we have $a_{ij} = 0$,", "\\item $w_1 = w$, $w_2 = \\ldots = w_{n - 1} = 2w$, $w_n = 4w$", "\\end{enumerate}", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-up-chain-equal-down}", "$n \\geq 6$ and we have a chain generalizing (\\ref{item-up-equal-equal-down}):", "\\begin{enumerate}", "\\item $m_1 = \\ldots = m_n = m$,", "\\item $a_{12} = \\ldots = a_{(n - 1) n} = w$,", "and for other $i < j$ we have $a_{ij} = 0$,", "\\item $w_1 = w$, $w_2 = \\ldots = w_{n - 1} = 2w$, $w_n = w$", "\\end{enumerate}", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-down-chain-equal-up}", "$n \\geq 6$ and we have a chain generalizing (\\ref{item-down-equal-equal-up}):", "\\begin{enumerate}", "\\item $m_1 = w$, $w_2 = \\ldots = m_{n - 1} = 2m$, $m_n = m$,", "\\item $a_{12} = 2w$, $a_{23} = \\ldots = a_{(n - 2) (n - 1)} = w$,", "$a_{(n - 1) n} = 2w$, and for other $i < j$ we have $a_{ij} = 0$,", "\\item $w_1 = 2w$, $w_2 = \\ldots = w_{n - 1} = w$, $w_n = 2w$", "\\end{enumerate}", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-Dn-extended-up}", "$n \\geq 6$ and we have a type generalizing (\\ref{item-triple-extended-up}):", "\\begin{enumerate}", "\\item $m_1 = m$, $m_2 = \\ldots = m_{n - 3} = 2m$, $m_{n - 1} = m_n = m$,", "\\item $a_{12} = 2w$, $a_{23} = \\ldots = a_{(n - 2) (n - 1)} = w$,", "$a_{(n - 2) n} = w$, and for other $i < j$ we have $a_{ij} = 0$,", "\\item $w_1 = 2w$, $w_2 = \\ldots = w_n = w$", "\\end{enumerate}", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-Dn-extended-down}", "$n \\geq 6$ and we have a type generalizing (\\ref{item-triple-extended-down}):", "\\begin{enumerate}", "\\item $m_1 = \\ldots = m_{n - 3} = 2m$, $m_{n - 1} = m_n = m$,", "\\item $a_{12} = \\ldots = a_{(n - 2) (n - 1)} = 2w$,", "$a_{(n - 2) n} = 2w$, and for other $i < j$ we have $a_{ij} = 0$,", "\\item $w_1 = w$, $w_2 = \\ldots = w_n = 2w$", "\\end{enumerate}", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-double-triple}", "$n \\geq 6$ and we have a type generalizing (\\ref{item-quadruple}):", "\\begin{enumerate}", "\\item $m_1 = m_2 = m$, $m_3 = \\ldots = m_{n - 2} = 2m$, $m_{n - 1} = m_n = m$,", "\\item $a_{13} = w$, $a_{23} = \\ldots = a_{(n - 2) (n - 1)} = w$,", "$a_{(n - 2) n} = w$, and for other $i < j$ we have $a_{ij} = 0$,", "\\item $w_1 = \\ldots = w_n = w$,", "\\end{enumerate}", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-E6-completed}", "$n = 7$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "2m \\\\", "3m \\\\", "m \\\\", "2m \\\\", "m \\\\", "2m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & 0 & 0 & 0 & 0 \\\\", "w & -2w & w & 0 & 0 & 0 & 0 \\\\", "0 & w & -2w & 0 & w & 0 & w \\\\", "0 & 0 & 0 & -2w & w & 0 & 0 \\\\", "0 & 0 & w & w & -2w & 0 & 0 \\\\", "0 & 0 & 0 & 0 & 0 & -2w & w \\\\", "0 & 0 & w & 0 & 0 & w & -2w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-E7-completed}", "$n = 8$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "2m \\\\", "3m \\\\", "4m \\\\", "3m \\\\", "2m \\\\", "m \\\\", "2m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\\\", "w & -2w & w & 0 & 0 & 0 & 0 & 0 \\\\", "0 & w & -2w & w & 0 & 0 & 0 & 0 \\\\", "0 & 0 & w & -2w & w & 0 & 0 & w \\\\", "0 & 0 & 0 & w & -2w & w & 0 & 0 \\\\", "0 & 0 & 0 & 0 & w & -2w & w & 0 \\\\", "0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\\\", "0 & 0 & 0 & w & 0 & 0 & 0 & -2w \\\\", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary,", "\\item", "\\label{item-E8-completed}", "$n = 9$, and $m_i, a_{ij}, w_i, g_i$ given by", "$$", "\\left(", "\\begin{matrix}", "m \\\\", "2m \\\\", "3m \\\\", "4m \\\\", "5m \\\\", "6m \\\\", "4m \\\\", "2m \\\\", "3m", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "-2w & w & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\", "w & -2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\\\", "0 & w & -2w & w & 0 & 0 & 0 & 0 & 0 \\\\", "0 & 0 & w & -2w & w & 0 & 0 & 0 & 0 \\\\", "0 & 0 & 0 & w & -2w & w & 0 & 0 & 0 \\\\", "0 & 0 & 0 & 0 & w & -2w & w & 0 & w \\\\", "0 & 0 & 0 & 0 & 0 & w & -2w & w & 0 \\\\", "0 & 0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\\\", "0 & 0 & 0 & 0 & 0 & w & 0 & 0 & -2w \\\\", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w \\\\", "w", "\\end{matrix}", "\\right),", "\\quad", "\\left(", "\\begin{matrix}", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0 \\\\", "0", "\\end{matrix}", "\\right)", "$$", "with $w$ and $m$ arbitrary.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is proved in Section \\ref{section-classify-proper-subgraphs}.", "See discussion at the", "start of Section \\ref{section-classify-proper-subgraphs}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9228, "type": "theorem", "label": "models-lemma-bound-neighbours", "categories": [ "models" ], "title": "models-lemma-bound-neighbours", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$.", "Given $i, j$ with $a_{ij} > 0$ we have", "$m_ia_{ij} \\leq m_j|a_{jj}|$ and $m_iw_i \\leq m_j|a_{jj}|$." ], "refs": [], "proofs": [ { "contents": [ "For every index $j$ we have $m_j a_{jj} + \\sum_{i \\not = j} m_ia_{ij} = 0$.", "Thus if we have an upper bound on $|a_{jj}|$ and $m_j$, then we also get an", "upper bound on the nonzero (and hence positive) $a_{ij}$ as well as", "$m_i$. Recalling that $w_i$ divides $a_{ij}$, the reader easily sees", "the lemma is correct." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9229, "type": "theorem", "label": "models-lemma-bound-heart", "categories": [ "models" ], "title": "models-lemma-bound-heart", "contents": [ "Fix $g \\geq 2$. For every minimal numerical type $n, m_i, a_{ij}, w_i, g_i$", "of genus $g$ with $n > 1$ we have", "\\begin{enumerate}", "\\item the set $J \\subset \\{1, \\ldots, n\\}$ of non-$(-2)$-indices", "has at most $2g - 2$ elements,", "\\item for $j \\in J$ we have $g_j < g$,", "\\item for $j \\in J$ we have $m_j|a_{jj}| \\leq 6g - 6$, and", "\\item for $j \\in J$ and $i \\in \\{1, \\ldots, n\\}$", "we have $m_ia_{ij} \\leq 6g - 6$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that $g = 1 + \\sum m_j(w_j(g_j - 1) - \\frac{1}{2} a_{jj})$.", "For $j \\in J$ the contribution $m_j(w_j(g_j - 1) - \\frac{1}{2} a_{jj})$", "to the genus $g$ is $> 0$ and hence $\\geq 1/2$. This uses", "Lemma \\ref{lemma-minus-one},", "Definition \\ref{definition-type-minus-one},", "Definition \\ref{definition-type-minimal},", "Lemma \\ref{lemma-minus-two}, and", "Definition \\ref{definition-type-minus-two}; we will use these", "results without further mention in the following.", "Thus $J$ has at most $2(g - 1)$ elements.", "This proves (1).", "\\medskip\\noindent", "Recall that $-a_{ii} > 0$ for all $i$ by Lemma \\ref{lemma-diagonal-negative}.", "Hence for $j \\in J$ the contribution $m_j(w_j(g_j - 1) - \\frac{1}{2} a_{jj})$", "to the genus $g$ is $> m_jw_j(g_j - 1)$. Thus", "$$", "g - 1 > m_jw_j(g_j - 1) \\Rightarrow g_j < (g - 1)/m_jw_j + 1", "$$", "This indeed implies $g_j < g$ which proves (2).", "\\medskip\\noindent", "For $j \\in J$ if $g_j > 0$, then the contribution", "$m_j(w_j(g_j - 1) - \\frac{1}{2} a_{jj})$ to the genus $g$", "is $\\geq -\\frac{1}{2}m_ja_{jj}$ and we immediately conclude", "that $m_j|a_{jj}| \\leq 2(g - 1)$. Otherwise $a_{jj} = -kw_j$", "for some integer $k \\geq 3$ (because $j \\in J$) and we get", "$$", "m_jw_j(-1 + \\frac{k}{2}) \\leq g - 1", "\\Rightarrow", "m_jw_j \\leq \\frac{2(g - 1)}{k - 2}", "$$", "Plugging this back into $a_{jj} = -km_jw_j$ we obtain", "$$", "m_j|a_{jj}| \\leq 2(g - 1) \\frac{k}{k - 2} \\leq 6(g - 1)", "$$", "This proves (3).", "\\medskip\\noindent", "Part (4) follows from Lemma \\ref{lemma-bound-neighbours} and (3)." ], "refs": [ "models-lemma-minus-one", "models-definition-type-minus-one", "models-definition-type-minimal", "models-lemma-minus-two", "models-definition-type-minus-two", "models-lemma-diagonal-negative", "models-lemma-bound-neighbours" ], "ref_ids": [ 9200, 9272, 9274, 9205, 9275, 9199, 9228 ] } ], "ref_ids": [] }, { "id": 9230, "type": "theorem", "label": "models-lemma-bound-wm", "categories": [ "models" ], "title": "models-lemma-bound-wm", "contents": [ "Fix $g \\geq 2$. For every minimal numerical type $n, m_i, a_{ij}, w_i, g_i$", "of genus $g$ we have $m_i|a_{ij}| \\leq 768g$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-bound-neighbours} it suffices to show", "$m_i|a_{ii}| \\leq 768g$ for all $i$.", "Let $J \\subset \\{1, \\ldots, n\\}$ be the set of non-$(-2)$-indices", "as in Lemma \\ref{lemma-bound-heart}. Observe that $J$ is", "nonempty as $g \\geq 2$. Also $m_j|a_{jj}| \\leq 6g$ for $j \\in J$", "by the lemma.", "\\medskip\\noindent", "Suppose we have $j \\in J$ and a sequence $i_1, \\ldots, i_7$", "of $(-2)$-indices such that $a_{ji_1}$ and $a_{i_1i_2}$,", "$a_{i_2i_3}$, $a_{i_3i_4}$, $a_{i_4i_5}$, $a_{i_5i_6}$, and", "$a_{i_6i_7}$ are nonzero. Then we see from", "Lemma \\ref{lemma-bound-neighbours}", "that $m_{i_1}w_{i_1} \\leq 6g$ and $m_{i_1}a_{ji_1} \\leq 6g$.", "Because $i_1$ is a $(-2)$-index, we have $a_{i_1i_1} = -2w_{i_1}$", "and we conclude that $m_{i_1}|a_{i_1i_1}| \\leq 12g$.", "Repeating the argument we conclude that", "$m_{i_2}w_{i_2} \\leq 12g$ and $m_{i_2}a_{i_1i_2} \\leq 12g$.", "Then $m_{i_2}|a_{i_2i_2}| \\leq 24g$ and so on.", "Eventually we conclude that", "$m_{i_k}|a_{i_ki_k}| \\leq 2^k(6g) \\leq 768g$ for $k = 1, \\ldots, 7$.", "\\medskip\\noindent", "Let $I \\subset \\{1, \\ldots, n\\} \\setminus J$ be a maximal connected subset.", "In other words, there does not exist a nonempty proper subset", "$I' \\subset I$ such that", "$a_{i'i} = 0$ for $i' \\in I'$ and $i \\in I \\setminus I'$", "and $I$ is maximal with this property. In particular, since", "a numerical type is connected by definition, we see that there", "exists a $j \\in J$ and $i \\in I$ with $a_{ij} > 0$.", "Looking at the classification", "of such $I$ in Proposition \\ref{proposition-classify-subgraphs}", "and using the result of the previous paragraph, we see that", "$w_i|a_{ii}| \\leq 768g$ for all $i \\in I$ unless $I$ is as described in", "Lemma \\ref{lemma-long} or", "Lemma \\ref{lemma-Dn}.", "Thus we may assume the nonvanishing of $a_{ii'}$, $i, i' \\in I$", "has either the shape", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{..}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet", "}", "$$", "(which has 3 subcases as detailed in Lemma \\ref{lemma-long})", "or the shape", "$$", "\\xymatrix{", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{..}[r] &", "\\bullet \\ar@{-}[r] &", "\\bullet \\ar@{-}[r] \\ar@{-}[d] &", "\\bullet \\\\", "& & & & \\bullet", "}", "$$", "We will prove the bound holds for the first subcase of", "Lemma \\ref{lemma-long} and leave the other cases to reader (the argument", "is almost exactly the same in those cases).", "\\medskip\\noindent", "After renumbering we may assume $I = \\{1, \\ldots, t\\} \\subset \\{1, \\ldots, n\\}$", "and there is an integer $w$ such that", "$$", "w = w_1 = \\ldots = w_t =", "a_{12} = \\ldots = a_{(t - 1) t} =", "-\\frac{1}{2} a_{i_1i_2} = \\ldots = -\\frac{1}{2} a_{(t - 1) t}", "$$", "The equalities $a_{ii}m_i + \\sum_{j \\not = i} a_{ij}m_j = 0$ imply", "that we have", "$$", "2m_2 \\geq m_1 + m_3, \\ldots, 2m_{t - 1} \\geq m_{t - 2} + m_t", "$$", "Equality holds in $2m_i \\geq m_{i - 1} + m_{i + 1}$", "if and only if $i$ does not ``meet'' any indices besides", "$i - 1$ and $i + 1$. And if $i$ does meet", "another index, then this index is in $J$ (by maximality of $I$).", "In particular, the map", "$\\{1, \\ldots, t\\} \\to \\mathbf{Z}$, $i \\mapsto m_i$ is concave.", "\\medskip\\noindent", "Let $m = \\max(m_i, i \\in \\{1, \\ldots, t\\})$. Then", "$m_i|a_{ii}| \\leq 2mw$ for $i \\leq t$ and", "our goal is to show that $2mw \\leq 768g$.", "Let $s$, resp.\\ $s'$ in $\\{1, \\ldots, t\\}$ be the", "smallest, resp.\\ biggest index with $m_s = m = m_{s'}$.", "By concavity we see that $m_i = m$ for $s \\leq i \\leq s'$.", "If $s > 1$, then we do not have equality in", "$2m_s \\geq m_{s - 1} + m_{s + 1}$", "and we see that $s$ meets an index from $J$.", "In this case $2mw \\leq 12g$ by the result of the second paragraph", "of the proof.", "Similarly, if $s' < t$, then $s'$ meets an index from $J$", "and we get $2mw \\leq 12g$ as well.", "But if $s = 1$ and $s' = t$, then we conclude", "that $a_{ij} = 0$ for all $j \\in J$ and $i \\in \\{2, \\ldots, t - 1\\}$.", "But as we've seen that there must be a pair $(i, j) \\in I \\times J$", "with $a_{ij} > 0$, we conclude that this happens either with", "$i = 1$ or with $i = t$ and we conclude $2mw \\leq 12g$", "in the same manner as before (as $m_1 = m = m_t$ in this case)." ], "refs": [ "models-lemma-bound-neighbours", "models-lemma-bound-heart", "models-lemma-bound-neighbours", "models-proposition-classify-subgraphs", "models-lemma-long", "models-lemma-Dn", "models-lemma-long", "models-lemma-long" ], "ref_ids": [ 9228, 9229, 9228, 9266, 9217, 9218, 9217, 9217 ] } ], "ref_ids": [] }, { "id": 9231, "type": "theorem", "label": "models-lemma-closure-is-model", "categories": [ "models" ], "title": "models-lemma-closure-is-model", "contents": [ "Let $V_1 \\to V_2$ be a closed immersion of algebraic schemes over $K$.", "If $X_2$ is a model for $V_2$, then the scheme theoretic image", "of $V_1 \\to X_2$ is a model for $V_1$." ], "refs": [], "proofs": [ { "contents": [ "Using", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-scheme-theoretic-image} and", "Example \\ref{morphisms-example-scheme-theoretic-image}", "this boils down to the following algebra statement.", "Let $A_1$ be a finite type $R$-algebra flat over $R$.", "Let $A_1 \\otimes_R K \\to B_2$ be a surjection. Then", "$A_2 = A_1 / \\Ker(A_1 \\to B_2)$ is a finite type $R$-algebra", "flat over $R$ such that $B_2 = A_2 \\otimes_R K$.", "We omit the detailed proof; use", "More on Algebra, Lemma \\ref{more-algebra-lemma-dedekind-torsion-free-flat}", "to prove that $A_2$ is flat." ], "refs": [ "morphisms-lemma-quasi-compact-scheme-theoretic-image", "more-algebra-lemma-dedekind-torsion-free-flat" ], "ref_ids": [ 5146, 9921 ] } ], "ref_ids": [] }, { "id": 9232, "type": "theorem", "label": "models-lemma-normalization", "categories": [ "models" ], "title": "models-lemma-normalization", "contents": [ "Let $X$ be a model of a geometrically normal variety $V$ over $K$.", "Then the normalization $\\nu : X^\\nu \\to X$ is finite and", "the base change of $X^\\nu$ to the completion $R^\\wedge$", "is the normalization of the base change of $X$. Moreover, for", "each $x \\in X^\\nu$ the completion of $\\mathcal{O}_{X^\\nu, x}$", "is normal." ], "refs": [], "proofs": [ { "contents": [ "Observe that $R^\\wedge$ is a discrete valuation ring", "(More on Algebra, Lemma \\ref{more-algebra-lemma-completion-dvr}).", "Set $Y = X \\times_{\\Spec(R)} \\Spec(R^\\wedge)$.", "Since $R^\\wedge$ is a discrete valuation ring, we see that", "$$", "Y \\setminus Y_k =", "Y \\times_{\\Spec(R^\\wedge)} \\Spec(K^\\wedge) =", "V \\times_{\\Spec(K)} \\Spec(K^\\wedge)", "$$", "where $K^\\wedge$ is the fraction field of $R^\\wedge$.", "Since $V$ is geometrically normal, we find that this is", "a normal scheme. Hence the first part of the lemma follows from", "Resolution of Surfaces, Lemma \\ref{resolve-lemma-normalization-completion}.", "\\medskip\\noindent", "To prove the second part we may assume $X$ and $Y$ are normal", "(by the first part). If $x$ is in the generic fibre, then", "$\\mathcal{O}_{X, x} = \\mathcal{O}_{V, x}$ is a normal local", "ring essentially of finite type over a field. Such a ring is", "excellent (More on Algebra, Proposition", "\\ref{more-algebra-proposition-ubiquity-excellent}).", "If $x$ is a point of the special fibre with image $y \\in Y$, then", "$\\mathcal{O}_{X, x}^\\wedge = \\mathcal{O}_{Y, y}^\\wedge$", "by Resolution of Surfaces, Lemma \\ref{resolve-lemma-iso-completions}.", "In this case $\\mathcal{O}_{Y, y}$ is a excellent normal local domain", "by the same reference as before as $R^\\wedge$ is excellent.", "If $B$ is a excellent local normal domain, then the completion", "$B^\\wedge$ is normal (as $B \\to B^\\wedge$ is regular and", "More on Algebra, Lemma \\ref{more-algebra-lemma-normal-goes-up} applies).", "This finishes the proof." ], "refs": [ "more-algebra-lemma-completion-dvr", "resolve-lemma-normalization-completion", "more-algebra-proposition-ubiquity-excellent", "resolve-lemma-iso-completions", "more-algebra-lemma-normal-goes-up" ], "ref_ids": [ 10046, 11684, 10584, 11679, 10041 ] } ], "ref_ids": [] }, { "id": 9233, "type": "theorem", "label": "models-lemma-regular", "categories": [ "models" ], "title": "models-lemma-regular", "contents": [ "Let $X$ be a model of a smooth curve $C$ over $K$. Then", "there exists a resolution of singularities of $X$", "and any resolution is a model of $C$." ], "refs": [], "proofs": [ { "contents": [ "We check condition (4) of Lipman's theorem", "(Resolution of Surfaces, Theorem \\ref{resolve-theorem-resolve}) hold.", "This is clear from Lemma \\ref{lemma-normalization}", "except for the statement that $X^\\nu$ has finitely many", "singular points. To see this we can use that $R$ is J-2 by", "More on Algebra, Proposition \\ref{more-algebra-proposition-ubiquity-J-2}", "and hence the nonsingular locus is open in $X^\\nu$.", "Since $X^\\nu$ is normal of dimension $\\leq 2$, the singular points", "are closed, hence closedness of the singular locus", "means there are finitely many of them (as $X$ is quasi-compact).", "Observe that any resolution of $X$ is a modification of $X$", "(Resolution of Surfaces, Definition \\ref{resolve-definition-resolution}).", "This will be an isomorphism over the normal locus of $X$ by Varieties, Lemma", "\\ref{varieties-lemma-modification-normal-iso-over-codimension-1}.", "Since the set of normal points includes", "$C = X_K$ we conclude any resolution is a model of $C$." ], "refs": [ "resolve-theorem-resolve", "models-lemma-normalization", "more-algebra-proposition-ubiquity-J-2", "resolve-definition-resolution", "varieties-lemma-modification-normal-iso-over-codimension-1" ], "ref_ids": [ 11635, 9232, 10578, 11714, 10979 ] } ], "ref_ids": [] }, { "id": 9234, "type": "theorem", "label": "models-lemma-pre-exists-minimal-model", "categories": [ "models" ], "title": "models-lemma-pre-exists-minimal-model", "contents": [ "\\begin{slogan}", "A regular proper model of a curve is obtained by successive blowups", "from a minimal model", "\\end{slogan}", "Let $C$ be a smooth projective curve over $K$ with", "$H^0(C, \\mathcal{O}_C) = K$. If $X$ is a regular proper", "model for $C$, then there exists a sequence of morphisms", "$$", "X = X_m \\to X_{m - 1} \\to \\ldots \\to X_1 \\to X_0", "$$", "of proper regular models of $C$, such that each morphism is a", "contraction of an exceptional curve of the first kind, and such", "that $X_0$ is a minimal model." ], "refs": [], "proofs": [ { "contents": [ "By Resolution of Surfaces, Lemma \\ref{resolve-lemma-regular-dim-2-projective}", "we see that $X$ is projective over $R$. Hence $X$ has an ample", "invertible sheaf by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-projective}", "(we will use this below).", "Let $E \\subset X$ be an exceptional curve of the first kind.", "See Resolution of Surfaces, Section \\ref{resolve-section-minus-one}.", "By Resolution of Surfaces, Lemma \\ref{resolve-lemma-contract-ample}", "we can contract $E$ by a morphism $X \\to X'$ such that $X'$ is", "regular and is projective over $R$. Clearly, the number of", "irreducible components of $X'_k$ is exactly one less than the", "number of irreducible components of $X_k$. Thus we can only", "perform a finite number of these contractions until we", "obtain a minimal model." ], "refs": [ "resolve-lemma-regular-dim-2-projective", "more-morphisms-lemma-projective", "resolve-lemma-contract-ample" ], "ref_ids": [ 11708, 13931, 11705 ] } ], "ref_ids": [] }, { "id": 9235, "type": "theorem", "label": "models-lemma-divisor-special-fiber", "categories": [ "models" ], "title": "models-lemma-divisor-special-fiber", "contents": [ "Let $X$ be a regular model of a smooth curve $C$ over $K$.", "\\begin{enumerate}", "\\item the special fibre $X_k$ is an effective Cartier divisor on $X$,", "\\item each irreducible component $C_i$ of $X_k$ is an effective", "Cartier divisor on $X$,", "\\item $X_k = \\sum m_i C_i$ (sum of effective Cartier divisors)", "where $m_i$ is the multiplicity of $C_i$ in $X_k$,", "\\item $\\mathcal{O}_X(X_k) \\cong \\mathcal{O}_X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that $R$ is a discrete valuation ring with uniformizer $\\pi$", "and residue field $k = R/(\\pi)$. Because $X \\to \\Spec(R)$ is flat,", "the element $\\pi$ is a nonzerodivisor affine locally on $X$", "(see More on Algebra, Lemma", "\\ref{more-algebra-lemma-dedekind-torsion-free-flat}). Thus", "if $U = \\Spec(A) \\subset X$ is an affine open, then", "$$", "X_K \\cap U = U_k = \\Spec(A \\otimes_R k) = \\Spec(A/\\pi A)", "$$", "and $\\pi$ is a nonzerodivisor in $A$.", "Hence $X_k = V(\\pi)$ is an effective Cartier divisor by", "Divisors, Lemma \\ref{divisors-lemma-characterize-effective-Cartier-divisor}.", "Hence (1) is true.", "\\medskip\\noindent", "The discussion above shows that the pair $(\\mathcal{O}_X(X_k), 1)$", "is isomorphic to the pair $(\\mathcal{O}_X, \\pi)$ which proves (4).", "\\medskip\\noindent", "By Divisors, Lemma \\ref{divisors-lemma-effective-Cartier-divisor-is-a-sum}", "there exist pairwise distinct integral effective Cartier divisors", "$D_i \\subset X$ and integers $a_i \\geq 0$ such that $X_k = \\sum a_i D_i$.", "We can throw out those divisors $D_i$ such that $a_i = 0$. Then it is", "clear (from the definition of addition of effective Cartier", "divisors) that $X_k = \\bigcup D_i$ set theoretically. Thus $C_i = D_i$", "are the irreducible components of $X_k$ which proves (2).", "Let $\\xi_i$ be the generic point of $C_i$.", "Then $\\mathcal{O}_{X, \\xi_i}$ is a discrete valuation ring", "(Divisors, Lemma \\ref{divisors-lemma-integral-effective-Cartier-divisor-dvr}).", "The uniformizer $\\pi_i \\in \\mathcal{O}_{X, \\xi_i}$ is a local equation", "for $C_i$ and the image of $\\pi$ is a local equation for $X_k$.", "Since $X_k = \\sum a_i C_i$ we see that $\\pi$ and $\\pi_i^{a_i}$", "generate the same ideal in $\\mathcal{O}_{X, \\xi_i}$.", "On the other hand, the multiplicity of $C_i$ in $X_k$ is", "$$", "m_i = \\text{length}_{\\mathcal{O}_{C_i, \\xi_i}} \\mathcal{O}_{X_k, \\xi_i} =", "\\text{length}_{\\mathcal{O}_{C_i, \\xi_i}} \\mathcal{O}_{X, \\xi_i}/(\\pi) =", "\\text{length}_{\\mathcal{O}_{C_i, \\xi_i}} \\mathcal{O}_{X, \\xi_i}/(\\pi_i^{a_i}) =", "a_i", "$$", "See Chow Homology, Definition", "\\ref{chow-definition-cycle-associated-to-closed-subscheme}.", "Thus $a_i = m_i$ and (3) is proved." ], "refs": [ "more-algebra-lemma-dedekind-torsion-free-flat", "divisors-lemma-characterize-effective-Cartier-divisor", "divisors-lemma-effective-Cartier-divisor-is-a-sum", "divisors-lemma-integral-effective-Cartier-divisor-dvr", "chow-definition-cycle-associated-to-closed-subscheme" ], "ref_ids": [ 9921, 7927, 7955, 7948, 5907 ] } ], "ref_ids": [] }, { "id": 9236, "type": "theorem", "label": "models-lemma-gorenstein", "categories": [ "models" ], "title": "models-lemma-gorenstein", "contents": [ "Let $X$ be a regular model of a smooth curve $C$ over $K$. Then", "\\begin{enumerate}", "\\item $X \\to \\Spec(R)$ is a Gorenstein morphism of relative dimension $1$,", "\\item each of the irreducible components $C_i$ of $X_k$ is Gorenstein.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $X \\to \\Spec(R)$ is flat, to prove (1)", "it suffices to show that the fibres are Gorenstein", "(Duality for Schemes, Lemma \\ref{duality-lemma-gorenstein-morphism}).", "The generic fibre is a smooth curve, which is regular and hence Gorenstein", "(Duality for Schemes, Lemma \\ref{duality-lemma-regular-gorenstein}).", "For the special fibre $X_k$ we use that it is an effective", "Cartier divisor on a regular (hence Gorenstein) scheme and hence", "Gorenstein for example by Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-gorenstein-divide-by-nonzerodivisor}.", "The curves $C_i$ are Gorenstein by the same argument." ], "refs": [ "duality-lemma-gorenstein-morphism", "duality-lemma-regular-gorenstein", "dualizing-lemma-gorenstein-divide-by-nonzerodivisor" ], "ref_ids": [ 13595, 13590, 2883 ] } ], "ref_ids": [] }, { "id": 9237, "type": "theorem", "label": "models-lemma-regular-model-connected", "categories": [ "models" ], "title": "models-lemma-regular-model-connected", "contents": [ "In Situation \\ref{situation-regular-model} the special fibre $X_k$ is connected." ], "refs": [], "proofs": [ { "contents": [ "Consequence of More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-geometrically-connected-fibres-towards-normal}." ], "refs": [ "more-morphisms-lemma-geometrically-connected-fibres-towards-normal" ], "ref_ids": [ 13945 ] } ], "ref_ids": [] }, { "id": 9238, "type": "theorem", "label": "models-lemma-regular-model-pic", "categories": [ "models" ], "title": "models-lemma-regular-model-pic", "contents": [ "In Situation \\ref{situation-regular-model} there is an exact sequence", "$$", "0 \\to \\mathbf{Z} \\to \\mathbf{Z}^{\\oplus n} \\to", "\\Pic(X) \\to \\Pic(C) \\to 0", "$$", "where the first map sends $1$ to $(m_1, \\ldots, m_n)$ and the second", "maps sends the $i$th basis vector to $\\mathcal{O}_X(C_i)$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $C \\subset X$ is an open subscheme. The restriction", "map $\\Pic(X) \\to \\Pic(C)$ is surjective by", "Divisors, Lemma \\ref{divisors-lemma-extend-invertible-module}.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module", "such that there is an isomorphism $s : \\mathcal{O}_C \\to \\mathcal{L}|_C$.", "Then $s$ is a regular meromorphic section of $\\mathcal{L}$", "and we see that $\\text{div}_\\mathcal{L}(s) = \\sum a_i C_i$", "for some $a_i \\in \\mathbf{Z}$", "(Divisors, Definition \\ref{divisors-definition-divisor-invertible-sheaf}).", "By Divisors, Lemma \\ref{divisors-lemma-normal-c1-injective}", "(and the fact that $X$ is normal)", "we conclude that $\\mathcal{L} = \\mathcal{O}_X(\\sum a_iC_i)$.", "Finally, suppose that $\\mathcal{O}_X(\\sum a_i C_i) \\cong \\mathcal{O}_X$.", "Then there exists an element $g$ of the function field of $X$", "with $\\text{div}_X(g) = \\sum a_i C_i$. In particular the rational", "function $g$ has no zeros or poles on the generic fibre $C$ of $X$.", "Since $C$ is a normal scheme this implies $g \\in H^0(C, \\mathcal{O}_C) = K$.", "Thus $g = \\pi^a u$ for some $a \\in \\mathbf{Z}$ and $u \\in R^*$.", "We conclude that $\\text{div}_X(g) = a \\sum m_i C_i$ and the proof", "is complete." ], "refs": [ "divisors-lemma-extend-invertible-module", "divisors-definition-divisor-invertible-sheaf", "divisors-lemma-normal-c1-injective" ], "ref_ids": [ 8032, 8111, 8028 ] } ], "ref_ids": [] }, { "id": 9239, "type": "theorem", "label": "models-lemma-intersection-pairing", "categories": [ "models" ], "title": "models-lemma-intersection-pairing", "contents": [ "In Situation \\ref{situation-regular-model} given $\\mathcal{L}$ an invertible", "$\\mathcal{O}_X$-module and", "$a = (a_1, \\ldots, a_n) \\in \\mathbf{Z}^{\\oplus n}$ we define", "$$", "\\langle a, \\mathcal{L} \\rangle = \\sum a_i\\deg(\\mathcal{L}|_{C_i})", "$$", "Then $\\langle , \\rangle$ is bilinear and for", "$b = (b_1, \\ldots, b_n) \\in \\mathbf{Z}^{\\oplus n}$ we have", "$$", "\\left\\langle a, \\mathcal{O}_X(\\sum b_i C_i) \\right\\rangle =", "\\left\\langle b, \\mathcal{O}_X(\\sum a_i C_i) \\right\\rangle", "$$" ], "refs": [], "proofs": [ { "contents": [ "Bilinearity is immediate from the definition and", "Varieties, Lemma \\ref{varieties-lemma-degree-tensor-product}.", "To prove symmetry it suffices to assume $a$ and $b$ are", "standard basis vectors in $\\mathbf{Z}^{\\oplus n}$.", "Hence it suffices to prove that", "$$", "\\deg(\\mathcal{O}_X(C_j)|_{C_i}) = \\deg(\\mathcal{O}_X(C_i)|_{C_j})", "$$", "for all $1 \\leq i, j \\leq n$. If $i = j$ there is nothing to prove.", "If $i \\not = j$, then the canonical section $1$ of $\\mathcal{O}_X(C_j)$", "restricts to a nonzero (hence regular) section of $\\mathcal{O}_X(C_j)|_{C_i}$", "whose zero scheme is exactly $C_i \\cap C_j$ (scheme theoretic intersection).", "In other words, $C_i \\cap C_j$ is an effective Cartier divisor on $C_i$", "and", "$$", "\\deg(\\mathcal{O}_X(C_j)|_{C_i}) = \\deg(C_i \\cap C_j)", "$$", "by Varieties, Lemma \\ref{varieties-lemma-degree-effective-Cartier-divisor}.", "By symmetry we obtain the same (!) formula for the other side", "and the proof is complete." ], "refs": [ "varieties-lemma-degree-tensor-product", "varieties-lemma-degree-effective-Cartier-divisor" ], "ref_ids": [ 11109, 11111 ] } ], "ref_ids": [] }, { "id": 9240, "type": "theorem", "label": "models-lemma-properties-form", "categories": [ "models" ], "title": "models-lemma-properties-form", "contents": [ "In Situation \\ref{situation-regular-model} the symmetric bilinear form", "(\\ref{equation-form}) has the following properties", "\\begin{enumerate}", "\\item $(C_i \\cdot C_j) \\geq 0$ if $i \\not = j$ with equality if and only", "if $C_i \\cap C_j = \\emptyset$,", "\\item $(\\sum m_i C_i \\cdot C_j) = 0$,", "\\item there is no nonempty proper subset $I \\subset \\{1, \\ldots, n\\}$", "such that $(C_i \\cdot C_j) = 0$ for $i \\in I$, $j \\not \\in I$.", "\\item $(\\sum a_i C_i \\cdot \\sum a_i C_i) \\leq 0$ with equality if and", "only if there exists a $q \\in \\mathbf{Q}$ such that $a_i = qm_i$", "for $i = 1, \\ldots, n$,", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-intersection-pairing} we saw that", "$(C_i \\cdot C_j) = \\deg(C_i \\cap C_j)$ if $i \\not = j$. This is", "$\\geq 0$ and $> 0 $ if and only if $C_i \\cap C_j \\not = \\emptyset$.", "This proves (1).", "\\medskip\\noindent", "Proof of (2). This is true because by Lemma \\ref{lemma-divisor-special-fiber}", "the invertible sheaf associated to $\\sum m_i C_i$", "is trivial and the trivial sheaf has degree zero.", "\\medskip\\noindent", "Proof of (3). This is expressing the fact that $X_k$ is connected", "(Lemma \\ref{lemma-regular-model-connected})", "via the description of the intersection products given in the proof of (1).", "\\medskip\\noindent", "Part (4) follows from (1), (2), and (3) by", "Lemma \\ref{lemma-recurring-symmetric-real}." ], "refs": [ "models-lemma-intersection-pairing", "models-lemma-divisor-special-fiber", "models-lemma-regular-model-connected", "models-lemma-recurring-symmetric-real" ], "ref_ids": [ 9239, 9235, 9237, 9193 ] } ], "ref_ids": [] }, { "id": 9241, "type": "theorem", "label": "models-lemma-multiple-fibre-normal-bundle", "categories": [ "models" ], "title": "models-lemma-multiple-fibre-normal-bundle", "contents": [ "In Situation \\ref{situation-regular-model} set $d = \\gcd(m_1, \\ldots, m_n)$", "and let $D = \\sum (m_i/d)C_i$ as an effective Cartier divisor.", "Then $\\mathcal{O}_X(D)$ has order dividing $d$ in $\\Pic(X)$", "and $\\mathcal{C}_{D/X}$ an invertible $\\mathcal{O}_D$-module", "of order dividing $d$ in $\\Pic(D)$." ], "refs": [], "proofs": [ { "contents": [ "We have", "$$", "\\mathcal{O}_X(D)^{\\otimes d} = \\mathcal{O}_X(dD) =", "\\mathcal{O}_X(X_k) = \\mathcal{O}_X", "$$", "by Lemma \\ref{lemma-divisor-special-fiber}.", "We conclude as $\\mathcal{C}_{D/X}$ is the pullback of", "$\\mathcal{O}_X(-D)$." ], "refs": [ "models-lemma-divisor-special-fiber" ], "ref_ids": [ 9235 ] } ], "ref_ids": [] }, { "id": 9242, "type": "theorem", "label": "models-lemma-regular-model-field", "categories": [ "models" ], "title": "models-lemma-regular-model-field", "contents": [ "\\begin{reference}", "\\cite[Lemma 2.6]{Artin-Winters}", "\\end{reference}", "In Situation \\ref{situation-regular-model} let $d = \\gcd(m_1, \\ldots, m_n)$.", "Let $D = \\sum (m_i/d) C_i$ as an effective Cartier divisor. Then there exists", "a sequence of effective Cartier divisors", "$$", "(X_k)_{red} = Z_0 \\subset Z_1 \\subset \\ldots \\subset Z_m = D", "$$", "such that $Z_j = Z_{j - 1} + C_{i_j}$ for some $i_j \\in \\{1, \\ldots, n\\}$", "for $j = 1, \\ldots, m$ and such that $H^0(Z_j, \\mathcal{O}_{Z_j})$", "is a field finite over $k$ for $j = 0, \\ldots m$." ], "refs": [], "proofs": [ { "contents": [ "The reduction $D_{red} = (X_k)_{red} = \\sum C_i$ is connected", "(Lemma \\ref{lemma-regular-model-connected}) and proper over $k$. Hence", "$H^0(D_{red}, \\mathcal{O})$ is a field and a finite extension of", "$k$ by Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}.", "Thus the result for $Z_0 = D_{red} = (X_k)_{red}$ is true.", "Suppose that we have already constructed", "$$", "(X_k)_{red} = Z_0 \\subset Z_1 \\subset \\ldots \\subset Z_t \\subset D", "$$", "with $Z_j = Z_{j - 1} + C_{i_j}$ for some $i_j \\in \\{1, \\ldots, n\\}$", "for $j = 1, \\ldots, t$ and such that $H^0(Z_j, \\mathcal{O}_{Z_j})$", "is a field finite over $k$ for $j = 0, \\ldots, t$.", "Write $Z_t = \\sum a_i C_i$ with $1 \\leq a_i \\leq m_i/d$.", "If $a_i = m_i/d$ for all $i$, then $Z_t = D$ and the lemma is proved.", "If not, then $a_i < m_i/d$ for some $i$ and it follows that", "$(Z_t \\cdot Z_t) < 0$ by Lemma \\ref{lemma-properties-form}. This means that", "$(D - Z_t \\cdot Z_t) > 0$ because $(D \\cdot Z_t) = 0$ by the lemma.", "Thus we can find an $i$ with $a_i < m_i/d$ such that", "$(C_i \\cdot Z_t) > 0$. Set $Z_{t + 1} = Z_t + C_i$ and $i_{t + 1} = i$.", "Consider the short exact sequence", "$$", "0 \\to \\mathcal{O}_X(-Z_t)|_{C_i} \\to \\mathcal{O}_{Z_{t + 1}} \\to", "\\mathcal{O}_{Z_t} \\to 0", "$$", "of Divisors, Lemma \\ref{divisors-lemma-ses-add-divisor}.", "By our choice of $i$ we see that", "$\\mathcal{O}_X(-Z_t)|_{C_i}$ is an invertible sheaf of negative degree", "on the proper curve $C_i$, hence it has no nonzero global sections", "(Varieties, Lemma \\ref{varieties-lemma-check-invertible-sheaf-trivial}).", "We conclude that $H^0(\\mathcal{O}_{Z_{t + 1}}) \\subset H^0(\\mathcal{O}_{Z_t})$", "is a field (this is clear but also follows from", "Algebra, Lemma \\ref{algebra-lemma-integral-under-field})", "and a finite extension of $k$. Thus we have extended the sequence.", "Since the process must stop, for example because $t \\leq \\sum (m_i/d - 1)$,", "this finishes the proof." ], "refs": [ "models-lemma-regular-model-connected", "varieties-lemma-proper-geometrically-reduced-global-sections", "models-lemma-properties-form", "divisors-lemma-ses-add-divisor", "varieties-lemma-check-invertible-sheaf-trivial", "algebra-lemma-integral-under-field" ], "ref_ids": [ 9237, 10948, 9240, 7939, 11114, 496 ] } ], "ref_ids": [] }, { "id": 9243, "type": "theorem", "label": "models-lemma-regular-model-genus", "categories": [ "models" ], "title": "models-lemma-regular-model-genus", "contents": [ "\\begin{reference}", "\\cite[Lemma 2.6]{Artin-Winters}", "\\end{reference}", "In Situation \\ref{situation-regular-model} let $d = \\gcd(m_1, \\ldots, m_n)$.", "Let $D = \\sum (m_i/d) C_i$ as an effective Cartier divisor on $X$. Then", "$$", "1 - g_C = d [\\kappa : k] (1 - g_D)", "$$", "where $g_C$ is the genus of $C$, $g_D$ is the genus of $D$, and", "$\\kappa = H^0(D, \\mathcal{O}_D)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-regular-model-field} we see that $\\kappa$ is a field", "and a finite extension of $k$. Since also $H^0(C, \\mathcal{O}_C) = K$", "we see that the genus of $C$ and $D$ are defined (see", "Algebraic Curves, Definition \\ref{curves-definition-genus}) and", "we have $g_C = \\dim_K H^1(C, \\mathcal{O}_C)$ and", "$g_D = \\dim_\\kappa H^1(D, \\mathcal{O}_D)$.", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-chi-locally-constant-geometric}", "we have", "$$", "1 - g_C = \\chi(C, \\mathcal{O}_C) =", "\\chi(X_k, \\mathcal{O}_{X_k}) = \\dim_k H^0(X_k, \\mathcal{O}_{X_k})", "- \\dim_k H^1(X_k, \\mathcal{O}_{X_k})", "$$", "We claim that", "$$", "\\chi(X_k, \\mathcal{O}_{X_k}) = d \\chi(D, \\mathcal{O}_D)", "$$", "This will prove the lemma because", "$$", "\\chi(D, \\mathcal{O}_D) =", "\\dim_k H^0(D, \\mathcal{O}_D) - \\dim_k H^1(D, \\mathcal{O}_D) =", "[\\kappa : k](1 - g_D)", "$$", "Observe that $X_k = dD$ as an effective Cartier divisor.", "To prove the claim we prove by induction on $1 \\leq r \\leq d$ that", "$\\chi(rD, \\mathcal{O}_{rD}) = r \\chi(D, \\mathcal{O}_D)$.", "The base case $r = 1$ is trivial. If $1 \\leq r < d$, then we consider", "the short exact sequence", "$$", "0 \\to \\mathcal{O}_X(rD)|_D \\to \\mathcal{O}_{(r + 1)D} \\to", "\\mathcal{O}_{rD} \\to 0", "$$", "of Divisors, Lemma \\ref{divisors-lemma-ses-add-divisor}. By additivity", "of Euler characteristics", "(Varieties, Lemma \\ref{varieties-lemma-euler-characteristic-additive})", "it suffices to prove that", "$\\chi(D, \\mathcal{O}_X(rD)|_D) = \\chi(D, \\mathcal{O}_D)$.", "This is true because $\\mathcal{O}_X(rD)|_D$ is a torsion", "element of $\\Pic(D)$ (Lemma \\ref{lemma-multiple-fibre-normal-bundle})", "and because the degree of a line bundle is additive", "(Varieties, Lemma \\ref{varieties-lemma-degree-tensor-product})", "hence zero for torsion invertible sheaves." ], "refs": [ "models-lemma-regular-model-field", "curves-definition-genus", "perfect-lemma-chi-locally-constant-geometric", "divisors-lemma-ses-add-divisor", "varieties-lemma-euler-characteristic-additive", "models-lemma-multiple-fibre-normal-bundle", "varieties-lemma-degree-tensor-product" ], "ref_ids": [ 9242, 6355, 7063, 7939, 11029, 9241, 11109 ] } ], "ref_ids": [] }, { "id": 9244, "type": "theorem", "label": "models-lemma-exceptional-curves-dont-meet", "categories": [ "models" ], "title": "models-lemma-exceptional-curves-dont-meet", "contents": [ "In Situation \\ref{situation-regular-model} given a pair of indices $i, j$", "such that $C_i$ and $C_j$ are exceptional curves of the first kind", "and $C_i \\cap C_j \\not = \\emptyset$, then", "$n = 2$, $m_1 = m_2 = 1$, $C_1 \\cong \\mathbf{P}^1_k$,", "$C_2 \\cong \\mathbf{P}^1_k$, $C_1$ and $C_2$ meet in a $k$-rational point,", "and $C$ has genus $0$." ], "refs": [], "proofs": [ { "contents": [ "Choose isomorphisms $C_i = \\mathbf{P}^1_{\\kappa_i}$ and", "$C_j = \\mathbf{P}^1_{\\kappa_j}$. The scheme $C_i \\cap C_j$", "is a nonempty effective Cartier divisor in both $C_i$ and $C_j$.", "Hence", "$$", "(C_i \\cdot C_j) = \\deg(C_i \\cap C_j) \\geq \\max([\\kappa_i: k], [\\kappa_j : k])", "$$", "The first equality was shown in the proof of", "Lemma \\ref{lemma-intersection-pairing}.", "On the other hand, the self intersection $(C_i \\cdot C_i)$ is equal", "to the degree of $\\mathcal{O}_X(C_i)$ on $C_i$ which is $-[\\kappa_i : k]$", "as $C_i$ is an exceptional curve of the first kind. Similarly for", "$C_j$. By Lemma \\ref{lemma-properties-form}", "$$", "0 \\geq (C_i + C_j)^2 = -[\\kappa_i : k] + 2(C_i \\cdot C_j) - [\\kappa_j : k]", "$$", "This implies that $[\\kappa_i : k] = \\deg(C_i \\cap C_j) = [\\kappa_j : k]$", "and that we have $(C_i + C_j)^2 = 0$. Looking at the lemma again", "we conclude that $n = 2$, $\\{1, 2\\} = \\{i, j\\}$, and $m_1 = m_2$.", "Moreover, the scheme theoretic intersection $C_i \\cap C_j$ consists of", "a single point $p$ with residue field $\\kappa$ and", "$\\kappa_i \\to \\kappa \\leftarrow \\kappa_j$ are isomorphisms.", "Let $D = C_1 + C_2$ as effective Cartier divisor on $X$.", "Observe that $D$ is the scheme theoretic union of $C_1$ and $C_2$", "(Divisors, Lemma \\ref{divisors-lemma-sum-effective-Cartier-divisors-union})", "hence we have a short exact sequence", "$$", "0 \\to \\mathcal{O}_D \\to \\mathcal{O}_{C_1} \\oplus \\mathcal{O}_{C_2} \\to", "\\mathcal{O}_p \\to 0", "$$", "by Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-union}.", "Since we know the cohomology of $C_i \\cong \\mathbf{P}^1_\\kappa$", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cohomology-projective-space-over-ring})", "we conclude from the long exact cohomology sequence that", "$H^0(D, \\mathcal{O}_D) = \\kappa$ and", "$H^1(D, \\mathcal{O}_D) = 0$. By Lemma \\ref{lemma-regular-model-genus}", "we conclude", "$$", "1 - g_C = d[\\kappa : k](1 - 0)", "$$", "where $d = m_1 = m_2$. It follows that $g_C = 0$ and $d = m_1 = m_2 = 1$", "and $\\kappa = k$." ], "refs": [ "models-lemma-intersection-pairing", "models-lemma-properties-form", "divisors-lemma-sum-effective-Cartier-divisors-union", "morphisms-lemma-scheme-theoretic-union", "coherent-lemma-cohomology-projective-space-over-ring", "models-lemma-regular-model-genus" ], "ref_ids": [ 9239, 9240, 7934, 5140, 3304, 9243 ] } ], "ref_ids": [] }, { "id": 9245, "type": "theorem", "label": "models-lemma-minimal-model-unique", "categories": [ "models" ], "title": "models-lemma-minimal-model-unique", "contents": [ "Let $C$ be a smooth projective curve over $K$ with", "$H^0(C, \\mathcal{O}_C) = K$ and genus $> 0$.", "There is a unique minimal model for $C$." ], "refs": [], "proofs": [ { "contents": [ "We have already proven the hard part of the lemma which is the existence", "of a minimal model (whose proof relies on", "resolution of surface singularities), see", "Proposition \\ref{proposition-exists-minimal-model}.", "To prove uniqueness, suppose that $X$ and $Y$ are two", "minimal models. By", "Resolution of Surfaces, Lemma \\ref{resolve-lemma-birational-regular-surfaces}", "there exists a diagram of $S$-morphisms", "$$", "X = X_0 \\leftarrow X_1 \\leftarrow \\ldots \\leftarrow X_n = Y_m", "\\to \\ldots \\to Y_1 \\to Y_0 = Y", "$$", "where each morphism is a blowup in a closed point. The", "exceptional fibre of the morphism $X_n \\to X_{n - 1}$ is an", "exceptional curve of the first kind $E$. We claim that $E$ is", "contracted to a point under the morphism $X_n = Y_m \\to Y$.", "If this is true, then $X_n \\to Y$ factors through $X_{n - 1}$ by", "Resolution of Surfaces, Lemma \\ref{resolve-lemma-factor-through-contraction}.", "In this case the morphism $X_{n - 1} \\to Y$ is still a sequence of", "contractions of exceptional curves by", "Resolution of Surfaces, Lemma", "\\ref{resolve-lemma-proper-birational-regular-surfaces}.", "Hence by induction on $n$ we conclude. (The base case $n = 0$ means", "that there is a sequence of contractions", "$X = Y_m \\to \\ldots \\to Y_1 \\to Y_0 = Y$", "ending with $Y$. However as $X$ is a minimal model it contains", "no exceptional curves of the first kind, hence $m = 0$ and $X = Y$.)", "\\medskip\\noindent", "Proof of the claim. We will show by induction on $m$ that any exceptional", "curve of the first kind $E \\subset Y_m$ is mapped to a point", "by the morphism $Y_m \\to Y$. If $m = 0$ this is clear because", "$Y$ is a minimal model. If $m > 0$, then either", "$Y_m \\to Y_{m - 1}$ contracts $E$ (and we're done) or", "the exceptional fibre $E' \\subset Y_m$ of $Y_m \\to Y_{m - 1}$", "is a second exceptional curve of the first kind.", "Since both $E$ and $E'$ are irreducible components of the special", "fibre and since $g_C > 0$ by assumption, we conclude that", "$E \\cap E' = \\emptyset$ by", "Lemma \\ref{lemma-exceptional-curves-dont-meet}.", "Then the image of $E$ in $Y_{m - 1}$ is an exceptional", "curve of the first kind (this is clear because the morphism", "$Y_m \\to Y_{m - 1}$ is an isomorphism in a neighbourhood of $E$).", "By induction we see that $Y_{m - 1} \\to Y$ contracts this curve", "and the proof is complete." ], "refs": [ "models-proposition-exists-minimal-model", "resolve-lemma-birational-regular-surfaces", "resolve-lemma-factor-through-contraction", "resolve-lemma-proper-birational-regular-surfaces", "models-lemma-exceptional-curves-dont-meet" ], "ref_ids": [ 9268, 11710, 11699, 11709, 9244 ] } ], "ref_ids": [] }, { "id": 9246, "type": "theorem", "label": "models-lemma-minimal-model-mapping-property", "categories": [ "models" ], "title": "models-lemma-minimal-model-mapping-property", "contents": [ "Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$", "and genus $> 0$. Let $X$ be the minimal model for $C$", "(Lemma \\ref{lemma-minimal-model-unique}).", "Let $Y$ be a regular proper model for $C$. Then there is a unique", "morphism of models $Y \\to X$ which is a sequence of contractions of", "exceptional curves of the first kind." ], "refs": [ "models-lemma-minimal-model-unique" ], "proofs": [ { "contents": [ "The existence and properties of the morphism $X \\to Y$", "follows immediately from Lemma \\ref{lemma-pre-exists-minimal-model}", "and the uniqueness of the minimal model.", "The morphism $Y \\to X$ is unique because", "$C \\subset Y$ is scheme theoretically dense and $X$ is separated", "(see Morphisms, Lemma \\ref{morphisms-lemma-equality-of-morphisms})." ], "refs": [ "models-lemma-pre-exists-minimal-model", "morphisms-lemma-equality-of-morphisms" ], "ref_ids": [ 9234, 5157 ] } ], "ref_ids": [ 9245 ] }, { "id": 9247, "type": "theorem", "label": "models-lemma-add-component", "categories": [ "models" ], "title": "models-lemma-add-component", "contents": [ "In Situation \\ref{situation-regular-model} suppose we have an", "effective Cartier divisors $D, D' \\subset X$ such that", "$D' = D + C_i$ for some $i \\in \\{1, \\ldots, n\\}$ and $D' \\subset X_k$.", "Then", "$$", "\\chi(X_k, \\mathcal{O}_{D'}) - \\chi(X_k, \\mathcal{O}_D) =", "\\chi(X_k, \\mathcal{O}_X(-D)|_{C_i}) =", "-(D \\cdot C_i) + \\chi(C_i, \\mathcal{O}_{C_i})", "$$" ], "refs": [], "proofs": [ { "contents": [ "The second equality follows from the definition of the bilinear form", "$(\\ \\cdot\\ )$ in (\\ref{equation-form}) and", "Lemma \\ref{lemma-intersection-pairing}. To see the first", "equality we distinguish two cases.", "Namely, if $C_i \\not \\subset D$, then $D'$ is the scheme", "theoretic union of $D$ and $C_i$ (by", "Divisors, Lemma \\ref{divisors-lemma-sum-effective-Cartier-divisors-union})", "and we get a short exact sequence", "$$", "0 \\to \\mathcal{O}_{D'} \\to", "\\mathcal{O}_D \\times \\mathcal{O}_{C_i} \\to", "\\mathcal{O}_{D \\cap C_i} \\to 0", "$$", "by Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-union}.", "Since we also have an exact sequence", "$$", "0 \\to \\mathcal{O}_X(-D)|_{C_i} \\to", "\\mathcal{O}_{C_i} \\to \\mathcal{O}_{D \\cap C_i} \\to 0", "$$", "(Divisors, Remark \\ref{divisors-remark-ses-regular-section})", "we conclude that the claim holds", "by additivity of euler characteristics", "(Varieties, Lemma \\ref{varieties-lemma-euler-characteristic-additive}).", "On the other hand, if $C_i \\subset D$ then we get an", "exact sequence", "$$", "0 \\to \\mathcal{O}_X(-D)|_{C_i} \\to \\mathcal{O}_{D'} \\to \\mathcal{O}_D \\to 0", "$$", "by Divisors, Lemma \\ref{divisors-lemma-ses-add-divisor}", "and we immediately see the lemma holds." ], "refs": [ "models-lemma-intersection-pairing", "divisors-lemma-sum-effective-Cartier-divisors-union", "morphisms-lemma-scheme-theoretic-union", "varieties-lemma-euler-characteristic-additive", "divisors-lemma-ses-add-divisor" ], "ref_ids": [ 9239, 7934, 5140, 11029, 7939 ] } ], "ref_ids": [] }, { "id": 9248, "type": "theorem", "label": "models-lemma-genus-formula", "categories": [ "models" ], "title": "models-lemma-genus-formula", "contents": [ "In Situation \\ref{situation-regular-model} we have", "$$", "g_C = 1 + \\sum\\nolimits_{i = 1, \\ldots, n}", "m_i\\left([\\kappa_i : k] (g_i - 1) - \\frac{1}{2}(C_i \\cdot C_i)\\right)", "$$", "where $\\kappa_i = H^0(C_i, \\mathcal{O}_{C_i})$,", "$g_i$ is the genus of $C_i$, and $g_C$ is the genus of $C$." ], "refs": [], "proofs": [ { "contents": [ "Our basic tool will be Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-chi-locally-constant-geometric}", "which shows that", "$$", "1 - g_C = \\chi(C, \\mathcal{O}_C) =", "\\chi(X_k, \\mathcal{O}_{X_k})", "$$", "Choose a sequence of effective Cartier divisors", "$$", "X_k = D_m \\supset D_{m - 1} \\supset \\ldots \\supset D_1 \\supset D_0 = \\emptyset", "$$", "such that $D_{j + 1} = D_j + C_{i_j}$ for each $j$. (It is clear that", "we can choose such a sequence by decreasing one nonzero multiplicity", "of $D_{j + 1}$ one step at a time.) Applying Lemma \\ref{lemma-add-component}", "starting with $\\chi(\\mathcal{O}_{D_0}) = 0$ we get", "\\begin{align*}", "1 - g_C", "& =", "\\chi(X_k, \\mathcal{O}_{X_k}) \\\\", "& =", "\\sum\\nolimits_j", "\\left(-(D_j \\cdot C_{i_j}) + \\chi(C_{i_j}, \\mathcal{O}_{C_{i_j}})\\right) \\\\", "& =", "- \\sum\\nolimits_j", "(C_{i_1} + C_{i_2} + \\ldots + C_{i_{j - 1}} \\cdot C_{i_j}) +", "\\sum\\nolimits_j \\chi(C_{i_j}, \\mathcal{O}_{C_{i_j}}) \\\\", "& =", "-\\frac{1}{2}\\sum\\nolimits_{j \\not = j'} (C_{i_{j'}} \\cdot C_{i_j}) +", "\\sum m_i \\chi(C_i, \\mathcal{O}_{C_i}) \\\\", "& =", "\\frac{1}{2} \\sum m_i(C_i \\cdot C_i) + \\sum m_i \\chi(C_i, \\mathcal{O}_{C_i})", "\\end{align*}", "Perhaps the last equality deserves some explanation. Namely, since", "$\\sum_j C_{i_j} = \\sum m_i C_i$ we have", "$(\\sum_j C_{i_j} \\cdot \\sum_j C_{i_j}) = 0$ by", "Lemma \\ref{lemma-properties-form}. Thus we see that", "$$", "0 = \\sum\\nolimits_{j \\not = j'} (C_{i_{j'}} \\cdot C_{i_j}) +", "\\sum m_i(C_i \\cdot C_i)", "$$", "by splitting this product into ``nondiagonal'' and ``diagonal'' terms.", "Note that $\\kappa_i$ is a field finite over $k$ by", "Varieties, Lemma \\ref{varieties-lemma-regular-functions-proper-variety}.", "Hence the genus of $C_i$ is defined and we have", "$\\chi(C_i, \\mathcal{O}_{C_i}) = [\\kappa_i : k](1 - g_i)$.", "Putting everything together and rearranging terms we get", "$$", "g_C = - \\frac{1}{2}\\sum m_i(C_i \\cdot C_i) +", "\\sum m_i[\\kappa_i : k](g_i - 1) + 1", "$$", "which is what the lemma says too." ], "refs": [ "perfect-lemma-chi-locally-constant-geometric", "models-lemma-add-component", "models-lemma-properties-form", "varieties-lemma-regular-functions-proper-variety" ], "ref_ids": [ 7063, 9247, 9240, 11012 ] } ], "ref_ids": [] }, { "id": 9249, "type": "theorem", "label": "models-lemma-numerical-type-of-model", "categories": [ "models" ], "title": "models-lemma-numerical-type-of-model", "contents": [ "In Situation \\ref{situation-regular-model} with", "$\\kappa_i = H^0(C_i, \\mathcal{O}_{C_i})$ and $g_i$ the genus of $C_i$", "the data", "$$", "n, m_i, (C_i \\cdot C_j), [\\kappa_i : k], g_i", "$$", "is a numerical type of genus equal to the genus of $C$." ], "refs": [], "proofs": [ { "contents": [ "(In the proof of Lemma \\ref{lemma-genus-formula}", "we have seen that the quantities", "used in the statement of the lemma are well defined.)", "We have to verify the conditions (1) -- (5) of", "Definition \\ref{definition-type}.", "\\medskip\\noindent", "Condition (1) is immediate.", "\\medskip\\noindent", "Condition (2). Symmetry of the matrix $(C_i \\cdot C_j)$ follows from", "Equation (\\ref{equation-form}) and", "Lemma \\ref{lemma-intersection-pairing}.", "Nonnegativity of $(C_i \\cdot C_j)$ for $i \\not = j$", "is part (1) of Lemma \\ref{lemma-properties-form}.", "\\medskip\\noindent", "Condition (3) is part (3) of Lemma \\ref{lemma-properties-form}.", "\\medskip\\noindent", "Condition (4) is part (2) of Lemma \\ref{lemma-properties-form}.", "\\medskip\\noindent", "Condition (5) follows from the fact that $(C_i \\cdot C_j)$ is", "the degree of an invertible module on $C_i$ which is divisible", "by $[\\kappa_i : k]$, see Varieties, Lemma \\ref{varieties-lemma-divisible}.", "\\medskip\\noindent", "The genus formula proved in Lemma \\ref{lemma-genus-formula}", "tells us that the numerical type has the genus as stated, see", "Definition \\ref{definition-genus}." ], "refs": [ "models-lemma-genus-formula", "models-definition-type", "models-lemma-intersection-pairing", "models-lemma-properties-form", "models-lemma-properties-form", "models-lemma-properties-form", "varieties-lemma-divisible", "models-lemma-genus-formula", "models-definition-genus" ], "ref_ids": [ 9248, 9269, 9239, 9240, 9240, 9240, 11112, 9248, 9271 ] } ], "ref_ids": [] }, { "id": 9250, "type": "theorem", "label": "models-lemma-numerical-type-minimal-model", "categories": [ "models" ], "title": "models-lemma-numerical-type-minimal-model", "contents": [ "In Situation \\ref{situation-regular-model}. The following", "are equivalent", "\\begin{enumerate}", "\\item $X$ is a minimal model, and", "\\item the numerical type associated to $X$ is minimal.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If the numerical type is minimal, then there is no $i$ with", "$g_i = 0$ and $(C_i \\cdot C_i) = -[\\kappa_i: k]$, see", "Definition \\ref{definition-type-minimal}.", "Certainly, this implies that none of the curves $C_i$", "are exceptional curves of the first kind.", "\\medskip\\noindent", "Conversely, suppose that the numerical type is not minimal.", "Then there exists an $i$ such that $g_i = 0$ and", "$(C_i \\cdot C_i) = -[\\kappa_i: k]$.", "We claim this implies that $C_i$ is an exceptional curve", "of the first kind. Namely, the invertible sheaf", "$\\mathcal{O}_X(-C_i)|_{C_i}$ has degree $-(C_i \\cdot C_i) = [\\kappa_i : k]$", "when $C_i$ is viewed as a proper curve over $k$, hence", "has degree $1$ when $C_i$ is viewed as a proper curve over $\\kappa_i$.", "Applying", "Algebraic Curves, Proposition \\ref{curves-proposition-projective-line}", "we conclude that $C_i \\cong \\mathbf{P}^1_{\\kappa_i}$ as schemes", "over $\\kappa_i$. Since the Picard group of $\\mathbf{P}^1$", "over a field is $\\mathbf{Z}$, we see that the normal sheaf", "of $C_i$ in $X$ is isomorphic to $\\mathcal{O}_{\\mathbf{P}_{\\kappa_i}}(-1)$", "and the proof is complete." ], "refs": [ "models-definition-type-minimal", "curves-proposition-projective-line" ], "ref_ids": [ 9274, 6350 ] } ], "ref_ids": [] }, { "id": 9251, "type": "theorem", "label": "models-lemma-numerical-type-rational-point", "categories": [ "models" ], "title": "models-lemma-numerical-type-rational-point", "contents": [ "In Situation \\ref{situation-regular-model} assume $C$ has a $K$-rational point.", "Then", "\\begin{enumerate}", "\\item $X_k$ has a $k$-rational point $x$ which is a smooth point of $X_k$", "over $k$,", "\\item if $x \\in C_i$, then $H^0(C_i, \\mathcal{O}_{C_i}) = k$ and", "$m_i = 1$, and", "\\item $H^0(X_k, \\mathcal{O}_{X_k}) = k$ and $X_k$ has genus equal to", "the genus of $C$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $X \\to \\Spec(R)$ is proper, the $K$-rational point extends to", "a morphism $a : \\Spec(R) \\to X$ by the valuative criterion of properness", "(Morphisms, Lemma \\ref{morphisms-lemma-characterize-proper}).", "Let $x \\in X$ be the image under $a$ of the closed point of $\\Spec(R)$.", "Then $a$ corresponds to an $R$-algebra homomorphism", "$\\psi : \\mathcal{O}_{X, x} \\to R$", "(see Schemes, Section \\ref{schemes-section-points}).", "It follows that $\\pi \\not \\in \\mathfrak m_x^2$ (since the image", "of $\\pi$ in $R$ is not in $\\mathfrak m_R^2$).", "Hence $\\mathcal{O}_{X_k, x} = \\mathcal{O}_{X, x}/\\pi \\mathcal{O}_{X, x}$", "is regular (Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}).", "Then $X_k \\to \\Spec(k)$ is smooth at $x$ by", "Algebra, Lemma \\ref{algebra-lemma-separable-smooth}.", "It follows that $x$ is contained in a unique irreducible component", "$C_i$ of $X_k$, that $\\mathcal{O}_{C_i, x} = \\mathcal{O}_{X_k, x}$,", "and that $m_i = 1$. The fact that $C_i$ has a", "$k$-rational point implies that the field", "$\\kappa_i = H^0(C_i, \\mathcal{O}_{C_i})$", "(Varieties, Lemma \\ref{varieties-lemma-regular-functions-proper-variety})", "is equal to $k$. This proves (1). We have", "$H^0(X_k, \\mathcal{O}_{X_k}) = k$", "because $H^0(X_k, \\mathcal{O}_{X_k})$ is a field", "extension of $k$ (Lemma \\ref{lemma-regular-model-field})", "which maps to $H^0(C_i, \\mathcal{O}_{C_i}) = k$.", "The genus equality follows from Lemma \\ref{lemma-regular-model-genus}." ], "refs": [ "morphisms-lemma-characterize-proper", "algebra-lemma-regular-ring-CM", "algebra-lemma-separable-smooth", "varieties-lemma-regular-functions-proper-variety", "models-lemma-regular-model-field", "models-lemma-regular-model-genus" ], "ref_ids": [ 5416, 941, 1225, 11012, 9242, 9243 ] } ], "ref_ids": [] }, { "id": 9252, "type": "theorem", "label": "models-lemma-genus-reduction-smaller", "categories": [ "models" ], "title": "models-lemma-genus-reduction-smaller", "contents": [ "In Situation \\ref{situation-regular-model} assume $X$ is a minimal model,", "$\\gcd(m_1, \\ldots, m_n) = 1$, and $H^0((X_k)_{red}, \\mathcal{O}) = k$. Then", "the map", "$$", "H^1(X_k, \\mathcal{O}_{X_k}) \\to H^1((X_k)_{red}, \\mathcal{O}_{(X_k)_{red}})", "$$", "is surjective and has a nontrivial kernel as soon as $(X_k)_{red} \\not = X_k$." ], "refs": [], "proofs": [ { "contents": [ "By vanishing of cohomology in degrees $\\geq 2$ over $X_k$", "(Cohomology, Proposition \\ref{cohomology-proposition-vanishing-Noetherian})", "any surjection of abelian sheaves on $X_k$ induces a surjection on $H^1$.", "Consider the sequence", "$$", "(X_k)_{red} = Z_0 \\subset Z_1 \\subset \\ldots \\subset Z_m = X_k", "$$", "of Lemma \\ref{lemma-regular-model-field}. Since the field maps", "$H^0(Z_j, \\mathcal{O}_{Z_j}) \\to", "H^0((X_k)_{red}, \\mathcal{O}_{(X_k)_{red}}) = k$", "are injective we conclude that $H^0(Z_j, \\mathcal{O}_{Z_j}) = k$ for", "$j = 0, \\ldots, m$. It follows that", "$H^0(X_k, \\mathcal{O}_{X_k}) \\to H^0(Z_{m - 1}, \\mathcal{O}_{Z_{m - 1}})$", "is surjective. Let $C = C_{i_m}$. Then $X_k = Z_{m - 1} + C$.", "Let $\\mathcal{L} = \\mathcal{O}_X(-Z_{m - 1})|_C$.", "Then $\\mathcal{L}$ is an invertible $\\mathcal{O}_C$-module.", "As in the proof of Lemma \\ref{lemma-regular-model-field}", "there is an exact sequence", "$$", "0 \\to \\mathcal{L} \\to \\mathcal{O}_{X_k} \\to \\mathcal{O}_{Z_{m - 1}} \\to 0", "$$", "of coherent sheaves on $X_k$. We conclude that", "we get a short exact sequence", "$$", "0 \\to", "H^1(C, \\mathcal{L}) \\to H^1(X_k, \\mathcal{O}_{X_k}) \\to", "H^1(Z_{m - 1}, \\mathcal{O}_{Z_{m - 1}}) \\to 0", "$$", "The degree of $\\mathcal{L}$ on $C$ over $k$ is", "$$", "(C \\cdot -Z_{m - 1}) = (C \\cdot C - X_k) = (C \\cdot C)", "$$", "Set $\\kappa = H^0(C, \\mathcal{O}_C)$ and $w = [\\kappa : k]$.", "By definition of the degree of an invertible sheaf we see that", "$$", "\\chi(C, \\mathcal{L}) =", "\\chi(C, \\mathcal{O}_C) + (C \\cdot C) =", "w(1 - g_C) + (C \\cdot C)", "$$", "where $g_C$ is the genus of $C$. This expression is $< 0$ as $X$ is minimal", "and hence $C$ is not an exceptional curve of the first kind", "(see proof of Lemma \\ref{lemma-numerical-type-minimal-model}).", "Thus $\\dim_k H^1(C, \\mathcal{L}) > 0$ which finishes the proof." ], "refs": [ "cohomology-proposition-vanishing-Noetherian", "models-lemma-regular-model-field", "models-lemma-regular-model-field", "models-lemma-numerical-type-minimal-model" ], "ref_ids": [ 2246, 9242, 9242, 9250 ] } ], "ref_ids": [] }, { "id": 9253, "type": "theorem", "label": "models-lemma-genus-reduction-bigger-than", "categories": [ "models" ], "title": "models-lemma-genus-reduction-bigger-than", "contents": [ "In Situation \\ref{situation-regular-model} assume $X_k$ has a $k$-rational", "point $x$ which is a smooth point of $X_k \\to \\Spec(k)$. Then", "$$", "\\dim_k H^1((X_k)_{red}, \\mathcal{O}_{(X_k)_{red}}) \\geq", "g_{top} + g_{geom}(X_k/k)", "$$", "where $g_{geom}$ is as in", "Algebraic Curves, Section \\ref{curves-section-genus-geometric-genus}", "and $g_{top}$ is the topological genus", "(Definition \\ref{definition-top-genus})", "of the numerical type associated to $X_k$", "(Definition \\ref{definition-numerical-type-model})." ], "refs": [ "models-definition-top-genus", "models-definition-numerical-type-model" ], "proofs": [ { "contents": [ "We are going to prove the inequality", "$$", "\\dim_k H^1(D, \\mathcal{O}_D) \\geq g_{top}(D) + g_{geom}(D/k)", "$$", "for all connected reduced effective Cartier divisors", "$D \\subset (X_k)_{red}$ containing $x$ by induction", "on the number of irreducible components of $D$.", "Here $g_{top}(D) = 1 - m + e$ where $m$ is the number of", "irreducible components of $D$ and $e$ is the number of", "unordered pairs of components of $D$ which meet.", "\\medskip\\noindent", "Base case: $D$ has one irreducible component. Then $D = C_i$", "is the unique irreducible component containing $x$.", "In this case $\\dim_k H^1(D, \\mathcal{O}_D) = g_i$", "and $g_{top}(D) = 0$. Since $C_i$ has a $k$-rational smooth point", "it is geometrically integral", "(Varieties, Lemma \\ref{varieties-lemma-variety-with-smooth-rational-point}).", "It follows that $g_i$ is the genus of $C_{i, \\overline{k}}$", "(Algebraic Curves, Lemma \\ref{curves-lemma-genus-base-change}).", "It also follows that $g_{geom}(D/k)$ is the genus of the normalization", "$C_{i, \\overline{k}}^\\nu$ of $C_{i, \\overline{k}}$. Applying", "Algebraic Curves, Lemma \\ref{curves-lemma-genus-normalization}", "to the normalization morphism $C_{i, \\overline{k}}^\\nu \\to C_{i, \\overline{k}}$", "we get", "\\begin{equation}", "\\label{equation-genus-change-special-component}", "\\text{genus of }C_{i, \\overline{k}} \\geq", "\\text{genus of }C_{i, \\overline{k}}^\\nu", "\\end{equation}", "Combining the above we conclude that", "$\\dim_k H^1(D, \\mathcal{O}_D) \\geq g_{top}(D) + g_{geom}(D/k)$", "in this case.", "\\medskip\\noindent", "Induction step. Suppose we have $D$ with more than $1$ irreducible", "component. Then we can write $D = C_i + D'$ where $x \\in D'$ and", "$D'$ is still connected. This is an exercise in graph theory we leave", "to the reader (hint: let $C_i$ be the component of $D$ which is", "farthest from $x$). We compute how the invariants change.", "As $x \\in D'$ we have $H^0(D, \\mathcal{O}_D) = H^0(D', \\mathcal{O}_{D'}) = k$.", "Looking at the short exact sequence of sheaves", "$$", "0 \\to \\mathcal{O}_D \\to \\mathcal{O}_{C_i} \\oplus \\mathcal{O}_{D'}", "\\to \\mathcal{O}_{C_i \\cap D'} \\to 0", "$$", "(Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-union})", "and using additivity of euler characteristics we find", "\\begin{align*}", "\\dim_k H^1(D, \\mathcal{O}_D) - \\dim_k H^1(D', \\mathcal{O}_{D'})", "& =", "-\\chi(\\mathcal{O}_{C_i}) + \\chi(\\mathcal{O}_{C_i \\cap D'}) \\\\", "& =", "w_i(g_i - 1) + \\sum\\nolimits_{C_j \\subset D'} a_{ij}", "\\end{align*}", "Here as in Lemma \\ref{lemma-numerical-type-of-model} we set", "$w_i = [\\kappa_i : k]$, $\\kappa_i = H^0(C_i, \\mathcal{O}_{C_i})$,", "$g_i$ is the genus of $C_i$, and $a_{ij} = (C_i \\cdot C_j)$.", "We have", "$$", "g_{top}(D) - g_{top}(D') = -1 +", "\\sum\\nolimits_{C_j \\subset D'\\text{ meeting }C_i} 1", "$$", "We have", "$$", "g_{geom}(D/k) - g_{geom}(D'/k) = g_{geom}(C_i/k)", "$$", "by Algebraic Curves, Lemma \\ref{curves-lemma-bound-geometric-genus}.", "Combining these with our induction hypothesis, we", "conclude that it suffices to show that", "$$", "w_i g_i - g_{geom}(C_i/k) +", "\\sum\\nolimits_{C_j \\subset D'\\text{ meets } C_i} (a_{ij} - 1) - (w_i - 1)", "$$", "is nonnegative. In fact, we have", "\\begin{equation}", "\\label{equation-genus-change}", "w_i g_i \\geq [\\kappa_i : k]_s g_i \\geq g_{geom}(C_i/k)", "\\end{equation}", "The second inequality by", "Algebraic Curves, Lemma \\ref{curves-lemma-bound-geometric-genus-curve}.", "On the other hand, since $w_i$ divides $a_{ij}$", "(Varieties, Lemma \\ref{varieties-lemma-divisible})", "it is clear that", "\\begin{equation}", "\\label{equation-change-intersections}", "\\sum\\nolimits_{C_j \\subset D'\\text{ meets } C_i} (a_{ij} - 1) - (w_i - 1)", "\\geq 0", "\\end{equation}", "because there is at least one $C_j \\subset D'$ which meets $C_i$." ], "refs": [ "varieties-lemma-variety-with-smooth-rational-point", "curves-lemma-genus-base-change", "curves-lemma-genus-normalization", "morphisms-lemma-scheme-theoretic-union", "models-lemma-numerical-type-of-model", "curves-lemma-bound-geometric-genus", "curves-lemma-bound-geometric-genus-curve", "varieties-lemma-divisible" ], "ref_ids": [ 11011, 6265, 6301, 5140, 9249, 6298, 6302, 11112 ] } ], "ref_ids": [ 9273, 9278 ] }, { "id": 9254, "type": "theorem", "label": "models-lemma-equality-genus-reduction-bigger-than", "categories": [ "models" ], "title": "models-lemma-equality-genus-reduction-bigger-than", "contents": [ "If equality holds in Lemma \\ref{lemma-genus-reduction-bigger-than}", "then", "\\begin{enumerate}", "\\item the unique irreducible component of $X_k$ containing", "$x$ is a smooth projective geometrically irreducible curve", "over $k$,", "\\item if $C \\subset X_k$ is another irreducible component, then", "$\\kappa = H^0(C, \\mathcal{O}_C)$ is a finite separable extension", "of $k$, $C$ has a $\\kappa$-rational point, and $C$ is smooth over $\\kappa$", "\\end{enumerate}" ], "refs": [ "models-lemma-genus-reduction-bigger-than" ], "proofs": [ { "contents": [ "Looking over the proof of Lemma \\ref{lemma-genus-reduction-bigger-than}", "we see that in order to get equality, the inequalities", "(\\ref{equation-genus-change-special-component}),", "(\\ref{equation-genus-change}), and", "(\\ref{equation-change-intersections})", "have to be equalities.", "\\medskip\\noindent", "Let $C_i$ be the irreducible component containing $x$.", "Equality in (\\ref{equation-genus-change-special-component})", "shows via", "Algebraic Curves, Lemma \\ref{curves-lemma-genus-normalization}", "that $C_{i, \\overline{k}}^\\nu \\to C_{i, \\overline{k}}$ is", "an isomorphism. Hence $C_{i, \\overline{k}}$ is smooth", "and part (1) holds.", "\\medskip\\noindent", "Next, let $C_i \\subset X_k$ be another irreducible component.", "Then we may assume we have $D = D' + C_i$ as in the induction step", "in the proof of Lemma \\ref{lemma-genus-reduction-bigger-than}.", "Equality in (\\ref{equation-genus-change}) immediately implies", "that $\\kappa_i/k$ is finite separable.", "Equality in (\\ref{equation-change-intersections})", "implies either $a_{ij} = 1$ for some $j$ or that there is a", "unique $C_j \\subset D'$ meeting $C_i$ and $a_{ij} = w_i$.", "In both cases we find that $C_i$ has a $\\kappa_i$-rational point", "$c$ and $c = C_i \\cap C_j$ scheme theoretically.", "Since $\\mathcal{O}_{X, c}$ is a regular local ring,", "this implies that the local equations of $C_i$ and $C_j$", "form a regular system of parameters in the local ring $\\mathcal{O}_{X, c}$.", "Then $\\mathcal{O}_{C_i, c}$ is regular by", "(Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}).", "We conclude that $C_i \\to \\Spec(\\kappa_i)$ is smooth at $c$", "(Algebra, Lemma \\ref{algebra-lemma-separable-smooth}).", "It follows that $C_i$ is geometrically integral over $\\kappa_i$", "(Varieties, Lemma \\ref{varieties-lemma-variety-with-smooth-rational-point}).", "To finish we have to show that $C_i$ is smooth over $\\kappa_i$. Observe that", "$$", "C_{i, \\overline{k}} = C_i \\times_{\\Spec(k)} \\Spec(\\overline{k})", "= \\coprod\\nolimits_{\\kappa_i \\to \\overline{k}}", "C_i \\times_{\\Spec(\\kappa_i)} \\Spec(\\overline{k})", "$$", "where there are $[\\kappa_i : k]$-summands. Thus if $C_i$ is not", "smooth over $\\kappa_i$, then each of these curves is not smooth, then", "these curves are not normal and the normalization morphism drops the genus", "(Algebraic Curves, Lemma \\ref{curves-lemma-genus-normalization})", "which is disallowed because it would drop the geometric genus", "of $C_i/k$ contradicting $[\\kappa_i : k] g_i = g_{geom}(C_i/k)$." ], "refs": [ "models-lemma-genus-reduction-bigger-than", "curves-lemma-genus-normalization", "models-lemma-genus-reduction-bigger-than", "algebra-lemma-regular-ring-CM", "algebra-lemma-separable-smooth", "varieties-lemma-variety-with-smooth-rational-point", "curves-lemma-genus-normalization" ], "ref_ids": [ 9253, 6301, 9253, 941, 1225, 11011, 6301 ] } ], "ref_ids": [ 9253 ] }, { "id": 9255, "type": "theorem", "label": "models-lemma-blowdown-regular-model", "categories": [ "models" ], "title": "models-lemma-blowdown-regular-model", "contents": [ "In Situation \\ref{situation-regular-model} assume that $C_n$ is", "an exceptional curve of the first kind. Let $f : X \\to X'$ be the", "contraction of $C_n$. Let $C'_i = f(C_i)$. Write $X'_k = \\sum m'_i C'_i$.", "Then $X'$, $C'_i$, $i = 1, \\ldots, n' = n - 1$, and $m'_i = m_i$", "is as in Situation \\ref{situation-regular-model} and we have", "\\begin{enumerate}", "\\item for $i, j < n$ we have", "$(C'_i \\cdot C'_j) =", "(C_i \\cdot C_j) - (C_i \\cdot C_n) (C_j \\cdot C_n) /(C_n \\cdot C_n)$,", "\\item for $i < n$ if $C_i \\cap C_n \\not = \\emptyset$, then there are", "maps $\\kappa_i \\leftarrow \\kappa'_i \\rightarrow \\kappa_n$.", "\\end{enumerate}", "Here $\\kappa_i = H^0(C_i, \\mathcal{O}_{C_i})$ and", "$\\kappa'_i = H^0(C'_i, \\mathcal{O}_{C'_i})$." ], "refs": [], "proofs": [ { "contents": [ "By Resolution of Surfaces, Lemma \\ref{resolve-lemma-contract-ample}", "we can contract $C_n$ by a morphism $f : X \\to X'$ such that $X'$ is", "regular and is projective over $R$. Thus we see that $X'$ is as in", "Situation \\ref{situation-regular-model}.", "Let $x \\in X'$ be the image of $C_n$.", "Since $f$ defines an isomorphism $X \\setminus C_n \\to X' \\setminus \\{x\\}$", "it is clear that $m'_i = m_i$ for $i < n$.", "\\medskip\\noindent", "Part (2) of the lemma is immediately clear from", "the existence of the morphisms $C_i \\to C'_i$ and $C_n \\to x \\to C'_i$.", "\\medskip\\noindent", "By Divisors, Lemma \\ref{divisors-lemma-blow-up-pullback-effective-Cartier}", "the pullback $f^{-1}C'_i$ is defined. By", "Divisors, Lemma \\ref{divisors-lemma-effective-Cartier-divisor-is-a-sum}", "we see that $f^{-1}C'_i = C_i + e_i C_n$ for some $e_i \\geq 0$. Since", "$\\mathcal{O}_X(C_i + e_i C_n) = \\mathcal{O}_X(f^{-1}C'_i) =", "f^*\\mathcal{O}_{X'}(C'_i)$", "(Divisors, Lemma \\ref{divisors-lemma-pullback-effective-Cartier-divisors})", "and since the pullback of an invertible sheaf restricts to the", "trivial invertible sheaf on $C_n$ we see that", "$$", "0 = \\deg_{C_n}(\\mathcal{O}_X(C_i + e_i C_n)) =", "(C_i + e_i C_n \\cdot C_n) = (C_i \\cdot C_n) + e_i(C_n \\cdot C_n)", "$$", "As $f_j = f|_{C_j} : C_j \\to C_j$ is a", "proper birational morphism of proper curves", "over $k$, we see that $\\deg_{C'_j}(\\mathcal{O}_{X'}(C'_i)|_{C'_j})$ is the", "same as $\\deg_{C_j}(f_j^*\\mathcal{O}_{X'}(C'_i)|_{C'_j})$", "(Varieties, Lemma \\ref{varieties-lemma-degree-birational-pullback}).", "Looking at the commutative diagram", "$$", "\\xymatrix{", "C_j \\ar[r] \\ar[d]_{f_j} & X \\ar[d]^f \\\\", "C'_j \\ar[r] & X'", "}", "$$", "and using Divisors, Lemma", "\\ref{divisors-lemma-pullback-effective-Cartier-divisors}", "we see that", "$$", "(C'_i \\cdot C'_j) = \\deg_{C'_j}(\\mathcal{O}_{X'}(C'_i)|_{C'_j}) =", "\\deg_{C_j}(\\mathcal{O}_X(C_i + e_i C_n)) = (C_i + e_i C_n \\cdot C_j)", "$$", "Plugging in the formula for $e_i$ found above we see that (1) holds." ], "refs": [ "resolve-lemma-contract-ample", "divisors-lemma-blow-up-pullback-effective-Cartier", "divisors-lemma-effective-Cartier-divisor-is-a-sum", "divisors-lemma-pullback-effective-Cartier-divisors", "varieties-lemma-degree-birational-pullback", "divisors-lemma-pullback-effective-Cartier-divisors" ], "ref_ids": [ 11705, 8061, 7955, 7941, 11106, 7941 ] } ], "ref_ids": [] }, { "id": 9256, "type": "theorem", "label": "models-lemma-nonuniqueness", "categories": [ "models" ], "title": "models-lemma-nonuniqueness", "contents": [ "Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$", "and genus $0$. If there is more than one minimal model for $C$, then", "the special fibre of every minimal model is isomorphic to $\\mathbf{P}^1_k$." ], "refs": [], "proofs": [ { "contents": [ "Let $X$ be some minimal model of $C$. The numerical type associated to $X$", "has genus $0$ and is minimal (Definition \\ref{definition-numerical-type-model}", "and Lemma \\ref{lemma-numerical-type-minimal-model}).", "Hence by Lemma \\ref{lemma-genus-zero} we see that", "$X_k$ is reduced, irreducible, has $H^0(X_k, \\mathcal{O}_{X_k}) = k$,", "and has genus $0$. Let $Y$ be a second minimal model for $C$", "which is not isomorphic to $X$. By", "Resolution of Surfaces, Lemma \\ref{resolve-lemma-birational-regular-surfaces}", "there exists a diagram of $S$-morphisms", "$$", "X = X_0 \\leftarrow X_1 \\leftarrow \\ldots \\leftarrow X_n = Y_m", "\\to \\ldots \\to Y_1 \\to Y_0 = Y", "$$", "where each morphism is a blowup in a closed point. We will prove the", "lemma by induction on $m$. The base case is $m = 0$; it is", "true in this case because we assumed that $Y$ is minimal", "hence this would mean $n = 0$, but $X$ is not isomorphic", "to $Y$, so this does not happen, i.e., there is nothing to check.", "\\medskip\\noindent", "Before we continue, note that $n + 1 = m + 1$ is equal to the number of", "irreducible components of the special fibre of $X_n = Y_m$ because", "both $X_k$ and $Y_k$ are irreducible. Another observation we", "will use below is that if $X' \\to X''$ is a morphism", "of regular proper models for $C$, then $X' \\to X''$ is an isomorphism", "over an open set of $X''$ whose complement is a finite set", "of closed points of the special fibre of $X''$, see", "Varieties, Lemma", "\\ref{varieties-lemma-modification-normal-iso-over-codimension-1}.", "In fact, any such $X' \\to X''$ is a sequence", "of blowing ups in closed points (Resolution of Surfaces, Lemma", "\\ref{resolve-lemma-proper-birational-regular-surfaces}) and the", "number of blowups is the difference in the number of irreducible", "components of the special fibres of $X'$ and $X''$.", "\\medskip\\noindent", "Let $E_i \\subset Y_i$, $m \\geq i \\geq 1$ be the curve which is contracted", "by the morphism $Y_i \\to Y_{i - 1}$. Let $i$ be the biggest index such", "that $E_i$ has multiplicity $> 1$ in the special fibre of $Y_i$.", "Then the further blowups $Y_m \\to \\ldots \\to Y_{i + 1} \\to Y_i$", "are isomorphisms over $E_i$ since otherwise $E_j$ for some $j > i$", "would have multiplicity $> 1$. Let $E \\subset Y_m$ be the inverse", "image of $E_i$. By what we just said $E \\subset Y_m$ is an", "exceptional curve of the first kind. Let $Y_m \\to Y'$ be the", "contraction of $E$ (which exists by Resolution of Surfaces, Lemma", "\\ref{resolve-lemma-contract-when-quasi-projective}). The morphism", "$Y_m \\to X$ has to contract $E$, because $X_k$ is reduced.", "Hence there are morphisms $Y' \\to Y$ and $Y' \\to X$ (by", "Resolution of Surfaces, Lemma", "\\ref{resolve-lemma-factor-through-contraction})", "which are compositions of at most $n - 1 = m - 1$ contractions", "of exceptional curves (see discussion above). We win by induction on $m$.", "Upshot: we may assume that the special fibres of all of", "the curves $X_i$ and $Y_i$ are reduced.", "\\medskip\\noindent", "Since the fibres of $X_i$ and $Y_i$ are reduced, it has to be the", "case that the blowups $X_i \\to X_{i - 1}$ and $Y_i \\to Y_{i - 1}$", "happen in closed points which are regular points of the special fibres.", "Namely, if $X''$ is a regular model for $C$ and if $x \\in X''$", "is a closed point of the special fibre, and", "$\\pi \\in \\mathfrak m_x^2$, then the exceptional fibre $E$ of the", "blowup $X' \\to X''$ at $x$ has multiplicity at least $2$ in the", "special fibre of $X'$ (local computation omitted).", "Hence $\\mathcal{O}_{X''_k, x} = \\mathcal{O}_{X'', x}/\\pi$ is", "regular (Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}) as claimed.", "In particular $x$ is a Cartier divisor on the unique", "irreducible component $Z'$ of $X''_k$ it lies on", "(Varieties, Lemma \\ref{varieties-lemma-regular-point-on-curve}).", "It follows that the strict transform $Z \\subset X'$ of $Z'$", "maps isomorphically to $Z'$", "(use Divisors, Lemmas \\ref{divisors-lemma-strict-transform} and", "\\ref{divisors-lemma-blow-up-effective-Cartier-divisor}).", "In other words, if an irreducible component $Z$ of $X_i$", "is not contracted under the map $X_i \\to X_j$ ($i > j$)", "then it maps isomorphically to its image.", "\\medskip\\noindent", "Now we are ready to prove the lemma.", "Let $E \\subset Y_m$ be the exceptional curve of the first kind", "which is contracted by the morphism $Y_m \\to Y_{m - 1}$. If $E$ is", "contracted by the morphism $Y_m = X_n \\to X$, then there is a factorization", "$Y_{m - 1} \\to X$ (Resolution of Surfaces, Lemma", "\\ref{resolve-lemma-factor-through-contraction})", "and moreover $Y_{m - 1} \\to X$ is a sequence of blowups", "in closed points (Resolution of Surfaces, Lemma", "\\ref{resolve-lemma-proper-birational-regular-surfaces}).", "In this case we lower $m$ and we win by induction.", "Finally, assume that $E$ is not contracted by the morphism $Y_m \\to X$.", "Then $E \\to X_k$ is surjective as $X_k$ is irreducible", "and by the above this means it is an isomorphism.", "Hence $X_k$ is isomorphic to a projective line as desired." ], "refs": [ "models-definition-numerical-type-model", "models-lemma-numerical-type-minimal-model", "models-lemma-genus-zero", "resolve-lemma-birational-regular-surfaces", "varieties-lemma-modification-normal-iso-over-codimension-1", "resolve-lemma-proper-birational-regular-surfaces", "resolve-lemma-contract-when-quasi-projective", "resolve-lemma-factor-through-contraction", "algebra-lemma-regular-ring-CM", "varieties-lemma-regular-point-on-curve", "divisors-lemma-strict-transform", "divisors-lemma-blow-up-effective-Cartier-divisor", "resolve-lemma-factor-through-contraction", "resolve-lemma-proper-birational-regular-surfaces" ], "ref_ids": [ 9278, 9250, 9226, 11710, 10979, 11709, 11706, 11699, 941, 11118, 8065, 8057, 11699, 11709 ] } ], "ref_ids": [] }, { "id": 9257, "type": "theorem", "label": "models-lemma-characterize-trivial", "categories": [ "models" ], "title": "models-lemma-characterize-trivial", "contents": [ "In Situation \\ref{situation-regular-model} let $d = \\gcd(m_1, \\ldots, m_n)$.", "If $\\mathcal{L}$ is an invertible $\\mathcal{O}_X$-module which", "\\begin{enumerate}", "\\item restricts to the trivial invertible module on $C$, and", "\\item has degree $0$ on each $C_i$,", "\\end{enumerate}", "then $\\mathcal{L}^{\\otimes d} \\cong \\mathcal{O}_X$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-regular-model-pic} we have", "$\\mathcal{L} \\cong \\mathcal{O}_X(\\sum a_i C_i)$ for some", "$a_i \\in \\mathbf{Z}$. The degree of $\\mathcal{L}|_{C_j}$", "is $\\sum_j a_j(C_i \\cdot C_j)$. In particular", "$(\\sum a_i C_i \\cdot \\sum a_i C_i) = 0$.", "Hence we see from Lemma \\ref{lemma-properties-form}", "that $(a_1, \\ldots, a_n) = q(m_1, \\ldots, m_n)$ for some", "$q \\in \\mathbf{Q}$. Thus $\\mathcal{L} = \\mathcal{O}_X(lD)$", "for some $l \\in \\mathbf{Z}$ where $D = \\sum (m_i/d) C_i$ is as in", "Lemma \\ref{lemma-multiple-fibre-normal-bundle}", "and we conclude." ], "refs": [ "models-lemma-regular-model-pic", "models-lemma-properties-form", "models-lemma-multiple-fibre-normal-bundle" ], "ref_ids": [ 9238, 9240, 9241 ] } ], "ref_ids": [] }, { "id": 9258, "type": "theorem", "label": "models-lemma-canonical-map-of-pic", "categories": [ "models" ], "title": "models-lemma-canonical-map-of-pic", "contents": [ "In Situation \\ref{situation-regular-model}", "let $T$ be the numerical type associated to $X$.", "There exists a canonical map", "$$", "\\Pic(C) \\to \\Pic(T)", "$$", "whose kernel is exactly those invertible modules on $C$", "which are the restriction of invertible modules $\\mathcal{L}$", "on $X$ with $\\deg_{C_i}(\\mathcal{L}|_{C_i}) = 0$ for", "$i = 1, \\ldots, n$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $w_i = [\\kappa_i : k]$ where", "$\\kappa_i = H^0(C_i, \\mathcal{O}_{C_i)})$ and recall", "that the degree of any invertible module on $C_i$ is divisible", "by $w_i$ (Varieties, Lemma \\ref{varieties-lemma-divisible}).", "Thus we can consider the map", "$$", "\\frac{\\deg}{w} : \\Pic(X) \\to \\mathbf{Z}^{\\oplus n}, \\quad", "\\mathcal{L} \\mapsto", "(\\frac{\\deg(\\mathcal{L}|_{C_1})}{w_1}, \\ldots,", "\\frac{\\deg(\\mathcal{L}|_{C_n})}{w_n})", "$$", "The image of $\\mathcal{O}_X(C_j)$ under this map is", "$$", "((C_j \\cdot C_1)/w_1, \\ldots, (C_j \\cdot C_n)/w_n) =", "(a_{1j}/w_1, \\ldots, a_{nj}/w_n)", "$$", "which is exactly the image of the $j$th basis vector under the map", "$(a_{ij}/w_i) : \\mathbf{Z}^{\\oplus n} \\to \\mathbf{Z}^{\\oplus n}$", "defining the Picard group of $T$, see", "Definition \\ref{definition-picard-group}.", "Thus the canonical map of the lemma comes from the commutative", "diagram", "$$", "\\xymatrix{", "\\mathbf{Z}^{\\oplus n} \\ar[r] \\ar[d]_{\\text{id}} &", "\\Pic(X) \\ar[r] \\ar[d]^{\\frac{\\deg}{w}} &", "\\Pic(C) \\ar[r] \\ar[d] & 0 \\\\", "\\mathbf{Z}^{\\oplus n} \\ar[r]^{(a_{ij}/w_i)} &", "\\mathbf{Z}^{\\oplus n} \\ar[r] &", "\\Pic(T) \\ar[r] & 0", "}", "$$", "with exact rows (top row by Lemma \\ref{lemma-regular-model-pic}).", "The description of the kernel is clear." ], "refs": [ "varieties-lemma-divisible", "models-definition-picard-group", "models-lemma-regular-model-pic" ], "ref_ids": [ 11112, 9276, 9238 ] } ], "ref_ids": [] }, { "id": 9259, "type": "theorem", "label": "models-lemma-sequence-torsion", "categories": [ "models" ], "title": "models-lemma-sequence-torsion", "contents": [ "In Situation \\ref{situation-regular-model} let $d = \\gcd(m_1, \\ldots, m_n)$", "and let $T$ be the numerical type associated to $X$.", "Let $h \\geq 1$ be an integer prime to $d$. There exists an exact sequence", "$$", "0 \\to \\Pic(X)[h] \\to \\Pic(C)[h] \\to \\Pic(T)[h]", "$$" ], "refs": [], "proofs": [ { "contents": [ "Taking $h$-torsion in the exact sequence of", "Lemma \\ref{lemma-regular-model-pic}", "we obtain the exactness of", "$0 \\to \\Pic(X)[h] \\to \\Pic(C)[h]$", "because $h$ is prime to $d$.", "Using the map Lemma \\ref{lemma-characterize-trivial}", "we get a map $\\Pic(C)[h] \\to \\Pic(T)[h]$", "which annihilates elements of $\\Pic(X)[h]$.", "Conversely, if $\\xi \\in \\Pic(C)[h]$", "maps to zero in $\\Pic(T)[h]$, then we can find", "an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$", "with $\\deg(\\mathcal{L}|_{C_i}) = 0$ for all $i$", "whose restriction to $C$ is $\\xi$.", "Then $\\mathcal{L}^{\\otimes h}$ is $d$-torsion by", "Lemma \\ref{lemma-characterize-trivial}.", "Let $d'$ be an integer such that $dd' \\equiv 1 \\bmod h$.", "Such an integer exists because $h$ and $d$ are coprime.", "Then $\\mathcal{L}^{\\otimes dd'}$ is an $h$-torsion", "invertible sheaf on $X$ whose restriction to $C$ is $\\xi$." ], "refs": [ "models-lemma-regular-model-pic", "models-lemma-characterize-trivial", "models-lemma-characterize-trivial" ], "ref_ids": [ 9238, 9257, 9257 ] } ], "ref_ids": [] }, { "id": 9260, "type": "theorem", "label": "models-lemma-torsion-embeds", "categories": [ "models" ], "title": "models-lemma-torsion-embeds", "contents": [ "In Situation \\ref{situation-regular-model} let $h$ be an integer", "prime to the characteristic of $k$. Then the map", "$$", "\\Pic(X)[h] \\longrightarrow \\Pic((X_k)_{red})[h]", "$$", "is injective." ], "refs": [], "proofs": [ { "contents": [ "Observe that $X \\times_{\\Spec(R)} \\Spec(R/\\pi^n)$ is a finite", "order thickening of $(X_k)_{red}$ (this follows for example from", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-power-ideal-kills-sheaf}).", "Thus the canonical map", "$\\Pic(X \\times_{\\Spec(R)} \\Spec(R/\\pi^n)) \\to \\Pic((X_k)_{red})$", "identifies $h$ torsion by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-torsion-pic-thickening}", "and our assumption on $h$.", "Thus if $\\mathcal{L}$ is an $h$-torsion invertible sheaf on $X$", "which restricts to the trivial sheaf on $(X_k)_{red}$ then", "$\\mathcal{L}$ restricts to the trivial sheaf on", "$X \\times_{\\Spec(R)} \\Spec(R/\\pi^n)$ for all $n$.", "We find", "\\begin{align*}", "H^0(X, \\mathcal{L})^\\wedge", "& =", "\\lim H^0(X \\times_{\\Spec(R)} \\Spec(R/\\pi^n),", "\\mathcal{L}|_{X \\times_{\\Spec(R)} \\Spec(R/\\pi^n)}) \\\\", "& \\cong", "\\lim H^0(X \\times_{\\Spec(R)} \\Spec(R/\\pi^n),", "\\mathcal{O}_{X \\times_{\\Spec(R)} \\Spec(R/\\pi^n)}) \\\\", "& =", "R^\\wedge", "\\end{align*}", "using the theorem on formal functions", "(Cohomology of Schemes, Theorem \\ref{coherent-theorem-formal-functions})", "for the first and last equality and for example", "More on Algebra, Lemma \\ref{more-algebra-lemma-isomorphic-completions}", "for the middle isomorphism. Since $H^0(X, \\mathcal{L})$ is a finite", "$R$-module and $R$ is a discrete valuation ring, this means that", "$H^0(X, \\mathcal{L})$ is free of rank $1$ as an $R$-module.", "Let $s \\in H^0(X, \\mathcal{L})$ be a basis element.", "Then tracing back through the isomorphisms above we see", "that $s|_{X \\times_{\\Spec(R)} \\Spec(R/\\pi^n)}$ is a trivialization", "for all $n$. Since the vanishing locus of $s$ is closed in $X$", "and $X \\to \\Spec(R)$ is proper we conclude that the vanishing", "locus of $s$ is empty as desired." ], "refs": [ "coherent-lemma-power-ideal-kills-sheaf", "more-morphisms-lemma-torsion-pic-thickening", "coherent-theorem-formal-functions", "more-algebra-lemma-isomorphic-completions" ], "ref_ids": [ 3320, 13687, 3278, 10423 ] } ], "ref_ids": [] }, { "id": 9261, "type": "theorem", "label": "models-lemma-etale-local-at-worst-nodal", "categories": [ "models" ], "title": "models-lemma-etale-local-at-worst-nodal", "contents": [ "Let $R$ be a discrete valuation ring. Let $X$ be a scheme which is", "at-worst-nodal of relative dimension $1$ over $R$.", "Let $x \\in X$ be a point of the special fibre", "of $X$ over $R$. Then there exists a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] &", "U \\ar[r] \\ar[d] \\ar[l] &", "\\Spec(A) \\ar[dl] \\\\", "\\Spec(R) &", "\\Spec(R') \\ar[l]", "}", "$$", "where $R \\subset R'$ is an \\'etale extension of discrete valuation rings,", "the morphism $U \\to X$ is \\'etale, the morphism $U \\to \\Spec(A)$ is \\'etale,", "there is a point $x' \\in U$ mapping to $x$, and", "$$", "A = R'[u, v]/(uv)", "\\quad\\text{or}\\quad", "A = R'[u, v]/(uv - \\pi^n)", "$$", "where $n \\geq 0$ and $\\pi \\in R'$ is a uniformizer." ], "refs": [], "proofs": [ { "contents": [ "We have already proved this lemma in much greater generality, see", "Algebraic Curves, Lemma \\ref{curves-lemma-etale-local-structure-nodal-family}.", "All we have to do here is to translate the statement", "given there into the statement given above.", "\\medskip\\noindent", "First, if the morphism $X \\to \\Spec(R)$ is smooth at $x$,", "then we can find an \\'etale morphism $U \\to \\mathbf{A}^1_R = \\Spec(R[u])$ for", "some affine open neighbourhood $U \\subset X$ of $x$. This is", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-etale-over-affine-space}.", "After replacing the coordinate $u$ by $u + 1$ if necessary, we may", "assume that $x$ maps to a point in the standard open", "$D(u) \\subset \\mathbf{A}^1_R$. Then $D(u) = \\Spec(A)$ with", "$A = R[u, v]/(uv - 1)$ and we see that the result is true in this case.", "\\medskip\\noindent", "Next, assume that $x$ is a singular point of the fibre. Then we may apply", "Algebraic Curves, Lemma \\ref{curves-lemma-etale-local-structure-nodal-family}", "to get a diagram", "$$", "\\xymatrix{", "X \\ar[d] &", "U \\ar[rr] \\ar[l] \\ar[rd] & &", "W \\ar[r] \\ar[ld] &", "\\Spec(\\mathbf{Z}[u, v, a]/(uv - a)) \\ar[d] \\\\", "\\Spec(R) & &", "V \\ar[ll] \\ar[rr] & & \\Spec(\\mathbf{Z}[a])", "}", "$$", "with all the properties mentioned in the statement of the cited lemma.", "Let $x' \\in U$ be the point mapping to $x$ promised by the lemma.", "First we shrink $V$ to an affine neighbourhood of the image of $x'$.", "Say $V = \\Spec(R')$. Then $R \\to R'$ is \\'etale. Since $R$ is a", "discrete valuation ring, we see that $R'$ is a finite", "product of quasi-local Dedekind domains (use", "More on Algebra, Lemma \\ref{more-algebra-lemma-Dedekind-etale-extension}).", "Hence (for example using prime avoidance) we find a standard", "open $D(f) \\subset V = \\Spec(R')$ containing the image of $x'$", "such that $R'_f$ is a discrete valuation ring.", "Replacing $R'$ by $R'_f$ we reach the situation where", "$V = \\Spec(R')$ with $R \\subset R'$ an \\'etale extension of", "discrete valuation rings (extensions of discrete valuation rings", "are defined in More on Algebra, Definition", "\\ref{more-algebra-definition-extension-discrete-valuation-rings}).", "\\medskip\\noindent", "The morphism $V \\to \\Spec(\\mathbf{Z}[a])$ is determined by", "the image $h$ of $a$ in $R'$. Then $W = \\Spec(R'[u, v]/(uv - h))$.", "Thus the lemma holds with", "$A = R'[u, v]/(uv - h)$. If $h = 0$ then we clearly", "obtain the first case mentioned in the lemma. If $h \\not = 0$", "then we may write $h = \\epsilon \\pi^n$ for some $n \\geq 0$", "where $\\epsilon$ is a unit of $R'$. Changing coordinates", "$u_{new} = \\epsilon u$ and $v_{new} = v$ we obtain the second", "isomorphism type of $A$ listed in the lemma." ], "refs": [ "curves-lemma-etale-local-structure-nodal-family", "morphisms-lemma-smooth-etale-over-affine-space", "curves-lemma-etale-local-structure-nodal-family", "more-algebra-lemma-Dedekind-etale-extension", "more-algebra-definition-extension-discrete-valuation-rings" ], "ref_ids": [ 6327, 5377, 6327, 10054, 10640 ] } ], "ref_ids": [] }, { "id": 9262, "type": "theorem", "label": "models-lemma-blowup-at-worst-nodal", "categories": [ "models" ], "title": "models-lemma-blowup-at-worst-nodal", "contents": [ "Let $R$ be a discrete valuation ring. Let $X$ be a quasi-compact scheme which", "is at-worst-nodal of relative dimension $1$ with smooth generic fibre over $R$.", "Then there exists $m \\geq 0$ and a sequence", "$$", "X_m \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "such that", "\\begin{enumerate}", "\\item $X_{i + 1} \\to X_i$ is the blowing up of a closed point", "$x_i$ where $X_i$ is singular,", "\\item $X_i \\to \\Spec(R)$ is at-worst-nodal of relative dimension $1$,", "\\item $X_m$ is regular.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is quasi-compact we see that the special fibre $X_k$ is quasi-compact.", "Since the singularities of $X_k$ are at-worst-nodal, we see", "that $X_k$ has a finite number of nodes and is otherwise", "smooth over $k$. As $X \\to \\Spec(R)$ is flat with smooth generic", "fibre it follows that $X$ is smooth over $R$ except at the", "finite number of nodes of $X_k$", "(use Morphisms, Lemma \\ref{morphisms-lemma-smooth-at-point}).", "It follows that $X$ is regular at every point except for possibly", "the nodes of its special fibre", "(see Algebra, Lemma \\ref{algebra-lemma-regular-goes-up}).", "Let $x \\in X$ be such a node.", "Choose a diagram", "$$", "\\xymatrix{", "X \\ar[d] &", "U \\ar[r] \\ar[d] \\ar[l] &", "\\Spec(A) \\ar[dl] \\\\", "\\Spec(R) &", "\\Spec(R') \\ar[l]", "}", "$$", "as in Lemma \\ref{lemma-etale-local-at-worst-nodal}.", "Observe that the case $A = R'[u, v]/(uv)$ cannot", "occur, as this would mean that the generic fibre of", "$X/R$ is singular (tiny detail omitted). Thus $A = R'[u, v]/(uv - \\pi^n)$", "for some $n \\geq 0$. Since $x$ is a singular point,", "we have $n \\geq 2$, see discussion in", "Example \\ref{example-blowup}.", "\\medskip\\noindent", "After shrinking $U$ we may assume there is", "a unique point $u \\in U$ mapping to $x$.", "Let $w \\in \\Spec(A)$ be the image of $u$.", "We may also assume that $u$ is the unique point of $U$", "mapping to $w$.", "Since the two horizontal arrows are \\'etale", "we see that $u$, viewed as a closed subscheme of $U$,", "is the scheme theoretic inverse image of $x \\in X$ and the", "scheme theoretic inverse image of $w \\in \\Spec(A)$.", "Since blowing up commutes with flat base change", "(Divisors, Lemma \\ref{divisors-lemma-flat-base-change-blowing-up})", "we find a commutative diagram", "$$", "\\xymatrix{", "X' \\ar[d] &", "U' \\ar[l] \\ar[d] \\ar[r] &", "W' \\ar[d] \\\\", "X & U \\ar[l] \\ar[r] & \\Spec(A)", "}", "$$", "with cartesian squares where the vertical arrows are the blowing", "up of $x, u, w$ in $X, U, \\Spec(A)$. The scheme $W'$ was described", "in Example \\ref{example-blowup}. We saw there that $W'$", "at-worst-nodal of relative dimension $1$ over $R'$. Thus", "$W'$ is at-worst-nodal of relative dimension $1$ over $R$", "(Algebraic Curves, Lemma \\ref{curves-lemma-nodal-family-postcompose-etale}).", "Hence $U'$ is at-worst-nodal of relative dimension $1$ over $R$ (see", "Algebraic Curves, Lemma \\ref{curves-lemma-nodal-family-etale-local-source}).", "Since $X' \\to X$ is an isomorphism over the complement of $x$,", "we conclude the same thing is true of $X'/R$ (by", "Algebraic Curves, Lemma \\ref{curves-lemma-nodal-family-etale-local-source}", "again).", "\\medskip\\noindent", "Finally, we need to argue that after doing a finite number", "of these blowups we arrive at a regular model $X_m$.", "This is rather clear because the ``invariant'' $n$ decreases by $2$", "under the blowup described above, see computation in", "Example \\ref{example-blowup}.", "However, as we want to avoid precisely defining this invariant", "and establishing its properties, we in stead argue as follows.", "If $n = 2$, then $W'$ is regular and hence $X'$", "is regular at all points lying over $x$ and we have", "decreased the number of singular points of $X$ by $1$.", "If $n > 2$, then the unique singular point $w'$ of $W'$ lying over $w$", "has $\\kappa(w) = \\kappa(w')$. Hence $U'$ has a unique", "singular point $u'$ lying over $u$ with $\\kappa(u) = \\kappa(u')$.", "Clearly, this implies that $X'$ has a unique singular point $x'$", "lying over $x$, namely the image of $u'$. Thus we can", "argue exactly as above that we get a commutative diagram", "$$", "\\xymatrix{", "X'' \\ar[d] &", "U'' \\ar[l] \\ar[d] \\ar[r] &", "W'' \\ar[d] \\\\", "X' & U' \\ar[l] \\ar[r] & W'", "}", "$$", "with cartesian squares where the vertical arrows are the blowing", "up of $x', u', w'$ in $X', U', W'$. Continuing like this we get", "a compatible sequence of blowups which stops after", "$\\lfloor n/2 \\rfloor$ steps. At the completion of this process", "the scheme $X^{(\\lfloor n/2 \\rfloor)}$ will have one fewer", "singular point than $X$. Induction on the number of singular", "points completes the proof." ], "refs": [ "morphisms-lemma-smooth-at-point", "algebra-lemma-regular-goes-up", "models-lemma-etale-local-at-worst-nodal", "divisors-lemma-flat-base-change-blowing-up", "curves-lemma-nodal-family-postcompose-etale", "curves-lemma-nodal-family-etale-local-source", "curves-lemma-nodal-family-etale-local-source" ], "ref_ids": [ 5335, 1369, 9261, 8053, 6322, 6323, 6323 ] } ], "ref_ids": [] }, { "id": 9263, "type": "theorem", "label": "models-lemma-blowdown-at-worst-nodal", "categories": [ "models" ], "title": "models-lemma-blowdown-at-worst-nodal", "contents": [ "Let $R$ be a discrete valuation ring with fraction field $K$", "and residue field $k$. Assume $X \\to \\Spec(R)$ is", "at-worst-nodal of relative dimension $1$ over $R$.", "Let $X \\to X'$ be the contraction of an", "exceptional curve $E \\subset X$ of the first kind.", "Then $X'$ is at-worst-nodal of relative dimension $1$ over $R$." ], "refs": [], "proofs": [ { "contents": [ "Namely, let $x' \\in X'$ be the image of $E$.", "Then the only issue is to see that $X' \\to \\Spec(R)$", "is at-worst-nodal of relative dimension $1$", "in a neighbourhood of $x'$.", "The closed fibre of $X \\to \\Spec(R)$ is reduced, hence", "$\\pi \\in R$ vanishes to order $1$ on $E$.", "This immediately implies that", "$\\pi$ viewed as an element of", "$\\mathfrak m_{x'} \\subset \\mathcal{O}_{X', x'}$ but", "is not in $\\mathfrak m_{x'}^2$.", "Since $\\mathcal{O}_{X', x'}$ is regular of", "dimension $2$ (by definition of contractions in", "Resolution of Surfaces, Section \\ref{resolve-section-minus-one}),", "this implies that $\\mathcal{O}_{X'_k, x'}$", "is regular of dimension $1$", "(Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}).", "On the other hand, the curve $E$ has to meet at", "least one other component, say $C$ of the closed fibre $X_k$.", "Say $x \\in E \\cap C$. Then $x$ is a node of the special", "fibre $X_k$ and hence $\\kappa(x)/k$ is finite separable,", "see Algebraic Curves, Lemma \\ref{curves-lemma-nodal}.", "Since $x \\mapsto x'$ we conclude that $\\kappa(x')/k$", "is finite separable.", "By Algebra, Lemma \\ref{algebra-lemma-separable-smooth}", "we conclude that $X'_k \\to \\Spec(k)$ is smooth", "in an open neighbourhood of $x'$.", "Combined with flatness, this proves that", "$X' \\to \\Spec(R)$ is smooth in a neighbourhood of $x'$", "(Morphisms, Lemma \\ref{morphisms-lemma-smooth-at-point}).", "This finishes the proof as a smooth morphism of", "relative dimension $1$ is at-worst-nodal of relative", "dimension $1$", "(Algebraic Curves, Lemma \\ref{curves-lemma-smooth-relative-dimension-1})." ], "refs": [ "algebra-lemma-regular-ring-CM", "curves-lemma-nodal", "algebra-lemma-separable-smooth", "morphisms-lemma-smooth-at-point", "curves-lemma-smooth-relative-dimension-1" ], "ref_ids": [ 941, 6309, 1225, 5335, 6318 ] } ], "ref_ids": [] }, { "id": 9264, "type": "theorem", "label": "models-lemma-semistable", "categories": [ "models" ], "title": "models-lemma-semistable", "contents": [ "Let $R$ be a discrete valuation ring with fraction field $K$.", "Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$.", "The following are equivalent", "\\begin{enumerate}", "\\item there exists a proper model of $C$ which is", "at-worst-nodal of relative dimension $1$ over $R$,", "\\item there exists a minimal model of $C$ which is at-worst-nodal", "of relative dimension $1$ over $R$, and", "\\item any minimal model of $C$ is at-worst-nodal", "of relative dimension $1$ over $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To make sense out of this statement, recall that a", "minimal model is defined as a regular proper model", "without exceptional curves of the first kind", "(Definition \\ref{definition-minimal-model}),", "that minimal models exist", "(Proposition \\ref{proposition-exists-minimal-model}), and", "that minimal models are unique if the genus", "of $C$ is $> 0$ (Lemma \\ref{lemma-minimal-model-unique}).", "Keeping this in mind the implications (2) $\\Rightarrow$ (1)", "and (3) $\\Rightarrow$ (2) are clear.", "\\medskip\\noindent", "Assume (1). Let $X$ be a proper model of $C$ which is", "at-worst-nodal of relative dimension $1$ over $R$.", "Applying Lemma \\ref{lemma-blowup-at-worst-nodal}", "we see that we may assume $X$ is regular as well.", "Let", "$$", "X = X_m \\to X_{m - 1} \\to \\ldots \\to X_1 \\to X_0", "$$", "be as in Lemma \\ref{lemma-pre-exists-minimal-model}.", "By Lemma \\ref{lemma-blowdown-at-worst-nodal} and induction", "this implies $X_0$ is at-worst-nodal of relative dimension $1$ over $R$.", "\\medskip\\noindent", "To finish the proof we have to show that (2) implies (3).", "This is clear if the genus of $C$ is $> 0$, since then", "the minimal model is unique (see discussion above).", "On the other hand, if the minimal model is not unique, then", "the morphism $X \\to \\Spec(R)$ is smooth for any minimal model", "as its special fibre will be isomorphic to $\\mathbf{P}^1_k$", "by Lemma \\ref{lemma-nonuniqueness}." ], "refs": [ "models-definition-minimal-model", "models-proposition-exists-minimal-model", "models-lemma-minimal-model-unique", "models-lemma-blowup-at-worst-nodal", "models-lemma-pre-exists-minimal-model", "models-lemma-blowdown-at-worst-nodal", "models-lemma-nonuniqueness" ], "ref_ids": [ 9277, 9268, 9245, 9262, 9234, 9263, 9256 ] } ], "ref_ids": [] }, { "id": 9265, "type": "theorem", "label": "models-lemma-good", "categories": [ "models" ], "title": "models-lemma-good", "contents": [ "Let $R$ be a discrete valuation ring with fraction field $K$.", "Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$.", "The following are equivalent", "\\begin{enumerate}", "\\item there exists a proper smooth model for $C$,", "\\item there exists a minimal model for $C$ which is smooth over $R$,", "\\item any minimal model is smooth over $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $X$ is a smooth proper model, then the special fibre is", "connected (Lemma \\ref{lemma-regular-model-connected})", "and smooth, hence irreducible. This immediately implies that", "it is minimal. Thus (1) implies (2). ", "To finish the proof we have to show that (2) implies (3).", "This is clear if the genus of $C$ is $> 0$, since then", "the minimal model is unique (Lemma \\ref{lemma-minimal-model-unique}).", "On the other hand, if the minimal model is not unique, then", "the morphism $X \\to \\Spec(R)$ is smooth for any minimal model", "as its special fibre will be isomorphic to $\\mathbf{P}^1_k$", "by Lemma \\ref{lemma-nonuniqueness}." ], "refs": [ "models-lemma-regular-model-connected", "models-lemma-minimal-model-unique", "models-lemma-nonuniqueness" ], "ref_ids": [ 9237, 9245, 9256 ] } ], "ref_ids": [] }, { "id": 9266, "type": "theorem", "label": "models-proposition-classify-subgraphs", "categories": [ "models" ], "title": "models-proposition-classify-subgraphs", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$.", "Let $I \\subset \\{1, \\ldots, n\\}$ be a proper subset of cardinality $\\geq 2$", "consisting of $(-2)$-indices such that there", "does not exist a nonempty proper subset $I' \\subset I$", "with $a_{i'i} = 0$ for $i' \\in I$, $i \\in I \\setminus I'$.", "Then up to reordering the $m_i$'s, $a_{ij}$'s, $w_i$'s", "for $i, j \\in I$ are as listed in", "Lemmas \\ref{lemma-two-by-two},", "\\ref{lemma-three-by-three},", "\\ref{lemma-four-by-four},", "\\ref{lemma-D4},", "\\ref{lemma-five-by-five},", "\\ref{lemma-D5},", "\\ref{lemma-long},", "\\ref{lemma-Dn},", "\\ref{lemma-E6},", "\\ref{lemma-E7}, or", "\\ref{lemma-E8}." ], "refs": [ "models-lemma-two-by-two", "models-lemma-three-by-three", "models-lemma-four-by-four", "models-lemma-D4", "models-lemma-five-by-five", "models-lemma-D5", "models-lemma-long", "models-lemma-Dn", "models-lemma-E6", "models-lemma-E7", "models-lemma-E8" ], "proofs": [ { "contents": [ "This follows from the discussion above; see discussion at the", "start of Section \\ref{section-classify-proper-subgraphs}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9210, 9211, 9212, 9213, 9214, 9216, 9217, 9218, 9219, 9222, 9223 ] }, { "id": 9267, "type": "theorem", "label": "models-proposition-bound-picard-group", "categories": [ "models" ], "title": "models-proposition-bound-picard-group", "contents": [ "Let $g \\geq 2$. For every numerical type $T$ of genus $g$", "and prime number $\\ell > 768g$ we have", "$$", "\\dim_{\\mathbf{F}_\\ell} \\Pic(T)[\\ell] \\leq g", "$$", "where $\\Pic(T)$ is as in Definition \\ref{definition-picard-group}.", "If $T$ is minimal, then we even have", "$$", "\\dim_{\\mathbf{F}_\\ell} \\Pic(T)[\\ell] \\leq g_{top} \\leq g", "$$", "where $g_{top}$ as in Definition \\ref{definition-top-genus}." ], "refs": [ "models-definition-picard-group", "models-definition-top-genus" ], "proofs": [ { "contents": [ "Say $T$ is given by $n, m_i, a_{ij}, w_i, g_i$.", "If $T$ is not minimal, then there exists a $(-1)$-index.", "After replacing $T$ by an equivalent type we may assume", "$n$ is a $(-1)$-index. Applying Lemma \\ref{lemma-contract-picard-group}", "we find $\\Pic(T) \\subset \\Pic(T')$ where $T'$", "is a numerical type of genus $g$ (Lemma \\ref{lemma-contract})", "with $n - 1$ indices. Thus we conclude by induction on $n$", "provided we prove the lemma for minimal numerical types.", "\\medskip\\noindent", "Assume that $T$ is a minimal numerical type of genus $\\geq 2$.", "Observe that $g_{top} \\leq g$ by Lemma \\ref{lemma-genus-nonnegative}.", "If $A = (a_{ij})$ then since $\\Pic(T) \\subset \\Coker(A)$", "by Lemma \\ref{lemma-picard-T-and-A}. Thus it suffices to prove", "the lemma for $\\Coker(A)$.", "By Lemma \\ref{lemma-bound-wm} we see that $m_i|a_{ij}| \\leq 768g$ for", "all $i, j$.", "Hence the result by Lemma \\ref{lemma-recurring-symmetric-integer}." ], "refs": [ "models-lemma-contract-picard-group", "models-lemma-contract", "models-lemma-genus-nonnegative", "models-lemma-picard-T-and-A", "models-lemma-bound-wm", "models-lemma-recurring-symmetric-integer" ], "ref_ids": [ 9208, 9201, 9204, 9207, 9230, 9196 ] } ], "ref_ids": [ 9276, 9273 ] }, { "id": 9268, "type": "theorem", "label": "models-proposition-exists-minimal-model", "categories": [ "models" ], "title": "models-proposition-exists-minimal-model", "contents": [ "Let $C$ be a smooth projective curve over $K$ with", "$H^0(C, \\mathcal{O}_C) = K$. A minimal model exists." ], "refs": [], "proofs": [ { "contents": [ "Choose a closed immersion $C \\to \\mathbf{P}^n_K$. Let", "$X$ be the scheme theoretic image of $C \\to \\mathbf{P}^n_R$.", "Then $X \\to \\Spec(R)$ is a projective model of $C$ by", "Lemma \\ref{lemma-closure-is-model}.", "By Lemma \\ref{lemma-regular} there exists a resolution", "of singularities $X' \\to X$ and $X'$ is a model for $C$.", "Then $X' \\to \\Spec(R)$ is proper as a composition of proper morphisms.", "Then we may apply Lemma \\ref{lemma-pre-exists-minimal-model}", "to obtain a minimal model." ], "refs": [ "models-lemma-closure-is-model", "models-lemma-regular", "models-lemma-pre-exists-minimal-model" ], "ref_ids": [ 9231, 9233, 9234 ] } ], "ref_ids": [] }, { "id": 9286, "type": "theorem", "label": "spaces-groupoids-lemma-restrict-relation", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-restrict-relation", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $U$ be an algebraic space over $B$.", "Let $j : R \\to U \\times_B U$ be a pre-relation.", "Let $g : U' \\to U$ be a morphism of algebraic spaces over $B$.", "Finally, set", "$$", "R' = (U' \\times_B U')\\times_{U \\times_B U} R", "\\xrightarrow{j'}", "U' \\times_B U'", "$$", "Then $j'$ is a pre-relation on $U'$ over $B$.", "If $j$ is a relation, then $j'$ is a relation.", "If $j$ is a pre-equivalence relation, then $j'$ is a pre-equivalence relation.", "If $j$ is an equivalence relation, then $j'$ is an equivalence relation." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9287, "type": "theorem", "label": "spaces-groupoids-lemma-pre-equivalence-equivalence-relation-points", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-pre-equivalence-equivalence-relation-points", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $j : R \\to U \\times_B U$ be a pre-relation of algebraic spaces over $B$.", "Consider the relation on $|U|$ defined by the rule", "$$", "x \\sim y", "\\Leftrightarrow", "\\exists\\ r \\in |R| :", "t(r) = x,", "s(r) = y.", "$$", "If $j$ is a pre-equivalence relation then this is an equivalence relation." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $x \\sim y$ and $y \\sim z$.", "Pick $r \\in |R|$ with $t(r) = x$, $s(r) = y$ and", "pick $r' \\in |R|$ with $t(r') = y$, $s(r') = z$.", "We may pick a field $K$ such that $r$ and $r'$ can be", "represented by morphisms $r, r' : \\Spec(K) \\to R$", "with $s \\circ r = t \\circ r'$.", "Denote $x = t \\circ r$, $y = s \\circ r = t \\circ r'$,", "and $z = s \\circ r'$, so $x, y, z : \\Spec(K) \\to U$.", "By construction $(x, y) \\in j(R(K))$ and", "$(y, z) \\in j(R(K))$. Since $j$ is a pre-equivalence relation", "we see that also $(x, z) \\in j(R(K))$.", "This clearly implies that $x \\sim z$.", "\\medskip\\noindent", "The proof that $\\sim$ is reflexive and symmetric is omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9288, "type": "theorem", "label": "spaces-groupoids-lemma-base-change-group-space", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-base-change-group-space", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(G, m)$ be a group algebraic space over $B$.", "Let $B' \\to B$ be a morphism of algebraic spaces.", "The pullback $(G_{B'}, m_{B'})$ is a group algebraic space over $B'$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9289, "type": "theorem", "label": "spaces-groupoids-lemma-group-scheme-separated", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-group-scheme-separated", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $G$ be a group algebraic space over $B$.", "Then $G \\to B$ is separated (resp.\\ quasi-separated, resp.\\ locally separated)", "if and only if the identity morphism $e : B \\to G$ is a closed immersion", "(resp.\\ quasi-compact, resp.\\ an immersion)." ], "refs": [], "proofs": [ { "contents": [ "We recall that by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-section-immersion}", "we have that $e$ is a closed immersion (resp.\\ quasi-compact, resp.\\ an", "immersion) if $G \\to B$ is separated (resp.\\ quasi-separated, resp.\\ locally", "separated).", "For the converse, consider the diagram", "$$", "\\xymatrix{", "G \\ar[r]_-{\\Delta_{G/B}} \\ar[d] &", "G \\times_B G \\ar[d]^{(g, g') \\mapsto m(i(g), g')} \\\\", "B \\ar[r]^e & G", "}", "$$", "It is an exercise in the functorial point of view in algebraic geometry", "to show that this diagram is cartesian. In other words, we see that", "$\\Delta_{G/B}$ is a base change of $e$. Hence if $e$ is a", "closed immersion (resp.\\ quasi-compact, resp.\\ an immersion) so is", "$\\Delta_{G/B}$, see", "Spaces, Lemma \\ref{spaces-lemma-base-change-immersions}", "(resp.\\ Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-quasi-compact},", "resp.\\ Spaces, Lemma \\ref{spaces-lemma-base-change-immersions})." ], "refs": [ "spaces-lemma-base-change-immersions", "spaces-morphisms-lemma-base-change-quasi-compact", "spaces-lemma-base-change-immersions" ], "ref_ids": [ 8161, 4738, 8161 ] } ], "ref_ids": [] }, { "id": 9290, "type": "theorem", "label": "spaces-groupoids-lemma-group-scheme-unramified-or-lqf", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-group-scheme-unramified-or-lqf", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $G$ be a group algebraic space over $B$. Assume $G \\to B$", "is locally of finite type. Then", "$G \\to B$ is unramified (resp.\\ locally quasi-finite)", "if and only if $G \\to B$ is unramified (resp.\\ quasi-finite)", "at $e(b)$ for all $b \\in |B|$." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-where-unramified}", "(resp.\\ Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-quasi-finite-locus})", "there is a maximal open subspace $U \\subset G$ such that $U \\to B$ is", "unramified (resp.\\ locally quasi-finite) and formation of $U$", "commutes with base change. Thus we reduce to the case where", "$B = \\Spec(k)$ is the spectrum of a field.", "Let $g \\in G(K)$ be a point with values in an extension $K/k$.", "Then to check whether or not $g$ is in $U$, we may base", "change to $K$. Hence it suffices to show", "$$", "G \\to \\Spec(k)\\text{ is unramified at }e", "\\Leftrightarrow", "G \\to \\Spec(k)\\text{ is unramified at }g", "$$", "for a $k$-rational point $g$ (resp.\\ similarly for", "quasi-finite at $g$ and $e$). Since translation by $g$", "is an automorphism of $G$ over $k$ this is clear." ], "refs": [ "spaces-morphisms-lemma-where-unramified", "spaces-morphisms-lemma-base-change-quasi-finite-locus" ], "ref_ids": [ 4903, 4830 ] } ], "ref_ids": [] }, { "id": 9291, "type": "theorem", "label": "spaces-groupoids-lemma-open-over-which-unramified-or-lqf", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-open-over-which-unramified-or-lqf", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $G$ be a group algebraic space over $B$. Assume $G \\to B$", "is locally of finite type.", "\\begin{enumerate}", "\\item There exists a maximal open subspace $U \\subset B$", "such that $G_U \\to U$ is unramified and formation of $U$", "commutes with base change.", "\\item There exists a maximal open subspace $U \\subset B$", "such that $G_U \\to U$ is locally quasi-finite and formation of $U$", "commutes with base change.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-where-unramified}", "(resp.\\ Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-quasi-finite-locus})", "there is a maximal open subspace $W \\subset G$ such that $W \\to B$ is", "unramified (resp.\\ locally quasi-finite). Moreover formation of $W$", "commutes with base change.", "By Lemma \\ref{lemma-group-scheme-unramified-or-lqf}", "we see that $U = e^{-1}(W)$ in either case." ], "refs": [ "spaces-morphisms-lemma-where-unramified", "spaces-morphisms-lemma-base-change-quasi-finite-locus", "spaces-groupoids-lemma-group-scheme-unramified-or-lqf" ], "ref_ids": [ 4903, 4830, 9290 ] } ], "ref_ids": [] }, { "id": 9292, "type": "theorem", "label": "spaces-groupoids-lemma-free-action", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-free-action", "contents": [ "Situation as in", "Definition \\ref{definition-free-action},", "The action $a$ is free if and only if", "$$", "G \\times_B X \\to X \\times_B X, \\quad (g, x) \\mapsto (a(g, x), x)", "$$", "is a monomorphism of algebraic spaces." ], "refs": [ "spaces-groupoids-definition-free-action" ], "proofs": [ { "contents": [ "Immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9343 ] }, { "id": 9293, "type": "theorem", "label": "spaces-groupoids-lemma-characterize-trivial-pseudo-torsors", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-characterize-trivial-pseudo-torsors", "contents": [ "In the situation of", "Definition \\ref{definition-pseudo-torsor}.", "\\begin{enumerate}", "\\item The algebraic space $X$ is a pseudo $G$-torsor if and only if for", "every scheme $T$ over $B$ the set $X(T)$ is either empty or the action", "of the group $G(T)$ on $X(T)$ is simply transitive.", "\\item A pseudo $G$-torsor $X$ is trivial if and only if the morphism", "$X \\to B$ has a section.", "\\end{enumerate}" ], "refs": [ "spaces-groupoids-definition-pseudo-torsor" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9344 ] }, { "id": 9294, "type": "theorem", "label": "spaces-groupoids-lemma-torsor", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-torsor", "contents": [ "Let $S$ be a scheme.", "Let $(G, m)$ be a group algebraic space over $S$.", "Let $X$ be an algebraic space over $S$, and let", "$a : G \\times_S X \\to X$ be an action of $G$ on $X$.", "Then", "$X$ is a $G$-torsor in the $fppf$-topology in the sense of", "Definition \\ref{definition-principal-homogeneous-space}", "if and only if", "$X$ is a $G$-torsor on $(\\Sch/S)_{fppf}$", "in the sense of", "Cohomology on Sites, Definition \\ref{sites-cohomology-definition-torsor}." ], "refs": [ "spaces-groupoids-definition-principal-homogeneous-space", "sites-cohomology-definition-torsor" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9345, 4411 ] }, { "id": 9295, "type": "theorem", "label": "spaces-groupoids-lemma-pseudo-torsor-implications", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-pseudo-torsor-implications", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $G$ be a group algebraic space over $B$.", "Let $X$ be a pseudo $G$-torsor over $B$.", "Assume $G$ and $X$ locally of finite type over $B$.", "\\begin{enumerate}", "\\item If $G \\to B$ is unramified, then $X \\to B$ is unramified.", "\\item If $G \\to B$ is locally quasi-finite, then $X \\to B$ is", "locally quasi-finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-where-unramified}", "we reduce to the case where $B$ is the spectrum of a field.", "If $X$ is empty, then the result holds. If $X$ is nonempty,", "then after increasing the field, we may assume $X$ has a point.", "Then $G \\cong X$ and the result holds.", "\\medskip\\noindent", "The proof of (2) works in exactly the same way using", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-quasi-finite-locus}." ], "refs": [ "spaces-morphisms-lemma-where-unramified", "spaces-morphisms-lemma-base-change-quasi-finite-locus" ], "ref_ids": [ 4903, 4830 ] } ], "ref_ids": [] }, { "id": 9296, "type": "theorem", "label": "spaces-groupoids-lemma-pullback-equivariant", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-pullback-equivariant", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $G$ be a group algebraic space over $B$.", "Let $f : X \\to Y$ be a $G$-equivariant morphism between", "algebraic spaces over $B$ endowed with $G$-actions.", "Then pullback $f^*$ given by", "$(\\mathcal{F}, \\alpha) \\mapsto (f^*\\mathcal{F}, (1_G \\times f)^*\\alpha)$", "defines a functor from the category of $G$-equivariant sheaves on", "$X$ to the category of quasi-coherent $G$-equivariant sheaves on $Y$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9297, "type": "theorem", "label": "spaces-groupoids-lemma-groupoid-pre-equivalence", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-groupoid-pre-equivalence", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Given a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$", "the morphism $j : R \\to U \\times_B U$ is a pre-equivalence", "relation." ], "refs": [], "proofs": [ { "contents": [ "Omitted.", "This is a nice exercise in the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9298, "type": "theorem", "label": "spaces-groupoids-lemma-equivalence-groupoid", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-equivalence-groupoid", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Given an equivalence relation $j : R \\to U \\times_B U$ over $B$", "there is a unique way to extend it to a groupoid in algebraic spaces", "$(U, R, s, t, c)$ over $B$." ], "refs": [], "proofs": [ { "contents": [ "Omitted.", "This is a nice exercise in the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9299, "type": "theorem", "label": "spaces-groupoids-lemma-diagram", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-diagram", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "In the commutative diagram", "$$", "\\xymatrix{", "& U & \\\\", "R \\ar[d]_s \\ar[ru]^t &", "R \\times_{s, U, t} R", "\\ar[l]^-{\\text{pr}_0} \\ar[d]^{\\text{pr}_1} \\ar[r]_-c &", "R \\ar[d]^s \\ar[lu]_t \\\\", "U & R \\ar[l]_t \\ar[r]^s & U", "}", "$$", "the two lower squares are fibre product squares.", "Moreover, the triangle on top (which is really a square)", "is also cartesian." ], "refs": [], "proofs": [ { "contents": [ "Omitted.", "Exercise in the definitions and the functorial point of", "view in algebraic geometry." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9300, "type": "theorem", "label": "spaces-groupoids-lemma-diagram-pull", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-diagram-pull", "contents": [ "Let $B \\to S$ be as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c, e, i)$ be a groupoid in algebraic spaces over $B$.", "The diagram", "\\begin{equation}", "\\label{equation-pull}", "\\xymatrix{", "R \\times_{t, U, t} R", "\\ar@<1ex>[r]^-{\\text{pr}_1} \\ar@<-1ex>[r]_-{\\text{pr}_0}", "\\ar[d]_{\\text{pr}_0 \\times c \\circ (i, 1)} &", "R \\ar[r]^t \\ar[d]^{\\text{id}_R} &", "U \\ar[d]^{\\text{id}_U} \\\\", "R \\times_{s, U, t} R", "\\ar@<1ex>[r]^-c \\ar@<-1ex>[r]_-{\\text{pr}_0} \\ar[d]_{\\text{pr}_1} &", "R \\ar[r]^t \\ar[d]^s &", "U \\\\", "R \\ar@<1ex>[r]^s \\ar@<-1ex>[r]_t &", "U", "}", "\\end{equation}", "is commutative. The two top rows are isomorphic via the vertical maps given.", "The two lower left squares are cartesian." ], "refs": [], "proofs": [ { "contents": [ "The commutativity of the diagram follows from the axioms of a groupoid.", "Note that, in terms of groupoids, the top left vertical arrow assigns to", "a pair of morphisms $(\\alpha, \\beta)$ with the same target, the pair", "of morphisms $(\\alpha, \\alpha^{-1} \\circ \\beta)$. In any groupoid", "this defines a bijection between", "$\\text{Arrows} \\times_{t, \\text{Ob}, t} \\text{Arrows}$", "and", "$\\text{Arrows} \\times_{s, \\text{Ob}, t} \\text{Arrows}$. Hence the second", "assertion of the lemma.", "The last assertion follows from Lemma \\ref{lemma-diagram}." ], "refs": [ "spaces-groupoids-lemma-diagram" ], "ref_ids": [ 9299 ] } ], "ref_ids": [] }, { "id": 9301, "type": "theorem", "label": "spaces-groupoids-lemma-base-change-groupoid", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-base-change-groupoid", "contents": [ "Let $B \\to S$ be as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "Let $B' \\to B$ be a morphism of algebraic spaces.", "Then the base changes $U' = B' \\times_B U$,", "$R' = B' \\times_B R$ endowed with the base changes $s'$, $t'$, $c'$", "of the morphisms $s, t, c$ form a groupoid in algebraic spaces", "$(U', R', s', t', c')$ over $B'$ and the projections", "determine a morphism", "$(U', R', s', t', c') \\to (U, R, s, t, c)$", "of groupoids in algebraic spaces over $B$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint:", "$R' \\times_{s', U', t'} R' = B' \\times_B (R \\times_{s, U, t} R)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9302, "type": "theorem", "label": "spaces-groupoids-lemma-isomorphism", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-isomorphism", "contents": [ "Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "If $(\\mathcal{F}, \\alpha)$ is a quasi-coherent module on $(U, R, s, t, c)$", "then $\\alpha$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Pull back the commutative diagram of", "Definition \\ref{definition-groupoid-module}", "by the morphism $(i, 1) : R \\to R \\times_{s, U, t} R$.", "Then we see that $i^*\\alpha \\circ \\alpha = s^*e^*\\alpha$.", "Pulling back by the morphism $(1, i)$ we obtain the relation", "$\\alpha \\circ i^*\\alpha = t^*e^*\\alpha$. By the second assumption ", "these morphisms are the identity. Hence $i^*\\alpha$ is an inverse of", "$\\alpha$." ], "refs": [ "spaces-groupoids-definition-groupoid-module" ], "ref_ids": [ 9348 ] } ], "ref_ids": [] }, { "id": 9303, "type": "theorem", "label": "spaces-groupoids-lemma-pullback", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-pullback", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Consider a morphism", "$f : (U, R, s, t, c) \\to (U', R', s', t', c')$", "of groupoid in algebraic spaces over $B$. Then pullback $f^*$ given by", "$$", "(\\mathcal{F}, \\alpha) \\mapsto (f^*\\mathcal{F}, f^*\\alpha)", "$$", "defines a functor from the category of quasi-coherent sheaves on", "$(U', R', s', t', c')$ to the category of quasi-coherent sheaves on", "$(U, R, s, t, c)$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9304, "type": "theorem", "label": "spaces-groupoids-lemma-colimits", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-colimits", "contents": [ "Let $B \\to S$ be as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "The category of quasi-coherent modules on $(U, R, s, t, c)$ has colimits." ], "refs": [], "proofs": [ { "contents": [ "Let $i \\mapsto (\\mathcal{F}_i, \\alpha_i)$ be a diagram over the index", "category $\\mathcal{I}$. We can form the colimit", "$\\mathcal{F} = \\colim \\mathcal{F}_i$", "which is a quasi-coherent sheaf on $U$, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-properties-quasi-coherent}.", "Since colimits commute with pullback we see that", "$s^*\\mathcal{F} = \\colim s^*\\mathcal{F}_i$ and similarly", "$t^*\\mathcal{F} = \\colim t^*\\mathcal{F}_i$. Hence we can set", "$\\alpha = \\colim \\alpha_i$. We omit the proof that $(\\mathcal{F}, \\alpha)$", "is the colimit of the diagram in the category of quasi-coherent modules", "on $(U, R, s, t, c)$." ], "refs": [ "spaces-properties-lemma-properties-quasi-coherent" ], "ref_ids": [ 11912 ] } ], "ref_ids": [] }, { "id": 9305, "type": "theorem", "label": "spaces-groupoids-lemma-abelian", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-abelian", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "If $s$, $t$ are flat, then the category of quasi-coherent modules on", "$(U, R, s, t, c)$ is abelian." ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi : (\\mathcal{F}, \\alpha) \\to (\\mathcal{G}, \\beta)$ be a", "homomorphism of quasi-coherent modules on $(U, R, s, t, c)$. Since $s$ is flat", "we see that", "$$", "0 \\to s^*\\Ker(\\varphi)", "\\to s^*\\mathcal{F} \\to s^*\\mathcal{G} \\to s^*\\Coker(\\varphi) \\to 0", "$$", "is exact and similarly for pullback by $t$. Hence $\\alpha$ and $\\beta$", "induce isomorphisms", "$\\kappa : t^*\\Ker(\\varphi) \\to s^*\\Ker(\\varphi)$ and", "$\\lambda : t^*\\Coker(\\varphi) \\to s^*\\Coker(\\varphi)$", "which satisfy the cocycle condition. Then it is straightforward to", "verify that $(\\Ker(\\varphi), \\kappa)$ and", "$(\\Coker(\\varphi), \\lambda)$ are a kernel and cokernel in the", "category of quasi-coherent modules on $(U, R, s, t, c)$. Moreover,", "the condition $\\Coim(\\varphi) = \\Im(\\varphi)$ follows", "because it holds over $U$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9306, "type": "theorem", "label": "spaces-groupoids-lemma-crystals-in-quasi-coherent-modules", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-crystals-in-quasi-coherent-modules", "contents": [ "In the situation above, if all the morphisms $f_\\phi$ are flat, then there", "exists a cardinal $\\kappa$ such that every object", "$(\\{\\mathcal{F}_i\\}_{i \\in I}, \\{\\alpha_\\phi\\}_{\\phi \\in \\Phi})$", "of $\\textit{CQC}(X)$ is the directed colimit of its", "$\\kappa$-generated submodules." ], "refs": [], "proofs": [ { "contents": [ "In the lemma and in this proof a {\\it submodule} of", "$(\\{\\mathcal{F}_i\\}_{i \\in I}, \\{\\alpha_\\phi\\}_{\\phi \\in \\Phi})$", "means the data of a quasi-coherent submodule", "$\\mathcal{G}_i \\subset \\mathcal{F}_i$ for all $i$ such that", "$\\alpha_\\phi(f_\\phi^*\\mathcal{G}_i) = \\mathcal{G}_{i'}$", "as subsheaves of $\\mathcal{F}_{i'}$ for all $\\phi \\in \\Phi$.", "This makes sense because since $f_\\phi$ is flat the", "pullback $f^*_\\phi$ is exact, i.e., preserves subsheaves.", "The proof will be a variant to the proof of", "Properties, Lemma \\ref{properties-lemma-colimit-kappa}.", "We urge the reader to read that proof first.", "\\medskip\\noindent", "We claim that it suffices to prove the lemma in case all the schemes", "$X_i$ are affine. To see this let", "$$", "J = \\coprod\\nolimits_{i \\in I} \\{U \\subset X_i\\text{ affine open}\\}", "$$", "and let", "\\begin{align*}", "\\Psi = & \\coprod\\nolimits_{\\phi \\in \\Phi}", "\\{", "(U, V) \\mid", "U \\subset X_i, V \\subset X_{i'}\\text{ affine open with } f_\\phi(U) \\subset V", "\\} \\\\", "&", "\\amalg \\coprod\\nolimits_{i \\in I}", "\\{", "(U, U') \\mid", "U, U' \\subset X_i\\text{ affine open with } U \\subset U'", "\\}", "\\end{align*}", "endowed with the obvious map $\\Psi \\to J \\times J$. Then our", "$(\\mathcal{F}, \\alpha)$ induces a crystal in quasi-coherent sheaves", "$(\\{\\mathcal{H}_j\\}_{j \\in J}, \\{\\beta_\\psi\\}_{\\psi \\in \\Psi})$", "on $Y = (J, \\Psi)$ by setting $\\mathcal{H}_{(i, U)} = \\mathcal{F}_i|_U$", "for $(i, U) \\in J$ and setting $\\beta_\\psi$ for $\\psi \\in \\Psi$", "equal to the restriction of $\\alpha_\\phi$ to $U$", "if $\\psi = (\\phi, U, V)$ and", "equal to $\\text{id} : (\\mathcal{F}_i|_{U'})|_U \\to \\mathcal{F}_i|_U$", "when $\\psi = (i, U, U')$. Moreover, submodules of", "$(\\{\\mathcal{H}_j\\}_{j \\in J}, \\{\\beta_\\psi\\}_{\\psi \\in \\Psi})$", "correspond $1$-to-$1$ with submodules of", "$(\\{\\mathcal{F}_i\\}_{i \\in I}, \\{\\alpha_\\phi\\}_{\\phi \\in \\Phi})$.", "We omit the proof (hint: use", "Sheaves, Section \\ref{sheaves-section-bases}).", "Moreover, it is clear that if $\\kappa$ works for $Y$, then", "the same $\\kappa$ works for $X$ (by the definition of $\\kappa$-generated", "modules). Hence it suffices to proof the lemma for crystals in", "quasi-coherent sheaves on $Y$.", "\\medskip\\noindent", "Assume that all the schemes $X_i$ are affine. Let $\\kappa$ be an infinite", "cardinal larger than the cardinality of $I$ or $\\Phi$. Let", "$(\\{\\mathcal{F}_i\\}_{i \\in I}, \\{\\alpha_\\phi\\}_{\\phi \\in \\Phi})$", "be an object of $\\textit{CQC}(X)$. For each $i$ write", "$X_i = \\Spec(A_i)$ and $M_i = \\Gamma(X_i, \\mathcal{F}_i)$.", "For every $\\phi \\in \\Phi$ with $j(\\phi) = (i, i')$ the map", "$\\alpha_\\phi$ translates into an $A_{i'}$-module isomorphism", "$$", "\\alpha_\\phi : M_i \\otimes_{A_i} A_{i'} \\longrightarrow M_{i'}", "$$", "Using the axiom of choice choose a rule", "$$", "(\\phi, m) \\longmapsto S(\\phi, m')", "$$", "where the source is the collection of pairs $(\\phi, m')$ such that", "$\\phi \\in \\Phi$ with $j(\\phi) = (i, i')$ and $m' \\in M_{i'}$ and", "where the output is a finite subset $S(\\phi, m') \\subset M_i$ so that", "$$", "m' = \\alpha_\\phi\\left(\\sum\\nolimits_{m \\in S(\\phi, m')} m \\otimes a'_m\\right)", "$$", "for some $a'_m \\in A_{i'}$.", "\\medskip\\noindent", "Having made these choices we claim that any section of any $\\mathcal{F}_i$", "over any $X_i$ is in a $\\kappa$-generated submodule. To see this suppose", "that we are given a collection $\\mathcal{S} = \\{S_i\\}_{i \\in I}$ of subsets", "$S_i \\subset M_i$ each with cardinality at most $\\kappa$. Then we define", "a new collection $\\mathcal{S}' = \\{S'_i\\}_{i \\in I}$ with", "$$", "S'_i = S_i \\cup", "\\bigcup\\nolimits_{(\\phi, m'),\\ j(\\phi) = (i, i'),\\ m' \\in S_{i'}} S(\\phi, m')", "$$", "Note that each $S'_i$ still has cardinality at most $\\kappa$.", "Set $\\mathcal{S}^{(0)} = \\mathcal{S}$,", "$\\mathcal{S}^{(1)} = \\mathcal{S}'$ and by induction", "$\\mathcal{S}^{(n + 1)} = (\\mathcal{S}^{(n)})'$. Then set", "$S_i^{(\\infty)} = \\bigcup_{n \\geq 0} S_i^{(n)}$ and", "$\\mathcal{S}^{(\\infty)} = \\{S_i^{(\\infty)}\\}_{i \\in I}$.", "By construction, for every $\\phi \\in \\Phi$ with $j(\\phi) = (i, i')$", "and every $m' \\in S^{(\\infty)}_{i'}$ we can write $m'$", "as a finite linear combination of images $\\alpha_\\phi(m \\otimes 1)$", "with $m \\in S_i^{(\\infty)}$. Thus we see that setting $N_i$ equal", "to the $A_i$-submodule of $M_i$ generated by $S_i^{(\\infty)}$", "the corresponding quasi-coherent submodules", "$\\widetilde{N_i} \\subset \\mathcal{F}_i$ form a $\\kappa$-generated submodule.", "This finishes the proof." ], "refs": [ "properties-lemma-colimit-kappa" ], "ref_ids": [ 3032 ] } ], "ref_ids": [] }, { "id": 9307, "type": "theorem", "label": "spaces-groupoids-lemma-set-generators", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-set-generators", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "If $s$, $t$ are flat, then there exists a set $T$ and a family of objects", "$(\\mathcal{F}_t, \\alpha_t)_{t \\in T}$ of $\\QCoh(U, R, s, t, c)$", "such that every object $(\\mathcal{F}, \\alpha)$ is the directed colimit", "of its submodules isomorphic to one of the objects $(\\mathcal{F}_t, \\alpha_t)$." ], "refs": [], "proofs": [ { "contents": [ "This lemma is a generalization of", "Groupoids, Lemma \\ref{groupoids-lemma-colimit-kappa}", "which deals with the case of a groupoid in schemes.", "We can't quite use the same argument, so we use the", "material on ``crystals of quasi-coherent sheaves'' we developed above.", "\\medskip\\noindent", "Choose a scheme $W$ and a surjective \\'etale morphism $W \\to U$.", "Choose a scheme $V$ and a surjective \\'etale morphism", "$V \\to W \\times_{U, s} R$. Choose a scheme $V'$", "and a surjective \\'etale morphism $V' \\to R \\times_{t, U} W$.", "Consider the collection of schemes", "$$", "I = \\{W, W \\times_U W, V, V', V \\times_R V'\\}", "$$", "and the set of morphisms of schemes", "$$", "\\Phi = \\{\\text{pr}_i : W \\times_U W \\to W, V \\to W, V' \\to W,", "V \\times_R V' \\to V, V \\times_R V' \\to V'\\}", "$$", "Set $X = (I, \\Phi)$. Recall that we have defined a category $\\textit{CQC}(X)$", "of crystals of quasi-coherent sheaves on $X$. There is a functor", "$$", "\\QCoh(U, R, s, t, c) \\longrightarrow \\textit{CQC}(X)", "$$", "which assigns to $(\\mathcal{F}, \\alpha)$ the sheaf $\\mathcal{F}|_W$ on $W$,", "the sheaf $\\mathcal{F}|_{W \\times_U W}$ on $W \\times_U W$, the", "pullback of $\\mathcal{F}$ via $V \\to W \\times_{U, s} R \\to W \\to U$", "on $V$, the pullback of $\\mathcal{F}$ via", "$V' \\to R \\times_{t, U} W \\to W \\to U$ on $V'$, and finally the", "pullback of $\\mathcal{F}$ via", "$V \\times_R V' \\to V \\to W \\times_{U, s} R \\to W \\to U$ on $V \\times_R V'$.", "As comparison maps $\\{\\alpha_\\phi\\}_{\\phi \\in \\Phi}$", "we use the obvious ones (coming from associativity of pullbacks)", "except for the map", "$\\phi = \\text{pr}_{V'} : V \\times_R V' \\to V'$ we use the pullback", "of $\\alpha : t^*\\mathcal{F} \\to s^*\\mathcal{F}$ to $V \\times_R V'$.", "This makes sense because of the following commutative diagram", "$$", "\\xymatrix{", "& V \\times_R V' \\ar[ld] \\ar[rd] \\\\", "V \\ar[rd] \\ar[dd] & & V' \\ar[ld] \\ar[dd] \\\\", "& R \\ar@<-1ex>[dd]_s \\ar@<1ex>[dd]^t \\\\", "W \\ar[rd] & & W \\ar[ld] \\\\", "& U", "}", "$$", "The functor displayed above isn't an equivalence of categories.", "However, since $W \\to U$ is surjective \\'etale it is faithful\\footnote{In", "fact the functor is fully faithful, but we won't need this.}.", "Since all the morphisms in the diagram above are flat we see that", "it is an exact functor of abelian categories. Moreover, we", "claim that given $(\\mathcal{F}, \\alpha)$ with image", "$(\\{\\mathcal{F}_i\\}_{i \\in I}, \\{\\alpha_\\phi\\}_{\\phi \\in \\Phi})$", "there is a $1$-to-$1$ correspondence between quasi-coherent submodules", "of $(\\mathcal{F}, \\alpha)$ and", "$(\\{\\mathcal{F}_i\\}_{i \\in I}, \\{\\alpha_\\phi\\}_{\\phi \\in \\Phi})$.", "Namely, given a submodule of", "$(\\{\\mathcal{F}_i\\}_{i \\in I}, \\{\\alpha_\\phi\\}_{\\phi \\in \\Phi})$", "compatibility of the submodule over $W$ with the projection maps", "$W \\times_U W \\to W$ will guarantee the submodule comes from a", "quasi-coherent submodule of $\\mathcal{F}$ (by", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-quasi-coherent})", "and compatibility with $\\alpha_{\\text{pr}_{V'}}$ will insure this", "subsheaf is compatible with $\\alpha$ (details omitted).", "\\medskip\\noindent", "Choose a cardinal $\\kappa$ as in", "Lemma \\ref{lemma-crystals-in-quasi-coherent-modules}", "for the system $X = (I, \\Phi)$. It is clear from", "Properties, Lemma \\ref{properties-lemma-set-of-iso-classes}", "that there is a set of isomorphism classes of $\\kappa$-generated", "crystals in quasi-coherent sheaves on $X$.", "Hence the result is clear." ], "refs": [ "groupoids-lemma-colimit-kappa", "spaces-properties-proposition-quasi-coherent", "spaces-groupoids-lemma-crystals-in-quasi-coherent-modules", "properties-lemma-set-of-iso-classes" ], "ref_ids": [ 9635, 11920, 9306, 3031 ] } ], "ref_ids": [] }, { "id": 9308, "type": "theorem", "label": "spaces-groupoids-lemma-groupoid-from-action", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-groupoid-from-action", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(G, m)$ be a group algebraic space over $B$ with", "identity $e_G$ and inverse $i_G$.", "Let $X$ be an algebraic space over $B$ and let $a : G \\times_B X \\to X$", "be an action of $G$ on $X$ over $B$.", "Then we get a groupoid in algebraic spaces $(U, R, s, t, c, e, i)$ over $B$", "in the following manner:", "\\begin{enumerate}", "\\item We set $U = X$, and $R = G \\times_B X$.", "\\item We set $s : R \\to U$ equal to $(g, x) \\mapsto x$.", "\\item We set $t : R \\to U$ equal to $(g, x) \\mapsto a(g, x)$.", "\\item We set $c : R \\times_{s, U, t} R \\to R$ equal to", "$((g, x), (g', x')) \\mapsto (m(g, g'), x')$.", "\\item We set $e : U \\to R$ equal to $x \\mapsto (e_G(x), x)$.", "\\item We set $i : R \\to R$ equal to $(g, x) \\mapsto (i_G(g), a(g, x))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: It is enough to show that this works on the set", "level. For this use the description above the lemma describing", "$g$ as an arrow from $v$ to $a(g, v)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9309, "type": "theorem", "label": "spaces-groupoids-lemma-action-groupoid-modules", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-action-groupoid-modules", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(G, m)$ be a group algebraic space over $B$.", "Let $X$ be an algebraic space over $B$ and let $a : G \\times_B X \\to X$", "be an action of $G$ on $X$ over $B$. Let $(U, R, s, t, c)$ be", "the groupoid in algebraic spaces constructed in", "Lemma \\ref{lemma-groupoid-from-action}.", "The rule", "$(\\mathcal{F}, \\alpha) \\mapsto (\\mathcal{F}, \\alpha)$ defines", "an equivalence of categories between $G$-equivariant", "$\\mathcal{O}_X$-modules and the category of quasi-coherent", "modules on $(U, R, s, t, c)$." ], "refs": [ "spaces-groupoids-lemma-groupoid-from-action" ], "proofs": [ { "contents": [ "The assertion makes sense because $t = a$ and $s = \\text{pr}_1$", "as morphisms $R = G \\times_B X \\to X$, see", "Definitions \\ref{definition-equivariant-module} and", "\\ref{definition-groupoid-module}.", "Using the translation in Lemma \\ref{lemma-groupoid-from-action}", "the commutativity requirements", "of the two definitions match up exactly." ], "refs": [ "spaces-groupoids-definition-equivariant-module", "spaces-groupoids-definition-groupoid-module", "spaces-groupoids-lemma-groupoid-from-action" ], "ref_ids": [ 9346, 9348, 9308 ] } ], "ref_ids": [ 9308 ] }, { "id": 9310, "type": "theorem", "label": "spaces-groupoids-lemma-groupoid-stabilizer", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-groupoid-stabilizer", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "The algebraic space $G$ defined by the cartesian square", "$$", "\\xymatrix{", "G \\ar[r] \\ar[d] & R \\ar[d]^{j = (t, s)} \\\\", "U \\ar[r]^-{\\Delta} & U \\times_B U", "}", "$$", "is a group algebraic space over $U$ with composition law", "$m$ induced by the composition law $c$." ], "refs": [], "proofs": [ { "contents": [ "This is true because in a groupoid category the", "set of self maps of any object forms a group." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9311, "type": "theorem", "label": "spaces-groupoids-lemma-groupoid-action-stabilizer", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-groupoid-action-stabilizer", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$, and let", "$G/U$ be its stabilizer. Denote $R_t/U$ the algebraic space $R$ seen as an", "algebraic space over $U$ via the morphism $t : R \\to U$. There is a", "canonical left action", "$$", "a : G \\times_U R_t \\longrightarrow R_t", "$$", "induced by the composition law $c$." ], "refs": [], "proofs": [ { "contents": [ "In terms of points over $T/B$ we define $a(g, r) = c(g, r)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9312, "type": "theorem", "label": "spaces-groupoids-lemma-restrict-groupoid", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-restrict-groupoid", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "Let $g : U' \\to U$ be a morphism of algebraic spaces.", "Consider the following diagram", "$$", "\\xymatrix{", "R' \\ar[d] \\ar[r] \\ar@/_3pc/[dd]_{t'} \\ar@/^1pc/[rr]^{s'}&", "R \\times_{s, U} U' \\ar[r] \\ar[d] &", "U' \\ar[d]^g \\\\", "U' \\times_{U, t} R \\ar[d] \\ar[r] &", "R \\ar[r]^s \\ar[d]_t &", "U \\\\", "U' \\ar[r]^g &", "U", "}", "$$", "where all the squares are fibre product squares. Then there is a", "canonical composition law $c' : R' \\times_{s', U', t'} R' \\to R'$", "such that $(U', R', s', t', c')$ is a groupoid in algebraic spaces over", "$B$ and such that $U' \\to U$, $R' \\to R$ defines a morphism", "$(U', R', s', t', c') \\to (U, R, s, t, c)$ of groupoids in algebraic spaces", "over $B$. Moreover, for any scheme $T$ over $B$ the functor of groupoids", "$$", "(U'(T), R'(T), s', t', c') \\to (U(T), R(T), s, t, c)", "$$", "is the restriction (see", "Groupoids, Section \\ref{groupoids-section-restrict-groupoid})", "of $(U(T), R(T), s, t, c)$ via the map $U'(T) \\to U(T)$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9313, "type": "theorem", "label": "spaces-groupoids-lemma-restrict-groupoid-relation", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-restrict-groupoid-relation", "contents": [ "The notions of restricting groupoids and", "(pre-)equivalence relations defined in Definitions", "\\ref{definition-restrict-groupoid} and \\ref{definition-restrict-relation}", "agree via the constructions of", "Lemmas \\ref{lemma-groupoid-pre-equivalence} and", "\\ref{lemma-equivalence-groupoid}." ], "refs": [ "spaces-groupoids-definition-restrict-groupoid", "spaces-groupoids-definition-restrict-relation", "spaces-groupoids-lemma-groupoid-pre-equivalence", "spaces-groupoids-lemma-equivalence-groupoid" ], "proofs": [ { "contents": [ "What we are saying here is that $R'$ of", "Lemma \\ref{lemma-restrict-groupoid} is also", "equal to", "$$", "R' = (U' \\times_B U')\\times_{U \\times_B U} R", "\\longrightarrow", "U' \\times_B U'", "$$", "In fact this might have been a clearer way to state that lemma." ], "refs": [ "spaces-groupoids-lemma-restrict-groupoid" ], "ref_ids": [ 9312 ] } ], "ref_ids": [ 9350, 9340, 9297, 9298 ] }, { "id": 9314, "type": "theorem", "label": "spaces-groupoids-lemma-constructing-invariant-opens", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-constructing-invariant-opens", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "\\begin{enumerate}", "\\item If $s$ and $t$ are open, then for every open $W \\subset U$", "the open $s(t^{-1}(W))$ is $R$-invariant.", "\\item If $s$ and $t$ are open and quasi-compact, then $U$ has an open", "covering consisting of $R$-invariant quasi-compact open subspaces.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $s$ and $t$ open and $W \\subset U$ open.", "Since $s$ is open we see that $W' = s(t^{-1}(W))$ is an open subspace of $U$.", "Now it is quite easy to using the functorial point of view", "that this is an $R$-invariant open subset of $U$, but we are going to argue", "this directly by some diagrams, since we think it is instructive.", "Note that $t^{-1}(W')$ is the image of the morphism", "$$", "A := t^{-1}(W) \\times_{s|_{t^{-1}(W)}, U, t} R", "\\xrightarrow{\\text{pr}_1} R", "$$", "and that $s^{-1}(W')$ is the image of the morphism", "$$", "B := R \\times_{s, U, s|_{t^{-1}(W)}} t^{-1}(W)", "\\xrightarrow{\\text{pr}_0} R.", "$$", "The algebraic spaces $A$, $B$", "on the left of the arrows above are open subspaces of", "$R \\times_{s, U, t} R$ and $R \\times_{s, U, s} R$ respectively.", "By Lemma \\ref{lemma-diagram} the diagram", "$$", "\\xymatrix{", "R \\times_{s, U, t} R \\ar[rd]_{\\text{pr}_1} \\ar[rr]_{(\\text{pr}_1, c)} & &", "R \\times_{s, U, s} R \\ar[ld]^{\\text{pr}_0} \\\\", "& R &", "}", "$$", "is commutative, and the horizontal arrow is an isomorphism. Moreover, it is", "clear that $(\\text{pr}_1, c)(A) = B$. Hence we conclude", "$s^{-1}(W') = t^{-1}(W')$, and $W'$ is $R$-invariant. This proves (1).", "\\medskip\\noindent", "Assume now that $s$, $t$ are both open and quasi-compact.", "Then, if $W \\subset U$ is a quasi-compact open, then also", "$W' = s(t^{-1}(W))$ is a quasi-compact open, and invariant by the", "discussion above. Letting $W$ range over images of affines \\'etale over $U$", "we see (2)." ], "refs": [ "spaces-groupoids-lemma-diagram" ], "ref_ids": [ 9299 ] } ], "ref_ids": [] }, { "id": 9315, "type": "theorem", "label": "spaces-groupoids-lemma-criterion-quotient-representable", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-criterion-quotient-representable", "contents": [ "In the situation of Definition \\ref{definition-quotient-sheaf}.", "Assume there is an algebraic space $M$ over $S$,", "and a morphism $U \\to M$ such that", "\\begin{enumerate}", "\\item the morphism $U \\to M$ equalizes $s, t$,", "\\item the map $U \\to M$ is a surjection of sheaves, and", "\\item the induced map $(t, s) : R \\to U \\times_M U$ is a", "surjection of sheaves.", "\\end{enumerate}", "In this case $M$ represents the quotient sheaf $U/R$." ], "refs": [ "spaces-groupoids-definition-quotient-sheaf" ], "proofs": [ { "contents": [ "Condition (1) says that $U \\to M$ factors through $U/R$.", "Condition (2) says that $U/R \\to M$ is surjective as a map of sheaves.", "Condition (3) says that $U/R \\to M$ is injective as a map of sheaves.", "Hence the lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9352 ] }, { "id": 9316, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-pre-equivalence", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-pre-equivalence", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-equivalence relation over $B$.", "For a scheme $S'$ over $S$ and $a, b \\in U(S')$ the following are equivalent:", "\\begin{enumerate}", "\\item $a$ and $b$ map to the same element of $(U/R)(S')$, and", "\\item there exists an fppf covering $\\{f_i : S_i \\to S'\\}$ of $S'$", "and morphisms $r_i : S_i \\to R$ such that", "$a \\circ f_i = s \\circ r_i$ and $b \\circ f_i = t \\circ r_i$.", "\\end{enumerate}", "In other words, in this case the map of sheaves", "$$", "R \\longrightarrow U \\times_{U/R} U", "$$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: The reason this works is that the presheaf", "(\\ref{equation-quotient-presheaf}) in this case is really given", "by $T \\mapsto U(T)/j(R(T))$ as $j(R(T)) \\subset U(T) \\times U(T)$", "is an equivalence relation, see", "Definition \\ref{definition-equivalence-relation}." ], "refs": [ "spaces-groupoids-definition-equivalence-relation" ], "ref_ids": [ 9339 ] } ], "ref_ids": [] }, { "id": 9317, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-pre-equivalence-relation-restrict", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-pre-equivalence-relation-restrict", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-relation over $B$", "and $g : U' \\to U$ a morphism of algebraic spaces over $B$.", "Let $j' : R' \\to U' \\times_B U'$ be the restriction of $j$ to $U'$.", "The map of quotient sheaves", "$$", "U'/R' \\longrightarrow U/R", "$$", "is injective. If $U' \\to U$ is surjective as a map of sheaves, for", "example if $\\{g : U' \\to U\\}$ is an fppf covering (see", "Topologies on Spaces,", "Definition \\ref{spaces-topologies-definition-fppf-covering}),", "then $U'/R' \\to U/R$ is an isomorphism of sheaves." ], "refs": [ "spaces-topologies-definition-fppf-covering" ], "proofs": [ { "contents": [ "Suppose $\\xi, \\xi' \\in (U'/R')(S')$ are sections which", "map to the same section of $U/R$.", "Then we can find an fppf covering $\\mathcal{S} = \\{S_i \\to S'\\}$ of $S'$", "such that $\\xi|_{S_i}, \\xi'|_{S_i}$ are given by $a_i, a_i' \\in U'(S_i)$. By", "Lemma \\ref{lemma-quotient-pre-equivalence}", "and the axioms of a site we may after refining", "$\\mathcal{T}$ assume there exist morphisms $r_i : S_i \\to R$", "such that $g \\circ a_i = s \\circ r_i$, $g \\circ a_i' = t \\circ r_i$.", "Since by construction", "$R' = R \\times_{U \\times_S U} (U' \\times_S U')$", "we see that $(r_i, (a_i, a_i')) \\in R'(S_i)$ and this", "shows that $a_i$ and $a_i'$ define the same section", "of $U'/R'$ over $S_i$. By the sheaf condition this implies", "$\\xi = \\xi'$.", "\\medskip\\noindent", "If $U' \\to U$ is a surjective map of sheaves, then $U'/R' \\to U/R$ is", "surjective also. Finally, if $\\{g : U' \\to U\\}$ is a fppf covering, then", "the map of sheaves $U' \\to U$ is surjective, see", "Topologies on Spaces,", "Lemma \\ref{spaces-topologies-lemma-fppf-covering-surjective}." ], "refs": [ "spaces-groupoids-lemma-quotient-pre-equivalence", "spaces-topologies-lemma-fppf-covering-surjective" ], "ref_ids": [ 9316, 3667 ] } ], "ref_ids": [ 3688 ] }, { "id": 9318, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-groupoid-restrict", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-groupoid-restrict", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "Let $g : U' \\to U$ a morphism of algebraic spaces over $B$.", "Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ to $U'$.", "The map of quotient sheaves", "$$", "U'/R' \\longrightarrow U/R", "$$", "is injective. If the composition", "$$", "\\xymatrix{", "U' \\times_{g, U, t} R \\ar[r]_-{\\text{pr}_1} \\ar@/^3ex/[rr]^h", "& R \\ar[r]_s & U", "}", "$$", "is a surjection of fppf sheaves then the map is bijective.", "This holds for example if $\\{h : U' \\times_{g, U, t} R \\to U\\}$ is an", "$fppf$-covering, or if $U' \\to U$ is a surjection of sheaves, or if", "$\\{g : U' \\to U\\}$ is a covering in the fppf topology." ], "refs": [], "proofs": [ { "contents": [ "Injectivity follows on combining", "Lemmas \\ref{lemma-groupoid-pre-equivalence} and", "\\ref{lemma-quotient-pre-equivalence-relation-restrict}.", "To see surjectivity (see", "Sites, Section \\ref{sites-section-sheaves-injective}", "for a characterization of surjective maps of sheaves) we argue as follows.", "Suppose that $T$ is a scheme and $\\sigma \\in U/R(T)$.", "There exists a covering $\\{T_i \\to T\\}$ such that $\\sigma|_{T_i}$", "is the image of some element $f_i \\in U(T_i)$. Hence we", "may assume that $\\sigma$ if the image of $f \\in U(T)$.", "By the assumption that $h$ is a surjection of sheaves, we", "can find an fppf covering $\\{\\varphi_i : T_i \\to T\\}$ and morphisms", "$f_i : T_i \\to U' \\times_{g, U, t} R$ such that", "$f \\circ \\varphi_i = h \\circ f_i$. Denote", "$f'_i = \\text{pr}_0 \\circ f_i : T_i \\to U'$. Then we see that", "$f'_i \\in U'(T_i)$ maps to $g \\circ f'_i \\in U(T_i)$ and", "that $g \\circ f'_i \\sim_{T_i} h \\circ f_i = f \\circ \\varphi_i$", "notation as in (\\ref{equation-quotient-presheaf}). Namely, the", "element of $R(T_i)$ giving the relation is $\\text{pr}_1 \\circ f_i$.", "This means that the restriction", "of $\\sigma$ to $T_i$ is in the image of $U'/R'(T_i) \\to U/R(T_i)$", "as desired.", "\\medskip\\noindent", "If $\\{h\\}$ is an fppf covering, then it induces a surjection of sheaves, see", "Topologies on Spaces,", "Lemma \\ref{spaces-topologies-lemma-fppf-covering-surjective}.", "If $U' \\to U$ is surjective, then also $h$ is surjective as $s$ has a section", "(namely the neutral element $e$ of the groupoid scheme)." ], "refs": [ "spaces-groupoids-lemma-groupoid-pre-equivalence", "spaces-groupoids-lemma-quotient-pre-equivalence-relation-restrict", "spaces-topologies-lemma-fppf-covering-surjective" ], "ref_ids": [ 9297, 9317, 3667 ] } ], "ref_ids": [] }, { "id": 9319, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-stack-arrows", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-stack-arrows", "contents": [ "Assume $B \\to S$ and $(U, R, s, t, c)$ as in", "Definition \\ref{definition-quotient-stack} (1).", "There are canonical $1$-morphisms", "$\\pi : \\mathcal{S}_U \\to [U/R]$, and $[U/R] \\to \\mathcal{S}_B$", "of stacks in groupoids over $(\\Sch/S)_{fppf}$.", "The composition $\\mathcal{S}_U \\to \\mathcal{S}_B$ is the $1$-morphism", "associated to the structure morphism $U \\to B$." ], "refs": [ "spaces-groupoids-definition-quotient-stack" ], "proofs": [ { "contents": [ "During this proof let us denote $[U/_{\\!p}R]$ the category fibred in", "groupoids associated to the presheaf in groupoids", "(\\ref{equation-quotient-stack}). By construction of the stackification", "there is a $1$-morphism $[U/_{\\!p}R] \\to [U/R]$.", "The $1$-morphism $\\mathcal{S}_U \\to [U/R]$ is simply the composition", "$\\mathcal{S}_U \\to [U/_{\\!p}R] \\to [U/R]$, where the first arrow", "associates to the scheme $S'/S$ and morphism $x : S' \\to U$ over $S$", "the object $x \\in U(S')$ of the fibre category of $[U/_{\\!p}R]$", "over $S'$.", "\\medskip\\noindent", "To construct the $1$-morphism $[U/R] \\to \\mathcal{S}_B$ it is enough to", "construct the $1$-morphism $[U/_{\\!p}R] \\to \\mathcal{S}_B$, see", "Stacks, Lemma \\ref{stacks-lemma-stackify-groupoids-universal-property}.", "On objects over $S'/S$ we just use the map", "$$", "U(S') \\longrightarrow B(S')", "$$", "coming from the structure morphism $U \\to B$.", "And clearly, if $a \\in R(S')$ is an ``arrow'' with source", "$s(a) \\in U(S')$ and target $t(a) \\in U(S')$, then since", "$s$ and $t$ are morphisms {\\it over} $B$ these both", "map to the same element $\\overline{a}$ of $B(S')$. Hence we can map an arrow", "$a \\in R(S')$ to the identity morphism of $\\overline{a}$. (This is", "good because the fibre category $(\\mathcal{S}_B)_{S'}$ only contains", "identities.) We omit the verification that this rule is compatible with", "pullback on these split fibred categories, and hence defines a", "$1$-morphism $[U/_{\\!p}R] \\to \\mathcal{S}_B$ as desired.", "\\medskip\\noindent", "We omit the verification of the last statement." ], "refs": [ "stacks-lemma-stackify-groupoids-universal-property" ], "ref_ids": [ 8967 ] } ], "ref_ids": [ 9354 ] }, { "id": 9320, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-stack-2-arrow", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-stack-2-arrow", "contents": [ "Assumptions and notation as in Lemma \\ref{lemma-quotient-stack-arrows}.", "There exists a canonical $2$-morphism", "$\\alpha : \\pi \\circ s \\to \\pi \\circ t$ making the diagram", "$$", "\\xymatrix{", "\\mathcal{S}_R \\ar[r]_s \\ar[d]_t & \\mathcal{S}_U \\ar[d]^\\pi \\\\", "\\mathcal{S}_U \\ar[r]^-\\pi & [U/R]", "}", "$$", "$2$-commutative." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-arrows" ], "proofs": [ { "contents": [ "Let $S'$ be a scheme over $S$. Let $r : S' \\to R$ be a morphism over $S$.", "Then $r \\in R(S')$ is an isomorphism between the objects", "$s \\circ r, t \\circ r \\in U(S')$. Moreover, this construction is", "compatible with pullbacks. This gives a canonical", "$2$-morphism $\\alpha_p : \\pi_p \\circ s \\to \\pi_p \\circ t$", "where $\\pi_p : \\mathcal{S}_U \\to [U/_{\\!p}R]$ is as in the", "proof of", "Lemma \\ref{lemma-quotient-stack-arrows}. Thus even the diagram", "$$", "\\xymatrix{", "\\mathcal{S}_R \\ar[r]_s \\ar[d]^t & \\mathcal{S}_U \\ar[d]^{\\pi_p} \\\\", "\\mathcal{S}_U \\ar[r]^-{\\pi_p} & [U/_{\\!p}R]", "}", "$$", "is $2$-commutative. Thus a fortiori the diagram", "of the lemma is $2$-commutative." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-arrows" ], "ref_ids": [ 9319 ] } ], "ref_ids": [ 9319 ] }, { "id": 9321, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-stack-functorial", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-stack-functorial", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $f : (U, R, s, t, c) \\to (U', R', s', t', c')$ be a morphism of", "groupoids in algebraic spaces over $B$.", "Then $f$ induces a canonical $1$-morphism of quotient stacks", "$$", "[f] : [U/R] \\longrightarrow [U'/R'].", "$$" ], "refs": [], "proofs": [ { "contents": [ "Denote $[U/_{\\!p}R]$ and $[U'/_{\\!p}R']$ the categories fibred", "in groupoids over the base site $(\\Sch/S)_{fppf}$ associated to the", "functors (\\ref{equation-quotient-stack}). It is clear that $f$ defines", "a $1$-morphism $[U/_{\\!p}R] \\to [U'/_{\\!p}R']$ which we can compose", "with the stackyfication functor for $[U'/R']$ to get $[U/_{\\!p}R] \\to [U'/R']$.", "Then, by the universal property of the stackyfication functor", "$[U/_{\\!p}R] \\to [U/R]$, see", "Stacks, Lemma \\ref{stacks-lemma-stackify-groupoids-universal-property}", "we get $[U/R] \\to [U'/R']$." ], "refs": [ "stacks-lemma-stackify-groupoids-universal-property" ], "ref_ids": [ 8967 ] } ], "ref_ids": [] }, { "id": 9322, "type": "theorem", "label": "spaces-groupoids-lemma-cartesian-square-of-morphism", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-cartesian-square-of-morphism", "contents": [ "Notation and assumption as in", "Lemma \\ref{lemma-quotient-stack-functorial}.", "Let $(U'', R'', s'', t'', c'')$ be the groupoid in algebraic spaces over $B$", "constructed above.", "There is a $2$-commutative square", "$$", "\\xymatrix{", "[U''/R''] \\ar[d] \\ar[r]_{[g]} & [U/R] \\ar[d]^{[f]} \\\\", "\\mathcal{S}_{U'} \\ar[r] & [U'/R']", "}", "$$", "which identifies $[U''/R'']$ with the $2$-fibre product." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-functorial" ], "proofs": [ { "contents": [ "The maps $[f]$ and $[g]$ come from an application of", "Lemma \\ref{lemma-quotient-stack-functorial}", "and the other two maps come from", "Lemma \\ref{lemma-quotient-stack-arrows}", "(and the fact that $(U'', R'', s'', t'', c'')$ lives over $U'$).", "To show the $2$-fibre product property, it suffices to prove the lemma", "for the diagram", "$$", "\\xymatrix{", "[U''/_{\\!p}R''] \\ar[d] \\ar[r]_{[g]} & [U/_{\\!p}R] \\ar[d]^{[f]} \\\\", "\\mathcal{S}_{U'} \\ar[r] & [U'/_{\\!p}R']", "}", "$$", "of categories fibred in groupoids, see", "Stacks, Lemma", "\\ref{stacks-lemma-stackification-fibre-product-categories-fibred-in-groupoids}.", "In other words, it suffices to show that an object of the $2$-fibre product", "$\\mathcal{S}_U \\times_{[U'/_{\\!p}R']} [U/_{\\!p}R]$ over $T$ corresponds", "to a $T$-valued point of $U''$ and similarly for morphisms. And of course", "this is exactly how we constructed $U''$ and $R''$ in the first place.", "\\medskip\\noindent", "In detail, an object of $\\mathcal{S}_U \\times_{[U'/_{\\!p}R']} [U/_{\\!p}R]$", "over $T$ is a triple $(u', u, r')$ where $u'$ is a $T$-valued point of $U'$,", "$u$ is a $T$-valued point of $U$, and $r'$ is a morphism from $u'$ to", "$f(u)$ in $[U'/R']_T$, i.e., $r'$ is a $T$-valued point of $R$ with", "$s'(r') = u'$ and $t'(r') = f(u)$. Clearly we can forget about $u'$", "without losing information and we see that these objects are in one-to-one", "correspondence with $T$-valued points of $R''$.", "\\medskip\\noindent", "Similarly for morphisms: Let $(u'_1, u_1, r'_1)$ and $(u'_2, u_2, r'_2)$ be", "two objects of the fibre product over $T$. Then a morphism from", "$(u'_2, u_2, r'_2)$ to $(u'_1, u_1, r'_1)$", "is given by $(1, r)$ where $1 : u'_1 \\to u'_2$ means simply $u'_1 = u'_2$", "(this is so because $\\mathcal{S}_U$ is fibred in sets), and $r$ is a", "$T$-valued point of $R$ with $s(r) = u_2$, $t(r) = u_1$ and moreover", "$c'(f(r), r'_2) = r'_1$. Hence the arrow", "$$", "(1, r) : (u'_2, u_2, r'_2) \\to (u'_1, u_1, r'_1)", "$$", "is completely determined by knowing the pair $(r, r'_2)$. Thus the functor", "of arrows is represented by $R''$, and moreover the morphisms $s''$, $t''$,", "and $c''$ clearly correspond to source, target and composition in", "the $2$-fibre product $\\mathcal{S}_U \\times_{[U'/_{\\!p}R']} [U/_{\\!p}R]$." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-functorial", "spaces-groupoids-lemma-quotient-stack-arrows", "stacks-lemma-stackification-fibre-product-categories-fibred-in-groupoids" ], "ref_ids": [ 9321, 9319, 8968 ] } ], "ref_ids": [ 9321 ] }, { "id": 9323, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-stack-morphisms", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-stack-morphisms", "contents": [ "Assume $B \\to S$, $(U, R, s, t, c)$ and $\\pi : \\mathcal{S}_U \\to [U/R]$", "are as in", "Lemma \\ref{lemma-quotient-stack-arrows}.", "Let $S'$ be a scheme over $S$.", "Let $x, y \\in \\Ob([U/R]_{S'})$ be objects of the", "quotient stack over $S'$. If $x = \\pi(x')$ and $y = \\pi(y')$ for", "some morphisms $x', y' : S' \\to U$, then", "$$", "\\mathit{Isom}(x, y) = S' \\times_{(y', x'), U \\times_S U} R", "$$", "as sheaves over $S'$." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-arrows" ], "proofs": [ { "contents": [ "Let $[U/_{\\!p}R]$ be the category fibred in groupoids associated to", "the presheaf in groupoids (\\ref{equation-quotient-stack}) as in the proof of", "Lemma \\ref{lemma-quotient-stack-arrows}.", "By construction the sheaf $\\mathit{Isom}(x, y)$ is the sheaf associated", "to the presheaf $\\mathit{Isom}(x', y')$. On the other hand, by definition", "of morphisms in $[U/_{\\!p}R]$ we have", "$$", "\\mathit{Isom}(x', y') = S' \\times_{(y', x'), U \\times_S U} R", "$$", "and the right hand side is an algebraic space, therefore a sheaf." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-arrows" ], "ref_ids": [ 9319 ] } ], "ref_ids": [ 9319 ] }, { "id": 9324, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-stack-2-cartesian", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-stack-2-cartesian", "contents": [ "Assume $B \\to S$, $(U, R, s, t, c)$, and $\\pi : \\mathcal{S}_U \\to [U/R]$", "are as in", "Lemma \\ref{lemma-quotient-stack-arrows}.", "The $2$-commutative square", "$$", "\\xymatrix{", "\\mathcal{S}_R \\ar[r]_s \\ar[d]_t & \\mathcal{S}_U \\ar[d]^\\pi \\\\", "\\mathcal{S}_U \\ar[r]^-\\pi & [U/R]", "}", "$$", "of", "Lemma \\ref{lemma-quotient-stack-2-arrow}", "is a $2$-fibre product of stacks in groupoids of $(\\Sch/S)_{fppf}$." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-arrows", "spaces-groupoids-lemma-quotient-stack-2-arrow" ], "proofs": [ { "contents": [ "According to", "Stacks, Lemma \\ref{stacks-lemma-2-product-stacks-in-groupoids}", "the lemma makes sense. It also tells us that we have to show that", "the functor", "$$", "\\mathcal{S}_R \\longrightarrow \\mathcal{S}_U \\times_{[U/R]} \\mathcal{S}_U", "$$", "which maps $r : T \\to R$ to $(T, t(r), s(r), \\alpha(r))$ is an equivalence,", "where the right hand side is the $2$-fibre product as described in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.", "This is, after spelling out the definitions, exactly the content of", "Lemma \\ref{lemma-quotient-stack-morphisms}. (Alternative proof: Work out", "the meaning of", "Lemma \\ref{lemma-cartesian-square-of-morphism}", "in this situation will give you the result also.)" ], "refs": [ "stacks-lemma-2-product-stacks-in-groupoids", "categories-lemma-2-product-categories-over-C", "spaces-groupoids-lemma-quotient-stack-morphisms", "spaces-groupoids-lemma-cartesian-square-of-morphism" ], "ref_ids": [ 8949, 12280, 9323, 9322 ] } ], "ref_ids": [ 9319, 9320 ] }, { "id": 9325, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-stack-isom", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-stack-isom", "contents": [ "Assume $B \\to S$ and $(U, R, s, t, c)$ are as in", "Definition \\ref{definition-quotient-stack} (1).", "For any scheme $T$ over $S$ and objects $x, y$ of $[U/R]$ over $T$", "the sheaf $\\mathit{Isom}(x, y)$ on $(\\Sch/T)_{fppf}$ has", "the following property: There exists a fppf covering", "$\\{T_i \\to T\\}_{i \\in I}$ such that", "$\\mathit{Isom}(x, y)|_{(\\Sch/T_i)_{fppf}}$", "is representable by an algebraic space." ], "refs": [ "spaces-groupoids-definition-quotient-stack" ], "proofs": [ { "contents": [ "Follows immediately from", "Lemma \\ref{lemma-quotient-stack-morphisms}", "and the fact that both $x$ and $y$ locally in the fppf", "topology come from objects of $\\mathcal{S}_U$ by construction", "of the quotient stack." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-morphisms" ], "ref_ids": [ 9323 ] } ], "ref_ids": [ 9354 ] }, { "id": 9326, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-stack-cocycle", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-stack-cocycle", "contents": [ "Assumptions and notation as in", "Lemmas \\ref{lemma-quotient-stack-arrows} and", "\\ref{lemma-quotient-stack-2-arrow}.", "The vertical composition of", "$$", "\\xymatrix@C=15pc{", "\\mathcal{S}_{R \\times_{s, U, t} R}", "\\ruppertwocell^{\\pi \\circ s \\circ \\text{pr}_1 = \\pi \\circ s \\circ c}{\\ \\ \\ \\ \\ \\ \\alpha \\star \\text{id}_{\\text{pr}_1}}", "\\ar[r]_(.3){\\pi \\circ t \\circ \\text{pr}_1 = \\pi \\circ s \\circ \\text{pr}_0}", "\\rlowertwocell_{\\pi \\circ t \\circ \\text{pr}_0 = \\pi \\circ t \\circ c}{\\ \\ \\ \\ \\ \\ \\alpha \\star \\text{id}_{\\text{pr}_0}}", "&", "[U/R]", "}", "$$", "is the $2$-morphism $\\alpha \\star \\text{id}_c$. In a formula", "$\\alpha \\star \\text{id}_c =", "(\\alpha \\star \\text{id}_{\\text{pr}_0})", "\\circ", "(\\alpha \\star \\text{id}_{\\text{pr}_1})", "$." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-arrows", "spaces-groupoids-lemma-quotient-stack-2-arrow" ], "proofs": [ { "contents": [ "We make two remarks:", "\\begin{enumerate}", "\\item The formula", "$\\alpha \\star \\text{id}_c = (\\alpha \\star \\text{id}_{\\text{pr}_0}) \\circ", "(\\alpha \\star \\text{id}_{\\text{pr}_1})$ only makes sense if you realize", "the {\\it equalities} $\\pi \\circ s \\circ \\text{pr}_1 = \\pi \\circ s \\circ c$,", "$\\pi \\circ t \\circ \\text{pr}_1 = \\pi \\circ s \\circ \\text{pr}_0$, and", "$\\pi \\circ t \\circ \\text{pr}_0 = \\pi \\circ t \\circ c$. Namely, the second", "one implies the vertical composition $\\circ$ makes sense, and the other", "two guarantee the two sides of the formula are $2$-morphisms with the", "same source and target.", "\\item The reason the lemma holds is that composition in the", "category fibred in groupoids $[U/_{\\!p}R]$ associated to the presheaf", "in groupoids (\\ref{equation-quotient-stack}) comes from the composition", "law $c : R \\times_{s, U, t} R \\to R$.", "\\end{enumerate}", "We omit the proof of the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9319, 9320 ] }, { "id": 9327, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-stack-2-coequalizer", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-stack-2-coequalizer", "contents": [ "Assumptions and notation as in", "Lemmas \\ref{lemma-quotient-stack-arrows} and", "\\ref{lemma-quotient-stack-2-arrow}.", "The $2$-commutative diagram of Lemma \\ref{lemma-quotient-stack-2-arrow}", "is a $2$-coequalizer in the following sense:", "Given", "\\begin{enumerate}", "\\item a stack in groupoids $\\mathcal{X}$ over $(\\Sch/S)_{fppf}$,", "\\item a $1$-morphism $f : \\mathcal{S}_U \\to \\mathcal{X}$, and", "\\item a $2$-arrow $\\beta : f \\circ s \\to f \\circ t$", "\\end{enumerate}", "such that", "$$", "\\beta \\star \\text{id}_c", "=", "(\\beta \\star \\text{id}_{\\text{pr}_0})", "\\circ", "(\\beta \\star \\text{id}_{\\text{pr}_1})", "$$", "then there exists a $1$-morphism $[U/R] \\to \\mathcal{X}$ which makes the", "diagram", "$$", "\\xymatrix{", "\\mathcal{S}_R \\ar[r]_s \\ar[d]^t & \\mathcal{S}_U \\ar[d] \\ar[ddr]^f \\\\", "\\mathcal{S}_U \\ar[r] \\ar[rrd]_f & [U/R] \\ar[rd] \\\\", "& & \\mathcal{X}", "}", "$$", "$2$-commute." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-arrows", "spaces-groupoids-lemma-quotient-stack-2-arrow", "spaces-groupoids-lemma-quotient-stack-2-arrow" ], "proofs": [ { "contents": [ "Suppose given $\\mathcal{X}$, $f$ and $\\beta$ as in the lemma. By", "Stacks, Lemma \\ref{stacks-lemma-stackify-groupoids-universal-property}", "it suffices to construct a $1$-morphism $g : [U/_{\\!p}R] \\to \\mathcal{X}$.", "First we note that the $1$-morphism", "$\\mathcal{S}_U \\to [U/_{\\!p}R]$ is bijective on objects.", "Hence on objects we can set $g(x) = f(x)$ for", "$x \\in \\Ob(\\mathcal{S}_U) = \\Ob([U/_{\\!p}R])$.", "A morphism $\\varphi : x \\to y$ of $[U/_{\\!p}R]$ arises from a", "commutative diagram", "$$", "\\xymatrix{", "S_2 \\ar[dd]_h \\ar[r]_x \\ar[dr]_\\varphi & U \\\\", "& R \\ar[u]_s \\ar[d]^t \\\\", "S_1 \\ar[r]^y & U.", "}", "$$", "Thus we can set $g(\\varphi)$ equal to the composition", "$$", "\\xymatrix{", "f(x) \\ar@{=}[r] \\ar[rrrrrd] &", "f(s \\circ \\varphi) \\ar@{=}[r] &", "(f \\circ s)(\\varphi) \\ar[r]^\\beta &", "(f \\circ t)(\\varphi) \\ar@{=}[r] &", "f(t \\circ \\varphi) \\ar@{=}[r] &", "f(y \\circ h) \\ar[d] \\\\", "& & & & & f(y).", "}", "$$", "The vertical arrow is the result of applying the functor $f$ to the", "canonical morphism $y \\circ h \\to y$ in $\\mathcal{S}_U$ (namely, the", "strongly cartesian morphism lifting $h$ with target $y$).", "Let us verify that $f$ so defined is compatible with composition, at least", "on fibre categories. So let $S'$ be a scheme over $S$, and let", "$a : S' \\to R \\times_{s, U, t} R$ be a morphism. In this situation", "we set $x = s \\circ \\text{pr}_1 \\circ a = s \\circ c \\circ a$,", "$y = t \\circ \\text{pr}_1 \\circ a = s \\circ \\text{pr}_0 \\circ a$, and", "$z = t \\circ \\text{pr}_0 \\circ a = t \\circ \\text{pr}_0 \\circ c$ to", "get a commutative diagram", "$$", "\\xymatrix{", "x \\ar[rr]_{c \\circ a} \\ar[rd]_{\\text{pr}_1 \\circ a} & & z \\\\", "& y \\ar[ru]_{\\text{pr}_0 \\circ a}", "}", "$$", "in the fibre category $[U/_{\\!p}R]_{S'}$. Moreover, any commutative", "triangle in this fibre category has this form. Then we see by our definitions", "above that $f$ maps this to a commutative diagram if and only if", "the diagram", "$$", "\\xymatrix{", "& (f \\circ s)(c \\circ a) \\ar[r]_-{\\beta} &", "(f \\circ t)(c \\circ a) \\ar@{=}[rd] & \\\\", "(f \\circ s)(\\text{pr}_1 \\circ a) \\ar[rd]^\\beta \\ar@{=}[ru] & & &", "(f \\circ t)(\\text{pr}_0 \\circ a) \\\\", "& (f \\circ t)(\\text{pr}_1 \\circ a) \\ar@{=}[r] &", "(f \\circ s)(\\text{pr}_0 \\circ a) \\ar[ru]^\\beta", "}", "$$", "is commutative which is exactly the condition expressed by the formula", "in the lemma. We omit the", "verification that $f$ maps identities to identities and is compatible", "with composition for arbitrary morphisms." ], "refs": [ "stacks-lemma-stackify-groupoids-universal-property" ], "ref_ids": [ 8967 ] } ], "ref_ids": [ 9319, 9320, 9320 ] }, { "id": 9328, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-stack-objects", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-stack-objects", "contents": [ "Assume $B \\to S$ and $(U, R, s, t, c)$ are as in", "Definition \\ref{definition-quotient-stack} (1).", "Let $\\pi : \\mathcal{S}_U \\to [U/R]$ be as in", "Lemma \\ref{lemma-quotient-stack-arrows}.", "Let $T$ be a scheme over $S$.", "\\begin{enumerate}", "\\item for every object $x$ of the fibre category $[U/R]_T$", "there exists an fppf covering $\\{f_i : T_i \\to T\\}_{i \\in I}$ such that", "$f_i^*x \\cong \\pi(u_i)$ for some $u_i \\in U(T_i)$,", "\\item the composition of the isomorphisms", "$$", "\\pi(u_i \\circ \\text{pr}_0)", "=", "\\text{pr}_0^*\\pi(u_i)", "\\cong", "\\text{pr}_0^*f_i^*x", "\\cong", "\\text{pr}_1^*f_j^*x", "\\cong", "\\text{pr}_1^*\\pi(u_j)", "=", "\\pi(u_j \\circ \\text{pr}_1)", "$$", "are of the form $\\pi(r_{ij})$ for certain morphisms", "$r_{ij} : T_i \\times_T T_j \\to R$,", "\\item the system $(u_i, r_{ij})$ forms", "a $[U/R]$-descent datum as defined above,", "\\item any $[U/R]$-descent datum $(u_i, r_{ij})$ arises in this manner,", "\\item if $x$ corresponds to $(u_i, r_{ij})$ as above, and", "$y \\in \\Ob([U/R]_T)$ corresponds to $(u'_i, r'_{ij})$", "then there is a canonical bijection", "$$", "\\Mor_{[U/R]_T}(x, y)", "\\longleftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{morphisms }(u_i, r_{ij}) \\to (u'_i, r'_{ij})\\\\", "\\text{of }[U/R]\\text{-descent data}", "\\end{matrix}", "\\right\\}", "$$", "\\item this correspondence is compatible with refinements of fppf coverings.", "\\end{enumerate}" ], "refs": [ "spaces-groupoids-definition-quotient-stack", "spaces-groupoids-lemma-quotient-stack-arrows" ], "proofs": [ { "contents": [ "Statement (1) is part of the construction of the stackyfication.", "Part (2) follows from", "Lemma \\ref{lemma-quotient-stack-morphisms}.", "We omit the verification of (3).", "Part (4) is a translation of the fact that in a stack all descent data", "are effective.", "We omit the verifications of (5) and (6)." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-morphisms" ], "ref_ids": [ 9323 ] } ], "ref_ids": [ 9354, 9319 ] }, { "id": 9329, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-stack-restrict", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-stack-restrict", "contents": [ "Notation and assumption as in", "Lemma \\ref{lemma-quotient-stack-functorial}.", "The morphism of quotient stacks", "$$", "[f] : [U/R] \\longrightarrow [U'/R']", "$$", "is fully faithful if and only if $R$ is the restriction of", "$R'$ via the morphism $f : U \\to U'$." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-functorial" ], "proofs": [ { "contents": [ "Let $x, y$ be objects of $[U/R]$ over a scheme $T/S$.", "Let $x', y'$ be the images of $x, y$ in the category $[U'/R']_T$.", "The functor $[f]$ is fully faithful if and only if the map of sheaves", "$$", "\\mathit{Isom}(x, y) \\longrightarrow \\mathit{Isom}(x', y')", "$$", "is an isomorphism for every $T, x, y$. We may test this locally on $T$", "(in the fppf topology). Hence, by", "Lemma \\ref{lemma-quotient-stack-objects}", "we may assume that $x, y$ come from $a, b \\in U(T)$.", "In that case we see that $x', y'$ correspond to $f \\circ a, f \\circ b$. By", "Lemma \\ref{lemma-quotient-stack-morphisms}", "the displayed map of sheaves in this case becomes", "$$", "T \\times_{(a, b), U \\times_B U} R", "\\longrightarrow", "T \\times_{f \\circ a, f \\circ b, U' \\times_B U'} R'.", "$$", "This is an isomorphism if $R$ is the restriction, because in that case", "$R = (U \\times_B U) \\times_{U' \\times_B U'} R'$, see", "Lemma \\ref{lemma-restrict-groupoid-relation}", "and its proof. Conversely, if the last displayed map is an isomorphism", "for all $T, a, b$, then it follows that", "$R = (U \\times_B U) \\times_{U' \\times_B U'} R'$, i.e.,", "$R$ is the restriction of $R'$." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-objects", "spaces-groupoids-lemma-quotient-stack-morphisms", "spaces-groupoids-lemma-restrict-groupoid-relation" ], "ref_ids": [ 9328, 9323, 9313 ] } ], "ref_ids": [ 9321 ] }, { "id": 9330, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-stack-restrict-equivalence", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-stack-restrict-equivalence", "contents": [ "Notation and assumption as in", "Lemma \\ref{lemma-quotient-stack-functorial}.", "The morphism of quotient stacks", "$$", "[f] : [U/R] \\longrightarrow [U'/R']", "$$", "is an equivalence if and only if", "\\begin{enumerate}", "\\item $(U, R, s, t, c)$ is the restriction of $(U', R', s', t', c')$", "via $f : U \\to U'$, and", "\\item the map", "$$", "\\xymatrix{", "U \\times_{f, U', t'} R' \\ar[r]_-{\\text{pr}_1} \\ar@/^3ex/[rr]^h", "& R' \\ar[r]_{s'} & U'", "}", "$$", "is a surjection of sheaves.", "\\end{enumerate}", "Part (2) holds for example if $\\{h : U \\times_{f, U', t'} R' \\to U'\\}$", "is an fppf covering, or if $f : U \\to U'$ is a surjection of sheaves, or if", "$\\{f : U \\to U'\\}$ is an fppf covering." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-functorial" ], "proofs": [ { "contents": [ "We already know that part (1) is equivalent to", "fully faithfulness by", "Lemma \\ref{lemma-quotient-stack-restrict}.", "Hence we may assume that (1) holds and that $[f]$ is fully faithful.", "Our goal is to show, under these assumptions, that $[f]$ is an equivalence", "if and only if (2) holds. We may use", "Stacks, Lemma \\ref{stacks-lemma-characterize-essentially-surjective-when-ff}", "which characterizes equivalences.", "\\medskip\\noindent", "Assume (2). We will use", "Stacks, Lemma \\ref{stacks-lemma-characterize-essentially-surjective-when-ff}", "to prove $[f]$ is an equivalence.", "Suppose that $T$ is a scheme and $x' \\in \\Ob([U'/R']_T)$.", "There exists a covering $\\{g_i : T_i \\to T\\}$ such that $g_i^*x'$", "is the image of some element $a'_i \\in U'(T_i)$, see", "Lemma \\ref{lemma-quotient-stack-objects}.", "Hence we may assume that $x'$ is the image of $a' \\in U'(T)$.", "By the assumption that $h$ is a surjection of sheaves, we", "can find an fppf covering $\\{\\varphi_i : T_i \\to T\\}$ and morphisms", "$b_i : T_i \\to U \\times_{g, U', t'} R'$ such that", "$a' \\circ \\varphi_i = h \\circ b_i$. Denote", "$a_i = \\text{pr}_0 \\circ b_i : T_i \\to U$. Then we see that", "$a_i \\in U(T_i)$ maps to $f \\circ a_i \\in U'(T_i)$ and", "that $f \\circ a_i \\cong_{T_i} h \\circ b_i = a' \\circ \\varphi_i$,", "where $\\cong_{T_i}$ denotes isomorphism in the fibre category", "$[U'/R']_{T_i}$. Namely, the element of $R'(T_i)$ giving the isomorphism", "is $\\text{pr}_1 \\circ b_i$. This means that the restriction", "of $x$ to $T_i$ is in the essential image of the functor", "$[U/R]_{T_i} \\to [U'/R']_{T_i}$ as desired.", "\\medskip\\noindent", "Assume $[f]$ is an equivalence. Let $\\xi' \\in [U'/R']_{U'}$ denote the", "object corresponding to the identity morphism of $U'$. Applying", "Stacks, Lemma \\ref{stacks-lemma-characterize-essentially-surjective-when-ff}", "we see there exists an fppf covering $\\mathcal{U}' = \\{g'_i : U'_i \\to U'\\}$", "such that $(g'_i)^*\\xi' \\cong [f](\\xi_i)$ for some $\\xi_i$ in $[U/R]_{U'_i}$.", "After refining the covering $\\mathcal{U}'$ (using", "Lemma \\ref{lemma-quotient-stack-objects})", "we may assume $\\xi_i$ comes from a morphism $a_i : U'_i \\to U$.", "The fact that $[f](\\xi_i) \\cong (g'_i)^*\\xi'$ means that, after possibly", "refining the covering $\\mathcal{U}'$ once more, there exist morphisms", "$r'_i : U'_i \\to R'$ with $t' \\circ r'_i = f \\circ a_i$ and", "$s' \\circ r'_i = \\text{id}_{U'} \\circ g'_i$. Picture", "$$", "\\xymatrix{", "U \\ar[d]^f & & U'_i \\ar[ll]^{a_i} \\ar[ld]_{r'_i} \\ar[d]^{g'_i} \\\\", "U' & R' \\ar[l]_{t'} \\ar[r]^{s'} & U'", "}", "$$", "Thus $(a_i, r'_i) : U'_i \\to U \\times_{g, U', t'} R'$ are morphisms", "such that $h \\circ (a_i, r'_i) = g'_i$ and we conclude that", "$\\{h : U \\times_{g, U', t'} R' \\to U'\\}$ can be refined by the", "fppf covering $\\mathcal{U}'$ which means that $h$ induces a surjection", "of sheaves, see", "Topologies on Spaces, Lemma", "\\ref{spaces-topologies-lemma-fppf-covering-surjective}.", "\\medskip\\noindent", "If $\\{h\\}$ is an fppf covering, then it induces a surjection of sheaves, see", "Topologies on Spaces,", "Lemma \\ref{spaces-topologies-lemma-fppf-covering-surjective}.", "If $U' \\to U$ is surjective, then also $h$ is surjective as $s$ has a section", "(namely the neutral element $e$ of the groupoid in algebraic spaces)." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-restrict", "stacks-lemma-characterize-essentially-surjective-when-ff", "stacks-lemma-characterize-essentially-surjective-when-ff", "spaces-groupoids-lemma-quotient-stack-objects", "stacks-lemma-characterize-essentially-surjective-when-ff", "spaces-groupoids-lemma-quotient-stack-objects", "spaces-topologies-lemma-fppf-covering-surjective", "spaces-topologies-lemma-fppf-covering-surjective" ], "ref_ids": [ 9329, 8945, 8945, 9328, 8945, 9328, 3667, 3667 ] } ], "ref_ids": [ 9321 ] }, { "id": 9331, "type": "theorem", "label": "spaces-groupoids-lemma-criterion-fibre-product", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-criterion-fibre-product", "contents": [ "Notation and assumption as in", "Lemma \\ref{lemma-quotient-stack-functorial}.", "Assume that", "$$", "\\xymatrix{", "R \\ar[d]_s \\ar[r]_f & R' \\ar[d]^{s'} \\\\", "U \\ar[r]^f & U'", "}", "$$", "is cartesian. Then", "$$", "\\xymatrix{", "\\mathcal{S}_U \\ar[d] \\ar[r] & [U/R] \\ar[d]^{[f]} \\\\", "\\mathcal{S}_{U'} \\ar[r] & [U'/R']", "}", "$$", "is a $2$-fibre product square." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-functorial" ], "proofs": [ { "contents": [ "Applying the inverse isomorphisms $i : R \\to R$ and $i' : R' \\to R'$", "to the (first) cartesian diagram of the statement of the lemma we see", "that", "$$", "\\xymatrix{", "R \\ar[d]_t \\ar[r]_f & R' \\ar[d]^{t'} \\\\", "U \\ar[r]^f & U'", "}", "$$", "is cartesian as well. By", "Lemma \\ref{lemma-cartesian-square-of-morphism}", "we have a $2$-fibre square", "$$", "\\xymatrix{", "[U''/R''] \\ar[d] \\ar[r] & [U/R] \\ar[d] \\\\", "\\mathcal{S}_{U'} \\ar[r] & [U'/R']", "}", "$$", "where $U'' = U \\times_{f, U', t'} R'$ and", "$R'' = R \\times_{f \\circ s, U', t'} R'$. By the above", "we see that $(t, f) : R \\to U''$ is an isomorphism, and that", "$$", "R'' =", "R \\times_{f \\circ s, U', t'} R' =", "R \\times_{s, U} U \\times_{f, U', t'} R' =", "R \\times_{s, U, t} \\times R.", "$$", "Explicitly the isomorphism $R \\times_{s, U, t} R \\to R''$ is given by", "the rule $(r_0, r_1) \\mapsto (r_0, f(r_1))$.", "Moreover, $s'', t'', c''$ translate into the maps", "$$", "R \\times_{s, U, t} R \\to R,", "\\quad", "s''(r_0, r_1) = r_1, \\quad t''(r_0, r_1) = c(r_0, r_1)", "$$", "and", "$$", "\\begin{matrix}", "c'' : &", "(R \\times_{s, U, t} R) \\times_{s'', R, t''} (R \\times_{s, U, t} R)", "&", "\\longrightarrow", "&", "R \\times_{s, U, t} R, \\\\", "&", "((r_0, r_1), (r_2, r_3)) &", "\\longmapsto & (c(r_0, r_2), r_3).", "\\end{matrix}", "$$", "Precomposing with the isomorphism", "$$", "R \\times_{s, U, s} R \\longrightarrow R \\times_{s, U, t} R,", "\\quad", "(r_0, r_1) \\longmapsto (c(r_0, i(r_1)), r_1)", "$$", "we see that $t''$ and $s''$ turn into", "$\\text{pr}_0$ and $\\text{pr}_1$ and that $c''$ turns into", "$\\text{pr}_{02} :", "R \\times_{s, U, s} R \\times_{s, U, s} R \\to R \\times_{s, U, s} R$.", "Hence we see that there is an isomorphism", "$[U''/R''] \\cong [R/R \\times_{s, U, s} R]$ where as a groupoid", "in algebraic spaces $(R, R \\times_{s, U, s} R, s'', t'', c'')$", "is the restriction of the trivial groupoid", "$(U, U, \\text{id}, \\text{id}, \\text{id})$ via $s : R \\to U$.", "Since $s : R \\to U$ is a surjection of fppf sheaves (as it has a", "right inverse) the morphism", "$$", "[U''/R''] \\cong [R/R \\times_{s, U, s} R]", "\\longrightarrow", "[U/U] = \\mathcal{S}_U", "$$", "is an equivalence by", "Lemma \\ref{lemma-quotient-stack-restrict-equivalence}.", "This proves the lemma." ], "refs": [ "spaces-groupoids-lemma-cartesian-square-of-morphism", "spaces-groupoids-lemma-quotient-stack-restrict-equivalence" ], "ref_ids": [ 9322, 9330 ] } ], "ref_ids": [ 9321 ] }, { "id": 9332, "type": "theorem", "label": "spaces-groupoids-lemma-presentation-inertia", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-presentation-inertia", "contents": [ "Assume $B \\to S$ and $(U, R, s, t, c)$ as in", "Definition \\ref{definition-quotient-stack} (1).", "Let $G/U$ be the stabilizer group algebraic space of the groupoid", "$(U, R, s, t, c, e, i)$, see", "Definition \\ref{definition-stabilizer-groupoid}.", "Set $R' = R \\times_{s, U} G$ and set", "\\begin{enumerate}", "\\item $s' : R' \\to G$, $(r, g) \\mapsto g$,", "\\item $t' : R' \\to G$, $(r, g) \\mapsto c(r, c(g, i(r)))$,", "\\item $c' : R' \\times_{s', G, t'} R' \\to R'$,", "$((r_1, g_1), (r_2, g_2) \\mapsto (c(r_1, r_2), g_1)$.", "\\end{enumerate}", "Then $(G, R', s', t', c')$ is a groupoid in algebraic spaces over $B$", "and", "$$", "\\mathcal{I}_{[U/R]} = [G/ R'].", "$$", "i.e., the associated quotient stack is the inertia stack of $[U/R]$." ], "refs": [ "spaces-groupoids-definition-quotient-stack", "spaces-groupoids-definition-stabilizer-groupoid" ], "proofs": [ { "contents": [ "By", "Stacks, Lemma \\ref{stacks-lemma-stackification-inertia}", "it suffices to prove that $\\mathcal{I}_{[U/_{\\!p}R]} = [G/_{\\!p} R']$.", "Let $T$ be a scheme over $S$. Recall that an object of the inertia fibred", "category of $[U/_{\\!p}R]$ over $T$ is given by a pair", "$(x, g)$ where $x$ is an object of $[U/_{\\!\\!p}R]$ over $T$", "and $g$ is an automorphism of $x$ in its fibre category over $T$.", "In other words, $x : T \\to U$ and $g : T \\to R$ such that", "$x = s \\circ g = t \\circ g$. This means exactly that", "$g : T \\to G$. A morphism in the inertia fibred category", "from $(x, g) \\to (y, h)$ over $T$ is given by", "$r : T \\to R$ such that $s(r) = x$, $t(r) = y$", "and $c(r, g) = c(h, r)$, see the commutative diagram in", "Categories, Lemma \\ref{categories-lemma-inertia-fibred-category}.", "In a formula", "$$", "h = c(r, c(g, i(r))) = c(c(r, g), i(r)).", "$$", "The notation $s(r)$, etc is a short hand for $s \\circ r$, etc.", "The composition of $r_1 : (x_2, g_2) \\to (x_1, g_1)$", "and $r_2 : (x_1, g_1) \\to (x_2, g_2)$ is", "$c(r_1, r_2) : (x_1, g_1) \\to (x_3, g_3)$.", "\\medskip\\noindent", "Note that in the above we could have written $g$ in stead of $(x, g)$", "for an object of $\\mathcal{I}_{[U/_{\\!p}R]}$ over $T$ as $x$ is the", "image of $g$ under the structure morphism $G \\to U$. Then the morphisms", "$g \\to h$ in $\\mathcal{I}_{[U/_{\\!p}R]}$ over $T$ correspond exactly", "to morphisms $r' : T \\to R'$ with $s'(r') = g$ and $t'(r') = h$.", "Moreover, the composition corresponds to the rule explained in (3).", "Thus the lemma is proved." ], "refs": [ "stacks-lemma-stackification-inertia", "categories-lemma-inertia-fibred-category" ], "ref_ids": [ 8965, 12292 ] } ], "ref_ids": [ 9354, 9349 ] }, { "id": 9333, "type": "theorem", "label": "spaces-groupoids-lemma-2-cartesian-inertia", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-2-cartesian-inertia", "contents": [ "Assume $B \\to S$ and $(U, R, s, t, c)$ as in", "Definition \\ref{definition-quotient-stack} (1).", "Let $G/U$ be the stabilizer group algebraic space of the groupoid", "$(U, R, s, t, c, e, i)$, see", "Definition \\ref{definition-stabilizer-groupoid}.", "There is a canonical $2$-cartesian diagram", "$$", "\\xymatrix{", "\\mathcal{S}_G \\ar[r] \\ar[d] & \\mathcal{S}_U \\ar[d] \\\\", "\\mathcal{I}_{[U/R]} \\ar[r] & [U/R]", "}", "$$", "of stacks in groupoids of $(\\Sch/S)_{fppf}$." ], "refs": [ "spaces-groupoids-definition-quotient-stack", "spaces-groupoids-definition-stabilizer-groupoid" ], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-criterion-fibre-product}", "it suffices to prove that the morphism", "$s' : R' \\to G$ of", "Lemma \\ref{lemma-presentation-inertia}", "isomorphic to the base change of $s$ by the structure", "morphism $G \\to U$. This base change property is", "clear from the construction of $s'$." ], "refs": [ "spaces-groupoids-lemma-criterion-fibre-product", "spaces-groupoids-lemma-presentation-inertia" ], "ref_ids": [ 9331, 9332 ] } ], "ref_ids": [ 9354, 9349 ] }, { "id": 9334, "type": "theorem", "label": "spaces-groupoids-lemma-when-gerbe", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-when-gerbe", "contents": [ "Notation and assumption as in", "Lemma \\ref{lemma-quotient-stack-functorial}.", "The morphism of quotient stacks", "$$", "[f] : [U/R] \\longrightarrow [U'/R']", "$$", "turns $[U/R]$ into a gerbe over $[U'/R']$ if $f : U \\to U'$ and", "$R \\to R'|_U$ are surjective maps of fppf sheaves. Here $R'|_U$ is", "the restriction of $R'$ to $U$ via $f : U \\to U'$." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-functorial" ], "proofs": [ { "contents": [ "We will verify that", "Stacks, Lemma \\ref{stacks-lemma-when-gerbe}", "properties (2) (a) and (2) (b) hold. Property (2)(a) holds because $U \\to U'$", "is a surjective map of sheaves (use", "Lemma \\ref{lemma-quotient-stack-objects}", "to see that objects in $[U'/R']$ locally come from $U'$).", "To prove (2)(b) let $x, y$ be objects of $[U/R]$ over a scheme $T/S$.", "Let $x', y'$ be the images of $x, y$ in the category $[U'/'R]_T$.", "Condition (2)(b) requires us to check the map of sheaves", "$$", "\\mathit{Isom}(x, y) \\longrightarrow \\mathit{Isom}(x', y')", "$$", "on $(\\Sch/T)_{fppf}$ is surjective. To see this we may work", "fppf locally on $T$ and assume that come from $a, b \\in U(T)$.", "In that case we see that $x', y'$ correspond to $f \\circ a, f \\circ b$. By", "Lemma \\ref{lemma-quotient-stack-morphisms}", "the displayed map of sheaves in this case becomes", "$$", "T \\times_{(a, b), U \\times_B U} R", "\\longrightarrow", "T \\times_{f \\circ a, f \\circ b, U' \\times_B U'} R' =", "T \\times_{(a, b), U \\times_B U} R'|_U.", "$$", "Hence the assumption that $R \\to R'|_U$ is a surjective map of fppf sheaves", "on $(\\Sch/S)_{fppf}$ implies the desired surjectivity." ], "refs": [ "stacks-lemma-when-gerbe", "spaces-groupoids-lemma-quotient-stack-objects", "spaces-groupoids-lemma-quotient-stack-morphisms" ], "ref_ids": [ 8975, 9328, 9323 ] } ], "ref_ids": [ 9321 ] }, { "id": 9335, "type": "theorem", "label": "spaces-groupoids-lemma-group-quotient-gerbe", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-group-quotient-gerbe", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let", "$G$ be a group algebraic space over $B$. Endow $B$ with the trivial", "action of $G$. The morphism", "$$", "[B/G] \\longrightarrow \\mathcal{S}_B", "$$", "(Lemma \\ref{lemma-quotient-stack-arrows})", "turns $[B/G]$ into a gerbe over $B$." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-arrows" ], "proofs": [ { "contents": [ "Immediate from", "Lemma \\ref{lemma-when-gerbe}", "as the morphisms $B \\to B$ and $B \\times_B G \\to B$ are surjective", "as morphisms of sheaves." ], "refs": [ "spaces-groupoids-lemma-when-gerbe" ], "ref_ids": [ 9334 ] } ], "ref_ids": [ 9319 ] }, { "id": 9336, "type": "theorem", "label": "spaces-groupoids-lemma-quotient-stack-change-big-site", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-quotient-stack-change-big-site", "contents": [ "Suppose given big sites $\\Sch_{fppf}$ and $\\Sch'_{fppf}$.", "Assume that $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$,", "see Topologies, Section \\ref{topologies-section-change-alpha}.", "Let $S \\in \\Ob(\\Sch_{fppf})$.", "Let $B, U, R \\in \\Sh((\\Sch/S)_{fppf})$ be algebraic spaces,", "and let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "Let $f : (\\Sch'/S)_{fppf} \\to (\\Sch/S)_{fppf}$ the morphism", "of sites corresponding to the inclusion functor", "$u : \\Sch_{fppf} \\to \\Sch'_{fppf}$.", "Then we have a canonical equivalence", "$$", "[f^{-1}U/f^{-1}R]", "\\longrightarrow", "f^{-1}[U/R]", "$$", "of stacks in groupoids over $(\\Sch'/S)_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "Note that $f^{-1}B, f^{-1}U, f^{-1}R \\in \\Sh((\\Sch'/S)_{fppf})$", "are algebraic spaces by", "Spaces, Lemma \\ref{spaces-lemma-change-big-site}", "and hence $(f^{-1}U, f^{-1}R, f^{-1}s, f^{-1}t, f^{-1}c)$", "is a groupoid in algebraic spaces over $f^{-1}B$. Thus the statement makes", "sense.", "\\medskip\\noindent", "The category $u_p[U/_{\\!p}R]$ is the localization of the category", "$u_{pp}[U/_{\\!p}R]$ at right multiplicative system $I$ of morphisms.", "An object of $u_{pp}[U/_{\\!p}R]$ is a triple", "$$", "(T', \\phi : T' \\to T, x)", "$$", "where", "$T' \\in \\Ob((\\Sch'/S)_{fppf})$,", "$T \\in \\Ob((\\Sch/S)_{fppf})$, $\\phi$ is a morphism of schemes", "over $S$, and $x : T \\to U$ is a morphism of sheaves on", "$(\\Sch/S)_{fppf}$. Note that the morphism of", "schemes $\\phi : T' \\to T$ is the same thing as a morphism", "$\\phi : T' \\to u(T)$, and since $u(T)$ represents $f^{-1}T$ it is the", "same thing as a morphism $T' \\to f^{-1}T$. Moreover, as $f^{-1}$ on", "algebraic spaces is fully faithful, see", "Spaces, Lemma \\ref{spaces-lemma-fully-faithful},", "we may think of $x$ as a morphism $x : f^{-1}T \\to f^{-1}U$ as well.", "From now on we will make such identifications without further mention.", "A morphism", "$$", "(a, a', \\alpha) :", "(T'_1, \\phi_1 : T'_1 \\to T_1, x_1)", "\\longrightarrow", "(T'_2, \\phi_2 : T'_2 \\to T_2, x_2)", "$$", "of $u_{pp}[U/_{\\!p}R]$ is a commutative diagram", "$$", "\\xymatrix{", "& & U \\\\", "T'_1 \\ar[d]_{a'} \\ar[r]_{\\phi_1} &", "T_1 \\ar[d]_a \\ar[ru]^{x_1} \\ar[r]_\\alpha &", "R \\ar[d]^t \\ar[u]_s \\\\", "T'_2 \\ar[r]^{\\phi_2} &", "T_2 \\ar[r]^{x_2} &", "U", "}", "$$", "and such a morphism is an element of $I$ if and only if", "$T'_1 = T'_2$ and $a' = \\text{id}$.", "We define a functor", "$$", "u_{pp}[U/_{\\!p}R] \\longrightarrow [f^{-1}U/_{\\!p}f^{-1}R]", "$$", "by the rules", "$$", "(T', \\phi : T' \\to T, x) \\longmapsto (x \\circ \\phi : T' \\to f^{-1}U)", "$$", "on objects and", "$$", "(a, a', \\alpha) \\longmapsto (\\alpha \\circ \\phi_1 : T'_1 \\to f^{-1}R)", "$$", "on morphisms as above. It is clear that elements of $I$ are transformed", "into isomorphisms as $(f^{-1}U, f^{-1}R, f^{-1}s, f^{-1}t, f^{-1}c)$", "is a groupoid in algebraic spaces over $f^{-1}B$. Hence this functor", "factors in a canonical way through a functor", "$$", "u_p[U/_{\\!p}R] \\longrightarrow [f^{-1}U/_{\\!p}f^{-1}R]", "$$", "Applying stackification we obtain a functor of stacks", "$$", "f^{-1}[U/R] \\longrightarrow [f^{-1}U/f^{-1}R]", "$$", "over $(\\Sch'/S)_{fppf}$, as by", "Stacks, Lemma \\ref{stacks-lemma-technical-up}", "the stack $f^{-1}[U/R]$ is the stackification of $u_p[U/_{\\!p}R]$.", "\\medskip\\noindent", "At this point we have a morphism of stacks, and to verify that it is an", "equivalence it suffices to show that it is fully faithful and that", "objects are locally in the essential image, see", "Stacks, Lemmas \\ref{stacks-lemma-characterize-ff} and", "\\ref{stacks-lemma-characterize-essentially-surjective-when-ff}.", "The statement on objects holds as $f^{-1}R$ admits a surjective \\'etale", "morphism $f^{-1}W \\to f^{-1}R$ for some object $W$ of", "$(\\Sch/S)_{fppf}$. To show that the functor is ``full'', it", "suffices to show that morphisms are locally in the image", "of the functor which holds as $f^{-1}U$ admits a surjective \\'etale morphism", "$f^{-1}W \\to f^{-1}U$ for some object $W$ of $(\\Sch/S)_{fppf}$.", "We omit the proof that the functor is faithful." ], "refs": [ "spaces-lemma-change-big-site", "spaces-lemma-fully-faithful", "stacks-lemma-technical-up", "stacks-lemma-characterize-ff", "stacks-lemma-characterize-essentially-surjective-when-ff" ], "ref_ids": [ 8167, 8168, 8988, 8944, 8945 ] } ], "ref_ids": [] }, { "id": 9337, "type": "theorem", "label": "spaces-groupoids-lemma-diagram-diagonal", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-diagram-diagonal", "contents": [ "Let $B \\to S$ be as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "Let $G \\to U$ be the stabilizer group algebraic space.", "The commutative diagram", "$$", "\\xymatrix{", "R \\ar[d]^{\\Delta_{R/U \\times_B U}} \\ar[rrr]_{f \\mapsto (f, s(f))} & & &", "R \\times_{s, U} U \\ar[d] \\ar[r] & U \\ar[d] \\\\", "R \\times_{(U \\times_B U)} R \\ar[rrr]^{(f, g) \\mapsto (f, f^{-1} \\circ g)} & & &", "R \\times_{s, U} G \\ar[r] & G", "}", "$$", "the two left horizontal arrows are isomorphisms", "and the right square is a fibre product square." ], "refs": [], "proofs": [ { "contents": [ "Omitted.", "Exercise in the definitions and the functorial point of", "view in algebraic geometry." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9338, "type": "theorem", "label": "spaces-groupoids-lemma-diagonal", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-lemma-diagonal", "contents": [ "Let $B \\to S$ be as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "Let $G \\to U$ be the stabilizer group algebraic space.", "\\begin{enumerate}", "\\item The following are equivalent", "\\begin{enumerate}", "\\item $j : R \\to U \\times_B U$ is separated,", "\\item $G \\to U$ is separated, and", "\\item $e : U \\to G$ is a closed immersion.", "\\end{enumerate}", "\\item The following are equivalent", "\\begin{enumerate}", "\\item $j : R \\to U \\times_B U$ is locally separated,", "\\item $G \\to U$ is locally separated, and", "\\item $e : U \\to G$ is an immersion.", "\\end{enumerate}", "\\item The following are equivalent", "\\begin{enumerate}", "\\item $j : R \\to U \\times_B U$ is quasi-separated,", "\\item $G \\to U$ is quasi-separated, and", "\\item $e : U \\to G$ is quasi-compact.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The group algebraic space $G \\to U$ is the base change of $R \\to U \\times_B U$", "by the diagonal morphism $U \\to U \\times_B U$, see", "Lemma \\ref{lemma-groupoid-stabilizer}. Hence if", "$j$ is separated (resp.\\ locally separated, resp.\\ quasi-separated),", "then $G \\to U$ is separated (resp.\\ locally separated, resp.\\ quasi-separated).", "See", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-separated}.", "Thus (a) $\\Rightarrow$ (b) in (1), (2), and (3).", "\\medskip\\noindent", "Conversely, if $G \\to U$ is separated", "(resp.\\ locally separated, resp.\\ quasi-separated), then the morphism", "$e : U \\to G$, as a section of the structure morphism $G \\to U$ is a closed", "immersion (resp.\\ an immersion, resp.\\ quasi-compact), see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-section-immersion}.", "Thus (b) $\\Rightarrow$ (c) in (1), (2), and (3).", "\\medskip\\noindent", "If $e$ is a closed immersion (resp.\\ an immersion, resp.\\ quasi-compact)", "then by the result of", "Lemma \\ref{lemma-diagram-diagonal}", "(and", "Spaces, Lemma \\ref{spaces-lemma-base-change-immersions}, and", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-base-change-quasi-compact})", "we see that $\\Delta_{R/U \\times_B U}$ is a closed", "immersion (resp.\\ an immersion, resp.\\ quasi-compact).", "Thus (c) $\\Rightarrow$ (a) in (1), (2), and (3)." ], "refs": [ "spaces-groupoids-lemma-groupoid-stabilizer", "spaces-morphisms-lemma-base-change-separated", "spaces-groupoids-lemma-diagram-diagonal", "spaces-lemma-base-change-immersions", "spaces-morphisms-lemma-base-change-quasi-compact" ], "ref_ids": [ 9310, 4714, 9337, 8161, 4738 ] } ], "ref_ids": [] }, { "id": 9357, "type": "theorem", "label": "spaces-descent-lemma-map-families", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-map-families", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ and", "$\\mathcal{V} = \\{V_j \\to V\\}_{j \\in J}$", "be families of morphisms of algebraic spaces over $S$ with fixed targets.", "Let $(g, \\alpha : I \\to J, (g_i)) : \\mathcal{U} \\to \\mathcal{V}$", "be a morphism of families of maps with fixed target, see", "Sites, Definition \\ref{sites-definition-morphism-coverings}.", "Let $(\\mathcal{F}_j, \\varphi_{jj'})$ be a descent", "datum for quasi-coherent sheaves with respect to the", "family $\\{V_j \\to V\\}_{j \\in J}$. Then", "\\begin{enumerate}", "\\item The system", "$$", "\\left(g_i^*\\mathcal{F}_{\\alpha(i)},", "(g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')}\\right)", "$$", "is a descent datum with respect to the family $\\{U_i \\to U\\}_{i \\in I}$.", "\\item This construction is functorial in the descent datum", "$(\\mathcal{F}_j, \\varphi_{jj'})$.", "\\item Given a second morphism", "$(g', \\alpha' : I \\to J, (g'_i))$", "of families of maps with fixed target", "with $g = g'$ there exists a functorial isomorphism of descent data", "$$", "(g_i^*\\mathcal{F}_{\\alpha(i)},", "(g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')})", "\\cong", "((g'_i)^*\\mathcal{F}_{\\alpha'(i)},", "(g'_i \\times g'_{i'})^*\\varphi_{\\alpha'(i)\\alpha'(i')}).", "$$", "\\end{enumerate}" ], "refs": [ "sites-definition-morphism-coverings" ], "proofs": [ { "contents": [ "Omitted. Hint: The maps", "$g_i^*\\mathcal{F}_{\\alpha(i)} \\to (g'_i)^*\\mathcal{F}_{\\alpha'(i)}$", "which give the isomorphism of descent data in part (3)", "are the pullbacks of the maps $\\varphi_{\\alpha(i)\\alpha'(i)}$ by the", "morphisms $(g_i, g'_i) : U_i \\to V_{\\alpha(i)} \\times_V V_{\\alpha'(i)}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8656 ] }, { "id": 9358, "type": "theorem", "label": "spaces-descent-lemma-zariski-descent-effective", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-zariski-descent-effective", "contents": [ "Let $S$ be a scheme. Let $U$ be an algebraic space over $S$.", "Let $\\{U_i \\to U\\}$ be a Zariski covering of $U$, see", "Topologies on Spaces,", "Definition \\ref{spaces-topologies-definition-zariski-covering}.", "Any descent datum on quasi-coherent sheaves", "for the family $\\mathcal{U} = \\{U_i \\to U\\}$ is", "effective. Moreover, the functor from the category of", "quasi-coherent $\\mathcal{O}_U$-modules to the category", "of descent data with respect to $\\{U_i \\to U\\}$ is fully faithful." ], "refs": [ "spaces-topologies-definition-zariski-covering" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 3681 ] }, { "id": 9359, "type": "theorem", "label": "spaces-descent-lemma-finite-type-descends", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-finite-type-descends", "contents": [ "Let $X$ be an algebraic space over a scheme $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that", "each $f_i^*\\mathcal{F}$ is a finite type $\\mathcal{O}_{X_i}$-module.", "Then $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "This follows from the case of schemes, see", "Descent, Lemma \\ref{descent-lemma-finite-type-descends},", "by \\'etale localization." ], "refs": [ "descent-lemma-finite-type-descends" ], "ref_ids": [ 14612 ] } ], "ref_ids": [] }, { "id": 9360, "type": "theorem", "label": "spaces-descent-lemma-finite-presentation-descends", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-finite-presentation-descends", "contents": [ "Let $X$ be an algebraic space over a scheme $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that", "each $f_i^*\\mathcal{F}$ is an $\\mathcal{O}_{X_i}$-module of finite", "presentation. Then $\\mathcal{F}$ is an $\\mathcal{O}_X$-module", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "This follows from the case of schemes, see", "Descent, Lemma \\ref{descent-lemma-finite-presentation-descends},", "by \\'etale localization." ], "refs": [ "descent-lemma-finite-presentation-descends" ], "ref_ids": [ 14614 ] } ], "ref_ids": [] }, { "id": 9361, "type": "theorem", "label": "spaces-descent-lemma-flat-descends", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-flat-descends", "contents": [ "Let $X$ be an algebraic space over a scheme $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that", "each $f_i^*\\mathcal{F}$ is a flat $\\mathcal{O}_{X_i}$-module.", "Then $\\mathcal{F}$ is a flat $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "This follows from the case of schemes, see", "Descent, Lemma \\ref{descent-lemma-flat-descends},", "by \\'etale localization." ], "refs": [ "descent-lemma-flat-descends" ], "ref_ids": [ 14616 ] } ], "ref_ids": [] }, { "id": 9362, "type": "theorem", "label": "spaces-descent-lemma-finite-locally-free-descends", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-finite-locally-free-descends", "contents": [ "Let $X$ be an algebraic space over a scheme $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that", "each $f_i^*\\mathcal{F}$ is a finite locally free $\\mathcal{O}_{X_i}$-module.", "Then $\\mathcal{F}$ is a finite locally free $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "This follows from the case of schemes, see", "Descent, Lemma \\ref{descent-lemma-finite-locally-free-descends},", "by \\'etale localization." ], "refs": [ "descent-lemma-finite-locally-free-descends" ], "ref_ids": [ 14617 ] } ], "ref_ids": [] }, { "id": 9363, "type": "theorem", "label": "spaces-descent-lemma-locally-projective-descends", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-locally-projective-descends", "contents": [ "Let $X$ be an algebraic space over a scheme $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that", "each $f_i^*\\mathcal{F}$ is a locally projective $\\mathcal{O}_{X_i}$-module.", "Then $\\mathcal{F}$ is a locally projective $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "This follows from the case of schemes, see", "Descent, Lemma \\ref{descent-lemma-locally-projective-descends},", "by \\'etale localization." ], "refs": [ "descent-lemma-locally-projective-descends" ], "ref_ids": [ 14618 ] } ], "ref_ids": [] }, { "id": 9364, "type": "theorem", "label": "spaces-descent-lemma-finite-over-finite-module", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-finite-over-finite-module", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume $f$ is a finite morphism.", "Then $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type", "if and only if $f_*\\mathcal{F}$ is an $\\mathcal{O}_Y$-module of finite", "type." ], "refs": [], "proofs": [ { "contents": [ "As $f$ is finite it is representable. Choose a scheme $V$ and a surjective", "\\'etale morphism $V \\to Y$. Then $U = V \\times_Y X$ is a scheme with", "a surjective \\'etale morphism towards $X$ and a finite morphism", "$\\psi : U \\to V$ (the base change of $f$). Since", "$\\psi_*(\\mathcal{F}|_U) = f_*\\mathcal{F}|_V$", "the result of the lemma follows immediately from the schemes version which", "is", "Descent, Lemma \\ref{descent-lemma-finite-over-finite-module}." ], "refs": [ "descent-lemma-finite-over-finite-module" ], "ref_ids": [ 14619 ] } ], "ref_ids": [] }, { "id": 9365, "type": "theorem", "label": "spaces-descent-lemma-finite-finitely-presented-module", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-finite-finitely-presented-module", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume $f$ is finite and of finite presentation.", "Then $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation", "if and only if $f_*\\mathcal{F}$ is an $\\mathcal{O}_Y$-module of finite", "presentation." ], "refs": [], "proofs": [ { "contents": [ "As $f$ is finite it is representable. Choose a scheme $V$ and a surjective", "\\'etale morphism $V \\to Y$. Then $U = V \\times_Y X$ is a scheme with", "a surjective \\'etale morphism towards $X$ and a finite morphism", "$\\psi : U \\to V$ (the base change of $f$). Since", "$\\psi_*(\\mathcal{F}|_U) = f_*\\mathcal{F}|_V$", "the result of the lemma follows immediately from the schemes version which", "is", "Descent, Lemma \\ref{descent-lemma-finite-finitely-presented-module}." ], "refs": [ "descent-lemma-finite-finitely-presented-module" ], "ref_ids": [ 14620 ] } ], "ref_ids": [] }, { "id": 9366, "type": "theorem", "label": "spaces-descent-lemma-open-fpqc-covering", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-open-fpqc-covering", "contents": [ "Let $S$ be a scheme.", "Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be an fpqc covering", "of algebraic spaces over $S$.", "Suppose that for each $i$ we have an open subspace $W_i \\subset T_i$", "such that for all $i, j \\in I$ we have", "$\\text{pr}_0^{-1}(W_i) = \\text{pr}_1^{-1}(W_j)$ as open", "subspaces of $T_i \\times_T T_j$. Then there exists a unique open subspace", "$W \\subset T$ such that $W_i = f_i^{-1}(W)$ for each $i$." ], "refs": [], "proofs": [ { "contents": [ "By", "Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-refine-fpqc-schemes}", "we may assume each $T_i$ is a scheme.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to T$.", "Then $\\{T_i \\times_T U \\to U\\}$ is an fpqc covering of $U$", "and $T_i \\times_T U$ is a scheme for each $i$. Hence we", "see that the collection of opens $W_i \\times_T U$ comes from a unique", "open subscheme $W' \\subset U$ by", "Descent, Lemma \\ref{descent-lemma-open-fpqc-covering}.", "As $U \\to X$ is open we can define $W \\subset X$ the Zariski", "open which is the image of $W'$, see", "Properties of Spaces, Section \\ref{spaces-properties-section-points}.", "We omit the verification that this works, i.e., that", "$W_i$ is the inverse image of $W$ for each $i$." ], "refs": [ "spaces-topologies-lemma-refine-fpqc-schemes", "descent-lemma-open-fpqc-covering" ], "ref_ids": [ 3680, 14637 ] } ], "ref_ids": [] }, { "id": 9367, "type": "theorem", "label": "spaces-descent-lemma-fpqc-universal-effective-epimorphisms", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-fpqc-universal-effective-epimorphisms", "contents": [ "Let $S$ be a scheme. Let $\\{T_i \\to T\\}$ be an fpqc covering of algebraic", "spaces over $S$, see Topologies on Spaces, Definition", "\\ref{spaces-topologies-definition-fpqc-covering}.", "Then given an algebraic space $B$ over $S$ the sequence", "$$", "\\xymatrix{", "\\Mor_S(T, B) \\ar[r] &", "\\prod\\nolimits_i \\Mor_S(T_i, B) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\prod\\nolimits_{i, j} \\Mor_S(T_i \\times_T T_j, B)", "}", "$$", "is an equalizer diagram.", "In other words, every representable functor on the category of", "algebraic spaces over $S$ satisfies the sheaf condition for", "fpqc coverings." ], "refs": [ "spaces-topologies-definition-fpqc-covering" ], "proofs": [ { "contents": [ "We know this is true if $\\{T_i \\to T\\}$ is an fpqc covering of", "schemes, see Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-sheaf-fpqc}.", "This is the key fact and we encourage the reader to skip the rest", "of the proof which is formal. Choose a scheme $U$ and a surjective", "\\'etale morphism", "$U \\to T$. Let $U_i$ be a scheme and let $U_i \\to T_i \\times_T U$", "be a surjective \\'etale morphism. Then $\\{U_i \\to U\\}$ is an", "fpqc covering. This follows from", "Topologies on Spaces, Lemmas \\ref{spaces-topologies-lemma-fpqc} and", "\\ref{spaces-topologies-lemma-recognize-fpqc-covering}.", "By the above we have the result for $\\{U_i \\to U\\}$.", "\\medskip\\noindent", "What this means is the following: Suppose that $b_i : T_i \\to B$", "is a family of morphisms with", "$b_i \\circ \\text{pr}_0 = b_j \\circ \\text{pr}_1$ as morphisms", "$T_i \\times_T T_j \\to B$. Then we let $a_i : U_i \\to B$ be the", "composition of $U_i \\to T_i$ with $b_i$. By what was said above", "we find a unique morphism $a : U \\to B$ such that", "$a_i$ is the composition of $a$ with $U_i \\to U$.", "The uniqueness guarantees that $a \\circ \\text{pr}_0 = a \\circ \\text{pr}_1$", "as morphisms $U \\times_T U \\to B$. Then since $T = U/(U \\times_T U)$", "as a sheaf, we find that $a$ comes from a unique morphism $b : T \\to B$.", "Chasing diagrams we find that $b$ is the morphism we are looking for." ], "refs": [ "spaces-properties-proposition-sheaf-fpqc", "spaces-topologies-lemma-fpqc", "spaces-topologies-lemma-recognize-fpqc-covering" ], "ref_ids": [ 11919, 3678, 3679 ] } ], "ref_ids": [ 3694 ] }, { "id": 9368, "type": "theorem", "label": "spaces-descent-lemma-curiosity", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-curiosity", "contents": [ "Let $S$ be a scheme. Let $X \\to Y \\to Z$ be morphism of algebraic spaces.", "Let $P$ be one of the following properties of morphisms of algebraic spaces", "over $S$:", "flat, locally finite type, locally finite presentation.", "Assume that $X \\to Z$ has $P$ and that", "$X \\to Y$ is a surjection of sheaves on $(\\Sch/S)_{fppf}$.", "Then $Y \\to Z$ is $P$." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $W$ and a surjective \\'etale morphism $W \\to Z$.", "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to W \\times_Z Y$.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to V \\times_Y X$.", "By assumption we can find an fppf covering $\\{V_i \\to V\\}$ and", "lifts $V_i \\to X$ of the morphism $V_i \\to Y$. Since $U \\to X$ is surjective", "\\'etale we see that over the members of the fppf covering", "$\\{V_i \\times_X U \\to V\\}$ we have lifts into $U$. Hence $U \\to V$ induces", "a surjection of sheaves on $(\\Sch/S)_{fppf}$.", "By our definition of what it means to have property $P$ for a", "morphism of algebraic spaces (see", "Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-flat},", "Definition \\ref{spaces-morphisms-definition-locally-finite-type}, and", "Definition \\ref{spaces-morphisms-definition-locally-finite-presentation})", "we see that $U \\to W$ has $P$ and we have to show $V \\to W$ has $P$.", "Thus we reduce the question to the case of morphisms of schemes", "which is treated in", "Descent, Lemma \\ref{descent-lemma-curiosity}." ], "refs": [ "spaces-morphisms-definition-flat", "spaces-morphisms-definition-locally-finite-type", "spaces-morphisms-definition-locally-finite-presentation", "descent-lemma-curiosity" ], "ref_ids": [ 5007, 5003, 5006, 14646 ] } ], "ref_ids": [] }, { "id": 9369, "type": "theorem", "label": "spaces-descent-lemma-flat-finitely-presented-permanence", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-flat-finitely-presented-permanence", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & &", "Y \\ar[dl]^q \\\\", "& B", "}", "$$", "be a commutative diagram of morphisms of algebraic spaces over $S$.", "Assume that $f$ is surjective, flat, and locally of finite presentation", "and assume that $p$ is locally of finite presentation (resp.\\ locally", "of finite type). Then $q$ is locally of finite presentation", "(resp.\\ locally of finite type)." ], "refs": [], "proofs": [ { "contents": [ "Since $\\{X \\to Y\\}$ is an fppf covering, it induces a surjection of", "fppf sheaves (Topologies on Spaces, Lemma", "\\ref{spaces-topologies-lemma-fppf-covering-surjective}) and the", "lemma is a special case of Lemma \\ref{lemma-curiosity}.", "On the other hand, an easier argument is to deduce it from", "the analogue for schemes. Namely, the problem is \\'etale local", "on $B$ and $Y$ (Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-finite-type-local} and", "\\ref{spaces-morphisms-lemma-finite-presentation-local}).", "Hence we may assume that $B$ and $Y$ are affine", "schemes. Since $|X| \\to |Y|$ is open", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-fppf-open}),", "we can choose an affine", "scheme $U$ and an \\'etale morphism $U \\to X$ such that the", "composition $U \\to Y$ is surjective. In this case the result", "follows from Descent, Lemma", "\\ref{descent-lemma-flat-finitely-presented-permanence}." ], "refs": [ "spaces-topologies-lemma-fppf-covering-surjective", "spaces-descent-lemma-curiosity", "spaces-morphisms-lemma-finite-type-local", "spaces-morphisms-lemma-finite-presentation-local", "spaces-morphisms-lemma-fppf-open", "descent-lemma-flat-finitely-presented-permanence" ], "ref_ids": [ 3667, 9368, 4816, 4841, 4855, 14642 ] } ], "ref_ids": [] }, { "id": 9370, "type": "theorem", "label": "spaces-descent-lemma-syntomic-smooth-etale-permanence", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-syntomic-smooth-etale-permanence", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & &", "Y \\ar[dl]^q \\\\", "& B", "}", "$$", "be a commutative diagram of morphisms of algebraic spaces over $S$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is surjective, and syntomic (resp.\\ smooth, resp.\\ \\'etale),", "\\item $p$ is syntomic (resp.\\ smooth, resp.\\ \\'etale).", "\\end{enumerate}", "Then $q$ is syntomic (resp.\\ smooth, resp.\\ \\'etale)." ], "refs": [], "proofs": [ { "contents": [ "We deduce this from the analogue for schemes.", "Namely, the problem is \\'etale local on $B$ and $Y$", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-syntomic-local},", "\\ref{spaces-morphisms-lemma-smooth-local}, and", "\\ref{spaces-morphisms-lemma-etale-local}).", "Hence we may assume that $B$ and $Y$ are affine", "schemes. Since $|X| \\to |Y|$ is open", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-fppf-open}),", "we can choose an affine", "scheme $U$ and an \\'etale morphism $U \\to X$ such that the", "composition $U \\to Y$ is surjective. In this case the result", "follows from Descent, Lemma", "\\ref{descent-lemma-syntomic-smooth-etale-permanence}." ], "refs": [ "spaces-morphisms-lemma-syntomic-local", "spaces-morphisms-lemma-smooth-local", "spaces-morphisms-lemma-etale-local", "spaces-morphisms-lemma-fppf-open", "descent-lemma-syntomic-smooth-etale-permanence" ], "ref_ids": [ 4882, 4888, 4905, 4855, 14643 ] } ], "ref_ids": [] }, { "id": 9371, "type": "theorem", "label": "spaces-descent-lemma-smooth-permanence", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-smooth-permanence", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & &", "Y \\ar[dl]^q \\\\", "& B", "}", "$$", "be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that", "\\begin{enumerate}", "\\item $f$ is surjective, flat, and locally of finite presentation,", "\\item $p$ is smooth (resp.\\ \\'etale).", "\\end{enumerate}", "Then $q$ is smooth (resp.\\ \\'etale)." ], "refs": [], "proofs": [ { "contents": [ "We deduce this from the analogue for schemes.", "Namely, the problem is \\'etale local on $B$ and $Y$", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-smooth-local} and", "\\ref{spaces-morphisms-lemma-etale-local}).", "Hence we may assume that $B$ and $Y$ are affine", "schemes. Since $|X| \\to |Y|$ is open", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-fppf-open}),", "we can choose an affine", "scheme $U$ and an \\'etale morphism $U \\to X$ such that the", "composition $U \\to Y$ is surjective. In this case the result", "follows from Descent, Lemma", "\\ref{descent-lemma-smooth-permanence}." ], "refs": [ "spaces-morphisms-lemma-smooth-local", "spaces-morphisms-lemma-etale-local", "spaces-morphisms-lemma-fppf-open", "descent-lemma-smooth-permanence" ], "ref_ids": [ 4888, 4905, 4855, 14644 ] } ], "ref_ids": [] }, { "id": 9372, "type": "theorem", "label": "spaces-descent-lemma-syntomic-permanence", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-syntomic-permanence", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & &", "Y \\ar[dl]^q \\\\", "& B", "}", "$$", "be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that", "\\begin{enumerate}", "\\item $f$ is surjective, flat, and locally of finite presentation,", "\\item $p$ is syntomic.", "\\end{enumerate}", "Then both $q$ and $f$ are syntomic." ], "refs": [], "proofs": [ { "contents": [ "We deduce this from the analogue for schemes.", "Namely, the problem is \\'etale local on $B$ and $Y$", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-syntomic-local}).", "Hence we may assume that $B$ and $Y$ are affine", "schemes. Since $|X| \\to |Y|$ is open", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-fppf-open}),", "we can choose an affine", "scheme $U$ and an \\'etale morphism $U \\to X$ such that the", "composition $U \\to Y$ is surjective. In this case the result", "follows from Descent, Lemma", "\\ref{descent-lemma-syntomic-permanence}." ], "refs": [ "spaces-morphisms-lemma-syntomic-local", "spaces-morphisms-lemma-fppf-open", "descent-lemma-syntomic-permanence" ], "ref_ids": [ 4882, 4855, 14645 ] } ], "ref_ids": [] }, { "id": 9373, "type": "theorem", "label": "spaces-descent-lemma-descend-unibranch", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descend-unibranch", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $x \\in |X|$.", "If $f$ is flat at $x$ and $X$ is geometrically unibranch at $x$, then $Y$ is", "geometrically unibranch at $f(x)$." ], "refs": [], "proofs": [ { "contents": [ "Consider the map of \\'etale local rings", "$\\mathcal{O}_{Y, f(\\overline{x})} \\to \\mathcal{O}_{X, \\overline{x}}$.", "By", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-flat-at-point-etale-local-rings}", "this is flat. Hence if $\\mathcal{O}_{X, \\overline{x}}$ has a unique minimal", "prime, so does $\\mathcal{O}_{Y, f(\\overline{x})}$ (by going down, see", "Algebra, Lemma \\ref{algebra-lemma-flat-going-down})." ], "refs": [ "spaces-morphisms-lemma-flat-at-point-etale-local-rings", "algebra-lemma-flat-going-down" ], "ref_ids": [ 4857, 539 ] } ], "ref_ids": [] }, { "id": 9374, "type": "theorem", "label": "spaces-descent-lemma-descend-reduced", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descend-reduced", "contents": [ "\\begin{slogan}", "A flat and surjective morphism of algebraic spaces with a reduced source", "has a reduced target.", "\\end{slogan}", "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $f$ is flat and surjective and $X$ is reduced, then $Y$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Choose a scheme $U$ and a surjective \\'etale morphism", "$U \\to X \\times_Y V$. As $f$ is surjective and flat, the morphism of", "schemes $U \\to V$ is surjective and flat. In this way we reduce the", "problem to the case of schemes (as reducedness of $X$ and $Y$ is defined", "in terms of reducedness of $U$ and $V$, see", "Properties of Spaces,", "Section \\ref{spaces-properties-section-types-properties}).", "The case of schemes is", "Descent, Lemma \\ref{descent-lemma-descend-reduced}." ], "refs": [ "descent-lemma-descend-reduced" ], "ref_ids": [ 14658 ] } ], "ref_ids": [] }, { "id": 9375, "type": "theorem", "label": "spaces-descent-lemma-descend-locally-Noetherian", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descend-locally-Noetherian", "contents": [ "Let $f : X \\to Y$ be a morphism of algebraic spaces.", "If $f$ is locally of finite presentation, flat, and surjective and", "$X$ is locally Noetherian, then $Y$ is locally Noetherian." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Choose a scheme $U$ and a surjective \\'etale morphism", "$U \\to X \\times_Y V$. As $f$ is surjective, flat, and locally of", "finite presentation the morphism of schemes $U \\to V$ is surjective, flat, and", "locally of finite presentation. In this way we reduce the", "problem to the case of schemes (as being locally Noetherian for $X$ and $Y$", "is defined in terms of being locally Noetherian of $U$ and $V$, see", "Properties of Spaces,", "Section \\ref{spaces-properties-section-types-properties}).", "In the case of schemes the result follows from", "Descent, Lemma \\ref{descent-lemma-Noetherian-local-fppf}." ], "refs": [ "descent-lemma-Noetherian-local-fppf" ], "ref_ids": [ 14648 ] } ], "ref_ids": [] }, { "id": 9376, "type": "theorem", "label": "spaces-descent-lemma-descend-regular", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descend-regular", "contents": [ "Let $f : X \\to Y$ be a morphism of algebraic spaces.", "If $f$ is locally of finite presentation, flat, and surjective and", "$X$ is regular, then $Y$ is regular." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-descend-locally-Noetherian}", "we know that $Y$ is locally Noetherian.", "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "It suffices to prove that the local rings of $V$ are all regular local", "rings, see", "Properties, Lemma \\ref{properties-lemma-characterize-regular}.", "Choose a scheme $U$ and a surjective \\'etale morphism", "$U \\to X \\times_Y V$. As $f$ is surjective and flat the morphism of schemes", "$U \\to V$ is surjective and flat. By assumption $U$ is a regular scheme", "in particular all of its local rings are regular (by the lemma above).", "Hence the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-flat-under-regular}." ], "refs": [ "spaces-descent-lemma-descend-locally-Noetherian", "properties-lemma-characterize-regular", "algebra-lemma-flat-under-regular" ], "ref_ids": [ 9375, 2975, 981 ] } ], "ref_ids": [] }, { "id": 9377, "type": "theorem", "label": "spaces-descent-lemma-reduced-local-smooth", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-reduced-local-smooth", "contents": [ "Let $f : X \\to Y$ be a smooth morphism of algebraic spaces.", "If $Y$ is reduced, then $X$ is reduced. If $f$ is surjective", "and $X$ is reduced, then $Y$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$ and $V$ are schemes, the vertical arrows are surjective and", "\\'etale, and $U \\to X \\times_Y V$ is surjective \\'etale. Observe that $X$ is", "a reduced algebraic space if and only if $U$ is a reduced scheme", "by our definition of reduced algebraic spaces in", "Properties of Spaces, Section \\ref{spaces-properties-section-types-properties}.", "Similarly for $Y$ and $V$. ", "The morphism $U \\to V$ is a smooth morphism of schemes, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-smooth-local}.", "Since being reduced is local for the smooth topology for", "schemes (Descent, Lemma \\ref{descent-lemma-reduced-local-smooth})", "we see that $U$ is reduced if $V$ is reduced.", "On the other hand, if $X \\to Y$ is surjective, then $U \\to V$ is", "surjective and in this case if $U$ is reduced, then $V$ is reduced." ], "refs": [ "spaces-morphisms-lemma-smooth-local", "descent-lemma-reduced-local-smooth" ], "ref_ids": [ 4888, 14653 ] } ], "ref_ids": [] }, { "id": 9378, "type": "theorem", "label": "spaces-descent-lemma-pullback-property-local-target", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-pullback-property-local-target", "contents": [ "Let $S$ be a scheme.", "Let $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale\\}$.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$", "which is $\\tau$ local on the target. Let $f : X \\to Y$ have property", "$\\mathcal{P}$. For any morphism $Y' \\to Y$ which is", "flat, resp.\\ flat and locally of finite presentation, resp.\\ syntomic,", "resp.\\ \\'etale, the base change", "$f' : Y' \\times_Y X \\to Y'$ of $f$ has property $\\mathcal{P}$." ], "refs": [], "proofs": [ { "contents": [ "This is true because we can fit $Y' \\to Y$ into a family of", "morphisms which forms a $\\tau$-covering." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9379, "type": "theorem", "label": "spaces-descent-lemma-largest-open-of-the-base", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-largest-open-of-the-base", "contents": [ "Let $S$ be a scheme.", "Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale\\}$.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$", "which is $\\tau$ local on the target. For any morphism of algebraic spaces", "$f : X \\to Y$ over $S$ there exists a largest open subspace", "$W(f) \\subset Y$ such that the restriction $X_{W(f)} \\to W(f)$ has", "$\\mathcal{P}$. Moreover,", "\\begin{enumerate}", "\\item if $g : Y' \\to Y$ is a morphism of algebraic spaces which is", "flat and locally of finite presentation, syntomic, smooth, or \\'etale", "and the base change $f' : X_{Y'} \\to Y'$ has $\\mathcal{P}$, then", "$g$ factors through $W(f)$,", "\\item if $g : Y' \\to Y$ is flat and locally of finite presentation,", "syntomic, smooth, or \\'etale, then $W(f') = g^{-1}(W(f))$, and", "\\item if $\\{g_i : Y_i \\to Y\\}$ is a $\\tau$-covering, then", "$g_i^{-1}(W(f)) = W(f_i)$, where $f_i$ is the base change of $f$", "by $Y_i \\to Y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the union $W_{set} \\subset |Y|$ of the images", "$g(|Y'|) \\subset |Y|$ of morphisms $g : Y' \\to Y$ with the properties:", "\\begin{enumerate}", "\\item $g$ is flat and locally of finite presentation, syntomic,", "smooth, or \\'etale, and", "\\item the base change $Y' \\times_{g, Y} X \\to Y'$ has property", "$\\mathcal{P}$.", "\\end{enumerate}", "Since such a morphism $g$ is open (see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-fppf-open})", "we see that $W_{set}$ is an open subset of $|Y|$. Denote $W \\subset Y$", "the open subspace whose underlying set of points is $W_{set}$, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-open-subspaces}.", "Since $\\mathcal{P}$ is local in the $\\tau$ topology the restriction", "$X_W \\to W$ has property $\\mathcal{P}$ because we are given a covering", "$\\{Y' \\to W\\}$ of $W$ such that the pullbacks have $\\mathcal{P}$.", "This proves the existence and proves that $W(f)$ has property (1).", "To see property (2) note that $W(f') \\supset g^{-1}(W(f))$ because", "$\\mathcal{P}$ is stable under base change by flat and locally of finite", "presentation, syntomic, smooth, or \\'etale morphisms, see", "Lemma \\ref{lemma-pullback-property-local-target}.", "On the other hand, if $Y'' \\subset Y'$ is an open such that", "$X_{Y''} \\to Y''$ has property $\\mathcal{P}$, then $Y'' \\to Y$ factors", "through $W$ by construction, i.e., $Y'' \\subset g^{-1}(W(f))$. This", "proves (2). Assertion (3) follows from (2) because each morphism", "$Y_i \\to Y$ is flat and locally of finite presentation, syntomic,", "smooth, or \\'etale by our definition of a $\\tau$-covering." ], "refs": [ "spaces-morphisms-lemma-fppf-open", "spaces-properties-lemma-open-subspaces", "spaces-descent-lemma-pullback-property-local-target" ], "ref_ids": [ 4855, 11823, 9378 ] } ], "ref_ids": [] }, { "id": 9380, "type": "theorem", "label": "spaces-descent-lemma-descending-properties-morphisms", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-properties-morphisms", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of", "algebraic spaces over $S$. Assume", "\\begin{enumerate}", "\\item if $X_i \\to Y_i$, $i = 1, 2$ have property $\\mathcal{P}$ so", "does $X_1 \\amalg X_2 \\to Y_1 \\amalg Y_2$,", "\\item a morphism of algebraic spaces $f : X \\to Y$ has property", "$\\mathcal{P}$ if and only if for every affine scheme $Z$ and", "morphism $Z \\to Y$ the base change $Z \\times_Y X \\to Z$ of $f$", "has property $\\mathcal{P}$, and", "\\item for any surjective flat morphism of affine schemes", "$Z' \\to Z$ over $S$ and a morphism $f : X \\to Z$ from an algebraic space", "to $Z$ we have", "$$", "f' : Z' \\times_Z X \\to Z'\\text{ has }\\mathcal{P}", "\\Rightarrow", "f\\text{ has }\\mathcal{P}.", "$$", "\\end{enumerate}", "Then $\\mathcal{P}$ is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{P}$ has property (2), then it is automatically", "stable under any base change. Hence the direct implication in", "Definition \\ref{definition-property-morphisms-local}.", "\\medskip\\noindent", "Let $\\{Y_i \\to Y\\}_{i \\in I}$ be an fpqc covering of algebraic spaces over $S$.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume each base change $f_i : Y_i \\times_Y X \\to Y_i$ has property", "$\\mathcal{P}$. Our goal is to show that $f$ has $\\mathcal{P}$.", "Let $Z$ be an affine scheme, and let $Z \\to Y$ be a morphism.", "By (2) it suffices to show that the morphism of algebraic spaces", "$Z \\times_Y X \\to Z$ has $\\mathcal{P}$.", "Since $\\{Y_i \\to Y\\}_{i \\in I}$ is an fpqc covering we know there", "exists a standard fpqc covering $\\{Z_j \\to Z\\}_{j = 1, \\ldots , n}$", "and morphisms $Z_j \\to Y_{i_j}$ over $Y$ for suitable indices $i_j \\in I$.", "Since $f_{i_j}$ has $\\mathcal{P}$ we see that", "$$", "Z_j \\times_Y X", "=", "Z_j \\times_{Y_{i_j}} (Y_{i_j} \\times_Y X)", "\\longrightarrow", "Z_j", "$$", "has $\\mathcal{P}$ as a base change of $f_{i_j}$ (see first remark of the", "proof). Set $Z' = \\coprod_{j = 1, \\ldots, n} Z_j$, so that $Z' \\to Z$ is", "a flat and surjective morphism of affine schemes over $S$. By (1)", "we conclude that $Z' \\times_Y X \\to Z'$ has property $\\mathcal{P}$.", "Since this is the base change of the morphism $Z \\times_Y X \\to Z$", "by the morphism $Z' \\to Z$ we conclude that", "$Z \\times_Y X \\to Z$ has property $\\mathcal{P}$ as desired." ], "refs": [ "spaces-descent-definition-property-morphisms-local" ], "ref_ids": [ 9440 ] } ], "ref_ids": [] }, { "id": 9381, "type": "theorem", "label": "spaces-descent-lemma-descending-property-quasi-compact", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-quasi-compact", "contents": [ "Let $S$ be a scheme.", "The property $\\mathcal{P}(f) =$``$f$ is quasi-compact''", "is fpqc local on the base on algebraic spaces over $S$." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-quasi-compact-local}.", "Let $Z' \\to Z$ be a surjective flat morphism of affine schemes over $S$.", "Let $f : X \\to Z$ be a morphism of algebraic spaces, and assume", "that the base change $f' : Z' \\times_Z X \\to Z'$ is quasi-compact. We have", "to show that $f$ is quasi-compact. To see this, using", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-quasi-compact-local}", "again, it is enough to show that for every affine scheme $Y$ and", "morphism $Y \\to Z$ the fibre product $Y \\times_Z X$ is quasi-compact.", "Here is a picture:", "\\begin{equation}", "\\label{equation-cube}", "\\vcenter{", "\\xymatrix{", "Y \\times_Z Z' \\times_Z X \\ar[dd] \\ar[rr] \\ar[rd] & &", "Z' \\times_Z X \\ar'[d][dd]^{f'} \\ar[rd] \\\\", "& Y \\times_Z X \\ar[dd] \\ar[rr] & & X \\ar[dd]^f \\\\", "Y \\times_Z Z' \\ar'[r][rr] \\ar[rd] & & Z' \\ar[rd] \\\\", "& Y \\ar[rr] & & Z", "}", "}", "\\end{equation}", "Note that all squares are cartesian and the bottom square consists", "of affine schemes. The assumption that $f'$ is quasi-compact combined with", "the fact that $Y \\times_Z Z'$ is affine implies that", "$Y \\times_Z Z' \\times_Z X$ is quasi-compact. Since", "$$", "Y \\times_Z Z' \\times_Z X \\longrightarrow Y \\times_Z X", "$$", "is surjective as a base change of $Z' \\to Z$", "we conclude that $Y \\times_Z X$ is quasi-compact, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-surjection-from-quasi-compact}.", "This finishes the proof." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-quasi-compact-local", "spaces-morphisms-lemma-quasi-compact-local", "spaces-morphisms-lemma-surjection-from-quasi-compact" ], "ref_ids": [ 9380, 4742, 4742, 4740 ] } ], "ref_ids": [] }, { "id": 9382, "type": "theorem", "label": "spaces-descent-lemma-descending-property-quasi-separated", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-quasi-separated", "contents": [ "Let $S$ be a scheme.", "The property $\\mathcal{P}(f) =$``$f$ is quasi-separated''", "is fpqc local on the base on algebraic spaces over $S$." ], "refs": [], "proofs": [ { "contents": [ "A base change of a quasi-separated morphism is quasi-separated, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-base-change-separated}.", "Hence the direct implication in", "Definition \\ref{definition-property-morphisms-local}.", "\\medskip\\noindent", "Let $\\{Y_i \\to Y\\}_{i \\in I}$ be an fpqc covering of algebraic spaces over $S$.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume each base change $X_i := Y_i \\times_Y X \\to Y_i$ is quasi-separated.", "This means that each of the morphisms", "$$", "\\Delta_i :", "X_i", "\\longrightarrow", "X_i \\times_{Y_i} X_i = Y_i \\times_Y (X \\times_Y X)", "$$", "is quasi-compact. The base change of a fpqc covering is an fpqc covering, see", "Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-fpqc}", "hence $\\{Y_i \\times_Y (X \\times_Y X) \\to X \\times_Y X\\}$", "is an fpqc covering of algebraic spaces. Moreover, each", "$\\Delta_i$ is the base change of the morphism", "$\\Delta : X \\to X \\times_Y X$. Hence it follows from", "Lemma \\ref{lemma-descending-property-quasi-compact}", "that $\\Delta$ is quasi-compact, i.e., $f$ is quasi-separated." ], "refs": [ "spaces-morphisms-lemma-base-change-separated", "spaces-descent-definition-property-morphisms-local", "spaces-topologies-lemma-fpqc", "spaces-descent-lemma-descending-property-quasi-compact" ], "ref_ids": [ 4714, 9440, 3678, 9381 ] } ], "ref_ids": [] }, { "id": 9383, "type": "theorem", "label": "spaces-descent-lemma-descending-property-universally-closed", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-universally-closed", "contents": [ "Let $S$ be a scheme.", "The property $\\mathcal{P}(f) =$``$f$ is universally closed''", "is fpqc local on the base on algebraic spaces over $S$." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-universally-closed-local}.", "Let $Z' \\to Z$ be a surjective flat morphism of affine schemes over $S$.", "Let $f : X \\to Z$ be a morphism of algebraic spaces, and assume", "that the base change $f' : Z' \\times_Z X \\to Z'$ is universally closed.", "We have to show that $f$ is universally closed. To see this, using", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-universally-closed-local}", "again, it is enough to show that for every affine scheme $Y$ and", "morphism $Y \\to Z$ the map $|Y \\times_Z X| \\to |Y|$ is closed.", "Consider the cube (\\ref{equation-cube}).", "The assumption that $f'$ is universally closed implies that", "$|Y \\times_Z Z' \\times_Z X| \\to |Y \\times_Z Z'|$ is closed.", "As $Y \\times_Z Z' \\to Y$ is quasi-compact, surjective, and flat", "as a base change of $Z' \\to Z$", "we see the map $|Y \\times_Z Z'| \\to |Y|$ is submersive, see", "Morphisms, Lemma \\ref{morphisms-lemma-fpqc-quotient-topology}.", "Moreover the map", "$$", "|Y \\times_Z Z' \\times_Z X|", "\\longrightarrow", "|Y \\times_Z Z'| \\times_{|Y|} |Y \\times_Z X|", "$$", "is surjective, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}.", "It follows by elementary topology that $|Y \\times_Z X| \\to |Y|$ is closed." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-universally-closed-local", "spaces-morphisms-lemma-universally-closed-local", "morphisms-lemma-fpqc-quotient-topology", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 9380, 4748, 4748, 5269, 11819 ] } ], "ref_ids": [] }, { "id": 9384, "type": "theorem", "label": "spaces-descent-lemma-descending-property-universally-open", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-universally-open", "contents": [ "Let $S$ be a scheme.", "The property $\\mathcal{P}(f) =$``$f$ is universally open''", "is fpqc local on the base on algebraic spaces over $S$." ], "refs": [], "proofs": [ { "contents": [ "The proof is the same as the proof of", "Lemma \\ref{lemma-descending-property-universally-closed}." ], "refs": [ "spaces-descent-lemma-descending-property-universally-closed" ], "ref_ids": [ 9383 ] } ], "ref_ids": [] }, { "id": 9385, "type": "theorem", "label": "spaces-descent-lemma-descending-property-universally-submersive", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-universally-submersive", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is universally submersive''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "The proof is the same as the proof of", "Lemma \\ref{lemma-descending-property-universally-closed}." ], "refs": [ "spaces-descent-lemma-descending-property-universally-closed" ], "ref_ids": [ 9383 ] } ], "ref_ids": [] }, { "id": 9386, "type": "theorem", "label": "spaces-descent-lemma-descending-property-surjective", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-surjective", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is surjective''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Omitted. (Hint: Use", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}.)" ], "refs": [ "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 11819 ] } ], "ref_ids": [] }, { "id": 9387, "type": "theorem", "label": "spaces-descent-lemma-descending-property-universally-injective", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-universally-injective", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is universally injective''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-universally-closed-local}.", "Let $Z' \\to Z$ be a flat surjective morphism of affine schemes", "over $S$ and let $f : X \\to Z$ be a morphism from an algebraic space to $Z$.", "Assume that the base change $f' : X' \\to Z'$ is universally injective.", "Let $K$ be a field, and let $a, b : \\Spec(K) \\to X$", "be two morphisms such that $f \\circ a = f \\circ b$.", "As $Z' \\to Z$ is surjective there exists a field", "extension $K \\subset K'$ and a morphism", "$\\Spec(K') \\to Z'$", "such that the following solid diagram commutes", "$$", "\\xymatrix{", "\\Spec(K') \\ar[rrd] \\ar@{-->}[rd]_{a', b'} \\ar[dd] \\\\", " &", "X' \\ar[r] \\ar[d] &", "Z' \\ar[d] \\\\", "\\Spec(K) \\ar[r]^{a, b} &", "X \\ar[r] &", "Z", "}", "$$", "As the square is cartesian we get the two dotted arrows $a'$, $b'$ making the", "diagram commute. Since $X' \\to Z'$ is universally injective we get $a' = b'$.", "This forces $a = b$ as $\\{\\Spec(K') \\to \\Spec(K)\\}$", "is an fpqc covering, see", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-sheaf-fpqc}.", "Hence $f$ is universally injective as desired." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-universally-closed-local", "spaces-properties-proposition-sheaf-fpqc" ], "ref_ids": [ 9380, 4748, 11919 ] } ], "ref_ids": [] }, { "id": 9388, "type": "theorem", "label": "spaces-descent-lemma-descending-property-universal-homeomorphism", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-universal-homeomorphism", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is a universal homeomorphism''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "This can be proved in exactly the same manner as", "Lemma \\ref{lemma-descending-property-universally-closed}.", "Alternatively, one can use that a map of topological spaces is a", "homeomorphism if and only if it is injective, surjective, and open.", "Thus a universal homeomorphism is the same thing as a", "surjective, universally injective, and universally open morphism.", "See Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-surjective} and", "Morphisms of Spaces, Definitions", "\\ref{spaces-morphisms-definition-universally-injective},", "\\ref{spaces-morphisms-definition-surjective},", "\\ref{spaces-morphisms-definition-open},", "\\ref{spaces-morphisms-definition-universal-homeomorphism}.", "Thus the lemma follows from", "Lemmas \\ref{lemma-descending-property-surjective},", "\\ref{lemma-descending-property-universally-injective}, and", "\\ref{lemma-descending-property-universally-open}." ], "refs": [ "spaces-descent-lemma-descending-property-universally-closed", "spaces-morphisms-lemma-base-change-surjective", "spaces-morphisms-definition-universally-injective", "spaces-morphisms-definition-surjective", "spaces-morphisms-definition-open", "spaces-morphisms-definition-universal-homeomorphism", "spaces-descent-lemma-descending-property-surjective", "spaces-descent-lemma-descending-property-universally-injective", "spaces-descent-lemma-descending-property-universally-open" ], "ref_ids": [ 9383, 4727, 4997, 4985, 4986, 5028, 9386, 9387, 9384 ] } ], "ref_ids": [] }, { "id": 9389, "type": "theorem", "label": "spaces-descent-lemma-descending-property-locally-finite-type", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-locally-finite-type", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is locally of finite type''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-finite-type-local}.", "Let $Z' \\to Z$ be a surjective flat morphism of affine schemes over $S$.", "Let $f : X \\to Z$ be a morphism of algebraic spaces, and assume", "that the base change $f' : Z' \\times_Z X \\to Z'$ is locally of finite type.", "We have to show that $f$ is locally of finite type. Let $U$ be a scheme", "and let $U \\to X$ be surjective and \\'etale. By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-finite-type-local}", "again, it is enough to show that $U \\to Z$ is locally of finite type.", "Since $f'$ is locally of finite type, and since $Z' \\times_Z U$ is a", "scheme \\'etale over $Z' \\times_Z X$ we conclude (by the same lemma again) that", "$Z' \\times_Z U \\to Z'$ is locally of finite type.", "As $\\{Z' \\to Z\\}$ is an fpqc covering we conclude that", "$U \\to Z$ is locally of finite type by", "Descent, Lemma \\ref{descent-lemma-descending-property-locally-finite-type}", "as desired." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-finite-type-local", "spaces-morphisms-lemma-finite-type-local", "descent-lemma-descending-property-locally-finite-type" ], "ref_ids": [ 9380, 4816, 4816, 14675 ] } ], "ref_ids": [] }, { "id": 9390, "type": "theorem", "label": "spaces-descent-lemma-descending-property-locally-finite-presentation", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-locally-finite-presentation", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is locally of finite presentation''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-finite-presentation-local}.", "Let $Z' \\to Z$ be a surjective flat morphism of affine schemes over $S$.", "Let $f : X \\to Z$ be a morphism of algebraic spaces, and assume", "that the base change $f' : Z' \\times_Z X \\to Z'$ is locally of", "finite presentation.", "We have to show that $f$ is locally of finite presentation. Let $U$ be a scheme", "and let $U \\to X$ be surjective and \\'etale. By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-finite-presentation-local}", "again, it is enough to show that $U \\to Z$ is locally of finite presentation.", "Since $f'$ is locally of finite presentation, and since $Z' \\times_Z U$ is a", "scheme \\'etale over $Z' \\times_Z X$ we conclude (by the same lemma again) that", "$Z' \\times_Z U \\to Z'$ is locally of finite presentation.", "As $\\{Z' \\to Z\\}$ is an fpqc covering we conclude that", "$U \\to Z$ is locally of finite presentation by", "Descent,", "Lemma \\ref{descent-lemma-descending-property-locally-finite-presentation}", "as desired." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-finite-presentation-local", "spaces-morphisms-lemma-finite-presentation-local", "descent-lemma-descending-property-locally-finite-presentation" ], "ref_ids": [ 9380, 4841, 4841, 14676 ] } ], "ref_ids": [] }, { "id": 9391, "type": "theorem", "label": "spaces-descent-lemma-descending-property-finite-type", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-finite-type", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is of finite type''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-descending-property-quasi-compact}", "and \\ref{lemma-descending-property-locally-finite-type}." ], "refs": [ "spaces-descent-lemma-descending-property-quasi-compact", "spaces-descent-lemma-descending-property-locally-finite-type" ], "ref_ids": [ 9381, 9389 ] } ], "ref_ids": [] }, { "id": 9392, "type": "theorem", "label": "spaces-descent-lemma-descending-property-finite-presentation", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-finite-presentation", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is of finite presentation''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-descending-property-quasi-compact},", "\\ref{lemma-descending-property-quasi-separated} and", "\\ref{lemma-descending-property-locally-finite-presentation}." ], "refs": [ "spaces-descent-lemma-descending-property-quasi-compact", "spaces-descent-lemma-descending-property-quasi-separated", "spaces-descent-lemma-descending-property-locally-finite-presentation" ], "ref_ids": [ 9381, 9382, 9390 ] } ], "ref_ids": [] }, { "id": 9393, "type": "theorem", "label": "spaces-descent-lemma-descending-property-flat", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-flat", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is flat''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-flat-local}.", "Let $Z' \\to Z$ be a surjective flat morphism of affine schemes over $S$.", "Let $f : X \\to Z$ be a morphism of algebraic spaces, and assume", "that the base change $f' : Z' \\times_Z X \\to Z'$ is flat.", "We have to show that $f$ is flat. Let $U$ be a scheme", "and let $U \\to X$ be surjective and \\'etale. By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-flat-local}", "again, it is enough to show that $U \\to Z$ is flat.", "Since $f'$ is flat, and since $Z' \\times_Z U$ is a", "scheme \\'etale over $Z' \\times_Z X$ we conclude (by the same lemma again) that", "$Z' \\times_Z U \\to Z'$ is flat.", "As $\\{Z' \\to Z\\}$ is an fpqc covering we conclude that", "$U \\to Z$ is flat by", "Descent, Lemma \\ref{descent-lemma-descending-property-flat}", "as desired." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-flat-local", "spaces-morphisms-lemma-flat-local", "descent-lemma-descending-property-flat" ], "ref_ids": [ 9380, 4854, 4854, 14680 ] } ], "ref_ids": [] }, { "id": 9394, "type": "theorem", "label": "spaces-descent-lemma-descending-property-open-immersion", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-open-immersion", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is an open immersion''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-closed-immersion-local}.", "Consider a cartesian diagram", "$$", "\\xymatrix{", "X' \\ar[r] \\ar[d] & X \\ar[d] \\\\", "Z' \\ar[r] & Z", "}", "$$", "of algebraic spaces over $S$", "where $Z' \\to Z$ is a surjective flat morphism of affine schemes,", "and $X' \\to Z'$ is an open immersion. We have to show that $X \\to Z$", "is an open immersion. Note that $|X'| \\subset |Z'|$ corresponds to an", "open subscheme $U' \\subset Z'$ (isomorphic to $X'$)", "with the property that $\\text{pr}_0^{-1}(U') = \\text{pr}_1^{-1}(U')$", "as open subschemes of $Z' \\times_Z Z'$. Hence there exists an open", "subscheme $U \\subset Z$ such that $X' = (Z' \\to Z)^{-1}(U)$, see", "Descent, Lemma \\ref{descent-lemma-open-fpqc-covering}.", "By Properties of Spaces,", "Proposition \\ref{spaces-properties-proposition-sheaf-fpqc}", "we see that $X$ satisfies the sheaf condition for the fpqc topology.", "Now we have the fpqc covering $\\mathcal{U} = \\{U' \\to U\\}$", "and the element $U' \\to X' \\to X \\in \\check{H}^0(\\mathcal{U}, X)$.", "By the sheaf condition we obtain a morphism $U \\to X$ such that", "$$", "\\xymatrix{", "U' \\ar[r] \\ar[d]^{\\cong} \\ar@/_3ex/[dd] & U \\ar[d] \\ar@/^3ex/[dd] \\\\", "X' \\ar[r] \\ar[d] & X \\ar[d] \\\\", "Z' \\ar[r] & Z", "}", "$$", "is commutative. On the other hand, we know that for any scheme $T$ over $S$", "and $T$-valued point $T \\to X$ the composition $T \\to X \\to Z$ is a", "morphism such that $Z' \\times_Z T \\to Z'$ factors through $U'$. Clearly", "this means that $T \\to Z$ factors through $U$. In other words the map", "of sheaves $U \\to X$ is bijective and we win." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-closed-immersion-local", "descent-lemma-open-fpqc-covering", "spaces-properties-proposition-sheaf-fpqc" ], "ref_ids": [ 9380, 4761, 14637, 11919 ] } ], "ref_ids": [] }, { "id": 9395, "type": "theorem", "label": "spaces-descent-lemma-descending-property-isomorphism", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-isomorphism", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is an isomorphism''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-descending-property-surjective}", "and \\ref{lemma-descending-property-open-immersion}." ], "refs": [ "spaces-descent-lemma-descending-property-surjective", "spaces-descent-lemma-descending-property-open-immersion" ], "ref_ids": [ 9386, 9394 ] } ], "ref_ids": [] }, { "id": 9396, "type": "theorem", "label": "spaces-descent-lemma-descending-property-affine", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-affine", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is affine''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-affine-local}.", "Let $Z' \\to Z$ be a surjective flat morphism of affine schemes over $S$.", "Let $f : X \\to Z$ be a morphism of algebraic spaces, and assume", "that the base change $f' : Z' \\times_Z X \\to Z'$ is affine.", "Let $X'$ be a scheme representing $Z' \\times_Z X$.", "We obtain a canonical isomorphism", "$$", "\\varphi : X' \\times_Z Z' \\longrightarrow Z' \\times_Z X'", "$$", "since both schemes represent the algebraic space $Z' \\times_Z Z' \\times_Z X$.", "This is a descent datum for $X'/Z'/Z$, see", "Descent, Definition \\ref{descent-definition-descent-datum}", "(verification omitted, compare with", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves}).", "Since $X' \\to Z'$ is affine this descent datum is effective, see", "Descent, Lemma \\ref{descent-lemma-affine}.", "Thus there exists a scheme $Y \\to Z$ over $Z$ and an", "isomorphism $\\psi : Z' \\times_Z Y \\to X'$ compatible with descent data.", "Of course $Y \\to Z$ is affine (by construction or by", "Descent, Lemma \\ref{descent-lemma-descending-property-affine}).", "Note that $\\mathcal{Y} = \\{Z' \\times_Z Y \\to Y\\}$ is a", "fpqc covering, and interpreting $\\psi$ as an element of", "$X(Z' \\times_Z Y)$ we see that $\\psi \\in \\check{H}^0(\\mathcal{Y}, X)$.", "By the sheaf condition for $X$ with respect to this covering (see", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-sheaf-fpqc})", "we obtain a morphism $Y \\to X$.", "By construction the base change of this to $Z'$ is an isomorphism, hence", "an isomorphism by", "Lemma \\ref{lemma-descending-property-isomorphism}.", "This proves that $X$ is representable by an affine scheme and we win." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-affine-local", "descent-definition-descent-datum", "descent-lemma-descent-data-sheaves", "descent-lemma-affine", "descent-lemma-descending-property-affine", "spaces-properties-proposition-sheaf-fpqc", "spaces-descent-lemma-descending-property-isomorphism" ], "ref_ids": [ 9380, 4798, 14776, 14751, 14748, 14683, 11919, 9395 ] } ], "ref_ids": [] }, { "id": 9397, "type": "theorem", "label": "spaces-descent-lemma-descending-property-closed-immersion", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-closed-immersion", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is a closed immersion''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-closed-immersion-local}.", "Consider a cartesian diagram", "$$", "\\xymatrix{", "X' \\ar[r] \\ar[d] & X \\ar[d] \\\\", "Z' \\ar[r] & Z", "}", "$$", "of algebraic spaces over $S$", "where $Z' \\to Z$ is a surjective flat morphism of affine schemes,", "and $X' \\to Z'$ is a closed immersion. We have to show that $X \\to Z$", "is a closed immersion. The morphism $X' \\to Z'$ is affine. Hence by", "Lemma \\ref{lemma-descending-property-affine}", "we see that $X$ is a scheme and $X \\to Z$ is affine.", "It follows from", "Descent, Lemma \\ref{descent-lemma-descending-property-closed-immersion}", "that $X \\to Z$ is a closed immersion as desired." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-closed-immersion-local", "spaces-descent-lemma-descending-property-affine", "descent-lemma-descending-property-closed-immersion" ], "ref_ids": [ 9380, 4761, 9396, 14684 ] } ], "ref_ids": [] }, { "id": 9398, "type": "theorem", "label": "spaces-descent-lemma-descending-property-separated", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-separated", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is separated''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "A base change of a separated morphism is separated, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-base-change-separated}.", "Hence the direct implication in", "Definition \\ref{definition-property-morphisms-local}.", "\\medskip\\noindent", "Let $\\{Y_i \\to Y\\}_{i \\in I}$ be an fpqc covering of algebraic spaces over $S$.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume each base change $X_i := Y_i \\times_Y X \\to Y_i$ is separated.", "This means that each of the morphisms", "$$", "\\Delta_i :", "X_i", "\\longrightarrow", "X_i \\times_{Y_i} X_i = Y_i \\times_Y (X \\times_Y X)", "$$", "is a closed immersion. The base change of a fpqc covering is an", "fpqc covering, see", "Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-fpqc}", "hence $\\{Y_i \\times_Y (X \\times_Y X) \\to X \\times_Y X\\}$", "is an fpqc covering of algebraic spaces. Moreover, each", "$\\Delta_i$ is the base change of the morphism", "$\\Delta : X \\to X \\times_Y X$. Hence it follows from", "Lemma \\ref{lemma-descending-property-closed-immersion}", "that $\\Delta$ is a closed immersion, i.e., $f$ is separated." ], "refs": [ "spaces-morphisms-lemma-base-change-separated", "spaces-descent-definition-property-morphisms-local", "spaces-topologies-lemma-fpqc", "spaces-descent-lemma-descending-property-closed-immersion" ], "ref_ids": [ 4714, 9440, 3678, 9397 ] } ], "ref_ids": [] }, { "id": 9399, "type": "theorem", "label": "spaces-descent-lemma-descending-property-proper", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-proper", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is proper''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "The lemma follows by combining", "Lemmas \\ref{lemma-descending-property-universally-closed},", "\\ref{lemma-descending-property-separated}", "and \\ref{lemma-descending-property-finite-type}." ], "refs": [ "spaces-descent-lemma-descending-property-universally-closed", "spaces-descent-lemma-descending-property-separated", "spaces-descent-lemma-descending-property-finite-type" ], "ref_ids": [ 9383, 9398, 9391 ] } ], "ref_ids": [] }, { "id": 9400, "type": "theorem", "label": "spaces-descent-lemma-descending-property-quasi-affine", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-quasi-affine", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is quasi-affine''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-quasi-affine-local}.", "Let $Z' \\to Z$ be a surjective flat morphism of affine schemes over $S$.", "Let $f : X \\to Z$ be a morphism of algebraic spaces, and assume", "that the base change $f' : Z' \\times_Z X \\to Z'$ is quasi-affine.", "Let $X'$ be a scheme representing $Z' \\times_Z X$.", "We obtain a canonical isomorphism", "$$", "\\varphi : X' \\times_Z Z' \\longrightarrow Z' \\times_Z X'", "$$", "since both schemes represent the algebraic space $Z' \\times_Z Z' \\times_Z X$.", "This is a descent datum for $X'/Z'/Z$, see", "Descent, Definition \\ref{descent-definition-descent-datum}", "(verification omitted, compare with", "Descent, Lemma \\ref{descent-lemma-descent-data-sheaves}).", "Since $X' \\to Z'$ is quasi-affine this descent datum is effective, see", "Descent, Lemma \\ref{descent-lemma-quasi-affine}.", "Thus there exists a scheme $Y \\to Z$ over $Z$ and an", "isomorphism $\\psi : Z' \\times_Z Y \\to X'$ compatible with descent data.", "Of course $Y \\to Z$ is quasi-affine (by construction or by", "Descent, Lemma \\ref{descent-lemma-descending-property-quasi-affine}).", "Note that $\\mathcal{Y} = \\{Z' \\times_Z Y \\to Y\\}$ is a", "fpqc covering, and interpreting $\\psi$ as an element of", "$X(Z' \\times_Z Y)$ we see that $\\psi \\in \\check{H}^0(\\mathcal{Y}, X)$.", "By the sheaf condition for $X$ (see", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-sheaf-fpqc})", "we obtain a morphism $Y \\to X$.", "By construction the base change of this to $Z'$ is an isomorphism, hence", "an isomorphism by", "Lemma \\ref{lemma-descending-property-isomorphism}.", "This proves that $X$ is representable by a quasi-affine scheme and we win." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-quasi-affine-local", "descent-definition-descent-datum", "descent-lemma-descent-data-sheaves", "descent-lemma-quasi-affine", "descent-lemma-descending-property-quasi-affine", "spaces-properties-proposition-sheaf-fpqc", "spaces-descent-lemma-descending-property-isomorphism" ], "ref_ids": [ 9380, 4807, 14776, 14751, 14750, 14685, 11919, 9395 ] } ], "ref_ids": [] }, { "id": 9401, "type": "theorem", "label": "spaces-descent-lemma-descending-property-quasi-compact-immersion", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-quasi-compact-immersion", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is a quasi-compact immersion''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-closed-immersion-local} and", "\\ref{spaces-morphisms-lemma-quasi-compact-local}.", "Consider a cartesian diagram", "$$", "\\xymatrix{", "X' \\ar[r] \\ar[d] & X \\ar[d] \\\\", "Z' \\ar[r] & Z", "}", "$$", "of algebraic spaces over $S$", "where $Z' \\to Z$ is a surjective flat morphism of affine schemes,", "and $X' \\to Z'$ is a quasi-compact immersion. We have to show that $X \\to Z$", "is a closed immersion. The morphism $X' \\to Z'$ is quasi-affine. Hence by", "Lemma \\ref{lemma-descending-property-quasi-affine}", "we see that $X$ is a scheme and $X \\to Z$ is quasi-affine.", "It follows from", "Descent, Lemma \\ref{descent-lemma-descending-property-quasi-compact-immersion}", "that $X \\to Z$ is a quasi-compact immersion as desired." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-closed-immersion-local", "spaces-morphisms-lemma-quasi-compact-local", "spaces-descent-lemma-descending-property-quasi-affine", "descent-lemma-descending-property-quasi-compact-immersion" ], "ref_ids": [ 9380, 4761, 4742, 9400, 14686 ] } ], "ref_ids": [] }, { "id": 9402, "type": "theorem", "label": "spaces-descent-lemma-descending-property-integral", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-integral", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is integral''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "An integral morphism is the same thing as an affine,", "universally closed morphism. See", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-integral-universally-closed}.", "Hence the lemma follows on combining", "Lemmas \\ref{lemma-descending-property-universally-closed}", "and \\ref{lemma-descending-property-affine}." ], "refs": [ "spaces-morphisms-lemma-integral-universally-closed", "spaces-descent-lemma-descending-property-universally-closed", "spaces-descent-lemma-descending-property-affine" ], "ref_ids": [ 4944, 9383, 9396 ] } ], "ref_ids": [] }, { "id": 9403, "type": "theorem", "label": "spaces-descent-lemma-descending-property-finite", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-finite", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is finite''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "An finite morphism is the same thing as an integral,", "morphism which is locally of finite type. See", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-integral}.", "Hence the lemma follows on combining", "Lemmas \\ref{lemma-descending-property-locally-finite-type}", "and \\ref{lemma-descending-property-integral}." ], "refs": [ "spaces-morphisms-lemma-finite-integral", "spaces-descent-lemma-descending-property-locally-finite-type", "spaces-descent-lemma-descending-property-integral" ], "ref_ids": [ 4943, 9389, 9402 ] } ], "ref_ids": [] }, { "id": 9404, "type": "theorem", "label": "spaces-descent-lemma-descending-property-quasi-finite", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-quasi-finite", "contents": [ "The properties", "$\\mathcal{P}(f) =$``$f$ is locally quasi-finite''", "and", "$\\mathcal{P}(f) =$``$f$ is quasi-finite''", "are fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We have already seen that ``quasi-compact'' is fpqc local on the base, see", "Lemma \\ref{lemma-descending-property-quasi-compact}. Hence it is enough", "to prove the lemma for ``locally quasi-finite''. We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-quasi-finite-local}.", "Let $Z' \\to Z$ be a surjective flat morphism of affine schemes over $S$.", "Let $f : X \\to Z$ be a morphism of algebraic spaces, and assume", "that the base change $f' : Z' \\times_Z X \\to Z'$ is locally quasi-finite.", "We have to show that $f$ is locally quasi-finite. Let $U$ be a scheme", "and let $U \\to X$ be surjective and \\'etale. By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-quasi-finite-local}", "again, it is enough to show that $U \\to Z$ is locally quasi-finite.", "Since $f'$ is locally quasi-finite, and since $Z' \\times_Z U$ is a", "scheme \\'etale over $Z' \\times_Z X$ we conclude (by the same lemma again) that", "$Z' \\times_Z U \\to Z'$ is locally quasi-finite.", "As $\\{Z' \\to Z\\}$ is an fpqc covering we conclude that", "$U \\to Z$ is locally quasi-finite by", "Descent, Lemma \\ref{descent-lemma-descending-property-quasi-finite}", "as desired." ], "refs": [ "spaces-descent-lemma-descending-property-quasi-compact", "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-quasi-finite-local", "spaces-morphisms-lemma-quasi-finite-local", "descent-lemma-descending-property-quasi-finite" ], "ref_ids": [ 9381, 9380, 4834, 4834, 14689 ] } ], "ref_ids": [] }, { "id": 9405, "type": "theorem", "label": "spaces-descent-lemma-descending-property-syntomic", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-syntomic", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is syntomic''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-syntomic-local}.", "Let $Z' \\to Z$ be a surjective flat morphism of affine schemes over $S$.", "Let $f : X \\to Z$ be a morphism of algebraic spaces, and assume", "that the base change $f' : Z' \\times_Z X \\to Z'$ is syntomic.", "We have to show that $f$ is syntomic. Let $U$ be a scheme", "and let $U \\to X$ be surjective and \\'etale. By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-syntomic-local}", "again, it is enough to show that $U \\to Z$ is syntomic.", "Since $f'$ is syntomic, and since $Z' \\times_Z U$ is a", "scheme \\'etale over $Z' \\times_Z X$ we conclude (by the same lemma again) that", "$Z' \\times_Z U \\to Z'$ is syntomic.", "As $\\{Z' \\to Z\\}$ is an fpqc covering we conclude that", "$U \\to Z$ is syntomic by", "Descent, Lemma \\ref{descent-lemma-descending-property-syntomic}", "as desired." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-syntomic-local", "spaces-morphisms-lemma-syntomic-local", "descent-lemma-descending-property-syntomic" ], "ref_ids": [ 9380, 4882, 4882, 14691 ] } ], "ref_ids": [] }, { "id": 9406, "type": "theorem", "label": "spaces-descent-lemma-descending-property-smooth", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-smooth", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is smooth''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-smooth-local}.", "Let $Z' \\to Z$ be a surjective flat morphism of affine schemes over $S$.", "Let $f : X \\to Z$ be a morphism of algebraic spaces, and assume", "that the base change $f' : Z' \\times_Z X \\to Z'$ is smooth.", "We have to show that $f$ is smooth. Let $U$ be a scheme", "and let $U \\to X$ be surjective and \\'etale. By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-smooth-local}", "again, it is enough to show that $U \\to Z$ is smooth.", "Since $f'$ is smooth, and since $Z' \\times_Z U$ is a", "scheme \\'etale over $Z' \\times_Z X$ we conclude (by the same lemma again) that", "$Z' \\times_Z U \\to Z'$ is smooth.", "As $\\{Z' \\to Z\\}$ is an fpqc covering we conclude that", "$U \\to Z$ is smooth by", "Descent, Lemma \\ref{descent-lemma-descending-property-smooth}", "as desired." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-smooth-local", "spaces-morphisms-lemma-smooth-local", "descent-lemma-descending-property-smooth" ], "ref_ids": [ 9380, 4888, 4888, 14692 ] } ], "ref_ids": [] }, { "id": 9407, "type": "theorem", "label": "spaces-descent-lemma-descending-property-unramified", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-unramified", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is unramified''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-unramified-local}.", "Let $Z' \\to Z$ be a surjective flat morphism of affine schemes over $S$.", "Let $f : X \\to Z$ be a morphism of algebraic spaces, and assume", "that the base change $f' : Z' \\times_Z X \\to Z'$ is unramified.", "We have to show that $f$ is unramified. Let $U$ be a scheme", "and let $U \\to X$ be surjective and \\'etale. By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-unramified-local}", "again, it is enough to show that $U \\to Z$ is unramified.", "Since $f'$ is unramified, and since $Z' \\times_Z U$ is a", "scheme \\'etale over $Z' \\times_Z X$ we conclude (by the same lemma again) that", "$Z' \\times_Z U \\to Z'$ is unramified.", "As $\\{Z' \\to Z\\}$ is an fpqc covering we conclude that", "$U \\to Z$ is unramified by", "Descent, Lemma \\ref{descent-lemma-descending-property-unramified}", "as desired." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-unramified-local", "spaces-morphisms-lemma-unramified-local", "descent-lemma-descending-property-unramified" ], "ref_ids": [ 9380, 4898, 4898, 14693 ] } ], "ref_ids": [] }, { "id": 9408, "type": "theorem", "label": "spaces-descent-lemma-descending-property-etale", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-etale", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is \\'etale''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. Assumptions (1) and (2) of that lemma follow from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-etale-local}.", "Let $Z' \\to Z$ be a surjective flat morphism of affine schemes over $S$.", "Let $f : X \\to Z$ be a morphism of algebraic spaces, and assume", "that the base change $f' : Z' \\times_Z X \\to Z'$ is \\'etale.", "We have to show that $f$ is \\'etale. Let $U$ be a scheme", "and let $U \\to X$ be surjective and \\'etale. By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-etale-local}", "again, it is enough to show that $U \\to Z$ is \\'etale.", "Since $f'$ is \\'etale, and since $Z' \\times_Z U$ is a", "scheme \\'etale over $Z' \\times_Z X$ we conclude (by the same lemma again) that", "$Z' \\times_Z U \\to Z'$ is \\'etale.", "As $\\{Z' \\to Z\\}$ is an fpqc covering we conclude that", "$U \\to Z$ is \\'etale by", "Descent, Lemma \\ref{descent-lemma-descending-property-etale}", "as desired." ], "refs": [ "spaces-descent-lemma-descending-properties-morphisms", "spaces-morphisms-lemma-etale-local", "spaces-morphisms-lemma-etale-local", "descent-lemma-descending-property-etale" ], "ref_ids": [ 9380, 4905, 4905, 14694 ] } ], "ref_ids": [] }, { "id": 9409, "type": "theorem", "label": "spaces-descent-lemma-descending-property-finite-locally-free", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-finite-locally-free", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is finite locally free''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Being finite locally free is equivalent to being", "finite, flat and locally of finite presentation", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-flat}).", "Hence this follows from Lemmas", "\\ref{lemma-descending-property-finite},", "\\ref{lemma-descending-property-flat}, and", "\\ref{lemma-descending-property-locally-finite-presentation}." ], "refs": [ "spaces-morphisms-lemma-finite-flat", "spaces-descent-lemma-descending-property-finite", "spaces-descent-lemma-descending-property-flat", "spaces-descent-lemma-descending-property-locally-finite-presentation" ], "ref_ids": [ 4954, 9403, 9393, 9390 ] } ], "ref_ids": [] }, { "id": 9410, "type": "theorem", "label": "spaces-descent-lemma-descending-property-monomorphism", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-monomorphism", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is a monomorphism''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a morphism of algebraic spaces.", "Let $\\{Y_i \\to Y\\}$ be an fpqc covering, and assume", "each of the base changes $f_i : X_i \\to Y_i$ of $f$ is", "a monomorphism. We have to show that $f$ is a monomorphism.", "\\medskip\\noindent", "First proof. Note that $f$ is a monomorphism if and only if", "$\\Delta : X \\to X \\times_Y X$ is an isomorphism. By applying this to", "$f_i$ we see that each of the morphisms", "$$", "\\Delta_i :", "X_i", "\\longrightarrow", "X_i \\times_{Y_i} X_i = Y_i \\times_Y (X \\times_Y X)", "$$", "is an isomorphism. The base change of an fpqc covering is an fpqc covering, see", "Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-fpqc}", "hence $\\{Y_i \\times_Y (X \\times_Y X) \\to X \\times_Y X\\}$", "is an fpqc covering of algebraic spaces. Moreover, each", "$\\Delta_i$ is the base change of the morphism", "$\\Delta : X \\to X \\times_Y X$. Hence it follows from", "Lemma \\ref{lemma-descending-property-isomorphism}", "that $\\Delta$ is an isomorphism, i.e., $f$ is a monomorphism.", "\\medskip\\noindent", "Second proof.", "Let $V$ be a scheme, and let $V \\to Y$ be a surjective \\'etale morphism.", "If we can show that $V \\times_Y X \\to V$ is a monomorphism, then it", "follows that $X \\to Y$ is a monomorphism. Namely, given any", "cartesian diagram of sheaves", "$$", "\\vcenter{", "\\xymatrix{", "\\mathcal{F} \\ar[r]_a \\ar[d]_b & \\mathcal{G} \\ar[d]^c \\\\", "\\mathcal{H} \\ar[r]^d & \\mathcal{I}", "}", "}", "\\quad", "\\quad", "\\mathcal{F} = \\mathcal{H} \\times_\\mathcal{I} \\mathcal{G}", "$$", "if $c$ is a surjection of sheaves, and $a$ is injective, then also", "$d$ is injective. This reduces the problem to the case where $Y$ is", "a scheme. Moreover, in this case we may assume that the algebraic spaces", "$Y_i$ are schemes also, since we can always refine the covering to place", "ourselves in this situation, see", "Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-refine-fpqc-schemes}.", "\\medskip\\noindent", "Assume $\\{Y_i \\to Y\\}$ is an fpqc covering of schemes.", "Let $a, b : T \\to X$ be two morphisms", "such that $f \\circ a = f \\circ b$. We have to show that $a = b$.", "Since $f_i$ is a monomorphism we see that $a_i = b_i$, where", "$a_i, b_i : Y_i \\times_Y T \\to X_i$ are", "the base changes. In particular the compositions", "$Y_i \\times_Y T \\to T \\to X$ are equal.", "Since $\\{Y_i \\times_Y T \\to T\\}$ is an fpqc covering we", "deduce that $a = b$ from Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-sheaf-fpqc}." ], "refs": [ "spaces-topologies-lemma-fpqc", "spaces-descent-lemma-descending-property-isomorphism", "spaces-topologies-lemma-refine-fpqc-schemes", "spaces-properties-proposition-sheaf-fpqc" ], "ref_ids": [ 3678, 9395, 3680, 11919 ] } ], "ref_ids": [] }, { "id": 9411, "type": "theorem", "label": "spaces-descent-lemma-descending-fppf-property-immersion", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-fppf-property-immersion", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is an immersion''", "is fppf local on the base." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a morphism of algebraic spaces.", "Let $\\{Y_i \\to Y\\}_{i \\in I}$ be an fppf covering of $Y$.", "Let $f_i : X_i \\to Y_i$ be the base change of $f$.", "\\medskip\\noindent", "If $f$ is an immersion, then each $f_i$ is an immersion by", "Spaces, Lemma \\ref{spaces-lemma-base-change-immersions}.", "This proves the direct implication in", "Definition \\ref{definition-property-morphisms-local}.", "\\medskip\\noindent", "Conversely, assume each $f_i$ is an immersion. By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-immersions-monomorphisms}", "this implies each $f_i$ is separated. By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-immersion-quasi-finite}", "this implies each $f_i$ is locally quasi-finite.", "Hence we see that $f$ is locally quasi-finite and separated, by applying", "Lemmas \\ref{lemma-descending-property-separated}", "and \\ref{lemma-descending-property-quasi-finite}.", "By", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable}", "this implies that $f$ is representable!", "\\medskip\\noindent", "By", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-closed-immersion-local}", "it suffices to show that for every scheme $Z$ and morphism $Z \\to Y$", "the base change $Z \\times_Y X \\to Z$ is an immersion. By", "Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-refine-fppf-schemes}", "we can find an fppf covering $\\{Z_i \\to Z\\}$ by schemes which refines", "the pullback of the covering $\\{Y_i \\to Y\\}$ to $Z$.", "Hence we see that $Z \\times_Y X \\to Z$ (which is a morphism of schemes", "according to the result of the preceding paragraph) becomes an immersion", "after pulling back to the members of an fppf (by schemes) of $Z$.", "Hence $Z \\times_Y X \\to Z$ is an immersion by the result for schemes, see", "Descent, Lemma \\ref{descent-lemma-descending-fppf-property-immersion}." ], "refs": [ "spaces-lemma-base-change-immersions", "spaces-descent-definition-property-morphisms-local", "spaces-morphisms-lemma-immersions-monomorphisms", "spaces-morphisms-lemma-immersion-quasi-finite", "spaces-descent-lemma-descending-property-separated", "spaces-descent-lemma-descending-property-quasi-finite", "spaces-morphisms-lemma-locally-quasi-finite-separated-representable", "spaces-morphisms-lemma-closed-immersion-local", "spaces-topologies-lemma-refine-fppf-schemes", "descent-lemma-descending-fppf-property-immersion" ], "ref_ids": [ 8161, 9440, 4756, 4835, 9398, 9404, 4972, 4761, 3666, 14698 ] } ], "ref_ids": [] }, { "id": 9412, "type": "theorem", "label": "spaces-descent-lemma-descending-fppf-property-locally-separated", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-fppf-property-locally-separated", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is locally separated''", "is fppf local on the base." ], "refs": [], "proofs": [ { "contents": [ "A base change of a locally separated morphism is locally separated, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-base-change-separated}.", "Hence the direct implication in", "Definition \\ref{definition-property-morphisms-local}.", "\\medskip\\noindent", "Let $\\{Y_i \\to Y\\}_{i \\in I}$ be an fppf covering of algebraic spaces over $S$.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume each base change $X_i := Y_i \\times_Y X \\to Y_i$ is locally separated.", "This means that each of the morphisms", "$$", "\\Delta_i :", "X_i", "\\longrightarrow", "X_i \\times_{Y_i} X_i = Y_i \\times_Y (X \\times_Y X)", "$$", "is an immersion. The base change of a fppf covering is an", "fppf covering, see", "Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-fppf}", "hence $\\{Y_i \\times_Y (X \\times_Y X) \\to X \\times_Y X\\}$", "is an fppf covering of algebraic spaces. Moreover, each", "$\\Delta_i$ is the base change of the morphism", "$\\Delta : X \\to X \\times_Y X$. Hence it follows from", "Lemma \\ref{lemma-descending-fppf-property-immersion}", "that $\\Delta$ is a immersion, i.e., $f$ is locally separated." ], "refs": [ "spaces-morphisms-lemma-base-change-separated", "spaces-descent-definition-property-morphisms-local", "spaces-topologies-lemma-fppf", "spaces-descent-lemma-descending-fppf-property-immersion" ], "ref_ids": [ 4714, 9440, 3665, 9411 ] } ], "ref_ids": [] }, { "id": 9413, "type": "theorem", "label": "spaces-descent-lemma-descending-property-ample", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descending-property-ample", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $\\{g_i : Y_i \\to Y\\}_{i \\in I}$ be an fpqc covering.", "Let $f_i : X_i \\to Y_i$ be the base change of $f$ and let $\\mathcal{L}_i$", "be the pullback of $\\mathcal{L}$ to $X_i$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is ample on $X/Y$, and", "\\item $\\mathcal{L}_i$ is ample on $X_i/Y_i$", "for every $i \\in I$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) follows from", "Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-ample-base-change}.", "Assume (2). To check $\\mathcal{L}$ is ample on $X/Y$ we may", "work \\'etale localy on $Y$, see", "Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-relatively-ample-local}.", "Thus we may assume that $Y$ is a scheme and then we may", "in turn assume each $Y_i$ is a scheme too, see", "Topologies on Spaces, Lemma \\ref{spaces-topologies-lemma-refine-fpqc-schemes}.", "In other words, we may assume that", "$\\{Y_i \\to Y\\}$ is an fpqc covering of schemes.", "\\medskip\\noindent", "By Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-relatively-ample-properties}", "we see that $X_i \\to Y_i$ is representable (i.e., $X_i$ is a scheme),", "quasi-compact, and separated. Hence $f$ is quasi-compact and separated by", "Lemmas \\ref{lemma-descending-property-quasi-compact} and", "\\ref{lemma-descending-property-separated}.", "This means that", "$\\mathcal{A} = \\bigoplus_{d \\geq 0} f_*\\mathcal{L}^{\\otimes d}$", "is a quasi-coherent graded $\\mathcal{O}_Y$-algebra", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-pushforward}).", "Moreover, the formation of $\\mathcal{A}$ commutes with flat", "base change by", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-flat-base-change-cohomology}.", "In particular, if we set", "$\\mathcal{A}_i = \\bigoplus_{d \\geq 0} f_{i, *}\\mathcal{L}_i^{\\otimes d}$", "then we have $\\mathcal{A}_i = g_i^*\\mathcal{A}$.", "It follows that the natural maps", "$\\psi_d : f^*\\mathcal{A}_d \\to \\mathcal{L}^{\\otimes d}$", "of $\\mathcal{O}_X$", "pullback to give the natural maps", "$\\psi_{i, d} : f_i^*(\\mathcal{A}_i)_d \\to \\mathcal{L}_i^{\\otimes d}$", "of $\\mathcal{O}_{X_i}$-modules. Since $\\mathcal{L}_i$ is ample on $X_i/Y_i$", "we see that for any point $x_i \\in X_i$, there exists a $d \\geq 1$", "such that $f_i^*(\\mathcal{A}_i)_d \\to \\mathcal{L}_i^{\\otimes d}$", "is surjective on stalks at $x_i$. This follows either directly", "from the definition of a relatively ample module or from", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-relatively-ample}.", "If $x \\in |X|$, then we can choose an $i$ and an $x_i \\in X_i$", "mapping to $x$. Since", "$\\mathcal{O}_{X, \\overline{x}} \\to \\mathcal{O}_{X_i, \\overline{x}_i}$", "is flat hence faithfully flat, we conclude that for every $x \\in |X|$", "there exists a $d \\geq 1$ such that", "$f^*\\mathcal{A}_d \\to \\mathcal{L}^{\\otimes d}$", "is surjective on stalks at $x$.", "This implies that the open subset $U(\\psi) \\subset X$ of", "Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-invertible-map-into-relative-proj}", "corresponding to the map", "$\\psi : f^*\\mathcal{A} \\to \\bigoplus_{d \\geq 0} \\mathcal{L}^{\\otimes d}$", "of graded $\\mathcal{O}_X$-algebras", "is equal to $X$. Consider the corresponding morphism", "$$", "r_{\\mathcal{L}, \\psi} : X \\longrightarrow \\underline{\\text{Proj}}_Y(\\mathcal{A})", "$$", "It is clear from the above that the base change of", "$r_{\\mathcal{L}, \\psi}$ to $Y_i$ is the morphism", "$r_{\\mathcal{L}_i, \\psi_i}$ which is an open immersion by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-relatively-ample}.", "Hence $r_{\\mathcal{L}, \\psi}$ is an open immersion", "by Lemma \\ref{lemma-descending-property-open-immersion}.", "Hence $X$ is a scheme", "and we conclude $\\mathcal{L}$ is ample on $X/Y$ by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-relatively-ample}." ], "refs": [ "spaces-divisors-lemma-ample-base-change", "spaces-divisors-lemma-relatively-ample-local", "spaces-topologies-lemma-refine-fpqc-schemes", "spaces-divisors-lemma-relatively-ample-properties", "spaces-descent-lemma-descending-property-quasi-compact", "spaces-descent-lemma-descending-property-separated", "spaces-morphisms-lemma-pushforward", "spaces-cohomology-lemma-flat-base-change-cohomology", "morphisms-lemma-characterize-relatively-ample", "spaces-divisors-lemma-invertible-map-into-relative-proj", "morphisms-lemma-characterize-relatively-ample", "spaces-descent-lemma-descending-property-open-immersion", "morphisms-lemma-characterize-relatively-ample" ], "ref_ids": [ 12978, 12981, 3680, 12979, 9381, 9398, 4760, 11296, 5380, 12976, 5380, 9394, 5380 ] } ], "ref_ids": [] }, { "id": 9414, "type": "theorem", "label": "spaces-descent-lemma-ample-in-neighbourhood", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-ample-in-neighbourhood", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a proper morphism of algebraic spaces over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "There exists an open subspace $V \\subset Y$ characterized by", "the following property:", "A morphism $Y' \\to Y$ of algebraic spaces factors", "through $V$ if and only if the pullback $\\mathcal{L}'$", "of $\\mathcal{L}$ to $X' = Y' \\times_Y X$ is ample on $X'/Y'$", "(as in Divisors on Spaces, Definition", "\\ref{spaces-divisors-definition-relatively-ample})." ], "refs": [ "spaces-divisors-definition-relatively-ample" ], "proofs": [ { "contents": [ "Suppose that the lemma holds whenever $Y$ is a scheme.", "Let $U$ be a scheme and let $U \\to Y$ be a surjective \\'etale morphism.", "Let $R = U \\times_Y U$ with projections $t, s : R \\to U$.", "Denote $X_U = U \\times_Y X$ and $\\mathcal{L}_U$ the pullback.", "Then we get an open subscheme $V' \\subset U$ as in the lemma", "for $(X_U \\to U, \\mathcal{L}_U)$. By the functorial characterization", "we see that $s^{-1}(V') = t^{-1}(V')$.", "Thus there is an open subspace $V \\subset Y$ such that", "$V'$ is the inverse image of $V$ in $U$.", "In particular $V' \\to V$ is surjective \\'etale and we", "conclude that $\\mathcal{L}_V$ is ample on $X_V/V$", "(Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-relatively-ample-local}).", "Now, if $Y' \\to Y$ is a morphism such that", "$\\mathcal{L}'$ is ample on $X'/Y'$, then", "$U \\times_Y Y' \\to Y'$ must factor through $V'$", "and we conclude that $Y' \\to Y$ factors through $V$.", "Hence $V \\subset Y$ is as in the statement of the lemma.", "In this way we reduce to the case dealt with in the next", "paragraph.", "\\medskip\\noindent", "Assume $Y$ is a scheme. Since the question is local on $Y$", "we may assume $Y$ is an affine scheme. We will show the", "following:", "\\begin{enumerate}", "\\item[(A)] If $\\Spec(k) \\to Y$ is a morphism such that", "$\\mathcal{L}_k$ is ample on $X_k/k$, then there is an", "open neighbourhood $V \\subset Y$ of the image of $\\Spec(k) \\to Y$", "such that $\\mathcal{L}_V$ is ample on $X_V/V$.", "\\end{enumerate}", "It is clear that (A) implies the truth of the lemma.", "\\medskip\\noindent", "Let $X \\to Y$, $\\mathcal{L}$, $\\Spec(k) \\to Y$ be as in (A).", "By Lemma \\ref{lemma-descending-property-ample}", "we may assume that $k = \\kappa(y)$ is the residue field of a point $y$", "of $Y$.", "\\medskip\\noindent", "As $Y$ is affine we can find a directed set $I$ and an", "inverse system of morphisms $X_i \\to Y_i$ of algebraic spaces", "with $Y_i$ of finite presentation over $\\mathbf{Z}$, with affine", "transition morphisms $X_i \\to X_{i'}$ and $Y_i \\to Y_{i'}$,", "with $X_i \\to Y_i$ proper and of finite presentation, and such that", "$X \\to Y = \\lim (X_i \\to Y_i)$. See Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-proper-limit-of-proper-finite-presentation-noetherian}.", "After shrinking $I$ we may assume $Y_i$ is an (affine) scheme for all $i$,", "see Limits of Spaces, Lemma \\ref{spaces-limits-lemma-limit-is-affine}.", "After shrinking $I$ we can assume we have a compatible system of", "invertible $\\mathcal{O}_{X_i}$-modules $\\mathcal{L}_i$", "pulling back to $\\mathcal{L}$, see", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-descend-invertible-modules}.", "Let $y_i \\in Y_i$ be the image of $y$.", "Then $\\kappa(y) = \\colim \\kappa(y_i)$.", "Hence $X_y = \\lim X_{i, y_i}$ and after shrinking $I$ we", "may assume $X_{i, y_i}$ is a scheme for all $i$, see", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-limit-is-scheme}.", "Hence for some $i$ we have $\\mathcal{L}_{i, y_i}$", "is ample on $X_{i, y_i}$ by", "Limits, Lemma \\ref{limits-lemma-limit-ample}.", "By Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-ample-in-neighbourhood}", "we find an open neigbourhood", "$V_i \\subset Y_i$ of $y_i$ such that", "$\\mathcal{L}_i$ restricted to $f_i^{-1}(V_i)$", "is ample relative to $V_i$.", "Letting $V \\subset Y$ be the inverse image of", "$V_i$ finishes the proof (hints: use", "Morphisms, Lemma \\ref{morphisms-lemma-ample-base-change} and", "the fact that $X \\to Y \\times_{Y_i} X_i$ is affine", "and the fact that the pullback of an", "ample invertible sheaf by an affine morphism is ample by", "Morphisms, Lemma \\ref{morphisms-lemma-pullback-ample-tensor-relatively-ample})." ], "refs": [ "spaces-divisors-lemma-relatively-ample-local", "spaces-descent-lemma-descending-property-ample", "spaces-limits-lemma-proper-limit-of-proper-finite-presentation-noetherian", "spaces-limits-lemma-limit-is-affine", "spaces-limits-lemma-descend-invertible-modules", "spaces-limits-lemma-limit-is-scheme", "limits-lemma-limit-ample", "spaces-divisors-lemma-ample-in-neighbourhood", "morphisms-lemma-ample-base-change", "morphisms-lemma-pullback-ample-tensor-relatively-ample" ], "ref_ids": [ 12981, 9413, 4617, 4578, 4600, 4579, 15045, 12984, 5385, 5383 ] } ], "ref_ids": [ 13028 ] }, { "id": 9415, "type": "theorem", "label": "spaces-descent-lemma-precompose-property-local-source", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-precompose-property-local-source", "contents": [ "Let $S$ be a scheme.", "Let $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0] syntomic, \\linebreak[0]", "smooth, \\linebreak[0] \\etale\\}$.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$", "which is $\\tau$ local on the source. Let $f : X \\to Y$ have property", "$\\mathcal{P}$. For any morphism $a : X' \\to X$ which is", "flat, resp.\\ flat and locally of finite presentation, resp.\\ syntomic,", "resp.\\ smooth, resp.\\ \\'etale, the composition $f \\circ a : X' \\to Y$ has", "property $\\mathcal{P}$." ], "refs": [], "proofs": [ { "contents": [ "This is true because we can fit $X' \\to X$ into a family of", "morphisms which forms a $\\tau$-covering." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9416, "type": "theorem", "label": "spaces-descent-lemma-transfer-from-schemes", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-transfer-from-schemes", "contents": [ "Let $S$ be a scheme.", "Let $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0] syntomic, \\linebreak[0]", "smooth, \\linebreak[0] \\etale\\}$.", "Suppose that $\\mathcal{P}$ is a property of morphisms of schemes over $S$", "which is \\'etale local on the source-and-target. Denote $\\mathcal{P}_{spaces}$", "the corresponding property of morphisms of algebraic spaces over $S$, see", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-P}.", "If $\\mathcal{P}$ is local on the source for the $\\tau$-topology, then", "$\\mathcal{P}_{spaces}$ is local on the source for the $\\tau$-topology." ], "refs": [ "spaces-morphisms-definition-P" ], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\{X_i \\to X\\}_{i \\in I}$ be a $\\tau$-covering of algebraic spaces.", "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X \\times_Y V$.", "For each $i$ choose a scheme $U_i$ and a surjective \\'etale morphism", "$U_i \\to X_i \\times_X U$.", "\\medskip\\noindent", "Note that $\\{X_i \\times_X U \\to U\\}_{i \\in I}$ is a $\\tau$-covering.", "Note that each $\\{U_i \\to X_i \\times_X U\\}$ is an \\'etale covering,", "hence a $\\tau$-covering. Hence $\\{U_i \\to U\\}_{i \\in I}$ is a", "$\\tau$-covering of algebraic spaces over $S$. But since $U$ and each $U_i$", "is a scheme we see that $\\{U_i \\to U\\}_{i \\in I}$ is a", "$\\tau$-covering of schemes over $S$.", "\\medskip\\noindent", "Now we have", "\\begin{align*}", "f \\text{ has }\\mathcal{P}_{spaces}", "& \\Leftrightarrow", "U \\to V \\text{ has }\\mathcal{P} \\\\", "& \\Leftrightarrow", "\\text{each }U_i \\to V \\text{ has }\\mathcal{P} \\\\", "& \\Leftrightarrow", "\\text{each }X_i \\to Y\\text{ has }\\mathcal{P}_{spaces}.", "\\end{align*}", "the first and last equivalence by the definition of", "$\\mathcal{P}_{spaces}$ the middle equivalence because we assumed", "$\\mathcal{P}$ is local on the source in the $\\tau$-topology." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 5001 ] }, { "id": 9417, "type": "theorem", "label": "spaces-descent-lemma-flat-fpqc-local-source", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-flat-fpqc-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is flat'' is fpqc local on the source." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-transfer-from-schemes}", "using", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-flat}", "and", "Descent, Lemma \\ref{descent-lemma-flat-fpqc-local-source}." ], "refs": [ "spaces-descent-lemma-transfer-from-schemes", "spaces-morphisms-definition-flat", "descent-lemma-flat-fpqc-local-source" ], "ref_ids": [ 9416, 5007, 14708 ] } ], "ref_ids": [] }, { "id": 9418, "type": "theorem", "label": "spaces-descent-lemma-locally-finite-presentation-fppf-local-source", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-locally-finite-presentation-fppf-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is locally of finite presentation''", "is fppf local on the source." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-transfer-from-schemes}", "using", "Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-locally-finite-presentation}", "and", "Descent,", "Lemma \\ref{descent-lemma-locally-finite-presentation-fppf-local-source}." ], "refs": [ "spaces-descent-lemma-transfer-from-schemes", "spaces-morphisms-definition-locally-finite-presentation", "descent-lemma-locally-finite-presentation-fppf-local-source" ], "ref_ids": [ 9416, 5006, 14710 ] } ], "ref_ids": [] }, { "id": 9419, "type": "theorem", "label": "spaces-descent-lemma-locally-finite-type-fppf-local-source", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-locally-finite-type-fppf-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is locally of finite type''", "is fppf local on the source." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-transfer-from-schemes}", "using", "Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-locally-finite-type}", "and", "Descent, Lemma \\ref{descent-lemma-locally-finite-type-fppf-local-source}." ], "refs": [ "spaces-descent-lemma-transfer-from-schemes", "spaces-morphisms-definition-locally-finite-type", "descent-lemma-locally-finite-type-fppf-local-source" ], "ref_ids": [ 9416, 5003, 14711 ] } ], "ref_ids": [] }, { "id": 9420, "type": "theorem", "label": "spaces-descent-lemma-open-fppf-local-source", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-open-fppf-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is open''", "is fppf local on the source." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-transfer-from-schemes}", "using", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-open}", "and", "Descent, Lemma \\ref{descent-lemma-open-fppf-local-source}." ], "refs": [ "spaces-descent-lemma-transfer-from-schemes", "spaces-morphisms-definition-open", "descent-lemma-open-fppf-local-source" ], "ref_ids": [ 9416, 4986, 14712 ] } ], "ref_ids": [] }, { "id": 9421, "type": "theorem", "label": "spaces-descent-lemma-universally-open-fppf-local-source", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-universally-open-fppf-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is universally open''", "is fppf local on the source." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-transfer-from-schemes}", "using", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-open}", "and", "Descent, Lemma \\ref{descent-lemma-universally-open-fppf-local-source}." ], "refs": [ "spaces-descent-lemma-transfer-from-schemes", "spaces-morphisms-definition-open", "descent-lemma-universally-open-fppf-local-source" ], "ref_ids": [ 9416, 4986, 14713 ] } ], "ref_ids": [] }, { "id": 9422, "type": "theorem", "label": "spaces-descent-lemma-syntomic-syntomic-local-source", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-syntomic-syntomic-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is syntomic''", "is syntomic local on the source." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-transfer-from-schemes}", "using", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-syntomic}", "and", "Descent, Lemma \\ref{descent-lemma-syntomic-syntomic-local-source}." ], "refs": [ "spaces-descent-lemma-transfer-from-schemes", "spaces-morphisms-definition-syntomic", "descent-lemma-syntomic-syntomic-local-source" ], "ref_ids": [ 9416, 5011, 14714 ] } ], "ref_ids": [] }, { "id": 9423, "type": "theorem", "label": "spaces-descent-lemma-smooth-smooth-local-source", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-smooth-smooth-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is smooth''", "is smooth local on the source." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-transfer-from-schemes}", "using", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-smooth}", "and", "Descent, Lemma \\ref{descent-lemma-smooth-smooth-local-source}." ], "refs": [ "spaces-descent-lemma-transfer-from-schemes", "spaces-morphisms-definition-smooth", "descent-lemma-smooth-smooth-local-source" ], "ref_ids": [ 9416, 5012, 14715 ] } ], "ref_ids": [] }, { "id": 9424, "type": "theorem", "label": "spaces-descent-lemma-etale-etale-local-source", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-etale-etale-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is \\'etale''", "is \\'etale local on the source." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-transfer-from-schemes}", "using", "Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-etale}", "and", "Descent, Lemma \\ref{descent-lemma-etale-etale-local-source}." ], "refs": [ "spaces-descent-lemma-transfer-from-schemes", "spaces-morphisms-definition-etale", "descent-lemma-etale-etale-local-source" ], "ref_ids": [ 9416, 5014, 14716 ] } ], "ref_ids": [] }, { "id": 9425, "type": "theorem", "label": "spaces-descent-lemma-locally-quasi-finite-etale-local-source", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-locally-quasi-finite-etale-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is locally quasi-finite''", "is \\'etale local on the source." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-transfer-from-schemes}", "using", "Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-locally-quasi-finite}", "and", "Descent, Lemma \\ref{descent-lemma-locally-quasi-finite-etale-local-source}." ], "refs": [ "spaces-descent-lemma-transfer-from-schemes", "spaces-morphisms-definition-locally-quasi-finite", "descent-lemma-locally-quasi-finite-etale-local-source" ], "ref_ids": [ 9416, 5005, 14717 ] } ], "ref_ids": [] }, { "id": 9426, "type": "theorem", "label": "spaces-descent-lemma-unramified-etale-local-source", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-unramified-etale-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is unramified''", "is \\'etale local on the source." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-transfer-from-schemes}", "using", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-unramified}", "and", "Descent, Lemma \\ref{descent-lemma-unramified-etale-local-source}." ], "refs": [ "spaces-descent-lemma-transfer-from-schemes", "spaces-morphisms-definition-unramified", "descent-lemma-unramified-etale-local-source" ], "ref_ids": [ 9416, 5013, 14718 ] } ], "ref_ids": [] }, { "id": 9427, "type": "theorem", "label": "spaces-descent-lemma-local-source-target-implies", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-local-source-target-implies", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$", "which is smooth local on source-and-target. Then", "\\begin{enumerate}", "\\item $\\mathcal{P}$ is smooth local on the source,", "\\item $\\mathcal{P}$ is smooth local on the target,", "\\item $\\mathcal{P}$ is stable under postcomposing with smooth morphisms:", "if $f : X \\to Y$ has $\\mathcal{P}$ and $g : Y \\to Z$ is smooth, then", "$g \\circ f$ has $\\mathcal{P}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We write everything out completely.", "\\medskip\\noindent", "Proof of (1). Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\{X_i \\to X\\}_{i \\in I}$ be a smooth covering of $X$. If each composition", "$h_i : X_i \\to Y$ has $\\mathcal{P}$, then for each $|x| \\in X$ we can find", "an $i \\in I$ and a point $x_i \\in |X_i|$ mapping to $x$. Then", "$(X_i, x_i) \\to (X, x)$ is a smooth morphism of pairs, and", "$\\text{id}_Y : Y \\to Y$ is a smooth morphism, and $h_i$ is as in part (3) of", "Definition \\ref{definition-local-source-target}.", "Thus we see that $f$ has $\\mathcal{P}$.", "Conversely, if $f$ has $\\mathcal{P}$ then each $X_i \\to Y$ has", "$\\mathcal{P}$ by", "Definition \\ref{definition-local-source-target} part (1).", "\\medskip\\noindent", "Proof of (2). Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\{Y_i \\to Y\\}_{i \\in I}$ be a smooth covering of $Y$.", "Write $X_i = Y_i \\times_Y X$ and $h_i : X_i \\to Y_i$ for the base change", "of $f$. If each $h_i : X_i \\to Y_i$ has $\\mathcal{P}$, then for each", "$x \\in |X|$ we pick an $i \\in I$ and a point $x_i \\in |X_i|$ mapping to $x$.", "Then $(X_i, x_i) \\to (X, x)$ is a smooth morphism of pairs, $Y_i \\to Y$ is", "smooth, and $h_i$ is as in part (3) of", "Definition \\ref{definition-local-source-target}.", "Thus we see that $f$ has $\\mathcal{P}$.", "Conversely, if $f$ has $\\mathcal{P}$, then each $X_i \\to Y_i$ has", "$\\mathcal{P}$ by", "Definition \\ref{definition-local-source-target} part (2).", "\\medskip\\noindent", "Proof of (3). Assume $f : X \\to Y$ has $\\mathcal{P}$ and $g : Y \\to Z$ is", "smooth. For every $x \\in |X|$ we can think of $(X, x) \\to (X, x)$ as a", "smooth morphism of pairs, $Y \\to Z$ is a smooth morphism, and $h = f$ is as", "in part (3) of", "Definition \\ref{definition-local-source-target}.", "Thus we see that $g \\circ f$ has $\\mathcal{P}$." ], "refs": [ "spaces-descent-definition-local-source-target", "spaces-descent-definition-local-source-target", "spaces-descent-definition-local-source-target", "spaces-descent-definition-local-source-target", "spaces-descent-definition-local-source-target" ], "ref_ids": [ 9442, 9442, 9442, 9442, 9442 ] } ], "ref_ids": [] }, { "id": 9428, "type": "theorem", "label": "spaces-descent-lemma-local-source-target-characterize", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-local-source-target-characterize", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$", "which is smooth local on source-and-target. Let $f : X \\to Y$ be a morphism", "of algebraic spaces over $S$. The following are equivalent:", "\\begin{enumerate}", "\\item[(a)] $f$ has property $\\mathcal{P}$,", "\\item[(b)] for every $x \\in |X|$ there exists a smooth morphism of pairs", "$a : (U, u) \\to (X, x)$, a smooth morphism $b : V \\to Y$, and", "a morphism $h : U \\to V$ such that $f \\circ a = b \\circ h$ and", "$h$ has $\\mathcal{P}$,", "\\item[(c)] for some commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "with $a$, $b$ smooth and $a$ surjective the morphism $h$ has $\\mathcal{P}$,", "\\item[(d)] for any commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "with $b$ smooth and $U \\to X \\times_Y V$ smooth", "the morphism $h$ has $\\mathcal{P}$,", "\\item[(e)] there exists a smooth covering $\\{Y_i \\to Y\\}_{i \\in I}$ such", "that each base change $Y_i \\times_Y X \\to Y_i$ has $\\mathcal{P}$,", "\\item[(f)] there exists a smooth covering $\\{X_i \\to X\\}_{i \\in I}$ such", "that each composition $X_i \\to Y$ has $\\mathcal{P}$,", "\\item[(g)] there exists a smooth covering $\\{Y_i \\to Y\\}_{i \\in I}$ and", "for each $i \\in I$ a smooth covering", "$\\{X_{ij} \\to Y_i \\times_Y X\\}_{j \\in J_i}$ such that each morphism", "$X_{ij} \\to Y_i$ has $\\mathcal{P}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (a) and (b) is part of", "Definition \\ref{definition-local-source-target}.", "The equivalence of (a) and (e) is", "Lemma \\ref{lemma-local-source-target-implies} part (2).", "The equivalence of (a) and (f) is", "Lemma \\ref{lemma-local-source-target-implies} part (1).", "As (a) is now equivalent to (e) and (f) it follows that", "(a) equivalent to (g).", "\\medskip\\noindent", "It is clear that (c) implies (b). If (b) holds, then for any", "$x \\in |X|$ we can choose a smooth morphism of pairs", "$a_x : (U_x, u_x) \\to (X, x)$, a smooth morphism $b_x : V_x \\to Y$, and", "a morphism $h_x : U_x \\to V_x$ such that $f \\circ a_x = b_x \\circ h_x$ and", "$h_x$ has $\\mathcal{P}$. Then $h = \\coprod h_x : \\coprod U_x \\to \\coprod V_x$", "with $a = \\coprod a_x$ and $b = \\coprod b_x$ is a diagram as in (c).", "(Note that $h$ has property $\\mathcal{P}$ as $\\{V_x \\to \\coprod V_x\\}$", "is a smooth covering and $\\mathcal{P}$ is smooth local on the target.)", "Thus (b) is equivalent to (c).", "\\medskip\\noindent", "Now we know that (a), (b), (c), (e), (f), and (g) are equivalent.", "Suppose (a) holds. Let $U, V, a, b, h$ be as in (d). Then", "$X \\times_Y V \\to V$ has $\\mathcal{P}$ as $\\mathcal{P}$ is stable under", "smooth base change, whence $U \\to V$ has $\\mathcal{P}$ as $\\mathcal{P}$", "is stable under precomposing with smooth morphisms. Conversely, if (d)", "holds, then setting $U = X$ and $V = Y$ we see that $f$ has $\\mathcal{P}$." ], "refs": [ "spaces-descent-definition-local-source-target", "spaces-descent-lemma-local-source-target-implies", "spaces-descent-lemma-local-source-target-implies" ], "ref_ids": [ 9442, 9427, 9427 ] } ], "ref_ids": [] }, { "id": 9429, "type": "theorem", "label": "spaces-descent-lemma-smooth-local-source-target", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-smooth-local-source-target", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$.", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{P}$ is smooth local on the source,", "\\item $\\mathcal{P}$ is smooth local on the target, and", "\\item $\\mathcal{P}$ is stable under postcomposing with smooth morphisms:", "if $f : X \\to Y$ has $\\mathcal{P}$ and $Y \\to Z$ is a smooth morphism", "then $X \\to Z$ has $\\mathcal{P}$.", "\\end{enumerate}", "Then $\\mathcal{P}$ is smooth local on the source-and-target." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which", "satisfies conditions (1), (2) and (3) of the lemma. By", "Lemma \\ref{lemma-precompose-property-local-source}", "we see that $\\mathcal{P}$ is stable under precomposing with", "smooth morphisms. By", "Lemma \\ref{lemma-pullback-property-local-target}", "we see that $\\mathcal{P}$ is stable under smooth base change.", "Hence it suffices to prove part (3) of", "Definition \\ref{definition-local-source-target}", "holds.", "\\medskip\\noindent", "More precisely, suppose that $f : X \\to Y$ is a morphism", "of algebraic spaces over $S$ which satisfies", "Definition \\ref{definition-local-source-target} part (3)(b).", "In other words, for every $x \\in X$ there exists a smooth", "morphism $a_x : U_x \\to X$, a point $u_x \\in |U_x|$ mapping to $x$,", "a smooth morphism $b_x : V_x \\to Y$, and a morphism $h_x : U_x \\to V_x$", "such that $f \\circ a_x = b_x \\circ h_x$ and $h_x$ has $\\mathcal{P}$.", "The proof of the lemma is complete once we show that $f$ has $\\mathcal{P}$.", "Set $U = \\coprod U_x$, $a = \\coprod a_x$, $V = \\coprod V_x$,", "$b = \\coprod b_x$, and $h = \\coprod h_x$. We obtain a", "commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "with $a$, $b$ smooth, $a$ surjective. Note that $h$ has $\\mathcal{P}$", "as each $h_x$ does and $\\mathcal{P}$ is smooth local on the target.", "Because $a$ is surjective and $\\mathcal{P}$ is smooth local on the source,", "it suffices to prove that $b \\circ h$ has $\\mathcal{P}$.", "This follows as we assumed that $\\mathcal{P}$ is stable under", "postcomposing with a smooth morphism and as $b$ is smooth." ], "refs": [ "spaces-descent-lemma-precompose-property-local-source", "spaces-descent-lemma-pullback-property-local-target", "spaces-descent-definition-local-source-target", "spaces-descent-definition-local-source-target" ], "ref_ids": [ 9415, 9378, 9442, 9442 ] } ], "ref_ids": [] }, { "id": 9430, "type": "theorem", "label": "spaces-descent-lemma-etale-smooth-local-source-target-implies", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-etale-smooth-local-source-target-implies", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$", "which is \\'etale-smooth local on source-and-target. Then", "\\begin{enumerate}", "\\item $\\mathcal{P}$ is \\'etale local on the source,", "\\item $\\mathcal{P}$ is smooth local on the target,", "\\item $\\mathcal{P}$ is stable under postcomposing with \\'etale morphisms:", "if $f : X \\to Y$ has $\\mathcal{P}$ and $g : Y \\to Z$ is \\'etale, then", "$g \\circ f$ has $\\mathcal{P}$, and", "\\item $\\mathcal{P}$ has a permanence property: given $f : X \\to Y$ and", "$g : Y \\to Z$ \\'etale such that $g \\circ f$ has $\\mathcal{P}$, then", "$f$ has $\\mathcal{P}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We write everything out completely.", "\\medskip\\noindent", "Proof of (1). Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\{X_i \\to X\\}_{i \\in I}$ be an \\'etale covering of $X$. If each composition", "$h_i : X_i \\to Y$ has $\\mathcal{P}$, then for each $|x| \\in X$ we can find", "an $i \\in I$ and a point $x_i \\in |X_i|$ mapping to $x$. Then", "$(X_i, x_i) \\to (X, x)$ is an \\'etale morphism of pairs, and", "$\\text{id}_Y : Y \\to Y$ is a smooth morphism, and $h_i$ is as in part (3) of", "Definition \\ref{definition-etale-smooth-local-source-target}.", "Thus we see that $f$ has $\\mathcal{P}$.", "Conversely, if $f$ has $\\mathcal{P}$ then each $X_i \\to Y$ has", "$\\mathcal{P}$ by", "Definition \\ref{definition-etale-smooth-local-source-target} part (1).", "\\medskip\\noindent", "Proof of (2). Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\{Y_i \\to Y\\}_{i \\in I}$ be a smooth covering of $Y$.", "Write $X_i = Y_i \\times_Y X$ and $h_i : X_i \\to Y_i$ for the base change", "of $f$. If each $h_i : X_i \\to Y_i$ has $\\mathcal{P}$, then for each", "$x \\in |X|$ we pick an $i \\in I$ and a point $x_i \\in |X_i|$ mapping to $x$.", "Then $X_i \\to X \\times_Y Y_i$ is an \\'etale morphism", "(because it is an isomorphism), $Y_i \\to Y$ is", "smooth, and $h_i$ is as in part (3) of", "Definition \\ref{definition-local-source-target}.", "Thus we see that $f$ has $\\mathcal{P}$.", "Conversely, if $f$ has $\\mathcal{P}$, then each $X_i \\to Y_i$ has", "$\\mathcal{P}$ by", "Definition \\ref{definition-local-source-target} part (2).", "\\medskip\\noindent", "Proof of (3). Assume $f : X \\to Y$ has $\\mathcal{P}$ and $g : Y \\to Z$ is", "\\'etale. The morphism $X \\to Y \\times_Z X$ is \\'etale as a morphism", "between algebraic spaces \\'etale over $X$ (", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-permanence}).", "Also $Y \\to Z$ is \\'etale hence a smooth morphism.", "Thus the diagram", "$$", "\\xymatrix{", "X \\ar[d] \\ar[r]_f & Y \\ar[d] \\\\", "X \\ar[r]^{g \\circ f} & Z", "}", "$$", "works for every $x \\in |X|$ in part (3) of", "Definition \\ref{definition-local-source-target}", "and we conclude that $g \\circ f$ has $\\mathcal{P}$.", "\\medskip\\noindent", "Proof of (4). Let $f : X \\to Y$ be a morphism and $g : Y \\to Z$ \\'etale", "such that $g \\circ f$ has $\\mathcal{P}$. Then by", "Definition \\ref{definition-etale-smooth-local-source-target} part (2)", "we see that $\\text{pr}_Y : Y \\times_Z X \\to Y$ has $\\mathcal{P}$. But", "the morphism $(f, 1) : X \\to Y \\times_Z X$ is \\'etale as a section to the", "\\'etale projection $\\text{pr}_X : Y \\times_Z X \\to X$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-etale-permanence}.", "Hence $f = \\text{pr}_Y \\circ (f, 1)$ has $\\mathcal{P}$ by", "Definition \\ref{definition-etale-smooth-local-source-target} part (1)." ], "refs": [ "spaces-descent-definition-etale-smooth-local-source-target", "spaces-descent-definition-etale-smooth-local-source-target", "spaces-descent-definition-local-source-target", "spaces-descent-definition-local-source-target", "spaces-properties-lemma-etale-permanence", "spaces-descent-definition-local-source-target", "spaces-descent-definition-etale-smooth-local-source-target", "spaces-morphisms-lemma-etale-permanence", "spaces-descent-definition-etale-smooth-local-source-target" ], "ref_ids": [ 9443, 9443, 9442, 9442, 11859, 9442, 9443, 4914, 9443 ] } ], "ref_ids": [] }, { "id": 9431, "type": "theorem", "label": "spaces-descent-lemma-etale-smooth-local-source-target-characterize", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-etale-smooth-local-source-target-characterize", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$", "which is etale-smooth local on source-and-target.", "Let $f : X \\to Y$ be a morphism", "of algebraic spaces over $S$. The following are equivalent:", "\\begin{enumerate}", "\\item[(a)] $f$ has property $\\mathcal{P}$,", "\\item[(b)] for every $x \\in |X|$ there exists a smooth morphism $b : V \\to Y$,", "an \\'etale morphism $a : U \\to V \\times_Y X$, and a point $u \\in |U|$", "mapping to $x$ such that $U \\to V$ has $\\mathcal{P}$,", "\\item[(c)] for some commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "with $b$ smooth, $U \\to V \\times_Y X$ \\'etale, and $a$ surjective", "the morphism $h$ has $\\mathcal{P}$,", "\\item[(d)] for any commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "with $b$ smooth and $U \\to X \\times_Y V$ \\'etale, the morphism $h$", "has $\\mathcal{P}$,", "\\item[(e)] there exists a smooth covering $\\{Y_i \\to Y\\}_{i \\in I}$ such", "that each base change $Y_i \\times_Y X \\to Y_i$ has $\\mathcal{P}$,", "\\item[(f)] there exists an \\'etale covering $\\{X_i \\to X\\}_{i \\in I}$ such", "that each composition $X_i \\to Y$ has $\\mathcal{P}$,", "\\item[(g)] there exists a smooth covering $\\{Y_i \\to Y\\}_{i \\in I}$ and", "for each $i \\in I$ an \\'etale covering", "$\\{X_{ij} \\to Y_i \\times_Y X\\}_{j \\in J_i}$ such that each morphism", "$X_{ij} \\to Y_i$ has $\\mathcal{P}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (a) and (b) is part of", "Definition \\ref{definition-etale-smooth-local-source-target}.", "The equivalence of (a) and (e) is", "Lemma \\ref{lemma-etale-smooth-local-source-target-implies} part (2).", "The equivalence of (a) and (f) is", "Lemma \\ref{lemma-etale-smooth-local-source-target-implies} part (1).", "As (a) is now equivalent to (e) and (f) it follows that", "(a) equivalent to (g).", "\\medskip\\noindent", "It is clear that (c) implies (b). If (b) holds, then for any", "$x \\in |X|$ we can choose a smooth morphism a smooth morphism", "$b_x : V_x \\to Y$, an \\'etale morphism $U_x \\to V_x \\times_Y X$,", "and $u_x \\in |U_x|$ mapping to $x$", "such that $U_x \\to V_x$ has $\\mathcal{P}$.", "Then $h = \\coprod h_x : \\coprod U_x \\to \\coprod V_x$", "with $a = \\coprod a_x$ and $b = \\coprod b_x$ is a diagram as in (c).", "(Note that $h$ has property $\\mathcal{P}$ as $\\{V_x \\to \\coprod V_x\\}$", "is a smooth covering and $\\mathcal{P}$ is smooth local on the target.)", "Thus (b) is equivalent to (c).", "\\medskip\\noindent", "Now we know that (a), (b), (c), (e), (f), and (g) are equivalent.", "Suppose (a) holds. Let $U, V, a, b, h$ be as in (d). Then", "$X \\times_Y V \\to V$ has $\\mathcal{P}$ as $\\mathcal{P}$ is stable under", "smooth base change, whence $U \\to V$ has $\\mathcal{P}$ as $\\mathcal{P}$", "is stable under precomposing with \\'etale morphisms. Conversely, if (d)", "holds, then setting $U = X$ and $V = Y$ we see that $f$ has $\\mathcal{P}$." ], "refs": [ "spaces-descent-definition-etale-smooth-local-source-target", "spaces-descent-lemma-etale-smooth-local-source-target-implies", "spaces-descent-lemma-etale-smooth-local-source-target-implies" ], "ref_ids": [ 9443, 9430, 9430 ] } ], "ref_ids": [] }, { "id": 9432, "type": "theorem", "label": "spaces-descent-lemma-etale-smooth-local-source-target", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-etale-smooth-local-source-target", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$.", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{P}$ is \\'etale local on the source,", "\\item $\\mathcal{P}$ is smooth local on the target, and", "\\item $\\mathcal{P}$ is stable under postcomposing with open immersions:", "if $f : X \\to Y$ has $\\mathcal{P}$ and $Y \\subset Z$ is an open embedding", "then $X \\to Z$ has $\\mathcal{P}$.", "\\end{enumerate}", "Then $\\mathcal{P}$ is \\'etale-smooth local on the source-and-target." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which", "satisfies conditions (1), (2) and (3) of the lemma. By", "Lemma \\ref{lemma-precompose-property-local-source}", "we see that $\\mathcal{P}$ is stable under precomposing with", "\\'etale morphisms. By", "Lemma \\ref{lemma-pullback-property-local-target}", "we see that $\\mathcal{P}$ is stable under smooth base change.", "Hence it suffices to prove part (3) of", "Definition \\ref{definition-local-source-target}", "holds.", "\\medskip\\noindent", "More precisely, suppose that $f : X \\to Y$ is a morphism", "of algebraic spaces over $S$ which satisfies", "Definition \\ref{definition-local-source-target} part (3)(b).", "In other words, for every $x \\in X$ there exists", "a smooth morphism $b_x : V_x \\to Y$,", "an \\'etale morphism $U_x \\to V_x \\times_Y X$, and", "a point $u_x \\in |U_x|$ mapping to $x$", "such that $h_x : U_x \\to V_x$ has $\\mathcal{P}$.", "The proof of the lemma is complete once we show that $f$ has $\\mathcal{P}$.", "\\medskip\\noindent", "Let $a_x : U_x \\to X$ be the composition $U_x \\to V_x \\times_Y X \\to X$.", "Set $U = \\coprod U_x$, $a = \\coprod a_x$, $V = \\coprod V_x$,", "$b = \\coprod b_x$, and $h = \\coprod h_x$. We obtain a", "commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "with $b$ smooth, $U \\to V \\times_Y X$ \\'etale, $a$ surjective.", "Note that $h$ has $\\mathcal{P}$ as each $h_x$ does and $\\mathcal{P}$", "is smooth local on the target. In the next paragraph we prove", "that we may assume $U, V, X, Y$ are schemes; we encourage the reader", "to skip it.", "\\medskip\\noindent", "Let $X, Y, U, V, a, b, f, h$ be as in the previous paragraph. We have", "to show $f$ has $\\mathcal{P}$. Let $X' \\to X$ be a surjective \\'etale", "morphism with $X_i$ a scheme. Set $U' = X' \\times_X U$. Then", "$U' \\to X'$ is surjective and $U' \\to X' \\times_Y V$ is \\'etale.", "Since $\\mathcal{P}$ is \\'etale local on the source, we see that", "$U' \\to V$ has $\\mathcal{P}$ and that it suffices to show that", "$X' \\to Y$ has $\\mathcal{P}$. In other words, we may assume", "that $X$ is a scheme. Next, choose a surjective \\'etale morphism", "$Y' \\to Y$ with $Y'$ a scheme. Set $V' = V \\times_Y Y'$,", "$X' = X \\times_Y Y'$, and $U' = U \\times_Y Y'$. Then", "$U' \\to X'$ is surjective and $U' \\to X' \\times_{Y'} V'$ is \\'etale.", "Since $\\mathcal{P}$ is smooth local on the target, we see that $U' \\to V'$ has", "$\\mathcal{P}$ and that it suffices to prove $X' \\to Y'$ has $\\mathcal{P}$.", "Thus we may assume both $X$ and $Y$ are schemes.", "Choose a surjective \\'etale morphism $V' \\to V$", "with $V'$ a scheme. Set $U' = U \\times_V V'$.", "Then $U' \\to X$ is surjective and $U' \\to X \\times_Y V'$ is \\'etale.", "Since $\\mathcal{P}$ is smooth local on the source, we see that", "$U' \\to V'$ has $\\mathcal{P}$. Thus we may replace $U, V$ by", "$U', V'$ and assume $X, Y, V$ are schemes.", "Finally, we replace $U$ by a scheme surjective \\'etale over $U$", "and we see that we may assume $U, V, X, Y$ are all schemes.", "\\medskip\\noindent", "If $U, V, X, Y$ are schemes, then $f$ has $\\mathcal{P}$", "by Descent, Lemma \\ref{descent-lemma-etale-tau-local-source-target}." ], "refs": [ "spaces-descent-lemma-precompose-property-local-source", "spaces-descent-lemma-pullback-property-local-target", "spaces-descent-definition-local-source-target", "spaces-descent-definition-local-source-target", "descent-lemma-etale-tau-local-source-target" ], "ref_ids": [ 9415, 9378, 9442, 9442, 14724 ] } ], "ref_ids": [] }, { "id": 9433, "type": "theorem", "label": "spaces-descent-lemma-family-is-one", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-family-is-one", "contents": [ "Let $S$ be a scheme.", "Let $\\{X_i \\to X\\}_{i \\in I}$ be a family of morphisms", "of algebraic spaces over $S$ with fixed target $X$.", "Set $Y = \\coprod_{i \\in I} X_i$.", "There is a canonical equivalence of categories", "$$", "\\begin{matrix}", "\\text{category of descent data } \\\\", "\\text{relative to the family } \\{X_i \\to X\\}_{i \\in I}", "\\end{matrix}", "\\longrightarrow", "\\begin{matrix}", "\\text{ category of descent data} \\\\", "\\text{ relative to } Y/X", "\\end{matrix}", "$$", "which maps $(V_i, \\varphi_{ij})$ to $(V, \\varphi)$ with", "$V = \\coprod_{i\\in I} V_i$ and $\\varphi = \\coprod \\varphi_{ij}$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $Y \\times_X Y = \\coprod_{ij} X_i \\times_X X_j$", "and similarly for higher fibre products.", "Giving a morphism $V \\to Y$ is exactly the same as", "giving a family $V_i \\to X_i$. And giving a descent datum", "$\\varphi$ is exactly the same as giving a family $\\varphi_{ij}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9434, "type": "theorem", "label": "spaces-descent-lemma-pullback", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-pullback", "contents": [ "Pullback of descent data. Let $S$ be a scheme.", "\\begin{enumerate}", "\\item Let", "$$", "\\xymatrix{", "Y' \\ar[r]_f \\ar[d]_{a'} & Y \\ar[d]^a \\\\", "X' \\ar[r]^h & X", "}", "$$", "be a commutative diagram of algebraic spaces over $S$.", "The construction", "$$", "(V \\to Y, \\varphi) \\longmapsto f^*(V \\to Y, \\varphi) = (V' \\to Y', \\varphi')", "$$", "where $V' = Y' \\times_Y V$ and where", "$\\varphi'$ is defined as the composition", "$$", "\\xymatrix{", "V' \\times_{X'} Y' \\ar@{=}[r] &", "(Y' \\times_Y V) \\times_{X'} Y' \\ar@{=}[r] &", "(Y' \\times_{X'} Y') \\times_{Y \\times_X Y} (V \\times_X Y)", "\\ar[d]^{\\text{id} \\times \\varphi} \\\\", "Y' \\times_{X'} V' \\ar@{=}[r] &", "Y' \\times_{X'} (Y' \\times_Y V) &", "(Y' \\times_X Y') \\times_{Y \\times_X Y} (Y \\times_X V) \\ar@{=}[l]", "}", "$$", "defines a functor from the category of descent data", "relative to $Y \\to X$ to the category of descent data", "relative to $Y' \\to X'$.", "\\item Given two morphisms $f_i : Y' \\to Y$, $i = 0, 1$ making the", "diagram commute the functors $f_0^*$ and $f_1^*$ are", "canonically isomorphic.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We omit the proof of (1), but we remark that the morphism", "$\\varphi'$ is the morphism $(f \\times f)^*\\varphi$ in the", "notation introduced in Remark \\ref{remark-easier}.", "For (2) we indicate which morphism", "$f_0^*V \\to f_1^*V$ gives the functorial isomorphism. Namely,", "since $f_0$ and $f_1$ both fit into the commutative diagram", "we see there is a unique morphism $r : Y' \\to Y \\times_X Y$", "with $f_i = \\text{pr}_i \\circ r$. Then we take", "\\begin{eqnarray*}", "f_0^*V & = &", "Y' \\times_{f_0, Y} V \\\\", "& = &", "Y' \\times_{\\text{pr}_0 \\circ r, Y} V \\\\", "& = &", "Y' \\times_{r, Y \\times_X Y} (Y \\times_X Y) \\times_{\\text{pr}_0, Y} V \\\\", "& \\xrightarrow{\\varphi} &", "Y' \\times_{r, Y \\times_X Y} (Y \\times_X Y) \\times_{\\text{pr}_1, Y} V \\\\", "& = &", "Y' \\times_{\\text{pr}_1 \\circ r, Y} V \\\\", "& = &", "Y' \\times_{f_1, Y} V \\\\", "& = & f_1^*V", "\\end{eqnarray*}", "We omit the verification that this works." ], "refs": [ "spaces-descent-remark-easier" ], "ref_ids": [ 9452 ] } ], "ref_ids": [] }, { "id": 9435, "type": "theorem", "label": "spaces-descent-lemma-pullback-family", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-pullback-family", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{U}' = \\{X'_i \\to X'\\}_{i \\in I'}$ and", "$\\mathcal{U} = \\{X_j \\to X\\}_{i \\in I}$ be families of morphisms with", "fixed target. Let $\\alpha : I' \\to I$, $g : X' \\to X$ and", "$g_i : X'_i \\to X_{\\alpha(i)}$ be a morphism of families", "of maps with fixed target, see", "Sites, Definition \\ref{sites-definition-morphism-coverings}.", "\\begin{enumerate}", "\\item Let $(V_i, \\varphi_{ij})$ be a descent datum relative to the", "family $\\mathcal{U}$. The system", "$$", "\\left(", "g_i^*V_{\\alpha(i)}, (g_i \\times g_j)^*\\varphi_{\\alpha(i) \\alpha(j)}", "\\right)", "$$", "(with notation as in Remark \\ref{remark-easier-family})", "is a descent datum relative to $\\mathcal{U}'$.", "\\item This construction defines a functor between the category of", "descent data relative to $\\mathcal{U}$ and the category of", "descent data relative to $\\mathcal{U}'$.", "\\item Given a second $\\beta : I' \\to I$, $h : X' \\to X$ and", "$h'_i : X'_i \\to X_{\\beta(i)}$ morphism of families", "of maps with fixed target, then if $g = h$ the two resulting functors", "between descent data are canonically isomorphic.", "\\item These functors agree, via Lemma \\ref{lemma-family-is-one},", "with the pullback functors constructed in Lemma \\ref{lemma-pullback}.", "\\end{enumerate}" ], "refs": [ "sites-definition-morphism-coverings", "spaces-descent-remark-easier-family", "spaces-descent-lemma-family-is-one", "spaces-descent-lemma-pullback" ], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-pullback} via the", "correspondence of Lemma \\ref{lemma-family-is-one}." ], "refs": [ "spaces-descent-lemma-pullback", "spaces-descent-lemma-family-is-one" ], "ref_ids": [ 9434, 9433 ] } ], "ref_ids": [ 8656, 9453, 9433, 9434 ] }, { "id": 9436, "type": "theorem", "label": "spaces-descent-lemma-descent-data-sheaves", "categories": [ "spaces-descent" ], "title": "spaces-descent-lemma-descent-data-sheaves", "contents": [ "Let $S$ be a scheme. Let $\\{X_i \\to X\\}_{i \\in I}$ be an fppf", "covering of algebraic spaces over $S$ (Topologies on Spaces,", "Definition \\ref{spaces-topologies-definition-fppf-covering}).", "There is an equivalence of categories", "$$", "\\left\\{", "\\begin{matrix}", "\\text{descent data }(V_i, \\varphi_{ij})\\\\", "\\text{relative to }\\{X_i \\to X\\}", "\\end{matrix}", "\\right\\}", "\\leftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{sheaves }F\\text{ on }(\\Sch/S)_{fppf}\\text{ endowed}\\\\", "\\text{with a map }F \\to X\\text{ such that each}\\\\", "X_i \\times_X F\\text{ is an algebraic space}", "\\end{matrix}", "\\right\\}.", "$$", "Moreover,", "\\begin{enumerate}", "\\item the algebraic space $X_i \\times_X F$ on the right hand side", "corresponds to $V_i$ on the left hand side, and", "\\item the sheaf $F$ is an algebraic space\\footnote{We will see", "later that this is always the case if $I$ is not too large, see", "Bootstrap, Lemma \\ref{bootstrap-lemma-descend-algebraic-space}.}", "if and only if the", "corresponding descent datum $(X_i, \\varphi_{ij})$ is effective.", "\\end{enumerate}" ], "refs": [ "spaces-topologies-definition-fppf-covering", "bootstrap-lemma-descend-algebraic-space" ], "proofs": [ { "contents": [ "Let us construct the functor from right to left.", "Let $F \\to X$ be a map of sheaves on $(\\Sch/S)_{fppf}$ such that each", "$V_i = X_i \\times_X F$ is an algebraic space. We have the", "projection $V_i \\to X_i$.", "Then both $V_i \\times_X X_j$ and $X_i \\times_X V_j$", "represent the sheaf $X_i \\times_X F \\times_X X_j$", "and hence we obtain an isomorphism", "$$", "\\varphi_{ii'} : V_i \\times_X X_j \\to X_i \\times_X V_j", "$$", "It is straightforward to see that the maps $\\varphi_{ij}$", "are morphisms over $X_i \\times_X X_j$ and satisfy the", "cocycle condition. The functor from right to left is given", "by this construction $F \\mapsto (V_i, \\varphi_{ij})$.", "\\medskip\\noindent", "Let us construct a functor from left to right.", "The isomorphisms $\\varphi_{ij}$ give isomorphisms", "$$", "\\varphi_{ij} : V_i \\times_X X_j \\longrightarrow X_i \\times_X V_j", "$$", "over $X_i \\times X_j$. Set $F$ equal to the coequalizer in the", "following diagram", "$$", "\\xymatrix{", "\\coprod_{i, i'} V_i \\times_X X_j", "\\ar@<1ex>[rr]^-{\\text{pr}_0}", "\\ar@<-1ex>[rr]_-{\\text{pr}_1 \\circ \\varphi_{ij}}", "& &", "\\coprod_i V_i \\ar[r]", "&", "F", "}", "$$", "The cocycle condition guarantees that $F$ comes with a map", "$F \\to X$ and that $X_i \\times_X F$ is isomorphic to $V_i$.", "The functor from left to right is given", "by this construction $(V_i, \\varphi_{ij}) \\mapsto F$.", "\\medskip\\noindent", "We omit the verification that these constructions", "are mutually quasi-inverse functors. The final statements", "(1) and (2) follow from the constructions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 3688, 2627 ] }, { "id": 9437, "type": "theorem", "label": "spaces-descent-proposition-fpqc-descent-quasi-coherent", "categories": [ "spaces-descent" ], "title": "spaces-descent-proposition-fpqc-descent-quasi-coherent", "contents": [ "Let $S$ be a scheme.", "Let $\\{X_i \\to X\\}$ be an fpqc covering of algebraic spaces over $S$, see", "Topologies on Spaces,", "Definition \\ref{spaces-topologies-definition-fpqc-covering}.", "Any descent datum on quasi-coherent sheaves", "for $\\{X_i \\to X\\}$ is effective.", "Moreover, the functor from the category of", "quasi-coherent $\\mathcal{O}_X$-modules to the category", "of descent data with respect to $\\{X_i \\to X\\}$ is fully faithful." ], "refs": [ "spaces-topologies-definition-fpqc-covering" ], "proofs": [ { "contents": [ "This is more or less a formal consequence of", "the corresponding result for schemes, see", "Descent, Proposition \\ref{descent-proposition-fpqc-descent-quasi-coherent}.", "Here is a strategy for a proof:", "\\begin{enumerate}", "\\item The fact that $\\{X_i \\to X\\}$ is a refinement of the trivial", "covering $\\{X \\to X\\}$ gives, via", "Lemma \\ref{lemma-map-families},", "a functor $\\QCoh(\\mathcal{O}_X) \\to DD(\\{X_i \\to X\\})$ from the", "category of quasi-coherent $\\mathcal{O}_X$-modules to the category of", "descent data for the given family.", "\\item In order to prove the proposition we will construct a", "quasi-inverse functor", "$back : DD(\\{X_i \\to X\\}) \\to \\QCoh(\\mathcal{O}_X)$.", "\\item Applying again", "Lemma \\ref{lemma-map-families}", "we see that there is a functor", "$DD(\\{X_i \\to X\\}) \\to DD(\\{T_j \\to X\\})$", "if $\\{T_j \\to X\\}$ is a refinement of the given family.", "Hence in order to construct the functor $back$ we may assume that", "each $X_i$ is a scheme, see", "Topologies on Spaces,", "Lemma \\ref{spaces-topologies-lemma-refine-fpqc-schemes}.", "This reduces us to the case where all the $X_i$ are schemes.", "\\item A quasi-coherent sheaf on $X$ is by definition a quasi-coherent", "$\\mathcal{O}_X$-module on $X_\\etale$. Now for any", "$U \\in \\Ob(X_\\etale)$ we get an fppf covering", "$\\{U_i \\times_X X_i \\to U\\}$ by schemes and a morphism", "$g : \\{U_i \\times_X X_i \\to U\\} \\to \\{X_i \\to X\\}$ of coverings", "lying over $U \\to X$. Given a descent datum", "$\\xi = (\\mathcal{F}_i, \\varphi_{ij})$ we obtain a quasi-coherent", "$\\mathcal{O}_U$-module $\\mathcal{F}_{\\xi, U}$ corresponding", "to the pullback $g^*\\xi$ of", "Lemma \\ref{lemma-map-families}", "to the covering of $U$ and using effectivity for fppf covering of schemes, see", "Descent, Proposition \\ref{descent-proposition-fpqc-descent-quasi-coherent}.", "\\item Check that $\\xi \\mapsto \\mathcal{F}_{\\xi, U}$ is functorial in $\\xi$.", "Omitted.", "\\item Check that $\\xi \\mapsto \\mathcal{F}_{\\xi, U}$ is compatible", "with morphisms $U \\to U'$ of the site $X_\\etale$, so that", "the system of sheaves $\\mathcal{F}_{\\xi, U}$ corresponds to a quasi-coherent", "$\\mathcal{F}_\\xi$ on $X_\\etale$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-characterize-quasi-coherent-small-etale}.", "Details omitted.", "\\item Check that $back : \\xi \\mapsto \\mathcal{F}_\\xi$ is quasi-inverse", "to the functor constructed in (1). Omitted.", "\\end{enumerate}", "This finishes the proof." ], "refs": [ "descent-proposition-fpqc-descent-quasi-coherent", "spaces-descent-lemma-map-families", "spaces-descent-lemma-map-families", "spaces-topologies-lemma-refine-fpqc-schemes", "spaces-descent-lemma-map-families", "descent-proposition-fpqc-descent-quasi-coherent", "spaces-properties-lemma-characterize-quasi-coherent-small-etale" ], "ref_ids": [ 14753, 9357, 9357, 3680, 9357, 14753, 11908 ] } ], "ref_ids": [ 3694 ] }, { "id": 9454, "type": "theorem", "label": "decent-spaces-theorem-decent-open-dense-scheme", "categories": [ "decent-spaces" ], "title": "decent-spaces-theorem-decent-open-dense-scheme", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "If $X$ is decent, then there exists a dense open subspace", "of $X$ which is a scheme." ], "refs": [], "proofs": [ { "contents": [ "Assume $X$ is a decent algebraic space for which the theorem is false. By", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme}", "there exists a largest open subspace $X' \\subset X$ which is a scheme.", "Since $X'$ is not dense in $X$, there exists an open subspace", "$X'' \\subset X$ such that $|X''| \\cap |X'| = \\emptyset$. Replacing $X$", "by $X''$ we get a nonempty decent algebraic space $X$ which does not", "contain {\\it any} open subspace which is a scheme.", "\\medskip\\noindent", "Choose a nonempty affine scheme $U$ and an \\'etale morphism $U \\to X$.", "We may and do replace $X$ by the open subscheme corresponding to the", "image of $|U| \\to |X|$. Consider the sequence of open subspaces", "$$", "X = X_0 \\supset X_1 \\supset X_2 \\ldots", "$$", "constructed in Lemma \\ref{lemma-stratify-flat-fp-lqf}", "for the morphism $U \\to X$. Note that $X_0 = X_1$ as $U \\to X$", "is surjective. Let $U = U_0 = U_1 \\supset U_2 \\ldots$ be the induced", "sequence of open subschemes of $U$.", "\\medskip\\noindent", "Choose a nonempty open affine $V_1 \\subset U_1$ (for example $V_1 = U_1$).", "By induction we will construct a sequence of nonempty affine opens", "$V_1 \\supset V_2 \\supset \\ldots$ with $V_n \\subset U_n$. Namely, having", "constructed $V_1, \\ldots, V_{n - 1}$ we can always choose $V_n$ unless", "$V_{n - 1} \\cap U_n = \\emptyset$. But if $V_{n - 1} \\cap U_n = \\emptyset$,", "then the open subspace $X' \\subset X$ with", "$|X'| = \\Im(|V_{n - 1}| \\to |X|)$ is contained in $|X| \\setminus |X_n|$.", "Hence $V_{n - 1} \\to X'$ is an \\'etale morphism whose fibres have degree", "bounded by $n - 1$. In other words, $X'$ is reasonable (by definition),", "hence $X'$ contains a nonempty open subscheme by", "Proposition \\ref{proposition-reasonable-open-dense-scheme}.", "This is a contradiction which shows that we can pick $V_n$.", "\\medskip\\noindent", "By Limits, Lemma \\ref{limits-lemma-limit-nonempty}", "the limit $V_\\infty = \\lim V_n$ is a nonempty scheme. Pick a morphism", "$\\Spec(k) \\to V_\\infty$. The composition $\\Spec(k) \\to V_\\infty \\to U \\to X$", "has image contained in all $X_d$ by construction. In other words, the", "fibred $U \\times_X \\Spec(k)$ has infinite degree which contradicts", "the definition of a decent space. This contradiction finishes the proof", "of the theorem." ], "refs": [ "spaces-properties-lemma-subscheme", "decent-spaces-lemma-stratify-flat-fp-lqf", "decent-spaces-proposition-reasonable-open-dense-scheme", "limits-lemma-limit-nonempty" ], "ref_ids": [ 11848, 9476, 9558, 15034 ] } ], "ref_ids": [] }, { "id": 9455, "type": "theorem", "label": "decent-spaces-lemma-composition-universally-bounded", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-composition-universally-bounded", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $V \\to U$ be a morphism of schemes over $S$, and let", "$U \\to X$ be a morphism from $U$ to $X$. If the fibres of", "$V \\to U$ and $U \\to X$ are universally bounded, then so", "are the fibres of $V \\to X$." ], "refs": [], "proofs": [ { "contents": [ "Let $n$ be an integer which works for $V \\to U$, and let $m$ be", "an integer which works for $U \\to X$ in", "Definition \\ref{definition-universally-bounded}.", "Let $\\Spec(k) \\to X$ be a morphism, where $k$ is a field.", "Consider the morphisms", "$$", "\\Spec(k) \\times_X V", "\\longrightarrow", "\\Spec(k) \\times_X U", "\\longrightarrow", "\\Spec(k).", "$$", "By assumption the scheme $\\Spec(k) \\times_X U$", "is finite of degree at most $m$ over $k$, and $n$ is an integer which", "bounds the degree of the fibres of the first morphism. Hence by", "Morphisms, Lemma \\ref{morphisms-lemma-composition-universally-bounded}", "we conclude that $\\Spec(k) \\times_X V$ is finite over $k$", "of degree at most $nm$." ], "refs": [ "decent-spaces-definition-universally-bounded", "morphisms-lemma-composition-universally-bounded" ], "ref_ids": [ 9561, 5526 ] } ], "ref_ids": [] }, { "id": 9456, "type": "theorem", "label": "decent-spaces-lemma-base-change-universally-bounded", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-base-change-universally-bounded", "contents": [ "Let $S$ be a scheme.", "Let $Y \\to X$ be a representable morphism of algebraic spaces over $S$.", "Let $U \\to X$ be a morphism from a scheme to $X$.", "If the fibres of $U \\to X$ are universally bounded, then the fibres", "of $U \\times_X Y \\to Y$ are universally bounded." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definition, and properties of fibre products.", "(Note that $U \\times_X Y$ is a scheme", "as we assumed $Y \\to X$ representable, so the definition applies.)" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9457, "type": "theorem", "label": "decent-spaces-lemma-descent-universally-bounded", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-descent-universally-bounded", "contents": [ "Let $S$ be a scheme. Let $g : Y \\to X$ be a representable morphism of", "algebraic spaces over $S$. Let $f : U \\to X$ be a morphism from a scheme", "towards $X$. Let $f' : U \\times_X Y \\to Y$ be the base change of $f$.", "If", "$$", "\\Im(|f| : |U| \\to |X|) \\subset \\Im(|g| : |Y| \\to |X|)", "$$", "and $f'$ has universally bounded fibres, then $f$ has universally", "bounded fibres." ], "refs": [], "proofs": [ { "contents": [ "Let $n \\geq 0$ be an integer bounding the degrees of the fibre", "products $\\Spec(k) \\times_Y (U \\times_X Y)$ as in", "Definition \\ref{definition-universally-bounded} for the morphism $f'$.", "We claim that $n$ works for $f$ also. Namely, suppose that", "$x : \\Spec(k) \\to X$ is a morphism from the spectrum of", "a field. Then either $\\Spec(k) \\times_X U$ is empty (and there", "is nothing to prove), or $x$ is in the image of $|f|$. By", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-points-cartesian}", "and the assumption of the lemma we see", "that this means there exists a field extension $k \\subset k'$ and a", "commutative diagram", "$$", "\\xymatrix{", "\\Spec(k') \\ar[r] \\ar[d] & Y \\ar[d] \\\\", "\\Spec(k) \\ar[r] & X", "}", "$$", "Hence we see that", "$$", "\\Spec(k') \\times_Y (U \\times_X Y) =", "\\Spec(k') \\times_{\\Spec(k)} (\\Spec(k) \\times_X U)", "$$", "Since the scheme $\\Spec(k') \\times_Y (U \\times_X Y)$ is assumed finite", "of degree $\\leq n$ over $k'$ it follows that also $\\Spec(k) \\times_X U$", "is finite of degree $\\leq n$ over $k$ as desired. (Some details omitted.)" ], "refs": [ "decent-spaces-definition-universally-bounded", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 9561, 11819 ] } ], "ref_ids": [] }, { "id": 9458, "type": "theorem", "label": "decent-spaces-lemma-universally-bounded-permanence", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-universally-bounded-permanence", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Consider a commutative diagram", "$$", "\\xymatrix{", "U \\ar[rd]_g \\ar[rr]_f & & V \\ar[ld]^h \\\\", "& X &", "}", "$$", "where $U$ and $V$ are schemes. If $g$ has universally bounded fibres,", "and $f$ is surjective and flat, then also $h$ has universally bounded fibres." ], "refs": [], "proofs": [ { "contents": [ "Assume $g$ has universally bounded fibres, and $f$ is surjective and flat.", "Say $n \\geq 0$ is an integer which bounds the degrees of the schemes", "$\\Spec(k) \\times_X U$ as in", "Definition \\ref{definition-universally-bounded}.", "We claim $n$ also works for $h$.", "Let $\\Spec(k) \\to X$ be a morphism from the spectrum of a", "field to $X$. Consider the morphism of schemes", "$$", "\\Spec(k) \\times_X V \\longrightarrow \\Spec(k) \\times_X U", "$$", "It is flat and surjective. By assumption the scheme", "on the left is finite of degree $\\leq n$ over $\\Spec(k)$.", "It follows from", "Morphisms, Lemma \\ref{morphisms-lemma-universally-bounded-permanence}", "that the degree of the scheme on the right is also bounded by $n$", "as desired." ], "refs": [ "decent-spaces-definition-universally-bounded", "morphisms-lemma-universally-bounded-permanence" ], "ref_ids": [ 9561, 5532 ] } ], "ref_ids": [] }, { "id": 9459, "type": "theorem", "label": "decent-spaces-lemma-universally-bounded-finite-fibres", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-universally-bounded-finite-fibres", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$, and let $U$ be a scheme over $S$.", "Let $\\varphi : U \\to X$ be a morphism over $S$.", "If the fibres of $\\varphi$ are universally bounded, then there exists an", "integer $n$ such that each fibre of $|U| \\to |X|$ has at most", "$n$ elements." ], "refs": [], "proofs": [ { "contents": [ "The integer $n$ of Definition \\ref{definition-universally-bounded} works.", "Namely, pick $x \\in |X|$. Represent $x$ by a morphism", "$x : \\Spec(k) \\to X$. Then we get a commutative diagram", "$$", "\\xymatrix{", "\\Spec(k) \\times_X U \\ar[r] \\ar[d] & U \\ar[d] \\\\", "\\Spec(k) \\ar[r]^x & X", "}", "$$", "which shows (via", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-points-cartesian})", "that the inverse image of $x$ in $|U|$ is the image of", "the top horizontal arrow. Since $\\Spec(k) \\times_X U$ is finite", "of degree $\\leq n$ over $k$ it has at most $n$ points." ], "refs": [ "decent-spaces-definition-universally-bounded", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 9561, 11819 ] } ], "ref_ids": [] }, { "id": 9460, "type": "theorem", "label": "decent-spaces-lemma-U-finite-above-x", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-U-finite-above-x", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$. The following are equivalent:", "\\begin{enumerate}", "\\item there exists a family of schemes $U_i$ and", "\\'etale morphisms $\\varphi_i : U_i \\to X$ such that", "$\\coprod \\varphi_i : \\coprod U_i \\to X$ is surjective,", "and such that for each $i$ the fibre of", "$|U_i| \\to |X|$ over $x$ is finite, and", "\\item for every affine scheme $U$ and \\'etale morphism $\\varphi : U \\to X$", "the fibre of $|U| \\to |X|$ over $x$ is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is trivial.", "Let $\\varphi_i : U_i \\to X$ be a family of \\'etale morphisms as in (1).", "Let $\\varphi : U \\to X$ be an \\'etale morphism from an affine scheme", "towards $X$. Consider the fibre product diagrams", "$$", "\\xymatrix{", "U \\times_X U_i \\ar[r]_-{p_i} \\ar[d]_{q_i} & U_i \\ar[d]^{\\varphi_i} \\\\", "U \\ar[r]^\\varphi & X", "}", "\\quad \\quad", "\\xymatrix{", "\\coprod U \\times_X U_i \\ar[r]_-{\\coprod p_i} \\ar[d]_{\\coprod q_i} &", "\\coprod U_i \\ar[d]^{\\coprod \\varphi_i} \\\\", "U \\ar[r]^\\varphi & X", "}", "$$", "Since $q_i$ is \\'etale it is open (see Remark \\ref{remark-recall}).", "Moreover, the morphism $\\coprod q_i$ is surjective.", "Hence there exist finitely many indices $i_1, \\ldots, i_n$ and", "a quasi-compact opens $W_{i_j} \\subset U \\times_X U_{i_j}$", "which surject onto $U$.", "The morphism $p_i$ is \\'etale, hence locally quasi-finite (see remark on", "\\'etale morphisms above). Thus we may apply", "Morphisms, Lemma", "\\ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded}", "to see the fibres of $p_{i_j}|_{W_{i_j}} : W_{i_j} \\to U_i$ are finite.", "Hence by", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-points-cartesian}", "and the assumption on $\\varphi_i$ we conclude that the fibre", "of $\\varphi$ over $x$ is finite. In other words (2) holds." ], "refs": [ "decent-spaces-remark-recall", "morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 9573, 5531, 11819 ] } ], "ref_ids": [] }, { "id": 9461, "type": "theorem", "label": "decent-spaces-lemma-R-finite-above-x", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-R-finite-above-x", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$. The following are equivalent:", "\\begin{enumerate}", "\\item there exists a scheme $U$, an \\'etale morphism", "$\\varphi : U \\to X$, and points $u, u' \\in U$ mapping to", "$x$ such that setting $R = U \\times_X U$ the fibre of", "$$", "|R| \\to |U| \\times_{|X|} |U|", "$$", "over $(u, u')$ is finite,", "\\item for every scheme $U$, \\'etale morphism $\\varphi : U \\to X$ and", "any points $u, u' \\in U$ mapping to", "$x$ setting $R = U \\times_X U$ the fibre of", "$$", "|R| \\to |U| \\times_{|X|} |U|", "$$", "over $(u, u')$ is finite,", "\\item there exists a morphism $\\Spec(k) \\to X$ with $k$ a field", "in the equivalence class of $x$ such that the projections", "$\\Spec(k) \\times_X \\Spec(k) \\to \\Spec(k)$ are", "\\'etale and quasi-compact, and", "\\item there exists a monomorphism $\\Spec(k) \\to X$ with $k$ a field", "in the equivalence class of $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1), i.e., let $\\varphi : U \\to X$ be an \\'etale morphism from a scheme", "towards $X$, and let $u, u'$ be points of $U$ lying over $x$", "such that the fibre of $|R| \\to |U| \\times_{|X|} |U|$ over $(u, u')$", "is a finite set. In this proof we think of a point $u = \\Spec(\\kappa(u))$", "as a scheme. Note that $u \\to U$, $u' \\to U$ are monomorphisms (see", "Schemes, Lemma \\ref{schemes-lemma-injective-points-surjective-stalks}),", "hence $u \\times_X u' \\to R = U \\times_X U$ is a monomorphism.", "In this language the assumption really means that", "$u \\times_X u'$ is a scheme whose underlying topological space has", "finitely many points.", "Let $\\psi : W \\to X$ be an \\'etale morphism from a scheme towards $X$.", "Let $w, w' \\in W$ be points of $W$ mapping to $x$.", "We have to show that $w \\times_X w'$ is a scheme whose underlying topological", "space has finitely many points.", "Consider the fibre product diagram", "$$", "\\xymatrix{", "W \\times_X U \\ar[r]_p \\ar[d]_q & U \\ar[d]^\\varphi \\\\", "W \\ar[r]^\\psi & X", "}", "$$", "As $x$ is the image of $u$ and $u'$ we may pick points", "$\\tilde w, \\tilde w'$ in $W \\times_X U$ with $q(\\tilde w) = w$,", "$q(\\tilde w') = w'$, $u = p(\\tilde w)$ and $u' = p(\\tilde w')$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-points-cartesian}.", "As $p$, $q$ are \\'etale the field extensions", "$\\kappa(w) \\subset \\kappa(\\tilde w) \\supset \\kappa(u)$ and", "$\\kappa(w') \\subset \\kappa(\\tilde w') \\supset \\kappa(u')$ are", "finite separable, see Remark \\ref{remark-recall}.", "Then we get a commutative diagram", "$$", "\\xymatrix{", "w \\times_X w' \\ar[d] &", "\\tilde w \\times_X \\tilde w' \\ar[l] \\ar[d] \\ar[r] &", "u \\times_X u' \\ar[d] \\\\", "w \\times_X w' &", "\\tilde w \\times_S \\tilde w' \\ar[l] \\ar[r] &", "u \\times_S u'", "}", "$$", "where the squares are fibre product squares. The lower horizontal", "morphisms are \\'etale and quasi-compact, as any scheme of the form", "$\\Spec(k) \\times_S \\Spec(k')$ is affine, and by our", "observations about the field extensions above.", "Thus we see that the top horizontal arrows are \\'etale and quasi-compact", "and hence have finite fibres.", "We have seen above that $|u \\times_X u'|$ is finite, so we conclude that", "$|w \\times_X w'|$ is finite. In other words, (2) holds.", "\\medskip\\noindent", "Assume (2). Let $U \\to X$ be an \\'etale morphism from a scheme $U$", "such that $x$ is in the image of $|U| \\to |X|$. Let $u \\in U$ be", "a point mapping to $x$. Then we have seen in the previous", "paragraph that $u = \\Spec(\\kappa(u)) \\to X$ has the property that", "$u \\times_X u$ has a finite underlying topological space. On the other", "hand, the projection maps $u \\times_X u \\to u$ are the composition", "$$", "u \\times_X u \\longrightarrow", "u \\times_X U \\longrightarrow", "u \\times_X X = u,", "$$", "i.e., the composition of a monomorphism (the base change of the monomorphism", "$u \\to U$) by an \\'etale morphism (the base change of the \\'etale morphism", "$U \\to X$). Hence $u \\times_X U$ is a disjoint union of spectra of fields", "finite separable over $\\kappa(u)$ (see", "Remark \\ref{remark-recall}). Since $u \\times_X u$ is finite the image", "of it in $u \\times_X U$ is a finite disjoint union of spectra of fields", "finite separable over $\\kappa(u)$. By", "Schemes, Lemma \\ref{schemes-lemma-mono-towards-spec-field}", "we conclude that $u \\times_X u$ is a finite disjoint union of spectra", "of fields finite separable over $\\kappa(u)$. In other words, we see that", "$u \\times_X u \\to u$ is quasi-compact and \\'etale. This means that (3) holds.", "\\medskip\\noindent", "Let us prove that (3) implies (4). Let $\\Spec(k) \\to X$ be a morphism", "from the spectrum of a field into $X$, in the equivalence class of $x$", "such that the two projections", "$t, s : R = \\Spec(k) \\times_X \\Spec(k) \\to \\Spec(k)$", "are quasi-compact and \\'etale.", "This means in particular", "that $R$ is an \\'etale equivalence relation on $\\Spec(k)$.", "By Spaces, Theorem \\ref{spaces-theorem-presentation}", "we know that the quotient sheaf", "$X' = \\Spec(k)/R$ is an algebraic space. By", "Groupoids, Lemma \\ref{groupoids-lemma-quotient-groupoid-restrict}", "the map $X' \\to X$ is a monomorphism.", "Since $s, t$ are quasi-compact, we see that $R$ is quasi-compact and hence", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-point-like-spaces}", "applies to $X'$, and we see that", "$X' = \\Spec(k')$ for some field $k'$. Hence we get a factorization", "$$", "\\Spec(k) \\longrightarrow", "\\Spec(k') \\longrightarrow X", "$$", "which shows that $\\Spec(k') \\to X$ is a monomorphism mapping", "to $x \\in |X|$. In other words (4) holds.", "\\medskip\\noindent", "Finally, we prove that (4) implies (1). Let $\\Spec(k) \\to X$", "be a monomorphism with $k$ a field in the equivalence class of $x$.", "Let $U \\to X$ be a surjective \\'etale morphism from a scheme $U$ to $X$.", "Let $u \\in U$ be a point over $x$. Since $\\Spec(k) \\times_X u$", "is nonempty, and since $\\Spec(k) \\times_X u \\to u$ is a monomorphism", "we conclude that $\\Spec(k) \\times_X u = u$ (see", "Schemes, Lemma \\ref{schemes-lemma-mono-towards-spec-field}).", "Hence $u \\to U \\to X$ factors through $\\Spec(k) \\to X$, here is", "a picture", "$$", "\\xymatrix{", "u \\ar[r] \\ar[d] & U \\ar[d] \\\\", "\\Spec(k) \\ar[r] & X", "}", "$$", "Since the right vertical arrow is \\'etale this implies that", "$k \\subset \\kappa(u)$ is a finite separable extension. Hence we conclude that", "$$", "u \\times_X u = u \\times_{\\Spec(k)} u", "$$", "is a finite scheme, and we win by the discussion of the meaning of property", "(1) in the first paragraph of this proof." ], "refs": [ "schemes-lemma-injective-points-surjective-stalks", "spaces-properties-lemma-points-cartesian", "decent-spaces-remark-recall", "decent-spaces-remark-recall", "schemes-lemma-mono-towards-spec-field", "spaces-theorem-presentation", "groupoids-lemma-quotient-groupoid-restrict", "spaces-properties-lemma-point-like-spaces", "schemes-lemma-mono-towards-spec-field" ], "ref_ids": [ 7726, 11819, 9573, 9573, 7729, 8124, 9651, 11854, 7729 ] } ], "ref_ids": [] }, { "id": 9462, "type": "theorem", "label": "decent-spaces-lemma-weak-UR-finite-above-x", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-weak-UR-finite-above-x", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$.", "Let $U$ be a scheme and let $\\varphi : U \\to X$ be an \\'etale morphism.", "The following are equivalent:", "\\begin{enumerate}", "\\item $x$ is in the image of $|U| \\to |X|$, and", "setting $R = U \\times_X U$ the fibres of both", "$$", "|U| \\longrightarrow |X|", "\\quad\\text{and}\\quad", "|R| \\longrightarrow |X|", "$$", "over $x$ are finite,", "\\item there exists a monomorphism $\\Spec(k) \\to X$ with $k$ a field", "in the equivalence class of $x$, and", "the fibre product $\\Spec(k) \\times_X U$ is", "a finite nonempty scheme over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). This clearly implies the first condition of", "Lemma \\ref{lemma-R-finite-above-x} and hence we obtain a monomorphism", "$\\Spec(k) \\to X$ in the class of $x$. Taking the fibre product", "we see that $\\Spec(k) \\times_X U \\to \\Spec(k)$ is a scheme", "\\'etale over $\\Spec(k)$ with finitely many points, hence a finite", "nonempty scheme over $k$, i.e., (2) holds.", "\\medskip\\noindent", "Assume (2). By assumption $x$ is in the image of", "$|U| \\to |X|$. The finiteness of the fibre of", "$|U| \\to |X|$ over $x$ is clear since this fibre is equal to", "$|\\Spec(k) \\times_X U|$ by", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-points-cartesian}.", "The finiteness of the fibre of $|R| \\to |X|$ above $x$ is also clear", "since it is equal to the set underlying the scheme", "$$", "(\\Spec(k) \\times_X U) \\times_{\\Spec(k)} (\\Spec(k) \\times_X U)", "$$", "which is finite over $k$. Thus (1) holds." ], "refs": [ "decent-spaces-lemma-R-finite-above-x", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 9461, 11819 ] } ], "ref_ids": [] }, { "id": 9463, "type": "theorem", "label": "decent-spaces-lemma-UR-finite-above-x", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-UR-finite-above-x", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$. The following are equivalent:", "\\begin{enumerate}", "\\item for every affine scheme $U$, any \\'etale morphism", "$\\varphi : U \\to X$ setting $R = U \\times_X U$ the fibres of both", "$$", "|U| \\longrightarrow |X|", "\\quad\\text{and}\\quad", "|R| \\longrightarrow |X|", "$$", "over $x$ are finite,", "\\item there exist schemes $U_i$ and \\'etale morphisms", "$U_i \\to X$ such that $\\coprod U_i \\to X$ is surjective and for each", "$i$, setting $R_i = U_i \\times_X U_i$ the fibres of both", "$$", "|U_i| \\longrightarrow |X|", "\\quad\\text{and}\\quad", "|R_i| \\longrightarrow |X|", "$$", "over $x$ are finite,", "\\item there exists a monomorphism $\\Spec(k) \\to X$ with $k$ a field", "in the equivalence class of $x$, and for any affine scheme $U$ and \\'etale", "morphism $U \\to X$ the fibre product $\\Spec(k) \\times_X U$ is", "a finite scheme over $k$,", "\\item there exists a quasi-compact monomorphism $\\Spec(k) \\to X$", "with $k$ a field in the equivalence class of $x$, and", "\\item there exists a quasi-compact morphism $\\Spec(k) \\to X$", "with $k$ a field in the equivalence class of $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (3) follows on applying", "Lemma \\ref{lemma-weak-UR-finite-above-x}", "to every \\'etale morphism $U \\to X$ with $U$ affine.", "It is clear that (3) implies (2).", "Assume $U_i \\to X$ and $R_i$ are as in (2). We conclude from", "Lemma \\ref{lemma-U-finite-above-x}", "that for any affine scheme $U$ and \\'etale morphism $U \\to X$", "the fibre of $|U| \\to |X|$ over $x$ is finite.", "Say this fibre is $\\{u_1, \\ldots, u_n\\}$. Then, as", "Lemma \\ref{lemma-R-finite-above-x} (1)", "applies to $U_i \\to X$ for some $i$ such that $x$ is in the image of", "$|U_i| \\to |X|$, we see that the fibre of", "$|R = U \\times_X U| \\to |U| \\times_{|X|} |U|$", "is finite over $(u_a, u_b)$, $a, b \\in \\{1, \\ldots, n\\}$.", "Hence the fibre of $|R| \\to |X|$ over $x$ is finite.", "In this way we see that (1) holds. At this point we know that", "(1), (2), and (3) are equivalent.", "\\medskip\\noindent", "If (4) holds, then for any affine scheme $U$ and \\'etale morphism", "$U \\to X$ the scheme $\\Spec(k) \\times_X U$ is on the one hand", "\\'etale over $k$ (hence a disjoint union of spectra of finite separable", "extensions of $k$ by", "Remark \\ref{remark-recall})", "and on the other hand quasi-compact over $U$ (hence quasi-compact).", "Thus we see that (3) holds.", "Conversely, if $U_i \\to X$ is as in (2) and $\\Spec(k) \\to X$", "is a monomorphism as in (3), then", "$$", "\\coprod \\Spec(k) \\times_X U_i", "\\longrightarrow", "\\coprod U_i", "$$", "is quasi-compact (because over each $U_i$ we see that", "$\\Spec(k) \\times_X U_i$ is a finite disjoint union spectra of fields).", "Thus $\\Spec(k) \\to X$ is quasi-compact by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-quasi-compact-local}.", "\\medskip\\noindent", "It is immediate that (4) implies (5). Conversely, let $\\Spec(k) \\to X$", "be a quasi-compact morphism in the equivalence class of $x$. Let $U \\to X$", "be an \\'etale morphism with $U$ affine. Consider the fibre product", "$$", "\\xymatrix{", "F \\ar[r] \\ar[d] & U \\ar[d] \\\\", "\\Spec(k) \\ar[r] & X", "}", "$$", "Then $F \\to U$ is quasi-compact, hence $F$ is quasi-compact.", "On the other hand, $F \\to \\Spec(k)$ is \\'etale, hence $F$ is a", "finite disjoint union of spectra of finite separable extensions of $k$", "(Remark \\ref{remark-recall}). Since the image of $|F| \\to |U|$", "is the fibre of $|U| \\to |X|$ over $x$ (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-points-cartesian}), we conclude that", "the fibre of $|U| \\to |X|$ over $x$ is finite. The scheme", "$F \\times_{\\Spec(k)} F$ is also a finite union of spectra of fields", "because it is also quasi-compact and \\'etale over $\\Spec(k)$.", "There is a monomorphism", "$F \\times_X F \\to F \\times_{\\Spec(k)} F$, hence $F \\times_X F$ is", "a finite disjoint union of spectra of fields", "(Schemes, Lemma \\ref{schemes-lemma-mono-towards-spec-field}).", "Thus the image of $F \\times_X F \\to U \\times_X U = R$ is finite.", "Since this image is the fibre of $|R| \\to |X|$ over $x$ by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}", "we conclude that (1) holds." ], "refs": [ "decent-spaces-lemma-weak-UR-finite-above-x", "decent-spaces-lemma-U-finite-above-x", "decent-spaces-lemma-R-finite-above-x", "decent-spaces-remark-recall", "spaces-morphisms-lemma-quasi-compact-local", "decent-spaces-remark-recall", "spaces-properties-lemma-points-cartesian", "schemes-lemma-mono-towards-spec-field", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 9462, 9460, 9461, 9573, 4742, 9573, 11819, 7729, 11819 ] } ], "ref_ids": [] }, { "id": 9464, "type": "theorem", "label": "decent-spaces-lemma-U-universally-bounded", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-U-universally-bounded", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item there exist schemes $U_i$ and \\'etale morphisms", "$U_i \\to X$ such that $\\coprod U_i \\to X$ is surjective and", "each $U_i \\to X$ has universally bounded fibres, and", "\\item for every affine scheme $U$ and \\'etale morphism $\\varphi : U \\to X$", "the fibres of $U \\to X$ are universally bounded.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is trivial.", "Assume (1). Let $(\\varphi_i : U_i \\to X)_{i \\in I}$ be a collection of", "\\'etale morphisms from schemes towards $X$, covering $X$, such that", "each $\\varphi_i$ has universally bounded fibres.", "Let $\\psi : U \\to X$ be an \\'etale morphism from an affine scheme towards $X$.", "For each $i$ consider the fibre product diagram", "$$", "\\xymatrix{", "U \\times_X U_i \\ar[r]_{p_i} \\ar[d]_{q_i} & U_i \\ar[d]^{\\varphi_i} \\\\", "U \\ar[r]^\\psi & X", "}", "$$", "Since $q_i$ is \\'etale it is open (see Remark \\ref{remark-recall}).", "Moreover, we have $U = \\bigcup \\Im(q_i)$, since the family", "$(\\varphi_i)_{i \\in I}$ is surjective. Since $U$ is affine, hence quasi-compact", "we can finite finitely many $i_1, \\ldots, i_n \\in I$ and quasi-compact", "opens $W_j \\subset U \\times_X U_{i_j}$ such that", "$U = \\bigcup p_{i_j}(W_j)$.", "The morphism $p_{i_j}$ is \\'etale, hence locally quasi-finite", "(see remark on \\'etale morphisms above). Thus we may apply", "Morphisms, Lemma", "\\ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded}", "to see the fibres of $p_{i_j}|_{W_j} : W_j \\to U_{i_j}$ are universally", "bounded. Hence by", "Lemma \\ref{lemma-composition-universally-bounded}", "we see that the fibres of $W_j \\to X$ are universally bounded.", "Thus also $\\coprod_{j = 1, \\ldots, n} W_j \\to X$ has universally", "bounded fibres. Since $\\coprod_{j = 1, \\ldots, n} W_j \\to X$ factors", "through the surjective \\'etale map", "$\\coprod q_{i_j}|_{W_j} : \\coprod_{j = 1, \\ldots, n} W_j \\to U$ we", "see that the fibres of $U \\to X$ are universally bounded by", "Lemma \\ref{lemma-universally-bounded-permanence}.", "In other words (2) holds." ], "refs": [ "decent-spaces-remark-recall", "morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded", "decent-spaces-lemma-composition-universally-bounded", "decent-spaces-lemma-universally-bounded-permanence" ], "ref_ids": [ 9573, 5531, 9455, 9458 ] } ], "ref_ids": [] }, { "id": 9465, "type": "theorem", "label": "decent-spaces-lemma-characterize-very-reasonable", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-characterize-very-reasonable", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item there exists a Zariski covering $X = \\bigcup X_i$ and for", "each $i$ a scheme $U_i$ and a quasi-compact surjective \\'etale", "morphism $U_i \\to X_i$, and", "\\item there exist schemes $U_i$ and \\'etale morphisms $U_i \\to X$", "such that the projections $U_i \\times_X U_i \\to U_i$ are quasi-compact", "and $\\coprod U_i \\to X$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (1) holds then the morphisms $U_i \\to X_i \\to X$ are \\'etale (combine", "Morphisms, Lemma \\ref{morphisms-lemma-composition-etale}", "and", "Spaces, Lemmas", "\\ref{spaces-lemma-composition-representable-transformations-property} and", "\\ref{spaces-lemma-morphism-schemes-gives-representable-transformation-property}", ").", "Moreover, as $U_i \\times_X U_i = U_i \\times_{X_i} U_i$,", "both projections $U_i \\times_X U_i \\to U_i$ are quasi-compact.", "\\medskip\\noindent", "If (2) holds then let $X_i \\subset X$ be the open subspace corresponding", "to the image of the open map $|U_i| \\to |X|$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-etale-image-open}.", "The morphisms $U_i \\to X_i$ are surjective.", "Hence $U_i \\to X_i$ is surjective \\'etale, and the projections", "$U_i \\times_{X_i} U_i \\to U_i$ are quasi-compact, because", "$U_i \\times_{X_i} U_i = U_i \\times_X U_i$. Thus by", "Spaces, Lemma \\ref{spaces-lemma-representable-morphisms-spaces-property}", "the morphisms $U_i \\to X_i$ are quasi-compact." ], "refs": [ "morphisms-lemma-composition-etale", "spaces-lemma-composition-representable-transformations-property", "spaces-lemma-morphism-schemes-gives-representable-transformation-property", "spaces-properties-lemma-etale-image-open", "spaces-lemma-representable-morphisms-spaces-property" ], "ref_ids": [ 5360, 8132, 8131, 11825, 8157 ] } ], "ref_ids": [] }, { "id": 9466, "type": "theorem", "label": "decent-spaces-lemma-bounded-fibres", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-bounded-fibres", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Consider the following conditions on $X$:", "\\begin{itemize}", "\\item[] $(\\alpha)$ For every $x \\in |X|$, the equivalent conditions of", "Lemma \\ref{lemma-U-finite-above-x}", "hold.", "\\item[] $(\\beta)$ For every $x \\in |X|$, the equivalent conditions of", "Lemma \\ref{lemma-R-finite-above-x}", "hold.", "\\item[] $(\\gamma)$ For every $x \\in |X|$, the equivalent conditions of", "Lemma \\ref{lemma-UR-finite-above-x}", "hold.", "\\item[] $(\\delta)$ The equivalent conditions of", "Lemma \\ref{lemma-U-universally-bounded}", "hold.", "\\item[] $(\\epsilon)$ The equivalent conditions of", "Lemma \\ref{lemma-characterize-very-reasonable}", "hold.", "\\item[] $(\\zeta)$ The space $X$ is Zariski locally quasi-separated.", "\\item[] $(\\eta)$ The space $X$ is quasi-separated", "\\item[] $(\\theta)$ The space $X$ is representable, i.e., $X$ is a scheme.", "\\item[] $(\\iota)$ The space $X$ is a quasi-separated scheme.", "\\end{itemize}", "We have", "$$", "\\xymatrix{", "& (\\theta) \\ar@{=>}[rd] & & & & \\\\", "(\\iota) \\ar@{=>}[ru] \\ar@{=>}[rd] & &", "(\\zeta) \\ar@{=>}[r] &", "(\\epsilon) \\ar@{=>}[r] &", "(\\delta) \\ar@{=>}[r] &", "(\\gamma) \\ar@{<=>}[r] & (\\alpha) + (\\beta) \\\\", "& (\\eta) \\ar@{=>}[ru] & & & &", "}", "$$" ], "refs": [ "decent-spaces-lemma-U-finite-above-x", "decent-spaces-lemma-R-finite-above-x", "decent-spaces-lemma-UR-finite-above-x", "decent-spaces-lemma-U-universally-bounded", "decent-spaces-lemma-characterize-very-reasonable" ], "proofs": [ { "contents": [ "The implication $(\\gamma) \\Leftrightarrow (\\alpha) + (\\beta)$ is immediate.", "The implications in the diamond on the left are clear from the", "definitions.", "\\medskip\\noindent", "Assume $(\\zeta)$, i.e., that $X$ is Zariski locally quasi-separated.", "Then $(\\epsilon)$ holds by", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-separated-quasi-compact-pieces}.", "\\medskip\\noindent", "Assume $(\\epsilon)$. By", "Lemma \\ref{lemma-characterize-very-reasonable}", "there exists", "a Zariski open covering $X = \\bigcup X_i$ such that for each $i$", "there exists a scheme $U_i$ and a quasi-compact surjective \\'etale morphism", "$U_i \\to X_i$. Choose an $i$ and an affine open subscheme $W \\subset U_i$.", "It suffices to show that $W \\to X$ has universally bounded fibres, since then", "the family of all these morphisms $W \\to X$ covers $X$.", "To do this we consider the diagram", "$$", "\\xymatrix{", "W \\times_X U_i \\ar[r]_-p \\ar[d]_q & U_i \\ar[d] \\\\", "W \\ar[r] & X", "}", "$$", "Since $W \\to X$ factors through $X_i$ we see that", "$W \\times_X U_i = W \\times_{X_i} U_i$, and hence $q$ is quasi-compact.", "Since $W$ is affine this implies that the scheme $W \\times_X U_i$", "is quasi-compact. Thus we may apply", "Morphisms, Lemma", "\\ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded}", "and we conclude that $p$ has universally bounded fibres. From", "Lemma \\ref{lemma-descent-universally-bounded}", "we conclude that $W \\to X$ has universally bounded fibres as well.", "\\medskip\\noindent", "Assume $(\\delta)$. Let $U$ be an affine scheme, and let $U \\to X$ be an \\'etale", "morphism. By assumption the fibres of the morphism $U \\to X$ are universally", "bounded. Thus also the fibres of both projections $R = U \\times_X U \\to U$", "are universally bounded, see", "Lemma \\ref{lemma-base-change-universally-bounded}.", "And by", "Lemma \\ref{lemma-composition-universally-bounded}", "also the fibres of $R \\to X$ are universally bounded.", "Hence for any $x \\in X$ the fibres of $|U| \\to |X|$ and $|R| \\to |X|$", "over $x$ are finite, see", "Lemma \\ref{lemma-universally-bounded-finite-fibres}.", "In other words, the equivalent conditions of", "Lemma \\ref{lemma-UR-finite-above-x}", "hold. This proves that $(\\delta) \\Rightarrow (\\gamma)$." ], "refs": [ "spaces-properties-lemma-quasi-separated-quasi-compact-pieces", "decent-spaces-lemma-characterize-very-reasonable", "morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded", "decent-spaces-lemma-descent-universally-bounded", "decent-spaces-lemma-base-change-universally-bounded", "decent-spaces-lemma-composition-universally-bounded", "decent-spaces-lemma-universally-bounded-finite-fibres", "decent-spaces-lemma-UR-finite-above-x" ], "ref_ids": [ 11835, 9465, 5531, 9457, 9456, 9455, 9459, 9463 ] } ], "ref_ids": [ 9460, 9461, 9463, 9464, 9465 ] }, { "id": 9467, "type": "theorem", "label": "decent-spaces-lemma-properties-local", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-properties-local", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{P}$ be one of the properties", "$(\\alpha)$, $(\\beta)$, $(\\gamma)$, $(\\delta)$, $(\\epsilon)$, $(\\zeta)$, or", "$(\\theta)$ of algebraic spaces listed in", "Lemma \\ref{lemma-bounded-fibres}.", "Then if $X$ is an algebraic space over $S$, and $X = \\bigcup X_i$ is a", "Zariski open covering such that each $X_i$ has $\\mathcal{P}$,", "then $X$ has $\\mathcal{P}$." ], "refs": [ "decent-spaces-lemma-bounded-fibres" ], "proofs": [ { "contents": [ "Let $X$ be an algebraic space over $S$, and let $X = \\bigcup X_i$ is a", "Zariski open covering such that each $X_i$ has $\\mathcal{P}$.", "\\medskip\\noindent", "The case $\\mathcal{P} = (\\alpha)$. The condition $(\\alpha)$ for $X_i$", "means that for every $x \\in |X_i|$ and every affine scheme $U$, and", "\\'etale morphism $\\varphi : U \\to X_i$ the fibre of $\\varphi : |U| \\to |X_i|$", "over $x$ is finite. Consider $x \\in X$, an affine scheme $U$ and", "an \\'etale morphism $U \\to X$. Since $X = \\bigcup X_i$ is a", "Zariski open covering there exits a finite affine open covering", "$U = U_1 \\cup \\ldots \\cup U_n$ such that each $U_j \\to X$ factors through", "some $X_{i_j}$. By assumption the fibres of $|U_j | \\to |X_{i_j}|$", "over $x$ are finite for $j = 1, \\ldots, n$. Clearly this means that", "the fibre of $|U| \\to |X|$ over $x$ is finite.", "This proves the result for $(\\alpha)$.", "\\medskip\\noindent", "The case $\\mathcal{P} = (\\beta)$. The condition $(\\beta)$ for $X_i$ means", "that every $x \\in |X_i|$ is represented by a monomorphism from the", "spectrum of a field towards $X_i$. Hence the same follows for $X$", "as $X_i \\to X$ is a monomorphism and $X = \\bigcup X_i$.", "\\medskip\\noindent", "The case $\\mathcal{P} = (\\gamma)$.", "Note that $(\\gamma) = (\\alpha) + (\\beta)$ by", "Lemma \\ref{lemma-bounded-fibres}", "hence the lemma for $(\\gamma)$ follows from the cases treated above.", "\\medskip\\noindent", "The case $\\mathcal{P} = (\\delta)$. The condition $(\\delta)$ for $X_i$ means", "there exist schemes $U_{ij}$ and \\'etale morphisms $U_{ij} \\to X_i$ with", "universally bounded fibres which cover $X_i$. These schemes also give an", "\\'etale surjective morphism $\\coprod U_{ij} \\to X$ and $U_{ij} \\to X$", "still has universally bounded fibres.", "\\medskip\\noindent", "The case $\\mathcal{P} = (\\epsilon)$. The condition $(\\epsilon)$ for $X_i$ means", "we can find a set $J_i$ and morphisms", "$\\varphi_{ij} : U_{ij} \\to X_i$ such that each $\\varphi_{ij}$", "is \\'etale, both projections $U_{ij} \\times_{X_i} U_{ij} \\to U_{ij}$", "are quasi-compact, and $\\coprod_{j \\in J_i} U_{ij} \\to X_i$ is surjective.", "In this case the compositions $U_{ij} \\to X_i \\to X$ are \\'etale", "(combine", "Morphisms, Lemmas", "\\ref{morphisms-lemma-composition-etale} and", "\\ref{morphisms-lemma-open-immersion-etale}", "and", "Spaces, Lemmas", "\\ref{spaces-lemma-composition-representable-transformations-property} and", "\\ref{spaces-lemma-morphism-schemes-gives-representable-transformation-property}", ").", "Since $X_i \\subset X$ is a subspace we see that", "$U_{ij} \\times_{X_i} U_{ij} = U_{ij} \\times_X U_{ij}$, and hence the", "condition on fibre products is preserved. And clearly", "$\\coprod_{i, j} U_{ij} \\to X$ is surjective. Hence $X$", "satisfies $(\\epsilon)$.", "\\medskip\\noindent", "The case $\\mathcal{P} = (\\zeta)$. The condition $(\\zeta)$ for $X_i$", "means that $X_i$ is Zariski locally quasi-separated. It is immediately", "clear that this means $X$ is Zariski locally quasi-separated.", "\\medskip\\noindent", "For $(\\theta)$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-subscheme}." ], "refs": [ "decent-spaces-lemma-bounded-fibres", "morphisms-lemma-composition-etale", "morphisms-lemma-open-immersion-etale", "spaces-lemma-composition-representable-transformations-property", "spaces-lemma-morphism-schemes-gives-representable-transformation-property", "spaces-properties-lemma-subscheme" ], "ref_ids": [ 9466, 5360, 5366, 8132, 8131, 11848 ] } ], "ref_ids": [ 9466 ] }, { "id": 9468, "type": "theorem", "label": "decent-spaces-lemma-representable-properties", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-representable-properties", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{P}$ be one of the properties", "$(\\beta)$, $(\\gamma)$, $(\\delta)$, $(\\epsilon)$, or", "$(\\theta)$ of algebraic spaces listed in", "Lemma \\ref{lemma-bounded-fibres}.", "Let $X$, $Y$ be algebraic spaces over $S$.", "Let $X \\to Y$ be a representable morphism.", "If $Y$ has property $\\mathcal{P}$, so does $X$." ], "refs": [ "decent-spaces-lemma-bounded-fibres" ], "proofs": [ { "contents": [ "Assume $f : X \\to Y$ is a representable morphism of algebraic spaces,", "and assume that $Y$ has $\\mathcal{P}$. Let $x \\in |X|$, and set", "$y = f(x) \\in |Y|$.", "\\medskip\\noindent", "The case $\\mathcal{P} = (\\beta)$. Condition $(\\beta)$ for $Y$ means", "there exists a monomorphism $\\Spec(k) \\to Y$ representing $y$.", "The fibre product $X_y = \\Spec(k) \\times_Y X$ is a scheme, and", "$x$ corresponds to a point of $X_y$, i.e., to a monomorphism", "$\\Spec(k') \\to X_y$. As $X_y \\to X$ is a monomorphism also we see", "that $x$ is represented by the monomorphism $\\Spec(k') \\to X_y \\to X$.", "In other words $(\\beta)$ holds for $X$.", "\\medskip\\noindent", "The case $\\mathcal{P} = (\\gamma)$. Since $(\\gamma) \\Rightarrow (\\beta)$", "we have seen in the preceding paragraph that $y$ and $x$ can be represented", "by monomorphisms as in the following diagram", "$$", "\\xymatrix{", "\\Spec(k') \\ar[r]_-x \\ar[d] & X \\ar[d] \\\\", "\\Spec(k) \\ar[r]^-y & Y", "}", "$$", "Also, by definition of property $(\\gamma)$ via", "Lemma \\ref{lemma-UR-finite-above-x} (2)", "there exist schemes", "$V_i$ and \\'etale morphisms $V_i \\to Y$ such that $\\coprod V_i \\to Y$", "is surjective and for each $i$, setting $R_i = V_i \\times_Y V_i$", "the fibres of both", "$$", "|V_i| \\longrightarrow |Y|", "\\quad\\text{and}\\quad", "|R_i| \\longrightarrow |Y|", "$$", "over $y$ are finite. This means that the schemes", "$(V_i)_y$ and $(R_i)_y$ are finite schemes over $y = \\Spec(k)$.", "As $X \\to Y$ is representable, the fibre products $U_i = V_i \\times_Y X$", "are schemes. The morphisms $U_i \\to X$ are \\'etale, and", "$\\coprod U_i \\to X$ is surjective. Finally, for each $i$ we have", "$$", "(U_i)_x =", "(V_i \\times_Y X)_x =", "(V_i)_y \\times_{\\Spec(k)} \\Spec(k')", "$$", "and", "$$", "(U_i \\times_X U_i)_x =", "\\left((V_i \\times_Y X) \\times_X (V_i \\times_Y X)\\right)_x =", "(R_i)_y \\times_{\\Spec(k)} \\Spec(k')", "$$", "hence these are finite over $k'$ as base changes of the finite", "schemes $(V_i)_y$ and $(R_i)_y$. This implies that $(\\gamma)$ holds for $X$,", "again via the second condition of", "Lemma \\ref{lemma-UR-finite-above-x}.", "\\medskip\\noindent", "The case $\\mathcal{P} = (\\delta)$. Let $V \\to Y$ be an \\'etale morphism with", "$V$ an affine scheme. Since $Y$ has property $(\\delta)$ this morphism has", "universally bounded fibres. By", "Lemma \\ref{lemma-base-change-universally-bounded}", "the base change $V \\times_Y X \\to X$ also has universally bounded fibres.", "Hence the first part of", "Lemma \\ref{lemma-U-universally-bounded}", "applies and we see that $Y$ also has property $(\\delta)$.", "\\medskip\\noindent", "The case $\\mathcal{P} = (\\epsilon)$. We will repeatedly use", "Spaces, Lemma", "\\ref{spaces-lemma-base-change-representable-transformations-property}.", "Let $V_i \\to Y$ be as in", "Lemma \\ref{lemma-characterize-very-reasonable} (2).", "Set $U_i = X \\times_Y V_i$. The morphisms $U_i \\to X$ are \\'etale,", "and $\\coprod U_i \\to X$ is surjective. Because", "$U_i \\times_X U_i = X \\times_Y (V_i \\times_Y V_i)$ we see", "that the projections $U_i \\times_Y U_i \\to U_i$ are", "base changes of the projections $V_i \\times_Y V_i \\to V_i$, and so", "quasi-compact as well. Hence $X$ satisfies", "Lemma \\ref{lemma-characterize-very-reasonable} (2).", "\\medskip\\noindent", "The case $\\mathcal{P} = (\\theta)$. In this case the result is", "Categories, Lemma \\ref{categories-lemma-representable-over-representable}." ], "refs": [ "decent-spaces-lemma-UR-finite-above-x", "decent-spaces-lemma-UR-finite-above-x", "decent-spaces-lemma-base-change-universally-bounded", "decent-spaces-lemma-U-universally-bounded", "spaces-lemma-base-change-representable-transformations-property", "decent-spaces-lemma-characterize-very-reasonable", "decent-spaces-lemma-characterize-very-reasonable", "categories-lemma-representable-over-representable" ], "ref_ids": [ 9463, 9463, 9456, 9464, 8133, 9465, 9465, 12207 ] } ], "ref_ids": [ 9466 ] }, { "id": 9469, "type": "theorem", "label": "decent-spaces-lemma-fun-property-reasonable", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-fun-property-reasonable", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact reasonable algebraic space.", "Then there exists a directed system of quasi-compact and quasi-separated", "algebraic spaces $X_i$ such that $X = \\colim_i X_i$", "(colimit in the category of sheaves)." ], "refs": [], "proofs": [ { "contents": [ "We sketch the proof. By", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}", "we have $X = U/R$ with $U$ affine.", "In this case, reasonable means $U \\to X$ is universally bounded.", "Hence there exists an integer $N$ such that the ``fibres'' of $U \\to X$", "have degree at most $N$, see", "Definition \\ref{definition-universally-bounded}.", "Denote $s, t : R \\to U$ and $c : R \\times_{s, U, t} R \\to R$ the", "groupoid structural maps.", "\\medskip\\noindent", "Claim: for every quasi-compact open $A \\subset R$ there exists", "an open $R' \\subset R$ such that", "\\begin{enumerate}", "\\item $A \\subset R'$,", "\\item $R'$ is quasi-compact, and", "\\item $(U, R', s|_{R'}, t|_{R'}, c|_{R' \\times_{s, U, t} R'})$ is", "a groupoid scheme.", "\\end{enumerate}", "Note that $e : U \\to R$ is open as it is a section of the \\'etale morphism", "$s : R \\to U$, see", "\\'Etale Morphisms, Proposition \\ref{etale-proposition-properties-sections}.", "Moreover $U$ is affine hence quasi-compact. Hence we may replace $A$ by", "$A \\cup e(U) \\subset R$, and assume that $A$ contains $e(U)$. Next, we", "define inductively $A^1 = A$, and", "$$", "A^n = c(A^{n - 1} \\times_{s, U, t} A) \\subset R", "$$", "for $n \\geq 2$. Arguing inductively, we see that $A^n$ is quasi-compact for", "all $n \\geq 2$, as the image of the quasi-compact fibre product", "$A^{n - 1} \\times_{s, U, t} A$. If $k$ is an algebraically", "closed field over $S$, and we consider $k$-points then", "$$", "A^n(k) = \\left\\{(u, u') \\in U(k)", ":", "\\begin{matrix}", "\\text{there exist } u = u_1, u_2, \\ldots, u_n \\in U(k)\\text{ with} \\\\", "(u_i , u_{i + 1}) \\in A \\text{ for all }i = 1, \\ldots, n - 1.", "\\end{matrix}", "\\right\\}", "$$", "But as the fibres of $U(k) \\to X(k)$ have size at most $N$ we see that if", "$n > N$ then we get a repeat in the sequence above, and we can shorten it", "proving $A^N = A^n$ for all $n \\geq N$.", "This implies that $R' = A^N$ gives a groupoid scheme", "$(U, R', s|_{R'}, t|_{R'}, c|_{R' \\times_{s, U, t} R'})$, proving the claim", "above.", "\\medskip\\noindent", "Consider the map of sheaves on $(\\Sch/S)_{fppf}$", "$$", "\\colim_{R' \\subset R} U/R' \\longrightarrow U/R", "$$", "where $R' \\subset R$ runs over the quasi-compact open subschemes", "of $R$ which give \\'etale equivalence relations as above. Each of the", "quotients $U/R'$ is an algebraic space", "(see Spaces, Theorem \\ref{spaces-theorem-presentation}).", "Since $R'$ is quasi-compact, and $U$ affine the morphism", "$R' \\to U \\times_{\\Spec(\\mathbf{Z})} U$ is quasi-compact,", "and hence $U/R'$ is quasi-separated. Finally, if $T$ is a quasi-compact", "scheme, then", "$$", "\\colim_{R' \\subset R} U(T)/R'(T) \\longrightarrow U(T)/R(T)", "$$", "is a bijection, since every morphism from $T$ into $R$ ends up in one", "of the open subrelations $R'$ by the claim above. This clearly implies", "that the colimit of the sheaves $U/R'$ is $U/R$. In other words", "the algebraic space $X = U/R$ is the colimit of the quasi-separated", "algebraic spaces $U/R'$." ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "decent-spaces-definition-universally-bounded", "etale-proposition-properties-sections", "spaces-theorem-presentation" ], "ref_ids": [ 11832, 9561, 10726, 8124 ] } ], "ref_ids": [] }, { "id": 9470, "type": "theorem", "label": "decent-spaces-lemma-representable-named-properties", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-representable-named-properties", "contents": [ "Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$.", "Let $X \\to Y$ be a representable morphism.", "If $Y$ is decent (resp.\\ reasonable), then so is $X$." ], "refs": [], "proofs": [ { "contents": [ "Translation of Lemma \\ref{lemma-representable-properties}." ], "refs": [ "decent-spaces-lemma-representable-properties" ], "ref_ids": [ 9468 ] } ], "ref_ids": [] }, { "id": 9471, "type": "theorem", "label": "decent-spaces-lemma-etale-named-properties", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-etale-named-properties", "contents": [ "Let $S$ be a scheme. Let $X \\to Y$ be an \\'etale morphism of", "algebraic spaces over $S$. If $Y$ is decent, resp.\\ reasonable,", "then so is $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be an affine scheme and $U \\to X$ an \\'etale morphism.", "Set $R = U \\times_X U$ and $R' = U \\times_Y U$. Note that", "$R \\to R'$ is a monomorphism.", "\\medskip\\noindent", "Let $x \\in |X|$. To show that $X$ is decent, we have to show that", "the fibres of $|U| \\to |X|$ and $|R| \\to |X|$ over $x$ are finite.", "But if $Y$ is decent, then the fibres of $|U| \\to |Y|$ and", "$|R'| \\to |Y|$ are finite. Hence the result for ``decent''.", "\\medskip\\noindent", "To show that $X$ is reasonable, we have to show that the fibres of", "$U \\to X$ are universally bounded. However, if $Y$ is reasonable,", "then the fibres of $U \\to Y$ are universally bounded, which immediately", "implies the same thing for the fibres of $U \\to X$.", "Hence the result for ``reasonable''." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9472, "type": "theorem", "label": "decent-spaces-lemma-no-specializations-map-to-same-point", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-no-specializations-map-to-same-point", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $U \\to X$ be an \\'etale morphism from a scheme to $X$.", "Assume $u, u' \\in |U|$ map to the same point $x$ of $|X|$, and", "$u' \\leadsto u$. If the pair $(X, x)$ satisfies the", "equivalent conditions of", "Lemma \\ref{lemma-U-finite-above-x}", "then $u = u'$." ], "refs": [ "decent-spaces-lemma-U-finite-above-x" ], "proofs": [ { "contents": [ "Assume the pair $(X, x)$ satisfies the", "equivalent conditions for Lemma \\ref{lemma-U-finite-above-x}.", "Let $U$ be a scheme, $U \\to X$ \\'etale, and", "let $u, u' \\in |U|$ map to $x$ of $|X|$, and", "$u' \\leadsto u$. We may and do replace $U$ by an affine", "neighbourhood of $u$. Let $t, s : R = U \\times_X U \\to U$", "be the \\'etale projection maps.", "\\medskip\\noindent", "Pick a point $r \\in R$ with $t(r) = u$ and $s(r) = u'$.", "This is possible by", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-points-presentation}.", "Because generalizations lift along the \\'etale morphism $t$", "(Remark \\ref{remark-recall}) we can find a specialization $r' \\leadsto r$ with", "$t(r') = u'$. Set $u'' = s(r')$. Then $u'' \\leadsto u'$.", "Thus we may repeat and find $r'' \\leadsto r'$ with", "$t(r'') = u''$. Set $u''' = s(r'')$, and so on.", "Here is a picture:", "$$", "\\xymatrix{", "& r'' \\ar[rd]^s \\ar[ld]_t \\ar@{~>}[d] & \\\\", "u'' \\ar@{~>}[d] & r' \\ar[rd]^s \\ar[ld]_t \\ar@{~>}[d] & u''' \\ar@{~>}[d] \\\\", "u' \\ar@{~>}[d] & r \\ar[rd]^s \\ar[ld]_t & u'' \\ar@{~>}[d] \\\\", "u & & u'", "}", "$$", "In Remark \\ref{remark-recall} we have seen that there are no specializations", "among points in the fibres of the \\'etale morphism $s$. Hence if", "$u^{(n + 1)} = u^{(n)}$ for some $n$, then also $r^{(n)} = r^{(n - 1)}$ and", "hence also (by taking $t$) $u^{(n)} = u^{(n - 1)}$. This then forces the", "whole tower to collapse, in particular $u = u'$. Thus we see that if", "$u \\not = u'$, then all the specializations are strict and", "$\\{u, u', u'', \\ldots\\}$ is an infinite set of points in $U$ which map to the", "point $x$ in $|X|$. As we chose $U$ affine this contradicts the second part of", "Lemma \\ref{lemma-U-finite-above-x}, as desired." ], "refs": [ "decent-spaces-lemma-U-finite-above-x", "spaces-properties-lemma-points-presentation", "decent-spaces-remark-recall", "decent-spaces-remark-recall", "decent-spaces-lemma-U-finite-above-x" ], "ref_ids": [ 9460, 11821, 9573, 9573, 9460 ] } ], "ref_ids": [ 9460 ] }, { "id": 9473, "type": "theorem", "label": "decent-spaces-lemma-specialization", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-specialization", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $x, x' \\in |X|$ and assume $x' \\leadsto x$, i.e., $x$ is a", "specialization of $x'$.", "Assume the pair $(X, x')$ satisfies the equivalent conditions", "of Lemma \\ref{lemma-UR-finite-above-x}.", "Then for every \\'etale morphism $\\varphi : U \\to X$ from a scheme $U$ and any", "$u \\in U$ with $\\varphi(u) = x$, exists a point $u'\\in U$,", "$u' \\leadsto u$ with $\\varphi(u') = x'$." ], "refs": [ "decent-spaces-lemma-UR-finite-above-x" ], "proofs": [ { "contents": [ "We may replace $U$ by an affine open neighbourhood of $u$.", "Hence we may assume that $U$ is affine. As $x$ is in the", "image of the open map $|U| \\to |X|$, so is $x'$. Thus we may", "replace $X$ by the Zariski open subspace corresponding to", "the image of $|U| \\to |X|$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-etale-image-open}.", "In other words we may assume that", "$U \\to X$ is surjective and \\'etale.", "Let $s, t : R = U \\times_X U \\to U$ be the projections.", "By our assumption that $(X, x')$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-UR-finite-above-x}", "we see that the fibres of $|U| \\to |X|$ and $|R| \\to |X|$", "over $x'$ are finite. Say $\\{u'_1, \\ldots, u'_n\\} \\subset U$ and", "$\\{r'_1, \\ldots, r'_m\\} \\subset R$ form the complete inverse image", "of $\\{x'\\}$.", "Consider the closed sets", "$$", "T = \\overline{\\{u'_1\\}} \\cup \\ldots \\cup \\overline{\\{u'_n\\}} \\subset |U|,", "\\quad", "T' = \\overline{\\{r'_1\\}} \\cup \\ldots \\cup \\overline{\\{r'_m\\}} \\subset |R|.", "$$", "Trivially we have $s(T') \\subset T$. Because $R$ is an equivalence", "relation we also have $t(T') = s(T')$ as the set $\\{r_j'\\}$", "is invariant under the inverse of $R$ by construction. Let $w \\in T$", "be any point. Then $u'_i \\leadsto w$ for some $i$. Choose $r \\in R$", "with $s(r) = w$. Since generalizations lift along $s : R \\to U$, see", "Remark \\ref{remark-recall}, we can find $r' \\leadsto r$ with", "$s(r') = u_i'$. Then $r' = r'_j$ for some $j$ and we conclude that", "$w \\in s(T')$. Hence $T = s(T') = t(T')$ is an $|R|$-invariant closed", "set in $|U|$. This means $T$ is the inverse image of a closed (!)", "subset $T'' = \\varphi(T)$ of $|X|$, see", "Properties of Spaces,", "Lemmas \\ref{spaces-properties-lemma-points-presentation} and", "\\ref{spaces-properties-lemma-topology-points}.", "Hence $T'' = \\overline{\\{x'\\}}$.", "Thus $T$ contains some point $u_1$ mapping to $x$ as $x \\in T''$.", "I.e., we see that for some $i$ there exists a specialization", "$u'_i \\leadsto u_1$ which maps to the given specialization", "$x' \\leadsto x$.", "\\medskip\\noindent", "To finish the proof, choose a point $r \\in R$ such that", "$s(r) = u$ and $t(r) = u_1$ (using", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-points-cartesian}).", "As generalizations lift along $t$, and $u'_i \\leadsto u_1$", "we can find a specialization $r' \\leadsto r$ such that $t(r') = u'_i$.", "Set $u' = s(r')$. Then $u' \\leadsto u$ and $\\varphi(u') = x'$ as", "desired." ], "refs": [ "spaces-properties-lemma-etale-image-open", "decent-spaces-lemma-UR-finite-above-x", "decent-spaces-remark-recall", "spaces-properties-lemma-points-presentation", "spaces-properties-lemma-topology-points", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 11825, 9463, 9573, 11821, 11822, 11819 ] } ], "ref_ids": [ 9463 ] }, { "id": 9474, "type": "theorem", "label": "decent-spaces-lemma-generalizations-lift-flat", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-generalizations-lift-flat", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a flat morphism of algebraic spaces", "over $S$. Let $x, x' \\in |X|$ and assume $x' \\leadsto x$, i.e., $x$ is a", "specialization of $x'$. Assume the pair $(X, x')$ satisfies the equivalent", "conditions of Lemma \\ref{lemma-UR-finite-above-x} (for example if", "$X$ is decent, $X$ is quasi-separated, or $X$ is representable).", "Then for every $y \\in |Y|$ with $f(y) = x$, there exists a point $y' \\in |Y|$,", "$y' \\leadsto y$ with $f(y') = x'$." ], "refs": [ "decent-spaces-lemma-UR-finite-above-x" ], "proofs": [ { "contents": [ "(The parenthetical statement holds by the definition of decent spaces", "and the implications between the different separation conditions", "mentioned in Section \\ref{section-reasonable-decent}.)", "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Choose $v \\in V$ mapping to $y$. Then we see that it suffices to", "prove the lemma for $V \\to X$. Thus we may assume $Y$ is a scheme.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "Choose $u \\in U$ mapping to $x$. By Lemma \\ref{lemma-specialization}", "we may choose $u' \\leadsto u$ mapping to $x'$. By", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}", "we may choose $z \\in U \\times_X Y$ mapping to $y$ and $u$.", "Thus we reduce to the case of the flat morphism of", "schemes $U \\times_X Y \\to U$ which is", "Morphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat}." ], "refs": [ "decent-spaces-lemma-specialization", "spaces-properties-lemma-points-cartesian", "morphisms-lemma-generalizations-lift-flat" ], "ref_ids": [ 9473, 11819, 5266 ] } ], "ref_ids": [ 9463 ] }, { "id": 9475, "type": "theorem", "label": "decent-spaces-lemma-quasi-compact-reasonable-stratification", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-quasi-compact-reasonable-stratification", "contents": [ "Let $S$ be a scheme. Let $W \\to X$ be a morphism of a scheme $W$", "to an algebraic space $X$ which is flat, locally of finite presentation,", "separated, locally quasi-finite with universally bounded fibres. There exist", "reduced closed subspaces", "$$", "\\emptyset = Z_{-1} \\subset Z_0 \\subset Z_1 \\subset Z_2 \\subset", "\\ldots \\subset Z_n = X", "$$", "such that with $X_r = Z_r \\setminus Z_{r - 1}$ the stratification", "$X = \\coprod_{r = 0, \\ldots, n} X_r$ is characterized by the following", "universal property: Given $g : T \\to X$ the projection", "$W \\times_X T \\to T$ is finite locally free of degree $r$ if and only if", "$g(|T|) \\subset |X_r|$." ], "refs": [], "proofs": [ { "contents": [ "Let $n$ be an integer bounding the degrees of the fibres of $W \\to X$.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "Apply More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-stratify-flat-fp-lqf-universally-bounded}", "to $W \\times_X U \\to U$. We obtain closed subsets", "$$", "\\emptyset = Y_{-1} \\subset Y_0 \\subset Y_1 \\subset Y_2 \\subset", "\\ldots \\subset Y_n = U", "$$", "characterized by the property stated in the lemma for the morphism", "$W \\times_X U \\to U$. Clearly, the formation of these closed subsets commutes", "with base change. Setting $R = U \\times_X U$ with projection maps", "$s, t : R \\to U$ we conclude that", "$$", "s^{-1}(Y_r) = t^{-1}(Y_r)", "$$", "as closed subsets of $R$. In other words the closed subsets $Y_r \\subset U$", "are $R$-invariant. This means that $|Y_r|$ is the inverse image of a closed", "subset $Z_r \\subset |X|$. Denote $Z_r \\subset X$ also the reduced induced", "algebraic space structure, see", "Properties of Spaces, Definition", "\\ref{spaces-properties-definition-reduced-induced-space}.", "\\medskip\\noindent", "Let $g : T \\to X$ be a morphism of algebraic spaces. Choose a scheme $V$", "and a surjective \\'etale morphism $V \\to T$. To prove the final", "assertion of the lemma it suffices to prove the assertion for the composition", "$V \\to X$ (by our definition of finite locally free morphisms, see", "Morphisms of Spaces, Section", "\\ref{spaces-morphisms-section-finite-locally-free}).", "Similarly, the morphism of schemes $W \\times_X V \\to V$ is finite", "locally free of degree $r$ if and only if the morphism of schemes", "$$", "W \\times_X (U \\times_X V)", "\\longrightarrow", "U \\times_X V", "$$", "is finite locally free of degree $r$ (see", "Descent, Lemma \\ref{descent-lemma-descending-property-finite-locally-free}).", "By construction this happens if and only if $|U \\times_X V| \\to |U|$", "maps into $|Y_r|$, which is true if and only if $|V| \\to |X|$ maps", "into $|Z_r|$." ], "refs": [ "more-morphisms-lemma-stratify-flat-fp-lqf-universally-bounded", "spaces-properties-definition-reduced-induced-space", "descent-lemma-descending-property-finite-locally-free" ], "ref_ids": [ 13908, 11932, 14695 ] } ], "ref_ids": [] }, { "id": 9476, "type": "theorem", "label": "decent-spaces-lemma-stratify-flat-fp-lqf", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-stratify-flat-fp-lqf", "contents": [ "Let $S$ be a scheme. Let $W \\to X$ be a morphism of a scheme $W$ to an", "algebraic space $X$ which is flat, locally of finite presentation,", "separated, and locally quasi-finite. Then there", "exist open subspaces", "$$", "X = X_0 \\supset X_1 \\supset X_2 \\supset \\ldots", "$$", "such that a morphism $\\Spec(k) \\to X$ factors through $X_d$ if and", "only if $W \\times_X \\Spec(k)$ has degree $\\geq d$ over $k$." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$. Apply", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-stratify-flat-fp-lqf}", "to $W \\times_X U \\to U$. We obtain open subschemes", "$$", "U = U_0 \\supset U_1 \\supset U_2 \\supset \\ldots", "$$", "characterized by the property stated in the lemma for the morphism", "$W \\times_X U \\to U$. Clearly, the formation of these closed subsets commutes", "with base change. Setting $R = U \\times_X U$ with projection maps", "$s, t : R \\to U$ we conclude that", "$$", "s^{-1}(U_d) = t^{-1}(U_d)", "$$", "as open subschemes of $R$. In other words the open subschemes $U_d \\subset U$", "are $R$-invariant. This means that $U_d$ is the inverse image of an", "open subspace $X_d \\subset X$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-subspaces-presentation})." ], "refs": [ "more-morphisms-lemma-stratify-flat-fp-lqf", "spaces-properties-lemma-subspaces-presentation" ], "ref_ids": [ 13910, 11845 ] } ], "ref_ids": [] }, { "id": 9477, "type": "theorem", "label": "decent-spaces-lemma-filter-quasi-compact", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-filter-quasi-compact", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact algebraic space", "over $S$. There exist open subspaces", "$$", "\\ldots \\subset U_4 \\subset U_3 \\subset U_2 \\subset U_1 = X", "$$", "with the following properties:", "\\begin{enumerate}", "\\item setting $T_p = U_p \\setminus U_{p + 1}$ (with reduced induced subspace", "structure) there exists a separated scheme $V_p$ and a surjective \\'etale", "morphism $f_p : V_p \\to U_p$ such that $f_p^{-1}(T_p) \\to T_p$ is an", "isomorphism,", "\\item if $x \\in |X|$ can be represented by a quasi-compact morphism", "$\\Spec(k) \\to X$ from a field, then $x \\in T_p$ for some $p$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}", "we can choose an affine scheme $U$ and a surjective \\'etale morphism", "$U \\to X$. For $p \\geq 0$ set", "$$", "W_p = U \\times_X \\ldots \\times_X U \\setminus \\text{all diagonals}", "$$", "where the fibre product has $p$ factors. Since $U$ is separated,", "the morphism $U \\to X$ is separated and all fibre products", "$U \\times_X \\ldots \\times_X U$ are separated schemes. Since $U \\to X$ is", "separated the diagonal $U \\to U \\times_X U$ is a closed immersion. Since", "$U \\to X$ is \\'etale the diagonal $U \\to U \\times_X U$ is an open", "immersion, see Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-etale-unramified} and", "\\ref{spaces-morphisms-lemma-diagonal-unramified-morphism}.", "Similarly, all the diagonal morphisms are open and closed immersions and", "$W_p$ is an open and closed subscheme of $U \\times_X \\ldots \\times_X U$.", "Moreover, the morphism", "$$", "U \\times_X \\ldots \\times_X U \\longrightarrow", "U \\times_{\\Spec(\\mathbf{Z})} \\ldots \\times_{\\Spec(\\mathbf{Z})} U", "$$", "is locally quasi-finite and separated (Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-fibre-product-after-map})", "and its target is an affine scheme. Hence every finite set of points of", "$U \\times_X \\ldots \\times_X U$ is contained in an affine open, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-over-affine}.", "Therefore, the same is true for $W_p$.", "There is a free action of the symmetric group $S_p$ on $W_p$ over $X$", "(because we threw out the fix point locus from", "$U \\times_X \\ldots \\times_X U$). By the above and", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-finite-flat-equivalence-global}", "the quotient $V_p = W_p/S_p$ is a scheme. Since the action of", "$S_p$ on $W_p$ was over $X$, there is a morphism $V_p \\to X$.", "Since $W_p \\to X$ is \\'etale and since $W_p \\to V_p$ is surjective", "\\'etale, it follows that also $V_p \\to X$ is \\'etale, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-local}.", "Observe that $V_p$ is a separated scheme by", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quotient-separated}.", "\\medskip\\noindent", "We let $U_p \\subset X$ be the open subspace which is the", "image of $V_p \\to X$. By construction a morphism $\\Spec(k) \\to X$ with", "$k$ algebraically closed, factors through $U_p$ if and only if", "$U \\times_X \\Spec(k)$ has $\\geq p$ points; as usual observe that", "$U \\times_X \\Spec(k)$ is scheme theoretically a disjoint union of", "(possibly infinitely many) copies of $\\Spec(k)$, see", "Remark \\ref{remark-recall}. It follows that", "the $U_p$ give a filtration of $X$ as stated in the lemma.", "Moreover, our morphism $\\Spec(k) \\to X$ factors through $T_p$", "if and only if $U \\times_X \\Spec(k)$ has exactly $p$ points.", "In this case we see that $V_p \\times_X \\Spec(k)$ has exactly one point.", "Set $Z_p = f_p^{-1}(T_p) \\subset V_p$. This is a closed subscheme of $V_p$.", "Then $Z_p \\to T_p$ is an \\'etale morphism between", "algebraic spaces which induces a bijection on $k$-valued", "points for any algebraically closed field $k$. To be sure this", "implies that $Z_p \\to T_p$ is universally injective, whence an", "open immersion by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-etale-universally-injective-open}", "hence an isomorphism and (1) has been proved.", "\\medskip\\noindent", "Let $x : \\Spec(k) \\to X$ be a quasi-compact morphism where $k$ is a field.", "Then the composition $\\Spec(\\overline{k}) \\to \\Spec(k) \\to X$ is quasi-compact", "as well (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-composition-quasi-compact}).", "In this case the scheme $U \\times_X \\Spec(\\overline{k})$ is", "quasi-compact. In view of the fact (seen above) that it is a disjoint union", "of copies of $\\Spec(\\overline{k})$ we find that it has finitely many points.", "If the number of points is $p$, then we see that indeed $x \\in T_p$ and", "the proof is finished." ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-morphisms-lemma-etale-unramified", "spaces-morphisms-lemma-diagonal-unramified-morphism", "spaces-morphisms-lemma-fibre-product-after-map", "more-morphisms-lemma-separated-locally-quasi-finite-over-affine", "spaces-properties-proposition-finite-flat-equivalence-global", "spaces-properties-lemma-etale-local", "spaces-properties-lemma-quotient-separated", "decent-spaces-remark-recall", "spaces-morphisms-lemma-etale-universally-injective-open", "spaces-morphisms-lemma-composition-quasi-compact" ], "ref_ids": [ 11832, 4913, 4902, 4715, 13906, 11918, 11856, 11851, 9573, 4973, 4739 ] } ], "ref_ids": [] }, { "id": 9478, "type": "theorem", "label": "decent-spaces-lemma-filter-reasonable", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-filter-reasonable", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact, reasonable algebraic space", "over $S$. There exist an integer $n$ and open subspaces", "$$", "\\emptyset = U_{n + 1} \\subset", "U_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X", "$$", "with the following property: setting $T_p = U_p \\setminus U_{p + 1}$", "(with reduced induced subspace structure) there exists a separated scheme", "$V_p$ and a surjective \\'etale morphism $f_p : V_p \\to U_p$ such that", "$f_p^{-1}(T_p) \\to T_p$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The proof of this lemma is identical to the proof of", "Lemma \\ref{lemma-filter-quasi-compact}.", "Let $n$ be an integer bounding the degrees of", "the fibres of $U \\to X$ which exists as $X$ is reasonable, see", "Definition \\ref{definition-very-reasonable}.", "Then we see that $U_{n + 1} = \\emptyset$ and the proof is complete." ], "refs": [ "decent-spaces-lemma-filter-quasi-compact", "decent-spaces-definition-very-reasonable" ], "ref_ids": [ 9477, 9562 ] } ], "ref_ids": [] }, { "id": 9479, "type": "theorem", "label": "decent-spaces-lemma-stratify-reasonable", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-stratify-reasonable", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact, reasonable algebraic space", "over $S$. There exist an integer $n$ and open subspaces", "$$", "\\emptyset = U_{n + 1} \\subset", "U_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X", "$$", "such that each $T_p = U_p \\setminus U_{p + 1}$ (with reduced induced subspace", "structure) is a scheme." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of Lemma \\ref{lemma-filter-reasonable}." ], "refs": [ "decent-spaces-lemma-filter-reasonable" ], "ref_ids": [ 9478 ] } ], "ref_ids": [] }, { "id": 9480, "type": "theorem", "label": "decent-spaces-lemma-filter-quasi-compact-quasi-separated", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-filter-quasi-compact-quasi-separated", "contents": [ "\\begin{reference}", "This result is almost identical to \\cite[Proposition 5.7.8]{GruRay}.", "\\end{reference}", "Let $X$ be a quasi-compact and quasi-separated algebraic space over", "$\\Spec(\\mathbf{Z})$. There exist an integer $n$ and open subspaces", "$$", "\\emptyset = U_{n + 1} \\subset", "U_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X", "$$", "with the following property: setting $T_p = U_p \\setminus U_{p + 1}$", "(with reduced induced subspace structure) there exists a quasi-compact", "separated scheme $V_p$ and a surjective \\'etale morphism $f_p : V_p \\to U_p$", "such that $f_p^{-1}(T_p) \\to T_p$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The proof of this lemma is identical to the proof of", "Lemma \\ref{lemma-filter-quasi-compact}.", "Observe that a quasi-separated space is reasonable, see", "Lemma \\ref{lemma-bounded-fibres} and", "Definition \\ref{definition-very-reasonable}.", "Hence we find that $U_{n + 1} = \\emptyset$ as in", "Lemma \\ref{lemma-filter-reasonable}.", "At the end of the argument we add that since $X$ is quasi-separated", "the schemes $U \\times_X \\ldots \\times_X U$ are all quasi-compact.", "Hence the schemes $W_p$ are quasi-compact. Hence the", "quotients $V_p = W_p/S_p$ by the symmetric group $S_p$ are quasi-compact", "schemes." ], "refs": [ "decent-spaces-lemma-filter-quasi-compact", "decent-spaces-lemma-bounded-fibres", "decent-spaces-definition-very-reasonable", "decent-spaces-lemma-filter-reasonable" ], "ref_ids": [ 9477, 9466, 9562, 9478 ] } ], "ref_ids": [] }, { "id": 9481, "type": "theorem", "label": "decent-spaces-lemma-locally-constructible", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-locally-constructible", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$.", "Let $E \\subset |X|$ be a subset. Then $E$ is \\'etale locally constructible", "(Properties of Spaces, Definition", "\\ref{spaces-properties-definition-locally-constructible})", "if and only if $E$ is a locally constructible subset of", "the topological space $|X|$", "(Topology, Definition \\ref{topology-definition-constructible})." ], "refs": [ "spaces-properties-definition-locally-constructible", "topology-definition-constructible" ], "proofs": [ { "contents": [ "Assume $E \\subset |X|$ is a locally constructible subset of", "the topological space $|X|$. Let $f : U \\to X$ be an", "\\'etale morphism where $U$ is a scheme. We have to show that", "$f^{-1}(E)$ is locally constructible in $U$. The question is", "local on $U$ and $X$, hence we may assume that $X$ is quasi-compact,", "$E \\subset |X|$ is constructible, and $U$ is affine.", "In this case $U \\to X$ is quasi-compact, hence", "$f : |U| \\to |X|$ is quasi-compact. Observe that retrocompact", "opens of $|X|$, resp.\\ $U$ are the same thing as quasi-compact opens", "of $|X|$, resp.\\ $U$, see", "Topology, Lemma \\ref{topology-lemma-topology-quasi-separated-scheme}.", "Thus $f^{-1}(E)$ is constructible by Topology, Lemma", "\\ref{topology-lemma-inverse-images-constructibles}.", "\\medskip\\noindent", "Conversely, assume $E$ is \\'etale locally constructible.", "We want to show that $E$ is locally constructible in the", "topological space $|X|$.", "The question is local on $X$, hence we may assume that $X$ is", "quasi-compact as well as quasi-separated. We will show that", "in this case $E$ is constructible in $|X|$.", "Choose open subspaces", "$$", "\\emptyset = U_{n + 1} \\subset", "U_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X", "$$", "and surjective \\'etale morphisms $f_p : V_p \\to U_p$", "inducing isomorphisms $f_p^{-1}(T_p) \\to T_p = U_p \\setminus U_{p + 1}$", "where $V_p$ is a quasi-compact separated scheme as in", "Lemma \\ref{lemma-filter-quasi-compact-quasi-separated}.", "By definition the inverse image $E_p \\subset V_p$ of $E$ is", "locally constructible in $V_p$. Then $E_p$ is constructible in $V_p$", "by Properties, Lemma", "\\ref{properties-lemma-constructible-quasi-compact-quasi-separated}.", "Thus $E_p \\cap |f_p^{-1}(T_p)| = E \\cap |T_p|$ is constructible", "in $|T_p|$ by", "Topology, Lemma \\ref{topology-lemma-intersect-constructible-with-closed}", "(observe that $V_p \\setminus f_p^{-1}(T_p)$ is quasi-compact as it is the", "inverse image of the quasi-compact space $U_{p + 1}$ by the", "quasi-compact morphism $f_p$).", "Thus", "$$", "E = (|T_n| \\cap E) \\cup (|T_{n - 1}| \\cap E) \\cup \\ldots \\cup", "(|T_1| \\cap E)", "$$", "is constructible by", "Topology, Lemma \\ref{topology-lemma-collate-constructible-from-constructible}.", "Here we use that $|T_p|$ is constructible in $|X|$ which is clear from", "what was said above." ], "refs": [ "topology-lemma-topology-quasi-separated-scheme", "topology-lemma-inverse-images-constructibles", "decent-spaces-lemma-filter-quasi-compact-quasi-separated", "properties-lemma-constructible-quasi-compact-quasi-separated", "topology-lemma-intersect-constructible-with-closed", "topology-lemma-collate-constructible-from-constructible" ], "ref_ids": [ 8333, 8254, 9480, 2942, 8258, 8265 ] } ], "ref_ids": [ 11928, 8362 ] }, { "id": 9482, "type": "theorem", "label": "decent-spaces-lemma-extend-integral-morphism", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-extend-integral-morphism", "contents": [ "Let $S$ be a scheme. Let $j : V \\to Y$ be a quasi-compact open immersion", "of algebraic spaces over $S$. Let $\\pi : Z \\to V$ be an integral morphism.", "Then there exists an integral morphism $\\nu : Y' \\to Y$ such that", "$Z$ is $V$-isomorphic to the inverse image of $V$ in $Y'$." ], "refs": [], "proofs": [ { "contents": [ "Since both $j$ and $\\pi$ are quasi-compact and separated, so is", "$j \\circ \\pi$. Let $\\nu : Y' \\to Y$ be the normalization of $Y$ in $Z$, see", "Morphisms of Spaces, Section", "\\ref{spaces-morphisms-section-normalization-X-in-Y}.", "Of course $\\nu$ is integral, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-characterize-normalization}.", "The final statement follows formally from", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-properties-normalization} and", "\\ref{spaces-morphisms-lemma-normalization-in-integral}." ], "refs": [ "spaces-morphisms-lemma-characterize-normalization", "spaces-morphisms-lemma-properties-normalization", "spaces-morphisms-lemma-normalization-in-integral" ], "ref_ids": [ 4959, 4958, 4964 ] } ], "ref_ids": [] }, { "id": 9483, "type": "theorem", "label": "decent-spaces-lemma-there-is-a-scheme-integral-over", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-there-is-a-scheme-integral-over", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$.", "\\begin{enumerate}", "\\item There exists a surjective integral morphism $Y \\to X$ where $Y$", "is a scheme,", "\\item given a surjective \\'etale morphism $U \\to X$ we may choose", "$Y \\to X$ such that for every $y \\in Y$ there is an open neighbourhood", "$V \\subset Y$ such that $V \\to X$ factors through $U$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is the special case of part (2) where $U = X$.", "Choose a surjective \\'etale morphism $U' \\to U$", "where $U'$ is a scheme. It is clear that we may replace $U$ by $U'$", "and hence we may assume $U$ is a scheme. Since $X$ is quasi-compact,", "there exist finitely many affine opens $U_i \\subset U$ such that", "$U' = \\coprod U_i \\to X$ is surjective.", "After replacing $U$ by $U'$ again, we see that we may assume $U$ is affine.", "Since $X$ is quasi-separated, hence reasonable, there exists an integer", "$d$ bounding the degree of the geometric fibres of $U \\to X$", "(see Lemma \\ref{lemma-bounded-fibres}).", "We will prove the lemma by induction on $d$ for all quasi-compact", "and separated schemes $U$ mapping surjective and \\'etale onto $X$.", "If $d = 1$, then $U = X$ and the result holds with $Y = U$.", "Assume $d > 1$.", "\\medskip\\noindent", "We apply Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-finite-separated-quasi-affine}", "and we obtain a factorization", "$$", "\\xymatrix{", "U \\ar[rr]_j \\ar[rd] & & Y \\ar[ld]^\\pi \\\\", "& X", "}", "$$", "with $\\pi$ integral and $j$ a quasi-compact open immersion. We may and do", "assume that $j(U)$ is scheme theoretically dense in $Y$. Note that", "$$", "U \\times_X Y = U \\amalg W", "$$", "where the first summand is the image of $U \\to U \\times_X Y$", "(which is closed by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-semi-diagonal}", "and open because it is \\'etale as a morphism between", "algebraic spaces \\'etale over $Y$) and", "the second summand is the (open and closed) complement.", "The image $V \\subset Y$ of $W$ is an open subspace containing", "$Y \\setminus U$.", "\\medskip\\noindent", "The \\'etale morphism $W \\to Y$ has geometric fibres of cardinality $< d$.", "Namely, this is clear for geometric points of $U \\subset Y$ by inspection.", "Since $|U| \\subset |Y|$ is dense, it holds for all geometric points of $Y$", "by Lemma \\ref{lemma-quasi-compact-reasonable-stratification}", "(the degree of the fibres of a quasi-compact \\'etale morphism", "does not go up under specialization). Thus we may apply the induction", "hypothesis to $W \\to V$ and find a surjective integral morphism", "$Z \\to V$ with $Z$ a scheme, which Zariski locally factors through $W$.", "Choose a factorization $Z \\to Z' \\to Y$ with $Z' \\to Y$ integral and", "$Z \\to Z'$ open immersion", "(Lemma \\ref{lemma-extend-integral-morphism}).", "After replacing $Z'$ by the scheme theoretic closure of $Z$ in $Z'$", "we may assume that $Z$ is scheme theoretically dense in $Z'$.", "After doing this we have $Z' \\times_Y V = Z$. Finally,", "let $T \\subset Y$ be the induced closed subspace structure on $Y \\setminus V$.", "Consider the morphism", "$$", "Z' \\amalg T \\longrightarrow X", "$$", "This is a surjective integral morphism by construction.", "Since $T \\subset U$ it is clear that the morphism $T \\to X$", "factors through $U$. On the other hand, let $z \\in Z'$", "be a point. If $z \\not \\in Z$, then $z$ maps to a point of", "$Y \\setminus V \\subset U$ and we find a neighbourhood of $z$", "on which the morphism factors through $U$.", "If $z \\in Z$, then we have an open neighbourhood of $z$ in $Z$", "(which is also an open neighbourhood of $z$ in $Z'$)", "which factors through $W \\subset U \\times_X Y$ and hence through $U$." ], "refs": [ "decent-spaces-lemma-bounded-fibres", "spaces-morphisms-lemma-quasi-finite-separated-quasi-affine", "spaces-morphisms-lemma-semi-diagonal", "decent-spaces-lemma-quasi-compact-reasonable-stratification", "decent-spaces-lemma-extend-integral-morphism" ], "ref_ids": [ 9466, 4975, 4716, 9475, 9482 ] } ], "ref_ids": [] }, { "id": 9484, "type": "theorem", "label": "decent-spaces-lemma-when-quotient-scheme-at-point", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-when-quotient-scheme-at-point", "contents": [ "Let $S$ be a scheme. Let $X \\to Y$ be a surjective finite locally free", "morphism of algebraic spaces over $S$. For $y \\in |Y|$ the following are", "equivalent", "\\begin{enumerate}", "\\item $y$ is in the schematic locus of $Y$, and", "\\item there exists an affine open $U \\subset X$", "containing the preimage of $y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $y \\in Y$ is in the schematic locus, then it has an affine open", "neighbourhood $V \\subset Y$ and the inverse image $U$ of $V$ in $X$", "is an open finite over $V$, hence affine. Thus (1) implies (2).", "\\medskip\\noindent", "Conversely, assume that $U \\subset X$ as in (2) is given.", "Set $R = X \\times_Y X$ and denote the projections $s, t : R \\to X$.", "Consider $Z = R \\setminus s^{-1}(U) \\cap t^{-1}(U)$.", "This is a closed subset of $R$. The image $t(Z)$ is a closed", "subset of $X$ which can loosely be described as the set of", "points of $X$ which are $R$-equivalent to a point of", "$X \\setminus U$. Hence $U' = X \\setminus t(Z)$ is an $R$-invariant,", "open subspace of $X$ contained in $U$ which contains", "the fibre of $X \\to Y$ over $y$. Since $X \\to Y$ is open", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-fppf-open})", "the image of $U'$ is an open subspace $V' \\subset Y$.", "Since $U'$ is $R$-invariant and $R = X \\times_Y X$, we see that $U'$ is the", "inverse image of $V'$ (use", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}).", "After replacing $Y$ by $V'$ and $X$ by $U'$ we see that we may assume", "$X$ is a scheme isomorphic to an open subscheme of an affine scheme.", "\\medskip\\noindent", "Assume $X$ is a scheme isomorphic to an open subscheme of an affine scheme.", "In this case the fppf quotient sheaf $X/R$ is a scheme, see", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-finite-flat-equivalence-global}.", "Since $Y$ is a sheaf in the fppf topology, obtain a canonical", "map $X/R \\to Y$ factoring $X \\to Y$. Since $X \\to Y$ is surjective", "finite locally free, it is surjective as a map of sheaves", "(Spaces, Lemma \\ref{spaces-lemma-surjective-flat-locally-finite-presentation}).", "We conclude that $X/R \\to Y$ is surjective as a map of sheaves.", "On the other hand, since $R = X \\times_Y X$ as sheaves we conclude that", "$X/R \\to Y$ is injective as a map of sheaves. Hence $X/R \\to Y$", "is an isomorphism and we see that $Y$ is representable." ], "refs": [ "spaces-morphisms-lemma-fppf-open", "spaces-properties-lemma-points-cartesian", "spaces-properties-proposition-finite-flat-equivalence-global", "spaces-lemma-surjective-flat-locally-finite-presentation" ], "ref_ids": [ 4855, 11819, 11918, 8137 ] } ], "ref_ids": [] }, { "id": 9485, "type": "theorem", "label": "decent-spaces-lemma-finite-etale-cover-dense-open-scheme", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-finite-etale-cover-dense-open-scheme", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "If there exists a finite, \\'etale, surjective morphism", "$U \\to X$ where $U$ is a scheme, then there exists a dense open subspace", "of $X$ which is a scheme." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "The morphism $U \\to X$ is finite locally free. Hence there is a decomposition", "of $X$ into open and closed subspaces $X_d \\subset X$ such that", "$U \\times_X X_d \\to X_d$ is finite locally free of degree $d$.", "Thus we may assume $U \\to X$ is finite locally free of degree $d$.", "In this case, let $U_i \\subset U$, $i \\in I$ be the set of affine opens.", "For each $i$ the morphism $U_i \\to X$ is \\'etale and has", "universally bounded fibres (namely, bounded by $d$).", "In other words, $X$ is reasonable and", "the result follows from", "Proposition \\ref{proposition-reasonable-open-dense-scheme}." ], "refs": [ "decent-spaces-proposition-reasonable-open-dense-scheme" ], "ref_ids": [ 9558 ] } ], "ref_ids": [] }, { "id": 9486, "type": "theorem", "label": "decent-spaces-lemma-decent-points-monomorphism", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-decent-points-monomorphism", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Consider the map", "$$", "\\{\\Spec(k) \\to X \\text{ monomorphism where }k\\text{ is a field}\\}", "\\longrightarrow", "|X|", "$$", "This map is always injective. If $X$ is decent then this map", "is a bijection." ], "refs": [], "proofs": [ { "contents": [ "We have seen in", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-points-monomorphism}", "that the map is an injection in general.", "By Lemma \\ref{lemma-bounded-fibres} it is surjective when $X$ is", "decent (actually one can say this is part of the definition", "of being decent)." ], "refs": [ "spaces-properties-lemma-points-monomorphism", "decent-spaces-lemma-bounded-fibres" ], "ref_ids": [ 11826, 9466 ] } ], "ref_ids": [] }, { "id": 9487, "type": "theorem", "label": "decent-spaces-lemma-identifies-residue-fields", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-identifies-residue-fields", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of decent", "algebraic spaces over $S$. Let $x \\in |X|$ be a point", "with image $y = f(x) \\in |Y|$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ induces an isomorphism $\\kappa(y) \\to \\kappa(x)$, and", "\\item the induced morphism $\\Spec(\\kappa(x)) \\to Y$ is a monomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Immediate from the discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9488, "type": "theorem", "label": "decent-spaces-lemma-decent-space-elementary-etale-neighbourhood", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-decent-space-elementary-etale-neighbourhood", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "For every point $x \\in |X|$ there exists an \\'etale morphism", "$$", "(U, u) \\longrightarrow (X, x)", "$$", "where $U$ is an affine scheme, $u$ is the only point of $U$ lying", "over $x$, and the induced homomorphism $\\kappa(x) \\to \\kappa(u)$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $X$ is quasi-compact by replacing $X$ with a", "quasi-compact open containing $x$. Recall that $x$ can be", "represented by a quasi-compact (mono)morphism", "from the spectrum a field (by definition of decent spaces). Thus the", "lemma follows from Lemma \\ref{lemma-filter-quasi-compact}." ], "refs": [ "decent-spaces-lemma-filter-quasi-compact" ], "ref_ids": [ 9477 ] } ], "ref_ids": [] }, { "id": 9489, "type": "theorem", "label": "decent-spaces-lemma-elementary-etale-neighbourhoods", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-elementary-etale-neighbourhoods", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $x$ be a point of $X$.", "The category of elementary \\'etale neighborhoods of $(X, x)$", "is cofiltered (see", "Categories, Definition \\ref{categories-definition-codirected})." ], "refs": [ "categories-definition-codirected" ], "proofs": [ { "contents": [ "The category is nonempty by", "Lemma \\ref{lemma-decent-space-elementary-etale-neighbourhood}.", "Suppose that we have two elementary \\'etale neighbourhoods", "$(U_i, u_i) \\to (X, x)$.", "Then consider $U = U_1 \\times_X U_2$. Since", "$\\Spec(\\kappa(u_i)) \\to X$, $i = 1, 2$ are both monomorphisms", "in the class of $x$ (Lemma \\ref{lemma-identifies-residue-fields})", ", we see that", "$$", "u = \\Spec(\\kappa(u_1)) \\times_X \\Spec(\\kappa(u_2))", "$$", "is the spectrum of a field $\\kappa(u)$ such that the induced maps", "$\\kappa(u_i) \\to \\kappa(u)$ are isomorphisms. Then $u \\to U$ is a point", "of $U$ and we see that $(U, u) \\to (X, x)$ is an elementary", "\\'etale neighbourhood dominating $(U_i, u_i)$.", "If $a, b : (U_1, u_1) \\to (U_2, u_2)$ are two morphisms between", "our elementary \\'etale neighbourhoods, then we consider the scheme", "$$", "U = U_1 \\times_{(a, b), (U_2 \\times_X U_2), \\Delta} U_2", "$$", "Using Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-etale-permanence}", "we see that $U \\to X$ is \\'etale. Moreover, in exactly the same manner", "as before we see that $U$ has a point $u$", "such that $(U, u) \\to (X, x)$ is an elementary", "\\'etale neighbourhood. Finally, $U \\to U_1$ equalizes $a$ and $b$", "and the proof is finished." ], "refs": [ "decent-spaces-lemma-decent-space-elementary-etale-neighbourhood", "decent-spaces-lemma-identifies-residue-fields", "spaces-properties-lemma-etale-permanence" ], "ref_ids": [ 9488, 9487, 11859 ] } ], "ref_ids": [ 12364 ] }, { "id": 9490, "type": "theorem", "label": "decent-spaces-lemma-describe-henselian-local-ring", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-describe-henselian-local-ring", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $x \\in |X|$. Let $(U, u) \\to (X, x)$ be an elementary", "\\'etale neighbourhood. Then", "$$", "\\mathcal{O}_{X, x}^h = \\mathcal{O}_{U, u}^h", "$$", "In words: the henselian local ring of $X$ at $x$", "is equal to the henselization $\\mathcal{O}_{U, u}^h$", "of the local ring $\\mathcal{O}_{U, u}$ of $U$ at $u$." ], "refs": [], "proofs": [ { "contents": [ "Since the category of elementary \\'etale neighbourhood of $(X, x)$", "is cofiltered (Lemma \\ref{lemma-elementary-etale-neighbourhoods})", "we see that", "the category of elementary \\'etale neighbourhoods of $(U, u)$", "is initial in", "the category of elementary \\'etale neighbourhood of $(X, x)$.", "Then the equality follows from", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-describe-henselization}", "and", "Categories, Lemma \\ref{categories-lemma-cofinal}", "(initial is turned into cofinal because the colimit", "definining henselian local rings is over the", "opposite of the category of elementary", "\\'etale neighbourhoods)." ], "refs": [ "decent-spaces-lemma-elementary-etale-neighbourhoods", "more-morphisms-lemma-describe-henselization", "categories-lemma-cofinal" ], "ref_ids": [ 9489, 13869, 12217 ] } ], "ref_ids": [] }, { "id": 9491, "type": "theorem", "label": "decent-spaces-lemma-henselian-local-ring-strict", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-henselian-local-ring-strict", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $\\overline{x}$ be a geometric point of $X$ lying over $x \\in |X|$. ", "The \\'etale local ring $\\mathcal{O}_{X, \\overline{x}}$ of $X$ at $\\overline{x}$", "(Properties of Spaces, Definition", "\\ref{spaces-properties-definition-etale-local-rings})", "is the strict henselization", "of the henselian local ring $\\mathcal{O}_{X, x}^h$ of $X$ at $x$." ], "refs": [ "spaces-properties-definition-etale-local-rings" ], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-describe-henselian-local-ring},", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring}", "and the fact that $(R^h)^{sh} = R^{sh}$", "for a local ring $(R, \\mathfrak m, \\kappa)$ and a given", "separable algebraic closure $\\kappa^{sep}$ of $\\kappa$.", "This equality follows from", "Algebra, Lemma \\ref{algebra-lemma-uniqueness-henselian}." ], "refs": [ "decent-spaces-lemma-describe-henselian-local-ring", "spaces-properties-lemma-describe-etale-local-ring", "algebra-lemma-uniqueness-henselian" ], "ref_ids": [ 9490, 11884, 1292 ] } ], "ref_ids": [ 11943 ] }, { "id": 9492, "type": "theorem", "label": "decent-spaces-lemma-residue-field-henselian-local-ring", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-residue-field-henselian-local-ring", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $x \\in |X|$. The residue field of the", "henselian local ring of $X$ at $x$", "(Definition \\ref{definition-henselian-local-ring})", "is the residue field of $X$ at $x$", "(Definition \\ref{definition-residue-field})." ], "refs": [ "decent-spaces-definition-henselian-local-ring", "decent-spaces-definition-residue-field" ], "proofs": [ { "contents": [ "Choose an elementary \\'etale neighbourhood $(U, u) \\to (X, x)$.", "Then $\\kappa(u) = \\kappa(x)$ and", "$\\mathcal{O}_{X, x}^h = \\mathcal{O}_{U, u}^h$", "(Lemma \\ref{lemma-describe-henselian-local-ring}).", "The residue field of $\\mathcal{O}_{U, u}^h$", "is $\\kappa(u)$ by Algebra, Lemma \\ref{algebra-lemma-henselization}", "(the output of this lemma is the construction/definition", "of the henselization of a local ring, see", "Algebra, Definition \\ref{algebra-definition-henselization})." ], "refs": [ "decent-spaces-lemma-describe-henselian-local-ring", "algebra-lemma-henselization", "algebra-definition-henselization" ], "ref_ids": [ 9490, 1294, 1546 ] } ], "ref_ids": [ 9565, 9563 ] }, { "id": 9493, "type": "theorem", "label": "decent-spaces-lemma-decent-no-specializations-map-to-same-point", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-decent-no-specializations-map-to-same-point", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $U \\to X$ be an \\'etale morphism from a scheme to $X$.", "If $u, u' \\in |U|$ map to the same point of $|X|$, and", "$u' \\leadsto u$, then $u = u'$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-bounded-fibres} and", "\\ref{lemma-no-specializations-map-to-same-point}." ], "refs": [ "decent-spaces-lemma-bounded-fibres", "decent-spaces-lemma-no-specializations-map-to-same-point" ], "ref_ids": [ 9466, 9472 ] } ], "ref_ids": [] }, { "id": 9494, "type": "theorem", "label": "decent-spaces-lemma-decent-specialization", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-decent-specialization", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $x, x' \\in |X|$ and assume $x' \\leadsto x$, i.e., $x$ is a", "specialization of $x'$. Then for every \\'etale morphism", "$\\varphi : U \\to X$ from a scheme $U$ and any $u \\in U$ with", "$\\varphi(u) = x$, exists a point $u'\\in U$, $u' \\leadsto u$ with", "$\\varphi(u') = x'$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-bounded-fibres} and", "\\ref{lemma-specialization}." ], "refs": [ "decent-spaces-lemma-bounded-fibres", "decent-spaces-lemma-specialization" ], "ref_ids": [ 9466, 9473 ] } ], "ref_ids": [] }, { "id": 9495, "type": "theorem", "label": "decent-spaces-lemma-kolmogorov", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-kolmogorov", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Then $|X|$ is Kolmogorov (see", "Topology, Definition \\ref{topology-definition-generic-point})." ], "refs": [ "topology-definition-generic-point" ], "proofs": [ { "contents": [ "Let $x_1, x_2 \\in |X|$ with $x_1 \\leadsto x_2$ and $x_2 \\leadsto x_1$.", "We have to show that $x_1 = x_2$. Pick a scheme $U$ and an \\'etale morphism", "$U \\to X$ such that $x_1, x_2$ are both in the image of $|U| \\to |X|$.", "By Lemma \\ref{lemma-decent-specialization} we can find a specialization", "$u_1 \\leadsto u_2$ in $U$ mapping to $x_1 \\leadsto x_2$.", "By Lemma \\ref{lemma-decent-specialization} we can find", "$u_2' \\leadsto u_1$ mapping to $x_2 \\leadsto x_1$. This means that", "$u_2' \\leadsto u_2$ is a specialization between points of $U$ mapping to", "the same point of $X$, namely $x_2$. This is not possible, unless", "$u_2' = u_2$, see", "Lemma \\ref{lemma-decent-no-specializations-map-to-same-point}. Hence", "also $u_1 = u_2$ as desired." ], "refs": [ "decent-spaces-lemma-decent-specialization", "decent-spaces-lemma-decent-specialization", "decent-spaces-lemma-decent-no-specializations-map-to-same-point" ], "ref_ids": [ 9494, 9494, 9493 ] } ], "ref_ids": [ 8354 ] }, { "id": 9496, "type": "theorem", "label": "decent-spaces-lemma-dimension-decent-space", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-dimension-decent-space", "contents": [ "Let $S$ be a scheme. Dimension as defined in", "Properties of Spaces, Section \\ref{spaces-properties-section-dimension}", "behaves well on decent algebraic spaces $X$ over $S$.", "\\begin{enumerate}", "\\item If $x \\in |X|$, then $\\dim_x(|X|) = \\dim_x(X)$, and", "\\item $\\dim(|X|) = \\dim(X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Choose a scheme $U$ with a point $u \\in U$", "and an \\'etale morphism $h : U \\to X$ mapping $u$ to $x$.", "By definition the dimension of $X$ at $x$ is $\\dim_u(|U|)$.", "Thus we may pick $U$ such that $\\dim_x(X) = \\dim(|U|)$.", "Let $d$ be an integer. If $\\dim(U) \\geq d$, then", "there exists a sequence of nontrivial specializations", "$u_d \\leadsto \\ldots \\leadsto u_0$ in $U$. Taking the image", "we find a corresponding sequence", "$h(u_d) \\leadsto \\ldots \\leadsto h(u_0)$", "each of which is nontrivial by", "Lemma \\ref{lemma-decent-no-specializations-map-to-same-point}.", "Hence we see that the image of $|U|$ in $|X|$ has dimension at least $d$.", "Conversely, suppose that $x_d \\leadsto \\ldots \\leadsto x_0$ is a", "sequence of specializations in $|X|$ with $x_0$ in the image of", "$|U| \\to |X|$. Then we can lift this to a sequence of specializations", "in $U$ by Lemma \\ref{lemma-decent-specialization}.", "\\medskip\\noindent", "Part (2) is an immediate consequence of part (1),", "Topology, Lemma \\ref{topology-lemma-dimension-supremum-local-dimensions},", "and Properties of Spaces, Section \\ref{spaces-properties-section-dimension}." ], "refs": [ "decent-spaces-lemma-decent-no-specializations-map-to-same-point", "decent-spaces-lemma-decent-specialization", "topology-lemma-dimension-supremum-local-dimensions" ], "ref_ids": [ 9493, 9494, 8224 ] } ], "ref_ids": [] }, { "id": 9497, "type": "theorem", "label": "decent-spaces-lemma-dimension-local-ring-quasi-finite", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-dimension-local-ring-quasi-finite", "contents": [ "Let $S$ be a scheme. Let $X \\to Y$ be a locally quasi-finite morphism", "of algebraic spaces over $S$. Let $x \\in |X|$ with image $y \\in |Y|$.", "Then the dimension of the local ring of $Y$ at $y$ is $\\geq$ to the", "dimension of the local ring of $X$ at $x$." ], "refs": [], "proofs": [ { "contents": [ "The definition of the dimension of the local ring of a point on an", "algebraic space is given in Properties of Spaces, Definition", "\\ref{spaces-properties-definition-dimension-local-ring}.", "Choose an \\'etale morphism $(V, v) \\to (Y, y)$ where $V$ is a scheme.", "Choose an \\'etale morphism $U \\to V \\times_Y X$ and a point $u \\in U$", "mapping to $x \\in |X|$ and $v \\in V$. Then $U \\to V$ is locally", "quasi-finite and we have to prove that", "$$", "\\dim(\\mathcal{O}_{V, v}) \\geq \\dim(\\mathcal{O}_{U, u})", "$$", "This is Algebra, Lemma \\ref{algebra-lemma-dimension-inequality-quasi-finite}." ], "refs": [ "spaces-properties-definition-dimension-local-ring", "algebra-lemma-dimension-inequality-quasi-finite" ], "ref_ids": [ 11931, 1073 ] } ], "ref_ids": [] }, { "id": 9498, "type": "theorem", "label": "decent-spaces-lemma-dimension-quasi-finite", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-dimension-quasi-finite", "contents": [ "Let $S$ be a scheme. Let $X \\to Y$ be a locally quasi-finite morphism", "of algebraic spaces over $S$. Then $\\dim(X) \\leq \\dim(Y)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-dimension-local-ring-quasi-finite}", "and Properties of Spaces, Lemma \\ref{spaces-properties-lemma-dimension}." ], "refs": [ "decent-spaces-lemma-dimension-local-ring-quasi-finite", "spaces-properties-lemma-dimension" ], "ref_ids": [ 9497, 11841 ] } ], "ref_ids": [] }, { "id": 9499, "type": "theorem", "label": "decent-spaces-lemma-decent-point-like-spaces", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-decent-point-like-spaces", "contents": [ "Let $S$ be a scheme. Let $k$ be a field. Let $X$ be an algebraic space", "over $S$ and assume that there exists a surjective \\'etale morphism", "$\\Spec(k) \\to X$. If $X$ is decent, then $X \\cong \\Spec(k')$", "where $k' \\subset k$ is a finite separable extension." ], "refs": [], "proofs": [ { "contents": [ "The assumption implies that $|X| = \\{x\\}$ is a singleton. Since", "$X$ is decent we can find a quasi-compact monomorphism $\\Spec(k') \\to X$", "whose image is $x$. Then the projection", "$U = \\Spec(k') \\times_X \\Spec(k) \\to \\Spec(k)$", "is a monomorphism, whence $U = \\Spec(k)$, see", "Schemes, Lemma \\ref{schemes-lemma-mono-towards-spec-field}.", "Hence the projection $\\Spec(k) = U \\to \\Spec(k')$ is \\'etale and", "we win." ], "refs": [ "schemes-lemma-mono-towards-spec-field" ], "ref_ids": [ 7729 ] } ], "ref_ids": [] }, { "id": 9500, "type": "theorem", "label": "decent-spaces-lemma-flat-cover-by-field", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-flat-cover-by-field", "contents": [ "Let $S$ be a scheme. Let $Z$ be an algebraic space over $S$.", "Let $k$ be a field and let $\\Spec(k) \\to Z$ be surjective and flat.", "Then any morphism $\\Spec(k') \\to Z$ where $k'$ is a field is", "surjective and flat." ], "refs": [], "proofs": [ { "contents": [ "Consider the fibre square", "$$", "\\xymatrix{", "T \\ar[d] \\ar[r] & \\Spec(k) \\ar[d] \\\\", "\\Spec(k') \\ar[r] & Z", "}", "$$", "Note that $T \\to \\Spec(k')$ is flat and surjective hence $T$", "is not empty. On the other hand $T \\to \\Spec(k)$ is flat as", "$k$ is a field. Hence $T \\to Z$ is flat and surjective.", "It follows from", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-permanence}", "that $\\Spec(k') \\to Z$ is flat. It is surjective as by assumption", "$|Z|$ is a singleton." ], "refs": [ "spaces-morphisms-lemma-flat-permanence" ], "ref_ids": [ 4865 ] } ], "ref_ids": [] }, { "id": 9501, "type": "theorem", "label": "decent-spaces-lemma-unique-point", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-unique-point", "contents": [ "Let $S$ be a scheme.", "Let $Z$ be an algebraic space over $S$. The following are equivalent", "\\begin{enumerate}", "\\item $Z$ is reduced and $|Z|$ is a singleton,", "\\item there exists a surjective flat morphism $\\Spec(k) \\to Z$", "where $k$ is a field, and", "\\item there exists a locally of finite type, surjective, flat morphism", "$\\Spec(k) \\to Z$ where $k$ is a field.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Let $W$ be a scheme and", "let $W \\to Z$ be a surjective \\'etale morphism. Then $W$ is", "a reduced scheme. Let $\\eta \\in W$ be a generic point of an irreducible", "component of $W$. Since $W$ is reduced we have", "$\\mathcal{O}_{W, \\eta} = \\kappa(\\eta)$. It follows that the canonical", "morphism $\\eta = \\Spec(\\kappa(\\eta)) \\to W$ is flat. We see that the", "composition $\\eta \\to Z$ is flat (see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-composition-flat}).", "It is also surjective as $|Z|$ is a singleton. In other words", "(2) holds.", "\\medskip\\noindent", "Assume (2). Let $W$ be a scheme and", "let $W \\to Z$ be a surjective \\'etale morphism. Choose a field", "$k$ and a surjective flat morphism $\\Spec(k) \\to Z$.", "Then $W \\times_Z \\Spec(k)$ is a scheme \\'etale over $k$.", "Hence $W \\times_Z \\Spec(k)$ is a disjoint union of spectra of fields", "(see Remark \\ref{remark-recall}), in particular reduced. Since", "$W \\times_Z \\Spec(k) \\to W$", "is surjective and flat we conclude that $W$ is reduced", "(Descent, Lemma \\ref{descent-lemma-descend-reduced}).", "In other words (1) holds.", "\\medskip\\noindent", "It is clear that (3) implies (2). Finally, assume (2). Pick a nonempty", "affine scheme $W$ and an \\'etale morphism $W \\to Z$. Pick a closed", "point $w \\in W$ and set $k = \\kappa(w)$. The composition", "$$", "\\Spec(k) \\xrightarrow{w} W \\longrightarrow Z", "$$", "is locally of finite type by", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-composition-finite-type} and", "\\ref{spaces-morphisms-lemma-etale-locally-finite-type}.", "It is also flat and surjective by", "Lemma \\ref{lemma-flat-cover-by-field}.", "Hence (3) holds." ], "refs": [ "spaces-morphisms-lemma-composition-flat", "decent-spaces-remark-recall", "descent-lemma-descend-reduced", "spaces-morphisms-lemma-composition-finite-type", "spaces-morphisms-lemma-etale-locally-finite-type", "decent-spaces-lemma-flat-cover-by-field" ], "ref_ids": [ 4852, 9573, 14658, 4814, 4912, 9500 ] } ], "ref_ids": [] }, { "id": 9502, "type": "theorem", "label": "decent-spaces-lemma-unique-point-better", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-unique-point-better", "contents": [ "Let $S$ be a scheme. Let $Z$ be an algebraic space over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $Z$ is reduced, locally Noetherian, and $|Z|$", "is a singleton, and", "\\item there exists a locally finitely presented, surjective, flat morphism", "$\\Spec(k) \\to Z$ where $k$ is a field.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (2) holds. By", "Lemma \\ref{lemma-unique-point}", "we see that $Z$ is reduced and $|Z|$ is a singleton.", "Let $W$ be a scheme and let $W \\to Z$ be a surjective \\'etale", "morphism. Choose a field $k$ and a locally finitely presented, surjective,", "flat morphism $\\Spec(k) \\to Z$.", "Then $W \\times_Z \\Spec(k)$ is a scheme", "\\'etale over $k$, hence a disjoint union of spectra of fields", "(see Remark \\ref{remark-recall}),", "hence locally Noetherian. Since $W \\times_Z \\Spec(k) \\to W$", "is flat, surjective, and locally of finite presentation, we see", "that $\\{W \\times_Z \\Spec(k) \\to W\\}$ is an fppf covering", "and we conclude that $W$ is locally Noetherian", "(Descent, Lemma", "\\ref{descent-lemma-Noetherian-local-fppf}).", "In other words (1) holds.", "\\medskip\\noindent", "Assume (1). Pick a nonempty affine scheme $W$ and an \\'etale morphism", "$W \\to Z$. Pick a closed point $w \\in W$ and set", "$k = \\kappa(w)$. Because $W$ is locally Noetherian the morphism", "$w : \\Spec(k) \\to W$ is of finite presentation, see", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-finite-presentation}.", "Hence the composition", "$$", "\\Spec(k) \\xrightarrow{w} W \\longrightarrow Z", "$$", "is locally of finite presentation by", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-composition-finite-presentation} and", "\\ref{spaces-morphisms-lemma-etale-locally-finite-presentation}.", "It is also flat and surjective by", "Lemma \\ref{lemma-flat-cover-by-field}.", "Hence (2) holds." ], "refs": [ "decent-spaces-lemma-unique-point", "decent-spaces-remark-recall", "descent-lemma-Noetherian-local-fppf", "morphisms-lemma-closed-immersion-finite-presentation", "spaces-morphisms-lemma-composition-finite-presentation", "spaces-morphisms-lemma-etale-locally-finite-presentation", "decent-spaces-lemma-flat-cover-by-field" ], "ref_ids": [ 9501, 9573, 14648, 5243, 4839, 4911, 9500 ] } ], "ref_ids": [] }, { "id": 9503, "type": "theorem", "label": "decent-spaces-lemma-monomorphism-into-point", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-monomorphism-into-point", "contents": [ "Let $S$ be a scheme.", "Let $Z' \\to Z$ be a monomorphism of algebraic spaces over $S$.", "Assume there exists a field $k$ and a locally finitely presented, surjective,", "flat morphism $\\Spec(k) \\to Z$. Then either $Z'$", "is empty or $Z' = Z$." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $Z'$ is nonempty. In this case the", "fibre product $T = Z' \\times_Z \\Spec(k)$", "is nonempty, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}.", "Now $T$ is an algebraic space and the projection $T \\to \\Spec(k)$", "is a monomorphism. Hence $T = \\Spec(k)$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-monomorphism-toward-field}.", "We conclude that $\\Spec(k) \\to Z$ factors through $Z'$.", "But as $\\Spec(k) \\to Z$ is surjective, flat and locally of finite", "presentation, we see that $\\Spec(k) \\to Z$ is surjective as a", "map of sheaves on $(\\Sch/S)_{fppf}$ (see", "Spaces, Remark \\ref{spaces-remark-warning})", "and we conclude that $Z' = Z$." ], "refs": [ "spaces-properties-lemma-points-cartesian", "spaces-morphisms-lemma-monomorphism-toward-field", "spaces-remark-warning" ], "ref_ids": [ 11819, 4757, 8187 ] } ], "ref_ids": [] }, { "id": 9504, "type": "theorem", "label": "decent-spaces-lemma-find-singleton-from-point", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-find-singleton-from-point", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$. Then there exists a unique monomorphism", "$Z \\to X$ of algebraic spaces", "over $S$ such that $Z$ is an algebraic space which satisfies the equivalent", "conditions of", "Lemma \\ref{lemma-unique-point-better}", "and such that the image of $|Z| \\to |X|$ is $\\{x\\}$." ], "refs": [ "decent-spaces-lemma-unique-point-better" ], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "Set $R = U \\times_X U$ so that $X = U/R$ is a presentation (see", "Spaces, Section \\ref{spaces-section-presentations}).", "Set", "$$", "U' = \\coprod\\nolimits_{u \\in U\\text{ lying over }x} \\Spec(\\kappa(u)).", "$$", "The canonical morphism $U' \\to U$ is a monomorphism. Let", "$$", "R' = U' \\times_X U' = R \\times_{(U \\times_S U)} (U' \\times_S U').", "$$", "Because $U' \\to U$ is a monomorphism we see that the projections", "$s', t' : R' \\to U'$ factor as a monomorphism followed by an", "\\'etale morphism. Hence, as $U'$ is a disjoint union of spectra", "of fields, using", "Remark \\ref{remark-recall},", "and using", "Schemes, Lemma \\ref{schemes-lemma-mono-towards-spec-field}", "we conclude that $R'$ is a disjoint union of spectra of fields and", "that the morphisms $s', t' : R' \\to U'$ are \\'etale. Hence", "$Z = U'/R'$ is an algebraic space by", "Spaces, Theorem \\ref{spaces-theorem-presentation}.", "As $R'$ is the restriction of $R$ by $U' \\to U$ we see", "$Z \\to X$ is a monomorphism by", "Groupoids, Lemma \\ref{groupoids-lemma-quotient-groupoid-restrict}.", "Since $Z \\to X$ is a monomorphism we see that $|Z| \\to |X|$ is injective, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-monomorphism-injective-points}.", "By", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}", "we see that", "$$", "|U'| = |Z \\times_X U'| \\to |Z| \\times_{|X|} |U'|", "$$", "is surjective which implies (by our choice of $U'$) that", "$|Z| \\to |X|$ has image $\\{x\\}$. We conclude that $|Z|$ is a singleton.", "Finally, by construction $U'$ is locally Noetherian and reduced, i.e.,", "we see that $Z$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-unique-point-better}.", "\\medskip\\noindent", "Let us prove uniqueness of $Z \\to X$. Suppose that", "$Z' \\to X$ is a second such monomorphism of algebraic spaces.", "Then the projections", "$$", "Z' \\longleftarrow Z' \\times_X Z \\longrightarrow Z", "$$", "are monomorphisms. The algebraic space in the middle is nonempty by", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-points-cartesian}.", "Hence the two projections are isomorphisms by", "Lemma \\ref{lemma-monomorphism-into-point}", "and we win." ], "refs": [ "decent-spaces-remark-recall", "schemes-lemma-mono-towards-spec-field", "spaces-theorem-presentation", "groupoids-lemma-quotient-groupoid-restrict", "spaces-morphisms-lemma-monomorphism-injective-points", "spaces-properties-lemma-points-cartesian", "decent-spaces-lemma-unique-point-better", "spaces-properties-lemma-points-cartesian", "decent-spaces-lemma-monomorphism-into-point" ], "ref_ids": [ 9573, 7729, 8124, 9651, 4758, 11819, 9502, 11819, 9503 ] } ], "ref_ids": [ 9502 ] }, { "id": 9505, "type": "theorem", "label": "decent-spaces-lemma-residual-space-regular", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-residual-space-regular", "contents": [ "A reduced, locally Noetherian singleton algebraic space $Z$ is regular." ], "refs": [], "proofs": [ { "contents": [ "Let $Z$ be a reduced, locally Noetherian singleton algebraic space", "over a scheme $S$. Let $W \\to Z$ be a surjective \\'etale morphism where $W$", "is a scheme. Let $k$ be a field and let $\\Spec(k) \\to Z$", "be surjective, flat, and locally of finite presentation (see", "Lemma \\ref{lemma-unique-point-better}).", "The scheme $T = W \\times_Z \\Spec(k)$ is", "\\'etale over $k$ in particular regular, see", "Remark \\ref{remark-recall}.", "Since $T \\to W$ is locally of finite presentation, flat, and surjective it", "follows that $W$ is regular, see", "Descent, Lemma \\ref{descent-lemma-descend-regular}.", "By definition this means that $Z$ is regular." ], "refs": [ "decent-spaces-lemma-unique-point-better", "decent-spaces-remark-recall", "descent-lemma-descend-regular" ], "ref_ids": [ 9502, 9573, 14659 ] } ], "ref_ids": [] }, { "id": 9506, "type": "theorem", "label": "decent-spaces-lemma-locally-Noetherian-decent-quasi-separated", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-locally-Noetherian-decent-quasi-separated", "contents": [ "Any locally Noetherian decent algebraic space is quasi-separated." ], "refs": [], "proofs": [ { "contents": [ "Namely, let $X$ be an algebraic space (over some base scheme, for", "example over $\\mathbf{Z}$) which is decent and locally Noetherian.", "Let $U \\to X$ and $V \\to X$ be \\'etale morphisms with $U$ and $V$", "affine schemes. We have to show that $W = U \\times_X V$ is quasi-compact", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-characterize-quasi-separated}).", "Since $X$ is locally Noetherian, the schemes $U$, $V$ are Noetherian", "and $W$ is locally Noetherian. Since $X$ is decent, the fibres", "of the morphism $W \\to U$ are finite. Namely, we can represent", "any $x \\in |X|$ by a quasi-compact monomorphism $\\Spec(k) \\to X$.", "Then $U_k$ and $V_k$ are finite disjoint unions of spectra of", "finite separable extensions of $k$ (Remark \\ref{remark-recall})", "and we see that $W_k = U_k \\times_{\\Spec(k)} V_k$ is finite.", "Let $n$ be the maximum degree of a fibre of $W \\to U$ at a generic", "point of an irreducible component of $U$. Consider the stratification", "$$", "U = U_0 \\supset U_1 \\supset U_2 \\supset \\ldots", "$$", "associated to $W \\to U$ in", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-stratify-flat-fp-lqf}.", "By our choice of $n$ above we conclude that $U_{n + 1}$ is empty.", "Hence we see that the fibres of $W \\to U$ are universally bounded.", "Then we can apply More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-stratify-flat-fp-lqf-universally-bounded}", "to find a stratification", "$$", "\\emptyset = Z_{-1} \\subset Z_0 \\subset Z_1 \\subset Z_2 \\subset", "\\ldots \\subset Z_n = U", "$$", "by closed subsets such that with $S_r = Z_r \\setminus Z_{r - 1}$", "the morphism $W \\times_U S_r \\to S_r$ is finite locally free.", "Since $U$ is Noetherian, the schemes $S_r$ are Noetherian,", "whence the schemes $W \\times_U S_r$ are Noetherian, whence", "$W = \\coprod W \\times_U S_r$ is quasi-compact as desired." ], "refs": [ "spaces-properties-lemma-characterize-quasi-separated", "decent-spaces-remark-recall", "more-morphisms-lemma-stratify-flat-fp-lqf", "more-morphisms-lemma-stratify-flat-fp-lqf-universally-bounded" ], "ref_ids": [ 11816, 9573, 13910, 13908 ] } ], "ref_ids": [] }, { "id": 9507, "type": "theorem", "label": "decent-spaces-lemma-when-field", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-when-field", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "\\begin{enumerate}", "\\item If $|X|$ is a singleton then $X$ is a scheme.", "\\item If $|X|$ is a singleton and $X$ is reduced, then", "$X \\cong \\Spec(k)$ for some field $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $|X|$ is a singleton. It follows immediately from", "Theorem \\ref{theorem-decent-open-dense-scheme} that $X$ is a scheme,", "but we can also argue directly as follows.", "Choose an affine scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "Set $R = U \\times_X U$. Then $U$ and $R$ have finitely many points by", "Lemma \\ref{lemma-UR-finite-above-x} (and the definition of a decent space).", "All of these points are closed in $U$ and $R$ by", "Lemma \\ref{lemma-decent-no-specializations-map-to-same-point}.", "It follows that $U$ and $R$ are affine schemes.", "We may shrink $U$ to a singleton space. Then $U$ is", "the spectrum of a henselian local ring, see", "Algebra, Lemma \\ref{algebra-lemma-local-dimension-zero-henselian}.", "The projections $R \\to U$ are \\'etale, hence finite \\'etale because", "$U$ is the spectrum of a $0$-dimensional henselian local ring, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-henselian}.", "It follows that $X$ is a scheme by", "Groupoids, Proposition \\ref{groupoids-proposition-finite-flat-equivalence}.", "\\medskip\\noindent", "Part (2) follows from (1) and the fact that a reduced singleton", "scheme is the spectrum of a field." ], "refs": [ "decent-spaces-theorem-decent-open-dense-scheme", "decent-spaces-lemma-UR-finite-above-x", "decent-spaces-lemma-decent-no-specializations-map-to-same-point", "algebra-lemma-local-dimension-zero-henselian", "algebra-lemma-characterize-henselian", "groupoids-proposition-finite-flat-equivalence" ], "ref_ids": [ 9454, 9463, 9493, 1283, 1276, 9669 ] } ], "ref_ids": [] }, { "id": 9508, "type": "theorem", "label": "decent-spaces-lemma-algebraic-residue-field-extension-closed-point", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-algebraic-residue-field-extension-closed-point", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Consider a commutative diagram", "$$", "\\xymatrix{", "\\Spec(k) \\ar[rr] \\ar[rd] & & X \\ar[ld] \\\\", "& S", "}", "$$", "Assume that the image point $s \\in S$ of $\\Spec(k) \\to S$ is", "a closed point and that $\\kappa(s) \\subset k$ is algebraic.", "Then the image $x$ of $\\Spec(k) \\to X$ is a closed point of $|X|$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $x \\leadsto x'$ for some $x' \\in |X|$. Choose an", "\\'etale morphism $U \\to X$ where $U$ is a scheme and a point $u' \\in U'$", "mapping to $x'$. Choose a specialization $u \\leadsto u'$ in $U$ with $u$", "mapping to $x$ in $X$, see Lemma \\ref{lemma-decent-specialization}.", "Then $u$ is the image of a point $w$ of the scheme", "$W = \\Spec(k) \\times_X U$. Since the projection $W \\to \\Spec(k)$ is \\'etale", "we see that $\\kappa(w) \\supset k$ is finite. Hence", "$\\kappa(w) \\supset \\kappa(s)$ is algebraic. Hence $\\kappa(u) \\supset \\kappa(s)$", "is algebraic. Thus $u$ is a closed point of $U$ by", "Morphisms, Lemma", "\\ref{morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre}.", "Thus $u = u'$, whence $x = x'$." ], "refs": [ "decent-spaces-lemma-decent-specialization", "morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre" ], "ref_ids": [ 9494, 5222 ] } ], "ref_ids": [] }, { "id": 9509, "type": "theorem", "label": "decent-spaces-lemma-finite-residue-field-extension-finite", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-finite-residue-field-extension-finite", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Consider a commutative diagram", "$$", "\\xymatrix{", "\\Spec(k) \\ar[rr] \\ar[rd] & & X \\ar[ld] \\\\", "& S", "}", "$$", "Assume that the image point $s \\in S$ of $\\Spec(k) \\to S$ is", "a closed point and that $\\kappa(s) \\subset k$ is finite.", "Then $\\Spec(k) \\to X$ is finite morphism. If $\\kappa(s) = k$", "then $\\Spec(k) \\to X$ is closed immersion." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-algebraic-residue-field-extension-closed-point}", "the image point $x \\in |X|$ is closed. Let $Z \\subset X$ be the", "reduced closed subspace with $|Z| = \\{x\\}$ (Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-reduced-closed-subspace}).", "Note that $Z$ is a decent algebraic space by", "Lemma \\ref{lemma-representable-named-properties}.", "By Lemma \\ref{lemma-when-field} we see that $Z = \\Spec(k')$", "for some field $k'$. Of course $k \\supset k' \\supset \\kappa(s)$.", "Then $\\Spec(k) \\to Z$ is a finite morphism of schemes", "and $Z \\to X$ is a finite morphism as it is a closed immersion.", "Hence $\\Spec(k) \\to X$ is finite (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-composition-integral}).", "If $k = \\kappa(s)$, then $\\Spec(k) = Z$ and $\\Spec(k) \\to X$", "is a closed immersion." ], "refs": [ "decent-spaces-lemma-algebraic-residue-field-extension-closed-point", "spaces-properties-lemma-reduced-closed-subspace", "decent-spaces-lemma-representable-named-properties", "decent-spaces-lemma-when-field", "spaces-morphisms-lemma-composition-integral" ], "ref_ids": [ 9508, 11846, 9470, 9507, 4941 ] } ], "ref_ids": [] }, { "id": 9510, "type": "theorem", "label": "decent-spaces-lemma-decent-space-closed-point", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-decent-space-closed-point", "contents": [ "Let $S$ be a scheme. Suppose $X$ is a decent algebraic space over $S$.", "Let $x \\in |X|$ be a closed point. Then $x$ can be represented by a", "closed immersion $i : \\Spec(k) \\to X$ from the spectrum of a field." ], "refs": [], "proofs": [ { "contents": [ "We know that $x$ can be represented by a quasi-compact monomorphism", "$i : \\Spec(k) \\to X$ where $k$ is a field", "(Definition \\ref{definition-very-reasonable}).", "Let $U \\to X$ be an \\'etale morphism where $U$ is an affine scheme.", "As $x$ is closed and $X$ decent, the fibre $F$ of $|U| \\to |X|$ over $x$", "consists of closed points", "(Lemma \\ref{lemma-decent-no-specializations-map-to-same-point}).", "As $i$ is a monomorphism, so is $U_k = U \\times_X \\Spec(k) \\to U$.", "In particular, the map $|U_k| \\to F$ is injective. Since $U_k$", "is quasi-compact and \\'etale over a field, we see that $U_k$ is a", "finite disjoint union of spectra of fields (Remark \\ref{remark-recall}).", "Say $U_k = \\Spec(k_1) \\amalg \\ldots \\amalg \\Spec(k_r)$.", "Since $\\Spec(k_i) \\to U$ is a monomorphism, we see that", "its image $u_i$ has residue field $\\kappa(u_i) = k_i$.", "Since $u_i \\in F$ is a closed point we conclude the morphism", "$\\Spec(k_i) \\to U$ is a closed immersion. As the $u_i$ are pairwise distinct,", "$U_k \\to U$ is a closed immersion. Hence $i$ is a closed immersion", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-closed-immersion-local}). This finishes the proof." ], "refs": [ "decent-spaces-definition-very-reasonable", "decent-spaces-lemma-decent-no-specializations-map-to-same-point", "decent-spaces-remark-recall", "spaces-morphisms-lemma-closed-immersion-local" ], "ref_ids": [ 9562, 9493, 9573, 4761 ] } ], "ref_ids": [] }, { "id": 9511, "type": "theorem", "label": "decent-spaces-lemma-infinite-number", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-infinite-number", "contents": [ "Let $A$ be a ring. Let $k$ be a field. Let $\\mathfrak p_n$, $n \\geq 1$", "be a sequence of pairwise distinct primes of $A$. Moreover, for each", "$n$ let $k \\to \\kappa(\\mathfrak p_n)$ be an embedding. Then the closure", "of the image of", "$$", "\\coprod\\nolimits_{n \\not = m}", "\\Spec(\\kappa(\\mathfrak p_n) \\otimes_k \\kappa(\\mathfrak p_m))", "\\longrightarrow", "\\Spec(A \\otimes A)", "$$", "meets the diagonal." ], "refs": [], "proofs": [ { "contents": [ "Set $k_n = \\kappa(\\mathfrak p_n)$. We may assume that $A = \\prod k_n$.", "Denote $x_n = \\Spec(k_n)$ the open and closed point corresponding to", "$A \\to k_n$. Then $\\Spec(A) = Z \\amalg \\{x_n\\}$ where $Z$ is a nonempty", "closed subset. Namely, $Z = V(e_n; n \\geq 1)$ where $e_n$", "is the idempotent of $A$ corresponding to the factor $k_n$", "and $Z$ is nonempty as the ideal generated by the $e_n$ is not", "equal to $A$. We will show that the closure of the image", "contains $\\Delta(Z)$. The kernel of the map", "$$", "(\\prod k_n) \\otimes_k (\\prod k_m)", "\\longrightarrow", "\\prod\\nolimits_{n \\not = m} k_n \\otimes_k k_m", "$$", "is the ideal generated by $e_n \\otimes e_n$, $n \\geq 1$.", "Hence the closure of the image of the map on spectra is ", "$V(e_n \\otimes e_n; n \\geq 1)$ whose intersection with $\\Delta(\\Spec(A))$", "is $\\Delta(Z)$. Thus it suffices to show that", "$$", "\\coprod\\nolimits_{n \\not = m} \\Spec(k_n \\otimes_k k_m)", "\\longrightarrow", "\\Spec(\\prod\\nolimits_{n \\not = m} k_n \\otimes_k k_m)", "$$", "has dense image. This follows as the family of ring maps", "$\\prod_{n \\not = m} k_n \\otimes_k k_m \\to k_n \\otimes_k k_m$", "is jointly injective." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9512, "type": "theorem", "label": "decent-spaces-lemma-locally-separated-decent", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-locally-separated-decent", "contents": [ "A locally separated algebraic space is decent." ], "refs": [], "proofs": [ { "contents": [ "Let $S$ be a scheme and let $X$ be a locally separated algebraic space", "over $S$. We may assume $S = \\Spec(\\mathbf{Z})$, see", "Properties of Spaces, Definition \\ref{spaces-properties-definition-separated}.", "Unadorned fibre products will be over $\\mathbf{Z}$.", "Let $x \\in |X|$. Choose a scheme $U$, an \\'etale", "morphism $U \\to X$, and a point $u \\in U$ mapping to $x$ in $|X|$.", "As usual we identify $u = \\Spec(\\kappa(u))$.", "As $X$ is locally separated the morphism", "$$", "u \\times_X u \\to u \\times u", "$$", "is an immersion (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-fibre-product-after-map}).", "Hence More on Groupoids, Lemma", "\\ref{more-groupoids-lemma-locally-closed-image-is-closed}", "tells us that it is a closed immersion (use", "Schemes, Lemma \\ref{schemes-lemma-immersion-when-closed}).", "As $u \\times_X u \\to u \\times_X U$ is a monomorphism (base change", "of $u \\to U$) and as $u \\times_X U \\to u$ is \\'etale we conclude that", "$u \\times_X u$ is a disjoint union of spectra of fields", "(see Remark \\ref{remark-recall} and", "Schemes, Lemma \\ref{schemes-lemma-mono-towards-spec-field}).", "Since it is also closed in the affine scheme $u \\times u$ we", "conclude $u \\times_X u$ is a finite disjoint union of spectra of fields.", "Thus $x$ can be represented by a monomorphism $\\Spec(k) \\to X$ where $k$", "is a field, see", "Lemma \\ref{lemma-R-finite-above-x}.", "\\medskip\\noindent", "Next, let $U = \\Spec(A)$ be an affine scheme and let $U \\to X$ be an", "\\'etale morphism. To finish the proof it suffices to show that", "$F = U \\times_X \\Spec(k)$ is finite. Write $F = \\coprod_{i \\in I} \\Spec(k_i)$", "as the disjoint union of finite separable extensions of $k$.", "We have to show that $I$ is finite.", "Set $R = U \\times_X U$. As $X$ is locally separated, the morphism", "$j : R \\to U \\times U$ is an immersion. Let $U' \\subset U \\times U$", "be an open such that $j$ factors through a closed immersion $j' : R \\to U'$.", "Let $e : U \\to R$ be the diagonal map. Using that $e$ is a morphism between", "schemes \\'etale over $U$ such that $\\Delta = j \\circ e$ is a", "closed immersion, we conclude that $R = e(U) \\amalg W$ for some", "open and closed subscheme $W \\subset R$. Since $j'$ is a closed immersion", "we conclude that $j'(W) \\subset U'$ is closed and disjoint from", "$j'(e(U))$. Therefore", "$\\overline{j(W)} \\cap \\Delta(U) = \\emptyset$ in $U \\times U$.", "Note that $W$ contains $\\Spec(k_i \\otimes_k k_{i'})$ for all", "$i \\not = i'$, $i, i' \\in I$. By Lemma \\ref{lemma-infinite-number}", "we conclude that $I$ is finite as desired." ], "refs": [ "spaces-properties-definition-separated", "spaces-morphisms-lemma-fibre-product-after-map", "more-groupoids-lemma-locally-closed-image-is-closed", "schemes-lemma-immersion-when-closed", "decent-spaces-remark-recall", "schemes-lemma-mono-towards-spec-field", "decent-spaces-lemma-R-finite-above-x", "decent-spaces-lemma-infinite-number" ], "ref_ids": [ 11922, 4715, 2483, 7671, 9573, 7729, 9461, 9511 ] } ], "ref_ids": [] }, { "id": 9513, "type": "theorem", "label": "decent-spaces-lemma-properties-trivial-implications", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-properties-trivial-implications", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "We have the following implications among the conditions on $f$:", "$$", "\\xymatrix{", "\\text{representable} \\ar@{=>}[rd] & & & & \\\\", "& \\text{very reasonable} \\ar@{=>}[r] & \\text{reasonable} \\ar@{=>}[r] &", "\\text{decent} \\ar@{=>}[r] & (\\beta) \\\\", "\\text{quasi-separated} \\ar@{=>}[ru] & & & &", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions,", "Lemma \\ref{lemma-bounded-fibres}", "and", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-separated-local}." ], "refs": [ "decent-spaces-lemma-bounded-fibres", "spaces-morphisms-lemma-separated-local" ], "ref_ids": [ 9466, 4722 ] } ], "ref_ids": [] }, { "id": 9514, "type": "theorem", "label": "decent-spaces-lemma-property-for-morphism-out-of-property", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-property-for-morphism-out-of-property", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. If $X$ is decent (resp.\\ is reasonable, resp.\\ has property", "$(\\beta)$ of Lemma \\ref{lemma-bounded-fibres}), then $f$ is", "decent (resp.\\ reasonable, resp.\\ has property $(\\beta)$)." ], "refs": [ "decent-spaces-lemma-bounded-fibres" ], "proofs": [ { "contents": [ "Let $T$ be a scheme and let $T \\to Y$ be a morphism. Then $T \\to Y$", "is representable, hence the base change $T \\times_Y X \\to X$ is representable.", "Hence if $X$ is decent (or reasonable), then so is $T \\times_Y X$, see", "Lemma \\ref{lemma-representable-named-properties}.", "Similarly, for property $(\\beta)$, see", "Lemma \\ref{lemma-representable-properties}." ], "refs": [ "decent-spaces-lemma-representable-named-properties", "decent-spaces-lemma-representable-properties" ], "ref_ids": [ 9470, 9468 ] } ], "ref_ids": [ 9466 ] }, { "id": 9515, "type": "theorem", "label": "decent-spaces-lemma-base-change-relative-conditions", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-base-change-relative-conditions", "contents": [ "Having property $(\\beta)$, being decent, or being reasonable", "is preserved under arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9516, "type": "theorem", "label": "decent-spaces-lemma-property-over-property", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-property-over-property", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\omega \\in \\{\\beta, decent, reasonable\\}$.", "Suppose that $Y$ has property $(\\omega)$ and $f : X \\to Y$ has $(\\omega)$.", "Then $X$ has $(\\omega)$." ], "refs": [], "proofs": [ { "contents": [ "Let us prove the lemma in case $\\omega = \\beta$. In this case we have to show", "that any $x \\in |X|$ is represented by a monomorphism from the spectrum", "of a field into $X$. Let $y = f(x) \\in |Y|$. By assumption there exists", "a field $k$ and a monomorphism $\\Spec(k) \\to Y$ representing $y$.", "Then $x$ corresponds to a point $x'$ of $\\Spec(k) \\times_Y X$.", "By assumption $x'$ is represented by a monomorphism", "$\\Spec(k') \\to \\Spec(k) \\times_Y X$. Clearly the composition", "$\\Spec(k') \\to X$ is a monomorphism representing $x$.", "\\medskip\\noindent", "Let us prove the lemma in case $\\omega = decent$.", "Let $x \\in |X|$ and $y = f(x) \\in |Y|$. By the result of the preceding", "paragraph we can choose a diagram", "$$", "\\xymatrix{", "\\Spec(k') \\ar[r]_x \\ar[d] & X \\ar[d]^f \\\\", "\\Spec(k) \\ar[r]^y & Y", "}", "$$", "whose horizontal arrows monomorphisms. As $Y$ is decent the morphism", "$y$ is quasi-compact. As $f$ is decent the algebraic space", "$\\Spec(k) \\times_Y X$ is decent. Hence the monomorphism", "$\\Spec(k') \\to \\Spec(k) \\times_Y X$ is quasi-compact.", "Then the monomorphism $x : \\Spec(k') \\to X$ is quasi-compact", "as a composition of quasi-compact morphisms (use", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-quasi-compact} and", "\\ref{spaces-morphisms-lemma-composition-quasi-compact}).", "As the point $x$ was arbitrary this implies $X$ is decent.", "\\medskip\\noindent", "Let us prove the lemma in case $\\omega = reasonable$.", "Choose $V \\to Y$ \\'etale with $V$ an affine scheme.", "Choose $U \\to V \\times_Y X$ \\'etale with $U$ an affine scheme.", "By assumption $V \\to Y$ has universally bounded fibres. By", "Lemma \\ref{lemma-base-change-universally-bounded}", "the morphism $V \\times_Y X \\to X$ has universally bounded fibres.", "By assumption on $f$ we see that $U \\to V \\times_Y X$ has", "universally bounded fibres. By", "Lemma \\ref{lemma-composition-universally-bounded}", "the composition $U \\to X$ has universally bounded fibres.", "Hence there exists sufficiently many \\'etale morphisms $U \\to X$", "from schemes with universally bounded fibres, and we conclude", "that $X$ is reasonable." ], "refs": [ "spaces-morphisms-lemma-base-change-quasi-compact", "spaces-morphisms-lemma-composition-quasi-compact", "decent-spaces-lemma-base-change-universally-bounded", "decent-spaces-lemma-composition-universally-bounded" ], "ref_ids": [ 4738, 4739, 9456, 9455 ] } ], "ref_ids": [] }, { "id": 9517, "type": "theorem", "label": "decent-spaces-lemma-composition-relative-conditions", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-composition-relative-conditions", "contents": [ "Having property $(\\beta)$, being decent, or being reasonable", "is preserved under compositions." ], "refs": [], "proofs": [ { "contents": [ "Let $\\omega \\in \\{\\beta, decent, reasonable\\}$.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces", "over the scheme $S$. Assume $f$ and $g$ both have property", "$(\\omega)$. Then we have to show", "that for any scheme $T$ and morphism $T \\to Z$ the space $T \\times_Z X$", "has $(\\omega)$. By", "Lemma \\ref{lemma-base-change-relative-conditions}", "this reduces us to the following claim: Suppose that $Y$ is an algebraic", "space having property $(\\omega)$, and that $f : X \\to Y$ is a morphism", "with $(\\omega)$. Then $X$ has $(\\omega)$.", "This is the content of Lemma \\ref{lemma-property-over-property}." ], "refs": [ "decent-spaces-lemma-base-change-relative-conditions", "decent-spaces-lemma-property-over-property" ], "ref_ids": [ 9515, 9516 ] } ], "ref_ids": [] }, { "id": 9518, "type": "theorem", "label": "decent-spaces-lemma-fibre-product-conditions", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-fibre-product-conditions", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$, $g : Z \\to Y$ be morphisms", "of algebraic spaces over $S$. If $X$ and $Z$ are decent", "(resp.\\ reasonable, resp.\\ have property ", "$(\\beta)$ of Lemma \\ref{lemma-bounded-fibres}), then so does $X \\times_Y Z$." ], "refs": [ "decent-spaces-lemma-bounded-fibres" ], "proofs": [ { "contents": [ "Namely, by Lemma \\ref{lemma-property-for-morphism-out-of-property}", "the morphism $X \\to Y$ has the property. Then the base change", "$X \\times_Y Z \\to Z$ has the property by", "Lemma \\ref{lemma-base-change-relative-conditions}.", "And finally this implies $X \\times_Y Z$ has the", "property by Lemma \\ref{lemma-property-over-property}." ], "refs": [ "decent-spaces-lemma-property-for-morphism-out-of-property", "decent-spaces-lemma-base-change-relative-conditions", "decent-spaces-lemma-property-over-property" ], "ref_ids": [ 9514, 9515, 9516 ] } ], "ref_ids": [ 9466 ] }, { "id": 9519, "type": "theorem", "label": "decent-spaces-lemma-descent-conditions", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-descent-conditions", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{P} \\in \\{(\\beta), decent, reasonable\\}$.", "Assume", "\\begin{enumerate}", "\\item $f$ is quasi-compact,", "\\item $f$ is \\'etale,", "\\item $|f| : |X| \\to |Y|$ is surjective, and", "\\item the algebraic space $X$ has property $\\mathcal{P}$.", "\\end{enumerate}", "Then $Y$ has property $\\mathcal{P}$." ], "refs": [], "proofs": [ { "contents": [ "Let us prove this in case $\\mathcal{P} = (\\beta)$. Let $y \\in |Y|$ be", "a point. We have to show that $y$ can be represented by a monomorphism", "from a field. Choose a point $x \\in |X|$ with $f(x) = y$.", "By assumption we may represent $x$ by a monomorphism", "$\\Spec(k) \\to X$, with $k$ a field. By", "Lemma \\ref{lemma-R-finite-above-x}", "it suffices to show that the projections", "$\\Spec(k) \\times_Y \\Spec(k) \\to \\Spec(k)$", "are \\'etale and quasi-compact. We can factor the first projection as", "$$", "\\Spec(k) \\times_Y \\Spec(k)", "\\longrightarrow", "\\Spec(k) \\times_Y X", "\\longrightarrow", "\\Spec(k)", "$$", "The first morphism is a monomorphism, and the second is \\'etale and", "quasi-compact. By", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-etale-over-field-scheme}", "we see that $\\Spec(k) \\times_Y X$ is a scheme. Hence it is a", "finite disjoint union of spectra of finite separable field extensions", "of $k$. By", "Schemes, Lemma \\ref{schemes-lemma-mono-towards-spec-field}", "we see that the first arrow identifies", "$\\Spec(k) \\times_Y \\Spec(k)$ with a finite disjoint", "union of spectra of finite separable field extensions of $k$.", "Hence the projection morphism is \\'etale and quasi-compact.", "\\medskip\\noindent", "Let us prove this in case $\\mathcal{P} = decent$.", "We have already seen in the first paragraph of the proof that this implies", "that every $y \\in |Y|$ can be represented by a monomorphism", "$y : \\Spec(k) \\to Y$. Pick such a $y$. Pick an affine", "scheme $U$ and an \\'etale morphism $U \\to X$ such that the image", "of $|U| \\to |Y|$ contains $y$. By", "Lemma \\ref{lemma-UR-finite-above-x}", "it suffices to show that $U_y$ is a finite scheme over $k$. The fibre", "product $X_y = \\Spec(k) \\times_Y X$ is a quasi-compact \\'etale", "algebraic space over $k$. Hence by", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-etale-over-field-scheme}", "it is a scheme. So it is a finite disjoint union of spectra of", "finite separable extensions of $k$. Say $X_y = \\{x_1, \\ldots, x_n\\}$", "so $x_i$ is given by $x_i : \\Spec(k_i) \\to X$ with", "$[k_i : k] < \\infty$. By assumption $X$ is decent, so the schemes", "$U_{x_i} = \\Spec(k_i) \\times_X U$ are finite over $k_i$.", "Finally, we note that $U_y = \\coprod U_{x_i}$ as a scheme and we conclude", "that $U_y$ is finite over $k$ as desired.", "\\medskip\\noindent", "Let us prove this in case $\\mathcal{P} = reasonable$.", "Pick an affine scheme $V$ and an \\'etale morphism $V \\to Y$.", "We have the show the fibres of $V \\to Y$ are universally bounded.", "The algebraic space $V \\times_Y X$ is quasi-compact.", "Thus we can find an affine scheme $W$ and a surjective \\'etale morphism", "$W \\to V \\times_Y X$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-quasi-compact-affine-cover}.", "Here is a picture (solid diagram)", "$$", "\\xymatrix{", "W \\ar[r] \\ar[rd] &", "V \\times_Y X \\ar[r] \\ar[d] &", "X \\ar[d]_f & \\Spec(k) \\ar@{..>}[l]^x \\ar@{..>}[ld]^y \\\\", " & V \\ar[r] & Y", "}", "$$", "The morphism $W \\to X$ is universally bounded by our assumption that", "the space $X$ is reasonable. Let $n$ be an integer bounding", "the degrees of the fibres of $W \\to X$. We claim that the same integer", "works for bounding the fibres of $V \\to Y$. Namely, suppose $y \\in |Y|$", "is a point. Then there exists a $x \\in |X|$ with $f(x) = y$ (see above).", "This means we can find a field $k$ and morphisms $x, y$ given as dotted", "arrows in the diagram above. In particular we get a surjective \\'etale", "morphism", "$$", "\\Spec(k) \\times_{x, X} W", "\\to", "\\Spec(k) \\times_{x, X} (V \\times_Y X) = \\Spec(k) \\times_{y, Y} V", "$$", "which shows that the degree of $\\Spec(k) \\times_{y, Y} V$ over $k$", "is less than or equal to the degree of $\\Spec(k) \\times_{x, X} W$", "over $k$, i.e., $\\leq n$, and we win. (This last part of the argument", "is the same as the argument in the proof of", "Lemma \\ref{lemma-descent-universally-bounded}.", "Unfortunately that lemma is not general enough because it only applies", "to representable morphisms.)" ], "refs": [ "decent-spaces-lemma-R-finite-above-x", "spaces-properties-lemma-etale-over-field-scheme", "schemes-lemma-mono-towards-spec-field", "decent-spaces-lemma-UR-finite-above-x", "spaces-properties-lemma-etale-over-field-scheme", "spaces-properties-lemma-quasi-compact-affine-cover", "decent-spaces-lemma-descent-universally-bounded" ], "ref_ids": [ 9461, 11861, 7729, 9463, 11861, 11832, 9457 ] } ], "ref_ids": [] }, { "id": 9520, "type": "theorem", "label": "decent-spaces-lemma-relative-conditions-local", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-relative-conditions-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{P} \\in \\{(\\beta), decent, reasonable, very\\ reasonable\\}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is $\\mathcal{P}$,", "\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the", "base change $Z \\times_Y X \\to Z$ of $f$ is $\\mathcal{P}$,", "\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the", "algebraic space $Z \\times_Y X$ is $\\mathcal{P}$, and", "\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such", "that each morphism $f^{-1}(Y_i) \\to Y_i$ has $\\mathcal{P}$.", "\\end{enumerate}", "If $\\mathcal{P} \\in \\{(\\beta), decent, reasonable\\}$, then this is also", "equivalent to", "\\begin{enumerate}", "\\item[(5)] there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that the base change $V \\times_Y X \\to V$ has", "$\\mathcal{P}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implications (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3) $\\Rightarrow$ (4)", "are trivial.", "The implication (3) $\\Rightarrow$ (1) can be seen as follows.", "Let $Z \\to Y$ be a morphism whose source is a scheme over $S$.", "Consider the algebraic space $Z \\times_Y X$. If we assume (3), then", "for any affine open $W \\subset Z$, the open subspace", "$W \\times_Y X$ of $Z \\times_Y X$ has property $\\mathcal{P}$. Hence by", "Lemma \\ref{lemma-properties-local}", "the space $Z \\times_Y X$ has property $\\mathcal{P}$, i.e., (1) holds.", "A similar argument (omitted) shows that (4) implies (1).", "\\medskip\\noindent", "The implication (1) $\\Rightarrow$ (5) is trivial. Let $V \\to Y$ be", "an \\'etale morphism from a scheme as in (5). Let $Z$ be an affine scheme,", "and let $Z \\to Y$ be a morphism. Consider the diagram", "$$", "\\xymatrix{", "Z \\times_Y V \\ar[r]_q \\ar[d]_p & V \\ar[d] \\\\", "Z \\ar[r] & Y", "}", "$$", "Since $p$ is \\'etale, and hence open, we can choose finitely many affine open", "subschemes $W_i \\subset Z \\times_Y V$ such that $Z = \\bigcup p(W_i)$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "V \\times_Y X \\ar[d] &", "(\\coprod W_i) \\times_Y X \\ar[l] \\ar[d] \\ar[r] &", "Z \\times_Y X \\ar[d] \\\\", "V &", "\\coprod W_i \\ar[l] \\ar[r] &", "Z", "}", "$$", "We know $V \\times_Y X$ has property $\\mathcal{P}$. By", "Lemma \\ref{lemma-representable-properties}", "we see that $(\\coprod W_i) \\times_Y X$ has property $\\mathcal{P}$.", "Note that the morphism $(\\coprod W_i) \\times_Y X \\to Z \\times_Y X$", "is \\'etale and quasi-compact as the base change of $\\coprod W_i \\to Z$.", "Hence by Lemma \\ref{lemma-descent-conditions}", "we conclude that $Z \\times_Y X$ has property $\\mathcal{P}$." ], "refs": [ "decent-spaces-lemma-properties-local", "decent-spaces-lemma-representable-properties", "decent-spaces-lemma-descent-conditions" ], "ref_ids": [ 9467, 9468, 9519 ] } ], "ref_ids": [] }, { "id": 9521, "type": "theorem", "label": "decent-spaces-lemma-re-characterize-universally-closed", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-re-characterize-universally-closed", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume $f$ is quasi-compact and decent.", "(For example if $f$ is representable, or quasi-separated, see", "Lemma \\ref{lemma-properties-trivial-implications}.)", "Then $f$ is universally closed if and only if the", "existence part of the valuative criterion holds." ], "refs": [ "decent-spaces-lemma-properties-trivial-implications" ], "proofs": [ { "contents": [ "In", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-quasi-compact-existence-universally-closed}", "we proved that any quasi-compact morphism which satisfies the existence", "part of the valuative criterion is universally closed.", "To prove the other, assume that $f$ is universally closed.", "In the proof of", "Proposition \\ref{proposition-characterize-universally-closed}", "we have seen that it suffices to show, for any valuation ring $A$,", "and any morphism $\\Spec(A) \\to Y$, that the base change", "$f_A : X_A \\to \\Spec(A)$ satisfies the existence part of the valuative", "criterion. By definition the algebraic space $X_A$ has property $(\\gamma)$", "and hence", "Proposition \\ref{proposition-characterize-universally-closed}", "applies to the morphism $f_A$ and we win." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-existence-universally-closed", "decent-spaces-proposition-characterize-universally-closed", "decent-spaces-proposition-characterize-universally-closed" ], "ref_ids": [ 4930, 9560, 9560 ] } ], "ref_ids": [ 9513 ] }, { "id": 9522, "type": "theorem", "label": "decent-spaces-lemma-surjective-on-fibres", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-surjective-on-fibres", "contents": [ "In the situation of (\\ref{equation-points-fibres}) if $Z' \\to Z$", "is a morphism and $z' \\in |Z'|$ maps to $z$, then the induced map", "$F_{x, z'} \\to F_{x, z}$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Set $W' = X \\times_Y Z' = W \\times_Z Z'$. Then", "$|W'| \\to |W| \\times_{|Z|} |Z'|$ is surjective by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}.", "Hence the surjectivity of $F_{x, z'} \\to F_{x, z}$." ], "refs": [ "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 11819 ] } ], "ref_ids": [] }, { "id": 9523, "type": "theorem", "label": "decent-spaces-lemma-qf-and-qc-finite-fibre", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-qf-and-qc-finite-fibre", "contents": [ "In diagram (\\ref{equation-points-fibres}) the set (\\ref{equation-fibre})", "is finite if $f$ is of finite type and $f$ is quasi-finite at $x$." ], "refs": [], "proofs": [ { "contents": [ "The morphism $q$ is quasi-finite at every $w \\in F_{x, z}$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-quasi-finite-locus}.", "Hence the lemma follows from", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-finite-at-a-finite-number-of-points}." ], "refs": [ "spaces-morphisms-lemma-base-change-quasi-finite-locus", "spaces-morphisms-lemma-quasi-finite-at-a-finite-number-of-points" ], "ref_ids": [ 4830, 4837 ] } ], "ref_ids": [] }, { "id": 9524, "type": "theorem", "label": "decent-spaces-lemma-decent-finite-fibre", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-decent-finite-fibre", "contents": [ "In diagram (\\ref{equation-points-fibres}) the set (\\ref{equation-fibre})", "is finite if $y$ can be represented by a monomorphism $\\Spec(k) \\to Y$", "where $k$ is a field and $g$ is quasi-finite at $z$.", "(Special case: $Y$ is decent and $g$ is \\'etale.)" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-surjective-on-fibres} applied twice", "we may replace $Z$ by $Z_k = \\Spec(k) \\times_Y Z$ and", "$X$ by $X_k = \\Spec(k) \\times_Y X$. We may and do", "replace $Y$ by $\\Spec(k)$ as well. Note that $Z_k \\to \\Spec(k)$", "is quasi-finite at $z$ by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-quasi-finite-locus}.", "Choose a scheme $V$, a point $v \\in V$, and an \\'etale morphism", "$V \\to Z_k$ mapping $v$ to $z$. Choose a scheme $U$, a point $u \\in U$,", "and an \\'etale morphism $U \\to X_k$ mapping $u$ to $x$.", "Again by Lemma \\ref{lemma-surjective-on-fibres}", "it suffices to show $F_{u, v}$ is finite for the diagram", "$$", "\\xymatrix{", "U \\times_{\\Spec(k)} V \\ar[r] \\ar[d] & V \\ar[d] \\\\", "U \\ar[r] & \\Spec(k)", "}", "$$", "The morphism $V \\to \\Spec(k)$ is quasi-finite at $v$", "(follows from the general discussion in", "Morphisms of Spaces, Section \\ref{spaces-morphisms-section-local-source-target}", "and the definition of being quasi-finite at a point).", "At this point the finiteness follows from Example \\ref{example-schemes}.", "The parenthetical remark of the statement of the lemma follows", "from the fact that on decent spaces points are represented by", "monomorphisms from fields and from the fact that an \\'etale", "morphism of algebraic spaces is locally quasi-finite." ], "refs": [ "decent-spaces-lemma-surjective-on-fibres", "spaces-morphisms-lemma-base-change-quasi-finite-locus", "decent-spaces-lemma-surjective-on-fibres" ], "ref_ids": [ 9522, 4830, 9522 ] } ], "ref_ids": [] }, { "id": 9525, "type": "theorem", "label": "decent-spaces-lemma-topology-fibre", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-topology-fibre", "contents": [ "\\begin{slogan}", "Fibers of field points of algebraic spaces have the", "expected Zariski topologies.", "\\end{slogan}", "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$.", "Let $y \\in |Y|$ and assume that $y$ is represented by a quasi-compact", "monomorphism $\\Spec(k) \\to Y$. Then $|X_k| \\to |X|$ is a", "homeomorphism onto $f^{-1}(\\{y\\}) \\subset |X|$ with induced topology." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-open}", "and", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-monomorphism-injective-points}", "without further mention.", "Let $V \\to Y$ be an \\'etale morphism with $V$ affine such that there", "exists a $v \\in V$ mapping to $y$. Since $\\Spec(k) \\to Y$ is quasi-compact", "there are a finite number of points of $V$ mapping to $y$", "(Lemma \\ref{lemma-UR-finite-above-x}). After shrinking", "$V$ we may assume $v$ is the only one. Choose a scheme $U$ and", "a surjective \\'etale morphism $U \\to X$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] & U_V \\ar[l] \\ar[d] & U_v \\ar[l] \\ar[d] \\\\", "X \\ar[d] & X_V \\ar[l] \\ar[d] & X_v \\ar[l] \\ar[d] \\\\", "Y & V \\ar[l] & v \\ar[l]", "}", "$$", "Since $U_v \\to U_V$ identifies $U_v$ with a subset of $U_V$ with", "the induced topology (Schemes, Lemma \\ref{schemes-lemma-fibre-topological}),", "and since $|U_V| \\to |X_V|$ and $|U_v| \\to |X_v|$ are surjective and open,", "we see that $|X_v| \\to |X_V|$ is a homeomorphism onto its image (with", "induced topology).", "On the other hand, the inverse image of $f^{-1}(\\{y\\})$", "under the open map $|X_V| \\to |X|$ is equal to $|X_v|$.", "We conclude that $|X_v| \\to f^{-1}(\\{y\\})$ is open.", "The morphism $X_v \\to X$ factors through $X_k$", "and $|X_k| \\to |X|$ is injective with image $f^{-1}(\\{y\\})$", "by Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-points-cartesian}. Using", "$|X_v| \\to |X_k| \\to f^{-1}(\\{y\\})$ the lemma follows because", "$X_v \\to X_k$ is surjective." ], "refs": [ "spaces-properties-lemma-etale-open", "spaces-morphisms-lemma-monomorphism-injective-points", "decent-spaces-lemma-UR-finite-above-x", "schemes-lemma-fibre-topological", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 11860, 4758, 9463, 7696, 11819 ] } ], "ref_ids": [] }, { "id": 9526, "type": "theorem", "label": "decent-spaces-lemma-conditions-on-point-on-space-over-field", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-conditions-on-point-on-space-over-field", "contents": [ "Let $X$ be an algebraic space locally of finite type over a field $k$.", "Let $x \\in |X|$. Consider the conditions", "\\begin{enumerate}", "\\item $\\dim_x(|X|) = 0$,", "\\item $x$ is closed in $|X|$ and if $x' \\leadsto x$ in $|X|$ then $x' = x$,", "\\item $x$ is an isolated point of $|X|$,", "\\item $\\dim_x(X) = 0$,", "\\item $X \\to \\Spec(k)$ is quasi-finite at $x$.", "\\end{enumerate}", "Then (2), (3), (4), and (5) are equivalent.", "If $X$ is decent, then (1) is equivalent to the others." ], "refs": [], "proofs": [ { "contents": [ "Parts (4) and (5) are equivalent for example by", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part} and", "\\ref{spaces-morphisms-lemma-quasi-finite-at-point}.", "\\medskip\\noindent", "Let $U \\to X$ be an \\'etale morphism where $U$ is an affine scheme and let", "$u \\in U$ be a point mapping to $x$. Moreover, if $x$ is a closed", "point, e.g., in case (2) or (3), then we may and do assume that $u$", "is a closed point. Observe that $\\dim_u(U) = \\dim_x(X)$ by definition", "and that this is equal to $\\dim(\\mathcal{O}_{U, u})$ if $u$ is a closed", "point, see Algebra, Lemma", "\\ref{algebra-lemma-dimension-closed-point-finite-type-field}.", "\\medskip\\noindent", "If $\\dim_x(X) > 0$ and $u$ is closed, by the arguments above", "we can choose a nontrivial", "specialization $u' \\leadsto u$ in $U$. Then the transcendence degree", "of $\\kappa(u')$ over $k$ exceeds the transcendence degree of", "$\\kappa(u)$ over $k$. It follows that the images $x$ and $x'$ in $X$", "are distinct, because the transcendence degree of $x/k$ and $x'/k$", "are well defined, see Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-dimension-fibre}.", "This applies in particular in cases (2) and (3) and we", "conclude that (2) and (3) imply (4).", "\\medskip\\noindent", "Conversely, if $X \\to \\Spec(k)$ is locally quasi-finite at $x$, then", "$U \\to \\Spec(k)$ is locally quasi-finite at $u$, hence $u$ is an", "isolated point of $U$", "(Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}).", "It follows that (5) implies (2) and (3) as", "$|U| \\to |X|$ is continuous and open.", "\\medskip\\noindent", "Assume $X$ is decent and (1) holds. Then $\\dim_x(X) = \\dim_x(|X|)$", "by Lemma \\ref{lemma-dimension-decent-space} and the proof is complete." ], "refs": [ "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part", "spaces-morphisms-lemma-quasi-finite-at-point", "algebra-lemma-dimension-closed-point-finite-type-field", "spaces-morphisms-definition-dimension-fibre", "morphisms-lemma-quasi-finite-at-point-characterize", "decent-spaces-lemma-dimension-decent-space" ], "ref_ids": [ 4876, 4877, 996, 5009, 5226, 9496 ] } ], "ref_ids": [] }, { "id": 9527, "type": "theorem", "label": "decent-spaces-lemma-conditions-on-space-over-field", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-conditions-on-space-over-field", "contents": [ "Let $X$ be an algebraic space locally of finite type over a field $k$.", "Consider the conditions", "\\begin{enumerate}", "\\item $|X|$ is a finite set,", "\\item $|X|$ is a discrete space,", "\\item $\\dim(|X|) = 0$,", "\\item $\\dim(X) = 0$,", "\\item $X \\to \\Spec(k)$ is locally quasi-finite,", "\\end{enumerate}", "Then (2), (3), (4), and (5) are equivalent.", "If $X$ is decent, then (1) implies the others." ], "refs": [], "proofs": [ { "contents": [ "Parts (4) and (5) are equivalent for example by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part}.", "\\medskip\\noindent", "Let $U \\to X$ be a surjective \\'etale morphism where $U$ is a scheme.", "\\medskip\\noindent", "If $\\dim(U) > 0$, then choose a nontrivial specialization", "$u \\leadsto u'$ in $U$ and the transcendence degree of $\\kappa(u)$", "over $k$ exceeds the transcendence degree of $\\kappa(u')$ over $k$.", "It follows that the images $x$ and $x'$ in $X$ are distinct, because", "the transcendence degree of $x/k$ and $x'/k$ is well defined, see", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-dimension-fibre}.", "We conclude that (2) and (3) imply (4).", "\\medskip\\noindent", "Conversely, if $X \\to \\Spec(k)$ is locally quasi-finite, then $U$ is", "locally Noetherian", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian})", "of dimension $0$", "(Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0})", "and hence is a disjoint union of spectra of Artinian local rings", "(Properties, Lemma \\ref{properties-lemma-locally-Noetherian-dimension-0}).", "Hence $U$ is a discrete topological space, and since $|U| \\to |X|$", "is continuous and open, the same is true for $|X|$.", "In other words, (4) implies (2) and (3).", "\\medskip\\noindent", "Assume $X$ is decent and (1) holds. Then we may choose $U$ above to", "be affine. The fibres of $|U| \\to |X|$ are finite (this is a part of the", "defining property of decent spaces). Hence $U$ is a finite type scheme", "over $k$ with finitely many points. Hence $U$ is quasi-finite over $k$", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-fibre})", "which by definition means that $X \\to \\Spec(k)$ is locally quasi-finite." ], "refs": [ "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part", "spaces-morphisms-definition-dimension-fibre", "morphisms-lemma-finite-type-noetherian", "morphisms-lemma-locally-quasi-finite-rel-dimension-0", "properties-lemma-locally-Noetherian-dimension-0", "morphisms-lemma-finite-fibre" ], "ref_ids": [ 4876, 5009, 5202, 5287, 2981, 5227 ] } ], "ref_ids": [] }, { "id": 9528, "type": "theorem", "label": "decent-spaces-lemma-conditions-on-point-in-fibre-and-qf", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-conditions-on-point-in-fibre-and-qf", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ which is locally of finite type. Let $x \\in |X|$ with image", "$y \\in |Y|$. Let $F = f^{-1}(\\{y\\})$ with induced topology from $|X|$.", "Let $k$ be a field and let $\\Spec(k) \\to Y$ be in the", "equivalence class defining $y$. Set $X_k = \\Spec(k) \\times_Y X$.", "Let $\\tilde x \\in |X_k|$ map to $x \\in |X|$.", "Consider the following conditions", "\\begin{enumerate}", "\\item", "\\label{item-fibre-at-x-dim-0}", "$\\dim_x(F) = 0$,", "\\item", "\\label{item-isolated-in-fibre}", "$x$ is isolated in $F$,", "\\item", "\\label{item-no-specializations-in-fibre}", "$x$ is closed in $F$ and if $x' \\leadsto x$ in $F$, then $x = x'$,", "\\item", "\\label{item-dimension-top-k-fibre}", "$\\dim_{\\tilde x}(|X_k|) = 0$,", "\\item", "\\label{item-isolated-in-k-fibre}", "$\\tilde x$ is isolated in $|X_k|$,", "\\item", "\\label{item-no-specializations-in-k-fibre}", "$\\tilde x$ is closed in $|X_k|$ and if $\\tilde x' \\leadsto \\tilde x$", "in $|X_k|$, then $\\tilde x = \\tilde x'$,", "\\item", "\\label{item-k-fibre-at-x-dim-0}", "$\\dim_{\\tilde x}(X_k) = 0$,", "\\item", "\\label{item-quasi-finite-at-x}", "$f$ is quasi-finite at $x$.", "\\end{enumerate}", "Then we have", "$$", "\\xymatrix{", "(\\ref{item-dimension-top-k-fibre}) \\ar@{=>}[r]_{f\\text{ decent}} &", "(\\ref{item-isolated-in-k-fibre}) \\ar@{<=>}[r] &", "(\\ref{item-no-specializations-in-k-fibre}) \\ar@{<=>}[r] &", "(\\ref{item-k-fibre-at-x-dim-0}) \\ar@{<=>}[r] &", "(\\ref{item-quasi-finite-at-x})", "}", "$$", "If $Y$ is decent, then conditions (\\ref{item-isolated-in-fibre}) and", "(\\ref{item-no-specializations-in-fibre}) are equivalent to each other", "and to conditions", "(\\ref{item-isolated-in-k-fibre}),", "(\\ref{item-no-specializations-in-k-fibre}),", "(\\ref{item-k-fibre-at-x-dim-0}), and", "(\\ref{item-quasi-finite-at-x}).", "If $Y$ and $X$ are decent, then all conditions are equivalent." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-conditions-on-point-on-space-over-field} conditions", "(\\ref{item-isolated-in-k-fibre}),", "(\\ref{item-no-specializations-in-k-fibre}), and (\\ref{item-k-fibre-at-x-dim-0})", "are equivalent to each other and to the condition that", "$X_k \\to \\Spec(k)$ is quasi-finite at $\\tilde x$.", "Thus by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-quasi-finite-locus}", "they are also equivalent to (\\ref{item-quasi-finite-at-x}).", "If $f$ is decent, then $X_k$ is a decent algebraic space and", "Lemma \\ref{lemma-conditions-on-point-on-space-over-field}", "shows that (\\ref{item-dimension-top-k-fibre}) implies", "(\\ref{item-isolated-in-k-fibre}).", "\\medskip\\noindent", "If $Y$ is decent, then we can pick a quasi-compact monomorphism", "$\\Spec(k') \\to Y$ in the equivalence class of $y$. In this case", "Lemma \\ref{lemma-topology-fibre}", "tells us that $|X_{k'}| \\to F$ is a homeomorphism.", "Combined with the arguments given above this implies", "the remaining statements of the lemma; details omitted." ], "refs": [ "decent-spaces-lemma-conditions-on-point-on-space-over-field", "spaces-morphisms-lemma-base-change-quasi-finite-locus", "decent-spaces-lemma-conditions-on-point-on-space-over-field", "decent-spaces-lemma-topology-fibre" ], "ref_ids": [ 9526, 4830, 9526, 9525 ] } ], "ref_ids": [] }, { "id": 9529, "type": "theorem", "label": "decent-spaces-lemma-conditions-on-fibre-and-qf", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-conditions-on-fibre-and-qf", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ which is locally of finite type. Let $y \\in |Y|$. Let $k$ be a field", "and let $\\Spec(k) \\to Y$ be in the equivalence class defining $y$.", "Set $X_k = \\Spec(k) \\times_Y X$ and let $F = f^{-1}(\\{y\\})$ with the", "induced topology from $|X|$. Consider the following conditions", "\\begin{enumerate}", "\\item", "\\label{item-fibre-finite}", "$F$ is finite,", "\\item", "\\label{item-fibre-discrete}", "$F$ is a discrete topological space,", "\\item", "\\label{item-fibre-no-specializations}", "$\\dim(F) = 0$,", "\\item", "\\label{item-k-fibre-finite}", "$|X_k|$ is a finite set,", "\\item", "\\label{item-k-fibre-discrete}", "$|X_k|$ is a discrete space,", "\\item", "\\label{item-k-fibre-no-specializations}", "$\\dim(|X_k|) = 0$,", "\\item", "\\label{item-k-fibre-dim-0}", "$\\dim(X_k) = 0$,", "\\item", "\\label{item-quasi-finite-at-points-fibre}", "$f$ is quasi-finite at all points of $|X|$ lying over $y$.", "\\end{enumerate}", "Then we have", "$$", "\\xymatrix{", "(\\ref{item-fibre-finite}) &", "(\\ref{item-k-fibre-finite}) \\ar@{=>}[l] \\ar@{=>}[r]_{f\\text{ decent}} &", "(\\ref{item-k-fibre-discrete}) \\ar@{<=>}[r] &", "(\\ref{item-k-fibre-no-specializations}) \\ar@{<=>}[r] &", "(\\ref{item-k-fibre-dim-0}) \\ar@{<=>}[r] &", "(\\ref{item-quasi-finite-at-points-fibre})", "}", "$$", "If $Y$ is decent, then conditions (\\ref{item-fibre-discrete}) and", "(\\ref{item-fibre-no-specializations})", "are equivalent to each other and to conditions (\\ref{item-k-fibre-discrete}),", "(\\ref{item-k-fibre-no-specializations}), (\\ref{item-k-fibre-dim-0}), and", "(\\ref{item-quasi-finite-at-points-fibre}).", "If $Y$ and $X$ are decent, then (\\ref{item-fibre-finite}) implies", "all the other conditions." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-conditions-on-space-over-field}", "conditions (\\ref{item-k-fibre-discrete}),", "(\\ref{item-k-fibre-no-specializations}), and (\\ref{item-k-fibre-dim-0})", "are equivalent to each other and to the condition that", "$X_k \\to \\Spec(k)$ is locally quasi-finite.", "Thus by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-quasi-finite-locus}", "they are also equivalent to (\\ref{item-quasi-finite-at-points-fibre}).", "If $f$ is decent, then $X_k$ is a decent algebraic space and", "Lemma \\ref{lemma-conditions-on-space-over-field}", "shows that (\\ref{item-k-fibre-finite}) implies (\\ref{item-k-fibre-discrete}).", "\\medskip\\noindent", "The map $|X_k| \\to F$ is surjective by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}", "and we see", "(\\ref{item-k-fibre-finite}) $\\Rightarrow$ (\\ref{item-fibre-finite}).", "\\medskip\\noindent", "If $Y$ is decent, then we can pick a quasi-compact monomorphism", "$\\Spec(k') \\to Y$ in the equivalence class of $y$. In this case", "Lemma \\ref{lemma-topology-fibre}", "tells us that $|X_{k'}| \\to F$ is a homeomorphism.", "Combined with the arguments given above this implies", "the remaining statements of the lemma; details omitted." ], "refs": [ "decent-spaces-lemma-conditions-on-space-over-field", "spaces-morphisms-lemma-base-change-quasi-finite-locus", "decent-spaces-lemma-conditions-on-space-over-field", "spaces-properties-lemma-points-cartesian", "decent-spaces-lemma-topology-fibre" ], "ref_ids": [ 9527, 4830, 9527, 11819, 9525 ] } ], "ref_ids": [] }, { "id": 9530, "type": "theorem", "label": "decent-spaces-lemma-monomorphism-toward-disjoint-union-dim-0-rings", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-monomorphism-toward-disjoint-union-dim-0-rings", "contents": [ "Let $S$ be a scheme. Let $Y$ be a disjoint union of spectra of", "zero dimensional local rings over $S$.", "Let $f : X \\to Y$ be a monomorphism of algebraic spaces over $S$.", "Then $f$ is representable, i.e., $X$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "This immediately reduces to the case $Y = \\Spec(A)$ where", "$A$ is a zero dimensional local ring, i.e.,", "$\\Spec(A) = \\{\\mathfrak m_A\\}$", "is a singleton. If $X = \\emptyset$, then there is nothing to prove.", "If not, choose a nonempty affine scheme $U = \\Spec(B)$", "and an \\'etale morphism $U \\to X$. As $|X|$ is a singleton (as a", "subset of $|Y|$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-monomorphism-injective-points})", "we see that $U \\to X$ is surjective. Note that", "$U \\times_X U = U \\times_Y U = \\Spec(B \\otimes_A B)$.", "Thus we see that the ring maps $B \\to B \\otimes_A B$ are \\'etale.", "Since", "$$", "(B \\otimes_A B)/\\mathfrak m_A(B \\otimes_A B)", "=", "(B/\\mathfrak m_AB) \\otimes_{A/\\mathfrak m_A} (B/\\mathfrak m_AB)", "$$", "we see that", "$B/\\mathfrak m_AB \\to (B \\otimes_A B)/\\mathfrak m_A(B \\otimes_A B)$", "is flat and in fact free of rank equal to the dimension of", "$B/\\mathfrak m_AB$ as a $A/\\mathfrak m_A$-vector space. Since", "$B \\to B \\otimes_A B$ is \\'etale, this can only happen if this", "dimension is finite (see for example", "Morphisms, Lemmas \\ref{morphisms-lemma-etale-universally-bounded} and", "\\ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded}).", "Every prime of $B$ lies over $\\mathfrak m_A$ (the unique prime of $A$).", "Hence $\\Spec(B) = \\Spec(B/\\mathfrak m_A)$ as a topological", "space, and this space is a finite discrete set as $B/\\mathfrak m_A B$", "is an Artinian ring, see", "Algebra, Lemmas \\ref{algebra-lemma-finite-dimensional-algebra} and", "\\ref{algebra-lemma-artinian-finite-length}.", "Hence all prime ideals of $B$ are maximal and", "$B = B_1 \\times \\ldots \\times B_n$ is a product of finitely many", "local rings of dimension zero, see", "Algebra, Lemma \\ref{algebra-lemma-product-local}.", "Thus $B \\to B \\otimes_A B$ is finite \\'etale as all the local rings", "$B_i$ are henselian by", "Algebra, Lemma \\ref{algebra-lemma-local-dimension-zero-henselian}.", "Thus $X$ is an affine scheme by", "Groupoids, Proposition \\ref{groupoids-proposition-finite-flat-equivalence}." ], "refs": [ "spaces-morphisms-lemma-monomorphism-injective-points", "morphisms-lemma-etale-universally-bounded", "morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded", "algebra-lemma-finite-dimensional-algebra", "algebra-lemma-artinian-finite-length", "algebra-lemma-product-local", "algebra-lemma-local-dimension-zero-henselian", "groupoids-proposition-finite-flat-equivalence" ], "ref_ids": [ 4758, 5530, 5531, 642, 646, 645, 1283, 9669 ] } ], "ref_ids": [] }, { "id": 9531, "type": "theorem", "label": "decent-spaces-lemma-decent-generic-points", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-decent-generic-points", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $x \\in |X|$. The following are equivalent", "\\begin{enumerate}", "\\item $x$ is a generic point of an irreducible component of $|X|$,", "\\item for any \\'etale morphism $(Y, y) \\to (X, x)$ of pointed algebraic", "spaces, $y$ is a generic point of an irreducible component of $|Y|$,", "\\item for some \\'etale morphism $(Y, y) \\to (X, x)$ of pointed algebraic", "spaces, $y$ is a generic point of an irreducible component of $|Y|$,", "\\item the dimension of the local ring of $X$ at $x$ is zero, and", "\\item $x$ is a point of codimension $0$ on $X$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Conditions (4) and (5) are equivalent for any algebraic space", "by definition, see Properties of Spaces, Definition", "\\ref{spaces-properties-definition-dimension-local-ring}.", "Observe that any $Y$ as in (2) and (3) is decent by", "Lemma \\ref{lemma-etale-named-properties}.", "Thus it suffices to prove the equivalence of (1) and (4)", "as then the equivalence with (2) and (3) follows since the dimension", "of the local ring of $Y$ at $y$ is equal to the dimension", "of the local ring of $X$ at $x$.", "Let $f : U \\to X$ be an \\'etale morphism from an affine scheme and let", "$u \\in U$ be a point mapping to $x$.", "\\medskip\\noindent", "Assume (1). Let $u' \\leadsto u$ be a specialization in $U$.", "Then $f(u') = f(u) = x$. By", "Lemma \\ref{lemma-decent-no-specializations-map-to-same-point}", "we see that $u' = u$. Hence $u$ is a generic point of an irreducible component", "of $U$. Thus $\\dim(\\mathcal{O}_{U, u}) = 0$ and we see that (4) holds.", "\\medskip\\noindent", "Assume (4). The point $x$ is contained in an irreducible component", "$T \\subset |X|$. Since $|X|$ is sober", "(Proposition \\ref{proposition-reasonable-sober})", "we $T$ has a generic point $x'$. Of course $x' \\leadsto x$.", "Then we can lift this specialization to $u' \\leadsto u$ in $U$", "(Lemma \\ref{lemma-decent-specialization}). This contradicts the assumption", "that $\\dim(\\mathcal{O}_{U, u}) = 0$ unless $u' = u$, i.e., $x' = x$." ], "refs": [ "spaces-properties-definition-dimension-local-ring", "decent-spaces-lemma-etale-named-properties", "decent-spaces-lemma-decent-no-specializations-map-to-same-point", "decent-spaces-proposition-reasonable-sober", "decent-spaces-lemma-decent-specialization" ], "ref_ids": [ 11931, 9471, 9493, 9559, 9494 ] } ], "ref_ids": [] }, { "id": 9532, "type": "theorem", "label": "decent-spaces-lemma-codimension-local-ring", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-codimension-local-ring", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $T \\subset |X|$ be an irreducible closed subset. Let $\\xi \\in T$", "be the generic point (Proposition \\ref{proposition-reasonable-sober}).", "Then $\\text{codim}(T, |X|)$", "(Topology, Definition \\ref{topology-definition-codimension})", "is the dimension of the local ring of $X$ at $\\xi$", "(Properties of Spaces, Definition", "\\ref{spaces-properties-definition-dimension-local-ring})." ], "refs": [ "decent-spaces-proposition-reasonable-sober", "topology-definition-codimension", "spaces-properties-definition-dimension-local-ring" ], "proofs": [ { "contents": [ "Choose a scheme $U$, a point $u \\in U$, and an \\'etale morphism", "$U \\to X$ sending $u$ to $\\xi$. Then any sequence of nontrivial", "specializations $\\xi_e \\leadsto \\ldots \\leadsto \\xi_0 = \\xi$", "can be lifted to a sequence $u_e \\leadsto \\ldots \\leadsto u_0 = u$ in $U$", "by Lemma \\ref{lemma-decent-specialization}.", "Conversely, any sequence of nontrivial specializations", "$u_e \\leadsto \\ldots \\leadsto u_0 = u$ in $U$", "maps to a sequence of nontrivial specializations", "$\\xi_e \\leadsto \\ldots \\leadsto \\xi_0 = \\xi$ by", "Lemma \\ref{lemma-decent-no-specializations-map-to-same-point}.", "Because $|X|$ and $U$ are sober topological spaces", "we conclude that the codimension of $T$ in $|X|$", "and of $\\overline{\\{u\\}}$ in $U$ are the same.", "In this way the lemma reduces to the schemes case which", "is Properties, Lemma \\ref{properties-lemma-codimension-local-ring}." ], "refs": [ "decent-spaces-lemma-decent-specialization", "decent-spaces-lemma-decent-no-specializations-map-to-same-point", "properties-lemma-codimension-local-ring" ], "ref_ids": [ 9494, 9493, 2979 ] } ], "ref_ids": [ 9559, 8358, 11931 ] }, { "id": 9533, "type": "theorem", "label": "decent-spaces-lemma-get-reasonable", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-get-reasonable", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume", "\\begin{enumerate}", "\\item every quasi-compact scheme \\'etale over $X$ has finitely many", "irreducible components, and", "\\item every $x \\in |X|$ of codimension $0$ on $X$ can be represented", "by a monomorphism $\\Spec(k) \\to X$.", "\\end{enumerate}", "Then $X$ is a reasonable algebraic space." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be an affine scheme and let $a : U \\to X$ be an \\'etale morphism.", "We have to show that the fibres of $a$ are universally bounded. By", "assumption (1) the scheme $U$ has finitely many irreducible components.", "Let $u_1, \\ldots, u_n \\in U$ be the generic points of these irreducible", "components. Let $\\{x_1, \\ldots, x_m\\} \\subset |X|$ be the image", "of $\\{u_1, \\ldots, u_n\\}$. Each $x_j$ is a point of codimension $0$.", "By assumption (2) we may choose a monomorphism $\\Spec(k_j) \\to X$", "representing $x_j$. Then", "$$", "U \\times_X \\Spec(k_j) = \\coprod\\nolimits_{a(u_i) = x_j} \\Spec(\\kappa(u_i))", "$$", "is finite over $\\Spec(k_j)$ of degree", "$d_j = \\sum_{a(u_i) = x_j} [\\kappa(u_i) : k_j]$. Set $n = \\max d_j$.", "\\medskip\\noindent", "Observe that $a$ is separated", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-separated-cover}).", "Consider the stratification", "$$", "X = X_0 \\supset X_1 \\supset X_2 \\supset \\ldots", "$$", "associated to $U \\to X$ in Lemma \\ref{lemma-stratify-flat-fp-lqf}.", "By our choice of $n$ above we conclude that $X_{n + 1}$ is empty.", "Namely, if not, then $a^{-1}(X_{n + 1})$ is a nonempty open", "of $U$ and hence would contain one of the $x_i$. This would mean", "that $X_{n + 1}$ contains $x_j = a(u_i)$ which is impossible.", "Hence we see that the fibres of $U \\to X$ are universally bounded", "(in fact by the integer $n$)." ], "refs": [ "spaces-properties-lemma-separated-cover", "decent-spaces-lemma-stratify-flat-fp-lqf" ], "ref_ids": [ 11833, 9476 ] } ], "ref_ids": [] }, { "id": 9534, "type": "theorem", "label": "decent-spaces-lemma-finitely-many-irreducible-components", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-finitely-many-irreducible-components", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is decent and $|X|$ has finitely many irreducible components,", "\\item every quasi-compact scheme \\'etale over $X$ has finitely many", "irreducible components, there are finitely many $x \\in |X|$ of", "codimension $0$ on $X$, and each of these can be represented", "by a monomorphism $\\Spec(k) \\to X$,", "\\item there exists a dense open $X' \\subset X$ which is", "a scheme, $X'$ has finitely many irreducible components", "with generic points $\\{x'_1, \\ldots, x'_m\\}$, and", "the morphism $x'_j \\to X$ is quasi-compact for $j = 1, \\ldots, m$.", "\\end{enumerate}", "Moreover, if these conditions hold, then $X$ is reasonable and the", "points $x'_j \\in |X|$ are the generic points of the irreducible", "components of $|X|$." ], "refs": [], "proofs": [ { "contents": [ "In the proof we use Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-codimension-0-points}", "without further mention.", "Assume (1). Then $X$ has a dense open subscheme $X'$ by", "Theorem \\ref{theorem-decent-open-dense-scheme}.", "Since the closure of an irreducible component of $|X'|$", "is an irreducible component of $|X|$, we see that $|X'|$", "has finitely many irreducible components. Thus (3) holds.", "\\medskip\\noindent", "Assume $X' \\subset X$ is as in (3). Let $\\{x'_1, \\ldots, x'_m\\}$", "be the generic points of the irreducible components of $X'$.", "Let $a : U \\to X$ be an \\'etale morphism with $U$ a quasi-compact scheme.", "It suffices to show that $U$ has finitely many irreducible components", "whose generic points lie over $\\{x'_1, \\ldots, x'_m\\}$. It suffices", "to prove this for the members of a finite affine open cover of $U$,", "hence we may and do assume $U$ is affine.", "Note that $U' = a^{-1}(X') \\subset U$ is a dense open.", "The generic points of irreducible components of $U'$ are the points", "lying over $\\{x'_1, \\ldots, x'_m\\}$ and since $x'_j \\to X$ is", "quasi-compact there are finitely many points of $U$ lying over $x'_j$", "(Lemma \\ref{lemma-UR-finite-above-x}). Hence $U'$ has finitely", "many irreducible components, which implies that the closures", "of these irreducible components are the irreducible components of", "$U$. Thus (2) holds.", "\\medskip\\noindent", "Assume (2). This implies (1) and the final", "statement by Lemma \\ref{lemma-get-reasonable}.", "(We also use that a reasonable algebraic space is decent, see", "discussion following Definition \\ref{definition-very-reasonable}.)" ], "refs": [ "spaces-properties-lemma-codimension-0-points", "decent-spaces-theorem-decent-open-dense-scheme", "decent-spaces-lemma-UR-finite-above-x", "decent-spaces-lemma-get-reasonable", "decent-spaces-definition-very-reasonable" ], "ref_ids": [ 11842, 9454, 9463, 9533, 9562 ] } ], "ref_ids": [] }, { "id": 9535, "type": "theorem", "label": "decent-spaces-lemma-generically-finite", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-generically-finite", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume that $f$ is quasi-separated of finite type.", "Let $y \\in |Y|$ be a point of codimension $0$ on $Y$.", "The following are equivalent:", "\\begin{enumerate}", "\\item the space $|X_k|$ is finite where $\\Spec(k) \\to Y$ represents $y$,", "\\item $X \\to Y$ is quasi-finite at all points of $|X|$ over $y$,", "\\item there exists an open subspace $Y' \\subset Y$ with $y \\in |Y'|$", "such that $Y' \\times_Y X \\to Y'$ is finite.", "\\end{enumerate}", "If $Y$ is decent these are also equivalent to", "\\begin{enumerate}", "\\item[(4)] the set $f^{-1}(\\{y\\})$ is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows from", "Lemma \\ref{lemma-conditions-on-fibre-and-qf}", "(and the fact that a quasi-separated morphism is decent by", "Lemma \\ref{lemma-properties-trivial-implications}).", "\\medskip\\noindent", "Assume the equivalent conditions of (1) and (2). Choose an affine scheme $V$", "and an \\'etale morphism $V \\to Y$ mapping a point $v \\in V$ to $y$. Then $v$", "is a generic point of an irreducible component of $V$ by", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-codimension-0-points}.", "Choose an affine scheme $U$", "and a surjective \\'etale morphism $U \\to V \\times_Y X$. Then $U \\to V$ is of", "finite type. The morphism $U \\to V$ is quasi-finite at every point lying over", "$v$ by (2). It follows that the fibre of $U \\to V$ over $v$ is finite", "(Morphisms, Lemma", "\\ref{morphisms-lemma-quasi-finite-at-a-finite-number-of-points}). By", "Morphisms, Lemma \\ref{morphisms-lemma-generically-finite}", "after shrinking $V$ we may assume that $U \\to V$ is finite.", "Let", "$$", "R = U \\times_{V \\times_Y X} U", "$$", "Since $f$ is quasi-separated, we see that $V \\times_Y X$ is quasi-separated", "and hence $R$ is a quasi-compact scheme. Moreover the morphisms", "$R \\to V$ is quasi-finite as the composition of an \\'etale morphism", "$R \\to U$ and a finite morphism $U \\to V$. Hence we may apply", "Morphisms, Lemma \\ref{morphisms-lemma-generically-finite}", "once more and after shrinking $V$ we may assume that $R \\to V$ is", "finite as well. This of course implies that the two projections", "$R \\to V$ are finite \\'etale. It follows that", "$V/R = V \\times_Y X$ is an affine scheme, see", "Groupoids, Proposition \\ref{groupoids-proposition-finite-flat-equivalence}.", "By Morphisms, Lemma \\ref{morphisms-lemma-image-proper-is-proper}", "we conclude that $V \\times_Y X \\to V$ is proper and by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}", "we conclude that $V \\times_Y X \\to V$ is finite.", "Finally, we let $Y' \\subset Y$ be the open subspace of $Y$", "corresponding to the image of $|V| \\to |Y|$.", "By Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-integral-local}", "we conclude that $Y' \\times_Y X \\to Y'$ is finite as the base", "change to $V$ is finite and as $V \\to Y'$ is a surjective \\'etale", "morphism.", "\\medskip\\noindent", "If $Y$ is decent and $f$ is quasi-separated, then we see that", "$X$ is decent too; use Lemmas", "\\ref{lemma-properties-trivial-implications} and", "\\ref{lemma-property-over-property}.", "Hence Lemma \\ref{lemma-conditions-on-fibre-and-qf}", "applies to show that (4) implies (1) and (2). On the other hand,", "we see that (2) implies (4) by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-finite-at-a-finite-number-of-points}." ], "refs": [ "decent-spaces-lemma-conditions-on-fibre-and-qf", "decent-spaces-lemma-properties-trivial-implications", "spaces-properties-lemma-codimension-0-points", "morphisms-lemma-quasi-finite-at-a-finite-number-of-points", "morphisms-lemma-generically-finite", "morphisms-lemma-generically-finite", "groupoids-proposition-finite-flat-equivalence", "morphisms-lemma-image-proper-is-proper", "morphisms-lemma-finite-proper", "spaces-morphisms-lemma-integral-local", "decent-spaces-lemma-properties-trivial-implications", "decent-spaces-lemma-property-over-property", "decent-spaces-lemma-conditions-on-fibre-and-qf", "spaces-morphisms-lemma-quasi-finite-at-a-finite-number-of-points" ], "ref_ids": [ 9529, 9513, 11842, 5234, 5487, 5487, 9669, 5413, 5445, 4940, 9513, 9516, 9529, 4837 ] } ], "ref_ids": [] }, { "id": 9536, "type": "theorem", "label": "decent-spaces-lemma-generically-finite-reprise", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-generically-finite-reprise", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume that $f$ is quasi-separated and locally of finite type", "and $Y$ quasi-separated. Let $y \\in |Y|$ be a point of codimension $0$ on $Y$.", "The following are equivalent:", "\\begin{enumerate}", "\\item the set $f^{-1}(\\{y\\})$ is finite,", "\\item the space $|X_k|$ is finite where $\\Spec(k) \\to Y$ represents $y$,", "\\item there exist open subspaces $X' \\subset X$ and $Y' \\subset Y$", "with $f(X') \\subset Y'$, $y \\in |Y'|$, and $f^{-1}(\\{y\\}) \\subset |X'|$", "such that $f|_{X'} : X' \\to Y'$ is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since quasi-separated algebraic spaces are decent, the equivalence", "of (1) and (2) follows from", "Lemma \\ref{lemma-conditions-on-fibre-and-qf}.", "To prove that (1) and (2) imply (3)", "we may and do replace $Y$ by a quasi-compact open containing $y$.", "Since $f^{-1}(\\{y\\})$ is finite, we can find a quasi-compact", "open subspace of $X' \\subset X$ containing the fibre.", "The restriction $f|_{X'} : X' \\to Y$ is quasi-compact and quasi-separated", "by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence}", "(this is where we use that $Y$ is quasi-separated).", "Applying Lemma \\ref{lemma-generically-finite}", "to $f|_{X'} : X' \\to Y$ we see that (3) holds.", "We omit the proof that (3) implies (2)." ], "refs": [ "decent-spaces-lemma-conditions-on-fibre-and-qf", "spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence", "decent-spaces-lemma-generically-finite" ], "ref_ids": [ 9529, 4744, 9535 ] } ], "ref_ids": [] }, { "id": 9537, "type": "theorem", "label": "decent-spaces-lemma-quasi-finiteness-over-generic-point", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-quasi-finiteness-over-generic-point", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is locally of finite type.", "Let $X^0 \\subset |X|$, resp.\\ $Y^0 \\subset |Y|$ denote the set of", "codimension $0$ points of $X$, resp.\\ $Y$. Let $y \\in Y^0$. The following are", "equivalent", "\\begin{enumerate}", "\\item $f^{-1}(\\{y\\}) \\subset X^0$,", "\\item $f$ is quasi-finite at all points lying over $y$,", "\\item $f$ is quasi-finite at all $x \\in X^0$ lying over $y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $V$ be a scheme and let $V \\to Y$ be a surjective \\'etale morphism.", "Let $U$ be a scheme and let $U \\to V \\times_Y X$ be a surjective \\'etale", "morphism. Then $f$ is quasi-finite at the image $x$ of a point $u \\in U$", "if and only if $U \\to V$ is quasi-finite at $u$. Moreover, $x \\in X^0$", "if and only if $u$ is the generic point of an irreducible component", "of $U$ (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-codimension-0-points}).", "Thus the lemma reduces to the case of the morphism $U \\to V$, i.e., to", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finiteness-over-generic-point}." ], "refs": [ "spaces-properties-lemma-codimension-0-points", "morphisms-lemma-quasi-finiteness-over-generic-point" ], "ref_ids": [ 11842, 5488 ] } ], "ref_ids": [] }, { "id": 9538, "type": "theorem", "label": "decent-spaces-lemma-finite-over-dense-open", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-finite-over-dense-open", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is locally of finite type.", "Let $X^0 \\subset |X|$, resp.\\ $Y^0 \\subset |Y|$ denote the set of", "codimension $0$ points of $X$, resp.\\ $Y$. Assume", "\\begin{enumerate}", "\\item $Y$ is decent,", "\\item $X^0$ and $Y^0$ are finite and $f^{-1}(Y^0) = X^0$,", "\\item either $f$ is quasi-compact or $f$ is separated.", "\\end{enumerate}", "Then there exists a dense open $V \\subset Y$", "such that $f^{-1}(V) \\to V$ is finite." ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-finitely-many-irreducible-components} and", "\\ref{lemma-decent-generic-points} we may assume $Y$ is a scheme", "with finitely many irreducible components. Shrinking further we", "may assume $Y$ is an irreducible affine scheme with generic point $y$.", "Then the fibre of $f$ over $y$ is finite.", "\\medskip\\noindent", "Assume $f$ is quasi-compact and $Y$ affine irreducible. Then $X$ is", "quasi-compact and we may choose an affine scheme $U$ and a", "surjective \\'etale morphism $U \\to X$. Then $U \\to Y$ is of finite type", "and the fibre of $U \\to Y$ over $y$ is the set $U^0$ of generic points of", "irreducible components of $U$ (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-codimension-0-points}).", "Hence $U^0$ is finite", "(Morphisms, Lemma", "\\ref{morphisms-lemma-quasi-finite-at-a-finite-number-of-points})", "and after shrinking $Y$ we may assume that $U \\to Y$ is finite", "(Morphisms, Lemma \\ref{morphisms-lemma-generically-finite}).", "Next, consider $R = U \\times_X U$. Since the projection", "$s : R \\to U$ is \\'etale we see that $R^0 = s^{-1}(U^0)$", "lies over $y$. Since $R \\to U \\times_Y U$ is a monomorphism,", "we conclude that $R^0$ is finite as $U \\times_Y U \\to Y$ is finite.", "And $R$ is separated", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-separated-cover}).", "Thus we may shrink $Y$ once more to reach the situation", "where $R$ is finite over $Y$", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-over-dense-open}).", "In this case it follows that $X = U/R$ is finite over $Y$", "by exactly the same arguments as given in the proof of", "Lemma \\ref{lemma-generically-finite}", "(or we can simply apply that lemma because", "it follows immediately that $X$ is quasi-separated as well).", "\\medskip\\noindent", "Assume $f$ is separated and $Y$ affine irreducible. Choose $V \\subset Y$", "and $U \\subset X$ as in Lemma \\ref{lemma-generically-finite-reprise}.", "Since $f|_U : U \\to V$ is finite, we see that $U \\subset f^{-1}(V)$", "is closed as well as open", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-universally-closed-permanence} and", "\\ref{spaces-morphisms-lemma-finite-proper}).", "Thus $f^{-1}(V) = U \\amalg W$ for some", "open subspace $W$ of $X$. However, since $U$ contains all the codimension", "$0$ points of $X$ we conclude that $W = \\emptyset$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-codimension-0-points-dense})", "as desired." ], "refs": [ "decent-spaces-lemma-finitely-many-irreducible-components", "decent-spaces-lemma-decent-generic-points", "spaces-properties-lemma-codimension-0-points", "morphisms-lemma-quasi-finite-at-a-finite-number-of-points", "morphisms-lemma-generically-finite", "spaces-properties-lemma-separated-cover", "morphisms-lemma-finite-over-dense-open", "decent-spaces-lemma-generically-finite", "decent-spaces-lemma-generically-finite-reprise", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-morphisms-lemma-finite-proper", "spaces-properties-lemma-codimension-0-points-dense" ], "ref_ids": [ 9534, 9531, 11842, 5234, 5487, 11833, 5489, 9535, 9536, 4920, 4946, 11843 ] } ], "ref_ids": [] }, { "id": 9539, "type": "theorem", "label": "decent-spaces-lemma-birational-dominant", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-birational-dominant", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which", "are decent and have finitely many irreducible components. If $f$ is", "birational then $f$ is dominant." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from the definitions. See", "Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-dominant}." ], "refs": [ "spaces-morphisms-definition-dominant" ], "ref_ids": [ 4996 ] } ], "ref_ids": [] }, { "id": 9540, "type": "theorem", "label": "decent-spaces-lemma-birational-generic-fibres", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-birational-generic-fibres", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a birational morphism of", "algebraic spaces over $S$ which are decent and have finitely", "many irreducible components. If $y \\in Y$ is the generic point of", "an irreducible component, then the base change", "$X \\times_Y \\Spec(\\mathcal{O}_{Y, y}) \\to \\Spec(\\mathcal{O}_{Y, y})$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Let $X' \\subset X$ and $Y' \\subset Y$ be the maximal open subspaces", "which are representable, see", "Lemma \\ref{lemma-finitely-many-irreducible-components}. By", "Lemma \\ref{lemma-quasi-finiteness-over-generic-point}", "the fibre of $f$ over $y$ is consists", "of points of codimension $0$ of $X$ and is therefore contained", "in $X'$. Hence $X \\times_Y \\Spec(\\mathcal{O}_{Y, y}) =", "X' \\times_{Y'} \\Spec(\\mathcal{O}_{Y', y})$ and the result follows", "from Morphisms, Lemma \\ref{morphisms-lemma-birational-generic-fibres}." ], "refs": [ "decent-spaces-lemma-finitely-many-irreducible-components", "decent-spaces-lemma-quasi-finiteness-over-generic-point", "morphisms-lemma-birational-generic-fibres" ], "ref_ids": [ 9534, 9537, 5483 ] } ], "ref_ids": [] }, { "id": 9541, "type": "theorem", "label": "decent-spaces-lemma-birational-birational", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-birational-birational", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a birational morphism of", "algebraic spaces over $S$ which are decent and have finitely many", "irreducible components. Assume one of the following conditions is satisfied", "\\begin{enumerate}", "\\item $f$ is locally of finite type and $Y$ reduced (i.e., integral),", "\\item $f$ is locally of finite presentation.", "\\end{enumerate}", "Then there exist dense opens $U \\subset X$ and $V \\subset Y$", "such that $f(U) \\subset V$ and $f|_U : U \\to V$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finitely-many-irreducible-components} we may assume", "that $X$ and $Y$ are schemes. In this case the result is", "Morphisms, Lemma \\ref{morphisms-lemma-birational-birational}." ], "refs": [ "decent-spaces-lemma-finitely-many-irreducible-components", "morphisms-lemma-birational-birational" ], "ref_ids": [ 9534, 5484 ] } ], "ref_ids": [] }, { "id": 9542, "type": "theorem", "label": "decent-spaces-lemma-birational-isomorphism-over-dense-open", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-birational-isomorphism-over-dense-open", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a birational morphism of", "algebraic spaces over $S$ which are decent and have finitely", "many irreducible components. Assume", "\\begin{enumerate}", "\\item either $f$ is quasi-compact or $f$ is separated, and", "\\item either $f$ is locally of finite type and $Y$ is reduced or", "$f$ is locally of finite presentation.", "\\end{enumerate}", "Then there exists a dense open $V \\subset Y$", "such that $f^{-1}(V) \\to V$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finitely-many-irreducible-components} we may assume", "$Y$ is a scheme. By Lemma \\ref{lemma-finite-over-dense-open} we may assume", "that $f$ is finite. Then $X$ is a scheme too and the result follows from", "Morphisms, Lemma \\ref{morphisms-lemma-birational-isomorphism-over-dense-open}." ], "refs": [ "decent-spaces-lemma-finitely-many-irreducible-components", "decent-spaces-lemma-finite-over-dense-open", "morphisms-lemma-birational-isomorphism-over-dense-open" ], "ref_ids": [ 9534, 9538, 5490 ] } ], "ref_ids": [] }, { "id": 9543, "type": "theorem", "label": "decent-spaces-lemma-birational-etale-localization", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-birational-etale-localization", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$ which are decent and have finitely many irreducible", "components. If $f$ is birational and $V \\to Y$ is an \\'etale morphism", "with $V$ affine, then $X \\times_Y V$ is decent with finitely", "many irreducible components and $X \\times_Y V \\to V$ is birational." ], "refs": [], "proofs": [ { "contents": [ "The algebraic space $U = X \\times_Y V$ is decent", "(Lemma \\ref{lemma-etale-named-properties}).", "The generic points of $V$ and $U$ are the elements of $|V|$ and $|U|$", "which lie over generic points of $|Y|$ and $|X|$", "(Lemma \\ref{lemma-decent-generic-points}).", "Since $Y$ is decent we conclude there are finitely many generic points", "on $V$. Let $\\xi \\in |X|$ be a generic point of an irreducible component.", "By the discussion following Definition \\ref{definition-birational}", "we have a cartesian square", "$$", "\\xymatrix{", "\\Spec(\\mathcal{O}_{X, \\xi}) \\ar[d] \\ar[r] & X \\ar[d] \\\\", "\\Spec(\\mathcal{O}_{Y, f(\\xi)}) \\ar[r] & Y", "}", "$$", "whose horizontal morphisms are monomorphisms identifying local rings", "and where the left vertical arrow is an isomorphism. It follows that", "in the diagram", "$$", "\\xymatrix{", "\\Spec(\\mathcal{O}_{X, \\xi}) \\times_X U \\ar[d] \\ar[r] & U \\ar[d] \\\\", "\\Spec(\\mathcal{O}_{Y, f(\\xi)}) \\times_Y V \\ar[r] & V", "}", "$$", "the vertical arrow on the left is an isomorphism. The horizonal arrows", "have image contained in the schematic locus of $U$ and $V$ and", "identify local rings (some details omitted). Since the image of", "the horizontal arrows are the points of $|U|$, resp.\\ $|V|$", "lying over $\\xi$, resp.\\ $f(\\xi)$ we conclude." ], "refs": [ "decent-spaces-lemma-etale-named-properties", "decent-spaces-lemma-decent-generic-points", "decent-spaces-definition-birational" ], "ref_ids": [ 9471, 9531, 9568 ] } ], "ref_ids": [] }, { "id": 9544, "type": "theorem", "label": "decent-spaces-lemma-birational-induced-morphism-normalizations", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-birational-induced-morphism-normalizations", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a birational morphism between", "algebraic spaces over $S$ which are decent and have finitely many irreducible", "components. Then the normalizations $X^\\nu \\to X$ and $Y^\\nu \\to Y$ exist", "and there is a commutative diagram", "$$", "\\xymatrix{", "X^\\nu \\ar[r] \\ar[d] & Y^\\nu \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "of algebraic spaces over $S$. The morphism $X^\\nu \\to Y^\\nu$ is birational." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finitely-many-irreducible-components} we see that", "$X$ and $Y$ satisfy the equivalent conditions of", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-prepare-normalization}", "and the normalizations are defined. By", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-normalization-normal}", "the algebraic space $X^\\nu$ is normal and maps codimension $0$ points", "to codimension $0$ points. Since $f$ maps codimension $0$ points to", "codimension $0$ points (this is the same as generic points on decent", "spaces by Lemma \\ref{lemma-decent-generic-points})", "we obtain from", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-normalization-normal}", "a factorization of the composition $X^\\nu \\to X \\to Y$ through $Y^\\nu$.", "\\medskip\\noindent", "Observe that $X^\\nu$ and $Y^\\nu$ are decent for example by", "Lemma \\ref{lemma-representable-named-properties}.", "Moreover the maps $X^\\nu \\to X$ and $Y^\\nu \\to Y$", "induce bijections on irreducible components (see references above)", "hence $X^\\nu$ and $Y^\\nu$ both have a finite number of irreducible", "components and the map $X^\\nu \\to Y^\\nu$ induces a bijection", "between their generic points.", "To prove that $X^\\nu \\to Y^\\nu$ is birational, it therefore", "suffices to show it induces an isomorphism on local rings at", "these points. To do this we may replace $X$ and $Y$ by open neighbourhoods", "of their generic points, hence we may assume $X$ and $Y$ are affine", "irreducible schemes with generic points $x$ and $y$. Since", "$f$ is birational the map $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{Y, y}$", "is an isomorphism. Let $x^\\nu \\in X^\\nu$ and $y^\\nu \\in Y^\\nu$ be", "the points lying over $x$ and $y$.", "By construction of the normalization", "we see that $\\mathcal{O}_{X^\\nu, x^\\nu} = \\mathcal{O}_{X, x}/\\mathfrak m_x$", "and similarly on $Y$. Thus the map", "$\\mathcal{O}_{X^\\nu, x^\\nu} \\to \\mathcal{O}_{Y^\\nu, y^\\nu}$", "is an isomorphism as well." ], "refs": [ "decent-spaces-lemma-finitely-many-irreducible-components", "spaces-morphisms-lemma-prepare-normalization", "spaces-morphisms-lemma-normalization-normal", "decent-spaces-lemma-decent-generic-points", "spaces-morphisms-lemma-normalization-normal", "decent-spaces-lemma-representable-named-properties" ], "ref_ids": [ 9534, 4966, 4969, 9531, 4969, 9470 ] } ], "ref_ids": [] }, { "id": 9545, "type": "theorem", "label": "decent-spaces-lemma-finite-birational-over-normal", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-finite-birational-over-normal", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. Assume", "\\begin{enumerate}", "\\item $X$ and $Y$ are decent and have finitely many irreducible components,", "\\item $f$ is integral and birational,", "\\item $Y$ is normal, and", "\\item $X$ is reduced.", "\\end{enumerate}", "Then $f$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\to Y$ be an \\'etale morphism with $V$ affine. It suffices to show that", "$U = X \\times_Y V \\to V$ is an isomorphism. By", "Lemma \\ref{lemma-birational-etale-localization} and its proof", "we see that $U$ and $V$ are decent and have finitely many", "irreducible components and that $U \\to V$ is birational.", "By Properties, Lemma", "\\ref{properties-lemma-normal-locally-finite-nr-irreducibles}", "$V$ is a finite disjoint union of integral schemes.", "Thus we may assume $V$ is integral. As $f$ is birational, we", "see that $U$ is irreducible and reduced, i.e., integral", "(note that $U$ is a scheme as $f$ is integral, hence representable).", "Thus we may assume that $X$ and $Y$ are integral schemes", "and the result follows from the case of schemes, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-birational-over-normal}." ], "refs": [ "decent-spaces-lemma-birational-etale-localization", "properties-lemma-normal-locally-finite-nr-irreducibles", "morphisms-lemma-finite-birational-over-normal" ], "ref_ids": [ 9543, 2969, 5518 ] } ], "ref_ids": [] }, { "id": 9546, "type": "theorem", "label": "decent-spaces-lemma-normalization-normal", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-normalization-normal", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an integral birational morphism of", "decent algebraic spaces over $S$ which have finitely many irreducible", "components. Then there exists a factorization $Y^\\nu \\to X \\to Y$ and", "$Y^\\nu \\to X$ is the normalization of $X$." ], "refs": [], "proofs": [ { "contents": [ "Consider the map $X^\\nu \\to Y^\\nu$ of", "Lemma \\ref{lemma-birational-induced-morphism-normalizations}.", "This map is integral by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-permanence}.", "Hence it is an isomorphism by", "Lemma \\ref{lemma-finite-birational-over-normal}." ], "refs": [ "decent-spaces-lemma-birational-induced-morphism-normalizations", "spaces-morphisms-lemma-finite-permanence", "decent-spaces-lemma-finite-birational-over-normal" ], "ref_ids": [ 9544, 4949, 9545 ] } ], "ref_ids": [] }, { "id": 9547, "type": "theorem", "label": "decent-spaces-lemma-Jacobson-universally-Jacobson", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-Jacobson-universally-Jacobson", "contents": [ "Let $S$ be a scheme. Let $X$ be a Jacobson algebraic space over $S$.", "Any algebraic space locally of finite type over $X$ is Jacobson." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be a surjective \\'etale morphism where $U$ is a scheme.", "Then $U$ is Jacobson (by definition) and for a morphism of schemes $V \\to U$", "which is locally of finite type we see that $V$ is Jacobson by the", "corresponding result for schemes (Morphisms, Lemma", "\\ref{morphisms-lemma-Jacobson-universally-Jacobson}).", "Thus if $Y \\to X$ is a morphism of algebraic spaces which is locally", "of finite type, then setting $V = U \\times_X Y$ we see that", "$Y$ is Jacobson by definition." ], "refs": [ "morphisms-lemma-Jacobson-universally-Jacobson" ], "ref_ids": [ 5212 ] } ], "ref_ids": [] }, { "id": 9548, "type": "theorem", "label": "decent-spaces-lemma-Jacobson-ft-points-lift-to-closed", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-Jacobson-ft-points-lift-to-closed", "contents": [ "Let $S$ be a scheme. Let $X$ be a Jacobson algebraic space over $S$.", "For $x \\in X_{\\text{ft-pts}}$ and $g : W \\to X$ locally of finite type", "with $W$ a scheme, if $x \\in \\Im(|g|)$, then there exists a closed", "point of $W$ mapping to $x$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be an \\'etale morphism with $U$ a scheme and with $u \\in U$", "closed mapping to $x$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-identify-finite-type-points}.", "Observe that $W$, $W \\times_X U$, and $U$ are Jacobson schemes", "by Lemma \\ref{lemma-Jacobson-universally-Jacobson}.", "Hence finite type points on these schemes", "are the same thing as closed points by", "Morphisms, Lemma \\ref{morphisms-lemma-jacobson-finite-type-points}.", "The inverse image $T \\subset W \\times_X U$ of $u$ is a nonempty", "(as $x$ in the image of $W \\to X$) closed subset.", "By Morphisms, Lemma \\ref{morphisms-lemma-enough-finite-type-points}", "there is a closed point $t$ of $W \\times_X U$ which maps to $u$.", "As $W \\times_X U \\to W$ is locally of finite type", "the image of $t$ in $W$ is closed by", "Morphisms, Lemma \\ref{morphisms-lemma-jacobson-finite-type-points}." ], "refs": [ "spaces-morphisms-lemma-identify-finite-type-points", "decent-spaces-lemma-Jacobson-universally-Jacobson", "morphisms-lemma-jacobson-finite-type-points", "morphisms-lemma-enough-finite-type-points", "morphisms-lemma-jacobson-finite-type-points" ], "ref_ids": [ 4823, 9547, 5211, 5210, 5211 ] } ], "ref_ids": [] }, { "id": 9549, "type": "theorem", "label": "decent-spaces-lemma-decent-Jacobson-ft-pts", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-decent-Jacobson-ft-pts", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent Jacobson algebraic space over $S$.", "Then $X_{\\text{ft-pts}} \\subset |X|$ is the set of closed points." ], "refs": [], "proofs": [ { "contents": [ "If $x \\in |X|$ is closed, then we can represent $x$ by a closed", "immersion $\\Spec(k) \\to X$, see Lemma \\ref{lemma-decent-space-closed-point}.", "Hence $x$ is certainly a finite type point.", "\\medskip\\noindent", "Conversely, let $x \\in |X|$ be a finite type point. We know that", "$x$ can be represented by a quasi-compact monomorphism", "$\\Spec(k) \\to X$ where $k$ is a field", "(Definition \\ref{definition-very-reasonable}). On the other hand,", "by definition, there exists a morphism $\\Spec(k') \\to X$", "which is locally of finite type and represents $x$", "(Morphisms, Definition \\ref{morphisms-definition-finite-type-point}).", "We obtain a factorization $\\Spec(k') \\to \\Spec(k) \\to X$.", "Let $U \\to X$ be any \\'etale morphism with $U$ affine and consider", "the morphisms", "$$", "\\Spec(k') \\times_X U \\to \\Spec(k) \\times_X U \\to U", "$$", "The quasi-compact scheme $\\Spec(k) \\times_X U$ is \\'etale over", "$\\Spec(k)$ hence is a finite disjoint union", "of spectra of fields (Remark \\ref{remark-recall}).", "Moreover, the first morphism is surjective and locally of finite type", "(Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type})", "hence surjective on finite type points", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-type-points-surjective-morphism})", "and the composition (which is locally of finite type) sends", "finite type points to closed points as $U$ is Jacobson", "(Morphisms, Lemma \\ref{morphisms-lemma-jacobson-finite-type-points}).", "Thus the image of", "$\\Spec(k) \\times_X U \\to U$ is a finite set of closed points hence", "closed. Since this is true for every affine $U$ and \\'etale morphism", "$U \\to X$, we conclude that $x \\in |X|$ is closed." ], "refs": [ "decent-spaces-lemma-decent-space-closed-point", "decent-spaces-definition-very-reasonable", "morphisms-definition-finite-type-point", "decent-spaces-remark-recall", "morphisms-lemma-permanence-finite-type", "morphisms-lemma-finite-type-points-surjective-morphism", "morphisms-lemma-jacobson-finite-type-points" ], "ref_ids": [ 9510, 9562, 5550, 9573, 5204, 5209, 5211 ] } ], "ref_ids": [] }, { "id": 9550, "type": "theorem", "label": "decent-spaces-lemma-decent-Jacobson", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-decent-Jacobson", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Then $X$ is Jacobson if and only if $|X|$ is Jacobson." ], "refs": [], "proofs": [ { "contents": [ "Assume $X$ is Jacobson and that $T \\subset |X|$ is a closed subset.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-enough-finite-type-points}", "we see that $T \\cap X_{\\text{ft-pts}}$ is dense in $T$.", "By Lemma \\ref{lemma-decent-Jacobson-ft-pts} we see that", "$X_{\\text{ft-pts}}$ are the", "closed points of $|X|$. Thus $|X|$ is indeed Jacobson.", "\\medskip\\noindent", "Assume $|X|$ is Jacobson. Let $f : U \\to X$ be an \\'etale", "morphism with $U$ an affine scheme. We have to show that $U$", "is Jacobson. If $x \\in |X|$ is closed,", "then the fibre $F = f^{-1}(\\{x\\})$ is a finite (by definition of", "decent) closed (by construction of the topology on $|X|$) subset of $U$.", "Since there are no specializations between points of $F$", "(Lemma \\ref{lemma-decent-no-specializations-map-to-same-point})", "we conclude that every point of $F$ is closed in $U$.", "If $U$ is not Jacobson, then there exists a non-closed point", "$u \\in U$ such that $\\{u\\}$ is locally closed (Topology, Lemma", "\\ref{topology-lemma-non-jacobson-Noetherian-characterize}).", "We will show that $f(u) \\in |X|$ is closed; by the above $u$", "is closed in $U$ which is a contradiction and finishes", "the proof. To prove this we may replace $U$ by an affine open", "neighbourhood of $u$.", "Thus we may assume that $\\{u\\}$ is closed in $U$.", "Let $R = U \\times_X U$ with projections $s, t : R \\to U$.", "Then $s^{-1}(\\{u\\}) = \\{r_1, \\ldots, r_m\\}$ is finite (by", "definition of decent spaces). After replacing $U$ by a smaller affine", "open neighbourhood of $u$ we may assume that $t(r_j) = u$ for", "$j = 1, \\ldots, m$. It follows that $\\{u\\}$ is an $R$-invariant", "closed subset of $U$. Hence $\\{f(u)\\}$ is a locally closed subset", "of $X$ as it is closed in the open $|f|(|U|)$ of $|X|$. Since $|X|$", "is Jacobson we conclude that $f(u)$ is closed in $|X|$ as desired." ], "refs": [ "spaces-morphisms-lemma-enough-finite-type-points", "decent-spaces-lemma-decent-Jacobson-ft-pts", "decent-spaces-lemma-decent-no-specializations-map-to-same-point", "topology-lemma-non-jacobson-Noetherian-characterize" ], "ref_ids": [ 4826, 9549, 9493, 8277 ] } ], "ref_ids": [] }, { "id": 9551, "type": "theorem", "label": "decent-spaces-lemma-punctured-spec", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-punctured-spec", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent locally Noetherian algebraic", "space over $S$. Let $x \\in |X|$. Then", "$$", "W = \\{x' \\in |X| : x' \\leadsto x,\\ x' \\not = x\\}", "$$", "is a Noetherian, spectral, sober, Jacobson topological space." ], "refs": [], "proofs": [ { "contents": [ "We may replace by any open subspace containing $x$.", "Thus we may assume that $X$ is quasi-compact.", "Then $|X|$ is a Noetherian topological space", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-Noetherian-topology}).", "Thus $W$ is a Noetherian topological space", "(Topology, Lemma \\ref{topology-lemma-Noetherian}).", "\\medskip\\noindent", "Combining Lemma \\ref{lemma-locally-Noetherian-decent-quasi-separated} with", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-quasi-separated-spectral}", "we see that $|X|$ is a spectral toplogical space.", "By Topology, Lemma \\ref{topology-lemma-make-spectral-space}", "we see that $W \\cup \\{x\\}$ is a spectral topological space.", "Now $W$ is a quasi-compact open of $W \\cup \\{x\\}$ and hence $W$ is", "spectral by Topology, Lemma \\ref{topology-lemma-spectral-sub}.", "\\medskip\\noindent", "Let $E \\subset W$ be an irreducible closed subset. Then if $Z \\subset |X|$", "is the closure of $E$ we see that $x \\in Z$. There is a unique generic", "point $\\eta \\in Z$ by Proposition \\ref{proposition-reasonable-sober}.", "Of course $\\eta \\in W$ and hence $\\eta \\in E$. We conclude that $E$", "has a unique generic point, i.e., $W$ is sober.", "\\medskip\\noindent", "Let $x' \\in W$ be a point such that $\\{x'\\}$ is locally closed in $W$.", "To finish the proof we have to show that $x'$ is a closed point of $W$.", "If not, then there exists a nontrivial specialization $x' \\leadsto x'_1$", "in $W$. Let $U$ be an affine scheme, $u \\in U$ a point, and let $U \\to X$", "be an \\'etale morphism mapping $u$ to $x$. By", "Lemma \\ref{lemma-decent-specialization}", "we can choose specializations $u' \\leadsto u'_1 \\leadsto u$", "mapping to $x' \\leadsto x'_1 \\leadsto x$.", "Let $\\mathfrak p' \\subset \\mathcal{O}_{U, u}$ be the prime ideal", "corresponding to $u'$. The existence of the specializations", "implies that $\\dim(\\mathcal{O}_{U, u}/\\mathfrak p') \\geq 2$.", "Hence every nonempty open of $\\Spec(\\mathcal{O}_{U, u}/\\mathfrak p')$", "is infinite by Algebra, Lemma", "\\ref{algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens}.", "By Lemma \\ref{lemma-decent-no-specializations-map-to-same-point}", "we obtain a continuous map", "$$", "\\Spec(\\mathcal{O}_{U, u}/\\mathfrak p')", "\\setminus \\{\\mathfrak m_u/\\mathfrak p'\\}", "\\longrightarrow", "W", "$$", "Since the generic point of the LHS maps to $x'$ the image is", "contained in $\\overline{\\{x'\\}}$. We conclude the inverse image of $\\{x'\\}$", "under the displayed arrow is nonempty open hence infinite.", "However, the fibres of $U \\to X$ are finite as $X$", "is decent and we conclude that $\\{x'\\}$ is infinite.", "This contradiction finishes the proof." ], "refs": [ "spaces-properties-lemma-Noetherian-topology", "topology-lemma-Noetherian", "decent-spaces-lemma-locally-Noetherian-decent-quasi-separated", "spaces-properties-lemma-quasi-compact-quasi-separated-spectral", "topology-lemma-make-spectral-space", "topology-lemma-spectral-sub", "decent-spaces-proposition-reasonable-sober", "decent-spaces-lemma-decent-specialization", "algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens", "decent-spaces-lemma-decent-no-specializations-map-to-same-point" ], "ref_ids": [ 11891, 8220, 9506, 11853, 8323, 8306, 9559, 9494, 687, 9493 ] } ], "ref_ids": [] }, { "id": 9552, "type": "theorem", "label": "decent-spaces-lemma-irreducible-local-ring", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-irreducible-local-ring", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $x \\in |X|$ be a point. The following are equivalent", "\\begin{enumerate}", "\\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$", "the local ring $\\mathcal{O}_{U, u}$ has a unique minimal prime,", "\\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$", "there is a unique irreducible component of $U$ through $u$,", "\\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$", "the local ring $\\mathcal{O}_{U, u}$ is unibranch,", "\\item the henselian local ring", "$\\mathcal{O}_{X, x}^h$ has a unique minimal prime.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows from the fact that irreducible", "components of $U$ passing through $u$ are in $1$-$1$ correspondence with", "minimal primes of the local ring of $U$ at $u$. The ring", "$\\mathcal{O}_{X, x}^h$ is the henselization of $\\mathcal{O}_{U, u}$, see", "discussion following Definition \\ref{definition-henselian-local-ring}.", "In particular (3) and (4) are equivalent by", "More on Algebra, Lemma \\ref{more-algebra-lemma-unibranch}.", "The equivalence of (2) and (3) follows from", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-nr-branches}." ], "refs": [ "decent-spaces-definition-henselian-local-ring", "more-algebra-lemma-unibranch", "more-morphisms-lemma-nr-branches" ], "ref_ids": [ 9565, 10466, 13872 ] } ], "ref_ids": [] }, { "id": 9553, "type": "theorem", "label": "decent-spaces-lemma-nr-branches-local-ring", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-nr-branches-local-ring", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $x \\in |X|$ be a point. Let $n \\in \\{1, 2, \\ldots\\}$ be an integer.", "The following are equivalent", "\\begin{enumerate}", "\\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$", "the number of minimal primes of the local ring $\\mathcal{O}_{U, u}$", "is $\\leq n$ and for at least one choice of $(U, u)$ it is $n$,", "\\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$", "the number irreducible components of $U$ passing through $u$ is $\\leq n$", "and for at least one choice of $(U, u)$ it is $n$,", "\\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$", "the number of branches of $U$ at $u$ is $\\leq n$", "and for at least one choice of $(U, u)$ it is $n$,", "\\item the number of minimal prime ideals of", "$\\mathcal{O}_{X, x}^h$ is $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows from the fact that irreducible", "components of $U$ passing through $u$ are in $1$-$1$ correspondence with", "minimal primes of the local ring of $U$ at $u$.", "The ring $\\mathcal{O}_{X, x}$ is the henselization of $\\mathcal{O}_{U, u}$, see", "discussion following Definition \\ref{definition-henselian-local-ring}.", "In particular (3) and (4) are equivalent by", "More on Algebra, Lemma \\ref{more-algebra-lemma-unibranch}.", "The equivalence of (2) and (3) follows from", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-nr-branches}." ], "refs": [ "decent-spaces-definition-henselian-local-ring", "more-algebra-lemma-unibranch", "more-morphisms-lemma-nr-branches" ], "ref_ids": [ 9565, 10466, 13872 ] } ], "ref_ids": [] }, { "id": 9554, "type": "theorem", "label": "decent-spaces-lemma-scheme-with-dimension-function", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-scheme-with-dimension-function", "contents": [ "Let $S$ be a locally Noetherian and universally catenary scheme.", "Let $\\delta : S \\to \\mathbf{Z}$ be a dimension function.", "Let $X$ be a decent algebraic space over $S$ such that", "the structure morphism $X \\to S$ is locally of", "finite type. Let $\\delta_X : |X| \\to \\mathbf{Z}$ be the map", "sending $x$ to $\\delta(f(x))$ plus the transcendence degree", "of $x/f(x)$. Then $\\delta_X$ is a dimension function on $|X|$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi : U \\to X$ be a surjective \\'etale morphism where $U$ is a scheme.", "Then the similarly defined function $\\delta_U$ is a", "dimension function on $U$ by", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-function-propagates}.", "On the other hand, by the definition of relative transcendence degree in", "(Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-dimension-fibre}) we see", "that $\\delta_U(u) = \\delta_X(\\varphi(u))$.", "\\medskip\\noindent", "Let $x \\leadsto x'$ be a specialization of points in $|X|$.", "by Lemma \\ref{lemma-decent-specialization} we can find", "a specialization $u \\leadsto u'$ of points of $U$ with", "$\\varphi(u) = x$ and $\\varphi(u') = x'$. Moreover, we see", "that $x = x'$ if and only if $u = u'$, see", "Lemma \\ref{lemma-decent-no-specializations-map-to-same-point}.", "Thus the fact that $\\delta_U$ is a dimension function implies that", "$\\delta_X$ is a dimension function, see", "Topology, Definition \\ref{topology-definition-dimension-function}." ], "refs": [ "morphisms-lemma-dimension-function-propagates", "spaces-morphisms-definition-dimension-fibre", "decent-spaces-lemma-decent-specialization", "decent-spaces-lemma-decent-no-specializations-map-to-same-point", "topology-definition-dimension-function" ], "ref_ids": [ 5495, 5009, 9494, 9493, 8367 ] } ], "ref_ids": [] }, { "id": 9555, "type": "theorem", "label": "decent-spaces-lemma-universally-catenary-scheme", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-universally-catenary-scheme", "contents": [ "Let $S$ be a locally Noetherian and universally catenary scheme.", "Let $X$ be an algebraic space over $S$ such that $X$ is decent", "and such that the structure morphism $X \\to S$ is locally of", "finite type. Then $X$ is catenary." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $S$ (use", "Topology, Lemma \\ref{topology-lemma-catenary}).", "Thus we may assume that $S$ has a", "dimension function, see Topology, Lemma", "\\ref{topology-lemma-locally-dimension-function}.", "Then we conclude that $|X|$ has a dimension function by", "Lemma \\ref{lemma-scheme-with-dimension-function}.", "Since $|X|$ is sober (Proposition \\ref{proposition-reasonable-sober})", "we conclude that $|X|$ is catenary by", "Topology, Lemma \\ref{topology-lemma-dimension-function-catenary}." ], "refs": [ "topology-lemma-catenary", "topology-lemma-locally-dimension-function", "decent-spaces-lemma-scheme-with-dimension-function", "decent-spaces-proposition-reasonable-sober", "topology-lemma-dimension-function-catenary" ], "ref_ids": [ 8226, 8293, 9554, 9559, 8291 ] } ], "ref_ids": [] }, { "id": 9556, "type": "theorem", "label": "decent-spaces-lemma-universally-catenary", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-universally-catenary", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent, locally Noetherian, and", "universally catenary algebraic space over $S$. Then any decent algebraic", "space locally of finite type over $X$ is universally catenary." ], "refs": [], "proofs": [ { "contents": [ "This is formal from the definitions and the fact that", "compositions of morphisms locally of finite type are", "locally of finite type (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-composition-finite-type})." ], "refs": [ "spaces-morphisms-lemma-composition-finite-type" ], "ref_ids": [ 4814 ] } ], "ref_ids": [] }, { "id": 9557, "type": "theorem", "label": "decent-spaces-lemma-check-dimension-function-finite-cover", "categories": [ "decent-spaces" ], "title": "decent-spaces-lemma-check-dimension-function-finite-cover", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a surjective finite morphism of", "decent and locally Noetherian algebraic spaces. Let", "$\\delta : |X| \\to \\mathbf{Z}$ be a function. If $\\delta \\circ |f|$ is a", "dimension function, then $\\delta$ is a dimension function." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\mapsto x'$, $x \\not = x'$ be a specialization in $|X|$.", "Choose $y \\in |Y|$ with $|f|(y) = x$. Since $|f|$ is closed", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-proper})", "we find a specialization $y \\leadsto y'$ with $|f|(y') = x'$.", "Thus we conclude that", "$\\delta(x) = \\delta(|f|(y)) > \\delta(|f|(y')) = \\delta(x')$", "(see Topology, Definition \\ref{topology-definition-dimension-function}).", "If $x \\leadsto x'$ is an immediate specialization, then", "$y \\leadsto y'$ is an immediate specialization too:", "namely if $y \\leadsto y'' \\leadsto y'$, then $|f|(y'')$", "must be either $x$ or $x'$ and there are no nontrivial", "specializations between points of fibres of $|f|$ by", "Lemma \\ref{lemma-conditions-on-fibre-and-qf}." ], "refs": [ "spaces-morphisms-lemma-finite-proper", "topology-definition-dimension-function", "decent-spaces-lemma-conditions-on-fibre-and-qf" ], "ref_ids": [ 4946, 8367, 9529 ] } ], "ref_ids": [] }, { "id": 9558, "type": "theorem", "label": "decent-spaces-proposition-reasonable-open-dense-scheme", "categories": [ "decent-spaces" ], "title": "decent-spaces-proposition-reasonable-open-dense-scheme", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "If $X$ is reasonable, then there exists a dense open subspace", "of $X$ which is a scheme." ], "refs": [], "proofs": [ { "contents": [ "By Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-subscheme}", "the question is local on $X$. Hence we may assume there exists an affine", "scheme $U$ and a surjective \\'etale morphism $U \\to X$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-cover-by-union-affines}).", "Let $n$ be an integer bounding the degrees of the fibres of $U \\to X$", "which exists as $X$ is reasonable, see", "Definition \\ref{definition-very-reasonable}.", "We will argue by induction on $n$ that whenever", "\\begin{enumerate}", "\\item $U \\to X$ is a surjective \\'etale morphism whose fibres have", "degree $\\leq n$, and", "\\item $U$ is isomorphic to a locally closed subscheme of an affine scheme", "\\end{enumerate}", "then the schematic locus is dense in $X$.", "\\medskip\\noindent", "Let $X_n \\subset X$ be the open subspace which is the complement of the", "closed subspace $Z_{n - 1} \\subset X$ constructed in", "Lemma \\ref{lemma-quasi-compact-reasonable-stratification}", "using the morphism $U \\to X$.", "Let $U_n \\subset U$ be the inverse image of $X_n$. Then", "$U_n \\to X_n$ is finite locally free of degree $n$.", "Hence $X_n$ is a scheme by", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-finite-flat-equivalence-global}", "(and the fact that any finite set of points of $U_n$ is contained in", "an affine open of $U_n$, see", "Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-affine}).", "\\medskip\\noindent", "Let $X' \\subset X$ be the open subspace such that $|X'|$ is the", "interior of $|Z_{n - 1}|$ in $|X|$ (see", "Topology, Definition \\ref{topology-definition-nowhere-dense}).", "Let $U' \\subset U$ be the inverse image. Then $U' \\to X'$ is surjective", "\\'etale and has degrees of fibres bounded by $n - 1$. By induction", "we see that the schematic locus of $X'$ is an open dense $X'' \\subset X'$.", "By elementary topology we see that $X'' \\cup X_n \\subset X$ is", "open and dense and we win." ], "refs": [ "spaces-properties-lemma-subscheme", "spaces-properties-lemma-cover-by-union-affines", "decent-spaces-definition-very-reasonable", "decent-spaces-lemma-quasi-compact-reasonable-stratification", "spaces-properties-proposition-finite-flat-equivalence-global", "properties-lemma-ample-finite-set-in-affine", "topology-definition-nowhere-dense" ], "ref_ids": [ 11848, 11830, 9562, 9475, 11918, 3062, 8368 ] } ], "ref_ids": [] }, { "id": 9559, "type": "theorem", "label": "decent-spaces-proposition-reasonable-sober", "categories": [ "decent-spaces" ], "title": "decent-spaces-proposition-reasonable-sober", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Then the topological space $|X|$ is sober (see", "Topology, Definition \\ref{topology-definition-generic-point})." ], "refs": [ "topology-definition-generic-point" ], "proofs": [ { "contents": [ "We have seen in Lemma \\ref{lemma-kolmogorov} that $|X|$ is Kolmogorov.", "Hence it remains to show that every irreducible closed subset", "$T \\subset |X|$ has a generic point. By", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-reduced-closed-subspace}", "there exists a closed subspace $Z \\subset X$ with $|Z| = |T|$.", "By definition this means that $Z \\to X$ is a representable morphism", "of algebraic spaces. Hence $Z$ is a decent algebraic space", "by Lemma \\ref{lemma-representable-properties}. By", "Theorem \\ref{theorem-decent-open-dense-scheme}", "we see that there exists an open dense subspace $Z' \\subset Z$", "which is a scheme. This means that $|Z'| \\subset T$ is open dense.", "Hence the topological space $|Z'|$ is irreducible, which means that", "$Z'$ is an irreducible scheme. By", "Schemes, Lemma \\ref{schemes-lemma-scheme-sober}", "we conclude that $|Z'|$ is the closure of a single point $\\eta \\in T$", "and hence also $T = \\overline{\\{\\eta\\}}$, and we win." ], "refs": [ "decent-spaces-lemma-kolmogorov", "spaces-properties-lemma-reduced-closed-subspace", "decent-spaces-lemma-representable-properties", "decent-spaces-theorem-decent-open-dense-scheme", "schemes-lemma-scheme-sober" ], "ref_ids": [ 9495, 11846, 9468, 9454, 7672 ] } ], "ref_ids": [ 8354 ] }, { "id": 9560, "type": "theorem", "label": "decent-spaces-proposition-characterize-universally-closed", "categories": [ "decent-spaces" ], "title": "decent-spaces-proposition-characterize-universally-closed", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is quasi-compact, and $X$ is decent. Then $f$ is", "universally closed if and only if the existence part of the valuative", "criterion holds." ], "refs": [], "proofs": [ { "contents": [ "In", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-quasi-compact-existence-universally-closed}", "we have seen one of the implications.", "To prove the other, assume that $f$ is universally closed. Let", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] & Y", "}", "$$", "be a diagram as in", "Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-valuative-criterion}.", "Let $X_A = \\Spec(A) \\times_Y X$, so that we have", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[rd] & X_A \\ar[d] \\\\", " & \\Spec(A)", "}", "$$", "By", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-base-change-quasi-compact}", "we see that $X_A \\to \\Spec(A)$ is quasi-compact. Since $X_A \\to X$", "is representable, we see that $X_A$ is decent also, see", "Lemma \\ref{lemma-representable-properties}.", "Moreover, as $f$ is universally closed, we see that $X_A \\to \\Spec(A)$", "is universally closed.", "Hence we may and do replace $X$ by $X_A$ and $Y$ by $\\Spec(A)$.", "\\medskip\\noindent", "Let $x' \\in |X|$ be the equivalence class of", "$\\Spec(K) \\to X$. Let $y \\in |Y| = |\\Spec(A)|$ be", "the closed point. Set $y' = f(x')$; it is the generic point of", "$\\Spec(A)$. Since $f$ is universally closed we see that", "$f(\\overline{\\{x'\\}})$ contains $\\overline{\\{y'\\}}$, and hence", "contains $y$. Let $x \\in \\overline{\\{x'\\}}$ be a point such that", "$f(x) = y$. Let $U$ be a scheme, and $\\varphi : U \\to X$", "an \\'etale morphism such that there exists a $u \\in U$ with", "$\\varphi(u) = x$. By", "Lemma \\ref{lemma-specialization}", "and our assumption that $X$ is decent", "there exists a specialization $u' \\leadsto u$ on $U$ with $\\varphi(u') = x'$.", "This means that there exists a common field extension", "$K \\subset K' \\supset \\kappa(u')$ such that", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & U \\ar[d] \\\\", "\\Spec(K) \\ar[r] \\ar[rd] & X \\ar[d] \\\\", " & \\Spec(A)", "}", "$$", "is commutative. This gives the following commutative diagram of rings", "$$", "\\xymatrix{", "K' & \\mathcal{O}_{U, u} \\ar[l] \\\\", "K \\ar[u] & \\\\", " & A \\ar[lu] \\ar[uu]", "}", "$$", "By", "Algebra, Lemma \\ref{algebra-lemma-dominate}", "we can find a valuation ring $A' \\subset K'$ dominating the image of", "$\\mathcal{O}_{U, u}$ in $K'$. Since by construction $\\mathcal{O}_{U, u}$", "dominates $A$ we see that $A'$ dominates $A$ also. Hence we obtain a diagram", "resembling the second diagram of", "Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-valuative-criterion}", "and the proposition is proved." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-existence-universally-closed", "spaces-morphisms-definition-valuative-criterion", "spaces-morphisms-lemma-base-change-quasi-compact", "decent-spaces-lemma-representable-properties", "decent-spaces-lemma-specialization", "algebra-lemma-dominate", "spaces-morphisms-definition-valuative-criterion" ], "ref_ids": [ 4930, 5016, 4738, 9468, 9473, 608, 5016 ] } ], "ref_ids": [] }, { "id": 9578, "type": "theorem", "label": "groupoids-lemma-restrict-relation", "categories": [ "groupoids" ], "title": "groupoids-lemma-restrict-relation", "contents": [ "Let $S$ be a scheme.", "Let $U$ be a scheme over $S$.", "Let $j : R \\to U \\times_S U$ be a pre-relation.", "Let $g : U' \\to U$ be a morphism of schemes.", "Finally, set", "$$", "R' = (U' \\times_S U')\\times_{U \\times_S U} R", "\\xrightarrow{j'}", "U' \\times_S U'", "$$", "Then $j'$ is a pre-relation on $U'$ over $S$.", "If $j$ is a relation, then $j'$ is a relation.", "If $j$ is a pre-equivalence relation, then $j'$ is a pre-equivalence relation.", "If $j$ is an equivalence relation, then $j'$ is an equivalence relation." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9579, "type": "theorem", "label": "groupoids-lemma-pre-equivalence-equivalence-relation-points", "categories": [ "groupoids" ], "title": "groupoids-lemma-pre-equivalence-equivalence-relation-points", "contents": [ "Let $j : R \\to U \\times_S U$ be a pre-relation.", "Consider the relation on points of the scheme $U$ defined by", "the rule", "$$", "x \\sim y", "\\Leftrightarrow", "\\exists\\ r \\in R :", "t(r) = x,", "s(r) = y.", "$$", "If $j$ is a pre-equivalence relation then this is an", "equivalence relation." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $x \\sim y$ and $y \\sim z$.", "Pick $r \\in R$ with $t(r) = x$, $s(r) = y$ and", "pick $r' \\in R$ with $t(r') = y$, $s(r') = z$.", "Pick a field $K$ fitting into the following commutative", "diagram", "$$", "\\xymatrix{", "\\kappa(r) \\ar[r] & K \\\\", "\\kappa(y) \\ar[u] \\ar[r] & \\kappa(r') \\ar[u]", "}", "$$", "Denote $x_K, y_K, z_K : \\Spec(K) \\to U$", "the morphisms", "$$", "\\begin{matrix}", "\\Spec(K) \\to \\Spec(\\kappa(r))", "\\to", "\\Spec(\\kappa(x)) \\to U \\\\", "\\Spec(K) \\to \\Spec(\\kappa(r))", "\\to", "\\Spec(\\kappa(y)) \\to U \\\\", "\\Spec(K) \\to \\Spec(\\kappa(r'))", "\\to", "\\Spec(\\kappa(z)) \\to U", "\\end{matrix}", "$$", "By construction $(x_K, y_K) \\in j(R(K))$ and", "$(y_K, z_K) \\in j(R(K))$. Since $j$ is a pre-equivalence relation", "we see that also $(x_K, z_K) \\in j(R(K))$.", "This clearly implies that $x \\sim z$.", "\\medskip\\noindent", "The proof that $\\sim$ is reflexive and symmetric is omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9580, "type": "theorem", "label": "groupoids-lemma-etale-equivalence-relation", "categories": [ "groupoids" ], "title": "groupoids-lemma-etale-equivalence-relation", "contents": [ "Let $j : R \\to U \\times_S U$ be a pre-relation. Assume", "\\begin{enumerate}", "\\item $s, t$ are unramified,", "\\item for any algebraically closed field $k$ over $S$", "the map $R(k) \\to U(k) \\times U(k)$ is an equivalence relation,", "\\item there are morphisms $e : U \\to R$, $i : R \\to R$,", "$c : R \\times_{s, U, t} R \\to R$ such that", "$$", "\\xymatrix{", "U \\ar[r]_e \\ar[d]_\\Delta &", "R \\ar[d]_j &", "R \\ar[d]^j \\ar[r]_i &", "R \\ar[d]^j &", "R \\times_{s, U, t} R \\ar[d]^{j \\times j} \\ar[r]_c &", "R \\ar[d]^j \\\\", "U \\times_S U \\ar[r] &", "U \\times_S U &", "U \\times_S U \\ar[r]^{flip} &", "U \\times_S U &", "U \\times_S U \\times_S U \\ar[r]^{\\text{pr}_{02}} &", "U \\times_S U", "}", "$$", "are commutative.", "\\end{enumerate}", "Then $j$ is an equivalence relation." ], "refs": [], "proofs": [ { "contents": [ "By condition (1) and", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-permanence}", "we see that $j$ is a unramified. Then", "$\\Delta_j : R \\to R \\times_{U \\times_S U} R$ is an open immersion by", "Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism}.", "However, then condition (2) says $\\Delta_j$ is bijective on", "$k$-valued points, hence $\\Delta_j$ is an isomorphism, hence $j$", "is a monomorphism. Then it easily follows from the commutative", "diagrams that $R(T) \\subset U(T) \\times U(T)$ is an equivalence", "relation for all schemes $T$ over $S$." ], "refs": [ "morphisms-lemma-unramified-permanence", "morphisms-lemma-diagonal-unramified-morphism" ], "ref_ids": [ 5357, 5354 ] } ], "ref_ids": [] }, { "id": 9581, "type": "theorem", "label": "groupoids-lemma-base-change-group-scheme", "categories": [ "groupoids" ], "title": "groupoids-lemma-base-change-group-scheme", "contents": [ "Let $(G, m)$ be a group scheme over $S$.", "Let $S' \\to S$ be a morphism of schemes.", "The pullback $(G_{S'}, m_{S'})$ is a group scheme over $S'$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9582, "type": "theorem", "label": "groupoids-lemma-closed-subgroup-scheme", "categories": [ "groupoids" ], "title": "groupoids-lemma-closed-subgroup-scheme", "contents": [ "Let $S$ be a scheme. Let $(G, m, e, i)$ be a group scheme over $S$.", "\\begin{enumerate}", "\\item A closed subscheme $H \\subset G$ is a closed subgroup scheme", "if and only if $e : S \\to G$, $m|_{H \\times_S H} : H \\times_S H \\to G$,", "and $i|_H : H \\to G$ factor through $H$.", "\\item An open subscheme $H \\subset G$ is an open subgroup scheme", "if and only if $e : S \\to G$, $m|_{H \\times_S H} : H \\times_S H \\to G$,", "and $i|_H : H \\to G$ factor through $H$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Looking at $T$-valued points this translates into the well known", "conditions characterizing subsets of groups as subgroups." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9583, "type": "theorem", "label": "groupoids-lemma-group-scheme-separated", "categories": [ "groupoids" ], "title": "groupoids-lemma-group-scheme-separated", "contents": [ "Let $S$ be a scheme.", "Let $G$ be a group scheme over $S$.", "Then $G \\to S$ is separated (resp.\\ quasi-separated) if and only if", "the identity morphism $e : S \\to G$ is a closed immersion", "(resp.\\ quasi-compact)." ], "refs": [], "proofs": [ { "contents": [ "We recall that by", "Schemes, Lemma \\ref{schemes-lemma-section-immersion}", "we have that $e$ is an immersion which is a closed immersion", "(resp.\\ quasi-compact) if $G \\to S$ is separated (resp.\\ quasi-separated).", "For the converse, consider the diagram", "$$", "\\xymatrix{", "G \\ar[r]_-{\\Delta_{G/S}} \\ar[d] &", "G \\times_S G \\ar[d]^{(g, g') \\mapsto m(i(g), g')} \\\\", "S \\ar[r]^e & G", "}", "$$", "It is an exercise in the functorial point of view in algebraic geometry", "to show that this diagram is cartesian. In other words, we see that", "$\\Delta_{G/S}$ is a base change of $e$. Hence if $e$ is a", "closed immersion (resp.\\ quasi-compact) so is $\\Delta_{G/S}$, see", "Schemes, Lemma \\ref{schemes-lemma-base-change-immersion}", "(resp.\\ Schemes, Lemma", "\\ref{schemes-lemma-quasi-compact-preserved-base-change})." ], "refs": [ "schemes-lemma-base-change-immersion", "schemes-lemma-quasi-compact-preserved-base-change" ], "ref_ids": [ 7695, 7698 ] } ], "ref_ids": [] }, { "id": 9584, "type": "theorem", "label": "groupoids-lemma-flat-action-on-group-scheme", "categories": [ "groupoids" ], "title": "groupoids-lemma-flat-action-on-group-scheme", "contents": [ "Let $S$ be a scheme.", "Let $G$ be a group scheme over $S$.", "Let $T$ be a scheme over $S$ and let $\\psi : T \\to G$ be a morphism over $S$.", "If $T$ is flat over $S$, then the morphism", "$$", "T \\times_S G \\longrightarrow G, \\quad", "(t, g) \\longmapsto m(\\psi(t), g)", "$$", "is flat. In particular, if $G$ is flat over $S$, then", "$m : G \\times_S G \\to G$ is flat." ], "refs": [], "proofs": [ { "contents": [ "Consider the diagram", "$$", "\\xymatrix{", "T \\times_S G \\ar[rrr]_{(t, g) \\mapsto (t, m(\\psi(t), g))} & & &", "T \\times_S G \\ar[r]_{\\text{pr}} \\ar[d] &", "G \\ar[d] \\\\", "& & &", "T \\ar[r] &", "S", "}", "$$", "The left top horizontal arrow is an isomorphism and the", "square is cartesian. Hence the lemma follows from", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat}." ], "refs": [ "morphisms-lemma-base-change-flat" ], "ref_ids": [ 5265 ] } ], "ref_ids": [] }, { "id": 9585, "type": "theorem", "label": "groupoids-lemma-group-scheme-module-differentials", "categories": [ "groupoids" ], "title": "groupoids-lemma-group-scheme-module-differentials", "contents": [ "\\begin{reference}", "\\cite[Proposition 3.15]{BookAV}", "\\end{reference}", "Let $(G, m, e, i)$ be a group scheme over the scheme $S$.", "Denote $f : G \\to S$ the structure morphism.", "Then there exist canonical isomorphisms", "$$", "\\Omega_{G/S} \\cong f^*\\mathcal{C}_{S/G} \\cong f^*e^*\\Omega_{G/S}", "$$", "where $\\mathcal{C}_{S/G}$ denotes the conormal sheaf of the", "immersion $e$. In particular, if $S$ is the spectrum of a field, then", "$\\Omega_{G/S}$ is a free $\\mathcal{O}_G$-module." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms, Lemma \\ref{morphisms-lemma-base-change-differentials} we have", "$$", "\\Omega_{G \\times_S G/G} = \\text{pr}_0^*\\Omega_{G/S}", "$$", "where on the left hand side we view $G \\times_S G$ as a scheme over $G$", "using $\\text{pr}_1$.", "Let $\\tau : G \\times_S G \\to G \\times_S G$ be the ``shearing map''", "given by $(g, h) \\mapsto (m(g, h), h)$ on points. This map is an automorphism", "of $G \\times_S G$ viewed as a scheme over $G$ via the projection $\\text{pr}_1$.", "Combining these two remarks we obtain an isomorphism", "$$", "\\tau^*\\text{pr}_0^*\\Omega_{G/S} \\to \\text{pr}_0^*\\Omega_{G/S}", "$$", "Since $\\text{pr}_0 \\circ \\tau = m$ this can be rewritten as an isomorphism", "$$", "m^*\\Omega_{G/S} \\to \\text{pr}_0^*\\Omega_{G/S}", "$$", "Pulling back this isomorphism by", "$(e \\circ f, \\text{id}_G) : G \\to G \\times_S G$", "and using that $m \\circ (e \\circ f, \\text{id}_G) = \\text{id}_G$", "and $\\text{pr}_0 \\circ (e \\circ f, \\text{id}_G) = e \\circ f$", "we obtain an isomorphism", "$$", "\\Omega_{G/S} \\to f^*e^*\\Omega_{G/S}", "$$", "as desired. By", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-relative-immersion-section}", "we have $\\mathcal{C}_{S/G} \\cong e^*\\Omega_{G/S}$.", "If $S$ is the spectrum of a field, then", "any $\\mathcal{O}_S$-module on $S$ is free", "and the final statement follows." ], "refs": [ "morphisms-lemma-base-change-differentials" ], "ref_ids": [ 5314 ] } ], "ref_ids": [] }, { "id": 9586, "type": "theorem", "label": "groupoids-lemma-group-scheme-addition-tangent-vectors", "categories": [ "groupoids" ], "title": "groupoids-lemma-group-scheme-addition-tangent-vectors", "contents": [ "Let $S$ be a scheme. Let $G$ be a group scheme over $S$.", "Let $s \\in S$. Then the composition", "$$", "T_{G/S, e(s)} \\oplus T_{G/S, e(s)} = T_{G \\times_S G/S, (e(s), e(s))}", "\\rightarrow T_{G/S, e(s)}", "$$", "is addition of tangent vectors. Here the $=$ comes from", "Varieties, Lemma \\ref{varieties-lemma-tangent-space-product}", "and the right arrow is induced from $m : G \\times_S G \\to G$ via", "Varieties, Lemma \\ref{varieties-lemma-map-tangent-spaces}." ], "refs": [ "varieties-lemma-tangent-space-product", "varieties-lemma-map-tangent-spaces" ], "proofs": [ { "contents": [ "We will use Varieties, Equation (\\ref{varieties-equation-tangent-space-fibre})", "and work with tangent vectors in fibres.", "An element $\\theta$ in the first factor $T_{G_s/s, e(s)}$", "is the image of $\\theta$ via the map", "$T_{G_s/s, e(s)} \\to T_{G_s \\times G_s/s, (e(s), e(s))}$", "coming from $(1, e) : G_s \\to G_s \\times G_s$.", "Since $m \\circ (1, e) = 1$ we see that $\\theta$ maps to $\\theta$", "by functoriality. Since the map is linear we see that", "$(\\theta_1, \\theta_2)$ maps to $\\theta_1 + \\theta_2$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10975, 10974 ] }, { "id": 9587, "type": "theorem", "label": "groupoids-lemma-group-scheme-over-field-open-multiplication", "categories": [ "groupoids" ], "title": "groupoids-lemma-group-scheme-over-field-open-multiplication", "contents": [ "If $(G, m)$ is a group scheme over a field $k$, then the", "multiplication map $m : G \\times_k G \\to G$ is open." ], "refs": [], "proofs": [ { "contents": [ "The multiplication map is isomorphic to the projection map", "$\\text{pr}_0 : G \\times_k G \\to G$", "because the diagram", "$$", "\\xymatrix{", "G \\times_k G \\ar[d]^m \\ar[rrr]_{(g, g') \\mapsto (m(g, g'), g')} & & &", "G \\times_k G \\ar[d]^{(g, g') \\mapsto g} \\\\", "G \\ar[rrr]^{\\text{id}} & & & G", "}", "$$", "is commutative with isomorphisms as horizontal arrows. The projection", "is open by", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open}." ], "refs": [ "morphisms-lemma-scheme-over-field-universally-open" ], "ref_ids": [ 5254 ] } ], "ref_ids": [] }, { "id": 9588, "type": "theorem", "label": "groupoids-lemma-group-scheme-over-field-translate-open", "categories": [ "groupoids" ], "title": "groupoids-lemma-group-scheme-over-field-translate-open", "contents": [ "If $(G, m)$ is a group scheme over a field $k$. Let $U \\subset G$", "open and $T \\to G$ a morphism of schemes. Then the image of the", "composition $T \\times_k U \\to G \\times_k G \\to G$ is open." ], "refs": [], "proofs": [ { "contents": [ "For any field extension $k \\subset K$ the morphism $G_K \\to G$ is open", "(Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open}).", "Every point $\\xi$ of $T \\times_k U$ is the image of a morphism", "$(t, u) : \\Spec(K) \\to T \\times_k U$ for some $K$. Then the image of", "$T_K \\times_K U_K = (T \\times_k U)_K \\to G_K$ contains the translate", "$t \\cdot U_K$ which is open. Combining these facts we see that the", "image of $T \\times_k U \\to G$ contains an open neighbourhood of", "the image of $\\xi$. Since $\\xi$ was arbitrary we win." ], "refs": [ "morphisms-lemma-scheme-over-field-universally-open" ], "ref_ids": [ 5254 ] } ], "ref_ids": [] }, { "id": 9589, "type": "theorem", "label": "groupoids-lemma-group-scheme-over-field-separated", "categories": [ "groupoids" ], "title": "groupoids-lemma-group-scheme-over-field-separated", "contents": [ "Let $G$ be a group scheme over a field.", "Then $G$ is a separated scheme." ], "refs": [], "proofs": [ { "contents": [ "Say $S = \\Spec(k)$ with $k$ a field, and let $G$ be a group scheme", "over $S$. By", "Lemma \\ref{lemma-group-scheme-separated}", "we have to show that $e : S \\to G$ is a closed immersion. By", "Morphisms, Lemma", "\\ref{morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre}", "the image of $e : S \\to G$ is a closed point of $G$.", "It is clear that $\\mathcal{O}_G \\to e_*\\mathcal{O}_S$ is surjective,", "since $e_*\\mathcal{O}_S$ is a skyscraper sheaf supported at the neutral", "element of $G$ with value $k$. We conclude that $e$ is a closed immersion by", "Schemes, Lemma \\ref{schemes-lemma-characterize-closed-immersions}." ], "refs": [ "groupoids-lemma-group-scheme-separated", "morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre", "schemes-lemma-characterize-closed-immersions" ], "ref_ids": [ 9583, 5222, 7731 ] } ], "ref_ids": [] }, { "id": 9590, "type": "theorem", "label": "groupoids-lemma-group-scheme-field-geometrically-irreducible", "categories": [ "groupoids" ], "title": "groupoids-lemma-group-scheme-field-geometrically-irreducible", "contents": [ "Let $G$ be a group scheme over a field $k$.", "Then", "\\begin{enumerate}", "\\item every local ring $\\mathcal{O}_{G, g}$ of $G$ has a unique", "minimal prime ideal,", "\\item there is exactly one irreducible component $Z$ of $G$", "passing through $e$, and", "\\item $Z$ is geometrically irreducible over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For any point $g \\in G$ there exists a field extension", "$k \\subset K$ and a $K$-valued point $g' \\in G(K)$ mapping to", "$g$. If we think of $g'$ as a $K$-rational point of the", "group scheme $G_K$, then we see that", "$\\mathcal{O}_{G, g} \\to \\mathcal{O}_{G_K, g'}$ is a faithfully flat", "local ring map (as $G_K \\to G$ is flat, and a local flat ring map", "is faithfully flat, see", "Algebra, Lemma \\ref{algebra-lemma-local-flat-ff}).", "The result for $\\mathcal{O}_{G_K, g'}$ implies the", "result for $\\mathcal{O}_{G, g}$, see", "Algebra, Lemma \\ref{algebra-lemma-injective-minimal-primes-in-image}.", "Hence in order to prove (1) it suffices to", "prove it for $k$-rational points $g$ of $G$. In this case", "translation by $g$ defines an automorphism $G \\to G$", "which maps $e$ to $g$. Hence $\\mathcal{O}_{G, g} \\cong \\mathcal{O}_{G, e}$.", "In this way we see that (2) implies (1), since irreducible components", "passing through $e$ correspond one to one with minimal prime ideals", "of $\\mathcal{O}_{G, e}$.", "\\medskip\\noindent", "In order to prove (2) and (3) it suffices to prove (2) when $k$", "is algebraically closed. In this case, let $Z_1$, $Z_2$ be two", "irreducible components of $G$ passing through $e$.", "Since $k$ is algebraically closed the closed subscheme", "$Z_1 \\times_k Z_2 \\subset G \\times_k G$ is irreducible too, see", "Varieties, Lemma \\ref{varieties-lemma-bijection-irreducible-components}.", "Hence $m(Z_1 \\times_k Z_2)$ is contained in an irreducible", "component of $G$. On the other hand it contains", "$Z_1$ and $Z_2$ since $m|_{e \\times G} = \\text{id}_G$ and", "$m|_{G \\times e} = \\text{id}_G$. We conclude $Z_1 = Z_2$ as desired." ], "refs": [ "algebra-lemma-local-flat-ff", "algebra-lemma-injective-minimal-primes-in-image", "varieties-lemma-bijection-irreducible-components" ], "ref_ids": [ 537, 445, 10934 ] } ], "ref_ids": [] }, { "id": 9591, "type": "theorem", "label": "groupoids-lemma-reduced-subgroup-scheme-perfect", "categories": [ "groupoids" ], "title": "groupoids-lemma-reduced-subgroup-scheme-perfect", "contents": [ "Let $G$ be a group scheme over a perfect field $k$.", "Then the reduction $G_{red}$ of $G$ is a closed subgroup scheme of $G$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use that $G_{red} \\times_k G_{red}$ is reduced by", "Varieties, Lemmas \\ref{varieties-lemma-perfect-reduced} and", "\\ref{varieties-lemma-geometrically-reduced-any-base-change}." ], "refs": [ "varieties-lemma-perfect-reduced", "varieties-lemma-geometrically-reduced-any-base-change" ], "ref_ids": [ 10907, 10911 ] } ], "ref_ids": [] }, { "id": 9592, "type": "theorem", "label": "groupoids-lemma-open-subgroup-closed-over-field", "categories": [ "groupoids" ], "title": "groupoids-lemma-open-subgroup-closed-over-field", "contents": [ "Let $k$ be a field. Let $\\psi : G' \\to G$ be a morphism of group schemes", "over $k$. If $\\psi(G')$ is open in $G$, then $\\psi(G')$ is closed in $G$." ], "refs": [], "proofs": [ { "contents": [ "Let $U = \\psi(G') \\subset G$. Let $Z = G \\setminus \\psi(G') = G \\setminus U$", "with the reduced induced closed subscheme structure. By", "Lemma \\ref{lemma-group-scheme-over-field-translate-open}", "the image of", "$$", "Z \\times_k G' \\longrightarrow", "Z \\times_k U \\longrightarrow G", "$$", "is open (the first arrow is surjective). On the other hand, since $\\psi$", "is a homomorphism of group schemes, the image of $Z \\times_k G' \\to G$", "is contained in $Z$ (because translation by $\\psi(g')$ preserves", "$U$ for all points $g'$ of $G'$; small detail omitted).", "Hence $Z \\subset G$ is an open subset (although not", "necessarily an open subscheme). Thus $U = \\psi(G')$ is closed." ], "refs": [ "groupoids-lemma-group-scheme-over-field-translate-open" ], "ref_ids": [ 9588 ] } ], "ref_ids": [] }, { "id": 9593, "type": "theorem", "label": "groupoids-lemma-immersion-group-schemes-closed-over-field", "categories": [ "groupoids" ], "title": "groupoids-lemma-immersion-group-schemes-closed-over-field", "contents": [ "Let $i : G' \\to G$ be an immersion of group schemes over a field $k$.", "Then $i$ is a closed immersion, i.e., $i(G')$ is a closed subgroup scheme", "of $G$." ], "refs": [], "proofs": [ { "contents": [ "To show that $i$ is a closed immersion it suffices to show that", "$i(G')$ is a closed subset of $G$. Let $k \\subset k'$ be a perfect", "extension of $k$. If $i(G'_{k'}) \\subset G_{k'}$ is closed, then", "$i(G') \\subset G$ is closed by", "Morphisms, Lemma \\ref{morphisms-lemma-fpqc-quotient-topology}", "(as $G_{k'} \\to G$ is flat, quasi-compact and surjective).", "Hence we may and do assume $k$ is perfect. We will use without further", "mention that products of reduced schemes over $k$ are reduced.", "We may replace $G'$ and $G$ by their reductions, see", "Lemma \\ref{lemma-reduced-subgroup-scheme-perfect}.", "Let $\\overline{G'} \\subset G$ be the closure of $i(G')$ viewed", "as a reduced closed subscheme. By", "Varieties, Lemma \\ref{varieties-lemma-closure-of-product}", "we conclude that $\\overline{G'} \\times_k \\overline{G'}$", "is the closure of the image of $G' \\times_k G' \\to G \\times_k G$. Hence", "$$", "m\\Big(\\overline{G'} \\times_k \\overline{G'}\\Big)", "\\subset \\overline{G'}", "$$", "as $m$ is continuous. It follows that $\\overline{G'} \\subset G$", "is a (reduced) closed subgroup scheme. By", "Lemma \\ref{lemma-open-subgroup-closed-over-field}", "we see that $i(G') \\subset \\overline{G'}$ is also closed", "which implies that $i(G') = \\overline{G'}$ as desired." ], "refs": [ "morphisms-lemma-fpqc-quotient-topology", "groupoids-lemma-reduced-subgroup-scheme-perfect", "varieties-lemma-closure-of-product", "groupoids-lemma-open-subgroup-closed-over-field" ], "ref_ids": [ 5269, 9591, 10999, 9592 ] } ], "ref_ids": [] }, { "id": 9594, "type": "theorem", "label": "groupoids-lemma-irreducible-group-scheme-over-field-qc", "categories": [ "groupoids" ], "title": "groupoids-lemma-irreducible-group-scheme-over-field-qc", "contents": [ "Let $G$ be a group scheme over a field $k$. If $G$ is irreducible,", "then $G$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $k \\subset K$ is a field extension. If $G_K$", "is quasi-compact, then $G$ is too as $G_K \\to G$ is surjective.", "By Lemma \\ref{lemma-group-scheme-field-geometrically-irreducible}", "we see that $G_K$ is irreducible. Hence it suffices to prove the lemma", "after replacing $k$ by some extension. Choose $K$ to be an algebraically", "closed field extension of very large cardinality. Then by", "Varieties, Lemma \\ref{varieties-lemma-make-Jacobson},", "we see that $G_K$ is a Jacobson scheme all of whose closed points have residue", "field equal to $K$. In other words we may assume $G$ is a Jacobson", "scheme all of whose closed points have residue field $k$.", "\\medskip\\noindent", "Let $U \\subset G$ be a nonempty affine open. Let $g \\in G(k)$. Then", "$gU \\cap U \\not = \\emptyset$. Hence we see that $g$ is in the image", "of the morphism", "$$", "U \\times_{\\Spec(k)} U \\longrightarrow G, \\quad", "(u_1, u_2) \\longmapsto u_1u_2^{-1}", "$$", "Since the image of this morphism is open", "(Lemma \\ref{lemma-group-scheme-over-field-open-multiplication})", "we see that the image is all of $G$ (because $G$ is Jacobson", "and closed points are $k$-rational).", "Since $U$ is affine, so is $U \\times_{\\Spec(k)} U$. Hence $G$ is the", "image of a quasi-compact scheme, hence quasi-compact." ], "refs": [ "groupoids-lemma-group-scheme-field-geometrically-irreducible", "varieties-lemma-make-Jacobson", "groupoids-lemma-group-scheme-over-field-open-multiplication" ], "ref_ids": [ 9590, 10965, 9587 ] } ], "ref_ids": [] }, { "id": 9595, "type": "theorem", "label": "groupoids-lemma-connected-group-scheme-over-field-irreducible", "categories": [ "groupoids" ], "title": "groupoids-lemma-connected-group-scheme-over-field-irreducible", "contents": [ "Let $G$ be a group scheme over a field $k$. If $G$ is connected,", "then $G$ is irreducible." ], "refs": [], "proofs": [ { "contents": [ "By Varieties, Lemma \\ref{varieties-lemma-geometrically-connected-criterion}", "we see that $G$ is geometrically connected. If we show that $G_K$", "is irreducible for some field extension $k \\subset K$, then", "the lemma follows. Hence we may apply", "Varieties, Lemma \\ref{varieties-lemma-make-Jacobson}", "to reduce to the case where $k$ is algebraically closed,", "$G$ is a Jacobson scheme, and all the closed points are $k$-rational.", "\\medskip\\noindent", "Let $Z \\subset G$ be the unique irreducible component of $G$ passing", "through the neutral element, see", "Lemma \\ref{lemma-group-scheme-field-geometrically-irreducible}.", "Endowing $Z$ with the reduced induced closed subscheme structure,", "we see that $Z \\times_k Z$ is reduced and irreducible", "(Varieties, Lemmas", "\\ref{varieties-lemma-geometrically-reduced-any-base-change} and", "\\ref{varieties-lemma-bijection-irreducible-components}).", "We conclude that $m|_{Z \\times_k Z} : Z \\times_k Z \\to G$ factors", "through $Z$. Hence $Z$ becomes a closed subgroup scheme of $G$.", "\\medskip\\noindent", "To get a contradiction, assume there exists another irreducible", "component $Z' \\subset G$. Then $Z \\cap Z' = \\emptyset$ by", "Lemma \\ref{lemma-group-scheme-field-geometrically-irreducible}.", "By Lemma \\ref{lemma-irreducible-group-scheme-over-field-qc}", "we see that $Z$ is quasi-compact. Thus we may choose a quasi-compact open", "$U \\subset G$ with $Z \\subset U$ and $U \\cap Z' = \\emptyset$.", "The image $W$ of $Z \\times_k U \\to G$ is open in $G$ by", "Lemma \\ref{lemma-group-scheme-over-field-translate-open}.", "On the other hand, $W$ is quasi-compact as the image of a", "quasi-compact space. We claim that $W$ is closed.", "If the claim is true, then $W \\subset G \\setminus Z'$ is a proper open", "and closed subset of $G$, which contradicts the assumption that", "$G$ is connected.", "\\medskip\\noindent", "Proof of the claim. Since $W$ is quasi-compact, we see that", "points in the closure of $W$ are specializations of points of $W$", "(Morphisms, Lemma \\ref{morphisms-lemma-reach-points-scheme-theoretic-image}).", "Thus we have to show that any irreducible", "component $Z'' \\subset G$ of $G$ which meets $W$ is contained in $W$.", "As $G$ is Jacobson and closed points are rational,", "$Z'' \\cap W$ has a rational point", "$g \\in Z''(k) \\cap W(k)$ and hence $Z'' = Zg$. But $W = m(Z \\times_k W)$", "by construction, so $Z'' \\cap W \\not = \\emptyset$ implies", "$Z'' \\subset W$." ], "refs": [ "varieties-lemma-geometrically-connected-criterion", "varieties-lemma-make-Jacobson", "groupoids-lemma-group-scheme-field-geometrically-irreducible", "varieties-lemma-geometrically-reduced-any-base-change", "varieties-lemma-bijection-irreducible-components", "groupoids-lemma-group-scheme-field-geometrically-irreducible", "groupoids-lemma-irreducible-group-scheme-over-field-qc", "groupoids-lemma-group-scheme-over-field-translate-open", "morphisms-lemma-reach-points-scheme-theoretic-image" ], "ref_ids": [ 10925, 10965, 9590, 10911, 10934, 9590, 9594, 9588, 5147 ] } ], "ref_ids": [] }, { "id": 9596, "type": "theorem", "label": "groupoids-lemma-profinite-product-over-field", "categories": [ "groupoids" ], "title": "groupoids-lemma-profinite-product-over-field", "contents": [ "Let $k$ be a field. Let $T = \\Spec(A)$ where $A$ is a directed colimit of", "algebras which are finite products of copies of $k$. For any scheme $X$", "over $k$ we have $|T \\times_k X| = |T| \\times |X|$ as topological spaces." ], "refs": [], "proofs": [ { "contents": [ "By taking an affine open covering we reduce to the case of an affine $X$.", "Say $X = \\Spec(B)$.", "Write $A = \\colim A_i$ with $A_i = \\prod_{t \\in T_i} k$ and $T_i$ finite.", "Then $T_i = |\\Spec(A_i)|$ with the discrete topology and the transition", "morphisms $A_i \\to A_{i'}$ are given by set maps $T_{i'} \\to T_i$. Thus", "$|T| = \\lim T_i$ as a topological space, see", "Limits, Lemma \\ref{limits-lemma-topology-limit}. Similarly we have", "\\begin{align*}", "|T \\times_k X| & =", "|\\Spec(A \\otimes_k B)| \\\\", "& =", "|\\Spec(\\colim A_i \\otimes_k B)| \\\\", "& =", "\\lim |\\Spec(A_i \\otimes_k B)| \\\\", "& =", "\\lim |\\Spec(\\prod\\nolimits_{t \\in T_i} B)| \\\\", "& =", "\\lim T_i \\times |X| \\\\", "& =", "(\\lim T_i) \\times |X| \\\\", "& =", "|T| \\times |X|", "\\end{align*}", "by the lemma above and the fact that limits commute with limits." ], "refs": [ "limits-lemma-topology-limit" ], "ref_ids": [ 15036 ] } ], "ref_ids": [] }, { "id": 9597, "type": "theorem", "label": "groupoids-lemma-compact-set-in-affine", "categories": [ "groupoids" ], "title": "groupoids-lemma-compact-set-in-affine", "contents": [ "Let $k$ be an algebraically closed field. Let $G$ be a group scheme over $k$.", "Assume that $G$ is Jacobson and that all closed points are $k$-rational.", "Let $T = \\Spec(A)$ where $A$ is a directed colimit of algebras which", "are finite products of copies of $k$. For any morphism $f : T \\to G$", "there exists an affine open $U \\subset G$ containing $f(T)$." ], "refs": [], "proofs": [ { "contents": [ "Let $G^0 \\subset G$ be the closed subgroup scheme found in", "Proposition \\ref{proposition-connected-component}. The first two paragraphs", "serve to reduce to the case $G = G^0$.", "\\medskip\\noindent", "Observe that $T$ is a directed inverse limit of finite topological spaces", "(Limits, Lemma \\ref{limits-lemma-topology-limit}), hence profinite as a", "topological space (Topology, Definition \\ref{topology-definition-profinite}).", "Let $W \\subset G$ be a quasi-compact open containing the image of $T \\to G$.", "After replacing $W$ by the image of $G^0 \\times W \\to G \\times G \\to G$ we may", "assume that $W$ is invariant under the action of left translation by $G^0$, see", "Lemma \\ref{lemma-group-scheme-over-field-translate-open}.", "Consider the composition", "$$", "\\psi = \\pi \\circ f : T \\xrightarrow{f} W \\xrightarrow{\\pi} \\pi_0(W)", "$$", "The space $\\pi_0(W)$ is profinite", "(Topology, Lemma \\ref{topology-lemma-spectral-pi0} and", "Properties, Lemma", "\\ref{properties-lemma-quasi-compact-quasi-separated-spectral}).", "Let $F_\\xi \\subset T$ be the fibre of $T \\to \\pi_0(W)$ over $\\xi \\in \\pi_0(W)$.", "Assume that for all $\\xi$ we can find an affine open $U_\\xi \\subset W$ with", "$F \\subset U$. Since $\\psi : T \\to \\pi_0(W)$ is proper as a map of", "topological spaces (Topology, Lemma \\ref{topology-lemma-closed-map}),", "we can find a quasi-compact open $V_\\xi \\subset \\pi_0(W)$ such that", "$\\psi^{-1}(V_\\xi) \\subset f^{-1}(U_\\xi)$ (easy topological argument omitted).", "After replacing $U_\\xi$ by $U_\\xi \\cap \\pi^{-1}(V_\\xi)$, which is open and", "closed in $U_\\xi$ hence affine, we see that $U_\\xi \\subset \\pi^{-1}(V_\\xi)$", "and $U_\\xi \\cap T = \\psi^{-1}(V_\\xi)$.", "By Topology, Lemma \\ref{topology-lemma-profinite-refine-open-covering}", "we can find a finite disjoint union decomposition", "$\\pi_0(W) = \\bigcup_{i = 1, \\ldots, n} V_i$ by quasi-compact opens such that", "$V_i \\subset V_{\\xi_i}$ for some $i$. Then we see that", "$$", "f(T) \\subset \\bigcup\\nolimits_{i = 1, \\ldots, n} U_{\\xi_i} \\cap \\pi^{-1}(V_i)", "$$", "the right hand side of which is a finite disjoint union of affines, therefore", "affine.", "\\medskip\\noindent", "Let $Z$ be a connected component of $G$ which meets $f(T)$. Then $Z$", "has a $k$-rational point $z$ (because all residue fields of the scheme $T$", "are isomorphic to $k$). Hence $Z = G^0 z$. By our choice of $W$, we see", "that $Z \\subset W$. The argument in the preceding paragraph reduces us to", "the problem of finding an affine open neighbourhood of $f(T) \\cap Z$ in $W$.", "After translation by a rational point we may assume that $Z = G^0$", "(details omitted). Observe that the scheme theoretic inverse image", "$T' = f^{-1}(G^0) \\subset T$ is a closed subscheme, which has the same type.", "After replacing $T$ by $T'$ we may assume that $f(T) \\subset G^0$.", "Choose an affine open neighbourhood $U \\subset G$", "of $e \\in G$, so that in particular $U \\cap G^0$ is nonempty. We will show", "there exists a $g \\in G^0(k)$ such that $f(T) \\subset g^{-1}U$.", "This will finish the proof as $g^{-1}U \\subset W$ by the left", "$G^0$-invariance of $W$.", "\\medskip\\noindent", "The arguments in the preceding two paragraphs allow us to pass to $G^0$", "and reduce the problem to the following:", "Assume $G$ is irreducible and $U \\subset G$ an affine", "open neighbourhood of $e$. Show that $f(T) \\subset g^{-1}U$", "for some $g \\in G(k)$. Consider the morphism", "$$", "U \\times_k T \\longrightarrow G \\times_k T,\\quad", "(t, u) \\longrightarrow (uf(t)^{-1}, t)", "$$", "which is an open immersion (because the extension of this morphism to", "$G \\times_k T \\to G \\times_k T$ is an isomorphism).", "By our assumption on $T$ we see that we have $|U \\times_k T| = |U| \\times |T|$", "and similarly for $G \\times_k T$, see", "Lemma \\ref{lemma-profinite-product-over-field}.", "Hence the image of the displayed open immersion is a finite union", "of boxes $\\bigcup_{i = 1, \\ldots, n} U_i \\times V_i$ with", "$V_i \\subset T$ and $U_i \\subset G$ quasi-compact open. This means that", "the possible opens $Uf(t)^{-1}$, $t \\in T$ are finite in number, say", "$Uf(t_1)^{-1}, \\ldots, Uf(t_r)^{-1}$. Since $G$ is irreducible the", "intersection", "$$", "Uf(t_1)^{-1} \\cap \\ldots \\cap Uf(t_r)^{-1}", "$$", "is nonempty and since $G$ is Jacobson with closed points $k$-rational,", "we can choose a $k$-valued point $g \\in G(k)$ of this intersection.", "Then we see that $g \\in Uf(t)^{-1}$ for all $t \\in T$ which", "means that $f(t) \\in g^{-1}U$ as desired." ], "refs": [ "groupoids-proposition-connected-component", "limits-lemma-topology-limit", "topology-definition-profinite", "groupoids-lemma-group-scheme-over-field-translate-open", "topology-lemma-spectral-pi0", "properties-lemma-quasi-compact-quasi-separated-spectral", "topology-lemma-closed-map", "topology-lemma-profinite-refine-open-covering", "groupoids-lemma-profinite-product-over-field" ], "ref_ids": [ 9667, 15036, 8369, 9588, 8310, 2941, 8274, 8301, 9596 ] } ], "ref_ids": [] }, { "id": 9598, "type": "theorem", "label": "groupoids-lemma-group-scheme-field-countable-affine", "categories": [ "groupoids" ], "title": "groupoids-lemma-group-scheme-field-countable-affine", "contents": [ "Let $G$ be a group scheme over a field.", "There exists an open and closed subscheme $G' \\subset G$", "which is a countable union of affines." ], "refs": [], "proofs": [ { "contents": [ "Let $e \\in U(k)$ be a quasi-compact open neighbourhood of the identity", "element. By replacing $U$ by $U \\cap i(U)$ we may assume that", "$U$ is invariant under the inverse map. As $G$ is separated this is", "still a quasi-compact set. Set", "$$", "G' = \\bigcup\\nolimits_{n \\geq 1} m_n(U \\times_k \\ldots \\times_k U)", "$$", "where $m_n : G \\times_k \\ldots \\times_k G \\to G$ is the $n$-slot", "multiplication map", "$(g_1, \\ldots, g_n) \\mapsto m(m(\\ldots (m(g_1, g_2), g_3), \\ldots ), g_n)$.", "Each of these maps are open (see", "Lemma \\ref{lemma-group-scheme-over-field-open-multiplication})", "hence $G'$ is an open subgroup scheme. By", "Lemma \\ref{lemma-open-subgroup-closed-over-field}", "it is also a closed subgroup scheme." ], "refs": [ "groupoids-lemma-group-scheme-over-field-open-multiplication", "groupoids-lemma-open-subgroup-closed-over-field" ], "ref_ids": [ 9587, 9592 ] } ], "ref_ids": [] }, { "id": 9599, "type": "theorem", "label": "groupoids-lemma-group-scheme-finite-type-field", "categories": [ "groupoids" ], "title": "groupoids-lemma-group-scheme-finite-type-field", "contents": [ "Let $k$ be a field. Let $G$ be a locally algebraic group scheme over $k$.", "Then $G$ is equidimensional and $\\dim(G) = \\dim_g(G)$ for all $g \\in G$.", "For any closed point $g \\in G$ we have $\\dim(G) = \\dim(\\mathcal{O}_{G, g})$." ], "refs": [], "proofs": [ { "contents": [ "Let us first prove that $\\dim_g(G) = \\dim_{g'}(G)$ for any", "pair of points $g, g' \\in G$. By", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}", "we may extend the ground field at will. Hence we may assume that", "both $g$ and $g'$ are defined over $k$. Hence there exists an", "automorphism of $G$ mapping $g$ to $g'$, whence the equality.", "By", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}", "we have", "$\\dim_g(G) = \\dim(\\mathcal{O}_{G, g}) +", "\\text{trdeg}_k(\\kappa(g))$.", "On the other hand, the dimension of $G$ (or any open subset of $G$)", "is the supremum of the dimensions of the local rings of $G$, see", "Properties, Lemma \\ref{properties-lemma-codimension-local-ring}.", "Clearly this is maximal for closed points $g$ in which case", "$\\text{trdeg}_k(\\kappa(g)) = 0$ (by the Hilbert Nullstellensatz, see", "Morphisms, Section \\ref{morphisms-section-points-finite-type}).", "Hence the lemma follows." ], "refs": [ "morphisms-lemma-dimension-fibre-after-base-change", "morphisms-lemma-dimension-fibre-at-a-point", "properties-lemma-codimension-local-ring" ], "ref_ids": [ 5279, 5277, 2979 ] } ], "ref_ids": [] }, { "id": 9600, "type": "theorem", "label": "groupoids-lemma-group-scheme-characteristic-zero-smooth", "categories": [ "groupoids" ], "title": "groupoids-lemma-group-scheme-characteristic-zero-smooth", "contents": [ "Let $k$ be a field of characteristic $0$. Let $G$ be a", "locally algebraic group scheme over $k$. Then the structure", "morphism $G \\to \\Spec(k)$ is smooth, i.e., $G$ is a smooth", "group scheme." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-group-scheme-module-differentials}", "the module of differentials of $G$ over $k$ is free.", "Hence smoothness follows from", "Varieties, Lemma \\ref{varieties-lemma-char-zero-differentials-free-smooth}." ], "refs": [ "groupoids-lemma-group-scheme-module-differentials", "varieties-lemma-char-zero-differentials-free-smooth" ], "ref_ids": [ 9585, 11002 ] } ], "ref_ids": [] }, { "id": 9601, "type": "theorem", "label": "groupoids-lemma-reduced-group-scheme-prefect-field-characteristic-p-smooth", "categories": [ "groupoids" ], "title": "groupoids-lemma-reduced-group-scheme-prefect-field-characteristic-p-smooth", "contents": [ "Let $k$ be a perfect field of characteristic $p > 0$ (see", "Lemma \\ref{lemma-group-scheme-characteristic-zero-smooth}", "for the characteristic zero case).", "Let $G$ be a locally algebraic group scheme over $k$.", "If $G$ is reduced then the structure", "morphism $G \\to \\Spec(k)$ is smooth, i.e., $G$ is a smooth", "group scheme." ], "refs": [ "groupoids-lemma-group-scheme-characteristic-zero-smooth" ], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-group-scheme-module-differentials}", "the sheaf $\\Omega_{G/k}$ is free. Hence the lemma follows from", "Varieties, Lemma \\ref{varieties-lemma-char-p-differentials-free-smooth}." ], "refs": [ "groupoids-lemma-group-scheme-module-differentials", "varieties-lemma-char-p-differentials-free-smooth" ], "ref_ids": [ 9585, 11003 ] } ], "ref_ids": [ 9600 ] }, { "id": 9602, "type": "theorem", "label": "groupoids-lemma-points-in-affine", "categories": [ "groupoids" ], "title": "groupoids-lemma-points-in-affine", "contents": [ "Let $k$ be an algebraically closed field.", "Let $G$ be a locally algebraic group scheme over $k$.", "Let $g_1, \\ldots, g_n \\in G(k)$ be $k$-rational points.", "Then there exists an affine open $U \\subset G$ containing $g_1, \\ldots, g_n$." ], "refs": [], "proofs": [ { "contents": [ "We first argue by induction on $n$ that we may assume all $g_i$ are", "on the same connected component of $G$. Namely, if not, then we can", "find a decomposition $G = W_1 \\amalg W_2$ with $W_i$ open in $G$ and", "(after possibly renumbering) $g_1, \\ldots, g_r \\in W_1$ and", "$g_{r + 1}, \\ldots, g_n \\in W_2$ for some $0 < r < n$. By", "induction we can find affine opens $U_1$ and $U_2$ of $G$ with", "$g_1, \\ldots, g_r \\in U_1$ and $g_{r + 1}, \\ldots, g_n \\in U_2$.", "Then", "$$", "g_1, \\ldots, g_n \\in (U_1 \\cap W_1) \\cup (U_2 \\cap W_2)", "$$", "is a solution to the problem. Thus we may assume $g_1, \\ldots, g_n$", "are all on the same connected component of $G$. Translating by $g_1^{-1}$", "we may assume $g_1, \\ldots, g_n \\in G^0$ where $G^0 \\subset G$ is as in", "Proposition \\ref{proposition-connected-component}. Choose an affine", "open neighbourhood $U$ of $e$, in particular $U \\cap G^0$ is nonempty.", "Since $G^0$ is irreducible we see that", "$$", "G^0 \\cap (Ug_1^{-1} \\cap \\ldots \\cap Ug_n^{-1})", "$$", "is nonempty. Since $G \\to \\Spec(k)$ is locally of finite type, also", "$G^0 \\to \\Spec(k)$ is locally of finite type, hence any nonempty", "open has a $k$-rational point. Thus we can pick $g \\in G^0(k)$ with", "$g \\in Ug_i^{-1}$ for all $i$. Then $g_i \\in g^{-1}U$ for all $i$", "and $g^{-1}U$ is the affine open we were looking for." ], "refs": [ "groupoids-proposition-connected-component" ], "ref_ids": [ 9667 ] } ], "ref_ids": [] }, { "id": 9603, "type": "theorem", "label": "groupoids-lemma-algebraic-quasi-projective", "categories": [ "groupoids" ], "title": "groupoids-lemma-algebraic-quasi-projective", "contents": [ "Let $k$ be a field. Let $G$ be an algebraic group scheme over $k$.", "Then $G$ is quasi-projective over $k$." ], "refs": [], "proofs": [ { "contents": [ "By Varieties, Lemma \\ref{varieties-lemma-ample-after-field-extension}", "we may assume that $k$ is algebraically closed. Let $G^0 \\subset G$", "be the connected component of $G$ as in", "Proposition \\ref{proposition-connected-component}.", "Then every other connected component of $G$ has a $k$-rational", "point and hence is isomorphic to $G^0$ as a scheme.", "Since $G$ is quasi-compact and Noetherian, there are finitely many of these", "connected components. Thus we reduce to the case discussed in", "the next paragraph.", "\\medskip\\noindent", "Let $G$ be a connected algebraic group scheme over an algebraically closed", "field $k$. If the characteristic of $k$ is zero, then $G$ is smooth over", "$k$ by Lemma \\ref{lemma-group-scheme-characteristic-zero-smooth}.", "If the characteristic of $k$ is $p > 0$, then we let $H = G_{red}$", "be the reduction of $G$. By", "Divisors, Proposition \\ref{divisors-proposition-push-down-ample}", "it suffices to show that $H$ has an ample invertible sheaf.", "(For an algebraic scheme over $k$ having an ample invertible", "sheaf is equivalent to being quasi-projective over $k$, see", "for example the very general", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-quasi-projective}.)", "By Lemma \\ref{lemma-reduced-subgroup-scheme-perfect}", "we see that $H$ is a group scheme over $k$.", "By Lemma \\ref{lemma-reduced-group-scheme-prefect-field-characteristic-p-smooth}", "we see that $H$ is smooth over $k$.", "This reduces us to the situation discussed in the next", "paragraph.", "\\medskip\\noindent", "Let $G$ be a quasi-compact irreducible smooth group scheme over an", "algebraically closed field $k$. Observe that the local rings of $G$", "are regular and hence UFDs", "(Varieties, Lemma \\ref{varieties-lemma-smooth-regular} and", "More on Algebra, Lemma \\ref{more-algebra-lemma-regular-local-UFD}).", "The complement of a nonempty affine open of $G$", "is the support of an effective Cartier divisor $D$.", "This follows from Divisors, Lemma", "\\ref{divisors-lemma-complement-open-affine-effective-cartier-divisor}.", "(Observe that $G$ is separated by", "Lemma \\ref{lemma-group-scheme-over-field-separated}.)", "We conclude there exists an effective Cartier divisor $D \\subset G$", "such that $G \\setminus D$ is affine. We will use below that", "for any $n \\geq 1$ and $g_1, \\ldots, g_n \\in G(k)$ the complement", "$G \\setminus \\bigcup D g_i$ is affine. Namely, it is the intersection", "of the affine opens $G \\setminus Dg_i \\cong G \\setminus D$", "in the separated scheme $G$.", "\\medskip\\noindent", "We may choose the top row of the diagram", "$$", "\\xymatrix{", "G & U \\ar[l]_j \\ar[r]^\\pi & \\mathbf{A}^d_k \\\\", "& W \\ar[r]^{\\pi'} \\ar[u] & V \\ar[u]", "}", "$$", "such that $U \\not = \\emptyset$, $j : U \\to G$ is an open immersion, and", "$\\pi$ is \\'etale, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-etale-over-affine-space}.", "There is a nonempty affine open $V \\subset \\mathbf{A}^d_k$ such that", "with $W = \\pi^{-1}(V)$ the morphism $\\pi' = \\pi|_W : W \\to V$ is finite \\'etale.", "In particular $\\pi'$ is finite locally free, say of degree $n$.", "Consider the effective Cartier divisor", "$$", "\\mathcal{D} = \\{(g, w) \\mid m(g, j(w)) \\in D\\} \\subset G \\times W", "$$", "(This is the restriction to $G \\times W$ of the pullback of $D \\subset G$", "under the flat morphism $m : G \\times G \\to G$.)", "Consider the closed subset\\footnote{Using the material", "in Divisors, Section \\ref{divisors-section-norms}", "we could take as effective Cartier", "divisor $E$ the norm of the effective Cartier divisor $\\mathcal{D}$", "along the finite locally free morphism $1 \\times \\pi'$ bypassing", "some of the arguments.}", "$T = (1 \\times \\pi')(\\mathcal{D}) \\subset G \\times V$.", "Since $\\pi'$ is finite locally free, every irreducible component", "of $T$ has codimension $1$ in $G \\times V$. Since $G \\times V$", "is smooth over $k$ we conclude these components are effective Cartier", "divisors (Divisors, Lemma \\ref{divisors-lemma-weil-divisor-is-cartier-UFD}", "and lemmas cited above)", "and hence $T$ is the support of an effective Cartier divisor", "$E$ in $G \\times V$. If $v \\in V(k)$, then", "$(\\pi')^{-1}(v) = \\{w_1, \\ldots, w_n\\} \\subset W(k)$ and we see that", "$$", "E_v = \\bigcup\\nolimits_{i = 1, \\ldots, n} D j(w_i)^{-1}", "$$", "in $G$ set theoretically. In particular we see that $G \\setminus E_v$", "is affine open (see above).", "Moreover, if $g \\in G(k)$, then there exists a $v \\in V$", "such that $g \\not \\in E_v$. Namely, the set $W'$ of $w \\in W$ such that", "$g \\not \\in Dj(w)^{-1}$ is nonempty open and it suffices to pick $v$", "such that the fibre of $W' \\to V$ over $v$ has $n$ elements.", "\\medskip\\noindent", "Consider the invertible sheaf", "$\\mathcal{M} = \\mathcal{O}_{G \\times V}(E)$ on $G \\times V$.", "By Varieties, Lemma \\ref{varieties-lemma-rational-equivalence-for-Pic}", "the isomorphism class $\\mathcal{L}$ of the restriction", "$\\mathcal{M}_v = \\mathcal{O}_G(E_v)$ is independent of $v \\in V(k)$.", "On the other hand, for every $g \\in G(k)$ we can find a $v$", "such that $g \\not \\in E_v$ and such that $G \\setminus E_v$", "is affine. Thus the canonical section", "(Divisors, Definition", "\\ref{divisors-definition-invertible-sheaf-effective-Cartier-divisor})", "of $\\mathcal{O}_G(E_v)$", "corresponds to a section $s_v$ of $\\mathcal{L}$ which does not", "vanish at $g$ and such that $G_{s_v}$ is affine.", "This means that $\\mathcal{L}$ is ample by definition", "(Properties, Definition \\ref{properties-definition-ample})." ], "refs": [ "varieties-lemma-ample-after-field-extension", "groupoids-proposition-connected-component", "groupoids-lemma-group-scheme-characteristic-zero-smooth", "divisors-proposition-push-down-ample", "more-morphisms-lemma-quasi-projective", "groupoids-lemma-reduced-subgroup-scheme-perfect", "groupoids-lemma-reduced-group-scheme-prefect-field-characteristic-p-smooth", "varieties-lemma-smooth-regular", "more-algebra-lemma-regular-local-UFD", "divisors-lemma-complement-open-affine-effective-cartier-divisor", "groupoids-lemma-group-scheme-over-field-separated", "morphisms-lemma-smooth-etale-over-affine-space", "divisors-lemma-weil-divisor-is-cartier-UFD", "varieties-lemma-rational-equivalence-for-Pic", "divisors-definition-invertible-sheaf-effective-Cartier-divisor", "properties-definition-ample" ], "ref_ids": [ 10966, 9667, 9600, 8081, 13928, 9591, 9601, 11004, 10544, 7961, 9589, 5377, 7951, 11028, 8092, 3088 ] } ], "ref_ids": [] }, { "id": 9604, "type": "theorem", "label": "groupoids-lemma-algebraic-center", "categories": [ "groupoids" ], "title": "groupoids-lemma-algebraic-center", "contents": [ "Let $k$ be a field. Let $G$ be a locally algebraic group scheme over $k$.", "Then the center of $G$ is a closed subgroup scheme of $G$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\text{Aut}(G)$ denote the contravariant functor on the category of", "schemes over $k$ which associates to $S/k$ the set of automorphisms", "of the base change $G_S$ as a group scheme over $S$. There is a natural", "transformation", "$$", "G \\longrightarrow \\text{Aut}(G),\\quad", "g \\longmapsto \\text{inn}_g", "$$", "sending an $S$-valued point $g$ of $G$ to the inner automorphism of $G$", "determined by $g$. The center $C$ of $G$ is by definition the kernel of", "this transformation, i.e., the functor which to $S$ associates those", "$g \\in G(S)$ whose associated inner automorphism is trivial. The statement", "of the lemma is that this functor is representable by a closed subgroup", "scheme of $G$.", "\\medskip\\noindent", "Choose an integer $n \\geq 1$. Let $G_n \\subset G$ be the $n$th infinitesimal", "neighbourhood of the identity element $e$ of $G$. For every scheme $S/k$", "the base change $G_{n, S}$ is the $n$th infinitesimal neighbourhood of", "$e_S : S \\to G_S$. Thus we see that there is a natural transformation", "$\\text{Aut}(G) \\to \\text{Aut}(G_n)$ where the right hand side is the", "functor of automorphisms of $G_n$ as a scheme ($G_n$ isn't in general", "a group scheme). Observe that $G_n$ is the spectrum of an artinian", "local ring $A_n$ with residue field $k$ which has finite dimension", "as a $k$-vector space", "(Varieties, Lemma \\ref{varieties-lemma-algebraic-scheme-dim-0}).", "Since every automorphism of $G_n$ induces in particular an invertible", "linear map $A_n \\to A_n$, we obtain transformations of functors", "$$", "G \\to \\text{Aut}(G) \\to \\text{Aut}(G_n) \\to \\text{GL}(A_n)", "$$", "The final group valued functor is representable, see", "Example \\ref{example-general-linear-group}, and the", "last arrow is visibly injective.", "Thus for every $n$ we obtain a closed subgroup scheme", "$$", "H_n = \\Ker(G \\to \\text{Aut}(G_n)) = \\Ker(G \\to \\text{GL}(A_n)).", "$$", "As a first approximation we set $H = \\bigcap_{n \\geq 1} H_n$", "(scheme theoretic intersection). This is a closed subgroup scheme", "which contains the center $C$.", "\\medskip\\noindent", "Let $h$ be an $S$-valued point of $H$ with $S$ locally Noetherian.", "Then the automorphism $\\text{inn}_h$ induces the identity on all", "the closed subschemes $G_{n, S}$. Consider the kernel", "$K = \\Ker(\\text{inn}_h : G_S \\to G_S)$.", "This is a closed subgroup scheme of $G_S$ over $S$ containing the", "closed subschemes $G_{n, S}$ for $n \\geq 1$.", "This implies that $K$ contains an open neighbourhood of", "$e(S) \\subset G_S$, see", "Algebra, Remark \\ref{algebra-remark-intersection-powers-ideal}.", "Let $G^0 \\subset G$ be as in Proposition \\ref{proposition-connected-component}.", "Since $G^0$ is geometrically irreducible, we conclude that $K$ contains", "$G^0_S$ (for any nonempty open $U \\subset G^0_{k'}$ and any field extension", "$k'/k$ we have $U \\cdot U^{-1} = G^0_{k'}$, see proof of", "Lemma \\ref{lemma-irreducible-group-scheme-over-field-qc}).", "Applying this with $S = H$ we find that $G^0$ and $H$", "are subgroup schemes of $G$ whose points commute: for any scheme $S$", "and any $S$-valued points $g \\in G^0(S)$, $h \\in H(S)$ we have", "$gh = hg$ in $G(S)$.", "\\medskip\\noindent", "Assume that $k$ is algebraically closed. Then we can pick a $k$-valued", "point $g_i$ in each irreducible component $G_i$ of $G$. Observe that in", "this case the connected components of $G$ are the irreducible components", "of $G$ are the translates of $G^0$ by our $g_i$. We claim that", "$$", "C = H \\cap \\bigcap\\nolimits_i \\Ker(\\text{inn}_{g_i} : G \\to G)", "\\quad (\\text{scheme theoretic intersection})", "$$", "Namely, $C$ is contained in the right hand side. On the other hand, every", "$S$-valued point $h$ of the right hand side commutes with $G^0$", "and with $g_i$ hence with everything in $G = \\bigcup G^0g_i$.", "\\medskip\\noindent", "The case of a general base field $k$ follows from the result for the", "algebraic closure $\\overline{k}$ by descent. Namely, let", "$A \\subset G_{\\overline{k}}$ the closed subgroup scheme representing", "the center of $G_{\\overline{k}}$. Then we have", "$$", "A \\times_{\\Spec(k)} \\Spec(\\overline{k}) =", "\\Spec(\\overline{k}) \\times_{\\Spec(k)} A", "$$", "as closed subschemes of $G_{\\overline{k} \\otimes_k \\overline{k}}$ by", "the functorial nature of the center. Hence we see that $A$ descends", "to a closed subgroup scheme $Z \\subset G$ by", "Descent, Lemma \\ref{descent-lemma-closed-immersion}", "(and Descent, Lemma \\ref{descent-lemma-descending-property-closed-immersion}).", "Then $Z$ represents $C$ (small argument omitted) and the proof is complete." ], "refs": [ "varieties-lemma-algebraic-scheme-dim-0", "algebra-remark-intersection-powers-ideal", "groupoids-proposition-connected-component", "groupoids-lemma-irreducible-group-scheme-over-field-qc", "descent-lemma-closed-immersion", "descent-lemma-descending-property-closed-immersion" ], "ref_ids": [ 10988, 1560, 9667, 9594, 14749, 14684 ] } ], "ref_ids": [] }, { "id": 9605, "type": "theorem", "label": "groupoids-lemma-abelian-variety-projective", "categories": [ "groupoids" ], "title": "groupoids-lemma-abelian-variety-projective", "contents": [ "Let $k$ be a field. Let $A$ be an abelian variety over $k$.", "Then $A$ is projective." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-algebraic-quasi-projective} and", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-projective}." ], "refs": [ "groupoids-lemma-algebraic-quasi-projective", "more-morphisms-lemma-projective" ], "ref_ids": [ 9603, 13931 ] } ], "ref_ids": [] }, { "id": 9606, "type": "theorem", "label": "groupoids-lemma-abelian-variety-change-field", "categories": [ "groupoids" ], "title": "groupoids-lemma-abelian-variety-change-field", "contents": [ "Let $k$ be a field. Let $A$ be an abelian variety over $k$.", "For any field extension $K/k$ the base change $A_K$ is an", "abelian variety over $K$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Note that this is why we insisted on $A$ being", "geometrically integral; without that condition this lemma", "(and many others below) would be wrong." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9607, "type": "theorem", "label": "groupoids-lemma-abelian-variety-smooth", "categories": [ "groupoids" ], "title": "groupoids-lemma-abelian-variety-smooth", "contents": [ "Let $k$ be a field. Let $A$ be an abelian variety over $k$.", "Then $A$ is smooth over $k$." ], "refs": [], "proofs": [ { "contents": [ "If $k$ is perfect then this follows from", "Lemma \\ref{lemma-group-scheme-characteristic-zero-smooth}", "(characteristic zero) and", "Lemma \\ref{lemma-reduced-group-scheme-prefect-field-characteristic-p-smooth}", "(positive characteristic).", "We can reduce the general case to this case by descent for smoothness", "(Descent, Lemma \\ref{descent-lemma-descending-property-smooth})", "and going to the perfect closure using", "Lemma \\ref{lemma-abelian-variety-change-field}." ], "refs": [ "groupoids-lemma-group-scheme-characteristic-zero-smooth", "groupoids-lemma-reduced-group-scheme-prefect-field-characteristic-p-smooth", "descent-lemma-descending-property-smooth", "groupoids-lemma-abelian-variety-change-field" ], "ref_ids": [ 9600, 9601, 14692, 9606 ] } ], "ref_ids": [] }, { "id": 9608, "type": "theorem", "label": "groupoids-lemma-abelian-variety-abelian", "categories": [ "groupoids" ], "title": "groupoids-lemma-abelian-variety-abelian", "contents": [ "An abelian variety is an abelian group scheme, i.e., the group", "law is commutative." ], "refs": [], "proofs": [ { "contents": [ "Let $k$ be a field. Let $A$ be an abelian variety over $k$.", "By Lemma \\ref{lemma-abelian-variety-change-field} we may replace", "$k$ by its algebraic closure. Consider the morphism", "$$", "h : A \\times_k A \\longrightarrow A \\times_k A,\\quad", "(x, y) \\longmapsto (x, xyx^{-1}y^{-1})", "$$", "This is a morphism over $A$ via the first projection on either side.", "Let $e \\in A(k)$ be the unit. Then we see that $h|_{e \\times A}$ is", "constant with value $(e, e)$. By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-flat-proper-family-cannot-collapse-fibre}", "there exists an open neighbourhood $U \\subset A$ of $e$", "such that $h|_{U \\times A}$ factors through some $Z \\subset U \\times A$", "finite over $U$. This means that for $x \\in U(k)$ the morphism", "$A \\to A$, $y \\mapsto xyx^{-1}y^{-1}$ takes finitely many values.", "Of course this means it is constant with value $e$. Thus", "$(x, y) \\mapsto xyx^{-1}y^{-1}$ is", "constant with value $e$ on $U \\times A$ which implies", "that the group law on $A$ is abelian." ], "refs": [ "groupoids-lemma-abelian-variety-change-field", "more-morphisms-lemma-flat-proper-family-cannot-collapse-fibre" ], "ref_ids": [ 9606, 13905 ] } ], "ref_ids": [] }, { "id": 9609, "type": "theorem", "label": "groupoids-lemma-apply-cube", "categories": [ "groupoids" ], "title": "groupoids-lemma-apply-cube", "contents": [ "Let $k$ be a field. Let $A$ be an abelian variety over $k$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_A$-module.", "Then there is an isomorphism", "$$", "m_{1, 2, 3}^*\\mathcal{L} \\otimes", "m_1^*\\mathcal{L} \\otimes", "m_2^*\\mathcal{L} \\otimes", "m_3^*\\mathcal{L} \\cong", "m_{1, 2}^*\\mathcal{L} \\otimes", "m_{1, 3}^*\\mathcal{L} \\otimes", "m_{2, 3}^*\\mathcal{L}", "$$", "of invertible modules on $A \\times_k A \\times_k A$", "where $m_{i_1, \\ldots, i_t} : A \\times_k A \\times_k A \\to A$", "is the morphism $(x_1, x_2, x_3) \\mapsto \\sum x_{i_j}$." ], "refs": [], "proofs": [ { "contents": [ "Apply the theorem of the cube", "(More on Morphisms, Theorem \\ref{more-morphisms-theorem-of-the-cube})", "to the difference", "$$", "\\mathcal{M} =", "m_{1, 2, 3}^*\\mathcal{L} \\otimes", "m_1^*\\mathcal{L} \\otimes", "m_2^*\\mathcal{L} \\otimes", "m_3^*\\mathcal{L} \\otimes", "m_{1, 2}^*\\mathcal{L}^{\\otimes -1} \\otimes", "m_{1, 3}^*\\mathcal{L}^{\\otimes -1} \\otimes", "m_{2, 3}^*\\mathcal{L}^{\\otimes -1}", "$$", "This works because the restriction of $\\mathcal{M}$", "to $A \\times A \\times e = A \\times A$ is equal to", "$$", "n_{1, 2}^*\\mathcal{L} \\otimes", "n_1^*\\mathcal{L} \\otimes", "n_2^*\\mathcal{L} \\otimes", "n_{1, 2}^*\\mathcal{L}^{\\otimes -1} \\otimes", "n_1^*\\mathcal{L}^{\\otimes -1} \\otimes", "n_2^*\\mathcal{L}^{\\otimes -1} \\cong \\mathcal{O}_{A \\times_k A}", "$$", "where $n_{i_1, \\ldots, i_t} : A \\times_k A \\to A$", "is the morphism $(x_1, x_2) \\mapsto \\sum x_{i_j}$.", "Similarly for $A \\times e \\times A$ and $e \\times A \\times A$." ], "refs": [ "more-morphisms-theorem-of-the-cube" ], "ref_ids": [ 13673 ] } ], "ref_ids": [] }, { "id": 9610, "type": "theorem", "label": "groupoids-lemma-pullbacks-by-n", "categories": [ "groupoids" ], "title": "groupoids-lemma-pullbacks-by-n", "contents": [ "Let $k$ be a field. Let $A$ be an abelian variety over $k$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_A$-module.", "Then", "$$", "[n]^*\\mathcal{L} \\cong", "\\mathcal{L}^{\\otimes n(n + 1)/2} \\otimes", "([-1]^*\\mathcal{L})^{\\otimes n(n - 1)/2}", "$$", "where $[n] : A \\to A$ sends $x$ to $x + x + \\ldots + x$ with $n$ summands", "and where $[-1] : A \\to A$ is the inverse of $A$." ], "refs": [], "proofs": [ { "contents": [ "Consider the morphism $A \\to A \\times_k A \\times_k A$,", "$x \\mapsto (x, x, -x)$ where $-x = [-1](x)$. Pulling back", "the relation of Lemma \\ref{lemma-apply-cube} we obtain", "$$", "\\mathcal{L} \\otimes", "\\mathcal{L} \\otimes", "\\mathcal{L} \\otimes", "[-1]^*\\mathcal{L} \\cong", "[2]^*\\mathcal{L}", "$$", "which proves the result for $n = 2$. By induction assume the result holds", "for $1, 2, \\ldots, n$. Then consider the morphism", "$A \\to A \\times_k A \\times_k A$, $x \\mapsto (x, x, [n - 1]x)$.", "Pulling back", "the relation of Lemma \\ref{lemma-apply-cube} we obtain", "$$", "[n + 1]^*\\mathcal{L} \\otimes", "\\mathcal{L} \\otimes", "\\mathcal{L} \\otimes", "[n - 1]^*\\mathcal{L} \\cong", "[2]^*\\mathcal{L} \\otimes", "[n]^*\\mathcal{L} \\otimes", "[n]^*\\mathcal{L}", "$$", "and the result follows by elementary arithmetic." ], "refs": [ "groupoids-lemma-apply-cube", "groupoids-lemma-apply-cube" ], "ref_ids": [ 9609, 9609 ] } ], "ref_ids": [] }, { "id": 9611, "type": "theorem", "label": "groupoids-lemma-degree-multiplication-by-d", "categories": [ "groupoids" ], "title": "groupoids-lemma-degree-multiplication-by-d", "contents": [ "Let $k$ be a field. Let $A$ be an abelian variety over $k$.", "Let $[d] : A \\to A$ be the multiplication by $d$.", "Then $[d]$ is finite locally free of degree $d^{2\\dim(A)}$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-abelian-variety-projective}", "(and More on Morphisms, Lemma \\ref{more-morphisms-lemma-projective})", "we see that $A$ has an ample invertible module $\\mathcal{L}$.", "Since $[-1] : A \\to A$ is an automorphism, we see that", "$[-1]^*\\mathcal{L}$ is an ample invertible $\\mathcal{O}_X$-module", "as well. Thus $\\mathcal{N} = \\mathcal{L} \\otimes [-1]^*\\mathcal{L}$", "is ample, see", "Properties, Lemma \\ref{properties-lemma-ample-tensor-globally-generated}.", "Since $\\mathcal{N} \\cong [-1]^*\\mathcal{N}$ we see that", "$[d]^*\\mathcal{N} \\cong \\mathcal{N}^{\\otimes d^2}$ by", "Lemma \\ref{lemma-pullbacks-by-n}.", "\\medskip\\noindent", "To get a contradiction $C \\subset X$ be a proper curve contained in a", "fibre of $[d]$. Then $\\mathcal{N}^{\\otimes d^2}|_C \\cong \\mathcal{O}_C$", "is an ample invertible $\\mathcal{O}_C$-module of degree $0$ which", "contradicts Varieties, Lemma \\ref{varieties-lemma-ample-curve} for example.", "(You can also use Varieties, Lemma \\ref{varieties-lemma-ample-positive}.)", "Thus every fibre of $[d]$ has dimension $0$ and hence $[d]$ is finite", "for example by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-characterize-finite}.", "Moreover, since $A$ is smooth over $k$ by", "Lemma \\ref{lemma-abelian-variety-smooth}", "we see that $[d] : A \\to A$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-CM-over-regular-flat}", "(we also use that schemes smooth over fields are regular and that", "regular rings are Cohen-Macaulay, see", "Varieties, Lemma \\ref{varieties-lemma-smooth-regular} and", "Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}).", "Thus $[d]$ is finite flat hence finite locally free by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}.", "\\medskip\\noindent", "Finally, we come to the formula for the degree. By", "Varieties, Lemma \\ref{varieties-lemma-degree-finite-morphism-in-terms-degrees}", "we see that", "$$", "\\deg_{\\mathcal{N}^{\\otimes d^2}}(A) = \\deg([d]) \\deg_\\mathcal{N}(A)", "$$", "Since the degree of $A$ with respect to", "$\\mathcal{N}^{\\otimes d^2}$, respectively $\\mathcal{N}$", "is the coefficient of $n^{\\dim(A)}$ in the polynomial", "$$", "n \\longmapsto \\chi(A, \\mathcal{N}^{\\otimes nd^2}),\\quad", "\\text{respectively}\\quad n \\longmapsto \\chi(A, \\mathcal{N}^{\\otimes n})", "$$", "we see that $\\deg([d]) = d^{2 \\dim(A)}$." ], "refs": [ "groupoids-lemma-abelian-variety-projective", "more-morphisms-lemma-projective", "properties-lemma-ample-tensor-globally-generated", "groupoids-lemma-pullbacks-by-n", "varieties-lemma-ample-curve", "varieties-lemma-ample-positive", "coherent-lemma-characterize-finite", "groupoids-lemma-abelian-variety-smooth", "algebra-lemma-CM-over-regular-flat", "varieties-lemma-smooth-regular", "algebra-lemma-regular-ring-CM", "morphisms-lemma-finite-flat", "varieties-lemma-degree-finite-morphism-in-terms-degrees" ], "ref_ids": [ 9605, 13931, 3043, 9610, 11116, 11128, 3365, 9607, 1107, 11004, 941, 5471, 11129 ] } ], "ref_ids": [] }, { "id": 9612, "type": "theorem", "label": "groupoids-lemma-abelian-variety-multiplication-by-d-etale", "categories": [ "groupoids" ], "title": "groupoids-lemma-abelian-variety-multiplication-by-d-etale", "contents": [ "\\begin{slogan}", "Multiplication by an integer on an abelian variety is an etale morphism", "if and only if the integer is invertible in the base field.", "\\end{slogan}", "Let $k$ be a field. Let $A$ be a nonzero abelian variety over $k$.", "Then $[d] : A \\to A$ is \\'etale if and only if $d$ is invertible in $k$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $[d](x + y) = [d](x) + [d](y)$. Since translation by a", "point is an automorphism of $A$, we see that the set of points where", "$[d] : A \\to A$ is \\'etale is either empty or equal to $A$ (some details", "omitted). Thus it suffices to check whether $[d]$ is \\'etale at", "the unit $e \\in A(k)$. Since we know that $[d]$ is finite locally free", "(Lemma \\ref{lemma-degree-multiplication-by-d})", "to see that it is \\'etale at $e$ is equivalent to", "proving that $\\text{d}[d] : T_{A/k, e} \\to T_{A/k, e}$ is injective. See", "Varieties, Lemma \\ref{varieties-lemma-injective-tangent-spaces-unramified} and", "Morphisms, Lemma \\ref{morphisms-lemma-flat-unramified-etale}.", "By Lemma \\ref{lemma-group-scheme-addition-tangent-vectors} we see that", "$\\text{d}[d]$ is given by multiplication by $d$ on $T_{A/k, e}$." ], "refs": [ "groupoids-lemma-degree-multiplication-by-d", "varieties-lemma-injective-tangent-spaces-unramified", "morphisms-lemma-flat-unramified-etale", "groupoids-lemma-group-scheme-addition-tangent-vectors" ], "ref_ids": [ 9611, 10976, 5373, 9586 ] } ], "ref_ids": [] }, { "id": 9613, "type": "theorem", "label": "groupoids-lemma-abelian-variety-multiplication-by-p", "categories": [ "groupoids" ], "title": "groupoids-lemma-abelian-variety-multiplication-by-p", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $A$ be an abelian variety", "over $k$. The fibre of $[p] : A \\to A$ over $0$ has at most", "$p^g$ distinct points." ], "refs": [], "proofs": [ { "contents": [ "To prove this, we may and do replace $k$ by the algebraic closure.", "By Lemma \\ref{lemma-group-scheme-addition-tangent-vectors}", "the derivative of $[p]$ is multiplication by $p$ as a map", "$T_{A/k, e} \\to T_{A/k, e}$ and hence is zero (compare", "with proof of Lemma \\ref{lemma-abelian-variety-multiplication-by-d-etale}).", "Since $[p]$ commutes with translation we conclude that the derivative of $[p]$", "is everywhere zero, i.e., that the induced map", "$[p]^*\\Omega_{A/k} \\to \\Omega_{A/k}$ is zero.", "Looking at generic points, we find that", "the corresponding map $[p]^* : k(A) \\to k(A)$", "of function fields induces the zero map on $\\Omega_{k(A)/k}$.", "Let $t_1, \\ldots, t_g$ be a p-basis of $k(A)$ over $k$", "(More on Algebra, Definition \\ref{more-algebra-definition-p-basis} and", "Lemma \\ref{more-algebra-lemma-p-basis}). Then $[p]^*(t_i)$", "has a $p$th root by", "Algebra, Lemma \\ref{algebra-lemma-derivative-zero-pth-power}.", "We conclude that", "$k(A)[x_1, \\ldots, x_g]/(x_1^p - t_1, \\ldots, x_g^p - t_g)$ is a subextension", "of $[p]^* : k(A) \\to k(A)$.", "Thus we can find an affine open $U \\subset A$ such that", "$t_i \\in \\mathcal{O}_A(U)$ and $x_i \\in \\mathcal{O}_A([p]^{-1}(U))$.", "We obtain a factorization", "$$", "[p]^{-1}(U)", "\\xrightarrow{\\pi_1}", "\\Spec(\\mathcal{O}(U)[x_1, \\ldots, x_g]/(x_1^p - t_1, \\ldots, x_g^p - t_g))", "\\xrightarrow{\\pi_2}", "U", "$$", "of $[p]$ over $U$. After shrinking $U$ we may assume that $\\pi_1$", "is finite locally free (for example by generic flatness -- actually it is", "already finite locally free in our case).", "By Lemma \\ref{lemma-degree-multiplication-by-d} we see that", "$[p]$ has degree $p^{2g}$. Since $\\pi_2$", "has degree $p^g$ we see that $\\pi_1$ has degree $p^g$ as well.", "The morphism $\\pi_2$ is a universal homeomorphism hence the fibres are", "singletons. We conclude that the (set theoretic) fibres of $[p]^{-1}(U) \\to U$", "are the fibres of $\\pi_1$. Hence they", "have at most $p^g$ elements. Since $[p]$ is a homomorphism of group", "schemes over $k$, the fibre of $[p] : A(k) \\to A(k)$ has the", "same cardinality for every $a \\in A(k)$ and the proof is complete." ], "refs": [ "groupoids-lemma-group-scheme-addition-tangent-vectors", "groupoids-lemma-abelian-variety-multiplication-by-d-etale", "more-algebra-definition-p-basis", "more-algebra-lemma-p-basis", "algebra-lemma-derivative-zero-pth-power", "groupoids-lemma-degree-multiplication-by-d" ], "ref_ids": [ 9586, 9612, 10614, 10068, 1315, 9611 ] } ], "ref_ids": [] }, { "id": 9614, "type": "theorem", "label": "groupoids-lemma-free-action", "categories": [ "groupoids" ], "title": "groupoids-lemma-free-action", "contents": [ "Situation as in Definition \\ref{definition-free-action},", "The action $a$ is free if and only if", "$$", "G \\times_S X \\to X \\times_S X, \\quad (g, x) \\mapsto (a(g, x), x)", "$$", "is a monomorphism." ], "refs": [ "groupoids-definition-free-action" ], "proofs": [ { "contents": [ "Immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9677 ] }, { "id": 9615, "type": "theorem", "label": "groupoids-lemma-characterize-trivial-pseudo-torsors", "categories": [ "groupoids" ], "title": "groupoids-lemma-characterize-trivial-pseudo-torsors", "contents": [ "In the situation of", "Definition \\ref{definition-pseudo-torsor}.", "\\begin{enumerate}", "\\item The scheme $X$ is a pseudo $G$-torsor if and only if for every scheme", "$T$ over $S$ the set $X(T)$ is either empty or the action of the group $G(T)$", "on $X(T)$ is simply transitive.", "\\item A pseudo $G$-torsor $X$ is trivial if and only if the morphism", "$X \\to S$ has a section.", "\\end{enumerate}" ], "refs": [ "groupoids-definition-pseudo-torsor" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9678 ] }, { "id": 9616, "type": "theorem", "label": "groupoids-lemma-torsor", "categories": [ "groupoids" ], "title": "groupoids-lemma-torsor", "contents": [ "Let $S$ be a scheme.", "Let $(G, m)$ be a group scheme over $S$.", "Let $X$ be a scheme over $S$, and let", "$a : G \\times_S X \\to X$ be an action of $G$ on $X$.", "Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.", "Then $X$ is a $G$-torsor in the $\\tau$-topology if and only if", "$\\underline{X}$ is a $\\underline{G}$-torsor on $(\\Sch/S)_\\tau$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9617, "type": "theorem", "label": "groupoids-lemma-pullback-equivariant", "categories": [ "groupoids" ], "title": "groupoids-lemma-pullback-equivariant", "contents": [ "Let $S$ be a scheme. Let $G$ be a group scheme over $S$.", "Let $f : Y \\to X$ be a $G$-equivariant morphism between $S$-schemes", "endowed with $G$-actions. Then pullback $f^*$ given by", "$(\\mathcal{F}, \\alpha) \\mapsto (f^*\\mathcal{F}, (1_G \\times f)^*\\alpha)$", "defines a functor from the category of $G$-equivariant quasi-coherent", "$\\mathcal{O}_X$-modules to the category of", "$G$-equivariant quasi-coherent $\\mathcal{O}_Y$-modules." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9618, "type": "theorem", "label": "groupoids-lemma-complete-reducibility-Gm", "categories": [ "groupoids" ], "title": "groupoids-lemma-complete-reducibility-Gm", "contents": [ "Let $a : \\mathbf{G}_m \\times X \\to X$ be an action on an affine scheme.", "Then $X$ is the spectrum of a $\\mathbf{Z}$-graded ring", "and the action is as in Example \\ref{example-Gm-on-affine}." ], "refs": [], "proofs": [ { "contents": [ "Let $f \\in A = \\Gamma(X, \\mathcal{O}_X)$. Then we can write", "$$", "a^\\sharp(f) = \\sum\\nolimits_{n \\in \\mathbf{Z}} f_n \\otimes x^n", "\\quad\\text{in}\\quad", "A \\otimes \\mathbf{Z}[x, x^{-1}] =", "\\Gamma(\\mathbf{G}_m \\times X, \\mathcal{O}_{\\mathbf{G}_m \\times X})", "$$", "as a finite sum with $f_n$ in $A$ uniquely determined.", "Thus we obtain maps $A \\to A$, $f \\mapsto f_n$.", "Since $a$ is an action, if we evaluate at $x = 1$,", "we see $f = \\sum f_n$. Since $a$ is an action", "we find that", "$$", "\\sum (f_n)_m \\otimes x^m \\otimes x^n = \\sum f_n x^n \\otimes x^n", "$$", "(compare with computation in Example \\ref{example-Gm-on-affine}).", "Thus $(f_n)_m = 0$ if $n \\not = m$ and $(f_n)_n = f_n$.", "Thus if we set", "$$", "A_n = \\{f \\in A \\mid f_n = f\\}", "$$", "then we get $A = \\sum A_n$. On the other hand, the sum has to be", "direct since $f = 0$ implies $f_n = 0$ in the situation above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9619, "type": "theorem", "label": "groupoids-lemma-Gm-equivariant-module", "categories": [ "groupoids" ], "title": "groupoids-lemma-Gm-equivariant-module", "contents": [ "Let $A$ be a graded ring. Let $X = \\Spec(A)$ with action", "$a : \\mathbf{G}_m \\times X \\to X$ as in Example \\ref{example-Gm-on-affine}.", "Let $\\mathcal{F}$ be a $\\mathbf{G}_m$-equivariant quasi-coherent", "$\\mathcal{O}_X$-module. Then $M = \\Gamma(X, \\mathcal{F})$", "has a canonical grading such that it is a graded $A$-module", "and such that the isomorphism $\\widetilde{M} \\to \\mathcal{F}$", "(Schemes, Lemma \\ref{schemes-lemma-quasi-coherent-affine})", "is an isomorphism of $\\mathbf{G}_m$-equivariant modules where", "the $\\mathbf{G}_m$-equivariant structure on $\\widetilde{M}$", "is the one from Example \\ref{example-Gm-on-affine}." ], "refs": [ "schemes-lemma-quasi-coherent-affine" ], "proofs": [ { "contents": [ "You can either prove this by repeating the arguments of", "Lemma \\ref{lemma-complete-reducibility-Gm} for the module $M$.", "Alternatively, you can consider the scheme", "$(X', \\mathcal{O}_{X'}) = (X, \\mathcal{O}_X \\oplus \\mathcal{F})$", "where $\\mathcal{F}$ is viewed as an ideal of square zero.", "There is a natural action $a' : \\mathbf{G}_m \\times X' \\to X'$", "defined using the action on $X$ and on $\\mathcal{F}$. Then apply", "Lemma \\ref{lemma-complete-reducibility-Gm} to $X'$ and conclude.", "(The nice thing about this argument is that it immediately shows", "that the grading on $A$ and $M$ are compatible, i.e., that $M$", "is a graded $A$-module.)", "Details omitted." ], "refs": [ "groupoids-lemma-complete-reducibility-Gm", "groupoids-lemma-complete-reducibility-Gm" ], "ref_ids": [ 9618, 9618 ] } ], "ref_ids": [ 7663 ] }, { "id": 9620, "type": "theorem", "label": "groupoids-lemma-groupoid-pre-equivalence", "categories": [ "groupoids" ], "title": "groupoids-lemma-groupoid-pre-equivalence", "contents": [ "Given a groupoid scheme $(U, R, s, t, c)$ over $S$", "the morphism $j : R \\to U \\times_S U$ is a pre-equivalence", "relation." ], "refs": [], "proofs": [ { "contents": [ "Omitted.", "This is a nice exercise in the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9621, "type": "theorem", "label": "groupoids-lemma-equivalence-groupoid", "categories": [ "groupoids" ], "title": "groupoids-lemma-equivalence-groupoid", "contents": [ "Given an equivalence relation $j : R \\to U$ over $S$", "there is a unique way to extend it to a groupoid", "$(U, R, s, t, c)$ over $S$." ], "refs": [], "proofs": [ { "contents": [ "Omitted.", "This is a nice exercise in the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9622, "type": "theorem", "label": "groupoids-lemma-diagram", "categories": [ "groupoids" ], "title": "groupoids-lemma-diagram", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid over $S$.", "In the commutative diagram", "$$", "\\xymatrix{", "& U & \\\\", "R \\ar[d]_s \\ar[ru]^t &", "R \\times_{s, U, t} R", "\\ar[l]^-{\\text{pr}_0} \\ar[d]^{\\text{pr}_1} \\ar[r]_-c &", "R \\ar[d]^s \\ar[lu]_t \\\\", "U & R \\ar[l]_t \\ar[r]^s & U", "}", "$$", "the two lower squares are fibre product squares.", "Moreover, the triangle on top (which is really a square)", "is also cartesian." ], "refs": [], "proofs": [ { "contents": [ "Omitted.", "Exercise in the definitions and the functorial point of", "view in algebraic geometry." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9623, "type": "theorem", "label": "groupoids-lemma-diagram-pull", "categories": [ "groupoids" ], "title": "groupoids-lemma-diagram-pull", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c, e, i)$ be a groupoid over $S$.", "The diagram", "\\begin{equation}", "\\label{equation-pull}", "\\xymatrix{", "R \\times_{t, U, t} R", "\\ar@<1ex>[r]^-{\\text{pr}_1} \\ar@<-1ex>[r]_-{\\text{pr}_0}", "\\ar[d]_{(\\text{pr}_0, c \\circ (i, 1))} &", "R \\ar[r]^t \\ar[d]^{\\text{id}_R} &", "U \\ar[d]^{\\text{id}_U} \\\\", "R \\times_{s, U, t} R", "\\ar@<1ex>[r]^-c \\ar@<-1ex>[r]_-{\\text{pr}_0} \\ar[d]_{\\text{pr}_1} &", "R \\ar[r]^t \\ar[d]^s &", "U \\\\", "R \\ar@<1ex>[r]^s \\ar@<-1ex>[r]_t &", "U", "}", "\\end{equation}", "is commutative. The two top rows are isomorphic via the vertical maps given.", "The two lower left squares are cartesian." ], "refs": [], "proofs": [ { "contents": [ "The commutativity of the diagram follows from the axioms of a groupoid.", "Note that, in terms of groupoids, the top left vertical arrow assigns to", "a pair of morphisms $(\\alpha, \\beta)$ with the same target, the pair", "of morphisms $(\\alpha, \\alpha^{-1} \\circ \\beta)$. In any groupoid", "this defines a bijection between", "$\\text{Arrows} \\times_{t, \\text{Ob}, t} \\text{Arrows}$", "and", "$\\text{Arrows} \\times_{s, \\text{Ob}, t} \\text{Arrows}$. Hence the second", "assertion of the lemma.", "The last assertion follows from Lemma \\ref{lemma-diagram}." ], "refs": [ "groupoids-lemma-diagram" ], "ref_ids": [ 9622 ] } ], "ref_ids": [] }, { "id": 9624, "type": "theorem", "label": "groupoids-lemma-base-change-groupoid", "categories": [ "groupoids" ], "title": "groupoids-lemma-base-change-groupoid", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid over a scheme $S$.", "Let $S' \\to S$ be a morphism. Then the base changes $U' = S' \\times_S U$,", "$R' = S' \\times_S R$ endowed with the base changes $s'$, $t'$, $c'$", "of the morphisms $s, t, c$ form a groupoid scheme", "$(U', R', s', t', c')$ over $S'$ and the projections", "determine a morphism", "$(U', R', s', t', c') \\to (U, R, s, t, c)$", "of groupoid schemes over $S$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint:", "$R' \\times_{s', U', t'} R' = S' \\times_S (R \\times_{s, U, t} R)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9625, "type": "theorem", "label": "groupoids-lemma-isomorphism", "categories": [ "groupoids" ], "title": "groupoids-lemma-isomorphism", "contents": [ "Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "If $(\\mathcal{F}, \\alpha)$ is a quasi-coherent module on $(U, R, s, t, c)$", "then $\\alpha$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Pull back the commutative diagram of", "Definition \\ref{definition-groupoid-module}", "by the morphism $(i, 1) : R \\to R \\times_{s, U, t} R$.", "Then we see that $i^*\\alpha \\circ \\alpha = s^*e^*\\alpha$.", "Pulling back by the morphism $(1, i)$ we obtain the relation", "$\\alpha \\circ i^*\\alpha = t^*e^*\\alpha$. By the second assumption ", "these morphisms are the identity. Hence $i^*\\alpha$ is an inverse of", "$\\alpha$." ], "refs": [ "groupoids-definition-groupoid-module" ], "ref_ids": [ 9682 ] } ], "ref_ids": [] }, { "id": 9626, "type": "theorem", "label": "groupoids-lemma-pullback", "categories": [ "groupoids" ], "title": "groupoids-lemma-pullback", "contents": [ "Let $S$ be a scheme. Consider a morphism", "$f : (U, R, s, t, c) \\to (U', R', s', t', c')$", "of groupoid schemes over $S$. Then pullback $f^*$ given by", "$$", "(\\mathcal{F}, \\alpha) \\mapsto (f^*\\mathcal{F}, f^*\\alpha)", "$$", "defines a functor from the category of quasi-coherent sheaves on", "$(U', R', s', t', c')$ to the category of quasi-coherent sheaves on", "$(U, R, s, t, c)$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9627, "type": "theorem", "label": "groupoids-lemma-pushforward", "categories": [ "groupoids" ], "title": "groupoids-lemma-pushforward", "contents": [ "Let $S$ be a scheme. Consider a morphism", "$f : (U, R, s, t, c) \\to (U', R', s', t', c')$", "of groupoid schemes over $S$. Assume that", "\\begin{enumerate}", "\\item $f : U \\to U'$ is quasi-compact and quasi-separated,", "\\item the square", "$$", "\\xymatrix{", "R \\ar[d]_t \\ar[r]_f & R' \\ar[d]^{t'} \\\\", "U \\ar[r]^f & U'", "}", "$$", "is cartesian, and", "\\item $s'$ and $t'$ are flat.", "\\end{enumerate}", "Then pushforward $f_*$ given by", "$$", "(\\mathcal{F}, \\alpha) \\mapsto (f_*\\mathcal{F}, f_*\\alpha)", "$$", "defines a functor from the category of quasi-coherent sheaves on", "$(U, R, s, t, c)$ to the category of quasi-coherent sheaves on", "$(U', R', s', t', c')$ which is right adjoint to pullback as defined in", "Lemma \\ref{lemma-pullback}." ], "refs": [ "groupoids-lemma-pullback" ], "proofs": [ { "contents": [ "Since $U \\to U'$ is quasi-compact and quasi-separated we see that", "$f_*$ transforms quasi-coherent sheaves into quasi-coherent sheaves", "(Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}).", "Moreover, since the squares", "$$", "\\vcenter{", "\\xymatrix{", "R \\ar[d]_t \\ar[r]_f & R' \\ar[d]^{t'} \\\\", "U \\ar[r]^f & U'", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "R \\ar[d]_s \\ar[r]_f & R' \\ar[d]^{s'} \\\\", "U \\ar[r]^f & U'", "}", "}", "$$", "are cartesian we find that $(t')^*f_*\\mathcal{F} = f_*t^*\\mathcal{F}$", "and $(s')^*f_*\\mathcal{F} = f_*s^*\\mathcal{F}$ , see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology}.", "Thus it makes sense to think of $f_*\\alpha$ as a map", "$(t')^*f_*\\mathcal{F} \\to (s')^*f_*\\mathcal{F}$. A similar argument", "shows that $f_*\\alpha$ satisfies the cocycle condition.", "The functor is adjoint to the pullback functor since pullback", "and pushforward on modules on ringed spaces are adjoint.", "Some details omitted." ], "refs": [ "schemes-lemma-push-forward-quasi-coherent", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 7730, 3298 ] } ], "ref_ids": [ 9626 ] }, { "id": 9628, "type": "theorem", "label": "groupoids-lemma-colimits", "categories": [ "groupoids" ], "title": "groupoids-lemma-colimits", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "The category of quasi-coherent modules on $(U, R, s, t, c)$ has colimits." ], "refs": [], "proofs": [ { "contents": [ "Let $i \\mapsto (\\mathcal{F}_i, \\alpha_i)$ be a diagram over the index", "category $\\mathcal{I}$. We can form the colimit", "$\\mathcal{F} = \\colim \\mathcal{F}_i$", "which is a quasi-coherent sheaf on $U$, see", "Schemes, Section \\ref{schemes-section-quasi-coherent}.", "Since colimits commute with pullback we see that", "$s^*\\mathcal{F} = \\colim s^*\\mathcal{F}_i$ and similarly", "$t^*\\mathcal{F} = \\colim t^*\\mathcal{F}_i$. Hence we can set", "$\\alpha = \\colim \\alpha_i$. We omit the proof that $(\\mathcal{F}, \\alpha)$", "is the colimit of the diagram in the category of quasi-coherent modules", "on $(U, R, s, t, c)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9629, "type": "theorem", "label": "groupoids-lemma-abelian", "categories": [ "groupoids" ], "title": "groupoids-lemma-abelian", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "If $s$, $t$ are flat, then the category of quasi-coherent modules on", "$(U, R, s, t, c)$ is abelian." ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi : (\\mathcal{F}, \\alpha) \\to (\\mathcal{G}, \\beta)$ be a", "homomorphism of quasi-coherent modules on $(U, R, s, t, c)$. Since", "$s$ is flat we see that", "$$", "0 \\to s^*\\Ker(\\varphi)", "\\to s^*\\mathcal{F} \\to s^*\\mathcal{G} \\to s^*\\Coker(\\varphi) \\to 0", "$$", "is exact and similarly for pullback by $t$. Hence $\\alpha$ and $\\beta$", "induce isomorphisms", "$\\kappa : t^*\\Ker(\\varphi) \\to s^*\\Ker(\\varphi)$ and", "$\\lambda : t^*\\Coker(\\varphi) \\to s^*\\Coker(\\varphi)$", "which satisfy the cocycle condition. Then it is straightforward to", "verify that $(\\Ker(\\varphi), \\kappa)$ and", "$(\\Coker(\\varphi), \\lambda)$ are a kernel and cokernel in the", "category of quasi-coherent modules on $(U, R, s, t, c)$. Moreover,", "the condition $\\Coim(\\varphi) = \\Im(\\varphi)$ follows", "because it holds over $U$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9630, "type": "theorem", "label": "groupoids-lemma-construct-quasi-coherent", "categories": [ "groupoids" ], "title": "groupoids-lemma-construct-quasi-coherent", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Assume $s, t$ are flat, quasi-compact, and quasi-separated.", "For any quasi-coherent module $\\mathcal{G}$ on $U$, there exists", "a canonical isomorphism", "$\\alpha : t^*t_*s^*\\mathcal{G} \\to s^*t_*s^*\\mathcal{G}$", "which turns $(t_*s^*\\mathcal{G}, \\alpha)$ into a quasi-coherent module", "on $(U, R, s, t, c)$. This construction defines a functor", "$$", "\\QCoh(\\mathcal{O}_U) \\longrightarrow \\QCoh(U, R, s, t, c)", "$$", "which is a right adjoint to the forgetful functor", "$(\\mathcal{F}, \\beta) \\mapsto \\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "The pushforward of a quasi-coherent module along a quasi-compact and", "quasi-separated morphism is quasi-coherent, see Schemes, Lemma", "\\ref{schemes-lemma-push-forward-quasi-coherent}. Hence $t_*s^*\\mathcal{G}$", "is quasi-coherent. With notation as in Lemma \\ref{lemma-diagram} we have", "$$", "t^*t_*s^*\\mathcal{G} =", "\\text{pr}_{0, *}c^* s^*\\mathcal{G} =", "\\text{pr}_{0, *}\\text{pr}_1^*s^*\\mathcal{G} =", "s^*t_*s^*\\mathcal{G}", "$$", "The middle equality because $s \\circ c = s \\circ \\text{pr}_1$ as", "morphisms $R \\times_{s, U, t} R \\to U$, and the first and the last", "equality because we know that base change and pushforward commute in", "these steps by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology}.", "\\medskip\\noindent", "To verify the cocycle condition of Definition \\ref{definition-groupoid-module}", "for $\\alpha$ and the adjointness property we describe the construction", "$\\mathcal{G} \\mapsto (\\mathcal{G}, \\alpha)$ in another way.", "Consider the groupoid scheme", "$(R, R \\times_{s, U, s} R, \\text{pr}_0, \\text{pr}_1, \\text{pr}_{02})$", "associated to the equivalence relation $R \\times_{s, U, s} R$", "on $R$, see Lemma \\ref{lemma-equivalence-groupoid}.", "There is a morphism", "$$", "f :", "(R, R \\times_{s, U, s} R, \\text{pr}_1, \\text{pr}_0, \\text{pr}_{02})", "\\longrightarrow", "(U, R, s, t, c)", "$$", "of groupoid schemes given by $t : R \\to U$ and $R \\times_{t, U, t} R \\to R$", "given by $(r_0, r_1) \\mapsto r_0 \\circ r_1^{-1}$ (we omit the verification", "of the commutativity of the required diagrams). Since", "$t, s : R \\to U$ are quasi-compact, quasi-separated, and flat,", "and since we have a cartesian square", "$$", "\\xymatrix{", "R \\times_{s, U, s} R \\ar[d]_{\\text{pr}_0}", "\\ar[rr]_-{(r_0, r_1) \\mapsto r_0 \\circ r_1^{-1}} & & R \\ar[d]^t \\\\", "R \\ar[rr]^t & & U", "}", "$$", "by Lemma \\ref{lemma-diagram-pull} it follows that", "Lemma \\ref{lemma-pushforward} applies to $f$. Note that", "$$", "\\QCoh(R, R \\times_{s, U, s} R, \\text{pr}_1, \\text{pr}_0, \\text{pr}_{02})", "= \\QCoh(\\mathcal{O}_U)", "$$", "by the theory of descent of quasi-coherent sheaves as $\\{t : R \\to U\\}$", "is an fpqc covering, see", "Descent, Proposition \\ref{descent-proposition-fpqc-descent-quasi-coherent}.", "Observe that pullback along $f$ agrees with the forgetful functor and", "that pushforward agrees with the construction that assigns to", "$\\mathcal{G}$ the pair $(\\mathcal{G}, \\alpha)$. We omit the precise", "verifications. Thus the lemma follows from Lemma \\ref{lemma-pushforward}." ], "refs": [ "schemes-lemma-push-forward-quasi-coherent", "groupoids-lemma-diagram", "coherent-lemma-flat-base-change-cohomology", "groupoids-definition-groupoid-module", "groupoids-lemma-equivalence-groupoid", "groupoids-lemma-diagram-pull", "groupoids-lemma-pushforward", "descent-proposition-fpqc-descent-quasi-coherent", "groupoids-lemma-pushforward" ], "ref_ids": [ 7730, 9622, 3298, 9682, 9621, 9623, 9627, 14753, 9627 ] } ], "ref_ids": [] }, { "id": 9631, "type": "theorem", "label": "groupoids-lemma-push-pull", "categories": [ "groupoids" ], "title": "groupoids-lemma-push-pull", "contents": [ "Let $f : Y \\to X$ be a morphism of schemes. Let $\\mathcal{F}$", "be a quasi-coherent $\\mathcal{O}_X$-module, let $\\mathcal{G}$", "be a quasi-coherent $\\mathcal{O}_Y$-module, and let", "$\\varphi : \\mathcal{G} \\to f^*\\mathcal{F}$ be a module map. Assume", "\\begin{enumerate}", "\\item $\\varphi$ is injective,", "\\item $f$ is quasi-compact, quasi-separated, flat, and surjective,", "\\item $X$, $Y$ are locally Noetherian, and", "\\item $\\mathcal{G}$ is a coherent $\\mathcal{O}_Y$-module.", "\\end{enumerate}", "Then $\\mathcal{F} \\cap f_*\\mathcal{G}$ defined as the pullback", "$$", "\\xymatrix{", "\\mathcal{F} \\ar[r] & f_*f^*\\mathcal{F} \\\\", "\\mathcal{F} \\cap f_*\\mathcal{G} \\ar[u] \\ar[r] &", "f_*\\mathcal{G} \\ar[u]", "}", "$$", "is a coherent $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "We will freely use the characterization of coherent modules of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-Noetherian}", "as well as the fact that coherent modules form a Serre subcategory", "of $\\QCoh(\\mathcal{O}_X)$, see", "Cohomology of Schemes,", "Lemma \\ref{coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient}.", "If $f$ has a section $\\sigma$, then we see that", "$\\mathcal{F} \\cap f_*\\mathcal{G}$ is contained in the image of", "$\\sigma^*\\mathcal{G} \\to \\sigma^*f^*\\mathcal{F} = \\mathcal{F}$,", "hence coherent. In general, to show that $\\mathcal{F} \\cap f_*\\mathcal{G}$", "is coherent, it suffices the show that", "$f^*(\\mathcal{F} \\cap f_*\\mathcal{G})$ is coherent (see", "Descent, Lemma \\ref{descent-lemma-finite-type-descends}).", "Since $f$ is flat this is equal to $f^*\\mathcal{F} \\cap f^*f_*\\mathcal{G}$.", "Since $f$ is flat, quasi-compact, and quasi-separated we see", "$f^*f_*\\mathcal{G} = p_*q^*\\mathcal{G}$ where $p, q : Y \\times_X Y \\to Y$", "are the projections, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}.", "Since $p$ has a section we win." ], "refs": [ "coherent-lemma-coherent-Noetherian", "coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient", "descent-lemma-finite-type-descends", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 3308, 3310, 14612, 3298 ] } ], "ref_ids": [] }, { "id": 9632, "type": "theorem", "label": "groupoids-lemma-colimit-coherent", "categories": [ "groupoids" ], "title": "groupoids-lemma-colimit-coherent", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Assume that", "\\begin{enumerate}", "\\item $U$, $R$ are Noetherian,", "\\item $s, t$ are flat, quasi-compact, and quasi-separated.", "\\end{enumerate}", "Then every quasi-coherent module $(\\mathcal{F}, \\alpha)$ on $(U, R, s, t, c)$", "is a filtered colimit of coherent modules." ], "refs": [], "proofs": [ { "contents": [ "We will use the characterization of Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-coherent-Noetherian} of coherent modules on locally", "Noetherian scheme without further mention. Write", "$\\mathcal{F} = \\colim \\mathcal{H}_i$ with $\\mathcal{H}_i$ coherent, see", "Properties, Lemma \\ref{properties-lemma-directed-colimit-finite-presentation}.", "Given a quasi-coherent sheaf $\\mathcal{H}$ on $U$ we denote $t_*s^*\\mathcal{H}$", "the quasi-coherent sheaf on $(U, R, s, t, c)$ of", "Lemma \\ref{lemma-construct-quasi-coherent}. There is an adjunction map", "$\\mathcal{F} \\to t_*s^*\\mathcal{F}$ in $\\QCoh(U, R, s, t, c)$.", "Consider the pullback diagram", "$$", "\\xymatrix{", "\\mathcal{F} \\ar[r] & t_*s^*\\mathcal{F} \\\\", "\\mathcal{F}_i \\ar[r] \\ar[u] & t_*s^*\\mathcal{H}_i \\ar[u]", "}", "$$", "in other words $\\mathcal{F}_i = \\mathcal{F} \\cap t_*s^*\\mathcal{H}_i$.", "Then $\\mathcal{F}_i$ is coherent by Lemma \\ref{lemma-push-pull}.", "On the other hand, the diagram above is a pullback diagram in", "$\\QCoh(U, R, s, t, c)$ also as restriction to $U$ is an", "exact functor by (the proof of) Lemma \\ref{lemma-abelian}. Finally,", "because $t$ is quasi-compact and quasi-separated we see that", "$t_*$ commutes with colimits (see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-colimit-cohomology}).", "Hence $t_*s^*\\mathcal{F} = \\colim t_*\\mathcal{H}_i$ and hence", "$\\mathcal{F} = \\colim \\mathcal{F}_i$ as desired." ], "refs": [ "coherent-lemma-coherent-Noetherian", "properties-lemma-directed-colimit-finite-presentation", "groupoids-lemma-construct-quasi-coherent", "groupoids-lemma-push-pull", "groupoids-lemma-abelian", "coherent-lemma-colimit-cohomology" ], "ref_ids": [ 3308, 3024, 9630, 9631, 9629, 3300 ] } ], "ref_ids": [] }, { "id": 9633, "type": "theorem", "label": "groupoids-lemma-colimit-finite-type", "categories": [ "groupoids" ], "title": "groupoids-lemma-colimit-finite-type", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Assume that", "\\begin{enumerate}", "\\item $U$, $R$ are affine,", "\\item there exist $e_i \\in \\mathcal{O}_R(R)$ such that", "every element $g \\in \\mathcal{O}_R(R)$ can be uniquely written as", "$\\sum s^*(f_i)e_i$ for some $f_i \\in \\mathcal{O}_U(U)$.", "\\end{enumerate}", "Then every quasi-coherent module $(\\mathcal{F}, \\alpha)$ on $(U, R, s, t, c)$", "is a filtered colimit of finite type quasi-coherent modules." ], "refs": [], "proofs": [ { "contents": [ "The assumption means that $\\mathcal{O}_R(R)$ is a free", "$\\mathcal{O}_U(U)$-module via $s$ with basis $e_i$. Hence", "for any quasi-coherent $\\mathcal{O}_U$-module $\\mathcal{G}$", "we see that $s^*\\mathcal{G}(R) = \\bigoplus_i \\mathcal{G}(U)e_i$.", "We will write $s(-)$ to indicate pullback of sections by $s$ and", "similarly for other morphisms.", "Let $(\\mathcal{F}, \\alpha)$ be a quasi-coherent module on", "$(U, R, s, t, c)$. Let $\\sigma \\in \\mathcal{F}(U)$. By the above", "we can write", "$$", "\\alpha(t(\\sigma)) = \\sum s(\\sigma_i) e_i", "$$", "for some unique $\\sigma_i \\in \\mathcal{F}(U)$ (all but finitely many", "are zero of course). We can also write", "$$", "c(e_i) = \\sum \\text{pr}_1(f_{ij}) \\text{pr}_0(e_j)", "$$", "as functions on $R \\times_{s, U, t}R$. Then the commutativity of the diagram", "in Definition \\ref{definition-groupoid-module} means that", "$$", "\\sum \\text{pr}_1(\\alpha(t(\\sigma_i))) \\text{pr}_0(e_i)", "=", "\\sum \\text{pr}_1(s(\\sigma_i)f_{ij}) \\text{pr}_0(e_j)", "$$", "(calculation omitted). Picking off the coefficients of $\\text{pr}_0(e_l)$", "we see that $\\alpha(t(\\sigma_l)) = \\sum s(\\sigma_i)f_{il}$. Hence", "the submodule $\\mathcal{G} \\subset \\mathcal{F}$ generated by the", "elements $\\sigma_i$ defines a finite type quasi-coherent module", "preserved by $\\alpha$. Hence it is a subobject of $\\mathcal{F}$ in", "$\\QCoh(U, R, s, t, c)$. This submodule contains $\\sigma$", "(as one sees by pulling back the first relation by $e$). Hence we win." ], "refs": [ "groupoids-definition-groupoid-module" ], "ref_ids": [ 9682 ] } ], "ref_ids": [] }, { "id": 9634, "type": "theorem", "label": "groupoids-lemma-set-of-iso-classes", "categories": [ "groupoids" ], "title": "groupoids-lemma-set-of-iso-classes", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Let $\\kappa$ be a cardinal.", "There exists a set $T$ and a family $(\\mathcal{F}_t, \\alpha_t)_{t \\in T}$ of", "$\\kappa$-generated quasi-coherent modules on $(U, R, s, t, c)$", "such that every $\\kappa$-generated quasi-coherent module on", "$(U, R, s, t, c)$ is isomorphic to one of the $(\\mathcal{F}_t, \\alpha_t)$." ], "refs": [], "proofs": [ { "contents": [ "For each quasi-coherent module $\\mathcal{F}$ on $U$ there is a", "(possibly empty) set of maps $\\alpha : t^*\\mathcal{F} \\to s^*\\mathcal{F}$", "such that $(\\mathcal{F}, \\alpha)$ is a quasi-coherent modules on", "$(U, R, s, t, c)$. By", "Properties, Lemma \\ref{properties-lemma-set-of-iso-classes}", "there exists a set of isomorphism classes of $\\kappa$-generated", "quasi-coherent $\\mathcal{O}_U$-modules." ], "refs": [ "properties-lemma-set-of-iso-classes" ], "ref_ids": [ 3031 ] } ], "ref_ids": [] }, { "id": 9635, "type": "theorem", "label": "groupoids-lemma-colimit-kappa", "categories": [ "groupoids" ], "title": "groupoids-lemma-colimit-kappa", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Assume that $s, t$ are flat. There exists a", "cardinal $\\kappa$ such that every quasi-coherent module", "$(\\mathcal{F}, \\alpha)$ on $(U, R, s, t, c)$", "is the directed colimit of its $\\kappa$-generated", "quasi-coherent submodules." ], "refs": [], "proofs": [ { "contents": [ "In the statement of the lemma and in this proof", "a {\\it submodule} of a quasi-coherent module $(\\mathcal{F}, \\alpha)$", "is a quasi-coherent submodule $\\mathcal{G} \\subset \\mathcal{F}$", "such that $\\alpha(t^*\\mathcal{G}) = s^*\\mathcal{G}$ as subsheaves of", "$s^*\\mathcal{F}$. This makes sense because since $s, t$ are flat the", "pullbacks $s^*$ and $t^*$ are exact, i.e., preserve subsheaves.", "The proof will be a repeat of the proof of", "Properties, Lemma \\ref{properties-lemma-colimit-kappa}.", "We urge the reader to read that proof first.", "\\medskip\\noindent", "Choose an affine open covering $U = \\bigcup_{i \\in I} U_i$.", "For each pair $i, j$ choose affine open coverings", "$$", "U_i \\cap U_j = \\bigcup\\nolimits_{k \\in I_{ij}} U_{ijk}", "\\quad\\text{and}\\quad", "s^{-1}(U_i) \\cap t^{-1}(U_j) = \\bigcup\\nolimits_{k \\in J_{ij}} W_{ijk}.", "$$", "Write $U_i = \\Spec(A_i)$, $U_{ijk} = \\Spec(A_{ijk})$,", "$W_{ijk} = \\Spec(B_{ijk})$.", "Let $\\kappa$ be any infinite cardinal $\\geq$ than the cardinality", "of any of the sets $I$, $I_{ij}$, $J_{ij}$.", "\\medskip\\noindent", "Let $(\\mathcal{F}, \\alpha)$ be a quasi-coherent module on $(U, R, s, t, c)$.", "Set $M_i = \\mathcal{F}(U_i)$, $M_{ijk} = \\mathcal{F}(U_{ijk})$.", "Note that", "$$", "M_i \\otimes_{A_i} A_{ijk} = M_{ijk} = M_j \\otimes_{A_j} A_{ijk}", "$$", "and that $\\alpha$ gives isomorphisms", "$$", "\\alpha|_{W_{ijk}} :", "M_i \\otimes_{A_i, t} B_{ijk}", "\\longrightarrow", "M_j \\otimes_{A_j, s} B_{ijk}", "$$", "see", "Schemes, Lemma \\ref{schemes-lemma-widetilde-pullback}.", "Using the axiom of choice we choose a map", "$$", "(i, j, k, m) \\mapsto S(i, j, k, m)", "$$", "which associates to every $i, j \\in I$, $k \\in I_{ij}$ or $k \\in J_{ij}$", "and $m \\in M_i$ a finite subset $S(i, j, k, m) \\subset M_j$", "such that we have", "$$", "m \\otimes 1 = \\sum\\nolimits_{m' \\in S(i, j, k, m)} m' \\otimes a_{m'}", "\\quad\\text{or}\\quad", "\\alpha(m \\otimes 1) = \\sum\\nolimits_{m' \\in S(i, j, k, m)} m' \\otimes b_{m'}", "$$", "in $M_{ijk}$ for some $a_{m'} \\in A_{ijk}$ or $b_{m'} \\in B_{ijk}$.", "Moreover, let's agree that $S(i, i, k, m) = \\{m\\}$ for all", "$i, j = i, k, m$ when $k \\in I_{ij}$. Fix such a collection $S(i, j, k, m)$", "\\medskip\\noindent", "Given a family $\\mathcal{S} = (S_i)_{i \\in I}$ of subsets", "$S_i \\subset M_i$ of cardinality at most $\\kappa$ we set", "$\\mathcal{S}' = (S'_i)$ where", "$$", "S'_j = \\bigcup\\nolimits_{(i, j, k, m)\\text{ such that }m \\in S_i}", "S(i, j, k, m)", "$$", "Note that $S_i \\subset S'_i$. Note that $S'_i$ has cardinality at most", "$\\kappa$ because it is a union over a set of cardinality at most $\\kappa$", "of finite sets. Set $\\mathcal{S}^{(0)} = \\mathcal{S}$,", "$\\mathcal{S}^{(1)} = \\mathcal{S}'$ and by induction", "$\\mathcal{S}^{(n + 1)} = (\\mathcal{S}^{(n)})'$. Then set", "$\\mathcal{S}^{(\\infty)} = \\bigcup_{n \\geq 0} \\mathcal{S}^{(n)}$.", "Writing $\\mathcal{S}^{(\\infty)} = (S^{(\\infty)}_i)$ we see that", "for any element $m \\in S^{(\\infty)}_i$ the image of $m$ in", "$M_{ijk}$ can be written as a finite sum $\\sum m' \\otimes a_{m'}$", "with $m' \\in S_j^{(\\infty)}$. In this way we see that setting", "$$", "N_i = A_i\\text{-submodule of }M_i\\text{ generated by }S^{(\\infty)}_i", "$$", "we have", "$$", "N_i \\otimes_{A_i} A_{ijk} = N_j \\otimes_{A_j} A_{ijk}", "\\quad\\text{and}\\quad", "\\alpha(N_i \\otimes_{A_i, t} B_{ijk}) = N_j \\otimes_{A_j, s} B_{ijk}", "$$", "as submodules of $M_{ijk}$ or $M_j \\otimes_{A_j, s} B_{ijk}$.", "Thus there exists a quasi-coherent submodule", "$\\mathcal{G} \\subset \\mathcal{F}$ with $\\mathcal{G}(U_i) = N_i$", "such that $\\alpha(t^*\\mathcal{G}) = s^*\\mathcal{G}$ as submodules", "of $s^*\\mathcal{F}$. In other words,", "$(\\mathcal{G}, \\alpha|_{t^*\\mathcal{G}})$ is a submodule of", "$(\\mathcal{F}, \\alpha)$.", "Moreover, by construction $\\mathcal{G}$ is $\\kappa$-generated.", "\\medskip\\noindent", "Let $\\{(\\mathcal{G}_t, \\alpha_t)\\}_{t \\in T}$ be the set of", "$\\kappa$-generated quasi-coherent submodules of $(\\mathcal{F}, \\alpha)$.", "If $t, t' \\in T$ then $\\mathcal{G}_t + \\mathcal{G}_{t'}$ is also a", "$\\kappa$-generated quasi-coherent submodule as it is the image of the map", "$\\mathcal{G}_t \\oplus \\mathcal{G}_{t'} \\to \\mathcal{F}$.", "Hence the system (ordered by inclusion) is directed.", "The arguments above show that every section of $\\mathcal{F}$ over $U_i$", "is in one of the $\\mathcal{G}_t$ (because we can start with $\\mathcal{S}$", "such that the given section is an element of $S_i$). Hence", "$\\colim_t \\mathcal{G}_t \\to \\mathcal{F}$ is both injective and surjective", "as desired." ], "refs": [ "properties-lemma-colimit-kappa", "schemes-lemma-widetilde-pullback" ], "ref_ids": [ 3032, 7662 ] } ], "ref_ids": [] }, { "id": 9636, "type": "theorem", "label": "groupoids-lemma-groupoid-from-action", "categories": [ "groupoids" ], "title": "groupoids-lemma-groupoid-from-action", "contents": [ "Let $S$ be a scheme.", "Let $Y$ be a scheme over $S$.", "Let $(G, m)$ be a group scheme over $Y$ with", "identity $e_G$ and inverse $i_G$.", "Let $X/Y$ be a scheme over $Y$ and let $a : G \\times_Y X \\to X$", "be an action of $G$ on $X/Y$.", "Then we get a groupoid scheme $(U, R, s, t, c, e, i)$ over $S$", "in the following manner:", "\\begin{enumerate}", "\\item We set $U = X$, and $R = G \\times_Y X$.", "\\item We set $s : R \\to U$ equal to $(g, x) \\mapsto x$.", "\\item We set $t : R \\to U$ equal to $(g, x) \\mapsto a(g, x)$.", "\\item We set $c : R \\times_{s, U, t} R \\to R$ equal to", "$((g, x), (g', x')) \\mapsto (m(g, g'), x')$.", "\\item We set $e : U \\to R$ equal to $x \\mapsto (e_G(x), x)$.", "\\item We set $i : R \\to R$ equal to $(g, x) \\mapsto (i_G(g), a(g, x))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: It is enough to show that this works on the set", "level. For this use the description above the lemma describing", "$g$ as an arrow from $v$ to $a(g, v)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9637, "type": "theorem", "label": "groupoids-lemma-action-groupoid-modules", "categories": [ "groupoids" ], "title": "groupoids-lemma-action-groupoid-modules", "contents": [ "Let $S$ be a scheme.", "Let $Y$ be a scheme over $S$.", "Let $(G, m)$ be a group scheme over $Y$.", "Let $X$ be a scheme over $Y$ and let $a : G \\times_Y X \\to X$", "be an action of $G$ on $X$ over $Y$. Let $(U, R, s, t, c)$ be", "the groupoid scheme constructed in Lemma \\ref{lemma-groupoid-from-action}.", "The rule", "$(\\mathcal{F}, \\alpha) \\mapsto (\\mathcal{F}, \\alpha)$ defines", "an equivalence of categories between $G$-equivariant", "$\\mathcal{O}_X$-modules and the category of quasi-coherent", "modules on $(U, R, s, t, c)$." ], "refs": [ "groupoids-lemma-groupoid-from-action" ], "proofs": [ { "contents": [ "The assertion makes sense because $t = a$ and $s = \\text{pr}_1$", "as morphisms $R = G \\times_Y X \\to X$, see", "Definitions \\ref{definition-equivariant-module} and", "\\ref{definition-groupoid-module}.", "Using the translation in Lemma \\ref{lemma-groupoid-from-action}", "the commutativity requirements", "of the two definitions match up exactly." ], "refs": [ "groupoids-definition-equivariant-module", "groupoids-definition-groupoid-module", "groupoids-lemma-groupoid-from-action" ], "ref_ids": [ 9680, 9682, 9636 ] } ], "ref_ids": [ 9636 ] }, { "id": 9638, "type": "theorem", "label": "groupoids-lemma-groupoid-stabilizer", "categories": [ "groupoids" ], "title": "groupoids-lemma-groupoid-stabilizer", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid over $S$.", "The scheme $G$ defined by the cartesian square", "$$", "\\xymatrix{", "G \\ar[r] \\ar[d] & R \\ar[d]^{j = (t, s)} \\\\", "U \\ar[r]^-{\\Delta} & U \\times_S U", "}", "$$", "is a group scheme over $U$ with composition law", "$m$ induced by the composition law $c$." ], "refs": [], "proofs": [ { "contents": [ "This is true because in a groupoid category the", "set of self maps of any object forms a group." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9639, "type": "theorem", "label": "groupoids-lemma-groupoid-action-stabilizer", "categories": [ "groupoids" ], "title": "groupoids-lemma-groupoid-action-stabilizer", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid over $S$, and let $G/U$ be its stabilizer.", "Denote $R_t/U$ the scheme $R$ seen as a scheme over $U$ via the", "morphism $t : R \\to U$.", "There is a canonical left action", "$$", "a : G \\times_U R_t \\longrightarrow R_t", "$$", "induced by the composition law $c$." ], "refs": [], "proofs": [ { "contents": [ "In terms of points over $T/S$ we define $a(g, r) = c(g, r)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9640, "type": "theorem", "label": "groupoids-lemma-groupoid-action-stabilizer-pseudo-torsor", "categories": [ "groupoids" ], "title": "groupoids-lemma-groupoid-action-stabilizer-pseudo-torsor", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme", "over $S$. Let $G$ be the stabilizer group scheme of $R$.", "Let", "$$", "G_0 = G \\times_{U, \\text{pr}_0} (U \\times_S U) = G \\times_S U", "$$", "as a group scheme over $U \\times_S U$. The action of $G$ on $R$ of", "Lemma \\ref{lemma-groupoid-action-stabilizer}", "induces an action of $G_0$ on $R$ over $U \\times_S U$", "which turns $R$ into a pseudo $G_0$-torsor over $U \\times_S U$." ], "refs": [ "groupoids-lemma-groupoid-action-stabilizer" ], "proofs": [ { "contents": [ "This is true because in a groupoid category $\\mathcal{C}$ the set", "$\\Mor_\\mathcal{C}(x, y)$ is a principal homogeneous set", "under the group $\\Mor_\\mathcal{C}(y, y)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9639 ] }, { "id": 9641, "type": "theorem", "label": "groupoids-lemma-fibres-j", "categories": [ "groupoids" ], "title": "groupoids-lemma-fibres-j", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Let $p \\in U \\times_S U$ be a point. Denote", "$R_p$ the scheme theoretic fibre of $j = (t, s) : R \\to U \\times_S U$.", "If $R_p \\not = \\emptyset$, then the action", "$$", "G_{0, \\kappa(p)} \\times_{\\kappa(p)} R_p \\longrightarrow R_p", "$$", "(see", "Lemma \\ref{lemma-groupoid-action-stabilizer-pseudo-torsor})", "which turns $R_p$ into a $G_{\\kappa(p)}$-torsor over $\\kappa(p)$." ], "refs": [ "groupoids-lemma-groupoid-action-stabilizer-pseudo-torsor" ], "proofs": [ { "contents": [ "The action is a pseudo-torsor by the lemma cited in the statement.", "And if $R_p$ is not the empty scheme, then $\\{R_p \\to p\\}$", "is an fpqc covering which trivializes the pseudo-torsor." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9640 ] }, { "id": 9642, "type": "theorem", "label": "groupoids-lemma-restrict-groupoid", "categories": [ "groupoids" ], "title": "groupoids-lemma-restrict-groupoid", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Let $g : U' \\to U$ be a morphism of schemes.", "Consider the following diagram", "$$", "\\xymatrix{", "R' \\ar[d] \\ar[r] \\ar@/_3pc/[dd]_{t'} \\ar@/^1pc/[rr]^{s'}&", "R \\times_{s, U} U' \\ar[r] \\ar[d] &", "U' \\ar[d]^g \\\\", "U' \\times_{U, t} R \\ar[d] \\ar[r] &", "R \\ar[r]^s \\ar[d]_t &", "U \\\\", "U' \\ar[r]^g &", "U", "}", "$$", "where all the squares are fibre product squares. Then there is a", "canonical composition law $c' : R' \\times_{s', U', t'} R' \\to R'$", "such that $(U', R', s', t', c')$ is a groupoid scheme over", "$S$ and such that $U' \\to U$, $R' \\to R$ defines a morphism", "$(U', R', s', t', c') \\to (U, R, s, t, c)$ of groupoid schemes over $S$.", "Moreover, for any scheme $T$ over $S$ the functor of groupoids", "$$", "(U'(T), R'(T), s', t', c') \\to (U(T), R(T), s, t, c)", "$$", "is the restriction (see above) of $(U(T), R(T), s, t, c)$ via the map", "$U'(T) \\to U(T)$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9643, "type": "theorem", "label": "groupoids-lemma-restrict-groupoid-relation", "categories": [ "groupoids" ], "title": "groupoids-lemma-restrict-groupoid-relation", "contents": [ "The notions of restricting groupoids and", "(pre-)equivalence relations defined in Definitions", "\\ref{definition-restrict-groupoid} and \\ref{definition-restrict-relation}", "agree via the constructions of", "Lemmas \\ref{lemma-groupoid-pre-equivalence} and", "\\ref{lemma-equivalence-groupoid}." ], "refs": [ "groupoids-definition-restrict-groupoid", "groupoids-definition-restrict-relation", "groupoids-lemma-groupoid-pre-equivalence", "groupoids-lemma-equivalence-groupoid" ], "proofs": [ { "contents": [ "What we are saying here is that $R'$ of", "Lemma \\ref{lemma-restrict-groupoid} is also", "equal to", "$$", "R' = (U' \\times_S U')\\times_{U \\times_S U} R", "\\longrightarrow", "U' \\times_S U'", "$$", "In fact this might have been a clearer way to state that lemma." ], "refs": [ "groupoids-lemma-restrict-groupoid" ], "ref_ids": [ 9642 ] } ], "ref_ids": [ 9684, 9671, 9620, 9621 ] }, { "id": 9644, "type": "theorem", "label": "groupoids-lemma-restrict-stabilizer", "categories": [ "groupoids" ], "title": "groupoids-lemma-restrict-stabilizer", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Let $g : U' \\to U$ be a morphism of schemes.", "Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$.", "Let $G$ be the stabilizer of $(U, R, s, t, c)$ and let", "$G'$ be the stabilizer of $(U', R', s', t', c')$.", "Then $G'$ is the base change of $G$ by $g$, i.e.,", "there is a canonical identification $G' = U' \\times_{g, U} G$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9645, "type": "theorem", "label": "groupoids-lemma-constructing-invariant-opens", "categories": [ "groupoids" ], "title": "groupoids-lemma-constructing-invariant-opens", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "\\begin{enumerate}", "\\item For any subset $W \\subset U$ the subset $t(s^{-1}(W))$", "is set-theoretically $R$-invariant.", "\\item If $s$ and $t$ are open, then for every open $W \\subset U$", "the open $t(s^{-1}(W))$ is an $R$-invariant open subscheme.", "\\item If $s$ and $t$ are open and quasi-compact, then $U$ has an open", "covering consisting of $R$-invariant quasi-compact open subschemes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from", "Lemmas \\ref{lemma-pre-equivalence-equivalence-relation-points} and", "\\ref{lemma-groupoid-pre-equivalence}, namely, $t(s^{-1}(W))$", "is the set of points of $U$ equivalent to a point of $W$.", "Next, assume $s$ and $t$ open and $W \\subset U$ open.", "Since $s$ is open the set $W' = t(s^{-1}(W))$ is an open subset of $U$.", "Finally, assume that $s$, $t$ are both open and quasi-compact.", "Then, if $W \\subset U$ is a quasi-compact open, then also", "$W' = t(s^{-1}(W))$ is a quasi-compact open, and invariant by the", "discussion above. Letting $W$ range over all affine opens of $U$", "we see (3)." ], "refs": [ "groupoids-lemma-pre-equivalence-equivalence-relation-points", "groupoids-lemma-groupoid-pre-equivalence" ], "ref_ids": [ 9579, 9620 ] } ], "ref_ids": [] }, { "id": 9646, "type": "theorem", "label": "groupoids-lemma-first-observation", "categories": [ "groupoids" ], "title": "groupoids-lemma-first-observation", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Assume $s$ and $t$ quasi-compact and flat and $U$ quasi-separated.", "Let $W \\subset U$ be quasi-compact open. Then $t(s^{-1}(W))$", "is an intersection of a nonempty family of quasi-compact open subsets of $U$." ], "refs": [], "proofs": [ { "contents": [ "Note that $s^{-1}(W)$ is quasi-compact open in $R$.", "As a continuous map $t$ maps the quasi-compact subset", "$s^{-1}(W)$ to a quasi-compact subset $t(s^{-1}(W))$.", "As $t$ is flat and $s^{-1}(W)$ is closed under generalization,", "so is $t(s^{-1}(W))$, see", "(Morphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat} and", "Topology, Lemma \\ref{topology-lemma-lift-specializations-images}).", "Pick a quasi-compact open $W' \\subset U$ containing $t(s^{-1}(W))$. By", "Properties, Lemma \\ref{properties-lemma-quasi-compact-quasi-separated-spectral}", "we see that $W'$ is a spectral space (here we use that $U$ is quasi-separated).", "Then the lemma follows from", "Topology, Lemma \\ref{topology-lemma-make-spectral-space}", "applied to $t(s^{-1}(W)) \\subset W'$." ], "refs": [ "morphisms-lemma-generalizations-lift-flat", "topology-lemma-lift-specializations-images", "properties-lemma-quasi-compact-quasi-separated-spectral", "topology-lemma-make-spectral-space" ], "ref_ids": [ 5266, 8286, 2941, 8323 ] } ], "ref_ids": [] }, { "id": 9647, "type": "theorem", "label": "groupoids-lemma-second-observation", "categories": [ "groupoids" ], "title": "groupoids-lemma-second-observation", "contents": [ "Assumptions and notation as in Lemma \\ref{lemma-first-observation}.", "There exists an $R$-invariant open $V \\subset U$ and a quasi-compact", "open $W'$ such that $W \\subset V \\subset W' \\subset U$." ], "refs": [ "groupoids-lemma-first-observation" ], "proofs": [ { "contents": [ "Set $E = t(s^{-1}(W))$. Recall that $E$ is set-theoretically $R$-invariant", "(Lemma \\ref{lemma-constructing-invariant-opens}).", "By Lemma \\ref{lemma-first-observation} there exists a quasi-compact", "open $W'$ containing $E$. Let $Z = U \\setminus W'$ and consider", "$T = t(s^{-1}(Z))$. Observe that $Z \\subset T$ and that", "$E \\cap T = \\emptyset$ because $s^{-1}(E) = t^{-1}(E)$ is disjoint", "from $s^{-1}(Z)$. Since $T$ is the image of the closed subset", "$s^{-1}(Z) \\subset R$ under the quasi-compact morphism $t : R \\to U$", "we see that any point $\\xi$ in the closure $\\overline{T}$", "is the specialization of a point of $T$, see", "Morphisms, Lemma \\ref{morphisms-lemma-reach-points-scheme-theoretic-image} (and", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}", "to see that the scheme theoretic image is the closure of the image).", "Say $\\xi' \\leadsto \\xi$ with $\\xi' \\in T$. Suppose that $r \\in R$ and", "$s(r) = \\xi$. Since $s$ is flat we can find a specialization $r' \\leadsto r$", "in $R$ such that $s(r') = \\xi'$", "(Morphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat}).", "Then $t(r') \\leadsto t(r)$. We conclude that $t(r') \\in T$ as $T$", "is set-theoretically invariant by", "Lemma \\ref{lemma-constructing-invariant-opens}.", "Thus $\\overline{T}$ is a set-theoretically $R$-invariant closed subset", "and $V = U \\setminus \\overline{T}$ is the open we are", "looking for. It is contained in $W'$ which finishes the proof." ], "refs": [ "groupoids-lemma-constructing-invariant-opens", "groupoids-lemma-first-observation", "morphisms-lemma-reach-points-scheme-theoretic-image", "morphisms-lemma-quasi-compact-scheme-theoretic-image", "morphisms-lemma-generalizations-lift-flat", "groupoids-lemma-constructing-invariant-opens" ], "ref_ids": [ 9645, 9646, 5147, 5146, 5266, 9645 ] } ], "ref_ids": [ 9646 ] }, { "id": 9648, "type": "theorem", "label": "groupoids-lemma-criterion-quotient-representable", "categories": [ "groupoids" ], "title": "groupoids-lemma-criterion-quotient-representable", "contents": [ "In the situation of Definition \\ref{definition-quotient-sheaf}.", "Assume there is a scheme $M$, and a morphism $U \\to M$ such that", "\\begin{enumerate}", "\\item the morphism $U \\to M$ equalizes $s, t$,", "\\item the morphism $U \\to M$ induces a surjection of sheaves", "$h_U \\to h_M$ in the $\\tau$-topology, and", "\\item the induced map $(t, s) : R \\to U \\times_M U$ induces a", "surjection of sheaves $h_R \\to h_{U \\times_M U}$ in the $\\tau$-topology.", "\\end{enumerate}", "In this case $M$ represents the quotient sheaf $U/R$." ], "refs": [ "groupoids-definition-quotient-sheaf" ], "proofs": [ { "contents": [ "Condition (1) says that $h_U \\to h_M$ factors through $U/R$.", "Condition (2) says that $U/R \\to h_M$ is surjective as a map of sheaves.", "Condition (3) says that $U/R \\to h_M$ is injective as a map of sheaves.", "Hence the lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9686 ] }, { "id": 9649, "type": "theorem", "label": "groupoids-lemma-quotient-pre-equivalence", "categories": [ "groupoids" ], "title": "groupoids-lemma-quotient-pre-equivalence", "contents": [ "Let $\\tau \\in \\{Zariski, \\etale, fppf, smooth, syntomic\\}$.", "Let $S$ be a scheme.", "Let $j : R \\to U \\times_S U$ be a pre-equivalence relation over $S$.", "Assume $U, R, S$ are objects of a $\\tau$-site $\\Sch_\\tau$.", "For $T \\in \\Ob((\\Sch/S)_\\tau)$ and", "$a, b \\in U(T)$ the following are equivalent:", "\\begin{enumerate}", "\\item $a$ and $b$ map to the same element of $(U/R)(T)$, and", "\\item there exists a $\\tau$-covering $\\{f_i : T_i \\to T\\}$ of $T$", "and morphisms $r_i : T_i \\to R$ such that", "$a \\circ f_i = s \\circ r_i$ and $b \\circ f_i = t \\circ r_i$.", "\\end{enumerate}", "In other words, in this case the map of $\\tau$-sheaves", "$$", "h_R \\longrightarrow h_U \\times_{U/R} h_U", "$$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: The reason this works is that the presheaf", "(\\ref{equation-quotient-presheaf}) in this case is really given", "by $T \\mapsto U(T)/j(R(T))$ as $j(R(T)) \\subset U(T) \\times U(T)$", "is an equivalence relation, see", "Definition \\ref{definition-equivalence-relation}." ], "refs": [ "groupoids-definition-equivalence-relation" ], "ref_ids": [ 9670 ] } ], "ref_ids": [] }, { "id": 9650, "type": "theorem", "label": "groupoids-lemma-quotient-pre-equivalence-relation-restrict", "categories": [ "groupoids" ], "title": "groupoids-lemma-quotient-pre-equivalence-relation-restrict", "contents": [ "Let $\\tau \\in \\{Zariski, \\etale, fppf, smooth, syntomic\\}$.", "Let $S$ be a scheme.", "Let $j : R \\to U \\times_S U$ be a pre-equivalence relation over $S$", "and $g : U' \\to U$ a morphism of schemes over $S$.", "Let $j' : R' \\to U' \\times_S U'$ be the restriction of $j$ to $U'$.", "Assume $U, U', R, S$ are objects of a $\\tau$-site $\\Sch_\\tau$.", "The map of quotient sheaves", "$$", "U'/R' \\longrightarrow U/R", "$$", "is injective. If $g$ defines a surjection $h_{U'} \\to h_U$ of sheaves", "in the $\\tau$-topology (for example if $\\{g : U' \\to U\\}$ is a", "$\\tau$-covering), then $U'/R' \\to U/R$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Suppose $\\xi, \\xi' \\in (U'/R')(T)$ are sections which", "map to the same section of $U/R$.", "Then we can find a $\\tau$-covering $\\mathcal{T} = \\{T_i \\to T\\}$ of $T$", "such that $\\xi|_{T_i}, \\xi'|_{T_i}$ are given by $a_i, a_i' \\in U'(T_i)$. By", "Lemma \\ref{lemma-quotient-pre-equivalence}", "and the axioms of a site we may after refining", "$\\mathcal{T}$ assume there exist morphisms $r_i : T_i \\to R$", "such that $g \\circ a_i = s \\circ r_i$, $g \\circ a_i' = t \\circ r_i$.", "Since by construction", "$R' = R \\times_{U \\times_S U} (U' \\times_S U')$", "we see that $(r_i, (a_i, a_i')) \\in R'(T_i)$ and this", "shows that $a_i$ and $a_i'$ define the same section", "of $U'/R'$ over $T_i$. By the sheaf condition this implies", "$\\xi = \\xi'$.", "\\medskip\\noindent", "If $h_{U'} \\to h_U$ is a surjection", "of sheaves, then of course $U'/R' \\to U/R$ is surjective also.", "If $\\{g : U' \\to U\\}$ is a $\\tau$-covering, then", "the map of sheaves $h_{U'} \\to h_U$ is surjective, see", "Sites, Lemma \\ref{sites-lemma-covering-surjective-after-sheafification}.", "Hence $U'/R' \\to U/R$ is surjective also in this case." ], "refs": [ "groupoids-lemma-quotient-pre-equivalence", "sites-lemma-covering-surjective-after-sheafification" ], "ref_ids": [ 9649, 8519 ] } ], "ref_ids": [] }, { "id": 9651, "type": "theorem", "label": "groupoids-lemma-quotient-groupoid-restrict", "categories": [ "groupoids" ], "title": "groupoids-lemma-quotient-groupoid-restrict", "contents": [ "Let $\\tau \\in \\{Zariski, \\etale, fppf, smooth, syntomic\\}$.", "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Let $g : U' \\to U$ a morphism of schemes over $S$.", "Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ to $U'$.", "Assume $U, U', R, S$ are objects of a $\\tau$-site $\\Sch_\\tau$.", "The map of quotient sheaves", "$$", "U'/R' \\longrightarrow U/R", "$$", "is injective. If the composition", "$$", "\\xymatrix{", "U' \\times_{g, U, t} R \\ar[r]_-{\\text{pr}_1} \\ar@/^3ex/[rr]^h", "& R \\ar[r]_s & U", "}", "$$", "defines a surjection of sheaves in the $\\tau$-topology then", "the map is bijective. This holds for example if", "$\\{h : U' \\times_{g, U, t} R \\to U\\}$ is a $\\tau$-covering, or", "if $U' \\to U$ defines a surjection of sheaves in the $\\tau$-topology, or if", "$\\{g : U' \\to U\\}$ is a covering in the $\\tau$-topology." ], "refs": [], "proofs": [ { "contents": [ "Injectivity follows on combining", "Lemmas \\ref{lemma-groupoid-pre-equivalence} and", "\\ref{lemma-quotient-pre-equivalence-relation-restrict}.", "To see surjectivity (see", "Sites, Section \\ref{sites-section-sheaves-injective}", "for a characterization of surjective maps of sheaves) we argue as follows.", "Suppose that $T$ is a scheme and $\\sigma \\in U/R(T)$.", "There exists a covering $\\{T_i \\to T\\}$ such that $\\sigma|_{T_i}$", "is the image of some element $f_i \\in U(T_i)$. Hence we", "may assume that $\\sigma$ is the image of $f \\in U(T)$.", "By the assumption that $h$ is a surjection of sheaves, we", "can find a $\\tau$-covering $\\{\\varphi_i : T_i \\to T\\}$ and morphisms", "$f_i : T_i \\to U' \\times_{g, U, t} R$ such that", "$f \\circ \\varphi_i = h \\circ f_i$. Denote", "$f'_i = \\text{pr}_0 \\circ f_i : T_i \\to U'$. Then we see that", "$f'_i \\in U'(T_i)$ maps to $g \\circ f'_i \\in U(T_i)$ and", "that $g \\circ f'_i \\sim_{T_i} h \\circ f_i = f \\circ \\varphi_i$", "notation as in (\\ref{equation-quotient-presheaf}). Namely, the", "element of $R(T_i)$ giving the relation is $\\text{pr}_1 \\circ f_i$.", "This means that the restriction", "of $\\sigma$ to $T_i$ is in the image of $U'/R'(T_i) \\to U/R(T_i)$", "as desired.", "\\medskip\\noindent", "If $\\{h\\}$ is a $\\tau$-covering, then it induces a surjection of sheaves, see", "Sites, Lemma \\ref{sites-lemma-covering-surjective-after-sheafification}.", "If $U' \\to U$ is surjective, then also $h$ is surjective as $s$ has a section", "(namely the neutral element $e$ of the groupoid scheme)." ], "refs": [ "groupoids-lemma-groupoid-pre-equivalence", "groupoids-lemma-quotient-pre-equivalence-relation-restrict", "sites-lemma-covering-surjective-after-sheafification" ], "ref_ids": [ 9620, 9650, 8519 ] } ], "ref_ids": [] }, { "id": 9652, "type": "theorem", "label": "groupoids-lemma-criterion-fibre-product", "categories": [ "groupoids" ], "title": "groupoids-lemma-criterion-fibre-product", "contents": [ "Let $S$ be a scheme. Let $f : (U, R, j) \\to (U', R', j')$ be a morphism", "between equivalence relations over $S$. Assume that", "$$", "\\xymatrix{", "R \\ar[d]_s \\ar[r]_f & R' \\ar[d]^{s'} \\\\", "U \\ar[r]^f & U'", "}", "$$", "is cartesian. For any", "$\\tau \\in \\{Zariski, \\etale, fppf, smooth, syntomic\\}$", "the diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & U/R \\ar[d]^f \\\\", "U' \\ar[r] & U'/R'", "}", "$$", "is a fibre product square of $\\tau$-sheaves." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-quotient-pre-equivalence} the quotient sheaves", "have a simple description which we will use below without further mention.", "We first show that", "$$", "U \\longrightarrow U' \\times_{U'/R'} U/R", "$$", "is injective. Namely, assume $a, b \\in U(T)$ map to the same element", "on the right hand side. Then $f(a) = f(b)$. After replacing $T$ by the", "members of a $\\tau$-covering we may assume that there exists an", "$r \\in R(T)$ such that $a = s(r)$ and $b = t(r)$. Then $r' = f(r)$", "is a $T$-valued point of $R'$ with $s'(r') = t'(r')$. Hence", "$r' = e'(f(a))$ (where $e'$ is the identity of the groupoid", "scheme associated to $j'$, see Lemma \\ref{lemma-equivalence-groupoid}).", "Because the first diagram of the lemma is cartesian this implies", "that $r$ has to equal $e(a)$. Thus $a = b$.", "\\medskip\\noindent", "Finally, we show that the displayed arrow is surjective. Let", "$T$ be a scheme over $S$ and let $(a', \\overline{b})$ be a section", "of the sheaf $U' \\times_{U'/R'} U/R$ over $T$. After replacing $T$", "by the members of a $\\tau$-covering we may assume that $\\overline{b}$", "is the class of an element $b \\in U(T)$. After replacing $T$", "by the members of a $\\tau$-covering we may assume that there exists", "an $r' \\in R'(T)$ such that $a' = t(r')$ and $s'(r') = f(b)$.", "Because the first diagram of the lemma is cartesian we can find", "$r \\in R(T)$ such that $s(r) = b$ and $f(r) = r'$. Then it is clear", "that $a = t(r) \\in U(T)$ is a section which maps to", "$(a', \\overline{b})$." ], "refs": [ "groupoids-lemma-quotient-pre-equivalence", "groupoids-lemma-equivalence-groupoid" ], "ref_ids": [ 9649, 9621 ] } ], "ref_ids": [] }, { "id": 9653, "type": "theorem", "label": "groupoids-lemma-characterize-cartesian-schemes", "categories": [ "groupoids" ], "title": "groupoids-lemma-characterize-cartesian-schemes", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "The category of groupoid schemes cartesian over $(U, R, s, t, c)$", "is equivalent to the category of pairs $(V, \\varphi)$ where $V$ is a", "scheme over $U$ and", "$$", "\\varphi :", "V \\times_{U, t} R", "\\longrightarrow", "R \\times_{s, U} V", "$$", "is an isomorphism over $R$ such that $e^*\\varphi = \\text{id}_V$ and such that", "$$", "c^*\\varphi = \\text{pr}_1^*\\varphi \\circ \\text{pr}_0^*\\varphi", "$$", "as morphisms of schemes over $R \\times_{s, U, t} R$." ], "refs": [], "proofs": [ { "contents": [ "The pullback notation in the lemma signifies base change. The displayed", "formula makes sense because", "$$", "(R \\times_{s, U, t} R) \\times_{\\text{pr}_1, R, \\text{pr}_1} (V \\times_{U, t} R)", "=", "(R \\times_{s, U, t} R) \\times_{\\text{pr}_0, R, \\text{pr}_0} (R \\times_{s, U} V)", "$$", "as schemes over $R \\times_{s, U, t} R$.", "\\medskip\\noindent", "Given $(V, \\varphi)$ we set $U' = V$ and $R' = V \\times_{U, t} R$.", "We set $t' : R' \\to U'$ equal to the projection $V \\times_{U, t} R \\to V$.", "We set $s'$ equal to $\\varphi$ followed by the projection", "$R \\times_{s, U} V \\to V$. We set $c'$ equal to the composition", "\\begin{align*}", "R' \\times_{s', U', t'} R'", "& \\xrightarrow{\\varphi, 1}", "(R \\times_{s, U} V) \\times_V (V \\times_{U, t} R) \\\\", "& \\xrightarrow{}", "R \\times_{s, U} V \\times_{U, t} R \\\\", "& \\xrightarrow{\\varphi^{-1}, 1}", "V \\times_{U, t} (R \\times_{s, U, t} R) \\\\", "& \\xrightarrow{1, c}", "V \\times_{U, t} R = R'", "\\end{align*}", "A computation, which we omit shows that we obtain a groupoid scheme", "over $(U, R, s, t, c)$. It is clear that this groupoid scheme is", "cartesian over $(U, R, s, t, c)$.", "\\medskip\\noindent", "Conversely, given $f : (U', R', s', t', c') \\to (U, R, s, t, c)$", "cartesian then the morphisms", "$$", "U' \\times_{U, t} R \\xleftarrow{t', f} R' \\xrightarrow{f, s'} R \\times_{s, U} U'", "$$", "are isomorphisms and we can set $V = U'$ and $\\varphi$ equal to the", "composition $(f, s') \\circ (t', f)^{-1}$. We omit the proof that", "$\\varphi$ satisfies the conditions in the lemma. We omit the proof that", "these constructions are mutually inverse." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9654, "type": "theorem", "label": "groupoids-lemma-cartesian-equivalent-descent-datum", "categories": [ "groupoids" ], "title": "groupoids-lemma-cartesian-equivalent-descent-datum", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$.", "The construction of Lemma \\ref{lemma-characterize-cartesian-schemes}", "determines an equivalence", "$$", "\\begin{matrix}", "\\text{category of groupoid schemes} \\\\", "\\text{cartesian over } (X, X \\times_Y X, \\ldots)", "\\end{matrix}", "\\longrightarrow", "\\begin{matrix}", "\\text{ category of descent data} \\\\", "\\text{ relative to } X/Y", "\\end{matrix}", "$$" ], "refs": [ "groupoids-lemma-characterize-cartesian-schemes" ], "proofs": [ { "contents": [ "This is clear from", "Lemma \\ref{lemma-characterize-cartesian-schemes}", "and the definition of descent data on schemes in", "Descent, Definition \\ref{descent-definition-descent-datum}." ], "refs": [ "groupoids-lemma-characterize-cartesian-schemes", "descent-definition-descent-datum" ], "ref_ids": [ 9653, 14776 ] } ], "ref_ids": [ 9653 ] }, { "id": 9655, "type": "theorem", "label": "groupoids-lemma-diagram-diagonal", "categories": [ "groupoids" ], "title": "groupoids-lemma-diagram-diagonal", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid over $S$.", "Let $G \\to U$ be the stabilizer group scheme.", "The commutative diagram", "$$", "\\xymatrix{", "R \\ar[d]^{\\Delta_{R/U \\times_S U}} \\ar[rrr]_{f \\mapsto (f, s(f))} & & &", "R \\times_{s, U} U \\ar[d] \\ar[r] & U \\ar[d] \\\\", "R \\times_{(U \\times_S U)} R \\ar[rrr]^{(f, g) \\mapsto (f, f^{-1} \\circ g)} & & &", "R \\times_{s, U} G \\ar[r] & G", "}", "$$", "the two left horizontal arrows are isomorphisms", "and the right square is a fibre product square." ], "refs": [], "proofs": [ { "contents": [ "Omitted.", "Exercise in the definitions and the functorial point of", "view in algebraic geometry." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9656, "type": "theorem", "label": "groupoids-lemma-diagonal", "categories": [ "groupoids" ], "title": "groupoids-lemma-diagonal", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid over $S$.", "Let $G \\to U$ be the stabilizer group scheme.", "\\begin{enumerate}", "\\item The following are equivalent", "\\begin{enumerate}", "\\item $j : R \\to U \\times_S U$ is separated,", "\\item $G \\to U$ is separated, and", "\\item $e : U \\to G$ is a closed immersion.", "\\end{enumerate}", "\\item The following are equivalent", "\\begin{enumerate}", "\\item $j : R \\to U \\times_S U$ is quasi-separated,", "\\item $G \\to U$ is quasi-separated, and", "\\item $e : U \\to G$ is quasi-compact.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The group scheme $G \\to U$ is the base change of $R \\to U \\times_S U$", "by the diagonal morphism $U \\to U \\times_S U$, see", "Lemma \\ref{lemma-groupoid-stabilizer}. Hence if", "$j$ is separated (resp.\\ quasi-separated),", "then $G \\to U$ is separated (resp.\\ quasi-separated).", "(See Schemes, Lemma", "\\ref{schemes-lemma-separated-permanence}).", "Thus (a) $\\Rightarrow$ (b) in both (1) and (2).", "\\medskip\\noindent", "If $G \\to U$ is separated (resp.\\ quasi-separated), then the morphism", "$U \\to G$, as a section of the structure morphism $G \\to U$ is a closed", "immersion (resp.\\ quasi-compact), see", "Schemes, Lemma \\ref{schemes-lemma-section-immersion}.", "Thus (b) $\\Rightarrow$ (a) in both (1) and (2).", "\\medskip\\noindent", "By the result of", "Lemma \\ref{lemma-diagram-diagonal}", "(and Schemes, Lemmas \\ref{schemes-lemma-base-change-immersion}", "and \\ref{schemes-lemma-quasi-compact-preserved-base-change})", "we see that if $e$ is a closed immersion (resp.\\ quasi-compact)", "$\\Delta_{R/U \\times_S U}$ is a closed", "immersion (resp.\\ quasi-compact).", "Thus (c) $\\Rightarrow$ (a) in both (1) and (2)." ], "refs": [ "groupoids-lemma-groupoid-stabilizer", "schemes-lemma-separated-permanence", "groupoids-lemma-diagram-diagonal", "schemes-lemma-base-change-immersion", "schemes-lemma-quasi-compact-preserved-base-change" ], "ref_ids": [ 9638, 7714, 9655, 7695, 7698 ] } ], "ref_ids": [] }, { "id": 9657, "type": "theorem", "label": "groupoids-lemma-determinant-trick", "categories": [ "groupoids" ], "title": "groupoids-lemma-determinant-trick", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Assume $U = \\Spec(A)$ and $R = \\Spec(B)$ are affine and", "$s, t : R \\to U$ finite locally free.", "Let $C$ be as in (\\ref{equation-invariants}).", "Let $f \\in A$. Then $\\text{Norm}_{s^\\sharp}(t^\\sharp(f)) \\in C$." ], "refs": [], "proofs": [ { "contents": [ "Consider the commutative diagram", "$$", "\\xymatrix{", "& U & \\\\", "R \\ar[d]_s \\ar[ru]^t &", "R \\times_{s, U, t} R", "\\ar[l]^-{\\text{pr}_0} \\ar[d]^{\\text{pr}_1} \\ar[r]_-c &", "R \\ar[d]^s \\ar[lu]_t \\\\", "U & R \\ar[l]_t \\ar[r]^s & U", "}", "$$", "of Lemma \\ref{lemma-diagram}.", "Think of $f \\in \\Gamma(U, \\mathcal{O}_U)$. The commutativity of the", "top part of the diagram shows that", "$\\text{pr}_0^\\sharp(t^\\sharp(f)) = c^\\sharp(t^\\sharp(f))$ as elements of", "$\\Gamma(R \\times_{S, U, t} R, \\mathcal{O})$.", "Looking at the right lower cartesian square", "the compatibility of the norm construction with base change shows that", "$s^\\sharp(\\text{Norm}_{s^\\sharp}(t^\\sharp(f))) =", "\\text{Norm}_{\\text{pr}_1^\\sharp}(c^\\sharp(t^\\sharp(f)))$.", "Similarly we get", "$t^\\sharp(\\text{Norm}_{s^\\sharp}(t^\\sharp(f))) =", "\\text{Norm}_{\\text{pr}_1^\\sharp}(\\text{pr}_0^\\sharp(t^\\sharp(f)))$.", "Hence by the first equality of this proof we see that", "$s^\\sharp(\\text{Norm}_{s^\\sharp}(t^\\sharp(f))) =", "t^\\sharp(\\text{Norm}_{s^\\sharp}(t^\\sharp(f)))$ as desired." ], "refs": [ "groupoids-lemma-diagram" ], "ref_ids": [ 9622 ] } ], "ref_ids": [] }, { "id": 9658, "type": "theorem", "label": "groupoids-lemma-finite-locally-free-disjoint-free", "categories": [ "groupoids" ], "title": "groupoids-lemma-finite-locally-free-disjoint-free", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Assume $s, t : R \\to U$ finite locally free.", "Then", "$$", "U = \\coprod\\nolimits_{r \\geq 1} U_r", "$$", "is a disjoint union of $R$-invariant opens such that the restriction $R_r$ of", "$R$ to $U_r$ has the property that $s, t : R_r \\to U_r$ are finite locally", "free of rank $r$." ], "refs": [], "proofs": [ { "contents": [ "By", "Morphisms, Lemma \\ref{morphisms-lemma-finite-locally-free}", "there exists a decomposition", "$U = \\coprod\\nolimits_{r \\geq 0} U_r$", "such that $s : s^{-1}(U_r) \\to U_r$ is finite locally free of rank $r$.", "As $s$ is surjective we see that $U_0 = \\emptyset$.", "Note that $u \\in U_r \\Leftrightarrow$ if and only if the scheme theoretic fibre", "$s^{-1}(u)$ has degree $r$ over $\\kappa(u)$. Now, if $z \\in R$ with $s(z) = u$", "and $t(z) = u'$ then using notation as in Lemma \\ref{lemma-diagram}", "$$", "\\text{pr}_1^{-1}(z) \\to \\Spec(\\kappa(z))", "$$", "is the base change of both", "$s^{-1}(u) \\to \\Spec(\\kappa(u))$ and $s^{-1}(u') \\to \\Spec(\\kappa(u'))$", "by the lemma cited. Hence $u \\in U_r \\Leftrightarrow u' \\in U_r$,", "in other words, the open subsets $U_r$ are $R$-invariant.", "In particular the restriction of $R$ to $U_r$ is just", "$s^{-1}(U_r)$ and $s : R_r \\to U_r$ is finite locally free of rank $r$.", "As $t : R_r \\to U_r$ is isomorphic to $s$ by the inverse of $R_r$", "we see that it has also rank $r$." ], "refs": [ "morphisms-lemma-finite-locally-free", "groupoids-lemma-diagram" ], "ref_ids": [ 5474, 9622 ] } ], "ref_ids": [] }, { "id": 9659, "type": "theorem", "label": "groupoids-lemma-integral-over-invariants", "categories": [ "groupoids" ], "title": "groupoids-lemma-integral-over-invariants", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Assume $U = \\Spec(A)$ and $R = \\Spec(B)$ are affine and", "$s, t : R \\to U$ finite locally free.", "Let $C \\subset A$ be as in (\\ref{equation-invariants}).", "Then $A$ is integral over $C$." ], "refs": [], "proofs": [ { "contents": [ "First, by Lemma \\ref{lemma-finite-locally-free-disjoint-free}", "we know that $(U, R, s, t, c)$ is a disjoint union", "of groupoid schemes $(U_r, R_r, s, t, c)$ such that each $s, t : R_r \\to U_r$", "has constant rank $r$. As $U$ is quasi-compact, we have $U_r = \\emptyset$ for", "almost all $r$. It suffices to prove the lemma for each $(U_r, R_r, s, t, c)$", "and hence we may assume that $s, t$ are finite locally free of rank $r$.", "\\medskip\\noindent", "Assume that $s, t$ are finite locally free of rank $r$.", "Let $f \\in A$. Consider the element $x - f \\in A[x]$, where we think", "of $x$ as the coordinate on $\\mathbf{A}^1$.", "Since", "$$", "(U \\times \\mathbf{A}^1, R \\times \\mathbf{A}^1,", "s \\times \\text{id}_{\\mathbf{A}^1},", "t \\times \\text{id}_{\\mathbf{A}^1},", "c \\times \\text{id}_{\\mathbf{A}^1})", "$$", "is also a groupoid scheme with finite source and target, we may apply", "Lemma \\ref{lemma-determinant-trick} to it and we see that", "$P(x) = \\text{Norm}_{s^\\sharp}(t^\\sharp(x - f))$", "is an element of $C[x]$. Because $s^\\sharp : A \\to B$ is finite locally", "free of rank $r$ we see that $P$ is monic of degree $r$.", "Moreover $P(f) = 0$ by Cayley-Hamilton", "(Algebra, Lemma \\ref{algebra-lemma-charpoly})." ], "refs": [ "groupoids-lemma-finite-locally-free-disjoint-free", "groupoids-lemma-determinant-trick", "algebra-lemma-charpoly" ], "ref_ids": [ 9658, 9657, 385 ] } ], "ref_ids": [] }, { "id": 9660, "type": "theorem", "label": "groupoids-lemma-invariants-base-change", "categories": [ "groupoids" ], "title": "groupoids-lemma-invariants-base-change", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Assume $U = \\Spec(A)$ and $R = \\Spec(B)$ are affine and", "$s, t : R \\to U$ finite locally free. Let $C \\subset A$ be as in", "(\\ref{equation-invariants}). Let $C \\to C'$ be a ring map, and set", "$U' = \\Spec(A \\otimes_C C')$,", "$R' = \\Spec(B \\otimes_C C')$.", "Then", "\\begin{enumerate}", "\\item The maps $s, t, c$ induce maps $s', t', c'$ such that", "$(U', R', s', t', c')$ is a groupoid scheme. Let $C^1 \\subset A'$", "be the $R'$-invariant functions on $U'$.", "\\item The canonical map $\\varphi : C' \\to C^1$ satisfies", "\\begin{enumerate}", "\\item for every $f \\in C^1$ there exists an $n > 0$ and a", "polynomial $P \\in C'[x]$ whose image in $C^1[x]$ is", "$(x - f)^n$, and", "\\item for every $f \\in \\Ker(\\varphi)$ there exists", "an $n > 0$ such that $f^n = 0$.", "\\end{enumerate}", "\\item If $C \\to C'$ is flat then $\\varphi$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The proof of part (1) is omitted. Let us denote $A' = A \\otimes_C C'$ and", "$B' = B \\otimes_C C'$. Then we have", "$$", "C^1", "= \\{a \\in A' \\mid (t')^\\sharp(a) = (s')^\\sharp(a) \\}", "= \\{a \\in A \\otimes_C C' \\mid t^\\sharp \\otimes 1(a) = s^\\sharp \\otimes 1(a) \\}.", "$$", "In other words, $C^1$ is the kernel of the difference map", "$(t^\\sharp - s^\\sharp) \\otimes 1$ which is just the base change", "of the $C$-linear map $t^\\sharp - s^\\sharp : A \\to B$ by $C \\to C'$.", "Hence (3) follows.", "\\medskip\\noindent", "Proof of part (2)(b). Since $C \\to A$ is integral", "(Lemma \\ref{lemma-integral-over-invariants}) and injective we see that", "$\\Spec(A) \\to \\Spec(C)$ is surjective, see", "Algebra, Lemma \\ref{algebra-lemma-integral-overring-surjective}.", "Thus also $\\Spec(A') \\to \\Spec(C')$ is surjective", "as a base change of a surjective morphism", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-surjective}).", "Hence $\\Spec(C^1) \\to \\Spec(C')$ is surjective also.", "This implies (2)(b) holds for example by", "Algebra, Lemma \\ref{algebra-lemma-image-dense-generic-points}.", "\\medskip\\noindent", "Proof of part (2)(a). By Lemma \\ref{lemma-finite-locally-free-disjoint-free}", "our groupoid scheme $(U, R, s, t, c)$ decomposes as a finite disjoint union", "of groupoid schemes $(U_r, R_r, s, t, c)$ such that $s, t : R_r \\to U_r$", "are finite locally free of rank $r$. Pulling back by $U' = \\Spec(C') \\to U$", "we obtain a similar decomposition of $U'$ and $U^1 = \\Spec(C^1)$.", "We will show in the next paragraph that (2)(a) holds for the corresponding", "system of rings $A_r, B_r, C_r, C'_r, C^1_r$ with $n = r$.", "Then given $f \\in C^1$ let $P_r \\in C_r[x]$ be the polynomial", "whose image in $C^1_r[x]$ is the image of $(x - f)^r$.", "Choosing a sufficiently divisible integer $n$ we see that", "there is a polynomial $P \\in C'[x]$ whose image in $C^1[x]$ is", "$(x - f)^n$; namely, we take $P$ to be the unique element of", "$C'[x]$ whose image in $C'_r[x]$ is $P_r^{n/r}$.", "\\medskip\\noindent", "In this paragraph we prove (2)(a) in case the ring maps", "$s^\\sharp, t^\\sharp : A \\to B$ are finite locally free of a fixed rank $r$.", "Let $f \\in C^1 \\subset A' = A \\otimes_C C'$. Choose a flat", "$C$-algebra $D$ and a surjection $D \\to C'$. Choose a lift", "$g \\in A \\otimes_C D$ of $f$.", "Consider the polynomial", "$$", "P = \\text{Norm}_{s^\\sharp \\otimes 1}(t^\\sharp \\otimes 1(x - g))", "$$", "in $(A \\otimes_C D)[x]$. By Lemma \\ref{lemma-determinant-trick}", "and part (3) of the current lemma the coefficients of $P$ are in $D$", "(compare with the proof of Lemma \\ref{lemma-integral-over-invariants}).", "On the other hand, the image of $P$ in $(A \\otimes_C C')[x]$ is", "$(x - f)^r$ because $t^\\sharp \\otimes 1(x - f) = s^\\sharp(x - f)$", "and $s^\\sharp$ is finite locally free of rank $r$.", "This proves what we want with $P$ as in the statement (2)(a)", "given by the image of our $P$ under the map $D[x] \\to C'[x]$." ], "refs": [ "groupoids-lemma-integral-over-invariants", "algebra-lemma-integral-overring-surjective", "morphisms-lemma-base-change-surjective", "algebra-lemma-image-dense-generic-points", "groupoids-lemma-finite-locally-free-disjoint-free", "groupoids-lemma-determinant-trick", "groupoids-lemma-integral-over-invariants" ], "ref_ids": [ 9659, 495, 5165, 446, 9658, 9657, 9659 ] } ], "ref_ids": [] }, { "id": 9661, "type": "theorem", "label": "groupoids-lemma-points", "categories": [ "groupoids" ], "title": "groupoids-lemma-points", "contents": [ "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Assume $U = \\Spec(A)$ and $R = \\Spec(B)$ are affine and", "$s, t : R \\to U$ finite locally free. Let $C \\subset A$ be as in", "(\\ref{equation-invariants}). Then $U \\to M = \\Spec(C)$ has", "the following properties:", "\\begin{enumerate}", "\\item the map on points $|U| \\to |M|$ is surjective and", "$u_0, u_1 \\in |U|$ map to the same point if and only if", "there exists a $r \\in |R|$ with $t(r) = u_0$ and $s(r) = u_1$, in", "a formula", "$$", "|M| = |U|/|R|", "$$", "\\item for any algebraically closed field $k$ we have", "$$", "M(k) = U(k)/R(k)", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $k$ be an algebraically closed field.", "Since $C \\to A$ is integral (Lemma \\ref{lemma-integral-over-invariants})", "and injective we see that", "$\\Spec(A) \\to \\Spec(C)$ is surjective, see", "Algebra, Lemma \\ref{algebra-lemma-integral-overring-surjective}.", "Thus $|U| \\to |M|$ is surjective.", "Let $C \\to k$ be a ring map. Since surjective morphisms are", "preserved under base change", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-surjective}) we see that", "$A \\otimes_C k$ is not zero. Now $k \\subset A \\otimes_C k$ is a", "nonzero integral extension. Hence any residue field of $A \\otimes_C k$", "is an algebraic extension of $k$, hence equal to $k$. Thus we see that", "$U(k) \\to M(k)$ is surjective.", "\\medskip\\noindent", "Let $a_0, a_1 : A \\to k$ be ring maps. If there exists a ring map", "$b : B \\to k$ such that $a_0 = b \\circ t^\\sharp$ and $a_1 = b \\circ s^\\sharp$", "then we see that $a_0|_C = a_1|_C$ by definition.", "Conversely, suppose that $a_0|_C = a_1|_C$. Let us name this algebra", "map $c : C \\to k$. Consider the diagram", "$$", "\\xymatrix{", "& &", "B \\ar@{-->}[lld] \\\\", "k & &", "A", "\\ar@<0.5ex>[ll]^{a_0}", "\\ar@<-0.5ex>[ll]_{a_1}", "\\ar@<1ex>[u]", "\\ar@<-1ex>[u] \\\\", "& &", "C \\ar[u] \\ar[llu]^c", "}", "$$", "We are trying to construct the dotted arrow, and if we do then", "part (2) follows, which in turn implies part (1).", "Since $A \\to B$ is finite and faithfully flat", "there exist finitely many ring maps", "$b_1, \\ldots, b_n : B \\to k$ such that $b_i \\circ s^\\sharp = a_1$.", "If the dotted arrow does not exist, then we see that none of the", "$a'_i = b_i \\circ t^\\sharp$, $i = 1, \\ldots, n$ is equal to $a_0$.", "Hence the maximal ideals", "$$", "\\mathfrak m'_i = \\Ker(a_i' \\otimes 1 : A \\otimes_C k \\to k)", "$$", "of $A \\otimes_C k$ are distinct from", "$\\mathfrak m = \\Ker(a_0 \\otimes 1 : A \\otimes_C k \\to k)$.", "By Algebra, Lemma \\ref{algebra-lemma-silly} we would get an element", "$f \\in A \\otimes_C k$ with $f \\in \\mathfrak m$, but", "$f \\not \\in \\mathfrak m_i'$ for $i = 1, \\ldots, n$.", "Consider the norm", "$$", "g = \\text{Norm}_{s^\\sharp \\otimes 1}(t^\\sharp \\otimes 1(f))", "\\in", "A \\otimes_C k", "$$", "By Lemma \\ref{lemma-determinant-trick} this lies in the invariants", "$C^1 \\subset A \\otimes_C k$ of the base change", "groupoid (base change via the map $c : C \\to k$). On the one hand,", "$a_1(g) \\in k^*$ since", "the value of $t^\\sharp(f)$ at all the points (which correspond to", "$b_1, \\ldots, b_n$) lying over $a_1$ is", "invertible (insert future reference on property determinant here).", "On the other hand, since $f \\in \\mathfrak m$, we see that", "$f$ is not a unit, hence $t^\\sharp(f)$ is not a unit", "(as $t^\\sharp \\otimes 1$ is faithfully flat),", "hence its norm is not a unit (insert future reference", "on property determinant here). We conclude that $C^1$ contains", "an element which is not nilpotent", "and not a unit. We will now show that this leads to a contradiction.", "Namely, apply Lemma \\ref{lemma-invariants-base-change}", "to the map $c : C \\to C' = k$, then", "we see that the map of $k$ into the invariants $C^1$ is injective", "and moreover, that for any element $x \\in C^1$ there exists an integer", "$n > 0$ such that $x^n \\in k$. Hence every element of $C^1$ is", "either a unit or nilpotent." ], "refs": [ "groupoids-lemma-integral-over-invariants", "algebra-lemma-integral-overring-surjective", "morphisms-lemma-base-change-surjective", "algebra-lemma-silly", "groupoids-lemma-determinant-trick", "groupoids-lemma-invariants-base-change" ], "ref_ids": [ 9659, 495, 5165, 378, 9657, 9660 ] } ], "ref_ids": [] }, { "id": 9662, "type": "theorem", "label": "groupoids-lemma-etale", "categories": [ "groupoids" ], "title": "groupoids-lemma-etale", "contents": [ "Let $S$ be a scheme. Let $f : (U', R', s', t') \\to (U, R, s, t, c)$ be a", "morphism of groupoid schemes over $S$.", "\\begin{enumerate}", "\\item $U$, $R$, $U'$, $R'$ are affine,", "\\item $s, t, s', t'$ are finite locally free,", "\\item the diagrams", "$$", "\\xymatrix{", "R' \\ar[d]_{s'} \\ar[r]_f & R \\ar[d]^s \\\\", "U' \\ar[r]^f & U", "}", "\\quad", "\\quad", "\\xymatrix{", "R' \\ar[d]_{t'} \\ar[r]_f & R \\ar[d]^t \\\\", "U' \\ar[r]^f & U", "}", "\\quad", "\\quad", "\\xymatrix{", "G' \\ar[d] \\ar[r]_f & G \\ar[d] \\\\", "U' \\ar[r]^f & U", "}", "$$", "are cartesian where $G$ and $G'$ are the stabilizer group schemes, and", "\\item $f : U' \\to U$ is \\'etale.", "\\end{enumerate}", "Then the map $C \\to C'$ from the $R$-invariant functions on $U$", "to the $R'$-invariant functions on $U'$ is \\'etale and", "$U' = \\Spec(C') \\times_{\\Spec(C)} U$." ], "refs": [], "proofs": [ { "contents": [ "Set $M = \\Spec(C)$ and $M' = \\Spec(C')$.", "Write $U = \\Spec(A)$, $U' = \\Spec(A')$, $R = \\Spec(B)$, and", "$R' = \\Spec(B')$. We will use the results of", "Lemmas \\ref{lemma-integral-over-invariants},", "\\ref{lemma-invariants-base-change}, and", "\\ref{lemma-points}", "without further mention.", "\\medskip\\noindent", "Assume $C$ is a strictly henselian local ring. Let $p \\in M$", "be the closed point and let $p' \\in M'$ map to $p$.", "Claim: in this case there is a disjoint union decomposition", "$(U', R', s', t', c') = (U, R, s, t, c) \\amalg (U'', R'', s'', t'', c'')$", "over $(U, R, s, t, c)$ such that for the corresponding", "disjoint union decomposition $M' = M \\amalg M''$ over $M$", "the point $p'$ corresponds to $p \\in M$.", "\\medskip\\noindent", "The claim implies the lemma. Suppose that $M_1 \\to M$ is a flat morphism", "of affine schemes. Then we can base change everything to $M_1$", "without affecting the hypotheses (1) -- (4).", "From Lemma \\ref{lemma-invariants-base-change}", "we see $M_1$, resp.\\ $M_1'$ is the spectrum of the", "$R_1$-invariant functions on $U_1$,", "resp.\\ the $R'_1$-invariant functions on $U'_1$.", "Suppose that $p' \\in M'$ maps to $p \\in M$.", "Let $M_1$ be the spectrum of the strict henselization of", "$\\mathcal{O}_{M, p}$ with closed point $p_1 \\in M_1$.", "Choose a point $p'_1 \\in M'_1$ mapping to $p_1$ and $p'$.", "From the claim we get", "$$", "(U'_1, R'_1, s'_1, t'_1, c'_1) =", "(U_1, R_1, s_1, t_1, c_1) \\amalg", "(U''_1, R''_1, s''_1, t''_1, c''_1)", "$$", "and correspondingly $M'_1 = M_1 \\amalg M''_1$ as a scheme over $M_1$.", "Write $M_1 = \\Spec(C_1)$ and write $C_1 = \\colim C_i$ as a filtered", "colimit of \\'etale $C$-algebras. Set $M_i = \\Spec(C_i)$.", "The $M_1 = \\lim M_i$ and similarly for the other schemes.", "By Limits, Lemmas \\ref{limits-lemma-descend-opens} and", "\\ref{limits-lemma-descend-isomorphism}", "we can find an $i$ such that", "$$", "(U'_i, R'_i, s'_i, t'_i, c'_i) =", "(U_i, R_i, s_i, t_i, c_i) \\amalg", "(U''_i, R''_i, s''_i, t''_i, c''_i)", "$$", "We conclude that $M'_i = M_i \\amalg M''_i$. In particular", "$M' \\to M$ becomes \\'etale at a point over $p'$ after an", "\\'etale base change. This implies that $M' \\to M$ is \\'etale at $p'$", "(for example by Morphisms, Lemma", "\\ref{morphisms-lemma-set-points-where-fibres-etale}).", "We will prove $U' \\cong M' \\times_M U$ after we prove the claim.", "\\medskip\\noindent", "Proof of the claim. Observe that $U_p$ and $U'_{p'}$ have finitely many points.", "For $u \\in U_p$ we have $\\kappa(u)/\\kappa(p)$ is algebraic,", "hence $\\kappa(u)$ is separably closed.", "As $U' \\to U$ is \\'etale, we conclude the morphism $U'_{p'} \\to U_p$", "induces isomorphisms on residue field extensions.", "Let $u' \\in U'_{p'}$ with image $u \\in U_p$.", "By assumption (3) the morphism of scheme theoretic fibres", "$(s')^{-1}(u') \\to s^{-1}(u)$,", "$(t')^{-1}(u') \\to t^{-1}(u)$, and", "$G'_{u'} \\to G_u$ are isomorphisms. Observing that $U_p = t(s^{-1}(u))$", "(set theoretically) we conclude that the points of $U'_{p'}$", "surject onto the points of $U_p$.", "Suppose that $u'_1$ and $u'_2$ are points of $U'_{p'}$ mapping", "to the same point $u$ of $U_p$. Then there exists a point", "$r' \\in R'_{p'}$ with $s'(r') = u'_1$ and $t'(r') = u'_2$.", "Consider the two towers of fields", "$$", "\\kappa(r')/\\kappa(u'_1)/\\kappa(u)/\\kappa(p) \\quad", "\\kappa(r')/\\kappa(u'_2)/\\kappa(u)/\\kappa(p)", "$$", "whose ``ends'' are the same as the two ``ends'' of the two towers", "$$", "\\kappa(r')/\\kappa(u'_1)/\\kappa(p')/\\kappa(p) \\quad", "\\kappa(r')/\\kappa(u'_2)/\\kappa(p')/\\kappa(p)", "$$", "These two induce the same maps $\\kappa(p') \\to \\kappa(r')$ as", "$(U'_{p'}, R'_{p'}, s', t', c')$ is a groupoid over $p'$.", "Since $\\kappa(u)/\\kappa(p)$ is purely inseparable,", "we conclude that the two induced maps", "$\\kappa(u) \\to \\kappa(r')$ are the same.", "Therefore $r'$ maps to a point of the fibre $G_u$.", "By assumption (3) we conclude that $r' \\in (G')_{u'_1}$.", "Namely, we may think of $G$ as a closed subscheme of $R$", "viewed as a scheme over $U$ via $s$ and use that", "the base change to $U'$ gives $G' \\subset R'$.", "In particular we have $u'_1 = u'_2$.", "We conclude that $U'_{p'} \\to U_p$ is a bijective", "map on points inducing isomorphisms on residue fields.", "It follows that $U'_{p'}$ is a finite set of closed points", "(Algebra, Lemma \\ref{algebra-lemma-finite-residue-extension-closed})", "and hence $U'_{p'}$ is closed in $U'$.", "Let $J' \\subset A'$ be the radical ideal cutting out $U'_{p'}$", "set theoretically.", "\\medskip\\noindent", "Second part proof of the claim.", "Let $\\mathfrak m \\subset C$ be the maximal ideal.", "Observe that $(A, \\mathfrak m A)$ is a henselian pair by", "More on Algebra, Lemma \\ref{more-algebra-lemma-integral-over-henselian-pair}.", "Let $J = \\sqrt{\\mathfrak m A}$.", "Then $(A, J)$ is a henselian pair", "(More on Algebra, Lemma \\ref{more-algebra-lemma-change-ideal-henselian-pair})", "and the \\'etale ring map", "$A \\to A'$ induces an isomorphism $A/J \\to A'/J'$", "by our deliberations above.", "We conclude that $A' = A \\times A''$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-characterize-henselian-pair}.", "Consider the corresponding disjoint union", "decomposition $U' = U \\amalg U''$. The open $(s')^{-1}(U)$ is the", "set of points of $R'$ specializing to a point of $R'_{p'}$.", "Similarly for $(t')^{-1}(U)$. Similarly we have", "$(s')^{-1}(U'') = (t')^{-1}(U'')$ as this is the set of", "points which do not specialize to $R'_{p'}$.", "Hence we obtain a disjoint union decomposition", "$$", "(U', R', s', t', c') =", "(U, R, s, t, c) \\amalg", "(U'', R'', s'', t'', c'')", "$$", "This immediately gives $M' = M \\amalg M''$ and the proof of the claim", "is complete.", "\\medskip\\noindent", "We still have to prove that the canonical map $U' \\to M' \\times_M U$", "is an isomorphism. It is an \\'etale morphism", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-permanence}).", "On the other hand, by base changing to strictly henselian local rings", "(as in the third paragraph of the proof) and using the bijectivity", "$U'_{p'} \\to U_p$ established in the course of the proof of the claim,", "we see that $U' \\to M' \\times_M U$ is universally bijective", "(some details omitted). However, a universally bijective", "\\'etale morphism is an isomorphism", "(Descent, Lemma \\ref{descent-lemma-universally-injective-etale-open-immersion})", "and the proof is complete." ], "refs": [ "groupoids-lemma-integral-over-invariants", "groupoids-lemma-invariants-base-change", "groupoids-lemma-points", "groupoids-lemma-invariants-base-change", "limits-lemma-descend-opens", "limits-lemma-descend-isomorphism", "morphisms-lemma-set-points-where-fibres-etale", "algebra-lemma-finite-residue-extension-closed", "more-algebra-lemma-integral-over-henselian-pair", "more-algebra-lemma-change-ideal-henselian-pair", "more-algebra-lemma-characterize-henselian-pair", "morphisms-lemma-etale-permanence", "descent-lemma-universally-injective-etale-open-immersion" ], "ref_ids": [ 9659, 9660, 9661, 9660, 15041, 15066, 5374, 472, 9863, 9862, 9861, 5375, 14700 ] } ], "ref_ids": [] }, { "id": 9663, "type": "theorem", "label": "groupoids-lemma-basis", "categories": [ "groupoids" ], "title": "groupoids-lemma-basis", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Assume", "\\begin{enumerate}", "\\item $U = \\Spec(A)$, and $R = \\Spec(B)$ are affine, and", "\\item there exist elements $x_i \\in A$, $i \\in I$ such that", "$B = \\bigoplus_{i \\in I} s^\\sharp(A)t^\\sharp(x_i)$.", "\\end{enumerate}", "Then $A = \\bigoplus_{i\\in I} Cx_i$, and $B \\cong A \\otimes_C A$", "where $C \\subset A$ is the $R$-invariant", "functions on $U$ as in (\\ref{equation-invariants})." ], "refs": [], "proofs": [ { "contents": [ "During this proof we will write $s, t : A \\to B$ instead of", "$s^\\sharp, t^\\sharp$, and similarly $c : B \\to B \\otimes_{s, A, t} B$.", "We write $p_0 : B \\to B \\otimes_{s, A, t} B$, $b \\mapsto b \\otimes 1$ and", "$p_1 : B \\to B \\otimes_{s, A, t} B$, $b \\mapsto 1 \\otimes b$. By", "Lemma \\ref{lemma-diagram-pull}", "and the definition of $C$ we have the following", "commutative diagram", "$$", "\\xymatrix{", "B \\otimes_{s, A, t} B &", "B \\ar@<-1ex>[l]_-c \\ar@<1ex>[l]^-{p_0} &", "A \\ar[l]^t \\\\", "B \\ar[u]^{p_1} &", "A \\ar@<-1ex>[l]_s \\ar@<1ex>[l]^t \\ar[u]_s &", "C \\ar[u] \\ar[l]", "}", "$$", "Moreover the tow left squares are cocartesian in the category of rings, and", "the top row is isomorphic to the diagram", "$$", "\\xymatrix{", "B \\otimes_{t, A, t} B &", "B \\ar@<-1ex>[l]_-{p_1} \\ar@<1ex>[l]^-{p_0} &", "A \\ar[l]^t", "}", "$$", "which is an equalizer diagram according to", "Descent, Lemma \\ref{descent-lemma-ff-exact} because condition (2) implies", "in particular that $s$ (and hence also then isomorphic arrow $t$)", "is faithfully flat.", "The lower row is an equalizer diagram by definition of $C$.", "We can use the $x_i$ and get a commutative diagram", "$$", "\\xymatrix{", "B \\otimes_{s, A, t} B &", "B \\ar@<-1ex>[l]_-c \\ar@<1ex>[l]^-{p_0} &", "A \\ar[l]^t \\\\", "\\bigoplus_{i \\in I} B x_i \\ar[u]^{p_1} &", "\\bigoplus_{i \\in I} A x_i \\ar@<-1ex>[l]_s \\ar@<1ex>[l]^t \\ar[u]_s &", "\\bigoplus_{i \\in I} C x_i \\ar[u] \\ar[l]", "}", "$$", "where in the right vertical arrow we map $x_i$ to $x_i$,", "in the middle vertical arrow we map $x_i$ to $t(x_i)$ and", "in the left vertical arrow we map $x_i$ to", "$c(t(x_i)) = t(x_i) \\otimes 1 = p_0(t(x_i))$ (equality by the commutativity", "of the top part of the diagram in Lemma \\ref{lemma-diagram}). Then the diagram", "commutes. Moreover the middle vertical arrow is an isomorphism", "by assumption. Since the left two squares are cocartesian we", "conclude that also the left vertical arrow is an isomorphism.", "On the other hand, the horizontal rows are exact (i.e., they are", "equalizers). Hence we conclude that also the right vertical arrow", "is an isomorphism." ], "refs": [ "groupoids-lemma-diagram-pull", "descent-lemma-ff-exact", "groupoids-lemma-diagram" ], "ref_ids": [ 9623, 14598, 9622 ] } ], "ref_ids": [] }, { "id": 9664, "type": "theorem", "label": "groupoids-lemma-find-invariant-affine", "categories": [ "groupoids" ], "title": "groupoids-lemma-find-invariant-affine", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Assume $s$, $t$ are finite locally free.", "Let $u \\in U$ be a point such that $t(s^{-1}(\\{u\\}))$", "is contained in an affine open of $U$.", "Then there exists an $R$-invariant affine open neighbourhood", "of $u$ in $U$." ], "refs": [], "proofs": [ { "contents": [ "Since $s$ is finite locally free it has finite fibres. Hence", "$t(s^{-1}(\\{u\\})) = \\{u_1, \\ldots, u_n\\}$ is a finite set.", "Note that $u \\in \\{u_1, \\ldots, u_n\\}$.", "Let $W \\subset U$ be an affine open containing $\\{u_1, \\ldots, u_n\\}$,", "in particular $u \\in W$. Consider", "$Z = R \\setminus s^{-1}(W) \\cap t^{-1}(W)$. This is a closed subset", "of $R$. The image $t(Z)$ is a closed subset of $U$ which can be loosely", "described as the set of points of $U$ which are $R$-equivalent to a point", "of $U \\setminus W$. Hence $W' = U \\setminus t(Z)$ is an $R$-invariant, open", "subscheme of $U$ contained in $W$, and $\\{u_1, \\ldots, u_n\\} \\subset W'$.", "Picture", "$$", "\\{u_1, \\ldots, u_n\\} \\subset W' \\subset W \\subset U.", "$$", "Let $f \\in \\Gamma(W, \\mathcal{O}_W)$ be an element such that", "$\\{u_1, \\ldots, u_n\\} \\subset D(f) \\subset W'$. Such an $f$ exists by", "Algebra, Lemma \\ref{algebra-lemma-silly}. By our choice of $W'$ we", "have $s^{-1}(W') \\subset t^{-1}(W)$, and hence we get a diagram", "$$", "\\xymatrix{", "s^{-1}(W') \\ar[d]_s \\ar[r]_-t & W \\\\", "W'", "}", "$$", "The vertical arrow is finite locally free by assumption. Set", "$$", "g = \\text{Norm}_s(t^\\sharp f) \\in \\Gamma(W', \\mathcal{O}_{W'})", "$$", "By construction $g$ is a function on $W'$ which is", "nonzero in $u$, as $t^\\sharp(f)$ is nonzero in each of the points of", "$R$ lying over $u$, since $f$ is nonzero in $u_1, \\ldots, u_n$.", "Similarly, $D(g) \\subset W'$ is equal to the", "set of points $w$ such that $f$ is not zero in any of the points", "equivalent to $w$. This means that $D(g)$ is an", "$R$-invariant affine open of $W'$. The final picture is", "$$", "\\{u_1, \\ldots, u_n\\} \\subset D(g) \\subset D(f) \\subset W' \\subset W \\subset U", "$$", "and hence we win." ], "refs": [ "algebra-lemma-silly" ], "ref_ids": [ 378 ] } ], "ref_ids": [] }, { "id": 9665, "type": "theorem", "label": "groupoids-lemma-descend-along-finite", "categories": [ "groupoids" ], "title": "groupoids-lemma-descend-along-finite", "contents": [ "Let $X \\to Y$ be a surjective finite locally free morphism.", "Let $V$ be a scheme over $X$ such that for all", "$(y, v_1, \\ldots, v_d)$ where $y \\in Y$ and", "$v_1, \\ldots, v_d \\in V_y$ there exists an affine open", "$U \\subset V$ with $v_1, \\ldots, v_d \\in U$.", "Then any descent datum on $V/X/Y$ is effective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi$ be a descent datum as in", "Descent, Definition \\ref{descent-definition-descent-datum}.", "Recall that the functor from schemes over $Y$ to descent data", "relative to $\\{X \\to Y\\}$ is fully faithful, see", "Descent, Lemma \\ref{descent-lemma-refine-coverings-fully-faithful}.", "Thus using Constructions, Lemma \\ref{constructions-lemma-relative-glueing}", "it suffices to prove the lemma in the case that $Y$ is affine.", "Some details omitted (this argument can be avoided if $Y$ is", "separated or has affine diagonal, because then every morphism from", "an affine scheme to $X$ is affine).", "\\medskip\\noindent", "Assume $Y$ is affine. If $V$ is also affine, then we have effectivity by", "Descent, Lemma \\ref{descent-lemma-affine}. Hence by", "Descent, Lemma \\ref{descent-lemma-effective-for-fpqc-is-local-upstairs}", "it suffices to prove that every point $v$ of $V$ has a $\\varphi$-invariant", "affine open neighbourhood. Consider the groupoid", "$(X, X \\times_Y X, \\text{pr}_1, \\text{pr}_0, \\text{pr}_{02})$.", "By Lemma \\ref{lemma-cartesian-equivalent-descent-datum}", "the descent datum $\\varphi$ determines and is determined by", "a cartesian morphism of groupoid schemes", "$$", "(V, R, s, t, c)", "\\longrightarrow", "(X, X \\times_Y X, \\text{pr}_1, \\text{pr}_0, \\text{pr}_{02})", "$$", "over $\\Spec(\\mathbf{Z})$.", "Since $X \\to Y$ is finite locally free, we see that", "$\\text{pr}_i : X \\times_Y X \\to X$ and hence $s$ and $t$", "are finite locally free. In particular the $R$-orbit", "$t(s^{-1}(\\{v\\}))$ of our point $v \\in V$", "is finite. Using the equivalence of categories of", "Lemma \\ref{lemma-cartesian-equivalent-descent-datum}", "once more we see that $\\varphi$-invariant opens of $V$", "are the same thing as $R$-invariant opens of $V$.", "Our assumption shows there exists an affine open of $V$", "containing the orbit $t(s^{-1}(\\{v\\}))$ as all the points", "in this orbit map to the same point of $Y$.", "Thus Lemma \\ref{lemma-find-invariant-affine}", "provides an $R$-invariant affine open containing $v$." ], "refs": [ "descent-definition-descent-datum", "descent-lemma-refine-coverings-fully-faithful", "constructions-lemma-relative-glueing", "descent-lemma-affine", "descent-lemma-effective-for-fpqc-is-local-upstairs", "groupoids-lemma-cartesian-equivalent-descent-datum", "groupoids-lemma-cartesian-equivalent-descent-datum", "groupoids-lemma-find-invariant-affine" ], "ref_ids": [ 14776, 14745, 12581, 14748, 14746, 9654, 9654, 9664 ] } ], "ref_ids": [] }, { "id": 9666, "type": "theorem", "label": "groupoids-lemma-descend-along-finite-quasi-projective", "categories": [ "groupoids" ], "title": "groupoids-lemma-descend-along-finite-quasi-projective", "contents": [ "Let $X \\to Y$ be a surjective finite locally free morphism.", "Let $V$ be a scheme over $X$ such that one of the following holds", "\\begin{enumerate}", "\\item $V \\to X$ is projective,", "\\item $V \\to X$ is quasi-projective,", "\\item there exists an ample invertible sheaf on $V$,", "\\item there exists an $X$-ample invertible sheaf on $V$,", "\\item there exists an $X$-very ample invertible sheaf on $V$.", "\\end{enumerate}", "Then any descent datum on $V/X/Y$ is effective." ], "refs": [], "proofs": [ { "contents": [ "We check the condition in Lemma \\ref{lemma-descend-along-finite}.", "Let $y \\in Y$ and $v_1, \\ldots, v_d \\in V$ points over $y$.", "Case (1) is a special case of (2), see", "Morphisms, Lemma \\ref{morphisms-lemma-projective-quasi-projective}.", "Case (2) is a special case of (4), see", "Morphisms, Definition \\ref{morphisms-definition-quasi-projective}.", "If there exists an ample invertible sheaf on $V$, then", "there exists an affine open containing $v_1, \\ldots, v_d$ by", "Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-affine}.", "Thus (3) is true.", "In cases (4) and (5) it is harmless to replace $Y$ by an", "affine open neighbourhood of $y$.", "Then $X$ is affine too.", "In case (4) we see that $V$ has an ample invertible sheaf", "by Morphisms, Definition \\ref{morphisms-definition-relatively-ample}", "and the result follows from case (3).", "In case (5) we can replace $V$ by a quasi-compact open containing", "$v_1, \\ldots, v_d$ and we reduce to case (4) by", "Morphisms, Lemma \\ref{morphisms-lemma-ample-very-ample}." ], "refs": [ "groupoids-lemma-descend-along-finite", "morphisms-lemma-projective-quasi-projective", "morphisms-definition-quasi-projective", "properties-lemma-ample-finite-set-in-affine", "morphisms-definition-relatively-ample", "morphisms-lemma-ample-very-ample" ], "ref_ids": [ 9665, 5427, 5570, 3062, 5568, 5387 ] } ], "ref_ids": [] }, { "id": 9667, "type": "theorem", "label": "groupoids-proposition-connected-component", "categories": [ "groupoids" ], "title": "groupoids-proposition-connected-component", "contents": [ "Let $G$ be a group scheme over a field $k$. There exists a canonical closed", "subgroup scheme $G^0 \\subset G$ with the following properties", "\\begin{enumerate}", "\\item $G^0 \\to G$ is a flat closed immersion,", "\\item $G^0 \\subset G$ is the connected component of the identity,", "\\item $G^0$ is geometrically irreducible, and", "\\item $G^0$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $G^0$ be the connected component of the identity with its canonical", "scheme structure (Morphisms, Definition", "\\ref{morphisms-definition-scheme-structure-connected-component}).", "To show that $G^0$ is a closed subsgroup scheme we will use the", "criterion of Lemma \\ref{lemma-closed-subgroup-scheme}.", "The morphism $e : \\Spec(k) \\to G$ factors through $G^0$ as we chose", "$G^0$ to be the connected component of $G$ containing $e$.", "Since $i : G \\to G$ is an automorphism fixing $e$, we see that", "$i$ sends $G^0$ into itself.", "By Varieties, Lemma \\ref{varieties-lemma-geometrically-connected-criterion}", "the scheme $G^0$ is geometrically connected over $k$.", "Thus $G^0 \\times_k G^0$ is connected", "(Varieties, Lemma \\ref{varieties-lemma-bijection-connected-components}).", "Thus $m(G^0 \\times_k G^0) \\subset G^0$ set theoretically.", "Thus $m|_{G^0 \\times_k G^0} : G^0 \\times_k G^0 \\to G$", "factors through $G^0$ by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-flat-closed-immersions}.", "Hence $G^0$ is a closed subgroup scheme of $G$.", "By Lemma \\ref{lemma-connected-group-scheme-over-field-irreducible}", "we see that $G^0$ is irreducible. By", "Lemma \\ref{lemma-group-scheme-field-geometrically-irreducible}", "we see that $G^0$ is geometrically irreducible. By", "Lemma \\ref{lemma-irreducible-group-scheme-over-field-qc}", "we see that $G^0$ is quasi-compact." ], "refs": [ "morphisms-definition-scheme-structure-connected-component", "groupoids-lemma-closed-subgroup-scheme", "varieties-lemma-geometrically-connected-criterion", "varieties-lemma-bijection-connected-components", "morphisms-lemma-characterize-flat-closed-immersions", "groupoids-lemma-connected-group-scheme-over-field-irreducible", "groupoids-lemma-group-scheme-field-geometrically-irreducible", "groupoids-lemma-irreducible-group-scheme-over-field-qc" ], "ref_ids": [ 5558, 9582, 10925, 10916, 5274, 9595, 9590, 9594 ] } ], "ref_ids": [] }, { "id": 9668, "type": "theorem", "label": "groupoids-proposition-review-abelian-varieties", "categories": [ "groupoids" ], "title": "groupoids-proposition-review-abelian-varieties", "contents": [ "\\begin{reference}", "Wonderfully explained in \\cite{AVar}.", "\\end{reference}", "Let $A$ be an abelian variety over a field $k$. Then", "\\begin{enumerate}", "\\item $A$ is projective over $k$,", "\\item $A$ is a commutative group scheme,", "\\item the morphism $[n] : A \\to A$ is surjective for all $n \\geq 1$,", "\\item if $k$ is algebraically closed, then $A(k)$ is a divisible abelian group,", "\\item $A[n] = \\Ker([n] : A \\to A)$ is a finite group scheme of degree", "$n^{2\\dim A}$ over $k$,", "\\item $A[n]$ is \\'etale over $k$ if and only if $n \\in k^*$,", "\\item if $n \\in k^*$ and $k$ is algebraically closed,", "then $A(k)[n] \\cong (\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2\\dim(A)}$,", "\\item if $k$ is algebraically closed of characteristic $p > 0$, then", "there exists an integer $0 \\leq f \\leq \\dim(A)$ such that", "$A(k)[p^m] \\cong (\\mathbf{Z}/p^m\\mathbf{Z})^{\\oplus f}$", "for all $m \\geq 1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from Lemma \\ref{lemma-abelian-variety-projective}.", "Part (2) follows from Lemma \\ref{lemma-abelian-variety-abelian}.", "Part (3) follows from Lemma \\ref{lemma-degree-multiplication-by-d}.", "If $k$ is algebraically closed then surjective morphisms of varieties", "over $k$ induce surjective maps on $k$-rational points, hence", "(4) follows from (3).", "Part (5) follows from Lemma \\ref{lemma-degree-multiplication-by-d}", "and the fact that a base change of a finite locally free morphism", "of degree $N$ is a finite locally free morphism of degree $N$.", "Part (6) follows from ", "Lemma \\ref{lemma-abelian-variety-multiplication-by-d-etale}.", "Namely, if $n$ is invertible in $k$, then $[n]$ is \\'etale", "and hence $A[n]$ is \\'etale over $k$.", "On the other hand, if $n$ is not invertible in $k$, then", "$[n]$ is not \\'etale at $e$ and it follows that $A[n]$", "is not \\'etale over $k$ at $e$ (use", "Morphisms, Lemmas \\ref{morphisms-lemma-flat-unramified-etale} and", "\\ref{morphisms-lemma-set-points-where-fibres-unramified}).", "\\medskip\\noindent", "Assume $k$ is algebraically closed. Set $g = \\dim(A)$. Proof of (7).", "Let $\\ell$ be a prime number which is invertible in $k$. Then we see that", "$$", "A[\\ell](k) = A(k)[\\ell]", "$$", "is a finite abelian group, annihilated by $\\ell$, of order $\\ell^{2g}$.", "It follows that it is isomorphic to $(\\mathbf{Z}/\\ell\\mathbf{Z})^{2g}$", "by the structure theory for finite abelian groups. Next, we consider", "the short exact sequence", "$$", "0 \\to A(k)[\\ell] \\to A(k)[\\ell^2] \\xrightarrow{\\ell} A(k)[\\ell] \\to 0", "$$", "Arguing similarly as above we conclude that ", "$A(k)[\\ell^2] \\cong (\\mathbf{Z}/\\ell^2\\mathbf{Z})^{2g}$.", "By induction on the exponent we find that", "$A(k)[\\ell^m] \\cong (\\mathbf{Z}/\\ell^m\\mathbf{Z})^{2g}$.", "For composite integers $n$ prime to the characteristic of $k$", "we take primary parts and we find the correct shape of the", "$n$-torsion in $A(k)$.", "The proof of (8) proceeds in exactly the same way, using that", "Lemma \\ref{lemma-abelian-variety-multiplication-by-p} gives", "$A(k)[p] \\cong (\\mathbf{Z}/p\\mathbf{Z})^{\\oplus f}$ for some $0 \\leq f \\leq g$." ], "refs": [ "groupoids-lemma-abelian-variety-projective", "groupoids-lemma-abelian-variety-abelian", "groupoids-lemma-degree-multiplication-by-d", "groupoids-lemma-degree-multiplication-by-d", "groupoids-lemma-abelian-variety-multiplication-by-d-etale", "morphisms-lemma-flat-unramified-etale", "morphisms-lemma-set-points-where-fibres-unramified", "groupoids-lemma-abelian-variety-multiplication-by-p" ], "ref_ids": [ 9605, 9608, 9611, 9611, 9612, 5373, 5356, 9613 ] } ], "ref_ids": [] }, { "id": 9669, "type": "theorem", "label": "groupoids-proposition-finite-flat-equivalence", "categories": [ "groupoids" ], "title": "groupoids-proposition-finite-flat-equivalence", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Assume", "\\begin{enumerate}", "\\item $U = \\Spec(A)$, and $R = \\Spec(B)$ are affine,", "\\item $s, t : R \\to U$ finite locally free, and", "\\item $j = (t, s)$ is an equivalence.", "\\end{enumerate}", "In this case, let $C \\subset A$ be as in", "(\\ref{equation-invariants}). Then $U \\to M = \\Spec(C)$", "is finite locally free and $R = U \\times_M U$.", "Moreover, $M$ represents the quotient sheaf $U/R$", "in the fppf topology (see Definition \\ref{definition-quotient-sheaf})." ], "refs": [ "groupoids-definition-quotient-sheaf" ], "proofs": [ { "contents": [ "During this proof we use the notation $s, t : A \\to B$", "instead of the notation $s^\\sharp, t^\\sharp$.", "By Lemma \\ref{lemma-criterion-quotient-representable}", "it suffices to show that $C \\to A$ is finite locally free", "and that the map", "$$", "t \\otimes s : A \\otimes_C A \\longrightarrow B", "$$", "is an isomorphism. First, note that $j$ is a monomorphism, and", "also finite (since already $s$ and $t$ are finite). Hence we see", "that $j$ is a closed immersion by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-monomorphism-closed}.", "Hence $A \\otimes_C A \\to B$ is surjective.", "\\medskip\\noindent", "We will perform base change by flat ring maps $C \\to C'$ as in", "Lemma \\ref{lemma-invariants-base-change}, and we will use that", "formation of invariants commutes with flat base change, see", "part (3) of the lemma cited.", "We will show below that for every prime $\\mathfrak p \\subset C$, there exists", "a local flat ring map $C_{\\mathfrak p} \\to C_{\\mathfrak p}'$", "such that the result holds after a base change to $C_{\\mathfrak p}'$.", "This implies immediately", "that $A \\otimes_C A \\to B$ is injective (use", "Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}).", "It also implies that $C \\to A$ is flat, by combining", "Algebra, Lemmas \\ref{algebra-lemma-local-flat-ff},", "\\ref{algebra-lemma-flat-localization}, and", "\\ref{algebra-lemma-flatness-descends}. Then since $U \\to \\Spec(C)$", "is surjective also (Lemma \\ref{lemma-points}) we conclude that $C \\to A$", "is faithfully flat. Then the isomorphism $B \\cong A \\otimes_C A$", "implies that $A$ is a finitely presented $C$-module, see", "Algebra, Lemma \\ref{algebra-lemma-descend-properties-modules}.", "Hence $A$ is finite locally free over $C$, see", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}.", "\\medskip\\noindent", "By Lemma \\ref{lemma-finite-locally-free-disjoint-free}", "we know that $A$ is a finite", "product of rings $A_r$ and $B$ is a finite product of rings $B_r$", "such that the groupoid scheme decomposes accordingly (see the proof", "of Lemma \\ref{lemma-integral-over-invariants}).", "Then also $C$ is a product of rings $C_r$ and", "correspondingly $C'$ decomposes as a product. Hence we may and do", "assume that the ring maps $s, t : A \\to B$ are finite", "locally free of a fixed rank $r$.", "\\medskip\\noindent", "The local ring maps $C_{\\mathfrak p} \\to C_{\\mathfrak p}'$ we are going", "to use are any local flat ring maps such that the residue field of", "$C_{\\mathfrak p}'$ is infinite.", "By Algebra, Lemma \\ref{algebra-lemma-flat-local-given-residue-field}", "such local ring maps exist.", "\\medskip\\noindent", "Assume $C$ is a local ring with maximal ideal $\\mathfrak m$ and", "infinite residue field, and assume that $s, t : A \\to B$ is", "finite locally free of constant rank $r > 0$.", "Since $C \\subset A$ is integral (Lemma \\ref{lemma-integral-over-invariants})", "all primes lying over $\\mathfrak m$ are maximal, and all maximal", "ideals of $A$ lie over $\\mathfrak m$. Similarly for $C \\subset B$.", "Pick a maximal ideal $\\mathfrak m'$", "of $A$ lying over $\\mathfrak m$ (exists by Lemma \\ref{lemma-points}).", "Since $t : A \\to B$ is finite locally free there exist at most finitely", "many maximal ideals of $B$ lying over $\\mathfrak m'$. Hence we conclude", "(by Lemma \\ref{lemma-points} again)", "that $A$ has finitely many maximal ideals, i.e.,", "$A$ is semi-local. This in turn implies that $B$ is semi-local as", "well. OK, and now, because $t \\otimes s : A \\otimes_C A \\to B$ is surjective,", "we can apply", "Algebra, Lemma \\ref{algebra-lemma-semi-local-module-basis-in-submodule}", "to the ring map $C \\to A$, the $A$-module $M = B$ (seen as an $A$-module", "via $t$) and the $C$-submodule $s(A) \\subset B$. This lemma implies that there", "exist $x_1, \\ldots, x_r \\in A$ such that $M$ is free over $A$", "on the basis $s(x_1), \\ldots, s(x_r)$. Hence we conclude that $C \\to A$", "is finite free and $B \\cong A \\otimes_C A$ by applying", "Lemma \\ref{lemma-basis}." ], "refs": [ "groupoids-lemma-criterion-quotient-representable", "morphisms-lemma-finite-monomorphism-closed", "groupoids-lemma-invariants-base-change", "algebra-lemma-characterize-zero-local", "algebra-lemma-local-flat-ff", "algebra-lemma-flat-localization", "algebra-lemma-flatness-descends", "groupoids-lemma-points", "algebra-lemma-descend-properties-modules", "algebra-lemma-finite-projective", "groupoids-lemma-finite-locally-free-disjoint-free", "groupoids-lemma-integral-over-invariants", "algebra-lemma-flat-local-given-residue-field", "groupoids-lemma-integral-over-invariants", "groupoids-lemma-points", "groupoids-lemma-points", "algebra-lemma-semi-local-module-basis-in-submodule", "groupoids-lemma-basis" ], "ref_ids": [ 9648, 5449, 9660, 410, 537, 538, 528, 9661, 819, 795, 9658, 9659, 1324, 9659, 9661, 9661, 800, 9663 ] } ], "ref_ids": [ 9686 ] }, { "id": 9694, "type": "theorem", "label": "local-cohomology-theorem-finiteness", "categories": [ "local-cohomology" ], "title": "local-cohomology-theorem-finiteness", "contents": [ "\\begin{reference}", "This is a special case of \\cite[Satz 2]{Faltings-finiteness}.", "\\end{reference}", "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.", "Set $Z = V(I) \\subset \\Spec(A)$. Let $M$ be a finite $A$-module.", "Set $s = s_{A, I}(M)$ as in (\\ref{equation-cutoff}).", "Assume that", "\\begin{enumerate}", "\\item $A$ is universally catenary,", "\\item the formal fibres of the local rings of $A$ are Cohen-Macaulay.", "\\end{enumerate}", "Then $H^i_Z(M)$ is finite for $0 \\leq i < s$ and", "$H^s_Z(M)$ is not finite." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-check-finiteness-local-cohomology-locally}", "we may assume that $A$ is a local ring.", "\\medskip\\noindent", "If $A$ is a Noetherian complete local ring, then we can write $A$", "as the quotient of a regular complete local ring $B$ by", "Cohen's structure theorem", "(Algebra, Theorem \\ref{algebra-theorem-cohen-structure-theorem}).", "Using Lemma \\ref{lemma-cutoff} and", "Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-local-cohomology-and-restriction}", "we reduce to the case", "of a regular local ring which is a consequence of", "Lemma \\ref{lemma-local-annihilator}", "because a regular local ring is Gorenstein", "(Dualizing Complexes, Lemma \\ref{dualizing-lemma-regular-gorenstein}).", "\\medskip\\noindent", "Let $A$ be a Noetherian local ring. Let $\\mathfrak m$ be the maximal ideal.", "We may assume $I \\subset \\mathfrak m$, otherwise the lemma is trivial.", "Let $A^\\wedge$ be the completion of $A$, let $Z^\\wedge = V(IA^\\wedge)$, and", "let $M^\\wedge = M \\otimes_A A^\\wedge$ be the completion of $M$", "(Algebra, Lemma \\ref{algebra-lemma-completion-tensor}).", "Then $H^i_Z(M) \\otimes_A A^\\wedge = H^i_{Z^\\wedge}(M^\\wedge)$ by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-torsion-change-rings}", "and flatness of $A \\to A^\\wedge$", "(Algebra, Lemma \\ref{algebra-lemma-completion-flat}).", "Hence it suffices to show that $H^i_{Z^\\wedge}(M^\\wedge)$ is", "finite for $i < s$ and not finite for $i = s$, see", "Algebra, Lemma \\ref{algebra-lemma-descend-properties-modules}.", "Since we know the result is true for $A^\\wedge$ it suffices", "to show that $s_{A, I}(M) = s_{A^\\wedge, I^\\wedge}(M^\\wedge)$.", "This follows from Lemma \\ref{lemma-cutoff-completion}." ], "refs": [ "local-cohomology-lemma-check-finiteness-local-cohomology-locally", "algebra-theorem-cohen-structure-theorem", "local-cohomology-lemma-cutoff", "dualizing-lemma-local-cohomology-and-restriction", "local-cohomology-lemma-local-annihilator", "dualizing-lemma-regular-gorenstein", "algebra-lemma-completion-tensor", "dualizing-lemma-torsion-change-rings", "algebra-lemma-completion-flat", "algebra-lemma-descend-properties-modules", "local-cohomology-lemma-cutoff-completion" ], "ref_ids": [ 9725, 327, 9739, 2816, 9742, 2880, 869, 2817, 870, 819, 9741 ] } ], "ref_ids": [] }, { "id": 9695, "type": "theorem", "label": "local-cohomology-lemma-local-cohomology-is-local-cohomology", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-local-cohomology-is-local-cohomology", "contents": [ "Let $A$ be a ring and let $I$ be a finitely generated ideal.", "Set $Z = V(I) \\subset X = \\Spec(A)$. For $K \\in D(A)$ corresponding", "to $\\widetilde{K} \\in D_\\QCoh(\\mathcal{O}_X)$ via", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-affine-compare-bounded}", "there is a functorial isomorphism", "$$", "R\\Gamma_Z(K) = R\\Gamma_Z(X, \\widetilde{K})", "$$", "where on the left we have", "Dualizing Complexes, Equation (\\ref{dualizing-equation-local-cohomology})", "and on the right we have the functor of", "Cohomology, Section \\ref{cohomology-section-cohomology-support-bis}." ], "refs": [ "perfect-lemma-affine-compare-bounded" ], "proofs": [ { "contents": [ "By Cohomology, Lemma \\ref{cohomology-lemma-triangle-sections-with-support}", "there exists a distinguished triangle", "$$", "R\\Gamma_Z(X, \\widetilde{K})", "\\to R\\Gamma(X, \\widetilde{K})", "\\to R\\Gamma(U, \\widetilde{K})", "\\to R\\Gamma_Z(X, \\widetilde{K})[1]", "$$", "where $U = X \\setminus Z$. We know that $R\\Gamma(X, \\widetilde{K}) = K$", "by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded}.", "Say $I = (f_1, \\ldots, f_r)$. Then we obtain a finite affine", "open covering $\\mathcal{U} : U = D(f_1) \\cup \\ldots \\cup D(f_r)$.", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-alternating-cech-complex-complex-computes-cohomology}", "the alternating {\\v C}ech complex", "$\\text{Tot}(\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U},", "\\widetilde{K^\\bullet}))$", "computes $R\\Gamma(U, \\widetilde{K})$ where $K^\\bullet$ is any", "complex of $A$-modules representing $K$. Working through the", "definitions we find", "$$", "R\\Gamma(U, \\widetilde{K}) =", "\\text{Tot}\\left(", "K^\\bullet \\otimes_A", "(\\prod\\nolimits_{i_0} A_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to A_{f_1\\ldots f_r})\\right)", "$$", "It is clear that", "$K^\\bullet = R\\Gamma(X, \\widetilde{K^\\bullet}) \\to", "R\\Gamma(U, \\widetilde{K}^\\bullet)$", "is induced by the diagonal map from $A$ into $\\prod A_{f_i}$.", "Hence we conclude that", "$$", "R\\Gamma_Z(X, \\mathcal{F}^\\bullet) =", "\\text{Tot}\\left(", "K^\\bullet \\otimes_A", "(A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to A_{f_1\\ldots f_r})\\right)", "$$", "By Dualizing Complexes, Lemma \\ref{dualizing-lemma-local-cohomology-adjoint}", "this complex computes $R\\Gamma_Z(K)$ and we see the lemma holds." ], "refs": [ "cohomology-lemma-triangle-sections-with-support", "perfect-lemma-affine-compare-bounded", "perfect-lemma-alternating-cech-complex-complex-computes-cohomology", "dualizing-lemma-local-cohomology-adjoint" ], "ref_ids": [ 2154, 6941, 6972, 2815 ] } ], "ref_ids": [ 6941 ] }, { "id": 9696, "type": "theorem", "label": "local-cohomology-lemma-local-cohomology", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-local-cohomology", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "Set $X = \\Spec(A)$, $Z = V(I)$, $U = X \\setminus Z$, and $j : U \\to X$", "the inclusion morphism. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_U$-module. Then", "\\begin{enumerate}", "\\item there exists an $A$-module $M$ such that $\\mathcal{F}$ is the", "restriction of $\\widetilde{M}$ to $U$,", "\\item given $M$ there is an exact sequence", "$$", "0 \\to H^0_Z(M) \\to M \\to H^0(U, \\mathcal{F}) \\to H^1_Z(M) \\to 0", "$$", "and isomorphisms $H^p(U, \\mathcal{F}) = H^{p + 1}_Z(M)$ for $p \\geq 1$,", "\\item we may take $M = H^0(U, \\mathcal{F})$ in which case", "we have $H^0_Z(M) = H^1_Z(M) = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The existence of $M$ follows from", "Properties, Lemma \\ref{properties-lemma-extend-trivial}", "and the fact that quasi-coherent sheaves on $X$ correspond", "to $A$-modules (Schemes, Lemma \\ref{schemes-lemma-equivalence-quasi-coherent}).", "Then we look at the distinguished triangle", "$$", "R\\Gamma_Z(X, \\widetilde{M}) \\to R\\Gamma(X, \\widetilde{M}) \\to", "R\\Gamma(U, \\widetilde{M}|_U) \\to R\\Gamma_Z(X, \\widetilde{M})[1]", "$$", "of Cohomology, Lemma \\ref{cohomology-lemma-triangle-sections-with-support}.", "Since $X$ is affine we have $R\\Gamma(X, \\widetilde{M}) = M$", "by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "By our choice of $M$ we have $\\mathcal{F} = \\widetilde{M}|_U$", "and hence this produces an exact sequence", "$$", "0 \\to H^0_Z(X, \\widetilde{M}) \\to M \\to H^0(U, \\mathcal{F}) \\to", "H^1_Z(X, \\widetilde{M}) \\to 0", "$$", "and isomorphisms $H^p(U, \\mathcal{F}) = H^{p + 1}_Z(X, \\widetilde{M})$", "for $p \\geq 1$. By Lemma \\ref{lemma-local-cohomology-is-local-cohomology}", "we have $H^i_Z(M) = H^i_Z(X, \\widetilde{M})$ for all $i$.", "Thus (1) and (2) do hold.", "Finally, setting $M' = H^0(U, \\mathcal{F})$ we see that", "the kernel and cokernel of $M \\to M'$ are $I$-power torsion.", "Therefore $\\widetilde{M}|_U \\to \\widetilde{M'}|_U$ is an isomorphism", "and we can indeed use $M'$ as predicted in (3). It goes without saying", "that we obtain zero for both $H^0_Z(M')$ and $H^0_Z(M')$." ], "refs": [ "properties-lemma-extend-trivial", "schemes-lemma-equivalence-quasi-coherent", "cohomology-lemma-triangle-sections-with-support", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "local-cohomology-lemma-local-cohomology-is-local-cohomology" ], "ref_ids": [ 3018, 7664, 2154, 3282, 9695 ] } ], "ref_ids": [] }, { "id": 9697, "type": "theorem", "label": "local-cohomology-lemma-already-torsion", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-already-torsion", "contents": [ "Let $I, J \\subset A$ be finitely generated ideals of a ring $A$.", "If $M$ is an $I$-power torsion module, then the", "canonical map", "$$", "H^i_{V(I) \\cap V(J)}(M) \\to H^i_{V(J)}(M)", "$$", "is an isomorphism for all $i$." ], "refs": [], "proofs": [ { "contents": [ "Use the spectral sequence of", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-local-cohomology-ss}", "to reduce to the statement $R\\Gamma_I(M) = M$ which is immediate", "from the construction of local cohomology", "in Dualizing Complexes, Section \\ref{dualizing-section-local-cohomology}." ], "refs": [ "dualizing-lemma-local-cohomology-ss" ], "ref_ids": [ 2820 ] } ], "ref_ids": [] }, { "id": 9698, "type": "theorem", "label": "local-cohomology-lemma-multiplicative", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-multiplicative", "contents": [ "Let $S \\subset A$ be a multiplicative set of a ring $A$.", "Let $M$ be an $A$-module with $S^{-1}M = 0$. Then", "$\\colim_{f \\in S} H^0_{V(f)}(M) = M$ and", "$\\colim_{f \\in S} H^1_{V(f)}(M) = 0$." ], "refs": [], "proofs": [ { "contents": [ "The statement on $H^0$ follows directly from the definitions.", "To see the statement on $H^1$ observe that $R\\Gamma_{V(f)}$", "and $H^1_{V(f)}$ commute with colimits. Hence we may assume", "$M$ is annihilated by some $f \\in S$. Then", "$H^1_{V(ff')}(M) = 0$ for all $f' \\in S$ (for example by", "Lemma \\ref{lemma-already-torsion})." ], "refs": [ "local-cohomology-lemma-already-torsion" ], "ref_ids": [ 9697 ] } ], "ref_ids": [] }, { "id": 9699, "type": "theorem", "label": "local-cohomology-lemma-elements-come-from-bigger", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-elements-come-from-bigger", "contents": [ "Let $I \\subset A$ be a finitely generated ideal of a ring $A$.", "Let $\\mathfrak p$ be a prime ideal. Let $M$ be an $A$-module.", "Let $i \\geq 0$ be an integer and consider the map", "$$", "\\Psi :", "\\colim_{f \\in A, f \\not \\in \\mathfrak p} H^i_{V((I, f))}(M)", "\\longrightarrow", "H^i_{V(I)}(M)", "$$", "Then", "\\begin{enumerate}", "\\item $\\Im(\\Psi)$ is the set of elements which map to zero in", "$H^i_{V(I)}(M)_\\mathfrak p$,", "\\item if $H^{i - 1}_{V(I)}(M)_\\mathfrak p = 0$, then $\\Psi$ is injective,", "\\item if $H^{i - 1}_{V(I)}(M)_\\mathfrak p = H^i_{V(I)}(M)_\\mathfrak p = 0$,", "then $\\Psi$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For $f \\in A$, $f \\not \\in \\mathfrak p$ the spectral sequence of", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-local-cohomology-ss}", "degenerates to give short exact sequences", "$$", "0 \\to H^1_{V(f)}(H^{i - 1}_{V(I)}(M)) \\to", "H^i_{V((I, f))}(M) \\to H^0_{V(f)}(H^i_{V(I)}(M)) \\to 0", "$$", "This proves (1) and part (2) follows from this and", "Lemma \\ref{lemma-multiplicative}.", "Part (3) is a formal consequence." ], "refs": [ "dualizing-lemma-local-cohomology-ss", "local-cohomology-lemma-multiplicative" ], "ref_ids": [ 2820, 9698 ] } ], "ref_ids": [] }, { "id": 9700, "type": "theorem", "label": "local-cohomology-lemma-isomorphism", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-isomorphism", "contents": [ "Let $I \\subset I' \\subset A$ be finitely generated ideals of a", "Noetherian ring $A$. Let $M$ be an $A$-module. Let $i \\geq 0$ be an integer.", "Consider the map", "$$", "\\Psi : H^i_{V(I')}(M) \\to H^i_{V(I)}(M)", "$$", "The following are true:", "\\begin{enumerate}", "\\item if $H^i_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) = 0$", "for all $\\mathfrak p \\in V(I) \\setminus V(I')$, then", "$\\Psi$ is surjective,", "\\item if $H^{i - 1}_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) = 0$", "for all $\\mathfrak p \\in V(I) \\setminus V(I')$, then", "$\\Psi$ is injective,", "\\item if $H^i_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) =", "H^{i - 1}_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) = 0$", "for all $\\mathfrak p \\in V(I) \\setminus V(I')$, then", "$\\Psi$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1).", "Let $\\xi \\in H^i_{V(I)}(M)$. Since $A$ is Noetherian, there exists a", "largest ideal $I \\subset I'' \\subset I'$ such that $\\xi$ is the image", "of some $\\xi'' \\in H^i_{V(I'')}(M)$. If $V(I'') = V(I')$, then we are", "done. If not, choose a generic point $\\mathfrak p \\in V(I'')$ not in $V(I')$.", "Then we have $H^i_{V(I'')}(M)_\\mathfrak p =", "H^i_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) = 0$ by assumption.", "By Lemma \\ref{lemma-elements-come-from-bigger} we can increase $I''$", "which contradicts maximality.", "\\medskip\\noindent", "Proof of (2). Let $\\xi' \\in H^i_{V(I')}(M)$ be in the kernel of $\\Psi$.", "Since $A$ is Noetherian, there exists a", "largest ideal $I \\subset I'' \\subset I'$ such that $\\xi'$", "maps to zero in $H^i_{V(I'')}(M)$. If $V(I'') = V(I')$, then we are", "done. If not, then choose a generic point $\\mathfrak p \\in V(I'')$", "not in $V(I')$. Then we have $H^{i - 1}_{V(I'')}(M)_\\mathfrak p =", "H^{i - 1}_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) = 0$ by assumption.", "By Lemma \\ref{lemma-elements-come-from-bigger} we can increase $I''$", "which contradicts maximality.", "\\medskip\\noindent", "Part (3) is formal from parts (1) and (2)." ], "refs": [ "local-cohomology-lemma-elements-come-from-bigger", "local-cohomology-lemma-elements-come-from-bigger" ], "ref_ids": [ 9699, 9699 ] } ], "ref_ids": [] }, { "id": 9701, "type": "theorem", "label": "local-cohomology-lemma-depth-2-connected-punctured-spectrum", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-depth-2-connected-punctured-spectrum", "contents": [ "\\begin{reference}", "\\cite[Proposition 2.1]{Hartshorne-connectedness}", "\\end{reference}", "\\begin{slogan}", "Hartshorne's connectedness", "\\end{slogan}", "Let $A$ be a Noetherian local ring of depth $\\geq 2$.", "Then the punctured spectra of $A$, $A^h$, and $A^{sh}$ are connected." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be the punctured spectrum of $A$.", "If $U$ is disconnected then we see that", "$\\Gamma(U, \\mathcal{O}_U)$ has a nontrivial idempotent.", "But $A$, being local, does not have a nontrivial idempotent.", "Hence $A \\to \\Gamma(U, \\mathcal{O}_U)$ is not an isomorphism.", "By Lemma \\ref{lemma-local-cohomology}", "we conclude that either $H^0_\\mathfrak m(A)$ or $H^1_\\mathfrak m(A)$", "is nonzero. Thus $\\text{depth}(A) \\leq 1$ by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth}.", "To see the result for $A^h$ and $A^{sh}$ use", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-depth}." ], "refs": [ "local-cohomology-lemma-local-cohomology", "dualizing-lemma-depth", "more-algebra-lemma-henselization-depth" ], "ref_ids": [ 9696, 2826, 10062 ] } ], "ref_ids": [] }, { "id": 9702, "type": "theorem", "label": "local-cohomology-lemma-catenary-S2-equidimensional", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-catenary-S2-equidimensional", "contents": [ "\\begin{reference}", "\\cite[Corollary 5.10.9]{EGA}", "\\end{reference}", "Let $A$ be a Noetherian local ring which is catenary and $(S_2)$.", "Then $\\Spec(A)$ is equidimensional." ], "refs": [], "proofs": [ { "contents": [ "Set $X = \\Spec(A)$. Say $d = \\dim(A) = \\dim(X)$. Inside $X$ consider the", "union $X_1$ of the irreducible components of dimension $d$ and the union", "$X_2$ of the irreducible components of dimension $< d$. Of course", "$X = X_1 \\cup X_2$. If $X_2 = \\emptyset$,", "then the lemma holds. If not, then $Z = X_1 \\cap X_2$ is a nonempty closed", "subset of $X$ because it contains at least the closed point of $X$.", "Hence we can choose a generic point $z \\in Z$ of an irreducible component", "of $Z$. Recall that the spectrum of $\\mathcal{O}_{Z, z}$ is the set of points", "of $X$ specializing to $z$. Since $z$ is both contained in an", "irreducible component of dimension $d$ and in an irreducible component", "of dimension $< d$ we obtain nontrivial specializations $x_1 \\leadsto z$ and", "$x_2 \\leadsto z$ such that the closures of $x_1$ and $x_2$ have different", "dimensions. Since $X$ is catenary, this can only happen if at least", "one of the specializations $x_1 \\leadsto z$ and $x_2 \\leadsto z$ is not", "immediate! Thus $\\dim(\\mathcal{O}_{Z, z}) \\geq 2$. Therefore", "$\\text{depth}(\\mathcal{O}_{Z, z}) \\geq 2$ because $A$ is $(S_2)$.", "However, the punctured spectrum $U$ of $\\mathcal{O}_{Z, z}$ is disconnected", "because the closed subsets $U \\cap X_1$ and $U \\cap X_2$ are disjoint", "(by our choice of $z$) and cover $U$. This is a contradiction with", "Lemma \\ref{lemma-depth-2-connected-punctured-spectrum}", "and the proof is complete." ], "refs": [ "local-cohomology-lemma-depth-2-connected-punctured-spectrum" ], "ref_ids": [ 9701 ] } ], "ref_ids": [] }, { "id": 9703, "type": "theorem", "label": "local-cohomology-lemma-cd", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cd", "contents": [ "Let $I \\subset A$ be a finitely generated ideal of a ring $A$.", "Set $Y = V(I) \\subset X = \\Spec(A)$. Let $d \\geq -1$ be an integer.", "The following are equivalent", "\\begin{enumerate}", "\\item $H^i_Y(A) = 0$ for $i > d$,", "\\item $H^i_Y(M) = 0$ for $i > d$ for every $A$-module $M$, and", "\\item if $d = -1$, then $Y = \\emptyset$, if $d = 0$, then", "$Y$ is open and closed in $X$, and if $d > 0$ then", "$H^i(X \\setminus Y, \\mathcal{F}) = 0$ for $i \\geq d$", "for every quasi-coherent $\\mathcal{O}_{X \\setminus Y}$-module $\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $R\\Gamma_Y(-)$ has finite cohomological dimension by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-local-cohomology-adjoint}", "for example. Hence there exists an integer $i_0$ such that", "$H^i_Y(M) = 0$ for all $A$-modules $M$ and $i \\geq i_0$.", "\\medskip\\noindent", "Let us prove that (1) and (2) are equivalent. It is immediate that", "(2) implies (1). Assume (1). By descending induction on $i > d$", "we will show that $H^i_Y(M) = 0$ for all $A$-modules $M$.", "For $i \\geq i_0$ we have seen this above. To do the induction step,", "let $i_0 > i > d$. Choose any $A$-module $M$ and fit it into", "a short exact sequence $0 \\to N \\to F \\to M \\to 0$ where $F$ is a", "free $A$-module. Since $R\\Gamma_Y$ is a right adjoint, we see that", "$H^i_Y(-)$ commutes with direct sums. Hence $H^i_Y(F) = 0$", "as $i > d$ by assumption (1). Then we see that", "$H^i_Y(M) = H^{i + 1}_Y(N) = 0$ as desired.", "\\medskip\\noindent", "Assume $d = -1$ and (2) holds. Then $0 = H^0_Y(A/I) = A/I \\Rightarrow A = I", "\\Rightarrow Y = \\emptyset$. Thus (3) holds. We omit the proof of the converse.", "\\medskip\\noindent", "Assume $d = 0$ and (2) holds. Set", "$J = H^0_I(A) = \\{x \\in A \\mid I^nx = 0 \\text{ for some }n > 0\\}$.", "Then", "$$", "H^1_Y(A) = \\Coker(A \\to \\Gamma(X \\setminus Y, \\mathcal{O}_{X \\setminus Y}))", "\\quad\\text{and}\\quad", "H^1_Y(I) = \\Coker(I \\to \\Gamma(X \\setminus Y, \\mathcal{O}_{X \\setminus Y}))", "$$", "and the kernel of the first map is equal to $J$. See", "Lemma \\ref{lemma-local-cohomology}.", "We conclude from (2) that $I(A/J) = A/J$.", "Thus we may pick $f \\in I$", "mapping to $1$ in $A/J$. Then $1 - f \\in J$ so $I^n(1 - f) = 0$ for some", "$n > 0$. Hence $f^n = f^{n + 1}$. Then $e = f^n \\in I$ is an idempotent.", "Consider the complementary idempotent $e' = 1 - f^n \\in J$.", "For any element $g \\in I$ we have $g^m e' = 0$ for some $m > 0$.", "Thus $I$ is contained in the radical of ideal $(e) \\subset I$.", "This means $Y = V(I) = V(e)$ is open and closed in $X$ as predicted in (3).", "Conversely, if $Y = V(I)$ is open and closed, then the functor", "$H^0_Y(-)$ is exact and has vanshing higher derived functors.", "\\medskip\\noindent", "If $d > 0$, then we see immediately from", "Lemma \\ref{lemma-local-cohomology} that (2) is equivalent to (3)." ], "refs": [ "dualizing-lemma-local-cohomology-adjoint", "local-cohomology-lemma-local-cohomology", "local-cohomology-lemma-local-cohomology" ], "ref_ids": [ 2815, 9696, 9696 ] } ], "ref_ids": [] }, { "id": 9704, "type": "theorem", "label": "local-cohomology-lemma-bound-cd", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-bound-cd", "contents": [ "Let $I \\subset A$ be a finitely generated ideal of a ring $A$.", "Then", "\\begin{enumerate}", "\\item $\\text{cd}(A, I)$ is at most equal to the number of", "generators of $I$,", "\\item $\\text{cd}(A, I) \\leq r$ if there exist $f_1, \\ldots, f_r \\in A$", "such that $V(f_1, \\ldots, f_r) = V(I)$,", "\\item $\\text{cd}(A, I) \\leq c$ if $\\Spec(A) \\setminus V(I)$", "can be covered by $c$ affine opens.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The explicit description for $R\\Gamma_Y(-)$ given in", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-local-cohomology-adjoint}", "shows that (1) is true. We can deduce (2) from (1) using the", "fact that $R\\Gamma_Z$ depends only on the closed subset", "$Z$ and not on the choice of the finitely generated ideal", "$I \\subset A$ with $V(I) = Z$. This follows either from the", "construction of local cohomology in", "Dualizing Complexes, Section \\ref{dualizing-section-local-cohomology}", "combined with", "More on Algebra, Lemma \\ref{more-algebra-lemma-local-cohomology-closed}", "or it follows from Lemma \\ref{lemma-local-cohomology-is-local-cohomology}.", "To see (3) we use Lemma \\ref{lemma-cd}", "and the vanishing result of Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-vanishing-nr-affines}." ], "refs": [ "dualizing-lemma-local-cohomology-adjoint", "more-algebra-lemma-local-cohomology-closed", "local-cohomology-lemma-local-cohomology-is-local-cohomology", "local-cohomology-lemma-cd", "coherent-lemma-vanishing-nr-affines" ], "ref_ids": [ 2815, 10337, 9695, 9703, 3292 ] } ], "ref_ids": [] }, { "id": 9705, "type": "theorem", "label": "local-cohomology-lemma-cd-sum", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cd-sum", "contents": [ "Let $I, J \\subset A$ be finitely generated ideals of a ring $A$.", "Then $\\text{cd}(A, I + J) \\leq \\text{cd}(A, I) + \\text{cd}(A, J)$." ], "refs": [], "proofs": [ { "contents": [ "Use the definition and Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-local-cohomology-ss}." ], "refs": [ "dualizing-lemma-local-cohomology-ss" ], "ref_ids": [ 2820 ] } ], "ref_ids": [] }, { "id": 9706, "type": "theorem", "label": "local-cohomology-lemma-cd-change-rings", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cd-change-rings", "contents": [ "Let $A \\to B$ be a ring map. Let $I \\subset A$ be a finitely generated ideal.", "Then $\\text{cd}(B, IB) \\leq \\text{cd}(A, I)$. If $A \\to B$ is faithfully", "flat, then equality holds." ], "refs": [], "proofs": [ { "contents": [ "Use the definition and", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-torsion-change-rings}." ], "refs": [ "dualizing-lemma-torsion-change-rings" ], "ref_ids": [ 2817 ] } ], "ref_ids": [] }, { "id": 9707, "type": "theorem", "label": "local-cohomology-lemma-cd-local", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cd-local", "contents": [ "Let $I \\subset A$ be a finitely generated ideal of a ring $A$.", "Then $\\text{cd}(A, I) = \\max \\text{cd}(A_\\mathfrak p, I_\\mathfrak p)$." ], "refs": [], "proofs": [ { "contents": [ "Let $Y = V(I)$ and $Y' = V(I_\\mathfrak p) \\subset \\Spec(A_\\mathfrak p)$.", "Recall that", "$R\\Gamma_Y(A) \\otimes_A A_\\mathfrak p = R\\Gamma_{Y'}(A_\\mathfrak p)$", "by Dualizing Complexes, Lemma \\ref{dualizing-lemma-torsion-change-rings}.", "Thus we conclude by Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}." ], "refs": [ "dualizing-lemma-torsion-change-rings", "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 2817, 410 ] } ], "ref_ids": [] }, { "id": 9708, "type": "theorem", "label": "local-cohomology-lemma-cd-dimension", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cd-dimension", "contents": [ "Let $I \\subset A$ be a finitely generated ideal of a ring $A$.", "If $M$ is a finite $A$-module, then", "$H^i_{V(I)}(M) = 0$ for $i > \\dim(\\text{Supp}(M))$.", "In particular, we have $\\text{cd}(A, I) \\leq \\dim(A)$." ], "refs": [], "proofs": [ { "contents": [ "We first prove the second statement.", "Recall that $\\dim(A)$ denotes the Krull dimension. By", "Lemma \\ref{lemma-cd-local} we may assume $A$ is local.", "If $V(I) = \\emptyset$, then the result is true.", "If $V(I) \\not = \\emptyset$, then", "$\\dim(\\Spec(A) \\setminus V(I)) < \\dim(A)$ because", "the closed point is missing. Observe that", "$U = \\Spec(A) \\setminus V(I)$ is a quasi-compact", "open of the spectral space $\\Spec(A)$, hence a spectral space itself.", "See Algebra, Lemma \\ref{algebra-lemma-spec-spectral} and", "Topology, Lemma \\ref{topology-lemma-spectral-sub}.", "Thus Cohomology, Proposition", "\\ref{cohomology-proposition-cohomological-dimension-spectral}", "implies $H^i(U, \\mathcal{F}) = 0$ for $i \\geq \\dim(A)$", "which implies what we want by Lemma \\ref{lemma-cd}.", "In the Noetherian case the reader may use", "Grothendieck's Cohomology, Proposition", "\\ref{cohomology-proposition-vanishing-Noetherian}.", "\\medskip\\noindent", "We will deduce the first statement from the second.", "Let $\\mathfrak a$ be the annihilator of the finite $A$-module $M$.", "Set $B = A/\\mathfrak a$. Recall that $\\Spec(B) = \\text{Supp}(M)$, see", "Algebra, Lemma \\ref{algebra-lemma-support-closed}.", "Set $J = IB$. Then $M$ is a $B$-module", "and $H^i_{V(I)}(M) = H^i_{V(J)}(M)$, see", "Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-local-cohomology-and-restriction}.", "Since $\\text{cd}(B, J) \\leq \\dim(B) = \\dim(\\text{Supp}(M))$", "by the first part we conclude." ], "refs": [ "local-cohomology-lemma-cd-local", "algebra-lemma-spec-spectral", "topology-lemma-spectral-sub", "cohomology-proposition-cohomological-dimension-spectral", "local-cohomology-lemma-cd", "cohomology-proposition-vanishing-Noetherian", "algebra-lemma-support-closed", "dualizing-lemma-local-cohomology-and-restriction" ], "ref_ids": [ 9707, 423, 8306, 2247, 9703, 2246, 543, 2816 ] } ], "ref_ids": [] }, { "id": 9709, "type": "theorem", "label": "local-cohomology-lemma-cd-is-one", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cd-is-one", "contents": [ "Let $I \\subset A$ be a finitely generated ideal of a ring $A$. If", "$\\text{cd}(A, I) = 1$ then $\\Spec(A) \\setminus V(I)$ is nonempty affine." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-cd} and", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-compact-h1-zero-covering}." ], "refs": [ "local-cohomology-lemma-cd", "coherent-lemma-quasi-compact-h1-zero-covering" ], "ref_ids": [ 9703, 3287 ] } ], "ref_ids": [] }, { "id": 9710, "type": "theorem", "label": "local-cohomology-lemma-cd-maximal", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cd-maximal", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring of dimension $d$.", "Then $H^d_\\mathfrak m(A)$ is nonzero and $\\text{cd}(A, \\mathfrak m) = d$." ], "refs": [], "proofs": [ { "contents": [ "By one of the characterizations of dimension, there exists", "an ideal of definition for $A$ generated by $d$ elements, see", "Algebra, Proposition \\ref{algebra-proposition-dimension}.", "Hence $\\text{cd}(A, \\mathfrak m) \\leq d$ by", "Lemma \\ref{lemma-bound-cd}. Thus $H^d_\\mathfrak m(A)$ is", "nonzero if and only if $\\text{cd}(A, \\mathfrak m) = d$ if and only if", "$\\text{cd}(A, \\mathfrak m) \\geq d$.", "\\medskip\\noindent", "Let $A \\to A^\\wedge$ be the map from $A$ to its completion.", "Observe that $A^\\wedge$ is a Noetherian local ring of the", "same dimension as $A$ with maximal ideal $\\mathfrak m A^\\wedge$.", "See Algebra, Lemmas", "\\ref{algebra-lemma-completion-Noetherian-Noetherian},", "\\ref{algebra-lemma-completion-complete}, and", "\\ref{algebra-lemma-completion-faithfully-flat} and", "More on Algebra, Lemma \\ref{more-algebra-lemma-completion-dimension}.", "By Lemma \\ref{lemma-cd-change-rings}", "it suffices to prove the lemma for $A^\\wedge$.", "\\medskip\\noindent", "By the previous paragraph we may assume that $A$ is", "a complete local ring. Then $A$ has a normalized dualizing complex", "$\\omega_A^\\bullet$ (Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-ubiquity-dualizing}).", "The local duality theorem (in the form of", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-special-case-local-duality})", "tells us $H^d_\\mathfrak m(A)$ is Matlis dual to", "$\\text{Ext}^{-d}(A, \\omega_A^\\bullet) = H^{-d}(\\omega_A^\\bullet)$", "which is nonzero for example by", "Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-nonvanishing-generically-local}." ], "refs": [ "algebra-proposition-dimension", "local-cohomology-lemma-bound-cd", "algebra-lemma-completion-Noetherian-Noetherian", "algebra-lemma-completion-complete", "algebra-lemma-completion-faithfully-flat", "more-algebra-lemma-completion-dimension", "local-cohomology-lemma-cd-change-rings", "dualizing-lemma-ubiquity-dualizing", "dualizing-lemma-special-case-local-duality", "dualizing-lemma-nonvanishing-generically-local" ], "ref_ids": [ 1411, 9704, 874, 872, 871, 10042, 9706, 2890, 2873, 2866 ] } ], "ref_ids": [] }, { "id": 9711, "type": "theorem", "label": "local-cohomology-lemma-cd-bound-dim-local", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cd-bound-dim-local", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "Let $I \\subset A$ be a proper ideal.", "Let $\\mathfrak p \\subset A$ be a prime ideal", "such that $V(\\mathfrak p) \\cap V(I) = \\{\\mathfrak m\\}$.", "Then $\\dim(A/\\mathfrak p) \\leq \\text{cd}(A, I)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-cd-change-rings} we have", "$\\text{cd}(A, I) \\geq \\text{cd}(A/\\mathfrak p, I(A/\\mathfrak p))$.", "Since $V(I) \\cap V(\\mathfrak p) = \\{\\mathfrak m\\}$ we have", "$\\text{cd}(A/\\mathfrak p, I(A/\\mathfrak p)) =", "\\text{cd}(A/\\mathfrak p, \\mathfrak m/\\mathfrak p)$.", "By Lemma \\ref{lemma-cd-maximal} this is equal to $\\dim(A/\\mathfrak p)$." ], "refs": [ "local-cohomology-lemma-cd-change-rings", "local-cohomology-lemma-cd-maximal" ], "ref_ids": [ 9706, 9710 ] } ], "ref_ids": [] }, { "id": 9712, "type": "theorem", "label": "local-cohomology-lemma-cd-blowup", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cd-blowup", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.", "Let $b : X' \\to X = \\Spec(A)$ be the blowing up of $I$.", "If the fibres of $b$ have dimension $\\leq d - 1$, then", "$\\text{cd}(A, I) \\leq d$." ], "refs": [], "proofs": [ { "contents": [ "Set $U = X \\setminus V(I)$. Denote $j : U \\to X'$ the canonical open", "immersion, see Divisors, Section \\ref{divisors-section-blowing-up}.", "Since the exceptional divisor is an effective Cartier divisor", "(Divisors, Lemma", "\\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor})", "we see that $j$ is affine, see", "Divisors, Lemma", "\\ref{divisors-lemma-complement-locally-principal-closed-subscheme}.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_U$-module.", "Then $R^pj_*\\mathcal{F} = 0$ for $p > 0$, see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-relative-affine-vanishing}.", "On the other hand, we have $R^qb_*(j_*\\mathcal{F}) = 0$ for", "$q \\geq d$ by Limits, Lemma", "\\ref{limits-lemma-higher-direct-images-zero-above-dimension-fibre}.", "Thus by the Leray spectral sequence", "(Cohomology, Lemma \\ref{cohomology-lemma-relative-Leray})", "we conclude that $R^n(b \\circ j)_*\\mathcal{F} = 0$ for", "$n \\geq d$. Thus $H^n(U, \\mathcal{F}) = 0$ for $n \\geq d$", "(by Cohomology, Lemma \\ref{cohomology-lemma-apply-Leray}).", "This means that $\\text{cd}(A, I) \\leq d$ by definition." ], "refs": [ "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "divisors-lemma-complement-locally-principal-closed-subscheme", "coherent-lemma-relative-affine-vanishing", "limits-lemma-higher-direct-images-zero-above-dimension-fibre", "cohomology-lemma-relative-Leray", "cohomology-lemma-apply-Leray" ], "ref_ids": [ 8054, 7928, 3283, 15109, 2073, 2071 ] } ], "ref_ids": [] }, { "id": 9713, "type": "theorem", "label": "local-cohomology-lemma-support", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-support", "contents": [ "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable", "under specialization. For an $A$-module $M$ the following are equivalent", "\\begin{enumerate}", "\\item $H^0_T(M) = M$, and", "\\item $\\text{Supp}(M) \\subset T$.", "\\end{enumerate}", "The category of such $A$-modules is a Serre subcategory", "of the category $A$-modules closed under direct sums." ], "refs": [], "proofs": [ { "contents": [ "The equivalence holds because the support of an element of $M$", "is contained in the support of $M$ and conversely the support of", "$M$ is the union of the supports of its elements.", "The category of these modules is a Serre subcategory", "(Homology, Definition \\ref{homology-definition-serre-subcategory})", "of $\\text{Mod}_A$ by", "Algebra, Lemma \\ref{algebra-lemma-support-quotient}.", "We omit the proof of the statement on direct sums." ], "refs": [ "homology-definition-serre-subcategory", "algebra-lemma-support-quotient" ], "ref_ids": [ 12146, 547 ] } ], "ref_ids": [] }, { "id": 9714, "type": "theorem", "label": "local-cohomology-lemma-adjoint", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-adjoint", "contents": [ "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$", "be a subset stable under specialization. The functor", "$RH^0_T$ is the right adjoint to the functor", "$D(\\text{Mod}_{A, T}) \\to D(A)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that the functor $H^0_T(-)$ is", "the right adjoint to the inclusion functor", "$\\text{Mod}_{A, T} \\to \\text{Mod}_A$, see", "Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors}." ], "refs": [ "derived-lemma-derived-adjoint-functors" ], "ref_ids": [ 1907 ] } ], "ref_ids": [] }, { "id": 9715, "type": "theorem", "label": "local-cohomology-lemma-adjoint-ext", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-adjoint-ext", "contents": [ "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$", "be a subset stable under specialization.", "For any object $K$ of $D(A)$ we have", "$$", "H^i(RH^0_T(K)) = \\colim_{Z \\subset T\\text{ closed}} H^i_Z(K)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $J^\\bullet$ be a K-injective complex representing $K$.", "By definition $RH^0_T$ is represented by the complex", "$$", "H^0_T(J^\\bullet) = \\colim H^0_Z(J^\\bullet)", "$$", "where the equality follows from our definition of $H^0_T$.", "Since filtered colimits are exact the cohomology of this", "complex in degree $i$ is", "$\\colim H^i(H^0_Z(J^\\bullet)) = \\colim H^i_Z(K)$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9716, "type": "theorem", "label": "local-cohomology-lemma-equal-plus", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-equal-plus", "contents": [ "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable", "under specialization. The functor $D^+(\\text{Mod}_{A, T}) \\to D^+_T(A)$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be an object of $\\text{Mod}_{A, T}$. Choose an embedding", "$M \\to J$ into an injective $A$-module. By", "Dualizing Complexes, Proposition", "\\ref{dualizing-proposition-structure-injectives-noetherian}", "the module $J$ is a direct sum of injective hulls of residue fields.", "Let $E$ be an injective hull of the residue field of $\\mathfrak p$.", "Since $E$ is $\\mathfrak p$-power torsion we see that", "$H^0_T(E) = 0$ if $\\mathfrak p \\not \\in T$ and", "$H^0_T(E) = E$ if $\\mathfrak p \\in T$.", "Thus $H^0_T(J)$ is injective as a direct sum of injective hulls", "(by the proposition) and we have an embedding $M \\to H^0_T(J)$.", "Thus every object $M$ of $\\text{Mod}_{A, T}$ has an injective resolution", "$M \\to J^\\bullet$ with $J^n$ also in $\\text{Mod}_{A, T}$. It follows", "that $RH^0_T(M) = M$.", "\\medskip\\noindent", "Next, suppose that $K \\in D_T^+(A)$. Then the spectral sequence", "$$", "R^qH^0_T(H^p(K)) \\Rightarrow R^{p + q}H^0_T(K)", "$$", "(Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor})", "converges and above we have seen that only the terms with $q = 0$", "are nonzero. Thus we see that $RH^0_T(K) \\to K$ is an isomorphism.", "Thus the functor $D^+(\\text{Mod}_{A, T}) \\to D^+_T(A)$", "is an equivalence with quasi-inverse given by $RH^0_T$." ], "refs": [ "dualizing-proposition-structure-injectives-noetherian", "derived-lemma-two-ss-complex-functor" ], "ref_ids": [ 2923, 1871 ] } ], "ref_ids": [] }, { "id": 9717, "type": "theorem", "label": "local-cohomology-lemma-equal-full", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-equal-full", "contents": [ "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable", "under specialization. If $\\dim(A) < \\infty$, then functor", "$D(\\text{Mod}_{A, T}) \\to D_T(A)$ is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Say $\\dim(A) = d$. Then we see that $H^i_Z(M) = 0$ for $i > d$", "for every closed subset $Z$ of $\\Spec(A)$, see", "Lemma \\ref{lemma-cd-dimension}.", "By Lemma \\ref{lemma-adjoint-ext} we find that $H^0_T$ has bounded", "cohomological dimension.", "\\medskip\\noindent", "Let $K \\in D_T(A)$. We claim that $RH^0_T(K) \\to K$ is an", "isomorphism. We know this is true when $K$ is bounded below, see", "Lemma \\ref{lemma-equal-plus}. However, since $H^0_T$ has bounded", "cohomological dimension, we see that the $i$th cohomology of", "$RH_T^0(K)$ only depends on $\\tau_{\\geq -d + i}K$ and we conclude.", "Thus $D(\\text{Mod}_{A, T}) \\to D_T(A)$ is an equivalence with", "quasi-inverse $RH^0_T$." ], "refs": [ "local-cohomology-lemma-cd-dimension", "local-cohomology-lemma-adjoint-ext", "local-cohomology-lemma-equal-plus" ], "ref_ids": [ 9708, 9715, 9716 ] } ], "ref_ids": [] }, { "id": 9718, "type": "theorem", "label": "local-cohomology-lemma-torsion-change-rings", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-torsion-change-rings", "contents": [ "Let $A \\to B$ be a flat homomorphism of Noetherian rings.", "Let $T \\subset \\Spec(A)$ be a subset stable under specialization.", "Let $T' \\subset \\Spec(B)$ be the inverse image of $T$.", "Then the canonical map", "$$", "R\\Gamma_T(K) \\otimes_A^\\mathbf{L} B", "\\longrightarrow", "R\\Gamma_{T'}(K \\otimes_A^\\mathbf{L} B)", "$$", "is an isomorphism for $K \\in D^+(A)$. If $A$ and $B$ have finite", "dimension, then this is true for $K \\in D(A)$." ], "refs": [], "proofs": [ { "contents": [ "From the map $R\\Gamma_T(K) \\to K$ we get a map", "$R\\Gamma_T(K) \\otimes_A^\\mathbf{L} B \\to K \\otimes_A^\\mathbf{L} B$.", "The cohomology modules of $R\\Gamma_T(K) \\otimes_A^\\mathbf{L} B$", "are supported on $T'$ and hence we get the arrow of the lemma.", "This arrow is an isomorphism if $T$ is a closed subset of $\\Spec(A)$ by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-torsion-change-rings}.", "Recall that $H^i_T(K)$ is the colimit of $H^i_Z(K)$ where $Z$ runs over", "the (directed set of) closed subsets of $T$, see", "Lemma \\ref{lemma-adjoint-ext}.", "Correspondingly", "$H^i_{T'}(K \\otimes_A^\\mathbf{L} B) =", "\\colim H^i_{Z'}(K \\otimes_A^\\mathbf{L} B)$ where $Z'$ is the inverse", "image of $Z$. Thus the result because $\\otimes_A B$ commutes", "with filtered colimits and there are no higher Tors." ], "refs": [ "dualizing-lemma-torsion-change-rings", "local-cohomology-lemma-adjoint-ext" ], "ref_ids": [ 2817, 9715 ] } ], "ref_ids": [] }, { "id": 9719, "type": "theorem", "label": "local-cohomology-lemma-local-cohomology-ss", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-local-cohomology-ss", "contents": [ "Let $A$ be a ring and let $T, T' \\subset \\Spec(A)$ subsets", "stable under specialization. For $K \\in D^+(A)$", "there is a spectral sequence", "$$", "E_2^{p, q} = H^p_T(H^p_{T'}(K)) \\Rightarrow H^{p + q}_{T \\cap T'}(K)", "$$", "as in Derived Categories, Lemma", "\\ref{derived-lemma-grothendieck-spectral-sequence}." ], "refs": [ "derived-lemma-grothendieck-spectral-sequence" ], "proofs": [ { "contents": [ "Let $E$ be an object of $D_{T \\cap T'}(A)$. Then we have", "$$", "\\Hom(E, R\\Gamma_T(R\\Gamma_{T'}(K))) =", "\\Hom(E, R\\Gamma_{T'}(K)) =", "\\Hom(E, K)", "$$", "The first equality by the adjointness property of $R\\Gamma_T$", "and the second by the adjointness property of $R\\Gamma_{T'}$.", "On the other hand, if $J^\\bullet$ is a bounded below complex", "of injectives representing $K$, then $H^0_{T'}(J^\\bullet)$", "is a complex of injective $A$-modules representing $R\\Gamma_{T'}(K)$", "and hence $H^0_T(H^0_{T'}(J^\\bullet))$ is a complex representing", "$R\\Gamma_T(R\\Gamma_{T'}(K))$. Thus $R\\Gamma_T(R\\Gamma_{T'}(K))$", "is an object of $D^+_{T \\cap T'}(A)$. Combining these two", "facts we find that $R\\Gamma_{T \\cap T'} = R\\Gamma_T \\circ R\\Gamma_{T'}$.", "This produces the spectral sequence by the lemma referenced", "in the statement." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1873 ] }, { "id": 9720, "type": "theorem", "label": "local-cohomology-lemma-torsion-tensor-product", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-torsion-tensor-product", "contents": [ "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset", "stable under specialization. Assume $A$ has finite dimension. Then", "$$", "R\\Gamma_T(K) = R\\Gamma_T(A) \\otimes_A^\\mathbf{L} K", "$$", "for $K \\in D(A)$. For $K, L \\in D(A)$ we have", "$$", "R\\Gamma_T(K \\otimes_A^\\mathbf{L} L) =", "K \\otimes_A^\\mathbf{L} R\\Gamma_T(L) =", "R\\Gamma_T(K) \\otimes_A^\\mathbf{L} L =", "R\\Gamma_T(K) \\otimes_A^\\mathbf{L} R\\Gamma_T(L)", "$$", "If $K$ or $L$ is in $D_T(A)$ then so is $K \\otimes_A^\\mathbf{L} L$." ], "refs": [], "proofs": [ { "contents": [ "By construction we may represent $R\\Gamma_T(A)$ by a complex $J^\\bullet$ in", "$\\text{Mod}_{A, T}$. Thus if we represent $K$ by a K-flat complex $K^\\bullet$", "then we see that $R\\Gamma_T(A) \\otimes_A^\\mathbf{L} K$ is represented", "by the complex $\\text{Tot}(J^\\bullet \\otimes_A K^\\bullet)$ in", "$\\text{Mod}_{A, T}$. Using the map $R\\Gamma_T(A) \\to A$ we obtain", "a map $R\\Gamma_T(A) \\otimes_A^\\mathbf{L} K\\to K$. Thus by the adjointness", "property of $R\\Gamma_T$ we obtain a canonical map", "$$", "R\\Gamma_T(A) \\otimes_A^\\mathbf{L} K \\longrightarrow R\\Gamma_T(K)", "$$", "factoring the just constructed map. Observe that $R\\Gamma_T$ commutes", "with direct sums in $D(A)$ for example by Lemma \\ref{lemma-adjoint-ext},", "the fact that directed colimits commute with direct sums, and the", "fact that usual local cohomology commutes with direct sums", "(for example by Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-local-cohomology-adjoint}).", "Thus by More on Algebra, Remark \\ref{more-algebra-remark-P-resolution}", "it suffices to check the map is an isomorphism for", "$K = A[k]$ where $k \\in \\mathbf{Z}$. This is clear.", "\\medskip\\noindent", "The final statements follow from the result we've just shown", "by transitivity of derived tensor products." ], "refs": [ "local-cohomology-lemma-adjoint-ext", "dualizing-lemma-local-cohomology-adjoint", "more-algebra-remark-P-resolution" ], "ref_ids": [ 9715, 2815, 10653 ] } ], "ref_ids": [] }, { "id": 9721, "type": "theorem", "label": "local-cohomology-lemma-filter-local-cohomology", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-filter-local-cohomology", "contents": [ "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$", "be a subset stable under specialization. Let $T' \\subset T$ be", "the set of nonminimal primes in $T$. Then $T'$", "is a subset of $\\Spec(A)$ stable under specialization", "and for every $A$-module $M$ there is an exact sequence", "$$", "0 \\to", "\\colim_{Z, f} H^1_f(H^{i - 1}_Z(M)) \\to", "H^i_{T'}(M) \\to H^i_T(M) \\to", "\\bigoplus\\nolimits_{\\mathfrak p \\in T \\setminus T'}", "H^i_{\\mathfrak p A_\\mathfrak p}(M_\\mathfrak p)", "$$", "where the colimit is over closed subsets $Z \\subset T$", "and $f \\in A$ with $V(f) \\cap Z \\subset T'$." ], "refs": [], "proofs": [ { "contents": [ "For every $Z$ and $f$ the spectral sequence of", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-local-cohomology-ss}", "degenerates to give short exact sequences", "$$", "0 \\to H^1_f(H^{i - 1}_Z(M)) \\to", "H^i_{Z \\cap V(f)}(M) \\to H^0_f(H^i_Z(M)) \\to 0", "$$", "We will use this without further mention below.", "\\medskip\\noindent", "Let $\\xi \\in H^i_T(M)$ map to zero in the direct sum.", "Then we first write $\\xi$ as the image of some $\\xi' \\in H^i_Z(M)$", "for some closed subset $Z \\subset T$, see Lemma \\ref{lemma-adjoint-ext}.", "Then $\\xi'$ maps to zero in $H^i_{\\mathfrak p A_\\mathfrak p}(M_\\mathfrak p)$", "for every $\\mathfrak p \\in Z$, $\\mathfrak p \\not \\in T'$.", "Since there are finitely many of these primes,", "we may choose $f \\in A$ not contained in any of these", "such that $f$ annihilates $\\xi'$. Then $\\xi'$", "is the image of some $\\xi'' \\in H^i_{Z'}(M)$", "where $Z' = Z \\cap V(f)$. By our choice of $f$ we have", "$Z' \\subset T'$ and we get exactness at the penultimate spot.", "\\medskip\\noindent", "Let $\\xi \\in H^i_{T'}(M)$ map to zero in $H^i_T(M)$.", "Choose closed subsets $Z' \\subset Z$ with $Z' \\subset T'$", "and $Z \\subset T$ such that $\\xi$ comes from $\\xi' \\in H^i_{Z'}(M)$", "and maps to zero in $H^i_Z(M)$. Then we can find $f \\in A$", "with $V(f) \\cap Z = Z'$ and we conclude." ], "refs": [ "dualizing-lemma-local-cohomology-ss", "local-cohomology-lemma-adjoint-ext" ], "ref_ids": [ 2820, 9715 ] } ], "ref_ids": [] }, { "id": 9722, "type": "theorem", "label": "local-cohomology-lemma-zero", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-zero", "contents": [ "Let $A$ be a Noetherian ring of finite dimension.", "Let $T \\subset \\Spec(A)$ be a subset stable under specialization.", "Let $\\{M_n\\}_{n \\geq 0}$ be an inverse system of $A$-modules.", "Let $i \\geq 0$ be an integer. Assume that for every $m$ there", "exists an integer $m'(m) \\geq m$ such that for all", "$\\mathfrak p \\in T$ the induced map", "$$", "H^i_{\\mathfrak p A_\\mathfrak p}(M_{k, \\mathfrak p})", "\\longrightarrow", "H^i_{\\mathfrak p A_\\mathfrak p}(M_{m, \\mathfrak p})", "$$", "is zero for $k \\geq m'(m)$. Let $m'' : \\mathbf{N} \\to \\mathbf{N}$", "be the $2^{\\dim(T)}$-fold self-composition of $m'$. Then the map", "$H^i_T(M_k) \\to H^i_T(M_m)$ is zero for all $k \\geq m''(m)$." ], "refs": [], "proofs": [ { "contents": [ "We first make a general remark: suppose we have an exact", "sequence", "$$", "(A_n) \\to (B_n) \\to (C_n)", "$$", "of inverse systems of abelian groups. Suppose that for every", "$m$ there exists an integer $m'(m) \\geq m$ such that", "$$", "A_k \\to A_m", "\\quad\\text{and}\\quad", "C_k \\to C_m", "$$", "are zero for $k \\geq m'(m)$. Then for $k \\geq m'(m'(m))$", "the map $B_k \\to B_m$ is zero.", "\\medskip\\noindent", "We will prove the lemma by induction on $\\dim(T)$ which is", "finite because $\\dim(A)$ is finite. Let $T' \\subset T$ be", "the set of nonminimal primes in $T$. Then $T'$", "is a subset of $\\Spec(A)$ stable under specialization", "and the hypotheses of the lemma apply to $T'$.", "Since $\\dim(T') < \\dim(T)$ we know the lemma holds for $T'$.", "For every $A$-module $M$ there is an exact sequence", "$$", "H^i_{T'}(M) \\to H^i_T(M) \\to", "\\bigoplus\\nolimits_{\\mathfrak p \\in T \\setminus T'}", "H^i_{\\mathfrak p A_\\mathfrak p}(M_\\mathfrak p)", "$$", "by Lemma \\ref{lemma-filter-local-cohomology}.", "Thus we conclude by the initial remark of the proof." ], "refs": [ "local-cohomology-lemma-filter-local-cohomology" ], "ref_ids": [ 9721 ] } ], "ref_ids": [] }, { "id": 9723, "type": "theorem", "label": "local-cohomology-lemma-essential-image", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-essential-image", "contents": [ "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset", "stable under specialization. Let $\\{M_n\\}_{n \\geq 0}$ be an inverse system", "of $A$-modules. Let $i \\geq 0$ be an integer. Assume the dimension of $A$", "is finite and that for every $m$ there exists an integer $m'(m) \\geq m$", "such that for all $\\mathfrak p \\in T$ we have", "\\begin{enumerate}", "\\item $H^{i - 1}_{\\mathfrak p A_\\mathfrak p}(M_{k, \\mathfrak p})", "\\to H^{i - 1}_{\\mathfrak p A_\\mathfrak p}(M_{m, \\mathfrak p})$", "is zero for $k \\geq m'(m)$, and", "\\item $ H^i_{\\mathfrak p A_\\mathfrak p}(M_{k, \\mathfrak p}) \\to", "H^i_{\\mathfrak p A_\\mathfrak p}(M_{m, \\mathfrak p})$", "has image $G(\\mathfrak p, m)$ independent of $k \\geq m'(m)$ and moreover", "$G(\\mathfrak p, m)$ maps injectively into", "$H^i_{\\mathfrak p A_\\mathfrak p}(M_{0, \\mathfrak p})$.", "\\end{enumerate}", "Then there exists an integer $m_0$ such that for every $m \\geq m_0$", "there exists an integer $m''(m) \\geq m$ such that", "for $k \\geq m''(m)$ the image of $H^i_T(M_k) \\to H^i_T(M_m)$", "maps injectively into $H^i_T(M_{m_0})$." ], "refs": [], "proofs": [ { "contents": [ "We first make a general remark: suppose we have an exact", "sequence", "$$", "(A_n) \\to (B_n) \\to (C_n) \\to (D_n)", "$$", "of inverse systems of abelian groups. Suppose that there exists", "an integer $m_0$ such that for every $m \\geq m_0$", "there exists an integer $m'(m) \\geq m$ such that the maps", "$$", "\\Im(B_k \\to B_m) \\longrightarrow B_{m_0}", "\\quad\\text{and}\\quad", "\\Im(D_k \\to D_m) \\longrightarrow D_{m_0}", "$$", "are injective for $k \\geq m'(m)$ and $A_k \\to A_m$ is zero", "for $k \\geq m'(m)$. Then for $m \\geq m'(m_0)$ and $k \\geq m'(m'(m))$", "the map", "$$", "\\Im(C_k \\to C_m) \\to C_{m'(m_0)}", "$$", "is injective. Namely, let $c_0 \\in C_m$ be the image of $c_3 \\in C_k$", "and say $c_0$ maps to zero in $C_{m'(m_0)}$. Picture", "$$", "C_k \\to C_{m'(m'(m))} \\to C_{m'(m)} \\to C_m \\to C_{m'(m_0)},\\quad", "c_3 \\mapsto c_2 \\mapsto c_1 \\mapsto c_0 \\mapsto 0", "$$", "We have to show $c_0 = 0$.", "The image $d_3$ of $c_3$ maps to zero in $C_{m_0}$ and hence", "we see that the image $d_1 \\in D_{m'(m)}$ is zero.", "Thus we can choose $b_1 \\in B_{m'(m)}$ mapping to", "the image $c_1$. Since $c_3$ maps to zero in", "$C_{m'(m_0)}$ we find an element $a_{-1} \\in A_{m'(m_0)}$", "which maps to the image $b_{-1} \\in B_{m'(m_0)}$ of $b_1$.", "Since $a_{-1}$ maps to zero in $A_{m_0}$ we conclude that", "$b_1$ maps to zero in $B_{m_0}$. Thus the image $b_0 \\in B_m$", "is zero which of course implies $c_0 = 0$ as desired.", "\\medskip\\noindent", "We will prove the lemma by induction on $\\dim(T)$ which is", "finite because $\\dim(A)$ is finite. Let $T' \\subset T$ be", "the set of nonminimal primes in $T$. Then $T'$ is a subset", "of $\\Spec(A)$ stable under specialization and the hypotheses", "of the lemma apply to $T'$. Since $\\dim(T') < \\dim(T)$ we know", "the lemma holds for $T'$. For every $A$-module $M$ there is an", "exact sequence", "$$", "0 \\to \\colim_{Z, f} H^1_f(H^{i - 1}_Z(M)) \\to", "H^i_{T'}(M) \\to H^i_T(M) \\to", "\\bigoplus\\nolimits_{\\mathfrak p \\in T \\setminus T'}", "H^i_{\\mathfrak p A_\\mathfrak p}(M_\\mathfrak p)", "$$", "by Lemma \\ref{lemma-filter-local-cohomology}.", "Thus we conclude by the initial remark of the proof", "and the fact that we've seen the system of groups", "$$", "\\left\\{\\colim_{Z, f} H^1_f(H^{i - 1}_Z(M_n))\\right\\}_{n \\geq 0}", "$$", "is pro-zero in Lemma \\ref{lemma-zero}; this uses that the function", "$m''(m)$ in that lemma for $H^{i - 1}_Z(M)$ is independent of $Z$." ], "refs": [ "local-cohomology-lemma-filter-local-cohomology", "local-cohomology-lemma-zero" ], "ref_ids": [ 9721, 9722 ] } ], "ref_ids": [] }, { "id": 9724, "type": "theorem", "label": "local-cohomology-lemma-check-finiteness-local-cohomology-by-annihilator", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-check-finiteness-local-cohomology-by-annihilator", "contents": [ "\\begin{reference}", "\\cite[Lemma 3]{Faltings-annulators}", "\\end{reference}", "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable", "under specialization. Let $M$ be a finite $A$-module. Let $n \\geq 0$.", "The following are equivalent", "\\begin{enumerate}", "\\item $H^i_T(M)$ is finite for $i \\leq n$,", "\\item there exists an ideal $J \\subset A$ with $V(J) \\subset T$", "such that $J$ annihilates $H^i_T(M)$ for $i \\leq n$.", "\\end{enumerate}", "If $T = V(I) = Z$ for an ideal $I \\subset A$, then these are also", "equivalent to", "\\begin{enumerate}", "\\item[(3)] there exists an $e \\geq 0$ such that $I^e$ annihilates", "$H^i_Z(M)$ for $i \\leq n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We prove the equivalence of (1) and (2) by induction on $n$.", "For $n = 0$ we have $H^0_T(M) \\subset M$ is finite. Hence (1) is true.", "Since $H^0_T(M) = \\colim H^0_{V(J)}(M)$ with $J$ as in (2) we see", "that (2) is true. Assume that $n > 0$.", "\\medskip\\noindent", "Assume (1) is true. Recall that $H^i_J(M) = H^i_{V(J)}(M)$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-local-cohomology-noetherian}.", "Thus $H^i_T(M) = \\colim H^i_J(M)$ where the colimit is over ideals", "$J \\subset A$ with $V(J) \\subset T$, see", "Lemma \\ref{lemma-adjoint-ext}. Since $H^i_T(M)$ is finitely generated", "for $i \\leq n$ we can find a $J \\subset A$ as in (2) such that", "$H^i_J(M) \\to H^i_T(M)$ is surjective for $i \\leq n$.", "Thus the finite list of generators are $J$-power torsion elements", "and we see that (2) holds with $J$ replaced by some power.", "\\medskip\\noindent", "Assume we have $J$ as in (2). Let $N = H^0_T(M)$ and $M' = M/N$.", "By construction of $R\\Gamma_T$ we find that", "$H^i_T(N) = 0$ for $i > 0$ and $H^0_T(N) = N$, see", "Remark \\ref{remark-upshot}. Thus we find that", "$H^0_T(M') = 0$ and $H^i_T(M') = H^i_T(M)$ for $i > 0$.", "We conclude that we may replace $M$ by $M'$.", "Thus we may assume that $H^0_T(M) = 0$.", "This means that the finite set of associated primes of $M$", "are not in $T$. By prime avoidance (Algebra, Lemma \\ref{algebra-lemma-silly})", "we can find $f \\in J$ not contained in any of the associated primes of $M$.", "Then the long exact local cohomology sequence associated to the short", "exact sequence", "$$", "0 \\to M \\to M \\to M/fM \\to 0", "$$", "turns into short exact sequences", "$$", "0 \\to H^i_T(M) \\to H^i_T(M/fM) \\to H^{i + 1}_T(M) \\to 0", "$$", "for $i < n$. We conclude that $J^2$ annihilates $H^i_T(M/fM)$", "for $i < n$. By induction hypothesis we see that $H^i_T(M/fM)$", "is finite for $i < n$. Using the short exact sequence once more", "we see that $H^{i + 1}_T(M)$ is finite for $i < n$ as desired.", "\\medskip\\noindent", "We omit the proof of the equivalence of (2) and (3)", "in case $T = V(I)$." ], "refs": [ "dualizing-lemma-local-cohomology-noetherian", "local-cohomology-lemma-adjoint-ext", "local-cohomology-remark-upshot", "algebra-lemma-silly" ], "ref_ids": [ 2823, 9715, 9794, 378 ] } ], "ref_ids": [] }, { "id": 9725, "type": "theorem", "label": "local-cohomology-lemma-check-finiteness-local-cohomology-locally", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-check-finiteness-local-cohomology-locally", "contents": [ "\\begin{reference}", "This is a special case of \\cite[Satz 1]{Faltings-finiteness}.", "\\end{reference}", "Let $A$ be a Noetherian ring, $I \\subset A$ an ideal, $M$ a finite", "$A$-module, and $n \\geq 0$ an integer. Let $Z = V(I)$.", "The following are equivalent", "\\begin{enumerate}", "\\item the modules $H^i_Z(M)$ are finite for $i \\leq n$, and", "\\item for all $\\mathfrak p \\in \\Spec(A)$ the modules", "$H^i_Z(M)_\\mathfrak p$, $i \\leq n$ are finite $A_\\mathfrak p$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) is immediate. We prove the converse", "by induction on $n$. The case $n = 0$ is clear because both (1) and", "(2) are always true in that case.", "\\medskip\\noindent", "Assume $n > 0$ and that (2) is true. Let $N = H^0_Z(M)$ and $M' = M/N$.", "By Dualizing Complexes, Lemma \\ref{dualizing-lemma-divide-by-torsion}", "we may replace $M$ by $M'$.", "Thus we may assume that $H^0_Z(M) = 0$.", "This means that $\\text{depth}_I(M) > 0$", "(Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth}).", "Pick $f \\in I$ a nonzerodivisor on $M$ and consider the short", "exact sequence", "$$", "0 \\to M \\to M \\to M/fM \\to 0", "$$", "which produces a long exact sequence", "$$", "0 \\to H^0_Z(M/fM) \\to H^1_Z(M) \\to H^1_Z(M) \\to H^1_Z(M/fM) \\to", "H^2_Z(M) \\to \\ldots", "$$", "and similarly after localization. Thus assumption (2) implies that", "the modules $H^i_Z(M/fM)_\\mathfrak p$ are finite for $i < n$. Hence", "by induction assumption $H^i_Z(M/fM)$ are finite for $i < n$.", "\\medskip\\noindent", "Let $\\mathfrak p$ be a prime of $A$ which is associated to", "$H^i_Z(M)$ for some $i \\leq n$. Say $\\mathfrak p$ is the annihilator", "of the element $x \\in H^i_Z(M)$. Then $\\mathfrak p \\in Z$, hence", "$f \\in \\mathfrak p$. Thus $fx = 0$ and hence $x$ comes from an", "element of $H^{i - 1}_Z(M/fM)$ by the boundary map $\\delta$ in the long", "exact sequence above. It follows that $\\mathfrak p$ is an associated", "prime of the finite module $\\Im(\\delta)$. We conclude that", "$\\text{Ass}(H^i_Z(M))$ is finite for $i \\leq n$, see", "Algebra, Lemma \\ref{algebra-lemma-finite-ass}.", "\\medskip\\noindent", "Recall that", "$$", "H^i_Z(M) \\subset", "\\prod\\nolimits_{\\mathfrak p \\in \\text{Ass}(H^i_Z(M))}", "H^i_Z(M)_\\mathfrak p", "$$", "by Algebra, Lemma \\ref{algebra-lemma-zero-at-ass-zero}. Since by", "assumption the modules on the right hand side are finite and $I$-power", "torsion, we can find integers $e_{\\mathfrak p, i} \\geq 0$, $i \\leq n$,", "$\\mathfrak p \\in \\text{Ass}(H^i_Z(M))$ such that", "$I^{e_{\\mathfrak p, i}}$ annihilates $H^i_Z(M)_\\mathfrak p$. We conclude", "that $I^e$ with $e = \\max\\{e_{\\mathfrak p, i}\\}$ annihilates $H^i_Z(M)$", "for $i \\leq n$. By", "Lemma \\ref{lemma-check-finiteness-local-cohomology-by-annihilator}", "we see that $H^i_Z(M)$ is finite for $i \\leq n$." ], "refs": [ "dualizing-lemma-divide-by-torsion", "dualizing-lemma-depth", "algebra-lemma-finite-ass", "algebra-lemma-zero-at-ass-zero", "local-cohomology-lemma-check-finiteness-local-cohomology-by-annihilator" ], "ref_ids": [ 2831, 2826, 701, 713, 9724 ] } ], "ref_ids": [] }, { "id": 9726, "type": "theorem", "label": "local-cohomology-lemma-annihilate-local-cohomology", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-annihilate-local-cohomology", "contents": [ "Let $A$ be a ring and let $J \\subset I \\subset A$ be finitely generated ideals.", "Let $i \\geq 0$ be an integer. Set $Z = V(I)$. If", "$H^i_Z(A)$ is annihilated by $J^n$ for some $n$, then", "$H^i_Z(M)$ annihilated by $J^m$ for some $m = m(M)$", "for every finitely presented $A$-module $M$ such that", "$M_f$ is a finite locally free $A_f$-module for all $f \\in I$." ], "refs": [], "proofs": [ { "contents": [ "Consider the annihilator $\\mathfrak a$ of $H^i_Z(M)$.", "Let $\\mathfrak p \\subset A$ with $\\mathfrak p \\not \\in Z$.", "By assumption there exists an $f \\in I$, $f \\not \\in \\mathfrak p$", "and an isomorphism $\\varphi : A_f^{\\oplus r} \\to M_f$", "of $A_f$-modules. Clearing denominators (and using that", "$M$ is of finite presentation) we find maps", "$$", "a : A^{\\oplus r} \\longrightarrow M", "\\quad\\text{and}\\quad", "b : M \\longrightarrow A^{\\oplus r}", "$$", "with $a_f = f^N \\varphi$ and $b_f = f^N \\varphi^{-1}$ for some $N$.", "Moreover we may assume that $a \\circ b$ and $b \\circ a$ are equal to", "multiplication by $f^{2N}$. Thus we see that $H^i_Z(M)$ is annihilated by", "$f^{2N}J^n$, i.e., $f^{2N}J^n \\subset \\mathfrak a$.", "\\medskip\\noindent", "As $U = \\Spec(A) \\setminus Z$ is quasi-compact we can find finitely many", "$f_1, \\ldots, f_t$ and $N_1, \\ldots, N_t$ such that $U = \\bigcup D(f_j)$ and", "$f_j^{2N_j}J^n \\subset \\mathfrak a$. Then $V(I) = V(f_1, \\ldots, f_t)$", "and since $I$ is finitely generated we conclude", "$I^M \\subset (f_1, \\ldots, f_t)$ for some $M$.", "All in all we see that $J^m \\subset \\mathfrak a$ for", "$m \\gg 0$, for example $m = M (2N_1 + \\ldots + 2N_t) n$ will do." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9727, "type": "theorem", "label": "local-cohomology-lemma-local-finiteness-for-finite-locally-free", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-local-finiteness-for-finite-locally-free", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Set $Z = V(I)$.", "Let $n \\geq 0$ be an integer. If $H^i_Z(A)$ is finite for $0 \\leq i \\leq n$,", "then the same is true for $H^i_Z(M)$, $0 \\leq i \\leq n$ for", "any finite $A$-module $M$ such that $M_f$ is a finite locally free", "$A_f$-module for all $f \\in I$." ], "refs": [], "proofs": [ { "contents": [ "The assumption that $H^i_Z(A)$ is finite for $0 \\leq i \\leq n$", "implies there exists an $e \\geq 0$ such that $I^e$ annihilates", "$H^i_Z(A)$ for $0 \\leq i \\leq n$, see", "Lemma \\ref{lemma-check-finiteness-local-cohomology-by-annihilator}.", "Then Lemma \\ref{lemma-annihilate-local-cohomology}", "implies that $H^i_Z(M)$, $0 \\leq i \\leq n$ is annihilated", "by $I^m$ for some $m = m(M, i)$. We may take the same $m$", "for all $0 \\leq i \\leq n$. Then", "Lemma \\ref{lemma-check-finiteness-local-cohomology-by-annihilator}", "implies that $H^i_Z(M)$ is finite for $0 \\leq i \\leq n$", "as desired." ], "refs": [ "local-cohomology-lemma-check-finiteness-local-cohomology-by-annihilator", "local-cohomology-lemma-annihilate-local-cohomology", "local-cohomology-lemma-check-finiteness-local-cohomology-by-annihilator" ], "ref_ids": [ 9724, 9726, 9724 ] } ], "ref_ids": [] }, { "id": 9728, "type": "theorem", "label": "local-cohomology-lemma-check-finiteness-pushforward-on-associated-points", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-check-finiteness-pushforward-on-associated-points", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $j : U \\to X$ be the inclusion", "of an open subscheme with complement $Z$. For $x \\in U$ let", "$i_x : W_x \\to U$ be the integral closed subscheme with generic point $x$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item for all $x \\in \\text{Ass}(\\mathcal{F})$ the", "$\\mathcal{O}_X$-module $j_*i_{x, *}\\mathcal{O}_{W_x}$ is coherent,", "\\item $j_*\\mathcal{F}$ is coherent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We first prove that (1) implies (2). Assume (1) holds.", "The statement is local on $X$, hence we may assume $X$ is affine.", "Then $U$ is quasi-compact, hence $\\text{Ass}(\\mathcal{F})$ is finite", "(Divisors, Lemma \\ref{divisors-lemma-finite-ass}). Thus we may argue by", "induction on the number of associated points. Let $x \\in U$ be a generic", "point of an irreducible component of the support of $\\mathcal{F}$.", "By Divisors, Lemma \\ref{divisors-lemma-finite-ass} we have", "$x \\in \\text{Ass}(\\mathcal{F})$. By our choice of $x$ we have", "$\\dim(\\mathcal{F}_x) = 0$ as $\\mathcal{O}_{X, x}$-module.", "Hence $\\mathcal{F}_x$ has finite length as an $\\mathcal{O}_{X, x}$-module", "(Algebra, Lemma \\ref{algebra-lemma-support-point}).", "Thus we may use induction on this length.", "\\medskip\\noindent", "Set $\\mathcal{G} = j_*i_{x, *}\\mathcal{O}_{W_x}$. This is a coherent", "$\\mathcal{O}_X$-module by assumption. We have $\\mathcal{G}_x = \\kappa(x)$.", "Choose a nonzero map", "$\\varphi_x : \\mathcal{F}_x \\to \\kappa(x) = \\mathcal{G}_x$.", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-map-stalks-local-map}", "there is an open $x \\in V \\subset U$ and a map", "$\\varphi_V : \\mathcal{F}|_V \\to \\mathcal{G}|_V$ whose stalk", "at $x$ is $\\varphi_x$. Choose $f \\in \\Gamma(X, \\mathcal{O}_X)$", "which does not vanish at $x$ such that $D(f) \\subset V$. By", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-homs-over-open}", "(for example) we see that $\\varphi_V$ extends to", "$f^n\\mathcal{F} \\to \\mathcal{G}|_U$ for some $n$.", "Precomposing with multiplication by $f^n$ we obtain a map", "$\\mathcal{F} \\to \\mathcal{G}|_U$ whose stalk at $x$ is nonzero.", "Let $\\mathcal{F}' \\subset \\mathcal{F}$ be the kernel.", "Note that $\\text{Ass}(\\mathcal{F}') \\subset \\text{Ass}(\\mathcal{F})$, see", "Divisors, Lemma \\ref{divisors-lemma-ses-ass}.", "Since", "$\\text{length}_{\\mathcal{O}_{X, x}}(\\mathcal{F}'_x) =", "\\text{length}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) - 1$", "we may apply the", "induction hypothesis to conclude $j_*\\mathcal{F}'$ is coherent.", "Since $\\mathcal{G} = j_*(\\mathcal{G}|_U) = j_*i_{x, *}\\mathcal{O}_{W_x}$", "is coherent, we can consider the exact sequence", "$$", "0 \\to j_*\\mathcal{F}' \\to j_*\\mathcal{F} \\to \\mathcal{G}", "$$", "By Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "the sheaf $j_*\\mathcal{F}$ is quasi-coherent.", "Hence the image of $j_*\\mathcal{F}$ in $j_*(\\mathcal{G}|_U)$", "is coherent by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient}.", "Finally, $j_*\\mathcal{F}$ is coherent by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-abelian-Noetherian}.", "\\medskip\\noindent", "Assume (2) holds. Exactly in the same manner as above we reduce", "to the case $X$ affine. We pick $x \\in \\text{Ass}(\\mathcal{F})$", "and we set $\\mathcal{G} = j_*i_{x, *}\\mathcal{O}_{W_x}$.", "Then we choose a nonzero map", "$\\varphi_x : \\mathcal{G}_x = \\kappa(x) \\to \\mathcal{F}_x$", "which exists exactly because $x$ is an associated point of $\\mathcal{F}$.", "Arguing exactly as above we may assume $\\varphi_x$", "extends to an $\\mathcal{O}_U$-module map", "$\\varphi : \\mathcal{G}|_U \\to \\mathcal{F}$.", "Then $\\varphi$ is injective (for example by", "Divisors, Lemma \\ref{divisors-lemma-check-injective-on-ass})", "and we find an injective map", "$\\mathcal{G} = j_*(\\mathcal{G}|_V) \\to j_*\\mathcal{F}$.", "Thus (1) holds." ], "refs": [ "divisors-lemma-finite-ass", "divisors-lemma-finite-ass", "algebra-lemma-support-point", "coherent-lemma-map-stalks-local-map", "coherent-lemma-homs-over-open", "divisors-lemma-ses-ass", "schemes-lemma-push-forward-quasi-coherent", "coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient", "coherent-lemma-coherent-abelian-Noetherian", "divisors-lemma-check-injective-on-ass" ], "ref_ids": [ 7859, 7859, 693, 3313, 3322, 7858, 7730, 3310, 3309, 7863 ] } ], "ref_ids": [] }, { "id": 9729, "type": "theorem", "label": "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "contents": [ "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.", "Set $X = \\Spec(A)$, $Z = V(I)$, $U = X \\setminus Z$, and $j : U \\to X$", "the inclusion morphism. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module.", "Then", "\\begin{enumerate}", "\\item there exists a finite $A$-module $M$ such that $\\mathcal{F}$ is the", "restriction of $\\widetilde{M}$ to $U$,", "\\item given $M$ there is an exact sequence", "$$", "0 \\to H^0_Z(M) \\to M \\to H^0(U, \\mathcal{F}) \\to H^1_Z(M) \\to 0", "$$", "and isomorphisms $H^p(U, \\mathcal{F}) = H^{p + 1}_Z(M)$ for $p \\geq 1$,", "\\item given $M$ and $p \\geq 0$ the following are equivalent", "\\begin{enumerate}", "\\item $R^pj_*\\mathcal{F}$ is coherent,", "\\item $H^p(U, \\mathcal{F})$ is a finite $A$-module,", "\\item $H^{p + 1}_Z(M)$ is a finite $A$-module,", "\\end{enumerate}", "\\item if the equivalent conditions in (3) hold for $p = 0$, we may take", "$M = \\Gamma(U, \\mathcal{F})$ in which case we have $H^0_Z(M) = H^1_Z(M) = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Properties, Lemma \\ref{properties-lemma-lift-finite-presentation}", "there exists a coherent $\\mathcal{O}_X$-module $\\mathcal{F}'$", "whose restriction to $U$ is isomorphic to $\\mathcal{F}$.", "Say $\\mathcal{F}'$ corresponds to the finite $A$-module $M$", "as in (1).", "Note that $R^pj_*\\mathcal{F}$ is quasi-coherent", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images})", "and corresponds to the $A$-module $H^p(U, \\mathcal{F})$.", "By Lemma \\ref{lemma-local-cohomology-is-local-cohomology}", "and the discussion in", "Cohomology, Sections \\ref{cohomology-section-cohomology-support} and", "\\ref{cohomology-section-cohomology-support-bis}", "we obtain an exact sequence", "$$", "0 \\to H^0_Z(M) \\to M \\to H^0(U, \\mathcal{F}) \\to H^1_Z(M) \\to 0", "$$", "and isomorphisms $H^p(U, \\mathcal{F}) = H^{p + 1}_Z(M)$ for $p \\geq 1$.", "Here we use that $H^j(X, \\mathcal{F}') = 0$ for $j > 0$ as $X$ is affine", "and $\\mathcal{F}'$ is quasi-coherent (Cohomology of Schemes,", "Lemma \\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}).", "This proves (2).", "Parts (3) and (4) are straightforward from (2); see also", "Lemma \\ref{lemma-local-cohomology}." ], "refs": [ "properties-lemma-lift-finite-presentation", "coherent-lemma-quasi-coherence-higher-direct-images", "local-cohomology-lemma-local-cohomology-is-local-cohomology", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "local-cohomology-lemma-local-cohomology" ], "ref_ids": [ 3022, 3295, 9695, 3282, 9696 ] } ], "ref_ids": [] }, { "id": 9730, "type": "theorem", "label": "local-cohomology-lemma-finiteness-pushforward", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-finiteness-pushforward", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $j : U \\to X$ be the inclusion of an", "open subscheme with complement $Z$. Let $\\mathcal{F}$ be a coherent", "$\\mathcal{O}_U$-module. Assume", "\\begin{enumerate}", "\\item $X$ is Nagata,", "\\item $X$ is universally catenary, and", "\\item for $x \\in \\text{Ass}(\\mathcal{F})$ and", "$z \\in Z \\cap \\overline{\\{x\\}}$ we have", "$\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) \\geq 2$.", "\\end{enumerate}", "Then $j_*\\mathcal{F}$ is coherent." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-check-finiteness-pushforward-on-associated-points}", "it suffices to prove $j_*i_{x, *}\\mathcal{O}_{W_x}$ is coherent", "for $x \\in \\text{Ass}(\\mathcal{F})$.", "Let $\\pi : Y \\to X$ be the normalization of $X$ in $\\Spec(\\kappa(x))$, see", "Morphisms, Section \\ref{morphisms-section-normalization}. By", "Morphisms, Lemma \\ref{morphisms-lemma-nagata-normalization-finite-general}", "the morphism $\\pi$ is finite. Since $\\pi$ is finite", "$\\mathcal{G} = \\pi_*\\mathcal{O}_Y$ is a coherent $\\mathcal{O}_X$-module by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-finite-pushforward-coherent}.", "Observe that $W_x = U \\cap \\pi(Y)$. Thus", "$\\pi|_{\\pi^{-1}(U)} : \\pi^{-1}(U) \\to U$ factors through $i_x : W_x \\to U$", "and we obtain a canonical map", "$$", "i_{x, *}\\mathcal{O}_{W_x}", "\\longrightarrow", "(\\pi|_{\\pi^{-1}(U)})_*(\\mathcal{O}_{\\pi^{-1}(U)}) =", "(\\pi_*\\mathcal{O}_Y)|_U = \\mathcal{G}|_U", "$$", "This map is injective (for example by Divisors, Lemma", "\\ref{divisors-lemma-check-injective-on-ass}). Hence", "$j_*i_{x, *}\\mathcal{O}_{W_x} \\subset j_*\\mathcal{G}|_U$", "and it suffices to show that $j_*\\mathcal{G}|_U$ is coherent.", "\\medskip\\noindent", "It remains to prove that $j_*(\\mathcal{G}|_U)$ is coherent. We claim", "Divisors, Lemma \\ref{divisors-lemma-depth-2-hartog}", "applies to", "$$", "\\mathcal{G} \\longrightarrow j_*(\\mathcal{G}|_U)", "$$", "which finishes the proof. It suffices to show that", "$\\text{depth}(\\mathcal{G}_z) \\geq 2$ for $z \\in Z$.", "Let $y_1, \\ldots, y_n \\in Y$ be the points mapping to $z$.", "By Algebra, Lemma \\ref{algebra-lemma-depth-goes-down-finite}", "it suffices to show that", "$\\text{depth}(\\mathcal{O}_{Y, y_i}) \\geq 2$ for $i = 1, \\ldots, n$.", "If not, then by Properties, Lemma \\ref{properties-lemma-criterion-normal}", "we see that $\\dim(\\mathcal{O}_{Y, y_i}) = 1$ for some $i$.", "This is impossible by the dimension formula", "(Morphisms, Lemma \\ref{morphisms-lemma-dimension-formula})", "for $\\pi : Y \\to \\overline{\\{x\\}}$ and assumption (3)." ], "refs": [ "local-cohomology-lemma-check-finiteness-pushforward-on-associated-points", "morphisms-lemma-nagata-normalization-finite-general", "coherent-lemma-finite-pushforward-coherent", "divisors-lemma-check-injective-on-ass", "divisors-lemma-depth-2-hartog", "algebra-lemma-depth-goes-down-finite", "properties-lemma-criterion-normal", "morphisms-lemma-dimension-formula" ], "ref_ids": [ 9728, 5509, 3316, 7863, 7881, 778, 2989, 5493 ] } ], "ref_ids": [] }, { "id": 9731, "type": "theorem", "label": "local-cohomology-lemma-sharp-finiteness-pushforward", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-sharp-finiteness-pushforward", "contents": [ "Let $X$ be an integral locally Noetherian scheme. Let $j : U \\to X$", "be the inclusion of a nonempty open subscheme with complement $Z$. Assume", "that for all $z \\in Z$ and any associated prime $\\mathfrak p$ of", "the completion $\\mathcal{O}_{X, z}^\\wedge$", "we have $\\dim(\\mathcal{O}_{X, z}^\\wedge/\\mathfrak p) \\geq 2$.", "Then $j_*\\mathcal{O}_U$ is coherent." ], "refs": [], "proofs": [ { "contents": [ "We may assume $X$ is affine.", "Using Lemmas \\ref{lemma-check-finiteness-local-cohomology-locally} and", "\\ref{lemma-finiteness-pushforwards-and-H1-local} we reduce to", "$X = \\Spec(A)$ where $(A, \\mathfrak m)$ is a Noetherian local domain", "and $\\mathfrak m \\in Z$.", "Then we can use induction on $d = \\dim(A)$.", "(The base case is $d = 0, 1$ which do not happen by", "our assumption on the local rings.)", "Set $V = \\Spec(A) \\setminus \\{\\mathfrak m\\}$.", "Observe that the local rings of $V$ have dimension strictly smaller than $d$.", "Repeating the arguments for $j' : U \\to V$ we", "and using induction we conclude that $j'_*\\mathcal{O}_U$ is", "a coherent $\\mathcal{O}_V$-module.", "Pick a nonzero $f \\in A$ which vanishes on $Z$.", "Since $D(f) \\cap V \\subset U$ we find an $n$ such that", "multiplication by $f^n$ on $U$ extends to a map", "$f^n : j'_*\\mathcal{O}_U \\to \\mathcal{O}_V$ over $V$", "(for example by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-homs-over-open}). This map is injective", "hence there is an injective map", "$$", "j_*\\mathcal{O}_U = j''_* j'_* \\mathcal{O}_U \\to j''_*\\mathcal{O}_V", "$$", "on $X$ where $j'' : V \\to X$ is the inclusion morphism.", "Hence it suffices to show that $j''_*\\mathcal{O}_V$ is coherent.", "In other words, we may assume that $X$ is the spectrum", "of a local Noetherian domain and that $Z$", "consists of the closed point.", "\\medskip\\noindent", "Assume $X = \\Spec(A)$ with $(A, \\mathfrak m)$ local and $Z = \\{\\mathfrak m\\}$.", "Let $A^\\wedge$ be the completion of $A$.", "Set $X^\\wedge = \\Spec(A^\\wedge)$, $Z^\\wedge = \\{\\mathfrak m^\\wedge\\}$,", "$U^\\wedge = X^\\wedge \\setminus Z^\\wedge$, and", "$\\mathcal{F}^\\wedge = \\mathcal{O}_{U^\\wedge}$.", "The ring $A^\\wedge$ is universally catenary and Nagata (Algebra, Remark", "\\ref{algebra-remark-Noetherian-complete-local-ring-universally-catenary} and", "Lemma \\ref{algebra-lemma-Noetherian-complete-local-Nagata}).", "Moreover, condition (3) of Lemma \\ref{lemma-finiteness-pushforward}", "for $X^\\wedge, Z^\\wedge, U^\\wedge, \\mathcal{F}^\\wedge$", "holds by assumption! Thus we see that", "$(U^\\wedge \\to X^\\wedge)_*\\mathcal{O}_{U^\\wedge}$", "is coherent. Since the morphism $c : X^\\wedge \\to X$", "is flat we conclude that the pullback of $j_*\\mathcal{O}_U$ is", "$(U^\\wedge \\to X^\\wedge)_*\\mathcal{O}_{U^\\wedge}$", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology}).", "Finally, since $c$ is faithfully flat we conclude that", "$j_*\\mathcal{O}_U$ is coherent by", "Descent, Lemma \\ref{descent-lemma-finite-type-descends}." ], "refs": [ "local-cohomology-lemma-check-finiteness-local-cohomology-locally", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "coherent-lemma-homs-over-open", "algebra-remark-Noetherian-complete-local-ring-universally-catenary", "algebra-lemma-Noetherian-complete-local-Nagata", "local-cohomology-lemma-finiteness-pushforward", "coherent-lemma-flat-base-change-cohomology", "descent-lemma-finite-type-descends" ], "ref_ids": [ 9725, 9729, 3322, 1582, 1353, 9730, 3298, 14612 ] } ], "ref_ids": [] }, { "id": 9732, "type": "theorem", "label": "local-cohomology-lemma-kollar-finiteness-H1-local", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-kollar-finiteness-H1-local", "contents": [ "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.", "Set $Z = V(I)$. Let $M$ be a finite $A$-module. The following", "are equivalent", "\\begin{enumerate}", "\\item $H^1_Z(M)$ is a finite $A$-module, and", "\\item for all $\\mathfrak p \\in \\text{Ass}(M)$, $\\mathfrak p \\not \\in Z$", "and all $\\mathfrak q \\in V(\\mathfrak p + I)$ the completion of", "$(A/\\mathfrak p)_\\mathfrak q$ does not have associated primes", "of dimension $1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Proposition \\ref{proposition-kollar}", "via Lemma \\ref{lemma-finiteness-pushforwards-and-H1-local}." ], "refs": [ "local-cohomology-proposition-kollar", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local" ], "ref_ids": [ 9785, 9729 ] } ], "ref_ids": [] }, { "id": 9733, "type": "theorem", "label": "local-cohomology-lemma-finiteness-pushforward-general", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-finiteness-pushforward-general", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $j : U \\to X$ be the inclusion of an", "open subscheme with complement $Z$. Let $\\mathcal{F}$ be a coherent", "$\\mathcal{O}_U$-module. Assume", "\\begin{enumerate}", "\\item $X$ is universally catenary,", "\\item for every $z \\in Z$ the formal fibres of $\\mathcal{O}_{X, z}$", "are $(S_1)$.", "\\end{enumerate}", "In this situation the following are equivalent", "\\begin{enumerate}", "\\item[(a)] for $x \\in \\text{Ass}(\\mathcal{F})$ and", "$z \\in Z \\cap \\overline{\\{x\\}}$ we have", "$\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) \\geq 2$, and", "\\item[(b)] $j_*\\mathcal{F}$ is coherent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in \\text{Ass}(\\mathcal{F})$. By Proposition \\ref{proposition-kollar}", "it suffices to check that $A = \\mathcal{O}_{\\overline{\\{x\\}}, z}$ satisfies", "the condition of the proposition on associated primes of its completion", "if and only if $\\dim(A) \\geq 2$.", "Observe that $A$ is universally catenary (this is clear)", "and that its formal fibres are $(S_1)$ as follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-formal-fibres-normal} and", "Proposition \\ref{more-algebra-proposition-finite-type-over-P-ring}.", "Let $\\mathfrak p' \\subset A^\\wedge$ be an associated prime.", "As $A \\to A^\\wedge$ is flat,", "by Algebra, Lemma \\ref{algebra-lemma-bourbaki},", "we find that $\\mathfrak p'$ lies over $(0) \\subset A$.", "The formal fibre $A^\\wedge \\otimes_A F$ is $(S_1)$ where $F$ is", "the fraction field of $A$. We conclude that $\\mathfrak p'$ is a", "minimal prime, see", "Algebra, Lemma \\ref{algebra-lemma-criterion-no-embedded-primes}.", "Since $A$ is universally catenary it is formally catenary", "by More on Algebra, Proposition \\ref{more-algebra-proposition-ratliff}.", "Hence $\\dim(A^\\wedge/\\mathfrak p') = \\dim(A)$ which", "proves the equivalence." ], "refs": [ "local-cohomology-proposition-kollar", "more-algebra-lemma-formal-fibres-normal", "more-algebra-proposition-finite-type-over-P-ring", "algebra-lemma-bourbaki", "algebra-lemma-criterion-no-embedded-primes", "more-algebra-proposition-ratliff" ], "ref_ids": [ 9785, 10102, 10583, 717, 1309, 10591 ] } ], "ref_ids": [] }, { "id": 9734, "type": "theorem", "label": "local-cohomology-lemma-ideal-depth-function", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-ideal-depth-function", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.", "Let $M$ be a finite $A$-module. Let $\\mathfrak p \\in V(I)$", "be a prime ideal. Assume", "$e = \\text{depth}_{IA_\\mathfrak p}(M_\\mathfrak p) < \\infty$.", "Then there exists a nonempty open $U \\subset V(\\mathfrak p)$", "such that $\\text{depth}_{IA_\\mathfrak q}(M_\\mathfrak q) \\geq e$", "for all $\\mathfrak q \\in U$." ], "refs": [], "proofs": [ { "contents": [ "By definition of depth we have $IM_\\mathfrak p \\not = M_\\mathfrak p$", "and there exists an $M_\\mathfrak p$-regular sequence", "$f_1, \\ldots, f_e \\in IA_\\mathfrak p$. After replacing $A$ by", "a principal localization we may assume $f_1, \\ldots, f_e \\in I$", "form an $M$-regular sequence, see", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-neighbourhood}.", "Consider the module $M' = M/IM$. Since $\\mathfrak p \\in \\text{Supp}(M')$", "and since the support of a finite module is closed, we find", "$V(\\mathfrak p) \\subset \\text{Supp}(M')$. Thus", "for $\\mathfrak q \\in V(\\mathfrak p)$ we get", "$IM_\\mathfrak q \\not = M_\\mathfrak q$. Hence, using that", "localization is exact, we see that", "$\\text{depth}_{IA_\\mathfrak q}(M_\\mathfrak q) \\geq e$", "for any $\\mathfrak q \\in V(I)$ by definition of depth." ], "refs": [ "algebra-lemma-regular-sequence-in-neighbourhood" ], "ref_ids": [ 741 ] } ], "ref_ids": [] }, { "id": 9735, "type": "theorem", "label": "local-cohomology-lemma-depth-function", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-depth-function", "contents": [ "Let $A$ be a Noetherian ring. Let $M$ be a finite $A$-module.", "Let $\\mathfrak p$ be a prime ideal. Assume", "$e = \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) < \\infty$.", "Then there exists a nonempty open $U \\subset V(\\mathfrak p)$", "such that $\\text{depth}_{A_\\mathfrak q}(M_\\mathfrak q) \\geq e$", "for all $\\mathfrak q \\in U$ and", "for all but finitely many $\\mathfrak q \\in U$ we have", "$\\text{depth}_{A_\\mathfrak q}(M_\\mathfrak q) > e$." ], "refs": [], "proofs": [ { "contents": [ "By definition of depth we have $\\mathfrak p M_\\mathfrak p \\not = M_\\mathfrak p$", "and there exists an $M_\\mathfrak p$-regular sequence", "$f_1, \\ldots, f_e \\in \\mathfrak p A_\\mathfrak p$. After replacing $A$ by", "a principal localization we may assume $f_1, \\ldots, f_e \\in \\mathfrak p$", "form an $M$-regular sequence, see", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-neighbourhood}.", "Consider the module $M' = M/(f_1, \\ldots, f_e)M$.", "Since $\\mathfrak p \\in \\text{Supp}(M')$", "and since the support of a finite module is closed, we find", "$V(\\mathfrak p) \\subset \\text{Supp}(M')$. Thus", "for $\\mathfrak q \\in V(\\mathfrak p)$ we get", "$\\mathfrak q M_\\mathfrak q \\not = M_\\mathfrak q$. Hence, using that", "localization is exact, we see that", "$\\text{depth}_{A_\\mathfrak q}(M_\\mathfrak q) \\geq e$", "for any $\\mathfrak q \\in V(I)$ by definition of depth.", "Moreover, as soon as $\\mathfrak q$ is not an associated", "prime of the module $M'$, then the depth goes up.", "Thus we see that the final statement holds by", "Algebra, Lemma \\ref{algebra-lemma-finite-ass}." ], "refs": [ "algebra-lemma-regular-sequence-in-neighbourhood", "algebra-lemma-finite-ass" ], "ref_ids": [ 741, 701 ] } ], "ref_ids": [] }, { "id": 9736, "type": "theorem", "label": "local-cohomology-lemma-finite-nr-points-next-S", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-finite-nr-points-next-S", "contents": [ "Let $X$ be a Noetherian scheme with dualizing complex $\\omega_X^\\bullet$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $k \\geq 0$", "be an integer. Assume $\\mathcal{F}$ is $(S_k)$.", "Then there is a finite number of points $x \\in X$ such that", "$$", "\\text{depth}(\\mathcal{F}_x) = k", "\\quad\\text{and}\\quad", "\\dim(\\text{Supp}(\\mathcal{F}_x)) > k", "$$" ], "refs": [], "proofs": [ { "contents": [ "We will prove this lemma by induction on $k$. The base case $k = 0$", "says that $\\mathcal{F}$ has a finite number of embedded associated points,", "which follows from Divisors, Lemma \\ref{divisors-lemma-finite-ass}.", "\\medskip\\noindent", "Assume $k > 0$ and the result holds for all smaller $k$.", "We can cover $X$ by finitely many affine opens, hence we may", "assume $X = \\Spec(A)$ is affine. Then $\\mathcal{F}$ is the", "coherent $\\mathcal{O}_X$-module associated to a finite $A$-module $M$", "which satisfies $(S_k)$. We will use", "Algebra, Lemmas \\ref{algebra-lemma-one-equation-module} and", "\\ref{algebra-lemma-depth-drops-by-one}", "without further mention.", "\\medskip\\noindent", "Let $f \\in A$ be a nonzerodivisor on $M$. Then $M/fM$ has $(S_{k - 1})$.", "By induction we see that there are finitely many", "primes $\\mathfrak p \\in V(f)$ with", "$\\text{depth}((M/fM)_\\mathfrak p) = k - 1$ and", "$\\dim(\\text{Supp}((M/fM)_\\mathfrak p)) > k - 1$.", "These are exactly the primes $\\mathfrak p \\in V(f)$ with", "$\\text{depth}(M_\\mathfrak p) = k$ and", "$\\dim(\\text{Supp}(M_\\mathfrak p)) > k$.", "Thus we may replace $A$ by $A_f$ and $M$ by $M_f$", "in trying to prove the finiteness statement.", "\\medskip\\noindent", "Since $M$ satisfies $(S_k)$ and $k > 0$ we see that $M$ has no", "embedded associated primes", "(Algebra, Lemma \\ref{algebra-lemma-criterion-no-embedded-primes}).", "Thus $\\text{Ass}(M)$ is the set of generic points of the support", "of $M$. Thus Dualizing Complexes, Lemma \\ref{dualizing-lemma-CM-open}", "shows the set", "$U = \\{\\mathfrak q \\mid M_\\mathfrak q\\text{ is Cohen-Macaulay}\\}$", "is an open containing $\\text{Ass}(M)$.", "By prime avoidance (Algebra, Lemma \\ref{algebra-lemma-silly})", "we can pick $f \\in A$ with", "$f \\not \\in \\mathfrak p$ for $\\mathfrak p \\in \\text{Ass}(M)$", "such that $D(f) \\subset U$.", "Then $f$ is a nonzerodivisor on $M$", "(Algebra, Lemma \\ref{algebra-lemma-ass-zero-divisors}).", "After replacing $A$ by $A_f$ and $M$ by $M_f$ (see above) we", "find that $M$ is Cohen-Macaulay.", "Thus for all $\\mathfrak q \\subset A$ we have", "$\\dim(M_\\mathfrak q) = \\text{depth}(M_\\mathfrak q)$", "and hence the set described in the lemma is empty", "and a fortiori finite." ], "refs": [ "divisors-lemma-finite-ass", "algebra-lemma-depth-drops-by-one", "algebra-lemma-criterion-no-embedded-primes", "dualizing-lemma-CM-open", "algebra-lemma-silly", "algebra-lemma-ass-zero-divisors" ], "ref_ids": [ 7859, 774, 1309, 2877, 378, 704 ] } ], "ref_ids": [] }, { "id": 9737, "type": "theorem", "label": "local-cohomology-lemma-sitting-in-degrees", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-sitting-in-degrees", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring with", "normalized dualizing complex $\\omega_A^\\bullet$.", "Let $M$ be a finite $A$-module.", "Set $E^i = \\text{Ext}_A^{-i}(M, \\omega_A^\\bullet)$.", "Then", "\\begin{enumerate}", "\\item $E^i$ is a finite $A$-module nonzero only for", "$0 \\leq i \\leq \\dim(\\text{Supp}(M))$,", "\\item $\\dim(\\text{Supp}(E^i)) \\leq i$,", "\\item $\\text{depth}(M)$ is the smallest integer $\\delta \\geq 0$ such that", "$E^\\delta \\not = 0$,", "\\item $\\mathfrak p \\in \\text{Supp}(E^0 \\oplus \\ldots \\oplus E^i)", "\\Leftrightarrow", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) \\leq i$,", "\\item the annihilator of $E^i$ is equal to the annihilator", "of $H^i_\\mathfrak m(M)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1), (2), and (3) are copies of the statements in", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-sitting-in-degrees}.", "For a prime $\\mathfrak p$ of $A$ we have that", "$(\\omega_A^\\bullet)_\\mathfrak p[-\\dim(A/\\mathfrak p)]$", "is a normalized dualzing complex for $A_\\mathfrak p$.", "See Dualizing Complexes, Lemma \\ref{dualizing-lemma-dimension-function}.", "Thus", "$$", "E^i_\\mathfrak p =", "\\text{Ext}^{-i}_A(M, \\omega_A^\\bullet)_\\mathfrak p =", "\\text{Ext}^{-i + \\dim(A/\\mathfrak p)}_{A_\\mathfrak p}", "(M_\\mathfrak p, (\\omega_A^\\bullet)_\\mathfrak p[-\\dim(A/\\mathfrak p)])", "$$", "is zero for", "$i - \\dim(A/\\mathfrak p) < \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p)$", "and nonzero for", "$i = \\dim(A/\\mathfrak p) + \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p)$", "by part (3) over $A_\\mathfrak p$.", "This proves part (4).", "If $E$ is an injective hull of the residue field of $A$, then we have", "$$", "\\Hom_A(H^i_\\mathfrak m(M), E) =", "\\text{Ext}^{-i}_A(M, \\omega_A^\\bullet)^\\wedge =", "(E^i)^\\wedge =", "E^i \\otimes_A A^\\wedge", "$$", "by the local duality theorem (in the form of", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-special-case-local-duality}).", "Since $A \\to A^\\wedge$ is faithfully flat, we find (5) is true by", "Matlis duality", "(Dualizing Complexes, Proposition \\ref{dualizing-proposition-matlis})." ], "refs": [ "dualizing-lemma-sitting-in-degrees", "dualizing-lemma-dimension-function", "dualizing-lemma-special-case-local-duality", "dualizing-proposition-matlis" ], "ref_ids": [ 2861, 2869, 2873, 2924 ] } ], "ref_ids": [] }, { "id": 9738, "type": "theorem", "label": "local-cohomology-lemma-kill-local-cohomology-at-prime", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-kill-local-cohomology-at-prime", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$.", "Let $M$ be a finite $A$-module, let $\\mathfrak p \\subset A$ be a prime", "ideal, and let $s \\geq 0$ be an integer. Assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex,", "\\item $\\mathfrak p \\not \\in V(I)$, and", "\\item for all primes $\\mathfrak p' \\subset \\mathfrak p$", "and $\\mathfrak q \\in V(I)$ with $\\mathfrak p' \\subset \\mathfrak q$ we have", "$$", "\\text{depth}_{A_{\\mathfrak p'}}(M_{\\mathfrak p'}) +", "\\dim((A/\\mathfrak p')_\\mathfrak q) > s", "$$", "\\end{enumerate}", "Then there exists an $f \\in A$, $f \\not \\in \\mathfrak p$ which annihilates", "$H^i_{V(I)}(M)$ for $i \\leq s$." ], "refs": [], "proofs": [ { "contents": [ "Consider the sets", "$$", "T = V(I)", "\\quad\\text{and}\\quad", "T' = \\bigcup\\nolimits_{f \\in A, f \\not \\in \\mathfrak p} V(f)", "$$", "These are subsets of $\\Spec(A)$ stable under specialization.", "Observe that $T \\subset T'$ and $\\mathfrak p \\not \\in T'$.", "Assumption (3) says that hypothesis (2) of", "Proposition \\ref{proposition-annihilator} holds.", "Hence we can find $J \\subset A$ with $V(J) \\subset T'$", "such that $J H^i_{V(I)}(M) = 0$ for $i \\leq s$.", "Choose $f \\in A$, $f \\not \\in \\mathfrak p$ with $V(J) \\subset V(f)$.", "A power of $f$ annihilates $H^i_{V(I)}(M)$ for $i \\leq s$." ], "refs": [ "local-cohomology-proposition-annihilator" ], "ref_ids": [ 9786 ] } ], "ref_ids": [] }, { "id": 9739, "type": "theorem", "label": "local-cohomology-lemma-cutoff", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cutoff", "contents": [ "Let $A \\to B$ be a finite homomorphism of Noetherian rings.", "Let $I \\subset A$ be an ideal and set $J = IB$. Let $M$ be", "a finite $B$-module. If $A$ is universally catenary, then", "$s_{B, J}(M) = s_{A, I}(M)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p \\subset \\mathfrak q \\subset A$ be primes with", "$I \\subset \\mathfrak q$ and $I \\not \\subset \\mathfrak p$.", "Since $A \\to B$ is finite there are finitely many primes", "$\\mathfrak p_i$ lying over $\\mathfrak p$. By", "Algebra, Lemma \\ref{algebra-lemma-depth-goes-down-finite}", "we have", "$$", "\\text{depth}(M_\\mathfrak p) = \\min \\text{depth}(M_{\\mathfrak p_i})", "$$", "Let $\\mathfrak p_i \\subset \\mathfrak q_{ij}$ be primes lying", "over $\\mathfrak q$. By going up for $A \\to B$", "(Algebra, Lemma \\ref{algebra-lemma-integral-going-up})", "there is at least one $\\mathfrak q_{ij}$ for each $i$.", "Then we see that", "$$", "\\dim((B/\\mathfrak p_i)_{\\mathfrak q_{ij}}) =", "\\dim((A/\\mathfrak p)_\\mathfrak q)", "$$", "by the dimension formula, see", "Algebra, Lemma \\ref{algebra-lemma-dimension-formula}.", "This implies that the minimum of the quantities", "used to define $s_{B, J}(M)$", "for the pairs $(\\mathfrak p_i, \\mathfrak q_{ij})$", "is equal to the quantity for the pair $(\\mathfrak p, \\mathfrak q)$.", "This proves the lemma." ], "refs": [ "algebra-lemma-depth-goes-down-finite", "algebra-lemma-integral-going-up", "algebra-lemma-dimension-formula" ], "ref_ids": [ 778, 500, 990 ] } ], "ref_ids": [] }, { "id": 9740, "type": "theorem", "label": "local-cohomology-lemma-change-completion", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-change-completion", "contents": [ "Let $A$ be a Noetherian ring which has a dualizing complex.", "Let $I \\subset A$ be an ideal.", "Let $M$ be a finite $A$-module. Let $A', M'$ be the $I$-adic", "completions of $A, M$. Let $\\mathfrak p' \\subset \\mathfrak q'$", "be prime ideals of $A'$ with $\\mathfrak q' \\in V(IA')$", "lying over $\\mathfrak p \\subset \\mathfrak q$ in $A$. Then", "$$", "\\text{depth}_{A_{\\mathfrak p'}}(M'_{\\mathfrak p'})", "\\geq", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p)", "$$", "and", "$$", "\\text{depth}_{A_{\\mathfrak p'}}(M'_{\\mathfrak p'}) +", "\\dim((A'/\\mathfrak p')_{\\mathfrak q'}) =", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q)", "$$" ], "refs": [], "proofs": [ { "contents": [ "We have", "$$", "\\text{depth}(M'_{\\mathfrak p'}) =", "\\text{depth}(M_\\mathfrak p) +", "\\text{depth}(A'_{\\mathfrak p'}/\\mathfrak p A'_{\\mathfrak p'})", "\\geq \\text{depth}(M_\\mathfrak p)", "$$", "by flatness of $A \\to A'$, see", "Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck-module}.", "Since the fibres of $A \\to A'$ are Cohen-Macaulay", "(Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-dualizing-gorenstein-formal-fibres} and", "More on Algebra, Section", "\\ref{more-algebra-section-properties-formal-fibres})", "we see that", "$\\text{depth}(A'_{\\mathfrak p'}/\\mathfrak p A'_{\\mathfrak p'}) =", "\\dim(A'_{\\mathfrak p'}/\\mathfrak p A'_{\\mathfrak p'})$.", "Thus we obtain", "\\begin{align*}", "\\text{depth}(M'_{\\mathfrak p'}) +", "\\dim((A'/\\mathfrak p')_{\\mathfrak q'})", "& =", "\\text{depth}(M_\\mathfrak p) +", "\\dim(A'_{\\mathfrak p'}/\\mathfrak p A'_{\\mathfrak p'}) +", "\\dim((A'/\\mathfrak p')_{\\mathfrak q'}) \\\\", "& =", "\\text{depth}(M_\\mathfrak p) +", "\\dim((A'/\\mathfrak p A')_{\\mathfrak q'}) \\\\", "& =", "\\text{depth}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q)", "\\end{align*}", "Second equality because $A'$ is catenary and third equality by", "More on Algebra, Lemma \\ref{more-algebra-lemma-completion-dimension}", "as $(A/\\mathfrak p)_\\mathfrak q$ and $(A'/\\mathfrak p A')_{\\mathfrak q'}$", "have the same $I$-adic completions." ], "refs": [ "algebra-lemma-apply-grothendieck-module", "dualizing-lemma-dualizing-gorenstein-formal-fibres", "more-algebra-lemma-completion-dimension" ], "ref_ids": [ 1360, 2892, 10042 ] } ], "ref_ids": [] }, { "id": 9741, "type": "theorem", "label": "local-cohomology-lemma-cutoff-completion", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cutoff-completion", "contents": [ "Let $A$ be a universally catenary Noetherian local ring.", "Let $I \\subset A$ be an ideal. Let $M$ be", "a finite $A$-module. Then", "$$", "s_{A, I}(M) \\geq s_{A^\\wedge, I^\\wedge}(M^\\wedge)", "$$", "If the formal fibres of $A$ are $(S_n)$, then", "$\\min(n + 1, s_{A, I}(M)) \\leq s_{A^\\wedge, I^\\wedge}(M^\\wedge)$." ], "refs": [], "proofs": [ { "contents": [ "Write $X = \\Spec(A)$, $X^\\wedge = \\Spec(A^\\wedge)$, $Z = V(I) \\subset X$, and", "$Z^\\wedge = V(I^\\wedge)$.", "Let $\\mathfrak p' \\subset \\mathfrak q' \\subset A^\\wedge$", "be primes with $\\mathfrak p' \\not \\in Z^\\wedge$ and", "$\\mathfrak q' \\in Z^\\wedge$. Let $\\mathfrak p \\subset \\mathfrak q$", "be the corresponding primes of $A$. Then $\\mathfrak p \\not \\in Z$", "and $\\mathfrak q \\in Z$. Picture", "$$", "\\xymatrix{", "\\mathfrak p' \\ar[r] & \\mathfrak q' \\ar[r] & A^\\wedge \\\\", "\\mathfrak p \\ar[r] \\ar@{-}[u] &", "\\mathfrak q \\ar[r] \\ar@{-}[u] & A \\ar[u]", "}", "$$", "Let us write", "\\begin{align*}", "a & = \\dim(A/\\mathfrak p) = \\dim(A^\\wedge/\\mathfrak pA^\\wedge),\\\\", "b & = \\dim(A/\\mathfrak q) = \\dim(A^\\wedge/\\mathfrak qA^\\wedge),\\\\", "a' & = \\dim(A^\\wedge/\\mathfrak p'),\\\\", "b' & = \\dim(A^\\wedge/\\mathfrak q')", "\\end{align*}", "Equalities by", "More on Algebra, Lemma \\ref{more-algebra-lemma-completion-dimension}.", "We also write", "\\begin{align*}", "p & = \\dim(A^\\wedge_{\\mathfrak p'}/\\mathfrak p A^\\wedge_{\\mathfrak p'}) =", "\\dim((A^\\wedge/\\mathfrak p A^\\wedge)_{\\mathfrak p'}) \\\\", "q & = \\dim(A^\\wedge_{\\mathfrak q'}/\\mathfrak p A^\\wedge_{\\mathfrak q'}) =", "\\dim((A^\\wedge/\\mathfrak q A^\\wedge)_{\\mathfrak q'})", "\\end{align*}", "Since $A$ is universally catenary we see that", "$A^\\wedge/\\mathfrak pA^\\wedge = (A/\\mathfrak p)^\\wedge$", "is equidimensional of dimension $a$", "(More on Algebra, Proposition \\ref{more-algebra-proposition-ratliff}).", "Hence $a = a' + p$. Similarly $b = b' + q$.", "By Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck-module}", "applied to the flat local ring map", "$A_\\mathfrak p \\to A^\\wedge_{\\mathfrak p'}$", "we have", "$$", "\\text{depth}(M^\\wedge_{\\mathfrak p'})", "=", "\\text{depth}(M_\\mathfrak p) +", "\\text{depth}(A^\\wedge_{\\mathfrak p'} / \\mathfrak p A^\\wedge_{\\mathfrak p'})", "$$", "The quantity we are minimizing for $s_{A, I}(M)$ is", "$$", "s(\\mathfrak p, \\mathfrak q) =", "\\text{depth}(M_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) =", "\\text{depth}(M_\\mathfrak p) + a - b", "$$", "(last equality as $A$ is catenary). The quantity we are minimizing", "for $s_{A^\\wedge, I^\\wedge}(M^\\wedge)$", "is", "$$", "s(\\mathfrak p', \\mathfrak q') =", "\\text{depth}(M^\\wedge_{\\mathfrak p'}) +", "\\dim((A^\\wedge/\\mathfrak p')_{\\mathfrak q'}) =", "\\text{depth}(M^\\wedge_{\\mathfrak p'}) + a' - b'", "$$", "(last equality as $A^\\wedge$ is catenary).", "Now we have enough notation in place to start the proof.", "\\medskip\\noindent", "Let $\\mathfrak p \\subset \\mathfrak q \\subset A$ be primes", "with $\\mathfrak p \\not \\in Z$ and $\\mathfrak q \\in Z$ such that", "$s_{A, I}(M) = s(\\mathfrak p, \\mathfrak q)$.", "Then we can pick $\\mathfrak q'$ minimal over $\\mathfrak q A^\\wedge$", "and $\\mathfrak p' \\subset \\mathfrak q'$ minimal over", "$\\mathfrak p A^\\wedge$ (using going down for $A \\to A^\\wedge$).", "Then we have four primes as above with $p = 0$ and $q = 0$.", "Moreover, we have", "$\\text{depth}(A^\\wedge_{\\mathfrak p'} / \\mathfrak p A^\\wedge_{\\mathfrak p'})=0$", "also because $p = 0$. This means that", "$s(\\mathfrak p', \\mathfrak q') = s(\\mathfrak p, \\mathfrak q)$.", "Thus we get the first inequality.", "\\medskip\\noindent", "Assume that the formal fibres of $A$ are $(S_n)$. Then", "$\\text{depth}(A^\\wedge_{\\mathfrak p'} / \\mathfrak p A^\\wedge_{\\mathfrak p'})", "\\geq \\min(n, p)$.", "Hence", "$$", "s(\\mathfrak p', \\mathfrak q') \\geq", "s(\\mathfrak p, \\mathfrak q) + q + \\min(n, p) - p \\geq", "s_{A, I}(M) + q + \\min(n, p) - p", "$$", "Thus the only way we can get in trouble is if $p > n$. If this happens", "then", "\\begin{align*}", "s(\\mathfrak p', \\mathfrak q')", "& =", "\\text{depth}(M^\\wedge_{\\mathfrak p'}) +", "\\dim((A^\\wedge/\\mathfrak p')_{\\mathfrak q'}) \\\\", "& =", "\\text{depth}(M_\\mathfrak p) +", "\\text{depth}(A^\\wedge_{\\mathfrak p'} / \\mathfrak p A^\\wedge_{\\mathfrak p'}) +", "\\dim((A^\\wedge/\\mathfrak p')_{\\mathfrak q'}) \\\\", "& \\geq", "0 + n + 1", "\\end{align*}", "because $(A^\\wedge/\\mathfrak p')_{\\mathfrak q'}$ has at least two primes.", "This proves the second inequality." ], "refs": [ "more-algebra-lemma-completion-dimension", "more-algebra-proposition-ratliff", "algebra-lemma-apply-grothendieck-module" ], "ref_ids": [ 10042, 10591, 1360 ] } ], "ref_ids": [] }, { "id": 9742, "type": "theorem", "label": "local-cohomology-lemma-local-annihilator", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-local-annihilator", "contents": [ "\\begin{reference}", "This is a special case of", "\\cite[Satz 1]{Faltings-annulators}.", "\\end{reference}", "Let $A$ be a Gorenstein Noetherian local ring. Let $I \\subset A$", "be an ideal and set $Z = V(I) \\subset \\Spec(A)$.", "Let $M$ be a finite $A$-module. Let $s = s_{A, I}(M)$ as in", "(\\ref{equation-cutoff}). Then $H^i_Z(M)$ is finite for $i < s$,", "but $H^s_Z(M)$ is not finite." ], "refs": [], "proofs": [ { "contents": [ "Since a Gorenstein local ring has a dualizing complex,", "this is a special case of Proposition \\ref{proposition-finiteness}.", "It would be helpful to have a short proof of this special case,", "which will be used in the proof of a general finiteness theorem below." ], "refs": [ "local-cohomology-proposition-finiteness" ], "ref_ids": [ 9787 ] } ], "ref_ids": [] }, { "id": 9743, "type": "theorem", "label": "local-cohomology-lemma-finiteness-Rjstar", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-finiteness-Rjstar", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $j : U \\to X$ be the inclusion", "of an open subscheme with complement $Z$. Let $\\mathcal{F}$ be a coherent", "$\\mathcal{O}_U$-module. Let $n \\geq 0$ be an integer. Assume", "\\begin{enumerate}", "\\item $X$ is universally catenary,", "\\item for every $z \\in Z$ the formal fibres of", "$\\mathcal{O}_{X, z}$ are $(S_n)$.", "\\end{enumerate}", "In this situation the following are equivalent", "\\begin{enumerate}", "\\item[(a)] for $x \\in \\text{Supp}(\\mathcal{F})$ and", "$z \\in Z \\cap \\overline{\\{x\\}}$ we have", "$\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) +", "\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > n$,", "\\item[(b)] $R^pj_*\\mathcal{F}$ is coherent for $0 \\leq p < n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The statement is local on $X$, hence we may assume $X$ is affine.", "Say $X = \\Spec(A)$ and $Z = V(I)$. Let $M$ be a finite $A$-module", "whose associated coherent $\\mathcal{O}_X$-module restricts", "to $\\mathcal{F}$ over $U$, see", "Lemma \\ref{lemma-finiteness-pushforwards-and-H1-local}.", "This lemma also tells us that $R^pj_*\\mathcal{F}$ is coherent", "if and only if $H^{p + 1}_Z(M)$ is a finite $A$-module.", "Observe that the minimum of the expressions", "$\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) +", "\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z})$", "is the number $s_{A, I}(M)$ of (\\ref{equation-cutoff}).", "Having said this the lemma follows from", "Theorem \\ref{theorem-finiteness}", "as elucidated by Remark \\ref{remark-astute-reader}." ], "refs": [ "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "local-cohomology-theorem-finiteness", "local-cohomology-remark-astute-reader" ], "ref_ids": [ 9729, 9694, 9797 ] } ], "ref_ids": [] }, { "id": 9744, "type": "theorem", "label": "local-cohomology-lemma-finiteness-for-finite-locally-free", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-finiteness-for-finite-locally-free", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $j : U \\to X$ be the inclusion", "of an open subscheme with complement $Z$. Let $n \\geq 0$ be an integer.", "If $R^pj_*\\mathcal{O}_U$ is coherent for $0 \\leq p < n$, then", "the same is true for $R^pj_*\\mathcal{F}$, $0 \\leq p < n$", "for any finite locally free $\\mathcal{O}_U$-module $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$, hence we may assume $X$ is affine.", "Say $X = \\Spec(A)$ and $Z = V(I)$. Via", "Lemma \\ref{lemma-finiteness-pushforwards-and-H1-local}", "our lemma follows from", "Lemma \\ref{lemma-local-finiteness-for-finite-locally-free}." ], "refs": [ "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "local-cohomology-lemma-local-finiteness-for-finite-locally-free" ], "ref_ids": [ 9729, 9727 ] } ], "ref_ids": [] }, { "id": 9745, "type": "theorem", "label": "local-cohomology-lemma-annihilate-Hp", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-annihilate-Hp", "contents": [ "\\begin{reference}", "\\cite[Lemma 1.9]{Bhatt-local}", "\\end{reference}", "Let $A$ be a ring and let $J \\subset I \\subset A$ be finitely generated ideals.", "Let $p \\geq 0$ be an integer. Set $U = \\Spec(A) \\setminus V(I)$. If", "$H^p(U, \\mathcal{O}_U)$ is annihilated by $J^n$ for some $n$, then", "$H^p(U, \\mathcal{F})$ annihilated by $J^m$ for some $m = m(\\mathcal{F})$", "for every finite locally free $\\mathcal{O}_U$-module $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the annihilator $\\mathfrak a$ of $H^p(U, \\mathcal{F})$.", "Let $u \\in U$. There exists an open neighbourhood $u \\in U' \\subset U$", "and an isomorphism", "$\\varphi : \\mathcal{O}_{U'}^{\\oplus r} \\to \\mathcal{F}|_{U'}$.", "Pick $f \\in A$ such that $u \\in D(f) \\subset U'$.", "There exist maps", "$$", "a : \\mathcal{O}_U^{\\oplus r} \\longrightarrow \\mathcal{F}", "\\quad\\text{and}\\quad", "b : \\mathcal{F} \\longrightarrow \\mathcal{O}_U^{\\oplus r}", "$$", "whose restriction to $D(f)$ are equal to $f^N \\varphi$", "and $f^N \\varphi^{-1}$ for some $N$. Moreover we may assume that", "$a \\circ b$ and $b \\circ a$ are equal to multiplication by $f^{2N}$.", "This follows from Properties, Lemma", "\\ref{properties-lemma-section-maps-backwards}", "since $U$ is quasi-compact ($I$ is finitely generated), separated, and", "$\\mathcal{F}$ and $\\mathcal{O}_U^{\\oplus r}$ are finitely presented.", "Thus we see that $H^p(U, \\mathcal{F})$ is annihilated by", "$f^{2N}J^n$, i.e., $f^{2N}J^n \\subset \\mathfrak a$.", "\\medskip\\noindent", "As $U$ is quasi-compact we can find finitely many $f_1, \\ldots, f_t$", "and $N_1, \\ldots, N_t$ such that $U = \\bigcup D(f_i)$ and", "$f_i^{2N_i}J^n \\subset \\mathfrak a$. Then $V(I) = V(f_1, \\ldots, f_t)$", "and since $I$ is finitely generated we conclude", "$I^M \\subset (f_1, \\ldots, f_t)$ for some $M$.", "All in all we see that $J^m \\subset \\mathfrak a$ for", "$m \\gg 0$, for example $m = M (2N_1 + \\ldots + 2N_t) n$ will do." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9746, "type": "theorem", "label": "local-cohomology-lemma-check-finiteness-local-cohomology-by-annihilator-complex", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-check-finiteness-local-cohomology-by-annihilator-complex", "contents": [ "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable", "under specialization. Let $K$ be an object of $D_{\\textit{Coh}}^+(A)$.", "Let $n \\in \\mathbf{Z}$. The following are equivalent", "\\begin{enumerate}", "\\item $H^i_T(K)$ is finite for $i \\leq n$,", "\\item there exists an ideal $J \\subset A$ with $V(J) \\subset T$", "such that $J$ annihilates $H^i_T(K)$ for $i \\leq n$.", "\\end{enumerate}", "If $T = V(I) = Z$ for an ideal $I \\subset A$, then these are also", "equivalent to", "\\begin{enumerate}", "\\item[(3)] there exists an $e \\geq 0$ such that $I^e$ annihilates", "$H^i_Z(K)$ for $i \\leq n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma is the natural generalization of", "Lemma \\ref{lemma-check-finiteness-local-cohomology-by-annihilator}", "whose proof the reader should read first.", "Assume (1) is true. Recall that $H^i_J(K) = H^i_{V(J)}(K)$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-local-cohomology-noetherian}.", "Thus $H^i_T(K) = \\colim H^i_J(K)$ where the colimit is over ideals", "$J \\subset A$ with $V(J) \\subset T$, see", "Lemma \\ref{lemma-adjoint-ext}. Since $H^i_T(K)$ is finitely generated", "for $i \\leq n$ we can find a $J \\subset A$ as in (2) such that", "$H^i_J(K) \\to H^i_T(K)$ is surjective for $i \\leq n$.", "Thus the finite list of generators are $J$-power torsion elements", "and we see that (2) holds with $J$ replaced by some power.", "\\medskip\\noindent", "Let $a \\in \\mathbf{Z}$ be an integer such that $H^i(K) = 0$ for $i < a$.", "We prove (2) $\\Rightarrow$ (1) by descending induction on $a$.", "If $a > n$, then we have $H^i_T(K) = 0$ for $i \\leq n$ hence both", "(1) and (2) are true and there is nothing to prove.", "\\medskip\\noindent", "Assume we have $J$ as in (2). Observe that $N = H^a_T(K) = H^0_T(H^a(K))$", "is finite as a submodule of the finite $A$-module $H^a(K)$.", "If $n = a$ we are done; so assume $a < n$ from now on. By construction of", "$R\\Gamma_T$ we find that $H^i_T(N) = 0$ for $i > 0$ and $H^0_T(N) = N$, see", "Remark \\ref{remark-upshot}. Choose a distinguished triangle", "$$", "N[-a] \\to K \\to K' \\to N[-a + 1]", "$$", "Then we see that $H^a_T(K') = 0$ and $H^i_T(K) = H^i_T(K')$ for $i > a$.", "We conclude that we may replace $K$ by $K'$. Thus we may assume that", "$H^a_T(K) = 0$. This means that the finite set of associated primes of", "$H^a(K)$ are not in $T$. By prime avoidance", "(Algebra, Lemma \\ref{algebra-lemma-silly}) we can find $f \\in J$", "not contained in any of the associated primes of $H^a(K)$.", "Choose a distinguished triangle", "$$", "L \\to K \\xrightarrow{f} K \\to L[1]", "$$", "By construction we see that $H^i(L) = 0$ for $i \\leq a$.", "On the other hand we have a long exact cohomology sequence", "$$", "0 \\to H^{a + 1}_T(L) \\to H^{a + 1}_T(K) \\xrightarrow{f}", "H^{a + 1}_T(K) \\to H^{a + 2}_T(L) \\to H^{a + 2}_T(K) \\xrightarrow{f} \\ldots", "$$", "which breaks into the identification $H^{a + 1}_T(L) = H^{a + 1}_T(K)$", "and short exact sequences", "$$", "0 \\to H^{i - 1}_T(K) \\to H^i_T(L) \\to H^i_T(K) \\to 0", "$$", "for $i \\leq n$ since $f \\in J$.", "We conclude that $J^2$ annihilates $H^i_T(L)$ for $i \\leq n$.", "By induction hypothesis applied to $L$ we see that $H^i_T(L)$", "is finite for $i \\leq n$. Using the short exact sequence once more", "we see that $H^i_T(K)$ is finite for $i \\leq n$ as desired.", "\\medskip\\noindent", "We omit the proof of the equivalence of (2) and (3)", "in case $T = V(I)$." ], "refs": [ "local-cohomology-lemma-check-finiteness-local-cohomology-by-annihilator", "dualizing-lemma-local-cohomology-noetherian", "local-cohomology-lemma-adjoint-ext", "local-cohomology-remark-upshot", "algebra-lemma-silly" ], "ref_ids": [ 9724, 2823, 9715, 9794, 378 ] } ], "ref_ids": [] }, { "id": 9747, "type": "theorem", "label": "local-cohomology-lemma-get-depth-1-along-Z", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-get-depth-1-along-Z", "contents": [ "Let $X$ be a Noetherian scheme. Let $T \\subset X$ be a subset", "stable under specialization. Let $\\mathcal{F}$ be a coherent", "$\\mathcal{O}_X$-module. Then there is a unique map", "$\\mathcal{F} \\to \\mathcal{F}'$ of coherent $\\mathcal{O}_X$-modules", "such that", "\\begin{enumerate}", "\\item $\\mathcal{F} \\to \\mathcal{F}'$ is surjective,", "\\item $\\mathcal{F}_x \\to \\mathcal{F}'_x$ is an isomorphism for $x \\not \\in T$,", "\\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}'_x) \\geq 1$ for $x \\in T$.", "\\end{enumerate}", "If $f : Y \\to X$ is a flat morphism with $Y$ Noetherian, then", "$f^*\\mathcal{F} \\to f^*\\mathcal{F}'$ is the corresponding", "quotient for $f^{-1}(T) \\subset Y$ and $f^*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Condition (3) just means that $\\text{Ass}(\\mathcal{F}') \\cap T = \\emptyset$.", "Thus $\\mathcal{F} \\to \\mathcal{F}'$ is the quotient of $\\mathcal{F}$", "by the subsheaf of sections whose support is contained in $T$.", "This proves uniqueness. The statement on pullbacks follows from", "Divisors, Lemma \\ref{divisors-lemma-bourbaki}", "and the uniqueness.", "\\medskip\\noindent", "Existence of $\\mathcal{F} \\to \\mathcal{F}'$.", "By the uniqueness it suffices to prove the", "existence and uniqueness locally on $X$; small detail omitted.", "Thus we may assume $X = \\Spec(A)$ is affine and $\\mathcal{F}$", "is the coherent module associated to the finite $A$-module $M$.", "Set $M' = M / H^0_T(M)$ with $H^0_T(M)$ as in Section \\ref{section-supports}.", "Then $M_\\mathfrak p = M'_\\mathfrak p$ for $\\mathfrak p \\not \\in T$", "which proves (1). On the other hand, we have", "$H^0_T(M) = \\colim H^0_Z(M)$ where $Z$ runs over the closed", "subsets of $X$ contained in $T$. Thus by", "Dualizing Complexes, Lemmas \\ref{dualizing-lemma-divide-by-torsion}", "we have $H^0_T(M') = 0$, i.e., no associated prime", "of $M'$ is in $T$. Therefore $\\text{depth}(M'_\\mathfrak p) \\geq 1$", "for $\\mathfrak p \\in T$." ], "refs": [ "divisors-lemma-bourbaki", "dualizing-lemma-divide-by-torsion" ], "ref_ids": [ 7865, 2831 ] } ], "ref_ids": [] }, { "id": 9748, "type": "theorem", "label": "local-cohomology-lemma-get-depth-2-along-Z", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-get-depth-2-along-Z", "contents": [ "Let $j : U \\to X$ be an open immersion of Noetherian schemes.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Assume $\\mathcal{F}' = j_*(\\mathcal{F}|_U)$ is coherent.", "Then $\\mathcal{F} \\to \\mathcal{F}'$ is the unique map", "of coherent $\\mathcal{O}_X$-modules such that", "\\begin{enumerate}", "\\item $\\mathcal{F}|_U \\to \\mathcal{F}'|_U$", "is an isomorphism,", "\\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}'_x) \\geq 2$", "for $x \\in X$, $x \\not \\in U$.", "\\end{enumerate}", "If $f : Y \\to X$ is a flat morphism with $Y$ Noetherian, then", "$f^*\\mathcal{F} \\to f^*\\mathcal{F}'$ is the corresponding", "map for $f^{-1}(U) \\subset Y$." ], "refs": [], "proofs": [ { "contents": [ "We have $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}'_x) \\geq 2$", "by Divisors, Lemma \\ref{divisors-lemma-depth-pushforward} part (3).", "The uniqueness of $\\mathcal{F} \\to \\mathcal{F}'$ follows from", "Divisors, Lemma \\ref{divisors-lemma-depth-2-hartog}.", "The compatibility with flat pullbacks follows from", "flat base change, see Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology}." ], "refs": [ "divisors-lemma-depth-pushforward", "divisors-lemma-depth-2-hartog", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 7888, 7881, 3298 ] } ], "ref_ids": [] }, { "id": 9749, "type": "theorem", "label": "local-cohomology-lemma-make-S2-along-Z", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-make-S2-along-Z", "contents": [ "Let $X$ be a Noetherian scheme. Let $Z \\subset X$ be a closed subscheme.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Assume", "$X$ is universally catenary and the formal fibres of local rings have $(S_1)$.", "Then there exists a unique map $\\mathcal{F} \\to \\mathcal{F}''$", "of coherent $\\mathcal{O}_X$-modules such that", "\\begin{enumerate}", "\\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$", "is an isomorphism for $x \\in X \\setminus Z$,", "\\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$ is surjective and", "$\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) = 1$", "for $x \\in Z$ such that there exists an immediate specialization", "$x' \\leadsto x$ with $x' \\not \\in Z$ and $x' \\in \\text{Ass}(\\mathcal{F})$,", "\\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) \\geq 2$", "for the remaining $x \\in Z$.", "\\end{enumerate}", "If $f : Y \\to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian,", "then $f^*\\mathcal{F} \\to f^*\\mathcal{F}''$ satisfies the same properties", "with respect to $f^{-1}(Z) \\subset Y$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F} \\to \\mathcal{F}'$ be the map constructed in", "Lemma \\ref{lemma-get-depth-1-along-Z} for the subset $Z$ of $X$.", "Recall that $\\mathcal{F}'$ is the quotient of $\\mathcal{F}$", "by the subsheaf of sections supported on $Z$.", "\\medskip\\noindent", "We first prove uniqueness. Let $\\mathcal{F} \\to \\mathcal{F}''$", "be as in the lemma. We get a factorization", "$\\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{F}''$", "since $\\text{Ass}(\\mathcal{F}'') \\cap Z = \\emptyset$", "by conditions (2) and (3). Let $U \\subset X$ be a maximal open", "subscheme such that $\\mathcal{F}'|_U \\to \\mathcal{F}''|_U$", "is an isomorphism. We see that $U$ contains all the points", "as in (2). Then by Divisors, Lemma \\ref{divisors-lemma-depth-2-hartog}", "we conclude that $\\mathcal{F}'' = j_*(\\mathcal{F}'|_U)$.", "In this way we get uniqueness (small detail: if we have two", "of these $\\mathcal{F}''$ then we take the intersection of the opens $U$", "we get from either).", "\\medskip\\noindent", "Proof of existence. Recall that", "$\\text{Ass}(\\mathcal{F}') = \\{x_1, \\ldots, x_n\\}$", "is finite and $x_i \\not \\in Z$.", "Let $Y_i$ be the closure of $\\{x_i\\}$. Let", "$Z_{i, j}$ be the irreducible components of $Z \\cap Y_i$.", "Observe that $\\text{Supp}(\\mathcal{F}') \\cap Z = \\bigcup Z_{i, j}$.", "Let $z_{i, j} \\in Z_{i, j}$ be the generic point.", "Let", "$$", "d_{i, j} = \\dim(\\mathcal{O}_{\\overline{\\{x_i\\}}, z_{i, j}})", "$$", "If $d_{i, j} = 1$, then $z_{i, j}$ is one of the points as in (2).", "Thus we do not need to modify $\\mathcal{F}'$ at these points.", "Furthermore, still assuming $d_{i, j} = 1$, using", "Lemma \\ref{lemma-depth-function}", "we can find an open neighbourhood", "$z_{i, j} \\in V_{i, j} \\subset X$ such that", "$\\text{depth}_{\\mathcal{O}_{X, z}}(\\mathcal{F}'_z) \\geq 2$", "for $z \\in Z_{i, j} \\cap V_{i, j}$, $z \\not = z_{i, j}$.", "Set", "$$", "Z' = X \\setminus", "\\left(", "X \\setminus Z \\cup \\bigcup\\nolimits_{d_{i, j} = 1} V_{i, j})", "\\right)", "$$", "Denote $j' : X \\setminus Z' \\to X$. By our choice of $Z'$", "the assumptions of Lemma \\ref{lemma-finiteness-pushforward-general}", "are satisfied.", "We conclude by setting $\\mathcal{F}'' = j'_*(\\mathcal{F}'|_{X \\setminus Z'})$", "and applying Lemma \\ref{lemma-get-depth-2-along-Z}.", "\\medskip\\noindent", "The final statement follows from the formula for the change in", "depth along a flat local homomorphism, see", "Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck-module}", "and the assumption on the fibres of $f$ inherent in $f$ being", "Cohen-Macaulay. Details omitted." ], "refs": [ "local-cohomology-lemma-get-depth-1-along-Z", "divisors-lemma-depth-2-hartog", "local-cohomology-lemma-depth-function", "local-cohomology-lemma-finiteness-pushforward-general", "local-cohomology-lemma-get-depth-2-along-Z", "algebra-lemma-apply-grothendieck-module" ], "ref_ids": [ 9747, 7881, 9735, 9733, 9748, 1360 ] } ], "ref_ids": [] }, { "id": 9750, "type": "theorem", "label": "local-cohomology-lemma-make-S2-along-T-simple", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-make-S2-along-T-simple", "contents": [ "Let $X$ be a Noetherian scheme which locally has a dualizing complex.", "Let $T' \\subset X$ be a subset stable under specialization.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Assume that if $x \\leadsto x'$ is an immediate specialization", "of points in $X$ with $x' \\in T'$ and $x \\not \\in T'$, then", "$\\text{depth}(\\mathcal{F}_x) \\geq 1$.", "Then there exists a unique map $\\mathcal{F} \\to \\mathcal{F}''$", "of coherent $\\mathcal{O}_X$-modules such that", "\\begin{enumerate}", "\\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$ is an isomorphism", "for $x \\not \\in T'$,", "\\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) \\geq 2$", "for $x \\in T'$.", "\\end{enumerate}", "If $f : Y \\to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian,", "then $f^*\\mathcal{F} \\to f^*\\mathcal{F}''$ satisfies the same properties", "with respect to $f^{-1}(T') \\subset Y$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F} \\to \\mathcal{F}'$ be the quotient of $\\mathcal{F}$", "constructed in Lemma \\ref{lemma-get-depth-1-along-Z} using $T'$.", "Recall that $\\mathcal{F}'$ is the quotient of $\\mathcal{F}$", "by the subsheaf of sections supported on $T'$.", "\\medskip\\noindent", "Proof of uniqueness. Let $\\mathcal{F} \\to \\mathcal{F}''$", "be as in the lemma. We get a factorization", "$\\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{F}''$", "since $\\text{Ass}(\\mathcal{F}'') \\cap T' = \\emptyset$", "by condition (2). Let $U \\subset X$ be a maximal open", "subscheme such that $\\mathcal{F}'|_U \\to \\mathcal{F}''|_U$", "is an isomorphism. We see that $U$ contains all the points of $T'$.", "Then by Divisors, Lemma \\ref{divisors-lemma-depth-2-hartog}", "we conclude that $\\mathcal{F}'' = j_*(\\mathcal{F}'|_U)$.", "In this way we get uniqueness (small detail: if we have two", "of these $\\mathcal{F}''$ then we take the intersection of the opens $U$", "we get from either).", "\\medskip\\noindent", "Proof of existence. We will define", "$$", "\\mathcal{F}'' = \\colim j_*(\\mathcal{F}'|_V)", "$$", "where $j : V \\to X$ runs over the open subschemes such that", "$X \\setminus V \\subset T'$. Observe that the colimit is filtered", "as $T'$ is stable under specialization. Each of the", "maps $\\mathcal{F}' \\to j_*(\\mathcal{F}'|_V)$ is injective", "as $\\text{Ass}(\\mathcal{F}')$ is disjoint from $T'$.", "Thus $\\mathcal{F}' \\to \\mathcal{F}''$ is injective.", "\\medskip\\noindent", "Suppose $X = \\Spec(A)$ is affine and $\\mathcal{F}$", "corresponds to the finite $A$-module $M$. Then $\\mathcal{F}'$", "corresponds to $M' = M / H^0_{T'}(M)$, see proof of", "Lemma \\ref{lemma-get-depth-1-along-Z}. Applying", "Lemmas \\ref{lemma-local-cohomology} and \\ref{lemma-adjoint-ext}", "we see that $\\mathcal{F}''$ corresponds to an $A$-module", "$M''$ which fits into the short exact sequence", "$$", "0 \\to M' \\to M'' \\to H^1_{T'}(M') \\to 0", "$$", "By Proposition \\ref{proposition-finiteness} and our condition", "on immediate specializations in the statement of the lemma", "we see that $M''$ is a finite $A$-module. In this way", "we see that $\\mathcal{F}''$ is coherent.", "\\medskip\\noindent", "The final statement follows from the formula for the change in", "depth along a flat local homomorphism, see", "Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck-module}", "and the assumption on the fibres of $f$ inherent in $f$ being", "Cohen-Macaulay. Details omitted." ], "refs": [ "local-cohomology-lemma-get-depth-1-along-Z", "divisors-lemma-depth-2-hartog", "local-cohomology-lemma-get-depth-1-along-Z", "local-cohomology-lemma-local-cohomology", "local-cohomology-lemma-adjoint-ext", "local-cohomology-proposition-finiteness", "algebra-lemma-apply-grothendieck-module" ], "ref_ids": [ 9747, 7881, 9747, 9696, 9715, 9787, 1360 ] } ], "ref_ids": [] }, { "id": 9751, "type": "theorem", "label": "local-cohomology-lemma-make-S2-along-T", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-make-S2-along-T", "contents": [ "Let $X$ be a Noetherian scheme which locally has a dualizing complex.", "Let $T' \\subset T \\subset X$ be subsets stable under specialization", "such that if $x \\leadsto x'$ is an immediate specialization", "of points in $X$ and $x' \\in T'$, then $x \\in T$. Let $\\mathcal{F}$", "be a coherent $\\mathcal{O}_X$-module.", "Then there exists a unique map $\\mathcal{F} \\to \\mathcal{F}''$", "of coherent $\\mathcal{O}_X$-modules such that", "\\begin{enumerate}", "\\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$ is an isomorphism", "for $x \\not \\in T$,", "\\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$ is surjective", "and $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) \\geq 1$", "for $x \\in T$, $x \\not \\in T'$, and", "\\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) \\geq 2$", "for $x \\in T'$.", "\\end{enumerate}", "If $f : Y \\to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian,", "then $f^*\\mathcal{F} \\to f^*\\mathcal{F}''$ satisfies the same properties", "with respect to $f^{-1}(T') \\subset f^{-1}(T) \\subset Y$." ], "refs": [], "proofs": [ { "contents": [ "First, let $\\mathcal{F} \\to \\mathcal{F}'$ be the quotient of $\\mathcal{F}$", "constructed in Lemma \\ref{lemma-get-depth-1-along-Z} using $T$.", "Second, let $\\mathcal{F}' \\to \\mathcal{F}''$ be the unique", "map of coherent modules construction in", "Lemma \\ref{lemma-make-S2-along-T-simple} using $T'$.", "Then $\\mathcal{F} \\to \\mathcal{F}''$ is as desired." ], "refs": [ "local-cohomology-lemma-get-depth-1-along-Z", "local-cohomology-lemma-make-S2-along-T-simple" ], "ref_ids": [ 9747, 9750 ] } ], "ref_ids": [] }, { "id": 9752, "type": "theorem", "label": "local-cohomology-lemma-cd-top-vanishing", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cd-top-vanishing", "contents": [ "Let $A$ be a Noetherian ring of dimension $d$. Let $I \\subset I' \\subset A$", "be ideals. If $I'$ is contained in the Jacobson radical", "of $A$ and $\\text{cd}(A, I') < d$, then $\\text{cd}(A, I) < d$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-cd-dimension} we know $\\text{cd}(A, I) \\leq d$.", "We will use Lemma \\ref{lemma-isomorphism} to show", "$$", "H^d_{V(I')}(A) \\to H^d_{V(I)}(A)", "$$", "is surjective which will finish the proof. Pick", "$\\mathfrak p \\in V(I) \\setminus V(I')$. By our assumption", "on $I'$ we see that $\\mathfrak p$ is not a maximal ideal of $A$.", "Hence $\\dim(A_\\mathfrak p) < d$. Then", "$H^d_{\\mathfrak pA_\\mathfrak p}(A_\\mathfrak p) = 0$", "by Lemma \\ref{lemma-cd-dimension}." ], "refs": [ "local-cohomology-lemma-cd-dimension", "local-cohomology-lemma-isomorphism", "local-cohomology-lemma-cd-dimension" ], "ref_ids": [ 9708, 9700, 9708 ] } ], "ref_ids": [] }, { "id": 9753, "type": "theorem", "label": "local-cohomology-lemma-cd-top-vanishing-some-module", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cd-top-vanishing-some-module", "contents": [ "Let $A$ be a Noetherian ring of dimension $d$. Let $I \\subset A$", "be an ideal. If $H^d_{V(I)}(M) = 0$ for some finite $A$-module", "whose support contains all the irreducible components of", "dimension $d$, then $\\text{cd}(A, I) < d$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-cd-dimension} we know $\\text{cd}(A, I) \\leq d$.", "Thus for any finite $A$-module $N$ we have $H^i_{V(I)}(N) = 0$", "for $i > d$. Let us say property $\\mathcal{P}$ holds for the", "finite $A$-module $N$ if $H^d_{V(I)}(N) = 0$.", "One of our assumptions is that $\\mathcal{P}(M)$ holds.", "Observe that $\\mathcal{P}(N_1 \\oplus N_2)", "\\Leftrightarrow (\\mathcal{P}(N_1) \\wedge \\mathcal{P}(N_2))$.", "Observe that if $N \\to N'$ is surjective, then", "$\\mathcal{P}(N) \\Rightarrow \\mathcal{P}(N')$ as we", "have the vanishing of $H^{d + 1}_{V(I)}$ (see above).", "Let $\\mathfrak p_1, \\ldots, \\mathfrak p_n$ be the", "minimal primes of $A$ with $\\dim(A/\\mathfrak p_i) = d$.", "Observe that $\\mathcal{P}(N)$ holds if the support", "of $N$ is disjoint from $\\{\\mathfrak p_1, \\ldots, \\mathfrak p_n\\}$", "for dimension reasons, see Lemma \\ref{lemma-cd-dimension}.", "For each $i$ set $M_i = M/\\mathfrak p_i M$.", "This is a finite $A$-module annihilated by $\\mathfrak p_i$", "whose support is equal to", "$V(\\mathfrak p_i)$ (here we use the assumption on the support of $M$).", "Finally, if $J \\subset A$ is an ideal, then we have $\\mathcal{P}(JM_i)$", "as $JM_i$ is a quotient of a direct sum of copies of $M$.", "Thus it follows from Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-property-higher-rank-cohomological}", "that $\\mathcal{P}$ holds for every finite $A$-module." ], "refs": [ "local-cohomology-lemma-cd-dimension", "local-cohomology-lemma-cd-dimension", "coherent-lemma-property-higher-rank-cohomological" ], "ref_ids": [ 9708, 9708, 3334 ] } ], "ref_ids": [] }, { "id": 9754, "type": "theorem", "label": "local-cohomology-lemma-top-coh-divisible", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-top-coh-divisible", "contents": [ "Let $A$ be a Noetherian local ring of dimension $d$. Let $f \\in A$", "be an element which is not contained in any minimal prime of", "dimension $d$. Then $f : H^d_{V(I)}(M) \\to H^d_{V(I)}(M)$", "is surjective for any finite $A$-module $M$ and any ideal $I \\subset A$." ], "refs": [], "proofs": [ { "contents": [ "The support of $M/fM$ has dimension $< d$ by our assumption on $f$.", "Thus $H^d_{V(I)}(M/fM) = 0$ by Lemma \\ref{lemma-cd-dimension}.", "Thus $H^d_{V(I)}(fM) \\to H^d_{V(I)}(M)$ is surjective.", "Since by Lemma \\ref{lemma-cd-dimension} we know $\\text{cd}(A, I) \\leq d$", "we also see that the surjection $M \\to fM$, $x \\mapsto fx$", "induces a surjection $H^d_{V(I)}(M) \\to H^d_{V(I)}(fM)$." ], "refs": [ "local-cohomology-lemma-cd-dimension", "local-cohomology-lemma-cd-dimension" ], "ref_ids": [ 9708, 9708 ] } ], "ref_ids": [] }, { "id": 9755, "type": "theorem", "label": "local-cohomology-lemma-cd-bound-dualizing", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cd-bound-dualizing", "contents": [ "Let $A$ be a Noetherian local ring with", "normalized dualizing complex $\\omega_A^\\bullet$.", "Let $I \\subset A$ be an ideal.", "If $H^0_{V(I)}(\\omega_A^\\bullet) = 0$, then $\\text{cd}(A, I) < \\dim(A)$." ], "refs": [], "proofs": [ { "contents": [ "Set $d = \\dim(A)$. Let $\\mathfrak p_1, \\ldots, \\mathfrak p_n \\subset A$", "be the minimal primes of dimension $d$.", "Recall that the finite $A$-module", "$H^{-i}(\\omega_A^\\bullet)$ is nonzero only for", "$i \\in \\{0, \\ldots, d\\}$ and that the support", "of $H^{-i}(\\omega_A^\\bullet)$ has dimension $\\leq i$, see", "Lemma \\ref{lemma-sitting-in-degrees}.", "Set $\\omega_A = H^{-d}(\\omega_A^\\bullet)$.", "By prime avoidence (Algebra, Lemma \\ref{algebra-lemma-silly})", "we can find $f \\in A$, $f \\not \\in \\mathfrak p_i$", "which annihilates $H^{-i}(\\omega_A^\\bullet)$ for $i < d$.", "Consider the distinguished triangle", "$$", "\\omega_A[d] \\to \\omega_A^\\bullet \\to", "\\tau_{\\geq -d + 1}\\omega_A^\\bullet \\to \\omega_A[d + 1]", "$$", "See Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}.", "By Derived Categories, Lemma \\ref{derived-lemma-trick-vanishing-composition}", "we see that $f^d$ induces the zero endomorphism of", "$\\tau_{\\geq -d + 1}\\omega_A^\\bullet$.", "Using the axioms of a triangulated category, we find a map", "$$", "\\omega_A^\\bullet \\to \\omega_A[d]", "$$", "whose composition with $\\omega_A[d] \\to \\omega_A^\\bullet$ is", "multiplication by $f^d$ on $\\omega_A[d]$.", "Thus we conclude that $f^d$ annihilates $H^d_{V(I)}(\\omega_A)$.", "By Lemma \\ref{lemma-top-coh-divisible} we conlude $H^d_{V(I)}(\\omega_A) = 0$.", "Then we conclude by Lemma \\ref{lemma-cd-top-vanishing-some-module}", "and the fact that $(\\omega_A)_{\\mathfrak p_i}$ is nonzero", "(see for example", "Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-nonvanishing-generically-local})." ], "refs": [ "local-cohomology-lemma-sitting-in-degrees", "algebra-lemma-silly", "derived-remark-truncation-distinguished-triangle", "derived-lemma-trick-vanishing-composition", "local-cohomology-lemma-top-coh-divisible", "local-cohomology-lemma-cd-top-vanishing-some-module", "dualizing-lemma-nonvanishing-generically-local" ], "ref_ids": [ 9737, 378, 2016, 1817, 9754, 9753, 2866 ] } ], "ref_ids": [] }, { "id": 9756, "type": "theorem", "label": "local-cohomology-lemma-inverse-system-symbolic-powers", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-inverse-system-symbolic-powers", "contents": [ "Let $(A, \\mathfrak m)$ be a complete Noetherian local domain. Let", "$\\mathfrak p \\subset A$ be a prime ideal of dimension $1$.", "For every $n \\geq 1$ there is an $m \\geq n$ such that", "$\\mathfrak p^{(m)} \\subset \\mathfrak p^n$." ], "refs": [], "proofs": [ { "contents": [ "Recall that the symbolic power $\\mathfrak p^{(m)}$ is defined as the", "kernel of $A \\to A_\\mathfrak p/\\mathfrak p^mA_\\mathfrak p$.", "Since localization is exact we conclude that in the short exact sequence", "$$", "0 \\to \\mathfrak a_n \\to A/\\mathfrak p^n \\to A/\\mathfrak p^{(n)} \\to 0", "$$", "the support of $\\mathfrak a_n$ is contained in $\\{\\mathfrak m\\}$.", "In particular, the inverse system $(\\mathfrak a_n)$ is Mittag-Leffler", "as each $\\mathfrak a_n$ is an Artinian $A$-module.", "We conclude that the lemma is equivalent to the requirement", "that $\\lim \\mathfrak a_n = 0$. Let $f \\in \\lim \\mathfrak a_n$.", "Then $f$ is an element of $A = \\lim A/\\mathfrak p^n$", "(here we use that $A$ is complete)", "which maps to zero in the completion $A_\\mathfrak p^\\wedge$", "of $A_\\mathfrak p$. Since $A_\\mathfrak p \\to A_\\mathfrak p^\\wedge$", "is faithfully flat, we see that $f$ maps to zero in $A_\\mathfrak p$.", "Since $A$ is a domain we see that $f$ is zero as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9757, "type": "theorem", "label": "local-cohomology-lemma-affine-complement", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-affine-complement", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "Let $I \\subset A$ be an ideal. Assume $A$ is excellent,", "normal, and $\\dim V(I) \\geq 1$. Then $\\text{cd}(A, I) < \\dim(A)$.", "In particular, if $\\dim(A) = 2$, then $\\Spec(A) \\setminus V(I)$ is affine." ], "refs": [], "proofs": [ { "contents": [ "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-completion-normal-local-ring}", "the completion $A^\\wedge$ is normal and hence a domain.", "Thus the assumption of", "Proposition \\ref{proposition-Hartshorne-Lichtenbaum-vanishing}", "holds and we conclude. The statement on affineness", "follows from Lemma \\ref{lemma-cd-is-one}." ], "refs": [ "more-algebra-lemma-completion-normal-local-ring", "local-cohomology-proposition-Hartshorne-Lichtenbaum-vanishing", "local-cohomology-lemma-cd-is-one" ], "ref_ids": [ 10109, 9790, 9709 ] } ], "ref_ids": [] }, { "id": 9758, "type": "theorem", "label": "local-cohomology-lemma-annihilator-frobenius-module", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-annihilator-frobenius-module", "contents": [ "Let $p$ be a prime number. Let $(A, \\mathfrak m, \\kappa)$", "be a Noetherian local ring", "with $p = 0$ in $A$. Let $M$ be a finite $A$-module", "such that $M \\otimes_{A, F} A \\cong M$. Then $M$ is finite free." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $A^{\\oplus m} \\to A^{\\oplus n} \\to M$", "which induces an isomorphism $\\kappa^{\\oplus n} \\to M/\\mathfrak m M$.", "Let $T = (a_{ij})$ be the matrix of the map $A^{\\oplus m} \\to A^{\\oplus n}$.", "Observe that $a_{ij} \\in \\mathfrak m$. Applying base change by", "$F$, using right exactness of base change, we get a presentation", "$A^{\\oplus m} \\to A^{\\oplus n} \\to M$ where the matrix is", "$T = (a_{ij}^p)$. Thus we have a presentation with", "$a_{ij} \\in \\mathfrak m^p$. Repeating this construction we", "find that for each $e \\geq 1$ there exists a presentation with", "$a_{ij} \\in \\mathfrak m^e$. This implies the fitting ideals", "(More on Algebra, Definition \\ref{more-algebra-definition-fitting-ideal})", "$\\text{Fit}_k(M)$ for $k < n$ are contained in", "$\\bigcap_{e \\geq 1} \\mathfrak m^e$. Since this is zero by", "Krull's intersection theorem", "(Algebra, Lemma \\ref{algebra-lemma-intersect-powers-ideal-module-zero})", "we conclude that", "$M$ is free of rank $n$ by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-fitting-ideal-finite-locally-free}." ], "refs": [ "more-algebra-definition-fitting-ideal", "algebra-lemma-intersect-powers-ideal-module-zero", "more-algebra-lemma-fitting-ideal-finite-locally-free" ], "ref_ids": [ 10595, 627, 9836 ] } ], "ref_ids": [] }, { "id": 9759, "type": "theorem", "label": "local-cohomology-lemma-1", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-1", "contents": [ "\\begin{reference}", "See \\cite{Lech-inequalities} and \\cite[Lemma 1 page 299]{MatCA}.", "\\end{reference}", "Let $A$ be a ring. If $f_1, \\ldots, f_{r - 1}, f_rg_r$", "are independent, then $f_1, \\ldots, f_r$ are independent." ], "refs": [], "proofs": [ { "contents": [ "Say $\\sum a_if_i = 0$. Then $\\sum a_ig_rf_i = 0$.", "Hence $a_r \\in (f_1, \\ldots, f_{r - 1}, f_rg_r)$.", "Write $a_r = \\sum_{i < r} b_i f_i + b f_rg_r$.", "Then $0 = \\sum_{i < r} (a_i + b_if_r)f_i + bf_r^2g_r$.", "Thus $a_i + b_i f_r \\in (f_1, \\ldots, f_{r - 1}, f_rg_r)$", "which implies $a_i \\in (f_1, \\ldots, f_r)$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9760, "type": "theorem", "label": "local-cohomology-lemma-2", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-2", "contents": [ "\\begin{reference}", "See \\cite{Lech-inequalities} and \\cite[Lemma 2 page 300]{MatCA}.", "\\end{reference}", "Let $A$ be a ring. If $f_1, \\ldots, f_{r - 1}, f_rg_r$", "are independent and if the $A$-module", "$A/(f_1, \\ldots, f_{r - 1}, f_rg_r)$ has finite length, then", "\\begin{align*}", "& \\text{length}_A(A/(f_1, \\ldots, f_{r - 1}, f_rg_r)) \\\\", "& =", "\\text{length}_A(A/(f_1, \\ldots, f_{r - 1}, f_r)) +", "\\text{length}_A(A/(f_1, \\ldots, f_{r - 1}, g_r))", "\\end{align*}" ], "refs": [], "proofs": [ { "contents": [ "We claim there is an exact sequence", "$$", "0 \\to", "A/(f_1, \\ldots, f_{r - 1}, g_r) \\xrightarrow{f_r}", "A/(f_1, \\ldots, f_{r - 1}, f_rg_r) \\to", "A/(f_1, \\ldots, f_{r - 1}, f_r) \\to 0", "$$", "Namely, if $a f_r \\in (f_1, \\ldots, f_{r - 1}, f_rg_r)$, then", "$\\sum_{i < r} a_i f_i + (a + bg_r)f_r = 0$", "for some $b, a_i \\in A$. Hence", "$\\sum_{i < r} a_i g_r f_i + (a + bg_r)g_rf_r = 0$", "which implies $a + bg_r \\in (f_1, \\ldots, f_{r - 1}, f_rg_r)$", "which means that $a$ maps to zero in $A/(f_1, \\ldots, f_{r - 1}, g_r)$.", "This proves the claim.", "To finish use additivity of lengths", "(Algebra, Lemma \\ref{algebra-lemma-length-additive})." ], "refs": [ "algebra-lemma-length-additive" ], "ref_ids": [ 631 ] } ], "ref_ids": [] }, { "id": 9761, "type": "theorem", "label": "local-cohomology-lemma-3", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-3", "contents": [ "\\begin{reference}", "See \\cite{Lech-inequalities} and \\cite[Lemma 3 page 300]{MatCA}.", "\\end{reference}", "Let $(A, \\mathfrak m)$ be a local ring. If $\\mathfrak m = (x_1, \\ldots, x_r)$", "and $x_1^{e_1}, \\ldots, x_r^{e_r}$ are independent for some $e_i > 0$,", "then $\\text{length}_A(A/(x_1^{e_1}, \\ldots, x_r^{e_r})) = e_1\\ldots e_r$." ], "refs": [], "proofs": [ { "contents": [ "Use Lemmas \\ref{lemma-1} and \\ref{lemma-2} and induction." ], "refs": [ "local-cohomology-lemma-1", "local-cohomology-lemma-2" ], "ref_ids": [ 9759, 9760 ] } ], "ref_ids": [] }, { "id": 9762, "type": "theorem", "label": "local-cohomology-lemma-flat-extension-independent", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-flat-extension-independent", "contents": [ "Let $\\varphi : A \\to B$ be a flat ring map.", "If $f_1, \\ldots, f_r \\in A$ are independent, then", "$\\varphi(f_1), \\ldots, \\varphi(f_r) \\in B$ are independent." ], "refs": [], "proofs": [ { "contents": [ "Let $I = (f_1, \\ldots, f_r)$ and $J = F(I)B$. By flatness we have", "$I/I^2 \\otimes_A B = J/J^2$. Hence freeness of $I/I^2$ over $A/I$", "implies freeness of $J/J^2$ over $B/J$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9763, "type": "theorem", "label": "local-cohomology-lemma-frobenius-flat-regular", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-frobenius-flat-regular", "contents": [ "\\begin{reference}", "\\cite{Kunz-flat}", "\\end{reference}", "Let $p$ be a prime number.", "Let $A$ be a Noetherian ring with $p = 0$.", "The following are equivalent", "\\begin{enumerate}", "\\item $A$ is regular, and", "\\item $F : A \\to A$, $a \\mapsto a^p$ is flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\Spec(F) : \\Spec(A) \\to \\Spec(A)$ is the identity map.", "Being regular is defined in terms of the local rings and being flat", "is something about local rings, see", "Algebra, Lemma \\ref{algebra-lemma-flat-localization}.", "Thus we may and do assume $A$ is a Noetherian", "local ring with maximal ideal $\\mathfrak m$.", "\\medskip\\noindent", "Assume $A$ is regular. Let $x_1, \\ldots, x_d$ be a", "system of parameters for $A$. Applying $F$ we find", "$F(x_1), \\ldots, F(x_d) = x_1^p, \\ldots, x_d^p$,", "which is a system of parameters for $A$. Hence $F$ is flat, see", "Algebra, Lemmas \\ref{algebra-lemma-CM-over-regular-flat} and", "\\ref{algebra-lemma-regular-ring-CM}.", "\\medskip\\noindent", "Conversely, assume $F$ is flat. Write $\\mathfrak m = (x_1, \\ldots, x_r)$", "with $r$ minimal. Then $x_1, \\ldots, x_r$ are independent in the sense", "defined above. Since $F$ is flat, we see that $x_1^p, \\ldots, x_r^p$", "are independent, see Lemma \\ref{lemma-flat-extension-independent}.", "Hence $\\text{length}_A(A/(x_1^p, \\ldots, x_r^p)) = p^r$ by", "Lemma \\ref{lemma-3}.", "Let $\\chi(n) = \\text{length}_A(A/\\mathfrak m^n)$ and recall", "that this is a numerical polynomial of degree $\\dim(A)$, see", "Algebra, Proposition \\ref{algebra-proposition-dimension}.", "Choose $n \\gg 0$. Observe that", "$$", "\\mathfrak m^{pn + pr} \\subset F(\\mathfrak m^n)A \\subset \\mathfrak m^{pn}", "$$", "as can be seen by looking at monomials in $x_1, \\ldots, x_r$. We have", "$$", "A/F(\\mathfrak m^n)A = A/\\mathfrak m^n \\otimes_{A, F} A", "$$", "By flatness of $F$ this has length $\\chi(n) \\text{length}_A(A/F(\\mathfrak m)A)$", "(Algebra, Lemma \\ref{algebra-lemma-pullback-module})", "which is equal to $p^r\\chi(n)$ by the above. We conclude", "$$", "\\chi(pn + pr) \\geq p^r\\chi(n) \\geq \\chi(pn)", "$$", "Looking at the leading terms this implies $r = \\dim(A)$, i.e., $A$ is regular." ], "refs": [ "algebra-lemma-flat-localization", "algebra-lemma-CM-over-regular-flat", "algebra-lemma-regular-ring-CM", "local-cohomology-lemma-flat-extension-independent", "local-cohomology-lemma-3", "algebra-proposition-dimension", "algebra-lemma-pullback-module" ], "ref_ids": [ 538, 1107, 941, 9762, 9761, 1411, 640 ] } ], "ref_ids": [] }, { "id": 9764, "type": "theorem", "label": "local-cohomology-lemma-structure-torsion-D-module-regular", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-structure-torsion-D-module-regular", "contents": [ "\\begin{reference}", "Special case of \\cite[Theorem 2.4]{Lyubeznik}", "\\end{reference}", "Let $k$ be a field of characteristic $0$. Let $d \\geq 1$.", "Let $A = k[[x_1, \\ldots, x_d]]$ with maximal ideal $\\mathfrak m$.", "Let $M$ be an $\\mathfrak m$-power torsion $A$-module endowed with", "additive operators $D_1, \\ldots, D_d$ satisfying the leibniz rule", "$$", "D_i(fz) = \\partial_i(f) z + f D_i(z)", "$$", "for $f \\in A$ and $z \\in M$. Here $\\partial_i$ is", "differentiation with respect to $x_i$.", "Then $M$ is isomorphic to a direct sum", "of copies of the injective hull $E$ of $k$." ], "refs": [], "proofs": [ { "contents": [ "Choose a set $J$ and an isomorphism $M[\\mathfrak m] \\to \\bigoplus_{j \\in J} k$.", "Since $\\bigoplus_{j \\in J} E$ is injective", "(Dualizing Complexes, Lemma \\ref{dualizing-lemma-sum-injective-modules})", "we can extend this isomorphism to an $A$-module homomorphism", "$\\varphi : M \\to \\bigoplus_{j \\in J} E$.", "We claim that $\\varphi$ is an isomorphism, i.e., bijective.", "\\medskip\\noindent", "Injective. Let $z \\in M$ be nonzero. Since $M$ is $\\mathfrak m$-power torsion", "we can choose an element $f \\in A$ such that $fz \\in M[\\mathfrak m]$ and", "$fz \\not = 0$. Then $\\varphi(fz) = f\\varphi(z)$ is nonzero, hence", "$\\varphi(z)$ is nonzero.", "\\medskip\\noindent", "Surjective. Let $z \\in M$. Then $x_1^n z = 0$ for some $n \\geq 0$.", "We will prove that $z \\in x_1M$ by induction on $n$.", "If $n = 0$, then $z = 0$ and the result is true.", "If $n > 0$, then applying $D_1$ we find $0 = n x_1^{n - 1} z + x_1^nD_1(z)$.", "Hence $x_1^{n - 1}(nz + x_1D_1(z)) = 0$. By induction we get", "$nz + x_1D_1(z) \\in x_1M$. Since $n$ is invertible, we conclude", "$z \\in x_1M$. Thus we see that $M$ is $x_1$-divisible.", "If $\\varphi$ is not surjective, then we can choose", "$e \\in \\bigoplus_{j \\in J} E$ not in $M$.", "Arguing as above we may assume $\\mathfrak m e \\subset M$,", "in particular $x_1 e \\in M$. There exists an element", "$z_1 \\in M$ with $x_1 z_1 = x_1 e$. Hence", "$x_1(z_1 - e) = 0$. Replacing $e$ by $e - z_1$", "we may assume $e$ is annihilated by $x_1$.", "Thus it suffices to prove that", "$$", "\\varphi[x_1] :", "M[x_1]", "\\longrightarrow", "\\left(\\bigoplus\\nolimits_{j \\in J} E\\right)[x_1] =", "\\bigoplus\\nolimits_{j \\in J} E[x_1]", "$$", "is surjective. If $d = 1$, this is true by construction of $\\varphi$.", "If $d > 1$, then we observe that $E[x_1]$ is the injective hull", "of the residue field of $k[[x_2, \\ldots, x_d]]$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-quotient}.", "Observe that $M[x_1]$ as a module over $k[[x_2, \\ldots, x_d]]$", "is $\\mathfrak m/(x_1)$-power torsion and comes", "equipped with operators $D_2, \\ldots, D_d$ satisfying", "the displayed Leibniz rule.", "Thus by induction on $d$ we conclude that $\\varphi[x_1]$", "is surjective as desired." ], "refs": [ "dualizing-lemma-sum-injective-modules", "dualizing-lemma-quotient" ], "ref_ids": [ 2789, 2804 ] } ], "ref_ids": [] }, { "id": 9765, "type": "theorem", "label": "local-cohomology-lemma-structure-torsion-Frobenius-regular", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-structure-torsion-Frobenius-regular", "contents": [ "\\begin{reference}", "Follows from \\cite[Corollary 3.6]{Huneke-Sharp} with a", "little bit of work. Also follows directly from", "\\cite[Theorem 1.4]{Lyubeznik2}.", "\\end{reference}", "Let $p$ be a prime number. Let $(A, \\mathfrak m, k)$", "be a regular local ring with $p = 0$. Denote $F : A \\to A$, $a \\mapsto a^p$", "be the Frobenius endomorphism. Let $M$ be a $\\mathfrak m$-power torsion module", "such that $M \\otimes_{A, F} A \\cong M$. Then $M$ is isomorphic to a direct sum", "of copies of the injective hull $E$ of $k$." ], "refs": [], "proofs": [ { "contents": [ "Choose a set $J$ and an $A$-module homorphism", "$\\varphi : M \\to \\bigoplus_{j \\in J} E$ which maps", "$M[\\mathfrak m]$ isomorphically onto", "$(\\bigoplus_{j \\in J} E)[\\mathfrak m] = \\bigoplus_{j \\in J} k$.", "We claim that $\\varphi$ is an isomorphism, i.e., bijective.", "\\medskip\\noindent", "Injective. Let $z \\in M$ be nonzero. Since $M$ is $\\mathfrak m$-power torsion", "we can choose an element $f \\in A$ such that $fz \\in M[\\mathfrak m]$ and", "$fz \\not = 0$. Then $\\varphi(fz) = f\\varphi(z)$ is nonzero, hence", "$\\varphi(z)$ is nonzero.", "\\medskip\\noindent", "Surjective. Recall that $F$ is flat, see", "Lemma \\ref{lemma-frobenius-flat-regular}.", "Let $x_1, \\ldots, x_d$ be a minimal system of generators of", "$\\mathfrak m$. Denote", "$$", "M_n = M[x_1^{p^n}, \\ldots, x_d^{p^n}]", "$$", "the submodule of $M$ consisting of elements killed by", "$x_1^{p^n}, \\ldots, x_d^{p^n}$. So $M_0 = M[\\mathfrak m]$", "is a vector space over $k$. Also $M = \\bigcup M_n$ by our", "assumption that $M$ is $\\mathfrak m$-power torsion. Since $F^n$ is flat and", "$F^n(x_i) = x_i^{p^n}$ we have", "$$", "M_n \\cong (M \\otimes_{A, F^n} A)[x_1^{p^n}, \\ldots, x_d^{p^n}] =", "M[x_1, \\ldots, x_d] \\otimes_{A, F} A =", "M_0 \\otimes_k A/(x_1^{p^n}, \\ldots, x_d^{p^n})", "$$", "Thus $M_n$ is free over $A/(x_1^{p^n}, \\ldots, x_d^{p^n})$.", "A computation shows that every element of $A/(x_1^{p^n}, \\ldots, x_d^{p^n})$", "annihilated by $x_1^{p^n - 1}$ is divisible by $x_1$; for example", "you can use that $A/(x_1^{p^n}, \\ldots, x_d^{p^n}) \\cong", "k[x_1, \\ldots, x_d]/(x_1^{p^n}, \\ldots, x_d^{p^n})$ by Algebra, Lemma", "\\ref{algebra-lemma-regular-complete-containing-coefficient-field}.", "Thus the same is true for every element of $M_n$.", "Since every element of $M$ is in $M_n$ for all $n \\gg 0$", "and since every element of $M$ is killed by some power of", "$x_1$, we conclude that $M$ is $x_1$-divisible.", "\\medskip\\noindent", "Let $x = x_1$. Above we have seen that $M$ is $x$-divisible.", "If $\\varphi$ is not surjective, then we can choose", "$e \\in \\bigoplus_{j \\in J} E$ not in $M$.", "Arguing as above we may assume $\\mathfrak m e \\subset M$,", "in particular $x e \\in M$. There exists an element", "$z_1 \\in M$ with $x z_1 = x e$. Hence", "$x(z_1 - e) = 0$. Replacing $e$ by $e - z_1$", "we may assume $e$ is annihilated by $x$.", "Thus it suffices to prove that", "$$", "\\varphi[x] :", "M[x]", "\\longrightarrow", "\\left(\\bigoplus\\nolimits_{j \\in J} E\\right)[x] =", "\\bigoplus\\nolimits_{j \\in J} E[x]", "$$", "is surjective. If $d = 1$, this is true by construction of $\\varphi$.", "If $d > 1$, then we observe that $E[x]$ is the injective hull", "of the residue field of the regular ring $A/xA$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-quotient}.", "Observe that $M[x]$ as a module over $A/xA$", "is $\\mathfrak m/(x)$-power torsion and we have", "\\begin{align*}", "M[x] \\otimes_{A/xA, F} A/xA", "& = M[x] \\otimes_{A, F} A \\otimes_A A/xA \\\\", "& = (M \\otimes_{A, F} A)[x^p] \\otimes_A A/xA \\\\", "& \\cong M[x^p] \\otimes_A A/xA", "\\end{align*}", "Argue using flatness of $F$ as before.", "We claim that $M[x^p] \\otimes_A A/xA \\to M[x]$,", "$z \\otimes 1 \\mapsto x^{p - 1}z$ is an isomorphism.", "This can be seen by proving it for", "each of the modules $M_n$, $n > 0$ defined above", "where it follows by the same result for", "$A/(x_1^{p^n}, \\ldots, x_d^{p^n})$ and $x = x_1$.", "Thus by induction on $\\dim(A)$ we conclude that $\\varphi[x]$", "is surjective as desired." ], "refs": [ "local-cohomology-lemma-frobenius-flat-regular", "algebra-lemma-regular-complete-containing-coefficient-field", "dualizing-lemma-quotient" ], "ref_ids": [ 9763, 1331, 2804 ] } ], "ref_ids": [] }, { "id": 9766, "type": "theorem", "label": "local-cohomology-lemma-derivation", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-derivation", "contents": [ "Let $A$ be a ring. Let $I \\subset A$ be a finitely generated ideal.", "Set $Z = V(I)$.", "For each derivation $\\theta : A \\to A$ there exists a canonical", "additive operator $D$ on the local cohomology modules", "$H^i_Z(A)$ satisfying the Leibniz rule with respect to $\\theta$." ], "refs": [], "proofs": [ { "contents": [ "Let $f_1, \\ldots, f_r$ be elements generating $I$.", "Recall that $R\\Gamma_Z(A)$ is computed by the complex", "$$", "A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}}", "\\to \\ldots \\to A_{f_1\\ldots f_r}", "$$", "See Dualizing Complexes, Lemma \\ref{dualizing-lemma-local-cohomology-adjoint}.", "Since $\\theta$ extends uniquely to an additive operator on", "any localization of $A$ satisfying the Leibniz rule with", "respect to $\\theta$, the lemma is clear." ], "refs": [ "dualizing-lemma-local-cohomology-adjoint" ], "ref_ids": [ 2815 ] } ], "ref_ids": [] }, { "id": 9767, "type": "theorem", "label": "local-cohomology-lemma-frobenius", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-frobenius", "contents": [ "Let $p$ be a prime number. Let $A$ be a ring with $p = 0$.", "Denote $F : A \\to A$, $a \\mapsto a^p$ the Frobenius endomorphism.", "Let $I \\subset A$ be a finitely generated ideal. Set $Z = V(I)$.", "There exists an isomorphism", "$R\\Gamma_Z(A) \\otimes_{A, F}^\\mathbf{L} A \\cong R\\Gamma_Z(A)$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-torsion-change-rings}", "and the fact that $Z = V(f_1^p, \\ldots, f_r^p)$", "if $I = (f_1, \\ldots, f_r)$." ], "refs": [ "dualizing-lemma-torsion-change-rings" ], "ref_ids": [ 2817 ] } ], "ref_ids": [] }, { "id": 9768, "type": "theorem", "label": "local-cohomology-lemma-etale-derivation", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-etale-derivation", "contents": [ "Let $A$ be a ring. Let $V \\to \\Spec(A)$ be quasi-compact, quasi-separated,", "and \\'etale. For each derivation $\\theta : A \\to A$ there exists a canonical", "additive operator $D$ on $H^i(V, \\mathcal{O}_V)$", "satisfying the Leibniz rule with respect to $\\theta$." ], "refs": [], "proofs": [ { "contents": [ "If $V$ is separated, then we can argue using an affine open covering", "$V = \\bigcup_{j = 1, \\ldots m} V_j$. Namely, because $V$ is separated", "we may write $V_{j_0 \\ldots j_p} = \\Spec(B_{j_0 \\ldots j_p})$.", "See Schemes, Lemma \\ref{schemes-lemma-characterize-separated}. Then we", "find that the $A$-module $H^i(V, \\mathcal{O}_V)$", "is the $i$th cohomology group of the {\\v C}ech complex", "$$", "\\prod B_{j_0} \\to", "\\prod B_{j_0j_1} \\to", "\\prod B_{j_0j_1j_2} \\to \\ldots", "$$", "See Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cech-cohomology-quasi-coherent}.", "Each $B = B_{j_0 \\ldots j_p}$ is an \\'etale $A$-algebra.", "Hence $\\Omega_B = \\Omega_A \\otimes_A B$ and we conclude", "$\\theta$ extends uniquely to a derivation $\\theta_B : B \\to B$.", "These maps define an endomorphism of the {\\v C}ech complex", "and define the desired operators on the cohomology groups.", "\\medskip\\noindent", "In the general case we use a hypercovering of $V$ by", "affine opens, exactly as in the first part of the proof of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-hypercoverings}.", "We omit the details." ], "refs": [ "schemes-lemma-characterize-separated", "coherent-lemma-cech-cohomology-quasi-coherent", "coherent-lemma-hypercoverings" ], "ref_ids": [ 7710, 3286, 3303 ] } ], "ref_ids": [] }, { "id": 9769, "type": "theorem", "label": "local-cohomology-lemma-etale-frobenius", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-etale-frobenius", "contents": [ "Let $p$ be a prime number. Let $A$ be a ring with $p = 0$.", "Denote $F : A \\to A$, $a \\mapsto a^p$ the Frobenius endomorphism.", "If $V \\to \\Spec(A)$ is quasi-compact, quasi-separated,", "and \\'etale, then there exists an isomorphism", "$R\\Gamma(V, \\mathcal{O}_V) \\otimes_{A, F}^\\mathbf{L} A \\cong", "R\\Gamma(V, \\mathcal{O}_V)$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the relative Frobenius morphism", "$$", "V \\longrightarrow V \\times_{\\Spec(A), \\Spec(F)} \\Spec(A)", "$$", "of $V$ over $A$ is an isomorphism, see", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-relative-frobenius-etale}.", "Thus the lemma follows from cohomology and base change, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-compare-base-change}.", "Observe that since $V$ is \\'etale over $A$, it is flat over $A$." ], "refs": [ "etale-lemma-relative-frobenius-etale", "perfect-lemma-compare-base-change" ], "ref_ids": [ 10708, 7028 ] } ], "ref_ids": [] }, { "id": 9770, "type": "theorem", "label": "local-cohomology-lemma-map-tor-1-zero", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-map-tor-1-zero", "contents": [ "Let $R$ be a ring. Let $M \\to M'$ be a map of $R$-modules with", "$M$ of finite presentation", "such that $\\text{Tor}_1^R(M, N) \\to \\text{Tor}_1^R(M', N)$ is zero for all", "$R$-modules $N$. Then $M \\to M'$ factors through a free $R$-module." ], "refs": [], "proofs": [ { "contents": [ "We may choose a map of short exact sequences", "$$", "\\xymatrix{", "0 \\ar[r] &", "K \\ar[r] \\ar[d] &", "R^{\\oplus r} \\ar[r] \\ar[d] &", "M \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "K' \\ar[r] &", "\\bigoplus_{i \\in I} R \\ar[r] &", "M' \\ar[r] &", "0", "}", "$$", "whose right vertical arrow is the given map.", "We can factor this map through the short exact sequence", "\\begin{equation}", "\\label{equation-pushout}", "0 \\to K' \\to E \\to M \\to 0", "\\end{equation}", "which is the pushout of the first short exact sequence by $K \\to K'$.", "By a diagram chase we see that the assumption in the lemma", "implies that the boundary map $\\text{Tor}_1^R(M, N) \\to K' \\otimes_R N$", "induced by (\\ref{equation-pushout}) is zero, i.e., the sequence", "(\\ref{equation-pushout}) is universally exact. This implies by", "Algebra, Lemma \\ref{algebra-lemma-universally-exact-split}", "that (\\ref{equation-pushout}) is split (this is where we use that", "$M$ is of finite presentation). Hence the map $M \\to M'$", "factors through $\\bigoplus_{i \\in I} R$ and we win." ], "refs": [ "algebra-lemma-universally-exact-split" ], "ref_ids": [ 808 ] } ], "ref_ids": [] }, { "id": 9771, "type": "theorem", "label": "local-cohomology-lemma-characterize-vanishing-tor-ext-above-e", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-characterize-vanishing-tor-ext-above-e", "contents": [ "Let $R$ be a ring. Let $\\alpha : M \\to M'$ be a map of $R$-modules.", "Let $P_\\bullet \\to M$ and $P'_\\bullet \\to M'$ be resolutions by", "projective $R$-modules. Let $e \\geq 0$ be an integer.", "Consider the following conditions", "\\begin{enumerate}", "\\item We can find a map of complexes $a_\\bullet : P_\\bullet \\to P'_\\bullet$", "inducing $\\alpha$ on cohomology with $a_i = 0$ for $i > e$.", "\\item We can find a map of complexes $a_\\bullet : P_\\bullet \\to P'_\\bullet$", "inducing $\\alpha$ on cohomology with $a_{e + 1} = 0$.", "\\item The map $\\Ext^i_R(M', N) \\to \\Ext^i_R(M, N)$ is zero", "for all $R$-modules $N$ and $i > e$.", "\\item The map $\\Ext^{e + 1}_R(M', N) \\to \\Ext^{e + 1}_R(M, N)$ is zero", "for all $R$-modules $N$.", "\\item Let $N = \\Im(P'_{e + 1} \\to P'_e)$ and denote", "$\\xi \\in \\Ext^{e + 1}_R(M', N)$ the canonical element (see proof).", "Then $\\xi$ maps to zero in $\\Ext^{e + 1}_R(M, N)$.", "\\item The map $\\text{Tor}_i^R(M, N) \\to \\text{Tor}_i^R(M', N)$", "is zero for all $R$-modules $N$ and $i > e$.", "\\item The map $\\text{Tor}_{e + 1}^R(M, N) \\to \\text{Tor}_{e + 1}^R(M', N)$", "is zero for all $R$-modules $N$.", "\\end{enumerate}", "Then we always have the implications", "$$", "(1) \\Leftrightarrow (2) \\Leftrightarrow (3) \\Leftrightarrow (4)", "\\Leftrightarrow (5) \\Rightarrow (6) \\Leftrightarrow (7)", "$$", "If $M$ is $(-e - 1)$-pseudo-coherent (for example if $R$ is Noetherian", "and $M$ is a finite $R$-module), then all conditions", "are equivalent." ], "refs": [], "proofs": [ { "contents": [ "It is clear that (2) implies (1). If $a_\\bullet$ is as in (1), then", "we can consider the map of complexes $a'_\\bullet : P_\\bullet \\to P'_\\bullet$", "with $a'_i = a_i$ for $i \\leq e + 1$ and $a'_i = 0$ for $i \\geq e + 1$", "to get a map of complexes as in (2). Thus (1) is equivalent to (2).", "\\medskip\\noindent", "By the construction of the $\\Ext$ and $\\text{Tor}$ functors using", "resolutions (Algebra, Sections \\ref{algebra-section-ext}", "and \\ref{algebra-section-tor}) we see ", "that (1) and (2) imply all of the other conditions.", "\\medskip\\noindent", "It is clear that (3) implies (4) implies (5). Let $N$ be as in (5).", "The canonical map $\\tilde \\xi : P'_{e + 1} \\to N$ precomposed with", "$P'_{e + 2} \\to P'_{e + 1}$ is zero. Hence we may consider the class", "$\\xi$ of $\\tilde \\xi$ in", "$$", "\\Ext^{e + 1}_R(M', N) =", "\\frac{\\Ker(\\Hom(P'_{e + 1}, N \\to \\Hom(P'_{e + 2}, N)}{", "\\Im(\\Hom(P'_e, N \\to \\Hom(P'_{e + 1}, N)}", "$$", "Choose a map of complexes $a_\\bullet : P_\\bullet \\to P'_\\bullet$", "lifting $\\alpha$, see Derived Categories, Lemma", "\\ref{derived-lemma-morphisms-lift-projective}.", "If $\\xi$ maps to zero in $\\Ext^{e + 1}_R(M', N)$, then", "we find a map $\\varphi : P_e \\to N$ such that", "$\\tilde \\xi \\circ a_{e + 1} = \\varphi \\circ d$.", "Thus we obtain a map of complexes", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "P_{e + 1} \\ar[r] \\ar[d]^0 &", "P_e \\ar[r] \\ar[d]^{a_e - \\varphi} &", "P_{e - 1} \\ar[r] \\ar[d]^{a_{e - 1}} &", "\\ldots \\\\", "\\ldots \\ar[r] &", "P'_{e + 1} \\ar[r] &", "P'_e \\ar[r] &", "P'_{e - 1} \\ar[r] &", "\\ldots", "}", "$$", "as in (2). Hence (1) -- (5) are equivalent.", "\\medskip\\noindent", "The equivalence of (6) and (7) follows from dimension shifting;", "we omit the details.", "\\medskip\\noindent", "Assume $M$ is $(-e - 1)$-pseudo-coherent. (The parenthetical statement", "in the lemma follows from More on Algebra, Lemma", "\\ref{more-algebra-lemma-Noetherian-pseudo-coherent}.)", "We will show that (7) implies (4) which", "finishes the proof. We will use induction on $e$.", "The base case is $e = 0$. Then $M$ is of finite presentation by", "More on Algebra, Lemma \\ref{more-algebra-lemma-n-pseudo-module}", "and we conclude from Lemma \\ref{lemma-map-tor-1-zero} that", "$M \\to M'$ factors through a free module. Of course if $M \\to M'$", "factors through a free module, then $\\Ext^i_R(M', N) \\to \\Ext^i_R(M, N)$", "is zero for all $i > 0$ as desired.", "Assume $e > 0$. We may choose a map of short exact sequences", "$$", "\\xymatrix{", "0 \\ar[r] &", "K \\ar[r] \\ar[d] &", "R^{\\oplus r} \\ar[r] \\ar[d] &", "M \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "K' \\ar[r] &", "\\bigoplus_{i \\in I} R \\ar[r] &", "M' \\ar[r] &", "0", "}", "$$", "whose right vertical arrow is the given map. We obtain", "$\\text{Tor}_{i + 1}^R(M, N) = \\text{Tor}^R_i(K, N)$", "and $\\Ext^{i + 1}_R(M, N) = \\Ext^i_R(K, N)$ for $i \\geq 1$", "and all $R$-modules $N$ and similarly for $M', K'$.", "Hence we see that $\\text{Tor}_e^R(K, N) \\to \\text{Tor}_e^R(K', N)$", "is zero for all $R$-modules $N$. By More on Algebra, Lemma", "\\ref{more-algebra-lemma-cone-pseudo-coherent} we see that $K$", "is $(-e)$-pseudo-coherent. By induction we conclude that", "$\\Ext^e(K', N) \\to \\Ext^e(K, N)$ is zero for all $R$-modules", "$N$, which gives what we want." ], "refs": [ "derived-lemma-morphisms-lift-projective", "more-algebra-lemma-Noetherian-pseudo-coherent", "more-algebra-lemma-n-pseudo-module", "local-cohomology-lemma-map-tor-1-zero", "more-algebra-lemma-cone-pseudo-coherent" ], "ref_ids": [ 1860, 10160, 10147, 9770, 10145 ] } ], "ref_ids": [] }, { "id": 9772, "type": "theorem", "label": "local-cohomology-lemma-cd-sequence-Koszul", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-cd-sequence-Koszul", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$.", "For all $n \\geq 1$ there exists an $m > n$ such that the map", "$A/I^m \\to A/I^n$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-characterize-vanishing-tor-ext-above-e} with", "$e = \\text{cd}(A, I)$." ], "refs": [ "local-cohomology-lemma-characterize-vanishing-tor-ext-above-e" ], "proofs": [ { "contents": [ "Let $\\xi \\in \\Ext^{e + 1}_A(A/I^n, N)$ be the element constructed in", "Lemma \\ref{lemma-characterize-vanishing-tor-ext-above-e} part (5).", "Since $e = \\text{cd}(A, I)$ we have", "$0 = H^{e + 1}_Z(N) = H^{e + 1}_I(N) = \\colim \\Ext^{e + 1}(A/I^m, N)$", "by Dualizing Complexes, Lemmas", "\\ref{dualizing-lemma-local-cohomology-noetherian} and", "\\ref{dualizing-lemma-local-cohomology-ext}.", "Thus we may pick $m \\geq n$ such that $\\xi$ maps to", "zero in $\\Ext^{e + 1}_A(A/I^m, N)$ as desired." ], "refs": [ "local-cohomology-lemma-characterize-vanishing-tor-ext-above-e", "dualizing-lemma-local-cohomology-noetherian", "dualizing-lemma-local-cohomology-ext" ], "ref_ids": [ 9771, 2823, 2812 ] } ], "ref_ids": [ 9771 ] }, { "id": 9773, "type": "theorem", "label": "local-cohomology-lemma-maps-zero-fixed-torsion", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-maps-zero-fixed-torsion", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$. For every $m \\geq 0$", "and $i > 0$ there exist a $c = c(A, I, m, i) \\geq 0$ such that", "for every $A$-module $M$ annihilated by $I^m$ the map", "$$", "\\text{Tor}^A_i(M, A/I^n) \\to \\text{Tor}^A_i(M, A/I^{n - c})", "$$", "is zero for all $n \\geq c$." ], "refs": [], "proofs": [ { "contents": [ "By induction on $i$. Base case $i = 1$. The short exact sequence", "$0 \\to I^n \\to A \\to A/I^n \\to 0$ determines an injection", "$\\text{Tor}_1^A(M, A/I^n) \\subset I^n \\otimes_A M$, see", "Algebra, Remark \\ref{algebra-remark-Tor-ring-mod-ideal}.", "As $M$ is annihilated by $I^m$ we see that the map", "$I^n \\otimes_A M \\to I^{n - m} \\otimes_A M$ is", "zero for $n \\geq m$. Hence the result holds with $c = m$.", "\\medskip\\noindent", "Induction step. Let $i > 1$ and assume $c$ works for $i - 1$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-tor-strictly-pro-zero}", "applied to $M = A/I^m$ we can choose $c' \\geq 0$ such that", "$\\text{Tor}_i(A/I^m, A/I^n) \\to \\text{Tor}_i(A/I^m, A/I^{n - c'})$", "is zero for $n \\geq c'$. Let $M$ be annihilated by $I^m$. Choose a short", "exact sequence", "$$", "0 \\to S \\to \\bigoplus\\nolimits_{i \\in I} A/I^m \\to M \\to 0", "$$", "The corresponding long exact sequence of tors gives an exact sequence", "$$", "\\text{Tor}_i^A(\\bigoplus\\nolimits_{i \\in I} A/I^m, A/I^n) \\to", "\\text{Tor}_i^A(M, A/I^n) \\to", "\\text{Tor}_{i - 1}^A(S, A/I^n)", "$$", "for all integers $n \\geq 0$. If $n \\geq c + c'$, then the map", "$\\text{Tor}_{i - 1}^A(S, A/I^n) \\to \\text{Tor}_{i - 1}^A(S, A/I^{n - c})$", "is zero and the map $\\text{Tor}_i^A(A/I^m, A/I^{n - c}) \\to ", "\\text{Tor}_i^A(A/I^m, A/I^{n - c - c'})$ is zero. Combined with the", "short exact sequences this implies the result holds for $i$ with", "constant $c + c'$." ], "refs": [ "algebra-remark-Tor-ring-mod-ideal", "more-algebra-lemma-tor-strictly-pro-zero" ], "ref_ids": [ 1570, 9954 ] } ], "ref_ids": [] }, { "id": 9774, "type": "theorem", "label": "local-cohomology-lemma-annihilates-affine", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-annihilates-affine", "contents": [ "Let $I = (a_1, \\ldots, a_t)$ be an ideal of a Noetherian ring $A$.", "Set $a = a_1$ and denote $B = A[\\frac{I}{a}]$ the affine blowup algebra.", "There exists a $c > 0$ such that $\\text{Tor}_i^A(B, M)$ is annihilated", "by $I^c$ for all $A$-modules $M$ and $i \\geq t$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $B$ is the quotient of", "$A[x_2, \\ldots, x_t]/(a_1x_2 - a_2, \\ldots, a_1x_t - a_t)$", "by its $a_1$-torsion, see", "Algebra, Lemma \\ref{algebra-lemma-affine-blowup-quotient-description}. Let", "$$", "B_\\bullet = \\text{Koszul complex on }a_1x_2 - a_2, \\ldots, a_1x_t - a_t", "\\text{ over }A[x_2, \\ldots, x_t]", "$$", "viewed as a chain complex sitting in degrees $(t - 1), \\ldots, 0$.", "The complex $B_\\bullet[1/a_1]$ is isomorphic to the Koszul complex", "on $x_2 - a_2/a_1, \\ldots, x_t - a_t/a_1$ which is a regular sequence", "in $A[1/a_1][x_2, \\ldots, x_t]$. Since regular sequences are", "Koszul regular, we conclude that the augmentation", "$$", "\\epsilon : B_\\bullet \\longrightarrow B", "$$", "is a quasi-isomorphism after inverting $a_1$. Since the homology modules", "of the cone $C_\\bullet$ on $\\epsilon$ are finite $A[x_2, \\ldots, x_n]$-modules", "and since $C_\\bullet$ is bounded,", "we conclude that there exists a $c \\geq 0$ such that $a_1^c$", "annihilates all of these. By", "Derived Categories, Lemma \\ref{derived-lemma-trick-vanishing-composition}", "this implies that, after possibly replacing $c$ by a larger integer,", "that $a_1^c$ is zero on $C_\\bullet$ in $D(A)$.", "The proof is finished once the reader contemplates", "the distinguished triangle", "$$", "B_\\bullet \\otimes_A^\\mathbf{L} M \\to", "B \\otimes_A^\\mathbf{L} M \\to", "C_\\bullet \\otimes_A^\\mathbf{L} M", "$$", "Namely, the first term is represented by $B_\\bullet \\otimes_A M$ which", "is sitting in homological degrees $(t - 1), \\ldots, 0$", "in view of the fact that the terms in the Koszul complex $B_\\bullet$", "are free (and hence flat) $A$-modules. Whence", "$\\text{Tor}_i^A(B, M) = H_i(C_\\bullet \\otimes_A^\\mathbf{L} M)$", "for $i > t - 1$ and this is annihilated by $a_1^c$.", "Since $a_1^cB = I^cB$ and since the tor module is a module", "over $B$ we conclude." ], "refs": [ "algebra-lemma-affine-blowup-quotient-description", "derived-lemma-trick-vanishing-composition" ], "ref_ids": [ 754, 1817 ] } ], "ref_ids": [] }, { "id": 9775, "type": "theorem", "label": "local-cohomology-lemma-compute-tor-Iq", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-compute-tor-Iq", "contents": [ "In the situation above, for $q \\geq q(A, I)$ and any $A$-module $M$ we have", "$$", "R\\Gamma(X, Lp^*\\widetilde{M}(q)) \\cong M \\otimes_A^\\mathbf{L} I^q", "$$", "in $D(A)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a free resolution $F_\\bullet \\to M$. Then $\\widetilde{F}_\\bullet$", "is a flat resolution of $\\widetilde{M}$. Hence $Lp^*\\widetilde{M}$", "is given by the complex $p^*\\widetilde{F}_\\bullet$. Thus", "$Lp^*\\widetilde{M}(q)$ is given by the complex $p^*\\widetilde{F}_\\bullet(q)$.", "Since $p^*\\widetilde{F}_i(q)$ are right acyclic for $\\Gamma(X, -)$ by", "our choice of $q \\geq q(A, I)$ and since we have", "$\\Gamma(X, p^*\\widetilde{F}_i(q)) = I^qF_i$ by our choice of $q \\geq q(A, I)$,", "we get that $R\\Gamma(X, Lp^*\\widetilde{M}(q))$ is given by the complex", "with terms $I^qF_i$ by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-acyclicity-lemma-global}.", "The result follows as the complex $I^qF_\\bullet$ computes", "$M \\otimes_A^\\mathbf{L} I^q$ by definition." ], "refs": [ "perfect-lemma-acyclicity-lemma-global" ], "ref_ids": [ 6948 ] } ], "ref_ids": [] }, { "id": 9776, "type": "theorem", "label": "local-cohomology-lemma-annihilates", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-annihilates", "contents": [ "In the situation above, let $t$ be an upper bound on the number of", "generators for $I$. There exists an integer $c = c(A, I) \\geq 0$", "such that for any $A$-module $M$ the cohomology sheaves", "$H^j(Lp^*\\widetilde{M})$ are annihilated by $I^c$ for $j \\leq -t$." ], "refs": [], "proofs": [ { "contents": [ "Say $I = (a_1, \\ldots, a_t)$. The question is affine local on $X$.", "For $1 \\leq i \\leq t$ let $B_i = A[\\frac{I}{a_i}]$ be the affine", "blowup algebra. Then $X$ has an affine open covering by", "the spectra of the rings $B_i$, see", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-affine}.", "By the description of derived pullback given in", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-pullback}", "we conclude it suffices to prove that for each $i$ there exists a $c \\geq 0$", "such that", "$$", "\\text{Tor}_j^A(B_i, M)", "$$", "is annihilated by $I^c$ for $j \\geq t$. This is", "Lemma \\ref{lemma-annihilates-affine}." ], "refs": [ "divisors-lemma-blowing-up-affine", "perfect-lemma-quasi-coherence-pullback", "local-cohomology-lemma-annihilates-affine" ], "ref_ids": [ 8052, 6944, 9774 ] } ], "ref_ids": [] }, { "id": 9777, "type": "theorem", "label": "local-cohomology-lemma-annihilates-tors", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-annihilates-tors", "contents": [ "In the situation above, let $t$ be an upper bound on the number of", "generators for $I$. There exists an integer $c = c(A, I) \\geq 0$", "such that for any $A$-module $M$ the tor modules", "$\\text{Tor}_i^A(M, A/I^q)$ are annihilated by $I^c$ for $i > t$", "and all $q \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $q(A, I)$ be as above. For $q \\geq q(A, I)$ we have", "$$", "R\\Gamma(X, Lp^*\\widetilde{M}(q)) = M \\otimes_A^\\mathbf{L} I^q", "$$", "by Lemma \\ref{lemma-compute-tor-Iq}.", "We have a bounded and convergent spectral sequence", "$$", "H^a(X, H^b(Lp^*\\widetilde{M}(q))) \\Rightarrow", "\\text{Tor}_{-a - b}^A(M, I^q)", "$$", "by Derived Categories of Schemes, Lemma \\ref{perfect-lemma-spectral-sequence}.", "Let $d$ be an integer as in Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-vanishing-nr-affines-quasi-separated}", "(actually we can take $d = t$, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-vanishing-nr-affines}).", "Then we see that $H^{-i}(X, Lp^*\\widetilde{M}(q)) = \\text{Tor}_i^A(M, I^q)$", "has a finite filtration with at most $d$ steps whose graded are", "subquotients of the modules", "$$", "H^a(X, H^{- i - a}(Lp^*\\widetilde{M})(q)),\\quad", "a = 0, 1, \\ldots, d - 1", "$$", "If $i \\geq t$ then all of these modules are annihilated", "by $I^c$ where $c = c(A, I)$ is as in Lemma \\ref{lemma-annihilates}", "because the cohomology sheaves $H^{- i - a}(Lp^*\\widetilde{M})$", "are all annihilated by $I^c$ by the lemma. Hence we see that", "$\\text{Tor}_i^A(M, I^q)$ is annihilated by $I^{dc}$ for", "$q \\geq q(A, I)$ and $i \\geq t$. Using the short exact sequence", "$0 \\to I^q \\to A \\to A/I^q \\to 0$ we find that", "$\\text{Tor}_i(M, A/I^q)$ is annihilated by $I^{dc}$ for $q \\geq q(A, I)$", "and $i > t$. We conclude that $I^m$ with $m = \\max(dc, q(A, I) - 1)$", "annihilates $\\text{Tor}_i^A(M, A/I^q)$ for all $q \\geq 0$", "and $i > t$ as desired." ], "refs": [ "local-cohomology-lemma-compute-tor-Iq", "perfect-lemma-spectral-sequence", "coherent-lemma-vanishing-nr-affines-quasi-separated", "coherent-lemma-vanishing-nr-affines", "local-cohomology-lemma-annihilates" ], "ref_ids": [ 9775, 6949, 3294, 3292, 9776 ] } ], "ref_ids": [] }, { "id": 9778, "type": "theorem", "label": "local-cohomology-lemma-tor-maps-vanish", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-tor-maps-vanish", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$. Let $t \\geq 0$", "be an upper bound on the number of generators of $I$.", "There exist $N, c \\geq 0$ such that the maps", "$$", "\\text{Tor}_{t + 1}^A(M, A/I^n) \\to \\text{Tor}_{t + 1}^A(M, A/I^{n - c})", "$$", "are zero for any $A$-module $M$ and all $n \\geq N$." ], "refs": [], "proofs": [ { "contents": [ "Let $c_1$ be the constant found in Lemma \\ref{lemma-annihilates-tors}.", "Please keep in mind that this constant $c_1$ works for $\\text{Tor}_i$ for all", "$i > t$ simultaneously.", "\\medskip\\noindent", "Say $I = (a_1, \\ldots, a_t)$. For an $A$-module $M$ we set", "$$", "\\ell(M) =", "\\#\\{i \\mid 1 \\leq i \\leq t,\\ a_i^{c_1}\\text{ is zero on }M\\}", "$$", "This is an element of $\\{0, 1, \\ldots, t\\}$.", "We will prove by descending induction on $0 \\leq s \\leq t$", "the following statement $H_s$: there exist $N, c \\geq 0$ such that", "for every module $M$ with $\\ell(M) \\geq s$ the maps", "$$", "\\text{Tor}_{t + 1 + i}^A(M, A/I^n) \\to \\text{Tor}_{t + 1 + i}^A(M, A/I^{n - c})", "$$", "are zero for $i = 0, \\ldots, s$ for all $n \\geq N$.", "\\medskip\\noindent", "Base case: $s = t$. If $\\ell(M) = t$, then $M$ is annihilated by", "$(a_1^{c_1}, \\ldots, a_t^{c_1}\\}$ and hence by", "$I^{t(c_1 - 1) + 1}$. We conclude from", "Lemma \\ref{lemma-maps-zero-fixed-torsion}", "that $H_t$ holds by taking $c = N$ to be the maximum of the integers", "$c(A, I, t(c_1 - 1) + 1, t + 1), \\ldots, c(A, I, t(c_1 - 1) + 1, 2t + 1)$", "found in the lemma.", "\\medskip\\noindent", "Induction step. Say $0 \\leq s < t$ we have $N, c$ as in $H_{s + 1}$.", "Consider a module $M$ with $\\ell(M) = s$. Then we can choose an $i$", "such that $a_i^{c_1}$ is nonzero on $M$. It follows that", "$\\ell(M[a_i^c]) \\geq s + 1$ and $\\ell(M/a_i^{c_1}M) \\geq s + 1$", "and the induction hypothesis applies to them.", "Consider the exact sequence", "$$", "0 \\to M[a_i^{c_1}] \\to M \\xrightarrow{a_i^{c_1}} M \\to M/a_i^{c_1}M \\to 0", "$$", "Denote $E \\subset M$ the image of the middle arrow.", "Consider the corresponding diagram of Tor modules", "$$", "\\xymatrix{", "&", "&", "\\text{Tor}_{i + 1}(M/a_i^{c_1}M, A/I^q) \\ar[d] \\\\", "\\text{Tor}_i(M[a_i^{c_1}], A/I^q) \\ar[r] &", "\\text{Tor}_i(M, A/I^q) \\ar[r] \\ar[rd]^0 &", "\\text{Tor}_i(E, A/I^q) \\ar[d] \\\\", "&", "&", "\\text{Tor}_i(M, A/I^q)", "}", "$$", "with exact rows and columns (for every $q$). The south-east arrow", "is zero by our choice of $c_1$. We conclude that the", "module $\\text{Tor}_i(M, A/I^q)$ is sandwiched between", "a quotient module of $\\text{Tor}_i(M[a_i^{c_1}], A/I^q)$", "and a submodule of $\\text{Tor}_{i + 1}(M/a_i^{c_1}M, A/I^q)$.", "Hence we conclude $H_s$ holds with $N$ replaced by $N + c$", "and $c$ replaced by $2c$. Some details omitted." ], "refs": [ "local-cohomology-lemma-annihilates-tors", "local-cohomology-lemma-maps-zero-fixed-torsion" ], "ref_ids": [ 9777, 9773 ] } ], "ref_ids": [] }, { "id": 9779, "type": "theorem", "label": "local-cohomology-lemma-bound-q-and-d", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-bound-q-and-d", "contents": [ "With $q_0 = q(S)$ and $d = d(S)$ as above, we have", "\\begin{enumerate}", "\\item for $n \\geq 1$, $q \\geq q_0$, and $i > 0$ we have", "$H^i(X, \\mathcal{O}_{Y_n}(q)) = 0$,", "\\item for $n \\geq 1$ and $q \\geq q_0$ we have", "$H^0(X, \\mathcal{O}_{Y_n}(q)) = I^q/I^{q + n}$,", "\\item for $q \\geq q_0$ and $i > 0$ we have", "$H^i(X, \\mathcal{O}_X(q)) = 0$,", "\\item for $q \\geq q_0$ we have", "$H^0(X, \\mathcal{O}_X(q)) = I^q$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $I = A$, then $X$ is affine and the statements are trivial.", "Hence we may and do assume $I \\not = A$.", "Thus $Y$ and $X$ are nonempty schemes.", "\\medskip\\noindent", "Let us prove (1) and (2) by induction on $n$. The base case $n = 1$", "is our definition of $q_0$ as $Y_1 = Y$.", "Recall that $\\mathcal{O}_X(1) = \\mathcal{O}_X(-Y)$.", "Hence we have a short exact sequence", "$$", "0 \\to \\mathcal{O}_{Y_n}(1) \\to \\mathcal{O}_{Y_{n + 1}} \\to \\mathcal{O}_Y \\to 0", "$$", "Hence for $i > 0$ we find", "$$", "H^i(X, \\mathcal{O}_{Y_n}(q + 1)) \\to", "H^i(X, \\mathcal{O}_{Y_{n + 1}}(q)) \\to", "H^i(X, \\mathcal{O}_{Y}(q))", "$$", "and we obtain the desired vanishing of the middle term from the given", "vanishing of the outer terms. For $i = 0$ we obtain a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "I^{q + 1}/I^{q + 1 + n} \\ar[d] \\ar[r] &", "I^q/I^{q + 1 + n} \\ar[d] \\ar[r] &", "I^q/I^{q + 1} \\ar[d] \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "H^0(X, \\mathcal{O}_{Y_n}(q + 1)) \\ar[r] &", "H^0(X, \\mathcal{O}_{Y_{n + 1}}(q)) \\ar[r] &", "H^0(Y, \\mathcal{O}_Y(q)) \\ar[r] &", "0", "}", "$$", "with exact rows for $q \\geq q_0$ (for the bottom row observe that", "the next term in the long exact cohomology sequence vanishes for", "$q \\geq q_0$). Since $q \\geq q_0$ the left and right vertical arrows", "are isomorphisms and we conclude the middle one is too.", "\\medskip\\noindent", "We omit the proofs of (3) and (4) which are similar.", "In fact, one can deduce (3) and (4) from (1) and (2)", "using the theorem on formal functors (but this would be", "overkill)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9780, "type": "theorem", "label": "local-cohomology-lemma-almost-exactness", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-almost-exactness", "contents": [ "Let $0 \\to K \\to L \\to M \\to 0$ be a short exact sequence of $A$-modules", "such that $K$ and $L$ are annihilated by $I^n$ and $M$ is an", "$(A, n, c)$-module. Then the kernel of $p^*K \\to p^*L$", "is scheme theoretically supported on $Y_c$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\Spec(B) \\subset X$ be an affine open. The restriction of the exact", "sequence over $\\Spec(B)$ corresponds to the sequence of $B$-modules", "$$", "K \\otimes_A B \\to L \\otimes_A B \\to M \\otimes_A B \\to 0", "$$", "which is isomorphismic to the sequence", "$$", "K \\otimes_{A/I^n} B/I^nB \\to", "L \\otimes_{A/I^n} B/I^nB \\to", "M \\otimes_{A/I^n} B/I^nB \\to 0", "$$", "Hence the kernel of the first map is the image of the module", "$\\text{Tor}_1^{A/I^n}(M, B/I^nB)$. Recall that the exceptional", "divisor $Y$ is cut out by $I\\mathcal{O}_X$. ", "Hence it suffices to show that", "$\\text{Tor}_1^{A/I^n}(M, B/I^nB)$ is annihilated by $I^c$. Since", "multiplication by $a \\in I^c$ on $M$ factors through a finite", "free $A/I^n$-module, this is clear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9781, "type": "theorem", "label": "local-cohomology-lemma-annihilated", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-annihilated", "contents": [ "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Then $\\mathcal{F}$ is scheme theoretically", "supported on $Y_c$ if and only if the canonical map", "$\\mathcal{F} \\to \\mathcal{F}(c)$ is zero." ], "refs": [], "proofs": [ { "contents": [ "This is true because $\\mathcal{O}_X \\to \\mathcal{O}_X(1)$", "vanishes exactly along $Y$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9782, "type": "theorem", "label": "local-cohomology-lemma-vanishing-coh-almost-projective", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-vanishing-coh-almost-projective", "contents": [ "With $q_0 = q(S)$ and $d = d(S)$ as above, suppose we have", "integers $n \\geq c \\geq 0$, an $(A, n, c)$-module $M$,", "an index $i \\in \\{0, 1, \\ldots, d\\}$, and an integer $q$.", "Then we distinguish the following cases", "\\begin{enumerate}", "\\item In the case $i = d \\geq 1$ and $q \\geq q_0$ we have", "$H^d(X, p^*M(q)) = 0$.", "\\item In the case $i = d - 1 \\geq 1$ and $q \\geq q_0$ we have", "$H^{d - 1}(X, p^*M(q)) = 0$.", "\\item In the case $d - 1 > i > 0$ and $q \\geq q_0 + (d - 1 - i)c$", "the map", "$H^i(X, p^*M(q)) \\to H^i(X, p^*M(q - (d - 1 - i)c))$", "is zero.", "\\item In the case $i = 0$, $d \\in \\{0, 1\\}$, and $q \\geq q_0$, there", "is a surjection", "$$", "I^qM \\longrightarrow H^0(X, p^*M(q))", "$$", "\\item In the case $i = 0$, $d > 1$, and $q \\geq q_0 + (d - 1)c$ the map", "$$", "H^0(X, p^*M(q)) \\to H^0(X, p^*M(q - (d - 1)c))", "$$", "has image contained in the image of the canonical map", "$I^{q - (d - 1)c}M \\to H^0(X, p^*M(q - (d - 1)c))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be an $(A, n, c)$-module. Choose a short exact sequence", "$$", "0 \\to K \\to (A/I^n)^{\\oplus r} \\to M \\to 0", "$$", "We will use below that $K$ is an $(A, n, c)$-module, see More on Algebra,", "Lemma \\ref{more-algebra-lemma-ses-near-projective}.", "Consider the corresponding exact sequence", "$$", "p^*K \\to (\\mathcal{O}_{Y_n})^{\\oplus r} \\to p^*M \\to 0", "$$", "We split this into short exact sequences", "$$", "0 \\to \\mathcal{F} \\to p^*K \\to \\mathcal{G} \\to 0", "\\quad\\text{and}\\quad", "0 \\to \\mathcal{G} \\to (\\mathcal{O}_{Y_n})^{\\oplus r} \\to p^*M \\to 0", "$$", "By Lemma \\ref{lemma-almost-exactness} the coherent module $\\mathcal{F}$", "is scheme theoretically supported on $Y_c$.", "\\medskip\\noindent", "Proof of (1). Assume $d > 0$. We have to prove", "$H^d(X, p^*M(q)) = 0$ for $q \\geq q_0$.", "By the vanishing of the cohomology of twists of $\\mathcal{G}$ in degrees $> d$", "and the long exact cohomology sequence associated to the second", "short exact sequence above, it suffices to prove that", "$H^d(X, \\mathcal{O}_{Y_n}(q)) = 0$.", "This is true by Lemma \\ref{lemma-bound-q-and-d}.", "\\medskip\\noindent", "Proof of (2). Assume $d > 1$. We have to prove", "$H^{d - 1}(X, p^*M(q)) = 0$ for $q \\geq q_0$.", "Arguing as in the previous paragraph, we see that it suffices", "to show that $H^d(X, \\mathcal{G}(q)) = 0$. Using the first", "short exact sequence and the vanishing of the cohomology", "of twists of $\\mathcal{F}$ in degrees $> d$ we see that it suffices", "to show $H^d(X, p^*K(q))$ is zero which is", "true by (1) and the fact that $K$ is an $(A, n, c)$-module (see above).", "\\medskip\\noindent", "Proof of (3). Let $0 < i < d - 1$ and assume the statement holds for $i + 1$", "except in the case $i = d - 2$ we have statement (2).", "Using the long exact sequence of cohomology associated to the second", "short exact sequence above we find an injection", "$$", "H^i(X, p^*M(q - (d - 1 - i)c)) \\subset", "H^{i + 1}(X, \\mathcal{G}(q - (d - 1 - i)c))", "$$", "as $q - (d - 1 - i)c \\geq q_0$ gives the vanishing of", "$H^i(X, \\mathcal{O}_{Y_n}(q - (d - 1 - i)c))$", "(see above). Thus it suffices to show that the map", "$H^{i + 1}(X, \\mathcal{G}(q)) \\to H^{i + 1}(X, \\mathcal{G}(q - (d - 1 - i)c))$", "is zero. To study this, we consider the maps of exact sequences", "$$", "\\xymatrix{", "H^{i + 1}(X, p^*K(q)) \\ar[r] \\ar[d] &", "H^{i + 1}(X, \\mathcal{G}(q)) \\ar[r] \\ar[d] \\ar@{..>}[ld] &", "H^{i + 2}(X, \\mathcal{F}(q)) \\ar[d] \\\\", "H^{i + 1}(X, p^*K(q - c)) \\ar[r] \\ar[d] &", "H^{i + 1}(X, \\mathcal{G}(q - c)) \\ar[r] \\ar[d] &", "H^{i + 2}(X, \\mathcal{F}(q - c)) \\\\", "H^{i + 1}(X, p^*K(q - (d - 1 - i)c)) \\ar[r] &", "H^{i + 1}(X, \\mathcal{G}(q - (d - 1 - i)c))", "}", "$$", "Since $\\mathcal{F}$ is scheme theoretically supported on $Y_c$", "we see that the canonical map", "$\\mathcal{G}(q) \\to \\mathcal{G}(q - c)$ factors through", "$p^*K(q - c)$ by Lemma \\ref{lemma-annihilated}.", "This gives the dotted arrow in the diagram. (In fact, for the proof it", "suffices to observe that the vertical arrow on the extreme right is", "zero in order to get the dotted arrow as a map of sets.)", "Thus it suffices to show that", "$H^{i + 1}(X, p^*K(q - c)) \\to", "H^{i + 1}(X, p^*K(q - (d - 1 - i)c))$", "is zero. If $i = d - 2$, then the source of this arrow", "is zero by (2) as $q - c \\geq q_0$ and $K$ is an $(A, n, c)$-module.", "If $i < d - 2$, then as $K$ is an $(A, n, c)$-module, we get from the", "induction hypothesis that the map is indeed zero", "since $q - c - (q - (d - 1 - i)c) = (d - 2 - i)c = (d - 1 - (i + 1))c$", "and since $q - c \\geq q_0 + (d - 1 - (i + 1))c$.", "In this way we conclude the proof of (3).", "\\medskip\\noindent", "Proof of (4). Assume $d \\in \\{0, 1\\}$ and $q \\geq q_0$.", "Then the first short exact sequence gives a surjection", "$H^1(X, p^*K(q)) \\to H^1(X, \\mathcal{G}(q))$", "and the source of this arrow is zero by case (1). Hence", "for all $q \\in \\mathbf{Z}$ we see that the map", "$$", "H^0(X, (\\mathcal{O}_{Y_n})^{\\oplus r}(q))", "\\longrightarrow", "H^0(X, p^*M(q))", "$$", "is surjective. For $q \\geq q_0$ the", "source is equal to $(I^q/I^{q + n})^{\\oplus r}$ by", "Lemma \\ref{lemma-bound-q-and-d} and this easily proves the statement.", "\\medskip\\noindent", "Proof of (5). Assume $d > 1$. Arguing as in the proof of (4) we see that", "it suffices to show that the image of", "$$", "H^0(X, p^*M(q))", "\\longrightarrow", "H^0(X, p^*M(q - (d - 1)c))", "$$", "is contained in the image of", "$$", "H^0(X, (\\mathcal{O}_{Y_n})^{\\oplus r}(q - (d - 1)c))", "\\longrightarrow", "H^0(X, p^*M(q - (d - 1)c))", "$$", "To show the inclusion above, it suffices to show that for", "$\\sigma \\in H^0(X, p^*M(q))$ with boundary", "$\\xi \\in H^1(X, \\mathcal{G}(q))$ the image of $\\xi$ in", "$H^1(X, \\mathcal{G}(q - (d - 1)c))$ is zero. This follows by the", "exact same arguments as in the proof of (3)." ], "refs": [ "more-algebra-lemma-ses-near-projective", "local-cohomology-lemma-almost-exactness", "local-cohomology-lemma-bound-q-and-d", "local-cohomology-lemma-annihilated", "local-cohomology-lemma-bound-q-and-d" ], "ref_ids": [ 10196, 9780, 9779, 9781, 9779 ] } ], "ref_ids": [] }, { "id": 9783, "type": "theorem", "label": "local-cohomology-lemma-factor-hom", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-factor-hom", "contents": [ "With $q_0 = q(S)$ and $d = d(S)$ as above, let $M$ be an $(A, n, c)$-module", "and let $\\varphi : M \\to I^n/I^{2n}$ be an $A$-linear map. Assume", "$n \\geq \\max(q_0 + (1 + d)c, (2 + d)c)$ and if $d = 0$ assume", "$n \\geq q_0 + 2c$. Then the composition", "$$", "M \\xrightarrow{\\varphi} I^n/I^{2n} \\to", "I^{n - (1 + d)c}/I^{2n - (1 + d)c}", "$$", "is of the form $\\sum a_i \\psi_i$ with $a_i \\in I^c$ and", "$\\psi_i : M \\to I^{n - (2 + d)c}/I^{2n - (2 + d)c}$." ], "refs": [], "proofs": [ { "contents": [ "The case $d > 1$. Since we have a compatible system of maps", "$p^*(I^q) \\to \\mathcal{O}_X(q)$ for $q \\geq 0$ there are canonical maps", "$p^*(I^q/I^{q + \\nu}) \\to \\mathcal{O}_{Y_\\nu}(q)$ for $\\nu \\geq 0$.", "Using this and pulling back $\\varphi$ we obtain a map", "$$", "\\chi : p^*M \\longrightarrow \\mathcal{O}_{Y_n}(n)", "$$", "such that the composition", "$M \\to H^0(X, p^*M) \\to H^0(X, \\mathcal{O}_{Y_n}(n))$", "is the given homomorphism $\\varphi$ combined with the", "map $I^n/I^{2n} \\to H^0(X, \\mathcal{O}_{Y_n}(n))$.", "Since $\\mathcal{O}_{Y_n}(n)$ is invertible on $Y_n$ the", "linear map $\\chi$ determines a section", "$$", "\\sigma \\in \\Gamma(X, (p^*M)^\\vee(n))", "$$", "with notation as in Remark \\ref{remark-duals}.", "The discussion in Remark \\ref{remark-duals} shows", "the cokernel and kernel of $can : p^*(M^\\vee) \\to (p^*M)^\\vee$", "are scheme theoretically supported on $Y_c$. By", "Lemma \\ref{lemma-annihilated} the map", "$(p^*M)^\\vee(n) \\to (p^*M)^\\vee(n - 2c)$ factors", "through $p^*(M^\\vee)(n - 2c)$; small detail omitted.", "Hence the image of $\\sigma$ in $\\Gamma(X, (p^*M)^\\vee(n - 2c))$", "comes from an element", "$$", "\\sigma' \\in \\Gamma(X, p^*(M^\\vee)(n - 2c))", "$$", "By Lemma \\ref{lemma-vanishing-coh-almost-projective} part (5),", "the fact that $M^\\vee$ is an $(A, n, c)$-module by", "More on Algebra, Lemma \\ref{more-algebra-lemma-dual-near-projective},", "and the fact that $n \\geq q_0 + (1 + d)c$ so $n - 2c \\geq q_0 + (d - 1)c$", "we see that the image of $\\sigma'$ in $H^0(X, p^*M^\\vee(n - (1 + d)c))$", "is the image of an element $\\tau$ in $I^{n - (1 + d)c}M^\\vee$.", "Write $\\tau = \\sum a_i \\tau_i$ with $\\tau_i \\in I^{n - (2 + d)c}M^\\vee$;", "this makes sense as $n - (2 + d)c \\geq 0$.", "Then $\\tau_i$ determines a homomorphism of modules", "$\\psi_i : M \\to I^{n - (2 + d)c}/I^{2n - (2 + d)c}$", "using the evaluation map $M \\otimes M^\\vee \\to A/I^n$.", "\\medskip\\noindent", "Let us prove that this works\\footnote{We hope some reader will suggest", "a less dirty proof of this fact.}. Pick $z \\in M$ and let us show that", "$\\varphi(z)$ and $\\sum a_i \\psi_i(z)$ have the same image in", "$I^{n - (1 + d)c}/I^{2n - (1 + d)c}$.", "First, the element $z$ determines a map", "$p^*z : \\mathcal{O}_{Y_n} \\to p^*M$ whose composition with", "$\\chi$ is equal to the map $\\mathcal{O}_{Y_n} \\to \\mathcal{O}_{Y_n}(n)$", "corresponding to $\\varphi(z)$ via the map", "$I^n/I^{2n} \\to \\Gamma(\\mathcal{O}_{Y_n}(n))$.", "Next $z$ and $p^*z$ determine evaluation maps", "$e_z : M^\\vee \\to A/I^n$ and $e_{p^*z} : (p^*M)^\\vee \\to \\mathcal{O}_{Y_n}$.", "Since $\\chi(p^*z)$ is the section corresponding to $\\varphi(z)$", "we see that $e_{p^*z}(\\sigma)$ is the section corresponding to $\\varphi(z)$.", "Here and below we abuse notation: for a map", "$a : \\mathcal{F} \\to \\mathcal{G}$ of modules on $X$", "we also denote $a : \\mathcal{F}(t) \\to \\mathcal{F}(t)$ the corresponding", "map of twisted modules. The diagram", "$$", "\\xymatrix{", "p^*(M^\\vee) \\ar[d]_{can} \\ar[r]_{p^*e_z} & \\mathcal{O}_{Y_n} \\ar@{=}[d] \\\\", "(p^*M)^\\vee \\ar[r]^{e_{p^*z}} & \\mathcal{O}_{Y_n}", "}", "$$", "commutes by functoriality of the construction $can$. Hence", "$(p^*e_z)(\\sigma')$ in $\\Gamma(Y_n, \\mathcal{O}_{Y_n}(n - 2c))$", "is the section corresponding to the image of $\\varphi(z)$", "in $I^{n - 2c}/I^{2n - 2c}$.", "The next step is that $\\sigma'$ maps to the image", "of $\\sum a_i \\tau_i$ in $H^0(X, p^*M^\\vee(n - (1 + d)c))$.", "This implies that $(p^*e_z)(\\sum a_i \\tau_i) = \\sum a_i p^*e_z(\\tau_i)$", "in $\\Gamma(Y_n, \\mathcal{O}_{Y_n}(n - (1 + d)c))$ is the section corresponding", "to the image of $\\varphi(z)$ in $I^{n - (1 + d)c}/I^{2n - (1 + d)c}$.", "Recall that $\\psi_i$ is defined from $\\tau_i$ using an evaluation", "map. Hence if we denote", "$$", "\\chi_i : p^*M \\longrightarrow \\mathcal{O}_{Y_n}(n - (2 + d)c)", "$$", "the map we get from $\\psi_i$, then we see by the same reasoning", "as above that the section corresponding to $\\psi_i(z)$ is", "$\\chi_i(p^*z) = e_{p^*z}(\\chi_i) = p^*e_z(\\tau_i)$. Hence we conclude that", "the image of $\\varphi(z)$ in $\\Gamma(Y_n, \\mathcal{O}_{Y_n}(n - (1 + d)c))$", "is equal to the image of $\\sum a_i\\psi_i(z)$.", "Since $n - (1 + d)c \\geq q_0$ we have", "$\\Gamma(Y_n, \\mathcal{O}_{Y_n}(n - (1 + d)c)) =", "I^{n - (1 + d)c}/I^{2n - (1 + d)c}$ by Lemma \\ref{lemma-bound-q-and-d} and", "we conclude the desired compatibility is true.", "\\medskip\\noindent", "The case $d = 1$. Here we argue as above that we get", "$$", "\\chi : p^*M \\longrightarrow \\mathcal{O}_{Y_n}(n),\\quad", "\\sigma \\in \\Gamma(X, (p^*M)^\\vee(n)),\\quad", "\\sigma' \\in \\Gamma(X, p^*(M^\\vee)(n - 2c)),", "$$", "and then we use", "Lemma \\ref{lemma-vanishing-coh-almost-projective} part (4)", "to see that $\\sigma'$ is the image of some element", "$\\tau \\in I^{n - 2c}M^\\vee$. The rest of the argument is the same.", "\\medskip\\noindent", "The case $d = 0$. Argument is exactly the same as in the", "case $d = 1$." ], "refs": [ "local-cohomology-remark-duals", "local-cohomology-remark-duals", "local-cohomology-lemma-annihilated", "local-cohomology-lemma-vanishing-coh-almost-projective", "more-algebra-lemma-dual-near-projective", "local-cohomology-lemma-bound-q-and-d", "local-cohomology-lemma-vanishing-coh-almost-projective" ], "ref_ids": [ 9801, 9801, 9781, 9782, 10197, 9779, 9782 ] } ], "ref_ids": [] }, { "id": 9784, "type": "theorem", "label": "local-cohomology-lemma-bound-two-term-complex", "categories": [ "local-cohomology" ], "title": "local-cohomology-lemma-bound-two-term-complex", "contents": [ "With $d = d(S)$ and $q_0 = q(S)$ as above. Then", "\\begin{enumerate}", "\\item for integers $n \\geq c \\geq 0$ with", "$n \\geq \\max(q_0 + (1 + d)c, (2 + d)c)$,", "\\item for $K$ of $D(A/I^n)$ with $H^i(K) = 0$ for $i \\not = -1, 0$", "and $H^i(K)$ finite for $i = -1, 0$ such that $\\Ext^1_{A/I^c}(K, N)$", "is annihilated by $I^c$ for all finite $A/I^c$-modules $N$", "\\end{enumerate}", "the map", "$$", "\\Ext^1_{A/I^n}(K, I^n/I^{2n})", "\\longrightarrow", "\\Ext^1_{A/I^n}(K, I^{n - (1 + d)c}/I^{2n - 2(1 + d)c})", "$$", "is zero." ], "refs": [], "proofs": [ { "contents": [ "The case $d > 0$. Let $K^{-1} \\to K^0$ be a complex representing $K$ as in", "More on Algebra, Lemma \\ref{more-algebra-lemma-ext-1-annihilated-definite}", "part (5) with respect to the ideal $I^c/I^n$ in the ring $A/I^n$.", "In particular $K^{-1}$ is $I^c/I^n$-projective as multiplication", "by elements of $I^c/I^n$ even factor through $K^0$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-map-out-of-almost-free} part (1)", "we have", "$$", "\\Ext^1_{A/I^n}(K, I^n/I^{2n}) =", "\\Coker(\\Hom_{A/I^n}(K^0, I^n/I^{2n}) \\to \\Hom_{A/I^n}(K^{-1}, I^n/I^{2n}))", "$$", "and similarly for other Ext groups. Hence any class $\\xi$ in", "$\\Ext^1_{A/I^n}(K, I^n/I^{2n})$", "comes from an element $\\varphi \\in \\Hom_{A/I^n}(K^{-1}, I^n/I^{2n})$.", "Denote $\\varphi'$ the image of $\\varphi$ in", "$\\Hom_{A/I^n}(K^{-1}, I^{n - (1 + d)c}/I^{2n - (1 + d)c})$.", "By Lemma \\ref{lemma-factor-hom}", "we can write $\\varphi' = \\sum a_i \\psi_i$ with $a_i \\in I^c$ and", "$\\psi_i \\in \\Hom_{A/I^n}(M, I^{n - (2 + d)c}/I^{2n - (2 + d)c})$.", "Choose $h_i : K^0 \\to K^{-1}$ such that", "$a_i \\text{id}_{K^{-1}} = h_i \\circ d_K^{-1}$. Set", "$\\psi = \\sum \\psi_i \\circ h_i : K^0 \\to I^{n - (2 + d)c}/I^{2n - (2 + d)c}$.", "Then $\\varphi' = \\psi \\circ \\text{d}_K^{-1}$ and we conclude that", "$\\xi$ already maps to zero in", "$\\Ext^1_{A/I^n}(K, I^{n - (1 + d)c}/I^{2n - (1 + d)c})$", "and a fortiori in", "$\\Ext^1_{A/I^n}(K, I^{n - (1 + d)c}/I^{2n - 2(1 + d)c})$.", "\\medskip\\noindent", "The case $d = 0$\\footnote{The argument given for $d > 0$ works but gives", "a slightly weaker result.}. Let $\\xi$ and $\\varphi$ be as above.", "We consider the diagram", "$$", "\\xymatrix{", "K^0 \\\\", "K^{-1} \\ar[u] \\ar[r]^\\varphi & I^n/I^{2n} \\ar[r] & I^{n - c}/I^{2n - c}", "}", "$$", "Pulling back to $X$ and using the map", "$p^*(I^n/I^{2n}) \\to \\mathcal{O}_{Y_n}(n)$", "we find a solid diagram", "$$", "\\xymatrix{", "p^*K^0 \\ar@{..>}[rrd] \\\\", "p^*K^{-1} \\ar[u] \\ar[r] &", "\\mathcal{O}_{Y_n}(n) \\ar[r] &", "\\mathcal{O}_{Y_n}(n - c)", "}", "$$", "We can cover $X$ by affine opens $U = \\Spec(B)$ such that there", "exists an $a \\in I$ with the following property: $IB = aB$", "and $a$ is a nonzerodivisor on $B$. Namely, we can cover", "$X$ by spectra of affine blowup algebras, see", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-affine}.", "The restriction of $\\mathcal{O}_{Y_n}(n) \\to \\mathcal{O}_{Y_n}(n - c)$", "to $U$ is isomorphic to the map of quasi-coherent $\\mathcal{O}_U$-modules", "corresponding to the $B$-module map $a^c : B/a^nB \\to B/a^nB$.", "Since $a^c : K^{-1} \\to K^{-1}$ factors through $K^0$ we see that", "the dotted arrow exists over $U$. In other words,", "locally on $X$ we can find the dotted arrow! Now the sheaf of dotted", "arrows fitting into the diagram is principal homogeneous under", "$$", "\\mathcal{F} = \\SheafHom_{\\mathcal{O}_X}(", "\\Coker(p^*K^{-1} \\to p^*K^0), \\mathcal{O}_{Y_n}(n - c))", "$$", "which is a coherent $\\mathcal{O}_X$-module.", "Hence the obstruction for finding the dotted arrow is an element", "of $H^1(X, \\mathcal{F})$. This cohomology group is zero as", "$1 > d = 0$, see discussion following the definition of $d = d(S)$.", "This proves that we can find a dotted arrow", "$\\psi : p^*K^0 \\to \\mathcal{O}_{Y_n}(n - c)$", "fitting into the diagram. Since $n - c \\geq q_0$", "we find that $\\psi$ induces a map $K^0 \\to I^{n - c}/I^{2n - c}$.", "Chasing the diagram we conclude that $\\varphi' = \\psi \\circ \\text{d}_K^{-1}$", "and the proof is finished as before." ], "refs": [ "more-algebra-lemma-ext-1-annihilated-definite", "more-algebra-lemma-map-out-of-almost-free", "local-cohomology-lemma-factor-hom", "divisors-lemma-blowing-up-affine" ], "ref_ids": [ 10300, 10299, 9783, 8052 ] } ], "ref_ids": [] }, { "id": 9785, "type": "theorem", "label": "local-cohomology-proposition-kollar", "categories": [ "local-cohomology" ], "title": "local-cohomology-proposition-kollar", "contents": [ "\\begin{reference}", "See \\cite{k-coherent} and see \\cite[IV, Proposition 7.2.2]{EGA}", "for a special case.", "\\end{reference}", "\\begin{slogan}", "Weak analogue of Hartogs' Theorem: On Noetherian schemes, the", "restriction of a coherent sheaf to an open set with complement", "of codimension 2 in the sheaf's support, is coherent.", "\\end{slogan}", "Let $j : U \\to X$ be an open immersion of locally Noetherian schemes", "with complement $Z$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $j_*\\mathcal{F}$ is coherent,", "\\item for $x \\in \\text{Ass}(\\mathcal{F})$ and", "$z \\in Z \\cap \\overline{\\{x\\}}$ and any associated prime", "$\\mathfrak p$ of the completion $\\mathcal{O}_{\\overline{\\{x\\}}, z}^\\wedge$", "we have $\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}^\\wedge/\\mathfrak p) \\geq 2$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (2) holds we get (1) by a combination of", "Lemmas \\ref{lemma-check-finiteness-pushforward-on-associated-points},", "Remark \\ref{remark-closure}, and", "Lemma \\ref{lemma-sharp-finiteness-pushforward}.", "If (2) does not hold, then $j_*i_{x, *}\\mathcal{O}_{W_x}$ is not finite", "for some $x \\in \\text{Ass}(\\mathcal{F})$ by the discussion in", "Remark \\ref{remark-no-finiteness-pushforward}", "(and Remark \\ref{remark-closure}).", "Thus $j_*\\mathcal{F}$ is not coherent by", "Lemma \\ref{lemma-check-finiteness-pushforward-on-associated-points}." ], "refs": [ "local-cohomology-lemma-check-finiteness-pushforward-on-associated-points", "local-cohomology-remark-closure", "local-cohomology-lemma-sharp-finiteness-pushforward", "local-cohomology-remark-no-finiteness-pushforward", "local-cohomology-remark-closure", "local-cohomology-lemma-check-finiteness-pushforward-on-associated-points" ], "ref_ids": [ 9728, 9795, 9731, 9796, 9795, 9728 ] } ], "ref_ids": [] }, { "id": 9786, "type": "theorem", "label": "local-cohomology-proposition-annihilator", "categories": [ "local-cohomology" ], "title": "local-cohomology-proposition-annihilator", "contents": [ "\\begin{reference}", "\\cite{Faltings-annulators}.", "\\end{reference}", "Let $A$ be a Noetherian ring which has a dualizing complex.", "Let $T \\subset T' \\subset \\Spec(A)$ be subsets stable under", "specialization. Let $s \\geq 0$ an integer. Let $M$ be a finite $A$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item there exists an ideal $J \\subset A$ with $V(J) \\subset T'$", "such that $J$ annihilates $H^i_T(M)$ for $i \\leq s$, and", "\\item for all $\\mathfrak p \\not \\in T'$,", "$\\mathfrak q \\in T$ with $\\mathfrak p \\subset \\mathfrak q$", "we have", "$$", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > s", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\omega_A^\\bullet$ be a dualizing complex. Let $\\delta$ be its", "dimension function, see Dualizing Complexes, Section", "\\ref{dualizing-section-dimension-function}.", "An important role will be played by the finite $A$-modules", "$$", "E^i = \\Ext_A^i(M, \\omega_A^\\bullet)", "$$", "For $\\mathfrak p \\subset A$ we will write $H^i_\\mathfrak p$ to denote the", "local cohomology of an $A_\\mathfrak p$-module with respect to", "$\\mathfrak pA_\\mathfrak p$. Then we see that", "the $\\mathfrak pA_\\mathfrak p$-adic completion of", "$$", "(E^i)_\\mathfrak p =", "\\Ext^{\\delta(\\mathfrak p) + i}_{A_\\mathfrak p}(M_\\mathfrak p,", "(\\omega_A^\\bullet)_\\mathfrak p[-\\delta(\\mathfrak p)])", "$$", "is Matlis dual to", "$$", "H^{-\\delta(\\mathfrak p) - i}_{\\mathfrak p}(M_\\mathfrak p)", "$$", "by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-special-case-local-duality}.", "In particular we deduce from this the", "following fact: an ideal $J \\subset A$ annihilates", "$(E^i)_\\mathfrak p$ if and only if $J$ annihilates", "$H^{-\\delta(\\mathfrak p) - i}_{\\mathfrak p}(M_\\mathfrak p)$.", "\\medskip\\noindent", "Set $T_n = \\{\\mathfrak p \\in T \\mid \\delta(\\mathfrak p) \\leq n\\}$.", "As $\\delta$ is a bounded function, we see that", "$T_a = \\emptyset$ for $a \\ll 0$ and $T_b = T$ for $b \\gg 0$.", "\\medskip\\noindent", "Assume (2). Let us prove the existence of $J$ as in (1).", "We will use a double induction to do this. For $i \\leq s$", "consider the induction hypothesis $IH_i$:", "$H^a_T(M)$ is annihilated by some $J \\subset A$ with $V(J) \\subset T'$", "for $0 \\leq a \\leq i$. The case $IH_0$ is trivial", "because $H^0_T(M)$ is a submodule of $M$ and hence finite", "and hence is annihilated by some ideal $J$ with $V(J) \\subset T$.", "\\medskip\\noindent", "Induction step. Assume $IH_{i - 1}$ holds for some $0 < i \\leq s$.", "Pick $J'$ with $V(J') \\subset T'$ annihilating $H^a_T(M)$ for", "$0 \\leq a \\leq i - 1$ (the induction hypothesis guarantees we can", "do this). We will show by descending induction on $n$", "that there exists an ideal $J$ with $V(J) \\subset T'$ such that the", "associated primes of $J H^i_T(M)$ are in $T_n$.", "For $n \\ll 0$ this implies $JH^i_T(M) = 0$ ", "(Algebra, Lemma \\ref{algebra-lemma-ass-zero})", "and hence $IH_i$ will hold.", "The base case $n \\gg 0$ is trivial because $T = T_n$ in this case", "and all associated primes of $H^i_T(M)$ are in $T$.", "\\medskip\\noindent", "Thus we assume given $J$ with the property for $n$.", "Let $\\mathfrak q \\in T_n$. Let $T_\\mathfrak q \\subset \\Spec(A_\\mathfrak q)$", "be the inverse image of $T$. We have", "$H^j_T(M)_\\mathfrak q = H^j_{T_\\mathfrak q}(M_\\mathfrak q)$", "by Lemma \\ref{lemma-torsion-change-rings}.", "Consider the spectral sequence", "$$", "H_\\mathfrak q^p(H^q_{T_\\mathfrak q}(M_\\mathfrak q))", "\\Rightarrow", "H^{p + q}_\\mathfrak q(M_\\mathfrak q)", "$$", "of Lemma \\ref{lemma-local-cohomology-ss}.", "Below we will find an ideal $J'' \\subset A$ with $V(J'') \\subset T'$", "such that $H^i_\\mathfrak q(M_\\mathfrak q)$ is annihilated by $J''$ for all", "$\\mathfrak q \\in T_n \\setminus T_{n - 1}$.", "Claim: $J (J')^i J''$ will work for $n - 1$.", "Namely, let $\\mathfrak q \\in T_n \\setminus T_{n - 1}$.", "The spectral sequence above defines a filtration", "$$", "E_\\infty^{0, i} = E_{i + 2}^{0, i} \\subset \\ldots \\subset E_3^{0, i} \\subset", "E_2^{0, i} = H^0_\\mathfrak q(H^i_{T_\\mathfrak q}(M_\\mathfrak q))", "$$", "The module $E_\\infty^{0, i}$ is annihilated by $J''$.", "The subquotients $E_j^{0, i}/E_{j + 1}^{0, i}$ for $i + 1 \\geq j \\geq 2$", "are annihilated by $J'$ because the target of $d_j^{0, i}$", "is a subquotient of", "$$", "H^j_\\mathfrak q(H^{i - j + 1}_{T_\\mathfrak q}(M_\\mathfrak q)) =", "H^j_\\mathfrak q(H^{i - j + 1}_T(M)_\\mathfrak q)", "$$", "and $H^{i - j + 1}_T(M)_\\mathfrak q$ is annihilated by $J'$ by choice of $J'$.", "Finally, by our choice of $J$ we have", "$J H^i_T(M)_\\mathfrak q \\subset H^0_\\mathfrak q(H^i_T(M)_\\mathfrak q)$", "since the non-closed points of $\\Spec(A_\\mathfrak q)$ have higher", "$\\delta$ values. Thus $\\mathfrak q$ cannot be an associated prime of", "$J(J')^iJ'' H^i_T(M)$ as desired.", "\\medskip\\noindent", "By our initial remarks we see that $J''$ should annihilate", "$$", "(E^{-\\delta(\\mathfrak q) - i})_\\mathfrak q =", "(E^{-n - i})_\\mathfrak q", "$$", "for all $\\mathfrak q \\in T_n \\setminus T_{n - 1}$.", "But if $J''$ works for one $\\mathfrak q$, then it works for all", "$\\mathfrak q$ in an open neighbourhood of $\\mathfrak q$", "as the modules $E^{-n - i}$ are finite.", "Since every subset of $\\Spec(A)$ is Noetherian with the induced", "topology (Topology, Lemma \\ref{topology-lemma-Noetherian}),", "we conclude that it suffices", "to prove the existence of $J''$ for one $\\mathfrak q$.", "\\medskip\\noindent", "Since the ext modules are finite the existence of $J''$ is", "equivalent to", "$$", "\\text{Supp}(E^{-n - i}) \\cap \\Spec(A_\\mathfrak q) \\subset T'.", "$$", "This is equivalent to showing the localization of $E^{-n - i}$ at every", "$\\mathfrak p \\subset \\mathfrak q$, $\\mathfrak p \\not \\in T'$", "is zero. Using local duality over $A_\\mathfrak p$ we find that we need", "to prove that", "$$", "H^{i + n - \\delta(\\mathfrak p)}_\\mathfrak p(M_\\mathfrak p) =", "H^{i - \\dim((A/\\mathfrak p)_\\mathfrak q)}_\\mathfrak p(M_\\mathfrak p)", "$$", "is zero (this uses that $\\delta$ is a dimension function).", "This vanishes by the assumption in the lemma and $i \\leq s$ and", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth}.", "\\medskip\\noindent", "To prove the converse implication we assume (2) does not hold", "and we work backwards through the arguments above. First, we pick a", "$\\mathfrak q \\in T$, $\\mathfrak p \\subset \\mathfrak q$", "with $\\mathfrak p \\not \\in T'$ such that", "$$", "i = \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) \\leq s", "$$", "is minimal. Then", "$H^{i - \\dim((A/\\mathfrak p)_\\mathfrak q)}_\\mathfrak p(M_\\mathfrak p)$", "is nonzero by the nonvanishing in", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth}.", "Set $n = \\delta(\\mathfrak q)$. Then", "there does not exist an ideal $J \\subset A$ with $V(J) \\subset T'$", "such that $J(E^{-n - i})_\\mathfrak q = 0$.", "Thus $H^i_\\mathfrak q(M_\\mathfrak q)$ is not", "annihilated by an ideal $J \\subset A$ with $V(J) \\subset T'$.", "By minimality of $i$ it follows from the spectral sequence displayed above", "that the module $H^i_T(M)_\\mathfrak q$", "is not annihilated by an ideal $J \\subset A$", "with $V(J) \\subset T'$. Thus $H^i_T(M)$", "is not annihilated by an ideal $J \\subset A$", "with $V(J) \\subset T'$. This finishes the proof of the proposition." ], "refs": [ "dualizing-lemma-special-case-local-duality", "algebra-lemma-ass-zero", "local-cohomology-lemma-torsion-change-rings", "local-cohomology-lemma-local-cohomology-ss", "topology-lemma-Noetherian", "dualizing-lemma-depth", "dualizing-lemma-depth" ], "ref_ids": [ 2873, 702, 9718, 9719, 8220, 2826, 2826 ] } ], "ref_ids": [] }, { "id": 9787, "type": "theorem", "label": "local-cohomology-proposition-finiteness", "categories": [ "local-cohomology" ], "title": "local-cohomology-proposition-finiteness", "contents": [ "\\begin{reference}", "\\cite{Faltings-annulators}.", "\\end{reference}", "Let $A$ be a Noetherian ring which has a dualizing complex.", "Let $T \\subset \\Spec(A)$ be a subset stable under specialization.", "Let $s \\geq 0$ an integer. Let $M$ be a finite $A$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $H^i_T(M)$ is a finite $A$-module for $i \\leq s$, and", "\\item for all $\\mathfrak p \\not \\in T$, $\\mathfrak q \\in T$ with", "$\\mathfrak p \\subset \\mathfrak q$ we have", "$$", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > s", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Formal consequence of Proposition \\ref{proposition-annihilator} and", "Lemma \\ref{lemma-check-finiteness-local-cohomology-by-annihilator}." ], "refs": [ "local-cohomology-proposition-annihilator", "local-cohomology-lemma-check-finiteness-local-cohomology-by-annihilator" ], "ref_ids": [ 9786, 9724 ] } ], "ref_ids": [] }, { "id": 9788, "type": "theorem", "label": "local-cohomology-proposition-annihilator-complex", "categories": [ "local-cohomology" ], "title": "local-cohomology-proposition-annihilator-complex", "contents": [ "Let $A$ be a Noetherian ring which has a dualizing complex.", "Let $T \\subset T' \\subset \\Spec(A)$ be subsets stable under", "specialization. Let $s \\in \\mathbf{Z}$. Let $K$ be an object of", "$D_{\\textit{Coh}}^+(A)$. The following are equivalent", "\\begin{enumerate}", "\\item there exists an ideal $J \\subset A$ with $V(J) \\subset T'$", "such that $J$ annihilates $H^i_T(K)$ for $i \\leq s$, and", "\\item for all $\\mathfrak p \\not \\in T'$,", "$\\mathfrak q \\in T$ with $\\mathfrak p \\subset \\mathfrak q$", "we have", "$$", "\\text{depth}_{A_\\mathfrak p}(K_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > s", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma is the natural generalization of", "Proposition \\ref{proposition-annihilator}", "whose proof the reader should read first.", "Let $\\omega_A^\\bullet$ be a dualizing complex. Let $\\delta$ be its", "dimension function, see Dualizing Complexes, Section", "\\ref{dualizing-section-dimension-function}.", "An important role will be played by the finite $A$-modules", "$$", "E^i = \\Ext_A^i(K, \\omega_A^\\bullet)", "$$", "For $\\mathfrak p \\subset A$ we will write $H^i_\\mathfrak p$ to denote the", "local cohomology of an object of $D(A_\\mathfrak p)$ with respect to", "$\\mathfrak pA_\\mathfrak p$. Then we see that", "the $\\mathfrak pA_\\mathfrak p$-adic completion of", "$$", "(E^i)_\\mathfrak p =", "\\Ext^{\\delta(\\mathfrak p) + i}_{A_\\mathfrak p}(K_\\mathfrak p,", "(\\omega_A^\\bullet)_\\mathfrak p[-\\delta(\\mathfrak p)])", "$$", "is Matlis dual to", "$$", "H^{-\\delta(\\mathfrak p) - i}_{\\mathfrak p}(K_\\mathfrak p)", "$$", "by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-special-case-local-duality}.", "In particular we deduce from this the", "following fact: an ideal $J \\subset A$ annihilates", "$(E^i)_\\mathfrak p$ if and only if $J$ annihilates", "$H^{-\\delta(\\mathfrak p) - i}_{\\mathfrak p}(K_\\mathfrak p)$.", "\\medskip\\noindent", "Set $T_n = \\{\\mathfrak p \\in T \\mid \\delta(\\mathfrak p) \\leq n\\}$.", "As $\\delta$ is a bounded function, we see that", "$T_a = \\emptyset$ for $a \\ll 0$ and $T_b = T$ for $b \\gg 0$.", "\\medskip\\noindent", "Assume (2). Let us prove the existence of $J$ as in (1).", "We will use a double induction to do this. For $i \\leq s$", "consider the induction hypothesis $IH_i$:", "$H^a_T(K)$ is annihilated by some $J \\subset A$ with $V(J) \\subset T'$", "for $a \\leq i$. The case $IH_i$ is trivial for $i$ small", "enough because $K$ is bounded below.", "\\medskip\\noindent", "Induction step. Assume $IH_{i - 1}$ holds for some $i \\leq s$.", "Pick $J'$ with $V(J') \\subset T'$ annihilating $H^a_T(K)$ for", "$a \\leq i - 1$ (the induction hypothesis guarantees we can", "do this). We will show by descending induction on $n$", "that there exists an ideal $J$ with $V(J) \\subset T'$ such that the", "associated primes of $J H^i_T(K)$ are in $T_n$.", "For $n \\ll 0$ this implies $JH^i_T(K) = 0$ ", "(Algebra, Lemma \\ref{algebra-lemma-ass-zero})", "and hence $IH_i$ will hold.", "The base case $n \\gg 0$ is trivial because $T = T_n$ in this case", "and all associated primes of $H^i_T(K)$ are in $T$.", "\\medskip\\noindent", "Thus we assume given $J$ with the property for $n$.", "Let $\\mathfrak q \\in T_n$. Let $T_\\mathfrak q \\subset \\Spec(A_\\mathfrak q)$", "be the inverse image of $T$. We have", "$H^j_T(K)_\\mathfrak q = H^j_{T_\\mathfrak q}(K_\\mathfrak q)$", "by Lemma \\ref{lemma-torsion-change-rings}.", "Consider the spectral sequence", "$$", "H_\\mathfrak q^p(H^q_{T_\\mathfrak q}(K_\\mathfrak q))", "\\Rightarrow", "H^{p + q}_\\mathfrak q(K_\\mathfrak q)", "$$", "of Lemma \\ref{lemma-local-cohomology-ss}.", "Below we will find an ideal $J'' \\subset A$ with $V(J'') \\subset T'$", "such that $H^i_\\mathfrak q(K_\\mathfrak q)$ is annihilated by $J''$ for all", "$\\mathfrak q \\in T_n \\setminus T_{n - 1}$.", "Claim: $J (J')^i J''$ will work for $n - 1$.", "Namely, let $\\mathfrak q \\in T_n \\setminus T_{n - 1}$.", "The spectral sequence above defines a filtration", "$$", "E_\\infty^{0, i} = E_{i + 2}^{0, i} \\subset \\ldots \\subset E_3^{0, i} \\subset", "E_2^{0, i} = H^0_\\mathfrak q(H^i_{T_\\mathfrak q}(K_\\mathfrak q))", "$$", "The module $E_\\infty^{0, i}$ is annihilated by $J''$.", "The subquotients $E_j^{0, i}/E_{j + 1}^{0, i}$ for $i + 1 \\geq j \\geq 2$", "are annihilated by $J'$ because the target of $d_j^{0, i}$", "is a subquotient of", "$$", "H^j_\\mathfrak q(H^{i - j + 1}_{T_\\mathfrak q}(K_\\mathfrak q)) =", "H^j_\\mathfrak q(H^{i - j + 1}_T(K)_\\mathfrak q)", "$$", "and $H^{i - j + 1}_T(K)_\\mathfrak q$ is annihilated by $J'$ by choice of $J'$.", "Finally, by our choice of $J$ we have", "$J H^i_T(K)_\\mathfrak q \\subset H^0_\\mathfrak q(H^i_T(K)_\\mathfrak q)$", "since the non-closed points of $\\Spec(A_\\mathfrak q)$ have higher", "$\\delta$ values. Thus $\\mathfrak q$ cannot be an associated prime of", "$J(J')^iJ'' H^i_T(K)$ as desired.", "\\medskip\\noindent", "By our initial remarks we see that $J''$ should annihilate", "$$", "(E^{-\\delta(\\mathfrak q) - i})_\\mathfrak q =", "(E^{-n - i})_\\mathfrak q", "$$", "for all $\\mathfrak q \\in T_n \\setminus T_{n - 1}$.", "But if $J''$ works for one $\\mathfrak q$, then it works for all", "$\\mathfrak q$ in an open neighbourhood of $\\mathfrak q$", "as the modules $E^{-n - i}$ are finite.", "Since every subset of $\\Spec(A)$ is Noetherian with the induced", "topology (Topology, Lemma \\ref{topology-lemma-Noetherian}),", "we conclude that it suffices", "to prove the existence of $J''$ for one $\\mathfrak q$.", "\\medskip\\noindent", "Since the ext modules are finite the existence of $J''$ is", "equivalent to", "$$", "\\text{Supp}(E^{-n - i}) \\cap \\Spec(A_\\mathfrak q) \\subset T'.", "$$", "This is equivalent to showing the localization of $E^{-n - i}$ at every", "$\\mathfrak p \\subset \\mathfrak q$, $\\mathfrak p \\not \\in T'$", "is zero. Using local duality over $A_\\mathfrak p$ we find that we need", "to prove that", "$$", "H^{i + n - \\delta(\\mathfrak p)}_\\mathfrak p(K_\\mathfrak p) =", "H^{i - \\dim((A/\\mathfrak p)_\\mathfrak q)}_\\mathfrak p(K_\\mathfrak p)", "$$", "is zero (this uses that $\\delta$ is a dimension function).", "This vanishes by the assumption in the lemma and $i \\leq s$ and", "our definition of depth in Definition \\ref{definition-depth-complex}.", "\\medskip\\noindent", "To prove the converse implication we assume (2) does not hold", "and we work backwards through the arguments above. First, we pick a", "$\\mathfrak q \\in T$, $\\mathfrak p \\subset \\mathfrak q$", "with $\\mathfrak p \\not \\in T'$ such that", "$$", "i = \\text{depth}_{A_\\mathfrak p}(K_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) \\leq s", "$$", "is minimal. Then", "$H^{i - \\dim((A/\\mathfrak p)_\\mathfrak q)}_\\mathfrak p(K_\\mathfrak p)$", "is nonzero by the our definition of depth in", "Definition \\ref{definition-depth-complex}.", "Set $n = \\delta(\\mathfrak q)$. Then", "there does not exist an ideal $J \\subset A$ with $V(J) \\subset T'$", "such that $J(E^{-n - i})_\\mathfrak q = 0$.", "Thus $H^i_\\mathfrak q(K_\\mathfrak q)$ is not", "annihilated by an ideal $J \\subset A$ with $V(J) \\subset T'$.", "By minimality of $i$ it follows from the spectral sequence displayed above", "that the module $H^i_T(K)_\\mathfrak q$", "is not annihilated by an ideal $J \\subset A$", "with $V(J) \\subset T'$. Thus $H^i_T(K)$", "is not annihilated by an ideal $J \\subset A$", "with $V(J) \\subset T'$. This finishes the proof of the proposition." ], "refs": [ "local-cohomology-proposition-annihilator", "dualizing-lemma-special-case-local-duality", "algebra-lemma-ass-zero", "local-cohomology-lemma-torsion-change-rings", "local-cohomology-lemma-local-cohomology-ss", "topology-lemma-Noetherian", "local-cohomology-definition-depth-complex", "local-cohomology-definition-depth-complex" ], "ref_ids": [ 9786, 2873, 702, 9718, 9719, 8220, 9793, 9793 ] } ], "ref_ids": [] }, { "id": 9789, "type": "theorem", "label": "local-cohomology-proposition-finiteness-complex", "categories": [ "local-cohomology" ], "title": "local-cohomology-proposition-finiteness-complex", "contents": [ "Let $A$ be a Noetherian ring which has a dualizing complex.", "Let $T \\subset \\Spec(A)$ be a subset stable under specialization.", "Let $s \\in \\mathbf{Z}$. Let $K \\in D_{\\textit{Coh}}^+(A)$.", "The following are equivalent", "\\begin{enumerate}", "\\item $H^i_T(K)$ is a finite $A$-module for $i \\leq s$, and", "\\item for all $\\mathfrak p \\not \\in T$, $\\mathfrak q \\in T$ with", "$\\mathfrak p \\subset \\mathfrak q$ we have", "$$", "\\text{depth}_{A_\\mathfrak p}(K_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > s", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Formal consequence of", "Proposition \\ref{proposition-annihilator-complex} and", "Lemma \\ref{lemma-check-finiteness-local-cohomology-by-annihilator-complex}." ], "refs": [ "local-cohomology-proposition-annihilator-complex", "local-cohomology-lemma-check-finiteness-local-cohomology-by-annihilator-complex" ], "ref_ids": [ 9788, 9746 ] } ], "ref_ids": [] }, { "id": 9790, "type": "theorem", "label": "local-cohomology-proposition-Hartshorne-Lichtenbaum-vanishing", "categories": [ "local-cohomology" ], "title": "local-cohomology-proposition-Hartshorne-Lichtenbaum-vanishing", "contents": [ "\\begin{reference}", "\\cite[Theorem 3.1]{CD}", "\\end{reference}", "Let $A$ be a Noetherian local ring with completion $A^\\wedge$.", "Let $I \\subset A$ be an ideal such that", "$$", "\\dim V(IA^\\wedge + \\mathfrak p) \\geq 1", "$$", "for every minimal prime $\\mathfrak p \\subset A^\\wedge$ of dimension $\\dim(A)$.", "Then $\\text{cd}(A, I) < \\dim(A)$." ], "refs": [], "proofs": [ { "contents": [ "Since $A \\to A^\\wedge$ is faithfully flat we have", "$H^d_{V(I)}(A) \\otimes_A A^\\wedge = H^d_{V(IA^\\wedge)}(A^\\wedge)$", "by Dualizing Complexes, Lemma \\ref{dualizing-lemma-torsion-change-rings}.", "Thus we may assume $A$ is complete.", "\\medskip\\noindent", "Assume $A$ is complete. Let $\\mathfrak p_1, \\ldots, \\mathfrak p_n \\subset A$", "be the minimal primes of dimension $d$. Consider the complete local ring", "$A_i = A/\\mathfrak p_i$. We have $H^d_{V(I)}(A_i) = H^d_{V(IA_i)}(A_i)$", "by Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-local-cohomology-and-restriction}.", "By Lemma \\ref{lemma-cd-top-vanishing-some-module}", "it suffices to prove the lemma for $(A_i, IA_i)$.", "Thus we may assume $A$ is a complete local domain.", "\\medskip\\noindent", "Assume $A$ is a complete local domain. We can choose a prime ideal", "$\\mathfrak p \\supset I$ with $\\dim(A/\\mathfrak p) = 1$.", "By Lemma \\ref{lemma-cd-top-vanishing}", "it suffices to prove the lemma for $\\mathfrak p$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-cd-bound-dualizing} it suffices to show that", "$H^0_{V(\\mathfrak p)}(\\omega_A^\\bullet) = 0$.", "Recall that", "$$", "H^0_{V(\\mathfrak p)}(\\omega_A^\\bullet) =", "\\colim \\text{Ext}^0_A(A/\\mathfrak p^n, \\omega_A^\\bullet)", "$$", "By Lemma \\ref{lemma-inverse-system-symbolic-powers}", "we see that the colimit is the same as", "$$", "\\colim \\text{Ext}^0_A(A/\\mathfrak p^{(n)}, \\omega_A^\\bullet)", "$$", "Since $\\text{depth}(A/\\mathfrak p^{(n)}) = 1$ we see that", "these ext groups are zero by Lemma \\ref{lemma-sitting-in-degrees}", "as desired." ], "refs": [ "dualizing-lemma-torsion-change-rings", "dualizing-lemma-local-cohomology-and-restriction", "local-cohomology-lemma-cd-top-vanishing-some-module", "local-cohomology-lemma-cd-top-vanishing", "local-cohomology-lemma-cd-bound-dualizing", "local-cohomology-lemma-inverse-system-symbolic-powers", "local-cohomology-lemma-sitting-in-degrees" ], "ref_ids": [ 2817, 2816, 9753, 9752, 9755, 9756, 9737 ] } ], "ref_ids": [] }, { "id": 9791, "type": "theorem", "label": "local-cohomology-proposition-uniform-artin-rees", "categories": [ "local-cohomology" ], "title": "local-cohomology-proposition-uniform-artin-rees", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$. Let $t \\geq 0$", "be an upper bound on the number of generators of $I$.", "There exist $N, c \\geq 0$ such that for $n \\geq N$ the maps", "$$", "A/I^n \\to A/I^{n - c}", "$$", "satisfy the equivalent conditions of", "Lemma \\ref{lemma-characterize-vanishing-tor-ext-above-e} with $e = t$." ], "refs": [ "local-cohomology-lemma-characterize-vanishing-tor-ext-above-e" ], "proofs": [ { "contents": [ "Immediate consequence of Lemmas", "\\ref{lemma-tor-maps-vanish} and", "\\ref{lemma-characterize-vanishing-tor-ext-above-e}." ], "refs": [ "local-cohomology-lemma-tor-maps-vanish", "local-cohomology-lemma-characterize-vanishing-tor-ext-above-e" ], "ref_ids": [ 9778, 9771 ] } ], "ref_ids": [ 9771 ] }, { "id": 9802, "type": "theorem", "label": "more-algebra-theorem-regular-fs", "categories": [ "more-algebra" ], "title": "more-algebra-theorem-regular-fs", "contents": [ "Let $k$ be a field. Let $(A, \\mathfrak m, K)$ be a Noetherian local", "$k$-algebra. If the characteristic of $k$ is zero then the following", "are equivalent", "\\begin{enumerate}", "\\item $A$ is a regular local ring, and", "\\item $k \\to A$ is formally smooth in the $\\mathfrak m$-adic topology.", "\\end{enumerate}", "If the characteristic of $k$ is $p > 0$ then the following are equivalent", "\\begin{enumerate}", "\\item $A$ is geometrically regular over $k$,", "\\item $k \\to A$ is formally smooth in the $\\mathfrak m$-adic topology.", "\\item for all $k \\subset k' \\subset k^{1/p}$", "finite over $k$ the ring $A \\otimes_k k'$ is regular,", "\\item $A$ is regular and the canonical map", "$H_1(L_{K/k}) \\to \\mathfrak m/\\mathfrak m^2$ is injective, and", "\\item $A$ is regular and the map", "$\\Omega_{k/\\mathbf{F}_p} \\otimes_k K \\to \\Omega_{A/\\mathbf{F}_p} \\otimes_A K$", "is injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If the characteristic of $k$ is zero, then the equivalence of (1) and (2)", "follows from", "Lemmas \\ref{lemma-fs-implies-regular} and \\ref{lemma-regular-implies-fs}.", "\\medskip\\noindent", "If the characteristic of $k$ is $p > 0$, then it follows from", "Proposition \\ref{proposition-characterization-geometrically-regular}", "that (1), (3), (4), and (5) are equivalent. Assume (2) holds.", "By Lemma \\ref{lemma-base-change-fs} we see that", "$k' \\to A' = A \\otimes_k k'$ is formally smooth for the", "$\\mathfrak m' = \\mathfrak mA$-adic topology. Hence if $k \\subset k'$ is", "finite purely inseparable, then $A'$ is a regular local ring by", "Lemma \\ref{lemma-fs-implies-regular}. Thus we see that (1) holds.", "\\medskip\\noindent", "Finally, we will prove that (5) implies (2). Choose a solid diagram", "$$", "\\xymatrix{", "A \\ar[r]_{\\bar\\psi} \\ar@{-->}[rd] & B/J \\\\", "k \\ar[u]^i \\ar[r]^\\varphi & B \\ar[u]_\\pi", "}", "$$", "as in Definition \\ref{definition-formally-smooth}. As $J^2 = 0$ we see", "that $J$ has a canonical $B/J$ module structure and via $\\bar\\psi$ an", "$A$-module structure. As $\\bar\\psi$ is continuous for the", "$\\mathfrak m$-adic topology we see that $\\mathfrak m^nJ = 0$ for some $n$.", "Hence we can filter $J$ by $B/J$-submodules", "$0 \\subset J_1 \\subset J_2 \\subset \\ldots \\subset J_n = J$", "such that each quotient $J_{t + 1}/J_t$ is annihilated by $\\mathfrak m$.", "Considering the sequence of ring maps", "$B \\to B/J_1 \\to B/J_2 \\to \\ldots \\to B/J$", "we see that it suffices to prove the existence of the dotted arrow when", "$J$ is annihilated by $\\mathfrak m$, i.e., when $J$ is a $K$-vector space.", "\\medskip\\noindent", "Assume given a diagram as above such that $J$ is annihilated by $\\mathfrak m$.", "By Lemma \\ref{lemma-regular-implies-fs} we see that $\\mathbf{F}_p \\to A$ is", "formally smooth in the $\\mathfrak m$-adic topology. Hence we can find a ring", "map $\\psi : A \\to B$ such that $\\pi \\circ \\psi = \\bar \\psi$. Then", "$\\psi \\circ i, \\varphi : k \\to B$ are two maps whose compositions with $\\pi$", "are equal. Hence $D = \\psi \\circ i - \\varphi : k \\to J$ is a derivation.", "By Algebra, Lemma \\ref{algebra-lemma-universal-omega} we can write", "$D = \\xi \\circ \\text{d}$ for some $k$-linear map", "$\\xi : \\Omega_{k/\\mathbf{F}_p} \\to J$. Using the $K$-vector space structure", "on $J$ we extend $\\xi$ to a $K$-linear map", "$\\xi' : \\Omega_{k/\\mathbf{F}_p} \\otimes_k K \\to J$.", "Using (5) we can find a $K$-linear map", "$\\xi'' : \\Omega_{A/\\mathbf{F}_p} \\otimes_A K$ whose restriction to", "$\\Omega_{k/\\mathbf{F}_p} \\otimes_k K$ is $\\xi'$. Write", "$$", "D' : A \\xrightarrow{\\text{d}} \\Omega_{A/\\mathbf{F}_p}", "\\to \\Omega_{A/\\mathbf{F}_p} \\otimes_A K \\xrightarrow{\\xi''} J.", "$$", "Finally, set $\\psi' = \\psi - D' : A \\to B$. The reader verifies that $\\psi'$", "is a ring map such that $\\pi \\circ \\psi' = \\bar \\psi$ and such that", "$\\psi' \\circ i = \\varphi$ as desired." ], "refs": [ "more-algebra-lemma-fs-implies-regular", "more-algebra-lemma-regular-implies-fs", "more-algebra-proposition-characterization-geometrically-regular", "more-algebra-lemma-base-change-fs", "more-algebra-lemma-fs-implies-regular", "more-algebra-definition-formally-smooth", "more-algebra-lemma-regular-implies-fs", "algebra-lemma-universal-omega" ], "ref_ids": [ 10022, 10025, 10576, 10019, 10022, 10611, 10025, 1129 ] } ], "ref_ids": [] }, { "id": 9803, "type": "theorem", "label": "more-algebra-theorem-formal-glueing", "categories": [ "more-algebra" ], "title": "more-algebra-theorem-formal-glueing", "contents": [ "Let $R$ be a ring, and let $f \\in R$.", "Let $\\varphi : R \\to S$ be a flat ring map inducing an isomorphism", "$R/fR \\to S/fS$. Then the functor", "$$", "\\text{Mod}_R", "\\longrightarrow", "\\text{Mod}_S \\times_{\\text{Mod}_{S_f}} \\text{Mod}_{R_f},", "\\quad", "M", "\\longmapsto", "(M \\otimes_R S, M_f, \\text{can})", "$$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "The category appearing on the right side of the arrow", "is the category of triples $(M', M_1, \\alpha_1)$ where $M'$ is an", "$S$-module, $M_1$ is a $R_f$-module, and", "$\\alpha_1 : M'_f \\to M_1 \\otimes_R S$ is a $S_f$-isomorphism, see", "Categories, Example \\ref{categories-example-2-fibre-product-categories}.", "Hence this theorem is a special case of", "Proposition \\ref{proposition-equivalence}." ], "refs": [ "more-algebra-proposition-equivalence" ], "ref_ids": [ 10587 ] } ], "ref_ids": [] }, { "id": 9804, "type": "theorem", "label": "more-algebra-theorem-BL-glueing", "categories": [ "more-algebra" ], "title": "more-algebra-theorem-BL-glueing", "contents": [ "\\begin{reference}", "Slight generalization of the main theorem of \\cite{Beauville-Laszlo}.", "\\end{reference}", "Let $(R \\to R',f)$ be a glueing pair. The functor", "$\\text{Can} : \\text{Mod}_R \\longrightarrow \\text{Glue}(R \\to R', f)$", "determines an equivalence of the category of $R$-modules glueable", "for $(R \\to R', f)$ and the category $\\text{Glue}(R \\to R', f)$", "of glueing data." ], "refs": [], "proofs": [ { "contents": [ "The functor is fully faithful due to the exactness of", "(\\ref{equation-BL-cech-mod-re}) for glueable modules, which tells", "us exactly that $H^0 \\circ \\text{Can} = \\text{id}$ on the", "full subcategory of glueable modules.", "Hence it suffices to check essential surjectivity.", "That is, we must show that an arbitrary glueing datum", "$(M', M_1, \\alpha_1)$ arises from some glueable $R$-module.", "\\medskip\\noindent", "We first check that the map $\\text{d} : M' \\oplus M_1 \\to (M')_f$ used in the", "definition of the functor $H^0$ is surjective. Observe that", "$(x, y) \\in M' \\oplus M_1$ maps to", "$\\text{d}(x, y) = x/1 - \\alpha_1^{-1}(y \\otimes 1)$", "in $(M')_f$. If $z \\in (M')_f$, then we can write", "$\\alpha_1(z) = \\sum y_i \\otimes g_i$ with $g_i \\in R'$", "and $y_i \\in M_1$. Write $\\alpha_1^{-1}(y_i \\otimes 1) = y_i'/f^n$", "for some $y'_i \\in M'$ and $n \\geq 0$ (we can pick the same $n$", "for all $i$). Write $g_i = a_i + f^n b_i$ with $a_i \\in R$ and", "$b_i \\in R'$. Then with $y = \\sum a_i y_i \\in M_1$ and", "$x = \\sum b_i y'_i \\in M'$ we have $\\text{d}(x, -y) = z$", "as desired.", "\\medskip\\noindent", "Put $M = H^0((M', M_1, \\alpha_1)) = \\Ker(\\text{d})$. We obtain", "an exact sequence of $R$-modules", "\\begin{equation}", "\\label{equation-define-M}", "0 \\to M \\to M' \\oplus M_1 \\to (M')_f \\to 0.", "\\end{equation}", "We will prove that the maps $M \\to M'$ and $M \\to M_1$ induce isomorphisms", "$M \\otimes_R R' \\to M'$ and $M \\otimes_R R_f \\to M_1$.", "This will imply that $M$ is glueable for $(R \\to R', f)$ and", "gives rise to the original glueing datum.", "\\medskip\\noindent", "Since $f$ is a nonzerodivisor on $M_1$, we have", "$M[f^\\infty] \\cong M'[f^\\infty]$. This yields an exact sequence", "\\begin{equation}", "\\label{equation-exact-mod-torsion}", "0 \\to M/M[f^\\infty] \\to M_1 \\to (M')_f/M' \\to 0.", "\\end{equation}", "Since $R \\to R_f$ is flat, we may tensor this exact sequence with $R_f$", "to deduce that $M \\otimes_R R_f = (M/M[f^\\infty]) \\otimes_R R_f \\to M_1$", "is an isomorphism.", "\\medskip\\noindent", "By Lemma \\ref{lemma-BL3} we have", "$\\text{Tor}_1^R(R', \\Coker(M' \\to (M')_f)) = 0$.", "The sequence (\\ref{equation-exact-mod-torsion})", "thus remains exact upon tensoring", "over $R$ with $R'$. Using $\\alpha_1$ and", "Lemma \\ref{lemma-torsion-completion}", "the resulting exact sequence can be written as", "\\begin{equation}", "\\label{equation-mod-torsion-sequence}", "0 \\to (M/M[f^\\infty]) \\otimes_R R' \\to", "(M')_f \\to (M')_f/M' \\to 0", "\\end{equation}", "This yields an isomorphism", "$(M/M[f^\\infty]) \\otimes_R R' \\cong M'/M'[f^\\infty]$.", "This implies that in the diagram", "$$", "\\xymatrix{", "& M[f^\\infty] \\otimes_R R' \\ar[r] \\ar[d] &", "M \\otimes_R R' \\ar[r] \\ar[d] &", "(M/M[f^\\infty]) \\otimes_R R' \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "M'[f^\\infty] \\ar[r] &", "M' \\ar[r] &", "M'/M'[f^\\infty] \\ar[r] & 0,", "}", "$$", "the third vertical arrow is an isomorphism. Since the rows are exact and the", "first vertical arrow is an isomorphism by", "Lemma \\ref{lemma-torsion-completion} and $M[f^\\infty] = M'[f^\\infty]$,", "the five lemma implies that $M \\otimes_R R' \\to M'$ is an isomorphism.", "This completes the proof." ], "refs": [ "more-algebra-lemma-BL3", "more-algebra-lemma-torsion-completion", "more-algebra-lemma-torsion-completion" ], "ref_ids": [ 10362, 10354, 10354 ] } ], "ref_ids": [] }, { "id": 9805, "type": "theorem", "label": "more-algebra-theorem-olivier", "categories": [ "more-algebra" ], "title": "more-algebra-theorem-olivier", "contents": [ "Let $A \\to B$ be a local homomorphism of local rings.", "If $A$ is strictly henselian and $A \\to B$ is weakly \\'etale, then", "$A = B$." ], "refs": [], "proofs": [ { "contents": [ "We will show that for all $\\mathfrak p \\subset A$ there is a unique", "prime $\\mathfrak q \\subset B$ lying over $\\mathfrak p$ and", "$\\kappa(\\mathfrak p) = \\kappa(\\mathfrak q)$.", "This implies that $B \\otimes_A B \\to B$ is bijective on spectra", "as well as surjective and flat. Hence it is an isomorphism", "for example by the description of pure ideals in", "Algebra, Lemma \\ref{algebra-lemma-pure-open-closed-specializations}.", "Hence $A \\to B$ is a faithfully flat epimorphism of rings. We get", "$A = B$ by", "Algebra, Lemma \\ref{algebra-lemma-faithfully-flat-epimorphism}.", "\\medskip\\noindent", "Note that the fibre ring $B \\otimes_A \\kappa(\\mathfrak p)$", "is a colimit of \\'etale extensions of $\\kappa(\\mathfrak p)$ by", "Lemmas \\ref{lemma-base-change-weakly-etale} and", "\\ref{lemma-absolutely-flat-over-field}.", "Hence, if there exists more than one prime lying over $\\mathfrak p$", "or if $\\kappa(\\mathfrak p) \\not = \\kappa(\\mathfrak q)$ for some $\\mathfrak q$,", "then $B \\otimes_A L$ has a nontrivial idempotent for some (separable)", "algebraic field extension $L \\supset \\kappa(\\mathfrak p)$.", "\\medskip\\noindent", "Let $\\kappa(\\mathfrak p) \\subset L$ be an algebraic field extension.", "Let $A' \\subset L$ be the integral closure of $A/\\mathfrak p$ in $L$.", "By Lemma \\ref{lemma-integral-over-henselian}", "we see that $A'$ is a strictly henselian local ring", "whose residue field is a purely inseparable extension of the residue", "field of $A$. Thus $B \\otimes_A A'$ is a local ring by", "Lemma \\ref{lemma-local-tensor-with-integral}.", "On the other hand, $B \\otimes_A A'$ is integrally closed in", "$B \\otimes_A L$ by Lemma \\ref{lemma-normality-goes-up}.", "Since $B \\otimes_A A'$ is local, it follows that the ring", "$B \\otimes_A L$ does not have nontrivial", "idempotents which is what we wanted to prove." ], "refs": [ "algebra-lemma-pure-open-closed-specializations", "algebra-lemma-faithfully-flat-epimorphism", "more-algebra-lemma-base-change-weakly-etale", "more-algebra-lemma-absolutely-flat-over-field", "more-algebra-lemma-integral-over-henselian", "more-algebra-lemma-local-tensor-with-integral", "more-algebra-lemma-normality-goes-up" ], "ref_ids": [ 962, 953, 10443, 10452, 10459, 10460, 10458 ] } ], "ref_ids": [] }, { "id": 9806, "type": "theorem", "label": "more-algebra-theorem-epp", "categories": [ "more-algebra" ], "title": "more-algebra-theorem-epp", "contents": [ "Let $A \\subset B$ be an extension of discrete valuation rings with fraction", "fields $K \\subset L$. If the characteristic of $\\kappa_A$ is $p > 0$,", "assume that every element of", "$$", "\\bigcap\\nolimits_{n \\geq 1} \\kappa_B^{p^n}", "$$", "is separable algebraic over $\\kappa_A$. Then there exists a finite extension", "$K \\subset K_1$ which is a weak solution for $A \\to B$ as defined in", "Definition \\ref{definition-solution}." ], "refs": [ "more-algebra-definition-solution" ], "proofs": [ { "contents": [ "If the characteristic of $\\kappa_A$ is zero or if the residue characteristic", "is $p$, the ramification index is prime to $p$, and the residue field", "extension is separable, then this follows from Abhyankar's lemma", "(Lemma \\ref{lemma-abhyankar}). Namely, suppose the ramification index", "is $e$. Choose a uniformizer $\\pi \\in A$. Let $K_1/K$", "be the extension obtained by adjoining an $e$th root of $\\pi$.", "By Lemma \\ref{lemma-pull-root-uniformizer} we see that the integral", "closure $A_1$ of $A$ in $K_1$ is a discrete valuation ring with", "ramification index over $A$. Thus $A_1 \\to (B_1)_\\mathfrak m$", "is formally smooth in the $\\mathfrak m$-adic topology", "for all maximal ideals $\\mathfrak m$ of $B_1$", "by Lemma \\ref{lemma-abhyankar} and a fortiori these are weakly", "unramified extensions of discrete valuation rings.", "\\medskip\\noindent", "From now on we let $p$ be a prime number and we assume that $\\kappa_A$ has", "characteristic $p$. We first apply", "Lemma \\ref{lemma-solution-after-strict-henselization}", "to reduce to the case that $A$ and $B$ have separably closed residue fields.", "Since $\\kappa_A$ and $\\kappa_B$ are replaced by their separable algebraic", "closures by this procedure we see that we obtain", "$$", "\\kappa_A \\supset \\bigcap\\nolimits_{n \\geq 1} \\kappa_B^{p^n}", "$$", "from the condition of the theorem.", "\\medskip\\noindent", "Let $\\pi \\in A$ be a uniformizer. Let $A^\\wedge$ and $B^\\wedge$ be the", "completions of $A$ and $B$. We have a commutative diagram", "$$", "\\xymatrix{", "B \\ar[r] & B^\\wedge \\\\", "A \\ar[u] \\ar[r] & A^\\wedge \\ar[u]", "}", "$$", "of extensions of discrete valuation rings. Let $K^\\wedge$ be the fraction", "field of $A^\\wedge$. Suppose that we can find a finite extension", "$K^\\wedge \\subset M$ which is (a) a weak solution for $A^\\wedge \\to B^\\wedge$", "and (b) a compositum of a separable extension and an extension obtained", "by adjoining a $p$-power root of $\\pi$. Then by", "Lemma \\ref{lemma-approximate-separable-extension}", "we can find a finite extension $K \\subset K_1$ such that", "$K^\\wedge \\otimes_K K_1 = M$. Let $A_1$, resp.\\ $A_1^\\wedge$", "be the integral closure of $A$, resp.\\ $A^\\wedge$ in $K_1$, resp.\\ $M$.", "Since $A \\to A^\\wedge$ is formally smooth in the $\\mathfrak m^\\wedge$-adic", "topology", "(Lemma \\ref{lemma-extension-dvrs-formally-smooth})", "we see that $A_1 \\to A_1^\\wedge$ is formally smooth in the", "$\\mathfrak m_1^\\wedge$-adic topology", "(Lemma \\ref{lemma-formally-smooth-goes-up} and", "$A_1$ and $A_1^\\wedge$ are discrete valuation", "rings by discussion in Remark \\ref{remark-construction}).", "We conclude from Lemma \\ref{lemma-solutions-go-down} part (2)", "that $K \\subset K_1$ is a weak solution for $A \\to B^\\wedge$.", "Applying Lemma \\ref{lemma-solutions-go-down} part (1)", "we see that $K \\subset K_1$ is a weak solution for $A \\to B$.", "\\medskip\\noindent", "Thus we may assume $A$ and $B$ are complete discrete valuation rings", "with separably closed residue fields of characteristic $p$ and", "with $\\kappa_A \\supset \\bigcap\\nolimits_{n \\geq 1} \\kappa_B^{p^n}$.", "We are also given a uniformizer $\\pi \\in A$ and we have to find a", "weak solution for $A \\to B$ which is a compositum of a separable", "extension and a field obtained by taking $p$-power roots of $\\pi$.", "Note that the second condition is automatic if $A$ has mixed characteristic.", "\\medskip\\noindent", "Set $k = \\bigcap\\nolimits_{n \\geq 1} \\kappa_B^{p^n}$.", "Observe that $k$ is an algebraically closed field of characteristic $p$.", "If $A$ has mixed characteristic let $\\Lambda$ be a Cohen", "ring for $k$ and in the equicharacteristic case set $\\Lambda = k[[t]]$.", "We can choose a ring map $\\Lambda \\to A$ which maps $t$ to $\\pi$ in the", "equicharacteristic case. In the equicharacteristic case this follows", "from the Cohen structure theorem", "(Algebra, Theorem \\ref{algebra-theorem-cohen-structure-theorem}) and", "in the mixed characteristic case this follows as", "$\\mathbf{Z}_p \\to \\Lambda$ is formally smooth in the adic topology", "(Lemmas \\ref{lemma-extension-dvrs-formally-smooth} and", "\\ref{lemma-lift-continuous}).", "Applying Lemma \\ref{lemma-solutions-go-down} we see that it suffices to prove", "the existence of a weak solution for $\\Lambda \\to B$ which in", "the equicharacteristic $p$ case is a compositum of a separable", "extension and a field obtained by taking $p$-power roots of $t$.", "However, since $\\Lambda = k[[t]]$ in the equicharacteristic case", "and any extension of $k((t))$ is such a compositum, we can now", "drop this requirement!", "\\medskip\\noindent", "Thus we arrive at the situation where $A$ and $B$ are complete, the residue", "field $k$ of $A$ is algebraically closed of characteristic $p > 0$,", "we have $k = \\bigcap \\kappa_B^{p^n}$, and in the mixed characteristic", "case $p$ is a uniformizer of $A$ (i.e., $A$ is a Cohen ring for $k$).", "If $A$ has mixed characteristic choose a Cohen ring", "$\\Lambda$ for $\\kappa_B$ and in the equicharacteristic case set", "$\\Lambda = \\kappa_B[[t]]$. Arguing as above we may choose a ring map", "$A \\to \\Lambda$ lifting $k \\to \\kappa_B$ and mapping a uniformizer", "to a uniformizer. Since $k \\subset \\kappa_B$ is separable the ring", "map $A \\to \\Lambda$ is formally smooth in the adic topology", "(Lemma \\ref{lemma-extension-dvrs-formally-smooth}). Hence", "we can find a ring map $\\Lambda \\to B$ such that the composition", "$A \\to \\Lambda \\to B$ is the given ring map $A \\to B$ (see", "Lemma \\ref{lemma-lift-continuous}).", "Since $\\Lambda$ and $B$ are complete discrete valuation rings with the same", "residue field, $B$ is finite over $\\Lambda$", "(Algebra, Lemma \\ref{algebra-lemma-finite-over-complete-ring}).", "This reduces us to the special case discussed in", "Lemma \\ref{lemma-special-case}." ], "refs": [ "more-algebra-lemma-abhyankar", "more-algebra-lemma-pull-root-uniformizer", "more-algebra-lemma-abhyankar", "more-algebra-lemma-solution-after-strict-henselization", "more-algebra-lemma-approximate-separable-extension", "more-algebra-lemma-extension-dvrs-formally-smooth", "more-algebra-lemma-formally-smooth-goes-up", "more-algebra-remark-construction", "more-algebra-lemma-solutions-go-down", "more-algebra-lemma-solutions-go-down", "algebra-theorem-cohen-structure-theorem", "more-algebra-lemma-extension-dvrs-formally-smooth", "more-algebra-lemma-lift-continuous", "more-algebra-lemma-solutions-go-down", "more-algebra-lemma-extension-dvrs-formally-smooth", "more-algebra-lemma-lift-continuous", "algebra-lemma-finite-over-complete-ring", "more-algebra-lemma-special-case" ], "ref_ids": [ 10509, 10507, 10509, 10518, 10506, 10495, 10508, 10679, 10516, 10516, 327, 10495, 10016, 10516, 10495, 10016, 868, 10530 ] } ], "ref_ids": [ 10645 ] }, { "id": 9807, "type": "theorem", "label": "more-algebra-lemma-exact-category-stably-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-exact-category-stably-free", "contents": [ "Let $R$ be a ring. Let $0 \\to P' \\to P \\to P'' \\to 0$ be a short", "exact sequence of finite projective $R$-modules. If $2$ out of $3$", "of these modules are stably free, then so is the third." ], "refs": [], "proofs": [ { "contents": [ "Since the modules are projective, the sequence is split. Thus we can", "choose an isomorphism $P = P' \\oplus P''$. If $P' \\oplus R^{\\oplus n}$", "and $P'' \\oplus R^{\\oplus m}$ are free, then we see that", "$P \\oplus R^{\\oplus n + m}$ is free. Suppose that $P'$ and $P$ are", "stably free, say $P \\oplus R^{\\oplus n}$ is free and $P' \\oplus R^{\\oplus m}$", "is free. Then", "$$", "P'' \\oplus (P' \\oplus R^{\\oplus m}) \\oplus R^{\\oplus n} =", "(P'' \\oplus P') \\oplus R^{\\oplus m} \\oplus R^{\\oplus n} =", "(P \\oplus R^{\\oplus n}) \\oplus R^{\\oplus m}", "$$", "is free. Thus $P''$ is stably free. By symmetry we get the last of the", "three cases." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9808, "type": "theorem", "label": "more-algebra-lemma-lift-stably-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-stably-free", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Assume that", "every element of $1 + I$ is a unit (in other words $I$ is contained", "in the Jacobson radical of $R$). For every finite stably free $R/I$-module $E$", "there exists a finite stably free $R$-module $M$ such that $M/IM \\cong E$." ], "refs": [], "proofs": [ { "contents": [ "Choose a $n$ and $m$ and an isomorphism", "$E \\oplus (R/I)^{\\oplus n} \\cong (R/I)^{\\oplus m}$.", "Choose $R$-linear maps $\\varphi : R^{\\oplus m} \\to R^{\\oplus n}$", "and $\\psi : R^{\\oplus n} \\to R^{\\oplus m}$ lifting the", "projection $(R/I)^{\\oplus m} \\to (R/I)^{\\oplus n}$", "and injection $(R/I)^{\\oplus n} \\to (R/I)^{\\oplus m}$.", "Then $\\varphi \\circ \\psi : R^{\\oplus n} \\to R^{\\oplus n}$", "reduces to the identity modulo $I$. Thus the determinant of", "this map is invertible by our assumption on $I$. Hence", "$P = \\Ker(\\varphi)$ is stably free and lifts $E$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9809, "type": "theorem", "label": "more-algebra-lemma-lift-projective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-projective", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal.", "Assume that every element of $1 + I$ is a unit", "(in other words $I$ is contained in the Jacobson radical of $R$).", "Let $M$ be a finite flat $R$-module such that", "$M/IM$ is a projective $R/I$-module.", "Then $M$ is a finite projective $R$-module." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-finite-flat-local}", "we see that $M_\\mathfrak p$ is finite free for all prime ideals", "$\\mathfrak p \\subset R$.", "By", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}", "it suffices to show that the function $\\rho_M : \\mathfrak p \\mapsto", "\\dim_{\\kappa(\\mathfrak p)} M \\otimes_R \\kappa(\\mathfrak p)$", "is locally constant on $\\Spec(R)$. Because $M/IM$ is finite projective, this", "is true on $V(I) \\subset \\Spec(R)$. Since every closed point", "of $\\Spec(R)$ is in $V(I)$ and since", "$\\rho_M(\\mathfrak p) = \\rho_M(\\mathfrak q)$", "whenever $\\mathfrak p \\subset \\mathfrak q \\subset R$", "are prime ideals, we conclude by", "an elementary argument on topological spaces which we omit." ], "refs": [ "algebra-lemma-finite-flat-local", "algebra-lemma-finite-projective" ], "ref_ids": [ 797, 795 ] } ], "ref_ids": [] }, { "id": 9810, "type": "theorem", "label": "more-algebra-lemma-isomorphic-finite-projective-lifts", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-isomorphic-finite-projective-lifts", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Assume that", "every element of $1 + I$ is a unit (in other words $I$ is contained", "in the Jacobson radical of $R$). If $P$ and $P'$ are finite", "projective $R$-modules, then", "\\begin{enumerate}", "\\item if $\\varphi : P \\to P'$ is an $R$-module map inducing an", "isomorphism $\\overline{\\varphi} : P/IP \\to P'/IP'$, then $\\varphi$", "is an isomorphism,", "\\item if $P/IP \\cong P'/IP'$, then $P \\cong P'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). As $P'$ is projective as an $R$-module we may", "choose a lift $\\psi : P' \\to P$ of the map", "$P' \\to P'/IP' \\xrightarrow{\\overline{\\varphi}^{-1}} P/IP$.", "By Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "$\\psi \\circ \\varphi$ and $\\varphi \\circ \\psi$ are surjective.", "Hence these maps are isomorphisms (Algebra, Lemma \\ref{algebra-lemma-fun}).", "Thus $\\varphi$ is an isomorphism.", "\\medskip\\noindent", "Proof of (2). Choose an isomorphism $P/IP \\cong P'/IP'$.", "Since $P$ is projective we can choose a lift $\\varphi : P \\to P'$ of the map", "$P \\to P/IP \\to P'/IP'$. Then $\\varphi$ is an isomorphism by (1)." ], "refs": [ "algebra-lemma-NAK", "algebra-lemma-fun" ], "ref_ids": [ 401, 388 ] } ], "ref_ids": [] }, { "id": 9811, "type": "theorem", "label": "more-algebra-lemma-approximate-complex", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-approximate-complex", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal contained in", "the Jacobson radical of $A$. Let", "$$", "S : L \\xrightarrow{f} M \\xrightarrow{g} N", "\\quad\\text{and}\\quad", "S' : L \\xrightarrow{f'} M \\xrightarrow{g'} N", "$$", "be two complexes of finite $A$-modules as shown. Assume that", "\\begin{enumerate}", "\\item $c$ works in the Artin-Rees lemma for $f$ and $g$,", "\\item the complex $S$ is exact, and", "\\item $f' = f \\bmod I^{c + 1}M$ and $g' = g \\bmod I^{c + 1}N$.", "\\end{enumerate}", "Then $c$ works in the Artin-Rees lemma for $g'$ and the", "complex $S'$ is exact." ], "refs": [], "proofs": [ { "contents": [ "We first show that $g'(M) \\cap I^nN \\subset g'(I^{n - c}M)$ for $n \\geq c$.", "Let $a$ be an element of $M$ such that $g'(a) \\in I^nN$. We want to", "adjust $a$ by an element of $f'(L)$, i.e, without changing $g'(a)$, so", "that $a \\in I^{n-c}M$. Assume that $a \\in I^rM$, where $r < n - c$.", "Then", "$$", "g(a) = g'(a) + (g - g')(a) \\in", "I^n N + I^{r + c + 1}N = I^{r + c + 1}N.", "$$", "By Artin-Rees for $g$ we have $g(a) \\in g(I^{r + 1}M)$. Say $g(a) = g(a_1)$", "with $a_1 \\in I^{r + 1}M$. Since the sequence $S$ is exact, $a - a_1 \\in f(L)$.", "Accordingly, we write $a = f(b) + a_1$ for some $b \\in L$.", "Then $f(b) = a - a_1 \\in I^rM$. Artin-Rees for $f$ shows that", "if $r \\geq c$, we may replace $b$ by an element of $I^{r - c}L$.", "Then in all cases, $a = f'(b) + a_2$, where", "$a_2 = (f - f')(b) + a_1 \\in I^{r + 1}M$. (Namely, either $c \\geq r$", "and $(f - f')(b) \\in I^{r + 1}M$ by assumption, or $c < r$ and", "$b \\in I^{r - c}$, whence again $(f - f')(b) \\in I^{c + 1} I^{r - c} M =", "I^{r + 1}M$.) So we can adjust $a$ by the element $f'(b) \\in f'(L)$ to", "increase $r$ by $1$.", "\\medskip\\noindent", "In fact, the argument above shows that", "$(g')^{-1}(I^nN) \\subset f'(L) + I^{n - c}M$ for all $n \\geq c$.", "Hence $S'$ is exact because", "$$", "(g')^{-1}(0) = (g')^{-1}(\\bigcap I^nN) \\subset", "\\bigcap f'(L) + I^{n - c}M = f'(L)", "$$", "as $I$ is contained in the Jacobson radical of $A$, see Algebra, Lemma", "\\ref{algebra-lemma-intersection-powers-ideal-module}." ], "refs": [ "algebra-lemma-intersection-powers-ideal-module" ], "ref_ids": [ 628 ] } ], "ref_ids": [] }, { "id": 9812, "type": "theorem", "label": "more-algebra-lemma-approximate-complex-graded", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-approximate-complex-graded", "contents": [ "Assumptions as in Lemma \\ref{lemma-approximate-complex}.", "Let $Q = \\Coker(g)$ and $Q' = \\Coker(g')$. Then", "$\\text{Gr}_I(Q) \\cong \\text{Gr}_I(Q')$", "as graded $\\text{Gr}_I(A)$-modules." ], "refs": [ "more-algebra-lemma-approximate-complex" ], "proofs": [ { "contents": [ "In degree $n$ we have", "$\\text{Gr}_I(Q)_n = I^nN/(I^{n + 1}N + g(M) \\cap I^nN)$", "and similarly for $Q'$. We claim that", "$$", "g(M) \\cap I^nN \\subset I^{n + 1}N + g'(M) \\cap I^nN.", "$$", "By symmetry (the proof of the claim will only use that $c$ works", "for $g$ which also holds for $g'$ by the lemma) this will imply that", "$$", "I^{n + 1}N + g(M) \\cap I^nN = I^{n + 1}N + g'(M) \\cap I^nN", "$$", "whence $\\text{Gr}_I(Q)_n$ and $\\text{Gr}_I(Q')_n$ agree as subquotients", "of $N$, implying the lemma. Observe that the claim is clear for", "$n \\leq c$ as $f = f' \\bmod I^{c + 1}N$. If $n > c$, then suppose", "$b \\in g(M) \\cap I^nN$. Write $b = g(a)$ for $a \\in I^{n - c}M$.", "Set $b' = g'(a)$. We have $b - b' = (g - g')(a) \\in I^{n + 1}N$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9811 ] }, { "id": 9813, "type": "theorem", "label": "more-algebra-lemma-works-flat-extension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-works-flat-extension", "contents": [ "Let $A \\to B$ be a flat map of Noetherian rings. Let $I \\subset A$ be", "an ideal. Let $f : M \\to N$ be a homomorphism of finite $A$-modules.", "Assume that $c$ works for $f$ in the Artin-Rees lemma. Then $c$ works for", "$f \\otimes 1 : M \\otimes_A B \\to N \\otimes_A B$ in the Artin-Rees lemma", "for the ideal $IB$." ], "refs": [], "proofs": [ { "contents": [ "Note that", "$$", "(f \\otimes 1)(M) \\cap I^n N \\otimes_A B", "= (f \\otimes 1)\\left((f \\otimes 1)^{-1}(I^n N \\otimes_A B)\\right)", "$$", "On the other hand,", "\\begin{align*}", "(f \\otimes 1)^{-1}(I^n N \\otimes_A B) &", "= \\Ker(M \\otimes_A B \\to N \\otimes_A B/(I^n N \\otimes_A B)) \\\\", "& =", "\\Ker(M \\otimes_A B \\to (N/I^nN) \\otimes_A B)", "\\end{align*}", "As $A \\to B$ is flat taking kernels and cokernels commutes with", "tensoring with $B$, whence this is equal to", "$f^{-1}(I^nN) \\otimes_A B$. By assumption $f^{-1}(I^nN)$ is contained in", "$\\Ker(f) + I^{n - c}M$. Thus the lemma holds." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9814, "type": "theorem", "label": "more-algebra-lemma-fibre-product-finite-type", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-fibre-product-finite-type", "contents": [ "Let $R$ be a ring. Let $A \\to B$ and $C \\to B$ be $R$-algebra maps.", "Assume", "\\begin{enumerate}", "\\item $R$ is Noetherian,", "\\item $A$, $B$, $C$ are of finite type over $R$,", "\\item $A \\to B$ is surjective, and", "\\item $B$ is finite over $C$.", "\\end{enumerate}", "Then $A \\times_B C$ is of finite type over $R$." ], "refs": [], "proofs": [ { "contents": [ "Set $D = A \\times_B C$. There is a commutative diagram", "$$", "\\xymatrix{", "0 &", "B \\ar[l] &", "A \\ar[l] &", "I \\ar[l] &", "0 \\ar[l] \\\\", "0 &", "C \\ar[l] \\ar[u] &", "D \\ar[l] \\ar[u] &", "I \\ar[l] \\ar[u] &", "0 \\ar[l]", "}", "$$", "with exact rows. Choose $y_1, \\ldots, y_n \\in B$ which are generators for", "$B$ as a $C$-module. Choose $x_i \\in A$ mapping to $y_i$.", "Then $1, x_1, \\ldots, x_n$ are generators for $A$ as a $D$-module.", "The map $D \\to A \\times C$ is injective, and the ring $A \\times C$ is finite", "as a $D$-module (because it is the direct sum of the finite $D$-modules", "$A$ and $C$). Hence the lemma follows from the Artin-Tate lemma", "(Algebra, Lemma \\ref{algebra-lemma-Artin-Tate})." ], "refs": [ "algebra-lemma-Artin-Tate" ], "ref_ids": [ 629 ] } ], "ref_ids": [] }, { "id": 9815, "type": "theorem", "label": "more-algebra-lemma-formal-consequence", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-formal-consequence", "contents": [ "Let $R$ be a Noetherian ring. Let $I$ be a finite set. Suppose given a", "cartesian diagram", "$$", "\\xymatrix{", "\\prod B_i &", "\\prod A_i \\ar[l]^{\\prod \\varphi_i} \\\\", "Q \\ar[u]^{\\prod \\psi_i} &", "P \\ar[u] \\ar[l]", "}", "$$", "with $\\psi_i$ and $\\varphi_i$ surjective, and $Q$, $A_i$, $B_i$ of", "finite type over $R$. Then $P$ is of finite type over $R$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-fibre-product-finite-type}", "and induction on the size of $I$.", "Namely, let $I = I' \\amalg \\{i_0\\}$. Let $P'$ be the ring defined", "by the diagram of the lemma using $I'$. Then $P'$ is of finite type", "by induction hypothesis. Finally, $P$ sits in a fibre product diagram", "$$", "\\xymatrix{", "B_{i_0} &", "A_{i_0} \\ar[l] \\\\", "P' \\ar[u] &", "P \\ar[u] \\ar[l] &", "}", "$$", "to which the lemma applies." ], "refs": [ "more-algebra-lemma-fibre-product-finite-type" ], "ref_ids": [ 9814 ] } ], "ref_ids": [] }, { "id": 9816, "type": "theorem", "label": "more-algebra-lemma-diagram-localize", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-diagram-localize", "contents": [ "Suppose given a cartesian diagram of rings", "$$", "\\xymatrix{", "R &", "R' \\ar[l]^t \\\\", "B \\ar[u]_s &", "B'\\ar[u] \\ar[l]", "}", "$$", "i.e., $B' = B \\times_R R'$. If $h \\in B'$ corresponds to $g \\in B$", "and $f \\in R'$ such that $s(g) = t(f)$, then the diagram", "$$", "\\xymatrix{", "R_{s(g)} = R_{t(f)} &", "(R')_f \\ar[l]^-t \\\\", "B_g \\ar[u]_s &", "(B')_h \\ar[u] \\ar[l]", "}", "$$", "is cartesian too." ], "refs": [], "proofs": [ { "contents": [ "The equality $B' = B \\times_R R'$ tells us that", "$$", "0 \\to B' \\to B \\oplus R' \\xrightarrow{s, -t} R", "$$", "is an exact sequence of $B'$-modules. We have $B_g = B_h$,", "$R'_f = R'_h$, and $R_{s(g)} = R_{t(f)} = R_h$ as $B'$-modules.", "By exactness of localization", "(Algebra, Proposition \\ref{algebra-proposition-localization-exact})", "we find that", "$$", "0 \\to B'_h \\to B_g \\oplus R'_f \\xrightarrow{s, -t} R_{s(g)} = R_{t(f)}", "$$", "is an exact sequence. This proves the lemma." ], "refs": [ "algebra-proposition-localization-exact" ], "ref_ids": [ 1402 ] } ], "ref_ids": [] }, { "id": 9817, "type": "theorem", "label": "more-algebra-lemma-modules", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-modules", "contents": [ "Given a commutative diagram of rings", "$$", "\\xymatrix{", "R &", "R' \\ar[l] \\\\", "B \\ar[u] &", "B' \\ar[u] \\ar[l]", "}", "$$", "the functor (\\ref{equation-modules}) has a right adjoint,", "namely the functor", "$$", "F : (N, M', \\varphi) \\longmapsto N \\times_\\varphi M'", "$$", "(see proof for elucidation)." ], "refs": [], "proofs": [ { "contents": [ "Given an object $(N, M', \\varphi)$ of the category", "$\\text{Mod}_B \\times_{\\text{Mod}_R} \\text{Mod}_{R'}$", "we set", "$$", "N \\times_\\varphi M' = \\{(n, m') \\in N \\times M' \\mid", "\\varphi(n \\otimes 1) = m' \\otimes 1\\text{ in }M' \\otimes_{R'} R\\}", "$$", "viewed as a $B'$-module.", "The adjointness statement is that for a $B'$-module $L'$ and", "a triple $(N, M', \\varphi)$ we have", "$$", "\\Hom_{B'}(L', N \\times_\\varphi M') =", "\\Hom_B(L' \\otimes_{B'} B, N)", "\\times_{\\Hom_R(L' \\otimes_{B'} R, M' \\otimes_{R'} R)}", "\\Hom_{R'}(L' \\otimes_{B'} R', M')", "$$", "By Algebra, Lemma \\ref{algebra-lemma-adjoint-tensor-restrict}", "the right hand side is equal to", "$$", "\\Hom_{B'}(L', N) \\times_{\\Hom_{B'}(L', M' \\otimes_{R'} R)} \\Hom_{B'}(L', M')", "$$", "Thus it is clear that for a pair $(g, f')$ of elements", "of this fibre product we get an $B'$-linear map", "$L' \\to N \\times_\\varphi M'$, $l' \\mapsto (g(l'), f'(l'))$.", "Conversely, given a $B'$ linear map $g' : L' \\to N \\times_\\varphi M'$", "we can set $g$ equal to the composition $L' \\to N \\times_\\varphi M' \\to N$", "and $f'$ equal to the composition $L' \\to N \\times_\\varphi M' \\to M'$.", "These constructions are mutually inverse to each other and", "define the desired isomorphism." ], "refs": [ "algebra-lemma-adjoint-tensor-restrict" ], "ref_ids": [ 374 ] } ], "ref_ids": [] }, { "id": 9818, "type": "theorem", "label": "more-algebra-lemma-points-of-fibre-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-points-of-fibre-product", "contents": [ "In Situation \\ref{situation-module-over-fibre-product}", "we have", "$$", "\\Spec(B') = \\Spec(B) \\amalg_{\\Spec(A)} \\Spec(A')", "$$", "as topological spaces." ], "refs": [], "proofs": [ { "contents": [ "Since $B' = B \\times_A A'$ we obtain a commutative square of", "spectra, which induces a continuous map", "$$", "can : \\Spec(B) \\amalg_{\\Spec(A)} \\Spec(A') \\longrightarrow \\Spec(B')", "$$", "as the source is a pushout in the category", "of topological spaces (which exists by", "Topology, Section \\ref{topology-section-colimits}).", "\\medskip\\noindent", "To show the map $can$ is surjective, let $\\mathfrak q' \\subset B'$ be a prime", "ideal. If $\\mathfrak q' \\cap I = 0$ (here and below we take the liberty", "of considering $I$ as an ideal of $B'$ as well as an ideal of $A$),", "then $\\mathfrak q'$ corresponds to a prime ideal of $B$ and is in the image.", "If not, then pick $h \\in I \\cap \\mathfrak q'$. In this case $B_h = A_h = 0$", "and the ring map $B'_h \\to A'_h$ is an isomorphism, see", "Lemma \\ref{lemma-diagram-localize}. Thus we see that", "$\\mathfrak q'$ corresponds to a unique prime ideal $\\mathfrak p' \\subset A'$", "which meets $I$.", "\\medskip\\noindent", "Since $B' \\to B$ is surjective, we see that $can$ is injective on", "the summand $\\Spec(B)$. We have seen above that $\\Spec(A') \\to \\Spec(B')$", "is injective on the complement of $V(I) \\subset \\Spec(A')$. Since", "$V(I) \\subset \\Spec(A')$ is exactly the image of $\\Spec(A) \\to \\Spec(A')$", "a trivial set theoretic argument shows that $can$ is injective.", "\\medskip\\noindent", "To finish the proof we have to show that $can$ is open. To do this, observe", "that an open of the pushout is of the form $V \\amalg U'$ where", "$V \\subset \\Spec(B)$ and $U' \\subset \\Spec(A')$ are opens whose inverse", "images in $\\Spec(A)$ agree. Let $v \\in V$. We can find a $g \\in B$", "such that $v \\in D(g) \\subset V$. Let $f \\in A$ be the image.", "Pick $f' \\in A'$ mapping to $f$. Then", "$D(f') \\cap U' \\cap V(I) = D(f') \\cap V(I)$.", "Hence $V(I) \\cap D(f')$ and $D(f') \\cap (U')^c$ are disjoint closed", "subsets of $D(f') = \\Spec(A'_{f'})$. Write $(U')^c = V(J)$", "for some ideal $J \\subset A'$. Since", "$A'_{f'} \\to'_{f'}/IA'_{f'} \\times A'_{f'}/J'A'_{f'}$", "is surjective by the disjointness just shown, we can find an $a'' \\in A'_{f'}$", "mapping to $1$ in $A'_{f'}/IA'_{f'}$ and mapping to zero in", "$A'_{f'}/J'A'_{f'}$. Clearing denominators, we find an element", "$a' \\in J$ mapping to $f^n$ in $A$. Then $D(a'f') \\subset U'$.", "Let $h' = (g^n, a'f') \\in B'$.", "Since $B'_{h'} = B_{g^n} \\times_{A_{f^n}} A'_{a'f'}$ by a previously", "cited lemma, we see that $D(h)$ pulls back to an open neighbourhood", "of $v$ in the pushout, i.e., the image of $V \\amalg U$ contains", "an open neighbourhood of the image of $v$. We omit the (easier) proof that", "the same thing is true for $u' \\in U'$ with $u' \\not \\in V(I)$." ], "refs": [ "more-algebra-lemma-diagram-localize" ], "ref_ids": [ 9816 ] } ], "ref_ids": [] }, { "id": 9819, "type": "theorem", "label": "more-algebra-lemma-fibre-product-integral", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-fibre-product-integral", "contents": [ "In Situation \\ref{situation-module-over-fibre-product}", "if $B \\to A$ is integral, then $B' \\to A'$ is integral." ], "refs": [], "proofs": [ { "contents": [ "Let $a' \\in A'$ with image $a \\in A$. Let $x^d + b_1 x^{d - 1} + \\ldots + b_d$", "be a monical polynomial with coefficients in $B$ satisfied by $a$.", "Choose $b'_i \\in B'$ mapping to $b_i \\in B$ (possible).", "Then $(a')^d + b'_1 (a')^{d - 1} + \\ldots + b'_d$ is in the kernel", "of $A' \\to A$. Since $\\Ker(B' \\to B) = \\Ker(A' \\to A)$ we can", "modify our choice of $b'_d$ to get", "$(a')^d + b'_1 (a')^{d - 1} + \\ldots + b'_d = 0$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9820, "type": "theorem", "label": "more-algebra-lemma-module-over-fibre-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-module-over-fibre-product", "contents": [ "In Situation \\ref{situation-module-over-fibre-product}", "the functor (\\ref{equation-functor}) has a right adjoint, namely", "the functor", "$$", "F : (N, M', \\varphi) \\longmapsto N \\times_{\\varphi, M} M'", "$$", "where $M = M'/IM'$. Moreover, the composition of $F$ with", "(\\ref{equation-functor}) is the identity functor on", "$\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$. In other words,", "setting $N' = N \\times_{\\varphi, M} M'$ we have", "$N' \\otimes_{B'} B = N$ and $N' \\otimes_{B'} A' = M'$." ], "refs": [], "proofs": [ { "contents": [ "The adjointness statement follows from the more general", "Lemma \\ref{lemma-modules}.", "To prove the final assertion, recall that", "$B' = B \\times_A A'$ and $N' = N \\times_{\\varphi, M} M'$ and extend", "these equalities to ", "$$", "\\vcenter{", "\\xymatrix{", "A & A' \\ar[l] & I \\ar[l] \\\\", "B \\ar[u] & B' \\ar[l] \\ar[u] & J \\ar[l] \\ar[u]", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "M & M' \\ar[l] & K \\ar[l] \\\\", "N \\ar[u]_\\varphi & N' \\ar[l] \\ar[u] & L \\ar[l] \\ar[u]", "}", "}", "$$", "where $I, J, K, L$ are the kernels of the horizontal maps of the original", "diagrams. We present the proof as a sequence of observations:", "\\begin{enumerate}", "\\item $K = IM'$ (see statement lemma),", "\\item $B' \\to B$ is surjective with kernel $J$ and $J \\to I$ is bijective,", "\\item $N' \\to N$ is surjective with kernel $L$ and $L \\to K$ is bijective,", "\\item $JN' \\subset L$,", "\\item $\\Im(N \\to M)$ generates $M$ as an $A$-module", "(because $N \\otimes_B A = M$),", "\\item $\\Im(N' \\to M')$ generates $M'$ as an $A'$-module", "(because it holds modulo $K$ and $L$ maps isomorphically to $K$),", "\\item $JN' = L$ (because $L \\cong K = I M'$", "is generated by images of elements $x n'$ with $x \\in I$ and", "$n' \\in N'$ by the previous statement),", "\\item $N' \\otimes_{B'} B = N$ (because $N = N'/L$, $B = B'/J$, and", "the previous statement),", "\\item there is a map $\\gamma : N' \\otimes_{B'} A' \\to M'$,", "\\item $\\gamma$ is surjective (see above),", "\\item the kernel of the composition $N' \\otimes_{B'} A' \\to M' \\to M$ ", "is generated by elements $l \\otimes 1$ and $n' \\otimes x$ with", "$l \\in K$, $n' \\in N'$, $x \\in I$ (because $M = N \\otimes_B A$ by assumption", "and because $N' \\to N$ and $A' \\to A$ are surjective with kernels", "$L$ and $I$),", "\\item any element of $N' \\otimes_{B'} A'$ in the submodule generated", "by the elements $l \\otimes 1$ and $n' \\otimes x$ with", "$l \\in L$, $n' \\in N'$, $x \\in I$ can be written as $l \\otimes 1$", "for some $l \\in L$ (because $J$ maps isomorphically to $I$ we see", "that $n' \\otimes x = n'x \\otimes 1$ in $N' \\otimes_{B'} A'$;", "similarly $x n' \\otimes a' = n' \\otimes xa' = n'(xa') \\otimes 1$", "in $N' \\otimes_{B'} A'$ when $n' \\in N'$, $x \\in J$ and $a' \\in A'$;", "since we have seen that $JN' = L$ this proves the assertion),", "\\item the kernel of $\\gamma$ is zero (because by (10) and (11) any element of", "the kernel is of the form $l \\otimes 1$ with $l \\in L$ which", "is mapped to $l \\in K \\subset M'$ by $\\gamma$).", "\\end{enumerate}", "This finishes the proof." ], "refs": [ "more-algebra-lemma-modules" ], "ref_ids": [ 9817 ] } ], "ref_ids": [] }, { "id": 9821, "type": "theorem", "label": "more-algebra-lemma-module-over-fibre-product-bis", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-module-over-fibre-product-bis", "contents": [ "In the situation of Lemma \\ref{lemma-module-over-fibre-product}", "for a $B'$-module $L'$ the adjunction map", "$$", "L' \\longrightarrow ", "(L' \\otimes_{B'} B) \\times_{(L' \\otimes_{B'} A)} (L' \\otimes_{B'} A')", "$$", "is surjective but in general not injective." ], "refs": [ "more-algebra-lemma-module-over-fibre-product" ], "proofs": [ { "contents": [ "As in the proof of Lemma \\ref{lemma-module-over-fibre-product}", "let $J \\subset B'$ be the kernel of the map $B' \\to B$.", "Then $L' \\otimes_{B'} B = L'/JL'$. Hence to prove surjectivity it suffices", "to show that elements of the form $(0, z)$ of the fibre product are in the", "image of the map of the lemma. The kernel of the map", "$L' \\otimes_{B'} A' \\to L' \\otimes_{B'} A$ is the image of", "$L' \\otimes_{B'} I \\to L' \\otimes_{B'} A'$. Since the map $J \\to I$", "induced by $B' \\to A'$ is an isomorphism", "the composition", "$$", "L' \\otimes_{B'} J \\to L' \\to", "(L' \\otimes_{B'} B) \\times_{(L' \\otimes_{B'} A)} (L' \\otimes_{B'} A')", "$$", "induces a surjection of $L' \\otimes_{B'} J$ onto the set of elements", "of the form $(0, z)$. To see the map is not injective in general we", "present a simple example. Namely, take a field $k$,", "set $B' = k[x, y]/(xy)$, $A' = B'/(x)$, $B = B'/(y)$, $A = B'/(x, y)$", "and $L' = B'/(x - y)$. In that case the class of $x$ in $L'$ is nonzero", "but is mapped to zero under the displayed arrow." ], "refs": [ "more-algebra-lemma-module-over-fibre-product" ], "ref_ids": [ 9820 ] } ], "ref_ids": [ 9820 ] }, { "id": 9822, "type": "theorem", "label": "more-algebra-lemma-surjection-module-over-fibre-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-surjection-module-over-fibre-product", "contents": [ "In Situation \\ref{situation-module-over-fibre-product}", "let $(N_1, M'_1, \\varphi_1) \\to (N_2, M'_2, \\varphi_2)$ be a morphism", "of $\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$", "with $N_1 \\to N_2$ and $M'_1 \\to M'_2$ surjective. Then", "$$", "N_1 \\times_{\\varphi_1, M_1} M'_1 \\to N_2 \\times_{\\varphi_2, M_2} M'_2", "$$", "where $M_1 = M'_1/IM'_1$ and $M_2 = M'_2/IM'_2$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Pick $(x_2, y_2) \\in N_2 \\times_{\\varphi_2, M_2} M'_2$. Choose $x_1 \\in N_1$", "mapping to $x_2$. Since $M'_1 \\to M_1$ is surjective we can find", "$y_1 \\in M'_1$ mapping to $\\varphi_1(x_1)$. Then $(x_1, y_1)$", "maps to $(x_2, y'_2)$ in $N_2 \\times_{\\varphi_2, M_2} M'_2$. Thus it suffices", "to show that elements of the form $(0, y_2)$ are in the image of the map.", "Here we see that $y_2 \\in IM'_2$. Write $y_2 = \\sum t_i y_{2, i}$", "with $t_i \\in I$. Choose $y_{1, i} \\in M'_1$ mapping to $y_{2, i}$.", "Then $y_1 = \\sum t_iy_{1, i} \\in IM'_1$ and the element $(0, y_1)$", "does the job." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9823, "type": "theorem", "label": "more-algebra-lemma-finite-module-over-fibre-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finite-module-over-fibre-product", "contents": [ "Let $A, A', B, B', I, M, M', N, \\varphi$ be as in", "Lemma \\ref{lemma-module-over-fibre-product}.", "If $N$ finite over $B$ and $M'$ finite over $A'$, then", "$N' = N \\times_{\\varphi, M} M'$ is finite over $B'$." ], "refs": [ "more-algebra-lemma-module-over-fibre-product" ], "proofs": [ { "contents": [ "We will use the results of", "Lemma \\ref{lemma-module-over-fibre-product}", "without further mention. Choose generators $y_1, \\ldots, y_r$ of $N$ over $B$", "and generators $x_1, \\ldots, x_s$ of $M'$ over $A'$. Using that", "$N = N' \\otimes_{B'} B$ and $B' \\to B$ is surjective we can find", "$u_1, \\ldots, u_r \\in N'$ mapping to $y_1, \\ldots, y_r$ in $N$.", "Using that $M' = N' \\otimes_{B'} A'$ we can find $v_1, \\ldots, v_t \\in N'$", "such that $x_i = \\sum v_j \\otimes a'_{ij}$ for some $a'_{ij} \\in A'$.", "In particular we see that the images $\\overline{v}_j \\in M'$", "of the $v_j$ generate $M'$ over $A'$.", "We claim that $u_1, \\ldots, u_r, v_1, \\ldots, v_t$", "generate $N'$ as a $B'$-module. Namely, pick $\\xi \\in N'$. We first choose", "$b'_1, \\ldots, b'_r \\in B'$ such that $\\xi$ and $\\sum b'_i u_i$ map", "to the same element of $N$. This is possible because $B' \\to B$", "is surjective and $y_1, \\ldots, y_r$ generate $N$ over $B$.", "The difference $\\xi - \\sum b'_i u_i$ is of the form $(0, \\theta)$", "for some $\\theta$ in $IM'$. Say $\\theta$ is $\\sum t_j\\overline{v}_j$", "with $t_j \\in I$. As $J = \\Ker(B' \\to B)$ maps isomorphically to $I$", "we can choose $s_j \\in J \\subset B'$ mapping to $t_j$.", "Because $N' = N \\times_{\\varphi, M} M'$ it follows", "that $\\xi = \\sum b'_i u_i + \\sum s_j v_j$ as desired." ], "refs": [ "more-algebra-lemma-module-over-fibre-product" ], "ref_ids": [ 9820 ] } ], "ref_ids": [ 9820 ] }, { "id": 9824, "type": "theorem", "label": "more-algebra-lemma-flat-module-over-fibre-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-module-over-fibre-product", "contents": [ "With $A, A', B, B', I$ as in", "Situation \\ref{situation-module-over-fibre-product}.", "\\begin{enumerate}", "\\item Let $(N, M', \\varphi)$ be an object of", "$\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$.", "If $M'$ is flat over $A'$ and $N$ is flat over $B$, then", "$N' = N \\times_{\\varphi, M} M'$ is flat over $B'$.", "\\item If $L'$ is a flat $B'$-module, then", "$L' = (L \\otimes_{B'} B) \\times_{(L \\otimes_{B'} A)} (L \\otimes_{B'} A')$.", "\\item The category of flat $B'$-modules is equivalent to the", "full subcategory of $\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$", "consisting of triples", "$(N, M', \\varphi)$ with $N$ flat over $B$ and $M'$ flat over $A'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In the proof we will use Lemma \\ref{lemma-module-over-fibre-product}", "without further mention.", "\\medskip\\noindent", "Proof of (1). Set $J = \\Ker(B' \\to B)$. This is an ideal of $B'$", "mapping isomorphically to $I = \\Ker(A' \\to A)$.", "Let $\\mathfrak b' \\subset B'$ be an ideal.", "We have to show that $\\mathfrak b' \\otimes_{B'} N' \\to N'$", "is injective, see Algebra, Lemma \\ref{algebra-lemma-flat}.", "We know that", "$$", "\\mathfrak b'/(\\mathfrak b' \\cap J) \\otimes_{B'} N' =", "\\mathfrak b'/(\\mathfrak b' \\cap J) \\otimes_B N \\to N", "$$", "is injective as $N$ is flat over $B$. As", "$\\mathfrak b' \\cap J \\to \\mathfrak b' \\to", "\\mathfrak b'/(\\mathfrak b' \\cap J) \\to 0$ is exact, we", "conclude that it suffices to show that", "$(\\mathfrak b' \\cap J) \\otimes_{B'} N' \\to N'$", "is injective. Thus we may assume that $\\mathfrak b' \\subset J$.", "Next, since $J \\to I$ is an isomorphism we have", "$$", "J \\otimes_{B'} N' =", "I \\otimes_{A'} A' \\otimes_{B'} N' =", "I \\otimes_{A'} M'", "$$", "which maps injectively into $M'$ as $M'$ is a flat $A'$-module.", "Hence $J \\otimes_{B'} N' \\to N'$ is injective and we conclude", "that $\\text{Tor}_1^{B'}(B'/J, N') = 0$, see", "Algebra, Remark \\ref{algebra-remark-Tor-ring-mod-ideal}.", "Thus we may apply Algebra, Lemma \\ref{algebra-lemma-what-does-it-mean}", "to $N'$ over $B'$ and the ideal $J$.", "Going back to our ideal $\\mathfrak b' \\subset J$, let", "$\\mathfrak b' \\subset \\mathfrak b'' \\subset J$ be the smallest ideal", "whose image in $I$ is an $A'$-submodule of $I$. In other words,", "we have $\\mathfrak b'' = A' \\mathfrak b'$ if we view $J = I$", "as $A'$-module. Then $\\mathfrak b''/\\mathfrak b'$ is killed", "by $J$ and we get a short exact sequence", "$$", "0 \\to \\mathfrak b' \\otimes_{B'} N' \\to", "\\mathfrak b'' \\otimes_{B'} N' \\to", "\\mathfrak b''/\\mathfrak b' \\otimes_{B'} N' \\to 0", "$$", "by the vanishing of $\\text{Tor}_1^{B'}(\\mathfrak b''/\\mathfrak b', N')$", "we get from the application of the lemma.", "Thus we may replace $\\mathfrak b'$ by $\\mathfrak b''$.", "In particular we may assume $\\mathfrak b'$ is an $A'$-module", "and maps to an ideal of $A'$. Then", "$$", "\\mathfrak b' \\otimes_{B'} N' =", "\\mathfrak b' \\otimes_{A'} A' \\otimes_{B'} N' =", "\\mathfrak b' \\otimes_{A'} M'", "$$", "This tensor product maps injectively into $M'$ by our assumption that", "$M'$ is flat over $A'$. We conclude that", "$\\mathfrak b' \\otimes_{B'} N' \\to N' \\to M'$ is injective", "and hence the first map is injective as desired.", "\\medskip\\noindent", "Proof of (2). This follows by tensoring the short exact sequence", "$0 \\to B' \\to B \\oplus A' \\to A \\to 0$ with $L'$ over $B'$.", "\\medskip\\noindent", "Proof of (3). Immediate consequence of (1) and (2)." ], "refs": [ "more-algebra-lemma-module-over-fibre-product", "algebra-lemma-flat", "algebra-remark-Tor-ring-mod-ideal", "algebra-lemma-what-does-it-mean" ], "ref_ids": [ 9820, 525, 1570, 890 ] } ], "ref_ids": [] }, { "id": 9825, "type": "theorem", "label": "more-algebra-lemma-finitely-presented-module-over-fibre-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finitely-presented-module-over-fibre-product", "contents": [ "Let $A, A', B, B', I$ be as in", "Situation \\ref{situation-module-over-fibre-product}.", "The category of finite projective $B'$-modules", "is equivalent to the full subcategory", "of $\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$", "consisting of triples $(N, M', \\varphi)$", "with $N$ finite projective over $B$ and $M'$ finite projective over $A'$." ], "refs": [], "proofs": [ { "contents": [ "Recall that a module is finite projective if and only if", "it is finitely presented and flat, see", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}.", "Using Lemmas \\ref{lemma-flat-module-over-fibre-product} and", "\\ref{lemma-finite-module-over-fibre-product}", "we reduce to showing that $N' = N \\times_{\\varphi, M} M'$ is", "a $B'$-module of finite presentation if", "$N$ finite projective over $B$", "and $M'$ finite projective over $A'$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-finite-module-over-fibre-product}", "the module $N'$ is finite over $B'$. Choose a surjection", "$(B')^{\\oplus n} \\to N'$ with kernel $K'$. By base change we obtain maps", "$B^{\\oplus n} \\to N$, $(A')^{\\oplus n} \\to M'$, and $A^{\\oplus n} \\to M$", "with kernels $K_B$, $K_{A'}$, and $K_A$. There is a canonical map", "$$", "K' \\longrightarrow K_B \\times_{K_A} K_{A'}", "$$", "On the other hand, since $N' = N \\times_{\\varphi, M} M'$ and", "$B' = B \\times_A A'$ there is also a", "canonical map $K_B \\times_{K_A} K_{A'} \\to K'$ inverse to the displayed", "arrow. Hence the displayed map is an isomorphism. By", "Algebra, Lemma \\ref{algebra-lemma-extension}", "the modules $K_B$ and $K_{A'}$ are finite. We conclude from", "Lemma \\ref{lemma-finite-module-over-fibre-product}", "that $K'$ is a finite $B'$-module provided that $K_B \\to K_A$ and", "$K_{A'} \\to K_A$ induce isomorphisms", "$K_B \\otimes_B A = K_A = K_{A'} \\otimes_{A'} A$.", "This is true because the flatness assumptions implies the sequences", "$$", "0 \\to K_B \\to B^{\\oplus n} \\to N \\to 0", "\\quad\\text{and}\\quad", "0 \\to K_{A'} \\to (A')^{\\oplus n} \\to M' \\to 0", "$$", "stay exact upon tensoring, see", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}." ], "refs": [ "algebra-lemma-finite-projective", "more-algebra-lemma-flat-module-over-fibre-product", "more-algebra-lemma-finite-module-over-fibre-product", "more-algebra-lemma-finite-module-over-fibre-product", "algebra-lemma-extension", "more-algebra-lemma-finite-module-over-fibre-product", "algebra-lemma-flat-tor-zero" ], "ref_ids": [ 795, 9824, 9823, 9823, 330, 9823, 532 ] } ], "ref_ids": [] }, { "id": 9826, "type": "theorem", "label": "more-algebra-lemma-relative-module-over-fibre-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-relative-module-over-fibre-product", "contents": [ "In Situation \\ref{situation-relative-module-over-fibre-product}", "the functor (\\ref{equation-relative-functor}) has a right adjoint, namely", "the functor", "$$", "F : (N, M', \\varphi) \\longmapsto N \\times_{\\varphi, M} M'", "$$", "where $M = M'/IM'$. Moreover, the composition of $F$ with", "(\\ref{equation-relative-functor}) is the identity functor on", "$\\text{Mod}_D \\times_{\\text{Mod}_C} \\text{Mod}_{C'}$. In other words,", "setting $N' = N \\times_{\\varphi, M} M'$ we have", "$N' \\otimes_{D'} D = N$ and $N' \\otimes_{D'} C' = M'$." ], "refs": [], "proofs": [ { "contents": [ "The adjointness statement follows from the more general", "Lemma \\ref{lemma-modules}.", "The final assertion follows from the corresponding assertion of", "Lemma \\ref{lemma-module-over-fibre-product}", "because", "$N' \\otimes_{D'} D = N' \\otimes_{D'} D' \\otimes_{B'} B = N' \\otimes_{B'} B$", "and", "$N' \\otimes_{D'} C' = N' \\otimes_{D'} D' \\otimes_{B'} A' = N' \\otimes_{B'} A'$." ], "refs": [ "more-algebra-lemma-modules", "more-algebra-lemma-module-over-fibre-product" ], "ref_ids": [ 9817, 9820 ] } ], "ref_ids": [] }, { "id": 9827, "type": "theorem", "label": "more-algebra-lemma-relative-surjection-ideals", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-relative-surjection-ideals", "contents": [ "In Situation \\ref{situation-relative-module-over-fibre-product}", "the map $JD' \\to IC'$ is surjective where $J = \\Ker(B' \\to B)$." ], "refs": [], "proofs": [ { "contents": [ "Since $C' = D' \\otimes_{B'} A'$ we have that $IC'$ is the image", "of $D' \\otimes_{B'} I = C' \\otimes_{A'} I \\to C'$. As the ring", "map $B' \\to A'$ induces an isomorphism $J \\to I$ the lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9828, "type": "theorem", "label": "more-algebra-lemma-relative-finite-module-over-fibre-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-relative-finite-module-over-fibre-product", "contents": [ "Let $A, A', B, B', C, C', D, D', I, M', M, N, \\varphi$ be as in", "Lemma \\ref{lemma-relative-module-over-fibre-product}.", "If $N$ finite over $D$ and $M'$ finite over $C'$, then", "$N' = N \\times_{\\varphi, M} M'$ is finite over $D'$." ], "refs": [ "more-algebra-lemma-relative-module-over-fibre-product" ], "proofs": [ { "contents": [ "Recall that $D' \\to D \\times_C C'$ is surjective by", "Lemma \\ref{lemma-module-over-fibre-product-bis}.", "Observe that $N' = N \\times_{\\varphi, M} M'$ is a module", "over $D \\times_C C'$. We can apply", "Lemma \\ref{lemma-finite-module-over-fibre-product}", "to the data $C, C', D, D', IC', M', M, N, \\varphi$", "to see that $N' = N \\times_{\\varphi, M} M'$ is finite", "over $D \\times_C C'$. Thus it is finite over $D'$." ], "refs": [ "more-algebra-lemma-module-over-fibre-product-bis", "more-algebra-lemma-finite-module-over-fibre-product" ], "ref_ids": [ 9821, 9823 ] } ], "ref_ids": [ 9826 ] }, { "id": 9829, "type": "theorem", "label": "more-algebra-lemma-relative-flat-module-over-fibre-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-relative-flat-module-over-fibre-product", "contents": [ "With $A, A', B, B', C, C', D, D', I$ as in", "Situation \\ref{situation-relative-module-over-fibre-product}.", "\\begin{enumerate}", "\\item Let $(N, M', \\varphi)$ be an object of", "$\\text{Mod}_D \\times_{\\text{Mod}_C} \\text{Mod}_{C'}$.", "If $M'$ is flat over $A'$ and $N$ is flat over $B$, then", "$N' = N \\times_{\\varphi, M} M'$ is flat over $B'$.", "\\item If $L'$ is a $D'$-module flat over $B'$, then", "$L' = (L \\otimes_{D'} D) \\times_{(L \\otimes_{D'} C)} (L \\otimes_{D'} C')$.", "\\item The category of $D'$-modules flat over $B'$", "is equivalent to the categories of objects $(N, M', \\varphi)$", "of $\\text{Mod}_D \\times_{\\text{Mod}_C} \\text{Mod}_{C'}$", "with $N$ flat over $B$ and $M'$ flat over $A'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from part (1) of", "Lemma \\ref{lemma-flat-module-over-fibre-product}.", "\\medskip\\noindent", "Part (2) follows from part (2) of", "Lemma \\ref{lemma-flat-module-over-fibre-product}", "using that $L' \\otimes_{D'} D = L' \\otimes_{B'} B$,", "$L' \\otimes_{D'} C' = L' \\otimes_{B'} A'$, and", "$L' \\otimes_{D'} C = L' \\otimes_{B'} A$, see discussion in", "Situation \\ref{situation-relative-module-over-fibre-product}.", "\\medskip\\noindent", "Part (3) is an immediate consequence of (1) and (2)." ], "refs": [ "more-algebra-lemma-flat-module-over-fibre-product", "more-algebra-lemma-flat-module-over-fibre-product" ], "ref_ids": [ 9824, 9824 ] } ], "ref_ids": [] }, { "id": 9830, "type": "theorem", "label": "more-algebra-lemma-relative-finitely-presented-module-over-fibre-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-relative-finitely-presented-module-over-fibre-product", "contents": [ "Let $A, A', B, B', C, C', D, D', I, M', M, N, \\varphi$ be as in", "Lemma \\ref{lemma-relative-module-over-fibre-product}. If", "\\begin{enumerate}", "\\item $N$ is finitely presented over $D$ and flat over $B$,", "\\item $M'$ finitely presented over $C'$ and flat over $A'$, and", "\\item the ring map $B' \\to D'$ factors as $B' \\to D'' \\to D'$", "with $B' \\to D''$ flat and $D'' \\to D'$ of finite presentation,", "\\end{enumerate}", "then $N' = N \\times_M M'$ is finitely presented over $D'$." ], "refs": [ "more-algebra-lemma-relative-module-over-fibre-product" ], "proofs": [ { "contents": [ "Choose a surjection $D''' = D''[x_1, \\ldots, x_n] \\to D'$ with", "finitely generated kernel $J$.", "By Algebra, Lemma \\ref{algebra-lemma-finite-finitely-presented-extension}", "it suffices to show that $N'$ is finitely presented as a", "$D'''$-module. Moreover, $D''' \\otimes_{B'} B \\to D' \\otimes_{B'} B = D$", "and $D''' \\otimes_{B'} A' \\to D' \\otimes_{B'} A' = C'$ are surjections", "whose kernels are generated by the image of $J$, hence $N$ is a", "finitely presented $D''' \\otimes_{B'} B$-module and", "$M'$ is a finitely presented $D''' \\otimes_{B'} A'$-module by", "Algebra, Lemma \\ref{algebra-lemma-finite-finitely-presented-extension}", "again. Thus we may replace $D'$ by $D'''$ and $D$ by", "$D''' \\otimes_{B'} B$, etc. Since $D'''$ is", "flat over $B'$, it follows that we may assume that $B' \\to D'$ is flat.", "\\medskip\\noindent", "Assume $B' \\to D'$ is flat.", "By Lemma \\ref{lemma-relative-finite-module-over-fibre-product}", "the module $N'$ is finite over $D'$. Choose a surjection", "$(D')^{\\oplus n} \\to N'$ with kernel $K'$. By base change we obtain maps", "$D^{\\oplus n} \\to N$, $(C')^{\\oplus n} \\to M'$, and $C^{\\oplus n} \\to M$", "with kernels $K_D$, $K_{C'}$, and $K_C$. There is a canonical map", "$$", "K' \\longrightarrow K_D \\times_{K_C} K_{C'}", "$$", "On the other hand, since $N' = N \\times_M M'$ and", "$D' = D \\times_C C'$ (by Lemma \\ref{lemma-flat-module-over-fibre-product};", "applied to the flat $B'$-module $D'$)", "there is also a", "canonical map $K_D \\times_{K_C} K_{C'} \\to K'$ inverse to the displayed", "arrow. Hence the displayed map is an isomorphism. By", "Algebra, Lemma \\ref{algebra-lemma-extension}", "the modules $K_D$ and $K_{C'}$ are finite. We conclude from", "Lemma \\ref{lemma-relative-finite-module-over-fibre-product}", "that $K'$ is a finite $D'$-module provided that $K_D \\to K_C$ and", "$K_{C'} \\to K_C$ induce isomorphisms", "$K_D \\otimes_B A = K_C = K_{C'} \\otimes_{A'} A$.", "This is true because the flatness assumptions implies the sequences", "$$", "0 \\to K_D \\to D^{\\oplus n} \\to N \\to 0", "\\quad\\text{and}\\quad", "0 \\to K_{C'} \\to (C')^{\\oplus n} \\to M' \\to 0", "$$", "stay exact upon tensoring, see", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}." ], "refs": [ "algebra-lemma-finite-finitely-presented-extension", "algebra-lemma-finite-finitely-presented-extension", "more-algebra-lemma-relative-finite-module-over-fibre-product", "more-algebra-lemma-flat-module-over-fibre-product", "algebra-lemma-extension", "more-algebra-lemma-relative-finite-module-over-fibre-product", "algebra-lemma-flat-tor-zero" ], "ref_ids": [ 501, 501, 9828, 9824, 330, 9828, 532 ] } ], "ref_ids": [ 9826 ] }, { "id": 9831, "type": "theorem", "label": "more-algebra-lemma-properties-algebras-over-fibre-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-properties-algebras-over-fibre-product", "contents": [ "Let $A, A', B, B', I$ be as in", "Situation \\ref{situation-module-over-fibre-product}.", "Let $(D, C', \\varphi)$ be a system consisting of an $B$-algebra $D$,", "a $A'$-algebra $C'$ and an isomorphism $D \\otimes_B A \\to C'/IC' = C$.", "Set $D' = D \\times_C C'$ (as in", "Lemma \\ref{lemma-module-over-fibre-product}). Then", "\\begin{enumerate}", "\\item $B' \\to D'$ is finite type if and only if $B \\to D$ and", "$A' \\to C'$ are finite type,", "\\item $B' \\to D'$ is flat if and only if $B \\to D$ and $A' \\to C'$ are flat,", "\\item $B' \\to D'$ is flat and of finite presentation if and only if", "$B \\to D$ and $A' \\to C'$ are flat and of finite presentation,", "\\item $B' \\to D'$ is smooth if and only if $B \\to D$ and $A' \\to C'$ are smooth,", "\\item $B' \\to D'$ is \\'etale if and only if $B \\to D$ and $A' \\to C'$", "are \\'etale.", "\\end{enumerate}", "Moreover, if $D'$ is a flat $B'$-algebra, then", "$D' \\to (D' \\otimes_{B'} B) \\times_{(D' \\otimes_{B'} A)} (D' \\otimes_{B'} A')$", "is an isomorphism. In this way the category of flat $B'$-algebras", "is equivalent to the categories of systems $(D, C', \\varphi)$ as above", "with $D$ flat over $B$ and $C'$ flat over $A'$." ], "refs": [ "more-algebra-lemma-module-over-fibre-product" ], "proofs": [ { "contents": [ "The implication ``$\\Rightarrow$'' follows from", "Algebra, Lemmas \\ref{algebra-lemma-base-change-finiteness},", "\\ref{algebra-lemma-flat-base-change},", "\\ref{algebra-lemma-base-change-smooth}, and", "\\ref{algebra-lemma-etale} because we have", "$D' \\otimes_{B'} B = D$ and $D' \\otimes_{B'} A' = C'$", "by Lemma \\ref{lemma-module-over-fibre-product}.", "Thus it suffices to prove the implications in the other direction.", "\\medskip\\noindent", "Ad (1). Assume $D$ of finite type over $B$ and $C'$ of finite type over $A'$.", "We will use the results of", "Lemma \\ref{lemma-module-over-fibre-product}", "without further mention. Choose generators $x_1, \\ldots, x_r$ of $D$ over $B$", "and generators $y_1, \\ldots, y_s$ of $C'$ over $A'$. Using that", "$D = D' \\otimes_{B'} B$ and $B' \\to B$ is surjective we can find", "$u_1, \\ldots, u_r \\in D'$ mapping to $x_1, \\ldots, x_r$ in $D$.", "Using that $C' = D' \\otimes_{B'} A'$ we can find $v_1, \\ldots, v_t \\in D'$", "such that $y_i = \\sum v_j \\otimes a'_{ij}$ for some $a'_{ij} \\in A'$.", "In particular, the images of $v_j$ in $C'$ generate $C'$ as an", "$A'$-algebra. Set $N = r + t$ and consider the cube of rings", "$$", "\\xymatrix{", "A[x_1, \\ldots, x_N] & & A'[x_1, \\ldots, x_N] \\ar[ll] \\\\", "& A \\ar[lu] & & A' \\ar[ll] \\ar[lu] \\\\", "B[x_1, \\ldots, x_N] \\ar[uu] & & B'[x_1, \\ldots, x_N] \\ar[uu] \\ar[ll] \\\\", "& B \\ar[uu] \\ar[lu] & & B' \\ar[ll] \\ar[uu] \\ar[lu]", "}", "$$", "Observe that the back square is cartesian as well.", "Consider the ring map", "$$", "B'[x_1, \\ldots, x_N] \\to D',\\quad", "x_i \\mapsto u_i \\quad\\text{and}\\quad x_{r + j} \\mapsto v_j.", "$$", "Then we see that the induced maps $B[x_1, \\ldots, x_N] \\to D$ and", "$A'[x_1, \\ldots, x_N] \\to C'$", "are surjective, in particular finite. We conclude from", "Lemma \\ref{lemma-relative-finite-module-over-fibre-product}", "that $B'[x_1, \\ldots, x_N] \\to D'$ is finite, which implies that $D'$", "is of finite type over $B'$ for example by", "Algebra, Lemma \\ref{algebra-lemma-compose-finite-type}.", "\\medskip\\noindent", "Ad (2). The implication ``$\\Leftarrow$'' follows from", "Lemma \\ref{lemma-relative-flat-module-over-fibre-product}.", "Moreover, the final statement follows from the final", "statement of Lemma \\ref{lemma-relative-flat-module-over-fibre-product}.", "\\medskip\\noindent", "Ad (3). Assume $B \\to D$ and $A' \\to C'$ are flat and of finite presentation.", "The flatness of $B' \\to D'$ we've seen in (2). We know $B' \\to D'$", "is of finite type by (1). Choose a surjection $B'[x_1, \\ldots, x_N] \\to D'$.", "By Algebra, Lemma \\ref{algebra-lemma-finite-presentation-independent}", "the ring $D$ is of finite presentation as a $B[x_1, \\ldots, x_N]$-module", "and the ring $C'$ is of finite presentation as a", "$A'[x_1, \\ldots, x_N]$-module. By", "Lemma \\ref{lemma-relative-finitely-presented-module-over-fibre-product}", "we see that $D'$ is of finite presentation as a $B'[x_1, \\ldots, x_N]$-module,", "i.e., $B' \\to D'$ is of finite presentation.", "\\medskip\\noindent", "Ad (4). Assume $B \\to D$ and $A' \\to C'$ smooth.", "By (3) we see that $B' \\to D'$ is flat and of finite presentation.", "By Algebra, Lemma \\ref{algebra-lemma-flat-fibre-smooth}", "it suffices to check that $D' \\otimes_{B'} k$ is smooth for any", "field $k$ over $B'$. If the composition $J \\to B' \\to k$ is zero,", "then $B' \\to k$ factors as $B' \\to B \\to k$ and we see that", "$$", "D' \\otimes_{B'} k = D' \\otimes_{B'} B \\otimes_B k", "= D \\otimes_B k", "$$", "is smooth as $B \\to D$ is smooth. If the composition $J \\to B' \\to k$", "is nonzero, then there exists an $h \\in J$ which does not map to zero", "in $k$. Then $B' \\to k$ factors as $B' \\to B'_h \\to k$.", "Observe that $h$ maps to zero in $B$, hence $B_h = 0$.", "Thus by Lemma \\ref{lemma-diagram-localize} we have $B'_h = A'_h$ and we get", "$$", "D' \\otimes_{B'} k = D' \\otimes_{B'} B'_h \\otimes_{B'_h} k", "= C'_h \\otimes_{A'_h} k", "$$", "is smooth as $A' \\to C'$ is smooth.", "\\medskip\\noindent", "Ad (5). Assume $B \\to D$ and $A' \\to C'$ are \\'etale. By (4) we see that", "$B' \\to D'$ is smooth. As we can read off whether or not a smooth", "map is \\'etale from the dimension of fibres we see that (5) holds", "(argue as in the proof of (4) to identify fibres -- some details omitted)." ], "refs": [ "algebra-lemma-base-change-finiteness", "algebra-lemma-flat-base-change", "algebra-lemma-base-change-smooth", "algebra-lemma-etale", "more-algebra-lemma-module-over-fibre-product", "more-algebra-lemma-module-over-fibre-product", "more-algebra-lemma-relative-finite-module-over-fibre-product", "algebra-lemma-compose-finite-type", "more-algebra-lemma-relative-flat-module-over-fibre-product", "more-algebra-lemma-relative-flat-module-over-fibre-product", "algebra-lemma-finite-presentation-independent", "more-algebra-lemma-relative-finitely-presented-module-over-fibre-product", "algebra-lemma-flat-fibre-smooth", "more-algebra-lemma-diagram-localize" ], "ref_ids": [ 373, 527, 1191, 1231, 9820, 9820, 9828, 333, 9829, 9829, 334, 9830, 1200, 9816 ] } ], "ref_ids": [ 9820 ] }, { "id": 9832, "type": "theorem", "label": "more-algebra-lemma-ideals-generated-by-minors", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ideals-generated-by-minors", "contents": [ "Let $R$ be a ring. Let $A$ be an $n \\times m$ matrix with coefficients", "in $R$. Let $I_r(A)$ be the ideal generated by the $r \\times r$-minors", "of $A$ with the convention that $I_0(A) = R$ and $I_r(A) = 0$ if", "$r > \\min(n, m)$. Then", "\\begin{enumerate}", "\\item $I_0(A) \\supset I_1(A) \\supset I_2(A) \\supset \\ldots$,", "\\item if $B$ is an $(n + n') \\times m$ matrix, and $A$ is the first", "$n$ rows of $B$, then $I_{r + n'}(B) \\subset I_r(A)$,", "\\item if $C$ is an $n \\times n$ matrix then $I_r(CA) \\subset I_r(A)$.", "\\item If $A$ is a block matrix", "$$", "\\left(", "\\begin{matrix}", "A_1 & 0 \\\\", "0 & A_2 ", "\\end{matrix}", "\\right)", "$$", "then $I_r(A) = \\sum_{r_1 + r_2 = r} I_{r_1}(A_1) I_{r_2}(A_2)$.", "\\item Add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. (Hint: Use that a determinant can be computed by expanding", "along a column or a row.)" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9833, "type": "theorem", "label": "more-algebra-lemma-fitting-ideal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-fitting-ideal", "contents": [ "Let $R$ be a ring. Let $M$ be a finite $R$-module. Choose a presentation", "$$", "\\bigoplus\\nolimits_{j \\in J} R \\longrightarrow R^{\\oplus n}", "\\longrightarrow M \\longrightarrow 0.", "$$", "of $M$. Let $A = (a_{ij})_{i = 1, \\ldots, n, j \\in J}$ be the matrix", "of the map $\\bigoplus_{j \\in J} R \\to R^{\\oplus n}$.", "The ideal $\\text{Fit}_k(M)$ generated by the", "$(n - k) \\times (n - k)$ minors of", "$A$ is independent of the choice of the presentation." ], "refs": [], "proofs": [ { "contents": [ "Let $K \\subset R^{\\oplus n}$ be the kernel of the surjection", "$R^{\\oplus n} \\to M$. Pick $z_1, \\ldots, z_{n - k} \\in K$", "and write $z_j = (z_{1j}, \\ldots, z_{nj})$.", "Another description of the ideal $\\text{Fit}_k(M)$", "is that it is the ideal generated by the $(n - k) \\times (n - k)$ minors of", "all the matrices $(z_{ij})$ we obtain in this way.", "\\medskip\\noindent", "Suppose we change the surjection into the surjection", "$R^{\\oplus n + n'} \\to M$ with kernel $K'$ where we use the original", "map on the first $n$ standard basis elements of $R^{\\oplus n + n'}$", "and $0$ on the last $n'$ basis vectors. Then the corresponding ideals", "are the same. Namely, if $z_1, \\ldots, z_{n - k} \\in K$ as above,", "let $z'_j = (z_{1j}, \\ldots, z_{nj}, 0, \\ldots, 0) \\in K'$ for", "$j = 1, \\ldots, n - k$ and", "$z'_{n + j'} = (0, \\ldots, 0, 1, 0, \\ldots, 0) \\in K'$. Then we see that", "the ideal of $(n - k) \\times (n - k)$ minors of $(z_{ij})$ agrees", "with the ideal of $(n + n' - k) \\times (n + n' - k)$ minors of", "$(z'_{ij})$. This gives one of the inclusions.", "Conversely, given $z'_1, \\ldots, z'_{n + n' - k}$", "in $K'$ we can project these to $R^{\\oplus n}$ to get", "$z_1, \\ldots, z_{n + n' - k}$ in $K$. By", "Lemma \\ref{lemma-ideals-generated-by-minors}", "we see that the ideal generated by the", "$(n + n' - k) \\times (n + n' - k)$ minors of", "$(z'_{ij})$ is contained in the ideal generated by the", "$(n - k) \\times (n - k)$ minors of $(z_{ij})$. This gives the", "other inclusion.", "\\medskip\\noindent", "Let $R^{\\oplus m} \\to M$ be another surjection with kernel $L$.", "By Schanuel's lemma (Algebra, Lemma \\ref{algebra-lemma-Schanuel})", "and the results of the previous paragraph, we may assume $m = n$", "and that there is an isomorphism $R^{\\oplus n} \\to R^{\\oplus m}$", "commuting with the surjections to $M$. Let $C = (c_{li})$ be the", "(invertible) matrix of this map (it is a square matrix as $n = m$).", "Then given $z'_1, \\ldots, z'_{n - k} \\in L$ as above we can find", "$z_1, \\ldots, z_{n - k} \\in K$ with", "$z_1' = Cz_1, \\ldots, z'_{n - k} = Cz_{n - k}$. By", "Lemma \\ref{lemma-ideals-generated-by-minors} we get one of the", "inclusions. By symmetry we get the other." ], "refs": [ "more-algebra-lemma-ideals-generated-by-minors", "algebra-lemma-Schanuel", "more-algebra-lemma-ideals-generated-by-minors" ], "ref_ids": [ 9832, 965, 9832 ] } ], "ref_ids": [] }, { "id": 9834, "type": "theorem", "label": "more-algebra-lemma-fitting-ideal-basics", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-fitting-ideal-basics", "contents": [ "Let $R$ be a ring. Let $M$ be a finite $R$-module.", "\\begin{enumerate}", "\\item If $M$ can be generated by $n$ elements, then", "$\\text{Fit}_n(M) = R$.", "\\item Given a second finite $R$-module $M'$ we have", "$$", "\\text{Fit}_l(M \\oplus M') =", "\\sum\\nolimits_{k + k' = l} \\text{Fit}_k(M)\\text{Fit}_{k'}(M')", "$$", "\\item If $R \\to R'$ is a ring map, then $\\text{Fit}_k(M \\otimes_R R')$", "is the ideal of $R'$ generated by the image of $\\text{Fit}_k(M)$.", "\\item If $M$ is of finite presentation, then $\\text{Fit}_k(M)$", "is a finitely generated ideal.", "\\item If $M \\to M'$ is a surjection, then", "$\\text{Fit}_k(M) \\subset \\text{Fit}_k(M')$.", "\\item We have $\\text{Fit}_0(M) \\subset \\text{Ann}_R(M)$.", "\\item We have $V(\\text{Fit}_0(M)) = \\text{Supp}(M)$.", "\\item Add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from the fact that $I_0(A) = R$ in", "Lemma \\ref{lemma-ideals-generated-by-minors}.", "\\medskip\\noindent", "Part (2) follows form the corresponding statement in", "Lemma \\ref{lemma-ideals-generated-by-minors}.", "\\medskip\\noindent", "Part (3) follows from the fact that $\\otimes_R R'$ is right exact,", "so the base change of a presentation of $M$ is a presentation of", "$M \\otimes_R R'$.", "\\medskip\\noindent", "Proof of (4). Let $R^{\\oplus m} \\xrightarrow{A} R^{\\oplus n} \\to M \\to 0$", "be a presentation. Then $\\text{Fit}_k(M)$ is the ideal generated by the", "$n - k \\times n - k$ minors of the matrix $A$.", "\\medskip\\noindent", "Part (5) is immediate from the definition.", "\\medskip\\noindent", "Proof of (6). Choose a presentation of $M$ with matrix $A$", "as in Lemma \\ref{lemma-fitting-ideal}.", "Let $J' \\subset J$ be a subset of cardinality $n$.", "It suffices to show that", "$f = \\det(a_{ij})_{i = 1, \\ldots, n, j \\in J'}$", "annihilates $M$.", "This is clear because the cokernel of", "$$", "R^{\\oplus n} \\xrightarrow{A' = (a_{ij})_{i = 1, \\ldots, n, j \\in J'}}", "R^{\\oplus n} \\to M \\to 0", "$$", "is killed by $f$ as there is a matrix $B$ with $A' B = f1_{n \\times n}$.", "\\medskip\\noindent", "Proof of (7). Choose a presentation of $M$ with matrix $A$", "as in Lemma \\ref{lemma-fitting-ideal}.", "By Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "we have", "$$", "M_\\mathfrak p \\not = 0", "\\Leftrightarrow", "M \\otimes_R \\kappa(\\mathfrak p) \\not = 0", "\\Leftrightarrow", "\\text{rank}(\\text{image }A\\text{ in }\\kappa(\\mathfrak p)) < n", "$$", "Clearly $\\text{Fit}_0(M)$ exactly cuts out the set of primes", "with this property." ], "refs": [ "more-algebra-lemma-ideals-generated-by-minors", "more-algebra-lemma-ideals-generated-by-minors", "more-algebra-lemma-fitting-ideal", "more-algebra-lemma-fitting-ideal", "algebra-lemma-NAK" ], "ref_ids": [ 9832, 9832, 9833, 9833, 401 ] } ], "ref_ids": [] }, { "id": 9835, "type": "theorem", "label": "more-algebra-lemma-fitting-ideal-generate-locally", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-fitting-ideal-generate-locally", "contents": [ "Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $k \\geq 0$.", "Let $\\mathfrak p \\subset R$ be a prime ideal. The following", "are equivalent", "\\begin{enumerate}", "\\item $\\text{Fit}_k(M) \\not \\subset \\mathfrak p$,", "\\item $\\dim_{\\kappa(\\mathfrak p)} M \\otimes_R \\kappa(\\mathfrak p) \\leq k$,", "\\item $M_\\mathfrak p$ can be generated by $k$ elements over $R_\\mathfrak p$, and", "\\item $M_f$ can be generated by $k$ elements over $R_f$", "for some $f \\in R$, $f \\not \\in \\mathfrak p$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK}) we see that", "$M_f$ can be generated by $k$ elements over $R_f$ for some", "$f \\in R$, $f \\not \\in \\mathfrak p$ if $M \\otimes_R \\kappa(\\mathfrak p)$", "can be generated by $k$ elements. Hence (2), (3), and (4)", "are equivalent. Using", "Lemma \\ref{lemma-fitting-ideal-basics} part (3)", "this reduces the problem to the", "case where $R$ is a field and $\\mathfrak p = (0)$. In this case", "the result follows from Example \\ref{example-fitting-free}." ], "refs": [ "algebra-lemma-NAK", "more-algebra-lemma-fitting-ideal-basics" ], "ref_ids": [ 401, 9834 ] } ], "ref_ids": [] }, { "id": 9836, "type": "theorem", "label": "more-algebra-lemma-fitting-ideal-finite-locally-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-fitting-ideal-finite-locally-free", "contents": [ "Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $r \\geq 0$.", "The following are equivalent", "\\begin{enumerate}", "\\item $M$ is finite locally free of rank $r$", "(Algebra, Definition \\ref{algebra-definition-locally-free}),", "\\item $\\text{Fit}_{r - 1}(M) = 0$ and $\\text{Fit}_r(M) = R$, and", "\\item $\\text{Fit}_k(M) = 0$ for $k < r$ and $\\text{Fit}_k(M) = R$", "for $k \\geq r$.", "\\end{enumerate}" ], "refs": [ "algebra-definition-locally-free" ], "proofs": [ { "contents": [ "It is immediate that (2) is equivalent to (3) because the Fitting ideals", "form an increasing sequence of ideals.", "Since the formation of $\\text{Fit}_k(M)$ commutes with base change", "(Lemma \\ref{lemma-fitting-ideal-basics}) we see that (1) implies (2) by", "Example \\ref{example-fitting-free}", "and glueing results (Algebra, Section \\ref{algebra-section-more-glueing}).", "Conversely, assume (2). By", "Lemma \\ref{lemma-fitting-ideal-generate-locally} we may assume that $M$", "is generated by $r$ elements. Thus a presentation", "$\\bigoplus_{j \\in J} R \\to R^{\\oplus r} \\to M \\to 0$.", "But now the assumption that $\\text{Fit}_{r - 1}(M) = 0$ implies", "that all entries of the matrix of the map", "$\\bigoplus_{j \\in J} R \\to R^{\\oplus r}$ are zero.", "Thus $M$ is free." ], "refs": [ "more-algebra-lemma-fitting-ideal-basics", "more-algebra-lemma-fitting-ideal-generate-locally" ], "ref_ids": [ 9834, 9835 ] } ], "ref_ids": [ 1492 ] }, { "id": 9837, "type": "theorem", "label": "more-algebra-lemma-principal-fitting-ideal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-principal-fitting-ideal", "contents": [ "Let $R$ be a local ring. Let $M$ be a finite $R$-module. Let $k \\geq 0$.", "Assume that $\\text{Fit}_k(M) = (f)$ for some $f \\in R$.", "Let $M'$ be the quotient of $M$ by $\\{x \\in M \\mid fx = 0\\}$. Then", "$M'$ can be generated by $k$ elements." ], "refs": [], "proofs": [ { "contents": [ "Choose generators $x_1, \\ldots, x_n \\in M$ corresponding to the", "surjection $R^{\\oplus n} \\to M$. Since $R$ is local if a set", "of elements $E \\subset (f)$ generates $(f)$, then some $e \\in E$ generates", "$(f)$, see Algebra, Lemma \\ref{algebra-lemma-NAK}. Hence we may pick", "$z_1, \\ldots, z_{n - k}$ in the kernel of $R^{\\oplus n} \\to M$ such", "that some $(n - k) \\times (n - k)$ minor of the $n \\times (n - k)$", "matrix $A = (z_{ij})$ generates $(f)$. After renumbering the $x_i$ we may", "assume the first minor $\\det(z_{ij})_{1 \\leq i, j \\leq n - k}$", "generates $(f)$, i.e., $\\det(z_{ij})_{1 \\leq i, j \\leq n - k} = uf$", "for some unit $u \\in R$. Every other minor is a multiple of $f$.", "By Algebra, Lemma \\ref{algebra-lemma-matrix-right-inverse} there exists a", "$n - k \\times n - k$ matrix $B$ such that", "$$", "AB = f", "\\left(", "\\begin{matrix}", "u 1_{n - k \\times n - k} \\\\", "C", "\\end{matrix}", "\\right)", "$$", "for some matrix $C$ with coefficients in $R$. This implies that for every", "$i \\leq n - k$ the element $y_i = ux_i + \\sum_j c_{ji}x_j$ is annihilated", "by $f$. Since $M/\\sum Ry_i$ is generated by the images of", "$x_{n - k + 1}, \\ldots, x_n$ we win." ], "refs": [ "algebra-lemma-NAK", "algebra-lemma-matrix-right-inverse" ], "ref_ids": [ 401, 382 ] } ], "ref_ids": [] }, { "id": 9838, "type": "theorem", "label": "more-algebra-lemma-fitting-ideals-and-pd1", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-fitting-ideals-and-pd1", "contents": [ "Let $R$ be a ring. Let $M$ be a finitely presented $R$-module. Let $k \\geq 0$.", "Assume that $\\text{Fit}_k(M) = (f)$ for some nonzerodivisor $f \\in R$", "and $\\text{Fit}_{k - 1}(M) = 0$. Then $M$ has projective dimension $\\leq 1$." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation", "$$", "R^{\\oplus m} \\xrightarrow{A} R^{\\oplus n} \\to M \\to 0", "$$", "We claim the image of $A$ is finite locally free of rank $n - k$.", "If the claim holds, then the lemma is true by definition of", "projective dimension. To prove the claim we may replace $R$", "by the localization at a prime, see", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}.", "This reduces us the the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $R$ is local. Set $M' = \\{x \\in M \\mid fx = 0\\}$. By", "Lemma \\ref{lemma-principal-fitting-ideal} we can choose", "$x_1, \\ldots, x_k \\in M$ which generate $M/M'$. Then", "$x_1, \\ldots, x_k$ generate $M_f = (M/M')_f$. Hence,", "if there is a relation $\\sum a_ix_i = 0$ in $M$,", "then we see that $a_1, \\ldots, a_k$ map to zero in $R_f$", "since otherwise $\\text{Fit}_{k - 1}(M) R_f = \\text{Fit}_{k - 1}(M_f)$", "would be nonzero.", "Since $f$ is a nonzerodivisor, we conclude $a_1 = \\ldots = a_k = 0$.", "Thus $M \\cong R^{\\oplus k} \\oplus M'$. After a change of basis", "in our presentation above, we may assume the first $n - k$", "basis vectors of $R^{\\oplus n}$ map into the summand $M'$ of $M$", "and the last $k$-basis vectors of $R^{\\oplus n}$ map to", "basis elements of the summand $R^{\\oplus k}$ of $M$.", "Having done so, the last $k$ rows of the matrix $A$ vanish.", "In this way we see that, replacing $M$ by $M'$, $k$ by $0$,", "$n$ by $n - k$, and $A$ by the submatrix where we", "delete the last $k$ rows, we reduce to the case discussed", "in the next paragraph.", "\\medskip\\noindent", "Assume $R$ is local, $k = 0$, and $M$ annihilated by $f$.", "Now the $0$th Fitting ideal of $M$ is $(f)$ and is generated by the", "$n \\times n$ minors of the matrix $A$ of size $n \\times m$.", "(This in particular implies $m \\geq n$.)", "Since $R$ is local, some $n \\times n$ minor of $A$ is $uf$", "for a unit $u \\in R$.", "After renumbering we may assume this minor is the first one.", "Moreover, we know all other $n \\times n$ minors of $A$ are", "divisible by $f$. Write $A = (A_1 A_2)$ in block form where", "$A_1$ is an $n \\times n$ matrix and $A_2$ is an", "$n \\times (m - n)$ matrix. By", "Algebra, Lemma \\ref{algebra-lemma-matrix-right-inverse}", "applied to the transpose of $A$ (!) we find there exists an", "$n \\times n$ matrix $B$ such that", "$$", "BA = B(A_1 A_2) = f", "\\left(", "\\begin{matrix}", "u 1_{n \\times n} & C", "\\end{matrix}", "\\right)", "$$", "for some $n \\times (m - n)$ matrix $C$ with coefficients in $R$.", "Then we first conclude $BA_1 = fu 1_{n \\times n}$.", "Thus", "$$", "BA_2 = fC = u^{-1}fuC = u^{-1}BA_1C", "$$", "Since the determinant of $B$ is a nonzerodivisor we conclude", "that $A_2 = u^{-1}A_1C$. Therefore the image of $A$ is equal", "to the image of $A_1$ which is isomorphic to $R^{\\oplus n}$", "because the determinant of $A_1$ is a nonzerodivisor.", "This finishes the proof." ], "refs": [ "algebra-lemma-finite-projective", "more-algebra-lemma-principal-fitting-ideal", "algebra-lemma-matrix-right-inverse" ], "ref_ids": [ 795, 9837, 382 ] } ], "ref_ids": [] }, { "id": 9839, "type": "theorem", "label": "more-algebra-lemma-lift-invertible-element", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-invertible-element", "contents": [ "Let $A$ be a ring, let $I \\subset A$ be an ideal, let $\\overline{u} \\in A/I$", "be an invertible element. There exists an \\'etale ring map $A \\to A'$ which", "induces an isomorphism $A/I \\to A'/IA'$ and an invertible element $u' \\in A'$", "lifting $\\overline{u}$." ], "refs": [], "proofs": [ { "contents": [ "Choose any lift $f \\in A$ of $\\overline{u}$ and set $A' = A_f$ and $u$", "the image of $f$ in $A'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9840, "type": "theorem", "label": "more-algebra-lemma-lift-idempotent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-idempotent", "contents": [ "Let $A$ be a ring, let $I \\subset A$ be an ideal, let $\\overline{e} \\in A/I$", "be an idempotent. There exists an \\'etale ring map $A \\to A'$ which", "induces an isomorphism $A/I \\to A'/IA'$ and an idempotent $e' \\in A'$", "lifting $\\overline{e}$." ], "refs": [], "proofs": [ { "contents": [ "Choose any lift $x \\in A$ of $\\overline{e}$. Set", "$$", "A' = A[t]/(t^2 - t)\\left[\\frac{1}{t - 1 + x}\\right].", "$$", "The ring map $A \\to A'$ is \\'etale because $(2t - 1)\\text{d}t = 0$", "and $(2t - 1)(2t - 1) = 1$ which is invertible. We have", "$A'/IA' = A/I[t]/(t^2 - t)[\\frac{1}{t - 1 + \\overline{e}}] \\cong A/I$", "the last map sending $t$ to $\\overline{e}$ which works as", "$\\overline{e}$ is a root of $t^2 - t$. This also shows that setting", "$e'$ equal to the class of $t$ in $A'$ works." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9841, "type": "theorem", "label": "more-algebra-lemma-lift-open-covering", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-open-covering", "contents": [ "Let $A$ be a ring, let $I \\subset A$ be an ideal. Let", "$\\Spec(A/I) = \\coprod_{j \\in J} \\overline{U}_j$ be a finite disjoint open", "covering. Then there exists an \\'etale ring map $A \\to A'$ which", "induces an isomorphism $A/I \\to A'/IA'$ and a finite disjoint open covering", "$\\Spec(A') = \\coprod_{j \\in J} U'_j$ lifting the given covering." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-lift-idempotent} and", "the fact that open and closed subsets of Spectra correspond", "to idempotents, see Algebra, Lemma \\ref{algebra-lemma-disjoint-decomposition}." ], "refs": [ "more-algebra-lemma-lift-idempotent", "algebra-lemma-disjoint-decomposition" ], "ref_ids": [ 9840, 405 ] } ], "ref_ids": [] }, { "id": 9842, "type": "theorem", "label": "more-algebra-lemma-localize-upstairs", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-localize-upstairs", "contents": [ "Let $A \\to B$ be a ring map and $J \\subset B$ an ideal. If", "$A \\to B$ is \\'etale at every prime of $V(J)$, then there exists", "a $g \\in B$ mapping to an invertible element", "of $B/J$ such that $A' = B_g$ is \\'etale over $A$." ], "refs": [], "proofs": [ { "contents": [ "The set of points of $\\Spec(B)$ where $A \\to B$ is not \\'etale is a", "closed subset of $\\Spec(B)$, see", "Algebra, Definition \\ref{algebra-definition-etale}.", "Write this as $V(J')$ for some ideal $J' \\subset B$. Then", "$V(J') \\cap V(J) = \\emptyset$ hence $J + J' = B$ by", "Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}.", "Write $1 = f + g$ with $f \\in J$ and $g \\in J'$.", "Then $g$ works." ], "refs": [ "algebra-definition-etale", "algebra-lemma-Zariski-topology" ], "ref_ids": [ 1539, 389 ] } ], "ref_ids": [] }, { "id": 9843, "type": "theorem", "label": "more-algebra-lemma-lift-factorization-monic", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-factorization-monic", "contents": [ "Let $A$ be a ring, let $I \\subset A$ be an ideal. Let $f \\in A[x]$ be a", "monic polynomial. Let $\\overline{f} = \\overline{g} \\overline{h}$ be a", "factorization of $f$ in $A/I[x]$ such that $\\overline{g}$ and $\\overline{h}$", "are monic and generate the unit ideal in $A/I[x]$. Then there exists an", "\\'etale ring map $A \\to A'$ which induces an isomorphism $A/I \\to A'/IA'$", "and a factorization $f = g' h'$ in $A'[x]$ with $g'$, $h'$ monic", "lifting the given factorization over $A/I$." ], "refs": [], "proofs": [ { "contents": [ "We will deduce this from results on the universal factorization proved", "earlier; however, we encourage the reader to find their own proof not", "using this trick. Say $\\deg(\\overline{g}) = n$ and $\\deg(\\overline{h}) = m$ so", "that $\\deg(f) = n + m$. Write $f = x^{n + m} + \\sum \\alpha_i x^{n + m - i}$", "for some $\\alpha_1, \\ldots, \\alpha_{n + m} \\in A$. Consider the ring map", "$$", "R = \\mathbf{Z}[a_1, \\ldots, a_{n + m}]", "\\longrightarrow", "S = \\mathbf{Z}[b_1, \\ldots, b_n, c_1, \\ldots, c_m]", "$$", "of Algebra, Example \\ref{algebra-example-factor-polynomials-etale}.", "Let $R \\to A$ be the ring map which sends $a_i$ to $\\alpha_i$.", "Set", "$$", "B = A \\otimes_R S", "$$", "By construction the image $f_B$ of $f$ in $B[x]$ factors, say", "$f_B = g_B h_B$ with $g_B = x^n + \\sum (1 \\otimes b_i) x^{n - i}$", "and similarly for $h_B$.", "Write $\\overline{g} = x^n + \\sum \\overline{\\beta}_i x^{n - i}$ and", "$\\overline{h} = x^m + \\sum \\overline{\\gamma}_i x^{m - i}$.", "The $A$-algebra map", "$$", "B \\longrightarrow A/I, \\quad", "1 \\otimes b_i \\mapsto \\overline{\\beta}_i, \\quad", "1 \\otimes c_i \\mapsto \\overline{\\gamma}_i", "$$", "maps $g_B$ and $h_B$ to $\\overline{g}$ and $\\overline{h}$ in $A/I[x]$.", "The displayed map is surjective; denote $J \\subset B$ its kernel.", "From the discussion in ", "Algebra, Example \\ref{algebra-example-factor-polynomials-etale}", "it is clear that $A \\to B$ is etale at all points of $V(J) \\subset \\Spec(B)$.", "Choose $g \\in B$ as in Lemma \\ref{lemma-localize-upstairs} and", "consider the $A$-algebra $B_g$. Since $g$ maps to a unit", "in $B/J = A/I$ we obtain also a map $B_g/I B_g \\to A/I$ of $A/I$-algebras.", "Since $A/I \\to B_g/I B_g$ is \\'etale, also $B_g/IB_g \\to A/I$ is \\'etale", "(Algebra, Lemma \\ref{algebra-lemma-map-between-etale}). Hence there exists an", "idempotent $e \\in B_g/I B_g$ such that $A/I = (B_g/I B_g)_e$", "(Algebra, Lemma \\ref{algebra-lemma-surjective-flat-finitely-presented}).", "Choose a lift $h \\in B_g$ of $e$. Then $A \\to A' = (B_g)_h$ with", "factorization given by the image of the factorization $f_B = g_B h_B$", "in $A'$ is a solution to the problem posed by the lemma." ], "refs": [ "more-algebra-lemma-localize-upstairs", "algebra-lemma-map-between-etale", "algebra-lemma-surjective-flat-finitely-presented" ], "ref_ids": [ 9842, 1236, 1237 ] } ], "ref_ids": [] }, { "id": 9844, "type": "theorem", "label": "more-algebra-lemma-lift-factorization-easy", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-factorization-easy", "contents": [ "Let $A$ be a ring, let $I \\subset A$ be an ideal. Let $f \\in A[x]$ be a", "monic polynomial. Let $\\overline{f} = \\overline{g} \\overline{h}$ be a", "factorization of $f$ in $A/I[x]$ and assume", "\\begin{enumerate}", "\\item the leading coefficient of $\\overline{g}$ is an invertible element", "of $A/I$, and", "\\item $\\overline{g}$, $\\overline{h}$ generate the unit ideal in $A/I[x]$.", "\\end{enumerate}", "Then there exists an \\'etale ring map $A \\to A'$ which induces an", "isomorphism $A/I \\to A'/IA'$ and a factorization $f = g' h'$ in $A'[x]$", "lifting the given factorization over $A/I$." ], "refs": [], "proofs": [ { "contents": [ "Applying Lemma \\ref{lemma-lift-invertible-element} we may assume that", "the leading coefficient of $\\overline{g}$ is the reduction of an", "invertible element $u \\in A$. Then we may replace $\\overline{g}$ by", "$\\overline{u}^{-1}\\overline{g}$ and $\\overline{h}$ by", "$\\overline{u}\\overline{h}$. Thus we may assume that $\\overline{g}$", "is monic. Since $f$ is monic we conclude that $\\overline{h}$ is monic", "too. In this case the result follows from", "Lemma \\ref{lemma-lift-factorization-monic}." ], "refs": [ "more-algebra-lemma-lift-invertible-element", "more-algebra-lemma-lift-factorization-monic" ], "ref_ids": [ 9839, 9843 ] } ], "ref_ids": [] }, { "id": 9845, "type": "theorem", "label": "more-algebra-lemma-lift-factorization", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-factorization", "contents": [ "Let $A$ be a ring, let $I \\subset A$ be an ideal.", "Let $f \\in A[x]$ be a monic polynomial.", "Let $\\overline{f} = \\overline{g} \\overline{h}$ be a factorization of $f$", "in $A/I[x]$ and assume that $\\overline{g}$, $\\overline{h}$ generate", "the unit ideal in $A/I[x]$. Then there exists an \\'etale ring map", "$A \\to A'$ which induces an isomorphism $A/I \\to A'/IA'$ and a factorization", "$f = g' h'$ in $A'[x]$ lifting the given factorization over $A/I$." ], "refs": [], "proofs": [ { "contents": [ "Say $f = x^d + a_1 x^{d - 1} + \\ldots + a_d$ has degree $d$.", "Write $\\overline{g} = \\sum \\overline{b}_j x^j$ and", "$\\overline{h} = \\sum \\overline{c}_j x^j$. Then we see that", "$1 = \\sum \\overline{b}_j \\overline{c}_{d - j}$. It follows that", "$\\Spec(A/I)$ is covered by the standard opens", "$D(\\overline{b}_j \\overline{c}_{d - j})$. However, each point", "$\\mathfrak p$ of $\\Spec(A/I)$ is contained in at most one of these as", "by looking at the induced factorization of $f$ over the field", "$\\kappa(\\mathfrak p)$ we see that $\\deg(\\overline{g} \\bmod \\mathfrak p) +", "\\deg(\\overline{h} \\bmod \\mathfrak p) = d$. Hence our open covering", "is a disjoint open covering. Applying Lemma \\ref{lemma-lift-open-covering}", "(and replacing $A$ by $A'$) we see that we may assume there is a", "corresponding disjoint open covering of $\\Spec(A)$. This disjoint open", "covering corresponds to a product decomposition of $A$, see", "Algebra, Lemma \\ref{algebra-lemma-disjoint-implies-product}. It follows that", "$$", "A = A_0 \\times \\ldots \\times A_d,", "\\quad", "I = I_0 \\times \\ldots \\times I_d,", "$$", "where the image of $\\overline{g}$, resp.\\ $\\overline{h}$ in $A_j/I_j$", "has degree $j$, resp.\\ $d - j$ with invertible leading coefficient.", "Clearly, it suffices to prove the result for each factor $A_j$", "separatedly. Hence the lemma follows from", "Lemma \\ref{lemma-lift-factorization-easy}." ], "refs": [ "more-algebra-lemma-lift-open-covering", "algebra-lemma-disjoint-implies-product", "more-algebra-lemma-lift-factorization-easy" ], "ref_ids": [ 9841, 415, 9844 ] } ], "ref_ids": [] }, { "id": 9846, "type": "theorem", "label": "more-algebra-lemma-separate-image-closed-from-closed", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-separate-image-closed-from-closed", "contents": [ "Let $R \\to S$ be a ring map. Let $I \\subset R$ be an ideal of $R$", "and let $J \\subset S$ be an ideal of $S$. If the closure of the image", "of $V(J)$ in $\\Spec(R)$ is disjoint from $V(I)$, then there exists", "an element $f \\in R$ which maps to $1$ in $R/I$ and to an element", "of $J$ in $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $I' \\subset R$ be an ideal such that $V(I')$ is the closure of", "the image of $V(J)$. Then $V(I) \\cap V(I') = \\emptyset$ by assumption", "and hence $I + I' = R$ by", "Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}.", "Write $1 = g + f$ with $g \\in I$ and $f \\in I'$.", "We have $V(f') \\supset V(J)$ where $f'$ is the image of $f$ in $S$.", "Hence $(f')^n \\in J$ for some $n$, see", "Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}.", "Replacing $f$ by $f^n$ we win." ], "refs": [ "algebra-lemma-Zariski-topology", "algebra-lemma-Zariski-topology" ], "ref_ids": [ 389, 389 ] } ], "ref_ids": [] }, { "id": 9847, "type": "theorem", "label": "more-algebra-lemma-helper-integral", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-helper-integral", "contents": [ "Let $I$ be an ideal of a ring $A$. Let $A \\to B$ be an integral ring map.", "Let $b \\in B$ map to an idempotent in $B/IB$. Then there exists a", "monic $f \\in A[x]$ with $f(b) = 0$ and $f \\bmod I = x^d(x - 1)^d$", "for some $d \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $z = b^2 - b$ is an element of $IB$. By", "Algebra, Lemma \\ref{algebra-lemma-integral-integral-over-ideal}", "there exist a monic polynomial", "$g(x) = x^d + \\sum a_j x^j$ of degree $d$ with $a_j \\in I$ such that", "$g(z) = 0$ in $B$. Hence $f(x) = g(x^2 - x) \\in A[x]$ is a monic", "polynomial such that $f(x) \\equiv x^d(x - 1)^d \\bmod I$", "and such that $f(b) = 0$ in $B$." ], "refs": [ "algebra-lemma-integral-integral-over-ideal" ], "ref_ids": [ 519 ] } ], "ref_ids": [] }, { "id": 9848, "type": "theorem", "label": "more-algebra-lemma-lift-idempotent-upstairs", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-idempotent-upstairs", "contents": [ "Let $A$ be a ring, let $I \\subset A$ be an ideal.", "Let $A \\to B$ be an integral ring map.", "Let $\\overline{e} \\in B/IB$ be an idempotent.", "Then there exists an \\'etale ring map $A \\to A'$", "which induces an isomorphism $A/I \\to A'/IA'$ and an idempotent", "$e' \\in B \\otimes_A A'$ lifting $\\overline{e}$." ], "refs": [], "proofs": [ { "contents": [ "Choose an element $y \\in B$ lifting $\\overline{e}$.", "Choose $f \\in A[x]$ as in Lemma \\ref{lemma-helper-integral} for $y$.", "By Lemma \\ref{lemma-lift-factorization-easy}", "we can find an \\'etale ring map $A \\to A'$ which induces", "an isomorphism $A/I \\to A'/IA'$ and such that $f = gh$", "in $A[x]$ with $g(x) = x^d \\bmod IA'$ and $h(x) = (x - 1)^d \\bmod IA'$.", "After replacing $A$ by $A'$ we may assume that the factorization", "is defined over $A$. In that case we see that", "$b_1 = g(y) \\in B$ is a lift of $\\overline{e}^d = \\overline{e}$ and", "$b_2 = h(y) \\in B$ is a lift of", "$(\\overline{e} - 1)^d = (-1)^d (1 - \\overline{e})^d = (-1)^d(1 - \\overline{e})$", "and moreover $b_1b_2 = 0$. Thus $(b_1, b_2)B/IB = B/IB$ and", "$V(b_1, b_2) \\subset \\Spec(B)$ is disjoint from $V(IB)$. Since", "$\\Spec(B) \\to \\Spec(A)$ is closed (see", "Algebra, Lemmas \\ref{algebra-lemma-integral-going-up} and", "\\ref{algebra-lemma-going-up-closed})", "we can find an $a \\in A$ which maps to an invertible element", "of $A/I$ whose image in $B$ lies in $(b_1, b_2)$, see", "Lemma \\ref{lemma-separate-image-closed-from-closed}.", "After replacing $A$ by the localization $A_a$ we get that", "$(b_1, b_2) = B$. Then $\\Spec(B) = D(b_1) \\amalg D(b_2)$;", "disjoint union because $b_1b_2 = 0$ and covers", "$\\Spec(B)$ because $(b_1, b_2) = B$. Let $e \\in B$ be the idempotent", "corresponding to the open and closed subset $D(b_1)$, see", "Algebra, Lemma \\ref{algebra-lemma-disjoint-decomposition}.", "Since $b_1$ is a lift of $\\overline{e}$ and $b_2$ is a", "lift of $\\pm (1 - \\overline{e})$ we conclude that $e$ is", "a lift of $\\overline{e}$ by the uniqueness statement in", "Algebra, Lemma \\ref{algebra-lemma-disjoint-decomposition}." ], "refs": [ "more-algebra-lemma-helper-integral", "more-algebra-lemma-lift-factorization-easy", "algebra-lemma-integral-going-up", "algebra-lemma-going-up-closed", "more-algebra-lemma-separate-image-closed-from-closed", "algebra-lemma-disjoint-decomposition", "algebra-lemma-disjoint-decomposition" ], "ref_ids": [ 9847, 9844, 500, 552, 9846, 405, 405 ] } ], "ref_ids": [] }, { "id": 9849, "type": "theorem", "label": "more-algebra-lemma-lift-projective-module", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-projective-module", "contents": [ "Let $A$ be a ring, let $I \\subset A$ be an ideal.", "Let $\\overline{P}$ be a finite projective $A/I$-module.", "Then there exists an \\'etale ring map $A \\to A'$ which induces", "an isomorphism $A/I \\to A'/IA'$ and a finite projective", "$A'$-module $P'$ lifting $\\overline{P}$." ], "refs": [], "proofs": [ { "contents": [ "We can choose an integer $n$ and a direct sum decomposition", "$(A/I)^{\\oplus n} = \\overline{P} \\oplus \\overline{K}$", "for some $R/I$-module $\\overline{K}$. Choose a lift", "$\\varphi : A^{\\oplus n} \\to A^{\\oplus n}$ of the projector $\\overline{p}$", "associated to the direct summand $\\overline{P}$.", "Let $f \\in A[x]$ be the characteristic polynomial of $\\varphi$.", "Set $B = A[x]/(f)$. By Cayley-Hamilton", "(Algebra, Lemma \\ref{algebra-lemma-charpoly}) there is a map", "$B \\to \\text{End}_A(A^{\\oplus n})$ mapping $x$ to $\\varphi$.", "For every prime $\\mathfrak p \\supset I$ the image of $f$ in", "$\\kappa(\\mathfrak p)$ is $(x - 1)^rx^{n - r}$ where $r$ is the", "dimension of $\\overline{P} \\otimes_{A/I} \\kappa(\\mathfrak p)$.", "Hence $(x - 1)^nx^n$ maps to zero in $B \\otimes_A \\kappa(\\mathfrak p)$", "for all $\\mathfrak p \\supset I$. Thus $x(1 - x)$ is contained", "in every prime ideal of $B/IB$. Hence $x^N(1 - x)^N$ is", "contained in $IB$ for some $N \\geq 1$.", "It follows that $x^N + (1 - x)^N$ is a unit in $B/IB$ and that", "$$", "\\overline{e} = \\text{image of }\\frac{x^N}{x^N + (1 - x)^N}\\text{ in }B/IB", "$$", "is an idempotent as both assertions hold in $\\mathbf{Z}[x]/(x^N(x - 1)^N)$.", "The image of $\\overline{e}$ in $\\text{End}_{A/I}((A/I)^{\\oplus n})$ is", "$$", "\\frac{\\overline{p}^N}{\\overline{p}^N + (1 - \\overline{p})^N} = \\overline{p}", "$$", "as $\\overline{p}$ is an idempotent. After replacing $A$ by an \\'etale", "extension $A'$ as in the lemma, we may assume there exists an idempotent", "$e \\in B$ which maps to $\\overline{e}$ in $B/IB$, see", "Lemma \\ref{lemma-lift-idempotent-upstairs}.", "Then the image of $e$ under the map", "$$", "B = A[x]/(f) \\longrightarrow \\text{End}_A(A^{\\oplus n}).", "$$", "is an idempotent element $p$ which lifts $\\overline{p}$.", "Setting $P = \\Im(p)$ we win." ], "refs": [ "algebra-lemma-charpoly", "more-algebra-lemma-lift-idempotent-upstairs" ], "ref_ids": [ 385, 9848 ] } ], "ref_ids": [] }, { "id": 9850, "type": "theorem", "label": "more-algebra-lemma-cotangent-complex-symmetric-algebra", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cotangent-complex-symmetric-algebra", "contents": [ "Let $A$ be a ring. Let $0 \\to K \\to A^{\\oplus m} \\to M \\to 0$", "be a sequence of $A$-modules. Consider the $A$-algebra", "$C = \\text{Sym}^*_A(M)$ with its presentation", "$\\alpha : A[y_1, \\ldots, y_m] \\to C$", "coming from the surjection $A^{\\oplus m} \\to M$. Then", "$$", "\\NL(\\alpha) =", "(K \\otimes_A C \\to \\bigoplus\\nolimits_{j = 1, \\ldots, m} C \\text{d}y_j)", "$$", "(see Algebra, Section \\ref{algebra-section-netherlander})", "in particular $\\Omega_{C/A} = M \\otimes_A C$." ], "refs": [], "proofs": [ { "contents": [ "Let $J = \\Ker(\\alpha)$. The lemma asserts that", "$J/J^2 \\cong K \\otimes_A C$. Note that $\\alpha$ is a homomorphism", "of graded algebras. We will prove that in degree $d$ we have", "$(J/J^2)_d = K \\otimes_A C_{d - 1}$. Note that", "$$", "J_d = \\Ker(\\text{Sym}^d_A(A^{\\oplus m}) \\to \\text{Sym}^d_A(M))", "= \\Im(K \\otimes_A \\text{Sym}^{d - 1}_A(A^{\\oplus m})", "\\to \\text{Sym}^d_A(A^{\\oplus m})),", "$$", "see Algebra, Lemma \\ref{algebra-lemma-presentation-sym-exterior}.", "It follows that $(J^2)_d = \\sum_{a + b = d} J_a \\cdot J_b$ is the image of", "$$", "K \\otimes_A K \\otimes_A \\text{Sym}^{d - 2}_A(A^{\\otimes m})", "\\to \\text{Sym}^d_A(A^{\\oplus m}).", "$$", "The cokernel of the map $K \\otimes_A \\text{Sym}^{d - 2}_A(A^{\\otimes m}) \\to", "\\text{Sym}^{d - 1}_A(A^{\\oplus m})$ is $\\text{Sym}^{d - 1}_A(M)$ by", "the lemma referenced above.", "Hence it is clear that $(J/J^2)_d = J_d/(J^2)_d$ is equal to", "\\begin{align*}", "\\Coker(", "K \\otimes_A K \\otimes_A \\text{Sym}^{d - 2}_A(A^{\\otimes m})", "\\to K \\otimes_A \\text{Sym}^{d - 1}_A(A^{\\otimes m}))", "& = K \\otimes_A \\text{Sym}^{d - 1}_A(M) \\\\", "& = K \\otimes_A C_{d -1}", "\\end{align*}", "as desired." ], "refs": [ "algebra-lemma-presentation-sym-exterior" ], "ref_ids": [ 369 ] } ], "ref_ids": [] }, { "id": 9851, "type": "theorem", "label": "more-algebra-lemma-symmetric-algebra-smooth", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-symmetric-algebra-smooth", "contents": [ "Let $A$ be a ring. Let $M$ be an $A$-module. Then $C = \\text{Sym}_A^*(M)$", "is smooth over $A$ if and only if $M$ is a finite projective $A$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $\\sigma : C \\to A$ be the projection onto the degree $0$ part of $C$.", "Then $J = \\Ker(\\sigma)$ is the part of degree $> 0$ and we see that", "$J/J^2 = M$ as an $A$-module. Hence if $A \\to C$ is smooth then $M$ is", "a finite projective $A$-module by", "Algebra, Lemma \\ref{algebra-lemma-section-smooth}.", "\\medskip\\noindent", "Conversely, assume that $M$ is finite projective and choose a surjection", "$A^{\\oplus n} \\to M$ with kernel $K$. Of course the sequence", "$0 \\to K \\to A^{\\oplus n} \\to M \\to 0$ is split as $M$ is projective.", "In particular we see that $K$ is a finite $A$-module and hence", "$C$ is of finite presentation over $A$ as $C$ is a quotient of", "$A[x_1, \\ldots, x_n]$ by the ideal generated by $K \\subset \\bigoplus Ax_i$.", "The computation of Lemma \\ref{lemma-cotangent-complex-symmetric-algebra}", "shows that $\\NL_{C/A}$ is homotopy equivalent to $(K \\to M) \\otimes_A C$.", "Hence $\\NL_{C/A}$ is quasi-isomorphic to $C \\otimes_A M$ placed in degree", "$0$ which means that $C$ is smooth over $A$ by", "Algebra, Definition \\ref{algebra-definition-smooth}." ], "refs": [ "more-algebra-lemma-cotangent-complex-symmetric-algebra", "algebra-definition-smooth" ], "ref_ids": [ 9850, 1534 ] } ], "ref_ids": [] }, { "id": 9852, "type": "theorem", "label": "more-algebra-lemma-lift-section-smooth-morphism", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-section-smooth-morphism", "contents": [ "Let $A$ be a ring, let $I \\subset A$ be an ideal. Consider a commutative", "diagram", "$$", "\\xymatrix{", "B \\ar[rd] \\\\", "A \\ar[u] \\ar[r] & A/I", "}", "$$", "where $B$ is a smooth $A$-algebra. Then there exists an \\'etale ring", "map $A \\to A'$ which induces an isomorphism $A/I \\to A'/IA'$ and an", "$A$-algebra map $B \\to A'$ lifting the ring map $B \\to A/I$." ], "refs": [], "proofs": [ { "contents": [ "Let $J \\subset B$ be the kernel of $B \\to A/I$ so that $B/J = A/I$. By", "Algebra, Lemma \\ref{algebra-lemma-application-NL-smooth} the sequence", "$$", "0 \\to I/I^2 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B B/J \\to 0", "$$", "is split exact. Thus $\\overline{P} = J/(J^2 + IB) = \\Omega_{B/A} \\otimes_B B/J$", "is a finite projective $A/I$-module. Choose an integer $n$ and a direct sum", "decomposition $A/I^{\\oplus n} = \\overline{P} \\oplus \\overline{K}$.", "By Lemma \\ref{lemma-lift-projective-module} we can find an", "\\'etale ring map $A \\to A'$ which induces an isomorphism", "$A/I \\to A'/IA'$ and a finite projective $A$-module $K$ which", "lifts $\\overline{K}$. We may and do replace $A$ by $A'$.", "Set $B' = B \\otimes_A \\text{Sym}_A^*(K)$. Since $A \\to \\text{Sym}_A^*(K)$", "is smooth by Lemma \\ref{lemma-symmetric-algebra-smooth} we see that", "$B \\to B'$ is smooth which in turn implies that $A \\to B'$ is smooth (see", "Algebra, Lemmas \\ref{algebra-lemma-base-change-smooth} and", "\\ref{algebra-lemma-locally-smooth}).", "Moreover the section $\\text{Sym}^*_A(K) \\to A$ determines a section", "$B' \\to B$ and we let $B' \\to A/I$ be the composition $B' \\to B \\to A/I$.", "Let $J' \\subset B'$ be the kernel of $B' \\to A/I$. We have", "$JB' \\subset J'$ and $B \\otimes_A K \\subset J'$. These maps combine", "to give an isomorphism", "$$", "(A/I)^{\\oplus n} \\cong J/J^2 \\oplus \\overline{K}", "\\longrightarrow", "J'/((J')^2 + IB')", "$$", "Thus, after replacing $B$ by $B'$ we may assume that", "$J/(J^2 + IB) = \\Omega_{B/A} \\otimes_B B/J$ is a free", "$A/I$-module of rank $n$.", "\\medskip\\noindent", "In this case, choose $f_1, \\ldots, f_n \\in J$ which map to a", "basis of $J/(J^2 + IB)$. Consider the finitely presented $A$-algebra", "$C = B/(f_1, \\ldots, f_n)$. Note that we have an exact sequence", "$$", "0 \\to H_1(L_{C/A}) \\to (f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2", "\\to \\Omega_{B/A} \\otimes_B C \\to \\Omega_{C/A} \\to 0", "$$", "see Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL} (note that", "$H_1(L_{B/A}) = 0$ and that $\\Omega_{B/A}$ is finite projective,", "in particular flat so the Tor group vanishes). For any prime", "$\\mathfrak q \\supset J$ of $B$ the module $\\Omega_{B/A, \\mathfrak q}$", "is free of rank $n$ because $\\Omega_{B/A}$ is finite projective and", "because $\\Omega_{B/A} \\otimes_B B/J$ is free of rank $n$", "(see Algebra, Lemma \\ref{algebra-lemma-finite-projective}). By our choice", "of $f_1, \\ldots, f_n$ the map", "$$", "\\left((f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2\\right)_{\\mathfrak q}", "\\to", "\\Omega_{B/A, \\mathfrak q}", "$$", "is surjective modulo $J$. Hence we see that this map of modules over", "the local ring $C_{\\mathfrak q}$ has to be an isomorphism", "(this is because by Nakayama's Algebra, Lemma \\ref{algebra-lemma-NAK}", "the map is surjective and then for example by", "Algebra, Lemma \\ref{algebra-lemma-fun}", "because $((f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2)_{\\mathfrak q}$", "is generated by $n$ elements the map is injective). Thus", "$H_1(L_{C/A})_{\\mathfrak q} = 0$ and $\\Omega_{C/A, \\mathfrak q} = 0$. By", "Algebra, Lemma \\ref{algebra-lemma-smooth-at-point}", "we see that $A \\to C$ is smooth at the prime $\\overline{\\mathfrak q}$", "of $C$ corresponding to $\\mathfrak q$. Since", "$\\Omega_{C/A, \\mathfrak q} = 0$ it is actually \\'etale at", "$\\overline{\\mathfrak q}$. Thus $A \\to C$ is \\'etale at all primes", "of $C$ containing $JC$. By Lemma \\ref{lemma-localize-upstairs}", "we can find an $f \\in C$ mapping to an invertible element of $C/JC$ such that", "$A \\to C_f$ is \\'etale. By our choice of $f$ it is still true that", "$C_f/JC_f = A/I$. The map $C_f/IC_f \\to A/I$ is surjective and", "\\'etale by Algebra, Lemma \\ref{algebra-lemma-map-between-etale}.", "Hence $A/I$ is isomorphic to the localization of $C_f/IC_f$ at", "some element $g \\in C$, see", "Algebra, Lemma \\ref{algebra-lemma-surjective-flat-finitely-presented}.", "Set $A' = C_{fg}$ to conclude the proof." ], "refs": [ "algebra-lemma-application-NL-smooth", "more-algebra-lemma-lift-projective-module", "more-algebra-lemma-symmetric-algebra-smooth", "algebra-lemma-base-change-smooth", "algebra-lemma-locally-smooth", "algebra-lemma-exact-sequence-NL", "algebra-lemma-finite-projective", "algebra-lemma-NAK", "algebra-lemma-fun", "algebra-lemma-smooth-at-point", "more-algebra-lemma-localize-upstairs", "algebra-lemma-map-between-etale", "algebra-lemma-surjective-flat-finitely-presented" ], "ref_ids": [ 1219, 9849, 9851, 1191, 1197, 1153, 795, 401, 388, 1196, 9842, 1236, 1237 ] } ], "ref_ids": [] }, { "id": 9853, "type": "theorem", "label": "more-algebra-lemma-idempotents-determined-modulo-radical", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-idempotents-determined-modulo-radical", "contents": [ "Let $(A, I)$ be a Zariski pair. Then the map from", "idempotents of $A$ to idempotents of $A/I$ is injective." ], "refs": [], "proofs": [ { "contents": [ "An idempotent of a local ring is either $0$ or $1$.", "Thus an idempotent is determined by the set of maximal ideals", "where it vanishes, by", "Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}." ], "refs": [ "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 410 ] } ], "ref_ids": [] }, { "id": 9854, "type": "theorem", "label": "more-algebra-lemma-check-isomorphism-zariski", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-check-isomorphism-zariski", "contents": [ "Let $(A, I)$ be a Zariski pair. Let $A \\to B$ be a flat,", "integral, finitely presented ring map such that $A/I \\to B/IB$", "is an isomorphism. Then $A \\to B$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The ring map $A \\to B$ is finite by", "Algebra, Lemma \\ref{algebra-lemma-characterize-finite-in-terms-of-integral}.", "Hence $B$ is finitely presented as an $A$-module by", "Algebra, Lemma \\ref{algebra-lemma-finite-finitely-presented-extension}.", "Hence $B$ is a finite locally free $A$-module by", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}.", "Since the module $B$ has rank $1$", "along $V(I)$ (see rank function described in", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}),", "and as $(A, I)$ is a Zariski pair, we conclude that the", "rank is $1$ everywhere.", "It follows that $A \\to B$ is an isomorphism:", "it is a pleasant exercise to show that", "a ring map $R \\to S$ such that $S$ is a locally free $R$-module", "of rank $1$ is an isomorphism (hint: look at local rings)." ], "refs": [ "algebra-lemma-characterize-finite-in-terms-of-integral", "algebra-lemma-finite-finitely-presented-extension", "algebra-lemma-finite-projective", "algebra-lemma-finite-projective" ], "ref_ids": [ 484, 501, 795, 795 ] } ], "ref_ids": [] }, { "id": 9855, "type": "theorem", "label": "more-algebra-lemma-helper-finite", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-helper-finite", "contents": [ "Let $(A, I)$ be a Zariski pair. Let $A \\to B$ be a finite ring map.", "Assume", "\\begin{enumerate}", "\\item $B/IB = B_1 \\times B_2$ is a product of $A/I$-algebras", "\\item $A/I \\to B_1/IB_1$ is surjective,", "\\item $b \\in B$ maps to $(1, 0)$ in the product.", "\\end{enumerate}", "Then there exists a monic $f \\in A[x]$ with $f(b) = 0$ and", "$f \\bmod I = (x - 1)x^d$ for some $d \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-lift-idempotent-upstairs} we can find an", "\\'etale ring map $A \\to A'$ inducing an isomorphism $A/I \\to A'/IA'$", "such that $B' = B \\otimes_A A'$ contains", "an idempotent $e'$ lifting the image of $b$ in $B'/IB'$.", "Consider the corresponding $A'$-algebra decomposition", "$$", "B' = B'_1 \\times B'_2", "$$", "which is compatible with the one given in the lemma upon", "reduction modulo $I$.", "The map $A' \\to B'_1$ is surjective modulo $IA'$. By Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK})", "we can find $i \\in IA'$ such that after replacing $A'$", "by $A'_{1 + i}$ the map $A' \\to B'_1$ is surjective.", "Observe that the image $b'_1 \\in B'_1$ of $b$ ", "satisfies $b'_1 - 1 \\in IB'_1$.", "Thus we may pick $a' \\in IA'$ mapping to $b'_1 - 1$.", "On the other hand, the image $b'_2 \\in B'_2$ of $b$ is in $IB'_2$. By", "Algebra, Lemma \\ref{algebra-lemma-integral-integral-over-ideal}", "there exist a monic polynomial", "$g(x) = x^d + \\sum a'_j x^j$ of degree $d$ with $a'_j \\in IA'$ such that", "$g(b'_2) = 0$ in $B'_2$. Thus the image $b' = (b'_1, b'_2) \\in B'$", "of $b$ is a root of the polynomial $(x - 1 - a')g(x)$. We conclude that", "$$", "(b' - 1)(b')^d \\in \\sum\\nolimits_{j = 0, \\ldots, d} IA' \\cdot (b')^j", "$$", "We claim that this implies", "$$", "(b - 1)b^d \\in \\sum\\nolimits_{j = 0, \\ldots, d} I \\cdot b^j", "$$", "in $B$. For this it is enough to see that the ring map", "$A \\to A'$ is faithfully flat, because the condition is that", "the image of $(b - 1)b^d$ is zero in $B/\\sum_{j = 0, \\ldots, d} Ib^j$", "(use Algebra, Lemma \\ref{algebra-lemma-faithfully-flat-universally-injective}).", "The map $A \\to A'$ flat because it is \\'etale", "(Algebra, Lemma \\ref{algebra-lemma-etale}).", "On the other hand, the induced map on spectra is open", "(see Algebra, Proposition \\ref{algebra-proposition-fppf-open} and use", "previous lemma referenced) and the image", "contains $V(I)$. Since $I$ is contained in the Jacobson radical of $A$", "we conclude." ], "refs": [ "more-algebra-lemma-lift-idempotent-upstairs", "algebra-lemma-NAK", "algebra-lemma-integral-integral-over-ideal", "algebra-lemma-faithfully-flat-universally-injective", "algebra-lemma-etale", "algebra-proposition-fppf-open" ], "ref_ids": [ 9848, 401, 519, 814, 1231, 1407 ] } ], "ref_ids": [] }, { "id": 9856, "type": "theorem", "label": "more-algebra-lemma-noetherian-zariski-jacobson-complement", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-noetherian-zariski-jacobson-complement", "contents": [ "Let $(A, I)$ be a Zariski pair with $A$ Noetherian. Let $f \\in I$.", "Then $A_f$ is a Jacobson ring." ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion of", "Algebra, Lemma \\ref{algebra-lemma-noetherian-dim-1-Jacobson}.", "Let $\\mathfrak p \\subset A$ be a prime ideal such that", "$\\mathfrak p_f = \\mathfrak p A_f$ is prime and not maximal. We have to", "show that $A_f/\\mathfrak p_f = (A/\\mathfrak p)_f$", "has infinitely many prime ideals. After replacing $A$ by $A/\\mathfrak p$", "we may assume $A$ is a domain, $\\dim A_f > 0$, and our goal is to", "show that $\\Spec(A_f)$ is infinite. Since $\\dim A_f > 0$ we can", "find a nonzero prime ideal $\\mathfrak q \\subset A$ not containing $f$.", "Choose a maximal ideal $\\mathfrak m \\subset A$ containing $\\mathfrak q$.", "Since $(A, I)$ is a Zariski pair, we see $I \\subset \\mathfrak m$.", "Hence $\\mathfrak m \\not = \\mathfrak q$ and $\\dim(A_\\mathfrak m) > 1$.", "Hence $\\Spec((A_\\mathfrak m)_f) \\subset \\Spec(A_f)$ is infinite by", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens}", "and we win." ], "refs": [ "algebra-lemma-noetherian-dim-1-Jacobson", "algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens" ], "ref_ids": [ 690, 687 ] } ], "ref_ids": [] }, { "id": 9857, "type": "theorem", "label": "more-algebra-lemma-locally-nilpotent-henselian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-locally-nilpotent-henselian", "contents": [ "Let $(A, I)$ be a pair with $I$ locally nilpotent. Then the functor", "$B \\mapsto B/IB$ induces an equivalence between the category of", "\\'etale algebras over $A$ and the category of \\'etale algebras over $A/I$.", "Moreover, the pair is henselian." ], "refs": [], "proofs": [ { "contents": [ "Essential surjectivity holds by Algebra, Lemma \\ref{algebra-lemma-lift-etale}.", "If $B$, $B'$ are \\'etale over $A$ and $B/IB \\to B'/IB'$ is a morphism", "of $A/I$-algebras, then we can lift this by", "Algebra, Lemma \\ref{algebra-lemma-smooth-strong-lift}.", "Finally, suppose that $f, g : B \\to B'$ are two $A$-algebra", "maps with $f \\mod I = g \\mod I$. Choose an idempotent $e \\in B \\otimes_A B$", "generating the kernel of the multiplication map $B \\otimes_A B \\to B$,", "see Algebra, Lemmas \\ref{algebra-lemma-diagonal-unramified}", "and \\ref{algebra-lemma-unramified} (to see that \\'etale is unramified).", "Then $(f \\otimes g)(e) \\in IB$. Since $IB$ is locally nilpotent", "(Algebra, Lemma \\ref{algebra-lemma-locally-nilpotent}) this implies", "$(f \\otimes g)(e) = 0$ by Algebra, Lemma \\ref{algebra-lemma-lift-idempotents}.", "Thus $f = g$.", "\\medskip\\noindent", "It is clear that $I$ is contained in the Jacobson radical of $A$.", "Let $f \\in A[T]$ be a monic polynomial and let", "$\\overline{f} = g_0h_0$ be a factorization", "of $\\overline{f} = f \\bmod I$ with $g_0, h_0 \\in A/I[T]$ monic", "generating the unit ideal in $A/I[T]$. By", "Lemma \\ref{lemma-lift-factorization-monic}", "there exists an \\'etale ring map $A \\to A'$ which", "induces an isomorphism $A/I \\to A'/IA'$ such that", "the factorization lifts to a factorization into monic polynomials", "over $A'$. By the above we have $A = A'$ and the factorization", "is over $A$." ], "refs": [ "algebra-lemma-lift-etale", "algebra-lemma-smooth-strong-lift", "algebra-lemma-diagonal-unramified", "algebra-lemma-unramified", "algebra-lemma-locally-nilpotent", "algebra-lemma-lift-idempotents", "more-algebra-lemma-lift-factorization-monic" ], "ref_ids": [ 1238, 1216, 1267, 1266, 458, 461, 9843 ] } ], "ref_ids": [] }, { "id": 9858, "type": "theorem", "label": "more-algebra-lemma-limit-henselian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-limit-henselian", "contents": [ "Let $A = \\lim A_n$ where $(A_n)$ is an inverse system of rings", "whose transition maps are surjective and have locally nilpotent kernels.", "Then $(A, I_n)$ is a henselian pair, where $I_n = \\Ker(A \\to A_n)$." ], "refs": [], "proofs": [ { "contents": [ "Fix $n$. Let $a \\in A$ be an element which maps to $1$ in $A_n$.", "By Algebra, Lemma \\ref{algebra-lemma-locally-nilpotent-unit}", "we see that $a$ maps to a unit in $A_m$ for all $m \\geq n$.", "Hence $a$ is a unit in $A$. Thus by", "Algebra, Lemma \\ref{algebra-lemma-contained-in-radical}", "the ideal $I_n$ is contained in the Jacobson radical of $A$.", "Let $f \\in A[T]$ be a monic polynomial and let", "$\\overline{f} = g_nh_n$ be a factorization", "of $\\overline{f} = f \\bmod I_n$ with $g_n, h_n \\in A_n[T]$ monic", "generating the unit ideal in $A_n[T]$. By", "Lemma \\ref{lemma-locally-nilpotent-henselian}", "we can successively lift this factorization to", "$f \\bmod I_m = g_m h_m$ with $g_m, h_m$ monic", "in $A_m[T]$ for all $m \\geq n$. At each step we have", "to verify that our lifts $g_m, h_m$ generate the unit ideal in", "$A_n[T]$; this follows from the corresponding fact for", "$g_n, h_n$ and the fact that $\\Spec(A_n[T]) = \\Spec(A_m[T])$", "because the kernel of $A_m \\to A_n$ is locally nilpotent.", "As $A = \\lim A_m$ this finishes the proof." ], "refs": [ "algebra-lemma-locally-nilpotent-unit", "algebra-lemma-contained-in-radical", "more-algebra-lemma-locally-nilpotent-henselian" ], "ref_ids": [ 459, 399, 9857 ] } ], "ref_ids": [] }, { "id": 9859, "type": "theorem", "label": "more-algebra-lemma-complete-henselian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-complete-henselian", "contents": [ "Let $(A, I)$ be a pair. If $A$ is $I$-adically complete, then", "the pair is henselian." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-radical-completion}", "the ideal $I$ is contained in the Jacobson radical of $A$.", "Let $f \\in A[T]$ be a monic polynomial and let", "$\\overline{f} = g_0h_0$ be a factorization", "of $\\overline{f} = f \\bmod I$ with $g_0, h_0 \\in A/I[T]$ monic", "generating the unit ideal in $A/I[T]$. By", "Lemma \\ref{lemma-locally-nilpotent-henselian}", "we can successively lift this factorization to", "$f \\bmod I^n = g_n h_n$ with $g_n, h_n$ monic", "in $A/I^n[T]$ for all $n \\geq 1$.", "As $A = \\lim A/I^n$ this finishes the proof." ], "refs": [ "algebra-lemma-radical-completion", "more-algebra-lemma-locally-nilpotent-henselian" ], "ref_ids": [ 862, 9857 ] } ], "ref_ids": [] }, { "id": 9860, "type": "theorem", "label": "more-algebra-lemma-helper-finite-type", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-helper-finite-type", "contents": [ "Let $(A, I)$ be a pair. Let $A \\to B$ be a finite type ring map", "such that $B/IB = C_1 \\times C_2$ with $A/I \\to C_1$ finite.", "Let $B'$ be the integral closure of $A$ in $B$.", "Then we can write $B'/IB' = C_1 \\times C'_2$ such that", "the map $B'/IB' \\to B/IB$ preserves product decompositions", "and there exists a $g \\in B'$ mapping to $(1, 0)$ in", "$C_1 \\times C'_2$ with $B'_g \\to B_g$ an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Observe that $A \\to B$ is quasi-finite at every prime of the", "closed subset $T = \\Spec(C_1) \\subset \\Spec(B)$ (this follows", "by looking at fibre rings, see", "Algebra, Definition \\ref{algebra-definition-quasi-finite}).", "Consider the diagram of topological spaces", "$$", "\\xymatrix{", "\\Spec(B) \\ar[rr]_\\phi \\ar[rd]_\\psi & & \\Spec(B') \\ar[ld]^{\\psi'} \\\\", "& \\Spec(A)", "}", "$$", "By Algebra, Theorem \\ref{algebra-theorem-main-theorem}", "for every $\\mathfrak p \\in T$ there is a $h_\\mathfrak p \\in B'$,", "$h_\\mathfrak p \\not \\in \\mathfrak p$ such that $B'_h \\to B_h$ is", "an isomorphism. The union $U = \\bigcup D(h_\\mathfrak p)$ gives an open", "$U \\subset \\Spec(B')$ such that $\\phi^{-1}(U) \\to U$ is a homeomorphism", "and $T \\subset \\phi^{-1}(U)$. Since $T$ is open in $\\psi^{-1}(V(I))$", "we conclude that $\\phi(T)$ is open in $U \\cap (\\psi')^{-1}(V(I))$.", "Thus $\\phi(T)$ is open in $(\\psi')^{-1}(V(I))$.", "On the other hand, since $C_1$ is finite over $A/I$ it is", "finite over $B'$. Hence $\\phi(T)$ is a closed subset of $\\Spec(B')$", "by Algebra, Lemmas \\ref{algebra-lemma-going-up-closed} and", "\\ref{algebra-lemma-integral-going-up}. We conclude that", "$\\Spec(B'/IB') \\supset \\phi(T)$ is open and closed. By", "Algebra, Lemma \\ref{algebra-lemma-disjoint-implies-product}", "we get a corresponding product decomposition $B'/IB' = C'_1 \\times C'_2$.", "The map $B'/IB' \\to B/IB$ maps $C'_1$ into $C_1$ and $C'_2$ into $C_2$", "as one sees by looking at what happens on spectra (hint: the inverse", "image of $\\phi(T)$ is exactly $T$; some details omitted).", "Pick a $g \\in B'$ mapping to $(1, 0)$ in $C'_1 \\times C'_2$", "such that $D(g) \\subset U$; this is possible because $\\Spec(C'_1)$", "and $\\Spec(C'_2)$ are disjoint and closed in $\\Spec(B')$ and", "$\\Spec(C'_1)$ is contained in $U$. Then $B'_g \\to B_g$ defines a homeomorphism", "on spectra and an isomorphism on local rings (by our choice of $U$ above).", "Hence it is an isomorphism, as follows for example from", "Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}.", "Finally, it follows that $C'_1 = C_1$ and the proof is complete." ], "refs": [ "algebra-definition-quasi-finite", "algebra-theorem-main-theorem", "algebra-lemma-going-up-closed", "algebra-lemma-integral-going-up", "algebra-lemma-disjoint-implies-product", "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 1522, 325, 552, 500, 415, 410 ] } ], "ref_ids": [] }, { "id": 9861, "type": "theorem", "label": "more-algebra-lemma-characterize-henselian-pair", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-characterize-henselian-pair", "contents": [ "\\begin{reference}", "\\cite[Chapter XI]{Henselian} and \\cite[Proposition 1]{Gabber-henselian}", "\\end{reference}", "Let $(A, I)$ be a pair. The following are equivalent", "\\begin{enumerate}", "\\item $(A, I)$ is a henselian pair,", "\\item given an \\'etale ring map $A \\to A'$ and an $A$-algebra map", "$\\sigma : A' \\to A/I$, there exists an $A$-algebra map $A' \\to A$", "lifting $\\sigma$,", "\\item for any finite $A$-algebra $B$ the map $B \\to B/IB$ induces", "a bijection on idempotents,", "\\item for any integral $A$-algebra $B$ the map $B \\to B/IB$ induces", "a bijection on idempotents, and", "\\item (Gabber) $I$ is contained in the Jacobson radical of $A$ and", "every monic polynomial $f(T) \\in A[T]$ of the form", "$$", "f(T) = T^n(T - 1) + a_n T^n + \\ldots + a_1 T + a_0", "$$", "with $a_n, \\ldots, a_0 \\in I$ and $n \\ge 1$ has a root $\\alpha \\in 1 + I$.", "\\end{enumerate}", "Moreover, in part (5) the root is unique." ], "refs": [], "proofs": [ { "contents": [ "Assume (2) holds. Then $I$ is contained in the Jacobson radical of $A$, since", "otherwise there would be a nonunit $f \\in A$ congruent to $1$ modulo $I$", "and the map $A \\to A_f$ would contradict (2). Hence $IB \\subset B$", "is contained in the Jacobson radical of $B$ for $B$ integral over $A$", "because $\\Spec(B) \\to \\Spec(A)$ is closed by", "Algebra, Lemmas \\ref{algebra-lemma-going-up-closed} and", "\\ref{algebra-lemma-integral-going-up}.", "Thus the map from idempotents of $B$ to idempotents of $B/IB$", "is injective by Lemma \\ref{lemma-idempotents-determined-modulo-radical}.", "On the other hand, since (2) holds, every idempotent", "of $B/IB$ lifts to an idempotent of $B$", "by Lemma \\ref{lemma-lift-idempotent-upstairs}.", "In this way we see that (2) implies (4).", "\\medskip\\noindent", "The implication (4) $\\Rightarrow$ (3) is trivial.", "\\medskip\\noindent", "Assume (3). Let $\\mathfrak m$ be a maximal ideal and consider the", "finite map $A \\to B = A/(I \\cap \\mathfrak m)$. The condition that", "$B \\to B/IB$ induces a bijection on idempotents implies that", "$I \\subset \\mathfrak m$ (if not, then $B = A/I \\times A/\\mathfrak m$", "and $B/IB = A/I$). Thus we see that $I$ is contained in the Jacobson", "radical of $A$. Let $f \\in A[T]$ be monic and suppose given a", "factorization $\\overline{f} = g_0h_0$ with $g_0, h_0 \\in A/I[T]$ monic.", "Set $B = A[T]/(f)$. Let $\\overline{e}$ be the idempotent", "of $B/IB$ corresponding to the decomposition", "$$", "B/IB = A/I[T]/(g_0) \\times A[T]/(h_0)", "$$", "of $A$-algebras. Let $e \\in B$ be an idempotent lifting $\\overline{e}$", "which exists as we assumed (3). This gives a product decomposition", "$$", "B = eB \\times (1 - e)B", "$$", "Note that $B$ is free of rank $\\deg(f)$ as an $A$-module.", "Hence $eB$ and $(1 - e)B$ are finite locally free $A$-modules.", "However, since $eB$ and $(1 - e)B$ have constant rank", "$\\deg(g_0)$ and $\\deg(h_0)$ over $A/I$ we find that the same", "is true over $\\Spec(A)$. We conclude that", "\\begin{align*}", "f & = \\text{CharPol}_A(T : B \\to B) \\\\", "& =", "\\text{CharPol}_A(T : eB \\to eB)", "\\text{CharPol}_A(T : (1 - e)B \\to (1 - e)B)", "\\end{align*}", "is a factorization into monic polynomials reducing to the given", "factorization modulo $I$. Here $\\text{CharPol}_A$ denotes the characteristic", "polynomial of an endomorphism of a finite locally free module over $A$.", "If the module is free the $\\text{CharPol}_A$ is defined as the", "characteristic polynomial of the corresponding matrix and in", "general one uses Algebra, Lemma \\ref{algebra-lemma-standard-covering} to", "glue. Details omitted. Thus (3) implies (1).", "\\medskip\\noindent", "Assume (1). Let $f$ be as in (5). The factorization of $f \\bmod I$", "as $T^n$ times $T - 1$ lifts to a factorization $f = gh$ with $g$", "and $h$ monic by Definition \\ref{definition-henselian-pair}.", "Then $h$ has to have degree $1$", "and we see that $f$ has a root reducing to $1$ modulo $1$.", "Finally, $I$ is contained in the Jacobson radical by", "the definition of a henselian pair.", "Thus (1) implies (5).", "\\medskip\\noindent", "Before we give the proof of the last step, let us show that", "the root $\\alpha$ in (5), if it exists, is unique. Namely,", "Due to the explicit shape of $f(T)$, we have", "$f'(\\alpha) \\in 1 + I$ where $f'$ is the derivative of $f$ with", "respect to $T$. An elementary argument shows that", "$$", "f(T) = f(\\alpha + T - \\alpha) =", "f(\\alpha) + f'(\\alpha) \\cdot (T - \\alpha) \\bmod", "(T - \\alpha)^2 A[T]", "$$", "This shows that any other root $\\alpha' \\in 1 + I$ of $f(T)$", "satisfies $0 = f(\\alpha') - f(\\alpha) = (\\alpha' - \\alpha)(1 + i)$", "for some $i \\in I$, so that, since $1 + i$ is a unit in $A$,", "we have $\\alpha = \\alpha'$.", "\\medskip\\noindent", "Assume (5). We will show that (2) holds, in other words, that", "for every \\'etale map $A \\to A'$, every section $\\sigma : A' \\to A/I$", "modulo $I$ lifts to a section $A' \\to A$.", "Since $A \\to A'$ is \\'etale, the section $\\sigma$", "determines a decomposition", "\\begin{equation}", "\\label{equation-GCHP}", "A'/IA' \\cong A/I \\times C", "\\end{equation}", "of $A/I$-algebras. Namely, the surjective ring map", "$A'/IA' \\to A/I$ is \\'etale by", "Algebra, Lemma \\ref{algebra-lemma-map-between-etale}", "and then we get the desired idempotent by", "Algebra, Lemma \\ref{algebra-lemma-surjective-flat-finitely-presented}.", "We will show that this decomposition lifts to a decomposition ", "\\begin{equation}", "\\label{equation-GCHP-want}", "A' \\cong A'_1 \\times A'_2", "\\end{equation}", "of $A$-algebras with $A'_1$ integral over $A$. Then $A \\to A'_1$", "is integral and \\'etale and $A/I \\to A'_1/IA'_1$ is an isomorphism,", "thus $A \\to A'_1$ is an isomorphism by", "Lemma \\ref{lemma-check-isomorphism-zariski}", "(here we also use that an \\'etale ring map is flat", "and of finite presentation, see Algebra, Lemma \\ref{algebra-lemma-etale}).", "\\medskip\\noindent", "Let $B'$ be the integral closure of $A$ in $A'$. By", "Lemma \\ref{lemma-helper-finite-type}", "we may decompose ", "\\begin{equation}", "\\label{equation-dec-mod-I}", "B'/IB' \\cong A/I \\times C'", "\\end{equation}", "as $A/I$-algebras compatibly with (\\ref{equation-GCHP})", "and we may find $b \\in B'$ that lifts $(1, 0)$ such that", "$B'_b \\to A'_b$ is an isomorphism. If the decomposition", "(\\ref{equation-dec-mod-I}) lifts to a decomposition", "\\begin{equation}", "\\label{equation-want-2}", "B' \\cong B'_1 \\times B'_2", "\\end{equation}", "of $A$-algebras, then the induced decomposition", "$A' = A'_1 \\times A'_2$ will give the", "desired (\\ref{equation-GCHP-want}): indeed, since $b$ is a unit in $B'_1$", "(details omitted),", "we will have $B'_1 \\cong A'_1$, so that $A'_1$ will be integral over $A$. ", "\\medskip\\noindent", "Choose a finite $A$-subalgebra $B'' \\subset B'$ containing $b$", "(observe that any finitely generated $A$-subalgebra of $B'$", "is finite over $A$). After enlarging $B''$ we may assume", "$b$ maps to an idempotent in $B''/IB''$ producing", "\\begin{equation}", "\\label{equation-again-dec-mod-I}", "B''/IB'' \\cong C''_1 \\times C''_2", "\\end{equation}", "Since $B'_b \\cong A'_b$ we see that $B'_b$ is of finite type over $A$.", "Say $B'_b$ is generated by $b_1/b^n, \\ldots, b_t/b^n$ over $A$", "and enlarge $B''$ so that $b_1, \\ldots, b_t \\in B''$. Then", "$B''_b \\to B'_b$ is surjective as well as injective, hence an isomorphism.", "In particular, we see that $C''_1 = A/I$! Therefore $A/I \\to C''_1$", "is an isomorphism, in particular surjective.", "By Lemma \\ref{lemma-helper-finite} we can find", "an $f(T) \\in A[T]$ of the form", "$$", "f(T) = T^n(T - 1) + a_n T^n + \\ldots + a_1 T + a_0", "$$", "with $a_n, \\ldots, a_0 \\in I$ and $n \\ge 1$ such that $f(b) = 0$.", "In particular, we find that $B'$ is a $A[T]/(f)$-algebra.", "By (5) we deduce there is a root $a \\in 1 + I$ of $f$.", "This produces a product decomposition $A[T]/(f) = A[T]/(T - a) \\times D$", "compatible with the splitting (\\ref{equation-dec-mod-I}) of $B'/IB'$.", "The induced splitting of $B'$ is then a desired (\\ref{equation-want-2})." ], "refs": [ "algebra-lemma-going-up-closed", "algebra-lemma-integral-going-up", "more-algebra-lemma-idempotents-determined-modulo-radical", "more-algebra-lemma-lift-idempotent-upstairs", "algebra-lemma-standard-covering", "more-algebra-definition-henselian-pair", "algebra-lemma-map-between-etale", "algebra-lemma-surjective-flat-finitely-presented", "more-algebra-lemma-check-isomorphism-zariski", "algebra-lemma-etale", "more-algebra-lemma-helper-finite-type", "more-algebra-lemma-helper-finite" ], "ref_ids": [ 552, 500, 9853, 9848, 414, 10597, 1236, 1237, 9854, 1231, 9860, 9855 ] } ], "ref_ids": [] }, { "id": 9862, "type": "theorem", "label": "more-algebra-lemma-change-ideal-henselian-pair", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-change-ideal-henselian-pair", "contents": [ "Let $A$ be a ring. Let $I, J \\subset A$ be ideals with $V(I) = V(J)$.", "Then $(A, I)$ is henselian if and only if $(A, J)$ is henselian." ], "refs": [], "proofs": [ { "contents": [ "For any integral ring map $A \\to B$ we see that $V(IB) = V(JB)$.", "Hence idempotents of $B/IB$ and $B/JB$ are in bijective correspondence", "(Algebra, Lemma \\ref{algebra-lemma-disjoint-decomposition}).", "It follows that $B \\to B/IB$ induces a bijection on sets of", "idempotents if and only if $B \\to B/JB$ induces a bijection on sets", "of idempotents. Thus we conclude by", "Lemma \\ref{lemma-characterize-henselian-pair}." ], "refs": [ "algebra-lemma-disjoint-decomposition", "more-algebra-lemma-characterize-henselian-pair" ], "ref_ids": [ 405, 9861 ] } ], "ref_ids": [] }, { "id": 9863, "type": "theorem", "label": "more-algebra-lemma-integral-over-henselian-pair", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-integral-over-henselian-pair", "contents": [ "Let $(A, I)$ be a henselian pair and let $A \\to B$ be an integral ring", "map. Then $(B, IB)$ is a henselian pair." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the fourth characterization of henselian pairs in", "Lemma \\ref{lemma-characterize-henselian-pair} and the fact that the", "composition of integral ring maps is integral." ], "refs": [ "more-algebra-lemma-characterize-henselian-pair" ], "ref_ids": [ 9861 ] } ], "ref_ids": [] }, { "id": 9864, "type": "theorem", "label": "more-algebra-lemma-henselian-henselian-pair", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselian-henselian-pair", "contents": [ "Let $I \\subset J \\subset A$ be ideals of a ring $A$.", "The following are equivalent", "\\begin{enumerate}", "\\item $(A, I)$ and $(A/I, J/I)$ are henselian pairs, and", "\\item $(A, J)$ is an henselian pair.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Let $B$ be an integral $A$-algebra. Consider the ring maps", "$$", "B \\to B/IB \\to B/JB", "$$", "By Lemma \\ref{lemma-characterize-henselian-pair} we find that", "both arrows induce bijections on idempotents. Hence so does the", "composition. Whence $(A, J)$ is a henselian pair", "by Lemma \\ref{lemma-characterize-henselian-pair}.", "\\medskip\\noindent", "Conversely, assume (2) holds. Then $(A/I, J/I)$ is a henselian pair", "by Lemma \\ref{lemma-integral-over-henselian-pair}. Let $B$ be an", "integral $A$-algebra. Consider the ring maps", "$$", "B \\to B/IB \\to B/JB", "$$", "By Lemma \\ref{lemma-characterize-henselian-pair} we find that the composition", "and the second arrow induce bijections on idempotents.", "Hence so does the first arrow. It follows that $(A, I)$ is a henselian", "pair (by the lemma again)." ], "refs": [ "more-algebra-lemma-characterize-henselian-pair", "more-algebra-lemma-characterize-henselian-pair", "more-algebra-lemma-integral-over-henselian-pair", "more-algebra-lemma-characterize-henselian-pair" ], "ref_ids": [ 9861, 9861, 9863, 9861 ] } ], "ref_ids": [] }, { "id": 9865, "type": "theorem", "label": "more-algebra-lemma-sum-henselian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-sum-henselian", "contents": [ "Let $A$ be a ring and let $(A, I)$ and $(A, I')$ be henselian pairs.", "Then $(A, I + I')$ is an henselian pair." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-integral-over-henselian-pair} the pair", "$(A/I, (I' + I)/I)$ is henselian. Thus we get the conclusion from", "Lemma \\ref{lemma-henselian-henselian-pair}." ], "refs": [ "more-algebra-lemma-integral-over-henselian-pair", "more-algebra-lemma-henselian-henselian-pair" ], "ref_ids": [ 9863, 9864 ] } ], "ref_ids": [] }, { "id": 9866, "type": "theorem", "label": "more-algebra-lemma-product-henselian-pairs", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-product-henselian-pairs", "contents": [ "Let $J$ be a set and let $\\{ (A_j, I_j)\\}_{j \\in J}$ be a collection", "of pairs. Then $(\\prod_{j \\in J} A_j, \\prod_{j\\in J} I_j)$ is Henselian", "if and only if so is each $(A_j, I_j)$." ], "refs": [], "proofs": [ { "contents": [ "For every $j \\in J$, the projection $\\prod_{j \\in J} A_j \\rightarrow A_j$", "is an integral ring map, so Lemma \\ref{lemma-integral-over-henselian-pair}", "proves that each $(A_j, I_j)$ is Henselian if", "$(\\prod_{j \\in J} A_j, \\prod_{j\\in J} I_j)$ is Henselian.", "\\medskip\\noindent", "Conversely, suppose that each $(A_j, I_j)$ is a Henselian pair.", "Then every $1 + x$ with $x \\in \\prod_{j \\in J} I_j$ is a unit", "in $\\prod_{j \\in J} A_j$ because it is so componentwise by", "Algebra, Lemma \\ref{algebra-lemma-contained-in-radical} and", "Definition \\ref{definition-henselian-pair}.", "Thus, by Algebra, Lemma \\ref{algebra-lemma-contained-in-radical}", "again, $\\prod_{j \\in J} I_j$ is contained in the Jacobson radical", "of $\\prod_{j \\in J} A_j$. Continuing to work componentwise, it", "likewise follows that for every monic $f \\in (\\prod_{j \\in J} A_j)[T]$", "and every factorization $\\overline{f} = g_0h_0$ with monic", "$g_0, h_0 \\in (\\prod_{j \\in J} A_j / \\prod_{j \\in J} I_j)[T] =", "(\\prod_{j \\in J} A_j/I_j)[T]$ that generate the unit ideal in", "$(\\prod_{j \\in J} A_j / \\prod_{j \\in J} I_j)[T]$, there exists a", "factorization $f = gh$ in $(\\prod_{j \\in J} A_j)[T]$ with $g$, $h$ monic", "and reducing to $g_0$, $h_0$. In conclusion, according to", "Definition \\ref{definition-henselian-pair}", "$(\\prod_{j \\in J} A_j, \\prod_{j\\in J} I_j)$ is a Henselian pair." ], "refs": [ "more-algebra-lemma-integral-over-henselian-pair", "algebra-lemma-contained-in-radical", "more-algebra-definition-henselian-pair", "algebra-lemma-contained-in-radical", "more-algebra-definition-henselian-pair" ], "ref_ids": [ 9863, 399, 10597, 399, 10597 ] } ], "ref_ids": [] }, { "id": 9867, "type": "theorem", "label": "more-algebra-lemma-limits-henselian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-limits-henselian", "contents": [ "The property of being Henselian is preserved under limits of pairs.", "More precisely, let $J$ be a preordered set and let $(A_j, I_j)$", "be an inverse system of henselian pairs over $J$.", "Then $A = \\lim A_j$ equipped with the ideal $I = \\lim I_j$", "is a henselian pair $(A, I)$." ], "refs": [], "proofs": [ { "contents": [ "By Categories, Lemma \\ref{categories-lemma-limits-products-equalizers},", "we only need to consider products and equalizers.", "For products, the claim follows from", "Lemma \\ref{lemma-product-henselian-pairs}.", "Thus, consider an equalizer diagram", "$$", "\\xymatrix{", "(A, I) \\ar[r] & (A', I')", "\\ar@<1ex>[r]^{\\varphi} \\ar@<-1ex>[r]_{\\psi}", "&", "(A'', I'') ", "}", "$$", "in which the pairs $(A', I')$ and $(A'', I'')$ are henselian.", "To check that the pair $(A, I)$ is also henselian, we will use the", "Gabber's criterion in Lemma \\ref{lemma-characterize-henselian-pair}.", "Every element of $1 + I$ is a unit in $A$ because,", "due to the uniqueness of the inverses of units,", "this may be checked in $(A', I')$.", "Thus $I$ is contained in the Jacobson radical of $A$, see", "Algebra, Lemma \\ref{algebra-lemma-contained-in-radical}.", "Thus, let ", "$$", "f(T) = T^{N - 1}(T - 1) + a_{N - 1} T^{N - 1} + \\dotsb + a_1 T + a_0", "$$", "be a polynomial in $A[T]$ with $a_{N - 1}, \\dotsc, a_0 \\in I$ and $N \\ge 1$.", "The image of $f(T)$ in $A'[T]$ has a unique root $\\alpha' \\in 1 + I'$", "and likewise for the further image in $A''[T]$.", "Thus, due to the uniqueness, $\\varphi(\\alpha') = \\psi(\\alpha')$,", "to the effect that $\\alpha'$ defines a root of $f(T)$ in $1 + I$, as desired." ], "refs": [ "categories-lemma-limits-products-equalizers", "more-algebra-lemma-product-henselian-pairs", "more-algebra-lemma-characterize-henselian-pair", "algebra-lemma-contained-in-radical" ], "ref_ids": [ 12213, 9866, 9861, 399 ] } ], "ref_ids": [] }, { "id": 9868, "type": "theorem", "label": "more-algebra-lemma-filtered-colimits-henselian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-filtered-colimits-henselian", "contents": [ "The property of being Henselian is preserved under filtered colimits of pairs.", "More precisely, let $J$ be a directed set and let $(A_j, I_j)$", "be a system of henselian pairs over $J$.", "Then $A = \\colim A_j$ equipped with the ideal $I = \\colim I_j$", "is a henselian pair $(A, I)$." ], "refs": [], "proofs": [ { "contents": [ "If $u \\in 1 + I$ then for some $j \\in J$ we see that $u$ is the", "image of some $u_j \\in 1 + I_j$. Then $u_j$ is invertible in $A_j$ by", "Algebra, Lemma \\ref{algebra-lemma-contained-in-radical}", "and the assumption that $I_j$ is contained in the Jacobson radical of $A_j$.", "Hence $u$ is invertible in $A$. Thus $I$", "is contained in the Jacobson radical of $A$ (by the lemma).", "\\medskip\\noindent", "Let $f \\in A[T]$ be a monic polynomial and let $\\overline{f} = g_0 h_0$", "be a factorization with $g_0, h_0 \\in A/I[T]$ monic generating the unit", "ideal in $A/I[T]$. Write $1 = g_0 g'_0 + h_0 h'_0$ for some", "$g'_0, h'_0 \\in A/I[T]$. Since $A = \\colim A_j$ and $A/I = \\colim A_j/I_j$", "are filterd colimits we can find a $j \\in J$ and $f_j \\in A_j$ and", "a factorization $\\overline{f}_j = g_{j, 0} h_{j, 0}$", "with $g_{j, 0}, h_{j, 0} \\in A_j/I_j[T]$ monic", "and $1 = g_{j, 0} g'_{j, 0} + h_{j, 0} h'_{j, 0}$", "for some $g'_{j, 0}, h'_{j, 0} \\in A_j/I_j[T]$ with", "$f_j, g_{j, 0}, h_{j, 0}, g'_{j, 0}, h'_{j, 0}$", "mapping to $f, g_0, h_0, g'_0, h'_0$.", "Since $(A_j, I_j)$ is a henselian pair, we can lift", "$\\overline{f}_j = g_{j, 0} h_{j, 0}$ to a factorization", "over $A_j$ and taking the image in $A$ we obtain a", "corresponding factorization in $A$. Hence $(A, I)$ is henselian." ], "refs": [ "algebra-lemma-contained-in-radical" ], "ref_ids": [ 399 ] } ], "ref_ids": [] }, { "id": 9869, "type": "theorem", "label": "more-algebra-lemma-largest-ideal-henselian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-largest-ideal-henselian", "contents": [ "Let $A$ be a ring. There exists a largest ideal $I \\subset A$ such that", "$(A, I)$ is a henselian pair." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-henselian-henselian-pair}, \\ref{lemma-sum-henselian},", "and \\ref{lemma-filtered-colimits-henselian}." ], "refs": [ "more-algebra-lemma-henselian-henselian-pair", "more-algebra-lemma-sum-henselian", "more-algebra-lemma-filtered-colimits-henselian" ], "ref_ids": [ 9864, 9865, 9868 ] } ], "ref_ids": [] }, { "id": 9870, "type": "theorem", "label": "more-algebra-lemma-irreducible-henselian-pair-connected", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-irreducible-henselian-pair-connected", "contents": [ "Let $(A, I)$ be a henselian pair. Let $\\mathfrak p \\subset A$", "be a prime ideal. Then $V(\\mathfrak p + I)$ is connected." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-integral-over-henselian-pair} we see that", "$(A/\\mathfrak p, I + \\mathfrak p/\\mathfrak p)$ is a henselian pair.", "Thus it suffices to prove: If $(A, I)$ is a henselian pair and", "$A$ is a domain, then $\\Spec(A/I) = V(I)$ is connected. If not,", "then $A/I$ has a nontrivial idempotent by", "Algebra, Lemma \\ref{algebra-lemma-characterize-spec-connected}.", "By Lemma \\ref{lemma-characterize-henselian-pair}", "this would imply $A$ has a nontrivial idempotent. This is a contradiction." ], "refs": [ "more-algebra-lemma-integral-over-henselian-pair", "algebra-lemma-characterize-spec-connected", "more-algebra-lemma-characterize-henselian-pair" ], "ref_ids": [ 9863, 406, 9861 ] } ], "ref_ids": [] }, { "id": 9871, "type": "theorem", "label": "more-algebra-lemma-henselization", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization", "contents": [ "The inclusion functor", "$$", "\\text{category of henselian pairs}", "\\longrightarrow", "\\text{category of pairs}", "$$", "has a left adjoint $(A, I) \\mapsto (A^h, I^h)$." ], "refs": [], "proofs": [ { "contents": [ "Let $(A, I)$ be a pair. Consider the category $\\mathcal{C}$ consisting", "of \\'etale ring maps $A \\to B$ such that $A/I \\to B/IB$ is an isomorphism.", "We will show that the category $\\mathcal{C}$ is directed and that", "$A^h = \\colim_{B \\in \\mathcal{C}} B$ with ideal $I^h = IA^h$ gives", "the desired adjoint.", "\\medskip\\noindent", "We first prove that $\\mathcal{C}$ is directed", "(Categories, Definition \\ref{categories-definition-directed}).", "It is nonempty because $\\text{id} : A \\to A$ is an object.", "If $B$ and $B'$ are two objects of $\\mathcal{C}$, then", "$B'' = B \\otimes_A B'$ is an object of $\\mathcal{C}$", "(use Algebra, Lemma \\ref{algebra-lemma-etale})", "and there are morphisms $B \\to B''$ and $B' \\to B''$.", "Suppose that $f, g : B \\to B'$ are two maps between", "objects of $\\mathcal{C}$. Then a coequalizer is", "$$", "(B' \\otimes_{f, B, g} B') \\otimes_{(B' \\otimes_A B')} B'", "$$", "which is \\'etale over $A$ by", "Algebra, Lemmas \\ref{algebra-lemma-etale} and", "\\ref{algebra-lemma-map-between-etale}.", "Thus the category $\\mathcal{C}$ is directed.", "\\medskip\\noindent", "Since $B/IB = A/I$ for all objects $B$ of $\\mathcal{C}$ we", "see that $A^h/I^h = A^h/IA^h = \\colim B/IB = \\colim A/I = A/I$.", "\\medskip\\noindent", "Next, we show that $A^h = \\colim_{B \\in \\mathcal{C}} B$ with", "$I^h = IA^h$ is a henselian pair. To do this we will verify", "condition (2) of Lemma \\ref{lemma-characterize-henselian-pair}.", "Namely, suppose given an \\'etale ring map $A^h \\to A'$", "and $A^h$-algebra map $\\sigma : A' \\to A^h/I^h$. Then there exists a", "$B \\in \\mathcal{C}$ and an \\'etale ring map $B \\to B'$ such that", "$A' = B' \\otimes_B A^h$. See Algebra, Lemma \\ref{algebra-lemma-etale}.", "Since $A^h/I^h = A/IB$, the map $\\sigma$ induces an $A$-algebra", "map $s : B' \\to A/I$. Then $B'/IB' = A/I \\times C$ as $A/I$-algebra,", "where $C$ is the kernel of the map $B'/IB' \\to A/I$ induced by $s$.", "Let $g \\in B'$ map to $(1, 0) \\in A/I \\times C$. Then $B \\to B'_g$", "is \\'etale and $A/I \\to B'_g/IB'_g$ is an isomorphism, i.e.,", "$B'_g$ is an object of $\\mathcal{C}$. Thus we obtain a canonical", "map $B'_g \\to A^h$ such that", "$$", "\\vcenter{", "\\xymatrix{", "B'_g \\ar[r] & A^h \\\\", "B \\ar[u] \\ar[ur]", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "B' \\ar[r] \\ar[rrd]_s & B'_g \\ar[r] & A^h \\ar[d] \\\\", "& & A/I", "}", "}", "$$", "commute. This induces a map $A' = B' \\otimes_B A^h \\to A^h$", "compatible with $\\sigma$ as desired.", "\\medskip\\noindent", "Let $(A, I) \\to (A', I')$ be a morphism of pairs with $(A', I')$ henselian.", "We will show there is a unique factorization $A \\to A^h \\to A'$ which will", "finish the proof. Namely, for each $A \\to B$ in $\\mathcal{C}$", "the ring map $A' \\to B' = A' \\otimes_A B$ is \\'etale and induces", "an isomorphism $A'/I' \\to B'/I'B'$. Hence there is a section", "$\\sigma_B : B' \\to A'$ by Lemma \\ref{lemma-characterize-henselian-pair}.", "Given a morphism $B_1 \\to B_2$ in $\\mathcal{C}$ we claim the diagram", "$$", "\\xymatrix{", "B'_1 \\ar[rr] \\ar[rd]_{\\sigma_{B_1}} & &", "B'_2 \\ar[ld]^{\\sigma_{B_2}} \\\\", "& A'", "}", "$$", "commutes. This follows once we prove that for every $B$ in $\\mathcal{C}$", "the section $\\sigma_B$ is the unique $A'$-algebra map $B' \\to A'$.", "We have $B' \\otimes_{A'} B' = B' \\times R$ for some ring $R$, see", "Algebra, Lemma \\ref{algebra-lemma-diagonal-unramified}. In our case", "$R/I'R = 0$ as $B'/I'B' = A'/I'$. Thus given two $A'$-algebra maps", "$\\sigma_B, \\sigma_B' : B' \\to A'$ then", "$e = (\\sigma_B \\otimes \\sigma_B')(0, 1) \\in A'$", "is an idempotent contained in $I'$. We conclude that $e = 0$", "by Lemma \\ref{lemma-idempotents-determined-modulo-radical}.", "Hence $\\sigma_B = \\sigma_B'$ as desired.", "Using the commutativity we obtain", "$$", "A^h = \\colim_{B \\in \\mathcal{C}} B \\to", "\\colim_{B \\in \\mathcal{C}} A' \\otimes_A B \\xrightarrow{\\colim \\sigma_B} A'", "$$", "as desired. The uniqueness of the maps $\\sigma_B$ also guarantees that", "this map is unique. Hence $(A, I) \\mapsto (A^h, I^h)$ is the desired adjoint." ], "refs": [ "categories-definition-directed", "algebra-lemma-etale", "algebra-lemma-etale", "algebra-lemma-map-between-etale", "more-algebra-lemma-characterize-henselian-pair", "algebra-lemma-etale", "more-algebra-lemma-characterize-henselian-pair", "algebra-lemma-diagonal-unramified", "more-algebra-lemma-idempotents-determined-modulo-radical" ], "ref_ids": [ 12363, 1231, 1231, 1236, 9861, 1231, 9861, 1267, 9853 ] } ], "ref_ids": [] }, { "id": 9872, "type": "theorem", "label": "more-algebra-lemma-henselization-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-flat", "contents": [ "Let $(A, I)$ be a pair. Let $(A^h, I^h)$ be as in ", "Lemma \\ref{lemma-henselization}. Then $A \\to A^h$ is flat,", "$I^h = IA^h$ and $A/I^n \\to A^h/I^nA^h$ is an isomorphism", "for all $n$." ], "refs": [ "more-algebra-lemma-henselization" ], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-henselization} we have seen that", "$A^h$ is a filtered colimit of \\'etale $A$-algebras $B$ such that", "$A/I \\to B/IB$ is an isomorphism and we have seen that", "$I^h = IA^h$. As an \\'etale ring map is flat", "(Algebra, Lemma \\ref{algebra-lemma-etale}) we conclude that", "$A \\to A^h$ is flat by Algebra, Lemma \\ref{algebra-lemma-colimit-flat}.", "Since each $A \\to B$ is flat we find that the maps", "$A/I^n \\to B/I^nB$ are isomorphisms as well (for example by", "Algebra, Lemma \\ref{algebra-lemma-lift-basis}).", "Taking the colimit we find that $A/I^n = A^h/I^nA^h$", "as desired." ], "refs": [ "more-algebra-lemma-henselization", "algebra-lemma-etale", "algebra-lemma-colimit-flat", "algebra-lemma-lift-basis" ], "ref_ids": [ 9871, 1231, 523, 902 ] } ], "ref_ids": [ 9871 ] }, { "id": 9873, "type": "theorem", "label": "more-algebra-lemma-henselization-local-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-local-ring", "contents": [ "\\begin{slogan}", "Compatibility henselization of pairs and of local rings.", "\\end{slogan}", "The functor of Lemma \\ref{lemma-henselization} associates to a local ring", "$(A, \\mathfrak m)$ its henselization." ], "refs": [ "more-algebra-lemma-henselization" ], "proofs": [ { "contents": [ "Let $(A^h, \\mathfrak m^h)$ be the henselization of the pair $(A, \\mathfrak m)$", "constructed in Lemma \\ref{lemma-henselization}. Then", "$\\mathfrak m^h = \\mathfrak m A^h$ is a maximal ideal", "by Lemma \\ref{lemma-henselization-flat} and since it", "is contained in the Jacobson radical, we conclude $A^h$ is", "local with maximal ideal $\\mathfrak m^h$. Having said this", "there are two ways to finish the proof.", "\\medskip\\noindent", "First proof: observe that the construction in the proof of", "Algebra, Lemma \\ref{algebra-lemma-henselization} as a colimit", "is the same as the colimit used to construct $A^h$ in", "Lemma \\ref{lemma-henselization}.", "Second proof: Both the henselization $A \\to S$ and $A \\to A^h$ of", "Lemma \\ref{lemma-henselization} are local ring homomorphisms,", "both $S$ and $A^h$ are filtered colimits of \\'etale $A$-algebras,", "both $S$ and $A^h$ are henselian local rings, and both $S$ and $A^h$", "have residue fields equal to $\\kappa(\\mathfrak m)$ (by", "Lemma \\ref{lemma-henselization-flat} for the second case).", "Hence they are canonically isomorphic by", "Algebra, Lemma \\ref{algebra-lemma-uniqueness-henselian}." ], "refs": [ "more-algebra-lemma-henselization", "more-algebra-lemma-henselization-flat", "algebra-lemma-henselization", "more-algebra-lemma-henselization", "more-algebra-lemma-henselization", "more-algebra-lemma-henselization-flat", "algebra-lemma-uniqueness-henselian" ], "ref_ids": [ 9871, 9872, 1294, 9871, 9871, 9872, 1292 ] } ], "ref_ids": [ 9871 ] }, { "id": 9874, "type": "theorem", "label": "more-algebra-lemma-henselization-Noetherian-pair", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-Noetherian-pair", "contents": [ "\\begin{slogan}", "The henselization of a Noetherian pair is a Noetherian pair", "with the same completion", "\\end{slogan}", "Let $(A, I)$ be a pair with $A$ Noetherian. Let $(A^h, I^h)$ be as in ", "Lemma \\ref{lemma-henselization}. Then the map of $I$-adic completions", "$$", "A^\\wedge \\to (A^h)^\\wedge", "$$", "is an isomorphism. Moreover, $A^h$ is Noetherian, the maps", "$A \\to A^h \\to A^\\wedge$ are flat, and $A^h \\to A^\\wedge$ is", "faithfully flat." ], "refs": [ "more-algebra-lemma-henselization" ], "proofs": [ { "contents": [ "The first statement is an immediate consequence of", "Lemma \\ref{lemma-henselization-flat}", "and in fact holds without assuming $A$ is Noetherian.", "In the proof of Lemma \\ref{lemma-henselization} we have seen that", "$A^h$ is a filtered colimit of \\'etale $A$-algebras $B$ such that", "$A/I \\to B/IB$ is an isomorphism. For each such $A \\to B$", "the induced map $A^\\wedge \\to B^\\wedge$ is an isomorphism", "(see proof of Lemma \\ref{lemma-henselization-flat}).", "By Algebra, Lemma \\ref{algebra-lemma-completion-flat} the ring map", "$B \\to A^\\wedge = B^\\wedge = (A^h)^\\wedge$ is flat for each $B$.", "Thus $A^h \\to A^\\wedge = (A^h)^\\wedge$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-colimit-rings-flat}.", "Since $I^h = IA^h$ is contained in the Jacobson radical of $A^h$", "and since $A^h \\to A^\\wedge$ induces an isomorphism $A^h/I^h \\to A/I$", "we see that $A^h \\to A^\\wedge$ is faithfully flat by", "Algebra, Lemma \\ref{algebra-lemma-ff}.", "By Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian-Noetherian}", "the ring $A^\\wedge$ is Noetherian.", "Hence we conclude that $A^h$ is Noetherian by", "Algebra, Lemma \\ref{algebra-lemma-descent-Noetherian}." ], "refs": [ "more-algebra-lemma-henselization-flat", "more-algebra-lemma-henselization", "more-algebra-lemma-henselization-flat", "algebra-lemma-completion-flat", "algebra-lemma-colimit-rings-flat", "algebra-lemma-ff", "algebra-lemma-completion-Noetherian-Noetherian", "algebra-lemma-descent-Noetherian" ], "ref_ids": [ 9872, 9871, 9872, 870, 526, 535, 874, 1370 ] } ], "ref_ids": [ 9871 ] }, { "id": 9875, "type": "theorem", "label": "more-algebra-lemma-henselization-colimit", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-colimit", "contents": [ "Let $(A, I) = \\colim (A_i, I_i)$ be a filtered colimit of pairs. The functor of", "Lemma \\ref{lemma-henselization} gives", "$A^h = \\colim A_i^h$ and $I^h = \\colim I_i^h$." ], "refs": [ "more-algebra-lemma-henselization" ], "proofs": [ { "contents": [ "By Categories, Lemma \\ref{categories-lemma-adjoint-exact}", "we see that $(A^h, I^h)$ is the colimit of the system $(A_i^h, I_i^h)$", "in the category of henselian pairs. Thus for a henselian pair $(B, J)$", "we have", "$$", "\\Mor((A^h, I^h), (B, J)) =", "\\lim \\Mor((A_i^h, I_i^h), (B, J)) =", "\\Mor(\\colim (A_i^h, I_i^h), (B, J))", "$$", "Here the colimit is in the category of pairs. Since the colimit is", "filtered we obtain $\\colim (A_i^h, I_i^h) = (\\colim A_i^h, \\colim I_i^h)$", "in the category of pairs; details omitted. Again using the colimit is filtered,", "this is a henselian pair (Lemma \\ref{lemma-filtered-colimits-henselian}).", "Hence by the Yoneda lemma we", "find $(A^h, I^h) = (\\colim A_i^h, \\colim I_i^h)$." ], "refs": [ "categories-lemma-adjoint-exact", "more-algebra-lemma-filtered-colimits-henselian" ], "ref_ids": [ 12249, 9868 ] } ], "ref_ids": [ 9871 ] }, { "id": 9876, "type": "theorem", "label": "more-algebra-lemma-henselization-change-ideal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-change-ideal", "contents": [ "\\begin{slogan}", "The henselization of a pair only depends on the radical of the ideal", "\\end{slogan}", "Let $A$ be a ring with ideals $I$ and $J$. If $V(I) = V(J)$ then the functor", "of Lemma \\ref{lemma-henselization} produces the same ring for the pair", "$(A, I)$ as for the pair $(A, J)$." ], "refs": [ "more-algebra-lemma-henselization" ], "proofs": [ { "contents": [ "Let $(A', IA')$ be the pair produced by Lemma \\ref{lemma-henselization}", "starting with the pair $(A, I)$, see Lemma \\ref{lemma-henselization-flat}.", "Let $(A'', JA'')$ be the pair produced by Lemma \\ref{lemma-henselization}", "starting with the pair $(A, J)$. By", "Lemma \\ref{lemma-change-ideal-henselian-pair} we see that", "$(A', JA')$ is a henselian pair and $(A'', IA'')$ is a henselian pair.", "By the universal property of the construction we obtain", "unique $A$-algebra maps $A'' \\to A'$ and $A' \\to A''$. The", "uniqueness shows that these are mutually inverse." ], "refs": [ "more-algebra-lemma-henselization", "more-algebra-lemma-henselization-flat", "more-algebra-lemma-henselization", "more-algebra-lemma-change-ideal-henselian-pair" ], "ref_ids": [ 9871, 9872, 9871, 9862 ] } ], "ref_ids": [ 9871 ] }, { "id": 9877, "type": "theorem", "label": "more-algebra-lemma-henselization-integral", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-integral", "contents": [ "\\begin{slogan}", "Henselization commutes with integral base change", "\\end{slogan}", "Let $(A, I) \\to (B, J)$ be a map of pairs such that $V(J) = V(IB)$.", "Let $(A^h , I^h) \\to (B^h, J^h)$ be the induced map", "on henselizations (Lemma \\ref{lemma-henselization}).", "If $A \\to B$ is integral, then the induced map", "$A^h \\otimes_A B \\to B^h$ is an isomorphism." ], "refs": [ "more-algebra-lemma-henselization" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-henselization-change-ideal} we may assume $J = IB$.", "By Lemma \\ref{lemma-integral-over-henselian-pair}", "the pair $(A^h \\otimes_A B, I^h(A^h \\otimes_A B))$ is henselian.", "By the universal property of $(B^h, IB^h)$ we obtain a map", "$B^h \\to A^h \\otimes_A B$. We omit the proof", "that this map is the inverse of the map in the lemma." ], "refs": [ "more-algebra-lemma-henselization-change-ideal", "more-algebra-lemma-integral-over-henselian-pair" ], "ref_ids": [ 9876, 9863 ] } ], "ref_ids": [ 9871 ] }, { "id": 9878, "type": "theorem", "label": "more-algebra-lemma-lift-finite-projective-module", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-finite-projective-module", "contents": [ "Let $(R, I)$ be a henselian pair. Let $\\overline{P}$ be a finite", "projective $R/I$-module. Then there exists a", "finite projective $R$-module $P$ such that $P/IP \\cong \\overline{P}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that we can lift the finite projective", "$R/I$-module $\\overline{P}$ to a finite projective module $P'$ over some $R'$", "\\'etale over $R$ with $R/I = R'/IR'$, see", "Lemma \\ref{lemma-lift-projective-module}.", "Then, since $(R, I)$ is a henselian pair, the \\'etale ring map $R \\to R'$ has", "a section $\\tau : R' \\to R$ (Lemma \\ref{lemma-characterize-henselian-pair}).", "Setting $P = P' \\otimes_{R', \\tau} R$ finishes the proof." ], "refs": [ "more-algebra-lemma-lift-projective-module", "more-algebra-lemma-characterize-henselian-pair" ], "ref_ids": [ 9849, 9861 ] } ], "ref_ids": [] }, { "id": 9879, "type": "theorem", "label": "more-algebra-lemma-finite-etale-equivalence", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finite-etale-equivalence", "contents": [ "Let $(A, I)$ be a henselian pair. The functor $B \\to B/IB$ determines", "an equivalence between finite \\'etale $A$-algebras and finite \\'etale", "$A/I$-algebras." ], "refs": [], "proofs": [ { "contents": [ "Let $B, B'$ be two $A$-algebras finite \\'etale over $A$.", "Then $B' \\to B'' = B \\otimes_A B'$ is finite \\'etale as well", "(Algebra, Lemmas \\ref{algebra-lemma-etale} and", "\\ref{algebra-lemma-base-change-integral}).", "Now we have $1$-to-$1$ correspondences between", "\\begin{enumerate}", "\\item $A$-algebra maps $B \\to B'$,", "\\item sections of $B' \\to B''$, and", "\\item idempotents $e$ of $B''$ such that $B' \\to B'' \\to eB''$ is", "an isomorphism.", "\\end{enumerate}", "The bijection between (2) and (3) sends $\\sigma : B'' \\to B'$", "to $e$ such that $(1 - e)$ is the idempotent", "that generates the kernel of $\\sigma$ which exists by", "Algebra, Lemmas \\ref{algebra-lemma-map-between-etale} and", "\\ref{algebra-lemma-surjective-flat-finitely-presented}.", "There is a similar correspondence between", "$A/I$-algebra maps $B/IB \\to B'/IB'$ and idempotents", "$\\overline{e}$ of $B''/IB''$ such that", "$B'/IB' \\to B''/IB'' \\to \\overline{e}(B''/IB'')$ is", "an isomorphism. However every idempotent $\\overline{e}$ of $B''/IB''$", "lifts uniquely to an idempotent $e$ of $B''$", "(Lemma \\ref{lemma-characterize-henselian-pair}).", "Moreover, if $B'/IB' \\to \\overline{e}(B''/IB'')$ is an isomorphism,", "then $B' \\to eB''$ is an isomorphism too by Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK}).", "In this way we see that the functor is fully faithful.", "\\medskip\\noindent", "Essential surjectivity. Let $A/I \\to C$ be a finite \\'etale map.", "By Algebra, Lemma \\ref{algebra-lemma-lift-etale}", "there exists an \\'etale map $A \\to B$ such that $B/IB \\cong C$.", "Let $B'$ be the integral closure of $A$ in $B$. ", "By Lemma \\ref{lemma-helper-finite-type} we have", "$B'/IB' = C \\times C'$ for some ring $C'$", "and $B'_g \\cong B_g$ for some $g \\in B'$ mapping to $(1, 0) \\in C \\times C'$.", "Since idempotents lift", "(Lemma \\ref{lemma-characterize-henselian-pair})", "we get $B' = B'_1 \\times B'_2$ with $C = B'_1/IB'_1$ and $C' = B'_2/IB'_2$.", "The image of $g$ in $B'_1$ is invertible.", "Then $B_g = B'_g = B'_1 \\times (B_2)_g$ and this implies", "that $A \\to B'_1$ is \\'etale.", "We conclude that $B'_1$ is finite \\'etale over $A$", "(integral \\'etale implies finite \\'etale by", "Algebra, Lemma \\ref{algebra-lemma-characterize-finite-in-terms-of-integral}", "for example)", "and the proof is done." ], "refs": [ "algebra-lemma-etale", "algebra-lemma-base-change-integral", "algebra-lemma-map-between-etale", "algebra-lemma-surjective-flat-finitely-presented", "more-algebra-lemma-characterize-henselian-pair", "algebra-lemma-NAK", "algebra-lemma-lift-etale", "more-algebra-lemma-helper-finite-type", "more-algebra-lemma-characterize-henselian-pair", "algebra-lemma-characterize-finite-in-terms-of-integral" ], "ref_ids": [ 1231, 491, 1236, 1237, 9861, 401, 1238, 9860, 9861, 484 ] } ], "ref_ids": [] }, { "id": 9880, "type": "theorem", "label": "more-algebra-lemma-lim-finite-projective-gives-finite-projective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lim-finite-projective-gives-finite-projective", "contents": [ "Let $A = \\lim A_n$ be a limit of an inverse system $(A_n)$ of rings.", "Suppose given $A_n$-modules $M_n$ and $A_{n + 1}$-module maps", "$M_{n + 1} \\to M_n$. Assume", "\\begin{enumerate}", "\\item the transition maps $A_{n + 1} \\to A_n$ are surjective", "with locally nilpotent kernels,", "\\item $M_1$ is a finite projective $A_1$-module,", "\\item $M_n$ is a finite flat $A_n$-module, and", "\\item the maps induce isomorphisms", "$M_{n + 1} \\otimes_{A_{n + 1}} A_n \\to M_n$.", "\\end{enumerate}", "Then $M = \\lim M_n$ is a finite projective $A$-module", "and $M \\otimes_A A_n \\to M_n$ is an isomorphism for all $n$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-limit-henselian} the pair $(A, \\Ker(A \\to A_1))$ is", "henselian. By Lemma \\ref{lemma-lift-finite-projective-module}", "we can choose a finite projective $A$-module $P$ and an isomorphism", "$P \\otimes_A A_1 \\to M_1$. Since $P$ is projective, we can successively lift", "the $A$-module map $P \\to M_1$ to $A$-module maps", "$P \\to M_2$, $P \\to M_3$, and so on. Thus we obtain a map", "$$", "P \\longrightarrow M", "$$", "Since $P$ is finite projective, we can write $A^{\\oplus m} = P \\oplus Q$", "for some $m \\geq 0$ and $A$-module $Q$. Since $A = \\lim A_n$ we conclude", "that $P = \\lim P \\otimes_A A_n$. Hence, in order to show that the displayed", "$A$-module map is an isomorphism, it suffices to show that the maps", "$P \\otimes_A A_n \\to M_n$ are isomorphisms.", "From Lemma \\ref{lemma-lift-projective} we see that", "$M_n$ is a finite projective module.", "By Lemma \\ref{lemma-isomorphic-finite-projective-lifts}", "the maps $P \\otimes_A A_n \\to M_n$ are isomorphisms." ], "refs": [ "more-algebra-lemma-limit-henselian", "more-algebra-lemma-lift-finite-projective-module", "more-algebra-lemma-lift-projective", "more-algebra-lemma-isomorphic-finite-projective-lifts" ], "ref_ids": [ 9858, 9878, 9809, 9810 ] } ], "ref_ids": [] }, { "id": 9881, "type": "theorem", "label": "more-algebra-lemma-absolutely-integrally-closed", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-absolutely-integrally-closed", "contents": [ "Let $A$ be a ring. The following are equivalent", "\\begin{enumerate}", "\\item $A$ is absolutely integrally closed, and", "\\item any monic $f \\in A[T]$ has a root in $A$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9882, "type": "theorem", "label": "more-algebra-lemma-absolutely-integrally-closed-quotient-localization", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-absolutely-integrally-closed-quotient-localization", "contents": [ "Let $A$ be absolutely integrally closed.", "\\begin{enumerate}", "\\item Any quotient ring $A/I$ of $A$ is absolutely integrally closed.", "\\item Any localization $S^{-1}A$ is absolutely integrally closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9883, "type": "theorem", "label": "more-algebra-lemma-integrally-closed-in-absolutely-integrally-closed", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-integrally-closed-in-absolutely-integrally-closed", "contents": [ "Let $A$ be a ring. Let $S \\subset A$ be a multiplicative subset.", "If $S^{-1}A$ is absolutely integrally closed and $A \\subset S^{-1}A$", "is integrally closed in $S^{-1}A$, then $A$ is absolutely integrally closed." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9884, "type": "theorem", "label": "more-algebra-lemma-normal-domain-absolutely-integrally-closed", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-normal-domain-absolutely-integrally-closed", "contents": [ "Let $A$ be a normal domain. Then $A$ is absolutely integrally closed", "if and only if its fraction field is algebraically closed." ], "refs": [], "proofs": [ { "contents": [ "Observe that a field is algebraically closed if and only if", "it is absolutely integrally closed as a ring. Hence the lemma follows", "from Lemmas \\ref{lemma-absolutely-integrally-closed-quotient-localization} and", "\\ref{lemma-integrally-closed-in-absolutely-integrally-closed}." ], "refs": [ "more-algebra-lemma-absolutely-integrally-closed-quotient-localization", "more-algebra-lemma-integrally-closed-in-absolutely-integrally-closed" ], "ref_ids": [ 9882, 9883 ] } ], "ref_ids": [] }, { "id": 9885, "type": "theorem", "label": "more-algebra-lemma-construct-absolute-integral-closure", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-construct-absolute-integral-closure", "contents": [ "For any ring $A$ there exists an extension $A \\subset B$ such that", "\\begin{enumerate}", "\\item $B$ is a filtered colimit of finite free $A$-algebras,", "\\item $B$ is free as an $A$-module, and", "\\item $B$ is absolutely integrally closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $I$ be the set of monic polynomials over $A$. For $i \\in I$", "denote $x_i$ a variable and $P_i$ the corresponding monic polynomial", "in the variable $x_i$. Then we set", "$$", "F(A) = A[x_i; i \\in I]/(P_i; i \\in I)", "$$", "As the notation suggests $F$ is a functor from the category", "of rings to itself. Note that $A \\subset F(A)$, that $F(A)$ is free", "as an $A$-module, and that $F(A)$ is a filtered colimit of", "finite free $A$-algebras. Then we take", "$$", "B = \\colim F^n(A)", "$$", "where the transition maps are the inclusions", "$F^n(A) \\subset F(F^n(A)) = F^{n + 1}(A)$.", "Any monic polynomial with coefficients in $B$", "actually has coefficients in $F^n(A)$ for some $n$", "and hence has a solution in $F^{n + 1}(A)$ by construction.", "This implies that $B$ is absolutely integrally closed by", "Lemma \\ref{lemma-absolutely-integrally-closed}.", "We omit the proof of the other properties." ], "refs": [ "more-algebra-lemma-absolutely-integrally-closed" ], "ref_ids": [ 9881 ] } ], "ref_ids": [] }, { "id": 9886, "type": "theorem", "label": "more-algebra-lemma-absolutely-integrally-closed-strictly-henselian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-absolutely-integrally-closed-strictly-henselian", "contents": [ "Let $A$ be absolutely integrally closed. Let $\\mathfrak p \\subset A$", "be a prime. Then the local ring $A_\\mathfrak p$ is strictly henselian." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-absolutely-integrally-closed-quotient-localization}", "we may assume $A$ is a local ring and $\\mathfrak p$ is its maximal ideal.", "The residue field is algebraically closed by", "Lemma \\ref{lemma-absolutely-integrally-closed-quotient-localization}.", "Every monic polynomial decomposes completely into linear factors", "hence Algebra, Definition \\ref{algebra-definition-henselian} applies directly." ], "refs": [ "more-algebra-lemma-absolutely-integrally-closed-quotient-localization", "more-algebra-lemma-absolutely-integrally-closed-quotient-localization", "algebra-definition-henselian" ], "ref_ids": [ 9882, 9882, 1545 ] } ], "ref_ids": [] }, { "id": 9887, "type": "theorem", "label": "more-algebra-lemma-absolutely-integrally-closed-henselian-pair", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-absolutely-integrally-closed-henselian-pair", "contents": [ "Let $A$ be absolutely integrally closed. Let $I \\subset A$ be an ideal.", "Then $(A, I)$ is a henselian pair if (and only if) the following", "conditions hold", "\\begin{enumerate}", "\\item $I$ is contained in the Jacobson radical of $A$,", "\\item $A \\to A/I$ induces a bijection on idempotents.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $f \\in A[T]$ be a monic polynomial and let", "$f \\mod I = g_0 h_0$ be a factorization over $A/I$", "with $g_0$, $h_0$ monic such that", "$g_0$ and $h_0$ generate the unit ideal of $A/I[T]$.", "This means that", "$$", "A/I[T]/(f) = A/I[T]/(g_0) \\times A/I[T]/(h_0)", "$$", "Denote $e \\in A/I[T]/(f)$ the element correspoing to the", "idempotent $(1, 0)$ in the ring on the right.", "Write $f = (T - a_1) \\ldots (T - a_d)$ with $a_i \\in A$.", "For each $i \\in \\{1, \\ldots, d\\}$ we obtain an $A$-algebra map", "$\\varphi_i : A[T]/(f) \\to A$, $T \\mapsto a_i$ which induces", "a similar $A/I$-algebra map $\\overline{\\varphi}_i : A/I[T]/(f) \\to A/I$.", "Denote $e_i = \\overline{\\varphi}_i(e) \\in A/I$. These are idempotents.", "By our assumption (2) we can lift $e_i$ to an idempotent in $A$.", "This means we can write $A = \\prod A_j$ as a finite product of", "rings such that in $A_j/IA_j$ each $e_i$ is either $0$ or $1$.", "Some details omitted. Observe that $A_j$ is absolutely integrally", "closed as a factor ring of $A$. It suffices to lift the factorization", "of $f$ over $A_j/IA_j$ to $A_j$.", "This reduces us to the situation discussed in the next paragraph.", "\\medskip\\noindent", "Assume $e_i = 1$ for $i = 1, \\ldots, r$ and $e_i = 0$ for", "$i = r + 1, \\ldots, d$. From $(g_0, h_0) = A/I[T]$ we have that there are", "$k_0, l_0 \\in A/I[T]$ such that $g_0 k_0 + h_0 l_0 = 1$.", "We see that $e = h_0 l_0$ and $e_i = h_0(a_i) l_0(a_i)$.", "We conclude that $h_0(a_i)$ is a unit for $i = 1, \\dots ,r$.", "Since $f(a_i) = 0$ we find $0 = h_0(a_i)g_0(a_i)$", "and we conclude that $g_0(a_i) = 0$ for $i = 1, \\ldots, r$.", "Thus $(T - a_1)$ divides $g_0$ in $A/I[T]$, say", "$g_0 = (T - a_1) g_0'$.", "Set $f' = (T - a_2) \\ldots (T - a_d)$ and $h'_0 = h_0$.", "By induction on $d$ we can lift the factorization $f' \\bmod I = g'_0 h'_0$", "to a factorization of $f' = g' h'$ over over $A$ which", "gives the factorization $f = (T - a_1) g' h'$", "lifting the factorization $f \\bmod I = g_0 h_0$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9888, "type": "theorem", "label": "more-algebra-lemma-auto-ass-implies-P", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-auto-ass-implies-P", "contents": [ "An auto-associated ring $R$ has the following property: (P)", "Every proper finitely generated ideal $I \\subset R$ has a nonzero", "annihilator." ], "refs": [], "proofs": [ { "contents": [ "By assumption there exists a nonzero element $x \\in R$ such that for every", "$f \\in \\mathfrak m$ we have $f^n x = 0$. Say $I = (f_1, \\ldots, f_r)$.", "Then $x$ is in the kernel of $R \\to \\bigoplus R_{f_i}$. Hence we see", "that there exists a nonzero $y \\in R$ such that $f_i y = 0$ for all $i$, see", "Algebra, Lemma \\ref{algebra-lemma-when-injective-covering}.", "As $y \\in \\text{Ann}_R(I)$ we win." ], "refs": [ "algebra-lemma-when-injective-covering" ], "ref_ids": [ 416 ] } ], "ref_ids": [] }, { "id": 9889, "type": "theorem", "label": "more-algebra-lemma-P-universally-injective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-P-universally-injective", "contents": [ "Let $R$ be a ring having property (P) of", "Lemma \\ref{lemma-auto-ass-implies-P}.", "Let $u : N \\to M$ be a homomorphism of projective $R$-modules.", "Then $u$ is universally injective if and only if $u$ is injective." ], "refs": [ "more-algebra-lemma-auto-ass-implies-P" ], "proofs": [ { "contents": [ "Assume $u$ is injective. Our goal is to show $u$ is universally injective.", "First we choose a module $Q$ such that $N \\oplus Q$ is free. On considering", "the map $N \\oplus Q \\to M \\oplus Q$ we see that it suffices to prove", "the lemma in case $N$ is free. In this case $N$ is a directed colimit of", "finite free $R$-modules. Thus we reduce to the case that $N$ is a finite", "free $R$-module, say $N = R^{\\oplus n}$. We prove the lemma by induction", "on $n$. The case $n = 0$ is trivial.", "\\medskip\\noindent", "Let $u : R^{\\oplus n} \\to M$ be an injective module map with $M$ projective.", "Choose an $R$-module $Q$ such that $M \\oplus Q$ is free. After replacing", "$u$ by the composition $R^{\\oplus n} \\to M \\to M \\oplus Q$ we see that", "we may assume that $M$ is free. Then we can find a direct summand", "$R^{\\oplus m} \\subset M$ such that $u(R^{\\oplus n}) \\subset R^{\\oplus m}$.", "Hence we may assume that $M = R^{\\oplus m}$.", "In this case $u$ is given by a matrix $A = (a_{ij})$ so that", "$u(x_1, \\ldots, x_n) = (\\sum x_i a_{i1}, \\ldots, \\sum x_i a_{im})$.", "As $u$ is injective, in particular", "$u(x, 0, \\ldots, 0) = (xa_{11}, xa_{12}, \\ldots, xa_{1m}) \\not = 0$ if", "$x \\not = 0$, and as $R$ has property (P) we see that", "$a_{11}R + a_{12}R + \\ldots + a_{1m}R = R$. Hence see that", "$R(a_{11}, \\ldots, a_{1m}) \\subset R^{\\oplus m}$ is a direct summand", "of $R^{\\oplus m}$, in particular $R^{\\oplus m}/R(a_{11}, \\ldots, a_{1m})$", "is a projective $R$-module. We get a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "R \\ar[rr] \\ar[d]^1 & & R^{\\oplus n} \\ar[r] \\ar[d]^u &", "R^{\\oplus n - 1} \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] & R \\ar[rr]^{(a_{11}, \\ldots, a_{1m})} & &", "R^{\\oplus m} \\ar[r] & R^{\\oplus m}/R(a_{11}, \\ldots, a_{1m}) \\ar[r] & 0", "}", "$$", "with split exact rows. Thus the right vertical arrow is injective", "and we may apply the induction hypothesis to conclude that", "the right vertical arrow is universally injective. It follows that the", "middle vertical arrow is universally injective." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9888 ] }, { "id": 9890, "type": "theorem", "label": "more-algebra-lemma-P-fPD-zero", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-P-fPD-zero", "contents": [ "Let $R$ be a ring. The following are equivalent", "\\begin{enumerate}", "\\item $R$ has property (P) of", "Lemma \\ref{lemma-auto-ass-implies-P},", "\\item any injective map of projective $R$-modules is", "universally injective,", "\\item if $u : N \\to M$ is injective and $N$, $M$ are finite projective", "$R$-modules then $\\Coker(u)$ is a finite projective $R$-module,", "\\item if $N \\subset M$ and $N$, $M$ are finite projective as $R$-modules, then", "$N$ is a direct summand of $M$, and", "\\item any injective map $R \\to R^{\\oplus n}$ is a split injection.", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-auto-ass-implies-P" ], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) is", "Lemma \\ref{lemma-P-universally-injective}.", "It is clear that (3) and (4) are equivalent.", "We have (2) $\\Rightarrow$ (3), (4) by", "Algebra, Lemma \\ref{algebra-lemma-universally-exact-split}.", "Part (5) is a special case of (4).", "Assume (5). Let $I = (a_1, \\ldots, a_n)$ be a proper finitely generated", "ideal of $R$. As $I \\not = R$ we see that", "$R \\to R^{\\oplus n}$, $x \\mapsto (xa_1, \\ldots, xa_n)$", "is not a split injection. Hence it has a nonzero kernel and we conclude", "that $\\text{Ann}_R(I) \\not = 0$. Thus (1) holds." ], "refs": [ "more-algebra-lemma-P-universally-injective", "algebra-lemma-universally-exact-split" ], "ref_ids": [ 9889, 808 ] } ], "ref_ids": [ 9888 ] }, { "id": 9891, "type": "theorem", "label": "more-algebra-lemma-exact-length-1", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-exact-length-1", "contents": [ "Let $R$ be a ring. Suppose that $\\varphi : R^m \\to R^n$ is a map", "of finite free modules. The following are equivalent", "\\begin{enumerate}", "\\item $\\varphi$ is injective,", "\\item the rank of $\\varphi$ is $m$ and the annihilator of", "$I(\\varphi)$ in $R$ is zero.", "\\end{enumerate}", "If $R$ is Noetherian these are also equivalent to", "\\begin{enumerate}", "\\item[(3)] the rank of $\\varphi$ is $m$ and", "either $I(\\varphi) = R$ or it contains a nonzerodivisor.", "\\end{enumerate}", "Here the rank of $\\varphi$ and $I(\\varphi)$ are defined", "as in Algebra, Definition \\ref{algebra-definition-rank}." ], "refs": [ "algebra-definition-rank" ], "proofs": [ { "contents": [ "If any matrix coefficient of $\\varphi$ is not in $\\mathfrak m$, then we apply", "Algebra, Lemma \\ref{algebra-lemma-add-trivial-complex}", "to write $\\varphi$ as the sum of $1 : R \\to R$ and a map", "$\\varphi' : R^{m-1} \\to R^{n-1}$. It is easy to see that", "the lemma for $\\varphi'$ implies the lemma for $\\varphi$.", "Thus we may assume from the outset that all the matrix", "coefficients of $\\varphi$ are in $\\mathfrak m$.", "\\medskip\\noindent", "Suppose $\\varphi$ is injective. We may assume $m > 0$.", "Let $\\mathfrak q \\in \\text{WeakAss}(R)$ so that $R_\\mathfrak q$", "is an auto-associated ring. Then $\\varphi$", "induces a injective map $R_\\mathfrak q^m \\to R_\\mathfrak q^n$", "which is universally injective by", "Lemmas \\ref{lemma-auto-ass-implies-P} and \\ref{lemma-P-universally-injective}.", "Thus $\\varphi : \\kappa(\\mathfrak q)^m \\to \\kappa(\\mathfrak q)^n$", "is injective. Hence the rank of $\\varphi \\bmod \\mathfrak q$ is $m$ and", "$I(\\varphi \\otimes \\kappa(\\mathfrak q))$ is not the zero ideal.", "Since $m$ is the maximum rank $\\varphi$ can have, we conclude", "that $\\varphi$ has rank $m$ as well (ranks of matrices", "can only drop after base change). Hence", "$I(\\varphi) \\cdot \\kappa(\\mathfrak q) =", "I(\\varphi \\otimes \\kappa(\\mathfrak q))$ is not zero.", "Thus $I(\\varphi)$ is not contained in $\\mathfrak q$.", "Thus none of the weakly associated primes of $R$", "are weakly associated primes of the $R$-module", "$\\text{Ann}_R I(\\varphi)$. Thus $\\text{Ann}_R I(\\varphi)$", "has no weakly associated primes, see", "Algebra, Lemma \\ref{algebra-lemma-weakly-ass}.", "It follows from Algebra, Lemma", "\\ref{algebra-lemma-weakly-ass-zero}", "that $\\text{Ann}_R I(\\varphi)$ is zero.", "\\medskip\\noindent", "Conversely, assume (2). The rank being $m$ implies $n \\geq m$.", "Write $I(\\varphi) = (f_1, \\ldots, f_r)$ which is possible as", "$I(\\varphi)$ is finitely generated. By", "Algebra, Lemma \\ref{algebra-lemma-matrix-left-inverse}", "we can find maps $\\psi_i : R^n \\to R^m$ such that", "$\\psi \\circ \\varphi = f_i \\text{id}_{R^m}$. Thus", "$\\varphi(x) = 0$ implies $f_i x = 0$ for $i = 1, \\ldots, r$.", "This implies $x = 0$ and hence $\\varphi$ is injective.", "\\medskip\\noindent", "For the equivalence of (1) and (3) in the Noetherian local case", "we refer to Algebra, Proposition \\ref{algebra-proposition-what-exact}.", "If the ring $R$ is Noetherian but not local, then the reader can", "deduce it from the local case; details omitted.", "Another option is to redo the argument above using associated", "primes, using that there are finitely many of these, using", "prime avoidance, and using the characterization of nonzerodivisors", "as elements of a Noetherian ring not contained in any associated prime." ], "refs": [ "algebra-lemma-add-trivial-complex", "more-algebra-lemma-auto-ass-implies-P", "more-algebra-lemma-P-universally-injective", "algebra-lemma-weakly-ass", "algebra-lemma-weakly-ass-zero", "algebra-lemma-matrix-left-inverse", "algebra-proposition-what-exact" ], "ref_ids": [ 908, 9888, 9889, 722, 723, 381, 1419 ] } ], "ref_ids": [ 1501 ] }, { "id": 9892, "type": "theorem", "label": "more-algebra-lemma-coker-injective-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-coker-injective-free", "contents": [ "Let $R$ be a ring. Suppose that $\\varphi : R^n \\to R^n$ be an injective", "map of finite free modules of the same rank. Then", "$\\Hom_R(\\Coker(\\varphi), R) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi^t : R^n \\to R^n$ be the transpose of $\\varphi$.", "The lemma claims that $\\varphi^t$ is injective.", "With notation as in Lemma \\ref{lemma-exact-length-1}", "we see that the rank of $\\varphi^t$ is $n$ and that", "$I(\\varphi) = I(\\varphi^t)$. Thus we conclude by", "the equivalence of (1) and (2) of the lemma." ], "refs": [ "more-algebra-lemma-exact-length-1" ], "ref_ids": [ 9891 ] } ], "ref_ids": [] }, { "id": 9893, "type": "theorem", "label": "more-algebra-lemma-intersection-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-intersection-flat", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module. Let $I_1$, $I_2$ be ideals of $R$.", "If $M/I_1M$ is flat over $R/I_1$ and $M/I_2M$ is flat over $R/I_2$,", "then $M/(I_1 \\cap I_2)M$ is flat over $R/(I_1 \\cap I_2)$." ], "refs": [], "proofs": [ { "contents": [ "By replacing $R$ with $R/(I_1 \\cap I_2)$ and $M$ by $M/(I_1 \\cap I_2)M$", "we may assume that $I_1 \\cap I_2 = 0$. Let $J \\subset R$ be an ideal.", "To prove that $M$ is flat over $R$ we have to show that", "$J \\otimes_R M \\to M$ is injective, see", "Algebra, Lemma \\ref{algebra-lemma-flat}.", "By flatness of $M/I_1M$ over $R/I_1$ the map", "$$", "J/(J \\cap I_1) \\otimes_R M =", "(J + I_1)/I_1 \\otimes_{R/I_1} M/I_1M", "\\longrightarrow M/I_1M", "$$", "is injective. As $0 \\to (J \\cap I_1) \\to J \\to J/(J \\cap I_1) \\to 0$", "is exact we obtain a diagram", "$$", "\\xymatrix{", "(J \\cap I_1) \\otimes_R M \\ar[r] \\ar[d] &", "J \\otimes_R M \\ar[r] \\ar[d] &", "J/(J \\cap I_1) \\otimes_R M \\ar[r] \\ar[d] & 0 \\\\", "M \\ar@{=}[r] &", "M \\ar[r] &", "M/I_1M", "}", "$$", "hence it suffices to show that $(J \\cap I_1) \\otimes_R M \\to M$ is", "injective. Since $I_1 \\cap I_2 = 0$ the ideal $J \\cap I_1$", "maps isomorphically to an ideal $J' \\subset R/I_2$ and we see that", "$(J \\cap I_1) \\otimes_R M = J' \\otimes_{R/I_2} M/I_2M$. By flatness", "of $M/I_2M$ over $R/I_2$ the map $J' \\otimes_{R/I_2} M/I_2M \\to M/I_2M$", "is injective, which clearly implies that $(J \\cap I_1) \\otimes_R M \\to M$ is", "injective." ], "refs": [ "algebra-lemma-flat" ], "ref_ids": [ 525 ] } ], "ref_ids": [] }, { "id": 9894, "type": "theorem", "label": "more-algebra-lemma-flattening-artinian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flattening-artinian", "contents": [ "Let $R$ be an Artinian ring.", "Let $M$ be an $R$-module.", "Then there exists a smallest ideal $I \\subset R$ such that", "$M/IM$ is flat over $R/I$." ], "refs": [], "proofs": [ { "contents": [ "This follows directly from", "Lemma \\ref{lemma-intersection-flat}", "and the Artinian property." ], "refs": [ "more-algebra-lemma-intersection-flat" ], "ref_ids": [ 9893 ] } ], "ref_ids": [] }, { "id": 9895, "type": "theorem", "label": "more-algebra-lemma-flattening-artinian-universal-property", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flattening-artinian-universal-property", "contents": [ "Let $R$ be an Artinian ring. Let $M$ be an $R$-module.", "Let $I \\subset R$ be the smallest ideal $I \\subset R$ such that", "$M/IM$ is flat over $R/I$.", "Then $I$ has the following universal property:", "For every ring map $\\varphi : R \\to R'$ we have", "$$", "R' \\otimes_R M\\text{ is flat over }R'", "\\Leftrightarrow", "\\text{we have }\\varphi(I) = 0.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Note that $I$ exists by", "Lemma \\ref{lemma-flattening-artinian}.", "The implication $\\Rightarrow$ follows from", "Algebra, Lemma \\ref{algebra-lemma-flat-base-change}.", "Let $\\varphi : R \\to R'$ be such that $M \\otimes_R R'$ is flat over $R'$.", "Let $J = \\Ker(\\varphi)$. By", "Algebra,", "Lemma \\ref{algebra-lemma-descent-flatness-injective-map-artinian-rings}", "and as $R' \\otimes_R M = R' \\otimes_{R/J} M/JM$ is", "flat over $R'$ we conclude that $M/JM$ is flat over $R/J$.", "Hence $I \\subset J$ as desired." ], "refs": [ "more-algebra-lemma-flattening-artinian", "algebra-lemma-flat-base-change", "algebra-lemma-descent-flatness-injective-map-artinian-rings" ], "ref_ids": [ 9894, 527, 906 ] } ], "ref_ids": [] }, { "id": 9896, "type": "theorem", "label": "more-algebra-lemma-base-change-flat-at-primes-over", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-flat-at-primes-over", "contents": [ "Let $R \\to S$ be a ring map.", "Let $I \\subset R$ be an ideal.", "Let $M$ be an $S$-module.", "Let $R \\to R'$ be a ring map and $IR' \\subset I' \\subset R'$ an ideal.", "If (\\ref{equation-flat-at-primes-over}) holds for", "$(R \\to S, I, M)$, then (\\ref{equation-flat-at-primes-over})", "holds for $(R' \\to S \\otimes_R R', I', M \\otimes_R R')$." ], "refs": [], "proofs": [ { "contents": [ "Assume (\\ref{equation-flat-at-primes-over}) holds for", "$(R \\to S, I \\subset R, M)$.", "Let $I'(S \\otimes_R R') \\subset \\mathfrak q'$ be a prime of $S \\otimes_R R'$.", "Let $\\mathfrak q \\subset S$ be the corresponding prime of $S$.", "Then $IS \\subset \\mathfrak q$. Note that $(M \\otimes_R R')_{\\mathfrak q'}$", "is a localization of the base change $M_{\\mathfrak q} \\otimes_R R'$.", "Hence $(M \\otimes_R R')_{\\mathfrak q'}$ is flat over $R'$ as a localization", "of a flat module, see", "Algebra, Lemmas \\ref{algebra-lemma-flat-base-change} and", "\\ref{algebra-lemma-flat-localization}." ], "refs": [ "algebra-lemma-flat-base-change", "algebra-lemma-flat-localization" ], "ref_ids": [ 527, 538 ] } ], "ref_ids": [] }, { "id": 9897, "type": "theorem", "label": "more-algebra-lemma-flat-descent-flat-at-primes-over", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-descent-flat-at-primes-over", "contents": [ "Let $R \\to S$ be a ring map.", "Let $I \\subset R$ be an ideal.", "Let $M$ be an $S$-module.", "Let $R \\to R'$ be a ring map and $IR' \\subset I' \\subset R'$ an ideal", "such that", "\\begin{enumerate}", "\\item the map $V(I') \\to V(I)$ induced by", "$\\Spec(R') \\to \\Spec(R)$ is surjective, and", "\\item $R'_{\\mathfrak p'}$ is flat over $R$ for all primes", "$\\mathfrak p' \\in V(I')$.", "\\end{enumerate}", "If (\\ref{equation-flat-at-primes-over}) holds for", "$(R' \\to S \\otimes_R R', I', M \\otimes_R R')$, then", "(\\ref{equation-flat-at-primes-over}) holds for $(R \\to S, I, M)$." ], "refs": [], "proofs": [ { "contents": [ "Assume (\\ref{equation-flat-at-primes-over}) holds for", "$(R' \\to S \\otimes_R R', IR', M \\otimes_R R')$. Pick a prime", "$IS \\subset \\mathfrak q \\subset S$. Let $I \\subset \\mathfrak p \\subset R$", "be the corresponding prime of $R$. By assumption there exists", "a prime $\\mathfrak p' \\in V(I')$ of $R'$ lying over $\\mathfrak p$ and", "$R_{\\mathfrak p} \\to R'_{\\mathfrak p'}$ is flat. Choose a prime", "$\\overline{\\mathfrak q}' \\subset", "\\kappa(\\mathfrak q) \\otimes_{\\kappa(\\mathfrak p)} \\kappa(\\mathfrak p')$", "which corresponds to a prime $\\mathfrak q' \\subset S \\otimes_R R'$ which", "lies over $\\mathfrak q$ and over $\\mathfrak p'$. Note that", "$(S \\otimes_R R')_{\\mathfrak q'}$ is a localization of", "$S_{\\mathfrak q} \\otimes_{R_{\\mathfrak p}} R'_{\\mathfrak p'}$.", "By assumption the module $(M \\otimes_R R')_{\\mathfrak q'}$ is", "flat over $R'_{\\mathfrak p'}$. Hence", "Algebra, Lemma \\ref{algebra-lemma-base-change-flat-up-down}", "implies that $M_{\\mathfrak q}$ is flat over $R_{\\mathfrak p}$", "which is what we wanted to prove." ], "refs": [ "algebra-lemma-base-change-flat-up-down" ], "ref_ids": [ 898 ] } ], "ref_ids": [] }, { "id": 9898, "type": "theorem", "label": "more-algebra-lemma-limit-preserving-flat-at-primes-over", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-limit-preserving-flat-at-primes-over", "contents": [ "Let $R \\to S$ be a ring map of finite presentation.", "Let $M$ be an $S$-module of finite presentation.", "Let $R' = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$", "be a directed colimit of $R$-algebras. Let $I_\\lambda \\subset R_\\lambda$", "be ideals such that $I_\\lambda R_\\mu \\subset I_\\mu$ for all $\\mu \\geq \\lambda$", "and set $I' = \\colim_\\lambda I_\\lambda$.", "If (\\ref{equation-flat-at-primes-over}) holds for", "$(R' \\to S \\otimes_R R', I', M \\otimes_R R')$, then there exists", "a $\\lambda \\in \\Lambda$ such that", "(\\ref{equation-flat-at-primes-over}) holds for", "$(R_\\lambda \\to S \\otimes_R R_\\lambda, I_\\lambda, M \\otimes_R R_\\lambda)$." ], "refs": [], "proofs": [ { "contents": [ "We are going to write $S_\\lambda = S \\otimes_R R_\\lambda$,", "$S' = S \\otimes_R R'$, $M_\\lambda = M \\otimes_R R_\\lambda$, and", "$M' = M \\otimes_R R'$.", "The base change $S'$ is of finite presentation over $R'$ and", "$M'$ is of finite presentation over $S'$ and similarly for the", "versions with subscript $\\lambda$, see", "Algebra, Lemma \\ref{algebra-lemma-base-change-finiteness}.", "By", "Algebra, Theorem \\ref{algebra-theorem-openness-flatness}", "the set", "$$", "U' = \\{\\mathfrak q' \\in \\Spec(S') \\mid", "M'_{\\mathfrak q'}\\text{ is flat over }R'\\}", "$$", "is open in $\\Spec(S')$. Note that $V(I'S')$ is a quasi-compact space", "which is contained in $U'$ by assumption. Hence there exist finitely many", "$g'_j \\in S'$, $j = 1, \\ldots, m$ such that $D(g'_j) \\subset U'$ and such", "that $V(I'S') \\subset \\bigcup D(g'_j)$.", "Note that in particular $(M')_{g'_j}$ is a flat module over $R'$.", "\\medskip\\noindent", "We are going to pick increasingly large elements $\\lambda \\in \\Lambda$.", "First we pick it large enough so that we can find", "$g_{j, \\lambda} \\in S_{\\lambda}$ mapping to $g'_j$.", "The inclusion $V(I'S') \\subset \\bigcup D(g'_j)$ means that", "$I'S' + (g'_1, \\ldots, g'_m) = S'$ which can be expressed as", "$1 = \\sum z_sh_s + \\sum f_jg'_j$ for some $z_s \\in I'$, $h_s, f_j \\in S'$.", "After increasing $\\lambda$ we may assume such an equation holds in", "$S_\\lambda$. Hence we may assume that", "$V(I_\\lambda S_\\lambda) \\subset \\bigcup D(g_{j, \\lambda})$. By", "Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "we see that for some sufficiently large $\\lambda$ the modules", "$(M_\\lambda)_{g_{j, \\lambda}}$ are flat over $R_\\lambda$.", "In particular the module $M_\\lambda$ is flat over $R_\\lambda$", "at all the primes lying over the ideal $I_\\lambda$." ], "refs": [ "algebra-lemma-base-change-finiteness", "algebra-theorem-openness-flatness", "algebra-lemma-flat-finite-presentation-limit-flat" ], "ref_ids": [ 373, 326, 1389 ] } ], "ref_ids": [] }, { "id": 9899, "type": "theorem", "label": "more-algebra-lemma-base-change-flat-at-primes", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-flat-at-primes", "contents": [ "In Situation \\ref{situation-flattening-general}", "let $R' \\to R''$ be an $R$-algebra map.", "Let $I' \\subset R'$ and $I'R'' \\subset I'' \\subset R''$ be ideals.", "If (\\ref{equation-flat-at-primes}) holds for", "$(R', I')$, then (\\ref{equation-flat-at-primes})", "holds for $(R'', I'')$." ], "refs": [], "proofs": [ { "contents": [ "Assume (\\ref{equation-flat-at-primes}) holds for $(R', I')$.", "Let $I''S'' + JS'' \\subset \\mathfrak q''$ be a prime of $S''$.", "Let $\\mathfrak q' \\subset S'$ be the corresponding prime of $S'$.", "Then both $I'S' \\subset \\mathfrak q'$ and $JS' \\subset \\mathfrak q'$", "because the corresponding conditions hold for $\\mathfrak q''$.", "Note that $(M'')_{\\mathfrak q''}$", "is a localization of the base change $M'_{\\mathfrak q'} \\otimes_R R''$.", "Hence $(M'')_{\\mathfrak q''}$ is flat over $R''$ as a localization", "of a flat module, see", "Algebra, Lemmas \\ref{algebra-lemma-flat-base-change} and", "\\ref{algebra-lemma-flat-localization}." ], "refs": [ "algebra-lemma-flat-base-change", "algebra-lemma-flat-localization" ], "ref_ids": [ 527, 538 ] } ], "ref_ids": [] }, { "id": 9900, "type": "theorem", "label": "more-algebra-lemma-flat-descent-flat-at-primes", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-descent-flat-at-primes", "contents": [ "In Situation \\ref{situation-flattening-general}", "let $R' \\to R''$ be an $R$-algebra map.", "Let $I' \\subset R'$ and $I'R'' \\subset I'' \\subset R''$ be ideals.", "Assume", "\\begin{enumerate}", "\\item the map $V(I'') \\to V(I')$ induced by", "$\\Spec(R'') \\to \\Spec(R')$ is surjective, and", "\\item $R''_{\\mathfrak p''}$ is flat over $R'$ for all primes", "$\\mathfrak p'' \\in V(I'')$.", "\\end{enumerate}", "If (\\ref{equation-flat-at-primes}) holds for", "$(R'', I'')$, then (\\ref{equation-flat-at-primes}) holds for $(R', I')$." ], "refs": [], "proofs": [ { "contents": [ "Assume (\\ref{equation-flat-at-primes}) holds for $(R'', I'')$. Pick a prime", "$I'S' + JS' \\subset \\mathfrak q' \\subset S'$. Let", "$I' \\subset \\mathfrak p' \\subset R'$ be the corresponding prime of $R'$.", "By assumption there exists a prime $\\mathfrak p'' \\in V(I'')$ of $R''$", "lying over $\\mathfrak p'$ and $R'_{\\mathfrak p'} \\to R''_{\\mathfrak p''}$", "is flat. Choose a prime", "$\\overline{\\mathfrak q}'' \\subset", "\\kappa(\\mathfrak q') \\otimes_{\\kappa(\\mathfrak p')} \\kappa(\\mathfrak p'')$.", "This corresponds to a prime $\\mathfrak q'' \\subset S'' = S' \\otimes_{R'} R''$", "which lies over $\\mathfrak q'$ and over $\\mathfrak p''$. In particular", "we see that $I''S'' \\subset \\mathfrak q''$ and that", "$JS'' \\subset \\mathfrak q''$. Note that", "$(S' \\otimes_{R'} R'')_{\\mathfrak q''}$ is a localization of", "$S'_{\\mathfrak q'} \\otimes_{R'_{\\mathfrak p'}} R''_{\\mathfrak p''}$.", "By assumption the module $(M' \\otimes_{R'} R'')_{\\mathfrak q''}$ is", "flat over $R''_{\\mathfrak p''}$. Hence", "Algebra, Lemma \\ref{algebra-lemma-base-change-flat-up-down}", "implies that $M'_{\\mathfrak q'}$ is flat over $R'_{\\mathfrak p'}$", "which is what we wanted to prove." ], "refs": [ "algebra-lemma-base-change-flat-up-down" ], "ref_ids": [ 898 ] } ], "ref_ids": [] }, { "id": 9901, "type": "theorem", "label": "more-algebra-lemma-limit-preserving-flat-at-primes", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-limit-preserving-flat-at-primes", "contents": [ "In Situation \\ref{situation-flattening-general}", "assume $R \\to S$ is essentially of finite presentation", "and $M$ is an $S$-module of finite presentation. Let", "$R' = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$", "be a directed colimit of $R$-algebras. Let $I_\\lambda \\subset R_\\lambda$", "be ideals such that $I_\\lambda R_\\mu \\subset I_\\mu$ for all", "$\\mu \\geq \\lambda$ and set $I' = \\colim_\\lambda I_\\lambda$.", "If (\\ref{equation-flat-at-primes}) holds for", "$(R', I')$, then there exists a $\\lambda \\in \\Lambda$ such that", "(\\ref{equation-flat-at-primes}) holds for $(R_\\lambda, I_\\lambda)$." ], "refs": [], "proofs": [ { "contents": [ "We first prove the lemma in case $R \\to S$ is of finite presentation", "and then we explain what needs to be changed in the general case.", "We are going to write $S_\\lambda = S \\otimes_R R_\\lambda$,", "$S' = S \\otimes_R R'$, $M_\\lambda = M \\otimes_R R_\\lambda$, and", "$M' = M \\otimes_R R'$.", "The base change $S'$ is of finite presentation over $R'$ and", "$M'$ is of finite presentation over $S'$ and similarly for the", "versions with subscript $\\lambda$, see", "Algebra, Lemma \\ref{algebra-lemma-base-change-finiteness}.", "By", "Algebra, Theorem \\ref{algebra-theorem-openness-flatness}", "the set", "$$", "U' = \\{\\mathfrak q' \\in \\Spec(S') \\mid", "M'_{\\mathfrak q'}\\text{ is flat over }R'\\}", "$$", "is open in $\\Spec(S')$. Note that $V(I'S' + JS')$", "is a quasi-compact space which is contained in $U'$ by assumption.", "Hence there exist finitely many $g'_j \\in S'$, $j = 1, \\ldots, m$", "such that $D(g'_j) \\subset U'$ and such", "that $V(I'S' + JS') \\subset \\bigcup D(g'_j)$.", "Note that in particular $(M')_{g'_j}$ is a flat module over $R'$.", "\\medskip\\noindent", "We are going to pick increasingly large elements $\\lambda \\in \\Lambda$.", "First we pick it large enough so that we can find", "$g_{j, \\lambda} \\in S_{\\lambda}$ mapping to $g'_j$.", "The inclusion $V(I'S' + JS') \\subset \\bigcup D(g'_j)$ means that", "$I'S' + JS' + (g'_1, \\ldots, g'_m) = S'$ which can be expressed as", "$$", "1 = \\sum y_tk_t + \\sum z_sh_s + \\sum f_jg'_j", "$$", "for some $z_s \\in I'$, $y_t \\in J$, $k_t, h_s, f_j \\in S'$.", "After increasing $\\lambda$ we may assume such an equation holds in", "$S_\\lambda$. Hence we may assume that", "$V(I_\\lambda S_\\lambda + J S_\\lambda) \\subset \\bigcup D(g_{j, \\lambda})$. By", "Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "we see that for some sufficiently large $\\lambda$ the modules", "$(M_\\lambda)_{g_{j, \\lambda}}$ are flat over $R_\\lambda$.", "In particular the module $M_\\lambda$ is flat over $R_\\lambda$", "at all the primes corresponding to points of", "$V(I_\\lambda S_\\lambda + J S_\\lambda)$.", "\\medskip\\noindent", "In the case that $S$ is essentially of finite presentation, we can write", "$S = \\Sigma^{-1}C$ where $R \\to C$ is of finite presentation and", "$\\Sigma \\subset C$ is a multiplicative subset. We can also write", "$M = \\Sigma^{-1}N$ for some finitely presented $C$-module $N$, see", "Algebra, Lemma \\ref{algebra-lemma-construct-fp-module}.", "At this point we introduce $C_\\lambda$, $C'$, $N_\\lambda$, $N'$. Then in", "the discussion above we obtain an open $U' \\subset \\Spec(C')$", "over which $N'$ is flat over $R'$. The assumption that", "(\\ref{equation-flat-at-primes}) is true means that $V(I'S' + JS')$ maps", "into $U'$, because for a prime $\\mathfrak q' \\subset S'$, corresponding", "to a prime $\\mathfrak r' \\subset C'$ we have", "$M'_{\\mathfrak q'} = N'_{\\mathfrak r'}$. Thus we can find", "$g'_j \\in C'$ such that $\\bigcup D(g'_j)$ contains the image of", "$V(I'S' + JS')$. The rest of the proof is exactly the same as before." ], "refs": [ "algebra-lemma-base-change-finiteness", "algebra-theorem-openness-flatness", "algebra-lemma-flat-finite-presentation-limit-flat", "algebra-lemma-construct-fp-module" ], "ref_ids": [ 373, 326, 1389, 1081 ] } ], "ref_ids": [] }, { "id": 9902, "type": "theorem", "label": "more-algebra-lemma-flat-module-powers-variant", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-module-powers-variant", "contents": [ "In Situation \\ref{situation-flattening-general}.", "Let $I \\subset R$ be an ideal. Assume", "\\begin{enumerate}", "\\item $R$ is a Noetherian ring,", "\\item $S$ is a Noetherian ring,", "\\item $M$ is a finite $S$-module, and", "\\item for each $n \\geq 1$ and any prime", "$\\mathfrak q \\in V(J + IS)$ the module $(M/I^n M)_{\\mathfrak q}$", "is flat over $R/I^n$.", "\\end{enumerate}", "Then (\\ref{equation-flat-at-primes}) holds for $(R, I)$, i.e.,", "for every prime $\\mathfrak q \\in V(J + IS)$", "the localization $M_{\\mathfrak q}$ is flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q \\in V(J + IS)$. Then", "Algebra, Lemma \\ref{algebra-lemma-flat-module-powers}", "applied to $R \\to S_{\\mathfrak q}$ and $M_{\\mathfrak q}$", "implies that $M_{\\mathfrak q}$ is flat over $R$." ], "refs": [ "algebra-lemma-flat-module-powers" ], "ref_ids": [ 893 ] } ], "ref_ids": [] }, { "id": 9903, "type": "theorem", "label": "more-algebra-lemma-flattening-complete-local-noetherian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flattening-complete-local-noetherian", "contents": [ "Let $R \\to S$ be a ring map.", "Let $M$ be an $S$-module.", "Assume", "\\begin{enumerate}", "\\item $(R, \\mathfrak m)$ is a complete local Noetherian ring,", "\\item $S$ is a Noetherian ring, and", "\\item $M$ is finite over $S$.", "\\end{enumerate}", "Then there exists an ideal $I \\subset \\mathfrak m$ such that", "\\begin{enumerate}", "\\item $(M/IM)_{\\mathfrak q}$ is flat over $R/I$ for all", "primes $\\mathfrak q$ of $S/IS$ lying over $\\mathfrak m$, and", "\\item if $J \\subset R$ is an ideal such that $(M/JM)_{\\mathfrak q}$", "is flat over $R/J$ for all primes $\\mathfrak q$ lying over", "$\\mathfrak m$, then $I \\subset J$.", "\\end{enumerate}", "In other words, $I$ is the smallest ideal of $R$ such that", "(\\ref{equation-flat-at-primes-over}) holds for", "$(\\overline{R} \\to \\overline{S}, \\overline{\\mathfrak m}, \\overline{M})$", "where $\\overline{R} = R/I$, $\\overline{S} = S/IS$,", "$\\overline{\\mathfrak m} = \\mathfrak m/I$ and $\\overline{M} = M/IM$." ], "refs": [], "proofs": [ { "contents": [ "Let $J \\subset R$ be an ideal. Apply", "Algebra, Lemma \\ref{algebra-lemma-flat-module-powers}", "to the module $M/JM$ over the ring $R/J$.", "Then we see that $(M/JM)_{\\mathfrak q}$", "is flat over $R/J$ for all primes $\\mathfrak q$ of $S/JS$", "if and only if $M/(J + \\mathfrak m^n)M$ is flat over", "$R/(J + \\mathfrak m^n)$ for all $n \\geq 1$.", "We will use this remark below.", "\\medskip\\noindent", "For every $n \\geq 1$ the local ring $R/\\mathfrak m^n$ is Artinian.", "Hence, by", "Lemma \\ref{lemma-flattening-artinian}", "there exists a smallest ideal $I_n \\supset \\mathfrak m^n$ such that", "$M/I_nM$ is flat over $R/I_n$. It is clear that $I_{n + 1} + \\mathfrak m^n$", "is contains $I_n$ and applying", "Lemma \\ref{lemma-intersection-flat}", "we see that $I_n = I_{n + 1} + \\mathfrak m^n$. Since", "$R = \\lim_n\\ R/\\mathfrak m^n$ we see that $I = \\lim_n\\ I_n/\\mathfrak m^n$", "is an ideal in $R$ such that $I_n = I + \\mathfrak m^n$ for all $n \\geq 1$.", "By the initial remarks of the proof we see that $I$ verifies (1)", "and (2). Some details omitted." ], "refs": [ "algebra-lemma-flat-module-powers", "more-algebra-lemma-flattening-artinian", "more-algebra-lemma-intersection-flat" ], "ref_ids": [ 893, 9894, 9893 ] } ], "ref_ids": [] }, { "id": 9904, "type": "theorem", "label": "more-algebra-lemma-flattening-complete-local-noetherian-property-by-finite-type", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flattening-complete-local-noetherian-property-by-finite-type", "contents": [ "With notation $R \\to S$, $M$, and $I$ and assumptions as in", "Lemma \\ref{lemma-flattening-complete-local-noetherian}.", "Consider a local homomorphism of local rings", "$\\varphi : (R, \\mathfrak m) \\to (R', \\mathfrak m')$", "such that $R'$ is Noetherian. Then the following are equivalent", "\\begin{enumerate}", "\\item condition (\\ref{equation-flat-at-primes-over}) holds", "for $(R' \\to S \\otimes_R R', \\mathfrak m', M \\otimes_R R')$, and", "\\item $\\varphi(I) = 0$.", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-flattening-complete-local-noetherian" ], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) follows from", "Lemma \\ref{lemma-base-change-flat-at-primes-over}.", "Let $\\varphi : R \\to R'$ be as in the lemma satisfying (1).", "We have to show that $\\varphi(I) = 0$.", "This is equivalent to the condition that $\\varphi(I)R' = 0$.", "By Artin-Rees in the Noetherian local ring $R'$ (see", "Algebra, Lemma \\ref{algebra-lemma-intersect-powers-ideal-module-zero})", "this is equivalent to the condition that", "$\\varphi(I)R' + (\\mathfrak m')^n = (\\mathfrak m')^n$ for all $n > 0$.", "Hence this is equivalent to the condition that the composition", "$\\varphi_n : R \\to R' \\to R'/(\\mathfrak m')^n$ annihilates $I$ for each $n$.", "Now assumption (1) for $\\varphi$ implies assumption (1) for", "$\\varphi_n$ by", "Lemma \\ref{lemma-base-change-flat-at-primes-over}.", "This reduces us to the case where $R'$ is Artinian local.", "\\medskip\\noindent", "Assume $R'$ Artinian. Let $J = \\Ker(\\varphi)$. We have to show that", "$I \\subset J$. By the construction of $I$ in", "Lemma \\ref{lemma-flattening-complete-local-noetherian}", "it suffices to show that $(M/JM)_{\\mathfrak q}$ is flat over $R/J$", "for every prime $\\mathfrak q$ of $S/JS$ lying over $\\mathfrak m$.", "As $R'$ is Artinian, condition (1) signifies that $M \\otimes_R R'$", "is flat over $R'$. As $R'$ is Artinian and $R/J \\to R'$ is a local", "injective ring map, it follows that $R/J$ is Artinian", "too. Hence the flatness of $M \\otimes_R R' = M/JM \\otimes_{R/J} R'$ over", "$R'$ implies that $M/JM$ is flat over $R/J$ by", "Algebra,", "Lemma \\ref{algebra-lemma-descent-flatness-injective-map-artinian-rings}.", "This concludes the proof." ], "refs": [ "more-algebra-lemma-base-change-flat-at-primes-over", "algebra-lemma-intersect-powers-ideal-module-zero", "more-algebra-lemma-base-change-flat-at-primes-over", "more-algebra-lemma-flattening-complete-local-noetherian", "algebra-lemma-descent-flatness-injective-map-artinian-rings" ], "ref_ids": [ 9896, 627, 9896, 9903, 906 ] } ], "ref_ids": [ 9903 ] }, { "id": 9905, "type": "theorem", "label": "more-algebra-lemma-flattening-complete-local-universal-property", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flattening-complete-local-universal-property", "contents": [ "With notation $R \\to S$, $M$, and $I$ and assumptions as in", "Lemma \\ref{lemma-flattening-complete-local-noetherian}.", "In addition assume that $R \\to S$ is of finite type.", "Then for any local homomorphism of local rings", "$\\varphi : (R, \\mathfrak m) \\to (R', \\mathfrak m')$", "the following are equivalent", "\\begin{enumerate}", "\\item condition (\\ref{equation-flat-at-primes-over}) holds", "for $(R' \\to S \\otimes_R R', \\mathfrak m', M \\otimes_R R')$, and", "\\item $\\varphi(I) = 0$.", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-flattening-complete-local-noetherian" ], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) follows from", "Lemma \\ref{lemma-base-change-flat-at-primes-over}.", "Let $\\varphi : R \\to R'$ be as in the lemma satisfying (1).", "As $R$ is Noetherian we see that $R \\to S$ is of finite presentation", "and $M$ is an $S$-module of finite presentation.", "Write $R' = \\colim_\\lambda R_\\lambda$", "as a directed colimit of local $R$-subalgebras $R_\\lambda \\subset R'$,", "with maximal ideals $\\mathfrak m_\\lambda = R_\\lambda \\cap \\mathfrak m'$", "such that each $R_\\lambda$ is essentially of finite type over $R$. By", "Lemma \\ref{lemma-limit-preserving-flat-at-primes-over}", "we see that condition (\\ref{equation-flat-at-primes-over}) holds for", "$(R_\\lambda \\to S \\otimes_R R_\\lambda, \\mathfrak m_\\lambda,", "M \\otimes_R R_\\lambda)$ for some $\\lambda$. Hence", "Lemma \\ref{lemma-flattening-complete-local-noetherian-property-by-finite-type}", "applies to the ring map $R \\to R_\\lambda$ and we see that", "$I$ maps to zero in $R_\\lambda$, a fortiori it maps to zero in $R'$." ], "refs": [ "more-algebra-lemma-base-change-flat-at-primes-over", "more-algebra-lemma-limit-preserving-flat-at-primes-over", "more-algebra-lemma-flattening-complete-local-noetherian-property-by-finite-type" ], "ref_ids": [ 9896, 9898, 9904 ] } ], "ref_ids": [ 9903 ] }, { "id": 9906, "type": "theorem", "label": "more-algebra-lemma-have-one-root", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-have-one-root", "contents": [ "Let $R$ be a ring. Let $P(T)$ be a monic polynomial with coefficients", "in $R$. Let $\\alpha \\in R$ be such that $P(\\alpha) = 0$. Then", "$P(T) = (T - \\alpha)Q(T)$ for some monic polynomial $Q(T) \\in R[T]$." ], "refs": [], "proofs": [ { "contents": [ "By induction on the degree of $P$. If $\\deg(P) = 1$, then", "$P(T) = T - \\alpha$ and the result is true. If $\\deg(P) > 1$, then", "we can write $P(T) = (T - \\alpha)Q(T) + r$ for some polynomial", "$Q \\in R[T]$ of degree $< \\deg(P)$ and some $r \\in R$ by long", "division. By assumption $0 = P(\\alpha) = (\\alpha - \\alpha)Q(\\alpha) + r = r$", "and we conclude that $r = 0$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9907, "type": "theorem", "label": "more-algebra-lemma-adjoin-one-root", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-adjoin-one-root", "contents": [ "Let $R$ be a ring. Let $P(T)$ be a monic polynomial with coefficients", "in $R$. There exists a finite free ring map $R \\to R'$ such that", "$P(T) = (T - \\alpha)Q(T)$ for some $\\alpha \\in R'$ and some", "monic polynomial $Q(T) \\in R'[T]$." ], "refs": [], "proofs": [ { "contents": [ "Write $P(T) = T^d + a_1T^{d - 1} + \\ldots + a_0$.", "Set $R' = R[x]/(x^d + a_1x^{d - 1} + \\ldots + a_0)$.", "Set $\\alpha$ equal to the congruence class of $x$.", "Then it is clear that $P(\\alpha) = 0$. Thus we win by", "Lemma \\ref{lemma-have-one-root}." ], "refs": [ "more-algebra-lemma-have-one-root" ], "ref_ids": [ 9906 ] } ], "ref_ids": [] }, { "id": 9908, "type": "theorem", "label": "more-algebra-lemma-finite-split", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finite-split", "contents": [ "Let $R \\to S$ be a finite ring map.", "There exists a finite free ring extension $R \\subset R'$ such", "that $S \\otimes_R R'$ is a quotient of a ring of the form", "$$", "R'[T_1, \\ldots, T_n]/(P_1(T_1), \\ldots, P_n(T_n))", "$$", "with $P_i(T) = \\prod_{j = 1, \\ldots, d_i} (T - \\alpha_{ij})$ for some", "$\\alpha_{ij} \\in R'$." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1, \\ldots, x_n \\in S$ be generators of $S$ over $R$.", "For each $i$ we can choose a monic polynomial $P_i(T) \\in R[T]$", "such that $P(x_i) = 0$ in $S$, see", "Algebra, Lemma \\ref{algebra-lemma-finite-is-integral}.", "Say $\\deg(P_i) = d_i$. By", "Lemma \\ref{lemma-adjoin-one-root}", "(applied $\\sum d_i$ times) there exists a finite free ring", "extension $R \\subset R'$ such that each $P_i$ splits completely:", "$$", "P_i(T) = \\prod\\nolimits_{j = 1, \\ldots, d_i} (T - \\alpha_{ij})", "$$", "for certain $\\alpha_{ik} \\in R'$. Let", "$R'[T_1, \\ldots, T_n] \\to S \\otimes_R R'$ be the $R'$-algebra map", "which maps $T_i$ to $x_i \\otimes 1$. As this maps $P_i(T_i)$ to zero,", "this induces the desired surjection." ], "refs": [ "algebra-lemma-finite-is-integral", "more-algebra-lemma-adjoin-one-root" ], "ref_ids": [ 482, 9907 ] } ], "ref_ids": [] }, { "id": 9909, "type": "theorem", "label": "more-algebra-lemma-split-image", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-split-image", "contents": [ "Let $R$ be a ring.", "Let $S = R[T_1, \\ldots, T_n]/J$.", "Assume $J$ contains elements of the form $P_i(T_i)$", "with $P_i(T) = \\prod_{j = 1, \\ldots, d_i} (T - \\alpha_{ij})$ for some", "$\\alpha_{ij} \\in R$. For $\\underline{k} = (k_1, \\ldots, k_n)$", "with $1 \\leq k_i \\leq d_i$ consider the ring map", "$$", "\\Phi_{\\underline{k}} : R[T_1, \\ldots, T_n] \\to R,", "\\quad", "T_i \\longmapsto \\alpha_{ik_i}", "$$", "Set $J_{\\underline{k}} = \\Phi_{\\underline{k}}(J)$.", "Then the image of $\\Spec(S) \\to \\Spec(R)$ is equal to", "$V(\\bigcap J_{\\underline{k}})$." ], "refs": [], "proofs": [ { "contents": [ "This lemma proves itself. Hint:", "$V(\\bigcap J_{\\underline{k}}) = \\bigcup V(J_{\\underline{k}})$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9910, "type": "theorem", "label": "more-algebra-lemma-descent-flatness-injective-finite-Noetherian-rings", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-descent-flatness-injective-finite-Noetherian-rings", "contents": [ "Let $R \\to S$ be a finite injective homomorphism of Noetherian rings.", "Let $M$ be an $R$-module. If $M \\otimes_R S$ is a flat $S$-module,", "then $M$ is a flat $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be an $R$-module such that $M \\otimes_R S$ is flat over $S$. By", "Algebra, Lemma \\ref{algebra-lemma-flatness-descends}", "in order to prove that $M$ is flat we may replace $R$ by any faithfully flat", "ring extension. By", "Lemma \\ref{lemma-finite-split}", "we can find a finite locally free ring extension $R \\subset R'$ such", "that $S' = S \\otimes_R R' = R'[T_1, \\ldots, T_n]/J$ for some ideal", "$J \\subset R'[T_1, \\ldots, T_n]$ which contains the elements of the form", "$P_i(T_i)$ with $P_i(T) = \\prod_{j = 1, \\ldots, d_i} (T - \\alpha_{ij})$", "for some $\\alpha_{ij} \\in R'$. Note that $R'$ is Noetherian", "and that $R' \\subset S'$ is a finite extension of rings. Hence we may", "replace $R$ by $R'$ and assume that $S$ has a presentation as in", "Lemma \\ref{lemma-split-image}.", "Note that $\\Spec(S) \\to \\Spec(R)$ is surjective, see", "Algebra, Lemma \\ref{algebra-lemma-integral-overring-surjective}.", "Thus, using", "Lemma \\ref{lemma-split-image}", "we conclude that $I = \\bigcap J_{\\underline{k}}$ is an ideal", "such that $V(I) = \\Spec(R)$. This means that", "$I \\subset \\sqrt{(0)}$, and since $R$ is Noetherian that $I$", "is nilpotent. The maps $\\Phi_{\\underline{k}}$ induce commutative", "diagrams", "$$", "\\xymatrix{", "S \\ar[rr] & & R/J_{\\underline{k}} \\\\", "& R \\ar[lu] \\ar[ru]", "}", "$$", "from which we conclude that $M/J_{\\underline{k}}M$ is flat over", "$R/J_{\\underline{k}}$. By", "Lemma \\ref{lemma-intersection-flat}", "we see that $M/IM$ is flat over $R/I$. Finally, applying", "Algebra, Lemma \\ref{algebra-lemma-lift-flatness}", "we conclude that $M$ is flat over $R$." ], "refs": [ "algebra-lemma-flatness-descends", "more-algebra-lemma-finite-split", "more-algebra-lemma-split-image", "algebra-lemma-integral-overring-surjective", "more-algebra-lemma-split-image", "more-algebra-lemma-intersection-flat", "algebra-lemma-lift-flatness" ], "ref_ids": [ 528, 9908, 9909, 495, 9909, 9893, 904 ] } ], "ref_ids": [] }, { "id": 9911, "type": "theorem", "label": "more-algebra-lemma-descent-flatness-injective-integral", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-descent-flatness-injective-integral", "contents": [ "Let $R \\to S$ be an injective integral ring map.", "Let $M$ be a finitely presented module over $R[x_1, \\ldots, x_n]$.", "If $M \\otimes_R S$ is flat over $S$, then $M$ is flat over $R$." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation", "$$", "R[x_1, \\ldots, x_n]^{\\oplus t} \\to R[x_1, \\ldots, x_n]^{\\oplus r} \\to M \\to 0.", "$$", "Let's say that the first map is given by the $r \\times t$-matrix", "$T = (f_{ij})$ with $f_{ij} \\in R[x_1, \\ldots, x_n]$. Write", "$f_{ij} = \\sum f_{ij, I} x^I$ with $f_{ij, I} \\in R$ (multi-index notation).", "Consider diagrams", "$$", "\\xymatrix{", "R \\ar[r] & S \\\\", "R_\\lambda \\ar[u] \\ar[r] & S_\\lambda \\ar[u]", "}", "$$", "where $R_\\lambda$ is a finitely generated $\\mathbf{Z}$-subalgebra of", "$R$ containing all $f_{ij, I}$ and $S_\\lambda$ is a finite", "$R_\\lambda$-subalgebra of $S$. Let $M_\\lambda$ be the finite", "$R_\\lambda[x_1, \\ldots, x_n]$-module defined by a presentation", "as above, using the same matrix $T$ but now viewed as a matrix", "over $R_\\lambda[x_1, \\ldots, x_n]$. Note that $S$ is the directed colimit", "of the $S_\\lambda$ (details omitted). By", "Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "we see that for some $\\lambda$ the module", "$M_\\lambda \\otimes_{R_\\lambda} S_\\lambda$ is flat over $S_\\lambda$. By", "Lemma \\ref{lemma-descent-flatness-injective-finite-Noetherian-rings}", "we conclude that $M_\\lambda$ is flat over $R_\\lambda$. Since", "$M = M_\\lambda \\otimes_{R_\\lambda} R$ we win by", "Algebra, Lemma \\ref{algebra-lemma-flat-base-change}." ], "refs": [ "algebra-lemma-flat-finite-presentation-limit-flat", "more-algebra-lemma-descent-flatness-injective-finite-Noetherian-rings", "algebra-lemma-flat-base-change" ], "ref_ids": [ 1389, 9910, 527 ] } ], "ref_ids": [] }, { "id": 9912, "type": "theorem", "label": "more-algebra-lemma-torsion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-torsion", "contents": [ "Let $R$ be a domain. Let $M$ be an $R$-module.", "The set of torsion elements of $M$ forms a submodule $M_{tors} \\subset M$.", "The quotient module $M/M_{tors}$ is torsion free." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9913, "type": "theorem", "label": "more-algebra-lemma-localize-torsion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-localize-torsion", "contents": [ "Let $R$ be a domain. Let $M$ be a torsion free $R$-module.", "For any multiplicative set $S \\subset R$ the module", "$S^{-1}M$ is a torsion free $S^{-1}R$-module." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9914, "type": "theorem", "label": "more-algebra-lemma-flat-pullback-torsion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-pullback-torsion", "contents": [ "Let $R \\to R'$ be a flat homomorphism of domains. If $M$ is a torsion", "free $R$-module, then $M \\otimes_R R'$ is a torsion free $R'$-module." ], "refs": [], "proofs": [ { "contents": [ "If $M$ is torsion free, then $M \\subset M \\otimes_R K$ is injective", "where $K$ is the fraction field of $R$. Since $R'$ is flat over $R$", "we see that $M \\otimes_R R' \\to (M \\otimes_R K) \\otimes_R R'$ is injective.", "Since $M \\otimes_R K$ is isomorphic to a direct sum of copies of $K$,", "it suffices to see that $K \\otimes_R R'$ is torsion free. This is true", "because it is a localization of $R'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9915, "type": "theorem", "label": "more-algebra-lemma-extension-torsion-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-extension-torsion-free", "contents": [ "Let $R$ be a domain. Let $0 \\to M \\to M' \\to M'' \\to 0$", "be a short exact sequence of $R$-modules. If $M$ and $M''$", "are torsion free, then $M'$ is torsion free." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9916, "type": "theorem", "label": "more-algebra-lemma-check-torsion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-check-torsion", "contents": [ "Let $R$ be a domain. Let $M$ be an $R$-module.", "Then $M$ is torsion free if and only if $M_\\mathfrak m$ is a", "torsion free $R_\\mathfrak m$-module for all maximal ideals", "$\\mathfrak m$ of $R$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use Lemma \\ref{lemma-localize-torsion} and", "Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}." ], "refs": [ "more-algebra-lemma-localize-torsion", "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 9913, 410 ] } ], "ref_ids": [] }, { "id": 9917, "type": "theorem", "label": "more-algebra-lemma-finite-torsion-free-submodule-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finite-torsion-free-submodule-free", "contents": [ "Let $R$ be a domain. Let $M$ be a finite $R$-module.", "Then $M$ is torsion free if and only if $M$ is a", "submodule of a finite free module." ], "refs": [], "proofs": [ { "contents": [ "If $M$ is a submodule of $R^{\\oplus n}$, then $M$ is torsion free.", "For the converse, assume $M$ is torsion free. Let $K$ be the", "fraction field of $R$. Then $M \\otimes_R K$ is a finite dimensional", "$K$-vector space. Choose a basis $e_1, \\ldots, e_r$ for this vector", "space. Let $x_1, \\ldots, x_n$ be generators of $M$. Write", "$x_i = \\sum (a_{ij}/b_{ij}) e_j$ for some $a_{ij}, b_{ij} \\in R$", "with $b_{ij} \\not = 0$. Set $b = \\prod_{i, j} b_{ij}$.", "Since $M$ is torsion free the map", "$M \\to M \\otimes_R K$ is injective and the image is contained", "in $R^{\\oplus r} = R e_1/b \\oplus \\ldots \\oplus Re_r/b$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9918, "type": "theorem", "label": "more-algebra-lemma-torsion-free-finite-noetherian-domain", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-torsion-free-finite-noetherian-domain", "contents": [ "Let $R$ be a Noetherian domain. Let $M$ be a nonzero finite $R$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $M$ is torsion free,", "\\item $M$ is a submodule of a finite free module,", "\\item $(0)$ is the only associated prime of $M$,", "\\item $(0)$ is in the support of $M$ and $M$ has property $(S_1)$, and", "\\item $(0)$ is in the support of $M$ and $M$ has no embedded associated prime.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have seen the equivalence of (1) and (2) in", "Lemma \\ref{lemma-finite-torsion-free-submodule-free}.", "We have seen the equivalence of (4) and (5) in", "Algebra, Lemma \\ref{algebra-lemma-criterion-no-embedded-primes}.", "The equivalence between (3) and (5) is immediate from the definition.", "A localization of a torsion free module is torsion free", "(Lemma \\ref{lemma-localize-torsion}), hence it is clear that a", "$M$ has no associated primes different from $(0)$. Thus (1)", "implies (5). Conversely, assume (5). If $M$ has torsion,", "then there exists an embedding $R/I \\subset M$ for some nonzero", "ideal $I$ of $R$. Hence $M$ has an associated prime different", "from $(0)$", "(see Algebra, Lemmas \\ref{algebra-lemma-ass} and \\ref{algebra-lemma-ass-zero}).", "This is an embedded associated prime which contradicts the assumption." ], "refs": [ "more-algebra-lemma-finite-torsion-free-submodule-free", "algebra-lemma-criterion-no-embedded-primes", "more-algebra-lemma-localize-torsion", "algebra-lemma-ass", "algebra-lemma-ass-zero" ], "ref_ids": [ 9917, 1309, 9913, 699, 702 ] } ], "ref_ids": [] }, { "id": 9919, "type": "theorem", "label": "more-algebra-lemma-flat-torsion-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-torsion-free", "contents": [ "Let $R$ be a domain. Any flat $R$-module is torsion free." ], "refs": [], "proofs": [ { "contents": [ "If $x \\in R$ is nonzero, then $x : R \\to R$ is injective, and hence if $M$", "is flat over $R$, then $x : M \\to M$ is injective. Thus if $M$ is flat over", "$R$, then $M$ is torsion free." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9920, "type": "theorem", "label": "more-algebra-lemma-valuation-ring-torsion-free-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-valuation-ring-torsion-free-flat", "contents": [ "Let $A$ be a valuation ring.", "An $A$-module $M$ is flat over $A$ if and only if $M$ is torsion free." ], "refs": [], "proofs": [ { "contents": [ "The implication ``flat $\\Rightarrow$ torsion free'' is", "Lemma \\ref{lemma-flat-torsion-free}.", "For the converse, assume $M$ is torsion free.", "By the equational criterion of flatness (see", "Algebra, Lemma \\ref{algebra-lemma-flat-eq})", "we have to show that every relation in $M$ is trivial. To do this assume that", "$\\sum_{i = 1, \\ldots, n} a_i x_i = 0$ with $x_i \\in M$ and $a_i \\in A$.", "After renumbering we may assume that $v(a_1) \\leq v(a_i)$ for all $i$.", "Hence we can write $a_i = a'_i a_1$ for some $a'_i \\in A$. Note that", "$a'_1 = 1$. As $M$ is torsion free we see that", "$x_1 = - \\sum_{i \\geq 2} a'_i x_i$. Thus, if we choose", "$y_i = x_i$, $i = 2, \\ldots, n$ then", "$$", "x_1 = \\sum\\nolimits_{j \\geq 2} -a'_j y_j, \\quad", "x_i = y_i, (i \\geq 2)\\quad", "0 = a_1 \\cdot (-a'_j) + a_j \\cdot 1 (j \\geq 2)", "$$", "shows that the relation was trivial (to be explicit the elements", "$a_{ij}$ are defined by setting $a_{11} = 0$, $a_{1j} = -a'_j$", "for $j > 1$, and $a_{ij} = \\delta_{ij}$ for $i, j \\geq 2$)." ], "refs": [ "more-algebra-lemma-flat-torsion-free", "algebra-lemma-flat-eq" ], "ref_ids": [ 9919, 531 ] } ], "ref_ids": [] }, { "id": 9921, "type": "theorem", "label": "more-algebra-lemma-dedekind-torsion-free-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-dedekind-torsion-free-flat", "contents": [ "Let $A$ be a Dedekind domain (for example a discrete valuation ring", "or more generally a PID).", "\\begin{enumerate}", "\\item An $A$-module is flat if and only if it is torsion free.", "\\item A finite torsion free $A$-module is finite locally free.", "\\item A finite torsion free $A$-module is finite free if", "$A$ is a PID.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Since a PID is a Dedekind domain", "(Algebra, Lemma \\ref{algebra-lemma-PID-dedekind}),", "it suffices to prove this for Dedekind domains.", "By Lemma \\ref{lemma-check-torsion}", "and Algebra, Lemma \\ref{algebra-lemma-flat-localization}", "it suffices to check the statement over $A_\\mathfrak m$", "for $\\mathfrak m \\subset A$ maximal. Since $A_\\mathfrak m$", "is a discrete valuation ring", "(Algebra, Lemma \\ref{algebra-lemma-characterize-Dedekind})", "we win by Lemma \\ref{lemma-valuation-ring-torsion-free-flat}.", "\\medskip\\noindent", "Proof of (2). Follows from", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}", "and (1).", "\\medskip\\noindent", "Proof of (3). Let $A$ be a PID and let $M$ be a finite", "torsion free module. By Lemma \\ref{lemma-finite-torsion-free-submodule-free}", "we see that $M \\subset A^{\\oplus n}$ for some $n$. We argue that", "$M$ is free by induction on $n$. The case $n = 1$ expresses exactly the", "fact that $A$ is a PID. If $n > 1$ let $M' \\subset R^{\\oplus n - 1}$", "be the image of the projection onto the last $n - 1$ summands of", "$R^{\\oplus n}$. Then we obtain a short exact sequence", "$0 \\to I \\to M \\to M' \\to 0$ where $I$ is the intersection of $M$", "with the first summand $R$ of $R^{\\oplus n}$. By induction we", "see that $M$ is an extension of finite free $R$-modules, whence", "finite free." ], "refs": [ "algebra-lemma-PID-dedekind", "more-algebra-lemma-check-torsion", "algebra-lemma-flat-localization", "algebra-lemma-characterize-Dedekind", "more-algebra-lemma-valuation-ring-torsion-free-flat", "algebra-lemma-finite-projective", "more-algebra-lemma-finite-torsion-free-submodule-free" ], "ref_ids": [ 1039, 9916, 538, 1041, 9920, 795, 9917 ] } ], "ref_ids": [] }, { "id": 9922, "type": "theorem", "label": "more-algebra-lemma-hom-into-torsion-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-hom-into-torsion-free", "contents": [ "Let $R$ be a domain. Let $M$, $N$ be $R$-modules.", "If $N$ is torsion free, so is $\\Hom_R(M, N)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjection $\\bigoplus_{i \\in I} R \\to M$.", "Then $\\Hom_R(M, N) \\subset \\prod_{i \\in I} N$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9923, "type": "theorem", "label": "more-algebra-lemma-reflexive-torsion-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-reflexive-torsion-free", "contents": [ "Let $R$ be a domain and let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item If $M$ is reflexive, then $M$ is torsion free.", "\\item If $M$ is finite, then $j : M \\to \\Hom_R(\\Hom_R(M, R), R)$ is injective", "if and only if $M$ is torsion free", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemmas \\ref{lemma-hom-into-torsion-free} and", "\\ref{lemma-finite-torsion-free-submodule-free}." ], "refs": [ "more-algebra-lemma-hom-into-torsion-free", "more-algebra-lemma-finite-torsion-free-submodule-free" ], "ref_ids": [ 9922, 9917 ] } ], "ref_ids": [] }, { "id": 9924, "type": "theorem", "label": "more-algebra-lemma-cokernel-map-double-dual-dvr", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cokernel-map-double-dual-dvr", "contents": [ "Let $R$ be a discrete valuation ring and let $M$ be a finite $R$-module.", "Then the map $j : M \\to \\Hom_R(\\Hom_R(M, R), R)$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $M_{tors} \\subset M$ be the torsion submodule. Then we have", "$\\Hom_R(M, R) = \\Hom_R(M/M_{tors}, R)$ (holds over any domain).", "Hence we may assume that $M$ is torsion free. Then $M$ is free", "by Lemma \\ref{lemma-dedekind-torsion-free-flat} and the lemma is clear." ], "refs": [ "more-algebra-lemma-dedekind-torsion-free-flat" ], "ref_ids": [ 9921 ] } ], "ref_ids": [] }, { "id": 9925, "type": "theorem", "label": "more-algebra-lemma-check-reflexive", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-check-reflexive", "contents": [ "Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module.", "The following are equivalent:", "\\begin{enumerate}", "\\item $M$ is reflexive,", "\\item $M_\\mathfrak p$ is a reflexive $R_\\mathfrak p$-module", "for all primes $\\mathfrak p \\subset R$, and", "\\item $M_\\mathfrak m$ is a reflexive $R_\\mathfrak m$-module", "for all maximal ideals $\\mathfrak m$ of $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The localization of $j : M \\to \\Hom_R(\\Hom_R(M, R), R)$", "at a prime $\\mathfrak p$ is the corresponding map for the module", "$M_\\mathfrak p$ over the Noetherian local domain $R_\\mathfrak p$.", "See Algebra, Lemma \\ref{algebra-lemma-hom-from-finitely-presented}.", "Thus the lemma holds by", "Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}." ], "refs": [ "algebra-lemma-hom-from-finitely-presented", "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 353, 410 ] } ], "ref_ids": [] }, { "id": 9926, "type": "theorem", "label": "more-algebra-lemma-sequence-reflexive", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-sequence-reflexive", "contents": [ "Let $R$ be a Noetherian domain. Let $0 \\to M \\to M' \\to M''$", "an exact sequence of finite $R$-modules. If $M'$ is reflexive", "and $M''$ is torsion free, then $M$ is reflexive." ], "refs": [], "proofs": [ { "contents": [ "We will use without further mention that $\\Hom_R(N, N')$ is a finite", "$R$-module for any finite $R$-modules $N$ and $N'$, see", "Algebra, Lemma \\ref{algebra-lemma-ext-noetherian}.", "We take duals to get a sequence", "$$", "\\Hom_R(M, R) \\leftarrow \\Hom_R(M', R) \\leftarrow \\Hom_R(M'', R)", "$$", "Dualizing again we obtain a commutative diagram", "$$", "\\xymatrix{", "\\Hom_R(\\Hom_R(M, R), R) \\ar[r]_j &", "\\Hom_R(\\Hom_R(M', R), R) \\ar[r] &", "\\Hom_R(\\Hom_R(M'', R), R) \\\\", "M \\ar[u] \\ar[r] & M' \\ar[u] \\ar[r] & M'' \\ar[u]", "}", "$$", "We do not know the top row is exact. But, by assumption", "the middle vertical arrow is an isomorphism and the right", "vertical arrow is injective (Lemma \\ref{lemma-reflexive-torsion-free}).", "We claim $j$ is injective. Assuming the claim", "a diagram chase shows that the left vertical", "arrow is an isomorphism, i.e., $M$ is reflexive.", "\\medskip\\noindent", "Proof of the claim. Consider the exact sequence", "$\\Hom_R(M',R)\\to \\Hom_R(M,R)\\to Q \\to 0$", "defining $Q$. One applies", "Algebra, Lemma \\ref{algebra-lemma-hom-from-finitely-presented}", "to obtain", "$$", "\\Hom_K(M'\\otimes_R K,K) \\to", "\\Hom_K(M\\otimes_R K,K) \\to", "Q\\otimes_R K\\to 0", "$$", "But $M \\otimes_R K \\to M' \\otimes_R K$ is an injective map of vector spaces,", "hence split injective, so $Q \\otimes_R K = 0$, that is, $Q$ is torsion.", "Then one gets the exact sequence ", "$$", "0 \\to \\Hom_R(Q,R) \\to", "\\Hom_R(\\Hom_R(M,R),R) \\to", "\\Hom_R(\\Hom_R(M',R),R)", "$$", "and $\\Hom_R(Q,R)=0$ because $Q$ is torsion." ], "refs": [ "algebra-lemma-ext-noetherian", "more-algebra-lemma-reflexive-torsion-free", "algebra-lemma-hom-from-finitely-presented" ], "ref_ids": [ 768, 9923, 353 ] } ], "ref_ids": [] }, { "id": 9927, "type": "theorem", "label": "more-algebra-lemma-characterize-reflexive", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-characterize-reflexive", "contents": [ "Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $M$ is reflexive,", "\\item there exists a short exact sequence $0 \\to M \\to F \\to N \\to 0$", "with $F$ finite free and $N$ torsion free.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that a finite free module is reflexive.", "By Lemma \\ref{lemma-sequence-reflexive} we see", "that (2) implies (1). Assume $M$ is reflexive. Choose a presentation", "$R^{\\oplus m} \\to R^{\\oplus n} \\to \\Hom_R(M, R) \\to 0$.", "Dualizing we get an exact sequence", "$$", "0 \\to \\Hom_R(\\Hom_R(M, R), R) \\to R^{\\oplus n} \\to N \\to 0", "$$", "with $N = \\Im(R^{\\oplus n} \\to R^{\\oplus m})$ a torsion free module.", "As $M = \\Hom_R(\\Hom_R(M, R), R)$ we get an exact sequence as in (2)." ], "refs": [ "more-algebra-lemma-sequence-reflexive" ], "ref_ids": [ 9926 ] } ], "ref_ids": [] }, { "id": 9928, "type": "theorem", "label": "more-algebra-lemma-flat-pullback-reflexive", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-pullback-reflexive", "contents": [ "Let $R \\to R'$ be a flat homomorphism of Noetherian domains.", "If $M$ is a finite reflexive $R$-module, then $M \\otimes_R R'$", "is a finite reflexive $R'$-module." ], "refs": [], "proofs": [ { "contents": [ "Choose a short exact sequence $0 \\to M \\to F \\to N \\to 0$", "with $F$ finite free and $N$ torsion free, see", "Lemma \\ref{lemma-characterize-reflexive}. Since $R \\to R'$ is flat", "we obtain a short exact sequence", "$0 \\to M \\otimes_R R' \\to F \\otimes_R R' \\to N \\otimes_R R' \\to 0$", "with $F \\otimes_R R'$ finite free and $N \\otimes_R R'$ torsion", "free (Lemma \\ref{lemma-flat-pullback-torsion}). Thus $M \\otimes_R R'$", "is reflexive by Lemma \\ref{lemma-characterize-reflexive}." ], "refs": [ "more-algebra-lemma-characterize-reflexive", "more-algebra-lemma-flat-pullback-torsion", "more-algebra-lemma-characterize-reflexive" ], "ref_ids": [ 9927, 9914, 9927 ] } ], "ref_ids": [] }, { "id": 9929, "type": "theorem", "label": "more-algebra-lemma-dual-reflexive", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-dual-reflexive", "contents": [ "Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module.", "Let $N$ be a finite reflexive $R$-module. Then $\\Hom_R(M, N)$ is reflexive." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $R^{\\oplus m} \\to R^{\\oplus n} \\to M \\to 0$.", "Then we obtain", "$$", "0 \\to \\Hom_R(M, N) \\to N^{\\oplus n} \\to N' \\to 0", "$$", "with $N' = \\Im(N^{\\oplus n} \\to N^{\\oplus m})$ torsion free.", "We conclude by Lemma \\ref{lemma-sequence-reflexive}." ], "refs": [ "more-algebra-lemma-sequence-reflexive" ], "ref_ids": [ 9926 ] } ], "ref_ids": [] }, { "id": 9930, "type": "theorem", "label": "more-algebra-lemma-hom-into-depth", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-hom-into-depth", "contents": [ "Let $R$ be a Noetherian local ring. Let $M$, $N$ be finite $R$-modules.", "\\begin{enumerate}", "\\item If $N$ has depth $\\geq 1$, then $\\Hom_R(M, N)$ has depth $\\geq 1$.", "\\item If $N$ has depth $\\geq 2$, then $\\Hom_R(M, N)$ has depth $\\geq 2$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $R^{\\oplus m} \\to R^{\\oplus n} \\to M \\to 0$.", "Dualizing we get an exact sequence", "$$", "0 \\to \\Hom_R(M, N) \\to N^{\\oplus n} \\to N' \\to 0", "$$", "with $N' = \\Im(N^{\\oplus n} \\to N^{\\oplus m})$. A submodule of a module", "with depth $\\geq 1$ has depth $\\geq 1$; this follows immediately from", "the definition. Thus part (1) is clear. For (2) note that here the", "assumption and the previous remark implies $N'$ has depth $\\geq 1$.", "The module $N^{\\oplus n}$ has depth $\\geq 2$.", "From Algebra, Lemma \\ref{algebra-lemma-depth-in-ses}", "we conclude $\\Hom_R(M, N)$ has depth $\\geq 2$." ], "refs": [ "algebra-lemma-depth-in-ses" ], "ref_ids": [ 773 ] } ], "ref_ids": [] }, { "id": 9931, "type": "theorem", "label": "more-algebra-lemma-hom-into-S2", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-hom-into-S2", "contents": [ "Let $R$ be a Noetherian ring. Let $M$, $N$ be finite $R$-modules.", "\\begin{enumerate}", "\\item If $N$ has property $(S_1)$, then $\\Hom_R(M, N)$ has property $(S_1)$.", "\\item If $N$ has property $(S_2)$, then $\\Hom_R(M, N)$ has property $(S_2)$.", "\\item If $R$ is a domain, $N$ is torsion free and $(S_2)$, then", "$\\Hom_R(M, N)$ is torsion free and has property $(S_2)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since localizing at primes commutes with taking $\\Hom_R$ for finite", "$R$-modules (Algebra, Lemma \\ref{algebra-lemma-ext-noetherian})", "parts (1) and (2) follow immediately from Lemma \\ref{lemma-hom-into-depth}.", "Part (3) follows from (2) and", "Lemma \\ref{lemma-hom-into-torsion-free}." ], "refs": [ "algebra-lemma-ext-noetherian", "more-algebra-lemma-hom-into-depth", "more-algebra-lemma-hom-into-torsion-free" ], "ref_ids": [ 768, 9930, 9922 ] } ], "ref_ids": [] }, { "id": 9932, "type": "theorem", "label": "more-algebra-lemma-check-injective-on-ass", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-check-injective-on-ass", "contents": [ "Let $R$ be a Noetherian ring. Let $\\varphi : M \\to N$ be a map of", "$R$-modules. Assume that for every prime $\\mathfrak p$", "of $R$ at least one of the following happens", "\\begin{enumerate}", "\\item $M_\\mathfrak p \\to N_\\mathfrak p$ is injective, or", "\\item $\\mathfrak p \\not \\in \\text{Ass}(M)$.", "\\end{enumerate}", "Then $\\varphi$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p$ be an associated prime of $\\Ker(\\varphi)$.", "Then there exists an element $x \\in M_\\mathfrak p$ which is", "in the kernel of $M_\\mathfrak p \\to N_\\mathfrak p$ and is", "annihilated by $\\mathfrak pR_\\mathfrak p$", "(Algebra, Lemma \\ref{algebra-lemma-associated-primes-localize}).", "This is impossible in both cases. Hence", "$\\text{Ass}(\\Ker(\\varphi)) = \\emptyset$ and we conclude $\\Ker(\\varphi) = 0$ by", "Algebra, Lemma \\ref{algebra-lemma-ass-zero}." ], "refs": [ "algebra-lemma-associated-primes-localize", "algebra-lemma-ass-zero" ], "ref_ids": [ 709, 702 ] } ], "ref_ids": [] }, { "id": 9933, "type": "theorem", "label": "more-algebra-lemma-check-isomorphism-via-depth-and-ass", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-check-isomorphism-via-depth-and-ass", "contents": [ "Let $R$ be a Noetherian ring. Let $\\varphi : M \\to N$ be a map of", "$R$-modules. Assume $M$ is finite and that for every prime $\\mathfrak p$", "of $R$ one of the following happens", "\\begin{enumerate}", "\\item $M_\\mathfrak p \\to N_\\mathfrak p$ is an isomorphism, or", "\\item $\\text{depth}(M_\\mathfrak p) \\geq 2$ and", "$\\mathfrak p \\not \\in \\text{Ass}(N)$.", "\\end{enumerate}", "Then $\\varphi$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-check-injective-on-ass} we see that $\\varphi$ is injective.", "Let $N' \\subset N$ be an finitely generated $R$-module containing", "the image of $M$. Then $\\text{Ass}(N_\\mathfrak p) = \\emptyset$ implies", "$\\text{Ass}(N'_\\mathfrak p) = \\emptyset$.", "Hence the assumptions of the lemma hold for $M \\to N'$.", "In order to prove that $\\varphi$ is an isomorphism, it suffices", "to prove the same thing for every such $N' \\subset N$. Thus we may", "assume $N$ is a finite $R$-module. In this case,", "$\\mathfrak p \\not \\in \\text{Ass}(N) \\Rightarrow", "\\text{depth}(N_\\mathfrak p) \\geq 1$, see", "Algebra, Lemma \\ref{algebra-lemma-ideal-nonzerodivisor}.", "Consider the short exact sequence", "$$", "0 \\to M \\to N \\to Q \\to 0", "$$", "defining $Q$.", "Looking at the conditions we see that either $Q_\\mathfrak p = 0$", "in case (1) or $\\text{depth}(Q_\\mathfrak p) \\geq 1$ in case (2)", "by Algebra, Lemma \\ref{algebra-lemma-depth-in-ses}.", "This implies that $Q$ does not have any associated primes, hence $Q = 0$ by", "Algebra, Lemma \\ref{algebra-lemma-ass-zero}." ], "refs": [ "more-algebra-lemma-check-injective-on-ass", "algebra-lemma-ideal-nonzerodivisor", "algebra-lemma-depth-in-ses", "algebra-lemma-ass-zero" ], "ref_ids": [ 9932, 712, 773, 702 ] } ], "ref_ids": [] }, { "id": 9934, "type": "theorem", "label": "more-algebra-lemma-isom-depth-2-torsion-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-isom-depth-2-torsion-free", "contents": [ "Let $R$ be a Noetherian domain. Let $\\varphi : M \\to N$ be a map of", "$R$-modules. Assume $M$ is finite, $N$ is torsion free, and", "that for every prime $\\mathfrak p$ of $R$ one of the following happens", "\\begin{enumerate}", "\\item $M_\\mathfrak p \\to N_\\mathfrak p$ is an isomorphism, or", "\\item $\\text{depth}(M_\\mathfrak p) \\geq 2$.", "\\end{enumerate}", "Then $\\varphi$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-check-isomorphism-via-depth-and-ass}." ], "refs": [ "more-algebra-lemma-check-isomorphism-via-depth-and-ass" ], "ref_ids": [ 9933 ] } ], "ref_ids": [] }, { "id": 9935, "type": "theorem", "label": "more-algebra-lemma-reflexive-depth-2", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-reflexive-depth-2", "contents": [ "Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $M$ is reflexive,", "\\item for every prime $\\mathfrak p$ of $R$ one of the following happens", "\\begin{enumerate}", "\\item $M_\\mathfrak p$ is a reflexive $R_\\mathfrak p$-module, or", "\\item $\\text{depth}(M_\\mathfrak p) \\geq 2$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (1) is true, then $M_\\mathfrak p$ is a reflexive module", "for all primes of $\\mathfrak p$ by Lemma \\ref{lemma-check-reflexive}.", "Thus (1) $\\Rightarrow$ (2). Assume (2). Set $N = \\Hom_R(\\Hom_R(M, R), R)$", "so that", "$$", "N_\\mathfrak p =", "\\Hom_{R_\\mathfrak p}(", "\\Hom_{R_\\mathfrak p}(M_\\mathfrak p, R_\\mathfrak p), R_\\mathfrak p)", "$$", "for every prime $\\mathfrak p$ of $R$.", "See Algebra, Lemma \\ref{algebra-lemma-hom-from-finitely-presented}.", "We apply Lemma \\ref{lemma-isom-depth-2-torsion-free} to the map", "$j : M \\to N$. This is allowed because $M$ is finite and", "$N$ is torsion free by Lemma \\ref{lemma-hom-into-torsion-free}.", "In case (2)(a) the map $M_\\mathfrak p \\to N_\\mathfrak p$ is an", "isomorphism and in case (2)(b) we have $\\text{depth}(M_\\mathfrak p) \\geq 2$." ], "refs": [ "more-algebra-lemma-check-reflexive", "algebra-lemma-hom-from-finitely-presented", "more-algebra-lemma-isom-depth-2-torsion-free", "more-algebra-lemma-hom-into-torsion-free" ], "ref_ids": [ 9925, 353, 9934, 9922 ] } ], "ref_ids": [] }, { "id": 9936, "type": "theorem", "label": "more-algebra-lemma-reflexive-S2", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-reflexive-S2", "contents": [ "Let $R$ be a Noetherian domain. Let $M$ be a finite reflexive $R$-module.", "Let $\\mathfrak p \\subset R$ be a prime ideal.", "\\begin{enumerate}", "\\item If $\\text{depth}(R_\\mathfrak p) \\geq 2$, then", "$\\text{depth}(M_\\mathfrak p) \\geq 2$.", "\\item If $R$ is $(S_2)$, then $M$ is $(S_2)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since formation of reflexive hull $\\Hom_R(\\Hom_R(M, R), R)$", "commutes with localization", "(Algebra, Lemma \\ref{algebra-lemma-hom-from-finitely-presented})", "part (1) follows from Lemma \\ref{lemma-hom-into-depth}.", "Part (2) is immediate from Lemma \\ref{lemma-hom-into-S2}." ], "refs": [ "algebra-lemma-hom-from-finitely-presented", "more-algebra-lemma-hom-into-depth", "more-algebra-lemma-hom-into-S2" ], "ref_ids": [ 353, 9930, 9931 ] } ], "ref_ids": [] }, { "id": 9937, "type": "theorem", "label": "more-algebra-lemma-reflexive-over-normal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-reflexive-over-normal", "contents": [ "Let $R$ be a Noetherian normal domain with fraction field $K$.", "Let $M$ be a finite $R$-module. The following are equivalent", "\\begin{enumerate}", "\\item $M$ is reflexive,", "\\item $M$ is torsion free and has property $(S_2)$,", "\\item $M$ is torsion free and", "$M = \\bigcap_{\\text{height}(\\mathfrak p) = 1} M_{\\mathfrak p}$", "where the intersection happens in $M_K = M \\otimes_R K$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-criterion-normal}", "we see that $R$ satisfies $(R_1)$ and $(S_2)$.", "\\medskip\\noindent", "Assume (1). Then $M$ is torsion free by Lemma \\ref{lemma-reflexive-torsion-free}", "and satisfies $(S_2)$ by Lemma \\ref{lemma-reflexive-S2}.", "Thus (2) holds.", "\\medskip\\noindent", "Assume (2). By definition", "$M' = \\bigcap_{\\text{height}(\\mathfrak p) = 1} M_{\\mathfrak p}$", "is the kernel of the map", "$$", "M_K", "\\longrightarrow", "\\bigoplus\\nolimits_{\\text{height}(\\mathfrak p) = 1} M_K/M_\\mathfrak p", "\\subset", "\\prod\\nolimits_{\\text{height}(\\mathfrak p) = 1} M_K/M_\\mathfrak p", "$$", "Observe that our map indeed factors through the direct sum as indicated", "since given", "$a/b \\in K$ there are at most finitely many height $1$ primes $\\mathfrak p$", "with $b \\in \\mathfrak p$. Let $\\mathfrak p_0$ be a prime of height $1$.", "Then $(M_K/M_\\mathfrak p)_{\\mathfrak p_0} = 0$ unless", "$\\mathfrak p = \\mathfrak p_0$ in which case we get", "$(M_K/M_\\mathfrak p)_{\\mathfrak p_0} = M_K/M_{\\mathfrak p_0}$.", "Thus by exactness of localization and the fact that localization", "commutes with direct sums, we see that", "$M'_{\\mathfrak p_0} = M_{\\mathfrak p_0}$.", "Since $M$ has depth $\\geq 2$ at primes of height $> 1$,", "we see that $M \\to M'$ is an isomorphism by", "Lemma \\ref{lemma-isom-depth-2-torsion-free}. Hence (3) holds.", "\\medskip\\noindent", "Assume (3). Let $\\mathfrak p$ be a prime of height $1$.", "Then $R_\\mathfrak p$ is a discrete valuation ring by $(R_1)$.", "By Lemma \\ref{lemma-dedekind-torsion-free-flat}", "we see that $M_\\mathfrak p$ is finite free, in particular", "reflexive. Hence the map $M \\to M^{**}$ induces an isomorphism at all", "the primes $\\mathfrak p$ of height $1$. Thus the condition", "$M = \\bigcap_{\\text{height}(\\mathfrak p) = 1} M_{\\mathfrak p}$", "implies that $M = M^{**}$ and (1) holds." ], "refs": [ "algebra-lemma-criterion-normal", "more-algebra-lemma-reflexive-torsion-free", "more-algebra-lemma-reflexive-S2", "more-algebra-lemma-isom-depth-2-torsion-free", "more-algebra-lemma-dedekind-torsion-free-flat" ], "ref_ids": [ 1311, 9923, 9936, 9934, 9921 ] } ], "ref_ids": [] }, { "id": 9938, "type": "theorem", "label": "more-algebra-lemma-describe-reflexive-hull", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-describe-reflexive-hull", "contents": [ "Let $R$ be a Noetherian normal domain. Let $M$ be a finite $R$-module.", "Then the reflexive hull of $M$ is the intersection", "$$", "M^{**} =", "\\bigcap\\nolimits_{\\text{height}(\\mathfrak p) = 1}", "M_{\\mathfrak p}/(M_\\mathfrak p)_{tors} =", "\\bigcap\\nolimits_{\\text{height}(\\mathfrak p) = 1}", "(M/M_{tors})_\\mathfrak p", "$$", "taken in $M \\otimes_R K$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p$ be a prime of height $1$.", "The kernel of $M_\\mathfrak p \\to M \\otimes_R K$ is the", "torsion submodule $(M_\\mathfrak p)_{tors}$ of $M_\\mathfrak p$.", "Moreover, we have", "$(M/M_{tors})_\\mathfrak p = M_\\mathfrak p/(M_\\mathfrak p)_{tors}$", "and this is a finite free module over the discrete valuation ring", "$R_\\mathfrak p$", "(Lemma \\ref{lemma-dedekind-torsion-free-flat}).", "Then $M_\\mathfrak p/(M_\\mathfrak p)_{tors} \\to", "(M_\\mathfrak p)^{**} = (M^{**})_\\mathfrak p$", "is an isomorphism, hence the lemma is a consequence of", "Lemma \\ref{lemma-reflexive-over-normal}." ], "refs": [ "more-algebra-lemma-dedekind-torsion-free-flat", "more-algebra-lemma-reflexive-over-normal" ], "ref_ids": [ 9921, 9937 ] } ], "ref_ids": [] }, { "id": 9939, "type": "theorem", "label": "more-algebra-lemma-integral-closure-reflexive", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-integral-closure-reflexive", "contents": [ "Let $A$ be a Noetherian normal domain with fraction field $K$.", "Let $L$ be a finite extension of $K$. If the integral closure", "$B$ of $A$ in $L$ is finite over $A$, then $B$ is reflexive as an $A$-module." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that $B = \\bigcap B_\\mathfrak p$ where the intersection", "is over height $1$ primes $\\mathfrak p \\subset A$, see", "Lemma \\ref{lemma-reflexive-over-normal}.", "Let $b \\in \\bigcap B_\\mathfrak p$. Let $x^d + a_1x^{d - 1} + \\ldots + a_d$", "be the minimal polynomial of $b$ over $K$.", "We want to show $a_i \\in A$.", "By Algebra, Lemma \\ref{algebra-lemma-minimal-polynomial-normal-domain}", "we see that $a_i \\in A_\\mathfrak p$ for all $i$ and all height one primes", "$\\mathfrak p$. Hence we get what we want from", "Algebra, Lemma", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}", "(or the lemma already cited as $A$ is a reflexive module over itself)." ], "refs": [ "more-algebra-lemma-reflexive-over-normal", "algebra-lemma-minimal-polynomial-normal-domain", "algebra-lemma-normal-domain-intersection-localizations-height-1" ], "ref_ids": [ 9937, 521, 1313 ] } ], "ref_ids": [] }, { "id": 9940, "type": "theorem", "label": "more-algebra-lemma-content-finitely-generated", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-content-finitely-generated", "contents": [ "Let $A$ be a ring. Let $M$ be a flat $A$-module. Let $x \\in M$.", "The content ideal of $x$, if it exists, is finitely generated." ], "refs": [], "proofs": [ { "contents": [ "Say $x \\in IM$. Then we can write $x = \\sum_{i = 1, \\ldots, n} f_i x_i$ with", "$f_i \\in I$ and $x_i \\in M$. Hence $x \\in I'M$ with", "$I' = (f_1, \\ldots, f_n)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9941, "type": "theorem", "label": "more-algebra-lemma-equal-content", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-equal-content", "contents": [ "Let $(A, \\mathfrak m)$ be a local ring. Let $u : M \\to N$ be a map of flat", "$A$-modules such that $\\overline{u} : M/\\mathfrak m M \\to N/\\mathfrak m N$", "is injective. If $x \\in M$ has content ideal $I$, then $u(x)$ has content", "ideal $I$ as well." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $u(x) \\in IN$. If $u(x) \\in I'N$, then", "$u(x) \\in (I' \\cap I)N$, see discussion following", "Definition \\ref{definition-content-ideal}. Hence it suffices to", "show: if $x \\in I'N$ and $I' \\subset I$, $I' \\not = I$, then", "$u(x) \\not \\in I'N$. Since $I/I'$ is a nonzero finite $A$-module", "(Lemma \\ref{lemma-content-finitely-generated}) there is a nonzero map", "$\\chi : I/I' \\to A/\\mathfrak m$ of $A$-modules", "by Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK}).", "Since $I$ is the content ideal of $x$ we see that", "$x \\not \\in I''M$ where $I'' = \\Ker(\\chi)$.", "Hence $x$ is not in the kernel of the map", "$$", "IM = I \\otimes_A M \\xrightarrow{\\chi \\otimes 1}", "A/\\mathfrak m \\otimes M \\cong M/\\mathfrak m M", "$$", "Applying our hypothesis on $\\overline{u}$ we conclude that", "$u(x)$ does not map to zero under the map", "$$", "IN = I \\otimes_A N \\xrightarrow{\\chi \\otimes 1}", "A/\\mathfrak m \\otimes N \\cong N/\\mathfrak m N", "$$", "and we conclude." ], "refs": [ "more-algebra-definition-content-ideal", "more-algebra-lemma-content-finitely-generated", "algebra-lemma-NAK" ], "ref_ids": [ 10603, 9940, 401 ] } ], "ref_ids": [] }, { "id": 9942, "type": "theorem", "label": "more-algebra-lemma-content-exists-flat-Mittag-Leffler", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-content-exists-flat-Mittag-Leffler", "contents": [ "Let $A$ be a ring. Let $M$ be a flat Mittag-Leffler module.", "Then every element of $M$ has a content ideal." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Algebra, Lemma \\ref{algebra-lemma-flat-ML-criterion}." ], "refs": [ "algebra-lemma-flat-ML-criterion" ], "ref_ids": [ 845 ] } ], "ref_ids": [] }, { "id": 9943, "type": "theorem", "label": "more-algebra-lemma-flat-finite-type-finite-presentation-local-module", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-finite-type-finite-presentation-local-module", "contents": [ "Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]$ be a polynomial", "ring over $R$. Let $M$ be an $S$-module.", "Assume", "\\begin{enumerate}", "\\item there exist finitely many primes $\\mathfrak p_1, \\ldots, \\mathfrak p_m$", "of $R$ such that the map $R \\to \\prod R_{\\mathfrak p_j}$ is injective,", "\\item $M$ is a finite $S$-module,", "\\item $M$ flat over $R$, and", "\\item for every prime $\\mathfrak p$ of $R$ the module $M_{\\mathfrak p}$", "is of finite presentation over $S_{\\mathfrak p}$.", "\\end{enumerate}", "Then $M$ is of finite presentation over $S$." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation", "$$", "0 \\to K \\to S^{\\oplus r} \\to M \\to 0", "$$", "of $M$ as an $S$-module. Let $\\mathfrak q$ be a prime ideal of $S$", "lying over a prime $\\mathfrak p$ of $R$. By assumption there exist", "finitely many elements $k_1, \\ldots, k_t \\in K$ such that if we set", "$K' = \\sum Sk_j \\subset K$ then", "$K'_{\\mathfrak p} = K_{\\mathfrak p}$ and", "$K'_{\\mathfrak p_j} = K_{\\mathfrak p_j}$ for $j = 1, \\ldots, m$.", "Setting $M' = S^{\\oplus r}/K'$ we deduce that in particular", "$M'_{\\mathfrak q} = M_{\\mathfrak q}$. By openness of flatness, see", "Algebra, Theorem \\ref{algebra-theorem-openness-flatness}", "we conclude that there exists a $g \\in S$, $g \\not \\in \\mathfrak q$", "such that $M'_g$ is flat over $R$. Thus $M'_g \\to M_g$ is a surjective", "map of flat $R$-modules. Consider the commutative diagram", "$$", "\\xymatrix{", "M'_g \\ar[r] \\ar[d] & M_g \\ar[d] \\\\", "\\prod (M'_g)_{\\mathfrak p_j} \\ar[r] & \\prod (M_g)_{\\mathfrak p_j}", "}", "$$", "The bottom arrow is an isomorphism by choice of $k_1, \\ldots, k_t$.", "The left vertical arrow is an injective map as", "$R \\to \\prod R_{\\mathfrak p_j}$ is injective and $M'_g$ is flat over $R$.", "Hence the top horizontal arrow is injective, hence an isomorphism.", "This proves that $M_g$ is of finite presentation over $S_g$.", "We conclude by applying", "Algebra, Lemma \\ref{algebra-lemma-cover}." ], "refs": [ "algebra-theorem-openness-flatness", "algebra-lemma-cover" ], "ref_ids": [ 326, 411 ] } ], "ref_ids": [] }, { "id": 9944, "type": "theorem", "label": "more-algebra-lemma-flat-finite-type-finite-presentation-local", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-finite-type-finite-presentation-local", "contents": [ "Let $R \\to S$ be a ring homomorphism.", "Assume", "\\begin{enumerate}", "\\item there exist finitely many primes", "$\\mathfrak p_1, \\ldots, \\mathfrak p_m$ of $R$ such that", "the map $R \\to \\prod R_{\\mathfrak p_j}$ is injective,", "\\item $R \\to S$ is of finite type,", "\\item $S$ flat over $R$, and", "\\item for every prime $\\mathfrak p$ of $R$ the ring $S_{\\mathfrak p}$", "is of finite presentation over $R_{\\mathfrak p}$.", "\\end{enumerate}", "Then $S$ is of finite presentation over $R$." ], "refs": [], "proofs": [ { "contents": [ "By assumption $S$ is a quotient of a polynomial ring over $R$.", "Thus the result follows directly from", "Lemma \\ref{lemma-flat-finite-type-finite-presentation-local-module}." ], "refs": [ "more-algebra-lemma-flat-finite-type-finite-presentation-local-module" ], "ref_ids": [ 9943 ] } ], "ref_ids": [] }, { "id": 9945, "type": "theorem", "label": "more-algebra-lemma-flat-graded-finite-type-finite-presentation-module", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-graded-finite-type-finite-presentation-module", "contents": [ "Let $R$ be a ring.", "Let $S = R[x_1, \\ldots, x_n]$ be a graded polynomial algebra over $R$,", "i.e., $\\deg(x_i) > 0$ but not necessarily equal to $1$.", "Let $M$ be a graded $S$-module.", "Assume", "\\begin{enumerate}", "\\item $R$ is a local ring,", "\\item $M$ is a finite $S$-module, and", "\\item $M$ is flat over $R$.", "\\end{enumerate}", "Then $M$ is finitely presented as an $S$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $M = \\bigoplus M_d$ be the grading on $M$.", "Pick homogeneous generators $m_1, \\ldots, m_r \\in M$ of $M$.", "Say $\\deg(m_i) = d_i \\in \\mathbf{Z}$. This gives us a presentation", "$$", "0 \\to K \\to \\bigoplus\\nolimits_{i = 1, \\ldots, r} S(-d_i) \\to M \\to 0", "$$", "which in each degree $d$ leads to the short exact sequence", "$$", "0 \\to K_d \\to \\bigoplus\\nolimits_{i = 1, \\ldots, r} S_{d - d_i} \\to", "M_d \\to 0.", "$$", "By assumption each $M_d$ is a finite flat $R$-module. By", "Algebra, Lemma \\ref{algebra-lemma-finite-flat-local}", "this implies each $M_d$ is a finite free $R$-module. Hence", "we see each $K_d$ is a finite $R$-module. Also each $K_d$ is flat", "over $R$ by", "Algebra, Lemma \\ref{algebra-lemma-flat-ses}.", "Hence we conclude that each $K_d$ is finite free by", "Algebra, Lemma \\ref{algebra-lemma-finite-flat-local}", "again.", "\\medskip\\noindent", "Let $\\mathfrak m$ be the maximal ideal of $R$. By the flatness of $M$", "over $R$ the short exact sequences above remain short exact after tensoring", "with $\\kappa = \\kappa(\\mathfrak m)$. As the ring $S \\otimes_R \\kappa$ is", "Noetherian we see that there exist homogeneous elements", "$k_1, \\ldots, k_t \\in K$ such that the images $\\overline{k}_j$", "generate $K \\otimes_R \\kappa$ over $S \\otimes_R \\kappa$. Say $\\deg(k_j) = e_j$.", "Thus for any $d$ the map", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, t} S_{d - e_j}", "\\longrightarrow", "K_d", "$$", "becomes surjective after tensoring with $\\kappa$. By", "Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK})", "this implies the map is surjective over $R$. Hence $K$ is generated", "by $k_1, \\ldots, k_t$ over $S$ and we win." ], "refs": [ "algebra-lemma-finite-flat-local", "algebra-lemma-flat-ses", "algebra-lemma-finite-flat-local", "algebra-lemma-NAK" ], "ref_ids": [ 797, 533, 797, 401 ] } ], "ref_ids": [] }, { "id": 9946, "type": "theorem", "label": "more-algebra-lemma-flat-graded-finite-type-finite-presentation", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-graded-finite-type-finite-presentation", "contents": [ "Let $R$ be a ring. Let $S = \\bigoplus_{n \\geq 0} S_n$ be a graded $R$-algebra.", "Let $M = \\bigoplus_{d \\in \\mathbf{Z}} M_d$ be a graded $S$-module.", "Assume $S$ is finitely generated as an $R$-algebra, assume $S_0$ is a finite", "$R$-algebra, and assume there exist finitely many primes", "$\\mathfrak p_j$, $i = 1, \\ldots, m$ such that", "$R \\to \\prod R_{\\mathfrak p_j}$ is injective.", "\\begin{enumerate}", "\\item If $S$ is flat over $R$, then $S$ is a finitely presented $R$-algebra.", "\\item If $M$ is flat as an $R$-module and finite as an $S$-module,", "then $M$ is finitely presented as an $S$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As $S$ is finitely generated as an $R$-algebra, it is finitely generated", "as an $S_0$ algebra, say by homogeneous elements $t_1, \\ldots, t_n \\in S$", "of degrees $d_1, \\ldots, d_n > 0$. Set $P = R[x_1, \\ldots, x_n]$ with", "$\\deg(x_i) = d_i$. The ring map $P \\to S$, $x_i \\to t_i$ is finite", "as $S_0$ is a finite $R$-module. To prove (1) it suffices to prove", "that $S$ is a finitely presented $P$-module. To prove (2) it suffices", "to prove that $M$ is a finitely presented $P$-module. Thus it suffices", "to prove that if $S = P$ is a graded polynomial ring and $M$ is a finite", "$S$-module flat over $R$, then $M$ is finitely presented as an $S$-module. By", "Lemma \\ref{lemma-flat-graded-finite-type-finite-presentation-module}", "we see $M_{\\mathfrak p}$ is a finitely presented $S_{\\mathfrak p}$-module", "for every prime $\\mathfrak p$ of $R$. Thus the result follows from", "Lemma \\ref{lemma-flat-finite-type-finite-presentation-local-module}." ], "refs": [ "more-algebra-lemma-flat-graded-finite-type-finite-presentation-module", "more-algebra-lemma-flat-finite-type-finite-presentation-local-module" ], "ref_ids": [ 9945, 9943 ] } ], "ref_ids": [] }, { "id": 9947, "type": "theorem", "label": "more-algebra-lemma-flat-finite-type-valuation-ring-finite-presentation", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-finite-type-valuation-ring-finite-presentation", "contents": [ "\\begin{reference}", "\\cite[Theorem 3]{Nagata-Finitely}", "\\end{reference}", "Let $A$ be a valuation ring. Let $A \\to B$ be a ring map of finite type.", "Let $M$ be a finite $B$-module.", "\\begin{enumerate}", "\\item If $B$ is flat over $A$, then $B$ is a finitely presented $A$-algebra.", "\\item If $M$ is flat as an $A$-module, then $M$ is finitely presented", "as a $B$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We are going to use that an $A$-module is flat if and only if it is", "torsion free, see", "Lemma \\ref{lemma-valuation-ring-torsion-free-flat}.", "By", "Algebra, Lemma \\ref{algebra-lemma-homogenize}", "we can find a graded $A$-algebra $S$ with $S_0 = A$ and generated", "by finitely many elements in degree $1$, an element $f \\in S_1$ and a", "finite graded $S$-module $N$ such that $B \\cong S_{(f)}$ and", "$M \\cong N_{(f)}$. If $M$ is torsion free, then we can take $N$ torsion", "free by replacing it by $N/N_{tors}$, see", "Lemma \\ref{lemma-torsion}.", "Similarly, if $B$ is torsion free, then we can take", "$S$ torsion free by replacing it by $S/S_{tors}$.", "Hence in case (1), we may apply", "Lemma \\ref{lemma-flat-graded-finite-type-finite-presentation}", "to see that $S$ is a finitely presented", "$A$-algebra, which implies that $B = S_{(f)}$ is a finitely", "presented $A$-algebra. To see (2) we may first replace $S$ by", "a graded polynomial ring, and then we may apply", "Lemma \\ref{lemma-flat-graded-finite-type-finite-presentation-module}", "to conclude." ], "refs": [ "more-algebra-lemma-valuation-ring-torsion-free-flat", "algebra-lemma-homogenize", "more-algebra-lemma-torsion", "more-algebra-lemma-flat-graded-finite-type-finite-presentation", "more-algebra-lemma-flat-graded-finite-type-finite-presentation-module" ], "ref_ids": [ 9920, 666, 9912, 9946, 9945 ] } ], "ref_ids": [] }, { "id": 9948, "type": "theorem", "label": "more-algebra-lemma-flatten-on-affine-blowup", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flatten-on-affine-blowup", "contents": [ "Let $(R, \\mathfrak m)$ be a local domain with fraction field $K$.", "Let $S$ be a finite type $R$-algebra.", "Let $M$ be a finite $S$-module.", "For every valuation ring $A \\subset K$ dominating $R$", "there exists an ideal $I \\subset \\mathfrak m$ and a nonzero", "element $a \\in I$ such that", "\\begin{enumerate}", "\\item $I$ is finitely generated,", "\\item $A$ has center on $R[\\frac{I}{a}]$,", "\\item the fibre ring of $R \\to R[\\frac{I}{a}]$ at $\\mathfrak m$", "is not zero, and", "\\item the strict transform $S_{I, a}$ of $S$ along $R \\to R[\\frac{I}{a}]$", "is flat and of finite presentation over $R$, and the strict transform", "$M_{I, a}$ of $M$ along $R \\to R[\\frac{I}{a}]$ is flat over $R$ and", "finitely presented over $S_{I, a}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $S = R[x_1, \\ldots, x_n]/J$ and denote $N = S \\oplus M$", "viewed as a module over $P = R[x_1, \\ldots, x_n]$. If we can prove the", "lemma in case $S$ is a polynomial algebra over $R$, then we can", "find $I, a$ satisfying (1), (2), (3) such that the strict transform", "$N_{I, a}$ of $N$ along $R \\to R[\\frac{I}{a}]$ is flat over", "$R$ and finitely presented as a module over the strict transform $P_{I, a}]$", "of $P$. Since $P_{I, a} = R[\\frac{I}{a}][x_1, \\ldots, x_n]$", "(small detail omitted)", "we find that the summand $S_{I, a} \\subset N_{I, a}$ is flat over $R$", "and finitely presented as a module over $R[\\frac{I}{a}][x_1, \\ldots, x_n]$.", "Hence $S_{I, a}$ is finitely presented as an $R[\\frac{I}{a}]$-algebra.", "Moreover, the summand $M_{I, a} \\subset N_{I, a}$ is flat over $R$", "and finitely presented as a module over $P_{I, a}$ hence also", "finitely presented as a module over $S_{I, a}$, see", "Algebra, Lemma \\ref{algebra-lemma-finitely-presented-over-subring}.", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $S = R[x_1, \\ldots, x_n]$. Choose a presentation", "$$", "0 \\to K \\to S^{\\oplus r} \\to M \\to 0.", "$$", "Let $M_A$ be the quotient of $M \\otimes_R A$ by its torsion submodule, see", "Lemma \\ref{lemma-torsion}. Then $M_A$ is a finite module over", "$S_A = A[x_1, \\ldots, x_n]$. By", "Lemma \\ref{lemma-valuation-ring-torsion-free-flat}", "we see that $M_A$ is flat over $A$. By", "Lemma \\ref{lemma-flat-finite-type-valuation-ring-finite-presentation}", "we see that $M_A$ is finitely presented. Hence there exist finitely many", "elements $k_1, \\ldots, k_t \\in S_A^{\\oplus r}$ which generate the", "kernel of the presentation $S_A^{\\oplus r} \\to M_A$ as", "an $S_A$-module. For any choice of $a \\in I \\subset \\mathfrak m$", "satisfying (1), (2), and (3) we denote $M_{I, a}$ the strict transform of", "$M$ along $R \\to R[\\frac{I}{a}]$. It is a finite module over", "$S_{I, a} = R[\\frac{I}{a}][x_1, \\ldots, x_n]$. By", "Algebra, Lemma \\ref{algebra-lemma-valuation-ring-colimit-affine-blowups}", "we have $A = \\colim_{I, a} R[\\frac{I}{a}]$.", "This implies that $S_A = \\colim S_{I, a}$ and", "$$", "\\colim M \\otimes_R R[\\textstyle{\\frac{I}{a}}] = M \\otimes_R A", "$$", "Choose $I, a$ and lifts $k_1, \\ldots, k_t \\in S_{I, a}^{\\oplus r}$.", "Since $M_A$ is the quotient of $M \\otimes_R A$", "by torsion, we see that the images of $k_1, \\ldots, k_t$ in $M \\otimes_R A$", "are annihilated by a nonzero element $\\alpha \\in A$.", "After replacing $I, a$ by a different pair", "(recall that the colimit is filtered),", "we may assume $\\alpha = x/a^n$ for some $x \\in I^n$ nonzero.", "Then we find that $x k_1, \\ldots, x k_t$ map to zero in $M \\otimes_R A$.", "Hence after replacing $I, a$ by a different pair", "we may assume $x k_1, \\ldots, x k_t$ map to zero in", "$M \\otimes_R R[\\frac{I}{a}]$ for some nonzero $x \\in R$.", "Then finally replacing $I, a$ by $xI, xa$ we find that we may assume", "$k_1, \\ldots, k_t$ map to $a$-power torsion elements of", "$M \\otimes_R R[\\frac{I}{a}]$. For any such pair $(I, a)$ we set", "$$", "M'_{I, a} = S_{I, a}^{\\oplus r}/ \\sum S_{I, a}k_j.", "$$", "Since $M_A = S_A^{\\oplus r}/ \\sum S_Ak_j$ we see that", "$M_A = \\colim_{I, a} M'_{I, a}$. At this point we finally apply", "Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat} (3)", "to conclude that $M'_{I, a}$ is flat for some pair $(I, a)$ as above.", "This lemma does not apply a priori to the system of strict transforms", "$$", "M_{I, a} = (M \\otimes_R R[\\textstyle{\\frac{I}{a}}])/a\\text{-power torsion}", "$$", "as the transition maps may not satisfy the assumptions of the lemma.", "But now, since flatness implies torsion free", "(Lemma \\ref{lemma-flat-torsion-free}) and since $M_{I, a}$", "is the quotient of $M'_{I, a}$ (because we arranged it so the", "elements $k_1, \\ldots, k_t$ map to zero in $M_{I, a}$)", "by the $a$-power torsion submodule", "we also conclude that $M'_{I, a} = M_{I, a}$ for such a pair and we win." ], "refs": [ "algebra-lemma-finitely-presented-over-subring", "more-algebra-lemma-torsion", "more-algebra-lemma-valuation-ring-torsion-free-flat", "more-algebra-lemma-flat-finite-type-valuation-ring-finite-presentation", "algebra-lemma-valuation-ring-colimit-affine-blowups", "algebra-lemma-flat-finite-presentation-limit-flat", "more-algebra-lemma-flat-torsion-free" ], "ref_ids": [ 335, 9912, 9920, 9947, 760, 1389, 9919 ] } ], "ref_ids": [] }, { "id": 9949, "type": "theorem", "label": "more-algebra-lemma-blowup-fitting-ideal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-blowup-fitting-ideal", "contents": [ "Let $R$ be a ring. Let $M$ be a finite $R$-module.", "Let $k \\geq 0$ and $I = \\text{Fit}_k(M)$. For every $a \\in I$", "with $R' = R[\\frac{I}{a}]$ the strict transform", "$$", "M' = (M \\otimes_R R')/a\\text{-power torsion}", "$$", "has $\\text{Fit}_k(M') = R'$." ], "refs": [], "proofs": [ { "contents": [ "First observe that $\\text{Fit}_k(M \\otimes_R R') = IR' = aR'$.", "The first equality by Lemma \\ref{lemma-fitting-ideal-basics} part (3)", "and the second equality by", "Algebra, Lemma \\ref{algebra-lemma-affine-blowup}.", "From Lemma \\ref{lemma-principal-fitting-ideal}", "and exactness of localization", "we see that $M'_{\\mathfrak p'}$", "can be generated by $\\leq k$ elements for every prime $\\mathfrak p'$", "of $R'$. Then $\\text{Fit}_k(M') = R'$ for example by", "Lemma \\ref{lemma-fitting-ideal-generate-locally}." ], "refs": [ "more-algebra-lemma-fitting-ideal-basics", "algebra-lemma-affine-blowup", "more-algebra-lemma-principal-fitting-ideal", "more-algebra-lemma-fitting-ideal-generate-locally" ], "ref_ids": [ 9834, 752, 9837, 9835 ] } ], "ref_ids": [] }, { "id": 9950, "type": "theorem", "label": "more-algebra-lemma-blowup-fitting-ideal-locally-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-blowup-fitting-ideal-locally-free", "contents": [ "Let $R$ be a ring. Let $M$ be a finite $R$-module.", "Let $k \\geq 0$ and $I = \\text{Fit}_k(M)$. Asssume that", "$M_\\mathfrak p$ is free of rank $k$ for every", "$\\mathfrak p \\not \\in V(I)$. Then for every $a \\in I$", "with $R' = R[\\frac{I}{a}]$ the strict transform", "$$", "M' = (M \\otimes_R R')/a\\text{-power torsion}", "$$", "is locally free of rank $k$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-blowup-fitting-ideal} we have", "$\\text{Fit}_k(M') = R'$. By Lemma \\ref{lemma-fitting-ideal-finite-locally-free}", "it suffices to show that $\\text{Fit}_{k - 1}(M') = 0$.", "Recall that $R' \\subset R'_a = R_a$, see", "Algebra, Lemma \\ref{algebra-lemma-affine-blowup}.", "Hence it suffices to prove that $\\text{Fit}_{k - 1}(M')$", "maps to zero in $R'_a = R_a$.", "Since clearly $(M')_a = M_a$ this reduces us to showing", "that $\\text{Fit}_{k - 1}(M_a) = 0$", "because formation of Fitting ideals commutes with base", "change according to Lemma \\ref{lemma-fitting-ideal-basics} part (3).", "This is true by our assumption that", "$M_a$ is finite locally free of rank $k$", "(see Algebra, Lemma \\ref{algebra-lemma-finite-projective})", "and the already cited Lemma \\ref{lemma-fitting-ideal-finite-locally-free}." ], "refs": [ "more-algebra-lemma-blowup-fitting-ideal", "more-algebra-lemma-fitting-ideal-finite-locally-free", "algebra-lemma-affine-blowup", "more-algebra-lemma-fitting-ideal-basics", "algebra-lemma-finite-projective", "more-algebra-lemma-fitting-ideal-finite-locally-free" ], "ref_ids": [ 9949, 9836, 752, 9834, 795, 9836 ] } ], "ref_ids": [] }, { "id": 9951, "type": "theorem", "label": "more-algebra-lemma-blowup-module", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-blowup-module", "contents": [ "Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $f \\in R$", "be an element such that $M_f$ is finite locally free of rank $r$.", "Then there exists a finitely generated ideal $I \\subset R$ with", "$V(f) = V(I)$ such that for all $a \\in I$ with $R' = R[\\frac{I}{a}]$", "the strict transform", "$$", "M' = (M \\otimes_R R')/a\\text{-power torsion}", "$$", "is locally free of rank $r$." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjection $R^{\\oplus n} \\to M$. Choose a finite submodule", "$K \\subset \\Ker(R^{\\oplus n} \\to M)$ such that $R^{\\oplus n}/K \\to M$", "becomes an isomorphism after inverting $f$. This is possible because", "$M_f$ is of finite presentation for example by", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}.", "Set $M_1 = R^{\\oplus n}/K$", "and suppose we can prove the lemma for $M_1$. Say $I \\subset R$ is the", "corresponding ideal. Then for $a \\in I$ the map", "$$", "M_1' = (M_1 \\otimes_R R')/a\\text{-power torsion}", "\\longrightarrow", "M' = (M \\otimes_R R')/a\\text{-power torsion}", "$$", "is surjective. It is also an isomorphism after inverting $a$ in $R'$", "as $R'_a = R_f$, see Algebra, Lemma \\ref{algebra-lemma-blowup-in-principal}.", "But $a$ is a nonzerodivisor on $M'_1$, whence the displayed map is an", "isomorphism. Thus it suffices to prove the lemma in case $M$ is a finitely", "presented $R$-module.", "\\medskip\\noindent", "Assume $M$ is a finitely presented $R$-module.", "Then $J = \\text{Fit}_r(M) \\subset S$ is a finitely generated ideal.", "We claim that $I = fJ$ works.", "\\medskip\\noindent", "We first check that $V(f) = V(I)$. The inclusion $V(f) \\subset V(I)$ is", "clear. Conversely, if $f \\not \\in \\mathfrak p$, then", "$\\mathfrak p$ is not an element of $V(J)$ by ", "Lemma \\ref{lemma-fitting-ideal-generate-locally}.", "Thus $\\mathfrak p \\not \\in V(fJ) = V(I)$.", "\\medskip\\noindent", "Let $a \\in I$ and set $R' = R[\\frac{I}{a}]$. We may write $a = fb$", "for some $b \\in J$. By Algebra, Lemmas \\ref{algebra-lemma-affine-blowup} and", "\\ref{algebra-lemma-blowup-add-principal} we see that $J R' = b R'$", "and $b$ is a nonzerodivisor in $R'$. ", "Let $\\mathfrak p' \\subset R' = R[\\frac{I}{a}]$ be", "a prime ideal. Then $JR'_{\\mathfrak p'}$ is generated by $b$.", "It follows from", "Lemma \\ref{lemma-principal-fitting-ideal}", "that $M'_{\\mathfrak p'}$ can be generated by $r$ elements.", "Since $M'$ is finite, there exist $m_1, \\ldots, m_r \\in M'$ and", "$g \\in R'$, $g \\not \\in \\mathfrak p'$ such that the corresponding map", "$(R')^{\\oplus r} \\to M'$ becomes surjective after inverting $g$.", "\\medskip\\noindent", "Finally, consider the ideal $J' = \\text{Fit}_{k - 1}(M')$.", "Note that $J' R'_g$ is generated by the coefficients of relations between", "$m_1, \\ldots, m_r$ (compatibility of Fitting ideal with base change).", "Thus it suffices to show that $J' = 0$, see", "Lemma \\ref{lemma-fitting-ideal-finite-locally-free}.", "Since $R'_a = R_f$ (Algebra, Lemma \\ref{algebra-lemma-blowup-in-principal})", "and $M'_a = M_f$ is free of rank $r$ we see that $J'_a = 0$.", "Since $a$ is a nonzerodivisor in $R'$ we", "conclude that $J' = 0$ and we win." ], "refs": [ "algebra-lemma-finite-projective", "algebra-lemma-blowup-in-principal", "more-algebra-lemma-fitting-ideal-generate-locally", "algebra-lemma-affine-blowup", "algebra-lemma-blowup-add-principal", "more-algebra-lemma-principal-fitting-ideal", "more-algebra-lemma-fitting-ideal-finite-locally-free", "algebra-lemma-blowup-in-principal" ], "ref_ids": [ 795, 755, 9835, 752, 756, 9837, 9836, 755 ] } ], "ref_ids": [] }, { "id": 9952, "type": "theorem", "label": "more-algebra-lemma-ui-completion-direct-sum-into-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ui-completion-direct-sum-into-product", "contents": [ "Let $R$ be a ring.", "Let $I \\subset R$ be an ideal.", "Let $A$ be a set.", "Assume $R$ is Noetherian and complete with respect to $I$. There is a", "canonical map", "$$", "\\left(\\bigoplus\\nolimits_{\\alpha \\in A} R\\right)^\\wedge", "\\longrightarrow", "\\prod\\nolimits_{\\alpha \\in A} R", "$$", "from the $I$-adic completion of the direct sum into the product", "which is universally injective." ], "refs": [], "proofs": [ { "contents": [ "By definition an element $x$ of the left hand side is $x = (x_n)$ where", "$x_n = (x_{n, \\alpha}) \\in \\bigoplus\\nolimits_{\\alpha \\in A} R/I^n$", "such that $x_{n, \\alpha} = x_{n + 1, \\alpha} \\bmod I^n$.", "As $R = R^\\wedge$ we see that for any $\\alpha$ there exists a $y_\\alpha \\in R$", "such that $x_{n, \\alpha} = y_\\alpha \\bmod I^n$. Note that for each $n$ there", "are only finitely many $\\alpha$ such that the elements $x_{n, \\alpha}$ are", "nonzero. Conversely, given $(y_\\alpha) \\in \\prod_\\alpha R$ such that for each", "$n$ there are only finitely many $\\alpha$ such that $y_{\\alpha} \\bmod I^n$", "is nonzero, then this defines an element of the left hand side.", "Hence we can think of an element of the left hand side as infinite", "``convergent sums'' $\\sum_\\alpha y_\\alpha$ with $y_\\alpha \\in R$", "such that for each $n$ there are only finitely many $y_\\alpha$", "which are nonzero modulo $I^n$. The displayed map maps this element", "to the element to $(y_\\alpha)$ in the product.", "In particular the map is injective.", "\\medskip\\noindent", "Let $Q$ be a finite $R$-module. We have to show that the map", "$$", "Q \\otimes_R \\left(\\bigoplus\\nolimits_{\\alpha \\in A} R\\right)^\\wedge", "\\longrightarrow", "Q \\otimes_R \\left(\\prod\\nolimits_{\\alpha \\in A} R\\right)", "$$", "is injective, see", "Algebra, Theorem \\ref{algebra-theorem-universally-exact-criteria}.", "Choose a presentation $R^{\\oplus k} \\to R^{\\oplus m} \\to Q \\to 0$", "and denote $q_1, \\ldots, q_m \\in Q$ the corresponding generators for $Q$.", "By Artin-Rees", "(Algebra, Lemma \\ref{algebra-lemma-Artin-Rees})", "there exists a constant $c$ such that", "$\\Im(R^{\\oplus k} \\to R^{\\oplus m}) \\cap (I^N)^{\\oplus m}", "\\subset \\Im((I^{N - c})^{\\oplus k} \\to R^{\\oplus m})$.", "Let us contemplate the diagram", "$$", "\\xymatrix{", "\\bigoplus_{l = 1}^k \\left(\\bigoplus\\nolimits_{\\alpha \\in A} R\\right)^\\wedge", "\\ar[r] \\ar[d] &", "\\bigoplus_{j = 1}^m \\left(\\bigoplus\\nolimits_{\\alpha \\in A} R\\right)^\\wedge", "\\ar[r] \\ar[d] &", "Q \\otimes_R \\left(\\bigoplus\\nolimits_{\\alpha \\in A} R\\right)^\\wedge", "\\ar[r] \\ar[d] &", "0 \\\\", "\\bigoplus_{l = 1}^k \\left(\\prod\\nolimits_{\\alpha \\in A} R\\right)", "\\ar[r] &", "\\bigoplus_{j = 1}^m \\left(\\prod\\nolimits_{\\alpha \\in A} R\\right)", "\\ar[r] &", "Q \\otimes_R \\left(\\prod\\nolimits_{\\alpha \\in A} R\\right)", "\\ar[r] &", "0", "}", "$$", "with exact rows. Pick an element $\\sum_j \\sum_\\alpha y_{j, \\alpha}$ of", "$\\bigoplus_{j = 1, \\ldots, m}", "\\left(\\bigoplus\\nolimits_{\\alpha \\in A} R\\right)^\\wedge$.", "If this element maps to zero in the module", "$Q \\otimes_R \\left(\\prod\\nolimits_{\\alpha \\in A} R\\right)$,", "then we see in particular that", "$\\sum_j q_j \\otimes y_{j, \\alpha} = 0$ in $Q$ for each $\\alpha$.", "Thus we can find an element", "$(z_{1, \\alpha}, \\ldots, z_{k, \\alpha}) \\in \\bigoplus_{l = 1, \\ldots, k} R$", "which maps to", "$(y_{1, \\alpha}, \\ldots, y_{m, \\alpha}) \\in \\bigoplus_{j = 1, \\ldots, m} R$.", "Moreover, if $y_{j, \\alpha} \\in I^{N_\\alpha}$ for $j = 1, \\ldots, m$, then", "we may assume that $z_{l, \\alpha} \\in I^{N_\\alpha - c}$ for", "$l = 1, \\ldots, k$.", "Hence the sum $\\sum_l \\sum_\\alpha z_{l, \\alpha}$ is ``convergent'' and", "defines an element of", "$\\bigoplus_{l = 1, \\ldots, k}", "\\left(\\bigoplus\\nolimits_{\\alpha \\in A} R\\right)^\\wedge$", "which maps to the element $\\sum_j \\sum_\\alpha y_{j, \\alpha}$ we started", "out with. Thus the right vertical arrow is injective and we win." ], "refs": [ "algebra-theorem-universally-exact-criteria", "algebra-lemma-Artin-Rees" ], "ref_ids": [ 319, 625 ] } ], "ref_ids": [] }, { "id": 9953, "type": "theorem", "label": "more-algebra-lemma-completed-direct-sum-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-completed-direct-sum-flat", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $A$ be a set.", "Assume $R$ is Noetherian. The completion", "$(\\bigoplus\\nolimits_{\\alpha \\in A} R)^\\wedge$", "is a flat $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Denote $R^\\wedge$ the completion of $R$ with respect to $I$. As", "$R \\to R^\\wedge$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}", "it suffices to prove that", "$(\\bigoplus\\nolimits_{\\alpha \\in A} R)^\\wedge$ is a flat", "$R^\\wedge$-module (use", "Algebra, Lemma \\ref{algebra-lemma-composition-flat}).", "Since", "$$", "(\\bigoplus\\nolimits_{\\alpha \\in A} R)^\\wedge", "=", "(\\bigoplus\\nolimits_{\\alpha \\in A} R^\\wedge)^\\wedge", "$$", "we may replace $R$ by $R^\\wedge$ and assume that $R$ is complete with", "respect to $I$ (see", "Algebra, Lemma \\ref{algebra-lemma-completion-complete}).", "In this case", "Lemma \\ref{lemma-ui-completion-direct-sum-into-product}", "tells us the map", "$(\\bigoplus\\nolimits_{\\alpha \\in A} R)^\\wedge \\to \\prod_{\\alpha \\in A} R$", "is universally injective. Thus, by", "Algebra, Lemma \\ref{algebra-lemma-ui-flat-domain}", "it suffices to show that $\\prod_{\\alpha \\in A} R$ is flat. By", "Algebra, Proposition \\ref{algebra-proposition-characterize-coherent}", "(and", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-coherent})", "we see that $\\prod_{\\alpha \\in A} R$ is flat." ], "refs": [ "algebra-lemma-completion-flat", "algebra-lemma-composition-flat", "algebra-lemma-completion-complete", "more-algebra-lemma-ui-completion-direct-sum-into-product", "algebra-lemma-ui-flat-domain", "algebra-proposition-characterize-coherent", "algebra-lemma-Noetherian-coherent" ], "ref_ids": [ 870, 524, 872, 9952, 810, 1418, 844 ] } ], "ref_ids": [] }, { "id": 9954, "type": "theorem", "label": "more-algebra-lemma-tor-strictly-pro-zero", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-tor-strictly-pro-zero", "contents": [ "\\begin{reference}", "This is \\cite[Lemma 9.9]{quillenhomology}; note that", "the author forgot the word ``strict'' in the statement", "although it was clearly intended.", "\\end{reference}", "Let $A$ be a Noetherian ring. Let $I$ be an ideal of $A$.", "Let $M$ be a finite $A$-module. For every $p > 0$ there exists a $c > 0$", "such that $\\text{Tor}_p^A(M, A/I^n) \\to \\text{Tor}_p^A(M, A/I^{n - c})$", "is zero for all $n \\geq c$." ], "refs": [], "proofs": [ { "contents": [ "Proof for $p = 1$. Choose a short exact sequence", "$0 \\to K \\to R^{\\oplus t} \\to M \\to 0$. Then", "$\\text{Tor}_1^A(M, A/I^n) = K \\cap (I^n)^{\\oplus t}/I^nK$.", "By Artin-Rees (Algebra, Lemma \\ref{algebra-lemma-Artin-Rees})", "there is a constant $c \\geq 0$ such that", "$K \\cap (I^n)^{\\oplus t} \\subset I^{n - c}K$ for $n \\geq c$. Thus the result", "for $p = 1$. For $p > 1$ we have", "$\\text{Tor}_p^A(M, A/I^n) = \\text{Tor}^A_{p - 1}(K, A/I^n)$.", "Thus the lemma follows by induction." ], "refs": [ "algebra-lemma-Artin-Rees" ], "ref_ids": [ 625 ] } ], "ref_ids": [] }, { "id": 9955, "type": "theorem", "label": "more-algebra-lemma-limit-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-limit-flat", "contents": [ "Let $A$ be a Noetherian ring. Let $I$ be an ideal of $A$. Let", "$(M_n)$ be an inverse system of $A$-modules such that", "\\begin{enumerate}", "\\item $M_n$ is a flat $A/I^n$-module,", "\\item $M_{n + 1} \\to M_n$ is surjective.", "\\end{enumerate}", "Then $M = \\lim M_n$ is a flat $A$-module and", "$Q \\otimes_A M = \\lim Q \\otimes_A M_n$ for every finite $A$-module $Q$." ], "refs": [], "proofs": [ { "contents": [ "We first show that $Q \\otimes_A M = \\lim Q \\otimes_A M_n$ for every finite", "$A$-module $Q$. Choose a resolution $F_2 \\to F_1 \\to F_0 \\to Q \\to 0$", "by finite free $A$-modules $F_i$. Then", "$$", "F_2 \\otimes_A M_n \\to F_1 \\otimes_A M_n \\to F_0 \\otimes_A M_n", "$$", "is a chain complex whose homology in degree $0$ is $Q \\otimes_A M_n$", "and whose homology in degree $1$ is", "$$", "\\text{Tor}_1^A(Q, M_n) = \\text{Tor}_1^A(Q, A/I^n) \\otimes_{A/I^n} M_n", "$$", "as $M_n$ is flat over $A/I^n$. By Lemma \\ref{lemma-tor-strictly-pro-zero}", "we see that this system is essentially constant (with value $0$).", "It follows from Homology, Lemma \\ref{homology-lemma-apply-Mittag-Leffler-again}", "that $\\lim Q \\otimes_A A/I^n =", "\\Coker(\\lim F_1 \\otimes_A M_n \\to \\lim F_0 \\otimes_A M_n)$.", "Since $F_i$ is finite free this equals", "$\\Coker(F_1 \\otimes_A M \\to F_0 \\otimes_A M) = Q \\otimes_A M$.", "\\medskip\\noindent", "Next, let $Q \\to Q'$ be an injective map of finite $A$-modules.", "We have to show that $Q \\otimes_A M \\to Q' \\otimes_A M$ is injective", "(Algebra, Lemma \\ref{algebra-lemma-flat}). By the above we see", "$$", "\\Ker(Q \\otimes_A M \\to Q' \\otimes_A M) =", "\\Ker(\\lim Q \\otimes_A M_n \\to \\lim Q' \\otimes_A M_n).", "$$", "For each $n$ we have an exact sequence", "$$", "\\text{Tor}_1^A(Q', M_n) \\to \\text{Tor}_1^A(Q'', M_n) \\to", "Q \\otimes_A M_n \\to Q' \\otimes_A M_n", "$$", "where $Q'' = \\Coker(Q \\to Q')$. Above we have seen that the", "inverse systems of Tor's are essentially constant with value $0$.", "It follows from", "Homology, Lemma \\ref{homology-lemma-apply-Mittag-Leffler-again}", "that the inverse limit of the right most maps is injective." ], "refs": [ "more-algebra-lemma-tor-strictly-pro-zero", "homology-lemma-apply-Mittag-Leffler-again", "algebra-lemma-flat", "homology-lemma-apply-Mittag-Leffler-again" ], "ref_ids": [ 9954, 12128, 525, 12128 ] } ], "ref_ids": [] }, { "id": 9956, "type": "theorem", "label": "more-algebra-lemma-flat-after-completion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-after-completion", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be", "an $R$-module. Assume", "\\begin{enumerate}", "\\item $I$ is finitely generated,", "\\item $R/I$ is Noetherian,", "\\item $M/IM$ is flat over $R/I$,", "\\item $\\text{Tor}_1^R(M, R/I) = 0$.", "\\end{enumerate}", "Then the $I$-adic completion $R^\\wedge$", "is a Noetherian ring and $M^\\wedge$ is flat over $R^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-what-does-it-mean}", "the modules $M/I^nM$ are flat over $R/I^n$ for all $n$.", "By Algebra, Lemma \\ref{algebra-lemma-hathat-finitely-generated} we have", "(a) $R^\\wedge$ and $M^\\wedge$ are $I$-adically complete and", "(b) $R/I^n = R^\\wedge/I^nR^\\wedge$ for all $n$.", "By Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian}", "the ring $R^\\wedge$ is Noetherian.", "Applying Lemma \\ref{lemma-limit-flat} we conclude that", "$M^\\wedge = \\lim M/I^nM$ is flat as an $R^\\wedge$-module." ], "refs": [ "algebra-lemma-what-does-it-mean", "algebra-lemma-hathat-finitely-generated", "algebra-lemma-completion-Noetherian", "more-algebra-lemma-limit-flat" ], "ref_ids": [ 890, 859, 873, 9955 ] } ], "ref_ids": [] }, { "id": 9957, "type": "theorem", "label": "more-algebra-lemma-functorial", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-functorial", "contents": [ "Let $\\varphi : E \\to R$ and $\\varphi' : E' \\to R$ be $R$-module maps.", "Let $\\psi : E \\to E'$ be an $R$-module map such that", "$\\varphi' \\circ \\psi = \\varphi$. Then $\\psi$ induces a", "homomorphism of differential graded algebras", "$K_\\bullet(\\varphi) \\to K_\\bullet(\\varphi')$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9958, "type": "theorem", "label": "more-algebra-lemma-change-basis", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-change-basis", "contents": [ "Let $f_1, \\ldots, f_r \\in R$ be a sequence.", "Let $(x_{ij})$ be an invertible $r \\times r$-matrix with", "coefficients in $R$. Then the complexes", "$K_\\bullet(f_\\bullet)$ and", "$$", "K_\\bullet(\\sum x_{1j}f_j, \\sum x_{2j}f_j, \\ldots, \\sum x_{rj}f_j)", "$$", "are isomorphic." ], "refs": [], "proofs": [ { "contents": [ "Set $g_i = \\sum x_{ij}f_j$.", "The matrix $(x_{ji})$ gives an isomorphism $x : R^{\\oplus r} \\to R^{\\oplus r}$", "such that $(g_1, \\ldots, g_r) = (f_1, \\ldots, f_r) \\circ x$.", "Hence this follows from the functoriality of the Koszul complex", "described in", "Lemma \\ref{lemma-functorial}." ], "refs": [ "more-algebra-lemma-functorial" ], "ref_ids": [ 9957 ] } ], "ref_ids": [] }, { "id": 9959, "type": "theorem", "label": "more-algebra-lemma-homotopy-koszul-abstract", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-homotopy-koszul-abstract", "contents": [ "Let $R$ be a ring. Let $\\varphi : E \\to R$ be an $R$-module map.", "Let $e \\in E$ with image $f = \\varphi(e)$ in $R$. Then", "$$", "f = de + ed", "$$", "as endomorphisms of $K_\\bullet(\\varphi)$." ], "refs": [], "proofs": [ { "contents": [ "This is true because $d(ea) = d(e)a - ed(a) = fa - ed(a)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9960, "type": "theorem", "label": "more-algebra-lemma-homotopy-koszul", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-homotopy-koszul", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be a sequence.", "Multiplication by $f_i$ on $K_\\bullet(f_\\bullet)$ is homotopic to", "zero, and in particular the cohomology modules $H_i(K_\\bullet(f_\\bullet))$", "are annihilated by the ideal $(f_1, \\ldots, f_r)$." ], "refs": [], "proofs": [ { "contents": [ "Special case of", "Lemma \\ref{lemma-homotopy-koszul-abstract}." ], "refs": [ "more-algebra-lemma-homotopy-koszul-abstract" ], "ref_ids": [ 9959 ] } ], "ref_ids": [] }, { "id": 9961, "type": "theorem", "label": "more-algebra-lemma-cone-koszul-abstract", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cone-koszul-abstract", "contents": [ "Let $R$ be a ring. Let $\\varphi : E \\to R$ be an $R$-module map.", "Let $f \\in R$. Set $E' = E \\oplus R$ and define $\\varphi' : E' \\to R$", "by $\\varphi$ on $E$ and multiplication by $f$ on $R$.", "The complex $K_\\bullet(\\varphi')$ is isomorphic to the", "cone of the map of complexes", "$$", "f :", "K_\\bullet(\\varphi)", "\\longrightarrow", "K_\\bullet(\\varphi).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Denote $e_0 \\in E'$ the element $1 \\in R \\subset R \\oplus E$.", "By our definition of the cone above we see that", "$$", "C(f)_n = K_n(\\varphi) \\oplus K_{n - 1}(\\varphi) =", "\\wedge^n(E) \\oplus \\wedge^{n - 1}(E) = \\wedge^n(E')", "$$", "where in the last $=$ we map $(0, e_1 \\wedge \\ldots \\wedge e_{n - 1})$", "to $e_0 \\wedge e_1 \\wedge \\ldots \\wedge e_{n - 1}$ in $\\wedge^n(E')$.", "A computation shows that this isomorphism is compatible with", "differentials. Namely, this is clear for elements of the first", "summand as $\\varphi'|_E = \\varphi$ and $d_{C(f)}$ restricted to", "the first summand is just $d_{K_\\bullet(\\varphi)}$.", "On the other hand, if $e_1 \\wedge \\ldots \\wedge e_{n - 1}$", "is in the second summand, then", "$$", "d_{C(f)}(0, e_1 \\wedge \\ldots \\wedge e_{n - 1}) =", "fe_1 \\wedge \\ldots \\wedge e_{n - 1}", "- d_{K_\\bullet(\\varphi)}(e_1 \\wedge \\ldots \\wedge e_{n - 1})", "$$", "and on the other hand", "\\begin{align*}", "& d_{K_\\bullet(\\varphi')}(0, e_0 \\wedge e_1 \\wedge \\ldots \\wedge e_{n - 1}) \\\\", "& =", "\\sum\\nolimits_{i = 0, \\ldots, n - 1}", "(-1)^i \\varphi'(e_i)e_0 \\wedge \\ldots \\wedge \\widehat{e_i}", "\\wedge \\ldots \\wedge e_{n - 1} \\\\", "& =", "fe_1 \\wedge \\ldots \\wedge e_{n - 1} +", "\\sum\\nolimits_{i = 1, \\ldots, n - 1}", "(-1)^i \\varphi(e_i)e_0 \\wedge \\ldots \\wedge \\widehat{e_i}", "\\wedge \\ldots \\wedge e_{n - 1} \\\\", "& =", "fe_1 \\wedge \\ldots \\wedge e_{n - 1} -", "e_0 \\left(\\sum\\nolimits_{i = 1, \\ldots, n - 1}", "(-1)^{i + 1} \\varphi(e_i)e_1 \\wedge \\ldots \\wedge \\widehat{e_i}", "\\wedge \\ldots \\wedge e_{n - 1}\\right)", "\\end{align*}", "which is the image of the result of the previous computation." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9962, "type": "theorem", "label": "more-algebra-lemma-cone-koszul", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cone-koszul", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_r$ be a sequence of elements", "of $R$. The complex $K_\\bullet(f_1, \\ldots, f_r)$ is isomorphic to the", "cone of the map of complexes", "$$", "f_r :", "K_\\bullet(f_1, \\ldots, f_{r - 1})", "\\longrightarrow", "K_\\bullet(f_1, \\ldots, f_{r - 1}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Special case of", "Lemma \\ref{lemma-cone-koszul-abstract}." ], "refs": [ "more-algebra-lemma-cone-koszul-abstract" ], "ref_ids": [ 9961 ] } ], "ref_ids": [] }, { "id": 9963, "type": "theorem", "label": "more-algebra-lemma-cone-squared", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cone-squared", "contents": [ "Let $R$ be a ring. Let $A_\\bullet$ be a complex of $R$-modules.", "Let $f, g \\in R$. Let $C(f)_\\bullet$ be the cone of", "$f : A_\\bullet \\to A_\\bullet$. Define similarly $C(g)_\\bullet$ and", "$C(fg)_\\bullet$. Then $C(fg)_\\bullet$ is homotopy equivalent to the", "cone of a map", "$$", "C(f)_\\bullet[1] \\longrightarrow C(g)_\\bullet", "$$" ], "refs": [], "proofs": [ { "contents": [ "We first prove this if $A_\\bullet$ is the complex consisting of $R$ placed", "in degree $0$. In this case the complex $C(f)_\\bullet$ is the", "complex", "$$", "\\ldots \\to 0 \\to R \\xrightarrow{f} R \\to 0 \\to \\ldots", "$$", "with $R$ placed in (homological) degrees $1$ and $0$. The map", "of complexes we use is", "$$", "\\xymatrix{", "0 \\ar[r] \\ar[d] &", "0 \\ar[r] \\ar[d] &", "R \\ar[r]^f \\ar[d]^1 &", "R \\ar[r] \\ar[d] & 0 \\ar[d] \\\\", "0 \\ar[r] & R \\ar[r]^g & R \\ar[r] & 0 \\ar[r] & 0", "}", "$$", "The cone of this is the chain complex consisting of $R^{\\oplus 2}$ placed in", "degrees $1$ and $0$ and differential (\\ref{equation-differential-cone})", "$$", "\\left(", "\\begin{matrix}", "g & 1 \\\\", "0 & -f", "\\end{matrix}", "\\right) :", "R^{\\oplus 2} \\longrightarrow R^{\\oplus 2}", "$$", "To see this chain complex is homotopic to", "$C(fg)_\\bullet$, i.e., to $R \\xrightarrow{fg} R$,", "consider the maps of complexes", "$$", "\\xymatrix{", "R \\ar[d]_{(1, -g)} \\ar[r]_{fg} & R \\ar[d]^{(0, 1)} \\\\", "R^{\\oplus 2} \\ar[r] & R^{\\oplus 2}", "}", "\\quad\\quad", "\\xymatrix{", "R^{\\oplus 2} \\ar[d]_{(1, 0)} \\ar[r] & R^{\\oplus 2} \\ar[d]^{(f, 1)} \\\\", "R \\ar[r]^{fg} & R", "}", "$$", "with obvious notation.", "The composition of these two maps in one direction is the", "identity on $C(fg)_\\bullet$, but in the other direction", "it isn't the identity. We omit writing out the required homotopy.", "\\medskip\\noindent", "To see the result holds in general, we use that we have a functor", "$K_\\bullet \\mapsto \\text{Tot}(A_\\bullet \\otimes_R K_\\bullet)$", "on the category of complexes which is compatible with homotopies", "and cones. Then we write $C(f)_\\bullet$ and $C(g)_\\bullet$", "as the total complex of the double complexes", "$$", "(R \\xrightarrow{f} R) \\otimes_R A_\\bullet", "\\quad\\text{and}\\quad", "(R \\xrightarrow{g} R) \\otimes_R A_\\bullet", "$$", "and in this way we deduce the result from the special case discussed above.", "Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9964, "type": "theorem", "label": "more-algebra-lemma-koszul-mult-abstract", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-koszul-mult-abstract", "contents": [ "Let $R$ be a ring. Let $\\varphi : E \\to R$ be an $R$-module map.", "Let $f, g \\in R$. Set $E' = E \\oplus R$ and define", "$\\varphi'_f, \\varphi'_g, \\varphi'_{fg} : E' \\to R$", "by $\\varphi$ on $E$ and multiplication by $f, g, fg$ on $R$.", "The complex $K_\\bullet(\\varphi'_{fg})$ is isomorphic to the", "cone of a map of complexes", "$$", "K_\\bullet(\\varphi'_f)[1]", "\\longrightarrow", "K_\\bullet(\\varphi'_g).", "$$" ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-cone-koszul-abstract}", "the complex $K_\\bullet(\\varphi'_f)$ is isomorphic to the cone of", "multiplication by $f$ on $K_\\bullet(\\varphi)$ and similarly", "for the other two cases. Hence the lemma follows from", "Lemma \\ref{lemma-cone-squared}." ], "refs": [ "more-algebra-lemma-cone-koszul-abstract", "more-algebra-lemma-cone-squared" ], "ref_ids": [ 9961, 9963 ] } ], "ref_ids": [] }, { "id": 9965, "type": "theorem", "label": "more-algebra-lemma-koszul-mult", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-koszul-mult", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_{r - 1}$ be a sequence of elements", "of $R$. Let $f, g \\in R$. The complex", "$K_\\bullet(f_1, \\ldots, f_{r - 1}, fg)$", "is homotopy equivalent to the cone of a map of complexes", "$$", "K_\\bullet(f_1, \\ldots, f_{r - 1}, f)[1]", "\\longrightarrow", "K_\\bullet(f_1, \\ldots, f_{r - 1}, g)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Special case of", "Lemma \\ref{lemma-koszul-mult-abstract}." ], "refs": [ "more-algebra-lemma-koszul-mult-abstract" ], "ref_ids": [ 9964 ] } ], "ref_ids": [] }, { "id": 9966, "type": "theorem", "label": "more-algebra-lemma-join-sequences-koszul-complex", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-join-sequences-koszul-complex", "contents": [ "Let $R$ be a ring.", "Let $f_1, \\ldots, f_r$, $g_1, \\ldots, g_s$ be elements of $R$.", "Then there is an isomorphism of Koszul complexes", "$$", "K_\\bullet(R, f_1, \\ldots, f_r, g_1, \\ldots, g_s) =", "\\text{Tot}(K_\\bullet(R, f_1, \\ldots, f_r) \\otimes_R", "K_\\bullet(R, g_1, \\ldots, g_s)).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: If $K_\\bullet(R, f_1, \\ldots, f_r)$ is generated as a", "differential graded algebra by $x_1, \\ldots, x_r$ with $\\text{d}(x_i) = f_i$", "and $K_\\bullet(R, g_1, \\ldots, g_s)$ is generated as a", "differential graded algebra by $y_1, \\ldots, y_s$ with $\\text{d}(y_j) = g_j$,", "then we can think of $K_\\bullet(R, f_1, \\ldots, f_r, g_1, \\ldots, g_s)$", "as the differential graded algebra generated by the sequence of elements", "$x_1, \\ldots, x_r, y_1, \\ldots, y_s$ with $\\text{d}(x_i) = f_i$", "and $\\text{d}(y_j) = g_j$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9967, "type": "theorem", "label": "more-algebra-lemma-extended-alternating-is-complex", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-extended-alternating-is-complex", "contents": [ "The extended alternating {\\v C}ech complexes defined above", "are complexes of $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9968, "type": "theorem", "label": "more-algebra-lemma-extended-alternating-form-module", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-extended-alternating-form-module", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. Let $M$ be an $R$-module.", "The extended alternating {\\v C}ech complex of $M$ is the tensor product", "over $R$ of $M$ with the extended alternating {\\v C}ech complex of $R$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9969, "type": "theorem", "label": "more-algebra-lemma-extended-alternating-base-change", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-extended-alternating-base-change", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. Let $M$ be an $R$-module.", "Let $R \\to S$ be a ring map, denote $g_1, \\ldots, g_r \\in S$ the images", "of $f_1, \\ldots, f_r$, and set $N = M \\otimes_R S$.", "The extended alternating {\\v C}ech complex constructed using", "$S$, $g_1, \\ldots, g_r$, and $N$ is the tensor product of the", "extended alternating {\\v C}ech complex of $M$ with $S$ over $R$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9970, "type": "theorem", "label": "more-algebra-lemma-extended-alternating-homotopy-zero", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-extended-alternating-homotopy-zero", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. Let $M$ be an $R$-module.", "If there exists an $i \\in \\{1, \\ldots, r\\}$ such that $f_i$ is a unit, then", "the extended alternating {\\v C}ech", "complex of $M$ is homotopy equivalent to $0$." ], "refs": [], "proofs": [ { "contents": [ "We will use the following notation: a cochain $x$ of degree $p + 1$", "in the extended alternating {\\v C}ech complex of $M$ is", "$x = (x_{i_0 \\ldots i_p})$ where $x_{i_0 \\ldots i_p}$ is in", "$M_{f_{i_0} \\ldots f_{i_p}}$. With this notation we have", "$$", "d(x)_{i_0 \\ldots i_{p + 1}} =", "\\sum\\nolimits_j (-1)^j x_{i_0 \\ldots \\hat i_j \\ldots i_p}", "$$", "As homotopy we use the maps", "$$", "h : \\text{cochains of degree }p + 2 \\to \\text{cochains of degree }p + 1", "$$", "given by the rule", "$$", "h(x)_{i_0 \\ldots i_p} = 0 \\text{ if } i \\in \\{i_0, \\ldots, i_p\\}", "\\text{ and }", "h(x)_{i_0 \\ldots i_p} = ", "(-1)^j x_{i_0 \\ldots i_j i i_{j + 1} \\ldots i_p} \\text{ if not}", "$$", "Here $j$ is the unique index such that $i_j < i < i_{j + 1}$ in the", "second case; also, since $f_i$ is a unit we have the equality", "$$", "M_{f_{i_0} \\ldots f_{i_p}} =", "M_{f_{i_0} \\ldots f_{i_j} f_i f_{i_{j + 1}} \\ldots f_{i_p}}", "$$", "which we can use to make sense of thinking of", "$(-1)^j x_{i_0 \\ldots i_j i i_{j + 1} \\ldots i_p}$", "as an element of $M_{f_{i_0} \\ldots f_{i_p}}$.", "We will show by a computation that $d h + h d$ equals", "the negative of the identity map which finishes the proof.", "To do this fix $x$ a cochain of degree $p + 1$ and let", "$1 \\leq i_0 < \\ldots < i_p \\leq r$.", "\\medskip\\noindent", "Case I: $i \\in \\{i_0, \\ldots, i_p\\}$. Say $i = i_t$. Then we have", "$h(d(x))_{i_0 \\ldots i_p} = 0$. On the other hand we have", "$$", "d(h(x))_{i_0 \\ldots i_p} =", "\\sum (-1)^j h(x)_{i_0 \\ldots \\hat i_j \\ldots i_p} =", "(-1)^t h(x)_{i_0 \\ldots \\hat i \\ldots i_p} =", "(-1)^t (-1)^{t - 1} x_{i_0 \\ldots i_p}", "$$", "Thus $(dh + hd)(x)_{i_0 \\ldots i_p} = -x_{i_0 \\ldots i_p}$ as desired.", "\\medskip\\noindent", "Case II: $i \\not \\in \\{i_0, \\ldots, i_p\\}$. Let $j$ be such that", "$i_j < i < i_{j + 1}$. Then we see that", "\\begin{align*}", "h(d(x))_{i_0 \\ldots i_p}", "& =", "(-1)^j d(x)_{i_0 \\ldots i_j i i_{j + 1} \\ldots i_p} \\\\", "& =", "\\sum\\nolimits_{j' \\leq j} (-1)^{j + j'}", "x_{i_0 \\ldots \\hat i_{j'} \\ldots i_j i i_{j + 1} \\ldots i_p} -", "x_{i_0 \\ldots i_p} \\\\", "&", "+ \\sum\\nolimits_{j' > j} (-1)^{j + j' + 1}", "x_{i_0 \\ldots i_j i i_{j + 1} \\ldots \\hat i_{j'} \\ldots i_p}", "\\end{align*}", "On the other hand we have", "\\begin{align*}", "d(h(x))_{i_0 \\ldots i_p}", "& =", "\\sum\\nolimits_{j'} (-1)^{j'} h(x)_{i_0 \\ldots \\hat i_{j'} \\ldots i_p} \\\\", "& =", "\\sum\\nolimits_{j' \\leq j} (-1)^{j' + j - 1}", "x_{i_0 \\ldots \\hat i_{j'} \\ldots i_j i i_{j + 1} \\ldots i_p} \\\\", "& +", "\\sum\\nolimits_{j' > j} (-1)^{j' + j}", "x_{i_0 \\ldots i_j i i_{j + 1} \\ldots \\hat i_{j'} \\ldots i_p}", "\\end{align*}", "Adding these up we obtain", "$(dh + hd)(x)_{i_0 \\ldots i_p} = - x_{i_0 \\ldots i_p}$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9971, "type": "theorem", "label": "more-algebra-lemma-extended-alternating-torsion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-extended-alternating-torsion", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. Let $M$ be an $R$-module.", "Let $H^q$ be the $q$th cohomology module of the extended alternation", "{\\v C}ech complex of $M$. Then", "\\begin{enumerate}", "\\item $H^q = 0$ if $q \\not \\in [0, r]$,", "\\item for $x \\in H^i$ there exists an $n \\geq 1$ such that $f_i^n x = 0$", "for $i = 1, \\ldots, r$,", "\\item the support of $H^q$ is contained in $V(f_1, \\ldots, f_r)$,", "\\item if there is an $f \\in (f_1, \\ldots, f_r)$ which acts invertibly", "on $M$, then $H^q = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from the fact that the extended alternating {\\v C}ech complex", "is zero in degrees $< 0$ and $> r$. To prove (2) it suffices to show that", "for each $i$ there exists an $n \\geq 1$ such that $f_i^n x = 0$. To see", "this it suffices to show that $(H^q)_{f_i} = 0$. Since localization is", "exact, $(H^q)_{f_i}$ is the $q$th cohomology module of the localization", "of the extended alternating complex of $M$ at $f_i$. By", "Lemma \\ref{lemma-extended-alternating-base-change}", "this localization is the extended alternating {\\v C}ech complex", "of $M_{f_i}$ over $R_{f_i}$ with respect to the images of", "$f_1, \\ldots, f_r$ in $R_{f_i}$. Thus", "we reduce to showing that $H^q$ is zero if $f_i$ is invertible, which", "follows from Lemma \\ref{lemma-extended-alternating-homotopy-zero}.", "Part (3) follows from the observation that $(H^q)_{f_i} = 0$", "for all $i$ that we just proved. To see part (4) note that in this", "case $f$ acts invertibly on $H^q$ and $H^q$ is supported on $V(f)$", "by (3). This forces $H^q$ to be zero (small detail omitted)." ], "refs": [ "more-algebra-lemma-extended-alternating-base-change", "more-algebra-lemma-extended-alternating-homotopy-zero" ], "ref_ids": [ 9969, 9970 ] } ], "ref_ids": [] }, { "id": 9972, "type": "theorem", "label": "more-algebra-lemma-extended-alternating-Cech-is-colimit-koszul", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-extended-alternating-Cech-is-colimit-koszul", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. The extended alternating", "{\\v C}ech complex", "$$", "R \\to \\bigoplus\\nolimits_{i_0} R_{f_{i_0}} \\to", "\\bigoplus\\nolimits_{i_0 < i_1} R_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to R_{f_1\\ldots f_r}", "$$", "is a colimit of the Koszul complexes $K(R, f_1^n, \\ldots, f_r^n)$; see", "proof for a precise statement." ], "refs": [], "proofs": [ { "contents": [ "We urge the reader to prove this for themselves.", "Denote $K(R, f_1^n, \\ldots, f_r^n)$ the Koszul complex of", "Definition \\ref{definition-koszul-complex} viewed as a cochain complex", "sitting in degrees $0, \\ldots, r$. Thus we have", "$$", "K(R, f_1^n, \\ldots, f_r^n) :", "0 \\to \\wedge^r(R^{\\oplus r}) \\to", "\\wedge^{r - 1}(R^{\\oplus r}) \\to \\ldots \\to", "R^{\\oplus r} \\to R \\to 0", "$$", "with the term $\\wedge^r(R^{\\oplus r})$ sitting in degree $0$.", "Let $e^n_1, \\ldots, e^n_r$ be the standard basis of $R^{\\oplus r}$.", "Then the elements $e^n_{j_1} \\wedge \\ldots \\wedge e^n_{j_{r - p}}$ for", "$1 \\leq j_1 < \\ldots < j_{r - p} \\leq r$ form a basis for the term in", "degree $p$ of the Koszul complex. Further, observe that", "$$", "d(e^n_{j_1} \\wedge \\ldots \\wedge e^n_{j_{r - p}}) =", "\\sum (-1)^{a + 1} f_{j_a}^n e^n_{j_1} \\wedge \\ldots \\wedge \\hat e^n_{j_a} \\wedge", "\\ldots \\wedge e^n_{j_{r - p}}", "$$", "by our construction of the Koszul complex in Section \\ref{section-koszul}.", "The transition maps of our system", "$$", "K(R, f_1^n, \\ldots, f_r^n) \\to K(R, f_1^{n + 1}, \\ldots, f_r^{n + 1})", "$$", "are given by the rule", "$$", "e^n_{j_1} \\wedge \\ldots \\wedge e^n_{j_{r - p}}", "\\longmapsto", "f_{i_0} \\ldots f_{i_{p - 1}}", "e^{n + 1}_{j_1} \\wedge \\ldots \\wedge e^{n + 1}_{j_{r - p}}", "$$", "where the indices $1 \\leq i_0 < \\ldots < i_{p - 1} \\leq r$ are such that", "$\\{1, \\ldots r\\} =", "\\{i_0, \\ldots, i_{p - 1}\\} \\amalg \\{j_1, \\ldots, j_{r - p}\\}$.", "We omit the short computation that shows this is compatible with differentials.", "Observe that the transition maps are always $1$ in degree $0$", "and equal to $f_1 \\ldots f_r$ in degree $r$.", "\\medskip\\noindent", "Denote $K^p(R, f_1^n, \\ldots, f_r^n)$ the term of degree $p$ in the Koszul", "complex. Observe that for any $f \\in R$ we have", "$$", "R_f = \\colim (R \\xrightarrow{f} R \\xrightarrow{f} R \\to \\ldots )", "$$", "Hence we see that in degree $p$ we obtain", "$$", "\\colim K^p(R, f_1^n, \\ldots f_r^n) =", "\\bigoplus\\nolimits_{1 \\leq i_0 < \\ldots < i_{p - 1} \\leq r}", "R_{f_{i_0} \\ldots f_{i_{p - 1}}}", "$$", "Here the element $e^n_{j_1} \\wedge \\ldots \\wedge e^n_{j_{r - p}}$", "of the Koszul complex above maps in the colimit", "to the element $(f_{i_0} \\ldots f_{i_{p - 1}})^{-n}$ in the", "summand $R_{f_{i_0} \\ldots f_{i_{p - 1}}}$", "where the indices are chosen such that", "$\\{1, \\ldots r\\} = \\{i_0, \\ldots, i_{p - 1}\\}", "\\amalg \\{j_1, \\ldots, j_{r - p - 2}\\}$.", "Thus the differential on this complex is given by", "$$", "d(1\\text{ in }R_{f_{i_0} \\ldots f_{i_{p - 1}}}) =", "\\sum\\nolimits_{i \\not \\in \\{i_0, \\ldots, i_{p - 1}\\}}", "(-1)^{i - t}\\text{ in }", "R_{f_{i_0} \\ldots f_{i_t} f_i f_{i_{t + 1}} \\ldots f_{i_{p - 1}}}", "$$", "Thus if we consider the map of complexes given in degree $p$", "by the map", "$$", "\\bigoplus\\nolimits_{1 \\leq i_0 < \\ldots < i_{p - 1} \\leq r}", "R_{f_{i_0} \\ldots f_{i_{p - 1}}}", "\\longrightarrow", "\\bigoplus\\nolimits_{1 \\leq i_0 < \\ldots < i_{p - 1} \\leq r}", "R_{f_{i_0} \\ldots f_{i_{p - 1}}}", "$$", "determined by the rule", "$$", "1\\text{ in }R_{f_{i_0} \\ldots f_{i_{p - 1}}}", "\\longmapsto", "(-1)^{i_0 + \\ldots + i_{p - 1} + p}\\text{ in }R_{f_{i_0} \\ldots f_{i_{p - 1}}}", "$$", "then we get an isomorphism of complexes from", "$\\colim K(R, f_1^n, \\ldots, f_r^n)$ to the", "extended alternating {\\v C}ech complex defined in this section.", "We omit the verification that the signs work out." ], "refs": [ "more-algebra-definition-koszul-complex" ], "ref_ids": [ 10606 ] } ], "ref_ids": [] }, { "id": 9973, "type": "theorem", "label": "more-algebra-lemma-regular-koszul-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-regular-koszul-regular", "contents": [ "An $M$-regular sequence is $M$-Koszul-regular.", "A regular sequence is Koszul-regular." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a ring and let $M$ be an $R$-module.", "It is immediate that an $M$-regular sequence of length $1$ is", "$M$-Koszul-regular.", "Let $f_1, \\ldots, f_r$ be an $M$-regular sequence.", "Then $f_1$ is a nonzerodivisor on $M$. Hence", "$$", "0 \\to K_\\bullet(f_2, \\ldots, f_r) \\otimes M", "\\xrightarrow{f_1}", "K_\\bullet(f_2, \\ldots, f_r) \\otimes M \\to", "K_\\bullet(\\overline{f}_2, \\ldots, \\overline{f}_r) \\otimes M/f_1M \\to 0", "$$", "is a short exact sequence of complexes where $\\overline{f}_i$", "is the image of $f_i$ in $R/(f_1)$. By", "Lemma \\ref{lemma-cone-koszul}", "the complex $K_\\bullet(R, f_1, \\ldots, f_r)$", "is isomorphic to the cone of multiplication by $f_1$", "on $K_\\bullet(f_2, \\ldots, f_r)$. Thus", "$K_\\bullet(R, f_1, \\ldots, f_r) \\otimes M$ is isomorphic", "to the cone on the first map. Hence", "$K_\\bullet(\\overline{f}_2, \\ldots, \\overline{f}_r) \\otimes M/f_1M$", "is quasi-isomorphic to $K_\\bullet(f_1, \\ldots, f_r) \\otimes M$.", "As $\\overline{f}_2, \\ldots, \\overline{f}_r$ is an $M/f_1M$-regular sequence", "in $R/(f_1)$ the result follows from the case $r = 1$ and induction." ], "refs": [ "more-algebra-lemma-cone-koszul" ], "ref_ids": [ 9962 ] } ], "ref_ids": [] }, { "id": 9974, "type": "theorem", "label": "more-algebra-lemma-koszul-regular-H1-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-koszul-regular-H1-regular", "contents": [ "A $M$-Koszul-regular sequence is $M$-$H_1$-regular.", "A Koszul-regular sequence is $H_1$-regular." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9975, "type": "theorem", "label": "more-algebra-lemma-mult-koszul-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-mult-koszul-regular", "contents": [ "Let $f_1, \\ldots, f_{r - 1} \\in R$ be a sequence and $f, g \\in R$.", "Let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item If $f_1, \\ldots, f_{r - 1}, f$ and $f_1, \\ldots, f_{r - 1}, g$", "are $M$-$H_1$-regular then $f_1, \\ldots, f_{r - 1}, fg$ is", "$M$-$H_1$-regular too.", "\\item If $f_1, \\ldots, f_{r - 1}, f$ and $f_1, \\ldots, f_{r - 1}, g$", "are $M$-Koszul-regular then $f_1, \\ldots, f_{r - 1}, fg$ is", "$M$-Koszul-regular too.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-koszul-mult}", "we have exact sequences", "$$", "H_i(K_\\bullet(f_1, \\ldots, f_{r - 1}, f) \\otimes M) \\to", "H_i(K_\\bullet(f_1, \\ldots, f_{r - 1}, fg) \\otimes M) \\to", "H_i(K_\\bullet(f_1, \\ldots, f_{r - 1}, g) \\otimes M)", "$$", "for all $i$." ], "refs": [ "more-algebra-lemma-koszul-mult" ], "ref_ids": [ 9965 ] } ], "ref_ids": [] }, { "id": 9976, "type": "theorem", "label": "more-algebra-lemma-koszul-regular-flat-base-change", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-koszul-regular-flat-base-change", "contents": [ "Let $\\varphi : R \\to S$ be a flat ring map. Let $f_1, \\ldots, f_r \\in R$.", "Let $M$ be an $R$-module and set $N = M \\otimes_R S$.", "\\begin{enumerate}", "\\item If $f_1, \\ldots, f_r$ in $R$ is an $M$-$H_1$-regular sequence, then", "$\\varphi(f_1), \\ldots, \\varphi(f_r)$ is an $N$-$H_1$-regular", "sequence in $S$.", "\\item If $f_1, \\ldots, f_r$ is an $M$-Koszul-regular sequence in $R$, then", "$\\varphi(f_1), \\ldots, \\varphi(f_r)$ is an $N$-Koszul-regular", "sequence in $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$K_\\bullet(f_1, \\ldots, f_r) \\otimes_R S =", "K_\\bullet(\\varphi(f_1), \\ldots, \\varphi(f_r))$", "and therefore", "$(K_\\bullet(f_1, \\ldots, f_r) \\otimes_R M) \\otimes_R S =", "K_\\bullet(\\varphi(f_1), \\ldots, \\varphi(f_r)) \\otimes_S N$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9977, "type": "theorem", "label": "more-algebra-lemma-H1-regular-quasi-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-H1-regular-quasi-regular", "contents": [ "An $M$-$H_1$-regular sequence is $M$-quasi-regular." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a ring and let $M$ be an $R$-module.", "Let $f_1, \\ldots, f_r$ be an $M$-$H_1$-regular sequence.", "Denote $J = (f_1, \\ldots, f_r)$. The assumption means that we have", "an exact sequence", "$$", "\\wedge^2(R^r) \\otimes M \\to R^{\\oplus r} \\otimes M \\to JM \\to 0", "$$", "where the first arrow is given by", "$e_i \\wedge e_j \\otimes m \\mapsto (f_ie_j - f_je_i) \\otimes m$.", "In particular this implies that", "$$", "JM/J^2M = JM \\otimes_R R/J = (M/JM)^{\\oplus r}", "$$", "is a finite free module. To finish the proof we have to prove", "for every $n \\geq 2$ the following: if", "$$", "\\xi = \\sum\\nolimits_{|I| = n, I = (i_1, \\ldots, i_r)}", "m_I f_1^{i_1} \\ldots f_r^{i_r} \\in J^{n + 1}M", "$$", "then $m_I \\in JM$ for all $I$. Note that $f_1, \\ldots, f_{r - 1}, f_r^n$", "is an $M$-$H_1$-regular sequence by", "Lemma \\ref{lemma-mult-koszul-regular}.", "Hence we see that the required result holds for", "the multi-index $I = (0, \\ldots, 0, n)$. It turns out that we can", "reduce the general case to this case as follows.", "\\medskip\\noindent", "Let $S = R[x_1, x_2, \\ldots, x_r, 1/x_r]$. The ring map $R \\to S$ is faithfully", "flat, hence $f_1, \\ldots, f_r$ is an $M$-$H_1$-regular sequence in $S$, see", "Lemma \\ref{lemma-koszul-regular-flat-base-change}.", "By", "Lemma \\ref{lemma-change-basis}", "we see that", "$$", "g_1 = f_1 - x_1/x_r f_r, \\ldots", "g_{r - 1} = f_{r - 1} - x_{r - 1}/x_r f_r,", "g_r = (1/x_r)f_r", "$$", "is an $M$-$H_1$-regular sequence in $S$. Finally, note that our element", "$\\xi$ can be rewritten", "$$", "\\xi = \\sum\\nolimits_{|I| = n, I = (i_1, \\ldots, i_r)}", "m_I (g_1 + x_r g_r)^{i_1} \\ldots (g_{r - 1} + x_r g_r)^{i_{r - 1}}", "(x_rg_r)^{i_r}", "$$", "and the coefficient of $g_r^n$ in this expression is", "$$", "\\sum m_I x_1^{i_1} \\ldots x_r^{i_r} \\in J(M \\otimes_R S).", "$$", "Since the monomials $x_1^{i_1} \\ldots x_r^{i_r}$ form part of an $R$-basis", "of $S$ over $R$ we conclude that $m_I \\in J$ for all $I$ as desired." ], "refs": [ "more-algebra-lemma-mult-koszul-regular", "more-algebra-lemma-koszul-regular-flat-base-change", "more-algebra-lemma-change-basis" ], "ref_ids": [ 9975, 9976, 9958 ] } ], "ref_ids": [] }, { "id": 9978, "type": "theorem", "label": "more-algebra-lemma-noetherian-finite-all-equivalent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-noetherian-finite-all-equivalent", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring. Let $M$ be a nonzero", "finite $R$-module. Let $f_1, \\ldots, f_r \\in \\mathfrak m$. The following", "are equivalent", "\\begin{enumerate}", "\\item $f_1, \\ldots, f_r$ is an $M$-regular sequence,", "\\item $f_1, \\ldots, f_r$ is a $M$-Koszul-regular sequence,", "\\item $f_1, \\ldots, f_r$ is an $M$-$H_1$-regular sequence,", "\\item $f_1, \\ldots, f_r$ is an $M$-quasi-regular sequence.", "\\end{enumerate}", "In particular the sequence $f_1, \\ldots, f_r$ is a regular sequence", "in $R$ if and only if it is a Koszul regular sequence, if and only if", "it is a $H_1$-regular sequence, if and only if it is a quasi-regular sequence." ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) is ", "Lemma \\ref{lemma-regular-koszul-regular}.", "The implication (2) $\\Rightarrow$ (3) is", "Lemma \\ref{lemma-koszul-regular-H1-regular}.", "The implication (3) $\\Rightarrow$ (4) is ", "Lemma \\ref{lemma-H1-regular-quasi-regular}.", "The implication (4) $\\Rightarrow$ (1) is", "Algebra, Lemma \\ref{algebra-lemma-quasi-regular-regular}." ], "refs": [ "more-algebra-lemma-regular-koszul-regular", "more-algebra-lemma-koszul-regular-H1-regular", "more-algebra-lemma-H1-regular-quasi-regular", "algebra-lemma-quasi-regular-regular" ], "ref_ids": [ 9973, 9974, 9977, 750 ] } ], "ref_ids": [] }, { "id": 9979, "type": "theorem", "label": "more-algebra-lemma-H1-regular-in-quotient", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-H1-regular-in-quotient", "contents": [ "Let $A$ be a ring. Let $I \\subset A$ be an ideal.", "Let $g_1, \\ldots, g_m$ be a sequence in $A$ whose image in", "$A/I$ is $H_1$-regular. Then $I \\cap (g_1, \\ldots, g_m) =", "I(g_1, \\ldots, g_m)$." ], "refs": [], "proofs": [ { "contents": [ "Consider the exact sequence of complexes", "$$", "0 \\to I \\otimes_A K_\\bullet(A, g_1, \\ldots, g_m)", "\\to K_\\bullet(A, g_1, \\ldots, g_m) \\to", "K_\\bullet(A/I, g_1, \\ldots, g_m) \\to 0", "$$", "Since the complex on the right has $H_1 = 0$ by assumption we", "see that", "$$", "\\Coker(I^{\\oplus m} \\to I)", "\\longrightarrow", "\\Coker(A^{\\oplus m} \\to A)", "$$", "is injective. This is equivalent to the assertion of the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9980, "type": "theorem", "label": "more-algebra-lemma-conormal-sequence-H1-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-conormal-sequence-H1-regular", "contents": [ "Let $A$ be a ring. Let $I \\subset J \\subset A$ be ideals.", "Assume that $J/I \\subset A/I$ is generated by an $H_1$-regular sequence.", "Then $I \\cap J^2 = IJ$." ], "refs": [], "proofs": [ { "contents": [ "To prove this choose $g_1, \\ldots, g_m \\in J$", "whose images in $A/I$ form a $H_1$-regular sequence which generates $J/I$.", "In particular $J = I + (g_1, \\ldots, g_m)$.", "Suppose that $x \\in I \\cap J^2$. Because $x \\in J^2$ can write", "$$", "x =", "\\sum a_{ij} g_ig_j +", "\\sum a_j g_j +", "a", "$$", "with $a_{ij} \\in A$, $a_j \\in I$ and $a \\in I^2$.", "Then $\\sum a_{ij}g_ig_j \\in I \\cap (g_1, \\ldots, g_m)$", "hence by", "Lemma \\ref{lemma-H1-regular-in-quotient}", "we see that $\\sum a_{ij}g_ig_j \\in I(g_1, \\ldots, g_m)$.", "Thus $x \\in IJ$ as desired." ], "refs": [ "more-algebra-lemma-H1-regular-in-quotient" ], "ref_ids": [ 9979 ] } ], "ref_ids": [] }, { "id": 9981, "type": "theorem", "label": "more-algebra-lemma-join-quasi-regular-H1-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-join-quasi-regular-H1-regular", "contents": [ "Let $A$ be a ring. Let $I$ be an ideal generated by a quasi-regular", "sequence $f_1, \\ldots, f_n$ in $A$. Let $g_1, \\ldots, g_m \\in A$ be", "elements whose images $\\overline{g}_1, \\ldots, \\overline{g}_m$ form an", "$H_1$-regular sequence in $A/I$. Then $f_1, \\ldots, f_n, g_1, \\ldots, g_m$", "is a quasi-regular sequence in $A$." ], "refs": [], "proofs": [ { "contents": [ "We claim that $g_1, \\ldots, g_m$ forms an $H_1$-regular sequence in", "$A/I^d$ for every $d$. By induction assume that this holds in", "$A/I^{d - 1}$. We have a short exact sequence of complexes", "$$", "0 \\to K_\\bullet(A, g_\\bullet) \\otimes_A I^{d - 1}/I^d", "\\to K_\\bullet(A/I^d, g_\\bullet) \\to", "K_\\bullet(A/I^{d - 1}, g_\\bullet) \\to 0", "$$", "Since $f_1, \\ldots, f_n$ is quasi-regular we see that the first complex", "is a direct sum of copies of $K_\\bullet(A/I, g_1, \\ldots, g_m)$", "hence acyclic in degree $1$. By induction hypothesis the last complex is", "acyclic in degree $1$. Hence also the middle complex is.", "In particular, the sequence $g_1, \\ldots, g_m$ forms a quasi-regular", "sequence in $A/I^d$ for every $d \\geq 1$, see", "Lemma \\ref{lemma-H1-regular-quasi-regular}.", "Now we are ready to prove that $f_1, \\ldots, f_n, g_1, \\ldots, g_m$", "is a quasi-regular sequence in $A$.", "Namely, set $J = (f_1, \\ldots, f_n, g_1, \\ldots, g_m)$ and suppose", "that (with multinomial notation)", "$$", "\\sum\\nolimits_{|N| + |M| = d} a_{N, M} f^N g^M \\in J^{d + 1}", "$$", "for some $a_{N, M} \\in A$. We have to show that $a_{N, M} \\in J$", "for all $N, M$. Let $e \\in \\{0, 1, \\ldots, d\\}$. Then", "$$", "\\sum\\nolimits_{|N| = d - e, \\ |M| = e} a_{N, M} f^N g^M \\in", "(g_1, \\ldots, g_m)^{e + 1} + I^{d - e + 1}", "$$", "Because $g_1, \\ldots, g_m$ is a quasi-regular sequence in $A/I^{d - e + 1}$", "we deduce", "$$", "\\sum\\nolimits_{|N| = d - e} a_{N, M} f^N \\in", "(g_1, \\ldots, g_m) + I^{d - e + 1}", "$$", "for each $M$ with $|M| = e$. By", "Lemma \\ref{lemma-H1-regular-in-quotient}", "applied to $I^{d - e}/I^{d - e + 1}$ in the ring $A/I^{d - e + 1}$", "this implies $\\sum_{|N| = d - e} a_{N, M} f^N \\in I^{d - e}(g_1, \\ldots, g_m)$.", "Since $f_1, \\ldots, f_n$ is quasi-regular in $A$ this implies", "that $a_{N, M} \\in J$ for each $N, M$ with $|N| = d - e$ and $|M| = e$.", "This proves the lemma." ], "refs": [ "more-algebra-lemma-H1-regular-quasi-regular", "more-algebra-lemma-H1-regular-in-quotient" ], "ref_ids": [ 9977, 9979 ] } ], "ref_ids": [] }, { "id": 9982, "type": "theorem", "label": "more-algebra-lemma-join-H1-regular-sequences", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-join-H1-regular-sequences", "contents": [ "Let $A$ be a ring. Let $I$ be an ideal generated by an", "$H_1$-regular sequence $f_1, \\ldots, f_n$ in $A$.", "Let $g_1, \\ldots, g_m \\in A$ be elements whose images", "$\\overline{g}_1, \\ldots, \\overline{g}_m$ form an $H_1$-regular sequence", "in $A/I$. Then $f_1, \\ldots, f_n, g_1, \\ldots, g_m$ is an $H_1$-regular", "sequence in $A$." ], "refs": [], "proofs": [ { "contents": [ "We have to show that $H_1(A, f_1, \\ldots, f_n, g_1, \\ldots, g_m) = 0$.", "To do this consider the commutative diagram", "$$", "\\xymatrix{", "\\wedge^2(A^{\\oplus n + m}) \\ar[r] \\ar[d] &", "A^{\\oplus n + m} \\ar[r] \\ar[d] &", "A \\ar[r] \\ar[d] & 0 \\\\", "\\wedge^2(A/I^{\\oplus m}) \\ar[r] &", "A/I^{\\oplus m} \\ar[r] &", "A/I \\ar[r] & 0", "}", "$$", "Consider an element $(a_1, \\ldots, a_{n + m}) \\in A^{\\oplus n + m}$", "which maps to zero in $A$. Because $\\overline{g}_1, \\ldots, \\overline{g}_m$", "form an $H_1$-regular sequence in $A/I$ we see that", "$(\\overline{a}_{n + 1}, \\ldots, \\overline{a}_{n + m})$ is the image", "of some element $\\overline{\\alpha}$ of $\\wedge^2(A/I^{\\oplus m})$.", "We can lift $\\overline{\\alpha}$ to an element", "$\\alpha \\in \\wedge^2(A^{\\oplus n + m})$ and substract the image of it", "in $A^{\\oplus n + m}$ from our element $(a_1, \\ldots, a_{n + m})$.", "Thus we may assume that $a_{n + 1}, \\ldots, a_{n + m} \\in I$.", "Since $I = (f_1, \\ldots, f_n)$ we can modify our element", "$(a_1, \\ldots, a_{n + m})$ by linear combinations of the elements", "$$", "(0, \\ldots, g_j, 0, \\ldots, 0, f_i, 0, \\ldots, 0)", "$$", "in the image of the top left horizontal arrow to reduce to the case", "that $a_{n + 1}, \\ldots, a_{n + m}$ are zero. In this case", "$(a_1, \\ldots, a_n, 0, \\ldots, 0)$ defines an element of", "$H_1(A, f_1, \\ldots, f_n)$ which we assumed to be zero." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9983, "type": "theorem", "label": "more-algebra-lemma-truncate-H1-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-truncate-H1-regular", "contents": [ "Let $A$ be a ring. Let $f_1, \\ldots, f_n, g_1, \\ldots, g_m \\in A$", "be an $H_1$-regular sequence. Then the images", "$\\overline{g}_1, \\ldots, \\overline{g}_m$ in $A/(f_1, \\ldots, f_n)$", "form an $H_1$-regular sequence." ], "refs": [], "proofs": [ { "contents": [ "Set $I = (f_1, \\ldots, f_n)$. We have to show that any relation", "$\\sum_{j = 1, \\ldots, m} \\overline{a}_j \\overline{g}_j$ in $A/I$", "is a linear combination of trivial relations. Because", "$I = (f_1, \\ldots, f_n)$ we can lift this relation to a relation", "$$", "\\sum\\nolimits_{j = 1, \\ldots, m} a_j g_j +", "\\sum\\nolimits_{i = 1, \\ldots, n} b_if_i = 0", "$$", "in $A$. By assumption this relation in $A$ is a linear combination of", "trivial relations. Taking the image in $A/I$ we obtain what we want." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 9984, "type": "theorem", "label": "more-algebra-lemma-join-koszul-regular-sequences", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-join-koszul-regular-sequences", "contents": [ "Let $A$ be a ring. Let $I$ be an ideal generated by a Koszul-regular", "sequence $f_1, \\ldots, f_n$ in $A$. Let $g_1, \\ldots, g_m \\in A$ be", "elements whose images $\\overline{g}_1, \\ldots, \\overline{g}_m$ form a", "Koszul-regular sequence in $A/I$. Then $f_1, \\ldots, f_n, g_1, \\ldots, g_m$", "is a Koszul-regular sequence in $A$." ], "refs": [], "proofs": [ { "contents": [ "Our assumptions say that $K_\\bullet(A, f_1, \\ldots, f_n)$ is a finite free", "resolution of $A/I$ and", "$K_\\bullet(A/I, \\overline{g}_1, \\ldots, \\overline{g}_m)$ is a", "finite free resolution of $A/(f_i, g_j)$ over $A/I$. Then", "\\begin{align*}", "K_\\bullet(A, f_1, \\ldots, f_n, g_1, \\ldots, g_m)", "& = \\text{Tot}(K_\\bullet(A, f_1, \\ldots, f_n) \\otimes_A", "K_\\bullet(A, g_1, \\ldots, g_m)) \\\\", "& \\cong A/I \\otimes_A K_\\bullet(A, g_1, \\ldots, g_m) \\\\", "& = K_\\bullet(A/I, \\overline{g}_1, \\ldots, \\overline{g}_m) \\\\", "& \\cong A/(f_i, g_j)", "\\end{align*}", "The first equality by", "Lemma \\ref{lemma-join-sequences-koszul-complex}.", "The first quasi-isomorphism $\\cong$ by (the dual of)", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}", "as the $q$th row of the double complex", "$K_\\bullet(A, f_1, \\ldots, f_n) \\otimes_A K_\\bullet(A, g_1, \\ldots, g_m)$", "is a resolution of $A/I \\otimes_A K_q(A, g_1, \\ldots, g_m)$.", "The second equality is clear. The last quasi-isomorphism by assumption.", "Hence we win." ], "refs": [ "more-algebra-lemma-join-sequences-koszul-complex", "homology-lemma-double-complex-gives-resolution" ], "ref_ids": [ 9966, 12106 ] } ], "ref_ids": [] }, { "id": 9985, "type": "theorem", "label": "more-algebra-lemma-truncate-koszul-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-truncate-koszul-regular", "contents": [ "Let $A$ be a ring. Let $f_1, \\ldots, f_n, g_1, \\ldots, g_m \\in A$.", "If both $f_1, \\ldots, f_n$ and $f_1, \\ldots, f_n, g_1, \\ldots, g_m$", "are Koszul-regular sequences in $A$, then", "$\\overline{g}_1, \\ldots, \\overline{g}_m$ in $A/(f_1, \\ldots, f_n)$", "form a Koszul-regular sequence." ], "refs": [], "proofs": [ { "contents": [ "Set $I = (f_1, \\ldots, f_n)$.", "Our assumptions say that $K_\\bullet(A, f_1, \\ldots, f_n)$ is a finite free", "resolution of $A/I$ and", "$K_\\bullet(A, f_1, \\ldots, f_n, g_1, \\ldots, g_m)$ is a", "finite free resolution of $A/(f_i, g_j)$ over $A$. Then", "\\begin{align*}", "A/(f_i, g_j) & \\cong K_\\bullet(A, f_1, \\ldots, f_n, g_1, \\ldots, g_m) \\\\", "& = \\text{Tot}(K_\\bullet(A, f_1, \\ldots, f_n) \\otimes_A", "K_\\bullet(A, g_1, \\ldots, g_m)) \\\\", "& \\cong A/I \\otimes_A K_\\bullet(A, g_1, \\ldots, g_m) \\\\", "& = K_\\bullet(A/I, \\overline{g}_1, \\ldots, \\overline{g}_m)", "\\end{align*}", "The first quasi-isomorphism $\\cong$ by assumption. The first equality by", "Lemma \\ref{lemma-join-sequences-koszul-complex}.", "The second quasi-isomorphism by (the dual of)", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}", "as the $q$th row of the double complex", "$K_\\bullet(A, f_1, \\ldots, f_n) \\otimes_A K_\\bullet(A, g_1, \\ldots, g_m)$", "is a resolution of $A/I \\otimes_A K_q(A, g_1, \\ldots, g_m)$.", "The second equality is clear. Hence we win." ], "refs": [ "more-algebra-lemma-join-sequences-koszul-complex", "homology-lemma-double-complex-gives-resolution" ], "ref_ids": [ 9966, 12106 ] } ], "ref_ids": [] }, { "id": 9986, "type": "theorem", "label": "more-algebra-lemma-independence-of-generators", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-independence-of-generators", "contents": [ "Let $R$ be a ring. Let $I$ be an ideal generated by $f_1, \\ldots, f_r \\in R$.", "\\begin{enumerate}", "\\item If $I$ can be generated by a quasi-regular sequence of length $r$,", "then $f_1, \\ldots, f_r$ is a quasi-regular sequence.", "\\item If $I$ can be generated by an $H_1$-regular sequence of length $r$,", "then $f_1, \\ldots, f_r$ is an $H_1$-regular sequence.", "\\item If $I$ can be generated by a Koszul-regular sequence of length $r$,", "then $f_1, \\ldots, f_r$ is a Koszul-regular sequence.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $I$ can be generated by a quasi-regular sequence of length $r$,", "then $I/I^2$ is free of rank $r$ over $R/I$. Since $f_1, \\ldots, f_r$", "generate by assumption we see that the images $\\overline{f}_i$ form a basis of", "$I/I^2$ over $R/I$. It follows that $f_1, \\ldots, f_r$ is a quasi-regular", "sequence as all this means, besides the freeness of $I/I^2$, is that the maps", "$\\text{Sym}^n_{R/I}(I/I^2) \\to I^n/I^{n + 1}$ are isomorphisms.", "\\medskip\\noindent", "We continue to assume that $I$ can be generated by a", "quasi-regular sequence, say", "$g_1, \\ldots, g_r$. Write $g_j = \\sum a_{ij}f_i$. As $f_1, \\ldots, f_r$", "is quasi-regular according to the previous paragraph, we see that", "$\\det(a_{ij})$ is invertible mod $I$. The matrix", "$a_{ij}$ gives a map $R^{\\oplus r} \\to R^{\\oplus r}$ which induces", "a map of Koszul complexes", "$\\alpha : K_\\bullet(R, f_1, \\ldots, f_r) \\to K_\\bullet(R, g_1, \\ldots, g_r)$,", "see", "Lemma \\ref{lemma-functorial}.", "This map becomes an isomorphism on inverting $\\det(a_{ij})$.", "Since the cohomology modules of both $K_\\bullet(R, f_1, \\ldots, f_r)$ and", "$K_\\bullet(R, g_1, \\ldots, g_r)$ are annihilated by $I$, see", "Lemma \\ref{lemma-homotopy-koszul},", "we see that $\\alpha$ is a quasi-isomorphism.", "\\medskip\\noindent", "Now assume that $g_1, \\ldots, g_r$ is a $H_1$-regular sequence generating $I$.", "Then $g_1, \\ldots, g_r$ is a quasi-regular sequence by", "Lemma \\ref{lemma-H1-regular-quasi-regular}. By the previous paragraph", "we conclude that $f_1, \\ldots, f_r$ is a $H_1$-regular sequence.", "Similarly for Koszul-regular sequences." ], "refs": [ "more-algebra-lemma-functorial", "more-algebra-lemma-homotopy-koszul", "more-algebra-lemma-H1-regular-quasi-regular" ], "ref_ids": [ 9957, 9960, 9977 ] } ], "ref_ids": [] }, { "id": 9987, "type": "theorem", "label": "more-algebra-lemma-make-nonzero-divisor", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-make-nonzero-divisor", "contents": [ "\\begin{reference}", "This is a particular case of \\cite[Corollary]{McCoy}", "\\end{reference}", "Let $R$ be a ring. Let $a_1, \\ldots, a_n \\in R$ be elements such", "that $R \\to R^{\\oplus n}$, $x \\mapsto (xa_1, \\ldots, xa_n)$ is injective.", "Then the element $\\sum a_i t_i$ of the polynomial ring $R[t_1, \\ldots, t_n]$", "is a nonzerodivisor." ], "refs": [], "proofs": [ { "contents": [ "If one of the $a_i$ is a unit this is just the statement that any", "element of the form $t_1 + a_2 t_2 + \\ldots + a_n t_n$ is a nonzerodivisor", "in the polynomial ring over $R$.", "\\medskip\\noindent", "Case I: $R$ is Noetherian. Let $\\mathfrak q_j$, $j = 1, \\ldots, m$", "be the associated primes of $R$. We have to show that", "each of the maps", "$$", "\\sum a_i t_i :", "\\text{Sym}^d(R^{\\oplus n})", "\\longrightarrow", "\\text{Sym}^{d + 1}(R^{\\oplus n})", "$$", "is injective. As $\\text{Sym}^d(R^{\\oplus n})$ is a free $R$-module its", "associated primes are $\\mathfrak q_j$, $j = 1, \\ldots, m$. For each $j$", "there exists an $i = i(j)$ such that $a_i \\not \\in \\mathfrak q_j$ because", "there exists an $x \\in R$ with $\\mathfrak q_jx = 0$ but $a_i x \\not = 0$", "for some $i$ by assumption. Hence $a_i$ is a unit in $R_{\\mathfrak q_j}$", "and the map is injective after localizing at $\\mathfrak q_j$. Thus the map", "is injective, see", "Algebra, Lemma \\ref{algebra-lemma-zero-at-ass-zero}.", "\\medskip\\noindent", "Case II: $R$ general. We can write $R$ as the union of Noetherian", "rings $R_\\lambda$ with $a_1, \\ldots, a_n \\in R_\\lambda$. For each $R_\\lambda$", "the result holds, hence the result holds for $R$." ], "refs": [ "algebra-lemma-zero-at-ass-zero" ], "ref_ids": [ 713 ] } ], "ref_ids": [] }, { "id": 9988, "type": "theorem", "label": "more-algebra-lemma-Koszul-regular-flat-locally-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Koszul-regular-flat-locally-regular", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_n$ be a Koszul-regular sequence", "in $R$ such that $(f_1, \\ldots, f_n) \\not = R$.", "Consider the faithfully flat, smooth ring map", "$$", "R \\longrightarrow", "S = R[\\{t_{ij}\\}_{i \\leq j}, t_{11}^{-1}, t_{22}^{-1}, \\ldots, t_{nn}^{-1}]", "$$", "For $1 \\leq i \\leq n$ set", "$$", "g_i = \\sum\\nolimits_{i \\leq j} t_{ij} f_j \\in S.", "$$", "Then $g_1, \\ldots, g_n$ is a regular sequence in $S$ and", "$(f_1, \\ldots, f_n)S = (g_1, \\ldots, g_n)$." ], "refs": [], "proofs": [ { "contents": [ "The equality of ideals is obvious as the matrix", "$$", "\\left(", "\\begin{matrix}", "t_{11} & t_{12} & t_{13} & \\ldots \\\\", "0 & t_{22} & t_{23} & \\ldots \\\\", "0 & 0 & t_{33} & \\ldots \\\\", "\\ldots & \\ldots & \\ldots & \\ldots", "\\end{matrix}", "\\right)", "$$", "is invertible in $S$.", "Because $f_1, \\ldots, f_n$ is a Koszul-regular sequence we see that", "the kernel of", "$R \\to R^{\\oplus n}$, $x \\mapsto (xf_1, \\ldots, xf_n)$ is zero (as it", "computes the $n$the Koszul homology of $R$ w.r.t.\\ $f_1, \\ldots, f_n$).", "Hence by", "Lemma \\ref{lemma-make-nonzero-divisor}", "we see that $g_1 = f_1 t_{11} + \\ldots + f_n t_{1n}$ is a nonzerodivisor", "in $S' = R[t_{11}, t_{12}, \\ldots, t_{1n}, t_{11}^{-1}]$. We see that", "$g_1, f_2, \\ldots, f_n$ is a Koszul-sequence in $S'$ by", "Lemma \\ref{lemma-koszul-regular-flat-base-change} and", "\\ref{lemma-independence-of-generators}.", "We conclude that", "$\\overline{f}_2, \\ldots, \\overline{f}_n$ is a Koszul-regular sequence", "in $S'/(g_1)$ by", "Lemma \\ref{lemma-truncate-koszul-regular}.", "Hence by induction on $n$ we see that the images", "$\\overline{g}_2, \\ldots, \\overline{g}_n$ of $g_2, \\ldots, g_n$ in", "$S'/(g_1)[\\{t_{ij}\\}_{2 \\leq i \\leq j}, t_{22}^{-1}, \\ldots, t_{nn}^{-1}]$", "form a regular sequence. This in turn means that", "$g_1, \\ldots, g_n$ forms a regular sequence in $S$." ], "refs": [ "more-algebra-lemma-make-nonzero-divisor", "more-algebra-lemma-koszul-regular-flat-base-change", "more-algebra-lemma-independence-of-generators", "more-algebra-lemma-truncate-koszul-regular" ], "ref_ids": [ 9987, 9976, 9986, 9985 ] } ], "ref_ids": [] }, { "id": 9989, "type": "theorem", "label": "more-algebra-lemma-vanishing-extended-alternating-koszul", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-vanishing-extended-alternating-koszul", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be an", "Koszul-regular sequence. Then the extended alternating", "{\\v C}ech complex $R \\to \\bigoplus\\nolimits_{i_0} R_{f_{i_0}} \\to", "\\bigoplus\\nolimits_{i_0 < i_1} R_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to R_{f_1\\ldots f_r}$ from Section \\ref{section-alternating-cech}", "only has cohomology in degree $r$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-mult-koszul-regular} and induction the sequence", "$f_1, \\ldots, f_{r - 1}, f_r^n$ is Koszul regular for all $n \\geq 1$.", "By Lemma \\ref{lemma-change-basis} any permutation of a Koszul regular", "sequence is a Koszul regular sequence. Hence we see that we may replace", "any (or all) $f_i$ by its $n$th power and still have a Koszul regular", "sequence. Thus $K_\\bullet(R, f_1^n, \\ldots, f_r^n)$ has nonzero", "cohomology only in homological degree $0$. This implies what we want by", "Lemma \\ref{lemma-extended-alternating-Cech-is-colimit-koszul}." ], "refs": [ "more-algebra-lemma-mult-koszul-regular", "more-algebra-lemma-change-basis", "more-algebra-lemma-extended-alternating-Cech-is-colimit-koszul" ], "ref_ids": [ 9975, 9958, 9972 ] } ], "ref_ids": [] }, { "id": 9990, "type": "theorem", "label": "more-algebra-lemma-blowup-regular-sequence", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-blowup-regular-sequence", "contents": [ "Let $a, a_2, \\ldots, a_r$ be an $H_1$-regular sequence in a ring $R$", "(for example a Koszul regular sequence or a regular sequence, see", "Lemmas \\ref{lemma-regular-koszul-regular} and", "\\ref{lemma-koszul-regular-H1-regular}).", "With $I = (a, a_2, \\ldots, a_r)$ the blowup algebra $R' = R[\\frac{I}{a}]$", "is isomorphic to $R'' = R[y_2, \\ldots, y_r]/(a y_i - a_i)$." ], "refs": [ "more-algebra-lemma-regular-koszul-regular", "more-algebra-lemma-koszul-regular-H1-regular" ], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-affine-blowup-quotient-description}", "it suffices to show that $R''$ is $a$-torsion free.", "\\medskip\\noindent", "We claim $a, ay_2 - a_2, \\ldots, ay_n - a_r$ is a $H_1$-regular", "sequence in $R[y_2, \\ldots, y_r]$. Namely, the map", "$$", "(a, ay_2 - a_2, \\ldots, ay_n - a_r) :", "R[y_2, \\ldots, y_r]^{\\oplus r}", "\\longrightarrow", "R[y_2, \\ldots, y_r]", "$$", "used to define the Koszul complex on $a, ay_2 - a_2, \\ldots, ay_n - a_r$", "is isomorphic to the map", "$$", "(a, a_2, \\ldots, a_r) :", "R[y_2, \\ldots, y_r]^{\\oplus r} \\longrightarrow", "R[y_2, \\ldots, y_r]", "$$", "used to the define the Koszul complex on $a, a_2, \\ldots, a_r$", "via the isomorphism", "$$", "R[y_2, \\ldots, y_r]^{\\oplus r}", "\\longrightarrow", "R[y_2, \\ldots, y_r]^{\\oplus r}", "$$", "sending $(b_1, \\ldots, b_r)$ to", "$(b_1 - b_2y_2 \\ldots - b_ry_r, -b_2, \\ldots, - b_r)$.", "By Lemma \\ref{lemma-functorial} these Koszul complexes are isomorphic.", "By Lemma \\ref{lemma-koszul-regular-flat-base-change}", "applied to the flat ring map $R \\to R[y_2, \\ldots, y_r]$", "we conclude our claim is true. By Lemma \\ref{lemma-cone-koszul}", "we see that the Koszul complex $K$ on $a, ay_2 - a_2, \\ldots, ay_n - a_r$", "is the cone on $a : L \\to L$ where $L$ is the Koszul complex", "on $ay_2 - a_2, \\ldots, ay_n - a_r$. Since $H_1(K) = 0$ by the claim,", "we conclude that $a : H_0(L) \\to H_0(L)$ is injective, in other words that", "$R'' = R[y_2, \\ldots, y_r]/(a y_i - a_i)$", "has no nonzero $a$-torsion elements as desired." ], "refs": [ "algebra-lemma-affine-blowup-quotient-description", "more-algebra-lemma-functorial", "more-algebra-lemma-koszul-regular-flat-base-change", "more-algebra-lemma-cone-koszul" ], "ref_ids": [ 754, 9957, 9976, 9962 ] } ], "ref_ids": [ 9973, 9974 ] }, { "id": 9991, "type": "theorem", "label": "more-algebra-lemma-base-change-H1-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-H1-regular", "contents": [ "Let $A \\to B$ be a ring map.", "Let $f_1, \\ldots, f_r$ be a sequence in $B$ such that $B/(f_1, \\ldots, f_r)$", "is $A$-flat. Let $A \\to A'$ be a ring map. Then the canonical map", "$$", "H_1(K_\\bullet(B, f_1, \\ldots, f_r)) \\otimes_A A'", "\\longrightarrow", "H_1(K_\\bullet(B', f'_1, \\ldots, f'_r))", "$$", "is an isomorphism. Here $B' = B \\otimes_A A'$ and $f_i' \\in B'$ is the image", "of $f_i$." ], "refs": [], "proofs": [ { "contents": [ "The sequence", "$$", "\\wedge^2(B^{\\oplus r}) \\to B^{\\oplus r} \\to B \\to B/J \\to 0", "$$", "is a complex of $A$-modules with $B/J$ flat over $A$ and", "cohomology group $H_1 = H_1(K_\\bullet(B, f_1, \\ldots, f_r))$ in the spot", "$B^{\\oplus r}$. If we tensor this with $A'$ we obtain a complex", "$$", "\\wedge^2((B')^{\\oplus r}) \\to (B')^{\\oplus r} \\to B' \\to B'/J' \\to 0", "$$", "which is exact at $B'$ and $B'/J'$. In order to compute its", "cohomology group $H'_1 = H_1(K_\\bullet(B', f'_1, \\ldots, f'_r))$", "at $(B')^{\\oplus r}$ we split the first sequence above into", "the exact sequences $0 \\to J \\to B \\to B/J \\to 0$,", "$0 \\to K \\to B^{\\oplus r} \\to J \\to 0$ and", "$\\wedge^2(B^{\\oplus r}) \\to K \\to H_1 \\to 0$.", "Tensoring over $A$ with $A'$ we obtain the exact sequences", "$$", "\\begin{matrix}", "0 \\to J \\otimes_A A' \\to B \\otimes_A A' \\to (B/J) \\otimes_A A' \\to 0 \\\\", "K \\otimes_A A' \\to B^{\\oplus r} \\otimes_A A' \\to J \\otimes_A A' \\to 0 \\\\", "\\wedge^2(B^{\\oplus r}) \\otimes_A A' \\to K \\otimes_A A' \\to H_1 \\otimes_A A'", "\\to 0", "\\end{matrix}", "$$", "where the first one is exact as $B/J$ is flat over $A$, see", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}. We conclude", "that $J' = J \\otimes_A A'$ and", "$\\Ker((B')^{\\oplus r} \\to B') = K \\otimes_A A'$. Thus", "\\begin{align*}", "H'_1 & =", "\\Coker\\left(", "\\wedge^2((B')^{\\oplus r}) \\to \\Ker((B')^{\\oplus r} \\to B')", "\\right) \\\\", "& =", "\\Coker\\left(\\wedge^2(B^{\\oplus r}) \\otimes_A A' \\to K \\otimes_A A'\\right) \\\\", "& =", "H_1 \\otimes_A A'", "\\end{align*}", "as $-\\otimes_A A'$ is right exact. This proves the lemma." ], "refs": [ "algebra-lemma-flat-tor-zero" ], "ref_ids": [ 532 ] } ], "ref_ids": [] }, { "id": 9992, "type": "theorem", "label": "more-algebra-lemma-relative-regular-immersion-algebra", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-relative-regular-immersion-algebra", "contents": [ "Let $A \\to B$ and $A \\to A'$ be ring maps. Set $B' = B \\otimes_A A'$.", "Let $f_1, \\ldots, f_r \\in B$. Assume $B/(f_1, \\ldots, f_r)B$ is flat over $A$", "\\begin{enumerate}", "\\item If $f_1, \\ldots, f_r$ is a quasi-regular sequence, then", "the image in $B'$ is a quasi-regular sequence.", "\\item If $f_1, \\ldots, f_r$ is a $H_1$-regular sequence, then", "the image in $B'$ is a $H_1$-regular sequence.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f_1, \\ldots, f_r$ is quasi-regular. Set $J = (f_1, \\ldots, f_r)$.", "By assumption $J^n/J^{n + 1}$ is isomorphic to a direct sum of copies of", "$B/J$ hence flat over $A$. By induction and", "Algebra, Lemma \\ref{algebra-lemma-flat-ses}", "we conclude that $B/J^n$ is flat over $A$. The ideal $(J')^n$ is equal to", "$J^n \\otimes_A A'$, see", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}.", "Hence $(J')^n/(J')^{n + 1} = J^n/J^{n + 1} \\otimes_A A'$ which clearly", "implies that $f_1, \\ldots, f_r$ is a quasi-regular sequence in $B'$.", "\\medskip\\noindent", "Assume $f_1, \\ldots, f_r$ is $H_1$-regular. By", "Lemma \\ref{lemma-base-change-H1-regular}", "the vanishing of the Koszul homology group", "$H_1(K_\\bullet(B, f_1, \\ldots, f_r))$", "implies the vanishing of $H_1(K_\\bullet(B', f'_1, \\ldots, f'_r))$", "and we win." ], "refs": [ "algebra-lemma-flat-ses", "algebra-lemma-flat-tor-zero", "more-algebra-lemma-base-change-H1-regular" ], "ref_ids": [ 533, 532, 9991 ] } ], "ref_ids": [] }, { "id": 9993, "type": "theorem", "label": "more-algebra-lemma-cut-by-koszul-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cut-by-koszul-flat", "contents": [ "Let $A' \\to B'$ be a ring map. Let $I \\subset A'$ be an ideal.", "Set $A = A'/I$ and $B = B'/IB'$. Let $f'_1, \\ldots, f'_r \\in B'$. Assume", "\\begin{enumerate}", "\\item $A' \\to B'$ is flat and of finite presentation,", "\\item $I$ is locally nilpotent,", "\\item the images $f_1, \\ldots, f_r \\in B$ form a quasi-regular sequence,", "\\item $B/(f_1, \\ldots, f_r)$ is flat over $A$.", "\\end{enumerate}", "Then $B'/(f'_1, \\ldots, f'_r)$ is flat over $A'$." ], "refs": [], "proofs": [ { "contents": [ "Set $C' = B'/(f'_1, \\ldots, f'_r)$. We have to show $A' \\to C'$ is flat.", "Let $\\mathfrak r' \\subset C'$ be a prime ideal lying over", "$\\mathfrak p' \\subset A'$. We let $\\mathfrak q' \\subset B'$", "be the inverse image of $\\mathfrak r'$.", "By Algebra, Lemma \\ref{algebra-lemma-flat-localization}", "it suffices to show that $A'_{\\mathfrak p'} \\to C'_{\\mathfrak q'}$ is flat.", "Algebra, Lemma \\ref{algebra-lemma-grothendieck-regular-sequence-general}", "tells us it suffices to show that $f'_1, \\ldots, f'_r$ map to", "a regular sequence in", "$$", "B'_{\\mathfrak q'}/\\mathfrak p'B'_{\\mathfrak q'} =", "B_\\mathfrak q/\\mathfrak p B_\\mathfrak q =", "(B \\otimes_A \\kappa(\\mathfrak p))_\\mathfrak q", "$$", "with obvious notation. What we know is that $f_1, \\ldots, f_r$", "is a quasi-regular sequence in $B$ and that $B/(f_1, \\ldots, f_r)$", "is flat over $A$. By Lemma \\ref{lemma-relative-regular-immersion-algebra}", "the images $\\overline{f}_1, \\ldots, \\overline{f}_r$", "of $f'_1, \\ldots, f'_r$ in $B \\otimes_A \\kappa(\\mathfrak p)$ form a", "quasi-regular sequence. Since $(B \\otimes_A \\kappa(\\mathfrak p))_\\mathfrak q$", "is a Noetherian local ring, we conclude by Lemma", "\\ref{lemma-noetherian-finite-all-equivalent}." ], "refs": [ "algebra-lemma-flat-localization", "algebra-lemma-grothendieck-regular-sequence-general", "more-algebra-lemma-relative-regular-immersion-algebra", "more-algebra-lemma-noetherian-finite-all-equivalent" ], "ref_ids": [ 538, 1112, 9992, 9978 ] } ], "ref_ids": [] }, { "id": 9994, "type": "theorem", "label": "more-algebra-lemma-cut-by-koszul", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cut-by-koszul", "contents": [ "Let $A' \\to B'$ be a ring map. Let $I \\subset A'$ be an ideal.", "Set $A = A'/I$ and $B = B'/IB'$. Let $f'_1, \\ldots, f'_r \\in B'$. Assume", "\\begin{enumerate}", "\\item $A' \\to B'$ is flat and of finite presentation (for example smooth),", "\\item $I$ is locally nilpotent,", "\\item the images $f_1, \\ldots, f_r \\in B$ form a quasi-regular sequence,", "\\item $B/(f_1, \\ldots, f_r)$ is smooth over $A$.", "\\end{enumerate}", "Then $B'/(f'_1, \\ldots, f'_r)$ is smooth over $A'$." ], "refs": [], "proofs": [ { "contents": [ "Set $C' = B'/(f'_1, \\ldots, f'_r)$ and $C = B/(f_1, \\ldots, f_r)$.", "Then $A' \\to C'$ is of finite presentation.", "By Lemma \\ref{lemma-cut-by-koszul-flat} we see that $A' \\to C'$ is flat.", "The fibre rings of $A' \\to C'$ are equal to the fibre rings of $A \\to C$", "and hence smooth by assumption (4). It follows that", "$A' \\to C'$ is smooth by", "Algebra, Lemma \\ref{algebra-lemma-flat-fibre-smooth}." ], "refs": [ "more-algebra-lemma-cut-by-koszul-flat", "algebra-lemma-flat-fibre-smooth" ], "ref_ids": [ 9993, 1200 ] } ], "ref_ids": [] }, { "id": 9995, "type": "theorem", "label": "more-algebra-lemma-quasi-regular-ideal-finite", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-quasi-regular-ideal-finite", "contents": [ "A quasi-regular ideal is finitely generated." ], "refs": [], "proofs": [ { "contents": [ "Let $I \\subset R$ be a quasi-regular ideal. Since $V(I)$ is quasi-compact,", "there exist $g_1, \\ldots, g_m \\in R$ such that", "$V(I) \\subset D(g_1) \\cup \\ldots \\cup D(g_m)$", "and such that $I_{g_j}$ is generated by a quasi-regular sequence", "$g_{j1}, \\ldots, g_{jr_j} \\in R_{g_j}$. Write $g_{ji} = g'_{ji}/g_j^{e_{ij}}$", "for some $g'_{ij} \\in I$. Write $1 + x = \\sum g_j h_j$", "for some $x \\in I$ which is possible as", "$V(I) \\subset D(g_1) \\cup \\ldots \\cup D(g_m)$.", "Note that $\\Spec(R) = D(g_1) \\cup \\ldots \\cup D(g_m) \\bigcup D(x)$", "Then $I$ is generated by the elements $g'_{ij}$ and $x$ as", "these generate on each of the pieces of the cover, see", "Algebra, Lemma \\ref{algebra-lemma-cover}." ], "refs": [ "algebra-lemma-cover" ], "ref_ids": [ 411 ] } ], "ref_ids": [] }, { "id": 9996, "type": "theorem", "label": "more-algebra-lemma-quasi-regular-ideal-finite-projective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-quasi-regular-ideal-finite-projective", "contents": [ "Let $I \\subset R$ be a quasi-regular ideal of a ring.", "Then $I/I^2$ is a finite projective $R/I$-module." ], "refs": [], "proofs": [ { "contents": [ "This follows from Algebra, Lemma \\ref{algebra-lemma-finite-projective}", "and the definitions." ], "refs": [ "algebra-lemma-finite-projective" ], "ref_ids": [ 795 ] } ], "ref_ids": [] }, { "id": 9997, "type": "theorem", "label": "more-algebra-lemma-flat-descent-regular-ideal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-descent-regular-ideal", "contents": [ "Let $A \\to B$ be a faithfully flat ring map. Let $I \\subset A$ be an ideal.", "If $IB$ is a Koszul-regular", "(resp.\\ $H_1$-regular, resp.\\ quasi-regular) ideal in $B$, then", "$I$ is a Koszul-regular (resp.\\ $H_1$-regular, resp.\\ quasi-regular)", "ideal in $A$." ], "refs": [], "proofs": [ { "contents": [ "We fix the prime $\\mathfrak p \\supset I$ throughout the proof.", "Assume $IB$ is quasi-regular. By", "Lemma \\ref{lemma-quasi-regular-ideal-finite}", "$IB$ is a finite module, hence $I$ is a finite $A$-module by", "Algebra, Lemma \\ref{algebra-lemma-descend-properties-modules}.", "As $A \\to B$ is flat we see that", "$$", "I/I^2 \\otimes_{A/I} B/IB = I/I^2 \\otimes_A B = IB/(IB)^2.", "$$", "As $IB$ is quasi-regular, the $B/IB$-module $IB/(IB)^2$ is finite", "locally free. Hence $I/I^2$ is finite projective, see", "Algebra, Proposition \\ref{algebra-proposition-ffdescent-finite-projectivity}.", "In particular, after replacing $A$ by $A_f$ for some", "$f \\in A$, $f \\not \\in \\mathfrak p$ we may assume that $I/I^2$ is free of", "rank $r$. Pick $f_1, \\ldots, f_r \\in I$ which", "give a basis of $I/I^2$. By Nakayama's lemma (see", "Algebra, Lemma \\ref{algebra-lemma-NAK})", "we see that, after another replacement $A \\leadsto A_f$ as above,", "$I$ is generated by $f_1, \\ldots, f_r$.", "\\medskip\\noindent", "Proof of the ``quasi-regular'' case. Above we have seen that", "$I/I^2$ is free on the $r$-generators $f_1, \\ldots, f_r$.", "To finish the proof in this case we have to show that the maps", "$\\text{Sym}^d(I/I^2) \\to I^d/I^{d + 1}$ are isomorphisms", "for each $d \\geq 2$. This is clear as the faithfully flat", "base changes $\\text{Sym}^d(IB/(IB)^2) \\to (IB)^d/(IB)^{d + 1}$", "are isomorphisms locally on $B$ by assumption.", "Details omitted.", "\\medskip\\noindent", "Proof of the ``$H_1$-regular'' and ``Koszul-regular'' case.", "Consider the sequence of elements $f_1, \\ldots, f_r$ generating", "$I$ we constructed above. By", "Lemma \\ref{lemma-independence-of-generators}", "we see that $f_1, \\ldots, f_r$ map to a $H_1$-regular or Koszul-regular", "sequence in $B_g$ for any $g \\in B$ such that $IB$ is generated by", "an $H_1$-regular or Koszul-regular sequence. Hence", "$K_\\bullet(A, f_1, \\ldots, f_r) \\otimes_A B_g$ has vanishing", "$H_1$ or $H_i$, $i > 0$. Since the homology of", "$K_\\bullet(B, f_1, \\ldots, f_r) = K_\\bullet(A, f_1, \\ldots, f_r) \\otimes_A B$", "is annihilated by $IB$ (see", "Lemma \\ref{lemma-homotopy-koszul})", "and since $V(IB) \\subset \\bigcup_{g\\text{ as above}} D(g)$ we conclude that", "$K_\\bullet(A, f_1, \\ldots, f_r) \\otimes_A B$ has vanishing homology in", "degree $1$ or all positive degrees. Using that $A \\to B$ is faithfully", "flat we conclude that the same is true for", "$K_\\bullet(A, f_1, \\ldots, f_r)$." ], "refs": [ "more-algebra-lemma-quasi-regular-ideal-finite", "algebra-lemma-descend-properties-modules", "algebra-proposition-ffdescent-finite-projectivity", "algebra-lemma-NAK", "more-algebra-lemma-independence-of-generators", "more-algebra-lemma-homotopy-koszul" ], "ref_ids": [ 9995, 819, 1413, 401, 9986, 9960 ] } ], "ref_ids": [] }, { "id": 9998, "type": "theorem", "label": "more-algebra-lemma-conormal-sequence-H1-regular-ideal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-conormal-sequence-H1-regular-ideal", "contents": [ "Let $A$ be a ring. Let $I \\subset J \\subset A$ be ideals.", "Assume that $J/I \\subset A/I$ is a $H_1$-regular ideal.", "Then $I \\cap J^2 = IJ$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-conormal-sequence-H1-regular}", "by localizing." ], "refs": [ "more-algebra-lemma-conormal-sequence-H1-regular" ], "ref_ids": [ 9980 ] } ], "ref_ids": [] }, { "id": 9999, "type": "theorem", "label": "more-algebra-lemma-koszul-independence-presentation", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-koszul-independence-presentation", "contents": [ "Let $A \\to B$ be a finite type ring map. If for some presentation", "$\\alpha : A[x_1, \\ldots, x_n] \\to B$ the kernel $I$ is a Koszul-regular ideal", "then for any presentation $\\beta : A[y_1, \\ldots, y_m] \\to B$ the kernel", "$J$ is a Koszul-regular ideal." ], "refs": [], "proofs": [ { "contents": [ "Choose $f_j \\in A[x_1, \\ldots, x_n]$ with $\\alpha(f_j) = \\beta(y_j)$", "and $g_i \\in A[y_1, \\ldots, y_m]$ with $\\beta(g_i) = \\alpha(x_i)$.", "Then we get a commutative diagram", "$$", "\\xymatrix{", "A[x_1, \\ldots, x_n, y_1, \\ldots, y_m]", "\\ar[d]^{x_i \\mapsto g_i} \\ar[rr]_-{y_j \\mapsto f_j} & &", "A[x_1, \\ldots, x_n] \\ar[d] \\\\", "A[y_1, \\ldots, y_m] \\ar[rr] & & B", "}", "$$", "Note that the kernel $K$ of $A[x_i, y_j] \\to B$ is equal to", "$K = (I, y_j - f_j) = (J, x_i - f_i)$. In particular, as", "$I$ is finitely generated by", "Lemma \\ref{lemma-quasi-regular-ideal-finite}", "we see that $J = K/(x_i - f_i)$ is finitely generated too.", "\\medskip\\noindent", "Pick a prime $\\mathfrak q \\subset B$. Since", "$I/I^2 \\oplus B^{\\oplus m} = J/J^2 \\oplus B^{\\oplus n}$", "(Algebra, Lemma \\ref{algebra-lemma-conormal-module})", "we see that", "$$", "\\dim J/J^2 \\otimes_B \\kappa(\\mathfrak q) + n =", "\\dim I/I^2 \\otimes_B \\kappa(\\mathfrak q) + m.", "$$", "Pick $p_1, \\ldots, p_t \\in I$ which map to a basis of", "$I/I^2 \\otimes \\kappa(\\mathfrak q) = I \\otimes_{A[x_i]} \\kappa(\\mathfrak q)$.", "Pick $q_1, \\ldots, q_s \\in J$ which map to a basis of", "$J/J^2 \\otimes \\kappa(\\mathfrak q) = J \\otimes_{A[y_j]} \\kappa(\\mathfrak q)$.", "So $s + n = t + m$. By Nakayama's lemma there exist $h \\in A[x_i]$ and", "$h' \\in A[y_j]$ both mapping to a nonzero element of $\\kappa(\\mathfrak q)$", "such that $I_h = (p_1, \\ldots, p_t)$ in $A[x_i, 1/h]$ and", "$J_{h'} = (q_1, \\ldots, q_s)$ in $A[y_j, 1/h']$.", "As $I$ is Koszul-regular we may also assume that $I_h$ is generated", "by a Koszul regular sequence. This sequence must necessarily have length", "$t = \\dim I/I^2 \\otimes_B \\kappa(\\mathfrak q)$, hence we see that", "$p_1, \\ldots, p_t$ is a Koszul-regular sequence by", "Lemma \\ref{lemma-independence-of-generators}.", "As also $y_1 - f_1, \\ldots, y_m - f_m$ is a regular sequence we", "conclude", "$$", "y_1 - f_1, \\ldots, y_m - f_m, p_1, \\ldots, p_t", "$$", "is a Koszul-regular sequence in $A[x_i, y_j, 1/h]$", "(see Lemma \\ref{lemma-join-koszul-regular-sequences}).", "This sequence generates the ideal $K_h$. Hence the", "ideal $K_{hh'}$ is generated by a Koszul-regular sequence", "of length $m + t = n + s$. But it is also generated by the sequence", "$$", "x_1 - g_1, \\ldots, x_n - g_n, q_1, \\ldots, q_s", "$$", "of the same length which is thus a Koszul-regular sequence by", "Lemma \\ref{lemma-independence-of-generators}.", "Finally, by", "Lemma \\ref{lemma-truncate-koszul-regular}", "we conclude that the images of $q_1, \\ldots, q_s$ in", "$$", "A[x_i, y_j, 1/hh']/(x_1 - g_1, \\ldots, x_n - g_n)", "\\cong A[y_j, 1/h'']", "$$", "form a Koszul-regular sequence generating $J_{h''}$. Since $h''$", "is the image of $hh'$ it doesn't map to zero in $\\kappa(\\mathfrak q)$", "and we win." ], "refs": [ "more-algebra-lemma-quasi-regular-ideal-finite", "algebra-lemma-conormal-module", "more-algebra-lemma-independence-of-generators", "more-algebra-lemma-join-koszul-regular-sequences", "more-algebra-lemma-independence-of-generators", "more-algebra-lemma-truncate-koszul-regular" ], "ref_ids": [ 9995, 1163, 9986, 9984, 9986, 9985 ] } ], "ref_ids": [] }, { "id": 10000, "type": "theorem", "label": "more-algebra-lemma-lci-local", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lci-local", "contents": [ "Let $R \\to S$ be a ring map. Let $g_1, \\ldots, g_m \\in S$", "generate the unit ideal. If each $R \\to S_{g_j}$ is a local", "complete intersection so is $R \\to S$." ], "refs": [], "proofs": [ { "contents": [ "Let $S = R[x_1, \\ldots, x_n]/I$ be a presentation. Pick", "$h_j \\in R[x_1, \\ldots, x_n]$ mapping to $g_j$ in $S$.", "Then $R[x_1, \\ldots, x_n, x_{n + 1}]/(I, x_{n + 1}h_j - 1)$", "is a presentation of $S_{g_j}$. Hence", "$I_j = (I, x_{n + 1}h_j - 1)$ is a Koszul-regular ideal in", "$R[x_1, \\ldots, x_n, x_{n + 1}]$. Pick a prime", "$I \\subset \\mathfrak q \\subset R[x_1, \\ldots, x_n]$.", "Then $h_j \\not \\in \\mathfrak q$ for some $j$ and", "$\\mathfrak q_j = (\\mathfrak q, x_{n + 1}h_j - 1)$ is a prime", "ideal of $V(I_j)$ lying over $\\mathfrak q$.", "Pick $f_1, \\ldots, f_r \\in I$ which map to a basis of", "$I/I^2 \\otimes \\kappa(\\mathfrak q)$. Then", "$x_{n + 1}h_j - 1, f_1, \\ldots, f_r$ is a sequence of elements of $I_j$", "which map to a basis of $I_j \\otimes \\kappa(\\mathfrak q_j)$.", "By Nakayama's lemma there exists an $h \\in R[x_1, \\ldots, x_n, x_{n + 1}]$", "such that $(I_j)_h$ is generated by $x_{n + 1}h_j - 1, f_1, \\ldots, f_r$.", "We may also assume that $(I_j)_h$ is generated by a Koszul regular", "sequence of some length $e$. Looking at the dimension of", "$I_j \\otimes \\kappa(\\mathfrak q_j)$ we see that $e = r + 1$.", "Hence by", "Lemma \\ref{lemma-independence-of-generators}", "we see that $x_{n + 1}h_j - 1, f_1, \\ldots, f_r$ is a", "Koszul-regular sequence generating $(I_j)_h$ for some", "$h \\in R[x_1, \\ldots, x_n, x_{n + 1}]$, $h \\not \\in \\mathfrak q_j$. By", "Lemma \\ref{lemma-truncate-koszul-regular}", "we see that $I_{h'}$ is generated by a Koszul-regular sequence for some", "$h' \\in R[x_1, \\ldots, x_n]$, $h' \\not \\in \\mathfrak q$ as desired." ], "refs": [ "more-algebra-lemma-independence-of-generators", "more-algebra-lemma-truncate-koszul-regular" ], "ref_ids": [ 9986, 9985 ] } ], "ref_ids": [] }, { "id": 10001, "type": "theorem", "label": "more-algebra-lemma-relative-global-complete-intersection-koszul", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-relative-global-complete-intersection-koszul", "contents": [ "Let $R$ be a ring. If $R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "is a relative global complete intersection, then $f_1, \\ldots, f_c$", "is a Koszul regular sequence." ], "refs": [], "proofs": [ { "contents": [ "Recall that the homology groups $H_i(K_\\bullet(f_\\bullet))$ are", "annihilated by the ideal $(f_1, \\ldots, f_c)$. Hence it suffices", "to show that $H_i(K_\\bullet(f_\\bullet))_\\mathfrak q$ is zero for", "all primes $\\mathfrak q \\subset R[x_1, \\ldots, x_n]$", "containing $(f_1, \\ldots, f_c)$. This follows from", "Algebra, Lemma", "\\ref{algebra-lemma-relative-global-complete-intersection-conormal}", "and the fact that a regular sequence is Koszul regular", "(Lemma \\ref{lemma-regular-koszul-regular})." ], "refs": [ "algebra-lemma-relative-global-complete-intersection-conormal", "more-algebra-lemma-regular-koszul-regular" ], "ref_ids": [ 1183, 9973 ] } ], "ref_ids": [] }, { "id": 10002, "type": "theorem", "label": "more-algebra-lemma-syntomic-lci", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-syntomic-lci", "contents": [ "Let $R \\to S$ be a ring map. The following are equivalent", "\\begin{enumerate}", "\\item $R \\to S$ is syntomic", "(Algebra, Definition \\ref{algebra-definition-lci}), and", "\\item $R \\to S$ is flat and a local complete intersection.", "\\end{enumerate}" ], "refs": [ "algebra-definition-lci" ], "proofs": [ { "contents": [ "Assume (1). Then $R \\to S$ is flat by definition.", "By Algebra, Lemma \\ref{algebra-lemma-syntomic} and", "Lemma \\ref{lemma-lci-local} we see that it suffices to", "show a relative global complete intersection is", "a local complete intersection homomorphism which is", "Lemma \\ref{lemma-relative-global-complete-intersection-koszul}.", "\\medskip\\noindent", "Assume (2). A local complete intersection is of finite presentation", "because a Koszul-regular ideal is finitely generated.", "Let $R \\to k$ be a map to a field. It suffices to show that", "$S' = S \\otimes_R k$ is a local complete intersection over $k$, see", "Algebra, Definition \\ref{algebra-definition-lci-field}.", "Choose a prime $\\mathfrak q' \\subset S'$.", "Write $S = R[x_1, \\ldots, x_n]/I$.", "Then $S' = k[x_1, \\ldots, x_n]/I'$ where", "$I' \\subset k[x_1, \\ldots, x_n]$ is the image of $I$.", "Let $\\mathfrak p' \\subset k[x_1, \\ldots, x_n]$,", "$\\mathfrak q \\subset S$,", "and $\\mathfrak p \\subset R[x_1, \\ldots, x_n]$", "be the corresponding primes.", "By Definition \\ref{definition-regular-ideal}", "exists an $g \\in R[x_1, \\ldots, x_n]$, $g \\not \\in \\mathfrak p$", "and $f_1, \\ldots, f_r \\in R[x_1, \\ldots, x_n]_g$ which form", "a Koszul-regular sequence generating $I_g$.", "Since $S$ and hence $S_g$ is flat over $R$", "we see that the images $f'_1, \\ldots, f'_r$ in", "$k[x_1, \\ldots, x_n]_g$ form a $H_1$-regular sequence", "generating $I'_g$, see", "Lemma \\ref{lemma-relative-regular-immersion-algebra}.", "Thus $f'_1, \\ldots, f'_r$ map to a regular sequence in", "$k[x_1, \\ldots, x_n]_{\\mathfrak p'}$ generating", "$I'_{\\mathfrak p'}$ by Lemma \\ref{lemma-noetherian-finite-all-equivalent}.", "Applying Algebra, Lemma \\ref{algebra-lemma-lci}", "we conclude $S'_{gg'}$ for some $g' \\in S$, $g' \\not \\in \\mathfrak q'$", "is a global complete intersection over $k$ as desired." ], "refs": [ "algebra-lemma-syntomic", "more-algebra-lemma-lci-local", "more-algebra-lemma-relative-global-complete-intersection-koszul", "algebra-definition-lci-field", "more-algebra-definition-regular-ideal", "more-algebra-lemma-relative-regular-immersion-algebra", "more-algebra-lemma-noetherian-finite-all-equivalent", "algebra-lemma-lci" ], "ref_ids": [ 1185, 10000, 10001, 1530, 10608, 9992, 9978, 1167 ] } ], "ref_ids": [ 1532 ] }, { "id": 10003, "type": "theorem", "label": "more-algebra-lemma-transitive-lci-at-end", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-transitive-lci-at-end", "contents": [ "Let $A \\to B \\to C$ be ring maps. Assume $B \\to C$ is a local complete", "intersection homomorphism. Choose a presentation", "$\\alpha : A[x_s, s \\in S] \\to B$ with kernel $I$. Choose a presentation", "$\\beta : B[y_1, \\ldots, y_m] \\to C$ with kernel $J$. Let", "$\\gamma : A[x_s, y_t] \\to C$ be the induced presentation of $C$ with kernel", "$K$. Then we get a canonical commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Omega_{A[x_s]/A} \\otimes C \\ar[r] &", "\\Omega_{A[x_s, y_t]/A} \\otimes C \\ar[r] &", "\\Omega_{B[y_t]/B} \\otimes C \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "I/I^2 \\otimes C \\ar[r] \\ar[u] &", "K/K^2 \\ar[r] \\ar[u] &", "J/J^2 \\ar[r] \\ar[u] &", "0", "}", "$$", "with exact rows. In particular, the six term exact sequence of", "Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL}", "can be completed with a zero on the left, i.e., the sequence", "$$", "0 \\to H_1(\\NL_{B/A} \\otimes_B C) \\to", "H_1(L_{C/A}) \\to", "H_1(L_{C/B}) \\to", "\\Omega_{B/A} \\otimes_B C \\to", "\\Omega_{C/A} \\to", "\\Omega_{C/B} \\to 0", "$$", "is exact." ], "refs": [ "algebra-lemma-exact-sequence-NL" ], "proofs": [ { "contents": [ "The only thing to prove is the injectivity of the map", "$I/I^2 \\otimes C \\to K/K^2$.", "By assumption the ideal $J$ is Koszul-regular.", "Hence we have $IA[x_s, y_j] \\cap K^2 = IK$ by", "Lemma \\ref{lemma-conormal-sequence-H1-regular-ideal}.", "This means that the kernel of $K/K^2 \\to J/J^2$ is", "isomorphic to $IA[x_s, y_j]/IK$. Since", "$I/I^2 \\otimes_A C = IA[x_s, y_j]/IK$ by right exactness", "of tensor product, this provides us with the desired injectivity of", "$I/I^2 \\otimes_A C \\to K/K^2$." ], "refs": [ "more-algebra-lemma-conormal-sequence-H1-regular-ideal" ], "ref_ids": [ 9998 ] } ], "ref_ids": [ 1153 ] }, { "id": 10004, "type": "theorem", "label": "more-algebra-lemma-transitive-colimit-lci-at-end", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-transitive-colimit-lci-at-end", "contents": [ "Let $A \\to B \\to C$ be ring maps.", "If $B \\to C$ is a filtered colimit of local complete intersection", "homomorphisms then the conclusion of", "Lemma \\ref{lemma-transitive-lci-at-end}", "remains valid." ], "refs": [ "more-algebra-lemma-transitive-lci-at-end" ], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-transitive-lci-at-end}", "and", "Algebra, Lemma \\ref{algebra-lemma-colimits-NL}." ], "refs": [ "more-algebra-lemma-transitive-lci-at-end", "algebra-lemma-colimits-NL" ], "ref_ids": [ 10003, 1157 ] } ], "ref_ids": [ 10003 ] }, { "id": 10005, "type": "theorem", "label": "more-algebra-lemma-henselization-NL", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-NL", "contents": [ "Let $A \\to B$ be a local homomorphism of local rings.", "Let $A^h \\to B^h$, resp.\\ $A^{sh} \\to B^{sh}$ be the induced", "map on henselizations, resp.\\ strict henselizations", "(Algebra, Lemma \\ref{algebra-lemma-henselian-functorial},", "resp.\\ Lemma \\ref{algebra-lemma-strictly-henselian-functorial}).", "Then $\\NL_{B/A} \\otimes_B B^h \\to \\NL_{B^h/A^h}$ and", "$\\NL_{B/A} \\otimes_B B^{sh} \\to \\NL_{B^{sh}/A^{sh}}$", "induce isomorphisms on cohomology groups." ], "refs": [ "algebra-lemma-henselian-functorial", "algebra-lemma-strictly-henselian-functorial" ], "proofs": [ { "contents": [ "Since $A^h$ is a filtered colimit of \\'etale algebras over $A$", "we see that $\\NL_{A^h/A}$ is an acyclic complex by", "Algebra, Lemma \\ref{algebra-lemma-colimits-NL} and", "Algebra, Definition \\ref{algebra-definition-etale}.", "The same is true for $B^h/B$.", "Using the Jacobi-Zariski sequence", "(Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL})", "for $A \\to A^h \\to B^h$ we find that", "$\\NL_{B^h/A} \\to \\NL_{B^h/A^h}$ induces isomorphisms", "on cohomology groups. Moreover, an \\'etale ring map is", "a local complete intersection as it is even a global complete", "intersection, see Algebra, Lemma \\ref{algebra-lemma-etale-standard-smooth}.", "By Lemma \\ref{lemma-transitive-colimit-lci-at-end}", "we get a six term exact Jacobi-Zariski sequence", "associated to $A \\to B \\to B^h$ which proves that", "$\\NL_{B/A} \\otimes_B B^h \\to \\NL_{B^h/A}$ induces isomorphisms", "on cohomology groups. This finishes the proof in the", "case of the map on henselizations. The case of strict henselization", "is proved in exactly the same manner." ], "refs": [ "algebra-lemma-colimits-NL", "algebra-definition-etale", "algebra-lemma-exact-sequence-NL", "algebra-lemma-etale-standard-smooth", "more-algebra-lemma-transitive-colimit-lci-at-end" ], "ref_ids": [ 1157, 1539, 1153, 1230, 10004 ] } ], "ref_ids": [ 1297, 1303 ] }, { "id": 10006, "type": "theorem", "label": "more-algebra-lemma-cartier-equality", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cartier-equality", "contents": [ "Let $K/k$ be a finitely generated field extension.", "Then $\\Omega_{K/k}$ and $H_1(L_{K/k})$ are finite dimensional and", "$\\text{trdeg}_k(K) = \\dim_K \\Omega_{K/k} - \\dim_K H_1(L_{K/k})$." ], "refs": [], "proofs": [ { "contents": [ "We can find a global complete intersection", "$A = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "over $k$ such that", "$K$ is isomorphic to the fraction field of $A$, see", "Algebra, Lemma \\ref{algebra-lemma-colimit-syntomic}", "and its proof. In this case we see that $\\NL_{K/k}$ is homotopy", "equivalent to the complex", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, c} K \\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} K\\text{d}x_i", "$$", "by Algebra, Lemmas \\ref{algebra-lemma-NL-homotopy} and", "\\ref{algebra-lemma-localize-NL}.", "The transcendence degree of $K$ over $k$ is the dimension of $A$", "(by Algebra, Lemma \\ref{algebra-lemma-dimension-prime-polynomial-ring})", "which is $n - c$ and we win." ], "refs": [ "algebra-lemma-colimit-syntomic", "algebra-lemma-NL-homotopy", "algebra-lemma-localize-NL", "algebra-lemma-dimension-prime-polynomial-ring" ], "ref_ids": [ 1323, 1151, 1161, 1005 ] } ], "ref_ids": [] }, { "id": 10007, "type": "theorem", "label": "more-algebra-lemma-transitivity-gamma", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-transitivity-gamma", "contents": [ "Let $K \\subset L \\subset M$ be field extensions. Then the Jacobi-Zariski", "sequence", "$$", "0 \\to H_1(L_{L/K}) \\otimes_L M \\to", "H_1(L_{M/K}) \\to", "H_1(L_{M/L}) \\to", "\\Omega_{L/K} \\otimes_L M \\to", "\\Omega_{M/K} \\to", "\\Omega_{M/L} \\to 0", "$$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemma \\ref{lemma-transitive-colimit-lci-at-end}", "with", "Algebra, Lemma \\ref{algebra-lemma-colimit-syntomic}." ], "refs": [ "more-algebra-lemma-transitive-colimit-lci-at-end", "algebra-lemma-colimit-syntomic" ], "ref_ids": [ 10004, 1323 ] } ], "ref_ids": [] }, { "id": 10008, "type": "theorem", "label": "more-algebra-lemma-gamma-commutative-diagram", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-gamma-commutative-diagram", "contents": [ "Given a commutative diagram of fields", "$$", "\\xymatrix{", "K \\ar[r] & K' \\\\", "k \\ar[u] \\ar[r] & k' \\ar[u]", "}", "$$", "with $k \\subset k'$ and $K \\subset K'$ finitely generated field extensions", "the kernel and cokernel of the maps", "$$", "\\alpha : \\Omega_{K/k} \\otimes_K K' \\to \\Omega_{K'/k'}", "\\quad\\text{and}\\quad", "\\beta : H_1(L_{K/k}) \\otimes_K K' \\to H_1(L_{K'/k'})", "$$", "are finite dimensional and", "$$", "\\dim \\Ker(\\alpha) - \\dim \\Coker(\\alpha)", "-\\dim \\Ker(\\beta) + \\dim \\Coker(\\beta)", "=", "\\text{trdeg}_k(k') - \\text{trdeg}_K(K')", "$$" ], "refs": [], "proofs": [ { "contents": [ "The Jacobi-Zariski sequences for $k \\subset k' \\subset K'$ and", "$k \\subset K \\subset K'$ are", "$$", "0 \\to H_1(L_{k'/k}) \\otimes K' \\to", "H_1(L_{K'/k}) \\to", "H_1(L_{K'/k'}) \\to", "\\Omega_{k'/k} \\otimes K' \\to", "\\Omega_{K'/k} \\to", "\\Omega_{K'/k} \\to 0", "$$", "and", "$$", "0 \\to H_1(L_{K/k}) \\otimes K' \\to", "H_1(L_{K'/k}) \\to", "H_1(L_{K'/K}) \\to", "\\Omega_{K/k} \\otimes K' \\to", "\\Omega_{K'/k} \\to", "\\Omega_{K'/K} \\to 0", "$$", "By", "Lemma \\ref{lemma-cartier-equality}", "the vector spaces $\\Omega_{k'/k}$, $\\Omega_{K'/K}$, $H_1(L_{K'/K})$, and", "$H_1(L_{k'/k})$ are finite dimensional and the alternating sum of their", "dimensions is $\\text{trdeg}_k(k') - \\text{trdeg}_K(K')$.", "The lemma follows." ], "refs": [ "more-algebra-lemma-cartier-equality" ], "ref_ids": [ 10006 ] } ], "ref_ids": [] }, { "id": 10009, "type": "theorem", "label": "more-algebra-lemma-geometrically-regular-over-field", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-geometrically-regular-over-field", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $(A, \\mathfrak m, K)$", "be a Noetherian local $k$-algebra. Assume $A$ is geometrically regular", "over $k$. Let $k \\subset F \\subset K$ be a finitely generated subextension.", "Let $\\varphi : k[y_1, \\ldots, y_m] \\to A$ be a $k$-algebra map", "such that $y_i$ maps to an element of $F$ in $K$ and such that", "$\\text{d}y_1, \\ldots, \\text{d}y_m$ map to a basis of $\\Omega_{F/k}$.", "Set $\\mathfrak p = \\varphi^{-1}(\\mathfrak m)$. Then", "$$", "k[y_1, \\ldots, y_m]_\\mathfrak p \\to A", "$$", "is flat and $A/\\mathfrak pA$ is regular." ], "refs": [], "proofs": [ { "contents": [ "Set $A_0 = k[y_1, \\ldots, y_m]_\\mathfrak p$ with maximal ideal", "$\\mathfrak m_0$ and residue field $K_0$. Note that", "$\\Omega_{A_0/k}$ is free of rank $m$ and", "$\\Omega_{A_0/k} \\otimes K_0 \\to \\Omega_{K_0/k}$ is an isomorphism.", "It is clear that $A_0$ is geometrically regular over $k$. Hence", "$H_1(L_{K_0/k}) \\to \\mathfrak m_0/\\mathfrak m_0^2$ is an isomorphism, see", "Proposition \\ref{proposition-characterization-geometrically-regular}.", "Now consider", "$$", "\\xymatrix{", "H_1(L_{K_0/k}) \\otimes K \\ar[d] \\ar[r] &", "\\mathfrak m_0/\\mathfrak m_0^2 \\otimes K \\ar[d] \\\\", "H_1(L_{K/k}) \\ar[r] & \\mathfrak m/\\mathfrak m^2", "}", "$$", "Since the left vertical arrow is injective by", "Lemma \\ref{lemma-transitivity-gamma}", "and the lower horizontal by", "Proposition \\ref{proposition-characterization-geometrically-regular}", "we conclude that the right vertical one is too.", "Hence a regular system of parameters in $A_0$ maps to", "part of a regular system of parameters in $A$.", "We win by", "Algebra, Lemmas \\ref{algebra-lemma-flat-over-regular} and", "\\ref{algebra-lemma-regular-ring-CM}." ], "refs": [ "more-algebra-proposition-characterization-geometrically-regular", "more-algebra-lemma-transitivity-gamma", "more-algebra-proposition-characterization-geometrically-regular", "algebra-lemma-flat-over-regular", "algebra-lemma-regular-ring-CM" ], "ref_ids": [ 10576, 10007, 10576, 1108, 941 ] } ], "ref_ids": [] }, { "id": 10010, "type": "theorem", "label": "more-algebra-lemma-continuous", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-continuous", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "Let $I \\subset R$ and $J \\subset S$ be ideals", "and endow $R$ with the $I$-adic topology and $S$ with the $J$-adic", "topology. Then $\\varphi$ is a homomorphism of topological rings", "if and only if $\\varphi(I^n) \\subset J$ for some $n \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10011, "type": "theorem", "label": "more-algebra-lemma-baire-category-complete-module", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-baire-category-complete-module", "contents": [ "Let $M$ be a topological abelian group. Assume $M$ is linearly", "topologized, complete, and has a countable fundamental system of", "neighbourhoods of $0$. If $U_n \\subset M$, $n \\geq 1$", "are open dense subsets, then $\\bigcap_{n \\geq 1} U_n$ is dense." ], "refs": [], "proofs": [ { "contents": [ "Let $U_n$ be as in the statement of the lemma. After replacing $U_n$ by", "$U_1 \\cap \\ldots \\cap U_n$, we may assume that", "$U_1 \\supset U_2 \\supset \\ldots$.", "Let $M_n$, $n \\in \\mathbf{N}$ be a fundamental", "system of neighbourhoods of $0$. We may assume that", "$M_{n + 1} \\subset M_n$. Pick $x \\in M$. We will show that", "for every $k \\geq 1$ there exists a $y \\in \\bigcap_{n \\geq 1} U_n$", "with $x - y \\in M_k$.", "\\medskip\\noindent", "To construct $y$ we argue as follows.", "First, we pick a $y_1 \\in U_1$ with $y_1 \\in x + M_k$.", "This is possible because $U_1$ is dense and $x + M_k$ is open.", "Then we pick a $k_1 > k$ such that $y_1 + M_{k_1} \\subset U_1$.", "This is possible because $U_1$ is open.", "Next, we pick a $y_2 \\in U_2$ with $y_2 \\in y_1 + M_{k_1}$.", "This is possible because $U_2$ is dense and $y_2 + M_{k_1}$ is open.", "Then we pick a $k_2 > k_1$ such that $y_2 + M_{k_2} \\subset U_2$.", "This is possible because $U_2$ is open.", "\\medskip\\noindent", "Continuing in this fashion we get a converging sequence", "$y_i$ of elements of $M$ with limit $y$. By construction", "$x - y \\in M_k$. Since", "$$", "y - y_i = (y_{i + 1} - y_i) + (y_{i + 2} - y_{i + 1}) + \\ldots", "$$", "is in $M_{k_i}$ we see that $y \\in y_i + M_{k_i} \\subset U_i$", "for all $i$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10012, "type": "theorem", "label": "more-algebra-lemma-consequence-baire-complete-module", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-consequence-baire-complete-module", "contents": [ "With same assumptions as Lemma \\ref{lemma-baire-category-complete-module}", "if $M = \\bigcup_{n \\geq 1} N_n$ for some closed subgroups $N_n$,", "then $N_n$ is open for some $n$." ], "refs": [ "more-algebra-lemma-baire-category-complete-module" ], "proofs": [ { "contents": [ "If not, then $U_n = M \\setminus N_n$ is dense for all $n$ and", "we get a contradiction with Lemma \\ref{lemma-baire-category-complete-module}." ], "refs": [ "more-algebra-lemma-baire-category-complete-module" ], "ref_ids": [ 10011 ] } ], "ref_ids": [ 10011 ] }, { "id": 10013, "type": "theorem", "label": "more-algebra-lemma-open-mapping", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-open-mapping", "contents": [ "Let $u : N \\to M$ be a continuous map of linearly topologized", "abelian groups. Assume that $N$ is complete, $M$ separated,", "and $N$ has a countable fundamental system of neighbourhoods of $0$.", "Then exactly one of the following holds", "\\begin{enumerate}", "\\item $u$ is open, or", "\\item for some open subgroup $N' \\subset N$ the image", "$u(N')$ is nowhere dense in $M$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $N_n$, $n \\in \\mathbf{N}$ be a fundamental system of neighbourhoods of $0$.", "We may assume that $N_{n + 1} \\subset N_n$. If (2) does not hold, then", "the closure $M_n$ of $u(N_n)$ is an open subgroup for $n = 1, 2, 3, \\ldots$.", "Since $u$ is continuous, we see that $M_n$, $n \\in \\mathbf{N}$", "must be a fundamental system of open neighbourhoods of $0$ in $M$.", "Also, since $M_n$ is the closure of $u(N_n)$ we see that", "$$", "u(N_n) + M_{n + 1} = M_n", "$$", "for all $n \\geq 1$. Pick $x_1 \\in M_1$. Then we can inductively choose", "$y_i \\in N_i$ and $x_{i + 1} \\in M_{i + 1}$ such that", "$$", "u(y_i) + x_{i + 1} = x_i", "$$", "The element $y = y_1 + y_2 + y_3 + \\ldots$ of $N$ exists because", "$N$ is complete. Whereupon we see that $x = u(y)$ because $M$ is separated.", "Thus $M_1 = u(N_1)$. In exactly the same way the reader shows that", "$M_i = u(N_i)$ for all $i \\geq 2$ and we see that $u$ is open." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10014, "type": "theorem", "label": "more-algebra-lemma-formally-smooth", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-formally-smooth", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "\\begin{enumerate}", "\\item If $R \\to S$ is formally smooth in", "the sense of Algebra, Definition \\ref{algebra-definition-formally-smooth},", "then $R \\to S$ is formally smooth for any linear topology on $R$ and", "any pre-adic topology on $S$ such that $R \\to S$ is continuous.", "\\item Let $\\mathfrak n \\subset S$ and $\\mathfrak m \\subset R$", "ideals such that $\\varphi$ is continuous for the $\\mathfrak m$-adic", "topology on $R$ and the $\\mathfrak n$-adic topology", "on $S$. Then the following are equivalent", "\\begin{enumerate}", "\\item $\\varphi$ is formally smooth for the $\\mathfrak m$-adic topology on", "$R$ and the $\\mathfrak n$-adic topology on $S$, and", "\\item $\\varphi$ is formally smooth for the discrete topology", "on $R$ and the $\\mathfrak n$-adic topology on $S$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [ "algebra-definition-formally-smooth" ], "proofs": [ { "contents": [ "Assume $R \\to S$ is formally smooth in", "the sense of Algebra, Definition \\ref{algebra-definition-formally-smooth}.", "If $S$ has a pre-adic topology, then", "there exists an ideal $\\mathfrak n \\subset S$ such that $S$ has the", "$\\mathfrak n$-adic topology. Suppose given a solid commutative diagram as in", "Definition \\ref{definition-formally-smooth}.", "Continuity of $S \\to A/J$ means that $\\mathfrak n^k$ maps to zero", "in $A/J$ for some $k \\geq 1$, see Lemma \\ref{lemma-continuous}.", "We obtain a ring map $\\psi : S \\to A$ from the assumed formal smoothness of", "$S$ over $R$. Then $\\psi(\\mathfrak n^k) \\subset J$ hence", "$\\psi(\\mathfrak n^{2k}) = 0$ as $J^2 = 0$. Hence $\\psi$ is continuous by", "Lemma \\ref{lemma-continuous}. This proves (1).", "\\medskip\\noindent", "The proof of (2)(b) $\\Rightarrow$ (2)(a) is the same as the proof of (1).", "Assume (2)(a). Suppose given a solid commutative diagram as in", "Definition \\ref{definition-formally-smooth} where we use the discrete", "topology on $R$. Since $\\varphi$ is continuous we see that", "$\\varphi(\\mathfrak m^n) \\subset \\mathfrak n$ for some $m \\geq 1$.", "As $S \\to A/J$ is continuous we see that $\\mathfrak n^k$ maps to", "zero in $A/J$ for some $k \\geq 1$. Hence $\\mathfrak m^{nk}$ maps", "into $J$ under the map $R \\to A$. Thus $\\mathfrak m^{2nk}$ maps to zero", "in $A$ and we see that $R \\to A$ is continuous in the $\\mathfrak m$-adic", "topology. Thus (2)(a) gives a dotted arrow as desired." ], "refs": [ "algebra-definition-formally-smooth", "more-algebra-definition-formally-smooth", "more-algebra-lemma-continuous", "more-algebra-lemma-continuous", "more-algebra-definition-formally-smooth" ], "ref_ids": [ 1537, 10611, 10010, 10010, 10611 ] } ], "ref_ids": [ 1537 ] }, { "id": 10015, "type": "theorem", "label": "more-algebra-lemma-formally-smooth-completion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-formally-smooth-completion", "contents": [ "Let $(R, \\mathfrak m)$ and $(S, \\mathfrak n)$ be rings endowed", "with finitely generated ideals. Endow $R$ and $S$ with the", "$\\mathfrak m$-adic and $\\mathfrak n$-adic topologies.", "Let $R \\to S$ be a homomorphism of topological rings.", "The following are equivalent", "\\begin{enumerate}", "\\item $R \\to S$ is formally smooth for the $\\mathfrak n$-adic topology,", "\\item $R \\to S^\\wedge$ is formally smooth for the $\\mathfrak n^\\wedge$-adic", "topology,", "\\item $R^\\wedge \\to S^\\wedge$ is formally smooth for the", "$\\mathfrak n^\\wedge$-adic topology.", "\\end{enumerate}", "Here $R^\\wedge$ and $S^\\wedge$ are the $\\mathfrak m$-adic and", "$\\mathfrak n$-adic completions of $R$ and $S$." ], "refs": [], "proofs": [ { "contents": [ "The assumption that $\\mathfrak m$ is finitely generated implies that", "$R^\\wedge$ is $\\mathfrak mR^\\wedge$-adically complete, that", "$\\mathfrak mR^\\wedge = \\mathfrak m^\\wedge$ and that", "$R^\\wedge/\\mathfrak m^nR^\\wedge = R/\\mathfrak m^n$,", "see Algebra, Lemma \\ref{algebra-lemma-hathat-finitely-generated}", "and its proof. Similarly for $(S, \\mathfrak n)$. Thus it is clear that", "diagrams as in Definition \\ref{definition-formally-smooth}", "for the cases (1), (2), and (3) are in 1-to-1 correspondence." ], "refs": [ "algebra-lemma-hathat-finitely-generated", "more-algebra-definition-formally-smooth" ], "ref_ids": [ 859, 10611 ] } ], "ref_ids": [] }, { "id": 10016, "type": "theorem", "label": "more-algebra-lemma-lift-continuous", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-continuous", "contents": [ "Let $R \\to S$ be a ring map. Let $\\mathfrak n$ be an ideal of $S$.", "Assume that $R \\to S$ is formally smooth in the $\\mathfrak n$-adic", "topology. Consider a solid commutative diagram", "$$", "\\xymatrix{", "S \\ar[r]_\\psi \\ar@{-->}[rd] & A/J \\\\", "R \\ar[r] \\ar[u] & A \\ar[u]", "}", "$$", "of homomorphisms of topological rings where $A$ is adic", "and $A/J$ is the quotient (as topological ring) of $A$ by a closed ideal", "$J \\subset A$ such that $J^t$ is contained in an ideal of definition", "of $A$ for some $t \\geq 1$. Then there exists a dotted arrow in the category of", "topological rings which makes the diagram commute." ], "refs": [], "proofs": [ { "contents": [ "Let $I \\subset A$ be an ideal of definition so that $I \\supset J^t$", "for some $n$. Then $A = \\lim A/I^n$ and $A/J = \\lim A/J + I^n$", "because $J$ is assumed closed. Consider the following diagram of", "discrete $R$ algebras $A_{n, m} = A/J^n + I^m$:", "$$", "\\xymatrix{", "A/J^3 + I^3 \\ar[r] \\ar[d] &", "A/J^2 + I^3 \\ar[r] \\ar[d] &", "A/J + I^3 \\ar[d] \\\\", "A/J^3 + I^2 \\ar[r] \\ar[d] &", "A/J^2 + I^2 \\ar[r] \\ar[d] &", "A/J + I^2 \\ar[d] \\\\", "A/J^3 + I \\ar[r] &", "A/J^2 + I \\ar[r] &", "A/J + I", "}", "$$", "Note that each of the commutative squares defines a surjection", "$$", "A_{n + 1, m + 1} \\longrightarrow A_{n + 1, m} \\times_{A_{n, m}} A_{n, m + 1}", "$$", "of $R$-algebras whose kernel has square zero.", "We will inductively construct $R$-algebra maps", "$\\varphi_{n, m} : S \\to A_{n, m}$.", "Namely, we have the maps $\\varphi_{1, m} = \\psi \\bmod J + I^m$.", "Note that each of these maps is continuous as $\\psi$ is.", "We can inductively choose the maps $\\varphi_{n, 1}$ by starting", "with our choice of $\\varphi_{1, 1}$ and lifting up, using the", "formal smoothness of $S$ over $R$, along the right column of the", "diagram above. We construct the remaining maps $\\varphi_{n, m}$", "by induction on $n + m$. Namely, we choose $\\varphi_{n + 1, m + 1}$", "by lifting the pair $(\\varphi_{n + 1, m}, \\varphi_{n, m + 1})$", "along the displayed surjection above (again using the formal smoothness", "of $S$ over $R$). In this way all of the maps $\\varphi_{n, m}$ are", "compatible with the transition maps of the system.", "As $J^t \\subset I$ we see that for example", "$\\varphi_n = \\varphi_{nt, n} \\bmod I^n$ induces a map $S \\to A/I^n$.", "Taking the limit $\\varphi = \\lim \\varphi_n$ we obtain a map", "$S \\to A = \\lim A/I^n$. The composition into $A/J$ agrees", "with $\\psi$ as we have seen that $A/J = \\lim A/J + I^n$.", "Finally we show that $\\varphi$ is continuous. Namely, we know that", "$\\psi(\\mathfrak n^r) \\subset J + I^r/J$ for some $r$ by our", "assumption that $\\psi$ is a morphism of topological rings, see", "Lemma \\ref{lemma-continuous}. Hence $\\varphi(\\mathfrak n^r) \\subset J + I$", "hence $\\varphi(\\mathfrak n^{rt}) \\subset I$ as desired." ], "refs": [ "more-algebra-lemma-continuous" ], "ref_ids": [ 10010 ] } ], "ref_ids": [] }, { "id": 10017, "type": "theorem", "label": "more-algebra-lemma-increase-ideal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-increase-ideal", "contents": [ "Let $R \\to S$ be a ring map. Let $\\mathfrak n \\subset \\mathfrak n' \\subset S$", "be ideals. If $R \\to S$ is formally smooth for the $\\mathfrak n$-adic", "topology, then $R \\to S$ is formally smooth for the $\\mathfrak n'$-adic", "topology." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10018, "type": "theorem", "label": "more-algebra-lemma-compose-formally-smooth", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-compose-formally-smooth", "contents": [ "A composition of formally smooth continuous homomorphisms of linearly", "topologized rings is formally smooth." ], "refs": [], "proofs": [ { "contents": [ "Omitted. (Hint: This is completely formal, and follows from considering", "a suitable diagram.)" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10019, "type": "theorem", "label": "more-algebra-lemma-base-change-fs", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-fs", "contents": [ "Let $R$, $S$ be rings. Let $\\mathfrak n \\subset S$ be an ideal.", "Let $R \\to S$ be formally smooth for the $\\mathfrak n$-adic topology.", "Let $R \\to R'$ be any ring map. Then $R' \\to S' = S \\otimes_R R'$", "is formally smooth in the $\\mathfrak n' = \\mathfrak nS'$-adic", "topology." ], "refs": [], "proofs": [ { "contents": [ "Let a solid diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar@{-->}[rrd] & S' \\ar[r] \\ar@{-->}[rd] & A/J \\\\", "R \\ar[u] \\ar[r] & R' \\ar[r] \\ar[u] & A \\ar[u]", "}", "$$", "as in Definition \\ref{definition-formally-smooth} be given.", "Then the composition $S \\to S' \\to A/J$ is continuous.", "By assumption the longer dotted arrow exists. By the universal", "property of tensor product we obtain the shorter dotted arrow." ], "refs": [ "more-algebra-definition-formally-smooth" ], "ref_ids": [ 10611 ] } ], "ref_ids": [] }, { "id": 10020, "type": "theorem", "label": "more-algebra-lemma-descent-fs", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-descent-fs", "contents": [ "Let $R$, $S$ be rings. Let $\\mathfrak n \\subset S$ be an ideal.", "Let $R \\to R'$ be a ring map. Set $S' = S \\otimes_R R'$ and", "$\\mathfrak n' = \\mathfrak nS$. If", "\\begin{enumerate}", "\\item the map $R \\to R'$ embeds $R$ as a direct summand of $R'$", "as an $R$-module, and", "\\item $R' \\to S'$ is formally smooth for the $\\mathfrak n'$-adic topology,", "\\end{enumerate}", "then $R \\to S$ is formally smooth in the $\\mathfrak n$-adic topology." ], "refs": [], "proofs": [ { "contents": [ "Let a solid diagram", "$$", "\\xymatrix{", "S \\ar[r] & A/J \\\\", "R \\ar[u] \\ar[r] & A \\ar[u]", "}", "$$", "as in Definition \\ref{definition-formally-smooth} be given.", "Set $A' = A \\otimes_R R'$ and $J' = \\Im(J \\otimes_R R' \\to A')$.", "The base change of the diagram above is the diagram", "$$", "\\xymatrix{", "S' \\ar[r] \\ar@{-->}[rd]^{\\psi'} & A'/J' \\\\", "R' \\ar[u] \\ar[r] & A' \\ar[u]", "}", "$$", "with continuous arrows. By condition (2) we obtain the dotted arrow", "$\\psi' : S' \\to A'$. Using condition (1) choose a direct summand decomposition", "$R' = R \\oplus C$ as $R$-modules. (Warning: $C$ isn't an ideal in $R'$.)", "Then $A' = A \\oplus A \\otimes_R C$. Set", "$$", "J'' = \\Im(J \\otimes_R C \\to A \\otimes_R C) \\subset J' \\subset A'.", "$$", "Then $J' = J \\oplus J''$ as $A$-modules. The image of the composition", "$\\psi : S \\to A'$ of $\\psi'$ with $S \\to S'$ is contained in", "$A + J' = A \\oplus J''$. However, in the ring $A + J' = A \\oplus J''$", "the $A$-submodule $J''$ is an ideal! (Use that $J^2 = 0$.) Hence the", "composition $S \\to A + J' \\to (A + J')/J'' = A$ is the arrow we were", "looking for." ], "refs": [ "more-algebra-definition-formally-smooth" ], "ref_ids": [ 10611 ] } ], "ref_ids": [] }, { "id": 10021, "type": "theorem", "label": "more-algebra-lemma-fs-local", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-fs-local", "contents": [ "Let $(R, \\mathfrak m) \\to (S, \\mathfrak n)$ be a local homomorphism", "of local rings. The following are equivalent", "\\begin{enumerate}", "\\item $R \\to S$ is formally smooth in the $\\mathfrak n$-adic topology,", "\\item for every solid commutative diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar@{-->}[rd] & A/J \\\\", "R \\ar[r] \\ar[u] & A \\ar[u]", "}", "$$", "of local homomorphisms of local rings where $J \\subset A$ is", "an ideal of square zero, $\\mathfrak m_A^n = 0$ for some $n > 0$, and", "$S \\to A/J$ induces an isomorphism on residue fields, a dotted", "arrow exists which makes the diagram commute.", "\\end{enumerate}", "If $S$ is Noetherian these conditions are also equivalent to", "\\begin{enumerate}", "\\item[(3)] same as in (2) but only for diagrams where in addition", "$A \\to A/J$ is a small extension", "(Algebra, Definition \\ref{algebra-definition-small-extension}).", "\\end{enumerate}" ], "refs": [ "algebra-definition-small-extension" ], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) follows from the definitions.", "Consider a diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar@{-->}[rd] & A/J \\\\", "R \\ar[r] \\ar[u] & A \\ar[u]", "}", "$$", "as in Definition \\ref{definition-formally-smooth} for the", "$\\mathfrak m$-adic topology on $R$ and the $\\mathfrak n$-adic topology on $S$.", "Pick $m > 0$ with $\\mathfrak n^m(A/J) = 0$", "(possible by continuity of maps in diagram).", "Consider the subring $A'$ of $A$ which is the inverse image", "of the image of $S$ in $A/J$. Set $J' = J$ viewed as an ideal in $A'$.", "Then $J'$ is an ideal of square zero in $A'$ and $A'/J'$", "is a quotient of $S/\\mathfrak n^m$. Hence $A'$ is local", "and $\\mathfrak m_{A'}^{2m} = 0$. Thus we get a diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar@{-->}[rd] & A'/J' \\\\", "R \\ar[r] \\ar[u] & A' \\ar[u]", "}", "$$", "as in (2). If we can construct the dotted arrow in this diagram,", "then we obtain the dotted arrow in the original one by composing", "with $A' \\to A$. In this way we see that (2) implies (1).", "\\medskip\\noindent", "Assume $S$ Noetherian. The implication (1) $\\Rightarrow$ (3) is immediate.", "Assume (3) and suppose a diagram as in (2) is given.", "Then $\\mathfrak m_A^n J = 0$ for some $n > 0$.", "Considering the maps", "$$", "A \\to A/\\mathfrak m_A^{n - 1}J \\to \\ldots \\to A/\\mathfrak mJ \\to A/J", "$$", "we see that it suffices to produce the lifting if $\\mathfrak m_A J = 0$.", "Assume $\\mathfrak m_A J = 0$ and let $A' \\subset A$ be the ring", "constructed above. Then $A'/J'$ is Artinian", "as a quotient of the Artinian local ring $S/\\mathfrak n^m$.", "Thus it suffices to show that given property (3) we can find the dotted", "arrow in diagrams as in (2) with $A/J$ Artinian and", "$\\mathfrak m_A J = 0$.", "Let $\\kappa$ be the common residue field of $A$,", "$A/J$, and $S$. By (3), if $J_0 \\subset J$ is an ideal with", "$\\dim_\\kappa(J/J_0) = 1$, then we can produce a dotted arrow", "$S \\to A/J_0$. Taking the product we obtain", "$$", "S \\longrightarrow \\prod\\nolimits_{J_0 \\text{ as above}} A/J_0", "$$", "Clearly the image of this arrow is contained in the sub $R$-algebra", "$A'$ of elements which map into the small diagonal", "$A/J \\subset \\prod_{J_0} A/J$.", "Let $J' \\subset A'$ be the elements mapping to zero in $A/J$.", "Then $J'$ is an ideal of square zero and as $\\kappa$-vector space", "equal to", "$$", "J' = \\prod\\nolimits_{J_0 \\text{ as above}} J/J_0", "$$", "Thus the map $J \\to J'$ is injective. By the theory of vector spaces", "we can choose a splitting $J' = J \\oplus M$. It follows that", "$$", "A' = A \\oplus M", "$$", "as an $R$-algebra. Hence the map $S \\to A'$ can be composed with", "the projection $A' \\to A$ to give the desired dotted arrow thereby", "finishing the proof of the lemma." ], "refs": [ "more-algebra-definition-formally-smooth" ], "ref_ids": [ 10611 ] } ], "ref_ids": [ 1538 ] }, { "id": 10022, "type": "theorem", "label": "more-algebra-lemma-fs-implies-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-fs-implies-regular", "contents": [ "Let $k$ be a field and let $(A, \\mathfrak m, K)$ be a Noetherian", "local $k$-algebra. If $k \\to A$ is formally smooth for the", "$\\mathfrak m$-adic topology, then $A$ is a regular local ring." ], "refs": [], "proofs": [ { "contents": [ "Let $k_0 \\subset k$ be the prime field. Then $k_0$ is perfect, hence", "$k / k_0$ is separable, hence formally smooth by", "Algebra, Lemma \\ref{algebra-lemma-formally-smooth-extensions-easy}. By", "Lemmas \\ref{lemma-formally-smooth} and \\ref{lemma-compose-formally-smooth}", "we see that $k_0 \\to A$ is formally smooth for the $\\mathfrak m$-adic", "topology on $A$. Hence we may assume $k = \\mathbf{Q}$ or $k = \\mathbf{F}_p$.", "\\medskip\\noindent", "By Algebra, Lemmas \\ref{algebra-lemma-completion-faithfully-flat} and", "\\ref{algebra-lemma-flat-under-regular} it", "suffices to prove the completion $A^\\wedge$ is regular.", "By Lemma \\ref{lemma-formally-smooth-completion} we may replace", "$A$ by $A^\\wedge$. Thus we may assume that $A$ is a Noetherian", "complete local ring. By the Cohen structure theorem", "(Algebra, Theorem \\ref{algebra-theorem-cohen-structure-theorem})", "there exist a map $K \\to A$. As $k$ is the prime field we see that", "$K \\to A$ is a $k$-algebra map.", "\\medskip\\noindent", "Let $x_1, \\ldots, x_n \\in \\mathfrak m$ be elements whose images", "form a basis of $\\mathfrak m/\\mathfrak m^2$.", "Set $T = K[[X_1, \\ldots, X_n]]$. Note that", "$$", "A/\\mathfrak m^2 \\cong K[x_1, \\ldots, x_n]/(x_ix_j)", "$$", "and", "$$", "T/\\mathfrak m_T^2 \\cong K[X_1, \\ldots, X_n]/(X_iX_j).", "$$", "Let $A/\\mathfrak m^2 \\to T/m_T^2$ be the local $K$-algebra isomorphism", "given by mapping the class of $x_i$ to the class of $X_i$.", "Denote $f_1 : A \\to T/\\mathfrak m_T^2$ the composition of this", "isomorphism with the quotient map $A \\to A/\\mathfrak m^2$.", "The assumption that $k \\to A$ is formally smooth in the $\\mathfrak m$-adic", "topology means we can lift $f_1$ to a map", "$f_2 : A \\to T/\\mathfrak{m}_T^3$, then to a map", "$f_3 : A \\to T/\\mathfrak{m}_T^4$, and so on, for all $n \\geq 1$.", "Warning: the maps $f_n$ are continuous $k$-algebra maps and may not", "be $K$-algebra maps. We get an induced map", "$f : A \\to T = \\lim T/\\mathfrak m_T^n$ of local $k$-algebras.", "By our choice of $f_1$, the map $f$ induces an", "isomorphism $\\mathfrak m/\\mathfrak m^2 \\to \\mathfrak m_T/\\mathfrak m_T^2$", "hence each $f_n$ is surjective and we conclude $f$ is surjective as $A$ is", "complete. This implies $\\dim(A) \\geq \\dim(T) = n$. Hence $A$ is regular", "by definition. (It also follows that $f$ is an isomorphism.)" ], "refs": [ "algebra-lemma-formally-smooth-extensions-easy", "more-algebra-lemma-formally-smooth", "more-algebra-lemma-compose-formally-smooth", "algebra-lemma-completion-faithfully-flat", "algebra-lemma-flat-under-regular", "more-algebra-lemma-formally-smooth-completion", "algebra-theorem-cohen-structure-theorem" ], "ref_ids": [ 1320, 10014, 10018, 871, 981, 10015, 327 ] } ], "ref_ids": [] }, { "id": 10023, "type": "theorem", "label": "more-algebra-lemma-lift-residue-field", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-residue-field", "contents": [ "Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be a complete", "local $k$-algebra. If $\\kappa/k$ is separable, then there exists", "a $k$-algebra map $\\kappa \\to A$ such that $\\kappa \\to A \\to \\kappa$", "is $\\text{id}_\\kappa$." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Proposition", "\\ref{algebra-proposition-characterize-separable-field-extensions}", "the extension $\\kappa/k$ is formally smooth. By", "Lemma \\ref{lemma-formally-smooth}", "$k \\to \\kappa$ is formally smooth in the sense of", "Definition \\ref{definition-formally-smooth}.", "Then we get $\\kappa \\to A$ from Lemma \\ref{lemma-lift-continuous}." ], "refs": [ "algebra-proposition-characterize-separable-field-extensions", "more-algebra-lemma-formally-smooth", "more-algebra-definition-formally-smooth", "more-algebra-lemma-lift-continuous" ], "ref_ids": [ 1429, 10014, 10611, 10016 ] } ], "ref_ids": [] }, { "id": 10024, "type": "theorem", "label": "more-algebra-lemma-power-series-over-residue-field", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-power-series-over-residue-field", "contents": [ "Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be a complete", "local $k$-algebra. If $\\kappa/k$ is separable and $A$ regular, then", "there exists an isomorphism of $A \\cong \\kappa[[t_1, \\ldots, t_d]]$", "as $k$-algebras." ], "refs": [], "proofs": [ { "contents": [ "Choose $\\kappa \\to A$ as in Lemma \\ref{lemma-lift-residue-field}", "and apply Algebra, Lemma", "\\ref{algebra-lemma-regular-complete-containing-coefficient-field}." ], "refs": [ "more-algebra-lemma-lift-residue-field", "algebra-lemma-regular-complete-containing-coefficient-field" ], "ref_ids": [ 10023, 1331 ] } ], "ref_ids": [] }, { "id": 10025, "type": "theorem", "label": "more-algebra-lemma-regular-implies-fs", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-regular-implies-fs", "contents": [ "Let $k$ be a field. Let $(A, \\mathfrak m, K)$ be a regular local", "$k$-algebra such that $K/k$ is separable. Then $k \\to A$", "is formally smooth in the $\\mathfrak m$-adic topology." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove that the completion of $A$ is formally", "smooth over $k$, see Lemma \\ref{lemma-formally-smooth-completion}.", "Hence we may assume that $A$ is a complete local regular $k$-algebra", "with residue field $K$ separable over $k$.", "By Lemma \\ref{lemma-power-series-over-residue-field} we see that", "$A = K[[x_1, \\ldots, x_n]]$.", "\\medskip\\noindent", "The power series ring $K[[x_1, \\ldots, x_n]]$ is formally", "smooth over $k$. Namely, $K$ is formally smooth over $k$ and", "$K[x_1, \\ldots, x_n]$ is formally smooth over $K$ as a polynomial algebra.", "Hence $K[x_1, \\ldots, x_n]$ is formally smooth over $k$ by", "Algebra, Lemma \\ref{algebra-lemma-compose-formally-smooth}.", "It follows that $k \\to K[x_1, \\ldots, x_n]$ is formally smooth", "for the $(x_1, \\ldots, x_n)$-adic topology by", "Lemma \\ref{lemma-formally-smooth}.", "Finally, it follows that $k \\to K[[x_1, \\ldots, x_n]]$ is formally", "smooth for the $(x_1, \\ldots, x_n)$-adic topology by", "Lemma \\ref{lemma-formally-smooth-completion}." ], "refs": [ "more-algebra-lemma-formally-smooth-completion", "more-algebra-lemma-power-series-over-residue-field", "algebra-lemma-compose-formally-smooth", "more-algebra-lemma-formally-smooth", "more-algebra-lemma-formally-smooth-completion" ], "ref_ids": [ 10015, 10024, 1205, 10014, 10015 ] } ], "ref_ids": [] }, { "id": 10026, "type": "theorem", "label": "more-algebra-lemma-formally-smooth-finite-type", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-formally-smooth-finite-type", "contents": [ "Let $A \\to B$ be a finite type ring map with $A$ Noetherian.", "Let $\\mathfrak q \\subset B$ be a prime ideal lying over", "$\\mathfrak p \\subset A$. The following are equivalent", "\\begin{enumerate}", "\\item $A \\to B$ is smooth at $\\mathfrak q$, and", "\\item $A_\\mathfrak p \\to B_\\mathfrak q$ is formally smooth in", "the $\\mathfrak q$-adic topology.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) follows from", "Algebra, Lemma \\ref{algebra-lemma-smooth-test-artinian}.", "Conversely, if $A \\to B$ is smooth at $\\mathfrak q$, then", "$A \\to B_g$ is smooth for some $g \\in B$, $g \\not \\in \\mathfrak q$.", "Then $A \\to B_g$ is formally smooth by", "Algebra, Proposition \\ref{algebra-proposition-smooth-formally-smooth}.", "Hence $A_\\mathfrak p \\to B_\\mathfrak q$ is formally smooth", "as localization preserves formal smoothness (for example by", "the criterion of Algebra, Proposition", "\\ref{algebra-proposition-characterize-formally-smooth}", "and the fact that the cotangent complex behaves well with respect to", "localization, see", "Algebra, Lemmas \\ref{algebra-lemma-NL-localize-bottom} and", "\\ref{algebra-lemma-localize-NL}).", "Finally, Lemma \\ref{lemma-formally-smooth} implies that", "$A_\\mathfrak p \\to B_\\mathfrak q$ is formally smooth in the", "$\\mathfrak q$-adic topology." ], "refs": [ "algebra-lemma-smooth-test-artinian", "algebra-proposition-smooth-formally-smooth", "algebra-proposition-characterize-formally-smooth", "algebra-lemma-NL-localize-bottom", "algebra-lemma-localize-NL", "more-algebra-lemma-formally-smooth" ], "ref_ids": [ 1229, 1426, 1425, 1159, 1161, 10014 ] } ], "ref_ids": [] }, { "id": 10027, "type": "theorem", "label": "more-algebra-lemma-power-series-ring-over-Cohen-fs", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-power-series-ring-over-Cohen-fs", "contents": [ "Let $K$ be a field of characteristic $0$ and $A = K[[x_1, \\ldots, x_n]]$.", "Let $L$ be a field of characteristic $p > 0$ and $B = L[[x_1, \\ldots, x_n]]$.", "Let $\\Lambda$ be a Cohen ring. Let $C = \\Lambda[[x_1, \\ldots, x_n]]$.", "\\begin{enumerate}", "\\item $\\mathbf{Q} \\to A$ is formally smooth in the ", "$\\mathfrak m$-adic topology.", "\\item $\\mathbf{F}_p \\to B$ is formally smooth in the ", "$\\mathfrak m$-adic topology.", "\\item $\\mathbf{Z} \\to C$ is formally smooth in the", "$\\mathfrak m$-adic topology.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By the universal property of power series rings it suffices to prove:", "\\begin{enumerate}", "\\item $\\mathbf{Q} \\to K$ is formally smooth.", "\\item $\\mathbf{F}_p \\to L$ is formally smooth.", "\\item $\\mathbf{Z} \\to \\Lambda$ is formally smooth in the", "$\\mathfrak m$-adic topology.", "\\end{enumerate}", "The first two are", "Algebra, Proposition", "\\ref{algebra-proposition-characterize-separable-field-extensions}.", "The third follows from", "Algebra, Lemma \\ref{algebra-lemma-cohen-ring-formally-smooth}", "since for any test diagram as in Definition \\ref{definition-formally-smooth}", "some power of $p$ will be zero in $A/J$ and hence some power of $p$ will", "be zero in $A$." ], "refs": [ "algebra-proposition-characterize-separable-field-extensions", "algebra-lemma-cohen-ring-formally-smooth", "more-algebra-definition-formally-smooth" ], "ref_ids": [ 1429, 1330, 10611 ] } ], "ref_ids": [] }, { "id": 10028, "type": "theorem", "label": "more-algebra-lemma-quotient-power-series-ring-over-Cohen", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-quotient-power-series-ring-over-Cohen", "contents": [ "Let $K$ be a field and $A = K[[x_1, \\ldots, x_n]]$.", "Let $\\Lambda$ be a Cohen ring and let $B = \\Lambda[[x_1, \\ldots, x_n]]$.", "\\begin{enumerate}", "\\item If $y_1, \\ldots, y_n \\in A$ is a regular system of parameters", "then $K[[y_1, \\ldots, y_n]] \\to A$ is an isomorphism.", "\\item If $z_1, \\ldots, z_r \\in A$ form part of a regular system of", "parameters for $A$, then $r \\leq n$ and", "$A/(z_1, \\ldots, z_r) \\cong K[[y_1, \\ldots, y_{n - r}]]$.", "\\item If $p, y_1, \\ldots, y_n \\in B$ is a regular system of parameters", "then $\\Lambda[[y_1, \\ldots, y_n]] \\to B$ is an isomorphism.", "\\item If $p, z_1, \\ldots, z_r \\in B$ form part of a regular system of", "parameters for $B$, then $r \\leq n$ and", "$B/(z_1, \\ldots, z_r) \\cong \\Lambda[[y_1, \\ldots, y_{n - r}]]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Set $A' = K[[y_1, \\ldots, y_n]]$. It is clear that", "the map $A' \\to A$ induces an isomorphism", "$A'/\\mathfrak m_{A'}^n \\to A/\\mathfrak m_A^n$ for all $n \\geq 1$.", "Since $A$ and $A'$ are both complete we deduce that $A' \\to A$ is an", "isomorphism.", "Proof of (2). Extend $z_1, \\ldots, z_r$ to a regular system of parameters", "$z_1, \\ldots, z_r, y_1, \\ldots, y_{n - r}$ of $A$. Consider the map", "$A' = K[[z_1, \\ldots, z_r, y_1, \\ldots, y_{n - r}]] \\to A$.", "This is an isomorphism by (1). Hence (2) follows as it is clear that", "$A'/(z_1, \\ldots, z_r) \\cong K[[y_1, \\ldots, y_{n - r}]]$.", "The proofs of (3) and (4) are exactly the same as the proofs of (1) and (2)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10029, "type": "theorem", "label": "more-algebra-lemma-embed-map-Noetherian-complete-local-rings", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-embed-map-Noetherian-complete-local-rings", "contents": [ "Let $A \\to B$ be a local homomorphism of Noetherian complete local rings.", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "S \\ar[r] & B \\\\", "R \\ar[u] \\ar[r] & A \\ar[u]", "}", "$$", "with the following properties:", "\\begin{enumerate}", "\\item the horizontal arrows are surjective,", "\\item if the characteristic of $A/\\mathfrak m_A$ is zero, then $S$ and $R$", "are power series rings over fields,", "\\item if the characteristic of $A/\\mathfrak m_A$ is $p > 0$, then $S$ and $R$", "are power series rings over Cohen rings, and", "\\item $R \\to S$ maps a regular system of parameters of $R$ to part of a", "regular system of parameters of $S$.", "\\end{enumerate}", "In particular $R \\to S$ is flat (see Algebra,", "Lemma \\ref{algebra-lemma-flat-over-regular}) with regular fibre", "$S/\\mathfrak m_R S$ (see Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM})." ], "refs": [ "algebra-lemma-flat-over-regular", "algebra-lemma-regular-ring-CM" ], "proofs": [ { "contents": [ "Use the Cohen structure theorem", "(Algebra, Theorem \\ref{algebra-theorem-cohen-structure-theorem})", "to choose a surjection $S \\to B$ as in the statement of the lemma", "where we choose $S$ to be a power series over a Cohen ring if the", "residue characteristic is $p > 0$ and a power series over a field else.", "Let $J \\subset S$ be the kernel of $S \\to B$.", "Next, choose a surjection $R = \\Lambda[[x_1, \\ldots, x_n]] \\to A$ where", "we choose $\\Lambda$ to be a Cohen ring if the residue characteristic of", "$A$ is $p > 0$ and $\\Lambda$ equal to the residue field of $A$ otherwise.", "We lift the composition $\\Lambda[[x_1, \\ldots, x_n]] \\to A \\to B$", "to a map $\\varphi : R \\to S$. This is possible because", "$\\Lambda[[x_1, \\ldots, x_n]]$ is formally smooth over $\\mathbf{Z}$", "in the $\\mathfrak m$-adic topology (see", "Lemma \\ref{lemma-power-series-ring-over-Cohen-fs})", "by an application of Lemma \\ref{lemma-lift-continuous}.", "Finally, we replace $\\varphi$ by the map", "$\\varphi' : R = \\Lambda[[x_1, \\ldots, x_n]] \\to S' = S[[y_1, \\ldots, y_n]]$", "with $\\varphi'|_\\Lambda = \\varphi|_\\Lambda$ and", "$\\varphi'(x_i) = \\varphi(x_i) + y_i$. We also replace $S \\to B$", "by the map $S' \\to B$ which maps $y_i$ to zero. After this replacement", "it is clear that a regular system of parameters of $R$ maps to part of a", "regular sequence in $S'$ and we win." ], "refs": [ "algebra-theorem-cohen-structure-theorem", "more-algebra-lemma-power-series-ring-over-Cohen-fs", "more-algebra-lemma-lift-continuous" ], "ref_ids": [ 327, 10027, 10016 ] } ], "ref_ids": [ 1108, 941 ] }, { "id": 10030, "type": "theorem", "label": "more-algebra-lemma-dominate-two-surjections", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-dominate-two-surjections", "contents": [ "Let $S \\to R$ and $S' \\to R$ be surjective maps of complete Noetherian local", "rings. Then $S \\times_R S'$ is a complete Noetherian local ring." ], "refs": [], "proofs": [ { "contents": [ "Let $k$ be the residue field of $R$. If the characteristic of", "$k$ is $p > 0$, then we denote $\\Lambda$ a Cohen ring", "(Algebra, Definition \\ref{algebra-definition-cohen-ring})", "with residue field $k$ (Algebra, Lemma \\ref{algebra-lemma-cohen-rings-exist}).", "If the characteristic of $k$ is $0$ we set $\\Lambda = k$.", "Choose a surjection $\\Lambda[[x_1, \\ldots, x_n]] \\to R$", "(as in the Cohen structure theorem, see", "Algebra, Theorem \\ref{algebra-theorem-cohen-structure-theorem})", "and lift this to maps $\\Lambda[[x_1, \\ldots, x_n]] \\to S$ and", "$\\varphi : \\Lambda[[x_1, \\ldots, x_n]] \\to S$ and", "$\\varphi' : \\Lambda[[x_1, \\ldots, x_n]] \\to S'$ using", "Lemmas \\ref{lemma-power-series-ring-over-Cohen-fs} and", "\\ref{lemma-lift-continuous}.", "Next, choose $f_1, \\ldots, f_m \\in S$ generating the kernel", "of $S \\to R$ and $f'_1, \\ldots, f'_{m'} \\in S'$ generating the", "kernel of $S' \\to R$. Then the map", "$$", "\\Lambda[[x_1, \\ldots, x_n, y_1, \\ldots, y_m, z_1, \\ldots, z_{m'}]]", "\\longrightarrow S \\times_R S,", "$$", "which sends $x_i$ to $(\\varphi(x_i), \\varphi'(x_i))$ and", "$y_j$ to $(f_j, 0)$ and", "$z_{j'}$ to $(0, f'_j)$", "is surjective. Thus $S \\times_R S'$ is a quotient of a complete", "local ring, whence complete." ], "refs": [ "algebra-definition-cohen-ring", "algebra-lemma-cohen-rings-exist", "algebra-theorem-cohen-structure-theorem", "more-algebra-lemma-power-series-ring-over-Cohen-fs", "more-algebra-lemma-lift-continuous" ], "ref_ids": [ 1550, 1329, 327, 10027, 10016 ] } ], "ref_ids": [] }, { "id": 10031, "type": "theorem", "label": "more-algebra-lemma-formally-smooth-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-formally-smooth-flat", "contents": [ "Let $A \\to B$ be a local homomorphism of Noetherian local rings.", "Assume $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic", "topology. Then $A \\to B$ is flat." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $A$ and $B$ a Noetherian complete local rings", "by Lemma \\ref{lemma-formally-smooth-completion} and", "Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian-Noetherian}", "(this also uses", "Algebra, Lemma \\ref{algebra-lemma-flatness-descends-more-general} and", "\\ref{algebra-lemma-completion-faithfully-flat}", "to see that flatness of the map on completions implies flatness of", "$A \\to B$).", "Choose a commutative diagram", "$$", "\\xymatrix{", "S \\ar[r] & B \\\\", "R \\ar[u] \\ar[r] & A \\ar[u]", "}", "$$", "as in Lemma \\ref{lemma-embed-map-Noetherian-complete-local-rings}", "with $R \\to S$ flat. Let $I \\subset R$ be the kernel of $R \\to A$.", "Because $B$ is formally smooth over $A$ we see that the $A$-algebra map", "$$", "S/IS \\longrightarrow B", "$$", "has a section, see Lemma \\ref{lemma-lift-continuous}.", "Hence $B$ is a direct summand of the flat $A$-module $S/IS$", "(by base change of flatness, see", "Algebra, Lemma \\ref{algebra-lemma-flat-base-change}),", "whence flat." ], "refs": [ "more-algebra-lemma-formally-smooth-completion", "algebra-lemma-completion-Noetherian-Noetherian", "algebra-lemma-flatness-descends-more-general", "algebra-lemma-completion-faithfully-flat", "more-algebra-lemma-embed-map-Noetherian-complete-local-rings", "more-algebra-lemma-lift-continuous", "algebra-lemma-flat-base-change" ], "ref_ids": [ 10015, 874, 529, 871, 10029, 10016, 527 ] } ], "ref_ids": [] }, { "id": 10032, "type": "theorem", "label": "more-algebra-lemma-formally-smooth-JZ", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-formally-smooth-JZ", "contents": [ "Let $A \\to B$ be a local homomorphism of Noetherian local rings.", "Assume $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic", "topology. Let $K$ be the residue field of $B$. Then", "the Jacobi-Zariski sequence for $A \\to B \\to K$ gives an exact sequence", "$$", "0 \\to H_1(\\NL_{K/A}) \\to \\mathfrak m_B/\\mathfrak m_B^2", "\\to \\Omega_{B/A} \\otimes_B K \\to \\Omega_{K/A} \\to 0", "$$" ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\mathfrak m_B/\\mathfrak m_B^2 = H_1(\\NL_{K/B})$", "by Algebra, Lemma \\ref{algebra-lemma-NL-surjection}.", "By Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL}", "it remains to show injectivity of", "$H_1(\\NL_{K/A}) \\to \\mathfrak m_B/\\mathfrak m_B^2$.", "With $k$ the residue field of $A$, the Jacobi-Zariski sequence", "for $A \\to k \\to K$ gives $\\Omega_{K/A} = \\Omega_{K/k}$ and", "an exact sequence", "$$", "\\mathfrak m_A/\\mathfrak m_A^2 \\otimes_k K \\to", "H_1(\\NL_{K/A}) \\to", "H_1(\\NL_{K/k}) \\to 0", "$$", "Set $\\overline{B} = B \\otimes_A k$. Since $\\overline{B}$ is", "regular the ideal $\\mathfrak m_{\\overline{B}}$ is generated", "by a regular sequence. Applying", "Lemmas \\ref{lemma-conormal-sequence-H1-regular} and", "\\ref{lemma-noetherian-finite-all-equivalent}", "to $\\mathfrak m_A B \\subset \\mathfrak m_B$", "we find $\\mathfrak m_A B / (\\mathfrak m_AB \\cap \\mathfrak m_B^2) =", "\\mathfrak m_A B / \\mathfrak m_A \\mathfrak m_B$ which is", "equal to $\\mathfrak m_A/\\mathfrak m_A^2 \\otimes_k K$", "as $A \\to B$ is flat by Lemma \\ref{lemma-formally-smooth-flat}.", "Thus we obtain a short exact sequence", "$$", "0 \\to", "\\mathfrak m_A/\\mathfrak m_A^2 \\otimes_k K \\to", "\\mathfrak m_B/\\mathfrak m_B^2 \\to", "\\mathfrak m_{\\overline{B}}/\\mathfrak m_{\\overline{B}}^2 \\to 0", "$$", "Functoriality of the Jacobi-Zariski sequences shows that", "we obtain a commutative diagram", "$$", "\\xymatrix{", "&", "\\mathfrak m_A/\\mathfrak m_A^2 \\otimes_k K \\ar[d] \\ar[r] &", "H_1(\\NL_{K/A}) \\ar[d] \\ar[r] &", "H_1(\\NL_{K/k}) \\ar[d] \\ar[r] & 0 \\\\", "0 \\ar[r] &", "\\mathfrak m_A/\\mathfrak m_A^2 \\otimes_k K \\ar[r] &", "\\mathfrak m_B/\\mathfrak m_B^2 \\ar[r] &", "\\mathfrak m_{\\overline{B}}/\\mathfrak m_{\\overline{B}}^2 \\ar[r] & 0", "}", "$$", "The left vertical arrow is injective by Theorem \\ref{theorem-regular-fs}", "as $k \\to \\overline{B}$ is formally smooth in", "the $\\mathfrak m_{\\overline{B}}$-adic topology by", "Lemma \\ref{lemma-base-change-fs}.", "This finishes the proof by the snake lemma." ], "refs": [ "algebra-lemma-NL-surjection", "algebra-lemma-exact-sequence-NL", "more-algebra-lemma-conormal-sequence-H1-regular", "more-algebra-lemma-noetherian-finite-all-equivalent", "more-algebra-lemma-formally-smooth-flat", "more-algebra-theorem-regular-fs", "more-algebra-lemma-base-change-fs" ], "ref_ids": [ 1154, 1153, 9980, 9978, 10031, 9802, 10019 ] } ], "ref_ids": [] }, { "id": 10033, "type": "theorem", "label": "more-algebra-lemma-lift-fs", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-fs", "contents": [ "Let $A$ be a Noetherian complete local ring with residue field $k$.", "Let $B$ be a Noetherian complete local $k$-algebra. Assume $k \\to B$", "is formally smooth in the $\\mathfrak m_B$-adic topology.", "Then there exists a Noetherian complete local ring $C$", "and a local homomorphism $A \\to C$ which is formally smooth", "in the $\\mathfrak m_C$-adic topology such that $C \\otimes_A k \\cong B$." ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram", "$$", "\\xymatrix{", "S \\ar[r] & B \\\\", "R \\ar[u] \\ar[r] & A \\ar[u]", "}", "$$", "as in Lemma \\ref{lemma-embed-map-Noetherian-complete-local-rings}.", "Let $t_1, \\ldots, t_d$ be a regular system of parameters for $R$", "with $t_1 = p$ in case the characteristic of $k$ is $p > 0$.", "As $B$ and $\\overline{S} = S \\otimes_A k$", "are regular we see that $\\Ker(\\overline{S} \\to B)$ is generated by", "elements $\\overline{x}_1, \\ldots, \\overline{x}_r$ which form part of a", "regular system of parameters of $\\overline{S}$, see", "Algebra, Lemma \\ref{algebra-lemma-regular-quotient-regular}.", "Lift these elements to $x_1, \\ldots, x_r \\in S$. Then", "$t_1, \\ldots, t_d, x_1, \\ldots, x_r$ is part of a regular system of", "parameters for $S$. Hence $S/(x_1, \\ldots, x_r)$ is a power", "series ring over a field (if the characteristic of $k$ is zero)", "or a power series ring over a Cohen ring (if the characteristic of", "$k$ is $p > 0$), see", "Lemma \\ref{lemma-quotient-power-series-ring-over-Cohen}.", "Moreover, it is still the case that $R \\to S/(x_1, \\ldots, x_r)$", "maps $t_1, \\ldots, t_d$ to a part of a regular system of parameters", "of $S/(x_1, \\ldots, x_r)$. In other words, we may replace $S$ by", "$S/(x_1, \\ldots, x_r)$ and assume we have a diagram", "$$", "\\xymatrix{", "S \\ar[r] & B \\\\", "R \\ar[u] \\ar[r] & A \\ar[u]", "}", "$$", "as in Lemma \\ref{lemma-embed-map-Noetherian-complete-local-rings}", "with moreover $\\overline{S} = B$. In this case $R \\to S$ is", "formally smooth in the $\\mathfrak m_S$-adic topology by", "Proposition \\ref{proposition-fs-flat-fibre-fs}.", "Hence the base change $C = S \\otimes_R A$ is formally smooth", "over $A$ in the $\\mathfrak m_C$-adic topology by", "Lemma \\ref{lemma-base-change-fs}." ], "refs": [ "more-algebra-lemma-embed-map-Noetherian-complete-local-rings", "algebra-lemma-regular-quotient-regular", "more-algebra-lemma-quotient-power-series-ring-over-Cohen", "more-algebra-lemma-embed-map-Noetherian-complete-local-rings", "more-algebra-proposition-fs-flat-fibre-fs", "more-algebra-lemma-base-change-fs" ], "ref_ids": [ 10029, 942, 10028, 10029, 10577, 10019 ] } ], "ref_ids": [] }, { "id": 10034, "type": "theorem", "label": "more-algebra-lemma-regular-local", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-regular-local", "contents": [ "Let $R \\to \\Lambda$ be a ring map with $\\Lambda$ Noetherian.", "The following are equivalent", "\\begin{enumerate}", "\\item $R \\to \\Lambda$ is regular,", "\\item $R_\\mathfrak p \\to \\Lambda_\\mathfrak q$ is regular for all", "$\\mathfrak q \\subset \\Lambda$ lying over $\\mathfrak p \\subset R$, and", "\\item $R_\\mathfrak m \\to \\Lambda_{\\mathfrak m'}$ is regular for", "all maximal ideals $\\mathfrak m' \\subset \\Lambda$", "lying over $\\mathfrak m$ in $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is true because a Noetherian ring is regular if and only if", "all the local rings are regular local rings, see", "Algebra, Definition \\ref{algebra-definition-regular}", "and a ring map is flat if and only if all the induced maps of local", "rings are flat, see", "Algebra, Lemma \\ref{algebra-lemma-flat-localization}." ], "refs": [ "algebra-definition-regular", "algebra-lemma-flat-localization" ], "ref_ids": [ 1512, 538 ] } ], "ref_ids": [] }, { "id": 10035, "type": "theorem", "label": "more-algebra-lemma-regular-base-change", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-regular-base-change", "contents": [ "Let $R \\to \\Lambda$ be a regular ring map.", "For any finite type ring map $R \\to R'$ the base change", "$R' \\to \\Lambda \\otimes_R R'$ is regular too." ], "refs": [], "proofs": [ { "contents": [ "Flatness is preserved under any base change, see", "Algebra, Lemma \\ref{algebra-lemma-flat-base-change}.", "Consider a prime $\\mathfrak p' \\subset R'$ lying over", "$\\mathfrak p \\subset R$. The residue field extension", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak p')$ is", "finitely generated as $R'$ is of finite type over $R$.", "Hence the fibre ring", "$$", "(\\Lambda \\otimes_R R') \\otimes_{R'} \\kappa(\\mathfrak p') =", "\\Lambda \\otimes_R \\kappa(\\mathfrak p) \\otimes_{\\kappa(\\mathfrak p)} ", "\\kappa(\\mathfrak p')", "$$", "is Noetherian by", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-field-extension}", "and the assumption on the fibre rings of $R \\to \\Lambda$.", "Geometric regularity of the fibres is preserved by", "Algebra, Lemma \\ref{algebra-lemma-geometrically-regular}." ], "refs": [ "algebra-lemma-flat-base-change", "algebra-lemma-Noetherian-field-extension", "algebra-lemma-geometrically-regular" ], "ref_ids": [ 527, 455, 1382 ] } ], "ref_ids": [] }, { "id": 10036, "type": "theorem", "label": "more-algebra-lemma-regular-composition", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-regular-composition", "contents": [ "Let $A \\to B \\to C$ be regular ring maps.", "If the fibre rings of $A \\to C$ are Noetherian, then", "$A \\to C$ is regular." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p \\subset A$ be a prime. Let $\\kappa(\\mathfrak p) \\subset k$", "be a finite purely inseparable extension. We have to show that", "$C \\otimes_A k$ is regular. By Lemma \\ref{lemma-regular-base-change}", "we may assume that $A = k$ and we reduce to proving that $C$ is regular.", "The assumption is that $B$ is regular and that $B \\to C$ is flat", "with regular fibres. Then $C$ is regular by Algebra, Lemma", "\\ref{algebra-lemma-flat-over-regular-with-regular-fibre}.", "Some details omitted." ], "refs": [ "more-algebra-lemma-regular-base-change", "algebra-lemma-flat-over-regular-with-regular-fibre" ], "ref_ids": [ 10035, 988 ] } ], "ref_ids": [] }, { "id": 10037, "type": "theorem", "label": "more-algebra-lemma-colimit-smooth-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-colimit-smooth-regular", "contents": [ "Let $R$ be a ring. Let $(A_i, \\varphi_{ii'})$ be a directed system", "of smooth $R$-algebras. Set $\\Lambda = \\colim A_i$. If the fibre", "rings $\\Lambda \\otimes_R \\kappa(\\mathfrak p)$ are Noetherian for all", "$\\mathfrak p \\subset R$, then $R \\to \\Lambda$ is regular." ], "refs": [], "proofs": [ { "contents": [ "Note that $\\Lambda$ is flat over $R$ by", "Algebra, Lemmas \\ref{algebra-lemma-colimit-flat} and", "\\ref{algebra-lemma-smooth-syntomic}.", "Let $\\kappa(\\mathfrak p) \\subset k$ be a finite purely inseparable", "extension. Note that", "$$", "\\Lambda \\otimes_R \\kappa(\\mathfrak p) \\otimes_{\\kappa(\\mathfrak p)} k =", "\\Lambda \\otimes_R k = \\colim A_i \\otimes_R k", "$$", "is a colimit of smooth $k$-algebras, see", "Algebra, Lemma \\ref{algebra-lemma-base-change-smooth}.", "Since each local ring of a smooth $k$-algebra is regular by", "Algebra, Lemma \\ref{algebra-lemma-characterize-smooth-over-field}", "we conclude that all local rings of $\\Lambda \\otimes_R k$ are", "regular by", "Algebra, Lemma \\ref{algebra-lemma-colimit-regular}.", "This proves the lemma." ], "refs": [ "algebra-lemma-colimit-flat", "algebra-lemma-smooth-syntomic", "algebra-lemma-base-change-smooth", "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-colimit-regular" ], "ref_ids": [ 523, 1195, 1191, 1223, 946 ] } ], "ref_ids": [] }, { "id": 10038, "type": "theorem", "label": "more-algebra-lemma-regular-field-extension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-regular-field-extension", "contents": [ "Let $k \\subset K$ be a field extension. Then $k \\to K$ is a regular", "ring map if and only if $K$ is a separable field extension of $k$." ], "refs": [], "proofs": [ { "contents": [ "If $k \\to K$ is regular, then $K$ is geometrically reduced over $k$,", "hence $K$ is separable over $k$ by", "Algebra, Proposition", "\\ref{algebra-proposition-characterize-separable-field-extensions}.", "Conversely, if $K/k$ is separable, then $K$ is a colimit of smooth", "$k$-algebras, see", "Algebra, Lemma \\ref{algebra-lemma-colimit-syntomic}", "hence is regular by", "Lemma \\ref{lemma-colimit-smooth-regular}." ], "refs": [ "algebra-proposition-characterize-separable-field-extensions", "algebra-lemma-colimit-syntomic", "more-algebra-lemma-colimit-smooth-regular" ], "ref_ids": [ 1429, 1323, 10037 ] } ], "ref_ids": [] }, { "id": 10039, "type": "theorem", "label": "more-algebra-lemma-regular-permanence", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-regular-permanence", "contents": [ "Let $A \\to B \\to C$ be ring maps. If $A \\to C$ is regular and $B \\to C$", "is flat and surjective on spectra, then $A \\to B$ is regular." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-flat-permanence} we see that", "$A \\to B$ is flat. Let $\\mathfrak p \\subset A$ be a prime. The ring", "map $B \\otimes_A \\kappa(\\mathfrak p) \\to C \\otimes_A \\kappa(\\mathfrak p)$", "is flat and surjective on spectra. Hence $B \\otimes_A \\kappa(\\mathfrak p)$", "is geometrically regular by", "Algebra, Lemma \\ref{algebra-lemma-geometrically-regular-descent}." ], "refs": [ "algebra-lemma-flat-permanence", "algebra-lemma-geometrically-regular-descent" ], "ref_ids": [ 530, 1383 ] } ], "ref_ids": [] }, { "id": 10040, "type": "theorem", "label": "more-algebra-lemma-reduced-goes-up", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-reduced-goes-up", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Assume", "\\begin{enumerate}", "\\item $\\varphi$ is regular,", "\\item $S$ is Noetherian, and", "\\item $R$ is Noetherian and reduced.", "\\end{enumerate}", "Then $S$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "For Noetherian rings being reduced is the same as having properties", "$(S_1)$ and $(R_0)$, see", "Algebra, Lemma \\ref{algebra-lemma-criterion-reduced}.", "Hence we may apply", "Algebra, Lemmas \\ref{algebra-lemma-Sk-goes-up} and", "\\ref{algebra-lemma-Rk-goes-up}." ], "refs": [ "algebra-lemma-criterion-reduced", "algebra-lemma-Sk-goes-up", "algebra-lemma-Rk-goes-up" ], "ref_ids": [ 1310, 1363, 1364 ] } ], "ref_ids": [] }, { "id": 10041, "type": "theorem", "label": "more-algebra-lemma-normal-goes-up", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-normal-goes-up", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Assume", "\\begin{enumerate}", "\\item $\\varphi$ is regular,", "\\item $S$ is Noetherian, and", "\\item $R$ is Noetherian and normal.", "\\end{enumerate}", "Then $S$ is normal." ], "refs": [], "proofs": [ { "contents": [ "For Noetherian rings being normal is the same as having properties", "$(S_2)$ and $(R_1)$, see", "Algebra, Lemma \\ref{algebra-lemma-criterion-normal}.", "Hence we may apply", "Algebra, Lemmas \\ref{algebra-lemma-Sk-goes-up} and", "\\ref{algebra-lemma-Rk-goes-up}." ], "refs": [ "algebra-lemma-criterion-normal", "algebra-lemma-Sk-goes-up", "algebra-lemma-Rk-goes-up" ], "ref_ids": [ 1311, 1363, 1364 ] } ], "ref_ids": [] }, { "id": 10042, "type": "theorem", "label": "more-algebra-lemma-completion-dimension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-completion-dimension", "contents": [ "Let $A$ be a Noetherian local ring.", "Then $\\dim(A) = \\dim(A^\\wedge)$." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-completion-complete} the map", "$A \\to A^\\wedge$ induces isomorphisms", "$A/\\mathfrak m^n = A^\\wedge/(\\mathfrak m^\\wedge)^n$ for $n \\geq 1$.", "By Algebra, Lemma \\ref{algebra-lemma-pushdown-module} this implies that", "$$", "\\text{length}_A(A/\\mathfrak m^n) =", "\\text{length}_{A^\\wedge}(A^\\wedge/(\\mathfrak m^\\wedge)^n)", "$$", "for all $n \\geq 1$. Thus $d(A) = d(A^\\wedge)$ and we conclude", "by Algebra, Proposition \\ref{algebra-proposition-dimension}.", "An alternative proof is to use", "Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}." ], "refs": [ "algebra-lemma-completion-complete", "algebra-lemma-pushdown-module", "algebra-proposition-dimension", "algebra-lemma-dimension-base-fibre-equals-total" ], "ref_ids": [ 872, 639, 1411, 987 ] } ], "ref_ids": [] }, { "id": 10043, "type": "theorem", "label": "more-algebra-lemma-completion-depth", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-completion-depth", "contents": [ "Let $A$ be a Noetherian local ring. Then", "$\\text{depth}(A) = \\text{depth}(A^\\wedge)$." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck}." ], "refs": [ "algebra-lemma-apply-grothendieck" ], "ref_ids": [ 1361 ] } ], "ref_ids": [] }, { "id": 10044, "type": "theorem", "label": "more-algebra-lemma-completion-CM", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-completion-CM", "contents": [ "Let $A$ be a Noetherian local ring.", "Then $A$ is Cohen-Macaulay if and only if $A^\\wedge$ is so." ], "refs": [], "proofs": [ { "contents": [ "A local ring $A$ is Cohen-Macaulay if and only if $\\dim(A) = \\text{depth}(A)$.", "As both of these invariants are preserved under completion", "(Lemmas \\ref{lemma-completion-dimension} and \\ref{lemma-completion-depth})", "the claim follows." ], "refs": [ "more-algebra-lemma-completion-dimension", "more-algebra-lemma-completion-depth" ], "ref_ids": [ 10042, 10043 ] } ], "ref_ids": [] }, { "id": 10045, "type": "theorem", "label": "more-algebra-lemma-completion-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-completion-regular", "contents": [ "Let $A$ be a Noetherian local ring.", "Then $A$ is regular if and only if $A^\\wedge$ is so." ], "refs": [], "proofs": [ { "contents": [ "If $A^\\wedge$ is regular, then $A$ is regular by", "Algebra, Lemma \\ref{algebra-lemma-flat-under-regular}.", "Assume $A$ is regular. Let $\\mathfrak m$ be the maximal ideal", "of $A$. Then $\\dim_{\\kappa(\\mathfrak m)} \\mathfrak m/\\mathfrak m^2 =", "\\dim(A) = \\dim(A^\\wedge)$ (Lemma \\ref{lemma-completion-dimension}).", "On the other hand, $\\mathfrak mA^\\wedge$ is the maximal ideal of", "$A^\\wedge$ and hence $\\mathfrak m_{A^\\wedge}$", "is generated by at most $\\dim(A^\\wedge)$ elements. Thus $A^\\wedge$ is regular.", "(You can also use", "Algebra, Lemma \\ref{algebra-lemma-flat-over-regular-with-regular-fibre}.)" ], "refs": [ "algebra-lemma-flat-under-regular", "more-algebra-lemma-completion-dimension", "algebra-lemma-flat-over-regular-with-regular-fibre" ], "ref_ids": [ 981, 10042, 988 ] } ], "ref_ids": [] }, { "id": 10046, "type": "theorem", "label": "more-algebra-lemma-completion-dvr", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-completion-dvr", "contents": [ "Let $A$ be a Noetherian local ring.", "Then $A$ is a discrete valuation ring if and only if $A^\\wedge$ is so." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-completion-dimension} and", "\\ref{lemma-completion-regular} and", "Algebra, Lemma \\ref{algebra-lemma-characterize-dvr}." ], "refs": [ "more-algebra-lemma-completion-dimension", "more-algebra-lemma-completion-regular", "algebra-lemma-characterize-dvr" ], "ref_ids": [ 10042, 10045, 1023 ] } ], "ref_ids": [] }, { "id": 10047, "type": "theorem", "label": "more-algebra-lemma-completion-reduced", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-completion-reduced", "contents": [ "Let $A$ be a Noetherian local ring.", "\\begin{enumerate}", "\\item If $A^\\wedge$ is reduced, then so is $A$.", "\\item In general $A$ reduced does not imply $A^\\wedge$ is reduced.", "\\item If $A$ is Nagata, then $A$ is reduced if and only if $A^\\wedge$", "is reduced.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As $A \\to A^\\wedge$ is faithfully flat we have (1) by", "Algebra, Lemma \\ref{algebra-lemma-descent-reduced}.", "For (2) see Algebra, Example \\ref{algebra-example-bad-dvr-char-p}", "(there are also examples in characteristic zero, see", "Algebra, Remark \\ref{algebra-remark-resolution-dim-1}).", "For (3) see Algebra, Lemmas", "\\ref{algebra-lemma-local-nagata-domain-analytically-unramified} and", "\\ref{algebra-lemma-analytically-unramified-easy}." ], "refs": [ "algebra-lemma-descent-reduced", "algebra-remark-resolution-dim-1", "algebra-lemma-local-nagata-domain-analytically-unramified", "algebra-lemma-analytically-unramified-easy" ], "ref_ids": [ 1371, 1576, 1357, 1354 ] } ], "ref_ids": [] }, { "id": 10048, "type": "theorem", "label": "more-algebra-lemma-completion-normal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-completion-normal", "contents": [ "Let $A$ be a Noetherian local ring. If $A^\\wedge$ is normal, then so is $A$." ], "refs": [], "proofs": [ { "contents": [ "As $A \\to A^\\wedge$ is faithfully flat this follows from", "Algebra, Lemma \\ref{algebra-lemma-descent-normal}." ], "refs": [ "algebra-lemma-descent-normal" ], "ref_ids": [ 1372 ] } ], "ref_ids": [] }, { "id": 10049, "type": "theorem", "label": "more-algebra-lemma-flat-completion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-completion", "contents": [ "Let $A \\to B$ be a local homomorphism of Noetherian local rings.", "Then the induced map of completions $A^\\wedge \\to B^\\wedge$", "is flat if and only if $A \\to B$ is flat." ], "refs": [], "proofs": [ { "contents": [ "Consider the commutative diagram", "$$", "\\xymatrix{", "A^\\wedge \\ar[r] & B^\\wedge \\\\", "A \\ar[r] \\ar[u] & B \\ar[u]", "}", "$$", "The vertical arrows are faithfully flat.", "Assume that $A^\\wedge \\to B^\\wedge$ is flat. Then $A \\to B^\\wedge$ is flat.", "Hence $B$ is flat over $A$ by", "Algebra, Lemma \\ref{algebra-lemma-flatness-descends-more-general}.", "\\medskip\\noindent", "Assume that $A \\to B$ is flat. Then $A \\to B^\\wedge$ is flat.", "Hence $B^\\wedge/\\mathfrak m_A^n B^\\wedge$ is flat over", "$A/\\mathfrak m_A^n$ for all $n \\geq 1$. Note that", "$\\mathfrak m_A^n A^\\wedge$ is the $n$th power of", "the maximal ideal $\\mathfrak m_A^\\wedge$ of $A^\\wedge$ and", "$A/\\mathfrak m_A^n = A^\\wedge/(\\mathfrak m_A^\\wedge)^n$.", "Thus we see that $B^\\wedge$ is flat over $A^\\wedge$ by applying", "Algebra, Lemma \\ref{algebra-lemma-flat-module-powers}", "(with $R = A^\\wedge$, $I = \\mathfrak m_A^\\wedge$,", "$S = B^\\wedge$, $M = S$)." ], "refs": [ "algebra-lemma-flatness-descends-more-general", "algebra-lemma-flat-module-powers" ], "ref_ids": [ 529, 893 ] } ], "ref_ids": [] }, { "id": 10050, "type": "theorem", "label": "more-algebra-lemma-flat-unramified", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-unramified", "contents": [ "Let $A \\to B$ be a flat local homomorphism of Noetherian local rings", "such that $\\mathfrak m_A B = \\mathfrak m_B$ and", "$\\kappa(\\mathfrak m_A) = \\kappa(\\mathfrak m_B)$.", "Then $A \\to B$ induces an isomorphism $A^\\wedge \\to B^\\wedge$", "of completions." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-finite-after-completion} we see that", "$B^\\wedge$ is the $\\mathfrak m_A$-adic", "completion of $B$ and that $A^\\wedge \\to B^\\wedge$ is finite.", "Since $A \\to B$ is flat we have $\\text{Tor}_1^A(B, \\kappa(\\mathfrak m_A)) = 0$.", "Hence we see that $B^\\wedge$ is flat over $A^\\wedge$ by", "Lemma \\ref{lemma-flat-after-completion}.", "Thus $B^\\wedge$ is a free $A^\\wedge$-module by", "Algebra, Lemma \\ref{algebra-lemma-finite-flat-local}.", "Since $A^\\wedge \\to B^\\wedge$ induces an isomorphism", "$\\kappa(\\mathfrak m_A) = A^\\wedge/\\mathfrak m_A A^\\wedge \\to", "B^\\wedge/\\mathfrak m_A B^\\wedge = B^\\wedge/\\mathfrak m_B B^\\wedge =", "\\kappa(\\mathfrak m_B)$ by our assumptions", "(and Algebra, Lemma \\ref{algebra-lemma-hathat-finitely-generated}),", "we see that $B^\\wedge$ is free of rank $1$. Thus $A^\\wedge \\to B^\\wedge$", "is an isomorphism." ], "refs": [ "algebra-lemma-finite-after-completion", "more-algebra-lemma-flat-after-completion", "algebra-lemma-finite-flat-local", "algebra-lemma-hathat-finitely-generated" ], "ref_ids": [ 875, 9956, 797, 859 ] } ], "ref_ids": [] }, { "id": 10051, "type": "theorem", "label": "more-algebra-lemma-Noetherian-etale-extension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Noetherian-etale-extension", "contents": [ "If $A \\to B$ is an \\'etale ring map and $\\mathfrak q$ is a prime of", "$B$ lying over $\\mathfrak p \\subset A$, then", "$A_{\\mathfrak p}$ is Noetherian if and only if $B_{\\mathfrak q}$ is", "Noetherian." ], "refs": [], "proofs": [ { "contents": [ "Since $A_\\mathfrak p \\to B_\\mathfrak q$ is faithfully flat", "we see that $B_\\mathfrak q$ Noetherian implies that $A_\\mathfrak p$", "is Noetherian, see", "Algebra, Lemma \\ref{algebra-lemma-descent-Noetherian}.", "Conversely, if $A_\\mathfrak p$ is Noetherian, then $B_\\mathfrak q$", "is Noetherian as it is a localization of a finite type", "$A_\\mathfrak p$-algebra." ], "refs": [ "algebra-lemma-descent-Noetherian" ], "ref_ids": [ 1370 ] } ], "ref_ids": [] }, { "id": 10052, "type": "theorem", "label": "more-algebra-lemma-dimension-etale-extension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-dimension-etale-extension", "contents": [ "If $A \\to B$ is an \\'etale ring map and $\\mathfrak q$ is a prime of", "$B$ lying over $\\mathfrak p \\subset A$, then", "$\\dim(A_{\\mathfrak p}) = \\dim(B_{\\mathfrak q})$." ], "refs": [], "proofs": [ { "contents": [ "Namely, because $A_{\\mathfrak p} \\to B_{\\mathfrak q}$ is flat we have", "going down, and hence the inequality", "$\\dim(A_{\\mathfrak p}) \\leq \\dim(B_{\\mathfrak q})$, see", "Algebra, Lemma \\ref{algebra-lemma-dimension-going-up}.", "On the other hand, suppose that", "$\\mathfrak q_0 \\subset \\mathfrak q_1 \\subset \\ldots \\subset \\mathfrak q_n$", "is a chain of primes in $B_{\\mathfrak q}$. Then the corresponding", "sequence of primes", "$\\mathfrak p_0 \\subset \\mathfrak p_1 \\subset \\ldots \\subset \\mathfrak p_n$", "(with $\\mathfrak p_i = \\mathfrak q_i \\cap A_{\\mathfrak p}$) is chain", "also (i.e., no equalities in the sequence) as an", "\\'etale ring map is quasi-finite (see", "Algebra, Lemma \\ref{algebra-lemma-etale-quasi-finite})", "and a quasi-finite ring map induces a map of spectra with", "discrete fibres (by definition).", "This means that $\\dim(A_{\\mathfrak p}) \\geq \\dim(B_{\\mathfrak q})$ as", "desired." ], "refs": [ "algebra-lemma-dimension-going-up", "algebra-lemma-etale-quasi-finite" ], "ref_ids": [ 982, 1234 ] } ], "ref_ids": [] }, { "id": 10053, "type": "theorem", "label": "more-algebra-lemma-regular-etale-extension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-regular-etale-extension", "contents": [ "If $A \\to B$ is an \\'etale ring map and $\\mathfrak q$ is a prime of", "$B$ lying over $\\mathfrak p \\subset A$, then", "$A_{\\mathfrak p}$ is regular if and only if $B_{\\mathfrak q}$ is regular." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-Noetherian-etale-extension}", "we may assume both $A_\\mathfrak p$ and $B_\\mathfrak q$", "are Noetherian in order to prove the equivalence.", "Let $x_1, \\ldots, x_t \\in \\mathfrak pA_\\mathfrak p$", "be a minimal set of generators. As $A_\\mathfrak p \\to B_\\mathfrak q$", "is faithfully flat we see that the images $y_1, \\ldots, y_t$", "in $B_\\mathfrak q$ form a minimal system of generators for", "$\\mathfrak pB_\\mathfrak q = \\mathfrak q B_\\mathfrak q$", "(Algebra, Lemma \\ref{algebra-lemma-etale-at-prime}).", "Regularity of $A_\\mathfrak p$ by definition means $t = \\dim(A_\\mathfrak p)$", "and similarly for $B_\\mathfrak q$. Hence the lemma follows from", "the equality $\\dim(A_\\mathfrak p) = \\dim(B_\\mathfrak q)$", "of Lemma \\ref{lemma-dimension-etale-extension}." ], "refs": [ "more-algebra-lemma-Noetherian-etale-extension", "algebra-lemma-etale-at-prime", "more-algebra-lemma-dimension-etale-extension" ], "ref_ids": [ 10051, 1233, 10052 ] } ], "ref_ids": [] }, { "id": 10054, "type": "theorem", "label": "more-algebra-lemma-Dedekind-etale-extension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Dedekind-etale-extension", "contents": [ "If $A \\to B$ is an \\'etale ring map and $A$ is a Dedekind domain, then", "$B$ is a finite product of Dedekind domains. In particular, the", "localizations $B_\\mathfrak q$ for $\\mathfrak q \\subset B$ maximal", "are discrete valuation rings." ], "refs": [], "proofs": [ { "contents": [ "The statement on the local rings follows from", "Lemmas \\ref{lemma-dimension-etale-extension} and", "\\ref{lemma-regular-etale-extension}", "and Algebra, Lemma \\ref{algebra-lemma-characterize-dvr}.", "It follows that $B$ is a Noetherian normal ring of dimension $1$.", "By Algebra, Lemma \\ref{algebra-lemma-characterize-reduced-ring-normal}", "we conclude that $B$ is a finite product of normal domains of", "dimension $1$. These are Dedekind domains by", "Algebra, Lemma \\ref{algebra-lemma-characterize-Dedekind}." ], "refs": [ "more-algebra-lemma-dimension-etale-extension", "more-algebra-lemma-regular-etale-extension", "algebra-lemma-characterize-dvr", "algebra-lemma-characterize-reduced-ring-normal", "algebra-lemma-characterize-Dedekind" ], "ref_ids": [ 10052, 10053, 1023, 515, 1041 ] } ], "ref_ids": [] }, { "id": 10055, "type": "theorem", "label": "more-algebra-lemma-dumb-properties-henselization", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-dumb-properties-henselization", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Then we have the following", "\\begin{enumerate}", "\\item $R \\to R^h \\to R^{sh}$ are faithfully flat ring maps,", "\\item $\\mathfrak m R^h = \\mathfrak m^h$ and", "$\\mathfrak m R^{sh} = \\mathfrak m^h R^{sh} = \\mathfrak m^{sh}$,", "\\item $R/\\mathfrak m^n = R^h/\\mathfrak m^nR^h$ for all $n$,", "\\item there exist elements $x_i \\in R^{sh}$ such that", "$R^{sh}/\\mathfrak m^nR^{sh}$ is a free $R/\\mathfrak m^n$-module", "on $x_i \\bmod \\mathfrak m^nR^{sh}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By construction $R^h$ is a colimit of \\'etale $R$-algebras, see", "Algebra, Lemma \\ref{algebra-lemma-henselization}. Since \\'etale", "ring maps are flat (Algebra, Lemma \\ref{algebra-lemma-etale}) we see that", "$R^h$ is flat over $R$ by", "Algebra, Lemma \\ref{algebra-lemma-colimit-flat}.", "As a flat local ring homomorphism is faithfully flat", "(Algebra, Lemma \\ref{algebra-lemma-local-flat-ff})", "we see that $R \\to R^h$ is faithfully flat.", "The ring map $R^h \\to R^{sh}$ is a colimit of finite \\'etale ring", "maps, see proof of Algebra, Lemma \\ref{algebra-lemma-strict-henselization}.", "Hence the same arguments as above show that $R^h \\to R^{sh}$ is", "faithfully flat.", "\\medskip\\noindent", "Part (2) follows from Algebra, Lemmas \\ref{algebra-lemma-henselization} and", "\\ref{algebra-lemma-strict-henselization}. Part (3) follows from", "Algebra, Lemma \\ref{algebra-lemma-local-artinian-basis-when-flat}", "because $R/\\mathfrak m \\to R^h/\\mathfrak mR^h$ is an isomorphism and", "$R/\\mathfrak m^n \\to R^h/\\mathfrak m^nR^h$ is flat as a base change of", "the flat ring map $R \\to R^h$", "(Algebra, Lemma \\ref{algebra-lemma-flat-base-change}).", "Let $\\kappa^{sep}$ be the residue field of $R^{sh}$ (it is a", "separable algebraic closure of $\\kappa$). Choose $x_i \\in R^{sh}$", "mapping to a basis of $\\kappa^{sep}$ as a $\\kappa$-vector space.", "Then (4) follows from", "Algebra, Lemma \\ref{algebra-lemma-local-artinian-basis-when-flat}", "in exactly the same way as above." ], "refs": [ "algebra-lemma-henselization", "algebra-lemma-etale", "algebra-lemma-colimit-flat", "algebra-lemma-local-flat-ff", "algebra-lemma-strict-henselization", "algebra-lemma-henselization", "algebra-lemma-strict-henselization", "algebra-lemma-local-artinian-basis-when-flat", "algebra-lemma-flat-base-change", "algebra-lemma-local-artinian-basis-when-flat" ], "ref_ids": [ 1294, 1231, 523, 537, 1295, 1294, 1295, 900, 527, 900 ] } ], "ref_ids": [] }, { "id": 10056, "type": "theorem", "label": "more-algebra-lemma-henselization-formally-smooth", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-formally-smooth", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Then", "\\begin{enumerate}", "\\item $R \\to R^h$, $R^h \\to R^{sh}$, and $R \\to R^{sh}$ are formally \\'etale,", "\\item $R \\to R^h$, $R^h \\to R^{sh}$, resp.\\ $R \\to R^{sh}$ are formally", "smooth in the $\\mathfrak m^h$, $\\mathfrak m^{sh}$,", "resp.\\ $\\mathfrak m^{sh}$-topology.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from the fact that $R^h$ and $R^{sh}$ are directed", "colimits of \\'etale algebras (by construction), that \\'etale algebras", "are formally \\'etale", "(Algebra, Lemma \\ref{algebra-lemma-formally-etale-etale}),", "and that colimits of formally \\'etale algebras are formally \\'etale", "(Algebra, Lemma \\ref{algebra-lemma-colimit-formally-etale}).", "Part (2) follows from the fact that a formally \\'etale ring", "map is formally smooth and Lemma \\ref{lemma-formally-smooth}." ], "refs": [ "algebra-lemma-formally-etale-etale", "algebra-lemma-colimit-formally-etale", "more-algebra-lemma-formally-smooth" ], "ref_ids": [ 1262, 1263, 10014 ] } ], "ref_ids": [] }, { "id": 10057, "type": "theorem", "label": "more-algebra-lemma-henselization-noetherian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-noetherian", "contents": [ "\\begin{reference}", "\\cite[IV, Theorem 18.6.6 and Proposition 18.8.8]{EGA}", "\\end{reference}", "Let $R$ be a local ring. The following are equivalent", "\\begin{enumerate}", "\\item $R$ is Noetherian,", "\\item $R^h$ is Noetherian, and", "\\item $R^{sh}$ is Noetherian.", "\\end{enumerate}", "In this case we have", "\\begin{enumerate}", "\\item[(a)] $(R^h)^\\wedge$ and $(R^{sh})^\\wedge$ are Noetherian complete", "local rings,", "\\item[(b)] $R^\\wedge \\to (R^h)^\\wedge$ is an isomorphism,", "\\item[(c)] $R^h \\to (R^h)^\\wedge$ and $R^{sh} \\to (R^{sh})^\\wedge$ are flat,", "\\item[(d)] $R^\\wedge \\to (R^{sh})^\\wedge$ is formally smooth in", "the $\\mathfrak m_{(R^{sh})^\\wedge}$-adic topology,", "\\item[(e)] $(R^\\wedge)^{sh} = R^\\wedge \\otimes_{R^h} R^{sh}$, and", "\\item[(f)] $((R^\\wedge)^{sh})^\\wedge = (R^{sh})^\\wedge$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $R \\to R^h \\to R^{sh}$ are faithfully flat", "(Lemma \\ref{lemma-dumb-properties-henselization}),", "we see that $R^h$ or $R^{sh}$ being Noetherian implies that $R$", "is Noetherian, see Algebra, Lemma \\ref{algebra-lemma-descent-Noetherian}.", "In the rest of the proof we assume $R$ is Noetherian.", "\\medskip\\noindent", "As $\\mathfrak m \\subset R$ is finitely generated it follows that", "$\\mathfrak m^h = \\mathfrak m R^h$ and $\\mathfrak m^{sh} = \\mathfrak mR^{sh}$", "are finitely generated, see Lemma \\ref{lemma-dumb-properties-henselization}.", "Hence $(R^h)^\\wedge$ and $(R^{sh})^\\wedge$ are Noetherian by", "Algebra, Lemma \\ref{algebra-lemma-complete-local-ring-Noetherian}.", "This proves (a).", "\\medskip\\noindent", "Note that (b) is immediate from", "Lemma \\ref{lemma-dumb-properties-henselization}.", "In particular we see that $(R^h)^\\wedge$ is flat over $R$, see", "Algebra, Lemma \\ref{algebra-lemma-completion-faithfully-flat}.", "\\medskip\\noindent", "Next, we show that $R^h \\to (R^h)^\\wedge$ is flat.", "Write $R^h = \\colim_i R_i$ as a directed colimit of", "localizations of \\'etale $R$-algebras. By", "Algebra, Lemma \\ref{algebra-lemma-colimit-rings-flat}", "if $(R^h)^\\wedge$ is flat over each $R_i$, then $R^h \\to (R^h)^\\wedge$ is", "flat. Note that $R^h = R_i^h$ (by construction).", "Hence $R_i^\\wedge = (R^h)^\\wedge$", "by part (b) is flat over $R_i$ as desired. To finish the proof of (c)", "we show that $R^{sh} \\to (R^{sh})^\\wedge$ is flat. To do this, by a", "limit argument as above, it suffices to show that $(R^{sh})^\\wedge$", "is flat over $R$. Note that it follows from", "Lemma \\ref{lemma-dumb-properties-henselization}", "that $(R^{sh})^\\wedge$ is the completion of a free $R$-module.", "By Lemma \\ref{lemma-completed-direct-sum-flat}", "we see this is flat over $R$ as desired. This finishes the proof of (c).", "\\medskip\\noindent", "At this point we know (c) is true and that $(R^h)^\\wedge$ and", "$(R^{sh})^\\wedge$ are Noetherian. It follows from", "Algebra, Lemma \\ref{algebra-lemma-descent-Noetherian}", "that $R^h$ and $R^{sh}$ are Noetherian.", "\\medskip\\noindent", "Part (d) follows from Lemma \\ref{lemma-henselization-formally-smooth}", "and Lemma \\ref{lemma-formally-smooth-completion}.", "\\medskip\\noindent", "Part (e) follows from Algebra, Lemma \\ref{algebra-lemma-sh-from-h-map}", "and the fact that $R^\\wedge$ is henselian by", "Algebra, Lemma \\ref{algebra-lemma-complete-henselian}.", "\\medskip\\noindent", "Proof of (f). Using (e) there is a map $R^{sh} \\to (R^\\wedge)^{sh}$", "which induces a map $(R^{sh})^\\wedge \\to ((R^\\wedge)^{sh})^\\wedge$", "upon completion. Using (e) there is a map $R^\\wedge \\to (R^{sh})^\\wedge$.", "Since $(R^{sh})^\\wedge$ is strictly henselian (see above) this map", "induces a map $(R^\\wedge)^{sh} \\to (R^{sh})^\\wedge$ by", "Algebra, Lemma \\ref{algebra-lemma-strictly-henselian-functorial}.", "Completing we obtain a map $((R^\\wedge)^{sh})^\\wedge \\to (R^{sh})^\\wedge$.", "We omit the verification that these two maps are mutually inverse." ], "refs": [ "more-algebra-lemma-dumb-properties-henselization", "algebra-lemma-descent-Noetherian", "more-algebra-lemma-dumb-properties-henselization", "algebra-lemma-complete-local-ring-Noetherian", "more-algebra-lemma-dumb-properties-henselization", "algebra-lemma-completion-faithfully-flat", "algebra-lemma-colimit-rings-flat", "more-algebra-lemma-dumb-properties-henselization", "more-algebra-lemma-completed-direct-sum-flat", "algebra-lemma-descent-Noetherian", "more-algebra-lemma-henselization-formally-smooth", "more-algebra-lemma-formally-smooth-completion", "algebra-lemma-sh-from-h-map", "algebra-lemma-complete-henselian", "algebra-lemma-strictly-henselian-functorial" ], "ref_ids": [ 10055, 1370, 10055, 1328, 10055, 871, 526, 10055, 9953, 1370, 10056, 10015, 1308, 1282, 1303 ] } ], "ref_ids": [] }, { "id": 10058, "type": "theorem", "label": "more-algebra-lemma-henselization-reduced", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-reduced", "contents": [ "\\begin{slogan}", "Reducedness passes to the (strict) henselization.", "\\end{slogan}", "Let $R$ be a local ring.", "The following are equivalent: $R$ is reduced,", "the henselization $R^h$ of $R$ is reduced, and", "the strict henselization $R^{sh}$ of $R$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "The ring maps $R \\to R^h \\to R^{sh}$ are faithfully flat.", "Hence one direction of the implications follows from", "Algebra, Lemma \\ref{algebra-lemma-descent-reduced}.", "Conversely, assume $R$ is reduced. Since $R^h$ and $R^{sh}$", "are filtered colimits of \\'etale, hence smooth $R$-algebras, the", "result follows from", "Algebra, Lemma \\ref{algebra-lemma-reduced-goes-up}." ], "refs": [ "algebra-lemma-descent-reduced", "algebra-lemma-reduced-goes-up" ], "ref_ids": [ 1371, 1366 ] } ], "ref_ids": [] }, { "id": 10059, "type": "theorem", "label": "more-algebra-lemma-henselization-nil", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-nil", "contents": [ "Let $R$ be a local ring. Let $nil(R)$ denote the ideal of", "nilpotent elements of $R$. Then $nil(R)R^h = nil(R^h)$ and", "$nil(R)R^{sh} = nil(R^{sh})$." ], "refs": [], "proofs": [ { "contents": [ "Note that $nil(R)$ is the biggest ideal consisting of nilpotent elements", "such that the quotient $R/nil(R)$ is reduced. Note that $nil(R)R^h$", "consists of nilpotent elements by", "Algebra, Lemma \\ref{algebra-lemma-locally-nilpotent}.", "Also, note that $R^h/nil(R) R^h$ is the", "henselization of $R/nil(R)$ by", "Algebra, Lemma \\ref{algebra-lemma-quotient-henselization}.", "Hence $R^h/nil(R)R^h$ is reduced by", "Lemma \\ref{lemma-henselization-reduced}.", "We conclude that $nil(R) R^h = nil(R^h)$ as desired.", "Similarly for the strict henselization but using", "Algebra, Lemma \\ref{algebra-lemma-quotient-strict-henselization}." ], "refs": [ "algebra-lemma-locally-nilpotent", "algebra-lemma-quotient-henselization", "more-algebra-lemma-henselization-reduced", "algebra-lemma-quotient-strict-henselization" ], "ref_ids": [ 458, 1301, 10058, 1307 ] } ], "ref_ids": [] }, { "id": 10060, "type": "theorem", "label": "more-algebra-lemma-henselization-normal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-normal", "contents": [ "Let $R$ be a local ring.", "The following are equivalent: $R$ is a normal domain,", "the henselization $R^h$ of $R$ is a normal domain, and", "the strict henselization $R^{sh}$ of $R$ is a normal domain." ], "refs": [], "proofs": [ { "contents": [ "A preliminary remark is that a local ring is normal if and only if it is", "a normal domain (see", "Algebra, Definition \\ref{algebra-definition-ring-normal}).", "The ring maps $R \\to R^h \\to R^{sh}$ are faithfully flat.", "Hence one direction of the implications follows from", "Algebra, Lemma \\ref{algebra-lemma-descent-normal}.", "Conversely, assume $R$ is normal. Since $R^h$ and $R^{sh}$", "are filtered colimits of \\'etale hence smooth $R$-algebras, the", "result follows from", "Algebra, Lemmas \\ref{algebra-lemma-normal-goes-up} and", "\\ref{algebra-lemma-colimit-normal-ring}." ], "refs": [ "algebra-definition-ring-normal", "algebra-lemma-descent-normal", "algebra-lemma-normal-goes-up", "algebra-lemma-colimit-normal-ring" ], "ref_ids": [ 1454, 1372, 1368, 516 ] } ], "ref_ids": [] }, { "id": 10061, "type": "theorem", "label": "more-algebra-lemma-henselization-dimension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-dimension", "contents": [ "Given any local ring $R$ we have $\\dim(R) = \\dim(R^h) = \\dim(R^{sh})$." ], "refs": [], "proofs": [ { "contents": [ "Since $R \\to R^{sh}$ is faithfully flat", "(Lemma \\ref{lemma-dumb-properties-henselization})", "we see that $\\dim(R^{sh}) \\geq \\dim(R)$ by going down, see", "Algebra, Lemma \\ref{algebra-lemma-dimension-going-up}.", "For the converse, we write $R^{sh} = \\colim R_i$ as", "a directed colimit of local rings $R_i$ each of which is a", "localization of an \\'etale $R$-algebra. Now if", "$\\mathfrak q_0 \\subset \\mathfrak q_1 \\subset \\ldots \\subset \\mathfrak q_n$", "is a chain of prime ideals in $R^{sh}$, then for some sufficiently", "large $i$ the sequence", "$$", "R_i \\cap \\mathfrak q_0 \\subset", "R_i \\cap \\mathfrak q_1 \\subset \\ldots \\subset", "R_i \\cap \\mathfrak q_n", "$$", "is a chain of primes in $R_i$. Thus we see that", "$\\dim(R^{sh}) \\leq \\sup_i \\dim(R_i)$.", "But by the result of", "Lemma \\ref{lemma-dimension-etale-extension}", "we have $\\dim(R_i) = \\dim(R)$ for each $i$ and we win." ], "refs": [ "more-algebra-lemma-dumb-properties-henselization", "algebra-lemma-dimension-going-up", "more-algebra-lemma-dimension-etale-extension" ], "ref_ids": [ 10055, 982, 10052 ] } ], "ref_ids": [] }, { "id": 10062, "type": "theorem", "label": "more-algebra-lemma-henselization-depth", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-depth", "contents": [ "Given a Noetherian local ring $R$ we have", "$\\text{depth}(R) = \\text{depth}(R^h) = \\text{depth}(R^{sh})$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-henselization-noetherian}", "we know that $R^h$ and $R^{sh}$ are Noetherian. Hence the lemma follows", "from", "Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck}." ], "refs": [ "more-algebra-lemma-henselization-noetherian", "algebra-lemma-apply-grothendieck" ], "ref_ids": [ 10057, 1361 ] } ], "ref_ids": [] }, { "id": 10063, "type": "theorem", "label": "more-algebra-lemma-henselization-CM", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-CM", "contents": [ "Let $R$ be a Noetherian local ring. The following are equivalent:", "$R$ is Cohen-Macaulay, the henselization $R^h$ of $R$ is Cohen-Macaulay,", "and the strict henselization $R^{sh}$ of $R$ is Cohen-Macaulay." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-henselization-noetherian}", "we know that $R^h$ and $R^{sh}$ are Noetherian, hence the lemma makes", "sense. Since we have", "$\\text{depth}(R) = \\text{depth}(R^h) = \\text{depth}(R^{sh})$", "and", "$\\dim(R) = \\dim(R^h) = \\dim(R^{sh})$", "by", "Lemmas \\ref{lemma-henselization-depth} and", "\\ref{lemma-henselization-dimension}", "we conclude." ], "refs": [ "more-algebra-lemma-henselization-noetherian", "more-algebra-lemma-henselization-depth", "more-algebra-lemma-henselization-dimension" ], "ref_ids": [ 10057, 10062, 10061 ] } ], "ref_ids": [] }, { "id": 10064, "type": "theorem", "label": "more-algebra-lemma-henselization-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-regular", "contents": [ "Let $R$ be a Noetherian local ring. The following are equivalent:", "$R$ is a regular local ring, the henselization $R^h$ of $R$ is a regular", "local ring, and the strict henselization $R^{sh}$ of $R$ is a regular", "local ring." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-henselization-noetherian}", "we know that $R^h$ and $R^{sh}$ are Noetherian, hence the lemma makes", "sense. Let $\\mathfrak m$ be the maximal ideal of $R$.", "Let $x_1, \\ldots, x_t \\in \\mathfrak m$ be a minimal system of", "generators of $\\mathfrak m$, i.e., such that the images in", "$\\mathfrak m/\\mathfrak m^2$ form a basis over $\\kappa = R/\\mathfrak m$.", "Because $R \\to R^h$ and $R \\to R^{sh}$ are faithfully flat, it follows", "that the images $x_1^h, \\ldots, x_t^h$ in $R^h$,", "resp.\\ $x_1^{sh}, \\ldots, x_t^{sh}$ in $R^{sh}$", "are a minimal system of generators for", "$\\mathfrak m^h = \\mathfrak mR^h$,", "resp.\\ $\\mathfrak m^{sh} = \\mathfrak mR^{sh}$.", "Regularity of $R$ by definition means $t = \\dim(R)$ and similarly", "for $R^h$ and $R^{sh}$. Hence the lemma follows from the equality", "of dimensions $\\dim(R) = \\dim(R^h) = \\dim(R^{sh})$ of", "Lemma \\ref{lemma-henselization-dimension}" ], "refs": [ "more-algebra-lemma-henselization-noetherian", "more-algebra-lemma-henselization-dimension" ], "ref_ids": [ 10057, 10061 ] } ], "ref_ids": [] }, { "id": 10065, "type": "theorem", "label": "more-algebra-lemma-henselization-dvr", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-dvr", "contents": [ "Let $R$ be a Noetherian local ring. Then $R$ is a discrete valuation ring", "if and only if $R^h$ is a discrete valuation ring if and only if", "$R^{sh}$ is a discrete valuation ring." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-henselization-dimension} and", "\\ref{lemma-henselization-regular} and", "Algebra, Lemma \\ref{algebra-lemma-characterize-dvr}." ], "refs": [ "more-algebra-lemma-henselization-dimension", "more-algebra-lemma-henselization-regular", "algebra-lemma-characterize-dvr" ], "ref_ids": [ 10061, 10064, 1023 ] } ], "ref_ids": [] }, { "id": 10066, "type": "theorem", "label": "more-algebra-lemma-filtered-colimit-etale-noetherian-fibres", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-filtered-colimit-etale-noetherian-fibres", "contents": [ "Let $A$ be a ring. Let $B$ be a filtered colimit of \\'etale $A$-algebras.", "Let $\\mathfrak p$ be a prime of $A$. If $B$ is Noetherian, then", "there are finitely many primes $\\mathfrak q_1, \\ldots, \\mathfrak q_r$", "lying over $\\mathfrak p$, we have", "$B \\otimes_A \\kappa(\\mathfrak p) = \\prod \\kappa(\\mathfrak q_i)$, and", "each of the field extensions", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q_i)$ is separable algebraic." ], "refs": [], "proofs": [ { "contents": [ "Write $B$ as a filtered colimit $B = \\colim B_i$ with $A \\to B_i$ \\'etale.", "Then on the one hand", "$B \\otimes_A \\kappa(\\mathfrak p) = \\colim B_i \\otimes_A \\kappa(\\mathfrak p)$", "is a filtered colimit of \\'etale $\\kappa(\\mathfrak p)$-algebras, and", "on the other hand it is Noetherian. An \\'etale", "$\\kappa(\\mathfrak p)$-algebra is a finite product of finite separable field", "extensions (Algebra, Lemma \\ref{algebra-lemma-etale-over-field}).", "Hence there are no nontrivial specializations between the primes", "(which are all maximal and minimal primes) of the algebras", "$B_i \\otimes_A \\kappa(\\mathfrak p)$ and hence there are no", "nontrivial specializations between the primes of", "$B \\otimes_A \\kappa(\\mathfrak p)$. Thus", "$B \\otimes_A \\kappa(\\mathfrak p)$ is reduced and", "has finitely many primes which all minimal.", "Thus it is a finite product of fields (use", "Algebra, Lemma \\ref{algebra-lemma-total-ring-fractions-no-embedded-points}", "or", "Algebra, Proposition \\ref{algebra-proposition-dimension-zero-ring}).", "Each of these fields is a colimit of finite separable extensions", "and hence the final statement of the lemma follows." ], "refs": [ "algebra-lemma-etale-over-field", "algebra-lemma-total-ring-fractions-no-embedded-points", "algebra-proposition-dimension-zero-ring" ], "ref_ids": [ 1232, 421, 1410 ] } ], "ref_ids": [] }, { "id": 10067, "type": "theorem", "label": "more-algebra-lemma-fibres-henselization", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-fibres-henselization", "contents": [ "Let $R$ be a Noetherian local ring. Let $\\mathfrak p \\subset R$ be a prime.", "Then", "$$", "R^h \\otimes_R \\kappa(\\mathfrak p) =", "\\prod\\nolimits_{i = 1, \\ldots, t} \\kappa(\\mathfrak q_i)", "\\quad\\text{resp.}\\quad", "R^{sh} \\otimes_R \\kappa(\\mathfrak p) =", "\\prod\\nolimits_{i = 1, \\ldots, s} \\kappa(\\mathfrak r_i)", "$$", "where $\\mathfrak q_1, \\ldots, \\mathfrak q_t$,", "resp.\\ $\\mathfrak r_1, \\ldots, \\mathfrak r_s$", "are the prime of $R^h$, resp.\\ $R^{sh}$ lying over $\\mathfrak p$.", "Moreover, the field extensions", "$\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q_i)$", "resp.\\ $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak r_i)$", "are separable algebraic." ], "refs": [], "proofs": [ { "contents": [ "This can be deduced from the more general", "Lemma \\ref{lemma-filtered-colimit-etale-noetherian-fibres}", "using that the henselization and strict henselization are Noetherian", "(as we've seen above). But we also give a direct proof as follows.", "\\medskip\\noindent", "We will use without further mention the results of", "Lemmas \\ref{lemma-dumb-properties-henselization} and", "\\ref{lemma-henselization-noetherian}.", "Note that $R^h/\\mathfrak pR^h$, resp.\\ $R^{sh}/\\mathfrak pR^{sh}$", "is the henselization, resp.\\ strict henselization of $R/\\mathfrak p$,", "see Algebra, Lemma \\ref{algebra-lemma-quotient-henselization}", "resp.\\ Algebra, Lemma \\ref{algebra-lemma-quotient-strict-henselization}.", "Hence we may replace $R$ by $R/\\mathfrak p$ and assume that $R$", "is a Noetherian local domain and that $\\mathfrak p = (0)$.", "Since $R^h$, resp.\\ $R^{sh}$ is Noetherian, it has finitely many", "minimal primes $\\mathfrak q_1, \\ldots, \\mathfrak q_t$,", "resp.\\ $\\mathfrak r_1, \\ldots, \\mathfrak r_s$.", "Since $R \\to R^h$, resp.\\ $R \\to R^{sh}$ is flat these are exactly", "the primes lying over $\\mathfrak p = (0)$ (by going down).", "Finally, as $R$ is a domain, we see that $R^h$, resp.\\ $R^{sh}$", "is reduced, see Lemma \\ref{lemma-henselization-reduced}.", "Thus we see that $R^h \\otimes_R \\kappa(\\mathfrak p)$", "resp.\\ $R^{sh} \\otimes_R \\kappa(\\mathfrak p)$", "is a reduced Noetherian ring with finitely many primes, all of which", "are minimal (and hence maximal). Thus these rings are Artinian and are", "products of their localizations at maximal ideals, each necessarily a field", "(see Algebra, Proposition \\ref{algebra-proposition-dimension-zero-ring} and", "Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring}).", "\\medskip\\noindent", "The final statement follows from the fact that $R \\to R^h$,", "resp.\\ $R \\to R^{sh}$ is a colimit of \\'etale ring maps and hence", "the induced residue field extensions are colimits of finite separable", "extensions, see", "Algebra, Lemma \\ref{algebra-lemma-etale-at-prime}." ], "refs": [ "more-algebra-lemma-filtered-colimit-etale-noetherian-fibres", "more-algebra-lemma-dumb-properties-henselization", "more-algebra-lemma-henselization-noetherian", "algebra-lemma-quotient-henselization", "algebra-lemma-quotient-strict-henselization", "more-algebra-lemma-henselization-reduced", "algebra-proposition-dimension-zero-ring", "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-etale-at-prime" ], "ref_ids": [ 10066, 10055, 10057, 1301, 1307, 10058, 1410, 418, 1233 ] } ], "ref_ids": [] }, { "id": 10068, "type": "theorem", "label": "more-algebra-lemma-p-basis", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-p-basis", "contents": [ "Let $k \\subset K$ be a field extension. Assume $k$ has characteristic", "$p > 0$. Let $\\{x_i\\}$ be a subset of $K$. The following are equivalent", "\\begin{enumerate}", "\\item the elements $\\{x_i\\}$ are $p$-independent over $k$, and", "\\item the elements $\\text{d}x_i$ are $K$-linearly independent", "in $\\Omega_{K/k}$.", "\\end{enumerate}", "Any $p$-independent collection can be extended to a $p$-basis of $K$ over $k$.", "In particular, the field $K$ has a $p$-basis over $k$.", "Moreover, the following are equivalent:", "\\begin{enumerate}", "\\item[(a)] $\\{x_i\\}$ is a $p$-basis of $K$ over $k$, and", "\\item[(b)] $\\text{d}x_i$ is a basis of the $K$-vector space $\\Omega_{K/k}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (2) and suppose that $\\sum a_E x^E = 0$ is a linear relation", "with $a_E \\in k K^p$. Let $\\theta_i : K \\to K$ be a $k$-derivation such that", "$\\theta_i(x_j) = \\delta_{ij}$ (Kronecker delta). Note that any $k$-derivation", "of $K$ annihilates $kK^p$. Applying $\\theta_i$ to the given relation we", "obtain new relations", "$$", "\\sum\\nolimits_{E, e_i > 0}", "e_i a_E x_1^{e_1}\\ldots x_i^{e_i - 1} \\ldots x_n^{e_n} = 0", "$$", "Hence if we pick $\\sum a_E x^E$ as the relation with minimal", "total degree $|E| = \\sum e_i$ for some $a_E \\not = 0$, then we", "get a contradiction. Hence (2) holds.", "\\medskip\\noindent", "If $\\{x_i\\}$ is a $p$-basis for $K$ over $k$, then", "$K \\cong kK^p[X_i]/(X_i^p - x_i^p)$. Hence we see that", "$\\text{d}x_i$ forms a basis for $\\Omega_{K/k}$ over $K$.", "Thus (a) implies (b).", "\\medskip\\noindent", "Let $\\{x_i\\}$ be a $p$-independent subset of $K$ over $k$. An application", "of Zorn's lemma shows that we can enlarge this to a maximal $p$-independent", "subset of $K$ over $k$. We claim that any maximal $p$-independent subset", "$\\{x_i\\}$ of $K$ is a $p$-basis of $K$ over $k$. The claim will imply", "that (1) implies (2) and establish the existence of $p$-bases.", "To prove the claim let $L$ be the subfield of $K$ generated by", "$kK^p$ and the $x_i$. We have to show that $L = K$. If $x \\in K$", "but $x \\not \\in L$, then $x^p \\in L$ and $L(x) \\cong L[z]/(z^p - x)$.", "Hence $\\{x_i\\} \\cup \\{x\\}$ is $p$-independent over $k$, a contradiction.", "\\medskip\\noindent", "Finally, we have to show that (b) implies (a). By the equivalence of (1)", "and (2) we see that $\\{x_i\\}$ is a maximal $p$-independent subset", "of $K$ over $k$. Hence by the claim above it is a $p$-basis." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10069, "type": "theorem", "label": "more-algebra-lemma-intersection-subfields-subspace", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-intersection-subfields-subspace", "contents": [ "Let $k \\subset K$ be a field extension. Let $\\{K_\\alpha\\}_{\\alpha \\in A}$", "be a collection of subfields of $K$ with the following properties", "\\begin{enumerate}", "\\item $k \\subset K_\\alpha$ for all $\\alpha \\in A$,", "\\item $k = \\bigcap_{\\alpha \\in A} K_\\alpha$,", "\\item for $\\alpha, \\alpha' \\in A$ there exists an $\\alpha'' \\in A$", "such that $K_{\\alpha''} \\subset K_\\alpha \\cap K_{\\alpha'}$.", "\\end{enumerate}", "Then for $n \\geq 1$ and $V \\subset K^{\\oplus n}$ a $K$-vector space", "we have $V \\cap k^{\\oplus n} \\not = 0$ if and only if", "$V \\cap K_\\alpha^{\\oplus n} \\not = 0$ for all $\\alpha \\in A$." ], "refs": [], "proofs": [ { "contents": [ "By induction on $n$. The case $n = 1$ follows from the assumptions.", "Assume the result proven for subspaces of $K^{\\oplus n - 1}$.", "Assume that $V \\subset K^{\\oplus n}$ has nonzero intersection with", "$K_\\alpha^{\\oplus n}$ for all $\\alpha \\in A$. If", "$V \\cap 0 \\oplus k^{\\oplus n - 1}$ is nonzero then we win. Hence we may", "assume this is not the case. By induction hypothesis we can find", "an $\\alpha$ such that $V \\cap 0 \\oplus K_\\alpha^{\\oplus n - 1}$", "is zero. Let $v = (x_1, \\ldots, x_n) \\in V \\cap K_\\alpha$ be a nonzero element.", "By our choice of $\\alpha$ we see that $x_1$ is not zero.", "Replace $v$ by $x_1^{-1}v$ so that $v = (1, x_2, \\ldots, x_n)$.", "Note that if $v' = (x_1', \\ldots, x'_n) \\in V \\cap K_\\alpha$, then", "$v' - x_1'v = 0$ by our choice of $\\alpha$. Hence we see that", "$V \\cap K_\\alpha^{\\oplus n} = K_\\alpha v$. If we choose some", "$\\alpha'$ such that $K_{\\alpha'} \\subset K_\\alpha$, then we", "see that necessarily $v \\in V \\cap K_{\\alpha'}^{\\oplus n}$ (by the", "same arguments applied to $\\alpha'$). Hence", "$$", "x_2, \\ldots, x_n \\in", "\\bigcap\\nolimits_{\\alpha' \\in A, K_{\\alpha'} \\subset K_\\alpha} K_{\\alpha'}", "$$", "which equals $k$ by (2) and (3)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10070, "type": "theorem", "label": "more-algebra-lemma-intersection-subfields", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-intersection-subfields", "contents": [ "Let $K$ be a field of characteristic $p$. Let $\\{K_\\alpha\\}_{\\alpha \\in A}$", "be a collection of subfields of $K$ with the following properties", "\\begin{enumerate}", "\\item $K^p \\subset K_\\alpha$ for all $\\alpha \\in A$,", "\\item $K^p = \\bigcap_{\\alpha \\in A} K_\\alpha$,", "\\item for $\\alpha, \\alpha' \\in A$ there exists an $\\alpha'' \\in A$", "such that $K_{\\alpha''} \\subset K_\\alpha \\cap K_{\\alpha'}$.", "\\end{enumerate}", "Then", "\\begin{enumerate}", "\\item the intersection of the kernels of the maps", "$\\Omega_{K/\\mathbf{F}_p} \\to \\Omega_{K/K_\\alpha}$ is zero,", "\\item for any finite extension $K \\subset L$ we have", "$L^p = \\bigcap_{\\alpha \\in A} L^pK_\\alpha$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1).", "Choose a $p$-basis $\\{x_i\\}$ for $K$ over $\\mathbf{F}_p$.", "Suppose that $\\eta = \\sum_{i \\in I'} y_i \\text{d}x_i$ maps to zero in", "$\\Omega_{K/K_\\alpha}$ for every $\\alpha \\in A$. Here the index set", "$I'$ is finite. By Lemma \\ref{lemma-p-basis}", "this means that for every $\\alpha$ there exists a relation", "$$", "\\sum\\nolimits_E a_{E, \\alpha} x^E,\\quad a_{E, \\alpha} \\in K_\\alpha", "$$", "where $E$ runs over multi-indices $E = (e_i)_{i \\in I'}$ with", "$0 \\leq e_i < p$. On the other hand, Lemma \\ref{lemma-p-basis}", "guarantees there is no such relation $\\sum a_E x^E = 0$ with", "$a_E \\in K^p$. This is a contradiction by", "Lemma \\ref{lemma-intersection-subfields-subspace}.", "\\medskip\\noindent", "Proof of (2). Suppose that we have a tower $K \\subset M \\subset L$", "of finite extensions of fields. Set $M_\\alpha = M^p K_\\alpha$", "and $L_\\alpha = L^p K_\\alpha = L^p M_\\alpha$. Then we can first prove that", "$M^p = \\bigcap_{\\alpha \\in A} M_\\alpha$, and after that prove", "that $L^p = \\bigcap_{\\alpha \\in A} L_\\alpha$. Hence it suffices to", "prove (2) for primitive field extensions having no nontrivial subfields.", "First, assume that $L = K(\\theta)$ is separable over $K$. Then", "$L$ is generated by $\\theta^p$ over $K$, hence we may assume that", "$\\theta \\in L^p$. In this case we see that", "$$", "L^p = K^p \\oplus K^p\\theta \\oplus \\ldots K^p\\theta^{d - 1}", "\\quad\\text{and}\\quad", "L^pK_\\alpha =", "K_\\alpha \\oplus K_\\alpha \\theta \\oplus \\ldots K_\\alpha\\theta^{d - 1}", "$$", "where $d = [L : K]$. Thus the conclusion is clear in this case. The other", "case is where $L = K(\\theta)$ with $\\theta^p = t \\in K$, $t \\not \\in K^p$.", "In this case we have", "$$", "L^p = K^p \\oplus K^pt \\oplus \\ldots K^pt^{p - 1}", "\\quad\\text{and}\\quad", "L^pK_\\alpha =", "K_\\alpha \\oplus K_\\alpha t \\oplus \\ldots K_\\alpha t^{p - 1}", "$$", "Again the result is clear." ], "refs": [ "more-algebra-lemma-p-basis", "more-algebra-lemma-p-basis", "more-algebra-lemma-intersection-subfields-subspace" ], "ref_ids": [ 10068, 10068, 10069 ] } ], "ref_ids": [] }, { "id": 10071, "type": "theorem", "label": "more-algebra-lemma-power-series-ring-subfields", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-power-series-ring-subfields", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $n, m \\geq 0$.", "Let $K$ be the fraction field of $k[[x_1, \\ldots, x_n]][y_1, \\ldots, y_m]$.", "As $k'$ ranges through all subfields $k/k'/k^p$", "with $[k : k'] < \\infty$ the subfields", "$$", "\\text{fraction field of }", "k'[[x_1^p, \\ldots, x_n^p]][y_1^p, \\ldots, y_m^p]", "\\subset", "K", "$$", "form a family of subfields as in Lemma \\ref{lemma-intersection-subfields}.", "Moreover, each of the ring extensions", "$k'[[x_1^p, \\ldots, x_n^p]][y_1^p, \\ldots, y_m^p] \\subset", "k[[x_1, \\ldots, x_n]][y_1, \\ldots, y_m]$ is finite." ], "refs": [ "more-algebra-lemma-intersection-subfields" ], "proofs": [ { "contents": [ "Write $A = k[[x_1, \\ldots, x_n]][y_1, \\ldots, y_m]$", "and $A' = k'[[x_1^p, \\ldots, x_n^p]][y_1^p, \\ldots, y_m^p]$.", "We also denote $K'$ the fraction field of $A'$. The ring extension", "$k'[[x_1^p, \\ldots, x_d^p]] \\subset k[[x_1, \\ldots, x_d]]$ is finite", "by Algebra, Lemma \\ref{algebra-lemma-finite-after-completion}", "which implies that $A \\to A'$ is finite.", "For $f \\in A$ we see that $f^p \\in A'$. Hence $K^p \\subset K'$.", "Any element of $K'$ can be written as $a/b^p$ with $a \\in A'$ and $b \\in A$", "nonzero. Suppose that $f/g^p \\in K$, $f, g \\in A$, $g \\not = 0$", "is contained in $K'$ for every choice of $k'$.", "Fix a choice of $k'$ for the moment. By the above we see", "$f/g^p = a/b^p$ for some $a \\in A'$ and some nonzero $b \\in A$.", "Hence $b^p f \\in A'$. For any $A'$-derivation $D : A \\to A$ we see", "that $0 = D(b^pf) = b^p D(f)$ hence $D(f) = 0$ as $A$ is a domain.", "Taking $D = \\partial_{x_i}$ and $D = \\partial_{y_j}$ we conclude", "that $f \\in k[[x_1^p, \\ldots, x_n^p]][y_1^p, \\ldots, y_d^p]$.", "Applying a $k'$-derivation $\\theta : k \\to k$", "we similarly conclude that all coefficients of $f$ are in $k'$, i.e.,", "$f \\in A'$. Since it is clear that", "$A^p = \\bigcap\\nolimits_{k'} A'$ where $k'$ ranges over all subfields", "as in the lemma we win." ], "refs": [ "algebra-lemma-finite-after-completion" ], "ref_ids": [ 875 ] } ], "ref_ids": [ 10070 ] }, { "id": 10072, "type": "theorem", "label": "more-algebra-lemma-J-1", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-J-1", "contents": [ "Let $R$ be a Noetherian ring. Let $X = \\Spec(R)$.", "The ring $R$ is J-1 if and only if $V(\\mathfrak p) \\cap \\text{Reg}(X)$", "contains a nonempty open subset of $V(\\mathfrak p)$ for all", "$\\mathfrak p \\in \\text{Reg}(X)$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from", "Topology, Lemma \\ref{topology-lemma-characterize-open-Noetherian}." ], "refs": [ "topology-lemma-characterize-open-Noetherian" ], "ref_ids": [ 8271 ] } ], "ref_ids": [] }, { "id": 10073, "type": "theorem", "label": "more-algebra-lemma-intersection-regular-with-closed", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-intersection-regular-with-closed", "contents": [ "Let $R$ be a Noetherian ring. Let $X = \\Spec(R)$. Assume that for all primes", "$\\mathfrak p \\subset R$ the ring $R/\\mathfrak p$ is J-0.", "Then $R$ is J-1." ], "refs": [], "proofs": [ { "contents": [ "We will show that the criterion of Lemma \\ref{lemma-J-1} applies.", "Let $\\mathfrak p \\in \\text{Reg}(X)$ be a prime of height $r$.", "Pick $f_1, \\ldots, f_r \\in \\mathfrak p$ which map to generators", "of $\\mathfrak pR_\\mathfrak p$. Since $\\mathfrak p \\in \\text{Reg}(X)$", "we see that $f_1, \\ldots, f_r$ maps to a regular sequence in $R_\\mathfrak p$,", "see Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}. Thus by", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-neighbourhood}", "we see that after replacing $R$ by $R_g$ for some $g \\in R$,", "$g \\not \\in \\mathfrak p$ the sequence $f_1, \\ldots, f_r$ is a", "regular sequence in $R$. After another replacement we may also", "assume $f_1, \\ldots, f_r$ generate $\\mathfrak p$.", "Next, let $\\mathfrak p \\subset \\mathfrak q$", "be a prime ideal such that $(R/\\mathfrak p)_\\mathfrak q$ is", "a regular local ring. By the assumption of the lemma there", "exists a non-empty open subset of $V(\\mathfrak p)$ consisting", "of such primes, hence it suffices to prove $R_\\mathfrak q$ is regular.", "Note that $f_1, \\ldots, f_r$ is a regular sequence in $R_\\mathfrak q$", "such that $R_\\mathfrak q/(f_1, \\ldots, f_r)R_\\mathfrak q$ is regular.", "Hence $R_\\mathfrak q$ is regular by", "Algebra, Lemma \\ref{algebra-lemma-regular-mod-x}." ], "refs": [ "more-algebra-lemma-J-1", "algebra-lemma-regular-ring-CM", "algebra-lemma-regular-sequence-in-neighbourhood", "algebra-lemma-regular-mod-x" ], "ref_ids": [ 10072, 941, 741, 945 ] } ], "ref_ids": [] }, { "id": 10074, "type": "theorem", "label": "more-algebra-lemma-J-0-goes-down", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-J-0-goes-down", "contents": [ "Let $R \\to S$ be a ring map. Assume that", "\\begin{enumerate}", "\\item $R$ is a Noetherian domain,", "\\item $R \\to S$ is injective and of finite type, and", "\\item $S$ is a domain and J-0.", "\\end{enumerate}", "Then $R$ is J-0." ], "refs": [], "proofs": [ { "contents": [ "After replacing $S$ by $S_g$ for some nonzero $g \\in S$ we may assume", "that $S$ is a regular ring. By generic flatness we may assume that also", "$R \\to S$ is faithfully flat, see", "Algebra, Lemma \\ref{algebra-lemma-generic-flatness-Noetherian}.", "Then $R$ is regular by", "Algebra, Lemma \\ref{algebra-lemma-descent-regular}." ], "refs": [ "algebra-lemma-generic-flatness-Noetherian", "algebra-lemma-descent-regular" ], "ref_ids": [ 1013, 1373 ] } ], "ref_ids": [] }, { "id": 10075, "type": "theorem", "label": "more-algebra-lemma-J-0-goes-up", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-J-0-goes-up", "contents": [ "Let $R \\to S$ be a ring map. Assume that", "\\begin{enumerate}", "\\item $R$ is a Noetherian domain and J-0,", "\\item $R \\to S$ is injective and of finite type, and", "\\item $S$ is a domain, and", "\\item the induced extension of fraction fields is separable.", "\\end{enumerate}", "Then $S$ is J-0." ], "refs": [], "proofs": [ { "contents": [ "We may replace $R$ by a principal localization and assume $R$ is", "a regular ring. By Algebra, Lemma \\ref{algebra-lemma-smooth-at-generic-point}", "the ring map $R \\to S$ is smooth at $(0)$.", "Hence after replacing $S$ by a principal localization", "we may assume that $S$ is smooth over $R$.", "Then $S$ is regular too, see", "Algebra, Lemma \\ref{algebra-lemma-regular-goes-up}." ], "refs": [ "algebra-lemma-smooth-at-generic-point", "algebra-lemma-regular-goes-up" ], "ref_ids": [ 1228, 1369 ] } ], "ref_ids": [] }, { "id": 10076, "type": "theorem", "label": "more-algebra-lemma-J-2", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-J-2", "contents": [ "Let $R$ be a Noetherian ring. The following are equivalent", "\\begin{enumerate}", "\\item $R$ is J-2,", "\\item every finite type $R$-algebra which is a domain is J-0,", "\\item every finite $R$-algebra is J-1,", "\\item for every prime $\\mathfrak p$ and every finite purely inseparable", "extension $\\kappa(\\mathfrak p) \\subset L$ there exists a finite", "$R$-algebra $R'$ which is a domain, which is J-0, and whose field", "of fractions is $L$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that we have the implications (1) $\\Rightarrow$ (2) and", "(2) $\\Rightarrow$ (4). Recall that a domain which is", "J-1 is J-0. Hence we also have the implications", "(1) $\\Rightarrow$ (3) and (3) $\\Rightarrow$ (4).", "\\medskip\\noindent", "Let $R \\to S$ be a finite type ring map and let's try to show $S$ is J-1. By", "Lemma \\ref{lemma-intersection-regular-with-closed} it suffices", "to prove that $S/\\mathfrak q$ is J-0 for every prime $\\mathfrak q$", "of $S$. In this way we see (2) $\\Rightarrow$ (1).", "\\medskip\\noindent", "Assume (4). We will show that (2) holds which will finish the proof.", "Let $R \\to S$ be a finite type ring map with $S$ a domain.", "Let $\\mathfrak p = \\Ker(R \\to S)$. Let $K$ be the fraction field of $S$.", "There exists a diagram of fields", "$$", "\\xymatrix{", "K \\ar[r] & K' \\\\", "\\kappa(\\mathfrak p) \\ar[u] \\ar[r] & L \\ar[u]", "}", "$$", "where the horizontal arrows are finite purely inseparable field extensions", "and where $K'/L$ is separable, see", "Algebra, Lemma \\ref{algebra-lemma-make-separably-generated}.", "Choose $R' \\subset L$ as in (4) and let", "$S'$ be the image of the map $S \\otimes_R R' \\to K'$.", "Then $S'$ is a domain whose fraction field is $K'$, hence", "$S'$ is J-0 by Lemma \\ref{lemma-J-0-goes-up} and our choice of $R'$.", "Then we apply Lemma \\ref{lemma-J-0-goes-down} to see that $S$", "is J-0 as desired." ], "refs": [ "more-algebra-lemma-intersection-regular-with-closed", "algebra-lemma-make-separably-generated", "more-algebra-lemma-J-0-goes-up", "more-algebra-lemma-J-0-goes-down" ], "ref_ids": [ 10073, 560, 10075, 10074 ] } ], "ref_ids": [] }, { "id": 10077, "type": "theorem", "label": "more-algebra-lemma-derivation-extends", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derivation-extends", "contents": [ "Let $R$ be a ring. Let $D : R \\to R$ be a derivation.", "\\begin{enumerate}", "\\item For any ideal $I \\subset R$ the derivation $D$ extends", "canonically to a derivation $D^\\wedge : R^\\wedge \\to R^\\wedge$", "on the $I$-adic completion.", "\\item For any multiplicative subset $S \\subset R$ the derivation", "$D$ extends uniquely to the localization $S^{-1}R$ of $R$.", "\\end{enumerate}", "If $R \\subset R'$ is a finite type extension of rings such that", "$R_g \\cong R'_g$ for some $g \\in R$ which is a nonzerodivisor in $R'$,", "then $g^ND$ extends to $R'$ for some $N \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). For $n \\geq 2$ we have $D(I^n) \\subset I^{n - 1}$", "by the Leibniz rule. Hence $D$ induces maps $D_n : R/I^n \\to R/I^{n - 1}$.", "Taking the limit we obtain $D^\\wedge$. We omit the verification that", "$D^\\wedge$ is a derivation.", "\\medskip\\noindent", "Proof of (2). To extend $D$ to $S^{-1}R$ just set", "$D(r/s) = D(r)/s - rD(s)/s^2$ and check the axioms.", "\\medskip\\noindent", "Proof of the final statement. Let $x_1, \\ldots, x_n \\in R'$ be generators", "of $R'$ over $R$. Choose an $N$ such that $g^Nx_i \\in R$.", "Consider $g^{N + 1}D$. By (2) this extends to $R_g$. Moreover, by", "the Leibniz rule and our construction of the extension above we have", "$$", "g^{N + 1}D(x_i) = g^{N + 1}D(g^{-N} g^Nx_i) = -Ng^Nx_iD(g) +", "gD(g^Nx_i)", "$$", "and both terms are in $R$. This implies that", "$$", "g^{N + 1}D(x_1^{e_1} \\ldots x_n^{e_n}) =", "\\sum e_i x_1^{e_1} \\ldots x_i^{e_i - 1} \\ldots x_n^{e_n} g^{N + 1}D(x_i)", "$$", "is an element of $R'$. Hence every element of $R'$ (which can be written", "as a sum of monomials in the $x_i$ with coefficients in $R$) is mapped to an", "element of $R'$ by $g^{N + 1}D$ and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10078, "type": "theorem", "label": "more-algebra-lemma-quotient-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-quotient-regular", "contents": [ "\\begin{slogan}", "The Jacobian criterion for hypersurfaces, done right.", "\\end{slogan}", "Let $R$ be a regular ring. Let $f \\in R$. Assume there exists a", "derivation $D : R \\to R$ such that $D(f)$ is a unit of $R/(f)$.", "Then $R/(f)$ is regular." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove this when $R$ is a local ring with maximal ideal", "$\\mathfrak m$ and residue field $\\kappa$. In this case it suffices", "to prove that $f \\not \\in \\mathfrak m^2$, see", "Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}.", "However, if $f \\in \\mathfrak m^2$ then $D(f) \\in \\mathfrak m$", "by the Leibniz rule, a contradiction." ], "refs": [ "algebra-lemma-regular-ring-CM" ], "ref_ids": [ 941 ] } ], "ref_ids": [] }, { "id": 10079, "type": "theorem", "label": "more-algebra-lemma-quotient-sequence-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-quotient-sequence-regular", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a regular local ring. Let $m \\geq 1$. Let", "$f_1, \\ldots, f_m \\in \\mathfrak m$. Assume there exist derivations", "$D_1, \\ldots, D_m : R \\to R$ such that", "$\\det_{1 \\leq i, j \\leq m}(D_i(f_j))$ is a unit of $R$.", "Then $R/(f_1, \\ldots, f_m)$ is regular and $f_1, \\ldots, f_m$", "is a regular sequence." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove that $f_1, \\ldots, f_m$ are $\\kappa$-linearly", "independent in $\\mathfrak m/\\mathfrak m^2$, see", "Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}.", "However, if there is a nontrivial linear relation the we get", "$\\sum a_i f_i \\in \\mathfrak m^2$ for some $a_i \\in R$", "but not all $a_i \\in \\mathfrak m$. Observe that", "$D_i(\\mathfrak m^2) \\subset \\mathfrak m$ and", "$D_i(a_j f_j) \\equiv a_j D_i(f_j) \\bmod \\mathfrak m$", "by the Leibniz rule for derivations. Hence", "this would imply", "$$", "\\sum a_j D_i(f_j) \\in \\mathfrak m", "$$", "which would contradict the assumption on the determinant." ], "refs": [ "algebra-lemma-regular-ring-CM" ], "ref_ids": [ 941 ] } ], "ref_ids": [] }, { "id": 10080, "type": "theorem", "label": "more-algebra-lemma-degree-p-extension-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-degree-p-extension-regular", "contents": [ "Let $R$ be a regular ring. Let $f \\in R$.", "Assume there exists a derivation $D : R \\to R$ such that $D(f)$ is a unit", "of $R$. Then $R[z]/(z^n - f)$ is regular for any integer $n \\geq 1$.", "More generally, $R[z]/(p(z) - f)$ is regular for any $p \\in \\mathbf{Z}[z]$." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-regular-goes-up} we see that", "$R[z]$ is a regular ring. Apply Lemma \\ref{lemma-quotient-regular}", "to the extension of $D$ to $R[z]$ which maps $z$ to zero.", "This works because $D$ annihilates any polynomial with", "integer coefficients and sends $f$ to a unit." ], "refs": [ "algebra-lemma-regular-goes-up", "more-algebra-lemma-quotient-regular" ], "ref_ids": [ 1369, 10078 ] } ], "ref_ids": [] }, { "id": 10081, "type": "theorem", "label": "more-algebra-lemma-find-D", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-find-D", "contents": [ "Let $p$ be a prime number. Let $B$ be a domain with $p = 0$ in $B$.", "Let $f \\in B$ be an element which is not a $p$th power in the fraction", "field of $B$. If $B$ is of finite type over a Noetherian complete", "local ring, then there exists a derivation $D : B \\to B$ such that $D(f)$", "is not zero." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a Noetherian complete local ring such that there exists", "a finite type ring map $R \\to B$. Of course we may replace $R$ by", "its image in $B$, hence we may assume $R$ is a domain of characteristic", "$p > 0$ (as well as Noetherian complete local). By Algebra, Lemma", "\\ref{algebra-lemma-complete-local-Noetherian-domain-finite-over-regular}", "we can write $R$ as a finite extension of $k[[x_1, \\ldots, x_n]]$ for some", "field $k$ and integer $n$. Hence we may replace $R$ by $k[[x_1, \\ldots, x_n]]$.", "Next, we use", "Algebra, Lemma \\ref{algebra-lemma-Noether-normalization-over-a-domain}", "to factor $R \\to B$ as", "$$", "R \\subset R[y_1, \\ldots, y_d] \\subset B' \\subset B", "$$", "with $B'$ finite over $R[y_1, \\ldots, y_d]$ and $B'_g \\cong B_g$", "for some nonzero $g \\in R$. Note that $f' = g^{pN} f \\in B'$ for some", "large integer $N$. It is clear that $f'$ is not a $p$th power in", "the fraction field of $B'$. If we can find a derivation", "$D' : B' \\to B'$ with $D'(f') \\not = 0$, then", "Lemma \\ref{lemma-derivation-extends}", "guarantees that $D = g^MD'$ extends to $B$ for some $M > 0$. Then", "$D(f) = g^ND'(f) = g^MD'(g^{-pN}f') = g^{M - pN}D'(f')$ is nonzero.", "Thus it suffices to prove the lemma in case", "$B$ is a finite extension of $A = k[[x_1, \\ldots, x_n]][y_1, \\ldots, y_m]$.", "\\medskip\\noindent", "Assume $B$ is a finite extension of", "$A = k[[x_1, \\ldots, x_n]][y_1, \\ldots, y_m]$.", "Denote $L$ the fraction field of $B$.", "Note that $\\text{d}f$ is not zero in $\\Omega_{L/\\mathbf{F}_p}$, see", "Algebra, Lemma \\ref{algebra-lemma-derivative-zero-pth-power}.", "We apply Lemma \\ref{lemma-power-series-ring-subfields} to find a subfield", "$k' \\subset k$ of finite index such that with", "$A' = k'[[x_1^p, \\ldots, x_n^p]][y_1^p, \\ldots, y_m^p]$", "the element $\\text{d}f$ does not map to zero in $\\Omega_{L/K'}$", "where $K'$ is the fraction field of $A'$.", "Thus we can choose a $K'$-derivation $D' : L \\to L$", "with $D'(f) \\not = 0$. Since $A' \\subset A$ and $A \\subset B$ are", "finite by construction we see that $A' \\subset B$ is finite.", "Choose $b_1, \\ldots, b_t \\in B$ which generate $B$ as an $A'$-module.", "Then $D'(b_i) = f_i/g_i$ for some $f_i, g_i \\in B$ with $g_i \\not = 0$.", "Setting $D = g_1 \\ldots g_t D'$ we win." ], "refs": [ "algebra-lemma-complete-local-Noetherian-domain-finite-over-regular", "algebra-lemma-Noether-normalization-over-a-domain", "more-algebra-lemma-derivation-extends", "algebra-lemma-derivative-zero-pth-power", "more-algebra-lemma-power-series-ring-subfields" ], "ref_ids": [ 1332, 1004, 10077, 1315, 10071 ] } ], "ref_ids": [] }, { "id": 10082, "type": "theorem", "label": "more-algebra-lemma-complete-Noetherian-domain-J-0", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-complete-Noetherian-domain-J-0", "contents": [ "Let $A$ be a Noetherian complete local domain. Then $A$ is J-0." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma", "\\ref{algebra-lemma-complete-local-Noetherian-domain-finite-over-regular}", "we can find a regular subring $A_0 \\subset A$ with $A$ finite over $A_0$.", "The induced extension $K/K_0$ of fraction fields is finite.", "If $K/K_0$ is separable, then we are done by", "Lemma \\ref{lemma-J-0-goes-up}. If not, then $A_0$ and $A$", "have characteristic $p > 0$. For any subextension $K/M/K_0$", "there exists a finite subextension $A_0 \\subset B \\subset A$", "whose fraction field is $M$. Hence, arguing by induction on", "$[K : K_0]$ we may assume there exists $A_0 \\subset B \\subset A$ such that", "$B$ is J-0 and $K/M$ has no nontrivial subextensions.", "In this case, if $K/M$ is separable, then we", "see that $A$ is J-0 by Lemma \\ref{lemma-J-0-goes-up}.", "If not, then $K = M[z]/(z^p - b)$ for some $b \\in B$", "which is not a $p$th power in $M$. By Lemma \\ref{lemma-find-D}", "we can find a derivation $D : B \\to B$ with $D(f) \\not = 0$.", "Applying Lemma \\ref{lemma-degree-p-extension-regular}", "we see that $A_\\mathfrak p$ is regular for any prime", "$\\mathfrak p$ of $A$ lying over a regular prime of $B$", "and not containing $D(f)$. As $B$ is J-0 we conclude $A$ is too." ], "refs": [ "algebra-lemma-complete-local-Noetherian-domain-finite-over-regular", "more-algebra-lemma-J-0-goes-up", "more-algebra-lemma-J-0-goes-up", "more-algebra-lemma-find-D", "more-algebra-lemma-degree-p-extension-regular" ], "ref_ids": [ 1332, 10075, 10075, 10081, 10080 ] } ], "ref_ids": [] }, { "id": 10083, "type": "theorem", "label": "more-algebra-lemma-lift-derivation-through-fs", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-derivation-through-fs", "contents": [ "Let $A \\to B$ be a local homomorphism of Noetherian local rings.", "Let $D : A \\to A$ be a derivation. Assume that $B$ is complete", "and $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic topology.", "Then there exists an extension $D' : B \\to B$ of $D$." ], "refs": [], "proofs": [ { "contents": [ "Denote $B[\\epsilon] = B[x]/(x^2)$ the ring of dual numbers over $B$.", "Consider the ring map $\\psi : A \\to B[\\epsilon]$,", "$a \\mapsto a + \\epsilon D(a)$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "B \\ar[r]_1 & B \\\\", "A \\ar[u] \\ar[r]^\\psi & B[\\epsilon] \\ar[u]", "}", "$$", "By Lemma \\ref{lemma-lift-continuous} and the assumption of formal", "smoothness of $B/A$ we find a map $\\varphi : B \\to B[\\epsilon]$ fitting into", "the diagram. Write $\\varphi(b) = b + \\epsilon D'(b)$. Then $D' : B \\to B$", "is the desired extension." ], "refs": [ "more-algebra-lemma-lift-continuous" ], "ref_ids": [ 10016 ] } ], "ref_ids": [] }, { "id": 10084, "type": "theorem", "label": "more-algebra-lemma-check-G-ring-easy", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-check-G-ring-easy", "contents": [ "Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if", "for every pair of primes $\\mathfrak q \\subset \\mathfrak p \\subset R$", "the algebra", "$$", "(R/\\mathfrak q)_\\mathfrak p^\\wedge \\otimes_{R/\\mathfrak q} \\kappa(\\mathfrak q)", "$$", "is geometrically regular over $\\kappa(\\mathfrak q)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that", "$$", "R_\\mathfrak p^\\wedge \\otimes_R \\kappa(\\mathfrak q) =", "(R/\\mathfrak q)_\\mathfrak p^\\wedge \\otimes_{R/\\mathfrak q} \\kappa(\\mathfrak q)", "$$", "as algebras over $\\kappa(\\mathfrak q)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10085, "type": "theorem", "label": "more-algebra-lemma-G-ring-goes-up-quasi-finite", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-G-ring-goes-up-quasi-finite", "contents": [ "Let $R \\to R'$ be a finite type map of Noetherian rings and let", "$$", "\\xymatrix{", "\\mathfrak q' \\ar[r] & \\mathfrak p' \\ar[r] & R' \\\\", "\\mathfrak q \\ar[r] \\ar@{-}[u] &", "\\mathfrak p \\ar[r] \\ar@{-}[u] & R \\ar[u]", "}", "$$", "be primes. Assume $R \\to R'$ is quasi-finite at $\\mathfrak p'$.", "\\begin{enumerate}", "\\item If the formal fibre $R_\\mathfrak p^\\wedge \\otimes_R \\kappa(\\mathfrak q)$", "is geometrically regular over $\\kappa(\\mathfrak q)$, then the formal fibre", "$R'_{\\mathfrak p'} \\otimes_{R'} \\kappa(\\mathfrak q')$ is geometrically regular", "over $\\kappa(\\mathfrak q')$.", "\\item If the formal fibres of $R_\\mathfrak p$ are geometrically regular,", "then the formal fibres of $R'_{\\mathfrak p'}$ are geometrically regular.", "\\item If $R \\to R'$ is quasi-finite and $R$ is a G-ring, then $R'$ is", "a G-ring.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3).", "Assume $R_\\mathfrak p^\\wedge \\otimes_R \\kappa(\\mathfrak q)$", "is geometrically regular over $\\kappa(\\mathfrak q)$.", "By Algebra, Lemma \\ref{algebra-lemma-completion-at-quasi-finite-prime}", "we see that", "$$", "R_\\mathfrak p^\\wedge \\otimes_R R'", "=", "(R'_{\\mathfrak p'})^\\wedge \\times B", "$$", "for some $R_\\mathfrak p^\\wedge$-algebra $B$. Hence", "$R'_{\\mathfrak p'} \\to (R'_{\\mathfrak p'})^\\wedge$ is a factor of", "a base change of the map $R_\\mathfrak p \\to R_\\mathfrak p^\\wedge$.", "It follows that $(R'_{\\mathfrak p'})^\\wedge \\otimes_{R'} \\kappa(\\mathfrak q')$", "is a factor of", "$$", "R_\\mathfrak p^\\wedge \\otimes_R R' \\otimes_{R'} \\kappa(\\mathfrak q') =", "R_\\mathfrak p^\\wedge \\otimes_R \\kappa(\\mathfrak q)", "\\otimes_{\\kappa(\\mathfrak q)} \\kappa(\\mathfrak q').", "$$", "Thus the result follows as extension of base field preserves", "geometric regularity, see", "Algebra, Lemma \\ref{algebra-lemma-geometrically-regular}." ], "refs": [ "algebra-lemma-completion-at-quasi-finite-prime", "algebra-lemma-geometrically-regular" ], "ref_ids": [ 1070, 1382 ] } ], "ref_ids": [] }, { "id": 10086, "type": "theorem", "label": "more-algebra-lemma-check-G-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-check-G-ring", "contents": [ "Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if", "for every finite free ring map $R \\to S$ the formal fibres of $S$", "are regular rings." ], "refs": [], "proofs": [ { "contents": [ "Assume that for any finite free ring map $R \\to S$ the ring $S$ has", "regular formal fibres. Let $\\mathfrak q \\subset \\mathfrak p \\subset R$", "be primes and let $\\kappa(\\mathfrak q) \\subset L$ be a finite purely", "inseparable extension. To show that $R$ is a G-ring it suffices to", "show that", "$$", "R_\\mathfrak p^\\wedge \\otimes_R \\kappa(\\mathfrak q)", "\\otimes_{\\kappa(\\mathfrak q)} L", "$$", "is a regular ring. Choose a finite free extension $R \\to R'$ such that", "$\\mathfrak q' = \\mathfrak qR'$ is a prime and such that $\\kappa(\\mathfrak q')$", "is isomorphic to $L$ over $\\kappa(\\mathfrak q)$, see", "Algebra, Lemma \\ref{algebra-lemma-finite-free-given-residue-field-extension}.", "By", "Algebra, Lemma \\ref{algebra-lemma-completion-finite-extension}", "we have", "$$", "R_\\mathfrak p^\\wedge \\otimes_R R' = \\prod (R'_{\\mathfrak p_i'})^\\wedge", "$$", "where $\\mathfrak p_i'$ are the primes of $R'$ lying over $\\mathfrak p$.", "Thus we have", "$$", "R_\\mathfrak p^\\wedge \\otimes_R \\kappa(\\mathfrak q)", "\\otimes_{\\kappa(\\mathfrak q)} L =", "R_\\mathfrak p^\\wedge \\otimes_R R'", "\\otimes_{R'} \\kappa(\\mathfrak q')", "=", "\\prod (R'_{\\mathfrak p_i'})^\\wedge", "\\otimes_{R'_{\\mathfrak p'_i}} \\kappa(\\mathfrak q')", "$$", "Our assumption is that the rings on the right are regular, hence the", "ring on the left is regular too. Thus $R$ is a G-ring. The converse", "follows from Lemma \\ref{lemma-G-ring-goes-up-quasi-finite}." ], "refs": [ "algebra-lemma-finite-free-given-residue-field-extension", "algebra-lemma-completion-finite-extension", "more-algebra-lemma-G-ring-goes-up-quasi-finite" ], "ref_ids": [ 1326, 876, 10085 ] } ], "ref_ids": [] }, { "id": 10087, "type": "theorem", "label": "more-algebra-lemma-helper-G-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-helper-G-ring", "contents": [ "Let $k$ be a field of characteristic $p$.", "Let $A = k[[x_1, \\ldots, x_n]][y_1, \\ldots, y_n]$ and denote $K$", "the fraction field of $A$.", "Let $\\mathfrak p \\subset A$ be a prime. Then", "$A_\\mathfrak p^\\wedge \\otimes_A K$ is geometrically regular over $K$." ], "refs": [], "proofs": [ { "contents": [ "Let $L \\supset K$ be a finite purely inseparable field extension.", "We will show by induction on $[L : K]$ that $A_\\mathfrak p^\\wedge \\otimes L$", "is regular. The base case is $L = K$: as $A$ is regular,", "$A_\\mathfrak p^\\wedge$ is regular (Lemma \\ref{lemma-completion-regular}),", "hence the localization $A_\\mathfrak p^\\wedge \\otimes K$ is regular.", "Let $K \\subset M \\subset L$ be a subfield such that", "$L$ is a degree $p$ extension of $M$ obtained by adjoining a $p$th root", "of an element $f \\in M$. Let $B$ be a finite $A$-subalgebra", "of $M$ with fraction field $M$. Clearing denominators, we may and do assume", "$f \\in B$. Set $C = B[z]/(z^p -f)$ and note that $B \\subset C$", "is finite and that the fraction field of $C$ is $L$. Since", "$A \\subset B \\subset C$ are finite and $L/M/K$ are purely", "inseparable we see that for every element of $B$ or $C$ some power of", "it lies in $A$. Hence there is a unique prime $\\mathfrak r \\subset B$,", "resp.\\ $\\mathfrak q \\subset C$ lying over $\\mathfrak p$. Note that", "$$", "A_\\mathfrak p^\\wedge \\otimes_A M = B_\\mathfrak r^\\wedge \\otimes_B M", "$$", "see Algebra, Lemma \\ref{algebra-lemma-completion-finite-extension}.", "By induction we know that this ring is regular. In the same manner we have", "$$", "A_\\mathfrak p^\\wedge \\otimes_A L =", "C_\\mathfrak r^\\wedge \\otimes_C L =", "B_\\mathfrak r^\\wedge \\otimes_B M[z]/(z^p - f)", "$$", "the last equality because the completion of", "$C = B[z]/(z^p - f)$ equals $B_\\mathfrak r^\\wedge[z]/(z^p -f)$.", "By Lemma \\ref{lemma-find-D} we know there exists a derivation", "$D : B \\to B$ such that $D(f) \\not = 0$. In other words, $g = D(f)$", "is a unit in $M$! By Lemma \\ref{lemma-derivation-extends}", "$D$ extends to a derivation of $B_\\mathfrak r$, $B_\\mathfrak r^\\wedge$", "and $B_\\mathfrak r^\\wedge \\otimes_B M$ (successively extending through a", "localization, a completion, and a localization). Since it is an", "extension we end up with a derivation of $B_\\mathfrak r^\\wedge \\otimes_B M$", "which maps $f$ to $g$ and $g$ is a unit of the ring", "$B_\\mathfrak r^\\wedge \\otimes_B M$.", "Hence $A_\\mathfrak p^\\wedge \\otimes_A L$ is regular by", "Lemma \\ref{lemma-degree-p-extension-regular} and we win." ], "refs": [ "more-algebra-lemma-completion-regular", "algebra-lemma-completion-finite-extension", "more-algebra-lemma-find-D", "more-algebra-lemma-derivation-extends", "more-algebra-lemma-degree-p-extension-regular" ], "ref_ids": [ 10045, 876, 10081, 10077, 10080 ] } ], "ref_ids": [] }, { "id": 10088, "type": "theorem", "label": "more-algebra-lemma-check-G-ring-maximal-ideals", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-check-G-ring-maximal-ideals", "contents": [ "Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if", "$R_\\mathfrak m$ has geometrically regular formal fibres for every", "maximal ideal $\\mathfrak m$ of $R$." ], "refs": [], "proofs": [ { "contents": [ "Assume $R_\\mathfrak m \\to R_\\mathfrak m^\\wedge$ is regular for every", "maximal ideal $\\mathfrak m$ of $R$. Let $\\mathfrak p$ be a prime of", "$R$ and choose a maximal ideal $\\mathfrak p \\subset \\mathfrak m$.", "Since $R_\\mathfrak m \\to R_\\mathfrak m^\\wedge$ is faithfully flat", "we can choose a prime $\\mathfrak p'$ if $R_\\mathfrak m^\\wedge$", "lying over $\\mathfrak pR_\\mathfrak m$. Consider the commutative diagram", "$$", "\\xymatrix{", "R_\\mathfrak m^\\wedge \\ar[r] &", "(R_\\mathfrak m^\\wedge)_{\\mathfrak p'} \\ar[r] &", "(R_\\mathfrak m^\\wedge)_{\\mathfrak p'}^\\wedge", "\\\\", "R_\\mathfrak m \\ar[u] \\ar[r] & R_\\mathfrak p \\ar[u] \\ar[r] &", "R_\\mathfrak p^\\wedge \\ar[u]", "}", "$$", "By assumption the ring map $R_\\mathfrak m \\to R_\\mathfrak m^\\wedge$ is", "regular. By Proposition \\ref{proposition-Noetherian-complete-G-ring}", "$(R_\\mathfrak m^\\wedge)_{\\mathfrak p'} \\to", "(R_\\mathfrak m^\\wedge)_{\\mathfrak p'}^\\wedge$ is regular.", "The localization", "$R_\\mathfrak m^\\wedge \\to (R_\\mathfrak m^\\wedge)_{\\mathfrak p'}$ is regular.", "Hence $R_\\mathfrak m \\to (R_\\mathfrak m^\\wedge)_{\\mathfrak p'}^\\wedge$", "is regular by Lemma \\ref{lemma-regular-composition}.", "Since it factors through the localization $R_\\mathfrak p$, also the ring map", "$R_\\mathfrak p \\to (R_\\mathfrak m^\\wedge)_{\\mathfrak p'}^\\wedge$", "is regular. Thus we may apply Lemma \\ref{lemma-regular-permanence} to see that", "$R_\\mathfrak p \\to R_\\mathfrak p^\\wedge$ is regular." ], "refs": [ "more-algebra-proposition-Noetherian-complete-G-ring", "more-algebra-lemma-regular-composition", "more-algebra-lemma-regular-permanence" ], "ref_ids": [ 10580, 10036, 10039 ] } ], "ref_ids": [] }, { "id": 10089, "type": "theorem", "label": "more-algebra-lemma-henselization-G-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-G-ring", "contents": [ "Let $R$ be a Noetherian local ring which is a G-ring.", "Then the henselization $R^h$ and the strict henselization $R^{sh}$", "are G-rings." ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion of Lemma \\ref{lemma-check-G-ring-maximal-ideals}.", "Let $\\mathfrak q \\subset R^h$ be a prime and set", "$\\mathfrak p = R \\cap \\mathfrak q$. Set $\\mathfrak q_1 = \\mathfrak q$", "and let $\\mathfrak q_2, \\ldots, \\mathfrak q_t$", "be the other primes of $R^h$ lying over $\\mathfrak p$, so that", "$R^h \\otimes_R \\kappa(\\mathfrak p) =", "\\prod\\nolimits_{i = 1, \\ldots, t} \\kappa(\\mathfrak q_i)$, see", "Lemma \\ref{lemma-fibres-henselization}.", "Using that $(R^h)^\\wedge = R^\\wedge$", "(Lemma \\ref{lemma-henselization-noetherian}) we see", "$$", "\\prod\\nolimits_{i = 1, \\ldots, t}", "(R^h)^\\wedge \\otimes_{R^h} \\kappa(\\mathfrak q_i) =", "(R^h)^\\wedge \\otimes_{R^h} (R^h \\otimes_R \\kappa(\\mathfrak p)) =", "R^\\wedge \\otimes_R \\kappa(\\mathfrak p)", "$$", "Hence $(R^h)^\\wedge \\otimes_{R^h} \\kappa(\\mathfrak q_i)$", "is geometrically regular over $\\kappa(\\mathfrak p)$ by assumption.", "Since $\\kappa(\\mathfrak q_i)$ is separable algebraic over $\\kappa(\\mathfrak p)$", "it follows from Algebra, Lemma", "\\ref{algebra-lemma-geometrically-regular-over-separable-algebraic} that", "$(R^h)^\\wedge \\otimes_{R^h} \\kappa(\\mathfrak q_i)$ is", "geometrically regular over $\\kappa(\\mathfrak q_i)$.", "\\medskip\\noindent", "Let $\\mathfrak r \\subset R^{sh}$ be a prime and set", "$\\mathfrak p = R \\cap \\mathfrak r$. Set $\\mathfrak r_1 = \\mathfrak r$", "and let $\\mathfrak r_2, \\ldots, \\mathfrak r_s$", "be the other primes of $R^{sh}$ lying over $\\mathfrak p$, so that", "$R^{sh} \\otimes_R \\kappa(\\mathfrak p) =", "\\prod\\nolimits_{i = 1, \\ldots, s} \\kappa(\\mathfrak r_i)$, see", "Lemma \\ref{lemma-fibres-henselization}.", "Then we see that", "$$", "\\prod\\nolimits_{i = 1, \\ldots, s}", "(R^{sh})^\\wedge \\otimes_{R^{sh}} \\kappa(\\mathfrak r_i) =", "(R^{sh})^\\wedge \\otimes_{R^{sh}} (R^{sh} \\otimes_R \\kappa(\\mathfrak p)) =", "(R^{sh})^\\wedge \\otimes_R \\kappa(\\mathfrak p)", "$$", "Note that $R^\\wedge \\to (R^{sh})^\\wedge$ is formally smooth", "in the $\\mathfrak m_{(R^{sh})^\\wedge}$-adic topology, see", "Lemma \\ref{lemma-henselization-noetherian}.", "Hence $R^\\wedge \\to (R^{sh})^\\wedge$ is regular by", "Proposition \\ref{proposition-fs-regular}.", "We conclude that $(R^{sh})^\\wedge \\otimes_{R^{sh}} \\kappa(\\mathfrak r_i)$", "is regular over $\\kappa(\\mathfrak p)$ by", "Lemma \\ref{lemma-regular-composition} as", "$R^\\wedge \\otimes_R \\kappa(\\mathfrak p)$ is regular over $\\kappa(\\mathfrak p)$", "by assumption. Since $\\kappa(\\mathfrak r_i)$ is separable algebraic over", "$\\kappa(\\mathfrak p)$", "it follows from Algebra, Lemma", "\\ref{algebra-lemma-geometrically-regular-over-separable-algebraic} that", "$(R^{sh})^\\wedge \\otimes_{R^{sh}} \\kappa(\\mathfrak r_i)$ is", "geometrically regular over $\\kappa(\\mathfrak r_i)$." ], "refs": [ "more-algebra-lemma-check-G-ring-maximal-ideals", "more-algebra-lemma-fibres-henselization", "more-algebra-lemma-henselization-noetherian", "algebra-lemma-geometrically-regular-over-separable-algebraic", "more-algebra-lemma-fibres-henselization", "more-algebra-lemma-henselization-noetherian", "more-algebra-proposition-fs-regular", "more-algebra-lemma-regular-composition", "algebra-lemma-geometrically-regular-over-separable-algebraic" ], "ref_ids": [ 10088, 10067, 10057, 1386, 10067, 10057, 10579, 10036, 1386 ] } ], "ref_ids": [] }, { "id": 10090, "type": "theorem", "label": "more-algebra-lemma-another-helper-G-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-another-helper-G-ring", "contents": [ "Let $p$ be a prime number. Let $A$ be a Noetherian complete local domain", "with fraction field $K$ of characteristic $p$. Let $\\mathfrak q \\subset A[x]$", "be a maximal ideal lying over the maximal ideal of $A$ and let", "$(0) \\not = \\mathfrak r \\subset \\mathfrak q$ be a prime lying over", "$(0) \\subset A$. Then", "$A[x]_\\mathfrak q^\\wedge \\otimes_{A[x]} \\kappa(\\mathfrak r)$ is geometrically", "regular over $\\kappa(\\mathfrak r)$." ], "refs": [], "proofs": [ { "contents": [ "Note that $K \\subset \\kappa(\\mathfrak r)$ is finite.", "Hence, given a finite purely", "inseparable extension $\\kappa(\\mathfrak r) \\subset L$ there exists a finite", "extension of Noetherian complete local domains $A \\subset B$ such that", "$\\kappa(\\mathfrak r) \\otimes_A B$ surjects onto $L$.", "Namely, you take $B \\subset L$", "a finite $A$-subalgebra whose field of fractions is $L$. Denote", "$\\mathfrak r' \\subset B[x]$ the kernel of the map", "$B[x] = A[x] \\otimes_A B \\to \\kappa(\\mathfrak r) \\otimes_A B \\to L$", "so that $\\kappa(\\mathfrak r') = L$. Then", "$$", "A[x]_\\mathfrak q^\\wedge \\otimes_{A[x]} L =", "A[x]_\\mathfrak q^\\wedge \\otimes_{A[x]} B[x]", "\\otimes_{B[x]} \\kappa(\\mathfrak r') =", "\\prod B[x]_{\\mathfrak q_i}^\\wedge \\otimes_{B[x]} \\kappa(\\mathfrak r')", "$$", "where $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ are the primes of $B[x]$", "lying over $\\mathfrak q$, see", "Algebra, Lemma \\ref{algebra-lemma-completion-finite-extension}.", "Thus we see that it suffices to prove the rings", "$B[x]_{\\mathfrak q_i}^\\wedge \\otimes_{B[x]} \\kappa(\\mathfrak r')$", "are regular. This reduces us to showing that", "$A[x]_\\mathfrak q^\\wedge \\otimes_{A[x]} \\kappa(\\mathfrak r)$", "is regular in the special case that $K = \\kappa(\\mathfrak r)$.", "\\medskip\\noindent", "Assume $K = \\kappa(\\mathfrak r)$. In this case we see that", "$\\mathfrak r K[x]$ is generated by $x - f$ for some $f \\in K$", "and", "$$", "A[x]_\\mathfrak q^\\wedge \\otimes_{A[x]} \\kappa(\\mathfrak r)", "=", "(A[x]_\\mathfrak q^\\wedge \\otimes_A K)/(x - f)", "$$", "The derivation $D = \\text{d}/\\text{d}x$ of $A[x]$ extends to $K[x]$ and", "maps $x - f$ to a unit of $K[x]$. Moreover $D$ extends to", "$A[x]_\\mathfrak q^\\wedge \\otimes_A K$ by Lemma \\ref{lemma-derivation-extends}.", "As $A \\to A[x]_\\mathfrak q^\\wedge$ is formally smooth (see", "Lemmas \\ref{lemma-formally-smooth} and", "\\ref{lemma-formally-smooth-completion})", "the ring $A[x]_\\mathfrak q^\\wedge \\otimes_A K$ is regular by", "Proposition \\ref{proposition-fs-regular} (the arguments of the", "proof of that proposition simplify significantly in this particular case).", "We conclude by Lemma \\ref{lemma-quotient-regular}." ], "refs": [ "algebra-lemma-completion-finite-extension", "more-algebra-lemma-derivation-extends", "more-algebra-lemma-formally-smooth", "more-algebra-lemma-formally-smooth-completion", "more-algebra-proposition-fs-regular", "more-algebra-lemma-quotient-regular" ], "ref_ids": [ 876, 10077, 10014, 10015, 10579, 10078 ] } ], "ref_ids": [] }, { "id": 10091, "type": "theorem", "label": "more-algebra-lemma-henselian-local-limit-G-rings", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselian-local-limit-G-rings", "contents": [ "Let $(A, \\mathfrak m)$ be a henselian local ring.", "Then $A$ is a filtered colimit of a system", "of henselian local G-rings with local", "transition maps." ], "refs": [], "proofs": [ { "contents": [ "Write $A = \\colim A_i$ as a filtered colimit of finite type", "$\\mathbf{Z}$-algebras. Let $\\mathfrak p_i$ be the prime ideal of", "$A_i$ lying under $\\mathfrak m$. We may replace $A_i$ by the", "localization of $A_i$ at $\\mathfrak p_i$. Then $A_i$ is a", "Noetherian local G-ring (Proposition \\ref{proposition-ubiquity-G-ring}).", "By Lemma \\ref{lemma-henselization-colimit}", "we see that $A = \\colim A_i^h$. By", "Lemma \\ref{lemma-henselization-G-ring}", "the rings $A_i^h$ are G-rings." ], "refs": [ "more-algebra-proposition-ubiquity-G-ring", "more-algebra-lemma-henselization-colimit", "more-algebra-lemma-henselization-G-ring" ], "ref_ids": [ 10582, 9875, 10089 ] } ], "ref_ids": [] }, { "id": 10092, "type": "theorem", "label": "more-algebra-lemma-map-G-ring-to-completion-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-map-G-ring-to-completion-regular", "contents": [ "\\begin{reference}", "\\cite[Theorem 79]{MatCA}", "\\end{reference}", "Let $A$ be a G-ring. Let $I \\subset A$ be an ideal", "and let $A^\\wedge$ be the completion of $A$ with respect to $I$.", "Then $A \\to A^\\wedge$ is regular." ], "refs": [], "proofs": [ { "contents": [ "The ring map $A \\to A^\\wedge$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}.", "The ring $A^\\wedge$ is Noetherian by", "Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian-Noetherian}.", "Thus it suffices to check the third condition of", "Lemma \\ref{lemma-regular-local}.", "Let $\\mathfrak m' \\subset A^\\wedge$ be a maximal ideal lying over", "$\\mathfrak m \\subset A$.", "By Algebra, Lemma \\ref{algebra-lemma-radical-completion}", "we have $IA^\\wedge \\subset \\mathfrak m'$.", "Since $A^\\wedge/IA^\\wedge = A/I$ we see that", "$I \\subset \\mathfrak m$, $\\mathfrak m/I = \\mathfrak m'/IA^\\wedge$, and", "$A/\\mathfrak m = A^\\wedge/\\mathfrak m'$. Since $A^\\wedge/\\mathfrak m'$", "is a field, we conclude that $\\mathfrak m$ is a maximal ideal as well.", "Then $A_\\mathfrak m \\to A^\\wedge_{\\mathfrak m'}$ is a flat local", "ring homomorphism of Noetherian local rings", "which identifies residue fields and such that", "$\\mathfrak m A^\\wedge_{\\mathfrak m'} = \\mathfrak m'A^\\wedge_{\\mathfrak m'}$.", "Thus it induces an isomorphism on complete local rings, see", "Lemma \\ref{lemma-flat-unramified}.", "Let $(A_\\mathfrak m)^\\wedge$ be the completion of $A_\\mathfrak m$", "with respect to its maximal ideal.", "The ring map", "$$", "(A^\\wedge)_{\\mathfrak m'} \\to", "((A^\\wedge)_{\\mathfrak m'})^\\wedge = (A_\\mathfrak m)^\\wedge", "$$", "is faithfully flat (Algebra, Lemma", "\\ref{algebra-lemma-completion-faithfully-flat}). Thus we can apply", "Lemma \\ref{lemma-regular-permanence} to the ring maps", "$$", "A_\\mathfrak m \\to (A^\\wedge)_{\\mathfrak m'} \\to (A_\\mathfrak m)^\\wedge", "$$", "to conclude because $A_\\mathfrak m \\to (A_\\mathfrak m)^\\wedge$", "is regular as $A$ is a G-ring." ], "refs": [ "algebra-lemma-completion-flat", "algebra-lemma-completion-Noetherian-Noetherian", "more-algebra-lemma-regular-local", "algebra-lemma-radical-completion", "more-algebra-lemma-flat-unramified", "algebra-lemma-completion-faithfully-flat", "more-algebra-lemma-regular-permanence" ], "ref_ids": [ 870, 874, 10034, 862, 10050, 871, 10039 ] } ], "ref_ids": [] }, { "id": 10093, "type": "theorem", "label": "more-algebra-lemma-henselization-pair-G-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-pair-G-ring", "contents": [ "\\begin{slogan}", "Being a G-ring is stable under Henselizations along ideals", "\\end{slogan}", "\\begin{reference}", "\\cite[Theorem 5.3 i)]{Greco}", "\\end{reference}", "Let $A$ be a G-ring. Let $I \\subset A$ be an ideal.", "Let $(A^h, I^h)$ be the henselization of the pair $(A, I)$, see", "Lemma \\ref{lemma-henselization}.", "Then $A^h$ is a G-ring." ], "refs": [ "more-algebra-lemma-henselization" ], "proofs": [ { "contents": [ "Let $\\mathfrak m^h \\subset A^h$ be a maximal ideal. We have to show", "that the map from $A^h_{\\mathfrak m^h}$ to its completion has", "geometrically regular fibres, see", "Lemma \\ref{lemma-check-G-ring-maximal-ideals}.", "Let $\\mathfrak m$ be the inverse image of $\\mathfrak m^h$ in $A$.", "Note that $I^h \\subset \\mathfrak m^h$ and hence $I \\subset \\mathfrak m$", "as $(A^h, I^h)$ is a henselian pair. Recall that $A^h$ is Noetherian,", "$I^h = IA^h$, and that $A \\to A^h$ induces an isomorphism on", "$I$-adic completions, see", "Lemma \\ref{lemma-henselization-Noetherian-pair}.", "Then the local homomorphism of Noetherian local rings", "$$", "A_\\mathfrak m \\to A^h_{\\mathfrak m^h}", "$$", "induces an isomorphism on completions at maximal ideals by", "Lemma \\ref{lemma-flat-unramified} (details omitted).", "Let $\\mathfrak q^h$ be a prime of $A^h_{\\mathfrak m^h}$ lying", "over $\\mathfrak q \\subset A_\\mathfrak m$.", "Set $\\mathfrak q_1 = \\mathfrak q^h$", "and let $\\mathfrak q_2, \\ldots, \\mathfrak q_t$", "be the other primes of $A^h$ lying over $\\mathfrak q$, so that", "$A^h \\otimes_A \\kappa(\\mathfrak q) =", "\\prod\\nolimits_{i = 1, \\ldots, t} \\kappa(\\mathfrak q_i)$, see", "Lemma \\ref{lemma-filtered-colimit-etale-noetherian-fibres}.", "Using that $(A^h)_{\\mathfrak m^h}^\\wedge = (A_\\mathfrak m)^\\wedge$", "as discussed above we see", "$$", "\\prod\\nolimits_{i = 1, \\ldots, t}", "(A^h_{\\mathfrak m^h})^\\wedge \\otimes_{A^h_{\\mathfrak m^h}}", "\\kappa(\\mathfrak q_i) =", "(A^h_{\\mathfrak m^h})^\\wedge \\otimes_{A^h_{\\mathfrak m^h}}", "(A^h_{\\mathfrak m^h} \\otimes_{A_{\\mathfrak m}} \\kappa(\\mathfrak q)) =", "(A_{\\mathfrak m})^\\wedge \\otimes_{A_{\\mathfrak m}} \\kappa(\\mathfrak q)", "$$", "Hence, as one of the components, the ring", "$$", "(A^h_{\\mathfrak m^h})^\\wedge \\otimes_{A^h_{\\mathfrak m^h}}", "\\kappa(\\mathfrak q^h)", "$$", "is geometrically regular over $\\kappa(\\mathfrak q)$ by assumption on $A$.", "Since $\\kappa(\\mathfrak q^h)$ is separable algebraic over", "$\\kappa(\\mathfrak q)$", "it follows from Algebra, Lemma", "\\ref{algebra-lemma-geometrically-regular-over-separable-algebraic} that", "$$", "(A^h_{\\mathfrak m^h})^\\wedge \\otimes_{A^h_{\\mathfrak m^h}}", "\\kappa(\\mathfrak q^h)", "$$", "is geometrically regular over $\\kappa(\\mathfrak q^h)$ as desired." ], "refs": [ "more-algebra-lemma-check-G-ring-maximal-ideals", "more-algebra-lemma-henselization-Noetherian-pair", "more-algebra-lemma-flat-unramified", "more-algebra-lemma-filtered-colimit-etale-noetherian-fibres", "algebra-lemma-geometrically-regular-over-separable-algebraic" ], "ref_ids": [ 10088, 9874, 10050, 10066, 1386 ] } ], "ref_ids": [ 9871 ] }, { "id": 10094, "type": "theorem", "label": "more-algebra-lemma-check-P-ring-easy", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-check-P-ring-easy", "contents": [ "Let $R$ be a Noetherian ring. Let $P$ be a property as above.", "Then $R$ is a $P$-ring if and only if", "for every pair of primes $\\mathfrak q \\subset \\mathfrak p \\subset R$", "the $\\kappa(\\mathfrak q)$-algebra", "$$", "(R/\\mathfrak q)_\\mathfrak p^\\wedge \\otimes_{R/\\mathfrak q} \\kappa(\\mathfrak q)", "$$", "has property $P$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that", "$$", "R_\\mathfrak p^\\wedge \\otimes_R \\kappa(\\mathfrak q) =", "(R/\\mathfrak q)_\\mathfrak p^\\wedge \\otimes_{R/\\mathfrak q} \\kappa(\\mathfrak q)", "$$", "as algebras over $\\kappa(\\mathfrak q)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10095, "type": "theorem", "label": "more-algebra-lemma-P-local", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-P-local", "contents": [ "Let $R \\to \\Lambda$ be a homomorphism of Noetherian rings.", "Assume $P$ has property (B). The following are equivalent", "\\begin{enumerate}", "\\item the fibres of $R \\to \\Lambda$ have $P$,", "\\item the fibres of $R_\\mathfrak p \\to \\Lambda_\\mathfrak q$ have $P$", "for all $\\mathfrak q \\subset \\Lambda$ lying over $\\mathfrak p \\subset R$, and", "\\item the fibres of $R_\\mathfrak m \\to \\Lambda_{\\mathfrak m'}$ have $P$", "for all maximal ideals $\\mathfrak m' \\subset \\Lambda$", "lying over $\\mathfrak m$ in $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p \\subset R$ be a prime. Then the fibre over", "$\\mathfrak p$ is the ring $\\Lambda \\otimes_R \\kappa(\\mathfrak p)$", "whose spectrum maps bijectively onto the subset of $\\Spec(\\Lambda)$", "consisting of primes $\\mathfrak q$ lying over $\\mathfrak p$, see", "Algebra, Remark \\ref{algebra-remark-fundamental-diagram}.", "For such a prime $\\mathfrak q$ choose a maximal", "ideal $\\mathfrak q \\subset \\mathfrak m'$ and set", "$\\mathfrak m = R \\cap \\mathfrak m'$.", "Then $\\mathfrak p \\subset \\mathfrak m$ and we have", "$$", "(\\Lambda \\otimes_R \\kappa(\\mathfrak p))_\\mathfrak q \\cong", "(\\Lambda_{\\mathfrak m'} \\otimes_{R_\\mathfrak m}", "\\kappa(\\mathfrak p))_\\mathfrak q", "$$", "as $\\kappa(\\mathfrak q)$-algebras. Thus (1), (2), and (3) are equivalent", "because by (B) we can check property $P$ on local rings." ], "refs": [ "algebra-remark-fundamental-diagram" ], "ref_ids": [ 1558 ] } ], "ref_ids": [] }, { "id": 10096, "type": "theorem", "label": "more-algebra-lemma-P-ring-goes-up-quasi-finite", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-P-ring-goes-up-quasi-finite", "contents": [ "Let $R \\to R'$ be a finite type map of Noetherian rings and let", "$$", "\\xymatrix{", "\\mathfrak q' \\ar[r] & \\mathfrak p' \\ar[r] & R' \\\\", "\\mathfrak q \\ar[r] \\ar@{-}[u] &", "\\mathfrak p \\ar[r] \\ar@{-}[u] & R \\ar[u]", "}", "$$", "be primes. Assume $R \\to R'$ is quasi-finite at $\\mathfrak p'$.", "Assume $P$ satisfies (A) and (B).", "\\begin{enumerate}", "\\item If $\\kappa(\\mathfrak q) \\to", "R_\\mathfrak p^\\wedge \\otimes_R \\kappa(\\mathfrak q)$", "has $P$, then", "$\\kappa(\\mathfrak q') \\to R'_{\\mathfrak p'} \\otimes_{R'} \\kappa(\\mathfrak q')$", "has $P$.", "\\item If the formal fibres of $R_\\mathfrak p$ have $P$,", "then the formal fibres of $R'_{\\mathfrak p'}$ have $P$.", "\\item If $R \\to R'$ is quasi-finite and $R$ is a $P$-ring, then $R'$ is", "a $P$-ring.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3).", "Assume $P$ holds for", "$\\kappa(\\mathfrak q) \\to R_\\mathfrak p^\\wedge \\otimes_R \\kappa(\\mathfrak q)$.", "By Algebra, Lemma \\ref{algebra-lemma-completion-at-quasi-finite-prime}", "we see that", "$$", "R_\\mathfrak p^\\wedge \\otimes_R R'", "=", "(R'_{\\mathfrak p'})^\\wedge \\times B", "$$", "for some $R_\\mathfrak p^\\wedge$-algebra $B$. Hence", "$R'_{\\mathfrak p'} \\to (R'_{\\mathfrak p'})^\\wedge$ is a factor of", "a base change of the map $R_\\mathfrak p \\to R_\\mathfrak p^\\wedge$.", "It follows that $(R'_{\\mathfrak p'})^\\wedge \\otimes_{R'} \\kappa(\\mathfrak q')$", "is a factor of", "$$", "R_\\mathfrak p^\\wedge \\otimes_R R' \\otimes_{R'} \\kappa(\\mathfrak q') =", "R_\\mathfrak p^\\wedge \\otimes_R \\kappa(\\mathfrak q)", "\\otimes_{\\kappa(\\mathfrak q)} \\kappa(\\mathfrak q').", "$$", "Thus the result follows from the assumptions on $P$." ], "refs": [ "algebra-lemma-completion-at-quasi-finite-prime" ], "ref_ids": [ 1070 ] } ], "ref_ids": [] }, { "id": 10097, "type": "theorem", "label": "more-algebra-lemma-check-P-ring-maximal-ideals", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-check-P-ring-maximal-ideals", "contents": [ "Let $R$ be a Noetherian ring. Assume $P$ satisfies (C) and (D).", "Then $R$ is a $P$-ring if and only if the formal fibres of", "$R_\\mathfrak m$ have $P$ for every", "maximal ideal $\\mathfrak m$ of $R$." ], "refs": [], "proofs": [ { "contents": [ "Assume the formal fibres of $R_\\mathfrak m$ have $P$ for all", "maximal ideals $\\mathfrak m$ of $R$. Let $\\mathfrak p$ be a prime of", "$R$ and choose a maximal ideal $\\mathfrak p \\subset \\mathfrak m$.", "Since $R_\\mathfrak m \\to R_\\mathfrak m^\\wedge$ is faithfully flat", "we can choose a prime $\\mathfrak p'$ if $R_\\mathfrak m^\\wedge$", "lying over $\\mathfrak pR_\\mathfrak m$. Consider the commutative diagram", "$$", "\\xymatrix{", "R_\\mathfrak m^\\wedge \\ar[r] &", "(R_\\mathfrak m^\\wedge)_{\\mathfrak p'} \\ar[r] &", "(R_\\mathfrak m^\\wedge)_{\\mathfrak p'}^\\wedge", "\\\\", "R_\\mathfrak m \\ar[u] \\ar[r] & R_\\mathfrak p \\ar[u] \\ar[r] &", "R_\\mathfrak p^\\wedge \\ar[u]", "}", "$$", "By assumption the fibres of the ring map", "$R_\\mathfrak m \\to R_\\mathfrak m^\\wedge$ have $P$.", "By Proposition \\ref{proposition-Noetherian-complete-G-ring}", "$(R_\\mathfrak m^\\wedge)_{\\mathfrak p'} \\to", "(R_\\mathfrak m^\\wedge)_{\\mathfrak p'}^\\wedge$ is regular.", "The localization", "$R_\\mathfrak m^\\wedge \\to (R_\\mathfrak m^\\wedge)_{\\mathfrak p'}$ is regular.", "Hence $R_\\mathfrak m^\\wedge \\to (R_\\mathfrak m^\\wedge)_{\\mathfrak p'}^\\wedge$", "is regular by Lemma \\ref{lemma-regular-composition}.", "Hence the fibres of", "$R_\\mathfrak m \\to (R_\\mathfrak m^\\wedge)_{\\mathfrak p'}^\\wedge$", "have $P$ by (C). Since", "$R_\\mathfrak m \\to (R_\\mathfrak m^\\wedge)_{\\mathfrak p'}^\\wedge$", "factors through the localization", "$R_\\mathfrak p$, also the fibres of", "$R_\\mathfrak p \\to (R_\\mathfrak m^\\wedge)_{\\mathfrak p'}^\\wedge$", "have $P$. Thus we may apply (D) to see that the fibres of", "$R_\\mathfrak p \\to R_\\mathfrak p^\\wedge$ have $P$." ], "refs": [ "more-algebra-proposition-Noetherian-complete-G-ring", "more-algebra-lemma-regular-composition" ], "ref_ids": [ 10580, 10036 ] } ], "ref_ids": [] }, { "id": 10098, "type": "theorem", "label": "more-algebra-lemma-map-P-ring-to-completion-P", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-map-P-ring-to-completion-P", "contents": [ "Let $A$ be a $P$-ring where $P$ satisfies (B) and (D).", "Let $I \\subset A$ be an ideal and let $A^\\wedge$ be the completion of $A$", "with respect to $I$. Then the fibres of $A \\to A^\\wedge$ have $P$." ], "refs": [], "proofs": [ { "contents": [ "The ring map $A \\to A^\\wedge$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}.", "The ring $A^\\wedge$ is Noetherian by", "Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian-Noetherian}.", "Thus it suffices to check the third condition of", "Lemma \\ref{lemma-P-local}.", "Let $\\mathfrak m' \\subset A^\\wedge$ be a maximal ideal lying over", "$\\mathfrak m \\subset A$.", "By Algebra, Lemma \\ref{algebra-lemma-radical-completion}", "we have $IA^\\wedge \\subset \\mathfrak m'$.", "Since $A^\\wedge/IA^\\wedge = A/I$ we see that", "$I \\subset \\mathfrak m$, $\\mathfrak m/I = \\mathfrak m'/IA^\\wedge$, and", "$A/\\mathfrak m = A^\\wedge/\\mathfrak m'$. Since $A^\\wedge/\\mathfrak m'$", "is a field, we conclude that $\\mathfrak m$ is a maximal ideal as well.", "Then $A_\\mathfrak m \\to A^\\wedge_{\\mathfrak m'}$ is a flat local", "ring homomorphism of Noetherian local rings", "which identifies residue fields and such that", "$\\mathfrak m A^\\wedge_{\\mathfrak m'} = \\mathfrak m'A^\\wedge_{\\mathfrak m'}$.", "Thus it induces an isomorphism on complete local rings, see", "Lemma \\ref{lemma-flat-unramified}.", "Let $(A_\\mathfrak m)^\\wedge$ be the completion of $A_\\mathfrak m$", "with respect to its maximal ideal.", "The ring map", "$$", "(A^\\wedge)_{\\mathfrak m'} \\to", "((A^\\wedge)_{\\mathfrak m'})^\\wedge = (A_\\mathfrak m)^\\wedge", "$$", "is faithfully flat (Algebra, Lemma", "\\ref{algebra-lemma-completion-faithfully-flat}). Thus we can apply", "(D) to the ring maps", "$$", "A_\\mathfrak m \\to (A^\\wedge)_{\\mathfrak m'} \\to (A_\\mathfrak m)^\\wedge", "$$", "to conclude because the fibres of", "$A_\\mathfrak m \\to (A_\\mathfrak m)^\\wedge$", "have $P$ as $A$ is a $P$-ring." ], "refs": [ "algebra-lemma-completion-flat", "algebra-lemma-completion-Noetherian-Noetherian", "more-algebra-lemma-P-local", "algebra-lemma-radical-completion", "more-algebra-lemma-flat-unramified", "algebra-lemma-completion-faithfully-flat" ], "ref_ids": [ 870, 874, 10095, 862, 10050, 871 ] } ], "ref_ids": [] }, { "id": 10099, "type": "theorem", "label": "more-algebra-lemma-henselization-pair-P-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-pair-P-ring", "contents": [ "\\begin{slogan}", "Henselization of a ring inherits good properties of formal fibers", "\\end{slogan}", "Let $A$ be a $P$-ring where $P$ satisfies (B), (C), (D), and (E).", "Let $I \\subset A$ be an ideal. Let $(A^h, I^h)$ be the henselization", "of the pair $(A, I)$, see Lemma \\ref{lemma-henselization}.", "Then $A^h$ is a $P$-ring." ], "refs": [ "more-algebra-lemma-henselization" ], "proofs": [ { "contents": [ "Let $\\mathfrak m^h \\subset A^h$ be a maximal ideal. We have to show", "that the fibres of $A^h_{\\mathfrak m^h} \\to (A^h_{\\mathfrak m^h})^\\wedge$", "have $P$, see Lemma \\ref{lemma-check-P-ring-maximal-ideals}.", "Let $\\mathfrak m$ be the inverse image of $\\mathfrak m^h$ in $A$.", "Note that $I^h \\subset \\mathfrak m^h$ and hence $I \\subset \\mathfrak m$", "as $(A^h, I^h)$ is a henselian pair. Recall that $A^h$ is Noetherian,", "$I^h = IA^h$, and that $A \\to A^h$ induces an isomorphism on", "$I$-adic completions, see", "Lemma \\ref{lemma-henselization-Noetherian-pair}.", "Then the local homomorphism of Noetherian local rings", "$$", "A_\\mathfrak m \\to A^h_{\\mathfrak m^h}", "$$", "induces an isomorphism on completions at maximal ideals by", "Lemma \\ref{lemma-flat-unramified} (details omitted).", "Let $\\mathfrak q^h$ be a prime of $A^h_{\\mathfrak m^h}$ lying", "over $\\mathfrak q \\subset A_\\mathfrak m$.", "Set $\\mathfrak q_1 = \\mathfrak q^h$", "and let $\\mathfrak q_2, \\ldots, \\mathfrak q_t$", "be the other primes of $A^h$ lying over $\\mathfrak q$, so that", "$A^h \\otimes_A \\kappa(\\mathfrak q) =", "\\prod\\nolimits_{i = 1, \\ldots, t} \\kappa(\\mathfrak q_i)$, see", "Lemma \\ref{lemma-filtered-colimit-etale-noetherian-fibres}.", "Using that $(A^h)_{\\mathfrak m^h}^\\wedge = (A_\\mathfrak m)^\\wedge$", "as discussed above we see", "$$", "\\prod\\nolimits_{i = 1, \\ldots, t}", "(A^h_{\\mathfrak m^h})^\\wedge \\otimes_{A^h_{\\mathfrak m^h}}", "\\kappa(\\mathfrak q_i) =", "(A^h_{\\mathfrak m^h})^\\wedge \\otimes_{A^h_{\\mathfrak m^h}}", "(A^h_{\\mathfrak m^h} \\otimes_{A_{\\mathfrak m}} \\kappa(\\mathfrak q)) =", "(A_{\\mathfrak m})^\\wedge \\otimes_{A_{\\mathfrak m}} \\kappa(\\mathfrak q)", "$$", "Hence, looking at local rings and using (B), we see that", "$$", "\\kappa(\\mathfrak q) \\longrightarrow", "(A^h_{\\mathfrak m^h})^\\wedge \\otimes_{A^h_{\\mathfrak m^h}}", "\\kappa(\\mathfrak q^h)", "$$", "has $P$ as", "$\\kappa(\\mathfrak q) \\to", "(A_\\mathfrak m)^\\wedge \\otimes_{A_\\mathfrak m} \\kappa(\\mathfrak q)$", "does by assumption on $A$. Since $\\kappa(\\mathfrak q^h)/\\kappa(\\mathfrak q)$", "is separable algebraic, by (E) we find that", "$\\kappa(\\mathfrak q^h) \\to", "(A^h_{\\mathfrak m^h})^\\wedge \\otimes_{A^h_{\\mathfrak m^h}}", "\\kappa(\\mathfrak q^h)$ has $P$ as desired." ], "refs": [ "more-algebra-lemma-check-P-ring-maximal-ideals", "more-algebra-lemma-henselization-Noetherian-pair", "more-algebra-lemma-flat-unramified", "more-algebra-lemma-filtered-colimit-etale-noetherian-fibres" ], "ref_ids": [ 10097, 9874, 10050, 10066 ] } ], "ref_ids": [ 9871 ] }, { "id": 10100, "type": "theorem", "label": "more-algebra-lemma-henselization-P-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-P-ring", "contents": [ "Let $R$ be a Noetherian local ring which is a $P$-ring where $P$", "satisfies (B), (C), (D), and (E). Then the henselization $R^h$", "and the strict henselization $R^{sh}$ are $P$-rings." ], "refs": [], "proofs": [ { "contents": [ "We have seen this for the henselization in", "Lemma \\ref{lemma-henselization-pair-P-ring}.", "To prove it for the strict henselization, it suffices to", "show that the formal fibres of $R^{sh}$ have $P$, see", "Lemma \\ref{lemma-check-P-ring-maximal-ideals}.", "Let $\\mathfrak r \\subset R^{sh}$ be a prime and set", "$\\mathfrak p = R \\cap \\mathfrak r$. Set $\\mathfrak r_1 = \\mathfrak r$", "and let $\\mathfrak r_2, \\ldots, \\mathfrak r_s$", "be the other primes of $R^{sh}$ lying over $\\mathfrak p$, so that", "$R^{sh} \\otimes_R \\kappa(\\mathfrak p) =", "\\prod\\nolimits_{i = 1, \\ldots, s} \\kappa(\\mathfrak r_i)$, see", "Lemma \\ref{lemma-fibres-henselization}.", "Then we see that", "$$", "\\prod\\nolimits_{i = 1, \\ldots, t}", "(R^{sh})^\\wedge \\otimes_{R^{sh}} \\kappa(\\mathfrak r_i) =", "(R^{sh})^\\wedge \\otimes_{R^{sh}} (R^{sh} \\otimes_R \\kappa(\\mathfrak p)) =", "(R^{sh})^\\wedge \\otimes_R \\kappa(\\mathfrak p)", "$$", "Note that $R^\\wedge \\to (R^{sh})^\\wedge$ is formally smooth", "in the $\\mathfrak m_{(R^{sh})^\\wedge}$-adic topology, see", "Lemma \\ref{lemma-henselization-noetherian}.", "Hence $R^\\wedge \\to (R^{sh})^\\wedge$ is regular by", "Proposition \\ref{proposition-fs-regular}.", "We conclude that property $P$ holds for", "$\\kappa(\\mathfrak p) \\to (R^{sh})^\\wedge \\otimes_R \\kappa(\\mathfrak p)$", "by (C) and our assumption on $R$. Using property (B), using the", "decomposition above, and looking", "at local rings we conclude that property $P$ holds for", "$\\kappa(\\mathfrak p) \\to (R^{sh})^\\wedge \\otimes_{R^{sh}} \\kappa(\\mathfrak r)$.", "Since $\\kappa(\\mathfrak r)/\\kappa(\\mathfrak p)$ is separable algebraic,", "it follows from (E) that $P$ holds for", "$\\kappa(\\mathfrak r) \\to (R^{sh})^\\wedge \\otimes_{R^{sh}} \\kappa(\\mathfrak r)$." ], "refs": [ "more-algebra-lemma-henselization-pair-P-ring", "more-algebra-lemma-check-P-ring-maximal-ideals", "more-algebra-lemma-fibres-henselization", "more-algebra-lemma-henselization-noetherian", "more-algebra-proposition-fs-regular" ], "ref_ids": [ 10099, 10097, 10067, 10057, 10579 ] } ], "ref_ids": [] }, { "id": 10101, "type": "theorem", "label": "more-algebra-lemma-formal-fibres-reduced", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-formal-fibres-reduced", "contents": [ "Properties (A), (B), (C), (D), and (E) hold for", "$P(k \\to R) =$``$R$ is geometrically reduced over $k$''." ], "refs": [], "proofs": [ { "contents": [ "Part (A) follows from the definition of geometrically reduced", "algebras (Algebra, Definition \\ref{algebra-definition-geometrically-reduced}).", "Part (B) follows too: a ring is reduced if and only if", "all local rings are reduced.", "Part (C). This follows from Lemma \\ref{lemma-reduced-goes-up}.", "Part (D). This follows from Algebra, Lemma \\ref{algebra-lemma-descent-reduced}.", "Part (E). This follows from Algebra, Lemma", "\\ref{algebra-lemma-geometrically-reduced-over-separable-algebraic}." ], "refs": [ "algebra-definition-geometrically-reduced", "more-algebra-lemma-reduced-goes-up", "algebra-lemma-descent-reduced", "algebra-lemma-geometrically-reduced-over-separable-algebraic" ], "ref_ids": [ 1461, 10040, 1371, 568 ] } ], "ref_ids": [] }, { "id": 10102, "type": "theorem", "label": "more-algebra-lemma-formal-fibres-normal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-formal-fibres-normal", "contents": [ "Properties (A), (B), (C), (D), and (E) hold for", "$P(k \\to R) =$``$R$ is geometrically normal over $k$''." ], "refs": [], "proofs": [ { "contents": [ "Part (A) follows from the definition of geometrically normal", "algebras (Algebra, Definition \\ref{algebra-definition-geometrically-normal}).", "Part (B) follows too: a ring is normal if and only if all of its", "local rings are normal.", "Part (C). This follows from Lemma \\ref{lemma-normal-goes-up}.", "Part (D). This follows from Algebra, Lemma \\ref{algebra-lemma-descent-normal}.", "Part (E). This follows from Algebra, Lemma", "\\ref{algebra-lemma-geometrically-normal-over-separable-algebraic}." ], "refs": [ "algebra-definition-geometrically-normal", "more-algebra-lemma-normal-goes-up", "algebra-lemma-descent-normal", "algebra-lemma-geometrically-normal-over-separable-algebraic" ], "ref_ids": [ 1554, 10041, 1372, 1381 ] } ], "ref_ids": [] }, { "id": 10103, "type": "theorem", "label": "more-algebra-lemma-formal-fibres-Sk", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-formal-fibres-Sk", "contents": [ "Fix $n \\geq 1$. Properties (A), (B), (C), (D), and (E) hold for", "$P(k \\to R) =$``$R$ has $(S_n)$''." ], "refs": [], "proofs": [ { "contents": [ "Let $k \\to R$ be a ring map where $k$ is a field and $R$ a Noetherian", "ring. Let $k \\subset k'$ be a finitely generated field extension.", "Then the fibres of the ring map $R \\to R \\otimes_k k'$ are", "Cohen-Macaulay by Algebra, Lemma \\ref{algebra-lemma-tensor-fields-CM}.", "Hence we may apply Algebra, Lemma \\ref{algebra-lemma-Sk-goes-up}", "to the ring map $R \\to R \\otimes_k k'$ to see that if $R$ has $(S_n)$", "so does $R \\otimes_k k'$. This proves (A).", "Part (B) follows too: a Noetherian rings has $(S_n)$ if and only if", "all of its local rings have $(S_n)$.", "Part (C). This follows from", "Algebra, Lemma \\ref{algebra-lemma-Sk-goes-up}", "as the fibres of a regular homomorphism are regular and in particular", "Cohen-Macaulay.", "Part (D). This follows from", "Algebra, Lemma \\ref{algebra-lemma-descent-Sk}.", "Part (E). This is immediate as the condition does not refer to", "the ground field." ], "refs": [ "algebra-lemma-tensor-fields-CM", "algebra-lemma-Sk-goes-up", "algebra-lemma-Sk-goes-up", "algebra-lemma-descent-Sk" ], "ref_ids": [ 1387, 1363, 1363, 1374 ] } ], "ref_ids": [] }, { "id": 10104, "type": "theorem", "label": "more-algebra-lemma-formal-fibres-CM", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-formal-fibres-CM", "contents": [ "Properties (A), (B), (C), (D), and (E) hold for", "$P(k \\to R) =$``$R$ is Cohen-Macaulay''." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-formal-fibres-Sk}", "and the fact that a Noetherian ring is Cohen-Macaulay if", "and only if it satisfies conditions $(S_n)$ for all $n$." ], "refs": [ "more-algebra-lemma-formal-fibres-Sk" ], "ref_ids": [ 10103 ] } ], "ref_ids": [] }, { "id": 10105, "type": "theorem", "label": "more-algebra-lemma-formal-fibres-Rk", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-formal-fibres-Rk", "contents": [ "Fix $n \\geq 0$. Properties (A), (B), (C), (D), and (E) hold for", "$P(k \\to R) =$``$R \\otimes_k k'$ has $(R_n)$ for all finite", "extensions $k'/k$''." ], "refs": [], "proofs": [ { "contents": [ "Let $k \\to R$ be a ring map where $k$ is a field and $R$ a Noetherian", "ring. Assume $P(k \\to R)$ is true.", "Let $k \\subset K$ be a finitely generated field extension.", "By Algebra, Lemma \\ref{algebra-lemma-make-separable} we can find a diagram", "$$", "\\xymatrix{", "K \\ar[r] & K' \\\\", "k \\ar[u] \\ar[r] & k' \\ar[u]", "}", "$$", "where $k \\subset k'$, $K \\subset K'$ are finite purely inseparable field", "extensions such that $k' \\subset K'$ is separable. By", "Algebra, Lemma \\ref{algebra-lemma-localization-smooth-separable}", "there exists a smooth $k'$-algebra $B$ such that $K'$ is the", "fraction field of $B$. Now we can argue as follows:", "Step 1: $R \\otimes_k k'$ satisfies $(S_n)$ because we assumed $P$", "for $k \\to R$.", "Step 2: $R \\otimes_k k' \\to R \\otimes_k k' \\otimes_{k'} B$ is a", "smooth ring map (Algebra, Lemma \\ref{algebra-lemma-base-change-smooth})", "and we conclude $R \\otimes_k k' \\otimes_{k'} B$ satisfies $(S_n)$", "by Algebra, Lemma \\ref{algebra-lemma-Rk-goes-up}", "(and using Algebra, Lemma \\ref{algebra-lemma-characterize-smooth-over-field}", "to see that the hypotheses are satisfied).", "Step 3. $R \\otimes_k k' \\otimes_{k'} K' = R \\otimes_k K'$ satisfies", "$(R_n)$ as it is a localization of a ring having $(R_n)$.", "Step 4. Finally $R \\otimes_k K$ satisfies $(R_n)$ by descent of", "$(R_n)$ along the faithfully flat ring map", "$K \\otimes_k A \\to K' \\otimes_k A$", "(Algebra, Lemma \\ref{algebra-lemma-descent-Rk}).", "This proves (A).", "Part (B) follows too: a Noetherian ring has $(R_n)$ if and only if", "all of its local rings have $(R_n)$.", "Part (C). This follows from Algebra, Lemma \\ref{algebra-lemma-Rk-goes-up}", "as the fibres of a regular homomorphism are regular (small detail omitted).", "Part (D). This follows from Algebra, Lemma \\ref{algebra-lemma-descent-Rk}", "(small detail omitted).", "\\medskip\\noindent", "Part (E). Let $l/k$ be a separable algebraic extension of fields and", "let $l \\to R$ be a ring map with $R$ Noetherian. Assume that", "$k \\to R$ has $P$. We have to show that $l \\to R$ has $P$.", "Let $l'/l$ be a finite extension. First observe that there exists a", "finite subextension $l/m/k$ and a finite extension $m'/m$", "such that $l' = l \\otimes_m m'$. Then $R \\otimes_l l' = R \\otimes_m m'$.", "Hence it suffices to prove that $m \\to R$ has property $P$, i.e.,", "we may assume that $l/k$ is finite. If $l/k$ is finite, then $l'/k$", "is finite and we see that", "$$", "l' \\otimes_l R = (l' \\otimes_k R) \\otimes_{l \\otimes_k l} l", "$$", "is a localization", "(by Algebra, Lemma \\ref{algebra-lemma-separable-algebraic-diagonal})", "of the Noetherian ring $l' \\otimes_k R$ which has property $(R_n)$", "by assumption $P$ for $k \\to R$. This proves that $l' \\otimes_l R$", "has property $(R_n)$ as desired." ], "refs": [ "algebra-lemma-make-separable", "algebra-lemma-localization-smooth-separable", "algebra-lemma-base-change-smooth", "algebra-lemma-Rk-goes-up", "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-descent-Rk", "algebra-lemma-Rk-goes-up", "algebra-lemma-descent-Rk", "algebra-lemma-separable-algebraic-diagonal" ], "ref_ids": [ 573, 1322, 1191, 1364, 1223, 1375, 1364, 1375, 567 ] } ], "ref_ids": [] }, { "id": 10106, "type": "theorem", "label": "more-algebra-lemma-finite-type-over-excellent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finite-type-over-excellent", "contents": [ "Any localization of a finite type ring over a (quasi-)excellent ring", "is (quasi-)excellent." ], "refs": [], "proofs": [ { "contents": [ "For finite type algebras this follows from the definitions for", "the properties J-2 and universally catenary. For G-rings, see", "Proposition \\ref{proposition-finite-type-over-G-ring}. We omit", "the proof that localization preserves (quasi-)excellency." ], "refs": [ "more-algebra-proposition-finite-type-over-G-ring" ], "ref_ids": [ 10581 ] } ], "ref_ids": [] }, { "id": 10107, "type": "theorem", "label": "more-algebra-lemma-Nagata-local-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Nagata-local-ring", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "The following are equivalent", "\\begin{enumerate}", "\\item $A$ is Nagata, and", "\\item the formal fibres of $A$ are geometrically reduced.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (2). By", "Algebra, Lemma \\ref{algebra-lemma-local-nagata-and-analytically-unramified}", "we have to show that if $A \\to B$ is finite, $B$ is a domain,", "and $\\mathfrak m' \\subset B$ is a maximal ideal, then $B_{\\mathfrak m'}$", "is analytically unramified.", "Combining Lemmas \\ref{lemma-formal-fibres-reduced} and", "\\ref{lemma-check-P-ring-maximal-ideals} and", "Proposition \\ref{proposition-finite-type-over-P-ring}", "we see that the formal fibres of $B_{\\mathfrak m'}$ are", "geometrically reduced. In particular", "$B_{\\mathfrak m'}^\\wedge \\otimes_B L$ is reduced", "where $L$ is the fraction field of $B$.", "It follows that $B_{\\mathfrak m'}^\\wedge$ is reduced, i.e.,", "$B_{\\mathfrak m'}$ is analytically unramified.", "\\medskip\\noindent", "Assume (1). Let $\\mathfrak q \\subset A$ be a prime ideal", "and let $\\kappa(\\mathfrak q) \\subset K$ be a finite extension.", "We have to show that $A^\\wedge \\otimes_A K$ is reduced.", "Let $A/\\mathfrak q \\subset B \\subset K$ be a local subring", "finite over $A$ whose fraction field is $K$.", "To construct $B$ choose $x_1, \\ldots, x_n \\in K$", "which generate $K$ over $\\kappa(\\mathfrak q)$", "and which satisfy monic polynomials", "$P_i(T) = T^{d_i} + a_{i, 1} T^{d_i - 1} + \\ldots + a_{i, d_i} = 0$", "with $a_{i, j} \\in \\mathfrak m$. Then let $B$ be the $A$-subalgebra", "of $K$ generated by $x_1, \\ldots, x_n$. (For more details see", "the proof of Algebra, Lemma", "\\ref{algebra-lemma-local-nagata-and-analytically-unramified}.)", "Then", "$$", "A^\\wedge \\otimes_A K =", "(A^\\wedge \\otimes_A B)_\\mathfrak q =", "B^\\wedge_\\mathfrak q", "$$", "Since $B^\\wedge$ is reduced by Algebra, Lemma", "\\ref{algebra-lemma-local-nagata-and-analytically-unramified}", "the proof is complete." ], "refs": [ "algebra-lemma-local-nagata-and-analytically-unramified", "more-algebra-lemma-formal-fibres-reduced", "more-algebra-lemma-check-P-ring-maximal-ideals", "more-algebra-proposition-finite-type-over-P-ring", "algebra-lemma-local-nagata-and-analytically-unramified", "algebra-lemma-local-nagata-and-analytically-unramified" ], "ref_ids": [ 1358, 10101, 10097, 10583, 1358, 1358 ] } ], "ref_ids": [] }, { "id": 10108, "type": "theorem", "label": "more-algebra-lemma-quasi-excellent-nagata", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-quasi-excellent-nagata", "contents": [ "A quasi-excellent ring is Nagata." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be quasi-excellent.", "Using that a finite type algebra over $R$ is quasi-excellent", "(Lemma \\ref{lemma-finite-type-over-excellent}) we see that", "it suffices to show that any quasi-excellent domain is N-1, see", "Algebra, Lemma \\ref{algebra-lemma-check-universally-japanese}.", "Applying Algebra, Lemma \\ref{algebra-lemma-characterize-N-1}", "(and using that a quasi-excellent ring is J-2) we reduce", "to showing that a quasi-excellent local domain $R$ is N-1.", "As $R \\to R^\\wedge$ is regular we see that $R^\\wedge$", "is reduced by Lemma \\ref{lemma-reduced-goes-up}.", "In other words, $R$ is analytically unramified.", "Hence $R$ is N-1 by", "Algebra, Lemma \\ref{algebra-lemma-analytically-unramified-easy}." ], "refs": [ "more-algebra-lemma-finite-type-over-excellent", "algebra-lemma-check-universally-japanese", "algebra-lemma-characterize-N-1", "more-algebra-lemma-reduced-goes-up", "algebra-lemma-analytically-unramified-easy" ], "ref_ids": [ 10106, 1348, 1344, 10040, 1354 ] } ], "ref_ids": [] }, { "id": 10109, "type": "theorem", "label": "more-algebra-lemma-completion-normal-local-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-completion-normal-local-ring", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring. If $A$ is normal", "and the formal fibres of $A$ are normal (for example if $A$", "is excellent or quasi-excellent), then $A^\\wedge$ is normal." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Algebra, Lemma \\ref{algebra-lemma-normal-goes-up-noetherian}." ], "refs": [ "algebra-lemma-normal-goes-up-noetherian" ], "ref_ids": [ 1367 ] } ], "ref_ids": [] }, { "id": 10110, "type": "theorem", "label": "more-algebra-lemma-injective-abelian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-injective-abelian", "contents": [ "An abelian group $J$ is an injective object in", "the category of abelian groups if and only if $J$", "is divisible." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $J$ is not divisible. Then there exists", "an $x \\in J$ and $n \\in \\mathbf{N}$ such that there", "is no $y \\in J$ with $n y = x$. Then the morphism", "$\\mathbf{Z} \\to J$, $m \\mapsto mx$ does not extend", "to $\\frac{1}{n}\\mathbf{Z} \\supset \\mathbf{Z}$. Hence", "$J$ is not injective.", "\\medskip\\noindent", "Let $A \\subset B$ be abelian groups.", "Assume that $J$ is a divisible abelian group.", "Let $\\varphi : A \\to J$ be a morphism.", "Consider the set of homomorphisms $\\varphi' : A' \\to J$", "with $A \\subset A' \\subset B$ and $\\varphi'|_A = \\varphi$.", "Define $(A', \\varphi') \\geq (A'', \\varphi'')$ if", "and only if $A' \\supset A''$ and $\\varphi'|_{A''} = \\varphi''$.", "If $(A_i, \\varphi_i)_{i \\in I}$ is a totally", "ordered collection of such pairs, then we obtain a map", "$\\bigcup_{i \\in I} A_i \\to J$ defined by $a \\in A_i$", "maps to $\\varphi_i(a)$. Thus Zorn's lemma applies.", "To conclude we have to show that if the pair", "$(A', \\varphi')$ is maximal then $A' = B$.", "In other words, it suffices to show, given", "any subgroup $A \\subset B$, $A \\not = B$ and", "any $\\varphi : A \\to J$, then we can find", "$\\varphi' : A' \\to J$ with $A \\subset A' \\subset B$", "such that (a) the inclusion $A \\subset A'$ is strict, and", "(b) the morphism $\\varphi'$ extends $\\varphi$.", "\\medskip\\noindent", "To prove this, pick $x \\in B$, $x \\not \\in A$.", "If there exists no $n\\in \\mathbf{N}$ such that", "$nx \\in A$, then $A \\oplus \\mathbf{Z} \\cong A + \\mathbf{Z}x$.", "Hence we can extend $\\varphi$ to $A' = A + \\mathbf{Z}x$", "by using $\\varphi$ on $A$ and mapping $x$ to zero for example.", "If there does exist an $n \\in \\mathbf{N}$ such that", "$nx \\in A$, then let $n$ be the minimal such integer.", "Let $z \\in J$ be an element such that $nz = \\varphi(nx)$.", "Define a morphism $\\tilde\\varphi : A \\oplus \\mathbf{Z} \\to J$ by", "$(a, m) \\mapsto \\varphi(a) + mz$. By our choice of", "$z$ the kernel of $\\tilde \\varphi$ contains the kernel", "of the map $A \\oplus \\mathbf{Z} \\to B$,", "$(a, m) \\mapsto a + mx$. Hence $\\tilde \\varphi$ factors", "through the image $A' = A + \\mathbf{Z}x$, and this extends the morphism", "$\\varphi$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10111, "type": "theorem", "label": "more-algebra-lemma-relation-ext-ext", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-relation-ext-ext", "contents": [ "Let $R$ be a ring. Let $\\mathcal{A}$ be the abelian category of", "$R$-modules. There is a canonical isomorphism", "$\\Ext_\\mathcal{A}(M, N) = \\Ext^1_R(M, N)$", "compatible with the long exact sequences of", "Algebra, Lemmas \\ref{algebra-lemma-long-exact-seq-ext} and", "\\ref{algebra-lemma-reverse-long-exact-seq-ext}", "and the $6$-term exact sequences of", "Homology, Lemma \\ref{homology-lemma-six-term-sequence-ext}." ], "refs": [ "algebra-lemma-long-exact-seq-ext", "algebra-lemma-reverse-long-exact-seq-ext", "homology-lemma-six-term-sequence-ext" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 765, 766, 12032 ] }, { "id": 10112, "type": "theorem", "label": "more-algebra-lemma-characterize-injective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-characterize-injective", "contents": [ "Let $R$ be a ring. Let $J$ be an $R$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $J$ is injective,", "\\item $\\Ext^1_R(M, J) = 0$ for every $R$-module $M$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $0 \\to M'' \\to M' \\to M \\to 0$ be a short exact sequence of $R$-modules.", "Consider the long exact sequence", "$$", "\\begin{matrix}", "0", "\\to \\Hom_R(M, J)", "\\to \\Hom_R(M', J)", "\\to \\Hom_R(M'', J)", "\\\\", "\\phantom{0\\ }", "\\to \\Ext^1_R(M, J)", "\\to \\Ext^1_R(M', J)", "\\to \\Ext^1_R(M'', J)", "\\to \\ldots", "\\end{matrix}", "$$", "of Algebra, Lemma \\ref{algebra-lemma-reverse-long-exact-seq-ext}.", "Thus we see that (2) implies (1). Conversely, if $J$ is injective", "then the $\\Ext$-group is zero by", "Homology, Lemma \\ref{homology-lemma-characterize-injectives} and", "Lemma \\ref{lemma-relation-ext-ext}." ], "refs": [ "algebra-lemma-reverse-long-exact-seq-ext", "homology-lemma-characterize-injectives", "more-algebra-lemma-relation-ext-ext" ], "ref_ids": [ 766, 12112, 10111 ] } ], "ref_ids": [] }, { "id": 10113, "type": "theorem", "label": "more-algebra-lemma-characterize-injective-bis", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-characterize-injective-bis", "contents": [ "Let $R$ be a ring. Let $J$ be an $R$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $J$ is injective,", "\\item $\\Ext^1_R(R/I, J) = 0$ for every ideal $I \\subset R$, and", "\\item for an ideal $I \\subset R$ and module map $I \\to J$", "there exists an extension $R \\to J$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $I \\subset R$ is an ideal, then the short exact sequence", "$0 \\to I \\to R \\to R/I \\to 0$ gives an exact sequence", "$$", "\\Hom_R(R, J) \\to", "\\Hom_R(I, J) \\to", "\\Ext^1_R(R/I, J) \\to 0", "$$", "by Algebra, Lemma \\ref{algebra-lemma-reverse-long-exact-seq-ext}", "and the fact that $\\Ext^1_R(R, J) = 0$ as $R$ is projective", "(Algebra, Lemma \\ref{algebra-lemma-characterize-projective}).", "Thus (2) and (3) are equivalent. In this proof", "we will show that (1) $\\Leftrightarrow$ (3) which is known", "as Baer's criterion.", "\\medskip\\noindent", "Assume (1). Given a module map $I \\to J$ as in (3) we find", "the extension $R \\to J$ because the map", "$\\Hom_R(R, J) \\to \\Hom_R(I, J)$ is surjective", "by definition.", "\\medskip\\noindent", "Assume (3). Let $M \\subset N$ be an inclusion of $R$-modules.", "Let $\\varphi : M \\to J$ be a homomorphism. We will show that $\\varphi$", "extends to $N$ which finishes the proof of the lemma.", "Consider the set of homomorphisms $\\varphi' : M' \\to J$", "with $M \\subset M' \\subset N$ and $\\varphi'|_M = \\varphi$.", "Define $(M', \\varphi') \\geq (M'', \\varphi'')$ if", "and only if $M' \\supset M''$ and $\\varphi'|_{M''} = \\varphi''$.", "If $(M_i, \\varphi_i)_{i \\in I}$ is a totally", "ordered collection of such pairs, then we obtain a map", "$\\bigcup_{i \\in I} M_i \\to J$ defined by $a \\in M_i$", "maps to $\\varphi_i(a)$. Thus Zorn's lemma applies.", "To conclude we have to show that if the pair", "$(M', \\varphi')$ is maximal then $M' = N$.", "In other words, it suffices to show, given", "any subgroup $M \\subset N$, $M \\not = N$ and", "any $\\varphi : M \\to J$, then we can find", "$\\varphi' : M' \\to J$ with $M \\subset M' \\subset N$", "such that (a) the inclusion $M \\subset M'$ is strict, and", "(b) the morphism $\\varphi'$ extends $\\varphi$.", "\\medskip\\noindent", "To prove this, pick $x \\in N$, $x \\not \\in M$.", "Let $I = \\{f \\in R \\mid fx \\in M\\}$. This is an ideal of $R$.", "Define a homomorphism $\\psi : I \\to J$ by $f \\mapsto \\varphi(fx)$.", "Extend to a map $\\tilde\\psi : R \\to J$ which is possible by assumption (3).", "By our choice of $I$ the kernel of", "$M \\oplus R \\to J$, $(y, f) \\mapsto y - \\tilde\\psi(f)$", "contains the kernel of the map $M \\oplus R \\to N$,", "$(y, f) \\mapsto y + fx$. Hence this homomorphism factors", "through the image $M' = M + Rx$ and this extends the given homomorphism", "as desired." ], "refs": [ "algebra-lemma-reverse-long-exact-seq-ext", "algebra-lemma-characterize-projective" ], "ref_ids": [ 766, 789 ] } ], "ref_ids": [] }, { "id": 10114, "type": "theorem", "label": "more-algebra-lemma-vee-exact", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-vee-exact", "contents": [ "Let $R$ be a ring.", "The functor $M \\mapsto M^\\vee$ is exact." ], "refs": [], "proofs": [ { "contents": [ "This because $\\mathbf{Q}/\\mathbf{Z}$", "is an injective abelian group by Lemma \\ref{lemma-injective-abelian}." ], "refs": [ "more-algebra-lemma-injective-abelian" ], "ref_ids": [ 10110 ] } ], "ref_ids": [] }, { "id": 10115, "type": "theorem", "label": "more-algebra-lemma-ev-injective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ev-injective", "contents": [ "For any $R$-module $M$ the evaluation map", "$ev : M \\to (M^\\vee)^\\vee$ is injective." ], "refs": [], "proofs": [ { "contents": [ "You can check this using that $\\mathbf{Q}/\\mathbf{Z}$ is an injective", "abelian group. Namely, if $x \\in M$ is not zero, then let", "$M' \\subset M$ be the cyclic group it generates. There exists", "a nonzero map $M' \\to \\mathbf{Q}/\\mathbf{Z}$ which necessarily does", "not annihilate $x$. This extends to", "a map $\\varphi : M \\to \\mathbf{Q}/\\mathbf{Z}$", "and then $ev(x)(\\varphi) = \\varphi(x) \\not = 0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10116, "type": "theorem", "label": "more-algebra-lemma-JM-injective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-JM-injective", "contents": [ "Let $R$ be a ring. For every $R$-module $M$ the", "$R$-module $J(M)$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Note that $J(M) \\cong \\prod_{\\varphi \\in M^\\vee} R^\\vee$ as an $R$-module.", "As the product of injective modules is injective, it suffices to", "show that $R^\\vee$ is injective. For this we use that", "$$", "\\Hom_R(N, R^\\vee) =", "\\Hom_R(N, \\Hom_{\\mathbf{Z}}(R, \\mathbf{Q}/\\mathbf{Z})) =", "N^\\vee", "$$", "and the", "fact that $(-)^\\vee$ is an exact functor by Lemma", "\\ref{lemma-vee-exact}." ], "refs": [ "more-algebra-lemma-vee-exact" ], "ref_ids": [ 10114 ] } ], "ref_ids": [] }, { "id": 10117, "type": "theorem", "label": "more-algebra-lemma-injectives-modules", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-injectives-modules", "contents": [ "Let $R$ be a ring.", "The construction above defines a covariant functor", "$M \\mapsto (M \\to J(M))$ from the category of", "$R$-modules to the category of arrows of $R$-modules", "such that for every module $M$ the output", "$M \\to J(M)$ is an injective map of $M$ into", "an injective $R$-module $J(M)$." ], "refs": [], "proofs": [ { "contents": [ "Follows from the above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10118, "type": "theorem", "label": "more-algebra-lemma-K-injective-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-K-injective-flat", "contents": [ "Let $R \\to S$ be a flat ring map. If $I^\\bullet$ is a K-injective", "complex of $S$-modules, then $I^\\bullet$ is K-injective as a", "complex of $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$\\Hom_{K(R)}(M^\\bullet, I^\\bullet) =", "\\Hom_{K(S)}(M^\\bullet \\otimes_R S, I^\\bullet)$", "by Algebra, Lemma \\ref{algebra-lemma-adjoint-tensor-restrict}", "and the fact that tensoring with $S$ is exact." ], "refs": [ "algebra-lemma-adjoint-tensor-restrict" ], "ref_ids": [ 374 ] } ], "ref_ids": [] }, { "id": 10119, "type": "theorem", "label": "more-algebra-lemma-K-injective-epimorphism", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-K-injective-epimorphism", "contents": [ "Let $R \\to S$ be an epimorphism of rings. Let $I^\\bullet$ be a complex", "of $S$-modules. If $I^\\bullet$ is K-injective as a complex of", "$R$-modules, then $I^\\bullet$ is a K-injective complex of $S$-modules." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$\\Hom_{K(R)}(N^\\bullet, I^\\bullet) =", "\\Hom_{K(S)}(N^\\bullet, I^\\bullet)$ for any complex of $S$-modules", "$N^\\bullet$,", "see Algebra, Lemma \\ref{algebra-lemma-epimorphism-modules}." ], "refs": [ "algebra-lemma-epimorphism-modules" ], "ref_ids": [ 959 ] } ], "ref_ids": [] }, { "id": 10120, "type": "theorem", "label": "more-algebra-lemma-hom-K-injective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-hom-K-injective", "contents": [ "Let $A \\to B$ be a ring map. If $I^\\bullet$ is a K-injective complex of", "$A$-modules, then $\\Hom_A(B, I^\\bullet)$ is a K-injective complex of", "$B$-modules." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$\\Hom_{K(B)}(N^\\bullet, \\Hom_A(B, I^\\bullet)) =", "\\Hom_{K(A)}(N^\\bullet, I^\\bullet)$", "by Algebra, Lemma \\ref{algebra-lemma-adjoint-hom-restrict}." ], "refs": [ "algebra-lemma-adjoint-hom-restrict" ], "ref_ids": [ 375 ] } ], "ref_ids": [] }, { "id": 10121, "type": "theorem", "label": "more-algebra-lemma-derived-tor-homotopy", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-tor-homotopy", "contents": [ "Let $R$ be a ring.", "Let $P^\\bullet$ be a complex of $R$-modules.", "Let $\\alpha, \\beta : L^\\bullet \\to M^\\bullet$ be homotopy equivalent", "maps of complexes. Then $\\alpha$ and $\\beta$ induce homotopy equivalent", "maps", "$$", "\\text{Tot}(\\alpha \\otimes \\text{id}_P),", "\\text{Tot}(\\beta \\otimes \\text{id}_P) :", "\\text{Tot}(L^\\bullet \\otimes_R P^\\bullet)", "\\longrightarrow", "\\text{Tot}(M^\\bullet \\otimes_R P^\\bullet).", "$$", "In particular the construction", "$L^\\bullet \\mapsto \\text{Tot}(L^\\bullet \\otimes_R P^\\bullet)$", "defines an endo-functor of the homotopy category of complexes." ], "refs": [], "proofs": [ { "contents": [ "Say $\\alpha = \\beta + dh + hd$ for some homotopy $h$ defined by", "$h^n : L^n \\to M^{n - 1}$. Set", "$$", "H^n = \\bigoplus\\nolimits_{a + b = n} h^a \\otimes \\text{id}_{P^b} :", "\\bigoplus\\nolimits_{a + b = n} L^a \\otimes_R P^b", "\\longrightarrow", "\\bigoplus\\nolimits_{a + b = n} M^{a - 1} \\otimes_R P^b", "$$", "Then a straightforward computation shows that", "$$", "\\text{Tot}(\\alpha \\otimes \\text{id}_P) =", "\\text{Tot}(\\beta \\otimes \\text{id}_P) + dH + Hd", "$$", "as maps $\\text{Tot}(L^\\bullet \\otimes_R P^\\bullet) \\to", "\\text{Tot}(M^\\bullet \\otimes_R P^\\bullet)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10122, "type": "theorem", "label": "more-algebra-lemma-derived-tor-exact", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-tor-exact", "contents": [ "Let $R$ be a ring.", "Let $P^\\bullet$ be a complex of $R$-modules.", "The functors", "$$", "K(\\text{Mod}_R) \\longrightarrow K(\\text{Mod}_R), \\quad", "L^\\bullet \\longmapsto \\text{Tot}(P^\\bullet \\otimes_R L^\\bullet)", "$$", "and", "$$", "K(\\text{Mod}_R) \\longrightarrow K(\\text{Mod}_R), \\quad", "L^\\bullet \\longmapsto \\text{Tot}(L^\\bullet \\otimes_R P^\\bullet)", "$$", "are exact functors of triangulated categories." ], "refs": [], "proofs": [ { "contents": [ "This follows from Derived Categories, Remark", "\\ref{derived-remark-double-complex-as-tensor-product-of}." ], "refs": [ "derived-remark-double-complex-as-tensor-product-of" ], "ref_ids": [ 2014 ] } ], "ref_ids": [] }, { "id": 10123, "type": "theorem", "label": "more-algebra-lemma-K-flat-quasi-isomorphism", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-K-flat-quasi-isomorphism", "contents": [ "Let $R$ be a ring. Let $K^\\bullet$ be a K-flat complex.", "Then the functor", "$$", "K(\\text{Mod}_R) \\longrightarrow K(\\text{Mod}_R), \\quad", "L^\\bullet \\longmapsto \\text{Tot}(L^\\bullet \\otimes_R K^\\bullet)", "$$", "transforms quasi-isomorphisms into quasi-isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-derived-tor-exact}", "and the fact that quasi-isomorphisms in $K(\\text{Mod}_R)$ and", "$K(\\text{Mod}_A)$ are characterized by having acyclic cones." ], "refs": [ "more-algebra-lemma-derived-tor-exact" ], "ref_ids": [ 10122 ] } ], "ref_ids": [] }, { "id": 10124, "type": "theorem", "label": "more-algebra-lemma-base-change-K-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-K-flat", "contents": [ "Let $R \\to R'$ be a ring map. If $K^\\bullet$ is a K-flat complex", "of $R$-modules, then $K^\\bullet \\otimes_R R'$ is a K-flat complex", "of $R'$-modules." ], "refs": [], "proofs": [ { "contents": [ "Follows from the definitions and the fact that", "$(K^\\bullet \\otimes_R R') \\otimes_{R'} L^\\bullet =", "K^\\bullet \\otimes_R L^\\bullet$ for any complex", "$L^\\bullet$ of $R'$-modules." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10125, "type": "theorem", "label": "more-algebra-lemma-tensor-product-K-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-tensor-product-K-flat", "contents": [ "Let $R$ be a ring. If $K^\\bullet$, $L^\\bullet$ are K-flat complexes", "of $R$-modules, then $\\text{Tot}(K^\\bullet \\otimes_R L^\\bullet)$ is a", "K-flat complex of $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "Follows from the isomorphism", "$$", "\\text{Tot}(M^\\bullet \\otimes_R \\text{Tot}(K^\\bullet \\otimes_R L^\\bullet))", "=", "\\text{Tot}(\\text{Tot}(M^\\bullet \\otimes_R K^\\bullet) \\otimes_R L^\\bullet)", "$$", "and the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10126, "type": "theorem", "label": "more-algebra-lemma-K-flat-two-out-of-three", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-K-flat-two-out-of-three", "contents": [ "Let $R$ be a ring. Let $(K_1^\\bullet, K_2^\\bullet, K_3^\\bullet)$ be", "a distinguished triangle in $K(\\text{Mod}_R)$. If two out of three", "of $K_i^\\bullet$ are K-flat, so is the third." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-derived-tor-exact}", "and the fact that in a distinguished triangle in", "$K(\\text{Mod}_A)$ if two out of three are acyclic, so is the third." ], "refs": [ "more-algebra-lemma-derived-tor-exact" ], "ref_ids": [ 10122 ] } ], "ref_ids": [] }, { "id": 10127, "type": "theorem", "label": "more-algebra-lemma-K-flat-two-out-of-three-ses", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-K-flat-two-out-of-three-ses", "contents": [ "Let $R$ be a ring. Let", "$0 \\to K_1^\\bullet \\to K_2^\\bullet \\to K_3^\\bullet \\to 0$ be", "a short exact sequence of complexes. If $K_3^n$ is flat for", "all $n \\in \\mathbf{Z}$ and two out of three", "of $K_i^\\bullet$ are K-flat, so is the third." ], "refs": [], "proofs": [ { "contents": [ "Let $L^\\bullet$ be a complex of $R$-modules. Then", "$$", "0 \\to", "\\text{Tot}(L^\\bullet \\otimes_R K_1^\\bullet) \\to", "\\text{Tot}(L^\\bullet \\otimes_R K_2^\\bullet) \\to", "\\text{Tot}(L^\\bullet \\otimes_R K_3^\\bullet) \\to 0", "$$", "is a short exact sequence of complexes. Namely, for each", "$n, m$ the sequence of modules", "$0 \\to L^n \\otimes_R K_1^m \\to", "L^n \\otimes_R K_2^m \\to", "L^n \\otimes_R K_3^m \\to 0$", "is exact by Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}", "and the sequence of complexes is a direct sum of these.", "Thus the lemma follows from this", "and the fact that in a short exact sequence of complexes", "if two out of three are acyclic, so is the third." ], "refs": [ "algebra-lemma-flat-tor-zero" ], "ref_ids": [ 532 ] } ], "ref_ids": [] }, { "id": 10128, "type": "theorem", "label": "more-algebra-lemma-derived-tor-quasi-isomorphism", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-tor-quasi-isomorphism", "contents": [ "Let $R$ be a ring. Let $P^\\bullet$ be a bounded above complex of", "flat $R$-modules. Then $P^\\bullet$ is K-flat." ], "refs": [], "proofs": [ { "contents": [ "Let $L^\\bullet$ be an acyclic complex of $R$-modules.", "Let $\\xi \\in H^n(\\text{Tot}(L^\\bullet \\otimes_R P^\\bullet))$.", "We have to show that $\\xi = 0$.", "Since $\\text{Tot}^n(L^\\bullet \\otimes_R P^\\bullet)$ is a direct", "sum with terms $L^a \\otimes_R P^b$ we see that $\\xi$ comes from", "an element in $H^n(\\text{Tot}(\\tau_{\\leq m}L^\\bullet \\otimes_R P^\\bullet))$", "for some $m \\in \\mathbf{Z}$. Since $\\tau_{\\leq m}L^\\bullet$ is also", "acyclic we may replace $L^\\bullet$ by $\\tau_{\\leq m}L^\\bullet$.", "Hence we may assume that $L^\\bullet$ is bounded above.", "In this case the spectral sequence of", "Homology, Lemma \\ref{homology-lemma-first-quadrant-ss}", "has", "$$", "{}'E_1^{p, q} = H^p(L^\\bullet \\otimes_R P^q)", "$$", "which is zero as $P^q$ is flat and $L^\\bullet$ acyclic. Hence", "$H^*(\\text{Tot}(L^\\bullet \\otimes_R P^\\bullet)) = 0$." ], "refs": [ "homology-lemma-first-quadrant-ss" ], "ref_ids": [ 12105 ] } ], "ref_ids": [] }, { "id": 10129, "type": "theorem", "label": "more-algebra-lemma-colimit-K-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-colimit-K-flat", "contents": [ "Let $R$ be a ring.", "Let $K_1^\\bullet \\to K_2^\\bullet \\to \\ldots$", "be a system of K-flat complexes.", "Then $\\colim_i K_i^\\bullet$ is K-flat.", "More generally any filtered colimit of K-flat complexes", "is K-flat." ], "refs": [], "proofs": [ { "contents": [ "Because we are taking termwise colimits we have", "$$", "\\colim_i \\text{Tot}(M^\\bullet \\otimes_R K_i^\\bullet) =", "\\text{Tot}(M^\\bullet \\otimes_R \\colim_i K_i^\\bullet)", "$$", "by Algebra, Lemma \\ref{algebra-lemma-tensor-products-commute-with-limits}.", "Hence the lemma follows from the fact that filtered colimits are", "exact, see Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}." ], "refs": [ "algebra-lemma-tensor-products-commute-with-limits", "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 363, 343 ] } ], "ref_ids": [] }, { "id": 10130, "type": "theorem", "label": "more-algebra-lemma-universally-acyclic-K-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-universally-acyclic-K-flat", "contents": [ "Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules.", "If $K^\\bullet \\otimes_R M$ is acyclic for all finitely presented", "$R$-modules $M$, then $K^\\bullet$ is K-flat." ], "refs": [], "proofs": [ { "contents": [ "We will use repeatedly that tensor product commute with colimits", "(Algebra, Lemma \\ref{algebra-lemma-tensor-products-commute-with-limits}).", "Thus we see that $K^\\bullet \\otimes_R M$ is acyclic for", "any $R$-module $M$, because any $R$-module is a filtered colimit", "of finitely presented $R$-modules $M$, see", "Algebra, Lemma \\ref{algebra-lemma-module-colimit-fp}.", "Let $M^\\bullet$ be an acyclic complex of $R$-modules.", "We have to show that $\\text{Tot}(M^\\bullet \\otimes_R K^\\bullet)$", "is acyclic. Since $M^\\bullet = \\colim \\tau_{\\leq n} M^\\bullet$ (termwise", "colimit) we have", "$$", "\\text{Tot}(M^\\bullet \\otimes_R K^\\bullet) =", "\\colim \\text{Tot}(\\tau_{\\leq n} M^\\bullet \\otimes_R K^\\bullet)", "$$", "with truncations as in Homology, Section \\ref{homology-section-truncations}.", "As filtered colimits are exact", "(Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact})", "we may replace $M^\\bullet$ by $\\tau_{\\leq n}M^\\bullet$ and", "assume that $M^\\bullet$ is bounded above.", "In the bounded above case, we can write", "$M^\\bullet = \\colim \\sigma_{\\geq -n} M^\\bullet$", "where the complexes $\\sigma_{\\geq -n} M^\\bullet$ are bounded", "but possibly no longer acyclic.", "Arguing as above we reduce to the case where $M^\\bullet$", "is a bounded complex.", "Finally, for a bounded complex $M^a \\to \\ldots \\to M^b$", "we can argue by induction on the length $b - a$ of the complex.", "The case $b - a = 1$ we have seen above.", "For $b - a > 1$ we consider the split short exact sequence", "of complexes", "$$", "0 \\to \\sigma_{\\geq a + 1}M^\\bullet \\to M^\\bullet \\to M^a[-a] \\to 0", "$$", "and we apply Lemma \\ref{lemma-derived-tor-exact}", "to do the induction step. Some details omitted." ], "refs": [ "algebra-lemma-tensor-products-commute-with-limits", "algebra-lemma-module-colimit-fp", "algebra-lemma-directed-colimit-exact", "more-algebra-lemma-derived-tor-exact" ], "ref_ids": [ 363, 355, 343, 10122 ] } ], "ref_ids": [] }, { "id": 10131, "type": "theorem", "label": "more-algebra-lemma-K-flat-resolution", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-K-flat-resolution", "contents": [ "Let $R$ be a ring. For any complex $M^\\bullet$ there exists a", "K-flat complex $K^\\bullet$ whose terms are flat $R$-modules", "and a quasi-isomorphism $K^\\bullet \\to M^\\bullet$ which is termwise", "surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{P} \\subset \\Ob(\\text{Mod}_R)$ be the", "class of flat $R$-modules. By", "Derived Categories, Lemma \\ref{derived-lemma-special-direct-system}", "there exists a system", "$K_1^\\bullet \\to K_2^\\bullet \\to \\ldots$", "and a diagram", "$$", "\\xymatrix{", "K_1^\\bullet \\ar[d] \\ar[r] &", "K_2^\\bullet \\ar[d] \\ar[r] & \\ldots \\\\", "\\tau_{\\leq 1}M^\\bullet \\ar[r] &", "\\tau_{\\leq 2}M^\\bullet \\ar[r] & \\ldots", "}", "$$", "with the properties (1), (2), (3) listed in that lemma.", "These properties imply each complex $K_i^\\bullet$ is a bounded", "above complex of flat modules. Hence $K_i^\\bullet$ is K-flat by", "Lemma \\ref{lemma-derived-tor-quasi-isomorphism}.", "The induced map $\\colim_i K_i^\\bullet \\to M^\\bullet$", "is a quasi-isomorphism and termwise surjective by construction. The complex", "$\\colim_i K_i^\\bullet$ is K-flat by", "Lemma \\ref{lemma-colimit-K-flat}.", "The terms $\\colim K_i^n$ are flat because filtered colimits of", "flat modules are flat, see", "Algebra, Lemma \\ref{algebra-lemma-colimit-flat}." ], "refs": [ "derived-lemma-special-direct-system", "more-algebra-lemma-derived-tor-quasi-isomorphism", "more-algebra-lemma-colimit-K-flat", "algebra-lemma-colimit-flat" ], "ref_ids": [ 1903, 10128, 10129, 523 ] } ], "ref_ids": [] }, { "id": 10132, "type": "theorem", "label": "more-algebra-lemma-derived-tor-quasi-isomorphism-other-side", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-tor-quasi-isomorphism-other-side", "contents": [ "Let $R$ be a ring. Let", "$\\alpha : P^\\bullet \\to Q^\\bullet$ be a quasi-isomorphism of", "K-flat complexes of $R$-modules. For every complex $L^\\bullet$", "of $R$-modules the induced map", "$$", "\\text{Tot}(\\text{id}_L \\otimes \\alpha) :", "\\text{Tot}(L^\\bullet \\otimes_R P^\\bullet)", "\\longrightarrow", "\\text{Tot}(L^\\bullet \\otimes_R Q^\\bullet)", "$$", "is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Choose a quasi-isomorphism $K^\\bullet \\to L^\\bullet$ with", "$K^\\bullet$ a K-flat complex, see", "Lemma \\ref{lemma-K-flat-resolution}.", "Consider the commutative diagram", "$$", "\\xymatrix{", "\\text{Tot}(K^\\bullet \\otimes_R P^\\bullet) \\ar[r] \\ar[d] &", "\\text{Tot}(K^\\bullet \\otimes_R Q^\\bullet) \\ar[d] \\\\", "\\text{Tot}(L^\\bullet \\otimes_R P^\\bullet) \\ar[r] &", "\\text{Tot}(L^\\bullet \\otimes_R Q^\\bullet)", "}", "$$", "The result follows as by", "Lemma \\ref{lemma-K-flat-quasi-isomorphism}", "the vertical arrows and the top horizontal arrow are quasi-isomorphisms." ], "refs": [ "more-algebra-lemma-K-flat-resolution", "more-algebra-lemma-K-flat-quasi-isomorphism" ], "ref_ids": [ 10131, 10123 ] } ], "ref_ids": [] }, { "id": 10133, "type": "theorem", "label": "more-algebra-lemma-flip-douoble-tensor-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flip-douoble-tensor-product", "contents": [ "Let $R$ be a ring. Let $K^\\bullet, L^\\bullet$ be complexes of $R$-modules.", "There is a canonical isomorphism", "$$", "K^\\bullet \\otimes_R^\\mathbf{L} L^\\bullet \\longrightarrow", "L^\\bullet \\otimes_R^\\mathbf{L} K^\\bullet", "$$", "functorial in both complexes which uses a sign of $(-1)^{pq}$", "for the map $K^p \\otimes_R L^q \\to L^q \\otimes_R K^p$ (see proof", "for explanation)." ], "refs": [], "proofs": [ { "contents": [ "Replace the complexes by K-flat complexes $K^\\bullet, L^\\bullet$.", "Then we consider the map", "$$", "\\text{Tot}(K^\\bullet \\otimes_R L^\\bullet)", "\\longrightarrow", "\\text{Tot}(L^\\bullet \\otimes_R K^\\bullet)", "$$", "given by using $(-1)^{pq}$ times the canonical map", "$K^p \\otimes_R L^q \\to L^q \\otimes_R K^p$. This is an isomorphism.", "To see that it is a map of complexes we compute for", "$x \\in K^p$ and $y \\in L^q$ that", "$$", "\\text{d}(x \\otimes y) =", "\\text{d}_K(x) \\otimes y + (-1)^px \\otimes \\text{d}_L(y)", "$$", "Our rule says the right hand side is mapped to", "$$", "(-1)^{(p + 1)q}y \\otimes \\text{d}_K(x) +", "(-1)^{p + p(q + 1)} \\text{d}_L(y) \\otimes x", "$$", "On the other hand, we see that", "$$", "\\text{d}((-1)^{pq}y \\otimes x) =", "(-1)^{pq} \\text{d}_L(y) \\otimes x +", "(-1)^{pq + q} y \\otimes \\text{d}_K(x)", "$$", "These two expressions agree by inspection and the lemma is proved." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10134, "type": "theorem", "label": "more-algebra-lemma-triple-tensor-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-triple-tensor-product", "contents": [ "Let $R$ be a ring. Let $K^\\bullet, L^\\bullet, M^\\bullet$", "be complexes of $R$-modules. There is a canonical isomorphism", "$$", "(K^\\bullet \\otimes_R^\\mathbf{L} L^\\bullet) \\otimes_R^\\mathbf{L} M^\\bullet", "=", "K^\\bullet \\otimes_R^\\mathbf{L} (L^\\bullet \\otimes_R^\\mathbf{L} M^\\bullet)", "$$", "functorial in all three complexes." ], "refs": [], "proofs": [ { "contents": [ "Replace the complexes by K-flat complexes and apply", "Homology, Remark \\ref{homology-remark-triple-complex}." ], "refs": [ "homology-remark-triple-complex" ], "ref_ids": [ 12191 ] } ], "ref_ids": [] }, { "id": 10135, "type": "theorem", "label": "more-algebra-lemma-factor-through-K-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-factor-through-K-flat", "contents": [ "Let $R$ be a ring. Let $a : K^\\bullet \\to L^\\bullet$ be a map of complexes", "of $R$-modules. If $K^\\bullet$ is K-flat, then there exist a complex", "$N^\\bullet$ and maps of complexes $b : K^\\bullet \\to N^\\bullet$", "and $c : N^\\bullet \\to L^\\bullet$ such that", "\\begin{enumerate}", "\\item $N^\\bullet$ is K-flat,", "\\item $c$ is a quasi-isomorphism,", "\\item $a$ is homotopic to $c \\circ b$.", "\\end{enumerate}", "If the terms of $K^\\bullet$ are flat, then we may choose", "$N^\\bullet$, $b$, and $c$", "such that the same is true for $N^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "We will use that the homotopy category $K(\\text{Mod}_R)$", "is a triangulated category, see Derived Categories, Proposition", "\\ref{derived-proposition-homotopy-category-triangulated}.", "Choose a distinguished triangle", "$K^\\bullet \\to L^\\bullet \\to C^\\bullet \\to K^\\bullet[1]$.", "Choose a quasi-isomorphism $M^\\bullet \\to C^\\bullet$ with", "$M^\\bullet$ K-flat with flat terms, see Lemma \\ref{lemma-K-flat-resolution}.", "By the axioms of triangulated categories,", "we may fit the composition $M^\\bullet \\to C^\\bullet \\to K^\\bullet[1]$", "into a distinguished triangle", "$K^\\bullet \\to N^\\bullet \\to M^\\bullet \\to K^\\bullet[1]$.", "By Lemma \\ref{lemma-K-flat-two-out-of-three} we see that $N^\\bullet$ is K-flat.", "Again using the axioms of triangulated categories,", "we can choose a map $N^\\bullet \\to L^\\bullet$ fitting into", "the following morphism of distinghuised triangles", "$$", "\\xymatrix{", "K^\\bullet \\ar[r] \\ar[d] &", "N^\\bullet \\ar[r] \\ar[d] &", "M^\\bullet \\ar[r] \\ar[d] &", "K^\\bullet[1] \\ar[d] \\\\", "K^\\bullet \\ar[r] &", "L^\\bullet \\ar[r] &", "C^\\bullet \\ar[r] &", "K^\\bullet[1]", "}", "$$", "Since two out of three of the arrows are quasi-isomorphisms, so is", "the third arrow $N^\\bullet \\to L^\\bullet$ by the long exact sequences", "of cohomology associated to these distinguished triangles", "(or you can look at the image of this diagram in $D(R)$ and use", "Derived Categories, Lemma \\ref{derived-lemma-third-isomorphism-triangle}", "if you like). This finishes the proof of (1), (2), and (3).", "To prove the final assertion, we may choose $N^\\bullet$", "such that $N^n \\cong M^n \\oplus K^n$, see", "Derived Categories, Lemma", "\\ref{derived-lemma-improve-distinguished-triangle-homotopy}.", "Hence we get the desired flatness", "if the terms of $K^\\bullet$ are flat." ], "refs": [ "derived-proposition-homotopy-category-triangulated", "more-algebra-lemma-K-flat-resolution", "more-algebra-lemma-K-flat-two-out-of-three", "derived-lemma-third-isomorphism-triangle", "derived-lemma-improve-distinguished-triangle-homotopy" ], "ref_ids": [ 1960, 10131, 10126, 1759, 1809 ] } ], "ref_ids": [] }, { "id": 10136, "type": "theorem", "label": "more-algebra-lemma-derived-base-change", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-base-change", "contents": [ "The construction above is independent of choices and defines an exact", "functor of triangulated categories", "$- \\otimes_R^\\mathbf{L} N^\\bullet : D(R) \\to D(A)$.", "There is a functorial isomorphism", "$$", "E^\\bullet \\otimes_R^\\mathbf{L} N^\\bullet =", "(E^\\bullet \\otimes_R^\\mathbf{L} A) \\otimes_A^\\mathbf{L} N^\\bullet", "$$", "for $E^\\bullet$ in $D(R)$." ], "refs": [], "proofs": [ { "contents": [ "To prove the existence of the derived functor", "$- \\otimes_R^\\mathbf{L} N^\\bullet$", "we use the general theory developed in", "Derived Categories, Section \\ref{derived-section-derived-functors}.", "Set $\\mathcal{D} = K(\\text{Mod}_R)$ and $\\mathcal{D}' = D(A)$.", "Let us write $F : \\mathcal{D} \\to \\mathcal{D}'$ the exact functor", "of triangulated categories defined by the rule", "$F(M^\\bullet) = \\text{Tot}(M^\\bullet \\otimes_R N^\\bullet)$.", "To prove the stated properties of $F$ use", "Lemmas \\ref{lemma-derived-tor-homotopy} and \\ref{lemma-derived-tor-exact}.", "We let $S$ be the set of", "quasi-isomorphisms in $\\mathcal{D} = K(\\text{Mod}_R)$.", "This gives a situation as in", "Derived Categories, Situation \\ref{derived-situation-derived-functor}", "so that", "Derived Categories, Definition", "\\ref{derived-definition-right-derived-functor-defined}", "applies. We claim that $LF$ is everywhere defined.", "This follows from", "Derived Categories, Lemma \\ref{derived-lemma-find-existence-computes}", "with $\\mathcal{P} \\subset \\Ob(\\mathcal{D})$ the collection", "of K-flat complexes: (1) follows from", "Lemma \\ref{lemma-K-flat-resolution}", "and (2) follows from", "Lemma \\ref{lemma-derived-tor-quasi-isomorphism-other-side}.", "Thus we obtain a derived functor", "$$", "LF : D(R) = S^{-1}\\mathcal{D} \\longrightarrow \\mathcal{D}' = D(A)", "$$", "see", "Derived Categories, Equation (\\ref{derived-equation-everywhere}).", "Finally,", "Derived Categories, Lemma \\ref{derived-lemma-find-existence-computes}", "guarantees that $LF(K^\\bullet) = F(K^\\bullet) =", "\\text{Tot}(K^\\bullet \\otimes_R N^\\bullet)$", "when $K^\\bullet$ is K-flat, i.e., $LF$ is indeed computed in the way", "described above. Moreover, by Lemma \\ref{lemma-base-change-K-flat}", "the complex $K^\\bullet \\otimes_R A$ is a K-flat complex of $A$-modules.", "Hence", "$$", "(K^\\bullet \\otimes_R^\\mathbf{L} A) \\otimes_A^\\mathbf{L} N^\\bullet =", "\\text{Tot}((K^\\bullet \\otimes_R A) \\otimes_A N^\\bullet) =", "\\text{Tot}(K^\\bullet \\otimes_A N^\\bullet) =", "K^\\bullet \\otimes_A^\\mathbf{L} N^\\bullet", "$$", "which proves the final statement of the lemma." ], "refs": [ "more-algebra-lemma-derived-tor-homotopy", "more-algebra-lemma-derived-tor-exact", "derived-definition-right-derived-functor-defined", "derived-lemma-find-existence-computes", "more-algebra-lemma-K-flat-resolution", "more-algebra-lemma-derived-tor-quasi-isomorphism-other-side", "derived-lemma-find-existence-computes", "more-algebra-lemma-base-change-K-flat" ], "ref_ids": [ 10121, 10122, 1987, 1832, 10131, 10132, 1832, 10124 ] } ], "ref_ids": [] }, { "id": 10137, "type": "theorem", "label": "more-algebra-lemma-functoriality-derived-base-change", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-functoriality-derived-base-change", "contents": [ "Let $R \\to A$ be a ring map. Let $f : L^\\bullet \\to N^\\bullet$ be a", "map of complexes of $A$-modules. Then $f$ induces a transformation", "of functors", "$$", "1 \\otimes f :", "- \\otimes_A^\\mathbf{L} L^\\bullet", "\\longrightarrow", "- \\otimes_A^\\mathbf{L} N^\\bullet", "$$", "If $f$ is a quasi-isomorphism, then $1 \\otimes f$ is an isomorphism", "of functors." ], "refs": [], "proofs": [ { "contents": [ "Since the functors are computing by evaluating on K-flat complexes", "$K^\\bullet$ we can simply use the functoriality", "$$", "\\text{Tot}(K^\\bullet \\otimes_R L^\\bullet) \\to", "\\text{Tot}(K^\\bullet \\otimes_R N^\\bullet)", "$$", "to define the transformation. The last statement follows from", "Lemma \\ref{lemma-K-flat-quasi-isomorphism}." ], "refs": [ "more-algebra-lemma-K-flat-quasi-isomorphism" ], "ref_ids": [ 10123 ] } ], "ref_ids": [] }, { "id": 10138, "type": "theorem", "label": "more-algebra-lemma-double-base-change", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-double-base-change", "contents": [ "Let $A \\to B \\to C$ be ring maps. Let $N^\\bullet$ be a complex of", "$B$-modules and $K^\\bullet$ a complex of $C$-modules.", "The compositions of the functors", "$$", "D(A) \\xrightarrow{- \\otimes_A^\\mathbf{L} N^\\bullet}", "D(B) \\xrightarrow{- \\otimes_B^\\mathbf{L} K^\\bullet} D(C)", "$$", "is the functor", "$- \\otimes_A^\\mathbf{L} (N^\\bullet \\otimes_B^\\mathbf{L} K^\\bullet) :", "D(A) \\to D(C)$. If $M$, $N$, $K$ are modules over $A$, $B$, $C$, then", "we have", "$$", "(M \\otimes_A^\\mathbf{L} N) \\otimes_B^\\mathbf{L} K =", "M \\otimes_A^\\mathbf{L} (N \\otimes_B^\\mathbf{L} K) =", "(M \\otimes_A^\\mathbf{L} C) \\otimes_C^\\mathbf{L} (N \\otimes_B^\\mathbf{L} K)", "$$", "in $D(C)$. We also have a canonical isomorphism", "$$", "(M \\otimes_A^\\mathbf{L} N) \\otimes_B^\\mathbf{L} K \\longrightarrow", "(M \\otimes_A^\\mathbf{L} K) \\otimes_C^\\mathbf{L} (N \\otimes_B^\\mathbf{L} C)", "$$", "using signs. Similar results holds for complexes." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-flat complex $P^\\bullet$ of $B$-modules and a quasi-isomorphism", "$P^\\bullet \\to N^\\bullet$ (Lemma \\ref{lemma-K-flat-resolution}).", "Let $M^\\bullet$ be a K-flat complex of $A$-modules representing an", "arbitrary object of $D(A)$. Then we see that", "$$", "(M^\\bullet \\otimes_A^\\mathbf{L} P^\\bullet) \\otimes_B^\\mathbf{L} K^\\bullet", "\\longrightarrow", "(M^\\bullet \\otimes_A^\\mathbf{L} N^\\bullet) \\otimes_B^\\mathbf{L} K^\\bullet", "$$", "is an isomorphism by Lemma \\ref{lemma-functoriality-derived-base-change}", "applied to the material inside the brackets. By", "Lemmas \\ref{lemma-base-change-K-flat} and \\ref{lemma-tensor-product-K-flat}", "the complex", "$$", "\\text{Tot}(M^\\bullet \\otimes_A P^\\bullet) =", "\\text{Tot}((M^\\bullet \\otimes_R A) \\otimes_A P^\\bullet", "$$", "is K-flat as a complex of $B$-modules and it represents", "the derived tensor product in $D(B)$ by construction.", "Hence we see that", "$(M^\\bullet \\otimes_A^\\mathbf{L} P^\\bullet) \\otimes_B^\\mathbf{L} K^\\bullet$", "is represented by the complex", "$$", "\\text{Tot}(\\text{Tot}(M^\\bullet \\otimes_A P^\\bullet)\\otimes_B K^\\bullet) =", "\\text{Tot}(M^\\bullet \\otimes_A \\text{Tot}(P^\\bullet \\otimes_B K^\\bullet))", "$$", "of $C$-modules. Equality by", "Homology, Remark \\ref{homology-remark-triple-complex}.", "Going back the way we came we see that this is equal to", "$$", "M^\\bullet \\otimes_A^\\mathbf{L} (P^\\bullet \\otimes_B^\\mathbf{L} K^\\bullet)", "\\longleftarrow", "M^\\bullet \\otimes_A^\\mathbf{L} (N^\\bullet \\otimes_B^\\mathbf{L} K^\\bullet)", "$$", "The arrow is an isomorphism by definition of the functor", "$-\\otimes_B^\\mathbf{L} K^\\bullet$. All of these constructions", "are functorial in the complex $M^\\bullet$ and hence we obtain", "our isomorphism of functors.", "\\medskip\\noindent", "By the above we have the first equality in", "$$", "(M \\otimes_A^\\mathbf{L} N) \\otimes_B^\\mathbf{L} K =", "M \\otimes_A^\\mathbf{L} (N \\otimes_B^\\mathbf{L} K) =", "(M \\otimes_A^\\mathbf{L} C) \\otimes_C^\\mathbf{L} (N \\otimes_B^\\mathbf{L} K)", "$$", "The second equality follows from the final statement of", "Lemma \\ref{lemma-derived-base-change}.", "The same thing allows us to write", "$N \\otimes_B^\\mathbf{L} K = (N \\otimes_B^\\mathbf{L} C) \\otimes_C^\\mathbf{L} K$", "and substituting we get", "\\begin{align*}", "(M \\otimes_A^\\mathbf{L} N) \\otimes_B^\\mathbf{L} K", "& =", "(M \\otimes_A^\\mathbf{L} C) \\otimes_C^\\mathbf{L}", "((N \\otimes_B^\\mathbf{L} C) \\otimes_C^\\mathbf{L} K) \\\\", "& =", "(M \\otimes_A^\\mathbf{L} C) \\otimes_C^\\mathbf{L}", "(K \\otimes_C^\\mathbf{L} (N \\otimes_B^\\mathbf{L} C)) \\\\", "& =", "((M \\otimes_A^\\mathbf{L} C) \\otimes_C^\\mathbf{L} K)", "\\otimes_C^\\mathbf{L} (N \\otimes_B^\\mathbf{L} C)) \\\\", "& =", "(M \\otimes_C^\\mathbf{L} K)", "\\otimes_C^\\mathbf{L} (N \\otimes_B^\\mathbf{L} C)", "\\end{align*}", "by Lemmas \\ref{lemma-flip-douoble-tensor-product} and", "\\ref{lemma-triple-tensor-product}", "as well as the previously mentioned lemma." ], "refs": [ "more-algebra-lemma-K-flat-resolution", "more-algebra-lemma-functoriality-derived-base-change", "more-algebra-lemma-base-change-K-flat", "more-algebra-lemma-tensor-product-K-flat", "homology-remark-triple-complex", "more-algebra-lemma-derived-base-change", "more-algebra-lemma-flip-douoble-tensor-product", "more-algebra-lemma-triple-tensor-product" ], "ref_ids": [ 10131, 10137, 10124, 10125, 12191, 10136, 10133, 10134 ] } ], "ref_ids": [] }, { "id": 10139, "type": "theorem", "label": "more-algebra-lemma-base-change-comparison", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-comparison", "contents": [ "The comparison map (\\ref{equation-comparison-map}) is an isomorphism", "if $A' = A \\otimes_R R'$ and $A$ and $R'$ are Tor independent over $R$." ], "refs": [], "proofs": [ { "contents": [ "To prove this we choose a free resolution $F^\\bullet \\to R'$", "of $R'$ as an $R$-module. Because $A$ and $R'$ are Tor independent over $R$", "we see that $F^\\bullet \\otimes_R A$ is a free $A$-module resolution of $A'$", "over $A$. By our general construction of the derived tensor product", "above we see that", "$$", "K^\\bullet \\otimes_A A' \\cong", "\\text{Tot}(K^\\bullet \\otimes_A (F^\\bullet \\otimes_R A)) =", "\\text{Tot}(K^\\bullet \\otimes_R F^\\bullet) \\cong", "\\text{Tot}(E^\\bullet \\otimes_R F^\\bullet) \\cong", "E^\\bullet \\otimes_R R'", "$$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10140, "type": "theorem", "label": "more-algebra-lemma-tor-independent-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-tor-independent-flat", "contents": [ "Consider a commutative diagram of rings", "$$", "\\xymatrix{", "A' & R' \\ar[r] \\ar[l] & B' \\\\", "A \\ar[u] & R \\ar[l] \\ar[u] \\ar[r] & B \\ar[u]", "}", "$$", "Assume that $R'$ is flat over $R$ and $A'$ is flat over $A \\otimes_R R'$", "and $B'$ is flat over $R' \\otimes_R B$. Then", "$$", "\\text{Tor}_i^R(A, B) \\otimes_{(A \\otimes_R B)} (A' \\otimes_{R'} B') =", "\\text{Tor}_i^{R'}(A', B')", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Section \\ref{algebra-section-functoriality-tor} there are", "canonical maps", "$$", "\\text{Tor}_i^R(A, B) \\longrightarrow", "\\text{Tor}_i^{R'}(A \\otimes_R R', B \\otimes_R R') \\longrightarrow", "\\text{Tor}_i^{R'}(A', B')", "$$", "These induce a map from left to right in the formula of the lemma.", "\\medskip\\noindent", "Take a free resolution $F_\\bullet \\to A$ of $A$ as an $R$-module.", "Then we see that $F_\\bullet \\otimes_R R'$ is a resolution of $A \\otimes_R R'$.", "Hence $\\text{Tor}_i^{R'}(A \\otimes_R R', B \\otimes_R R')$ is computed", "by $F_\\bullet \\otimes_R B \\otimes_R R'$. By our assumption that $R'$", "is flat over $R$, this computes $\\text{Tor}_i^R(A, B) \\otimes_R R'$.", "Thus $\\text{Tor}_i^{R'}(A \\otimes_R R', B \\otimes_R R') =", "\\text{Tor}_i^R(A, B) \\otimes_R R'$ (uses only flatness of $R'$ over $R$).", "\\medskip\\noindent", "By Lazard's theorem (Algebra, Theorem \\ref{algebra-theorem-lazard})", "we can write $A'$, resp.\\ $B'$ as a filtered colimit of finite free", "$A \\otimes_R R'$, resp.\\ $B \\otimes_R R'$-modules. Say", "$A' = \\colim M_i$ and $B' = \\colim N_j$. The result above gives", "$$", "\\text{Tor}_i^{R'}(M_i, N_j) =", "\\text{Tor}_i^R(A, B) \\otimes_{A \\otimes_R B} (M_i \\otimes_{R'} N_j)", "$$", "as one can see by writing everything out in terms of bases.", "Taking the colimit we get the result of the lemma." ], "refs": [ "algebra-theorem-lazard" ], "ref_ids": [ 318 ] } ], "ref_ids": [] }, { "id": 10141, "type": "theorem", "label": "more-algebra-lemma-flat-base-change-tor-independent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-base-change-tor-independent", "contents": [ "Let $R \\to A$ and $R \\to B$ be ring maps. Let $R \\to R'$ be a", "ring map and set $A' = A \\otimes_R R'$ and $B' = B \\otimes_R R'$.", "If $A$ and $B$ are tor independent over $R$ and $R \\to R'$ is flat,", "then $A'$ and $B'$ are tor independent over $R'$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-tor-independent-flat}", "and Definition \\ref{definition-tor-independent}." ], "refs": [ "more-algebra-lemma-tor-independent-flat", "more-algebra-definition-tor-independent" ], "ref_ids": [ 10140, 10622 ] } ], "ref_ids": [] }, { "id": 10142, "type": "theorem", "label": "more-algebra-lemma-lemma-tor-independent-flat-compare", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lemma-tor-independent-flat-compare", "contents": [ "Assumptions as in Lemma \\ref{lemma-tor-independent-flat}.", "For $M \\in D(A)$ there are canonical isomorphisms", "$$", "H^i((M \\otimes_A^\\mathbf{L} A') \\otimes_{R'}^\\mathbf{L} B') =", "H^i(M \\otimes_R^\\mathbf{L} B) \\otimes_{(A \\otimes_R B)} (A' \\otimes_{R'} B')", "$$", "of $A' \\otimes_{R'} B'$-modules." ], "refs": [ "more-algebra-lemma-tor-independent-flat" ], "proofs": [ { "contents": [ "Let us elucidate the two sides of the equation. On the left hand side", "we have the composition of the functors", "$D(A) \\to D(A') \\to D(R') \\to D(B')$ with the functor", "$H^i : D(B') \\to \\text{Mod}_{B'}$. Since there is", "a map from $A'$ to the endomorphisms of the object", "$(M \\otimes_A^\\mathbf{L} A') \\otimes_{R'}^\\mathbf{L} B'$", "in $D(B')$, we see that the left hand side is indeed", "an $A' \\otimes_{R'} B'$-module. By the same arguments", "we see that $H^i(M \\otimes_R^\\mathbf{L} B)$", "has an $A \\otimes_R B$-module structure.", "\\medskip\\noindent", "We first prove the result in case $B' = R' \\otimes_R B$.", "In this case we choose a resolution $F^\\bullet \\to B$", "by free $R$-modules. We also choose a K-flat complex", "$M^\\bullet$ of $A$-modules representing $M$.", "Then the left hand side is represented by", "\\begin{align*}", "H^i(\\text{Tot}((M^\\bullet \\otimes_A A') \\otimes_{R'} (R' \\otimes_R F^\\bullet)))", "& =", "H^i(\\text{Tot}(M^\\bullet \\otimes_A A' \\otimes_R F^\\bullet)) \\\\", "& =", "H^i(\\text{Tot}(M^\\bullet \\otimes_R F^\\bullet) \\otimes_A A') \\\\", "& =", "H^i(M \\otimes_R^\\mathbf{L} B) \\otimes_A A'", "\\end{align*}", "The final equality because $A \\to A'$ is flat. The final module", "is the desired module because $A' \\otimes_{R'} B' = A' \\otimes_R B$", "since we've assumed $B' = R' \\otimes_R B$ in this paragraph.", "\\medskip\\noindent", "General case. Suppose that $B' \\to B''$ is a flat ring map.", "Then it is easy to see that", "$$", "H^i((M \\otimes_A^\\mathbf{L} A') \\otimes_{R'}^\\mathbf{L} B'') =", "H^i((M \\otimes_A^\\mathbf{L} A') \\otimes_{R'}^\\mathbf{L} B')", "\\otimes_{B'} B''", "$$", "and", "$$", "H^i(M \\otimes_R^\\mathbf{L} B) \\otimes_{(A \\otimes_R B)} (A' \\otimes_{R'} B'')", "=", "\\left(", "H^i(M \\otimes_R^\\mathbf{L} B) \\otimes_{(A \\otimes_R B)} (A' \\otimes_{R'} B')", "\\right) \\otimes_{B'} B''", "$$", "Thus the result for $B'$ implies the result for $B''$. Since we've", "proven the result for $R' \\otimes_R B$ in the previous paragraph,", "this implies the result in general." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10140 ] }, { "id": 10143, "type": "theorem", "label": "more-algebra-lemma-tor-independent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-tor-independent", "contents": [ "Let $R$ be a ring. Let $A$, $B$ be $R$-algebras. The following are equivalent", "\\begin{enumerate}", "\\item $A$ and $B$ are Tor independent over $R$,", "\\item for every pair of primes $\\mathfrak p \\subset A$ and", "$\\mathfrak q \\subset B$ lying over the same prime $\\mathfrak r \\subset R$", "the rings $A_\\mathfrak p$ and $B_\\mathfrak q$ are Tor independent over", "$R_\\mathfrak r$, and", "\\item For every prime $\\mathfrak s$ of $A \\otimes_R B$ the module", "$$", "\\text{Tor}_i^R(A, B)_\\mathfrak s =", "\\text{Tor}_i^{R_\\mathfrak r}(A_\\mathfrak p, B_\\mathfrak q)_\\mathfrak s", "$$", "(where $\\mathfrak p = A \\cap \\mathfrak s$, $\\mathfrak q = B \\cap \\mathfrak s$", "and $\\mathfrak r = R \\cap \\mathfrak s$) is zero.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak s$ be a prime of $A \\otimes_R B$ as in (3).", "The equality", "$$", "\\text{Tor}_i^R(A, B)_\\mathfrak s =", "\\text{Tor}_i^{R_\\mathfrak r}(A_\\mathfrak p, B_\\mathfrak q)_\\mathfrak s", "$$", "where $\\mathfrak p = A \\cap \\mathfrak s$, $\\mathfrak q = B \\cap \\mathfrak s$", "and $\\mathfrak r = R \\cap \\mathfrak s$ follows from", "Lemma \\ref{lemma-tor-independent-flat}.", "Hence (2) implies (3).", "Since we can test the vanishing of modules by localizing at primes", "(Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local})", "we conclude that (3) implies (1). For", "(1) $\\Rightarrow$ (2) we use that", "$$", "\\text{Tor}_i^{R_\\mathfrak r}(A_\\mathfrak p, B_\\mathfrak q) =", "\\text{Tor}_i^R(A, B) \\otimes_{(A \\otimes_R B)}", "(A_\\mathfrak p \\otimes_{R_{\\mathfrak r}} B_\\mathfrak q)", "$$", "again by Lemma \\ref{lemma-tor-independent-flat}." ], "refs": [ "more-algebra-lemma-tor-independent-flat", "algebra-lemma-characterize-zero-local", "more-algebra-lemma-tor-independent-flat" ], "ref_ids": [ 10140, 410, 10140 ] } ], "ref_ids": [] }, { "id": 10144, "type": "theorem", "label": "more-algebra-lemma-functoriality-product-tor", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-functoriality-product-tor", "contents": [ "Let $R$ be a ring. Let $A, B, C$ be $R$-algebras and let $B \\to C$ be an", "$R$-algebra map. Then the induced map", "$$", "\\text{Tor}^R_{\\star}(B, A)", "\\longrightarrow", "\\text{Tor}^R_{\\star}(C, A)", "$$", "is an $A$-algebra homomorphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: You can prove this by working through the definitions,", "writing all the complexes explicitly." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10145, "type": "theorem", "label": "more-algebra-lemma-cone-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cone-pseudo-coherent", "contents": [ "Let $R$ be a ring and $m \\in \\mathbf{Z}$.", "Let $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$ be a distinguished", "triangle in $D(R)$.", "\\begin{enumerate}", "\\item If $K^\\bullet$ is $(m + 1)$-pseudo-coherent and", "$L^\\bullet$ is $m$-pseudo-coherent then $M^\\bullet$ is", "$m$-pseudo-coherent.", "\\item If $K^\\bullet, M^\\bullet$ are $m$-pseudo-coherent, then", "$L^\\bullet$ is $m$-pseudo-coherent.", "\\item If $L^\\bullet$ is $(m + 1)$-pseudo-coherent and $M^\\bullet$", "is $m$-pseudo-coherent, then $K^\\bullet$ is $(m + 1)$-pseudo-coherent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Choose $\\alpha : P^\\bullet \\to K^\\bullet$", "with $P^\\bullet$ a bounded complex of finite free modules", "such that $H^i(\\alpha)$ is an isomorphism for $i > m + 1$ and", "surjective for $i = m + 1$. We may replace $P^\\bullet$ by", "$\\sigma_{\\geq m + 1}P^\\bullet$ and hence we may assume that $P^i = 0$", "for $i < m + 1$. Choose $\\beta : E^\\bullet \\to L^\\bullet$ with $E^\\bullet$", "a bounded complex of finite free modules such that", "$H^i(\\beta)$ is an isomorphism for $i > m$ and", "surjective for $i = m$. By", "Derived Categories,", "Lemma \\ref{derived-lemma-lift-map}", "we can find a map $\\alpha : P^\\bullet \\to E^\\bullet$ such that the diagram", "$$", "\\xymatrix{", "K^\\bullet \\ar[r] & L^\\bullet \\\\", "P^\\bullet \\ar[u] \\ar[r]^\\alpha & E^\\bullet \\ar[u]", "}", "$$", "is commutative in $D(R)$. The cone $C(\\alpha)^\\bullet$ is a bounded", "complex of finite free $R$-modules, and the commutativity of the", "diagram implies that there exists a morphism of distinguished triangles", "$$", "(P^\\bullet, E^\\bullet, C(\\alpha)^\\bullet)", "\\longrightarrow", "(K^\\bullet, L^\\bullet, M^\\bullet).", "$$", "It follows from the induced map on long exact cohomology sequences and", "Homology, Lemmas \\ref{homology-lemma-four-lemma} and", "\\ref{homology-lemma-five-lemma}", "that $C(\\alpha)^\\bullet \\to M^\\bullet$ induces an isomorphism", "on cohomology in degrees $> m$ and a surjection in degree $m$.", "Hence $M^\\bullet$ is $m$-pseudo-coherent.", "\\medskip\\noindent", "Assertions (2) and (3) follow from (1) by rotating the distinguished", "triangle." ], "refs": [ "derived-lemma-lift-map", "homology-lemma-four-lemma", "homology-lemma-five-lemma" ], "ref_ids": [ 1865, 12029, 12030 ] } ], "ref_ids": [] }, { "id": 10146, "type": "theorem", "label": "more-algebra-lemma-finite-cohomology", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finite-cohomology", "contents": [ "Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules.", "Let $m \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item If $K^\\bullet$ is $m$-pseudo-coherent and $H^i(K^\\bullet) = 0$", "for $i > m$, then $H^m(K^\\bullet)$ is a finite type $R$-module.", "\\item If $K^\\bullet$ is $m$-pseudo-coherent and $H^i(K^\\bullet) = 0$", "for $i > m + 1$, then $H^{m + 1}(K^\\bullet)$ is a finitely presented", "$R$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Choose a bounded complex $E^\\bullet$ of finite projective", "$R$-modules and a map $\\alpha : E^\\bullet \\to K^\\bullet$ which induces", "an isomorphism on cohomology in degrees $> m$ and a surjection in degree $m$.", "It is clear that it suffices to prove the result for $E^\\bullet$.", "Let $n$ be the largest integer such that $E^n \\not = 0$.", "If $n = m$, then the result is clear.", "If $n > m$, then $E^{n - 1} \\to E^n$ is surjective as", "$H^n(E^\\bullet) = 0$. As $E^n$ is finite projective we see that", "$E^{n - 1} = E' \\oplus E^n$. Hence it suffices to prove the result", "for the complex $(E')^\\bullet$ which is the same as $E^\\bullet$", "except has $E'$ in degree $n - 1$ and $0$ in degree $n$.", "We win by induction on $n$.", "\\medskip\\noindent", "Proof of (2). Choose a bounded complex $E^\\bullet$ of finite projective", "$R$-modules and a map $\\alpha : E^\\bullet \\to K^\\bullet$ which induces", "an isomorphism on cohomology in degrees $> m$ and a surjection in degree $m$.", "As in the proof of (1) we can reduce to the case that $E^i = 0$ for", "$i > m + 1$. Then we see that", "$H^{m + 1}(K^\\bullet) \\cong", "H^{m + 1}(E^\\bullet) = \\Coker(E^m \\to E^{m + 1})$", "which is of finite presentation." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10147, "type": "theorem", "label": "more-algebra-lemma-n-pseudo-module", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-n-pseudo-module", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Then", "\\begin{enumerate}", "\\item $M$ is $0$-pseudo-coherent if and only if $M$ is a finite", "$R$-module,", "\\item $M$ is $(-1)$-pseudo-coherent if and only if $M$ is a finitely", "presented $R$-module,", "\\item $M$ is $(-d)$-pseudo-coherent if and only if there exists a", "resolution", "$$", "R^{\\oplus a_d} \\to R^{\\oplus a_{d - 1}} \\to \\ldots \\to R^{\\oplus a_0} \\to", "M \\to 0", "$$", "of length $d$, and", "\\item $M$ is pseudo-coherent if and only if there exists an", "infinite resolution", "$$", "\\ldots \\to R^{\\oplus a_1} \\to R^{\\oplus a_0} \\to M \\to 0", "$$", "by finite free $R$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $M$ is of finite type (resp.\\ of finite presentation), then $M$", "is $0$-pseudo-coherent (resp.\\ $(-1)$-pseudo-coherent) as follows from the", "discussion preceding", "Definition \\ref{definition-pseudo-coherent}.", "Conversely, if $M$ is $0$-pseudo-coherent, then $M = H^0(M[0])$", "is of finite type by", "Lemma \\ref{lemma-finite-cohomology}.", "If $M$ is $(-1)$-pseudo-coherent, then it is $0$-pseudo-coherent hence", "of finite type. Choose a surjection $R^{\\oplus a} \\to M$ and denote", "$K = \\Ker(R^{\\oplus a} \\to M)$. By", "Lemma \\ref{lemma-cone-pseudo-coherent}", "we see that $K$ is $0$-pseudo-coherent, hence of finite type, whence", "$M$ is of finite presentation.", "\\medskip\\noindent", "To prove the third and fourth statement use", "induction and an argument similar to the above (details omitted)." ], "refs": [ "more-algebra-definition-pseudo-coherent", "more-algebra-lemma-finite-cohomology", "more-algebra-lemma-cone-pseudo-coherent" ], "ref_ids": [ 10623, 10146, 10145 ] } ], "ref_ids": [] }, { "id": 10148, "type": "theorem", "label": "more-algebra-lemma-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pseudo-coherent", "contents": [ "Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules.", "The following are equivalent", "\\begin{enumerate}", "\\item $K^\\bullet$ is pseudo-coherent,", "\\item $K^\\bullet$ is $m$-pseudo-coherent for every $m \\in \\mathbf{Z}$, and", "\\item $K^\\bullet$ is quasi-isomorphic to a bounded above complex of finite", "projective $R$-modules.", "\\end{enumerate}", "If (1), (2), and (3) hold and $H^i(K^\\bullet) = 0$ for $i > b$, then", "we can find a quasi-isomorphism $F^\\bullet \\to K^\\bullet$ with", "$F^i$ finite free $R$-modules and $F^i = 0$ for $i > b$." ], "refs": [], "proofs": [ { "contents": [ "We see that (1) $\\Rightarrow$ (3) as a finite free module is a finite", "projective $R$-module. Conversely, suppose $P^\\bullet$ is a bounded", "above complex of finite projective $R$-modules. Say $P^i = 0$ for", "$i > n_0$. We choose a direct sum decompositions", "$F^{n_0} = P^{n_0} \\oplus C^{n_0}$ with $F^{n_0}$ a finite free", "$R$-module, and inductively", "$$", "F^{n - 1} = P^{n - 1} \\oplus C^n \\oplus C^{n - 1}", "$$", "for $n \\leq n_0$ with $F^{n_0}$ a finite free $R$-module. As a complex", "$F^\\bullet$ has maps $F^{n - 1} \\to F^n$ which agree with $P^{n - 1} \\to P^n$,", "induce the identity $C^n \\to C^n$, and are zero on $C^{n - 1}$. The map", "$F^\\bullet \\to P^\\bullet$ is a quasi-isomorphism (even a homotopy equivalence)", "and hence (3) implies (1).", "\\medskip\\noindent", "Assume (1). Let $E^\\bullet$ be a bounded above complex of finite free", "$R$-modules and let $E^\\bullet \\to K^\\bullet$ be a", "quasi-isomorphism. Then the induced maps", "$\\sigma_{\\geq m}E^\\bullet \\to K^\\bullet$ from the stupid truncation", "of $E^\\bullet$ to $K^\\bullet$ show that $K^\\bullet$ is $m$-pseudo-coherent.", "Hence (1) implies (2).", "\\medskip\\noindent", "Assume (2). Since $K^\\bullet$ is $0$-pseudo-coherent we see in particular", "that $K^\\bullet$ is bounded above. Let $b$ be an integer such that", "$H^i(K^\\bullet) = 0$ for $i > b$. By descending induction on", "$n \\in \\mathbf{Z}$ we are going to construct finite free $R$-modules", "$F^i$ for $i \\geq n$, differentials $d^i : F^i \\to F^{i + 1}$ for", "$i \\geq n$, maps $\\alpha : F^i \\to K^i$ compatible with differentials,", "such that (1) $H^i(\\alpha)$ is an isomorphism for $i > n$ and surjective for", "$i = n$, and (2) $F^i = 0$ for $i > b$. Picture", "$$", "\\xymatrix{", "& F^n \\ar[r] \\ar[d]^\\alpha & F^{n + 1} \\ar[d]^\\alpha \\ar[r] & \\ldots \\\\", "K^{n - 1} \\ar[r] & K^n \\ar[r] & K^{n + 1} \\ar[r] & \\ldots", "}", "$$", "The base case is $n = b + 1$ where we can take $F^i = 0$ for all $i$.", "Induction step. Let $C^\\bullet$ be the cone on $\\alpha$", "(Derived Categories, Definition \\ref{derived-definition-cone}).", "The long exact sequence", "of cohomology shows that $H^i(C^\\bullet) = 0$ for $i \\geq n$.", "By Lemma \\ref{lemma-cone-pseudo-coherent} we see that $C^\\bullet$", "is $(n - 1)$-pseudo-coherent. By Lemma \\ref{lemma-finite-cohomology}", "we see that $H^{n - 1}(C^\\bullet)$ is a finite $R$-module.", "Choose a finite free $R$-module $F^{n - 1}$ and a map", "$\\beta : F^{n - 1} \\to C^{n - 1}$ such that the composition", "$F^{n - 1} \\to C^{n - 1} \\to C^n$ is zero and such that $F^{n - 1}$", "surjects onto $H^{n - 1}(C^\\bullet)$. Since $C^{n - 1} = K^{n - 1} \\oplus F^n$", "we can write $\\beta = (\\alpha^{n - 1}, -d^{n - 1})$. The vanishing of the", "composition $F^{n - 1} \\to C^{n - 1} \\to C^n$ implies", "these maps fit into a morphism of complexes", "$$", "\\xymatrix{", "& F^{n - 1} \\ar[d]^{\\alpha^{n - 1}} \\ar[r]_{d^{n - 1}} &", "F^n \\ar[r] \\ar[d]^\\alpha &", "F^{n + 1} \\ar[d]^\\alpha \\ar[r] & \\ldots \\\\", "\\ldots \\ar[r] &", "K^{n - 1} \\ar[r] & K^n \\ar[r] & K^{n + 1} \\ar[r] & \\ldots", "}", "$$", "Moreover, these maps define a morphism of distinguished triangles", "$$", "\\xymatrix{", "(F^n \\to \\ldots) \\ar[r] \\ar[d] &", "(F^{n - 1} \\to \\ldots) \\ar[r] \\ar[d] &", "F^{n - 1} \\ar[r] \\ar[d]_\\beta &", "(F^n \\to \\ldots)[1] \\ar[d] \\\\", "(F^n \\to \\ldots) \\ar[r] &", "K^\\bullet \\ar[r] &", "C^\\bullet \\ar[r] &", "(F^n \\to \\ldots)[1]", "}", "$$", "Hence our choice of $\\beta$ implies that the map of complexes", "$(F^{n - 1} \\to \\ldots) \\to K^\\bullet$ induces an isomorphism on", "cohomology in degrees $\\geq n$ and a surjection in degree $n - 1$.", "This finishes the proof of the lemma." ], "refs": [ "derived-definition-cone", "more-algebra-lemma-cone-pseudo-coherent", "more-algebra-lemma-finite-cohomology" ], "ref_ids": [ 1978, 10145, 10146 ] } ], "ref_ids": [] }, { "id": 10149, "type": "theorem", "label": "more-algebra-lemma-two-out-of-three-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-two-out-of-three-pseudo-coherent", "contents": [ "Let $R$ be a ring. Let $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$", "be a distinguished triangle in $D(R)$. If two out of three of", "$K^\\bullet, L^\\bullet, M^\\bullet$ are", "pseudo-coherent then the third is also pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemmas \\ref{lemma-cone-pseudo-coherent} and \\ref{lemma-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-cone-pseudo-coherent", "more-algebra-lemma-pseudo-coherent" ], "ref_ids": [ 10145, 10148 ] } ], "ref_ids": [] }, { "id": 10150, "type": "theorem", "label": "more-algebra-lemma-recognize-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-recognize-pseudo-coherent", "contents": [ "Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules.", "Let $m \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item If $H^i(K^\\bullet) = 0$ for all $i \\geq m$, then", "$K^\\bullet$ is $m$-pseudo-coherent.", "\\item If $H^i(K^\\bullet) = 0$ for $i > m$ and $H^m(K^\\bullet)$ is", "a finite $R$-module, then $K^\\bullet$ is $m$-pseudo-coherent.", "\\item If $H^i(K^\\bullet) = 0$ for $i > m + 1$, the module", "$H^{m + 1}(K^\\bullet)$ is of finite presentation, and", "$H^m(K^\\bullet)$ is of finite type, then $K^\\bullet$ is", "$m$-pseudo-coherent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove (3). Set $M = H^{m + 1}(K^\\bullet)$.", "Note that $\\tau_{\\geq m + 1}K^\\bullet$ is quasi-isomorphic to", "$M[- m - 1]$. By", "Lemma \\ref{lemma-n-pseudo-module}", "we see that $M[- m - 1]$ is $m$-pseudo-coherent. Since we have", "the distinguished triangle", "$$", "(\\tau_{\\leq m}K^\\bullet, K^\\bullet, \\tau_{\\geq m + 1}K^\\bullet)", "$$", "(Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}) by", "Lemma \\ref{lemma-cone-pseudo-coherent}", "it suffices to prove that $\\tau_{\\leq m}K^\\bullet$ is pseudo-coherent.", "By assumption $H^m(\\tau_{\\leq m}K^\\bullet)$ is a finite type $R$-module.", "Hence we can find a finite free $R$-module $E$ and a map", "$E \\to \\Ker(d_K^m)$ such that the composition", "$E \\to \\Ker(d_K^m) \\to H^m(\\tau_{\\leq m}K^\\bullet)$ is surjective.", "Then $E[-m] \\to \\tau_{\\leq m}K^\\bullet$ witnesses the fact", "that $\\tau_{\\leq m}K^\\bullet$ is $m$-pseudo-coherent." ], "refs": [ "more-algebra-lemma-n-pseudo-module", "derived-remark-truncation-distinguished-triangle", "more-algebra-lemma-cone-pseudo-coherent" ], "ref_ids": [ 10147, 2016, 10145 ] } ], "ref_ids": [] }, { "id": 10151, "type": "theorem", "label": "more-algebra-lemma-summands-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-summands-pseudo-coherent", "contents": [ "Let $R$ be a ring. Let $m \\in \\mathbf{Z}$. If $K^\\bullet \\oplus L^\\bullet$", "is $m$-pseudo-coherent (resp.\\ pseudo-coherent)", "so are $K^\\bullet$ and $L^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "In this proof we drop the superscript ${}^\\bullet$.", "Assume that $K \\oplus L$ is $m$-pseudo-coherent.", "It is clear that $K, L \\in D^{-}(R)$.", "Note that there is a distinguished triangle", "$$", "(K \\oplus L, K \\oplus L, L \\oplus L[1]) =", "(K, K, 0) \\oplus (L, L, L \\oplus L[1])", "$$", "see", "Derived Categories, Lemma \\ref{derived-lemma-direct-sum-triangles}.", "By", "Lemma \\ref{lemma-cone-pseudo-coherent}", "we see that $L \\oplus L[1]$ is $m$-pseudo-coherent.", "Hence also $L[1] \\oplus L[2]$ is $m$-pseudo-coherent.", "By induction $L[n] \\oplus L[n + 1]$ is $m$-pseudo-coherent.", "By", "Lemma \\ref{lemma-recognize-pseudo-coherent}", "we see that $L[n]$ is $m$-pseudo-coherent for large $n$.", "Hence working backwards, using the distinguished triangles", "$$", "(L[n], L[n] \\oplus L[n - 1], L[n - 1])", "$$", "we conclude that $L[n], L[n - 1], \\ldots, L$ are $m$-pseudo-coherent", "as desired. The pseudo-coherent case follows from this and", "Lemma \\ref{lemma-pseudo-coherent}." ], "refs": [ "derived-lemma-direct-sum-triangles", "more-algebra-lemma-cone-pseudo-coherent", "more-algebra-lemma-recognize-pseudo-coherent", "more-algebra-lemma-pseudo-coherent" ], "ref_ids": [ 1765, 10145, 10150, 10148 ] } ], "ref_ids": [] }, { "id": 10152, "type": "theorem", "label": "more-algebra-lemma-complex-pseudo-coherent-modules", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-complex-pseudo-coherent-modules", "contents": [ "Let $R$ be a ring. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a bounded", "above complex of $R$-modules such that $K^i$ is $(m - i)$-pseudo-coherent", "for all $i$. Then $K^\\bullet$ is $m$-pseudo-coherent.", "In particular, if $K^\\bullet$ is a bounded above complex of", "pseudo-coherent $R$-modules, then $K^\\bullet$ is pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "We may replace $K^\\bullet$ by $\\sigma_{\\geq m - 1}K^\\bullet$ (for example) and", "hence assume that $K^\\bullet$ is bounded.", "Then the complex $K^\\bullet$ is $m$-pseudo-coherent as each", "$K^i[-i]$ is $m$-pseudo-coherent by induction on the length of the", "complex: use Lemma \\ref{lemma-cone-pseudo-coherent}", "and the stupid truncations.", "For the final statement, it suffices to prove that", "$K^\\bullet$ is $m$-pseudo-coherent for all $m \\in \\mathbf{Z}$, see", "Lemma \\ref{lemma-pseudo-coherent}.", "This follows from the first part." ], "refs": [ "more-algebra-lemma-cone-pseudo-coherent", "more-algebra-lemma-pseudo-coherent" ], "ref_ids": [ 10145, 10148 ] } ], "ref_ids": [] }, { "id": 10153, "type": "theorem", "label": "more-algebra-lemma-cohomology-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cohomology-pseudo-coherent", "contents": [ "Let $R$ be a ring. Let $m \\in \\mathbf{Z}$.", "Let $K^\\bullet \\in D^{-}(R)$ such that $H^i(K^\\bullet)$ is", "$(m - i)$-pseudo-coherent (resp.\\ pseudo-coherent) for all $i$.", "Then $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)." ], "refs": [], "proofs": [ { "contents": [ "Assume $K^\\bullet$ is an object of $D^{-}(R)$ such that", "each $H^i(K^\\bullet)$ is $(m - i)$-pseudo-coherent.", "Let $n$ be the largest integer such that $H^n(K^\\bullet)$ is nonzero.", "We will prove the lemma by induction on $n$.", "If $n < m$, then $K^\\bullet$ is $m$-pseudo-coherent by", "Lemma \\ref{lemma-recognize-pseudo-coherent}.", "If $n \\geq m$, then we have the distinguished triangle", "$$", "(\\tau_{\\leq n - 1}K^\\bullet, K^\\bullet, H^n(K^\\bullet)[-n])", "$$", "(Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle})", "Since $H^n(K^\\bullet)[-n]$ is $m$-pseudo-coherent by assumption, we", "can use", "Lemma \\ref{lemma-cone-pseudo-coherent}", "to see that it suffices to prove that $\\tau_{\\leq n - 1}K^\\bullet$", "is $m$-pseudo-coherent. By induction on $n$ we win. (The pseudo-coherent", "case follows from this and", "Lemma \\ref{lemma-pseudo-coherent}.)" ], "refs": [ "more-algebra-lemma-recognize-pseudo-coherent", "derived-remark-truncation-distinguished-triangle", "more-algebra-lemma-cone-pseudo-coherent", "more-algebra-lemma-pseudo-coherent" ], "ref_ids": [ 10150, 2016, 10145, 10148 ] } ], "ref_ids": [] }, { "id": 10154, "type": "theorem", "label": "more-algebra-lemma-finite-push-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finite-push-pseudo-coherent", "contents": [ "Let $A \\to B$ be a ring map. Assume that $B$ is pseudo-coherent as an", "$A$-module. Let $K^\\bullet$ be a complex of $B$-modules.", "The following are equivalent", "\\begin{enumerate}", "\\item $K^\\bullet$ is $m$-pseudo-coherent", "as a complex of $B$-modules, and", "\\item $K^\\bullet$ is $m$-pseudo-coherent", "as a complex of $A$-modules.", "\\end{enumerate}", "The same equivalence holds for pseudo-coherence." ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Choose a bounded complex of finite free $B$-modules", "$E^\\bullet$ and a map $\\alpha : E^\\bullet \\to K^\\bullet$ which is", "an isomorphism on cohomology in degrees $> m$ and a surjection in degree $m$.", "Consider the distinguished triangle", "$(E^\\bullet, K^\\bullet, C(\\alpha)^\\bullet)$. By", "Lemma \\ref{lemma-recognize-pseudo-coherent}", "$C(\\alpha)^\\bullet$ is $m$-pseudo-coherent as a complex of", "$A$-modules. Hence it suffices to prove that $E^\\bullet$ is", "pseudo-coherent as a complex of $A$-modules, which follows from", "Lemma \\ref{lemma-complex-pseudo-coherent-modules}.", "The pseudo-coherent case of (1) $\\Rightarrow$ (2) follows from this and", "Lemma \\ref{lemma-pseudo-coherent}.", "\\medskip\\noindent", "Assume (2). Let $n$ be the largest integer such that $H^n(K^\\bullet) \\not = 0$.", "We will prove that $K^\\bullet$ is $m$-pseudo-coherent as a complex", "of $B$-modules by induction on $n - m$. The case $n < m$ follows from", "Lemma \\ref{lemma-recognize-pseudo-coherent}.", "Choose a bounded complex of finite free $A$-modules $E^\\bullet$ and a", "map $\\alpha : E^\\bullet \\to K^\\bullet$ which is an isomorphism on", "cohomology in degrees $> m$ and a surjection in degree $m$.", "Consider the induced map of complexes", "$$", "\\alpha \\otimes 1 : E^\\bullet \\otimes_A B \\to K^\\bullet.", "$$", "Note that $C(\\alpha \\otimes 1)^\\bullet$ is acyclic in degrees", "$\\geq n$ as $H^n(E) \\to H^n(E^\\bullet \\otimes_A B) \\to H^n(K^\\bullet)$", "is surjective by construction and since $H^i(E^\\bullet \\otimes_A B) = 0$", "for $i > n$ by the spectral sequence of", "Example \\ref{example-tor}.", "On the other hand, $C(\\alpha \\otimes 1)^\\bullet$", "is $m$-pseudo-coherent as a complex of $A$-modules because", "both $K^\\bullet$ and $E^\\bullet \\otimes_A B$ (see", "Lemma \\ref{lemma-complex-pseudo-coherent-modules})", "are so, see", "Lemma \\ref{lemma-cone-pseudo-coherent}.", "Hence by induction we see that $C(\\alpha \\otimes 1)^\\bullet$", "is $m$-pseudo-coherent as a complex of $B$-modules. Finally", "another application of", "Lemma \\ref{lemma-cone-pseudo-coherent}", "shows that $K^\\bullet$ is $m$-pseudo-coherent as a complex of $B$-modules", "(as clearly $E^\\bullet \\otimes_A B$ is pseudo-coherent as a complex", "of $B$-modules). The pseudo-coherent case", "of (2) $\\Rightarrow$ (1) follows from this and", "Lemma \\ref{lemma-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-recognize-pseudo-coherent", "more-algebra-lemma-complex-pseudo-coherent-modules", "more-algebra-lemma-pseudo-coherent", "more-algebra-lemma-recognize-pseudo-coherent", "more-algebra-lemma-complex-pseudo-coherent-modules", "more-algebra-lemma-cone-pseudo-coherent", "more-algebra-lemma-cone-pseudo-coherent", "more-algebra-lemma-pseudo-coherent" ], "ref_ids": [ 10150, 10152, 10148, 10150, 10152, 10145, 10145, 10148 ] } ], "ref_ids": [] }, { "id": 10155, "type": "theorem", "label": "more-algebra-lemma-pull-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pull-pseudo-coherent", "contents": [ "Let $A \\to B$ be a ring map.", "Let $K^\\bullet$ be an $m$-pseudo-coherent (resp.\\ pseudo-coherent)", "complex of $A$-modules. Then $K^\\bullet \\otimes_A^{\\mathbf{L}} B$", "is an $m$-pseudo-coherent (resp.\\ pseudo-coherent) complex of $B$-modules." ], "refs": [], "proofs": [ { "contents": [ "First we note that the statement of the lemma makes sense as", "$K^\\bullet$ is bounded above and hence $K^\\bullet \\otimes_A^{\\mathbf{L}} B$", "is defined by Equation (\\ref{equation-derived-tensor-algebra}).", "Having said this, choose a bounded complex $E^\\bullet$", "of finite free $A$-modules and $\\alpha : E^\\bullet \\to K^\\bullet$", "with $H^i(\\alpha)$ an isomorphism for $i > m$ and surjective for", "$i = m$. Then the cone $C(\\alpha)^\\bullet$ is acyclic in degrees", "$\\geq m$. Since $-\\otimes_A^{\\mathbf{L}} B$ is an exact functor", "we get a distinguished triangle", "$$", "(E^\\bullet \\otimes_A^{\\mathbf{L}} B, K^\\bullet \\otimes_A^{\\mathbf{L}} B,", "C(\\alpha)^\\bullet \\otimes_A^{\\mathbf{L}} B)", "$$", "of complexes of $B$-modules. By the dual to", "Derived Categories, Lemma \\ref{derived-lemma-negative-vanishing}", "we see that $H^i(C(\\alpha)^\\bullet \\otimes_A^{\\mathbf{L}} B) = 0$", "for $i \\geq m$. Since $E^\\bullet$ is a complex of projective $R$-modules", "we see that $E^\\bullet \\otimes_A^{\\mathbf{L}} B = E^\\bullet \\otimes_A B$", "and hence", "$$", "E^\\bullet \\otimes_A B", "\\longrightarrow", "K^\\bullet \\otimes_A^{\\mathbf{L}} B", "$$", "is a morphism of complexes of $B$-modules that witnesses the", "fact that $K^\\bullet \\otimes_A^{\\mathbf{L}} B$ is $m$-pseudo-coherent.", "The case of pseudo-coherent complexes follows from the case", "of $m$-pseudo-coherent complexes via", "Lemma \\ref{lemma-pseudo-coherent}." ], "refs": [ "derived-lemma-negative-vanishing", "more-algebra-lemma-pseudo-coherent" ], "ref_ids": [ 1839, 10148 ] } ], "ref_ids": [] }, { "id": 10156, "type": "theorem", "label": "more-algebra-lemma-flat-base-change-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-base-change-pseudo-coherent", "contents": [ "Let $A \\to B$ be a flat ring map.", "Let $M$ be an $m$-pseudo-coherent (resp.\\ pseudo-coherent)", "$A$-module. Then $M \\otimes_A B$", "is an $m$-pseudo-coherent (resp.\\ pseudo-coherent) $B$-module." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Lemma \\ref{lemma-pull-pseudo-coherent}", "and the fact that $M \\otimes_A^{\\mathbf{L}} B = M \\otimes_A B$", "because $B$ is flat over $A$." ], "refs": [ "more-algebra-lemma-pull-pseudo-coherent" ], "ref_ids": [ 10155 ] } ], "ref_ids": [] }, { "id": 10157, "type": "theorem", "label": "more-algebra-lemma-glue-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-glue-pseudo-coherent", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be elements which", "generate the unit ideal. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$", "be a complex of $R$-modules. If for each $i$ the complex", "$K^\\bullet \\otimes_R R_{f_i}$ is $m$-pseudo-coherent", "(resp.\\ pseudo-coherent), then $K^\\bullet$ is $m$-pseudo-coherent", "(resp.\\ pseudo-coherent)." ], "refs": [], "proofs": [ { "contents": [ "We will use without further mention that $- \\otimes_R R_{f_i}$ is", "an exact functor and that therefore", "$$", "H^i(K^\\bullet)_{f_i} =", "H^i(K^\\bullet) \\otimes_R R_{f_i} = H^i(K^\\bullet \\otimes_R R_{f_i}).", "$$", "Assume $K^\\bullet \\otimes_R R_{f_i}$ is $m$-pseudo-coherent", "for $i = 1, \\ldots, r$. Let $n \\in \\mathbf{Z}$ be the largest", "integer such that $H^n(K^\\bullet \\otimes_R R_{f_i})$ is nonzero", "for some $i$. This implies in particular that $H^i(K^\\bullet) = 0$", "for $i > n$ (and that $H^n(K^\\bullet) \\not = 0$) see", "Algebra, Lemma \\ref{algebra-lemma-cover}.", "We will prove the lemma by induction on $n - m$.", "If $n < m$, then the lemma is true by", "Lemma \\ref{lemma-recognize-pseudo-coherent}.", "If $n \\geq m$, then $H^n(K^\\bullet)_{f_i}$ is a finite $R_{f_i}$-module", "for each $i$, see", "Lemma \\ref{lemma-finite-cohomology}.", "Hence $H^n(K^\\bullet)$ is a finite $R$-module, see", "Algebra, Lemma \\ref{algebra-lemma-cover}.", "Choose a finite free $R$-module $E$ and a surjection $E \\to H^n(K^\\bullet)$.", "As $E$ is projective we can lift this to a map of complexes", "$\\alpha : E[-n] \\to K^\\bullet$. Then the cone $C(\\alpha)^\\bullet$ has", "vanishing cohomology in degrees $\\geq n$. On the other hand, the", "complexes $C(\\alpha)^\\bullet \\otimes_R R_{f_i}$ are $m$-pseudo-coherent", "for each $i$, see", "Lemma \\ref{lemma-cone-pseudo-coherent}.", "Hence by induction we see that $C(\\alpha)^\\bullet$ is $m$-pseudo-coherent", "as a complex of $R$-modules. Applying", "Lemma \\ref{lemma-cone-pseudo-coherent}", "once more we conclude." ], "refs": [ "algebra-lemma-cover", "more-algebra-lemma-recognize-pseudo-coherent", "more-algebra-lemma-finite-cohomology", "algebra-lemma-cover", "more-algebra-lemma-cone-pseudo-coherent", "more-algebra-lemma-cone-pseudo-coherent" ], "ref_ids": [ 411, 10150, 10146, 411, 10145, 10145 ] } ], "ref_ids": [] }, { "id": 10158, "type": "theorem", "label": "more-algebra-lemma-flat-descent-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-descent-pseudo-coherent", "contents": [ "Let $R$ be a ring. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$", "be a complex of $R$-modules. Let $R \\to R'$ be a faithfully flat", "ring map. If the complex $K^\\bullet \\otimes_R R'$ is $m$-pseudo-coherent", "(resp.\\ pseudo-coherent), then $K^\\bullet$ is $m$-pseudo-coherent", "(resp.\\ pseudo-coherent)." ], "refs": [], "proofs": [ { "contents": [ "We will use without further mention that $- \\otimes_R R'$ is", "an exact functor and that therefore", "$$", "H^i(K^\\bullet) \\otimes_R R' = H^i(K^\\bullet \\otimes_R R').", "$$", "Assume $K^\\bullet \\otimes_R R'$ is $m$-pseudo-coherent.", "Let $n \\in \\mathbf{Z}$ be the largest integer such that", "$H^n(K^\\bullet)$ is nonzero; then $n$ is also the largest integer", "such that $H^n(K^\\bullet \\otimes_R R')$ is nonzero.", "We will prove the lemma by induction on $n - m$.", "If $n < m$, then the lemma is true by", "Lemma \\ref{lemma-recognize-pseudo-coherent}.", "If $n \\geq m$, then $H^n(K^\\bullet) \\otimes_R R'$ is a finite", "$R'$-module, see", "Lemma \\ref{lemma-finite-cohomology}.", "Hence $H^n(K^\\bullet)$ is a finite $R$-module, see", "Algebra, Lemma \\ref{algebra-lemma-descend-properties-modules}.", "Choose a finite free $R$-module $E$ and a surjection $E \\to H^n(K^\\bullet)$.", "As $E$ is projective we can lift this to a map of complexes", "$\\alpha : E[-n] \\to K^\\bullet$. Then the cone $C(\\alpha)^\\bullet$ has", "vanishing cohomology in degrees $\\geq n$. On the other hand, the", "complex $C(\\alpha)^\\bullet \\otimes_R R'$ is $m$-pseudo-coherent, see", "Lemma \\ref{lemma-cone-pseudo-coherent}.", "Hence by induction we see that $C(\\alpha)^\\bullet$ is $m$-pseudo-coherent", "as a complex of $R$-modules. Applying", "Lemma \\ref{lemma-cone-pseudo-coherent}", "once more we conclude." ], "refs": [ "more-algebra-lemma-recognize-pseudo-coherent", "more-algebra-lemma-finite-cohomology", "algebra-lemma-descend-properties-modules", "more-algebra-lemma-cone-pseudo-coherent", "more-algebra-lemma-cone-pseudo-coherent" ], "ref_ids": [ 10150, 10146, 819, 10145, 10145 ] } ], "ref_ids": [] }, { "id": 10159, "type": "theorem", "label": "more-algebra-lemma-tensor-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-tensor-pseudo-coherent", "contents": [ "Let $R$ be a ring. Let $K, L$ be objects of $D(R)$.", "\\begin{enumerate}", "\\item If $K$ is $n$-pseudo-coherent and $H^i(K) = 0$ for $i > a$", "and $L$ is $m$-pseudo-coherent and $H^j(L) = 0$ for $j > b$, then", "$K \\otimes_R^\\mathbf{L} L$ is $t$-pseudo-coherent with $t = \\max(m + a, n + b)$.", "\\item If $K$ and $L$ are pseudo-coherent, then", "$K \\otimes_R^\\mathbf{L} L$ is pseudo-coherent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). We may assume there exist bounded complexes $K^\\bullet$", "and $L^\\bullet$ of finite free $R$-modules and maps", "$\\alpha : K^\\bullet \\to K$ and $\\beta : L^\\bullet \\to L$ with", "$H^i(\\alpha)$ and isomorphism for $i > n$ and surjective for $i = n$ and with", "$H^i(\\beta)$ and isomorphism for $i > m$ and surjective for $i = m$.", "Then the map", "$$", "\\alpha \\otimes^\\mathbf{L} \\beta :", "\\text{Tot}(K^\\bullet \\otimes_R L^\\bullet)", "\\to K \\otimes_R^\\mathbf{L} L", "$$", "induces isomorphisms on cohomology in degree $i$ for", "$i > t$ and a surjection for $i = t$. This follows from the", "spectral sequence of tors (details omitted). Part (2) follows", "from part (1) and Lemma \\ref{lemma-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-pseudo-coherent" ], "ref_ids": [ 10148 ] } ], "ref_ids": [] }, { "id": 10160, "type": "theorem", "label": "more-algebra-lemma-Noetherian-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Noetherian-pseudo-coherent", "contents": [ "Let $R$ be a Noetherian ring. Then", "\\begin{enumerate}", "\\item A complex of $R$-modules $K^\\bullet$ is $m$-pseudo-coherent", "if and only if $K^\\bullet \\in D^{-}(R)$ and", "$H^i(K^\\bullet)$ is a finite $R$-module for $i \\geq m$.", "\\item A complex of $R$-modules $K^\\bullet$ is pseudo-coherent", "if and only if $K^\\bullet \\in D^{-}(R)$ and", "$H^i(K^\\bullet)$ is a finite $R$-module for all $i$.", "\\item An $R$-module is pseudo-coherent if and only if it is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In", "Algebra, Lemma \\ref{algebra-lemma-resolution-by-finite-free}", "we have seen that any finite $R$-module is pseudo-coherent.", "On the other hand, a pseudo-coherent module is finite, see", "Lemma \\ref{lemma-n-pseudo-module}.", "Hence (3) holds. Suppose that $K^\\bullet$ is an $m$-pseudo-coherent complex.", "Then there exists a bounded complex of finite free $R$-modules $E^\\bullet$", "such that $H^i(K^\\bullet)$ is isomorphic to $H^i(E^\\bullet)$ for", "$i > m$ and such that $H^m(K^\\bullet)$ is a quotient of $H^m(E^\\bullet)$.", "Thus it is clear that each $H^i(K^\\bullet)$, $i \\geq m$ is a finite module.", "The converse implication in (1) follows from", "Lemma \\ref{lemma-cohomology-pseudo-coherent}", "and part (3).", "Part (2) follows from (1) and", "Lemma \\ref{lemma-pseudo-coherent}." ], "refs": [ "algebra-lemma-resolution-by-finite-free", "more-algebra-lemma-n-pseudo-module", "more-algebra-lemma-cohomology-pseudo-coherent", "more-algebra-lemma-pseudo-coherent" ], "ref_ids": [ 761, 10147, 10153, 10148 ] } ], "ref_ids": [] }, { "id": 10161, "type": "theorem", "label": "more-algebra-lemma-coherent-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-coherent-pseudo-coherent", "contents": [ "Let $R$ be a coherent ring", "(Algebra, Definition \\ref{algebra-definition-coherent}).", "Let $K \\in D^-(R)$. The following are equivalent", "\\begin{enumerate}", "\\item $K$ is $m$-pseudo-coherent,", "\\item $H^m(K)$ is a finite $R$-module and $H^i(K)$ is coherent for $i > m$, and", "\\item $H^m(K)$ is a finite $R$-module and", "$H^i(K)$ is finitely presented for $i > m$.", "\\end{enumerate}", "Thus $K$ is pseudo-coherent if and only if $H^i(K)$", "is a coherent module for all $i$." ], "refs": [ "algebra-definition-coherent" ], "proofs": [ { "contents": [ "Recall that an $R$-module $M$ is coherent if and only if it is of finite", "presentation (Algebra, Lemma \\ref{algebra-lemma-coherent-ring}).", "This explains the equivalence of (2) and (3). If so and if we choose an exact", "sequence $0 \\to N \\to R^{\\oplus m} \\to M \\to 0$, then $N$ is coherent by", "Algebra, Lemma \\ref{algebra-lemma-coherent}. Thus in this case, repeating", "this procedure with $N$ we find a resolution", "$$", "\\ldots \\to R^{\\oplus n} \\to R^{\\oplus m} \\to M \\to 0", "$$", "by finite free $R$-modules. In other words, $M$ is pseudo-coherent.", "The equivalence of (1) and (2) follows from this and", "Lemmas \\ref{lemma-cohomology-pseudo-coherent} and", "\\ref{lemma-n-pseudo-module}.", "The final assertion follows from the equivalence of (1) and (2)", "combined with Lemma \\ref{lemma-pseudo-coherent}." ], "refs": [ "algebra-lemma-coherent-ring", "algebra-lemma-coherent", "more-algebra-lemma-cohomology-pseudo-coherent", "more-algebra-lemma-n-pseudo-module", "more-algebra-lemma-pseudo-coherent" ], "ref_ids": [ 843, 842, 10153, 10147, 10148 ] } ], "ref_ids": [ 1499 ] }, { "id": 10162, "type": "theorem", "label": "more-algebra-lemma-pseudo-coherence-colimit-ext", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pseudo-coherence-colimit-ext", "contents": [ "Let $R$ be a ring. Let $M = \\colim M_i$ be a filtered colimit of $R$-modules.", "Let $K \\in D(R)$ be $m$-pseudo-coherent. Then", "$\\colim \\Ext^n_R(K, M_i) = \\Ext^n_R(K, M)$ for $n < -m$ and", "$\\colim \\Ext^{-m}_R(K, M_i) \\to \\Ext^{-m}_R(K, M)$ is injective." ], "refs": [], "proofs": [ { "contents": [ "By definition we can find a distinguished triangle", "$$", "E \\to K \\to L \\to E[1]", "$$", "in $D(R)$ such that $E$ is represented by a bounded complex of finite free", "$R$-modules and such that $H^i(L) = 0$ for $i \\geq m$. Then", "$\\Ext^n_R(L, N) = 0$ for any $R$-module $N$ and $n \\leq -m$, see", "Derived Categories, Lemma \\ref{derived-lemma-negative-exts}.", "By the long exact sequence of $\\Ext$ associated to the distinguished", "triangle we see that $\\Ext^n_R(K, N) \\to \\Ext^n_R(E, N)$ is an isomorphism", "for $n < -m$ and injective for $n = -m$. Thus it suffices to prove that", "$M \\mapsto \\Ext_R^n(E, M)$ commutes with filtered colimits", "when $E$ can be represented by a bounded complex of finite free $R$-modules", "$E^\\bullet$. The modules $\\Ext^n_R(E, M)$ are computed by", "the complex $\\Hom_R(E^\\bullet, M)$, see Derived Categories, Lemma", "\\ref{derived-lemma-morphisms-from-projective-complex}.", "The functor $M \\mapsto \\Hom_R(E^p, M)$ commutes with filtered colimits", "as $E^p$ is finite free. Thus", "$\\Hom_R(E^\\bullet, M) = \\colim \\Hom_R(E^\\bullet, M_i)$ as complexes.", "Since filtered colimits are exact (Algebra, Lemma", "\\ref{algebra-lemma-directed-colimit-exact}) we conclude." ], "refs": [ "derived-lemma-negative-exts", "derived-lemma-morphisms-from-projective-complex", "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 1893, 1862, 343 ] } ], "ref_ids": [] }, { "id": 10163, "type": "theorem", "label": "more-algebra-lemma-characterize-pseudo-coherent-colimit-ext", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-characterize-pseudo-coherent-colimit-ext", "contents": [ "Let $R$ be a ring. Let $K \\in D^-(R)$. Let $m \\in \\mathbf{Z}$.", "Then $K$ is $m$-pseudo-coherent if and only if", "for any filtered colimit $M = \\colim M_i$ of $R$-modules we have", "$\\colim \\Ext^n_R(K, M_i) = \\Ext^n_R(K, M)$ for $n < -m$ and", "$\\colim \\Ext^{-m}_R(K, M_i) \\to \\Ext^{-m}_R(K, M)$ is injective." ], "refs": [], "proofs": [ { "contents": [ "One implication was shown in Lemma \\ref{lemma-pseudo-coherence-colimit-ext}.", "Assume for any filtered colimit $M = \\colim M_i$ of $R$-modules we have", "$\\colim \\Ext^n_R(K, M_i) = \\Ext^n_R(K, M)$ for $n < -m$ and", "$\\colim \\Ext^{-m}_R(K, M_i) \\to \\Ext^{-m}_R(K, M)$ is injective.", "We will show $K$ is $m$-pseudo-coherent.", "\\medskip\\noindent", "Let $t$ be the maximal integer such that $H^t(K)$ is nonzero.", "We will use induction on $t$. If $t < m$, then $K$ is $m$-pseudo-coherent", "by Lemma \\ref{lemma-recognize-pseudo-coherent}.", "If $t \\geq m$, then since $\\Hom_R(H^t(K), M) = \\Ext^{-t}_R(K, M)$", "we conclude that $\\colim \\Hom_R(H^t(K), M_i) \\to \\Hom_R(H^t(K), M)$", "is injective for any filtered colimit $M = \\colim M_i$.", "This implies that $H^t(K)$ is a finite $R$-module by Algebra,", "Lemma \\ref{algebra-lemma-characterize-finite-module-hom}.", "Choose a finite free $R$-module $F$ and a surjection", "$F \\to H^t(K)$. We can lift this to a morphism $F[-t] \\to K$", "in $D(R)$ and choose a distinguished triangle", "$$", "F[-t] \\to K \\to L \\to F[-t + 1]", "$$", "in $D(R)$. Then $H^i(L) = 0$ for $i \\geq t$. Moreover, the long", "exact sequence of $\\Ext$ associated to this distinguished triangle", "shows that $L$ inherts the assumption we made on $K$ by a small", "argument we omit. By induction on $t$ we conclude that", "$L$ is $m$-pseudo-coherent. Hence $K$ is $m$-pseudo-coherent", "by Lemma \\ref{lemma-cone-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-pseudo-coherence-colimit-ext", "more-algebra-lemma-recognize-pseudo-coherent", "algebra-lemma-characterize-finite-module-hom", "more-algebra-lemma-cone-pseudo-coherent" ], "ref_ids": [ 10162, 10150, 354, 10145 ] } ], "ref_ids": [] }, { "id": 10164, "type": "theorem", "label": "more-algebra-lemma-pseudo-coherence-and-ext", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pseudo-coherence-and-ext", "contents": [ "Let $R$ be a ring. Let $L$, $M$, $N$ be $R$-modules.", "\\begin{enumerate}", "\\item If $M$ is finitely presented and $L$ is flat, then the canonical map", "$\\Hom_R(M, N) \\otimes_R L \\to \\Hom_R(M, N \\otimes_R L)$", "is an isomorphism.", "\\item If $M$ is $(-m)$-pseudo-coherent and $L$ is flat, then the canonical map", "$\\Ext^i_R(M, N) \\otimes_R L \\to \\Ext^i_R(M, N \\otimes_R L)$", "is an isomorphism for $i < m$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a resolution $F_\\bullet \\to M$ whose terms are free $R$-modules, see", "Algebra, Lemma \\ref{algebra-lemma-resolution-by-finite-free}. The complex", "$\\Hom_R(F_\\bullet, N)$ computes $\\Ext^i_R(M, N)$ and the complex", "$\\Hom_R(F_\\bullet, N \\otimes_R L)$ computes $\\Ext^i_R(M, N \\otimes_R L)$.", "There always is a map of cochain complexes", "$$", "\\Hom_R(F_\\bullet, N) \\otimes_R L", "\\longrightarrow", "\\Hom_R(F_\\bullet, N \\otimes_R L)", "$$", "which induces canonical maps", "$\\Ext^i_R(M, N) \\otimes_R L \\to \\Ext^i_R(M, N \\otimes_R L)$", "for all $i \\geq 0$ (canonical for example in the sense that these maps", "do not depend on the choice of the resolution $F_\\bullet$).", "If $L$ is flat, then the complex $\\Hom_R(F_\\bullet, N) \\otimes_R L$", "computes $\\Ext^i_R(M, N) \\otimes_R L$ since taking cohomology commutes", "with tensoring by $L$.", "\\medskip\\noindent", "Having said all of the above, if $M$ is $(-m)$-pseudo-coherent, then we", "may choose $F_\\bullet$ such that $F_i$ is finite free for $i = 0, \\ldots, m$.", "Then the map of cochain complexes displayed above is an isomorphism in degrees", "$\\leq m$ and hence an isomorphism on cohomology groups in degrees", "$< m$. This proves (2). If $M$ is finitely presented, then $M$ is", "$(-1)$-pseudo-coherent by Lemma \\ref{lemma-n-pseudo-module}", "and we get the result because $\\Hom = \\Ext^0$." ], "refs": [ "algebra-lemma-resolution-by-finite-free", "more-algebra-lemma-n-pseudo-module" ], "ref_ids": [ 761, 10147 ] } ], "ref_ids": [] }, { "id": 10165, "type": "theorem", "label": "more-algebra-lemma-pseudo-coherence-and-base-change-ext", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pseudo-coherence-and-base-change-ext", "contents": [ "Let $R \\to R'$ be a flat ring map. Let $M$, $N$ be $R$-modules.", "\\begin{enumerate}", "\\item If $M$ is a finitely presented $R$-module, then", "$\\Hom_R(M, N) \\otimes_R R' = \\Hom_{R'}(M \\otimes_R R', N \\otimes_R R')$.", "\\item If $M$ is $(-m)$-pseudo-coherent, then", "$\\Ext^i_R(M, N) \\otimes_R R' = \\Ext^i_{R'}(M \\otimes_R R', N \\otimes_R R')$", "for $i < m$.", "\\end{enumerate}", "In particular if $R$ is Noetherian and $M$ is a finite module this", "holds for all $i$." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-flat-base-change-ext} we have", "$\\Ext^i_{R'}(M \\otimes_R R', N \\otimes_R R') =", "\\Ext^i_R(M, N \\otimes_R R')$.", "Combined with Lemma \\ref{lemma-pseudo-coherence-and-ext}", "we conclude (1) and (2) holds. The final statement follows from this", "and Lemma \\ref{lemma-Noetherian-pseudo-coherent}." ], "refs": [ "algebra-lemma-flat-base-change-ext", "more-algebra-lemma-pseudo-coherence-and-ext", "more-algebra-lemma-Noetherian-pseudo-coherent" ], "ref_ids": [ 779, 10164, 10160 ] } ], "ref_ids": [] }, { "id": 10166, "type": "theorem", "label": "more-algebra-lemma-pseudo-coherent-tensor", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pseudo-coherent-tensor", "contents": [ "Let $R$ be a ring. Let $K \\in D^-(R)$. The following are equivalent:", "\\begin{enumerate}", "\\item $K$ is pseudo-coherent,", "\\item for every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules, the", "canonical map", "$$", "\\alpha :", "K \\otimes_R^\\mathbf{L} \\left( \\prod\\nolimits_\\alpha Q_{\\alpha} \\right)", "\\longrightarrow", "\\prod\\nolimits_\\alpha (K \\otimes_R^\\mathbf{L} Q_{\\alpha})", "$$", "is an isomorphism in $D(A)$,", "\\item for every $R$-module $Q$ and every set $A$, the canonical map", "$$", "\\beta : K \\otimes_R^\\mathbf{L} Q^A \\longrightarrow (K \\otimes_R^\\mathbf{L} Q)^A", "$$", "is an isomorphism in $D(A)$, and", "\\item for every set $A$, the canonical map", "$$", "\\gamma : K \\otimes_R^\\mathbf{L} R^A \\longrightarrow K^A", "$$", "is an isomorphism in $D(A)$.", "\\end{enumerate}", "Given $m \\in \\mathbf{Z}$ the following are equivalent", "\\begin{enumerate}", "\\item[(a)] $K$ is $m$-pseudo-coherent,", "\\item[(b)] for every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules,", "with $\\alpha$ as above", "$H^i(\\alpha)$ is an isomorphism for $i > m$ and surjective for $i = m$,", "\\item[(c)] for every $R$-module $Q$ and every set $A$, with $\\beta$ as above", "$H^i(\\beta)$ is an isomorphism for $i > m$ and surjective for $i = m$,", "\\item[(d)] for every set $A$, with $\\gamma$ as above", "$H^i(\\gamma)$ is an isomorphism for $i > m$ and surjective for $i = m$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $K$ is pseudo-coherent, then $K$ can be represented by", "a bounded above complex of finite free $R$-modules.", "Then the derived tensor products are computed by", "tensoring with this complex. Also, products in $D(A)$", "are given by taking products of any choices of representative", "complexes. Hence (1) implies (2), (3), (4) by the corresponding", "fact for modules, see Algebra, Proposition \\ref{algebra-proposition-fp-tensor}.", "\\medskip\\noindent", "In the same way (using the tensor product is right exact)", "the reader shows that (a) implies (b), (c), and (d).", "\\medskip\\noindent", "Assume (4) holds. To show that $K$ is pseudo-coherent", "it suffices to show that $K$ is $m$-pseudo-coherent for", "all $m$ (Lemma \\ref{lemma-pseudo-coherent}). Hence to finish then proof it", "suffices to prove that (d) implies (a).", "\\medskip\\noindent", "Assume (d). Let $i$ be the largest integer such that", "$H^i(K)$ is nonzero. If $i < m$, then we are done.", "If not, then from (d) and the description of products in $D(A)$", "given above we find that $H^i(K) \\otimes_R R^A \\to H^i(K)^A$", "is surjective. Hence $H^i(K)$ is a finitely generated $R$-module by", "Algebra, Proposition \\ref{algebra-proposition-fg-tensor}.", "Thus we may choose a complex $L$ consisting of a single finite", "free module sitting in degree $i$ and a map of complexes $L \\to K$", "such that $H^i(L) \\to H^i(K)$ is surjective. In particular $L$", "satisfies (1), (2), (3), and (4). Choose a distinguished", "triangle", "$$", "L \\to K \\to M \\to L[1]", "$$", "Then we see that $H^j(M) = 0$ for $j \\geq i$.", "On the other hand, $M$ still has property (d) by a small argument", "which we omit. By induction on $i$ we find that $M$ is", "$m$-pseudo-coherent. Hence $K$ is $m$-pseudo-coherent", "by Lemma \\ref{lemma-cone-pseudo-coherent}." ], "refs": [ "algebra-proposition-fp-tensor", "more-algebra-lemma-pseudo-coherent", "algebra-proposition-fg-tensor", "more-algebra-lemma-cone-pseudo-coherent" ], "ref_ids": [ 1416, 10148, 1415, 10145 ] } ], "ref_ids": [] }, { "id": 10167, "type": "theorem", "label": "more-algebra-lemma-detect-cohomology-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-detect-cohomology-pseudo-coherent", "contents": [ "Let $R$ be a ring. Let $K \\in D(R)$ be pseudo-coherent.", "Let $i \\in \\mathbf{Z}$. There exists a finitely presented", "$R$-module $M$ and a map $K \\to M[-i]$ in $D(R)$ which induces", "an injection $H^i(K) \\to M$." ], "refs": [], "proofs": [ { "contents": [ "By Definition \\ref{definition-pseudo-coherent} we may represent", "$K$ by a complex $P^\\bullet$", "of finite free $R$-modules. Set $M = \\Coker(P^{i - 1} \\to P^i)$." ], "refs": [ "more-algebra-definition-pseudo-coherent" ], "ref_ids": [ 10623 ] } ], "ref_ids": [] }, { "id": 10168, "type": "theorem", "label": "more-algebra-lemma-detect-cohomology", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-detect-cohomology", "contents": [ "Let $A$ be a Noetherian ring. Let $K \\in D(A)$ be pseudo-coherent,", "i.e., $K \\in D^-(A)$ with finite cohomology modules.", "Let $\\mathfrak m$ be a maximal ideal of $A$.", "If $H^i(K)/\\mathfrak m H^i(K) \\not = 0$, then there exists a finite", "$A$-module $E$ annihilated by a power of $\\mathfrak m$", "and a map $K \\to E[-i]$ which is nonzero on $H^i(K)$." ], "refs": [], "proofs": [ { "contents": [ "(The equivalent formulation of pseudo-coherence in the statement of the", "lemma is Lemma \\ref{lemma-Noetherian-pseudo-coherent}.)", "Choose $K \\to M[-i]$ as in Lemma \\ref{lemma-detect-cohomology-pseudo-coherent}.", "By Artin-Rees (Algebra, Lemma \\ref{algebra-lemma-Artin-Rees})", "we can find an $n$ such that", "$H^i(K) \\cap \\mathfrak m^n M \\subset \\mathfrak m H^i(K)$.", "Take $E = M/\\mathfrak m^n M$." ], "refs": [ "more-algebra-lemma-Noetherian-pseudo-coherent", "more-algebra-lemma-detect-cohomology-pseudo-coherent", "algebra-lemma-Artin-Rees" ], "ref_ids": [ 10160, 10167, 625 ] } ], "ref_ids": [] }, { "id": 10169, "type": "theorem", "label": "more-algebra-lemma-last-one-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-last-one-flat", "contents": [ "Let $R$ be a ring. Let $K^\\bullet$ be a bounded above complex of", "flat $R$-modules with tor-amplitude in $[a, b]$.", "Then $\\Coker(d_K^{a - 1})$ is a flat $R$-module." ], "refs": [], "proofs": [ { "contents": [ "As $K^\\bullet$ is a bounded above complex of flat modules we see", "that $K^\\bullet \\otimes_R M = K^\\bullet \\otimes_R^{\\mathbf{L}} M$.", "Hence for every $R$-module $M$ the sequence", "$$", "K^{a - 2} \\otimes_R M \\to K^{a - 1} \\otimes_R M \\to K^a \\otimes_R M", "$$", "is exact in the middle. Since", "$K^{a - 2} \\to K^{a - 1} \\to K^a \\to \\Coker(d_K^{a - 1}) \\to 0$", "is a flat resolution this implies that", "$\\text{Tor}_1^R(\\Coker(d_K^{a - 1}), M) = 0$", "for all $R$-modules $M$. This means that", "$\\Coker(d_K^{a - 1})$ is flat, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-flat}." ], "refs": [ "algebra-lemma-characterize-flat" ], "ref_ids": [ 786 ] } ], "ref_ids": [] }, { "id": 10170, "type": "theorem", "label": "more-algebra-lemma-tor-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-tor-amplitude", "contents": [ "Let $R$ be a ring. Let $K^\\bullet$ be an object of $D(R)$.", "Let $a, b \\in \\mathbf{Z}$. The following are equivalent", "\\begin{enumerate}", "\\item $K^\\bullet$ has tor-amplitude in $[a, b]$.", "\\item $K^\\bullet$ is quasi-isomorphic to a complex", "$E^\\bullet$ of flat $R$-modules with $E^i = 0$ for $i \\not \\in [a, b]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (2) holds, then we may compute", "$K^\\bullet \\otimes_R^\\mathbf{L} M = E^\\bullet \\otimes_R M$", "and it is clear that (1) holds.", "Assume that (1) holds. We may replace $K^\\bullet$ by", "a projective resolution with $K^i = 0$ for $i > b$.", "See Derived Categories, Lemma \\ref{derived-lemma-projective-resolutions-exist}.", "Set $E^\\bullet = \\tau_{\\geq a}K^\\bullet$. Everything is clear except", "that $E^a$ is flat which follows immediately from", "Lemma \\ref{lemma-last-one-flat}", "and the definitions." ], "refs": [ "derived-lemma-projective-resolutions-exist", "more-algebra-lemma-last-one-flat" ], "ref_ids": [ 1858, 10169 ] } ], "ref_ids": [] }, { "id": 10171, "type": "theorem", "label": "more-algebra-lemma-bounded-below-tor-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-bounded-below-tor-amplitude", "contents": [ "Let $R$ be a ring. Let $a \\in \\mathbf{Z}$ and let $K$ be an object of $D(R)$.", "The following are equivalent", "\\begin{enumerate}", "\\item $K$ has tor-amplitude in $[a, \\infty]$, and", "\\item $K$ is quasi-isomorphic to a K-flat complex", "$E^\\bullet$ whose terms are flat $R$-modules with", "$E^i = 0$ for $i \\not \\in [a, \\infty]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is immediate. Assume (1) holds.", "First we choose a K-flat complex $K^\\bullet$ with flat terms representing", "$K$, see Lemma \\ref{lemma-K-flat-resolution}.", "For any $R$-module $M$ the cohomology of", "$$", "K^{n - 1} \\otimes_R M \\to K^n \\otimes_R M \\to K^{n + 1} \\otimes_R M", "$$", "computes $H^n(K \\otimes_R^\\mathbf{L} M)$. This is always zero", "for $n < a$. Hence if we apply Lemma \\ref{lemma-last-one-flat}", "to the complex $\\ldots \\to K^{a - 1} \\to K^a \\to K^{a + 1}$", "we conclude that $N = \\Coker(K^{a - 1} \\to K^a)$ is a flat $R$-module.", "We set", "$$", "E^\\bullet = \\tau_{\\geq a}K^\\bullet =", "(\\ldots \\to 0 \\to N \\to K^{a + 1} \\to \\ldots )", "$$", "The kernel $L^\\bullet$ of $K^\\bullet \\to E^\\bullet$ is the complex", "$$", "L^\\bullet = (\\ldots \\to K^{a - 1} \\to I \\to 0 \\to \\ldots)", "$$", "where $I \\subset K^a$ is the image of $K^{a - 1} \\to K^a$.", "Since we have the short exact sequence $0 \\to I \\to K^a \\to N \\to 0$", "we see that $I$ is a flat $R$-module. Thus $L^\\bullet$ is a bounded", "above complex of flat modules, hence K-flat by", "Lemma \\ref{lemma-derived-tor-quasi-isomorphism}.", "It follows that $E^\\bullet$ is K-flat by", "Lemma \\ref{lemma-K-flat-two-out-of-three-ses}." ], "refs": [ "more-algebra-lemma-K-flat-resolution", "more-algebra-lemma-last-one-flat", "more-algebra-lemma-derived-tor-quasi-isomorphism", "more-algebra-lemma-K-flat-two-out-of-three-ses" ], "ref_ids": [ 10131, 10169, 10128, 10127 ] } ], "ref_ids": [] }, { "id": 10172, "type": "theorem", "label": "more-algebra-lemma-cone-tor-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cone-tor-amplitude", "contents": [ "Let $R$ be a ring.", "Let $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$ be a distinguished", "triangle in $D(R)$. Let $a, b \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item If $K^\\bullet$ has tor-amplitude in $[a + 1, b + 1]$ and", "$L^\\bullet$ has tor-amplitude in $[a, b]$ then $M^\\bullet$ has", "tor-amplitude in $[a, b]$.", "\\item If $K^\\bullet, M^\\bullet$ have tor-amplitude in $[a, b]$, then", "$L^\\bullet$ has tor-amplitude in $[a, b]$.", "\\item If $L^\\bullet$ has tor-amplitude in $[a + 1, b + 1]$", "and $M^\\bullet$ has tor-amplitude in $[a, b]$, then", "$K^\\bullet$ has tor-amplitude in $[a + 1, b + 1]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This just follows from the long exact cohomology sequence", "associated to a distinguished triangle and the fact that", "$- \\otimes_R^{\\mathbf{L}} M$ preserves distinguished triangles.", "The easiest one to prove is (2) and the others follow from it by", "translation." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10173, "type": "theorem", "label": "more-algebra-lemma-tor-dimension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-tor-dimension", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Let $d \\geq 0$. The following are equivalent", "\\begin{enumerate}", "\\item $M$ has tor dimension $\\leq d$, and", "\\item there exists a resolution", "$$", "0 \\to F_d \\to \\ldots \\to F_1 \\to F_0 \\to M \\to 0", "$$", "with $F_i$ a flat $R$-module.", "\\end{enumerate}", "In particular an $R$-module has tor dimension $0$ if and only if", "it is a flat $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Assume (2). Then the complex $E^\\bullet$ with $E^{-i} = F_i$", "is quasi-isomorphic to $M$. Hence the Tor dimension of $M$ is", "at most $d$ by", "Lemma \\ref{lemma-tor-amplitude}.", "Conversely, assume (1). Let $P^\\bullet \\to M$ be a projective", "resolution of $M$. By", "Lemma \\ref{lemma-last-one-flat}", "we see that $\\tau_{\\geq -d}P^\\bullet$ is a flat resolution of", "$M$ of length $d$, i.e., (2) holds." ], "refs": [ "more-algebra-lemma-tor-amplitude", "more-algebra-lemma-last-one-flat" ], "ref_ids": [ 10170, 10169 ] } ], "ref_ids": [] }, { "id": 10174, "type": "theorem", "label": "more-algebra-lemma-summands-tor-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-summands-tor-amplitude", "contents": [ "Let $R$ be a ring. Let $a, b \\in \\mathbf{Z}$.", "If $K^\\bullet \\oplus L^\\bullet$ has tor amplitude in $[a, b]$", "so do $K^\\bullet$ and $L^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Clear from the fact that the Tor functors are additive." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10175, "type": "theorem", "label": "more-algebra-lemma-complex-finite-tor-dimension-modules", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-complex-finite-tor-dimension-modules", "contents": [ "Let $R$ be a ring. Let $K^\\bullet$ be a bounded complex of $R$-modules", "such that $K^i$ has tor amplitude in $[a - i, b - i]$ for all $i$.", "Then $K^\\bullet$ has tor amplitude in $[a, b]$. In particular", "if $K^\\bullet$ is a finite complex of $R$-modules of finite tor dimension,", "then $K^\\bullet$ has finite tor dimension." ], "refs": [], "proofs": [ { "contents": [ "Follows by induction on the length of the finite complex: use", "Lemma \\ref{lemma-cone-tor-amplitude}", "and the stupid truncations." ], "refs": [ "more-algebra-lemma-cone-tor-amplitude" ], "ref_ids": [ 10172 ] } ], "ref_ids": [] }, { "id": 10176, "type": "theorem", "label": "more-algebra-lemma-cohomology-tor-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cohomology-tor-amplitude", "contents": [ "Let $R$ be a ring. Let $a, b \\in \\mathbf{Z}$. Let $K^\\bullet \\in D^b(R)$", "such that $H^i(K^\\bullet)$ has tor amplitude in $[a - i, b - i]$", "for all $i$. Then $K^\\bullet$ has tor amplitude in $[a, b]$. In particular", "if $K^\\bullet \\in D^b(R)$ and all its cohomology groups have finite tor", "dimension then $K^\\bullet$ has finite tor dimension." ], "refs": [], "proofs": [ { "contents": [ "Follows by induction on the length of the finite complex: use", "Lemma \\ref{lemma-cone-tor-amplitude}", "and the canonical truncations." ], "refs": [ "more-algebra-lemma-cone-tor-amplitude" ], "ref_ids": [ 10172 ] } ], "ref_ids": [] }, { "id": 10177, "type": "theorem", "label": "more-algebra-lemma-push-tor-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-push-tor-amplitude", "contents": [ "Let $A \\to B$ be a ring map. Let $K^\\bullet$ and $L^\\bullet$ be complexes", "of $B$-modules. Let $a, b, c, d \\in \\mathbf{Z}$. If", "\\begin{enumerate}", "\\item $K^\\bullet$ as a complex of $B$-modules has tor amplitude in $[a, b]$,", "\\item $L^\\bullet$ as a complex of $A$-modules has tor amplitude in $[c, d]$,", "\\end{enumerate}", "then $K^\\bullet \\otimes^\\mathbf{L}_B L^\\bullet$ as a complex of $A$-modules", "has tor amplitude in $[a + c, b + d]$." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $K^\\bullet$ is a complex of flat $B$-modules with $K^i = 0$", "for $i \\not \\in [a, b]$, see Lemma \\ref{lemma-tor-amplitude}.", "Let $M$ be an $A$-module. Choose a free resolution $F^\\bullet \\to M$.", "Then", "$$", "(K^\\bullet \\otimes_B^\\mathbf{L} L^\\bullet) \\otimes_A^{\\mathbf{L}} M =", "\\text{Tot}(\\text{Tot}(K^\\bullet \\otimes_B L^\\bullet) \\otimes_A F^\\bullet) =", "\\text{Tot}(K^\\bullet \\otimes_B \\text{Tot}(L^\\bullet \\otimes_A F^\\bullet))", "$$", "see Homology, Remark \\ref{homology-remark-triple-complex} for the second", "equality. By assumption (2) the complex", "$\\text{Tot}(L^\\bullet \\otimes_A F^\\bullet)$", "has nonzero cohomology only in degrees $[c, d]$. Hence the spectral sequence of", "Homology, Lemma \\ref{homology-lemma-ss-double-complex}", "for the double complex", "$K^\\bullet \\otimes_B \\text{Tot}(L^\\bullet \\otimes_A F^\\bullet)$", "proves that", "$(K^\\bullet \\otimes_B^\\mathbf{L} L^\\bullet) \\otimes_A^{\\mathbf{L}} M$", "has nonzero cohomology only in degrees $[a + c, b + d]$." ], "refs": [ "more-algebra-lemma-tor-amplitude", "homology-remark-triple-complex", "homology-lemma-ss-double-complex" ], "ref_ids": [ 10170, 12191, 12104 ] } ], "ref_ids": [] }, { "id": 10178, "type": "theorem", "label": "more-algebra-lemma-flat-push-tor-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-push-tor-amplitude", "contents": [ "Let $A \\to B$ be a ring map. Assume that $B$ is flat as an", "$A$-module. Let $K^\\bullet$ be a complex of $B$-modules.", "Let $a, b \\in \\mathbf{Z}$. If $K^\\bullet$ as a complex of $B$-modules", "has tor amplitude in $[a, b]$, then $K^\\bullet$ as a complex of", "$A$-modules has tor amplitude in $[a, b]$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-push-tor-amplitude}, but can also", "be seen directly as follows. We have", "$K^\\bullet \\otimes_A^{\\mathbf{L}} M =", "K^\\bullet \\otimes_B^{\\mathbf{L}} (M \\otimes_A B)$", "since any projective resolution of $K^\\bullet$ as a complex of $B$-modules", "is a flat resolution of $K^\\bullet$ as a complex of $A$-modules and", "can be used to compute $K^\\bullet \\otimes_A^{\\mathbf{L}} M$." ], "refs": [ "more-algebra-lemma-push-tor-amplitude" ], "ref_ids": [ 10177 ] } ], "ref_ids": [] }, { "id": 10179, "type": "theorem", "label": "more-algebra-lemma-finite-tor-dimension-push-tor-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finite-tor-dimension-push-tor-amplitude", "contents": [ "Let $A \\to B$ be a ring map. Assume that $B$ has tor dimension $\\leq d$", "as an $A$-module. Let $K^\\bullet$ be a complex of $B$-modules.", "Let $a, b \\in \\mathbf{Z}$. If $K^\\bullet$ as a complex of $B$-modules", "has tor amplitude in $[a, b]$, then $K^\\bullet$ as a complex of", "$A$-modules has tor amplitude in $[a - d, b]$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-push-tor-amplitude}, but can also", "be seen directly as follows.", "Let $M$ be an $A$-module. Choose a free resolution $F^\\bullet \\to M$.", "Then", "$$", "K^\\bullet \\otimes_A^{\\mathbf{L}} M =", "\\text{Tot}(K^\\bullet \\otimes_A F^\\bullet) =", "\\text{Tot}(K^\\bullet \\otimes_B (F^\\bullet \\otimes_A B)) =", "K^\\bullet \\otimes_B^{\\mathbf{L}} (M \\otimes_A^{\\mathbf{L}} B).", "$$", "By our assumption on $B$ as an $A$-module we see that", "$M \\otimes_A^{\\mathbf{L}} B$ has cohomology only in degrees", "$-d, -d + 1, \\ldots, 0$. Because $K^\\bullet$ has tor amplitude in", "$[a, b]$ we see from the spectral sequence in", "Example \\ref{example-tor}", "that $K^\\bullet \\otimes_B^{\\mathbf{L}} (M \\otimes_A^{\\mathbf{L}} B)$", "has cohomology only in degrees $[-d + a, b]$ as desired." ], "refs": [ "more-algebra-lemma-push-tor-amplitude" ], "ref_ids": [ 10177 ] } ], "ref_ids": [] }, { "id": 10180, "type": "theorem", "label": "more-algebra-lemma-pull-tor-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pull-tor-amplitude", "contents": [ "Let $A \\to B$ be a ring map.", "Let $a, b \\in \\mathbf{Z}$.", "Let $K^\\bullet$ be a complex of $A$-modules with tor amplitude in $[a, b]$.", "Then $K^\\bullet \\otimes_A^{\\mathbf{L}} B$ as a complex of $B$-modules", "has tor amplitude in $[a, b]$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-tor-amplitude}", "we can find a quasi-isomorphism $E^\\bullet \\to K^\\bullet$ where", "$E^\\bullet$ is a complex of flat $A$-modules with $E^i = 0$ for", "$i \\not \\in [a, b]$. Then $E^\\bullet \\otimes_A B$ computes", "$K^\\bullet \\otimes_A ^{\\mathbf{L}} B$ by construction and", "each $E^i \\otimes_A B$ is a flat $B$-module by", "Algebra, Lemma \\ref{algebra-lemma-flat-base-change}.", "Hence we conclude by", "Lemma \\ref{lemma-tor-amplitude}." ], "refs": [ "more-algebra-lemma-tor-amplitude", "algebra-lemma-flat-base-change", "more-algebra-lemma-tor-amplitude" ], "ref_ids": [ 10170, 527, 10170 ] } ], "ref_ids": [] }, { "id": 10181, "type": "theorem", "label": "more-algebra-lemma-flat-base-change-finite-tor-dimension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-base-change-finite-tor-dimension", "contents": [ "Let $A \\to B$ be a flat ring map. Let $d \\geq 0$.", "Let $M$ be an $A$-module of tor dimension $\\leq d$.", "Then $M \\otimes_A B$ is a $B$-module of tor dimension $\\leq d$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Lemma \\ref{lemma-pull-tor-amplitude}", "and the fact that $M \\otimes_A^{\\mathbf{L}} B = M \\otimes_A B$", "because $B$ is flat over $A$." ], "refs": [ "more-algebra-lemma-pull-tor-amplitude" ], "ref_ids": [ 10180 ] } ], "ref_ids": [] }, { "id": 10182, "type": "theorem", "label": "more-algebra-lemma-tor-amplitude-localization", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-tor-amplitude-localization", "contents": [ "Let $A \\to B$ be a ring map. Let $K^\\bullet$ be a complex of $B$-modules.", "Let $a, b \\in \\mathbf{Z}$. The following are equivalent", "\\begin{enumerate}", "\\item $K^\\bullet$ has tor amplitude in $[a, b]$ as a complex of $A$-modules,", "\\item $K^\\bullet_\\mathfrak q$ has tor amplitude in $[a, b]$ as a complex", "of $A_\\mathfrak p$-modules for every prime $\\mathfrak q \\subset B$", "with $\\mathfrak p = A \\cap \\mathfrak q$,", "\\item $K^\\bullet_\\mathfrak m$ has tor amplitude in $[a, b]$ as a complex", "of $A_\\mathfrak p$-modules for every maximal ideal $\\mathfrak m \\subset B$", "with $\\mathfrak p = A \\cap \\mathfrak m$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (3) and let $M$ be an $A$-module. Then", "$H^i = H^i(K^\\bullet \\otimes_A^\\mathbf{L} M)$ is a $B$-module and", "$(H^i)_\\mathfrak m =", "H^i(K^\\bullet_\\mathfrak m \\otimes_{A_\\mathfrak p}^\\mathbf{L} M_\\mathfrak p)$.", "Hence $H^i = 0$ for $i \\not \\in [a, b]$ by", "Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}. Thus", "(3) $\\Rightarrow$ (1). We omit the proofs of (1) $\\Rightarrow$ (2)", "and (2) $\\Rightarrow$ (3)." ], "refs": [ "algebra-lemma-characterize-zero-local" ], "ref_ids": [ 410 ] } ], "ref_ids": [] }, { "id": 10183, "type": "theorem", "label": "more-algebra-lemma-glue-tor-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-glue-tor-amplitude", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be elements which", "generate the unit ideal. Let $a, b \\in \\mathbf{Z}$. Let $K^\\bullet$", "be a complex of $R$-modules. If for each $i$ the complex", "$K^\\bullet \\otimes_R R_{f_i}$ has tor amplitude in $[a, b]$,", "then $K^\\bullet$ has tor amplitude in $[a, b]$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from Lemma \\ref{lemma-tor-amplitude-localization}", "but can also be seen directly as follows.", "Note that $- \\otimes_R R_{f_i}$ is an exact functor and that therefore", "$$", "H^i(K^\\bullet)_{f_i} =", "H^i(K^\\bullet) \\otimes_R R_{f_i} = H^i(K^\\bullet \\otimes_R R_{f_i}).", "$$", "and similarly for every $R$-module $M$ we have", "$$", "H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} M)_{f_i} =", "H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} M) \\otimes_R R_{f_i} =", "H^i(K^\\bullet \\otimes_R R_{f_i} \\otimes_{R_{f_i}}^{\\mathbf{L}} M_{f_i}).", "$$", "Hence the result follows from the fact that an $R$-module $N$", "is zero if and only if $N_{f_i}$ is zero for each $i$, see", "Algebra, Lemma \\ref{algebra-lemma-cover}." ], "refs": [ "more-algebra-lemma-tor-amplitude-localization", "algebra-lemma-cover" ], "ref_ids": [ 10182, 411 ] } ], "ref_ids": [] }, { "id": 10184, "type": "theorem", "label": "more-algebra-lemma-flat-descent-tor-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-descent-tor-amplitude", "contents": [ "Let $R$ be a ring. Let $a, b \\in \\mathbf{Z}$. Let $K^\\bullet$", "be a complex of $R$-modules. Let $R \\to R'$ be a faithfully flat", "ring map. If the complex $K^\\bullet \\otimes_R R'$ has tor amplitude", "in $[a, b]$, then $K^\\bullet$ has tor amplitude in $[a, b]$." ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be an $R$-module. Since $R \\to R'$ is flat we see that", "$$", "(M \\otimes_R^{\\mathbf{L}} K^\\bullet) \\otimes_R R'", "=", "((M \\otimes_R R') \\otimes_{R'}^{\\mathbf{L}} (K^\\bullet \\otimes_R R')", "$$", "and taking cohomology commutes with tensoring with $R'$.", "Hence $\\text{Tor}_i^R(M, K^\\bullet) =", "\\text{Tor}_i^{R'}(M \\otimes_R R', K^\\bullet \\otimes_R R')$.", "Since $R \\to R'$ is faithfully flat, the vanishing of", "$\\text{Tor}_i^{R'}(M \\otimes_R R', K^\\bullet \\otimes_R R')$ for", "$i \\not \\in [a, b]$ implies the same thing for", "$\\text{Tor}_i^R(M, K^\\bullet)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10185, "type": "theorem", "label": "more-algebra-lemma-no-change-tor-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-no-change-tor-amplitude", "contents": [ "Given ring maps $R \\to A \\to B$ with $A \\to B$ faithfully flat", "and $K \\in D(A)$ the tor amplitude of $K$ over $R$", "is the same as the tor amplitude of $K \\otimes_A^\\mathbf{L} B$", "over $R$." ], "refs": [], "proofs": [ { "contents": [ "This is true because for an $R$-module $M$ we have", "$H^i(K \\otimes_R^\\mathbf{L} M) \\otimes_A B =", "H^i((K \\otimes_A^\\mathbf{L} B) \\otimes_R^\\mathbf{L} M)$", "for all $i$. Namely, represent $K$ by a complex", "$K^\\bullet$ of $A$-modules and choose a free resolution", "$F^\\bullet \\to M$. Then we have the equality", "$$", "\\text{Tot}(K^\\bullet \\otimes_A B \\otimes_R F^\\bullet) =", "\\text{Tot}(K^\\bullet \\otimes_R F^\\bullet) \\otimes_A B", "$$", "The cohomology groups of the left hand side are", "$H^i((K \\otimes_A^\\mathbf{L} B) \\otimes_R^\\mathbf{L} M)$", "and on the right hand side we obtain", "$H^i(K \\otimes_R^\\mathbf{L} M) \\otimes_A B$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10186, "type": "theorem", "label": "more-algebra-lemma-finite-gl-dim-tor-dimension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finite-gl-dim-tor-dimension", "contents": [ "Let $R$ be a ring of finite global dimension $d$. Then", "\\begin{enumerate}", "\\item every module has tor dimension $\\leq d$,", "\\item a complex of $R$-modules $K^\\bullet$ with $H^i(K^\\bullet) \\not = 0$", "only if $i \\in [a, b]$ has tor amplitude in $[a - d, b]$, and", "\\item a complex of $R$-modules $K^\\bullet$ has finite tor dimension if and only", "if $K^\\bullet \\in D^b(R)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The assumption on $R$ means that every module has a finite projective", "resolution of length at most $d$, in particular every module has", "tor dimension $\\leq d$. The second statement follows from", "Lemma \\ref{lemma-cohomology-tor-amplitude}", "and the definitions. The third statement is a rephrasing of the second." ], "refs": [ "more-algebra-lemma-cohomology-tor-amplitude" ], "ref_ids": [ 10176 ] } ], "ref_ids": [] }, { "id": 10187, "type": "theorem", "label": "more-algebra-lemma-projective-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-projective-amplitude", "contents": [ "Let $R$ be a ring. Let $K$ be an object of $D(R)$. Let $a, b \\in \\mathbf{Z}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $K$ has projective-amplitude in $[a, b]$,", "\\item $\\Ext^i_R(K, N) = 0$ for all $R$-modules $N$ and all", "$i \\not \\in [-b, -a]$,", "\\item $H^n(K) = 0$ for $n > b$ and", "$\\Ext^i_R(K, N) = 0$ for all $R$-modules $N$ and all $i > -a$, and", "\\item $H^n(K) = 0$ for $n \\not \\in [a - 1, b]$ and", "$\\Ext^{-a + 1}_R(K, N) = 0$ for all $R$-modules $N$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). We may assume $K$ is the complex", "$$", "\\ldots \\to 0 \\to P^a \\to P^{a + 1} \\to \\ldots \\to", "P^{b - 1} \\to P^b \\to 0 \\to \\ldots", "$$", "where $P^i$ is a projective $R$-module for all $i \\in \\mathbf{Z}$.", "In this case we can compute the ext groups by the complex", "$$", "\\ldots \\to 0 \\to \\Hom_R(P^b, N) \\to \\ldots \\to", "\\Hom_R(P^a, N) \\to 0 \\to \\ldots", "$$", "and we obtain (2).", "\\medskip\\noindent", "Assume (2) holds. Choose an injection $H^n(K) \\to I$ where $I$", "is an injective $R$-module. Since $\\Hom_R(-, I)$ is an exact functor,", "we see that $\\Ext^{-n}(K, I) = \\Hom_R(H^n(K), I)$.", "We conclude in particular that $H^n(K)$ is zero for $n > b$.", "Thus (2) implies (3).", "\\medskip\\noindent", "By the same argument as in (2) implies (3) gives that (3) implies (4).", "\\medskip\\noindent", "Assume (4). The same argument as in (2) implies (3) shows that", "$H^{a - 1}(K) = 0$, i.e., we have $H^i(K) = 0$ unless $i \\in [a, b]$.", "In particular, $K$ is bounded above and we can choose a", "a complex $P^\\bullet$ representing $K$ with $P^i$ projective", "(for example free) for all $i \\in \\mathbf{Z}$ and $P^i = 0$ for $i > b$.", "See Derived Categories, Lemma", "\\ref{derived-lemma-subcategory-left-resolution}.", "Let $Q = \\Coker(P^{a - 1} \\to P^a)$. Then $K$ is quasi-isomorphic", "to the complex", "$$", "\\ldots \\to 0 \\to Q \\to P^{a + 1} \\to \\ldots \\to P^b \\to 0 \\to \\ldots", "$$", "as $H^i(K) = 0$ for $i < a$.", "Denote $K' = (P^{a + 1} \\to \\ldots \\to P^b)$ the corresponding object of", "$D(R)$. We obtain a distinguished triangle", "$$", "K' \\to K \\to Q[-a] \\to K'[1]", "$$", "in $D(R)$. Thus for every $R$-module $N$ an exact sequence", "$$", "\\Ext^{-a}(K', N) \\to \\Ext^1(Q, N) \\to \\Ext^{1 - a}(K, N)", "$$", "By assumption the term on the right vanishes. By the implication", "(1) $\\Rightarrow$ (2) the term on the left vanishes. Thus $Q$", "is a projective $R$-module by", "Algebra, Lemma \\ref{algebra-lemma-characterize-projective}.", "Hence (1) holds and the proof is complete." ], "refs": [ "derived-lemma-subcategory-left-resolution", "algebra-lemma-characterize-projective" ], "ref_ids": [ 1835, 789 ] } ], "ref_ids": [] }, { "id": 10188, "type": "theorem", "label": "more-algebra-lemma-injective-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-injective-amplitude", "contents": [ "Let $R$ be a ring. Let $K$ be an object of $D(R)$. Let $a, b \\in \\mathbf{Z}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $K$ has injective-amplitude in $[a, b]$,", "\\item $\\Ext^i_R(N, K) = 0$ for all $R$-modules $N$ and all", "$i \\not \\in [a, b]$,", "\\item $\\Ext^i(R/I, K) = 0$ for all ideals $I \\subset R$ and", "all $i \\not \\in [a, b]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). We may assume $K$ is the complex", "$$", "\\ldots \\to 0 \\to I^a \\to I^{a + 1} \\to \\ldots \\to", "I^{b - 1} \\to I^b \\to 0 \\to \\ldots", "$$", "where $I^i$ is a injective $R$-module for all $i \\in \\mathbf{Z}$.", "In this case we can compute the ext groups by the complex", "$$", "\\ldots \\to 0 \\to \\Hom_R(N, I^a) \\to \\ldots \\to", "\\Hom_R(N, I^b) \\to 0 \\to \\ldots", "$$", "and we obtain (2). It is clear that (2) implies (3).", "\\medskip\\noindent", "Assume (3) holds. Choose a nonzero map $R \\to H^n(K)$. Since $\\Hom_R(R, -)$", "is an exact functor, we see that", "$\\Ext^n_R(R, K) = \\Hom_R(R, H^n(K)) = H^n(K)$.", "We conclude that $H^n(K)$ is zero for $n \\not \\in [a, b]$.", "In particular, $K$ is bounded below and we can choose a quasi-isomorphism", "$$", "K \\to I^\\bullet", "$$", "with $I^i$ injective for all $i \\in \\mathbf{Z}$ and", "$I^i = 0$ for $i < a$. See Derived Categories, Lemma", "\\ref{derived-lemma-subcategory-right-resolution}.", "Let $J = \\Ker(I^b \\to I^{b + 1})$. Then $K$ is quasi-isomorphic", "to the complex", "$$", "\\ldots \\to 0 \\to I^a \\to \\ldots \\to I^{b - 1} \\to J \\to 0 \\to \\ldots", "$$", "Denote $K' = (I^a \\to \\ldots \\to I^{b - 1})$ the corresponding object of", "$D(R)$. We obtain a distinguished triangle", "$$", "J[-b] \\to K \\to K' \\to J[1 - b]", "$$", "in $D(R)$. Thus for every ideal $I \\subset R$ an exact sequence", "$$", "\\Ext^b(R/I, K') \\to \\Ext^1(R/I, J) \\to \\Ext^{1 + b}(R/I, K)", "$$", "By assumption the term on the right vanishes. By the implication", "(1) $\\Rightarrow$ (2) the term on the left vanishes. Thus $J$", "is a injective $R$-module by", "Lemma \\ref{lemma-characterize-injective-bis}." ], "refs": [ "derived-lemma-subcategory-right-resolution", "more-algebra-lemma-characterize-injective-bis" ], "ref_ids": [ 1836, 10113 ] } ], "ref_ids": [] }, { "id": 10189, "type": "theorem", "label": "more-algebra-lemma-finite-injective-dimension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finite-injective-dimension", "contents": [ "Let $R$ be a ring. Let $K \\in D(R)$.", "\\begin{enumerate}", "\\item If $K$ is in $D^b(R)$ and $H^i(K)$ has finite injective dimension", "for all $i$, then $K$ has finite injective dimension.", "\\item If $K^\\bullet$ represents $K$, is a bounded complex of $R$-modules,", "and $K^i$ has finite injective dimension for all $i$, then $K$ has finite", "injective dimension.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Apply the spectral sequences of", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}", "to the functor $F = \\Hom_R(N, -)$ to get a computation of", "$\\Ext^i_A(N, K)$ and use the criterion of", "Lemma \\ref{lemma-injective-amplitude}." ], "refs": [ "derived-lemma-two-ss-complex-functor", "more-algebra-lemma-injective-amplitude" ], "ref_ids": [ 1871, 10188 ] } ], "ref_ids": [] }, { "id": 10190, "type": "theorem", "label": "more-algebra-lemma-finite-injective-dimension-Noetherian-radical", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finite-injective-dimension-Noetherian-radical", "contents": [ "Let $R$ be a Noetherian ring. Let $I \\subset R$ be an ideal contained", "in the Jacobson radical of $R$. Let $K \\in D^+(R)$ have", "finite cohomology modules. Then the following are equivalent", "\\begin{enumerate}", "\\item $K$ has finite injective dimension, and", "\\item there exists a $b$ such that $\\Ext^i_R(R/J, K) = 0$ for $i > b$", "and any ideal $J \\supset I$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) is immediate. Assume (2).", "Say $H^i(K) = 0$ for $i < a$. Then $\\Ext^i(M, K) = 0$ for $i < a$ and", "all $R$-modules $M$. Thus it suffices to show that", "$\\Ext^i(M, K) = 0$ for $i > b$ any finite $R$-module $M$, see", "Lemma \\ref{lemma-injective-amplitude}.", "By Algebra, Lemma \\ref{algebra-lemma-filter-Noetherian-module}", "the module $M$ has a finite filtration whose successive", "quotients are of the form $R/\\mathfrak p$ where $\\mathfrak p$ is", "a prime ideal. If $0 \\to M_1 \\to M \\to M_2 \\to 0$", "is a short exact sequence and $\\Ext^i(M_j, K) = 0$ for $i > b$", "and $j = 1, 2$, then $\\Ext^i(M, K) = 0$ for $i > b$.", "Thus we may assume $M = R/\\mathfrak p$.", "If $I \\subset \\mathfrak p$, then the vanishing follows", "from the assumption. If not, then choose $f \\in I$, $f \\not \\in \\mathfrak p$.", "Consider the short exact sequence", "$$", "0 \\to R/\\mathfrak p \\xrightarrow{f} R/\\mathfrak p \\to R/(\\mathfrak p, f) \\to 0", "$$", "The $R$-module $R/(\\mathfrak p, f)$ has a filtration whose successive", "quotients are $R/\\mathfrak q$ with $(\\mathfrak p, f) \\subset \\mathfrak q$.", "Thus by Noetherian induction and the argument above", "we may assume the vanishing holds for $R/(\\mathfrak p, f)$.", "On the other hand, the modules", "$E^i = \\Ext^i(R/\\mathfrak p, K)$ are finite by our assumption on $K$", "(bounded below with finite cohomology modules), the spectral", "sequence (\\ref{equation-first-ss-ext}), and", "Algebra, Lemma \\ref{algebra-lemma-ext-noetherian}.", "Thus $E^i$ for $i > b$ is a finite $R$-module", "such that $E^i/fE^i = 0$. We conclude by Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK})", "that $E^i$ is zero." ], "refs": [ "more-algebra-lemma-injective-amplitude", "algebra-lemma-filter-Noetherian-module", "algebra-lemma-ext-noetherian", "algebra-lemma-NAK" ], "ref_ids": [ 10188, 691, 768, 401 ] } ], "ref_ids": [] }, { "id": 10191, "type": "theorem", "label": "more-algebra-lemma-finite-injective-dimension-Noetherian-local", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finite-injective-dimension-Noetherian-local", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local Noetherian ring.", "Let $K \\in D^+(R)$ have finite cohomology modules.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $K$ has finite injective dimension, and", "\\item $\\Ext^i_R(\\kappa, K) = 0$ for $i \\gg 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-finite-injective-dimension-Noetherian-radical}." ], "refs": [ "more-algebra-lemma-finite-injective-dimension-Noetherian-radical" ], "ref_ids": [ 10190 ] } ], "ref_ids": [] }, { "id": 10192, "type": "theorem", "label": "more-algebra-lemma-ideal-factor-through-projective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ideal-factor-through-projective", "contents": [ "Let $R$ be a ring. Let $M$, $N$ be $R$-modules.", "\\begin{enumerate}", "\\item Given an $R$-module map $\\varphi : M \\to N$ the following are", "equivalent: (a) $\\varphi$ factors through a projective $R$-module, and", "(b) $\\varphi$ factors through a free $R$-module.", "\\item The set of $\\varphi : M \\to N$ satisfying the equivalent", "conditions of (1) is an $R$-submodule of $\\Hom_R(M, N)$.", "\\item Given maps $\\psi : M' \\to M$ and $\\xi : N \\to N'$, if", "$\\varphi : M \\to N$ satisfies the equivalent conditions", "of (1), then $\\xi \\circ \\varphi \\circ \\psi : M' \\to N'$ does too.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1)(a) and (1)(b) follows from", "Algebra, Lemma \\ref{algebra-lemma-characterize-projective}.", "If $\\varphi : M \\to N$ and $\\varphi' : M \\to N$ factor through", "the modules $P$ and $P'$ then $\\varphi + \\varphi'$", "factors through $P \\oplus P'$ and $\\lambda \\varphi$ factors through", "$P$ for all $\\lambda \\in R$. This proves (2). If $\\varphi : M \\to N$", "factors through the module $P$ and $\\psi$ and $\\xi$ are as in (3),", "then $\\xi \\circ \\varphi \\circ \\psi$ factors through $P$. This proves (3)." ], "refs": [ "algebra-lemma-characterize-projective" ], "ref_ids": [ 789 ] } ], "ref_ids": [] }, { "id": 10193, "type": "theorem", "label": "more-algebra-lemma-factor-through-finite-projective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-factor-through-finite-projective", "contents": [ "Let $R$ be a ring. Let $\\varphi : M \\to N$ be an $R$-module map.", "If $\\varphi$ factors through a projective module and $M$ is a", "finite $R$-module, then $\\varphi$ factors through a finite projective", "module." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-ideal-factor-through-projective} we can", "factor $\\varphi = \\tau \\circ \\sigma$ where the target of $\\sigma$", "is $\\bigoplus_{i \\in I} R$ for some set $I$. Choose generators", "$x_1, \\ldots, x_n$ for $M$. Write $\\sigma(x_j) = (a_{ji})_{i \\in I}$.", "For each $j$ only a finite number of $a_{ij}$ are nonzero.", "Hence $\\sigma$ has image contained in a finite free $R$-module", "and we conclude." ], "refs": [ "more-algebra-lemma-ideal-factor-through-projective" ], "ref_ids": [ 10192 ] } ], "ref_ids": [] }, { "id": 10194, "type": "theorem", "label": "more-algebra-lemma-near-projective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-near-projective", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module.", "The following conditions are equivalent", "\\begin{enumerate}", "\\item for every $a \\in I$ the map $a : M \\to M$ factors through a projective", "$R$-module,", "\\item for every $a \\in I$ the map $a : M \\to M$ factors through a free", "$R$-module, and", "\\item $\\Ext^1_R(M, N)$ is annihilated by $I$ for every $R$-module $N$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows from", "Lemma \\ref{lemma-ideal-factor-through-projective}.", "If (1) holds, then (3) holds because $\\Ext^1_R(P, N)$", "for any $N$ and any projective module $P$.", "Conversely, assume (3) holds. Choose a short exact sequence", "$0 \\to N \\to P \\to M \\to 0$ with $P$ projective (or even free).", "By assumption the corresponding element of $\\Ext^1_R(M, N)$", "is annihilated by $I$. Hence for every $a \\in I$ the map", "$a : M \\to M$ can be factored through the surjection $P \\to M$", "and we conclude (1) holds." ], "refs": [ "more-algebra-lemma-ideal-factor-through-projective" ], "ref_ids": [ 10192 ] } ], "ref_ids": [] }, { "id": 10195, "type": "theorem", "label": "more-algebra-lemma-torsion-near-projective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-torsion-near-projective", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module.", "If $M$ is annihilated by $I$, then $M$ is $I$-projective." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definition and the fact that the zero module is projective." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10196, "type": "theorem", "label": "more-algebra-lemma-ses-near-projective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ses-near-projective", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let", "$$", "0 \\to K \\to P \\to M \\to 0", "$$", "be a short exact sequence of $R$-modules.", "If $M$ is $I$-projective and $P$ is projective, then $K$ is $I$-projective." ], "refs": [], "proofs": [ { "contents": [ "The element $\\text{id}_K \\in \\Hom_R(K, K)$ maps to the class of", "the given extension in $\\Ext^1_R(M, K)$. Since by assumption this", "class is annihilated by any $a \\in I$ we see that $a : K \\to K$", "factors through $K \\to P$ and we conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10197, "type": "theorem", "label": "more-algebra-lemma-dual-near-projective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-dual-near-projective", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal.", "If $M$ is a finite, $I$-projective $R$-module, then", "$M^\\vee = \\Hom_R(M, R)$ is $I$-projective." ], "refs": [], "proofs": [ { "contents": [ "Assume $M$ is finite and $I$-projective.", "Choose a short exact sequence $0 \\to K \\to R^{\\oplus r} \\to M \\to 0$.", "This produces an injection $M^\\vee \\to R^{\\oplus r} = (R^{\\oplus r})^\\vee$.", "Since the extension class in $\\Ext^1_R(M, K)$ corresponding to", "the short exact sequence is annihilated by $I$, we see that", "for any $a \\in I$ we can find a map $M \\to R^{\\oplus r}$ such that", "the composition with the given map $R^{\\oplus r} \\to M$ is equal", "to $a : M \\to M$. Taking duals we find that $a : M^\\vee \\to M^\\vee$", "factors through the map $M^\\vee \\to R^{\\oplus r}$ given above and", "we conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10198, "type": "theorem", "label": "more-algebra-lemma-compose", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-compose", "contents": [ "Let $R$ be a ring. Given complexes $K^\\bullet, L^\\bullet, M^\\bullet$", "of $R$-modules there is a canonical isomorphism", "$$", "\\Hom^\\bullet(K^\\bullet, \\Hom^\\bullet(L^\\bullet, M^\\bullet))", "=", "\\Hom^\\bullet(\\text{Tot}(K^\\bullet \\otimes_R L^\\bullet), M^\\bullet)", "$$", "of complexes of $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha$ be an element of degree $n$ on the left hand side.", "Thus", "$$", "\\alpha = (\\alpha^{p, q}) \\in", "\\prod\\nolimits_{p + q = n} \\Hom_R(K^{-q}, \\Hom^p(L^\\bullet, M^\\bullet))", "$$", "Each $\\alpha^{p, q}$ is an element", "$$", "\\alpha^{p, q} = (\\alpha^{r, s, q}) \\in", "\\prod\\nolimits_{r + s + q = n} \\Hom_R(K^{-q}, \\Hom_R(L^{-s}, M^r))", "$$", "If we make the identifications", "\\begin{equation}", "\\label{equation-identification}", "\\Hom_R(K^{-q}, \\Hom_R(L^{-s}, M^r)) = \\Hom_R(K^{-q} \\otimes_R L^{-s}, M^r)", "\\end{equation}", "then by our sign rules we get", "\\begin{align*}", "\\text{d}(\\alpha^{r, s, q})", "& =", "\\text{d}_{\\Hom^\\bullet(L^\\bullet, M^\\bullet)} \\circ \\alpha^{r, s, q}", "- (-1)^n \\alpha^{r, s, q} \\circ \\text{d}_K \\\\", "& =", "\\text{d}_M \\circ \\alpha^{r, s, q}", "- (-1)^{r + s} \\alpha^{r, s, q} \\circ \\text{d}_L", "- (-1)^{r + s + q} \\alpha^{r, s, q} \\circ \\text{d}_K", "\\end{align*}", "On the other hand, if $\\beta$ is an element of degree $n$ of", "the right hand side, then", "$$", "\\beta = (\\beta^{r, s, q}) \\in \\prod\\nolimits_{r + s + q = n}", "\\Hom_R(K^{-q} \\otimes_R L^{-s}, M^r)", "$$", "and by our sign rule (Homology, Definition", "\\ref{homology-definition-associated-simple-complex}) we get", "\\begin{align*}", "\\text{d}(\\beta^{r, s, q})", "& =", "\\text{d}_M \\circ \\beta^{r, s, q}", "- (-1)^n \\beta^{r, s, q} \\circ", "\\text{d}_{\\text{Tot}(K^\\bullet \\otimes L^\\bullet)} \\\\", "& =", "\\text{d}_M \\circ \\beta^{r, s, q}", "- (-1)^{r + s + q} \\left(", "\\beta^{r, s, q} \\circ \\text{d}_K + (-1)^{-q} \\beta^{r, s, q} \\circ \\text{d}_L", "\\right)", "\\end{align*}", "Thus we see that the map induced by the identifications", "(\\ref{equation-identification}) indeed is a morphism of complexes." ], "refs": [ "homology-definition-associated-simple-complex" ], "ref_ids": [ 12164 ] } ], "ref_ids": [] }, { "id": 10199, "type": "theorem", "label": "more-algebra-lemma-composition", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-composition", "contents": [ "Let $R$ be a ring. Given complexes", "$K^\\bullet, L^\\bullet, M^\\bullet$", "of $R$-modules there is a canonical morphism", "$$", "\\text{Tot}\\left(", "\\Hom^\\bullet(L^\\bullet, M^\\bullet) \\otimes_R \\Hom^\\bullet(K^\\bullet, L^\\bullet)", "\\right)", "\\longrightarrow", "\\Hom^\\bullet(K^\\bullet, M^\\bullet)", "$$", "of complexes of $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "An element $\\alpha$ of degree $n$ of the left hand side is", "$$", "\\alpha = (\\alpha^{p, q}) \\in \\bigoplus\\nolimits_{p + q = n}", "\\Hom^p(L^\\bullet, M^\\bullet) \\otimes_R \\Hom^q(K^\\bullet, L^\\bullet)", "$$", "The element $\\alpha^{p, q}$ is a finite sum", "$\\alpha^{p, q} = \\sum \\beta^p_i \\otimes \\gamma^q_i$", "with", "$$", "\\beta^p_i = (\\beta^{r, s}_i)", "\\in \\prod\\nolimits_{r + s = p} \\Hom_R(L^{-s}, M^r)", "$$", "and", "$$", "\\gamma^q_i = (\\gamma^{u, v}_i)", "\\in \\prod\\nolimits_{u + v = q} \\Hom_R(K^{-v}, L^u)", "$$", "The map is given by sending $\\alpha$ to", "$\\delta = (\\delta^{r, v})$ with", "$$", "\\delta^{r, v} =", "\\sum\\nolimits_{i, s} \\beta^{r, s}_i", "\\circ \\gamma^{-s, v}_i", "\\in \\Hom_R(K^{-v}, M^r)", "$$", "For given $r + v = n$ this sum is finite as there are only finitely many", "nonzero $\\alpha^{p, q}$, hence only finitely many nonzero", "$\\beta^p_i$ and $\\gamma^q_i$.", "By our sign rules we have", "\\begin{align*}", "\\text{d}(\\alpha^{p, q})", "& =", "\\text{d}_{\\Hom^\\bullet(L^\\bullet, M^\\bullet)}(\\alpha^{p, q})", "+ (-1)^p \\text{d}_{\\Hom^\\bullet(K^\\bullet, L^\\bullet)}(\\alpha^{p, q}) \\\\", "& =", "\\sum \\Big( \\text{d}_M \\circ \\beta^p_i \\circ \\gamma^q_i", "- (-1)^p \\beta^p_i \\circ \\text{d}_L \\circ \\gamma^q_i \\Big) \\\\", "&", "\\quad + (-1)^p \\sum \\Big( \\beta^p_i \\circ \\text{d}_L \\circ \\gamma^q_i", "- (-1)^q \\beta^p_i \\circ \\gamma^q_i \\circ \\text{d}_K \\Big) \\\\", "& =", "\\sum \\Big( \\text{d}_M \\circ \\beta^p_i \\circ \\gamma^q_i ", "-(-1)^n \\beta^p_i \\circ \\gamma^q_i \\circ \\text{d}_K \\Big)", "\\end{align*}", "It follows that the rules $\\alpha \\mapsto \\delta$ is compatible", "with differentials and the lemma is proved." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10200, "type": "theorem", "label": "more-algebra-lemma-diagonal-better", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-diagonal-better", "contents": [ "Let $R$ be a ring. Given complexes $K^\\bullet, L^\\bullet, M^\\bullet$", "of $R$-modules there is a canonical morphism", "$$", "\\text{Tot}(K^\\bullet \\otimes_R \\Hom^\\bullet(M^\\bullet, L^\\bullet))", "\\longrightarrow", "\\Hom^\\bullet(M^\\bullet, \\text{Tot}(K^\\bullet \\otimes_R L^\\bullet))", "$$", "of complexes of $R$-modules functorial in all three complexes." ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha$ be an element of degree $n$ of the right hand side.", "Thus", "$$", "\\alpha = (\\alpha^{p, q}) \\in \\prod\\nolimits_{p + q = n}", "\\Hom_R(M^{-q}, \\text{Tot}^p(K^\\bullet \\otimes_R L^\\bullet))", "$$", "Each $\\alpha^{p, q}$ is an element", "$$", "\\alpha^{p, q} = (\\alpha^{r, s, q}) \\in", "\\Hom_R(M^{-q}, \\bigoplus\\nolimits_{r + s + q = n} K^r \\otimes_R L^s)", "$$", "where we think of $\\alpha^{r, s, q}$ as a family of maps such that", "for every $x \\in M^{-q}$ only a finite number of", "$\\alpha^{r, s, q}(x)$ are nonzero. By our sign rules we get", "\\begin{align*}", "\\text{d}(\\alpha^{r, s, q})", "& =", "\\text{d}_{\\text{Tot}(K^\\bullet \\otimes_R L^\\bullet)} \\circ \\alpha^{r, s, q}", "- (-1)^n \\alpha^{r, s, q} \\circ \\text{d}_M \\\\", "& =", "\\text{d}_K \\circ \\alpha^{r, s, q} + (-1)^r \\text{d}_L \\circ \\alpha^{r, s, q}", "- (-1)^n \\alpha^{r, s, q} \\circ \\text{d}_M", "\\end{align*}", "On the other hand, if $\\beta$ is an element of degree $n$ of the", "left hand side, then", "$$", "\\beta = (\\beta^{p, q}) \\in", "\\bigoplus\\nolimits_{p + q = n} K^p \\otimes_R \\Hom^q(M^\\bullet, L^\\bullet)", "$$", "and we can write $\\beta^{p, q} = \\sum \\gamma_i^p \\otimes \\delta_i^q$ with", "$\\gamma_i^p \\in K^p$ and", "$$", "\\delta_i^q = (\\delta_i^{r, s}) \\in", "\\prod\\nolimits_{r + s = q} \\Hom_R(M^{-s}, L^r)", "$$", "By our sign rules we have", "\\begin{align*}", "\\text{d}(\\beta^{p, q})", "& =", "\\text{d}_K(\\beta^{p, q}) +", "(-1)^p \\text{d}_{\\Hom^\\bullet(M^\\bullet, L^\\bullet)}(\\beta^{p, q}) \\\\", "& =", "\\sum \\text{d}_K(\\gamma_i^p) \\otimes \\delta_i^q +", "(-1)^p \\sum \\gamma_i^p \\otimes", "(\\text{d}_L \\circ \\delta_i^q - (-1)^q \\delta_i^q \\circ \\text{d}_M)", "\\end{align*}", "We send the element $\\beta$ to $\\alpha$ with", "$$", "\\alpha^{r, s, q} = c^{r, s, q}(\\sum \\gamma_i^r \\otimes \\delta_i^{s, q})", "$$", "where $c^{r, s, q} : K^r \\otimes_R \\Hom_R(M^{-q}, L^s) \\to", "\\Hom_R(M^{-q}, K^r \\otimes_R L^s)$ is the canonical map.", "For a given $\\beta$ and $r$ there are only finitely many nonzero", "$\\gamma_i^r$ hence only finitely many nonzero $\\alpha^{r, s, q}$", "are nonzero (for a given $r$). Thus this family of maps satisfies", "the conditions above and the map is well defined.", "Comparing signs we see that this is compatible with differentials." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10201, "type": "theorem", "label": "more-algebra-lemma-diagonal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-diagonal", "contents": [ "Let $R$ be a ring. Given complexes $K^\\bullet, L^\\bullet$", "of $R$-modules there is a canonical morphism", "$$", "K^\\bullet", "\\longrightarrow", "\\Hom^\\bullet(L^\\bullet, \\text{Tot}(K^\\bullet \\otimes_R L^\\bullet))", "$$", "of complexes of $R$-modules functorial in both complexes." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-diagonal-better}", "but we will also construct it directly here.", "Let $\\alpha$ be an element of degree $n$ of the right hand side.", "Thus", "$$", "\\alpha = (\\alpha^{p, q}) \\in \\prod\\nolimits_{p + q = n}", "\\Hom_R(L^{-q}, \\text{Tot}^p(K^\\bullet \\otimes_R L^\\bullet))", "$$", "Each $\\alpha^{p, q}$ is an element", "$$", "\\alpha^{p, q} = (\\alpha^{r, s, q}) \\in", "\\Hom_R(L^{-q}, \\bigoplus\\nolimits_{r + s + q = n} K^r \\otimes_R L^s)", "$$", "where we think of $\\alpha^{r, s, q}$ as a family of maps such that", "for every $x \\in L^{-q}$ only a finite number of", "$\\alpha^{r, s, q}(x)$ are nonzero. By our sign rules we get", "\\begin{align*}", "\\text{d}(\\alpha^{r, s, q})", "& =", "\\text{d}_{\\text{Tot}(K^\\bullet \\otimes_R L^\\bullet)} \\circ \\alpha^{r, s, q}", "- (-1)^n \\alpha^{r, s, q} \\circ \\text{d}_L \\\\", "& =", "\\text{d}_K \\circ \\alpha^{r, s, q} + (-1)^r \\text{d}_L \\circ \\alpha^{r, s, q}", "- (-1)^n \\alpha^{r, s, q} \\circ \\text{d}_L", "\\end{align*}", "Now an element $\\beta \\in K^n$ we send to $\\alpha$ with", "$\\alpha^{n, -q, q} = \\beta \\otimes \\text{id}_{L^{-q}}$", "and $\\alpha^{r, s, q} = 0$ if $r \\not = n$. This is indeed", "an element as above, as for fixed $q$ there is only one nonzero", "$\\alpha^{r, s, q}$. The description of", "the differential shows this is compatible with differentials." ], "refs": [ "more-algebra-lemma-diagonal-better" ], "ref_ids": [ 10200 ] } ], "ref_ids": [] }, { "id": 10202, "type": "theorem", "label": "more-algebra-lemma-evaluate-and-more", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-evaluate-and-more", "contents": [ "Let $R$ be a ring. Given complexes $K^\\bullet, L^\\bullet, M^\\bullet$", "of $R$-modules there is a canonical morphism", "$$", "\\text{Tot}(\\Hom^\\bullet(L^\\bullet, M^\\bullet) \\otimes_R K^\\bullet)", "\\longrightarrow", "\\Hom^\\bullet(\\Hom^\\bullet(K^\\bullet, L^\\bullet), M^\\bullet)", "$$", "of complexes of $R$-modules functorial in all three complexes." ], "refs": [], "proofs": [ { "contents": [ "Consider an element $\\beta$ of degree $n$ of the right hand side.", "Then", "$$", "\\beta = (\\beta^{p, s}) \\in", "\\prod\\nolimits_{p + s = n} \\Hom_R(\\Hom^{-s}(K^\\bullet, L^\\bullet), M^p)", "$$", "Our sign rules tell us that", "\\begin{align*}", "\\text{d}(\\beta^{p, s})", "& =", "\\text{d}_M \\circ \\beta^{p, s}", "- (-1)^n \\beta^{p, s} \\circ", "\\text{d}_{\\Hom^\\bullet(K^\\bullet, L^\\bullet)}", "\\end{align*}", "We can describe the last term as follows", "$$", "(\\beta^{p, s} \\circ \\text{d}_{\\Hom^\\bullet(K^\\bullet, L^\\bullet)})(f) =", "\\beta^{p, s}(\\text{d}_L \\circ f - (-1)^{s + 1} f \\circ \\text{d}_K)", "$$", "if $f \\in \\Hom^{-s - 1}(K^\\bullet, L^\\bullet)$. We conclude that in", "some unspecified sense $\\text{d}(\\beta^{p, s})$ is a sum of three terms", "with signs as follows", "\\begin{equation}", "\\label{equation-beta}", "\\text{d}(\\beta^{p, s}) =", "\\text{d}_M(\\beta^{p, s})", "-(-1)^n\\text{d}_L(\\beta^{p, s}) +", "(-1)^{p + 1}\\text{d}_K(\\beta^{p, s})", "\\end{equation}", "\\noindent", "Next, we consider an element $\\alpha$ of degree $n$ of the left hand side.", "We can write it like so", "$$", "\\alpha = (\\alpha^{t, r}) \\in", "\\bigoplus\\nolimits_{t + r = n} \\Hom^t(L^\\bullet, M^\\bullet) \\otimes K^r", "$$", "Each $\\alpha^{t, r}$ maps to an element", "$$", "\\alpha^{t, r} \\mapsto (\\alpha^{p, q, r}) \\in", "\\prod\\nolimits_{p + q = t} \\Hom_R(L^{-q}, M^p) \\otimes_R K^r", "$$", "Our sign rules tell us that", "\\begin{align*}", "\\text{d}(\\alpha^{p, q, r})", "& =", "\\text{d}_{\\Hom^\\bullet(L^\\bullet, M^\\bullet)}(\\alpha^{p, q, r})", "+ (-1)^{p + q} \\text{d}_K(\\alpha^{p, q, r})", "\\end{align*}", "where if we further write $\\alpha^{p, q, r} = \\sum g_i^{p, q} \\otimes k_i^r$", "then we have", "$$", "\\text{d}_{\\Hom^\\bullet(L^\\bullet, M^\\bullet)}(\\alpha^{p, q, r}) =", "\\sum (\\text{d}_M \\circ g_i^{p, q}) \\otimes k_i^r", "- (-1)^{p + q} \\sum (g_i^{p, q} \\circ \\text{d}_L) \\otimes k_i^r", "$$", "We conclude that in some unspecified sense $\\text{d}(\\alpha^{p, q, r})$", "is a sum of three terms with signs as follows", "\\begin{equation}", "\\label{equation-alpha}", "\\text{d}(\\alpha^{p, q, r}) =", "\\text{d}_M(\\alpha^{p, q, r})", "-(-1)^{p + q}\\text{d}_L(\\alpha^{p, q, r}) +", "(-1)^{p + q}\\text{d}_K(\\alpha^{p, q, r})", "\\end{equation}", "\\noindent", "To define our map we will use the canonical maps", "$$", "c_{p, q, r} : ", "\\Hom_R(L^{-q}, M^p) \\otimes_R K^r", "\\longrightarrow", "\\Hom_R(\\Hom_R(K^r, L^{-q}), M^p)", "$$", "which sends $\\varphi \\otimes k$ to the map $\\psi \\mapsto \\varphi(\\psi(k))$.", "This is functorial in all three variables. With $s = q + r$", "there is an inclusion", "$$", "\\Hom_R(\\Hom_R(K^r, L^{-q}), M^p) \\subset", "\\Hom_R(\\Hom^{-s}(K^\\bullet, L^\\bullet), M^p)", "$$", "coming from the projection", "$\\Hom^{-s}(K^\\bullet, L^\\bullet) \\to \\Hom_R(K^r, L^{-q})$.", "Since $\\alpha^{p, q, r}$", "is nonzero only for a finite number of $r$ we see that for a given", "$s$ there is only a finite number of $q, r$ with $q + r = s$.", "Thus we can send $\\alpha$ to the element $\\beta$ with", "$$", "\\beta^{p, s} =", "\\sum\\nolimits_{q + r = s} \\epsilon_{p, q, r} c_{p, q, r}(\\alpha^{p, q, r})", "$$", "where where the sum uses the inclusions given above and where", "$\\epsilon_{p, q, r} \\in \\{\\pm 1\\}$. Comparing signs in the", "equations (\\ref{equation-beta}) and (\\ref{equation-alpha})", "we see that", "\\begin{enumerate}", "\\item $\\epsilon_{p, q, r} = \\epsilon_{p + 1, q, r}$", "\\item $-(-1)^n\\epsilon_{p, q, r} = -(-1)^{p + q}\\epsilon_{p, q - 1, r}$", "or equivalently $\\epsilon_{p, q, r} = (-1)^r\\epsilon_{p, q - 1, r}$", "\\item $(-1)^{p + 1}\\epsilon_{p, q, r} = (-1)^{p + q}\\epsilon_{p, q, r + 1}$", "or equivalently", "$(-1)^{q + 1}\\epsilon_{p, q, r} = \\epsilon_{p, q, r + 1}$.", "\\end{enumerate}", "A good solution is to take", "$$", "\\epsilon_{p, r, s} = (-1)^{r + qr}", "$$", "The choice of this sign is explained in the remark following", "the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10203, "type": "theorem", "label": "more-algebra-lemma-symmetric-monoidal-cat-complexes", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-symmetric-monoidal-cat-complexes", "contents": [ "Let $R$ be a ring. The category of complexes of $R$-modules with", "tensor product defined by", "$K^\\bullet \\otimes M^\\bullet = \\text{Tot}(K^\\bullet \\otimes_R M^\\bullet)$", "is a symmetric monoidal category with the associativity and", "commutativity constraints given above." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints: as unit $\\mathbf{1}$ we take the complex having", "$R$ in degree $0$ and zero in other degrees with obvious isomorphisms", "$\\text{Tot}(\\mathbf{1} \\otimes_R M^\\bullet) = M^\\bullet$ and", "$\\text{Tot}(K^\\bullet \\otimes_R \\mathbf{1}) = K^\\bullet$.", "to prove the lemma you have to check the commutativity", "of various diagrams, see Categories, Definitions", "\\ref{categories-definition-monoidal-category} and", "\\ref{categories-definition-symmetric-monoidal-category}.", "The verifications are straightforward in each case." ], "refs": [ "categories-definition-monoidal-category", "categories-definition-symmetric-monoidal-category" ], "ref_ids": [ 12404, 12408 ] } ], "ref_ids": [] }, { "id": 10204, "type": "theorem", "label": "more-algebra-lemma-left-dual-module", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-left-dual-module", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module. Let $N, \\eta, \\epsilon$", "be a left dual of $M$ in the monoidal category of $R$-modules, see", "Categories, Definition \\ref{categories-definition-dual}. Then", "\\begin{enumerate}", "\\item $M$ and $N$ are finite projective $R$-modules,", "\\item the map", "$e : \\Hom_R(M, R) \\to N$, $\\lambda \\mapsto (\\lambda \\otimes 1)(\\eta)$", "is an isomorphism,", "\\item we have $\\epsilon(n, m) = e^{-1}(n)(m)$ for $n \\in N$ and $m \\in M$.", "\\end{enumerate}" ], "refs": [ "categories-definition-dual" ], "proofs": [ { "contents": [ "The assumptions mean that", "$$", "M \\xrightarrow{\\eta \\otimes 1} M \\otimes_R N \\otimes_R M", "\\xrightarrow{1 \\otimes \\epsilon} M", "\\quad\\text{and}\\quad", "N \\xrightarrow{1 \\otimes \\eta} N \\otimes_R M \\otimes_R N", "\\xrightarrow{\\epsilon \\otimes 1} N", "$$", "are the identity map. We can choose a finite free module $F$, an $R$-module", "map $F \\to M$, and a lift $\\tilde \\eta : R \\to F \\otimes_R N$", "of $\\eta$. We obtain a commutative diagram", "$$", "\\xymatrix{", "M \\ar[rr]_-{\\eta \\otimes 1} \\ar[rrd]_-{\\tilde \\eta \\otimes 1} & &", "M \\otimes_R N \\otimes_R M \\ar[r]_-{1 \\otimes \\epsilon} &", "M \\\\", "& &", "F \\otimes_R N \\otimes_R M \\ar[u] \\ar[r]^-{1 \\otimes \\epsilon} &", "F \\ar[u]", "}", "$$", "This shows that the identity on $M$ factors through a finite free module", "and hence $M$ is finite projective. By symmetry we see that $N$ is", "finite projective. This proves part (1). Part (2) follows from", "Categories, Lemma \\ref{categories-lemma-left-dual} and its proof.", "Part (3) follows from the first equality of the proof." ], "refs": [ "categories-lemma-left-dual" ], "ref_ids": [ 12325 ] } ], "ref_ids": [ 12407 ] }, { "id": 10205, "type": "theorem", "label": "more-algebra-lemma-left-dual-complex", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-left-dual-complex", "contents": [ "Let $R$ be a ring. Let $M^\\bullet$ be a complex of $R$-modules.", "Let $N^\\bullet, \\eta, \\epsilon$ be a left dual of $M^\\bullet$", "in the monoidal category of complexes of $R$-modules.", "Then", "\\begin{enumerate}", "\\item $M^\\bullet$ and $N^\\bullet$ are bounded,", "\\item $M^n$ and $N^n$ are finite projective $R$-modules,", "\\item writing $\\epsilon = \\sum \\epsilon_n$", "with $\\epsilon_n : N^{-n} \\otimes_R M^n \\to R$ and $\\eta = \\sum \\eta_n$", "with $\\eta_n : R \\to M^n \\otimes_R N^{-n}$ then $(N^{-n}, \\eta_n, \\epsilon_n)$", "is the left dual of $M^n$ as in Lemma \\ref{lemma-left-dual-module},", "\\item the differential $d_N^n : N^n \\to N^{n + 1}$ is equal", "to $-(-1)^n$ times the map", "$$", "N^n = \\Hom_R(M^{-n}, R)", "\\xrightarrow{d_M^{-n - 1}}", "\\Hom_R(M^{-n - 1}, R) = N^{n + 1}", "$$", "where the equality signs are the identifications from", "Lemma \\ref{lemma-left-dual-module} part (2).", "\\end{enumerate}", "Conversely, given a bounded complex $M^\\bullet$ of finite projective", "$R$-modules, setting $N^n = \\Hom_R(M^{-n}, R)$ with differentials as above,", "setting $\\epsilon = \\sum \\epsilon_n$ with", "$\\epsilon_n : N^{-n} \\otimes_R M^n \\to R$ given by evaluation, and", "setting $\\eta = \\sum \\eta_n$ with $\\eta_n : R \\to M^n \\otimes_R N^{-n}$", "mapping $1$ to $\\text{id}_{M_n}$ we obtain a left dual of $M^\\bullet$", "in the monoidal category of complexes of $R$-modules." ], "refs": [ "more-algebra-lemma-left-dual-module", "more-algebra-lemma-left-dual-module" ], "proofs": [ { "contents": [ "Since $(1 \\otimes \\epsilon) \\circ (\\eta \\otimes 1) = \\text{id}_{M^\\bullet}$", "and $(\\epsilon \\otimes 1) \\circ (1 \\otimes \\eta) = \\text{id}_{N^\\bullet}$ by", "Categories, Definition \\ref{categories-definition-dual} we see immediately", "that we have", "$(1 \\otimes \\epsilon_n) \\circ (\\eta_n \\otimes 1) = \\text{id}_{M^n}$", "and $(\\epsilon_n \\otimes 1) \\circ (1 \\otimes \\eta_n) = \\text{id}_{N^{-n}}$", "which proves (3). By Lemma \\ref{lemma-left-dual-module} we have (2).", "Since the sum $\\eta = \\sum \\eta_n$ is finite, we get (1).", "Since $\\eta = \\sum \\eta_n$ is a map of complexes", "$R \\to \\text{Tot}(M^\\bullet \\otimes_R N^\\bullet)$ we see that", "$$", "(d_M^{-n - 1} \\otimes 1) \\circ \\eta_{-n - 1} +", "(-1)^n (1 \\otimes d_N^{-n}) \\circ \\eta_{-n} = 0", "$$", "by our choice of signs for the differential on", "$\\text{Tot}(M^\\bullet \\otimes_R N^\\bullet)$.", "Unwinding definitions, this proves (4). To see the final statement", "of the lemma one reads the above backwards." ], "refs": [ "categories-definition-dual", "more-algebra-lemma-left-dual-module" ], "ref_ids": [ 12407, 10204 ] } ], "ref_ids": [ 10204, 10204 ] }, { "id": 10206, "type": "theorem", "label": "more-algebra-lemma-internal-hom", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-internal-hom", "contents": [ "Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical", "isomorphism", "$$", "R\\Hom_R(K, R\\Hom_R(L, M)) = R\\Hom_R(K \\otimes_R^\\mathbf{L} L, M)", "$$", "in $D(R)$ functorial in $K, L, M$ which recovers", "(\\ref{equation-internal-hom}) by taking $H^0$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $I^\\bullet$ representing", "$M$ and a K-flat complex of $R$-modules $L^\\bullet$", "representing $L$. For any complex of $R$-modules $K^\\bullet$", "we have", "$$", "\\Hom^\\bullet(K^\\bullet, \\Hom^\\bullet(L^\\bullet, I^\\bullet)) =", "\\Hom^\\bullet(\\text{Tot}(K^\\bullet \\otimes_R L^\\bullet), I^\\bullet)", "$$", "by Lemma \\ref{lemma-compose}. The lemma follows by the definition", "of $R\\Hom$ and because", "$\\text{Tot}(K^\\bullet \\otimes_R L^\\bullet)$", "represents the derived tensor product." ], "refs": [ "more-algebra-lemma-compose" ], "ref_ids": [ 10198 ] } ], "ref_ids": [] }, { "id": 10207, "type": "theorem", "label": "more-algebra-lemma-RHom-out-of-projective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-RHom-out-of-projective", "contents": [ "Let $R$ be a ring. Let $P^\\bullet$ be a bounded above complex", "of projective $R$-modules. Let $L^\\bullet$ be a complex of $R$-modules.", "Then $R\\Hom_R(P^\\bullet, L^\\bullet)$ is represented by the complex", "$\\Hom^\\bullet(P^\\bullet, L^\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "By (\\ref{equation-cohomology-hom-complex}) and", "Derived Categories, Lemma \\ref{derived-lemma-morphisms-from-projective-complex}", "the cohomology groups of the complex are ``correct''.", "Hence if we choose a quasi-isomorphism $L^\\bullet \\to I^\\bullet$", "with $I^\\bullet$ a K-injective complex of $R$-modules", "then the induced map", "$$", "\\Hom^\\bullet(P^\\bullet, L^\\bullet)", "\\longrightarrow", "\\Hom^\\bullet(P^\\bullet, I^\\bullet)", "$$", "is a quasi-isomorphism. As the right hand side is our definition", "of $R\\Hom_R(P^\\bullet, L^\\bullet)$ we win." ], "refs": [ "derived-lemma-morphisms-from-projective-complex" ], "ref_ids": [ 1862 ] } ], "ref_ids": [] }, { "id": 10208, "type": "theorem", "label": "more-algebra-lemma-internal-hom-evaluate", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-internal-hom-evaluate", "contents": [ "Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$.", "There is a canonical morphism", "$$", "R\\Hom_R(L, M) \\otimes_R^\\mathbf{L} K", "\\longrightarrow", "R\\Hom_R(R\\Hom_R(K, L), M)", "$$", "in $D(R)$ functorial in $K, L, M$." ], "refs": [], "proofs": [ { "contents": [ "Choose", "a K-injective complex $I^\\bullet$ representing $M$,", "a K-injective complex $J^\\bullet$ representing $L$, and", "a K-flat complex $K^\\bullet$ representing $K$.", "The map is defined using the map", "$$", "\\text{Tot}(\\Hom^\\bullet(J^\\bullet, I^\\bullet) \\otimes_R K^\\bullet)", "\\longrightarrow", "\\Hom^\\bullet(\\Hom^\\bullet(K^\\bullet, J^\\bullet), I^\\bullet)", "$$", "of Lemma \\ref{lemma-evaluate-and-more}. We omit the proof that", "this is functorial in all three objects of $D(R)$." ], "refs": [ "more-algebra-lemma-evaluate-and-more" ], "ref_ids": [ 10202 ] } ], "ref_ids": [] }, { "id": 10209, "type": "theorem", "label": "more-algebra-lemma-internal-hom-composition", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-internal-hom-composition", "contents": [ "Let $R$ be a ring. Given $K, L, M$ in $D(R)$ there is a canonical morphism", "$$", "R\\Hom_R(L, M) \\otimes_R^\\mathbf{L} R\\Hom_R(K, L) \\longrightarrow R\\Hom_R(K, M)", "$$", "in $D(R)$ functorial in $K, L, M$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex $I^\\bullet$ representing $M$,", "a K-injective complex $J^\\bullet$ representing $L$, and", "any complex of $R$-modules $K^\\bullet$ representing $K$.", "By Lemma \\ref{lemma-composition} there is a map of complexes", "$$", "\\text{Tot}\\left(", "\\Hom^\\bullet(J^\\bullet, I^\\bullet) \\otimes_R \\Hom^\\bullet(K^\\bullet, J^\\bullet)", "\\right)", "\\longrightarrow", "\\Hom^\\bullet(K^\\bullet, I^\\bullet)", "$$", "The complexes of $R$-modules $\\Hom^\\bullet(J^\\bullet, I^\\bullet)$,", "$\\Hom^\\bullet(K^\\bullet, J^\\bullet)$, and $\\Hom^\\bullet(K^\\bullet, I^\\bullet)$", "represent $R\\Hom_R(L, M)$, $R\\Hom_R(K, L)$, and $R\\Hom_R(K, M)$.", "If we choose a K-flat complex $H^\\bullet$ and a quasi-isomorphism", "$H^\\bullet \\to \\Hom^\\bullet(K^\\bullet, J^\\bullet)$, then there is a map", "$$", "\\text{Tot}\\left(", "\\Hom^\\bullet(J^\\bullet, I^\\bullet) \\otimes_R H^\\bullet", "\\right)", "\\longrightarrow", "\\text{Tot}\\left(", "\\Hom^\\bullet(J^\\bullet, I^\\bullet) \\otimes_R \\Hom^\\bullet(K^\\bullet, J^\\bullet)", "\\right)", "$$", "whose source represents $R\\Hom_R(L, M) \\otimes_R^\\mathbf{L} R\\Hom_R(K, L)$.", "Composing the two displayed arrows gives the desired map. We omit the", "proof that the construction is functorial." ], "refs": [ "more-algebra-lemma-composition" ], "ref_ids": [ 10199 ] } ], "ref_ids": [] }, { "id": 10210, "type": "theorem", "label": "more-algebra-lemma-internal-hom-diagonal-better", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-internal-hom-diagonal-better", "contents": [ "Let $R$ be a ring. Given complexes $K, L, M$ in $D(R)$", "there is a canonical morphism", "$$", "K \\otimes_R^\\mathbf{L} R\\Hom_R(M, L)", "\\longrightarrow", "R\\Hom_R(M, K \\otimes_R^\\mathbf{L} L)", "$$", "in $D(R)$ functorial in $K$, $L$, $M$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-flat complex $K^\\bullet$ representing $K$,", "and a K-injective complex $I^\\bullet$ representing $L$, and", "choose any complex $M^\\bullet$ representing $M$.", "Choose a quasi-isomorphism", "$\\text{Tot}(K^\\bullet \\otimes_R I^\\bullet) \\to J^\\bullet$", "where $J^\\bullet$ is K-injective. Then we use the map", "$$", "\\text{Tot}\\left(", "K^\\bullet \\otimes_R \\Hom^\\bullet(M^\\bullet, I^\\bullet)", "\\right)", "\\to", "\\Hom^\\bullet(M^\\bullet, \\text{Tot}(K^\\bullet \\otimes_R I^\\bullet))", "\\to", "\\Hom^\\bullet(M^\\bullet, J^\\bullet)", "$$", "where the first map is the map from Lemma \\ref{lemma-diagonal-better}." ], "refs": [ "more-algebra-lemma-diagonal-better" ], "ref_ids": [ 10200 ] } ], "ref_ids": [] }, { "id": 10211, "type": "theorem", "label": "more-algebra-lemma-internal-hom-diagonal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-internal-hom-diagonal", "contents": [ "Let $R$ be a ring. Given complexes $K, L$ in $D(R)$", "there is a canonical morphism", "$$", "K \\longrightarrow R\\Hom_R(L, K \\otimes_R^\\mathbf{L} L)", "$$", "in $D(R)$ functorial in both $K$ and $L$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-internal-hom-diagonal-better}", "but we will also prove it directly.", "Choose a K-flat complex $K^\\bullet$ representing $K$ and", "any complex $L^\\bullet$ representing $L$. Choose a quasi-isomorphism", "$\\text{Tot}(K^\\bullet \\otimes_R L^\\bullet) \\to J^\\bullet$", "where $J^\\bullet$ is K-injective. Then we use the map", "$$", "K^\\bullet \\to", "\\Hom^\\bullet(L^\\bullet, \\text{Tot}(K^\\bullet \\otimes_R L^\\bullet))", "\\to \\Hom^\\bullet(L^\\bullet, J^\\bullet)", "$$", "where the first map is the map from Lemma \\ref{lemma-diagonal}." ], "refs": [ "more-algebra-lemma-internal-hom-diagonal-better", "more-algebra-lemma-diagonal" ], "ref_ids": [ 10210, 10201 ] } ], "ref_ids": [] }, { "id": 10212, "type": "theorem", "label": "more-algebra-lemma-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-perfect", "contents": [ "Let $K^\\bullet$ be an object of $D(R)$. The following are equivalent", "\\begin{enumerate}", "\\item $K^\\bullet$ is perfect, and", "\\item $K^\\bullet$ is pseudo-coherent and has finite tor dimension.", "\\end{enumerate}", "If (1) and (2) hold and $K^\\bullet$ has tor-amplitude", "in $[a, b]$, then $K^\\bullet$ is quasi-isomorphic to a complex", "$E^\\bullet$ of finite projective $R$-modules with $E^i = 0$", "for $i \\not \\in [a, b]$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2), see", "Lemmas \\ref{lemma-pseudo-coherent} and \\ref{lemma-tor-amplitude}.", "Assume (2) holds and that $K^\\bullet$ has tor-amplitude in $[a, b]$.", "In particular, $H^i(K^\\bullet) = 0$ for $i > b$.", "Choose a complex $F^\\bullet$ of finite free $R$-modules with", "$F^i = 0$ for $i > b$ and a quasi-isomorphism $F^\\bullet \\to K^\\bullet$", "(Lemma \\ref{lemma-pseudo-coherent}).", "Set $E^\\bullet = \\tau_{\\geq a}F^\\bullet$. Note that $E^i$ is finite", "free except $E^a$ which is a finitely presented $R$-module. By", "Lemma \\ref{lemma-last-one-flat} $E^a$ is flat. Hence by", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}", "we see that $E^a$ is finite projective." ], "refs": [ "more-algebra-lemma-pseudo-coherent", "more-algebra-lemma-tor-amplitude", "more-algebra-lemma-pseudo-coherent", "more-algebra-lemma-last-one-flat", "algebra-lemma-finite-projective" ], "ref_ids": [ 10148, 10170, 10148, 10169, 795 ] } ], "ref_ids": [] }, { "id": 10213, "type": "theorem", "label": "more-algebra-lemma-perfect-module", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-perfect-module", "contents": [ "Let $M$ be a module over a ring $R$. The following are equivalent", "\\begin{enumerate}", "\\item $M$ is a perfect module, and", "\\item there exists a resolution", "$$", "0 \\to F_d \\to \\ldots \\to F_1 \\to F_0 \\to M \\to 0", "$$", "with each $F_i$ a finite projective $R$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (2). Then the complex $E^\\bullet$ with $E^{-i} = F_i$", "is quasi-isomorphic to $M[0]$. Hence $M$ is perfect.", "Conversely, assume (1). By", "Lemmas \\ref{lemma-perfect} and \\ref{lemma-n-pseudo-module}", "we can find resolution $E^\\bullet \\to M$ with $E^{-i}$ a finite free", "$R$-module. By", "Lemma \\ref{lemma-last-one-flat}", "we see that $F_d = \\Coker(E^{d - 1} \\to E^d)$ is flat for", "some $d$ sufficiently large. By", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}", "we see that $F_d$ is finite projective.", "Hence", "$$", "0 \\to F_d \\to E^{-d+1} \\to \\ldots \\to E^0 \\to M \\to 0", "$$", "is the desired resolution." ], "refs": [ "more-algebra-lemma-perfect", "more-algebra-lemma-n-pseudo-module", "more-algebra-lemma-last-one-flat", "algebra-lemma-finite-projective" ], "ref_ids": [ 10212, 10147, 10169, 795 ] } ], "ref_ids": [] }, { "id": 10214, "type": "theorem", "label": "more-algebra-lemma-two-out-of-three-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-two-out-of-three-perfect", "contents": [ "Let $R$ be a ring. Let $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$", "be a distinguished triangle in $D(R)$. If two out of three of", "$K^\\bullet, L^\\bullet, M^\\bullet$ are", "perfect then the third is also perfect." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemmas \\ref{lemma-perfect}, \\ref{lemma-two-out-of-three-pseudo-coherent}, and", "\\ref{lemma-cone-tor-amplitude}." ], "refs": [ "more-algebra-lemma-perfect", "more-algebra-lemma-two-out-of-three-pseudo-coherent", "more-algebra-lemma-cone-tor-amplitude" ], "ref_ids": [ 10212, 10149, 10172 ] } ], "ref_ids": [] }, { "id": 10215, "type": "theorem", "label": "more-algebra-lemma-summands-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-summands-perfect", "contents": [ "Let $R$ be a ring. If $K^\\bullet \\oplus L^\\bullet$ is perfect, then", "so are $K^\\bullet$ and $L^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemmas \\ref{lemma-perfect}, \\ref{lemma-summands-pseudo-coherent}, and", "\\ref{lemma-summands-tor-amplitude}." ], "refs": [ "more-algebra-lemma-perfect", "more-algebra-lemma-summands-pseudo-coherent", "more-algebra-lemma-summands-tor-amplitude" ], "ref_ids": [ 10212, 10151, 10174 ] } ], "ref_ids": [] }, { "id": 10216, "type": "theorem", "label": "more-algebra-lemma-complex-perfect-modules", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-complex-perfect-modules", "contents": [ "Let $R$ be a ring. Let $K^\\bullet$ be a bounded complex of perfect", "$R$-modules. Then $K^\\bullet$ is a perfect complex." ], "refs": [], "proofs": [ { "contents": [ "Follows by induction on the length of the finite complex: use", "Lemma \\ref{lemma-two-out-of-three-perfect}", "and the stupid truncations." ], "refs": [ "more-algebra-lemma-two-out-of-three-perfect" ], "ref_ids": [ 10214 ] } ], "ref_ids": [] }, { "id": 10217, "type": "theorem", "label": "more-algebra-lemma-cohomology-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cohomology-perfect", "contents": [ "Let $R$ be a ring. If $K^\\bullet \\in D^b(R)$ and all its cohomology", "modules are perfect, then $K^\\bullet$ is perfect." ], "refs": [], "proofs": [ { "contents": [ "Follows by induction on the length of the finite complex: use", "Lemma \\ref{lemma-two-out-of-three-perfect}", "and the canonical truncations." ], "refs": [ "more-algebra-lemma-two-out-of-three-perfect" ], "ref_ids": [ 10214 ] } ], "ref_ids": [] }, { "id": 10218, "type": "theorem", "label": "more-algebra-lemma-perfect-push-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-perfect-push-perfect", "contents": [ "Let $A \\to B$ be a ring map. Assume that $B$ is perfect as", "an $A$-module. Let $K^\\bullet$ be a perfect complex of $B$-modules.", "Then $K^\\bullet$ is perfect as a complex of $A$-modules." ], "refs": [], "proofs": [ { "contents": [ "Using", "Lemma \\ref{lemma-perfect}", "this translates into the corresponding results for pseudo-coherent modules", "and modules of finite tor dimension. See", "Lemma \\ref{lemma-finite-tor-dimension-push-tor-amplitude}", "and", "Lemma \\ref{lemma-finite-push-pseudo-coherent}", "for those results." ], "refs": [ "more-algebra-lemma-perfect", "more-algebra-lemma-finite-tor-dimension-push-tor-amplitude", "more-algebra-lemma-finite-push-pseudo-coherent" ], "ref_ids": [ 10212, 10179, 10154 ] } ], "ref_ids": [] }, { "id": 10219, "type": "theorem", "label": "more-algebra-lemma-pull-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pull-perfect", "contents": [ "Let $A \\to B$ be a ring map.", "Let $K^\\bullet$ be a perfect", "complex of $A$-modules. Then $K^\\bullet \\otimes_A^{\\mathbf{L}} B$", "is a perfect complex of $B$-modules." ], "refs": [], "proofs": [ { "contents": [ "Using", "Lemma \\ref{lemma-perfect}", "this translates into the corresponding results for pseudo-coherent modules", "and modules of finite tor dimension. See", "Lemma \\ref{lemma-pull-tor-amplitude}", "and", "Lemma \\ref{lemma-pull-pseudo-coherent}", "for those results." ], "refs": [ "more-algebra-lemma-perfect", "more-algebra-lemma-pull-tor-amplitude", "more-algebra-lemma-pull-pseudo-coherent" ], "ref_ids": [ 10212, 10180, 10155 ] } ], "ref_ids": [] }, { "id": 10220, "type": "theorem", "label": "more-algebra-lemma-flat-base-change-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-base-change-perfect", "contents": [ "Let $A \\to B$ be a flat ring map. Let $M$ be a perfect $A$-module.", "Then $M \\otimes_A B$ is a perfect $B$-module." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-perfect-module}", "the assumption implies that $M$ has a finite resolution $F_\\bullet$ by", "finite projective $R$-modules. As $A \\to B$ is flat the complex", "$F_\\bullet \\otimes_A B$ is a finite length resolution of $M \\otimes_A B$", "by finite projective modules over $B$. Hence $M \\otimes_A B$ is perfect." ], "refs": [ "more-algebra-lemma-perfect-module" ], "ref_ids": [ 10213 ] } ], "ref_ids": [] }, { "id": 10221, "type": "theorem", "label": "more-algebra-lemma-glue-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-glue-perfect", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be elements which", "generate the unit ideal. Let $K^\\bullet$", "be a complex of $R$-modules. If for each $i$ the complex", "$K^\\bullet \\otimes_R R_{f_i}$ is perfect,", "then $K^\\bullet$ is perfect." ], "refs": [], "proofs": [ { "contents": [ "Using", "Lemma \\ref{lemma-perfect}", "this translates into the corresponding results for pseudo-coherent modules", "and modules of finite tor dimension. See", "Lemma \\ref{lemma-glue-tor-amplitude}", "and", "Lemma \\ref{lemma-glue-pseudo-coherent}", "for those results." ], "refs": [ "more-algebra-lemma-perfect", "more-algebra-lemma-glue-tor-amplitude", "more-algebra-lemma-glue-pseudo-coherent" ], "ref_ids": [ 10212, 10183, 10157 ] } ], "ref_ids": [] }, { "id": 10222, "type": "theorem", "label": "more-algebra-lemma-flat-descent-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-descent-perfect", "contents": [ "Let $R$ be a ring. Let $a, b \\in \\mathbf{Z}$. Let $K^\\bullet$", "be a complex of $R$-modules. Let $R \\to R'$ be a faithfully flat", "ring map. If the complex $K^\\bullet \\otimes_R R'$ is perfect, then", "$K^\\bullet$ is perfect." ], "refs": [], "proofs": [ { "contents": [ "Using", "Lemma \\ref{lemma-perfect}", "this translates into the corresponding results for pseudo-coherent modules", "and modules of finite tor dimension. See", "Lemma \\ref{lemma-flat-descent-tor-amplitude}", "and", "Lemma \\ref{lemma-flat-descent-pseudo-coherent}", "for those results." ], "refs": [ "more-algebra-lemma-perfect", "more-algebra-lemma-flat-descent-tor-amplitude", "more-algebra-lemma-flat-descent-pseudo-coherent" ], "ref_ids": [ 10212, 10184, 10158 ] } ], "ref_ids": [] }, { "id": 10223, "type": "theorem", "label": "more-algebra-lemma-regular-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-regular-perfect", "contents": [ "Let $R$ be a regular ring of finite dimension. Then", "\\begin{enumerate}", "\\item an $R$-module is perfect if and only if it is a finite $R$-module, and", "\\item a complex of $R$-modules $K^\\bullet$ is perfect if and only", "if $K^\\bullet \\in D^b(R)$ and each $H^i(K^\\bullet)$ is a finite $R$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By", "Algebra, Lemma \\ref{algebra-lemma-finite-gl-dim-finite-dim-regular}", "the assumption on $R$ means that $R$ has finite global dimension.", "Hence every module has finite tor dimension, see", "Lemma \\ref{lemma-finite-gl-dim-tor-dimension}.", "On the other hand, as $R$ is Noetherian, a module is pseudo-coherent", "if and only if it is finite, see", "Lemma \\ref{lemma-Noetherian-pseudo-coherent}.", "This proves part (1).", "\\medskip\\noindent", "Let $K^\\bullet$ be a complex of $R$-modules.", "If $K^\\bullet$ is perfect, then it is in $D^b(R)$ and it is", "quasi-isomorphic to a finite complex of finite projective $R$-modules", "so certainly each $H^i(K^\\bullet)$ is a finite $R$-module (as $R$ is", "Noetherian). Conversely, suppose that $K^\\bullet$ is in $D^b(R)$", "and each $H^i(K^\\bullet)$ is a finite $R$-module. Then by (1) each", "$H^i(K^\\bullet)$ is a perfect $R$-module, whence $K^\\bullet$ is", "perfect by", "Lemma \\ref{lemma-cohomology-perfect}" ], "refs": [ "algebra-lemma-finite-gl-dim-finite-dim-regular", "more-algebra-lemma-finite-gl-dim-tor-dimension", "more-algebra-lemma-Noetherian-pseudo-coherent", "more-algebra-lemma-cohomology-perfect" ], "ref_ids": [ 980, 10186, 10160, 10217 ] } ], "ref_ids": [] }, { "id": 10224, "type": "theorem", "label": "more-algebra-lemma-dual-perfect-complex", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-dual-perfect-complex", "contents": [ "Let $A$ be a ring. Let $K \\in D(A)$ be perfect. Then $K^\\vee = R\\Hom_A(K, A)$", "is a perfect complex and $K \\cong (K^\\vee)^\\vee$. There are functorial", "isomorphisms", "$$", "L \\otimes_A^\\mathbf{L} K^\\vee = R\\Hom_A(K, L)", "\\quad\\text{and}\\quad", "H^0(L \\otimes_A^\\mathbf{L} K^\\vee) = \\Ext_A^0(K, L)", "$$", "for $L \\in D(A)$." ], "refs": [], "proofs": [ { "contents": [ "We can represent $K$ by a complex $K^\\bullet$ of finite projective", "$A$-modules. By Lemma \\ref{lemma-RHom-out-of-projective} the object $K^\\vee$", "is represented by the complex $E^\\bullet = \\Hom^\\bullet(K^\\bullet, A)$.", "Note that $E^n = \\Hom_A(K^{-n}, A)$ and the differentials of", "$E^\\bullet$ are the transpose of the differentials of $K^\\bullet$", "up to sign. Observe that $E^\\bullet$ is the left dual of $K^\\bullet$", "in the symmetric monoidal category of complexes of $R$-modules, see", "Lemma \\ref{lemma-left-dual-complex}. There is a canonical map", "$$", "K^\\bullet = \\text{Tot}(\\Hom^\\bullet(A, A) \\otimes_A K^\\bullet)", "\\longrightarrow", "\\Hom^\\bullet(\\Hom^\\bullet(K^\\bullet, A), A)", "$$", "which up to sign uses the evaluation map in each degree,", "see Lemma \\ref{lemma-evaluate-and-more}. (For sign rules see", "Section \\ref{section-sign-rules}.)", "Thus this map defines a canonical isomorphism $(K^\\vee)^\\vee \\cong K$", "as the double dual of a finite projective module is itself.", "\\medskip\\noindent", "The second equality follows from the first by", "Lemma \\ref{lemma-internal-hom} and", "Derived Categories, Lemma \\ref{derived-lemma-morphisms-from-projective-complex}", "as well as the definition of Ext groups, see", "Derived Categories, Section \\ref{derived-section-ext}.", "Let $L^\\bullet$ be a complex of $A$-modules representing $L$.", "By Section \\ref{section-sign-rules} item (\\ref{item-compatible})", "there is a canonical isomorphism", "$$", "\\text{Tot}(L^\\bullet \\otimes_A E^\\bullet)", "\\longrightarrow", "\\Hom^\\bullet(K^\\bullet, L^\\bullet)", "$$", "of complexes of $A$-modules. This proves the first displayed equality", "and the proof is complete." ], "refs": [ "more-algebra-lemma-RHom-out-of-projective", "more-algebra-lemma-left-dual-complex", "more-algebra-lemma-evaluate-and-more", "more-algebra-lemma-internal-hom", "derived-lemma-morphisms-from-projective-complex" ], "ref_ids": [ 10207, 10205, 10202, 10206, 1862 ] } ], "ref_ids": [] }, { "id": 10225, "type": "theorem", "label": "more-algebra-lemma-colim-and-lim-of-duals", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-colim-and-lim-of-duals", "contents": [ "\\begin{slogan}", "Trivial duality for systems of perfect objects.", "\\end{slogan}", "Let $A$ be a ring. Let $(K_n)_{n \\in \\mathbf{N}}$ be a system of", "perfect objects of $D(A)$. Let $K = \\text{hocolim} K_n$ be the derived colimit", "(Derived Categories, Definition \\ref{derived-definition-derived-colimit}).", "Then for any object $E$ of $D(A)$ we have", "$$", "R\\Hom_A(K, E) = R\\lim E \\otimes^\\mathbf{L}_A K_n^\\vee", "$$", "where $(K_n^\\vee)$ is the inverse system of dual perfect complexes." ], "refs": [ "derived-definition-derived-colimit" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-dual-perfect-complex} we have", "$R\\lim E \\otimes^\\mathbf{L}_A K_n^\\vee =", "R\\lim R\\Hom_A(K_n, E)$", "which fits into the distinguished triangle", "$$", "R\\lim R\\Hom_A(K_n, E) \\to", "\\prod R\\Hom_A(K_n, E) \\to", "\\prod R\\Hom_A(K_n, E)", "$$", "Because $K$ similarly fits into the distinguished triangle", "$\\bigoplus K_n \\to \\bigoplus K_n \\to K$ it suffices to show that", "$\\prod R\\Hom_A(K_n, E) = R\\Hom_A(\\bigoplus K_n, E)$.", "This is a formal consequence of (\\ref{equation-internal-hom})", "and the fact that derived tensor product commutes with direct sums." ], "refs": [ "more-algebra-lemma-dual-perfect-complex" ], "ref_ids": [ 10224 ] } ], "ref_ids": [ 2001 ] }, { "id": 10226, "type": "theorem", "label": "more-algebra-lemma-colimit-perfect-complexes", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-colimit-perfect-complexes", "contents": [ "Let $R = \\colim_{i \\in I} R_i$ be a filtered colimit of rings.", "\\begin{enumerate}", "\\item Given a perfect $K$ in $D(R)$ there exists an $i \\in I$", "and a perfect $K_i$ in $D(R_i)$ such that", "$K \\cong K_i \\otimes_{R_i}^\\mathbf{L} R$ in $D(R)$.", "\\item Given $0 \\in I$ and $K_0, L_0 \\in D(R_0)$ with $K_0$ perfect,", "we have", "$$", "\\Hom_{D(R)}(K_0 \\otimes_{R_0}^\\mathbf{L} R, L_0 \\otimes_{R_0}^\\mathbf{L} R) =", "\\colim_{i \\geq 0}", "\\Hom_{D(R_i)}(K_0 \\otimes_{R_0}^\\mathbf{L} R_i,", "L_0 \\otimes_{R_0}^\\mathbf{L} R_i)", "$$", "\\end{enumerate}", "In other words, the triangulated category of perfect complexes over $R$", "is the colimit of the triangulated categories of perfect complexes over $R_i$." ], "refs": [], "proofs": [ { "contents": [ "We will use the results of", "Algebra, Lemmas \\ref{algebra-lemma-module-map-property-in-colimit} and", "\\ref{algebra-lemma-colimit-category-fp-modules}", "without further mention. These lemmas in particular say that the", "category of finitely presented $R$-modules is the colimit of the", "categories of finitely presented $R_i$-modules. Since finite projective", "modules can be characterized as summands of finite free modules", "(Algebra, Lemma \\ref{algebra-lemma-finite-projective}) we see that", "the same is true for the category of finite projective modules.", "This proves (1) by our definition of perfect objects of $D(R)$.", "\\medskip\\noindent", "To prove (2) we may represent $K_0$ by a bounded complex $K_0^\\bullet$ of", "finite projective $R_0$-modules. We may represent $L_0$ by a K-flat", "complex $L_0^\\bullet$ (Lemma \\ref{lemma-K-flat-resolution}).", "Then we have", "$$", "\\Hom_{D(R)}(K_0 \\otimes_{R_0}^\\mathbf{L} R, L_0 \\otimes_{R_0}^\\mathbf{L} R) =", "\\Hom_{K(R)}(K_0^\\bullet \\otimes_{R_0} R, L_0^\\bullet \\otimes_{R_0} R)", "$$", "by Derived Categories, Lemma", "\\ref{derived-lemma-morphisms-from-projective-complex}.", "Similarly for the $\\Hom$ with $R$ replaced by $R_i$. Since", "in the right hand side only a finite number of terms are involved,", "since", "$$", "\\Hom_R(K_0^p \\otimes_{R_0} R, L_0^q \\otimes_{R_0} R) =", "\\colim_{i \\geq 0}", "\\Hom_{R_i}(K_0^p \\otimes_{R_0} R_i, L_0^q \\otimes_{R_0} R_i)", "$$", "by the lemmas cited at the beginning of the proof, and since", "filtered colimits are exact", "(Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact})", "we conclude that (2) holds as well." ], "refs": [ "algebra-lemma-module-map-property-in-colimit", "algebra-lemma-colimit-category-fp-modules", "algebra-lemma-finite-projective", "more-algebra-lemma-K-flat-resolution", "derived-lemma-morphisms-from-projective-complex", "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 1094, 1095, 795, 10131, 1862, 343 ] } ], "ref_ids": [] }, { "id": 10227, "type": "theorem", "label": "more-algebra-lemma-lift-acyclic-complex", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-acyclic-complex", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $\\mathcal{P}$", "be a class of $R$-modules. Assume", "\\begin{enumerate}", "\\item each $P \\in \\mathcal{P}$ is a projective $R$-module,", "\\item if $P_1 \\in \\mathcal{P}$ and $P_1 \\oplus P_2 \\in \\mathcal{P}$, then", "$P_2 \\in \\mathcal{P}$, and", "\\item if $f : P_1 \\to P_2$, $P_1, P_2 \\in \\mathcal{P}$ is surjective", "modulo $I$, then $f$ is surjective.", "\\end{enumerate}", "Then given any bounded above acyclic complex $E^\\bullet$", "whose terms are of the form $P/IP$ for $P \\in \\mathcal{P}$ there", "exists a bounded above acyclic complex $P^\\bullet$ whose terms", "are in $\\mathcal{P}$ lifting $E^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Say $E^i = 0$ for $i > b$. Assume given $n$ and a morphism of complexes", "$$", "\\xymatrix{", "& & P^n \\ar[r] \\ar[d] & P^{n + 1} \\ar[r] \\ar[d] & \\ldots \\ar[r]", "& P^b \\ar[r] \\ar[d] & 0 \\ar[r] \\ar[d] & \\ldots \\\\", "\\ldots \\ar[r] & E^{n - 1} \\ar[r] &", "E^n \\ar[r] & E^{n + 1} \\ar[r] & \\ldots \\ar[r] &", "E^b \\ar[r] & 0 \\ar[r] & \\ldots", "}", "$$", "with $P^i \\in \\mathcal{P}$, with", "$P^n \\to P^{n + 1} \\to \\ldots \\to P^b$ acyclic in degrees $\\geq n + 1$,", "and with vertical maps inducing isomorphisms $P^i/IP^i \\to E^i$.", "In this situation one can inductively choose isomorphisms", "$P^i = Z^i \\oplus Z^{i + 1}$ such that the maps $P^i \\to P^{i + 1}$", "are given by", "$Z^i \\oplus Z^{i + 1} \\to Z^{i + 1} \\to Z^{i + 1} \\oplus Z^{i + 2}$.", "By property (2) and arguing inductively we see that $Z^i \\in \\mathcal{P}$.", "Choose $P^{n - 1} \\in \\mathcal{P}$ and an isomorphism", "$P^{n - 1}/IP^{n - 1} \\to E^{n - 1}$. Since", "$P^{n - 1}$ is projective and since $Z^n/IZ^n = \\Im(E^{n - 1} \\to E^n)$,", "we can lift the map $P^{n - 1} \\to E^{n - 1} \\to E^n$ to a map", "$P^{n - 1} \\to Z^n$. By property (3) the map $P^{n - 1} \\to Z^n$", "is surjective. Thus we obtain an extension of the diagram", "by adding $P^{n - 1}$ and the maps just constructed to the left", "of $P^n$. Since a diagram of the desired form exists for $n > b$", "we conclude by induction on $n$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10228, "type": "theorem", "label": "more-algebra-lemma-lift-complex", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-complex", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $\\mathcal{P}$", "be a class of $R$-modules.", "Let $K \\in D(R)$ and let $E^\\bullet$ be a complex of $R/I$-modules", "representing $K \\otimes_R^\\mathbf{L} R/I$. Assume", "\\begin{enumerate}", "\\item each $P \\in \\mathcal{P}$ is a projective $R$-module,", "\\item $P_1 \\in \\mathcal{P}$ and $P_1 \\oplus P_2 \\in \\mathcal{P}$", "if and only if $P_1, P_2 \\in \\mathcal{P}$,", "\\item if $f : P_1 \\to P_2$, $P_1, P_2 \\in \\mathcal{P}$ is surjective", "modulo $I$, then $f$ is surjective,", "\\item $E^\\bullet$ is bounded above and $E^i$ is of the form $P/IP$", "for $P \\in \\mathcal{P}$, and", "\\item $K$ can be represented by a bounded above complex", "whose terms are in $\\mathcal{P}$.", "\\end{enumerate}", "Then there exists a bounded above complex $P^\\bullet$ whose terms", "are in $\\mathcal{P}$ with $P^\\bullet/IP^\\bullet$ isomorphic to", "$E^\\bullet$ and representing $K$ in $D(R)$." ], "refs": [], "proofs": [ { "contents": [ "By assumption (5) we can represent $K$ by a bounded above complex ", "$K^\\bullet$ whose terms are in $\\mathcal{P}$. Then $K \\otimes_R^\\mathbf{L} R/I$", "is represented by $K^\\bullet/IK^\\bullet$. Since $E^\\bullet$ is", "a bounded above complex of projective $R/I$-modules by (4), ", "we can choose a quasi-isomorphism $\\delta : E^\\bullet \\to K^\\bullet/IK^\\bullet$", "(Derived Categories, Lemma", "\\ref{derived-lemma-morphisms-from-projective-complex}).", "Let $C^\\bullet$ be cone on $\\delta$", "(Derived Categories, Definition \\ref{derived-definition-cone}).", "The module $C^i$ is the direct sum $K^i/IK^i \\oplus E^{i + 1}$", "hence is of the form $P/IP$ for some $P \\in \\mathcal{P}$", "as (2) says in particular that $\\mathcal{P}$ is preserved under taking sums.", "Since $C^\\bullet$ is acyclic, we can apply", "Lemma \\ref{lemma-lift-acyclic-complex} and find a acyclic", "lift $A^\\bullet$ of $C^\\bullet$. The complex $A^\\bullet$ is", "bounded above and has terms in $\\mathcal{P}$. In", "$$", "\\xymatrix{", "K^\\bullet \\ar@{..>}[r] \\ar[d] & A^\\bullet \\ar[d] \\\\", "K^\\bullet/IK^\\bullet \\ar[r] & C^\\bullet \\ar[r] & E^\\bullet[1]", "}", "$$", "we can find the dotted arrow making the diagram commute", "by Derived Categories, Lemma", "\\ref{derived-lemma-morphisms-lift-projective}.", "We will show below that it follows from (1), (2), (3)", "that $K^i \\to A^i$ is the inclusion of a direct summand", "for every $i$. By property (2) we see that $P^i = \\Coker(K^i \\to A^i)$", "is in $\\mathcal{P}$. Thus we can take", "$P^\\bullet = \\Coker(K^\\bullet \\to A^\\bullet)[-1]$ to conclude.", "\\medskip\\noindent", "To finish the proof we have to show the following: Let $f : P_1 \\to P_2$,", "$P_1, P_2 \\in \\mathcal{P}$ and $P_1/IP_1 \\to P_2/IP_2$ is split", "injective with cokernel of the form $P_3/IP_3$ for some", "$P_3 \\in \\mathcal{P}$, then $f$ is split injective.", "Write $E_i = P_i/IP_i$. Then $E_2 = E_1 \\oplus E_3$.", "Since $P_2$ is projective we can choose a map $g : P_2 \\to P_3$", "lifting the map $E_2 \\to E_3$. By condition (3) the map $g$", "is surjective, hence split as $P_3$ is projective. Set $P_1' = \\Ker(g)$", "and choose a splitting $P_2 = P'_1 \\oplus P_3$. Then $P'_1 \\in \\mathcal{P}$", "by (2). We do not know that", "$g \\circ f = 0$, but we can consider the map", "$$", "P_1 \\xrightarrow{f} P_2 \\xrightarrow{projection} P'_1", "$$", "The composition modulo $I$ is an isomorphism. Since $P'_1$ is", "projective we can split $P_1 = T \\oplus P'_1$. If $T = 0$, then", "we are done, because then $P_2 \\to P'_1$ is a splitting of $f$.", "We see that $T \\in \\mathcal{P}$ by (2).", "Calculating modulo $I$ we see that $T/IT = 0$.", "Since $0 \\in \\mathcal{P}$ (as the summand of any $P$ in $\\mathcal{P}$)", "we see the map $0 \\to T$ is surjective and we conclude that $T = 0$", "as desired." ], "refs": [ "derived-lemma-morphisms-from-projective-complex", "derived-definition-cone", "more-algebra-lemma-lift-acyclic-complex", "derived-lemma-morphisms-lift-projective" ], "ref_ids": [ 1862, 1978, 10227, 1860 ] } ], "ref_ids": [] }, { "id": 10229, "type": "theorem", "label": "more-algebra-lemma-lift-complex-projectives", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-complex-projectives", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $E^\\bullet$", "be a complex of $R/I$-modules. Let $K$ be an object of $D(R)$. Assume that", "\\begin{enumerate}", "\\item $E^\\bullet$ is a bounded above complex of projective $R/I$-modules,", "\\item $K \\otimes_R^\\mathbf{L} R/I$ is represented by $E^\\bullet$ in", "$D(R/I)$, and", "\\item $I$ is a nilpotent ideal.", "\\end{enumerate}", "Then there exists a bounded above complex $P^\\bullet$ of projective", "$R$-modules representing $K$ in $D(R)$ such that $P^\\bullet \\otimes_R R/I$", "is isomorphic to $E^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "We apply Lemma \\ref{lemma-lift-complex} using the class $\\mathcal{P}$", "of all projective $R$-modules. Properties (1) and (2) of the lemma", "are immediate. Property (3) follows from Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK}).", "Property (4) follows from the fact that we can lift projective", "$R/I$-modules to projective $R$-modules, see", "Algebra, Lemma \\ref{algebra-lemma-lift-projective-module}.", "To see that (5) holds it suffices to show that $K$ is in $D^{-}(R)$.", "We are given that $K \\otimes_R^\\mathbf{L} R/I$ is in $D^{-}(R/I)$", "(because $E^\\bullet$ is bounded above).", "We will show by induction on $n$ that", "$K \\otimes_R^\\mathbf{L} R/I^n$ is in $D^{-}(R/I^n)$.", "This will finish the proof because $I$ being nilpotent exactly", "means that $I^n = 0$ for some $n$.", "We may represent $K$ by a K-flat complex $K^\\bullet$ with flat terms", "(Lemma \\ref{lemma-K-flat-resolution}).", "Then derived tensor products are represented by usual tensor products.", "Thus we consider the exact sequence", "$$", "0 \\to K^\\bullet \\otimes_R I^n/I^{n + 1} \\to", "K^\\bullet \\otimes_R R/I^{n + 1} \\to", "K^\\bullet \\otimes_R R/I^n \\to 0", "$$", "Thus the cohomology of $K \\otimes_R^\\mathbf{L} R/I^{n + 1}$", "sits in a long exact sequence with the cohomology of", "$K \\otimes_R^\\mathbf{L} R/I^n$ and the cohomology of", "$$", "K \\otimes_R^\\mathbf{L} I^n/I^{n + 1} =", "K \\otimes_R^\\mathbf{L} R/I \\otimes_{R/I}^\\mathbf{L} I^n/I^{n + 1}", "$$", "The first cohomologies vanish above a certain degree", "by induction assumption and the second cohomologies vanish", "above a certain degree because $K^\\bullet \\otimes_R^\\mathbf{L} R/I$", "is bounded above and $I^n/I^{n + 1}$ is in degree $0$." ], "refs": [ "more-algebra-lemma-lift-complex", "algebra-lemma-NAK", "algebra-lemma-lift-projective-module", "more-algebra-lemma-K-flat-resolution" ], "ref_ids": [ 10228, 401, 792, 10131 ] } ], "ref_ids": [] }, { "id": 10230, "type": "theorem", "label": "more-algebra-lemma-lift-complex-stably-frees", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-complex-stably-frees", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $E^\\bullet$", "be a complex of $R/I$-modules. Let $K$ be an object of $D(R)$. Assume that", "\\begin{enumerate}", "\\item $E^\\bullet$ is a bounded above complex of", "finite stably free $R/I$-modules,", "\\item $K \\otimes_R^\\mathbf{L} R/I$ is represented by $E^\\bullet$ in $D(R/I)$,", "\\item $K^\\bullet$ is pseudo-coherent, and", "\\item every element of $1 + I$ is invertible.", "\\end{enumerate}", "Then there exists a bounded above complex $P^\\bullet$ of finite stably free", "$R$-modules representing $K$ in $D(R)$ such that $P^\\bullet \\otimes_R R/I$", "is isomorphic to $E^\\bullet$. Moreover, if $E^i$ is free, then $P^i$ is free." ], "refs": [], "proofs": [ { "contents": [ "We apply Lemma \\ref{lemma-lift-complex} using the class $\\mathcal{P}$", "of all finite stably free $R$-modules. Property (1) of the lemma is immediate.", "Property (2) follows from Lemma \\ref{lemma-exact-category-stably-free}.", "Property (3) follows from Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK}).", "Property (4) follows from the fact that we can lift finite stably free", "$R/I$-modules to finite stably free $R$-modules, see", "Lemma \\ref{lemma-lift-stably-free}.", "Part (5) holds because a pseudo-coherent complex can be represented", "by a bounded above complex of finite free $R$-modules.", "The final assertion of the lemma follows from", "Lemma \\ref{lemma-isomorphic-finite-projective-lifts}." ], "refs": [ "more-algebra-lemma-lift-complex", "more-algebra-lemma-exact-category-stably-free", "algebra-lemma-NAK", "more-algebra-lemma-lift-stably-free", "more-algebra-lemma-isomorphic-finite-projective-lifts" ], "ref_ids": [ 10228, 9807, 401, 9808, 9810 ] } ], "ref_ids": [] }, { "id": 10231, "type": "theorem", "label": "more-algebra-lemma-lift-pseudo-coherent-from-residue-field", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-pseudo-coherent-from-residue-field", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Let $K \\in D(R)$", "be pseudo-coherent. Set", "$d_i = \\dim_\\kappa H^i(K \\otimes_R^\\mathbf{L} \\kappa)$.", "Then $d_i < \\infty$ and for some $b \\in \\mathbf{Z}$ we have", "$d_i = 0$ for $i > b$.", "Then there exists a complex", "$$", "\\ldots \\to", "R^{\\oplus d_{b - 2}} \\to", "R^{\\oplus d_{b - 1}} \\to", "R^{\\oplus d_b} \\to 0 \\to \\ldots", "$$", "representing $K$ in $D(R)$. Moreover, this complex is unique up to", "isomorphism(!)." ], "refs": [], "proofs": [ { "contents": [ "Observe that $K \\otimes_R^\\mathbf{L} \\kappa$ is pseudo-coherent", "as an object of $D(\\kappa)$, see Lemma \\ref{lemma-pull-pseudo-coherent}.", "Hence the cohomology spaces are finite dimensional and vanish above", "some cutoff. Every object of $D(\\kappa)$ is isomorphic", "in $D(\\kappa)$ to a complex $E^\\bullet$ with zero differentials.", "In particular $E^i \\cong \\kappa^{\\oplus d_i}$ is finite free.", "Applying Lemma \\ref{lemma-lift-complex-stably-frees} we obtain", "the existence.", "\\medskip\\noindent", "If we have two complexes $F^\\bullet$ and $G^\\bullet$", "with $F^i$ and $G^i$ free of rank $d_i$ representing $K$.", "Then we may choose a map of complexes $\\beta : F^\\bullet \\to G^\\bullet$", "representing the isomorphism $F^\\bullet \\cong K \\cong G^\\bullet$, see", "Derived Categories, Lemma", "\\ref{derived-lemma-morphisms-from-projective-complex}.", "The induced map of complexes", "$\\beta \\otimes 1 : F^\\bullet \\otimes_R^\\mathbf{L} \\kappa \\to", "G^\\bullet \\otimes_R^\\mathbf{L} \\kappa$", "must be an isomorphism of complexes as the differentials", "in $F^\\bullet \\otimes_R^\\mathbf{L} \\kappa$ and", "$G^\\bullet \\otimes_R^\\mathbf{L} \\kappa$ are zero.", "Thus $\\beta^i : F^i \\to G^i$ is a map of finite free $R$-modules", "whose reduction modulo $\\mathfrak m$ is an isomorphism.", "Hence $\\beta^i$ is an isomorphism and we win." ], "refs": [ "more-algebra-lemma-pull-pseudo-coherent", "more-algebra-lemma-lift-complex-stably-frees", "derived-lemma-morphisms-from-projective-complex" ], "ref_ids": [ 10155, 10230, 1862 ] } ], "ref_ids": [] }, { "id": 10232, "type": "theorem", "label": "more-algebra-lemma-lift-perfect-from-residue-field", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-perfect-from-residue-field", "contents": [ "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime. Let $K \\in D(R)$", "be perfect. Set", "$d_i =", "\\dim_{\\kappa(\\mathfrak p)} H^i(K \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p))$.", "Then $d_i < \\infty$ and only a finite number are nonzero.", "Then there exists an $f \\in R$, $f \\not \\in \\mathfrak p$ and a", "complex", "$$", "\\ldots \\to 0 \\to R_f^{\\oplus d_a} \\to R_f^{\\oplus d_{a + 1}} \\to", "\\ldots \\to", "R_f^{\\oplus d_{b - 1}} \\to", "R_f^{\\oplus d_b} \\to 0", "\\to \\ldots", "$$", "representing $K \\otimes_R^\\mathbf{L} R_f$ in $D(R_f)$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $K \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p)$", "is perfect as an object of $D(\\kappa(\\mathfrak p))$, see", "Lemma \\ref{lemma-pull-perfect}. Hence only a finite number of", "$d_i$ are nonzero and they are all finite. Applying", "Lemma \\ref{lemma-lift-pseudo-coherent-from-residue-field}", "we get a complex representing $K$", "having the desired shape over the local ring $R_\\mathfrak p$.", "We have $R_\\mathfrak p = \\colim R_f$ for", "$f \\in R$, $f \\not \\in \\mathfrak p$", "(Algebra, Lemma \\ref{algebra-lemma-localization-colimit}).", "We conclude by Lemma \\ref{lemma-colimit-perfect-complexes}.", "Some details omitted." ], "refs": [ "more-algebra-lemma-pull-perfect", "more-algebra-lemma-lift-pseudo-coherent-from-residue-field", "algebra-lemma-localization-colimit", "more-algebra-lemma-colimit-perfect-complexes" ], "ref_ids": [ 10219, 10231, 348, 10226 ] } ], "ref_ids": [] }, { "id": 10233, "type": "theorem", "label": "more-algebra-lemma-compare-representatives-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-compare-representatives-perfect", "contents": [ "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime. Let $M^\\bullet$", "and $N^\\bullet$ be bounded complexes of finite projective $R$-modules", "representing the same object of $D(R)$. Then there exists an $f \\in R$,", "$f \\not \\in \\mathfrak p$ such that there is an isomorphism (!)", "of complexes", "$$", "M^\\bullet_f \\oplus P^\\bullet \\cong N^\\bullet_f \\oplus Q^\\bullet", "$$", "where $P^\\bullet$ and $Q^\\bullet$ are finite direct sums of", "trivial complexes, i.e., complexes of the form", "the form $\\ldots \\to 0 \\to R_f \\xrightarrow{1} R_f \\to 0 \\to \\ldots$", "(placed in arbitrary degrees)." ], "refs": [], "proofs": [ { "contents": [ "If we have an isomorphism of the type described over the localization", "$R_\\mathfrak p$, then using that $R_\\mathfrak p = \\colim R_f$", "(Algebra, Lemma \\ref{algebra-lemma-localization-colimit}) we can", "descend the isomorphism to an isomorphism over $R_f$ for some $f$.", "Thus we may assume $R$ is local and $\\mathfrak p$ is the maximal ideal.", "In this case the result follows from the uniqueness of a ``minimal''", "complex representing a perfect object, see", "Lemma \\ref{lemma-lift-pseudo-coherent-from-residue-field}, and", "the fact that any complex is a direct sum of a trivial complex", "and a minimal one (Algebra, Lemma \\ref{algebra-lemma-add-trivial-complex})." ], "refs": [ "algebra-lemma-localization-colimit", "more-algebra-lemma-lift-pseudo-coherent-from-residue-field", "algebra-lemma-add-trivial-complex" ], "ref_ids": [ 348, 10231, 908 ] } ], "ref_ids": [] }, { "id": 10234, "type": "theorem", "label": "more-algebra-lemma-lift-complex-finite-projectives", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-complex-finite-projectives", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $E^\\bullet$", "be a complex of $R/I$-modules. Let $K$ be an object of $D(R)$. Assume that", "\\begin{enumerate}", "\\item $E^\\bullet$ is a bounded above complex of finite projective $R/I$-modules,", "\\item $K \\otimes_R^\\mathbf{L} R/I$ is represented by $E^\\bullet$ in $D(R/I)$,", "\\item $K$ is pseudo-coherent, and", "\\item $(R, I)$ is a henselian pair.", "\\end{enumerate}", "Then there exists a bounded above complex $P^\\bullet$ of finite projective", "$R$-modules representing $K$ in $D(R)$ such that $P^\\bullet \\otimes_R R/I$", "is isomorphic to $E^\\bullet$. Moreover, if $E^i$ is free, then $P^i$ is free." ], "refs": [], "proofs": [ { "contents": [ "We apply Lemma \\ref{lemma-lift-complex} using the class $\\mathcal{P}$", "of all finite projective $R$-modules. Properties (1) and (2)", "of the lemma are immediate.", "Property (3) follows from Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK}).", "Property (4) follows from the fact that we can lift finite projective", "$R/I$-modules to finite projective $R$-modules, see", "Lemma \\ref{lemma-lift-finite-projective-module}.", "Property (5) holds because a pseudo-coherent complex can be represented", "by a bounded above complex of finite free $R$-modules.", "Thus Lemma \\ref{lemma-lift-complex} applies and we find $P^\\bullet$", "as desired. The final assertion of the lemma follows from", "Lemma \\ref{lemma-isomorphic-finite-projective-lifts}." ], "refs": [ "more-algebra-lemma-lift-complex", "algebra-lemma-NAK", "more-algebra-lemma-lift-finite-projective-module", "more-algebra-lemma-lift-complex", "more-algebra-lemma-isomorphic-finite-projective-lifts" ], "ref_ids": [ 10228, 401, 9878, 10228, 9810 ] } ], "ref_ids": [] }, { "id": 10235, "type": "theorem", "label": "more-algebra-lemma-splitting-unique", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-splitting-unique", "contents": [ "Let $R$ be a ring. Let $K$ and $L$ be objects of $D(R)$.", "Assume $L$ has projective-amplitude in $[a, b]$, for example if $L$", "is perfect of tor-amplitude in $[a, b]$.", "\\begin{enumerate}", "\\item If $H^i(K) = 0$ for $i \\geq a$, then", "$\\Hom_{D(R)}(L, K) = 0$.", "\\item If $H^i(K) = 0$ for $i \\geq a + 1$, then given any distinguished", "triangle $K \\to M \\to L \\to K[1]$", "there is an isomorphism $M \\cong K \\oplus L$", "in $D(R)$ compatible with the maps in the distinguished triangle.", "\\item If $H^i(K) = 0$ for $i \\geq a$, then the isomorphism", "in (2) exists and is unique.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The assumption that $L$ has projective-amplitude in $[a, b]$ means we", "can represent $L$ by a complex $L^\\bullet$ of projective $R$-modules", "with $L^i = 0$ for $i \\not \\in [a, b]$, see Definition", "\\ref{definition-projective-dimension}.", "If $L$ is perfect of tor-amplitude in $[a, b]$, then", "we can represent $L$ by a complex $L^\\bullet$ of finite projective $R$-modules", "with $L^i = 0$ for $i \\not \\in [a, b]$, see Lemma \\ref{lemma-perfect}.", "If $H^i(K) = 0$ for $i \\geq a$, then $K$ is quasi-isomorphic", "to $\\tau_{\\leq a - 1}K$. Hence we can represent $K$ by a complex", "$K^\\bullet$ of $R$-modules with $K^i = 0$ for $i \\geq a$. Then we obtain", "$$", "\\Hom_{D(R)}(L, K) = \\Hom_{K(R)}(L^\\bullet, K^\\bullet) = 0", "$$", "by Derived Categories, Lemma", "\\ref{derived-lemma-morphisms-from-projective-complex}.", "This proves (1). Under the hypotheses of (2) we see that", "$\\Hom_{D(R)}(L, K[1]) = 0$ by (1), hence", "the distinguished triangle is split by", "Derived Categories, Lemma \\ref{derived-lemma-split}.", "The uniqueness of (3) follows from (1)." ], "refs": [ "more-algebra-definition-projective-dimension", "more-algebra-lemma-perfect", "derived-lemma-morphisms-from-projective-complex", "derived-lemma-split" ], "ref_ids": [ 10625, 10212, 1862, 1766 ] } ], "ref_ids": [] }, { "id": 10236, "type": "theorem", "label": "more-algebra-lemma-better-cut-complex-in-two", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-better-cut-complex-in-two", "contents": [ "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime ideal.", "Let $K^\\bullet$ be a pseudo-coherent complex of $R$-modules.", "Assume that for some $i \\in \\mathbf{Z}$ the map", "$$", "H^i(K^\\bullet) \\otimes_R \\kappa(\\mathfrak p)", "\\longrightarrow", "H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p))", "$$", "is surjective. Then there exists an $f \\in R$, $f \\not \\in \\mathfrak p$", "such that $\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f)$ is a perfect", "object of $D(R_f)$ with tor amplitude in $[i + 1, \\infty]$ and", "a canonical isomorphism", "$$", "K^\\bullet \\otimes_R R_f \\cong", "\\tau_{\\leq i}(K^\\bullet \\otimes_R R_f) \\oplus", "\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f)", "$$", "in $D(R_f)$." ], "refs": [], "proofs": [ { "contents": [ "In this proof all tensor products are over $R$ and we write", "$\\kappa = \\kappa(\\mathfrak p)$. We may assume that $K^\\bullet$", "is a bounded above complex of finite free $R$-modules. Let us", "inspect what is happening in degree $i$:", "$$", "\\ldots \\to K^{i - 1} \\xrightarrow{d^{i - 1}} K^i \\xrightarrow{d^i}", "K^{i + 1} \\to \\ldots", "$$", "Let $0 \\subset V \\subset W \\subset K^i \\otimes \\kappa$ be defined", "by the formulas", "$$", "V = \\Im\\left(", "K^{i - 1} \\otimes \\kappa \\to K^i \\otimes \\kappa", "\\right)", "\\quad\\text{and}\\quad", "W = \\Ker\\left(", "K^i \\otimes \\kappa \\to K^{i + 1} \\otimes \\kappa", "\\right)", "$$", "Set $\\dim(V) = r$, $\\dim(W/V) = s$, and $\\dim(K^i \\otimes \\kappa/W) = t$.", "We can pick $x_1, \\ldots, x_r \\in K^{i - 1}$ which map by $d^{i - 1}$", "to a basis of $V$. By our assumption we can pick", "$y_1, \\ldots, y_s \\in \\Ker(d^i)$ mapping to a basis of $W/V$.", "Finally, choose $z_1, \\ldots, z_t \\in K^i$ mapping to a basis of", "$K^i \\otimes \\kappa/W$. Then we see that the elements", "$d^i(z_1), \\ldots, d^i(z_t) \\in K^{i + 1}$ are linearly independent", "in $K^{i + 1} \\otimes_R \\kappa$.", "By Algebra, Lemma \\ref{algebra-lemma-cokernel-flat} we may after replacing", "$R$ by $R_f$ for some $f \\in R$, $f \\not \\in \\mathfrak p$ assume that", "\\begin{enumerate}", "\\item $d^i(x_a), y_b, z_c$ is an $R$-basis of $K^i$,", "\\item $d^i(z_1), \\ldots, d^i(z_t)$ are $R$-linearly independent in", "$K^{i + 1}$, and", "\\item the quotient $E^{i + 1} = K^{i + 1}/\\sum Rd^i(z_c)$ is finite projective.", "\\end{enumerate}", "Since $d^i$ annihilates $d^{i - 1}(x_a)$ and $y_b$, we deduce from", "condition (2) that $E^{i + 1} = \\Coker(d^i : K^i \\to K^{i + 1})$.", "Thus we see that", "$$", "\\tau_{\\geq i + 1}K^\\bullet =", "(\\ldots \\to 0 \\to E^{i + 1} \\to K^{i + 2} \\to \\ldots)", "$$", "is a bounded complex of finite projective modules sitting in degrees", "$[i + 1, b]$ for some $b$. Thus $\\tau_{\\geq i + 1}K^\\bullet$ is perfect", "of amplitude $[i + 1, b]$. Since $\\tau_{\\leq i}K^\\bullet$ has no", "cohomology in degrees $> i$, we may apply Lemma \\ref{lemma-splitting-unique}", "to the distinguished triangle", "$$", "\\tau_{\\leq i}K^\\bullet \\to K^\\bullet \\to \\tau_{\\geq i + 1}K^\\bullet \\to", "(\\tau_{\\leq i}K^\\bullet)[1]", "$$", "(Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}) to conclude." ], "refs": [ "algebra-lemma-cokernel-flat", "more-algebra-lemma-splitting-unique", "derived-remark-truncation-distinguished-triangle" ], "ref_ids": [ 804, 10235, 2016 ] } ], "ref_ids": [] }, { "id": 10237, "type": "theorem", "label": "more-algebra-lemma-isolate-a-cohomology-group", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-isolate-a-cohomology-group", "contents": [ "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime ideal.", "Let $K^\\bullet$ be a pseudo-coherent complex of $R$-modules.", "Assume that for some $i \\in \\mathbf{Z}$ the maps", "$$", "H^i(K^\\bullet) \\otimes_R \\kappa(\\mathfrak p)", "\\longrightarrow", "H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p))", "\\quad\\text{and}\\quad", "H^{i - 1}(K^\\bullet) \\otimes_R \\kappa(\\mathfrak p)", "\\longrightarrow", "H^{i - 1}(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p))", "$$", "are surjective. Then there exists an $f \\in R$, $f \\not \\in \\mathfrak p$", "such that", "\\begin{enumerate}", "\\item $\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f)$ is a perfect", "object of $D(R_f)$ with tor amplitude in $[i + 1, \\infty]$,", "\\item $H^i(K^\\bullet)_f$ is a finite free $R_f$-module, and", "\\item there is a canonical direct sum decomposition", "$$", "K^\\bullet \\otimes_R R_f \\cong", "\\tau_{\\leq i - 1}(K^\\bullet \\otimes_R R_f) \\oplus", "H^i(K^\\bullet)_f[-i] \\oplus", "\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f)", "$$", "in $D(R_f)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We get (1) from Lemma \\ref{lemma-better-cut-complex-in-two} as well", "as a splitting", "$K^\\bullet \\otimes_R R_f = \\tau_{\\leq i}K^\\bullet \\otimes_R R_f \\oplus", "\\tau_{\\geq i + 1}K^\\bullet \\otimes_R R_f$", "in $D(R_f)$. Applying Lemma \\ref{lemma-better-cut-complex-in-two} once", "more to $\\tau_{\\leq i}K^\\bullet \\otimes_R R_f$", "we obtain (after suitably choosing $f$) a splitting", "$\\tau_{\\leq i}K^\\bullet \\otimes_R R_f =", "\\tau_{\\leq i - 1}K^\\bullet \\otimes_R R_f \\oplus H^i(K^\\bullet)_f$ in $D(R_f)$", "as well as the conclusion that $H^i(K)_f$ is a flat perfect module, i.e.,", "finite projective." ], "refs": [ "more-algebra-lemma-better-cut-complex-in-two", "more-algebra-lemma-better-cut-complex-in-two" ], "ref_ids": [ 10236, 10236 ] } ], "ref_ids": [] }, { "id": 10238, "type": "theorem", "label": "more-algebra-lemma-cut-complex-in-two", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cut-complex-in-two", "contents": [ "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime ideal.", "Let $i \\in \\mathbf{Z}$. Let $K^\\bullet$ be a pseudo-coherent complex", "of $R$-modules such that", "$H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p)) = 0$.", "Then there exists an $f \\in R$, $f \\not \\in \\mathfrak p$", "and a canonical direct sum decomposition", "$$", "K^\\bullet \\otimes_R R_f =", "\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f) \\oplus", "\\tau_{\\leq i - 1}(K^\\bullet \\otimes_R R_f)", "$$", "in $D(R_f)$ with $\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f)$ a perfect", "complex with tor-amplitude in $[i + 1, \\infty]$." ], "refs": [], "proofs": [ { "contents": [ "This is an often used special case of", "Lemma \\ref{lemma-better-cut-complex-in-two}.", "A direct proof is as follows.", "We may assume that $K^\\bullet$ is a bounded above", "complex of finite free $R$-modules. Let us inspect what is happening", "in degree $i$:", "$$", "\\ldots \\to K^{i - 2} \\to R^{\\oplus l}", "\\to R^{\\oplus m} \\to R^{\\oplus n} \\to K^{i + 2} \\to \\ldots", "$$", "Let $A$ be the $m \\times l$ matrix corresponding to $K^{i - 1} \\to K^i$", "and let $B$ be the $n \\times m$ matrix corresponding to $K^i \\to K^{i + 1}$.", "The assumption is that $A \\bmod \\mathfrak p$ has rank $r$ and that", "$B \\bmod \\mathfrak p$ has rank $m - r$. In other words, there is some", "$r \\times r$ minor $a$ of $A$ which is not in $\\mathfrak p$ and there is", "some $(m - r) \\times (m - r)$-minor $b$ of $B$ which is not in $\\mathfrak p$.", "Set $f = ab$. Then after inverting $f$ we can find direct sum decompositions", "$K^{i - 1} = R^{\\oplus l - r} \\oplus R^{\\oplus r}$,", "$K^i = R^{\\oplus r} \\oplus R^{\\oplus m - r}$,", "$K^{i + 1} = R^{\\oplus m - r} \\oplus R^{\\oplus n - m + r}$", "such that the module map $K^{i - 1} \\to K^i$ kills of", "$R^{\\oplus l - r}$ and induces an isomorphism of $R^{\\oplus r}$ onto the", "corresponding summand of $K^i$ and such that the module map $K^i \\to K^{i + 1}$", "kills of $R^{\\oplus r}$ and induces an isomorphism of $R^{\\oplus m - r}$", "onto the corresponding summand of $K^{i + 1}$. Thus $K^\\bullet$ becomes", "quasi-isomorphic to", "$$", "\\ldots \\to K^{i - 2} \\to R^{\\oplus l - r}", "\\to 0 \\to R^{\\oplus n - m + r} \\to K^{i + 2} \\to \\ldots", "$$", "and everything is clear." ], "refs": [ "more-algebra-lemma-better-cut-complex-in-two" ], "ref_ids": [ 10236 ] } ], "ref_ids": [] }, { "id": 10239, "type": "theorem", "label": "more-algebra-lemma-split-using-ext-injective", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-split-using-ext-injective", "contents": [ "Let $R$ be a ring. Let $K \\in D^-(R)$. Let $a \\in \\mathbf{Z}$.", "Assume that for any injective $R$-module map $M \\to M'$ the map", "$\\Ext^{-a}_R(K, M) \\to \\Ext^{-a}_R(K, M')$ is injective.", "Then there is a unique direct sum decomposition", "$K \\cong \\tau_{\\leq a}K \\oplus \\tau_{\\geq a + 1}K$", "and $\\tau_{\\geq a + 1}K$ has projective-amplitude in $[a + 1, b]$", "for some $b$." ], "refs": [], "proofs": [ { "contents": [ "Consider the distinguished triangle", "$$", "\\tau_{\\leq a}K \\to K \\to \\tau_{\\geq a + 1}K \\to (\\tau_{\\leq a}K)[1]", "$$", "in $D(R)$, see Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}.", "Observe that", "$\\Ext^{-a}_R(\\tau_{\\leq a}K, M) = \\Hom_R(H^a(K), M)$", "and", "$\\Ext^{-a - 1}_R(\\tau_{\\leq a}K, M) = 0$, see", "Derived Categories, Lemma \\ref{derived-lemma-negative-exts}. Thus", "the long exact sequence of $\\Ext$ gives an exact sequence", "$$", "0 \\to", "\\Ext^{-a}_R(\\tau_{\\geq a + 1}K, M) \\to", "\\Ext^{-a}_R(K, M) \\to", "\\Hom_R(H^a(K), M)", "$$", "functorial in the $R$-module $M$.", "Now if $I$ is an injective $R$-module, then", "$\\Ext^{-a}_R(\\tau_{\\geq a + 1}K, I) = 0$ for example by", "Derived Categories, Lemma \\ref{derived-lemma-compute-ext-resolutions}.", "Since every module injects into an injective module,", "we conclude that $\\Ext^{-a}_R(\\tau_{\\geq a + 1}K, M) = 0$", "for every $R$-module $M$. By Lemma \\ref{lemma-projective-amplitude}", "we conclude that $\\tau_{\\geq a + 1}K$ has projective-amplitude", "in $[a + 1, b]$ for some $b$ (this is where we use that $K$", "is bounded above). We obtain the splitting by", "Lemma \\ref{lemma-splitting-unique}." ], "refs": [ "derived-remark-truncation-distinguished-triangle", "derived-lemma-negative-exts", "derived-lemma-compute-ext-resolutions", "more-algebra-lemma-projective-amplitude", "more-algebra-lemma-splitting-unique" ], "ref_ids": [ 2016, 1893, 1892, 10187, 10235 ] } ], "ref_ids": [] }, { "id": 10240, "type": "theorem", "label": "more-algebra-lemma-split-using-ext-zero", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-split-using-ext-zero", "contents": [ "Let $R$ be a ring. Let $K \\in D^-(R)$. Let $a \\in \\mathbf{Z}$.", "Assume $\\Ext^{-a}_R(K, M) = 0$ for any $R$-module $M$.", "Then there is a unique direct sum decomposition", "$K \\cong \\tau_{\\leq a - 1}K \\oplus \\tau_{\\geq a + 1}K$", "and $\\tau_{\\geq a + 1}K$ has projective-amplitude in $[a + 1, b]$", "for some $b$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-split-using-ext-injective}", "we have a direct sum decomposition", "$K \\cong \\tau_{\\leq a}K \\oplus \\tau_{\\geq a + 1}K$", "and $\\tau_{\\geq a + 1}K$ has projective-amplitude in $[a + 1, b]$", "for some $b$. Clearly, we must have $H^a(K) = 0$ and", "we conclude that $\\tau_{\\leq a}K = \\tau_{\\leq a - 1}K$", "in $D(R)$." ], "refs": [ "more-algebra-lemma-split-using-ext-injective" ], "ref_ids": [ 10239 ] } ], "ref_ids": [] }, { "id": 10241, "type": "theorem", "label": "more-algebra-lemma-lift-bounded-pseudo-coherent-to-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-bounded-pseudo-coherent-to-perfect", "contents": [ "Let $R$ be a ring and let $\\mathfrak p \\subset R$ be a prime.", "Let $K$ be pseudo-coherent and bounded below. Set", "$d_i = \\dim_{\\kappa(\\mathfrak p)}", "H^i(K \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p))$.", "If there exists an $a \\in \\mathbf{Z}$ such that $d_i = 0$ for $i < a$,", "then there exists an $f \\in R$, $f \\not \\in \\mathfrak p$ and a", "complex", "$$", "\\ldots \\to 0 \\to R_f^{\\oplus d_a} \\to R_f^{\\oplus d_{a + 1}} \\to", "\\ldots \\to", "R_f^{\\oplus d_{b - 1}} \\to", "R_f^{\\oplus d_b} \\to 0", "\\to \\ldots", "$$", "representing $K \\otimes_R^\\mathbf{L} R_f$ in $D(R_f)$.", "In particular $K \\otimes_R^\\mathbf{L} R_f$ is perfect." ], "refs": [], "proofs": [ { "contents": [ "After decreasing $a$ we may assume that also $H^i(K^\\bullet) = 0$ for", "$i < a$. By Lemma \\ref{lemma-cut-complex-in-two} after replacing $R$ by $R_f$", "for some $f \\in R$, $f \\not \\in \\mathfrak p$", "we can write $K^\\bullet = \\tau_{\\leq a - 1}K^\\bullet \\oplus", "\\tau_{\\geq a}K^\\bullet$ in $D(R)$ with $\\tau_{\\geq a}K^\\bullet$", "perfect. Since $H^i(K^\\bullet) = 0$ for $i < a$ we see that", "$\\tau_{\\leq a - 1}K^\\bullet = 0$ in $D(R)$. Hence $K^\\bullet$", "is perfect. Then we can conclude using", "Lemma \\ref{lemma-lift-perfect-from-residue-field}." ], "refs": [ "more-algebra-lemma-cut-complex-in-two", "more-algebra-lemma-lift-perfect-from-residue-field" ], "ref_ids": [ 10238, 10232 ] } ], "ref_ids": [] }, { "id": 10242, "type": "theorem", "label": "more-algebra-lemma-check-perfect-pointwise", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-check-perfect-pointwise", "contents": [ "Let $R$ be a ring. Let $a, b \\in \\mathbf{Z}$.", "Let $K^\\bullet$ be a pseudo-coherent complex of $R$-modules.", "The following are equivalent", "\\begin{enumerate}", "\\item $K^\\bullet$ is perfect with tor amplitude in $[a, b]$,", "\\item for every prime $\\mathfrak p$ we have", "$H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p)) = 0$ for all", "$i \\not \\in [a, b]$, and", "\\item for every maximal ideal $\\mathfrak m$ we have", "$H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak m)) = 0$ for all", "$i \\not \\in [a, b]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We omit the proof of the implications (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3).", "Assume (3). Let $i \\in \\mathbf{Z}$ with $i \\not \\in [a, b]$. By", "Lemma \\ref{lemma-cut-complex-in-two}", "we see that the assumption implies that $H^i(K^\\bullet)_{\\mathfrak m} = 0$", "for all maximal ideals of $R$. Hence $H^i(K^\\bullet) = 0$, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}.", "Moreover,", "Lemma \\ref{lemma-cut-complex-in-two}", "now also implies that for every maximal ideal", "$\\mathfrak m$ there exists an element $f \\in R$, $f \\not \\in \\mathfrak m$", "such that $K^\\bullet \\otimes_R R_f$ is perfect with tor amplitude in", "$[a, b]$. Hence we conclude by appealing to", "Lemmas \\ref{lemma-glue-perfect} and \\ref{lemma-glue-tor-amplitude}." ], "refs": [ "more-algebra-lemma-cut-complex-in-two", "algebra-lemma-characterize-zero-local", "more-algebra-lemma-cut-complex-in-two", "more-algebra-lemma-glue-perfect", "more-algebra-lemma-glue-tor-amplitude" ], "ref_ids": [ 10238, 410, 10238, 10221, 10183 ] } ], "ref_ids": [] }, { "id": 10243, "type": "theorem", "label": "more-algebra-lemma-check-perfect-stalks", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-check-perfect-stalks", "contents": [ "Let $R$ be a ring. Let $K^\\bullet$ be a pseudo-coherent", "complex of $R$-modules. Consider the following conditions", "\\begin{enumerate}", "\\item $K^\\bullet$ is perfect,", "\\item for every prime ideal $\\mathfrak p$ the complex", "$K^\\bullet \\otimes_R R_{\\mathfrak p}$ is perfect,", "\\item for every maximal ideal $\\mathfrak m$ the complex", "$K^\\bullet \\otimes_R R_{\\mathfrak m}$ is perfect,", "\\item for every prime $\\mathfrak p$ we have", "$H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p)) = 0$ for all", "$i \\ll 0$,", "\\item for every maximal ideal $\\mathfrak m$ we have", "$H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak m)) = 0$ for all", "$i \\ll 0$.", "\\end{enumerate}", "We always have the implications", "$$", "(1) \\Rightarrow (2) \\Leftrightarrow (3) \\Leftrightarrow (3)", "\\Leftrightarrow (4) \\Leftrightarrow (5)", "$$", "If $K^\\bullet$ is bounded below, then all conditions are equivalent." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-pull-perfect} we see that (1) implies (2).", "It is immediate that (2) $\\Rightarrow$ (3). Since every prime", "$\\mathfrak p$ is contained in a maximal ideal $\\mathfrak m$,", "we can apply Lemma \\ref{lemma-pull-perfect} to the map", "$R_\\mathfrak m \\to R_\\mathfrak p$ to see that (3) implies (2).", "Applying Lemma \\ref{lemma-pull-perfect} to the residue maps", "$R_\\mathfrak p \\to \\kappa(\\mathfrak p)$ and", "$R_\\mathfrak m \\to \\kappa(\\mathfrak m)$ we see that (2) implies", "(4) and (3) implies (5).", "\\medskip\\noindent", "Assume $R$ is local with maximal ideal $\\mathfrak m$ and", "residue field $\\kappa$. We will show that if", "$H^i(K^\\bullet \\otimes^\\mathbf{L} \\kappa) = 0$ for $i < a$", "for some $a$, then $K$ is perfect. This will show that", "(4) implies (2) and (5) implies (3) whence the first part", "of the lemma. First we apply Lemma \\ref{lemma-cut-complex-in-two}", "with $i = a - 1$ to see that", "$K^\\bullet = \\tau_{\\leq a - 1}K^\\bullet \\oplus \\tau_{\\geq a}K^\\bullet$", "in $D(R)$ with $\\tau_{\\geq a}K^\\bullet$ perfect of tor-amplitude", "contained in $[a, \\infty]$. To finish we need to show that", "$\\tau_{\\leq a - 1}K$ is zero, i.e., that its cohomology groups are zero.", "If not let $i$ be the largest index such that $M = H^i(\\tau_{\\leq a - 1}K)$", "is not zero. Then $M$ is a finite $R$-module because", "$\\tau_{\\leq a - 1}K^\\bullet$ is pseudo-coherent", "(Lemmas \\ref{lemma-finite-cohomology} and \\ref{lemma-summands-pseudo-coherent}).", "Thus by Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "we find that $M \\otimes_R \\kappa$ is nonzero.", "This implies that", "$$", "H^i((\\tau_{\\leq a - 1}K^\\bullet) \\otimes_R^\\mathbf{L} \\kappa) =", "H^i(K^\\bullet \\otimes_R^\\mathbf{L} \\kappa)", "$$", "is nonzero which is a contradiction.", "\\medskip\\noindent", "Assume the equivalent conditions (2) -- (5) hold and that", "$K^\\bullet$ is bounded below. Say $H^i(K^\\bullet) = 0$ for", "$i < a$. Pick a maximal ideal $\\mathfrak m$ of $R$.", "It suffices to show there exists an $f \\in R$, $f \\not \\in \\mathfrak m$", "such that $K^\\bullet \\otimes_R^\\mathbf{L} R_f$ is perfect", "(Lemma \\ref{lemma-glue-perfect} and", "Algebra, Lemma \\ref{algebra-lemma-quasi-compact}).", "This follows from Lemma \\ref{lemma-lift-bounded-pseudo-coherent-to-perfect}." ], "refs": [ "more-algebra-lemma-pull-perfect", "more-algebra-lemma-pull-perfect", "more-algebra-lemma-pull-perfect", "more-algebra-lemma-cut-complex-in-two", "more-algebra-lemma-finite-cohomology", "more-algebra-lemma-summands-pseudo-coherent", "algebra-lemma-NAK", "more-algebra-lemma-glue-perfect", "algebra-lemma-quasi-compact", "more-algebra-lemma-lift-bounded-pseudo-coherent-to-perfect" ], "ref_ids": [ 10219, 10219, 10219, 10238, 10146, 10151, 401, 10221, 395, 10241 ] } ], "ref_ids": [] }, { "id": 10244, "type": "theorem", "label": "more-algebra-lemma-projective-amplitude-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-projective-amplitude-pseudo-coherent", "contents": [ "Let $R$ be a ring. Let $K$ be a pseudo-coherent object of $D(R)$.", "Let $a, b \\in \\mathbf{Z}$. The following are equivalent", "\\begin{enumerate}", "\\item $K$ has projective-amplitude in $[a, b]$,", "\\item $K$ is perfect of tor-amplitude in $[a, b]$,", "\\item $\\Ext^i_R(K, N) = 0$ for all finitely presented $R$-modules $N$ and all", "$i \\not \\in [-b, -a]$,", "\\item $H^n(K) = 0$ for $n > b$ and", "$\\Ext^i_R(K, N) = 0$ for all finitely presented $R$-modules $N$ and", "all $i > -a$, and", "\\item $H^n(K) = 0$ for $n \\not \\in [a - 1, b]$ and", "$\\Ext^{-a + 1}_R(K, N) = 0$ for all finitely presented $R$-modules $N$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "From the final statement of Lemma \\ref{lemma-perfect} we see that", "(2) implies (1). If (1) holds, then $K$ can be represented by", "a complex of projective modules $P^i$ with $P^i = 0$ for $i \\not \\in [a, b]$.", "Since projective modules are flat (as summands of free modules), we see", "that $K$ has tor-amplitude in $[a, b]$, see Lemma \\ref{lemma-tor-amplitude}.", "Thus by Lemma \\ref{lemma-perfect} we see that (2) holds.", "\\medskip\\noindent", "In conditions (3), (4), (5) the assumed vanishing of ext groups", "$\\Ext^i_R(K, M)$ for $M$ of finite presentation is equivalent to the", "vanishing for all $R$-modules $M$ by", "Lemma \\ref{lemma-pseudo-coherence-colimit-ext}", "and Algebra, Lemma \\ref{algebra-lemma-module-colimit-fp}.", "Thus the equivalence of (1), (3), (4), and (5)", "follows from Lemma \\ref{lemma-projective-amplitude}." ], "refs": [ "more-algebra-lemma-perfect", "more-algebra-lemma-tor-amplitude", "more-algebra-lemma-perfect", "more-algebra-lemma-pseudo-coherence-colimit-ext", "algebra-lemma-module-colimit-fp", "more-algebra-lemma-projective-amplitude" ], "ref_ids": [ 10212, 10170, 10212, 10162, 355, 10187 ] } ], "ref_ids": [] }, { "id": 10245, "type": "theorem", "label": "more-algebra-lemma-perfect-over-polynomial-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-perfect-over-polynomial-ring", "contents": [ "Let $A \\to B$ be a ring map. Let $a, b \\in \\mathbf{Z}$. Let $d \\geq 0$.", "Let $K^\\bullet$ be a complex of $B$-modules. Assume", "\\begin{enumerate}", "\\item the ring map $A \\to B$ is flat,", "\\item for every prime $\\mathfrak p \\subset A$ the ring", "$B \\otimes_A \\kappa(\\mathfrak p)$ has finite global dimension $\\leq d$,", "\\item $K^\\bullet$ is pseudo-coherent as a complex of $B$-modules, and", "\\item $K^\\bullet$ has tor amplitude in $[a, b]$ as a complex", "of $A$-modules.", "\\end{enumerate}", "Then $K^\\bullet$ is perfect as a complex of $B$-modules", "with tor amplitude in $[a - d, b]$." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $K^\\bullet$ is a bounded above complex of", "finite free $B$-modules. In particular, $K^\\bullet$ is flat as a", "complex of $A$-modules and", "$K^\\bullet \\otimes_A M = K^\\bullet \\otimes_A^{\\mathbf{L}} M$ for any", "$A$-module $M$. For every prime $\\mathfrak p$ of $A$ the complex", "$$", "K^\\bullet \\otimes_A \\kappa(\\mathfrak p)", "$$", "is a bounded above complex of finite free modules over", "$B \\otimes_A \\kappa(\\mathfrak p)$ with vanishing $H^i$ except", "for $i \\in [a, b]$. As $B \\otimes_A \\kappa(\\mathfrak p)$", "has global dimension $d$ we see from", "Lemma \\ref{lemma-finite-gl-dim-tor-dimension}", "that $K^\\bullet \\otimes_A \\kappa(\\mathfrak p)$ has tor amplitude in", "$[a - d, b]$. Let $\\mathfrak q$ be a prime of $B$ lying over $\\mathfrak p$.", "Since $K^\\bullet \\otimes_A \\kappa(\\mathfrak p)$ is a bounded above", "complex of free $B \\otimes_A \\kappa(\\mathfrak p)$-modules we see", "that", "\\begin{align*}", "K^\\bullet \\otimes_B^{\\mathbf{L}} \\kappa(\\mathfrak q)", "& = K^\\bullet \\otimes_B \\kappa(\\mathfrak q) \\\\", "& = (K^\\bullet \\otimes_A \\kappa(\\mathfrak p))", "\\otimes_{B \\otimes_A \\kappa(\\mathfrak p)} \\kappa(\\mathfrak q) \\\\", "& = (K^\\bullet \\otimes_A \\kappa(\\mathfrak p))", "\\otimes^{\\mathbf{L}}_{B \\otimes_A \\kappa(\\mathfrak p)} \\kappa(\\mathfrak q)", "\\end{align*}", "Hence the arguments above imply that", "$H^i(K^\\bullet \\otimes_B^{\\mathbf{L}} \\kappa(\\mathfrak q)) = 0$", "for $i \\not \\in [a - d, b]$. We conclude by", "Lemma \\ref{lemma-check-perfect-pointwise}." ], "refs": [ "more-algebra-lemma-finite-gl-dim-tor-dimension", "more-algebra-lemma-check-perfect-pointwise" ], "ref_ids": [ 10186, 10242 ] } ], "ref_ids": [] }, { "id": 10246, "type": "theorem", "label": "more-algebra-lemma-perfect-over-regular-local-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-perfect-over-regular-local-ring", "contents": [ "Let $A \\to B$ be a local ring homomorphism.", "Let $a, b \\in \\mathbf{Z}$. Let $d \\geq 0$.", "Let $K^\\bullet$ be a complex of $B$-modules. Assume", "\\begin{enumerate}", "\\item the ring map $A \\to B$ is flat,", "\\item the ring $B/\\mathfrak m_AB$ is regular of dimension $d$,", "\\item $K^\\bullet$ is pseudo-coherent as a complex of $B$-modules, and", "\\item $K^\\bullet$ has tor amplitude in $[a, b]$ as a complex", "of $A$-modules, in fact it suffices if", "$H^i(K^\\bullet \\otimes_A^\\mathbf{L} \\kappa(\\mathfrak m_A))$", "is nonzero only for $i \\in [a, b]$.", "\\end{enumerate}", "Then $K^\\bullet$ is perfect as a complex of $B$-modules", "with tor amplitude in $[a - d, b]$." ], "refs": [], "proofs": [ { "contents": [ "By (3) we may assume that $K^\\bullet$ is a bounded above complex of finite free", "$B$-modules. We compute", "\\begin{align*}", "K^\\bullet \\otimes_B^{\\mathbf{L}} \\kappa(\\mathfrak m_B)", "& = K^\\bullet \\otimes_B \\kappa(\\mathfrak m_B) \\\\", "& = (K^\\bullet \\otimes_A \\kappa(\\mathfrak m_A))", "\\otimes_{B/\\mathfrak m_A B} \\kappa(\\mathfrak m_B) \\\\", "& = (K^\\bullet \\otimes_A \\kappa(\\mathfrak m_A))", "\\otimes^{\\mathbf{L}}_{B/\\mathfrak m_A B} \\kappa(\\mathfrak m_B)", "\\end{align*}", "The first equality because $K^\\bullet$ is a bounded above complex", "of flat $B$-modules. The second equality follows from basic ", "properties of the tensor product. The third equality holds because", "$K^\\bullet \\otimes_A \\kappa(\\mathfrak m_A) =", "K^\\bullet/ \\mathfrak m_A K^\\bullet$ is a bounded above complex", "of flat $B/\\mathfrak m_A B$-modules. Since $K^\\bullet$ is a bounded", "above complex of flat $A$-modules by (1), the cohomology modules $H^i$", "of the complex $K^\\bullet \\otimes_A \\kappa(\\mathfrak m_A)$ are nonzero only", "for $i \\in [a, b]$ by assumption (4). Thus the spectral sequence", "of Example \\ref{example-cohomology-complex-tensored} and the", "fact that $B/\\mathfrak m_AB$ has finite global dimension $d$", "(by (2) and", "Algebra, Proposition \\ref{algebra-proposition-regular-finite-gl-dim})", "shows that $H^j(K^\\bullet \\otimes_B^{\\mathbf{L}} \\kappa(\\mathfrak m_B))$", "is zero for $j \\not \\in [a - d, b]$.", "This finishes the proof by Lemma \\ref{lemma-check-perfect-pointwise}." ], "refs": [ "algebra-proposition-regular-finite-gl-dim", "more-algebra-lemma-check-perfect-pointwise" ], "ref_ids": [ 1421, 10242 ] } ], "ref_ids": [] }, { "id": 10247, "type": "theorem", "label": "more-algebra-lemma-perfect-ring-classical-generator", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-perfect-ring-classical-generator", "contents": [ "Let $R$ be a ring. The full subcategory $D_{perf}(R) \\subset D(R)$ of perfect", "objects is the smallest strictly full, saturated, triangulated subcategory", "containing $R = R[0]$. In other words $D_{perf}(R) = \\langle R \\rangle$.", "In particular, $R$ is a classical generator for $D_{perf}(R)$." ], "refs": [], "proofs": [ { "contents": [ "To see what the statement means, please look at", "Derived Categories, Definitions \\ref{derived-definition-saturated} and", "\\ref{derived-definition-generators}.", "It was shown in Lemmas \\ref{lemma-two-out-of-three-perfect} and", "\\ref{lemma-summands-perfect} that $D_{perf}(R) \\subset D(R)$", "is a strictly full, saturated, triangulated subcategory of $D(R)$.", "Of course $R \\in D_{perf}(R)$.", "\\medskip\\noindent", "Recall that $\\langle R \\rangle = \\bigcup \\langle R \\rangle_n$.", "To finish the proof we will show that if $M \\in D_{perf}(R)$", "is represented by", "$$", "\\ldots \\to 0 \\to M^a \\to M^{a + 1} \\to \\ldots \\to M^b \\to 0 \\to \\ldots", "$$", "with $M^i$ finite projective, then $M \\in \\langle R \\rangle_{b - a + 1}$.", "The proof is by induction on $b - a$.", "By definition $\\langle R \\rangle_1$ contains any finite projective", "$R$-module placed in any degree; this deals with the base case", "$b - a = 0$ of the induction. In general, we consider the distinguished", "triangle", "$$", "M_b[-b] \\to M^\\bullet \\to \\sigma_{\\leq b - 1}M^\\bullet \\to M_b[-b + 1]", "$$", "By induction the truncated complex $\\sigma_{\\leq b - 1}M^\\bullet$ is", "in $\\langle R \\rangle_{b - a}$ and $M_b[-b]$ is in $\\langle R \\rangle_1$.", "Hence $M^\\bullet \\in \\langle R \\rangle_{b - a + 1}$ by definition." ], "refs": [ "derived-definition-saturated", "derived-definition-generators", "more-algebra-lemma-two-out-of-three-perfect", "more-algebra-lemma-summands-perfect" ], "ref_ids": [ 1974, 2003, 10214, 10215 ] } ], "ref_ids": [] }, { "id": 10248, "type": "theorem", "label": "more-algebra-lemma-commutes-with-countable-sums", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-commutes-with-countable-sums", "contents": [ "Let $R$ be a ring. Let $K \\in D(R)$ be an object such that for every", "countable set of objects $E_n \\in D(R)$ the canonical map", "$$", "\\bigoplus \\Hom_{D(R)}(K, E_n) \\longrightarrow \\Hom_{D(R)}(K, \\bigoplus E_n)", "$$", "is a bijection. Then, given any system $L_n^\\bullet$ of complexes over", "$\\mathbf{N}$ we have that", "$$", "\\colim \\Hom_{D(R)}(K, L^\\bullet_n) \\longrightarrow \\Hom_{D(R)}(K, L^\\bullet)", "$$", "is a bijection, where $L^\\bullet$ is the termwise colimit, i.e.,", "$L^m = \\colim L_n^m$ for all $m \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the short exact sequence of complexes", "$$", "0 \\to \\bigoplus L_n^\\bullet \\to \\bigoplus L_n^\\bullet \\to L^\\bullet \\to 0", "$$", "where the first map is given by $1 - t_n$ in degree $n$ where", "$t_n : L_n^\\bullet \\to L_{n + 1}^\\bullet$ is the transition map.", "By", "Derived Categories, Lemma \\ref{derived-lemma-derived-canonical-delta-functor}", "this is a distinguished triangle in $D(R)$.", "Apply the homological functor $\\Hom_{D(R)}(K, -)$, see", "Derived Categories, Lemma \\ref{derived-lemma-representable-homological}.", "Thus a long exact cohomology sequence", "$$", "\\xymatrix{", "& \\ldots \\ar[r] & \\Hom_{D(R)}(K, \\colim L^\\bullet_n[-1]) \\ar[lld] \\\\", "\\Hom_{D(R)}(K, \\bigoplus L^\\bullet_n) \\ar[r] &", "\\Hom_{D(R)}(K, \\bigoplus L^\\bullet_n) \\ar[r] &", "\\Hom_{D(R)}(K, \\colim L^\\bullet_n) \\ar[lld] \\\\", "\\Hom_{D(R)}(K, \\bigoplus L^\\bullet_n[1]) \\ar[r] & \\ldots", "}", "$$", "Since we have assumed that $\\Hom_{D(R)}(K, \\bigoplus L^\\bullet_n)$", "is equal to $\\bigoplus \\Hom_{D(R)}(K, L^\\bullet_n)$ we see that the first", "map on every row of the diagram is injective (by the explicit description", "of this map as the sum of the maps induced by $1 - t_n$). Hence", "we conclude that $\\Hom_{D(R)}(K, \\colim L^\\bullet_n)$ is the cokernel", "of the first map of the middle row in the diagram above which is what", "we had to show." ], "refs": [ "derived-lemma-derived-canonical-delta-functor", "derived-lemma-representable-homological" ], "ref_ids": [ 1814, 1758 ] } ], "ref_ids": [] }, { "id": 10249, "type": "theorem", "label": "more-algebra-lemma-perfect-modulo-nilpotent-ideal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-perfect-modulo-nilpotent-ideal", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal.", "Let $K$ be an object of $D(R)$. Assume that", "\\begin{enumerate}", "\\item $K \\otimes_R^\\mathbf{L} R/I$ is perfect in $D(R/I)$, and", "\\item $I$ is a nilpotent ideal.", "\\end{enumerate}", "Then $K$ is perfect in $D(R)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite complex $\\overline{P}^\\bullet$ of finite projective", "$R/I$-modules representing $K \\otimes_R^\\mathbf{L} R/I$, see", "Definition \\ref{definition-perfect}. By", "Lemma \\ref{lemma-lift-complex-projectives}", "there exists a complex $P^\\bullet$ of projective $R$-modules", "representing $K$ such that $\\overline{P}^\\bullet = P^\\bullet/IP^\\bullet$.", "It follows from Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "that $P^\\bullet$ is a finite complex of finite projective", "$R$-modules." ], "refs": [ "more-algebra-definition-perfect", "more-algebra-lemma-lift-complex-projectives", "algebra-lemma-NAK" ], "ref_ids": [ 10628, 10229, 401 ] } ], "ref_ids": [] }, { "id": 10250, "type": "theorem", "label": "more-algebra-lemma-perfect-modulo-two-ideals", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-perfect-modulo-two-ideals", "contents": [ "Let $R$ be a ring. Let $I, J \\subset R$ be ideals.", "Let $K$ be an object of $D(R)$. Assume that", "\\begin{enumerate}", "\\item $K \\otimes_R^\\mathbf{L} R/I$ is perfect in $D(R/I)$, and", "\\item $K \\otimes_R^\\mathbf{L} R/J$ is perfect in $D(R/J)$.", "\\end{enumerate}", "Then $K \\otimes_R^\\mathbf{L} R/IJ$ is perfect in $D(R/IJ)$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that we may assume replace $R$ by $R/IJ$ and $K$ by", "$K \\otimes_R^\\mathbf{L} R/IJ$. Then $R \\to R/(I \\cap J)$ is", "a surjection whose kernel has square zero. Hence by", "Lemma \\ref{lemma-perfect-modulo-nilpotent-ideal}", "it suffices to prove that $K \\otimes_R^\\mathbf{L} R/(I \\cap J)$ is", "perfect. Thus we may assume that $I \\cap J = 0$.", "\\medskip\\noindent", "We prove the lemma in case $I \\cap J = 0$. First, we may represent $K$", "by a K-flat complex $K^\\bullet$ with all $K^n$ flat, see", "Lemma \\ref{lemma-K-flat-resolution}. Then we see that we have a short", "exact sequence of complexes", "$$", "0 \\to", "K^\\bullet \\to K^\\bullet/IK^\\bullet \\oplus K^\\bullet/JK^\\bullet \\to", "K^\\bullet/(I + J)K^\\bullet \\to 0", "$$", "Note that $K^\\bullet/IK^\\bullet$ represents $K \\otimes^\\mathbf{L}_R R/I$", "by construction of the derived tensor product. Similarly for", "$K^\\bullet/JK^\\bullet$ and $K^\\bullet/(I + J)K^\\bullet$.", "Note that $K^\\bullet/(I + J)K^\\bullet$ is a perfect complex", "of $R/(I + J)$-modules, see", "Lemma \\ref{lemma-pull-perfect}.", "Hence the complexes $K^\\bullet/IK^\\bullet$, and", "$K^\\bullet/JK^\\bullet$ and $K^\\bullet/(I + J)K^\\bullet$", "have finitely many nonzero cohomology groups", "(since a perfect complex has finite Tor-amplitude, see", "Lemma \\ref{lemma-perfect}). We conclude that $K \\in D^b(R)$ by the", "long exact cohomology sequence associated to short exact sequence", "of complexes displayed above. In particular we assume $K^\\bullet$", "is a bounded above complex of free $R$-modules (see", "Derived Categories, Lemma \\ref{derived-lemma-subcategory-left-resolution}).", "\\medskip\\noindent", "We will now show that $K$ is perfect using the criterion of", "Proposition \\ref{proposition-perfect-is-compact}. Thus we let", "$E_j \\in D(R)$ be a family of objects parametrized by a set $J$.", "We choose complexes $E_j^\\bullet$ with flat terms", "representing $E_j$, see for example Lemma \\ref{lemma-K-flat-resolution}.", "It is clear that", "$$", "0 \\to", "E_j^\\bullet \\to", "E_j^\\bullet/IE_j^\\bullet \\oplus E_j^\\bullet/JE_j^\\bullet \\to", "E_j^\\bullet/(I + J)E_j^\\bullet \\to 0", "$$", "is a short exact sequence of complexes. Taking direct sums we obtain a", "similar short exact sequence", "$$", "0 \\to", "\\bigoplus E_j^\\bullet \\to", "\\bigoplus E_j^\\bullet/IE_j^\\bullet \\oplus E_j^\\bullet/JE_j^\\bullet \\to", "\\bigoplus E_j^\\bullet/(I + J)E_j^\\bullet \\to 0", "$$", "(Note that $- \\otimes_R R/I$ commutes with direct sums.)", "This short exact sequence determines a distinguished triangle in $D(R)$, see", "Derived Categories, Lemma \\ref{derived-lemma-derived-canonical-delta-functor}.", "Apply the homological functor $\\Hom_{D(R)}(K, -)$ (see", "Derived Categories, Lemma \\ref{derived-lemma-representable-homological})", "to get a commutative diagram", "$$", "\\xymatrix{", "\\bigoplus \\Hom_{D(R)}(K^\\bullet, E_j^\\bullet/(I + J))[-1] \\ar[r] \\ar[d] &", "\\Hom_{D(R)}(K^\\bullet, \\bigoplus E_j^\\bullet/(I + J))[-1] \\ar[d] \\\\", "\\bigoplus \\Hom_{D(R)}(K^\\bullet, E_j^\\bullet/I \\oplus E_j^\\bullet/J)[-1]", "\\ar[r] \\ar[d] &", "\\Hom_{D(R)}(K^\\bullet, \\bigoplus E_j^\\bullet/I \\oplus E_j^\\bullet/J)[-1]", "\\ar[d] \\\\", "\\bigoplus \\Hom_{D(R)}(K^\\bullet, E_j^\\bullet) \\ar[r] \\ar[d] &", "\\Hom_{D(R)}(K^\\bullet, \\bigoplus E_j^\\bullet) \\ar[d] \\\\", "\\bigoplus \\Hom_{D(R)}(K^\\bullet, E_j^\\bullet/I \\oplus E_j^\\bullet/J)", "\\ar[r] \\ar[d] &", "\\Hom_{D(R)}(K^\\bullet, \\bigoplus E_j^\\bullet/I \\oplus E_j^\\bullet/J)", "\\ar[d] \\\\", "\\bigoplus \\Hom_{D(R)}(K^\\bullet, E_j^\\bullet/(I + J)) \\ar[r] &", "\\Hom_{D(R)}(K^\\bullet, \\bigoplus E_j^\\bullet/(I + J))", "}", "$$", "with exact columns. It is clear that, for any complex $E^\\bullet$", "of $R$-modules we have", "\\begin{align*}", "\\Hom_{D(R)}(K^\\bullet, E^\\bullet/I) & =", "\\Hom_{K(R)}(K^\\bullet, E^\\bullet/I) \\\\", "& =", "\\Hom_{K(R/I)}(K^\\bullet/IK^\\bullet, E^\\bullet/I) \\\\", "& =", "\\Hom_{D(R/I)}(K^\\bullet/IK^\\bullet, E^\\bullet/I)", "\\end{align*}", "and similarly for when dividing by $J$ or $I + J$, see", "Derived Categories,", "Lemma \\ref{derived-lemma-morphisms-from-projective-complex}.", "Derived Categories. Thus all the horizontal", "arrows, except for possibly the middle one, are isomorphisms as the complexes", "$K^\\bullet/IK^\\bullet$, $K^\\bullet/JK^\\bullet$, $K^\\bullet/(I + J)K^\\bullet$", "are perfect complexes of $R/I$, $R/J$, $R/(I + J)$-modules, see", "Proposition \\ref{proposition-perfect-is-compact}.", "It follows from the $5$-lemma (Homology, Lemma \\ref{homology-lemma-five-lemma})", "that the middle map is an isomorphism and the lemma follows by", "Proposition \\ref{proposition-perfect-is-compact}." ], "refs": [ "more-algebra-lemma-perfect-modulo-nilpotent-ideal", "more-algebra-lemma-K-flat-resolution", "more-algebra-lemma-pull-perfect", "more-algebra-lemma-perfect", "derived-lemma-subcategory-left-resolution", "more-algebra-proposition-perfect-is-compact", "more-algebra-lemma-K-flat-resolution", "derived-lemma-derived-canonical-delta-functor", "derived-lemma-representable-homological", "derived-lemma-morphisms-from-projective-complex", "more-algebra-proposition-perfect-is-compact", "homology-lemma-five-lemma", "more-algebra-proposition-perfect-is-compact" ], "ref_ids": [ 10249, 10131, 10219, 10212, 1835, 10585, 10131, 1814, 1758, 1862, 10585, 12030, 10585 ] } ], "ref_ids": [] }, { "id": 10251, "type": "theorem", "label": "more-algebra-lemma-ghost-lemma", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ghost-lemma", "contents": [ "\\begin{reference}", "\\cite{Kelly}", "\\end{reference}", "Let $R$ be a ring. Let $n \\geq 1$. Let $K \\in \\langle R \\rangle_n$ with", "notation as in Derived Categories, Section \\ref{derived-section-generators}.", "Consider maps", "$$", "K \\xrightarrow{f_1} K_1 \\xrightarrow{f_2} K_2", "\\xrightarrow{f_3} \\ldots \\xrightarrow{f_n} K_n", "$$", "in $D(R)$. If $H^i(f_j) = 0$ for all $i, j$, then", "$f_n \\circ \\ldots \\circ f_1 = 0$." ], "refs": [], "proofs": [ { "contents": [ "If $n = 1$, then $K$ is a direct summand in $D(R)$ of a bounded complex", "$P^\\bullet$ whose terms are finite free $R$-modules and whose differentials", "are zero. Thus it suffices to show any morphism $f : P^\\bullet \\to K_1$", "in $D(R)$ with $H^i(f) = 0$ for all $i$ is zero. Since $P^\\bullet$ is", "a finite direct sum $P^\\bullet = \\bigoplus R[m_j]$ it suffices to show", "any morphism $g : R[m] \\to K_1$ with $H^{-m}(g) = 0$ in $D(R)$ is zero.", "This follows from the fact that $\\Hom_{D(R)}(R[-m], K) = H^m(K)$.", "\\medskip\\noindent", "For $n > 1$ we proceed by induction on $n$. Namely, we know that $K$", "is a summand in $D(R)$ of an object $P$ which sits in a distinguished triangle", "$$", "P' \\xrightarrow{i} P \\xrightarrow{p} P'' \\to P'[1]", "$$", "with $P' \\in \\langle R \\rangle_1$ and $P'' \\in \\langle R \\rangle_{n - 1}$.", "As above we may replace $K$ by $P$ and assume that we have", "$$", "P \\xrightarrow{f_1} K_1 \\xrightarrow{f_2} K_2", "\\xrightarrow{f_3} \\ldots \\xrightarrow{f_n} K_n", "$$", "in $D(R)$ with $f_j$ zero on cohomology.", "By the case $n = 1$ the composition $f_1 \\circ i$ is zero.", "Hence by Derived Categories, Lemma \\ref{derived-lemma-representable-homological}", "we can find a morphism $h : P'' \\to K_1$ such that $f_1 = h \\circ p$.", "Observe that $f_2 \\circ h$ is zero on cohomology.", "Hence by induction we find that $f_n \\circ \\ldots \\circ f_2 \\circ h = 0$", "which implies $f_n \\circ \\ldots \\circ f_1 = ", "f_n \\circ \\ldots \\circ f_2 \\circ h \\circ p = 0$ as desired." ], "refs": [ "derived-lemma-representable-homological" ], "ref_ids": [ 1758 ] } ], "ref_ids": [] }, { "id": 10252, "type": "theorem", "label": "more-algebra-lemma-not-regular-not-strong", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-not-regular-not-strong", "contents": [ "Let $R$ be a Noetherian ring. If $R$ is", "a strong generator for $D_{perf}(R)$, then $R$ is regular", "of finite dimension." ], "refs": [], "proofs": [ { "contents": [ "Assume $D_{perf}(R) = \\langle R \\rangle_n$ for some $n \\geq 1$.", "For any finite $R$-module $M$ we can choose a complex", "$$", "P = (", "P^{-n - 1} \\xrightarrow{d^{-n - 1}}", "P^{-n} \\xrightarrow{d^{-n}}", "P^{-n + 1} \\xrightarrow{d^1}", "\\ldots \\xrightarrow{d^{-1}} P^0)", "$$", "of finite free $R$-modules with $H^i(P) = 0$ for $i = -n, \\ldots, - 1$", "and $M \\cong \\Coker(d^{-1})$. Note that $P$ is in $D_{perf}(R)$.", "For any $R$-module $N$ we can compute $\\Ext^n_R(M, N)$", "the finite free resolution $P$ of $M$, see", "Algebra, Section \\ref{algebra-section-ext} and compare with", "Derived Categories, Section \\ref{derived-section-ext}.", "In particular, the sequence above defines an element", "$$", "\\xi \\in \\Ext^n_R(\\Coker(d^{-1}), \\Coker(d^{-n - 1})) =", "\\Ext^n_R(M, \\Coker(d^{-n - 1}))", "$$", "and for any element $\\overline{\\xi}$ in $\\Ext^n_R(M, N)$", "there is a $R$-module map $\\varphi : \\Coker(d^{-n - 1}) \\to N$ such", "that $\\varphi$ maps $\\xi$ to $\\overline{\\xi}$.", "For $j = 1, \\ldots, n - 1$ consider the complexes", "$$", "K_j = (\\Coker(d^{-n - 1}) \\to P^{-n + 1} \\to \\ldots \\to P^{-j})", "$$", "with $\\Coker(d^{-n - 1})$ in degree $-n$ and $P^t$ in degree $t$.", "We also set $K_n = \\Coker(d^{-n - 1})[n]$. Then we have maps", "$$", "P \\to K_1 \\to K_2 \\to \\ldots \\to K_n", "$$", "which induce vanishing maps on cohomology. By Lemma \\ref{lemma-ghost-lemma}", "since $P \\in D_{perf}(R) = \\langle R \\rangle_n$ we find that the composition", "of this maps is zero in $D(R)$. Since", "$\\Hom_{D(R)}(P, K_n) = \\Hom_{K(R)}(P, K_n)$ by", "Derived Categories, Lemma \\ref{derived-lemma-morphisms-from-projective-complex}", "we conclude $\\xi = 0$. Hence $\\Ext^n_R(M, N) = 0$ for all $R$-modules", "$N$, see discussion above.", "It follows that $M$ has projective dimension $\\leq n - 1$ by", "Algebra, Lemma \\ref{algebra-lemma-projective-dimension-ext}.", "Since this holds for all finite $R$-modules $M$ we conclude that", "$R$ has finite global dimension, see", "Algebra, Lemma \\ref{algebra-lemma-finite-gl-dim}.", "We finally conclude by Algebra, Lemma", "\\ref{algebra-lemma-finite-gl-dim-finite-dim-regular}." ], "refs": [ "more-algebra-lemma-ghost-lemma", "derived-lemma-morphisms-from-projective-complex", "algebra-lemma-projective-dimension-ext", "algebra-lemma-finite-gl-dim", "algebra-lemma-finite-gl-dim-finite-dim-regular" ], "ref_ids": [ 10251, 1862, 971, 974, 980 ] } ], "ref_ids": [] }, { "id": 10253, "type": "theorem", "label": "more-algebra-lemma-ext-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ext-regular", "contents": [ "Let $R$ be a Noetherian regular ring of dimension $d < \\infty$.", "Let $K, L \\in D^-(R)$. Assume there exists an $k$ such that", "$H^i(K) = 0$ for $i \\leq k$ and $H^i(L) = 0$ for $i \\geq k - d + 1$.", "Then $\\Hom_{D(R)}(K, L) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $K^\\bullet$ be a bounded above complex representing $K$, say", "$K^i = 0$ for $i \\geq n + 1$. After replacing $K^\\bullet$ by", "$\\tau_{\\geq k + 1}K^\\bullet$ we may assume $K^i = 0$ for $i \\leq k$.", "Then we may use the distinguished triangle", "$$", "K^n[-n] \\to K^\\bullet \\to \\sigma_{\\leq n - 1}K^\\bullet", "$$", "to see it suffices to prove the lemma for $K^n[-n]$ and", "$\\sigma_{\\leq n - 1}K^\\bullet$. By induction on $n$, we conclude", "that it suffices to prove the lemma in case $K$ is represented by", "the complex $M[-m]$ for some $R$-module $M$ and some $m \\geq k + 1$.", "Since $R$ has global dimension $d$ by Algebra, Lemma", "\\ref{algebra-lemma-finite-gl-dim-finite-dim-regular}", "we see that $M$ has a projective resolution", "$0 \\to P_d \\to \\ldots \\to P_0 \\to M \\to 0$.", "Then the complex $P^\\bullet$ having $P_i$ in degree $m - i$", "is a bounded complex of projectives representing $M[-m]$.", "On the other hand, we can choose a complex $L^\\bullet$ representing", "$L$ with $L^i = 0$ for $i \\geq k - d + 1$.", "Hence any map of complexes $P^\\bullet \\to L^\\bullet$ is zero.", "This implies the lemma by Derived Categories, Lemma", "\\ref{derived-lemma-morphisms-from-projective-complex}." ], "refs": [ "algebra-lemma-finite-gl-dim-finite-dim-regular", "derived-lemma-morphisms-from-projective-complex" ], "ref_ids": [ 980, 1862 ] } ], "ref_ids": [] }, { "id": 10254, "type": "theorem", "label": "more-algebra-lemma-split-complex-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-split-complex-regular", "contents": [ "Let $R$ be a Noetherian regular ring of dimension $1 \\leq d < \\infty$.", "Let $K \\in D(R)$ be perfect and let $k \\in \\mathbf{Z}$ such that", "$H^i(K) = 0$ for $i = k - d + 2, \\ldots, k$ (empty condition if $d = 1$).", "Then $K = \\tau_{\\leq k - d + 1}K \\oplus \\tau_{\\geq k + 1}K$." ], "refs": [], "proofs": [ { "contents": [ "The vanishing of cohomology shows that we have a distinguished triangle", "$$", "\\tau_{\\leq k - d + 1}K \\to K \\to \\tau_{\\geq k + 1}K \\to", "(\\tau_{\\leq k - d + 1}K)[1]", "$$", "By Derived Categories, Lemma \\ref{derived-lemma-split} it suffices to show that", "the third arrow is zero. Thus it suffices to show that", "$\\Hom_{D(R)}(\\tau_{\\geq k + 1}K, (\\tau_{\\leq k - d + 1}K)[1]) = 0$", "which follows from Lemma \\ref{lemma-ext-regular}." ], "refs": [ "derived-lemma-split", "more-algebra-lemma-ext-regular" ], "ref_ids": [ 1766, 10253 ] } ], "ref_ids": [] }, { "id": 10255, "type": "theorem", "label": "more-algebra-lemma-regular-strong-generator", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-regular-strong-generator", "contents": [ "Let $R$ be a Noetherian regular ring of finite dimension.", "Then $R$ is a strong generator for the full subcategory", "$D_{perf}(R) \\subset D(R)$ of perfect objects." ], "refs": [], "proofs": [ { "contents": [ "To see what the statement means, please look at", "Derived Categories, Definition \\ref{derived-definition-generators}.", "Please keep in mind that since $R$ is Noetherian and regular", "objects of $D(R)$ are perfect if and only if they are bounded", "and have finite cohomology modules, see Lemma \\ref{lemma-regular-perfect}", "\\medskip\\noindent", "Let $d = \\dim(R)$. In the next two paragraphs we first handle the cases", "$d = 0, 1$ but we suggest the reader skip these.", "\\medskip\\noindent", "If $d = 0$, then $R$ is a finite product of fields. Then every $R$-module", "is projective. Thus every object of $D(R)$ is isomorphic to a direct sum of", "its cohomologies and every $R$-module is a summand of a free $R$-module,", "so $D_{perf}(R) = \\langle R \\rangle_1$ and the lemma is true.", "\\medskip\\noindent", "If $d = 1$, then every object of $D^b(R)$ is isomorphic to the", "direct sum of its cohomology objects as follows from", "Derived Categories, Lemma \\ref{derived-lemma-ext-2-zero} and", "Algebra, Lemmas \\ref{algebra-lemma-finite-gl-dim-finite-dim-regular} and", "\\ref{algebra-lemma-projective-dimension-ext}. Hence to prove", "$D_{perf}(R)$ is strongly generated by $R$ it suffices to prove", "that $M \\in \\langle R \\rangle_2$ for every finite $R$-module $M$.", "This is clear from the fact that $M$ has a finite projective", "resolution of length $1$.", "\\medskip\\noindent", "Assume $d \\geq 2$. We will prove that", "$D_{perf}(R) \\subset \\langle R \\rangle_{2(d^2 - 1)}$ which finishes the proof", "(we make no attempt to get a good bound).", "In the argument we will use both canonical and stupid truncations, see", "Homology, Section \\ref{homology-section-truncations} for our conventions.", "Let $M$ be an object of $D_{perf}(R)$. Choose $a \\leq b$ such that $H^i(M) = 0$", "for $i \\not \\in [a, b]$. For $k \\in 2(d - 1)\\mathbf{Z} \\cap [a, b + 2d]$", "choose a quasi-isomorphism", "$$", "P_k^\\bullet \\to \\tau_{\\leq k}M", "$$", "where $P_k^\\bullet$ is a complex of finite free $R$-modules with", "$P_k^i = 0$ for $i \\geq k + 1$.", "Then we consider the map", "$$", "N_k = \\sigma_{\\geq k - d + 2}P_k^\\bullet \\to", "P_k^\\bullet \\to \\tau_{\\leq k}M \\to M", "$$", "Clearly the maps $H^i(N_k) \\to H^i(M)$ are isomorphisms for", "$i = k - d + 3, \\ldots, k$ (no cases for $d = 2$) and surjective for", "$i = k - d + 2$. Observe that $H^i(N_k) = 0$ for $i \\not \\in [k - d + 2, k]$.", "Choose a distinguished triangle", "$$", "\\bigoplus\\nolimits_{k \\in 2(d - 1)\\mathbf{Z} \\cap [a, b + 2d]} N_k", "\\to M \\to L \\to", "\\left(", "\\bigoplus\\nolimits_{k \\in 2(d - 1)\\mathbf{Z} \\cap [a, b + 2d]} N_k", "\\right)[1]", "$$", "Examining the long exact sequence of cohomology we find that", "$H^i(L) = 0$ for $i \\in [k - d + 2, k]$ for", "$k \\in 2(d - 1)\\mathbf{Z} \\cap [a, b + 2d]$.", "By Lemma \\ref{lemma-split-complex-regular}", "this means that $L$ splits as a direct sum", "$$", "L = \\bigoplus\\nolimits_{k \\in 2(d - 1)\\mathbf{Z}} L_k", "$$", "with $H^i(L_k)$ nonzero only for $i \\in [k - 2(d - 1) + 1, k - (d - 1)]$", "and $L_k$ zero for almost all $k$. Details omitted.", "Each $L_k$ is a perfect complex.", "By the result of the next paragraph we see that", "$\\bigoplus N_k$ and $L = \\bigoplus L_k$ are in $\\langle R \\rangle_{d^2 - 1}$.", "Hence $M$ is in $\\langle R \\rangle_{2(d^2 - 1)}$ by", "Derived Categories, Lemma \\ref{derived-lemma-generated-by-E-explicit}", "as desired.", "\\medskip\\noindent", "Let $K$ be an object of $D_{perf}(R)$ with $H^i(K)$ nonzero only for", "$i = 0, \\ldots, d - 2$. Then we claim $K$ is in $\\langle R \\rangle_{d^2 - 1}$.", "First, using $d - 2$ distinguished triangles of canonical truncations", "(Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle})", "we reduce to proving that any finite $R$-module $M$ is in", "$\\langle R \\rangle_{d + 1}$.", "This is clear because $M$ has a resolution of length $d$", "by finite projective $R$-modules. See", "Derived Categories, Lemmas \\ref{derived-lemma-generated-by-E-explicit} and", "\\ref{derived-lemma-in-cone-n}." ], "refs": [ "derived-definition-generators", "more-algebra-lemma-regular-perfect", "derived-lemma-ext-2-zero", "algebra-lemma-finite-gl-dim-finite-dim-regular", "algebra-lemma-projective-dimension-ext", "more-algebra-lemma-split-complex-regular", "derived-lemma-generated-by-E-explicit", "derived-remark-truncation-distinguished-triangle", "derived-lemma-generated-by-E-explicit", "derived-lemma-in-cone-n" ], "ref_ids": [ 2003, 10223, 1897, 980, 971, 10254, 1935, 2016, 1935, 1933 ] } ], "ref_ids": [] }, { "id": 10256, "type": "theorem", "label": "more-algebra-lemma-relatively-finitely-presented", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-relatively-finitely-presented", "contents": [ "Let $R \\to A$ be a ring map of finite type.", "Let $M$ be an $A$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item for some presentation $\\alpha : R[x_1, \\ldots, x_n] \\to A$", "the module $M$ is a finitely presented $R[x_1, \\ldots, x_n]$-module,", "\\item for all presentations $\\alpha : R[x_1, \\ldots, x_n] \\to A$", "the module $M$ is a finitely presented $R[x_1, \\ldots, x_n]$-module, and", "\\item for any surjection $A' \\to A$ where $A'$ is a finitely presented", "$R$-algebra, the module $M$ is finitely presented as $A'$-module.", "\\end{enumerate}", "In this case $M$ is a finitely presented $A$-module." ], "refs": [], "proofs": [ { "contents": [ "If $\\alpha : R[x_1, \\ldots, x_n] \\to A$ and", "$\\beta : R[y_1, \\ldots, y_m] \\to A$ are presentations.", "Choose $f_j \\in R[x_1, \\ldots, x_n]$ with $\\alpha(f_j) = \\beta(y_j)$", "and $g_i \\in R[y_1, \\ldots, y_m]$ with $\\beta(g_i) = \\alpha(x_i)$.", "Then we get a commutative diagram", "$$", "\\xymatrix{", "R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]", "\\ar[d]^{x_i \\mapsto g_i} \\ar[rr]_-{y_j \\mapsto f_j} & &", "R[x_1, \\ldots, x_n] \\ar[d] \\\\", "R[y_1, \\ldots, y_m] \\ar[rr] & & A", "}", "$$", "Hence the equivalence of (1) and (2) follows by applying", "Algebra, Lemmas \\ref{algebra-lemma-finitely-presented-over-subring} and", "\\ref{algebra-lemma-finite-finitely-presented-extension}.", "The equivalence of (2) and (3) follows by choosing a presentation", "$A' = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$ and using", "Algebra, Lemma \\ref{algebra-lemma-finite-finitely-presented-extension}", "to show that $M$ is finitely presented as $A'$-module if and only if", "$M$ is finitely presented as a $R[x_1, \\ldots, x_n]$-module." ], "refs": [ "algebra-lemma-finitely-presented-over-subring", "algebra-lemma-finite-finitely-presented-extension", "algebra-lemma-finite-finitely-presented-extension" ], "ref_ids": [ 335, 501, 501 ] } ], "ref_ids": [] }, { "id": 10257, "type": "theorem", "label": "more-algebra-lemma-finite-extension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finite-extension", "contents": [ "Let $R$ be a ring. Let $A \\to B$ be a finite map of finite type $R$-algebras.", "Let $M$ be a $B$-module. Then", "$M$ is an $A$-module finitely presented relative to $R$", "if and only if", "$M$ is a $B$-module finitely presented relative to $R$." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjection $R[x_1, \\ldots, x_n] \\to A$.", "Choose $y_1, \\ldots, y_m \\in B$ which generate $B$ over $A$.", "As $A \\to B$ is finite each $y_i$ satisfies a monic equation with", "coefficients in $A$. Hence we can find monic polynomials", "$P_j(T) \\in R[x_1, \\ldots, x_n][T]$ such that $P_j(y_j) = 0$ in $B$.", "Then we get a commutative diagram", "$$", "\\xymatrix{", "R[x_1, \\ldots, x_n] \\ar[d] \\ar[r] &", "R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]/(P_j(y_j)) \\ar[d] \\\\", "A \\ar[r] & B", "}", "$$", "Since the top arrow is a finite and finitely presented ring map", "we conclude by", "Algebra, Lemma \\ref{algebra-lemma-finite-finitely-presented-extension}", "and the definition." ], "refs": [ "algebra-lemma-finite-finitely-presented-extension" ], "ref_ids": [ 501 ] } ], "ref_ids": [] }, { "id": 10258, "type": "theorem", "label": "more-algebra-lemma-localize-relative-finite-presentation", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-localize-relative-finite-presentation", "contents": [ "Let $R$ be a ring, $f \\in R$ an element, $R_f \\to A$ is a finite type ring map,", "$g \\in A$, and $M$ an $A$-module. If $M$ of finite presentation relative", "to $R_f$, then $M_g$ is an $A_g$-module of finite presentation relative", "to $R$." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $R_f[x_1, \\ldots, x_n] \\to A$. We write", "$R_f = R[x_0]/(fx_0 - 1)$. Consider the presentation", "$R[x_0, x_1, \\ldots, x_n, x_{n + 1}] \\to A_g$ which extends the given", "map, maps $x_0$ to the image of $1/f$, and maps $x_{n + 1}$ to $1/g$.", "Choose $g' \\in R[x_0, x_1, \\ldots, x_n]$ which maps to $g$ (this is", "possible). Suppose that", "$$", "R_f[x_1, \\ldots, x_n]^{\\oplus s} \\to", "R_f[x_1, \\ldots, x_n]^{\\oplus t} \\to M \\to 0", "$$", "is a presentation of $M$ given by a matrix $(h_{ij})$. Pick", "$h'_{ij} \\in R[x_0, x_1, \\ldots, x_n]$ which map to $h_{ij}$.", "Then", "$$", "R[x_0, x_1, \\ldots, x_n, x_{n + 1}]^{\\oplus s + 2t} \\to", "R[x_0, x_1, \\ldots, x_n, x_{n + 1}]^{\\oplus t} \\to M_g \\to 0", "$$", "is a presentation of $M_f$.", "Here the $t \\times (s + 2t)$ matrix defining the map has a first", "$t \\times s$ block consisting of the matrix $h'_{ij}$, a second", "$t \\times t$ block which is $(x_0f - )I_t$, and a third block", "which is $(x_{n + 1}g' - 1)I_t$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10259, "type": "theorem", "label": "more-algebra-lemma-base-change-relative-finite-presentation", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-relative-finite-presentation", "contents": [ "Let $R \\to A$ be a finite type ring map. Let $M$ be an $A$-module finitely", "presented relative to $R$. For any ring map $R \\to R'$ the", "$A \\otimes_R R'$-module", "$$", "M \\otimes_A A' = M \\otimes_R R'", "$$", "is finitely presented relative to $R'$." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjection $R[x_1, \\ldots, x_n] \\to A$. Choose a presentation", "$$", "R[x_1, \\ldots, x_n]^{\\oplus s} \\to", "R[x_1, \\ldots, x_n]^{\\oplus t} \\to M \\to 0", "$$", "Then", "$$", "R'[x_1, \\ldots, x_n]^{\\oplus s} \\to", "R'[x_1, \\ldots, x_n]^{\\oplus t} \\to M \\otimes_R R' \\to 0", "$$", "is a presentation of the base change and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10260, "type": "theorem", "label": "more-algebra-lemma-pull-relative-finite-presentation", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pull-relative-finite-presentation", "contents": [ "Let $R \\to A$ be a finite type ring map.", "Let $M$ be an $A$-module finitely presented relative to $R$.", "Let $A \\to A'$ be a ring map of finite presentation.", "The $A'$-module $M \\otimes_A A'$ is finitely presented relative to $R$." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjection $R[x_1, \\ldots, x_n] \\to A$. Choose a presentation", "$A' = A[y_1, \\ldots, y_m]/(g_1, \\ldots, g_l)$.", "Pick $g'_i \\in R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]$ mapping to $g_i$.", "Say", "$$", "R[x_1, \\ldots, x_n]^{\\oplus s} \\to", "R[x_1, \\ldots, x_n]^{\\oplus t} \\to M \\to 0", "$$", "is a presentation of $M$ given by a matrix $(h_{ij})$.", "Then", "$$", "R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]^{\\oplus s + tl} \\to", "R[x_0, x_1, \\ldots, x_n, y_1, \\ldots, y_m]^{\\oplus t} \\to M \\otimes_A A' \\to 0", "$$", "is a presentation of $M \\otimes_A A'$.", "Here the $t \\times (s + lt)$ matrix defining the map has a first", "$t \\times s$ block consisting of the matrix $h_{ij}$, followed", "by $l$ blocks of size $t \\times t$ which are $g'_iI_t$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10261, "type": "theorem", "label": "more-algebra-lemma-composition-relative-finite-presentation", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-composition-relative-finite-presentation", "contents": [ "Let $R \\to A \\to B$ be finite type ring maps. Let $M$ be a $B$-module.", "If $M$ is finitely presented relative to $A$ and $A$ is of finite presentation", "over $R$, then $M$ is finitely presented relative to $R$." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjection $A[x_1, \\ldots, x_n] \\to B$.", "Choose a presentation", "$$", "A[x_1, \\ldots, x_n]^{\\oplus s} \\to", "A[x_1, \\ldots, x_n]^{\\oplus t} \\to M \\to 0", "$$", "given by a matrix $(h_{ij})$. Choose a presentation", "$$", "A = R[y_1, \\ldots, y_m]/(g_1, \\ldots, g_u).", "$$", "Choose $h'_{ij} \\in R[y_1, \\ldots, y_m, x_1, \\ldots, x_n]$", "mapping to $h_{ij}$. Then we obtain the presentation", "$$", "R[y_1, \\ldots, y_m, x_1, \\ldots, x_n]^{\\oplus s + tu} \\to", "R[y_1, \\ldots, y_m, x_1, \\ldots, x_n]^{\\oplus t} \\to M \\to 0", "$$", "where the $t \\times (s + tu)$-matrix is given by a first $t \\times s$ block", "consisting of $h'_{ij}$ followed by $u$ blocks of size $t \\times t$ given", "by $g_iI_t$, $i = 1, \\ldots, u$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10262, "type": "theorem", "label": "more-algebra-lemma-glue-relative-finite-presentation", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-glue-relative-finite-presentation", "contents": [ "Let $R \\to A$ be a finite type ring map. Let $M$ be an $A$-module.", "Let $f_1, \\ldots, f_r \\in A$ generate the unit ideal.", "The following are equivalent", "\\begin{enumerate}", "\\item each $M_{f_i}$ is finitely presented relative to $R$, and", "\\item $M$ is finitely presented relative to $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is in", "Lemma \\ref{lemma-localize-relative-finite-presentation}.", "Assume (1). Write $1 = \\sum f_ig_i$ in $A$.", "Choose a surjection", "$R[x_1, \\ldots, x_n, y_1, \\ldots, y_r, z_1, \\ldots, z_r] \\to A$.", "such that $y_i$ maps to $f_i$ and $z_i$ maps to $g_i$. Then we", "see that there exists a surjection", "$$", "P = R[x_1, \\ldots, x_n, y_1, \\ldots, y_r, z_1, \\ldots, z_r]/(\\sum y_iz_i - 1)", "\\longrightarrow", "A.", "$$", "By", "Lemma \\ref{lemma-relatively-finitely-presented}", "we see that $M_{f_i}$ is a finitely presented $A_{f_i}$-module, hence by", "Algebra, Lemma \\ref{algebra-lemma-cover}", "we see that $M$ is a finitely presented $A$-module.", "Hence $M$ is a finite $P$-module (with $P$ as above).", "Choose a surjection $P^{\\oplus t} \\to M$.", "We have to show that the kernel $K$ of this map is a finite", "$P$-module. Since $P_{y_i}$ surjects onto", "$A_{f_i}$ we see by", "Lemma \\ref{lemma-relatively-finitely-presented}", "and", "Algebra, Lemma \\ref{algebra-lemma-extension}", "that the localization $K_{y_i}$ is a finitely generated", "$P_{y_i}$-module. Choose elements", "$k_{i, j} \\in K$, $i = 1, \\ldots, r$, $j = 1, \\ldots, s_i$ such", "that the images of $k_{i, j}$ in $K_{y_i}$ generate.", "Set $K' \\subset K$ equal to the $P$-module", "generated by the elements $k_{i, j}$. Then $K/K'$ is a module", "whose localization at $y_i$ is zero for all $i$. Since $(y_1, \\ldots, y_r) = P$", "we see that $K/K' = 0$ as desired." ], "refs": [ "more-algebra-lemma-localize-relative-finite-presentation", "more-algebra-lemma-relatively-finitely-presented", "algebra-lemma-cover", "more-algebra-lemma-relatively-finitely-presented", "algebra-lemma-extension" ], "ref_ids": [ 10258, 10256, 411, 10256, 330 ] } ], "ref_ids": [] }, { "id": 10263, "type": "theorem", "label": "more-algebra-lemma-ses-relatively-finite-presentation", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ses-relatively-finite-presentation", "contents": [ "Let $R \\to A$ be a finite type ring map. Let $0 \\to M' \\to M \\to M'' \\to 0$", "be a short exact sequence of $A$-modules.", "\\begin{enumerate}", "\\item If $M', M''$ are finitely presented relative to $R$, then so is $M$.", "\\item If $M'$ is a finite type $A$-module and $M$ is finitely presented", "relative to $R$, then $M''$ is finitely presented relative to $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Algebra, Lemma \\ref{algebra-lemma-extension}." ], "refs": [ "algebra-lemma-extension" ], "ref_ids": [ 330 ] } ], "ref_ids": [] }, { "id": 10264, "type": "theorem", "label": "more-algebra-lemma-sum-relatively-finite-presentation", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-sum-relatively-finite-presentation", "contents": [ "Let $R \\to A$ be a finite type ring map.", "Let $M, M'$ be $A$-modules. If $M \\oplus M'$ is", "finitely presented relative to $R$, then so are $M$ and $M'$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10265, "type": "theorem", "label": "more-algebra-lemma-pull-push", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pull-push", "contents": [ "Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules.", "Consider the $R$-algebra map $R[x] \\to R$ which maps $x$ to zero.", "Then", "$$", "K^\\bullet \\otimes_{R[x]}^{\\mathbf{L}} R \\cong K^\\bullet \\oplus K^\\bullet[1]", "$$", "in $D(R)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a K-flat resolution $P^\\bullet \\to K^\\bullet$ over $R$", "such that $P^n$ is a flat $R$-module for all $n$, see", "Lemma \\ref{lemma-K-flat-resolution}. Then $P^\\bullet \\otimes_R R[x]$", "is a K-flat complex of $R[x]$-modules whose terms are flat $R[x]$-modules, see", "Lemma \\ref{lemma-base-change-K-flat} and", "Algebra, Lemma \\ref{algebra-lemma-flat-base-change}.", "In particular $x : P^n \\otimes_R R[x] \\to P^n \\otimes_R R[x]$", "is injective with cokernel isomorphic to $P^n$. Thus", "$$", "P^\\bullet \\otimes_R R[x] \\xrightarrow{x} P^\\bullet \\otimes_R R[x]", "$$", "is a double complex of $R[x]$-modules whose associated", "total complex is quasi-isomorphic to $P^\\bullet$ and hence $K^\\bullet$.", "Moreover, this associated total complex is a K-flat complex", "of $R[x]$-modules for example by", "Lemma \\ref{lemma-tensor-product-K-flat} or by", "Lemma \\ref{lemma-K-flat-two-out-of-three}.", "Hence", "\\begin{align*}", "K^\\bullet \\otimes_{R[x]}^{\\mathbf{L}} R", "& \\cong", "\\text{Tot}(P^\\bullet \\otimes_R R[x] \\xrightarrow{x} P^\\bullet \\otimes_R R[x])", "\\otimes_{R[x]} R =", "\\text{Tot}(P^\\bullet \\xrightarrow{0} P^\\bullet) \\\\", "& = P^\\bullet \\oplus P^\\bullet[1] \\cong K^\\bullet \\oplus K^\\bullet[1]", "\\end{align*}", "as desired." ], "refs": [ "more-algebra-lemma-K-flat-resolution", "more-algebra-lemma-base-change-K-flat", "algebra-lemma-flat-base-change", "more-algebra-lemma-tensor-product-K-flat", "more-algebra-lemma-K-flat-two-out-of-three" ], "ref_ids": [ 10131, 10124, 527, 10125, 10126 ] } ], "ref_ids": [] }, { "id": 10266, "type": "theorem", "label": "more-algebra-lemma-add-variable-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-add-variable-pseudo-coherent", "contents": [ "Let $R$ be a ring and $K^\\bullet$ a complex of $R$-modules.", "Let $m \\in \\mathbf{Z}$. Consider the $R$-algebra map $R[x] \\to R$", "which maps $x$ to zero. Then $K^\\bullet$ is $m$-pseudo-coherent as", "a complex of $R$-modules if and only if $K^\\bullet$ is $m$-pseudo-coherent", "as a complex of $R[x]$-modules." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-finite-push-pseudo-coherent}.", "We also prove it in another way as follows.", "\\medskip\\noindent", "Note that $0 \\to R[x] \\to R[x] \\to R \\to 0$ is exact. Hence $R$ is", "pseudo-coherent as an $R[x]$-module. Thus one implication of the lemma", "follows from", "Lemma \\ref{lemma-finite-push-pseudo-coherent}.", "To prove the other implication, assume that $K^\\bullet$ is", "$m$-pseudo-coherent as a complex of $R[x]$-modules. By", "Lemma \\ref{lemma-pull-pseudo-coherent}", "we see that $K^\\bullet \\otimes^{\\mathbf{L}}_{R[x]} R$ is", "$m$-pseudo-coherent as a complex of $R$-modules. By", "Lemma \\ref{lemma-pull-push}", "we see that $K^\\bullet \\oplus K^\\bullet[1]$ is $m$-pseudo-coherent", "as a complex of $R$-modules.", "Finally, we conclude that $K^\\bullet$ is $m$-pseudo-coherent", "as a complex of $R$-modules from", "Lemma \\ref{lemma-summands-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-finite-push-pseudo-coherent", "more-algebra-lemma-finite-push-pseudo-coherent", "more-algebra-lemma-pull-pseudo-coherent", "more-algebra-lemma-pull-push", "more-algebra-lemma-summands-pseudo-coherent" ], "ref_ids": [ 10154, 10154, 10155, 10265, 10151 ] } ], "ref_ids": [] }, { "id": 10267, "type": "theorem", "label": "more-algebra-lemma-relatively-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-relatively-pseudo-coherent", "contents": [ "Let $R \\to A$ be a ring map of finite type.", "Let $K^\\bullet$ be a complex of $A$-modules.", "Let $m \\in \\mathbf{Z}$.", "The following are equivalent", "\\begin{enumerate}", "\\item for some presentation $\\alpha : R[x_1, \\ldots, x_n] \\to A$", "the complex $K^\\bullet$ is an $m$-pseudo-coherent complex of", "$R[x_1, \\ldots, x_n]$-modules,", "\\item for all presentations $\\alpha : R[x_1, \\ldots, x_n] \\to A$", "the complex $K^\\bullet$ is an $m$-pseudo-coherent complex of", "$R[x_1, \\ldots, x_n]$-modules.", "\\end{enumerate}", "In particular the same equivalence holds for pseudo-coherence." ], "refs": [], "proofs": [ { "contents": [ "If $\\alpha : R[x_1, \\ldots, x_n] \\to A$ and", "$\\beta : R[y_1, \\ldots, y_m] \\to A$ are presentations.", "Choose $f_j \\in R[x_1, \\ldots, x_n]$ with $\\alpha(f_j) = \\beta(y_j)$", "and $g_i \\in R[y_1, \\ldots, y_m]$ with $\\beta(g_i) = \\alpha(x_i)$.", "Then we get a commutative diagram", "$$", "\\xymatrix{", "R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]", "\\ar[d]^{x_i \\mapsto g_i} \\ar[rr]_-{y_j \\mapsto f_j} & &", "R[x_1, \\ldots, x_n] \\ar[d] \\\\", "R[y_1, \\ldots, y_m] \\ar[rr] & & A", "}", "$$", "After a change of coordinates the ring homomorphism", "$R[x_1, \\ldots, x_n, y_1, \\ldots, y_m] \\to R[x_1, \\ldots, x_n]$", "is isomorphic to the ring homomorphism which maps", "each $y_i$ to zero. Similarly for the left vertical map in the", "diagram. Hence, by induction on the number of variables this lemma follows from", "Lemma \\ref{lemma-add-variable-pseudo-coherent}.", "The pseudo-coherent case follows from this and", "Lemma \\ref{lemma-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-add-variable-pseudo-coherent", "more-algebra-lemma-pseudo-coherent" ], "ref_ids": [ 10266, 10148 ] } ], "ref_ids": [] }, { "id": 10268, "type": "theorem", "label": "more-algebra-lemma-finite-extension-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-finite-extension-pseudo-coherent", "contents": [ "Let $R$ be a ring. Let $A \\to B$ be a finite map of finite type $R$-algebras.", "Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a complex of $B$-modules.", "Then $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)", "relative to $R$ if and only if $K^\\bullet$ seen as a complex of $A$-modules", "is $m$-pseudo-coherent (pseudo-coherent) relative to $R$." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjection $R[x_1, \\ldots, x_n] \\to A$.", "Choose $y_1, \\ldots, y_m \\in B$ which generate $B$ over $A$.", "As $A \\to B$ is finite each $y_i$ satisfies a monic equation with", "coefficients in $A$. Hence we can find monic polynomials", "$P_j(T) \\in R[x_1, \\ldots, x_n][T]$ such that $P_j(y_j) = 0$ in $B$.", "Then we get a commutative diagram", "$$", "\\xymatrix{", "& R[x_1, \\ldots, x_n, y_1, \\ldots, y_m] \\ar[d] \\\\", "R[x_1, \\ldots, x_n] \\ar[d] \\ar[r] &", "R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]/(P_j(y_j)) \\ar[d] \\\\", "A \\ar[r] & B", "}", "$$", "The top horizontal arrow and the top right vertical arrow", "satisfy the assumptions of", "Lemma \\ref{lemma-finite-push-pseudo-coherent}.", "Hence $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) as a complex", "of $R[x_1, \\ldots, x_n]$-modules if and only if $K^\\bullet$ is", "$m$-pseudo-coherent (resp.\\ pseudo-coherent) as a complex of", "$R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]$-modules." ], "refs": [ "more-algebra-lemma-finite-push-pseudo-coherent" ], "ref_ids": [ 10154 ] } ], "ref_ids": [] }, { "id": 10269, "type": "theorem", "label": "more-algebra-lemma-cone-relatively-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cone-relatively-pseudo-coherent", "contents": [ "Let $R$ be a ring. Let $R \\to A$ be a finite type ring map.", "Let $m \\in \\mathbf{Z}$. Let $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$", "be a distinguished triangle in $D(A)$.", "\\begin{enumerate}", "\\item If $K^\\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$ and", "$L^\\bullet$ is $m$-pseudo-coherent relative to $R$ then $M^\\bullet$ is", "$m$-pseudo-coherent relative to $R$.", "\\item If $K^\\bullet, M^\\bullet$ are $m$-pseudo-coherent relative to $R$,", "then $L^\\bullet$ is $m$-pseudo-coherent relative to $R$.", "\\item If $L^\\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$", "and $M^\\bullet$ is $m$-pseudo-coherent relative to $R$, then", "$K^\\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$.", "\\end{enumerate}", "Moreover, if two out of three of $K^\\bullet, L^\\bullet, M^\\bullet$", "are pseudo-coherent relative to $R$, the so is the third." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Lemma \\ref{lemma-cone-pseudo-coherent}", "and the definitions." ], "refs": [ "more-algebra-lemma-cone-pseudo-coherent" ], "ref_ids": [ 10145 ] } ], "ref_ids": [] }, { "id": 10270, "type": "theorem", "label": "more-algebra-lemma-rel-n-pseudo-module", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-rel-n-pseudo-module", "contents": [ "Let $R \\to A$ be a finite type ring map. Let $M$ be an $A$-module.", "Then", "\\begin{enumerate}", "\\item $M$ is $0$-pseudo-coherent relative to $R$ if and only if", "$M$ is a finite type $A$-module,", "\\item $M$ is $(-1)$-pseudo-coherent relative to $R$ if and only if", "$M$ is a finitely presented relative to $R$,", "\\item $M$ is $(-d)$-pseudo-coherent relative to $R$ if and only if", "for every surjection $R[x_1, \\ldots, x_n] \\to A$ there exists a", "resolution", "$$", "R[x_1, \\ldots, x_n]^{\\oplus a_d} \\to R[x_1, \\ldots, x_n]^{\\oplus a_{d - 1}}", "\\to \\ldots \\to R[x_1, \\ldots, x_n]^{\\oplus a_0} \\to M \\to 0", "$$", "of length $d$, and", "\\item $M$ is pseudo-coherent relative to $R$ if and only if", "for every presentation $R[x_1, \\ldots, x_n] \\to A$ there exists an", "infinite resolution", "$$", "\\ldots \\to R[x_1, \\ldots, x_n]^{\\oplus a_1} \\to", "R[x_1, \\ldots, x_n]^{\\oplus a_0} \\to M \\to 0", "$$", "by finite free $R[x_1, \\ldots, x_n]$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Lemma \\ref{lemma-n-pseudo-module}", "and the definitions." ], "refs": [ "more-algebra-lemma-n-pseudo-module" ], "ref_ids": [ 10147 ] } ], "ref_ids": [] }, { "id": 10271, "type": "theorem", "label": "more-algebra-lemma-summands-relative-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-summands-relative-pseudo-coherent", "contents": [ "Let $R \\to A$ be a finite type ring map.", "Let $m \\in \\mathbf{Z}$. Let $K^\\bullet, L^\\bullet \\in D(A)$.", "If $K^\\bullet \\oplus L^\\bullet$", "is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$", "so are $K^\\bullet$ and $L^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from", "Lemma \\ref{lemma-summands-pseudo-coherent}", "and the definitions." ], "refs": [ "more-algebra-lemma-summands-pseudo-coherent" ], "ref_ids": [ 10151 ] } ], "ref_ids": [] }, { "id": 10272, "type": "theorem", "label": "more-algebra-lemma-complex-relative-pseudo-coherent-modules", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-complex-relative-pseudo-coherent-modules", "contents": [ "Let $R \\to A$ be a finite type ring map.", "Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a bounded", "above complex of $A$-modules such that $K^i$ is $(m - i)$-pseudo-coherent", "relative to $R$ for all $i$. Then $K^\\bullet$ is $m$-pseudo-coherent", "relative to $R$. In particular, if $K^\\bullet$ is a bounded above complex of", "$A$-modules pseudo-coherent relative to $R$, then $K^\\bullet$ is", "pseudo-coherent relative to $R$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from", "Lemma \\ref{lemma-complex-pseudo-coherent-modules}", "and the definitions." ], "refs": [ "more-algebra-lemma-complex-pseudo-coherent-modules" ], "ref_ids": [ 10152 ] } ], "ref_ids": [] }, { "id": 10273, "type": "theorem", "label": "more-algebra-lemma-cohomology-relative-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cohomology-relative-pseudo-coherent", "contents": [ "Let $R \\to A$ be a finite type ring map. Let $m \\in \\mathbf{Z}$.", "Let $K^\\bullet \\in D^{-}(A)$ such that $H^i(K^\\bullet)$ is", "$(m - i)$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$", "for all $i$. Then $K^\\bullet$ is $m$-pseudo-coherent", "(resp.\\ pseudo-coherent) relative to $R$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from", "Lemma \\ref{lemma-cohomology-pseudo-coherent}", "and the definitions." ], "refs": [ "more-algebra-lemma-cohomology-pseudo-coherent" ], "ref_ids": [ 10153 ] } ], "ref_ids": [] }, { "id": 10274, "type": "theorem", "label": "more-algebra-lemma-localize-relative-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-localize-relative-pseudo-coherent", "contents": [ "Let $R$ be a ring, $f \\in R$ an element, $R_f \\to A$ is a finite type ring map,", "$g \\in A$, and $K^\\bullet$ a complex of $A$-modules.", "If $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)", "relative to $R_f$, then $K^\\bullet \\otimes_A A_g$ is", "$m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$." ], "refs": [], "proofs": [ { "contents": [ "First we show that $K^\\bullet$ is $m$-pseudo-coherent relative to $R$.", "Namely, suppose $R_f[x_1, \\ldots, x_n] \\to A$ is surjective. Write", "$R_f = R[x_0]/(fx_0 - 1)$. Then $R[x_0, x_1, \\ldots, x_n] \\to A$", "is surjective, and $R_f[x_1, \\ldots, x_n]$ is pseudo-coherent as", "an $R[x_0, \\ldots, x_n]$-module. Hence by", "Lemma \\ref{lemma-finite-push-pseudo-coherent}", "we see that $K^\\bullet$ is $m$-pseudo-coherent as a complex of", "$R[x_0, x_1, \\ldots, x_n]$-modules.", "\\medskip\\noindent", "Choose an element $g' \\in R[x_0, x_1, \\ldots, x_n]$ which maps to $g \\in A$. By", "Lemma \\ref{lemma-pull-pseudo-coherent}", "we see that", "\\begin{align*}", "K^\\bullet \\otimes_{R[x_0, x_1, \\ldots, x_n]}^{\\mathbf{L}}", "R[x_0, x_1, \\ldots, x_n, \\frac{1}{g'}] & =", "K^\\bullet \\otimes_{R[x_0, x_1, \\ldots, x_n]}", "R[x_0, x_1, \\ldots, x_n, \\frac{1}{g'}] \\\\", "& = K^\\bullet \\otimes_A A_f", "\\end{align*}", "is $m$-pseudo-coherent as a complex of", "$R[x_0, x_1, \\ldots, x_n, \\frac{1}{g'}]$-modules.", "write", "$$", "R[x_0, x_1, \\ldots, x_n, \\frac{1}{g'}] =", "R[x_0, \\ldots, x_n, x_{n + 1}]/(x_{n + 1}g' - 1).", "$$", "As $R[x_0, x_1, \\ldots, x_n, \\frac{1}{g'}]$ is pseudo-coherent as a", "$R[x_0, \\ldots, x_n, x_{n + 1}]$-module we conclude (see", "Lemma \\ref{lemma-finite-push-pseudo-coherent})", "that $K^\\bullet \\otimes_A A_g$ is $m$-pseudo-coherent as a complex of", "$R[x_0, \\ldots, x_n, x_{n + 1}]$-modules as desired." ], "refs": [ "more-algebra-lemma-finite-push-pseudo-coherent", "more-algebra-lemma-pull-pseudo-coherent", "more-algebra-lemma-finite-push-pseudo-coherent" ], "ref_ids": [ 10154, 10155, 10154 ] } ], "ref_ids": [] }, { "id": 10275, "type": "theorem", "label": "more-algebra-lemma-base-change-relative-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-relative-pseudo-coherent", "contents": [ "Let $R \\to A$ be a finite type ring map. Let $m \\in \\mathbf{Z}$.", "Let $K^\\bullet$ be a complex of $A$-modules which is $m$-pseudo-coherent", "(resp.\\ pseudo-coherent) relative to $R$. Let $R \\to R'$ be a ring", "map such that $A$ and $R'$ are Tor independent over $R$. Set", "$A' = A \\otimes_R R'$. Then", "$K^\\bullet \\otimes_A^{\\mathbf{L}} A'$", "is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R'$." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjection $R[x_1, \\ldots, x_n] \\to A$.", "Note that", "$$", "K^\\bullet \\otimes_A^{\\mathbf{L}} A' =", "K^\\bullet \\otimes_R^{\\mathbf{L}} R' =", "K^\\bullet \\otimes_{R[x_1, \\ldots, x_n]}^{\\mathbf{L}} R'[x_1, \\ldots, x_n]", "$$", "by", "Lemma \\ref{lemma-base-change-comparison}", "applied twice. Hence we win by", "Lemma \\ref{lemma-pull-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-base-change-comparison", "more-algebra-lemma-pull-pseudo-coherent" ], "ref_ids": [ 10139, 10155 ] } ], "ref_ids": [] }, { "id": 10276, "type": "theorem", "label": "more-algebra-lemma-pull-relative-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pull-relative-pseudo-coherent", "contents": [ "Let $R \\to A \\to B$ be finite type ring maps.", "Let $m \\in \\mathbf{Z}$.", "Let $K^\\bullet$ be a complex of $A$-modules.", "Assume $B$ as a $B$-module is pseudo-coherent relative to $A$.", "If $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)", "relative to $R$, then $K^\\bullet \\otimes_A^{\\mathbf{L}} B$ is", "$m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjection $A[y_1, \\ldots, y_m] \\to B$.", "Choose a surjection $R[x_1, \\ldots, x_n] \\to A$.", "Combined we get a surjection $R[x_1, \\ldots, x_n, y_1, \\ldots y_m] \\to B$.", "Choose a resolution $E^\\bullet \\to B$ of $B$ by a complex of", "finite free $A[y_1, \\ldots, y_n]$-modules (which is possible", "by our assumption on the ring map $A \\to B$). We may assume", "that $K^\\bullet$ is a bounded above complex of flat $A$-modules. Then", "\\begin{align*}", "K^\\bullet \\otimes_A^{\\mathbf{L}} B & =", "\\text{Tot}(K^\\bullet \\otimes_A B[0]) \\\\", "& = \\text{Tot}(K^\\bullet \\otimes_A A[y_1, \\ldots, y_m]", "\\otimes_{A[y_1, \\ldots, y_m]} B[0]) \\\\", "& \\cong", "\\text{Tot}\\left(", "(K^\\bullet \\otimes_A A[y_1, \\ldots, y_m])", "\\otimes_{A[y_1, \\ldots, y_m]} E^\\bullet", "\\right) \\\\", "& =", "\\text{Tot}(K^\\bullet \\otimes_A E^\\bullet)", "\\end{align*}", "in $D(A[y_1, \\ldots, y_m])$. The quasi-isomorphism $\\cong$ comes from", "an application of", "Lemma \\ref{lemma-derived-tor-quasi-isomorphism}.", "Thus we have to show that", "$\\text{Tot}(K^\\bullet \\otimes_A E^\\bullet)$ is $m$-pseudo-coherent", "as a complex of $R[x_1, \\ldots, x_n, y_1, \\ldots y_m]$-modules.", "Note that $\\text{Tot}(K^\\bullet \\otimes_A E^\\bullet)$ has a filtration by", "subcomplexes with successive quotients the complexes", "$K^\\bullet \\otimes_A E^i[-i]$. Note that for $i \\ll 0$ the", "complexes $K^\\bullet \\otimes_A E^i[-i]$ have zero cohomology", "in degrees $\\leq m$ and hence are $m$-pseudo-coherent (over any ring).", "Hence, applying", "Lemma \\ref{lemma-cone-relatively-pseudo-coherent}", "and induction, it suffices to show that $K^\\bullet \\otimes_A E^i[-i]$ is", "pseudo-coherent relative to $R$ for all $i$. Note that $E^i = 0$ for", "$i > 0$. Since also $E^i$ is finite free this", "reduces to proving that $K^\\bullet \\otimes_A A[y_1, \\ldots, y_m]$ is", "$m$-pseudo-coherent relative to $R$ which follows from", "Lemma \\ref{lemma-base-change-relative-pseudo-coherent}", "for instance." ], "refs": [ "more-algebra-lemma-derived-tor-quasi-isomorphism", "more-algebra-lemma-cone-relatively-pseudo-coherent", "more-algebra-lemma-base-change-relative-pseudo-coherent" ], "ref_ids": [ 10128, 10269, 10275 ] } ], "ref_ids": [] }, { "id": 10277, "type": "theorem", "label": "more-algebra-lemma-pull-relative-pseudo-coherent-module", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pull-relative-pseudo-coherent-module", "contents": [ "Let $R \\to A \\to B$ be finite type ring maps.", "Let $m \\in \\mathbf{Z}$. Let $M$ be an $A$-module.", "Assume $B$ is flat over $A$ and $B$ as a $B$-module is", "pseudo-coherent relative to $A$.", "If $M$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)", "relative to $R$, then $M \\otimes_A B$ is", "$m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from", "Lemma \\ref{lemma-pull-relative-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-pull-relative-pseudo-coherent" ], "ref_ids": [ 10276 ] } ], "ref_ids": [] }, { "id": 10278, "type": "theorem", "label": "more-algebra-lemma-composition-relative-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-composition-relative-pseudo-coherent", "contents": [ "Let $R$ be a ring. Let $A \\to B$ be a map of finite type $R$-algebras.", "Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a complex of $B$-modules.", "Assume $A$ is pseudo-coherent relative to $R$. Then the following are", "equivalent", "\\begin{enumerate}", "\\item $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)", "relative to $A$, and", "\\item $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)", "relative to $R$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a surjection $R[x_1, \\ldots, x_n] \\to A$.", "Choose a surjection $A[y_1, \\ldots, y_m] \\to B$.", "Then we get a surjection", "$$", "R[x_1, \\ldots, x_n, y_1, \\ldots, y_m] \\to A[y_1, \\ldots, y_m]", "$$", "which is a flat base change of $R[x_1, \\ldots, x_n] \\to A$.", "By assumption $A$ is a pseudo-coherent module over $R[x_1, \\ldots, x_n]$", "hence by", "Lemma \\ref{lemma-flat-base-change-pseudo-coherent}", "we see that $A[y_1, \\ldots, y_m]$ is pseudo-coherent over", "$R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]$. Thus the lemma follows from", "Lemma \\ref{lemma-finite-push-pseudo-coherent}", "and the definitions." ], "refs": [ "more-algebra-lemma-flat-base-change-pseudo-coherent", "more-algebra-lemma-finite-push-pseudo-coherent" ], "ref_ids": [ 10156, 10154 ] } ], "ref_ids": [] }, { "id": 10279, "type": "theorem", "label": "more-algebra-lemma-glue-relative-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-glue-relative-pseudo-coherent", "contents": [ "Let $R \\to A$ be a finite type ring map.", "Let $K^\\bullet$ be a complex of $A$-modules.", "Let $m \\in \\mathbf{Z}$.", "Let $f_1, \\ldots, f_r \\in A$ generate the unit ideal.", "The following are equivalent", "\\begin{enumerate}", "\\item each $K^\\bullet \\otimes_A A_{f_i}$ is", "$m$-pseudo-coherent relative to $R$, and", "\\item $K^\\bullet$ is $m$-pseudo-coherent relative to $R$.", "\\end{enumerate}", "The same equivalence holds for pseudo-coherence relative to $R$." ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is in", "Lemma \\ref{lemma-localize-relative-pseudo-coherent}.", "Assume (1). Write $1 = \\sum f_ig_i$ in $A$.", "Choose a surjection", "$R[x_1, \\ldots, x_n, y_1, \\ldots, y_r, z_1, \\ldots, z_r] \\to A$.", "such that $y_i$ maps to $f_i$ and $z_i$ maps to $g_i$. Then we", "see that there exists a surjection", "$$", "P = R[x_1, \\ldots, x_n, y_1, \\ldots, y_r, z_1, \\ldots, z_r]/(\\sum y_iz_i - 1)", "\\longrightarrow", "A.", "$$", "Note that $P$ is pseudo-coherent as an", "$R[x_1, \\ldots, x_n, y_1, \\ldots, y_r, z_1, \\ldots, z_r]$-module", "and that $P[1/y_i]$ is pseudo-coherent as an", "$R[x_1, \\ldots, x_n, y_1, \\ldots, y_r, z_1, \\ldots, z_r, 1/y_i]$-module.", "Hence by", "Lemma \\ref{lemma-finite-push-pseudo-coherent}", "we see that", "$K^\\bullet \\otimes_A A_{f_i}$ is an $m$-pseudo-coherent complex", "of $P[1/y_i]$-modules for each $i$.", "Thus by", "Lemma \\ref{lemma-glue-pseudo-coherent}", "we see that $K^\\bullet$ is pseudo-coherent as a complex of", "$P$-modules, and", "Lemma \\ref{lemma-finite-push-pseudo-coherent}", "shows that $K^\\bullet$ is pseudo-coherent as a complex of", "$R[x_1, \\ldots, x_n, y_1, \\ldots, y_r, z_1, \\ldots, z_r]$-modules." ], "refs": [ "more-algebra-lemma-localize-relative-pseudo-coherent", "more-algebra-lemma-finite-push-pseudo-coherent", "more-algebra-lemma-glue-pseudo-coherent", "more-algebra-lemma-finite-push-pseudo-coherent" ], "ref_ids": [ 10274, 10154, 10157, 10154 ] } ], "ref_ids": [] }, { "id": 10280, "type": "theorem", "label": "more-algebra-lemma-Noetherian-relative-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Noetherian-relative-pseudo-coherent", "contents": [ "Let $R$ be a Noetherian ring. Let $R \\to A$ be a finite type ring map. Then", "\\begin{enumerate}", "\\item A complex of $A$-modules $K^\\bullet$ is $m$-pseudo-coherent", "relative to $R$ if and only if $K^\\bullet \\in D^{-}(A)$ and", "$H^i(K^\\bullet)$ is a finite $A$-module for $i \\geq m$.", "\\item A complex of $A$-modules $K^\\bullet$ is pseudo-coherent relative to $R$", "if and only if $K^\\bullet \\in D^{-}(A)$ and", "$H^i(K^\\bullet)$ is a finite $A$-module for all $i$.", "\\item An $A$-module is pseudo-coherent relative to $R$", "if and only if it is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Lemma \\ref{lemma-Noetherian-pseudo-coherent}", "and the definitions." ], "refs": [ "more-algebra-lemma-Noetherian-pseudo-coherent" ], "ref_ids": [ 10160 ] } ], "ref_ids": [] }, { "id": 10281, "type": "theorem", "label": "more-algebra-lemma-perfect-ring-map", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-perfect-ring-map", "contents": [ "A ring map $A \\to B$ is perfect if and only if $B = A[x_1, \\ldots, x_n]/I$", "and $B$ as an $A[x_1, \\ldots, x_n]$-module has a finite resolution by", "finite projective $A[x_1, \\ldots, x_n]$-modules." ], "refs": [], "proofs": [ { "contents": [ "If $A \\to B$ is perfect, then $B = A[x_1, \\ldots, x_n]/I$ and", "$B$ is pseudo-coherent as an $A[x_1, \\ldots, x_n]$-module and", "has finite tor dimension as an $A$-module. Hence", "Lemma \\ref{lemma-perfect-over-polynomial-ring}", "implies that $B$ is perfect as a $A[x_1, \\ldots, x_n]$-module, i.e.,", "it has a finite resolution by finite projective $A[x_1, \\ldots, x_n]$-modules", "(Lemma \\ref{lemma-perfect-module}).", "Conversely, if $B = A[x_1, \\ldots, x_n]/I$", "and $B$ as an $A[x_1, \\ldots, x_n]$-module has a finite resolution by", "finite projective $A[x_1, \\ldots, x_n]$-modules then", "$B$ is pseudo-coherent as an $A[x_1, \\ldots, x_n]$-module,", "hence $A \\to B$ is pseudo-coherent. Moreover, the given resolution", "over $A[x_1, \\ldots, x_n]$ is a finite resolution by flat", "$A$-modules and hence $B$ has finite tor dimension as an $A$-module." ], "refs": [ "more-algebra-lemma-perfect-over-polynomial-ring", "more-algebra-lemma-perfect-module" ], "ref_ids": [ 10245, 10213 ] } ], "ref_ids": [] }, { "id": 10282, "type": "theorem", "label": "more-algebra-lemma-Noetherian-pseudo-coherent-ring-map", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Noetherian-pseudo-coherent-ring-map", "contents": [ "A finite type ring map of Noetherian rings is pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "See", "Lemma \\ref{lemma-Noetherian-relative-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-Noetherian-relative-pseudo-coherent" ], "ref_ids": [ 10280 ] } ], "ref_ids": [] }, { "id": 10283, "type": "theorem", "label": "more-algebra-lemma-flat-finite-presentation-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-finite-presentation-perfect", "contents": [ "A ring map which is flat and of finite presentation is perfect." ], "refs": [], "proofs": [ { "contents": [ "Let $A \\to B$ be a ring map which is flat and of finite presentation.", "It is clear that $B$ has finite tor dimension. By", "Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "there exists a finite type $\\mathbf{Z}$-algebra $A_0 \\subset A$", "and a flat finite type ring map $A_0 \\to B_0$ such that", "$B = B_0 \\otimes_{A_0} A$. By", "Lemma \\ref{lemma-Noetherian-relative-pseudo-coherent}", "we see that $A_0 \\to B_0$ is pseudo-coherent.", "As $A_0 \\to B_0$ is flat we see that $B_0$ and $A$ are tor independent", "over $A_0$, hence we may use", "Lemma \\ref{lemma-base-change-relative-pseudo-coherent}", "to conclude that $A \\to B$ is pseudo-coherent." ], "refs": [ "algebra-lemma-flat-finite-presentation-limit-flat", "more-algebra-lemma-Noetherian-relative-pseudo-coherent", "more-algebra-lemma-base-change-relative-pseudo-coherent" ], "ref_ids": [ 1389, 10280, 10275 ] } ], "ref_ids": [] }, { "id": 10284, "type": "theorem", "label": "more-algebra-lemma-regular-perfect-ring-map", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-regular-perfect-ring-map", "contents": [ "Let $A \\to B$ be a finite type ring map with $A$ a regular ring", "of finite dimension. Then $A \\to B$ is perfect." ], "refs": [], "proofs": [ { "contents": [ "By", "Algebra, Lemma \\ref{algebra-lemma-finite-gl-dim-finite-dim-regular}", "the assumption on $A$ means that $A$ has finite global dimension.", "Hence every module has finite tor dimension, see", "Lemma \\ref{lemma-finite-gl-dim-tor-dimension},", "in particular $B$ does. By", "Lemma \\ref{lemma-Noetherian-pseudo-coherent-ring-map}", "the map is pseudo-coherent." ], "refs": [ "algebra-lemma-finite-gl-dim-finite-dim-regular", "more-algebra-lemma-finite-gl-dim-tor-dimension", "more-algebra-lemma-Noetherian-pseudo-coherent-ring-map" ], "ref_ids": [ 980, 10186, 10282 ] } ], "ref_ids": [] }, { "id": 10285, "type": "theorem", "label": "more-algebra-lemma-lci-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lci-perfect", "contents": [ "A local complete intersection homomorphism is perfect." ], "refs": [], "proofs": [ { "contents": [ "Let $A \\to B$ be a local complete intersection homomorphism.", "By Definition \\ref{definition-local-complete-intersection} this", "means that $B = A[x_1, \\ldots, x_n]/I$ where $I$ is a Koszul ideal", "in $A[x_1, \\ldots, x_n]$. ", "By Lemmas \\ref{lemma-perfect-ring-map} and \\ref{lemma-perfect-module}", "it suffices to show that $I$ is a perfect module over $A[x_1, \\ldots, x_n]$.", "By Lemma \\ref{lemma-glue-perfect} this is a local question. Hence we", "may assume that $I$ is generated by a Koszul-regular sequence (by", "Definition \\ref{definition-regular-ideal}).", "Of course this means that $I$ has a finite free resolution and we win." ], "refs": [ "more-algebra-definition-local-complete-intersection", "more-algebra-lemma-perfect-ring-map", "more-algebra-lemma-perfect-module", "more-algebra-lemma-glue-perfect", "more-algebra-definition-regular-ideal" ], "ref_ids": [ 10609, 10281, 10213, 10221, 10608 ] } ], "ref_ids": [] }, { "id": 10286, "type": "theorem", "label": "more-algebra-lemma-relative-pseudo-coherent-is-moot", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-relative-pseudo-coherent-is-moot", "contents": [ "Let $R \\to A$ be a pseudo-coherent ring map. Let $K \\in D(A)$.", "The following are equivalent", "\\begin{enumerate}", "\\item $K$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$, and", "\\item $K$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) in $D(A)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Reformulation of a special case of", "Lemma \\ref{lemma-composition-relative-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-composition-relative-pseudo-coherent" ], "ref_ids": [ 10278 ] } ], "ref_ids": [] }, { "id": 10287, "type": "theorem", "label": "more-algebra-lemma-more-relative-pseudo-coherent-is-moot", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-more-relative-pseudo-coherent-is-moot", "contents": [ "Let $R \\to B \\to A$ be ring maps with $\\varphi : B \\to A$ surjective and", "$R \\to B$ and $R \\to A$ flat and of finite presentation. For $K \\in D(A)$", "denote $\\varphi_*K \\in D(B)$ the restriction.", "The following are equivalent", "\\begin{enumerate}", "\\item $K$ is pseudo-coherent,", "\\item $K$ is pseudo-coherent relative to $R$,", "\\item $K$ is pseudo-coherent relative to $A$,", "\\item $\\varphi_*K$ is pseudo-coherent,", "\\item $\\varphi_*K$ is pseudo-coherent relative to $R$.", "\\end{enumerate}", "Similar holds for $m$-pseudo-coherence." ], "refs": [], "proofs": [ { "contents": [ "Observe that $R \\to A$ and $R \\to B$ are perfect ring maps", "(Lemma \\ref{lemma-flat-finite-presentation-perfect})", "hence a fortiori pseudo-coherent ring maps.", "Thus (1) $\\Leftrightarrow$ (2) and (4) $\\Leftrightarrow$ (5)", "by Lemma \\ref{lemma-relative-pseudo-coherent-is-moot}.", "\\medskip\\noindent", "Using that $A$ is pseudo-coherent relative to $R$ we use", "Lemma \\ref{lemma-composition-relative-pseudo-coherent}", "to see that (2) $\\Leftrightarrow$ (3).", "However, since $A \\to B$ is surjective, we see directly from", "Definition \\ref{definition-relatively-pseudo-coherent}", "that (3) is equivalent with (4)." ], "refs": [ "more-algebra-lemma-flat-finite-presentation-perfect", "more-algebra-lemma-relative-pseudo-coherent-is-moot", "more-algebra-lemma-composition-relative-pseudo-coherent", "more-algebra-definition-relatively-pseudo-coherent" ], "ref_ids": [ 10283, 10286, 10278, 10630 ] } ], "ref_ids": [] }, { "id": 10288, "type": "theorem", "label": "more-algebra-lemma-cone-relatively-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cone-relatively-perfect", "contents": [ "Let $R \\to A$ be a flat ring map of finite presentation.", "The $R$-perfect objects of $D(A)$ form a", "saturated\\footnote{Derived Categories, Definition", "\\ref{derived-definition-saturated}.} triangulated", "strictly full subcategory." ], "refs": [ "derived-definition-saturated" ], "proofs": [ { "contents": [ "This follows from", "Lemmas \\ref{lemma-cone-pseudo-coherent},", "\\ref{lemma-summands-pseudo-coherent},", "\\ref{lemma-cone-tor-amplitude}, and", "\\ref{lemma-summands-tor-amplitude}." ], "refs": [ "more-algebra-lemma-cone-pseudo-coherent", "more-algebra-lemma-summands-pseudo-coherent", "more-algebra-lemma-cone-tor-amplitude", "more-algebra-lemma-summands-tor-amplitude" ], "ref_ids": [ 10145, 10151, 10172, 10174 ] } ], "ref_ids": [ 1974 ] }, { "id": 10289, "type": "theorem", "label": "more-algebra-lemma-perfect-relatively-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-perfect-relatively-perfect", "contents": [ "Let $R \\to A$ be a flat ring map of finite presentation.", "A perfect object of $D(A)$ is $R$-perfect. If $K, M \\in D(A)$", "then $K \\otimes_A^\\mathbf{L} M$ is $R$-perfect if $K$ is perfect", "and $M$ is $R$-perfect." ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from the second by taking $M = A$.", "The second statement follows from Lemmas \\ref{lemma-perfect},", "\\ref{lemma-push-tor-amplitude}, and \\ref{lemma-tensor-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-perfect", "more-algebra-lemma-push-tor-amplitude", "more-algebra-lemma-tensor-pseudo-coherent" ], "ref_ids": [ 10212, 10177, 10159 ] } ], "ref_ids": [] }, { "id": 10290, "type": "theorem", "label": "more-algebra-lemma-structure-relatively-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-structure-relatively-perfect", "contents": [ "Let $R \\to A$ be a flat ring map of finite presentation.", "Let $K \\in D(A)$. The following are equivalent", "\\begin{enumerate}", "\\item $K$ is $R$-perfect, and", "\\item $K$ is isomorphic to a finite complex of $R$-flat,", "finitely presented $A$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove (2) implies (1) it suffices by", "Lemma \\ref{lemma-cone-relatively-perfect}", "to show that an $R$-flat, finitely presented $A$-module $M$ defines", "an $R$-perfect object of $D(A)$. Since $M$ has finite tor dimension", "over $R$, it suffices to show that $M$ is pseudo-coherent. By", "Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "there exists a finite type $\\mathbf{Z}$-algebra $R_0 \\subset R$", "and a flat finite type ring map $R_0 \\to A_0$ and", "a finite $A_0$-module $M_0$ flat over $R_0$ such that", "$A = A_0 \\otimes_{R_0} R$ and $M = M_0 \\otimes_{R_0} R$. By", "Lemma \\ref{lemma-Noetherian-pseudo-coherent}", "we see that $M_0$ is pseudo-coherent $A_0$-module.", "Choose a resolution $P_0^\\bullet \\to M_0$ by finite free", "$A_0$-modules $P_0^n$. Since $A_0$ is flat over $R_0$,", "this is a flat resolution. Since $M_0$ is flat over $R_0$", "we find that $P^\\bullet = P_0^\\bullet \\otimes_{R_0} R$", "still resolves $M = M_0 \\otimes_{R_0} R$. (You can use", "Lemma \\ref{lemma-base-change-comparison} to see this.)", "Hence $P^\\bullet$ is a finite free resolution of $M$", "over $A$ and we conclude that $M$ is pseudo-coherent.", "\\medskip\\noindent", "Assume (1). We can represent $K$ by a bounded above complex", "$P^\\bullet$ of finite free $A$-modules. Assume that", "$K$ viewed as an object of $D(R)$ has tor amplitude in $[a, b]$.", "By Lemma \\ref{lemma-last-one-flat} we see that", "$\\tau_{\\geq a}P^\\bullet$ is a complex of $R$-flat, finitely", "presented $A$-modules representing $K$." ], "refs": [ "more-algebra-lemma-cone-relatively-perfect", "algebra-lemma-flat-finite-presentation-limit-flat", "more-algebra-lemma-Noetherian-pseudo-coherent", "more-algebra-lemma-base-change-comparison", "more-algebra-lemma-last-one-flat" ], "ref_ids": [ 10288, 1389, 10160, 10139, 10169 ] } ], "ref_ids": [] }, { "id": 10291, "type": "theorem", "label": "more-algebra-lemma-base-change-relatively-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-relatively-perfect", "contents": [ "Let $R \\to A$ be a flat ring map of finite presentation.", "Let $R \\to R'$ be a ring map and set $A' = A \\otimes_R R'$.", "If $K \\in D(A)$ is $R$-perfect, then $K \\otimes_A^\\mathbf{L} A'$ is", "$R'$-perfect." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-pull-pseudo-coherent} we see that", "$K \\otimes_A^\\mathbf{L} A'$ is pseudo-coherent.", "By Lemma \\ref{lemma-base-change-comparison} we see that", "$K \\otimes_A^\\mathbf{L} A'$ is equal to $K \\otimes_R^\\mathbf{L} R'$ in $D(R')$.", "Then we can apply Lemma \\ref{lemma-pull-tor-amplitude}", "to see that $K \\otimes_R^\\mathbf{L} R'$ in $D(R')$ has finite", "tor dimension." ], "refs": [ "more-algebra-lemma-pull-pseudo-coherent", "more-algebra-lemma-base-change-comparison", "more-algebra-lemma-pull-tor-amplitude" ], "ref_ids": [ 10155, 10139, 10180 ] } ], "ref_ids": [] }, { "id": 10292, "type": "theorem", "label": "more-algebra-lemma-compute-RHom-relatively-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-compute-RHom-relatively-perfect", "contents": [ "Let $R \\to A$ be a flat ring map. Let $K, L \\in D(A)$ with $K$", "pseudo-coherent and $L$ finite tor dimension over $R$. We may choose", "\\begin{enumerate}", "\\item a bounded above complex $P^\\bullet$", "of finite free $A$-modules representing $K$, and", "\\item a bounded complex of $R$-flat $A$-modules", "$F^\\bullet$ representing $L$.", "\\end{enumerate}", "Given these choices we have", "\\begin{enumerate}", "\\item[(a)] $E^\\bullet = \\Hom^\\bullet(P^\\bullet, F^\\bullet)$", "is a bounded below complex", "of $R$-flat $A$-modules representing $R\\Hom_A(K, L)$,", "\\item[(b)] for any ring map $R \\to R'$ with $A' = A \\otimes_R R'$", "the complex $E^\\bullet \\otimes_R R'$ represents", "$R\\Hom_{A'}(K \\otimes_A^\\mathbf{L} A', L \\otimes_A^\\mathbf{L} A')$.", "\\end{enumerate}", "If in addition $R \\to A$ is of finite presentation and $L$ is", "$R$-perfect, then we may choose $F^p$ to be finitely presented", "$A$-modules and consequently $E^n$ will be finitely presented $A$-modules", "as well." ], "refs": [], "proofs": [ { "contents": [ "The existence of $P^\\bullet$ is the definition of a pseudo-coherent complex.", "We first represent $L$", "by a bounded above complex $F^\\bullet$ of free $A$-modules", "(this is possible because bounded tor dimension in particular", "implies bounded). Next, say $L$ viewed as an object of $D(R)$", "has tor amplitude in $[a, b]$. Then, after replacing", "$F^\\bullet$ by $\\tau_{\\geq a}F^\\bullet$, we get a complex", "as in (2). This follows from Lemma \\ref{lemma-last-one-flat}.", "\\medskip\\noindent", "Proof of (a).", "Since $F^\\bullet$ is bounded an since $P^\\bullet$ is bounded above,", "we see that $E^n = 0$ for $n \\ll 0$ and that", "$E^n$ is a finite (!) direct sum", "$$", "E^n = \\bigoplus\\nolimits_{p + q = n} \\Hom_A(P^{-q}, F^p)", "$$", "and since $P^{-q}$ is finite free, this is indeed an $R$-flat $A$-module.", "The fact that $E^\\bullet$ represents $R\\Hom_A(K, L)$", "follows from Lemma \\ref{lemma-RHom-out-of-projective}.", "\\medskip\\noindent", "Proof of (b).", "Let $R \\to R'$ be a ring map and $A' = A \\otimes_R R'$.", "By Lemma \\ref{lemma-base-change-comparison} the object", "$L \\otimes_A^\\mathbf{L} A'$ is represented by", "$F^\\bullet \\otimes_R R'$ viewed as a complex of $A'$-modules", "(by flatness of $F^p$ over $R$). Similarly for $P^\\bullet \\otimes_R R'$.", "As above $R\\Hom_{A'}(K \\otimes_A^\\mathbf{L} A', L \\otimes_A^\\mathbf{L} A')$", "is represented by", "$$", "\\Hom^\\bullet(P^\\bullet \\otimes_R R', F^\\bullet \\otimes_R R') =", "E^\\bullet \\otimes_R R'", "$$", "The equality holds by looking at the terms of the complex individually", "and using that $\\Hom_{A'}(P^{-q} \\otimes_R R', F^p \\otimes_R R') =", "\\Hom_A(P^{-q}, F^p) \\otimes_R R'$." ], "refs": [ "more-algebra-lemma-last-one-flat", "more-algebra-lemma-RHom-out-of-projective", "more-algebra-lemma-base-change-comparison" ], "ref_ids": [ 10169, 10207, 10139 ] } ], "ref_ids": [] }, { "id": 10293, "type": "theorem", "label": "more-algebra-lemma-colimit-relatively-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-colimit-relatively-perfect", "contents": [ "Let $R = \\colim_{i \\in I} R_i$ be a filtered colimit of rings.", "Let $0 \\in I$ and $R_0 \\to A_0$ be a flat ring map of", "finite presentation. For $i \\geq 0$ set $A_i = R_i \\otimes_{R_0} A_0$", "and set $A = R \\otimes_{R_0} A_0$.", "\\begin{enumerate}", "\\item Given an $R$-perfect $K$ in $D(A)$ there exists an $i \\in I$", "and an $R_i$-perfect $K_i$ in $D(A_i)$ such that", "$K \\cong K_i \\otimes_{A_i}^\\mathbf{L} A$ in $D(A)$.", "\\item Given $K_0, L_0 \\in D(A_0)$ with $K_0$ pseudo-coherent", "and $L_0$ finite tor dimension over $R_0$, then", "we have", "$$", "\\Hom_{D(A)}(K_0 \\otimes_{A_0}^\\mathbf{L} A, L_0 \\otimes_{A_0}^\\mathbf{L} A) =", "\\colim_{i \\geq 0}", "\\Hom_{D(A_i)}(K_0 \\otimes_{A_0}^\\mathbf{L} A_i,", "L_0 \\otimes_{A_0}^\\mathbf{L} A_i)", "$$", "\\end{enumerate}", "In particular, the triangulated category of $R$-perfect complexes over $A$", "is the colimit of the triangulated categories of", "$R_i$-perfect complexes over $A_i$." ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-colimit-category-fp-modules}", "the category of finitely presented $A$-modules is the colimit of", "the categories of finitely presented $A_i$-modules.", "Given this, Algebra, Lemma", "\\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "tells us that category of $R$-flat, finitely presented $A$-modules", "is the colimit of the categories of", "$R_i$-flat, finitely presented $A_i$-modules.", "Thus the characterization in", "Lemma \\ref{lemma-structure-relatively-perfect}", "proves that (1) is true.", "\\medskip\\noindent", "To prove (2) we choose $P_0^\\bullet$ representing $K_0$ and", "$F_0^\\bullet$ representing $L_0$ as in", "Lemma \\ref{lemma-compute-RHom-relatively-perfect}.", "Then $E_0^\\bullet = \\Hom^\\bullet(P_0^\\bullet, F_0^\\bullet)$", "satisfies", "$$", "H^0(E_0^\\bullet \\otimes_{R_0} R_i) =", "\\Hom_{D(A_i)}(K_0 \\otimes_{A_0}^\\mathbf{L} A_i,", "L_0 \\otimes_{A_0}^\\mathbf{L} A_i)", "$$", "and", "$$", "H^0(E_0^\\bullet \\otimes_{R_0} R) =", "\\Hom_{D(A)}(K_0 \\otimes_{A_0}^\\mathbf{L} A, L_0 \\otimes_{A_0}^\\mathbf{L} A)", "$$", "by the lemma. Thus the result because tensor product commutes", "with colimits and filtered colimits are exact", "(Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact})." ], "refs": [ "algebra-lemma-colimit-category-fp-modules", "algebra-lemma-flat-finite-presentation-limit-flat", "more-algebra-lemma-structure-relatively-perfect", "more-algebra-lemma-compute-RHom-relatively-perfect", "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 1095, 1389, 10290, 10292, 343 ] } ], "ref_ids": [] }, { "id": 10294, "type": "theorem", "label": "more-algebra-lemma-thickening-relatively-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-thickening-relatively-perfect", "contents": [ "Let $R' \\to A'$ be a flat ring map of finite presentation.", "Let $R' \\to R$ be a surjective ring map whose kernel is a nilpotent ideal.", "Set $A = A' \\otimes_{R'} R$. Let $K' \\in D(A')$ and set", "$K = K' \\otimes_{A'}^\\mathbf{L} A$ in $D(A)$.", "If $K$ is $R$-perfect, then $K'$ is $R'$-perfect." ], "refs": [], "proofs": [ { "contents": [ "We can represent $K$ by a bounded above complex of finite free", "$A$-modules $E^\\bullet$, see Lemma \\ref{lemma-pseudo-coherent}.", "By Lemma \\ref{lemma-lift-complex-projectives}", "we conclude that $K'$ is pseudo-coherent because it can be represented", "by a bounded above complex $P^\\bullet$ of finite free $A'$-modules", "with $P^\\bullet \\otimes_{A'} A = E^\\bullet$. Observe that", "this also means $P^\\bullet \\otimes_{R'} R = E^\\bullet$", "(since $A = A' \\otimes_{R'} R$).", "\\medskip\\noindent", "Let $I = \\Ker(R' \\to R)$. Then $I^n = 0$ for some $n$.", "Choose $[a, b]$ such that $K$ has tor amplitude in $[a, b]$", "as a complex of $R$-modules. We will show $K'$ has", "tor amplitude in $[a, b]$. To do this, let $M'$", "be an $R'$-module. If $IM' = 0$, then", "$$", "K' \\otimes_{R'}^\\mathbf{L} M' =", "P^\\bullet \\otimes_{R'} M' =", "E^\\bullet \\otimes_R M' = K \\otimes_R^\\mathbf{L} M'", "$$", "(because $A'$ is flat over $R'$ and $A$ is flat over $R$)", "which has nonzero cohomology only for degrees in $[a, b]$", "by choice of $a, b$.", "If $I^{t + 1}M' = 0$, then we consider the short exact sequence", "$$", "0 \\to IM' \\to M' \\to M'/IM' \\to 0", "$$", "with $M = M'/IM'$. By induction on $t$ we have that both", "$K' \\otimes_{R'}^\\mathbf{L} IM'$ and", "$K' \\otimes_{R'}^\\mathbf{L} M'/IM'$ have nonzero cohomology", "only for degrees in $[a, b]$. Then the distinguished", "triangle", "$$", "K' \\otimes_{R'}^\\mathbf{L} IM' \\to", "K' \\otimes_{R'}^\\mathbf{L} M' \\to", "K' \\otimes_{R'}^\\mathbf{L} M'/IM' \\to", "(K' \\otimes_{R'}^\\mathbf{L} IM')[1]", "$$", "proves the same is true for", "$K' \\otimes_{R'}^\\mathbf{L} M'$.", "This proves the desired bound for all $M'$ and hence the", "desired bound on the tor amplitude of $K'$." ], "refs": [ "more-algebra-lemma-pseudo-coherent", "more-algebra-lemma-lift-complex-projectives" ], "ref_ids": [ 10148, 10229 ] } ], "ref_ids": [] }, { "id": 10295, "type": "theorem", "label": "more-algebra-lemma-lift-from-fibre-relatively-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-from-fibre-relatively-perfect", "contents": [ "Let $R$ be a ring. Let $A = R[x_1, \\ldots, x_d]/I$", "be flat and of finite presentation over $R$.", "Let $\\mathfrak q \\subset A$ be a prime ideal lying over", "$\\mathfrak p \\subset R$. Let $K \\in D(A)$ be pseudo-coherent.", "Let $a, b \\in \\mathbf{Z}$. If", "$H^i(K_\\mathfrak q \\otimes_{R_\\mathfrak p}^\\mathbf{L} \\kappa(\\mathfrak p))$", "is nonzero only for $i \\in [a, b]$, then", "$K_\\mathfrak q$ has tor amplitude in $[a - d, b]$ over $R$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-more-relative-pseudo-coherent-is-moot}", "$K$ is pseudo-coherent as a complex of $R[x_1, \\ldots, x_d]$-modules.", "Therefore we may assume $A = R[x_1, \\ldots, x_d]$.", "Applying Lemma \\ref{lemma-perfect-over-regular-local-ring}", "to $R_\\mathfrak p \\to A_\\mathfrak q$ and the complex $K_\\mathfrak q$", "using our assumption, we find that $K_\\mathfrak q$ is perfect", "in $D(A_\\mathfrak q)$ with tor amplitude in $[a - d, b]$.", "Since $R_\\mathfrak p \\to A_\\mathfrak q$ is flat, we conclude", "by Lemma \\ref{lemma-flat-push-tor-amplitude}." ], "refs": [ "more-algebra-lemma-more-relative-pseudo-coherent-is-moot", "more-algebra-lemma-perfect-over-regular-local-ring", "more-algebra-lemma-flat-push-tor-amplitude" ], "ref_ids": [ 10287, 10246, 10178 ] } ], "ref_ids": [] }, { "id": 10296, "type": "theorem", "label": "more-algebra-lemma-bounded-on-fibres-relatively-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-bounded-on-fibres-relatively-perfect", "contents": [ "Let $R \\to A$ be a ring map which is flat and of finite presentation.", "Let $K \\in D(A)$ be pseudo-coherent. The following are equivalent", "\\begin{enumerate}", "\\item $K$ is $R$-perfect, and", "\\item $K$ is bounded below and for every prime ideal $\\mathfrak p \\subset R$", "the object $K \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p)$ is bounded below.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that (1) implies (2) as an $R$-perfect complex has bounded", "tor dimension as a complex of $R$-modules by definition. Let us prove", "the other implication.", "\\medskip\\noindent", "Write $A = R[x_1, \\ldots, x_d]/I$. Denote $L$ in $D(R[x_1, \\ldots, x_d])$", "the restriction of $K$. By", "Lemma \\ref{lemma-more-relative-pseudo-coherent-is-moot}", "we see that $L$ is pseudo-coherent.", "Since $L$ and $K$ have the same image in $D(R)$ we see that", "$L$ is $R$-perfect if and only if $K$ is $R$-perfect.", "Also $L \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p)$", "and $K \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p)$ are the same objects of", "$D(\\kappa(\\mathfrak p))$. This reduces us to the case", "$A = R[x_1, \\ldots, x_d]$.", "\\medskip\\noindent", "Say $A = R[x_1, \\ldots, x_d]$ and $K$ satisfies (2).", "Let $\\mathfrak q \\subset A$ be a prime lying over a", "prime $\\mathfrak p \\subset R$. By", "Lemma \\ref{lemma-perfect-over-regular-local-ring} applied", "to $R_\\mathfrak p \\to A_\\mathfrak q$ and the complex $K_\\mathfrak q$", "using our assumption, we find that $K_\\mathfrak q$ is perfect", "in $D(A_\\mathfrak q)$. Since $K$ is bounded below, we see that", "$K$ is perfect in $D(A)$ by ", "Lemma \\ref{lemma-check-perfect-stalks}.", "This implies that $K$ is $R$-perfect by", "Lemma \\ref{lemma-perfect-relatively-perfect}", "and the proof is complete." ], "refs": [ "more-algebra-lemma-more-relative-pseudo-coherent-is-moot", "more-algebra-lemma-perfect-over-regular-local-ring", "more-algebra-lemma-check-perfect-stalks", "more-algebra-lemma-perfect-relatively-perfect" ], "ref_ids": [ 10287, 10246, 10243, 10289 ] } ], "ref_ids": [] }, { "id": 10297, "type": "theorem", "label": "more-algebra-lemma-ext-1-zero", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ext-1-zero", "contents": [ "Let $R$ be a ring. Let $K \\in D(R)$ with $H^i(K) = 0$ for", "$i \\not \\in \\{-1, 0\\}$. The following are equivalent", "\\begin{enumerate}", "\\item $H^{-1}(K) = 0$ and $H^0(K)$ is a projective module and", "\\item $\\Ext^1_R(K, M) = 0$ for every $R$-module $M$.", "\\end{enumerate}", "If $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for", "$i = -1, 0$, then these are also equivalent to", "\\begin{enumerate}", "\\item[(3)] $\\Ext^1_R(K, M) = 0$ for every finite $R$-module $M$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows from", "Lemma \\ref{lemma-projective-amplitude}.", "If $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for", "$i = -1, 0$, then $K$ is pseudo-coherent, see", "Lemma \\ref{lemma-Noetherian-pseudo-coherent}.", "Thus the equivalence of (1) and (3) follows from", "Lemma \\ref{lemma-projective-amplitude-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-projective-amplitude", "more-algebra-lemma-Noetherian-pseudo-coherent", "more-algebra-lemma-projective-amplitude-pseudo-coherent" ], "ref_ids": [ 10187, 10160, 10244 ] } ], "ref_ids": [] }, { "id": 10298, "type": "theorem", "label": "more-algebra-lemma-represent-two-term-complex", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-represent-two-term-complex", "contents": [ "Let $R$ be a ring. Let $K$ be an object of $D(R)$ with $H^i(K) = 0$", "for $i \\not \\in \\{-1, 0\\}$. Then", "\\begin{enumerate}", "\\item $K$ can be represented by a two term complex", "$K^{-1} \\to K^0$ with $K^0$ a free module, and", "\\item if $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for", "$i = -1, 0$, then $K$ can be represented by a two term complex", "$K^{-1} \\to K^0$ with $K^0$ a finite free module and $K^{-1}$ finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Suppose $K$ is given by the complex of modules $M^\\bullet$.", "We may first replace $M^\\bullet$ by $\\tau_{\\leq 0}M^\\bullet$. Thus we", "may assume $M^i = 0$ for $i > 0$, Next, we may choose a", "free resolution $P^\\bullet \\to M^\\bullet$ with $P^i = 0$ for $i > 0$, see", "Derived Categories, Lemma \\ref{derived-lemma-subcategory-left-resolution}.", "Finally, we can set $K^\\bullet = \\tau_{\\geq -1}P^\\bullet$.", "\\medskip\\noindent", "Proof of (2). Assume $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for", "$i = -1, 0$. By Lemma \\ref{lemma-pseudo-coherent} we can choose a", "quasi-isomorphism $F^\\bullet \\to M^\\bullet$ with $F^i = 0$ for $i > 0$", "and $F^i$ finite free. Then we can set $K^\\bullet = \\tau_{\\geq -1}F^\\bullet$." ], "refs": [ "derived-lemma-subcategory-left-resolution", "more-algebra-lemma-pseudo-coherent" ], "ref_ids": [ 1835, 10148 ] } ], "ref_ids": [] }, { "id": 10299, "type": "theorem", "label": "more-algebra-lemma-map-out-of-almost-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-map-out-of-almost-free", "contents": [ "Let $R$ be a ring. Let $M^\\bullet$ be a complex of modules over $R$", "with $M^i = 0$ for $i > 0$ and $M^0$ a projective $R$-module.", "Let $K^\\bullet$ be a second complex.", "\\begin{enumerate}", "\\item Assume $K^i = 0$ for $i \\leq -2$. Then", "$\\Hom_{D(R)}(M^\\bullet, K^\\bullet) = \\Hom_{K(R)}(M^\\bullet, K^\\bullet)$.", "\\item Assume $K^i = 0$ for $i \\not \\in [-1, 0]$ and", "$K^0$ a projective $R$-module. Then for a map of complexes", "$a^\\bullet : M^\\bullet \\to K^\\bullet$, the following are equivalent", "\\begin{enumerate}", "\\item $a^\\bullet$ induces the zero map $\\Ext^1_R(K^\\bullet, N) \\to", "\\Ext^1_R(M^\\bullet, N)$ for all $R$-modules $N$, and", "\\item there is a map $h^0 : M^0 \\to K^{-1}$ such that", "$a^{-1} + h^0 \\circ d^{-1}_K = 0$.", "\\end{enumerate}", "\\item Assume $K^i = 0$ for $i \\leq -3$. Let", "$\\alpha \\in \\Hom_{D(R)}(M^\\bullet, K^\\bullet)$. If the", "composition of $\\alpha$ with", "$K^\\bullet \\to K^{-2}[2]$ comes from an $R$-module map", "$a : M^{-2} \\to K^{-2}$ with $a \\circ d_M^{-3} = 0$, then", "$\\alpha$ can be represented by a map of complexes", "$a^\\bullet : M^\\bullet \\to K^\\bullet$ with $a^{-2} = a$.", "\\item In (2) for any second map of complexes", "$(a')^\\bullet : M^\\bullet \\to K^\\bullet$", "representing $\\alpha$ with $a = (a')^{-2}$", "there exist $h^i : M^i \\to K^{i - 1}$ for $i = 0, -1$ such that", "$$", "h^{-1} \\circ d_M^{-2} = 0, \\quad", "(a')^{-1} = a^{-1} + d_K^{-2} \\circ h^{-1} + h^0 \\circ d_M^{-1},\\quad", "(a')^0 = a^0 + d_K^{-1} \\circ h^0", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Set $F^0 = M^0$.", "Choose a free $R$-module $F^{-1}$ and a surjection $F^{-1} \\to M^{-1}$.", "Choose a free $R$-module $F^{-2}$ and a surjection", "$F^{-2} \\to M^{-2} \\times_{M^{-1}} F^{-1}$. Continuing in this", "way we obtain a quasi-isomorphism $p^\\bullet : F^\\bullet \\to M^\\bullet$", "which is termwise surjective and with $F^i$ projective for all $i$.", "\\medskip\\noindent", "Proof of (1). By", "Derived Categories, Lemma \\ref{derived-lemma-morphisms-from-projective-complex}", "we have", "$$", "\\Hom_{D(R)}(M^\\bullet, K^\\bullet) = \\Hom_{K(R)}(F^\\bullet, K^\\bullet)", "$$", "If $K^i = 0$ for $i \\leq -2$, then any morphism of complexes", "$F^\\bullet \\to K^\\bullet$ factors through $p^\\bullet$. Similarly, any", "homotopy $\\{h^i : F^i \\to K^{i - 1}\\}$ factors through $p^\\bullet$.", "Thus (1) holds.", "\\medskip\\noindent", "Proof of (2). If (2)(b) holds, then $a^\\bullet$ is homotopic to a map", "of complexes $(a')^\\bullet : M^\\bullet \\to K^\\bullet$", "which is zero in degree $-1$.", "On the other hand, let $N \\to I^\\bullet$ be an injective resolution.", "We have", "$$", "\\Ext^1_R(K^\\bullet, N) = \\Hom_{D(R)}(K^\\bullet, I^\\bullet[1]) =", "\\Hom_{K(R)}(K^\\bullet, I^\\bullet[1])", "$$", "by Derived Categories, Lemma", "\\ref{derived-lemma-morphisms-into-injective-complex}.", "Let $b^\\bullet : K^\\bullet \\to I^\\bullet[1]$ be a map of complexes.", "Since $K^1 = 0$ the map $b^0 : K^0 \\to I^1$ maps into the kernel of", "$I^1 \\to I^2$ which is the image of $I^0 \\to I^1$. Since $K^0$ is projective", "we can lift $b^0$ to a map $h : K^0 \\to I^0$. Thus we see that $b^\\bullet$", "is homotopic to a map of complexes $(b')^\\bullet$ with $(b')^0 = 0$. Since", "$K^i = 0$ for $i \\not \\in [-1, 0]$ it follows that", "$(b')^\\bullet \\circ (a')^\\bullet = 0$ as a map of complexes.", "Hence the map $\\Ext^1_R(K^\\bullet, N) \\to", "\\Ext^1_R(M^\\bullet, N)$ is zero. In this way we see that (2)(b)", "implies (2)(a). Conversely, assume (2)(a). We see that the", "canonical element in $\\Ext^1_R(K^\\bullet, K^{-1})$ maps to", "zero in $\\Ext^1_R(M^\\bullet, K^{-1})$. Using (1) we", "see immediately that we get a map $h^0$ as in (2)(b).", "\\medskip\\noindent", "Proof of (3). Choose $b^\\bullet : F^\\bullet \\to K^\\bullet$ representing", "$\\alpha$. The composition of $\\alpha$ with $K^\\bullet \\to K^{-2}[2]$ is", "represented by $b^{-2} : F^{-2} \\to K^{-2}$. As this is homotopic to", "$a \\circ p^{-2} : F^{-2} \\to M^{-2} \\to K^{-2}$, there is a map", "$h : F^{-1} \\to K^{-2}$ such that $b^{-2} = a \\circ p^{-2} + h \\circ d_F^{-2}$.", "Adjusting $b^\\bullet$ by $h$ viewed as a homotopy from $F^\\bullet$", "to $K^\\bullet$, we find that $b^{-2} = a \\circ p^{-2}$. Hence $b^{-2}$", "factors through $p^{-2}$. Since $F^0 = M^0$ the kernel of $p^{-2}$", "surjects onto the kernel of $p^{-1}$ (for example because the kernel", "of $p^\\bullet$ is an acyclic complex or by a diagram chase). Hence $b^{-1}$", "necessarily factors through $p^{-1}$ as well and we see that (3)", "holds for these factorizations and $a^0 = b^0$.", "\\medskip\\noindent", "Proof of (4) is omitted. Hint: There is a homotopy between", "$a^\\bullet \\circ p^\\bullet$ and $(a')^\\bullet \\circ p^\\bullet$", "and we argue as before that this homotopy factors through $p^\\bullet$." ], "refs": [ "derived-lemma-morphisms-from-projective-complex", "derived-lemma-morphisms-into-injective-complex" ], "ref_ids": [ 1862, 1855 ] } ], "ref_ids": [] }, { "id": 10300, "type": "theorem", "label": "more-algebra-lemma-ext-1-annihilated-definite", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ext-1-annihilated-definite", "contents": [ "Let $R$ be a ring and let $I \\subset R$ be an ideal.", "Let $K \\in D(R)$. Assume $H^i(K) = 0$ for $i \\not \\in \\{-1, 0\\}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\Ext^1_R(K, N)$ is annihilated by $I$ for all $R$-modules $N$,", "\\item $K$ can be represented by a complex $K^{-1} \\to K^0$", "with $K^0$ free such that for any $a \\in I$ the map", "$a : K^{-1} \\to K^{-1}$ factors through $d_K^{-1} : K^{-1} \\to K^0$,", "\\item whenever $K$ is represented by a two term complex", "$K^{-1} \\to K^0$ with $K^0$ projective, then for any $a \\in I$ the map", "$a : K^{-1} \\to K^{-1}$ factors through $d_K^{-1} : K^{-1} \\to K^0$.", "\\end{enumerate}", "If $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for $i = -1, 0$,", "then these are also equivalent to", "\\begin{enumerate}", "\\item[(4)] $\\Ext^1_R(K, N)$ is annihilated by $I$ for every finite", "$R$-module $N$,", "\\item[(5)] $K$ can be represented by a complex $K^{-1} \\to K^0$", "with $K^0$ finite free and $K^{-1}$ finite such that for any $a \\in I$ the map", "$a : K^{-1} \\to K^{-1}$ factors through $d_K^{-1} : K^{-1} \\to K^0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "\\medskip\\noindent", "Assume (1) and let $K^{-1} \\to K^0$", "be a two term complex representing $K$", "with $K^0$ projective.", "We will use the description of maps in $D(R)$ out of $K^\\bullet$ given in", "Lemma \\ref{lemma-map-out-of-almost-free} without further mention.", "Choosing $N = K^{-1}$ consider the element $\\xi$ of $\\Ext^1_R(K, N)$", "given by $\\text{id}_{K^{-1}} : K^{-1} \\to K^{-1}$. Since is annihilated", "by $a \\in I$ we see that we get the dotted arrow fitting into the", "following commutative diagram", "$$", "\\xymatrix{", "K^{-1} \\ar[d]_a \\ar[r]_{d_K^{-1}} & K^0 \\ar@{..>}[ld]^h \\\\", "K^{-1}", "}", "$$", "This proves that (3) holds. Part (3) implies (2) in view of", "Lemma \\ref{lemma-represent-two-term-complex} part (1). Assume", "$K^\\bullet$ is as in (2) and $N$ is an arbitrary $R$-module.", "Any element $\\xi$ of $\\Ext^1_R(K, N)$ is given as the class of", "a map $\\varphi : K^{-1} \\to N$. Then for $a \\in I$ by assumption", "we may choose a map $h$ as in the diagram above and we see that", "$a\\varphi = \\varphi \\circ a = \\varphi \\circ h \\circ \\text{d}_K^{-1}$", "which proves that $a \\xi$ is zero in $\\Ext^1_R(K, N)$.", "Thus (1), (2), and (3) are equivalent.", "\\medskip\\noindent", "Assume $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for $i = -1, 0$.", "Part (3) implies (5) in view of", "Lemma \\ref{lemma-represent-two-term-complex} part (2).", "It is clear that (5) implies (2). Trivially (1) implies (4).", "Thus to finish the proof it suffices to show that (4) implies", "any of the other conditions. Let $K^{-1} \\to K^0$ be a complex", "representing $K$ with $K^0$ finite free and $K^{-1}$ finite as in", "Lemma \\ref{lemma-represent-two-term-complex} part (2). The argument", "given in the proof of (2) $\\Rightarrow$ (1)", "shows that if $\\Ext^1_R(K, K^{-1})$ is annihilated by $I$,", "then (1) holds. In this way we see that (4) implies (1) and the", "proof is complete." ], "refs": [ "more-algebra-lemma-map-out-of-almost-free", "more-algebra-lemma-represent-two-term-complex", "more-algebra-lemma-represent-two-term-complex", "more-algebra-lemma-represent-two-term-complex" ], "ref_ids": [ 10299, 10298, 10298, 10298 ] } ], "ref_ids": [] }, { "id": 10301, "type": "theorem", "label": "more-algebra-lemma-two-term-base-change", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-two-term-base-change", "contents": [ "Let $R$ be a ring. Let $K$ be an object of $D(R)$ with $H^i(K) = 0$", "for $i \\not \\in \\{-1, 0\\}$. Let $K^{-1} \\to K^0$ be a two term complex", "of $R$-modules representing $K$ such that $K^0$ is a flat $R$-module", "(for example projective or free). Let $R \\to R'$ be a ring map.", "Then the complex $K^\\bullet \\otimes_R R'$ represents", "$\\tau_{\\geq -1}(K \\otimes_R^\\mathbf{L} R')$." ], "refs": [], "proofs": [ { "contents": [ "We have a distinguished triangle", "$$", "K^0 \\to K^\\bullet \\to K^{-1}[1] \\to K^0[1]", "$$", "in $D(R)$. This determines a map of distinguished triangles", "$$", "\\xymatrix{", "K^0 \\otimes_R^\\mathbf{L} R' \\ar[d] \\ar[r] &", "K^\\bullet \\otimes_R^\\mathbf{L} R' \\ar[r] \\ar[d] &", "K^{-1} \\otimes_R^\\mathbf{L} R'[1] \\ar[r] \\ar[d] &", "K^0 \\otimes_R^\\mathbf{L} R'[1] \\ar[d] \\\\", "K^0 \\otimes_R R' \\ar[r] &", "K^\\bullet \\otimes_R R' \\ar[r] &", "K^{-1} \\otimes_R R'[1] \\ar[r] &", "K^0 \\otimes_R R'[1]", "}", "$$", "The left and right vertical arrows are isomorphisms as $K^0$ is flat.", "Since $K^{-1} \\otimes_R^\\mathbf{L} R' \\to K^{-1} \\otimes_R R'$", "is an isomorphism on cohomology in degree $0$ we conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10302, "type": "theorem", "label": "more-algebra-lemma-base-change-property-ext-1-annihilated", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-property-ext-1-annihilated", "contents": [ "Let $I$ be an ideal of a ring $R$. Let $K$ be an object of $D(R)$ with", "$H^i(K) = 0$ for $i \\not \\in \\{-1, 0\\}$. Let $R \\to R'$ be a ring map.", "If $K$ satisfies the equivalent conditions (1), (2), and (3)", "of Lemma \\ref{lemma-ext-1-annihilated-definite} with respect to $(R, I)$,", "then $\\tau_{\\geq -1}(K \\otimes_R^\\mathbf{L} R')$", "satisfies the equivalent conditions (1), (2), and (3)", "of Lemma \\ref{lemma-ext-1-annihilated-definite} with respect to $(R', IR')$" ], "refs": [ "more-algebra-lemma-ext-1-annihilated-definite", "more-algebra-lemma-ext-1-annihilated-definite" ], "proofs": [ { "contents": [ "We may assume $K$ is represented by a two term complex $K^{-1} \\to K^0$", "with $K^0$ free such that for any $a \\in I$ the map $a : K^{-1} \\to K^{-1}$", "is equal to $h_a \\circ d_K^{-1}$ for some map $h_a : K^0 \\to K^{-1}$.", "By Lemma \\ref{lemma-two-term-base-change} we see that", "$\\tau_{\\geq -1}(K \\otimes_R^\\mathbf{L} R')$ is represented by", "$K^\\bullet \\otimes_R R'$. Then of course for every $a \\in I$", "we see that $a \\otimes 1 : K^{-1} \\otimes_R R' \\to K^{-1} \\otimes_R R'$", "is equal to $(h_a \\otimes 1) \\circ (d_K^{-1} \\otimes 1)$.", "Since the collection of maps $K^{-1} \\otimes_R R' \\to K^{-1} \\otimes_R R'$", "which factor through $\\text{d}_K^{-1} \\otimes 1$", "forms an $R'$-module we conclude." ], "refs": [ "more-algebra-lemma-two-term-base-change" ], "ref_ids": [ 10301 ] } ], "ref_ids": [ 10300, 10300 ] }, { "id": 10303, "type": "theorem", "label": "more-algebra-lemma-two-term-surjection-map-zero", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-two-term-surjection-map-zero", "contents": [ "Let $R$ be a ring. Let $\\alpha : K \\to K'$ be a morphism of $D(R)$. Assume", "\\begin{enumerate}", "\\item $H^i(K) = H^i(K') = 0$ for $i \\not \\in \\{-1, 0\\}$", "\\item $H^0(\\alpha)$ is an isomorphism and $H^{-1}(\\alpha)$ is surjective.", "\\end{enumerate}", "For any $f \\in R$ if $f : K \\to K$ is $0$, then $f : K' \\to K'$ is $0$." ], "refs": [], "proofs": [ { "contents": [ "Set $M = \\Ker(H^{-1}(\\alpha))$. Then $\\alpha$ fits into a distinguished", "triangle", "$$", "M[1] \\to K \\to K' \\to M[2]", "$$", "Since $K \\to K' \\xrightarrow{f} K'$ is zero by our assumption, we see", "that $f : K' \\to K'$ factors over a map $M[2] \\to K'$. However", "$\\Hom(M[2], K') = 0$ for example by", "Derived Categories, Lemma \\ref{derived-lemma-negative-exts}." ], "refs": [ "derived-lemma-negative-exts" ], "ref_ids": [ 1893 ] } ], "ref_ids": [] }, { "id": 10304, "type": "theorem", "label": "more-algebra-lemma-surjection-property-ext-1-annihilated", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-surjection-property-ext-1-annihilated", "contents": [ "Let $I$ be an ideal of a ring $R$. Let $\\alpha : K \\to K'$", "be a morphism of $D(R)$. Assume", "\\begin{enumerate}", "\\item $H^i(K) = H^i(K') = 0$ for $i \\not \\in \\{-1, 0\\}$", "\\item $H^0(\\alpha)$ is an isomorphism and $H^{-1}(\\alpha)$ is surjective.", "\\end{enumerate}", "If $K$ satisfies the equivalent conditions (1), (2), and (3)", "of Lemma \\ref{lemma-ext-1-annihilated-definite},", "then $K'$ does too." ], "refs": [ "more-algebra-lemma-ext-1-annihilated-definite" ], "proofs": [ { "contents": [ "Set $M = \\Ker(H^{-1}(\\alpha))$. Then $\\alpha$ fits into a distinguished", "trangle", "$$", "M[1] \\to K \\to K' \\to M[2]", "$$", "For any $R$-module $N$ this determines an exact sequence", "$$", "\\Ext^0_R(M[1], N) \\to", "\\Ext^1_R(K', N) \\to", "\\Ext^1_R(K, N)", "$$", "Since $\\Ext^0_R(M[1], N) = \\Ext^{-1}_R(M, N) = 0$ we see that", "$\\Ext^1_R(K', N)$ is a submodule of $\\Ext^1_R(K, N)$. Hence if", "$\\Ext^1_R(K, N)$ is annihilated by $I$ so is $\\Ext^1_R(K', N)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10300 ] }, { "id": 10305, "type": "theorem", "label": "more-algebra-lemma-ext-1-annihilated", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ext-1-annihilated", "contents": [ "Let $R$ be ring and let $I \\subset R$ be an ideal.", "Let $K \\in D(R)$ with $H^i(K) = 0$ for $i \\not \\in \\{-1, 0\\}$.", "The following are equivalent", "\\begin{enumerate}", "\\item there exists a $c \\geq 0$ such that the equivalent", "conditions (1), (2), (3) of Lemma \\ref{lemma-ext-1-annihilated-definite}", "hold for $K$ and the ideal $I^c$,", "\\item there exists a $c \\geq 0$ such that (a) $I^c$ annihilates", "$H^{-1}(K)$ and (b) $H^0(K)$ is an $I^c$-projective module (see", "Section \\ref{section-near-projective}).", "\\end{enumerate}", "If $R$ is Noetherian and $H^i(K)$ is a finite $R$-module", "for $i = -1, 0$, then these are also equivalent to", "\\begin{enumerate}", "\\item[(3)] there exists a $c \\geq 0$ such that the equivalent", "conditions (4), (5) of Lemma \\ref{lemma-ext-1-annihilated-definite}", "hold for $K$ and the ideal $I^c$,", "\\item[(4)] $H^{-1}(K)$ is $I$-power torsion and there exist", "$f_1, \\ldots, f_s \\in R$ with $V(f_1, \\ldots, f_s) \\subset V(I)$", "such that the localizations $H^0(K)_{f_i}$ are projective", "$R_{f_i}$-modules,", "\\item[(5)] $H^{-1}(K)$ is $I$-power torsion and there exist", "$f_1, \\ldots, f_s \\in I$ with $V(f_1, \\ldots, f_s) = V(I)$", "such that the localizations $H^0(K)_{f_i}$ are projective", "$R_{f_i}$-modules.", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-ext-1-annihilated-definite", "more-algebra-lemma-ext-1-annihilated-definite" ], "proofs": [ { "contents": [ "The distinguished triangle $H^{-1}(K)[1] \\to K \\to H^0(K)[0] \\to H^{-1}(K)[2]$", "determines an exact sequence", "$$", "0 \\to \\Ext^1_R(H^0(K), N) \\to \\Ext^1_R(K, N) \\to \\Hom_R(H^{-1}(K), N) \\to", "\\Ext^2_R(H^0(K), N)", "$$", "Thus (2) implies that $I^{2c}$ annihilates $\\Ext^1_R(K, N)$ for every", "$R$-module $N$. Assuming (1) we immediately see that $H^0(K)$ is", "$I^c$-projective. On the other hand, we may choose an injective map", "$H^{-1}(K) \\to N$ for some injective $R$-module $N$. Then this map", "is the image of an element of $\\Ext^1_R(K, N)$ by the vanishing", "of the $\\Ext^2$ in the sequence and we conclude $H^{-1}(K)$", "is annihilated by $I^c$.", "\\medskip\\noindent", "Assume $R$ is Noetherian and $H^i(K)$ is a finite $R$-module", "for $i = -1, 0$. By Lemma \\ref{lemma-ext-1-annihilated-definite}", "we see that (3) is equivalent to (1) and (2). Also, if (3)", "holds then for $f \\in I$ the multiplication by $f$ on", "$H^0(K)$ factors through a projective module, which implies", "that $H^0(K)_f$ is a summand of a projective $R_f$-module and hence", "itself a projective $R_f$-module. Choosing $f_1, \\ldots, f_s$", "to be generators of $I$ we find", "the equivalent conditions (1), (2), and (3) imply (5).", "Of course (5) trivially implies (4).", "\\medskip\\noindent", "Assume (4). Since $H^{-1}(K)$ is a finite $R$-module and $I$-power torsion", "we see that $I^{c_1}$ annihilates $H^{-1}(K)$ for some $c_1 \\geq 0$.", "Choose a short exact sequence", "$$", "0 \\to M \\to R^{\\oplus r} \\to H^0(K) \\to 0", "$$", "which determines an element $\\xi \\in \\Ext^1_R(H^0(K), M)$.", "For any $f \\in I$ we have $\\Ext^1_R(H^0(K), M)_f = \\Ext^1_{R_f}(H^0(K)_f, M_f)$", "by Lemma \\ref{lemma-pseudo-coherence-and-base-change-ext}.", "Hence if $H^0(K)_f$ is projective, then a power of $f$ annihilates $\\xi$.", "We conclude that $\\xi$ is annihilated by $(f_1, \\ldots, f_s)^{c_2}$", "for some $c_2 \\geq 0$. Since $V(f_1, \\ldots, f_s) \\subset V(I)$ we have", "$\\sqrt{I} \\subset (f_1, \\ldots, f_s)$", "(Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}).", "Since $R$ is Noetherian we find $I^{c_3} \\subset (f_1, \\ldots, f_s)$", "for some $c_3 \\geq 0$ (Algebra, Lemma \\ref{algebra-lemma-Noetherian-power}).", "Hence $I^{c2c3}$ annihilates $\\xi$.", "This in turn says that $H^0(K)$ is $I^{c_2c_3}$-projective (as multiplication", "by $a \\in I$ which annihilate $\\xi$ factor through $R^{\\oplus r}$).", "Hence taking $c = \\max(c_1, c_2c_3)$ we see that (2) holds." ], "refs": [ "more-algebra-lemma-ext-1-annihilated-definite", "more-algebra-lemma-pseudo-coherence-and-base-change-ext", "algebra-lemma-Zariski-topology", "algebra-lemma-Noetherian-power" ], "ref_ids": [ 10300, 10165, 389, 460 ] } ], "ref_ids": [ 10300, 10300 ] }, { "id": 10306, "type": "theorem", "label": "more-algebra-lemma-zero-in-derived", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-zero-in-derived", "contents": [ "Let $R$ be a ring. Let $K_j \\in D(R)$, $j = 1, 2, 3$ with $H^i(K_j) = 0$", "for $i \\not \\in \\{-1, 0\\}$. Let $\\varphi : K_1 \\to K_2$ and", "$\\psi : K_2 \\to K_3$ be maps in $D(R)$.", "If $H^0(\\varphi) = 0$ and $H^{-1}(\\psi) = 0$, then", "$\\varphi \\circ \\psi = 0$." ], "refs": [], "proofs": [ { "contents": [ "Apply Derived Categories, Lemma \\ref{derived-lemma-trick-vanishing-composition}", "to see that $\\varphi \\circ \\psi$ factors through $\\tau_{\\leq -2}K_2 = 0$." ], "refs": [ "derived-lemma-trick-vanishing-composition" ], "ref_ids": [ 1817 ] } ], "ref_ids": [] }, { "id": 10307, "type": "theorem", "label": "more-algebra-lemma-silly", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-silly", "contents": [ "Let $R$ be a ring. Let $K \\in D(R)$ be given by a two term complex", "of the form $R^{\\oplus n} \\to R^{\\oplus n}$. Denote", "$A \\in \\text{Mat}(n \\times n, R)$ the matrix of the differential.", "Then $\\det(a) : K \\to K$ is zero in $D(R)$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Good exercise." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10308, "type": "theorem", "label": "more-algebra-lemma-tensor-NL", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-tensor-NL", "contents": [ "Let $R \\to S$ and $S \\to S'$ be ring maps. The canonical map", "$\\NL_{S/R} \\otimes_S^\\mathbf{L} S' \\to \\NL_{S/R} \\otimes_S S'$", "induces an isomorphism", "$\\tau_{\\geq -1}(\\NL_{S/R} \\otimes_S^\\mathbf{L} S') \\to \\NL_{S/R} \\otimes_S S'$", "in $D(S')$. Similarly, given a presentation $\\alpha$ of $S$ over $R$", "the canonical map", "$\\NL(\\alpha) \\otimes_S^\\mathbf{L} S' \\to \\NL(\\alpha) \\otimes_S S'$", "induces an isomorphism $\\tau_{\\geq -1}(\\NL(\\alpha) \\otimes_S^\\mathbf{L} S') \\to", "\\NL(\\alpha) \\otimes_S S'$ in $D(S')$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-two-term-base-change}." ], "refs": [ "more-algebra-lemma-two-term-base-change" ], "ref_ids": [ 10301 ] } ], "ref_ids": [] }, { "id": 10309, "type": "theorem", "label": "more-algebra-lemma-base-change-NL", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-NL", "contents": [ "Let $R \\to S$ and $R \\to R'$ be ring maps.", "Let $\\alpha : P \\to S$ be a presentation of $S$ over $R$.", "Then $\\alpha' : P \\otimes_R R' \\to S \\otimes_R R'$ is a", "presentation of $S' = S \\otimes_R R'$ over $R'$.", "The canonical map", "$$", "NL(\\alpha) \\otimes_S S' \\to \\NL(\\alpha')", "$$", "is an isomorphism on $H^0$ and surjective on $H^{-1}$. In particular,", "the canonical map", "$$", "\\NL_{S/R} \\otimes_S S' \\to \\NL_{S'/R'}", "$$", "is an isomorphism on $H^0$ and surjective on $H^{-1}$." ], "refs": [], "proofs": [ { "contents": [ "Denote $I = \\Ker(P \\to S)$. Denote $P' = P \\otimes_R R'$ and", "$I' = \\Ker(P' \\to S')$. Suppose $P$ is a polynomial algebra", "on $x_j$ for $j \\in J$. The map displayed in the lemma becomes", "$$", "\\xymatrix{", "\\bigoplus_{j \\in J} S' \\text{d}x_j \\ar[r] &", "\\bigoplus_{j \\in J} S' \\text{d}x_j \\\\", "I/I^2 \\otimes_S S' \\ar[r] \\ar[u] &", "I'/(I')^2 \\ar[u]", "}", "$$", "where the left column is $\\NL(\\alpha) \\otimes_S S'$ and the right", "column is $\\NL(\\alpha')$. By right exactness of tensor product we", "see that $I \\otimes_R R' \\to I'$ is surjective. Hence the bottom arrow", "is a surjection. This proves the first statement of the lemma.", "The statement for $\\NL_{S/R} \\otimes_S S' \\to \\NL_{S'/R'}$", "follows as these complexes are homotopic to $\\NL(\\alpha) \\otimes_S S'$ and", "$\\NL(\\alpha')$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10310, "type": "theorem", "label": "more-algebra-lemma-base-change-NL-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-NL-flat", "contents": [ "Consider a cocartesian diagram of rings", "$$", "\\xymatrix{", "B \\ar[r] & B' \\\\", "A \\ar[r] \\ar[u] & A' \\ar[u]", "}", "$$", "If $B$ is flat over $A$, then the canonical map", "$\\NL_{B/A} \\otimes_B B' \\to \\NL_{B'/A'}$ is a quasi-isomorphism.", "If in addition $\\NL_{B/A}$ has tor-amplitude in $[-1, 0]$", "then $\\NL_{B/A} \\otimes_B^\\mathbf{L} B' \\to \\NL_{B'/A'}$", "is a quasi-isomorphism too." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $\\alpha : P \\to B$ as in", "Algebra, Section \\ref{algebra-section-netherlander}.", "Let $I = \\Ker(\\alpha)$. Set $P' = P \\otimes_A A'$ and denote", "$\\alpha' : P' \\to B'$ the corresponding presentation of $B'$ over $A'$.", "As $B$ is flat over $A$ we see that $I' = \\Ker(\\alpha')$ is equal", "to $I \\otimes_A A'$. Hence", "$$", "I'/(I')^2 = \\Coker(I^2 \\otimes_A A' \\to I \\otimes_A A') =", "I/I^2 \\otimes_A A' = I/I^2 \\otimes_B B'", "$$", "We have $\\Omega_{P'/A'} = \\Omega_{P/A} \\otimes_A A'$", "because both sides have the same basis. It follows that", "$\\Omega_{P'/A'} \\otimes_{P'} B' = \\Omega_{P/A} \\otimes_P B \\otimes_B B'$.", "This proves that $\\NL(\\alpha) \\otimes_B B' \\to \\NL(\\alpha')$", "is an isomorphism of complexes and hence the first statement holds.", "\\medskip\\noindent", "We have", "$$", "\\NL(\\alpha) = I/I^2 \\longrightarrow \\Omega_{P/A} \\otimes_P B", "$$", "as a complex of $B$-modules with $I/I^2$ placed in degree $-1$.", "Since the term in degree $0$ is free, this complex has tor-amplitude", "in $[-1, 0]$ if and only if $I/I^2$ is a flat $B$-module, see", "Lemma \\ref{lemma-last-one-flat}.", "If this holds, then $\\NL(\\alpha) \\otimes_B^\\mathbf{L} B' =", "\\NL(\\alpha) \\otimes_B B'$ and we get the second statement." ], "refs": [ "more-algebra-lemma-last-one-flat" ], "ref_ids": [ 10169 ] } ], "ref_ids": [] }, { "id": 10311, "type": "theorem", "label": "more-algebra-lemma-lci-NL", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lci-NL", "contents": [ "Let $A \\to B$ be a local complete intersection as in", "Definition \\ref{definition-local-complete-intersection}.", "Then $\\NL_{B/A}$ is a perfect object of", "$D(B)$ with tor amplitude in $[-1, 0]$." ], "refs": [ "more-algebra-definition-local-complete-intersection" ], "proofs": [ { "contents": [ "Write $B = A[x_1, \\ldots, x_n]/I$. Then $\\NL_{B/A}$ is represented by", "the complex", "$$", "I/I^2 \\longrightarrow \\bigoplus B \\text{d}x_i", "$$", "of $B$-modules with $I/I^2$ placed in degree $-1$. Since the term in", "degree $0$ is finite free, this complex has tor-amplitude in $[-1, 0]$ if and", "only if $I/I^2$ is a flat $B$-module, see", "Lemma \\ref{lemma-last-one-flat}. By definition $I$ is a Koszul regular", "ideal and hence a quasi-regular ideal, see Section \\ref{section-ideals}.", "Thus $I/I^2$ is a finite projective $B$-module", "(Lemma \\ref{lemma-quasi-regular-ideal-finite-projective})", "and we conclude both that $\\NL_{B/A}$ is perfect and that it has tor amplitude", "in $[-1, 0]$." ], "refs": [ "more-algebra-lemma-last-one-flat", "more-algebra-lemma-quasi-regular-ideal-finite-projective" ], "ref_ids": [ 10169, 9996 ] } ], "ref_ids": [ 10609 ] }, { "id": 10312, "type": "theorem", "label": "more-algebra-lemma-base-change-lci-bis", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-lci-bis", "contents": [ "Consider a cocartesian diagram of rings", "$$", "\\xymatrix{", "B \\ar[r] & B' \\\\", "A \\ar[r] \\ar[u] & A' \\ar[u]", "}", "$$", "If $A \\to B$ and $A' \\to B'$ are local complete intersections as in", "Definition \\ref{definition-local-complete-intersection}, then", "the kernel of $H^{-1}(\\NL_{B/A} \\otimes_B B') \\to H^{-1}(\\NL_{B'/A'})$", "is a finite projective $B'$-module." ], "refs": [ "more-algebra-definition-local-complete-intersection" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-lci-NL} the complexes $\\NL_{B/A}$ and $\\NL_{B'/A'}$", "are perfect of tor-amplitude in $[-1, 0]$.", "Combining Lemmas \\ref{lemma-tensor-NL}, \\ref{lemma-pull-perfect}, and", "\\ref{lemma-pull-tor-amplitude} we have", "$\\NL_{B/A} \\otimes_B B' = \\NL_{B/A} \\otimes_B^\\mathbf{L} B'$", "and this complex is also perfect of tor-amplitude in $[-1, 0]$.", "Choose a distinguished triangle", "$$", "C \\to \\NL_{B/A} \\otimes_B B' \\to \\NL_{B'/A'} \\to C[1]", "$$", "in $D(B')$. By Lemmas \\ref{lemma-two-out-of-three-perfect} and", "\\ref{lemma-cone-tor-amplitude} we conclude that $C$ is perfect", "with tor-amplitude in $[-1, 1]$. By Lemma \\ref{lemma-base-change-NL}", "the complex $C$ has only one nonzero cohomology module, namely the module", "of the lemma sitting in degree $-1$. This module is of finite presentation", "(Lemma \\ref{lemma-n-pseudo-module}) and flat", "(Lemma \\ref{lemma-tor-dimension}). Hence it is finite projective by", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}." ], "refs": [ "more-algebra-lemma-lci-NL", "more-algebra-lemma-tensor-NL", "more-algebra-lemma-pull-perfect", "more-algebra-lemma-pull-tor-amplitude", "more-algebra-lemma-two-out-of-three-perfect", "more-algebra-lemma-cone-tor-amplitude", "more-algebra-lemma-base-change-NL", "more-algebra-lemma-n-pseudo-module", "more-algebra-lemma-tor-dimension", "algebra-lemma-finite-projective" ], "ref_ids": [ 10311, 10308, 10219, 10180, 10214, 10172, 10309, 10147, 10173, 795 ] } ], "ref_ids": [ 10609 ] }, { "id": 10313, "type": "theorem", "label": "more-algebra-lemma-compute-Rlim", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-compute-Rlim", "contents": [ "The functor $\\lim : \\textit{Ab}(\\mathbf{N}) \\to \\textit{Ab}$", "has a right derived functor", "\\begin{equation}", "\\label{equation-Rlim}", "R\\lim : D(\\textit{Ab}(\\mathbf{N})) \\longrightarrow D(\\textit{Ab})", "\\end{equation}", "As usual we set $R^p\\lim(K) = H^p(R\\lim(K))$. Moreover, we have", "\\begin{enumerate}", "\\item for any $(A_n)$ in $\\textit{Ab}(\\mathbf{N})$ we have", "$R^p\\lim A_n = 0$ for $p > 1$,", "\\item the object $R\\lim A_n$ of $D(\\textit{Ab})$ is represented", "by the complex", "$$", "\\prod A_n \\to \\prod A_n,\\quad (x_n) \\mapsto (x_n - f_{n + 1}(x_{n + 1}))", "$$", "sitting in degrees $0$ and $1$,", "\\item if $(A_n)$ is ML, then $R^1\\lim A_n = 0$, i.e., $(A_n)$", "is right acyclic for $\\lim$,", "\\item every $K^\\bullet \\in D(\\textit{Ab}(\\mathbf{N}))$ is quasi-isomorphic", "to a complex whose terms are right acyclic for $\\lim$, and", "\\item if each $K^p = (K^p_n)$ is right acyclic for $\\lim$, i.e.,", "of $R^1\\lim_n K^p_n = 0$, then $R\\lim K$ is represented by the", "complex whose term in degree $p$ is $\\lim_n K_n^p$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $(A_n)$ be an arbitrary inverse system. Let $(B_n)$ be the inverse", "system with", "$$", "B_n = A_n \\oplus A_{n - 1} \\oplus \\ldots \\oplus A_1", "$$", "and transition maps given by projections. Let $A_n \\to B_n$ be given", "by $(1, f_n, f_{n - 1} \\circ f_n, \\ldots, f_2 \\circ \\ldots \\circ f_n$", "where $f_i : A_i \\to A_{i - 1}$ are the transition maps.", "In this way we see that every inverse system is a subobject of a", "ML system (Homology, Section \\ref{homology-section-inverse-systems}).", "It follows from", "Derived Categories, Lemma \\ref{derived-lemma-subcategory-right-acyclics}", "using Homology, Lemma \\ref{homology-lemma-Mittag-Leffler}", "that every ML system is right acyclic for $\\lim$, i.e., (3) holds.", "This already implies that $RF$ is defined on $D^+(\\textit{Ab}(\\mathbf{N}))$,", "see Derived Categories, Proposition \\ref{derived-proposition-enough-acyclics}.", "Set $C_n = A_{n - 1} \\oplus \\ldots \\oplus A_1$ for $n > 1$ and", "$C_1 = 0$ with transition maps given by projections as well.", "Then there is a short exact sequence of inverse systems", "$0 \\to (A_n) \\to (B_n) \\to (C_n) \\to 0$ where $B_n \\to C_n$", "is given by $(x_i) \\mapsto (x_i - f_{i + 1}(x_{i + 1}))$.", "Since $(C_n)$ is ML as well, we conclude that (2) holds", "(by proposition reference above) which also implies (1).", "Finally, this implies by Derived Categories, Lemma", "\\ref{derived-lemma-unbounded-right-derived}", "that $R\\lim$ is in fact defined on all of $D(\\textit{Ab}(\\mathbf{N}))$.", "In fact, the proof of Derived Categories, Lemma", "\\ref{derived-lemma-unbounded-right-derived}", "proceeds by proving assertions (4) and (5)." ], "refs": [ "derived-lemma-subcategory-right-acyclics", "homology-lemma-Mittag-Leffler", "derived-proposition-enough-acyclics", "derived-lemma-unbounded-right-derived", "derived-lemma-unbounded-right-derived" ], "ref_ids": [ 1837, 12124, 1962, 1917, 1917 ] } ], "ref_ids": [] }, { "id": 10314, "type": "theorem", "label": "more-algebra-lemma-apply-Mittag-Leffler-again", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-apply-Mittag-Leffler-again", "contents": [ "Let", "$$", "(A^{-2}_n \\to A^{-1}_n \\to A^0_n \\to A^1_n)", "$$", "be an inverse system of complexes of abelian groups and denote", "$A^{-2} \\to A^{-1} \\to A^0 \\to A^1$ its limit. Denote", "$(H_n^{-1})$, $(H_n^0)$ the inverse systems of cohomologies, and", "denote $H^{-1}$, $H^0$ the cohomologies of $A^{-2} \\to A^{-1} \\to A^0 \\to A^1$.", "If", "\\begin{enumerate}", "\\item $(A^{-2}_n)$ and $(A^{-1}_n)$ have vanishing $R^1\\lim$,", "\\item $(H^{-1}_n)$ has vanishing $R^1\\lim$,", "\\end{enumerate}", "then $H^0 = \\lim H_n^0$." ], "refs": [], "proofs": [ { "contents": [ "Let $K \\in D(\\textit{Ab}(\\mathbf{N}))$ be the object represented", "by the system of complexes whose $n$th constituent", "is the complex $A^{-2}_n \\to A^{-1}_n \\to A^0_n \\to A^1_n$.", "We will compute $H^0(R\\lim K)$ using both spectral", "sequences\\footnote{To use these spectral sequences we have to", "show that $\\textit{Ab}(\\mathbf{N})$ has enough injectives.", "A inverse system $(I_n)$ of abelian groups is injective if and only", "if each $I_n$ is an injective abelian group and the transition maps are", "split surjections. Every system embeds in one of these. Details omitted.} of", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "The first has $E_1$-page", "$$", "\\begin{matrix}", "0 & 0 & R^1\\lim A^0_n & R^1\\lim A^1_n \\\\", "A^{-2} & A^{-1} & A^0 & A^1", "\\end{matrix}", "$$", "with horizontal differentials and all higher differentials are zero.", "The second has $E_2$ page", "$$", "\\begin{matrix}", "R^1\\lim H^{-2}_n & 0 & R^1\\lim H^0_n & R^1 \\lim H^1_n \\\\", "\\lim H^{-2}_n &", "\\lim H^{-1}_n &", "\\lim H^0_n &", "\\lim H^1_n", "\\end{matrix}", "$$", "and degenerates at this point. The result follows." ], "refs": [ "derived-lemma-two-ss-complex-functor" ], "ref_ids": [ 1871 ] } ], "ref_ids": [] }, { "id": 10315, "type": "theorem", "label": "more-algebra-lemma-map-into-Rlim", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-map-into-Rlim", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let", "$(K_n)$ be an inverse system of objects of $\\mathcal{D}$.", "Let $K$ be a derived limit of the system $(K_n)$.", "Then for every $L$ in $\\mathcal{D}$ we have a short exact sequence", "$$", "0 \\to R^1\\lim \\Hom_\\mathcal{D}(L, K_n[-1]) \\to", "\\Hom_\\mathcal{D}(L, K) \\to", "\\lim \\Hom_\\mathcal{D}(L, K_n) \\to 0", "$$" ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Derived Categories, Definition \\ref{derived-definition-derived-limit} and", "Lemma \\ref{derived-lemma-representable-homological},", "and the description of $\\lim$ and $R^1\\lim$ in", "Lemma \\ref{lemma-compute-Rlim} above." ], "refs": [ "derived-definition-derived-limit", "derived-lemma-representable-homological", "more-algebra-lemma-compute-Rlim" ], "ref_ids": [ 2002, 1758, 10313 ] } ], "ref_ids": [] }, { "id": 10316, "type": "theorem", "label": "more-algebra-lemma-map-from-hocolim", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-map-from-hocolim", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let", "$(K_n)$ be a system of objects of $\\mathcal{D}$.", "Let $K$ be a derived colimit of the system $(K_n)$.", "Then for every $L$ in $\\mathcal{D}$ we have a short exact sequence", "$$", "0 \\to R^1\\lim \\Hom_\\mathcal{D}(K_n, L[-1]) \\to", "\\Hom_\\mathcal{D}(K, L) \\to", "\\lim \\Hom_\\mathcal{D}(K_n, L) \\to 0", "$$" ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Derived Categories, Definition \\ref{derived-definition-derived-colimit} and", "Lemma \\ref{derived-lemma-representable-homological},", "and the description of $\\lim$ and $R^1\\lim$ in", "Lemma \\ref{lemma-compute-Rlim} above." ], "refs": [ "derived-definition-derived-colimit", "derived-lemma-representable-homological", "more-algebra-lemma-compute-Rlim" ], "ref_ids": [ 2001, 1758, 10313 ] } ], "ref_ids": [] }, { "id": 10317, "type": "theorem", "label": "more-algebra-lemma-distinguished-triangle-Rlim", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-distinguished-triangle-Rlim", "contents": [ "Let $K = (K_n^\\bullet)$ be an object of $D(\\textit{Ab}(\\mathbf{N}))$.", "There exists a canonical distinguished triangle", "$$", "R\\lim K \\to \\prod\\nolimits_n K_n^\\bullet \\to \\prod\\nolimits_n K_n^\\bullet", "\\to R\\lim K[1]", "$$", "in $D(\\textit{Ab})$. In other words, $R\\lim K$ is a derived limit", "of the inverse system $(K_n^\\bullet)$ of $D(\\textit{Ab})$, see", "Derived Categories, Definition \\ref{derived-definition-derived-limit}." ], "refs": [ "derived-definition-derived-limit" ], "proofs": [ { "contents": [ "Suppose that for each $p$ the inverse system $(K_n^p)$ is right", "acyclic for $\\lim$. By Lemma \\ref{lemma-compute-Rlim}", "this gives a short exact sequence", "$$", "0 \\to \\lim_n K^p_n \\to \\prod\\nolimits_n K^p_n \\to \\prod\\nolimits_n K^p_n \\to 0", "$$", "for each $p$. Since the complex consisting of $\\lim_n K^p_n$", "computes $R\\lim K$ by Lemma \\ref{lemma-compute-Rlim} we see that the", "lemma holds in this case.", "\\medskip\\noindent", "Next, assume $K = (K_n^\\bullet)$ is general. By Lemma \\ref{lemma-compute-Rlim}", "there is a quasi-isomorphism $K \\to L$ in $D(\\textit{Ab}(\\mathbf{N}))$", "such that $(L_n^p)$ is acyclic for each $p$. Then $\\prod K_n^\\bullet$", "is quasi-isomorphic to $\\prod L_n^\\bullet$ as products are exact in", "$\\textit{Ab}$, whence the result for $L$ (proved above) implies the", "result for $K$." ], "refs": [ "more-algebra-lemma-compute-Rlim", "more-algebra-lemma-compute-Rlim", "more-algebra-lemma-compute-Rlim" ], "ref_ids": [ 10313, 10313, 10313 ] } ], "ref_ids": [ 2002 ] }, { "id": 10318, "type": "theorem", "label": "more-algebra-lemma-break-long-exact-sequence", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-break-long-exact-sequence", "contents": [ "With notation as in Lemma \\ref{lemma-distinguished-triangle-Rlim}", "the long exact cohomology sequence associated to the distinguished", "triangle breaks up into short exact sequences", "$$", "0 \\to R^1\\lim_n H^{p - 1}(K_n^\\bullet) \\to", "H^p(R\\lim K) \\to", "\\lim_n H^p(K_n^\\bullet) \\to 0", "$$" ], "refs": [ "more-algebra-lemma-distinguished-triangle-Rlim" ], "proofs": [ { "contents": [ "The long exact sequence of the distinguished triangle is", "$$", "\\ldots \\to H^p(R\\lim K) \\to \\prod\\nolimits_n H^p(K_n^\\bullet)", "\\to \\prod\\nolimits_n H^p(K_n^\\bullet) \\to", "H^{p + 1}(R\\lim K) \\to \\ldots", "$$", "The map in the middle has kernel $\\lim_n H^p(K_n^\\bullet)$ by its", "explicit description given in the lemma.", "The cokernel of this map is $R^1\\lim_n H^p(K_n^\\bullet)$", "by Lemma \\ref{lemma-compute-Rlim}." ], "refs": [ "more-algebra-lemma-compute-Rlim" ], "ref_ids": [ 10313 ] } ], "ref_ids": [ 10317 ] }, { "id": 10319, "type": "theorem", "label": "more-algebra-lemma-lift-to-system-complexes-Ab", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-to-system-complexes-Ab", "contents": [ "Let $(K_n)$ be an inverse system of objects of $D(\\textit{Ab})$.", "Then there exists an object $M = (M_n^\\bullet)$", "of $D(\\textit{Ab}(\\mathbf{N}))$ and isomorphisms", "$M_n^\\bullet \\to K_n$ in $D(\\textit{Ab})$ such that the diagrams", "$$", "\\xymatrix{", "M_{n + 1}^\\bullet \\ar[d] \\ar[r] &", "M_n^\\bullet \\ar[d] \\\\", "K_{n + 1} \\ar[r] & K_n", "}", "$$", "commute in $D(\\textit{Ab})$." ], "refs": [], "proofs": [ { "contents": [ "Namely, let $M_1^\\bullet$ be a complex of abelian groups representing $K_1$.", "Suppose we have constructed", "$M_e^\\bullet \\to M_{e - 1}^\\bullet \\to \\ldots \\to M_1^\\bullet$", "and maps $\\psi_i : M_i^\\bullet \\to K_i$ such that", "the diagrams in the statement of the lemma commute for all $n < e$.", "Then we consider the diagram", "$$", "\\xymatrix{", "& M_n^\\bullet \\ar[d]^{\\psi_n} \\\\", "K_{n + 1} \\ar[r] & K_n", "}", "$$", "in $D(\\textit{Ab})$. By the definition of morphisms in $D(\\textit{Ab})$", "we can find a complex $M_{n + 1}^\\bullet$ of abelian groups, an isomorphism", "$M_{n + 1}^\\bullet \\to K_{n + 1}$ in $D(\\textit{Ab})$, and a", "morphism of complexes $M_{n + 1}^\\bullet \\to M_n^\\bullet$", "representing the composition", "$$", "K_{n + 1} \\to K_n \\xrightarrow{\\psi_n^{-1}} M_n^\\bullet", "$$", "in $D(\\textit{Ab})$.", "Thus the lemma holds by induction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10320, "type": "theorem", "label": "more-algebra-lemma-Rlim-pro-equal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Rlim-pro-equal", "contents": [ "Let $E \\to D$ be a morphism of $D(\\textit{Ab}(\\mathbf{N}))$.", "Let $(E_n)$, resp.\\ $(D_n)$ be the system of objects of", "$D(\\textit{Ab})$ associated to $E$, resp.\\ $D$.", "If $(E_n) \\to (D_n)$ is an isomorphism of pro-objects, then", "$R\\lim E \\to R\\lim D$ is an isomorphism in $D(\\textit{Ab})$." ], "refs": [], "proofs": [ { "contents": [ "The assumption in particular implies that the pro-objects", "$H^p(E_n)$ and $H^p(D_n)$ are isomorphic. By the short exact sequences of", "Lemma \\ref{lemma-break-long-exact-sequence}", "it suffices to show that given a map $(A_n) \\to (B_n)$ of inverse", "systems of abelian groupsc which induces an isomorphism", "of pro-objects, then $\\lim A_n \\cong \\lim B_n$ and", "$R^1\\lim A_n \\cong R^1\\lim B_n$.", "\\medskip\\noindent", "The assumption implies there are $1 \\leq m_1 < m_2 < m_3 < \\ldots$", "and maps $\\varphi_n : B_{m_n} \\to A_n$ such that", "$(\\varphi_n) : (B_{m_n}) \\to (A_n)$ is a map of systems", "which is inverse to the given map $\\psi = (\\psi_n) : (A_n) \\to (B_n)$", "as a morphism of pro-objects. What this means is that", "(after possibly replacing $m_n$ by larger integers) we may", "assume that the compositions $A_{m_n} \\to B_{m_n} \\to A_n$ and", "$B_{m_n} \\to A_n \\to B_n$ are equal to the transition maps", "of the inverse systems. Now, if $(b_n) \\in \\lim B_n$ we can set", "$a_n = \\varphi_{m_n}(b_{m_n})$. This defines an inverse", "$\\lim B_n \\to \\lim A_n$ (computation omitted). Let us use the", "cokernel of the map", "$$", "\\prod B_n \\longrightarrow \\prod B_n", "$$", "as an avatar of $R^1\\lim B_n$ (Lemma \\ref{lemma-compute-Rlim}).", "Any element in this cokernel can be represented by an element", "$(b_i)$ with $b_i = 0$ if $i \\not = m_n$ for some $n$ (computation omitted).", "We can define a map $R^1\\lim B_n \\to R^1\\lim A_n$ by mapping the class", "of such a special element $(b_n)$ to the class of $(\\varphi_n(b_{m_n}))$.", "We omit the verification this map is inverse to the map", "$R^1\\lim A_n \\to R^1\\lim B_n$." ], "refs": [ "more-algebra-lemma-break-long-exact-sequence", "more-algebra-lemma-compute-Rlim" ], "ref_ids": [ 10318, 10313 ] } ], "ref_ids": [] }, { "id": 10321, "type": "theorem", "label": "more-algebra-lemma-emmanouil", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-emmanouil", "contents": [ "\\begin{reference}", "Taken from \\cite{Emmanouil}.", "\\end{reference}", "Let $(A_n)$ be an inverse system of abelian groups.", "The following are equivalent", "\\begin{enumerate}", "\\item $(A_n)$ is Mittag-Leffler,", "\\item $R^1\\lim A_n = 0$ and", "the same holds for $\\bigoplus_{i \\in \\mathbf{N}} (A_n)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Set $B = \\bigoplus_{i \\in \\mathbf{N}} (A_n)$ and hence", "$B = (B_n)$ with $B_n = \\bigoplus_{i \\in \\mathbf{N}} A_n$.", "If $(A_n)$ is ML, then $B$ is ML and hence $R^1\\lim A_n = 0$ and", "$R^1\\lim B_n = 0$ by Lemma \\ref{lemma-compute-Rlim}.", "\\medskip\\noindent", "Conversely, assume $(A_n)$ is not ML. Then we can pick an $m$", "and a sequence of integers $m < m_1 < m_2 < \\ldots$ and elements", "$x_i \\in A_{m_i}$ whose image $y_i \\in A_m$ is not in the", "image of $A_{m_i + 1} \\to A_m$.", "We will use the elements $x_i$ and $y_i$ to", "show that $R^1\\lim B_n \\not = 0$ in two ways.", "This will finish the proof of the lemma.", "\\medskip\\noindent", "First proof.", "Set $C = (C_n)$ with $C_n = \\prod_{i \\in \\mathbf{N}} A_n$.", "There is a canonical injective map $B_n \\to C_n$ with cokernel", "$Q_n$. Set $Q = (Q_n)$. We may and do think of elements $q_n$ of $Q_n$", "as sequences of elements $q_n = (q_{n, 1}, q_{n, 2}, \\ldots)$", "with $q_{n, i} \\in A_n$ modulo sequences whose tail is zero", "(in other words, we identify sequences which differ in", "finitely many places).", "We have a short exact sequence of inverse systems", "$$", "0 \\to (B_n) \\to (C_n) \\to (Q_n) \\to 0", "$$", "Consider the element $q_n \\in Q_n$ given by", "$$", "q_{n, i} =", "\\left\\{", "\\begin{matrix}", "\\text{image of }x_i &\\text{if}& m_i \\geq n \\\\", "0 & \\text{else}", "\\end{matrix}", "\\right.", "$$", "Then it is clear that $q_{n + 1}$ maps to $q_n$.", "Hence we obtain $q = (q_n) \\in \\lim Q_n$.", "On the other hand, we claim that $q$ is not in the image", "of $\\lim C_n \\to \\lim Q_n$. Namely, say that $c = (c_n)$", "maps to $q$. Then we can write $c_n = (c_{n, i})$ and since", "$c_{n', i} \\mapsto c_{n, i}$ for $n' \\geq n$, we see that", "$c_{n, i} \\in \\Im(C_{n'} \\to C_n)$ for all $n, i, n' \\geq n$.", "In particular, the image of $c_{m, i}$ in $A_m$ is in", "$\\Im(A_{m_i + 1} \\to A_m)$ whence cannot be equal to $y_i$.", "Thus $c_m$ and $q_m = (y_1, y_2, y_3, \\ldots)$", "differ in infinitely many spots, which", "is a contradiction. Considering the long exact cohomology sequence", "$$", "0 \\to \\lim B_n \\to \\lim C_n \\to \\lim Q_n \\to R^1\\lim B_n", "$$", "we conclude that the last group is nonzero as desired.", "\\medskip\\noindent", "Second proof. For $n' \\geq n$ we denote $A_{n, n'} = \\Im(A_{n'} \\to A_n)$.", "Then we have $y_i \\in A_m$, $y_i \\not \\in A_{m, m_i + 1}$.", "Let $\\xi = (\\xi_n) \\in \\prod B_n$ be the element with", "$\\xi_n = 0$ unless $n = m_i$ and $\\xi_{m_i} = (0, \\ldots, 0, x_i, 0, \\ldots)$", "with $x_i$ placed in the $i$th summand. We claim that $\\xi$ is not in the", "image of the map $\\prod B_n \\to \\prod B_n$ of Lemma \\ref{lemma-compute-Rlim}.", "This shows that $R^1\\lim B_n$ is nonzero and finishes the proof.", "Namely, suppose that $\\xi$ is the image of $\\eta = (z_1, z_2, \\ldots)$", "with $z_n = \\sum z_{n, i} \\in \\bigoplus_i A_n$.", "Observe that $x_i = z_{m_i, i} \\bmod A_{m_i, m_i + 1}$.", "Then $z_{m_i - 1, i}$ is the image of $z_{m_i, i}$ under", "$A_{m_i} \\to A_{m_i - 1}$, and so on, and we conclude that", "$z_{m, i}$ is the image of $z_{m_i, i}$ under $A_{m_i} \\to A_m$.", "We conclude that $z_{m, i}$ is congruent to $y_i$ modulo", "$A_{m, m_i + 1}$. In particular $z_{m, i} \\not = 0$.", "This is impossible as $\\sum z_{m, i} \\in \\bigoplus_i A_m$", "hence only a finite number of $z_{m, i}$ can be nonzero." ], "refs": [ "more-algebra-lemma-compute-Rlim", "more-algebra-lemma-compute-Rlim" ], "ref_ids": [ 10313, 10313 ] } ], "ref_ids": [] }, { "id": 10322, "type": "theorem", "label": "more-algebra-lemma-Mittag-Leffler", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Mittag-Leffler", "contents": [ "Let", "$$", "0 \\to (A_i) \\to (B_i) \\to (C_i) \\to 0", "$$", "be a short exact sequence of inverse systems of abelian groups.", "If $(A_i)$ and $(C_i)$ are ML, then so is $(B_i)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-emmanouil}, the fact that", "taking infinite direct sums is exact, and the long exact sequence", "of cohomology associated to $R\\lim$." ], "refs": [ "more-algebra-lemma-emmanouil" ], "ref_ids": [ 10321 ] } ], "ref_ids": [] }, { "id": 10323, "type": "theorem", "label": "more-algebra-lemma-Rlim-zero-of-direct-sums", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Rlim-zero-of-direct-sums", "contents": [ "Let $(A_n)$ be an inverse system of abelian groups.", "The following are equivalent", "\\begin{enumerate}", "\\item $(A_n)$ is zero as a pro-object,", "\\item $\\lim A_n = 0$ and $R^1\\lim A_n = 0$ and", "the same holds for $\\bigoplus_{i \\in \\mathbf{N}} (A_n)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It follows from Lemma \\ref{lemma-Rlim-pro-equal} that (1) implies (2).", "Assume (2). Then $(A_n)$ is ML by Lemma \\ref{lemma-emmanouil}.", "For $m \\geq n$ let $A_{n, m} = \\Im(A_m \\to A_n)$", "so that $A_n = A_{n, n} \\supset A_{n, n + 1} \\supset \\ldots$.", "Note that $(A_n)$ is zero as a pro-object if and only if for every", "$n$ there is an $m \\geq n$ such that $A_{n, m} = 0$.", "Note that $(A_n)$ is ML if and only if for every $n$ there is an $m_n \\geq n$", "such that $A_{n, m} = A_{n, m + 1} = \\ldots$. In the ML case it is", "clear that $\\lim A_n = 0$ implies that $A_{n, m_n} = 0$", "because the maps $A_{n + 1, m_{n + 1}} \\to A_{n, m}$ are surjective.", "This finishes the proof." ], "refs": [ "more-algebra-lemma-Rlim-pro-equal", "more-algebra-lemma-emmanouil" ], "ref_ids": [ 10320, 10321 ] } ], "ref_ids": [] }, { "id": 10324, "type": "theorem", "label": "more-algebra-lemma-compute-Rlim-modules", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-compute-Rlim-modules", "contents": [ "In the situation above. The functor", "$\\lim : \\textit{Mod}(\\mathbf{N}, (A_n)) \\to \\text{Mod}_A$", "has a right derived functor", "$$", "R\\lim :", "D(\\textit{Mod}(\\mathbf{N}, (A_n)))", "\\longrightarrow", "D(A)", "$$", "As usual we set $R^p\\lim(K) = H^p(R\\lim(K))$. Moreover, we have", "\\begin{enumerate}", "\\item for any $(M_n)$ in $\\textit{Mod}(\\mathbf{N}, (A_n))$ we have", "$R^p\\lim M_n = 0$ for $p > 1$,", "\\item the object $R\\lim M_n$ of $D(\\text{Mod}_A)$ is represented", "by the complex", "$$", "\\prod M_n \\to \\prod M_n,\\quad", "(x_n) \\mapsto (x_n - f_{n + 1}(x_{n + 1}))", "$$", "sitting in degrees $0$ and $1$,", "\\item if $(M_n)$ is ML, then $R^1\\lim M_n = 0$, i.e., $(M_n)$", "is right acyclic for $\\lim$,", "\\item every $K^\\bullet \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$", "is quasi-isomorphic to a complex whose terms are right acyclic for $\\lim$, and", "\\item if each $K^p = (K^p_n)$ is right acyclic for $\\lim$, i.e.,", "of $R^1\\lim_n K^p_n = 0$, then $R\\lim K$ is represented by the", "complex whose term in degree $p$ is $\\lim_n K_n^p$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The proof of this is word for word the same as the proof of", "Lemma \\ref{lemma-compute-Rlim}." ], "refs": [ "more-algebra-lemma-compute-Rlim" ], "ref_ids": [ 10313 ] } ], "ref_ids": [] }, { "id": 10325, "type": "theorem", "label": "more-algebra-lemma-distinguished-triangle-Rlim-modules", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-distinguished-triangle-Rlim-modules", "contents": [ "Let $K = (K_n^\\bullet)$ be an object of", "$D(\\textit{Mod}(\\mathbf{N}, (A_n)))$.", "There exists a canonical distinguished triangle", "$$", "R\\lim K \\to \\prod\\nolimits_n K_n^\\bullet \\to \\prod\\nolimits_n K_n^\\bullet", "\\to R\\lim K[1]", "$$", "in $D(A)$. In other words, $R\\lim K$ is a derived limit", "of the inverse system $(K_n^\\bullet)$ of $D(A)$, see", "Derived Categories, Definition \\ref{derived-definition-derived-limit}." ], "refs": [ "derived-definition-derived-limit" ], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Lemma \\ref{lemma-distinguished-triangle-Rlim}", "using Lemma \\ref{lemma-compute-Rlim-modules} in stead of", "Lemma \\ref{lemma-compute-Rlim}." ], "refs": [ "more-algebra-lemma-distinguished-triangle-Rlim", "more-algebra-lemma-compute-Rlim-modules", "more-algebra-lemma-compute-Rlim" ], "ref_ids": [ 10317, 10324, 10313 ] } ], "ref_ids": [ 2002 ] }, { "id": 10326, "type": "theorem", "label": "more-algebra-lemma-break-long-exact-sequence-modules", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-break-long-exact-sequence-modules", "contents": [ "With notation as in Lemma \\ref{lemma-distinguished-triangle-Rlim-modules}", "the long exact cohomology sequence associated to the distinguished", "triangle breaks up into short exact sequences", "$$", "0 \\to R^1\\lim_n H^{p - 1}(K_n^\\bullet) \\to", "H^p(R\\lim K) \\to", "\\lim_n H^p(K_n^\\bullet) \\to 0", "$$", "of $A$-modules." ], "refs": [ "more-algebra-lemma-distinguished-triangle-Rlim-modules" ], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Lemma \\ref{lemma-break-long-exact-sequence}", "using Lemma \\ref{lemma-compute-Rlim-modules} in stead of", "Lemma \\ref{lemma-compute-Rlim}." ], "refs": [ "more-algebra-lemma-break-long-exact-sequence", "more-algebra-lemma-compute-Rlim-modules", "more-algebra-lemma-compute-Rlim" ], "ref_ids": [ 10318, 10324, 10313 ] } ], "ref_ids": [ 10325 ] }, { "id": 10327, "type": "theorem", "label": "more-algebra-lemma-lift-to-system-complexes", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-to-system-complexes", "contents": [ "Let $(A_n)$ be an inverse system of rings. Suppose that we are given", "\\begin{enumerate}", "\\item for every $n$ an object $K_n$ of $D(A_n)$, and", "\\item for every $n$ a map $\\varphi_n : K_{n + 1} \\to K_n$ of", "$D(A_{n + 1})$ where we think of $K_n$ as an object of $D(A_{n + 1})$", "by restriction via $A_{n + 1} \\to A_n$.", "\\end{enumerate}", "There exists an object", "$M = (M_n^\\bullet) \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$", "and isomorphisms $\\psi_n : M_n^\\bullet \\to K_n$ in $D(A_n)$", "such that the diagrams", "$$", "\\xymatrix{", "M_{n + 1}^\\bullet \\ar[d]_{\\psi_{n + 1}} \\ar[r] & M_n^\\bullet \\ar[d]^{\\psi_n} \\\\", "K_{n + 1} \\ar[r]^{\\varphi_n} & K_n", "}", "$$", "commute in $D(A_{n + 1})$." ], "refs": [], "proofs": [ { "contents": [ "We write out the proof in detail. For an $A_n$-module $T$ we write", "$T_{A_{n + 1}}$ for the same module viewd as an $A_{n + 1}$-module.", "Suppose that $K_n^\\bullet$ is a complex of $A_n$-modules representing $K_n$.", "Then $K_{n, A_{n + 1}}^\\bullet$ is the same complex, but viewed as a", "complex of $A_{n + 1}$-modules. By the construction of the derived", "category, the map $\\psi_n$ can be given as", "$$", "\\psi_n = \\tau_n \\circ \\sigma_n^{-1}", "$$", "where $\\sigma_n : L_{n + 1}^\\bullet \\to K_{n + 1}^\\bullet$", "is a quasi-isomorphism of complexes of $A_{n + 1}$-modules", "and $\\tau_n : L_{n + 1}^\\bullet \\to K_{n, A_{n + 1}}^\\bullet$", "is a map of complexes of $A_{n + 1}$-modules.", "\\medskip\\noindent", "Now we construct the complexes $M_n^\\bullet$ by induction. As base case", "we let $M_1^\\bullet = K_1^\\bullet$. Suppose we have already constructed", "$M_e^\\bullet \\to M_{e - 1}^\\bullet \\to \\ldots \\to M_1^\\bullet$", "and maps of complexes $\\psi_i : M_i^\\bullet \\to K_i^\\bullet$ such that", "the diagrams", "$$", "\\xymatrix{", "M_{n + 1}^\\bullet \\ar[d]_{\\psi_{n + 1}} \\ar[rr] & &", "M_{n, A_{n + 1}}^\\bullet \\ar[d]^{\\psi_{n, A_{n + 1}}} \\\\", "K_{n + 1}^\\bullet & L_{n + 1}^\\bullet \\ar[l]_{\\sigma_n} \\ar[r]^{\\tau_n} &", "K_{n, A_{n + 1}}^\\bullet", "}", "$$", "above commute in $D(A_{n + 1})$ for all $n < e$. Then we consider the diagram", "$$", "\\xymatrix{", "& & M_{e, A_{e + 1}}^\\bullet \\ar[d]^{\\psi_{e, A_{e + 1}}} \\\\", "K_{e + 1}^\\bullet & L_{e + 1}^\\bullet \\ar[r]^{\\tau_e} \\ar[l]_{\\sigma_e} &", "K_{e, A_{e + 1}}^\\bullet", "}", "$$", "in $D(A_{e + 1})$. Because $\\psi_e$ is a quasi-isomorphism,", "we see that $\\psi_{e, A_{e + 1}}$ is a quasi-isomorphism too.", "By the definition of morphisms in $D(A_{e + 1})$ we can", "find a quasi-isomorphism", "$\\psi_{e + 1} : M_{e + 1}^\\bullet \\to K_{e + 1}^\\bullet$", "of complexes of $A_{e + 1}$-modules such that there exists a morphism", "of complexes $M_{e + 1}^\\bullet \\to M_{e, A_{e + 1}}^\\bullet$", "of $A_{e + 1}$-modules representing the composition ", "$\\psi_{e, A_{e + 1}}^{-1} \\circ \\tau_e \\circ \\sigma_e^{-1}$", "in $D(A_{e + 1})$. Thus the lemma holds by induction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10328, "type": "theorem", "label": "more-algebra-lemma-get-ML-system", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-get-ML-system", "contents": [ "Let $(A_n)$ be an inverse system of rings. Every", "$K \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$", "can be represented by a system of complexes $(M_n^\\bullet)$", "such that all the transition maps $M_{n + 1}^\\bullet \\to M_n^\\bullet$", "are surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $K$ be represented by the system $(K_n^\\bullet)$. Set", "$M_1^\\bullet = K_1^\\bullet$. Suppose we have constructed surjective maps", "of complexes $M_n^\\bullet \\to M_{n - 1}^\\bullet \\to \\ldots \\to M_1^\\bullet$", "and homotopy equivalences $\\psi_e : K_e^\\bullet \\to M_e^\\bullet$ such that", "the diagrams", "$$", "\\xymatrix{", "K_{e + 1}^\\bullet \\ar[d] \\ar[r] & K_e^\\bullet \\ar[d] \\\\", "M_{e + 1}^\\bullet \\ar[r] & M_e^\\bullet", "}", "$$", "commute for all $e < n$. Then we consider the diagram", "$$", "\\xymatrix{", "K_{n + 1}^\\bullet \\ar[r] & K_n^\\bullet \\ar[d] \\\\", "& M_n^\\bullet", "}", "$$", "By Derived Categories, Lemma \\ref{derived-lemma-make-surjective}", "we can factor the composition $K_{n + 1}^\\bullet \\to M_n^\\bullet$ as", "$K_{n + 1}^\\bullet \\to M_{n + 1}^\\bullet \\to M_n^\\bullet$", "such that the first arrow is a homotopy equivalence and the", "second a termwise split surjection. The lemma follows", "from this and induction." ], "refs": [ "derived-lemma-make-surjective" ], "ref_ids": [ 1798 ] } ], "ref_ids": [] }, { "id": 10329, "type": "theorem", "label": "more-algebra-lemma-get-K-flat-system", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-get-K-flat-system", "contents": [ "Let $(A_n)$ be an inverse system of rings. Every", "$K \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$", "can be represented by a system of complexes $(K_n^\\bullet)$", "such that each $K_n^\\bullet$ is K-flat." ], "refs": [], "proofs": [ { "contents": [ "First use Lemma \\ref{lemma-get-ML-system} to represent $K$ by a", "system of complexes $(M_n^\\bullet)$ such that all the transition maps", "$M_{n + 1}^\\bullet \\to M_n^\\bullet$ are surjective.", "Next, let $K_1^\\bullet \\to M_1^\\bullet$ be a quasi-isomorphism", "with $K_1^\\bullet$ a K-flat complex of $A_1$-modules", "(Lemma \\ref{lemma-K-flat-resolution}).", "Suppose we have constructed", "$K_n^\\bullet \\to K_{n - 1}^\\bullet \\to \\ldots \\to K_1^\\bullet$", "and maps of complexes $\\psi_e : K_e^\\bullet \\to M_e^\\bullet$ such that", "$$", "\\xymatrix{", "K_{e + 1}^\\bullet \\ar[d] \\ar[r] & K_e^\\bullet \\ar[d] \\\\", "M_{e + 1}^\\bullet \\ar[r] & M_e^\\bullet", "}", "$$", "commutes for all $e < n$. Then we consider the diagram", "$$", "\\xymatrix{", "C^\\bullet \\ar@{..>}[d] \\ar@{..>}[r] & K_n^\\bullet \\ar[d]^{\\psi_n} \\\\", "M_{n + 1}^\\bullet \\ar[r]^{\\varphi_n} & M_n^\\bullet", "}", "$$", "in $D(A_{n + 1})$. As $M_{n + 1}^\\bullet \\to M_n^\\bullet$ is", "termwise surjective, the complex $C^\\bullet$ fitting into the left", "upper corner with terms", "$$", "C^p = M_{n + 1}^p \\times_{M_n^p} K_n^p", "$$", "is quasi-isomorphic to $M_{n + 1}^\\bullet$ (details omitted).", "Choose a quasi-isomorphism $K_{n + 1}^\\bullet \\to C^\\bullet$", "with $K_{n +1}^\\bullet$ K-flat.", "Thus the lemma holds by induction." ], "refs": [ "more-algebra-lemma-get-ML-system", "more-algebra-lemma-K-flat-resolution" ], "ref_ids": [ 10328, 10131 ] } ], "ref_ids": [] }, { "id": 10330, "type": "theorem", "label": "more-algebra-lemma-derived-tensor-product-systems", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-tensor-product-systems", "contents": [ "Let $(A_n)$ be an inverse system of rings. Given", "$K, L \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$ there is a canonical derived", "tensor product $K \\otimes^\\mathbf{L} L$ in $D(\\mathbf{N}, (A_n))$", "compatible with the maps to $D(A_n)$. The construction is symmetric", "in $K$ and $L$ and an exact functor of triangulated categories in", "each variable." ], "refs": [], "proofs": [ { "contents": [ "Choose a representative $(K_n^\\bullet)$ for $K$ such that each $K_n^\\bullet$", "is a K-flat complex (Lemma \\ref{lemma-get-K-flat-system}).", "Then you can define $K \\otimes^\\mathbf{L} L$ as the object represented by", "the system of complexes", "$$", "(\\text{Tot}(K_n^\\bullet \\otimes_{A_n} L_n^\\bullet))", "$$", "for any choice of representative $(L_n^\\bullet)$ for $L$.", "This is well defined in both variables by", "Lemmas \\ref{lemma-K-flat-quasi-isomorphism} and", "\\ref{lemma-derived-tor-quasi-isomorphism-other-side}.", "Compatibility with the map to $D(A_n)$ is clear.", "Exactness follows exactly as in", "Lemma \\ref{lemma-derived-tor-exact}." ], "refs": [ "more-algebra-lemma-get-K-flat-system", "more-algebra-lemma-K-flat-quasi-isomorphism", "more-algebra-lemma-derived-tor-quasi-isomorphism-other-side", "more-algebra-lemma-derived-tor-exact" ], "ref_ids": [ 10329, 10123, 10132, 10122 ] } ], "ref_ids": [] }, { "id": 10331, "type": "theorem", "label": "more-algebra-lemma-tensor-Rlim-exact", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-tensor-Rlim-exact", "contents": [ "Let $A$ be a ring. Let $E \\to D \\to F \\to E[1]$ be a distinguished", "triangle of $D(\\mathbf{N}, A)$. Let $(E_n)$, resp.\\ $(D_n)$, resp.\\ $(F_n)$", "be the system of objects of $D(A)$ associated to $E$, resp.\\ $D$, resp.\\ $F$.", "Then for every $K \\in D(A)$ there is a canonical distinguished triangle", "$$", "R\\lim (K \\otimes^\\mathbf{L}_A E_n) \\to", "R\\lim (K \\otimes^\\mathbf{L}_A D_n) \\to", "R\\lim (K \\otimes^\\mathbf{L}_A F_n) \\to", "R\\lim (K \\otimes^\\mathbf{L}_A E_n)[1]", "$$", "in $D(A)$ with notation as in", "Remark \\ref{remark-constructing-tensor-with-limits-functorially}." ], "refs": [ "more-algebra-remark-constructing-tensor-with-limits-functorially" ], "proofs": [ { "contents": [ "This is clear from the construction in", "Remark \\ref{remark-constructing-tensor-with-limits-functorially}", "and the fact that $\\Delta : D(A) \\to D(\\mathbf{N}, A)$,", "$- \\otimes^\\mathbf{L} -$, and $R\\lim$ are exact functors of triangulated", "categories." ], "refs": [ "more-algebra-remark-constructing-tensor-with-limits-functorially" ], "ref_ids": [ 10661 ] } ], "ref_ids": [ 10661 ] }, { "id": 10332, "type": "theorem", "label": "more-algebra-lemma-tensor-Rlim-pro-equal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-tensor-Rlim-pro-equal", "contents": [ "Let $A$ be a ring. Let $E \\to D$ be a morphism of", "$D(\\mathbf{N}, A)$. Let $(E_n)$, resp.\\ $(D_n)$", "be the system of objects of $D(A)$ associated to $E$, resp.\\ $D$.", "If $(E_n) \\to (D_n)$ is an isomorphism of pro-objects, then for every", "$K \\in D(A)$ the corresponding map", "$$", "R\\lim (K \\otimes^\\mathbf{L}_A E_n)", "\\longrightarrow", "R\\lim (K \\otimes^\\mathbf{L}_A D_n)", "$$", "in $D(A)$ is an isomorphism", "(notation as in", "Remark \\ref{remark-constructing-tensor-with-limits-functorially})." ], "refs": [ "more-algebra-remark-constructing-tensor-with-limits-functorially" ], "proofs": [ { "contents": [ "Follows from the definitions and Lemma \\ref{lemma-Rlim-pro-equal}." ], "refs": [ "more-algebra-lemma-Rlim-pro-equal" ], "ref_ids": [ 10320 ] } ], "ref_ids": [ 10661 ] }, { "id": 10333, "type": "theorem", "label": "more-algebra-lemma-I-power-torsion-presentation", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-I-power-torsion-presentation", "contents": [ "Let $R$ be a ring.", "Let $I$ be an ideal of $R$.", "Let $M$ be an $I$-power torsion module.", "Then $M$ admits a resolution", "$$", "\\ldots \\to K_2 \\to K_1 \\to K_0 \\to M \\to 0", "$$", "with each $K_i$ a direct sum of copies of $R/I^n$ for $n$ variable." ], "refs": [], "proofs": [ { "contents": [ "There is a canonical surjection", "$$", "\\oplus_{m \\in M} R/I^{n_m} \\to M \\to 0", "$$", "where $n_m$ is the smallest positive integer such that $I^{n_m} \\cdot m = 0$.", "The kernel of the preceding surjection is also an $I$-power torsion module.", "Proceeding inductively, we construct the desired resolution of $M$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10334, "type": "theorem", "label": "more-algebra-lemma-torsion-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-torsion-free", "contents": [ "Let $R$ be a ring. Let $I$ be an ideal of $R$.", "For any $R$-module $M$ set $M[I^n] = \\{m \\in M \\mid I^nm = 0\\}$.", "If $I$ is finitely generated then the following are equivalent", "\\begin{enumerate}", "\\item $M[I] = 0$,", "\\item $M[I^n] = 0$ for all $n \\geq 1$, and", "\\item if $I = (f_1, \\ldots, f_t)$, then the map", "$M \\to \\bigoplus M_{f_i}$ is injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Algebra, Lemma \\ref{algebra-lemma-when-injective-covering}." ], "refs": [ "algebra-lemma-when-injective-covering" ], "ref_ids": [ 416 ] } ], "ref_ids": [] }, { "id": 10335, "type": "theorem", "label": "more-algebra-lemma-divide-by-torsion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-divide-by-torsion", "contents": [ "Let $R$ be a ring. Let $I$ be a finitely generated ideal of $R$.", "\\begin{enumerate}", "\\item For any $R$-module $M$ we have $(M/M[I^\\infty])[I] = 0$.", "\\item An extension of $I$-power torsion modules is $I$-power torsion.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $m \\in M$. If $m$ maps to an element of $(M/M[I^\\infty])[I]$", "then $Im \\subset M[I^\\infty]$.", "Write $I = (f_1, \\ldots, f_t)$. Then we see that", "$f_i m \\in M[I^\\infty]$, i.e., $I^{n_i}f_i m = 0$ for some $n_i > 0$.", "Thus we see that $I^Nm = 0$ with $N = \\sum n_i + 2$.", "Hence $m$ maps to zero in $(M/M[I^\\infty])$ which proves the", "first statement of the lemma.", "\\medskip\\noindent", "For the second, suppose that $0 \\to M' \\to M \\to M'' \\to 0$ is a short", "exact sequence of modules with $M'$ and $M''$ both $I$-power torsion", "modules. Then $M[I^\\infty] \\supset M'$ and hence $M/M[I^\\infty]$ is a", "quotient of $M''$ and therefore $I$-power torsion. Combined with", "the first statement and Lemma \\ref{lemma-torsion-free} this implies", "that it is zero" ], "refs": [ "more-algebra-lemma-torsion-free" ], "ref_ids": [ 10334 ] } ], "ref_ids": [] }, { "id": 10336, "type": "theorem", "label": "more-algebra-lemma-I-power-torsion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-I-power-torsion", "contents": [ "Let $I$ be a finitely generated ideal of a ring $R$.", "The $I$-power torsion modules form a Serre subcategory of", "the abelian category $\\text{Mod}_R$, see", "Homology, Definition \\ref{homology-definition-serre-subcategory}." ], "refs": [ "homology-definition-serre-subcategory" ], "proofs": [ { "contents": [ "It is clear that a submodule and a quotient module of an $I$-power", "torsion module is $I$-power torsion. Moreover, the extension", "of two $I$-power torsion modules is $I$-power torsion by", "Lemma \\ref{lemma-divide-by-torsion}.", "Hence the statement of the lemma by", "Homology, Lemma \\ref{homology-lemma-characterize-serre-subcategory}." ], "refs": [ "more-algebra-lemma-divide-by-torsion", "homology-lemma-characterize-serre-subcategory" ], "ref_ids": [ 10335, 12045 ] } ], "ref_ids": [ 12146 ] }, { "id": 10337, "type": "theorem", "label": "more-algebra-lemma-local-cohomology-closed", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-local-cohomology-closed", "contents": [ "Let $R$ be a ring and let $I \\subset R$ be a finitely generated ideal.", "The subcategory $I^\\infty\\text{-torsion} \\subset \\text{Mod}_R$", "depends only on the closed subset $Z = V(I) \\subset \\Spec(R)$.", "In fact, an $R$-module $M$ is $I$-power torsion if and only if its", "support is contained in $Z$." ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be an $R$-module. Let $x \\in M$. If $x \\in M[I^\\infty]$, then $x$", "maps to zero in $M_f$ for all $f \\in I$. Hence $x$ maps to zero in", "$M_\\mathfrak p$ for all $\\mathfrak p \\not \\supset I$. Conversely, if $x$", "maps to zero in $M_\\mathfrak p$ for all $\\mathfrak p \\not \\supset I$,", "then $x$ maps to zero in $M_f$ for all $f \\in I$. Hence if", "$I = (f_1, \\ldots, f_r)$, then $f_i^{n_i}x = 0$ for some $n_i \\geq 1$.", "It follows that $x \\in M[I^{\\sum n_i}]$. Thus $M[I^\\infty]$ is", "the kernel of $M \\to \\prod_{\\mathfrak p \\not \\in Z} M_\\mathfrak p$.", "The second statement of the lemma follows and it implies the first." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10338, "type": "theorem", "label": "more-algebra-lemma-derived-vanishing-mod-I", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-vanishing-mod-I", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $K$ be an object of", "$D(R)$ such hat $K \\otimes_R^\\mathbf{L} R/I = 0$ in $D(R)$. Then", "\\begin{enumerate}", "\\item $K \\otimes_R^\\mathbf{L} R/I^n = 0$ for all $n \\geq 1$,", "\\item $K \\otimes_R^\\mathbf{L} N = 0$ for any $I$-power torsion", "$R$-module $N$,", "\\item $K \\otimes_R^\\mathbf{L} M = 0$ for any $M \\in D^b(R)$ whose", "cohomology modules are $I$-power torsion.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (2). We can write $N = \\bigcup N[I^n]$. We have", "$K \\otimes_R^\\mathbf{L} N = \\text{hocolim}_n K \\otimes_R^\\mathbf{L} N[I^n]$", "as tensor products commute with colimits (details omitted; hint: represent", "$K$ by a K-flat complex and compute directly).", "Hence we may assume $N$ is annihilated by $I^n$.", "Consider the $R$-algebra $R' = R/I^n \\oplus N$", "where $N$ is an ideal of square zero. It suffices to show that", "$K' = K \\otimes_R^\\mathbf{L} R'$ is $0$ in $D(R')$.", "We have a surjection $R' \\to R/I$ of $R$-algebras whose kernel $J$", "is nilpotent (any product of $n$ elements in the kernel is zero).", "We have", "$$", "0 = K \\otimes_R^\\mathbf{L} R/I =", "(K \\otimes_R^\\mathbf{L} R') \\otimes_{R'}^\\mathbf{L} R/I =", "K' \\otimes_{R'}^\\mathbf{L} R/I", "$$", "by Lemma \\ref{lemma-double-base-change}.", "Hence by Lemma \\ref{lemma-perfect-modulo-nilpotent-ideal}", "we find that $K'$", "is a perfect complex of $R'$-modules. In particular $K'$", "is bounded above and if $H^b(K')$ is the right-most nonvanishing", "cohomology module (if it exists), then $H^b(K')$ is a finite", "$R'$-module (use Lemmas \\ref{lemma-perfect} and", "\\ref{lemma-finite-cohomology})", "with $H^b(K') \\otimes_{R'} R'/J = H^b(K')/JH^b(K') = 0$ (because", "$K' \\otimes_{R'}^\\mathbf{L} R'/J = 0$).", "By Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "we find $H^b(K') = 0$, i.e., $K' = 0$ as desired.", "\\medskip\\noindent", "Part (1) follows trivially from part (2).", "Part (3) follows from part (2), induction on the number of nonzero", "cohomology modules of $M$, and the distinguished triangles of truncation", "from Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}. Details omitted." ], "refs": [ "more-algebra-lemma-double-base-change", "more-algebra-lemma-perfect-modulo-nilpotent-ideal", "more-algebra-lemma-perfect", "more-algebra-lemma-finite-cohomology", "algebra-lemma-NAK", "derived-remark-truncation-distinguished-triangle" ], "ref_ids": [ 10138, 10249, 10212, 10146, 401, 2016 ] } ], "ref_ids": [] }, { "id": 10339, "type": "theorem", "label": "more-algebra-lemma-characterize-flatness-on-torsion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-characterize-flatness-on-torsion", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Let $I \\subset R$ be an ideal.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\varphi$ is flat and $R/I \\to S/IS$ is faithfully flat,", "\\item $\\varphi$ is flat, and the map", "$\\Spec(S/IS) \\to \\Spec(R/I)$ is surjective.", "\\item $\\varphi$ is flat, and the base change functor", "$M \\mapsto M \\otimes_R S$ is faithful on modules annihilated by $I$, and", "\\item $\\varphi$ is flat, and the base change functor", "$M \\mapsto M \\otimes_R S$ is faithful on $I$-power torsion modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $R \\to S$ is flat, then $R/I^n \\to S/I^nS$ is flat for every $n$, see", "Algebra, Lemma \\ref{algebra-lemma-flat-base-change}.", "Hence (1) and (2) are equivalent by", "Algebra, Lemma \\ref{algebra-lemma-ff-rings}.", "The equivalence of (1) with (3) follows by identifying $I$-torsion", "$R$-modules with $R/I$-modules, using that", "$$", "M \\otimes_R S = M \\otimes_{R/I} S/IS", "$$", "for $R$-modules $M$ annihilated by $I$, and", "Algebra, Lemma \\ref{algebra-lemma-easy-ff}.", "The implication (4) $\\Rightarrow$ (3) is immediate. Assume (3). We have", "seen above that $R/I^n \\to S/I^nS$ is flat, and by assumption it induces", "a surjection on spectra, as $\\Spec(R/I^n) = \\Spec(R/I)$ and", "similarly for $S$. Hence the base change functor is faithful on modules", "annihilated by $I^n$. Since any $I$-power torsion module $M$ is the union", "$M = \\bigcup M_n$ where $M_n$ is annihilated by $I^n$ we see that the base", "change functor is faithful on the category of all $I$-power torsion modules", "(as tensor product commutes with colimits)." ], "refs": [ "algebra-lemma-flat-base-change", "algebra-lemma-ff-rings", "algebra-lemma-easy-ff" ], "ref_ids": [ 527, 536, 534 ] } ], "ref_ids": [] }, { "id": 10340, "type": "theorem", "label": "more-algebra-lemma-neighbourhood-isomorphism", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-neighbourhood-isomorphism", "contents": [ "Assume $(\\varphi : R \\to S, I)$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-characterize-flatness-on-torsion}.", "The following are equivalent", "\\begin{enumerate}", "\\item for any $I$-power torsion module $M$, the natural map", "$M \\to M \\otimes_R S$ is an isomorphism, and", "\\item $R/I \\to S/IS$ is an isomorphism.", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-characterize-flatness-on-torsion" ], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) is immediate.", "Assume (2). First assume that $M$ is annihilated by $I$.", "In this case, $M$ is an $R/I$-module. Hence, we have an isomorphism", "$$", "M \\otimes_R S = M \\otimes_{R/I} S/IS = M \\otimes_{R/I} R/I = M", "$$", "proving the claim. Next we prove by induction that $M \\to M \\otimes_R S$", "is an isomorphism for any module $M$ is annihilated by $I^n$. Assume", "the induction hypothesis holds for $n$ and assume $M$ is annihilated by", "$I^{n + 1}$. Then we have a short exact sequence", "$$", "0 \\to I^nM \\to M \\to M/I^nM \\to 0", "$$", "and as $R \\to S$ is flat this gives rise to a short exact sequence", "$$", "0 \\to I^nM \\otimes_R S \\to M \\otimes_R S \\to M/I^nM \\otimes_R S \\to 0", "$$", "Using that the canonical map is an isomorphism for $M' = I^nM$ and", "$M'' = M/I^nM$ (by induction hypothesis) we conclude the same thing is", "true for $M$. Finally, suppose that $M$ is a general $I$-power torsion", "module. Then $M = \\bigcup M_n$ where $M_n$ is annihilated by $I^n$", "and we conclude using that tensor products commute with colimits." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10339 ] }, { "id": 10341, "type": "theorem", "label": "more-algebra-lemma-neighbourhood-equivalence", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-neighbourhood-equivalence", "contents": [ "Assume $\\varphi : R \\to S$ is a flat ring map and $I \\subset R$ is a", "finitely generated ideal such that $R/I \\to S/IS$ is an isomorphism. Then", "\\begin{enumerate}", "\\item for any $R$-module $M$ the map $M \\to M \\otimes_R S$ induces", "an isomorphism", "$M[I^\\infty] \\to (M \\otimes_R S)[(IS)^\\infty]$ of $I$-power", "torsion submodules,", "\\item the natural map", "$$", "\\Hom_R(M, N) \\longrightarrow \\Hom_S(M \\otimes_R S, N \\otimes_R S)", "$$", "is an isomorphism if either $M$ or $N$ is $I$-power torsion, and", "\\item the base change functor $M \\mapsto M \\otimes_R S$ defines an", "equivalence of categories between $I$-power torsion modules", "and $IS$-power torsion modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that the equivalent conditions of both", "Lemma \\ref{lemma-characterize-flatness-on-torsion} and", "Lemma \\ref{lemma-neighbourhood-isomorphism}", "are satisfied. We will use these without further mention.", "We first prove (1). Let $M$ be any $R$-module.", "Set $M' = M/M[I^\\infty]$ and consider the exact sequence", "$$", "0 \\to M[I^\\infty] \\to M \\to M' \\to 0", "$$", "As $M[I^\\infty] = M[I^\\infty] \\otimes_R S$ we see that it suffices to", "show that $(M' \\otimes_R S)[(IS)^\\infty] = 0$.", "Write $I = (f_1, \\ldots, f_t)$. By", "Lemma \\ref{lemma-divide-by-torsion}", "we see that $M'[I^\\infty] = 0$. Hence for every $n > 0$ the map", "$$", "M' \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots t} M',", "\\quad", "x \\longmapsto (f_1^n x, \\ldots, f_t^n x)", "$$", "is injective. As $S$ is flat over $R$ also the corresponding map", "$M' \\otimes_R S \\to \\bigoplus_{i = 1, \\ldots t} M' \\otimes_R S$", "is injective. This means that $(M' \\otimes_R S)[I^n] = 0$ as desired.", "\\medskip\\noindent", "Next we prove (2). If $N$ is $I$-power torsion, then", "$N \\otimes_R S = N$ and the displayed map of (2) is an isomorphism by", "Algebra, Lemma \\ref{algebra-lemma-adjoint-tensor-restrict}.", "If $M$ is $I$-power torsion, then the image of any map", "$M \\to N$ factors through $M[I^\\infty]$ and the image of any map", "$M \\otimes_R S \\to N \\otimes_R S$ factors through", "$(N \\otimes_R S)[(IS)^\\infty]$. Hence in this case", "part (1) guarantees that we may replace $N$ by $N[I^\\infty]$", "and the result follows from the case where $N$ is $I$-power torsion", "we just discussed.", "\\medskip\\noindent", "Next we prove (3). The functor is fully faithful by (2).", "For essential surjectivity, we simply note that for any $IS$-power torsion", "$S$-module $N$, the natural map $N \\otimes_R S \\to N$ is an isomorphism." ], "refs": [ "more-algebra-lemma-characterize-flatness-on-torsion", "more-algebra-lemma-neighbourhood-isomorphism", "more-algebra-lemma-divide-by-torsion", "algebra-lemma-adjoint-tensor-restrict" ], "ref_ids": [ 10339, 10340, 10335, 374 ] } ], "ref_ids": [] }, { "id": 10342, "type": "theorem", "label": "more-algebra-lemma-map-identifies-koszul-and-cech-complexes", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-map-identifies-koszul-and-cech-complexes", "contents": [ "Assume $\\varphi : R \\to S$ is a flat ring map and $I \\subset R$ is a", "finitely generated ideal such that $R/I \\to S/IS$ is an isomorphism.", "For any $f_1, \\ldots, f_r \\in R$ such that $V(f_1, \\ldots, f_r) = V(I)$", "\\begin{enumerate}", "\\item the map of Koszul complexes", "$K(R, f_1, \\ldots, f_r) \\to K(S, f_1, \\ldots, f_r)$ is a quasi-isomorphism, and", "\\item The map of extended alternating {\\v C}ech complexes", "$$", "\\xymatrix{", "R \\to \\prod_{i_0} R_{f_{i_0}} \\to \\prod_{i_0 < i_1} R_{f_{i_0}f_{i_1}}", "\\to \\ldots \\to R_{f_1\\ldots f_r} \\ar[d] \\\\", "S \\to \\prod_{i_0} S_{f_{i_0}} \\to \\prod_{i_0 < i_1} S_{f_{i_0}f_{i_1}}", "\\to \\ldots \\to S_{f_1\\ldots f_r}", "}", "$$", "is a quasi-isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In both cases we have a complex $K_\\bullet$ of $R$ modules and we want", "to show that $K_\\bullet \\to K_\\bullet \\otimes_R S$ is a quasi-isomorphism.", "By Lemma \\ref{lemma-neighbourhood-isomorphism} and the flatness of", "$R \\to S$ this will hold as soon as all homology groups of $K$ are $I$-power", "torsion. This is true for the Koszul complex by", "Lemma \\ref{lemma-homotopy-koszul} and for the extended alternating {\\v C}ech", "complex by Lemma \\ref{lemma-extended-alternating-torsion}." ], "refs": [ "more-algebra-lemma-neighbourhood-isomorphism", "more-algebra-lemma-homotopy-koszul", "more-algebra-lemma-extended-alternating-torsion" ], "ref_ids": [ 10340, 9960, 9971 ] } ], "ref_ids": [] }, { "id": 10343, "type": "theorem", "label": "more-algebra-lemma-naive-Koszul-complex", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-naive-Koszul-complex", "contents": [ "Let $R$ be a ring. Let $I = (f_1, \\ldots, f_n)$ be a finitely generated ideal", "of $R$. Let $M$ be the $R$-module generated by elements", "$e_1, \\ldots, e_n$ subject to the relations $f_i e_j - f_j e_i = 0$.", "There exists a short exact sequence", "$$", "0 \\to K \\to M \\to I \\to 0", "$$", "such that $K$ is annihilated by $I$." ], "refs": [], "proofs": [ { "contents": [ "This is just a truncation of the Koszul complex.", "The map $M \\to I$ is determined by the rule $e_i \\mapsto f_i$. If", "$m = \\sum a_i e_i$ is in the kernel of $M \\to I$, i.e., $\\sum a_i f_i = 0$,", "then $f_j m = \\sum f_j a_i e_i = (\\sum f_i a_i) e_j = 0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10344, "type": "theorem", "label": "more-algebra-lemma-explicit-ext", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-explicit-ext", "contents": [ "Let $R$ be a ring. Let $I = (f_1, \\ldots, f_n)$ be a finitely generated ideal", "of $R$. For any $R$-module $N$ set", "$$", "H_1(N, f_\\bullet) =", "\\frac{\\{(x_1, \\ldots, x_n) \\in N^{\\oplus n} \\mid f_i x_j = f_j x_i \\}}", "{\\{f_1x, \\ldots, f_nx) \\mid x \\in N\\}}", "$$", "For any $R$-module $N$ there exists a canonical short exact sequence", "$$", "0 \\to \\Ext_R(R/I, N) \\to H_1(N, f_\\bullet) \\to \\Hom_R(K, N)", "$$", "where $K$ is as in", "Lemma \\ref{lemma-naive-Koszul-complex}." ], "refs": [ "more-algebra-lemma-naive-Koszul-complex" ], "proofs": [ { "contents": [ "The notation above indicates the $\\Ext$-groups in $\\text{Mod}_R$", "as defined in", "Homology, Section \\ref{homology-section-extensions}.", "These are denoted $\\Ext_R(M, N)$. Using the long exact sequence of", "Homology, Lemma \\ref{homology-lemma-six-term-sequence-ext}", "associated to the short exact sequence $0 \\to I \\to R \\to R/I \\to 0$", "and the fact that $\\Ext_R(R, N) = 0$ we see that", "$$", "\\Ext_R(R/I, N) =", "\\Coker(N \\longrightarrow \\Hom(I, N))", "$$", "Using the short exact sequence of", "Lemma \\ref{lemma-naive-Koszul-complex}", "we see that we get a complex", "$$", "N \\to \\Hom(M, N) \\to \\Hom_R(K, N)", "$$", "whose homology in the middle is canonically isomorphic to", "$\\Ext_R(R/I, N)$. The proof of the lemma is now complete", "as the cokernel of the first map", "is canonically isomorphic to $H_1(N, f_\\bullet)$." ], "refs": [ "homology-lemma-six-term-sequence-ext", "more-algebra-lemma-naive-Koszul-complex" ], "ref_ids": [ 12032, 10343 ] } ], "ref_ids": [ 10343 ] }, { "id": 10345, "type": "theorem", "label": "more-algebra-lemma-koszul-homology-annihilated", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-koszul-homology-annihilated", "contents": [ "Let $R$ be a ring. Let $I = (f_1, \\ldots, f_n)$ be a finitely generated ideal", "of $R$. For any $R$-module $N$ the Koszul homology group", "$H_1(N, f_\\bullet)$ defined in", "Lemma \\ref{lemma-explicit-ext}", "is annihilated by $I$." ], "refs": [ "more-algebra-lemma-explicit-ext" ], "proofs": [ { "contents": [ "Let $(x_1, \\ldots, x_n) \\in N^{\\oplus n}$ with $f_i x_j = f_j x_i$.", "Then we have $f_i(x_1, \\ldots, x_n) = (f_i x_i, \\ldots, f_i x_n)$.", "In other words $f_i$ annihilates $H_1(N, f_\\bullet)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10344 ] }, { "id": 10346, "type": "theorem", "label": "more-algebra-lemma-neighbourhood-extensions", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-neighbourhood-extensions", "contents": [ "Assume $\\varphi : R \\to S$ is a flat ring map and $I \\subset R$ is a", "finitely generated ideal such that $R/I \\to S/IS$ is an isomorphism.", "Let $M$, $N$ be $R$-modules. Assume $M$ is $I$-power torsion.", "Given an short exact sequence", "$$", "0 \\to N \\otimes_R S \\to \\tilde E \\to M \\otimes_R S \\to 0", "$$", "there exists a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "N \\ar[r] \\ar[d] &", "E \\ar[r] \\ar[d] &", "M \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "N \\otimes_R S \\ar[r] &", "\\tilde E \\ar[r] &", "M \\otimes_R S \\ar[r] &", "0", "}", "$$", "with exact rows." ], "refs": [], "proofs": [ { "contents": [ "As $M$ is $I$-power torsion we see that $M \\otimes_R S = M$, see", "Lemma \\ref{lemma-neighbourhood-isomorphism}.", "We will use this identification without further mention.", "As $R \\to S$ is flat, the base change functor is exact and we", "obtain a functorial map of $\\Ext$-groups", "$$", "\\Ext_R(M, N)", "\\longrightarrow", "\\Ext_S(M \\otimes_R S, N \\otimes_R S),", "$$", "see", "Homology, Lemma \\ref{homology-lemma-exact-functor-ext}.", "The claim of the lemma is that this map is surjective when", "$M$ is $I$-power torsion. In fact we will show that it is an", "isomorphism. By", "Lemma \\ref{lemma-I-power-torsion-presentation}", "we can find a surjection $M' \\to M$ with $M'$ a direct sum of", "modules of the form $R/I^n$. Using the long exact sequence of", "Homology, Lemma \\ref{homology-lemma-six-term-sequence-ext}", "we see that it suffices to prove the lemma for $M'$.", "Using compatibility of $\\Ext$ with direct sums (details omitted)", "we reduce to the case where $M = R/I^n$ for some $n$.", "\\medskip\\noindent", "Let $f_1, \\ldots, f_t$ be generators for $I^n$. By", "Lemma \\ref{lemma-explicit-ext}", "we have a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Ext_R(R/I^n, N) \\ar[r] \\ar[d] &", "H_1(N, f_\\bullet) \\ar[r] \\ar[d] &", "\\Hom_R(K, N) \\ar[d] \\\\", "0 \\ar[r] &", "\\Ext_S(S/I^nS, N \\otimes S) \\ar[r] &", "H_1(N \\otimes S, f_\\bullet) \\ar[r] &", "\\Hom_S(K \\otimes S, N \\otimes S)", "}", "$$", "with exact rows where $K$ is as in", "Lemma \\ref{lemma-naive-Koszul-complex}.", "Hence it suffices to prove that the two right vertical arrows are", "isomorphisms. Since $K$ is annihilated by $I^n$ we see that", "$\\Hom_R(K, N) = \\Hom_S(K \\otimes_R S, N \\otimes_R S)$ by", "Lemma \\ref{lemma-neighbourhood-equivalence}.", "As $R \\to S$ is flat we have", "$H_1(N, f_\\bullet) \\otimes_R S = H_1(N \\otimes_R S, f_\\bullet)$.", "As $H_1(N, f_\\bullet)$ is annihilated by $I^n$, see", "Lemma \\ref{lemma-koszul-homology-annihilated}", "we have $H_1(N, f_\\bullet) \\otimes_R S = H_1(N, f_\\bullet)$ by", "Lemma \\ref{lemma-neighbourhood-isomorphism}." ], "refs": [ "more-algebra-lemma-neighbourhood-isomorphism", "homology-lemma-exact-functor-ext", "more-algebra-lemma-I-power-torsion-presentation", "homology-lemma-six-term-sequence-ext", "more-algebra-lemma-explicit-ext", "more-algebra-lemma-naive-Koszul-complex", "more-algebra-lemma-neighbourhood-equivalence", "more-algebra-lemma-koszul-homology-annihilated", "more-algebra-lemma-neighbourhood-isomorphism" ], "ref_ids": [ 10340, 12035, 10333, 12032, 10344, 10343, 10341, 10345, 10340 ] } ], "ref_ids": [] }, { "id": 10347, "type": "theorem", "label": "more-algebra-lemma-recover-module-from-glueing-data", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-recover-module-from-glueing-data", "contents": [ "Assume $\\varphi : R \\to S$ is a flat ring map and", "$I = (f_1, \\ldots, f_t) \\subset R$ is an ideal such that", "$R/I \\to S/IS$ is an isomorphism.", "Let $M$ be an $R$-module. Then the", "complex (\\ref{equation-glueing-complex})", "is exact." ], "refs": [], "proofs": [ { "contents": [ "First proof. Denote $\\check{\\mathcal{C}}_R \\to \\check{\\mathcal{C}}_S$", "the quasi-isomorphism of extended alternating {\\v C}ech complexes of", "Lemma \\ref{lemma-map-identifies-koszul-and-cech-complexes}.", "Since these complexes are bounded with flat terms, we see that", "$M \\otimes_R \\check{\\mathcal{C}}_R \\to M \\otimes_R \\check{\\mathcal{C}}_S$", "is a quasi-isomorphism too (Lemmas", "\\ref{lemma-derived-tor-quasi-isomorphism} and", "\\ref{lemma-derived-tor-quasi-isomorphism-other-side}). Now the complex", "(\\ref{equation-glueing-complex}) is a truncation of the cone", "of the map", "$M \\otimes_R \\check{\\mathcal{C}}_R \\to M \\otimes_R \\check{\\mathcal{C}}_S$", "and we win.", "\\medskip\\noindent", "Second computational proof.", "Let $m \\in M$. If $\\alpha(m) = 0$, then $m \\in M[I^\\infty]$, see", "Lemma \\ref{lemma-torsion-free}. Pick $n$ such that $I^n m = 0$", "and consider the map $\\varphi : R/I^n \\to M$.", "If $m \\otimes 1 = 0$, then $\\varphi \\otimes 1_S = 0$, hence", "$\\varphi = 0$ (see", "Lemma \\ref{lemma-neighbourhood-equivalence})", "hence $m = 0$. In this way we see that $\\alpha$ is injective.", "\\medskip\\noindent", "Let $(m', m'_1, \\ldots, m'_t) \\in \\Ker(\\beta)$.", "Write $m'_i = m_i/f_i^n$ for some $n > 0$ and $m_i \\in M$.", "We may, after possibly enlarging $n$ assume that", "$f_i^n m' = m_i \\otimes 1$ in $M \\otimes_R S$ and", "$f_j^nm_i - f_i^nm_j = 0$ in $M$.", "In particular we see that", "$(m_1, \\ldots, m_t)$ defines an element $\\xi$ of", "$H_1(M, (f_1^n, \\ldots, f_t^n))$.", "Since $H_1(M, (f_1^n, \\ldots, f_t^n))$ is annihilated by $I^{tn + 1}$ (see", "Lemma \\ref{lemma-koszul-homology-annihilated})", "and since $R \\to S$ is flat we see that", "$$", "H_1(M, (f_1^n, \\ldots, f_t^n)) =", "H_1(M, (f_1^n, \\ldots, f_t^n)) \\otimes_R S =", "H_1(M \\otimes_R S, (f_1^n, \\ldots, f_t^n))", "$$", "by", "Lemma \\ref{lemma-neighbourhood-isomorphism}", "The existence of $m'$ implies that $\\xi$ maps to zero in the last group, i.e.,", "the element $\\xi$ is zero. Thus there exists an $m \\in M$ such that", "$m_i = f_i^n m$. Then $(m', m'_1, \\ldots, m'_t) - \\alpha(m)", "= (m'', 0, \\ldots, 0)$ for some $m'' \\in (M \\otimes_R S)[(IS)^\\infty]$.", "By", "Lemma \\ref{lemma-neighbourhood-equivalence}", "we conclude that $m'' \\in M[I^\\infty]$ and we win." ], "refs": [ "more-algebra-lemma-map-identifies-koszul-and-cech-complexes", "more-algebra-lemma-derived-tor-quasi-isomorphism", "more-algebra-lemma-derived-tor-quasi-isomorphism-other-side", "more-algebra-lemma-torsion-free", "more-algebra-lemma-neighbourhood-equivalence", "more-algebra-lemma-koszul-homology-annihilated", "more-algebra-lemma-neighbourhood-isomorphism", "more-algebra-lemma-neighbourhood-equivalence" ], "ref_ids": [ 10342, 10128, 10132, 10334, 10341, 10345, 10340, 10341 ] } ], "ref_ids": [] }, { "id": 10348, "type": "theorem", "label": "more-algebra-lemma-H0-inverse", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-H0-inverse", "contents": [ "Assume $\\varphi : R \\to S$ is a flat ring map and", "$I = (f_1, \\ldots, f_t) \\subset R$ is an ideal such that", "$R/I \\to S/IS$ is an isomorphism. Then the functor $H^0$", "is a left quasi-inverse to the functor $\\text{Can}$ of", "Remark \\ref{remark-glueing-data}." ], "refs": [ "more-algebra-remark-glueing-data" ], "proofs": [ { "contents": [ "This is a reformulation of", "Lemma \\ref{lemma-recover-module-from-glueing-data}." ], "refs": [ "more-algebra-lemma-recover-module-from-glueing-data" ], "ref_ids": [ 10347 ] } ], "ref_ids": [ 10662 ] }, { "id": 10349, "type": "theorem", "label": "more-algebra-lemma-exact", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-exact", "contents": [ "Assume $\\varphi : R \\to S$ is a flat ring map and let", "$I = (f_1, \\ldots, f_t) \\subset R$ be an ideal.", "Then $\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$ is an abelian category, and", "the functor $\\text{Can}$ is exact and commutes with arbitrary colimits." ], "refs": [], "proofs": [ { "contents": [ "Given a morphism", "$(\\varphi', \\varphi_i) :", "(M', M_i, \\alpha_i, \\alpha_{ij})", "\\to", "(N', N_i, \\beta_i, \\beta_{ij})$", "of the category $\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$", "we see that its kernel exists and is equal to the object", "$(\\Ker(\\varphi'), \\Ker(\\varphi_i), \\alpha_i, \\alpha_{ij})$", "and its cokernel exists and is equal to the object", "$(\\Coker(\\varphi'), \\Coker(\\varphi_i), \\beta_i, \\beta_{ij})$.", "This works because $R \\to S$ is flat, hence taking kernels/cokernels", "commutes with $- \\otimes_R S$. Details omitted.", "The exactness follows from the $R$-flatness of $R_{f_i}$ and $S$, while", "commuting with colimits follows as tensor products commute with colimits." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10350, "type": "theorem", "label": "more-algebra-lemma-equivalence-I-unit", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-equivalence-I-unit", "contents": [ "Let $\\varphi : R \\to S$ be a flat ring map and $(f_1, \\ldots, f_t) = R$.", "Then $\\text{Can}$ and $H^0$ are quasi-inverse equivalences of categories", "$$", "\\text{Mod}_R = \\text{Glue}(R \\to S, f_1, \\ldots, f_t)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Consider an object $\\mathbf{M} = (M', M_i, \\alpha_i, \\alpha_{ij})$", "of $\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$. By", "Algebra, Lemma \\ref{algebra-lemma-glue-modules}", "there exists a unique module $M$ and isomorphisms", "$M_{f_i} \\to M_i$ which recover the glueing data $\\alpha_{ij}$.", "Then both $M'$ and $M \\otimes_R S$ are $S$-modules", "which recover the modules $M_i \\otimes_R S$ upon localizing at $f_i$.", "Whence there is a canonical isomorphism $M \\otimes_R S \\to M'$.", "This shows that $\\mathbf{M}$ is in the essential image of $\\text{Can}$.", "Combined with", "Lemma \\ref{lemma-H0-inverse}", "the lemma follows." ], "refs": [ "algebra-lemma-glue-modules", "more-algebra-lemma-H0-inverse" ], "ref_ids": [ 417, 10348 ] } ], "ref_ids": [] }, { "id": 10351, "type": "theorem", "label": "more-algebra-lemma-base-change-glue", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-glue", "contents": [ "Let $\\varphi : R \\to S$ be a flat ring map and $I = (f_1, \\ldots, f_t)$", "and ideal. Let $R \\to R'$ be a flat ring map, and set $S' = S \\otimes_R R'$.", "Then we obtain a commutative diagram of categories and functors", "$$", "\\xymatrix{", "\\text{Mod}_R \\ar[r]_-{\\text{Can}} \\ar[d]_{-\\otimes_R R'} &", "\\text{Glue}(R \\to S, f_1, \\ldots, f_t) \\ar[r]_-{H^0} \\ar[d]^{-\\otimes_R R'} &", "\\text{Mod}_R \\ar[d]^{-\\otimes_R R'} \\\\", "\\text{Mod}_{R'} \\ar[r]^-{\\text{Can}} &", "\\text{Glue}(R' \\to S', f_1, \\ldots, f_t) \\ar[r]^-{H^0} &", "\\text{Mod}_{R'}", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10352, "type": "theorem", "label": "more-algebra-lemma-application-formal-glueing", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-application-formal-glueing", "contents": [ "Let $\\varphi : R \\to S$ be a flat ring map and let $I \\subset R$ be a", "finitely generated ideal such that $R/I \\to S/IS$ is an isomorphism. ", "\\begin{enumerate}", "\\item Given an $R$-module $N$, an $S$-module $M'$ and an $S$-module", "map $\\varphi : M' \\to N \\otimes_R S$ whose kernel and cokernel are", "$I$-power torsion, there exists an $R$-module map", "$\\psi : M \\to N$ and an isomorphism $M \\otimes_R S = M'$", "compatible with $\\varphi$ and $\\psi$.", "\\item Given an $R$-module $M$, an $S$-module $N'$ and an $S$-module", "map $\\varphi : M \\otimes_R S \\to N'$ whose kernel and cokernel are", "$I$-power torsion, there exists an $R$-module map", "$\\psi : M \\to N$ and an isomorphism $N \\otimes_R S = N'$", "compatible with $\\varphi$ and $\\psi$.", "\\end{enumerate}", "In both cases we have $\\Ker(\\varphi) \\cong \\Ker(\\psi)$ and", "$\\Coker(\\varphi) \\cong \\Coker(\\psi)$." ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Say $I = (f_1, \\ldots, f_t)$. It is clear that", "the localization $\\varphi_{f_i}$ is an isomorphism.", "Thus we see that $(M', N_{f_i}, \\varphi_{f_i}, can_{ij})$ is an", "object of $\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$, see", "Remark \\ref{remark-glueing-data}.", "By Proposition \\ref{proposition-equivalence}", "we conclude that there exists an $R$-module $M$ such that", "$M' = M \\otimes_R S$ and $N_{f_i} = M_{f_i}$ compatibly", "with the isomorphisms $\\varphi_{f_i}$ and $can_{ij}$. There is a", "morphism", "$$", "(M \\otimes_R S, M_{f_i}, can_i, can_{ij}) =", "(M', N_{f_i}, \\varphi_{f_i}, can_{ij})", "\\to", "(N \\otimes_R S, N_{f_i}, can_i, can_{ij})", "$$", "of $\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$", "which uses $\\varphi$ in the first component. This", "corresponds to an $R$-module map $\\psi : M \\to N$ (by the equivalence of", "categories of Proposition \\ref{proposition-equivalence}).", "The composition of the base change of $M \\to N$ with the", "isomorphism $M' \\cong M \\otimes_R S$ is $\\varphi$, in other words", "$M \\to N$ is compatible with $\\varphi$.", "\\medskip\\noindent", "Proof of (2). This is just the dual of the argument above.", "Namely, the localization $\\varphi_{f_i}$ is an isomorphism.", "Thus we see that $(N', M_{f_i}, \\varphi_{f_i}^{-1}, can_{ij})$ is an", "object of $\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$, see", "Remark \\ref{remark-glueing-data}.", "By Proposition \\ref{proposition-equivalence}", "we conclude that there exists an $R$-module $N$ such that", "$N' = N \\otimes_R S$ and $N_{f_i} = M_{f_i}$ compatibly", "with the isomorphisms $\\varphi_{f_i}^{-1}$ and $can_{ij}$. There is a", "morphism", "$$", "(M \\otimes_R S, M_{f_i}, can_i, can_{ij}) \\to", "(N', M_{f_i}, \\varphi_{f_i}, can_{ij}) =", "(N \\otimes_R S, N_{f_i}, can_i, can_{ij})", "$$", "of $\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$", "which uses $\\varphi$ in the first component. This", "corresponds to an $R$-module map $\\psi : M \\to N$ (by the equivalence of", "categories of Proposition \\ref{proposition-equivalence}).", "The composition of the base change of $M \\to N$ with the", "isomorphism $N' \\cong N \\otimes_R S$ is $\\varphi$, in other words", "$M \\to N$ is compatible with $\\varphi$.", "\\medskip\\noindent", "The final statement follows for example from", "Lemma \\ref{lemma-neighbourhood-equivalence}." ], "refs": [ "more-algebra-remark-glueing-data", "more-algebra-proposition-equivalence", "more-algebra-proposition-equivalence", "more-algebra-remark-glueing-data", "more-algebra-proposition-equivalence", "more-algebra-proposition-equivalence", "more-algebra-lemma-neighbourhood-equivalence" ], "ref_ids": [ 10662, 10587, 10587, 10662, 10587, 10587, 10341 ] } ], "ref_ids": [] }, { "id": 10353, "type": "theorem", "label": "more-algebra-lemma-same-quotients", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-same-quotients", "contents": [ "Let $R$ be a ring and let $f \\in R$. For every positive integer $n$ the map", "$R/f^nR \\to R^\\wedge/f^n R^\\wedge$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Algebra, Lemma", "\\ref{algebra-lemma-hathat-finitely-generated}." ], "refs": [ "algebra-lemma-hathat-finitely-generated" ], "ref_ids": [ 859 ] } ], "ref_ids": [] }, { "id": 10354, "type": "theorem", "label": "more-algebra-lemma-torsion-completion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-torsion-completion", "contents": [ "\\begin{reference}", "Slight generalization of \\cite[Lemme~1]{Beauville-Laszlo}.", "\\end{reference}", "Let $R$ be a ring, let $f \\in R$ be an element, and let $R \\to R'$ be a ring", "map which induces isomorphisms $R/f^nR \\to R'/f^nR'$ for $n > 0$.", "For any $f$-power torsion $R$-module $M$ the map $M \\to M \\otimes_R R'$", "is an isomorphism.", "For example, we have $M \\cong M \\otimes_R R^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "If $M$ is annihilated by $f^n$, then", "$$", "M \\otimes_R R' \\cong", "M \\otimes_{R/f^nR} R'/f^n R' \\cong", "M \\otimes_{R/f^nR} R/f^n R \\cong M.", "$$", "Since $M = \\bigcup M[f^n]$ and since tensor products commute", "with direct limits", "(Algebra, Lemma \\ref{algebra-lemma-tensor-products-commute-with-limits}),", "we obtain the desired isomorphism. The last statement is a special", "case of the first statement by Lemma \\ref{lemma-same-quotients}." ], "refs": [ "algebra-lemma-tensor-products-commute-with-limits", "more-algebra-lemma-same-quotients" ], "ref_ids": [ 363, 10353 ] } ], "ref_ids": [] }, { "id": 10355, "type": "theorem", "label": "more-algebra-lemma-BL-faithful", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-BL-faithful", "contents": [ "Let $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map", "which induces isomorphisms $R/f^nR \\to R'/f^nR'$ for $n > 0$.", "The $R$-module $R' \\oplus R_f$ is faithful: for every nonzero", "$R$-module $M$, the module $M \\otimes_R (R' \\oplus R_f)$", "is also nonzero. For example, if $M$ is nonzero, then", "$M \\otimes_R (R^\\wedge \\oplus R_f)$ is nonzero." ], "refs": [], "proofs": [ { "contents": [ "If $M \\neq 0$ but $M \\otimes_R R_f = 0$, then $M$ is $f$-power torsion.", "By Lemma \\ref{lemma-torsion-completion} we find that", "$M \\otimes_R R' \\cong M \\neq 0$. The last statement is a special", "case of the first statement by Lemma \\ref{lemma-same-quotients}." ], "refs": [ "more-algebra-lemma-torsion-completion", "more-algebra-lemma-same-quotients" ], "ref_ids": [ 10354, 10353 ] } ], "ref_ids": [] }, { "id": 10356, "type": "theorem", "label": "more-algebra-lemma-cover-spec", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cover-spec", "contents": [ "Let $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map", "which induces an isomorphism $R/fR \\to R'/fR'$.", "The map $\\Spec(R') \\amalg \\Spec(R_f) \\to \\Spec(R)$ is surjective.", "For example, the map", "$\\Spec(R^\\wedge) \\amalg \\Spec(R_f) \\to \\Spec(R)$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\Spec(R) = V(f) \\amalg D(f)$ where $V(f) = \\Spec(R/fR)$", "and $D(f) = \\Spec(R_f)$, see", "Algebra, Section \\ref{algebra-section-spectrum-ring}", "and especially Lemmas \\ref{algebra-lemma-spec-closed} and", "\\ref{algebra-lemma-standard-open}.", "Thus the lemma follows as the map $R \\to R/fR$ factors through $R'$.", "The last statement is a special", "case of the first statement by Lemma \\ref{lemma-same-quotients}." ], "refs": [ "algebra-lemma-spec-closed", "algebra-lemma-standard-open", "more-algebra-lemma-same-quotients" ], "ref_ids": [ 393, 392, 10353 ] } ], "ref_ids": [] }, { "id": 10357, "type": "theorem", "label": "more-algebra-lemma-faithful-descent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-faithful-descent", "contents": [ "\\begin{reference}", "Slight generalization of \\cite[Lemme~2(a)]{Beauville-Laszlo}.", "\\end{reference}", "Let $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map", "which induces isomorphisms $R/f^nR \\to R'/f^nR'$ for $n > 0$.", "An $R$-module $M$ is finitely generated if and only if the", "($R' \\oplus R_f$)-module $M \\otimes_R (R' \\oplus R_f)$ is finitely generated.", "For example, if $M \\otimes_R (R^\\wedge \\oplus R_f)$ is finitely generated", "as a module over $R^\\wedge \\oplus R_f$, then $M$ is a finitely generated", "$R$-module." ], "refs": [], "proofs": [ { "contents": [ "The `only if' is clear, so we assume that", "$M \\otimes_R (R' \\oplus R_f)$ is finitely generated. In this case, by", "writing each generator as a sum of simple tensors,", "$M \\otimes_R (R' \\oplus R_f)$ admits a finite", "generating set consisting of elements of $M$.", "That is, there exists a morphism from a finite free", "$R$-module to $M$ whose cokernel is killed by tensoring with $R' \\oplus R_f$;", "we may thus deduce $M$ is finite generated by applying", "Lemma \\ref{lemma-BL-faithful} to this cokernel.", "The last statement is a special", "case of the first statement by Lemma \\ref{lemma-same-quotients}." ], "refs": [ "more-algebra-lemma-BL-faithful", "more-algebra-lemma-same-quotients" ], "ref_ids": [ 10355, 10353 ] } ], "ref_ids": [] }, { "id": 10358, "type": "theorem", "label": "more-algebra-lemma-same-f-torsion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-same-f-torsion", "contents": [ "Let $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map", "which induces isomorphisms $R/f^nR \\to R'/f^nR'$ for $n > 0$.", "The sequence (\\ref{equation-BL-cech-re}) is", "\\begin{enumerate}", "\\item exact on the right,", "\\item exact on the left if and only if $R[f^\\infty] \\to R'[f^\\infty]$", "is injective, and", "\\item exact in the middle if and only if $R[f^\\infty] \\to R'[f^\\infty]$", "is surjective.", "\\end{enumerate}", "In particular, $(R \\to R', f)$ is a glueing pair if and only if", "$R[f^\\infty] \\to R'[f^\\infty]$ is bijective. For example, $(R, f)$", "is a glueing pair if and only if", "$R[f^\\infty] \\to R^\\wedge[f^\\infty]$ is bijective." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in R'_f$. Write $x = x'/f^n$ with $x' \\in R'$.", "Write $x' = x'' + f^n y$ with $x'' \\in R$ and $y \\in R'$.", "Then we see that $(y, -x''/f^n)$ maps to $x$. Thus (1) holds.", "\\medskip\\noindent", "Part (2) follows from the fact that $\\Ker(R \\to R_f) = R[f^\\infty]$.", "\\medskip\\noindent", "If the sequence is exact in the middle, then elements of the form", "$(x, 0)$ with $x \\in R'[f^\\infty]$ are in the image of", "the first arrow. This implies that $R[f^\\infty] \\to R'[f^\\infty]$", "is surjective. Conversely, assume that $R[f^\\infty] \\to R'[f^\\infty]$", "is surjective. Let $(x, y)$ be an element in the middle", "which maps to zero on the right. Write $y = y'/f^n$ for some $y' \\in R$.", "Then we see that $f^n x - y'$ is annihilated by some power of $f$ in", "$R'$. By assumption we can write $f^nx - y' = z$ for some", "$z \\in R[f^\\infty]$. Then $y = y''/f^n$ where $y'' = y' + z$", "is in the kernel of $R \\to R/f^nR$. Hence we see that $y$ can be", "represented as $y'''/1$ for some $y''' \\in R$. Then", "$x - y'''$ is in $R'[f^\\infty]$. Thus $x - y''' = z' \\in R[f^\\infty]$.", "Then $(x, y'''/1) = (y''' + z', (y''' + z')/1)$ as desired.", "\\medskip\\noindent", "The last statement of the lemma is a special case of the penultimate statement", "by Lemma \\ref{lemma-same-quotients}." ], "refs": [ "more-algebra-lemma-same-quotients" ], "ref_ids": [ 10353 ] } ], "ref_ids": [] }, { "id": 10359, "type": "theorem", "label": "more-algebra-lemma-same-f-torsion-module", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-same-f-torsion-module", "contents": [ "Let $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map", "which induces isomorphisms $R/f^nR \\to R'/f^nR'$ for $n > 0$.", "The sequence (\\ref{equation-BL-cech-mod-re}) is", "\\begin{enumerate}", "\\item exact on the right,", "\\item exact on the left if and only if", "$M[f^\\infty] \\to (M \\otimes_R R')[f^\\infty]$", "is injective, and", "\\item exact in the middle if and only if", "$M[f^\\infty] \\to (M \\otimes_R R')[f^\\infty]$", "is surjective.", "\\end{enumerate}", "Thus $M$ is glueable for $(R \\to R', f)$ if and only if", "$M[f^\\infty] \\to (M \\otimes_R R')[f^\\infty]$ is bijective.", "If $(R \\to R', f)$ is a glueing pair, then $M$ is glueable for $(R \\to R', f)$", "if and only if $M[f^\\infty] \\to (M \\otimes_R R')[f^\\infty]$ is injective.", "For example, if $(R, f)$ is a glueing pair, then $M$ is glueable", "if and only if $M[f^\\infty] \\to (M \\otimes_R R^\\wedge)[f^\\infty]$ is", "injective." ], "refs": [], "proofs": [ { "contents": [ "We will use the results of Lemma \\ref{lemma-same-f-torsion} without", "further mention. The functor $M \\otimes_R -$ is right exact", "(Algebra, Lemma \\ref{algebra-lemma-tensor-product-exact})", "hence we get (1).", "\\medskip\\noindent", "The kernel of $M \\to M \\otimes_R R_f = M_f$ is $M[f^\\infty]$.", "Thus (2) follows.", "\\medskip\\noindent", "If the sequence is exact in the middle, then elements of the form", "$(x, 0)$ with $x \\in (M \\otimes_R R')[f^\\infty]$ are in the image of", "the first arrow. This implies that", "$M[f^\\infty] \\to (M \\otimes_R R')[f^\\infty]$", "is surjective. Conversely, assume that", "$M[f^\\infty] \\to (M \\otimes_R R')[f^\\infty]$", "is surjective. Let $(x, y)$ be an element in the middle", "which maps to zero on the right. Write $y = y'/f^n$ for some $y' \\in M$.", "Then we see that $f^n x - y'$ is annihilated by some power of $f$ in", "$M \\otimes_R R'$. By assumption we can write $f^nx - y' = z$ for some", "$z \\in M[f^\\infty]$. Then $y = y''/f^n$ where $y'' = y' + z$", "is in the kernel of $M \\to M/f^nM$. Hence we see that $y$ can be", "represented as $y'''/1$ for some $y''' \\in M$. Then", "$x - y'''$ is in $(M \\otimes_R R')[f^\\infty]$.", "Thus $x - y''' = z' \\in M[f^\\infty]$.", "Then $(x, y'''/1) = (y''' + z', (y''' + z')/1)$ as desired.", "\\medskip\\noindent", "If $(R \\to R', f)$ is a glueing pair, then (\\ref{equation-BL-cech-mod-re})", "is exact in the middle for any $M$ by", "Algebra, Lemma \\ref{algebra-lemma-tensor-product-exact}.", "This gives the penultimate statement of the lemma.", "The final statement of the lemma follows from this and", "the fact that $(R, f)$ is a glueing pair if and only if", "$(R \\to R^\\wedge, f)$ is a glueing pair." ], "refs": [ "more-algebra-lemma-same-f-torsion", "algebra-lemma-tensor-product-exact", "algebra-lemma-tensor-product-exact" ], "ref_ids": [ 10358, 364, 364 ] } ], "ref_ids": [] }, { "id": 10360, "type": "theorem", "label": "more-algebra-lemma-first-tor", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-first-tor", "contents": [ "Let $(R \\to R', f)$ be a glueing pair. Then", "$\\text{Tor}^R_1(R', f^n R) = 0$ for each $n > 0$." ], "refs": [], "proofs": [ { "contents": [ "From the exact sequence $0 \\to R[f^n] \\to R \\to f^n R \\to 0$ we see that it", "suffices to check that $R[f^n] \\otimes_R R' \\to R'$ is injective.", "By Lemma \\ref{lemma-torsion-completion} we have", "$R[f^n] \\otimes_R R' = R[f^n]$ and by Lemma \\ref{lemma-same-f-torsion}", "we see that $R[f^n] \\to R'$ is injective as $(R \\to R', f)$ is a glueing pair." ], "refs": [ "more-algebra-lemma-torsion-completion", "more-algebra-lemma-same-f-torsion" ], "ref_ids": [ 10354, 10358 ] } ], "ref_ids": [] }, { "id": 10361, "type": "theorem", "label": "more-algebra-lemma-first-tor-total", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-first-tor-total", "contents": [ "Let $(R \\to R',f)$ be a glueing pair.", "Then $\\text{Tor}^R_1(R', R/R[f^\\infty]) = 0$." ], "refs": [], "proofs": [ { "contents": [ "We have $R/R[f^\\infty] = \\colim R/R[f^n] = \\colim f^nR$.", "As formation of Tor groups commutes with filtered colimits", "(Algebra, Lemma \\ref{algebra-lemma-tor-commutes-filtered-colimits})", "we may apply Lemma \\ref{lemma-first-tor}." ], "refs": [ "algebra-lemma-tor-commutes-filtered-colimits", "more-algebra-lemma-first-tor" ], "ref_ids": [ 788, 10360 ] } ], "ref_ids": [] }, { "id": 10362, "type": "theorem", "label": "more-algebra-lemma-BL3", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-BL3", "contents": [ "\\begin{reference}", "Slight generalization of \\cite[Lemme 3(a)]{Beauville-Laszlo}", "\\end{reference}", "Let $(R \\to R', f)$ be a glueing pair. For every $R$-module $M$, we have", "$\\text{Tor}^R_1(R', \\Coker(M \\to M_f)) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Set $\\overline{M} = M/M[f^\\infty]$. Then", "$\\Coker(M \\to M_f) \\cong \\Coker(\\overline{M} \\to \\overline{M}_f)$", "hence we may and do assume that $f$ is a nonzerodivisor on $M$.", "In this case $M \\subset M_f$ and $M_f/M = \\colim M/f^nM$ where the", "transition maps are given by multiplication by $f$. Since", "formation of Tor groups commutes with colimits", "(Algebra, Lemma \\ref{algebra-lemma-tor-commutes-filtered-colimits})", "it suffices to show that $\\text{Tor}^R_1(R', M/f^n M) = 0$.", "\\medskip\\noindent", "We first treat the case $M = R/R[f^\\infty]$. By", "Lemma \\ref{lemma-same-f-torsion}", "we have $M \\otimes_R R' = R'/R'[f^\\infty]$.", "From the short exact sequence $0 \\to M \\to M \\to M/f^nM \\to 0$", "we obtain the exact sequence", "$$", "\\xymatrix{", "\\text{Tor}_1^R(R', R/R[f^\\infty]) \\ar[r] &", "\\text{Tor}_1^R(R', M/f^n M) \\ar[r] &", "R'/R'[f^\\infty] \\ar[dll]_{f^n} \\\\", "R'/R'[f^\\infty] \\ar[r] &", "(R'/R'[f^\\infty])/(f^n", "(R'/R'[f^\\infty])) \\ar[r] & 0", "}", "$$", "by Algebra, Lemma \\ref{algebra-lemma-long-exact-sequence-tor}.", "Here the diagonal arrow is injective. Since the first group", "$\\text{Tor}_1^R(R', R/R[f^\\infty])$ is zero by", "Lemma \\ref{lemma-first-tor-total}, we deduce that", "$\\text{Tor}_1^R(R', M/f^nM) = 0$ as desired.", "\\medskip\\noindent", "To treat the general case, choose a surjection $F \\to M$ with $F$ a free", "$R/R[f^\\infty]$-module, and form an exact sequence", "$$", "0 \\to N \\to F/f^n F \\to M/f^n M \\to 0.", "$$", "By Lemma \\ref{lemma-torsion-completion}", "this sequence remains unchanged, and hence", "exact, upon tensoring with $R'$.", "Since $\\text{Tor}^R_1(R', F/f^n F) = 0$ by the", "previous paragraph, we deduce that", "$\\text{Tor}^R_1(R', M/f^n M) = 0$ as desired." ], "refs": [ "algebra-lemma-tor-commutes-filtered-colimits", "more-algebra-lemma-same-f-torsion", "algebra-lemma-long-exact-sequence-tor", "more-algebra-lemma-first-tor-total", "more-algebra-lemma-torsion-completion" ], "ref_ids": [ 788, 10358, 782, 10361, 10354 ] } ], "ref_ids": [] }, { "id": 10363, "type": "theorem", "label": "more-algebra-lemma-BL-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-BL-flat", "contents": [ "Let $(R \\to R', f)$ be a glueing pair. Let $M$ be an $R$-module", "which is not necessarily glueable for $(R \\to R', f)$. Then $M$", "is flat over $R$ if and only if $M \\otimes_R R'$ is flat over $R'$", "and $M_f$ is flat over $R_f$." ], "refs": [], "proofs": [ { "contents": [ "One direction of the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-flat-base-change}.", "For the other direction, assume $M \\otimes_R R'$", "is flat over $R'$ and $M_f$ is flat over $R_f$.", "Let $\\tilde M$ be as in Remark \\ref{remark-what-you-get-for-general-modules}.", "If $\\tilde M$ is flat over $R$, then applying", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}", "to the short exact sequence", "$0 \\to \\Ker(M \\to \\tilde M) \\to M \\to \\tilde M \\to 0$", "we find that $\\Ker(M \\to \\tilde M) \\otimes_R (R' \\oplus R_f)$ is zero.", "Hence $M = \\tilde M$ by Lemma \\ref{lemma-BL-faithful} and we conclude.", "In other words, we may replace $M$ by $\\tilde M$ and assume", "$M$ is glueable for $(R \\to R', f)$. Let $N$ be a second $R$-module.", "It suffices to prove that $\\text{Tor}_1^R(M, N) = 0$, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-flat}.", "\\medskip\\noindent", "The long the exact sequence of Tors associated to the", "short exact sequence $0 \\to R \\to R' \\oplus R_f \\to (R')_f \\to 0$", "and $N$ gives an exact sequence", "$$", "0 \\to \\text{Tor}_1^R(R', N) \\to \\text{Tor}_1^R((R')_f, N)", "$$", "and isomorphisms", "$\\text{Tor}_i^R(R', N) = \\text{Tor}_i^R((R')_f, N)$", "for $i \\geq 2$. Since", "$\\text{Tor}_i^R((R')_f, N) = \\text{Tor}_i^R(R', N)_f$", "we conclude that $f$ is a nonzerodivisor on $\\text{Tor}_1^R(R', N)$", "and invertible on $\\text{Tor}_i^R(R', N)$ for $i \\geq 2$.", "Since $M \\otimes_R R'$ is flat over $R'$ we have", "$$", "\\text{Tor}_i^R(M \\otimes_R R', N) =", "(M \\otimes_R R') \\otimes_{R'} \\text{Tor}_i^R(R', N)", "$$", "by the spectral sequence of Example \\ref{example-tor-change-rings}.", "Writing $M \\otimes_R R'$ as a filtered colimit of", "finite free $R'$-modules (Algebra, Theorem \\ref{algebra-theorem-lazard})", "we conclude that $f$ is a nonzerodivisor", "on $\\text{Tor}_1^R(M \\otimes_R R', N)$ and invertible on", "$\\text{Tor}_i^R(M \\otimes_R R', N)$. Next, we consider", "the exact sequence", "$0 \\to M \\to M \\otimes_R R' \\oplus M_f \\to M \\otimes_R (R')_f \\to 0$", "coming from the fact that $M$ is glueable and the associated long exact", "sequence of $\\text{Tor}$. The relevant part is", "$$", "\\xymatrix{", "\\text{Tor}_1^R(M, N) \\ar[r] &", "\\text{Tor}_1^R(M \\otimes_R R', N) \\ar[r] &", "\\text{Tor}_1^R(M \\otimes_R (R')_f, N) \\\\", "& \\text{Tor}_2^R(M \\otimes_R R', N) \\ar[r] &", "\\text{Tor}_2^R(M \\otimes_R (R')_f, N) \\ar[llu]", "}", "$$", "We conclude that $\\text{Tor}_1^R(M, N) = 0$ by our remarks above on the", "action on $f$ on $\\text{Tor}_i^R(M \\otimes_R R', N)$." ], "refs": [ "algebra-lemma-flat-base-change", "more-algebra-remark-what-you-get-for-general-modules", "algebra-lemma-flat-tor-zero", "more-algebra-lemma-BL-faithful", "algebra-lemma-characterize-flat", "algebra-theorem-lazard" ], "ref_ids": [ 527, 10669, 532, 10355, 786, 318 ] } ], "ref_ids": [] }, { "id": 10364, "type": "theorem", "label": "more-algebra-lemma-BL-properties", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-BL-properties", "contents": [ "Let $(R \\to R', f)$ be a glueing pair. Let $M$ be an $R$-module", "which is not necessarily glueable for $(R \\to R', f)$. Then", "$M$ is a finite projective $R$-module if and only if", "$M \\otimes_R R'$ is finite projective over $R'$ and", "$M_f$ is finite projective over $R_f$." ], "refs": [], "proofs": [ { "contents": [ "Assume that $M \\otimes_R R'$ is a finite projective module over", "$R'$ and that $M_f$ is a finite projective module over $R_f$.", "Our task is to prove that $M$ is finite projective over $R$.", "We will use Algebra, Lemma \\ref{algebra-lemma-finite-projective}", "without further mention.", "By Lemma \\ref{lemma-BL-flat} we see that $M$ is flat.", "By Lemma \\ref{lemma-faithful-descent} we see that $M$ is finite.", "Choose a short exact sequence $0 \\to K \\to R^{\\oplus n} \\to M \\to 0$.", "Since a finite projective module is of finite presentation", "and since the sequence remains exact after tensoring with", "$R'$ (by Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero})", "and $R_f$, we conclude that $K \\otimes_R R'$ and $K_f$", "are finite modules. Using the lemma above", "we conclude that $K$ is finitely generated.", "Hence $M$ is finitely presented and hence finite projective." ], "refs": [ "algebra-lemma-finite-projective", "more-algebra-lemma-BL-flat", "more-algebra-lemma-faithful-descent", "algebra-lemma-flat-tor-zero" ], "ref_ids": [ 795, 10363, 10357, 532 ] } ], "ref_ids": [] }, { "id": 10365, "type": "theorem", "label": "more-algebra-lemma-hom-from-Af", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-hom-from-Af", "contents": [ "Let $A$ be a ring. Let $f \\in A$. Let $K \\in D(A)$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\Ext^n_A(A_f, K) = 0$ for all $n$,", "\\item $\\Hom_{D(A)}(E, K) = 0$ for all $E$ in $D(A_f)$,", "\\item $T(K, f) = 0$,", "\\item for every $p \\in \\mathbf{Z}$ we have $T(H^p(K), f) = 0$,", "\\item for every $p \\in \\mathbf{Z}$ we have", "$\\Hom_A(A_f, H^p(K)) = 0$ and $\\Ext^1_A(A_f, H^p(K)) = 0$,", "\\item $R\\Hom_A(A_f, K) = 0$,", "\\item the map $\\prod_{n \\geq 0} K \\to \\prod_{n \\geq 0} K$,", "$(x_0, x_1, \\ldots) \\mapsto (x_0 - fx_1, x_1 - fx_2, \\ldots)$", "is an isomorphism in $D(A)$, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (2) implies (1) and that (1) is equivalent to (6). Assume (1).", "Let $I^\\bullet$ be a K-injective complex of $A$-modules representing $K$.", "Condition (1) signifies that $\\Hom_A(A_f, I^\\bullet)$ is acyclic.", "Let $M^\\bullet$ be a complex of $A_f$-modules representing $E$.", "Then", "$$", "\\Hom_{D(A)}(E, K) =", "\\Hom_{K(A)}(M^\\bullet, I^\\bullet) =", "\\Hom_{K(A_f)}(M^\\bullet, \\Hom_A(A_f, I^\\bullet))", "$$", "by Algebra, Lemma \\ref{algebra-lemma-adjoint-hom-restrict}.", "As $\\Hom_A(A_f, I^\\bullet)$ is a K-injective complex of", "$A_f$-modules by Lemma \\ref{lemma-hom-K-injective}", "the fact that it is acyclic implies that it is homotopy equivalent to zero", "(Derived Categories, Lemma \\ref{derived-lemma-K-injective}).", "Thus we get (2).", "\\medskip\\noindent", "A free resolution of the $A$-module $A_f$ is given by", "$$", "0 \\to \\bigoplus\\nolimits_{n \\in \\mathbf{N}} A \\to", "\\bigoplus\\nolimits_{n \\in \\mathbf{N}} A", "\\to A_f \\to 0", "$$", "where the first map sends the $(a_0, a_1, a_2, \\ldots)$ to", "$(a_0, a_1 - fa_0, a_2 - fa_1, \\ldots)$ and the second map sends", "$(a_0, a_1, a_2, \\ldots)$ to $a_0 + a_1/f + a_2/f^2 + \\ldots$.", "Applying $\\Hom_A(-, I^\\bullet)$ we get", "$$", "0 \\to \\Hom_A(A_f, I^\\bullet) \\to \\prod I^\\bullet \\to \\prod I^\\bullet \\to 0", "$$", "Since $\\prod I^\\bullet$ represents $\\prod_{n \\geq 0} K$", "this proves the equivalence of (1) and (7). On the other hand,", "by construction of derived limits in", "Derived Categories, Section \\ref{derived-section-derived-limit}", "the displayed exact sequence shows the object $T(K, f)$", "is a representative of $R\\Hom_A(A_f, K)$ in $D(A)$.", "Thus the equivalence of (1) and (3).", "\\medskip\\noindent", "There is a spectral sequence", "$$", "E_2^{p, q} = \\Ext^q_A(A_f, H^p(K)) \\Rightarrow", "\\Ext^{p + q}_A(A_f, K)", "$$", "(details omitted). This spectral sequence degenerates at $E_2$ because", "$A_f$ has a length $1$ resolution by projective $A$-modules (see above)", "hence the $E_2$-page has only 2 nonzero rows. Thus we obtain short exact", "sequences", "$$", "0 \\to \\Ext^1_A(A_f, H^{p - 1}(K)) \\to \\Ext^p_A(A_f, K)", "\\to \\Hom_A(A_f, H^p(K)) \\to 0", "$$", "This proves (4) and (5) are equivalent to (1)." ], "refs": [ "algebra-lemma-adjoint-hom-restrict", "more-algebra-lemma-hom-K-injective", "derived-lemma-K-injective" ], "ref_ids": [ 375, 10120, 1908 ] } ], "ref_ids": [] }, { "id": 10366, "type": "theorem", "label": "more-algebra-lemma-ideal-of-elements-complete-wrt", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ideal-of-elements-complete-wrt", "contents": [ "Let $A$ be a ring. Let $K \\in D(A)$. The set $I$ of $f \\in A$ such that", "$T(K, f) = 0$ is a radical ideal of $A$." ], "refs": [], "proofs": [ { "contents": [ "We will use the results of Lemma \\ref{lemma-hom-from-Af}", "without further mention.", "If $f \\in I$, and $g \\in A$, then $A_{gf}$ is an $A_f$-module", "hence $\\Ext^n_A(A_{gf}, K) = 0$ for all $n$, hence $gf \\in I$.", "Suppose $f, g \\in I$. Then there is a short exact sequence", "$$", "0 \\to A_{f + g} \\to A_{f(f + g)} \\oplus A_{g(f + g)} \\to A_{gf(f + g)} \\to 0", "$$", "because $f, g$ generate the unit ideal in $A_{f + g}$. This follows from", "Algebra, Lemma \\ref{algebra-lemma-standard-covering}", "and the easy fact that the last arrow is surjective.", "From the long exact sequence of $\\Ext$ and the vanishing of", "$\\Ext^n_A(A_{f(f + g)}, K)$,", "$\\Ext^n_A(A_{g(f + g)}, K)$, and", "$\\Ext^n_A(A_{gf(f + g)}, K)$ for all $n$", "we deduce the vanishing of $\\Ext^n_A(A_{f + g}, K)$ for all $n$.", "Finally, if $f^n \\in I$ for some $n > 0$, then $f \\in I$ because", "$T(K, f) = T(K, f^n)$ or because $A_f \\cong A_{f^n}$." ], "refs": [ "more-algebra-lemma-hom-from-Af", "algebra-lemma-standard-covering" ], "ref_ids": [ 10365, 414 ] } ], "ref_ids": [] }, { "id": 10367, "type": "theorem", "label": "more-algebra-lemma-complete-derived-complete", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-complete-derived-complete", "contents": [ "Let $A$ be a ring. Let $I \\subset A$ be an ideal. Let $M$ be an $A$-module.", "\\begin{enumerate}", "\\item If $M$ is $I$-adically complete, then $T(M, f) = 0$ for all $f \\in I$.", "\\item Conversely, if $T(M, f) = 0$ for all $f \\in I$ and $I$ is finitely", "generated, then $M \\to \\lim M/I^nM$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Assume $M$ is $I$-adically complete.", "By Lemma \\ref{lemma-hom-from-Af} it suffices to prove", "$\\Ext^1_A(A_f, M) = 0$ and $\\Hom_A(A_f, M) = 0$.", "Since $M = \\lim M/I^nM$ and since $\\Hom_A(A_f, M/I^nM) = 0$", "it follows that $\\Hom_A(A_f, M) = 0$. Suppose we have an extension", "$$", "0 \\to M \\to E \\to A_f \\to 0", "$$", "For $n \\geq 0$ pick $e_n \\in E$ mapping to $1/f^n$.", "Set $\\delta_n = fe_{n + 1} - e_n \\in M$ for $n \\geq 0$.", "Replace $e_n$ by", "$$", "e'_n = e_n + \\delta_n + f\\delta_{n + 1} + f^2 \\delta_{n + 2} + \\ldots", "$$", "The infinite sum exists as $M$ is complete with respect to $I$ and $f \\in I$.", "A simple calculation shows that $fe'_{n + 1} = e'_n$. Thus we get a splitting", "of the extension by mapping $1/f^n$ to $e'_n$.", "\\medskip\\noindent", "Proof of (2). Assume that $I = (f_1, \\ldots, f_r)$ and that $T(M, f_i) = 0$", "for $i = 1, \\ldots, r$. By", "Algebra, Lemma \\ref{algebra-lemma-when-surjective-to-completion}", "we may assume $I = (f)$ and $T(M, f) = 0$. Let $x_n \\in M$ for $n \\geq 0$.", "Consider the extension", "$$", "0 \\to M \\to E \\to A_f \\to 0", "$$", "given by", "$$", "E = M \\oplus \\bigoplus Ae_n\\Big/\\langle x_n - fe_{n + 1} + e_n\\rangle", "$$", "mapping $e_n$ to $1/f^n$ in $A_f$ (see above).", "By assumption and Lemma \\ref{lemma-hom-from-Af}", "this extension is split, hence we obtain an element", "$x + e_0$ which generates a copy of $A_f$ in $E$.", "Then", "$$", "x + e_0 = x - x_0 + fe_1 = x - x_0 - x_1 + f^2 e_2 = \\ldots", "$$", "Since $M/f^nM = E/f^nE$ by the snake lemma, we see that", "$x = x_0 + fx_1 + \\ldots + f^{n - 1}x_{n - 1}$ modulo $f^nM$.", "In other words, the map $M \\to \\lim M/f^nM$ is surjective as desired." ], "refs": [ "more-algebra-lemma-hom-from-Af", "algebra-lemma-when-surjective-to-completion", "more-algebra-lemma-hom-from-Af" ], "ref_ids": [ 10365, 863, 10365 ] } ], "ref_ids": [] }, { "id": 10368, "type": "theorem", "label": "more-algebra-lemma-serre-subcategory", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-serre-subcategory", "contents": [ "Let $I$ be an ideal of a ring $A$.", "\\begin{enumerate}", "\\item The derived complete $A$-modules form a weak Serre", "subcategory $\\mathcal{C}$ of $\\text{Mod}_A$.", "\\item $D_\\mathcal{C}(A) \\subset D(A)$ is the full subcategory", "of derived complete objects.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) is immediate from Lemma \\ref{lemma-hom-from-Af}", "and the definitions. For part (1), suppose that $M \\to N$ is", "a map of derived complete modules. Denote $K = (M \\to N)$", "the corresponding object of $D(A)$. Pick $f \\in I$. Then", "$\\Ext_A^n(A_f, K)$ is zero for all $n$ because", "$\\Ext_A^n(A_f, M)$ and $\\Ext_A^n(A_f, N)$ are zero for all $n$.", "Hence $K$ is derived complete. By (2) we see that $\\Ker(M \\to N)$ and", "$\\Coker(M \\to N)$ are objects of $\\mathcal{C}$.", "Finally, suppose that $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$", "is a short exact sequence of $A$-modules and", "$M_1$, $M_3$ are derived complete. Then it follows from", "the long exact sequence of $\\Ext$'s that $M_2$", "is derived complete. Thus $\\mathcal{C}$ is a weak Serre subcategory by", "Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}." ], "refs": [ "more-algebra-lemma-hom-from-Af", "homology-lemma-characterize-weak-serre-subcategory" ], "ref_ids": [ 10365, 12046 ] } ], "ref_ids": [] }, { "id": 10369, "type": "theorem", "label": "more-algebra-lemma-derived-complete-zero", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-complete-zero", "contents": [ "Let $I$ be a finitely generated ideal of a ring $A$.", "Let $M$ be a derived complete $A$-module.", "If $M/IM = 0$, then $M = 0$." ], "refs": [], "proofs": [ { "contents": [ "Assume that $M/IM$ is zero. Let $I = (f_1, \\ldots, f_r)$.", "Let $i < r$ be the largest integer such that $N = M/(f_1, \\ldots, f_i)M$", "is nonzero. If $i$ does not exist, then $M = 0$ which is what we", "want to show. Then $N$ is derived complete as a cokernel", "of a map between derived complete modules, see", "Lemma \\ref{lemma-serre-subcategory}.", "By our choice of $i$ we have that $f_{i + 1} : N \\to N$ is surjective.", "Hence", "$$", "\\lim (\\ldots \\to N \\xrightarrow{f_{i + 1}} N \\xrightarrow{f_{i + 1}} N)", "$$", "is nonzero, contradicting the derived completeness of $N$." ], "refs": [ "more-algebra-lemma-serre-subcategory" ], "ref_ids": [ 10368 ] } ], "ref_ids": [] }, { "id": 10370, "type": "theorem", "label": "more-algebra-lemma-pseudo-coherent-is-derived-complete", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pseudo-coherent-is-derived-complete", "contents": [ "Let $A$ be a ring and $I \\subset A$ an ideal. If $A$ is $I$-adically complete", "then any pseudo-coherent object of $D(A)$ is derived complete." ], "refs": [], "proofs": [ { "contents": [ "Let $K$ be a pseudo-coherent object of $D(A)$. By definition this", "means $K$ is represented by a bounded above complex $K^\\bullet$", "of finite free $A$-modules. Since $A$ is $I$-adically complete, it is", "derived complete (Lemma \\ref{lemma-complete-derived-complete}).", "It follows that $H^n(K)$ is derived complete for all $n$, by part (1)", "of Lemma \\ref{lemma-serre-subcategory}. This in turn implies that", "$K$ is derived complete by part (2) of the same lemma." ], "refs": [ "more-algebra-lemma-complete-derived-complete", "more-algebra-lemma-serre-subcategory" ], "ref_ids": [ 10367, 10368 ] } ], "ref_ids": [] }, { "id": 10371, "type": "theorem", "label": "more-algebra-lemma-double-localize", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-double-localize", "contents": [ "Let $A$ be a ring. Let $f, g \\in A$. Then for $K \\in D(A)$ we have", "$R\\Hom_A(A_f, R\\Hom_A(A_g, K)) = R\\Hom_A(A_{fg}, K)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-internal-hom}." ], "refs": [ "more-algebra-lemma-internal-hom" ], "ref_ids": [ 10206 ] } ], "ref_ids": [] }, { "id": 10372, "type": "theorem", "label": "more-algebra-lemma-derived-completion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-completion", "contents": [ "\\begin{slogan}", "Derived completions along finitely generated ideals exist, and can", "be computed by a {\\v C}ech procedure.", "\\end{slogan}", "Let $I$ be a finitely generated ideal of a ring $A$.", "The inclusion functor $D_{comp}(A, I) \\to D(A)$ has a", "left adjoint, i.e., given any object $K$ of $D(A)$ there", "exists a map $K \\to K^\\wedge$ of $K$ into a derived complete", "object of $D(A)$ such that the map", "$$", "\\Hom_{D(A)}(K^\\wedge, E) \\longrightarrow \\Hom_{D(A)}(K, E)", "$$", "is bijective whenever $E$ is a derived complete object of $D(A)$.", "In fact, if $I$ is generated by $f_1, \\ldots, f_r \\in A$, then we have", "$$", "K^\\wedge = R\\Hom\\left((A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}}", "\\to \\ldots \\to A_{f_1\\ldots f_r}), K\\right)", "$$", "functorially in $K$." ], "refs": [], "proofs": [ { "contents": [ "Define $K^\\wedge$ by the last displayed formula of the lemma.", "There is a map of complexes", "$$", "(A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to A_{f_1\\ldots f_r}) \\longrightarrow A", "$$", "which induces a map $K \\to K^\\wedge$. It suffices to prove that", "$K^\\wedge$ is derived complete and that $K \\to K^\\wedge$ is an", "isomorphism if $K$ is derived complete.", "\\medskip\\noindent", "Let $f \\in A$. By Lemma \\ref{lemma-double-localize}", "the object $R\\Hom_A(A_f, K^\\wedge)$ is equal to", "$$", "R\\Hom\\left((A_f \\to \\prod\\nolimits_{i_0} A_{ff_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{ff_{i_0}f_{i_1}} \\to", "\\ldots \\to A_{ff_1\\ldots f_r}), K\\right)", "$$", "If $f \\in I$, then $f_1, \\ldots, f_r$ generate the", "unit ideal in $A_f$, hence the extended alternating", "{\\v C}ech complex", "$$", "A_f \\to \\prod\\nolimits_{i_0} A_{ff_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{ff_{i_0}f_{i_1}} \\to", "\\ldots \\to A_{ff_1\\ldots f_r}", "$$", "is zero in $D(A)$ by", "Lemma \\ref{lemma-extended-alternating-torsion}.", "(In fact, if $f = f_i$ for some $i$, then this complex", "is homotopic to zero by", "Lemma \\ref{lemma-extended-alternating-homotopy-zero};", "this is the only case we need.)", "Hence $R\\Hom_A(A_f, K^\\wedge) = 0$ and we conclude that", "$K^\\wedge$ is derived complete by Lemma \\ref{lemma-hom-from-Af}.", "\\medskip\\noindent", "Conversely, if $K$ is derived complete, then $R\\Hom_A(A_f, K)$", "is zero for all $f = f_{i_0} \\ldots f_{i_p}$, $p \\geq 0$. Thus", "$K \\to K^\\wedge$ is an isomorphism in $D(A)$." ], "refs": [ "more-algebra-lemma-double-localize", "more-algebra-lemma-extended-alternating-torsion", "more-algebra-lemma-extended-alternating-homotopy-zero", "more-algebra-lemma-hom-from-Af" ], "ref_ids": [ 10371, 9971, 9970, 10365 ] } ], "ref_ids": [] }, { "id": 10373, "type": "theorem", "label": "more-algebra-lemma-derived-completion-vanishes", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-completion-vanishes", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "Let $K^\\bullet$ be a complex of $A$-modules such that", "$f : K^\\bullet \\to K^\\bullet$ is an isomorphism for some", "$f \\in I$, i.e., $K^\\bullet$ is a complex of $A_f$-modules. Then", "the derived completion of $K^\\bullet$ is zero." ], "refs": [], "proofs": [ { "contents": [ "Indeed, in this case the $R\\Hom_A(K, L)$ is zero for any derived complete", "complex $L$, see", "Lemma \\ref{lemma-hom-from-Af}. Hence $K^\\wedge$ is zero by the", "universal property in Lemma \\ref{lemma-derived-completion}." ], "refs": [ "more-algebra-lemma-hom-from-Af", "more-algebra-lemma-derived-completion" ], "ref_ids": [ 10365, 10372 ] } ], "ref_ids": [] }, { "id": 10374, "type": "theorem", "label": "more-algebra-lemma-completion-RHom", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-completion-RHom", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "Let $K, L \\in D(A)$. Then", "$$", "R\\Hom_A(K, L)^\\wedge = R\\Hom_A(K, L^\\wedge) = R\\Hom_A(K^\\wedge, L^\\wedge)", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-derived-completion} we know that derived completion is", "given by $R\\Hom_A(C, -)$ for some $C \\in D(A)$. Then", "\\begin{align*}", "R\\Hom_A(C, R\\Hom_A(K, L))", "& =", "R\\Hom_A(C \\otimes_A^\\mathbf{L} K, L) \\\\", "& =", "R\\Hom_A(K, R\\Hom_A(C, L))", "\\end{align*}", "by Lemma \\ref{lemma-internal-hom}. This proves the first equation.", "The map $K \\to K^\\wedge$ induces a map", "$$", "R\\Hom_A(K^\\wedge, L^\\wedge) \\to R\\Hom_A(K, L^\\wedge)", "$$", "which is an isomorphism in $D(A)$ by definition of the derived completion", "as the left adjoint to the inclusion functor." ], "refs": [ "more-algebra-lemma-derived-completion", "more-algebra-lemma-internal-hom" ], "ref_ids": [ 10372, 10206 ] } ], "ref_ids": [] }, { "id": 10375, "type": "theorem", "label": "more-algebra-lemma-naive-derived-completion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-naive-derived-completion", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be an ideal. Let $(K_n)$ be an inverse", "system of objects of $D(A)$ such that for all $f \\in I$ and $n$", "there exists an $e = e(n, f)$ such that $f^e$ is zero on $K_n$.", "Then for $K \\in D(A)$ the object $K' = R\\lim (K \\otimes_A^\\mathbf{L} K_n)$", "is derived complete with respect to $I$." ], "refs": [], "proofs": [ { "contents": [ "Since the category of derived complete objects is preserved under $R\\lim$", "it suffices to show that each $K \\otimes_A^\\mathbf{L} K_n$ is derived", "complete. By assumption for all $f \\in I$ there is an $e$ such", "that $f^e$ is zero on $K \\otimes_A^\\mathbf{L} K_n$. Of course this", "implies that $T(K \\otimes_A^\\mathbf{L} K_n, f) = 0$ and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10376, "type": "theorem", "label": "more-algebra-lemma-koszul-derived-completion-complete", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-koszul-derived-completion-complete", "contents": [ "In Situation \\ref{situation-koszul}. For", "$K \\in D(A)$ the object $K' = R\\lim (K \\otimes_A^\\mathbf{L} K_n^\\bullet)$", "is derived complete with respect to $I$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-naive-derived-completion}", "because $f_i^n$ acts by an endomorphism of $K_n^\\bullet$ which is", "homotopic to zero by Lemma \\ref{lemma-homotopy-koszul}." ], "refs": [ "more-algebra-lemma-naive-derived-completion", "more-algebra-lemma-homotopy-koszul" ], "ref_ids": [ 10375, 9960 ] } ], "ref_ids": [] }, { "id": 10377, "type": "theorem", "label": "more-algebra-lemma-characterize-derived-complete-Koszul", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-characterize-derived-complete-Koszul", "contents": [ "In Situation \\ref{situation-koszul}. Let $K \\in D(A)$.", "The following are equivalent", "\\begin{enumerate}", "\\item $K$ is derived complete with respect to $I$, and", "\\item the canonical map $K \\to R\\lim (K \\otimes_A^\\mathbf{L} K_n^\\bullet)$", "is an isomorphism of $D(A)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (2) holds, then $K$ is derived complete with respect to $I$", "by Lemma \\ref{lemma-koszul-derived-completion-complete}.", "Conversely, assume that $K$ is derived complete with respect to $I$.", "Consider the filtrations", "$$", "K_n^\\bullet \\supset", "\\sigma_{\\geq -r + 1}K_n^\\bullet \\supset", "\\sigma_{\\geq -r + 2}K_n^\\bullet \\supset \\ldots \\supset", "\\sigma_{\\geq -1}K_n^\\bullet \\supset", "\\sigma_{\\geq 0}K_n^\\bullet = A", "$$", "by stupid truncations (Homology, Section \\ref{homology-section-truncations}).", "Because the construction $R\\lim(K \\otimes E)$ is exact in", "the second variable (Lemma \\ref{lemma-tensor-Rlim-exact})", "we see that it suffices to show", "$$", "R\\lim \\left(", "K \\otimes_A^\\mathbf{L}", "(\\sigma_{\\geq p}K_n^\\bullet/ \\sigma_{\\geq p + 1}K_n^\\bullet)", "\\right) = 0", "$$", "for $p < 0$. The explicit description of the Koszul complexes above", "shows that", "$$", "R\\lim \\left(", "K \\otimes_A^\\mathbf{L}", "(\\sigma_{\\geq p}K_n^\\bullet/ \\sigma_{\\geq p + 1}K_n^\\bullet)", "\\right) =", "\\bigoplus\\nolimits_{i_1, \\ldots, i_{-p}}", "T(K, f_{i_1}\\ldots f_{i_{-p}})", "$$", "which is zero for $p < 0$ by assumption on $K$." ], "refs": [ "more-algebra-lemma-koszul-derived-completion-complete", "more-algebra-lemma-tensor-Rlim-exact" ], "ref_ids": [ 10376, 10331 ] } ], "ref_ids": [] }, { "id": 10378, "type": "theorem", "label": "more-algebra-lemma-derived-completion-koszul", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-completion-koszul", "contents": [ "In Situation \\ref{situation-koszul}.", "The functor which sends $K \\in D(A)$ to the derived limit", "$K' = R\\lim( K \\otimes_A^\\mathbf{L} K_n^\\bullet )$ is the left", "adjoint to the inclusion functor $D_{comp}(A) \\to D(A)$", "constructed in Lemma \\ref{lemma-derived-completion}." ], "refs": [ "more-algebra-lemma-derived-completion" ], "proofs": [ { "contents": [ "[First proof]", "The assignment $K \\leadsto K'$ is a functor and $K'$ is derived", "complete with respect to $I$ by", "Lemma \\ref{lemma-koszul-derived-completion-complete}.", "By a formal argument (omitted) we see that it suffices", "to show $K \\to K'$ is an isomorphism if $K$ is derived complete", "with respect to $I$. This is", "Lemma \\ref{lemma-characterize-derived-complete-Koszul}." ], "refs": [ "more-algebra-lemma-koszul-derived-completion-complete", "more-algebra-lemma-characterize-derived-complete-Koszul" ], "ref_ids": [ 10376, 10377 ] } ], "ref_ids": [ 10372 ] }, { "id": 10379, "type": "theorem", "label": "more-algebra-lemma-derived-complete-zero-bis", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-complete-zero-bis", "contents": [ "\\begin{slogan}", "Derived Nakayama", "\\end{slogan}", "\\begin{reference}", "A related result is \\cite[Proposition 6.5]{Dwyer-Greenlees}.", "The derived Nakayama lemma can for example be found in Bhatt's 3rd lecture", "on Prismatic cohomology at Columbia University in Fall 2018", "as Section 2 property (2). Leonid Positselski proposed a proof in", "\\url{https://mathoverflow.net/a/331501}. However, we follow the proof", "suggested by Anonymous in the comments.", "\\end{reference}", "Let $I$ be a finitely generated ideal of a ring $A$.", "Let $K$ be a derived complete object of $D(A)$.", "If $K \\otimes_A^\\mathbf{L} A/I = 0$, then $K = 0$." ], "refs": [], "proofs": [ { "contents": [ "Choose generators $f_1, \\ldots, f_r$ of $I$. Denote $K_n$ the", "Koszul complex on $f_1^n, \\ldots, f_r^n$ over $A$. Recall that", "$K_n$ is bounded and that the cohomology modules of $K_n$ are", "annihilated by $f_1^n, \\ldots, f_r^n$ and hence by $I^{nr}$.", "By Lemma \\ref{lemma-derived-vanishing-mod-I}", "we see that $K \\otimes_A^\\mathbf{L} K_n = 0$.", "Since $K$ is derived complete by Lemma \\ref{lemma-derived-completion-koszul}", "we have $K = R\\lim K \\otimes_A^\\mathbf{L} K_n = 0$", "as desired." ], "refs": [ "more-algebra-lemma-derived-vanishing-mod-I", "more-algebra-lemma-derived-completion-koszul" ], "ref_ids": [ 10338, 10378 ] } ], "ref_ids": [] }, { "id": 10380, "type": "theorem", "label": "more-algebra-lemma-derived-completion-finite-cohomological-dimension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-completion-finite-cohomological-dimension", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be an ideal which can be", "generated by $r$ elements. Then derived completion has finite", "cohomological dimension:", "\\begin{enumerate}", "\\item Let $K \\to L$ be a morphism in $D(A)$ such that $H^i(K) \\to H^i(L)$", "is an isomorphism for $i \\geq 1$ and surjective for $i = 0$.", "Then $H^i(K^\\wedge) \\to H^i(L^\\wedge)$ is an isomorphism for $i \\geq 1$", "and surjective for $i = 0$.", "\\item Let $K \\to L$ be a morphism of $D(A)$ such that $H^i(K) \\to H^i(L)$", "is an isomorphism for $i \\leq -1$ and injective for $i = 0$.", "Then $H^i(K^\\wedge) \\to H^i(L^\\wedge)$ is an isomorphism for $i \\leq -r - 1$", "and injective for $i = -r$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Say $I$ is generated by $f_1, \\ldots, f_r$. For any $K \\in D(A)$", "by Lemma \\ref{lemma-derived-completion-koszul} we have", "$K^\\wedge = R\\lim K \\otimes_A^\\mathbf{L} K_n$ where $K_n$", "is the Koszul complex on $f_1^n, \\ldots, f_r^n$", "and hence we obtain a short exact sequence", "$$", "0 \\to R^1\\lim H^{i - 1}(K \\otimes_A^\\mathbf{L} K_n)", "\\to H^i(K^\\wedge) \\to \\lim H^i(K \\otimes_A^\\mathbf{L} K_n) \\to 0", "$$", "by Lemma \\ref{lemma-break-long-exact-sequence-modules}.", "\\medskip\\noindent", "Proof of (1). Pick a distinguished triangle $K \\to L \\to C \\to K[1]$.", "Then $H^i(C) = 0$ for $i \\geq 0$. Since $K_n$ is sitting in degrees $\\leq 0$", "we see that $H^i(C \\otimes_A^\\mathbf{L} K_n) = 0$ for $i \\geq 0$", "and that $H^{-1}(C \\otimes_A^\\mathbf{L} K_n) =", "H^{-1}(C) \\otimes_A A/(f_1^n, \\ldots, f_r^n)$ is a system with", "surjective transition maps. The displayed equation above shows", "that $H^i(C^\\wedge) = 0$ for $i \\geq 0$. Applying the distinguished triangle", "$K^\\wedge \\to L^\\wedge \\to C^\\wedge \\to K^\\wedge[1]$ we get (1).", "\\medskip\\noindent", "Proof of (2). Pick a distinguished triangle $K \\to L \\to C \\to K[1]$.", "Then $H^i(C) = 0$ for $i < 0$. Since $K_n$ is sitting in degrees", "$-r, \\ldots, 0$ we see that $H^i(C \\otimes_A^\\mathbf{L} K_n) = 0$", "for $i < -r$. The displayed equation above shows", "that $H^i(C^\\wedge) = 0$ for $i < r$. Applying the distinguished triangle", "$K^\\wedge \\to L^\\wedge \\to C^\\wedge \\to K^\\wedge[1]$ we get (2)." ], "refs": [ "more-algebra-lemma-derived-completion-koszul", "more-algebra-lemma-break-long-exact-sequence-modules" ], "ref_ids": [ 10378, 10326 ] } ], "ref_ids": [] }, { "id": 10381, "type": "theorem", "label": "more-algebra-lemma-derived-completion-spectral-sequence", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-completion-spectral-sequence", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "Let $K^\\bullet$ be a filtered complex of $A$-modules. There exists a", "canonical spectral sequence $(E_r, \\text{d}_r)_{r \\geq 1}$", "of bigraded derived complete $A$-modules with $d_r$ of bidegree", "$(r, -r + 1)$ and with", "$$", "E_1^{p, q} = H^{p + q}((\\text{gr}^pK^\\bullet)^\\wedge)", "$$", "If the filtration on each $K^n$ is finite, then the spectral sequence is", "bounded and converges to $H^*((K^\\bullet)^\\wedge)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-derived-completion} we know that derived completion is", "given by $R\\Hom_A(C, -)$ for some $C \\in D^b(A)$. By", "Lemmas \\ref{lemma-derived-completion-finite-cohomological-dimension} and", "\\ref{lemma-projective-amplitude} we see that $C$ has finite projective", "dimension. Thus we may choose a bounded complex of projective", "modules $P^\\bullet$ representing $C$. Then", "$$", "M^\\bullet = \\Hom^\\bullet(P^\\bullet, K^\\bullet)", "$$", "is a complex of $A$-modules representing $(K^\\bullet)^\\wedge$.", "It comes with a filtration given by", "$F^pM^\\bullet = \\Hom^\\bullet(P^\\bullet, F^pK^\\bullet)$.", "We see that $F^pM^\\bullet$ represents $(F^pK^\\bullet)^\\wedge$", "and hence $\\text{gr}^pM^\\bullet$ represents $(\\text{gr}K^\\bullet)^\\wedge$.", "Thus we find our spectral sequence by taking the spectral sequence of", "the filtered complex $M^\\bullet$, see", "Homology, Section \\ref{homology-section-filtered-complex}.", "If the filtration on each $K^n$ is finite, then the filtration", "on each $M^n$ is finite because $P^\\bullet$ is a bounded complex.", "Hence the final statement follows from", "Homology, Lemma \\ref{homology-lemma-biregular-ss-converges}." ], "refs": [ "more-algebra-lemma-derived-completion", "more-algebra-lemma-derived-completion-finite-cohomological-dimension", "more-algebra-lemma-projective-amplitude", "homology-lemma-biregular-ss-converges" ], "ref_ids": [ 10372, 10380, 10187, 12101 ] } ], "ref_ids": [] }, { "id": 10382, "type": "theorem", "label": "more-algebra-lemma-restriction-derived-complete", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-restriction-derived-complete", "contents": [ "Let $A \\to B$ be a ring map. Let $I \\subset A$ be an ideal. The inverse", "image of $D_{comp}(A, I)$ under the restriction functor $D(B) \\to D(A)$ is", "$D_{comp}(B, IB)$." ], "refs": [], "proofs": [ { "contents": [ "Using Lemma \\ref{lemma-ideal-of-elements-complete-wrt}", "we see that $L \\in D(B)$ is in $D_{comp}(B, IB)$", "if and only if $T(L, f)$ is zero for every local section", "$f \\in I$. Observe that the cohomology of", "$T(L, f)$ is computed in the category of abelian groups,", "so it doesn't matter whether we think of $f$ as an element of $A$", "or take the image of $f$ in $B$.", "The lemma follows immediately from this and the", "definition of derived complete objects." ], "refs": [ "more-algebra-lemma-ideal-of-elements-complete-wrt" ], "ref_ids": [ 10366 ] } ], "ref_ids": [] }, { "id": 10383, "type": "theorem", "label": "more-algebra-lemma-restriction-derived-complete-equivalence", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-restriction-derived-complete-equivalence", "contents": [ "Let $A \\to B$ be a ring map. Let $I \\subset A$ be a finitely generated ideal.", "If $A \\to B$ is flat and $A/I \\cong B/IB$, then the restriction functor", "$D(B) \\to D(A)$ induces an equivalence", "$D_{comp}(B, IB) \\to D_{comp}(A, I)$." ], "refs": [], "proofs": [ { "contents": [ "Choose generators $f_1, \\ldots, f_r$ of $I$.", "Denote $\\check{\\mathcal{C}}^\\bullet_A \\to \\check{\\mathcal{C}}^\\bullet_B$", "the quasi-isomorphism of extended alternating {\\v C}ech complexes of", "Lemma \\ref{lemma-map-identifies-koszul-and-cech-complexes}.", "Let $K \\in D_{comp}(A, I)$. Let $I^\\bullet$ be a K-injective", "complex of $A$-modules representing $K$. Since $\\Ext^n_A(A_f, K)$", "and $\\Ext^n_A(B_f, K)$ are zero for all $f \\in I$ and", "$n \\in \\mathbf{Z}$ (Lemma \\ref{lemma-hom-from-Af}) we conclude that", "$\\check{\\mathcal{C}}^\\bullet_A \\to A$ and", "$\\check{\\mathcal{C}}^\\bullet_B \\to B$ induce quasi-isomorphisms", "$$", "I^\\bullet = \\Hom_A(A, I^\\bullet) \\longrightarrow", "\\text{Tot}(\\Hom_A(\\check{\\mathcal{C}}^\\bullet_A, I^\\bullet))", "$$", "and", "$$", "\\Hom_A(B, I^\\bullet) \\longrightarrow", "\\text{Tot}(\\Hom_A(\\check{\\mathcal{C}}^\\bullet_B, I^\\bullet))", "$$", "Some details omitted.", "Since $\\check{\\mathcal{C}}^\\bullet_A \\to \\check{\\mathcal{C}}^\\bullet_B$", "is a quasi-isomorphism and $I^\\bullet$ is K-injective we conclude", "that $\\Hom_A(B, I^\\bullet) \\to I^\\bullet$ is a quasi-isomorphism.", "As the complex $\\Hom_A(B, I^\\bullet)$ is a complex of $B$-modules", "we conclude that $K$ is in the image of the restriction map, i.e.,", "the functor is essentially surjective", "\\medskip\\noindent", "In fact, the argument shows that", "$F : D_{comp}(A, I) \\to D_{comp}(B, IB)$, $K \\mapsto \\Hom_A(B, I^\\bullet)$", "is a left inverse to restriction. Finally, suppose that", "$L \\in D_{comp}(B, IB)$. Represent $L$ by a K-injective complex", "$J^\\bullet$ of $B$-modules.", "Then $J^\\bullet$ is also K-injective as a complex of $A$-modules", "(Lemma \\ref{lemma-K-injective-flat}) hence", "$F(\\text{restriction of }L) = \\Hom_A(B, J^\\bullet)$.", "There is a map $J^\\bullet \\to \\Hom_A(B, J^\\bullet)$", "of complexes of $B$-modules, whose composition with", "$\\Hom_A(B, J^\\bullet) \\to J^\\bullet$ is the identity.", "We conclude that $F$ is also a right inverse to restriction", "and the proof is finished." ], "refs": [ "more-algebra-lemma-map-identifies-koszul-and-cech-complexes", "more-algebra-lemma-hom-from-Af", "more-algebra-lemma-K-injective-flat" ], "ref_ids": [ 10342, 10365, 10118 ] } ], "ref_ids": [] }, { "id": 10384, "type": "theorem", "label": "more-algebra-lemma-lift-universally", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lift-universally", "contents": [ "Let $A$ be a ring. Let $f \\in A$. If there exists an integer $c \\geq 1$", "such that $A[f^c] = A[f^{c + 1}] = A[f^{c + 2}] = \\ldots$ (for example", "if $A$ is Noetherian), then for all $n \\geq 1$ there exist maps", "$$", "(A \\xrightarrow{f^n} A) \\longrightarrow A/(f^n),", "\\quad\\text{and}\\quad", "A/(f^{n + c}) \\longrightarrow (A \\xrightarrow{f^n} A)", "$$", "in $D(A)$ inducing an isomorphism of the pro-objects $\\{A/(f^n)\\}$ and", "$\\{(f^n : A \\to A)\\}$ in $D(A)$." ], "refs": [], "proofs": [ { "contents": [ "The first displayed arrow is obvious. We can define the second arrow of", "the lemma by the diagram", "$$", "\\xymatrix{", "A/A[f^c] \\ar[r]_-{f^{n + c}} \\ar[d]_{f^c} & A \\ar[d]^1 \\\\", "A \\ar[r]^{f^n} & A", "}", "$$", "Since the top horizontal arrow is injective the complex", "in the top row is quasi-isomorphic to $A/f^{n + c}A$.", "We omit the calculation of compositions needed to show", "the statement on pro objects." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10385, "type": "theorem", "label": "more-algebra-lemma-when-does-it-work", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-when-does-it-work", "contents": [ "Let $A$ be a ring and $f \\in A$. Set $I = (f)$. In this situation", "we have the naive derived completion", "$K \\mapsto K' = R\\lim (K \\otimes_A^\\mathbf{L} A/f^nA)$ and the", "derived completion", "$$", "K \\mapsto K^\\wedge = R\\lim (K \\otimes_A^\\mathbf{L} (A \\xrightarrow{f^n} A))", "$$", "of Lemma \\ref{lemma-derived-completion-koszul}.", "The natural transformation of functors $K^\\wedge \\to K'$", "is an isomorphism if and only if the $f$-power torsion of $A$ is bounded." ], "refs": [ "more-algebra-lemma-derived-completion-koszul" ], "proofs": [ { "contents": [ "If the $f$-power torsion is bounded, then the pro-objects", "$\\{(f^n : A \\to A)\\}$ and $\\{A/f^nA\\}$ are isomorphic by", "Lemma \\ref{lemma-lift-universally}.", "Hence the functors are isomorphic by Lemma \\ref{lemma-Rlim-pro-equal}.", "Conversely, we see from Lemma \\ref{lemma-tensor-Rlim-exact}", "that the condition is exactly that", "$$", "R\\lim (K \\otimes_A^\\mathbf{L} A[f^n])", "$$", "is zero for all $K \\in D(A)$. Here the maps of the system $(A[f^n])$", "are given by multiplication by $f$. Taking $K = A$ and", "$K = \\bigoplus_{i \\in \\mathbf{N}} A$ we see from", "Lemma \\ref{lemma-Rlim-zero-of-direct-sums}", "this implies $(A[f^n])$ is zero as a pro-object, i.e.,", "$f^{n - 1}A[f^n] = 0$ for some $n$, i.e., $A[f^{n - 1}] = A[f^n]$, i.e.,", "the $f$-power torsion is bounded." ], "refs": [ "more-algebra-lemma-lift-universally", "more-algebra-lemma-Rlim-pro-equal", "more-algebra-lemma-tensor-Rlim-exact", "more-algebra-lemma-Rlim-zero-of-direct-sums" ], "ref_ids": [ 10384, 10320, 10331, 10323 ] } ], "ref_ids": [ 10378 ] }, { "id": 10386, "type": "theorem", "label": "more-algebra-lemma-torsion-and-derived-complete", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-torsion-and-derived-complete", "contents": [ "Let $I$ be a finitely generated ideal in a ring $A$.", "Let $M$ be a derived complete $A$-module. If $M$ is", "an $I$-power torsion module, then $I^nM = 0$ for some $n$." ], "refs": [], "proofs": [ { "contents": [ "Say $I = (f_1, \\ldots, f_r)$. It suffices to show", "that for each $i$ there is an $n_i$ such that $f_i^{n_i}M = 0$.", "Hence we may assume that $I = (f)$ is a principal ideal.", "Let $B = \\mathbf{Z}[x] \\to A$ be the ring map sending $x$ to $f$.", "By Lemma \\ref{lemma-restriction-derived-complete}", "we see that $M$ is derived complete as a $B$-module", "with respect to the ideal $(x)$. After replacing $A$ by", "$B$, we may assume that $f$ is a nonzerodivisor in $A$.", "\\medskip\\noindent", "Assume $I = (f)$ with $f \\in A$ a nonzerodivisor.", "According to Example \\ref{example-derived-complete-modules}", "there exists a short exact sequence", "$$", "0 \\to K \\xrightarrow{u} L \\to M \\to 0", "$$", "where $K$ and $L$ are $I$-adically complete $A$-modules", "whose $f$-torsion is zero\\footnote{For the proof it is enough", "to show that there exists a sequence $K \\xrightarrow{u} L \\to M \\to 0$", "where $K$ and $L$ are $I$-adically complete $A$-modules. This can", "be shown by choosing a presentation $F_1 \\to F_0 \\to M \\to 0$", "with $F_i$ free and then setting $K$ and $L$ equal to the", "$f$-adic completions of $F_1$ and $F_0$. Namely, as $f$", "is a nonzerodivisor these completions will be the", "derived completions and the sequence will remain exact.}.", "Consider $K$ and $L$ as", "topological modules with the $I$-adic topology. Then $u$ is continuous.", "Let", "$$", "L_n = \\{x \\in L \\mid f^n x \\in u(K)\\}", "$$", "Since $M$ is $f$-power torsion we see that", "$L = \\bigcup L_n$. Let $N_n$ be the closure of $L_n$ in $L$.", "By Lemma \\ref{lemma-consequence-baire-complete-module}", "we see that $N_n$ is open in $L$ for some $n$. Fix such an $n$.", "Since $f^{n + m} : L \\to L$ is a continuous open map, and since", "$f^{n + m} L_n \\subset u(f^m K)$ we conclude that", "the closure of $u(f^mK)$ is open for all $m \\geq 1$.", "Thus by Lemma \\ref{lemma-open-mapping}", "we conclude that $u$ is open. Hence $f^tL \\subset \\Im(u)$", "for some $t$ and we conclude that $f^t$ annihilates $M$", "as desired." ], "refs": [ "more-algebra-lemma-restriction-derived-complete", "more-algebra-lemma-consequence-baire-complete-module", "more-algebra-lemma-open-mapping" ], "ref_ids": [ 10382, 10012, 10013 ] } ], "ref_ids": [] }, { "id": 10387, "type": "theorem", "label": "more-algebra-lemma-kernel-to-completion-square-zero", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-kernel-to-completion-square-zero", "contents": [ "Let $f \\in A$ be an element of a ring. Set $J = \\bigcap f^nA$.", "Let $M$ be an $A$-module derived complete with respect to $f$.", "Then $JM' = 0$ where $M' = \\Ker(M \\to \\lim M/f^nM)$. In particular,", "if $A$ is derived complete then $J$ is an ideal of square zero." ], "refs": [], "proofs": [ { "contents": [ "Take $x \\in M'$ and $g \\in J$. For every $n \\geq 1$ we may write", "$x = f^n x_n$. Since $g$ is in $f^nA$ we see that the element", "$y_n = gx_n$ in $M'$ is independent of the choice of $x_n$.", "In particular, we may take $x_n = fx_{n + 1}$ and we find that", "$y_n = fy_{n + 1}$. Thus we obtain a map $A_f \\to M$ sending", "$1/f^n$ to $y_n$. This map has to be zero as $M$ is derived complete", "(Lemma \\ref{lemma-hom-from-Af})", "and hence $y_n = 0$ for all $n$. Since $gx = gfx_1 = fy_1$", "this completes the proof." ], "refs": [ "more-algebra-lemma-hom-from-Af" ], "ref_ids": [ 10365 ] } ], "ref_ids": [] }, { "id": 10388, "type": "theorem", "label": "more-algebra-lemma-derived-complete-henselian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-complete-henselian", "contents": [ "Let $A$ be a ring derived complete with respect to an ideal $I$.", "Then $(A, I)$ is a henselian pair." ], "refs": [], "proofs": [ { "contents": [ "Let $f \\in I$. By Lemma \\ref{lemma-largest-ideal-henselian}", "it suffices to show that $(A, fA)$ is a henselian pair.", "Observe that $A$ is derived complete with respect to $fA$", "(follows immediately from Definition \\ref{definition-derived-complete}).", "By Lemma \\ref{lemma-complete-derived-complete} the map from", "$A$ to the $f$-adic completion $A'$ of $A$ is surjective.", "By Lemma \\ref{lemma-complete-henselian} the pair $(A', fA')$", "is henselian. Thus it suffices to show that $(A, \\bigcap f^nA)$", "is a henselian pair, see Lemma \\ref{lemma-henselian-henselian-pair}.", "This follows from Lemmas \\ref{lemma-kernel-to-completion-square-zero}", "and \\ref{lemma-locally-nilpotent-henselian}." ], "refs": [ "more-algebra-lemma-largest-ideal-henselian", "more-algebra-definition-derived-complete", "more-algebra-lemma-complete-derived-complete", "more-algebra-lemma-complete-henselian", "more-algebra-lemma-henselian-henselian-pair", "more-algebra-lemma-kernel-to-completion-square-zero", "more-algebra-lemma-locally-nilpotent-henselian" ], "ref_ids": [ 9869, 10634, 10367, 9859, 9864, 10387, 9857 ] } ], "ref_ids": [] }, { "id": 10389, "type": "theorem", "label": "more-algebra-lemma-kernel-to-completion-nilpotent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-kernel-to-completion-nilpotent", "contents": [ "Let $A$ be a ring derived complete with respect to an ideal $I$.", "Set $J = \\bigcap I^n$. If $I$ can be generated by $r$ elements", "then $J^N = 0$ where $N = 2^r$." ], "refs": [], "proofs": [ { "contents": [ "When $r = 1$ this is Lemma \\ref{lemma-kernel-to-completion-square-zero}.", "Say $I = (f_1, \\ldots, f_r)$ with $r > 1$.", "By Lemma \\ref{lemma-serre-subcategory}", "the ring $A_t = A/f_r^tA$ is derived complete with", "respect to $I$ and hence a fortiori derived complete with respect to", "$I_t = (f_1, \\ldots, f_{r - 1})A_t$. Observe that $A \\to A_t$ sends $J$", "into $J_t = \\bigcap I_t^n$. By induction $J_t^{N/2} = 0$ with", "$N = 2^r$. The ideal $\\bigcap \\Ker(A \\to A_t) = \\bigcap f_r^t A$", "has square zero by the case $r = 1$. This finishes the proof." ], "refs": [ "more-algebra-lemma-kernel-to-completion-square-zero", "more-algebra-lemma-serre-subcategory" ], "ref_ids": [ 10387, 10368 ] } ], "ref_ids": [] }, { "id": 10390, "type": "theorem", "label": "more-algebra-lemma-reduced-derived-complete-complete", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-reduced-derived-complete-complete", "contents": [ "Let $A$ be a reduced ring derived complete with respect to", "a finitely generated ideal $I$. Then $A$ is $I$-adically complete." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-kernel-to-completion-nilpotent} and", "Proposition \\ref{proposition-derived-complete-modules}." ], "refs": [ "more-algebra-lemma-kernel-to-completion-nilpotent", "more-algebra-proposition-derived-complete-modules" ], "ref_ids": [ 10389, 10589 ] } ], "ref_ids": [] }, { "id": 10391, "type": "theorem", "label": "more-algebra-lemma-sequence-Koszul-complexes", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-sequence-Koszul-complexes", "contents": [ "In Situation \\ref{situation-koszul}. If $A$ is Noetherian, then the pro-objects", "$\\{K_n^\\bullet\\}$ and $\\{A/(f_1^n, \\ldots, f_r^n)\\}$ of $D(A)$ are", "isomorphic\\footnote{In particular, for every $n$ there exists an $m \\geq n$", "such that $K_m^\\bullet \\to K_n^\\bullet$ factors through the map", "$K_m^\\bullet \\to A/(f_1^m, \\ldots, f_r^m)$.}." ], "refs": [], "proofs": [ { "contents": [ "We have an inverse system of distinguished triangles", "$$", "\\tau_{\\leq -1}K_n^\\bullet \\to K_n^\\bullet \\to A/(f_1^m, \\ldots, f_r^m) \\to", "(\\tau_{\\leq -1}K_n^\\bullet)[1]", "$$", "See Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}.", "By Derived Categories, Lemma \\ref{derived-lemma-pro-isomorphism}", "it suffices to show that the inverse system", "$\\tau_{\\leq -1}K_n^\\bullet$ is pro-zero.", "Recall that $K_n^\\bullet$ has nonzero terms only in degrees", "$i$ with $-r \\leq i \\leq 0$. Thus by", "Derived Categories, Lemma \\ref{derived-lemma-essentially-constant-cohomology}", "it suffices to show that $H^p(K_n^\\bullet)$ is pro-zero", "for $p \\leq -1$.", "In other words, for every $n \\in \\mathbf{N}$ we have to show", "there exists an $m \\geq n$ such that $H^p(K_m^\\bullet) \\to H^p(K_n^\\bullet)$", "is zero. Since $A$ is Noetherian, we see that", "$$", "H^p(K_n^\\bullet) =", "\\frac{\\Ker(K_n^p \\to K_n^{p + 1})}{\\Im(K_n^{p - 1} \\to K_n^p)}", "$$", "is a finite $A$-module. Moreover, the map $K_m^p \\to K_n^p$ is given", "by a diagonal matrix whose entries are in the ideal", "$(f_1^{m - n}, \\ldots, f_r^{m - n})$ as $p < 0$.", "Note that $H^p(K_n^\\bullet)$ is annihilated by", "$J = (f_1^n, \\ldots, f_r^n)$, see Lemma \\ref{lemma-homotopy-koszul}. Now", "$J^t \\subset (f_1^{m - n}, \\ldots, f_r^{m - n})$ for $m = n + tr$.", "Thus by Artin-Rees (Algebra, Lemma \\ref{algebra-lemma-Artin-Rees})", "for some $m$ large enough we see that", "the image of $K_m^p \\to K_n^p$ intersected with", "$\\Ker(K_n^p \\to K_n^{p + 1})$ is contained in", "$J \\Ker(K_n^p \\to K_n^{p + 1})$. For this $m$ we get the zero map." ], "refs": [ "derived-remark-truncation-distinguished-triangle", "derived-lemma-pro-isomorphism", "derived-lemma-essentially-constant-cohomology", "more-algebra-lemma-homotopy-koszul", "algebra-lemma-Artin-Rees" ], "ref_ids": [ 2016, 1956, 1955, 9960, 625 ] } ], "ref_ids": [] }, { "id": 10392, "type": "theorem", "label": "more-algebra-lemma-noetherian-calculate", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-noetherian-calculate", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$. Let $M$ be an $A$-module", "with derived completion $M^\\wedge$. Then there are short exact sequences", "$$", "0 \\to R^1\\lim \\text{Tor}_{i + 1}^A(M, A/I^n) \\to", "H^{-i}(M^\\wedge) \\to \\lim \\text{Tor}_i^A(M, A/I^n) \\to 0", "$$", "A similar result holds for $M \\in D^-(A)$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Proposition \\ref{proposition-noetherian-naive-completion-is-completion}", "and Lemma \\ref{lemma-break-long-exact-sequence-modules}." ], "refs": [ "more-algebra-proposition-noetherian-naive-completion-is-completion", "more-algebra-lemma-break-long-exact-sequence-modules" ], "ref_ids": [ 10590, 10326 ] } ], "ref_ids": [] }, { "id": 10393, "type": "theorem", "label": "more-algebra-lemma-derived-completion-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-completion-pseudo-coherent", "contents": [ "Let $A$ be a Noetherian ring and $I \\subset A$ an ideal. Let $K$ be an", "object of $D(A)$ such that $H^n(K)$ a finite $A$-module for all", "$n \\in \\mathbf{Z}$. Then the cohomology modules $H^n(K^\\wedge)$ of", "the derived completion are the $I$-adic", "completions of the cohomology modules $H^n(K)$." ], "refs": [], "proofs": [ { "contents": [ "The complex $\\tau_{\\leq m}K$ is pseudo-coherent for all $m$", "by Lemma \\ref{lemma-Noetherian-pseudo-coherent}.", "Thus $\\tau_{\\leq m}K$ is represented by a bounded above complex", "$P^\\bullet$ of finite free $A$-modules. Then", "$\\tau_{\\leq m}K \\otimes_A^\\mathbf{L} A/I^n = P^\\bullet/I^nP^\\bullet$.", "Hence $(\\tau_{\\leq m}K)^\\wedge = R\\lim P^\\bullet/I^nP^\\bullet$", "(Proposition \\ref{proposition-noetherian-naive-completion-is-completion})", "and since the $R\\lim$ is just given by termwise $\\lim$", "(Lemma \\ref{lemma-compute-Rlim-modules}) and since", "$I$-adic completion is an exact functor on finite $A$-modules", "(Algebra, Lemma \\ref{algebra-lemma-completion-flat}) we conclude", "the result holds for $\\tau_{\\leq m}K$. Hence the result holds for", "$K$ as derived completion has finite cohomological dimension, see", "Lemma \\ref{lemma-derived-completion-finite-cohomological-dimension}." ], "refs": [ "more-algebra-lemma-Noetherian-pseudo-coherent", "more-algebra-proposition-noetherian-naive-completion-is-completion", "more-algebra-lemma-compute-Rlim-modules", "algebra-lemma-completion-flat", "more-algebra-lemma-derived-completion-finite-cohomological-dimension" ], "ref_ids": [ 10160, 10590, 10324, 870, 10380 ] } ], "ref_ids": [] }, { "id": 10394, "type": "theorem", "label": "more-algebra-lemma-derived-complete-finite", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-complete-finite", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$.", "Let $M$ be a derived complete $A$-module.", "If $M/IM$ is a finite $A/I$-module, then", "$M = \\lim M/I^nM$ and $M$ is a finite $A^\\wedge$-module." ], "refs": [], "proofs": [ { "contents": [ "Assume $M/IM$ is finite. Pick $x_1, \\ldots, x_t \\in M$ which map to", "generators of $M/IM$. We obtain a map $A^{\\oplus t} \\to M$ mapping", "the $i$th basis vector to $x_i$. By", "Proposition \\ref{proposition-noetherian-naive-completion-is-completion}", "the derived completion", "of $A$ is $A^\\wedge = \\lim A/I^n$. As $M$ is derived complete, we", "see that our map factors through a map $q : (A^\\wedge)^{\\oplus t} \\to M$.", "The module $\\Coker(q)$ is zero by", "Lemma \\ref{lemma-derived-complete-zero}.", "Thus $M$ is a finite $A^\\wedge$-module.", "Since $A^\\wedge$ is Noetherian and complete with respect to $IA^\\wedge$,", "it follows that $M$ is $I$-adically complete (use", "Algebra, Lemmas \\ref{algebra-lemma-completion-Noetherian},", "\\ref{algebra-lemma-when-finite-module-complete-over-complete-ring}, and", "\\ref{algebra-lemma-Artin-Rees})." ], "refs": [ "more-algebra-proposition-noetherian-naive-completion-is-completion", "more-algebra-lemma-derived-complete-zero", "algebra-lemma-completion-Noetherian", "algebra-lemma-when-finite-module-complete-over-complete-ring", "algebra-lemma-Artin-Rees" ], "ref_ids": [ 10590, 10369, 873, 867, 625 ] } ], "ref_ids": [] }, { "id": 10395, "type": "theorem", "label": "more-algebra-lemma-when-derived-completion-is-completion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-when-derived-completion-is-completion", "contents": [ "Let $I$ be an ideal in a Noetherian ring $A$.", "\\begin{enumerate}", "\\item If $M$ is a finite $A$-module and $N$ is a flat $A$-module, then the", "derived $I$-adic completion of $M \\otimes_A N$ is the usual", "$I$-adic completion of $M \\otimes_A N$.", "\\item If $M$ is a finite $A$-module and $f \\in A$, then the derived", "$I$-adic completion of $M_f$ is the usual $I$-adic completion", "of $M_f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For an $A$-module $M$ denote $M^\\wedge$ the derived completion", "and $\\lim M/I^nM$ the usual completion.", "Assume $M$ is finite. The system $\\text{Tor}^A_i(M, A/I^n)$", "is pro-zero for $i > 0$, see", "Lemma \\ref{lemma-tor-strictly-pro-zero}.", "Since $\\text{Tor}_i^A(M \\otimes_A N, A/I^n) =", "\\text{Tor}_i^A(M, A/I^n) \\otimes_A N$ as $N$ is flat, the", "same is true for the system $\\text{Tor}^A_i(M \\otimes_A N, A/I^n)$.", "By Lemma \\ref{lemma-noetherian-calculate}", "we conclude $R\\lim (M \\otimes_A N) \\otimes_A^\\mathbf{L} A/I^n$", "only has cohomology in degree $0$ given by the usual completion", "$\\lim M \\otimes_A N/ I^n(M \\otimes_A N)$. This proves (1).", "Part (2) follows from (1) and the fact that $M_f = M \\otimes_A A_f$." ], "refs": [ "more-algebra-lemma-tor-strictly-pro-zero", "more-algebra-lemma-noetherian-calculate" ], "ref_ids": [ 9954, 10392 ] } ], "ref_ids": [] }, { "id": 10396, "type": "theorem", "label": "more-algebra-lemma-derived-completion-tensor-finite", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-completion-tensor-finite", "contents": [ "Let $I$ be an ideal in a Noetherian ring $A$.", "Let ${}^\\wedge$ denote derived completion with respect to $I$.", "Let $K \\in D^-(A)$.", "\\begin{enumerate}", "\\item If $M$ is a finite $A$-module, then", "$(K \\otimes_A^\\mathbf{L} M)^\\wedge = K^\\wedge \\otimes_A^\\mathbf{L} M$.", "\\item If $L \\in D(A)$ is pseudo-coherent, then", "$(K \\otimes_A^\\mathbf{L} L)^\\wedge = K^\\wedge \\otimes_A^\\mathbf{L} L$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $L$ be as in (2).", "We may represent $K$ by a bounded above complex $P^\\bullet$", "of free $A$-modules. We may represent $L$ by a bounded above complex", "$F^\\bullet$ of finite free $A$-modules.", "Since $\\text{Tot}(P^\\bullet \\otimes_A F^\\bullet)$", "represents $K \\otimes_A^\\mathbf{L} L$ we see that", "$(K \\otimes_A^\\mathbf{L} L)^\\wedge$ is represented by", "$$", "\\text{Tot}((P^\\bullet)^\\wedge \\otimes_A F^\\bullet)", "$$", "where $(P^\\bullet)^\\wedge$ is the complex whose terms are", "the usual $=$ derived completions $(P^n)^\\wedge$, see for example", "Proposition \\ref{proposition-noetherian-naive-completion-is-completion}", "and Lemma \\ref{lemma-when-derived-completion-is-completion}.", "This proves (2). Part (1) is a special case of (2)." ], "refs": [ "more-algebra-proposition-noetherian-naive-completion-is-completion", "more-algebra-lemma-when-derived-completion-is-completion" ], "ref_ids": [ 10590, 10395 ] } ], "ref_ids": [] }, { "id": 10397, "type": "theorem", "label": "more-algebra-lemma-eta-first-property", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-eta-first-property", "contents": [ "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let $M^\\bullet$ be", "a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$.", "There is a canonical isomorphism", "$$", "f^i : H^i(M^\\bullet)/H^i(M^\\bullet)[f] \\longrightarrow H^i(\\eta_fM^\\bullet)", "$$", "given by multiplication by $f^i$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\Ker(d^i : (\\eta_fM)^i \\to (\\eta_fM)^{i + 1})$", "is equal to", "$\\Ker(d^i : f^iM^i \\to f^iM^{i + 1}) = f^i\\Ker(d^i : M^i \\to M^{i + 1})$.", "This we get a surjection", "$f^i : H^i(M^\\bullet) \\to H^i(\\eta_fM^\\bullet)$", "by sending the class of $z \\in \\Ker(d^i : M^i \\to M^{i + 1})$", "to the class of $f^iz$. If we obtain the zero class in", "$H^i(\\eta_fM^\\bullet)$ then we see that $f^i z = d^{i - 1}(f^{i - 1}y)$", "for some $y \\in M^{i - 1}$. Since $f$ is a nonzerodivisor on all", "the modules involved, this means $f z = d^{i - 1}(y)$ which", "exactly means that the class of $z$ is $f$-torsion as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10398, "type": "theorem", "label": "more-algebra-lemma-eta-qis", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-eta-qis", "contents": [ "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor.", "If $M^\\bullet \\to N^\\bullet$ is a quasi-isomorphism of complexes of", "$A$-modules such that $f$ is a nonzerodivisor on all $M^i$ and $N^i$,", "then the induced map $\\eta_fM^\\bullet \\to \\eta_fN^\\bullet$", "is a quasi-isomorphism too." ], "refs": [], "proofs": [ { "contents": [ "This is true because the isomorphisms of Lemma \\ref{lemma-eta-first-property}", "are compatible with maps of complexes." ], "refs": [ "more-algebra-lemma-eta-first-property" ], "ref_ids": [ 10397 ] } ], "ref_ids": [] }, { "id": 10399, "type": "theorem", "label": "more-algebra-lemma-eta-second-property", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-eta-second-property", "contents": [ "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let $M^\\bullet$ be", "a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$.", "There is a canonical map of complexes", "$$", "\\eta_fM^\\bullet \\otimes_A A/fA", "\\longrightarrow", "H^\\bullet(M^\\bullet/f)", "$$", "which is a quasi-isomorphism where the right hand side is as constructed above." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in (\\eta_fM)^i$. Then $x = f^ix' \\in f^iM$ and", "$d^i(x) = f^{i + 1}y \\in f^{i + 1}M^{i + 1}$. Thus $d^i$ maps $x' \\otimes f^i$", "to zero in $M^{i + 1} \\otimes f^iA/f^{i + 1}A$. All tensor products", "are over $A$ in this proof.", "Hence we may map $x$ to the class of $x' \\otimes f^i$ in", "$H^i(M^\\bullet \\otimes f^iA/f^{i + 1}A)$. It is clear that this", "rule defines a map", "$$", "(\\eta_fM)^i \\otimes A/fA", "\\longrightarrow", "H^i(M^\\bullet \\otimes f^iA/f^{i + 1}A)", "$$", "of $A/fA$-modules. Observe that in the situation above, we may view", "$x' \\otimes f^i$ as an element of $M^i \\otimes f^iA/f^{i + 2}A$", "with differential $d^i(x' \\otimes f^i) = y \\otimes f^{i + 1}$.", "By the construction of $\\beta$ above we find that", "$\\beta(x' \\otimes f^i) = y \\otimes f^{i + 1}$ and we conclude that", "our maps are compatible with differentials, i.e., we have a map", "of complexes.", "\\medskip\\noindent", "To finish the proof, we observe that the construction given", "in the previous paragraph agrees with the maps", "$(\\eta_fM)^i \\otimes A/fA \\to Z^i/B^i$ discussed in", "Remark \\ref{remark-eta-BZ}.", "Since we have seen that the kernel of these maps is", "an acyclic subcomplex of $\\eta_fM^\\bullet \\otimes A/fA$, the lemma is proved." ], "refs": [ "more-algebra-remark-eta-BZ" ], "ref_ids": [ 10673 ] } ], "ref_ids": [] }, { "id": 10400, "type": "theorem", "label": "more-algebra-lemma-eta-third-property", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-eta-third-property", "contents": [ "Let $A$ be a ring and let $f, g \\in A$ be nonzerodivisors. Let $M^\\bullet$ be", "a complex of $A$-modules such that $fg$ is a nonzerodivisor on all $M^i$.", "Then $\\eta_f\\eta_gM^\\bullet = \\eta_{fg}M^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "The statement means that in degree $i$ we obtain the same submodule", "of the localization $M^i_{fg} = (M^i_g)_f$. We omit the details." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10401, "type": "theorem", "label": "more-algebra-lemma-eta-locally-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-eta-locally-free", "contents": [ "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let", "$M^\\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor", "on all $M^i$. Assume", "\\begin{enumerate}", "\\item $M^i$, $M^{i + 1}$ free of ranks $r_i, r_{i + 1}$,", "\\item the $r_i \\times r_i$ minors of", "$(f, d^i) : M^i \\to M^i \\oplus M^{i + 1}$ generate a principal ideal.", "\\end{enumerate}", "Then $(\\eta_fM)^i$ is locally free of rank $r_i$ and the canonical map", "$(\\eta_fM)^i \\to f^iM^i \\oplus f^{i + 1}M^{i + 1}$ is the inclusion", "of a direct summand." ], "refs": [], "proofs": [ { "contents": [ "Observe that $I$ contains the element $f^{r_i}$. Hence $I = (g)$", "where $g$ divides a power of $f$.", "Consider the complex of finite free $A$-modules", "$$", "0 \\to M^i \\xrightarrow{f, d^i} M^i \\oplus M^{i + 1}", "\\xrightarrow{d^i, -f} M^{i + 1} \\to 0", "$$", "which becomes split exact after localizing at $f$. By construction", "$(\\eta_fM)^i$ is isomorphic to the kernel of the second map.", "Hence $(\\eta_fM)^i$ is the kernel of the map", "$$", "M^i \\oplus M^{i + 1} \\longrightarrow Q = \\Coker(f, d^i)/f\\text{-power torsion}", "$$", "By Lemma \\ref{lemma-principal-fitting-ideal} the finite module $Q$", "can locally be generated by $r_{i + 1}$ elements. On the other hand we", "have $Q_f \\cong M^{i + 1}_f$ is free of rank $r_{i + 1}$.", "Hence $Q$ is locally free of rank $r_{i + 1}$. Thus $Q$ is a projective", "module and the surjection above is split." ], "refs": [ "more-algebra-lemma-principal-fitting-ideal" ], "ref_ids": [ 9837 ] } ], "ref_ids": [] }, { "id": 10402, "type": "theorem", "label": "more-algebra-lemma-eta-base-change", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-eta-base-change", "contents": [ "Let $A \\to B$ be a ring map. Let $f \\in A$ be a nonzerodivisor. Let", "$M^\\bullet$ be a complex of $A$-modules. Assume", "\\begin{enumerate}", "\\item $f$ maps to a nonzerodivisor $g$ in $B$,", "\\item $f$ is a nonzerodivisor on $M^i$,", "\\item $M^i$ is finite free of rank $r_i$,", "\\item the $r_i \\times r_i$ minors of", "$(f, d^i) : M^i \\to M^i \\oplus M^{i + 1}$ generate a principal ideal.", "\\end{enumerate}", "for all $i \\in \\mathbf{Z}$. Then there is a canonical isomorphism", "$\\eta_fM^\\bullet \\otimes_A B = \\eta_g(M^\\bullet \\otimes_A B)$." ], "refs": [], "proofs": [ { "contents": [ "Set $N^i = M^i \\otimes_A B$. Observe that $f^iM^i \\otimes_A B = g^iN^i$", "as submodules of $(N^i)_g$. The maps", "$$", "(\\eta_fM)^i \\otimes_A B \\to g^iN^i \\otimes g^{i + 1}N^{i + 1}", "\\quad\\text{and}\\quad", "(\\eta_gN)^i \\to g^iN^i \\otimes g^{i + 1}N^{i + 1}", "$$", "are inclusions of direct summands by Lemma \\ref{lemma-eta-locally-free}.", "Since their images agree after localizing at $g$ we conclude." ], "refs": [ "more-algebra-lemma-eta-locally-free" ], "ref_ids": [ 10401 ] } ], "ref_ids": [] }, { "id": 10403, "type": "theorem", "label": "more-algebra-lemma-vanishing-beta", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-vanishing-beta", "contents": [ "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let", "$M^\\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor", "on all $M^i$. For $i \\in \\mathbf{Z}$ the following are equivalent", "\\begin{enumerate}", "\\item $\\Ker(d^i \\bmod f^2)$ surjects onto $\\Ker(d^i \\bmod f)$,", "\\item $\\beta : H^i(M^\\bullet \\otimes_A f^iA/f^{i + 1}A) \\to", "H^{i + 1}(M^\\bullet \\otimes_A f^{i + 1}A/f^{i + 2}A)$ is zero.", "\\end{enumerate}", "These equivalent conditions are implied by the condition", "$H^{i + 1}(M^\\bullet)[f] = 0$." ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows immediately from the definitions.", "If $\\beta \\not = 0$, then $H^{i + 1}(M^\\bullet)[f] \\not = 0$ because", "$\\beta$ factors through a map $H^i(M^\\bullet \\otimes f^iA/f^{i + 1}A) \\to", "H^{i + 1}(M^\\bullet \\otimes f^{i + 1}A)$ (see discussion following the", "construction of the Bockstein operators)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10404, "type": "theorem", "label": "more-algebra-lemma-eta-vanishing-beta", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-eta-vanishing-beta", "contents": [ "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let", "$M^\\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor", "on all $M^i$. If $\\Ker(d^i \\bmod f^2)$ surjects onto $\\Ker(d^i \\bmod f)$,", "then the canonical map", "$$", "(\\eta_fM)^i / f(\\eta_fM)^i \\longrightarrow", "f^iM^i/f^{i + 1}M^i \\oplus f^{i + 1}M^{i + 1}/f^{i + 2}M^{i + 1},", "x \\longmapsto (x, d^i(x))", "$$", "identifies the left hand side with a direct sum of submodules of", "the right hand side." ], "refs": [], "proofs": [ { "contents": [ "With notation as in Remark \\ref{remark-eta-BZ}", "we define a map $t^{-1} : Z^i \\to (\\eta_fM)^i / f(\\eta_fM)^i$.", "Namely, for $x \\in M^i$ with $d^i(x) = f^2y$ we send the class of", "$x$ in $Z^i$ to the class of $f^ix$ in $(\\eta_fM)^i / f(\\eta_fM)^i$.", "We omit the verification that this is well defined; the assumption", "of the lemma exactly signifies that the domain of this operation", "is all of $Z^i$. Then", "$t \\circ t^{-1} = \\text{id}_{Z^i}$. Hence $t^{-1}$", "defines a splitting of the short exact sequence in", "Remark \\ref{remark-eta-BZ} and the resulting direct sum", "decomposition", "$$", "(\\eta_fM)^i / f(\\eta_fM)^i = Z^i \\oplus B^{i + 1}", "$$", "is compatible with the map displayed in the lemma." ], "refs": [ "more-algebra-remark-eta-BZ", "more-algebra-remark-eta-BZ" ], "ref_ids": [ 10673, 10673 ] } ], "ref_ids": [] }, { "id": 10405, "type": "theorem", "label": "more-algebra-lemma-ideal-direct-summand", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ideal-direct-summand", "contents": [ "Let $A$ be a ring. Let $M$, $N_1$, $N_2$ be finite projective $A$-modules.", "Let $s : M \\to N_1 \\oplus N_2$ be a split injection. There exists a", "finitely generated ideal $I \\subset A$ with the following property:", "a ring map $A \\to B$ factors through $A/I$ if and only if", "$s \\otimes \\text{id}_B$", "identifies $M \\otimes_A B$ with a direct sum of submodules of", "$N_1 \\otimes_A B \\oplus N_2 \\otimes_A B$." ], "refs": [], "proofs": [ { "contents": [ "Choose a splitting $\\pi : N_1 \\oplus N_2 \\to M$ of $s$. Denote", "$q_i : N_1 \\oplus N_2 \\to N_1 \\oplus N_2$ the projector onto $N_i$.", "Set $p_i = \\pi \\circ q_i \\circ s$. Observe that $p_1 + p_2 = \\text{id}_M$.", "We claim $M$ is a direct sum of submodules of $N_1 \\oplus N_2$", "if and only if $p_1$ and $p_2$ are orthogonal projectors.", "Thus $I$ is the smallest ideal of $A$ such that", "$p_1 \\circ p_1 - p_1$, $p_2 \\circ p_2 - p_2$, $p_1 \\circ p_2$, and", "$p_2 \\circ p_1$ are contained in $I \\otimes_A \\text{End}_A(M)$.", "Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10406, "type": "theorem", "label": "more-algebra-lemma-eta-vanishing-beta-plus", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-eta-vanishing-beta-plus", "contents": [ "Let $A$ be a ring and let $f \\in \\mathfrak m_A$ be a nonzerodivisor. Let", "$M^\\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor", "on all $M^i$. Assume", "\\begin{enumerate}", "\\item $M^i$ is finite free of rank $r_i$ and $M^i = 0$ for $|i| \\gg 0$,", "\\item the $r_i \\times r_i$ minors of $(f, d^i) : M^i \\to M^i \\oplus M^{i + 1}$", "generate a principal ideal", "\\end{enumerate}", "for all $i \\in \\mathbf{Z}$. Consider the set of prime ideals", "$$", "E = \\{f \\in \\mathfrak p \\subset A \\mid", "\\Ker(d^i \\bmod f^2)_\\mathfrak p \\text{ surjects onto }", "\\Ker(d^i \\bmod f)_\\mathfrak p \\text{ for all }i \\in \\mathbf{Z}\\}", "$$", "There exists a finitely generated ideal $f \\in I$", "such that $I_\\mathfrak p = fA_\\mathfrak p$ for all $\\mathfrak p \\in E$", "and such that the cohomology modules of $\\eta_f M^\\bullet \\otimes_A A/I$", "are finite free." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-eta-locally-free} we find that", "$(\\eta_fM)^i$ is free of rank $r_i$ for all $i$. Consider the map", "$$", "(\\eta_fM)^i / f(\\eta_fM)^i \\longrightarrow", "f^iM^i/f^{i + 1}M^i \\oplus f^{i + 1}M^{i + 1}/f^{i + 2}M^{i + 1}", "$$", "of Lemma \\ref{lemma-eta-vanishing-beta}.", "By Lemma \\ref{lemma-eta-locally-free}", "this map is split injective. Let $I_i \\subset A$ be the ideal containing", "$f$ such that the ideal $I_i/fA \\subset A/fA$ is the one found in", "Lemma \\ref{lemma-ideal-direct-summand} for the displayed arrow. We set", "$I = \\sum_{i \\in \\mathbf{Z}} I_i$. Since almost all $I_i = A$ this", "is a finitely generated ideal.", "\\medskip\\noindent", "Pick $\\mathfrak p \\in E$. By Lemma \\ref{lemma-eta-vanishing-beta}", "and the freeness of the modules $(\\eta_fM)^i$ we may write", "$$", "\\left((\\eta_fM)^i / f(\\eta_fM)^i\\right)_\\mathfrak p =", "(A/fA)_\\mathfrak p^{\\oplus m_i} \\oplus (A/fA)_\\mathfrak p^{\\oplus n_i}", "$$", "compatible with the arrow above. By the universal property of", "the ideal $I_i$ we conclude that $(I_i)_\\mathfrak p = fA_\\mathfrak p$.", "Hence $I_\\mathfrak p = fA_\\mathfrak p$ for $\\mathfrak p \\in E$.", "\\medskip\\noindent", "To finish the proof we need to show the assertion about cohomology.", "To see this, observe that the differential on $\\eta_fM^\\bullet$", "fits into a commutative diagram", "$$", "\\xymatrix{", "(\\eta_fM)^i \\ar[d] \\ar[r] &", "f^iM^i \\oplus f^{i + 1}M^{i + 1}", "\\ar[d]^{\\left(", "\\begin{matrix}", "0 & 1 \\\\", "0 & 0", "\\end{matrix}", "\\right)} \\\\", "(\\eta_fM)^{i + 1} \\ar[r] &", "f^{i + 1}M^i \\oplus f^{i + 2}M^{i + 2}", "}", "$$", "By construction, after tensoring with $A/I$, the modules on the", "left are direct sums of direct summands of the summands on the right.", "Picture", "$$", "\\xymatrix{", "(\\eta_fM)^i \\otimes A/I \\ar[d] \\ar@{=}[r] &", "K^i \\oplus L^i \\ar[r] \\ar[d] &", "f^iM^i \\otimes A/I \\oplus f^{i + 1}M^{i + 1} \\otimes A/I", "\\ar[d]^{\\left(", "\\begin{matrix}", "0 & 1 \\\\", "0 & 0", "\\end{matrix}", "\\right)} \\\\", "(\\eta_fM)^{i + 1} \\otimes A/I \\ar@{=}[r] &", "K^{i + 1} \\oplus L^{i + 1} \\ar[r] &", "f^{i + 1}M^i \\otimes A/I \\oplus f^{i + 2}M^{i + 2} \\otimes A/I", "}", "$$", "where the horizontal arrows are compatible with direct sum decompositions", "as well as inclusions of direct summands.", "It follows that the differential identifies $L^i$ with a direct summand", "of $K^{i + 1}$ and we conclude that the cohomology of", "$\\eta_fM^\\bullet \\otimes_A A/I$ in degree $i$ is the", "module $K^{i + 1}/L^i$ which is finite projective as desired." ], "refs": [ "more-algebra-lemma-eta-locally-free", "more-algebra-lemma-eta-vanishing-beta", "more-algebra-lemma-eta-locally-free", "more-algebra-lemma-ideal-direct-summand", "more-algebra-lemma-eta-vanishing-beta" ], "ref_ids": [ 10401, 10404, 10401, 10405, 10404 ] } ], "ref_ids": [] }, { "id": 10407, "type": "theorem", "label": "more-algebra-lemma-Rlim-pseudo-coherent-gives-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Rlim-pseudo-coherent-gives-pseudo-coherent", "contents": [ "Let $A = \\lim A_n$ be a limit of an inverse system $(A_n)$ of rings.", "Suppose given $K_n \\in D(A_n)$ and maps $K_{n + 1} \\to K_n$", "in $D(A_{n + 1})$. Assume", "\\begin{enumerate}", "\\item the transition maps $A_{n + 1} \\to A_n$ are surjective", "with locally nilpotent kernels,", "\\item $K_1$ is pseudo-coherent, and", "\\item the maps induce isomorphisms", "$K_{n + 1} \\otimes_{A_{n + 1}}^\\mathbf{L} A_n \\to K_n$.", "\\end{enumerate}", "Then $K = R\\lim K_n$ is a pseudo-coherent object of $D(A)$", "and $K \\otimes_A^\\mathbf{L} A_n \\to K_n$ is an isomorphism for all $n$." ], "refs": [], "proofs": [ { "contents": [ "By assumption we can find a bounded above complex of", "finite free $A_1$-modules $P_1^\\bullet$ representing $K_1$, see", "Definition \\ref{definition-pseudo-coherent}.", "By Lemma \\ref{lemma-lift-complex-stably-frees}", "we can, by induction on $n > 1$, find", "complexes $P_n^\\bullet$ of finite free $A_n$-modules representing $K_n$", "and maps $P_n^\\bullet \\to P_{n - 1}^\\bullet$ representing the maps", "$K_n \\to K_{n - 1}$ inducing isomorphisms (!)", "of complexes $P_n^\\bullet \\otimes_{A_n} A_{n - 1} \\to P_{n - 1}^\\bullet$.", "Thus $K = R\\lim K_n$ is represented by $P^\\bullet = \\lim P_n^\\bullet$, see", "Lemma \\ref{lemma-compute-Rlim-modules} and", "Remark \\ref{remark-how-unique}.", "Since $P_n^i$ is a finite free $A_n$-module for each $n$ and", "$A = \\lim A_n$ we see that $P^i$ is finite free of the same rank", "as $P_1^i$ for each $i$.", "This means that $K$ is pseudo-coherent.", "It also follows that $K \\otimes_A^\\mathbf{L} A_n$ is represented by", "$P^\\bullet \\otimes_A A_n = P_n^\\bullet$ which proves the final", "assertion." ], "refs": [ "more-algebra-definition-pseudo-coherent", "more-algebra-lemma-lift-complex-stably-frees", "more-algebra-lemma-compute-Rlim-modules", "more-algebra-remark-how-unique" ], "ref_ids": [ 10623, 10230, 10324, 10660 ] } ], "ref_ids": [] }, { "id": 10408, "type": "theorem", "label": "more-algebra-lemma-Rlim-pseudo-coherent-gives-complete-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Rlim-pseudo-coherent-gives-complete-pseudo-coherent", "contents": [ "Let $A$ be a ring and $I \\subset A$ an ideal.", "Suppose given $K_n \\in D(A/I^n)$ and maps $K_{n + 1} \\to K_n$", "in $D(A/I^{n + 1})$. Assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete,", "\\item $K_1$ is pseudo-coherent, and", "\\item the maps induce isomorphisms", "$K_{n + 1} \\otimes_{A/I^{n + 1}}^\\mathbf{L} A/I^n \\to K_n$.", "\\end{enumerate}", "Then $K = R\\lim K_n$ is a pseudo-coherent, derived complete object of $D(A)$", "and $K \\otimes_A^\\mathbf{L} A/I^n \\to K_n$ is an isomorphism for all $n$." ], "refs": [], "proofs": [ { "contents": [ "We already know that $K$ is pseudo-coherent and that", "$K \\otimes_A^\\mathbf{L} A/I^n \\to K_n$ is an isomorphism for all $n$, see", "Lemma \\ref{lemma-Rlim-pseudo-coherent-gives-pseudo-coherent}.", "To finish the proof it suffices to show that $K$ is derived complete.", "This follows from Lemma \\ref{lemma-pseudo-coherent-is-derived-complete}." ], "refs": [ "more-algebra-lemma-Rlim-pseudo-coherent-gives-pseudo-coherent", "more-algebra-lemma-pseudo-coherent-is-derived-complete" ], "ref_ids": [ 10407, 10370 ] } ], "ref_ids": [] }, { "id": 10409, "type": "theorem", "label": "more-algebra-lemma-Rlim-perfect-gives-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Rlim-perfect-gives-perfect", "contents": [ "\\begin{reference}", "\\cite[Lemma 4.2]{Bhatt-Algebraize}", "\\end{reference}", "Let $A = \\lim A_n$ be a limit of an inverse system $(A_n)$ of rings.", "Suppose given $K_n \\in D(A_n)$ and maps $K_{n + 1} \\to K_n$", "in $D(A_{n + 1})$. Assume", "\\begin{enumerate}", "\\item the transition maps $A_{n + 1} \\to A_n$ are surjective with", "locally nilpotent kernels,", "\\item $K_1$ is a perfect object, and", "\\item the maps induce isomorphisms", "$K_{n + 1} \\otimes_{A_{n + 1}}^\\mathbf{L} A_n \\to K_n$.", "\\end{enumerate}", "Then $K = R\\lim K_n$ is a perfect object of $D(A)$", "and $K \\otimes_A^\\mathbf{L} A_n \\to K_n$ is an isomorphism for all $n$." ], "refs": [], "proofs": [ { "contents": [ "We already know that $K$ is pseudo-coherent and that", "$K \\otimes_A^\\mathbf{L} A_n \\to K_n$ is an isomorphism for all $n$", "by Lemma \\ref{lemma-Rlim-pseudo-coherent-gives-pseudo-coherent}.", "Thus it suffices to show that $H^i(K \\otimes_A^\\mathbf{L} \\kappa) = 0$ for", "$i \\ll 0$ and every surjective map $A \\to \\kappa$ whose kernel is", "a maximal ideal $\\mathfrak m$, see", "Lemma \\ref{lemma-check-perfect-stalks}.", "Any element of $A$ which maps to a unit in $A_1$ is a unit", "in $A$ by Algebra, Lemma \\ref{algebra-lemma-locally-nilpotent-unit}", "and hence $\\Ker(A \\to A_1)$ is contained in the Jacobson radical of $A$ by", "Algebra, Lemma \\ref{algebra-lemma-contained-in-radical}.", "Hence $A \\to \\kappa$ factors as $A \\to A_1 \\to \\kappa$.", "Hence", "$$", "K \\otimes_A^\\mathbf{L} \\kappa =", "K \\otimes_A^\\mathbf{L} A_1 \\otimes_{A_1}^\\mathbf{L} \\kappa =", "K_1 \\otimes_{A_1}^\\mathbf{L} \\kappa", "$$", "and we get what we want as $K_1$ has finite tor dimension by", "Lemma \\ref{lemma-perfect}." ], "refs": [ "more-algebra-lemma-Rlim-pseudo-coherent-gives-pseudo-coherent", "more-algebra-lemma-check-perfect-stalks", "algebra-lemma-locally-nilpotent-unit", "algebra-lemma-contained-in-radical", "more-algebra-lemma-perfect" ], "ref_ids": [ 10407, 10243, 459, 399, 10212 ] } ], "ref_ids": [] }, { "id": 10410, "type": "theorem", "label": "more-algebra-lemma-Rlim-perfect-gives-complete", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Rlim-perfect-gives-complete", "contents": [ "Let $A$ be a ring and $I \\subset A$ an ideal.", "Suppose given $K_n \\in D(A/I^n)$ and maps $K_{n + 1} \\to K_n$", "in $D(A/I^{n + 1})$. Assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete,", "\\item $K_1$ is a perfect object, and", "\\item the maps induce isomorphisms", "$K_{n + 1} \\otimes_{A/I^{n + 1}}^\\mathbf{L} A/I^n \\to K_n$.", "\\end{enumerate}", "Then $K = R\\lim K_n$ is a perfect, derived complete object of $D(A)$", "and $K \\otimes_A^\\mathbf{L} A/I^n \\to K_n$ is an isomorphism for all $n$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-Rlim-perfect-gives-perfect} and", "\\ref{lemma-Rlim-pseudo-coherent-gives-complete-pseudo-coherent}", "(to get derived completeness)." ], "refs": [ "more-algebra-lemma-Rlim-perfect-gives-perfect", "more-algebra-lemma-Rlim-pseudo-coherent-gives-complete-pseudo-coherent" ], "ref_ids": [ 10409, 10408 ] } ], "ref_ids": [] }, { "id": 10411, "type": "theorem", "label": "more-algebra-lemma-Rlim-gives-complete", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-Rlim-gives-complete", "contents": [ "Let $A$ be a ring and $I \\subset A$ an ideal. Suppose", "given $K_n \\in D(A/I^n)$ and maps $K_{n + 1} \\to K_n$ in", "$D(A/I^{n + 1})$. If", "\\begin{enumerate}", "\\item $A$ is Noetherian,", "\\item $K_1$ is bounded above, and", "\\item the maps induce isomorphisms", "$K_{n + 1} \\otimes_{A/I^{n + 1}}^\\mathbf{L} A/I^n \\to K_n$,", "\\end{enumerate}", "then $K = R\\lim K_n$ is a derived complete object of $D^-(A)$ and", "$K \\otimes_A^\\mathbf{L} A/I^n \\to K_n$ is an isomorphism for all $n$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $H^i(K_1) = 0$ for $i > b$. Then we can find a complex of", "free $A/I$-modules $P_1^\\bullet$ representing $K_1$ with $P_1^i = 0$", "for $i > b$. By Lemma \\ref{lemma-lift-complex-projectives}", "we can, by induction on $n > 1$, find", "complexes $P_n^\\bullet$ of free $A/I^n$-modules representing $K_n$", "and maps $P_n^\\bullet \\to P_{n - 1}^\\bullet$ representing the maps", "$K_n \\to K_{n - 1}$ inducing isomorphisms (!)", "of complexes $P_n^\\bullet/I^{n - 1}P_n^\\bullet \\to P_{n - 1}^\\bullet$.", "\\medskip\\noindent", "Thus we have arrived at the situation where $R\\lim K_n$ is represented by", "$P^\\bullet = \\lim P_n^\\bullet$, see", "Lemma \\ref{lemma-compute-Rlim-modules} and", "Remark \\ref{remark-how-unique}.", "The complexes $P_n^\\bullet$ are uniformly bounded above complexes", "of flat $A/I^n$-modules and the transition maps are termwise surjective.", "Then $P^\\bullet$ is a bounded above complex of flat $A$-modules by", "Lemma \\ref{lemma-limit-flat}.", "It follows that $K \\otimes_A^\\mathbf{L} A/I^t$ is represented by", "$P^\\bullet \\otimes_A A/I^t$. We have", "$P^\\bullet \\otimes_A A/I^t = \\lim P_n^\\bullet \\otimes_A A/I^t$", "termwise by Lemma \\ref{lemma-limit-flat}.", "The transition maps", "$P_{n + 1}^\\bullet \\otimes_A A/I^t \\to P_n^\\bullet \\otimes_A A/I^t$", "are isomorphisms for $n \\geq t$. Hence we have", "$\\lim P_n^\\bullet \\otimes_A A/I^t = R\\lim P_n^\\bullet \\otimes_A A/I^t$.", "By assumption and our choice of $P_n^\\bullet$ the complex", "$P_n^\\bullet \\otimes_A A/I^t = P_n^\\bullet \\otimes_{A/I^n} A/I^t$", "represents $K_n \\otimes_{A/I^n}^\\mathbf{L} A/I^t = K_t$ for all $n \\geq t$.", "We conclude", "$$", "P^\\bullet \\otimes_A A/I^t =", "R\\lim P_n^\\bullet \\otimes_A A/I^t =", "R\\lim K_t = K_t", "$$", "In other words, we have $K \\otimes_A^\\mathbf{L} A/I^t = K_t$.", "This proves the lemma as it follows that $K$ is derived complete by", "Proposition \\ref{proposition-noetherian-naive-completion-is-completion}." ], "refs": [ "more-algebra-lemma-lift-complex-projectives", "more-algebra-lemma-compute-Rlim-modules", "more-algebra-remark-how-unique", "more-algebra-lemma-limit-flat", "more-algebra-lemma-limit-flat", "more-algebra-proposition-noetherian-naive-completion-is-completion" ], "ref_ids": [ 10229, 10324, 10660, 9955, 9955, 10590 ] } ], "ref_ids": [] }, { "id": 10412, "type": "theorem", "label": "more-algebra-lemma-kollar-kovacs", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-kollar-kovacs", "contents": [ "\\begin{reference}", "Email from Kovacs of 23/02/2018.", "\\end{reference}", "Let $I$ be an ideal of a Noetherian ring $A$. Let $K \\in D(A)$.", "Set $K_n = K \\otimes_A^\\mathbf{L} A/I^n$. Assume for all", "$i \\in \\mathbf{Z}$ we have", "\\begin{enumerate}", "\\item $H^i(K)$ is a finite $A$-module, and", "\\item the system $H^i(K_n)$ satisfies Mittag-Leffler.", "\\end{enumerate}", "Then $\\lim H^i(K)/I^nH^i(K)$ is equal to $\\lim H^i(K_n)$ for all", "$i \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $K^\\wedge = R\\lim K_n$ is the derived completion of $K$, see", "Proposition \\ref{proposition-noetherian-naive-completion-is-completion}.", "By Lemma \\ref{lemma-derived-completion-pseudo-coherent} we have", "$H^i(K^\\wedge) = \\lim H^i(K)/I^nH^i(K)$. By", "Lemma \\ref{lemma-break-long-exact-sequence-modules}", "we get short exact sequences", "$$", "0 \\to R^1\\lim H^{i - 1}(K_n) \\to H^i(K^\\wedge) \\to \\lim H^i(K_n) \\to 0", "$$", "The Mittag-Leffler condition guarantees that the left terms are zero", "(Lemma \\ref{lemma-compute-Rlim-modules}) and we conclude the lemma is true." ], "refs": [ "more-algebra-proposition-noetherian-naive-completion-is-completion", "more-algebra-lemma-derived-completion-pseudo-coherent", "more-algebra-lemma-break-long-exact-sequence-modules", "more-algebra-lemma-compute-Rlim-modules" ], "ref_ids": [ 10590, 10393, 10326, 10324 ] } ], "ref_ids": [] }, { "id": 10413, "type": "theorem", "label": "more-algebra-lemma-internal-hom-evaluate-isomorphism", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-internal-hom-evaluate-isomorphism", "contents": [ "Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$.", "the map", "$$", "R\\Hom_R(L, M) \\otimes_R^\\mathbf{L} K \\longrightarrow R\\Hom_R(R\\Hom_R(K, L), M)", "$$", "of Lemma \\ref{lemma-internal-hom-evaluate} is an isomorphism", "in the following two cases", "\\begin{enumerate}", "\\item $K$ perfect, or", "\\item $K$ is pseudo-coherent, $L \\in D^+(R)$, and $M$ finite injective", "dimension.", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-internal-hom-evaluate" ], "proofs": [ { "contents": [ "Choose", "a K-injective complex $I^\\bullet$ representing $M$,", "a K-injective complex $J^\\bullet$ representing $L$, and", "a bounded above complex of finite projective modules $K^\\bullet$", "representing $K$. Consider the map of complexes", "$$", "\\text{Tot}(\\Hom^\\bullet(J^\\bullet, I^\\bullet) \\otimes_R K^\\bullet)", "\\longrightarrow", "\\Hom^\\bullet(\\Hom^\\bullet(K^\\bullet, J^\\bullet), I^\\bullet)", "$$", "of Lemma \\ref{lemma-evaluate-and-more}. Note that", "$$", "\\left(\\prod\\nolimits_{p + r = t} \\Hom_R(J^{-r}, I^p)\\right) \\otimes_R K^s =", "\\prod\\nolimits_{p + r = t} \\Hom_R(J^{-r}, I^p) \\otimes_R K^s", "$$", "because $K^s$ is finite projective. The map is given by the maps", "$$", "c_{p, r, s} :", "\\Hom_R(J^{-r}, I^p) \\otimes_R K^s", "\\longrightarrow", "\\Hom_R(\\Hom_R(K^s, J^{-r}), I^p)", "$$", "which are isomorphisms as $K^s$ is finite projective.", "For every element $\\alpha = (\\alpha^{p, r, s})$", "of degree $n$ of the left hand side, there are only finitely", "many values of $s$ such that $\\alpha^{p, r, s}$ is nonzero", "(for some $p, r$ with $n = p + r + s$). Hence our map", "is an isomorphism if the same vanishing condition is forced", "on the elements $\\beta = (\\beta^{p, r, s})$ of the right hand side.", "If $K^\\bullet$ is a bounded complex of finite projective", "modules, this is clear. On the other hand, if we can choose", "$I^\\bullet$ bounded and $J^\\bullet$ bounded below, then", "$\\beta^{p, r, s}$ is zero for $p$ outside a fixed range, for", "$s \\gg 0$, and for $r \\gg 0$. Hence among solutions of $n = p + r + s$", "with $\\beta^{p, r, s}$ nonzero only a finite number of $s$ values", "occur." ], "refs": [ "more-algebra-lemma-evaluate-and-more" ], "ref_ids": [ 10202 ] } ], "ref_ids": [ 10208 ] }, { "id": 10414, "type": "theorem", "label": "more-algebra-lemma-internal-hom-evaluate-isomorphism-technical", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-internal-hom-evaluate-isomorphism-technical", "contents": [ "Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$.", "the map", "$$", "R\\Hom_R(L, M) \\otimes_R^\\mathbf{L} K \\longrightarrow R\\Hom_R(R\\Hom_R(K, L), M)", "$$", "of Lemma \\ref{lemma-internal-hom-evaluate} is an isomorphism", "if the following three conditions are satisfied", "\\begin{enumerate}", "\\item $L, M$ have finite injective dimension,", "\\item $R\\Hom_R(L, M)$ has finite tor dimension,", "\\item for every $n \\in \\mathbf{Z}$ the truncation $\\tau_{\\leq n}K$", "is pseudo-coherent", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-internal-hom-evaluate" ], "proofs": [ { "contents": [ "Pick an integer $n$ and consider the distinguished triangle", "$$", "\\tau_{\\leq n}K \\to K \\to \\tau_{\\geq n + 1}K \\to \\tau_{\\leq n}K[1]", "$$", "see Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}.", "By assumption (3) and Lemma \\ref{lemma-internal-hom-evaluate-isomorphism}", "the map is an isomorphism for $\\tau_{\\leq n}K$. Hence it", "suffices to show that both", "$$", "R\\Hom_R(L, M) \\otimes_R^\\mathbf{L} \\tau_{\\geq n + 1}K", "\\quad\\text{and}\\quad", "R\\Hom_R(R\\Hom_R(\\tau_{\\geq n + 1}K, L), M)", "$$", "have vanishing cohomology in degrees $\\leq n - c$ for some $c$.", "This follows immediately from assumptions (2) and (1)." ], "refs": [ "derived-remark-truncation-distinguished-triangle", "more-algebra-lemma-internal-hom-evaluate-isomorphism" ], "ref_ids": [ 2016, 10413 ] } ], "ref_ids": [ 10208 ] }, { "id": 10415, "type": "theorem", "label": "more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism", "contents": [ "Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. The map", "$$", "K \\otimes_R^\\mathbf{L} R\\Hom_R(M, L) \\longrightarrow", "R\\Hom_R(M, K \\otimes_R^\\mathbf{L} L)", "$$", "of Lemma \\ref{lemma-internal-hom-diagonal-better}", "is an isomorphism in the following cases", "\\begin{enumerate}", "\\item $M$ perfect, or", "\\item $K$ is perfect, or", "\\item $M$ is pseudo-coherent, $L \\in D^+(R)$, and $K$ has", "tor amplitude in $[a, \\infty]$.", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-internal-hom-diagonal-better" ], "proofs": [ { "contents": [ "Proof in case $M$ is perfect. Note that both sides of the arrow", "transform distinguished triangles in $M$ into distinguished triangles", "and commute with direct sums. Hence it suffices to check", "it holds when $M = R[n]$, see", "Derived Categories, Remark \\ref{derived-remark-check-on-generator}", "and Lemma \\ref{lemma-perfect-ring-classical-generator}.", "In this case the result is obvious.", "\\medskip\\noindent", "Proof in case $K$ is perfect. Same argument as in the previous case.", "\\medskip\\noindent", "Proof in case (3). We may represent $K$ and $L$ by", "bounded below complexes of $R$-modules $K^\\bullet$ and $L^\\bullet$.", "We may assume that $K^\\bullet$ is a K-flat complex", "consisting of flat $R$-modules, see", "Lemma \\ref{lemma-bounded-below-tor-amplitude}.", "We may represent $M$ by a bounded above complex $M^\\bullet$", "of finite free $R$-modules, see Definition \\ref{definition-pseudo-coherent}.", "Then the object on the LHS is represented by", "$$", "\\text{Tot}(K^\\bullet \\otimes_R \\Hom^\\bullet(M^\\bullet, L^\\bullet))", "$$", "and the object on the RHS by", "$$", "\\Hom^\\bullet(M^\\bullet, \\text{Tot}(K^\\bullet \\otimes_R L^\\bullet))", "$$", "This uses Lemma \\ref{lemma-RHom-out-of-projective}.", "Both complexes have in degree $n$ the module", "$$", "\\bigoplus\\nolimits_{p + q + r = n} K^p \\otimes \\Hom_R(M^{-r}, L^q) =", "\\bigoplus\\nolimits_{p + q + r = n} \\Hom_R(M^{-r}, K^p \\otimes_R L^q)", "$$", "because $M^{-r}$ is finite free (as well these are finite direct sums).", "The map defined in Lemma \\ref{lemma-internal-hom-diagonal-better}", "comes from the map of complexes defined in", "Lemma \\ref{lemma-diagonal-better} which uses", "the canonical isomorphisms between these modules." ], "refs": [ "derived-remark-check-on-generator", "more-algebra-lemma-perfect-ring-classical-generator", "more-algebra-lemma-bounded-below-tor-amplitude", "more-algebra-definition-pseudo-coherent", "more-algebra-lemma-RHom-out-of-projective", "more-algebra-lemma-internal-hom-diagonal-better", "more-algebra-lemma-diagonal-better" ], "ref_ids": [ 2030, 10247, 10171, 10623, 10207, 10210, 10200 ] } ], "ref_ids": [ 10210 ] }, { "id": 10416, "type": "theorem", "label": "more-algebra-lemma-hom-complex-K-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-hom-complex-K-flat", "contents": [ "Let $R$ be a ring. Let $P^\\bullet$ be a bounded above complex", "of projective $R$-modules. Let $K^\\bullet$ be a K-flat complex", "of $R$-modules. If $P^\\bullet$ is a perfect object of $D(R)$,", "then $\\Hom^\\bullet(P^\\bullet, K^\\bullet)$ is K-flat and", "represents $R\\Hom_R(P^\\bullet, K^\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "The last statement is Lemma \\ref{lemma-RHom-out-of-projective}.", "Since $P^\\bullet$ represents a perfect object, there exists a", "finite complex of finite projective $R$-modules $F^\\bullet$", "such that $P^\\bullet$ and $F^\\bullet$ are isomorphic in $D(R)$, see", "Definition \\ref{definition-perfect}.", "Then $P^\\bullet$ and $F^\\bullet$ are homotopy equivalent, see", "Derived Categories, Lemma", "\\ref{derived-lemma-morphisms-from-projective-complex}.", "Then $\\Hom^\\bullet(P^\\bullet, K^\\bullet)$", "and $\\Hom^\\bullet(F^\\bullet, K^\\bullet)$", "are homotopy equivalent. Hence the first is K-flat if and", "only if the second is (follows from", "Definition \\ref{definition-K-flat} and", "Lemma \\ref{lemma-derived-tor-homotopy}).", "It is clear that", "$$", "\\Hom^\\bullet(F^\\bullet, K^\\bullet) =", "\\text{Tot}(E^\\bullet \\otimes_R K^\\bullet)", "$$", "where $E^\\bullet$ is the dual complex to $F^\\bullet$", "with terms $E^n = \\Hom_R(F^{-n}, R)$, see", "Lemma \\ref{lemma-dual-perfect-complex}", "and its proof.", "Since $E^\\bullet$ is a bounded complex of projectives", "we find that it is K-flat by Lemma \\ref{lemma-derived-tor-quasi-isomorphism}.", "Then we conclude by Lemma \\ref{lemma-tensor-product-K-flat}." ], "refs": [ "more-algebra-lemma-RHom-out-of-projective", "more-algebra-definition-perfect", "derived-lemma-morphisms-from-projective-complex", "more-algebra-definition-K-flat", "more-algebra-lemma-derived-tor-homotopy", "more-algebra-lemma-dual-perfect-complex", "more-algebra-lemma-derived-tor-quasi-isomorphism", "more-algebra-lemma-tensor-product-K-flat" ], "ref_ids": [ 10207, 10628, 1862, 10620, 10121, 10224, 10128, 10125 ] } ], "ref_ids": [] }, { "id": 10417, "type": "theorem", "label": "more-algebra-lemma-upgrade-adjoint-tensor-RHom", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-upgrade-adjoint-tensor-RHom", "contents": [ "Let $R \\to R'$ be a ring map. For $K \\in D(R)$ and", "$M \\in D(R')$ there is a canonical isomorphism", "$$", "R\\Hom_R(K, M) = R\\Hom_{R'}(K \\otimes_R^\\mathbf{L} R', M)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose a K-injective complex of $R'$-modules $J^\\bullet$ representing $M$. ", "Choose a quasi-isomorphism $J^\\bullet \\to I^\\bullet$ where $I^\\bullet$", "is a K-injective complex of $R$-modules. Choose a K-flat complex", "$K^\\bullet$ of $R$-modules representing $K$. Consider the map", "$$", "\\Hom^\\bullet(K^\\bullet \\otimes_R R', J^\\bullet)", "\\longrightarrow", "\\Hom^\\bullet(K^\\bullet, I^\\bullet)", "$$", "The map on degree $n$ terms is given by the map", "$$", "\\prod\\nolimits_{n = p + q} \\Hom_{R'}(K^{-q} \\otimes_R R', J^p)", "\\longrightarrow", "\\prod\\nolimits_{n = p + q} \\Hom_R(K^{-q}, I^p)", "$$", "coming from precomposing by $K^{-q} \\to K^{-q} \\otimes_R R'$", "and postcomposing by $J^p \\to I^p$. To finish the proof it suffices", "to show that we get isomorphisms on cohomology groups:", "$$", "\\Hom_{D(R)}(K, M) = \\Hom_{D(R')}(K \\otimes_R^\\mathbf{L} R', M)", "$$", "which is true because base change $- \\otimes_R^\\mathbf{L} R' : D(R) \\to D(R')$", "is left adjoint to the restriction functor $D(R') \\to D(R)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10418, "type": "theorem", "label": "more-algebra-lemma-base-change-RHom", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-RHom", "contents": [ "Let $R \\to R'$ be a ring map. Let $K, M \\in D(R)$. The map", "(\\ref{equation-base-change-RHom})", "$$", "R\\Hom_R(K, M) \\otimes_R^\\mathbf{L} R'", "\\longrightarrow", "R\\Hom_{R'}(K \\otimes_R^\\mathbf{L} R', M \\otimes_R^\\mathbf{L} R')", "$$", "is an isomorphism in $D(R')$ in the following cases", "\\begin{enumerate}", "\\item $K$ is perfect,", "\\item $R'$ is perfect as an $R$-module,", "\\item $R \\to R'$ is flat, $K$ is pseudo-coherent, and $M \\in D^{+}(R)$, or", "\\item $R'$ has finite tor dimension as an $R$-module,", "$K$ is pseudo-coherent, and $M \\in D^{+}(R)$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may check the map is an isomorphism after applying the", "restriction functor $D(R') \\to D(R)$. After applying this", "functor our map becomes the map", "$$", "R\\Hom_R(K, L) \\otimes_R^\\mathbf{L} R'", "\\longrightarrow", "R\\Hom_R(K, L \\otimes_R^\\mathbf{L} R')", "$$", "of Lemma \\ref{lemma-internal-hom-diagonal-better}.", "See discussion above the lemma to match the left and right hand sides;", "in particular, this uses Lemma \\ref{lemma-upgrade-adjoint-tensor-RHom}.", "Thus we conclude by", "Lemma \\ref{lemma-internal-hom-evaluate-tensor-isomorphism}." ], "refs": [ "more-algebra-lemma-internal-hom-diagonal-better", "more-algebra-lemma-upgrade-adjoint-tensor-RHom", "more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism" ], "ref_ids": [ 10210, 10417, 10415 ] } ], "ref_ids": [] }, { "id": 10419, "type": "theorem", "label": "more-algebra-lemma-consequence-Artin-Rees", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-consequence-Artin-Rees", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$. Let", "$", "K \\xrightarrow{\\alpha} L \\xrightarrow{\\beta} M", "$", "be a complex of finite $A$-modules. Set $H = \\Ker(\\beta)/\\Im(\\alpha)$.", "For $n \\geq 0$ let", "$$", "K/I^nK \\xrightarrow{\\alpha_n} L/I^nL \\xrightarrow{\\beta_n} M/I^nM", "$$", "be the induced complex. Set $H_n = \\Ker(\\beta_n)/\\Im(\\alpha_n)$.", "Then there are canonical $A$-module maps giving a commutative diagram", "$$", "\\xymatrix{", "& & & H \\ar[lld] \\ar[ld] \\ar[d] \\\\", "\\ldots \\ar[r] & H_3 \\ar[r] & H_2 \\ar[r] & H_1", "}", "$$", "Moreover, there exists a $c > 0$ and canonical $A$-module maps", "$H_n \\to H/I^{n - c}H$ for $n \\geq c$ such that the compositions", "$$", "H/I^n H \\to H_n \\to H/I^{n - c}H", "\\quad\\text{and}\\quad", "H_n \\to H/I^{n - c}H \\to H_{n - c}", "$$", "are the canonical ones. Moreover, we have", "\\begin{enumerate}", "\\item $(H_n)$ and $(H/I^nH)$ are isomorphic as pro-objects of $\\text{Mod}_A$,", "\\item $\\lim H_n = \\lim H/I^n H$,", "\\item the inverse system $(H_n)$ is Mittag-Leffler,", "\\item the image of $H_{n + c} \\to H_n$ is equal to the image of $H \\to H_n$,", "\\item the composition $I^cH_n \\to H_n \\to H/I^{n - c}H \\to H_n/I^{n - c}H_n$", "is the inclusion $I^cH_n \\to H_n$ followed by the quotient map", "$H_n \\to H_n/I^{n - c}H_n$, and", "\\item the kernel and cokernel of $H/I^nH \\to H_n$ is annihilated by $I^c$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $H_n = \\beta^{-1}(I^nM)/\\Im(\\alpha) + I^nL$. For $n \\geq 2$", "we have $\\beta^{-1}(I^nM) \\subset \\beta^{-1}(I^{n - 1}M)$ and", "$\\Im(\\alpha) + I^nL \\subset \\Im(\\alpha) + I^{n - 1}L$. Thus", "we obtain our canonical map $H_n \\to H_{n - 1}$. Similarly, we have", "$\\Ker(\\beta) \\subset \\beta^{-1}(I^nM)$ and", "$\\Im(\\alpha) \\subset \\Im(\\alpha) + I^nL$ which produces the", "canonical map $H \\to H_n$. We omit the verification that the diagram", "commutes.", "\\medskip\\noindent", "By Artin-Rees we may choose $c_1, c_2 \\geq 0$ such that", "$\\beta^{-1}(I^nM) \\subset \\Ker(\\beta) + I^{n - c_1}L$ for $n \\geq c_1$ and", "$\\Ker(\\beta) \\cap I^nL \\subset I^{n - c_2}\\Ker(\\beta)$ for $n \\geq c_2$, see", "Algebra, Lemmas \\ref{algebra-lemma-map-AR} and", "\\ref{algebra-lemma-Artin-Rees}. Set $c = c_1 + c_2$.", "\\medskip\\noindent", "Let $n \\geq c$. We define $\\psi_n : H_n \\to H/I^{n - c}H$ as follows.", "Say $x \\in H_n$. Choose $y \\in \\beta^{-1}(I^nM)$ representing $x$.", "Write $y = z + w$ with $z \\in \\Ker(\\beta)$ and $w \\in I^{n - c_1}L$", "(this is possible by our choice of $c_1$). We set $\\psi_n(x)$", "equal to the class of $z$ in $H/I^{n - c}H$. To see this is well defined,", "suppose we have a second set of choices $y', z', w'$ as above for $x$", "with obvious notation.", "Then $y' - y \\in \\Im(\\alpha) + I^nL$, say $y' - y = \\alpha(v) + u$", "with $v \\in K$ and $u \\in I^nL$. Thus", "$$", "y' = z' + w' = \\alpha(v) + u + z + w", "\\Rightarrow", "z' = z + \\alpha(v) + u + w - w'", "$$", "Since $\\beta(z' - z - \\alpha(v)) = 0$ we find that", "$u + w - w' \\in \\Ker(\\beta) \\cap I^{n - c_1}L$", "which is contained in $I^{n - c_1 - c_2}\\Ker(\\beta) = I^{n - c}\\Ker(\\beta)$", "by our choice of $c_2$. Thus $z'$ and $z$ have the same image in", "$H/I^{n - c}H$ as desired.", "\\medskip\\noindent", "The composition $H/I^n H \\to H_n \\to H/I^{n - c}H$ is the canonical map", "because if $z \\in \\Ker(\\beta)$ represents an element $x$ in", "$H/I^nH = \\Ker(\\beta)/\\Im(\\alpha) + I^n\\Ker(\\beta)$ then it is clear", "from the above that $x$ maps to the class of $z$ in $H/I^{n - c}H$", "under the maps constructed above.", "\\medskip\\noindent", "Let us consider the composition $H_n \\to H/I^{n - c}H \\to H_{n - c}$.", "Given $x, y, z, w$ as in the construction of $\\psi_n$ above, we see", "that $x$ is mapped to the cass of $z$ in $H_{n - c}$. On the other hand,", "the canonical map $H_n \\to H_{n - c}$ from the first paragraph of the", "proof sends $x$ to the class of $y$. Thus we have to show that", "$y - z \\in \\Im(\\alpha) + I^{n - c}L$ which is the case because", "$y - z = w \\in I^{n - c_1}L \\subset I^{n - c}L$.", "\\medskip\\noindent", "Statements (1) -- (4) are formal consequences of what we just proved.", "Namely, (1) follows from the existence of the maps and the definition", "of morphisms of pro-objects in", "Categories, Remark \\ref{categories-remark-pro-category}.", "Part (2) holds because isomorphic pro-objects have isomorphic limits.", "Part (3) is immediate from part (4).", "Part (4) follows from the factorization", "$H_{n + c} \\to H/I^nH \\to H_n$ of the canonical map", "$H_{n + c} \\to H_n$.", "\\medskip\\noindent", "Proof of part (5). Let $x \\in I^cH_n$. Write $x = \\sum f_i x_i$ with", "$x_i \\in H_n$ and $f_i \\in I^c$. Choose $y_i, z_i, w_i$ as in the", "construction of $\\psi_n$ for $x_i$. Then for the computation of", "$\\psi_n$ of $x$ we may choose", "$y = \\sum f_iy_i$, $z = \\sum f_i z_i$ and $w = \\sum f_i w_i$", "and we see that $\\psi_n(x)$ is given by the class of $z$.", "The image of this in $H_n/I^{n - c}H_n$ is equal to the class", "of $y$ as $w = \\sum f_i w_i$ is in $I^nL$. This proves (5).", "\\medskip\\noindent", "Proof of part (6). Let $y \\in \\Ker(\\beta)$ whose class is $x$ in $H$.", "If $x$ maps to zero in $H_n$, then $y \\in I^nL + \\Im(\\alpha)$.", "Hence $y - \\alpha(v) \\in \\Ker(\\beta) \\cap I^nL$ for some $v \\in K$.", "Then $y - \\alpha(v) \\in I^{n - c_2}\\Ker(\\beta)$ and hence the class", "of $y$ in $H/I^nH$ is annihilated by $I^{c_2}$. Finally, let $x \\in H_n$", "be the class of $y \\in \\beta^{-1}(I^nM)$. Then we write $y = z + w$", "with $z \\in \\Ker(\\beta)$ and $w \\in I^{n - c_1}L$ as above. Clearly, if", "$f \\in I^{c_1}$ then $fx$ is the class of", "$fy + fw \\equiv fy$ modulo $\\Im(\\alpha) + I^nL$ and hence $fx$ is", "the image of the class of $fy$ in $H$ as desired." ], "refs": [ "algebra-lemma-map-AR", "algebra-lemma-Artin-Rees", "categories-remark-pro-category" ], "ref_ids": [ 626, 625, 12420 ] } ], "ref_ids": [] }, { "id": 10420, "type": "theorem", "label": "more-algebra-lemma-kollar-kovacs-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-kollar-kovacs-pseudo-coherent", "contents": [ "\\begin{reference}", "Email from Kovacs of 23/02/2018.", "\\end{reference}", "Let $I$ be an ideal of a Noetherian ring $A$. Let $K \\in D(A)$", "be pseudo-coherent. Set $K_n = K \\otimes_A^\\mathbf{L} A/I^n$.", "Then for all $i \\in \\mathbf{Z}$ the system $H^i(K_n)$", "satisfies Mittag-Leffler and $\\lim H^i(K)/I^nH^i(K)$ is equal to", "$\\lim H^i(K_n)$." ], "refs": [], "proofs": [ { "contents": [ "We may represent $K$ by a bounded above complex $P^\\bullet$ of", "finite free $A$-modules. Then $K_n$ is represented by", "$P^\\bullet/I^nP^\\bullet$. Hence the Mittag-Leffler property", "by Lemma \\ref{lemma-consequence-Artin-Rees}.", "The final statement follows then from", "Lemma \\ref{lemma-kollar-kovacs}." ], "refs": [ "more-algebra-lemma-consequence-Artin-Rees", "more-algebra-lemma-kollar-kovacs" ], "ref_ids": [ 10419, 10412 ] } ], "ref_ids": [] }, { "id": 10421, "type": "theorem", "label": "more-algebra-lemma-derived-completion-plain-completion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-derived-completion-plain-completion", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let", "$M^\\bullet$ be a bounded complex of finite $A$-modules. The", "inverse system of maps", "$$", "M^\\bullet \\otimes_A^\\mathbf{L} A/I^n \\longrightarrow M^\\bullet/I^nM^\\bullet", "$$", "defines an isomorphism of pro-objects of $D(A)$." ], "refs": [], "proofs": [ { "contents": [ "Say $I = (f_1, \\ldots, f_r)$.", "Let $K_n \\in D(A)$ be the object represented by the Koszul complex", "on $f_1^n, \\ldots, f_r^n$. Recall that we have maps $K_n \\to A/I^n$", "which induce a pro-isomorphism of inverse systems, see", "Lemma \\ref{lemma-sequence-Koszul-complexes}.", "Hence it suffices to show that", "$$", "M^\\bullet \\otimes_A^\\mathbf{L} K_n \\longrightarrow M^\\bullet/I^nM^\\bullet", "$$", "defines an isomorphism of pro-objects of $D(A)$. Since $K_n$ is represented", "by a complex of finite free $A$-modules sitting in degrees $-r, \\ldots, 0$", "there exist $a, b \\in \\mathbf{Z}$ such that", "the source and target of the displayed arrow have vanishing cohomology", "in degrees outside $[a, b]$ for all $n$. Thus we may apply", "Derived Categories, Lemma \\ref{derived-lemma-pro-isomorphism-bis}", "and we find that it suffices to show that the maps", "$$", "H^i(M^\\bullet \\otimes_A^\\mathbf{L} A/I^n) \\to H^i(M^\\bullet/I^nM^\\bullet)", "$$", "define isomorphisms of pro-systems of $A$-modules for any $i \\in \\mathbf{Z}$.", "To see this choose a quasi-isomorphism", "$P^\\bullet \\to M^\\bullet$ where $P^\\bullet$ is a bounded", "above complex of finite free $A$-modules. The arrows above are", "given by the maps", "$$", "H^i(P^\\bullet/I^nP^\\bullet) \\to H^i(M^\\bullet/I^nM^\\bullet)", "$$", "These define an isomorphism of pro-systems by", "Lemma \\ref{lemma-consequence-Artin-Rees}. Namely,", "the lemma shows both are isomorphic to the pro-system $H^i/I^nH^i$", "with $H^i = H^i(M^\\bullet) = H^i(P^\\bullet)$." ], "refs": [ "more-algebra-lemma-sequence-Koszul-complexes", "derived-lemma-pro-isomorphism-bis", "more-algebra-lemma-consequence-Artin-Rees" ], "ref_ids": [ 10391, 1957, 10419 ] } ], "ref_ids": [] }, { "id": 10422, "type": "theorem", "label": "more-algebra-lemma-hom-systems-ML", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-hom-systems-ML", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$, $N$ be", "finite $A$-modules. Set $M_n = M/I^nM$ and $N_n = N/I^nN$. Then", "\\begin{enumerate}", "\\item the systems $(\\Hom_A(M_n, N_n))$ and $(\\text{Isom}_A(M_n, N_n))$", "are Mittag-Leffler,", "\\item there exists a $c \\geq 0$ such that the kernels and cokernels of", "$$", "\\Hom_A(M, N)/I^n\\Hom_A(M, N) \\to \\Hom_A(M_n, N_n)", "$$", "are killed by $I^c$ for all $n$,", "\\item we have", "$\\lim \\Hom_A(M_n, N_n) =\\Hom_A(M, N)^\\wedge =", "\\Hom_{A^\\wedge}(M^\\wedge, N^\\wedge)$", "\\item $\\lim \\text{Isom}_A(M_n, N_n) =", "\\text{Isom}_{A^\\wedge}(M^\\wedge, N^\\wedge)$.", "\\end{enumerate}", "Here ${}^\\wedge$ denotes usual $I$-adic completion." ], "refs": [], "proofs": [ { "contents": [ "Note that $\\Hom_A(M_n, N_n) = \\Hom_A(M, N_n)$. Choose a presentation", "$$", "A^{\\oplus t} \\to A^{\\oplus s} \\to M \\to 0", "$$", "Applying the right exact functor $\\Hom_A(-, N)$ we obtain a complex", "$$", "0 \\xrightarrow{\\alpha} N^{\\oplus s} \\xrightarrow{\\beta} N^{\\oplus t}", "$$", "whose cohomology in the middle is $\\Hom_A(M, N)$ and such that for", "$n \\geq 0$ the cohomology of", "$$", "0 \\xrightarrow{\\alpha_n} N_n^{\\oplus s} \\xrightarrow{\\beta_n} N_n^{\\oplus t}", "$$", "is $\\Hom_A(M_n, N_n)$. Let $c \\geq 0$ be as in", "Lemma \\ref{lemma-consequence-Artin-Rees}", "for this $A$, $I$, $\\alpha$, and $\\beta$.", "By part (3) of the lemma", "we deduce the Mittag-Leffler property for $(\\Hom_A(M_n, N_n))$.", "The kernel and cokernel of the maps", "$\\Hom_A(M, N)/I^n\\Hom_A(M, N) \\to \\Hom_A(M_n, N_n)$", "are killed by $I^c$ by [art part (6) of the lemma.", "We find that $\\lim \\Hom_A(M_n, N_n) = \\Hom_A(M, N)^\\wedge$ by", "part (2) of the lemma. The equality", "$$", "\\Hom_{A^\\wedge}(M^\\wedge, N^\\wedge) = \\lim \\Hom_A(M_n, N_n)", "$$", "follows formally from the fact that $M^\\wedge = \\lim M_n$ and", "$M_n = M^\\wedge/I^nM^\\wedge$ and the corresponding facts for $N$, see", "Algebra, Lemma \\ref{algebra-lemma-completion-complete}.", "\\medskip\\noindent", "The result for isomorphisms follows from the case of homomorphisms", "applied to both $(\\Hom(M_n, N_n))$ and $(\\Hom(N_n, M_n))$", "and the following fact: for $n > m > 0$, if we have maps", "$\\alpha : M_n \\to N_n$ and $\\beta : N_n \\to M_n$ which", "induce an isomorphisms $M_m \\to N_m$ and $N_m \\to M_m$, then", "$\\alpha$ and $\\beta$ are isomorphisms. Namely, then $\\alpha \\circ \\beta$", "is surjective by Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "hence $\\alpha \\circ \\beta$ is an isomorphism by", "Algebra, Lemma \\ref{algebra-lemma-fun}." ], "refs": [ "more-algebra-lemma-consequence-Artin-Rees", "algebra-lemma-completion-complete", "algebra-lemma-NAK", "algebra-lemma-fun" ], "ref_ids": [ 10419, 872, 401, 388 ] } ], "ref_ids": [] }, { "id": 10423, "type": "theorem", "label": "more-algebra-lemma-isomorphic-completions", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-isomorphic-completions", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$, $N$ be", "finite $A$-modules. Set $M_n = M/I^nM$ and $N_n = N/I^nN$. If", "$M_n \\cong N_n$ for all $n$, then $M^\\wedge \\cong N^\\wedge$", "as $A^\\wedge$-modules." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-hom-systems-ML} the system $(\\text{Isom}_A(M_n, N_n))$", "is Mittag-Leffler. By assumption each of the sets", "$\\text{Isom}_A(M_n, N_n)$ is nonempty. Hence $\\lim \\text{Isom}_A(M_n, N_n)$", "is nonempty. Since", "$\\lim \\text{Isom}_A(M_n, N_n) = \\text{Isom}_{A^\\wedge}(M^\\wedge, N^\\wedge)$", "we obtain an isomorphism." ], "refs": [ "more-algebra-lemma-hom-systems-ML" ], "ref_ids": [ 10422 ] } ], "ref_ids": [] }, { "id": 10424, "type": "theorem", "label": "more-algebra-lemma-iso", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-iso", "contents": [ "A morphism $(c, \\varphi_n)$ of the category of", "Remark \\ref{remark-weird-systems} is an", "isomorphism if and only if there exists a $c' \\geq 0$ such that", "$\\Ker(\\varphi_n)$ and $\\Coker(\\varphi_n)$ are $I^{c'}$-torsion for", "all $n \\gg 0$." ], "refs": [ "more-algebra-remark-weird-systems" ], "proofs": [ { "contents": [ "We may and do assume $c' \\geq c$ and that the", "$\\Ker(\\varphi_n)$ and $\\Coker(\\varphi_n)$ are $I^{c'}$-torsion", "for all $n$. For $n \\geq c'$ and", "$x \\in I^{c'}E'_n$ we can choose $y \\in I^cE_n$ with", "$x = \\varphi_n(y) \\bmod E'_n[I^c]$ as $\\Coker(\\varphi_n)$", "is annihilated by $I^{c'}$. Set $\\psi_n(x)$ equal to the class of", "$y$ in $E_n/E_n[I^{c'}]$. For a different choice $y' \\in I^cE_n$", "with $x = \\varphi_n(y') \\bmod E'_n[I^c]$ the difference", "$y - y'$ maps to zero in $E'_n/E'_n[I^c]$ and hence is", "annihilated by $I^{c'}$ in $I^cE_n$. Thus the maps", "$\\psi_n : I^{c'}E'_n \\to E_n/E_n[I^{c'}]$", "are well defined.", "We omit the verification that $(c', \\psi_n)$ is the inverse of", "$(c, \\varphi_n)$ in the category." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10674 ] }, { "id": 10425, "type": "theorem", "label": "more-algebra-lemma-dejong-kollar-kovacs", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-dejong-kollar-kovacs", "contents": [ "\\begin{reference}", "Email correspondence between Janos Kollar, Sandor Kovacs, and", "Johan de Jong of 23/02/2018.", "\\end{reference}", "Let $I$ be an ideal of the Noetherian ring $A$. Let $M$ and $N$", "be finite $A$-modules. Write $A_n = A/I^n$, $M_n = M/I^nM$, and", "$N_n = N/I^nN$.", "For every $i \\geq 0$ the objects", "$$", "\\{\\Ext^i_A(M, N)/I^n\\Ext^i_A(M, N)\\}_{n \\geq 1}", "\\quad\\text{and}\\quad", "\\{\\Ext^i_{A_n}(M_n, N_n)\\}_{n \\geq 1}", "$$", "are isomorphic in the category $\\mathcal{C}$ of", "Remark \\ref{remark-weird-systems}." ], "refs": [ "more-algebra-remark-weird-systems" ], "proofs": [ { "contents": [ "Choose a short exact sequence", "$$", "0 \\to K \\to A^{\\oplus r} \\to M \\to 0", "$$", "and set $K_n = K/I^nK$.", "For $n \\geq 1$ define $K(n) = \\Ker(A_n^{\\oplus r} \\to M_n)$", "so that we have exact sequences", "$$", "0 \\to K(n) \\to A_n^{\\oplus r} \\to M_n \\to 0", "$$", "and surjections $K_n \\to K(n)$. In fact, by", "Lemma \\ref{lemma-consequence-Artin-Rees}", "there is a $c \\geq 0$ and maps $K(n) \\to K_n/I^{n - c}K_n$", "which are ``almost inverse''.", "Since $I^{n - c}K_n \\subset K_n[I^c]$ these maps which witness the fact that", "the systems $\\{K(n)\\}_{n \\geq 1}$ and $\\{K_n\\}_{n \\geq 1}$", "are isomorphic in $\\mathcal{C}$.", "\\medskip\\noindent", "We claim the systems", "$$", "\\{\\Ext^i_{A_n}(K(n), N_n)\\}_{n \\geq 1}", "\\quad\\text{and}\\quad", "\\{\\Ext^i_{A_n}(K_n, N_n)\\}_{n \\geq 1}", "$$", "are isomorphic in the category $\\mathcal{C}$. Namely, the surjective maps", "$K_n \\to K(n)$ have kernels annihilated by $I^c$ and therefore determine maps", "$$", "\\Ext^i_{A_n}(K(n), N_n) \\to \\Ext^i_{A_n}(K_n, N_n)", "$$", "whose kernel and cokernel are annihilated by $I^c$. Hence the claim", "by Lemma \\ref{lemma-iso}.", "\\medskip\\noindent", "For $i \\geq 2$ we have isomorphisms", "$$", "\\Ext^{i - 1}_A(K, N) = \\Ext^i_A(M, N)", "\\quad\\text{and}\\quad", "\\Ext^{i - 1}_{A_n}(K(n), N_n) = \\Ext^i_{A_n}(M_n, N_n)", "$$", "In this way we see that it suffices to prove the lemma", "for $i = 0, 1$.", "\\medskip\\noindent", "For $i = 0, 1$ we consider the commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Hom(M, N) \\ar[r] \\ar[dd] &", "N^{\\oplus r} \\ar[r]_-\\varphi \\ar[dd] &", "\\Hom(K, N) \\ar[r] \\ar[d] &", "\\Ext^1(M, N) \\ar[r] &", "0 \\\\", "& & &", "\\Hom(K_n, N_n)", "\\\\", "0 \\ar[r] &", "\\Hom(M_n, N_n) \\ar[r] &", "N_n^{\\oplus r} \\ar[r] &", "\\Hom(K(n), N_n) \\ar[r] \\ar[u] &", "\\Ext^1(M_n, N_n) \\ar[r] &", "0", "}", "$$", "By Lemma \\ref{lemma-hom-systems-ML} we see that the kernel and cokernel of", "$\\Hom(M, N)/I^n \\Hom(M, N) \\to \\Hom(M_n, N_n)$ and", "$\\Hom(K, N)/I^n \\Hom(K, N) \\to \\Hom(K_n, N_n)$ and", "are $I^c$-torsion for some $c \\geq 0$ independent of $n$.", "Above we have seen the cokernel of the injective maps", "$\\Hom(K(n), N_n) \\to \\Hom(K_n, N_n)$ are annihilated by $I^c$", "after possibly increasing $c$. For such a $c$ we obtain maps", "$\\delta_n : I^c\\Hom(K, N)/I^n\\Hom(K, N) \\to \\Hom(K(n), N_n)$", "fitting into the diagram (precise formulation omitted).", "The kernel and cokernel of $\\delta_n$ are annihilated by", "$I^c$ after possibly increasing $c$ since we know that the", "same thing is true for $\\Hom(K, N)/I^n \\Hom(K, N) \\to \\Hom(K_n, N_n)$", "and $\\Hom(K(n), N_n) \\to \\Hom(K_n, N_n)$.", "Then we can use commutativity of the solid diagram", "$$", "\\xymatrix{", "\\varphi^{-1}(I^c\\Hom(K, N)) \\ar[r]_-\\varphi \\ar[d] &", "I^c\\Hom(K, N)/I^n\\Hom(K, N) \\ar[r] \\ar[d]^{\\delta_n} &", "I^c\\Ext^1(M, N)/I^n\\Ext^1(M, N) \\ar[r] \\ar@{..>}[d] & 0 \\\\", "N_n^{\\oplus r} \\ar[r] &", "\\Hom(K(n), N_n) \\ar[r] &", "\\Ext^1(M_n, N_n) \\ar[r] & 0", "}", "$$", "to define the dotted arrow. A straightforward diagram chase", "(omitted) shows that the kernel and cokernel of the", "dotted arrow are annihilated buy $I^c$ after possibly", "increasing $c$ one final time." ], "refs": [ "more-algebra-lemma-consequence-Artin-Rees", "more-algebra-lemma-iso", "more-algebra-lemma-hom-systems-ML" ], "ref_ids": [ 10419, 10424, 10422 ] } ], "ref_ids": [ 10674 ] }, { "id": 10426, "type": "theorem", "label": "more-algebra-lemma-not-awkward", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-not-awkward", "contents": [ "\\begin{reference}", "Email correspondence between Janos Kollar, Sandor Kovacs, and", "Johan de Jong of 23/02/2018.", "\\end{reference}", "Let $A \\to B$ be a flat homomorphism of Noetherian rings.", "Let $I \\subset A$ be an ideal. Let $M, N$ be $A$-modules.", "Set $B_n = B/I^nB$, $M_n = M/I^nM$, $N_n = N/I^nN$.", "If $M$ is flat over $A$, then we have", "$$", "\\lim \\Ext^i_B(M, N)/I^n \\Ext^i_B(M, N) =", "\\lim \\Ext^i_{B_n}(M_n, N_n)", "$$", "for all $i \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Choose a resolution", "$$", "\\ldots \\to P_2 \\to P_1 \\to P_0 \\to M \\to 0", "$$", "by finite free $B$-modues $P_i$. Set $P_{i, n} = P_i/I^nP_i$.", "Since $M$ and $B$ are flat over $A$, the sequence", "$$", "\\ldots \\to P_{2, n} \\to P_{1, n} \\to P_{0, n} \\to M_n \\to 0", "$$", "is exact. We see that on the one hand the complex", "$$", "\\Hom_B(P_0, N) \\to \\Hom_B(P_1, N) \\to \\Hom_B(P_2, N) \\to \\ldots", "$$", "computes the modules $\\Ext^i_B(M, N)$ and on the other hand the complex", "$$", "\\Hom_{B_n}(P_{0, n}, N_n) \\to \\Hom_{B_n}(P_{1, n}, N_n) \\to", "\\Hom_{B_n}(P_{2, n}, N_n) \\to \\ldots", "$$", "computes the modules $\\Ext^i_{B_n}(M_n, N_n)$. Since", "$$", "\\Hom_{B_n}(P_{i, n}, N_n) = \\Hom_B(P_i, N)/I^n \\Hom_B(P_i, N)", "$$", "we obtain the result from", "Lemma \\ref{lemma-consequence-Artin-Rees} part (2)." ], "refs": [ "more-algebra-lemma-consequence-Artin-Rees" ], "ref_ids": [ 10419 ] } ], "ref_ids": [] }, { "id": 10427, "type": "theorem", "label": "more-algebra-lemma-consequence-Artin-Rees-bis", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-consequence-Artin-Rees-bis", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$. Let", "$", "K \\xrightarrow{\\alpha} L \\xrightarrow{\\beta} M", "$", "be a complex of finite $A$-modules. Set $H = \\Ker(\\beta)/\\Im(\\alpha)$.", "For $n \\geq 0$ let", "$$", "I^nK \\xrightarrow{\\alpha_n} I^nL \\xrightarrow{\\beta_n} I^nM", "$$", "be the induced complex. Set $H_n = \\Ker(\\beta_n)/\\Im(\\alpha_n)$.", "Then there are canonical $A$-module maps", "$$", "\\ldots \\to H_3 \\to H_2 \\to H_1 \\to H", "$$", "There exists a $c > 0$ such that for $n \\geq c$ the image of $H_n \\to H$ is", "contained in $I^{n - c}H$ and there is a canonical $A$-module map", "$I^nH \\to H_{n - c}$ such that the compositions", "$$", "I^n H \\to H_{n - c} \\to I^{n - 2c}H", "\\quad\\text{and}\\quad", "H_n \\to I^{n - c}H \\to H_{n - 2c}", "$$", "are the canonical ones. In particular, the inverse systems", "$(H_n)$ and $(I^nH)$ are isomorphic as pro-objects of $\\text{Mod}_A$." ], "refs": [], "proofs": [ { "contents": [ "We have $H_n = \\Ker(\\beta) \\cap I^nL/\\alpha(I^nK)$.", "Since $\\Ker(\\beta) \\cap I^nL \\subset \\Ker(\\beta) \\cap I^{n - 1}L$", "and $\\alpha(I^nK) \\subset \\alpha(I^{n - 1}K)$ we get the maps", "$H_n \\to H_{n - 1}$. Similarly for the map $H_1 \\to H$.", "\\medskip\\noindent", "By Artin-Rees we may choose $c_1, c_2 \\geq 0$ such that", "$\\Im(\\alpha) \\cap I^nL \\subset \\alpha(I^{n - c_1}K)$ for $n \\geq c_1$ and", "$\\Ker(\\beta) \\cap I^nL \\subset I^{n - c_2}\\Ker(\\beta)$ for $n \\geq c_2$, see", "Algebra, Lemmas \\ref{algebra-lemma-map-AR} and", "\\ref{algebra-lemma-Artin-Rees}. Set $c = c_1 + c_2$.", "\\medskip\\noindent", "It follows immediately from our choice of $c \\geq c_2$ that for $n \\geq c$", "the image of $H_n \\to H$ is contained in $I^{n - c}H$.", "\\medskip\\noindent", "Let $n \\geq c$. We define $\\psi_n : I^nH \\to H_{n - c}$ as follows.", "Say $x \\in I^nH$. Choose $y \\in I^n\\Ker(\\beta)$ representing $x$.", "We set $\\psi_n(x)$ equal to the class of $y$ in $H_{n - c}$.", "To see this is well defined, suppose we have a second choice $y'$", "as above for $x$. Then $y' - y \\in \\Im(\\alpha)$. By our choice of", "$c \\geq c_1$ we conclude that $y' - y \\in \\alpha(I^{n - c}K)$", "which implies that $y$ and $y'$ represent the same element of $H_{n - c}$.", "Thus $\\psi_n$ is well defined.", "\\medskip\\noindent", "The statements on the compositions $I^n H \\to H_{n - c} \\to I^{n - 2c}H$", "and $H_n \\to I^{n - c}H \\to H_{n - 2c}$ follow immediately from our", "definitions." ], "refs": [ "algebra-lemma-map-AR", "algebra-lemma-Artin-Rees" ], "ref_ids": [ 626, 625 ] } ], "ref_ids": [] }, { "id": 10428, "type": "theorem", "label": "more-algebra-lemma-ext-factors", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ext-factors", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.", "Let $M$, $N$ be $A$-modules with $M$ finite. For each $p > 0$ there exists a", "$c \\geq 0$ such that for $n \\geq c$ the map", "$\\Ext_A^p(M, N) \\to \\Ext_A^p(I^nM, N)$", "factors through $\\Ext^p_A(I^nM, I^{n - c}N) \\to \\Ext_A^p(I^nM, N)$." ], "refs": [], "proofs": [ { "contents": [ "For $p = 0$, if $\\varphi : M \\to N$ is an $A$-linear map, then", "$\\varphi(\\sum f_i m_i) = \\sum f_i \\varphi(m_i)$ for $f_i \\in A$", "and $m_i \\in M$. Hence $\\varphi$ induces a map $I^nM \\to I^nN$", "for all $n$ and the result is true with $c = 0$.", "\\medskip\\noindent", "Choose a short exact sequence $0 \\to K \\to A^{\\oplus t} \\to M \\to 0$.", "For each $n$ we pick a short exact sequence", "$0 \\to L_n \\to A^{\\oplus s_n} \\to I^nM \\to 0$.", "It is clear that we can construct a map of short exact sequences", "$$", "\\xymatrix{", "0 \\ar[r] &", "L_n \\ar[r] \\ar[d] &", "A^{\\oplus s_n} \\ar[r] \\ar[d] &", "I^nM \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "K \\ar[r] &", "A^{\\oplus t} \\ar[r] &", "M \\ar[r] & 0", "}", "$$", "such that $A^{\\oplus s_n} \\to A^{\\oplus t}$ has image in $(I^n)^{\\oplus t}$.", "By Artin-Rees (Algebra, Lemma \\ref{algebra-lemma-Artin-Rees}) there exists", "a $c \\geq 0$ such that $L_n \\to K$ factors through $I^{n - c}K$ if $n \\geq c$.", "\\medskip\\noindent", "For $p = 1$ our choices above induce a solid commutative diagram", "$$", "\\xymatrix{", "\\Hom_A(A^{\\oplus s_n}, N) \\ar[r] &", "\\Hom_A(L_n, N) \\ar[r] &", "\\Ext_A^1(I^nM, N) \\ar[r] & 0 \\\\", "\\Hom_A((I^n)^{\\oplus t}, I^{n - c}N) \\ar[r] \\ar[u] &", "\\Hom_A(K \\cap (I^n)^{\\oplus t}, I^{n - c}N) \\ar[r] \\ar[u] &", "\\Ext_A^1(I^nM, I^{n - c}N) \\ar[u] \\\\", "\\Hom_A(A^{\\oplus t}, N) \\ar[r] \\ar[u] &", "\\Hom_A(K, N) \\ar[r] \\ar[u] &", "\\Ext_A^1(M, N) \\ar@{..>}[u] \\ar[r] & 0", "}", "$$", "whose horizontal arrows are exact. The lower middle vertical arrow", "arises because $K \\cap (I^n)^{\\oplus t} \\subset I^{n - c}K$", "and hence any $A$-linear map $K \\to N$ induces an $A$-linear map", "$(I^n)^{\\oplus t} \\to I^{n - c}N$ by the argument of the first paragraph.", "Thus we obtain the dotted arrow as desired.", "\\medskip\\noindent", "For $p > 1$ we obtain a commutative diagram", "$$", "\\xymatrix{", "\\Ext^{p - 1}_A(I^{n - c}K, N) \\ar[r] &", "\\Ext^{p - 1}_A(L_n, N) \\ar[r] &", "\\Ext_A^p(I^nM, N) \\\\", "\\Ext^{p - 1}_A(K, N) \\ar[rr] \\ar[u] & &", "\\Ext_A^p(M, N) \\ar[u]", "}", "$$", "whose bottom horizontal arrow is an isomorphism. By induction", "on $p$ the left vertical map factors through", "$\\Ext^{p - 1}_A(I^{n - c}K, I^{n - c - c'}N)$ for some $c' \\geq 0$", "and all $n \\geq c + c'$. Using the composition", "$\\Ext^{p - 1}_A(I^{n - c}K, I^{n - c - c'}N) \\to", "\\Ext^{p - 1}_A(L_n, I^{n - c - c'}N) \\to \\Ext^p_A(I^nM, I^{n - c - c'}N)$", "we obtain the desired factorization (for $n \\geq c + c'$ and with $c$", "replaced by $c + c'$)." ], "refs": [ "algebra-lemma-Artin-Rees" ], "ref_ids": [ 625 ] } ], "ref_ids": [] }, { "id": 10429, "type": "theorem", "label": "more-algebra-lemma-ext-annihilated", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ext-annihilated", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$, $N$", "be $A$-modules with $M$ finite and $N$ annihilated by a power of $I$.", "For each $p > 0$ there exists an $n$ such that the map", "$\\Ext_A^p(M, N) \\to \\Ext_A^p(I^nM, N)$ is zero." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of Lemma \\ref{lemma-ext-factors} and the fact that", "$I^mN = 0$ for some $m > 0$." ], "refs": [ "more-algebra-lemma-ext-factors" ], "ref_ids": [ 10428 ] } ], "ref_ids": [] }, { "id": 10430, "type": "theorem", "label": "more-algebra-lemma-ext-induced-toplogy", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ext-induced-toplogy", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.", "Let $K \\in D(A)$ be pseudo-coherent and let $M$ be a finite", "$A$-module. For each $p \\in \\mathbf{Z}$ there exists an $c$", "such that the image of $\\Ext_A^p(K, I^nM) \\to \\Ext_A^p(K, M)$", "is contained in $I^{n - c}\\Ext_A^p(K, M)$ for $n \\geq c$." ], "refs": [], "proofs": [ { "contents": [ "Choose a bounded above complex $P^\\bullet$ of finite free $A$-modules", "representing $K$. Then $\\Ext_A^p(K, M)$ is the cohomology of", "$$", "\\Hom_A(F^{-p + 1}, M) \\xrightarrow{a}", "\\Hom_A(F^{-p}, M) \\xrightarrow{b}", "\\Hom_A(F^{-p - 1}, M)", "$$", "and $\\Ext_A^p(K, I^nM)$ is computed by replacing these finite $A$-modules", "by $I^n$ times themselves. Thus the result by", "Lemma \\ref{lemma-consequence-Artin-Rees-bis}", "(and much more is true)." ], "refs": [ "more-algebra-lemma-consequence-Artin-Rees-bis" ], "ref_ids": [ 10427 ] } ], "ref_ids": [] }, { "id": 10431, "type": "theorem", "label": "more-algebra-lemma-sequence-powers-pro-bounded", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-sequence-powers-pro-bounded", "contents": [ "In Situation \\ref{situation-koszul} assume $A$ is Noetherian. With", "notation as above, the inverse system $(I^n)$ is pro-isomorphic", "in $D(A)$ to the inverse system $(I_n^\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "It is elementary to show that the inverse system $I^n$ is pro-isomorphic", "to the inverse system $(f_1^n, \\ldots, f_r^n)$ in the category of $A$-modules.", "Consider the inverse system of distinguished triangles", "$$", "I_n^\\bullet \\to (f_1^n, \\ldots, f_r^n) \\to C_n^\\bullet \\to I_n^\\bullet[1]", "$$", "where $C_n^\\bullet$ is the cone of the first arrow. By", "Derived Categories, Lemma \\ref{derived-lemma-pro-isomorphism}", "it suffices to show that the inverse system $C_n^\\bullet$ is pro-zero.", "The complex $I_n^\\bullet$ has nonzero terms only in degrees", "$i$ with $-r + 1 \\leq i \\leq 0$ hence $C_n^\\bullet$ is bounded", "similarly. Thus by", "Derived Categories, Lemma \\ref{derived-lemma-essentially-constant-cohomology}", "it suffices to show that $H^p(C_n^\\bullet)$ is pro-zero. By the discussion", "above we have $H^p(C_n^\\bullet) = H^p(K_n^\\bullet)$ for $p \\leq -1$ and", "$H^p(C_n^\\bullet) = 0$ for $p \\geq 0$.", "The fact that the inverse systems $H^p(K_n^\\bullet)$ are pro-zero", "was shown in the proof of Lemma \\ref{lemma-sequence-Koszul-complexes}", "(and this is where the assumption that $A$ is Noetherian is used)." ], "refs": [ "derived-lemma-pro-isomorphism", "derived-lemma-essentially-constant-cohomology", "more-algebra-lemma-sequence-Koszul-complexes" ], "ref_ids": [ 1956, 1955, 10391 ] } ], "ref_ids": [] }, { "id": 10432, "type": "theorem", "label": "more-algebra-lemma-tensoring-Deligne-system", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-tensoring-Deligne-system", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let", "$M^\\bullet$ be a bounded complex of finite $A$-modules. The", "inverse system of maps", "$$", "I^n \\otimes_A^\\mathbf{L} M^\\bullet \\longrightarrow I^nM^\\bullet", "$$", "defines an isomorphism of pro-objects of $D(A)$." ], "refs": [], "proofs": [ { "contents": [ "Choose generators $f_1, \\ldots, f_r \\in I$ of $I$. The inverse system $I^n$", "is pro-isomorphic to the inverse system $(f_1^n, \\ldots, f_r^n)$", "in the category of $A$-modules. With notation as in", "Lemma \\ref{lemma-sequence-powers-pro-bounded} we find that", "it suffices to prove the inverse system of maps", "$$", "I_n^\\bullet \\otimes_A^\\mathbf{L} M^\\bullet", "\\longrightarrow", "(f_1^n, \\ldots, f_r^n)M^\\bullet", "$$", "defines an isomorphism of pro-objects of $D(A)$.", "Say we have $a \\leq b$ such that $M^i = 0$ if $i \\not \\in [a, b]$.", "Then source and target of the arrows above have cohomology", "only in degrees $[-r + a, b]$.", "Thus it suffices to show that for any $p \\in \\mathbf{Z}$ the", "inverse system of maps", "$$", "H^p(I_n^\\bullet \\otimes_A^\\mathbf{L} M^\\bullet)", "\\longrightarrow", "H^p((f_1^n, \\ldots, f_r^n)M^\\bullet)", "$$", "defines an isomorphism of pro-objects of $A$-modules, see", "Derived Categories, Lemma \\ref{derived-lemma-pro-isomorphism-bis}.", "Using the pro-isomorphism between", "$I_n^\\bullet \\otimes_A^\\mathbf{L} M^\\bullet$", "and $I^n \\otimes_A^\\mathbf{L} M^\\bullet$", "and the pro-isomorphism between", "$(f_1^n, \\ldots, f_r^n)M^\\bullet$ and $I^nM^\\bullet$", "this is equivalent to showing that the inverse system of maps", "$$", "H^p(I^n \\otimes_A^\\mathbf{L} M^\\bullet)", "\\longrightarrow", "H^p(I^nM^\\bullet)", "$$", "defines an isomorphism of pro-objects of $A$-modules", "Choose a bounded above complex of finite free $A$-modules", "$P^\\bullet$ and a quasi-isomorphism $P^\\bullet \\to M^\\bullet$.", "Then it suffices to show that the inverse system of maps", "$$", "H^p(I^nP^\\bullet)", "\\longrightarrow", "H^p(I^nM^\\bullet)", "$$", "is a pro-isomorphism. This follows from", "Lemma \\ref{lemma-consequence-Artin-Rees-bis}", "as $H^p(P^\\bullet) = H^p(M^\\bullet)$." ], "refs": [ "more-algebra-lemma-sequence-powers-pro-bounded", "derived-lemma-pro-isomorphism-bis", "more-algebra-lemma-consequence-Artin-Rees-bis" ], "ref_ids": [ 10431, 1957, 10427 ] } ], "ref_ids": [] }, { "id": 10433, "type": "theorem", "label": "more-algebra-lemma-factor-through-derived-tensor-product", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-factor-through-derived-tensor-product", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$", "be a finite $A$-module. There exists an integer $n > 0$ such that", "$I^nM \\to M$ factors through the map $I \\otimes_A^\\mathbf{L} M \\to M$", "in $D(A)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-tensoring-Deligne-system}.", "It can also been seen directly as follows. Consider the distinguished triangle", "$$", "I \\otimes_A^\\mathbf{L} M \\to M \\to A/I \\otimes_A^\\mathbf{L} M \\to", "I \\otimes_A^\\mathbf{L} M[1]", "$$", "By the axioms of a triangulated category it suffices to prove that", "$I^nM \\to A/I \\otimes_A^\\mathbf{L} M$ is zero in $D(A)$ for some $n$.", "Choose generators $f_1, \\ldots, f_r$ of $I$ and let", "$K = K_\\bullet(A, f_1, \\ldots, f_r)$ be the Koszul complex", "and consider the factorization $A \\to K \\to A/I$ of the quotient map.", "Then we see that it suffices to show that $I^nM \\to K \\otimes_A M$", "is zero in $D(A)$ for some $n > 0$. Suppose that we have found an $n > 0$", "such that $I^nM \\to K \\otimes_A M$ factors through", "$\\tau_{\\geq t}(K \\otimes_A M)$ in $D(A)$. Then the obstruction", "to factoring through $\\tau_{\\geq t + 1}(K \\otimes_A M)$ is an element", "in $\\Ext^t(I^nM, H_t(K \\otimes_A M))$. The finite $A$-module", "$H_t(K \\otimes_A M)$ is annihilated by $I$. Then by", "Lemma \\ref{lemma-ext-annihilated}", "we can after increasing $n$ assume this obstruction element is zero.", "Repeating this a finite number of times we find $n$ such that", "$I^nM \\to K \\otimes_A M$ factors through", "$0 = \\tau_{\\geq r + 1}(K \\otimes_A M)$ in $D(A)$ and we win." ], "refs": [ "more-algebra-lemma-tensoring-Deligne-system", "more-algebra-lemma-ext-annihilated" ], "ref_ids": [ 10432, 10429 ] } ], "ref_ids": [] }, { "id": 10434, "type": "theorem", "label": "more-algebra-lemma-ext-annihilated-into", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ext-annihilated-into", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.", "Let $K \\in D(A)$ be pseudo-coherent. Let $a \\in \\mathbf{Z}$.", "Assume that for every finite $A$-module $M$ the modules", "$\\Ext^i_A(K, M)$ are $I$-power torsion for $i \\geq a$.", "Then for $i \\geq a$ and $M$ finite", "the system $\\Ext^i_A(K, M/I^nM)$", "is essentially constant with value", "$$", "\\Ext^i_A(K, M) = \\lim \\Ext^i_A(K, M/I^nM)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be a finite $A$-module. Since $K$ is pseudo-coherent we see that", "$\\Ext^i_A(K, M)$ is a finite $A$-module. Thus for $i \\geq a$", "it is annihilated by $I^t$ for some $t \\geq 0$. By", "Lemma \\ref{lemma-ext-induced-toplogy} we see that the image of", "$\\Ext^i_A(K, I^nM) \\to \\Ext^i_A(K, M)$ is", "zero for some $n > 0$.", "The short exact sequence $0 \\to I^nM \\to M \\to M/I^n M \\to 0$", "gives a long exact sequence", "$$", "\\Ext^i_A(K, I^nM) \\to \\Ext^i_A(K, M) \\to", "\\Ext^i_A(K, M/I^nM) \\to \\Ext^{i + 1}_A(K, I^nM)", "$$", "The systems $\\Ext^i_A(K, I^nM)$ and $\\Ext^{i + 1}_A(K, I^nM)$", "are essentially constant with value $0$ by what we just said", "(applied to the finite $A$-modules $I^mM$). A diagram chase", "shows $\\Ext^i_A(K, M/I^nM)$ is essentially constant with value", "$\\Ext^i_A(K, M)$." ], "refs": [ "more-algebra-lemma-ext-induced-toplogy" ], "ref_ids": [ 10430 ] } ], "ref_ids": [] }, { "id": 10435, "type": "theorem", "label": "more-algebra-lemma-tor-annihilated", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-tor-annihilated", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$", "be a finite $A$-module. Let $N$ be an $A$-module annihilated by $I$.", "There exists an integer $n > 0$ such that", "$\\text{Tor}^A_p(I^nM, N) \\to \\text{Tor}^A_p(M, N)$ is zero", "for all $p \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-factor-through-derived-tensor-product}", "we can factor $I^nM \\to M$ as $I^nM \\to M \\otimes_A^\\mathbf{L} I \\to M$.", "We claim the composition", "$$", "I^nM \\otimes_A^\\mathbf{L} N \\to", "(M \\otimes_A^\\mathbf{L} I) \\otimes_A^\\mathbf{L} N", "\\to M \\otimes_A^\\mathbf{L} N", "$$", "is zero. Namely, the diagram", "$$", "\\xymatrix{", "(M \\otimes_A^\\mathbf{L} I) \\otimes_A^\\mathbf{L} N \\ar[rr] \\ar[rd] & &", "M \\otimes_A^\\mathbf{L} (I \\otimes_A^\\mathbf{L} N) \\ar[ld] \\\\", "& M \\otimes_A^\\mathbf{L} N", "}", "$$", "commutes (details omitted) and the map $I \\otimes_A^\\mathbf{L} N \\to N$", "is zero as $N$ is annihilated by $I$." ], "refs": [ "more-algebra-lemma-factor-through-derived-tensor-product" ], "ref_ids": [ 10433 ] } ], "ref_ids": [] }, { "id": 10436, "type": "theorem", "label": "more-algebra-lemma-pseudo-coherent-tensor-limit", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pseudo-coherent-tensor-limit", "contents": [ "Let $R$ be a ring. Let $K \\in D(R)$ be pseudo-coherent.", "Let $(M_n)$ be an inverse system of $R$-modules.", "Then $R\\lim K \\otimes_R^\\mathbf{L} M_n = K \\otimes_R^\\mathbf{L} R\\lim M_n$." ], "refs": [], "proofs": [ { "contents": [ "Consider the defining distinguished triangle", "$$", "R\\lim M_n \\to \\prod M_n \\to \\prod M_n \\to R\\lim M_n[1]", "$$", "and apply Lemma \\ref{lemma-pseudo-coherent-tensor}." ], "refs": [ "more-algebra-lemma-pseudo-coherent-tensor" ], "ref_ids": [ 10166 ] } ], "ref_ids": [] }, { "id": 10437, "type": "theorem", "label": "more-algebra-lemma-additivity-of-pd", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-additivity-of-pd", "contents": [ "Let $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal", "and let $E$ be a nonzero module over $R/I$. If $R/I$ has finite projective", "dimension and $E$ has finite projective dimension over $R/I$, then", "$E$ has finite projective dimension over $R$ and", "$$", "\\text{pd}_R(E) = \\text{pd}_R(R/I) + \\text{pd}_{R/I}(E)", "$$" ], "refs": [], "proofs": [ { "contents": [ "We will use that, for a finite module, having finite projective dimension", "over $R$, resp.\\ $R/I$ is the same as being a perfect module, see", "discussion following Definition \\ref{definition-perfect}.", "We see that $E$ has finite projective dimension", "over $R$ by Lemma \\ref{lemma-cohomology-perfect}.", "Thus we can apply Auslander-Buchsbaum (Algebra, Proposition", "\\ref{algebra-proposition-Auslander-Buchsbaum}) to see that", "$$", "\\text{pd}_R(E) + \\text{depth}(E) = \\text{depth}(R),\\quad", "\\text{pd}_{R/I}(E) + \\text{depth}(E) = \\text{depth}(R/I),", "$$", "and", "$$", "\\text{pd}_R(R/I) + \\text{depth}(R/I) = \\text{depth}(R)", "$$", "Note that in the first equation we take the depth of $E$", "as an $R$-module and in the second as an $R/I$-module.", "However these depths are the same (this is trivial but", "also follows from Algebra, Lemma \\ref{algebra-lemma-depth-goes-down-finite}).", "This concludes the proof." ], "refs": [ "more-algebra-definition-perfect", "more-algebra-lemma-cohomology-perfect", "algebra-proposition-Auslander-Buchsbaum", "algebra-lemma-depth-goes-down-finite" ], "ref_ids": [ 10628, 10217, 1423, 778 ] } ], "ref_ids": [] }, { "id": 10438, "type": "theorem", "label": "more-algebra-lemma-prod-qis-gives-qis", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-prod-qis-gives-qis", "contents": [ "Let", "$$", "(A_0^\\bullet \\to A_1^\\bullet \\to A_2^\\bullet \\to \\ldots)", "\\longrightarrow", "(B_0^\\bullet \\to B_1^\\bullet \\to B_2^\\bullet \\to \\ldots)", "$$", "be a map between two complexes of complexes of abelian groups.", "Set $A^{p, q} = A_p^q$, $B^{p, q} = B_p^q$ to obtain double complexes.", "Let $\\text{Tot}_\\pi(A^{\\bullet, \\bullet})$", "and $\\text{Tot}_\\pi(B^{\\bullet, \\bullet})$ be the", "product total complexes associated to the double complexes.", "If each $A_p^\\bullet \\to B_p^\\bullet$ is a", "quasi-isomorphism, then", "$\\text{Tot}_\\pi(A^{\\bullet, \\bullet}) \\to \\text{Tot}_\\pi(B^{\\bullet, \\bullet})$", "is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\text{Tot}_\\pi(A^{\\bullet, \\bullet})$ in degree", "$n$ is given by $\\prod_{p + q = n} A^{p, q} = \\prod_{p + 1 = n} A^q_p$.", "Let $C_p^\\bullet$ be the cone on the map $A_p^\\bullet \\to B_p^\\bullet$,", "see Derived Categories, Section \\ref{derived-section-cones}.", "By the functoriality of the cone construction we obtain a", "complex of complexes", "$$", "C_0^\\bullet \\to C_1^\\bullet \\to C_2^\\bullet \\to \\ldots", "$$", "Then we see $\\text{Tot}_\\pi(C^{\\bullet, \\bullet})$ in degree $n$", "is given by", "$$", "\\prod_{p + q = n} C^{p, q} = \\prod_{p + q = n} C^q_p =", "\\prod_{p + q = n} (B^q_p \\oplus A^{q + 1}_p) =", "\\prod_{p + q = n} B^q_p \\oplus \\prod_{p + q = n} A^{q + 1}_p", "$$", "We conclude that $\\text{Tot}_\\pi(C^{\\bullet, \\bullet})$", "is the cone of the map", "$\\text{Tot}_\\pi(A^{\\bullet, \\bullet}) \\to \\text{Tot}_\\pi(B^{\\bullet, \\bullet})$", "(We omit the verification that the differentials agree.)", "Thus it suffices to show $\\text{Tot}_\\pi(A^{\\bullet, \\bullet})$ is", "acyclic if each $A_p^\\bullet$ is acyclic.", "\\medskip\\noindent", "Denote $f_p : A_p^\\bullet \\to A_{p + 1}^\\bullet$ the given maps", "of complexes. Recall that the differential on", "$\\text{Tot}_\\pi(A^{\\bullet, \\bullet})$ is given by", "$$", "\\prod\\nolimits_{p + q = n} (f^q_p + (-1)^p\\text{d}^q_{A_p^\\bullet})", "$$", "on elements in degree $n$.", "Let $\\xi \\in H^0(\\text{Tot}_\\pi(A^{\\bullet, \\bullet}))$ be a cohomology", "class. We will show $\\xi$ is zero; the same argument works in other", "degrees. Represent $\\xi$ as the class of an cocycle", "$x = (x_p) \\in \\prod A^{p, -p}$.", "Since $\\text{d}(x) = 0$ we find that", "$\\text{d}_{A_0^\\bullet}(x_0) = 0$.", "Thus there exists a $y_{-1} \\in A^{0, -1}$ with", "$\\text{d}_{A_0^\\bullet}(y_{-1}) = x_0$.", "Then we see that $\\text{d}_{A_1^\\bullet}(x_1 + f_0(y_{-1})) = 0$.", "Thus we can find a $y_{-2} \\in A^{1, -2}$ such that", "$-\\text{d}_{A_1^\\bullet}(y_{-2}) = x_1 + f_0(y_{-1})$.", "By induction we can find", "$y_{-p - 1} \\in A^{p, -p - 1}$ such that", "$$", "(-1)^p\\text{d}_{A_p^\\bullet}(y_{-p - 1}) = x_p + f_{p - 1}(y_{-p})", "$$", "This implies that $\\text{d}(y) = x$ where $y = (y_{-p - 1})$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10439, "type": "theorem", "label": "more-algebra-lemma-key", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-key", "contents": [ "Let $A \\to B$ be a ring map such that $B \\otimes_A B \\to B$ is flat.", "Let $N$ be a $B$-module. If $N$ is flat as an $A$-module, then", "$N$ is flat as a $B$-module." ], "refs": [], "proofs": [ { "contents": [ "Assume $N$ is a flat as an $A$-module.", "Then the functor", "$$", "\\text{Mod}_B \\longrightarrow \\text{Mod}_{B \\otimes_A B},\\quad", "N' \\mapsto N \\otimes_A N'", "$$", "is exact. As $B \\otimes_A B \\to B$ is flat we conclude that the functor", "$$", "\\text{Mod}_B \\longrightarrow \\text{Mod}_B,\\quad", "N' \\mapsto (N \\otimes_A N') \\otimes_{B \\otimes_A B} B = N \\otimes_B N'", "$$", "is exact, hence $N$ is flat over $B$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10440, "type": "theorem", "label": "more-algebra-lemma-weak-dimension-goes-up", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-weak-dimension-goes-up", "contents": [ "Let $A \\to B$ be a weakly \\'etale ring map.", "If $A$ has weak dimension at most $d$, then so does $B$." ], "refs": [], "proofs": [ { "contents": [ "Let $N$ be a $B$-module. If $d = 0$, then $N$ is flat as an $A$-module,", "hence flat as a $B$-module by Lemma \\ref{lemma-key}.", "Assume $d > 0$. Choose a resolution $F_\\bullet \\to N$", "by free $B$-modules. Our assumption implies that", "$K = \\Im(F_d \\to F_{d - 1})$ is $A$-flat, see", "Lemma \\ref{lemma-last-one-flat}. Hence it is $B$-flat", "by Lemma \\ref{lemma-key}. Thus", "$0 \\to K \\to F_{d - 1} \\to \\ldots \\to F_0 \\to N \\to 0$", "is a flat resolution of length $d$ and we see that $N$ has", "tor dimension at most $d$." ], "refs": [ "more-algebra-lemma-key", "more-algebra-lemma-last-one-flat", "more-algebra-lemma-key" ], "ref_ids": [ 10439, 10169, 10439 ] } ], "ref_ids": [] }, { "id": 10441, "type": "theorem", "label": "more-algebra-lemma-absolutely-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-absolutely-flat", "contents": [ "Let $A$ be a ring. The following are equivalent", "\\begin{enumerate}", "\\item $A$ has weak dimension $\\leq 0$,", "\\item $A$ is absolutely flat, and", "\\item $A$ is reduced and every prime is maximal.", "\\end{enumerate}", "In this case every local ring of $A$ is a field." ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is immediate.", "Assume $A$ is absolutely flat. This implies every ideal of $A$ is pure, see", "Algebra, Definition \\ref{algebra-definition-pure-ideal}.", "Hence every finitely generated ideal is generated by an idempotent by", "Algebra, Lemma \\ref{algebra-lemma-finitely-generated-pure-ideal}.", "If $f \\in A$, then $(f) = (e)$ for some idempotent $e \\in A$", "and $D(f) = D(e)$ is open and closed", "(Algebra, Lemma \\ref{algebra-lemma-idempotent-spec}).", "This already implies every ideal of $A$ is maximal", "for example by", "Algebra, Lemma \\ref{algebra-lemma-ring-with-only-minimal-primes}.", "Moreover, if $f$ is nilpotent, then $e = 0$ hence $f = 0$.", "Thus $A$ is reduced.", "\\medskip\\noindent", "Assume $A$ is reduced and every prime of $A$ is maximal.", "Let $M$ be an $A$-module. Our goal is to show that $M$ is flat.", "We may write $M$ as a filtered colimit of finite $A$-modules, hence", "we may assume $M$ is finite", "(Algebra, Lemma \\ref{algebra-lemma-colimit-flat}).", "There is a finite filtration of $M$ by modules of the form", "$A/I$ (Algebra, Lemma \\ref{algebra-lemma-trivial-filter-finite-module}),", "hence we may assume that $M = A/I$", "(Algebra, Lemma \\ref{algebra-lemma-flat-ses}).", "Thus it suffices to show every ideal of $A$ is pure.", "Since every local ring of $A$ is a field", "(by Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring} and", "the fact that every prime of $A$ is minimal),", "we see that every ideal $I \\subset A$ is radical.", "Note that every closed subset of $\\Spec(A)$ is closed under generalization.", "Thus every (radical) ideal of $A$ is pure by", "Algebra, Lemma \\ref{algebra-lemma-pure-open-closed-specializations}." ], "refs": [ "algebra-definition-pure-ideal", "algebra-lemma-finitely-generated-pure-ideal", "algebra-lemma-idempotent-spec", "algebra-lemma-ring-with-only-minimal-primes", "algebra-lemma-colimit-flat", "algebra-lemma-trivial-filter-finite-module", "algebra-lemma-flat-ses", "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-pure-open-closed-specializations" ], "ref_ids": [ 1509, 963, 403, 426, 523, 331, 533, 418, 962 ] } ], "ref_ids": [] }, { "id": 10442, "type": "theorem", "label": "more-algebra-lemma-product-fields-absolutely-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-product-fields-absolutely-flat", "contents": [ "A product of fields is an absolutely flat ring." ], "refs": [], "proofs": [ { "contents": [ "Let $K_i$ be a family of fields. If $f = (f_i) \\in \\prod K_i$, then", "the ideal generated by $f$ is the same as the ideal generated by", "the idempotent $e = (e_i)$ with $e_i = 0, 1$ according to whether", "$f_i$ is $0$ or not. Thus $D(f) = D(e)$ is open and closed and we conclude", "by Lemma \\ref{lemma-absolutely-flat} and", "Algebra, Lemma \\ref{algebra-lemma-ring-with-only-minimal-primes}." ], "refs": [ "more-algebra-lemma-absolutely-flat", "algebra-lemma-ring-with-only-minimal-primes" ], "ref_ids": [ 10441, 426 ] } ], "ref_ids": [] }, { "id": 10443, "type": "theorem", "label": "more-algebra-lemma-base-change-weakly-etale", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-weakly-etale", "contents": [ "Let $A \\to B$ and $A \\to A'$ be ring maps. Let $B' = B \\otimes_A A'$", "be the base change of $B$.", "\\begin{enumerate}", "\\item If $B \\otimes_A B \\to B$ is flat, then $B' \\otimes_{A'} B' \\to B'$", "is flat.", "\\item If $A \\to B$ is weakly \\'etale, then $A' \\to B'$ is weakly \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $B \\otimes_A B \\to B$ is flat.", "The ring map $B' \\otimes_{A'} B' \\to B'$ is the base change of", "$B \\otimes_A B \\to B$ by $A \\to A'$. Hence it is flat by", "Algebra, Lemma \\ref{algebra-lemma-flat-base-change}. This proves (1).", "Part (2) follows from (1) and the fact (just used) that the", "base change of a flat ring map is flat." ], "refs": [ "algebra-lemma-flat-base-change" ], "ref_ids": [ 527 ] } ], "ref_ids": [] }, { "id": 10444, "type": "theorem", "label": "more-algebra-lemma-absolutely-flat-over-absolutely-flat", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-absolutely-flat-over-absolutely-flat", "contents": [ "Let $A \\to B$ be a ring map such that $B \\otimes_A B \\to B$ is flat.", "\\begin{enumerate}", "\\item If $A$ is an absolutely flat ring, then so is $B$.", "\\item If $A$ is reduced and $A \\to B$ is weakly \\'etale, then $B$ is reduced.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows immediately from Lemma \\ref{lemma-key} and the definitions.", "If $A$ is reduced, then there exists an injection", "$A \\to A' = \\prod_{\\mathfrak p \\subset A\\text{ minimal}} A_\\mathfrak p$", "of $A$ into an absolutely flat ring", "(Algebra, Lemma \\ref{algebra-lemma-reduced-ring-sub-product-fields} and", "Lemma \\ref{lemma-product-fields-absolutely-flat}).", "If $A \\to B$ is flat, then the induced map $B \\to B' = B \\otimes_A A'$", "is injective too. By Lemma \\ref{lemma-base-change-weakly-etale}", "the ring map $A' \\to B'$ is weakly \\'etale.", "By part (1) we see that $B'$ is absolutely flat.", "By Lemma \\ref{lemma-absolutely-flat} the ring $B'$ is reduced.", "Hence $B$ is reduced." ], "refs": [ "more-algebra-lemma-key", "algebra-lemma-reduced-ring-sub-product-fields", "more-algebra-lemma-product-fields-absolutely-flat", "more-algebra-lemma-base-change-weakly-etale", "more-algebra-lemma-absolutely-flat" ], "ref_ids": [ 10439, 419, 10442, 10443, 10441 ] } ], "ref_ids": [] }, { "id": 10445, "type": "theorem", "label": "more-algebra-lemma-composition-weakly-etale", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-composition-weakly-etale", "contents": [ "Let $A \\to B$ and $B \\to C$ be ring maps.", "\\begin{enumerate}", "\\item If $B \\otimes_A B \\to B$ and $C \\otimes_B C \\to C$", "are flat, then $C \\otimes_A C \\to C$ is flat.", "\\item If $A \\to B$ and $B \\to C$ are weakly \\'etale, then $A \\to C$", "is weakly \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from the factorization", "$$", "C \\otimes_A C \\longrightarrow C \\otimes_B C \\longrightarrow C", "$$", "of the multiplication map, the fact that", "$$", "C \\otimes_B C = (C \\otimes_A C) \\otimes_{B \\otimes_A B} B,", "$$", "the fact that a base change of a flat map is flat, and the", "fact that the composition of flat ring maps is flat.", "See Algebra, Lemmas \\ref{algebra-lemma-flat-base-change} and", "\\ref{algebra-lemma-composition-flat}.", "Part (2) follows from (1) and the fact (just used) that the", "composition of flat ring maps is flat." ], "refs": [ "algebra-lemma-flat-base-change", "algebra-lemma-composition-flat" ], "ref_ids": [ 527, 524 ] } ], "ref_ids": [] }, { "id": 10446, "type": "theorem", "label": "more-algebra-lemma-go-down", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-go-down", "contents": [ "Let $A \\to B \\to C$ be ring maps.", "\\begin{enumerate}", "\\item If $B \\to C$ is faithfully flat and $C \\otimes_A C \\to C$ is flat,", "then $B \\otimes_A B \\to B$ is flat.", "\\item If $B \\to C$ is faithfully flat and $A \\to C$ is weakly \\'etale,", "then $A \\to B$ is weakly \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $B \\to C$ is faithfully flat and $C \\otimes_A C \\to C$ is flat.", "Consider the commutative diagram", "$$", "\\xymatrix{", "C \\otimes_A C \\ar[r] & C \\\\", "B \\otimes_A B \\ar[r] \\ar[u] & B \\ar[u]", "}", "$$", "The vertical arrows are flat, the top horizontal arrow is flat.", "Hence $C$ is flat as a $B \\otimes_A B$-module. The map $B \\to C$ is", "faithfully flat and $C = B \\otimes_B C$. Hence $B$ is flat as a", "$B \\otimes_A B$-module by", "Algebra, Lemma \\ref{algebra-lemma-flatness-descends-more-general}.", "This proves (1). Part (2) follows from (1) and the fact that", "$A \\to B$ is flat if $A \\to C$ is flat and $B \\to C$ is faithfully flat", "(Algebra, Lemma \\ref{algebra-lemma-flatness-descends-more-general})." ], "refs": [ "algebra-lemma-flatness-descends-more-general", "algebra-lemma-flatness-descends-more-general" ], "ref_ids": [ 529, 529 ] } ], "ref_ids": [] }, { "id": 10447, "type": "theorem", "label": "more-algebra-lemma-weakly-etale-permanence", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-weakly-etale-permanence", "contents": [ "Let $A$ be a ring. Let $B \\to C$ be an $A$-algebra map of weakly \\'etale", "$A$-algebras. Then $B \\to C$ is weakly \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Write $B \\to C$ as the composition $B \\to B \\otimes_A C \\to C$.", "The first map is flat as the base change of the flat ring map $A \\to C$.", "The second is the base change of the flat ring map $B \\otimes_A B \\to B$", "by the ring map $B \\otimes_A B \\to B \\otimes_A C$, hence flat.", "Thus $B \\to C$ is flat. The ring map", "$C \\otimes_A C \\to C \\otimes_B C$ is surjective, hence an epimorphism.", "Thus Lemma \\ref{lemma-key} implies, that since", "$C$ is flat over $C \\otimes_A C$ it follows that $C$ is", "flat over $C \\otimes_B C$." ], "refs": [ "more-algebra-lemma-key" ], "ref_ids": [ 10439 ] } ], "ref_ids": [] }, { "id": 10448, "type": "theorem", "label": "more-algebra-lemma-formally-unramified", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-formally-unramified", "contents": [ "Let $A \\to B$ be a ring map such that $B \\otimes_A B \\to B$ is flat.", "Then $\\Omega_{B/A} = 0$, i.e., $B$ is formally unramified over $A$." ], "refs": [], "proofs": [ { "contents": [ "Let $I \\subset B \\otimes_A B$ be the kernel of the flat surjective map", "$B \\otimes_A B \\to B$. Then $I$ is a pure ideal", "(Algebra, Definition \\ref{algebra-definition-pure-ideal}),", "so $I^2 = I$ (Algebra, Lemma \\ref{algebra-lemma-pure}).", "Since $\\Omega_{B/A} = I/I^2$", "(Algebra, Lemma \\ref{algebra-lemma-differentials-diagonal})", "we obtain the vanishing. This means $B$ is formally unramified over", "$A$ by", "Algebra, Lemma \\ref{algebra-lemma-characterize-formally-unramified}." ], "refs": [ "algebra-definition-pure-ideal", "algebra-lemma-pure", "algebra-lemma-differentials-diagonal", "algebra-lemma-characterize-formally-unramified" ], "ref_ids": [ 1509, 960, 1139, 1254 ] } ], "ref_ids": [] }, { "id": 10449, "type": "theorem", "label": "more-algebra-lemma-weakly-etale-finite-type", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-weakly-etale-finite-type", "contents": [ "Let $A \\to B$ be a ring map such that $B \\otimes_A B \\to B$ is flat.", "\\begin{enumerate}", "\\item If $A \\to B$ is of finite type, then $A \\to B$ is unramified.", "\\item If $A \\to B$ is of finite presentation and flat, then", "$A \\to B$ is \\'etale.", "\\end{enumerate}", "In particular a weakly \\'etale ring map of finite presentation is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from Lemma \\ref{lemma-formally-unramified} and", "Algebra, Definition \\ref{algebra-definition-unramified}.", "Part (2) follows from part (1) and", "Algebra, Lemma \\ref{algebra-lemma-etale-flat-unramified-finite-presentation}." ], "refs": [ "more-algebra-lemma-formally-unramified", "algebra-definition-unramified", "algebra-lemma-etale-flat-unramified-finite-presentation" ], "ref_ids": [ 10448, 1544, 1271 ] } ], "ref_ids": [] }, { "id": 10450, "type": "theorem", "label": "more-algebra-lemma-when-weakly-etale", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-when-weakly-etale", "contents": [ "Let $A \\to B$ be a ring map. Then $A \\to B$ is weakly \\'etale in each", "of the following cases", "\\begin{enumerate}", "\\item $B = S^{-1}A$ is a localization of $A$,", "\\item $A \\to B$ is \\'etale,", "\\item $B$ is a filtered colimit of weakly \\'etale $A$-algebras.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "An \\'etale ring map is flat and the map $B \\otimes_A B \\to B$ is", "also \\'etale as a map between \\'etale $A$-algebras", "(Algebra, Lemma \\ref{algebra-lemma-map-between-etale}).", "This proves (2).", "\\medskip\\noindent", "Let $B_i$ be a directed system of weakly \\'etale $A$-algebras.", "Then $B = \\colim B_i$ is flat over $A$ by", "Algebra, Lemma \\ref{algebra-lemma-colimit-flat}.", "Note that the transition maps $B_i \\to B_{i'}$ are flat", "by Lemma \\ref{lemma-weakly-etale-permanence}.", "Hence $B$ is flat over $B_i$ for each $i$, and we see that $B$ is flat", "over $B_i \\otimes_A B_i$ by", "Algebra, Lemma \\ref{algebra-lemma-composition-flat}.", "Thus $B$ is flat over $B \\otimes_A B = \\colim B_i \\otimes_A B_i$", "by Algebra, Lemma \\ref{algebra-lemma-colimit-rings-flat}.", "\\medskip\\noindent", "Part (1) can be proved directly, but also follows by combining", "(2) and (3)." ], "refs": [ "algebra-lemma-map-between-etale", "algebra-lemma-colimit-flat", "more-algebra-lemma-weakly-etale-permanence", "algebra-lemma-composition-flat", "algebra-lemma-colimit-rings-flat" ], "ref_ids": [ 1236, 523, 10447, 524, 526 ] } ], "ref_ids": [] }, { "id": 10451, "type": "theorem", "label": "more-algebra-lemma-absolutely-flat-fields", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-absolutely-flat-fields", "contents": [ "Let $K \\subset L$ be an extension of fields. If $L \\otimes_K L \\to L$", "is flat, then $L$ is an algebraic separable extension of $K$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-go-down} we see that any subfield", "$K \\subset L' \\subset L$ the map $L' \\otimes_K L' \\to L'$ is flat.", "Thus we may assume $L$ is a finitely generated field extension of $K$.", "In this case the fact that $L/K$ is formally unramified", "(Lemma \\ref{lemma-formally-unramified})", "implies that $L/K$ is finite separable, see Algebra, Lemma", "\\ref{algebra-lemma-characterize-separable-algebraic-field-extensions}." ], "refs": [ "more-algebra-lemma-go-down", "more-algebra-lemma-formally-unramified", "algebra-lemma-characterize-separable-algebraic-field-extensions" ], "ref_ids": [ 10446, 10448, 1314 ] } ], "ref_ids": [] }, { "id": 10452, "type": "theorem", "label": "more-algebra-lemma-absolutely-flat-over-field", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-absolutely-flat-over-field", "contents": [ "Let $B$ be an algebra over a field $K$. The following are", "equivalent", "\\begin{enumerate}", "\\item $B \\otimes_K B \\to B$ is flat,", "\\item $K \\to B$ is weakly \\'etale, and", "\\item $B$ is a filtered colimit of \\'etale $K$-algebras.", "\\end{enumerate}", "Moreover, every finitely generated $K$-subalgebra of $B$", "is \\'etale over $K$." ], "refs": [], "proofs": [ { "contents": [ "Parts (1) and (2) are equivalent because every $K$-algebra is flat over $K$.", "Part (3) implies (1) and (2) by Lemma \\ref{lemma-when-weakly-etale}", "\\medskip\\noindent", "Assume (1) and (2) hold. We will prove (3) and the finite statement of", "the lemma. A field is absolutely flat ring, hence $B$ is a absolutely", "flat ring by Lemma \\ref{lemma-absolutely-flat-over-absolutely-flat}.", "Hence $B$ is reduced and every local", "ring is a field, see Lemma \\ref{lemma-absolutely-flat}.", "\\medskip\\noindent", "Let $\\mathfrak q \\subset B$ be a prime. The ring map", "$B \\to B_\\mathfrak q$ is weakly \\'etale, hence $B_\\mathfrak q$", "is weakly \\'etale over $K$ (Lemma \\ref{lemma-composition-weakly-etale}).", "Thus $B_\\mathfrak q$ is a separable algebraic extension of $K$ by", "Lemma \\ref{lemma-absolutely-flat-fields}.", "\\medskip\\noindent", "Let $K \\subset A \\subset B$ be a finitely generated $K$-sub algebra.", "We will show that $A$ is \\'etale over $K$ which will finish the proof", "of the lemma.", "Then every minimal prime $\\mathfrak p \\subset A$ is the image of a prime", "$\\mathfrak q$ of $B$, see", "Algebra, Lemma \\ref{algebra-lemma-injective-minimal-primes-in-image}.", "Thus $\\kappa(\\mathfrak p)$ as a subfield of", "$B_\\mathfrak q = \\kappa(\\mathfrak q)$ is separable algebraic over $K$.", "Hence every generic point of $\\Spec(A)$", "is closed (Algebra, Lemma \\ref{algebra-lemma-finite-residue-extension-closed}).", "Thus $\\dim(A) = 0$.", "Then $A$ is the product of its local rings, e.g., by", "Algebra, Proposition \\ref{algebra-proposition-dimension-zero-ring}.", "Moreover, since $A$ is reduced, all local rings are equal", "to their residue fields wich are finite separable over $K$.", "This means that $A$ is \\'etale over $K$ by", "Algebra, Lemma \\ref{algebra-lemma-etale-over-field}", "and finishes the proof." ], "refs": [ "more-algebra-lemma-when-weakly-etale", "more-algebra-lemma-absolutely-flat-over-absolutely-flat", "more-algebra-lemma-absolutely-flat", "more-algebra-lemma-composition-weakly-etale", "more-algebra-lemma-absolutely-flat-fields", "algebra-lemma-injective-minimal-primes-in-image", "algebra-lemma-finite-residue-extension-closed", "algebra-proposition-dimension-zero-ring", "algebra-lemma-etale-over-field" ], "ref_ids": [ 10450, 10444, 10441, 10445, 10451, 445, 472, 1410, 1232 ] } ], "ref_ids": [] }, { "id": 10453, "type": "theorem", "label": "more-algebra-lemma-weakly-etale-residue-field-extensions", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-weakly-etale-residue-field-extensions", "contents": [ "Let $A \\to B$ be a ring map. If $A \\to B$ is weakly \\'etale, then", "$A \\to B$ induces separable algebraic residue field extensions." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p$ be a prime of $A$. Then", "$\\kappa(\\mathfrak p) \\to B \\otimes_A \\kappa(\\mathfrak p)$ is weakly \\'etale by", "Lemma \\ref{lemma-base-change-weakly-etale}.", "Hence $B \\otimes_A \\kappa(\\mathfrak p)$ is a filtered colimit of", "\\'etale $\\kappa(\\mathfrak p)$-algebras by", "Lemma \\ref{lemma-absolutely-flat-over-field}.", "Hence for $\\mathfrak q \\subset B$ lying over $\\mathfrak p$ the", "extension $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$ is", "a filtered colimit of finite separable extensions by", "Algebra, Lemma \\ref{algebra-lemma-etale-over-field}." ], "refs": [ "more-algebra-lemma-base-change-weakly-etale", "more-algebra-lemma-absolutely-flat-over-field", "algebra-lemma-etale-over-field" ], "ref_ids": [ 10443, 10452, 1232 ] } ], "ref_ids": [] }, { "id": 10454, "type": "theorem", "label": "more-algebra-lemma-weak-dimension-at-most-1", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-weak-dimension-at-most-1", "contents": [ "Let $A$ be a ring. The following are equivalent", "\\begin{enumerate}", "\\item $A$ has weak dimension $\\leq 1$,", "\\item every ideal of $A$ is flat,", "\\item every finitely generated ideal of $A$ is flat,", "\\item every submodule of a flat $A$-module is flat, and", "\\item every local ring of $A$ is a valuation ring.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $A$ has weak dimension $\\leq 1$, then the resolution", "$0 \\to I \\to A \\to A/I \\to 0$ shows that every ideal $I$", "is flat by Lemma \\ref{lemma-last-one-flat}.", "Hence (1) $\\Rightarrow$ (2).", "\\medskip\\noindent", "Assume (4). Let $M$ be an $A$-module. Choose a surjection", "$F \\to M$ where $F$ is a free $A$-module. Then $\\Ker(F \\to M)$", "is flat by assumption, and we see that $M$ has tor dimension", "$\\leq 1$ by Lemma \\ref{lemma-tor-dimension}.", "Hence (4) $\\Rightarrow$ (1).", "\\medskip\\noindent", "Every ideal is the union of the finitely generated ideals", "contained in it. Hence (3) implies (2) by", "Algebra, Lemma \\ref{algebra-lemma-colimit-flat}.", "Thus (3) $\\Leftrightarrow$ (2).", "\\medskip\\noindent", "Assume (2). Suppose that $N \\subset M$ with $M$ a flat $A$-module.", "We will prove that $N$ is flat.", "We can write $M = \\colim M_i$ with each $M_i$ finite free, see", "Algebra, Theorem \\ref{algebra-theorem-lazard}.", "Setting $N_i \\subset M_i$ the inverse image of $N$ we see that", "$N = \\colim N_i$. By", "Algebra, Lemma \\ref{algebra-lemma-colimit-flat}.", "it suffices to prove $N_i$ is flat and we reduce", "to the case $M = R^{\\oplus n}$. In this case", "the module $N$ has a finite filtration by the submodules", "$R^{\\oplus j} \\cap N$ whose subquotients are ideals.", "By (2) these ideals are flat and hence $N$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-flat-ses}. Thus (2) $\\Rightarrow$ (4).", "\\medskip\\noindent", "Assume $A$ satisfies (1) and let $\\mathfrak p \\subset A$ be a", "prime ideal. By", "Lemmas \\ref{lemma-when-weakly-etale} and \\ref{lemma-weak-dimension-goes-up}", "we see that $A_\\mathfrak p$ satisfies (1). We will show $A$ is a valuation ring", "if $A$ is a local ring satisfying (3). Let $f \\in \\mathfrak m$", "be a nonzero element. Then $(f)$ is a flat nonzero module generated by", "one element. Hence it is a free $A$-module by", "Algebra, Lemma \\ref{algebra-lemma-finite-flat-local}.", "It follows that $f$ is a nonzerodivisor and $A$ is a domain.", "If $I \\subset A$ is a finitely generated ideal, then we similarly", "see that $I$ is a finite free $A$-module, hence (by considering the", "rank) free of rank $1$ and $I$ is a principal ideal. Thus $A$ is a", "valuation ring by", "Algebra, Lemma \\ref{algebra-lemma-characterize-valuation-ring}.", "Thus (1) $\\Rightarrow$ (5).", "\\medskip\\noindent", "Assume (5). Let $I \\subset A$ be a finitely generated ideal.", "Then $I_\\mathfrak p \\subset A_\\mathfrak p$ is a finitely generated ideal", "in a valuation ring, hence principal", "(Algebra, Lemma \\ref{algebra-lemma-characterize-valuation-ring}), hence flat.", "Thus $I$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-flat-localization}.", "Thus (5) $\\Rightarrow$ (3). This finishes the proof of the lemma." ], "refs": [ "more-algebra-lemma-last-one-flat", "more-algebra-lemma-tor-dimension", "algebra-lemma-colimit-flat", "algebra-theorem-lazard", "algebra-lemma-colimit-flat", "algebra-lemma-flat-ses", "more-algebra-lemma-when-weakly-etale", "more-algebra-lemma-weak-dimension-goes-up", "algebra-lemma-finite-flat-local", "algebra-lemma-characterize-valuation-ring", "algebra-lemma-characterize-valuation-ring", "algebra-lemma-flat-localization" ], "ref_ids": [ 10169, 10173, 523, 318, 523, 533, 10450, 10440, 797, 620, 620, 538 ] } ], "ref_ids": [] }, { "id": 10455, "type": "theorem", "label": "more-algebra-lemma-product-weak-dimension-at-most-1", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-product-weak-dimension-at-most-1", "contents": [ "Let $J$ be a set. For each $j \\in J$ let", "$A_j$ be a valuation ring with fraction field $K_j$.", "Set $A = \\prod A_j$ and $K = \\prod K_j$.", "Then $A$ has weak dimension at most $1$ and $A \\to K$ is", "a localization." ], "refs": [], "proofs": [ { "contents": [ "Let $I \\subset A$ be a finitely generated ideal.", "By Lemma \\ref{lemma-weak-dimension-at-most-1}", "it suffices to show that $I$ is a flat $A$-module.", "Let $I_j \\subset A_j$ be the image of $I$.", "Observe that $I_j = I \\otimes_A A_j$, hence", "$I \\to \\prod I_j$ is surjective by", "Algebra, Proposition \\ref{algebra-proposition-fg-tensor}.", "Thus $I = \\prod I_j$.", "Since $A_j$ is a valuation ring, the ideal $I_j$", "is generated by a single element", "(Algebra, Lemma \\ref{algebra-lemma-characterize-valuation-ring}).", "Say $I_j = (f_j)$. Then $I$ is generated by the element $f = (f_j)$.", "Let $e \\in A$ be the idempotent which has a $0$ or $1$", "in $A_j$ depending on whether $f_j$ is $0$ or not.", "Then $f = g e$ for some nonzerodivisor $g \\in A$:", "take $g = (g_j)$ with $g_j = 1$ if $f_j = 0$ and $g_j = f_j$ else.", "Thus $I \\cong (e)$ as a module. We conclude $I$ is flat as $(e)$ is a", "direct summand of $A$. The final statement is true because", "$K = S^{-1}A$ where $S = \\prod (A_j \\setminus \\{0\\})$." ], "refs": [ "more-algebra-lemma-weak-dimension-at-most-1", "algebra-proposition-fg-tensor", "algebra-lemma-characterize-valuation-ring" ], "ref_ids": [ 10454, 1415, 620 ] } ], "ref_ids": [] }, { "id": 10456, "type": "theorem", "label": "more-algebra-lemma-product-found-valuation-rings", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-product-found-valuation-rings", "contents": [ "Let $A$ be a normal domain with fraction field $K$.", "There exists a cartesian diagram", "$$", "\\xymatrix{", "A \\ar[d] \\ar[r] & K \\ar[d] \\\\", "V \\ar[r] & L", "}", "$$", "of rings where $V$ has weak dimension at most $1$", "and $V \\to L$ is a flat, injective, epimorphism of rings." ], "refs": [], "proofs": [ { "contents": [ "For every $x \\in K$, $x \\not \\in A$ pick $V_x \\subset K$ as in", "Algebra, Lemma \\ref{algebra-lemma-find-valuation-rings}.", "Set $V = \\prod_{x \\in K \\setminus A} V_x$ and", "$L = \\prod_{x \\in K \\setminus A} K$. The ring $V$", "has weak dimension at most $1$ by", "Lemma \\ref{lemma-product-weak-dimension-at-most-1}", "which also shows that $V \\to L$ is a localization.", "A localization is flat and an epimorphism, see", "Algebra, Lemmas \\ref{algebra-lemma-flat-localization} and", "\\ref{algebra-lemma-epimorphism-local}." ], "refs": [ "algebra-lemma-find-valuation-rings", "more-algebra-lemma-product-weak-dimension-at-most-1", "algebra-lemma-flat-localization", "algebra-lemma-epimorphism-local" ], "ref_ids": [ 617, 10455, 538, 951 ] } ], "ref_ids": [] }, { "id": 10457, "type": "theorem", "label": "more-algebra-lemma-weak-dimension-at-most-1-integrally-closed", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-weak-dimension-at-most-1-integrally-closed", "contents": [ "Let $A$ be a ring of weak dimension at most $1$.", "If $A \\to B$ is a flat, injective, epimorphism of rings, then", "$A$ is integrally closed in $B$." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in B$ be integral over $A$. Let $A' = A[x] \\subset B$.", "Then $A'$ is a finite ring extension of $A$ by", "Algebra, Lemma \\ref{algebra-lemma-characterize-finite-in-terms-of-integral}.", "To show $A = A'$ it suffices to show $A \\to A'$ is an epimorphism by", "Algebra, Lemma \\ref{algebra-lemma-finite-epimorphism-surjective}.", "Note that $A'$ is flat over $A$ by assumption on $A$ and the fact that", "$B$ is flat over $A$ (Lemma \\ref{lemma-weak-dimension-at-most-1}).", "Hence the composition", "$$", "A' \\otimes_A A' \\to B \\otimes_A A' \\to B \\otimes_A B \\to B", "$$", "is injective, i.e., $A' \\otimes_A A' \\cong A'$ and the lemma is proved." ], "refs": [ "algebra-lemma-characterize-finite-in-terms-of-integral", "algebra-lemma-finite-epimorphism-surjective", "more-algebra-lemma-weak-dimension-at-most-1" ], "ref_ids": [ 484, 952, 10454 ] } ], "ref_ids": [] }, { "id": 10458, "type": "theorem", "label": "more-algebra-lemma-normality-goes-up", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-normality-goes-up", "contents": [ "Let $A$ be a normal domain with fraction field $K$.", "Let $A \\to B$ be weakly \\'etale. Then", "$B$ is integrally closed in $B \\otimes_A K$." ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram as in Lemma \\ref{lemma-product-found-valuation-rings}.", "As $A \\to B$ is flat, the base change gives a cartesian diagram", "$$", "\\xymatrix{", "B \\ar[d] \\ar[r] & B \\otimes_A K \\ar[d] \\\\", "B \\otimes_A V \\ar[r] & B \\otimes_A L", "}", "$$", "of rings. Note that $V \\to B \\otimes_A V$ is weakly \\'etale", "(Lemma \\ref{lemma-base-change-weakly-etale}), hence $B \\otimes_A V$", "has weak dimension at most $1$ by Lemma \\ref{lemma-weak-dimension-goes-up}.", "Note that $B \\otimes_A V \\to B \\otimes_A L$ is a flat, injective,", "epimorphism of rings as a flat base change of such", "(Algebra, Lemmas \\ref{algebra-lemma-flat-base-change} and", "\\ref{algebra-lemma-base-change-epimorphism}).", "By Lemma \\ref{lemma-weak-dimension-at-most-1-integrally-closed}", "we see that $B \\otimes_A V$ is integrally closed in $B \\otimes_A L$.", "It follows from the cartesian property of the diagram", "that $B$ is integrally closed in $B \\otimes_A K$." ], "refs": [ "more-algebra-lemma-product-found-valuation-rings", "more-algebra-lemma-base-change-weakly-etale", "more-algebra-lemma-weak-dimension-goes-up", "algebra-lemma-flat-base-change", "algebra-lemma-base-change-epimorphism", "more-algebra-lemma-weak-dimension-at-most-1-integrally-closed" ], "ref_ids": [ 10456, 10443, 10440, 527, 949, 10457 ] } ], "ref_ids": [] }, { "id": 10459, "type": "theorem", "label": "more-algebra-lemma-integral-over-henselian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-integral-over-henselian", "contents": [ "Let $A \\to B$ be a ring homomorphism.", "Assume", "\\begin{enumerate}", "\\item $A$ is a henselian local ring,", "\\item $A \\to B$ is integral,", "\\item $B$ is a domain.", "\\end{enumerate}", "Then $B$ is a henselian local ring and $A \\to B$ is a local homomorphism.", "If $A$ is strictly henselian, then $B$ is a strictly henselian local ring", "and the extension $\\kappa(\\mathfrak m_A) \\subset \\kappa(\\mathfrak m_B)$", "of residue fields is purely inseparable." ], "refs": [], "proofs": [ { "contents": [ "Write $B$ as a filtered colimit $B = \\colim B_i$ of finite $A$-sub algebras.", "If we prove the results for each $B_i$, then the result follows for $B$.", "See Algebra, Lemma \\ref{algebra-lemma-colimit-henselian}.", "If $A \\to B$ is finite, then $B$ is a product of local henselian rings by", "Algebra, Lemma \\ref{algebra-lemma-finite-over-henselian}.", "Since $B$ is a domain we see that $B$ is a local ring.", "The maximal ideal of $B$ lies over the maximal ideal of $A$ by", "going up for $A \\to B$ (Algebra, Lemma \\ref{algebra-lemma-integral-going-up}).", "If $A$ is strictly henselian, then the field extension", "$\\kappa(\\mathfrak m_A) \\subset \\kappa(\\mathfrak m_B)$", "being algebraic, has to be purely inseparable.", "Of course, then $\\kappa(\\mathfrak m_B)$ is separably algebraically", "closed and $B$ is strictly henselian." ], "refs": [ "algebra-lemma-colimit-henselian", "algebra-lemma-finite-over-henselian", "algebra-lemma-integral-going-up" ], "ref_ids": [ 1293, 1277, 500 ] } ], "ref_ids": [] }, { "id": 10460, "type": "theorem", "label": "more-algebra-lemma-local-tensor-with-integral", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-local-tensor-with-integral", "contents": [ "Let $A \\to B$ and $A \\to C$ be local homomorphisms of local rings.", "If $A \\to C$ is integral and either", "$\\kappa(\\mathfrak m_A) \\subset \\kappa(\\mathfrak m_C)$ or", "$\\kappa(\\mathfrak m_A) \\subset \\kappa(\\mathfrak m_B)$ is purely", "inseparable, then $D = B \\otimes_A C$ is a local ring and", "$B \\to D$ and $C \\to D$ are local." ], "refs": [], "proofs": [ { "contents": [ "Any maximal ideal of $D$ lies over the maximal ideal of $B$ by", "going up for the integral ring map", "$B \\to D$ (Algebra, Lemma \\ref{algebra-lemma-integral-going-up}).", "Now $D/\\mathfrak m_B D = \\kappa(\\mathfrak m_B) \\otimes_A C =", "\\kappa(\\mathfrak m_B) \\otimes_{\\kappa(\\mathfrak m_A)} C/\\mathfrak m_A C$.", "The spectrum of $C/\\mathfrak m_A C$ consists of a", "single point, namely $\\mathfrak m_C$. Thus the spectrum of", "$D/\\mathfrak m_B D$ is the same as the spectrum of", "$\\kappa(\\mathfrak m_B) \\otimes_{\\kappa(\\mathfrak m_A)} \\kappa(\\mathfrak m_C)$", "which is a single point by our assumption that either", "$\\kappa(\\mathfrak m_A) \\subset \\kappa(\\mathfrak m_C)$ or", "$\\kappa(\\mathfrak m_A) \\subset \\kappa(\\mathfrak m_B)$ is purely", "inseparable. This proves that $D$ is local and that the ring maps", "$B \\to D$ and $C \\to D$ are local." ], "refs": [ "algebra-lemma-integral-going-up" ], "ref_ids": [ 500 ] } ], "ref_ids": [] }, { "id": 10461, "type": "theorem", "label": "more-algebra-lemma-class-weakly-etale-over-field", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-class-weakly-etale-over-field", "contents": [ "Let $K$ be a field. If $B$ is weakly \\'etale over $K$, then", "\\begin{enumerate}", "\\item $B$ is reduced,", "\\item $B$ is integral over $K$,", "\\item any finitely generated $K$-subalgebra of $B$ is a finite product", "of finite separable extensions of $K$,", "\\item $B$ is a field if and only if $B$ does not have nontrivial idempotents", "and in this case it is a separable algebraic extension of $K$,", "\\item any sub or quotient $K$-algebra of $B$ is weakly \\'etale over $K$,", "\\item if $B'$ is weakly \\'etale over $K$, then $B \\otimes_K B'$ is", "weakly \\'etale over $K$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from Lemma \\ref{lemma-absolutely-flat-over-absolutely-flat}", "but of course it follows from part (3) as well.", "Part (3) follows from Lemma \\ref{lemma-absolutely-flat-over-field}", "and the fact that \\'etale $K$-algebras are finite products of", "finite separable extensions of $K$, see", "Algebra, Lemma \\ref{algebra-lemma-etale-over-field}.", "Part (3) implies (2).", "Part (4) follows from (3) as a product of fields is", "a field if and only if it has no nontrivial idempotents.", "\\medskip\\noindent", "If $S \\subset B$ is a subalgebra, then it is the filtered", "colimit of its finitely generated subalgebras which are", "all \\'etale over $K$ by the above and hence", "$S$ is weakly \\'etale over $K$ by", "Lemma \\ref{lemma-absolutely-flat-over-field}.", "If $B \\to Q$ is a quotient algebra, then", "$Q$ is the filtered colimit of $K$-algebra quotients of", "finite products $\\prod_{i \\in I} L_i$ of finite separable extensions", "$L_i/K$. Such a quotient is of the form $\\prod_{i \\in J} L_i$", "for some subset $J \\subset I$ and hence the result", "holds for quotients by the same reasoning.", "\\medskip\\noindent", "The statement on tensor products follows in a similar manner", "or by combining Lemmas \\ref{lemma-base-change-weakly-etale} and", "\\ref{lemma-composition-weakly-etale}." ], "refs": [ "more-algebra-lemma-absolutely-flat-over-absolutely-flat", "more-algebra-lemma-absolutely-flat-over-field", "algebra-lemma-etale-over-field", "more-algebra-lemma-absolutely-flat-over-field", "more-algebra-lemma-base-change-weakly-etale", "more-algebra-lemma-composition-weakly-etale" ], "ref_ids": [ 10444, 10452, 1232, 10452, 10443, 10445 ] } ], "ref_ids": [] }, { "id": 10462, "type": "theorem", "label": "more-algebra-lemma-max-weakly-etale-subalgebra", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-max-weakly-etale-subalgebra", "contents": [ "Let $K$ be a field. Let $A$ be a $K$-algebra. There exists", "a maximal weakly \\'etale $K$-subalgebra $B_{max} \\subset A$." ], "refs": [], "proofs": [ { "contents": [ "Let $B_1, B_2 \\subset A$ be weakly \\'etale $K$-subalgebras.", "Then $B_1 \\otimes_K B_2$ is weakly \\'etale over $K$", "and so is the image of $B_1 \\otimes_K B_2 \\to A$", "(Lemma \\ref{lemma-class-weakly-etale-over-field}).", "Thus the collection $\\mathcal{B}$ of weakly \\'etale $K$-subalgebras", "$B \\subset A$ is directed and the colimit", "$B_{max} = \\colim_{B \\in \\mathcal{B}} B$ is", "a weakly \\'etale $K$-algebra by Lemma \\ref{lemma-when-weakly-etale}.", "Hence the image of $B_{max} \\to A$ is weakly \\'etale over $K$", "(previous lemma cited). It follows that this image is in $\\mathcal{B}$", "and hence $\\mathcal{B}$ has a maximal element", "(and the image is the same as $B_{max}$)." ], "refs": [ "more-algebra-lemma-class-weakly-etale-over-field", "more-algebra-lemma-when-weakly-etale" ], "ref_ids": [ 10461, 10450 ] } ], "ref_ids": [] }, { "id": 10463, "type": "theorem", "label": "more-algebra-lemma-properties-of-max-weakly-etale-subalgebra", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-properties-of-max-weakly-etale-subalgebra", "contents": [ "Let $K$ be a field. For a $K$-algebra $A$ denote $B_{max}(A)$", "the maximal weakly \\'etale $K$-subalgebra of $A$ as in", "Lemma \\ref{lemma-max-weakly-etale-subalgebra}. Then", "\\begin{enumerate}", "\\item any $K$-algebra map $A' \\to A$ induces a $K$-algebra map", "$B_{max}(A') \\to B_{max}(A)$,", "\\item if $A' \\subset A$, then $B_{max}(A') = B_{max}(A) \\cap A'$,", "\\item if $A = \\colim A_i$ is a filtered colimit, then", "$B_{max}(A) = \\colim B_{max}(A_i)$,", "\\item the map $B_{max}(A) \\to B_{max}(A_{red})$ is an isomorphism,", "\\item $B_{max}(A_1 \\times \\ldots \\times A_n) =", "B_{max}(A_1) \\times \\ldots \\times B_{max}(A_n)$,", "\\item if $A$ has no nontrivial idempotents, then $B_{max}(A)$ is a", "field and a separable algebraic extension of $K$,", "\\item add more here.", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-max-weakly-etale-subalgebra" ], "proofs": [ { "contents": [ "Proof of (1). This is true because the image of $B_{max}(A') \\to A$", "is weakly \\'etale over $K$ by Lemma \\ref{lemma-class-weakly-etale-over-field}.", "\\medskip\\noindent", "Proof of (2). By (1) we have $B_{max}(A') \\subset B_{max}(A)$.", "Conversely, $B_{max}(A) \\cap A'$ is a weakly \\'etale $K$-algebra", "by Lemma \\ref{lemma-class-weakly-etale-over-field} and hence", "contained in $B_{max}(A')$.", "\\medskip\\noindent", "Proof of (3). By (1) there is a map $\\colim B_{max}(A_i) \\to A$", "which is injective because the system is filtered and", "$B_{max}(A_i) \\subset A_i$. The colimit $\\colim B_{max}(A_i)$", "is weakly \\'etale over $K$ by", "Lemma \\ref{lemma-when-weakly-etale}.", "Hence we get an injective map $\\colim B_{max}(A_i) \\to B_{max}(A)$.", "Suppose that $a \\in B_{max}(A)$. Then $a$ generates a finitely presented", "$K$-subalgebra $B \\subset B_{max}(A)$.", "By Algebra, Lemma \\ref{algebra-lemma-characterize-finite-presentation}", "there is an $i$ and a $K$-algebra map $f : B \\to A_i$", "lifting the given map $B \\to A$.", "Since $B$ is weakly \\'etale by", "Lemma \\ref{lemma-class-weakly-etale-over-field},", "we see that $f(B) \\subset B_{max}(A_i)$ and we conclude that $a$", "is in the image of $\\colim B_{max}(A_i) \\to B_{max}(A)$.", "\\medskip\\noindent", "Proof of (4). Write $B_{max}(A_{red}) = \\colim B_i$", "as a filtered colimit of \\'etale $K$-algebras", "(Lemma \\ref{lemma-absolutely-flat-over-field}).", "By Algebra, Lemma \\ref{algebra-lemma-smooth-strong-lift}", "for each $i$ there is a $K$-algebra map $f_i : B_i \\to A$", "lifting the given map $B_i \\to A_{red}$. It follows that the", "canonical map $B_{max}(A_{red}) \\to B_{max}(A)$ is surjective.", "The kernel consists of nilpotent elements and hence is zero", "as $B_{max}(A_{red})$ is reduced", "(Lemma \\ref{lemma-class-weakly-etale-over-field}).", "\\medskip\\noindent", "Proof of (5). Omitted.", "\\medskip\\noindent", "Proof of (6). Follows from Lemma \\ref{lemma-class-weakly-etale-over-field}", "part (4)." ], "refs": [ "more-algebra-lemma-class-weakly-etale-over-field", "more-algebra-lemma-class-weakly-etale-over-field", "more-algebra-lemma-when-weakly-etale", "algebra-lemma-characterize-finite-presentation", "more-algebra-lemma-class-weakly-etale-over-field", "more-algebra-lemma-absolutely-flat-over-field", "algebra-lemma-smooth-strong-lift", "more-algebra-lemma-class-weakly-etale-over-field", "more-algebra-lemma-class-weakly-etale-over-field" ], "ref_ids": [ 10461, 10461, 10450, 1092, 10461, 10452, 1216, 10461, 10461 ] } ], "ref_ids": [ 10462 ] }, { "id": 10464, "type": "theorem", "label": "more-algebra-lemma-change-fields-max-weakly-etale-subalgebra", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-change-fields-max-weakly-etale-subalgebra", "contents": [ "Let $L/K$ be an extension of fields. Let $A$ be a $K$-algebra.", "Let $B \\subset A$ be the maximal weakly \\'etale $K$-subalgebra of", "$A$ as in Lemma \\ref{lemma-max-weakly-etale-subalgebra}.", "Then $B \\otimes_K L$ is the maximal weakly \\'etale $L$-subalgebra", "of $A \\otimes_K L$." ], "refs": [ "more-algebra-lemma-max-weakly-etale-subalgebra" ], "proofs": [ { "contents": [ "For an algebra $A$ over $K$ we write $B_{max}(A/K)$", "for the maximal weakly \\'etale $K$-subalgebra of $A$.", "Similarly we write $B_{max}(A'/L)$ for the maximal weakly \\'etale", "$L$-subalgebra of $A'$ if $A'$ is an $L$-algebra.", "Since $B_{max}(A/K) \\otimes_K L$ is weakly \\'etale over $L$", "(Lemma \\ref{lemma-base-change-weakly-etale})", "and since $B_{max}(A/K) \\otimes_K L \\subset A \\otimes_K L$", "we obtain a canonical injective map", "$$", "B_{max}(A/K) \\otimes_K L \\to B_{max}((A \\otimes_K L)/L)", "$$", "The lemma states that this map is an isomorphism.", "\\medskip\\noindent", "To prove the lemma for $L$ and our $K$-algebra $A$, it suffices to", "prove the lemma for any field extension $L'$ of $L$. Namely, we have", "the factorization", "$$", "B_{max}(A/K) \\otimes_K L' \\to", "B_{max}((A \\otimes_K L)/L) \\otimes_L L' \\to", "B_{max}((A \\otimes_K L')/L')", "$$", "hence the composition cannot be surjective without", "$B_{max}(A/K) \\otimes_K L \\to B_{max}((A \\otimes_K L)/L)$", "being surjective. Thus we may assume $L$ is algebraically closed.", "\\medskip\\noindent", "Reduction to finite type $K$-algebra. We may write $A$", "is the filtered colimit of its finite type $K$-subalgebras.", "Using Lemma \\ref{lemma-properties-of-max-weakly-etale-subalgebra}", "we see that it suffices to prove the lemma for finite type", "$K$-algebras.", "\\medskip\\noindent", "Assume $A$ is a finite type $K$-algebra. Since the kernel", "of $A \\to A_{red}$ is nilpotent, the same is true for", "$A \\otimes_K L \\to A_{red} \\otimes_K L$. Then", "$$", "B_{max}((A \\otimes_K L)/L) \\to B_{max}((A_{red} \\otimes_K L)/L)", "$$", "is injective because the kernel is nilpotent and the", "weakly \\'etale $L$-algebra $B_{max}((A \\otimes_K L)/L)$ is reduced", "(Lemma \\ref{lemma-class-weakly-etale-over-field}).", "Since $B_{max}(A/K) = B_{max}(A_{red}/K)$", "by Lemma \\ref{lemma-properties-of-max-weakly-etale-subalgebra}", "we conclude", "that it suffices to prove the lemma for $A_{red}$.", "\\medskip\\noindent", "Assume $A$ is a reduced finite type $K$-algebra.", "Let $Q = Q(A)$ be the total quotient ring of $A$.", "Then $A \\subset Q$ and $A \\otimes_K L \\subset Q \\otimes_A L$", "and hence", "$$", "B_{max}(A/K) = A \\cap B_{max}(Q/K)", "$$", "and", "$$", "B_{max}((A \\otimes_K L)/L) =", "(A \\otimes_K L) \\cap B_{max}((Q \\otimes_K L)/L)", "$$", "by Lemma \\ref{lemma-properties-of-max-weakly-etale-subalgebra}.", "Since $-\\otimes_K L$ is an exact functor, it follows that", "if we prove the result for $Q$, then the result follows for $A$.", "Since $Q$ is a finite product of fields (Algebra, Lemmas", "\\ref{algebra-lemma-total-ring-fractions-no-embedded-points},", "\\ref{algebra-lemma-minimal-prime-reduced-ring},", "\\ref{algebra-lemma-Noetherian-irreducible-components}, and", "\\ref{algebra-lemma-Noetherian-permanence})", "and since $B_{max}$ commutes with products", "(Lemma \\ref{lemma-properties-of-max-weakly-etale-subalgebra})", "it suffices to prove the lemma when $A$ is a field.", "\\medskip\\noindent", "Assume $A$ is a field. We reduce to $A$ being finitely generated", "over $K$ by the argument in the third paragraph of the proof.", "(In fact the way we reduced to the case of a field produces", "a finitely generated field extension of $K$.)", "\\medskip\\noindent", "Assume $A$ is a finitely generated field extension of $K$.", "Then $K' = B_{max}(A/K)$ is a field separable algebraic over $K$ by", "Lemma \\ref{lemma-properties-of-max-weakly-etale-subalgebra} part (6).", "Hence $K'$ is a finite separable field extension", "of $K$ and $A$ is geometrically irreducible over $K'$ by", "Algebra, Lemma \\ref{algebra-lemma-make-geometrically-irreducible}.", "Since $L$ is algebraically closed and $K'/K$ finite separable", "we see that", "$$", "K' \\otimes_K L \\to \\prod\\nolimits_{\\sigma \\in \\Hom_K(K', L)} L,\\quad", "\\alpha \\otimes \\beta \\mapsto (\\sigma(\\alpha)\\beta)_\\sigma", "$$", "is an isomorphism", "(Fields, Lemma \\ref{fields-lemma-finite-separable-tensor-alg-closed}).", "We conclude", "$$", "A \\otimes_K L = A \\otimes_{K'} (K' \\otimes_K L) =", "\\prod\\nolimits_{\\sigma \\in \\Hom_K(K', L)} A \\otimes_{K', \\sigma} L", "$$", "Since $A$ is geometrically irreducible over $K'$ we see that", "$A \\otimes_{K', \\sigma} L$ has a unique minimal prime.", "Since $L$ is algebraically closed it follows that", "$B_{max}((A \\otimes_{K', \\sigma} L)/L) = L$", "because this $L$-algebra is a field algebraic over $L$ by", "Lemma \\ref{lemma-properties-of-max-weakly-etale-subalgebra} part (6).", "It follows that the maximal weakly \\'etale $K' \\otimes_K L$-subalgebra", "of $A \\otimes_K L$ is $K' \\otimes_K L$ because we can decompose", "these subalgebras into products as above. Hence the inclusion", "$K' \\otimes_K L \\subset B_{max}((A \\otimes_K L)/L)$ is an", "equality: the ring map $K' \\otimes_K L \\to B_{max}((A \\otimes_K L)/L)$", "is weakly \\'etale by", "Lemma \\ref{lemma-weakly-etale-permanence}." ], "refs": [ "more-algebra-lemma-base-change-weakly-etale", "more-algebra-lemma-properties-of-max-weakly-etale-subalgebra", "more-algebra-lemma-class-weakly-etale-over-field", "more-algebra-lemma-properties-of-max-weakly-etale-subalgebra", "more-algebra-lemma-properties-of-max-weakly-etale-subalgebra", "algebra-lemma-total-ring-fractions-no-embedded-points", "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-Noetherian-irreducible-components", "algebra-lemma-Noetherian-permanence", "more-algebra-lemma-properties-of-max-weakly-etale-subalgebra", "more-algebra-lemma-properties-of-max-weakly-etale-subalgebra", "algebra-lemma-make-geometrically-irreducible", "fields-lemma-finite-separable-tensor-alg-closed", "more-algebra-lemma-properties-of-max-weakly-etale-subalgebra", "more-algebra-lemma-weakly-etale-permanence" ], "ref_ids": [ 10443, 10463, 10461, 10463, 10463, 421, 418, 453, 448, 10463, 10463, 598, 4477, 10463, 10447 ] } ], "ref_ids": [ 10462 ] }, { "id": 10465, "type": "theorem", "label": "more-algebra-lemma-branches", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-branches", "contents": [ "Let $A$ be a local ring. Assume $A$ has finitely many minimal prime ideals.", "Let $A'$ be the integral closure of $A$ in the total ring of fractions", "of $A_{red}$.", "Let $A^h$ be the henselization of $A$.", "Consider the maps", "$$", "\\Spec(A') \\leftarrow \\Spec((A')^h) \\rightarrow \\Spec(A^h)", "$$", "where $(A')^h = A' \\otimes_A A^h$. Then", "\\begin{enumerate}", "\\item the left arrow is bijective on maximal ideals,", "\\item the right arrow is bijective on minimal primes,", "\\item every minimal prime of $(A')^h$ is contained in a unique", "maximal ideal and every maximal ideal contains exact one minimal prime.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $I \\subset A$ be the ideal of nilpotents.", "We have $(A/I)^h = A^h/IA^h$ by", "(Algebra, Lemma \\ref{algebra-lemma-quotient-henselization}).", "The spectra of $A$, $A^h$, $A'$, and $(A')^h$ are the same", "as the spectra of $A/I$, $A^h/IA^h$, $A'$, and", "$(A')^h = A' \\otimes_{A/I} A^h/IA^h$. Thus", "we may replace $A$ by $A_{red} = A/I$ and assume $A$ is reduced.", "Then $A \\subset A'$ which we will use below without further mention.", "\\medskip\\noindent", "Proof of (1). As $A'$ is integral over $A$ we see that $(A')^h$", "is integral over $A^h$. By going up", "(Algebra, Lemma \\ref{algebra-lemma-integral-going-up})", "every maximal ideal of $A'$, resp.\\ $(A')^h$ lies over", "the maximal ideal $\\mathfrak m$, resp.\\ $\\mathfrak m^h$ of", "$A$, resp.\\ $A^h$. Thus (1) follows from the isomorphism", "$$", "(A')^h \\otimes_{A^h} \\kappa^h =", "A' \\otimes_A A^h \\otimes_{A^h} \\kappa^h = A' \\otimes_A \\kappa", "$$", "because the residue field extension $\\kappa \\subset \\kappa^h$", "induced by $A \\to A^h$ is trivial. We will use below that the", "displayed ring is integral over a field hence spectrum of", "this ring is a profinite space, see", "Algebra, Lemmas \\ref{algebra-lemma-integral-over-field} and", "\\ref{algebra-lemma-ring-with-only-minimal-primes}.", "\\medskip\\noindent", "Proof of (3). The ring $A'$ is a normal ring and in fact a", "finite product of normal domains, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-reduced-ring-normal}.", "Since $A^h$ is a filtered colimit of \\'etale $A$-algebras,", "$(A')^h$ is filtered colimit of \\'etale $A'$-algebras", "hence $(A')^h$ is a normal ring by", "Algebra, Lemmas \\ref{algebra-lemma-normal-goes-up} and", "\\ref{algebra-lemma-colimit-normal-ring}.", "Thus every local ring of $(A')^h$ is a normal domain", "and we see that every maximal ideal contains a unique minimal prime.", "By Lemma \\ref{lemma-integral-over-henselian-pair} applied", "to $A^h \\to (A')^h$ we see that $((A')^h, \\mathfrak m(A')^h)$ is", "a henselian pair. If $\\mathfrak q \\subset (A')^h$", "is a minimal prime (or any prime), then the intersection of", "$V(\\mathfrak q)$ with $V(\\mathfrak m (A')^h)$ is connected", "by Lemma \\ref{lemma-irreducible-henselian-pair-connected}", "Since $V(\\mathfrak m (A')^h) = \\Spec((A')^h \\otimes \\kappa^h)$", "is a profinite space by we see there is a unique", "maximal ideal containing $\\mathfrak q$.", "\\medskip\\noindent", "Proof of (2). The minimal primes of $A'$ are exactly the primes", "lying over a minimal prime of $A$ (by construction).", "Since $A' \\to (A')^h$ is flat by going down", "(Algebra, Lemma \\ref{algebra-lemma-flat-going-down})", "every minimal prime of $(A')^h$ lies over a minimal prime", "of $A'$. Conversely, any prime of $(A')^h$ lying over a", "minimal prime of $A'$ is minimal because $(A')^h$ is a filtered", "colimit of \\'etale hence quasi-finite algebras over $A'$ (small detail omitted).", "We conclude that the minimal primes of $(A')^h$", "are exactly the primes which lie over a minimal prime of $A$.", "Similarly, the minimal primes of $A^h$ are exactly the", "primes lying over minimal primes of $A$.", "By construction we have $A' \\otimes_A Q(A) = Q(A)$ where $Q(A)$", "is the total fraction ring of our reduced local ring $A$.", "Of course $Q(A)$ is the finite product of residue fields of", "the minimal primes of $A$. It follows that", "$$", "(A')^h \\otimes_A Q(A) = A^h \\otimes_A A' \\otimes_A Q(A) = A^h \\otimes_A Q(A)", "$$", "Our discussion above shows the spectrum of the ring", "on the left is the set of minimal primes of $(A')^h$", "and the spectrum of the ring on the right is the", "is the set of minimal primes of $A^h$. This finishes the proof." ], "refs": [ "algebra-lemma-quotient-henselization", "algebra-lemma-integral-going-up", "algebra-lemma-integral-over-field", "algebra-lemma-ring-with-only-minimal-primes", "algebra-lemma-characterize-reduced-ring-normal", "algebra-lemma-normal-goes-up", "algebra-lemma-colimit-normal-ring", "more-algebra-lemma-integral-over-henselian-pair", "more-algebra-lemma-irreducible-henselian-pair-connected", "algebra-lemma-flat-going-down" ], "ref_ids": [ 1301, 500, 497, 426, 515, 1368, 516, 9863, 9870, 539 ] } ], "ref_ids": [] }, { "id": 10466, "type": "theorem", "label": "more-algebra-lemma-unibranch", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-unibranch", "contents": [ "\\begin{reference}", "\\cite[Chapter IV Proposition 18.6.12]{EGA4}", "\\end{reference}", "Let $A$ be a local ring. Let $A^h$ be the henselization of $A$.", "The following are equivalent", "\\begin{enumerate}", "\\item $A$ is unibranch, and", "\\item $A^h$ has a unique minimal prime.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-branches} but we will also give a", "direct proof. Denote $\\mathfrak m$ the maximal ideal of the ring $A$.", "Recall that the residue field $\\kappa = A/\\mathfrak m$ is the same as", "the residue field of $A^h$.", "\\medskip\\noindent", "Assume (2). Let $\\mathfrak p^h$ be the unique minimal prime of", "$A^h$. The flatness of $A \\to A^h$ implies that", "$\\mathfrak p = A \\cap \\mathfrak p^h$ is the unique minimal", "prime of $A$ (by going down, see", "Algebra, Lemma \\ref{algebra-lemma-flat-going-down}).", "Also, since $A^h/\\mathfrak pA^h = (A/\\mathfrak p)^h$ (see", "Algebra, Lemma \\ref{algebra-lemma-quotient-henselization})", "is reduced by Lemma \\ref{lemma-henselization-reduced}", "we see that $\\mathfrak p^h = \\mathfrak pA^h$.", "Let $A'$ be the integral closure of $A/\\mathfrak p$ in its fraction", "field. We have to show that $A'$ is local.", "Since $A \\to A'$ is integral, every maximal ideal of $A'$ lies over", "$\\mathfrak m$ (by going up for integral ring maps, see", "Algebra, Lemma \\ref{algebra-lemma-integral-going-up}).", "If $A'$ is not local, then we can find distinct maximal ideals", "$\\mathfrak m_1$, $\\mathfrak m_2$. Choose elements $f_1, f_2 \\in A'$", "with $f_i \\in \\mathfrak m_i$ and $f_i \\not \\in \\mathfrak m_{3 - i}$.", "We find a finite subalgebra $B = A[f_1, f_2] \\subset A'$ with distinct maximal", "ideals $B \\cap \\mathfrak m_i$, $i = 1, 2$.", "Note that the inclusions", "$$", "A/\\mathfrak p \\subset B \\subset \\kappa(\\mathfrak p)", "$$", "give, on tensoring with the flat ring map $A \\to A^h$ the inclusions", "$$", "A^h/\\mathfrak p^h \\subset", "B \\otimes_A A^h \\subset", "\\kappa(\\mathfrak p) \\otimes_A A^h \\subset", "\\kappa(\\mathfrak p^h)", "$$", "the last inclusion because", "$\\kappa(\\mathfrak p) \\otimes_A A^h =", "\\kappa(\\mathfrak p) \\otimes_{A/\\mathfrak p} A^h/\\mathfrak p^h$", "is a localization of the domain $A^h/\\mathfrak p^h$.", "Note that $B \\otimes_A \\kappa$ has at least two maximal ideals", "because $B/\\mathfrak mB$ has two maximal ideals. Hence, as", "$A^h$ is henselian we see that", "$B \\otimes_A A^h$ is a product of $\\geq 2$ local rings, see", "Algebra, Lemma \\ref{algebra-lemma-mop-up}.", "But we've just seen that $B \\otimes_A A^h$ is a subring of a domain", "and we get a contradiction.", "\\medskip\\noindent", "Assume (1). Let $\\mathfrak p \\subset A$ be the unique minimal", "prime and let $A'$ be the integral closure of $A/\\mathfrak p$", "in its fraction field. Let $A \\to B$ be a local map of local rings", "inducing an isomorphism of residue fields which is a", "localization of an \\'etale $A$-algebra. In particular $\\mathfrak m_B$", "is the unique prime containing $\\mathfrak m B$. Then $B' = A' \\otimes_A B$", "is integral over $B$ and the assumption that $A \\to A'$ is local", "implies that $B'$ is local (Lemma \\ref{lemma-local-tensor-with-integral}).", "On the other hand, $A' \\to B'$ is the localization", "of an \\'etale ring map, hence $B'$ is normal, see", "Algebra, Lemma \\ref{algebra-lemma-normal-goes-up}.", "Thus $B'$ is a (local) normal domain. Finally, we have", "$$", "B/\\mathfrak pB \\subset B \\otimes_A \\kappa(\\mathfrak p)", "= B' \\otimes_{A'} (\\text{fraction field of }A') \\subset", "\\text{fraction field of }B'", "$$", "Hence $B/\\mathfrak pB$ is a domain, which implies that $B$ has a unique", "minimal prime (since by flatness of $A \\to B$ these all have to lie", "over $\\mathfrak p$). Since $A^h$ is a filtered colimit of", "the local rings $B$ it follows that $A^h$ has a unique minimal prime.", "Namely, if $fg = 0$ in $A^h$ for some non-nilpotent elements", "$f, g$, then we can find a $B$ as above containing both $f$ and $g$", "which leads to a contradiction." ], "refs": [ "more-algebra-lemma-branches", "algebra-lemma-flat-going-down", "algebra-lemma-quotient-henselization", "more-algebra-lemma-henselization-reduced", "algebra-lemma-integral-going-up", "algebra-lemma-mop-up", "more-algebra-lemma-local-tensor-with-integral", "algebra-lemma-normal-goes-up" ], "ref_ids": [ 10465, 539, 1301, 10058, 500, 1278, 10460, 1368 ] } ], "ref_ids": [] }, { "id": 10467, "type": "theorem", "label": "more-algebra-lemma-geometric-branches", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-geometric-branches", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a local ring.", "Assume $A$ has finitely many minimal prime ideals.", "Let $A'$ be the integral closure of $A$ in the total ring of fractions", "of $A_{red}$. Choose an algebraic closure $\\overline{\\kappa}$ of", "$\\kappa$ and denote $\\kappa^{sep} \\subset \\overline{\\kappa}$", "the separable algebraic closure of $\\kappa$.", "Let $A^{sh}$ be the strict henselization of $A$", "with respect to $\\kappa^{sep}$.", "Consider the maps", "$$", "\\Spec(A') \\xleftarrow{c} \\Spec((A')^{sh}) \\xrightarrow{e} \\Spec(A^{sh})", "$$", "where $(A')^{sh} = A' \\otimes_A A^{sh}$. Then", "\\begin{enumerate}", "\\item for $\\mathfrak m' \\subset A'$ maximal the residue field", "$\\kappa'$ is algebraic over $\\kappa$ and the fibre of $c$", "over $\\mathfrak m'$ can be canonically identified", "with $\\Hom_\\kappa(\\kappa', \\overline{\\kappa})$,", "\\item the right arrow is bijective on minimal primes,", "\\item every minimal prime of $(A')^{sh}$ is contained in a unique", "maximal ideal and every maximal ideal contains a unique minimal prime.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The proof is almost exactly the same as for Lemma \\ref{lemma-branches}.", "Let $I \\subset A$ be the ideal of nilpotents.", "We have $(A/I)^{sh} = A^{sh}/IA^{sh}$ by", "(Algebra, Lemma \\ref{algebra-lemma-quotient-henselization}).", "The spectra of $A$, $A^{sh}$, $A'$, and $(A')^h$ are the same", "as the spectra of $A/I$, $A^{sh}/IA^{sh}$, $A'$, and", "$(A')^{sh} = A' \\otimes_{A/I} A^{sh}/IA^{sh}$. Thus", "we may replace $A$ by $A_{red} = A/I$ and assume $A$ is reduced.", "Then $A \\subset A'$ which we will use below without further mention.", "\\medskip\\noindent", "Proof of (1). The field extension $\\kappa'/\\kappa$ is algebraic", "because $A'$ is integral over $A$. Since $A'$ is integral over $A$,", "we see that $(A')^{sh}$ is integral over $A^{sh}$. By going up", "(Algebra, Lemma \\ref{algebra-lemma-integral-going-up})", "every maximal ideal of $A'$, resp.\\ $(A')^{sh}$ lies over", "the maximal ideal $\\mathfrak m$, resp.\\ $\\mathfrak m^{sh}$ of", "$A$, resp.\\ $A^h$. We have", "$$", "(A')^{sh} \\otimes_{A^{sh}} \\kappa^{sep} =", "A' \\otimes_A A^h \\otimes_{A^h} \\kappa^{sep} =", "(A' \\otimes_A \\kappa) \\otimes_{\\kappa} \\kappa^{sep}", "$$", "because the residue field of $A^{sh}$ is $\\kappa^{sep}$.", "Thus the fibre of $c$ over $\\mathfrak m'$ is the spectrum", "of $\\kappa' \\otimes_\\kappa \\kappa^{sep}$.", "We conclude (1) is true because there is a bijection", "$$", "\\Hom_\\kappa(\\kappa', \\overline{\\kappa}) \\to", "\\Spec(\\kappa' \\otimes_\\kappa \\kappa^{sep}),\\quad", "\\sigma \\mapsto \\Ker(", "\\sigma \\otimes 1 : \\kappa' \\otimes_\\kappa \\kappa^{sep} \\to \\overline{\\kappa}", ")", "$$", "We will use below that the displayed ring is integral over a field", "hence spectrum of this ring is a profinite space, see", "Algebra, Lemmas \\ref{algebra-lemma-integral-over-field} and", "\\ref{algebra-lemma-ring-with-only-minimal-primes}.", "\\medskip\\noindent", "Proof of (3). The ring $A'$ is a normal ring and in fact a", "finite product of normal domains, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-reduced-ring-normal}.", "Since $A^{sh}$ is a filtered colimit of \\'etale $A$-algebras,", "$(A')^{sh}$ is filtered colimit of \\'etale $A'$-algebras", "hence $(A')^{sh}$ is a normal ring by", "Algebra, Lemmas \\ref{algebra-lemma-normal-goes-up} and", "\\ref{algebra-lemma-colimit-normal-ring}.", "Thus every local ring of $(A')^{sh}$ is a normal domain", "and we see that every maximal ideal contains a unique minimal prime.", "By Lemma \\ref{lemma-integral-over-henselian-pair} applied", "to $A^{sh} \\to (A')^{sh}$ to see that $((A')^{sh}, \\mathfrak m(A')^{sh})$", "is a henselian pair. If $\\mathfrak q \\subset (A')^{sh}$", "is a minimal prime (or any prime), then the intersection of", "$V(\\mathfrak q)$ with $V(\\mathfrak m (A')^{sh})$ is connected", "by Lemma \\ref{lemma-irreducible-henselian-pair-connected}", "Since $V(\\mathfrak m (A')^{sh}) = \\Spec((A')^{sh} \\otimes \\kappa^{sh})$", "is a profinite space by we see there is a unique", "maximal ideal containing $\\mathfrak q$.", "\\medskip\\noindent", "Proof of (2). The minimal primes of $A'$ are exactly the primes", "lying over a minimal prime of $A$ (by construction).", "Since $A' \\to (A')^{sh}$ is flat by going down", "(Algebra, Lemma \\ref{algebra-lemma-flat-going-down})", "every minimal prime of $(A')^{sh}$ lies over a minimal prime", "of $A'$. Conversely, any prime of $(A')^{sh}$ lying over a", "minimal prime of $A'$ is minimal because $(A')^{sh}$ is a filtered", "colimit of \\'etale hence quasi-finite algebras over $A'$ (small detail omitted).", "We conclude that the minimal primes of $(A')^{sh}$", "are exactly the primes which lie over a minimal prime of $A$.", "Similarly, the minimal primes of $A^{sh}$ are exactly the", "primes lying over minimal primes of $A$.", "By construction we have $A' \\otimes_A Q(A) = Q(A)$ where $Q(A)$", "is the total fraction ring of our reduced local ring $A$.", "Of course $Q(A)$ is the finite product of residue fields of", "the minimal primes of $A$. It follows that", "$$", "(A')^{sh} \\otimes_A Q(A) =", "A^{sh} \\otimes_A A' \\otimes_A Q(A) = A^{sh} \\otimes_A Q(A)", "$$", "Our discussion above shows the spectrum of the ring", "on the left is the set of minimal primes of $(A')^{sh}$", "and the spectrum of the ring on the right is the", "is the set of minimal primes of $A^{sh}$. This finishes the proof." ], "refs": [ "more-algebra-lemma-branches", "algebra-lemma-quotient-henselization", "algebra-lemma-integral-going-up", "algebra-lemma-integral-over-field", "algebra-lemma-ring-with-only-minimal-primes", "algebra-lemma-characterize-reduced-ring-normal", "algebra-lemma-normal-goes-up", "algebra-lemma-colimit-normal-ring", "more-algebra-lemma-integral-over-henselian-pair", "more-algebra-lemma-irreducible-henselian-pair-connected", "algebra-lemma-flat-going-down" ], "ref_ids": [ 10465, 1301, 500, 497, 426, 515, 1368, 516, 9863, 9870, 539 ] } ], "ref_ids": [] }, { "id": 10468, "type": "theorem", "label": "more-algebra-lemma-geometrically-unibranch", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-geometrically-unibranch", "contents": [ "\\begin{reference}", "\\cite[Lemma 2.2]{Etale-coverings} and", "\\cite[Chapter IV Proposition 18.8.15]{EGA4}", "\\end{reference}", "Let $A$ be a local ring. Let $A^{sh}$ be a strict henselization of $A$.", "The following are equivalent", "\\begin{enumerate}", "\\item $A$ is geometrically unibranch, and", "\\item $A^{sh}$ has a unique minimal prime.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-geometric-branches}", "but we will also give a direct proof; this direct", "proof is almost exactly the same as the direct proof of", "Lemma \\ref{lemma-unibranch}.", "Denote $\\mathfrak m$ the maximal ideal of the ring $A$.", "Denote $\\kappa$, $\\kappa^{sh}$ the residue field of $A$, $A^{sh}$.", "\\medskip\\noindent", "Assume (2). Let $\\mathfrak p^{sh}$ be the unique minimal prime of", "$A^{sh}$. The flatness of $A \\to A^{sh}$ implies that", "$\\mathfrak p = A \\cap \\mathfrak p^{sh}$ is the unique minimal", "prime of $A$ (by going down, see", "Algebra, Lemma \\ref{algebra-lemma-flat-going-down}).", "Also, since $A^{sh}/\\mathfrak pA^{sh} = (A/\\mathfrak p)^{sh}$ (see", "Algebra, Lemma \\ref{algebra-lemma-quotient-strict-henselization})", "is reduced by Lemma \\ref{lemma-henselization-reduced}", "we see that $\\mathfrak p^{sh} = \\mathfrak pA^{sh}$.", "Let $A'$ be the integral closure of $A/\\mathfrak p$ in its fraction", "field. We have to show that $A'$ is local and that its residue", "field is purely inseparable over $\\kappa$.", "Since $A \\to A'$ is integral, every maximal ideal of $A'$ lies over", "$\\mathfrak m$ (by going up for integral ring maps, see", "Algebra, Lemma \\ref{algebra-lemma-integral-going-up}).", "If $A'$ is not local, then we can find distinct maximal ideals", "$\\mathfrak m_1$, $\\mathfrak m_2$. Choosing elements $f_1, f_2 \\in A'$", "with $f_i \\in \\mathfrak m_i, f_i \\not \\in \\mathfrak m_{3 - i}$ we find", "a finite subalgebra $B = A[f_1, f_2] \\subset A'$ with distinct maximal", "ideals $B \\cap \\mathfrak m_i$, $i = 1, 2$. If $A'$ is local with maximal", "ideal $\\mathfrak m'$, but $A/\\mathfrak m \\subset A'/\\mathfrak m'$", "is not purely inseparable, then we can find $f \\in A'$ whose image in", "$A'/\\mathfrak m'$ generates a finite, not purely inseparable extension", "of $A/\\mathfrak m$ and we find a finite local subalgebra $B = A[f] \\subset A'$", "whose residue field is not a purely inseparable extension of $A/\\mathfrak m$.", "Note that the inclusions", "$$", "A/\\mathfrak p \\subset B \\subset \\kappa(\\mathfrak p)", "$$", "give, on tensoring with the flat ring map $A \\to A^{sh}$ the inclusions", "$$", "A^{sh}/\\mathfrak p^{sh} \\subset", "B \\otimes_A A^{sh} \\subset", "\\kappa(\\mathfrak p) \\otimes_A A^{sh} \\subset", "\\kappa(\\mathfrak p^{sh})", "$$", "the last inclusion because", "$\\kappa(\\mathfrak p) \\otimes_A A^{sh} =", "\\kappa(\\mathfrak p) \\otimes_{A/\\mathfrak p} A^{sh}/\\mathfrak p^{sh}$", "is a localization of the domain $A^{sh}/\\mathfrak p^{sh}$.", "Note that $B \\otimes_A \\kappa^{sh}$ has at least two maximal ideals", "because $B/\\mathfrak mB$ either has two maximal ideals or one whose", "residue field is not purely inseparable over $\\kappa$, and because", "$\\kappa^{sh}$ is separably algebraically closed. Hence, as", "$A^{sh}$ is strictly henselian we see that", "$B \\otimes_A A^{sh}$ is a product of $\\geq 2$ local rings, see", "Algebra, Lemma \\ref{algebra-lemma-mop-up-strictly-henselian}.", "But we've just seen that $B \\otimes_A A^{sh}$ is a subring of a domain", "and we get a contradiction.", "\\medskip\\noindent", "Assume (1). Let $\\mathfrak p \\subset A$ be the unique minimal", "prime and let $A'$ be the integral closure of $A/\\mathfrak p$", "in its fraction field. Let $A \\to B$ be a local map of local rings which is a", "localization of an \\'etale $A$-algebra. In particular $\\mathfrak m_B$", "is the unique prime containing $\\mathfrak m_AB$. Then $B' = A' \\otimes_A B$", "is integral over $B$ and the assumption that $A \\to A'$ is local", "with purely inseparable residue field extension implies that $B'$", "is local (Lemma \\ref{lemma-local-tensor-with-integral}).", "On the other hand, $A' \\to B'$ is the localization", "of an \\'etale ring map, hence $B'$ is normal, see", "Algebra, Lemma \\ref{algebra-lemma-normal-goes-up}.", "Thus $B'$ is a (local) normal domain. Finally, we have", "$$", "B/\\mathfrak pB \\subset B \\otimes_A \\kappa(\\mathfrak p)", "= B' \\otimes_{A'} (\\text{fraction field of }A') \\subset", "\\text{fraction field of }B'", "$$", "Hence $B/\\mathfrak pB$ is a domain, which implies that $B$ has a unique", "minimal prime (since by flatness of $A \\to B$ these all have to lie", "over $\\mathfrak p$). Since $A^{sh}$ is a filtered colimit of", "the local rings $B$ it follows that $A^{sh}$ has a unique minimal prime.", "Namely, if $fg = 0$ in $A^{sh}$ for some non-nilpotent elements", "$f, g$, then we can find a $B$ as above containing both $f$ and $g$", "which leads to a contradiction." ], "refs": [ "more-algebra-lemma-geometric-branches", "more-algebra-lemma-unibranch", "algebra-lemma-flat-going-down", "algebra-lemma-quotient-strict-henselization", "more-algebra-lemma-henselization-reduced", "algebra-lemma-integral-going-up", "algebra-lemma-mop-up-strictly-henselian", "more-algebra-lemma-local-tensor-with-integral", "algebra-lemma-normal-goes-up" ], "ref_ids": [ 10467, 10466, 539, 1307, 10058, 500, 1279, 10460, 1368 ] } ], "ref_ids": [] }, { "id": 10469, "type": "theorem", "label": "more-algebra-lemma-number-of-branches-1", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-number-of-branches-1", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a local ring.", "\\begin{enumerate}", "\\item If $A$ has infinitely many minimal prime ideals, then", "the number of (geometric) branches of $A$ is $\\infty$.", "\\item The number of branches of $A$ is $1$ if and only if", "$A$ is unibranch.", "\\item The number of geometric branches of $A$ is $1$ if and only if", "$A$ is geometrically unibranch.", "\\end{enumerate}", "Assume $A$ has finitely many minimal primes and let $A'$ be the", "integral closure of $A$ in the total ring of fractions of $A_{red}$.", "Then", "\\begin{enumerate}", "\\item[(4)] the number of branches of $A$ is the number of maximal ideals", "$\\mathfrak m'$ of $A'$,", "\\item[(5)] to get the number of geometric branches of $A$ we have to count", "each maximal ideal $\\mathfrak m'$ of $A'$ with multiplicity given by the", "separable degree of $\\kappa(\\mathfrak m')/\\kappa$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma follows immediately from the definitions,", "Lemma \\ref{lemma-branches},", "Lemma \\ref{lemma-geometric-branches}, and", "Fields, Lemma \\ref{fields-lemma-separable-degree}." ], "refs": [ "more-algebra-lemma-branches", "more-algebra-lemma-geometric-branches", "fields-lemma-separable-degree" ], "ref_ids": [ 10465, 10467, 4483 ] } ], "ref_ids": [] }, { "id": 10470, "type": "theorem", "label": "more-algebra-lemma-invariance-number-branches-smooth", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-invariance-number-branches-smooth", "contents": [ "Let $A \\to B$ be a local homomorphism of local rings which", "is the localization of a smooth ring map.", "\\begin{enumerate}", "\\item The number of geometric branches of $A$ is equal to the number", "of geometric branches of $B$.", "\\item If $A \\to B$ induces a purely inseparable extension of residue fields,", "then the number of branches of $A$ is the number of branches of $B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use that smooth ring maps are flat", "(Algebra, Lemma \\ref{algebra-lemma-smooth-syntomic}),", "that localizations are flat (Algebra, Lemma", "\\ref{algebra-lemma-flat-localization}),", "that compositions of flat ring maps are flat", "(Algebra, Lemma \\ref{algebra-lemma-composition-flat}),", "that base change of a flat ring map is flat", "(Algebra, Lemma \\ref{algebra-lemma-flat-base-change}),", "that flat local homomorphisms are faithfully flat", "(Algebra, Lemma \\ref{algebra-lemma-local-flat-ff}),", "that (strict) henselization is flat", "(Lemma \\ref{lemma-dumb-properties-henselization}), and", "Going down for flat ring maps", "(Algebra, Lemma \\ref{algebra-lemma-flat-going-down}).", "\\medskip\\noindent", "Proof of (2). Let $A^h$, $B^h$ be the henselizations of $A$, $B$. Then $B^h$", "is the henselization of $A^h \\otimes_A B$ at the unique", "maximal ideal lying over $\\mathfrak m_B$, see", "Algebra, Lemma \\ref{algebra-lemma-henselian-functorial-improve}.", "Thus we may and do assume $A$ is henselian.", "Since $A \\to B \\to B^h$ is flat, every minimal prime of $B^h$", "lies over a minimal prime of $A$ and since $A \\to B^h$ is faithfully flat,", "every minimal prime of $A$ does lie under a minimal prime of $B^h$;", "in both cases use going down for flat ring maps.", "Therefore it suffices to show that given a minimal prime", "$\\mathfrak p \\subset A$, there is at most one minimal prime", "of $B^h$ lying over $\\mathfrak p$.", "After replacing $A$ by $A/\\mathfrak p$ and $B$ by $B/\\mathfrak p B$", "we may assume that $A$ is a domain; the $A$ is still henselian by", "Algebra, Lemma \\ref{algebra-lemma-quotient-henselization}.", "By Lemma \\ref{lemma-unibranch} we see that the integral closure", "$A'$ of $A$ in its field of fractions is a local domain.", "Of course $A'$ is a normal domain. By", "Algebra, Lemma \\ref{algebra-lemma-normal-goes-up}", "we see that $A' \\otimes_A B^h$ is a normal ring", "(the lemma just gives it for $A' \\otimes_A B$, to go up to", "$A' \\otimes_A B^h$ use that $B^h$ is a colimit of \\'etale", "$B$-algebras and use Algebra, Lemma \\ref{algebra-lemma-colimit-normal-ring}).", "By Lemma \\ref{lemma-local-tensor-with-integral}", "we see that $A' \\otimes_A B^h$ is local (this is where we", "use the assumption on the residue fields of $A$ and $B$).", "Hence $A' \\otimes_A B^h$ is a local normal ring, hence a local domain.", "Since $B^h \\subset A' \\otimes_A B^h$ by flatness of $A \\to B^h$", "we conclude that $B^h$ is a domain as desired.", "\\medskip\\noindent", "Proof of (1). Let $A^{sh}$, $B^{sh}$ be strict henselizations", "of $A$, $B$. Then $B^{sh}$ is a strict henselization of", "$A^h \\otimes_A B$ at a maximal ideal lying over $\\mathfrak m_B$", "and $\\mathfrak m_{A^h}$, see", "Algebra, Lemma \\ref{algebra-lemma-strictly-henselian-functorial-improve}.", "Thus we may and do assume $A$ is strictly henselian.", "Since $A \\to B \\to B^{sh}$ is flat, every minimal prime of $B^{sh}$", "lies over a minimal prime of $A$ and since $A \\to B^{sh}$ is faithfully flat,", "every minimal prime of $A$ does lie under a minimal prime of $B^{sh}$;", "in both cases use going down for flat ring maps.", "Therefore it suffices to show that given a minimal prime", "$\\mathfrak p \\subset A$, there is at most one minimal prime", "of $B^{sh}$ lying over $\\mathfrak p$.", "After replacing $A$ by $A/\\mathfrak p$ and $B$ by $B/\\mathfrak p B$", "we may assume that $A$ is a domain; then $A$ is still strictly henselian by", "Algebra, Lemma \\ref{algebra-lemma-quotient-strict-henselization}.", "By Lemma \\ref{lemma-geometrically-unibranch} we see that the integral closure", "$A'$ of $A$ in its field of fractions is a local domain whose residue", "field is a purely inseparable extension of the residue field of $A$.", "Of course $A'$ is a normal domain. By", "Algebra, Lemma \\ref{algebra-lemma-normal-goes-up}", "we see that $A' \\otimes_A B^{sh}$ is a normal ring", "(the lemma just gives it for $A' \\otimes_A B$, to go up to", "$A' \\otimes_A B^{sh}$ use that $B^{sh}$ is a colimit of \\'etale", "$B$-algebras and use Algebra, Lemma \\ref{algebra-lemma-colimit-normal-ring}).", "By Lemma \\ref{lemma-local-tensor-with-integral}", "we see that $A' \\otimes_A B^{sh}$ is local (since $A \\subset A'$", "induces a purely inseparable residue field extension).", "Hence $A' \\otimes_A B^{sh}$ is a local normal ring, hence a local domain.", "Since $B^{sh} \\subset A' \\otimes_A B^{sh}$ by flatness of $A \\to B^{sh}$", "we conclude that $B^{sh}$ is a domain as desired." ], "refs": [ "algebra-lemma-smooth-syntomic", "algebra-lemma-flat-localization", "algebra-lemma-composition-flat", "algebra-lemma-flat-base-change", "algebra-lemma-local-flat-ff", "more-algebra-lemma-dumb-properties-henselization", "algebra-lemma-flat-going-down", "algebra-lemma-henselian-functorial-improve", "algebra-lemma-quotient-henselization", "more-algebra-lemma-unibranch", "algebra-lemma-normal-goes-up", "algebra-lemma-colimit-normal-ring", "more-algebra-lemma-local-tensor-with-integral", "algebra-lemma-strictly-henselian-functorial-improve", "algebra-lemma-quotient-strict-henselization", "more-algebra-lemma-geometrically-unibranch", "algebra-lemma-normal-goes-up", "algebra-lemma-colimit-normal-ring", "more-algebra-lemma-local-tensor-with-integral" ], "ref_ids": [ 1195, 538, 524, 527, 537, 10055, 539, 1299, 1301, 10466, 1368, 516, 10460, 1305, 1307, 10468, 1368, 516, 10460 ] } ], "ref_ids": [] }, { "id": 10471, "type": "theorem", "label": "more-algebra-lemma-minimal-primes-tensor-strictly-henselian", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-minimal-primes-tensor-strictly-henselian", "contents": [ "Let $k$ be an algebraically closed field. Let $A$, $B$ be strictly", "henselian local $k$-algebras with residue field equal to $k$.", "Let $C$ be the strict henselization of $A \\otimes_k B$ at the maximal", "ideal $\\mathfrak m_A \\otimes_k B + A \\otimes_k \\mathfrak m_B$.", "Then the minimal primes of $C$ correspond $1$-to-$1$ to pairs of", "minimal primes of $A$ and $B$." ], "refs": [], "proofs": [ { "contents": [ "First note that a minimal prime $\\mathfrak r$ of $C$ maps to a minimal", "prime $\\mathfrak p$ in $A$ and to a minimal prime $\\mathfrak q$ of $B$", "because the ring maps $A \\to C$ and $B \\to C$ are flat (by going down for", "flat ring map Algebra, Lemma \\ref{algebra-lemma-flat-going-down}).", "Hence it suffices to show that the strict henselization of", "$(A/\\mathfrak p \\otimes_k B/\\mathfrak q)_{", "\\mathfrak m_A \\otimes_k B + A \\otimes_k \\mathfrak m_B}$", "has a unique minimal prime ideal. By", "Algebra, Lemma \\ref{algebra-lemma-quotient-strict-henselization}", "the rings $A/\\mathfrak p$, $B/\\mathfrak q$ are strictly henselian.", "Hence we may assume that $A$ and $B$ are strictly henselian", "local domains and our goal is to show that $C$ has a unique minimal prime.", "By Lemma \\ref{lemma-geometrically-unibranch} the", "integral closure $A'$ of $A$ in its fraction field", "is a normal local domain with residue field $k$. Similarly for the", "integral closure $B'$ of $B$ into its fraction field. By", "Algebra, Lemma \\ref{algebra-lemma-geometrically-normal-tensor-normal}", "we see that $A' \\otimes_k B'$ is a normal ring. Hence its localization", "$$", "R = (A' \\otimes_k B')_{", "\\mathfrak m_{A'} \\otimes_k B' + A' \\otimes_k \\mathfrak m_{B'}}", "$$", "is a normal local domain. Note that $A \\otimes_k B \\to A' \\otimes_k B'$", "is integral (hence gong up holds --", "Algebra, Lemma \\ref{algebra-lemma-integral-going-up})", "and that $\\mathfrak m_{A'} \\otimes_k B' + A' \\otimes_k \\mathfrak m_{B'}$", "is the unique maximal ideal of $A' \\otimes_k B'$", "lying over $\\mathfrak m_A \\otimes_k B + A \\otimes_k \\mathfrak m_B$.", "Hence we see that", "$$", "R = (A' \\otimes_k B')_{", "\\mathfrak m_A \\otimes_k B + A \\otimes_k \\mathfrak m_B}", "$$", "by", "Algebra, Lemma \\ref{algebra-lemma-unique-prime-over-localize-below}.", "It follows that", "$$", "(A \\otimes_k B)_{", "\\mathfrak m_A \\otimes_k B + A \\otimes_k \\mathfrak m_B}", "\\longrightarrow", "R", "$$", "is integral. We conclude that $R$ is the integral closure of", "$(A \\otimes_k B)_{", "\\mathfrak m_A \\otimes_k B + A \\otimes_k \\mathfrak m_B}$", "in its fraction field, and by", "Lemma \\ref{lemma-geometrically-unibranch}", "once again we conclude that $C$ has a unique prime ideal." ], "refs": [ "algebra-lemma-flat-going-down", "algebra-lemma-quotient-strict-henselization", "more-algebra-lemma-geometrically-unibranch", "algebra-lemma-geometrically-normal-tensor-normal", "algebra-lemma-integral-going-up", "algebra-lemma-unique-prime-over-localize-below", "more-algebra-lemma-geometrically-unibranch" ], "ref_ids": [ 539, 1307, 10468, 1380, 500, 556, 10468 ] } ], "ref_ids": [] }, { "id": 10472, "type": "theorem", "label": "more-algebra-lemma-nr-branches-completion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-nr-branches-completion", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "\\begin{enumerate}", "\\item The map $A^h \\to A^\\wedge$ defines a surjective map from minimal", "primes of $A^\\wedge$ to minimal primes of $A^h$.", "\\item The number of branches of $A$ is at most the number of branches", "of $A^\\wedge$.", "\\item The number of geometric branches of $A$ is at most the number", "of geometric branches of $A^\\wedge$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-henselization-noetherian} the map $A^h \\to A^\\wedge$", "is flat and injective. Combining going down", "(Algebra, Lemma \\ref{algebra-lemma-flat-going-down}) and", "Algebra, Lemma \\ref{algebra-lemma-injective-minimal-primes-in-image}", "we see that part (1) holds. Part (2) follows from this,", "Definition \\ref{definition-number-of-branches}, and", "the fact that $A^\\wedge$ is henselian", "(Algebra, Lemma \\ref{algebra-lemma-complete-henselian}).", "By Lemma \\ref{lemma-henselization-noetherian}", "we have $(A^\\wedge)^{sh} = A^{sh} \\otimes_{A^h} A^\\wedge$.", "Thus we can repeat the arguments above using the flat injective", "map $A^{sh} \\to (A^\\wedge)^{sh}$ to prove (3)." ], "refs": [ "more-algebra-lemma-henselization-noetherian", "algebra-lemma-flat-going-down", "algebra-lemma-injective-minimal-primes-in-image", "more-algebra-definition-number-of-branches", "algebra-lemma-complete-henselian", "more-algebra-lemma-henselization-noetherian" ], "ref_ids": [ 10057, 539, 445, 10638, 1282, 10057 ] } ], "ref_ids": [] }, { "id": 10473, "type": "theorem", "label": "more-algebra-lemma-equal-nr-branches-completion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-equal-nr-branches-completion", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring. The number of", "branches of $A$ is the same as the number of branches of $A^\\wedge$", "if and only if $\\sqrt{\\mathfrak qA^\\wedge}$ is prime for every", "minimal prime $\\mathfrak q \\subset A^h$ of the henselization." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-nr-branches-completion}", "and the fact that there are only a finite number of branches", "for both $A$ and $A^\\wedge$ by", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-irreducible-components}", "and the fact that $A^h$ and $A^\\wedge$ are Noetherian", "(Lemma \\ref{lemma-henselization-noetherian})." ], "refs": [ "more-algebra-lemma-nr-branches-completion", "algebra-lemma-Noetherian-irreducible-components", "more-algebra-lemma-henselization-noetherian" ], "ref_ids": [ 10472, 453, 10057 ] } ], "ref_ids": [] }, { "id": 10474, "type": "theorem", "label": "more-algebra-lemma-glueing-sum-components-open", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-glueing-sum-components-open", "contents": [ "Let $A$ be a ring and let $I$ be a finitely generated ideal.", "Let $A \\to C$ be a ring map such that for all $f \\in I$", "the ring map $A_f \\to C_f$ is localization at an idempotent.", "Then there exists a surjection $A \\to C'$ such that", "$A_f \\to (C \\times C')_f$ is an isomorphism for all $f \\in I$." ], "refs": [], "proofs": [ { "contents": [ "Choose generators $f_1, \\ldots, f_r$ of $I$. Write", "$$", "C_{f_i} = (A_{f_i})_{e_i}", "$$", "for some idempotent $e_i \\in A_{f_i}$. Write $e_i = a_i/f_i^n$", "for some $a_i \\in A$ and $n \\geq 0$; we may use the same $n$", "for all $i = 1, \\ldots, r$. After replacing $a_i$ by $f_i^ma_i$", "and $n$ by $n + m$ for a suitable $m \\gg 0$, we may assume", "$a_i^2 = f_i^n a_i$ for all $i$. Since $e_i$ maps to $1$ in", "$C_{f_if_j} = (A_{f_if_j})_{e_j} = A_{f_if_ja_j}$ we see that", "$$", "(f_if_ja_j)^N(f_j^n a_i - f_i^na_j) = 0", "$$", "for some $N$ (we can pick the same $N$ for all pairs $i, j$). Using", "$a_j^2 = f_j^na_j$ this gives", "$$", "f_i^{N + n} f_j^{N + nN} a_j = f_i^N f_j^{N + n} a_ia_j^N", "$$", "After increasing $n$ to $n + N + nN$ and replacing $a_i$ by $f_i^{N + nN}a_i$", "we see that $f_i^n a_j$ is in the ideal of $a_i$ for all pairs $i, j$.", "Let $C' = A/(a_1, \\ldots, a_r)$. Then", "$$", "C'_{f_i} = A_{f_i}/(a_i) = A_{f_i}/(e_i)", "$$", "because $a_j$ is in the ideal generated by $a_i$ after inverting $f_i$.", "Since for an idempotent $e$ of a ring $B$ we have $B = B_e \\times B/(e)$", "we see that the conclusion of the lemma holds for $f$ equal to one", "of $f_1, \\ldots, f_r$. Using glueing of functions, in the form of", "Algebra, Lemma \\ref{algebra-lemma-cover},", "we conclude that the result holds for all $f \\in I$.", "Namely, for $f \\in I$ the elements $f_1, \\ldots, f_r$", "generate the unit ideal in $A_f$ so $A_f \\to (C \\times C')_f$", "is an isomorphism if and only if this is the case after localizing", "at $f_1, \\ldots, f_r$." ], "refs": [ "algebra-lemma-cover" ], "ref_ids": [ 411 ] } ], "ref_ids": [] }, { "id": 10475, "type": "theorem", "label": "more-algebra-lemma-quotient-by-idempotent", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-quotient-by-idempotent", "contents": [ "Let $A$ be a Noetherian ring and $I$ an ideal. Let $B$", "be a finite type $A$-algebra. Let $B^\\wedge \\to C$ be a surjective", "ring map with kernel $J$ where $B^\\wedge$ is the $I$-adic completion.", "If $J/J^2$ is annihilated by $I^c$ for some $c \\geq 0$, then $C$ is", "isomorphic to the completion of a finite type $A$-algebra." ], "refs": [], "proofs": [ { "contents": [ "Let $f \\in I$. Since $B^\\wedge$ is Noetherian (Algebra, Lemma", "\\ref{algebra-lemma-completion-Noetherian-Noetherian}),", "we see that $J$ is a finitely generated ideal.", "Hence we conclude from", "Algebra, Lemma \\ref{algebra-lemma-ideal-is-squared-union-connected}", "that", "$$", "C_f = ((B^\\wedge)_f)_e", "$$", "for some idempotent $e \\in (B^\\wedge)_f$. By", "Lemma \\ref{lemma-glueing-sum-components-open}", "we can find a surjection $B^\\wedge \\to C'$", "such that $B^\\wedge \\to C \\times C'$ becomes an", "isomorphism after inverting any $f \\in I$.", "Observe that $C \\times C'$ is a finite $B^\\wedge$-algebra.", "\\medskip\\noindent", "Choose generators $f_1, \\ldots, f_r \\in I$. Denote", "$\\alpha_i : (C \\times C')_{f_i} \\to B_{f_i} \\otimes_B B^\\wedge$", "the inverse of the isomorphism of $(B^\\wedge)_{f_i}$-algebras", "we obtained above. Denote $\\alpha_{ij} : (B_{f_i})_{f_j} \\to (B_{f_j})_{f_i}$", "the obvious $B$-algebra isomorphism.", "Consider the object", "$$", "(C \\times C', B_{f_i}, \\alpha_i, \\alpha_{ij})", "$$", "of the category $\\text{Glue}(B \\to B^\\wedge, f_1, \\ldots, f_r)$ introduced", "in Remark \\ref{remark-glueing-data}. We omit the verification of", "conditions (1)(a) and (1)(b).", "Since $B \\to B^\\wedge$ is a flat map", "(Algebra, Lemma \\ref{algebra-lemma-completion-flat}) inducing an isomorphism", "$B/IB \\to B^\\wedge/IB^\\wedge$ we may apply", "Proposition \\ref{proposition-equivalence}", "and Remark \\ref{remark-formal-glueing-algebras}.", "We conclude that $C \\times C'$ is isomorphic to $D \\otimes_B B^\\wedge$", "for some finite $B$-algebra $D$.", "Then $D/ID \\cong C/IC \\times C'/IC'$. Let $\\overline{e} \\in D/ID$", "be the idempotent corresponding to the factor $C/IC$.", "By Lemma \\ref{lemma-lift-idempotent-upstairs} there exists an", "\\'etale ring map $B \\to B'$ which induces an isomorphism", "$B/IB \\to B'/IB'$ such that $D' = D \\otimes_B B'$ contains an", "idempotent $e$ lifting $\\overline{e}$. Since $C \\times C'$", "is $I$-adically complete the pair $(C \\times C', IC \\times IC')$", "is henselian (Lemma \\ref{lemma-complete-henselian}).", "Thus we can factor the map $B \\to C \\times C'$ through $B'$.", "Doing so we may replace $B$ by $B'$ and $D$ by $D'$. Then", "we find that $D = D_e \\times D_{1 - e} = D/(1 - e) \\times D/(e)$", "is a product of finite type $A$-algebras and the completion of the", "first part is $C$ and the completion of the second part is $C'$." ], "refs": [ "algebra-lemma-completion-Noetherian-Noetherian", "algebra-lemma-ideal-is-squared-union-connected", "more-algebra-lemma-glueing-sum-components-open", "more-algebra-remark-glueing-data", "algebra-lemma-completion-flat", "more-algebra-proposition-equivalence", "more-algebra-remark-formal-glueing-algebras", "more-algebra-lemma-lift-idempotent-upstairs", "more-algebra-lemma-complete-henselian" ], "ref_ids": [ 874, 407, 10474, 10662, 870, 10587, 10663, 9848, 9859 ] } ], "ref_ids": [] }, { "id": 10476, "type": "theorem", "label": "more-algebra-lemma-one-dimensional-formal-branch", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-one-dimensional-formal-branch", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring with henselization $A^h$.", "Let $\\mathfrak q \\subset A^\\wedge$ be a minimal prime with", "$\\dim(A^\\wedge/\\mathfrak q) = 1$. Then there exists a minimal", "prime $\\mathfrak q^h$ of $A^h$ such that", "$\\mathfrak q = \\sqrt{\\mathfrak q^hA^\\wedge}$." ], "refs": [], "proofs": [ { "contents": [ "Since the completion of $A$ and $A^h$ are the same, we may assume", "that $A$ is henselian (Lemma \\ref{lemma-henselization-noetherian}).", "We will apply Lemma \\ref{lemma-quotient-by-idempotent}", "to $A^\\wedge \\to A^\\wedge/J$ where", "$J = \\Ker(A^\\wedge \\to (A^\\wedge)_{\\mathfrak q})$.", "Since $\\dim((A^\\wedge)_\\mathfrak q) = 0$ we see that", "$\\mathfrak q^n \\subset J$ for some $n$. Hence $J/J^2$ is", "annihilated by $\\mathfrak q^n$. On the other hand $(J/J^2)_\\mathfrak q = 0$", "because $J_\\mathfrak q = 0$. Hence $\\mathfrak m$ is the only", "associated prime of $J/J^2$ and we find that a power", "of $\\mathfrak m$ annihilates $J/J^2$. Thus the lemma applies", "and we find that $A^\\wedge/J = C^\\wedge$ for some finite type", "$A$-algebra $C$.", "\\medskip\\noindent", "Then $C/\\mathfrak m C = A/\\mathfrak m$ because $A^\\wedge/J$ has the same", "property. Hence $\\mathfrak m_C = \\mathfrak m C$ is a maximal ideal", "and $A \\to C$ is unramified at $\\mathfrak m_C$", "(Algebra, Lemma \\ref{algebra-lemma-characterize-unramified}).", "After replacing $C$ by a principal localization we may", "assume that $C$ is a quotient of an \\'etale $A$-algebra $B$, see", "Algebra, Proposition \\ref{algebra-proposition-unramified-locally-standard}.", "However, since the residue field extension of $A \\to C_{\\mathfrak m_C}$", "is trivial and $A$ is henselian, we conclude that $B = A$", "again after a localization.", "Thus $C = A/I$ for some ideal $I \\subset A$ and it follows that", "$J = IA^\\wedge$ (because completion is exact in our situation by", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}) and $I = J \\cap A$", "(by flatness of $A \\to A^\\wedge$). Since", "$\\mathfrak q^n \\subset J \\subset \\mathfrak q$ we see that", "$\\mathfrak p = \\mathfrak q \\cap A$ satisfies", "$\\mathfrak p^n \\subset I \\subset \\mathfrak p$.", "Then $\\sqrt{\\mathfrak p A^\\wedge} = \\mathfrak q$ and the proof is complete." ], "refs": [ "more-algebra-lemma-henselization-noetherian", "more-algebra-lemma-quotient-by-idempotent", "algebra-lemma-characterize-unramified", "algebra-proposition-unramified-locally-standard", "algebra-lemma-completion-flat" ], "ref_ids": [ 10057, 10475, 1270, 1428, 870 ] } ], "ref_ids": [] }, { "id": 10477, "type": "theorem", "label": "more-algebra-lemma-completion-disconnected", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-completion-disconnected", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring. The punctured spectrum", "of $A^\\wedge$ is disconnected if and only if the punctured spectrum of $A^h$", "is disconnected." ], "refs": [], "proofs": [ { "contents": [ "Since the completion of $A$ and $A^h$ are the same, we may assume", "that $A$ is henselian (Lemma \\ref{lemma-henselization-noetherian}).", "\\medskip\\noindent", "Since $A \\to A^\\wedge$ is faithfully flat (see reference just given)", "the map from the punctured spectrum of $A^\\wedge$ to the punctured", "spectrum of $A$ is surjective", "(see Algebra, Lemma \\ref{algebra-lemma-ff-rings}).", "Hence if the punctured spectrum of $A$ is disconnected, then", "the same is true for $A^\\wedge$.", "\\medskip\\noindent", "Assume the punctured spectrum of $A^\\wedge$ is disconnected.", "This means that", "$$", "\\Spec(A^\\wedge) \\setminus \\{\\mathfrak m^\\wedge\\} = Z \\amalg Z'", "$$", "with $Z$ and $Z'$ closed. Let", "$\\overline{Z}, \\overline{Z}' \\subset \\Spec(A^\\wedge)$ be the closures.", "Say $\\overline{Z} = V(J)$, $\\overline{Z}' = V(J')$ for some ideals", "$J, J' \\subset A^\\wedge$. Then $V(J + J') = \\{\\mathfrak m^\\wedge\\}$", "and $V(JJ') = \\Spec(A^\\wedge)$. The first equality means that", "$\\mathfrak m^\\wedge = \\sqrt{J + J'}$ which implies", "$(\\mathfrak m^\\wedge)^e \\subset J + J'$ for some $e \\geq 1$.", "The second equality implies every element", "of $JJ'$ is nilpotent hence $(JJ')^n = 0$ for some $n \\geq 1$.", "Combined this means that $J^n/J^{2n}$ is annihilated by", "$J^n$ and $(J')^n$ and hence by $(\\mathfrak m^\\wedge)^{2en}$.", "Thus we may apply Lemma \\ref{lemma-quotient-by-idempotent}", "to see that there is a finite type $A$-algebra $C$ and an", "isomorphism $A^\\wedge/J^n = C^\\wedge$.", "\\medskip\\noindent", "The rest of the proof is exactly the same as the second part", "of the proof of Lemma \\ref{lemma-one-dimensional-formal-branch};", "of course that lemma is a special case of this one!", "We have $C/\\mathfrak m C = A/\\mathfrak m$ because $A^\\wedge/J^n$ has the same", "property. Hence $\\mathfrak m_C = \\mathfrak m C$ is a maximal ideal", "and $A \\to C$ is unramified at $\\mathfrak m_C$", "(Algebra, Lemma \\ref{algebra-lemma-characterize-unramified}).", "After replacing $C$ by a principal localization we may", "assume that $C$ is a quotient of an \\'etale $A$-algebra $B$, see", "Algebra, Proposition \\ref{algebra-proposition-unramified-locally-standard}.", "However, since the residue field extension of $A \\to C_{\\mathfrak m_C}$", "is trivial and $A$ is henselian, we conclude that $B = A$", "again after a localization.", "Thus $C = A/I$ for some ideal $I \\subset A$ and it follows that", "$J^n = IA^\\wedge$ (because completion is exact in our situation by", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}) and $I = J^n \\cap A$", "(by flatness of $A \\to A^\\wedge$).", "By symmetry $I' = (J')^n \\cap A$ satisfies $(J')^n = I'A^\\wedge$.", "Then $\\mathfrak m^e \\subset I + I'$ and $II' = 0$", "and we conclude that $V(I)$ and $V(I')$ are closed subschemes", "which give the desired disjoint union decomposition of the", "punctured spectrum of $A$." ], "refs": [ "more-algebra-lemma-henselization-noetherian", "algebra-lemma-ff-rings", "more-algebra-lemma-quotient-by-idempotent", "more-algebra-lemma-one-dimensional-formal-branch", "algebra-lemma-characterize-unramified", "algebra-proposition-unramified-locally-standard", "algebra-lemma-completion-flat" ], "ref_ids": [ 10057, 536, 10475, 10476, 1270, 1428, 870 ] } ], "ref_ids": [] }, { "id": 10478, "type": "theorem", "label": "more-algebra-lemma-one-dimensional-number-of-branches", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-one-dimensional-number-of-branches", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring of dimension $1$.", "Then the number of (geometric) branches of $A$ and $A^\\wedge$ is the same." ], "refs": [], "proofs": [ { "contents": [ "To see this for the number of branches, combine", "Lemmas \\ref{lemma-nr-branches-completion},", "\\ref{lemma-equal-nr-branches-completion}, and", "\\ref{lemma-one-dimensional-formal-branch}", "and use that the dimension of $A^\\wedge$ is one, see", "Lemma \\ref{lemma-completion-dimension}.", "To see this is true for the number of geometric branches", "we use the result for branches, the fact that the dimension", "does not change under strict henselization", "(Lemma \\ref{lemma-henselization-dimension}), and the fact that", "$(A^{sh})^\\wedge = ((A^\\wedge)^{sh})^\\wedge$", "by Lemma \\ref{lemma-henselization-noetherian}." ], "refs": [ "more-algebra-lemma-nr-branches-completion", "more-algebra-lemma-equal-nr-branches-completion", "more-algebra-lemma-one-dimensional-formal-branch", "more-algebra-lemma-completion-dimension", "more-algebra-lemma-henselization-dimension", "more-algebra-lemma-henselization-noetherian" ], "ref_ids": [ 10472, 10473, 10476, 10042, 10061, 10057 ] } ], "ref_ids": [] }, { "id": 10479, "type": "theorem", "label": "more-algebra-lemma-geometrically-normal-formal-fibres-number-of-branches", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-geometrically-normal-formal-fibres-number-of-branches", "contents": [ "\\begin{reference}", "\\cite[Theorem 2.3]{Beddani}", "\\end{reference}", "Let $(A, \\mathfrak m)$ be a Noetherian local ring. If the formal", "fibres of $A$ are geometrically normal (for example if $A$ is", "excellent or quasi-excellent), then $A$ is Nagata", "and the number of (geometric) branches of $A$ and $A^\\wedge$ is the same." ], "refs": [], "proofs": [ { "contents": [ "Since a normal ring is reduced, we see that $A$ is Nagata by", "Lemma \\ref{lemma-Nagata-local-ring}. In the rest of the proof", "we will use Lemma \\ref{lemma-formal-fibres-normal},", "Proposition \\ref{proposition-finite-type-over-P-ring}, and", "Lemma \\ref{lemma-check-P-ring-maximal-ideals}. This tells us", "that $A$ is a P-ring where $P(k \\to R) = $``$R$ is geometrically", "normal over $k$'' and the same is true for any (essentially of) finite type", "$A$-algebra.", "\\medskip\\noindent", "Let $\\mathfrak q \\subset A$ be a minimal prime. Then", "$A^\\wedge/\\mathfrak q A^\\wedge = (A/\\mathfrak q)^\\wedge$", "and $A^h/\\mathfrak qA^h = (A/\\mathfrak q)^h$", "(Algebra, Lemma \\ref{algebra-lemma-quotient-henselization}).", "Hence the number of branches of $A$ is the sum of the", "number of branches of the rings $A/\\mathfrak q$ and", "similarly for $A^\\wedge$. In this way we reduce", "to the case that $A$ is a domain.", "\\medskip\\noindent", "Assume $A$ is a domain. Let $A'$ be the integral closure of $A$", "in the fraction field $K$ of $A$. Since $A$ is Nagata, we see that", "$A \\to A'$ is finite. Recall that the number of branches of", "$A$ is the number of maximal ideals $\\mathfrak m'$ of $A'$", "(Lemma \\ref{lemma-branches}). Also, recall that", "$$", "(A')^\\wedge = A' \\otimes_A A^\\wedge =", "\\prod\\nolimits_{\\mathfrak m' \\subset A'} (A'_{\\mathfrak m'})^\\wedge", "$$", "by Algebra, Lemma \\ref{algebra-lemma-completion-finite-extension}.", "Because $A'_{\\mathfrak m'}$ is a local ring whose formal", "fibres are geometrically normal, we see that", "$(A'_{\\mathfrak m'})^\\wedge$ is normal", "(Lemma \\ref{lemma-completion-normal-local-ring}).", "Hence the minimal primes of $A' \\otimes_A A^\\wedge$", "are in $1$-to-$1$ correspondence with the factors in the", "decomposition above. By flatness of $A \\to A^\\wedge$ we have", "$$", "A^\\wedge \\subset A' \\otimes_A A^\\wedge \\subset K \\otimes_A A^\\wedge", "$$", "Since the left and the right ring have the same set of minimal", "primes, the same is true for the ring in the middle (small", "detail omitted) and this finishes the proof.", "\\medskip\\noindent", "To see this is true for the number of geometric branches", "we use the result for branches, the fact that the formal", "fibres of $A^{sh}$ are geometrically normal", "(Lemmas \\ref{lemma-formal-fibres-normal} and", "\\ref{lemma-henselization-P-ring})", "and the fact that $(A^{sh})^\\wedge = ((A^\\wedge)^{sh})^\\wedge$", "by Lemma \\ref{lemma-henselization-noetherian}." ], "refs": [ "more-algebra-lemma-Nagata-local-ring", "more-algebra-lemma-formal-fibres-normal", "more-algebra-proposition-finite-type-over-P-ring", "more-algebra-lemma-check-P-ring-maximal-ideals", "algebra-lemma-quotient-henselization", "more-algebra-lemma-branches", "algebra-lemma-completion-finite-extension", "more-algebra-lemma-completion-normal-local-ring", "more-algebra-lemma-formal-fibres-normal", "more-algebra-lemma-henselization-P-ring", "more-algebra-lemma-henselization-noetherian" ], "ref_ids": [ 10107, 10102, 10583, 10097, 1301, 10465, 876, 10109, 10102, 10100, 10057 ] } ], "ref_ids": [] }, { "id": 10480, "type": "theorem", "label": "more-algebra-lemma-not-formally-catenary", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-not-formally-catenary", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring which is not", "formally catenary. Then $A$ is not universally catenary." ], "refs": [], "proofs": [ { "contents": [ "By assumption there exists a minimal prime $\\mathfrak p \\subset A$ such that", "the spectrum of $A^\\wedge /\\mathfrak p A^\\wedge$ is not equidimensional.", "After replacing $A$ by $A/\\mathfrak p$ we may assume that $A$", "is a domain and that the spectrum of $A^\\wedge$ is not equidimensional.", "Let $\\mathfrak q$ be a minimal prime of", "$A^\\wedge$ such that $d = \\dim(A^\\wedge/\\mathfrak q)$", "is minimal and hence $0 < d < \\dim(A)$. We prove the lemma by induction", "on $d$.", "\\medskip\\noindent", "The case $d = 1$. In this case $\\dim(A^\\wedge_\\mathfrak q) = 0$.", "Hence $A^\\wedge_\\mathfrak q$ is Artinian local and we see that", "for some $n > 0$ the ideal $J = \\mathfrak q^n$ maps to zero in", "$A^\\wedge_\\mathfrak q$. It follows that $\\mathfrak m$ is the", "only associated prime of $J/J^2$, whence $\\mathfrak m^m$ annihilates", "$J/J^2$ for some $m > 0$. Thus we can use", "Lemma \\ref{lemma-quotient-by-idempotent}", "to find $A \\to B$ of finite type such that $B^\\wedge \\cong A^\\wedge/J$.", "It follows that $\\mathfrak m_B = \\sqrt{\\mathfrak mB}$ is a maximal", "ideal with the same residue field as $\\mathfrak m$ and $B^\\wedge$", "is the $\\mathfrak m_B$-adic completion", "(Algebra, Lemma \\ref{algebra-lemma-finite-after-completion}).", "Then", "$$", "\\dim(B_{\\mathfrak m_B}) = \\dim(B^\\wedge) = 1 = d.", "$$", "Since we have the factorization $A \\to B \\to A^\\wedge/J$ the inverse image", "of $\\mathfrak q/J$ is a prime $\\mathfrak q' \\subset \\mathfrak m_B$ lying", "over $(0)$ in $A$. Thus, if $A$ were universally catenary, the dimension", "formula (Algebra, Lemma \\ref{algebra-lemma-dimension-formula}) would give", "\\begin{align*}", "\\dim(B_{\\mathfrak m_B})", "& \\geq", "\\dim((B/\\mathfrak q')_{\\mathfrak m_B}) \\\\", "& =", "\\dim(A) + \\text{trdeg}_A(B/\\mathfrak q') -", "\\text{trdeg}_{\\kappa(\\mathfrak m)}(\\kappa(\\mathfrak m_B)) \\\\", "& =", "\\dim(A) + \\text{trdeg}_A(B/\\mathfrak q')", "\\end{align*}", "This contradiction finishes the argument in case $d = 1$.", "\\medskip\\noindent", "Assume $d > 1$. Let $Z \\subset \\Spec(A^\\wedge)$ be the union of", "the irreducible components distinct from $V(\\mathfrak q)$.", "Let $\\mathfrak r_1, \\ldots, \\mathfrak r_m \\subset A^\\wedge$", "be the prime ideals corresponding to irreducible components of", "$V(\\mathfrak q) \\cap Z$ of dimension $> 0$.", "Choose $f \\in \\mathfrak m$, $f \\not \\in A \\cap \\mathfrak r_j$", "using prime avoidance (Algebra, Lemma \\ref{algebra-lemma-silly}).", "Then $\\dim(A/fA) = \\dim(A) - 1$ and there is some irreducible", "component of $V(\\mathfrak q, f)$ of dimension $d - 1$.", "Thus $A/fA$ is not formally catenary and the invariant $d$ has", "decreased. By induction $A/fA$ is not universally catenary, hence", "$A$ is not universally catenary." ], "refs": [ "more-algebra-lemma-quotient-by-idempotent", "algebra-lemma-finite-after-completion", "algebra-lemma-dimension-formula", "algebra-lemma-silly" ], "ref_ids": [ 10475, 875, 990, 378 ] } ], "ref_ids": [] }, { "id": 10481, "type": "theorem", "label": "more-algebra-lemma-flat-under-catenary-equidimensional", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-flat-under-catenary-equidimensional", "contents": [ "Let $A \\to B$ be a flat local ring map of local Noetherian rings.", "Assume $B$ is catenary and is $\\Spec(B)$ equidimensional. Then", "\\begin{enumerate}", "\\item $\\Spec(B/\\mathfrak p B)$ is equidimensional for all", "$\\mathfrak p \\subset A$ and", "\\item $A$ is catenary and $\\Spec(A)$ is equidimensional.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p \\subset A$ be a prime ideal. Let $\\mathfrak q \\subset B$", "be a prime minimal over $\\mathfrak pB$. Then $\\mathfrak q \\cap A = \\mathfrak p$", "by going down for $A \\to B$", "(Algebra, Lemma \\ref{algebra-lemma-flat-going-down}).", "Hence $A_\\mathfrak p \\to B_\\mathfrak q$ is a flat local ring map", "with special fibre of dimension $0$ and hence", "$$", "\\dim(A_\\mathfrak p) = \\dim(B_\\mathfrak q) = \\dim(B) - \\dim(B/\\mathfrak q)", "$$", "(Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}).", "The second equality because $\\Spec(B)$ is equidimensional and $B$ is catenary.", "Thus $\\dim(B/\\mathfrak q)$ is independent of the choice of $\\mathfrak q$", "and we conclude that $\\Spec(B/\\mathfrak p B)$ is equidimensional of", "dimension $\\dim(B) - \\dim(A_\\mathfrak p)$. On the other hand, we", "have", "$\\dim(B/\\mathfrak p B) = \\dim(A/\\mathfrak p) + \\dim(B/\\mathfrak m_A B)$", "and", "$\\dim(B) = \\dim(A) + \\dim(B/\\mathfrak m_A B)$", "by flatness (see lemma cited above) and we get", "$$", "\\dim(A_\\mathfrak p) = \\dim(A) - \\dim(A/\\mathfrak p)", "$$", "for all $\\mathfrak p$ in $A$. Applying this to all minimal primes in", "$A$ we see that $A$ is equidimensional.", "If $\\mathfrak p \\subset \\mathfrak p'$ is a strict inclusion", "with no primes in between, then we may apply the above to", "the prime $\\mathfrak p'/\\mathfrak p$ in $A/\\mathfrak p$", "because $A/\\mathfrak p \\to B/\\mathfrak p B$ is flat and", "$\\Spec(B/\\mathfrak p B)$ is equidimensional, to get", "$$", "1 = \\dim((A/\\mathfrak p)_{\\mathfrak p'}) =", "\\dim(A/\\mathfrak p) - \\dim(A/\\mathfrak p')", "$$", "Thus $\\mathfrak p \\mapsto \\dim(A/\\mathfrak p)$ is a dimension", "function and we conclude that $A$ is catenary." ], "refs": [ "algebra-lemma-flat-going-down", "algebra-lemma-dimension-base-fibre-equals-total" ], "ref_ids": [ 539, 987 ] } ], "ref_ids": [] }, { "id": 10482, "type": "theorem", "label": "more-algebra-lemma-formally-catenary", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-formally-catenary", "contents": [ "Let $A$ be a formally catenary Noetherian local ring.", "Then $A$ is universally catenary." ], "refs": [], "proofs": [ { "contents": [ "We may replace $A$ by $A/\\mathfrak p$ where $\\mathfrak p$ is a minimal prime", "of $A$, see Algebra, Lemma \\ref{algebra-lemma-catenary-check-irreducible}.", "Thus we may assume that the spectrum of $A^\\wedge$ is equidimensional.", "It suffices to show that every local ring essentially of finite type", "over $A$ is catenary (see for example", "Algebra, Lemma \\ref{algebra-lemma-catenary-check-local}).", "Hence it suffices to show that $A[x_1, \\ldots, x_n]_\\mathfrak m$ is catenary", "where $\\mathfrak m \\subset A[x_1, \\ldots, x_n]$ is a maximal", "ideal lying over $\\mathfrak m_A$, see", "Algebra, Lemma \\ref{algebra-lemma-localization-at-closed-point-special-fibre}", "(and Algebra, Lemmas \\ref{algebra-lemma-quotient-catenary} and", "\\ref{algebra-lemma-localization-catenary}).", "Let $\\mathfrak m' \\subset A^\\wedge[x_1, \\ldots, x_n]$ be the unique", "maximal ideal lying over $\\mathfrak m$. Then", "$$", "A[x_1, \\ldots, x_n]_\\mathfrak m \\to A^\\wedge[x_1, \\ldots, x_n]_{\\mathfrak m'}", "$$", "is local and flat (Algebra, Lemma \\ref{algebra-lemma-completion-flat}).", "Hence it suffices to show that the ring on the right", "hand side catenary with equidimensional spectrum, see", "Lemma \\ref{lemma-flat-under-catenary-equidimensional}.", "It is catenary because complete local rings are universally catenary", "(Algebra, Remark", "\\ref{algebra-remark-Noetherian-complete-local-ring-universally-catenary}).", "Pick any minimal prime $\\mathfrak q$ of", "$A^\\wedge[x_1, \\ldots, x_n]_{\\mathfrak m'}$. Then", "$\\mathfrak q = \\mathfrak p A^\\wedge[x_1, \\ldots, x_n]_{\\mathfrak m'}$", "for some minimal prime $\\mathfrak p$ of $A^\\wedge$ (small detail omitted).", "Hence", "$$", "\\dim(A^\\wedge[x_1, \\ldots, x_n]_{\\mathfrak m'}/\\mathfrak q) =", "\\dim(A^\\wedge/\\mathfrak p) + n = \\dim(A^\\wedge) + n", "$$", "the first equality by", "Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}", "and the second because the spectrum of $A^\\wedge$ is equidimensional.", "This finishes the proof." ], "refs": [ "algebra-lemma-catenary-check-irreducible", "algebra-lemma-catenary-check-local", "algebra-lemma-localization-at-closed-point-special-fibre", "algebra-lemma-quotient-catenary", "algebra-lemma-localization-catenary", "algebra-lemma-completion-flat", "more-algebra-lemma-flat-under-catenary-equidimensional", "algebra-remark-Noetherian-complete-local-ring-universally-catenary", "algebra-lemma-dimension-base-fibre-equals-total" ], "ref_ids": [ 936, 934, 650, 935, 932, 870, 10481, 1582, 987 ] } ], "ref_ids": [] }, { "id": 10483, "type": "theorem", "label": "more-algebra-lemma-geometrically-normal-fibres-universally-catenary", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-geometrically-normal-fibres-universally-catenary", "contents": [ "\\begin{reference}", "\\cite[Corollary 2.3]{Heinzer-Rotthaus-Wiegand}", "\\end{reference}", "Let $(A, \\mathfrak m)$ be a Noetherian local ring with", "geometrically normal formal fibres. Then", "\\begin{enumerate}", "\\item $A^h$ is universally catenary, and", "\\item if $A$ is unibranch (for example normal), then", "$A$ is universally catenary.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-geometrically-normal-formal-fibres-number-of-branches}", "the number of branches of $A$ and $A^\\wedge$ are the same,", "hence Lemma \\ref{lemma-equal-nr-branches-completion} applies.", "Then for any minimal prime $\\mathfrak q \\subset A^h$", "we see that $A^\\wedge/\\mathfrak q A^\\wedge$", "has a unique minimal prime. Thus $A^h$ is formally catenary", "(by definition) and hence universally catenary by", "Proposition \\ref{proposition-ratliff}. If $A$ is unibranch,", "then $A^h$ has a unique minimal prime, hence $A^\\wedge$ has", "a unique minimal prime, hence $A$ is formally catenary and", "we conclude in the same way." ], "refs": [ "more-algebra-lemma-geometrically-normal-formal-fibres-number-of-branches", "more-algebra-lemma-equal-nr-branches-completion", "more-algebra-proposition-ratliff" ], "ref_ids": [ 10479, 10473, 10591 ] } ], "ref_ids": [] }, { "id": 10484, "type": "theorem", "label": "more-algebra-lemma-pol-lifting", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pol-lifting", "contents": [ "Let $\\varphi : A \\to B$ be a surjection of rings. Let $G$ be a finite group", "of order $n$ acting on $\\varphi : A \\to B$. If $b \\in B^G$, then", "there exists a monic polynomial $P \\in A^G[T]$ which maps to", "$(T - b)^n$ in $B^G[T]$." ], "refs": [], "proofs": [ { "contents": [ "Choose $a \\in A$ lifting $b$ and set", "$P = \\prod_{\\sigma \\in G} (T - \\sigma(a))$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10485, "type": "theorem", "label": "more-algebra-lemma-invariants-modulo", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-invariants-modulo", "contents": [ "Let $R$ be a ring. Let $G$ be a finite group acting on $R$. Let $I \\subset R$", "be an ideal such that $\\sigma(I) \\subset I$ for all $\\sigma \\in G$.", "Then $R^G/I^G \\subset (R/I)^G$ is an integral extension of rings which", "induces homeomorphisms on spectra and purely inseparable extensions of", "residue fields." ], "refs": [], "proofs": [ { "contents": [ "Since $I^G = R^G \\cap I$ it is clear that the map is injective.", "Lemma \\ref{lemma-pol-lifting} shows that", "Algebra, Lemma \\ref{algebra-lemma-universally-bijective}", "applies." ], "refs": [ "more-algebra-lemma-pol-lifting", "algebra-lemma-universally-bijective" ], "ref_ids": [ 10484, 586 ] } ], "ref_ids": [] }, { "id": 10486, "type": "theorem", "label": "more-algebra-lemma-functor-invariants-tensor", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-functor-invariants-tensor", "contents": [ "Let $R$ be a ring. Let $G$ be a finite group of order $n$ acting on $R$.", "Let $A$ be an $R^G$-algebra.", "\\begin{enumerate}", "\\item for $b \\in (A \\otimes_{R^G} R)^G$ there exists a monic polynomial", "$P \\in A[T]$ whose image in $(A \\otimes_{R^G} R)^G[T]$ is $(T - b)^n$,", "\\item for $a \\in A$ mapping to zero in $(A \\otimes_{R^G} R)^G$ we have", "$(T - a)^{n^2} = T^{n^2}$ in $A[T]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $A$ as the quotient of a polynomial algebra $P$ over $R^G$.", "Then $(P \\otimes_{R^G} R)^G = P$ because $P$ is free as an $R^G$-module.", "Hence part (1) follows from Lemma \\ref{lemma-pol-lifting}.", "\\medskip\\noindent", "Let $J = \\Ker(P \\to A)$. Lift $a$ as in (2) to an element $f \\in P$.", "Then $f \\otimes 1$ maps to zero in $A \\otimes_{R^G} R$.", "Hence $f \\otimes 1$ is in $(J')^G$ where $J' \\subset P \\otimes_{R^G} R$", "is the image of the map $J \\otimes_{R^G} R \\to P \\otimes_{R^G} R$.", "Apply Lemma \\ref{lemma-pol-lifting} to $f \\otimes 1$", "and the surjective ring map", "$$", "\\text{Sym}^*_{R^G}(J) \\otimes_{R^G} R", "\\longrightarrow", "A' \\subset \\text{Sym}^*_{R^G}(P) \\otimes_{R^G} R", "$$", "which defines $A'$. We obtain", "$P \\in (\\text{Sym}^*_{R^G}(J) \\otimes_{R^G} R)^G[T]$", "mapping to $(T - f \\otimes 1)^n$ in $A'[T]$.", "Apply part (1) to see that there exists a", "$P' \\in \\text{Sym}^*_{R^G}(J)[T, T']$ whose image", "is $(T' - P)^n$. Since $\\text{Sym}_{R^G}^*(P)$ is still", "free over $R^G$ we conclude that $P'$ maps to $(T' - (T - f)^n)^n$", "in $\\text{Sym}_{R^G}^*(P)$. On the other hand, tracing through", "the construction of the polynomials $P$ and $P'$ in", "Lemma \\ref{lemma-pol-lifting}", "we see that $P'$ is congruent to $(T' - T^n)^n$ modulo", "the irrelevant ideal of the graded ring $\\text{Sym}^*_{R^G}(J)$.", "It follows that", "$$", "(T' - (T - a)^n)^n = (T' - T^n)^n", "$$", "in $A[T', T]$. Setting $T' = 0$ for example we obtain the statement", "of the lemma." ], "refs": [ "more-algebra-lemma-pol-lifting", "more-algebra-lemma-pol-lifting", "more-algebra-lemma-pol-lifting" ], "ref_ids": [ 10484, 10484, 10484 ] } ], "ref_ids": [] }, { "id": 10487, "type": "theorem", "label": "more-algebra-lemma-base-change-invariants", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-base-change-invariants", "contents": [ "Let $R$ be a ring. Let $G$ be a finite group acting on $R$.", "Let $R^G \\to A$ be a ring map. The map", "$$", "A \\to (A \\otimes_{R^G} R)^G", "$$", "is an isomorphism if $R^G \\to A$ is flat. In general the map", "is integral, induces a homeomorphism on spectra, and", "induces purely inseparable residue field extensions." ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from Lemma \\ref{lemma-functor-invariants-tensor}", "and Algebra, Lemma \\ref{algebra-lemma-universally-bijective}.", "To see the second consider the", "exact sequence $0 \\to R^G \\to R \\to \\bigoplus_{\\sigma \\in G} R$", "where the second map sends $x$ to $(\\sigma(x) - x)$. Tensoring with", "$A$ the sequence remains exact if $R^G \\to A$ is flat." ], "refs": [ "more-algebra-lemma-functor-invariants-tensor", "algebra-lemma-universally-bijective" ], "ref_ids": [ 10486, 586 ] } ], "ref_ids": [] }, { "id": 10488, "type": "theorem", "label": "more-algebra-lemma-one-orbit", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-one-orbit", "contents": [ "Let $G$ be a finite group acting on a ring $R$. For any two primes", "$\\mathfrak q, \\mathfrak q' \\subset R$ lying over the same prime in $R^G$", "there exists a $\\sigma \\in G$ with $\\sigma(\\mathfrak q) = \\mathfrak q'$." ], "refs": [], "proofs": [ { "contents": [ "The extension $R^G \\subset R$ is integral because every $x \\in R$", "is a root of the monic polynomial $\\prod_{\\sigma \\in G}(T - \\sigma(x))$", "in $R^G[T]$. Thus there are no inclusion relations among the primes", "lying over a given prime $\\mathfrak p$", "(Algebra, Lemma \\ref{algebra-lemma-integral-no-inclusion}).", "If the lemma is wrong, then", "we can choose $x \\in \\mathfrak q'$, $x \\not \\in \\sigma(\\mathfrak q)$", "for all $\\sigma \\in G$. See Algebra, Lemma \\ref{algebra-lemma-silly}.", "Then $y = \\prod_{\\sigma \\in G} \\sigma(x)$ is in $R^G$ and", "in $\\mathfrak p = R^G \\cap \\mathfrak q'$. On the other hand,", "$x \\not \\in \\sigma(\\mathfrak q)$ for all $\\sigma$ means", "$\\sigma(x) \\not \\in \\mathfrak q$ for all $\\sigma$. Hence", "$y \\not \\in \\mathfrak q$ as $\\mathfrak q$ is a prime ideal.", "This is impossible as $y \\in \\mathfrak p \\subset \\mathfrak q$." ], "refs": [ "algebra-lemma-integral-no-inclusion", "algebra-lemma-silly" ], "ref_ids": [ 498, 378 ] } ], "ref_ids": [] }, { "id": 10489, "type": "theorem", "label": "more-algebra-lemma-one-orbit-geometric", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-one-orbit-geometric", "contents": [ "Let $G$ be a finite group acting on a ring $R$. Let $\\mathfrak q \\subset R$", "be a prime lying over $\\mathfrak p \\subset R^G$. Then", "$\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$ is an algebraic normal", "extension and the map", "$$", "D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q) = \\mathfrak q\\}", "\\longrightarrow", "\\text{Aut}(\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p))", "$$", "is surjective\\footnote{Recall that we use the notation $\\text{Gal}$", "only in the case of Galois extensions.}." ], "refs": [], "proofs": [ { "contents": [ "With $A = (R^G)_\\mathfrak p$ and $B = A \\otimes_{R^G} R$ we see that $A = B^G$", "as localization is flat, see Lemma \\ref{lemma-base-change-invariants}.", "Observe that $\\mathfrak pA$ and $\\mathfrak qB$ are prime ideals,", "$D$ is the stabilizer of $\\mathfrak qB$, and", "$\\kappa(\\mathfrak p) = \\kappa(\\mathfrak pA)$ and", "$\\kappa(\\mathfrak q) = \\kappa(\\mathfrak qB)$. Thus we may replace", "$R$ by $B$ and assume that $\\mathfrak p$ is a maximal ideal.", "Since $R \\subset R^G$ is an integral ring extension, we find", "that the maximal ideals of $R$ are exactly the primes lying over", "$\\mathfrak p$ (follows from", "Algebra, Lemmas \\ref{algebra-lemma-integral-no-inclusion} and", "\\ref{algebra-lemma-integral-going-up}).", "By Lemma \\ref{lemma-one-orbit} there are finitely many of them", "$\\mathfrak q = \\mathfrak q_1, \\mathfrak q_2, \\ldots, \\mathfrak q_m$", "and they form a single orbit for $G$.", "By the Chinese remainder theorem", "(Algebra, Lemma \\ref{algebra-lemma-chinese-remainder}) the map", "$R \\to \\prod_{j = 1, \\ldots, m} R/\\sigma(\\mathfrak q_j)$ is surjective.", "\\medskip\\noindent", "First we prove that the extension is normal. Pick an element", "$\\alpha \\in \\kappa(\\mathfrak q)$. We have to show that the", "minimal polynomial $P$ of $\\alpha$ over $\\kappa(\\mathfrak p)$", "splits completely. By the above we can choose", "$a \\in \\mathfrak q_2 \\cap \\ldots \\cap \\mathfrak q_m$", "mapping to $\\alpha$ in $\\kappa(\\mathfrak q)$.", "Consider the polynomial $Q = \\prod_{\\sigma \\in G} (T - \\sigma(a))$", "in $R^G[T]$. The image of $Q$ in $R[T]$ splits completely", "into linear factors, hence the same is true for its", "image in $\\kappa(\\mathfrak q)[T]$. Since $P$ divides", "the image of $Q$ in $\\kappa(\\mathfrak p)[T]$ we conclude", "that $P$ splits completely into linear factors over", "$\\kappa(\\mathfrak q)$ as desired.", "\\medskip\\noindent", "Since $\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$ is normal we may assume", "$\\kappa(\\mathfrak q) = \\kappa_1 \\otimes_{\\kappa(\\mathfrak p)} \\kappa_2$", "with $\\kappa_1/\\kappa(\\mathfrak p)$ purely inseparable and", "$\\kappa_2/\\kappa(\\mathfrak p)$ Galois, see", "Fields, Lemma \\ref{fields-lemma-normal-case}.", "$\\alpha \\in \\kappa_2$ which generates $\\kappa_2$ over $\\kappa(\\mathfrak p)$", "if it is finite and a subfield of degree $> |G|$ if it is infinite", "(to get a contradiction).", "This is possible by Fields, Lemma \\ref{fields-lemma-primitive-element}.", "Pick $a$, $P$, and $Q$ as in the previous paragraph.", "If $\\alpha' \\in \\kappa_2$", "is a Galois conjugate of $\\alpha$, then the above shows there exists a", "$\\sigma \\in G$ such that $\\sigma(a)$ maps to $\\alpha'$. By our choice of", "$a$ (vanishing at other maximal ideals) this implies $\\sigma \\in D$ and", "that the image of $\\sigma$ in", "$\\text{Aut}(\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p))$", "maps $\\alpha$ to $\\alpha'$. Hence the surjectivity or the", "desired absurdity", "in case $\\alpha$ has degree $> |G|$ over $\\kappa(\\mathfrak p)$." ], "refs": [ "more-algebra-lemma-base-change-invariants", "algebra-lemma-integral-no-inclusion", "algebra-lemma-integral-going-up", "more-algebra-lemma-one-orbit", "algebra-lemma-chinese-remainder", "fields-lemma-normal-case", "fields-lemma-primitive-element" ], "ref_ids": [ 10487, 498, 500, 10488, 380, 4522, 4498 ] } ], "ref_ids": [] }, { "id": 10490, "type": "theorem", "label": "more-algebra-lemma-one-orbit-geometric-galois", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-one-orbit-geometric-galois", "contents": [ "Let $A$ be a normal domain with fraction field $K$.", "Let $L/K$ be a (possibly infinite) Galois extension.", "Let $G = \\text{Gal}(L/K)$ and let", "$B$ be the integral closure of $A$ in $L$.", "\\begin{enumerate}", "\\item For any two primes", "$\\mathfrak q, \\mathfrak q' \\subset B$ lying over the same prime in $A$", "there exists a $\\sigma \\in G$ with $\\sigma(\\mathfrak q) = \\mathfrak q'$.", "\\item Let $\\mathfrak q \\subset B$ be a prime lying over", "$\\mathfrak p \\subset A$. Then $\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$", "is an algebraic normal extension and the map", "$$", "D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q) = \\mathfrak q\\}", "\\longrightarrow", "\\text{Aut}(\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p))", "$$", "is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Consider pairs $(M, \\sigma)$ where $K \\subset M \\subset L$", "is a subfield such that $M/K$ is Galois, $\\sigma \\in \\text{Gal}(M/K)$", "with $\\sigma(\\mathfrak q \\cap M) = \\mathfrak q' \\cap M$.", "We say $(M', \\sigma') \\geq (M, \\sigma)$ if and only if", "$M \\subset M'$ and $\\sigma'|_M = \\sigma$.", "Observe that $(K, \\text{id}_K)$ is such a pair as $A = K \\cap B$", "since $A$ is a normal domain.", "The collection of these pairs satisfies the hypotheses of Zorn's lemma,", "hence there exists a maximal pair $(M, \\sigma)$.", "If $M \\not = L$, then we can find", "$M \\subset M' \\subset L$ with $M'/M$ nontrivial and finite and $M'/K$ Galois", "(Fields, Lemma \\ref{fields-lemma-normal-closure-inside-normal}).", "Choose $\\sigma' \\in \\text{Gal}(M'/K)$ whose restriction to $M$", "is $\\sigma$ (Fields, Lemma \\ref{fields-lemma-galois-infinite}).", "Then the primes $\\sigma'(\\mathfrak q \\cap M')$ and $\\mathfrak q' \\cap M'$", "restrict to the same prime of $B \\cap M$. Since", "$B \\cap M = (B \\cap M')^{\\text{Gal}(M'/M)}$ we can", "use Lemma \\ref{lemma-one-orbit} to find $\\tau \\in \\text{Gal}(M'/M)$", "with $\\tau(\\sigma'(\\mathfrak q \\cap M')) = \\mathfrak q' \\cap M'$.", "Hence $(M', \\tau \\circ \\sigma') > (M, \\sigma)$", "contradicting the maximality of $(M, \\sigma)$.", "\\medskip\\noindent", "Part (2) is proved in exactly the same manner as part (1). We", "write out the details. Pick", "$\\overline{\\sigma} \\in \\text{Aut}(\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p))$.", "Consider pairs $(M, \\sigma)$ where $K \\subset M \\subset L$", "is a subfield such that $M/K$ is Galois, $\\sigma \\in \\text{Gal}(M/K)$", "with $\\sigma(\\mathfrak q \\cap M) = \\mathfrak q \\cap M$ and", "$$", "\\xymatrix{", "\\kappa(\\mathfrak q \\cap M) \\ar[r] \\ar[d]_\\sigma &", "\\kappa(\\mathfrak q) \\ar[d]_{\\overline{\\sigma}} \\\\", "\\kappa(\\mathfrak q \\cap M) \\ar[r] & \\kappa(\\mathfrak q)", "}", "$$", "commutes. We say $(M', \\sigma') \\geq (M, \\sigma)$ if and only if", "$M \\subset M'$ and $\\sigma'|_M = \\sigma$.", "As above $(K, \\text{id}_K)$ is such a pair.", "The collection of these pairs satisfies the hypotheses of Zorn's lemma,", "hence there exists a maximal pair $(M, \\sigma)$.", "If $M \\not = L$, then we can find", "$M \\subset M' \\subset L$ with $M'/M$ finite and $M'/K$ Galois", "(Fields, Lemma \\ref{fields-lemma-normal-closure-inside-normal}).", "Choose $\\sigma' \\in \\text{Gal}(M'/K)$ whose restriction to $M$", "is $\\sigma$ (Fields, Lemma \\ref{fields-lemma-galois-infinite}).", "Then the primes $\\sigma'(\\mathfrak q \\cap M')$ and $\\mathfrak q \\cap M'$", "restrict to the same prime of $B \\cap M$. Adjusting the choice", "of $\\sigma'$ as in the first paragraph, we may assume that", "$\\sigma'(\\mathfrak q \\cap M') = \\mathfrak q \\cap M'$.", "Then $\\sigma'$ and $\\overline{\\sigma}$ define maps", "$\\kappa(\\mathfrak q \\cap M') \\to \\kappa(\\mathfrak q)$", "which agree on $\\kappa(\\mathfrak q \\cap M)$. Since", "$B \\cap M = (B \\cap M')^{\\text{Gal}(M'/M)}$ we can", "use Lemma \\ref{lemma-one-orbit-geometric}", "to find $\\tau \\in \\text{Gal}(M'/M)$ with", "$\\tau(\\mathfrak q \\cap M') = \\mathfrak q \\cap M'$", "such that $\\tau \\circ \\sigma$ and $\\overline{\\sigma}$", "induce the same map on $\\kappa(\\mathfrak q \\cap M')$.", "There is a small detail here in that the lemma first", "guarantees that $\\kappa(\\mathfrak q \\cap M')/\\kappa(\\mathfrak q \\cap M)$", "is normal, which then tells us that the difference between", "the maps is an automorphism of this extension", "(Fields, Lemma \\ref{fields-lemma-normal-embeddings-differ-by-aut}),", "to which we can", "apply the lemma to get $\\tau$. Hence $(M', \\tau \\circ \\sigma') > (M, \\sigma)$", "contradicting the maximality of $(M, \\sigma)$." ], "refs": [ "fields-lemma-normal-closure-inside-normal", "fields-lemma-galois-infinite", "more-algebra-lemma-one-orbit", "fields-lemma-normal-closure-inside-normal", "fields-lemma-galois-infinite", "more-algebra-lemma-one-orbit-geometric", "fields-lemma-normal-embeddings-differ-by-aut" ], "ref_ids": [ 4495, 4510, 10488, 4495, 4510, 10489, 4492 ] } ], "ref_ids": [] }, { "id": 10491, "type": "theorem", "label": "more-algebra-lemma-one-orbit-geometric-galois-compare", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-one-orbit-geometric-galois-compare", "contents": [ "Let $A$ be a normal domain with fraction field $K$.", "Let $M/L/K$ be a tower of (possibly infinite) Galois extensions of $K$.", "Let $H = \\text{Gal}(M/K)$ and $G = \\text{Gal}(L/K)$ and let", "$C$ and $B$ be the integral closure of $A$ in $M$ and $L$.", "Let $\\mathfrak r \\subset C$ and $\\mathfrak q = B \\cap \\mathfrak r$.", "Set", "$D_\\mathfrak r = \\{\\tau \\in H \\mid \\tau(\\mathfrak r) = \\mathfrak r\\}$", "and", "$I_\\mathfrak r = \\{\\tau \\in D_\\mathfrak r \\mid", "\\tau \\bmod \\mathfrak r = \\text{id}_{\\kappa(\\mathfrak r)}\\}$", "and similarly for $D_\\mathfrak q$ and $I_\\mathfrak q$.", "Under the map $H \\to G$ the induced maps", "$D_\\mathfrak r \\to D_\\mathfrak q$ and", "$I_\\mathfrak r \\to I_\\mathfrak q$ are surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\sigma \\in D_\\mathfrak q$. Pick $\\tau \\in H$ mapping to $\\sigma$.", "This is possible by Fields, Lemma \\ref{fields-lemma-galois-infinite}.", "Then $\\tau(\\mathfrak r)$ and $\\mathfrak r$ both lie over $\\mathfrak q$.", "Hence by Lemma \\ref{lemma-one-orbit-geometric-galois}", "there exists a $\\sigma' \\in \\text{Gal}(M/L)$ with", "$\\sigma'(\\tau(\\mathfrak r)) = \\mathfrak r$. Hence", "$\\sigma'\\tau \\in D_\\mathfrak r$ maps to $\\sigma$.", "The case of inertia groups is proved in exactly the same", "way using surjectivity onto automorphism groups." ], "refs": [ "fields-lemma-galois-infinite", "more-algebra-lemma-one-orbit-geometric-galois" ], "ref_ids": [ 4510, 10490 ] } ], "ref_ids": [] }, { "id": 10492, "type": "theorem", "label": "more-algebra-lemma-inequality", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-inequality", "contents": [ "Let $A \\subset B$ be an extension of discrete valuation rings with", "fraction fields $K \\subset L$. If the extension $L/K$", "is finite, then the residue field extension is finite and we have", "$ef \\leq [L : K]$." ], "refs": [], "proofs": [ { "contents": [ "Finiteness of the residue field extension is", "Algebra, Lemma \\ref{algebra-lemma-finite-extension-residue-fields-dimension-1}.", "The inequality follows from", "Algebra, Lemmas \\ref{algebra-lemma-finite-length} and", "\\ref{algebra-lemma-pushdown-module}." ], "refs": [ "algebra-lemma-finite-extension-residue-fields-dimension-1", "algebra-lemma-finite-length", "algebra-lemma-pushdown-module" ], "ref_ids": [ 1025, 1024, 639 ] } ], "ref_ids": [] }, { "id": 10493, "type": "theorem", "label": "more-algebra-lemma-multiplicative-e-f", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-multiplicative-e-f", "contents": [ "Let $A \\subset B \\subset C$ be extensions of discrete valuation rings.", "Then the ramification indices of $B/A$ and $C/B$ multiply to give", "the ramification index of $C/A$. In a formula $e_{C/A} = e_{B/A} e_{C/B}$.", "Similarly for the residual degrees in case they are finite." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions and", "Fields, Lemma \\ref{fields-lemma-multiplicativity-degrees}." ], "refs": [ "fields-lemma-multiplicativity-degrees" ], "ref_ids": [ 4450 ] } ], "ref_ids": [] }, { "id": 10494, "type": "theorem", "label": "more-algebra-lemma-ramification-index-a-power-of-p", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-ramification-index-a-power-of-p", "contents": [ "Let $A \\subset B$ be an extension of discrete valuation rings", "inducing the field extension $K \\subset L$. If the characteristic", "of $K$ is $p > 0$ and $L$ is purely inseparable over $K$, then", "the ramification index $e$ is a power of $p$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\pi_A = u \\pi_B^e$ for some $u \\in B^*$. On the other hand, we have", "$\\pi_B^q \\in K$ for some $p$-power $q$. Write", "$\\pi_B^q = v \\pi_A^k$ for some $v \\in A^*$ and $k \\in \\mathbf{Z}$.", "Then $\\pi_A^q = u^q \\pi_B^{qe} = u^q v^e \\pi_A^{ke}$.", "Taking valuations in $B$ we conclude that $ke = q$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10495, "type": "theorem", "label": "more-algebra-lemma-extension-dvrs-formally-smooth", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-extension-dvrs-formally-smooth", "contents": [ "Let $A \\subset B$ be an extension of discrete valuation rings.", "The following are equivalent", "\\begin{enumerate}", "\\item $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic topology, and", "\\item $A \\to B$ is weakly unramified and $\\kappa_A \\subset \\kappa_B$", "is a separable field extension.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from Proposition \\ref{proposition-fs-flat-fibre-fs} and", "Algebra, Proposition", "\\ref{algebra-proposition-characterize-separable-field-extensions}." ], "refs": [ "more-algebra-proposition-fs-flat-fibre-fs", "algebra-proposition-characterize-separable-field-extensions" ], "ref_ids": [ 10577, 1429 ] } ], "ref_ids": [] }, { "id": 10496, "type": "theorem", "label": "more-algebra-lemma-permanence-unramified", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-permanence-unramified", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "\\begin{enumerate}", "\\item If $M/L/K$ are finite separable extensions and", "$M$ is unramified with respect to $A$, then $L$ is unramified", "with respect to $A$.", "\\item If $L/K$ is a finite separable extension which is", "unramified with respect to $A$, then there exists a Galois", "extension $M/K$ containing $L$ which is unramified with respect to $A$.", "\\item If $L_1/K$, $L_2/K$ are finite separable extensions which are", "unramified with respect to $A$, then there exists a a finite", "separable extension $L/K$ which is unramified with respect", "to $A$ containing $L_1$ and $L_2$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use the results of the discussion in", "Remark \\ref{remark-finite-separable-extension}", "without further mention.", "\\medskip\\noindent", "Proof of (1). Let $C/B/A$ be the integral closures of $A$ in $M/L/K$.", "Since $C$ is a finite ring extension of $B$, we see that", "$\\Spec(C) \\to \\Spec(B)$ is surjective. Hence for ever maximal", "ideal $\\mathfrak m \\subset B$ there is a maximal ideal", "$\\mathfrak m' \\subset C$ lying over $\\mathfrak m$.", "By the multiplicativity of ramification indices", "(Lemma \\ref{lemma-multiplicative-e-f})", "and the assumption, we conclude that the ramification", "index of $B_\\mathfrak m$ over $A$ is $1$.", "Since $\\kappa(\\mathfrak m')/\\kappa_A$ is finite separable,", "the same is true for $\\kappa(\\mathfrak m)/\\kappa_A$.", "\\medskip\\noindent", "Proof of (2). Let $M$ be the normal closure of $L$ over $K$, see", "Fields, Definition \\ref{fields-definition-normal-closure}.", "Then $M/K$ is Galois by Fields, Lemma \\ref{fields-lemma-normal-closure-galois}.", "On the other hand, there is a surjection", "$$", "L \\otimes_K \\ldots \\otimes_K L \\longrightarrow M", "$$", "of $K$-algebras, see Fields, Lemma", "\\ref{fields-lemma-normal-closure-tensor-product}.", "Let $B$ be the integral closure of $A$ in $L$", "as in Remark \\ref{remark-finite-separable-extension}. The", "condition that $L$ is unramified with respect to $A$", "exactly means that $A \\to B$ is an \\'etale ring map, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-etale}.", "By permanence properties of \\'etale ring maps we see that", "$$", "B \\otimes_A \\ldots \\otimes_A B", "$$", "is \\'etale over $A$, see", "Algebra, Lemma \\ref{algebra-lemma-etale}.", "Hence the displayed ring is a product of Dedekind domains, see", "Lemma \\ref{lemma-Dedekind-etale-extension}.", "We conclude that $M$ is the fraction field of a", "Dedekind domain finite \\'etale over $A$.", "This means that $M$ is unramified with respect to $A$", "as desired.", "\\medskip\\noindent", "Proof of (3). Let $B_i \\subset L_i$ be the integral closure of $A$.", "Argue in the same manner as above to show that $B_1 \\otimes_A B_2$", "is finite \\'etale over $A$. Details omitted." ], "refs": [ "more-algebra-remark-finite-separable-extension", "more-algebra-lemma-multiplicative-e-f", "fields-definition-normal-closure", "fields-lemma-normal-closure-galois", "fields-lemma-normal-closure-tensor-product", "more-algebra-remark-finite-separable-extension", "algebra-lemma-characterize-etale", "algebra-lemma-etale", "more-algebra-lemma-Dedekind-etale-extension" ], "ref_ids": [ 10676, 10493, 4544, 4506, 4496, 10676, 1235, 1231, 10054 ] } ], "ref_ids": [] }, { "id": 10497, "type": "theorem", "label": "more-algebra-lemma-composition-unramified", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-composition-unramified", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "Let $M/L/K$ be finite separable extensions.", "Let $B$ be the integral closure of $A$ in $L$.", "If $L/K$ is unramified with respect to $A$", "and $M/L$ is unramified with respect to $B_\\mathfrak m$", "for every maximal ideal $\\mathfrak m$ of $B$, then", "$M/K$ is unramified with respect to $A$." ], "refs": [], "proofs": [ { "contents": [ "Let $C$ be the integral closure of $A$ in $M$.", "Every maximal ideal $\\mathfrak m'$ of $C$ lies over", "a maximal ideal $\\mathfrak m$ of $B$.", "Then the lemma follows from the multiplicativity", "of ramification indices (Lemma \\ref{lemma-multiplicative-e-f})", "and the fact that we have the tower", "$\\kappa(\\mathfrak m')/\\kappa(\\mathfrak m)/\\kappa_A$", "of finite extensions of fields." ], "refs": [ "more-algebra-lemma-multiplicative-e-f" ], "ref_ids": [ 10493 ] } ], "ref_ids": [] }, { "id": 10498, "type": "theorem", "label": "more-algebra-lemma-galois", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-galois", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "Let $L/K$ be a finite Galois extension with Galois group $G$.", "Then $G$ acts on the ring $B$ of Remark \\ref{remark-finite-separable-extension}", "and acts transitively on the set of maximal ideals of $B$." ], "refs": [ "more-algebra-remark-finite-separable-extension" ], "proofs": [ { "contents": [ "Observe that $A = B^G$ as $A$ is integrally closed in $K$ and $K = L^G$.", "Hence this lemma is a special case of Lemma \\ref{lemma-one-orbit}." ], "refs": [ "more-algebra-lemma-one-orbit" ], "ref_ids": [ 10488 ] } ], "ref_ids": [ 10676 ] }, { "id": 10499, "type": "theorem", "label": "more-algebra-lemma-galois-conclusion", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-galois-conclusion", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "Let $L/K$ be a finite Galois extension. Then there are $e \\geq 1$ and", "$f \\geq 1$ such that $e_i = e$ and $f_i = f$ for all $i$ (notation", "as in Remark \\ref{remark-finite-separable-extension}). In particular", "$[L : K] = n e f$." ], "refs": [ "more-algebra-remark-finite-separable-extension" ], "proofs": [ { "contents": [ "Immediate consequence of Lemma \\ref{lemma-galois} and the definitions." ], "refs": [ "more-algebra-lemma-galois" ], "ref_ids": [ 10498 ] } ], "ref_ids": [ 10676 ] }, { "id": 10500, "type": "theorem", "label": "more-algebra-lemma-galois-galois", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-galois-galois", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$ and residue field", "$\\kappa$. Let $L/K$ be a finite Galois extension with Galois group $G$.", "Let $B$ be the integral closure of $A$ in $L$. Let $\\mathfrak m$ be a maximal", "ideal of $B$. Then", "\\begin{enumerate}", "\\item the field extension $\\kappa(\\mathfrak m)/\\kappa$ is normal, and", "\\item $D \\to \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa)$ is surjective.", "\\end{enumerate}", "If for some (equivalently all) maximal ideal(s) $\\mathfrak m \\subset B$", "the field extension $\\kappa(\\mathfrak m)/\\kappa$ is separable, then", "\\begin{enumerate}", "\\item[(3)] $\\kappa(\\mathfrak m)/\\kappa$ is Galois, and", "\\item[(4)] $D \\to \\text{Gal}(\\kappa(\\mathfrak m)/\\kappa)$ is surjective.", "\\end{enumerate}", "Here $D \\subset G$ is the decomposition group of $\\mathfrak m$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $A = B^G$ as $A$ is integrally closed in $K$ and $K = L^G$.", "Thus parts (1) and (2) follow from Lemma \\ref{lemma-one-orbit-geometric}.", "The ``equivalently all'' part of the lemma follows from", "Lemma \\ref{lemma-galois}. Assume $\\kappa(\\mathfrak m)/\\kappa$", "is separable. Then parts (3) and (4) follow immediately from (1) and (2)." ], "refs": [ "more-algebra-lemma-one-orbit-geometric", "more-algebra-lemma-galois" ], "ref_ids": [ 10489, 10498 ] } ], "ref_ids": [] }, { "id": 10501, "type": "theorem", "label": "more-algebra-lemma-galois-inertia", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-galois-inertia", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "Let $L/K$ be a finite Galois extension with Galois group $G$.", "Let $B$ be the integral closure of $A$ in $L$. Let $\\mathfrak m \\subset B$", "be a maximal ideal. The inertia group $I$ of $\\mathfrak m$", "sits in a canonical exact sequence", "$$", "1 \\to P \\to I \\to I_t \\to 1", "$$", "such that", "\\begin{enumerate}", "\\item $P = \\{\\sigma \\in D \\mid", "\\sigma|_{B/\\mathfrak m^2} = \\text{id}_{B/\\mathfrak m^2}\\}$", "where $D$ is the decomposition group,", "\\item $P$ is a normal subgroup of $D$,", "\\item $P$ is a $p$-group if the characteristic of $\\kappa_A$ is", "$p > 0$ and $P = \\{1\\}$ if the characteristic of $\\kappa_A$ is zero,", "\\item $I_t$ is cyclic of order the prime to $p$ part of the integer $e$, and", "\\item there is a canonical isomorphism", "$\\theta : I_t \\to \\mu_e(\\kappa(\\mathfrak m))$.", "\\end{enumerate}", "Here $e$ is the integer of Lemma \\ref{lemma-galois-conclusion}." ], "refs": [ "more-algebra-lemma-galois-conclusion" ], "proofs": [ { "contents": [ "Recall that $|G| = [L : K] = nef$, see Lemma \\ref{lemma-galois-conclusion}.", "Since $G$ acts transitively on the set", "$\\{\\mathfrak m_1, \\ldots, \\mathfrak m_n\\}$ of maximal ideals of $B$", "(Lemma \\ref{lemma-galois})", "and since $D$ is the stabilizer of an element we see that $|D| = ef$.", "By Lemma \\ref{lemma-galois-galois} we have", "$$", "ef = |D| = |I| \\cdot |\\text{Aut}(\\kappa(\\mathfrak m)/\\kappa)|", "$$", "where $\\kappa$ is the residue field of $A$.", "As $\\kappa(\\mathfrak m)$ is normal over $\\kappa$ the order of", "$\\text{Aut}(\\kappa(\\mathfrak m)/\\kappa)$ differs from $f$ by", "a power of $p$ (see", "Fields, Lemma \\ref{fields-lemma-normal-and-automorphisms}", "and discussion following", "Fields, Definition \\ref{fields-definition-insep-degree}).", "Hence the prime to $p$ part", "of $|I|$ is equal to the prime to $p$ part of $e$.", "\\medskip\\noindent", "Set $C = B_\\mathfrak m$. Then $I$ acts on $C$ over $A$ and trivially", "on the residue field of $C$. Let $\\pi_A \\in A$ and $\\pi_C \\in C$ be", "uniformizers. Write $\\pi_A = u \\pi_C^e$ for some unit $u$ in $C$.", "For $\\sigma \\in I$ write $\\sigma(\\pi_C) = \\theta_\\sigma \\pi_C$ for some", "unit $\\theta_\\sigma$ in $C$. Then we have", "$$", "\\pi_A = \\sigma(\\pi_A) = \\sigma(u) (\\theta_\\sigma \\pi_C)^e", "= \\sigma(u) \\theta_\\sigma^e \\pi_C^e = \\frac{\\sigma(u)}{u} \\theta_\\sigma^e \\pi_A", "$$", "Since $\\sigma(u) \\equiv u \\bmod \\mathfrak m_C$ as $\\sigma \\in I$", "we see that the image $\\overline{\\theta}_\\sigma$", "of $\\theta_\\sigma$ in $\\kappa_C = \\kappa(\\mathfrak m)$", "is an $e$th root of unity.", "We obtain a map", "\\begin{equation}", "\\label{equation-inertia-character}", "\\theta : I \\longrightarrow \\mu_e(\\kappa(\\mathfrak m)),\\quad", "\\sigma \\mapsto \\overline{\\theta}_\\sigma", "\\end{equation}", "We claim that $\\theta$ is a homomorphism of groups and independent", "of the choice of uniformizer $\\pi_C$. Namely, if $\\tau$ is a second", "element of $I$, then", "$\\tau(\\sigma(\\pi_C)) = \\tau(\\theta_\\sigma \\pi_C) =", "\\tau(\\theta_\\sigma) \\theta_\\tau \\pi_C$, hence", "$\\theta_{\\tau \\sigma} = \\tau(\\theta_\\sigma) \\theta_\\tau$ and", "since $\\tau \\in I$ we conclude that", "$\\overline{\\theta}_{\\tau \\sigma} =", "\\overline{\\theta}_\\sigma \\overline{\\theta}_\\tau$.", "If $\\pi'_C$ is a second uniformizer, then we see", "that $\\pi'_C = w \\pi_C$ for some unit $w$ of $C$ and", "$\\sigma(\\pi'_C) = w^{-1}\\sigma(w)\\theta_\\sigma \\pi'_C$,", "hence $\\theta'_\\sigma = w^{-1}\\sigma(w)\\theta_\\sigma$,", "hence $\\theta'_\\sigma$ and $\\theta_\\sigma$", "map to the same element of the residue field as before.", "\\medskip\\noindent", "Since $\\kappa(\\mathfrak m)$ has characteristic $p$, the group", "$\\mu_e(\\kappa(\\mathfrak m))$ is cyclic of order at most the prime", "to $p$ part of $e$ (see Fields, Section \\ref{fields-section-roots-of-1}).", "\\medskip\\noindent", "Let $P = \\Ker(\\theta)$. The elements of $P$ are exactly the elements", "of $D$ acting trivially on $C/\\pi_C^2C \\cong B/\\mathfrak m^2$.", "Thus (a) is true. This implies (b) as $P$ is the kernel", "of the map $D \\to \\text{Aut}(B/\\mathfrak m^2)$.", "If we can prove (c), then parts (d) and (e) will follow as $I_t$", "will be isomorphic to $\\mu_e(\\kappa(\\mathfrak m))$ as the arguments above show", "that $|I_t| \\geq |\\mu_e(\\kappa(\\mathfrak m))|$.", "\\medskip\\noindent", "Thus it suffices to prove that the", "kernel $P$ of $\\theta$ is a $p$-group. Let $\\sigma$ be a nontrivial element of", "the kernel. Then $\\sigma - \\text{id}$", "sends $\\mathfrak m_C^i$ into $\\mathfrak m_C^{i + 1}$", "for all $i$. Let $m$ be the order of $\\sigma$. Pick $c \\in C$ such", "that $\\sigma(c) \\not = c$. Then $\\sigma(c) - c \\in \\mathfrak m_C^i$,", "$\\sigma(c) - c \\not \\in \\mathfrak m_C^{i + 1}$ for some $i$ and", "we have", "\\begin{align*}", "0", "& =", "\\sigma^m(c) - c \\\\", "& =", "\\sigma^m(c) - \\sigma^{m - 1}(c) + \\ldots + \\sigma(c) - c \\\\", "& =", "\\sum\\nolimits_{j = 0, \\ldots, m - 1} \\sigma^j(\\sigma(c) - c) \\\\", "& \\equiv", "m(\\sigma(c) - c) \\bmod \\mathfrak m_C^{i + 1}", "\\end{align*}", "It follows that $p | m$ (or $m = 0$ if $p = 1$). Thus every element of the", "kernel of $\\theta$ has order divisible by $p$, i.e., $\\Ker(\\theta)$", "is a $p$-group." ], "refs": [ "more-algebra-lemma-galois-conclusion", "more-algebra-lemma-galois", "more-algebra-lemma-galois-galois", "fields-lemma-normal-and-automorphisms", "fields-definition-insep-degree" ], "ref_ids": [ 10499, 10498, 10500, 4491, 4540 ] } ], "ref_ids": [ 10499 ] }, { "id": 10502, "type": "theorem", "label": "more-algebra-lemma-inertia-character", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-inertia-character", "contents": [ "With assumptions and notation as in Lemma \\ref{lemma-galois-inertia}.", "The inertia character $\\theta : I \\to \\mu_e(\\kappa(\\mathfrak m))$", "satisfies the following property", "$$", "\\theta(\\tau \\sigma \\tau^{-1}) = \\tau(\\theta(\\sigma))", "$$", "for $\\tau \\in D$ and $\\sigma \\in I$." ], "refs": [ "more-algebra-lemma-galois-inertia" ], "proofs": [ { "contents": [ "The formula makes sense as $I$ is a normal subgroup of $D$", "and as $\\tau$ acts on $\\kappa(\\mathfrak m)$ via the map", "$D \\to \\text{Aut}(\\kappa(\\mathfrak m))$ discussed in", "Lemma \\ref{lemma-galois-galois} for example.", "Recall the construction of $\\theta$. Choose", "a uniformizer $\\pi$ of $B_\\mathfrak m$ and for", "$\\sigma \\in I$ write $\\sigma(\\pi) = \\theta_\\sigma \\pi$.", "Then $\\theta(\\sigma)$ is the image $\\overline{\\theta}_\\sigma$", "of $\\theta_\\sigma$ in the residue field. For any $\\tau \\in D$", "we can write $\\tau(\\pi) = \\theta_\\tau \\pi$ for some unit $\\theta_\\tau$.", "Then $\\theta_{\\tau^{-1}} = \\tau^{-1}(\\theta_\\tau^{-1})$.", "We compute", "\\begin{align*}", "\\theta_{\\tau \\sigma \\tau^{-1}}", "& =", "\\tau(\\sigma(\\tau^{-1}(\\pi)))/\\pi \\\\", "& =", "\\tau(\\sigma(\\tau^{-1}(\\theta_\\tau^{-1}) \\pi))/\\pi \\\\", "& =", "\\tau(\\sigma(\\tau^{-1}(\\theta_\\tau^{-1})) \\theta_\\sigma \\pi)/\\pi \\\\", "& =", "\\tau(\\sigma(\\tau^{-1}(\\theta_\\tau^{-1}))) \\tau(\\theta_\\sigma) \\theta_\\tau", "\\end{align*}", "However, since $\\sigma$ acts trivially modulo $\\pi$ we see that", "the product $\\tau(\\sigma(\\tau^{-1}(\\theta_\\tau^{-1}))) \\theta_\\tau$", "maps to $1$ in the residue field. This proves the lemma." ], "refs": [ "more-algebra-lemma-galois-galois" ], "ref_ids": [ 10500 ] } ], "ref_ids": [ 10501 ] }, { "id": 10503, "type": "theorem", "label": "more-algebra-lemma-inertial-invariants-unramified", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-inertial-invariants-unramified", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "Let $L/K$ be a finite Galois extension. Let $\\mathfrak m \\subset B$", "be a maximal ideal of the integral closure of $A$ in $L$.", "Let $I \\subset G$ be the inertia group of $\\mathfrak m$.", "Then $B^I$ is the integral closure of $A$ in $L^I$ and", "$A \\to (B^I)_{B^I \\cap \\mathfrak m}$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Write $B' = B^I$. It follows from the definitions that $B' = B^I$", "is the integral closure of $A$ in $L^I$. Write", "$\\mathfrak m' = B^I \\cap \\mathfrak m = B' \\cap \\mathfrak m \\subset B'$.", "By Lemma \\ref{lemma-one-orbit} the maximal ideal $\\mathfrak m$ is the", "unique prime ideal of $B$ lying over $\\mathfrak m'$.", "As $I$ acts trivially on $\\kappa(\\mathfrak m)$ we see from", "Lemma \\ref{lemma-invariants-modulo} that the extension", "$\\kappa(\\mathfrak m)/\\kappa(\\mathfrak m')$ is purely inseparable", "(perhaps an easier alternative is to apply the result of", "Lemma \\ref{lemma-one-orbit-geometric}).", "Since $D/I$ acts faithfully on $\\kappa(\\mathfrak m')$,", "we conclude that $D/I$ acts faithfully on $\\kappa(\\mathfrak m)$.", "Of course the elements of the residue field $\\kappa$ of $A$", "are fixed by this action.", "By Galois theory we see that $[\\kappa(\\mathfrak m') : \\kappa] \\geq |D/I|$,", "see Fields, Lemma \\ref{fields-lemma-galois-over-fixed-field}.", "\\medskip\\noindent", "Let $\\pi$ be the uniformizer of $A$. Since", "$\\text{Norm}_{L/K}(\\pi) = \\pi^{[L : K]}$ we see from", "Algebra, Lemma \\ref{algebra-lemma-finite-extension-dim-1}", "that", "$$", "|G| = [L : K] = [L : K]\\ \\text{ord}_A(\\pi) =", "|G/D|\\ [\\kappa(\\mathfrak m) : \\kappa]\\ \\text{ord}_{B_\\mathfrak m}(\\pi)", "$$", "as there are $n = |G/D|$ maximal ideals of $B$ which are all", "conjugate under $G$, see", "Remark \\ref{remark-finite-separable-extension} and", "Lemma \\ref{lemma-galois}.", "Applying the same reasoning to the finite extension", "the finite extension $L/L^I$ of degree $|I|$ we find", "$$", "|I|\\ \\text{ord}_{B'_{\\mathfrak m'}}(\\pi) =", "[\\kappa(\\mathfrak m) : \\kappa(\\mathfrak m')]\\ \\text{ord}_{B_\\mathfrak m}(\\pi)", "$$", "We conclude that", "$$", "\\text{ord}_{B'_{\\mathfrak m'}}(\\pi) =", "\\frac{|D/I|}{[\\kappa(\\mathfrak m') : \\kappa]}", "$$", "Since the left hand side is a positive integer and since the right hand", "side is $\\leq 1$ by the above, we conclude that we have equality,", "$\\text{ord}_{B'_{\\mathfrak m'}}(\\pi) = 1$ and", "$\\kappa(\\mathfrak m')/\\kappa$ has degree $|D/I|$.", "Thus $\\pi B'_{\\mathfrak m'} = \\mathfrak m' B_\\mathfrak m'$ and", "$\\kappa(\\mathfrak m')$ is Galois over $\\kappa$ with", "Galois group $D/I$, in particular separable, see", "Fields, Lemma \\ref{fields-lemma-finite-Galois}.", "By Algebra, Lemma \\ref{algebra-lemma-characterize-etale}", "we find that $A \\to B'_{\\mathfrak m'}$ is \\'etale", "as desired." ], "refs": [ "more-algebra-lemma-one-orbit", "more-algebra-lemma-invariants-modulo", "more-algebra-lemma-one-orbit-geometric", "fields-lemma-galois-over-fixed-field", "algebra-lemma-finite-extension-dim-1", "more-algebra-remark-finite-separable-extension", "more-algebra-lemma-galois", "fields-lemma-finite-Galois", "algebra-lemma-characterize-etale" ], "ref_ids": [ 10488, 10485, 10489, 4507, 1047, 10676, 10498, 4504, 1235 ] } ], "ref_ids": [] }, { "id": 10504, "type": "theorem", "label": "more-algebra-lemma-compare-inertia", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-compare-inertia", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "Let $M/L/K$ be a tower with $M/K$ and $L/K$ finite Galois.", "Let $C$, $B$ be the integral closure of $A$ in $M$, $L$.", "Let $\\mathfrak m' \\subset C$ be a maximal ideal and set", "$\\mathfrak m = \\mathfrak m' \\cap B$. Let", "$$", "P \\subset I \\subset D \\subset \\text{Gal}(L/K)", "\\quad\\text{and}\\quad", "P' \\subset I' \\subset D' \\subset \\text{Gal}(M/K)", "$$", "be the wild inertia, inertia, decomposition group of", "$\\mathfrak m$ and $\\mathfrak m'$.", "Then the canonical surjection $\\text{Gal}(M/K) \\to \\text{Gal}(L/K)$", "induces surjections $P' \\to P$, $I' \\to I$, and $D' \\to D$. Moreover", "these fit into commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "D' \\ar[r] \\ar[d] &", "\\text{Aut}(\\kappa(\\mathfrak m')/\\kappa_A) \\ar[d] \\\\", "D \\ar[r] &", "\\text{Aut}(\\kappa(\\mathfrak m)/\\kappa_A)", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "I' \\ar[r]_-{\\theta'} \\ar[d] &", "\\mu_{e'}(\\kappa(\\mathfrak m')) \\ar[d]^{(-)^{e'/e}} \\\\", "I \\ar[r]^-\\theta &", "\\mu_e(\\kappa(\\mathfrak m))", "}", "}", "$$", "where $e'$ and $e$ are the ramification indices of", "$A \\to C_{\\mathfrak m'}$ and $A \\to B_\\mathfrak m$." ], "refs": [], "proofs": [ { "contents": [ "The fact that under the map $\\text{Gal}(M/K) \\to \\text{Gal}(L/K)$", "the groups $P', I', D'$ map into $P, I, D$ is immediate from the", "definitions of these groups. The commutativity of the first diagram", "is clear (observe that since $\\kappa(\\mathfrak m)/\\kappa_A$ is normal", "every automorphism of $\\kappa(\\mathfrak m')$ over $\\kappa_A$ indeed", "induces an automorphism of $\\kappa(\\mathfrak m)$ over $\\kappa_A$", "and hence we obtain the right vertical arrow in the first diagram, see", "Lemma \\ref{lemma-galois-galois} and", "Fields, Lemma \\ref{fields-lemma-lift-maps}).", "\\medskip\\noindent", "The maps $I' \\to I$ and $D' \\to D$ are surjective by", "Lemma \\ref{lemma-one-orbit-geometric-galois-compare}.", "The surjectivity of $P' \\to P$ follows as $P'$ and $P$", "are p-Sylow subgroups of $I'$ and $I$.", "\\medskip\\noindent", "To see the commutativity of the second diagram we choose a uniformizer", "$\\pi'$ of $C_{\\mathfrak m'}$ and a uniformizer $\\pi$ of $B_\\mathfrak m$.", "Then $\\pi = c' (\\pi')^{e'/e}$ for some unit $c'$ of $C_{\\mathfrak m'}$.", "For $\\sigma' \\in I'$ the image $\\sigma \\in I$ is simply the restriction", "of $\\sigma'$ to $L$. Write $\\sigma'(\\pi') = c \\pi'$ for a unit", "$c \\in C_{\\mathfrak m'}$ and write", "$\\sigma(\\pi) = b \\pi$ for a unit $b$ of $B_\\mathfrak m$.", "Then $\\sigma'(\\pi) = b \\pi$ and we obtain", "$$", "b \\pi = \\sigma'(\\pi) = \\sigma'(c' (\\pi')^{e'/e}) =", "\\sigma'(c') c^{e'/e} (\\pi')^{e'/e} =", "\\frac{\\sigma'(c')}{c'} c^{e'/e} \\pi", "$$", "As $\\sigma' \\in I'$ we see that $b$ and $c^{e'/e}$ have the same", "image in the residue field which proves what we want." ], "refs": [ "more-algebra-lemma-galois-galois", "fields-lemma-lift-maps", "more-algebra-lemma-one-orbit-geometric-galois-compare" ], "ref_ids": [ 10500, 4490, 10491 ] } ], "ref_ids": [] }, { "id": 10505, "type": "theorem", "label": "more-algebra-lemma-krasner", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-krasner", "contents": [ "Let $A$ be a complete local domain of dimension $1$. Let $P(t) \\in A[t]$", "be a polynomial with coefficients in $A$. Let $\\alpha \\in A$ be a root", "of $P$ but not a root of the derivative $P' = \\text{d}P/\\text{d}t$.", "For every $c \\geq 0$ there exists an integer $n$ such that for any", "$Q \\in A[t]$ whose coefficients are in $\\mathfrak m_A^n$ the polynomial", "$P + Q$ has a root $\\beta \\in A$ with $\\beta - \\alpha \\in \\mathfrak m_A^c$." ], "refs": [], "proofs": [ { "contents": [ "Choose a nonzero $\\pi \\in \\mathfrak m$. Since the dimension of $A$ is $1$", "we have $\\mathfrak m = \\sqrt{(\\pi)}$. By assumption we may write", "$P'(\\alpha)^{-1} = \\pi^{-m} a$ for some $m \\geq 0$ and $a \\in A$.", "We may and do assume that $c \\geq m + 1$.", "Pick $n$ such that $\\mathfrak m_A^n \\subset (\\pi^{c + m})$.", "Pick any $Q$ as in the statement. For later use we observe that we can write", "$$", "P(x + y) = P(x) + P'(x)y + R(x, y)y^2", "$$", "for some $R(x, y) \\in A[x, y]$. We will show by induction that we can find a", "sequence $\\alpha_m, \\alpha_{m + 1}, \\alpha_{m + 2}, \\ldots$ such that", "\\begin{enumerate}", "\\item $\\alpha_k \\equiv \\alpha \\bmod \\pi^c$,", "\\item $\\alpha_{k + 1} - \\alpha_k \\in (\\pi^k)$, and", "\\item $(P + Q)(\\alpha_k) \\in (\\pi^{m + k})$.", "\\end{enumerate}", "Setting $\\beta = \\lim \\alpha_k$ will finish the proof.", "\\medskip\\noindent", "Base case. Since the coefficients of $Q$ are in $(\\pi^{c + m})$ we have", "$(P + Q)(\\alpha) \\in (\\pi^{c + m})$. Hence $\\alpha_m = \\alpha$ works.", "This choice guarantees that $\\alpha_k \\equiv \\alpha \\bmod \\pi^c$", "for all $k \\geq m$.", "\\medskip\\noindent", "Induction step. Given $\\alpha_k$ we write", "$\\alpha_{k + 1} = \\alpha_k + \\delta$ for some $\\delta \\in (\\pi^k)$.", "Then we have", "$$", "(P + Q)(\\alpha_{k + 1}) =", "P(\\alpha_k + \\delta) + Q(\\alpha_k + \\delta)", "$$", "Because the coefficients of $Q$ are in $(\\pi^{c + m})$ we see that", "$Q(\\alpha_k + \\delta) \\equiv Q(\\alpha_k) \\bmod \\pi^{c + m + k}$.", "On the other hand we have", "$$", "P(\\alpha_k + \\delta) =", "P(\\alpha_k) + P'(\\alpha_k)\\delta + R(\\alpha_k, \\delta)\\delta^2", "$$", "Note that $P'(\\alpha_k) \\equiv P'(\\alpha) \\bmod (\\pi^{m + 1})$", "as $\\alpha_k \\equiv \\alpha \\bmod \\pi^{m + 1}$. Hence we obtain", "$$", "P(\\alpha_k + \\delta) \\equiv P(\\alpha_k) + P'(\\alpha) \\delta", "\\bmod \\pi^{k + m + 1}", "$$", "Recombining the two terms we see that", "$$", "(P + Q)(\\alpha_{k + 1}) \\equiv (P + Q)(\\alpha_k) + P'(\\alpha) \\delta", "\\bmod \\pi^{k + m + 1}", "$$", "Thus a solution is to take", "$\\delta = -P'(\\alpha)^{-1} (P + Q)(\\alpha_k) = - \\pi^{-m} a (P + Q)(\\alpha_k)$", "which is contained in $(\\pi^k)$ by induction assumption." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10506, "type": "theorem", "label": "more-algebra-lemma-approximate-separable-extension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-approximate-separable-extension", "contents": [ "Let $A$ be a discrete valuation ring with field of fractions $K$.", "Let $A^\\wedge$ be the completion of $A$ with fraction field $K^\\wedge$.", "If $M/K^\\wedge$ is a finite separable extension, then", "there exists a finite separable extension $L/K$", "such that $M = K^\\wedge \\otimes_K L$." ], "refs": [], "proofs": [ { "contents": [ "Note that $A^\\wedge$ is a discrete valuation ring too (by", "Lemmas \\ref{lemma-completion-regular} and \\ref{lemma-completion-dimension}).", "In particular $A^\\wedge$ is a domain. The proof will work more generally", "for Noetherian local rings $A$ such that $A^\\wedge$ is a local domain", "of dimension $1$.", "\\medskip\\noindent", "Let $\\theta \\in M$ be an element that generates $M$ over $K^\\wedge$.", "(Theorem of the primitive element.)", "Let $P(t) \\in K^\\wedge[t]$ be the minimal polynomial of $\\theta$ over", "$K^\\wedge$. Let $\\pi \\in \\mathfrak m_A$ be a nonzero element.", "After replacing $\\theta$ by $\\pi^n\\theta$ we may assume that", "the coefficients of $P(t)$ are in $A^\\wedge$. Let", "$B = A^\\wedge[\\theta] = A^\\wedge[t]/(P(t))$. Note that $B$ is", "a complete local domain of dimension $1$ because it is finite over $A$ and", "contained in $M$. Since $M$ is separable over $K$ the element $\\theta$", "is not a root of the derivative of $P$. For any integer $n$ we can find", "a monic polynomial $P_1 \\in A[t]$ such that $P - P_1$ has coefficients in", "$\\pi^nA^\\wedge[t]$. By Krasner's lemma (Lemma \\ref{lemma-krasner}) we see that", "$P_1$ has a root $\\beta$ in $B$ for $n$ sufficiently large.", "Moreover, we may assume (if $n$ is chosen large enough)", "that $\\theta - \\beta \\in \\pi B$. Consider the map", "$\\Phi : A^\\wedge[t]/(P_1) \\to B$ of $A^\\wedge$-algebras", "which maps $t$ to $\\beta$. Since", "$B = \\pi B + \\sum_{i < \\deg(P)} A^\\wedge \\theta^i$, the map $\\Phi$", "is surjective by Nakayama's lemma. As $\\deg(P_1) = \\deg(P)$ it", "follows that $\\Phi$ is an isomorphism. We conclude that the ring", "extension $L = K[t]/(P_1(t))$ satisfies $K^\\wedge \\otimes_K L \\cong M$.", "This implies that $L$ is a field and the proof is complete." ], "refs": [ "more-algebra-lemma-completion-regular", "more-algebra-lemma-completion-dimension", "more-algebra-lemma-krasner" ], "ref_ids": [ 10045, 10042, 10505 ] } ], "ref_ids": [] }, { "id": 10507, "type": "theorem", "label": "more-algebra-lemma-pull-root-uniformizer", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pull-root-uniformizer", "contents": [ "Let $A$ be a discrete valuation ring with uniformizer $\\pi$.", "Let $n \\geq 2$. Then $K_1 = K[\\pi^{1/n}]$ is a degree $n$ extension of $K$", "and the integral closure $A_1$ of $A$ in $K_1$ is the ring $A[\\pi^{1/n}]$", "which is a discrete valuation ring with ramification index $n$ over $A$." ], "refs": [], "proofs": [ { "contents": [ "This lemma proves itself." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10508, "type": "theorem", "label": "more-algebra-lemma-formally-smooth-goes-up", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-formally-smooth-goes-up", "contents": [ "Let $A \\to B$ be an extension of discrete valuation rings with fraction fields", "$K \\subset L$. Assume that $A \\to B$ is formally smooth in the", "$\\mathfrak m_B$-adic topology. Then for any finite extension $K \\subset K_1$", "we have $L_1 = L \\otimes_K K_1$, $B_1 = B \\otimes_A A_1$, and each extension", "$(A_1)_{\\mathfrak m_i} \\subset (B_1)_{\\mathfrak m_{ij}}$ (see", "Remark \\ref{remark-construction}) is formally smooth in the", "$\\mathfrak m_{ij}$-adic topology." ], "refs": [ "more-algebra-remark-construction" ], "proofs": [ { "contents": [ "We will use the equivalence of Lemma \\ref{lemma-extension-dvrs-formally-smooth}", "without further mention. Let $\\pi \\in A$ and $\\pi_i \\in (A_1)_{\\mathfrak m_i}$", "be uniformizers. As $\\kappa_A \\subset \\kappa_B$ is separable, the ring", "$$", "(B \\otimes_A (A_1)_{\\mathfrak m_i})/\\pi_i (B \\otimes_A (A_1)_{\\mathfrak m_i}) =", "B/\\pi B \\otimes_{A/\\pi A} (A_1)_{\\mathfrak m_i}/\\pi_i (A_1)_{\\mathfrak m_i}", "$$", "is a product of fields each separable over $\\kappa_{\\mathfrak m_i}$.", "Hence the element $\\pi_i$ in $B \\otimes_A (A_1)_{\\mathfrak m_i}$", "is a nonzerodivisor and the quotient by this element is a product of fields.", "It follows that $B \\otimes_A A_1$ is a Dedekind domain in particular", "reduced. Thus $B \\otimes_A A_1 \\subset B_1$ is an equality." ], "refs": [ "more-algebra-lemma-extension-dvrs-formally-smooth" ], "ref_ids": [ 10495 ] } ], "ref_ids": [ 10679 ] }, { "id": 10509, "type": "theorem", "label": "more-algebra-lemma-abhyankar", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-abhyankar", "contents": [ "Let $A \\subset B$ be an extension of discrete valuation rings.", "Assume that either the residue characteristic of $A$ is $0$", "or it is $p$, the ramification index $e$ is prime to $p$, and", "$\\kappa_B/\\kappa_A$ is a separable field extension.", "Let $K_1/K$ be a finite extension. Using the notation of", "Remark \\ref{remark-construction}", "assume $e$ divides the ramification index of $A \\subset (A_1)_{\\mathfrak m_i}$", "for some $i$. Then $(A_1)_{\\mathfrak m_i} \\subset (B_1)_{\\mathfrak m_{ij}}$", "is formally smooth $\\mathfrak m_{ij}$-adic topology", "for all $j = 1, \\ldots, m_i$." ], "refs": [ "more-algebra-remark-construction" ], "proofs": [ { "contents": [ "Let $\\pi \\in A$ be a uniformizer. Let $\\pi_1$ be a uniformizer", "of $(A_1)_{\\mathfrak m_i}$. Write $\\pi = u \\pi_1^{e_1}$ with $u$ a unit", "of $(A_1)_{\\mathfrak m_i}$ and $e_1$ the ramification index of", "$A \\subset (A_1)_{\\mathfrak m_i}$.", "\\medskip\\noindent", "Claim: we may assume that $u$ is an $e$th power in $K_1$.", "Namely, let $K_2$ be an extension of $K_1$ obtained by", "adjoining a root of $x^e = u$; thus $K_2$ is a factor", "of $K_1[x]/(x^e - u)$. Then $K_2/K_1$ is a finite", "separable extension (by our assumption on $e$)", "and hence $A_1 \\subset A_2$ is finite.", "Since $(A_1)_{\\mathfrak m_i} \\to (A_1)_{\\mathfrak m_i}[x]/(x^e - u)$", "is finite \\'etale", "(as $e$ is prime to the residue characteristic and $u$ a unit)", "we conclude that $(A_2)_{\\mathfrak m_i}$ is a factor of", "a finite \\'etale extension of $(A_1)_{\\mathfrak m_i}$ hence", "finite \\'etale over $(A_1)_{\\mathfrak m_i}$ itself.", "The same reasoning shows that $B_1 \\subset B_2$ induces", "finite \\'etale extensions", "$(B_1)_{\\mathfrak m_{ij}} \\subset (B_2)_{\\mathfrak m_{ij}}$.", "Pick a maximal ideal $\\mathfrak m'_{ij} \\subset B_2$", "lying over $\\mathfrak m_{ij} \\subset B_1$", "(of course there may be more than one) and consider", "$$", "\\xymatrix{", "(B_1)_{\\mathfrak m_{ij}} \\ar[r] & (B_2)_{\\mathfrak m'_{ij}} \\\\", "(A_1)_{\\mathfrak m_i} \\ar[u] \\ar[r] &", "(A_2)_{\\mathfrak m'_i} \\ar[u]", "}", "$$", "where $\\mathfrak m'_i \\subset A_2$ is the image.", "Now the horizontal arrows have ramification index $1$", "and induce finite separable residue field extensions.", "Thus, using the equivalence of", "Lemma \\ref{lemma-extension-dvrs-formally-smooth},", "we see that it suffices to show that the right vertical", "arrow is formally smooth in the $\\mathfrak m'_{ij}$-adic topology.", "Since $u$ has a $e$th root", "in $K_2$ we obtain the claim.", "\\medskip\\noindent", "Assume $u$ has an $e$th root in $K_1$.", "Since $e | e_1$ and since $u$ has a $e$th root in $K_1$", "we see that $\\pi = \\theta^e$ for some $\\theta \\in K_1$.", "Let $K[\\theta] \\subset K_1$ be the subfield generated by $\\theta$.", "By Lemma \\ref{lemma-pull-root-uniformizer} the integral closure of", "$A$ in $K[\\theta]$ is the discrete valuation ring $A[\\theta]$.", "If we can prove the lemma for the extension $K \\subset K[\\theta]$,", "then $K \\subset K[\\theta]$ is a solution for $A \\subset B$ and", "we conclude by", "Lemma \\ref{lemma-formally-smooth-goes-up}.", "\\medskip\\noindent", "Assume $K_1 = K[\\pi^{1/e}]$ and set $\\theta = \\pi^{1/e}$. Let $\\pi_B$ be a", "uniformizer for $B$ and write $\\pi = w \\pi_B^e$ for some unit $w$ of $B$.", "Then we see that $L_1 = L \\otimes_K K_1$ is obtained by adjoining", "$\\pi_B/\\theta$ which is an $e$th root of the unit $w$. Thus", "$B \\subset B_1$ is finite \\'etale. Thus for any maximal ideal", "$\\mathfrak m \\subset B_1$ consider the commutative diagram", "$$", "\\xymatrix{", "B \\ar[r]_1 & (B_1)_{\\mathfrak m} \\\\", "A \\ar[u]^e \\ar[r]^e & A_1 \\ar[u]_{e_\\mathfrak m}", "}", "$$", "Here the numbers along the arrows are the ramification indices.", "By multiplicativity of ramification indices", "(Lemma \\ref{lemma-multiplicative-e-f})", "we conclude $e_\\mathfrak m = 1$. Looking at the residue field extensions", "we find that $\\kappa(\\mathfrak m)$ is a finite separable extension", "of $\\kappa_B$ which is separable over $\\kappa_A$. Therefore", "$\\kappa(\\mathfrak m)$ is separable over $\\kappa_A$", "which is equal to the residue field of $A_1$ and we win by", "Lemma \\ref{lemma-extension-dvrs-formally-smooth}." ], "refs": [ "more-algebra-lemma-extension-dvrs-formally-smooth", "more-algebra-lemma-pull-root-uniformizer", "more-algebra-lemma-formally-smooth-goes-up", "more-algebra-lemma-multiplicative-e-f", "more-algebra-lemma-extension-dvrs-formally-smooth" ], "ref_ids": [ 10495, 10507, 10508, 10493, 10495 ] } ], "ref_ids": [ 10679 ] }, { "id": 10510, "type": "theorem", "label": "more-algebra-lemma-composition-tame", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-composition-tame", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "Let $M/L/K$ be finite separable extensions.", "Let $B$ be the integral closure of $A$ in $L$.", "If $L/K$ is tamely ramified with respect to $A$", "and $M/L$ is tamely ramified with respect to $B_\\mathfrak m$", "for every maximal ideal $\\mathfrak m$ of $B$, then", "$M/K$ is tamely ramified with respect to $A$." ], "refs": [], "proofs": [ { "contents": [ "Let $C$ be the integral closure of $A$ in $M$.", "Every maximal ideal $\\mathfrak m'$ of $C$ lies over", "a maximal ideal $\\mathfrak m$ of $B$.", "Then the lemma follows from the multiplicativity", "of ramification indices (Lemma \\ref{lemma-multiplicative-e-f})", "and the fact that we have the tower", "$\\kappa(\\mathfrak m')/\\kappa(\\mathfrak m)/\\kappa_A$", "of finite extensions of fields." ], "refs": [ "more-algebra-lemma-multiplicative-e-f" ], "ref_ids": [ 10493 ] } ], "ref_ids": [] }, { "id": 10511, "type": "theorem", "label": "more-algebra-lemma-subextension-tame", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-subextension-tame", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "If $M/L/K$ are finite separable extensions and", "$M$ is tamely ramified with respect to $A$, then", "$L$ is tamely ramified with respect to $A$." ], "refs": [], "proofs": [ { "contents": [ "We will use the results of the discussion in", "Remark \\ref{remark-finite-separable-extension}", "without further mention.", "Let $C/B/A$ be the integral closures of $A$ in $M/L/K$.", "Since $C$ is a finite ring extension of $B$, we see that", "$\\Spec(C) \\to \\Spec(B)$ is surjective. Hence for ever maximal", "ideal $\\mathfrak m \\subset B$ there is a maximal ideal", "$\\mathfrak m' \\subset C$ lying over $\\mathfrak m$.", "By the multiplicativity of ramification indices", "(Lemma \\ref{lemma-multiplicative-e-f})", "and the assumption, we conclude that the ramification", "index of $B_\\mathfrak m$ over $A$ is prime to the residue characteristic.", "Since $\\kappa(\\mathfrak m')/\\kappa_A$ is finite separable,", "the same is true for $\\kappa(\\mathfrak m)/\\kappa_A$." ], "refs": [ "more-algebra-remark-finite-separable-extension", "more-algebra-lemma-multiplicative-e-f" ], "ref_ids": [ 10676, 10493 ] } ], "ref_ids": [] }, { "id": 10512, "type": "theorem", "label": "more-algebra-lemma-characterize-tame", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-characterize-tame", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "Let $\\pi \\in A$ be a uniformizer.", "Let $L/K$ be a finite separable extension.", "The following are equivalent", "\\begin{enumerate}", "\\item $L$ is tamely ramified with respect to $A$,", "\\item there exists an $e \\geq 1$ invertible in $\\kappa_A$", "and an extension $L'/K' = K[\\pi^{1/e}]$ unramified with respect to", "$A' = A[\\pi^{1/e}]$ such that $L$ is contained in $L'$, and", "\\item there exists an $e_0 \\geq 1$ invertible in $\\kappa_A$", "such that for every $d \\geq 1$ invertible in $\\kappa_A$", "(2) holds with $e = de_0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $A'$ is a discrete valuation ring with fraction", "field $K'$, see Lemma \\ref{lemma-pull-root-uniformizer}.", "Of course the ramification index of $A'$ over $A$ is $e$.", "Thus if (2) holds, then $L'$ is tamely ramified with respect to $A$", "by Lemma \\ref{lemma-composition-tame}. Hence $L$ is tamely", "ramified with respect to $A$ by Lemma \\ref{lemma-subextension-tame}.", "\\medskip\\noindent", "The implication (3) $\\Rightarrow$ (2) is immediate.", "\\medskip\\noindent", "Assume that (1) holds. Let $B$ be the integral closure of $A$ in $L$", "and let $\\mathfrak m_1, \\ldots, \\mathfrak m_n$ be its maximal ideals.", "Denote $e_i$ the ramification index of $A \\to B_{\\mathfrak m_i}$.", "Let $e_0$ be the least common multiple of $e_1, \\ldots, e_r$.", "This is invertible in $\\kappa_A$ by our assumption (1).", "Let $e = de_0$ as in (3). Set $A' = A[\\pi^{1/e}]$.", "Then $A \\to A'$ is an extension of discrete valuation rings", "with fraction field $K' = K[\\pi^{1/e}]$, see", "Lemma \\ref{lemma-pull-root-uniformizer}.", "Choose a product decomposition", "$$", "L \\otimes_K K' = \\prod L'_j", "$$", "where $L'_j$ are fields. Let $B'_j$ be the integral closure", "of $A$ in $L'_j$. Let $\\mathfrak m_{ijk}$ be the maximal", "ideals of $B'_j$ lying over $\\mathfrak m_i$.", "Observe that $(B'_j)_{\\mathfrak m_i}$ is the integral", "closure of $B_{\\mathfrak m_i}$ in $L'_j$.", "By Abhyankar's lemma (Lemma \\ref{lemma-abhyankar})", "applied to $A \\subset B_{\\mathfrak m_i}$ and the extension $K \\subset K'$", "we see that $A' \\to (B'_j)_{\\mathfrak m_{ijk}}$", "is formally smooth in the $\\mathfrak m_{ijk}$-adic topology.", "This implies that the ramification index is $1$ and", "that the residue field extension is separable", "(Lemma \\ref{lemma-extension-dvrs-formally-smooth}).", "In this way we see that $L'_j$ is unramified with respect to $A'$.", "This finishes the proof: we take $L' = L'_j$ for some $j$." ], "refs": [ "more-algebra-lemma-pull-root-uniformizer", "more-algebra-lemma-composition-tame", "more-algebra-lemma-subextension-tame", "more-algebra-lemma-pull-root-uniformizer", "more-algebra-lemma-abhyankar", "more-algebra-lemma-extension-dvrs-formally-smooth" ], "ref_ids": [ 10507, 10510, 10511, 10507, 10509, 10495 ] } ], "ref_ids": [] }, { "id": 10513, "type": "theorem", "label": "more-algebra-lemma-permanence-tame", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-permanence-tame", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "\\begin{enumerate}", "\\item If $L/K$ is a finite separable extension which is tamely", "ramified with respect to $A$, then there exists a Galois", "extension $M/K$ containing $L$ which is tamely ramified", "with respect to $A$.", "\\item If $L_1/K$, $L_2/K$ are finite separable extensions which are tamely", "ramified with respect to $A$, then there exists a a finite", "separable extension $L/K$ which is tamely ramified with respect", "to $A$ containing $L_1$ and $L_2$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (2). Choose a uniformizer $\\pi \\in A$.", "We can choose an integer $e$ invertible", "in $\\kappa_A$ and extensions $L_i'/K' = K[\\pi^{1/e}]$", "unramified with respect to $A' = A[\\pi^{1/e}]$", "with $L_i \\subset L'_i$ as extensions of $K$, see", "Lemma \\ref{lemma-characterize-tame}.", "By Lemma \\ref{lemma-permanence-unramified}", "we can find an extension $L'/K'$ which is unramified", "with respect to $A'$ such that $L'_i/K$ is isomorphic", "to a subextension of $L'/K'$ for $i = 1, 2$.", "This finishes the proof of (3) as $L'/K$ is tamely ramified", "(use same lemma as above).", "\\medskip\\noindent", "Proof of (1). We may first replace $L$ by a larger extension", "and assume that $L$ is an extension of $K' = K[\\pi^{1/e}]$", "unramified with respect to $A' = A[\\pi^{1/e}]$ where $e$ is", "invertible in $\\kappa_A$, see Lemma \\ref{lemma-characterize-tame}.", "Let $M$ be the normal closure of $L$ over $K$, see", "Fields, Definition \\ref{fields-definition-normal-closure}.", "Then $M/K$ is Galois by Fields, Lemma \\ref{fields-lemma-normal-closure-galois}.", "On the other hand, there is a surjection", "$$", "L \\otimes_K \\ldots \\otimes_K L \\longrightarrow M", "$$", "of $K$-algebras, see Fields, Lemma", "\\ref{fields-lemma-normal-closure-tensor-product}.", "Let $B$ be the integral closure of $A$ in $L$", "as in Remark \\ref{remark-finite-separable-extension}. The", "condition that $L$ is unramified with respect to $A' = A[\\pi^{1/e}]$", "exactly means that $A' \\to B$ is an \\'etale ring map, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-etale}.", "Claim: ", "$$", "K' \\otimes_K \\ldots \\otimes_K K' = \\prod K'_i", "$$", "is a product of field extensions $K'_i/K$ tamely", "ramified with respect to $A$. Then if $A'_i$ is the integral", "closure of $A$ in $K'_i$ we see that", "$$", "\\prod A'_i \\otimes_{(A' \\otimes_A \\ldots \\otimes_A A')}", "(B \\otimes_A \\ldots \\otimes_A B)", "$$", "is finite \\'etale over $\\prod A'_i$ and hence a product of", "Dedekind domains (Lemma \\ref{lemma-Dedekind-etale-extension}).", "We conclude that $M$ is the fraction field", "of one of these Dedekind domains which is finite \\'etale", "over $A'_i$ for some $i$. It follows that $M/K'_i$", "is unramified with respect to every maximal ideal of $A'_i$", "and hence $M/K$ is tamely ramified by Lemma \\ref{lemma-composition-tame}.", "\\medskip\\noindent", "It remains the prove the claim. For this we write", "$A' = A[x]/(x^e - \\pi)$ and we see that", "$$", "A' \\otimes_A \\ldots \\otimes_A A' =", "A'[x_1, \\ldots, x_r]/(x_1^e - \\pi, \\ldots, x_r^e - \\pi)", "$$", "The normalization of this ring certainly contains the", "elements $y_i = x_i/x_1$ for $i = 2, \\ldots, r$ subject", "to the relations $y_i^e - 1 = 0$ and", "we obtain", "$$", "A[x_1, y_2, \\ldots, y_r]/(x_1^e - \\pi, y_2^e - 1, \\ldots, y_r - 1) =", "A'[y_2, \\ldots, y_r]/(y_2^e - 1, \\ldots, y_r^e - 1)", "$$", "This ring is finite \\'etale over $A'$ because $e$ is invertible in $A'$.", "Hence it is a product of Dedekind domains each unramified over $A'$", "as desired (see references given above in case of confusion)." ], "refs": [ "more-algebra-lemma-characterize-tame", "more-algebra-lemma-permanence-unramified", "more-algebra-lemma-characterize-tame", "fields-definition-normal-closure", "fields-lemma-normal-closure-galois", "fields-lemma-normal-closure-tensor-product", "more-algebra-remark-finite-separable-extension", "algebra-lemma-characterize-etale", "more-algebra-lemma-Dedekind-etale-extension", "more-algebra-lemma-composition-tame" ], "ref_ids": [ 10512, 10496, 10512, 4544, 4506, 4496, 10676, 1235, 10054, 10510 ] } ], "ref_ids": [] }, { "id": 10514, "type": "theorem", "label": "more-algebra-lemma-tame-goes-up", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-tame-goes-up", "contents": [ "Let $A \\subset B$ be an extension of discrete valuation rings.", "Denote $L/K$ the corresponding extension of fraction fields.", "Let $K'/K$ be a finite separable extension.", "Then", "$$", "K' \\otimes_K L = \\prod L'_i", "$$", "is a finite product of fields and the following is true", "\\begin{enumerate}", "\\item If $K'$ is unramified with respect to $A$, then", "each $L'_i$ is unramified with respect to $B$.", "\\item If $K'$ is tamely ramified with respect to $A$, then", "each $L'_i$ is tamely ramified with respect to $B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The algebra $K' \\otimes_K L$ is a finite product of fields as it is", "a finite \\'etale algebra over $L$.", "Let $A'$ be the integral closure of $A$ in $K'$.", "\\medskip\\noindent", "In case (1) the ring map $A \\to A'$ is finite \\'etale.", "Hence $B' = B \\otimes_A A'$ is finite \\'etale over $B$ and", "is a finite product of Dedekind domains", "(Lemma \\ref{lemma-Dedekind-etale-extension}).", "Hence $B'$ is the integral closure of $B$ in $K' \\otimes_K L$.", "It follows immediately that each $L'_i$ is unramified", "with respect to $B$.", "\\medskip\\noindent", "Choose a uniformizer $\\pi \\in A$. To prove (2) we may replace $K'$", "by a larger extension tame ramified with respect to $A$ (details omitted;", "hint: use Lemma \\ref{lemma-subextension-tame}).", "Thus by Lemma \\ref{lemma-characterize-tame} we may assume", "there exists some $e \\geq 1$ invertible in $\\kappa_A$ such that", "$K'$ contains $K[\\pi^{1/e}]$ and such that $K'$ is unramified with respect", "to $A[\\pi^{1/e}]$. Choose a product decomposition", "$$", "K[\\pi^{1/e}] \\otimes_K L = \\prod L_{e, j}", "$$", "For every $i$ there exists a $j_i$ such that", "$L'_i/L_{e, j_i}$ is a finite separable extension.", "Let $B_{e, j}$ be the integral closure of $B$ in $L_{e, j}$.", "By (1) applied to $K'/K[\\pi^{1/e}]$ and", "$A[\\pi^{1/e}] \\subset (B_{e, j_i})_\\mathfrak m$", "we see that $L'_i$ is unramified with respect to $(B_{e, j_i})_\\mathfrak m$", "for every maximal ideal $\\mathfrak m \\subset B_{e, j_i}$.", "Hence the proof will be complete if we can show that", "$L_{e, j}$ is tamely ramified with respect to $A[\\pi^{1/e}]$, see", "Lemma \\ref{lemma-composition-tame}.", "\\medskip\\noindent", "Choose a uniformizer $\\theta$ in $B$.", "Write $\\pi = u \\theta^t$ where $u$ is a unit of $B$ and $t \\geq 1$.", "Then we have", "$$", "A[\\pi^{1/e}] \\otimes_A B = B[x]/(x^e - u \\theta^t)", "\\subset B[y, z]/(y^{e'} - \\theta, z^e - u)", "$$", "where $e' = e/\\gcd(e, t)$. The map sends $x$ to", "$z y^{t/\\gcd(e, t)}$. Since the right hand side is", "a product of Dedekind domains each tamely ramified", "over $B$ the proof is complete (details omitted)." ], "refs": [ "more-algebra-lemma-Dedekind-etale-extension", "more-algebra-lemma-subextension-tame", "more-algebra-lemma-characterize-tame", "more-algebra-lemma-composition-tame" ], "ref_ids": [ 10054, 10511, 10512, 10510 ] } ], "ref_ids": [] }, { "id": 10515, "type": "theorem", "label": "more-algebra-lemma-weakly-unramified-goes-up-along-totally-ramified", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-weakly-unramified-goes-up-along-totally-ramified", "contents": [ "Let $A \\to B$ be an extension of discrete valuation rings with fraction", "fields $K \\subset L$. Assume that $A \\to B$ is weakly unramified. Then for", "any finite separable extension $K_1/K$ totally ramified with respect to $A$", "we have that $L_1 = L \\otimes_K K_1$ is a field, $A_1$ and", "$B_1 = B \\otimes_A A_1$ are discrete valuation rings, and the extension", "$A_1 \\subset B_1$ (see", "Remark \\ref{remark-construction}) is weakly unramified." ], "refs": [ "more-algebra-remark-construction" ], "proofs": [ { "contents": [ "Let $\\pi \\in A$ and $\\pi_1 \\in A_1$ be uniformizers. As $K_1/K$", "is totally ramified with respect to $A$", "we have $\\pi_1^e = u_1 \\pi$ for some unit $u_1$ in $A_1$.", "Hence $A_1$ is generated by $\\pi_1$ over $A$ and the minimal polynomial", "$P(t)$ of $\\pi_1$ over $K$ has the form", "$$", "P(t) = t^e + a_{e - 1} t^{e - 1} + \\ldots + a_0", "$$", "with $a_i \\in (\\pi)$ and $a_0 = u\\pi$ for some unit $u$ of $A$.", "Note that $e = [K_1 : K]$ as well. Since $A \\to B$ is weakly", "unramified we see that $\\pi$ is a uniformizer of $B$ and hence", "$B_1 = B[t]/(P(t))$ is a discrete valuation ring with uniformizer", "the class of $t$. Thus the lemma is clear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10679 ] }, { "id": 10516, "type": "theorem", "label": "more-algebra-lemma-solutions-go-down", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-solutions-go-down", "contents": [ "Let $A \\to B \\to C$ be extensions of discrete valuation rings with fraction", "fields $K \\subset L \\subset M$. Let $K \\subset K_1$ be a finite extension.", "\\begin{enumerate}", "\\item If $K_1$ is a (weak) solution for $A \\to C$, then $K_1$ is a (weak)", "solution for $A \\to B$.", "\\item If $K_1$ is a (weak) solution for $A \\to B$ and", "$L_1 = (L \\otimes_K K_1)_{red}$ is a product of fields which are", "(weak) solutions for $B \\to C$, then $K_1$ is a weak solution for $A \\to C$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $L_1 = (L \\otimes_K K_1)_{red}$ and $M_1 = (M \\otimes_K K_1)_{red}$", "and let $B_1 \\subset L_1$ and $C_1 \\subset M_1$ be the integral closure", "of $B$ and $C$. Note that $M_1 = (M \\otimes_L L_1)_{red}$ and that $L_1$", "is a (nonempty) finite product of finite extensions of $L$. Hence the", "ring map $B_1 \\to C_1$ is a finite product of ring maps of the form discussed", "in Remark \\ref{remark-construction}. In particular, the map", "$\\Spec(C_1) \\to \\Spec(B_1)$ is surjective. Choose a maximal ideal", "$\\mathfrak m \\subset C_1$ and consider the extensions of discrete", "valuation rings", "$$", "(A_1)_{A_1 \\cap \\mathfrak m} \\to", "(B_1)_{B_1 \\cap \\mathfrak m} \\to", "(C_1)_\\mathfrak m", "$$", "If the composition is weakly unramified, so is the map", "$(A_1)_{A_1 \\cap \\mathfrak m} \\to (B_1)_{B_1 \\cap \\mathfrak m}$.", "If the residue field extension", "$\\kappa_{A_1 \\cap \\mathfrak m} \\to \\kappa_\\mathfrak m$ is separable,", "so is the subextension", "$\\kappa_{A_1 \\cap \\mathfrak m} \\to \\kappa_{B_1 \\cap \\mathfrak m}$.", "Taking into account Lemma \\ref{lemma-extension-dvrs-formally-smooth}", "this proves (1). A similar argument works for (2)." ], "refs": [ "more-algebra-remark-construction", "more-algebra-lemma-extension-dvrs-formally-smooth" ], "ref_ids": [ 10679, 10495 ] } ], "ref_ids": [] }, { "id": 10517, "type": "theorem", "label": "more-algebra-lemma-separable-solution-separable-solution", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-separable-solution-separable-solution", "contents": [ "Let $A \\subset B$ be an extension of discrete valuation rings.", "Assume", "\\begin{enumerate}", "\\item the extension $K \\subset L$ of fraction fields is separable,", "\\item $B$ is Nagata, and", "\\item there exists a solution for $A \\subset B$.", "\\end{enumerate}", "Then there exists a separable solution for $A \\subset B$." ], "refs": [], "proofs": [ { "contents": [ "The lemma is trivial if the characteristic of $K$ is zero; thus we may", "and do assume that the characteristic of $K$ is $p > 0$.", "\\medskip\\noindent", "Let $K \\subset K_1$ be a finite extension. Since $L/K$ is separable,", "the algebra $L \\otimes_K K_1$ is reduced", "(Algebra, Lemma \\ref{algebra-lemma-separable-extension-preserves-reducedness}).", "Since $B$ is Nagata, the ring extension $B \\subset B_1$ is finite", "(Remark \\ref{remark-construction}) and $B_1$ is a Nagata ring.", "Moreover, if $K \\subset K_1 \\subset K_2$ is a tower of finite extensions,", "then the same thing is true, i.e., the ring extension $B_1 \\subset B_2$", "is finite too where $B_2$ is the integral closure of $B$ (or $B_1$)", "in $L \\otimes_K K_2$.", "\\medskip\\noindent", "Let $K \\subset K_2$ be a solution for $A \\to B$. There exists a subfield", "$K \\subset K_1 \\subset K_2$ such that $K_1/K$ is separable and", "$K_2/K_1$ is purely inseparable", "(Fields, Lemma \\ref{fields-lemma-separable-first}). Thus it suffices", "to show that if we have $K \\subset K_1 \\subset K_2$ with", "$K_2/K_1$ purely inseparable of degree $p$, then $K \\subset K_1$", "is a solution for $A \\subset B$. Using the remarks above, we may replace", "$A$ by a localization $(A_1)_{\\mathfrak m_i}$ and $B$ by", "$(B_1)_{\\mathfrak m_{ij}}$ (notation as in Remark \\ref{remark-construction})", "and reduce to the problem discussed in the following paragraph.", "\\medskip\\noindent", "Assume that $K \\subset K_1$ is a purely inseparable extension of degree", "$p$ which is a solution for $A \\subset B$. Problem: show that $A \\to B$ is", "formally smooth in the $\\mathfrak m_B$-adic topology.", "By the discussion in Remark \\ref{remark-construction}", "we see that $A_1$ and $B_1$ are discrete valuation rings", "and as discussed above $B \\subset B_1$ is finite. Consider the diagrams", "$$", "\\vcenter{", "\\xymatrix{", "B \\ar[r]_{e_u} & B_1 \\\\", "A \\ar[u]_e \\ar[r]^{e_d} & A_1 \\ar[u]^1", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "\\kappa_B \\ar[r]_{d_u} & \\kappa_{B_1} \\\\", "\\kappa_A \\ar[u]_d \\ar[r]^{d_d} & \\kappa_{A_1} \\ar[u]^1", "}", "}", "$$", "of extensions of discrete valuation rings and residue fields.", "Here $e, e_u, e_d, 1$ denote ramification indices, so $e e_u = e_d$.", "Also $d, d_u, d_d, 1$ denote the inseparable degrees", "(Fields, Definition \\ref{fields-definition-insep-degree}), so $dd_u = d_d$", "(Fields, Lemma \\ref{fields-lemma-multiplicativity-all-degrees}).", "By Algebra, Lemma \\ref{algebra-lemma-finite-extension-dim-1}", "and the fact that $L \\subset L \\otimes_K K_1$ is a degree $p$", "field extension, we see that $e_ud_u = p$", "(this is where we really use that $B$ is Nagata; this need", "not be true if the extension $B \\subset B_1$ is not finite).", "We have $e_d d_d \\leq p$ by Lemma \\ref{lemma-inequality}.", "Thus it follows that $e = d = 1$ as desired." ], "refs": [ "algebra-lemma-separable-extension-preserves-reducedness", "more-algebra-remark-construction", "fields-lemma-separable-first", "more-algebra-remark-construction", "more-algebra-remark-construction", "fields-definition-insep-degree", "fields-lemma-multiplicativity-all-degrees", "algebra-lemma-finite-extension-dim-1", "more-algebra-lemma-inequality" ], "ref_ids": [ 565, 10679, 4482, 10679, 10679, 4540, 4484, 1047, 10492 ] } ], "ref_ids": [] }, { "id": 10518, "type": "theorem", "label": "more-algebra-lemma-solution-after-strict-henselization", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-solution-after-strict-henselization", "contents": [ "Let $A \\to B$ be an extension of discrete valuation rings. There exists", "a commutative diagram", "$$", "\\xymatrix{", "B \\ar[r] & B' \\\\", "A \\ar[r] \\ar[u] & A' \\ar[u]", "}", "$$", "of extensions of discrete valuation rings such that", "\\begin{enumerate}", "\\item the extensions $K \\subset K'$ and $L \\subset L'$ of fraction fields", "are separable algebraic,", "\\item the residue fields of $A'$ and $B'$ are separable algebraic", "closures of the residue fields of $A$ and $B$, and", "\\item if a solution, weak solution, or separable solution exists for", "$A' \\to B'$, then a solution, weak solution, or separable solution exists", "for $A \\to B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Algebra, Lemma \\ref{algebra-lemma-colimit-finite-etale-given-residue-field}", "there exists an extension $A \\subset A'$ which is a filtered colimit of finite", "\\'etale extensions such that the residue field of $A'$ is a separable algebraic", "closure of the residue field of $A$. Then $A \\subset A'$ is an extension of", "discrete valuation rings such that the induced extension $K \\subset K'$ of", "fraction fields is separable algebraic.", "\\medskip\\noindent", "Let $B \\subset B'$ be a strict henselization of $B$. Then $B \\subset B'$ is an", "extension of discrete valuation rings whose fraction field extension is", "separable algebraic. By", "Algebra, Lemma \\ref{algebra-lemma-strictly-henselian-functorial-prepare}", "there exists a commutative diagram as in the statement of the lemma.", "Parts (1) and (2) of the lemma are clear.", "\\medskip\\noindent", "Let $K' \\subset K'_1$ be a (weak) solution for $A' \\to B'$. Since $A'$ is a", "colimit, we can find a finite \\'etale extension $A \\subset A_1'$ and a finite", "extension $K_1$ of the fraction field $F$ of $A_1'$ such that", "$K'_1 = K' \\otimes_F K_1$. As $A \\subset A_1'$ is finite \\'etale", "and $B'$ strictly henselian, it follows that $B' \\otimes_A A_1'$ is a finite", "product of rings isomorphic to $B'$. Hence", "$$", "L' \\otimes_K K_1 = L' \\otimes_K F \\otimes_F K_1", "$$", "is a finite product of rings isomorphic to $L' \\otimes_{K'} K'_1$.", "Thus we see that $K \\subset K_1$ is a (weak) solution for $A \\to B'$.", "Hence it is also a (weak) solution for $A \\to B$ by", "Lemma \\ref{lemma-solutions-go-down}." ], "refs": [ "algebra-lemma-colimit-finite-etale-given-residue-field", "algebra-lemma-strictly-henselian-functorial-prepare", "more-algebra-lemma-solutions-go-down" ], "ref_ids": [ 1325, 1302, 10516 ] } ], "ref_ids": [] }, { "id": 10519, "type": "theorem", "label": "more-algebra-lemma-galois-relative", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-galois-relative", "contents": [ "Let $A \\to B$ be an extension of discrete valuation rings with fraction fields", "$K \\subset L$. Let $K \\subset K_1$ be a normal extension. Say", "$G = \\text{Aut}(K_1/K)$. Then $G$ acts on the rings $K_1$, $L_1$,", "$A_1$ and $B_1$ of Remark \\ref{remark-construction}", "and acts transitively on the set of maximal ideals of $B_1$." ], "refs": [ "more-algebra-remark-construction" ], "proofs": [ { "contents": [ "Everything is clear apart from the last assertion. If there are two or", "more orbits of the action, then we can find an element $b \\in B_1$", "which vanishes at all the maximal ideals of one orbit and has residue", "$1$ at all the maximal ideals in another orbit. Then", "$b' = \\prod_{\\sigma \\in G} \\sigma(b)$ is a $G$-invariant element of", "$B_1 \\subset L_1 = (L \\otimes_K K_1)_{red}$ which is in some maximal", "ideals of $B_1$", "but not in all maximal ideals of $B_1$. Lifting it to an element of", "$L \\otimes_K K_1$ and raising to a high power we obtain a $G$-invariant", "element $b''$ of $L \\otimes_K K_1$ mapping to $(b')^N$ for some $N > 0$;", "in fact, we only need to do this in case the characteristic is $p > 0$ and", "in this case raising to a suitably large $p$-power $q$ defines a", "canonical map $(L \\otimes_K K_1)_{red} \\to L \\otimes_K K_1$.", "Since $K = (K_1)^G$ we conclude that $b'' \\in L$. Since $b''$ maps", "to an element of $B_1$ we see that $b'' \\in B$ (as $B$ is normal).", "Then on the one hand it must be true that $b'' \\in \\mathfrak m_B$", "as $b'$ is in some maximal ideal of $B_1$ and on the other hand it", "must be true that $b'' \\not \\in \\mathfrak m_B$ as $b'$ is not in", "all maximal ideals of $B_1$. This contradiction finishes the proof of the", "lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10679 ] }, { "id": 10520, "type": "theorem", "label": "more-algebra-lemma-make-degree-q-extension", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-make-degree-q-extension", "contents": [ "Let $A$ be a discrete valuation ring with uniformizer $\\pi$. If the residue", "characteristic of $A$ is $p > 0$, then for every $n > 1$ and $p$-power $q$", "there exists a degree $q$ separable extension $L/K$", "totally ramified with respect to $A$", "such that the integral closure $B$ of $A$ in $L$ has ramification index", "$q$ and a uniformizer $\\pi_B$ such that", "$\\pi_B^q = \\pi + \\pi^n b$ and $\\pi_B^q = \\pi + (\\pi_B)^{nq}b'$", "for some $b, b' \\in B$." ], "refs": [], "proofs": [ { "contents": [ "If the characteristic of $K$ is zero, then we can take the", "extension given by $\\pi_B^q = \\pi$, see", "Lemma \\ref{lemma-pull-root-uniformizer}.", "If the characteristic of $K$ is $p > 0$, then we can take the", "extension of $K$ given by $z^q - \\pi^n z = \\pi^{1 - q}$.", "Namely, then we see that $y^q - \\pi^{n + q - 1} y = \\pi$", "where $y = \\pi z$. Taking $\\pi_B = y$ we obtain the desired result." ], "refs": [ "more-algebra-lemma-pull-root-uniformizer" ], "ref_ids": [ 10507 ] } ], "ref_ids": [] }, { "id": 10521, "type": "theorem", "label": "more-algebra-lemma-pre-purely-inseparable-case", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pre-purely-inseparable-case", "contents": [ "Let $A$ be a discrete valuation ring. Assume the reside field $\\kappa_A$ has", "characteristic $p > 0$ and that $a \\in A$ is an element whose residue", "class in $\\kappa_A$ is not a $p$th power. Then $a$ is not a $p$th power in $K$", "and the integral closure of $A$ in $K[a^{1/p}]$ is the ring $A[a^{1/p}]$", "which is a discrete valuation ring weakly unramified over $A$." ], "refs": [], "proofs": [ { "contents": [ "This lemma proves itself." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10522, "type": "theorem", "label": "more-algebra-lemma-purely-inseparable-case", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-purely-inseparable-case", "contents": [ "Let $A \\subset B \\subset C$ be extensions of discrete valuation rings", "with fractions fields $K \\subset L \\subset M$. Let $\\pi \\in A$ be a", "uniformizer. Assume", "\\begin{enumerate}", "\\item $B$ is a Nagata ring,", "\\item $A \\subset B$ is weakly unramified,", "\\item $M$ is a degree $p$ purely inseparable extension of $L$.", "\\end{enumerate}", "Then either", "\\begin{enumerate}", "\\item $A \\to C$ is weakly unramified, or", "\\item $C = B[\\pi^{1/p}]$, or", "\\item there exists a degree $p$ separable extension $K_1/K$", "totally ramified with respect to $A$", "such that $L_1 = L \\otimes_K K_1$ and $M_1 = M \\otimes_K K_1$", "are fields and the maps of integral closures $A_1 \\to B_1 \\to C_1$", "are weakly unramified extensions of discrete valuation rings.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $e$ be the ramification index of $C$ over $B$. If $e = 1$, then we are", "done. If not, then $e = p$ by Lemmas \\ref{lemma-inequality} and", "\\ref{lemma-ramification-index-a-power-of-p}.", "This in turn implies that the residue fields of $B$ and $C$ agree.", "Choose a uniformizer $\\pi_C$ of $C$.", "Write $\\pi_C^p = u \\pi$ for some unit $u$ of $C$.", "Since $\\pi_C^p \\in L$, we see that $u \\in B^*$. Also $M = L[\\pi_C]$.", "\\medskip\\noindent", "Suppose there exists an integer $m \\geq 0$ such that", "$$", "u = \\sum\\nolimits_{0 \\leq i < m} b_i^p \\pi^i + b \\pi^m", "$$", "with $b_i \\in B$ and with $b \\in B$ an element whose image in $\\kappa_B$", "is not a $p$th power. Choose an extension $K \\subset K_1$ as in", "Lemma \\ref{lemma-make-degree-q-extension}", "with $n = m + 2$ and denote $\\pi'$ the uniformizer", "of the integral closure $A_1$ of $A$ in $K_1$ such that", "$\\pi = (\\pi')^p + (\\pi')^{np} a$ for some $a \\in A_1$.", "Let $B_1$ be the integral closure of $B$ in $L \\otimes_K K_1$.", "Observe that $A_1 \\to B_1$ is weakly unramified by", "Lemma \\ref{lemma-weakly-unramified-goes-up-along-totally-ramified}.", "In $B_1$ we have", "$$", "u \\pi =", "\\left(\\sum\\nolimits_{0 \\leq i < m} b_i (\\pi')^{i + 1}\\right)^p +", "b (\\pi')^{(m + 1)p} + (\\pi')^{np} b_1", "$$", "for some $b_1 \\in B_1$ (computation omitted).", "We conclude that $M_1$ is obtained from", "$L_1$ by adjoining a $p$th root of", "$$", "b + (\\pi')^{n - m - 1} b_1", "$$", "Since the residue field of $B_1$ equals the residue field of $B$", "we see from Lemma \\ref{lemma-pre-purely-inseparable-case}", "that $M_1/L_1$ has degree $p$ and", "the integral closure $C_1$ of $B_1$ is weakly unramified over $B_1$.", "Thus we conclude in this case.", "\\medskip\\noindent", "If there does not exist an integer $m$ as in the preceding paragraph,", "then $u$ is a $p$th power in the $\\pi$-adic completion of $B_1$.", "Since $B$ is Nagata, this means that $u$ is a $p$th power in $B_1$", "by Algebra, Lemma \\ref{algebra-lemma-nagata-pth-roots}.", "Whence the second case of the statement of the lemma holds." ], "refs": [ "more-algebra-lemma-inequality", "more-algebra-lemma-ramification-index-a-power-of-p", "more-algebra-lemma-make-degree-q-extension", "more-algebra-lemma-weakly-unramified-goes-up-along-totally-ramified", "more-algebra-lemma-pre-purely-inseparable-case", "algebra-lemma-nagata-pth-roots" ], "ref_ids": [ 10492, 10494, 10520, 10515, 10521, 1359 ] } ], "ref_ids": [] }, { "id": 10523, "type": "theorem", "label": "more-algebra-lemma-cohen", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-cohen", "contents": [ "Let $A$ be a local ring annihilated by a prime $p$ whose maximal ideal is", "nilpotent. There exists a ring map $\\sigma : \\kappa_A \\to A$", "which is a section to the residue map $A \\to \\kappa_A$. If $A \\to A'$ is", "a local homomorphism of local rings, then we can choose a similar", "ring map $\\sigma' : \\kappa_{A'} \\to A'$ compatible with $\\sigma$ provided", "that the extension $\\kappa_A \\subset \\kappa_{A'}$ is separable." ], "refs": [], "proofs": [ { "contents": [ "Separable extensions are formally smooth by Algebra, Proposition", "\\ref{algebra-proposition-characterize-separable-field-extensions}.", "Thus the existence of $\\sigma$ follows from the fact that", "$\\mathbf{F}_p \\to \\kappa_A$ is separable.", "Similarly for the existence of $\\sigma'$ compatible with $\\sigma$." ], "refs": [ "algebra-proposition-characterize-separable-field-extensions" ], "ref_ids": [ 1429 ] } ], "ref_ids": [] }, { "id": 10524, "type": "theorem", "label": "more-algebra-lemma-pre-characteristic-p-case", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-pre-characteristic-p-case", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$ of characteristic", "$p > 0$. Let $\\xi \\in K$. Let $L$ be an extension of $K$ obtained by", "adjoining a root of $z^p - z = \\xi$. Then $L/K$ is Galois and one of the", "following happens", "\\begin{enumerate}", "\\item $L = K$,", "\\item $L/K$ is unramified with respect to $A$ of degree $p$,", "\\item $L/K$ is totally ramified with respect to $A$", "with ramification index $p$, and", "\\item the integral closure $B$ of $A$ in $L$ is a discrete valuation ring,", "$A \\subset B$ is weakly unramified, and $A \\to B$ induces a purely inseparable", "residue field extension of degree $p$.", "\\end{enumerate}", "Let $\\pi$ be a uniformizer of $A$. We have the following implications:", "\\begin{enumerate}", "\\item[(A)] If $\\xi \\in A$, then we are in case (1) or (2).", "\\item[(B)] If $\\xi = \\pi^{-n}a$ where $n > 0$ is not divisible by", "$p$ and $a$ is a unit in $A$, then we are in case (3)", "\\item[(C)] If $\\xi = \\pi^{-n} a$ where $n > 0$ is divisible by $p$ and", "the image of $a$ in $\\kappa_A$ is not a $p$th power, then we are in case (4).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The extension is Galois of order dividing $p$ by the discussion in", "Fields, Section \\ref{fields-section-Artin-Schreier}.", "It immediately follows from the discussion in", "Section \\ref{section-ramification} that we are in one of the cases (1) -- (4)", "listed in the lemma.", "\\medskip\\noindent", "Case (A). Here we see that $A \\to A[x]/(x^p - x - \\xi)$ is a finite", "\\'etale ring extension. Hence we are in cases (1) or (2).", "\\medskip\\noindent", "Case (B). Write $\\xi = \\pi^{-n}a$ where $p$ does not divide $n$.", "Let $B \\subset L$ be the integral closure of $A$ in $L$.", "If $C = B_\\mathfrak m$ for some maximal ideal $\\mathfrak m$,", "then it is clear that $p \\text{ord}_C(z) = -n \\text{ord}_C(\\pi)$.", "In particular $A \\subset C$ has ramification index divisible by $p$.", "It follows that it is $p$ and that $B = C$.", "\\medskip\\noindent", "Case (C). Set $k = n/p$. Then we can rewrite the equation as", "$$", "(\\pi^kz)^p - \\pi^{n - k} (\\pi^kz) = a", "$$", "Since $A[y]/(y^p - \\pi^{n - k}y - a)$ is a discrete valuation ring", "weakly unramified over $A$, the lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10525, "type": "theorem", "label": "more-algebra-lemma-characteristic-p-case", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-characteristic-p-case", "contents": [ "Let $A \\subset B \\subset C$ be extensions of discrete valuation rings", "with fractions fields $K \\subset L \\subset M$. Assume", "\\begin{enumerate}", "\\item $A \\subset B$ weakly unramified,", "\\item the characteristic of $K$ is $p$,", "\\item $M$ is a degree $p$ Galois extension of $L$, and", "\\item $\\kappa_A = \\bigcap_{n \\geq 1} \\kappa_B^{p^n}$.", "\\end{enumerate}", "Then there exists a finite Galois extension $K_1/K$", "totally ramified with respect to $A$", "which is a weak solution for $A \\to C$." ], "refs": [], "proofs": [ { "contents": [ "Since the characteristic of $L$ is $p$ we know that $M$ is an Artin-Schreier", "extension of $L$ (Fields, Lemma \\ref{fields-lemma-Artin-Schreier}).", "Thus we may pick $z \\in M$, $z \\not \\in L$ such that", "$\\xi = z^p - z \\in L$. Choose $n \\geq 0$ such that $\\pi^n\\xi \\in B$.", "We pick $z$ such that $n$ is minimal. If $n = 0$, then $M/L$ is unramified", "with respect to $B$ (Lemma \\ref{lemma-pre-characteristic-p-case}) and", "we are done. Thus we have $n > 0$.", "\\medskip\\noindent", "Assumption (4) implies that $\\kappa_A$ is perfect. Thus we may", "choose compatible ring maps $\\overline{\\sigma} : \\kappa_A \\to A/\\pi^n A$ and", "$\\overline{\\sigma} : \\kappa_B \\to B/\\pi^n B$ as in", "Lemma \\ref{lemma-cohen}. We lift the second of these to a", "map of sets $\\sigma : \\kappa_B \\to B$\\footnote{If $B$ is complete, then", "we can choose $\\sigma$ to be a ring map. If $A$ is also complete and", "$\\sigma$ is a ring map, then $\\sigma$ maps $\\kappa_A$ into $A$.}.", "Then we can write", "$$", "\\xi = \\sum\\nolimits_{i = n, \\ldots, 1} \\sigma(\\lambda_i) \\pi^{-i} + b", "$$", "for some $\\lambda_i \\in \\kappa_B$ and $b \\in B$. Let", "$$", "I = \\{i \\in \\{n, \\ldots, 1\\} \\mid \\lambda_i \\in \\kappa_A\\}", "$$", "and", "$$", "J = \\{j \\in \\{n, \\ldots, 1\\} \\mid \\lambda_i \\not \\in \\kappa_A\\}", "$$", "We will argue by induction on the size of the finite set $J$.", "\\medskip\\noindent", "The case $J = \\emptyset$. Here for all $i \\in \\{n, \\ldots, 1\\}$ we have", "$\\sigma(\\lambda_i) = a_i + \\pi^n b_i$ for some $a_i \\in A$ and $b_i \\in B$", "by our choice of $\\sigma$. Thus", "$\\xi = \\pi^{-n} a + b$ for some $a \\in A$ and $b \\in B$.", "If $p | n$, then we write $a = a_0^p + \\pi a_1$ for some $a_0, a_1 \\in A$", "(as the residue field of $A$ is perfect). We compute", "$$", "(z - \\pi^{-n/p}a_0)^p - (z - \\pi^{-n/p}a_0) =", "\\pi^{-(n - 1)}(a_1 + \\pi^{n - 1 - n/p}a_0) + b'", "$$", "for some $b' \\in B$. This would contradict the minimality of $n$. Thus $p$", "does not divide $n$. Consider the degree $p$ extension $K_1$ of $K$ given", "by $w^p - w = \\pi^{-n}a$. By Lemma \\ref{lemma-pre-characteristic-p-case}", "this extension is Galois and totally ramified with respect to $A$.", "Thus $L_1 = L \\otimes_K K_1$ is a field and $A_1 \\subset B_1$", "is weakly unramified", "(Lemma \\ref{lemma-weakly-unramified-goes-up-along-totally-ramified}).", "By Lemma \\ref{lemma-pre-characteristic-p-case}", "the ring $M_1 = M \\otimes_K K_1$ is either a product of $p$ copies", "of $L_1$ (in which case we are done) or a field extension of $L_1$", "of degree $p$. Moreover, in the second case, either $C_1$ is weakly unramified", "over $B_1$ (in which case we are done) or $M_1/L_1$ is degree $p$,", "Galois, and totally ramified with respect to $B_1$.", "In this last case the extension $M_1/L_1$", "is generated by the element $z - w$ and", "$$", "(z - w)^p - (z - w) = z^p - z - (w^p - w) = b", "$$", "with $b \\in B$ (see above). Thus by Lemma \\ref{lemma-pre-characteristic-p-case}", "once more the extension $M_1/L_1$ is unramified with respect to $B_1$", "and we conclude that $K_1$ is a weak solution for $A \\to C$.", "From now on we assume $J \\not = \\emptyset$.", "\\medskip\\noindent", "Suppose that $j', j \\in J$ such that $j' = p^r j$ for some", "$r > 0$. Then we change our choice of $z$ into", "$$", "z' = z -", "(\\sigma(\\lambda_j) \\pi^{-j} + \\sigma(\\lambda_j^p) \\pi^{-pj} + \\ldots +", "\\sigma(\\lambda_j^{p^{r - 1}}) \\pi^{-p^{r - 1}j})", "$$", "Then $\\xi$ changes into $\\xi' = (z')^p - (z')$ as follows", "$$", "\\xi' =", "\\xi - \\sigma(\\lambda_j) \\pi^{-j} + \\sigma(\\lambda_j^{p^r}) \\pi^{-j'}", "+ \\text{something in }B", "$$", "Writing", "$\\xi' = \\sum\\nolimits_{i = n, \\ldots, 1} \\sigma(\\lambda'_i) \\pi^{-i} + b'$", "as before we find that", "$\\lambda'_i = \\lambda_i$ for $i \\not = j, j'$ and $\\lambda'_j = 0$.", "Thus the set $J$ has gotten smaller.", "By induction on the size of $J$ we may assume no such pair $j, j'$ exists.", "(Please observe that in this procedure we may get thrown back into the case", "that $J = \\emptyset$ we treated above.)", "\\medskip\\noindent", "For $j \\in J$ write $\\lambda_j = \\mu_j^{p^{r_j}}$ for some $r_j \\geq 0$ and", "$\\mu_j \\in \\kappa_B$ which is not a $p$th power. This is possible by our", "assumption (4). Let $j \\in J$ be the unique index such that $j p^{-r_j}$", "is maximal. (The index is unique by the result of the preceding paragraph.)", "Choose $r > \\max(r_j + 1)$ and such that $j p^{r - r_j} > n$ for $j \\in J$.", "Choose a separable extension $K_1/K$ totally ramified with respect to $A$", "of degree $p^r$ such that the corresponding discrete valuation ring", "$A_1 \\subset K_1$ has uniformizer $\\pi'$ with", "$(\\pi')^{p^r} = \\pi + \\pi^{n + 1}a$ for some $a \\in A_1$", "(Lemma \\ref{lemma-make-degree-q-extension}).", "Observe that $L_1 = L \\otimes_K K_1$ is a field and that", "$L_1/L$ is totally ramified with respect to $B$", "(Lemma \\ref{lemma-weakly-unramified-goes-up-along-totally-ramified}).", "Computing in the integral closure $B_1$ we get", "$$", "\\xi = \\sum\\nolimits_{i \\in I} \\sigma(\\lambda_i) (\\pi')^{-i p^r} +", "\\sum\\nolimits_{j \\in J} \\sigma(\\mu_j)^{p^{r_j}} (\\pi')^{-j p^r} + b_1", "$$", "for some $b_1 \\in B_1$. Note that $\\sigma(\\lambda_i)$ for $i \\in I$", "is a $q$th power modulo $\\pi^n$, i.e., modulo $(\\pi')^{n p^r}$.", "Hence we can rewrite the above as", "$$", "\\xi = \\sum\\nolimits_{i \\in I} x_i^{p^r} (\\pi')^{-i p^r} +", "\\sum\\nolimits_{j \\in J} \\sigma(\\mu_j)^{p^{r_j}} (\\pi')^{- j p^r}", "+ b_1", "$$", "As in the previous paragraph we change our choice of $z$ into", "\\begin{align*}", "z' & = z \\\\", "& -", "\\sum\\nolimits_{i \\in I}", "\\left(x_i (\\pi')^{-i} + \\ldots + x_i^{p^{r - 1}} (\\pi')^{-i p^{r - 1}}\\right)", "\\\\", "& -", "\\sum\\nolimits_{j \\in J}", "\\left(", "\\sigma(\\mu_j) (\\pi')^{- j p^{r - r_j}}", "+ \\ldots +", "\\sigma(\\mu_j)^{p^{r_j - 1}} (\\pi')^{- j p^{r - 1}}", "\\right)", "\\end{align*}", "to obtain", "$$", "(z')^p - z' =", "\\sum\\nolimits_{i \\in I} x_i (\\pi')^{-i} +", "\\sum\\nolimits_{j \\in J} \\sigma(\\mu_j) (\\pi')^{- j p^{r - r_j}} + b_1'", "$$", "for some $b'_1 \\in B_1$.", "Since there is a unique $j$ such that $j p^{r - r_j}$ is maximal", "and since $j p^{r - r_j}$ is bigger than $i \\in I$ and divisible", "by $p$, we see that $M_1 / L_1$ falls into case (C) of", "Lemma \\ref{lemma-pre-characteristic-p-case}.", "This finishes the proof." ], "refs": [ "fields-lemma-Artin-Schreier", "more-algebra-lemma-pre-characteristic-p-case", "more-algebra-lemma-cohen", "more-algebra-lemma-pre-characteristic-p-case", "more-algebra-lemma-weakly-unramified-goes-up-along-totally-ramified", "more-algebra-lemma-pre-characteristic-p-case", "more-algebra-lemma-pre-characteristic-p-case", "more-algebra-lemma-make-degree-q-extension", "more-algebra-lemma-weakly-unramified-goes-up-along-totally-ramified", "more-algebra-lemma-pre-characteristic-p-case" ], "ref_ids": [ 4517, 10524, 10523, 10524, 10515, 10524, 10524, 10520, 10515, 10524 ] } ], "ref_ids": [] }, { "id": 10526, "type": "theorem", "label": "more-algebra-lemma-prepare", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-prepare", "contents": [ "Let $A$ be a ring which contains a primitive $p$th root of unity $\\zeta$.", "Set $w = 1 - \\zeta$. Then", "$$", "P(z) = \\frac{(1 + wz)^p - 1}{w^p} =", "z^p - z + \\sum\\nolimits_{0 < i < p} a_i z^i", "$$", "is an element of $A[z]$ and in fact $a_i \\in (w)$. Moreover, we have", "$$", "P(z_1 + z_2 + w z_1 z_2) = P(z_1) + P(z_2) + w^p P(z_1) P(z_2)", "$$", "in the polynomial ring $A[z_1, z_2]$." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove this when", "$$", "A = \\mathbf{Z}[\\zeta] = \\mathbf{Z}[x]/(x^{p - 1} + \\ldots + x + 1)", "$$", "is the ring of integers of the cyclotomic field. The polynomial identity", "$t^p - 1 = (t - 1)(t - \\zeta) \\ldots (t - \\zeta^{p - 1})$", "(which is proved by looking at the roots on both sides)", "shows that", "$t^{p - 1} + \\ldots + t + 1 = (t - \\zeta) \\ldots (t - \\zeta^{p - 1})$.", "Substituting $t = 1$ we obtain", "$p = (1 - \\zeta)(1 - \\zeta^2) \\ldots (1 - \\zeta^{p - 1})$.", "The maximal ideal $(p, w) = (w)$ is the unique prime ideal of $A$", "lying over $p$ (as fields of characteristic $p$ do not have nontrivial", "$p$th roots of $1$). It follows that $p = u w^{p - 1}$ for some unit $u$.", "This implies that", "$$", "a_i = \\frac{1}{p} {p \\choose i} u w^{i - 1}", "$$", "for $p > i > 1$ and $- 1 + a_1 = pw/w^p = u$. Since $P(-1) = 0$ we", "see that $0 = (-1)^p - u$ modulo $(w)$. Hence $a_1 \\in (w)$ and the", "proof if the first part is done. The second part follows from a direct", "computation we omit." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10527, "type": "theorem", "label": "more-algebra-lemma-extension-defined-by-nice-polynial", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-extension-defined-by-nice-polynial", "contents": [ "Let $A$ be a discrete valuation ring of mixed characteristic $(0, p)$", "which contains a primitive $p$th root of $1$.", "Let $P(t) \\in A[t]$ be the polynomial of Lemma \\ref{lemma-prepare}.", "Let $\\xi \\in K$.", "Let $L$ be an extension of $K$ obtained by", "adjoining a root of $P(z) = \\xi$. Then $L/K$ is Galois and one of the", "following happens", "\\begin{enumerate}", "\\item $L = K$,", "\\item $L/K$ is unramified with respect to $A$ of degree $p$,", "\\item $L/K$ is totally ramified with respect to $A$", "with ramification index $p$, and", "\\item the integral closure $B$ of $A$ in $L$ is a discrete valuation ring,", "$A \\subset B$ is weakly unramified, and $A \\to B$ induces a purely inseparable", "residue field extension of degree $p$.", "\\end{enumerate}", "Let $\\pi$ be a uniformizer of $A$. We have the following implications:", "\\begin{enumerate}", "\\item[(A)] If $\\xi \\in A$, then we are in case (1) or (2).", "\\item[(B)] If $\\xi = \\pi^{-n}a$ where $n > 0$ is not divisible by", "$p$ and $a$ is a unit in $A$, then we are in case (3)", "\\item[(C)] If $\\xi = \\pi^{-n} a$ where $n > 0$ is divisible by $p$ and", "the image of $a$ in $\\kappa_A$ is not a $p$th power, then we are in case (4).", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-prepare" ], "proofs": [ { "contents": [ "Adjoining a root of $P(z) = \\xi$ is the same thing as adjoining a root", "of $y^p = w^p(1 + \\xi)$. Since $K$ contains a primitive $p$th root of $1$", "the extension is Galois of order dividing $p$ by the discussion in", "Fields, Section \\ref{fields-section-Kummer}.", "It immediately follows from the discussion in", "Section \\ref{section-ramification} that we are in one of the cases (1) -- (4)", "listed in the lemma.", "\\medskip\\noindent", "Case (A). Here we see that $A \\to A[x]/(P(x) - \\xi)$ is a finite", "\\'etale ring extension. Hence we are in cases (1) or (2).", "\\medskip\\noindent", "Case (B). Write $\\xi = \\pi^{-n}a$ where $p$ does not divide $n$.", "Let $B \\subset L$ be the integral closure of $A$ in $L$.", "If $C = B_\\mathfrak m$ for some maximal ideal $\\mathfrak m$,", "then it is clear that $p \\text{ord}_C(z) = -n \\text{ord}_C(\\pi)$.", "In particular $A \\subset C$ has ramification index divisible by $p$.", "It follows that it is $p$ and that $B = C$.", "\\medskip\\noindent", "Case (C). Set $k = n/p$. Then we can rewrite the equation as", "$$", "(\\pi^kz)^p - \\pi^{n - k} (\\pi^kz) + \\sum a_i \\pi^{n - ik} (\\pi^kz)^i = a", "$$", "Since $A[y]/(y^p - \\pi^{n - k}y - \\sum a_i \\pi^{n - ik} y^i - a)$", "is a discrete valuation ring weakly unramified over $A$, the lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10526 ] }, { "id": 10528, "type": "theorem", "label": "more-algebra-lemma-make-finite-level", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-make-finite-level", "contents": [ "Let $A \\subset B \\subset C$ be extensions of discrete valuation rings", "with fractions fields $K \\subset L \\subset M$. Assume that", "\\begin{enumerate}", "\\item $A$ has mixed characteristic $(0, p)$,", "\\item $A \\subset B$ is weakly unramified,", "\\item $B$ contains a primitive $p$th root of $1$, and", "\\item $M/L$ is Galois of degree $p$.", "\\end{enumerate}", "Then there exists a finite Galois extension $K_1/K$ totally ramified", "with respect to $A$ which is either a weak solution for $A \\to C$", "or is such that $M_1/L_1$ is a degree $p$ extension of finite level." ], "refs": [], "proofs": [ { "contents": [ "Let $\\pi \\in A$ be a uniformizer. By Kummer theory", "(Fields, Lemma \\ref{fields-lemma-Kummer}) $M$ is obtained", "from $L$ by adjoining the root of $y^p = b$ for some $b \\in L$.", "\\medskip\\noindent", "If $\\text{ord}_B(b)$ is prime to $p$, then we choose a degree $p$", "separable extension $K \\subset K_1$", "totally ramified with respect to $A$ (for example", "using Lemma \\ref{lemma-make-degree-q-extension}).", "Let $A_1$ be the integral closure of $A$ in $K_1$.", "By Lemma \\ref{lemma-weakly-unramified-goes-up-along-totally-ramified}", "the integral closure $B_1$ of $B$ in $L_1 = L \\otimes_K K_1$", "is a discrete valuation ring weakly unramified over $A_1$.", "If $K \\subset K_1$ is not a weak solution for $A \\to C$, then", "the integral closure $C_1$ of $C$ in $M_1 = M \\otimes_K K_1$ is a", "discrete valuation ring and $B_1 \\to C_1$ has ramification index $p$.", "In this case, the field $M_1$ is obtained from $L_1$ by adjoining the", "$p$th root of $b$ with $\\text{ord}_{B_1}(b)$ divisible by $p$.", "Replacing $A$ by $A_1$, etc we may assume that $b = \\pi^n u$ where", "$u \\in B$ is a unit and $n$ is divisible by $p$. Of course, in", "this case the extension $M$ is obtained from $L$ by adjoining", "the $p$th root of a unit.", "\\medskip\\noindent", "Suppose $M$ is obtained from $L$ by adjoining the root of", "$y^p = u$ for some unit $u$ of $B$. If the residue class of $u$", "in $\\kappa_B$ is not a $p$th power, then $B \\subset C$ is", "weakly unramified (Lemma \\ref{lemma-pre-purely-inseparable-case})", "and we are done. Otherwise, we can replace our choice of $y$ by", "$y/v$ where $v^p$ and $u$ have the same image in $\\kappa_B$.", "After such a replacement we have", "$$", "y^p = 1 + \\pi b", "$$", "for some $b \\in B$. Then we see that $P(z) = \\pi b/ w^p$ where", "$z = (y - 1)/w$. Thus we see that the extension is a degree $p$", "extension of finite level with $\\xi = \\pi b / w^p$." ], "refs": [ "fields-lemma-Kummer", "more-algebra-lemma-make-degree-q-extension", "more-algebra-lemma-weakly-unramified-goes-up-along-totally-ramified", "more-algebra-lemma-pre-purely-inseparable-case" ], "ref_ids": [ 4514, 10520, 10515, 10521 ] } ], "ref_ids": [] }, { "id": 10529, "type": "theorem", "label": "more-algebra-lemma-lowering-the-level", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-lowering-the-level", "contents": [ "Let $A \\subset B \\subset C$ be extensions of discrete valuation rings", "with fractions fields $K \\subset L \\subset M$. Assume", "\\begin{enumerate}", "\\item $A$ has mixed characteristic $(0, p)$,", "\\item $A \\subset B$ weakly unramified,", "\\item $B$ contains a primitive $p$th root of $1$,", "\\item $M/L$ is a degree $p$ extension of finite level $l > 0$,", "\\item $\\kappa_A = \\bigcap_{n \\geq 1} \\kappa_B^{p^n}$.", "\\end{enumerate}", "Then there exists a finite separable extension $K_1$ of $K$", "totally ramified with respect to $A$", "such that either $K_1$ is a weak solution for $A \\to C$, or the extension", "$M_1/L_1$ is a degree $p$ extension of finite level", "$\\leq \\max(0, l - 1, 2l - p)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\pi \\in A$ be a uniformizer.", "Let $w \\in B$ and $P \\in B[t]$ be as in Lemma \\ref{lemma-prepare} (for $B$).", "Set $e_1 = \\text{ord}_B(w)$, so that $w$ and $\\pi^{e_1}$ are associates in $B$.", "Pick $z \\in M$ generating $M$ over $L$ with $\\xi = P(z) \\in K$", "and $n$ such that $\\pi^n\\xi \\in B$ as in the definition of the level", "of $M$ over $L$, i.e., $l = n/e_1$.", "\\medskip\\noindent", "The proof of this lemma is completely similar to the proof of", "Lemma \\ref{lemma-characteristic-p-case}.", "To explain what is going on, observe that", "\\begin{equation}", "\\label{equation-first-congruence}", "P(z) \\equiv z^p - z \\bmod \\pi^{-n + e_1}B", "\\end{equation}", "for any $z \\in L$ such that $\\pi^{-n} P(z) \\in B$ (use that $z$ has valuation", "at worst $-n/p$ and the shape of the polynomial $P$). Moreover, we have", "\\begin{equation}", "\\label{equation-second-congruence}", "\\xi_1 + \\xi_2 + w^p \\xi_1 \\xi_2 \\equiv \\xi_1 + \\xi_2 \\bmod \\pi^{-2n + pe_1}B ", "\\end{equation}", "for $\\xi_1, \\xi_2 \\in \\pi^{-n}B$. Finally, observe that", "$n - e_1 = (l - 1)/e_1$ and $-2n + pe_1 = -(2l - p)e_1$.", "Write $m = n - e_1 \\max(0, l - 1, 2l - p)$. The above shows that doing", "calculations in $\\pi^{-n}B / \\pi^{-n + m}B$ the polynomial $P$ behaves exactly", "as the polynomial $z^p - z$. This explains why the lemma is true", "but we also give the details below.", "\\medskip\\noindent", "Assumption (4) implies that $\\kappa_A$ is perfect. Observe that", "$m \\leq e_1$ and hence $A/\\pi^m$ is annihilated by $w$ and hence $p$.", "Thus we may choose compatible ring maps", "$\\overline{\\sigma} : \\kappa_A \\to A/\\pi^mA$ and", "$\\overline{\\sigma} : \\kappa_B \\to B/\\pi^mB$ as in", "Lemma \\ref{lemma-cohen}. We lift the second of these to a", "map of sets $\\sigma : \\kappa_B \\to B$. Then we can write", "$$", "\\xi =", "\\sum\\nolimits_{i = n, \\ldots, n - m + 1} \\sigma(\\lambda_i) \\pi^{-i} +", "\\pi^{-n + m)} b", "$$", "for some $\\lambda_i \\in \\kappa_B$ and $b \\in B$. Let", "$$", "I = \\{i \\in \\{n, \\ldots, n - m + 1\\} \\mid \\lambda_i \\in \\kappa_A\\}", "$$", "and", "$$", "J = \\{j \\in \\{n, \\ldots, n - m + 1\\} \\mid \\lambda_i \\not \\in \\kappa_A\\}", "$$", "We will argue by induction on the size of the finite set $J$.", "\\medskip\\noindent", "The case $J = \\emptyset$. Here for all $i \\in \\{n, \\ldots, n - m + 1\\}$", "we have $\\sigma(\\lambda_i) = a_i + \\pi^{n - m}b_i$ for some $a_i \\in A$", "and $b_i \\in B$ by our choice of $\\overline{\\sigma}$. Thus", "$\\xi = \\pi^{-n} a + \\pi^{-n + m} b$ for some $a \\in A$ and $b \\in B$.", "If $p | n$, then we write $a = a_0^p + \\pi a_1$ for some $a_0, a_1 \\in A$", "(as the residue field of $A$ is perfect). Set $z_1 = - \\pi^{-n/p} a_0$.", "Note that $P(z_1) \\in \\pi^{-n}B$ and that $z + z_1 + w z z_1$ is an", "element generating $M$ over $L$ (note that $wz_1 \\not = -1$ as", "$n < pe_1$). Moreover, by Lemma \\ref{lemma-prepare} we have", "$$", "P(z + z_1 + w z z_1) = P(z) + P(z_1) + w^p P(z) P(z_1) \\in K", "$$", "and by equations (\\ref{equation-first-congruence}) and", "(\\ref{equation-second-congruence}) we have", "$$", "P(z) + P(z_1) + w^p P(z) P(z_1)", "\\equiv", "\\xi + z_1^p - z_1 \\bmod \\pi^{-n + m}B", "$$", "for some $b' \\in B$. This contradict the minimality of $n$! Thus $p$", "does not divide $n$. Consider the degree $p$ extension $K_1$ of $K$ given", "by $P(y) = -\\pi^{-n}a$. By Lemma \\ref{lemma-extension-defined-by-nice-polynial}", "this extension is separable and totally ramified with respect to $A$.", "Thus $L_1 = L \\otimes_K K_1$", "is a field and $A_1 \\subset B_1$ is weakly unramified", "(Lemma \\ref{lemma-weakly-unramified-goes-up-along-totally-ramified}).", "By Lemma \\ref{lemma-extension-defined-by-nice-polynial}", "the ring $M_1 = M \\otimes_K K_1$ is either a product of $p$ copies", "of $L_1$ (in which case we are done) or a field extension of $L_1$", "of degree $p$. Moreover, in the second case, either $C_1$ is weakly unramified", "over $B_1$ (in which case we are done) or $M_1/L_1$ is degree $p$,", "Galois, totally ramified with respect to $B_1$.", "In this last case the extension $M_1/L_1$", "is generated by the element $z + y + wzy$ and we see that", "$P(z + y + wzy) \\in L_1$ and", "\\begin{align*}", "P(z + y + wzy)", "& = P(z) + P(y) + w^p P(z) P(y) \\\\", "& \\equiv", "\\xi - \\pi^{-n}a \\bmod \\pi^{-n + m}B_1 \\\\", "& \\equiv", "0 \\bmod \\pi^{-n + m}B_1", "\\end{align*}", "in exactly the same manner as above. By our choice of $m$ this", "means exactly that $M_1/L_1$ has level at most $\\max(0, l - 1, 2l - p)$.", "From now on we assume that $J \\not = \\emptyset$.", "\\medskip\\noindent", "Suppose that $j', j \\in J$ such that $j' = p^r j$ for some $r > 0$.", "Then we set", "$$", "z_1 = - \\sigma(\\lambda_j) \\pi^{-j} - \\sigma(\\lambda_j^p) \\pi^{-pj} -", "\\ldots - \\sigma(\\lambda_j^{p^{r - 1}}) \\pi^{-p^{r - 1}j}", "$$", "and we change $z$ into $z' = z + z_1 + wzz_1$. Observe that $z' \\in M$", "generates $M$ over $L$ and that we have", "$\\xi' = P(z') = P(z) + P(z_1) + wP(z)P(z_1) \\in L$ with", "$$", "\\xi' \\equiv", "\\xi - \\sigma(\\lambda_j) \\pi^{-j} + \\sigma(\\lambda_j^{p^r}) \\pi^{-j'}", "\\bmod \\pi^{-n + m}B", "$$", "by using equations (\\ref{equation-first-congruence}) and", "(\\ref{equation-second-congruence}) as above. Writing", "$$", "\\xi' = \\sum\\nolimits_{i = n, \\ldots, n - m + 1} \\sigma(\\lambda'_i) \\pi^{-i}", "+ \\pi^{-n + m}b'", "$$", "as before we find that", "$\\lambda'_i = \\lambda_i$ for $i \\not = j, j'$ and $\\lambda'_j = 0$.", "Thus the set $J$ has gotten smaller.", "By induction on the size of $J$ we may assume there is no pair", "$j, j'$ of $J$ such that $j'/j$ is a power of $p$.", "(Please observe that in this procedure we may get thrown back into the case", "that $J = \\emptyset$ we treated above.)", "\\medskip\\noindent", "For $j \\in J$ write $\\lambda_j = \\mu_j^{p^{r_j}}$ for some $r_j \\geq 0$ and", "$\\mu_j \\in \\kappa_B$ which is not a $p$th power. This is possible by our", "assumption (4). Let $j \\in J$ be the unique index such that $j p^{-r_j}$", "is maximal. (The index is unique by the result of the preceding paragraph.)", "Choose $r > \\max(r_j + 1)$ and such that $j p^{r - r_j} > n$ for $j \\in J$.", "Let $K_1/K$ be the extension of degree $p^r$, totally ramified", "with respect to $A$, defined by $(\\pi')^{p^r} = \\pi$.", "Observe that $\\pi'$ is the uniformizer of the", "corresponding discrete valuation ring $A_1 \\subset K_1$.", "Observe that $L_1 = L \\otimes_K K_1$ is a field and $L_1/L$", "is totally ramified with respect to $B$", "(Lemma \\ref{lemma-weakly-unramified-goes-up-along-totally-ramified}).", "Computing in the integral closure $B_1$ we get", "$$", "\\xi = \\sum\\nolimits_{i \\in I} \\sigma(\\lambda_i) (\\pi')^{-i p^r} +", "\\sum\\nolimits_{j \\in J} \\sigma(\\mu_j)^{p^{r_j}} (\\pi')^{-j p^r} +", "\\pi^{-n + m} b_1", "$$", "for some $b_1 \\in B_1$. Note that $\\sigma(\\lambda_i)$ for $i \\in I$", "is a $q$th power modulo $\\pi^m$, i.e., modulo $(\\pi')^{m p^r}$.", "Hence we can rewrite the above as", "$$", "\\xi = \\sum\\nolimits_{i \\in I} x_i^{p^r} (\\pi')^{-i p^r} +", "\\sum\\nolimits_{j \\in J} \\sigma(\\mu_j)^{p^{r_j}} (\\pi')^{- j p^r}", "+ \\pi^{-n + m}b_1", "$$", "Similar to our choice in the previous paragraph we set", "\\begin{align*}", "z_1 & - \\sum\\nolimits_{i \\in I}", "\\left(x_i (\\pi')^{-i} + \\ldots + x_i^{p^{r - 1}} (\\pi')^{-i p^{r - 1}}\\right)", "\\\\", "& - \\sum\\nolimits_{j \\in J}", "\\left(", "\\sigma(\\mu_j) (\\pi')^{- j p^{r - r_j}}", "+ \\ldots +", "\\sigma(\\mu_j)^{p^{r_j - 1}} (\\pi')^{- j p^{r - 1}}", "\\right)", "\\end{align*}", "and we change our choice of $z$ into $z' = z + z_1 + wzz_1$.", "Then $z'$ generates $M_1$ over $L_1$ and", "$\\xi' = P(z') = P(z) + P(z_1) + w^p P(z) P(z_1) \\in L_1$", "and a calculation shows that", "$$", "\\xi' \\equiv", "\\sum\\nolimits_{i \\in I} x_i (\\pi')^{-i} +", "\\sum\\nolimits_{j \\in J} \\sigma(\\mu_j) (\\pi')^{- j p^{r - r_j}} +", "(\\pi')^{(-n + m)p^r}b'_1", "$$", "for some $b'_1 \\in B_1$. There is a unique $j$ such that $j p^{r - r_j}$", "is maximal and $j p^{r - r_j}$ is bigger than $i \\in I$. If", "$j p^{r - r_j} \\leq (n - m)p^r$ then the level of the extension $M_1/L_1$", "is less than $\\max(0, l - 1, 2l - p)$. If not, then, as $p$ divides", "$j p^{r - r_j}$, we see that $M_1 / L_1$ falls into case (C) of", "Lemma \\ref{lemma-extension-defined-by-nice-polynial}.", "This finishes the proof." ], "refs": [ "more-algebra-lemma-prepare", "more-algebra-lemma-characteristic-p-case", "more-algebra-lemma-cohen", "more-algebra-lemma-prepare", "more-algebra-lemma-extension-defined-by-nice-polynial", "more-algebra-lemma-weakly-unramified-goes-up-along-totally-ramified", "more-algebra-lemma-extension-defined-by-nice-polynial", "more-algebra-lemma-weakly-unramified-goes-up-along-totally-ramified", "more-algebra-lemma-extension-defined-by-nice-polynial" ], "ref_ids": [ 10526, 10525, 10523, 10526, 10527, 10515, 10527, 10515, 10527 ] } ], "ref_ids": [] }, { "id": 10530, "type": "theorem", "label": "more-algebra-lemma-special-case", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-special-case", "contents": [ "Let $A \\subset B \\subset C$ be extensions of discrete valuation rings", "with fraction fields $K \\subset L \\subset M$. Assume", "\\begin{enumerate}", "\\item the residue field $k$ of $A$ is algebraically closed of", "characteristic $p > 0$,", "\\item $A$ and $B$ are complete,", "\\item $A \\to B$ is weakly unramified,", "\\item $M$ is a finite extension of $L$,", "\\item $k = \\bigcap\\nolimits_{n \\geq 1} \\kappa_B^{p^n}$", "\\end{enumerate}", "Then there exists a finite extension $K \\subset K_1$ which", "is a weak solution for $A \\to C$." ], "refs": [], "proofs": [ { "contents": [ "Let $M'$ be any finite extension of $L$ and consider the integral closure", "$C'$ of $B$ in $M'$. Then $C'$ is finite over $B$ as $B$ is Nagata by", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-complete-local-Nagata}.", "Moreover, $C'$ is a discrete valuation ring, see discussion in", "Remark \\ref{remark-construction}. Moreover $C'$ is complete as a", "$B$-module, hence complete as a discrete valuation ring, see", "Algebra, Section \\ref{algebra-section-completion}.", "It follows in particular that $C$ is the integral", "closure of $B$ in $M$ (by definition of valuation rings as maximal", "for the relation of domination).", "\\medskip\\noindent", "Let $M \\subset M'$ be a finite extension and let $C' \\subset M'$", "be the integral closure of $B$ as above. By", "Lemma \\ref{lemma-solutions-go-down}", "it suffices to prove the result for $A \\to B \\to C'$.", "Hence we may assume that $M/L$ is normal, see", "Fields, Lemma \\ref{fields-lemma-normal-closure}.", "\\medskip\\noindent", "If $M / L$ is normal, we can find a chain of finite extensions", "$$", "L = L^0 \\subset L^1 \\subset L^2 \\subset \\ldots \\subset L^r = M", "$$", "such that each extension $L^{j + 1}/L^j$ is either:", "\\begin{enumerate}", "\\item[(a)] purely inseparable of degree $p$,", "\\item[(b)] totally ramified with respect to $B^j$ and Galois of degree $p$,", "\\item[(c)] totally ramified with respect to $B^j$ and Galois cyclic of", "order prime to $p$,", "\\item[(d)] Galois and unramified with respect to $B^j$.", "\\end{enumerate}", "Here $B^j$ is the integral closure of $B$ in $L^j$.", "Namely, since $M/L$ is normal we can write it as a compositum of", "a Galois extension and a purely inseparable extension", "(Fields, Lemma \\ref{fields-lemma-normal-case}).", "For the purely inseparable extension the existence of the filtration", "is clear. In the Galois case, note that $G$ is ``the'' decomposition group", "and let $I \\subset G$ be the inertia group. Then on the one hand", "$I$ is solvable by Lemma \\ref{lemma-galois-inertia} and on the other", "hand the extension $M^I/L$ is unramified with respect to $B$ by", "Lemma \\ref{lemma-inertial-invariants-unramified}.", "This proves we have a filtration as stated.", "\\medskip\\noindent", "We are going to argue by induction on the integer $r$. Suppose that we", "can find a finite extension $K \\subset K_1$ which is a weak solution", "for $A \\to B^1$ where $B^1$ is the integral closure of $B$ in $L^1$.", "Let $K'_1$ be the normal closure of $K_1/K$", "(Fields, Lemma \\ref{fields-lemma-normal-closure}).", "Since $A$ is complete and the residue field of $A$ is algebraically closed", "we see that $K'_1/K_1$ is separable and totally ramified with", "respect to $A_1$ (some details omitted).", "Hence $K \\subset K'_1$ is a weak solution for $A \\to B^1$ as well by", "Lemma \\ref{lemma-weakly-unramified-goes-up-along-totally-ramified}.", "In other words, we may and do assume that $K_1$ is a normal extension of $K$.", "Having done so we consider the sequence", "$$", "L^0_1 = (L^0 \\otimes_K K_1)_{red} \\subset", "L^1_1 = (L^1 \\otimes_K K_1)_{red} \\subset \\ldots \\subset", "L^r_1 = (L^r \\otimes_K K_1)_{red}", "$$", "and the corresponding integral closures $B^i_1$. Note that $C_1 = B^r_1$", "is a product of discrete valuation rings which are transitively permuted", "by $G = \\text{Aut}(K_1/K)$ by Lemma \\ref{lemma-galois-relative}.", "In particular all the extensions of discrete valuation rings", "$A_1 \\to (C_1)_\\mathfrak m$ are isomorphic and a solution for one", "will be a solution for all of them. We can apply the induction", "hypothesis to the sequence", "$$", "A_1 \\to (B^1_1)_{B^1_1 \\cap \\mathfrak m} \\to", "(B^2_1)_{B^2_1 \\cap \\mathfrak m} \\to", "\\ldots \\to", "(B^r_1)_{B^r_1 \\cap \\mathfrak m} =", "(C_1)_\\mathfrak m", "$$", "to get a solution $K_1 \\subset K_2$ for $A_1 \\to (C_1)_\\mathfrak m$.", "The extension $K \\subset K_2$ will then be a solution for $A \\to C$", "by what we said before. Note that the induction hypothesis applies:", "the ring map $A_1 \\to (B^1_1)_{B^1_1 \\cap \\mathfrak m}$", "is weakly unramified by our choice of $K_1$", "and the sequence of fraction field extensions", "each still have one of the properties (a), (b), (c), or (d)", "listed above. Moreover, observe that for any finite extension ", "$\\kappa_B \\subset \\kappa$ we still have $k = \\bigcap \\kappa^{p^n}$.", "\\medskip\\noindent", "Thus everything boils down to finding a weak solution for $A \\subset C$", "when the field extension $L \\subset M$ satisfies one of the properties", "(a), (b), (c), or (d).", "\\medskip\\noindent", "Case (d). This case is trivial as here $B \\to C$ is unramified already.", "\\medskip\\noindent", "Case (c). Say $M/L$ is cyclic of order $n$ prime to $p$. Because", "$M/L$ is totally ramified with respect to $B$ we see that the ramification", "index of $B \\subset C$ is $n$ and hence the ramification index of $A \\subset C$", "is $n$ as well. Choose a uniformizer $\\pi \\in A$ and set", "$K_1 = K[\\pi^{1/n}]$. Then $K_1/K$ is a solution for $A \\subset C$", "by Abhyankar's lemma (Lemma \\ref{lemma-abhyankar}).", "\\medskip\\noindent", "Case (b). We divide this case into the mixed characteristic case and the", "equicharacteristic case. In the equicharacteristic case this is", "Lemma \\ref{lemma-characteristic-p-case}. In the mixed characteristic", "case, we first replace $K$ by a finite extension to get to the", "situation where $M/L$ is a degree $p$ extension of finite level using", "Lemma \\ref{lemma-make-finite-level}.", "Then the level is a rational number $l \\in [0, p)$, see discussion", "preceding Lemma \\ref{lemma-lowering-the-level}. If the level is $0$,", "then $B \\to C$ is weakly unramified and we're done. If not, then we", "can replacing the field $K$ by a finite extension to obtain a new", "situation with level $l' \\leq \\max(0, l - 1, 2l - p)$ by", "Lemma \\ref{lemma-lowering-the-level}.", "If $l = p - \\epsilon$ for $\\epsilon < 1$ then we see that", "$l' \\leq p - 2\\epsilon$. Hence after a finite number of replacements", "we obtain a case with level $\\leq p - 1$. Then after at most $p - 1$", "more such replacements we reach the situation where the level is zero.", "\\medskip\\noindent", "Case (a) is Lemma \\ref{lemma-purely-inseparable-case}. This is the only case", "where we possibly need a purely inseparable extension of $K$, namely, in", "case (2) of the statement of the lemma we win by adjoining a $p$th power", "of the element $\\pi$. This finishes the proof of the lemma." ], "refs": [ "algebra-lemma-Noetherian-complete-local-Nagata", "more-algebra-remark-construction", "more-algebra-lemma-solutions-go-down", "fields-lemma-normal-closure", "fields-lemma-normal-case", "more-algebra-lemma-galois-inertia", "more-algebra-lemma-inertial-invariants-unramified", "fields-lemma-normal-closure", "more-algebra-lemma-weakly-unramified-goes-up-along-totally-ramified", "more-algebra-lemma-galois-relative", "more-algebra-lemma-abhyankar", "more-algebra-lemma-characteristic-p-case", "more-algebra-lemma-make-finite-level", "more-algebra-lemma-lowering-the-level", "more-algebra-lemma-lowering-the-level", "more-algebra-lemma-purely-inseparable-case" ], "ref_ids": [ 1353, 10679, 10516, 4494, 4522, 10501, 10503, 4494, 10515, 10519, 10509, 10525, 10528, 10529, 10529, 10522 ] } ], "ref_ids": [] }, { "id": 10531, "type": "theorem", "label": "more-algebra-lemma-big-extension-is-ok", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-big-extension-is-ok", "contents": [ "Let $A \\to B$ be an extension of discrete valuation rings with fraction", "fields $K \\subset L$. Assume $B$ is essentially of finite type over $A$.", "Let $K \\subset K'$ be an algebraic extension of fields such that", "the integral closure $A'$ of $A$ in $K'$ is Noetherian. Then the integral", "closure $B'$ of $B$ in $L' = (L \\otimes_K K')_{red}$ is Noetherian", "as well. Moreover, the map $\\Spec(B') \\to \\Spec(A')$", "is surjective and the corresponding residue field extensions are finitely", "generated field extensions." ], "refs": [], "proofs": [ { "contents": [ "Let $A \\to C$ be a finite type ring map such that $B$ is a localization of", "$C$ at a prime $\\mathfrak p$. Then $C' = C \\otimes_A A'$ is a finite type", "$A'$-algebra, in particular Noetherian. Since $A \\to A'$ is integral, so", "is $C \\to C'$. Thus $B = C_\\mathfrak p \\subset C'_\\mathfrak p$ is", "integral too. It follows that the dimension of $C'_\\mathfrak p$ is $1$", "(Algebra, Lemma \\ref{algebra-lemma-integral-sub-dim-equal}).", "Of course $C'_\\mathfrak p$ is Noetherian.", "Let $\\mathfrak q_1, \\ldots, \\mathfrak q_n$ be the minimal primes", "of $C'_\\mathfrak p$. Let $B'_i$ be the integral closure of", "$B = C_\\mathfrak p$, or equivalently by the above of $C'_\\mathfrak p$", "in the field of fractions of $C'_{\\mathfrak p'}/\\mathfrak q_i$.", "It follows from Krull-Akizuki", "(Algebra, Lemma \\ref{algebra-lemma-krull-akizuki} applied", "to the finitely many localizations of $C'_\\mathfrak p$ at its", "maximal ideals) that each $B'_i$ is Noetherian.", "Moreover the residue field extensions in $C'_\\mathfrak p \\to B'_i$", "are finite by", "Algebra, Lemma \\ref{algebra-lemma-finite-extension-residue-fields-dimension-1}.", "Finally, we observe that $B' = \\prod B'_i$ is the integral closure of $B$ in", "$L' = (L \\otimes_K K')_{red}$." ], "refs": [ "algebra-lemma-integral-sub-dim-equal", "algebra-lemma-krull-akizuki", "algebra-lemma-finite-extension-residue-fields-dimension-1" ], "ref_ids": [ 985, 1027, 1025 ] } ], "ref_ids": [] }, { "id": 10532, "type": "theorem", "label": "more-algebra-lemma-epp-essentially-finite-type-separable", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-epp-essentially-finite-type-separable", "contents": [ "Let $A \\to B$ be an extension of discrete valuation rings with fraction", "fields $K \\subset L$. Assume", "\\begin{enumerate}", "\\item $B$ is essentially of finite type over $A$,", "\\item either $A$ or $B$ is a Nagata ring, and", "\\item $L/K$ is separable.", "\\end{enumerate}", "Then there exists a separable solution for $A \\to B$", "(Definition \\ref{definition-solution})." ], "refs": [ "more-algebra-definition-solution" ], "proofs": [ { "contents": [ "Observe that if $A$ is Nagata, then so is $B$", "(Algebra, Lemma \\ref{algebra-lemma-nagata-localize} and", "Proposition \\ref{algebra-proposition-nagata-universally-japanese}).", "Thus the lemma follows on combining", "Proposition \\ref{proposition-epp-essentially-finite-type}", "and", "Lemma \\ref{lemma-separable-solution-separable-solution}." ], "refs": [ "algebra-lemma-nagata-localize", "algebra-proposition-nagata-universally-japanese", "more-algebra-proposition-epp-essentially-finite-type", "more-algebra-lemma-separable-solution-separable-solution" ], "ref_ids": [ 1351, 1430, 10592, 10517 ] } ], "ref_ids": [ 10645 ] }, { "id": 10533, "type": "theorem", "label": "more-algebra-lemma-invertible", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-invertible", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module. Equivalent are", "\\begin{enumerate}", "\\item $M$ is finite locally free module of rank $1$,", "\\item $M$ is invertible, and", "\\item there exists an $R$-module $N$ such that $M \\otimes_R N \\cong R$.", "\\end{enumerate}", "Moreover, in this case the module $N$ is (3) is isomorphic", "to $\\Hom_R(M, R)$." ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Consider the module $N = \\Hom_R(M, R)$ and the evaluation", "map $M \\otimes_R N = M \\otimes_R \\Hom_R(M, R) \\to R$. If $f \\in R$", "such that $M_f \\cong R_f$, then the evaluation map becomes an isomorphism", "after localization at $f$ (details omitted). Thus we see the evaluation", "map is an isomorphism by Algebra, Lemma \\ref{algebra-lemma-cover}.", "Thus (1) $\\Rightarrow$ (3).", "\\medskip\\noindent", "Assume (3). Then the functor $K \\mapsto K \\otimes_R N$ is a quasi-inverse", "to the functor $K \\mapsto K \\otimes_R M$. Thus (3) $\\Rightarrow$ (2).", "Conversely, if (2) holds, then $K \\mapsto K \\otimes_R M$ is essentially", "surjective and we see that (3) holds.", "\\medskip\\noindent", "Assume the equivalent conditions (2) and (3) hold. Denote", "$\\psi : M \\otimes_R N \\to R$ the isomorphism from (3).", "Choose an element $\\xi = \\sum_{i = 1, \\ldots, n} x_i \\otimes y_i$", "such that $\\psi(\\xi) = 1$. Consider the isomorphisms", "$$", "M \\to M \\otimes_R M \\otimes_R N \\to M", "$$", "where the first arrow sends $x$ to $\\sum x_i \\otimes x \\otimes y_i$", "and the second arrow sends $x \\otimes x' \\otimes y$ to $\\psi(x' \\otimes y)x$.", "We conclude that $x \\mapsto \\sum \\psi(x \\otimes y_i)x_i$ is", "an automorphism of $M$. This automorphism factors as", "$$", "M \\to R^{\\oplus n} \\to M", "$$", "where the first arrow is given by", "$x \\mapsto (\\psi(x \\otimes y_1), \\ldots, \\psi(x \\otimes y_n))$", "and the second arrow by $(a_1, \\ldots, a_n) \\mapsto \\sum a_i x_i$.", "In this way we conclude that $M$ is a direct summand of a finite free", "$R$-module. This means that $M$ is finite locally free", "(Algebra, Lemma \\ref{algebra-lemma-finite-projective}).", "Since the same is true for $N$ by symmetry and since", "$M \\otimes_R N \\cong R$, we see that", "$M$ and $N$ both have to have rank $1$." ], "refs": [ "algebra-lemma-cover", "algebra-lemma-finite-projective" ], "ref_ids": [ 411, 795 ] } ], "ref_ids": [] }, { "id": 10534, "type": "theorem", "label": "more-algebra-lemma-UFD-Pic-trivial", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-UFD-Pic-trivial", "contents": [ "Let $R$ be a UFD. Then $\\Pic(R)$ is trivial." ], "refs": [], "proofs": [ { "contents": [ "Let $L$ be an invertible $R$-module. By Lemma \\ref{lemma-invertible}", "we see that $L$ is a finite locally free $R$-module. In particular", "$L$ is torsion free and finite over $R$. Pick a nonzero element", "$\\varphi \\in \\Hom_R(L, R)$ of the dual invertible module.", "Then $I = \\varphi(L) \\subset R$ is an ideal which is an invertible module.", "Pick a nonzero $f \\in I$ and let", "$$", "f = u p_1^{e_1} \\ldots p_r^{e_r}", "$$", "be the factorization into prime elements with $p_i$ pairwise distinct.", "Since $L$ is finite locally free there exist $a_i \\in R$,", "$a_i \\not \\in (p_i)$ such that $I_{a_i} = (g_i)$ for some $g_i \\in R_{a_i}$.", "Then $p_i$ is still a prime element of the UFD $R_{a_i}$ and", "we can write $g_i = p_i^{c_i} g'_i$ for some $g'_i \\in R_{a_i}$", "not divisible by $p_i$. Since $f \\in I_{a_i}$ we see that $e_i \\geq c_i$.", "We claim that $I$ is generated by $h = p_1^{c_1} \\ldots p_r^{c_r}$ which", "finishes the proof.", "\\medskip\\noindent", "To prove the claim it suffices to show that $I_a$ is generated by $h$", "for any $a \\in R$ such that $I_a$ is a principal ideal", "(Algebra, Lemma \\ref{algebra-lemma-cover}). Say $I_a = (g)$.", "Let $J \\subset \\{1, \\ldots, r\\}$ be the set of $i$ such that", "$p_i$ is a nonunit (and hence a prime element) in $R_a$. Because", "$f \\in I_a = (g)$ we find the prime factorization", "$g = v \\prod_{i \\in J} p_j^{b_j}$", "with $v$ a unit and $b_j \\leq e_j$. For each $j \\in J$ we have", "$I_{aa_j} = g R_{aa_j} = g_j R_{aa_j}$, in other words", "$g$ and $g_j$ map to associates in $R_{aa_j}$. By uniqueness", "of factorization this implies that $b_j = c_j$ and the proof is complete." ], "refs": [ "more-algebra-lemma-invertible", "algebra-lemma-cover" ], "ref_ids": [ 10533, 411 ] } ], "ref_ids": [] }, { "id": 10535, "type": "theorem", "label": "more-algebra-lemma-det-ses", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-det-ses", "contents": [ "Let $R$ be a ring. Let", "$$", "0 \\to M' \\to M \\to M'' \\to 0", "$$", "be a short exact sequence of finite projective $R$-modules. Then there", "is a canonical isomorphism", "$$", "\\gamma : \\det(M') \\otimes \\det(M'') \\longrightarrow \\det(M)", "$$" ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "First proof. Decompose $R$ into a product of rings $R_{ij}$ such that", "$M' = \\prod M'_{ij}$ and $M'' = \\prod M''_{ij}$ where", "$M'_{ij}$ has rank $i$ and $M''_{ij}$ has rank $j$.", "Of course then $M = \\prod M_{ij}$ and $M_{ij}$ has rank $i + j$.", "This reduces us to the case where $M'$ and $M''$ have constant", "rank say $i$ and $j$. In this case we have to construct a canonical", "map", "$$", "\\wedge^i(M') \\otimes \\wedge^j(M'') \\longrightarrow \\wedge^{i + j}(M)", "$$", "To do this choose $m'_1, \\ldots, m'_i$ in $M'$ and", "$m''_1, \\ldots, m''_j$ in $M''$. Denote $m_1, \\ldots, m_i \\in M$", "the images of $m'_1, \\ldots, m'_i$ and denote", "$m_{i + 1}, \\ldots , m_{i + j} \\in M$ elements mapping to", "$m''_1, \\ldots, m''_j$ in $M''$. Our rule will be that", "$$", "m'_1 \\wedge \\ldots \\wedge m'_i \\otimes", "m''_1 \\wedge \\ldots \\wedge m''_j", "\\longmapsto", "m_1 \\wedge \\ldots \\wedge m_{i + j}", "$$", "We omit the detailed proof that this is well defined and an isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10536, "type": "theorem", "label": "more-algebra-lemma-det-ses-functorial", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-det-ses-functorial", "contents": [ "Let $R$ be a ring. Let", "$$", "\\xymatrix{", "0 \\ar[r] &", "M' \\ar[r] \\ar[d]^u &", "M \\ar[r] \\ar[d]^v &", "M'' \\ar[r] \\ar[d]^w &", "0 \\\\", "0 \\ar[r] &", "K' \\ar[r] &", "K \\ar[r] &", "K'' \\ar[r] &", "0", "}", "$$", "be a commutative diagram of finite projective $R$-modules", "whose vertical arrows are isomorphisms. Then we get a commutative", "diagram of isomorphisms", "$$", "\\xymatrix{", "\\det(M') \\otimes \\det(M'') \\ar[r]_-\\gamma \\ar[d]_{\\det(u) \\otimes \\det(w)} &", "\\det(M) \\ar[d]^{\\det(v)} \\\\", "\\det(K') \\otimes \\det(K'') \\ar[r]^-\\gamma & \\det(K)", "}", "$$", "where the horizontal arrows are the ones constructed", "in Lemma \\ref{lemma-det-ses}." ], "refs": [ "more-algebra-lemma-det-ses" ], "proofs": [ { "contents": [ "Omitted. Hint: use the second construction of the maps $\\gamma$", "in Lemma \\ref{lemma-det-ses}." ], "refs": [ "more-algebra-lemma-det-ses" ], "ref_ids": [ 10535 ] } ], "ref_ids": [ 10535 ] }, { "id": 10537, "type": "theorem", "label": "more-algebra-lemma-det-filtration", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-det-filtration", "contents": [ "Let $R$ be a ring. Let", "$$", "K \\subset L \\subset M", "$$", "be $R$-modules such that $K$, $L/K$, and $M/L$ are finite projective", "$R$-modules. Then the diagram", "$$", "\\xymatrix{", "\\det(K) \\otimes \\det(L/K) \\otimes \\det(M/L) \\ar[r] \\ar[d] &", "\\det(L) \\otimes \\det(M/L) \\ar[d] \\\\", "\\det(K) \\otimes \\det(M/K) \\ar[r] &", "\\det(M)", "}", "$$", "commutes where the maps are those of Lemma \\ref{lemma-det-ses}." ], "refs": [ "more-algebra-lemma-det-ses" ], "proofs": [ { "contents": [ "Omitted. Hint: after localizing at a prime of $R$ we can assume", "$K \\subset L \\subset M$ is isomorphic to", "$R^{\\oplus a} \\subset R^{\\oplus a + b} \\subset R^{\\oplus a + b + c}$", "and in this case the result is an evident computation." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10535 ] }, { "id": 10538, "type": "theorem", "label": "more-algebra-lemma-det-direct-sum", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-det-direct-sum", "contents": [ "Let $R$ be a ring. Let $M'$ and $M''$ be two finite projective", "$R$-modules. Then the diagram", "$$", "\\xymatrix{", "\\det(M') \\otimes \\det(M'') \\ar[r]", "\\ar[d]_{\\epsilon \\cdot (\\text{switch tensors})} &", "\\det(M' \\oplus M'') \\ar[d]^{\\det(\\text{swith summands})} \\\\", "\\det(M'') \\otimes \\det(M') \\ar[r] &", "\\det(M'' \\oplus M')", "}", "$$", "commutes where $\\epsilon = \\det( -\\text{id}_{M' \\otimes M''}) \\in R^*$", "and the horizontal arrows are those of Lemma \\ref{lemma-det-ses}." ], "refs": [ "more-algebra-lemma-det-ses" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10535 ] }, { "id": 10539, "type": "theorem", "label": "more-algebra-lemma-det-switch", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-det-switch", "contents": [ "Let $R$ be a ring. Let $M$, $N$ be finite projective $R$-modules.", "Let $a : M \\to N$ and $b : N \\to M$ be $R$-linear maps.", "Then", "$$", "\\det(\\text{id} + a \\circ b) = \\det(\\text{id} + b \\circ a)", "$$", "as elements of $R$." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove the assertion after replacing $R$ by a localization", "at a prime ideal. Thus we may assume $R$ is local and $M$ and $N$ are", "finite free. In this case we have to prove the equality", "$$", "\\det(I_n + AB) = \\det(I_m + BA)", "$$", "of usual determinants of matrices where $A$ has size $n \\times m$", "and $B$ has size $m \\times n$. This reduces to the case", "of the ring $R = \\mathbf{Z}[a_{ij}, b_{ji}; 1 \\leq i \\leq n, 1 \\leq j \\leq m]$", "where $a_{ij}$ and $b_{ij}$ are variables and the entries of the", "matrices $A$ and $B$. Taking the fraction field, this reduces to the", "case of a field of characteristic zero. In characteristic zero there", "is a universal polynomial expressing the determinant of a matrix", "of size $\\leq N$ in the traces of the powers of said matrix.", "Hence it suffices to prove", "$$", "\\text{Trace}((I_n + AB)^k) = \\text{Trace}((I_m + BA)^k)", "$$", "for all $k \\geq 1$. Expanding we see that it suffices to prove", "$\\text{Trace}((AB)^k) = \\text{Trace}((BA)^k)$ for all $k \\geq 0$.", "For $k = 1$ this is the well known fact that", "$\\text{Trace}(AB) = \\text{Trace}(BA)$. For $k > 1$", "it follows from this by writing $(AB)^k = A(BA)^{k - 1}B$", "and $(BA)^k = (BA)^{k - 1} A B$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10540, "type": "theorem", "label": "more-algebra-lemma-det", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-det", "contents": [ "Let $R$ be a ring. There is a map", "$$", "\\det : K_0(R) \\longrightarrow \\Pic(R)", "$$", "which maps $[M]$ to the class of the invertible module", "$\\wedge^n(M)$ if $M$ is a finite locally free module of rank $n$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the constructions above and in particular", "Lemma \\ref{lemma-det-ses} to see that the relations are mapped to $0$." ], "refs": [ "more-algebra-lemma-det-ses" ], "ref_ids": [ 10535 ] } ], "ref_ids": [] }, { "id": 10541, "type": "theorem", "label": "more-algebra-lemma-perfect-to-K-group", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-perfect-to-K-group", "contents": [ "Let $R$ be a ring. There is a map", "$$", "c : \\text{perfect complexes over }R \\longrightarrow K_0(R)", "$$", "with the following properties", "\\begin{enumerate}", "\\item $c(K[n]) = (-1)^nc(K)$ for a perfect complex $K$,", "\\item if $K \\to L \\to M \\to K[1]$ is a distinguished triangle of", "perfect complexes, then $c(L) = c(K) + c(M)$,", "\\item if $K$ is represented by a finite complex $M^\\bullet$", "consisting of finite projective modules, then", "$c(K) = \\sum (-1)^i[M_i]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $K$ be a perfect object of $D(R)$. By definition we can represent", "$K$ by a finite complex $M^\\bullet$ of finite projective $R$-modules.", "We define $c$ by setting", "$$", "c(K) = \\sum (-1)^n[M^n]", "$$", "in $K_0(R)$. Of course we have to show that this is well defined,", "but once it is well defined, then (1) and (3) are immediate.", "For the moment we view the map $c$ as defined on complexes of", "finite projective $R$-modules.", "\\medskip\\noindent", "Suppose that $L^\\bullet \\to M^\\bullet$ is a surjective map", "of finite complexes of finite projective $R$-modules.", "Let $K^\\bullet$ be the kernel. Then we obtain short exact", "sequences of $R$-modules", "$$", "0 \\to K^n \\to L^n \\to M^n \\to 0", "$$", "which are split because $M^n$ is projective. Hence $K^\\bullet$", "is also a finite complex of finite projective $R$-modules and", "$c(L^\\bullet) = c(K^\\bullet) + c(M^\\bullet)$ in $K_0(R)$.", "\\medskip\\noindent", "Suppose given finite complex $M^\\bullet$ of finite projective $R$-modules", "which is acyclic. Say $M^n = 0$ for $n \\not \\in [a, b]$. Then we", "can break $M^\\bullet$ into short exact sequences", "$$", "\\begin{matrix}", "0 \\to M^a \\to M^{a + 1} \\to N^{a + 1} \\to 0, \\\\", "0 \\to N^{a + 1} \\to M^{a + 2} \\to N^{a + 3} \\to 0, \\\\", "\\ldots \\\\", "0 \\to N^{b - 3} \\to M^{b - 2} \\to N^{b - 2} \\to 0, \\\\", "0 \\to N^{b - 2} \\to M^{b - 1} \\to M^b \\to 0", "\\end{matrix}", "$$", "Arguing by descending induction we see that $N^{b - 2}, \\ldots, N^{a + 1}$", "are finite projective $R$-modules, the sequences are split exact, and", "$$", "c(M^\\bullet) = \\sum (-1)[M^n] = \\sum (-1)^n([N^{n - 1}] + [N^n]) = 0", "$$", "Thus our construction gives zero on acyclic complexes.", "\\medskip\\noindent", "It follows formally from the results of the preceding two", "paragraphs that $c$ is well defined and satisfies (2). Namely,", "suppose the finite complexes $M^\\bullet$ and $L^\\bullet$ of", "finite projective $R$-modules represent the same object", "of $D(R)$. Then we can represent the isomorphism by a map", "$f : M^\\bullet \\to L^\\bullet$ of complexes, see", "Derived Categories, Lemma \\ref{derived-lemma-morphisms-from-projective-complex}.", "We obtain a short exact sequence of complexes", "$$", "0 \\to L^\\bullet \\to C(f)^\\bullet \\to K^\\bullet[1] \\to 0", "$$", "see Derived Categories, Definition \\ref{derived-definition-cone}.", "Since $f$ is a quasi-isomorphism, the cone $C(f)^\\bullet$ is", "acyclic (this follows for example from the discussion in", "Derived Categories, Section \\ref{derived-section-canonical-delta-functor}).", "Hence", "$$", "0 = c(C(f)^\\bullet) = c(L^\\bullet) + c(K^\\bullet[1]) =", "c(L^\\bullet) - c(K^\\bullet)", "$$", "as desired. We omit the proof of (2) which is similar." ], "refs": [ "derived-lemma-morphisms-from-projective-complex", "derived-definition-cone" ], "ref_ids": [ 1862, 1978 ] } ], "ref_ids": [] }, { "id": 10542, "type": "theorem", "label": "more-algebra-lemma-perfect-to-K-group-universal", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-perfect-to-K-group-universal", "contents": [ "Let $R$ be a ring. Let $D_{perf}(R)$ be the derived category of", "perfect objects, see Lemma \\ref{lemma-perfect-ring-classical-generator}.", "The map $c$ of Lemma \\ref{lemma-perfect-to-K-group} gives an isomorphism", "$K_0(D_{perf}(R)) = K_0(R)$." ], "refs": [ "more-algebra-lemma-perfect-ring-classical-generator", "more-algebra-lemma-perfect-to-K-group" ], "proofs": [ { "contents": [ "It follows from the definition of $K_0(D_{perf}(R))$", "(Derived Categories, Definition \\ref{derived-definition-K-zero})", "that $c$ induces a homomorphism $K_0(D_{perf}(R)) \\to K_0(R)$.", "\\medskip\\noindent", "Given a finite projective module $M$ over $R$ let us denote", "$M[0]$ the perfect complex over $R$ which has $M$ sitting in degree $0$", "and zero in other degrees. Given a short exact sequence", "$0 \\to M \\to M' \\to M'' \\to 0$ of finite projective modules", "we obtain a distinguished triangle $M[0] \\to M'[0] \\to M''[0] \\to M[1]$, see", "Derived Categories, Section \\ref{derived-section-canonical-delta-functor}.", "This shows that we obtain a map $K_0(R) \\to K_0(D_{perf}(R))$", "by sending $[M]$ to $[M[0]]$ with apologies for the horrendous notation.", "\\medskip\\noindent", "It is clear that $K_0(R) \\to K_0(D_{perf}(R)) \\to K_0(R)$ is the identity.", "On the other hand, if $M^\\bullet$ is a bounded complex of finite projective", "$R$-modules, then the the existence of the distinguished triangles", "of ``stupid truncations''", "(see Homology, Section \\ref{homology-section-truncations})", "$$", "\\sigma_{\\geq n}M^\\bullet \\to \\sigma_{\\geq n - 1}M^\\bullet \\to", "M^{n - 1}[-n + 1] \\to (\\sigma_{\\geq n}M^\\bullet)[1]", "$$", "and induction show that", "$$", "[M^\\bullet] = \\sum (-1)^i[M^i[0]]", "$$", "in $K_0(D_{perf}(R))$ (with again apologies for the notation).", "Hence the map $K_0(R) \\to K_0(D_{perf}(R))$ is surjective which", "finishes the proof." ], "refs": [ "derived-definition-K-zero" ], "ref_ids": [ 1999 ] } ], "ref_ids": [ 10247, 10541 ] }, { "id": 10543, "type": "theorem", "label": "more-algebra-lemma-regular-local-Pic-zero", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-regular-local-Pic-zero", "contents": [ "Let $R$ be a regular local ring. Let $f \\in R$.", "Then $\\Pic(R_f) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $L$ be an invertible $R_f$-module. In particular $L$ is", "a finite $R_f$-module. There exists a finite $R$-module", "$M$ such that $M_f \\cong L$, see", "Algebra, Lemma \\ref{algebra-lemma-construct-fp-module}.", "By Algebra, Proposition \\ref{algebra-proposition-regular-finite-gl-dim}", "we see that $M$ has a finite free resolution $F_\\bullet$ over $R$.", "It follows that $L$ is quasi-isomorphic to a finite complex", "of {\\it free} $R_f$-modules. Hence by", "Lemma \\ref{lemma-perfect-to-K-group} we see that", "$[L] = n[R_f]$ in $K_0(R)$ for some $n \\in \\mathbf{Z}$.", "Applying the map of Lemma \\ref{lemma-det}", "we see that $L$ is trivial." ], "refs": [ "algebra-lemma-construct-fp-module", "algebra-proposition-regular-finite-gl-dim", "more-algebra-lemma-perfect-to-K-group", "more-algebra-lemma-det" ], "ref_ids": [ 1081, 1421, 10541, 10540 ] } ], "ref_ids": [] }, { "id": 10544, "type": "theorem", "label": "more-algebra-lemma-regular-local-UFD", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-regular-local-UFD", "contents": [ "A regular local ring is a UFD." ], "refs": [], "proofs": [ { "contents": [ "Recall that a regular local ring is a domain, see", "Algebra, Lemma \\ref{algebra-lemma-regular-domain}.", "We will prove the unique factorization property", "by induction on the dimension of the regular local ring $R$.", "If $\\dim(R) = 0$, then $R$ is a field and in particular a UFD.", "Assume $\\dim(R) > 0$. Let $x \\in \\mathfrak m$, $x \\not \\in \\mathfrak m^2$.", "Then $R/(x)$ is regular by Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM},", "hence a domain by", "Algebra, Lemma \\ref{algebra-lemma-regular-domain},", "hence $x$ is a prime element.", "Let $\\mathfrak p \\subset R$ be a height $1$ prime. We have", "to show that $\\mathfrak p$ is principal, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-UFD-height-1}.", "We may assume $x \\not \\in \\mathfrak p$, since if $x \\in \\mathfrak p$,", "then $\\mathfrak p = (x)$ and we are done.", "For every nonmaximal prime $\\mathfrak q \\subset R$", "the local ring $R_\\mathfrak q$ is a regular local ring, see", "Algebra, Lemma \\ref{algebra-lemma-localization-of-regular-local-is-regular}.", "By induction we see that $\\mathfrak pR_\\mathfrak q$ is principal.", "In particular, the $R_x$-module $\\mathfrak p_x = \\mathfrak pR_x \\subset R_x$", "is a finitely presented $R_x$-module whose localization at", "any prime is free of rank $1$. ", "By Algebra, Lemma \\ref{algebra-lemma-finite-projective}", "we see that $\\mathfrak p_x$ is an invertible $R_x$-module.", "By Lemma \\ref{lemma-regular-local-Pic-zero} we see that", "$\\mathfrak p_x = (y)$ for some $y \\in R_x$.", "We can write $y = x^e f$ for some $f \\in \\mathfrak p$ and $e \\in \\mathbf{Z}$.", "Factor $f = a_1 \\ldots a_r$ into irreducible elements of $R$", "(Algebra, Lemma \\ref{algebra-lemma-factorization-exists}).", "Since $\\mathfrak p$ is prime, we see that $a_i \\in \\mathfrak p$", "for some $i$. Since $\\mathfrak p_x = (y)$ is prime and", "$a_i | y$ in $R_x$, it follows that $\\mathfrak p_x$ is generated by", "$a_i$ in $R_x$, i.e., the image of $a_i$ in $R_x$ is prime.", "As $x$ is a prime element, we find that $a_i$ is prime in $R$ by", "Algebra, Lemma \\ref{algebra-lemma-invert-prime-elements}.", "Since $(a_i) \\subset \\mathfrak p$ and $\\mathfrak p$ has height", "$1$ we conclude that $(a_i) = \\mathfrak p$ as desired." ], "refs": [ "algebra-lemma-regular-domain", "algebra-lemma-regular-ring-CM", "algebra-lemma-regular-domain", "algebra-lemma-characterize-UFD-height-1", "algebra-lemma-localization-of-regular-local-is-regular", "algebra-lemma-finite-projective", "more-algebra-lemma-regular-local-Pic-zero", "algebra-lemma-factorization-exists", "algebra-lemma-invert-prime-elements" ], "ref_ids": [ 940, 941, 940, 1032, 979, 795, 10543, 1030, 1033 ] } ], "ref_ids": [] }, { "id": 10545, "type": "theorem", "label": "more-algebra-lemma-picard-group-generic-fibre-regular", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-picard-group-generic-fibre-regular", "contents": [ "Let $R$ be a valuation ring with fraction field $K$", "and residue field $\\kappa$. Let $R \\to A$ be a", "homomorphism of rings such that", "\\begin{enumerate}", "\\item $A$ is local and $R \\to A$ is local,", "\\item $A$ is flat and essentially of finite type over $R$,", "\\item $A \\otimes_R \\kappa$ regular.", "\\end{enumerate}", "Then $\\Pic(A \\otimes_R K) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $L$ be an invertible $A \\otimes_R K$-module. In particular $L$ is", "a finite module. There exists a finite $A$-module", "$M$ such that $M \\otimes_R K \\cong L$, see", "Algebra, Lemma \\ref{algebra-lemma-construct-fp-module}.", "We may assume $M$ is torsion free as an $R$-module.", "Thus $M$ is flat as an $R$-module", "(Lemma \\ref{lemma-valuation-ring-torsion-free-flat}).", "From Lemma \\ref{lemma-flat-finite-type-valuation-ring-finite-presentation}", "we deduce that $M$ is of finite presentation as an $A$-module", "and $A$ is essentially of finite presentation as an $R$-algebra.", "By Lemma \\ref{lemma-structure-relatively-perfect}", "we see that $M$ is perfect relative to $R$,", "in particular $M$ is pseudo-coherent as an $A$-module.", "By Lemma \\ref{lemma-perfect-over-regular-local-ring}", "we see that $M$ is perfect, hence", "$M$ has a finite free resolution $F_\\bullet$ over $A$.", "It follows that $L$ is quasi-isomorphic to a finite complex", "of {\\it free} $A \\otimes_R K$-modules. Hence by", "Lemma \\ref{lemma-perfect-to-K-group} we see that", "$[L] = n[A \\otimes_R K]$ in $K_0(A \\otimes_R K)$", "for some $n \\in \\mathbf{Z}$.", "Applying the map of Lemma \\ref{lemma-det}", "we see that $L$ is trivial." ], "refs": [ "algebra-lemma-construct-fp-module", "more-algebra-lemma-valuation-ring-torsion-free-flat", "more-algebra-lemma-flat-finite-type-valuation-ring-finite-presentation", "more-algebra-lemma-structure-relatively-perfect", "more-algebra-lemma-perfect-over-regular-local-ring", "more-algebra-lemma-perfect-to-K-group", "more-algebra-lemma-det" ], "ref_ids": [ 1081, 9920, 9947, 10290, 10246, 10541, 10540 ] } ], "ref_ids": [] }, { "id": 10546, "type": "theorem", "label": "more-algebra-lemma-canonical-element-well-defined", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-canonical-element-well-defined", "contents": [ "Let $R$ be a ring. Let $a^\\bullet : K^\\bullet \\to L^\\bullet$ be a map of", "complexes of $R$-modules satisfying (1), (2), (3) above. If $L^\\bullet$", "has rank $0$, then $\\det(a^\\bullet)$ maps the", "canonical element $\\delta(L^\\bullet)$ to $\\delta(K^\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "Write $M^i = \\Ker(a^i)$. Thus we have a map of short", "exact sequences", "$$", "\\xymatrix{", "0 \\ar[r] &", "M^{-1} \\ar[r] \\ar[d]_{d_M} &", "K^{-1} \\ar[r] \\ar[d]_{d_K} &", "L^{-1} \\ar[r] \\ar[d]_{d_L} &", "0 \\\\", "0 \\ar[r] &", "M^0 \\ar[r] &", "K^0 \\ar[r] &", "L^0 \\ar[r] &", "0", "}", "$$", "By Lemma \\ref{lemma-det-ses-functorial} we know that", "$\\det(d_K)$ corresponds to $\\det(d_M) \\otimes \\det(d_L)$", "as maps. Unwinding the definitions this gives the required", "equality." ], "refs": [ "more-algebra-lemma-det-ses-functorial" ], "ref_ids": [ 10536 ] } ], "ref_ids": [] }, { "id": 10547, "type": "theorem", "label": "more-algebra-lemma-homotopic-surjections", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-homotopic-surjections", "contents": [ "Let $R$ be a ring. Let $a^\\bullet : K^\\bullet \\to L^\\bullet$ be a map of", "complexes of $R$-modules satisfying (1), (2), (3) above.", "Let $h : K^0 \\to L^{-1}$ be a map such that", "$b^0 = a^0 + d \\circ h$ and $b^{-1} = a^{-1} + h \\circ d$ are surjective.", "Then $\\det(a^\\bullet) = \\det(b^\\bullet)$ as maps", "$\\det(L^\\bullet) \\to \\det(K^\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "Suppose there exists a map $\\tilde h : K^0 \\to K^{-1}$ such that", "$h = a^{-1} \\circ \\tilde h$ and such that", "$k^0 = \\text{id} + d \\circ \\tilde h : K^0 \\to K^0$ and", "$k^1 = \\text{id} + \\tilde h \\circ d : K^{-1} \\to K^{-1}$ are isomorphisms.", "Then we obtain a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Ker(b^\\bullet) \\ar[r] \\ar[d]_{c^\\bullet} &", "K^\\bullet \\ar[r]_{b^\\bullet}", "\\ar[d]_{k^\\bullet} &", "L^\\bullet \\ar[r] \\ar[d]^{\\text{id}} &", "0 \\\\", "0 \\ar[r] &", "\\Ker(a^\\bullet) \\ar[r] &", "K^\\bullet \\ar[r]^{a^\\bullet} &", "L^\\bullet \\ar[r] &", "0", "}", "$$", "of complexes, where $c^\\bullet$ is the induced isomorphism of kernels.", "Using Lemma \\ref{lemma-det-ses-functorial} we see that", "$$", "\\xymatrix{", "\\det(\\Ker(b^i)) \\otimes \\det(L^i) \\ar[r] \\ar[d]_{\\det(c^i) \\otimes 1} &", "\\det(K^i) \\ar[d]^{\\det(k^i)} \\\\", "\\det(\\Ker(a^i)) \\otimes \\det(L^i) \\ar[r] &", "\\det(K^i)", "}", "$$", "commutes. Since $\\det(c^\\bullet)$ maps the canonical trivialization", "of $\\det(\\Ker(a^\\bullet))$ to the same for $b^\\bullet$", "(Lemma \\ref{lemma-canonical-element-well-defined})", "we see that we conclude if (and only if)", "$$", "\\det(k^0) = \\det(k^{-1})", "$$", "as elements of $R$ which follows from Lemma \\ref{lemma-det-switch}.", "\\medskip\\noindent", "Suppose there exists a direct summand $U \\subset K^{-1}$ such that both", "$a^{-1}|_U : U \\to L^{-1}$ and $b^{-1}|_U : U \\to L^{-1}$", "are isomorphisms. Define $\\tilde h$ as the composition of $h$", "with the inverse of $a^{-1}|_U$. We claim that $\\tilde h$ is a", "map as in the first paragraph of the proof. Namely, we have", "$h = a^{-1} \\circ \\tilde h$ by construction. ", "To show that $k^{-1} : K^{-1} \\to K^{-1}$", "is an isomorphism it suffices to show that it is surjective", "(Algebra, Lemma \\ref{algebra-lemma-fun}). Let $u \\in U$. We may", "choose $u' \\in U$ such that $b^{-1}(u') = a^{-1}(u)$.", "Then $u = k^{-1}(u')$. Namely, both $u$ and $k^{-1}(u')$", "are in $U$ and $a^{-1}(u) = a^{-1}(k^{-1}(u'))$ by a", "calculation\\footnote{$a^{-1}(k^{-1}(u')) =", "a^{-1}(u') + a^{-1}(\\tilde h(d(u'))) =", "a^{-1}(u') + h(d(u')) = b^{-1}(u') = a^{-1}(u)$} Since $a^{-1}|_U$", "is an isomorphism we get the equality. Thus $U \\subset \\Im(k^{-1})$.", "On the other hand, if $x \\in \\Ker(a^{-1})$ then", "$x = k^{-1}(x) \\bmod U$. Since $K^{-1} = \\Ker(a^{-1}) + U$ we conclude", "$k^{-1}$ is surjective.", "Finally, we show that $k^0 : K^0 \\to K^0$ is surjective.", "First, since $a^0 \\circ k^0 = b^0$ we see that $a^0 \\circ k^0$ is", "surjective. If $x \\in \\Ker(a^0)$, then $x = d(y)$ for some", "$y \\in \\Ker(a^{-1})$. We may write $y = k^{-1}(z)$ for some $z \\in K^{-1}$", "by the above. Then $x = k^0(d(z))$ and we conclude.", "\\medskip\\noindent", "Final step of the proof. It suffices to find $U$ as in the ", "preceding paragraph, but this may not always be possible. However,", "in order to show equality of two maps of $R$-modules, it suffices", "to do so after localization at primes of $R$. Hence we may assume", "$R$ is local. Then we get the following problem: suppose", "$$", "\\alpha, \\beta : R^{\\oplus n} \\longrightarrow R^{\\oplus m}", "$$", "are two surjective $R$-linear maps. Find a direct summand", "$U \\subset R^{\\oplus n}$ such that both $\\alpha|_U$ and $\\beta|_U$", "are isomorphisms. If $R$ is a field, this is possible by linear algebra.", "In general, one takes a solution over the residue field and lifts this", "to a solution over the local ring $R$. Some details omitted." ], "refs": [ "more-algebra-lemma-det-ses-functorial", "more-algebra-lemma-canonical-element-well-defined", "more-algebra-lemma-det-switch", "algebra-lemma-fun" ], "ref_ids": [ 10536, 10546, 10539, 388 ] } ], "ref_ids": [] }, { "id": 10548, "type": "theorem", "label": "more-algebra-lemma-compose-surjections", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-compose-surjections", "contents": [ "Let $R$ be a ring. Let $a^\\bullet : K^\\bullet \\to L^\\bullet$", "and $b^\\bullet : L^\\bullet \\to M^\\bullet$ be maps of", "complexes of $R$-modules satisfying (1), (2), (3) above.", "Then we have $\\det(a^\\bullet) \\circ \\det(b^\\bullet) = ", "\\det(b^\\bullet \\circ a^\\bullet)$ as maps", "$\\det(M^\\bullet) \\to \\det(K^\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints: Straightforward from Lemmas", "\\ref{lemma-det-ses}, \\ref{lemma-det-ses-functorial},", "and \\ref{lemma-det-filtration}." ], "refs": [ "more-algebra-lemma-det-ses", "more-algebra-lemma-det-ses-functorial", "more-algebra-lemma-det-filtration" ], "ref_ids": [ 10535, 10536, 10537 ] } ], "ref_ids": [] }, { "id": 10549, "type": "theorem", "label": "more-algebra-lemma-determinant-two-term-complexes", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-determinant-two-term-complexes", "contents": [ "Let $R$ be a ring. The constructions above determine a functor", "$$", "\\det :", "\\left\\{", "\\begin{matrix}", "\\text{category of perfect complexes} \\\\", "\\text{with tor amplitude in }[-1, 0] \\\\", "\\text{morphisms are isomorphisms}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{category of invertible modules} \\\\", "\\text{morphisms are isomorphisms}", "\\end{matrix}", "\\right\\}", "$$", "Moreover, given a rank $0$ perfect object $L$ of $D(R)$ with", "tor-amplitude in $[-1, 0]$ there is a canonical element", "$\\delta(L) \\in \\det(L)$ such that for any isomorphism", "$a : L \\to K$ in $D(R)$ we have $\\det(a)(\\delta(L)) = \\delta(K)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-perfect} every object of the source category", "may be represented by a complex", "$$", "L^\\bullet = \\ldots \\to 0 \\to L^{-1} \\to L^0 \\to 0 \\to \\ldots", "$$", "with $L^{-1}$ and $L^0$ finite projective $R$-modules. Let us temporarily", "call a complex of this type good. By ", "Derived Categories, Lemma \\ref{derived-lemma-morphisms-from-projective-complex}", "morphisms between good complexes in the derived category are", "homotopy classes of maps of complexes. Thus we may work with good complexes", "and we can use the determinant", "$\\det(L^\\bullet) = \\det(L^0) \\otimes \\det(L^{-1})^{\\otimes -1}$", "we investigated above.", "\\medskip\\noindent", "Let $a^\\bullet : L^\\bullet \\to K^\\bullet$ be a morphism of good complexes", "which is an isomorphism in $D(R)$, i.e., a quasi-isomorphism. We say that", "$$", "\\xymatrix{", "L^\\bullet \\ar[rr]_{a^\\bullet} & & K^\\bullet \\\\", "& M^\\bullet \\ar[lu]^{b^\\bullet} \\ar[ru]_{c^\\bullet}", "}", "$$", "is a good diagram if it commutes up to homotopy and $b^\\bullet$", "and $c^\\bullet$ satisfy conditions (1), (2), (3) above. Whenever", "we have such a diagram it makes sense to define", "$$", "\\det(a^\\bullet) = \\det(c^\\bullet)^{-1} \\circ \\det(b^\\bullet)", "$$", "where $\\det(c^\\bullet)$ and $\\det(b^\\bullet)$ are the isomorphisms", "constructed in the text above. We will show that good diagrams always", "exist and that the resulting map $\\det(a^\\bullet)$", "is independent of the choice of good diagram.", "\\medskip\\noindent", "Existence of good diagrams for a quasi-isomorphism", "$a^\\bullet : L^\\bullet \\to K^\\bullet$ of good complexes.", "Choose a", "surjection $p : R^{\\oplus n} \\to K^{-1}$. Then we can consider the new", "good complex", "$$", "M^\\bullet = \\ldots \\to 0 \\to", "L^{-1} \\oplus R^{\\oplus n} \\xrightarrow{d \\oplus 1}", "L^0 \\oplus R^{\\oplus n} \\to 0 \\to \\ldots", "$$", "with the projection map $b^\\bullet : M^\\bullet \\to L^\\bullet$ and the map", "$c^\\bullet : M^\\bullet \\to K^\\bullet$ using $a^{-1} \\oplus p$ in degree $-1$", "and using $a^0 \\oplus d \\circ p$ in degree $0$. The maps", "$b^\\bullet : M^\\bullet \\to L^\\bullet$ and", "$c^\\bullet : M^\\bullet \\to K^\\bullet$", "satisfy conditions (1), (2), (3) above and we get a good diagram.", "\\medskip\\noindent", "Suppose that we have a good diagram", "$$", "\\xymatrix{", "L^\\bullet \\ar[rr]_{\\text{id}^\\bullet} & & L^\\bullet \\\\", "& M^\\bullet \\ar[lu]^{b^\\bullet} \\ar[ru]_{c^\\bullet}", "}", "$$", "Then by Lemma \\ref{lemma-homotopic-surjections} we see that", "$\\det(c^\\bullet) = \\det(b^\\bullet)$. Thus we see that", "$\\det(\\text{id}^\\bullet) = \\text{id}$ is independent of the choice", "of good diagram.", "\\medskip\\noindent", "Before we prove independence in general, we think about", "composition. Suppose we have quasi-isomorphisms", "$L_1^\\bullet \\to L_2^\\bullet$ and $L_2^\\bullet \\to L_3^\\bullet$", "of good complexes and good diagrams", "$$", "\\vcenter{", "\\xymatrix{", "L_1^\\bullet \\ar[rr] & &", "L_2^\\bullet \\\\", "& M_{12}^\\bullet \\ar[lu] \\ar[ru]", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "L_2^\\bullet \\ar[rr] & &", "L_3^\\bullet \\\\", "& M_{23}^\\bullet \\ar[lu] \\ar[ru]", "}", "}", "$$", "We can extend this to a diagram", "$$", "\\xymatrix{", "L_1^\\bullet \\ar[rr] & &", "L_2^\\bullet \\ar[rr] & &", "L_3^\\bullet \\\\", "& M_{12}^\\bullet \\ar[lu] \\ar[ru]", "& & M_{23}^\\bullet \\ar[lu] \\ar[ru] \\\\", "& &", "M_{123}^\\bullet \\ar[lu] \\ar[ru]", "}", "$$", "where $M_{123}^\\bullet \\to M_{12}^\\bullet$ and", "$M_{123}^\\bullet \\to M_{23}^\\bullet$ have properties (1), (2), (3)", "and the square in the diagram commutes: we can just take", "$M_{123}^n = M_{12}^n \\times_{L_2^n} M_{23}^n$.", "Then Lemma \\ref{lemma-compose-surjections}", "shows that", "$$", "\\xymatrix{", "\\det(L_2^\\bullet) \\ar[r] \\ar[d] &", "\\det(M_{23}^\\bullet) \\ar[d] \\\\", "\\det(M_{12}^\\bullet) \\ar[r] &", "\\det(M_{123}^\\bullet)", "}", "$$", "commutes. A diagram chase shows that the composition", "$\\det(L_1^\\bullet) \\to \\det(L_2^\\bullet) \\to \\det(L_3^\\bullet)$", "of the maps associated to the two good diagrams using", "$M_{12}^\\bullet$ and $M_{23}^\\bullet$", "is equal to the map associated to the good diagram", "$$", "\\xymatrix{", "L_1^\\bullet \\ar[rr] & & L_3^\\bullet \\\\", "& M_{123}^\\bullet \\ar[lu] \\ar[ru]", "}", "$$", "Thus if we can show that these maps are independent of choices,", "then the composition law is satisfied too and we obtain our functor.", "\\medskip\\noindent", "Independence. Let a quasi-isomorphism", "$a^\\bullet : L^\\bullet \\to K^\\bullet$ of good complexes be given.", "Choose an inverse quasi-isomorphism $b^\\bullet : K^\\bullet \\to L^\\bullet$.", "Setting $L_1^\\bullet = L$, $L_2^\\bullet = K^\\bullet$ and", "$L_3^\\bullet = L^\\bullet$ may fix our choice of good diagram for", "$b^\\bullet$ and consider varying good diagrams for $a^\\bullet$.", "Then the result of the previous paragraphs is that no matter", "what choices, the composition always equals the identity", "map on $\\det(L^\\bullet)$. This clearly proves indepence of those choices.", "\\medskip\\noindent", "The statement on canonical elements follows immediately from", "Lemma \\ref{lemma-canonical-element-well-defined} and our construction." ], "refs": [ "more-algebra-lemma-perfect", "derived-lemma-morphisms-from-projective-complex", "more-algebra-lemma-homotopic-surjections", "more-algebra-lemma-compose-surjections", "more-algebra-lemma-canonical-element-well-defined" ], "ref_ids": [ 10212, 1862, 10547, 10548, 10546 ] } ], "ref_ids": [] }, { "id": 10550, "type": "theorem", "label": "more-algebra-lemma-inequality-general", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-inequality-general", "contents": [ "Let $A \\subset B$ be an extension of valuation rings with", "fraction fields $K \\subset L$. If the extension $K \\subset L$", "is finite, then the residue field extension is finite,", "the index of $\\Gamma_A$ in $\\Gamma_B$ is finite, and", "$$", "[\\Gamma_B : \\Gamma_A] [\\kappa_B : \\kappa_A] \\leq [L : K].", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $b_1, \\ldots, b_n \\in B$ be units whose images in $\\kappa_B$", "are linearly independent over $\\kappa_A$. Let $c_1, \\ldots, c_m \\in B$", "be nonzero elements whose images in $\\Gamma_B/\\Gamma_A$ are pairwise", "distinct. We claim that $b_i c_j$ are $K$-linearly independent", "in $L$. Namely, we claim a sum", "$$", "\\sum a_{ij} b_i c_j", "$$", "with $a_{ij} \\in K$ not all zero cannot be zero. Choose $(i_0, j_0)$ with", "$v(a_{i_0j_0}b_{i_0}c_{j_0})$ minimal. Replace $a_{ij}$ by", "$a_{ij}/a_{i_0j_0}$, so that $a_{i_0 j_0} = 1$. Let", "$$", "P = \\{(i, j) \\mid", "v(a_{ij}b_ic_j) = v(a_{i_0j_0}b_{i_0}c_{j_0}) \\}", "$$", "By our choice of $c_1, \\ldots, c_m$ we see that $(i, j) \\in P$ implies", "$j = j_0$. Hence if $(i, j) \\in P$, then $v(a_{ij}) = v(a_{i_0j_0}) = 0$,", "i.e., $a_{ij}$ is a unit. By our choice of $b_1, \\ldots, b_n$", "we see that", "$$", "\\sum\\nolimits_{(i, j) \\in P} a_{ij}b_i", "$$", "is a unit in $B$. Thus the valuation of", "$\\sum\\nolimits_{(i, j) \\in P} a_{ij}b_ic_j$ is", "$v(c_{j_0}) = v(a_{i_0j_0}b_{i_0}c_{j_0})$.", "Since the terms with $(i, j) \\not \\in P$ in the first displayed sum", "have strictly bigger valuation, we conclude that this sum cannot be", "zero, thereby proving the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10551, "type": "theorem", "label": "more-algebra-lemma-extension-normal-domains-and-roots", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-extension-normal-domains-and-roots", "contents": [ "Let $A \\to B$ be a flat local homomorphism of Noetherian local normal domains.", "Let $f \\in A$ and $h \\in B$ such that $f = w h^n$ for some $n > 1$ and some", "unit $w$ of $B$. Assume that for every height $1$ prime", "$\\mathfrak p \\subset A$ there is a height $1$ prime", "$\\mathfrak q \\subset B$ lying over $\\mathfrak p$", "such that the extension $A_\\mathfrak p \\subset B_\\mathfrak q$ is", "weakly unramified. Then $f = u g^n$ for some $g \\in A$ and unit $u$ of $A$." ], "refs": [], "proofs": [ { "contents": [ "The local rings of $A$ and $B$ at height $1$ primes are", "discrete valuation rings (Algebra, Lemma \\ref{algebra-lemma-characterize-dvr}).", "Thus the assumption makes sense (via", "Definition \\ref{definition-extension-discrete-valuation-rings}).", "Let $\\mathfrak p_1, \\ldots, \\mathfrak p_r$ be the primes of $A$ minimal", "over $f$. These have height $1$ by", "Algebra, Lemma \\ref{algebra-lemma-minimal-over-1}.", "For each $i$ let $\\mathfrak q_{i, j} \\subset B$, $j = 1, \\ldots, r_i$", "be the height $1$ primes of $B$ lying over $\\mathfrak p_i$.", "Say we number them so that $A_{\\mathfrak p_i} \\to B_{\\mathfrak q_{i, 1}}$", "is weakly unramified.", "Since $f$ maps to an $n$th power times a unit in $B_{\\mathfrak q_{i, 1}}$", "we see that the valuation $v_i$ of $f$ in $A_{\\mathfrak p_i}$ is", "divisible by $n$. Consider the exact sequence", "$$", "0 \\to I \\to A \\to", "\\prod\\nolimits_{i = 1, \\ldots, r}", "A_{\\mathfrak p_i}/\\mathfrak p_i^{v_i/n}A_{\\mathfrak p_i}", "$$", "Applying the exact functor $- \\otimes_A B$ we obtain", "$$", "0 \\to I \\otimes_A B \\to B \\to", "\\prod\\nolimits_{i = 1, \\ldots, r}", "\\prod\\nolimits_{j = 1, \\ldots, r_i}", "B_{\\mathfrak q_{i, j}}/\\mathfrak q_{i, j}^{e_{i, j}v_i/n}A_{\\mathfrak p_i}", "$$", "where $e_{i, j}$ is the ramification index of ", "$A_{\\mathfrak p_i} \\to B_{\\mathfrak q_{i, j}}$.", "It follows that $I \\otimes_A B$ is the set of elements $h'$ of $B$", "which have valuation $\\geq e_{i, j}v_i/n$ at $\\mathfrak q_{i, j}$.", "Since $f = wh^n$ in $B$ we see that $h$ has valuation", "$e_{i, j}v_i/n$ at $\\mathfrak q_{i, j}$. Thus $h'/h \\in B$", "by Algebra, Lemma", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}.", "It follows that $I \\otimes_A B$ is a free $B$-module of rank $1$.", "Therefore $I$ is a free $A$-module of rank $1$, see", "Algebra, Lemma \\ref{algebra-lemma-finite-projective-descends}.", "Let $g \\in I$ be a generator. Then we see that", "$g$ and $h$ differ by a unit in $B$. Working backwards we", "conclude that the valuation of $g$ in $A_{\\mathfrak p_i}$ is", "$v_i/n$. Hence $g^n$ and $f$ differ by a unit in $A$", "(by Algebra, Lemma", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1})", "as desired." ], "refs": [ "algebra-lemma-characterize-dvr", "more-algebra-definition-extension-discrete-valuation-rings", "algebra-lemma-minimal-over-1", "algebra-lemma-normal-domain-intersection-localizations-height-1", "algebra-lemma-finite-projective-descends", "algebra-lemma-normal-domain-intersection-localizations-height-1" ], "ref_ids": [ 1023, 10640, 683, 1313, 798, 1313 ] } ], "ref_ids": [] }, { "id": 10552, "type": "theorem", "label": "more-algebra-lemma-etale-extension-valuation-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-etale-extension-valuation-ring", "contents": [ "Let $A$ be a valuation ring. Let $A \\to B$ be an \\'etale ring map", "and let $\\mathfrak m \\subset B$ be a prime lying over the maximal", "ideal of $A$. Then $A \\subset B_\\mathfrak m$ is an extension of", "valuation rings which is weakly unramified." ], "refs": [], "proofs": [ { "contents": [ "The ring $A$ has weak dimension $\\leq 1$ by", "Lemma \\ref{lemma-weak-dimension-at-most-1}. Then $B$ has weak dimension", "$\\leq 1$ by Lemmas \\ref{lemma-weak-dimension-goes-up} and", "\\ref{lemma-when-weakly-etale}. hence the local ring $B_\\mathfrak m$", "is a valuation ring by Lemma \\ref{lemma-weak-dimension-at-most-1}.", "Since the extension $A \\subset B_\\mathfrak m$ induces a finite", "extension of fraction fields,", "we see that the $\\Gamma_A$ has finite index in the value group of", "$B_{\\mathfrak m}$. Thus for every $h \\in B_\\mathfrak m$ there exists", "an $n > 0$, an element $f \\in A$, and a unit $w \\in B_\\mathfrak m$", "such that $f = w h^n$ in $B_\\mathfrak m$. We will show that this implies", "$f = ug^n$ for some $g \\in A$ and unit $u \\in A$; this will show that", "the value groups of $A$ and $B_\\mathfrak m$ agree, as claimed in the lemma.", "\\medskip\\noindent", "Write $A = \\colim A_i$ as the colimit of its local subrings which", "are essentially of finite type over $\\mathbf{Z}$. Since $A$", "is a normal domain (Algebra, Lemma \\ref{algebra-lemma-valuation-ring-normal}),", "we may assume that each $A_i$ is normal (here we use that taking", "normalizations the local rings remain essentially of finite type", "over $\\mathbf{Z}$ by", "Algebra, Proposition \\ref{algebra-proposition-ubiquity-nagata}).", "For some $i$ we can find an \\'etale extension $A_i \\to B_i$", "such that $B = A \\otimes_{A_i} B_i$, see", "Algebra, Lemma \\ref{algebra-lemma-etale}.", "Let $\\mathfrak m_i$ be the intersection of $B_i$ with $\\mathfrak m$.", "Then we may apply Lemma \\ref{lemma-extension-normal-domains-and-roots}", "to the ring map $A_i \\to (B_i)_{\\mathfrak m_i}$ to conclude.", "The hypotheses of the lemma are satisfied because:", "\\begin{enumerate}", "\\item $A_i$ and $(B_i)_{\\mathfrak m_i}$ are Noetherian as they are", "essentially of finite type over $\\mathbf{Z}$,", "\\item $A_i \\to (B_i)_{\\mathfrak m_i}$ is flat as $A_i \\to B_i$ is \\'etale,", "\\item $B_i$ is normal as $A_i \\to B_i$ is \\'etale, see", "Algebra, Lemma \\ref{algebra-lemma-normal-goes-up},", "\\item for every height $1$ prime of $A_i$ there exists a height $1$", "prime of $(B_i)_{\\mathfrak m_i}$ lying over it by", "Algebra, Lemma \\ref{algebra-lemma-finite-in-codim-1} and the fact that", "$\\Spec((B_i)_{\\mathfrak m_i}) \\to \\Spec(A_i)$ is surjective,", "\\item the induced extensions $(A_i)_\\mathfrak p \\to (B_i)_\\mathfrak q$", "are unramified for every prime $\\mathfrak q$ lying over a prime", "$\\mathfrak p$ as $A_i \\to B_i$ is \\'etale.", "\\end{enumerate}", "This concludes the proof of the lemma." ], "refs": [ "more-algebra-lemma-weak-dimension-at-most-1", "more-algebra-lemma-weak-dimension-goes-up", "more-algebra-lemma-when-weakly-etale", "more-algebra-lemma-weak-dimension-at-most-1", "algebra-lemma-valuation-ring-normal", "algebra-proposition-ubiquity-nagata", "algebra-lemma-etale", "more-algebra-lemma-extension-normal-domains-and-roots", "algebra-lemma-normal-goes-up", "algebra-lemma-finite-in-codim-1" ], "ref_ids": [ 10454, 10440, 10450, 10454, 616, 1431, 1231, 10551, 1368, 991 ] } ], "ref_ids": [] }, { "id": 10553, "type": "theorem", "label": "more-algebra-lemma-henselization-valuation-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-henselization-valuation-ring", "contents": [ "Let $A$ be a valuation ring. Let $A^h$, resp.\\ $A^{sh}$ be its", "henselization, resp.\\ strict henselization. Then", "$$", "A \\subset A^h \\subset A^{sh}", "$$", "are extensions of valuation rings which induce bijections on", "value groups, i.e., which are weakly unramified." ], "refs": [], "proofs": [ { "contents": [ "Write $A^h = \\colim (B_i)_{\\mathfrak q_i}$ where $A \\to B_i$", "is \\'etale and $\\mathfrak q_i \\subset B_i$ is a prime ideal", "lying over $\\mathfrak m_A$, see", "Algebra, Lemma \\ref{algebra-lemma-henselization-different}.", "Then Lemma \\ref{lemma-etale-extension-valuation-ring}", "tells us that $(B_i)_{\\mathfrak q_i}$", "is a valuation ring and that the induced map", "$$", "(A \\setminus \\{0\\})/A^* \\longrightarrow", "((B_i)_{\\mathfrak q_i} \\setminus \\{0\\}) / (B_i)_{\\mathfrak q_i}^*", "$$", "is bijective. By Algebra, Lemma \\ref{algebra-lemma-colimit-valuation-rings}", "we conclude that $A^h$ is a valuation ring. It also follows that", "$(A \\setminus \\{0\\})/A^* \\to (A^h \\setminus \\{0\\})/(A^h)^*$", "is bijective. This proves the lemma for the inclusion $A \\subset A^h$.", "To prove it for $A \\subset A^{sh}$ we can use exactly the same argument", "except we replace Algebra, Lemma \\ref{algebra-lemma-henselization-different} by", "Algebra, Lemma \\ref{algebra-lemma-strict-henselization-different}.", "Since $A^{sh} = (A^h)^{sh}$ we see that this also proves the", "assertions of the lemma for the inclusion $A^h \\subset A^{sh}$." ], "refs": [ "algebra-lemma-henselization-different", "more-algebra-lemma-etale-extension-valuation-ring", "algebra-lemma-colimit-valuation-rings", "algebra-lemma-henselization-different", "algebra-lemma-strict-henselization-different" ], "ref_ids": [ 1298, 10552, 611, 1298, 1304 ] } ], "ref_ids": [] }, { "id": 10554, "type": "theorem", "label": "more-algebra-lemma-characterize-PD-modules", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-characterize-PD-modules", "contents": [ "\\begin{reference}", "\\cite[Corollary 1]{Warfield-Purity}", "\\end{reference}", "Let $P$ be a module over a ring $R$. The following are equivalent", "\\begin{enumerate}", "\\item $P$ is a direct summand of a direct sum of modules of the", "form $R/fR$, for $f \\in R$ varying.", "\\item for every short exact sequence $0 \\to A \\to B \\to C \\to 0$", "of $R$-modules such that $fA = A \\cap fB$ for all $f \\in R$", "the map $\\Hom_R(P, B) \\to \\Hom_R(P, C)$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $0 \\to A \\to B \\to C \\to 0$ be an exact sequence as in (2).", "To prove that (1) implies (2) it suffices to prove that", "$\\Hom_R(R/fR, B) \\to \\Hom_R(R/fR, C)$ is surjective for every $f \\in R$.", "Let $\\psi : R/fR \\to C$ be a map. Say $\\psi(1)$ is the image", "of $b \\in B$. Then $fb \\in A$. Hence there exists an $a \\in A$", "such that $fa = fb$. Then $f(b - a) = 0$ hence we get a morphism", "$\\varphi : R/fR \\to B$ mapping $1$ to $b - a$ which lifts $\\psi$.", "\\medskip\\noindent", "Conversely, assume that (2) holds. Let $I$ be the set of pairs", "$(f, \\varphi)$ where $f \\in R$ and $\\varphi : R/fR \\to P$. For", "$i \\in I$ denote $(f_i, \\varphi_i)$ the corresponding pair.", "Consider the map", "$$", "B = \\bigoplus\\nolimits_{i \\in I} R/f_iR \\longrightarrow P", "$$", "which sends the element $r$ in the summand $R/f_iR$ to $\\varphi_i(r)$ in $P$.", "Let $A = \\Ker(F \\to P)$. Then we see that (1) is true if the sequence", "$$", "0 \\to A \\to B \\to P \\to 0", "$$", "is an exact sequence as in (2). To see this suppose $f \\in R$ and", "$a \\in A$ maps to $f b$ in $B$. Write $b = (r_i)_{i \\in I}$ with", "almost all $r_i = 0$. Then we see that", "$$", "f\\sum \\varphi_i(r_i) = 0", "$$", "in $P$. Hence there is an $i_0 \\in I$ such that $f_{i_0} = f$ and", "$\\varphi_{i_0}(1) = \\sum \\varphi_i(r_i)$. Let $x_{i_0} \\in R/f_{i_0}R$", "be the class of $1$. Then we see that", "$$", "a = (r_i)_{i \\in I} - (0, \\ldots, 0, x_{i_0}, 0, \\ldots )", "$$", "is an element of $A$ and $fa = b$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10555, "type": "theorem", "label": "more-algebra-lemma-generalized-valuation-ring", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-generalized-valuation-ring", "contents": [ "\\begin{reference}", "\\cite{Warfield-Decomposition}", "\\end{reference}", "Let $R$ be a nonzero ring. The following are equivalent", "\\begin{enumerate}", "\\item For $a, b \\in R$ either $a$ divides $b$ or $b$ divides $a$.", "\\item Every finitely generated ideal is principal and $R$ is local.", "\\item The set of ideals of $R$ are linearly ordered by inclusion.", "\\end{enumerate}", "This holds in particular if $R$ is a valuation ring." ], "refs": [], "proofs": [ { "contents": [ "Assume (2) and let $a, b \\in R$. Then $(a, b) = (c)$. If $c = 0$,", "then $a = b = 0$ and $a$ divides $b$. Assume $c \\not = 0$. Write", "$c = ua + vb$ and $a = wc$ and $b = zc$. Then $c(1 - uw - vz) = 0$.", "Since $R$ is local, this implies that $1 - uw - vz \\in \\mathfrak m$.", "Hence either $w$ or $z$ is a unit, so either $a$ divides $b$ or", "$b$ divides $a$. Thus (2) implies (1).", "\\medskip\\noindent", "Assume (1). If $R$ has two maximal ideals $\\mathfrak m_i$", "we can choose $a \\in \\mathfrak m_1$ with $a \\not \\in \\mathfrak m_2$", "and $b \\in \\mathfrak m_2$ with $b \\not \\in \\mathfrak m_1$.", "Then $a$ does not divide $b$ and $b$ does not divide $a$.", "Hence $R$ has a unique maximal ideal and is local.", "It follows easily from condition (1) and induction that every", "finitely generated ideal is principal. Thus (1) implies (2).", "\\medskip\\noindent", "It is straightforward to prove that (1) and (3) are equivalent.", "The final statement is Algebra, Lemma", "\\ref{algebra-lemma-valuation-ring-x-or-x-inverse}." ], "refs": [ "algebra-lemma-valuation-ring-x-or-x-inverse" ], "ref_ids": [ 609 ] } ], "ref_ids": [] }, { "id": 10556, "type": "theorem", "label": "more-algebra-lemma-generalized-valuation-ring-modules", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-generalized-valuation-ring-modules", "contents": [ "\\begin{reference}", "\\cite[Theorem 1]{Warfield-Decomposition}", "\\end{reference}", "Let $R$ be a ring satisfying the equivalent conditions of", "Lemma \\ref{lemma-generalized-valuation-ring}.", "Then every finitely presented $R$-module", "is isomorphic to a finite direct sum of modules of the form $R/fR$." ], "refs": [ "more-algebra-lemma-generalized-valuation-ring" ], "proofs": [ { "contents": [ "Let $M$ be a finitely presented $R$-module. We will use all", "the equivalent properties of $R$ from", "Lemma \\ref{lemma-generalized-valuation-ring}", "without further mention. Denote $\\mathfrak m \\subset R$", "the maximal ideal and $\\kappa = R/\\mathfrak m$ the residue field.", "Let $I \\subset R$ be the annihilator of $M$.", "Choose a basis $y_1, \\ldots, y_n$ of the finite dimensional", "$\\kappa$-vector space $M/\\mathfrak m M$. We will argue", "by induction on $n$.", "\\medskip\\noindent", "By Nakayama's lemma any collection of elements $x_1, \\ldots, x_n \\in M$", "lifting the elements $y_1, \\ldots, y_n$ in $M/\\mathfrak m M$", "generate $M$, see Algebra, Lemma \\ref{algebra-lemma-NAK}.", "This immediately proves the base case $n = 0$ of the induction.", "\\medskip\\noindent", "We claim there exists an index $i$ such that for any choice", "of $x_i \\in M$ mapping to $y_i$ the annihilator of $x_i$ is $I$.", "Namely, if not, then we can choose $x_1, \\ldots, x_n$ such", "that $I_i = \\text{Ann}(x_i) \\not = I$ for all $i$. But as", "$I \\subset I_i$ for all $i$, ideals being totally ordered", "implies $I_i$ is strictly bigger than $I$ for $i = 1, \\ldots, n$,", "and by total ordering once more we would see that", "$\\text{Ann}(M) = I_1 \\cap \\ldots \\cap I_n$ is bigger than $I$", "which is a contradiction. After renumbering we may assume that", "$y_1$ has the property: for any $x_1 \\in M$ lifting $y_1$", "the annihilator of $x_1$ is $I$.", "\\medskip\\noindent", "We set $A = Rx_1 \\subset M$. Consider the exact sequence", "$0 \\to A \\to M \\to M/A \\to 0$. Since $A$ is finite, we see that", "$M/A$ is a finitely presented $R$-module", "(Algebra, Lemma \\ref{algebra-lemma-extension}) with fewer generators.", "Hence $M/A \\cong \\bigoplus_{j = 1, \\ldots, m} R/f_jR$ by induction.", "On the other hand, we claim that $A \\to M$ satisfies the property:", "if $f \\in R$, then $fA = A \\cap fM$. The inclusion $fA \\subset A \\cap fM$", "is trivial. Conversely, if $x \\in A \\cap fM$, then $x = gx_1 = f y$", "for some $g \\in R$ and $y \\in M$. If $f$ divides $g$, then $x \\in fA$", "as desired. If not, then we can write $f = hg$ for some $h \\in \\mathfrak m$.", "The element $x'_1 = x_1 - hy$ has annihilator $I$ by the previous", "paragraph. Thus $g \\in I$ and we see that $x = 0$ as desired.", "The claim and Lemma \\ref{lemma-characterize-PD-modules}", "imply the sequence $0 \\to A \\to M \\to M/A \\to 0$ is split", "and we find $M \\cong A \\oplus \\bigoplus_{j = 1, \\ldots, m} R/f_jR$.", "Then $A = R/I$ is finitely presented (as a summand of $M$)", "and hence $I$ is finitely generated, hence principal.", "This finishes the proof." ], "refs": [ "more-algebra-lemma-generalized-valuation-ring", "algebra-lemma-NAK", "algebra-lemma-extension", "more-algebra-lemma-characterize-PD-modules" ], "ref_ids": [ 10555, 401, 330, 10554 ] } ], "ref_ids": [ 10555 ] }, { "id": 10557, "type": "theorem", "label": "more-algebra-lemma-warfield", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-warfield", "contents": [ "\\begin{reference}", "\\cite[Theorem 3]{Warfield-Decomposition}", "\\end{reference}", "Let $R$ be a ring such that every local ring of $R$ at a maximal", "ideal satisfies the equivalent conditions of", "Lemma \\ref{lemma-generalized-valuation-ring}.", "Then every finitely presented $R$-module is a summand of a ", "finite direct sum of modules of the form $R/fR$ for $f$ in $R$ varying." ], "refs": [ "more-algebra-lemma-generalized-valuation-ring" ], "proofs": [ { "contents": [ "Let $M$ be a finitely presented $R$-module. We first show that $M$ is a", "summand of a direct sum of modules of the form $R/fR$ and at the end we", "argue the direct sum can be taken to be finite. Let", "$$", "0 \\to A \\to B \\to C \\to 0", "$$", "be a short exact sequence of $R$-modules such that $fA = A \\cap fB$", "for all $f \\in R$. By Lemma \\ref{lemma-characterize-PD-modules}", "we have to show that $\\Hom_R(M, B) \\to \\Hom_R(M, C)$ is surjective.", "It suffices to prove this after localization at maximal ideals", "$\\mathfrak m$, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}.", "Note that the localized", "sequences $0 \\to A_\\mathfrak m \\to B_\\mathfrak m \\to C_\\mathfrak m \\to 0$", "satisfy the condition that $fA_\\mathfrak m = A_\\mathfrak m \\cap fB_\\mathfrak m$", "for all $f \\in R_\\mathfrak m$ (because we can write $f = uf'$ with", "$u \\in R_\\mathfrak m$ a unit and $f' \\in R$ and because localization", "is exact). Since $M$ is finitely presented, we see that", "$$", "\\Hom_R(M, B)_\\mathfrak m = \\Hom_{R_\\mathfrak m}(M_\\mathfrak m, B_\\mathfrak m)", "\\quad\\text{and}\\quad", "\\Hom_R(M, C)_\\mathfrak m = \\Hom_{R_\\mathfrak m}(M_\\mathfrak m, C_\\mathfrak m)", "$$", "by Algebra, Lemma \\ref{algebra-lemma-hom-from-finitely-presented}.", "The module $M_\\mathfrak m$ is a finitely presented $R_\\mathfrak m$-module. By", "Lemma \\ref{lemma-generalized-valuation-ring-modules}", "we see that $M_\\mathfrak m$ is a direct sum of modules", "of the form $R_\\mathfrak m/fR_\\mathfrak m$. Thus we conclude by", "Lemma \\ref{lemma-characterize-PD-modules} that the map on", "localizations is surjective.", "\\medskip\\noindent", "At this point we know that $M$ is a summand of", "$\\bigoplus_{i \\in I} R/f_i R$. Consider the map", "$M \\to \\bigoplus_{i \\in I} R/f_i R$. Since $M$ is a finite $R$-module,", "the image is contained in $\\bigoplus_{i \\in I'} R/f_i R$ for some finite", "subset $I' \\subset I$. This finishes the proof." ], "refs": [ "more-algebra-lemma-characterize-PD-modules", "algebra-lemma-characterize-zero-local", "algebra-lemma-hom-from-finitely-presented", "more-algebra-lemma-generalized-valuation-ring-modules", "more-algebra-lemma-characterize-PD-modules" ], "ref_ids": [ 10554, 410, 353, 10556, 10554 ] } ], "ref_ids": [ 10555 ] }, { "id": 10558, "type": "theorem", "label": "more-algebra-lemma-elementary-divisor-is-bezout", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-elementary-divisor-is-bezout", "contents": [ "An elementary divisor domain is B\\'ezout." ], "refs": [], "proofs": [ { "contents": [ "Let $a, b \\in R$ be nonzero. Consider the $1 \\times 2$ matrix $A = (a\\ b)$.", "Then we see that $u(a\\ b)V = (f\\ 0)$ with $u \\in R$ invertible", "and $V = (g_{ij})$ an invertible $2 \\times 2$ matrix.", "Then $f = u a g_{11} + u b g_{2 1}$ and $(g_{11}, g_{2 1}) = R$.", "It follows that $(a, b) = (f)$. An induction argument (omitted)", "then shows any finitely generated ideal in $R$ is generated by one element." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10559, "type": "theorem", "label": "more-algebra-lemma-localize-bezout", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-localize-bezout", "contents": [ "The localization of a B\\'ezout domain is B\\'ezout.", "Every local ring of a B\\'ezout domain is a valuation ring.", "A local domain is B\\'ezout if and only if it is a valuation ring." ], "refs": [], "proofs": [ { "contents": [ "We omit the proof of the statement on localizations. The final", "statement is Algebra, Lemma \\ref{algebra-lemma-characterize-valuation-ring}.", "The second statement follows from the other two." ], "refs": [ "algebra-lemma-characterize-valuation-ring" ], "ref_ids": [ 620 ] } ], "ref_ids": [] }, { "id": 10560, "type": "theorem", "label": "more-algebra-lemma-split-off-free-part", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-split-off-free-part", "contents": [ "Let $R$ be a B\\'ezout domain.", "\\begin{enumerate}", "\\item Every finite submodule of a free module is finite free.", "\\item Every finitely presented $R$-module $M$ is a direct sum of a", "finite free module and a torsion module $M_{tors}$ which is a", "summand of a module of the form $\\bigoplus_{i = 1, \\ldots, n} R/f_iR$", "with $f_1, \\ldots, f_n \\in R$ nonzero.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $M \\subset F$ be a finite submodule of a free module $F$.", "Since $M$ is finite, we may assume $F$ is a finite free module", "(details omitted). Say $F = R^{\\oplus n}$. We argue by induction", "on $n$. If $n = 1$, then $M$ is a finitely generated ideal, hence", "principal by our assumption that $R$ is B\\'ezout. If $n > 1$, then", "we consider the image $I$ of $M$ under the projection", "$R^{\\oplus n} \\to R$ onto the last summand. If $I = (0)$, then", "$M \\subset R^{\\oplus n - 1}$ and we are done by induction.", "If $I \\not = 0$, then $I = (f) \\cong R$. Hence", "$M \\cong R \\oplus \\Ker(M \\to I)$ and we are done by induction as well.", "\\medskip\\noindent", "Let $M$ be a finitely presented $R$-module. Since the localizations", "of $R$ are maximal ideals are valuation rings", "(Lemma \\ref{lemma-localize-bezout})", "we may apply Lemma \\ref{lemma-warfield}.", "Thus $M$ is a summand of a module of the form", "$R^{\\oplus r} \\oplus \\bigoplus_{i = 1, \\ldots, n} R/f_iR$", "with $f_i \\not = 0$. Since taking the torsion submodule is", "a functor we see that $M_{tors}$ is", "a summand of the module $\\bigoplus_{i = 1, \\ldots, n} R/f_iR$", "and $M/M_{tors}$ is a summand of $R^{\\oplus r}$.", "By the first part of the proof we see that $M/M_{tors}$ is", "finite free. Hence $M \\cong M_{tors} \\oplus M/M_{tors}$", "as desired." ], "refs": [ "more-algebra-lemma-localize-bezout", "more-algebra-lemma-warfield" ], "ref_ids": [ 10559, 10557 ] } ], "ref_ids": [] }, { "id": 10561, "type": "theorem", "label": "more-algebra-lemma-modules-PID", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-modules-PID", "contents": [ "Let $R$ be a PID. Every finite $R$-module $M$ is of isomorphic", "to a module of the form", "$$", "R^{\\oplus r} \\oplus \\bigoplus\\nolimits_{i = 1, \\ldots, n} R/f_iR", "$$", "for some $r, n \\geq 0$ and $f_1, \\ldots, f_n \\in R$ nonzero." ], "refs": [], "proofs": [ { "contents": [ "A PID is a Noetherian B\\'ezout ring. By Lemma \\ref{lemma-split-off-free-part}", "it suffices to prove the result if $M$ is torsion. Since $M$ is finite, this", "means that the annihilator of $M$ is nonzero. Say $fM = 0$ for some", "$f \\in R$ nonzero. Then we can think of $M$ as a module over $R/fR$.", "Since $R/fR$ is Noetherian of dimension $0$ (small detail omitted)", "we see that $R/fR = \\prod R_j$ is a finite product of Artinian", "local rings $R_i$", "(Algebra, Proposition \\ref{algebra-proposition-dimension-zero-ring}).", "Each $R_i$, being a local ring and a quotient of a PID, is a generalized", "valuation ring in the sense of", "Lemma \\ref{lemma-generalized-valuation-ring} (small detail omitted).", "Write $M = \\prod M_j$ with $M_j = e_j M$ where $e_j \\in R/fR$ is", "the idempotent corresponding to the factor $R_j$.", "By Lemma \\ref{lemma-generalized-valuation-ring-modules}", "we see that $M_j = \\bigoplus_{i = 1, \\ldots, n_j} R_j/\\overline{f}_{ji}R_j$", "for some $\\overline{f}_{ji} \\in R_j$. Choose lifts $f_{ji} \\in R$", "and choose $g_{ji} \\in R$ with $(g_{ji}) = (f_j, f_{ji})$.", "Then we conclude that", "$$", "M \\cong \\bigoplus R/g_{ji}R", "$$", "as an $R$-module which finishes the proof." ], "refs": [ "more-algebra-lemma-split-off-free-part", "algebra-proposition-dimension-zero-ring", "more-algebra-lemma-generalized-valuation-ring", "more-algebra-lemma-generalized-valuation-ring-modules" ], "ref_ids": [ 10560, 1410, 10555, 10556 ] } ], "ref_ids": [] }, { "id": 10562, "type": "theorem", "label": "more-algebra-lemma-unimodular-vector", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-unimodular-vector", "contents": [ "Let $R$ be a B\\'ezout domain. Let $n \\geq 1$ and $f_1, \\ldots, f_n \\in R$", "generate the unit ideal. There exists an invertible $n \\times n$ matrix in", "$R$ whose first row is $f_1 \\ldots f_n$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-split-off-free-part}", "but we can also prove it directly as follows.", "By induction on $n$. The result holds for $n = 1$. Assume $n > 1$.", "We may assume $f_1 \\not = 0$ after renumbering.", "Choose $f \\in R$ such that $(f) = (f_1, \\ldots, f_{n - 1})$.", "Let $A$ be an $(n - 1) \\times (n - 1)$ matrix whose first row", "is $f_1/f, \\ldots, f_{n - 1}/f$. Choose $a, b \\in R$ such that", "$af - bf_n = 1$ which is possible because", "$1 \\in (f_1, \\ldots, f_n) = (f, f_n)$. Then a solution is", "the matrix", "$$", "\\left(", "\\begin{matrix}", "f & 0 & \\ldots & 0 & f_n \\\\", "0 & 1 & \\ldots & 0 & 0 \\\\", " & & \\ldots \\\\", "0 & 0 & \\ldots & 1 & 0 \\\\", "b & 0 & \\ldots & 0 & a", "\\end{matrix}", "\\right)", "\\left(", "\\begin{matrix}", " & & & 0 \\\\", " & A \\\\", " & & & 0 \\\\", "0 & \\ldots & 0 & 1", "\\end{matrix}", "\\right)", "$$", "Observe that the left matrix is invertible because it has determinant $1$." ], "refs": [ "more-algebra-lemma-split-off-free-part" ], "ref_ids": [ 10560 ] } ], "ref_ids": [] }, { "id": 10563, "type": "theorem", "label": "more-algebra-lemma-polypoly", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-polypoly", "contents": [ "Let $(R,\\mathfrak m)$ be a Noetherian local ring of dimension one, and", "let $x\\in\\mathfrak m$ be an element not contained in any minimal prime", "of $R$. Then", "\\begin{enumerate}", "\\item the function $P : n \\mapsto \\text{length}_R(R/x^n R)$", "satisfies $P(n) \\leq n P(1)$ for $n \\geq 0$,", "\\item if $x$ is a nonzerodivisor, then $P(n) = nP(1)$ for $n \\geq 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $\\dim(R) = 1$, we have $\\dim(R/x^n R) = 0$", "and so $\\text{length}_R(R/x^n R)$ is finite for each $n$", "(Algebra, Lemma \\ref{algebra-lemma-support-point}).", "To show the lemma we will induct on $n$. Since $x^0 R = R$, we have that", "$P(0) = \\text{length}_R(R/x^0R) = \\text{length}_R 0 = 0$.", "The statement also holds for $n = 1$.", "Now let $n \\geq 2$ and suppose the statement holds for $n - 1$.", "The following sequence is exact", "$$", "R/x^{n-1}R \\xrightarrow{x} R/x^nR \\to R/xR \\to 0", "$$", "where $x$ denotes the multiplication by $x$ map.", "Since length is additive", "(Algebra, Lemma \\ref{algebra-lemma-length-additive}),", "we have that $P(n) \\leq P(n - 1) + P(1)$. By induction", "$P(n - 1) \\leq (n - 1)P(1)$, whence $P(n) \\leq nP(1)$.", "This proves the induction step.", "\\medskip\\noindent", "If $x$ is a nonzerodivisor, then the displayed exact sequence", "above is exact on the left also. Hence we get", "$P(n) = P(n - 1) + P(1)$ for all $n \\geq 1$." ], "refs": [ "algebra-lemma-support-point", "algebra-lemma-length-additive" ], "ref_ids": [ 693, 631 ] } ], "ref_ids": [] }, { "id": 10564, "type": "theorem", "label": "more-algebra-lemma-minprimespoly", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-minprimespoly", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $1$.", "Let $x \\in \\mathfrak m$ be an element not contained in any minimal", "prime of $R$. Let $t$ be the number of minimal prime ideals of $R$.", "Then $t \\leq \\text{length}_R(R/xR)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak p_1, \\ldots, \\mathfrak p_t$ be the minimal prime ideals", "of $R$. Set $R' = R/\\sqrt{0} = R/(\\bigcap_{i = 1}^t \\mathfrak p_i)$.", "We claim it suffices to prove the lemma for $R'$. Namely, it is clear", "that $R'$ has $t$ minimal primes", "too and $\\text{length}_{R'}(R'/xR') = \\text{length}_R(R'/xR')$", "is less than $\\text{length}_R(R/xR)$ as there is a surjection", "$R/xR \\to R'/xR'$. Thus we may assume $R$ is reduced.", "\\medskip\\noindent", "Assume $R$ is reduced with minimal primes", "$\\mathfrak p_1, \\ldots, \\mathfrak p_t$.", "This means there is an exact sequence", "$$", "0 \\to R \\to", "\\prod\\nolimits_{i = 1}^t R/\\mathfrak p_i \\to Q \\to 0", "$$", "Here $Q$ is the cokernel of the first map.", "Write $M = \\prod_{i = 1}^t R/\\mathfrak p_i$.", "Localizing at $\\mathfrak p_j$ we see that", "$$", "R_{\\mathfrak p_j} \\to M_{\\mathfrak p_j} =", "\\left(\\prod\\nolimits_{i=1}^t R/\\mathfrak p_i\\right)_{\\mathfrak p_j} =", "(R/\\mathfrak p_j)_{\\mathfrak p_j}", "$$", "is surjective. Thus $Q_{\\mathfrak p_j} = 0$ for all $j$.", "We conclude that $\\text{Supp}(Q) = \\{\\mathfrak m\\}$ as $\\mathfrak m$", "is the only prime of $R$ different from the $\\mathfrak p_i$.", "It follows that $Q$ has finite length", "(Algebra, Lemma \\ref{algebra-lemma-support-point}).", "Since $\\text{Supp}(Q) = \\{\\mathfrak m\\}$ we", "can pick an $n \\gg 0$ such that $x^n$", "acts as $0$ on $Q$", "(Algebra, Lemma \\ref{algebra-lemma-Noetherian-power-ideal-kills-module}).", "Now consider the diagram", "$$", "\\xymatrix{", "0 \\ar[r] & R \\ar[r] \\ar[d]^-{x^n} & M", "\\ar[r] \\ar[d]^-{x^n} & Q \\ar[r] \\ar[d]^-{x^n} & 0 \\\\", "0 \\ar[r] & R \\ar[r] & M \\ar[r] & Q \\ar[r] & 0", "}", "$$", "where the vertical maps are multiplication by $x^n$. This is injective on", "$R$ and on $M$ since $x$ is not contained in any of the $\\mathfrak p_i$.", "By the snake lemma (Algebra, Lemma \\ref{algebra-lemma-snake}),", "the following sequence is exact:", "$$", "0 \\to Q \\to R/x^nR \\to M/x^nM \\to Q \\to 0", "$$", "Hence we find that $\\text{length}_R(R/x^nR) = \\text{length}_R(M/x^nM)$", "for large enough $n$. Writing $R_i = R/\\mathfrak p_i$ we see", "that $\\text{length}(M/x^nM) =", "\\sum_{i = 1}^t \\text{length}_R(R_i/x^nR_i)$.", "Applying Lemma \\ref{lemma-polypoly} and the fact that $x$ is a nonzerodivisor", "on $R$ and $R_i$, we conclude that", "$$", "n \\text{length}_R(R/xR) =", "\\sum\\nolimits_{i = 1}^t n \\text{length}_{R_i}(R_i/x R_i)", "$$", "Since $\\text{length}_{R_i}(R_i/x R_i) \\geq 1$ the lemma is proved." ], "refs": [ "algebra-lemma-support-point", "algebra-lemma-Noetherian-power-ideal-kills-module", "algebra-lemma-snake", "more-algebra-lemma-polypoly" ], "ref_ids": [ 693, 694, 328, 10563 ] } ], "ref_ids": [] }, { "id": 10565, "type": "theorem", "label": "more-algebra-lemma-sopexists", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-sopexists", "contents": [ "Let $(R,\\mathfrak m)$ be a Noetherian local ring of dimension $d > 1$,", "let $f \\in \\mathfrak m$ be an element not contained in any minimal prime", "ideal of $R$, and let $k\\in\\mathbf{N}$. Then there exist elements", "$g_1, \\ldots, g_{d - 1} \\in \\mathfrak m^k$ such that", "$f, g_1, \\ldots, g_{d - 1}$ is a system of parameters." ], "refs": [], "proofs": [ { "contents": [ "We have $\\dim(R/fR) = d - 1$ by", "Algebra, Lemma \\ref{algebra-lemma-one-equation}.", "Choose a system of parameters", "$\\overline{g}_1, \\ldots, \\overline{g}_{d - 1}$", "in $R/fR$ (Algebra, Proposition \\ref{algebra-proposition-dimension})", "and take lifts $g_1, \\ldots, g_{d - 1}$ in $R$.", "It is straightforward to see that", "$f, g_1, \\ldots, g_{d - 1}$ is a system of parameters in $R$.", "Then $f, g_1^k, \\ldots, g_{d - 1}^k$ is also a system of", "parameters and the proof is complete." ], "refs": [ "algebra-proposition-dimension" ], "ref_ids": [ 1411 ] } ], "ref_ids": [] }, { "id": 10566, "type": "theorem", "label": "more-algebra-lemma-syspar", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-syspar", "contents": [ "Let $(R,\\mathfrak m)$ be a Noetherian local ring of dimension", "two, and let $f \\in \\mathfrak m$ be an element not contained in", "any minimal prime ideal of $R$. Then there exist", "$g \\in \\mathfrak m$ and $N \\in \\mathbf{N}$ such that", "\\begin{enumerate}", "\\item[(a)] $f,g$ form a system of parameters for $R$.", "\\item[(b)] If $h \\in \\mathfrak m^N$, then $f + h, g$ is a", "system of parameters and", "$\\text{length}_R (R/(f, g)) = \\text{length}_R(R/(f + h, g))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-sopexists} there exists a $g \\in \\mathfrak m$", "such that $f, g$ is a system of parameters for $R$.", "Then $\\mathfrak m = \\sqrt{(f, g)}$. Thus there exists an $n$", "such that $\\mathfrak m^n \\subset (f, g)$, see", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-power}. We claim that", "$N = n + 1$ works.", "Namely, let $h \\in \\mathfrak m^N$. By our choice of $N$ we can write", "$h = af + bg$ with $a, b \\in \\mathfrak m$. Thus", "$$", "(f + h, g) = (f + af + bg, g) = ((1 + a)f, g) = (f, g)", "$$", "because $1 + a$ is a unit in $R$. This proves the equality", "of lengths and the fact that $f + h, g$ is a system of parameters." ], "refs": [ "more-algebra-lemma-sopexists", "algebra-lemma-Noetherian-power" ], "ref_ids": [ 10565, 460 ] } ], "ref_ids": [] }, { "id": 10567, "type": "theorem", "label": "more-algebra-lemma-radical-element", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-radical-element", "contents": [ "Let $R$ be a Noetherian local normal domain of dimension $2$.", "Let $\\mathfrak p_1, \\ldots, \\mathfrak p_r$ be pairwise distinct", "primes of height $1$. There exists a nonzero element", "$f \\in \\mathfrak p_1 \\cap \\ldots \\cap \\mathfrak p_r$ such", "that $R/fR$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "Let $f \\in \\mathfrak p_1 \\cap \\ldots \\cap \\mathfrak p_r$ be a nonzero element.", "We will modify $f$ slightly to obtain an element that generates a radical ideal.", "The localization $R_\\mathfrak p$ of $R$ at each height one prime", "ideal $\\mathfrak p$ is a discrete valuation ring, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-dvr} or", "Algebra, Lemma \\ref{algebra-lemma-criterion-normal}.", "We denote by $\\text{ord}_\\mathfrak p(f)$ the corresponding", "valuation of $f$ in $R_{\\mathfrak p}$. Let", "$\\mathfrak q_1, \\ldots, \\mathfrak q_s$", "be the distinct height one prime ideals containing $f$.", "Write $\\text{ord}_{\\mathfrak q_j}(f) = m_j \\geq 1$ for each $j$.", "Then we define $\\text{div}(f) = \\sum_{j = 1}^s m_j\\mathfrak q_j$", "as a formal linear combination of", "height one primes with integer coefficients.", "Note for later use that each of the primes $\\mathfrak p_i$", "occurs among the primes $\\mathfrak q_j$.", "The ring $R/fR$ is reduced if and only if", "$m_j = 1$ for $j = 1, \\ldots, s$. Namely, if $m_j$ is $1$ then", "$(R/fR)\\mathfrak q_j$ is reduced and", "$R/fR \\subset \\prod (R/fR)_{\\mathfrak q_j}$ as", "$\\mathfrak q_1, \\ldots, \\mathfrak q_j$ are the associated primes", "of $R/fR$, see Algebra, Lemmas", "\\ref{algebra-lemma-zero-at-ass-zero} and", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}.", "\\medskip\\noindent", "Choose and fix $g$ and $N$ as in Lemma \\ref{lemma-syspar}.", "For a nonzero $y \\in R$ denote $t(y)$ the number of primes minimal over $y$.", "Since $R$ is a normal domain, these primes", "are height one and correspond $1$-to-$1$ to the minimal primes of", "$R/yR$ (Algebra, Lemmas \\ref{algebra-lemma-minimal-over-1} and", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}).", "For example $t(f) = s$ is the number", "of primes $\\mathfrak q_j$ occurring in $\\text{div}(f)$.", "Let $h \\in \\mathfrak m^N$. By Lemma \\ref{lemma-minprimespoly} we have", "\\begin{align*}", "t(f + h) & \\leq \\text{length}_{R/(f + h)}(R/(f + h, g)) \\\\", "& = \\text{length}_R(R/(f + h, g)) \\\\", "& = \\text{length}_R(R/(f, g))", "\\end{align*}", "see Algebra, Lemma \\ref{algebra-lemma-length-independent}", "for the first equality.", "Therefore we see that $t(f + h)$ is bounded independent of", "$h \\in \\mathfrak m^N$.", "\\medskip\\noindent", "By the boundedness proved above we may pick", "$h \\in \\mathfrak m^N \\cap \\mathfrak p_1 \\cap \\ldots \\cap \\mathfrak p_r$", "such that $t(f + h)$ is maximal among such $h$. Set $f' = f + h$.", "Given $h' \\in \\mathfrak m^N \\cap \\mathfrak p_1 \\cap \\ldots \\cap \\mathfrak p_r$", "we see that the number $t(f' + h') \\leq t(f + h)$.", "Thus after replacing $f$ by $f'$ we may assume that for every", "$h \\in \\mathfrak m^N \\cap \\mathfrak p_1 \\cap \\ldots \\cap \\mathfrak p_r$", "we have $t(f + h) \\leq s$.", "\\medskip\\noindent", "Next, assume that we can find an element $h \\in \\mathfrak m^N$ such that", "for each $j$ we have $\\text{ord}_{\\mathfrak q_j}(h) \\geq 1$ and", "$\\text{ord}_{\\mathfrak q_j}(h) = 1 \\Leftrightarrow m_j > 1$.", "Observe that", "$h \\in \\mathfrak m^N \\cap \\mathfrak p_1 \\cap \\ldots \\cap \\mathfrak p_r$.", "Then $\\text{ord}_{\\mathfrak q_j}(f + h) = 1$", "for every $j$ by elementary properties of valuations.", "Thus", "$$", "\\text{div}(f + h) = \\sum\\nolimits_{j = 1}^s \\mathfrak q_j +", "\\sum\\nolimits_{k = 1}^v e_k \\mathfrak r_k", "$$", "for some pairwise distinct height one prime ideals", "$\\mathfrak r_1, \\ldots, \\mathfrak r_v$ and $e_k \\geq 1$.", "However, since $s = t(f) \\geq t(f + h)$ we see that $v = 0$", "and we have found the desired element.", "\\medskip\\noindent", "Now we will pick $h$ that satisfies the above criteria.", "By prime avoidance (Algebra, Lemma \\ref{algebra-lemma-silly})", "for each $1 \\leq j \\leq s$ we can find an element $a_j \\in \\mathfrak q_j$", "such that $a_j \\not \\in \\mathfrak q_{j'}$ for $j' \\not = j$", "and $a_j \\not \\in \\mathfrak q_j^{(2)}$. Here", "$\\mathfrak q_j^{(2)} = \\{x \\in R \\mid \\text{ord}_{\\mathfrak q_j}(x) \\geq 2\\}$", "is the second symbolic power of $\\mathfrak q_j$.", "Then we take", "$$", "h = \\prod\\nolimits_{m_j = 1} a_j^2 \\times", "\\prod\\nolimits_{m_j > 1} a_j", "$$", "Then $h$ clearly satisfies the conditions on valuations imposed above.", "If $h \\not \\in \\mathfrak m^N$, then we multiply by an element of", "$\\mathfrak m^N$ which is not contained in $\\mathfrak q_j$ for all $j$." ], "refs": [ "algebra-lemma-characterize-dvr", "algebra-lemma-criterion-normal", "algebra-lemma-zero-at-ass-zero", "algebra-lemma-normal-domain-intersection-localizations-height-1", "more-algebra-lemma-syspar", "algebra-lemma-minimal-over-1", "algebra-lemma-normal-domain-intersection-localizations-height-1", "more-algebra-lemma-minprimespoly", "algebra-lemma-length-independent", "algebra-lemma-silly" ], "ref_ids": [ 1023, 1311, 713, 1313, 10566, 683, 1313, 10564, 633, 378 ] } ], "ref_ids": [] }, { "id": 10568, "type": "theorem", "label": "more-algebra-lemma-divides-radical", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-divides-radical", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian normal local domain", "of dimension $2$. If $a \\in \\mathfrak m$ is nonzero, then there exists an", "element $c \\in A$ such that $A/cA$ is reduced and such that $a$ divides", "$c^n$ for some $n$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\text{div}(a) = \\sum_{i = 1, \\ldots, r} n_i \\mathfrak p_i$", "with notation as in the proof of Lemma \\ref{lemma-radical-element}.", "Choose $c \\in \\mathfrak p_1 \\cap \\ldots \\cap \\mathfrak p_r$ with $A/cA$", "reduced, see Lemma \\ref{lemma-radical-element}. For $n \\geq \\max(n_i)$", "we see that $-\\text{div}(a) + \\text{div}(c^n)$", "is an effective divisor (all coefficients nonnegative).", "Thus $c^n/a \\in A$ by Algebra, Lemma", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}." ], "refs": [ "more-algebra-lemma-radical-element", "more-algebra-lemma-radical-element", "algebra-lemma-normal-domain-intersection-localizations-height-1" ], "ref_ids": [ 10567, 10567, 1313 ] } ], "ref_ids": [] }, { "id": 10569, "type": "theorem", "label": "more-algebra-lemma-multiplicity", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-multiplicity", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $d$, let", "$g_1, \\ldots, g_d$ be a system of parameters, and let", "$I = (g_1, \\ldots, g_d)$. If $e_I/d!$ is the leading coefficient of the", "numerical polynomial", "$n \\mapsto \\text{length}_R(R/I^{n+1})$, then $e_I \\leq \\text{length}_R(R/I)$." ], "refs": [], "proofs": [ { "contents": [ "The function is a numerical polynomial by", "Algebra, Proposition \\ref{algebra-proposition-hilbert-function-polynomial}.", "It has degree $d$ by", "Algebra, Proposition \\ref{algebra-proposition-dimension}.", "If $d = 0$, then the result is trivial.", "If $d = 1$, then the result is Lemma \\ref{lemma-polypoly}.", "To prove it in general, observe that there is a surjection", "$$", "\\bigoplus\\nolimits_{i_1, \\ldots, i_d \\geq 0,\\ \\sum i_j = n} R/I", "\\longrightarrow", "I^n/I^{n + 1}", "$$", "sending the basis element corresponding to $i_1, \\ldots, i_d$", "to the class of $g_1^{i_1} \\ldots g_d^{i_d}$ in $I^n/I^{n + 1}$.", "Thus we see that", "$$", "\\text{length}_R(R/I^{n + 1}) - \\text{length}_R(R/I^n)", "\\leq \\text{length}_R(R/I) {n + d - 1 \\choose d - 1}", "$$", "Since $d \\geq 2$ the numerical polynomial on the left has", "degree $d - 1$ with leading coefficient $e_I / (d - 1)!$.", "The polynomial on the right has degree $d - 1$ and its", "leading coefficient is $\\text{length}_R(R/I)/ (d - 1)!$.", "This proves the lemma." ], "refs": [ "algebra-proposition-hilbert-function-polynomial", "algebra-proposition-dimension", "more-algebra-lemma-polypoly" ], "ref_ids": [ 1409, 1411, 10563 ] } ], "ref_ids": [] }, { "id": 10570, "type": "theorem", "label": "more-algebra-lemma-minprimespolyhigher", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-minprimespolyhigher", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $d$, let $t$", "be the number of minimal prime ideals of $R$ of dimension $d$, and let", "$(g_1,\\ldots,g_d)$ be a system of parameters. Then", "$t \\leq \\text{length}_R(R/(g_1,\\ldots,g_n))$." ], "refs": [], "proofs": [ { "contents": [ "If $d = 0$ the lemma is trivial. If $d = 1$ the lemma is", "Lemma \\ref{lemma-minprimespoly}. Thus we may assume $d > 1$.", "Let $\\mathfrak p_1, \\ldots, \\mathfrak p_s$ be the minimal prime ideals of", "$R$ where the first $t$ have dimension $d$, and denote", "$I = (g_1, \\ldots, g_n)$. Arguing in exactly the same way as in", "the proof of Lemma \\ref{lemma-minprimespoly} we can assume $R$ is reduced.", "\\medskip\\noindent", "Assume $R$ is reduced with minimal primes", "$\\mathfrak p_1, \\ldots, \\mathfrak p_t$.", "This means there is an exact sequence", "$$", "0 \\to R \\to", "\\prod\\nolimits_{i = 1}^t R/\\mathfrak p_i \\to Q \\to 0", "$$", "Here $Q$ is the cokernel of the first map.", "Write $M = \\prod_{i = 1}^t R/\\mathfrak p_i$.", "Localizing at $\\mathfrak p_j$ we see that", "$$", "R_{\\mathfrak p_j} \\to M_{\\mathfrak p_j} =", "\\left(\\prod\\nolimits_{i=1}^t R/\\mathfrak p_i\\right)_{\\mathfrak p_j} =", "(R/\\mathfrak p_j)_{\\mathfrak p_j}", "$$", "is surjective. Thus $Q_{\\mathfrak p_j} = 0$ for all $j$. Therefore no", "height $0$ prime of $R$ is in the support of $Q$. It follows that", "the degree of the numerical polynomial", "$n \\mapsto \\text{length}_R(Q/I^nQ)$ equals $\\dim(\\text{Supp}(Q)) < d$, see", "Algebra, Lemma \\ref{algebra-lemma-support-dimension-d}.", "By Algebra, Lemma \\ref{algebra-lemma-hilbert-ses-chi}", "(which applies as $R$ does not have finite length) the polynomial", "$$", "n \\longmapsto", "\\text{length}_R(M/I^nM) - \\text{length}_R(R/I^n) - \\text{length}_R(Q/I^nQ)", "$$", "has degree $< d$. Since $M = \\prod R/\\mathfrak p_i$ and since", "$n \\to \\text{length}_R(R/\\mathfrak p_i + I^n)$ is a numerical", "polynomial of degree exactly(!) $d$ for $i = 1, \\ldots, t$ (by", "Algebra, Lemma \\ref{algebra-lemma-support-dimension-d})", "we see that the leading coefficient of $n \\mapsto \\text{length}_R(M/I^nM)$", "is at least $t/d!$. Thus we conclude by Lemma \\ref{lemma-multiplicity}." ], "refs": [ "more-algebra-lemma-minprimespoly", "more-algebra-lemma-minprimespoly", "algebra-lemma-support-dimension-d", "algebra-lemma-hilbert-ses-chi", "algebra-lemma-support-dimension-d", "more-algebra-lemma-multiplicity" ], "ref_ids": [ 10564, 10564, 696, 678, 696, 10569 ] } ], "ref_ids": [] }, { "id": 10571, "type": "theorem", "label": "more-algebra-lemma-sysparhigher", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-sysparhigher", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $d$, and let", "$f \\in \\mathfrak m$ be an element not contained in any minimal", "prime ideal of $R$. Then there exist elements", "$g_1, \\ldots, g_{d - 1} \\in \\mathfrak m$ and $N \\in \\mathbf{N}$ such that", "\\begin{enumerate}", "\\item $f, g_1, \\ldots, g_{d - 1}$ form a system of parameters for $R$", "\\item If $h \\in \\mathfrak m^N$, then $f + h, g_1, \\ldots, g_{d - 1}$ is a", "system of parameters and we have", "$\\text{length}_R R/(f, g_1, \\ldots, g_{d-1}) =", "\\text{length}_R R/(f + h, g_1, \\ldots, g_{d-1})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-sopexists} there exist", "$g_1, \\ldots, g_{d - 1} \\in \\mathfrak m$", "such that $f, g_1, \\ldots, g_{d - 1}$ is a system of parameters for $R$.", "Then $\\mathfrak m = \\sqrt{(f, g_1, \\ldots, g_{d - 1})}$.", "Thus there exists an $n$ such that $\\mathfrak m^n \\subset (f, g)$, see", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-power}. We claim that", "$N = n + 1$ works.", "Namely, let $h \\in \\mathfrak m^N$. By our choice of $N$ we can write", "$h = af + \\sum b_ig_i$ with $a, b_i \\in \\mathfrak m$. Thus", "\\begin{align*}", "(f + h, g_1, \\ldots, g_{d - 1})", "& =", "(f + af + \\sum b_ig_i, g_1, \\ldots, g_{d - 1}) \\\\", "& =", "((1 + a)f, g_1, \\ldots, g_{d - 1}) \\\\", "& =", "(f, g_1, \\ldots, g_{d - 1})", "\\end{align*}", "because $1 + a$ is a unit in $R$. This proves the equality", "of lengths and the fact that $f + h, g_1, \\ldots, g_{d - 1}$", "is a system of parameters." ], "refs": [ "more-algebra-lemma-sopexists", "algebra-lemma-Noetherian-power" ], "ref_ids": [ 10565, 460 ] } ], "ref_ids": [] }, { "id": 10572, "type": "theorem", "label": "more-algebra-lemma-symmetric-monoidal-derived", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-symmetric-monoidal-derived", "contents": [ "Let $R$ be a ring. The derived category $D(R)$ of $R$", "is a symmetric monoidal category with tensor product", "given by derived tensor product and associativity and", "commutativity constraints as in Section \\ref{section-sign-rules}." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Compare with Lemma \\ref{lemma-symmetric-monoidal-cat-complexes}." ], "refs": [ "more-algebra-lemma-symmetric-monoidal-cat-complexes" ], "ref_ids": [ 10203 ] } ], "ref_ids": [] }, { "id": 10573, "type": "theorem", "label": "more-algebra-lemma-complex-bounded-above-free-colim-bounded-finite-free", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-complex-bounded-above-free-colim-bounded-finite-free", "contents": [ "Let $R$ be a ring. Let $F^\\bullet$ be a bounded above complex of", "free $R$-modules. Given pairs $(n_i, f_i)$, $i = 1, \\ldots, N$", "with $n_i \\in \\mathbf{Z}$ and $f_i \\in F^{n_i}$ there exists", "a subcomplex $G^\\bullet \\subset F^\\bullet$ containing", "all $f_i$ which is bounded and consists of finite free $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "By descending induction on $a = \\min(n_i; i = 1, \\ldots, N)$.", "If $F^n = 0$ for $n \\geq a$, then the result is true with", "$G^\\bullet$ equal to the zero complex. In general, after renumbering", "we may assume there exists an $1 \\leq r \\leq N$ such that", "$n_1 = \\ldots = n_r = a$ and $n_i > a$ for $i > r$.", "Choose a basis $b_j, j \\in J$ for $F^a$. We can choose a finite subset", "$J' \\subset J$ such that $f_i \\in \\bigoplus_{j \\in J'} Rb_j$", "for $i = 1, \\ldots, r$. Choose a basis $c_k, k \\in K$ for $F^{a + 1}$.", "We can choose a finite subset $K' \\subset K$ such that", "$\\text{d}_F^a(b_j) \\in \\bigoplus_{k \\in K'} Rc_k$ for", "$j \\in J'$. Then we can apply the induction", "hypothesis to find a subcomplex $H^\\bullet \\subset F^\\bullet$", "containing $c_k \\in F^{a + 1}$ for $k \\in K'$ and", "$f_i \\in F^{n_i}$ for $i > r$. Take $G^\\bullet$ equal to", "$H^\\bullet$ in degrees $> a$ and equal to $\\bigoplus_{j \\in J'} Rb_j$", "in degree $a$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10574, "type": "theorem", "label": "more-algebra-lemma-have-dual-derived", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-have-dual-derived", "contents": [ "Let $R$ be a ring. Let $M$ be an object of $D(R)$. The following", "are equivalent", "\\begin{enumerate}", "\\item $M$ has a left dual in $D(R)$ as in", "Categories, Definition \\ref{categories-definition-dual},", "\\item $M$ is a perfect object of $D(R)$.", "\\end{enumerate}", "Moreover, in this case the left dual of $M$ is the object", "$M^\\vee$ of Lemma \\ref{lemma-dual-perfect-complex}." ], "refs": [ "categories-definition-dual", "more-algebra-lemma-dual-perfect-complex" ], "proofs": [ { "contents": [ "If $M$ is perfect, then we can represent $M$ by a bounded complex", "$M^\\bullet$ of finite projective $R$-modules. In this case $M^\\bullet$", "has a left dual in the category of complexes by", "Lemma \\ref{lemma-left-dual-complex}", "which is a fortiori a left dual in $D(R)$.", "\\medskip\\noindent", "Assume (1). Say $N$, $\\eta : R \\to M \\otimes_R^\\mathbf{L} N$, and", "$\\epsilon : M \\otimes_R^\\mathbf{L} N \\to R$ is a left dual", "as in Categories, Definition \\ref{categories-definition-dual}.", "Choose a complex $M^\\bullet$ representing $M$. Choose a", "K-flat complexes $N^\\bullet$ with flat terms", "representing $N$, see Lemma \\ref{lemma-K-flat-resolution}.", "Then $\\eta$ is given by a map of complexes", "$$", "\\eta : R \\longrightarrow \\text{Tot}(M^\\bullet \\otimes_R N^\\bullet)", "$$", "We can write the image of $1$ as a finite sum", "$$", "\\eta(1) = \\sum\\nolimits_n \\sum\\nolimits_i m_{n, i} \\otimes n_{-n, i}", "$$", "with $m_{n, i} \\in M^n$ and $n_{-n, i} \\in N^{-n}$. Let", "$K^\\bullet \\subset M^\\bullet$ be the subcomplex", "generated by all the elements $m_{n, i}$ and $\\text{d}(m_{n, i})$.", "By our choice of $N^\\bullet$ we find that", "$\\text{Tot}(K^\\bullet \\otimes_R N^\\bullet) \\subset", "\\text{Tot}(M^\\bullet \\otimes_R N^\\bullet)$ and", "$\\eta(1)$ is in the subcomplex by our choice above.", "Denote $K$ the object of $D(R)$ represented by $K^\\bullet$.", "Then we see that $\\eta$ factors over a map", "$\\tilde \\eta : R \\longrightarrow K \\otimes_R^\\mathbf{L} N$.", "Since", "$(1 \\otimes \\epsilon) \\circ (\\eta \\otimes 1) = \\text{id}_M$", "we conclude that the identity on $M$ factors through $K$", "by the commutative diagram", "$$", "\\xymatrix{", "M \\ar[rr]_-{\\eta \\otimes 1} \\ar[rrd]_{\\tilde \\eta \\otimes 1} & &", "M \\otimes_R^\\mathbf{L} N \\otimes_R^\\mathbf{L} M", "\\ar[r]_-{1 \\otimes \\epsilon} &", "M \\\\", "& &", "K \\otimes_R^\\mathbf{L} N \\otimes_R^\\mathbf{L} M", "\\ar[u] \\ar[r]^-{1 \\otimes \\epsilon} &", "K \\ar[u]", "}", "$$", "Since $K$ is bounded above it follows that $M \\in D^-(R)$.", "Thus we can represent $M$ by a bounded above complex", "$M^\\bullet$ of free $R$-modules, see for example", "Derived Categories, Lemma \\ref{derived-lemma-subcategory-left-resolution}.", "Write $\\eta(1) = \\sum\\nolimits_n \\sum\\nolimits_i m_{n, i} \\otimes n_{-n, i}$", "as before.", "By Lemma \\ref{lemma-complex-bounded-above-free-colim-bounded-finite-free}", "we can find a subcomplex $K^\\bullet \\subset M^\\bullet$", "containing all the elements $m_{n, i}$", "which is bounded and consists of finite free $R$-modules.", "As above we find that the identity on $M$ factors through $K$.", "Since $K$ is perfect we conclude $M$ is perfect too, see", "Lemma \\ref{lemma-summands-perfect}." ], "refs": [ "more-algebra-lemma-left-dual-complex", "categories-definition-dual", "more-algebra-lemma-K-flat-resolution", "derived-lemma-subcategory-left-resolution", "more-algebra-lemma-complex-bounded-above-free-colim-bounded-finite-free", "more-algebra-lemma-summands-perfect" ], "ref_ids": [ 10205, 12407, 10131, 1835, 10573, 10215 ] } ], "ref_ids": [ 12407, 10224 ] }, { "id": 10575, "type": "theorem", "label": "more-algebra-lemma-invertible-derived", "categories": [ "more-algebra" ], "title": "more-algebra-lemma-invertible-derived", "contents": [ "Let $R$ be a ring. Let $M$ be an object of $D(R)$. The following", "are equivalent", "\\begin{enumerate}", "\\item $M$ is invertible in $D(R)$, see", "Categories, Definition \\ref{categories-definition-invertible}, and", "\\item for every prime ideal $\\mathfrak p \\subset R$ there", "exists an $f \\in R$, $f \\not \\in \\mathfrak p$ such that", "$M_f \\cong R_f[-n]$ for some $n \\in \\mathbf{Z}$.", "\\end{enumerate}", "Moreover, in this case", "\\begin{enumerate}", "\\item[(a)] $M$ is a perfect object of $D(R)$,", "\\item[(b)] $M = \\bigoplus H^n(M)[-n]$ in $D(R)$,", "\\item[(c)] each $H^n(M)$ is a finite projective $R$-module,", "\\item[(d)] we can write $R = \\prod_{a \\leq n \\leq b} R_n$", "such that $H^n(M)$ corresponds to an invertible $R_n$-module.", "\\end{enumerate}" ], "refs": [ "categories-definition-invertible" ], "proofs": [ { "contents": [ "Assume (2). Consider the object $R\\Hom_R(M, R)$ and the composition map", "$$", "R\\Hom(M, R) \\otimes_R^\\mathbf{L} M \\to R", "$$", "Checking locally we see that this is an isomorphism; we omit the details.", "Because $D(R)$ is symmetric monoidal we see that $M$ is invertible.", "\\medskip\\noindent", "Assume (1). Observe that an invertible object of a monoidal category", "has a left dual, namely, its inverse. Thus $M$ is perfect by", "Lemma \\ref{lemma-have-dual-derived}. Consider a prime ideal", "$\\mathfrak p \\subset R$ with residue field $\\kappa$.", "Then we see that $M \\otimes_R^\\mathbf{L} \\kappa$ is an", "invertible object of $D(\\kappa)$. Clearly this implies that", "$\\dim H^i(M \\otimes_R^\\mathbf{L} \\kappa)$ is nonzero exactly", "for one $i$ and equal to $1$ in that case. By", "Lemma \\ref{lemma-lift-perfect-from-residue-field}", "this gives (2).", "\\medskip\\noindent", "In the proof above we have seen that (a) holds. Let $U_n \\subset \\Spec(R)$", "be the union of the opens of the form $D(f)$ such that", "$M_f \\cong R_f[-n]$. Clearly, $U_n \\cap U_{n'} = \\emptyset$ if $n \\not = n'$.", "If $M$ has tor amplitude in $[a, b]$, then $U_n = \\emptyset$", "if $n \\not \\in [a, b]$. Hence we see that we have a product decomposition", "$R = \\prod_{a \\leq n \\leq b} R_n$ as in (d) such that", "$U_n$ corresponds to $\\Spec(R_n)$, see", "Algebra, Lemma \\ref{algebra-lemma-disjoint-implies-product}.", "Since $D(R) = \\prod_{a \\leq n \\leq b} D(R_n)$ and similary", "for the category of modules parts (b), (c), and (d) follow immediately." ], "refs": [ "more-algebra-lemma-have-dual-derived", "more-algebra-lemma-lift-perfect-from-residue-field", "algebra-lemma-disjoint-implies-product" ], "ref_ids": [ 10574, 10232, 415 ] } ], "ref_ids": [ 12406 ] }, { "id": 10576, "type": "theorem", "label": "more-algebra-proposition-characterization-geometrically-regular", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-characterization-geometrically-regular", "contents": [ "Let $k$ be a field of characteristic $p > 0$.", "Let $(A, \\mathfrak m, K)$ be a Noetherian local", "$k$-algebra. The following are equivalent", "\\begin{enumerate}", "\\item $A$ is geometrically regular over $k$,", "\\item for all $k \\subset k' \\subset k^{1/p}$", "finite over $k$ the ring $A \\otimes_k k'$ is regular,", "\\item $A$ is regular and the canonical map", "$H_1(L_{K/k}) \\to \\mathfrak m/\\mathfrak m^2$ is injective, and", "\\item $A$ is regular and the map", "$\\Omega_{k/\\mathbf{F}_p} \\otimes_k K \\to \\Omega_{A/\\mathbf{F}_p} \\otimes_A K$", "is injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (3) $\\Rightarrow$ (1).", "Assume (3). Let $k \\subset k'$ be a finite purely inseparable extension.", "Set $A' = A \\otimes_k k'$. This is a local ring with maximal ideal", "$\\mathfrak m'$. Set $K' = A'/\\mathfrak m'$. We get a commutative", "diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "H_1(L_{K/k}) \\otimes K' \\ar[r] \\ar[d]_\\beta &", "\\mathfrak m/\\mathfrak m^2 \\otimes K' \\ar[r] \\ar[d] &", "\\Omega_{A/k} \\otimes_A K' \\ar[r] \\ar[d]_{\\cong} &", "\\Omega_{K/k} \\otimes K' \\ar[r] \\ar[d]_\\alpha &", "0", "\\\\", " &", "H_1(L_{K'/k'}) \\ar[r] &", "\\mathfrak m'/(\\mathfrak m')^2 \\ar[r] &", "\\Omega_{A'/k'} \\otimes_{A'} K' \\ar[r] &", "\\Omega_{K'/k'} \\ar[r] &", "0", "}", "$$", "with exact rows. The third vertical arrow is an isomorphism by base", "change for modules of differentials", "(Algebra, Lemma \\ref{algebra-lemma-differentials-base-change}).", "Thus $\\alpha$ is surjective. By", "Lemma \\ref{lemma-gamma-commutative-diagram} we have", "$$", "\\dim \\Ker(\\alpha) - \\dim \\Ker(\\beta) + \\dim \\Coker(\\beta) = 0", "$$", "(and these dimensions are all finite). A diagram chase shows that", "$\\dim \\mathfrak m'/(\\mathfrak m')^2 \\leq \\dim \\mathfrak m/\\mathfrak m^2$.", "However, since $A \\to A'$ is finite flat we see that", "$\\dim(A) = \\dim(A')$, see", "Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-total}.", "Hence $A'$ is regular by definition.", "\\medskip\\noindent", "Equivalence of (3) and (4). Consider the Jacobi-Zariski sequences", "for rows of the commutative diagram", "$$", "\\xymatrix{", "\\mathbf{F}_p \\ar[r] & A \\ar[r] & K \\\\", "\\mathbf{F}_p \\ar[r] \\ar[u] & k \\ar[r] \\ar[u] & K \\ar[u]", "}", "$$", "to get a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathfrak m/\\mathfrak m^2 \\ar[r] &", "\\Omega_{A/\\mathbf{F}_p} \\otimes_A K \\ar[r] &", "\\Omega_{K/\\mathbf{F}_p} \\ar[r] & 0 & \\\\", "0 \\ar[r] &", "H_1(L_{K/k}) \\ar[r] \\ar[u] &", "\\Omega_{k/\\mathbf{F}_p} \\otimes_k K \\ar[r] \\ar[u] &", "\\Omega_{K/\\mathbf{F}_p} \\ar[r] \\ar[u] &", "\\Omega_{K/k} \\ar[r] \\ar[u] &", "0", "}", "$$", "with exact rows. We have used that $H_1(L_{K/A}) = \\mathfrak m/\\mathfrak m^2$", "and that $H_1(L_{K/\\mathbf{F}_p}) = 0$ as $K/\\mathbf{F}_p$ is separable, see", "Algebra, Proposition", "\\ref{algebra-proposition-characterize-separable-field-extensions}.", "Thus it is clear that the kernels of", "$H_1(L_{K/k}) \\to \\mathfrak m/\\mathfrak m^2$", "and", "$\\Omega_{k/\\mathbf{F}_p} \\otimes_k K \\to \\Omega_{A/\\mathbf{F}_p} \\otimes_A K$", "have the same dimension.", "\\medskip\\noindent", "Proof of (2) $\\Rightarrow$ (4) following Faltings, see", "\\cite{Faltings-einfacher}. Let $a_1, \\ldots, a_n \\in k$ be elements", "such that $\\text{d}a_1, \\ldots, \\text{d}a_n$ are linearly independent", "in $\\Omega_{k/\\mathbf{F}_p}$. Consider the field extension", "$k' = k(a_1^{1/p}, \\ldots, a_n^{1/p})$. By", "Algebra, Lemma \\ref{algebra-lemma-size-extension-pth-roots}", "we see that $k' = k[x_1, \\ldots, x_n]/(x_1^p - a_1, \\ldots, x_n^p - a_n)$.", "In particular we see that the naive cotangent complex of $k'/k$", "is homotopic to the complex", "$\\bigoplus_{j = 1, \\ldots, n} k' \\rightarrow \\bigoplus_{i = 1, \\ldots, n} k'$", "with the zero differential as", "$\\text{d}(x_j^p - a_j) = 0$ in $\\Omega_{k[x_1, \\ldots, x_n]/k}$.", "Set $A' = A \\otimes_k k'$ and $K' = A'/\\mathfrak m'$ as above.", "By Algebra, Lemma \\ref{algebra-lemma-change-base-NL}", "we see that $\\NL_{A'/A}$ is homotopy equivalent to the complex", "$\\bigoplus_{j = 1, \\ldots, n} A' \\rightarrow \\bigoplus_{i = 1, \\ldots, n} A'$", "with the zero differential, i.e., $H_1(L_{A'/A})$ and", "$\\Omega_{A'/A}$ are free of rank $n$. The Jacobi-Zariski sequence for", "$\\mathbf{F}_p \\to A \\to A'$ is", "$$", "H_1(L_{A'/A}) \\to \\Omega_{A/\\mathbf{F}_p} \\otimes_A A'", "\\to \\Omega_{A'/\\mathbf{F}_p} \\to \\Omega_{A'/A} \\to 0", "$$", "Using the presentation $A[x_1, \\ldots, x_n] \\to A'$ with", "kernel $(x_j^p - a_j)$ we see, unwinding the maps in", "Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL},", "that the $j$th basis vector of $H_1(L_{A'/A})$ maps to", "$\\text{d}a_j \\otimes 1$ in $\\Omega_{A/\\mathbf{F}_p} \\otimes A'$.", "As $\\Omega_{A'/A}$ is free (hence flat) we get on tensoring with $K'$", "an exact sequence", "$$", "K'^{\\oplus n} \\to \\Omega_{A/\\mathbf{F}_p} \\otimes_A K'", "\\xrightarrow{\\beta} \\Omega_{A'/\\mathbf{F}_p} \\otimes_{A'} K' \\to", "K'^{\\oplus n} \\to 0", "$$", "We conclude that the elements $\\text{d}a_j \\otimes 1$ generate", "$\\Ker(\\beta)$ and we have to show that are linearly independent, i.e.,", "we have to show $\\dim(\\Ker(\\beta)) = n$.", "Consider the following big diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathfrak m'/(\\mathfrak m')^2 \\ar[r] &", "\\Omega_{A'/\\mathbf{F}_p} \\otimes K' \\ar[r] &", "\\Omega_{K'/\\mathbf{F}_p} \\ar[r] & 0 \\\\", "0 \\ar[r] &", "\\mathfrak m/\\mathfrak m^2 \\otimes K' \\ar[r] \\ar[u]^\\alpha &", "\\Omega_{A/\\mathbf{F}_p} \\otimes K' \\ar[r] \\ar[u]^\\beta &", "\\Omega_{K/\\mathbf{F}_p} \\otimes K' \\ar[r] \\ar[u]^\\gamma & 0", "}", "$$", "By Lemma \\ref{lemma-cartier-equality} and the Jacobi-Zariski sequence for", "$\\mathbf{F}_p \\to K \\to K'$ we see that the kernel and cokernel of", "$\\gamma$ have the same finite dimension.", "By assumption $A'$ is regular (and of the same dimension as $A$, see", "above) hence the kernel and cokernel of $\\alpha$ have the same dimension.", "It follows that the kernel and cokernel of $\\beta$ have the same", "dimension which is what we wanted to show.", "\\medskip\\noindent", "The implication (1) $\\Rightarrow$ (2) is trivial. This finishes", "the proof of the proposition." ], "refs": [ "algebra-lemma-differentials-base-change", "more-algebra-lemma-gamma-commutative-diagram", "algebra-lemma-dimension-base-fibre-total", "algebra-proposition-characterize-separable-field-extensions", "algebra-lemma-size-extension-pth-roots", "algebra-lemma-change-base-NL", "algebra-lemma-exact-sequence-NL", "more-algebra-lemma-cartier-equality" ], "ref_ids": [ 1138, 10008, 986, 1429, 1316, 1156, 1153, 10006 ] } ], "ref_ids": [] }, { "id": 10577, "type": "theorem", "label": "more-algebra-proposition-fs-flat-fibre-fs", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-fs-flat-fibre-fs", "contents": [ "Let $A \\to B$ be a local homomorphism of Noetherian local rings.", "Let $k$ be the residue field of $A$ and $\\overline{B} = B \\otimes_A k$", "the special fibre. The following are equivalent", "\\begin{enumerate}", "\\item $A \\to B$ is flat and $\\overline{B}$ is geometrically regular", "over $k$,", "\\item $A \\to B$ is flat and $k \\to \\overline{B}$ is formally smooth", "in the $\\mathfrak m_{\\overline{B}}$-adic topology, and", "\\item $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic", "topology.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows from Theorem \\ref{theorem-regular-fs}.", "\\medskip\\noindent", "Assume (3). By Lemma \\ref{lemma-formally-smooth-flat} we see that", "$A \\to B$ is flat. By Lemma \\ref{lemma-base-change-fs} we see that", "$k \\to \\overline{B}$ is formally smooth in the", "$\\mathfrak m_{\\overline{B}}$-adic topology. Thus (2) holds.", "\\medskip\\noindent", "Assume (2). Lemma \\ref{lemma-formally-smooth-completion}", "tells us formal smoothness is preserved under completion. The same is true", "for flatness by Algebra, Lemma \\ref{algebra-lemma-completion-faithfully-flat}.", "Hence we may replace $A$ and $B$ by their respective completions and", "assume that $A$ and $B$ are Noetherian complete local rings.", "In this case choose a diagram", "$$", "\\xymatrix{", "S \\ar[r] & B \\\\", "R \\ar[u] \\ar[r] & A \\ar[u]", "}", "$$", "as in Lemma \\ref{lemma-embed-map-Noetherian-complete-local-rings}.", "We will use all of the properties of this diagram without further mention.", "Fix a regular system of parameters $t_1, \\ldots, t_d$ of $R$", "with $t_1 = p$ in case the characteristic of $k$ is $p > 0$.", "Set $\\overline{S} = S \\otimes_R k$. Consider the short exact sequence", "$$", "0 \\to J \\to S \\to B \\to 0", "$$", "Since $B$ is flat over $A$ we see that $J \\otimes_R k$ is the kernel", "of $\\overline{S} \\to \\overline{B}$. As $\\overline{B}$ and $\\overline{S}$", "are regular we see that $J \\otimes_R k$ is generated by elements", "$\\overline{x}_1, \\ldots, \\overline{x}_r$ which form part of a regular", "system of parameters of $\\overline{S}$, see", "Algebra, Lemma \\ref{algebra-lemma-regular-quotient-regular}.", "Lift these elements to $x_1, \\ldots, x_r \\in J$. Then", "$t_1, \\ldots, t_d, x_1, \\ldots, x_r$ is part of a regular system of", "parameters for $S$. Hence $S/(x_1, \\ldots, x_r)$ is a power", "series ring over a field (if the characteristic of $k$ is zero)", "or a power series ring over a Cohen ring (if the characteristic of", "$k$ is $p > 0$), see", "Lemma \\ref{lemma-quotient-power-series-ring-over-Cohen}.", "Moreover, it is still the case that $R \\to S/(x_1, \\ldots, x_r)$", "maps $t_1, \\ldots, t_d$ to a part of a regular system of parameters", "of $S/(x_1, \\ldots, x_r)$. In other words, we may replace $S$ by", "$S/(x_1, \\ldots, x_r)$ and assume we have a diagram", "$$", "\\xymatrix{", "S \\ar[r] & B \\\\", "R \\ar[u] \\ar[r] & A \\ar[u]", "}", "$$", "as in Lemma \\ref{lemma-embed-map-Noetherian-complete-local-rings}", "with moreover $\\overline{S} = \\overline{B}$. In this case the map", "$$", "S \\otimes_R A \\longrightarrow B", "$$", "is an isomorphism as it is surjective and an isomorphism on special", "fibres, see Algebra, Lemma \\ref{algebra-lemma-mod-injective}. Thus by", "Lemma \\ref{lemma-base-change-fs}", "it suffices to show that $R \\to S$ is formally", "smooth in the $\\mathfrak m_S$-adic topology.", "Of course, since $\\overline{S} = \\overline{B}$, we have", "that $\\overline{S}$ is formally smooth over $k = R/\\mathfrak m_R$.", "\\medskip\\noindent", "Choose elements $y_1, \\ldots, y_m \\in S$ such that", "$t_1, \\ldots, t_d, y_1, \\ldots, y_m$ is a regular system of parameters", "for $S$. If the characteristic of $k$ is zero, choose a coefficient", "field $K \\subset S$ and if the characteristic of $k$ is $p > 0$", "choose a Cohen ring $\\Lambda \\subset S$ with residue field $K$.", "At this point the map $K[[t_1, \\ldots, t_d, y_1, \\ldots, y_m]] \\to S$", "(characteristic zero case) or", "$\\Lambda[[t_2, \\ldots, t_d, y_1, \\ldots, y_m]] \\to S$", "(characteristic $p > 0$ case) is an isomorphism, see", "Lemma \\ref{lemma-quotient-power-series-ring-over-Cohen}.", "From now on we think of $S$ as the above power series ring.", "\\medskip\\noindent", "The rest of the proof is analogous to the argument", "in the proof of Theorem \\ref{theorem-regular-fs}. Choose a solid diagram", "$$", "\\xymatrix{", "S \\ar[r]_{\\bar\\psi} \\ar@{-->}[rd] & N/J \\\\", "R \\ar[u]^i \\ar[r]^\\varphi & N \\ar[u]_\\pi", "}", "$$", "as in Definition \\ref{definition-formally-smooth}. As $J^2 = 0$ we see", "that $J$ has a canonical $N/J$ module structure and via $\\bar\\psi$ a", "$S$-module structure. As $\\bar\\psi$ is continuous for the", "$\\mathfrak m_S$-adic topology we see that $\\mathfrak m_S^nJ = 0$ for some", "$n$. Hence we can filter $J$ by $N/J$-submodules", "$0 \\subset J_1 \\subset J_2 \\subset \\ldots \\subset J_n = J$", "such that each quotient $J_{t + 1}/J_t$ is annihilated by $\\mathfrak m_S$.", "Considering the sequence of ring maps", "$N \\to N/J_1 \\to N/J_2 \\to \\ldots \\to N/J$", "we see that it suffices to prove the existence of the dotted arrow when", "$J$ is annihilated by $\\mathfrak m_S$, i.e., when $J$ is a", "$K$-vector space.", "\\medskip\\noindent", "Assume given a diagram as above such that $J$ is annihilated by", "$\\mathfrak m_S$. As $\\mathbf{Q} \\to S$ (characteristic zero case)", "or $\\mathbf{Z} \\to S$ (characteristic $p > 0$ case)", "is formally smooth in the $\\mathfrak m_S$-adic topology (see", "Lemma \\ref{lemma-power-series-ring-over-Cohen-fs}), we can find", "a ring map $\\psi : S \\to N$ such that $\\pi \\circ \\psi = \\bar \\psi$.", "Since $S$ is a power series ring in $t_1, \\ldots, t_d$ (characteristic zero)", "or $t_2, \\ldots, t_d$ (characteristic $p > 0$) over", "a subring, it follows from the universal property of power series rings", "that we can change our choice of $\\psi$ so that $\\psi(t_i)$ equals", "$\\varphi(t_i)$ (automatic for $t_1 = p$ in the characteristic $p$ case).", "Then $\\psi \\circ i$ and $\\varphi : R \\to N$ are two maps whose", "compositions with $\\pi$ are equal and which agree on $t_1, \\ldots, t_d$.", "Hence $D = \\psi \\circ i - \\varphi : R \\to J$ is a derivation which", "annihilates $t_1, \\ldots, t_d$.", "By Algebra, Lemma \\ref{algebra-lemma-universal-omega} we can write", "$D = \\xi \\circ \\text{d}$ for some $R$-linear map", "$\\xi : \\Omega_{R/\\mathbf{Z}} \\to J$ which annihilates", "$\\text{d}t_1, \\ldots, \\text{d}t_d$ (by construction) and", "$\\mathfrak m_R \\Omega_{R/\\mathbf{Z}}$ (as $J$ is annihilated by", "$\\mathfrak m_R$). Hence $\\xi$ factors as a composition", "$$", "\\Omega_{R/\\mathbf{Z}} \\to \\Omega_{k/\\mathbf{Z}} \\xrightarrow{\\xi'} J", "$$", "where $\\xi'$ is $k$-linear. Using the $K$-vector space structure on $J$ we", "extend $\\xi'$ to a $K$-linear map", "$$", "\\xi'' : \\Omega_{k/\\mathbf{Z}} \\otimes_k K \\longrightarrow J.", "$$", "Using that $\\overline{S}/k$ is formally smooth we see that", "$$", "\\Omega_{k/\\mathbf{Z}} \\otimes_k K \\to", "\\Omega_{\\overline{S}/\\mathbf{Z}} \\otimes_S K", "$$", "is injective by Theorem \\ref{theorem-regular-fs} (this is true also", "in the characteristic zero case as it is even true that", "$\\Omega_{k/\\mathbf{Z}} \\to \\Omega_{K/\\mathbf{Z}}$ is injective", "in characteristic zero, see Algebra,", "Proposition \\ref{algebra-proposition-characterize-separable-field-extensions}).", "Hence we can find a $K$-linear map", "$\\xi''' : \\Omega_{\\overline{S}/\\mathbf{Z}} \\otimes_S K \\to J$ whose", "restriction to $\\Omega_{k/\\mathbf{Z}} \\otimes_k K$ is $\\xi''$. Write", "$$", "D' : S \\xrightarrow{\\text{d}} \\Omega_{S/\\mathbf{Z}}", "\\to \\Omega_{\\overline{S}/\\mathbf{Z}} \\to", "\\Omega_{\\overline{S}/\\mathbf{Z}} \\otimes_S K \\xrightarrow{\\xi'''} J.", "$$", "Finally, set $\\psi' = \\psi - D' : S \\to N$. The reader verifies that $\\psi'$", "is a ring map such that $\\pi \\circ \\psi' = \\bar \\psi$ and such that", "$\\psi' \\circ i = \\varphi$ as desired." ], "refs": [ "more-algebra-theorem-regular-fs", "more-algebra-lemma-formally-smooth-flat", "more-algebra-lemma-base-change-fs", "more-algebra-lemma-formally-smooth-completion", "algebra-lemma-completion-faithfully-flat", "more-algebra-lemma-embed-map-Noetherian-complete-local-rings", "algebra-lemma-regular-quotient-regular", "more-algebra-lemma-quotient-power-series-ring-over-Cohen", "more-algebra-lemma-embed-map-Noetherian-complete-local-rings", "algebra-lemma-mod-injective", "more-algebra-lemma-base-change-fs", "more-algebra-lemma-quotient-power-series-ring-over-Cohen", "more-algebra-theorem-regular-fs", "more-algebra-definition-formally-smooth", "more-algebra-lemma-power-series-ring-over-Cohen-fs", "algebra-lemma-universal-omega", "more-algebra-theorem-regular-fs", "algebra-proposition-characterize-separable-field-extensions" ], "ref_ids": [ 9802, 10031, 10019, 10015, 871, 10029, 942, 10028, 10029, 883, 10019, 10028, 9802, 10611, 10027, 1129, 9802, 1429 ] } ], "ref_ids": [] }, { "id": 10578, "type": "theorem", "label": "more-algebra-proposition-ubiquity-J-2", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-ubiquity-J-2", "contents": [ "The following types of rings are J-2:", "\\begin{enumerate}", "\\item fields,", "\\item Noetherian complete local rings,", "\\item $\\mathbf{Z}$,", "\\item Noetherian local rings of dimension $1$,", "\\item Nagata rings of dimension $1$,", "\\item Dedekind domains with fraction field of characteristic zero,", "\\item finite type ring extensions of any of the above.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For cases (1), (3), (5), and (6) this is proved by checking", "condition (4) of Lemma \\ref{lemma-J-2}. We will only do this", "in case $R$ is a Nagata ring of dimension $1$. Let $\\mathfrak p \\subset R$", "be a prime ideal and let $\\kappa(\\mathfrak p) \\subset L$ be a finite", "purely inseparable extension. If $\\mathfrak p \\subset R$ is a", "maximal ideal, then $R \\to L$ is finite and $L$ is a regular ring", "and we've checked the condition. If $\\mathfrak p \\subset R$ is a", "minimal prime, then the Nagata condition insures that the", "integral closure $R' \\subset L$ of $R$ in $L$ is finite over $R$.", "Then $R'$ is a normal domain of dimension $1$", "(Algebra, Lemma \\ref{algebra-lemma-integral-dim-up})", "hence regular (Algebra, Lemma \\ref{algebra-lemma-criterion-normal})", "and we've checked the condition in this case as well.", "\\medskip\\noindent", "For case (2), we will use condition (3) of Lemma \\ref{lemma-J-2}.", "Let $R$ be a Noetherian complete local ring.", "Note that if $R \\to R'$ is finite, then $R'$ is a product of", "Noetherian complete local rings, see", "Algebra, Lemma \\ref{algebra-lemma-quotient-complete-local}.", "Hence it suffices to prove that a Noetherian complete local ring", "which is a domain is J-0, which is", "Lemma \\ref{lemma-complete-Noetherian-domain-J-0}.", "\\medskip\\noindent", "For case (4), we also use condition (3) of Lemma \\ref{lemma-J-2}.", "Namely, if $R$ is a local Noetherian ring of dimension $1$ and", "$R \\to R'$ is finite, then $\\Spec(R')$ is finite. Since the", "regular locus is stable under generalization, we see", "that $R'$ is J-1." ], "refs": [ "more-algebra-lemma-J-2", "algebra-lemma-integral-dim-up", "algebra-lemma-criterion-normal", "more-algebra-lemma-J-2", "algebra-lemma-quotient-complete-local", "more-algebra-lemma-complete-Noetherian-domain-J-0", "more-algebra-lemma-J-2" ], "ref_ids": [ 10076, 984, 1311, 10076, 1327, 10082, 10076 ] } ], "ref_ids": [] }, { "id": 10579, "type": "theorem", "label": "more-algebra-proposition-fs-regular", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-fs-regular", "contents": [ "Let $A \\to B$ be a local homomorphism of Noetherian complete local rings.", "The following are equivalent", "\\begin{enumerate}", "\\item $A \\to B$ is regular,", "\\item $A \\to B$ is flat and $\\overline{B}$ is geometrically regular", "over $k$,", "\\item $A \\to B$ is flat and $k \\to \\overline{B}$ is formally smooth", "in the $\\mathfrak m_{\\overline{B}}$-adic topology, and", "\\item $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic", "topology.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have seen the equivalence of (2), (3), and (4) in", "Proposition \\ref{proposition-fs-flat-fibre-fs}.", "It is clear that (1) implies (2).", "Thus we assume the equivalent conditions (2), (3), and (4) hold", "and we prove (1).", "\\medskip\\noindent", "Let $\\mathfrak p$ be a prime of $A$. We will show that", "$B \\otimes_A \\kappa(\\mathfrak p)$ is geometrically regular", "over $\\kappa(\\mathfrak p)$.", "By Lemma \\ref{lemma-base-change-fs}", "we may replace $A$ by $A/\\mathfrak p$ and $B$ by $B/\\mathfrak pB$.", "Thus we may assume that $A$ is a domain and that $\\mathfrak p = (0)$.", "\\medskip\\noindent", "Choose $A_0 \\subset A$ as in Algebra, Lemma", "\\ref{algebra-lemma-complete-local-Noetherian-domain-finite-over-regular}.", "We will use all the properties stated in that lemma without further mention.", "As $A_0 \\to A$ induces an isomorphism on residue fields, and as", "$B/\\mathfrak m_A B$ is geometrically regular over $A/\\mathfrak m_A$", "we can find a diagram", "$$", "\\xymatrix{", "C \\ar[r] & B \\\\", "A_0 \\ar[r] \\ar[u] & A \\ar[u]", "}", "$$", "with $A_0 \\to C$ formally smooth in the $\\mathfrak m_C$-adic topology", "such that $B = C \\otimes_{A_0} A$, see Remark \\ref{remark-what-does-it-mean}.", "(Completion in the tensor product is not needed as $A_0 \\to A$ is", "finite, see Algebra, Lemma \\ref{algebra-lemma-completion-tensor}.)", "Hence it suffices to show that $C \\otimes_{A_0} K_0$", "is a geometrically regular algebra over the fraction field $K_0$ of $A_0$.", "\\medskip\\noindent", "The upshot of the preceding paragraph is that we may assume that", "$A = k[[x_1, \\ldots, x_n]]$ where $k$ is a field or", "$A = \\Lambda[[x_1, \\ldots, x_n]]$ where $\\Lambda$ is a Cohen ring.", "In this case $B$ is a regular ring, see", "Algebra, Lemma \\ref{algebra-lemma-flat-over-regular-with-regular-fibre}.", "Hence $B \\otimes_A K$ is a regular ring too (where $K$ is the", "fraction field of $A$) and we win", "if the characteristic of $K$ is zero.", "\\medskip\\noindent", "Thus we are left with the case where $A = k[[x_1, \\ldots, x_n]]$", "and $k$ is a field of characteristic $p > 0$.", "Let $L \\supset K$ be a finite purely inseparable field extension.", "We will show by induction on $[L : K]$ that $B \\otimes_A L$", "is regular. The base case is $L = K$ which we've seen above.", "Let $K \\subset M \\subset L$ be a subfield such that", "$L$ is a degree $p$ extension of $M$ obtained by adjoining a $p$th root", "of an element $f \\in M$. Let $A'$ be a finite $A$-subalgebra", "of $M$ with fraction field $M$. Clearing denominators, we may and do assume", "$f \\in A'$. Set $A'' = A'[z]/(z^p -f)$ and note that $A' \\subset A''$", "is finite and that the fraction field of $A''$ is $L$.", "By induction we know that $B \\otimes_A M$ ring is regular.", "We have", "$$", "B \\otimes_A L = B \\otimes_A M[z]/(z^p - f)", "$$", "By Lemma \\ref{lemma-find-D} we know there exists a derivation", "$D : A' \\to A'$ such that $D(f) \\not = 0$. As $A' \\to B \\otimes_A A'$", "is formally smooth in the $\\mathfrak m$-adic topology by", "Lemma \\ref{lemma-descent-fs}", "we can use", "Lemma \\ref{lemma-lift-derivation-through-fs}", "to extend $D$ to a derivation $D' : B \\otimes_A A' \\to B \\otimes_A A'$.", "Note that $D'(f) = D(f)$ is a unit in $B \\otimes_A M$ as $D(f)$", "is not zero in $A' \\subset M$. Hence $B \\otimes_A L$ is regular by", "Lemma \\ref{lemma-degree-p-extension-regular} and we win." ], "refs": [ "more-algebra-proposition-fs-flat-fibre-fs", "more-algebra-lemma-base-change-fs", "algebra-lemma-complete-local-Noetherian-domain-finite-over-regular", "more-algebra-remark-what-does-it-mean", "algebra-lemma-completion-tensor", "algebra-lemma-flat-over-regular-with-regular-fibre", "more-algebra-lemma-find-D", "more-algebra-lemma-descent-fs", "more-algebra-lemma-lift-derivation-through-fs", "more-algebra-lemma-degree-p-extension-regular" ], "ref_ids": [ 10577, 10019, 1332, 10651, 869, 988, 10081, 10020, 10083, 10080 ] } ], "ref_ids": [] }, { "id": 10580, "type": "theorem", "label": "more-algebra-proposition-Noetherian-complete-G-ring", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-Noetherian-complete-G-ring", "contents": [ "A Noetherian complete local ring is a G-ring." ], "refs": [], "proofs": [ { "contents": [ "Let $A$ be a Noetherian complete local ring. By", "Lemma \\ref{lemma-check-G-ring-easy}", "it suffices to check that $B = A/\\mathfrak q$ has geometrically regular", "formal fibres over the minimal prime $(0)$ of $B$. Thus we may assume", "that $A$ is a domain and it suffices to check the condition for", "the formal fibres over the minimal prime $(0)$ of $A$.", "Let $K$ be the fraction field of $A$.", "\\medskip\\noindent", "We can choose a subring $A_0 \\subset A$ which is a regular complete local", "ring such that $A$ is finite over $A_0$, see Algebra, Lemma", "\\ref{algebra-lemma-complete-local-Noetherian-domain-finite-over-regular}.", "Moreover, we may assume that $A_0$ is a power series ring over a", "field or a Cohen ring. By Lemma \\ref{lemma-G-ring-goes-up-quasi-finite}", "we see that it suffices to prove the result for $A_0$.", "\\medskip\\noindent", "Assume that $A$ is a power series ring over a field or a Cohen ring.", "Since $A$ is regular the localizations $A_\\mathfrak p$ are regular", "(see Algebra, Definition \\ref{algebra-definition-regular} and the", "discussion preceding it).", "Hence the completions $A_\\mathfrak p^\\wedge$ are regular, see", "Lemma \\ref{lemma-completion-regular}.", "Hence the fibre $A_{\\mathfrak p}^\\wedge \\otimes_A K$ is, as a localization", "of $A_\\mathfrak p^\\wedge$, also regular. Thus we are done if the", "characteristic of $K$ is $0$. The positive characteristic case", "is the case $A = k[[x_1, \\ldots, x_d]]$ which is a special case of", "Lemma \\ref{lemma-helper-G-ring}." ], "refs": [ "more-algebra-lemma-check-G-ring-easy", "algebra-lemma-complete-local-Noetherian-domain-finite-over-regular", "more-algebra-lemma-G-ring-goes-up-quasi-finite", "algebra-definition-regular", "more-algebra-lemma-completion-regular", "more-algebra-lemma-helper-G-ring" ], "ref_ids": [ 10084, 1332, 10085, 1512, 10045, 10087 ] } ], "ref_ids": [] }, { "id": 10581, "type": "theorem", "label": "more-algebra-proposition-finite-type-over-G-ring", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-finite-type-over-G-ring", "contents": [ "Let $R$ be a G-ring. If $R \\to S$ is essentially of finite type", "then $S$ is a G-ring." ], "refs": [], "proofs": [ { "contents": [ "Since being a G-ring is a property of the local rings it is clear", "that a localization of a G-ring is a G-ring. Conversely, if every", "localization at a prime is a G-ring, then the ring is a G-ring.", "Thus it suffices to show that $S_\\mathfrak q$ is a G-ring for every", "finite type $R$-algebra $S$ and every prime $\\mathfrak q$ of $S$.", "Writing $S$ as a quotient of $R[x_1, \\ldots, x_n]$ we see from", "Lemma \\ref{lemma-G-ring-goes-up-quasi-finite} that it suffices to prove", "that $R[x_1, \\ldots, x_n]$ is a G-ring. By induction on $n$ it", "suffices to prove that $R[x]$ is a G-ring. Let $\\mathfrak q \\subset R[x]$", "be a maximal ideal. By Lemma \\ref{lemma-check-G-ring-maximal-ideals}", "it suffices to show that", "$$", "R[x]_\\mathfrak q \\longrightarrow R[x]_\\mathfrak q^\\wedge", "$$", "is regular. If $\\mathfrak q$ lies over $\\mathfrak p \\subset R$, then", "we may replace $R$ by $R_\\mathfrak p$. Hence we may assume that $R$", "is a Noetherian local G-ring with maximal ideal $\\mathfrak m$ and", "that $\\mathfrak q \\subset R[x]$ lies over $\\mathfrak m$. Note that", "there is a unique prime $\\mathfrak q' \\subset R^\\wedge[x]$", "lying over $\\mathfrak q$. Consider the diagram", "$$", "\\xymatrix{", "R[x]_\\mathfrak q^\\wedge \\ar[r] &", "(R^\\wedge[x]_{\\mathfrak q'})^\\wedge \\\\", "R[x]_\\mathfrak q \\ar[r] \\ar[u] & R^\\wedge[x]_{\\mathfrak q'} \\ar[u]", "}", "$$", "Since $R$ is a G-ring the lower horizontal arrow is regular", "(as a localization of a base change of the regular ring map", "$R \\to R^\\wedge$). Suppose we can prove the right vertical arrow", "is regular. Then it follows that the composition", "$R[x]_\\mathfrak q \\to (R^\\wedge[x]_{\\mathfrak q'})^\\wedge$", "is regular, and hence the left vertical arrow is regular by", "Lemma \\ref{lemma-regular-permanence}.", "Hence we see that we may assume $R$ is a Noetherian complete", "local ring and $\\mathfrak q$ a prime lying over the maximal", "ideal of $R$.", "\\medskip\\noindent", "Let $R$ be a Noetherian complete local ring and let $\\mathfrak q \\subset R[x]$", "be a maximal ideal lying over the maximal ideal of $R$. Let", "$\\mathfrak r \\subset \\mathfrak q$ be a prime ideal. We want to show that", "$R[x]_\\mathfrak q^\\wedge \\otimes_{R[x]} \\kappa(\\mathfrak r)$ is", "a geometrically regular algebra over $\\kappa(\\mathfrak r)$.", "Set $\\mathfrak p = R \\cap \\mathfrak r$. Then we can replace $R$", "by $R/\\mathfrak p$ and $\\mathfrak q$ and $\\mathfrak r$ by their", "images in $R/\\mathfrak p[x]$, see", "Lemma \\ref{lemma-check-G-ring-easy}.", "Hence we may assume that $R$ is a domain and that $\\mathfrak r \\cap R = (0)$.", "\\medskip\\noindent", "By Algebra, Lemma", "\\ref{algebra-lemma-complete-local-Noetherian-domain-finite-over-regular}", "we can find $R_0 \\subset R$ which is regular and such that $R$ is", "finite over $R_0$. Applying Lemma \\ref{lemma-G-ring-goes-up-quasi-finite}", "we see that it suffices to prove", "$R[x]_\\mathfrak q^\\wedge \\otimes_{R[x]} \\kappa(\\mathfrak r)$", "is geometrically regular over $\\kappa(r)$ when, in addition to the above,", "$R$ is a regular complete local ring.", "\\medskip\\noindent", "Now $R$ is a regular complete local ring, we have", "$\\mathfrak q \\subset \\mathfrak r \\subset R[x]$, we have", "$(0) = R \\cap \\mathfrak r$ and $\\mathfrak q$ is a maximal ideal", "lying over the maximal ideal of $R$. Since $R$ is regular the", "ring $R[x]$ is regular (Algebra, Lemma \\ref{algebra-lemma-regular-goes-up}).", "Hence the localization $R[x]_\\mathfrak q$ is regular.", "Hence the completions $R[x]_\\mathfrak q^\\wedge$ are regular, see", "Lemma \\ref{lemma-completion-regular}.", "Hence the fibre $R[x]_{\\mathfrak q}^\\wedge \\otimes_{R[x]} \\kappa(\\mathfrak r)$", "is, as a localization of $R[x]_\\mathfrak q^\\wedge$, also regular.", "Thus we are done if the characteristic of the fraction field of $R$ is $0$.", "\\medskip\\noindent", "If the characteristic of $R$ is positive, then $R = k[[x_1, \\ldots, x_n]]$.", "In this case we split the argument in two subcases:", "\\begin{enumerate}", "\\item The case $\\mathfrak r = (0)$. The result is a direct consequence", "of Lemma \\ref{lemma-helper-G-ring}.", "\\item The case $\\mathfrak r \\not = (0)$. This is", "Lemma \\ref{lemma-another-helper-G-ring}.", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-G-ring-goes-up-quasi-finite", "more-algebra-lemma-check-G-ring-maximal-ideals", "more-algebra-lemma-regular-permanence", "more-algebra-lemma-check-G-ring-easy", "algebra-lemma-complete-local-Noetherian-domain-finite-over-regular", "more-algebra-lemma-G-ring-goes-up-quasi-finite", "algebra-lemma-regular-goes-up", "more-algebra-lemma-completion-regular", "more-algebra-lemma-helper-G-ring", "more-algebra-lemma-another-helper-G-ring" ], "ref_ids": [ 10085, 10088, 10039, 10084, 1332, 10085, 1369, 10045, 10087, 10090 ] } ], "ref_ids": [] }, { "id": 10582, "type": "theorem", "label": "more-algebra-proposition-ubiquity-G-ring", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-ubiquity-G-ring", "contents": [ "The following types of rings are G-rings:", "\\begin{enumerate}", "\\item fields,", "\\item Noetherian complete local rings,", "\\item $\\mathbf{Z}$,", "\\item Dedekind domains with fraction field of characteristic zero,", "\\item finite type ring extensions of any of the above.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For fields, $\\mathbf{Z}$ and Dedekind domains of characteristic zero", "this follows immediately from the definition and the fact that the", "completion of a discrete valuation ring is a discrete valuation ring.", "A Noetherian complete local ring is a G-ring by", "Proposition \\ref{proposition-Noetherian-complete-G-ring}.", "The statement on finite type overrings is", "Proposition \\ref{proposition-finite-type-over-G-ring}." ], "refs": [ "more-algebra-proposition-Noetherian-complete-G-ring", "more-algebra-proposition-finite-type-over-G-ring" ], "ref_ids": [ 10580, 10581 ] } ], "ref_ids": [] }, { "id": 10583, "type": "theorem", "label": "more-algebra-proposition-finite-type-over-P-ring", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-finite-type-over-P-ring", "contents": [ "Let $R$ be a $P$-ring where $P$ satisfies (A), (B), (C), and (D).", "If $R \\to S$ is essentially of finite type then $S$ is a $P$-ring." ], "refs": [], "proofs": [ { "contents": [ "Since being a $P$-ring is a property of the local rings it is clear", "that a localization of a $P$-ring is a $P$-ring. Conversely, if every", "localization at a prime is a $P$-ring, then the ring is a $P$-ring.", "Thus it suffices to show that $S_\\mathfrak q$ is a $P$-ring for every", "finite type $R$-algebra $S$ and every prime $\\mathfrak q$ of $S$.", "Writing $S$ as a quotient of $R[x_1, \\ldots, x_n]$ we see from", "Lemma \\ref{lemma-P-ring-goes-up-quasi-finite} that it suffices to prove", "that $R[x_1, \\ldots, x_n]$ is a $P$-ring. By induction on $n$ it", "suffices to prove that $R[x]$ is a $P$-ring. Let $\\mathfrak q \\subset R[x]$", "be a maximal ideal. By Lemma \\ref{lemma-check-P-ring-maximal-ideals}", "it suffices to show that the fibres of", "$$", "R[x]_\\mathfrak q \\longrightarrow R[x]_\\mathfrak q^\\wedge", "$$", "have $P$. If $\\mathfrak q$ lies over $\\mathfrak p \\subset R$, then", "we may replace $R$ by $R_\\mathfrak p$. Hence we may assume that $R$", "is a Noetherian local $P$-ring with maximal ideal $\\mathfrak m$ and", "that $\\mathfrak q \\subset R[x]$ lies over $\\mathfrak m$. Note that", "there is a unique prime $\\mathfrak q' \\subset R^\\wedge[x]$", "lying over $\\mathfrak q$. Consider the diagram", "$$", "\\xymatrix{", "R[x]_\\mathfrak q^\\wedge \\ar[r] &", "(R^\\wedge[x]_{\\mathfrak q'})^\\wedge \\\\", "R[x]_\\mathfrak q \\ar[r] \\ar[u] & R^\\wedge[x]_{\\mathfrak q'} \\ar[u]", "}", "$$", "Since $R$ is a $P$-ring the fibres of $R[x] \\to R^\\wedge[x]$ have", "$P$ because they are base changes of the fibres of $R \\to R^\\wedge$", "by a finitely generated field extension so (A) applies. Hence", "the fibres of the lower horizontal arrow have $P$ for example by", "Lemma \\ref{lemma-P-local}.", "The right vertical arrow is regular because $R^\\wedge$ is", "a G-ring (Propositions \\ref{proposition-Noetherian-complete-G-ring} and", "\\ref{proposition-finite-type-over-G-ring}).", "It follows that the fibres of the composition", "$R[x]_\\mathfrak q \\to (R^\\wedge[x]_{\\mathfrak q'})^\\wedge$", "have $P$ by (C). Hence", "the fibres of the left vertical arrow have $P$ by (D) and the", "proof is complete." ], "refs": [ "more-algebra-lemma-P-ring-goes-up-quasi-finite", "more-algebra-lemma-check-P-ring-maximal-ideals", "more-algebra-lemma-P-local", "more-algebra-proposition-Noetherian-complete-G-ring", "more-algebra-proposition-finite-type-over-G-ring" ], "ref_ids": [ 10096, 10097, 10095, 10580, 10581 ] } ], "ref_ids": [] }, { "id": 10584, "type": "theorem", "label": "more-algebra-proposition-ubiquity-excellent", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-ubiquity-excellent", "contents": [ "The following types of rings are excellent:", "\\begin{enumerate}", "\\item fields,", "\\item Noetherian complete local rings,", "\\item $\\mathbf{Z}$,", "\\item Dedekind domains with fraction field of characteristic zero,", "\\item finite type ring extensions of any of the above.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See Propositions \\ref{proposition-ubiquity-G-ring} and", "\\ref{proposition-ubiquity-J-2} to see that these rings are", "G-rings and have J-2. Any Cohen-Macaulay ring is universally", "catenary, see Algebra, Lemma \\ref{algebra-lemma-CM-ring-catenary}.", "In particular fields, Dedekind rings, and more generally", "regular rings are universally catenary. Via the Cohen structure theorem ", "we see that complete local rings are universally catenary, see", "Algebra, Remark", "\\ref{algebra-remark-Noetherian-complete-local-ring-universally-catenary}." ], "refs": [ "more-algebra-proposition-ubiquity-G-ring", "more-algebra-proposition-ubiquity-J-2", "algebra-lemma-CM-ring-catenary", "algebra-remark-Noetherian-complete-local-ring-universally-catenary" ], "ref_ids": [ 10582, 10578, 937, 1582 ] } ], "ref_ids": [] }, { "id": 10585, "type": "theorem", "label": "more-algebra-proposition-perfect-is-compact", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-perfect-is-compact", "contents": [ "Let $R$ be a ring. For an object $K$ of $D(R)$ the following are equivalent", "\\begin{enumerate}", "\\item $K$ is perfect, and", "\\item $K$ is a compact object of $D(R)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $K$ is perfect, i.e., $K$ is quasi-isomorphic to a bounded complex", "$P^\\bullet$ of finite projective modules, see", "Definition \\ref{definition-perfect}. If $E_i$ is represented by the complex", "$E_i^\\bullet$, then $\\bigoplus E_i$ is represented by the complex", "whose degree $n$ term is $\\bigoplus E_i^n$. On the other hand,", "as $P^n$ is projective for all $n$ we have", "$\\Hom_{D(R)}(P^\\bullet, K^\\bullet) = \\Hom_{K(R)}(P^\\bullet, K^\\bullet)$", "for every complex of $R$-modules $K^\\bullet$, see", "Derived Categories,", "Lemma \\ref{derived-lemma-morphisms-from-projective-complex}.", "Thus $\\Hom_{D(R)}(P^\\bullet, E^\\bullet)$ is the cohomology of the complex", "$$", "\\prod \\Hom_R(P^n, E^{n - 1}) \\to", "\\prod \\Hom_R(P^n, E^n) \\to", "\\prod \\Hom_R(P^n, E^{n + 1}).", "$$", "Since $P^\\bullet$ is bounded we see that we may replace the $\\prod$", "signs by $\\bigoplus$ signs in the complex above. Since each $P^n$ is a finite", "$R$-module we see that", "$\\Hom_R(P^n, \\bigoplus_i E_i^m) = \\bigoplus_i \\Hom_R(P^n, E_i^m)$", "for all $n, m$.", "Combining these remarks we see that the map of", "Derived Categories, Definition \\ref{derived-definition-compact-object}", "is a bijection.", "\\medskip\\noindent", "Conversely, assume $K$ is compact.", "Represent $K$ by a complex $K^\\bullet$ and consider the map", "$$", "K^\\bullet", "\\longrightarrow", "\\bigoplus\\nolimits_{n \\geq 0} \\tau_{\\geq n} K^\\bullet", "$$", "where we have used the canonical truncations, see", "Homology, Section \\ref{homology-section-truncations}.", "This makes sense as in each degree the direct sum on the right is finite.", "By assumption this map factors through a finite direct sum.", "We conclude that $K \\to \\tau_{\\geq n} K$ is zero for at least one $n$,", "i.e., $K$ is in $D^{-}(R)$.", "\\medskip\\noindent", "Since $K \\in D^{-}(R)$ and since every $R$-module is a quotient of a free", "module, we may represent $K$ by a bounded above complex $K^\\bullet$", "of free $R$-modules, see", "Derived Categories, Lemma \\ref{derived-lemma-subcategory-left-resolution}.", "Note that we have", "$$", "K^\\bullet = \\bigcup\\nolimits_{n \\leq 0} \\sigma_{\\geq n}K^\\bullet", "$$", "where we have used the stupid truncations, see", "Homology, Section \\ref{homology-section-truncations}.", "Hence by Lemma \\ref{lemma-commutes-with-countable-sums} we see that", "$1 : K^\\bullet \\to K^\\bullet$ factors through", "$\\sigma_{\\geq n}K^\\bullet \\to K^\\bullet$ in $D(R)$.", "Thus we see that $1 : K^\\bullet \\to K^\\bullet$ factors as", "$$", "K^\\bullet \\xrightarrow{\\varphi} L^\\bullet \\xrightarrow{\\psi} K^\\bullet", "$$", "in $D(R)$ for some complex $L^\\bullet$ which is bounded and whose terms", "are free $R$-modules. Say $L^i = 0$ for $i \\not \\in [a, b]$.", "Fix $a, b$ from now on. Let $c$ be the largest integer $\\leq b + 1$", "such that we can find a factorization of $1_{K^\\bullet}$ as above", "with $L^i$ is finite free for $i < c$. We will show by induction that", "$c = b + 1$. Namely, write $L^c = \\bigoplus_{\\lambda \\in \\Lambda} R$.", "Since $L^{c - 1}$ is finite free we can find a finite subset", "$\\Lambda' \\subset \\Lambda$ such that $L^{c - 1} \\to L^c$ factors", "through $\\bigoplus_{\\lambda \\in \\Lambda'} R \\subset L^c$. Consider the", "map of complexes", "$$", "\\pi :", "L^\\bullet", "\\longrightarrow", "(\\bigoplus\\nolimits_{\\lambda \\in \\Lambda \\setminus \\Lambda'} R)[-i]", "$$", "given by the projection onto the factors corresponding to", "$\\Lambda \\setminus \\Lambda'$ in degree $i$.", "By our assumption on $K$ we see that, after possibly replacing $\\Lambda'$ by", "a larger finite subset, we may assume that $\\pi \\circ \\varphi = 0$", "in $D(R)$. Let $(L')^\\bullet \\subset L^\\bullet$ be the kernel of $\\pi$.", "Since $\\pi$ is surjective we get a short exact sequence of complexes,", "which gives a distinguished triangle in $D(R)$ (see", "Derived Categories, Lemma \\ref{derived-lemma-derived-canonical-delta-functor}).", "Since $\\Hom_{D(R)}(K, -)$ is homological (see", "Derived Categories, Lemma \\ref{derived-lemma-representable-homological})", "and $\\pi \\circ \\varphi = 0$, we can find a morphism", "$\\varphi' : K^\\bullet \\to (L')^\\bullet$ in $D(R)$ whose", "composition with $(L')^\\bullet \\to L^\\bullet$ gives $\\varphi$.", "Setting $\\psi'$ equal to the composition of $\\psi$ with", "$(L')^\\bullet \\to L^\\bullet$ we obtain a new factorization.", "Since $(L')^\\bullet$ agrees with $L^\\bullet$ except in degree $c$", "and since $(L')^c = \\bigoplus_{\\lambda \\in \\Lambda'} R$ the", "induction step is proved.", "\\medskip\\noindent", "The conclusion of the discussion of the preceding paragraph is that", "$1_K : K \\to K$ factors as", "$$", "K \\xrightarrow{\\varphi} L \\xrightarrow{\\psi} K", "$$", "in $D(R)$ where $L$ can be represented by a finite", "complex of free $R$-modules. In particular we see that $L$ is", "perfect. Note that $e = \\varphi \\circ \\psi \\in \\text{End}_{D(R)}(L)$", "is an idempotent. By Derived Categories,", "Lemma \\ref{derived-lemma-projectors-have-images-triangulated}", "we see that $L = \\Ker(e) \\oplus \\Ker(1 - e)$.", "The map $\\varphi : K \\to L$ induces an isomorphism with", "$\\Ker(1 - e)$ in $D(R)$. Hence we finally conclude that", "$K$ is perfect by Lemma \\ref{lemma-summands-perfect}." ], "refs": [ "more-algebra-definition-perfect", "derived-lemma-morphisms-from-projective-complex", "derived-definition-compact-object", "derived-lemma-subcategory-left-resolution", "more-algebra-lemma-commutes-with-countable-sums", "derived-lemma-derived-canonical-delta-functor", "derived-lemma-representable-homological", "derived-lemma-projectors-have-images-triangulated", "more-algebra-lemma-summands-perfect" ], "ref_ids": [ 10628, 1862, 2004, 1835, 10248, 1814, 1758, 1769, 10215 ] } ], "ref_ids": [] }, { "id": 10586, "type": "theorem", "label": "more-algebra-proposition-regular-strong-generator", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-regular-strong-generator", "contents": [ "Let $R$ be a Noetherian ring. The following are equivalent", "\\begin{enumerate}", "\\item $R$ is regular of finite dimension,", "\\item $D_{perf}(R)$ has a strong generator, and", "\\item $R$ is a strong generator for $D_{perf}(R)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a formal consequence of", "Lemmas \\ref{lemma-perfect-ring-classical-generator},", "\\ref{lemma-not-regular-not-strong}, and", "\\ref{lemma-regular-strong-generator}", "as well as Derived Categories, Lemma", "\\ref{derived-lemma-classical-generator-strong-generator}." ], "refs": [ "more-algebra-lemma-perfect-ring-classical-generator", "more-algebra-lemma-not-regular-not-strong", "more-algebra-lemma-regular-strong-generator", "derived-lemma-classical-generator-strong-generator" ], "ref_ids": [ 10247, 10252, 10255, 1939 ] } ], "ref_ids": [] }, { "id": 10587, "type": "theorem", "label": "more-algebra-proposition-equivalence", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-equivalence", "contents": [ "Assume $\\varphi : R \\to S$ is a flat ring map and", "$I = (f_1, \\ldots, f_t) \\subset R$ is an ideal such that", "$R/I \\to S/IS$ is an isomorphism. Then $\\text{Can}$ and", "$H^0$ are quasi-inverse equivalences of categories", "$$", "\\text{Mod}_R = \\text{Glue}(R \\to S, f_1, \\ldots, f_t)", "$$" ], "refs": [], "proofs": [ { "contents": [ "We have already seen that $H^0 \\circ \\text{Can}$ is isomorphic to the", "identity functor, see", "Lemma \\ref{lemma-H0-inverse}.", "Consider an object $\\mathbf{M} = (M', M_i, \\alpha_i, \\alpha_{ij})$", "of $\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$.", "We get a natural morphism", "$$", "\\Psi :", "(H^0(\\mathbf{M}) \\otimes_R S, H^0(\\mathbf{M})_{f_i},", "\\text{can}_i, \\text{can}_{ij})", "\\longrightarrow", "(M', M_i, \\alpha_i, \\alpha_{ij}).", "$$", "Namely, by definition $H^0(\\mathbf{M})$ comes equipped with compatible", "$R$-module maps $H^0(\\mathbf{M}) \\to M'$ and $H^0(\\mathbf{M}) \\to M_i$.", "We have to show that this map is an isomorphism.", "\\medskip\\noindent", "Pick an index $i$ and set $R' = R_{f_i}$. Combining", "Lemmas \\ref{lemma-base-change-glue} and \\ref{lemma-equivalence-I-unit}", "we see that $\\Psi \\otimes_R R'$ is an isomorphism.", "Hence the kernel, resp.\\ cokernel of $\\Psi$ is a system of the form", "$(K, 0, 0, 0)$, resp.\\ $(Q, 0, 0, 0)$. Note that", "$H^0((K, 0, 0, 0)) = K$, that $H^0$ is left exact, and that by", "construction $H^0(\\Psi)$ is bijective. Hence we see $K = 0$, i.e.,", "the kernel of $\\Psi$ is zero.", "\\medskip\\noindent", "The conclusion of the above is that we obtain a short exact sequence", "$$", "0 \\to H^0(\\mathbf{M}) \\otimes_R S \\to M' \\to Q \\to 0", "$$", "and that $M_i = H^0(\\mathbf{M})_{f_i}$.", "Note that we may think of $Q$ as an $R$-module which is $I$-power", "torsion so that $Q = Q \\otimes_R S$. By", "Lemma \\ref{lemma-neighbourhood-extensions}", "we see that there exists a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "H^0(\\mathbf{M}) \\ar[r] \\ar[d] &", "E \\ar[r] \\ar[d] &", "Q \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "H^0(\\mathbf{M}) \\otimes_R S \\ar[r] &", "M' \\ar[r] &", "Q \\ar[r] &", "0", "}", "$$", "with exact rows. This clearly determines an isomorphism", "$\\text{Can}(E) \\to (M', M_i, \\alpha_i, \\alpha_{ij})$", "in the category $\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$", "and we win. (Of course, a posteriori we have $Q = 0$.)" ], "refs": [ "more-algebra-lemma-H0-inverse", "more-algebra-lemma-base-change-glue", "more-algebra-lemma-equivalence-I-unit", "more-algebra-lemma-neighbourhood-extensions" ], "ref_ids": [ 10348, 10351, 10350, 10346 ] } ], "ref_ids": [] }, { "id": 10588, "type": "theorem", "label": "more-algebra-proposition-formal-glueing", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-formal-glueing", "contents": [ "Let $R$ be a Noetherian ring.", "Let $f \\in R$ be an element.", "Let $R^\\wedge$ be the $f$-adic completion of $R$.", "Then the functor $M \\mapsto (M^\\wedge, M_f, \\text{can})$", "defines an equivalence", "$$", "\\text{Mod}^{fg}_R", "\\longrightarrow", "\\text{Mod}^{fg}_{R^\\wedge}", "\\times_{\\text{Mod}^{fg}_{(R^\\wedge)_f}}", "\\text{Mod}^{fg}_{R_f}", "$$" ], "refs": [], "proofs": [ { "contents": [ "The ring map $R \\to R^\\wedge$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}.", "It is clear that $R/fR = R^\\wedge/fR^\\wedge$.", "By", "Algebra, Lemma \\ref{algebra-lemma-completion-tensor}", "the completion of a finite $R$-module $M$ is equal to $M \\otimes_R R^\\wedge$.", "Hence the displayed functor of the proposition is equal to the", "functor occurring in", "Theorem \\ref{theorem-formal-glueing}.", "In particular it is fully faithful. Let $(M_1, M_2, \\psi)$ be an", "object of the right hand side. By", "Theorem \\ref{theorem-formal-glueing}", "there exists an $R$-module $M$ such that", "$M_1 = M \\otimes_R R^\\wedge$ and $M_2 = M_f$. As $R \\to R^\\wedge \\times R_f$", "is faithfully flat we conclude from", "Algebra, Lemma \\ref{algebra-lemma-cover}", "that $M$ is finitely generated, i.e., $M \\in \\text{Mod}^{fg}_R$.", "This proves the proposition." ], "refs": [ "algebra-lemma-completion-flat", "algebra-lemma-completion-tensor", "more-algebra-theorem-formal-glueing", "more-algebra-theorem-formal-glueing", "algebra-lemma-cover" ], "ref_ids": [ 870, 869, 9803, 9803, 411 ] } ], "ref_ids": [] }, { "id": 10589, "type": "theorem", "label": "more-algebra-proposition-derived-complete-modules", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-derived-complete-modules", "contents": [ "Let $I \\subset A$ be a finitely generated ideal of a ring $A$.", "Let $M$ be an $A$-module. The following are equivalent", "\\begin{enumerate}", "\\item $M$ is $I$-adically complete, and", "\\item $M$ is derived complete with respect to $I$ and $\\bigcap I^nM = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is clear from the results of", "Lemma \\ref{lemma-complete-derived-complete}." ], "refs": [ "more-algebra-lemma-complete-derived-complete" ], "ref_ids": [ 10367 ] } ], "ref_ids": [] }, { "id": 10590, "type": "theorem", "label": "more-algebra-proposition-noetherian-naive-completion-is-completion", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-noetherian-naive-completion-is-completion", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.", "The functor which sends $K \\in D(A)$ to the derived limit", "$K' = R\\lim( K \\otimes_A^\\mathbf{L} A/I^n )$ is the left", "adjoint to the inclusion functor $D_{comp}(A) \\to D(A)$", "constructed in Lemma \\ref{lemma-derived-completion}." ], "refs": [ "more-algebra-lemma-derived-completion" ], "proofs": [ { "contents": [ "Say $(f_1, \\ldots, f_r) = I$ and let $K_n^\\bullet$ be the Koszul complex", "with respect to $f_1^n, \\ldots, f_r^n$. By", "Lemma \\ref{lemma-derived-completion-koszul}", "it suffices to prove that", "$$", "R\\lim (K \\otimes_A^\\mathbf{L} K_n^\\bullet) =", "R\\lim (K \\otimes_A^\\mathbf{L} A/(f_1^n, \\ldots, f_r^n) ) =", "R\\lim (K \\otimes_A^\\mathbf{L} A/I^n ).", "$$", "By Lemma \\ref{lemma-sequence-Koszul-complexes} the pro-objects", "$\\{K_n^\\bullet\\}$ and $\\{A/(f_1^n, \\ldots, f_r^n)\\}$ of $D(A)$ are", "isomorphic. It is clear that the pro-objects", "$\\{A/(f_1^n, \\ldots, f_r^n)\\}$ and $\\{A/I^n\\}$ are isomorphic.", "Thus the map from left to right is an isomorphism by", "Lemma \\ref{lemma-tensor-Rlim-pro-equal}." ], "refs": [ "more-algebra-lemma-derived-completion-koszul", "more-algebra-lemma-sequence-Koszul-complexes", "more-algebra-lemma-tensor-Rlim-pro-equal" ], "ref_ids": [ 10378, 10391, 10332 ] } ], "ref_ids": [ 10372 ] }, { "id": 10591, "type": "theorem", "label": "more-algebra-proposition-ratliff", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-ratliff", "contents": [ "\\begin{reference}", "\\cite{Ratliff}", "\\end{reference}", "A Noetherian local ring is universally catenary if and only if", "it is formally catenary." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-not-formally-catenary} and", "\\ref{lemma-formally-catenary}." ], "refs": [ "more-algebra-lemma-not-formally-catenary", "more-algebra-lemma-formally-catenary" ], "ref_ids": [ 10480, 10482 ] } ], "ref_ids": [] }, { "id": 10592, "type": "theorem", "label": "more-algebra-proposition-epp-essentially-finite-type", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-epp-essentially-finite-type", "contents": [ "\\begin{reference}", "See \\cite[Lemma 2.13]{alterations} for a special case.", "\\end{reference}", "Let $A \\to B$ be an extension of discrete valuation rings with fraction", "fields $K \\subset L$. If $B$ is essentially of finite type over $A$, then", "there exists a finite extension $K \\subset K_1$ which is a solution for", "$A \\to B$ as defined in", "Definition \\ref{definition-solution}." ], "refs": [ "more-algebra-definition-solution" ], "proofs": [ { "contents": [ "Observe that a weak solution is a solution if the residue field of $A$", "is perfect, see Lemma \\ref{lemma-extension-dvrs-formally-smooth}.", "Thus the proposition follows immediately from Theorem \\ref{theorem-epp}", "if the residue characteristic of $A$ is $0$ (and in fact we do not need", "the assumption that $A \\to B$ is essentially of finite type).", "If the residue characteristic of $A$ is $p > 0$ we will also deduce it from", "Epp's theorem.", "\\medskip\\noindent", "Let $x_i \\in A$, $i \\in I$ be a set of elements mapping to a $p$-base of", "the residue field $\\kappa$ of $A$. Set", "$$", "A' = \\bigcup\\nolimits_{n \\geq 1} A[t_{i, n}]/(t_{i, n}^{p^n} - x_i)", "$$", "where the transition maps send $t_{i, n + 1}$ to $t_{i, n}^p$. Observe", "that $A'$ is a filtered colimit of weakly unramified finite extensions", "of discrete valuation rings over $A$. Thus $A'$ is a discrete valuation", "ring and $A \\to A'$ is weakly unramified. By construction the residue field", "$\\kappa' = A'/\\mathfrak m_A A'$ is the perfection of $\\kappa$.", "\\medskip\\noindent", "Let $K'$ be the fraction field of $A'$.", "We may apply Lemma \\ref{lemma-big-extension-is-ok}", "to the extension $K \\subset K'$. Thus $B'$ is a finite product of", "Dedekind domains. Let $\\mathfrak m_1, \\ldots, \\mathfrak m_n$ be the", "maximal ideals of $B'$. Using Epp's theorem (Theorem \\ref{theorem-epp})", "we find a weak solution $K' \\subset K'_i$ for each of the", "extensions $A' \\subset B'_{\\mathfrak m_i}$. Since the residue field", "of $A'$ is perfect, these are actually solutions. Let $K' \\subset K'_1$", "be a finite extension which contains each $K'_i$. Then $K' \\subset K'_1$", "is still a solution for each $A' \\subset B'_{\\mathfrak m_i}$ by", "Lemma \\ref{lemma-formally-smooth-goes-up}.", "\\medskip\\noindent", "Let $A'_1$ be the integral closure of $A$ in $K'_1$. Note that", "$A'_1$ is a Dedekind domain by the discussion in", "Remark \\ref{remark-construction} applied to $K' \\subset K'_1$.", "Thus Lemma \\ref{lemma-big-extension-is-ok} applies to $K \\subset K'_1$.", "Therefore the integral closure $B'_1$ of $B$ in", "$L'_1 = (L \\otimes_K K'_1)_{red}$ is a Dedekind domain and because", "$K' \\subset K'_1$ is a solution for each $A' \\subset B'_{\\mathfrak m_i}$", "we see that $(A'_1)_{A'_1 \\cap \\mathfrak m} \\to (B'_1)_{\\mathfrak m}$", "is formally smooth in the $\\mathfrak m$-adic topology", "for each maximal ideal $\\mathfrak m \\subset B'_1$.", "\\medskip\\noindent", "By construction, the field $K'_1$ is a filtered colimit of finite", "extensions of $K$. Say $K'_1 = \\colim_{i \\in I} K_i$. For each $i$ let", "$A_i$, resp.\\ $B_i$ be the integral closure of", "$A$, resp.\\ $B$ in $K_i$, resp.\\ $L_i = (L \\otimes_K K_i)_{red}$.", "Then it is clear that", "$$", "A'_1 = \\colim A_i\\quad\\text{and}\\quad B'_1 = \\colim B_i", "$$", "Since the ring maps $A_i \\to A'_1$ and $B_i \\to B'_1$ are injective", "integral ring maps and since $A'_1$ and $B'_1$ have finite spectra,", "we see that for all $i$ large enough the ring maps", "$A_i \\to A'_1$ and $B_i \\to B'_1$ are bijective on spectra.", "Once this is true, for all $i$ large enough the maps", "$A_i \\to A'_1$ and $B_i \\to B'_1$ will be weakly unramified", "(once the uniformizer is in the image). It follows from multiplicativity", "of ramification indices that $A_i \\to B_i$ induces weakly unramified maps", "on all localizations at maximal ideals of $B_i$ for such $i$.", "Increasing $i$ a bit more we see that", "$$", "B_i \\otimes_{A_i} A'_1 \\longrightarrow B'_1", "$$", "induces surjective maps on residue fields (because the residue fields", "of $B'_1$ are finitely generated over those of $A'_1$ by", "Lemma \\ref{lemma-big-extension-is-ok}). Picture of residue", "fields at maximal ideals lying under a chosen maximal ideal", "of $B'_1$:", "$$", "\\xymatrix{", "\\kappa_{B_i} \\ar[r] &", "\\kappa_{B_{i'}} \\ar[r] &", " & \\ldots &", "\\kappa_{B'_1} \\\\", "\\kappa_{A_i} \\ar[r] \\ar[u] &", "\\kappa_{A_{i'}} \\ar[r] \\ar[u] &", " & \\ldots &", "\\kappa_{A'_1} \\ar[u]", "}", "$$", "Thus $\\kappa_{B_i}$ is a finitely generated extension of", "$\\kappa_{A_i}$ such that the compositum of $\\kappa_{B_i}$", "and $\\kappa_{A'_1}$ in $\\kappa_{B'_1}$ is separable over", "$\\kappa_{A'_1}$. Then that happens already at a finite stage:", "for example, say $\\kappa_{B'_1}$ is finite separable over", "$\\kappa_{A'_1}(x_1, \\ldots, x_n)$, then just increase $i$", "such that $x_1, \\ldots, x_n$ are in $\\kappa_{B_i}$ and such that", "all generators satisfy separable polynomial equations over", "$\\kappa_{A_i}(x_1, \\ldots, x_n)$. This means that", "$A_i \\to (B_i)_\\mathfrak m$ is formally smooth in the $\\mathfrak m$-adic", "topology for all maximal ideals $\\mathfrak m$ of", "$B_i$ and the proof is complete." ], "refs": [ "more-algebra-lemma-extension-dvrs-formally-smooth", "more-algebra-theorem-epp", "more-algebra-lemma-big-extension-is-ok", "more-algebra-theorem-epp", "more-algebra-lemma-formally-smooth-goes-up", "more-algebra-remark-construction", "more-algebra-lemma-big-extension-is-ok", "more-algebra-lemma-big-extension-is-ok" ], "ref_ids": [ 10495, 9806, 10531, 9806, 10508, 10679, 10531, 10531 ] } ], "ref_ids": [ 10645 ] }, { "id": 10593, "type": "theorem", "label": "more-algebra-proposition-propdimd", "categories": [ "more-algebra" ], "title": "more-algebra-proposition-propdimd", "contents": [ "\\begin{reference}", "\\cite[Lemma 3.14]{Artin-Lipman} has this result without the", "assumption that the ring is catenary", "\\end{reference}", "Let $R$ be a catenary Noetherian local normal domain.", "Let $J \\subset R$ be a radical ideal.", "Then there exists a nonzero element $f \\in J$", "such that $R/fR$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "The proof is the same as that of", "Lemma \\ref{lemma-radical-element},", "using Lemma \\ref{lemma-minprimespolyhigher} instead of", "Lemma \\ref{lemma-minprimespoly} and", "Lemma \\ref{lemma-sysparhigher} instead of", "Lemma \\ref{lemma-syspar}.", "We can use Lemma \\ref{lemma-minprimespolyhigher} because $R$", "is a catenary domain, so every height one prime ideal of $R$", "has dimension $d - 1$, and hence the spectrum of $R/(f + h)$ is", "equidimensional. For the convenience of the reader we write out", "the details.", "\\medskip\\noindent", "Let $f \\in J$ be a nonzero element.", "We will modify $f$ slightly to obtain an element that generates a radical ideal.", "The localization $R_\\mathfrak p$ of $R$ at each height one prime", "ideal $\\mathfrak p$ is a discrete valuation ring, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-dvr} or", "Algebra, Lemma \\ref{algebra-lemma-criterion-normal}.", "We denote by $\\text{ord}_\\mathfrak p(f)$ the corresponding", "valuation of $f$ in $R_{\\mathfrak p}$. Let", "$\\mathfrak q_1, \\ldots, \\mathfrak q_s$", "be the distinct height one prime ideals containing $f$.", "Write $\\text{ord}_{\\mathfrak q_j}(f) = m_j \\geq 1$", "for each $j$. Then we define", "$\\text{div}(f) = \\sum_{j = 1}^s m_j\\mathfrak q_j$", "as a formal linear combination of", "height one primes with integer coefficients.", "The ring $R/fR$ is reduced if and only if", "$m_j = 1$ for $j = 1, \\ldots, s$. Namely, if $m_j$ is $1$ then", "$(R/fR)\\mathfrak q_j$ is reduced and", "$R/fR \\subset \\prod (R/fR)_{\\mathfrak q_j}$ as", "$\\mathfrak q_1, \\ldots, \\mathfrak q_j$ are the associated primes", "of $R/fR$, see Algebra, Lemmas", "\\ref{algebra-lemma-zero-at-ass-zero} and", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}.", "\\medskip\\noindent", "Choose and fix $g_2, \\ldots, g_{d - 1}$ and $N$ as in", "Lemma \\ref{lemma-sysparhigher}.", "For a nonzero $y \\in R$ denote $t(y)$ the number of primes minimal over $y$.", "Since $R$ is a normal domain, these primes", "are height one and correspond $1$-to-$1$ to the minimal primes of", "$R/yR$ (Algebra, Lemmas \\ref{algebra-lemma-minimal-over-1} and", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}).", "For example $t(f) = s$ is the number", "of primes $\\mathfrak q_j$ occurring in $\\text{div}(f)$.", "Let $h \\in \\mathfrak m^N$. Because $R$ is catenary, for each", "height one prime $\\mathfrak p$ of $R$ we have", "$\\dim(R/\\mathfrak p) = d$. Hence by", "Lemma \\ref{lemma-minprimespolyhigher}", "we have", "\\begin{align*}", "t(f + h) & \\leq \\text{length}_{R/(f + h)}(R/(f + h, g_1, \\ldots, g_{d - 1})) \\\\", "& = \\text{length}_R(R/(f + h, g_1, \\ldots, g_{d - 1})) \\\\", "& = \\text{length}_R(R/(f, g_1, \\ldots, g_{d - 1}))", "\\end{align*}", "see Algebra, Lemma \\ref{algebra-lemma-length-independent}", "for the first equality.", "Therefore we see that $t(f + h)$ is bounded independent of", "$h \\in \\mathfrak m^N$.", "\\medskip\\noindent", "By the boundedness proved above we may pick $h \\in \\mathfrak m^N \\cap J$", "such that $t(f + h)$ is maximal among such $h$. Set $f' = f + h$.", "Given $h' \\in \\mathfrak m^N \\cap J$", "we see that the number $t(f' + h') \\leq t(f + h)$.", "Thus after replacing $f$ by $f'$ we may assume that for every", "$h \\in \\mathfrak m^N \\cap J$ we have $t(f + h) \\leq s$.", "\\medskip\\noindent", "Next, assume that we can find an element $h \\in \\mathfrak m^N \\cap J$", "such that for each $j$ we have $\\text{ord}_{\\mathfrak q_j}(h) \\geq 1$ and", "$\\text{ord}_{\\mathfrak q_j}(h) = 1 \\Leftrightarrow m_j > 1$.", "Then $\\text{ord}_{\\mathfrak q_j}(f + h) = 1$", "for every $j$ by elementary properties of valuations.", "Thus", "$$", "\\text{div}(f + h) = \\sum\\nolimits_{j = 1}^s \\mathfrak q_j +", "\\sum\\nolimits_{k = 1}^v e_k \\mathfrak r_k", "$$", "for some pairwise distinct height one prime ideals", "$\\mathfrak r_1, \\ldots, \\mathfrak r_v$ and $e_k \\geq 1$.", "However, since $s = t(f) \\geq t(f + h)$ we see that $v = 0$", "and we have found the desired element.", "\\medskip\\noindent", "Now we will pick $h$ that satisfies the above criteria.", "By prime avoidance (Algebra, Lemma \\ref{algebra-lemma-silly})", "for each $1 \\leq j \\leq s$ we can find an element", "$a_j \\in \\mathfrak q_j \\cap J$", "such that $a_j \\not \\in \\mathfrak q_{j'}$ for $j' \\not = j$.", "Next, we can pick $b_j \\in J \\cap \\mathfrak q_1 \\cap \\ldots \\cap q_s$", "with $b_j \\not \\in \\mathfrak q_j^{(2)}$. Here", "$\\mathfrak q_j^{(2)} = \\{x \\in R \\mid \\text{ord}_{\\mathfrak q_j}(x) \\geq 2\\}$", "is the second symbolic power of $\\mathfrak q_j$.", "Prime avoidance applies because the ideal", "$J' = J \\cap \\mathfrak q_1 \\cap \\ldots \\cap q_s$", "is radical, hence $R/J'$ is reduced, hence $(R/J')_{\\mathfrak q_j}$", "is reduced, hence $J'$ contains an element $x$ with", "$\\text{ord}_{\\mathfrak q_j}(x) = 1$, hence", "$J' \\not \\subset \\mathfrak q_j^{(2)}$. Then the element", "$$", "c = \\sum\\nolimits_{j = 1, \\ldots, s}", "b_j \\times \\prod\\nolimits_{j' \\not = j} a_{j'}", "$$", "is an element of $J$", "with $\\text{ord}_{\\mathfrak q_j}(c) = 1$ for all $j = 1, \\ldots, s$", "by elementary properties of valuations. Finally, we let", "$$", "h = c \\times \\prod\\nolimits_{m_j = 1} a_j \\times y", "$$", "where $y \\in \\mathfrak m^N$ is an element", "which is not contained in $\\mathfrak q_j$ for all $j$." ], "refs": [ "more-algebra-lemma-radical-element", "more-algebra-lemma-minprimespolyhigher", "more-algebra-lemma-minprimespoly", "more-algebra-lemma-sysparhigher", "more-algebra-lemma-syspar", "more-algebra-lemma-minprimespolyhigher", "algebra-lemma-characterize-dvr", "algebra-lemma-criterion-normal", "algebra-lemma-zero-at-ass-zero", "algebra-lemma-normal-domain-intersection-localizations-height-1", "more-algebra-lemma-sysparhigher", "algebra-lemma-minimal-over-1", "algebra-lemma-normal-domain-intersection-localizations-height-1", "more-algebra-lemma-minprimespolyhigher", "algebra-lemma-length-independent", "algebra-lemma-silly" ], "ref_ids": [ 10567, 10570, 10564, 10571, 10566, 10570, 1023, 1311, 713, 1313, 10571, 683, 1313, 10570, 633, 378 ] } ], "ref_ids": [] }, { "id": 10681, "type": "theorem", "label": "etale-theorem-unramified-equivalence", "categories": [ "etale" ], "title": "etale-theorem-unramified-equivalence", "contents": [ "Let $Y$ be a locally Noetherian scheme.", "Let $f : X \\to Y$ be a morphism of schemes which is locally of finite type.", "Let $x$ be a point of $X$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is unramified at $x$,", "\\item the stalk $\\Omega_{X/Y, x}$ of the module of relative differentials", "at $x$ is trivial,", "\\item there exist open neighbourhoods $U$ of $x$ and $V$ of $f(x)$, and a", "commutative diagram", "$$", "\\xymatrix{", "U \\ar[rr]_i \\ar[rd] & & \\mathbf{A}^n_V \\ar[ld] \\\\", "& V", "}", "$$", "where $i$ is a closed immersion defined by a", "quasi-coherent sheaf of ideals $\\mathcal{I}$ such that the differentials", "$\\text{d}g$ for $g \\in \\mathcal{I}_{i(x)}$ generate", "$\\Omega_{\\mathbf{A}^n_V/V, i(x)}$, and", "\\item the diagonal $\\Delta_{X/Y} : X \\to X \\times_Y X$", "is a local isomorphism at $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is proved in", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-at-point}.", "\\medskip\\noindent", "If $f$ is unramified at $x$, then $f$ is unramified in an open", "neighbourhood of $x$; this does not follow immediately", "from Definition \\ref{definition-unramified-schemes} of this chapter", "but it does follow from", "Morphisms, Definition \\ref{morphisms-definition-unramified} which we", "proved to be equivalent in", "Lemma \\ref{lemma-unramified-definition}.", "Choose affine opens $V \\subset Y$, $U \\subset X$", "with $f(U) \\subset V$ and $x \\in U$, such that $f$ is", "unramified on $U$, i.e., $f|_U : U \\to V$ is unramified.", "By Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism}", "the morphism $U \\to U \\times_V U$", "is an open immersion. This proves that (1) implies (4).", "\\medskip\\noindent", "If $\\Delta_{X/Y}$ is a local isomorphism at $x$, then", "$\\Omega_{X/Y, x} = 0$ by", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-diagonal}.", "Hence we see that (4) implies (2).", "At this point we know that (1), (2) and (4) are all equivalent.", "\\medskip\\noindent", "Assume (3). The assumption on the diagram combined with", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-relative-immersion}", "show that $\\Omega_{U/V, x} = 0$. Since $\\Omega_{U/V, x} = \\Omega_{X/Y, x}$", "we conclude (2) holds.", "\\medskip\\noindent", "Finally, assume that (2) holds. To prove (3) we may localize on", "$X$ and $Y$ and assume that $X$ and $Y$ are affine.", "Say $X = \\Spec(B)$ and $Y = \\Spec(A)$.", "The point $x \\in X$ corresponds to a prime $\\mathfrak q \\subset B$.", "Our assumption is that $\\Omega_{B/A, \\mathfrak q} = 0$", "(see Morphisms, Lemma \\ref{morphisms-lemma-differentials-affine} for the", "relationship between differentials on schemes and modules", "of differentials in commutative algebra).", "Since $Y$ is locally Noetherian and $f$ locally of finite type", "we see that $A$ is Noetherian and", "$B \\cong A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$, see", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian} and", "Morphisms, Lemma \\ref{morphisms-lemma-locally-finite-type-characterize}.", "In particular, $\\Omega_{B/A}$ is a finite $B$-module. Hence we", "can find a single $g \\in B$, $g \\not \\in \\mathfrak q$ such that", "the principal localization $(\\Omega_{B/A})_g$ is zero. Hence after", "replacing $B$ by $B_g$ we see that $\\Omega_{B/A} = 0$ (formation", "of modules of differentials commutes with localization, see", "Algebra, Lemma \\ref{algebra-lemma-differentials-localize}). This means that", "$\\text{d}(f_j)$ generate the kernel of the canonical map", "$\\Omega_{A[x_1, \\ldots, x_n]/A} \\otimes_A B \\to \\Omega_{B/A}$.", "Thus the surjection $A[x_1, \\ldots, x_n] \\to B$ of $A$-algebras gives the", "commutative diagram of (3), and the theorem is proved." ], "refs": [ "morphisms-lemma-unramified-at-point", "etale-definition-unramified-schemes", "morphisms-definition-unramified", "etale-lemma-unramified-definition", "morphisms-lemma-diagonal-unramified-morphism", "morphisms-lemma-differentials-diagonal", "morphisms-lemma-differentials-relative-immersion", "morphisms-lemma-differentials-affine", "properties-lemma-locally-Noetherian", "morphisms-lemma-locally-finite-type-characterize", "algebra-lemma-differentials-localize" ], "ref_ids": [ 5355, 10735, 5566, 10699, 5354, 5311, 5319, 5310, 2951, 5198, 1134 ] } ], "ref_ids": [] }, { "id": 10682, "type": "theorem", "label": "etale-theorem-formally-unramified", "categories": [ "etale" ], "title": "etale-theorem-formally-unramified", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume $S$ is a locally Noetherian scheme, and $f$ is locally of finite type.", "Then the following are equivalent:", "\\begin{enumerate}", "\\item $f$ is unramified,", "\\item the morphism $f$ is formally unramified:", "for any affine $S$-scheme $T$ and subscheme $T_0$ of $T$", "defined by a square-zero ideal,", "the natural map", "$$", "\\Hom_S(T, X) \\longrightarrow \\Hom_S(T_0, X)", "$$", "is injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-unramified-formally-unramified}", "for a more general statement and proof.", "What follows is a sketch of the proof in the current case.", "\\medskip\\noindent", "Firstly, one checks both properties are local on the source and the target.", "This we may assume that $S$ and $X$ are affine.", "Say $X = \\Spec(B)$ and $S = \\Spec(R)$.", "Say $T = \\Spec(C)$. Let $J$ be the square-zero ideal of $C$", "with $T_0 = \\Spec(C/J)$. Assume that we are given the diagram", "$$", "\\xymatrix{", "& B \\ar[d]^\\phi \\ar[rd]^{\\bar{\\phi}}", "& \\\\", "R \\ar[r] \\ar[ur] & C \\ar[r]", "& C/J", "}", "$$", "Secondly, one checks that the association $\\phi' \\mapsto \\phi' - \\phi$", "gives a bijection between the set of liftings of $\\bar{\\phi}$ and the module", "$\\text{Der}_R(B, J)$. Thus, we obtain the implication (1) $\\Rightarrow$ (2)", "via the description of unramified morphisms having trivial module", "of differentials, see Theorem \\ref{theorem-unramified-equivalence}.", "\\medskip\\noindent", "To obtain the reverse implication, consider the surjection", "$q : C = (B \\otimes_R B)/I^2 \\to B = C/J$ defined by the square zero ideal", "$J = I/I^2$ where $I$ is the kernel of the multiplication map", "$B \\otimes_R B \\to B$. We already have a lifting $B \\to C$ defined by, say,", "$b \\mapsto b \\otimes 1$. Thus, by the same reasoning as above, we obtain a", "bijective correspondence between liftings of $\\text{id} : B \\to C/J$ and", "$\\text{Der}_R(B, J)$. The hypothesis therefore implies that the latter module is", "trivial. But we know that $J \\cong \\Omega_{B/R}$. Thus, $B/R$ is unramified." ], "refs": [ "more-morphisms-lemma-unramified-formally-unramified", "etale-theorem-unramified-equivalence" ], "ref_ids": [ 13696, 10681 ] } ], "ref_ids": [] }, { "id": 10683, "type": "theorem", "label": "etale-theorem-sections-unramified-maps", "categories": [ "etale" ], "title": "etale-theorem-sections-unramified-maps", "contents": [ "Let $Y$ be a connected scheme.", "Let $f : X \\to Y$ be unramified and separated.", "Every section of $f$ is an isomorphism onto a connected component.", "There exists a bijective correspondence", "$$", "\\text{sections of }f", "\\leftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{connected components }X'\\text{ of }X\\text{ such that}\\\\", "\\text{the induced map }X' \\to Y\\text{ is an isomorphism}", "\\end{matrix}", "\\right\\}", "$$", "In particular, given $x \\in X$ there is at most one", "section passing through $x$." ], "refs": [], "proofs": [ { "contents": [ "Direct from Proposition \\ref{proposition-properties-sections} part (3)." ], "refs": [ "etale-proposition-properties-sections" ], "ref_ids": [ 10726 ] } ], "ref_ids": [] }, { "id": 10684, "type": "theorem", "label": "etale-theorem-flatness-grothendieck", "categories": [ "etale" ], "title": "etale-theorem-flatness-grothendieck", "contents": [ "Let $A$, $B$ be Noetherian local rings.", "Let $f : A \\to B$ be a local homomorphism.", "If $M$ is a finite $B$-module that is flat as an $A$-module,", "and $t \\in \\mathfrak m_B$ is an element such that multiplication", "by $t$ is injective on $M/\\mathfrak m_AM$, then $M/tM$ is also $A$-flat." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-mod-injective}.", "See also \\cite[Section 20]{MatCA}." ], "refs": [ "algebra-lemma-mod-injective" ], "ref_ids": [ 883 ] } ], "ref_ids": [] }, { "id": 10685, "type": "theorem", "label": "etale-theorem-flat-open", "categories": [ "etale" ], "title": "etale-theorem-flat-open", "contents": [ "Let $Y$ be a locally Noetherian scheme.", "Let $f : X \\to Y$ be a morphism which is locally of finite type.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "The set of points in $X$ where $\\mathcal{F}$ is flat over $Y$ is an open set.", "In particular the set of points where $f$ is flat is open in $X$." ], "refs": [], "proofs": [ { "contents": [ "See More on Morphisms, Theorem \\ref{more-morphisms-theorem-openness-flatness}." ], "refs": [ "more-morphisms-theorem-openness-flatness" ], "ref_ids": [ 13670 ] } ], "ref_ids": [] }, { "id": 10686, "type": "theorem", "label": "etale-theorem-flat-map-open", "categories": [ "etale" ], "title": "etale-theorem-flat-map-open", "contents": [ "Let $Y$ be a locally Noetherian scheme.", "Let $f : X \\to Y$ be a morphism which is flat and locally of finite type.", "Then $f$ is (universally) open." ], "refs": [], "proofs": [ { "contents": [ "See Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}." ], "refs": [ "morphisms-lemma-fppf-open" ], "ref_ids": [ 5267 ] } ], "ref_ids": [] }, { "id": 10687, "type": "theorem", "label": "etale-theorem-flat-is-quotient", "categories": [ "etale" ], "title": "etale-theorem-flat-is-quotient", "contents": [ "A faithfully flat quasi-compact morphism is a quotient map for", "the Zariski topology." ], "refs": [], "proofs": [ { "contents": [ "See Morphisms, Lemma \\ref{morphisms-lemma-fpqc-quotient-topology}." ], "refs": [ "morphisms-lemma-fpqc-quotient-topology" ], "ref_ids": [ 5269 ] } ], "ref_ids": [] }, { "id": 10688, "type": "theorem", "label": "etale-theorem-structure-etale", "categories": [ "etale" ], "title": "etale-theorem-structure-etale", "contents": [ "Let $f : A \\to B$ be an \\'etale homomorphism of local rings.", "Then there exist $f, g \\in A[t]$ such that", "\\begin{enumerate}", "\\item $B' = A[t]_g/(f)$ is standard \\'etale -- see (a) and (b) above, and", "\\item $B$ is isomorphic to a localization of $B'$ at a prime.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $B = B'_{\\mathfrak q}$ for some finite type $A$-algebra $B'$", "(we can do this because $B$ is essentially of finite type over $A$).", "By Lemma \\ref{lemma-characterize-etale-Noetherian}", "we see that $A \\to B'$ is \\'etale at $\\mathfrak q$.", "Hence we may apply", "Algebra, Proposition \\ref{algebra-proposition-etale-locally-standard}", "to see that a principal localization of $B'$ is standard \\'etale." ], "refs": [ "etale-lemma-characterize-etale-Noetherian", "algebra-proposition-etale-locally-standard" ], "ref_ids": [ 10704, 1427 ] } ], "ref_ids": [] }, { "id": 10689, "type": "theorem", "label": "etale-theorem-structure-unramified", "categories": [ "etale" ], "title": "etale-theorem-structure-unramified", "contents": [ "Let $f : A \\to B$ be an unramified morphism of local rings.", "Then there exist $f, g \\in A[t]$ such that", "\\begin{enumerate}", "\\item $B' = A[t]_g/(f)$ is standard \\'etale -- see (a) and (b) above, and", "\\item $B$ is isomorphic to a quotient of a localization of $B'$ at a prime.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $B = B'_{\\mathfrak q}$ for some finite type $A$-algebra $B'$", "(we can do this because $B$ is essentially of finite type over $A$).", "By Lemma \\ref{lemma-characterize-unramified-Noetherian}", "we see that $A \\to B'$ is unramified at $\\mathfrak q$.", "Hence we may apply", "Algebra, Proposition \\ref{algebra-proposition-unramified-locally-standard}", "to see that a principal localization of $B'$ is a quotient of a", "standard \\'etale $A$-algebra." ], "refs": [ "etale-lemma-characterize-unramified-Noetherian", "algebra-proposition-unramified-locally-standard" ], "ref_ids": [ 10696, 1428 ] } ], "ref_ids": [] }, { "id": 10690, "type": "theorem", "label": "etale-theorem-geometric-structure", "categories": [ "etale" ], "title": "etale-theorem-geometric-structure", "contents": [ "Let $\\varphi : X \\to Y$ be a morphism of schemes. Let $x \\in X$.", "Let $V \\subset Y$ be an affine open neighbourhood of $\\varphi(x)$.", "If $\\varphi$ is \\'etale at $x$, then there exist exists an affine open", "$U \\subset X$ with $x \\in U$ and $\\varphi(U) \\subset V$", "such that we have the following diagram", "$$", "\\xymatrix{", "X \\ar[d] & U \\ar[l] \\ar[d] \\ar[r]_-j & \\Spec(R[t]_{f'}/(f)) \\ar[d] \\\\", "Y & V \\ar[l] \\ar@{=}[r] & \\Spec(R)", "}", "$$", "where $j$ is an open immersion, and $f \\in R[t]$ is monic." ], "refs": [], "proofs": [ { "contents": [ "This is equivalent to", "Morphisms, Lemma \\ref{morphisms-lemma-etale-locally-standard-etale}", "although the statements differ slightly.", "See also, Varieties, Lemma \\ref{varieties-lemma-geometric-structure-unramified}", "for a variant for unramified morphisms." ], "refs": [ "morphisms-lemma-etale-locally-standard-etale", "varieties-lemma-geometric-structure-unramified" ], "ref_ids": [ 5371, 10982 ] } ], "ref_ids": [] }, { "id": 10691, "type": "theorem", "label": "etale-theorem-smooth-etale-over-n-space", "categories": [ "etale" ], "title": "etale-theorem-smooth-etale-over-n-space", "contents": [ "Let $\\varphi : X \\to Y$ be a morphism of schemes.", "Let $x \\in X$.", "If $\\varphi$ is smooth at $x$, then", "there exist an integer $n \\geq 0$ and affine opens", "$V \\subset Y$ and $U \\subset X$ with $x \\in U$ and $\\varphi(U) \\subset V$", "such that there exists a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] & U \\ar[l] \\ar[d] \\ar[r]_-\\pi &", "\\mathbf{A}^n_R \\ar[d] \\ar@{=}[r] & \\Spec(R[x_1, \\ldots, x_n]) \\ar[dl] \\\\", "Y & V \\ar[l] \\ar@{=}[r] & \\Spec(R)", "}", "$$", "where $\\pi$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "See", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-etale-over-affine-space}." ], "refs": [ "morphisms-lemma-smooth-etale-over-affine-space" ], "ref_ids": [ 5377 ] } ], "ref_ids": [] }, { "id": 10692, "type": "theorem", "label": "etale-theorem-etale-radicial-open", "categories": [ "etale" ], "title": "etale-theorem-etale-radicial-open", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is an open immersion,", "\\item $f$ is universally injective and \\'etale, and", "\\item $f$ is a flat monomorphism, locally of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "An open immersion is universally injective", "since any base change of an open immersion", "is an open immersion. Moreover, it is \\'etale by", "Morphisms, Lemma \\ref{morphisms-lemma-open-immersion-etale}.", "Hence (1) implies (2).", "\\medskip\\noindent", "Assume $f$ is universally injective and \\'etale.", "Since $f$ is \\'etale it is flat and locally of finite presentation, see", "Morphisms, Lemmas \\ref{morphisms-lemma-etale-flat} and", "\\ref{morphisms-lemma-etale-locally-finite-presentation}.", "By", "Lemma \\ref{lemma-universally-injective-unramified}", "we see that $f$ is a monomorphism. Hence (2) implies (3).", "\\medskip\\noindent", "Assume $f$ is flat, locally of finite presentation, and a monomorphism.", "Then $f$ is open, see", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}.", "Thus we may replace $Y$ by $f(X)$ and we may assume $f$ is", "surjective. Then $f$ is open and bijective hence a homeomorphism.", "Hence $f$ is quasi-compact. Hence", "Descent, Lemma", "\\ref{descent-lemma-flat-surjective-quasi-compact-monomorphism-isomorphism}", "shows that $f$ is an isomorphism and we win." ], "refs": [ "morphisms-lemma-open-immersion-etale", "morphisms-lemma-etale-flat", "morphisms-lemma-etale-locally-finite-presentation", "etale-lemma-universally-injective-unramified", "morphisms-lemma-fppf-open", "descent-lemma-flat-surjective-quasi-compact-monomorphism-isomorphism" ], "ref_ids": [ 5366, 5369, 5368, 10701, 5267, 14699 ] } ], "ref_ids": [] }, { "id": 10693, "type": "theorem", "label": "etale-theorem-etale-topological", "categories": [ "etale" ], "title": "etale-theorem-etale-topological", "contents": [ "Let $X$ and $Y$ be two schemes over a base scheme $S$. Let $S_0$ be a closed", "subscheme of $S$ with the same underlying topological space", "(for example if the ideal sheaf of $S_0$ in $S$ has square zero).", "Denote $X_0$ (resp.\\ $Y_0$) the base change $S_0 \\times_S X$", "(resp.\\ $S_0 \\times_S Y$).", "If $X$ is \\'etale over $S$, then the map", "$$", "\\Mor_S(Y, X) \\longrightarrow \\Mor_{S_0}(Y_0, X_0)", "$$", "is bijective." ], "refs": [], "proofs": [ { "contents": [ "After base changing via $Y \\to S$, we may assume that $Y = S$.", "In this case the theorem states that any $S$-morphism $\\sigma_0 : S_0 \\to X$", "actually factors uniquely through a section $S \\to X$ of the", "\\'etale structure morphism $f : X \\to S$.", "\\medskip\\noindent", "Uniqueness. Suppose we have two sections $\\sigma, \\sigma'$", "through which $\\sigma_0$ factors. Because $X \\to S$ is \\'etale", "we see that $\\Delta : X \\to X \\times_S X$ is an open immersion", "(Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism}).", "The morphism $(\\sigma, \\sigma') : S \\to X \\times_S X$ factors", "through this open because for any $s \\in S$ we have", "$(\\sigma, \\sigma')(s) = (\\sigma_0(s), \\sigma_0(s))$. Thus", "$\\sigma = \\sigma'$.", "\\medskip\\noindent", "To prove existence we first reduce to the affine case", "(we suggest the reader skip this step).", "Let $X = \\bigcup X_i$ be an affine open covering such", "that each $X_i$ maps into an affine open $S_i$ of $S$.", "For every $s \\in S$ we can choose an $i$ such that", "$\\sigma_0(s) \\in X_i$.", "Choose an affine open neighbourhood $U \\subset S_i$ of $s$", "such that $\\sigma_0(U_0) \\subset X_{i, 0}$. Note that", "$X' = X_i \\times_S U = X_i \\times_{S_i} U$ is affine.", "If we can lift $\\sigma_0|_{U_0} : U_0 \\to X'_0$ to", "$U \\to X'$, then by uniqueness these local lifts will glue", "to a global morphism $S \\to X$. Thus we may assume $S$ and", "$X$ are affine.", "\\medskip\\noindent", "Existence when $S$ and $X$ are affine. Write $S = \\Spec(A)$", "and $X = \\Spec(B)$. Then $A \\to B$ is \\'etale and in particular", "smooth (of relative dimension $0$). As $|S_0| = |S|$ we see", "that $S_0 = \\Spec(A/I)$ with $I \\subset A$ locally nilpotent.", "Thus existence follows from", "Algebra, Lemma \\ref{algebra-lemma-smooth-strong-lift}." ], "refs": [ "morphisms-lemma-diagonal-unramified-morphism", "algebra-lemma-smooth-strong-lift" ], "ref_ids": [ 5354, 1216 ] } ], "ref_ids": [] }, { "id": 10694, "type": "theorem", "label": "etale-theorem-remarkable-equivalence", "categories": [ "etale" ], "title": "etale-theorem-remarkable-equivalence", "contents": [ "\\begin{reference}", "\\cite[IV, Theorem 18.1.2]{EGA}", "\\end{reference}", "Let $S$ be a scheme.", "Let $S_0 \\subset S$ be a closed subscheme with the same underlying", "topological space (for example if the ideal sheaf of $S_0$ in $S$", "has square zero). The functor", "$$", "X \\longmapsto X_0 = S_0 \\times_S X", "$$", "defines an equivalence of categories", "$$", "\\{", "\\text{schemes }X\\text{ \\'etale over }S", "\\}", "\\leftrightarrow", "\\{", "\\text{schemes }X_0\\text{ \\'etale over }S_0", "\\}", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Theorem \\ref{theorem-etale-topological}", "we see that this functor is fully faithful.", "It remains to show that the functor is essentially surjective.", "Let $Y \\to S_0$ be an \\'etale morphism of schemes.", "\\medskip\\noindent", "Suppose that the result holds if $S$ and $Y$ are affine.", "In that case, we choose an affine open covering", "$Y = \\bigcup V_j$ such that each $V_j$ maps", "into an affine open of $S$. By assumption (affine case) we can", "find \\'etale morphisms $W_j \\to S$ such that $W_{j, 0} \\cong V_j$", "(as schemes over $S_0$). Let $W_{j, j'} \\subset W_j$", "be the open subscheme whose underlying topological space", "corresponds to $V_j \\cap V_{j'}$. Because we have isomorphisms", "$$", "W_{j, j', 0} \\cong V_j \\cap V_{j'} \\cong W_{j', j, 0}", "$$", "as schemes over $S_0$ we see by fully faithfulness that we", "obtain isomorphisms", "$\\theta_{j, j'} : W_{j, j'} \\to W_{j', j}$ of schemes over $S$.", "We omit the verification that these isomorphisms satisfy the", "cocycle condition of Schemes, Section \\ref{schemes-section-glueing-schemes}.", "Applying Schemes, Lemma \\ref{schemes-lemma-glue-schemes}", "we obtain a scheme $X \\to S$ by", "glueing the schemes $W_j$ along the identifications $\\theta_{j, j'}$.", "It is clear that $X \\to S$ is \\'etale and $X_0 \\cong Y$ by construction.", "\\medskip\\noindent", "Thus it suffices to show the lemma in case $S$ and $Y$ are affine.", "Say $S = \\Spec(R)$ and $S_0 = \\Spec(R/I)$ with $I$ locally nilpotent.", "By Algebra, Lemma \\ref{algebra-lemma-etale-standard-smooth} we know that", "$Y$ is the spectrum of a ring $\\overline{A}$ with", "$$", "\\overline{A} = (R/I)[x_1, \\ldots, x_n]/(\\overline{f}_1, \\ldots, \\overline{f}_n)", "$$", "such that", "$$", "\\overline{g} =", "\\det", "\\left(", "\\begin{matrix}", "\\partial \\overline{f}_1/\\partial x_1 &", "\\partial \\overline{f}_2/\\partial x_1 &", "\\ldots &", "\\partial \\overline{f}_n/\\partial x_1 \\\\", "\\partial \\overline{f}_1/\\partial x_2 &", "\\partial \\overline{f}_2/\\partial x_2 &", "\\ldots &", "\\partial \\overline{f}_n/\\partial x_2 \\\\", "\\ldots & \\ldots & \\ldots & \\ldots \\\\", "\\partial \\overline{f}_1/\\partial x_n &", "\\partial \\overline{f}_2/\\partial x_n &", "\\ldots &", "\\partial \\overline{f}_n/\\partial x_n", "\\end{matrix}", "\\right)", "$$", "maps to an invertible element in $\\overline{A}$. Choose any lifts", "$f_i \\in R[x_1, \\ldots, x_n]$. Set", "$$", "A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)", "$$", "Since $I$ is locally nilpotent the ideal $IA$ is locally nilpotent", "(Algebra, Lemma \\ref{algebra-lemma-locally-nilpotent}).", "Observe that $\\overline{A} = A/IA$.", "It follows that the determinant of the matrix of partials of the", "$f_i$ is invertible in the algebra $A$ by", "Algebra, Lemma \\ref{algebra-lemma-locally-nilpotent-unit}.", "Hence $R \\to A$ is \\'etale and the proof is complete." ], "refs": [ "etale-theorem-etale-topological", "schemes-lemma-glue-schemes", "algebra-lemma-etale-standard-smooth", "algebra-lemma-locally-nilpotent", "algebra-lemma-locally-nilpotent-unit" ], "ref_ids": [ 10693, 7687, 1230, 458, 459 ] } ], "ref_ids": [] }, { "id": 10695, "type": "theorem", "label": "etale-theorem-formally-etale", "categories": [ "etale" ], "title": "etale-theorem-formally-etale", "contents": [ "Let $f : X \\to S$ be a morphism that is locally of finite presentation.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is \\'etale,", "\\item for all affine $S$-schemes $Y$, and closed subschemes $Y_0 \\subset Y$", "defined by square-zero ideals, the natural map", "$$", "\\Mor_S(Y, X) \\longrightarrow \\Mor_S(Y_0, X)", "$$", "is bijective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-etale-formally-etale}." ], "refs": [ "more-morphisms-lemma-etale-formally-etale" ], "ref_ids": [ 13715 ] } ], "ref_ids": [] }, { "id": 10696, "type": "theorem", "label": "etale-lemma-characterize-unramified-Noetherian", "categories": [ "etale" ], "title": "etale-lemma-characterize-unramified-Noetherian", "contents": [ "\\begin{slogan}", "Unramifiedness is a stalk local condition.", "\\end{slogan}", "Let $A \\to B$ be of finite type with $A$ a Noetherian ring.", "Let $\\mathfrak q$ be a prime of $B$ lying over $\\mathfrak p \\subset A$.", "Then $A \\to B$ is unramified at $\\mathfrak q$ if and only if", "$A_{\\mathfrak p} \\to B_{\\mathfrak q}$ is an unramified homomorphism", "of local rings." ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10697, "type": "theorem", "label": "etale-lemma-unramified-completions", "categories": [ "etale" ], "title": "etale-lemma-unramified-completions", "contents": [ "Let $A$, $B$ be Noetherian local rings.", "Let $A \\to B$ be a local homomorphism.", "\\begin{enumerate}", "\\item if $A \\to B$ is an unramified homomorphism of local rings,", "then $B^\\wedge$ is a finite $A^\\wedge$ module,", "\\item if $A \\to B$ is an unramified homomorphism of local rings and", "$\\kappa(\\mathfrak m_A) = \\kappa(\\mathfrak m_B)$,", "then $A^\\wedge \\to B^\\wedge$ is surjective,", "\\item if $A \\to B$ is an unramified homomorphism of local rings", "and $\\kappa(\\mathfrak m_A)$", "is separably closed, then $A^\\wedge \\to B^\\wedge$ is surjective,", "\\item if $A$ and $B$ are complete discrete valuation rings, then", "$A \\to B$ is an unramified homomorphism of local rings", "if and only if the uniformizer for $A$ maps to a uniformizer for $B$,", "and the residue field extension is finite separable (and $B$ is", "essentially of finite type over $A$).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is a special case of", "Algebra, Lemma \\ref{algebra-lemma-finite-after-completion}.", "For part (2), note that the $\\kappa(\\mathfrak m_A)$-vector space", "$B^\\wedge/\\mathfrak m_{A^\\wedge}B^\\wedge$", "is generated by $1$. Hence by Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK}) the map", "$A^\\wedge \\to B^\\wedge$ is surjective.", "Part (3) is a special case of part (2).", "Part (4) is immediate from the definitions." ], "refs": [ "algebra-lemma-finite-after-completion", "algebra-lemma-NAK" ], "ref_ids": [ 875, 401 ] } ], "ref_ids": [] }, { "id": 10698, "type": "theorem", "label": "etale-lemma-characterize-unramified-completions", "categories": [ "etale" ], "title": "etale-lemma-characterize-unramified-completions", "contents": [ "Let $A$, $B$ be Noetherian local rings.", "Let $A \\to B$ be a local homomorphism such that $B$ is", "essentially of finite type over $A$.", "The following are equivalent", "\\begin{enumerate}", "\\item $A \\to B$ is an unramified homomorphism of local rings", "\\item $A^\\wedge \\to B^\\wedge$ is an unramified homomorphism of local rings, and", "\\item $A^\\wedge \\to B^\\wedge$ is unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows from the fact that", "$\\mathfrak m_AA^\\wedge$ is the maximal ideal of $A^\\wedge$", "(and similarly for $B$) and faithful flatness of $B \\to B^\\wedge$.", "For example if $A^\\wedge \\to B^\\wedge$ is unramified, then", "$\\mathfrak m_AB^\\wedge = (\\mathfrak m_AB)B^\\wedge = \\mathfrak m_BB^\\wedge$", "and hence $\\mathfrak m_AB = \\mathfrak m_B$.", "\\medskip\\noindent", "Assume the equivalent conditions (1) and (2).", "By Lemma \\ref{lemma-unramified-completions}", "we see that $A^\\wedge \\to B^\\wedge$ is", "finite. Hence $A^\\wedge \\to B^\\wedge$ is of finite presentation, and by", "Algebra, Lemma \\ref{algebra-lemma-characterize-unramified}", "we conclude that $A^\\wedge \\to B^\\wedge$ is unramified at", "$\\mathfrak m_{B^\\wedge}$. Since $B^\\wedge$ is local we conclude", "that $A^\\wedge \\to B^\\wedge$ is unramified.", "\\medskip\\noindent", "Assume (3). By Algebra, Lemma \\ref{algebra-lemma-unramified-at-prime}", "we conclude that $A^\\wedge \\to B^\\wedge$ is an unramified homomorphism", "of local rings, i.e., (2) holds." ], "refs": [ "etale-lemma-unramified-completions", "algebra-lemma-characterize-unramified", "algebra-lemma-unramified-at-prime" ], "ref_ids": [ 10697, 1270, 1268 ] } ], "ref_ids": [] }, { "id": 10699, "type": "theorem", "label": "etale-lemma-unramified-definition", "categories": [ "etale" ], "title": "etale-lemma-unramified-definition", "contents": [ "Let $Y$ be a locally Noetherian scheme.", "Let $f : X \\to Y$ be locally of finite type.", "Let $x \\in X$. The morphism $f$ is unramified at $x$ in", "the sense of Definition \\ref{definition-unramified-schemes}", "if and only if it is unramified in", "the sense of Morphisms, Definition \\ref{morphisms-definition-unramified}." ], "refs": [ "etale-definition-unramified-schemes", "morphisms-definition-unramified" ], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-characterize-unramified-Noetherian}", "and the definitions." ], "refs": [ "etale-lemma-characterize-unramified-Noetherian" ], "ref_ids": [ 10696 ] } ], "ref_ids": [ 10735, 5566 ] }, { "id": 10700, "type": "theorem", "label": "etale-lemma-finitely-many-maps-to-unramified", "categories": [ "etale" ], "title": "etale-lemma-finitely-many-maps-to-unramified", "contents": [ "Let $S$ be a Noetherian scheme. Let $X \\to S$ be a quasi-compact unramified", "morphism. Let $Y \\to S$ be a morphism with $Y$ Noetherian. Then", "$\\Mor_S(Y, X)$ is a finite set." ], "refs": [], "proofs": [ { "contents": [ "Assume first $X \\to S$ is separated (which is often the case in practice).", "Since $Y$ is Noetherian it has finitely many connected components. Thus we", "may assume $Y$ is connected. Choose a point $y \\in Y$ with image $s \\in S$.", "Since $X \\to S$ is unramified and quasi-compact", "then fibre $X_s$ is finite, say $X_s = \\{x_1, \\ldots, x_n\\}$", "and $\\kappa(s) \\subset \\kappa(x_i)$ is a finite field extension.", "See Morphisms, Lemma \\ref{morphisms-lemma-unramified-quasi-finite},", "\\ref{morphisms-lemma-residue-field-quasi-finite}, and", "\\ref{morphisms-lemma-quasi-finite}.", "For each $i$ there are at most finitely many $\\kappa(s)$-algebra", "maps $\\kappa(x_i) \\to \\kappa(y)$ (by elementary field theory).", "Thus $\\Mor_S(Y, X)$ is finite by", "Proposition \\ref{proposition-equality}.", "\\medskip\\noindent", "General case. There exists a nonempty open $U \\subset X$ such", "that $X_U \\to U$ is finite (in particular separated), see", "Morphisms, Lemma \\ref{morphisms-lemma-generically-finite}", "(the lemma applies since we've already seen above that a quasi-compact", "unramified morphism is quasi-finite and since $X \\to S$ is quasi-separated by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-Noetherian-quasi-separated}).", "Let $Z \\subset S$ be the reduced closed subscheme supported on", "the complement of $U$. By Noetherian induction, we see that", "$\\Mor_Z(Y_Z, X_Z)$ is finite (details omitted).", "By the result of the first paragraph the set", "$\\Mor_U(Y_U, X_U)$ is finite. Thus it suffices to show that", "$$", "\\Mor_S(Y, X) \\longrightarrow \\Mor_Z(Y_Z, X_Z) \\times \\Mor_U(Y_U, X_U)", "$$", "is injective. This follows from the fact that the set of points where", "two morphisms $a, b : Y \\to X$ agree is open in $Y$, due to the fact", "that $\\Delta : X \\to X \\times_S X$ is open, see", "Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism}." ], "refs": [ "morphisms-lemma-unramified-quasi-finite", "morphisms-lemma-residue-field-quasi-finite", "morphisms-lemma-quasi-finite", "etale-proposition-equality", "morphisms-lemma-generically-finite", "morphisms-lemma-finite-type-Noetherian-quasi-separated", "morphisms-lemma-diagonal-unramified-morphism" ], "ref_ids": [ 5351, 5225, 5230, 10727, 5487, 5203, 5354 ] } ], "ref_ids": [] }, { "id": 10701, "type": "theorem", "label": "etale-lemma-universally-injective-unramified", "categories": [ "etale" ], "title": "etale-lemma-universally-injective-unramified", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is unramified and a monomorphism,", "\\item $f$ is unramified and universally injective,", "\\item $f$ is locally of finite type and a monomorphism,", "\\item $f$ is universally injective, locally of finite type, and", "formally unramified,", "\\item $f$ is locally of finite type and $X_s$ is either empty", "or $X_s \\to s$ is an isomorphism for all $s \\in S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have seen in", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-unramified-formally-unramified}", "that being formally unramified and locally of finite type is the same thing", "as being unramified. Hence (4) is equivalent to (2).", "A monomorphism is certainly universally injective and", "formally unramified hence (3) implies (4).", "It is clear that (1) implies (3). Finally, if (2) holds, then", "$\\Delta : X \\to X \\times_S X$ is both an open immersion", "(Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism})", "and surjective", "(Morphisms, Lemma \\ref{morphisms-lemma-universally-injective})", "hence an isomorphism, i.e., $f$ is a monomorphism. In this way we see that", "(2) implies (1).", "\\medskip\\noindent", "Condition (3) implies (5) because monomorphisms are preserved under", "base change", "(Schemes, Lemma \\ref{schemes-lemma-base-change-monomorphism})", "and because of the description of monomorphisms towards the spectra of fields", "in", "Schemes, Lemma \\ref{schemes-lemma-mono-towards-spec-field}.", "Condition (5) implies (4) by", "Morphisms, Lemmas \\ref{morphisms-lemma-universally-injective} and", "\\ref{morphisms-lemma-unramified-etale-fibres}." ], "refs": [ "more-morphisms-lemma-unramified-formally-unramified", "morphisms-lemma-diagonal-unramified-morphism", "morphisms-lemma-universally-injective", "schemes-lemma-base-change-monomorphism", "schemes-lemma-mono-towards-spec-field", "morphisms-lemma-universally-injective", "morphisms-lemma-unramified-etale-fibres" ], "ref_ids": [ 13696, 5354, 5167, 7724, 7729, 5167, 5353 ] } ], "ref_ids": [] }, { "id": 10702, "type": "theorem", "label": "etale-lemma-characterize-closed-immersion", "categories": [ "etale" ], "title": "etale-lemma-characterize-closed-immersion", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is a closed immersion,", "\\item $f$ is a proper monomorphism,", "\\item $f$ is proper, unramified, and universally injective,", "\\item $f$ is universally closed, unramified, and a monomorphism,", "\\item $f$ is universally closed, unramified, and universally injective,", "\\item $f$ is universally closed, locally of finite type, and a monomorphism,", "\\item $f$ is universally closed, universally injective, locally of", "finite type, and formally unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (4) -- (7) follows immediately from", "Lemma \\ref{lemma-universally-injective-unramified}.", "\\medskip\\noindent", "Let $f : X \\to S$ satisfy (6). Then $f$ is separated, see", "Schemes, Lemma \\ref{schemes-lemma-monomorphism-separated}", "and has finite fibres. Hence", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-characterize-finite}", "shows $f$ is finite. Then", "Morphisms, Lemma \\ref{morphisms-lemma-finite-monomorphism-closed}", "implies $f$ is a closed immersion, i.e., (1) holds.", "\\medskip\\noindent", "Note that (1) $\\Rightarrow$ (2) because a closed immersion is", "proper and a monomorphism", "(Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-proper}", "and", "Schemes, Lemma \\ref{schemes-lemma-immersions-monomorphisms}).", "By", "Lemma \\ref{lemma-universally-injective-unramified}", "we see that (2) implies (3). It is clear that (3) implies (5)." ], "refs": [ "etale-lemma-universally-injective-unramified", "schemes-lemma-monomorphism-separated", "more-morphisms-lemma-characterize-finite", "morphisms-lemma-finite-monomorphism-closed", "morphisms-lemma-closed-immersion-proper", "schemes-lemma-immersions-monomorphisms", "etale-lemma-universally-injective-unramified" ], "ref_ids": [ 10701, 7722, 13903, 5449, 5410, 7727, 10701 ] } ], "ref_ids": [] }, { "id": 10703, "type": "theorem", "label": "etale-lemma-finite-unramified-one-point", "categories": [ "etale" ], "title": "etale-lemma-finite-unramified-one-point", "contents": [ "Let $\\pi : X \\to S$ be a morphism of schemes. Let $s \\in S$.", "Assume that", "\\begin{enumerate}", "\\item $\\pi$ is finite,", "\\item $\\pi$ is unramified,", "\\item $\\pi^{-1}(\\{s\\}) = \\{x\\}$, and", "\\item $\\kappa(s) \\subset \\kappa(x)$ is purely", "inseparable\\footnote{In view of condition (2)", "this is equivalent to $\\kappa(s) = \\kappa(x)$.}.", "\\end{enumerate}", "Then there exists an open neighbourhood $U$ of $s$ such that", "$\\pi|_{\\pi^{-1}(U)} : \\pi^{-1}(U) \\to U$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $S$. Hence we may assume that $S = \\Spec(A)$.", "By definition of a finite morphism this implies $X = \\Spec(B)$.", "Note that the ring map $\\varphi : A \\to B$ defining $\\pi$", "is a finite unramified ring map.", "Let $\\mathfrak p \\subset A$ be the prime corresponding to $s$.", "Let $\\mathfrak q \\subset B$ be the prime corresponding to $x$.", "Conditions (2), (3) and (4) imply that", "$B_{\\mathfrak q}/\\mathfrak pB_{\\mathfrak q} = \\kappa(\\mathfrak p)$.", "By Algebra, Lemma \\ref{algebra-lemma-unique-prime-over-localize-below}", "we have $B_{\\mathfrak q} = B_{\\mathfrak p}$", "(note that a finite ring map satisfies going up, see", "Algebra, Section \\ref{algebra-section-going-up}.)", "Hence we see that", "$B_{\\mathfrak p}/\\mathfrak pB_{\\mathfrak p} = \\kappa(\\mathfrak p)$.", "As $B$ is a finite $A$-module we see from Nakayama's lemma (see", "Algebra, Lemma \\ref{algebra-lemma-NAK})", "that $B_{\\mathfrak p} = \\varphi(A_{\\mathfrak p})$. Hence (using the finiteness", "of $B$ as an $A$-module again) there exists a", "$f \\in A$, $f \\not \\in \\mathfrak p$ such that $B_f = \\varphi(A_f)$", "as desired." ], "refs": [ "algebra-lemma-unique-prime-over-localize-below", "algebra-lemma-NAK" ], "ref_ids": [ 556, 401 ] } ], "ref_ids": [] }, { "id": 10704, "type": "theorem", "label": "etale-lemma-characterize-etale-Noetherian", "categories": [ "etale" ], "title": "etale-lemma-characterize-etale-Noetherian", "contents": [ "Let $A \\to B$ be of finite type with $A$ a Noetherian ring.", "Let $\\mathfrak q$ be a prime of $B$ lying over $\\mathfrak p \\subset A$.", "Then $A \\to B$ is \\'etale at $\\mathfrak q$ if and only if", "$A_{\\mathfrak p} \\to B_{\\mathfrak q}$ is an \\'etale homomorphism", "of local rings." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemmas \\ref{algebra-lemma-etale} (flatness of \\'etale maps),", "\\ref{algebra-lemma-etale-at-prime} (\\'etale maps are unramified)", "and \\ref{algebra-lemma-characterize-etale} (flat and unramified maps", "are \\'etale)." ], "refs": [ "algebra-lemma-etale", "algebra-lemma-etale-at-prime", "algebra-lemma-characterize-etale" ], "ref_ids": [ 1231, 1233, 1235 ] } ], "ref_ids": [] }, { "id": 10705, "type": "theorem", "label": "etale-lemma-characterize-etale-completions", "categories": [ "etale" ], "title": "etale-lemma-characterize-etale-completions", "contents": [ "Let $A$, $B$ be Noetherian local rings.", "Let $A \\to B$ be a local homomorphism such that $B$ is essentially of", "finite type over $A$.", "The following are equivalent", "\\begin{enumerate}", "\\item $A \\to B$ is an \\'etale homomorphism of local rings", "\\item $A^\\wedge \\to B^\\wedge$ is an \\'etale homomorphism of local rings, and", "\\item $A^\\wedge \\to B^\\wedge$ is \\'etale.", "\\end{enumerate}", "Moreover, in this case $B^\\wedge \\cong (A^\\wedge)^{\\oplus n}$ as", "$A^\\wedge$-modules for some $n \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "To see the equivalences of (1), (2) and (3), as we have the corresponding", "results for unramified ring maps", "(Lemma \\ref{lemma-characterize-unramified-completions})", "it suffices to prove that", "$A \\to B$ is flat if and only if $A^\\wedge \\to B^\\wedge$ is flat.", "This is clear from our lists of properties of flat maps since", "the ring maps $A \\to A^\\wedge$ and $B \\to B^\\wedge$ are faithfully flat.", "For the final statement, by Lemma \\ref{lemma-unramified-completions}", "we see that $B^\\wedge$ is a finite flat $A^\\wedge$ module.", "Hence it is finite free by our list", "of properties on flat modules in Section \\ref{section-flat-morphisms}." ], "refs": [ "etale-lemma-characterize-unramified-completions", "etale-lemma-unramified-completions" ], "ref_ids": [ 10698, 10697 ] } ], "ref_ids": [] }, { "id": 10706, "type": "theorem", "label": "etale-lemma-etale-definition", "categories": [ "etale" ], "title": "etale-lemma-etale-definition", "contents": [ "Let $Y$ be a locally Noetherian scheme.", "Let $f : X \\to Y$ be locally of finite type.", "Let $x \\in X$. The morphism $f$ is \\'etale at $x$ in", "the sense of Definition \\ref{definition-etale-schemes-1}", "if and only if it is \\'etale at $x$ in", "the sense of Morphisms, Definition \\ref{morphisms-definition-etale}." ], "refs": [ "etale-definition-etale-schemes-1", "morphisms-definition-etale" ], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-characterize-etale-Noetherian}", "and the definitions." ], "refs": [ "etale-lemma-characterize-etale-Noetherian" ], "ref_ids": [ 10704 ] } ], "ref_ids": [ 10739, 5567 ] }, { "id": 10707, "type": "theorem", "label": "etale-lemma-finite-etale-one-point", "categories": [ "etale" ], "title": "etale-lemma-finite-etale-one-point", "contents": [ "Let $\\pi : X \\to S$ be a morphism of schemes. Let $s \\in S$.", "Assume that", "\\begin{enumerate}", "\\item $\\pi$ is finite,", "\\item $\\pi$ is \\'etale,", "\\item $\\pi^{-1}(\\{s\\}) = \\{x\\}$, and", "\\item $\\kappa(s) \\subset \\kappa(x)$ is purely", "inseparable\\footnote{In view of condition (2)", "this is equivalent to $\\kappa(s) = \\kappa(x)$.}.", "\\end{enumerate}", "Then there exists an open neighbourhood $U$ of $s$ such that", "$\\pi|_{\\pi^{-1}(U)} : \\pi^{-1}(U) \\to U$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-finite-unramified-one-point}", "there exists an open neighbourhood $U$ of $s$ such that", "$\\pi|_{\\pi^{-1}(U)} : \\pi^{-1}(U) \\to U$ is a closed immersion.", "But a morphism which is \\'etale and a closed immersion is an", "open immersion (for example by", "Theorem \\ref{theorem-etale-radicial-open}).", "Hence after shrinking $U$ we obtain an isomorphism." ], "refs": [ "etale-lemma-finite-unramified-one-point", "etale-theorem-etale-radicial-open" ], "ref_ids": [ 10703, 10692 ] } ], "ref_ids": [] }, { "id": 10708, "type": "theorem", "label": "etale-lemma-relative-frobenius-etale", "categories": [ "etale" ], "title": "etale-lemma-relative-frobenius-etale", "contents": [ "Let $U \\to X$ be an \\'etale morphism of schemes", "where $X$ is a scheme in characteristic $p$.", "Then the relative Frobenius $F_{U/X} : U \\to U \\times_{X, F_X} X$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The morphism $F_{U/X}$ is a universal homeomorphism by", "Varieties, Lemma \\ref{varieties-lemma-relative-frobenius}.", "The morphism $F_{U/X}$ is \\'etale as a", "morphism between schemes \\'etale over $X$", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-permanence}).", "Hence $F_{U/X}$ is an isomorphism by", "Theorem \\ref{theorem-etale-radicial-open}." ], "refs": [ "varieties-lemma-relative-frobenius", "morphisms-lemma-etale-permanence", "etale-theorem-etale-radicial-open" ], "ref_ids": [ 11050, 5375, 10692 ] } ], "ref_ids": [] }, { "id": 10709, "type": "theorem", "label": "etale-lemma-unramified-etale-local", "categories": [ "etale" ], "title": "etale-lemma-unramified-etale-local", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$.", "Assume $f$ is unramified at each $x_i$.", "Then there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$", "and opens $V_{i, j} \\subset X_U$, $i = 1, \\ldots, n$, $j = 1, \\ldots, m_i$", "such that", "\\begin{enumerate}", "\\item $V_{i, j} \\to U$ is a closed immersion passing through $u$,", "\\item $u$ is not in the image of $V_{i, j} \\cap V_{i', j'}$ unless", "$i = i'$ and $j = j'$, and", "\\item any point of $(X_U)_u$ mapping to $x_i$ is in some $V_{i, j}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By", "Morphisms, Definition \\ref{morphisms-definition-unramified}", "there exists an open neighbourhood of each $x_i$ which is locally of finite", "type over $S$. Replacing $X$ by an open neighbourhood of $\\{x_1, \\ldots, x_n\\}$", "we may assume $f$ is locally of finite type. Apply", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points-var}", "to get the \\'etale neighbourhood $(U, u)$ and the opens $V_{i, j}$ finite over", "$U$. By", "Lemma \\ref{lemma-finite-unramified-one-point}", "after possibly shrinking $U$ we get that $V_{i, j} \\to U$ is a closed", "immersion." ], "refs": [ "morphisms-definition-unramified", "more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points-var", "etale-lemma-finite-unramified-one-point" ], "ref_ids": [ 5566, 13894, 10703 ] } ], "ref_ids": [] }, { "id": 10710, "type": "theorem", "label": "etale-lemma-unramified-etale-local-technical", "categories": [ "etale" ], "title": "etale-lemma-unramified-etale-local-technical", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$.", "Assume $f$ is separated and $f$ is unramified at each $x_i$.", "Then there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$", "and a disjoint union decomposition", "$$", "X_U =", "W \\amalg \\coprod\\nolimits_{i, j} V_{i, j}", "$$", "such that", "\\begin{enumerate}", "\\item $V_{i, j} \\to U$ is a closed immersion passing through $u$,", "\\item the fibre $W_u$ contains no point mapping to any $x_i$.", "\\end{enumerate}", "In particular, if $f^{-1}(\\{s\\}) = \\{x_1, \\ldots, x_n\\}$, then", "the fibre $W_u$ is empty." ], "refs": [], "proofs": [ { "contents": [ "Apply", "Lemma \\ref{lemma-unramified-etale-local}.", "We may assume $U$ is affine, so $X_U$ is separated.", "Then $V_{i, j} \\to X_U$ is a closed map, see", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}.", "Suppose $(i, j) \\not = (i', j')$.", "Then $V_{i, j} \\cap V_{i', j'}$ is closed in $V_{i, j}$ and", "its image in $U$ does not contain $u$.", "Hence after shrinking $U$ we may assume that", "$V_{i, j} \\cap V_{i', j'} = \\emptyset$. Moreover, $\\bigcup V_{i, j}$ is", "a closed and open subscheme of $X_U$ and hence has an open and closed", "complement $W$. This finishes the proof." ], "refs": [ "etale-lemma-unramified-etale-local", "morphisms-lemma-image-proper-scheme-closed" ], "ref_ids": [ 10709, 5411 ] } ], "ref_ids": [] }, { "id": 10711, "type": "theorem", "label": "etale-lemma-finite-unramified-etale-local", "categories": [ "etale" ], "title": "etale-lemma-finite-unramified-etale-local", "contents": [ "Let $f : X \\to S$ be a finite unramified morphism of schemes.", "Let $s \\in S$.", "There exists an \\'etale neighbourhood $(U, u) \\to (S, s)$", "and a finite disjoint union decomposition", "$$", "X_U = \\coprod\\nolimits_j V_j", "$$", "such that each $V_j \\to U$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Since $X \\to S$ is finite the fibre over $s$ is a finite set", "$\\{x_1, \\ldots, x_n\\}$ of points of $X$. Apply", "Lemma \\ref{lemma-unramified-etale-local-technical}", "to this set (a finite morphism is separated, see", "Morphisms, Section \\ref{morphisms-section-integral}).", "The image of $W$ in $U$ is a closed", "subset (as $X_U \\to U$ is finite, hence proper) which does not", "contain $u$. After removing this from $U$ we see that $W = \\emptyset$", "as desired." ], "refs": [ "etale-lemma-unramified-etale-local-technical" ], "ref_ids": [ 10710 ] } ], "ref_ids": [] }, { "id": 10712, "type": "theorem", "label": "etale-lemma-etale-etale-local", "categories": [ "etale" ], "title": "etale-lemma-etale-etale-local", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$.", "Assume $f$ is \\'etale at each $x_i$.", "Then there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$", "and opens $V_{i, j} \\subset X_U$, $i = 1, \\ldots, n$, $j = 1, \\ldots, m_i$", "such that", "\\begin{enumerate}", "\\item $V_{i, j} \\to U$ is an isomorphism,", "\\item $u$ is not in the image of $V_{i, j} \\cap V_{i', j'}$ unless", "$i = i'$ and $j = j'$, and", "\\item any point of $(X_U)_u$ mapping to $x_i$ is in some $V_{i, j}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "An \\'etale morphism is unramified, hence we may apply", "Lemma \\ref{lemma-unramified-etale-local}.", "Now $V_{i, j} \\to U$ is a closed immersion and \\'etale.", "Hence it is an open immersion, for example by", "Theorem \\ref{theorem-etale-radicial-open}.", "Replace $U$ by the intersection of the images of $V_{i, j} \\to U$", "to get the lemma." ], "refs": [ "etale-lemma-unramified-etale-local", "etale-theorem-etale-radicial-open" ], "ref_ids": [ 10709, 10692 ] } ], "ref_ids": [] }, { "id": 10713, "type": "theorem", "label": "etale-lemma-etale-etale-local-technical", "categories": [ "etale" ], "title": "etale-lemma-etale-etale-local-technical", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$.", "Assume $f$ is separated and $f$ is \\'etale at each $x_i$.", "Then there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$", "and a disjoint union decomposition", "$$", "X_U =", "W \\amalg \\coprod\\nolimits_{i, j} V_{i, j}", "$$", "such that", "\\begin{enumerate}", "\\item $V_{i, j} \\to U$ is an isomorphism,", "\\item the fibre $W_u$ contains no point mapping to any $x_i$.", "\\end{enumerate}", "In particular, if $f^{-1}(\\{s\\}) = \\{x_1, \\ldots, x_n\\}$, then", "the fibre $W_u$ is empty." ], "refs": [], "proofs": [ { "contents": [ "An \\'etale morphism is unramified, hence we may apply", "Lemma \\ref{lemma-unramified-etale-local-technical}.", "As in the proof of", "Lemma \\ref{lemma-etale-etale-local}", "the morphisms $V_{i, j} \\to U$ are open immersions and", "we win after replacing $U$ by the intersection of their", "images." ], "refs": [ "etale-lemma-unramified-etale-local-technical", "etale-lemma-etale-etale-local" ], "ref_ids": [ 10710, 10712 ] } ], "ref_ids": [] }, { "id": 10714, "type": "theorem", "label": "etale-lemma-finite-etale-etale-local", "categories": [ "etale" ], "title": "etale-lemma-finite-etale-etale-local", "contents": [ "Let $f : X \\to S$ be a finite \\'etale morphism of schemes.", "Let $s \\in S$. There exists an \\'etale neighbourhood $(U, u) \\to (S, s)$", "and a disjoint union decomposition", "$$", "X_U = \\coprod\\nolimits_j V_j", "$$", "such that each $V_j \\to U$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "An \\'etale morphism is unramified, hence we may apply", "Lemma \\ref{lemma-finite-unramified-etale-local}.", "As in the proof of", "Lemma \\ref{lemma-etale-etale-local}", "we see that $V_{i, j} \\to U$ is an open immersion and we win", "after replacing $U$ by the intersection of their images." ], "refs": [ "etale-lemma-finite-unramified-etale-local", "etale-lemma-etale-etale-local" ], "ref_ids": [ 10711, 10712 ] } ], "ref_ids": [] }, { "id": 10715, "type": "theorem", "label": "etale-lemma-etale-dimension", "categories": [ "etale" ], "title": "etale-lemma-etale-dimension", "contents": [ "Let $A$, $B$ be Noetherian local rings.", "Let $A \\to B$ be a \\'etale homomorphism of local rings.", "Then $\\dim(A) = \\dim(B)$." ], "refs": [], "proofs": [ { "contents": [ "See for example", "Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}." ], "refs": [ "algebra-lemma-dimension-base-fibre-equals-total" ], "ref_ids": [ 987 ] } ], "ref_ids": [] }, { "id": 10716, "type": "theorem", "label": "etale-lemma-faithful", "categories": [ "etale" ], "title": "etale-lemma-faithful", "contents": [ "If $f : X \\to S$ is surjective, then the functor", "(\\ref{equation-descent-etale}) is faithful." ], "refs": [], "proofs": [ { "contents": [ "Let $a, b : U_1 \\to U_2$ be two morphisms between schemes \\'etale over $S$.", "Assume the base changes of $a$ and $b$ to $X$ agree.", "We have to show that $a = b$.", "By Proposition \\ref{proposition-equality} it suffices to", "show that $a$ and $b$ agree on points and residue fields.", "This is clear because for every $u \\in U_1$ we can find a point", "$v \\in X \\times_S U_1$ mapping to $u$." ], "refs": [ "etale-proposition-equality" ], "ref_ids": [ 10727 ] } ], "ref_ids": [] }, { "id": 10717, "type": "theorem", "label": "etale-lemma-fully-faithful", "categories": [ "etale" ], "title": "etale-lemma-fully-faithful", "contents": [ "Assume $f : X \\to S$ is submersive and any \\'etale base change", "of $f$ is submersive. Then the functor", "(\\ref{equation-descent-etale}) is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-faithful} the functor is faithful.", "Let $U_1 \\to S$ and $U_2 \\to S$ be \\'etale morphisms", "and let $a : X \\times_S U_1 \\to X \\times_S U_2$ be a", "morphism compatible with canonical descent data.", "We will prove that $a$ is the base change of a morphism $U_1 \\to U_2$.", "\\medskip\\noindent", "Let $U'_2 \\subset U_2$ be an open subscheme. Consider", "$W = a^{-1}(X \\times_S U'_2)$. This is an open subscheme", "of $X \\times_S U_1$ which is compatible with the canonical", "descent datum on $V_1 = X \\times_S U_1$. This means that the", "two inverse images of $W$ by the projections", "$V_1 \\times_{U_1} V_1 \\to V_1$ agree. Since $V_1 \\to U_1$", "is surjective (as the base change of $X \\to S$) we conclude", "that $W$ is the inverse image of some subset $U'_1 \\subset U_1$.", "Since $W$ is open, our assumption on $f$ implies that $U'_1 \\subset U_1$", "is open.", "\\medskip\\noindent", "Let $U_2 = \\bigcup U_{2, i}$ be an affine open covering.", "By the result of the preceding paragraph we obtain an open", "covering $U_1 = \\bigcup U_{1, i}$ such that", "$X \\times_S U_{1, i} = a^{-1}(X \\times_S U_{2, i})$.", "If we can prove there exists a morphism $U_{1, i} \\to U_{2, i}$", "whose base change is the morphism", "$a_i : X \\times_S U_{1, i} \\to X \\times_S U_{2, i}$", "then we can glue these morphisms to a morphism $U_1 \\to U_2$", "(using faithfulness). In this way we reduce to the case that", "$U_2$ is affine. In particular $U_2 \\to S$ is separated", "(Schemes, Lemma \\ref{schemes-lemma-compose-after-separated}).", "\\medskip\\noindent", "Assume $U_2 \\to S$ is separated. Then the graph $\\Gamma_a$ of $a$", "is a closed subscheme of", "$$", "V = (X \\times_S U_1) \\times_X (X \\times_S U_2) = X \\times_S U_1 \\times_S U_2", "$$", "by Schemes, Lemma \\ref{schemes-lemma-semi-diagonal}.", "On the other hand the graph is open for example", "because it is a section of an \\'etale morphism", "(Proposition \\ref{proposition-properties-sections}).", "Since $a$ is a morphism of descent data, the two inverse images of", "$\\Gamma_a \\subset V$ under the projections", "$V \\times_{U_1 \\times_S U_2} V \\to V$ are the same.", "Hence arguing as in the second paragraph of the proof we", "find an open and closed subscheme $\\Gamma \\subset U_1 \\times_S U_2$", "whose base change to $X$ gives $\\Gamma_a$. Then", "$\\Gamma \\to U_1$ is an \\'etale morphism whose base change", "to $X$ is an isomorphism. This means that $\\Gamma \\to U_1$", "is universally bijective, hence an isomorphism", "by Theorem \\ref{theorem-etale-radicial-open}.", "Thus $\\Gamma$ is the graph of a morphism $U_1 \\to U_2$", "and the base change of this morphism is $a$ as desired." ], "refs": [ "etale-lemma-faithful", "schemes-lemma-compose-after-separated", "schemes-lemma-semi-diagonal", "etale-proposition-properties-sections", "etale-theorem-etale-radicial-open" ], "ref_ids": [ 10716, 7715, 7712, 10726, 10692 ] } ], "ref_ids": [] }, { "id": 10718, "type": "theorem", "label": "etale-lemma-fully-faithful-cases", "categories": [ "etale" ], "title": "etale-lemma-fully-faithful-cases", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. In the following", "cases the functor (\\ref{equation-descent-etale}) is fully faithful:", "\\begin{enumerate}", "\\item $f$ is surjective and universally closed", "(e.g., finite, integral, or proper),", "\\item $f$ is surjective and universally open", "(e.g., locally of finite presentation and flat, smooth, or etale),", "\\item $f$ is surjective, quasi-compact, and flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-fully-faithful}.", "For example a closed surjective map of topological spaces", "is submersive (Topology, Lemma", "\\ref{topology-lemma-closed-morphism-quotient-topology}).", "Finite, integral, and proper morphisms are universally closed, see", "Morphisms, Lemmas \\ref{morphisms-lemma-integral-universally-closed} and", "\\ref{morphisms-lemma-finite-proper} and", "Definition \\ref{morphisms-definition-proper}.", "On the other hand an open surjective map of topological spaces", "is submersive (Topology, Lemma", "\\ref{topology-lemma-open-morphism-quotient-topology}).", "Flat locally finitely presented, smooth, and \\'etale morphisms are", "universally open, see", "Morphisms, Lemmas \\ref{morphisms-lemma-fppf-open},", "\\ref{morphisms-lemma-smooth-open}, and", "\\ref{morphisms-lemma-etale-open}.", "The case of surjective, quasi-compact, flat morphisms follows", "from Morphisms, Lemma \\ref{morphisms-lemma-fpqc-quotient-topology}." ], "refs": [ "etale-lemma-fully-faithful", "topology-lemma-closed-morphism-quotient-topology", "morphisms-lemma-integral-universally-closed", "morphisms-lemma-finite-proper", "morphisms-definition-proper", "topology-lemma-open-morphism-quotient-topology", "morphisms-lemma-fppf-open", "morphisms-lemma-smooth-open", "morphisms-lemma-etale-open", "morphisms-lemma-fpqc-quotient-topology" ], "ref_ids": [ 10717, 8204, 5441, 5445, 5571, 8203, 5267, 5332, 5370, 5269 ] } ], "ref_ids": [] }, { "id": 10719, "type": "theorem", "label": "etale-lemma-reduce-to-affine", "categories": [ "etale" ], "title": "etale-lemma-reduce-to-affine", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $(V, \\varphi)$ be a descent datum relative to $X/S$", "with $V \\to X$ \\'etale. Let $S = \\bigcup S_i$ be an", "open covering. Assume that", "\\begin{enumerate}", "\\item the pullback of the descent datum $(V, \\varphi)$", "to $X \\times_S S_i/S_i$ is effective,", "\\item the functor (\\ref{equation-descent-etale})", "for $X \\times_S (S_i \\cap S_j) \\to (S_i \\cap S_j)$ is fully faithful, and", "\\item the functor (\\ref{equation-descent-etale})", "for $X \\times_S (S_i \\cap S_j \\cap S_k) \\to (S_i \\cap S_j \\cap S_k)$", "is faithful.", "\\end{enumerate}", "Then $(V, \\varphi)$ is effective." ], "refs": [], "proofs": [ { "contents": [ "(Recall that pullbacks of descent data are defined in", "Descent, Definition \\ref{descent-definition-pullback-functor}.)", "Set $X_i = X \\times_S S_i$. Denote $(V_i, \\varphi_i)$ the pullback", "of $(V, \\varphi)$ to $X_i/S_i$.", "By assumption (1) we can find an \\'etale morphism $U_i \\to S_i$", "which comes with an isomorphism $X_i \\times_{S_i} U_i \\to V_i$ compatible with", "$can$ and $\\varphi_i$. By assumption (2) we obtain isomorphisms", "$\\psi_{ij} : U_i \\times_{S_i} (S_i \\cap S_j) \\to", "U_j \\times_{S_j} (S_i \\cap S_j)$.", "By assumption (3) these isomorphisms satisfy the cocycle condition", "so that $(U_i, \\psi_{ij})$ is a descend datum for the", "Zariski covering $\\{S_i \\to S\\}$. Then Descent, Lemma", "\\ref{descent-lemma-Zariski-refinement-coverings-equivalence}", "(which is essentially just a reformulation of", "Schemes, Section \\ref{schemes-section-glueing-schemes})", "tells us that there exists a morphism of schemes $U \\to S$", "and isomorphisms $U \\times_S S_i \\to U_i$ compatible", "with $\\psi_{ij}$. The isomorphisms $U \\times_S S_i \\to U_i$", "determine corresponding isomorphisms $X_i \\times_S U \\to V_i$", "which glue to a morphism $X \\times_S U \\to V$ compatible", "with the canonical descent datum and $\\varphi$." ], "refs": [ "descent-definition-pullback-functor", "descent-lemma-Zariski-refinement-coverings-equivalence" ], "ref_ids": [ 14778, 14744 ] } ], "ref_ids": [] }, { "id": 10720, "type": "theorem", "label": "etale-lemma-split-henselian", "categories": [ "etale" ], "title": "etale-lemma-split-henselian", "contents": [ "Let $(A, I)$ be a henselian pair. Let $U \\to \\Spec(A)$ be a", "quasi-compact, separated, \\'etale morphism such that", "$U \\times_{\\Spec(A)} \\Spec(A/I) \\to \\Spec(A/I)$ is finite.", "Then", "$$", "U = U_{fin} \\amalg U_{away}", "$$", "where $U_{fin} \\to \\Spec(A)$ is finite and $U_{away}$ has", "no points lying over $Z$." ], "refs": [], "proofs": [ { "contents": [ "By Zariski's main theorem, the scheme $U$ is quasi-affine.", "In fact, we can find an open immersion $U \\to T$ with $T$ affine and", "$T \\to \\Spec(A)$ finite, see More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite}.", "Write $Z = \\Spec(A/I)$ and denote $U_Z \\to T_Z$ the base change.", "Since $U_Z \\to Z$ is finite, we see that $U_Z \\to T_Z$ is closed", "as well as open. Hence by", "More on Algebra, Lemma \\ref{more-algebra-lemma-characterize-henselian-pair}", "we obtain a unique decomposition $T = T' \\amalg T''$ with $T'_Z = U_Z$.", "Set $U_{fin} = U \\cap T'$ and $U_{away} = U \\cap T''$. Since", "$T'_Z \\subset U_Z$ we see that all closed points of $T'$ are in $U$", "hence $T' \\subset U$, hence $U_{fin} = T'$, hence $U_{fin} \\to \\Spec(A)$", "is finite. We omit the proof", "of uniqueness of the decomposition." ], "refs": [ "more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "more-algebra-lemma-characterize-henselian-pair" ], "ref_ids": [ 13901, 9861 ] } ], "ref_ids": [] }, { "id": 10721, "type": "theorem", "label": "etale-lemma-strict-normal-crossings", "categories": [ "etale" ], "title": "etale-lemma-strict-normal-crossings", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be an", "effective Cartier divisor. Let $D_i \\subset D$, $i \\in I$ be its", "irreducible components viewed as reduced closed subschemes of $X$.", "The following are equivalent", "\\begin{enumerate}", "\\item $D$ is a strict normal crossings divisor, and", "\\item $D$ is reduced, each $D_i$ is an effective Cartier divisor, and", "for $J \\subset I$ finite the scheme theoretic", "intersection $D_J = \\bigcap_{j \\in J} D_j$ is a", "regular scheme each of whose irreducible components has", "codimension $|J|$ in $X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $D$ is a strict normal crossings divisor. Pick $p \\in D$", "and choose a regular system of parameters $x_1, \\ldots, x_d \\in \\mathfrak m_p$", "and $1 \\leq r \\leq d$ as in", "Definition \\ref{definition-strict-normal-crossings}.", "Since $\\mathcal{O}_{X, p}/(x_i)$ is a regular local ring", "(and in particular a domain) we see that the irreducible components", "$D_1, \\ldots, D_r$ of $D$ passing through $p$ correspond $1$-to-$1$", "to the height one primes $(x_1), \\ldots, (x_r)$ of $\\mathcal{O}_{X, p}$.", "By Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}", "we find that the intersections $D_{i_1} \\cap \\ldots \\cap D_{i_s}$", "have codimension $s$ in an open neighbourhood of $p$", "and that this intersection has a regular local ring at $p$.", "Since this holds for all $p \\in D$ we conclude that (2) holds.", "\\medskip\\noindent", "Assume (2). Let $p \\in D$. Since $\\mathcal{O}_{X, p}$ is finite", "dimensional we see that $p$ can be contained in at most", "$\\dim(\\mathcal{O}_{X, p})$ of the components $D_i$.", "Say $p \\in D_1, \\ldots, D_r$ for some $r \\geq 1$.", "Let $x_1, \\ldots, x_r \\in \\mathfrak m_p$ be local equations", "for $D_1, \\ldots, D_r$. Then $x_1$ is a nonzerodivisor in $\\mathcal{O}_{X, p}$", "and $\\mathcal{O}_{X, p}/(x_1) = \\mathcal{O}_{D_1, p}$ is regular.", "Hence $\\mathcal{O}_{X, p}$ is regular, see", "Algebra, Lemma \\ref{algebra-lemma-regular-mod-x}.", "Since $D_1 \\cap \\ldots \\cap D_r$ is a regular (hence normal) scheme", "it is a disjoint union of its irreducible components", "(Properties, Lemma \\ref{properties-lemma-normal-Noetherian}).", "Let $Z \\subset D_1 \\cap \\ldots \\cap D_r$", "be the irreducible component containing $p$.", "Then $\\mathcal{O}_{Z, p} = \\mathcal{O}_{X, p}/(x_1, \\ldots, x_r)$", "is regular of codimension $r$ (note that since we already know", "that $\\mathcal{O}_{X, p}$ is regular and hence Cohen-Macaulay,", "there is no ambiguity about codimension as the ring is catenary, see", "Algebra, Lemmas \\ref{algebra-lemma-regular-ring-CM} and", "\\ref{algebra-lemma-CM-dim-formula}).", "Hence $\\dim(\\mathcal{O}_{Z, p}) = \\dim(\\mathcal{O}_{X, p}) - r$.", "Choose additional $x_{r + 1}, \\ldots, x_n \\in \\mathfrak m_p$", "which map to a minimal system of generators of $\\mathfrak m_{Z, p}$.", "Then $\\mathfrak m_p = (x_1, \\ldots, x_n)$ by Nakayama's lemma", "and we see that $D$ is a normal crossings divisor." ], "refs": [ "etale-definition-strict-normal-crossings", "algebra-lemma-regular-ring-CM", "algebra-lemma-regular-mod-x", "properties-lemma-normal-Noetherian", "algebra-lemma-regular-ring-CM", "algebra-lemma-CM-dim-formula" ], "ref_ids": [ 10740, 941, 945, 2970, 941, 925 ] } ], "ref_ids": [] }, { "id": 10722, "type": "theorem", "label": "etale-lemma-smooth-pullback-strict-normal-crossings", "categories": [ "etale" ], "title": "etale-lemma-smooth-pullback-strict-normal-crossings", "contents": [ "\\begin{slogan}", "Pullback of a strict normal crossings divisor by a smooth", "morphism is a strict normal crossings divisor.", "\\end{slogan}", "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a", "strict normal crossings divisor. If $f : Y \\to X$ is a smooth", "morphism of schemes, then the pullback $f^*D$ is a", "strict normal crossings divisor on $Y$." ], "refs": [], "proofs": [ { "contents": [ "As $f$ is flat the pullback is defined by", "Divisors, Lemma \\ref{divisors-lemma-pullback-effective-Cartier-defined}", "hence the statement makes sense.", "Let $q \\in f^*D$ map to $p \\in D$. Choose a regular system", "of parameters $x_1, \\ldots, x_d \\in \\mathfrak m_p$", "and $1 \\leq r \\leq d$ as in", "Definition \\ref{definition-strict-normal-crossings}.", "Since $f$ is smooth the local ring homomorphism", "$\\mathcal{O}_{X, p} \\to \\mathcal{O}_{Y, q}$ is flat", "and the fibre ring", "$$", "\\mathcal{O}_{Y, q}/\\mathfrak m_p \\mathcal{O}_{Y, q} =", "\\mathcal{O}_{Y_p, q}", "$$", "is a regular local ring (see for example", "Algebra, Lemma \\ref{algebra-lemma-characterize-smooth-over-field}).", "Pick $y_1, \\ldots, y_n \\in \\mathfrak m_q$ which map to a regular", "system of parameters in $\\mathcal{O}_{Y_p, q}$.", "Then $x_1, \\ldots, x_d, y_1, \\ldots, y_n$ generate the", "maximal ideal $\\mathfrak m_q$. Hence $\\mathcal{O}_{Y, q}$", "is a regular local ring of dimension", "$d + n$ by Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-equals-total}", "and $x_1, \\ldots, x_d, y_1, \\ldots, y_n$", "is a regular system of parameters. Since $f^*D$ is cut", "out by $x_1 \\ldots x_r$ in $\\mathcal{O}_{Y, q}$ we conclude", "that the lemma is true." ], "refs": [ "divisors-lemma-pullback-effective-Cartier-defined", "etale-definition-strict-normal-crossings", "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-dimension-base-fibre-equals-total" ], "ref_ids": [ 7936, 10740, 1223, 987 ] } ], "ref_ids": [] }, { "id": 10723, "type": "theorem", "label": "etale-lemma-smooth-pullback-normal-crossings", "categories": [ "etale" ], "title": "etale-lemma-smooth-pullback-normal-crossings", "contents": [ "\\begin{slogan}", "Pullback of a normal crossings divisor by a smooth", "morphism is a normal crossings divisor.", "\\end{slogan}", "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a", "normal crossings divisor. If $f : Y \\to X$ is a smooth", "morphism of schemes, then the pullback $f^*D$ is a", "normal crossings divisor on $Y$." ], "refs": [], "proofs": [ { "contents": [ "As $f$ is flat the pullback is defined by", "Divisors, Lemma \\ref{divisors-lemma-pullback-effective-Cartier-defined}", "hence the statement makes sense.", "Let $q \\in f^*D$ map to $p \\in D$.", "Choose an \\'etale morphism $U \\to X$ whose image contains $p$", "such that $D \\times_X U \\subset U$ is a strict normal crossings", "divisor as in Definition \\ref{definition-normal-crossings}.", "Set $V = Y \\times_X U$. Then $V \\to Y$ is \\'etale as a base", "change of $U \\to X$", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-etale})", "and the pullback $D \\times_X V$ is a strict normal crossings", "divisor on $V$ by Lemma \\ref{lemma-smooth-pullback-strict-normal-crossings}.", "Thus we have checked the condition of", "Definition \\ref{definition-normal-crossings}", "for $q \\in f^*D$ and we conclude." ], "refs": [ "divisors-lemma-pullback-effective-Cartier-defined", "etale-definition-normal-crossings", "morphisms-lemma-base-change-etale", "etale-lemma-smooth-pullback-strict-normal-crossings", "etale-definition-normal-crossings" ], "ref_ids": [ 7936, 10741, 5361, 10722, 10741 ] } ], "ref_ids": [] }, { "id": 10724, "type": "theorem", "label": "etale-lemma-characterize-normal-crossings-normalization", "categories": [ "etale" ], "title": "etale-lemma-characterize-normal-crossings-normalization", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a closed", "subscheme. The following are equivalent", "\\begin{enumerate}", "\\item $D$ is a normal crossings divisor in $X$,", "\\item $D$ is reduced, the normalization $\\nu : D^\\nu \\to D$ is unramified,", "and for any $n \\geq 1$ the scheme", "$$", "Z_n = D^\\nu \\times_D \\ldots \\times_D D^\\nu", "\\setminus \\{(p_1, \\ldots, p_n) \\mid p_i = p_j\\text{ for some }i\\not = j\\}", "$$", "is regular, the morphism $Z_n \\to X$ is a local complete intersection", "morphism whose conormal sheaf is locally free of rank $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First we explain how to think about condition (2).", "The diagonal of an unramified morphism is open", "(Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism}).", "On the other hand $D^\\nu \\to D$ is separated, hence the", "diagonal $D^\\nu \\to D^\\nu \\times_D D^\\nu$ is closed.", "Thus $Z_n$ is an open and closed subscheme of", "$D^\\nu \\times_D \\ldots \\times_D D^\\nu$. On the other hand,", "$Z_n \\to X$ is unramified as it is the composition", "$$", "Z_n \\to D^\\nu \\times_D \\ldots \\times_D D^\\nu \\to \\ldots \\to", "D^\\nu \\times_D D^\\nu \\to D^\\nu \\to D \\to X", "$$", "and each of the arrows is unramified.", "Since an unramified morphism is formally unramified", "(More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-unramified-formally-unramified})", "we have a conormal sheaf", "$\\mathcal{C}_n = \\mathcal{C}_{Z_n/X}$ of $Z_n \\to X$, see", "More on Morphisms, Definition", "\\ref{more-morphisms-definition-universal-thickening}.", "\\medskip\\noindent", "Formation of normalization commutes with \\'etale localization by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-normalization-and-smooth}.", "Checking that local rings are regular, or that", "a morphism is unramified, or that a morphism is a", "local complete intersection or that a morphism is", "unramified and has a conormal sheaf which is", "locally free of a given rank, may be done \\'etale locally (see", "More on Algebra, Lemma \\ref{more-algebra-lemma-regular-etale-extension},", "Descent, Lemma \\ref{descent-lemma-descending-property-unramified},", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-descending-property-lci}", "and", "Descent, Lemma \\ref{descent-lemma-finite-locally-free-descends}).", "\\medskip\\noindent", "By the remark of the preceding paragraph and the definition", "of normal crossings divisor it suffices to prove that a", "strict normal crossings divisor $D = \\bigcup_{i \\in I} D_i$", "satisfies (2). In this case $D^\\nu = \\coprod D_i$", "and $D^\\nu \\to D$ is unramified (being unramified", "is local on the source and $D_i \\to D$ is a closed", "immersion which is unramified). Similarly, $Z_1 = D^\\nu \\to X$", "is a local complete intersection morphism because we may", "check this locally on the source and each morphism $D_i \\to X$", "is a regular immersion as it is the inclusion of a Cartier divisor", "(see Lemma \\ref{lemma-strict-normal-crossings} and", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-regular-immersion-lci}).", "Since an effective Cartier divisor has an invertible", "conormal sheaf, we conclude that the requirement on the", "conormal sheaf is satisfied.", "Similarly, the scheme $Z_n$ for $n \\geq 2$ is the disjoint union", "of the schemes $D_J = \\bigcap_{j \\in J} D_j$ where $J \\subset I$", "runs over the subsets of order $n$. Since $D_J \\to X$ is", "a regular immersion of codimension $n$", "(by the definition of strict normal crossings and the", "fact that we may check this on stalks by", "Divisors, Lemma \\ref{divisors-lemma-Noetherian-scheme-regular-ideal})", "it follows in the same manner that $Z_n \\to X$ has the required", "properties. Some details omitted.", "\\medskip\\noindent", "Assume (2). Let $p \\in D$. Since $D^\\nu \\to D$ is unramified, it is", "finite (by Morphisms, Lemma \\ref{morphisms-lemma-finite-integral}).", "Hence $D^\\nu \\to X$ is finite unramified.", "By Lemma \\ref{lemma-finite-unramified-etale-local}", "and \\'etale localization (permissible by the discussion", "in the second paragraph and the definition of normal", "crossings divisors) we reduce to the case where", "$D^\\nu = \\coprod_{i \\in I} D_i$", "with $I$ finite and $D_i \\to U$ a closed immersion.", "After shrinking $X$ if necessary, we may assume", "$p \\in D_i$ for all $i \\in I$. The condition that $Z_1 = D^\\nu \\to X$ is an", "unramified local complete intersection morphism", "with conormal sheaf locally free of rank $1$", "implies that $D_i \\subset X$ is an effective Cartier divisor, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-lci} and", "Divisors, Lemma \\ref{divisors-lemma-regular-immersion-noetherian}.", "To finish the proof we may assume $X = \\Spec(A)$ is affine", "and $D_i = V(f_i)$ with $f_i \\in A$ a nonzerodivisor.", "If $I = \\{1, \\ldots, r\\}$, then $p \\in Z_r = V(f_1, \\ldots, f_r)$.", "The same reference as above implies that", "$(f_1, \\ldots, f_r)$ is a Koszul regular ideal in $A$.", "Since the conormal sheaf has rank $r$, we see that", "$f_1, \\ldots, f_r$ is a minimal set of generators of", "the ideal defining $Z_r$ in $\\mathcal{O}_{X, p}$.", "This implies that $f_1, \\ldots, f_r$ is a regular sequence", "in $\\mathcal{O}_{X, p}$ such that $\\mathcal{O}_{X, p}/(f_1, \\ldots, f_r)$", "is regular. Thus we conclude by", "Algebra, Lemma \\ref{algebra-lemma-regular-mod-x}", "that $f_1, \\ldots, f_r$ can be extended to a regular system of parameters", "in $\\mathcal{O}_{X, p}$ and this finishes the proof." ], "refs": [ "morphisms-lemma-diagonal-unramified-morphism", "more-morphisms-lemma-unramified-formally-unramified", "more-morphisms-definition-universal-thickening", "more-morphisms-lemma-normalization-and-smooth", "more-algebra-lemma-regular-etale-extension", "descent-lemma-descending-property-unramified", "more-morphisms-lemma-descending-property-lci", "descent-lemma-finite-locally-free-descends", "etale-lemma-strict-normal-crossings", "more-morphisms-lemma-regular-immersion-lci", "divisors-lemma-Noetherian-scheme-regular-ideal", "morphisms-lemma-finite-integral", "etale-lemma-finite-unramified-etale-local", "more-morphisms-lemma-lci", "divisors-lemma-regular-immersion-noetherian", "algebra-lemma-regular-mod-x" ], "ref_ids": [ 5354, 13696, 14109, 13775, 10053, 14693, 14017, 14617, 10721, 14007, 7988, 5438, 10711, 14001, 7990, 945 ] } ], "ref_ids": [] }, { "id": 10725, "type": "theorem", "label": "etale-lemma-characterize-normal-crossings", "categories": [ "etale" ], "title": "etale-lemma-characterize-normal-crossings", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a closed", "subscheme. If $X$ is J-2 or Nagata, then following are equivalent", "\\begin{enumerate}", "\\item $D$ is a normal crossings divisor in $X$,", "\\item for every $p \\in D$ the pullback of $D$ to the spectrum of the", "strict henselization $\\mathcal{O}_{X, p}^{sh}$", "is a strict normal crossings divisor.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) is straightforward and", "does not need the assumption that $X$ is J-2 or Nagata.", "Namely, let $p \\in D$ and choose an \\'etale neighbourhood", "$(U, u) \\to (X, p)$ such that the pullback of $D$ is", "a strict normal crossings divisor on $U$.", "Then $\\mathcal{O}_{X, p}^{sh} = \\mathcal{O}_{U, u}^{sh}$", "and we see that the trace of $D$ on $\\Spec(\\mathcal{O}_{U, u}^{sh})$", "is cut out by part of a regular system of parameters", "as this is already the case in $\\mathcal{O}_{U, u}$.", "\\medskip\\noindent", "To prove the implication in the other direction", "we will use the criterion of", "Lemma \\ref{lemma-characterize-normal-crossings-normalization}.", "Observe that formation of the normalization $D^\\nu \\to D$", "commutes with strict henselization, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-normalization-and-henselization}.", "If we can show that $D^\\nu \\to D$ is finite,", "then we see that $D^\\nu \\to D$ and the schemes", "$Z_n$ satisfy all desired properties because these", "can all be checked on the level of local rings", "(but the finiteness of the morphism $D^\\nu \\to D$", "is not something we can check on local rings).", "We omit the detailed verifications.", "\\medskip\\noindent", "If $X$ is Nagata, then $D^\\nu \\to D$ is finite by", "Morphisms, Lemma \\ref{morphisms-lemma-nagata-normalization}.", "\\medskip\\noindent", "Assume $X$ is J-2. Choose a point $p \\in D$. We will show", "that $D^\\nu \\to D$ is finite over a neighbourhood of $p$.", "By assumption there exists a regular system of", "parameters $f_1, \\ldots, f_d$ of $\\mathcal{O}_{X, p}^{sh}$", "and $1 \\leq r \\leq d$ such that the trace of $D$ on", "$\\Spec(\\mathcal{O}_{X, p}^{sh})$ is cut out by $f_1 \\ldots f_r$.", "Then", "$$", "D^\\nu \\times_X \\Spec(\\mathcal{O}_{X, p}^{sh}) = ", "\\coprod\\nolimits_{i = 1, \\ldots, r} V(f_i)", "$$", "Choose an affine \\'etale neighbourhood", "$(U, u) \\to (X, p)$ such that $f_i$ comes from", "$f_i \\in \\mathcal{O}_U(U)$. Set $D_i = V(f_i) \\subset U$.", "The strict henselization of $\\mathcal{O}_{D_i, u}$", "is $\\mathcal{O}_{X, p}^{sh}/(f_i)$ which is regular.", "Hence $\\mathcal{O}_{D_i, u}$ is regular (for example by", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-regular}).", "Because $X$ is J-2 the regular locus is open in $D_i$.", "Thus after replacing $U$ by a Zariski open we may assume", "that $D_i$ is regular for each $i$. It follows that", "$$", "\\coprod\\nolimits_{i = 1, \\ldots, r} D_i = D^\\nu \\times_X U", "\\longrightarrow D \\times_X U", "$$", "is the normalization morphism and it is clearly finite.", "In other words, we have found", "an \\'etale neighbourhood $(U, u)$ of $(X, p)$ such that", "the base change of $D^\\nu \\to D$ to this neighbourhood is finite.", "This implies $D^\\nu \\to D$ is finite by descent", "(Descent, Lemma \\ref{descent-lemma-descending-property-finite})", "and the proof is complete." ], "refs": [ "etale-lemma-characterize-normal-crossings-normalization", "more-morphisms-lemma-normalization-and-henselization", "morphisms-lemma-nagata-normalization", "more-algebra-lemma-henselization-regular", "descent-lemma-descending-property-finite" ], "ref_ids": [ 10724, 13776, 5520, 10064, 14688 ] } ], "ref_ids": [] }, { "id": 10726, "type": "theorem", "label": "etale-proposition-properties-sections", "categories": [ "etale" ], "title": "etale-proposition-properties-sections", "contents": [ "Sections of unramified morphisms.", "\\begin{enumerate}", "\\item Any section of an unramified morphism is an open immersion.", "\\item Any section of a separated morphism is a closed immersion.", "\\item Any section of an unramified separated morphism is open and closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Fix a base scheme $S$.", "If $f : X' \\to X$ is any $S$-morphism, then the graph", "$\\Gamma_f : X' \\to X' \\times_S X$", "is obtained as the base change of the diagonal", "$\\Delta_{X/S} : X \\to X \\times_S X$ via the projection", "$X' \\times_S X \\to X \\times_S X$.", "If $g : X \\to S$ is separated (resp. unramified)", "then the diagonal is a closed immersion (resp. open immersion)", "by Schemes, Definition \\ref{schemes-definition-separated}", "(resp.\\ Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism}).", "Hence so is the graph as a base change (by", "Schemes, Lemma \\ref{schemes-lemma-base-change-immersion}).", "In the special case $X' = S$, we obtain (1), resp.\\ (2).", "Part (3) follows on combining (1) and (2)." ], "refs": [ "schemes-definition-separated", "morphisms-lemma-diagonal-unramified-morphism", "schemes-lemma-base-change-immersion" ], "ref_ids": [ 7756, 5354, 7695 ] } ], "ref_ids": [] }, { "id": 10727, "type": "theorem", "label": "etale-proposition-equality", "categories": [ "etale" ], "title": "etale-proposition-equality", "contents": [ "Let $S$ is be a scheme.", "Let $\\pi : X \\to S$ be unramified and separated.", "Let $Y$ be an $S$-scheme and $y \\in Y$ a point.", "Let $f, g : Y \\to X$ be two $S$-morphisms. Assume", "\\begin{enumerate}", "\\item $Y$ is connected", "\\item $x = f(y) = g(y)$, and", "\\item the induced maps $f^\\sharp, g^\\sharp : \\kappa(x) \\to \\kappa(y)$", "on residue fields are equal.", "\\end{enumerate}", "Then $f = g$." ], "refs": [], "proofs": [ { "contents": [ "The maps $f, g : Y \\to X$ define maps $f', g' : Y \\to X_Y = Y \\times_S X$", "which are sections of the structure map $X_Y \\to Y$.", "Note that $f = g$ if and only if $f' = g'$.", "The structure map $X_Y \\to Y$ is the base change of $\\pi$ and hence", "unramified and separated also (see", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-unramified} and", "Schemes, Lemma \\ref{schemes-lemma-separated-permanence}).", "Thus according to Theorem \\ref{theorem-sections-unramified-maps}", "it suffices to prove that $f'$ and $g'$ pass through the same", "point of $X_Y$. And this is exactly what the hypotheses (2) and (3)", "guarantee, namely $f'(y) = g'(y) \\in X_Y$." ], "refs": [ "morphisms-lemma-base-change-unramified", "schemes-lemma-separated-permanence", "etale-theorem-sections-unramified-maps" ], "ref_ids": [ 5346, 7714, 10683 ] } ], "ref_ids": [] }, { "id": 10728, "type": "theorem", "label": "etale-proposition-etale-depth", "categories": [ "etale" ], "title": "etale-proposition-etale-depth", "contents": [ "Let $A$, $B$ be Noetherian local rings.", "Let $f : A \\to B$ be an \\'etale homomorphism of local rings.", "Then $\\text{depth}(A) = \\text{depth}(B)$" ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck}." ], "refs": [ "algebra-lemma-apply-grothendieck" ], "ref_ids": [ 1361 ] } ], "ref_ids": [] }, { "id": 10729, "type": "theorem", "label": "etale-proposition-etale-CM", "categories": [ "etale" ], "title": "etale-proposition-etale-CM", "contents": [ "\\begin{slogan}", "Being Cohen-Macaulay ascends and descends along \\'etale maps.", "\\end{slogan}", "Let $A$, $B$ be Noetherian local rings.", "Let $f : A \\to B$ be an \\'etale homomorphism of local rings.", "Then $A$ is Cohen-Macaulay if and only if $B$ is so." ], "refs": [], "proofs": [ { "contents": [ "A local ring $A$ is Cohen-Macaulay if and only if $\\dim(A) = \\text{depth}(A)$.", "As both of these invariants is preserved under an \\'etale extension,", "the claim follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10730, "type": "theorem", "label": "etale-proposition-etale-regular", "categories": [ "etale" ], "title": "etale-proposition-etale-regular", "contents": [ "Let $A$, $B$ be Noetherian local rings.", "Let $f : A \\to B$ be an \\'etale homomorphism of local rings.", "Then $A$ is regular if and only if $B$ is so." ], "refs": [], "proofs": [ { "contents": [ "If $B$ is regular, then $A$ is regular by", "Algebra, Lemma \\ref{algebra-lemma-flat-under-regular}.", "Assume $A$ is regular. Let $\\mathfrak m$ be the maximal ideal", "of $A$. Then $\\dim_{\\kappa(\\mathfrak m)} \\mathfrak m/\\mathfrak m^2 =", "\\dim(A) = \\dim(B)$ (see Lemma \\ref{lemma-etale-dimension}).", "On the other hand, $\\mathfrak mB$ is the maximal ideal of", "$B$ and hence $\\mathfrak m_B/\\mathfrak m_B = \\mathfrak mB/\\mathfrak m^2B$", "is generated by at most $\\dim(B)$ elements. Thus $B$ is regular.", "(You can also use the slightly more general", "Algebra, Lemma \\ref{algebra-lemma-flat-over-regular-with-regular-fibre}.)" ], "refs": [ "algebra-lemma-flat-under-regular", "etale-lemma-etale-dimension", "algebra-lemma-flat-over-regular-with-regular-fibre" ], "ref_ids": [ 981, 10715, 988 ] } ], "ref_ids": [] }, { "id": 10731, "type": "theorem", "label": "etale-proposition-etale-reduced", "categories": [ "etale" ], "title": "etale-proposition-etale-reduced", "contents": [ "Let $A$, $B$ be Noetherian local rings.", "Let $f : A \\to B$ be an \\'etale homomorphism of local rings.", "Then $A$ is reduced if and only if $B$ is so." ], "refs": [], "proofs": [ { "contents": [ "It is clear from the faithful flatness of $A \\to B$ that if $B$ is reduced, so", "is $A$. See also Algebra, Lemma \\ref{algebra-lemma-descent-reduced}.", "Conversely, assume $A$ is reduced. By assumption $B$ is a localization", "of a finite type $A$-algebra $B'$ at some prime $\\mathfrak q$.", "After replacing $B'$ by a localization we may assume that $B'$", "is \\'etale over $A$, see Lemma \\ref{lemma-characterize-etale-Noetherian}.", "Then we see that Algebra, Lemma \\ref{algebra-lemma-reduced-goes-up} applies to", "$A \\to B'$ and $B'$ is reduced. Hence $B$ is reduced." ], "refs": [ "algebra-lemma-descent-reduced", "etale-lemma-characterize-etale-Noetherian", "algebra-lemma-reduced-goes-up" ], "ref_ids": [ 1371, 10704, 1366 ] } ], "ref_ids": [] }, { "id": 10732, "type": "theorem", "label": "etale-proposition-etale-normal", "categories": [ "etale" ], "title": "etale-proposition-etale-normal", "contents": [ "Let $A$, $B$ be Noetherian local rings.", "Let $f : A \\to B$ be an \\'etale homomorphism of local rings.", "Then $A$ is a normal domain if and only if $B$ is so." ], "refs": [], "proofs": [ { "contents": [ "See", "Algebra, Lemma \\ref{algebra-lemma-descent-normal}", "for descending normality. Conversely, assume $A$ is normal.", "By assumption $B$ is a localization of a finite type $A$-algebra", "$B'$ at some prime $\\mathfrak q$. After replacing $B'$ by a localization", "we may assume that $B'$ is \\'etale over $A$, see", "Lemma \\ref{lemma-characterize-etale-Noetherian}.", "Then we see that", "Algebra, Lemma \\ref{algebra-lemma-normal-goes-up}", "applies to $A \\to B'$ and we conclude that $B'$ is normal.", "Hence $B$ is a normal domain." ], "refs": [ "algebra-lemma-descent-normal", "etale-lemma-characterize-etale-Noetherian", "algebra-lemma-normal-goes-up" ], "ref_ids": [ 1372, 10704, 1368 ] } ], "ref_ids": [] }, { "id": 10733, "type": "theorem", "label": "etale-proposition-effective", "categories": [ "etale" ], "title": "etale-proposition-effective", "contents": [ "Let $f : X \\to S$ be a surjective integral morphism.", "The functor (\\ref{equation-descent-etale}) induces an equivalence", "$$", "\\begin{matrix}", "\\text{schemes quasi-compact,}\\\\", "\\text{separated, \\'etale over }S", "\\end{matrix}", "\\longrightarrow", "\\begin{matrix}", "\\text{descent data }(V, \\varphi)\\text{ relative to }X/S\\text{ with}\\\\", "V\\text{ quasi-compact, separated, \\'etale over }X", "\\end{matrix}", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-fully-faithful-cases} the", "functor (\\ref{equation-descent-etale})", "is fully faithful and the same remains the case after any", "base change $S \\to S'$. Let $(V, \\varphi)$ be a descent data", "relative to $X/S$ with $V \\to X$ quasi-compact, separated, and \\'etale.", "We can use Lemma \\ref{lemma-reduce-to-affine}", "to see that it suffices to prove the effectivity ", "Zariski locally on $S$. In particular we may and do", "assume that $S$ is affine.", "\\medskip\\noindent", "If $S$ is affine we can find a directed set $\\Lambda$ and", "an inverse system $X_\\lambda \\to S_\\lambda$", "of finite morphisms of affine schemes of finite type over", "$\\Spec(\\mathbf{Z})$ such that $(X \\to S) = \\lim (X_\\lambda \\to S_\\lambda)$.", "See Algebra, Lemma \\ref{algebra-lemma-limit-integral}.", "Since limits commute with limits we deduce that", "$X \\times_S X = \\lim X_\\lambda \\times_{S_\\lambda} X_\\lambda$", "and ", "$X \\times_S X \\times_S X = \\lim", "X_\\lambda \\times_{S_\\lambda} X_\\lambda \\times_{S_\\lambda} X_\\lambda$.", "Observe that $V \\to X$ is a morphism of finite presentation.", "Using Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation}", "we can find an $\\lambda$ and a descent datum $(V_\\lambda, \\varphi_\\lambda)$", "relative to $X_\\lambda/S_\\lambda$ whose pullback to $X/S$ is", "$(V, \\varphi)$. Of course it is enough to show that", "$(V_\\lambda, \\varphi_\\lambda)$ is effective. Note that $V_\\lambda$", "is quasi-compact by construction.", "After possibly increasing $\\lambda$ we may assume", "that $V_\\lambda \\to X_\\lambda$ is separated and \\'etale, see", "Limits, Lemma \\ref{limits-lemma-descend-separated-finite-presentation} and", "\\ref{limits-lemma-descend-etale}.", "Thus we may assume that $f$ is finite surjective and", "$S$ affine of finite type over $\\mathbf{Z}$.", "\\medskip\\noindent", "Consider an open $S' \\subset S$ such that the pullback $(V', \\varphi')$", "of $(V, \\varphi)$ to $X' = X \\times_S S'$ is effective. Below we will", "prove, that $S' \\not = S$ implies there is a strictly larger open over", "which the descent datum is effective. Since $S$ is Noetherian (and hence", "has a Noetherian underlying topological space) this will finish the proof.", "Let $\\xi \\in S$ be a generic point of an irreducible component of the", "closed subset $Z = S \\setminus S'$.", "If $\\xi \\in S'' \\subset S$ is an open over which the descent datum is", "effective, then the descent datum is effective over", "$S' \\cup S''$ by the glueing argument of the first paragraph. Thus", "in the rest of the proof we may replace $S$ by an affine open", "neighbourhood of $\\xi$.", "\\medskip\\noindent", "After a first such replacement we may assume that $Z$ is irreducible", "with generic point $Z$. Let us endow $Z$ with the reduced induced", "closed subscheme structure. After another shrinking we may assume", "$X_Z = X \\times_S Z = f^{-1}(Z) \\to Z$ is flat, see", "Morphisms, Proposition \\ref{morphisms-proposition-generic-flatness}.", "Let $(V_Z, \\varphi_Z)$ be the pullback of the descent datum to $X_Z/Z$.", "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend}", "this descent datum is effective and we obtain an \\'etale morphism", "$U_Z \\to Z$ whose base change is isomorphic to $V_Z$ in a manner", "compatible with descent data.", "Of course $U_Z \\to Z$ is quasi-compact and separated", "(Descent, Lemmas \\ref{descent-lemma-descending-property-quasi-compact} and", "\\ref{descent-lemma-descending-property-separated}).", "Thus after shrinking once more we may assume", "that $U_Z \\to Z$ is finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-generically-finite}.", "\\medskip\\noindent", "Let $S = \\Spec(A)$ and let $I \\subset A$ be the prime ideal corresponding", "to $Z \\subset S$. Let $(A^h, IA^h)$ be the henselization of the pair", "$(A, I)$. Denote $S^h = \\Spec(A^h)$ and $Z^h = V(IA^h) \\cong Z$.", "We claim that it suffices to show effectivity after base change to", "$S^h$. Namely, $\\{S^h \\to S, S' \\to S\\}$ is an fpqc covering", "($A \\to A^h$ is flat by More on Algebra, Lemma", "\\ref{more-algebra-lemma-henselization-flat}) and", "by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend}", "we have fpqc descent for separated \\'etale morphisms.", "Namely, if $U^h \\to S^h$ and $U' \\to S'$ are the objects", "corresponding to the pullbacks $(V^h, \\varphi^h)$ and", "$(V', \\varphi')$, then the required isomorphisms", "$$", "U^h \\times_S S^h \\to S^h \\times_S V^h", "\\quad\\text{and}\\quad", "U^h \\times_S S' \\to S^h \\times_S U'", "$$", "are obtained by the fully faithfulness pointed out in the first", "paragraph. In this way we reduce to the situation described in", "the next paragraph.", "\\medskip\\noindent", "Here $S = \\Spec(A)$, $Z = V(I)$, $S' = S \\setminus Z$ where", "$(A, I)$ is a henselian pair, we have $U' \\to S'$ corresponding", "to the descent datum $(V', \\varphi')$ and we have a finite \\'etale", "morphism $U_Z \\to Z$ corresponding to the descent datum", "$(V_Z, \\varphi_Z)$. We no longer have that $A$ is of finite type", "over $\\mathbf{Z}$; but the rest of the argument will not even use", "that $A$ is Noetherian.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-finite-etale-equivalence}", "we can find a finite \\'etale morphism $U_{fin} \\to S$ whose", "restriction to $Z$ is isomorphic to $U_Z \\to Z$.", "Write $X = \\Spec(B)$ and $Y = V(IB)$. Since $(B, IB)$ is a henselian pair", "(More on Algebra, Lemma \\ref{more-algebra-lemma-integral-over-henselian-pair})", "and since the restriction $V \\to X$ to $Y$", "is finite (as base change of $U_Z \\to Z$) we see that", "there is a canonical disjoint union decomposition", "$$", "V = V_{fin} \\amalg V_{away}", "$$", "were $V_{fin} \\to X$ is finite and where $V_{away}$ has no", "points lying over $Y$. See Lemma \\ref{lemma-split-henselian}.", "Using the uniqueness of this decomposition over $X \\times_S X$", "we see that $\\varphi$ preserves it and we obtain", "$$", "(V, \\varphi) = (V_{fin}, \\varphi_{fin}) \\amalg (V_{away}, \\varphi_{away})", "$$", "in the category of descent data.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-finite-etale-equivalence}", "there is a unique isomorphism", "$$", "X \\times_S U_{fin} \\longrightarrow V_{fin}", "$$", "compatible with the given isomorphism $Y \\times_Z U_Z \\to V \\times_X Y$", "over $Y$.", "By the uniqueness we see that this isomorphism is compatible", "with descent data, i.e.,", "$(X \\times_S U_{fin}, can) \\cong (V_{fin}, \\varphi_{fin})$.", "Denote $U'_{fin} = U_{fin} \\times_S S'$. By fully faithfulness", "we obtain a morphism $U'_{fin} \\to U'$ which is", "the inclusion of an open (and closed) subscheme.", "Then we set $U = U_{fin} \\amalg_{U'_{fin}} U'$ (glueing of schemes as", "in Schemes, Section \\ref{schemes-section-glueing-schemes}).", "The morphisms $X \\times_S U_{fin} \\to V$ and", "$X \\times_S U' \\to V$ glue to a morphism $X \\times_S U \\to V$", "which is the desired isomorphism." ], "refs": [ "etale-lemma-fully-faithful-cases", "etale-lemma-reduce-to-affine", "algebra-lemma-limit-integral", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-separated-finite-presentation", "limits-lemma-descend-etale", "morphisms-proposition-generic-flatness", "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "descent-lemma-descending-property-quasi-compact", "descent-lemma-descending-property-separated", "morphisms-lemma-generically-finite", "more-algebra-lemma-henselization-flat", "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "more-algebra-lemma-finite-etale-equivalence", "more-algebra-lemma-integral-over-henselian-pair", "etale-lemma-split-henselian", "more-algebra-lemma-finite-etale-equivalence" ], "ref_ids": [ 10718, 10719, 1103, 15077, 15061, 15065, 5533, 13949, 14666, 14671, 5487, 9872, 13949, 9879, 9863, 10720, 9879 ] } ], "ref_ids": [] }, { "id": 10743, "type": "theorem", "label": "crystalline-theorem-cohomology-F-crystal", "categories": [ "crystalline" ], "title": "crystalline-theorem-cohomology-F-crystal", "contents": [ "In Situation \\ref{situation-F-crystal} let $(\\mathcal{E}, F_\\mathcal{E})$", "be a nondegenerate $F$-crystal. Assume $A$ is a $p$-adically complete", "Noetherian ring and that $X \\to S_0$ is proper smooth. Then", "the canonical map", "$$", "F_\\mathcal{E} \\circ (F_X)_{\\text{cris}}^* :", "R\\Gamma(\\text{Cris}(X/S), \\mathcal{E}) \\otimes^\\mathbf{L}_{A, \\sigma} A", "\\longrightarrow", "R\\Gamma(\\text{Cris}(X/S), \\mathcal{E})", "$$", "becomes an isomorphism after inverting $p$." ], "refs": [], "proofs": [ { "contents": [ "We first write the arrow as a composition of three arrows.", "Namely, set", "$$", "X^{(1)} = X \\times_{S_0, F_{S_0}} S_0", "$$", "and denote $F_{X/S_0} : X \\to X^{(1)}$ the relative Frobenius morphism.", "Denote $\\mathcal{E}^{(1)}$ the base change of $\\mathcal{E}$", "by $\\Spec(\\sigma)$, in other words the pullback of $\\mathcal{E}$", "to $\\text{Cris}(X^{(1)}/S)$ by the morphism of crystalline topoi", "associated to the commutative diagram", "$$", "\\xymatrix{", "X^{(1)} \\ar[r] \\ar[d] & X \\ar[d] \\\\", "S \\ar[r]^{\\Spec(\\sigma)} & S", "}", "$$", "Then we have the base change map", "\\begin{equation}", "\\label{equation-base-change-sigma}", "R\\Gamma(\\text{Cris}(X/S), \\mathcal{E}) \\otimes^\\mathbf{L}_{A, \\sigma} A", "\\longrightarrow", "R\\Gamma(\\text{Cris}(X^{(1)}/S), \\mathcal{E}^{(1)})", "\\end{equation}", "see Remark \\ref{remark-base-change}. Note that the composition", "of $F_{X/S_0} : X \\to X^{(1)}$ with the projection $X^{(1)} \\to X$", "is the absolute Frobenius morphism $F_X$. Hence we see that", "$F_{X/S_0}^*\\mathcal{E}^{(1)} = (F_X)_{\\text{cris}}^*\\mathcal{E}$.", "Thus pullback by $F_{X/S_0}$ is a map", "\\begin{equation}", "\\label{equation-to-prove}", "F_{X/S_0}^* :", "R\\Gamma(\\text{Cris}(X^{(1)}/S), \\mathcal{E}^{(1)})", "\\longrightarrow", "R\\Gamma(\\text{Cris}(X/S), (F_X)^*_{\\text{cris}}\\mathcal{E})", "\\end{equation}", "Finally we can use $F_\\mathcal{E}$ to get a map", "\\begin{equation}", "\\label{equation-F-E}", "R\\Gamma(\\text{Cris}(X/S), (F_X)^*_{\\text{cris}}\\mathcal{E})", "\\longrightarrow", "R\\Gamma(\\text{Cris}(X/S), \\mathcal{E})", "\\end{equation}", "The map of the theorem is the composition of the three maps", "(\\ref{equation-base-change-sigma}), (\\ref{equation-to-prove}), and", "(\\ref{equation-F-E}) above. The first is a", "quasi-isomorphism modulo all powers of $p$ by", "Remark \\ref{remark-base-change-isomorphism}.", "Hence it is a quasi-isomorphism since the complexes involved are perfect", "in $D(A)$ see Remark \\ref{remark-complete-perfect}.", "The third map is a quasi-isomorphism after inverting $p$ simply", "because $F_\\mathcal{E}$ has an inverse up to a power of $p$, see", "Remark \\ref{remark-F-crystal-variants}.", "Finally, the second is an isomorphism after inverting $p$", "by Lemma \\ref{lemma-pullback-relative-frobenius}." ], "refs": [ "crystalline-remark-base-change", "crystalline-remark-base-change-isomorphism", "crystalline-remark-complete-perfect", "crystalline-remark-F-crystal-variants", "crystalline-lemma-pullback-relative-frobenius" ], "ref_ids": [ 10836, 10837, 10841, 10843, 10792 ] } ], "ref_ids": [] }, { "id": 10744, "type": "theorem", "label": "crystalline-lemma-divided-power-envelope", "categories": [ "crystalline" ], "title": "crystalline-lemma-divided-power-envelope", "contents": [ "Let $(A, I, \\gamma)$ be a divided power ring.", "Let $A \\to B$ be a ring map. Let $J \\subset B$ be an ideal", "with $IB \\subset J$. There exists a homomorphism of", "divided power rings", "$$", "(A, I, \\gamma) \\longrightarrow (D, \\bar J, \\bar \\gamma)", "$$", "such that", "$$", "\\Hom_{(A, I, \\gamma)}((D, \\bar J, \\bar \\gamma), (C, K, \\delta)) =", "\\Hom_{(A, I)}((B, J), (C, K))", "$$", "functorially in the divided power algebra $(C, K, \\delta)$ over", "$(A, I, \\gamma)$. Here the LHS is morphisms of divided", "power rings over $(A, I, \\gamma)$ and the RHS is morphisms of", "(ring, ideal) pairs over $(A, I)$." ], "refs": [], "proofs": [ { "contents": [ "Denote $\\mathcal{C}$ the category of divided power rings", "$(C, K, \\delta)$. Consider the functor", "$F : \\mathcal{C} \\longrightarrow \\textit{Sets}$ defined by", "$$", "F(C, K, \\delta) =", "\\left\\{", "(\\varphi, \\psi)", "\\middle|", "\\begin{matrix}", "\\varphi : (A, I, \\gamma) \\to (C, K, \\delta)", "\\text{ homomorphism of divided power rings} \\\\", "\\psi : (B, J) \\to (C, K)\\text{ an }", "A\\text{-algebra homomorphism with }\\psi(J) \\subset K", "\\end{matrix}", "\\right\\}", "$$", "We will show that", "Divided Power Algebra, Lemma \\ref{dpa-lemma-a-version-of-brown}", "applies to this functor which will", "prove the lemma. Suppose that $(\\varphi, \\psi) \\in F(C, K, \\delta)$.", "Let $C' \\subset C$ be the subring generated by $\\varphi(A)$,", "$\\psi(B)$, and $\\delta_n(\\psi(f))$ for all $f \\in J$.", "Let $K' \\subset K \\cap C'$ be the ideal of $C'$ generated by", "$\\varphi(I)$ and $\\delta_n(\\psi(f))$ for $f \\in J$.", "Then $(C', K', \\delta|_{K'})$ is a divided power ring and", "$C'$ has cardinality bounded by the cardinal", "$\\kappa = |A| \\otimes |B|^{\\aleph_0}$.", "Moreover, $\\varphi$ factors as $A \\to C' \\to C$ and $\\psi$ factors", "as $B \\to C' \\to C$. This proves assumption (1) of", "Divided Power Algebra, Lemma \\ref{dpa-lemma-a-version-of-brown}", "holds. Assumption (2) is clear", "as limits in the category of divided power rings commute with the", "forgetful functor $(C, K, \\delta) \\mapsto (C, K)$, see", "Divided Power Algebra, Lemma \\ref{dpa-lemma-limits} and its proof." ], "refs": [ "dpa-lemma-a-version-of-brown", "dpa-lemma-a-version-of-brown", "dpa-lemma-limits" ], "ref_ids": [ 1655, 1655, 1654 ] } ], "ref_ids": [] }, { "id": 10745, "type": "theorem", "label": "crystalline-lemma-divided-power-envelop-quotient", "categories": [ "crystalline" ], "title": "crystalline-lemma-divided-power-envelop-quotient", "contents": [ "Let $(A, I, \\gamma)$ be a divided power ring.", "Let $\\varphi : B' \\to B$ be a surjection of $A$-algebras with kernel $K$.", "Let $IB \\subset J \\subset B$ be an ideal. Let $J' \\subset B'$", "be the inverse image of $J$. Write", "$D_{B', \\gamma}(J') = (D', \\bar J', \\bar\\gamma)$.", "Then $D_{B, \\gamma}(J) = (D'/K', \\bar J'/K', \\bar\\gamma)$", "where $K'$ is the ideal generated by the elements $\\bar\\gamma_n(k)$", "for $n \\geq 1$ and $k \\in K$." ], "refs": [], "proofs": [ { "contents": [ "Write $D_{B, \\gamma}(J) = (D, \\bar J, \\bar \\gamma)$.", "The universal property of $D'$ gives us a homomorphism $D' \\to D$", "of divided power algebras. As $B' \\to B$ and $J' \\to J$ are surjective, we", "see that $D' \\to D$ is surjective (see remarks above). It is clear that", "$\\bar\\gamma_n(k)$ is in the kernel for $n \\geq 1$ and $k \\in K$, i.e.,", "we obtain a homomorphism $D'/K' \\to D$. Conversely, there exists a divided", "power structure on $\\bar J'/K' \\subset D'/K'$, see", "Divided Power Algebra, Lemma \\ref{dpa-lemma-kernel}.", "Hence the universal property of $D$ gives an inverse $D \\to D'/K'$ and we win." ], "refs": [ "dpa-lemma-kernel" ], "ref_ids": [ 1658 ] } ], "ref_ids": [] }, { "id": 10746, "type": "theorem", "label": "crystalline-lemma-describe-divided-power-envelope", "categories": [ "crystalline" ], "title": "crystalline-lemma-describe-divided-power-envelope", "contents": [ "Let $(B, I, \\gamma)$ be a divided power algebra. Let $I \\subset J \\subset B$", "be an ideal. Let $(D, \\bar J, \\bar \\gamma)$ be the divided power envelope", "of $J$ relative to $\\gamma$. Choose elements $f_t \\in J$, $t \\in T$ such", "that $J = I + (f_t)$. Then there exists a surjection", "$$", "\\Psi : B\\langle x_t \\rangle \\longrightarrow D", "$$", "of divided power rings mapping $x_t$ to the image of $f_t$ in $D$.", "The kernel of $\\Psi$ is generated by the elements $x_t - f_t$ and", "all", "$$", "\\delta_n\\left(\\sum r_t x_t - r_0\\right)", "$$", "whenever $\\sum r_t f_t = r_0$ in $B$ for some $r_t \\in B$, $r_0 \\in I$." ], "refs": [], "proofs": [ { "contents": [ "In the statement of the lemma we think of $B\\langle x_t \\rangle$", "as a divided power ring with ideal", "$J' = IB\\langle x_t \\rangle + B\\langle x_t \\rangle_{+}$, see", "Divided Power Algebra, Remark \\ref{dpa-remark-divided-power-polynomial-algebra}.", "The existence of $\\Psi$ follows from the universal property of", "divided power polynomial rings. Surjectivity of $\\Psi$ follows from", "the fact that its image is a divided power subring of $D$, hence equal to $D$", "by the universal property of $D$. It is clear that", "$x_t - f_t$ is in the kernel. Set", "$$", "\\mathcal{R} = \\{(r_0, r_t) \\in I \\oplus \\bigoplus\\nolimits_{t \\in T} B", "\\mid \\sum r_t f_t = r_0 \\text{ in }B\\}", "$$", "If $(r_0, r_t) \\in \\mathcal{R}$ then it is clear that", "$\\sum r_t x_t - r_0$ is in the kernel.", "As $\\Psi$ is a homomorphism of divided power rings", "and $\\sum r_tx_t - r_0 \\in J'$", "it follows that $\\delta_n(\\sum r_t x_t - r_0)$ is in the kernel as well.", "Let $K \\subset B\\langle x_t \\rangle$ be the ideal generated by", "$x_t - f_t$ and the elements $\\delta_n(\\sum r_t x_t - r_0)$ for", "$(r_0, r_t) \\in \\mathcal{R}$.", "To show that $K = \\Ker(\\Psi)$ it suffices to show that", "$\\delta$ extends to $B\\langle x_t \\rangle/K$. Namely, if so the universal", "property of $D$ gives a map $D \\to B\\langle x_t \\rangle/K$", "inverse to $\\Psi$. Hence we have to show that $K \\cap J'$ is", "preserved by $\\delta_n$, see", "Divided Power Algebra, Lemma \\ref{dpa-lemma-kernel}.", "Let $K' \\subset B\\langle x_t \\rangle$ be the ideal", "generated by the elements", "\\begin{enumerate}", "\\item $\\delta_m(\\sum r_t x_t - r_0)$ where $m > 0$ and", "$(r_0, r_t) \\in \\mathcal{R}$,", "\\item $x_{t'}^{[m]}(x_t - f_t)$ where $m > 0$ and $t', t \\in I$.", "\\end{enumerate}", "We claim that $K' = K \\cap J'$. The claim proves that $K \\cap J'$", "is preserved by $\\delta_n$, $n > 0$ by the criterion of", "Divided Power Algebra, Lemma \\ref{dpa-lemma-kernel} (2)(c)", "and a computation of $\\delta_n$", "of the elements listed which we leave to the reader.", "To prove the claim note that $K' \\subset K \\cap J'$.", "Conversely, if $h \\in K \\cap J'$ then, modulo $K'$ we can write", "$$", "h = \\sum r_t (x_t - f_t)", "$$", "for some $r_t \\in B$. As $h \\in K \\cap J' \\subset J'$", "we see that $r_0 = \\sum r_t f_t \\in I$. Hence $(r_0, r_t) \\in \\mathcal{R}$", "and we see that", "$$", "h = \\sum r_t x_t - r_0", "$$", "is in $K'$ as desired." ], "refs": [ "dpa-remark-divided-power-polynomial-algebra", "dpa-lemma-kernel", "dpa-lemma-kernel" ], "ref_ids": [ 1703, 1658, 1658 ] } ], "ref_ids": [] }, { "id": 10747, "type": "theorem", "label": "crystalline-lemma-divided-power-envelope-add-variables", "categories": [ "crystalline" ], "title": "crystalline-lemma-divided-power-envelope-add-variables", "contents": [ "Let $(A, I, \\gamma)$ be a divided power ring.", "Let $B$ be an $A$-algebra and $IB \\subset J \\subset B$ an ideal.", "Let $x_i$ be a set of variables. Then", "$$", "D_{B[x_i], \\gamma}(JB[x_i] + (x_i)) = D_{B, \\gamma}(J) \\langle x_i \\rangle", "$$" ], "refs": [], "proofs": [ { "contents": [ "One possible proof is to deduce this from", "Lemma \\ref{lemma-describe-divided-power-envelope}", "as any relation between $x_i$ in $B[x_i]$ is trivial.", "On the other hand, the lemma follows from the universal property", "of the divided power polynomial algebra and the universal property of", "divided power envelopes." ], "refs": [ "crystalline-lemma-describe-divided-power-envelope" ], "ref_ids": [ 10746 ] } ], "ref_ids": [] }, { "id": 10748, "type": "theorem", "label": "crystalline-lemma-flat-base-change-divided-power-envelope", "categories": [ "crystalline" ], "title": "crystalline-lemma-flat-base-change-divided-power-envelope", "contents": [ "Let $(A, I, \\gamma)$ be a divided power ring.", "Let $B \\to B'$ be a homomorphism of $A$-algebras.", "Assume that", "\\begin{enumerate}", "\\item $B/IB \\to B'/IB'$ is flat, and", "\\item $\\text{Tor}_1^B(B', B/IB) = 0$.", "\\end{enumerate}", "Then for any ideal $IB \\subset J \\subset B$ the canonical map", "$$", "D_B(J) \\otimes_B B' \\longrightarrow D_{B'}(JB')", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Set $D = D_B(J)$ and denote $\\bar J \\subset D$ its divided power ideal", "with divided power structure $\\bar\\gamma$. The universal property of", "$D$ produces a $B$-algebra map $D \\to D_{B'}(JB')$, whence a map as in", "the lemma. It suffices to show that", "the divided powers $\\bar\\gamma$ extend to $D \\otimes_B B'$ since then", "the universal property of $D_{B'}(JB')$ will produce a map", "$D_{B'}(JB') \\to D \\otimes_B B'$ inverse to the one in the lemma.", "\\medskip\\noindent", "Choose a surjection $P \\to B'$ where $P$ is a polynomial algebra over $B$.", "In particular $B \\to P$ is flat, hence $D \\to D \\otimes_B P$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-flat-base-change}.", "Then $\\bar\\gamma$ extends to $D \\otimes_B P$ by", "Divided Power Algebra, Lemma \\ref{dpa-lemma-gamma-extends}; we", "will denote this extension", "$\\bar\\gamma$ also. Set $\\mathfrak a = \\Ker(P \\to B')$ so that", "we have the short exact sequence", "$$", "0 \\to \\mathfrak a \\to P \\to B' \\to 0", "$$", "Thus $\\text{Tor}_1^B(B', B/IB) = 0$ implies that", "$\\mathfrak a \\cap IP = I\\mathfrak a$.", "Now we have the following commutative diagram", "$$", "\\xymatrix{", "B/J \\otimes_B \\mathfrak a \\ar[r]_\\beta &", "B/J \\otimes_B P \\ar[r] &", "B/J \\otimes_B B' \\\\", "D \\otimes_B \\mathfrak a \\ar[r]^\\alpha \\ar[u] &", "D \\otimes_B P \\ar[r] \\ar[u] &", "D \\otimes_B B' \\ar[u] \\\\", "\\bar J \\otimes_B \\mathfrak a \\ar[r] \\ar[u] &", "\\bar J \\otimes_B P \\ar[r] \\ar[u] &", "\\bar J \\otimes_B B' \\ar[u]", "}", "$$", "This diagram is exact even with $0$'s added at the top and the right.", "We have to show the divided powers on the ideal", "$\\bar J \\otimes_B P$ preserve the ideal", "$\\Im(\\alpha) \\cap \\bar J \\otimes_B P$, see", "Divided Power Algebra, Lemma \\ref{dpa-lemma-kernel}.", "Consider the exact sequence", "$$", "0 \\to \\mathfrak a/I\\mathfrak a \\to P/IP \\to B'/IB' \\to 0", "$$", "(which uses that $\\mathfrak a \\cap IP = I\\mathfrak a$ as seen above).", "As $B'/IB'$ is flat over $B/IB$ this sequence remains exact after", "applying $B/J \\otimes_{B/IB} -$, see", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}. Hence", "$$", "\\Ker(B/J \\otimes_{B/IB} \\mathfrak a/I\\mathfrak a \\to", "B/J \\otimes_{B/IB} P/IP) =", "\\Ker(\\mathfrak a/J\\mathfrak a \\to P/JP)", "$$", "is zero. Thus $\\beta$ is injective. It follows that", "$\\Im(\\alpha) \\cap \\bar J \\otimes_B P$ is the", "image of $\\bar J \\otimes \\mathfrak a$. Now if", "$f \\in \\bar J$ and $a \\in \\mathfrak a$, then", "$\\bar\\gamma_n(f \\otimes a) = \\bar\\gamma_n(f) \\otimes a^n$", "hence the result is clear." ], "refs": [ "algebra-lemma-flat-base-change", "dpa-lemma-gamma-extends", "dpa-lemma-kernel", "algebra-lemma-flat-tor-zero" ], "ref_ids": [ 527, 1657, 1658, 532 ] } ], "ref_ids": [] }, { "id": 10749, "type": "theorem", "label": "crystalline-lemma-flat-extension-divided-power-envelope", "categories": [ "crystalline" ], "title": "crystalline-lemma-flat-extension-divided-power-envelope", "contents": [ "Let $(B, I, \\gamma) \\to (B', I', \\gamma')$ be a homomorphism of", "divided power rings. Let $I \\subset J \\subset B$ and", "$I' \\subset J' \\subset B'$ be ideals. Assume", "\\begin{enumerate}", "\\item $B/I \\to B'/I'$ is flat, and", "\\item $J' = JB' + I'$.", "\\end{enumerate}", "Then the canonical map", "$$", "D_{B, \\gamma}(J) \\otimes_B B' \\longrightarrow D_{B', \\gamma'}(J')", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Set $D = D_{B, \\gamma}(J)$. Choose elements $f_t \\in J$ which generate $J/I$.", "Set $\\mathcal{R} = \\{(r_0, r_t) \\in I \\oplus \\bigoplus\\nolimits_{t \\in T} B", "\\mid \\sum r_t f_t = r_0 \\text{ in }B\\}$ as in the proof of", "Lemma \\ref{lemma-describe-divided-power-envelope}. This lemma shows that", "$$", "D = B\\langle x_t \\rangle/ K", "$$", "where $K$ is generated by the elements $x_t - f_t$ and", "$\\delta_n(\\sum r_t x_t - r_0)$ for $(r_0, r_t) \\in \\mathcal{R}$.", "Thus we see that", "\\begin{equation}", "\\label{equation-base-change}", "D \\otimes_B B' = B'\\langle x_t \\rangle/K'", "\\end{equation}", "where $K'$ is generated by the images in $B'\\langle x_t \\rangle$", "of the generators of $K$ listed above. Let $f'_t \\in B'$ be the image", "of $f_t$. By assumption (1) we see that the elements $f'_t \\in J'$", "generate $J'/I'$ and we see that $x_t - f'_t \\in K'$. Set", "$$", "\\mathcal{R}' =", "\\{(r'_0, r'_t) \\in I' \\oplus \\bigoplus\\nolimits_{t \\in T} B'", "\\mid \\sum r'_t f'_t = r'_0 \\text{ in }B'\\}", "$$", "To finish the proof we have to show that", "$\\delta'_n(\\sum r'_t x_t - r'_0) \\in K'$ for", "$(r'_0, r'_t) \\in \\mathcal{R}'$, because then the presentation", "(\\ref{equation-base-change}) of $D \\otimes_B B'$ is identical", "to the presentation of $D_{B', \\gamma'}(J')$ obtain in", "Lemma \\ref{lemma-describe-divided-power-envelope} from the generators $f'_t$.", "Suppose that $(r'_0, r'_t) \\in \\mathcal{R}'$. Then", "$\\sum r'_t f'_t = 0$ in $B'/I'$. As $B/I \\to B'/I'$ is flat by", "assumption (1) we can apply the equational criterion of flatness", "(Algebra, Lemma \\ref{algebra-lemma-flat-eq}) to see", "that there exist an $m > 0$ and", "$r_{jt} \\in B$ and $c_j \\in B'$, $j = 1, \\ldots, m$ such", "that", "$$", "r_{j0} = \\sum\\nolimits_t r_{jt} f_t \\in I \\text{ for } j = 1, \\ldots, m", "$$", "and", "$$", "i'_t = r'_t - \\sum\\nolimits_j c_j r_{jt} \\in I' \\text{ for all }t", "$$", "Note that this also implies that", "$r'_0 = \\sum_t i'_t f_t + \\sum_j c_j r_{j0}$.", "Then we have", "\\begin{align*}", "\\delta'_n(\\sum\\nolimits_t r'_t x_t - r'_0)", "& =", "\\delta'_n(", "\\sum\\nolimits_t i'_t x_t +", "\\sum\\nolimits_{t, j} c_j r_{jt} x_t -", "\\sum\\nolimits_t i'_t f_t -", "\\sum\\nolimits_j c_j r_{j0}) \\\\", "& =", "\\delta'_n(", "\\sum\\nolimits_t i'_t(x_t - f_t) +", "\\sum\\nolimits_j c_j (\\sum\\nolimits_t r_{jt} x_t - r_{j0}))", "\\end{align*}", "Since $\\delta_n(a + b) = \\sum_{m = 0, \\ldots, n} \\delta_m(a) \\delta_{n - m}(b)$", "and since $\\delta_m(\\sum i'_t(x_t - f_t))$ is in the ideal", "generated by $x_t - f_t \\in K'$ for $m > 0$, it suffices to prove that", "$\\delta_n(\\sum c_j (\\sum r_{jt} x_t - r_{j0}))$ is in $K'$.", "For this we use", "$$", "\\delta_n(\\sum\\nolimits_j c_j (\\sum\\nolimits_t r_{jt} x_t - r_{j0}))", "=", "\\sum c_1^{n_1} \\ldots c_m^{n_m}", "\\delta_{n_1}(\\sum r_{1t} x_t - r_{10}) \\ldots", "\\delta_{n_m}(\\sum r_{mt} x_t - r_{m0})", "$$", "where the sum is over $n_1 + \\ldots + n_m = n$. This proves what we want." ], "refs": [ "crystalline-lemma-describe-divided-power-envelope", "crystalline-lemma-describe-divided-power-envelope", "algebra-lemma-flat-eq" ], "ref_ids": [ 10746, 10746, 531 ] } ], "ref_ids": [] }, { "id": 10750, "type": "theorem", "label": "crystalline-lemma-divided-power-first-order-thickening", "categories": [ "crystalline" ], "title": "crystalline-lemma-divided-power-first-order-thickening", "contents": [ "Let $(A, I, \\gamma)$ be a divided power ring. Let $M$ be an $A$-module.", "Let $B = A \\oplus M$ as an $A$-algebra where $M$ is an ideal of square zero", "and set $J = I \\oplus M$. Set", "$$", "\\delta_n(x + z) = \\gamma_n(x) + \\gamma_{n - 1}(x)z", "$$", "for $x \\in I$ and $z \\in M$.", "Then $\\delta$ is a divided power structure and", "$A \\to B$ is a homomorphism of divided power rings from", "$(A, I, \\gamma)$ to $(B, J, \\delta)$." ], "refs": [], "proofs": [ { "contents": [ "We have to check conditions (1) -- (5) of", "Divided Power Algebra, Definition \\ref{dpa-definition-divided-powers}.", "We will prove this directly for this case, but please see the proof of", "the next lemma for a method which avoids calculations.", "Conditions (1) and (3) are clear. Condition (2) follows from", "\\begin{align*}", "\\delta_n(x + z)\\delta_m(x + z)", "& =", "(\\gamma_n(x) + \\gamma_{n - 1}(x)z)(\\gamma_m(x) + \\gamma_{m - 1}(x)z) \\\\", "& = \\gamma_n(x)\\gamma_m(x) + \\gamma_n(x)\\gamma_{m - 1}(x)z +", "\\gamma_{n - 1}(x)\\gamma_m(x)z \\\\", "& =", "\\frac{(n + m)!}{n!m!} \\gamma_{n + m}(x) +", "\\left(\\frac{(n + m - 1)!}{n!(m - 1)!} +", "\\frac{(n + m - 1)!}{(n - 1)!m!}\\right)", "\\gamma_{n + m - 1}(x) z \\\\", "& =", "\\frac{(n + m)!}{n!m!}\\delta_{n + m}(x + z)", "\\end{align*}", "Condition (5) follows from", "\\begin{align*}", "\\delta_n(\\delta_m(x + z))", "& =", "\\delta_n(\\gamma_m(x) + \\gamma_{m - 1}(x)z) \\\\", "& =", "\\gamma_n(\\gamma_m(x)) + \\gamma_{n - 1}(\\gamma_m(x))\\gamma_{m - 1}(x)z \\\\", "& =", "\\frac{(nm)!}{n! (m!)^n} \\gamma_{nm}(x) +", "\\frac{((n - 1)m)!}{(n - 1)! (m!)^{n - 1}}", "\\gamma_{(n - 1)m}(x) \\gamma_{m - 1}(x) z \\\\", "& = \\frac{(nm)!}{n! (m!)^n}(\\gamma_{nm}(x) + \\gamma_{nm - 1}(x) z)", "\\end{align*}", "by elementary number theory. To prove (4) we have to see that", "$$", "\\delta_n(x + x' + z + z')", "=", "\\gamma_n(x + x') + \\gamma_{n - 1}(x + x')(z + z')", "$$", "is equal to", "$$", "\\sum\\nolimits_{i = 0}^n", "(\\gamma_i(x) + \\gamma_{i - 1}(x)z)", "(\\gamma_{n - i}(x') + \\gamma_{n - i - 1}(x')z')", "$$", "This follows easily on collecting the coefficients of", "$1$, $z$, and $z'$ and using condition (4) for $\\gamma$." ], "refs": [ "dpa-definition-divided-powers" ], "ref_ids": [ 1696 ] } ], "ref_ids": [] }, { "id": 10751, "type": "theorem", "label": "crystalline-lemma-divided-power-second-order-thickening", "categories": [ "crystalline" ], "title": "crystalline-lemma-divided-power-second-order-thickening", "contents": [ "Let $(A, I, \\gamma)$ be a divided power ring. Let $M$, $N$ be $A$-modules.", "Let $q : M \\times M \\to N$ be an $A$-bilinear map.", "Let $B = A \\oplus M \\oplus N$ as an $A$-algebra with multiplication", "$$", "(x, z, w)\\cdot (x', z', w') = (xx', xz' + x'z, xw' + x'w + q(z, z') + q(z', z))", "$$", "and set $J = I \\oplus M \\oplus N$. Set", "$$", "\\delta_n(x, z, w) = (\\gamma_n(x), \\gamma_{n - 1}(x)z,", "\\gamma_{n - 1}(x)w + \\gamma_{n - 2}(x)q(z, z))", "$$", "for $(x, z, w) \\in J$.", "Then $\\delta$ is a divided power structure and", "$A \\to B$ is a homomorphism of divided power rings from", "$(A, I, \\gamma)$ to $(B, J, \\delta)$." ], "refs": [], "proofs": [ { "contents": [ "Suppose we want to prove that property (4) of", "Divided Power Algebra, Definition \\ref{dpa-definition-divided-powers}", "is satisfied. Pick $(x, z, w)$ and $(x', z', w')$ in $J$.", "Pick a map", "$$", "A_0 = \\mathbf{Z}\\langle s, s'\\rangle \\longrightarrow A,\\quad", "s \\longmapsto x,", "s' \\longmapsto x'", "$$", "which is possible by the universal property of divided power", "polynomial rings. Set $M_0 = A_0 \\oplus A_0$ and", "$N_0 = A_0 \\oplus A_0 \\oplus M_0 \\otimes_{A_0} M_0$.", "Let $q_0 : M_0 \\times M_0 \\to N_0$ be the obvious map.", "Define $M_0 \\to M$ as the $A_0$-linear map which sends", "the basis vectors of $M_0$ to $z$ and $z'$. Define $N_0 \\to N$", "as the $A_0$ linear map which sends the first two basis vectors", "of $N_0$ to $w$ and $w'$ and uses", "$M_0 \\otimes_{A_0} M_0 \\to M \\otimes_A M \\xrightarrow{q} N$", "on the last summand. Then we see that it suffices to prove the", "identity (4) for the situation $(A_0, M_0, N_0, q_0)$.", "Similarly for the other identities. This reduces us to the case", "of a $\\mathbf{Z}$-torsion free ring $A$ and $A$-torsion free modules.", "In this case all we have to do is show that", "$$", "n! \\delta_n(x, z, w) = (x, z, w)^n", "$$", "in the ring $A$, see Divided Power Algebra, Lemma \\ref{dpa-lemma-silly}.", "To see this note that", "$$", "(x, z, w)^2 = (x^2, 2xz, 2xw + 2q(z, z))", "$$", "and by induction", "$$", "(x, z, w)^n = (x^n, nx^{n - 1}z, nx^{n - 1}w + n(n - 1)x^{n - 2}q(z, z))", "$$", "On the other hand,", "$$", "n! \\delta_n(x, z, w) = (n!\\gamma_n(x), n!\\gamma_{n - 1}(x)z,", "n!\\gamma_{n - 1}(x)w + n!\\gamma_{n - 2}(x) q(z, z))", "$$", "which matches. This finishes the proof." ], "refs": [ "dpa-definition-divided-powers", "dpa-lemma-silly" ], "ref_ids": [ 1696, 1650 ] } ], "ref_ids": [] }, { "id": 10752, "type": "theorem", "label": "crystalline-lemma-affine-thickenings-colimits", "categories": [ "crystalline" ], "title": "crystalline-lemma-affine-thickenings-colimits", "contents": [ "In Situation \\ref{situation-affine}.", "\\begin{enumerate}", "\\item $\\text{CRIS}(C/A)$ has products,", "\\item $\\text{CRIS}(C/A)$ has all finite nonempty colimits and", "(\\ref{equation-forget-affine}) commutes with these, and", "\\item $\\text{Cris}(C/A)$ has all finite nonempty colimits and", "$\\text{Cris}(C/A) \\to \\text{CRIS}(C/A)$ commutes with them.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The empty product is $(C, 0, \\emptyset)$. If $(B_t, J_t, \\delta_t)$ is a", "family of objects of $\\text{CRIS}(C/A)$ then we can form the product", "$(\\prod B_t, \\prod J_t, \\prod \\delta_t)$ as in", "Divided Power Algebra, Lemma \\ref{dpa-lemma-colimits}.", "The map $C \\to \\prod B_t/\\prod J_t = \\prod B_t/J_t$ is clear.", "\\medskip\\noindent", "Given two objects $(B, J, \\gamma)$ and $(B', J', \\gamma')$ of", "$\\text{CRIS}(C/A)$ we can form a cocartesian diagram", "$$", "\\xymatrix{", "(B, J, \\delta) \\ar[r] & (B'', J'', \\delta'') \\\\", "(A, I, \\gamma) \\ar[r] \\ar[u] & (B', J', \\delta') \\ar[u]", "}", "$$", "in the category of divided power rings. Then we see that we have", "$$", "B''/J'' = B/J \\otimes_{A/I} B'/J' \\longleftarrow C \\otimes_{A/I} C", "$$", "see Divided Power Algebra, Remark \\ref{dpa-remark-forgetful}.", "Denote $J'' \\subset K \\subset B''$", "the ideal such that", "$$", "\\xymatrix{", "B''/J'' \\ar[r] & B''/K \\\\", "C \\otimes_{A/I} C \\ar[r] \\ar[u] & C \\ar[u]", "}", "$$", "is a pushout, i.e., $B''/K \\cong B/J \\otimes_C B'/J'$.", "Let $D_{B''}(K) = (D, \\bar K, \\bar \\delta)$", "be the divided power envelope of $K$ in $B''$ relative to", "$(B'', J'', \\delta'')$. Then it is easily verified that", "$(D, \\bar K, \\bar \\delta)$ is a coproduct of $(B, J, \\delta)$ and", "$(B', J', \\delta')$ in $\\text{CRIS}(C/A)$.", "\\medskip\\noindent", "Next, we come to coequalizers. Let", "$\\alpha, \\beta : (B, J, \\delta) \\to (B', J', \\delta')$ be morphisms of", "$\\text{CRIS}(C/A)$. Consider $B'' = B'/ (\\alpha(b) - \\beta(b))$. Let", "$J'' \\subset B''$ be the image of $J'$. Let", "$D_{B''}(J'') = (D, \\bar J, \\bar\\delta)$ be the divided power envelope of", "$J''$ in $B''$ relative to $(B', J', \\delta')$. Then it is easily verified", "that $(D, \\bar J, \\bar \\delta)$ is the coequalizer of $(B, J, \\delta)$ and", "$(B', J', \\delta')$ in $\\text{CRIS}(C/A)$.", "\\medskip\\noindent", "By Categories, Lemma \\ref{categories-lemma-almost-finite-colimits-exist}", "we have all finite nonempty colimits in $\\text{CRIS}(C/A)$. The constructions", "above shows that (\\ref{equation-forget-affine}) commutes with them.", "This formally implies part (3) as $\\text{Cris}(C/A)$ is the fibre category", "of (\\ref{equation-forget-affine}) over $C$." ], "refs": [ "dpa-lemma-colimits", "dpa-remark-forgetful", "categories-lemma-almost-finite-colimits-exist" ], "ref_ids": [ 1656, 1702, 12226 ] } ], "ref_ids": [] }, { "id": 10753, "type": "theorem", "label": "crystalline-lemma-list-properties", "categories": [ "crystalline" ], "title": "crystalline-lemma-list-properties", "contents": [ "In Situation \\ref{situation-affine}.", "Let $P \\to C$ be a surjection of $A$-algebras with kernel $J$.", "Write $D_{P, \\gamma}(J) = (D, \\bar J, \\bar\\gamma)$.", "Let $(D^\\wedge, J^\\wedge, \\bar\\gamma^\\wedge)$ be the", "$p$-adic completion of $D$, see", "Divided Power Algebra, Lemma \\ref{dpa-lemma-extend-to-completion}.", "For every $e \\geq 1$ set $P_e = P/p^eP$ and $J_e \\subset P_e$", "the image of $J$ and write", "$D_{P_e, \\gamma}(J_e) = (D_e, \\bar J_e, \\bar\\gamma)$.", "Then for all $e$ large enough we have", "\\begin{enumerate}", "\\item $p^eD \\subset \\bar J$ and $p^eD^\\wedge \\subset \\bar J^\\wedge$", "are preserved by divided powers,", "\\item $D^\\wedge/p^eD^\\wedge = D/p^eD = D_e$ as divided power rings,", "\\item $(D_e, \\bar J_e, \\bar\\gamma)$ is an object of $\\text{Cris}(C/A)$,", "\\item $(D^\\wedge, \\bar J^\\wedge, \\bar\\gamma^\\wedge)$ is equal to", "$\\lim_e (D_e, \\bar J_e, \\bar\\gamma)$, and", "\\item $(D^\\wedge, \\bar J^\\wedge, \\bar\\gamma^\\wedge)$ is an object of", "$\\text{Cris}^\\wedge(C/A)$.", "\\end{enumerate}" ], "refs": [ "dpa-lemma-extend-to-completion" ], "proofs": [ { "contents": [ "Part (1) follows from", "Divided Power Algebra, Lemma \\ref{dpa-lemma-extend-to-completion}.", "It is a general property of $p$-adic completion that", "$D/p^eD = D^\\wedge/p^eD^\\wedge$. Since $D/p^eD$ is a divided power ring", "and since $P \\to D/p^eD$ factors through $P_e$, the universal property of", "$D_e$ produces a map $D_e \\to D/p^eD$. Conversely, the universal property", "of $D$ produces a map $D \\to D_e$ which factors through $D/p^eD$. We omit", "the verification that these maps are mutually inverse. This proves (2).", "If $e$ is large enough, then $p^eC = 0$, hence we see (3) holds.", "Part (4) follows from", "Divided Power Algebra, Lemma \\ref{dpa-lemma-extend-to-completion}.", "Part (5) is clear from the definitions." ], "refs": [ "dpa-lemma-extend-to-completion", "dpa-lemma-extend-to-completion" ], "ref_ids": [ 1660, 1660 ] } ], "ref_ids": [ 1660 ] }, { "id": 10754, "type": "theorem", "label": "crystalline-lemma-set-generators", "categories": [ "crystalline" ], "title": "crystalline-lemma-set-generators", "contents": [ "In Situation \\ref{situation-affine}.", "Let $P$ be a polynomial algebra over $A$ and let", "$P \\to C$ be a surjection of $A$-algebras with kernel $J$.", "With $(D_e, \\bar J_e, \\bar\\gamma)$ as in Lemma \\ref{lemma-list-properties}:", "for every object $(B, J_B, \\delta)$ of $\\text{CRIS}(C/A)$ there", "exists an $e$ and a morphism $D_e \\to B$ of $\\text{CRIS}(C/A)$." ], "refs": [ "crystalline-lemma-list-properties" ], "proofs": [ { "contents": [ "We can find an $A$-algebra homomorphism $P \\to B$", "lifting the map $C \\to B/J_B$. By our definition of", "$\\text{CRIS}(C/A)$ we see that $p^eB = 0$ for", "some $e$ hence $P \\to B$ factors as $P \\to P_e \\to B$.", "By the universal property of the divided power envelope we", "conclude that $P_e \\to B$ factors through $D_e$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10753 ] }, { "id": 10755, "type": "theorem", "label": "crystalline-lemma-generator-completion", "categories": [ "crystalline" ], "title": "crystalline-lemma-generator-completion", "contents": [ "In Situation \\ref{situation-affine}.", "Let $P$ be a polynomial algebra over $A$ and let", "$P \\to C$ be a surjection of $A$-algebras with kernel $J$.", "Let $(D, \\bar J, \\bar\\gamma)$ be the $p$-adic completion of", "$D_{P, \\gamma}(J)$. For every object $(B \\to C, \\delta)$ of", "$\\text{Cris}^\\wedge(C/A)$ there", "exists a morphism $D \\to B$ of $\\text{Cris}^\\wedge(C/A)$." ], "refs": [], "proofs": [ { "contents": [ "We can find an $A$-algebra homomorphism $P \\to B$ compatible", "with maps to $C$. By our definition of", "$\\text{Cris}(C/A)$ we see that $P \\to B$ factors as", "$P \\to D_{P, \\gamma}(J) \\to B$. As $B$ is $p$-adically complete", "we can factor this map through $D$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10756, "type": "theorem", "label": "crystalline-lemma-omega", "categories": [ "crystalline" ], "title": "crystalline-lemma-omega", "contents": [ "Let $A$ be a ring. Let $(B, J, \\delta)$ be a divided power ring and", "$A \\to B$ a ring map. ", "\\begin{enumerate}", "\\item Consider $B[x]$ with divided power ideal $(JB[x], \\delta')$", "where $\\delta'$ is the extension of $\\delta$ to $B[x]$. Then", "$$", "\\Omega_{B[x]/A, \\delta'} =", "\\Omega_{B/A, \\delta} \\otimes_B B[x] \\oplus B[x]\\text{d}x.", "$$", "\\item Consider $B\\langle x \\rangle$ with divided power ideal", "$(JB\\langle x \\rangle + B\\langle x \\rangle_{+}, \\delta')$. Then", "$$", "\\Omega_{B\\langle x\\rangle/A, \\delta'} =", "\\Omega_{B/A, \\delta} \\otimes_B B\\langle x \\rangle \\oplus", "B\\langle x\\rangle \\text{d}x.", "$$", "\\item Let $K \\subset J$ be an ideal preserved by $\\delta_n$ for", "all $n > 0$. Set $B' = B/K$ and denote $\\delta'$ the induced", "divided power on $J/K$. Then $\\Omega_{B'/A, \\delta'}$ is the quotient", "of $\\Omega_{B/A, \\delta} \\otimes_B B'$ by the $B'$-submodule generated", "by $\\text{d}k$ for $k \\in K$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "These are proved directly from the construction of $\\Omega_{B/A, \\delta}$", "as the free $B$-module on the elements $\\text{d}b$ modulo the relations", "\\begin{enumerate}", "\\item $\\text{d}(b + b') = \\text{d}b + \\text{d}b'$, $b, b' \\in B$,", "\\item $\\text{d}a = 0$, $a \\in A$,", "\\item $\\text{d}(bb') = b \\text{d}b' + b' \\text{d}b$, $b, b' \\in B$,", "\\item $\\text{d}\\delta_n(f) = \\delta_{n - 1}(f)\\text{d}f$, $f \\in J$, $n > 1$.", "\\end{enumerate}", "Note that the last relation explains why we get ``the same'' answer for", "the divided power polynomial algebra and the usual polynomial algebra:", "in the first case $x$ is an element of the divided power ideal and hence", "$\\text{d}x^{[n]} = x^{[n - 1]}\\text{d}x$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10757, "type": "theorem", "label": "crystalline-lemma-diagonal-and-differentials", "categories": [ "crystalline" ], "title": "crystalline-lemma-diagonal-and-differentials", "contents": [ "Let $(A, I, \\gamma) \\to (B, J, \\delta)$ be a homomorphism", "of divided power rings. Let $(B(1), J(1), \\delta(1))$ be the coproduct", "of $(B, J, \\delta)$ with itself over $(A, I, \\gamma)$, i.e.,", "such that", "$$", "\\xymatrix{", "(B, J, \\delta) \\ar[r] & (B(1), J(1), \\delta(1)) \\\\", "(A, I, \\gamma) \\ar[r] \\ar[u] & (B, J, \\delta) \\ar[u]", "}", "$$", "is cocartesian. Denote $K = \\Ker(B(1) \\to B)$.", "Then $K \\cap J(1) \\subset J(1)$ is preserved by the divided power", "structure and", "$$", "\\Omega_{B/A, \\delta} = K/ \\left(K^2 + (K \\cap J(1))^{[2]}\\right)", "$$", "canonically." ], "refs": [], "proofs": [ { "contents": [ "The fact that $K \\cap J(1) \\subset J(1)$ is preserved by the divided power", "structure follows from the fact that $B(1) \\to B$ is a homomorphism of", "divided power rings.", "\\medskip\\noindent", "Recall that $K/K^2$ has a canonical $B$-module structure.", "Denote $s_0, s_1 : B \\to B(1)$ the two coprojections and consider", "the map $\\text{d} : B \\to K/K^2 +(K \\cap J(1))^{[2]}$ given by", "$b \\mapsto s_1(b) - s_0(b)$. It is clear that $\\text{d}$ is additive,", "annihilates $A$, and satisfies the Leibniz rule.", "We claim that $\\text{d}$ is a divided power $A$-derivation.", "Let $x \\in J$. Set $y = s_1(x)$ and $z = s_0(x)$.", "Denote $\\delta$ the divided power structure on $J(1)$.", "We have to show that $\\delta_n(y) - \\delta_n(z) = \\delta_{n - 1}(y)(y - z)$", "modulo $K^2 +(K \\cap J(1))^{[2]}$ for $n \\geq 1$.", "The equality holds for $n = 1$. Assume $n > 1$.", "Note that $\\delta_i(y - z)$ lies in $(K \\cap J(1))^{[2]}$ for $i > 1$.", "Calculating modulo $K^2 + (K \\cap J(1))^{[2]}$ we have", "$$", "\\delta_n(z) = \\delta_n(z - y + y) =", "\\sum\\nolimits_{i = 0}^n \\delta_i(z - y)\\delta_{n - i}(y) =", "\\delta_{n - 1}(y) \\delta_1(z - y) + \\delta_n(y)", "$$", "This proves the desired equality.", "\\medskip\\noindent", "Let $M$ be a $B$-module. Let $\\theta : B \\to M$ be a divided power", "$A$-derivation.", "Set $D = B \\oplus M$ where $M$ is an ideal of square zero. Define a", "divided power structure on $J \\oplus M \\subset D$ by setting", "$\\delta_n(x + m) = \\delta_n(x) + \\delta_{n - 1}(x)m$ for $n > 1$, see", "Lemma \\ref{lemma-divided-power-first-order-thickening}.", "There are two divided power algebra homomorphisms $B \\to D$: the first", "is given by the inclusion and the second by the map $b \\mapsto b + \\theta(b)$.", "Hence we get a canonical homomorphism $B(1) \\to D$ of divided power", "algebras over $(A, I, \\gamma)$. This induces a map $K \\to M$", "which annihilates $K^2$ (as $M$ is an ideal of square zero) and", "$(K \\cap J(1))^{[2]}$ as $M^{[2]} = 0$. The composition", "$B \\to K/K^2 + (K \\cap J(1))^{[2]} \\to M$ equals $\\theta$ by construction.", "It follows that $\\text{d}$", "is a universal divided power $A$-derivation and we win." ], "refs": [ "crystalline-lemma-divided-power-first-order-thickening" ], "ref_ids": [ 10750 ] } ], "ref_ids": [] }, { "id": 10758, "type": "theorem", "label": "crystalline-lemma-diagonal-and-differentials-affine-site", "categories": [ "crystalline" ], "title": "crystalline-lemma-diagonal-and-differentials-affine-site", "contents": [ "In Situation \\ref{situation-affine}.", "Let $(B, J, \\delta)$ be an object of $\\text{CRIS}(C/A)$.", "Let $(B(1), J(1), \\delta(1))$ be the coproduct of $(B, J, \\delta)$", "with itself in $\\text{CRIS}(C/A)$. Denote", "$K = \\Ker(B(1) \\to B)$. Then $K \\cap J(1) \\subset J(1)$", "is preserved by the divided power structure and", "$$", "\\Omega_{B/A, \\delta} = K/ \\left(K^2 + (K \\cap J(1))^{[2]}\\right)", "$$", "canonically." ], "refs": [], "proofs": [ { "contents": [ "Word for word the same as the proof of", "Lemma \\ref{lemma-diagonal-and-differentials}.", "The only point that has to be checked is that the", "divided power ring $D = B \\oplus M$ is an object of $\\text{CRIS}(C/A)$", "and that the two maps $B \\to C$ are morphisms of $\\text{CRIS}(C/A)$.", "Since $D/(J \\oplus M) = B/J$ we can use $C \\to B/J$ to view", "$D$ as an object of $\\text{CRIS}(C/A)$", "and the statement on morphisms is clear from the construction." ], "refs": [ "crystalline-lemma-diagonal-and-differentials" ], "ref_ids": [ 10757 ] } ], "ref_ids": [] }, { "id": 10759, "type": "theorem", "label": "crystalline-lemma-module-differentials-divided-power-envelope", "categories": [ "crystalline" ], "title": "crystalline-lemma-module-differentials-divided-power-envelope", "contents": [ "Let $(A, I, \\gamma)$ be a divided power ring. Let $A \\to B$ be a ring", "map and let $IB \\subset J \\subset B$ be an ideal. Let", "$D_{B, \\gamma}(J) = (D, \\bar J, \\bar \\gamma)$ be the divided power envelope.", "Then we have", "$$", "\\Omega_{D/A, \\bar\\gamma} = \\Omega_{B/A} \\otimes_B D", "$$" ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "Let $M$ be a $D$-module. We claim that an $A$-derivation", "$\\vartheta : B \\to M$ is the same thing as a divided power", "$A$-derivation $\\theta : D \\to M$. The claim implies the", "statement by the Yoneda lemma.", "\\medskip\\noindent", "Consider the square zero thickening $D \\oplus M$ of $D$.", "There is a divided power structure $\\delta$ on $\\bar J \\oplus M$", "if we set the higher divided power operations zero on $M$.", "In other words, we set", "$\\delta_n(x + m) = \\bar\\gamma_n(x) + \\bar\\gamma_{n - 1}(x)m$ for", "any $x \\in \\bar J$ and $m \\in M$, see", "Lemma \\ref{lemma-divided-power-first-order-thickening}.", "Consider the $A$-algebra map $B \\to D \\oplus M$ whose first", "component is given by the map $B \\to D$ and whose second component", "is $\\vartheta$. By the universal property we get a corresponding", "homomorphism $D \\to D \\oplus M$ of divided power algebras", "whose second component is the divided power", "$A$-derivation $\\theta$ corresponding to $\\vartheta$." ], "refs": [ "crystalline-lemma-divided-power-first-order-thickening" ], "ref_ids": [ 10750 ] } ], "ref_ids": [] }, { "id": 10760, "type": "theorem", "label": "crystalline-lemma-differentials-completion", "categories": [ "crystalline" ], "title": "crystalline-lemma-differentials-completion", "contents": [ "Let $A \\to B$ be a ring map and let $(J, \\delta)$ be a divided power", "structure on $B$. Let $p$ be a prime number. Assume that $A$ is a", "$\\mathbf{Z}_{(p)}$-algebra and that $p$ is nilpotent in $B/J$. Then", "we have", "$$", "\\lim_e \\Omega_{B_e/A, \\bar\\delta} =", "\\lim_e \\Omega_{B/A, \\delta}/p^e\\Omega_{B/A, \\delta} =", "\\lim_e \\Omega_{B^\\wedge/A, \\delta^\\wedge}/p^e \\Omega_{B^\\wedge/A, \\delta^\\wedge}", "$$", "see proof for notation and explanation." ], "refs": [], "proofs": [ { "contents": [ "By Divided Power Algebra, Lemma \\ref{dpa-lemma-extend-to-completion}", "we see that $\\delta$ extends", "to $B_e = B/p^eB$ for all sufficiently large $e$. Hence the first limit", "make sense. The lemma also produces a divided power structure $\\delta^\\wedge$", "on the completion $B^\\wedge = \\lim_e B_e$, hence the last limit makes", "sense. By Lemma \\ref{lemma-omega}", "and the fact that $\\text{d}p^e = 0$ (always)", "we see that the surjection", "$\\Omega_{B/A, \\delta} \\to \\Omega_{B_e/A, \\bar\\delta}$ has kernel", "$p^e\\Omega_{B/A, \\delta}$. Similarly for the kernel of ", "$\\Omega_{B^\\wedge/A, \\delta^\\wedge} \\to \\Omega_{B_e/A, \\bar\\delta}$.", "Hence the lemma is clear." ], "refs": [ "dpa-lemma-extend-to-completion", "crystalline-lemma-omega" ], "ref_ids": [ 1660, 10756 ] } ], "ref_ids": [] }, { "id": 10761, "type": "theorem", "label": "crystalline-lemma-fibre-product", "categories": [ "crystalline" ], "title": "crystalline-lemma-fibre-product", "contents": [ "Let $(U', T', \\delta') \\to (S'_0, S', \\gamma')$ and", "$(S_0, S, \\gamma) \\to (S'_0, S', \\gamma')$ be morphisms of", "divided power schemes. If $(U', T', \\delta')$ is a divided power", "thickening, then there exists a divided power scheme $(T_0, T, \\delta)$", "and", "$$", "\\xymatrix{", "T \\ar[r] \\ar[d] & T' \\ar[d] \\\\", "S \\ar[r] & S'", "}", "$$", "which is a cartesian diagram in the category of divided power schemes." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints: If $T$ exists, then $T_0 = S_0 \\times_{S'_0} U'$", "(argue as in Divided Power Algebra, Remark \\ref{dpa-remark-forgetful}).", "Since $T'$ is a divided power thickening, we see that $T$", "(if it exists) will be a divided power thickening too.", "Hence we can define $T$ as the scheme with underlying topological", "space the underlying topological space of $T_0 = S_0 \\times_{S'_0} U'$", "and as structure sheaf on affine pieces the ring given", "by Lemma \\ref{lemma-affine-thickenings-colimits}." ], "refs": [ "dpa-remark-forgetful", "crystalline-lemma-affine-thickenings-colimits" ], "ref_ids": [ 1702, 10752 ] } ], "ref_ids": [] }, { "id": 10762, "type": "theorem", "label": "crystalline-lemma-divided-power-thickening-fibre-products", "categories": [ "crystalline" ], "title": "crystalline-lemma-divided-power-thickening-fibre-products", "contents": [ "In Situation \\ref{situation-global}.", "The category $\\text{CRIS}(X/S)$ has all finite nonempty limits,", "in particular products of pairs and fibre products.", "The functor (\\ref{equation-forget}) commutes with limits." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: See Lemma \\ref{lemma-affine-thickenings-colimits}", "for the affine case. See also", "Divided Power Algebra, Remark \\ref{dpa-remark-forgetful}." ], "refs": [ "crystalline-lemma-affine-thickenings-colimits", "dpa-remark-forgetful" ], "ref_ids": [ 10752, 1702 ] } ], "ref_ids": [] }, { "id": 10763, "type": "theorem", "label": "crystalline-lemma-divided-power-thickening-base-change-flat", "categories": [ "crystalline" ], "title": "crystalline-lemma-divided-power-thickening-base-change-flat", "contents": [ "In Situation \\ref{situation-global}. Let", "$$", "\\xymatrix{", "(U_3, T_3, \\delta_3) \\ar[d] \\ar[r] & (U_2, T_2, \\delta_2) \\ar[d] \\\\", "(U_1, T_1, \\delta_1) \\ar[r] & (U, T, \\delta)", "}", "$$", "be a fibre square in the category of divided power thickenings of", "$X$ relative to $(S, \\mathcal{I}, \\gamma)$. If $T_2 \\to T$ is", "flat and $U_2 = T_2 \\times_T U$, then $T_3 = T_1 \\times_T T_2$ (as schemes)." ], "refs": [], "proofs": [ { "contents": [ "This is true because a divided power structure extends uniquely", "along a flat ring map. See", "Divided Power Algebra, Lemma \\ref{dpa-lemma-gamma-extends}." ], "refs": [ "dpa-lemma-gamma-extends" ], "ref_ids": [ 1657 ] } ], "ref_ids": [] }, { "id": 10764, "type": "theorem", "label": "crystalline-lemma-compare-big-small", "categories": [ "crystalline" ], "title": "crystalline-lemma-compare-big-small", "contents": [ "Assumptions as in Definition \\ref{definition-divided-power-thickening-X}.", "The inclusion functor", "$$", "\\text{Cris}(X/S) \\to \\text{CRIS}(X/S)", "$$", "commutes with finite nonempty limits, is fully faithful, continuous,", "and cocontinuous. There are morphisms of topoi", "$$", "(X/S)_{\\text{cris}} \\xrightarrow{i} (X/S)_{\\text{CRIS}}", "\\xrightarrow{\\pi} (X/S)_{\\text{cris}}", "$$", "whose composition is the identity and of which the first is induced", "by the inclusion functor. Moreover, $\\pi_* = i^{-1}$." ], "refs": [ "crystalline-definition-divided-power-thickening-X" ], "proofs": [ { "contents": [ "For the first assertion see", "Lemma \\ref{lemma-divided-power-thickening-fibre-products}.", "This gives us a morphism of topoi", "$i : (X/S)_{\\text{cris}} \\to (X/S)_{\\text{CRIS}}$ and a left adjoint", "$i_!$ such that $i^{-1}i_! = i^{-1}i_* = \\text{id}$, see", "Sites, Lemmas \\ref{sites-lemma-when-shriek},", "\\ref{sites-lemma-preserve-equalizers}, and", "\\ref{sites-lemma-back-and-forth}.", "We claim that $i_!$ is exact. If this is true, then we can define", "$\\pi$ by the rules $\\pi^{-1} = i_!$ and $\\pi_* = i^{-1}$", "and everything is clear. To prove the claim, note that we already know", "that $i_!$ is right exact and preserves fibre products (see references", "given). Hence it suffices to show that $i_! * = *$ where $*$ indicates", "the final object in the category of sheaves of sets. ", "To see this it suffices to produce a set of objects", "$(U_i, T_i, \\delta_i)$, $i \\in I$ of $\\text{Cris}(X/S)$ such that", "$$", "\\coprod\\nolimits_{i \\in I} h_{(U_i, T_i, \\delta_i)} \\to *", "$$", "is surjective in $(X/S)_{\\text{CRIS}}$ (details omitted; hint: use that", "$\\text{Cris}(X/S)$ has products and that the functor", "$\\text{Cris}(X/S) \\to \\text{CRIS}(X/S)$ commutes with them).", "In the affine case this", "follows from Lemma \\ref{lemma-set-generators}. We omit the proof", "in general." ], "refs": [ "crystalline-lemma-divided-power-thickening-fibre-products", "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers", "sites-lemma-back-and-forth", "crystalline-lemma-set-generators" ], "ref_ids": [ 10762, 8545, 8546, 8547, 10754 ] } ], "ref_ids": [ 10804 ] }, { "id": 10765, "type": "theorem", "label": "crystalline-lemma-localize", "categories": [ "crystalline" ], "title": "crystalline-lemma-localize", "contents": [ "In Situation \\ref{situation-global}.", "Let $X' \\subset X$ and $S' \\subset S$ be open subschemes such that", "$X'$ maps into $S'$. Then there is a fully faithful functor", "$\\text{Cris}(X'/S') \\to \\text{Cris}(X/S)$", "which gives rise to a morphism of topoi fitting into the commutative", "diagram", "$$", "\\xymatrix{", "(X'/S')_{\\text{cris}} \\ar[r] \\ar[d]_{u_{X'/S'}} &", "(X/S)_{\\text{cris}} \\ar[d]^{u_{X/S}} \\\\", "\\Sh(X'_{Zar}) \\ar[r] & \\Sh(X_{Zar})", "}", "$$", "Moreover, this diagram is an example of localization of morphisms of", "topoi as in Sites, Lemma \\ref{sites-lemma-localize-morphism-topoi}." ], "refs": [ "sites-lemma-localize-morphism-topoi" ], "proofs": [ { "contents": [ "The fully faithful functor comes from thinking of", "objects of $\\text{Cris}(X'/S')$ as divided power", "thickenings $(U, T, \\delta)$ of $X$ where $U \\to X$", "factors through $X' \\subset X$ (since then automatically $T \\to S$", "will factor through $S'$). This functor is clearly cocontinuous", "hence we obtain a morphism of topoi as indicated.", "Let $h_{X'} \\in \\Sh(X_{Zar})$ be the representable sheaf associated", "to $X'$ viewed as an object of $X_{Zar}$. It is clear that", "$\\Sh(X'_{Zar})$ is the localization $\\Sh(X_{Zar})/h_{X'}$.", "On the other hand, the category $\\text{Cris}(X/S)/u_{X/S}^{-1}h_{X'}$", "(see Sites, Lemma \\ref{sites-lemma-localize-topos-site})", "is canonically identified with $\\text{Cris}(X'/S')$ by the functor above.", "This finishes the proof." ], "refs": [ "sites-lemma-localize-topos-site" ], "ref_ids": [ 8585 ] } ], "ref_ids": [ 8589 ] }, { "id": 10766, "type": "theorem", "label": "crystalline-lemma-crystal-quasi-coherent-modules", "categories": [ "crystalline" ], "title": "crystalline-lemma-crystal-quasi-coherent-modules", "contents": [ "With notation $X/S, \\mathcal{I}, \\gamma, \\mathcal{C}, \\mathcal{F}$", "as in Definition \\ref{definition-modules}. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is quasi-coherent, and", "\\item $\\mathcal{F}$ is locally quasi-coherent and a crystal in", "$\\mathcal{O}_{X/S}$-modules.", "\\end{enumerate}" ], "refs": [ "crystalline-definition-modules" ], "proofs": [ { "contents": [ "Assume (1). Let $f : (U', T', \\delta') \\to (U, T, \\delta)$ be an object of", "$\\mathcal{C}$. We have to prove (a) $\\mathcal{F}_T$ is a quasi-coherent", "$\\mathcal{O}_T$-module and (b) $c_f : f^*\\mathcal{F}_T \\to \\mathcal{F}_{T'}$", "is an isomorphism. The assumption means that we can find a covering", "$\\{(T_i, U_i, \\delta_i) \\to (T, U, \\delta)\\}$ and for each $i$", "the restriction of $\\mathcal{F}$ to $\\mathcal{C}/(T_i, U_i, \\delta_i)$", "has a global presentation. Since it suffices to prove (a) and (b)", "Zariski locally, we may replace $f : (T', U', \\delta') \\to (T, U, \\delta)$", "by the base change to $(T_i, U_i, \\delta_i)$ and assume that $\\mathcal{F}$", "restricted to $\\mathcal{C}/(T, U, \\delta)$ has a global", "presentation", "$$", "\\bigoplus\\nolimits_{j \\in J}", "\\mathcal{O}_{X/S}|_{\\mathcal{C}/(U, T, \\delta)} \\longrightarrow", "\\bigoplus\\nolimits_{i \\in I}", "\\mathcal{O}_{X/S}|_{\\mathcal{C}/(U, T, \\delta)} \\longrightarrow", "\\mathcal{F}|_{\\mathcal{C}/(U, T, \\delta)}", "\\longrightarrow 0", "$$", "It is clear that this gives a presentation", "$$", "\\bigoplus\\nolimits_{j \\in J} \\mathcal{O}_T \\longrightarrow", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{O}_T \\longrightarrow", "\\mathcal{F}_T", "\\longrightarrow 0", "$$", "and hence (a) holds. Moreover, the presentation restricts to $T'$", "to give a similar presentation of $\\mathcal{F}_{T'}$, whence (b) holds.", "\\medskip\\noindent", "Assume (2). Let $(U, T, \\delta)$ be an object of $\\mathcal{C}$.", "We have to find a covering of $(U, T, \\delta)$ such that $\\mathcal{F}$ has a", "global presentation when we restrict to the localization of $\\mathcal{C}$", "at the members of the covering. Thus we may assume that $T$ is affine.", "In this case we can choose a presentation", "$$", "\\bigoplus\\nolimits_{j \\in J} \\mathcal{O}_T \\longrightarrow", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{O}_T \\longrightarrow", "\\mathcal{F}_T", "\\longrightarrow 0", "$$", "as $\\mathcal{F}_T$ is assumed to be a quasi-coherent $\\mathcal{O}_T$-module.", "Then by the crystal property of $\\mathcal{F}$ we see that this pulls back", "to a presentation of $\\mathcal{F}_{T'}$ for any morphism", "$f : (U', T', \\delta') \\to (U, T, \\delta)$ of $\\mathcal{C}$.", "Thus the desired presentation of $\\mathcal{F}|_{\\mathcal{C}/(U, T, \\delta)}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10807 ] }, { "id": 10767, "type": "theorem", "label": "crystalline-lemma-module-differentials-divided-power-scheme", "categories": [ "crystalline" ], "title": "crystalline-lemma-module-differentials-divided-power-scheme", "contents": [ "Let $(T, \\mathcal{J}, \\delta)$ be a divided power scheme.", "Let $T \\to S$ be a morphism of schemes.", "The quotient $\\Omega_{T/S} \\to \\Omega_{T/S, \\delta}$", "described above is a quasi-coherent $\\mathcal{O}_T$-module.", "For $W \\subset T$ affine open mapping into $V \\subset S$ affine open", "we have", "$$", "\\Gamma(W, \\Omega_{T/S, \\delta}) =", "\\Omega_{\\Gamma(W, \\mathcal{O}_W)/\\Gamma(V, \\mathcal{O}_V), \\delta}", "$$", "where the right hand side is", "as constructed in Section \\ref{section-differentials}." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10768, "type": "theorem", "label": "crystalline-lemma-module-of-differentials", "categories": [ "crystalline" ], "title": "crystalline-lemma-module-of-differentials", "contents": [ "In Situation \\ref{situation-global}.", "For $(U, T, \\delta)$ in $\\text{Cris}(X/S)$ the restriction", "$(\\Omega_{X/S})_T$ to $T$ is $\\Omega_{T/S, \\delta}$ and the restriction", "$\\text{d}_{X/S}|_T$ is equal to $\\text{d}_{T/S, \\delta}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10769, "type": "theorem", "label": "crystalline-lemma-module-of-differentials-on-affine", "categories": [ "crystalline" ], "title": "crystalline-lemma-module-of-differentials-on-affine", "contents": [ "In Situation \\ref{situation-global}.", "For any affine object $(U, T, \\delta)$ of $\\text{Cris}(X/S)$", "mapping into an affine open $V \\subset S$ we have", "$$", "\\Gamma((U, T, \\delta), \\Omega_{X/S}) =", "\\Omega_{\\Gamma(T, \\mathcal{O}_T)/\\Gamma(V, \\mathcal{O}_V), \\delta}", "$$", "where the right hand side is", "as constructed in Section \\ref{section-differentials}." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-module-differentials-divided-power-scheme} and", "\\ref{lemma-module-of-differentials}." ], "refs": [ "crystalline-lemma-module-differentials-divided-power-scheme", "crystalline-lemma-module-of-differentials" ], "ref_ids": [ 10767, 10768 ] } ], "ref_ids": [] }, { "id": 10770, "type": "theorem", "label": "crystalline-lemma-describe-omega-small", "categories": [ "crystalline" ], "title": "crystalline-lemma-describe-omega-small", "contents": [ "In Situation \\ref{situation-global}.", "Let $(U, T, \\delta)$ be an object of $\\text{Cris}(X/S)$.", "Let", "$$", "(U(1), T(1), \\delta(1)) = (U, T, \\delta) \\times (U, T, \\delta)", "$$", "in $\\text{Cris}(X/S)$. Let $\\mathcal{K} \\subset \\mathcal{O}_{T(1)}$", "be the quasi-coherent sheaf of ideals corresponding to the closed", "immersion $\\Delta : T \\to T(1)$. Then", "$\\mathcal{K} \\subset \\mathcal{J}_{T(1)}$ is preserved by the", "divided structure on $\\mathcal{J}_{T(1)}$ and we have", "$$", "(\\Omega_{X/S})_T = \\mathcal{K}/\\mathcal{K}^{[2]}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Note that $U = U(1)$ as $U \\to X$ is an open immersion and as", "(\\ref{equation-forget-small}) commutes with products. Hence we see that", "$\\mathcal{K} \\subset \\mathcal{J}_{T(1)}$. Given this fact the lemma follows", "by working affine locally on $T$ and using", "Lemmas \\ref{lemma-module-of-differentials-on-affine} and", "\\ref{lemma-diagonal-and-differentials-affine-site}." ], "refs": [ "crystalline-lemma-module-of-differentials-on-affine", "crystalline-lemma-diagonal-and-differentials-affine-site" ], "ref_ids": [ 10769, 10758 ] } ], "ref_ids": [] }, { "id": 10771, "type": "theorem", "label": "crystalline-lemma-omega-locally-quasi-coherent", "categories": [ "crystalline" ], "title": "crystalline-lemma-omega-locally-quasi-coherent", "contents": [ "In Situation \\ref{situation-global}.", "The sheaf of differentials $\\Omega_{X/S}$ has the following two", "properties:", "\\begin{enumerate}", "\\item $\\Omega_{X/S}$ is locally quasi-coherent, and", "\\item for any morphism $(U, T, \\delta) \\to (U', T', \\delta')$", "of $\\text{Cris}(X/S)$ where $f : T \\to T'$ is a closed immersion", "the map $c_f : f^*(\\Omega_{X/S})_{T'} \\to (\\Omega_{X/S})_T$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from a combination of", "Lemmas \\ref{lemma-module-differentials-divided-power-scheme} and", "\\ref{lemma-module-of-differentials}.", "Part (2) follows from the fact that", "$(\\Omega_{X/S})_T = \\Omega_{T/S, \\delta}$", "is a quotient of $\\Omega_{T/S}$ and that $f^*\\Omega_{T'/S} \\to \\Omega_{T/S}$", "is surjective." ], "refs": [ "crystalline-lemma-module-differentials-divided-power-scheme", "crystalline-lemma-module-of-differentials" ], "ref_ids": [ 10767, 10768 ] } ], "ref_ids": [] }, { "id": 10772, "type": "theorem", "label": "crystalline-lemma-automatic-connection", "categories": [ "crystalline" ], "title": "crystalline-lemma-automatic-connection", "contents": [ "In Situation \\ref{situation-global}.", "Let $\\mathcal{F}$ be a crystal in $\\mathcal{O}_{X/S}$-modules", "on $\\text{Cris}(X/S)$. Then $\\mathcal{F}$ comes equipped with a", "canonical integrable connection." ], "refs": [], "proofs": [ { "contents": [ "Say $(U, T, \\delta)$ is an object of $\\text{Cris}(X/S)$.", "Let $(U, T', \\delta')$ be the infinitesimal thickening of $T$", "by $(\\Omega_{X/S})_T = \\Omega_{T/S, \\delta}$", "constructed in Remark \\ref{remark-first-order-thickening}.", "It comes with projections $p_0, p_1 : T' \\to T$", "and a diagonal $i : T \\to T'$. By assumption we get", "isomorphisms", "$$", "p_0^*\\mathcal{F}_T \\xrightarrow{c_0}", "\\mathcal{F}_{T'} \\xleftarrow{c_1}", "p_1^*\\mathcal{F}_T", "$$", "of $\\mathcal{O}_{T'}$-modules. Pulling $c = c_1^{-1} \\circ c_0$", "back to $T$ by $i$ we obtain the identity map", "of $\\mathcal{F}_T$. Hence if $s \\in \\Gamma(T, \\mathcal{F}_T)$", "then $\\nabla(s) = p_1^*s - c(p_0^*s)$ is a section of", "$p_1^*\\mathcal{F}_T$ which vanishes on pulling back by $i$. Hence", "$\\nabla(s)$ is a section of", "$$", "\\mathcal{F}_T", "\\otimes_{\\mathcal{O}_T}", "\\Omega_{T/S, \\delta}", "$$", "because this is the kernel of $p_1^*\\mathcal{F}_T \\to \\mathcal{F}_T$", "as $\\mathcal{O}_{T'} = \\mathcal{O}_T \\oplus \\Omega_{T/S, \\delta}$", "by construction. It is easily verified that $\\nabla(fs) =", "f\\nabla(s) + s \\otimes \\text{d}(f)$ using the description of", "$\\text{d}$ in Remark \\ref{remark-first-order-thickening}.", "\\medskip\\noindent", "The collection of maps", "$$", "\\nabla : \\Gamma(T, \\mathcal{F}_T) \\to", "\\Gamma(T, \\mathcal{F}_T \\otimes_{\\mathcal{O}_T} \\Omega_{T/S, \\delta})", "$$", "so obtained is functorial in $T$ because the construction of $T'$", "is functorial in $T$. Hence we obtain a connection.", "\\medskip\\noindent", "To show that the connection is integrable we consider the", "object $(U, T'', \\delta'')$ constructed in", "Remark \\ref{remark-second-order-thickening}.", "Because $\\mathcal{F}$ is a sheaf we see that", "$$", "\\xymatrix{", "q_0^*\\mathcal{F}_T \\ar[rr]_{q_{01}^*c} \\ar[rd]_{q_{02}^*c} & &", "q_1^*\\mathcal{F}_T \\ar[ld]^{q_{12}^*c} \\\\", "& q_2^*\\mathcal{F}_T", "}", "$$", "is a commutative diagram of $\\mathcal{O}_{T''}$-modules. For", "$s \\in \\Gamma(T, \\mathcal{F}_T)$ we have", "$c(p_0^*s) = p_1^*s - \\nabla(s)$. Write", "$\\nabla(s) = \\sum p_1^*s_i \\cdot \\omega_i$ where $s_i$ is a local section", "of $\\mathcal{F}_T$ and $\\omega_i$ is a local section of $\\Omega_{T/S, \\delta}$.", "We think of $\\omega_i$ as a local section of the structure", "sheaf of $\\mathcal{O}_{T'}$ and hence we write product instead of tensor", "product. On the one hand", "\\begin{align*}", "q_{12}^*c \\circ q_{01}^*c(q_0^*s) & = ", "q_{12}^*c(q_1^*s - \\sum q_1^*s_i \\cdot q_{01}^*\\omega_i) \\\\", "& =", "q_2^*s - \\sum q_2^*s_i \\cdot q_{12}^*\\omega_i -", "\\sum q_2^*s_i \\cdot q_{01}^*\\omega_i +", "\\sum q_{12}^*\\nabla(s_i) \\cdot q_{01}^*\\omega_i", "\\end{align*}", "and on the other hand", "$$", "q_{02}^*c(q_0^*s) = q_2^*s - \\sum q_2^*s_i \\cdot q_{02}^*\\omega_i.", "$$", "From the formulae of Remark \\ref{remark-second-order-thickening} we see", "that", "$q_{01}^*\\omega_i + q_{12}^*\\omega_i - q_{02}^*\\omega_i = \\text{d}\\omega_i$.", "Hence the difference of the two expressions above is", "$$", "\\sum q_2^*s_i \\cdot \\text{d}\\omega_i -", "\\sum q_{12}^*\\nabla(s_i) \\cdot q_{01}^*\\omega_i", "$$", "Note that", "$q_{12}^*\\omega \\cdot q_{01}^*\\omega' = \\omega' \\wedge \\omega =", "- \\omega \\wedge \\omega'$ by the definition of the multiplication on", "$\\mathcal{O}_{T''}$. Thus the expression above is $\\nabla^2(s)$ viewed", "as a section of the subsheaf $\\mathcal{F}_T \\otimes \\Omega^2_{T/S, \\delta}$ of", "$q_2^*\\mathcal{F}$. Hence we get the integrability condition." ], "refs": [ "crystalline-remark-first-order-thickening", "crystalline-remark-first-order-thickening", "crystalline-remark-second-order-thickening", "crystalline-remark-second-order-thickening" ], "ref_ids": [ 10825, 10825, 10826, 10826 ] } ], "ref_ids": [] }, { "id": 10773, "type": "theorem", "label": "crystalline-lemma-homotopy-tensor", "categories": [ "crystalline" ], "title": "crystalline-lemma-homotopy-tensor", "contents": [ "Let $A_*$ be a cosimplicial ring. Let $\\varphi_*, \\psi_* : K_* \\to M_*$", "be homomorphisms of cosimplicial $A_*$-modules.", "\\begin{enumerate}", "\\item", "\\label{item-tensor}", "If $\\varphi_*$ and $\\psi_*$ are homotopic, then", "$$", "\\varphi_* \\otimes 1, \\psi_* \\otimes 1 :", "K_* \\otimes_{A_*} L_* \\longrightarrow M_* \\otimes_{A_*} L_*", "$$", "are homotopic for any cosimplicial $A_*$-module $L_*$.", "\\item", "\\label{item-wedge}", "If $\\varphi_*$ and $\\psi_*$ are homotopic, then", "$$", "\\wedge^i(\\varphi_*), \\wedge^i(\\psi_*) :", "\\wedge^i(K_*) \\longrightarrow \\wedge^i(M_*)", "$$", "are homotopic.", "\\item", "\\label{item-base-change}", "If $\\varphi_*$ and $\\psi_*$ are homotopic, and $A_* \\to B_*$", "is a homomorphism of cosimplicial rings, then", "$$", "\\varphi_* \\otimes 1, \\psi_* \\otimes 1 :", "K_* \\otimes_{A_*} B_* \\longrightarrow M_* \\otimes_{A_*} B_*", "$$", "are homotopic as homomorphisms of cosimplicial $B_*$-modules.", "\\item", "\\label{item-completion}", "If $I_* \\subset A_*$ is a cosimplicial ideal, then the induced", "maps", "$$", "\\varphi^\\wedge_*, \\psi^\\wedge_* :", "K_*^\\wedge \\longrightarrow M_*^\\wedge", "$$", "between completions are homotopic.", "\\item Add more here as needed, for example symmetric powers.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $h : M_* \\longrightarrow \\Hom(\\Delta[1], N_*)$ be the given", "homotopy. In degree $n$ we have", "$$", "h_n = (h_{n, \\alpha}) :", "K_n \\longrightarrow", "\\prod\\nolimits_{\\alpha \\in \\Delta[1]_n} K_n", "$$", "see Simplicial, Section \\ref{simplicial-section-homotopy-cosimplicial}.", "In order for a collection of $h_{n, \\alpha}$ to form a homotopy,", "it is necessary and sufficient if for every $f : [n] \\to [m]$ we", "have", "$$", "h_{m, \\alpha} \\circ M_*(f) = N_*(f) \\circ h_{n, \\alpha \\circ f}", "$$", "see", "Simplicial, Equation (\\ref{simplicial-equation-property-homotopy-cosimplicial}).", "We also should have that $\\psi_n = h_{n, 0 : [n] \\to [1]}$ and", "$\\varphi_n = h_{n, 1 : [n] \\to [1]}$.", "\\medskip\\noindent", "In each of the cases of the lemma we can produce the corresponding maps.", "Case (\\ref{item-tensor}). We can use the homotopy $h \\otimes 1$ defined", "in degree $n$ by setting", "$$", "(h \\otimes 1)_{n, \\alpha} = h_{n, \\alpha} \\otimes 1_{L_n} :", "K_n \\otimes_{A_n} L_n", "\\longrightarrow", "M_n \\otimes_{A_n} L_n.", "$$", "Case (\\ref{item-wedge}). We can use the homotopy $\\wedge^ih$ defined", "in degree $n$ by setting", "$$", "\\wedge^i(h)_{n, \\alpha} = \\wedge^i(h_{n, \\alpha}) :", "\\wedge_{A_n}(K_n)", "\\longrightarrow", "\\wedge^i_{A_n}(M_n).", "$$", "Case (\\ref{item-base-change}). We can use the homotopy $h \\otimes 1$ defined", "in degree $n$ by setting", "$$", "(h \\otimes 1)_{n, \\alpha} = h_{n, \\alpha} \\otimes 1 :", "K_n \\otimes_{A_n} B_n", "\\longrightarrow", "M_n \\otimes_{A_n} B_n.", "$$", "Case (\\ref{item-completion}). We can use the homotopy $h^\\wedge$ defined", "in degree $n$ by setting", "$$", "(h^\\wedge)_{n, \\alpha} = h_{n, \\alpha}^\\wedge :", "K_n^\\wedge", "\\longrightarrow", "M_n^\\wedge.", "$$", "This works because each $h_{n, \\alpha}$ is $A_n$-linear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10774, "type": "theorem", "label": "crystalline-lemma-structure-Dn", "categories": [ "crystalline" ], "title": "crystalline-lemma-structure-Dn", "contents": [ "Let $D$ and $D(n)$ be as in (\\ref{equation-D}) and (\\ref{equation-Dn}).", "The coprojection $P \\to P \\otimes_A \\ldots \\otimes_A P$,", "$f \\mapsto f \\otimes 1 \\otimes \\ldots \\otimes 1$", "induces an isomorphism", "\\begin{equation}", "\\label{equation-structure-Dn}", "D(n) = \\lim_e D\\langle \\xi_i(j) \\rangle/p^eD\\langle \\xi_i(j) \\rangle", "\\end{equation}", "of algebras over $D$ with", "$$", "\\xi_i(j) = x_i \\otimes 1 \\otimes \\ldots \\otimes 1 -", "1 \\otimes \\ldots \\otimes 1 \\otimes x_i \\otimes 1 \\otimes \\ldots \\otimes 1", "$$", "for $j = 1, \\ldots, n$ where the second $x_i$ is placed in the $j + 1$st", "slot; recall that $D(n)$ is constructed starting with the", "$n + 1$-fold tensor product of $P$ over $A$." ], "refs": [], "proofs": [ { "contents": [ "We have", "$$", "P \\otimes_A \\ldots \\otimes_A P = P[\\xi_i(j)]", "$$", "and $J(n)$ is generated by $J$ and the elements $\\xi_i(j)$.", "Hence the lemma follows from", "Lemma \\ref{lemma-divided-power-envelope-add-variables}." ], "refs": [ "crystalline-lemma-divided-power-envelope-add-variables" ], "ref_ids": [ 10747 ] } ], "ref_ids": [] }, { "id": 10775, "type": "theorem", "label": "crystalline-lemma-property-Dn", "categories": [ "crystalline" ], "title": "crystalline-lemma-property-Dn", "contents": [ "Let $D$ and $D(n)$ be as in (\\ref{equation-D}) and (\\ref{equation-Dn}).", "Then $(D, \\bar J, \\bar\\gamma)$ and $(D(n), \\bar J(n), \\bar\\gamma(n))$", "are objects of $\\text{Cris}^\\wedge(C/A)$, see", "Remark \\ref{remark-completed-affine-site}, and", "$$", "D(n) = \\coprod\\nolimits_{j = 0, \\ldots, n} D", "$$", "in $\\text{Cris}^\\wedge(C/A)$." ], "refs": [ "crystalline-remark-completed-affine-site" ], "proofs": [ { "contents": [ "The first assertion is clear. For the second, if $(B \\to C, \\delta)$ is an", "object of $\\text{Cris}^\\wedge(C/A)$, then we have", "$$", "\\Mor_{\\text{Cris}^\\wedge(C/A)}(D, B) = ", "\\Hom_A((P, J), (B, \\Ker(B \\to C)))", "$$", "and similarly for $D(n)$ replacing $(P, J)$ by", "$(P \\otimes_A \\ldots \\otimes_A P, J(n))$. The property on coproducts follows", "as $P \\otimes_A \\ldots \\otimes_A P$ is a coproduct." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10811 ] }, { "id": 10776, "type": "theorem", "label": "crystalline-lemma-crystals-on-affine", "categories": [ "crystalline" ], "title": "crystalline-lemma-crystals-on-affine", "contents": [ "In the situation above there is a functor", "$$", "\\begin{matrix}", "\\text{crystals in quasi-coherent} \\\\", "\\mathcal{O}_{X/S}\\text{-modules on }\\text{Cris}(X/S)", "\\end{matrix}", "\\longrightarrow", "\\begin{matrix}", "\\text{pairs }(M, \\nabla)\\text{ satisfying} \\\\", "\\text{(\\ref{item-complete}), (\\ref{item-connection}),", "(\\ref{item-integrable}), and (\\ref{item-topologically-quasi-nilpotent})}", "\\end{matrix}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a crystal in quasi-coherent modules on $X/S$.", "Set $T_e = \\Spec(D_e)$ so that $(X, T_e, \\bar\\gamma)$ is an object", "of $\\text{Cris}(X/S)$ for $e \\gg 0$. We have morphisms", "$$", "(X, T_e, \\bar\\gamma) \\to (X, T_{e + 1}, \\bar\\gamma) \\to \\ldots", "$$", "which are closed immersions. We set", "$$", "M =", "\\lim_e \\Gamma((X, T_e, \\bar\\gamma), \\mathcal{F}) =", "\\lim_e \\Gamma(T_e, \\mathcal{F}_{T_e}) = \\lim_e M_e", "$$", "Note that since $\\mathcal{F}$ is locally quasi-coherent we have", "$\\mathcal{F}_{T_e} = \\widetilde{M_e}$. Since $\\mathcal{F}$ is a", "crystal we have $M_e = M_{e + 1}/p^eM_{e + 1}$. Hence we see that", "$M_e = M/p^eM$ and that $M$ is $p$-adically complete, see", "Algebra, Lemma \\ref{algebra-lemma-limit-complete}.", "\\medskip\\noindent", "By Lemma \\ref{lemma-automatic-connection} we know that $\\mathcal{F}$", "comes endowed with a canonical integrable connection", "$\\nabla : \\mathcal{F} \\to \\mathcal{F} \\otimes \\Omega_{X/S}$.", "If we evaluate this connection on the objects $T_e$ constructed above", "we obtain a canonical integrable connection", "$$", "\\nabla : M \\longrightarrow M \\otimes^\\wedge_D \\Omega_D", "$$", "To see that this is topologically nilpotent we work out what this means.", "\\medskip\\noindent", "Now we can do the same procedure for the rings $D(n)$.", "This produces a $p$-adically complete $D(n)$-module $M(n)$. Again using", "the crystal property of $\\mathcal{F}$ we obtain isomorphisms", "$$", "M \\otimes^\\wedge_{D, p_0} D(1) \\rightarrow M(1)", "\\leftarrow M \\otimes^\\wedge_{D, p_1} D(1)", "$$", "compare with the proof of Lemma \\ref{lemma-automatic-connection}.", "Denote $c$ the composition from left to right. Pick $m \\in M$.", "Write $\\xi_i = x_i \\otimes 1 - 1 \\otimes x_i$.", "Using (\\ref{equation-structure-Dn}) we can write uniquely", "$$", "c(m \\otimes 1) = \\sum\\nolimits_K \\theta_K(m) \\otimes \\prod \\xi_i^{[k_i]}", "$$", "for some $\\theta_K(m) \\in M$ where the sum is over multi-indices", "$K = (k_i)$ with $k_i \\geq 0$ and $\\sum k_i < \\infty$. Set", "$\\theta_i = \\theta_K$ where $K$ has a $1$ in the $i$th spot and", "zeros elsewhere. We have", "$$", "\\nabla(m) = \\sum \\theta_i(m) \\text{d}x_i.", "$$", "as can be seen by comparing with the definition of", "$\\nabla$. Namely, the defining equation is", "$p_1^*m = \\nabla(m) - c(p_0^*m)$ in Lemma \\ref{lemma-automatic-connection}", "but the sign works out because in the Stacks project we consistently use", "$\\text{d}f = p_1(f) - p_0(f)$ modulo the ideal of the diagonal squared,", "and hence $\\xi_i = x_i \\otimes 1 - 1 \\otimes x_i$ maps to $-\\text{d}x_i$", "modulo the ideal of the diagonal squared.", "\\medskip\\noindent", "Denote $q_i : D \\to D(2)$ and $q_{ij} : D(1) \\to D(2)$ the coprojections", "corresponding to the indices $i, j$. As in the last paragraph of the proof of", "Lemma \\ref{lemma-automatic-connection}", "we see that", "$$", "q_{02}^*c = q_{12}^*c \\circ q_{01}^*c.", "$$", "This means that", "$$", "\\sum\\nolimits_{K''} \\theta_{K''}(m) \\otimes \\prod {\\zeta''_i}^{[k''_i]}", "=", "\\sum\\nolimits_{K', K} \\theta_{K'}(\\theta_K(m))", "\\otimes \\prod {\\zeta'_i}^{[k'_i]} \\prod \\zeta_i^{[k_i]}", "$$", "in $M \\otimes^\\wedge_{D, q_2} D(2)$ where", "\\begin{align*}", "\\zeta_i & = x_i \\otimes 1 \\otimes 1 - 1 \\otimes x_i \\otimes 1,\\\\", "\\zeta'_i & = 1 \\otimes x_i \\otimes 1 - 1 \\otimes 1 \\otimes x_i,\\\\", "\\zeta''_i & = x_i \\otimes 1 \\otimes 1 - 1 \\otimes 1 \\otimes x_i.", "\\end{align*}", "In particular $\\zeta''_i = \\zeta_i + \\zeta'_i$ and we have that", "$D(2)$ is the $p$-adic completion of the divided power polynomial", "ring in $\\zeta_i, \\zeta'_i$ over $q_2(D)$, see Lemma \\ref{lemma-structure-Dn}.", "Comparing coefficients in the expression above it follows immediately that", "$\\theta_i \\circ \\theta_j = \\theta_j \\circ \\theta_i$", "(this provides an alternative proof of the integrability of $\\nabla$) and that", "$$", "\\theta_K(m) = (\\prod \\theta_i^{k_i})(m).", "$$", "In particular, as the sum expressing $c(m \\otimes 1)$ above has to converge", "$p$-adically we conclude that for each $i$ and each $m \\in M$ only a finite", "number of $\\theta_i^k(m)$ are allowed to be nonzero modulo $p$." ], "refs": [ "algebra-lemma-limit-complete", "crystalline-lemma-automatic-connection", "crystalline-lemma-automatic-connection", "crystalline-lemma-automatic-connection", "crystalline-lemma-automatic-connection", "crystalline-lemma-structure-Dn" ], "ref_ids": [ 880, 10772, 10772, 10772, 10772, 10774 ] } ], "ref_ids": [] }, { "id": 10777, "type": "theorem", "label": "crystalline-lemma-crystals-on-affine-smooth", "categories": [ "crystalline" ], "title": "crystalline-lemma-crystals-on-affine-smooth", "contents": [ "In Situation \\ref{situation-affine}.", "Let $A \\to P' \\to C$ be ring maps with $A \\to P'$ smooth and $P' \\to C$", "surjective with kernel $J'$. Let $D'$ be the $p$-adic completion of", "$D_{P', \\gamma}(J')$. There are homomorphisms of divided power $A$-algebras", "$$", "a : D \\longrightarrow D',\\quad b : D' \\longrightarrow D", "$$", "compatible with the maps $D \\to C$ and $D' \\to C$ such that", "$a \\circ b = \\text{id}_{D'}$. These maps induce", "an equivalence of categories of pairs $(M, \\nabla)$ satisfying", "(\\ref{item-complete}), (\\ref{item-connection}),", "(\\ref{item-integrable}), and (\\ref{item-topologically-quasi-nilpotent})", "over $D$ and pairs $(M', \\nabla')$ satisfying", "(\\ref{item-complete}), (\\ref{item-connection}),", "(\\ref{item-integrable}), and (\\ref{item-topologically-quasi-nilpotent})", "over $D'$. In particular, the equivalence of categories of", "Proposition \\ref{proposition-crystals-on-affine}", "also holds for the corresponding functor towards pairs over $D'$." ], "refs": [ "crystalline-proposition-crystals-on-affine" ], "proofs": [ { "contents": [ "Before we embark on the proof we briefly explain how to formulate", "condition (\\ref{item-topologically-quasi-nilpotent}) for the case of", "pairs $(M', \\nabla')$ over $D'$. Since $A \\to P'$ is smooth", "the $P'$-module $\\Omega_{P'/A}$ is finite projective. Thus we can", "choose a map $\\Omega_{P'/A} \\to \\bigoplus_{i = 1, \\ldots, n} P'$", "wich identifies $\\Omega_{P'/A}$ with a direct summand of the target.", "This determines a map $\\Omega_{D'} \\to \\bigoplus_{i = 1, \\ldots, n} D'$", "identifying the source with a direct summand of the target.", "Thus for $m' \\in M'$ we can write $\\nabla'(m') = \\sum \\theta'_i(m')$", "with $\\theta'_i(m') \\in D'$. The topogical quasi-nilpotence of", "$\\nabla'$ means: for any $m' \\in M'$ there are only finitely", "many pairs $(i, k)$ such that $(\\theta'_i)^k(m') \\not \\in pM'$.", "(Since in this case there are only finitely many $i$ this just means", "that $\\theta_i^k(m') \\in pM'$ for all $k \\gg 0$.)", "\\medskip\\noindent", "We can pick the map $P = A[x_i] \\to C$ such that it factors through", "a surjection of $A$-algebras $P \\to P'$ (we may have to increase the", "number of variables in $P$ to do this). Hence we obtain a surjective", "map $a : D \\to D'$ by functoriality of divided power envelopes and", "completion. Pick $e$ large enough so that $D_e$ is a divided power", "thickening of $C$ over $A$. Then $D_e \\to C$ is a surjection whose kernel", "is locally nilpotent, see Divided Power Algebra, Lemma \\ref{dpa-lemma-nil}.", "Setting $D'_e = D'/p^eD'$", "we see that the kernel of $D_e \\to D'_e$ is locally nilpotent.", "Hence by Algebra, Lemma \\ref{algebra-lemma-smooth-strong-lift}", "we can find a lift $\\beta_e : P' \\to D_e$ of the map $P' \\to D'_e$.", "Note that $D_{e + i + 1} \\to D_{e + i} \\times_{D'_{e + i}} D'_{e + i + 1}$", "is surjective with square zero kernel for any $i \\geq 0$ because", "$p^{e + i}D \\to p^{e + i}D'$ is surjective. Applying the usual lifting", "property (Algebra, Proposition \\ref{algebra-proposition-smooth-formally-smooth})", "successively to the diagrams", "$$", "\\xymatrix{", "P' \\ar[r] & D_{e + i} \\times_{D'_{e + i}} D'_{e + i + 1} \\\\", "A \\ar[u] \\ar[r] & D_{e + i + 1} \\ar[u]", "}", "$$", "we see that we can find an $A$-algebra map $\\beta : P' \\to D$ whose", "composition with $a$ is the given map $P' \\to D'$.", "By the universal property of the divided power envelope we obtain a", "map $D_{P', \\gamma}(J') \\to D$. As $D$ is $p$-adically complete we", "obtain $b : D' \\to D$ such that $a \\circ b = \\text{id}_{D'}$.", "\\medskip\\noindent", "Consider the base change functor", "$$", "(M, \\nabla) \\longmapsto", "(M \\otimes^\\wedge_D D', \\nabla')", "$$", "from pairs for $D$ to pairs for $D'$, see", "Remark \\ref{remark-base-change-connection}.", "Similarly, we have the base change functor corresponding to the divided", "power homomorphism $D' \\to D$. To finish the proof of the lemma we have", "to show that the base change for the compositions $b \\circ a : D \\to D$", "and $a \\circ b : D' \\to D'$ are isomorphic to the identity functor.", "This is clear for the second as $a \\circ b = \\text{id}_{D'}$.", "To prove it for the first, we use the functorial isomorphism", "$$", "c_{\\text{id}_D, b \\circ a} :", "M \\otimes_{D, \\text{id}_D} D", "\\longrightarrow", "M \\otimes_{D, b \\circ a} D", "$$", "of the proof of Proposition \\ref{proposition-crystals-on-affine}.", "The only thing to prove is that these maps are horizontal, which we omit.", "\\medskip\\noindent", "The last statement of the proof now follows." ], "refs": [ "dpa-lemma-nil", "algebra-lemma-smooth-strong-lift", "algebra-proposition-smooth-formally-smooth", "crystalline-remark-base-change-connection", "crystalline-proposition-crystals-on-affine" ], "ref_ids": [ 1653, 1216, 1426, 10815, 10793 ] } ], "ref_ids": [ 10793 ] }, { "id": 10778, "type": "theorem", "label": "crystalline-lemma-vanishing-lqc", "categories": [ "crystalline" ], "title": "crystalline-lemma-vanishing-lqc", "contents": [ "In Situation \\ref{situation-global}.", "Let $\\mathcal{F}$ be a locally quasi-coherent $\\mathcal{O}_{X/S}$-module", "on $\\text{Cris}(X/S)$. Then we have", "$$", "H^p((U, T, \\delta), \\mathcal{F}) = 0", "$$", "for all $p > 0$ and all $(U, T, \\delta)$ with $T$ or $U$ affine." ], "refs": [], "proofs": [ { "contents": [ "As $U \\to T$ is a thickening we see that $U$ is affine if and only if $T$", "is affine, see Limits, Lemma \\ref{limits-lemma-affine}.", "Having said this, let us apply", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cech-vanish-collection}", "to the collection $\\mathcal{B}$ of affine objects $(U, T, \\delta)$ and the", "collection $\\text{Cov}$ of affine open coverings", "$\\mathcal{U} = \\{(U_i, T_i, \\delta_i) \\to (U, T, \\delta)\\}$. The", "{\\v C}ech complex", "${\\check C}^*(\\mathcal{U}, \\mathcal{F})$ for such a covering is simply", "the {\\v C}ech complex of the quasi-coherent $\\mathcal{O}_T$-module", "$\\mathcal{F}_T$", "(here we are using the assumption that $\\mathcal{F}$ is locally quasi-coherent)", "with respect to the affine open covering $\\{T_i \\to T\\}$ of the", "affine scheme $T$. Hence the {\\v C}ech cohomology is zero by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cech-cohomology-quasi-coherent} and", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "Thus the hypothesis of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cech-vanish-collection}", "are satisfied and we win." ], "refs": [ "limits-lemma-affine", "sites-cohomology-lemma-cech-vanish-collection", "coherent-lemma-cech-cohomology-quasi-coherent", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "sites-cohomology-lemma-cech-vanish-collection" ], "ref_ids": [ 15082, 4205, 3286, 3282, 4205 ] } ], "ref_ids": [] }, { "id": 10779, "type": "theorem", "label": "crystalline-lemma-compare", "categories": [ "crystalline" ], "title": "crystalline-lemma-compare", "contents": [ "In Situation \\ref{situation-global}.", "Assume moreover $X$ and $S$ are affine schemes.", "Consider the full subcategory $\\mathcal{C} \\subset \\text{Cris}(X/S)$", "consisting of divided power thickenings $(X, T, \\delta)$", "endowed with the chaotic topology (see", "Sites, Example \\ref{sites-example-indiscrete}).", "For any locally quasi-coherent $\\mathcal{O}_{X/S}$-module $\\mathcal{F}$", "we have", "$$", "R\\Gamma(\\mathcal{C}, \\mathcal{F}|_\\mathcal{C}) =", "R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})", "$$" ], "refs": [], "proofs": [ { "contents": [ "We will use without further mention that $\\mathcal{C}$ and $\\text{Cris}(X/S)$", "have products and fibre products, see", "Lemma \\ref{lemma-divided-power-thickening-fibre-products}.", "Note that the inclusion functor $u : \\mathcal{C} \\to \\text{Cris}(X/S)$", "is fully faithful, continuous and commutes with products and fibre products.", "We claim it defines a morphism of ringed sites", "$$", "f :", "(\\text{Cris}(X/S), \\mathcal{O}_{X/S})", "\\longrightarrow", "(\\Sh(\\mathcal{C}), \\mathcal{O}_{X/S}|_\\mathcal{C})", "$$", "To see this we will use Sites, Lemma \\ref{sites-lemma-directed-morphism}.", "Note that $\\mathcal{C}$ has fibre products and $u$ commutes with them", "so the categories $\\mathcal{I}^u_{(U, T, \\delta)}$ are disjoint unions", "of directed categories (by Sites, Lemma \\ref{sites-lemma-almost-directed} and", "Categories, Lemma \\ref{categories-lemma-split-into-directed}). Hence it", "suffices to show that $\\mathcal{I}^u_{(U, T, \\delta)}$ is connected.", "Nonempty follows from Lemma \\ref{lemma-set-generators}", "and connectedness follows from the fact that $\\mathcal{C}$ has products", "and that $u$ commutes with them (compare with the proof of", "Sites, Lemma \\ref{sites-lemma-directed}).", "\\medskip\\noindent", "Note that $f_*\\mathcal{F} = \\mathcal{F}|_\\mathcal{C}$. Hence the lemma", "follows if $R^pf_*\\mathcal{F} = 0$ for $p > 0$, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-apply-Leray}. By", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-higher-direct-images}", "it suffices to show that $H^p((X, T, \\delta), \\mathcal{F}) = 0$ for all", "$(X, T, \\delta)$. This follows from Lemma \\ref{lemma-vanishing-lqc}." ], "refs": [ "crystalline-lemma-divided-power-thickening-fibre-products", "sites-lemma-directed-morphism", "sites-lemma-almost-directed", "categories-lemma-split-into-directed", "crystalline-lemma-set-generators", "sites-lemma-directed", "sites-cohomology-lemma-apply-Leray", "sites-cohomology-lemma-higher-direct-images", "crystalline-lemma-vanishing-lqc" ], "ref_ids": [ 10762, 8526, 8497, 12234, 10754, 8498, 4221, 4189, 10778 ] } ], "ref_ids": [] }, { "id": 10780, "type": "theorem", "label": "crystalline-lemma-complete", "categories": [ "crystalline" ], "title": "crystalline-lemma-complete", "contents": [ "In Situation \\ref{situation-affine}.", "Set $\\mathcal{C} = (\\text{Cris}(C/A))^{opp}$ and", "$\\mathcal{C}^\\wedge = (\\text{Cris}^\\wedge(C/A))^{opp}$", "endowed with the chaotic topology, see", "Remark \\ref{remark-completed-affine-site} for notation.", "There is a morphism of topoi", "$$", "g : \\Sh(\\mathcal{C}) \\longrightarrow \\Sh(\\mathcal{C}^\\wedge)", "$$", "such that if $\\mathcal{F}$ is a sheaf of abelian groups on", "$\\mathcal{C}$, then", "$$", "R^pg_*\\mathcal{F}(B \\to C, \\delta) =", "\\left\\{", "\\begin{matrix}", "\\lim_e \\mathcal{F}(B_e \\to C, \\delta) & \\text{if }p = 0 \\\\", "R^1\\lim_e \\mathcal{F}(B_e \\to C, \\delta) & \\text{if }p = 1 \\\\", "0 & \\text{else}", "\\end{matrix}", "\\right.", "$$", "where $B_e = B/p^eB$ for $e \\gg 0$." ], "refs": [ "crystalline-remark-completed-affine-site" ], "proofs": [ { "contents": [ "Any functor between categories defines a morphism between chaotic", "topoi in the same direction, for example because such a functor", "can be considered as a cocontinuous functor between sites, see", "Sites, Section \\ref{sites-section-cocontinuous-morphism-topoi}.", "Proof of the description of $g_*\\mathcal{F}$ is omitted.", "Note that in the statement we take $(B_e \\to C, \\delta)$", "is an object of $\\text{Cris}(C/A)$ only for $e$ large enough.", "Let $\\mathcal{I}$ be an injective abelian sheaf on $\\mathcal{C}$.", "Then the transition maps", "$$", "\\mathcal{I}(B_e \\to C, \\delta) \\leftarrow", "\\mathcal{I}(B_{e + 1} \\to C, \\delta)", "$$", "are surjective as the morphisms", "$$", "(B_e \\to C, \\delta)", "\\longrightarrow", "(B_{e + 1} \\to C, \\delta)", "$$", "are monomorphisms in the category $\\mathcal{C}$. Hence for an injective", "abelian sheaf both sides of the displayed formula of the lemma agree.", "Taking an injective resolution of $\\mathcal{F}$ one easily obtains", "the result (sheaves are presheaves, so exactness is measured on the", "level of groups of sections over objects)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10811 ] }, { "id": 10781, "type": "theorem", "label": "crystalline-lemma-category-with-covering", "categories": [ "crystalline" ], "title": "crystalline-lemma-category-with-covering", "contents": [ "Let $\\mathcal{C}$ be a category endowed with the chaotic topology.", "Let $X$ be an object of $\\mathcal{C}$ such that every object of", "$\\mathcal{C}$ has a morphism towards $X$. Assume that $\\mathcal{C}$", "has products of pairs.", "Then for every abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}$", "the total cohomology $R\\Gamma(\\mathcal{C}, \\mathcal{F})$ is represented", "by the complex", "$$", "\\mathcal{F}(X) \\to \\mathcal{F}(X \\times X) \\to", "\\mathcal{F}(X \\times X \\times X) \\to \\ldots", "$$", "associated to the cosimplicial abelian group $[n] \\mapsto \\mathcal{F}(X^n)$." ], "refs": [], "proofs": [ { "contents": [ "Note that $H^q(X^p, \\mathcal{F}) = 0$ for all $q > 0$ as any presheaf is a", "sheaf on $\\mathcal{C}$. The assumption on $X$ is that $h_X \\to *$", "is surjective. Using that $H^q(X, \\mathcal{F}) = H^q(h_X, \\mathcal{F})$ and", "$H^q(\\mathcal{C}, \\mathcal{F}) = H^q(*, \\mathcal{F})$ we see that our", "statement is a special case of", "Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-cech-to-cohomology-sheaf-sets}." ], "refs": [ "sites-cohomology-lemma-cech-to-cohomology-sheaf-sets" ], "ref_ids": [ 4214 ] } ], "ref_ids": [] }, { "id": 10782, "type": "theorem", "label": "crystalline-lemma-vanishing-omega-1", "categories": [ "crystalline" ], "title": "crystalline-lemma-vanishing-omega-1", "contents": [ "With notation as in (\\ref{equation-omega-Dn}) the complex", "$$", "\\Omega_{D(0)} \\to \\Omega_{D(1)} \\to \\Omega_{D(2)} \\to \\ldots", "$$", "is homotopic to zero as a $D(*)$-cosimplicial module." ], "refs": [], "proofs": [ { "contents": [ "We are going to use the principle of", "Simplicial, Lemma \\ref{simplicial-lemma-functorial-homotopy}", "and more specifically", "Lemma \\ref{lemma-homotopy-tensor}", "which tells us that homotopic maps between (co)simplicial objects", "are transformed by any functor into homotopic maps.", "The complex of the lemma is equal to the $p$-adic completion of the", "base change of the cosimplicial module", "$$", "M_* = \\left(", "\\Omega_{P/A} \\to", "\\Omega_{P \\otimes_A P/A} \\to", "\\Omega_{P \\otimes_A P \\otimes_A P/A} \\to \\ldots", "\\right)", "$$", "via the cosimplicial ring map $P\\otimes_A \\ldots \\otimes_A P \\to D(n)$. This", "follows from Lemma \\ref{lemma-module-differentials-divided-power-envelope},", "see comments following (\\ref{equation-omega-D}). Hence it", "suffices to show that the cosimplicial module $M_*$ is homotopic to zero", "(uses base change and $p$-adic completion).", "We can even assume $A = \\mathbf{Z}$ and $P = \\mathbf{Z}[\\{x_i\\}_{i \\in I}]$", "as we can use base change with $\\mathbf{Z} \\to A$.", "In this case $P^{\\otimes n + 1}$ is the polynomial algebra on the elements", "$$", "x_i(e) = 1 \\otimes \\ldots \\otimes x_i \\otimes \\ldots \\otimes 1", "$$", "with $x_i$ in the $e$th slot. The modules of the complex are free on the", "generators $\\text{d}x_i(e)$. Note that if $f : [n] \\to [m]$ is a", "map then we see that", "$$", "M_*(f)(\\text{d}x_i(e)) = \\text{d}x_i(f(e))", "$$", "Hence we see that $M_*$ is a direct sum over $I$ of copies of the module", "studied in Example \\ref{example-cosimplicial-module} and we win." ], "refs": [ "simplicial-lemma-functorial-homotopy", "crystalline-lemma-homotopy-tensor", "crystalline-lemma-module-differentials-divided-power-envelope" ], "ref_ids": [ 14879, 10773, 10759 ] } ], "ref_ids": [] }, { "id": 10783, "type": "theorem", "label": "crystalline-lemma-vanishing", "categories": [ "crystalline" ], "title": "crystalline-lemma-vanishing", "contents": [ "With notation as in (\\ref{equation-Dn}) and (\\ref{equation-omega-Dn}),", "given any cosimplicial module $M_*$ over $D(*)$ and", "$i > 0$ the cosimplicial module", "$$", "M_0 \\otimes^\\wedge_{D(0)} \\Omega^i_{D(0)} \\to", "M_1 \\otimes^\\wedge_{D(1)} \\Omega^i_{D(1)} \\to", "M_2 \\otimes^\\wedge_{D(2)} \\Omega^i_{D(2)} \\to \\ldots", "$$", "is homotopic to zero, where $\\Omega^i_{D(n)}$ is the $p$-adic completion", "of the $i$th exterior power of $\\Omega_{D(n)}$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-vanishing-omega-1} the endomorphisms $0$ and $1$", "of $\\Omega_{D(*)}$ are homotopic.", "If we apply the functor $\\wedge^i$ we see that", "the same is true for the cosimplicial module $\\wedge^i\\Omega_{D(*)}$, see", "Lemma \\ref{lemma-homotopy-tensor}.", "Another application of the same lemma shows the $p$-adic completion", "$\\Omega^i_{D(*)}$ is homotopy equivalent to zero.", "Tensoring with $M_*$ we see that $M_* \\otimes_{D(*)} \\Omega^i_{D(*)}$", "is homotopic to zero, see Lemma \\ref{lemma-homotopy-tensor} again.", "A final application of the $p$-adic completion functor finishes the proof." ], "refs": [ "crystalline-lemma-vanishing-omega-1", "crystalline-lemma-homotopy-tensor", "crystalline-lemma-homotopy-tensor" ], "ref_ids": [ 10782, 10773, 10773 ] } ], "ref_ids": [] }, { "id": 10784, "type": "theorem", "label": "crystalline-lemma-poincare", "categories": [ "crystalline" ], "title": "crystalline-lemma-poincare", "contents": [ "Let $A$ be a ring. Let $P = A\\langle x_i \\rangle$ be a divided", "power polynomial ring over $A$. For any $A$-module $M$ the complex", "$$", "0 \\to M \\to", "M \\otimes_A P \\to", "M \\otimes_A \\Omega^1_{P/A, \\delta} \\to", "M \\otimes_A \\Omega^2_{P/A, \\delta} \\to \\ldots", "$$", "is exact. Let $D$ be the $p$-adic completion of $P$.", "Let $\\Omega^i_D$ be the $p$-adic completion of the $i$th exterior", "power of $\\Omega_{D/A, \\delta}$. For any $p$-adically complete", "$A$-module $M$ the complex", "$$", "0 \\to M \\to", "M \\otimes^\\wedge_A D \\to", "M \\otimes^\\wedge_A \\Omega^1_D \\to", "M \\otimes^\\wedge_A \\Omega^2_D \\to \\ldots", "$$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that the complex", "$$", "E :", "(0 \\to A \\to P \\to \\Omega^1_{P/A, \\delta} \\to", "\\Omega^2_{P/A, \\delta} \\to \\ldots)", "$$", "is homotopy equivalent to zero as a complex of $A$-modules.", "For every multi-index $K = (k_i)$ we can consider the subcomplex $E(K)$", "which in degree $j$ consists of", "$$", "\\bigoplus\\nolimits_{I = \\{i_1, \\ldots, i_j\\} \\subset \\text{Supp}(K)}", "A", "\\prod\\nolimits_{i \\not \\in I} x_i^{[k_i]}", "\\prod\\nolimits_{i \\in I} x_i^{[k_i - 1]}", "\\text{d}x_{i_1} \\wedge \\ldots \\wedge \\text{d}x_{i_j}", "$$", "Since $E = \\bigoplus E(K)$ we see that it suffices to prove each of the", "complexes $E(K)$ is homotopic to zero. If $K = 0$, then", "$E(K) : (A \\to A)$ is homotopic to zero. If $K$ has nonempty (finite)", "support $S$, then the complex $E(K)$ is isomorphic to the complex", "$$", "0 \\to A \\to", "\\bigoplus\\nolimits_{s \\in S} A \\to", "\\wedge^2(\\bigoplus\\nolimits_{s \\in S} A) \\to", "\\ldots \\to \\wedge^{\\# S}(\\bigoplus\\nolimits_{s \\in S} A) \\to 0", "$$", "which is homotopic to zero, for example by", "More on Algebra, Lemma \\ref{more-algebra-lemma-homotopy-koszul-abstract}." ], "refs": [ "more-algebra-lemma-homotopy-koszul-abstract" ], "ref_ids": [ 9959 ] } ], "ref_ids": [] }, { "id": 10785, "type": "theorem", "label": "crystalline-lemma-relative-poincare", "categories": [ "crystalline" ], "title": "crystalline-lemma-relative-poincare", "contents": [ "Let $A$ be a ring. Let $(B, J, \\delta)$ be a divided power ring.", "Let $P = B\\langle x_i \\rangle$ be a divided power polynomial", "ring over $B$ with divided power ideal $J = IP + B\\langle x_i \\rangle_{+}$", "as usual. Let $M$ be a $B$-module endowed with an integrable connection", "$\\nabla : M \\to M \\otimes_B \\Omega^1_{B/A, \\delta}$. Then the map of", "de Rham complexes", "$$", "M \\otimes_B \\Omega^*_{B/A, \\delta}", "\\longrightarrow", "M \\otimes_P \\Omega^*_{P/A, \\delta}", "$$", "is a quasi-isomorphism. Let $D$, resp.\\ $D'$ be the $p$-adic completion of", "$B$, resp.\\ $P$ and let $\\Omega^i_D$, resp.\\ $\\Omega^i_{D'}$ be the $p$-adic", "completion of $\\Omega^i_{B/A, \\delta}$,", "resp.\\ $\\Omega^i_{P/A, \\delta}$. Let $M$ be a $p$-adically complete", "$D$-module endowed with an integral connection", "$\\nabla : M \\to M \\otimes^\\wedge_D \\Omega^1_D$.", "Then the map of de Rham complexes", "$$", "M \\otimes^\\wedge_D \\Omega^*_D", "\\longrightarrow", "M \\otimes^\\wedge_D \\Omega^*_{D'}", "$$", "is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Consider the decreasing filtration $F^*$ on $\\Omega^*_{B/A, \\delta}$", "given by the subcomplexes", "$F^i(\\Omega^*_{B/A, \\delta}) = \\sigma_{\\geq i}\\Omega^*_{B/A, \\delta}$.", "See Homology, Section \\ref{homology-section-truncations}.", "This induces a decreasing filtration $F^*$ on $\\Omega^*_{P/A, \\delta}$", "by setting", "$$", "F^i(\\Omega^*_{P/A, \\delta}) =", "F^i(\\Omega^*_{B/A, \\delta}) \\wedge \\Omega^*_{P/A, \\delta}.", "$$", "We have a split short exact sequence", "$$", "0 \\to \\Omega^1_{B/A, \\delta} \\otimes_B P \\to", "\\Omega^1_{P/A, \\delta} \\to", "\\Omega^1_{P/B, \\delta} \\to 0", "$$", "and the last module is free on $\\text{d}x_i$. It follows from this that", "$F^i(\\Omega^*_{P/A, \\delta}) \\to \\Omega^*_{P/A, \\delta}$ is a termwise", "split injection and that", "$$", "\\text{gr}^i_F(\\Omega^*_{P/A, \\delta}) =", "\\Omega^i_{B/A, \\delta} \\otimes_B \\Omega^*_{P/B, \\delta}", "$$", "as complexes. Thus we can define a filtration $F^*$ on", "$M \\otimes_B \\Omega^*_{B/A, \\delta}$ by setting", "$$", "F^i(M \\otimes_B \\Omega^*_{P/A, \\delta}) =", "M \\otimes_B F^i(\\Omega^*_{P/A, \\delta})", "$$", "and we have", "$$", "\\text{gr}^i_F(M \\otimes_B \\Omega^*_{P/A, \\delta}) =", "M \\otimes_B \\Omega^i_{B/A, \\delta} \\otimes_B \\Omega^*_{P/B, \\delta}", "$$", "as complexes.", "By Lemma \\ref{lemma-poincare} each of these complexes is", "quasi-isomorphic to $M \\otimes_B \\Omega^i_{B/A, \\delta}$ placed in degree $0$.", "Hence we see that the first displayed map of the lemma is a morphism of", "filtered complexes which induces a quasi-isomorphism on graded pieces. This", "implies that it is a quasi-isomorphism, for example by the spectral sequence", "associated to a filtered complex, see", "Homology, Section \\ref{homology-section-filtered-complex}.", "\\medskip\\noindent", "The proof of the second quasi-isomorphism is exactly the same." ], "refs": [ "crystalline-lemma-poincare" ], "ref_ids": [ 10784 ] } ], "ref_ids": [] }, { "id": 10786, "type": "theorem", "label": "crystalline-lemma-cohomology-is-zero", "categories": [ "crystalline" ], "title": "crystalline-lemma-cohomology-is-zero", "contents": [ "Assumptions and notation as in", "Proposition \\ref{proposition-compute-cohomology}.", "Then", "$$", "H^j(\\text{Cris}(X/S), \\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^i_{X/S})", "= 0", "$$", "for all $i > 0$ and all $j \\geq 0$." ], "refs": [ "crystalline-proposition-compute-cohomology" ], "proofs": [ { "contents": [ "Using Lemma \\ref{lemma-omega-locally-quasi-coherent} it follows that", "$\\mathcal{H} = \\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^i_{X/S}$", "also satisfies assumptions (1) and (2) of", "Proposition \\ref{proposition-compute-cohomology}.", "Write $M(n)_e = \\Gamma((X, T(n)_e, \\delta(n)), \\mathcal{F})$", "so that $M(n) = \\lim_e M(n)_e$. Then", "\\begin{align*}", "\\lim_e \\Gamma((X, T(n)_e, \\delta(n)), \\mathcal{H}) & =", "\\lim_e M(n)_e \\otimes_{D(n)_e} \\Omega_{D(n)}/p^e\\Omega_{D(n)} \\\\", "& = \\lim_e M(n)_e \\otimes_{D(n)} \\Omega_{D(n)}", "\\end{align*}", "By", "Lemma \\ref{lemma-vanishing}", "the cosimplicial modules", "$$", "M(0)_e \\otimes_{D(0)} \\Omega^i_{D(0)} \\to", "M(1)_e \\otimes_{D(1)} \\Omega^i_{D(1)} \\to", "M(2)_e \\otimes_{D(2)} \\Omega^i_{D(2)} \\to \\ldots", "$$", "are homotopic to zero. Because the transition maps", "$M(n)_{e + 1} \\to M(n)_e$ are surjective, we see that", "the inverse limit of the associated complexes are acyclic\\footnote{Actually,", "they are even homotopic to zero as the homotopies fit together, but we don't", "need this. The reason for this roundabout argument is that", "the limit $\\lim_e M(n)_e \\otimes_{D(n)} \\Omega^i_{D(n)}$ isn't the", "$p$-adic completion of $M(n) \\otimes_{D(n)} \\Omega^i_{D(n)}$ as with", "the assumptions of the lemma we don't know that", "$M(n)_e = M(n)_{e + 1}/p^eM(n)_{e + 1}$. If $\\mathcal{F}$ is a crystal", "then this does hold.}.", "Hence the vanishing of cohomology of $\\mathcal{H}$ by", "Proposition \\ref{proposition-compute-cohomology}." ], "refs": [ "crystalline-lemma-omega-locally-quasi-coherent", "crystalline-proposition-compute-cohomology", "crystalline-lemma-vanishing", "crystalline-proposition-compute-cohomology" ], "ref_ids": [ 10771, 10794, 10783, 10794 ] } ], "ref_ids": [ 10794 ] }, { "id": 10787, "type": "theorem", "label": "crystalline-lemma-compute-cohomology-crystal-smooth", "categories": [ "crystalline" ], "title": "crystalline-lemma-compute-cohomology-crystal-smooth", "contents": [ "Assumptions as in Proposition \\ref{proposition-compute-cohomology-crystal}.", "Let $A \\to P' \\to C$ be ring maps with $A \\to P'$ smooth and $P' \\to C$", "surjective with kernel $J'$. Let $D'$ be the $p$-adic completion of", "$D_{P', \\gamma}(J')$. Let $(M', \\nabla')$ be the pair over $D'$", "corresponding to $\\mathcal{F}$, see", "Lemma \\ref{lemma-crystals-on-affine-smooth}. Then the complex", "$$", "M' \\otimes^\\wedge_{D'} \\Omega^*_{D'}", "$$", "computes $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$." ], "refs": [ "crystalline-proposition-compute-cohomology-crystal", "crystalline-lemma-crystals-on-affine-smooth" ], "proofs": [ { "contents": [ "Choose $a : D \\to D'$ and $b : D' \\to D$ as in", "Lemma \\ref{lemma-crystals-on-affine-smooth}.", "Note that the base change $M = M' \\otimes_{D', b} D$ with its", "connection $\\nabla$ corresponds to $\\mathcal{F}$. Hence we know", "that $M \\otimes^\\wedge_D \\Omega_D^*$ computes the crystalline", "cohomology of $\\mathcal{F}$, see", "Proposition \\ref{proposition-compute-cohomology-crystal}.", "Hence it suffices to show that the base change maps (induced", "by $a$ and $b$)", "$$", "M' \\otimes^\\wedge_{D'} \\Omega^*_{D'}", "\\longrightarrow", "M \\otimes^\\wedge_D \\Omega^*_D", "\\quad\\text{and}\\quad", "M \\otimes^\\wedge_D \\Omega^*_D", "\\longrightarrow", "M' \\otimes^\\wedge_{D'} \\Omega^*_{D'}", "$$", "are quasi-isomorphisms. Since $a \\circ b = \\text{id}_{D'}$ we see", "that the composition one way around is the identity on the complex", "$M' \\otimes^\\wedge_{D'} \\Omega^*_{D'}$. Hence it suffices to show that", "the map", "$$", "M \\otimes^\\wedge_D \\Omega^*_D", "\\longrightarrow", "M \\otimes^\\wedge_D \\Omega^*_D", "$$", "induced by $b \\circ a : D \\to D$ is a quasi-isomorphism. (Note that we", "have the same complex on both sides as $M = M' \\otimes^\\wedge_{D', b} D$,", "hence $M \\otimes^\\wedge_{D, b \\circ a} D =", "M' \\otimes^\\wedge_{D', b \\circ a \\circ b} D =", "M' \\otimes^\\wedge_{D', b} D = M$.) In fact, we claim that for any", "divided power $A$-algebra homomorphism $\\rho : D \\to D$ compatible", "with the augmentation to $C$ the induced map", "$M \\otimes^\\wedge_D \\Omega^*_D \\to M \\otimes^\\wedge_{D, \\rho} \\Omega^*_D$", "is a quasi-isomorphism.", "\\medskip\\noindent", "Write $\\rho(x_i) = x_i + z_i$. The elements $z_i$ are in the", "divided power ideal of $D$ because $\\rho$ is compatible with the", "augmentation to $C$. Hence we can factor the map $\\rho$", "as a composition", "$$", "D \\xrightarrow{\\sigma} D\\langle \\xi_i \\rangle^\\wedge \\xrightarrow{\\tau} D", "$$", "where the first map is given by $x_i \\mapsto x_i + \\xi_i$ and the", "second map is the divided power $D$-algebra map which maps $\\xi_i$ to $z_i$.", "(This uses the universal properties of polynomial algebra, divided", "power polynomial algebras, divided power envelopes, and $p$-adic completion.)", "Note that there exists an {\\it automorphism} $\\alpha$ of", "$D\\langle \\xi_i \\rangle^\\wedge$ with $\\alpha(x_i) = x_i - \\xi_i$", "and $\\alpha(\\xi_i) = \\xi_i$. Applying Lemma \\ref{lemma-relative-poincare}", "to $\\alpha \\circ \\sigma$ (which maps $x_i$ to $x_i$) and using that", "$\\alpha$ is an isomorphism we conclude that $\\sigma$ induces a", "quasi-isomorphism of $M \\otimes^\\wedge_D \\Omega^*_D$ with", "$M \\otimes^\\wedge_{D, \\sigma} \\Omega^*_{D\\langle x_i \\rangle^\\wedge}$.", "On the other hand the map $\\tau$ has as a left inverse the", "map $D \\to D\\langle x_i \\rangle^\\wedge$, $x_i \\mapsto x_i$", "and we conclude (using Lemma \\ref{lemma-relative-poincare} once more)", "that $\\tau$ induces a quasi-isomorphism of", "$M \\otimes^\\wedge_{D, \\sigma} \\Omega^*_{D\\langle x_i \\rangle^\\wedge}$", "with $M \\otimes^\\wedge_{D, \\tau \\circ \\sigma} \\Omega^*_D$. Composing these", "two quasi-isomorphisms we obtain that $\\rho$ induces a quasi-isomorphism", "$M \\otimes^\\wedge_D \\Omega^*_D \\to M \\otimes^\\wedge_{D, \\rho} \\Omega^*_D$", "as desired." ], "refs": [ "crystalline-lemma-crystals-on-affine-smooth", "crystalline-proposition-compute-cohomology-crystal", "crystalline-lemma-relative-poincare", "crystalline-lemma-relative-poincare" ], "ref_ids": [ 10777, 10795, 10785, 10785 ] } ], "ref_ids": [ 10795, 10777 ] }, { "id": 10788, "type": "theorem", "label": "crystalline-lemma-find-homotopy", "categories": [ "crystalline" ], "title": "crystalline-lemma-find-homotopy", "contents": [ "In the situation above there exists a map of complexes", "$$", "e_M^\\bullet :", "M \\otimes_B (\\Omega')^\\bullet", "\\longrightarrow", "M \\otimes_B \\Omega^\\bullet", "$$", "such that $c_M^\\bullet \\circ e_M^\\bullet$", "and $e_M^\\bullet \\circ c_M^\\bullet$ are homotopic to", "multiplication by $a$." ], "refs": [], "proofs": [ { "contents": [ "In this proof all tensor products are over $B$.", "Assumption (\\ref{item-direct-sum}) implies that", "$$", "M \\otimes (\\Omega')^i =", "(B' \\otimes M \\otimes \\Omega^i)", "\\oplus", "(B' \\text{d}z \\otimes M \\otimes \\Omega^{i - 1})", "$$", "for all $i \\geq 0$. A collection of additive generators for", "$M \\otimes (\\Omega')^i$ is formed by elements of the form", "$f \\omega$ and elements of the form $f \\text{d}z \\wedge \\eta$", "where $f \\in B'$, $\\omega \\in M \\otimes \\Omega^i$, and", "$\\eta \\in M \\otimes \\Omega^{i - 1}$.", "\\medskip\\noindent", "For $f \\in B'$ we write", "$$", "\\epsilon(f) = af - \\theta(\\partial_z(f))", "\\quad\\text{and}\\quad", "\\epsilon'(f) = (\\theta \\otimes 1)(\\text{d}_1(f)) - \\text{d}_1(\\theta(f))", "$$", "so that $\\epsilon(f) \\in B$ and $\\epsilon'(f) \\in \\Omega$ by", "assumptions (\\ref{item-factor}) and (\\ref{item-horizontal}).", "We define $e_M^\\bullet$ by the rules", "$e^i_M(f\\omega) = \\epsilon(f) \\omega$ and", "$e^i_M(f \\text{d}z \\wedge \\eta) = \\epsilon'(f) \\wedge \\eta$.", "We will see below that the collection of maps $e^i_M$ is a map of complexes.", "\\medskip\\noindent", "We define", "$$", "h^i : M \\otimes_B (\\Omega')^i \\longrightarrow M \\otimes_B (\\Omega')^{i - 1}", "$$", "by the rules $h^i(f \\omega) = 0$ and", "$h^i(f \\text{d}z \\wedge \\eta) = \\theta(f) \\eta$", "for elements as above. We claim that", "$$", "\\text{d} \\circ h + h \\circ \\text{d} = a - c_M^\\bullet \\circ e_M^\\bullet", "$$", "Note that multiplication by $a$ is a map of complexes", "by (\\ref{item-d-a-zero}). Hence, since $c_M^\\bullet$ is an injective map", "of complexes by assumption (\\ref{item-injective}), we conclude that", "$e_M^\\bullet$ is a map of complexes. To prove the claim we compute", "\\begin{align*}", "(\\text{d} \\circ h + h \\circ \\text{d})(f \\omega)", "& =", "h\\left(\\text{d}(f) \\wedge \\omega + f \\nabla(\\omega)\\right)", "\\\\", "& =", "\\theta(\\partial_z(f)) \\omega", "\\\\", "& =", "a f\\omega - \\epsilon(f)\\omega ", "\\\\", "& =", "a f \\omega - c^i_M(e^i_M(f\\omega))", "\\end{align*}", "The second equality because $\\text{d}z$ does not occur in $\\nabla(\\omega)$", "and the third equality by assumption (6). Similarly, we have", "\\begin{align*}", "(\\text{d} \\circ h + h \\circ \\text{d})(f \\text{d}z \\wedge \\eta)", "& =", "\\text{d}(\\theta(f) \\eta) +", "h\\left(\\text{d}(f) \\wedge \\text{d}z \\wedge \\eta -", "f \\text{d}z \\wedge \\nabla(\\eta)\\right)", "\\\\", "& =", "\\text{d}(\\theta(f)) \\wedge \\eta + \\theta(f) \\nabla(\\eta)", "- (\\theta \\otimes 1)(\\text{d}_1(f)) \\wedge \\eta", "- \\theta(f) \\nabla(\\eta)", "\\\\", "& =", "\\text{d}_1(\\theta(f)) \\wedge \\eta +", "\\partial_z(\\theta(f)) \\text{d}z \\wedge \\eta -", "(\\theta \\otimes 1)(\\text{d}_1(f)) \\wedge \\eta", "\\\\", "& =", "a f \\text{d}z \\wedge \\eta - \\epsilon'(f) \\wedge \\eta \\\\", "& = a f \\text{d}z \\wedge \\eta - c^i_M(e^i_M(f \\text{d}z \\wedge \\eta))", "\\end{align*}", "The second equality because", "$\\text{d}(f) \\wedge \\text{d}z \\wedge \\eta =", "- \\text{d}z \\wedge \\text{d}_1(f) \\wedge \\eta$.", "The fourth equality by assumption (\\ref{item-integrate}).", "On the other hand it is immediate from the definitions", "that $e^i_M(c^i_M(\\omega)) = \\epsilon(1) \\omega = a \\omega$.", "This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10789, "type": "theorem", "label": "crystalline-lemma-computation", "categories": [ "crystalline" ], "title": "crystalline-lemma-computation", "contents": [ "In Situation \\ref{situation-affine}. Assume $D$ and $\\Omega_D$ are as in", "(\\ref{equation-D}) and (\\ref{equation-omega-D}).", "Let $\\lambda \\in D$. Let $D'$ be the $p$-adic completion of", "$$", "D[z]\\langle \\xi \\rangle/(\\xi - (z^p - \\lambda))", "$$", "and let $\\Omega_{D'}$ be the $p$-adic completion of the module of", "divided power differentials of $D'$ over $A$. For any pair $(M, \\nabla)$", "over $D$ satisfying (\\ref{item-complete}), (\\ref{item-connection}),", "(\\ref{item-integrable}), and (\\ref{item-topologically-quasi-nilpotent})", "the canonical map of complexes (\\ref{equation-base-change-map-complexes})", "$$", "c_M^\\bullet : M \\otimes_D^\\wedge \\Omega^\\bullet_D", "\\longrightarrow", "M \\otimes_D^\\wedge \\Omega^\\bullet_{D'}", "$$", "has the following property: There exists a map $e_M^\\bullet$", "in the opposite direction such that both $c_M^\\bullet \\circ e_M^\\bullet$", "and $e_M^\\bullet \\circ c_M^\\bullet$ are homotopic to multiplication by $p$." ], "refs": [], "proofs": [ { "contents": [ "We will prove this using Lemma \\ref{lemma-find-homotopy} with $a = p$.", "Thus we have to find $\\theta : D' \\to D'$ and prove", "(\\ref{item-d-a-zero}), (\\ref{item-direct-sum}), (\\ref{item-theta-linear}),", "(\\ref{item-integrate}), (\\ref{item-injective}), (\\ref{item-factor}),", "(\\ref{item-horizontal}). We first collect some information about the rings", "$D$ and $D'$ and the modules $\\Omega_D$ and $\\Omega_{D'}$.", "\\medskip\\noindent", "Writing", "$$", "D[z]\\langle \\xi \\rangle/(\\xi - (z^p - \\lambda))", "=", "D\\langle \\xi \\rangle[z]/(z^p - \\xi - \\lambda)", "$$", "we see that $D'$ is the $p$-adic completion of the free $D$-module", "$$", "\\bigoplus\\nolimits_{i = 0, \\ldots, p - 1}", "\\bigoplus\\nolimits_{n \\geq 0}", "z^i \\xi^{[n]} D", "$$", "where $\\xi^{[0]} = 1$.", "It follows that $D \\to D'$ has a continuous $D$-linear section, in particular", "$D \\to D'$ is universally injective, i.e., (\\ref{item-injective}) holds.", "We think of $D'$ as a divided power algebra", "over $A$ with divided power ideal $\\overline{J}' = \\overline{J}D' + (\\xi)$.", "Then $D'$ is also the $p$-adic completion of the divided power envelope", "of the ideal generated by $z^p - \\lambda$ in $D$, see", "Lemma \\ref{lemma-describe-divided-power-envelope}. Hence", "$$", "\\Omega_{D'} = \\Omega_D \\otimes_D^\\wedge D' \\oplus D'\\text{d}z", "$$", "by Lemma \\ref{lemma-module-differentials-divided-power-envelope}.", "This proves (\\ref{item-direct-sum}). Note that (\\ref{item-d-a-zero})", "is obvious.", "\\medskip\\noindent", "At this point we construct $\\theta$. (We wrote a PARI/gp script theta.gp", "verifying some of the formulas in this proof which can be found in the", "scripts subdirectory of the Stacks project.) Before we do so we compute", "the derivative of the elements $z^i \\xi^{[n]}$. We have", "$\\text{d}z^i = i z^{i - 1} \\text{d}z$. For $n \\geq 1$ we have", "$$", "\\text{d}\\xi^{[n]} =", "\\xi^{[n - 1]} \\text{d}\\xi =", "- \\xi^{[n - 1]}\\text{d}\\lambda + p z^{p - 1} \\xi^{[n - 1]}\\text{d}z", "$$", "because $\\xi = z^p - \\lambda$. For $0 < i < p$ and $n \\geq 1$ we have", "\\begin{align*}", "\\text{d}(z^i\\xi^{[n]})", "& =", "iz^{i - 1}\\xi^{[n]}\\text{d}z + z^i\\xi^{[n - 1]}\\text{d}\\xi \\\\", "& =", "iz^{i - 1}\\xi^{[n]}\\text{d}z + z^i\\xi^{[n - 1]}\\text{d}(z^p - \\lambda) \\\\", "& =", "- z^i\\xi^{[n - 1]}\\text{d}\\lambda +", "(iz^{i - 1}\\xi^{[n]} + pz^{i + p - 1}\\xi^{[n - 1]})\\text{d}z \\\\", "& =", "- z^i\\xi^{[n - 1]}\\text{d}\\lambda +", "(iz^{i - 1}\\xi^{[n]} + pz^{i - 1}(\\xi + \\lambda)\\xi^{[n - 1]})\\text{d}z \\\\", "& =", "- z^i\\xi^{[n - 1]}\\text{d}\\lambda +", "((i + pn)z^{i - 1}\\xi^{[n]} + p\\lambda z^{i - 1}\\xi^{[n - 1]})\\text{d}z", "\\end{align*}", "the last equality because $\\xi \\xi^{[n - 1]} = n\\xi^{[n]}$.", "Thus we see that", "\\begin{align*}", "\\partial_z(z^i) & = i z^{i - 1} \\\\", "\\partial_z(\\xi^{[n]}) & = p z^{p - 1} \\xi^{[n - 1]} \\\\", "\\partial_z(z^i\\xi^{[n]}) & =", "(i + pn) z^{i - 1} \\xi^{[n]} + p \\lambda z^{i - 1}\\xi^{[n - 1]}", "\\end{align*}", "Motivated by these formulas we define $\\theta$ by the rules", "$$", "\\begin{matrix}", "\\theta(z^j)", "& = & p\\frac{z^{j + 1}}{j + 1}", "& j = 0, \\ldots p - 1, \\\\", "\\theta(z^{p - 1}\\xi^{[m]})", "& = & \\xi^{[m + 1]}", "& m \\geq 1, \\\\", "\\theta(z^j \\xi^{[m]})", "& = &", "\\frac{p z^{j + 1} \\xi^{[m]} - \\theta(p\\lambda z^j \\xi^{[m - 1]})}{(j + 1 + pm)}", "& 0 \\leq j < p - 1, m \\geq 1", "\\end{matrix}", "$$", "where in the last line we use induction on $m$ to define our choice of", "$\\theta$. Working this out we get (for $0 \\leq j < p - 1$ and $1 \\leq m$)", "$$", "\\theta(z^j \\xi^{[m]}) =", "\\textstyle{\\frac{p z^{j + 1} \\xi^{[m]}}{(j + 1 + pm)} -", "\\frac{p^2 \\lambda z^{j + 1} \\xi^{[m - 1]}}{(j + 1 + pm)(j + 1 + p(m - 1))} +", "\\ldots +", "\\frac{(-1)^m p^{m + 1} \\lambda^m z^{j + 1}}", "{(j + 1 + pm) \\ldots (j + 1)}}", "$$", "although we will not use this expression below. It is clear that $\\theta$", "extends uniquely to a $p$-adically continuous $D$-linear map on $D'$.", "By construction we have (\\ref{item-theta-linear}) and (\\ref{item-integrate}).", "It remains to prove (\\ref{item-factor}) and (\\ref{item-horizontal}).", "\\medskip\\noindent", "Proof of (\\ref{item-factor}) and (\\ref{item-horizontal}).", "As $\\theta$ is $D$-linear and continuous it suffices to prove that", "$p - \\theta \\circ \\partial_z$,", "resp.\\ $(\\theta \\otimes 1) \\circ \\text{d}_1 - \\text{d}_1 \\circ \\theta$", "gives an element of $D$, resp.\\ $\\Omega_D$ when evaluated on the", "elements $z^i\\xi^{[n]}$\\footnote{This can be done by direct computation:", "It turns out that $p - \\theta \\circ \\partial_z$ evaluated on", "$z^i\\xi^{[n]}$ gives zero except for $1$ which is mapped to $p$ and", "$\\xi$ which is mapped to $-p\\lambda$. It turns out that ", "$(\\theta \\otimes 1) \\circ \\text{d}_1 - \\text{d}_1 \\circ \\theta$", "evaluated on $z^i\\xi^{[n]}$ gives zero except for $z^{p - 1}\\xi$", "which is mapped to $-\\lambda$.}.", "Set $D_0 = \\mathbf{Z}_{(p)}[\\lambda]$ and", "$D_0' = \\mathbf{Z}_{(p)}[z, \\lambda]\\langle \\xi \\rangle/(\\xi - z^p + \\lambda)$.", "Observe that each of the expressions above is an element of", "$D_0'$ or $\\Omega_{D_0'}$. Hence it suffices to prove the result", "in the case of $D_0 \\to D_0'$. Note that $D_0$ and $D_0'$", "are torsion free rings and that $D_0 \\otimes \\mathbf{Q} = \\mathbf{Q}[\\lambda]$", "and $D'_0 \\otimes \\mathbf{Q} = \\mathbf{Q}[z, \\lambda]$.", "Hence $D_0 \\subset D'_0$ is the subring of elements annihilated", "by $\\partial_z$ and (\\ref{item-factor})", "follows from (\\ref{item-integrate}), see the discussion directly preceding", "Lemma \\ref{lemma-find-homotopy}. Similarly, we have", "$\\text{d}_1(f) = \\partial_\\lambda(f)\\text{d}\\lambda$ hence", "$$", "\\left((\\theta \\otimes 1) \\circ \\text{d}_1 - \\text{d}_1 \\circ \\theta\\right)(f)", "=", "\\left(\\theta(\\partial_\\lambda(f)) - \\partial_\\lambda(\\theta(f))\\right)", "\\text{d}\\lambda", "$$", "Applying $\\partial_z$ to the coefficient we obtain", "\\begin{align*}", "\\partial_z\\left(", "\\theta(\\partial_\\lambda(f)) - \\partial_\\lambda(\\theta(f))", "\\right)", "& =", "p \\partial_\\lambda(f) - \\partial_z(\\partial_\\lambda(\\theta(f))) \\\\", "& =", "p \\partial_\\lambda(f) - \\partial_\\lambda(\\partial_z(\\theta(f))) \\\\", "& =", "p \\partial_\\lambda(f) - \\partial_\\lambda(p f) = 0", "\\end{align*}", "whence the coefficient does not depend on $z$ as desired.", "This finishes the proof of the lemma." ], "refs": [ "crystalline-lemma-find-homotopy", "crystalline-lemma-describe-divided-power-envelope", "crystalline-lemma-module-differentials-divided-power-envelope", "crystalline-lemma-find-homotopy" ], "ref_ids": [ 10788, 10746, 10759, 10788 ] } ], "ref_ids": [] }, { "id": 10790, "type": "theorem", "label": "crystalline-lemma-pullback-along-p-power-cover", "categories": [ "crystalline" ], "title": "crystalline-lemma-pullback-along-p-power-cover", "contents": [ "Let $p$ be a prime number. Let $(S, \\mathcal{I}, \\gamma)$ be a divided power", "scheme over $\\mathbf{Z}_{(p)}$ with $p \\in \\mathcal{I}$. We set", "$S_0 = V(\\mathcal{I}) \\subset S$. Let $f : X' \\to X$ be an iterated", "$\\alpha_p$-cover of schemes over $S_0$ with constant degree $q$. Let", "$\\mathcal{F}$ be any crystal in quasi-coherent sheaves on $X$ and set", "$\\mathcal{F}' = f_{\\text{cris}}^*\\mathcal{F}$.", "In the distinguished triangle", "$$", "Ru_{X/S, *}\\mathcal{F}", "\\longrightarrow", "f_*Ru_{X'/S, *}\\mathcal{F}'", "\\longrightarrow", "E", "\\longrightarrow", "Ru_{X/S, *}\\mathcal{F}[1]", "$$", "the object $E$ has cohomology sheaves annihilated by $q$." ], "refs": [], "proofs": [ { "contents": [ "Note that $X' \\to X$ is a homeomorphism hence we can identify the underlying", "topological spaces of $X$ and $X'$. The question is clearly local on $X$,", "hence we may assume $X$, $X'$, and $S$ affine and $X' \\to X$ given as a", "composition", "$$", "X' = X_n \\to X_{n - 1} \\to X_{n - 2} \\to \\ldots \\to X_0 = X", "$$", "where each morphism $X_{i + 1} \\to X_i$ is an $\\alpha_p$-cover.", "Denote $\\mathcal{F}_i$ the pullback of $\\mathcal{F}$ to $X_i$.", "It suffices to prove that each of the maps", "$$", "R\\Gamma(\\text{Cris}(X_i/S), \\mathcal{F}_i)", "\\longrightarrow", "R\\Gamma(\\text{Cris}(X_{i + 1}/S), \\mathcal{F}_{i + 1})", "$$", "fits into a triangle whose third member has cohomology groups annihilated", "by $p$. (This uses axiom TR4 for the triangulated category $D(X)$. Details", "omitted.)", "\\medskip\\noindent", "Hence we may assume that $S = \\Spec(A)$, $X = \\Spec(C)$, $X' = \\Spec(C')$", "and $C' = C[z]/(z^p - c)$ for some $c \\in C$. Choose a polynomial algebra", "$P$ over $A$ and a surjection $P \\to C$. Let $D$ be the $p$-adically completed", "divided power envelop of $\\Ker(P \\to C)$ in $P$ as in (\\ref{equation-D}).", "Set $P' = P[z]$ with surjection $P' \\to C'$ mapping $z$ to the class of $z$", "in $C'$. Choose a lift $\\lambda \\in D$ of $c \\in C$. Then we see that", "the $p$-adically completed divided power envelope $D'$ of", "$\\Ker(P' \\to C')$ in $P'$ is isomorphic to the $p$-adic completion of", "$D[z]\\langle \\xi \\rangle/(\\xi - (z^p - \\lambda))$, see", "Lemma \\ref{lemma-computation} and its proof.", "Thus we see that the result follows from this lemma", "by the computation of cohomology of crystals in quasi-coherent modules in", "Proposition \\ref{proposition-compute-cohomology-crystal}." ], "refs": [ "crystalline-lemma-computation", "crystalline-proposition-compute-cohomology-crystal" ], "ref_ids": [ 10789, 10795 ] } ], "ref_ids": [] }, { "id": 10791, "type": "theorem", "label": "crystalline-lemma-pullback-along-p-power-cover-cohomology", "categories": [ "crystalline" ], "title": "crystalline-lemma-pullback-along-p-power-cover-cohomology", "contents": [ "With notations and assumptions as in", "Lemma \\ref{lemma-pullback-along-p-power-cover}", "the map", "$$", "f^* :", "H^i(\\text{Cris}(X/S), \\mathcal{F})", "\\longrightarrow", "H^i(\\text{Cris}(X'/S), \\mathcal{F}')", "$$", "has kernel and cokernel annihilated by $q^{i + 1}$." ], "refs": [ "crystalline-lemma-pullback-along-p-power-cover" ], "proofs": [ { "contents": [ "This follows from the fact that $E$ has nonzero cohomology sheaves in", "degrees $-1$ and up, so that the spectral sequence", "$H^a(\\mathcal{H}^b(E)) \\Rightarrow H^{a + b}(E)$ converges.", "This combined with the long exact cohomology sequence associated", "to a distinguished triangle gives the bound." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10790 ] }, { "id": 10792, "type": "theorem", "label": "crystalline-lemma-pullback-relative-frobenius", "categories": [ "crystalline" ], "title": "crystalline-lemma-pullback-relative-frobenius", "contents": [ "In the situation above, assume that $X \\to S_0$ is smooth of relative", "dimension $d$. Then $F_{X/S_0}$ is an iterated $\\alpha_p$-cover", "of degree $p^d$. Hence Lemmas \\ref{lemma-pullback-along-p-power-cover} and", "\\ref{lemma-pullback-along-p-power-cover-cohomology} apply to this", "situation. In particular, for any crystal in quasi-coherent modules", "$\\mathcal{G}$ on $\\text{Cris}(X^{(1)}/S)$ the map", "$$", "F_{X/S_0}^* : H^i(\\text{Cris}(X^{(1)}/S), \\mathcal{G})", "\\longrightarrow", "H^i(\\text{Cris}(X/S), F_{X/S_0, \\text{cris}}^*\\mathcal{G})", "$$", "has kernel and cokernel annihilated by $p^{d(i + 1)}$." ], "refs": [ "crystalline-lemma-pullback-along-p-power-cover", "crystalline-lemma-pullback-along-p-power-cover-cohomology" ], "proofs": [ { "contents": [ "It suffices to prove the first statement. To see this we may assume", "that $X$ is \\'etale over $\\mathbf{A}^d_{S_0}$, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-etale-over-affine-space}.", "Denote $\\varphi : X \\to \\mathbf{A}^d_{S_0}$ this \\'etale morphism.", "In this case the relative Frobenius of $X/S_0$ fits into a diagram", "$$", "\\xymatrix{", "X \\ar[d] \\ar[r] & X^{(1)} \\ar[d] \\\\", "\\mathbf{A}^d_{S_0} \\ar[r] & \\mathbf{A}^d_{S_0}", "}", "$$", "where the lower horizontal arrow is the relative frobenius morphism", "of $\\mathbf{A}^d_{S_0}$ over $S_0$. This is the morphism which raises", "all the coordinates to the $p$th power, hence it is an iterated", "$\\alpha_p$-cover. The proof is finished by observing that the diagram", "is a fibre square, see", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-relative-frobenius-etale}." ], "refs": [ "morphisms-lemma-smooth-etale-over-affine-space", "etale-lemma-relative-frobenius-etale" ], "ref_ids": [ 5377, 10708 ] } ], "ref_ids": [ 10790, 10791 ] }, { "id": 10793, "type": "theorem", "label": "crystalline-proposition-crystals-on-affine", "categories": [ "crystalline" ], "title": "crystalline-proposition-crystals-on-affine", "contents": [ "The functor", "$$", "\\begin{matrix}", "\\text{crystals in quasi-coherent} \\\\", "\\mathcal{O}_{X/S}\\text{-modules on }\\text{Cris}(X/S)", "\\end{matrix}", "\\longrightarrow", "\\begin{matrix}", "\\text{pairs }(M, \\nabla)\\text{ satisfying} \\\\", "\\text{(\\ref{item-complete}), (\\ref{item-connection}),", "(\\ref{item-integrable}), and (\\ref{item-topologically-quasi-nilpotent})}", "\\end{matrix}", "$$", "of Lemma \\ref{lemma-crystals-on-affine}", "is an equivalence of categories." ], "refs": [ "crystalline-lemma-crystals-on-affine" ], "proofs": [ { "contents": [ "Let $(M, \\nabla)$ be given. We are going to construct", "a crystal in quasi-coherent modules $\\mathcal{F}$.", "Write $\\nabla(m) = \\sum \\theta_i(m)\\text{d}x_i$.", "Then $\\theta_i \\circ \\theta_j = \\theta_j \\circ \\theta_i$ and we", "can set $\\theta_K(m) = (\\prod \\theta_i^{k_i})(m)$ for any multi-index", "$K = (k_i)$ with $k_i \\geq 0$ and $\\sum k_i < \\infty$.", "\\medskip\\noindent", "Let $(U, T, \\delta)$ be any object of $\\text{Cris}(X/S)$ with $T$ affine.", "Say $T = \\Spec(B)$ and the ideal of $U \\to T$ is $J_B \\subset B$.", "By Lemma \\ref{lemma-set-generators} there exists an integer $e$ and a morphism", "$$", "f : (U, T, \\delta) \\longrightarrow (X, T_e, \\bar\\gamma)", "$$", "where $T_e = \\Spec(D_e)$ as in the proof of", "Lemma \\ref{lemma-crystals-on-affine}.", "Choose such an $e$ and $f$; denote $f : D \\to B$ also the corresponding", "divided power $A$-algebra map. We will set $\\mathcal{F}_T$ equal to the", "quasi-coherent sheaf of $\\mathcal{O}_T$-modules associated to the $B$-module", "$$", "M \\otimes_{D, f} B.", "$$", "However, we have to show that this is independent of the choice of $f$.", "Suppose that $g : D \\to B$ is a second such morphism. Since $f$ and $g$", "are morphisms in $\\text{Cris}(X/S)$ we see that the image of", "$f - g : D \\to B$ is contained in the divided power ideal $J_B$.", "Write $\\xi_i = f(x_i) - g(x_i) \\in J_B$. By analogy with the proof", "of Lemma \\ref{lemma-crystals-on-affine} we define an isomorphism", "$$", "c_{f, g} : M \\otimes_{D, f} B \\longrightarrow M \\otimes_{D, g} B", "$$", "by the formula", "$$", "m \\otimes 1 \\longmapsto ", "\\sum\\nolimits_K \\theta_K(m) \\otimes \\prod \\xi_i^{[k_i]}", "$$", "which makes sense by our remarks above and the fact that $\\nabla$", "is topologically quasi-nilpotent (so the sum is finite!).", "A computation shows that", "$$", "c_{g, h} \\circ c_{f, g} = c_{f, h}", "$$", "if given a third morphism", "$h : (U, T, \\delta) \\longrightarrow (X, T_e, \\bar\\gamma)$.", "It is also true that $c_{f, f} = 1$.", "Hence these maps are all isomorphisms and we see that", "the module $\\mathcal{F}_T$ is independent of the choice of $f$.", "\\medskip\\noindent", "If $a : (U', T', \\delta') \\to (U, T, \\delta)$ is a morphism of affine objects", "of $\\text{Cris}(X/S)$, then choosing $f' = f \\circ a$ it is clear", "that there exists a canonical isomorphism", "$a^*\\mathcal{F}_T \\to \\mathcal{F}_{T'}$. We omit the verification that this", "map is independent of the choice of $f$. Using these maps as the restriction", "maps it is clear that we obtain a crystal in quasi-coherent modules", "on the full subcategory of $\\text{Cris}(X/S)$ consisting of affine objects.", "We omit the proof that this extends to a crystal on all of", "$\\text{Cris}(X/S)$. We also omit the proof that this procedure is a functor", "and that it is quasi-inverse to the functor constructed in", "Lemma \\ref{lemma-crystals-on-affine}." ], "refs": [ "crystalline-lemma-set-generators", "crystalline-lemma-crystals-on-affine", "crystalline-lemma-crystals-on-affine", "crystalline-lemma-crystals-on-affine" ], "ref_ids": [ 10754, 10776, 10776, 10776 ] } ], "ref_ids": [ 10776 ] }, { "id": 10794, "type": "theorem", "label": "crystalline-proposition-compute-cohomology", "categories": [ "crystalline" ], "title": "crystalline-proposition-compute-cohomology", "contents": [ "With notations as above assume that", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is locally quasi-coherent, and", "\\item for any morphism $(U, T, \\delta) \\to (U', T', \\delta')$", "of $\\text{Cris}(X/S)$ where $f : T \\to T'$ is a closed immersion", "the map $c_f : f^*\\mathcal{F}_{T'} \\to \\mathcal{F}_T$ is surjective.", "\\end{enumerate}", "Then the complex", "$$", "M(0) \\to M(1) \\to M(2) \\to \\ldots", "$$", "computes $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Using assumption (1) and Lemma \\ref{lemma-compare} we see that", "$R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$ is isomorphic to", "$R\\Gamma(\\mathcal{C}, \\mathcal{F})$. Note that the categories", "$\\mathcal{C}$ used in Lemmas \\ref{lemma-compare} and \\ref{lemma-complete}", "agree. Let $f : T \\to T'$ be a closed immersion as in (2). Surjectivity", "of $c_f : f^*\\mathcal{F}_{T'} \\to \\mathcal{F}_T$ is equivalent to", "surjectivity of $\\mathcal{F}_{T'} \\to f_*\\mathcal{F}_T$. Hence, if", "$\\mathcal{F}$ satisfies (1) and (2), then we obtain a short exact sequence", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{F}_{T'} \\to f_*\\mathcal{F}_T \\to 0", "$$", "of quasi-coherent $\\mathcal{O}_{T'}$-modules on $T'$, see", "Schemes, Section \\ref{schemes-section-quasi-coherent} and in particular", "Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}.", "Thus, if $T'$ is affine, then we conclude that the restriction map", "$\\mathcal{F}(U', T', \\delta') \\to \\mathcal{F}(U, T, \\delta)$", "is surjective by the vanishing of $H^1(T', \\mathcal{K})$, see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "Hence the transition maps of the inverse systems in Lemma \\ref{lemma-complete}", "are surjective. We conclude that", "$R^pg_*(\\mathcal{F}|_\\mathcal{C}) = 0$ for all $p \\geq 1$", "where $g$ is as in Lemma \\ref{lemma-complete}.", "The object $D$ of the category $\\mathcal{C}^\\wedge$", "satisfies the assumption of Lemma \\ref{lemma-category-with-covering} by", "Lemma \\ref{lemma-generator-completion}", "with", "$$", "D \\times \\ldots \\times D = D(n)", "$$", "in $\\mathcal{C}$ because $D(n)$ is the $n + 1$-fold coproduct of", "$D$ in $\\text{Cris}^\\wedge(C/A)$, see Lemma \\ref{lemma-property-Dn}.", "Thus we win." ], "refs": [ "crystalline-lemma-compare", "crystalline-lemma-compare", "crystalline-lemma-complete", "schemes-lemma-push-forward-quasi-coherent", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "crystalline-lemma-complete", "crystalline-lemma-complete", "crystalline-lemma-category-with-covering", "crystalline-lemma-generator-completion", "crystalline-lemma-property-Dn" ], "ref_ids": [ 10779, 10779, 10780, 7730, 3282, 10780, 10780, 10781, 10755, 10775 ] } ], "ref_ids": [] }, { "id": 10795, "type": "theorem", "label": "crystalline-proposition-compute-cohomology-crystal", "categories": [ "crystalline" ], "title": "crystalline-proposition-compute-cohomology-crystal", "contents": [ "Assumptions as in Proposition \\ref{proposition-compute-cohomology}", "but now assume that $\\mathcal{F}$ is a crystal in quasi-coherent modules.", "Let $(M, \\nabla)$ be the corresponding module with connection over $D$, see", "Proposition \\ref{proposition-crystals-on-affine}. Then the complex", "$$", "M \\otimes^\\wedge_D \\Omega^*_D", "$$", "computes $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$." ], "refs": [ "crystalline-proposition-compute-cohomology", "crystalline-proposition-crystals-on-affine" ], "proofs": [ { "contents": [ "We will prove this using the two spectral sequences associated to the", "double complex $K^{*, *}$ with terms", "$$", "K^{a, b} = M \\otimes_D^\\wedge \\Omega^a_{D(b)}", "$$", "What do we know so far? Well, Lemma \\ref{lemma-vanishing}", "tells us that each column $K^{a, *}$, $a > 0$ is acyclic.", "Proposition \\ref{proposition-compute-cohomology} tells us that", "the first column $K^{0, *}$ is quasi-isomorphic to", "$R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$.", "Hence the first spectral sequence associated to the double complex", "shows that there is a canonical quasi-isomorphism of", "$R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$ with", "$\\text{Tot}(K^{*, *})$.", "\\medskip\\noindent", "Next, let's consider the rows $K^{*, b}$. By", "Lemma \\ref{lemma-structure-Dn}", "each of the $b + 1$ maps $D \\to D(b)$ presents $D(b)$ as the $p$-adic", "completion of a divided power polynomial algebra over $D$.", "Hence Lemma \\ref{lemma-relative-poincare} shows that the map", "$$", "M \\otimes^\\wedge_D\\Omega^*_D", "\\longrightarrow", "M \\otimes^\\wedge_{D(b)} \\Omega^*_{D(b)} = K^{*, b}", "$$", "is a quasi-isomorphism. Note that each of these maps defines the {\\it same}", "map on cohomology (and even the same map in the derived category) as", "the inverse is given by the co-diagonal map $D(b) \\to D$ (corresponding", "to the multiplication map $P \\otimes_A \\ldots \\otimes_A P \\to P$).", "Hence if we look at the $E_1$ page of the second spectral sequence", "we obtain", "$$", "E_1^{a, b} = H^a(M \\otimes^\\wedge_D\\Omega^*_D)", "$$", "with differentials", "$$", "E_1^{a, 0} \\xrightarrow{0}", "E_1^{a, 1} \\xrightarrow{1}", "E_1^{a, 2} \\xrightarrow{0}", "E_1^{a, 3} \\xrightarrow{1} \\ldots", "$$", "as each of these is the alternation sum of the given identifications", "$H^a(M \\otimes^\\wedge_D\\Omega^*_D) = E_1^{a, 0} = E_1^{a, 1} = \\ldots$.", "Thus we see that the $E_2$ page is equal $H^a(M \\otimes^\\wedge_D\\Omega^*_D)$", "on the first row and zero elsewhere. It follows that the identification", "of $M \\otimes^\\wedge_D\\Omega^*_D$ with the first row induces a", "quasi-isomorphism of $M \\otimes^\\wedge_D\\Omega^*_D$ with", "$\\text{Tot}(K^{*, *})$." ], "refs": [ "crystalline-lemma-vanishing", "crystalline-proposition-compute-cohomology", "crystalline-lemma-structure-Dn", "crystalline-lemma-relative-poincare" ], "ref_ids": [ 10783, 10794, 10774, 10785 ] } ], "ref_ids": [ 10794, 10793 ] }, { "id": 10796, "type": "theorem", "label": "crystalline-proposition-compare-with-de-Rham", "categories": [ "crystalline" ], "title": "crystalline-proposition-compare-with-de-Rham", "contents": [ "In Situation \\ref{situation-global}.", "Let $\\mathcal{F}$ be a crystal in quasi-coherent modules on", "$\\text{Cris}(X/S)$. The truncation map of complexes", "$$", "(\\mathcal{F} \\to", "\\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^1_{X/S} \\to", "\\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^2_{X/S} \\to \\ldots)", "\\longrightarrow \\mathcal{F}[0],", "$$", "while not a quasi-isomorphism, becomes a quasi-isomorphism after applying", "$Ru_{X/S, *}$. In fact, for any $i > 0$, we have ", "$$", "Ru_{X/S, *}(\\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^i_{X/S}) = 0.", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-automatic-connection} we get a de Rham complex", "as indicated in the lemma. We abbreviate", "$\\mathcal{H} = \\mathcal{F} \\otimes \\Omega^i_{X/S}$.", "Let $X' \\subset X$ be an affine open", "subscheme which maps into an affine open subscheme $S' \\subset S$.", "Then", "$$", "(Ru_{X/S, *}\\mathcal{H})|_{X'_{Zar}} =", "Ru_{X'/S', *}(\\mathcal{H}|_{\\text{Cris}(X'/S')}),", "$$", "see Lemma \\ref{lemma-localize}. Thus", "Lemma \\ref{lemma-cohomology-is-zero}", "shows that $Ru_{X/S, *}\\mathcal{H}$ is a complex of sheaves on", "$X_{Zar}$ whose cohomology on any affine open is trivial.", "As $X$ has a basis for its topology consisting of affine opens", "this implies that $Ru_{X/S, *}\\mathcal{H}$ is quasi-isomorphic to zero." ], "refs": [ "crystalline-lemma-automatic-connection", "crystalline-lemma-localize", "crystalline-lemma-cohomology-is-zero" ], "ref_ids": [ 10772, 10765, 10786 ] } ], "ref_ids": [] }, { "id": 10844, "type": "theorem", "label": "spaces-pushouts-theorem-nagata", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-theorem-nagata", "contents": [ "\\begin{reference}", "\\cite{CLO}", "\\end{reference}", "Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $X \\to B$ be a separated, finite type morphism.", "Then $X$ has a compactification over $B$." ], "refs": [], "proofs": [ { "contents": [ "We first reduce to the Noetherian case. We strongly urge the reader", "to skip this paragraph. First, we may replace $S$ by $\\Spec(\\mathbf{Z})$.", "See Spaces, Section \\ref{spaces-section-change-base-scheme} and", "Properties of Spaces, Definition \\ref{spaces-properties-definition-separated}.", "There exists a closed immersion", "$X \\to X'$ with $X' \\to B$ of finite presentation and separated.", "See Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-separated-closed-in-finite-presentation}.", "If we find a compactification of $X'$ over $B$, then", "taking the scheme theoretic closure of $X$ in this will give", "a compactification of $X$ over $B$. Thus we may assume", "$X \\to B$ is separated and of finite presentation.", "We may write $B = \\lim B_i$ as a directed", "limit of a system of Noetherian algebraic spaces", "of finite type over $\\Spec(\\mathbf{Z})$", "with affine transition morphisms.", "See Limits of Spaces, Proposition \\ref{spaces-limits-proposition-approximate}.", "We can choose an $i$ and a morphism $X_i \\to B_i$ of finite", "presentation whose base change to $B$ is $X \\to B$, see", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-descend-finite-presentation}.", "After increasing $i$ we may assume $X_i \\to B_i$ is separated, see", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-separated-morphism}.", "If we can find a compactification of $X_i$ over $B_i$, then the", "base change of this to $B$ will be a compactification of $X$ over $B$.", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $B$ is of finite type over $\\mathbf{Z}$ in addition to being", "quasi-compact and quasi-separated. Let $U \\to X$ be an \\'etale", "morphism of algebraic spaces such that $U$ has a compactification", "$Y$ over $\\Spec(\\mathbf{Z})$. The morphism", "$$", "U \\longrightarrow B \\times_{\\Spec(\\mathbf{Z})} Y", "$$", "is separated and quasi-finite by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}", "(the displayed morphism factors into an immersion hence is a monomorphism).", "Hence by Zariski's main theorem (More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-quasi-finite-separated-pass-through-finite})", "there is an open immersion of $U$ into an algebraic space $Y'$", "finite over $B \\times_{\\Spec(\\mathbf{Z})} Y$. Then $Y' \\to B$ is proper", "as the composition $Y' \\to B \\times_{\\Spec(\\mathbf{Z})} Y \\to B$", "of two proper morphisms", "(use Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-finite-proper},", "\\ref{spaces-morphisms-lemma-composition-proper}, and", "\\ref{spaces-morphisms-lemma-base-change-proper}).", "We conclude that $U$ has a compactification over $B$.", "\\medskip\\noindent", "There is a dense open subspace $U \\subset X$ which is a scheme.", "(Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}).", "In fact, we may choose $U$ to be an affine scheme", "(Properties, Lemmas \\ref{properties-lemma-Noetherian-irreducible-components}", "and \\ref{properties-lemma-maximal-points-affine}).", "Thus $U$ has a compactification over $\\Spec(\\mathbf{Z})$;", "this is easily shown directly but also follows from the", "theorem for schemes, see", "More on Flatness, Theorem \\ref{flat-theorem-nagata}.", "By the previous paragraph $U$ has a compactification over $B$. ", "By Noetherian induction we can find a maximal dense open subspace", "$U \\subset X$ which has a compactification over $B$. We will show", "that the assumption that $U \\not = X$ leads to a contradiction.", "Namely, by Lemma \\ref{lemma-filter-Noetherian-space}", "we can find a strictly larger open $U \\subset W \\subset X$", "and a distinguished square $(U \\subset W, f : V \\to W)$", "with $V$ affine and $U \\times_W V$ dense image in $U$.", "Since $V$ is affine, as before it has a compactification over $B$. Hence", "Lemma \\ref{lemma-two-compactifications}", "applies to show that $W$ has a compactification over $B$", "which is the desired contradiction." ], "refs": [ "spaces-properties-definition-separated", "spaces-limits-proposition-separated-closed-in-finite-presentation", "spaces-limits-proposition-approximate", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-separated-morphism", "spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "spaces-more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "spaces-morphisms-lemma-finite-proper", "spaces-morphisms-lemma-composition-proper", "spaces-morphisms-lemma-base-change-proper", "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme", "properties-lemma-Noetherian-irreducible-components", "properties-lemma-maximal-points-affine", "flat-theorem-nagata", "spaces-pushouts-lemma-filter-Noetherian-space", "spaces-pushouts-lemma-two-compactifications" ], "ref_ids": [ 11922, 4657, 4656, 4598, 4592, 4838, 171, 4946, 4918, 4917, 11917, 2956, 3059, 5976, 10897, 10896 ] } ], "ref_ids": [] }, { "id": 10845, "type": "theorem", "label": "spaces-pushouts-lemma-colimit-agrees", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-colimit-agrees", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{I} \\to (\\Sch/S)_{fppf}$, $i \\mapsto X_i$", "be a diagram of schemes over $S$ as above. Assume that", "\\begin{enumerate}", "\\item $X = \\colim X_i$ exists in the category of schemes,", "\\item $\\coprod X_i \\to X$ is surjective,", "\\item if $U \\to X$ is \\'etale and $U_i = X_i \\times_X U$, then", "$U = \\colim U_i$ in the category of schemes, and", "\\item every object $(U_i \\to X_i)$ of $\\lim X_{i, \\etale}$", "with $U_i \\to X_i$ separated is in the essential image", "the functor $X_\\etale \\to \\lim X_{i, \\etale}$.", "\\end{enumerate}", "Then $X = \\colim X_i$ in the category of algebraic spaces over $S$ also." ], "refs": [], "proofs": [ { "contents": [ "Let $Z$ be an algebraic space over $S$. Suppose that $f_i : X_i \\to Z$ is", "a family of morphisms such that for each $i \\to j$ the composition", "$X_i \\to X_j \\to Z$ is equal to $f_i$. We have to construct a morphism", "of algebraic spaces $f : X \\to Z$ such that we can recover $f_i$ as", "the composition $X_i \\to X \\to Z$. Let $W \\to Z$ be a surjective", "\\'etale morphism of a scheme to $Z$. We may assume that $W$ is a", "disjoint union of affines and in particular we may assume that", "$W \\to Z$ is separated. For each $i$ set", "$U_i = W \\times_{Z, f_i} X_i$ and denote $h_i : U_i \\to W$ the projection.", "Then $U_i \\to X_i$ forms an object of $\\lim X_{i, \\etale}$", "with $U_i \\to X_i$ separated. By", "assumption (4) we can find an \\'etale morphism $U \\to X$ and (functorial)", "isomorphisms $U_i = X_i \\times_X U$. By assumption (3) there exists a morphism", "$h : U \\to W$ such that the compositions $U_i \\to U \\to W$ are $h_i$.", "Let $g : U \\to Z$ be the composition of $h$ with the map $W \\to Z$. To", "finish the proof we have to show that $g : U \\to Z$ descends to a morphism", "$X \\to Z$. To do this, consider the morphism", "$(h, h) : U \\times_X U \\to W \\times_S W$.", "Composing with $U_i \\times_{X_i} U_i \\to U \\times_X U$ we obtain", "$(h_i, h_i)$ which factors through $W \\times_Z W$. Since $U \\times_X U$", "is the colimit of the schemes $U_i \\times_{X_i} U_i$ by (3) we see", "that $(h, h)$ factors through $W \\times_Z W$. Hence the two compositions", "$U \\times_X U \\to U \\to W \\to Z$ are equal. Because each $U_i \\to X_i$ is", "surjective and assumption (2) we see that $U \\to X$ is surjective.", "As $Z$ is a sheaf for the \\'etale topology, we conclude that", "$g : U \\to Z$ descends to $f : X \\to Z$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10846, "type": "theorem", "label": "spaces-pushouts-lemma-pushout-fpqc-local", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-pushout-fpqc-local", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $\\mathcal{I} \\to (\\Sch/S)_{fppf}$, $i \\mapsto X_i$", "be a diagram of algebraic spaces over $B$. Let $(X, X_i \\to X)$", "be a cocone for the diagram in the category of algebraic spaces over $B$", "(Categories, Remark \\ref{categories-remark-cones-and-cocones}).", "If there exists a fpqc covering $\\{U_a \\to X\\}_{a \\in A}$ such that", "\\begin{enumerate}", "\\item for all $a \\in A$ we have", "$U_a = \\colim X_i \\times_X U_a$", "in the category of algebraic spaces over $B$, and", "\\item for all $a, b \\in A$ we have", "$U_a \\times_X U_b = \\colim X_i \\times_X U_a \\times_X U_b$", "in the category of algebraic spaces over $B$,", "\\end{enumerate}", "then $X = \\colim X_i$ in the category of algebraic spaces over $B$." ], "refs": [ "categories-remark-cones-and-cocones" ], "proofs": [ { "contents": [ "Namely, for an algebraic space $Y$ over $B$ a morphism $X \\to Y$ over $B$", "is the same thing as a collection of morphism $U_a \\to Y$ which agree on", "the overlaps $U_a \\times_X U_b$ for all $a, b \\in A$, see", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-fpqc-universal-effective-epimorphisms}." ], "refs": [ "spaces-descent-lemma-fpqc-universal-effective-epimorphisms" ], "ref_ids": [ 9367 ] } ], "ref_ids": [ 12416 ] }, { "id": 10847, "type": "theorem", "label": "spaces-pushouts-lemma-colimit-check-etale-locally", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-colimit-check-etale-locally", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $\\mathcal{I} \\to (\\Sch/S)_{fppf}$, $i \\mapsto X_i$", "be a diagram of algebraic spaces over $B$. Let $(X, X_i \\to X)$", "be a cocone for the diagram in the category of algebraic spaces over $B$", "(Categories, Remark \\ref{categories-remark-cones-and-cocones}).", "Assume that", "\\begin{enumerate}", "\\item the base change functor", "$X_{spaces, \\'etale} \\to \\lim X_{i, spaces, \\etale}$,", "sending $U$ to $U_i = X_i \\times_X U$ is an equivalence,", "\\item given", "\\begin{enumerate}", "\\item $B'$ affine and \\'etale over $B$,", "\\item $Z$ an affine scheme over $B'$,", "\\item $U \\to X \\times_B B'$ an \\'etale morphism of algebraic spaces", "with $U$ affine,", "\\item $f_i : U_i \\to Z$ a cocone over $B'$ of the diagram", "$i \\mapsto U_i = U \\times_X X_i$,", "\\end{enumerate}", "there exists a unique morphism $f : U \\to Z$ over $B'$", "such that $f_i$ equals the composition $U_i \\to U \\to Z$.", "\\end{enumerate}", "Then $X = \\colim X_i$ in the category of all algebraic spaces over $B$." ], "refs": [ "categories-remark-cones-and-cocones" ], "proofs": [ { "contents": [ "In this paragraph we reduce to the case where $B$ is an affine scheme.", "Let $B' \\to B$ be an \\'etale morphism of algebraic spaces.", "Observe that conditions (1) and (2) are preserved if we replace", "$B$, $X_i$, $X$ by $B'$, $X_i \\times_B B'$, $X \\times_B B'$.", "Let $\\{B_a \\to B\\}_{a \\in A}$ be an \\'etale covering with $B_a$", "affine, see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-cover-by-union-affines}.", "For $a \\in A$ denote $X_a$, $X_{a, i}$ the base changes of $X$ and the", "diagram to $B_a$. For $a, b \\in A$ denote", "$X_{a, b}$ and $X_{a, b, i}$ the base changes of $X$ and the", "diagram to $B_a \\times_B B_b$.", "By Lemma \\ref{lemma-pushout-fpqc-local}", "it suffices to prove that $X_a = \\colim X_{a, i}$ and", "$X_{a, b} = \\colim X_{a, b, i}$.", "This reduces us to the case where $B = B_a$ (an affine scheme) or", "$B = B_a \\times_B B_b$ (a separated scheme). Repeating the", "argument once more, we conclude that we may assume $B$ is", "an affine scheme (this uses that the intersection of affine", "opens in a separated scheme is affine).", "\\medskip\\noindent", "Assume $B$ is an affine scheme. Let $Z$ be an algebraic space over $B$.", "We have to show", "$$", "\\Mor_B(X, Z) \\longrightarrow \\lim \\Mor_B(X_i, Z)", "$$", "is a bijection.", "\\medskip\\noindent", "Proof of injectivity. Let $f, g : X \\to Z$ be morphisms", "such that the compositions $f_i, g_i : X_i \\to Z$ are the same for all $i$.", "Choose an affine scheme $Z'$ and an \\'etale morphism $Z' \\to Z$. By", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-cover-by-union-affines}", "we know we can cover $Z$ by such affines. Set $U = X \\times_{f, Z} Z'$", "and $U' = X \\times_{g, Z} Z'$ and denote $p : U \\to X$ and $p' : U' \\to X$", "the projections. Since $f_i = g_i$ for all $i$, we see that", "$$", "U_i = X_i \\times_{f_i, Z} Z' = X_i \\times_{g_i, Z} Z' = U'_i", "$$", "compatible with transition morphisms. By (1) there is a unique isomorphism", "$\\epsilon : U \\to U'$ as algebraic spaces over $X$, i.e., with", "$p = p' \\circ \\epsilon$ which is compatible with", "the displayed identifications. Choose an \\'etale covering", "$\\{h_a : U_a \\to U\\}$ with $U_a$ affine.", "By (2) we see that $f \\circ p \\circ h_a = g \\circ p' \\circ \\epsilon \\circ h_a =", "g \\circ p \\circ h_a$. Since $\\{h_a : U_a \\to U\\}$ is an \\'etale covering", "we conclude $f \\circ p = g \\circ p$.", "Since the collection of morphisms $p : U \\to X$ we obtain", "in this manner is an \\'etale covering, we conclude that $f = g$.", "\\medskip\\noindent", "Proof of surjectivity. Let $f_i : X_i \\to Z$ be an element of the", "right hand side of the displayed arrow in the first paragraph of the proof.", "It suffices to find an \\'etale covering $\\{U_c \\to X\\}_{c \\in C}$", "such that the families $f_{c, i} \\in \\lim_i \\Mor_B(X_i \\times_X U_c, Z)$", "come from morphisms $f_c : U_c \\to Z$. Namely, by the uniqueness", "proved above the morphisms $f_c$ will agree on $U_c \\times_X U_b$", "and hence will descend to give the desired morphism $f : X \\to Z$.", "To find our covering, we first choose an \\'etale covering", "$\\{g_a : Z_a \\to Z\\}_{a \\in A}$ where each $Z_a$ is affine. Then we let", "$U_{a, i} = X_i \\times_{f_i, Z} Z_a$. By (1) we find", "$U_{a, i} = X_i \\times_X U_a$ for some algebraic spaces $U_a$ \\'etale", "over $X$. Then we choose \\'etale coverings", "$\\{U_{a, b} \\to U_a\\}_{b \\in B_a}$", "with $U_{a, b}$ affine and we consider the morphisms", "$$", "U_{a, b, i} = X_i \\times_X U_{a, b} \\to", "X_i \\times_X U_a = X_i \\times_{f_i, Z} Z_a \\to Z_a", "$$", "By (2) we obtain morphisms $f_{a, b} : U_{a, b} \\to Z_a$ compatible with these", "morphisms. Setting $C = \\coprod_{a \\in A} B_a$ and for $c \\in C$", "corresponding to $b \\in B_a$ setting $U_c = U_{a, b}$ and", "$f_c = g_a \\circ f_{a, b} : U_c \\to Z$ we conclude." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-pushouts-lemma-pushout-fpqc-local", "spaces-properties-lemma-cover-by-union-affines" ], "ref_ids": [ 11830, 10846, 11830 ] } ], "ref_ids": [ 12416 ] }, { "id": 10848, "type": "theorem", "label": "spaces-pushouts-lemma-colimit-separated-enough", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-colimit-separated-enough", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $\\mathcal{I} \\to (\\Sch/S)_{fppf}$, $i \\mapsto X_i$", "be a diagram of algebraic spaces over $B$. Assume that", "\\begin{enumerate}", "\\item each $X_i$ is separated over $B$,", "\\item $X = \\colim X_i$ exists in the category of", "algebraic spaces separated over $B$,", "\\item $\\coprod X_i \\to X$ is surjective,", "\\item if $U \\to X$ is an \\'etale separated morphism of algebraic spaces and", "$U_i = X_i \\times_X U$, then $U = \\colim U_i$ in", "the category of algebraic spaces separated over $B$, and", "\\item every object $(U_i \\to X_i)$ of $\\lim X_{i, spaces, \\etale}$", "with $U_i \\to X_i$ separated is of the form $U_i = X_i \\times_X U$", "for some \\'etale separated morphism of algebraic spaces $U \\to X$.", "\\end{enumerate}", "Then $X = \\colim X_i$ in the category of all algebraic spaces over $B$." ], "refs": [], "proofs": [ { "contents": [ "We encourage the reader to look instead at", "Lemma \\ref{lemma-colimit-check-etale-locally}", "and its proof.", "\\medskip\\noindent", "Let $Z$ be an algebraic space over $B$. Suppose that $f_i : X_i \\to Z$ is", "a family of morphisms such that for each $i \\to j$ the composition", "$X_i \\to X_j \\to Z$ is equal to $f_i$. We have to construct a morphism", "of algebraic spaces $f : X \\to Z$ over $B$ such that we can recover $f_i$ as", "the composition $X_i \\to X \\to Z$. Let $W \\to Z$ be a surjective", "\\'etale morphism of a scheme to $Z$. We may assume that $W$ is a", "disjoint union of affines and in particular we may assume that", "$W \\to Z$ is separated and that $W$ is separated over $B$. For each $i$ set", "$U_i = W \\times_{Z, f_i} X_i$ and denote $h_i : U_i \\to W$ the projection.", "Then $U_i \\to X_i$ forms an object of $\\lim X_{i, spaces, \\etale}$", "with $U_i \\to X_i$ separated. By", "assumption (5) we can find a separated \\'etale morphism $U \\to X$", "of algebraic spaces and (functorial) isomorphisms $U_i = X_i \\times_X U$.", "By assumption (4) there exists a morphism $h : U \\to W$ over $B$", "such that the compositions $U_i \\to U \\to W$ are $h_i$.", "Let $g : U \\to Z$ be the composition of $h$ with the map $W \\to Z$. To", "finish the proof we have to show that $g : U \\to Z$", "descends to a morphism $X \\to Z$. To do this, consider the morphism", "$(h, h) : U \\times_X U \\to W \\times_S W$.", "Composing with $U_i \\times_{X_i} U_i \\to U \\times_X U$ we obtain", "$(h_i, h_i)$ which factors through $W \\times_Z W$. Since $U \\times_X U$", "is the colimit of the algebraic spaces $U_i \\times_{X_i} U_i$", "in the category of algebraic spaces separated over $B$ by (4) we see", "that $(h, h)$ factors through $W \\times_Z W$. Hence the two compositions", "$U \\times_X U \\to U \\to W \\to Z$ are equal. Because each $U_i \\to X_i$ is", "surjective and assumption (2) we see that $U \\to X$ is surjective.", "As $Z$ is a sheaf for the \\'etale topology, we conclude that", "$g : U \\to Z$ descends to $f : X \\to Z$ as desired." ], "refs": [ "spaces-pushouts-lemma-colimit-check-etale-locally" ], "ref_ids": [ 10847 ] } ], "ref_ids": [] }, { "id": 10849, "type": "theorem", "label": "spaces-pushouts-lemma-glue-etale-sheaf-etale", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-glue-etale-sheaf-etale", "contents": [ "Let $S$ be a scheme. Let $\\{f_i : X_i \\to X\\}$ be an \\'etale covering of", "algebraic spaces. The functor", "$$", "\\Sh(X_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{f_i : X_i \\to X\\}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "In Properties of Spaces, Section \\ref{spaces-properties-section-etale-site}", "we have defined a site $X_{spaces, \\etale}$ whose objects", "are algebraic spaces \\'etale over $X$ with \\'etale coverings.", "Moreover, we have a identifications", "$\\Sh(X_\\etale) = \\Sh(X_{spaces, \\etale})$ compatible", "with morphisms of algebraic spaces, i.e., compatible with", "pushforward and pullback. Hence the statement of the lemma follows", "from the much more general discussion in", "Sites, Section \\ref{sites-section-glueing-sheaves}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10850, "type": "theorem", "label": "spaces-pushouts-lemma-reduce-to-scheme-base", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-reduce-to-scheme-base", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "algebraic spaces over $S$. Let $\\{Y_i \\to Y\\}_{i \\in I}$ be an \\'etale", "covering of algebraic spaces. If for each $i \\in I$ the functor", "$$", "\\Sh(Y_{i, \\etale})", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{X \\times_Y Y_i \\to Y_i\\}", "$$", "is an equivalence of categories and for each $i, j \\in I$ the functor", "$$", "\\Sh((Y_i \\times_Y Y_j)_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }", "\\{X \\times_Y Y_i \\times_Y Y_j \\to Y_i \\times_Y Y_j\\}", "$$", "is an equivalence of categories, then", "$$", "\\Sh(Y_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "Formal consequence of Lemma \\ref{lemma-glue-etale-sheaf-etale}", "and the definitions." ], "refs": [ "spaces-pushouts-lemma-glue-etale-sheaf-etale" ], "ref_ids": [ 10849 ] } ], "ref_ids": [] }, { "id": 10851, "type": "theorem", "label": "spaces-pushouts-lemma-representable-case", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-representable-case", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is representable (by schemes) and $f$", "has one of the following properties:", "surjective and integral,", "surjective and proper, or", "surjective and flat and locally of finite presentation", "Then ", "$$", "\\Sh(Y_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "Each of the properties of morphisms of algebraic spaces", "mentioned in the statement of the lemma is preserved by", "arbitrary base change, see the lists in", "Spaces, Section \\ref{spaces-section-lists}.", "Thus we can apply Lemma \\ref{lemma-reduce-to-scheme-base}", "to see that we can work \\'etale locally on $Y$.", "In this way we reduce to the case where $Y$ is a scheme;", "some details omitted. In this case $X$ is also a scheme", "and the result follows from \\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-glue-etale-sheaf-integral-surjective},", "\\ref{etale-cohomology-lemma-glue-etale-sheaf-proper-surjective}, or", "\\ref{etale-cohomology-lemma-glue-etale-sheaf-fppf-cover}." ], "refs": [ "spaces-pushouts-lemma-reduce-to-scheme-base", "etale-cohomology-lemma-glue-etale-sheaf-integral-surjective", "etale-cohomology-lemma-glue-etale-sheaf-proper-surjective", "etale-cohomology-lemma-glue-etale-sheaf-fppf-cover" ], "ref_ids": [ 10850, 6683, 6684, 6686 ] } ], "ref_ids": [] }, { "id": 10852, "type": "theorem", "label": "spaces-pushouts-lemma-reduce-to-scheme-source", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-reduce-to-scheme-source", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $\\pi : X' \\to X$ be a morphism of algebraic spaces. Assume", "\\begin{enumerate}", "\\item $f \\circ \\pi$ is representable (by schemes),", "\\item $f \\circ \\pi$ has one of the following properties:", "surjective and integral,", "surjective and proper, or", "surjective and flat and locally of finite presentation.", "\\end{enumerate}", "Then ", "$$", "\\Sh(Y_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "Formal consequence of Lemma \\ref{lemma-representable-case}", "and Stacks, Lemma \\ref{stacks-lemma-compare-descent-condition}." ], "refs": [ "spaces-pushouts-lemma-representable-case", "stacks-lemma-compare-descent-condition" ], "ref_ids": [ 10851, 8939 ] } ], "ref_ids": [] }, { "id": 10853, "type": "theorem", "label": "spaces-pushouts-lemma-glue-etale-sheaf-proper-surjective", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-glue-etale-sheaf-proper-surjective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "algebraic spaces over $S$ which has one of the following", "properties: surjective and integral, surjective and proper, or", "surjective and flat and locally of finite presentation. Then the functor", "$$", "\\Sh(Y_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "Observe that the base change of a proper surjective morphism is", "proper and surjective, see", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-base-change-proper}", "and \\ref{spaces-morphisms-lemma-base-change-surjective}.", "Hence by Lemma \\ref{lemma-reduce-to-scheme-base}", "we may work \\'etale locally on $Y$. Hence", "we reduce to $Y$ being an affine scheme; some details omitted.", "\\medskip\\noindent", "Assume $Y$ is affine. By Lemma \\ref{lemma-reduce-to-scheme-source}", "it suffices to find a morphism $X' \\to X$ where $X'$ is a scheme such", "that $X' \\to Y$ is surjective and integral, surjective and proper, or", "surjective and flat and locally of finite presentation.", "\\medskip\\noindent", "In case $X \\to Y$ is integral and surjective, we can take $X = X'$", "as an integral morphism is representable.", "\\medskip\\noindent", "If $f$ is proper and surjective, then the algebraic space", "$X$ is quasi-compact and separated, see", "Morphisms of Spaces, Section \\ref{spaces-morphisms-section-quasi-compact} and", "Lemma \\ref{spaces-morphisms-lemma-separated-over-separated}.", "Choose a scheme $X'$ and a surjective finite morphism $X' \\to X$, see", "Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-there-is-a-scheme-finite-over}.", "Then $X' \\to Y$ is surjective and proper.", "\\medskip\\noindent", "Finally, if $X \\to Y$ is surjective and flat and locally of finite", "presentation then we can take an affine \\'etale covering $\\{U_i \\to X\\}$", "and set $X'$ equal to the disjoint $\\coprod U_i$." ], "refs": [ "spaces-morphisms-lemma-base-change-proper", "spaces-morphisms-lemma-base-change-surjective", "spaces-pushouts-lemma-reduce-to-scheme-base", "spaces-pushouts-lemma-reduce-to-scheme-source", "spaces-morphisms-lemma-separated-over-separated", "spaces-limits-proposition-there-is-a-scheme-finite-over" ], "ref_ids": [ 4917, 4727, 10850, 10852, 4719, 4659 ] } ], "ref_ids": [] }, { "id": 10854, "type": "theorem", "label": "spaces-pushouts-lemma-glue-etale-sheaf-fppf", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-glue-etale-sheaf-fppf", "contents": [ "Let $S$ be a scheme.", "Let $\\{f_i : X_i \\to X\\}$ be an fppf covering of algebraic spaces over $S$.", "The functor", "$$", "\\Sh(X_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{f_i : X_i \\to X\\}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "We have Lemma \\ref{lemma-glue-etale-sheaf-proper-surjective}", "for the morphism $f : \\coprod X_i \\to X$.", "Then a formal argument shows that descent data for $f$", "are the same thing as descent data for the covering, compare", "with Descent, Lemma \\ref{descent-lemma-family-is-one}.", "Details omitted." ], "refs": [ "spaces-pushouts-lemma-glue-etale-sheaf-proper-surjective", "descent-lemma-family-is-one" ], "ref_ids": [ 10853, 14732 ] } ], "ref_ids": [] }, { "id": 10855, "type": "theorem", "label": "spaces-pushouts-lemma-glue-etale-sheaf-modification", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-glue-etale-sheaf-modification", "contents": [ "Let $S$ be a scheme. Let $f : Y' \\to Y$ be a proper morphism of", "algebraic spaces over $S$. Let $i : Z \\to Y$", "be a closed immersion. Set $E = Z \\times_Y Y'$. Picture", "$$", "\\xymatrix{", "E \\ar[d]_g \\ar[r]_j & Y' \\ar[d]^f \\\\", "Z \\ar[r]^i & Y", "}", "$$", "If $f$ is an isomorphism over $Y \\setminus Z$, then the functor", "$$", "\\Sh(Y_\\etale)", "\\longrightarrow", "\\Sh(Y'_\\etale) \\times_{\\Sh(E_\\etale)} \\Sh(Z_\\etale)", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "Observe that $X = Y' \\coprod Z \\to Y$ is a proper surjective morphism.", "Thus it suffice to construct an equivalence of categories", "$$", "\\Sh(Y'_\\etale) \\times_{\\Sh(E_\\etale)} \\Sh(Z_\\etale)", "\\longrightarrow", "\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}", "$$", "compatible with pullback functors from $Y$", "because then we can use Lemma \\ref{lemma-glue-etale-sheaf-proper-surjective}", "to conclude. Thus let $(\\mathcal{G}', \\mathcal{G}, \\alpha)$ be an", "object of $\\Sh(Y'_\\etale) \\times_{\\Sh(E_\\etale)} \\Sh(Z_\\etale)$ with", "notation as in Categories, Example", "\\ref{categories-example-2-fibre-product-categories}.", "Then we can consider the sheaf $\\mathcal{F}$ on $X$ defined", "by taking $\\mathcal{G}'$ on the summand $Y'$ and $\\mathcal{G}$", "on the summand $Z$. We have", "$$", "X \\times_Y X = Y' \\times_Y Y' \\amalg", "Y' \\times_Y Z \\amalg Z \\times_Y Y' \\amalg Z \\times_Y Z =", "Y' \\times_Y Y' \\amalg E \\amalg E \\amalg Z", "$$", "The isomorphisms of the two pullbacks of $\\mathcal{F}$ to this algebraic", "space are obvious over the summands $E$, $E$, $Z$. The interesting", "part of the proof is to find an isomorphism", "$\\text{pr}_{0, small}^{-1}\\mathcal{G}' \\to", "\\text{pr}_{1, small}^{-1}\\mathcal{G}'$", "over $Y' \\times_Y Y'$ satisfying the cocycle condition.", "However, our assumption that $Y' \\to Y$ is an isomorphism", "over $Y \\setminus Z$ implies that", "$$", "h : Y \\coprod E \\times_Z E \\longrightarrow Y' \\times_Y Y'", "$$", "is a surjective proper morphism. (It is in fact a finite morphism", "as it is the disjoint union of two closed immersions.)", "Hence it suffices to construct an isomorphism of the pullbacks", "of $\\text{pr}_{0, small}^{-1}\\mathcal{G}'$and", "$\\text{pr}_{1, small}^{-1}\\mathcal{G}'$ by $h_{small}$ satisfying", "a certain cocycle condition. For the diagonal, it is clear", "how to do this. And for the pullback to $E \\times_Z E$", "we use that both sheaves pull back to the pullback of", "$\\mathcal{G}$ by the morphism $E \\times_Z E \\to Z$.", "We omit the details." ], "refs": [ "spaces-pushouts-lemma-glue-etale-sheaf-proper-surjective" ], "ref_ids": [ 10853 ] } ], "ref_ids": [] }, { "id": 10856, "type": "theorem", "label": "spaces-pushouts-lemma-descend-etale-proper-surjective", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-descend-etale-proper-surjective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper surjective morphism", "of algebraic spaces over $S$. Any descent datum $(U/X, \\varphi)$ relative to $f$", "(Descent on Spaces, Definition \\ref{spaces-descent-definition-descent-datum})", "with $U$ \\'etale over $X$ is effective", "(Descent on Spaces, Definition \\ref{spaces-descent-definition-effective}).", "More precisely, there exists an \\'etale morphism $V \\to Y$ of algebraic spaces", "whose corresponding canonical descent datum is isomorphic to $(U/X, \\varphi)$." ], "refs": [ "spaces-descent-definition-descent-datum", "spaces-descent-definition-effective" ], "proofs": [ { "contents": [ "Recall that $U$ gives rise to a representable sheaf", "$\\mathcal{F} = h_U$ in $\\Sh(X_{spaces, \\etale}) = \\Sh(X_\\etale)$, see", "Properties of Spaces, Section \\ref{spaces-properties-section-etale-site}.", "The descent datum on $U$ relative to $f$", "exactly gives a descent datum $(\\mathcal{F}, \\varphi)$", "for \\'etale sheaves with respect to $\\{X \\to Y\\}$.", "By Lemma \\ref{lemma-glue-etale-sheaf-proper-surjective}", "this descent datum is effective.", "Let $\\mathcal{G}$ be the corresponding sheaf on $Y_\\etale$.", "By Properties of Spaces, Lemma \\ref{spaces-properties-lemma-sheaf-gives-space}", "we obtain an \\'etale morphism $V \\to Y$ of algebraic spaces", "corresponding to $\\mathcal{G}$; we omit the verification of", "the set theoretic condition\\footnote{It follows from the", "fact that $\\mathcal{F}$ satisfies the corresponding condition.}.", "The given isomorphism $\\mathcal{F} \\to f_{small}^{-1}\\mathcal{G}$", "corresponds to an isomorphism $U \\to V \\times_Y X$ compatible", "with the descent datum." ], "refs": [ "spaces-pushouts-lemma-glue-etale-sheaf-proper-surjective", "spaces-properties-lemma-sheaf-gives-space" ], "ref_ids": [ 10853, 11902 ] } ], "ref_ids": [ 9444, 9448 ] }, { "id": 10857, "type": "theorem", "label": "spaces-pushouts-lemma-glue-etale-space-modification", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-glue-etale-space-modification", "contents": [ "Let $S$ be a scheme. Let $f : Y' \\to Y$ be a proper morphism of", "algebraic spaces over $S$. Let $i : Z \\to Y$", "be a closed immersion. Set $E = Z \\times_Y Y'$. Picture", "$$", "\\xymatrix{", "E \\ar[d]_g \\ar[r]_j & Y' \\ar[d]^f \\\\", "Z \\ar[r]^i & Y", "}", "$$", "If $f$ is an isomorphism over $Y \\setminus Z$, then the functor", "$$", "Y_{spaces, \\etale}", "\\longrightarrow", "Y'_{spaces, \\etale} \\times_{E_{spaces, \\etale}} \\Sh(Z_{spaces, \\etale})", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "Let $(V' \\to Y', W \\to Z, \\alpha)$ be an object of the right hand side.", "Recall that $V'$, resp.\\ $W$ gives rise to a representable sheaf", "$\\mathcal{G}' = h_{V'}$ in $\\Sh(Y'_{spaces, \\etale}) = \\Sh(Y'_\\etale)$,", "resp.\\ $\\mathcal{G} = h_W$ ", "in $\\Sh(Z_{spaces, \\etale}) = \\Sh(Z_\\etale)$, see", "Properties of Spaces, Section \\ref{spaces-properties-section-etale-site}.", "The isomorphism $\\alpha : V' \\times_{Y'} E \\to W \\times_Z E$", "determines an isomorphism", "$j_{small}^{-1}\\mathcal{G}' \\to g_{small}^{-1}\\mathcal{G}$ of", "sheaves on $E$.", "By Lemma \\ref{lemma-glue-etale-sheaf-modification}", "we obtain a unique sheaf $\\mathcal{F}$ on $Y$ pulling pack", "to $\\mathcal{G}'$ and $\\mathcal{G}$ compatibly with the isomorphism.", "By Properties of Spaces, Lemma \\ref{spaces-properties-lemma-sheaf-gives-space}", "we obtain an \\'etale morphism $V \\to Y$ of algebraic spaces", "corresponding to $\\mathcal{F}$; we omit the verification of", "the set theoretic condition\\footnote{It follows from the", "fact that $\\mathcal{G}$ and $\\mathcal{G}'$ satisfies the", "corresponding condition.}.", "The given isomorphism $\\mathcal{G}' \\to f_{small}^{-1}\\mathcal{F}$", "and $\\mathcal{G} \\to i_{small}^{-1}\\mathcal{F}$", "corresponds to isomorphisms $V' \\to V \\times_Y Y'$", "and $W \\to V \\times_Y Z$ compatible", "with $\\alpha$ as desired." ], "refs": [ "spaces-pushouts-lemma-glue-etale-sheaf-modification", "spaces-properties-lemma-sheaf-gives-space" ], "ref_ids": [ 10855, 11902 ] } ], "ref_ids": [] }, { "id": 10858, "type": "theorem", "label": "spaces-pushouts-lemma-pushout-along-thickening-schemes", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-pushout-along-thickening-schemes", "contents": [ "Let $S$ be a scheme. Let $X \\to X'$ be a thickening of schemes", "over $S$ and let $X \\to Y$ be an affine morphism of schemes over $S$.", "Let $Y' = Y \\amalg_X X'$ be the pushout in the category of schemes (see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}).", "Then $Y'$ is also a pushout in the category of algebraic spaces over $S$." ], "refs": [ "more-morphisms-lemma-pushout-along-thickening" ], "proofs": [ { "contents": [ "This is an immediate consequence of Lemma \\ref{lemma-colimit-agrees} and", "More on Morphisms, Lemmas", "\\ref{more-morphisms-lemma-pushout-along-thickening},", "\\ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout}, and", "\\ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat}." ], "refs": [ "spaces-pushouts-lemma-colimit-agrees", "more-morphisms-lemma-pushout-along-thickening", "more-morphisms-lemma-equivalence-categories-schemes-over-pushout", "more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat" ], "ref_ids": [ 10845, 13762, 13763, 13765 ] } ], "ref_ids": [ 13762 ] }, { "id": 10859, "type": "theorem", "label": "spaces-pushouts-lemma-pushout-along-thickening", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-pushout-along-thickening", "contents": [ "Let $S$ be a scheme. Let $X \\to X'$ be a thickening of algebraic spaces", "over $S$ and let $X \\to Y$ be an affine morphism of algebraic spaces over $S$.", "Then there exists a pushout", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d]_f", "&", "X' \\ar[d]^{f'}", "\\\\", "Y \\ar[r]", "&", "Y \\amalg_X X'", "}", "$$", "in the category of algebraic spaces over $S$. Moreover $Y' = Y \\amalg_X X'$", "is a thickening of $Y$ and", "$$", "\\mathcal{O}_{Y'} = \\mathcal{O}_Y \\times_{f_*\\mathcal{O}_X} f'_*\\mathcal{O}_{X'}", "$$", "as sheaves on $Y_\\etale = (Y')_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Set $U = V \\times_Y X$. This is a scheme affine over $V$ with a", "surjective \\'etale morphism $U \\to X$. By More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-thickening-equivalence}", "there exists a $U' \\to X'$", "surjective \\'etale with $U = U' \\times_{X'} X$. In particular the", "morphism of schemes $U \\to U'$ is a thickening too. Apply", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}", "to obtain a pushout $V' = V \\amalg_U U'$ in the category of schemes.", "\\medskip\\noindent", "We repeat this procedure to construct a pushout", "$$", "\\xymatrix{", "U \\times_X U \\ar[d] \\ar[r] & U' \\times_{X'} U' \\ar[d] \\\\", "V \\times_Y V \\ar[r] & R'", "}", "$$", "in the category of schemes. Consider the morphisms", "$$", "U \\times_X U \\to U \\to V',\\quad", "U' \\times_{X'} U' \\to U' \\to V',\\quad", "V \\times_Y V \\to V \\to V'", "$$", "where we use the first projection in each case. Clearly these glue to", "give a morphism $t' : R' \\to V'$ which is \\'etale by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat}.", "Similarly, we obtain $s' : R' \\to V'$ \\'etale.", "The morphism $j' = (t', s') : R' \\to V' \\times_S V'$ is unramified", "(as $t'$ is \\'etale) and a monomorphism when restricted to the closed", "subscheme $V \\times_Y V \\subset R'$. As $V \\times_Y V \\subset R'$ is", "a thickening it follows that $j'$ is a monomorphism too. Finally, $j'$", "is an equivalence relation as we can use the functoriality of pushouts", "of schemes to construct a morphism $c' : R' \\times_{s', V', t'} R' \\to R'$", "(details omitted). At this point we set $Y' = U'/R'$, see", "Spaces, Theorem \\ref{spaces-theorem-presentation}.", "\\medskip\\noindent", "We have morphisms $X' = U'/U' \\times_{X'} U' \\to V'/R' = Y'$ and", "$Y = V/V \\times_Y V \\to V'/R' = Y'$.", "By construction these fit into the commutative diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d]_f & X' \\ar[d]^{f'} \\\\", "Y \\ar[r] & Y'", "}", "$$", "Since $Y \\to Y'$ is a thickening we have", "$Y_\\etale = (Y')_\\etale$, see More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-thickening-equivalence}.", "The commutativity of the diagram gives a map of sheaves", "$$", "\\mathcal{O}_{Y'}", "\\longrightarrow", "\\mathcal{O}_Y \\times_{f_*\\mathcal{O}_X} f'_*\\mathcal{O}_{X'}", "$$", "on this set. By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-pushout-along-thickening}", "this map is an isomorphism when we restrict to", "the scheme $V'$, hence it is an isomorphism.", "\\medskip\\noindent", "To finish the proof we show that the diagram above is a pushout in", "the category of algebraic spaces. To see this, let $Z$ be an algebraic", "space and let $a' : X' \\to Z$ and $b : Y \\to Z$ be morphisms of", "algebraic spaces. By", "Lemma \\ref{lemma-pushout-along-thickening-schemes}", "we obtain a unique morphism $h : V' \\to Z$ fitting into the commutative", "diagrams", "$$", "\\vcenter{", "\\xymatrix{", "U' \\ar[d] \\ar[r] & V' \\ar[d]^h \\\\", "X' \\ar[r]^{a'} & Z", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "V \\ar[r] \\ar[d] & V' \\ar[d]^h \\\\", "Y \\ar[r]^b & Z", "}", "}", "$$", "The uniqueness shows that $h \\circ t' = h \\circ s'$. Hence $h$ factors", "uniquely as $V' \\to Y' \\to Z$ and we win." ], "refs": [ "spaces-more-morphisms-lemma-thickening-equivalence", "more-morphisms-lemma-pushout-along-thickening", "more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat", "spaces-theorem-presentation", "spaces-more-morphisms-lemma-thickening-equivalence", "more-morphisms-lemma-pushout-along-thickening", "spaces-pushouts-lemma-pushout-along-thickening-schemes" ], "ref_ids": [ 50, 13762, 13765, 8124, 50, 13762, 10858 ] } ], "ref_ids": [] }, { "id": 10860, "type": "theorem", "label": "spaces-pushouts-lemma-categories-spaces-over-pushout", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-categories-spaces-over-pushout", "contents": [ "Let $S$ be a base scheme. Let $X \\to X'$ be a thickening of algebraic spaces", "over $S$ and let $X \\to Y$ be an affine morphism of algebraic spaces over $S$.", "Let $Y' = Y \\amalg_X X'$ be the pushout (see", "Lemma \\ref{lemma-pushout-along-thickening}). Base change gives a functor", "$$", "F :", "(\\textit{Spaces}/Y')", "\\longrightarrow", "(\\textit{Spaces}/Y) \\times_{(\\textit{Spaces}/Y')} (\\textit{Spaces}/X')", "$$", "given by $V' \\longmapsto (V' \\times_{Y'} Y, V' \\times_{Y'} X', 1)$ which", "sends $(\\Sch/Y')$ into $(\\Sch/Y) \\times_{(\\Sch/Y')} (\\Sch/X')$.", "The functor $F$ has a left adjoint", "$$", "G :", "(\\textit{Spaces}/Y) \\times_{(\\textit{Spaces}/Y')} (\\textit{Spaces}/X')", "\\longrightarrow", "(\\textit{Spaces}/Y')", "$$", "which sends the triple $(V, U', \\varphi)$ to the pushout", "$V \\amalg_{(V \\times_Y X)} U'$ in the category of algebraic spaces over $S$.", "The functor $G$ sends $(\\Sch/Y) \\times_{(\\Sch/Y')} (\\Sch/X')$ into $(\\Sch/Y')$." ], "refs": [ "spaces-pushouts-lemma-pushout-along-thickening" ], "proofs": [ { "contents": [ "The proof is completely formal.", "Since the morphisms $X \\to X'$ and $X \\to Y$ are representable it", "is clear that $F$ sends $(\\Sch/Y')$ into", "$(\\Sch/Y) \\times_{(\\Sch/Y')} (\\Sch/X')$.", "\\medskip\\noindent", "Let us construct $G$. Let $(V, U', \\varphi)$ be an object of the fibre", "product category. Set $U = U' \\times_{X'} X$. Note that $U \\to U'$ is a", "thickening. Since $\\varphi : V \\times_Y X \\to U' \\times_{X'} X = U$ is an", "isomorphism we have a morphism $U \\to V$ over $X \\to Y$ which identifies", "$U$ with the fibre product $X \\times_Y V$. In particular $U \\to V$ is", "affine, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-affine}.", "Hence we can apply Lemma \\ref{lemma-pushout-along-thickening}", "to get a pushout $V' = V \\amalg_U U'$. Denote $V' \\to Y'$ the morphism", "we obtain in virtue of the fact that $V'$ is a pushout and because", "we are given morphisms $V \\to Y$ and $U' \\to X'$ agreeing on $U$", "as morphisms into $Y'$. Setting $G(V, U', \\varphi) = V'$", "gives the functor $G$.", "\\medskip\\noindent", "If $(V, U', \\varphi)$ is an object of $(\\Sch/Y) \\times_{(\\Sch/Y')} (\\Sch/X')$", "then $U = U' \\times_{X'} X$ is a scheme too and we can form the pushout", "$V' = V \\amalg_U U'$ in the category of schemes by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.", "By Lemma \\ref{lemma-pushout-along-thickening-schemes}", "this is also a pushout in the category of schemes, hence", "$G$ sends $(\\Sch/Y) \\times_{(\\Sch/Y')} (\\Sch/X')$ into $(\\Sch/Y')$.", "\\medskip\\noindent", "Let us prove that $G$ is a left adjoint to $F$. Let $Z$ be an algebraic space", "over $Y'$. We have to show that", "$$", "\\Mor(V', Z) = \\Mor((V, U', \\varphi), F(Z))", "$$", "where the morphism sets are taking in their respective categories.", "Let $g' : V' \\to Z$ be a morphism. Denote $\\tilde g$, resp.\\ $\\tilde f'$", "the composition of $g'$ with the morphism $V \\to V'$, resp.\\ $U' \\to V'$.", "Base change $\\tilde g$, resp.\\ $\\tilde f'$ by $Y \\to Y'$, resp.\\ $X' \\to Y'$", "to get a morphism $g : V \\to Z \\times_{Y'} Y$,", "resp.\\ $f' : U' \\to Z \\times_{Y'} X'$. Then $(g, f')$ is an element", "of the right hand side of the equation above (details omitted).", "Conversely, suppose that $(g, f') : (V, U', \\varphi) \\to F(Z)$ is an", "element of the right hand side.", "We may consider the composition $\\tilde g : V \\to Z$,", "resp.\\ $\\tilde f' : U' \\to Z$ of $g$, resp.\\ $f$ by", "$Z \\times_{Y'} X' \\to Z$, resp.\\ $Z \\times_{Y'} Y \\to Z$.", "Then $\\tilde g$ and $\\tilde f'$ agree as morphism from $U$ to $Z$.", "By the universal property of pushout, we obtain a morphism", "$g' : V' \\to Z$, i.e., an element of the left hand side.", "We omit the verification that these constructions are mutually inverse." ], "refs": [ "spaces-morphisms-lemma-base-change-affine", "spaces-pushouts-lemma-pushout-along-thickening", "more-morphisms-lemma-pushout-along-thickening", "spaces-pushouts-lemma-pushout-along-thickening-schemes" ], "ref_ids": [ 4800, 10859, 13762, 10858 ] } ], "ref_ids": [ 10859 ] }, { "id": 10861, "type": "theorem", "label": "spaces-pushouts-lemma-diagram", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-diagram", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "A \\ar[r] \\ar[d] & C \\ar[d] \\ar[r] & E \\ar[d] \\\\", "B \\ar[r] & D \\ar[r] & F", "}", "$$", "be a commutative diagram of algebraic spaces over $S$.", "Assume that $A, B, C, D$ and $A, B, E, F$ form cartesian squares", "and that $B \\to D$ is surjective \\'etale.", "Then $C, D, E, F$ is a cartesian square." ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10862, "type": "theorem", "label": "spaces-pushouts-lemma-equivalence-categories-spaces-over-pushout", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-equivalence-categories-spaces-over-pushout", "contents": [ "In the situation of Lemma \\ref{lemma-categories-spaces-over-pushout}", "the functor $F \\circ G$ is isomorphic to the identity functor." ], "refs": [ "spaces-pushouts-lemma-categories-spaces-over-pushout" ], "proofs": [ { "contents": [ "We will prove that $F \\circ G$ is isomorphic to the identity by", "reducing this to the corresponding statement of", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout}.", "\\medskip\\noindent", "Choose a scheme $Y_1$ and a surjective \\'etale morphism", "$Y_1 \\to Y$. Set $X_1 = Y_1 \\times_Y X$. This is a scheme affine over", "$Y_1$ with a surjective \\'etale morphism $X_1 \\to X$. By", "More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-thickening-equivalence}", "there exists a $X'_1 \\to X'$", "surjective \\'etale with $X_1 = X_1' \\times_{X'} X$. In particular the", "morphism of schemes $X_1 \\to X_1'$ is a thickening too. Apply", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}", "to obtain a pushout $Y_1' = Y_1 \\amalg_{X_1} X_1'$ in the category of", "schemes. In the proof of Lemma \\ref{lemma-pushout-along-thickening}", "we constructed", "$Y'$ as a quotient of an \\'etale equivalence relation on $Y_1'$", "such that we get a commutative diagram", "\\begin{equation}", "\\label{equation-cube}", "\\vcenter{", "\\xymatrix{", "& X \\ar[rr] \\ar'[d][dd] & & X' \\ar[dd] \\\\", "X_1 \\ar[rr] \\ar[dd] \\ar[ru] & & X_1' \\ar[dd] \\ar[ru] & \\\\", "& Y \\ar'[r][rr] & & Y' \\\\", "Y_1 \\ar[rr] \\ar[ru] & & Y_1' \\ar[ru]", "}", "}", "\\end{equation}", "where all squares except the front and back squares are cartesian", "(the front and back squares are pushouts) and the northeast arrows", "are surjective \\'etale. Denote $F_1$, $G_1$ the", "functors constructed in", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout}", "for the front square. Then the diagram of categories", "$$", "\\xymatrix{", "(\\Sch/Y_1') \\ar@<-1ex>[r]_-{F_1} \\ar[d] &", "(\\Sch/Y_1) \\times_{(\\Sch/Y_1')} (\\Sch/X_1') \\ar[d] \\ar@<-1ex>[l]_-{G_1} \\\\", "(\\textit{Spaces}/Y') \\ar@<-1ex>[r]_-F &", "(\\textit{Spaces}/Y) \\times_{(\\textit{Spaces}/Y')} (\\textit{Spaces}/X')", "\\ar@<-1ex>[l]_-G", "}", "$$", "is commutative by simple considerations regarding base change functors", "and the agreement of pushouts in schemes with pushouts in", "spaces of Lemma \\ref{lemma-pushout-along-thickening-schemes}.", "\\medskip\\noindent", "Let $(V, U', \\varphi)$ be an object of", "$(\\textit{Spaces}/Y) \\times_{(\\textit{Spaces}/Y')} (\\textit{Spaces}/X')$.", "Denote $U = U' \\times_{X'} X$ so that $G(V, U', \\varphi) = V \\amalg_U U'$.", "Choose a scheme $V_1$ and a surjective \\'etale morphism", "$V_1 \\to Y_1 \\times_Y V$. Set $U_1 = V_1 \\times_Y X$. Then", "$$", "U_1 = V_1 \\times_Y X", "\\longrightarrow", "(Y_1 \\times_Y V) \\times_Y X =", "X_1 \\times_Y V = X_1 \\times_X X \\times_Y V = X_1 \\times_X U", "$$", "is surjective \\'etale too. By", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-thickening-equivalence}", "there exists a thickening $U_1 \\to U_1'$ and a surjective \\'etale morphism", "$U_1' \\to X_1' \\times_{X'} U'$ whose base change to $X_1 \\times_X U$ is the", "displayed morphism. At this point $(V_1, U'_1, \\varphi_1)$ is an object of", "$(\\Sch/Y_1) \\times_{(\\Sch/Y_1')} (\\Sch/X_1')$. In the proof of", "Lemma \\ref{lemma-pushout-along-thickening} we constructed", "$G(V, U', \\varphi) = V \\amalg_U U'$ as a quotient of an \\'etale equivalence", "relation on $G_1(V_1, U_1', \\varphi_1) = V_1 \\amalg_{U_1} U_1'$", "such that we get a commutative diagram", "\\begin{equation}", "\\label{equation-cube-over}", "\\vcenter{", "\\xymatrix{", "& U \\ar[rr] \\ar'[d][dd] & & U' \\ar[dd] \\\\", "U_1 \\ar[rr] \\ar[dd] \\ar[ru] & & U_1' \\ar[dd] \\ar[ru] & \\\\", "& V \\ar'[r][rr] & & G(V, U', \\varphi) \\\\", "V_1 \\ar[rr] \\ar[ru] & & G_1(V_1, U_1', \\varphi_1) \\ar[ru]", "}", "}", "\\end{equation}", "where all squares except the front and back squares are cartesian", "(the front and back squares are pushouts) and the northeast arrows", "are surjective \\'etale. In particular", "$$", "G_1(V_1, U_1', \\varphi_1) \\to G(V, U', \\varphi)", "$$", "is surjective \\'etale.", "\\medskip\\noindent", "Finally, we come to the proof of the lemma. We have to show that the adjunction", "mapping $(V, U', \\varphi) \\to F(G(V, U', \\varphi))$ is an isomorphism. We know", "$(V_1, U_1', \\varphi_1) \\to F_1(G_1(V_1, U_1', \\varphi_1))$ is an isomorphism", "by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout}.", "Recall that $F$ and $F_1$ are given by base change.", "Using the properties of (\\ref{equation-cube-over})", "and Lemma \\ref{lemma-diagram}", "we see that", "$V \\to G(V, U', \\varphi) \\times_{Y'} Y$ and", "$U' \\to G(V, U', \\varphi) \\times_{Y'} X'$ are isomorphisms, i.e.,", "$(V, U', \\varphi) \\to F(G(V, U', \\varphi))$ is an isomorphism." ], "refs": [ "more-morphisms-lemma-equivalence-categories-schemes-over-pushout", "spaces-more-morphisms-lemma-thickening-equivalence", "more-morphisms-lemma-pushout-along-thickening", "spaces-pushouts-lemma-pushout-along-thickening", "more-morphisms-lemma-equivalence-categories-schemes-over-pushout", "spaces-pushouts-lemma-pushout-along-thickening-schemes", "spaces-more-morphisms-lemma-thickening-equivalence", "spaces-pushouts-lemma-pushout-along-thickening", "more-morphisms-lemma-equivalence-categories-schemes-over-pushout", "spaces-pushouts-lemma-diagram" ], "ref_ids": [ 13763, 50, 13762, 10859, 13763, 10858, 50, 10859, 13763, 10861 ] } ], "ref_ids": [ 10860 ] }, { "id": 10863, "type": "theorem", "label": "spaces-pushouts-lemma-space-over-pushout-flat-modules", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-space-over-pushout-flat-modules", "contents": [ "Let $S$ be a base scheme.", "Let $X \\to X'$ be a thickening of algebraic spaces over $S$", "and let $X \\to Y$ be an affine morphism of algebraic spaces over $S$.", "Let $Y' = Y \\amalg_X X'$ be the pushout", "(see Lemma \\ref{lemma-pushout-along-thickening}).", "Let $V' \\to Y'$ be a morphism of algebraic spaces over $S$. Set", "$V = Y \\times_{Y'} V'$, $U' = X' \\times_{Y'} V'$, and $U = X \\times_{Y'} V'$.", "There is an equivalence of categories between", "\\begin{enumerate}", "\\item quasi-coherent $\\mathcal{O}_{V'}$-modules flat over $Y'$, and", "\\item the category of triples $(\\mathcal{G}, \\mathcal{F}', \\varphi)$ where", "\\begin{enumerate}", "\\item $\\mathcal{G}$ is a quasi-coherent $\\mathcal{O}_V$-module flat over $Y$,", "\\item $\\mathcal{F}'$ is a quasi-coherent $\\mathcal{O}_{U'}$-module flat", "over $X$, and", "\\item $\\varphi : (U \\to V)^*\\mathcal{G} \\to (U \\to U')^*\\mathcal{F}'$", "is an isomorphism of $\\mathcal{O}_U$-modules.", "\\end{enumerate}", "\\end{enumerate}", "The equivalence maps $\\mathcal{G}'$ to", "$((V \\to V')^*\\mathcal{G}', (U' \\to V')^*\\mathcal{G}', can)$.", "Suppose $\\mathcal{G}'$ corresponds to the triple", "$(\\mathcal{G}, \\mathcal{F}', \\varphi)$. Then", "\\begin{enumerate}", "\\item[(a)] $\\mathcal{G}'$ is a finite type $\\mathcal{O}_{V'}$-module if and", "only if $\\mathcal{G}$ and $\\mathcal{F}'$ are finite type", "$\\mathcal{O}_Y$ and $\\mathcal{O}_{U'}$-modules.", "\\item[(b)] if $V' \\to Y'$ is locally of finite presentation, then", "$\\mathcal{G}'$ is an $\\mathcal{O}_{V'}$-module of finite", "presentation if and only if $\\mathcal{G}$ and $\\mathcal{F}'$ are", "$\\mathcal{O}_Y$ and $\\mathcal{O}_{U'}$-modules of finite presentation.", "\\end{enumerate}" ], "refs": [ "spaces-pushouts-lemma-pushout-along-thickening" ], "proofs": [ { "contents": [ "A quasi-inverse functor assigns to the triple", "$(\\mathcal{G}, \\mathcal{F}', \\varphi)$ the fibre product", "$$", "(V \\to V')_*\\mathcal{G}", "\\times_{(U \\to V')_*\\mathcal{F}}", "(U' \\to V')_*\\mathcal{F}'", "$$", "where $\\mathcal{F} = (U \\to U')^*\\mathcal{F}'$. This works, because on", "affines \\'etale over $V'$ and $Y'$ we recover the equivalence of", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-relative-flat-module-over-fibre-product}.", "Details omitted.", "\\medskip\\noindent", "Parts (a) and (b) reduce by \\'etale localization", "(Properties of Spaces, Section", "\\ref{spaces-properties-section-properties-modules})", "to the case where $V'$ and $Y'$ are affine in which case the result", "follows from", "More on Algebra, Lemmas", "\\ref{more-algebra-lemma-relative-finite-module-over-fibre-product} and", "\\ref{more-algebra-lemma-relative-finitely-presented-module-over-fibre-product}." ], "refs": [ "more-algebra-lemma-relative-flat-module-over-fibre-product", "more-algebra-lemma-relative-finite-module-over-fibre-product", "more-algebra-lemma-relative-finitely-presented-module-over-fibre-product" ], "ref_ids": [ 9829, 9828, 9830 ] } ], "ref_ids": [ 10859 ] }, { "id": 10864, "type": "theorem", "label": "spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat", "contents": [ "In the situation of", "Lemma \\ref{lemma-equivalence-categories-spaces-over-pushout}.", "If $V' = G(V, U', \\varphi)$ for some triple $(V, U', \\varphi)$, then", "\\begin{enumerate}", "\\item $V' \\to Y'$ is locally of finite type if and only if $V \\to Y$ and", "$U' \\to X'$ are locally of finite type,", "\\item $V' \\to Y'$ is flat if and only if $V \\to Y$ and $U' \\to X'$ are flat,", "\\item $V' \\to Y'$ is flat and locally of finite presentation if and only if", "$V \\to Y$ and $U' \\to X'$ are flat and locally of finite presentation,", "\\item $V' \\to Y'$ is smooth if and only if $V \\to Y$ and $U' \\to X'$ are smooth,", "\\item $V' \\to Y'$ is \\'etale if and only if $V \\to Y$ and $U' \\to X'$", "are \\'etale, and", "\\item add more here as needed.", "\\end{enumerate}", "If $W'$ is flat over $Y'$, then the adjunction mapping", "$G(F(W')) \\to W'$ is an isomorphism. Hence $F$ and $G$ define mutually", "quasi-inverse functors between the category of spaces flat over $Y'$", "and the category of triples $(V, U', \\varphi)$ with $V \\to Y$", "and $U' \\to X'$ flat." ], "refs": [ "spaces-pushouts-lemma-equivalence-categories-spaces-over-pushout" ], "proofs": [ { "contents": [ "Choose a diagram (\\ref{equation-cube}) as in the proof of", "Lemma \\ref{lemma-equivalence-categories-spaces-over-pushout}.", "\\medskip\\noindent", "Proof of (1) -- (5). Let $(V, U', \\varphi)$ be an object of", "$(\\textit{Spaces}/Y) \\times_{(\\textit{Spaces}/Y')} (\\textit{Spaces}/X')$.", "Construct a diagram (\\ref{equation-cube-over}) as in the proof of", "Lemma \\ref{lemma-equivalence-categories-spaces-over-pushout}.", "Then the base change of $G(V, U', \\varphi) \\to Y'$ to", "$Y'_1$ is $G_1(V_1, U_1', \\varphi_1) \\to Y_1'$. Hence (1) -- (5)", "follow immediately from the corresponding statements of", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat}", "for schemes.", "\\medskip\\noindent", "Suppose that $W' \\to Y'$ is flat. Choose a scheme $W'_1$ and a surjective", "\\'etale morphism $W'_1 \\to Y_1' \\times_{Y'} W'$. Observe that $W'_1 \\to W'$", "is surjective \\'etale as a composition of surjective \\'etale morphisms. We", "know that $G_1(F_1(W_1')) \\to W_1'$ is an isomorphism by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat}", "applied to $W'_1$ over $Y'_1$ and the front of the diagram (with functors", "$G_1$ and $F_1$ as in the proof of", "Lemma \\ref{lemma-equivalence-categories-spaces-over-pushout}).", "Then the construction of $G(F(W'))$ (as a pushout, i.e.,", "as constructed in Lemma \\ref{lemma-pushout-along-thickening}) shows that", "$G_1(F_1(W'_1)) \\to G(F(W))$ is surjective \\'etale. Whereupon we conclude", "that $G(F(W)) \\to W$ is \\'etale, see for example", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-local}.", "But $G(F(W)) \\to W$ is an isomorphism on underlying reduced", "algebraic spaces (by construction), hence it is an isomorphism." ], "refs": [ "spaces-pushouts-lemma-equivalence-categories-spaces-over-pushout", "spaces-pushouts-lemma-equivalence-categories-spaces-over-pushout", "more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat", "more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat", "spaces-pushouts-lemma-equivalence-categories-spaces-over-pushout", "spaces-pushouts-lemma-pushout-along-thickening", "spaces-properties-lemma-etale-local" ], "ref_ids": [ 10862, 10862, 13765, 13765, 10862, 10859, 11856 ] } ], "ref_ids": [ 10862 ] }, { "id": 10865, "type": "theorem", "label": "spaces-pushouts-lemma-pushout-along-closed-immersion-and-integral", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-pushout-along-closed-immersion-and-integral", "contents": [ "In More on Morphisms, Situation", "\\ref{more-morphisms-situation-pushout-along-closed-immersion-and-integral}", "let $Y \\amalg_Z X$ be the pushout in the category of schemes", "(More on Morphisms, Proposition", "\\ref{more-morphisms-proposition-pushout-along-closed-immersion-and-integral}).", "Then $Y \\amalg_Z X$", "is also a pushout in the category of algebraic spaces over $S$." ], "refs": [ "more-morphisms-proposition-pushout-along-closed-immersion-and-integral" ], "proofs": [ { "contents": [ "This is a consequence of Lemma \\ref{lemma-colimit-agrees}, the proposition", "mentioned in the lemma and More on Morphisms, Lemmas", "\\ref{more-morphisms-lemma-pushout-functor} and", "\\ref{more-morphisms-lemma-pushout-functor-equivalence-flat}.", "Conditions (1) and (2) of Lemma \\ref{lemma-colimit-agrees}", "follow immediately. To see (3) and (4) note that an \\'etale morphism", "is locally quasi-finite and use that the equivalence of categories of", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-pushout-functor-equivalence-flat}", "is constructed using the pushout construction of", "More on Morphisms, Lemmas \\ref{more-morphisms-lemma-pushout-functor}.", "Minor details omitted." ], "refs": [ "spaces-pushouts-lemma-colimit-agrees", "more-morphisms-lemma-pushout-functor", "more-morphisms-lemma-pushout-functor-equivalence-flat", "spaces-pushouts-lemma-colimit-agrees", "more-morphisms-lemma-pushout-functor-equivalence-flat", "more-morphisms-lemma-pushout-functor" ], "ref_ids": [ 10845, 14049, 14050, 10845, 14050, 14049 ] } ], "ref_ids": [ 14103 ] }, { "id": 10866, "type": "theorem", "label": "spaces-pushouts-lemma-pushout-along-thickening-derived", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-pushout-along-thickening-derived", "contents": [ "Let $S$ be a scheme. Consider a pushout", "$$", "\\xymatrix{", "X \\ar[r]_i \\ar[d]_f & X' \\ar[d]^{f'}", "\\\\", "Y \\ar[r]^j & Y'", "}", "$$", "in the category of algebraic spaces over $S$", "as in Lemma \\ref{lemma-pushout-along-thickening}.", "Assume $i$ is a thickening. Then the essential", "image of the functor\\footnote{All functors given by derived pullback.}", "$$", "D(\\mathcal{O}_{Y'}) \\longrightarrow", "D(\\mathcal{O}_Y) \\times_{D(\\mathcal{O}_X)} D(\\mathcal{O}_{X'})", "$$", "contains every triple $(M, K', \\alpha)$ where $M \\in D(\\mathcal{O}_Y)$", "and $K' \\in D(\\mathcal{O}_{X'})$ are pseudo-coherent." ], "refs": [ "spaces-pushouts-lemma-pushout-along-thickening" ], "proofs": [ { "contents": [ "Let $(M, K', \\alpha)$ be an object of the target of the functor", "of the lemma. Here $\\alpha : Lf^*M \\to Li^*K'$", "is an isomorphism which is adjoint to a map $\\beta : M \\to Rf_*Li^*K'$.", "Thus we obtain maps", "$$", "Rj_*M \\xrightarrow{Rj_*\\beta}", "Rj_*Rf_*Li^*K' = Rf'_*Ri_*Li^*K' \\leftarrow Rf'_*K'", "$$", "where the arrow pointing left comes from $K' \\to Ri_*Li^*K'$.", "Choose a distinguished triangle", "$$", "M' \\to Rj_*M \\oplus Rf'_*K' \\to Rj_*Rf_*Li^*K' \\to M'[1]", "$$", "in $D(\\mathcal{O}_{Y'})$. The first arrow defines canonical maps", "$Lj^*M' \\to M$ and $L(f')^*M' \\to K'$ compatible with $\\alpha$.", "Thus it suffices to show that the maps", "$Lj^*M' \\to M$ and $L(f')^*M' \\to K$ are isomorphisms.", "This we may check \\'etale locally on $Y'$, hence we may", "assume $Y'$ is \\'etale.", "\\medskip\\noindent", "Assume $Y'$ affine and $M \\in D(\\mathcal{O}_Y)$", "and $K' \\in D(\\mathcal{O}_{X'})$ are pseudo-coherent.", "Say our pushout corresponds to the fibre product", "$$", "\\xymatrix{", "B & B' \\ar[l] \\\\", "A \\ar[u] & A' \\ar[l] \\ar[u]", "}", "$$", "of rings where $B' \\to B$ is surjective with locally nilpotent kernel $I$", "(and hence $A' \\to A$ is surjective with locally nilpotent kernel $I$ as well).", "The assumption on $M$ and $K'$ imply that $M$ comes from a pseudo-coherent", "object of $D(A)$ and $K'$ comes from a pseudo-coherent object of $D(B')$, see", "Derived Categories of Spaces, Lemmas", "\\ref{spaces-perfect-lemma-pseudo-coherent},", "\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}, and", "\\ref{spaces-perfect-lemma-descend-pseudo-coherent}", "and", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded} and", "\\ref{perfect-lemma-pseudo-coherent-affine}.", "Moreover, pushforward and derived pullback agree with the", "corresponding operations on derived categories of modules, see", "Derived Categories of Spaces, Remark", "\\ref{spaces-perfect-remark-match-total-direct-images}", "and", "Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-quasi-coherence-pushforward} and", "\\ref{perfect-lemma-quasi-coherence-pullback}.", "This reduces us to the statement formulated in the next paragraph.", "(To be sure these references show", "the object $M'$ lies $D_\\QCoh(\\mathcal{O}_{Y'})$", "as this is a triangulated subcategory of $D(\\mathcal{O}_{Y'})$.)", "\\medskip\\noindent", "Given a diagram of rings as above and a triple", "$(M, K', \\alpha)$ where $M \\in D(A)$, $K' \\in D(B')$ are", "pseudo-coherent and", "$\\alpha : M \\otimes_A^\\mathbf{L} B \\to K' \\otimes_{B'}^\\mathbf{L} B$", "is an isomorphism suppose we have distinguished triangle", "$$", "M' \\to M \\oplus K' \\to K' \\otimes_{B'}^\\mathbf{L} B \\to M'[1]", "$$", "in $D(A')$. Goal: show that the induced maps", "$M' \\otimes_{A'}^\\mathbf{L} A \\to M$ and", "$M' \\otimes_{A'}^\\mathbf{L} B' \\to K'$ are isomorphisms.", "To do this, choose a bounded above complex", "$E^\\bullet$ of finite free $A$-modules representing $M$.", "Since $(B', I)$ is a henselian pair", "(More on Algebra, Lemma \\ref{more-algebra-lemma-locally-nilpotent-henselian})", "with $B = B'/I$ we may apply More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-complex-finite-projectives}", "to see that there exists a bounded above complex $P^\\bullet$", "of free $B'$-modules such that $\\alpha$ is represented", "by an isomorphism $E^\\bullet \\otimes_A B \\cong P^\\bullet \\otimes_{B'} B$.", "Then we can consider the short exact sequence", "$$", "0 \\to L^\\bullet \\to", "E^\\bullet \\oplus P^\\bullet \\to P^\\bullet \\otimes_{B'} B \\to 0", "$$", "of complexes of $B'$-modules.", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-finitely-presented-module-over-fibre-product}", "implies $L^\\bullet$ is a bounded above complex of", "finite projective $A'$-modules", "(in fact it is rather easy to show directly that $L^n$ is finite free", "in our case) and that we have", "$L^\\bullet \\otimes_{A'} A = E^\\bullet$ and", "$L^\\bullet \\otimes_{A'} B' = P^\\bullet$.", "The short exact sequence gives a distinguished triangle", "$$", "L^\\bullet \\to M \\oplus K' \\to K' \\otimes_{B'}^\\mathbf{L} B \\to (L^\\bullet)[1]", "$$", "in $D(A')$ (Derived Categories, Section", "\\ref{derived-section-canonical-delta-functor}) which is isomorphic", "to the given distinguished triangle by general properties of", "triangulated categories (Derived Categories, Section", "\\ref{derived-section-elementary-results}). In other words, $L^\\bullet$", "represents $M'$ compatibly with the given maps. Thus the maps", "$M' \\otimes_{A'}^\\mathbf{L} A \\to M$ and", "$M' \\otimes_{A'}^\\mathbf{L} B' \\to K'$ are", "isomorphisms because we just saw that the corresponding", "thing is true for $L^\\bullet$." ], "refs": [ "spaces-perfect-lemma-pseudo-coherent", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-descend-pseudo-coherent", "perfect-lemma-affine-compare-bounded", "perfect-lemma-pseudo-coherent-affine", "spaces-perfect-remark-match-total-direct-images", "perfect-lemma-quasi-coherence-pushforward", "perfect-lemma-quasi-coherence-pullback", "more-algebra-lemma-locally-nilpotent-henselian", "more-algebra-lemma-lift-complex-finite-projectives", "more-algebra-lemma-finitely-presented-module-over-fibre-product" ], "ref_ids": [ 2696, 2644, 2692, 6941, 6975, 2768, 6943, 6944, 9857, 10234, 9825 ] } ], "ref_ids": [ 10859 ] }, { "id": 10867, "type": "theorem", "label": "spaces-pushouts-lemma-elementary-distinguished-square-pushout", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-elementary-distinguished-square-pushout", "contents": [ "Let $S$ be a scheme. Let $(U \\subset W, f : V \\to W)$ be", "an elementary distinguished square. Then", "$$", "\\xymatrix{", "U \\times_W V \\ar[r] \\ar[d] &", "V \\ar[d]^f \\\\", "U \\ar[r] & W", "}", "$$", "is a pushout in the category of algebraic spaces over $S$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $U \\amalg V \\to W$ is a surjective \\'etale morphism.", "The fibre product", "$$", "(U \\amalg V) \\times_W (U \\amalg V)", "$$", "is the disjoint union of four pieces, namely", "$U = U \\times_W U$, $U \\times_W V$, $V \\times_W U$,", "and $V \\times_W V$.", "There is a surjective \\'etale morphism", "$$", "V \\amalg (U \\times_W V) \\times_U (U \\times_W V) \\longrightarrow V \\times_W V", "$$", "because $f$ induces an isomorphism over $W \\setminus U$", "(part of the definition of being an elementary distinguished square).", "Let $B$ be an algebraic space over $S$ and let", "$g : V \\to B$ and $h : U \\to B$ be morphisms over", "$S$ which agree after restricting to $U \\times_W V$.", "Then the description of", "$(U \\amalg V) \\times_W (U \\amalg V)$ given above", "shows that $h \\amalg g : U \\amalg V \\to B$", "equalizes the two projections. Since $B$ is a sheaf", "for the \\'etale topology we obtain a unique", "factorization of $h \\amalg g$ through $W$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10868, "type": "theorem", "label": "spaces-pushouts-lemma-construct-elementary-distinguished-square", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-construct-elementary-distinguished-square", "contents": [ "Let $S$ be a scheme. Let $V$, $U$ be algebraic spaces over $S$.", "Let $V' \\subset V$ be an open subspace and let $f' : V' \\to U$ be a", "separated \\'etale morphism of algebraic spaces over $S$.", "Then there exists a pushout", "$$", "\\xymatrix{", "V' \\ar[r] \\ar[d] &", "V \\ar[d]^f \\\\", "U \\ar[r] & W", "}", "$$", "in the category of algebraic spaces over $S$ and moreover", "$(U \\subset W, f : V \\to W)$ is an elementary distinguished square." ], "refs": [], "proofs": [ { "contents": [ "We are going to construct $W$ as the quotient of an \\'etale", "equivalence relation $R$ on $U \\amalg V$. Such a quotient is an", "algebraic space for example by", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}.", "Moreover, the proof of", "Lemma \\ref{lemma-elementary-distinguished-square-pushout} tells us to take", "$$", "R = U \\amalg V' \\amalg V' \\amalg V \\amalg", "(V' \\times_U V' \\setminus \\Delta_{V'/U}(V'))", "$$", "Since we assumed $V' \\to U$ is separated, the image of", "$\\Delta_{V'/U}$ is closed and hence the complement is an", "open subspace. The morphism $j : R \\to (U \\amalg V) \\times_S (U \\amalg V)$", "is given by", "$$", "u,\\ v',\\ v',\\ v,\\ (v'_1, v'_2) \\mapsto", "(u, u),\\ (f'(v'), v'),\\ (v', f'(v')),\\ (v, v),\\ (v'_1, v'_2)", "$$", "with obvious notation. It is immediately verified that this is a", "monomorphism, an equivalence relation, and that the induced morphisms", "$s, t : R \\to U \\amalg V$ are \\'etale. Let", "$W = (U \\amalg V)/R$ be the quotient algebraic space.", "We obtain a commutative diagram as in the statement of the lemma.", "To finish the proof it suffices to show that this diagram is", "an elementary distinguished square, since then", "Lemma \\ref{lemma-elementary-distinguished-square-pushout}", "implies that it is a pushout.", "Thus we have to show that $U \\to W$ is open and that", "$f$ is \\'etale and is an isomorphism over $W \\setminus U$.", "This follows from the choice of $R$; we omit the details." ], "refs": [ "bootstrap-theorem-final-bootstrap", "spaces-pushouts-lemma-elementary-distinguished-square-pushout", "spaces-pushouts-lemma-elementary-distinguished-square-pushout" ], "ref_ids": [ 2602, 10867, 10867 ] } ], "ref_ids": [] }, { "id": 10869, "type": "theorem", "label": "spaces-pushouts-lemma-stalk-pushforward-with-support", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-stalk-pushforward-with-support", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces", "over $S$. Let $Z \\subset X$ closed subspace such that $f^{-1}Z \\to Z$ is", "integral and universally injective. Let $\\overline{y}$ be a geometric point", "of $Y$ and $\\overline{x} = f(\\overline{y})$. We have", "$$", "(Rf_*Q)_{\\overline{x}} = Q_{\\overline{y}}", "$$", "in $D(\\textit{Ab})$ for any object $Q$ of $D(Y_\\etale)$ supported", "on $|f^{-1}Z|$." ], "refs": [], "proofs": [ { "contents": [ "Consider the commutative diagram of algebraic spaces", "$$", "\\xymatrix{", "f^{-1}Z \\ar[r]_{i'} \\ar[d]_{f'} & Y \\ar[d]_f \\\\", "Z \\ar[r]^i & X", "}", "$$", "By Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-complexes-with-support-on-closed} we can write", "$Q = Ri'_*K'$ for some object $K'$ of $D(f^{-1}Z_\\etale)$.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-integral-universally-injective-push-pull}", "we have $K' = (f')^{-1}K$ with $K = Rf'_*K'$.", "Then we have $Rf_*Q = Rf_*Ri'_*K' = Ri_*Rf'_*K' = Ri_*K$.", "Let $\\overline{z}$ be the geometric point of $Z$ corresponding", "to $\\overline{x}$ and let $\\overline{z}'$ be the geometric point", "of $f^{-1}Z$ corresponding to $\\overline{y}$. We obtain", "the result of the lemma as follows", "$$", "Q_{\\overline{y}} = (Ri'_*K')_{\\overline{y}} = K'_{\\overline{z}'} =", "(f')^{-1}K_{\\overline{z}'} = K_{\\overline{z}} = Ri_*K_{\\overline{x}} =", "Rf_*Q_{\\overline{x}}", "$$", "The middle equality holds because of the description of the stalk", "of a pullback given in", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-stalk-pullback}." ], "refs": [ "spaces-cohomology-lemma-complexes-with-support-on-closed", "spaces-morphisms-lemma-integral-universally-injective-push-pull", "spaces-properties-lemma-stalk-pullback" ], "ref_ids": [ 11293, 4980, 11875 ] } ], "ref_ids": [] }, { "id": 10870, "type": "theorem", "label": "spaces-pushouts-lemma-stalk-formal-glueing", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-stalk-formal-glueing", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces", "over $S$. Let $Z \\subset X$ closed subspace such that $f^{-1}Z \\to Z$ is", "integral and universally injective. Let $\\overline{y}$ be a geometric point", "of $Y$ and $\\overline{x} = f(\\overline{y})$. Let $\\mathcal{G}$", "be an abelian sheaf on $Y$. Then the map of two term complexes", "$$", "\\left(f_*\\mathcal{G}_{\\overline{x}} \\to", "(f \\circ j')_*(\\mathcal{G}|_V)_{\\overline{x}}\\right)", "\\longrightarrow", "\\left(\\mathcal{G}_{\\overline{y}} \\to j'_*(\\mathcal{G}|_V)_{\\overline{y}}\\right)", "$$", "induces an isomorphism on kernels and an injection on cokernels.", "Here $V = Y \\setminus f^{-1}Z$ and $j' : V \\to Y$ is the inclusion." ], "refs": [], "proofs": [ { "contents": [ "Choose a distinguished triangle", "$$", "\\mathcal{G} \\to Rj'_*\\mathcal{G}|_V \\to Q \\to \\mathcal{G}[1]", "$$", "n $D(Y_\\etale)$. The cohomology sheaves of $Q$", "are supported on $|f^{-1}Z|$. We apply $Rf_*$ and we obtain", "$$", "Rf_*\\mathcal{G} \\to Rf_*Rj'_*\\mathcal{G}|_V \\to Rf_*Q", "\\to Rf_*\\mathcal{G}[1]", "$$", "Taking stalks at $\\overline{x}$ we obtain an exact sequence", "$$", "0 \\to", "(R^{-1}f_*Q)_{\\overline{x}} \\to", "f_*\\mathcal{G}_{\\overline{x}} \\to", "(f \\circ j')_*(\\mathcal{G}|_V)_{\\overline{x}} \\to", "(R^0f_*Q)_{\\overline{x}}", "$$", "We can compare this with the exact sequence", "$$", "0 \\to", "H^{-1}(Q)_{\\overline{y}} \\to", "\\mathcal{G}_{\\overline{y}} \\to", "j'_*(\\mathcal{G}|_V)_{\\overline{y}} \\to", "H^0(Q)_{\\overline{y}}", "$$", "Thus we see that the lemma follows because", "$Q_{\\overline{y}} = Rf_*Q_{\\overline{x}}$ by", "Lemma \\ref{lemma-stalk-pushforward-with-support}." ], "refs": [ "spaces-pushouts-lemma-stalk-pushforward-with-support" ], "ref_ids": [ 10869 ] } ], "ref_ids": [] }, { "id": 10871, "type": "theorem", "label": "spaces-pushouts-lemma-stalk-of-pushforward", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-stalk-of-pushforward", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism.", "Let $\\overline{x}$ be a geometric point of $X$ and let", "$\\Spec(\\mathcal{O}_{X, \\overline{x}}) \\to X$", "be the canonical morphism. For a quasi-coherent module", "$\\mathcal{G}$ on $Y$ we have", "$$", "f_*\\mathcal{G}_{\\overline{x}} =", "\\Gamma(Y \\times_X \\Spec(\\mathcal{O}_{X, \\overline{x}}), p^*\\mathcal{F})", "$$", "where $p : Y \\times_X \\Spec(\\mathcal{O}_{X, \\overline{x}}) \\to Y$", "is the projection." ], "refs": [], "proofs": [ { "contents": [ "Observe that $f_*\\mathcal{G}_{\\overline{x}} =", "\\Gamma(\\Spec(\\mathcal{O}_{X, \\overline{x}}), h^*f_*\\mathcal{G})$", "where $h : \\Spec(\\mathcal{O}_{X, \\overline{x}}) \\to X$.", "Hence the result is true because $h$ is flat so that", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-flat-base-change-cohomology}", "applies." ], "refs": [ "spaces-cohomology-lemma-flat-base-change-cohomology" ], "ref_ids": [ 11296 ] } ], "ref_ids": [] }, { "id": 10872, "type": "theorem", "label": "spaces-pushouts-lemma-stalk-of-module-with-support", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-stalk-of-module-with-support", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $i : Z \\to X$ be a closed immersion of finite presentation.", "Let $Q \\in D_\\QCoh(\\mathcal{O}_X)$ be supported on $|Z|$.", "Let $\\overline{x}$ be a geometric point of $X$ and let", "$I_{\\overline{x}} \\subset \\mathcal{O}_{X, \\overline{x}}$ be the stalk of", "the ideal sheaf of $Z$. Then the cohomology modules", "$H^n(Q_{\\overline{x}})$ are $I_{\\overline{x}}$-power torsion", "(see More on Algebra, Definition", "\\ref{more-algebra-definition-f-power-torsion})." ], "refs": [ "more-algebra-definition-f-power-torsion" ], "proofs": [ { "contents": [ "Choose an affine scheme $U$ and an \\'etale morphism $U \\to X$ such", "that $\\overline{x}$ lifts to a geometric point $\\overline{u}$", "of $U$. Then we can replace $X$ by $U$, $Z$ by $U \\times_X Z$,", "$Q$ by the restriction $Q|_U$, and $\\overline{x}$ by $\\overline{u}$.", "Thus we may assume that $X = \\Spec(A)$ is affine. Let $I \\subset A$", "be the ideal defining $Z$. Since $i : Z \\to X$ is of finite presentation,", "the ideal $I = (f_1, \\ldots, f_r)$ is finitely generated.", "The object $Q$ comes from a complex of $A$-modules $M^\\bullet$, see", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}", "and", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded}.", "Since the cohomology sheaves of $Q$ are supported on $Z$", "we see that the localization $M^\\bullet_f$ is acyclic for each $f \\in I$.", "Take $x \\in H^p(M^\\bullet)$. By the above we can find $n_i$ such", "that $f_i^{n_i} x = 0$ in $H^p(M^\\bullet)$ for each $i$.", "Then with $n = \\sum n_i$ we see that $I^n$ annihilates $x$.", "Thus $H^p(M^\\bullet)$ is $I$-power torsion. Since the ring", "map $A \\to \\mathcal{O}_{X, \\overline{x}}$ is flat and since", "$I_{\\overline{x}} = I\\mathcal{O}_{X, \\overline{x}}$ we conclude." ], "refs": [ "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "perfect-lemma-affine-compare-bounded" ], "ref_ids": [ 2644, 6941 ] } ], "ref_ids": [ 10633 ] }, { "id": 10873, "type": "theorem", "label": "spaces-pushouts-lemma-formal-glueing-on-closed", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-formal-glueing-on-closed", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces", "over $S$. Let $Z \\subset X$ be a closed subspace. Assume $f^{-1}Z \\to Z$", "is an isomorphism and that $f$ is flat in every point of $f^{-1}Z$. For any", "$Q$ in $D_\\QCoh(\\mathcal{O}_Y)$ supported on $|f^{-1}Z|$ we have", "$Lf^*Rf_*Q = Q$." ], "refs": [], "proofs": [ { "contents": [ "We show the canonical map $Lf^*Rf_*Q \\to Q$ is an isomorphism", "by checking on stalks at $\\overline{y}$. If $\\overline{y}$ is not", "in $f^{-1}Z$, then both sides are zero and the result is true.", "Assume the image $\\overline{x}$ of $\\overline{y}$ is in $Z$.", "By Lemma \\ref{lemma-stalk-pushforward-with-support} we have", "$Rf_*Q_{\\overline{x}} = Q_{\\overline{y}}$ and since $f$ is flat", "at $\\overline{y}$ we see that", "$$", "(Lf^*Rf_*Q)_{\\overline{y}} =", "(Rf_*Q)_{\\overline{x}}", "\\otimes_{\\mathcal{O}_{X, \\overline{x}}}", "\\mathcal{O}_{Y, \\overline{y}} =", "Q_{\\overline{y}} \\otimes_{\\mathcal{O}_{X, \\overline{x}}}", "\\mathcal{O}_{Y, \\overline{y}}", "$$", "Thus we have to check that the canonical map", "$$", "Q_{\\overline{y}} \\otimes_{\\mathcal{O}_{X, \\overline{x}}}", "\\mathcal{O}_{Y, \\overline{y}}", "\\longrightarrow Q_{\\overline{y}}", "$$", "is an isomorphism in the derived category. Let", "$I_{\\overline{x}} \\subset \\mathcal{O}_{X, \\overline{x}}$ be the", "stalk of the ideal sheaf defining $Z$. Since $Z \\to X$ is locally of", "finite presentation this ideal is finitely generated and the", "cohomology groups of $Q_{\\overline{y}}$", "are $I_{\\overline{y}} = I_{\\overline{x}}\\mathcal{O}_{Y, \\overline{y}}$-power", "torsion by Lemma \\ref{lemma-stalk-of-module-with-support} applied to $Q$ on $Y$.", "It follows that they are also $I_{\\overline{x}}$-power torsion.", "The ring map", "$\\mathcal{O}_{X, \\overline{x}} \\to \\mathcal{O}_{Y, \\overline{y}}$", "is flat and induces an isomorphism after dividing by", "$I_{\\overline{x}}$ and $I_{\\overline{y}}$ because we assumed", "that $f^{-1}Z \\to Z$ is an isomorphism. Hence we see that", "the cohomology modules of", "$Q_{\\overline{y}} \\otimes_{\\mathcal{O}_{X, \\overline{x}}}", "\\mathcal{O}_{Y, \\overline{y}}$", "are equal to the cohomology modules of $Q_{\\overline{y}}$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-neighbourhood-isomorphism}", "which finishes the proof." ], "refs": [ "spaces-pushouts-lemma-stalk-pushforward-with-support", "spaces-pushouts-lemma-stalk-of-module-with-support", "more-algebra-lemma-neighbourhood-isomorphism" ], "ref_ids": [ 10869, 10872, 10340 ] } ], "ref_ids": [] }, { "id": 10874, "type": "theorem", "label": "spaces-pushouts-lemma-adjoint", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-adjoint", "contents": [ "In Situation \\ref{situation-formal-glueing}.", "The functor (\\ref{equation-reverse}) is right adjoint to", "the functor (\\ref{equation-formal-glueing-modules})." ], "refs": [], "proofs": [ { "contents": [ "This follows easily from the adjointness of $f^*$ to $f_*$", "and $j^*$ to $j_*$. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10875, "type": "theorem", "label": "spaces-pushouts-lemma-reverse-commutes-with-flat-base-change", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-reverse-commutes-with-flat-base-change", "contents": [ "In Situation \\ref{situation-formal-glueing}.", "Let $X' \\to X$ be a flat morphism of algebraic spaces.", "Set $Z' = X' \\times_X Z$ and $Y' = X' \\times_X Y$.", "The pullbacks $\\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_{X'})$", "and $\\QCoh(Y \\to X, Z) \\to \\QCoh(Y' \\to X', Z')$ are compatible", "with the functors (\\ref{equation-reverse}) and", "\\ref{equation-formal-glueing-modules})." ], "refs": [], "proofs": [ { "contents": [ "This is true because pullback commutes with pullback and because", "flat pullback commutes with pushforward along quasi-compact", "and quasi-separated morphisms, see", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-flat-base-change-cohomology}." ], "refs": [ "spaces-cohomology-lemma-flat-base-change-cohomology" ], "ref_ids": [ 11296 ] } ], "ref_ids": [] }, { "id": 10876, "type": "theorem", "label": "spaces-pushouts-lemma-derived-equivalent", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-derived-equivalent", "contents": [ "In Situation \\ref{situation-formal-glueing} the functor", "$Rf_*$ induces an equivalence between $D_{\\QCoh, |f^{-1}Z|}(\\mathcal{O}_Y)$", "and $D_{\\QCoh, |Z|}(\\mathcal{O}_X)$ with quasi-inverse given by", "$Lf^*$." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is quasi-compact and quasi-separated we see that $Rf_*$", "defines a functor from $D_{\\QCoh, |f^{-1}Z|}(\\mathcal{O}_Y)$", "to $D_{\\QCoh, |Z|}(\\mathcal{O}_X)$, see", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-quasi-coherence-direct-image}.", "By Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-quasi-coherence-pullback}", "we see that $Lf^*$ maps $D_{\\QCoh, |Z|}(\\mathcal{O}_X)$", "into $D_{\\QCoh, |f^{-1}Z|}(\\mathcal{O}_Y)$.", "In Lemma \\ref{lemma-formal-glueing-on-closed} we have seen that", "$Lf^*Rf_*Q = Q$ for $Q$ in $D_{\\QCoh, |f^{-1}Z|}(\\mathcal{O}_Y)$.", "By the dual of Derived Categories, Lemma", "\\ref{derived-lemma-fully-faithful-adjoint-kernel-zero}", "to finish the proof it suffices to show that $Lf^*K = 0$", "implies $K = 0$ for $K$ in $D_{\\QCoh, |Z|}(\\mathcal{O}_X)$.", "This follows from the fact that $f$ is flat at all points of", "$f^{-1}Z$ and the fact that $f^{-1}Z \\to Z$ is surjective." ], "refs": [ "spaces-perfect-lemma-quasi-coherence-direct-image", "spaces-perfect-lemma-quasi-coherence-pullback", "spaces-pushouts-lemma-formal-glueing-on-closed", "derived-lemma-fully-faithful-adjoint-kernel-zero" ], "ref_ids": [ 2652, 2648, 10873, 1793 ] } ], "ref_ids": [] }, { "id": 10877, "type": "theorem", "label": "spaces-pushouts-lemma-dominate-by-fpqc-covering", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-dominate-by-fpqc-covering", "contents": [ "In Situation \\ref{situation-formal-glueing} there exists an", "fpqc covering $\\{X_i \\to X\\}_{i \\in I}$ refining the", "family $\\{U \\to X, Y \\to X\\}$." ], "refs": [], "proofs": [ { "contents": [ "For the definition and general properties of fpqc coverings we refer to", "Topologies, Section \\ref{topologies-section-fpqc}. In particular, we can", "first choose an \\'etale covering $\\{X_i \\to X\\}$ with $X_i$ affine and by", "base changing $Y$, $Z$, and $U$ to each $X_i$ we reduce to the case where", "$X$ is affine. In this case $U$ is quasi-compact and hence a finite union", "$U = U_1 \\cup \\ldots \\cup U_n$ of affine opens. ", "Then $Z$ is quasi-compact hence also $f^{-1}Z$ is quasi-compact.", "Thus we can choose an affine scheme $W$ and an \\'etale morphism", "$h : W \\to Y$ such that $h^{-1}f^{-1}Z \\to f^{-1}Z$ is surjective.", "Say $W = \\Spec(B)$ and $h^{-1}f^{-1}Z = V(J)$ where $J \\subset B$", "is an ideal of finite type.", "By Pro-\\'etale Cohomology, Lemma \\ref{proetale-lemma-localization}", "there exists a localization $B \\to B'$ such that points of", "$\\Spec(B')$ correspond exactly to points of $W = \\Spec(B)$", "specializing to $h^{-1}f^{-1}Z = V(J)$. It follows that the", "composition $\\Spec(B') \\to \\Spec(B) = W \\to Y \\to X$ is flat", "as by assumption $f : Y \\to X$ is flat at all the points of $f^{-1}Z$. Then", "$\\{\\Spec(B') \\to X, U_1 \\to X, \\ldots, U_n \\to X\\}$", "is an fpqc covering by", "Topologies, Lemma \\ref{topologies-lemma-recognize-fpqc-covering}." ], "refs": [ "proetale-lemma-localization", "topologies-lemma-recognize-fpqc-covering" ], "ref_ids": [ 3712, 12493 ] } ], "ref_ids": [] }, { "id": 10878, "type": "theorem", "label": "spaces-pushouts-lemma-equivalence-on-affine", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-equivalence-on-affine", "contents": [ "In Situation \\ref{situation-formal-glueing} the functor", "(\\ref{equation-formal-glueing-spaces}) restricts to an", "equivalence", "\\begin{enumerate}", "\\item from the category of algebraic spaces affine over $X$", "to the full subcategory of $\\textit{Spaces}(Y \\to X, Z)$ consisting", "of $(U' \\leftarrow V' \\rightarrow Y')$ with $U' \\to U$, $V' \\to V$,", "and $Y' \\to Y$ affine,", "\\item from the category of closed immersions $X' \\to X$", "to the full subcategory of $\\textit{Spaces}(Y \\to X, Z)$ consisting", "of $(U' \\leftarrow V' \\rightarrow Y')$ with $U' \\to U$, $V' \\to V$,", "and $Y' \\to Y$ closed immersions, and", "\\item same statement as in (2) for finite morphisms.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The category of algebraic spaces affine over $X$ is equivalent to the", "category of quasi-coherent sheaves $\\mathcal{A}$ of $\\mathcal{O}_X$-algebras.", "The full subcategory of $\\textit{Spaces}(Y \\to X, Z)$ consisting of", "$(U' \\leftarrow V' \\rightarrow Y')$ with $U' \\to U$, $V' \\to V$,", "and $Y' \\to Y$ affine is equivalent to the category of", "algebra objects of $\\QCoh(Y \\to X, Z)$. In both cases this follows", "from Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-affine-equivalence-algebras}", "with quasi-inverse given by the relative spectrum construction", "(Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-relative-spec})", "which commutes with arbitrary base change. Thus part (1) of the", "lemma follows from Proposition \\ref{proposition-formal-glueing-modules}.", "\\medskip\\noindent", "Fully faithfulness in part (2) follows from part (1). For essential", "surjectivity, we reduce by part (1) to proving that $X' \\to X$", "is a closed immersion if and only if both $U \\times_X X' \\to U$ and", "$Y \\times_X X' \\to Y$ are closed immersions. By", "Lemma \\ref{lemma-dominate-by-fpqc-covering}", "$\\{U \\to X, Y \\to X\\}$ can be refined by an fpqc covering.", "Hence the result follows from", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-closed-immersion}.", "\\medskip\\noindent", "For (3) use the argument proving (2) and", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-finite}." ], "refs": [ "spaces-morphisms-lemma-affine-equivalence-algebras", "spaces-morphisms-definition-relative-spec", "spaces-pushouts-proposition-formal-glueing-modules", "spaces-pushouts-lemma-dominate-by-fpqc-covering", "spaces-descent-lemma-descending-property-closed-immersion", "spaces-descent-lemma-descending-property-finite" ], "ref_ids": [ 4802, 4999, 10898, 10877, 9397, 9403 ] } ], "ref_ids": [] }, { "id": 10879, "type": "theorem", "label": "spaces-pushouts-lemma-reflects-isomorphisms", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-reflects-isomorphisms", "contents": [ "In Situation \\ref{situation-formal-glueing} the functor", "(\\ref{equation-formal-glueing-spaces}) reflects isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "By a formal argument with base change, this reduces to the following", "question: A morphism $a : X' \\to X$ of algebraic spaces such that", "$U \\times_X X' \\to U$ and $Y \\times_X X' \\to Y$ are isomorphisms, is", "an isomorphism. The family $\\{U \\to X, Y \\to X\\}$ can be refined by", "an fpqc covering by Lemma \\ref{lemma-dominate-by-fpqc-covering}.", "Hence the result follows from", "Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-isomorphism}." ], "refs": [ "spaces-pushouts-lemma-dominate-by-fpqc-covering", "spaces-descent-lemma-descending-property-isomorphism" ], "ref_ids": [ 10877, 9395 ] } ], "ref_ids": [] }, { "id": 10880, "type": "theorem", "label": "spaces-pushouts-lemma-fully-faithful-on-separated", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-fully-faithful-on-separated", "contents": [ "In Situation \\ref{situation-formal-glueing} the functor", "(\\ref{equation-formal-glueing-spaces}) is fully faithful", "on algebraic spaces separated over $X$. More precisely, it induces", "a bijection", "$$", "\\Mor_X(X'_1, X'_2)", "\\longrightarrow", "\\Mor_{\\textit{Spaces}(Y \\to X, Z)}(F(X'_1), F(X'_2))", "$$", "whenever $X'_2 \\to X$ is separated." ], "refs": [], "proofs": [ { "contents": [ "Since $X'_2 \\to X$ is separated, the graph $i : X'_1 \\to X'_1 \\times_X X'_2$", "of a morphism $X'_1 \\to X'_2$ over $X$ is a closed immersion, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-semi-diagonal}.", "Moreover a closed immersion $i : T \\to X'_1 \\times_X X'_2$ is the graph of a", "morphism if and only if $\\text{pr}_1 \\circ i$ is an isomorphism.", "The same is true for", "\\begin{enumerate}", "\\item the graph of a morphism $U \\times_X X'_1 \\to U \\times_X X'_2$ over $U$,", "\\item the graph of a morphism $V \\times_X X'_1 \\to V \\times_X X'_2$ over $V$,", "and", "\\item the graph of a morphism $Y \\times_X X'_1 \\to Y \\times_X X'_2$ over $Y$.", "\\end{enumerate}", "Moreover, if morphisms as in (1), (2), (3) fit together to form a", "morphism in the category $\\textit{Spaces}(Y \\to X, Z)$, then these", "graphs fit together to give an object of", "$\\textit{Spaces}(Y \\times_X (X'_1 \\times_X X'_2) \\to X'_1 \\times_X X'_2,", "Z \\times_X (X'_1 \\times_X X'_2))$", "whose triple of morphisms are closed immersions. The proof is finished", "by applying Lemmas \\ref{lemma-equivalence-on-affine} and", "\\ref{lemma-reflects-isomorphisms}." ], "refs": [ "spaces-morphisms-lemma-semi-diagonal", "spaces-pushouts-lemma-equivalence-on-affine", "spaces-pushouts-lemma-reflects-isomorphisms" ], "ref_ids": [ 4716, 10878, 10879 ] } ], "ref_ids": [] }, { "id": 10881, "type": "theorem", "label": "spaces-pushouts-lemma-glueable", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-glueable", "contents": [ "Let $(R \\to R', f)$ be a glueing pair, see above. Let $Y$ be an algebraic", "space over $X$. The following are equivalent", "\\begin{enumerate}", "\\item there exists an \\'etale covering $\\{Y_i \\to Y\\}_{i \\in I}$", "with $Y_i$ affine and $\\Gamma(Y_i, \\mathcal{O}_{Y_i})$", "glueable as an $R$-module,", "\\item for every \\'etale morphism $W \\to Y$ with $W$ affine", "$\\Gamma(W, \\mathcal{O}_W)$ is a glueable $R$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is immediate that (2) implies (1). Assume $\\{Y_i \\to Y\\}$", "is as in (1) and let $W \\to Y$ be as in (2). Then", "$\\{Y_i \\times_Y W \\to W\\}_{i \\in I}$ is an \\'etale covering,", "which we may refine by an \\'etale covering", "$\\{W_j \\to W\\}_{j = 1, \\ldots, m}$ with $W_j$ affine", "(Topologies, Lemma \\ref{topologies-lemma-etale-affine}).", "Thus to finish the proof it suffices to show", "the following three algebraic statements:", "\\begin{enumerate}", "\\item if $R \\to A \\to B$ are ring maps with $A \\to B$ \\'etale", "and $A$ glueable as an $R$-module, then $B$ is glueable as an", "$R$-module,", "\\item finite products of glueable $R$-modules are glueable,", "\\item if $R \\to A \\to B$ are ring maps with $A \\to B$ faithfully \\'etale", "and $B$ glueable as an $R$-module, then $A$ is glueable as an", "$R$-module.", "\\end{enumerate}", "Namely, the first of these will imply that $\\Gamma(W_j, \\mathcal{O}_{W_j})$", "is a glueable $R$-module, the second will imply that", "$\\prod \\Gamma(W_j, \\mathcal{O}_{W_j})$ is a glueable $R$-module, and", "the third will imply that $\\Gamma(W, \\mathcal{O}_W)$ is a glueable", "$R$-module.", "\\medskip\\noindent", "Consider an \\'etale $R$-algebra homomorphism $A \\to B$. Set", "$A' = A \\otimes_R R'$ and $B' = B \\otimes_R R' = A' \\otimes_A B$.", "Statements (1) and (3) then follow from the following facts:", "(a) $A$, resp.\\ $B$ is glueable if and only if the sequence", "$$", "0 \\to A \\to A_f \\oplus A' \\to A'_f \\to 0,", "\\quad\\text{resp.}\\quad", "0 \\to B \\to B_f \\oplus B' \\to B'_f \\to 0,", "$$", "is exact, (b) the second sequence is equal to", "the functor $- \\otimes_A B$ applied to the first and", "(c) (faithful) flatness of $A \\to B$. We omit the proof of (2)." ], "refs": [ "topologies-lemma-etale-affine" ], "ref_ids": [ 12447 ] } ], "ref_ids": [] }, { "id": 10882, "type": "theorem", "label": "spaces-pushouts-lemma-glueing-affines", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-glueing-affines", "contents": [ "Let $(R \\to R', f)$ be a glueing pair, see above.", "The functor (\\ref{equation-beauville-laszlo-glueing-spaces})", "restricts to an equivalence between the category of affine", "$Y/X$ which are glueable for $(R \\to R', f)$ and the", "full subcategory of objects $(V, V', Y')$ of", "$\\textit{Spaces}(U \\leftarrow U' \\to X')$", "with $V$, $V'$, $Y'$ affine." ], "refs": [], "proofs": [ { "contents": [ "Let $(V, V', Y')$ be an object of", "$\\textit{Spaces}(U \\leftarrow U' \\to X')$", "with $V$, $V'$, $Y'$ affine.", "Write $V = \\Spec(A_1)$ and $Y' = \\Spec(A')$. By our definition of the", "category $\\textit{Spaces}(U \\leftarrow U' \\to X')$ we find that", "$V'$ is the spectrum of $A_1 \\otimes_{R_f} R'_f = A_1 \\otimes_R R'$", "and the spectrum of $A'_f$. Hence we get an isomorphism", "$\\varphi : A'_f \\to A_1 \\otimes_R R'$ of $R'_f$-algebras.", "By More on Algebra, Theorem \\ref{more-algebra-theorem-BL-glueing}", "there exists a unique glueable $R$-module $A$ and isomorphisms", "$A_f \\to A_1$ and $A \\otimes_R R' \\to A'$ of modules compatible with", "$\\varphi$. Since the sequence", "$$", "0 \\to A \\to A_1 \\oplus A' \\to A'_f \\to 0", "$$", "is short exact, the multiplications on $A_1$ and $A'$ define", "a unique $R$-algebra structure on $A$ such that the maps $A \\to A_1$", "and $A \\to A'$ are ring homomorphisms. We omit the verification", "that this construction defines a quasi-inverse to the functor", "(\\ref{equation-beauville-laszlo-glueing-spaces})", "restricted to the subcategories mentioned in the statement of the lemma." ], "refs": [ "more-algebra-theorem-BL-glueing" ], "ref_ids": [ 9804 ] } ], "ref_ids": [] }, { "id": 10883, "type": "theorem", "label": "spaces-pushouts-lemma-glueing-affines-etale", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-glueing-affines-etale", "contents": [ "Let $P$ be one of the following properties of morphisms:", "``finite'', ``closed immersion'', ``flat'', ``finite type'',", "``flat and finite presentation'', ``\\'etale''.", "Under the equivalence of Lemma \\ref{lemma-glueing-affines}", "the morphisms having $P$ correspond to morphisms of triples", "whose components have $P$." ], "refs": [ "spaces-pushouts-lemma-glueing-affines" ], "proofs": [ { "contents": [ "Let $P'$ be one of the following properties of homomorphisms of rings:", "``finite'', ``surjective'', ``flat'', ``finite type'',", "``flat and of finite presentation'', ``\\'etale''.", "Translated into algebra, the statement means the following:", "If $A \\to B$ is an $R$-algebra homomorphism and $A$ and $B$", "are glueable for $(R \\to R', f)$, then", "$A_f \\to B_f$ and $A \\otimes_R R' \\to B \\otimes_R R'$ have $P'$", "if and only if $A \\to B$ has $P'$.", "\\medskip\\noindent", "By More on Algebra, Lemmas \\ref{more-algebra-lemma-faithful-descent}", "and \\ref{more-algebra-lemma-BL-flat} the algebraic statement", "is true for $P'$ equal to ``finite'' or ``flat''.", "\\medskip\\noindent", "If $A_f \\to B_f$ and $A \\otimes_R R' \\to B \\otimes_R R'$ are surjective,", "then $N = B/A$ is an $R$-module with $N_f = 0$ and $N \\otimes_R R' = 0$ and", "hence vanishes by More on Algebra, Lemma", "\\ref{more-algebra-lemma-BL-faithful}. Thus $A \\to B$ is surjective.", "\\medskip\\noindent", "If $A_f \\to B_f$ and $A \\otimes_R R' \\to B \\otimes_R R'$ are finite type,", "then we can choose an $A$-algebra homomorphism $A[x_1, \\ldots, x_n] \\to B$", "such that $A_f[x_1, \\ldots, x_n] \\to B_f$ and", "$(A \\otimes_R R')[x_1, \\ldots, x_n] \\to B \\otimes_R R'$ are surjective", "(small detail omitted). We conclude that $A[x_1, \\ldots, x_n] \\to B$", "is surjective by the previous result. Thus $A \\to B$ is of finite type.", "\\medskip\\noindent", "If $A_f \\to B_f$ and $A \\otimes_R R' \\to B \\otimes_R R'$ are", "flat and of finite presentation, then we know that $A \\to B$ is flat and", "of finite type by what we have already shown. Choose a surjection", "$A[x_1, \\ldots, x_n] \\to B$ and denote $I$ the kernel.", "By flatness of $B$ over $A$ we see that $I_f$ is the kernel of", "$A_f[x_1, \\ldots, x_n] \\to B_f$ and $I \\otimes_R R'$ is the kernel of", "$A \\otimes_R R'[x_1, \\ldots, x_n] \\to B \\otimes_R R'$.", "Thus $I_f$ is a finite $A_f[x_1, \\ldots, x_n]$-module and", "$I \\otimes_R R'$ is a finite $(A \\otimes_R R')[x_1, \\ldots, x_n]$-module.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-faithful-descent}", "applied to $I$ viewed as a module over $A[x_1, \\ldots, x_n]$", "we conclude that $I$ is a finitely generated ideal and we conclude", "$A \\to B$ is flat and of finite presentation.", "\\medskip\\noindent", "If $A_f \\to B_f$ and $A \\otimes_R R' \\to B \\otimes_R R'$ are \\'etale,", "then we know that $A \\to B$ is flat and of finite presentation by what", "we have already shown. Since the fibres of $\\Spec(B) \\to \\Spec(A)$", "are isomorphic to fibres of $\\Spec(B_f) \\to \\Spec(A_f)$ or", "$\\Spec(B/fB) \\to \\Spec(A/fA)$, we conclude that $A \\to B$ is unramified,", "see Morphisms, Lemmas \\ref{morphisms-lemma-unramified-over-field}", "and \\ref{morphisms-lemma-unramified-etale-fibres}.", "We conclude that $A \\to B$ is \\'etale by", "Morphisms, Lemma \\ref{morphisms-lemma-flat-unramified-etale} for example." ], "refs": [ "more-algebra-lemma-faithful-descent", "more-algebra-lemma-BL-flat", "more-algebra-lemma-BL-faithful", "more-algebra-lemma-faithful-descent", "morphisms-lemma-unramified-over-field", "morphisms-lemma-unramified-etale-fibres", "morphisms-lemma-flat-unramified-etale" ], "ref_ids": [ 10357, 10363, 10355, 10357, 5352, 5353, 5373 ] } ], "ref_ids": [ 10882 ] }, { "id": 10884, "type": "theorem", "label": "spaces-pushouts-lemma-glueing-f", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-glueing-f", "contents": [ "Let $(R \\to R', f)$ be a glueing pair, see above.", "The functor (\\ref{equation-beauville-laszlo-glueing-spaces})", "is faithful on the full subcategory of", "algebraic spaces $Y/X$ glueable for $(R \\to R', f)$." ], "refs": [], "proofs": [ { "contents": [ "Let $f, g : Y \\to Z$ be two morphisms of algebraic spaces over $X$", "with $Y$ and $Z$ glueable for $(R \\to R', f)$ such that $f$ and $g$ are mapped", "to the same morphism in the category $\\textit{Spaces}(U \\leftarrow U' \\to X')$.", "We have to show the equalizer $E \\to Y$ of $f$ and $g$ is an isomorphism.", "Working \\'etale locally on $Y$ we may assume $Y$ is an affine scheme.", "Then $E$ is a scheme and the morphism $E \\to Y$ is a monomorphism", "and locally quasi-finite, see Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-properties-diagonal}.", "Moreover, the base change of $E \\to Y$ to $U$ and to $X'$ is", "an isomorphism. As $Y$ is the disjoint union of the affine open", "$V = U \\times_X Y$ and the affine closed $V(f) \\times_X Y$, we conclude", "$E$ is the disjoint union of their isomorphic inverse images.", "It follows in particular that $E$ is quasi-compact.", "By Zariski's main theorem (More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite})", "we conclude that $E$ is quasi-affine.", "Set $B = \\Gamma(E, \\mathcal{O}_E)$ and $A = \\Gamma(Y, \\mathcal{O}_Y)$", "so that we have an $R$-algebra homomorphism $A \\to B$.", "Since $E \\to Y$ becomes an isomorphism after base change to $U$ and $X'$", "we obtain ring maps $B \\to A_f$ and $B \\to A \\otimes_R R'$", "agreeing as maps into $A \\otimes_R R'_f$. Since $A$ is glueable", "for $(R \\to R', f)$ we get a ring map $B \\to A$ which is left inverse", "to the map $A \\to B$. The corresponding morphism $Y = \\Spec(A) \\to \\Spec(B)$", "maps into the open subscheme $E \\subset \\Spec(B)$ pointwise because", "this is true after base change to $U$ and $X'$. Hence we get a morphism", "$Y \\to E$ over $Y$. Since $E \\to Y$ is a monomorhism we conclude", "$Y \\to E$ is an isomorphism as desired." ], "refs": [ "spaces-morphisms-lemma-properties-diagonal", "more-morphisms-lemma-quasi-finite-separated-pass-through-finite" ], "ref_ids": [ 4712, 13901 ] } ], "ref_ids": [] }, { "id": 10885, "type": "theorem", "label": "spaces-pushouts-lemma-glueing-ff", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-glueing-ff", "contents": [ "Let $(R \\to R', f)$ be a glueing pair, see above.", "The functor (\\ref{equation-beauville-laszlo-glueing-spaces})", "is fully faithful on the full subcategory of", "algebraic spaces $Y/X$ which are (a) glueable for $(R \\to R', f)$", "and (b) have affine diagonal $Y \\to Y \\times_X Y$." ], "refs": [], "proofs": [ { "contents": [ "Let $Y, Z$ be two algebraic spaces over $X$ which are both glueable for", "$(R \\to R', f)$ and assume the diagonal of $Z$ is affine. Let", "$a : U \\times_X Y \\to U \\times_X Z$ over $U$ and", "$b : X' \\times_X Y \\to X' \\times_X Z$ over $X'$ be two morphisms", "of algebraic spaces", "which induce the same morphism $c : U' \\times_X Y \\to U' \\times_X Z$", "over $U'$.", "We want to construct a morphism $f : Y \\to Z$ over $X$", "which produces the morphisms $a$, $b$ on base change to $U$, $X'$.", "By the faithfulness of Lemma \\ref{lemma-glueing-f}, it suffices to construct", "the morphism $f$ \\'etale locally on $Y$ (details omitted).", "Thus we may and do assume $Y$ is affine.", "\\medskip\\noindent", "Let $y \\in |Y|$ be a point. If $y$ maps into the open $U \\subset X$,", "then $U \\times_X Y$ is an open of $Y$ on which the morphism $f$", "is defined (we can just take $a$).", "Thus we may assume $y$ maps into the closed subset", "$V(f)$ of $X$. Since $R/fR = R'/fR'$ there is a unique point", "$y' \\in |X' \\times_X Y|$ mapping to $y$. Denote", "$z' = b(y') \\in |X' \\times_X Z|$ and $z \\in |Z|$ the images of $y'$.", "Choose an \\'etale neighbourhood $(W, w) \\to (Z, z)$", "with $W$ affine. Observe that", "$$", "(U \\times_X W) \\times_{U \\times_X Z, a} (U \\times_X Y),\\quad", "(U' \\times_X W) \\times_{U' \\times_X Z, c} (U' \\times_X Y),", "$$", "and", "$$", "(X' \\times_X W) \\times_{X' \\times_X Z, b} (X' \\times_X Y)", "$$", "form an object of $\\textit{Spaces}(U \\leftarrow U' \\to X')$", "with affine parts (this is where we use that $Z$ has affine diagonal).", "Hence by Lemma \\ref{lemma-glueing-affines}", "there exists a unique affine scheme $V$ glueable for $(R \\to R', f)$ such that", "$$", "(U \\times_X V, U' \\times_X V, X' \\times_X V)", "$$", "is the triple displayed above. By fully faithfulness for the affine", "case (Lemma \\ref{lemma-glueing-affines}) we get a unique morphisms", "$V \\to W$ and $V \\to Y$ agreeing with the first and second projection", "morphisms over $U$ and $X'$ in the construction above.", "By Lemma \\ref{lemma-glueing-affines-etale} the morphism $V \\to Y$ is \\'etale.", "To finish the proof, it suffices to show that there is a point $v \\in |V|$", "mapping to $y$ (because then $f$ is defined on an \\'etale neighbourhood", "of $y$, namely $V$).", "There is a unique point $w' \\in |X' \\times_X W|$ mapping to $w$.", "By uniqueness $w'$ is mapped to $z'$ under the map", "$|X' \\times_X W| \\to |X' \\times_X Z|$. Then we consider the cartesian", "diagram", "$$", "\\xymatrix{", "X' \\times_X V \\ar[r] \\ar[d] & X' \\times_X W \\ar[d] \\\\", "X' \\times_X Y \\ar[r] & X' \\times_X Z", "}", "$$", "to see that there is a point $v' \\in |X' \\times_X V|$", "mapping to $y'$ and $w'$, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}.", "Of course the image $v$ of $v'$ in $|V|$ maps to $y$ and the proof is complete." ], "refs": [ "spaces-pushouts-lemma-glueing-f", "spaces-pushouts-lemma-glueing-affines", "spaces-pushouts-lemma-glueing-affines", "spaces-pushouts-lemma-glueing-affines-etale", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 10884, 10882, 10882, 10883, 11819 ] } ], "ref_ids": [] }, { "id": 10886, "type": "theorem", "label": "spaces-pushouts-lemma-glueing-quasi-affines", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-glueing-quasi-affines", "contents": [ "Let $(R \\to R', f)$ be a glueing pair, see above. Any object", "$(V, V', Y')$ of $\\textit{Spaces}(U \\leftarrow U' \\to X')$", "with $V$, $V'$, $Y'$ quasi-affine is isomorphic to", "the image under the functor (\\ref{equation-beauville-laszlo-glueing-spaces})", "of a separated algebraic space $Y$ over $X$." ], "refs": [], "proofs": [ { "contents": [ "Choose $n'$, $T' \\to Y'$ and $n_1$, $T_1 \\to V$ as in", "Properties, Lemma \\ref{properties-lemma-quasi-affine-presentation}.", "Picture", "$$", "\\xymatrix{", "& &", "T_1 \\times_V V' \\times_Y T' \\ar[ld] \\ar[rd] \\\\", "T_1 \\ar[d] &", "T_1 \\times_V V' \\ar[l] \\ar[dr] & &", "V' \\times_{Y'} T' \\ar[r] \\ar[dl] &", "T' \\ar[d] \\\\", "V & &", "V' \\ar[rr] \\ar[ll] & &", "Y'", "}", "$$", "Observe that $T_1 \\times_V V'$ and $V' \\times_{Y'} T'$", "are affine (namely the morphisms $V' \\to V$ and $V' \\to Y'$", "are affine as base changes of the affine morphisms $U' \\to U$", "and $U' \\to X'$). By construction we see that", "$$", "\\mathbf{A}^{n'}_{T_1 \\times_V V'} \\cong", "T_1 \\times_V V' \\times_{Y'} T' \\cong", "\\mathbf{A}^{n_1}_{V' \\times_{Y'} T'}", "$$", "In other words, the affine schemes $\\mathbf{A}^{n'}_{T_1}$", "and $\\mathbf{A}^{n_1}_{T'}$ are part of a triple making an affine object of", "$\\textit{Spaces}(U \\leftarrow U' \\to X')$.", "By Lemma \\ref{lemma-glueing-affines}", "there exists a morphism of affine schemes $T \\to X$", "and isomorphisms $U \\times_X T \\cong \\mathbf{A}^{n'}_{T_1}$", "and $X' \\times_X T \\cong \\mathbf{A}^{n_1}_{T'}$ compatible", "with the isomorphisms displayed above.", "These isomorphisms produce morphisms", "$$", "U \\times_X T \\longrightarrow V", "\\quad\\text{and}\\quad", "X' \\times_X T \\longrightarrow Y'", "$$", "satisfying the property of", "Properties, Lemma \\ref{properties-lemma-quasi-affine-presentation}", "with $n = n' + n_1$ and moreover define a morphism from the triple", "$(U \\times_X T, U' \\times_X T, X' \\times_X T)$ to", "our triple $(V, V', Y')$ in the category", "$\\textit{Spaces}(U \\leftarrow U' \\to X')$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-glueing-affines} there is an affine scheme $W$ whose", "image in $\\textit{Spaces}(U \\leftarrow U' \\to X')$ is isomorphic to", "the triple", "$$", "((U \\times_X T) \\times_V (U \\times_X T),", "(U' \\times_X T) \\times_{V'} (U' \\times_X T),", "(X' \\times_X T) \\times_{Y'} (X' \\times_X T))", "$$", "By fully faithfulness of this construction, we obtain", "two maps $p_0, p_1 : W \\to T$ whose base changes", "to $U, U', X'$ are the projection morphisms.", "By Lemma \\ref{lemma-glueing-affines-etale}", "the morphisms $p_0, p_1$ are flat and of finite presentation and", "the morphism $(p_0, p_1) : W \\to T \\times_X T$ is a closed immersion.", "In fact, $W \\to T \\times_X T$ is an equivalence relation: by the lemmas", "used above we may check symmetry, reflexivity, and transitivity", "after base change to $U$ and $X'$, where these are obvious (details omitted).", "Thus the quotient sheaf", "$$", "Y = T/W", "$$", "is an algebraic space for example by", "Bootstrap, Theorem \\ref{bootstrap-theorem-final-bootstrap}.", "Since it is clear that $Y/X$ is sent to the triple $(V, V', Y')$.", "The base change of the diagonal $\\Delta : Y \\to Y \\times_X Y$", "by the quasi-compact surjective flat morphism $T \\times_X T \\to Y \\times_X Y$", "is the closed immersion $W \\to T \\times_X T$. Thus $\\Delta$", "is a closed immersion by Descent on Spaces, Lemma", "\\ref{spaces-descent-lemma-descending-property-closed-immersion}.", "Thus the algebraic space $Y$ is separated and the proof is complete." ], "refs": [ "properties-lemma-quasi-affine-presentation", "spaces-pushouts-lemma-glueing-affines", "properties-lemma-quasi-affine-presentation", "spaces-pushouts-lemma-glueing-affines", "spaces-pushouts-lemma-glueing-affines-etale", "bootstrap-theorem-final-bootstrap", "spaces-descent-lemma-descending-property-closed-immersion" ], "ref_ids": [ 3011, 10882, 3011, 10882, 10883, 2602, 9397 ] } ], "ref_ids": [] }, { "id": 10887, "type": "theorem", "label": "spaces-pushouts-lemma-coequalizer", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-coequalizer", "contents": [ "Let $S$ be a scheme. Let", "$$", "g : Y \\longrightarrow X", "$$", "be a morphism of algebraic spaces over $S$. Assume $X$ is locally Noetherian,", "and $g$ is proper. Let $R = Y \\times_X Y$ with projection morphisms", "$t, s : R \\to Y$. There exists a coequalizer $X'$ of $s, t : R \\to Y$", "in the category of algebraic spaces over $S$. Moreover", "\\begin{enumerate}", "\\item The morphism $X' \\to X$ is finite.", "\\item The morphism $Y \\to X'$ is proper.", "\\item The morphism $Y \\to X'$ is surjective.", "\\item The morphism $X' \\to X$ is universally injective.", "\\item If $g$ is surjective, the morphism $X' \\to X$", "is a universal homeomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $h : R \\to X$ denote the composition of either $s$ or $t$", "with $g$. Then $h$ is proper by Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-proper} and", "\\ref{spaces-morphisms-lemma-composition-proper}.", "The sheaves", "$$", "g_*\\mathcal{O}_Y", "\\quad\\text{and}\\quad", "h_*\\mathcal{O}_R", "$$", "are coherent $\\mathcal{O}_X$-algebras by Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-pushforward-coherent}.", "The $X$-morphisms $s$, $t$ induce $\\mathcal{O}_X$-algebra maps", "$s^\\sharp, t^\\sharp$ from the first to the second.", "Set", "$$", "\\mathcal{A} = \\text{Equalizer}\\left(s^\\sharp, t^\\sharp :", "g_*\\mathcal{O}_Y \\longrightarrow h_*\\mathcal{O}_R\\right)", "$$", "Then $\\mathcal{A}$ is a coherent $\\mathcal{O}_X$-algebra and we", "can define", "$$", "X' = \\underline{\\Spec}_X(\\mathcal{A})", "$$", "as in Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-relative-spec}.", "By Morphisms of Spaces, Remark", "\\ref{spaces-morphisms-remark-factorization-quasi-compact-quasi-separated}", "and functoriality of the $\\underline{\\Spec}$ construction", "there is a factorization", "$$", "Y \\longrightarrow X' \\longrightarrow X", "$$", "and the morphism $g' : Y \\to X'$ equalizes $s$ and $t$.", "\\medskip\\noindent", "Before we show that $X'$ is the coequalizer of $s$ and $t$, we show", "that $Y \\to X'$ and $X' \\to X$ have the desired properties. Since $\\mathcal{A}$", "is a coherent $\\mathcal{O}_X$-module it is clear that $X' \\to X$ is a", "finite morphism of algebraic spaces. This proves (1).", "The morphism $Y \\to X'$ is proper by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-permanence}. This proves (2).", "Denote $Y \\to Y' \\to X$ with", "$Y' = \\underline{\\Spec}_X(g_*\\mathcal{O}_Y)$ the Stein factorization of $g$,", "see More on Morphisms of Spaces, Theorem", "\\ref{spaces-more-morphisms-theorem-stein-factorization-Noetherian}.", "Of course we obtain morphisms $Y \\to Y' \\to X' \\to X$ fitting", "with the morphisms studied above.", "Since $\\mathcal{O}_{X'} \\subset g_*\\mathcal{O}_Y$ is a finite extension", "we see that $Y' \\to X'$ is finite and surjective.", "Some details omitted; hint: use", "Algebra, Lemma \\ref{algebra-lemma-integral-overring-surjective}", "and reduce to the affine case by \\'etale localization.", "Since $Y \\to Y'$ is surjective (with geometrically connected fibres)", "we conclude that $Y \\to X'$ is surjective. This proves (3).", "To show that $X' \\to X$ is universally injective, we have to show", "that $X' \\to X' \\times_X X'$ is surjective, see", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-universally-injective} and", "Lemma \\ref{spaces-morphisms-lemma-universally-injective}.", "Since $Y \\to X'$ is surjective (see above) and since base changes", "and compositions of surjective morphisms are surjective by", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-surjective} and", "\\ref{spaces-morphisms-lemma-composition-surjective}", "we see that $Y \\times_X Y \\to X' \\times_X X'$ is surjective.", "However, since $Y \\to X'$ equalizes $s$ and $t$, we see that", "$Y \\times_X Y \\to X' \\times_X X'$ factors through $X' \\to X' \\times_X X'$", "and we conclude this latter map is surjective. This proves (4).", "Finally, if $g$ is surjective, then since $g$ factors through $X' \\to X$", "we see that $X' \\to X$ is surjective. Since a surjective, universally", "injective, finite morphism is a universal homeomorphism (because it", "is universally bijective and universally closed), this proves (5).", "\\medskip\\noindent", "In the rest of the proof we show that $Y \\to X'$ is the coequalizer", "of $s$ and $t$ in the category of algebraic spaces over $S$.", "Observe that $X'$ is locally Noetherian", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}).", "Moreover, observe that $Y \\times_{X'} Y \\to Y \\times_X Y$ is an", "isomorphism as $Y \\to X'$ equalizes $s$ and $t$ (this is a categorical", "statement). Hence in order to prove the statement that $Y \\to X'$", "is the coequalizer of $s$ and $t$, we may and do assume $X = X'$.", "In other words, $\\mathcal{O}_X$ is the equalizer of the maps", "$s^\\sharp, t^\\sharp : g_*\\mathcal{O}_Y \\to h_*\\mathcal{O}_R$.", "\\medskip\\noindent", "Let $X_1 \\to X$ be a flat morphism of algebraic spaces over $S$ with $X_1$", "locally Noetherian. Denote $g_1 : Y_1 \\to X_1$, $h_1 : R_1 \\to X_1$ and", "$s_1, t_1 : R_1 \\to Y_1$ the base changes of $g, h, s, t$ to $X_1$.", "Of course $g_1$ is proper and $R_1 = Y_1 \\times_{X_1} Y_1$.", "Since we have flat base change for pushforward of quasi-coherent modules,", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-flat-base-change-cohomology}, we see that", "$\\mathcal{O}_{X_1}$ is the equalizer of the maps", "$s_1^\\sharp, t_1^\\sharp : g_{1, *}\\mathcal{O}_{Y_1} \\to", "h_{1, *}\\mathcal{O}_{R_1}$. Hence all the assumptions we have", "are preserved by this base change.", "\\medskip\\noindent", "At this point we are going to check conditions (1) and (2) of", "Lemma \\ref{lemma-colimit-check-etale-locally}. Condition (1)", "follows from Lemma \\ref{lemma-descend-etale-proper-surjective}", "and the fact that $g$ is proper and surjective (because $X = X'$).", "To check condition (2), by the remarks on base change above,", "we reduce to the statement discussed and proved in the next paragraph.", "\\medskip\\noindent", "Assume $S = \\Spec(A)$ is an affine scheme, $X = X'$ is an affine scheme, and", "$Z$ is an affine scheme over $S$. We have to show that", "$$", "\\Mor_S(X, Z) \\longrightarrow", "\\text{Equalizer}(s, t : \\Mor_S(Y, Z) \\to \\Mor_S(R, Z))", "$$", "is bijective. However, this is clear from the fact that $X = X'$", "which implies $\\mathcal{O}_X$ is the equalizer of the maps", "$s^\\sharp, t^\\sharp : g_*\\mathcal{O}_Y \\to h_*\\mathcal{O}_R$", "which in turn implies", "$$", "\\Gamma(X, \\mathcal{O}_X) =", "\\text{Equalizer}\\left(", "s^\\sharp, t^\\sharp : \\Gamma(Y, \\mathcal{O}_Y) \\to", "\\Gamma(R, \\mathcal{O}_R)", "\\right)", "$$", "Namely, we have", "$$", "\\Mor_S(X, Z) = \\Hom_A(\\Gamma(Z, \\mathcal{O}_Z), \\Gamma(X, \\mathcal{O}_X))", "$$", "and similarly for $Y$ and $R$, see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-morphism-to-affine-scheme}." ], "refs": [ "spaces-morphisms-lemma-base-change-proper", "spaces-morphisms-lemma-composition-proper", "spaces-cohomology-lemma-proper-pushforward-coherent", "spaces-morphisms-definition-relative-spec", "spaces-morphisms-remark-factorization-quasi-compact-quasi-separated", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-more-morphisms-theorem-stein-factorization-Noetherian", "algebra-lemma-integral-overring-surjective", "spaces-morphisms-definition-universally-injective", "spaces-morphisms-lemma-universally-injective", "spaces-morphisms-lemma-base-change-surjective", "spaces-morphisms-lemma-composition-surjective", "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "spaces-cohomology-lemma-flat-base-change-cohomology", "spaces-pushouts-lemma-colimit-check-etale-locally", "spaces-pushouts-lemma-descend-etale-proper-surjective", "spaces-properties-lemma-morphism-to-affine-scheme" ], "ref_ids": [ 4917, 4918, 11331, 4999, 5032, 4920, 13, 495, 4997, 4793, 4727, 4726, 4817, 11296, 10847, 10856, 11915 ] } ], "ref_ids": [] }, { "id": 10888, "type": "theorem", "label": "spaces-pushouts-lemma-coequalizer-glue", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-coequalizer-glue", "contents": [ "In Situation \\ref{situation-coequalizer-glue} let $Y = X' \\amalg Z$ and", "$R = Y \\times_X Y$ with projections $t, s : R \\to Y$. There exists a", "coequalizer $X_1$ of $s, t : R \\to Y$ in the category of algebraic spaces", "over $S$. The morphism $X_1 \\to X$ is a finite universal homeomorphism,", "an isomorphism over $U$, and $Z \\to X$ lifts to $X_1$." ], "refs": [], "proofs": [ { "contents": [ "Existence of $X_1$ and the fact that $X_1 \\to X$ is a finite", "universal homeomorphism is a special case of Lemma \\ref{lemma-coequalizer}.", "The formation of $X_1$ commutes with \\'etale localization on $X$", "(see proof of Lemma \\ref{lemma-coequalizer}).", "Thus the morphism $X_1 \\to X$ is an isomorphism over $U$.", "It is immediate from the construction that $Z \\to X$ lifts to $X_1$." ], "refs": [ "spaces-pushouts-lemma-coequalizer", "spaces-pushouts-lemma-coequalizer" ], "ref_ids": [ 10887, 10887 ] } ], "ref_ids": [] }, { "id": 10889, "type": "theorem", "label": "spaces-pushouts-lemma-essentially-constant", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-essentially-constant", "contents": [ "In Situation \\ref{situation-coequalizer-glue} assume $X$ quasi-compact.", "In (\\ref{equation-system-coequalizers}) for all $n$ large enough, there", "exists an $m$ such that $X_n \\to X_{n + m}$ factors through a", "closed immersion $X \\to X_{n + m}$." ], "refs": [], "proofs": [ { "contents": [ "Let's look a bit more closely at the construction of $X_n$", "and how it changes as we increase $n$. We have", "$X_n = \\underline{\\Spec}(\\mathcal{A}_n)$", "where $\\mathcal{A}_n$ is the equalizer of $s_n^\\sharp$ and $t_n^\\sharp$", "going from $g_{n , *}\\mathcal{O}_{Y_n}$ to $h_{n, *}\\mathcal{O}_{R_n}$.", "Here $g_n : Y_n = X' \\amalg Z_n \\to X$ and $h_n : R_n = Y_n \\times_X Y_n \\to X$", "are the given morphisms. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the", "coherent sheaf of ideals corresponding to $Z$. Then", "$$", "g_{n, *}\\mathcal{O}_{Y_n} =", "f_*\\mathcal{O}_{X'} \\times \\mathcal{O}_X/\\mathcal{I}^n", "$$", "Similarly, we have a decomposition", "$$", "R_n = X' \\times_X X' \\amalg X' \\times_X Z_n \\amalg", "Z_n \\times_X X' \\amalg Z_n \\times_X Z_n", "$$", "As $Z_n \\to X$ is a monomorphism, we see that", "$X' \\times_X Z_n = Z_n \\times_X X'$ and that this identification", "is compatible with the two morphisms to $X$, with the two morphisms to", "$X'$, and with the two morphisms to $Z_n$.", "Denote $f_n : X' \\times_X Z_n \\to X$ the morphism to $X$.", "Denote", "$$", "\\mathcal{A} = \\text{Equalizer}(", "\\xymatrix{", "f_*\\mathcal{O}_{X'} \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "(f \\times f)_*\\mathcal{O}_{X' \\times_X X'}", "}", ")", "$$", "By the remarks above we find that", "$$", "\\mathcal{A}_n =", "\\text{Equalizer}(", "\\xymatrix{", "\\mathcal{A} \\times \\mathcal{O}_X/\\mathcal{I}^n \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "f_{n, *}\\mathcal{O}_{X' \\times_X Z_n}", "}", ")", "$$", "We have canonical maps", "$$", "\\mathcal{O}_X \\to \\ldots \\to \\mathcal{A}_3 \\to \\mathcal{A}_2 \\to \\mathcal{A}_1", "$$", "of coherent $\\mathcal{O}_X$-algebras. The statement of the lemma means that", "for $n$ large enough there exists an $m \\geq 0$ such that the image of", "$\\mathcal{A}_{n + m} \\to \\mathcal{A}_n$ is isomorphic to $\\mathcal{O}_X$.", "This we may check \\'etale locally on $X$. Hence by Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-quasi-compact-affine-cover}", "we may assume $X$ is an affine Noetherian scheme.", "\\medskip\\noindent", "Since $X_n \\to X$ is an isomorphism over $U$ we see that the kernel", "of $\\mathcal{O}_X \\to \\mathcal{A}_n$ is supported on $|Z|$.", "Since $X$ is Noetherian, the sequence of kernels", "$\\mathcal{J}_n = \\Ker(\\mathcal{O}_X \\to \\mathcal{A}_n)$ stabilizes", "(Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-acc-coherent}).", "Say $\\mathcal{J}_{n_0} = \\mathcal{J}_{n_0 + 1} = \\ldots = \\mathcal{J}$.", "By Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-power-ideal-kills-sheaf}", "we find that $\\mathcal{I}^t \\mathcal{J} = 0$ for some $t \\geq 0$.", "On the other hand, there is an $\\mathcal{O}_X$-algebra map", "$\\mathcal{A}_n \\to \\mathcal{O}_X/\\mathcal{I}^n$", "and hence $\\mathcal{J} \\subset \\mathcal{I}^n$ for all $n$.", "By Artin-Rees (Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-Artin-Rees}) we find that", "$\\mathcal{J} \\cap \\mathcal{I}^n \\subset \\mathcal{I}^{n - c}\\mathcal{J}$", "for some $c \\geq 0$ and all $n \\gg 0$. We conclude that $\\mathcal{J} = 0$.", "\\medskip\\noindent", "Pick $n \\geq n_0$ as in the previous paragraph. Then", "$\\mathcal{O}_X \\to \\mathcal{A}_n$ is injective. Hence it now", "suffices to find $m \\geq 0$ such that the image of", "$\\mathcal{A}_{n + m} \\to \\mathcal{A}_n$ is equal", "to the image of $\\mathcal{O}_X$. Observe that $\\mathcal{A}_n$", "sits in a short exact sequence", "$$", "0 \\to \\Ker(\\mathcal{A} \\to f_{n, *}\\mathcal{O}_{X' \\times_X Z_n})", "\\to \\mathcal{A}_n \\to \\mathcal{O}_X/\\mathcal{I}^n \\to 0", "$$", "and similarly for $\\mathcal{A}_{n + m}$. Hence it suffices to show", "$$", "\\Ker(\\mathcal{A} \\to f_{n + m, *}\\mathcal{O}_{X' \\times_X Z_{n + m}})", "\\subset", "\\Im(\\mathcal{I}^n \\to \\mathcal{A})", "$$", "for some $m \\geq 0$. To do this we may work \\'etale locally on", "$X$ and since $X$ is Noetherian we may assume that $X$ is", "a Noetherian affine scheme. Say $X = \\Spec(R)$ and $\\mathcal{I}$", "corresponds to the ideal $I \\subset R$. Let $\\mathcal{A} = \\widetilde{A}$", "for a finite $R$-algebra $A$. Let $f_*\\mathcal{O}_{X'} = \\widetilde{B}$", "for a finite $R$-algebra $B$. Then $R \\to A \\subset B$ and these maps", "become isomorphisms on inverting any element of $I$.", "\\medskip\\noindent", "Note that $f_{n, *}\\mathcal{O}_{X' \\times_X Z_n}$", "is equal to $f_*(\\mathcal{O}_{X'}/I^n\\mathcal{O}_{X'})$", "in the notation used in Cohomology of Spaces, Section", "\\ref{spaces-cohomology-section-theorem-formal-functions}.", "By Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-ML-cohomology-powers-ideal}", "we see that there exists a $c \\geq 0$ such that", "$$", "\\Ker(B \\to \\Gamma(X, f_*(\\mathcal{O}_{X'}/I^{n + m + c}\\mathcal{O}_{X'}))", "$$", "is contained in $I^{n + m}B$. On the other hand, as $R \\to B$ is", "finite and an isomorphism after inverting any element of $I$", "we see that $I^{n + m}B \\subset \\Im(I^n \\to B)$ for $m$ large enough", "(can be chosen independent of $n$). This finishes the proof as $A \\subset B$." ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-cohomology-lemma-acc-coherent", "spaces-cohomology-lemma-power-ideal-kills-sheaf", "spaces-cohomology-lemma-Artin-Rees", "spaces-cohomology-lemma-ML-cohomology-powers-ideal" ], "ref_ids": [ 11832, 11305, 11306, 11307, 11338 ] } ], "ref_ids": [] }, { "id": 10890, "type": "theorem", "label": "spaces-pushouts-lemma-check-separated", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-check-separated", "contents": [ "Let $S$ be a scheme. Let $X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $(U \\subset X, f : V \\to X)$ is an elementary distinguished square", "such that $U \\to Y$ and $V \\to Y$ are separated and", "$U \\times_X V \\to U \\times_Y V$ is closed, then $X \\to Y$ is separated." ], "refs": [], "proofs": [ { "contents": [ "We have to check that $\\Delta : X \\to X \\times_Y X$ is a closed immersion.", "There is an \\'etale covering of $X \\times_Y X$ given by the four parts", "$U \\times_Y U$, $U \\times_Y V$, $V \\times_Y U$, and $V \\times_Y V$.", "Observe that", "$(U \\times_Y U) \\times_{(X \\times_Y X), \\Delta} X = U$,", "$(U \\times_Y V) \\times_{(X \\times_Y X), \\Delta} X = U \\times_X V$,", "$(V \\times_Y U) \\times_{(X \\times_Y X), \\Delta} X = V \\times_X U$, and", "$(V \\times_Y V) \\times_{(X \\times_Y X), \\Delta} X = V$.", "Thus the assumptions of the lemma", "exactly tell us that $\\Delta$ is a closed immersion." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10891, "type": "theorem", "label": "spaces-pushouts-lemma-separate-disjoint-locally-closed-by-blowing-up", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-separate-disjoint-locally-closed-by-blowing-up", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $U \\subset X$ be a quasi-compact open.", "\\begin{enumerate}", "\\item If $Z_1, Z_2 \\subset X$ are closed subspaces of finite", "presentation such that $Z_1 \\cap Z_2 \\cap U = \\emptyset$, then", "there exists a $U$-admissible blowing up $X' \\to X$", "such that the strict transforms of $Z_1$ and $Z_2$ are disjoint.", "\\item If $T_1, T_2 \\subset |U|$ are disjoint constructible closed", "subsets, then there is a $U$-admissible blowing up $X' \\to X$", "such that the closures of $T_1$ and $T_2$ are disjoint.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). The assumption that $Z_i \\to X$ is of finite presentation", "signifies that the quasi-coherent ideal sheaf $\\mathcal{I}_i$ of $Z_i$", "is of finite type, see ", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-closed-immersion-finite-presentation}.", "Denote $Z \\subset X$ the closed subspace", "cut out by the product $\\mathcal{I}_1 \\mathcal{I}_2$.", "Observe that $Z \\cap U$ is the disjoint union", "of $Z_1 \\cap U$ and $Z_2 \\cap U$. By Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-separate-disjoint-opens-by-blowing-up}", "there is a $U \\cap Z$-admissible blowup $Z' \\to Z$ such that", "the strict transforms of $Z_1$ and $Z_2$ are disjoint.", "Denote $Y \\subset Z$ the center of this blowing up.", "Then $Y \\to X$ is a closed immersion of finite presentation as the composition", "of $Y \\to Z$ and $Z \\to X$ (Divisors on Spaces, Definition", "\\ref{spaces-divisors-definition-admissible-blowup} and", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-composition-finite-presentation}).", "Thus the blowing up $X' \\to X$ of $Y$ is a $U$-admissible blowing", "up. By general properties of strict transforms, the", "strict transform of $Z_1, Z_2$ with respect to $X' \\to X$", "is the same as the strict transform of $Z_1, Z_2$ with respect", "to $Z' \\to Z$, see", "Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-strict-transform}.", "Thus (1) is proved.", "\\medskip\\noindent", "Proof of (2). By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-quasi-coherent-finite-type-ideals}", "there exists a finite type quasi-coherent sheaf of ideals", "$\\mathcal{J}_i \\subset \\mathcal{O}_U$ such that", "$T_i = V(\\mathcal{J}_i)$ (set theoretically).", "By Limits of Spaces, Lemma \\ref{spaces-limits-lemma-extend}", "there exists a finite type quasi-coherent sheaf", "of ideals $\\mathcal{I}_i \\subset \\mathcal{O}_X$", "whose restriction to $U$ is $\\mathcal{J}_i$.", "Apply the result of part (1) to the closed", "subspaces $Z_i = V(\\mathcal{I}_i)$ to conclude." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-finite-presentation", "spaces-divisors-lemma-separate-disjoint-opens-by-blowing-up", "spaces-divisors-definition-admissible-blowup", "spaces-morphisms-lemma-composition-finite-presentation", "spaces-divisors-lemma-strict-transform", "spaces-limits-lemma-quasi-coherent-finite-type-ideals", "spaces-limits-lemma-extend" ], "ref_ids": [ 4849, 13012, 13031, 4839, 13001, 4621, 4608 ] } ], "ref_ids": [] }, { "id": 10892, "type": "theorem", "label": "spaces-pushouts-lemma-blowup-iso-along", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-blowup-iso-along", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of quasi-compact", "and quasi-separated algebraic spaces over $S$. Let $V \\subset Y$ be a", "quasi-compact open and $U = f^{-1}(V)$. Let $T \\subset |V|$ be a closed subset", "such that $f|_U : U \\to V$ is an isomorphism over an open neighbourhood of $T$", "in $V$. Then there exists a $V$-admissible blowing up $Y' \\to Y$ such that", "the strict transform $f' : X' \\to Y'$ of $f$ is an isomorphism over an open", "neighbourhood of the closure of $T$ in $|Y'|$." ], "refs": [], "proofs": [ { "contents": [ "Let $T' \\subset |V|$ be the complement of the maximal open over which", "$f|_U$ is an isomorphism. Then $T', T$ are closed in $|V|$ and", "$T \\cap T' = \\emptyset$. Since $|V|$ is a spectral topological space", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-quasi-separated-spectral})", "we can find constructible closed subsets $T_c, T'_c$ of $|V|$", "with $T \\subset T_c$, $T' \\subset T'_c$ such that", "$T_c \\cap T'_c = \\emptyset$ (choose a quasi-compact", "open $W$ of $|V|$ containing $T'$ not meeting $T$", "and set $T_c = |V| \\setminus W$, then choose a quasi-compact", "open $W'$ of $|V|$ containing $T_c$ not meeting $T'$", "and set $T'_c = |V| \\setminus W'$).", "By Lemma \\ref{lemma-separate-disjoint-locally-closed-by-blowing-up}", "we may, after replacing $Y$ by a $V$-admissible blowing up,", "assume that $T_c$ and $T'_c$ have disjoint closures in $|Y|$.", "Let $Y_0$ be the open subspace of $Y$ corresponding to the open", "$|Y| \\setminus \\overline{T}'_c$ and set $V_0 = V \\cap Y_0$,", "$U_0 = U \\times_V V_0$, and $X_0 = X \\times_Y Y_0$.", "Since $U_0 \\to V_0$ is an isomorphism, we can find a", "$V_0$-admissible blowing up $Y'_0 \\to Y_0$ such that the", "strict transform $X'_0$ of $X_0$ maps isomorphically to $Y'_0$, see", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-zariski-after-blowup}.", "By Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-extend-admissible-blowups}", "there exists a $V$-admissible blow up $Y' \\to Y$ whose restriction", "to $Y_0$ is $Y'_0 \\to Y_0$. If $f' : X' \\to Y'$ denotes the", "strict transform of $f$, then we see what we want is true because", "$f'$ restricts to an isomorphism over $Y'_0$." ], "refs": [ "spaces-properties-lemma-quasi-compact-quasi-separated-spectral", "spaces-pushouts-lemma-separate-disjoint-locally-closed-by-blowing-up", "spaces-more-morphisms-lemma-zariski-after-blowup", "spaces-divisors-lemma-extend-admissible-blowups" ], "ref_ids": [ 11853, 10891, 192, 13010 ] } ], "ref_ids": [] }, { "id": 10893, "type": "theorem", "label": "spaces-pushouts-lemma-blowup-etale-along", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-blowup-etale-along", "contents": [ "Let $S$ be a scheme. Consider a diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & U \\ar[l] \\ar[d]_{f|_U} & A \\ar[d] \\ar[l] \\\\", "Y & V \\ar[l] & B \\ar[l]", "}", "$$", "of quasi-compact and quasi-separated algebraic spaces over $S$.", "Assume", "\\begin{enumerate}", "\\item $f$ is proper,", "\\item $V$ is a quasi-compact open of $Y$, $U = f^{-1}(V)$,", "\\item $B \\subset V$ and $A \\subset U$ are closed subspaces,", "\\item $f|_A : A \\to B$ is an isomorphism, and", "$f$ is \\'etale at every point of $A$.", "\\end{enumerate}", "Then there exists a $V$-admissible blowing up $Y' \\to Y$ such that the strict", "transform $f' : X' \\to Y'$ satisfies: for every geometric point", "$\\overline{a}$ of the closure of $|A|$ in $|X'|$", "there exists a quotient $\\mathcal{O}_{X', \\overline{a}} \\to \\mathcal{O}$", "such that $\\mathcal{O}_{Y', f'(\\overline{a})} \\to \\mathcal{O}$", "is finite flat." ], "refs": [], "proofs": [ { "contents": [ "Let $T' \\subset |U|$ be the complement of the maximal open on which", "$f|_U$ is \\'etale. Then $T'$ is closed in $|U|$ and disjoint from $|A|$.", "Since $|U|$ is a spectral topological space (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-quasi-separated-spectral})", "we can find constructible closed subsets $T_c, T'_c$ of $|U|$", "with $|A| \\subset T_c$, $T' \\subset T'_c$ such that", "$T_c \\cap T'_c = \\emptyset$ (see proof of Lemma \\ref{lemma-blowup-iso-along}).", "By Lemma \\ref{lemma-separate-disjoint-locally-closed-by-blowing-up}", "there is a $U$-admissible blowing up $X_1 \\to X$ such that", "$T_c$ and $T'_c$ have disjoint closures in $|X_1|$.", "Let $X_{1, 0}$ be the open subspace of $X_1$ corresponding to the open", "$|X_1| \\setminus \\overline{T}'_c$ and set $U_0 = U \\cap X_{1, 0}$.", "Observe that the scheme theoretic image $\\overline{A}_1 \\subset X_1$", "of $A$ is contained in $X_{1, 0}$ by construction.", "\\medskip\\noindent", "After replacing $Y$ by a $V$-admissible blowing up and taking", "strict transforms, we may assume $X_{1, 0} \\to Y$ is flat, quasi-finite,", "and of finite presentation, see", "More on Morphisms of Spaces, Lemmas", "\\ref{spaces-more-morphisms-lemma-flat-after-blowing-up} and", "\\ref{spaces-more-morphisms-lemma-flat-fp-dimension-over-dense-open}.", "Consider the commutative diagram", "$$", "\\vcenter{", "\\xymatrix{", "X_1 \\ar[rr] \\ar[rd] & & X \\ar[ld] \\\\", "& Y", "}", "}", "\\quad\\text{and the diagram}\\quad", "\\vcenter{", "\\xymatrix{", "\\overline{A}_1 \\ar[rr] \\ar[rd] & & \\overline{A} \\ar[ld] \\\\", "& \\overline{B}", "}", "}", "$$", "of scheme theoretic images. The morphism $\\overline{A}_1 \\to \\overline{A}$", "is surjective because it is proper and hence the scheme theoretic", "image of $\\overline{A}_1 \\to \\overline{A}$ must be equal to $\\overline{A}$", "and then we can use Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-scheme-theoretic-image-is-proper}.", "The statement on \\'etale local rings follows", "by choosing a lift of the geometric point $\\overline{a}$", "to a geometric point $\\overline{a}_1$ of $\\overline{A}_1$ and setting", "$\\mathcal{O} = \\mathcal{O}_{X_1, \\overline{a}_1}$. Namely, since", "$X_1 \\to Y$ is flat and quasi-finite on", "$X_{1, 0} \\supset \\overline{A}_1$, the map", "$\\mathcal{O}_{Y', f'(\\overline{a})} \\to \\mathcal{O}_{X_1, \\overline{a}_1}$", "is finite flat, see Algebra, Lemmas", "\\ref{algebra-lemma-quasi-finite-strict-henselization}", "and \\ref{algebra-lemma-characterize-henselian}." ], "refs": [ "spaces-properties-lemma-quasi-compact-quasi-separated-spectral", "spaces-pushouts-lemma-blowup-iso-along", "spaces-pushouts-lemma-separate-disjoint-locally-closed-by-blowing-up", "spaces-more-morphisms-lemma-flat-after-blowing-up", "spaces-more-morphisms-lemma-flat-fp-dimension-over-dense-open", "spaces-morphisms-lemma-scheme-theoretic-image-is-proper", "algebra-lemma-quasi-finite-strict-henselization", "algebra-lemma-characterize-henselian" ], "ref_ids": [ 11853, 10892, 10891, 190, 186, 4922, 1306, 1276 ] } ], "ref_ids": [] }, { "id": 10894, "type": "theorem", "label": "spaces-pushouts-lemma-replaced-by-strict-transform", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-replaced-by-strict-transform", "contents": [ "Let $S$ be a scheme. Let $X \\to B$ and $Y \\to B$ be morphisms of", "algebraic spaces over $S$. Let $U \\subset X$ be an open subspace.", "Let $V \\to X \\times_B Y$ be a quasi-compact morphism", "whose composition with the first projection maps into $U$.", "Let $Z \\subset X \\times_B Y$ be the scheme theoretic image of", "$V \\to X \\times_B Y$. Let $X' \\to X$ be a $U$-admissible blowup.", "Then the scheme theoretic image of $V \\to X' \\times_B Y$ is the", "strict transform of $Z$ with respect to the blowing up." ], "refs": [], "proofs": [ { "contents": [ "Denote $Z' \\to Z$ the strict transform. The morphism $Z' \\to X'$", "induces a morphism $Z' \\to X' \\times_B Y$ which is a closed immersion", "(as $Z'$ is a closed subspace of $X' \\times_X Z$ by definition).", "Thus to finish the proof it suffices to show that the scheme theoretic", "image $Z''$ of $V \\to Z'$ is $Z'$. Observe that $Z'' \\subset Z'$", "is a closed subspace such that $V \\to Z'$ factors through $Z''$.", "Since both $V \\to X \\times_B Y$ and $V \\to X' \\times_B Y$ are", "quasi-compact (for the latter this follows from Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-permanence}", "and the fact that $X' \\times_B Y \\to X \\times_B Y$ is separated", "as a base change of a proper morphism), by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image}", "we see that $Z \\cap (U \\times_B Y) = Z'' \\cap (U \\times_B Y)$.", "Thus the inclusion morphism $Z'' \\to Z'$ is an isomorphism", "away from the exceptional divisor $E$ of $Z' \\to Z$. However, the", "structure sheaf of $Z'$ does not have any nonzero sections supported", "on $E$ (by definition of strict transforms) and we conclude that", "the surjection $\\mathcal{O}_{Z'} \\to \\mathcal{O}_{Z''}$ must be an isomorphism." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-permanence", "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image" ], "ref_ids": [ 4743, 4780 ] } ], "ref_ids": [] }, { "id": 10895, "type": "theorem", "label": "spaces-pushouts-lemma-compactification-dominates", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-compactification-dominates", "contents": [ "Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $U$ be an algebraic space", "of finite type and separated over $B$. Let $V \\to U$ be an \\'etale morphism.", "If $V$ has a compactification $V \\subset Y$ over $B$, then there", "exists a $V$-admissible blowing up $Y' \\to Y$ and an", "open $V \\subset V' \\subset Y'$ such that $V \\to U$", "extends to a proper morphism $V' \\to U$." ], "refs": [], "proofs": [ { "contents": [ "Consider the scheme theoretic image $Z \\subset Y \\times_B U$", "of the ``diagonal'' morphism $V \\to Y \\times_B U$. If we replace", "$Y$ by a $V$-admissible blowing up, then $Z$ is replaced by", "the strict transform with respect to this blowing up, see", "Lemma \\ref{lemma-replaced-by-strict-transform}. Hence by", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-zariski-after-blowup}", "we may assume $Z \\to Y$", "is an open immersion. If $V' \\subset Y$ denotes the image, then we", "see that the induced morphism $V' \\to U$ is proper because the", "projection $Y \\times_B U \\to U$ is proper and $V' \\cong Z$", "is a closed subspace of $Y \\times_B U$." ], "refs": [ "spaces-pushouts-lemma-replaced-by-strict-transform", "spaces-more-morphisms-lemma-zariski-after-blowup" ], "ref_ids": [ 10894, 192 ] } ], "ref_ids": [] }, { "id": 10896, "type": "theorem", "label": "spaces-pushouts-lemma-two-compactifications", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-two-compactifications", "contents": [ "Let $B$ be an algebraic space of finite type over $\\mathbf{Z}$.", "Let $U$ be an algebraic space of finite type and separated over $B$.", "Let $(U_2 \\subset U, f : U_1 \\to U)$ be an", "elementary distinguished square. Assume $U_1$ and $U_2$ have", "compactifications over $B$ and $U_1 \\times_U U_2 \\to U$ has dense image.", "Then $U$ has a compactification over $B$." ], "refs": [], "proofs": [ { "contents": [ "Choose a compactification $U_i \\subset X_i$ over $B$ for $i = 1, 2$. We may", "assume $U_i$ is scheme theoretically dense in $X_i$. We may assume there", "is an open $V_i \\subset X_i$ and a proper morphism", "$\\psi_i : V_i \\to U$ extending $U_i \\to U$, see", "Lemma \\ref{lemma-compactification-dominates}. Picture", "$$", "\\xymatrix{", "U_i \\ar[r] \\ar[d] & V_i \\ar[r] \\ar[dl]^{\\psi_i} & X_i \\\\", "U", "}", "$$", "Denote $Z_1 \\subset U$ the reduced closed subspace corresponding", "to the closed subset $|U| \\setminus |U_2|$. Recall that $f^{-1}Z_1$", "is a closed subspace of $U_1$ mapping isomorphically to $Z_1$.", "Denote $Z_2 \\subset U$ the reduced closed subspace corresponding", "to the closed subset $|U| \\setminus \\Im(|f|) =", "|U_2| \\setminus \\Im(|U_1 \\times_U U_2| \\to |U_2|)$.", "Thus we have", "$$", "U = U_2 \\amalg Z_1 = Z_2 \\amalg \\Im(f) =", "Z_2 \\amalg \\Im(U_1 \\times_U U_2 \\to U_2) \\amalg Z_1", "$$", "set theoretically. Denote $Z_{i, i} \\subset V_i$ the inverse image of $Z_i$", "under $\\psi_i$. Observe that $\\psi_2$ is an isomorphism over an open", "neighbourhood of $Z_2$. Observe that", "$Z_{1, 1} = \\psi_1^{-1}Z_1 = f^{-1}Z_1 \\amalg T$ for some", "closed subspace $T \\subset V_1$ disjoint from $f^{-1}Z_1$ and furthermore", "$\\psi_1$ is \\'etale along $f^{-1}Z_1$.", "Denote $Z_{i, j} \\subset V_i$ the inverse image of $Z_j$ under $\\psi_i$.", "Observe that $\\psi_i : Z_{i, j} \\to Z_j$ is a proper morphism.", "Since $Z_i$ and $Z_j$ are disjoint closed subspaces of $U$, we see that", "$Z_{i, i}$ and $Z_{i, j}$ are disjoint closed subspaces of $V_i$.", "\\medskip\\noindent", "Denote $\\overline{Z}_{i, i}$ and $\\overline{Z}_{i, j}$ the", "scheme theoretic images of $Z_{i, i}$ and $Z_{i, j}$ in $X_i$.", "We recall that $|Z_{i, j}|$ is dense in $|\\overline{Z}_{i, j}|$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-immersion}.", "After replacing $X_i$ by a $V_i$-admissible blowup we may assume that", "$\\overline{Z}_{i, i}$ and $\\overline{Z}_{i, j}$ are disjoint, see", "Lemma \\ref{lemma-separate-disjoint-locally-closed-by-blowing-up}.", "We assume this holds for both $X_1$ and $X_2$.", "Observe that this property is preserved if we replace $X_i$", "by a further $V_i$-admissible blowup. Hence we may replace $X_1$ by another", "$V_1$-admissible blowup and assume $|\\overline{Z}_{1, 1}|$", "is the disjoint union of the closures of $|T|$ and $|f^{-1}Z_1|$ in $|X_1|$.", "\\medskip\\noindent", "Set $V_{12} = V_1 \\times_U V_2$. We have an immersion", "$V_{12} \\to X_1 \\times_B X_2$ which is the composition of the closed", "immersion $V_{12} = V_1 \\times_U V_2 \\to V_1 \\times_B V_2$", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-fibre-product-after-map})", "and the open immersion $V_1 \\times_B V_2 \\to X_1 \\times_B X_2$.", "Let $X_{12} \\subset X_1 \\times_B X_2$ be the scheme theoretic image of", "$V_{12} \\to X_1 \\times_B X_2$. The projection morphisms", "$$", "p_1 : X_{12} \\to X_1", "\\quad\\text{and}\\quad", "p_2 : X_{12} \\to X_2", "$$", "are proper as $X_1$ and $X_2$ are proper over $B$. If we replace $X_1$ by a", "$V_1$-admissible blowing up, then $X_{12}$ is replaced by", "the strict transform with respect to this blowing up, see", "Lemma \\ref{lemma-replaced-by-strict-transform}.", "\\medskip\\noindent", "Denote $\\psi : V_{12} \\to U$ the compositions", "$\\psi = \\psi_1 \\circ p_1|_{V_{12}} = \\psi_2 \\circ p_2|_{V_{12}}$.", "Consider the closed subspace", "$$", "Z_{12, 2} =", "(p_1|_{V_{12}})^{-1}Z_{1, 2} =", "(p_2|_{V_{12}})^{-1}Z_{2, 2} =", "\\psi^{-1}Z_2 \\subset V_{12}", "$$", "The morphism $p_1|_{V_{12}} : V_{12} \\to V_1$ is an isomorphism", "over an open neighbourhood of $Z_{1, 2}$ because $\\psi_2 : V_2 \\to U$", "is an isomorphism over an open neighbourhood of $Z_2$ and", "$V_{12} = V_1 \\times_U V_2$. By Lemma \\ref{lemma-blowup-iso-along}", "there exists a $V_1$-admissible blowing up $X_1' \\to X_1$", "such that the strict tranform $p'_1 : X'_{12} \\to X'_1$", "of $p_1$ is an isomorphism over an open neighbourhood of", "the closure of $|Z_{1, 2}|$ in $|X'_1|$.", "After replacing $X_1$ by $X'_1$ and $X_{12}$ by $X'_{12}$", "we may assume that $p_1$ is an isomorphism over an open", "neighbourhood of $|\\overline{Z}_{1, 2}|$.", "\\medskip\\noindent", "The result of the previous paragraph tells us that", "$$", "X_{12} \\cap (\\overline{Z}_{1, 2} \\times_B \\overline{Z}_{2, 1}) = \\emptyset", "$$", "where the intersection taken in $X_1 \\times_B X_2$. Namely, the inverse", "image $p_1^{-1}\\overline{Z}_{1, 2}$ in $X_{12}$ maps isomorphically", "to $\\overline{Z}_{1, 2}$. In particular, we see that $|Z_{12, 2}|$", "is dense in $|p_1^{-1}\\overline{Z}_{1, 2}|$. Thus $p_2$ maps", "$|p_1^{-1}\\overline{Z}_{1, 2}|$ into $|\\overline{Z}_{2, 2}|$.", "Since $|\\overline{Z}_{2, 2}| \\cap |\\overline{Z}_{2, 1}| = \\emptyset$", "we conclude.", "\\medskip\\noindent", "It turns out that we need to do one additional blowing up before we", "can conclude the argument. Namely, let $V_2 \\subset W_2 \\subset X_2$", "be the open subspace with underlying topological space", "$$", "|W_2| =", "|V_2| \\cup (|X_2| \\setminus |\\overline{Z}_{2, 1}|) =", "|X_2| \\setminus \\left(|\\overline{Z}_{2, 1}| \\setminus |Z_{2, 1}|\\right)", "$$", "Since $p_2(p_1^{-1}\\overline{Z}_{1, 2})$ is contained in $W_2$ (see above)", "we see that replacing $X_2$ by a $W_2$-admissible blowup and $X_{21}$", "by the corresponding strict transform will preserve the property of", "$p_1$ being an isomorphism over an open neighbourhood of $\\overline{Z}_{1, 2}$.", "Since $\\overline{Z}_{2, 1} \\cap W_2 = \\overline{Z}_{2, 1} \\cap V_2 = Z_{2, 1}$", "we see that $Z_{2, 1}$ is a closed subspace of $W_2$ and $V_2$.", "Observe that $V_{12} = V_1 \\times_U V_2 = p_1^{-1}(V_1) = p_2^{-1}(V_2)$", "as open subspaces of $X_{12}$ as it is the largest open subspace of $X_{12}$", "over which the morphism $\\psi : V_{12} \\to U$ extends; details", "omitted\\footnote{Namely, $V_1 \\times_U V_2$ is proper over $U$ so if", "$\\psi$ extends to a larger open of $X_{12}$, then $V_1 \\times_U V_2$", "would be closed in this open by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-permanence}.", "Then we get equality as $V_{12} \\subset X_{12}$ is dense.}.", "We have the following equalities of closed subspaces of $V_{12}$:", "$$", "p_2^{-1}Z_{2, 1} = p_2^{-1} \\psi_2^{-1} Z_1 =", "p_1^{-1} \\psi_1^{-1} Z_1= p_1^{-1}Z_{1, 1} =", "p_1^{-1}f^{-1}Z_1 \\amalg p_1^{-1}T", "$$", "Here and below we use the slight abuse of notation of writing $p_2$ in", "stead of the restriction of $p_2$ to $V_{12}$, etc.", "Since $p_2^{-1}(Z_{2, 1})$ is a closed subspace of $p_2^{-1}(W_2)$", "as $Z_{2, 1}$ is a closed subspace of $W_2$ we conclude that also", "$p_1^{-1}f^{-1}Z_1$ is a closed subspace of $p_2^{-1}(W_2)$.", "Finally, the morphism $p_2 : X_{12} \\to X_2$ is \\'etale at points of", "$p_1^{-1}f^{-1}Z_1$ as $\\psi_1$ is \\'etale along $f^{-1}Z_1$", "and $V_{12} = V_1 \\times_U V_2$.", "Thus we may apply Lemma \\ref{lemma-blowup-etale-along} to the morphism", "$p_2 : X_{12} \\to X_2$, the open $W_2$, the closed subspace", "$Z_{2, 1} \\subset W_2$, and the closed subspace", "$p_1^{-1}f^{-1}Z_1 \\subset p_2^{-1}(W_2)$.", "Hence after replacing $X_2$ by a $W_2$-admissible blowup and $X_{12}$ by", "the corresponding strict transform, we obtain for every geometric", "point $\\overline{y}$ of the closure of $|p_1^{-1}f^{-1}Z_1|$ a local ring map", "$\\mathcal{O}_{X_{12}, \\overline{y}} \\to \\mathcal{O}$ such that", "$\\mathcal{O}_{X_2, p_2(\\overline{y})} \\to \\mathcal{O}$ is finite flat.", "\\medskip\\noindent", "Consider the algebraic space", "$$", "W_2 = U \\coprod\\nolimits_{U_2} (X_2 \\setminus \\overline{Z}_{2, 1}),", "$$", "and with $T \\subset V_1$ as in the first paragraph the algebraic space", "$$", "W_1 = U \\coprod\\nolimits_{U_1}", "(X_1 \\setminus \\overline{Z}_{1, 2} \\cup \\overline{T}),", "$$", "obtained by pushout, see", "Lemma \\ref{lemma-construct-elementary-distinguished-square}.", "Let us apply Lemma \\ref{lemma-check-separated}", "to see that $W_i \\to B$ is separated. First,", "$U \\to B$ and $X_i \\to B$ are separated. Let us check the quasi-compact", "immersion $U_i \\to U \\times_B (X_i \\setminus \\overline{Z}_{i, j})$", "is closed using the valuative criterion, see Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-existence-universally-closed}.", "Choose a valuation ring $A$ over $B$ with fraction field $K$ and", "compatible morphisms $(u, x_i) : \\Spec(A) \\to U \\times_B X_i$ and", "$u_i : \\Spec(K) \\to U_i$. Since $\\psi_i$ is proper, we", "can find a unique $v_i : \\Spec(A) \\to V_i$ compatible with", "$u$ and $u_i$. Since $X_i$ is proper over $B$ we see that $x_i = v_i$.", "If $v_i$ does not factor through $U_i \\subset V_i$, then we conclude", "that $x_i$ maps the closed point of $\\Spec(A)$ into $Z_{i, j}$ or", "$T$ when $i = 1$. This finishes the proof because we removed", "$\\overline{Z}_{i, j}$ and $\\overline{T}$ in the construction of $W_i$.", "\\medskip\\noindent", "On the other hand, for any valuation ring $A$ over $B$ with", "fraction field $K$ and any morphism", "$$", "\\gamma : \\Spec(K) \\to \\Im(U_1 \\times_U U_2 \\to U)", "$$", "over $B$, we claim that after replacing $A$ by an extension of valuation", "rings, there is an $i$ and an extension of $\\gamma$ to a morphism", "$h_i : \\Spec(A) \\to W_i$. Namely, we first extend $\\gamma$ to a", "morphism $g_2 : \\Spec(A) \\to X_2$ using the valuative criterion of", "properness. If the image of $g_2$ does not meet $\\overline{Z}_{2, 1}$,", "then we obtain our morphism into $W_2$.", "Otherwise, denote $\\overline{z} \\in \\overline{Z}_{2, 1}$ a geometric", "point lying over the image of the closed point under $g_2$.", "We may lift this to a geometric point $\\overline{y}$ of $X_{12}$", "in the closure of $|p_1^{-1}f^{-1}Z_1|$ because the map of", "spaces $|p_1^{-1}f^{-1}Z_1| \\to |\\overline{Z}_{2, 1}|$ is closed", "with image containing the dense open $|Z_{2, 1}|$. After replacing $A$", "by its strict henselization", "(More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-valuation-ring})", "we get the following diagram", "$$", "\\xymatrix{", "A \\ar@{..>}[rr] & & A' \\\\", "\\mathcal{O}_{X_2, \\overline{z}} \\ar[r] \\ar[u] &", "\\mathcal{O}_{X_{12}, \\overline{y}} \\ar[r] &", "\\mathcal{O} \\ar@{..>}[u]", "}", "$$", "where $\\mathcal{O}_{X_{12}, \\overline{y}} \\to \\mathcal{O}$ is the", "map we found in the 5th paragraph of the proof.", "Since the horizontal composition is finite and flat we can find an", "extension of valuation rings $A'/A$ and dotted arrow making the diagram", "commute. After replacing $A$ by $A'$ this means that we obtain a lift", "$g_{12} : \\Spec(A) \\to X_{12}$ whose closed point maps into", "the closure of $|p_1^{-1}f^{-1}Z_1|$.", "Then $g_1 = p_1 \\circ g_{12} : \\Spec(A) \\to X_1$ is a morphism whose", "closed point maps into the closure of $|f^{-1}Z_1|$. Since the closure", "of $|f^{-1}Z_1|$ is disjoint from the closure of $|T|$ and contained in", "$|\\overline{Z}_{1, 1}|$ which is disjoint from $|\\overline{Z}_{1, 2}|$", "we conclude that $g_1$ defines a morphism $h_1 : \\Spec(A) \\to W_1$", "as desired.", "\\medskip\\noindent", "Consider a diagram", "$$", "\\xymatrix{", "W_1' \\ar[d] \\ar[r] & W & W_2' \\ar[l] \\ar[d] \\\\", "W_1 & U \\ar[l] \\ar[lu] \\ar[u] \\ar[ru] \\ar[r] & W_2", "}", "$$", "as in More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-find-common-blowups}.", "By the previous paragraph for every solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_\\gamma \\ar[d] & W \\ar[d] \\\\", "\\Spec(A) \\ar@{..>}[ru] \\ar[r] & B", "}", "$$", "where $\\Im(\\gamma) \\subset \\Im(U_1 \\times_U U_2 \\to U)$", "there is an $i$ and an extension $h_i : \\Spec(A) \\to W_i$ of $\\gamma$", "after possibly replacing $A$ by an extension of valuation rings.", "Using the valuative criterion of properness for $W'_i \\to W_i$,", "we can then lift $h_i$ to $h'_i : \\Spec(A) \\to W'_i$.", "Hence the dotted arrow in the diagram exists after possibly", "extending $A$. Since $W$ is separated over $B$, we see that the choice", "of extension isn't needed and the arrow is unique as well, see", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-usual-enough}", "and \\ref{spaces-morphisms-lemma-separated-implies-valuative}.", "Then finally the existence of the dotted arrow", "implies that $W \\to B$ is universally closed by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-refined-valuative-criterion-universally-closed}.", "As $W \\to B$ is already of finite type and separated, we win." ], "refs": [ "spaces-pushouts-lemma-compactification-dominates", "spaces-morphisms-lemma-quasi-compact-immersion", "spaces-pushouts-lemma-separate-disjoint-locally-closed-by-blowing-up", "spaces-morphisms-lemma-fibre-product-after-map", "spaces-pushouts-lemma-replaced-by-strict-transform", "spaces-pushouts-lemma-blowup-iso-along", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-pushouts-lemma-blowup-etale-along", "spaces-pushouts-lemma-construct-elementary-distinguished-square", "spaces-pushouts-lemma-check-separated", "spaces-morphisms-lemma-quasi-compact-existence-universally-closed", "more-algebra-lemma-henselization-valuation-ring", "spaces-more-morphisms-lemma-find-common-blowups", "spaces-morphisms-lemma-usual-enough", "spaces-morphisms-lemma-separated-implies-valuative", "spaces-morphisms-lemma-refined-valuative-criterion-universally-closed" ], "ref_ids": [ 10895, 4790, 10891, 4715, 10894, 10892, 4920, 10893, 10868, 10890, 4930, 10553, 195, 4927, 4935, 4934 ] } ], "ref_ids": [] }, { "id": 10897, "type": "theorem", "label": "spaces-pushouts-lemma-filter-Noetherian-space", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-lemma-filter-Noetherian-space", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $U \\subset X$ be a proper dense open subspace. Then there exists an", "affine scheme $V$ and an \\'etale morphism $V \\to X$ such that", "\\begin{enumerate}", "\\item the open subspace $W = U \\cup \\Im(V \\to X)$ is strictly larger", "than $U$,", "\\item $(U \\subset W, V \\to W)$ is a distinguished square, and", "\\item $U \\times_W V \\to U$ has dense image.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a stratification", "$$", "\\emptyset = U_{n + 1} \\subset", "U_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X", "$$", "and morphisms $f_p : V_p \\to U_p$ as in Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated}.", "Let $p$ be the smallest integer such that $U_p \\not \\subset U$", "(this is possible as $U \\not = X$). Choose an affine open $V \\subset V_p$", "such that the \\'etale morphism $f_p|_V : V \\to X$ does not factor through $U$.", "Consider the open $W = U \\cup \\Im(V \\to X)$ and the", "reduced closed subspace $Z \\subset W$ with $|Z| = |W| \\setminus |U|$.", "Then $f^{-1}Z \\to Z$ is an isomorphism because we have the", "corresponding property for the morphism $f_p$, see the lemma cited above.", "Thus $(U \\subset W, f : V \\to W)$ is a distinguished square.", "It may not be true that the open $I = \\Im(U \\times_W V \\to U)$", "is dense in $U$. The algebraic space $U' \\subset U$ whose underlying", "set is $|U| \\setminus \\overline{|I|}$ is Noetherian", "and hence we can find a dense open subscheme $U'' \\subset U'$, see", "for example Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}.", "Then we can find a dense open affine $U''' \\subset U''$, see", "Properties, Lemmas \\ref{properties-lemma-Noetherian-irreducible-components}", "and \\ref{properties-lemma-maximal-points-affine}.", "After we replace $f$ by $V \\amalg U''' \\to X$ everything is clear." ], "refs": [ "decent-spaces-lemma-filter-quasi-compact-quasi-separated", "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme", "properties-lemma-Noetherian-irreducible-components", "properties-lemma-maximal-points-affine" ], "ref_ids": [ 9480, 11917, 2956, 3059 ] } ], "ref_ids": [] }, { "id": 10898, "type": "theorem", "label": "spaces-pushouts-proposition-formal-glueing-modules", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-proposition-formal-glueing-modules", "contents": [ "In Situation \\ref{situation-formal-glueing} the functor", "(\\ref{equation-formal-glueing-modules}) is an equivalence", "with quasi-inverse given by (\\ref{equation-reverse})." ], "refs": [], "proofs": [ { "contents": [ "We first treat the special case where $X$ and $Y$ are affine schemes", "and where the morphism $f$ is flat. Say $X = \\Spec(R)$ and $Y = \\Spec(S)$.", "Then $f$ corresponds to a flat ring map $R \\to S$. Moreover, $Z \\subset X$", "is cut out by a finitely generated ideal $I \\subset R$. Choose generators", "$f_1, \\ldots, f_t \\in I$. By the description of quasi-coherent modules", "in terms of modules", "(Schemes, Section \\ref{schemes-section-quasi-coherent-affine}),", "we see that the category $\\textit{QCoh}(Y \\to X, Z)$", "is canonically equivalent to the category", "$\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$", "of More on Algebra, Remark \\ref{more-algebra-remark-glueing-data}", "such that the functors", "(\\ref{equation-formal-glueing-modules}) and (\\ref{equation-reverse})", "correspond to the functors $\\text{Can}$ and $H^0$.", "Hence the result follows from", "More on Algebra, Proposition \\ref{more-algebra-proposition-equivalence}", "in this case.", "\\medskip\\noindent", "We return to the general case.", "Let $\\mathcal{F}$ be a quasi-coherent module on $X$.", "We will show that", "$$", "\\alpha :", "\\mathcal{F}", "\\longrightarrow", "\\Ker\\left(j_*\\mathcal{F}|_U \\oplus f_*f^*\\mathcal{F} \\to", "(f \\circ j')_*f^*\\mathcal{F}|_V\\right)", "$$", "is an isomorphism. Let $(\\mathcal{H}, \\mathcal{G}, \\varphi)$", "be an object of $\\QCoh(Y \\to X, Z)$. We will show that", "$$", "\\beta :", "f^*\\Ker\\left(", "j_*\\mathcal{H} \\oplus f_*\\mathcal{G} \\to (f \\circ j')_*\\mathcal{G}|_V", "\\right)", "\\longrightarrow", "\\mathcal{G}", "$$", "and", "$$", "\\gamma :", "j^*\\Ker\\left(", "j_*\\mathcal{H} \\oplus f_*\\mathcal{G} \\to (f \\circ j')_*\\mathcal{G}|_V", "\\right)", "\\longrightarrow", "\\mathcal{H}", "$$", "are isomorphisms. To see these statements are true it suffices to", "look at stalks. Let $\\overline{y}$ be a geometric point of $Y$ mapping", "to the geometric point $\\overline{x}$ of $X$.", "\\medskip\\noindent", "Fix an object $(\\mathcal{H}, \\mathcal{G}, \\varphi)$ of $\\QCoh(Y \\to X, Z)$.", "By Lemma \\ref{lemma-stalk-formal-glueing}", "and a diagram chase (omitted) the canonical map", "$$", "\\Ker(j_*\\mathcal{H} \\oplus f_*\\mathcal{G} \\to", "(f \\circ j')_*\\mathcal{G}|_V)_{\\overline{x}}", "\\longrightarrow", "\\Ker(", "j_*\\mathcal{H}_{\\overline{x}} \\oplus \\mathcal{G}_{\\overline{y}}", "\\to", "j'_*\\mathcal{G}_{\\overline{y}}", ")", "$$", "is an isomorphism.", "\\medskip\\noindent", "In particular, if $\\overline{y}$ is a geometric point of $V$, then", "we see that $j'_*\\mathcal{G}_{\\overline{y}} = \\mathcal{G}_{\\overline{y}}$", "and hence that this kernel is equal to $\\mathcal{H}_{\\overline{x}}$.", "This easily implies that $\\alpha_{\\overline{x}}$, $\\beta_{\\overline{x}}$,", "and $\\beta_{\\overline{y}}$ are isomorphisms in this case.", "\\medskip\\noindent", "Next, assume that $\\overline{y}$ is a point of $f^{-1}Z$.", "Let $I_{\\overline{x}} \\subset \\mathcal{O}_{X, \\overline{x}}$,", "resp.\\ $I_{\\overline{y}} \\subset \\mathcal{O}_{Y, \\overline{y}}$", "be the stalk of the ideal cutting out $Z$, resp.\\ $f^{-1}Z$.", "Then $I_{\\overline{x}}$ is a finitely generated ideal,", "$I_{\\overline{y}} = I_{\\overline{x}}\\mathcal{O}_{Y, \\overline{y}}$,", "and $\\mathcal{O}_{X, \\overline{x}} \\to \\mathcal{O}_{Y, \\overline{y}}$", "is a flat local homomorphism inducing an isomorphism", "$\\mathcal{O}_{X, \\overline{x}}/I_{\\overline{x}} =", "\\mathcal{O}_{Y, \\overline{y}}/I_{\\overline{y}}$.", "At this point we can bootstrap using the diagram of categories", "$$", "\\xymatrix{", "\\QCoh(\\mathcal{O}_X) \\ar[r]_-{(\\ref{equation-formal-glueing-modules})} \\ar[d] &", "\\QCoh(Y \\to X, Z) \\ar[d] \\ar@/_2pc/[l]^{(\\ref{equation-reverse})} \\\\", "\\text{Mod}_{\\mathcal{O}_{X, \\overline{x}}} \\ar[r]^-{\\text{Can}} &", "\\text{Glue}(\\mathcal{O}_{X, \\overline{x}} \\to \\mathcal{O}_{Y, \\overline{y}},", "f_1, \\ldots, f_t) \\ar@/^2pc/[l]_{H^0}", "}", "$$", "Namely, as in the first paragraph of the proof we identify", "$$", "\\text{Glue}(\\mathcal{O}_{X, \\overline{x}} \\to \\mathcal{O}_{Y, \\overline{y}},", "f_1, \\ldots, f_t)", "=", "\\QCoh(\\Spec(\\mathcal{O}_{Y, \\overline{y}}) \\to", "\\Spec(\\mathcal{O}_{X, \\overline{x}}), V(I_{\\overline{x}}))", "$$", "The right vertical functor is given by pullback, and it is clear that", "the inner square is commutative. Our computation of the stalk of the", "kernel in the third paragraph of the proof combined with", "Lemma \\ref{lemma-stalk-of-pushforward} implies that", "the outer square (using the curved arrows) commutes. Thus we", "conclude using the case of a flat morphism of affine schemes", "which we handled in the first paragraph of the proof." ], "refs": [ "more-algebra-remark-glueing-data", "more-algebra-proposition-equivalence", "spaces-pushouts-lemma-stalk-formal-glueing", "spaces-pushouts-lemma-stalk-of-pushforward" ], "ref_ids": [ 10662, 10587, 10870, 10871 ] } ], "ref_ids": [] }, { "id": 10900, "type": "theorem", "label": "varieties-theorem-varieties-rational-maps", "categories": [ "varieties" ], "title": "varieties-theorem-varieties-rational-maps", "contents": [ "Let $k$ be a field. The category of varieties and", "dominant rational maps is equivalent to the category of", "finitely generated field extensions $K/k$." ], "refs": [], "proofs": [ { "contents": [ "Let $X$ and $Y$ be varieties with generic points $x \\in X$ and $y \\in Y$.", "Recall that dominant rational maps from $X$ to $Y$ are exactly those", "rational maps which map $x$ to $y$", "(Morphisms, Definition \\ref{morphisms-definition-dominant-rational}", "and discussion following).", "Thus given a dominant rational map $X \\supset U \\to Y$ we obtain a map of", "function fields", "$$", "k(Y) = \\kappa(y) = \\mathcal{O}_{Y, y}", "\\longrightarrow", "\\mathcal{O}_{X, x} = \\kappa(x) = k(X)", "$$", "Conversely, such a $k$-algebra map (which is automatically local as the", "source and target are fields) determines (uniquely) a dominant rational", "map by Morphisms, Lemma \\ref{morphisms-lemma-rational-map-finite-presentation}.", "In this way we obtain a fully faithful functor.", "To finish the proof it suffices to show that every finitely generated", "field extension $K/k$ is in the essential image.", "Since $K/k$ is finitely generated, there exists a finite type", "$k$-algebra $A \\subset K$ such that $K$ is the fraction field of $A$.", "Then $X = \\Spec(A)$ is a variety whose function field is $K$." ], "refs": [ "morphisms-definition-dominant-rational", "morphisms-lemma-rational-map-finite-presentation" ], "ref_ids": [ 5584, 5477 ] } ], "ref_ids": [] }, { "id": 10901, "type": "theorem", "label": "varieties-lemma-product-varieties", "categories": [ "varieties" ], "title": "varieties-lemma-product-varieties", "contents": [ "\\begin{slogan}", "Products of varieties are varieties over algebraically closed fields.", "\\end{slogan}", "Let $k$ be an algebraically closed field.", "Let $X$, $Y$ be varieties over $k$.", "Then $X \\times_{\\Spec(k)} Y$ is a variety over $k$." ], "refs": [], "proofs": [ { "contents": [ "The morphism $X \\times_{\\Spec(k)} Y \\to \\Spec(k)$ is of", "finite type and separated because it is the composition of the", "morphisms $X \\times_{\\Spec(k)} Y \\to Y \\to \\Spec(k)$", "which are separated and of finite type, see", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-finite-type} and", "\\ref{morphisms-lemma-composition-finite-type}", "and", "Schemes, Lemma \\ref{schemes-lemma-separated-permanence}.", "To finish the proof it suffices to show that $X \\times_{\\Spec(k)} Y$", "is integral.", "Let $X = \\bigcup_{i = 1, \\ldots, n} U_i$,", "$Y = \\bigcup_{j = 1, \\ldots, m} V_j$ be finite affine open coverings.", "If we can show that each $U_i \\times_{\\Spec(k)} V_j$ is integral,", "then we are done by", "Properties, Lemmas \\ref{properties-lemma-characterize-reduced},", "\\ref{properties-lemma-characterize-irreducible}, and", "\\ref{properties-lemma-characterize-integral}.", "This reduces us to the affine case.", "\\medskip\\noindent", "The affine case translates into the following algebra statement: Suppose", "that $A$, $B$ are integral domains and finitely generated $k$-algebras.", "Then $A \\otimes_k B$ is an integral domain. To get a contradiction suppose that", "$$", "(\\sum\\nolimits_{i = 1, \\ldots, n} a_i \\otimes b_i)", "(\\sum\\nolimits_{j = 1, \\ldots, m} c_j \\otimes d_j) = 0", "$$", "in $A \\otimes_k B$ with both factors nonzero in $A \\otimes_k B$.", "We may assume that $b_1, \\ldots, b_n$ are $k$-linearly", "independent in $B$, and that $d_1, \\ldots, d_m$ are $k$-linearly independent", "in $B$. Of course we may also assume that $a_1$ and $c_1$ are nonzero", "in $A$. Hence $D(a_1c_1) \\subset \\Spec(A)$ is nonempty. By the", "Hilbert Nullstellensatz", "(Algebra, Theorem \\ref{algebra-theorem-nullstellensatz})", "we can find a maximal ideal $\\mathfrak m \\subset A$ contained in", "$D(a_1c_1)$ and $A/\\mathfrak m = k$ as $k$ is algebraically closed.", "Denote $\\overline{a}_i, \\overline{c}_j$ the residue classes of", "$a_i, c_j$ in $A/\\mathfrak m = k$. The equation above becomes", "$$", "(\\sum\\nolimits_{i = 1, \\ldots, n} \\overline{a}_i b_i)", "(\\sum\\nolimits_{j = 1, \\ldots, m} \\overline{c}_j d_j) = 0", "$$", "which is a contradiction with", "$\\mathfrak m \\in D(a_1c_1)$, the linear independence of", "$b_1, \\ldots, b_n$ and $d_1, \\ldots, d_m$, and the fact that $B$ is a domain." ], "refs": [ "morphisms-lemma-base-change-finite-type", "morphisms-lemma-composition-finite-type", "schemes-lemma-separated-permanence", "properties-lemma-characterize-reduced", "properties-lemma-characterize-irreducible", "properties-lemma-characterize-integral", "algebra-theorem-nullstellensatz" ], "ref_ids": [ 5200, 5199, 7714, 2945, 2946, 2947, 316 ] } ], "ref_ids": [] }, { "id": 10902, "type": "theorem", "label": "varieties-lemma-birational-varieties", "categories": [ "varieties" ], "title": "varieties-lemma-birational-varieties", "contents": [ "Let $X$ and $Y$ be varieties over a field $k$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ and $Y$ are birational varieties,", "\\item the function fields $k(X)$ and $k(Y)$ are isomorphic,", "\\item there exist nonempty opens of $X$ and $Y$ which are isomorphic", "as varieties,", "\\item there exists an open $U \\subset X$ and a birational morphism", "$U \\to Y$ of varieties.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Morphisms, Lemma \\ref{morphisms-lemma-criterion-birational-finite-presentation}." ], "refs": [ "morphisms-lemma-criterion-birational-finite-presentation" ], "ref_ids": [ 5485 ] } ], "ref_ids": [] }, { "id": 10903, "type": "theorem", "label": "varieties-lemma-change-fields-flat", "categories": [ "varieties" ], "title": "varieties-lemma-change-fields-flat", "contents": [ "Let $K/k$ be an extension of fields. Let $X$ be scheme over $k$", "and set $Y = X_K$. If $y \\in Y$ with image $x \\in X$, then", "\\begin{enumerate}", "\\item $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{Y, y}$ is a", "faithfully flat local ring homomorphism,", "\\item with $\\mathfrak p_0 = \\Ker(\\kappa(x) \\otimes_k K \\to \\kappa(y))$", "we have $\\kappa(y) = \\kappa(\\mathfrak p_0)$,", "\\item $\\mathcal{O}_{Y, y} = (\\mathcal{O}_{X, x} \\otimes_k K)_\\mathfrak p$", "where $\\mathfrak p \\subset \\mathcal{O}_{X, x} \\otimes_k K$ is the inverse", "image of $\\mathfrak p_0$.", "\\item we have", "$\\mathcal{O}_{Y, y}/\\mathfrak m_x\\mathcal{O}_{Y, y} =", "(\\kappa(x) \\otimes_k K)_{\\mathfrak p_0}$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may assume $X = \\Spec(A)$ is affine. Then $Y = \\Spec(A \\otimes_k K)$.", "Since $K$ is flat over $k$, we see that $A \\to A \\otimes_k K$ is flat.", "Hence $Y \\to X$ is flat and we get the first statement if we also", "use Algebra, Lemma \\ref{algebra-lemma-local-flat-ff}.", "The second statement follows from", "Schemes, Lemma \\ref{schemes-lemma-points-fibre-product}.", "Now $y$ corresponds to a prime ideal $\\mathfrak q \\subset A \\otimes_k K$", "and $x$ to $\\mathfrak r = A \\cap \\mathfrak q$. Then $\\mathfrak p_0$", "is the kernel of the induced map", "$\\kappa(\\mathfrak r) \\otimes_k K \\to \\kappa(\\mathfrak q)$.", "The map on local rings is", "$$", "A_\\mathfrak r \\longrightarrow (A \\otimes_k K)_\\mathfrak q", "$$", "We can factor this map through", "$A_\\mathfrak r \\otimes_k K = (A \\otimes_k K)_{\\mathfrak r}$", "to get", "$$", "A_\\mathfrak r \\longrightarrow A_\\mathfrak r \\otimes_k K", "\\longrightarrow (A \\otimes_k K)_\\mathfrak q", "$$", "and then the second arrow is a localization at some prime. This prime ideal", "is the inverse image of $\\mathfrak p_0$ (details omitted) and this", "proves (3). To see (4) use (3) and that localization and $- \\otimes_k K$", "are exact functors." ], "refs": [ "algebra-lemma-local-flat-ff", "schemes-lemma-points-fibre-product" ], "ref_ids": [ 537, 7693 ] } ], "ref_ids": [] }, { "id": 10904, "type": "theorem", "label": "varieties-lemma-change-fields-algebraic-dim", "categories": [ "varieties" ], "title": "varieties-lemma-change-fields-algebraic-dim", "contents": [ "Notation as in Lemma \\ref{lemma-change-fields-flat}.", "Assume $X$ is locally of finite type over $k$. Then", "$$", "\\dim(\\mathcal{O}_{Y, y}/\\mathfrak m_x\\mathcal{O}_{Y, y}) =", "\\text{trdeg}_k(\\kappa(x)) - \\text{trdeg}_K(\\kappa(y)) =", "\\dim(\\mathcal{O}_{Y, y}) - \\dim(\\mathcal{O}_{X, x})", "$$" ], "refs": [ "varieties-lemma-change-fields-flat" ], "proofs": [ { "contents": [ "This is a restatement of Algebra, Lemma", "\\ref{algebra-lemma-inequalities-under-field-extension}." ], "refs": [ "algebra-lemma-inequalities-under-field-extension" ], "ref_ids": [ 1011 ] } ], "ref_ids": [ 10903 ] }, { "id": 10905, "type": "theorem", "label": "varieties-lemma-change-fields-algebraic-unramified", "categories": [ "varieties" ], "title": "varieties-lemma-change-fields-algebraic-unramified", "contents": [ "Notation as in Lemma \\ref{lemma-change-fields-flat}.", "Assume $X$ is locally of finite type over $k$,", "that $\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y})$", "and that $\\kappa(x) \\otimes_k K$ is reduced", "(for example if $\\kappa(x)/k$ is separable or $K/k$ is separable).", "Then $\\mathfrak m_x \\mathcal{O}_{Y, y} = \\mathfrak m_y$." ], "refs": [ "varieties-lemma-change-fields-flat" ], "proofs": [ { "contents": [ "(The parenthetical statement follows from", "Algebra, Lemma \\ref{algebra-lemma-separable-extension-preserves-reducedness}.)", "Combining Lemmas \\ref{lemma-change-fields-flat} and", "\\ref{lemma-change-fields-algebraic-dim}", "we see that $\\mathcal{O}_{Y, y}/\\mathfrak m_x \\mathcal{O}_{Y, y}$", "has dimension $0$ and is reduced. Hence it is a field." ], "refs": [ "algebra-lemma-separable-extension-preserves-reducedness", "varieties-lemma-change-fields-flat", "varieties-lemma-change-fields-algebraic-dim" ], "ref_ids": [ 565, 10903, 10904 ] } ], "ref_ids": [ 10903 ] }, { "id": 10906, "type": "theorem", "label": "varieties-lemma-geometrically-reduced-at-point", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-reduced-at-point", "contents": [ "\\begin{slogan}", "Geometric reducedness can be checked on local rings.", "\\end{slogan}", "Let $k$ be a field.", "Let $X$ be a scheme over $k$.", "Let $x \\in X$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically reduced at $x$, and", "\\item the ring $\\mathcal{O}_{X, x}$ is geometrically", "reduced over $k$ (see", "Algebra, Definition \\ref{algebra-definition-geometrically-reduced}).", "\\end{enumerate}" ], "refs": [ "algebra-definition-geometrically-reduced" ], "proofs": [ { "contents": [ "Assume (1). This in particular implies that $\\mathcal{O}_{X, x}$", "is reduced. Let $k \\subset k'$ be a finite purely inseparable field", "extension. Consider the ring $\\mathcal{O}_{X, x} \\otimes_k k'$.", "By Algebra, Lemma \\ref{algebra-lemma-p-ring-map}", "its spectrum is the same as the spectrum of $\\mathcal{O}_{X, x}$.", "Hence it is a local ring also", "(Algebra, Lemma \\ref{algebra-lemma-characterize-local-ring}).", "Therefore there is a unique point $x' \\in X_{k'}$ lying over $x$", "and $\\mathcal{O}_{X_{k'}, x'} \\cong \\mathcal{O}_{X, x} \\otimes_k k'$.", "By assumption this is a reduced ring. Hence we deduce (2) by", "Algebra, Lemma", "\\ref{algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension}.", "\\medskip\\noindent", "Assume (2). Let $k \\subset k'$ be a field extension. Since", "$\\Spec(k') \\to \\Spec(k)$ is surjective, also", "$X_{k'} \\to X$ is surjective", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-surjective}).", "Let $x' \\in X_{k'}$ be any point lying over $x$.", "The local ring $\\mathcal{O}_{X_{k'}, x'}$", "is a localization of the ring $\\mathcal{O}_{X, x} \\otimes_k k'$.", "Hence it is reduced by assumption and (1) is proved." ], "refs": [ "algebra-lemma-p-ring-map", "algebra-lemma-characterize-local-ring", "algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension", "morphisms-lemma-base-change-surjective" ], "ref_ids": [ 582, 397, 571, 5165 ] } ], "ref_ids": [ 1461 ] }, { "id": 10907, "type": "theorem", "label": "varieties-lemma-perfect-reduced", "categories": [ "varieties" ], "title": "varieties-lemma-perfect-reduced", "contents": [ "Let $X$ be a scheme over a perfect field $k$ (e.g.\\ $k$ has", "characteristic zero). Let $x \\in X$. If $\\mathcal{O}_{X, x}$ is", "reduced, then $X$ is geometrically reduced at $x$.", "If $X$ is reduced, then $X$ is geometrically reduced over $k$." ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from", "Lemma \\ref{lemma-geometrically-reduced-at-point} and", "Algebra, Lemma \\ref{algebra-lemma-separable-extension-preserves-reducedness}", "and the definition of a perfect field", "(Algebra, Definition \\ref{algebra-definition-perfect}).", "The second statement follows from the first." ], "refs": [ "varieties-lemma-geometrically-reduced-at-point", "algebra-lemma-separable-extension-preserves-reducedness", "algebra-definition-perfect" ], "ref_ids": [ 10906, 565, 1462 ] } ], "ref_ids": [] }, { "id": 10908, "type": "theorem", "label": "varieties-lemma-geometrically-reduced", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-reduced", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically reduced,", "\\item $X_{k'}$ is reduced for every field extension $k \\subset k'$,", "\\item $X_{k'}$ is reduced for every finite purely inseparable field extension", "$k \\subset k'$,", "\\item $X_{k^{1/p}}$ is reduced,", "\\item $X_{k^{perf}}$ is reduced,", "\\item $X_{\\bar k}$ is reduced,", "\\item for every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$", "is geometrically reduced (see", "Algebra, Definition \\ref{algebra-definition-geometrically-reduced}).", "\\end{enumerate}" ], "refs": [ "algebra-definition-geometrically-reduced" ], "proofs": [ { "contents": [ "Assume (1). Then for every field extension $k \\subset k'$ and", "every point $x' \\in X_{k'}$ the local ring of $X_{k'}$ at $x'$", "is reduced. In other words $X_{k'}$ is reduced. Hence (2).", "\\medskip\\noindent", "Assume (2). Let $U \\subset X$ be an affine open. Then for", "every field extension $k \\subset k'$ the scheme $X_{k'}$ is reduced, hence", "$U_{k'} = \\Spec(\\mathcal{O}(U)\\otimes_k k')$ is reduced,", "hence $\\mathcal{O}(U)\\otimes_k k'$ is reduced (see Properties,", "Section \\ref{properties-section-integral}). In other words", "$\\mathcal{O}(U)$ is geometrically reduced, so (7) holds.", "\\medskip\\noindent", "Assume (7). For any field extension $k \\subset k'$ the base", "change $X_{k'}$ is gotten by gluing the spectra of the", "rings $\\mathcal{O}_X(U) \\otimes_k k'$ where $U$ is affine open", "in $X$ (see Schemes, Section \\ref{schemes-section-fibre-products}).", "Hence $X_{k'}$ is reduced. So (1) holds.", "\\medskip\\noindent", "This proves that (1), (2), and (7) are equivalent. These are equivalent", "to (3), (4), (5), and (6) because we can apply Algebra, Lemma", "\\ref{algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension}", "to $\\mathcal{O}_X(U)$ for $U \\subset X$ affine open." ], "refs": [ "algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension" ], "ref_ids": [ 571 ] } ], "ref_ids": [ 1461 ] }, { "id": 10909, "type": "theorem", "label": "varieties-lemma-check-only-finite-inseparable-extensions", "categories": [ "varieties" ], "title": "varieties-lemma-check-only-finite-inseparable-extensions", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$.", "Let $x \\in X$. The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically reduced at $x$,", "\\item $\\mathcal{O}_{X_{k'}, x'}$ is reduced for every", "finite purely inseparable field extension $k'$ of $k$ and", "$x' \\in X_{k'}$ the unique point lying over $x$,", "\\item $\\mathcal{O}_{X_{k^{1/p}}, x'}$ is reduced for", "$x' \\in X_{k^{1/p}}$ the unique point lying over $x$, and", "\\item $\\mathcal{O}_{X_{k^{perf}}, x'}$ is reduced for", "$x' \\in X_{k^{perf}}$ the unique point lying over $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that if $k \\subset k'$ is purely inseparable, then", "$X_{k'} \\to X$ induces a homeomorphism on underlying topological", "spaces, see Algebra, Lemma \\ref{algebra-lemma-p-ring-map}.", "Whence the uniqueness of $x'$ lying over $x$ mentioned in the", "statement. Moreover, in this case", "$\\mathcal{O}_{X_{k'}, x'} = \\mathcal{O}_{X, x} \\otimes_k k'$.", "Hence the lemma follows from Lemma \\ref{lemma-geometrically-reduced-at-point}", "above and Algebra, Lemma", "\\ref{algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension}." ], "refs": [ "algebra-lemma-p-ring-map", "varieties-lemma-geometrically-reduced-at-point", "algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension" ], "ref_ids": [ 582, 10906, 571 ] } ], "ref_ids": [] }, { "id": 10910, "type": "theorem", "label": "varieties-lemma-geometrically-reduced-upstairs", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-reduced-upstairs", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme over $k$.", "Let $k'/k$ be a field extension.", "Let $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying over $x$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically reduced at $x$,", "\\item $X_{k'}$ is geometrically reduced at $x'$.", "\\end{enumerate}", "In particular, $X$ is geometrically reduced over $k$ if and only if", "$X_{k'}$ is geometrically reduced over $k'$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2). Assume (2).", "Let $k \\subset k''$ be a finite purely inseparable field extension", "and let $x'' \\in X_{k''}$ be a point lying over $x$ (actually it is", "unique). We can find a common field extension $k \\subset k'''$", "(i.e.\\ with both $k' \\subset k'''$ and $k'' \\subset k'''$) and a point", "$x''' \\in X_{k'''}$ lying over both $x'$ and $x''$.", "Consider the map of local rings", "$$", "\\mathcal{O}_{X_{k''}, x''} \\longrightarrow \\mathcal{O}_{X_{k'''}, x''''}.", "$$", "This is a flat local ring homomorphism and hence faithfully flat.", "By (2) we see that the local ring on the right is reduced.", "Thus by Algebra, Lemma \\ref{algebra-lemma-descent-reduced}", "we conclude that $\\mathcal{O}_{X_{k''}, x''}$ is reduced.", "Thus by Lemma \\ref{lemma-check-only-finite-inseparable-extensions}", "we conclude that $X$ is geometrically reduced at $x$." ], "refs": [ "algebra-lemma-descent-reduced", "varieties-lemma-check-only-finite-inseparable-extensions" ], "ref_ids": [ 1371, 10909 ] } ], "ref_ids": [] }, { "id": 10911, "type": "theorem", "label": "varieties-lemma-geometrically-reduced-any-base-change", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-reduced-any-base-change", "contents": [ "Let $k$ be a field.", "Let $X$, $Y$ be schemes over $k$.", "\\begin{enumerate}", "\\item If $X$ is geometrically reduced at $x$, and $Y$ reduced,", "then $X \\times_k Y$ is reduced at every point lying over $x$.", "\\item If $X$ geometrically reduced over $k$ and $Y$ reduced.", "Then $X \\times_k Y$ is reduced.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Combine, Lemmas \\ref{lemma-geometrically-reduced-at-point}", "and \\ref{lemma-geometrically-reduced} and Algebra,", "Lemma \\ref{algebra-lemma-geometrically-reduced-any-reduced-base-change}." ], "refs": [ "varieties-lemma-geometrically-reduced-at-point", "varieties-lemma-geometrically-reduced", "algebra-lemma-geometrically-reduced-any-reduced-base-change" ], "ref_ids": [ 10906, 10908, 564 ] } ], "ref_ids": [] }, { "id": 10912, "type": "theorem", "label": "varieties-lemma-generic-points-geometrically-reduced", "categories": [ "varieties" ], "title": "varieties-lemma-generic-points-geometrically-reduced", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme over $k$.", "\\begin{enumerate}", "\\item If $x' \\leadsto x$ is a specialization and $X$ is geometrically", "reduced at $x$, then $X$ is geometrically reduced at $x'$.", "\\item If $x \\in X$ such that (a) $\\mathcal{O}_{X, x}$", "is reduced, and (b) for each specialization $x' \\leadsto x$ where", "$x'$ is a generic point of an irreducible component of $X$ the", "scheme $X$ is geometrically reduced at $x'$, then $X$ is geometrically", "reduced at $x$.", "\\item If $X$ is reduced and geometrically reduced at all generic", "points of irreducible components of $X$, then $X$ is geometrically", "reduced.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from", "Lemma \\ref{lemma-geometrically-reduced-at-point}", "and the fact that if $A$ is a geometrically reduced", "$k$-algebra, then $S^{-1}A$ is a geometrically reduced $k$-algebra for", "any multiplicative subset $S$ of $A$, see", "Algebra, Lemma \\ref{algebra-lemma-geometrically-reduced-permanence}.", "\\medskip\\noindent", "Let $A = \\mathcal{O}_{X, x}$. The assumptions (a) and (b) of (2) imply", "that $A$ is reduced, and that $A_{\\mathfrak q}$ is geometrically", "reduced over $k$ for every minimal prime $\\mathfrak q$ of $A$.", "Hence $A$ is geometrically reduced over $k$, see", "Algebra, Lemma \\ref{algebra-lemma-generic-points-geometrically-reduced}.", "Thus $X$ is geometrically reduced at $x$, see", "Lemma \\ref{lemma-geometrically-reduced-at-point}.", "\\medskip\\noindent", "Part (3) follows trivially from part (2)." ], "refs": [ "varieties-lemma-geometrically-reduced-at-point", "algebra-lemma-geometrically-reduced-permanence", "algebra-lemma-generic-points-geometrically-reduced", "varieties-lemma-geometrically-reduced-at-point" ], "ref_ids": [ 10906, 562, 566, 10906 ] } ], "ref_ids": [] }, { "id": 10913, "type": "theorem", "label": "varieties-lemma-Noetherian-geometrically-reduced-at-point", "categories": [ "varieties" ], "title": "varieties-lemma-Noetherian-geometrically-reduced-at-point", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme over $k$.", "Let $x \\in X$.", "Assume $X$ locally Noetherian and geometrically reduced at $x$.", "Then there exists an open neighbourhood $U \\subset X$ of $x$", "which is geometrically reduced over $k$." ], "refs": [], "proofs": [ { "contents": [ "Assume $X$ locally Noetherian and geometrically reduced at $x$.", "By Properties, Lemma", "\\ref{properties-lemma-ring-affine-open-injective-local-ring}", "we can find an affine open neighbourhood $U \\subset X$ of $x$ such that", "$R = \\mathcal{O}_X(U) \\to \\mathcal{O}_{X, x}$", "is injective. By", "Lemma \\ref{lemma-geometrically-reduced-at-point} the assumption", "means that $\\mathcal{O}_{X, x}$ is geometrically reduced over $k$.", "By Algebra, Lemma \\ref{algebra-lemma-subalgebra-separable}", "this implies that $R$ is geometrically reduced over $k$, which", "in turn implies that $U$ is geometrically reduced." ], "refs": [ "properties-lemma-ring-affine-open-injective-local-ring", "varieties-lemma-geometrically-reduced-at-point", "algebra-lemma-subalgebra-separable" ], "ref_ids": [ 3065, 10906, 561 ] } ], "ref_ids": [] }, { "id": 10914, "type": "theorem", "label": "varieties-lemma-finite-extension-geometrically-reduced", "categories": [ "varieties" ], "title": "varieties-lemma-finite-extension-geometrically-reduced", "contents": [ "Let $k$ be a field.", "Let $X \\to \\Spec(k)$ be locally of finite type.", "Assume $X$ has finitely many irreducible components.", "Then there exists a finite purely inseparable extension $k \\subset k'$", "such that $(X_{k'})_{red}$ is geometrically reduced over $k'$." ], "refs": [], "proofs": [ { "contents": [ "To prove this lemma we may replace $X$ by its reduction $X_{red}$.", "Hence we may assume that $X$ is reduced and locally of finite type", "over $k$.", "Let $x_1, \\ldots, x_n \\in X$ be the generic points of the irreducible", "components of $X$.", "Note that for every purely inseparable algebraic extension $k \\subset k'$", "the morphism $(X_{k'})_{red} \\to X$ is a homeomorphism, see", "Algebra, Lemma \\ref{algebra-lemma-p-ring-map}. Hence the points", "$x'_1, \\ldots, x'_n$ lying over $x_1, \\ldots, x_n$ are the generic", "points of the irreducible components of $(X_{k'})_{red}$.", "As $X$ is reduced the local rings $K_i = \\mathcal{O}_{X, x_i}$ are fields, see", "Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring}.", "As $X$ is locally of finite type over $k$ the field extensions", "$k \\subset K_i$ are finitely generated field extensions.", "Finally, the local rings $\\mathcal{O}_{(X_{k'})_{red}, x'_i}$ are the", "fields $(K_i \\otimes_k k')_{red}$. By", "Algebra, Lemma \\ref{algebra-lemma-make-separable}", "we can find a finite purely inseparable extension $k \\subset k'$", "such that $(K_i \\otimes_k k')_{red}$ are separable field", "extensions of $k'$. In particular each $(K_i \\otimes_k k')_{red}$", "is geometrically reduced over $k'$ by", "Algebra, Lemma \\ref{algebra-lemma-characterize-separable-field-extensions}.", "At this point", "Lemma \\ref{lemma-generic-points-geometrically-reduced} part (3)", "implies that $(X_{k'})_{red}$ is geometrically reduced." ], "refs": [ "algebra-lemma-p-ring-map", "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-make-separable", "algebra-lemma-characterize-separable-field-extensions", "varieties-lemma-generic-points-geometrically-reduced" ], "ref_ids": [ 582, 418, 573, 569, 10912 ] } ], "ref_ids": [] }, { "id": 10915, "type": "theorem", "label": "varieties-lemma-geometrically-connected-check-after-extension", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-connected-check-after-extension", "contents": [ "Let $X$ be a scheme over the field $k$.", "Let $k \\subset k'$ be a field extension.", "Then $X$ is geometrically connected over $k$ if and only if", "$X_{k'}$ is geometrically connected over $k'$." ], "refs": [], "proofs": [ { "contents": [ "If $X$ is geometrically connected over $k$, then it is clear that", "$X_{k'}$ is geometrically connected over $k'$. For the converse, note", "that for any field extension $k \\subset k''$ there exists a common", "field extension $k' \\subset k'''$ and $k'' \\subset k'''$. As the", "morphism $X_{k'''} \\to X_{k''}$ is surjective (as a base change of", "a surjective morphism between spectra of fields) we see that the", "connectedness of $X_{k'''}$ implies the connectedness of $X_{k''}$.", "Thus if $X_{k'}$ is geometrically connected over $k'$ then", "$X$ is geometrically connected over $k$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10916, "type": "theorem", "label": "varieties-lemma-bijection-connected-components", "categories": [ "varieties" ], "title": "varieties-lemma-bijection-connected-components", "contents": [ "Let $k$ be a field.", "Let $X$, $Y$ be schemes over $k$.", "Assume $X$ is geometrically connected over $k$.", "Then the projection morphism", "$$", "p : X \\times_k Y \\longrightarrow Y", "$$", "induces a bijection between connected components." ], "refs": [], "proofs": [ { "contents": [ "The scheme theoretic fibres of $p$ are connected, since they", "are base changes of the geometrically connected scheme $X$ by", "field extensions. Moreover the scheme theoretic fibres are", "homeomorphic to the set theoretic fibres, see", "Schemes, Lemma \\ref{schemes-lemma-fibre-topological}.", "By", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open}", "the map $p$ is open.", "Thus we may apply Topology,", "Lemma \\ref{topology-lemma-connected-fibres-connected-components}", "to conclude." ], "refs": [ "schemes-lemma-fibre-topological", "morphisms-lemma-scheme-over-field-universally-open", "topology-lemma-connected-fibres-connected-components" ], "ref_ids": [ 7696, 5254, 8208 ] } ], "ref_ids": [] }, { "id": 10917, "type": "theorem", "label": "varieties-lemma-affine-geometrically-connected", "categories": [ "varieties" ], "title": "varieties-lemma-affine-geometrically-connected", "contents": [ "Let $k$ be a field.", "Let $A$ be a $k$-algebra.", "Then $X = \\Spec(A)$ is geometrically connected over $k$", "if and only if $A$ is geometrically connected over $k$ (see", "Algebra, Definition \\ref{algebra-definition-geometrically-connected})." ], "refs": [ "algebra-definition-geometrically-connected" ], "proofs": [ { "contents": [ "Immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1465 ] }, { "id": 10918, "type": "theorem", "label": "varieties-lemma-separably-closed-field-connected-components", "categories": [ "varieties" ], "title": "varieties-lemma-separably-closed-field-connected-components", "contents": [ "Let $k \\subset k'$ be an extension of fields.", "Let $X$ be a scheme over $k$.", "Assume $k$ separably algebraically closed.", "Then the morphism $X_{k'} \\to X$ induces a bijection of connected", "components. In particular, $X$ is geometrically connected over $k$", "if and only if $X$ is connected." ], "refs": [], "proofs": [ { "contents": [ "Since $k$ is separably algebraically closed we see that", "$k'$ is geometrically connected over $k$, see", "Algebra,", "Lemma \\ref{algebra-lemma-separably-closed-connected-implies-geometric}.", "Hence $Z = \\Spec(k')$ is geometrically connected over $k$ by", "Lemma \\ref{lemma-affine-geometrically-connected}", "above. Since $X_{k'} = Z \\times_k X$ the result is a special case of", "Lemma \\ref{lemma-bijection-connected-components}." ], "refs": [ "algebra-lemma-separably-closed-connected-implies-geometric", "varieties-lemma-affine-geometrically-connected", "varieties-lemma-bijection-connected-components" ], "ref_ids": [ 602, 10917, 10916 ] } ], "ref_ids": [] }, { "id": 10919, "type": "theorem", "label": "varieties-lemma-characterize-geometrically-connected", "categories": [ "varieties" ], "title": "varieties-lemma-characterize-geometrically-connected", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme over $k$.", "Let $\\overline{k}$ be a separable algebraic closure of $k$.", "Then $X$ is geometrically connected if and only if the base change", "$X_{\\overline{k}}$ is connected." ], "refs": [], "proofs": [ { "contents": [ "Assume $X_{\\overline{k}}$ is connected.", "Let $k \\subset k'$ be a field extension.", "There exists a field extension $\\overline{k} \\subset \\overline{k}'$", "such that $k'$ embeds into $\\overline{k}'$ as an extension of $k$.", "By Lemma \\ref{lemma-separably-closed-field-connected-components}", "we see that $X_{\\overline{k}'}$ is connected.", "Since $X_{\\overline{k}'} \\to X_{k'}$ is surjective we conclude", "that $X_{k'}$ is connected as desired." ], "refs": [ "varieties-lemma-separably-closed-field-connected-components" ], "ref_ids": [ 10918 ] } ], "ref_ids": [] }, { "id": 10920, "type": "theorem", "label": "varieties-lemma-descend-open", "categories": [ "varieties" ], "title": "varieties-lemma-descend-open", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme over $k$.", "Let $A$ be a $k$-algebra.", "Let $V \\subset X_A$ be a quasi-compact open.", "Then there exists a finitely generated $k$-subalgebra $A' \\subset A$", "and a quasi-compact open $V' \\subset X_{A'}$", "such that $V = V'_A$." ], "refs": [], "proofs": [ { "contents": [ "We remark that if $X$ is also quasi-separated this follows from", "Limits, Lemma \\ref{limits-lemma-descend-opens}. Let", "$U_1, \\ldots, U_n$ be finitely many affine opens of $X$", "such that $V \\subset \\bigcup U_{i, A}$. Say $U_i = \\Spec(R_i)$.", "Since $V$ is quasi-compact we can find finitely many", "$f_{ij} \\in R_i \\otimes_k A$, $j = 1, \\ldots, n_i$", "such that $V = \\bigcup_i \\bigcup_{j = 1, \\ldots, n_i} D(f_{ij})$", "where $D(f_{ij}) \\subset U_{i, A}$ is the corresponding standard", "open. (We do not claim that $V \\cap U_{i, A}$ is the union", "of the $D(f_{ij})$, $j = 1, \\ldots, n_i$.)", "It is clear that we can find a finitely generated $k$-subalgebra", "$A' \\subset A$ such that $f_{ij}$ is the image of some", "$f_{ij}' \\in R_i \\otimes_k A'$.", "Set $V' = \\bigcup D(f_{ij}')$ which is a quasi-compact open of $X_{A'}$.", "Denote $\\pi : X_A \\to X_{A'}$ the canonical morphism.", "We have $\\pi(V) \\subset V'$ as $\\pi(D(f_{ij})) \\subset D(f_{ij}')$.", "If $x \\in X_A$ with $\\pi(x) \\in V'$, then $\\pi(x) \\in D(f_{ij}')$", "for some $i, j$ and we see that $x \\in D(f_{ij})$ as $f_{ij}'$", "maps to $f_{ij}$. Thus we see that $V = \\pi^{-1}(V')$ as desired." ], "refs": [ "limits-lemma-descend-opens" ], "ref_ids": [ 15041 ] } ], "ref_ids": [] }, { "id": 10921, "type": "theorem", "label": "varieties-lemma-Galois-action-quasi-compact-open", "categories": [ "varieties" ], "title": "varieties-lemma-Galois-action-quasi-compact-open", "contents": [ "Let $k$ be a field. Let $X$ be a scheme over $k$.", "Let $\\overline{k}$ be a (possibly infinite) Galois extension of $k$.", "Let $V \\subset X_{\\overline{k}}$ be a quasi-compact open.", "Then", "\\begin{enumerate}", "\\item there exists a finite subextension $k \\subset k' \\subset \\overline{k}$", "and a quasi-compact open $V' \\subset X_{k'}$ such that", "$V = (V')_{\\overline{k}}$,", "\\item there exists an open subgroup $H \\subset \\text{Gal}(\\overline{k}/k)$", "such that $\\sigma(V) = V$ for all $\\sigma \\in H$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-descend-open} there exists a finite subextension", "$k \\subset k' \\subset \\overline{k}$ and an open $V' \\subset X_{k'}$", "which pulls back to $V$. This proves (1). Since $\\text{Gal}(\\overline{k}/k')$", "is open in $\\text{Gal}(\\overline{k}/k)$ part (2) is clear as well." ], "refs": [ "varieties-lemma-descend-open" ], "ref_ids": [ 10920 ] } ], "ref_ids": [] }, { "id": 10922, "type": "theorem", "label": "varieties-lemma-closed-fixed-by-Galois", "categories": [ "varieties" ], "title": "varieties-lemma-closed-fixed-by-Galois", "contents": [ "Let $k$ be a field. Let $k \\subset \\overline{k}$ be a (possibly infinite)", "Galois extension. Let $X$ be a scheme over $k$. Let", "$\\overline{T} \\subset X_{\\overline{k}}$ have the following properties", "\\begin{enumerate}", "\\item $\\overline{T}$ is a closed subset of $X_{\\overline{k}}$,", "\\item for every $\\sigma \\in \\text{Gal}(\\overline{k}/k)$", "we have $\\sigma(\\overline{T}) = \\overline{T}$.", "\\end{enumerate}", "Then there exists a closed subset $T \\subset X$ whose inverse image", "in $X_{\\overline{k}}$ is $\\overline{T}$." ], "refs": [], "proofs": [ { "contents": [ "This lemma immediately reduces to the case where $X = \\Spec(A)$", "is affine. In this case, let $\\overline{I} \\subset A \\otimes_k \\overline{k}$", "be the radical ideal corresponding to $\\overline{T}$.", "Assumption (2) implies that $\\sigma(\\overline{I}) = \\overline{I}$", "for all $\\sigma \\in \\text{Gal}(\\overline{k}/k)$.", "Pick $x \\in \\overline{I}$. There exists a finite Galois extension", "$k \\subset k'$ contained in $\\overline{k}$ such that $x \\in A \\otimes_k k'$.", "Set $G = \\text{Gal}(k'/k)$. Set", "$$", "P(T) = \\prod\\nolimits_{\\sigma \\in G} (T - \\sigma(x)) \\in (A \\otimes_k k')[T]", "$$", "It is clear that $P(T)$ is monic and is actually an element of", "$(A \\otimes_k k')^G[T] = A[T]$ (by basic Galois theory).", "Moreover, if we write $P(T) = T^d + a_1T^{d - 1} + \\ldots + a_0$", "the we see that $a_i \\in I := A \\cap \\overline{I}$. By", "Algebra, Lemma \\ref{algebra-lemma-polynomials-divide}", "we see that $x$ is contained in the radical of $I(A \\otimes_k \\overline{k})$.", "Hence $\\overline{I}$ is the radical of $I(A \\otimes_k \\overline{k})$ and", "setting $T = V(I)$ is a solution." ], "refs": [ "algebra-lemma-polynomials-divide" ], "ref_ids": [ 520 ] } ], "ref_ids": [] }, { "id": 10923, "type": "theorem", "label": "varieties-lemma-characterize-geometrically-disconnected", "categories": [ "varieties" ], "title": "varieties-lemma-characterize-geometrically-disconnected", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme over $k$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically connected,", "\\item for every finite separable field extension $k \\subset k'$", "the scheme $X_{k'}$ is connected.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It follows immediately from the definition that (1) implies (2).", "Assume that $X$ is not geometrically connected.", "Let $k \\subset \\overline{k}$ be a separable algebraic", "closure of $k$. By", "Lemma \\ref{lemma-characterize-geometrically-connected}", "it follows that $X_{\\overline{k}}$ is disconnected.", "Say $X_{\\overline{k}} = \\overline{U} \\amalg \\overline{V}$", "with $\\overline{U}$ and $\\overline{V}$ open, closed, and nonempty.", "\\medskip\\noindent", "Suppose that $W \\subset X$ is any quasi-compact open.", "Then $W_{\\overline{k}} \\cap \\overline{U}$ and", "$W_{\\overline{k}} \\cap \\overline{V}$ are open and closed in", "$W_{\\overline{k}}$. In particular $W_{\\overline{k}} \\cap \\overline{U}$ and", "$W_{\\overline{k}} \\cap \\overline{V}$ are quasi-compact, and by", "Lemma \\ref{lemma-Galois-action-quasi-compact-open}", "both $W_{\\overline{k}} \\cap \\overline{U}$ and", "$W_{\\overline{k}} \\cap \\overline{V}$", "are defined over a finite subextension and invariant under an", "open subgroup of $\\text{Gal}(\\overline{k}/k)$.", "We will use this without further mention in the following.", "\\medskip\\noindent", "Pick $W_0 \\subset X$ quasi-compact open such that both", "$W_{0, \\overline{k}} \\cap \\overline{U}$ and", "$W_{0, \\overline{k}} \\cap \\overline{V}$ are nonempty.", "Choose a finite subextension $k \\subset k' \\subset \\overline{k}$", "and a decomposition $W_{0, k'} = U_0' \\amalg V_0'$ into open and closed", "subsets such that", "$W_{0, \\overline{k}} \\cap \\overline{U} = (U'_0)_{\\overline{k}}$ and", "$W_{0, \\overline{k}} \\cap \\overline{V} = (V'_0)_{\\overline{k}}$.", "Let $H = \\text{Gal}(\\overline{k}/k') \\subset \\text{Gal}(\\overline{k}/k)$.", "In particular", "$\\sigma(W_{0, \\overline{k}} \\cap \\overline{U}) =", "W_{0, \\overline{k}} \\cap \\overline{U}$ and similarly for", "$\\overline{V}$.", "\\medskip\\noindent", "Having chosen $W_0$, $k'$ as above, for every quasi-compact open", "$W \\subset X$ we set", "$$", "U_W =", "\\bigcap\\nolimits_{\\sigma \\in H} \\sigma(W_{\\overline{k}} \\cap \\overline{U}),", "\\quad", "V_W =", "\\bigcup\\nolimits_{\\sigma \\in H} \\sigma(W_{\\overline{k}} \\cap \\overline{V}).", "$$", "Now, since $W_{\\overline{k}} \\cap \\overline{U}$ and", "$W_{\\overline{k}} \\cap \\overline{V}$ are fixed by an open subgroup of", "$\\text{Gal}(\\overline{k}/k)$ we see that the union and intersection", "above are finite. Hence $U_W$ and $V_W$ are both open and closed.", "Also, by construction $W_{\\bar k} = U_W \\amalg V_W$.", "\\medskip\\noindent", "We claim that if $W \\subset W' \\subset X$ are quasi-compact", "open, then $W_{\\overline{k}} \\cap U_{W'} = U_W$ and", "$W_{\\overline{k}} \\cap V_{W'} = V_W$. Verification omitted.", "Hence we see that upon defining $U = \\bigcup_{W \\subset X} U_W$", "and $V = \\bigcup_{W \\subset X} V_W$ we obtain", "$X_{\\overline{k}} = U \\amalg V$ is a disjoint union of open", "and closed subsets.", "It is clear that $V$ is nonempty as it is constructed by taking", "unions (locally). On the other hand, $U$ is nonempty since it contains", "$W_0 \\cap \\overline{U}$ by construction. Finally, $U, V \\subset X_{\\bar k}$", "are closed and $H$-invariant by construction. Hence by", "Lemma \\ref{lemma-closed-fixed-by-Galois}", "we have $U = (U')_{\\bar k}$, and $V = (V')_{\\bar k}$ for some", "closed $U', V' \\subset X_{k'}$. Clearly $X_{k'} = U' \\amalg V'$", "and we see that $X_{k'}$ is disconnected as desired." ], "refs": [ "varieties-lemma-characterize-geometrically-connected", "varieties-lemma-Galois-action-quasi-compact-open", "varieties-lemma-closed-fixed-by-Galois" ], "ref_ids": [ 10919, 10921, 10922 ] } ], "ref_ids": [] }, { "id": 10924, "type": "theorem", "label": "varieties-lemma-tricky", "categories": [ "varieties" ], "title": "varieties-lemma-tricky", "contents": [ "Let $k$ be a field. Let $k \\subset \\overline{k}$ be a (possibly infinite)", "Galois extension. Let $f : T \\to X$ be a morphism of schemes over $k$.", "Assume $T_{\\overline{k}}$ connected and $X_{\\overline{k}}$", "disconnected. Then $X$ is disconnected." ], "refs": [], "proofs": [ { "contents": [ "Write $X_{\\overline{k}} = \\overline{U} \\amalg \\overline{V}$", "with $\\overline{U}$ and $\\overline{V}$ open and closed.", "Denote $\\overline{f} : T_{\\overline{k}} \\to X_{\\overline{k}}$ the base", "change of $f$. Since $T_{\\overline{k}}$ is connected we see that", "$T_{\\overline{k}}$ is contained in either $\\overline{f}^{-1}(\\overline{U})$", "or $\\overline{f}^{-1}(\\overline{V})$.", "Say $T_{\\overline{k}} \\subset \\overline{f}^{-1}(\\overline{U})$.", "\\medskip\\noindent", "Fix a quasi-compact open $W \\subset X$. There exists a", "finite Galois subextension $k \\subset k' \\subset \\overline{k}$", "such that $\\overline{U} \\cap W_{\\overline{k}}$ and", "$\\overline{V} \\cap W_{\\overline{k}}$ come from quasi-compact", "opens $U', V' \\subset W_{k'}$. Then also $W_{k'} = U' \\amalg V'$.", "Consider", "$$", "U'' = \\bigcap\\nolimits_{\\sigma \\in \\text{Gal}(k'/k)} \\sigma(U'),", "\\quad", "V'' = \\bigcup\\nolimits_{\\sigma \\in \\text{Gal}(k'/k)} \\sigma(V').", "$$", "These are Galois invariant, open and closed, and", "$W_{k'} = U'' \\amalg V''$.", "By Lemma \\ref{lemma-closed-fixed-by-Galois} we get open and closed subsets", "$U_W, V_W \\subset W$ such that", "$U'' = (U_W)_{k'}$, $V'' = (V_W)_{k'}$ and", "$W = U_W \\amalg V_W$.", "\\medskip\\noindent", "We claim that if $W \\subset W' \\subset X$ are quasi-compact", "open, then $W \\cap U_{W'} = U_W$ and $W \\cap V_{W'} = V_W$.", "Verification omitted.", "Hence we see that upon defining $U = \\bigcup_{W \\subset X} U_W$", "and $V = \\bigcup_{W \\subset X} V_W$ we obtain $X = U \\amalg V$.", "It is clear that $V$ is nonempty as it is constructed by taking", "unions (locally). On the other hand, $U$ is nonempty since it contains", "$f(T)$ by construction." ], "refs": [ "varieties-lemma-closed-fixed-by-Galois" ], "ref_ids": [ 10922 ] } ], "ref_ids": [] }, { "id": 10925, "type": "theorem", "label": "varieties-lemma-geometrically-connected-criterion", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-connected-criterion", "contents": [ "\\begin{reference}", "\\cite[IV Corollary 4.5.13.1(i)]{EGA}", "\\end{reference}", "Let $k$ be a field. Let $T \\to X$ be a morphism of schemes over $k$.", "Assume $T$ is geometrically connected and $X$ connected.", "Then $X$ is geometrically connected." ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of", "Lemma \\ref{lemma-tricky}." ], "refs": [ "varieties-lemma-tricky" ], "ref_ids": [ 10924 ] } ], "ref_ids": [] }, { "id": 10926, "type": "theorem", "label": "varieties-lemma-geometrically-connected-if-connected-and-point", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-connected-if-connected-and-point", "contents": [ "Let $k$ be a field. Let $X$ be a scheme over $k$.", "Assume $X$ is connected and has a point $x$ such that", "$k$ is algebraically closed in $\\kappa(x)$.", "Then $X$ is geometrically connected.", "In particular, if $X$ has a $k$-rational point and $X$ is connected,", "then $X$ is geometrically connected." ], "refs": [], "proofs": [ { "contents": [ "Set $T = \\Spec(\\kappa(x))$. Let $k \\subset \\overline{k}$ be a", "separable algebraic closure of $k$. The assumption on $k \\subset \\kappa(x)$", "implies that $T_{\\overline{k}}$ is irreducible, see", "Algebra, Lemma \\ref{algebra-lemma-field-extension-geometrically-irreducible}.", "Hence by", "Lemma \\ref{lemma-geometrically-connected-criterion}", "we see that $X_{\\overline{k}}$ is connected. By", "Lemma \\ref{lemma-characterize-geometrically-connected}", "we conclude that $X$ is geometrically connected." ], "refs": [ "algebra-lemma-field-extension-geometrically-irreducible", "varieties-lemma-geometrically-connected-criterion", "varieties-lemma-characterize-geometrically-connected" ], "ref_ids": [ 593, 10925, 10919 ] } ], "ref_ids": [] }, { "id": 10927, "type": "theorem", "label": "varieties-lemma-inverse-image-connected-component", "categories": [ "varieties" ], "title": "varieties-lemma-inverse-image-connected-component", "contents": [ "Let $k \\subset K$ be an extension of fields.", "Let $X$ be a scheme over $k$.", "For every connected component $T$ of $X$ the inverse image", "$T_K \\subset X_K$ is a union of connected components of $X_K$." ], "refs": [], "proofs": [ { "contents": [ "This is a purely topological statement.", "Denote $p : X_K \\to X$ the projection morphism.", "Let $T \\subset X$ be a connected component of $X$.", "Let $t \\in T_K = p^{-1}(T)$. Let $C \\subset X_K$ be a connected component", "containing $t$. Then $p(C)$ is a connected subset of $X$", "which meets $T$, hence $p(C) \\subset T$. Hence $C \\subset T_K$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10928, "type": "theorem", "label": "varieties-lemma-image-connected-component-finite-extension", "categories": [ "varieties" ], "title": "varieties-lemma-image-connected-component-finite-extension", "contents": [ "Let $k \\subset K$ be a finite extension of fields and let $X$ be a scheme over ", "$k$. Denote by $p : X_K \\to X$ the projection morphism. For every connected ", "component $T$ of $X_K$ the image $p(T)$ is a connected component of ", "$X$." ], "refs": [], "proofs": [ { "contents": [ "The image $p(T)$ is contained in some connected component $X'$ of $X$. Consider", "$X'$ as a closed subscheme of $X$ in any way. Then $T$ is also a connected", "component of $X'_K = p^{-1}(X')$ and we may therefore assume that $X$ is", "connected. The morphism $p$ is open", "(Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open}), ", "closed", "(Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed})", "and the fibers of $p$ are finite sets", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-quasi-finite}).", "Thus we may apply", "Topology, Lemma \\ref{topology-lemma-finite-fibre-connected-components}", "to conclude." ], "refs": [ "morphisms-lemma-scheme-over-field-universally-open", "morphisms-lemma-integral-universally-closed", "morphisms-lemma-finite-quasi-finite", "topology-lemma-finite-fibre-connected-components" ], "ref_ids": [ 5254, 5441, 5444, 8209 ] } ], "ref_ids": [] }, { "id": 10929, "type": "theorem", "label": "varieties-lemma-image-connected-component", "categories": [ "varieties" ], "title": "varieties-lemma-image-connected-component", "contents": [ "\\begin{reference}", "Email from Ofer Gabber dated June 4, 2016", "\\end{reference}", "Let $k \\subset K$ be an extension of fields. Let $X$ be a scheme over $k$.", "Denote $p : X_K \\to X$ the projection morphism.", "Let $\\overline{T} \\subset X_K$ be a connected component.", "Then $p(\\overline{T})$ is a connected component of $X$." ], "refs": [], "proofs": [ { "contents": [ "When $k \\subset K$ is finite this is", "Lemma \\ref{lemma-image-connected-component-finite-extension}.", "In general the proof is more difficult.", "\\medskip\\noindent", "Let $T \\subset X$ be the connected component of $X$ containing", "the image of $\\overline{T}$. We may replace $X$ by $T$", "(with the induced reduced subscheme structure). Thus we", "may assume $X$ is connected. Let $A = H^0(X, \\mathcal{O}_X)$.", "Let $L \\subset A$ be the maximal weakly \\'etale $k$-subalgebra, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-max-weakly-etale-subalgebra}.", "Since $A$ does not have any nontrivial idempotents we see", "that $L$ is a field and a separable algebraic extension of $k$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-class-weakly-etale-over-field}.", "Observe that $L$ is also the maximal weakly \\'etale $L$-subalgebra of $A$", "(because any weakly \\'etale $L$-algebra is weakly \\'etale over $k$", "by More on Algebra, Lemma \\ref{more-algebra-lemma-composition-weakly-etale}).", "By Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}", "we obtain a factorization $X \\to \\Spec(L) \\to \\Spec(k)$", "of the structure morphism.", "\\medskip\\noindent", "Let $L'/L$ be a finite separable extension. By", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-finite-locally-free-base-change-cohomology}", "we have", "$$", "A \\otimes_L L' =", "H^0(X \\times_{\\Spec(L)} \\Spec(L'), \\mathcal{O}_{X \\times_{\\Spec(L)} \\Spec(L')})", "$$", "The maximal weakly \\'etale $L'$-subalgebra of $A \\otimes_L L'$", "is $L \\otimes_L L' = L'$ by More on Algebra, Lemma", "\\ref{more-algebra-lemma-change-fields-max-weakly-etale-subalgebra}.", "In particular $A \\otimes_L L'$ does not have nontrivial idempotents", "(such an idempotent would generate a weakly \\'etale subalgebra)", "and we conclude that $X \\times_{\\Spec(L)} \\Spec(L')$ is connected.", "By Lemma \\ref{lemma-characterize-geometrically-disconnected}", "we conclude that $X$ is geometrically connected over $L$.", "\\medskip\\noindent", "Let's give $\\overline{T}$ the reduced induced scheme structure", "and consider the composition", "$$", "\\overline{T} \\xrightarrow{i} X_K = X \\times_{\\Spec(k)} \\Spec(K)", "\\xrightarrow{\\pi}", "\\Spec(L \\otimes_k K)", "$$", "The image is contained in a connected component of $\\Spec(L \\otimes_k K)$.", "Since $K \\to L \\otimes_k K$ is integral we see that", "the connected components of $\\Spec(L \\otimes_k K)$", "are points and all points are closed, see", "Algebra, Lemma \\ref{algebra-lemma-integral-over-field}.", "Thus we get a quotient field $L \\otimes_k K \\to E$", "such that $\\overline{T}$ maps into $\\Spec(E) \\subset \\Spec(L \\otimes_k K)$.", "Hence $i(\\overline{T}) \\subset \\pi^{-1}(\\Spec(E))$. But", "$$", "\\pi^{-1}(\\Spec(E)) =", "(X \\times_{\\Spec(k)} \\Spec(K)) \\times_{\\Spec(L \\otimes_k K)} \\Spec(E) =", "X \\times_{\\Spec(L)} \\Spec(E)", "$$", "which is connected because $X$ is geometrically connected over $L$.", "Then we get the equality", "$\\overline{T} = X \\times_{\\Spec(L)} \\Spec(E)$ (set theoretically)", "and we conclude that $\\overline{T} \\to X$ is surjective as desired." ], "refs": [ "varieties-lemma-image-connected-component-finite-extension", "more-algebra-lemma-max-weakly-etale-subalgebra", "more-algebra-lemma-class-weakly-etale-over-field", "more-algebra-lemma-composition-weakly-etale", "schemes-lemma-morphism-into-affine", "coherent-lemma-finite-locally-free-base-change-cohomology", "more-algebra-lemma-change-fields-max-weakly-etale-subalgebra", "varieties-lemma-characterize-geometrically-disconnected", "algebra-lemma-integral-over-field" ], "ref_ids": [ 10928, 10462, 10461, 10445, 7655, 3299, 10464, 10923, 497 ] } ], "ref_ids": [] }, { "id": 10930, "type": "theorem", "label": "varieties-lemma-galois-action-connected-components", "categories": [ "varieties" ], "title": "varieties-lemma-galois-action-connected-components", "contents": [ "Let $k$ be a field, with separable algebraic closure $\\overline{k}$.", "Let $X$ be a scheme over $k$.", "There is an action", "$$", "\\text{Gal}(\\overline{k}/k)^{opp} \\times \\pi_0(X_{\\overline{k}})", "\\longrightarrow", "\\pi_0(X_{\\overline{k}})", "$$", "with the following properties:", "\\begin{enumerate}", "\\item An element $\\overline{T} \\in \\pi_0(X_{\\overline{k}})$", "is fixed by the action if and only if there exists a connected component", "$T \\subset X$, which is geometrically connected over $k$,", "such that $T_{\\overline{k}} = \\overline{T}$.", "\\item For any field extension $k \\subset k'$ with separable", "algebraic closure $\\overline{k}'$ the diagram", "$$", "\\xymatrix{", "\\text{Gal}(\\overline{k}'/k') \\times \\pi_0(X_{\\overline{k}'})", "\\ar[r] \\ar[d] &", "\\pi_0(X_{\\overline{k}'}) \\ar[d] \\\\", "\\text{Gal}(\\overline{k}/k) \\times \\pi_0(X_{\\overline{k}})", "\\ar[r] &", "\\pi_0(X_{\\overline{k}})", "}", "$$", "is commutative (where the right vertical arrow is a bijection", "according to Lemma \\ref{lemma-separably-closed-field-connected-components}).", "\\end{enumerate}" ], "refs": [ "varieties-lemma-separably-closed-field-connected-components" ], "proofs": [ { "contents": [ "The action (\\ref{equation-galois-action-base-change-kbar})", "of $\\text{Gal}(\\overline{k}/k)$ on $X_{\\overline{k}}$", "induces an action on its connected components.", "Connected components are always closed", "(Topology, Lemma \\ref{topology-lemma-connected-components}).", "Hence if $\\overline{T}$ is as in (1), then by", "Lemma \\ref{lemma-closed-fixed-by-Galois} there exists a closed", "subset $T \\subset X$ such that $\\overline{T} = T_{\\overline{k}}$.", "Note that $T$ is geometrically connected over $k$, see", "Lemma \\ref{lemma-characterize-geometrically-connected}.", "To see that $T$ is a connected component of $X$, suppose that", "$T \\subset T'$, $T \\not = T'$ where $T'$ is a connected component of $X$.", "In this case $T'_{k'}$ strictly contains $\\overline{T}$ and hence is", "disconnected. By Lemma \\ref{lemma-tricky} this means that $T'$ is", "disconnected! Contradiction.", "\\medskip\\noindent", "We omit the proof of the functoriality in (2)." ], "refs": [ "topology-lemma-connected-components", "varieties-lemma-closed-fixed-by-Galois", "varieties-lemma-characterize-geometrically-connected", "varieties-lemma-tricky" ], "ref_ids": [ 8206, 10922, 10919, 10924 ] } ], "ref_ids": [ 10918 ] }, { "id": 10931, "type": "theorem", "label": "varieties-lemma-galois-action-connected-components-continuous", "categories": [ "varieties" ], "title": "varieties-lemma-galois-action-connected-components-continuous", "contents": [ "Let $k$ be a field, with separable algebraic closure $\\overline{k}$.", "Let $X$ be a scheme over $k$.", "Assume", "\\begin{enumerate}", "\\item $X$ is quasi-compact, and", "\\item the connected components of $X_{\\overline{k}}$ are open.", "\\end{enumerate}", "Then", "\\begin{enumerate}", "\\item[(a)] $\\pi_0(X_{\\overline{k}})$ is finite, and", "\\item[(b)] the action of $\\text{Gal}(\\overline{k}/k)$ on", "$\\pi_0(X_{\\overline{k}})$ is continuous.", "\\end{enumerate}", "Moreover, assumptions (1) and (2) are satisfied when $X$ is", "of finite type over $k$." ], "refs": [], "proofs": [ { "contents": [ "Since the connected components are open, cover $X_{\\overline{k}}$", "(Topology, Lemma \\ref{topology-lemma-connected-components}) and", "$X_{\\overline{k}}$ is quasi-compact, we conclude that there are only", "finitely many of them. Thus (a) holds.", "By Lemma \\ref{lemma-descend-open} these connected components", "are each defined over a finite subextension of $k \\subset \\overline{k}$", "and we get (b).", "If $X$ is of finite type over $k$, then $X_{\\overline{k}}$ is of finite", "type over $\\overline{k}$", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-type}).", "Hence $X_{\\overline{k}}$ is a Noetherian scheme", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}).", "Thus $X_{\\overline{k}}$ has finitely many irreducible components", "(Properties, Lemma \\ref{properties-lemma-Noetherian-irreducible-components})", "and a fortiori finitely many connected components (which are", "therefore open)." ], "refs": [ "topology-lemma-connected-components", "varieties-lemma-descend-open", "morphisms-lemma-base-change-finite-type", "morphisms-lemma-finite-type-noetherian", "properties-lemma-Noetherian-irreducible-components" ], "ref_ids": [ 8206, 10920, 5200, 5202, 2956 ] } ], "ref_ids": [] }, { "id": 10932, "type": "theorem", "label": "varieties-lemma-geometrically-irreducible-check-after-extension", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-irreducible-check-after-extension", "contents": [ "Let $X$ be a scheme over the field $k$.", "Let $k \\subset k'$ be a field extension.", "Then $X$ is geometrically irreducible over $k$ if and only if", "$X_{k'}$ is geometrically irreducible over $k'$." ], "refs": [], "proofs": [ { "contents": [ "If $X$ is geometrically irreducible over $k$, then it is clear that", "$X_{k'}$ is geometrically irreducible over $k'$. For the converse, note", "that for any field extension $k \\subset k''$ there exists a common", "field extension $k' \\subset k'''$ and $k'' \\subset k'''$. As the", "morphism $X_{k'''} \\to X_{k''}$ is surjective (as a base change of", "a surjective morphism between spectra of fields) we see that the", "irreducibility of $X_{k'''}$ implies the irreducibility of $X_{k''}$.", "Thus if $X_{k'}$ is geometrically irreducible over $k'$ then", "$X$ is geometrically irreducible over $k$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10933, "type": "theorem", "label": "varieties-lemma-separably-closed-irreducible", "categories": [ "varieties" ], "title": "varieties-lemma-separably-closed-irreducible", "contents": [ "Let $X$ be a scheme over a separably closed field $k$.", "If $X$ is irreducible, then $X_K$ is irreducible for any", "field extension $k \\subset K$. I.e., $X$ is geometrically", "irreducible over $k$." ], "refs": [], "proofs": [ { "contents": [ "Use Properties, Lemma \\ref{properties-lemma-characterize-irreducible}", "and Algebra, Lemma \\ref{algebra-lemma-separably-closed-irreducible}." ], "refs": [ "properties-lemma-characterize-irreducible", "algebra-lemma-separably-closed-irreducible" ], "ref_ids": [ 2946, 588 ] } ], "ref_ids": [] }, { "id": 10934, "type": "theorem", "label": "varieties-lemma-bijection-irreducible-components", "categories": [ "varieties" ], "title": "varieties-lemma-bijection-irreducible-components", "contents": [ "Let $k$ be a field.", "Let $X$, $Y$ be schemes over $k$.", "Assume $X$ is geometrically irreducible over $k$.", "Then the projection morphism", "$$", "p : X \\times_k Y \\longrightarrow Y", "$$", "induces a bijection between irreducible components." ], "refs": [], "proofs": [ { "contents": [ "First, note that the scheme theoretic fibres of $p$ are irreducible,", "since they are base changes of the geometrically irreducible scheme $X$", "by field extensions. Moreover the scheme theoretic fibres are", "homeomorphic to the set theoretic fibres, see", "Schemes, Lemma \\ref{schemes-lemma-fibre-topological}.", "By Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open}", "the map $p$ is open.", "Thus we may apply Topology,", "Lemma \\ref{topology-lemma-irreducible-fibres-irreducible-components}", "to conclude." ], "refs": [ "schemes-lemma-fibre-topological", "morphisms-lemma-scheme-over-field-universally-open", "topology-lemma-irreducible-fibres-irreducible-components" ], "ref_ids": [ 7696, 5254, 8218 ] } ], "ref_ids": [] }, { "id": 10935, "type": "theorem", "label": "varieties-lemma-geometrically-irreducible-local", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-irreducible-local", "contents": [ "\\begin{slogan}", "Geometric irreductibility is Zariski local modulo connectedness.", "\\end{slogan}", "Let $k$ be a field. Let $X$ be a scheme over $k$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically irreducible over $k$,", "\\item for every nonempty affine open $U$ the $k$-algebra $\\mathcal{O}_X(U)$", "is geometrically irreducible over $k$ (see", "Algebra, Definition \\ref{algebra-definition-geometrically-irreducible}),", "\\item $X$ is irreducible and there exists an affine open covering", "$X = \\bigcup U_i$ such that each $k$-algebra $\\mathcal{O}_X(U_i)$ is", "geometrically irreducible, and", "\\item there exists an open covering $X = \\bigcup_{i \\in I} X_i$", "with $I \\not = \\emptyset$ such", "that $X_i$ is geometrically irreducible for each $i$ and such that", "$X_i \\cap X_j \\not = \\emptyset$ for all $i, j \\in I$.", "\\end{enumerate}", "Moreover, if $X$ is geometrically irreducible so is every nonempty", "open subscheme of $X$." ], "refs": [ "algebra-definition-geometrically-irreducible" ], "proofs": [ { "contents": [ "An affine scheme $\\Spec(A)$ over $k$ is geometrically", "irreducible if and only if $A$ is geometrically irreducible over $k$;", "this is immediate from the definitions.", "Recall that if a scheme is irreducible so is every nonempty", "open subscheme of $X$, any two nonempty open subsets have", "a nonempty intersection. Also, if every affine open is irreducible", "then the scheme is irreducible, see Properties,", "Lemma \\ref{properties-lemma-characterize-irreducible}.", "Hence the final statement of the lemma", "is clear, as well as the implications (1) $\\Rightarrow$ (2),", "(2) $\\Rightarrow$ (3), and (3) $\\Rightarrow$ (4). If (4) holds,", "then for any field extension $k'/k$ the scheme $X_{k'}$", "has a covering by irreducible opens which pairwise intersect.", "Hence $X_{k'}$ is irreducible. Hence (4) implies (1)." ], "refs": [ "properties-lemma-characterize-irreducible" ], "ref_ids": [ 2946 ] } ], "ref_ids": [ 1464 ] }, { "id": 10936, "type": "theorem", "label": "varieties-lemma-geometrically-irreducible-function-field", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-irreducible-function-field", "contents": [ "Let $X$ be an irreducible scheme over the field $k$. Let $\\xi \\in X$", "be its generic point. The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically irreducible over $k$, and", "\\item $\\kappa(\\xi)$ is geometrically irreducible over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Recall that $\\mathcal{O}_{X, \\xi}$ is the filtered", "colimit of $\\mathcal{O}_X(U)$ where $U$ runs over the nonempty", "open affine subschemes of $X$. Combining", "Lemma \\ref{lemma-geometrically-irreducible-local}", "and", "Algebra, Lemma \\ref{algebra-lemma-subalgebra-geometrically-irreducible}", "we see that $\\mathcal{O}_{X, \\xi}$ is geometrically irreducible over $k$.", "Since $\\mathcal{O}_{X, \\xi} \\to \\kappa(\\xi)$ is a surjection with", "locally nilpotent kernel (see", "Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring})", "it follows that $\\kappa(\\xi)$ is geometrically irreducible, see", "Algebra, Lemma \\ref{algebra-lemma-p-ring-map}.", "\\medskip\\noindent", "Assume (2). We may assume that $X$ is reduced. Let $U \\subset X$ be", "a nonempty affine open. Then $U = \\Spec(A)$ where $A$ is a domain", "with fraction field $\\kappa(\\xi)$. Thus $A$ is a $k$-subalgebra", "of a geometrically irreducible $k$-algebra. Hence by", "Algebra, Lemma \\ref{algebra-lemma-subalgebra-geometrically-irreducible}", "we see that $A$ is geometrically irreducible over $k$.", "By Lemma \\ref{lemma-geometrically-irreducible-local} we conclude", "that $X$ is geometrically irreducible over $k$." ], "refs": [ "varieties-lemma-geometrically-irreducible-local", "algebra-lemma-subalgebra-geometrically-irreducible", "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-p-ring-map", "algebra-lemma-subalgebra-geometrically-irreducible", "varieties-lemma-geometrically-irreducible-local" ], "ref_ids": [ 10935, 591, 418, 582, 591, 10935 ] } ], "ref_ids": [] }, { "id": 10937, "type": "theorem", "label": "varieties-lemma-separably-closed-field-irreducible-components", "categories": [ "varieties" ], "title": "varieties-lemma-separably-closed-field-irreducible-components", "contents": [ "Let $k \\subset k'$ be an extension of fields.", "Let $X$ be a scheme over $k$. Set $X' = X_{k'}$.", "Assume $k$ separably algebraically closed.", "Then the morphism $X' \\to X$ induces a bijection of irreducible components." ], "refs": [], "proofs": [ { "contents": [ "Since $k$ is separably algebraically closed we see that", "$k'$ is geometrically irreducible over $k$, see Algebra,", "Lemma \\ref{algebra-lemma-separably-closed-irreducible-implies-geometric}.", "Hence $Z = \\Spec(k')$ is geometrically irreducible over $k$.", "by Lemma \\ref{lemma-geometrically-irreducible-local} above.", "Since $X' = Z \\times_k X$ the result is a special case", "of Lemma \\ref{lemma-bijection-irreducible-components}." ], "refs": [ "algebra-lemma-separably-closed-irreducible-implies-geometric", "varieties-lemma-geometrically-irreducible-local", "varieties-lemma-bijection-irreducible-components" ], "ref_ids": [ 590, 10935, 10934 ] } ], "ref_ids": [] }, { "id": 10938, "type": "theorem", "label": "varieties-lemma-characterize-geometrically-irreducible", "categories": [ "varieties" ], "title": "varieties-lemma-characterize-geometrically-irreducible", "contents": [ "\\begin{slogan}", "Geometric irreducibility can be tested over a separable algebraic", "closure of the base field.", "\\end{slogan}", "Let $k$ be a field. Let $X$ be a scheme over $k$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $X$ is geometrically irreducible over $k$,", "\\item for every finite separable field extension $k \\subset k'$", "the scheme $X_{k'}$ is irreducible, and", "\\item $X_{\\overline{k}}$ is irreducible, where $k \\subset \\overline{k}$", "is a separable algebraic closure of $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $X_{\\overline{k}}$ is irreducible, i.e., assume (3).", "Let $k \\subset k'$ be a field extension.", "There exists a field extension $\\overline{k} \\subset \\overline{k}'$", "such that $k'$ embeds into $\\overline{k}'$ as an extension of $k$.", "By Lemma \\ref{lemma-separably-closed-field-irreducible-components}", "we see that $X_{\\overline{k}'}$ is irreducible.", "Since $X_{\\overline{k}'} \\to X_{k'}$ is surjective we conclude", "that $X_{k'}$ is irreducible. Hence (1) holds.", "\\medskip\\noindent", "Let $k \\subset \\overline{k}$ be a separable algebraic closure of $k$.", "Assume not (3), i.e., assume $X_{\\overline{k}}$ is reducible.", "Our goal is to show that also $X_{k'}$ is", "reducible for some finite subextension", "$k \\subset k' \\subset \\overline{k}$.", "Let $X = \\bigcup_{i \\in I} U_i$ be an affine open covering", "with $U_i$ not empty. If for some $i$ the scheme", "$U_i$ is reducible, or if for some pair $i \\not = j$ the", "intersection $U_i \\cap U_j$ is empty, then $X$ is reducible", "(Properties, Lemma \\ref{properties-lemma-characterize-irreducible})", "and we are done.", "In particular we may assume that", "$U_{i, \\overline{k}} \\cap U_{j, \\overline{k}}$ for all $i, j \\in I$", "is nonempty and we conclude that $U_{i, \\overline{k}}$ has", "to be reducible for some $i$. According to", "Algebra, Lemma \\ref{algebra-lemma-geometrically-irreducible}", "this means that $U_{i, k'}$ is reducible for some", "finite separable field extension $k \\subset k'$.", "Hence also $X_{k'}$ is reducible. Thus we see that", "(2) implies (3).", "\\medskip\\noindent", "The implication (1) $\\Rightarrow$ (2) is immediate.", "This proves the lemma." ], "refs": [ "varieties-lemma-separably-closed-field-irreducible-components", "properties-lemma-characterize-irreducible", "algebra-lemma-geometrically-irreducible" ], "ref_ids": [ 10937, 2946, 589 ] } ], "ref_ids": [] }, { "id": 10939, "type": "theorem", "label": "varieties-lemma-inverse-image-irreducible", "categories": [ "varieties" ], "title": "varieties-lemma-inverse-image-irreducible", "contents": [ "Let $k \\subset K$ be an extension of fields.", "Let $X$ be a scheme over $k$.", "For every irreducible component $T$ of $X$ the inverse image", "$T_K \\subset X_K$ is a union of irreducible components of $X_K$." ], "refs": [], "proofs": [ { "contents": [ "Let $T \\subset X$ be an irreducible component of $X$.", "The morphism $T_K \\to T$ is flat, so generalizations lift", "along $T_K \\to T$. Hence every $\\xi \\in T_K$", "which is a generic point of an irreducible component of $T_K$", "maps to the generic point $\\eta$ of $T$. If $\\xi' \\leadsto \\xi$ is", "a specialization in $X_K$ then $\\xi'$ maps to $\\eta$ since there", "are no points specializing to $\\eta$ in $X$. Hence $\\xi' \\in T_K$", "and we conclude that $\\xi = \\xi'$. In other words $\\xi$ is the", "generic point of an irreducible component of $X_K$. This", "means that the irreducible components of $T_K$ are all irreducible", "components of $X_K$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10940, "type": "theorem", "label": "varieties-lemma-image-irreducible", "categories": [ "varieties" ], "title": "varieties-lemma-image-irreducible", "contents": [ "Let $k \\subset K$ be an extension of fields.", "Let $X$ be a scheme over $k$.", "For every irreducible component $\\overline{T} \\subset X_K$", "the image of $\\overline{T}$ in $X$ is an irreducible component in $X$.", "This defines a canonical map", "$$", "\\text{IrredComp}(X_K)", "\\longrightarrow", "\\text{IrredComp}(X)", "$$", "which is surjective." ], "refs": [], "proofs": [ { "contents": [ "Consider the diagram", "$$", "\\xymatrix{", "X_K \\ar[d] & X_{\\overline{K}} \\ar[d] \\ar[l] \\\\", "X & X_{\\overline{k}} \\ar[l]", "}", "$$", "where $\\overline{K}$ is the separable algebraic closure of $K$, and", "where $\\overline{k}$ is the separable algebraic closure of $k$. By", "Lemma \\ref{lemma-separably-closed-field-irreducible-components}", "the morphism $X_{\\overline{K}} \\to X_{\\overline{k}}$ induces", "a bijection between irreducible components. Hence it suffices", "to show the lemma for the morphisms", "$X_{\\overline{k}} \\to X$ and $X_{\\overline{K}} \\to X_K$.", "In other words we may assume that $K = \\overline{k}$.", "\\medskip\\noindent", "The morphism $p : X_{\\overline{k}} \\to X$ is integral, flat and surjective.", "Flatness implies that generalizations lift along $p$, see", "Morphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat}.", "Hence generic points of irreducible components of $X_{\\overline{k}}$", "map to generic points of irreducible components of $X$.", "Integrality implies that $p$ is universally closed, see", "Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed}.", "Hence we conclude that the image $p(\\overline{T})$ of an irreducible component", "is a closed irreducible subset which contains a generic point of an", "irreducible component of $X$, hence $p(\\overline{T})$", "is an irreducible component of $X$. This proves the first assertion.", "If $T \\subset X$ is an irreducible component, then $p^{-1}(T) =T_K$", "is a nonempty union of irreducible components, see", "Lemma \\ref{lemma-inverse-image-irreducible}.", "Each of these necessarily maps onto $T$ by the first part.", "Hence the map is surjective." ], "refs": [ "varieties-lemma-separably-closed-field-irreducible-components", "morphisms-lemma-generalizations-lift-flat", "morphisms-lemma-integral-universally-closed", "varieties-lemma-inverse-image-irreducible" ], "ref_ids": [ 10937, 5266, 5441, 10939 ] } ], "ref_ids": [] }, { "id": 10941, "type": "theorem", "label": "varieties-lemma-irreducible-dense-rational-points-geometrically-irreducible", "categories": [ "varieties" ], "title": "varieties-lemma-irreducible-dense-rational-points-geometrically-irreducible", "contents": [ "Let $k$ be a field. Let $X$ be a scheme over $k$. If $X$ is irreducible", "and has a dense set of $k$-rational points, then $X$ is geometrically", "irreducible." ], "refs": [], "proofs": [ { "contents": [ "Let $k'/k$ be a finite extension of fields and let $Z, Z' \\subset X_{k'}$", "be irreducible components. It suffices to show $Z = Z'$, see", "Lemma \\ref{lemma-characterize-geometrically-irreducible}.", "By Lemma \\ref{lemma-image-irreducible} we have $p(Z) = p(Z') = X$", "where $p : X_{k'} \\to X$ is the projection. If $Z \\not = Z'$", "then $Z \\cap Z'$ is nowhere dense in $X_{k'}$ and hence $p(Z \\cap Z')$", "is not dense by", "Morphisms, Lemma \\ref{morphisms-lemma-image-nowhere-dense-finite};", "here we also use that $p$ is a finite morphism as the base change", "of the finite morphism $\\Spec(k') \\to \\Spec(k)$, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite}.", "Thus we can pick a $k$-rational point $x \\in X$ with", "$x \\not \\in p(Z \\cap Z')$. Since the residue field of $x$ is $k$", "we see that $p^{-1}(\\{x\\}) = \\{x'\\}$ where $x' \\in X_{k'}$", "is a point whose residue field is $k'$. Since $x \\in p(Z) = p(Z')$", "we conclude that $x' \\in Z \\cap Z'$ which is the contradiction", "we were looking for." ], "refs": [ "varieties-lemma-characterize-geometrically-irreducible", "varieties-lemma-image-irreducible", "morphisms-lemma-image-nowhere-dense-finite", "morphisms-lemma-base-change-finite" ], "ref_ids": [ 10938, 10940, 5476, 5440 ] } ], "ref_ids": [] }, { "id": 10942, "type": "theorem", "label": "varieties-lemma-galois-action-irreducible-components", "categories": [ "varieties" ], "title": "varieties-lemma-galois-action-irreducible-components", "contents": [ "Let $k$ be a field, with separable algebraic closure $\\overline{k}$.", "Let $X$ be a scheme over $k$.", "There is an action", "$$", "\\text{Gal}(\\overline{k}/k)^{opp} \\times \\text{IrredComp}(X_{\\overline{k}})", "\\longrightarrow", "\\text{IrredComp}(X_{\\overline{k}})", "$$", "with the following properties:", "\\begin{enumerate}", "\\item An element $\\overline{T} \\in \\text{IrredComp}(X_{\\overline{k}})$", "is fixed by the action if and only if there exists an irreducible", "component $T \\subset X$, which is geometrically irreducible over $k$,", "such that $T_{\\overline{k}} = \\overline{T}$.", "\\item For any field extension $k \\subset k'$ with separable", "algebraic closure $\\overline{k}'$ the diagram", "$$", "\\xymatrix{", "\\text{Gal}(\\overline{k}'/k') \\times \\text{IrredComp}(X_{\\overline{k}'})", "\\ar[r] \\ar[d] &", "\\text{IrredComp}(X_{\\overline{k}'}) \\ar[d] \\\\", "\\text{Gal}(\\overline{k}/k) \\times \\text{IrredComp}(X_{\\overline{k}})", "\\ar[r] &", "\\text{IrredComp}(X_{\\overline{k}})", "}", "$$", "is commutative (where the right vertical arrow is a bijection", "according to Lemma \\ref{lemma-separably-closed-field-irreducible-components}).", "\\end{enumerate}" ], "refs": [ "varieties-lemma-separably-closed-field-irreducible-components" ], "proofs": [ { "contents": [ "The action (\\ref{equation-galois-action-base-change-kbar})", "of $\\text{Gal}(\\overline{k}/k)$ on $X_{\\overline{k}}$", "induces an action on its irreducible components.", "Irreducible components are always closed", "(Topology, Lemma \\ref{topology-lemma-connected-components}).", "Hence if $\\overline{T}$ is as in (1), then by", "Lemma \\ref{lemma-closed-fixed-by-Galois} there exists a closed", "subset $T \\subset X$ such that $\\overline{T} = T_{\\overline{k}}$.", "Note that $T$ is geometrically irreducible over $k$, see", "Lemma \\ref{lemma-characterize-geometrically-irreducible}.", "To see that $T$ is an irreducible component of $X$, suppose that", "$T \\subset T'$, $T \\not = T'$ where $T'$ is an irreducible", "component of $X$. Let $\\overline{\\eta}$ be the generic point of", "$\\overline{T}$. It maps to the generic point $\\eta$ of $T$.", "Then the generic point $\\xi \\in T'$ specializes to $\\eta$.", "As $X_{\\overline{k}} \\to X$ is flat there exists a point", "$\\overline{\\xi} \\in X_{\\overline{k}}$ which maps to $\\xi$ and", "specializes to $\\overline{\\eta}$. It follows that", "the closure of the singleton $\\{\\overline{\\xi}\\}$ is an", "irreducible closed subset of $X_{\\overline{\\xi}}$ which", "strictly contains $\\overline{T}$. This is the desired contradiction.", "\\medskip\\noindent", "We omit the proof of the functoriality in (2)." ], "refs": [ "topology-lemma-connected-components", "varieties-lemma-closed-fixed-by-Galois", "varieties-lemma-characterize-geometrically-irreducible" ], "ref_ids": [ 8206, 10922, 10938 ] } ], "ref_ids": [ 10937 ] }, { "id": 10943, "type": "theorem", "label": "varieties-lemma-orbit-irreducible-components", "categories": [ "varieties" ], "title": "varieties-lemma-orbit-irreducible-components", "contents": [ "Let $k$ be a field, with separable algebraic closure $\\overline{k}$.", "Let $X$ be a scheme over $k$.", "The fibres of the map", "$$", "\\text{IrredComp}(X_{\\overline{k}})", "\\longrightarrow", "\\text{IrredComp}(X)", "$$", "of", "Lemma \\ref{lemma-image-irreducible}", "are exactly the orbits of $\\text{Gal}(\\overline{k}/k)$ under the action of", "Lemma \\ref{lemma-galois-action-irreducible-components}." ], "refs": [ "varieties-lemma-image-irreducible", "varieties-lemma-galois-action-irreducible-components" ], "proofs": [ { "contents": [ "Let $T \\subset X$ be an irreducible component of $X$.", "Let $\\eta \\in T$ be its generic point. By", "Lemmas \\ref{lemma-inverse-image-irreducible} and", "\\ref{lemma-image-irreducible}", "the generic points of irreducible components of $\\overline{T}$", "which map into $T$ map to $\\eta$. By", "Algebra, Lemma \\ref{algebra-lemma-Galois-orbit}", "the Galois group acts transitively on", "all of the points of $X_{\\overline{k}}$ mapping to $\\eta$.", "Hence the lemma follows." ], "refs": [ "varieties-lemma-inverse-image-irreducible", "varieties-lemma-image-irreducible", "algebra-lemma-Galois-orbit" ], "ref_ids": [ 10939, 10940, 599 ] } ], "ref_ids": [ 10940, 10942 ] }, { "id": 10944, "type": "theorem", "label": "varieties-lemma-galois-action-irreducible-components-locally-finite-type", "categories": [ "varieties" ], "title": "varieties-lemma-galois-action-irreducible-components-locally-finite-type", "contents": [ "Let $k$ be a field.", "Assume $X \\to \\Spec(k)$ locally of finite type.", "In this case", "\\begin{enumerate}", "\\item the action", "$$", "\\text{Gal}(\\overline{k}/k)^{opp} \\times \\text{IrredComp}(X_{\\overline{k}})", "\\longrightarrow", "\\text{IrredComp}(X_{\\overline{k}})", "$$", "is continuous if we give $\\text{IrredComp}(X_{\\overline{k}})$ the discrete", "topology,", "\\item every irreducible component of $X_{\\overline{k}}$", "can be defined over a finite extension of $k$, and", "\\item given any irreducible component $T \\subset X$ the scheme", "$T_{\\overline{k}}$ is a finite union of irreducible components of", "$X_{\\overline{k}}$ which are all in the same", "$\\text{Gal}(\\overline{k}/k)$-orbit.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\overline{T}$ be an irreducible component of $X_{\\overline{k}}$.", "We may choose an affine open $U \\subset X$ such that", "$\\overline{T} \\cap U_{\\overline{k}}$ is not empty.", "Write $U = \\Spec(A)$, so $A$ is a finite type $k$-algebra, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-finite-type-characterize}.", "Hence $A_{\\overline{k}}$ is a finite type $\\overline{k}$-algebra,", "and in particular Noetherian. Let $\\mathfrak p = (f_1, \\ldots, f_n)$", "be the prime ideal corresponding to $\\overline{T} \\cap U_{\\overline{k}}$.", "Since $A_{\\overline{k}} = A \\otimes_k \\overline{k}$", "we see that there exists a finite subextension", "$k \\subset k' \\subset \\overline{k}$ such that each $f_i \\in A_{k'}$.", "It is clear that $\\text{Gal}(\\overline{k}/k')$", "fixes $\\overline{T}$, which proves (1).", "\\medskip\\noindent", "Part (2) follows by applying", "Lemma \\ref{lemma-galois-action-irreducible-components} (1)", "to the situation over $k'$ which implies the irreducible component", "$\\overline{T}$ is of the form $T'_{\\overline{k}}$ for some irreducible", "$T' \\subset X_{k'}$.", "\\medskip\\noindent", "To prove (3), let $T \\subset X$ be an irreducible component.", "Choose an irreducible component $\\overline{T} \\subset X_{\\overline{k}}$", "which maps to $T$, see", "Lemma \\ref{lemma-image-irreducible}.", "By the above the orbit of $\\overline{T}$ is finite, say it is", "$\\overline{T}_1, \\ldots, \\overline{T}_n$. Then", "$\\overline{T}_1 \\cup \\ldots \\cup \\overline{T}_n$", "is a $\\text{Gal}(\\overline{k}/k)$-invariant closed subset of $X_{\\overline{k}}$", "hence of the form $W_{\\overline{k}}$ for some $W \\subset X$ closed by", "Lemma \\ref{lemma-closed-fixed-by-Galois}.", "Clearly $W = T$ and we win." ], "refs": [ "morphisms-lemma-locally-finite-type-characterize", "varieties-lemma-galois-action-irreducible-components", "varieties-lemma-image-irreducible", "varieties-lemma-closed-fixed-by-Galois" ], "ref_ids": [ 5198, 10942, 10940, 10922 ] } ], "ref_ids": [] }, { "id": 10945, "type": "theorem", "label": "varieties-lemma-finite-extension-geometrically-irreducible-components", "categories": [ "varieties" ], "title": "varieties-lemma-finite-extension-geometrically-irreducible-components", "contents": [ "Let $k$ be a field.", "Let $X \\to \\Spec(k)$ be locally of finite type.", "Assume $X$ has finitely many irreducible components.", "Then there exists a finite separable extension $k \\subset k'$", "such that every irreducible component of $X_{k'}$", "is geometrically irreducible over $k'$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\overline{k}$ be a separable algebraic closure of $k$.", "The assumption that $X$ has finitely many irreducible components", "combined with", "Lemma \\ref{lemma-galois-action-irreducible-components-locally-finite-type} (3)", "shows that $X_{\\overline{k}}$ has finitely many irreducible components", "$\\overline{T}_1, \\ldots, \\overline{T}_n$. By", "Lemma \\ref{lemma-galois-action-irreducible-components-locally-finite-type} (2)", "there exists a finite extension $k \\subset k' \\subset \\overline{k}$ and", "irreducible components $T_i \\subset X_{k'}$ such that", "$\\overline{T}_i = T_{i, \\overline{k}}$ and we win." ], "refs": [ "varieties-lemma-galois-action-irreducible-components-locally-finite-type", "varieties-lemma-galois-action-irreducible-components-locally-finite-type" ], "ref_ids": [ 10944, 10944 ] } ], "ref_ids": [] }, { "id": 10946, "type": "theorem", "label": "varieties-lemma-irreducible-components-geometrically-irreducible", "categories": [ "varieties" ], "title": "varieties-lemma-irreducible-components-geometrically-irreducible", "contents": [ "Let $X$ be a scheme over the field $k$.", "Assume $X$ has finitely many irreducible components which are", "all geometrically irreducible.", "Then $X$ has finitely many connected components each of which is", "geometrically connected." ], "refs": [], "proofs": [ { "contents": [ "This is clear because a connected component is a union of irreducible", "components. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10947, "type": "theorem", "label": "varieties-lemma-geometrically-integral", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-integral", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme over $k$.", "Then $X$ is geometrically integral over $k$ if and only if", "$X$ is both geometrically reduced and geometrically irreducible", "over $k$." ], "refs": [], "proofs": [ { "contents": [ "See Properties, Lemma \\ref{properties-lemma-characterize-integral}." ], "refs": [ "properties-lemma-characterize-integral" ], "ref_ids": [ 2947 ] } ], "ref_ids": [] }, { "id": 10948, "type": "theorem", "label": "varieties-lemma-proper-geometrically-reduced-global-sections", "categories": [ "varieties" ], "title": "varieties-lemma-proper-geometrically-reduced-global-sections", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$.", "\\begin{enumerate}", "\\item $A = H^0(X, \\mathcal{O}_X)$ is a finite dimensional $k$-algebra,", "\\item $A = \\prod_{i = 1, \\ldots, n} A_i$ is a product of Artinian", "local $k$-algebras, one factor for each connected component of $X$,", "\\item if $X$ is reduced, then $A = \\prod_{i = 1, \\ldots, n} k_i$", "is a product of fields, each a finite extension of $k$,", "\\item if $X$ is geometrically reduced, then $k_i$ is finite separable", "over $k$,", "\\item if $X$ is geometrically connected, then $A$ is geometrically", "irreducible over $k$,", "\\item if $X$ is geometrically irreducible, then $A$ is geometrically", "irreducible over $k$,", "\\item if $X$ is geometrically reduced and connected, then $A = k$, and", "\\item if $X$ is geometrically integral, then $A = k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-over-affine-cohomology-finite}", "we see that $A = H^0(X, \\mathcal{O}_X)$ is a finite dimensional", "$k$-algebra. This proves (1).", "\\medskip\\noindent", "Then $A$ is a product of local Artinian $k$-algebras by", "Algebra, Lemma \\ref{algebra-lemma-finite-dimensional-algebra} and", "Proposition \\ref{algebra-proposition-dimension-zero-ring}.", "If $X = Y \\amalg Z$ with $Y$ and $Z$ open in $X$, then we obtain", "an idempotent $e \\in A$ by taking the section of $\\mathcal{O}_X$", "which is $1$ on $Y$ and $0$ on $Z$. Conversely, if $e \\in A$", "is an idempotent, then we get a corresponding decomposition of $X$.", "Finally, as $X$ has a Noetherian underlying topological space", "its connected components are open. Hence the connected components", "of $X$ correspond $1$-to-$1$ with primitive idempotents of $A$.", "This proves (2).", "\\medskip\\noindent", "If $X$ is reduced, then $A$ is reduced. Hence the local rings $A_i = k_i$", "are reduced and therefore fields (for example by", "Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring}).", "This proves (3).", "\\medskip\\noindent", "If $X$ is geometrically reduced, then", "$A \\otimes_k \\overline{k} =", "H^0(X_{\\overline{k}}, \\mathcal{O}_{X_{\\overline{k}}})$", "(equality by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology}) is reduced.", "This implies that $k_i \\otimes_k \\overline{k}$ is a product", "of fields and hence $k_i/k$ is separable for example by", "Algebra,", "Lemmas \\ref{algebra-lemma-characterize-separable-field-extensions} and", "\\ref{algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension}.", "This proves (4).", "\\medskip\\noindent", "If $X$ is geometrically connected, then $A \\otimes_k \\overline{k} =", "H^0(X_{\\overline{k}}, \\mathcal{O}_{X_{\\overline{k}}})$", "is a zero dimensional local ring by part (2) and hence its", "spectrum has one point, in particular it is irreducible.", "Thus $A$ is geometrically irreducible. This proves (5).", "Of course (5) implies (6).", "\\medskip\\noindent", "If $X$ is geometrically reduced and connected, then", "$A = k_1$ is a field and the extension $k_1/k$ is finite separable and", "geometrically irreducible. However, then $k_1 \\otimes_k \\overline{k}$", "is a product of $[k_1 : k]$ copies of $\\overline{k}$ and we conclude", "that $k_1 = k$. This proves (7). Of course (7) implies (8)." ], "refs": [ "coherent-lemma-proper-over-affine-cohomology-finite", "algebra-lemma-finite-dimensional-algebra", "algebra-proposition-dimension-zero-ring", "algebra-lemma-minimal-prime-reduced-ring", "coherent-lemma-flat-base-change-cohomology", "algebra-lemma-characterize-separable-field-extensions", "algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension" ], "ref_ids": [ 3355, 642, 1410, 418, 3298, 569, 571 ] } ], "ref_ids": [] }, { "id": 10949, "type": "theorem", "label": "varieties-lemma-baby-stein", "categories": [ "varieties" ], "title": "varieties-lemma-baby-stein", "contents": [ "Let $X$ be a proper scheme over a field $k$. Set", "$A = H^0(X, \\mathcal{O}_X)$. The fibres of the canonical", "morphism $X \\to \\Spec(A)$ are geometrically connected." ], "refs": [], "proofs": [ { "contents": [ "Set $S = \\Spec(A)$. The canonical morphism $X \\to S$", "is the morphism corresponding to", "$\\Gamma(S, \\mathcal{O}_S) = A = \\Gamma(X, \\mathcal{O}_X)$ via", "Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}.", "The $k$-algebra $A$ is a finite product $A = \\prod A_i$", "of local Artinian $k$-algebras finite over $k$, see", "Lemma \\ref{lemma-proper-geometrically-reduced-global-sections}.", "Denote $s_i \\in S$ the point corresponding to the maximal", "ideal of $A_i$. Choose an algebraic closure $\\overline{k}$ of $k$ and", "set $\\overline{A} = A \\otimes_k \\overline{k}$.", "Choose an embedding $\\kappa(s_i) \\to \\overline{k}$ over $k$; this", "determines a $\\overline{k}$-algebra map", "$$", "\\sigma_i : \\overline{A} = A \\otimes_k \\overline{k} \\to", "\\kappa(s_i) \\otimes_k \\overline{k} \\to \\overline{k}", "$$", "Consider the base change", "$$", "\\xymatrix{", "\\overline{X} \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\overline{S} \\ar[r] & S", "}", "$$", "of $X$ to $\\overline{S} = \\Spec(\\overline{A})$. By", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology} we have", "$\\Gamma(\\overline{X}, \\mathcal{O}_{\\overline{X}}) = \\overline{A}$.", "If $\\overline{s}_i \\in \\Spec(\\overline{A})$ denotes the", "$\\overline{k}$-rational point corresponding to $\\sigma_i$,", "then we see that $\\overline{s}_i$ maps to $s_i \\in S$", "and $\\overline{X}_{\\overline{s}_i}$ is the base change of", "$X_{s_i}$ by $\\Spec(\\sigma_i)$. Thus we see that it suffices to prove", "the lemma in case $k$ is algebraically closed.", "\\medskip\\noindent", "Assume $k$ is algebraically closed. In this case $\\kappa(s_i)$", "is algebraically closed and we have to show that $X_{s_i}$", "is connected. The product decomposition", "$A = \\prod A_i$ corresponds to a disjoint union decomposition", "$\\Spec(A) = \\coprod \\Spec(A_i)$, see", "Algebra, Lemma \\ref{algebra-lemma-spec-product}.", "Denote $X_i$ the inverse image of $\\Spec(A_i)$.", "It follows from", "Lemma \\ref{lemma-proper-geometrically-reduced-global-sections} part (2) that", "$A_i = \\Gamma(X_i, \\mathcal{O}_{X_i})$.", "Observe that $X_{s_i} \\to X_i$ is a closed immersion inducing", "an isomorphism on underlying topological spaces (because $\\Spec(A_i)$", "is a singleton). Hence if $X_{s_i}$ isn't connected, then neither is", "$X_i$. So either $X_i$ is empty and $A_i = 0$ or $X_i$ can be written", "as $U \\amalg V$ with $U$ and $V$ open and nonempty which would", "imply that $A_i$ has a nontrivial idempotent. Since $A_i$ is", "local this is a contradiction and the proof is complete." ], "refs": [ "schemes-lemma-morphism-into-affine", "varieties-lemma-proper-geometrically-reduced-global-sections", "coherent-lemma-flat-base-change-cohomology", "algebra-lemma-spec-product", "varieties-lemma-proper-geometrically-reduced-global-sections" ], "ref_ids": [ 7655, 10948, 3298, 404, 10948 ] } ], "ref_ids": [] }, { "id": 10950, "type": "theorem", "label": "varieties-lemma-geometrically-reduced-stein", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-reduced-stein", "contents": [ "Let $k$ be a field. Let $X$ be a proper geometrically reduced scheme over $k$.", "The following are equivalent", "\\begin{enumerate}", "\\item $H^0(X, \\mathcal{O}_X) = k$, and", "\\item $X$ is geometrically connected.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-baby-stein} we have (1) $\\Rightarrow$ (2).", "By Lemma \\ref{lemma-proper-geometrically-reduced-global-sections}", "we have (2) $\\Rightarrow$ (1)." ], "refs": [ "varieties-lemma-baby-stein", "varieties-lemma-proper-geometrically-reduced-global-sections" ], "ref_ids": [ 10949, 10948 ] } ], "ref_ids": [] }, { "id": 10951, "type": "theorem", "label": "varieties-lemma-geometrically-normal-at-point", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-normal-at-point", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme over $k$.", "Let $x \\in X$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically normal at $x$,", "\\item for every finite purely inseparable field extension $k'$ of $k$", "and $x' \\in X_{k'}$ lying over $x$ the local ring", "$\\mathcal{O}_{X_{k'}, x'}$ is normal, and", "\\item the ring $\\mathcal{O}_{X, x}$ is geometrically", "normal over $k$ (see", "Algebra, Definition \\ref{algebra-definition-geometrically-normal}).", "\\end{enumerate}" ], "refs": [ "algebra-definition-geometrically-normal" ], "proofs": [ { "contents": [ "It is clear that (1) implies (2). Assume (2). Let $k \\subset k'$ be a finite", "purely inseparable field extension (for example $k = k'$). Consider the ring", "$\\mathcal{O}_{X, x} \\otimes_k k'$.", "By Algebra, Lemma \\ref{algebra-lemma-p-ring-map}", "its spectrum is the same as the spectrum of $\\mathcal{O}_{X, x}$.", "Hence it is a local ring also", "(Algebra, Lemma \\ref{algebra-lemma-characterize-local-ring}).", "Therefore there is a unique point $x' \\in X_{k'}$ lying over $x$", "and $\\mathcal{O}_{X_{k'}, x'} \\cong \\mathcal{O}_{X, x} \\otimes_k k'$.", "By assumption this is a normal ring. Hence we deduce (3) by", "Algebra, Lemma", "\\ref{algebra-lemma-geometrically-normal}.", "\\medskip\\noindent", "Assume (3). Let $k \\subset k'$ be a field extension. Since", "$\\Spec(k') \\to \\Spec(k)$ is surjective, also", "$X_{k'} \\to X$ is surjective", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-surjective}).", "Let $x' \\in X_{k'}$ be any point lying over $x$.", "The local ring $\\mathcal{O}_{X_{k'}, x'}$", "is a localization of the ring $\\mathcal{O}_{X, x} \\otimes_k k'$.", "Hence it is normal by assumption and (1) is proved." ], "refs": [ "algebra-lemma-p-ring-map", "algebra-lemma-characterize-local-ring", "algebra-lemma-geometrically-normal", "morphisms-lemma-base-change-surjective" ], "ref_ids": [ 582, 397, 1377, 5165 ] } ], "ref_ids": [ 1554 ] }, { "id": 10952, "type": "theorem", "label": "varieties-lemma-geometrically-normal", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-normal", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme over $k$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically normal,", "\\item $X_{k'}$ is a normal scheme for every field extension $k'/k$,", "\\item $X_{k'}$ is a normal scheme for every finitely generated field", "extension $k'/k$,", "\\item $X_{k'}$ is a normal scheme for every finite purely inseparable", "field extension $k'/k$,", "\\item for every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$", "is geometrically normal (see", "Algebra, Definition \\ref{algebra-definition-geometrically-normal}), and", "\\item $X_{k^{perf}}$ is a normal scheme.", "\\end{enumerate}" ], "refs": [ "algebra-definition-geometrically-normal" ], "proofs": [ { "contents": [ "Assume (1). Then for every field extension $k \\subset k'$ and", "every point $x' \\in X_{k'}$ the local ring of $X_{k'}$ at $x'$", "is normal. By definition this means that $X_{k'}$ is normal.", "Hence (2).", "\\medskip\\noindent", "It is clear that (2) implies (3) implies (4).", "\\medskip\\noindent", "Assume (4) and let $U \\subset X$ be an affine open subscheme.", "Then $U_{k'}$ is a normal scheme for any finite purely inseparable", "extension $k \\subset k'$ (including $k = k'$). This means that", "$k' \\otimes_k \\mathcal{O}(U)$ is a normal ring for all", "finite purely inseparable extensions $k \\subset k'$. Hence", "$\\mathcal{O}(U)$ is a geometrically normal $k$-algebra by definition.", "Hence (4) implies (5).", "\\medskip\\noindent", "Assume (5). For any field extension $k \\subset k'$ the base", "change $X_{k'}$ is gotten by gluing the spectra of the", "rings $\\mathcal{O}_X(U) \\otimes_k k'$ where $U$ is affine open", "in $X$ (see Schemes, Section \\ref{schemes-section-fibre-products}).", "Hence $X_{k'}$ is normal. So (1) holds.", "\\medskip\\noindent", "The equivalence of (5) and (6) follows from the definition", "of geometrically normal algebras and the equivalence (just proved)", "of (3) and (4)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1554 ] }, { "id": 10953, "type": "theorem", "label": "varieties-lemma-geometrically-normal-upstairs", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-normal-upstairs", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme over $k$.", "Let $k'/k$ be a field extension.", "Let $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying over $x$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically normal at $x$,", "\\item $X_{k'}$ is geometrically normal at $x'$.", "\\end{enumerate}", "In particular, $X$ is geometrically normal over $k$ if and only if", "$X_{k'}$ is geometrically normal over $k'$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2). Assume (2).", "Let $k \\subset k''$ be a finite purely inseparable field extension", "and let $x'' \\in X_{k''}$ be a point lying over $x$ (actually it is", "unique). We can find a common field extension $k \\subset k'''$", "(i.e.\\ with both $k' \\subset k'''$ and $k'' \\subset k'''$) and a point", "$x''' \\in X_{k'''}$ lying over both $x'$ and $x''$.", "Consider the map of local rings", "$$", "\\mathcal{O}_{X_{k''}, x''} \\longrightarrow \\mathcal{O}_{X_{k'''}, x''''}.", "$$", "This is a flat local ring homomorphism and hence faithfully flat.", "By (2) we see that the local ring on the right is normal.", "Thus by Algebra, Lemma \\ref{algebra-lemma-descent-normal}", "we conclude that $\\mathcal{O}_{X_{k''}, x''}$ is normal.", "By Lemma \\ref{lemma-geometrically-normal-at-point} we see that $X$", "is geometrically normal at $x$." ], "refs": [ "algebra-lemma-descent-normal", "varieties-lemma-geometrically-normal-at-point" ], "ref_ids": [ 1372, 10951 ] } ], "ref_ids": [] }, { "id": 10954, "type": "theorem", "label": "varieties-lemma-fibre-product-normal", "categories": [ "varieties" ], "title": "varieties-lemma-fibre-product-normal", "contents": [ "Let $k$ be a field. Let $X$ be a geometrically normal scheme over $k$", "and let $Y$ be a normal scheme over $k$. Then $X \\times_k Y$ is a normal", "scheme." ], "refs": [], "proofs": [ { "contents": [ "This reduces to", "Algebra, Lemma \\ref{algebra-lemma-geometrically-normal-tensor-normal}", "by", "Lemma \\ref{lemma-geometrically-normal}." ], "refs": [ "algebra-lemma-geometrically-normal-tensor-normal", "varieties-lemma-geometrically-normal" ], "ref_ids": [ 1380, 10952 ] } ], "ref_ids": [] }, { "id": 10955, "type": "theorem", "label": "varieties-lemma-base-change-normal-by-separable", "categories": [ "varieties" ], "title": "varieties-lemma-base-change-normal-by-separable", "contents": [ "Let $k$ be a field. Let $X$ be a normal scheme over $k$. Let $K/k$", "be a separable field extension. Then $X_K$ is a normal scheme." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-fibre-product-normal} and", "Algebra, Lemma", "\\ref{algebra-lemma-separable-field-extension-geometrically-normal}." ], "refs": [ "varieties-lemma-fibre-product-normal", "algebra-lemma-separable-field-extension-geometrically-normal" ], "ref_ids": [ 10954, 1379 ] } ], "ref_ids": [] }, { "id": 10956, "type": "theorem", "label": "varieties-lemma-geometrically-normal-stein", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-normal-stein", "contents": [ "Let $k$ be a field. Let $X$ be a proper geometrically normal scheme over $k$.", "The following are equivalent", "\\begin{enumerate}", "\\item $H^0(X, \\mathcal{O}_X) = k$,", "\\item $X$ is geometrically connected,", "\\item $X$ is geometrically irreducible, and", "\\item $X$ is geometrically integral.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-geometrically-reduced-stein} we have the", "equivalence of (1) and (2). A locally Noetherian normal scheme", "(such as $X_{\\overline{k}}$) is a disjoint union of", "its irreducible components", "(Properties, Lemma \\ref{properties-lemma-normal-Noetherian}).", "Thus we see that (2) and (3) are equivalent.", "Since $X_{\\overline{k}}$ is assumed reduced, we see that", "(3) and (4) are equivalent too." ], "refs": [ "varieties-lemma-geometrically-reduced-stein", "properties-lemma-normal-Noetherian" ], "ref_ids": [ 10950, 2970 ] } ], "ref_ids": [] }, { "id": 10957, "type": "theorem", "label": "varieties-lemma-locally-Noetherian-base-change", "categories": [ "varieties" ], "title": "varieties-lemma-locally-Noetherian-base-change", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme over $k$.", "Let $k \\subset k'$ be a finitely generated field extension.", "Then $X$ is locally Noetherian if and only if $X_{k'}$ is locally", "Noetherian." ], "refs": [], "proofs": [ { "contents": [ "Using Properties, Lemma \\ref{properties-lemma-locally-Noetherian}", "we reduce to the case where $X$ is", "affine, say $X = \\Spec(A)$. In this case we have to prove that", "$A$ is Noetherian if and only if $A_{k'}$ is Noetherian.", "Since $A \\to A_{k'} = k' \\otimes_k A$ is faithfully flat, we see", "that if $A_{k'}$ is Noetherian, then so is $A$, by", "Algebra, Lemma \\ref{algebra-lemma-descent-Noetherian}.", "Conversely, if $A$ is Noetherian then $A_{k'}$ is Noetherian by", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-field-extension}." ], "refs": [ "properties-lemma-locally-Noetherian", "algebra-lemma-descent-Noetherian", "algebra-lemma-Noetherian-field-extension" ], "ref_ids": [ 2951, 1370, 455 ] } ], "ref_ids": [] }, { "id": 10958, "type": "theorem", "label": "varieties-lemma-geometrically-regular-at-point", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-regular-at-point", "contents": [ "Let $k$ be a field.", "Let $X$ be a locally Noetherian scheme over $k$.", "Let $x \\in X$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically regular at $x$,", "\\item for every finite purely inseparable field extension $k'$ of $k$", "and $x' \\in X_{k'}$ lying over $x$ the local ring", "$\\mathcal{O}_{X_{k'}, x'}$ is regular, and", "\\item the ring $\\mathcal{O}_{X, x}$ is geometrically", "regular over $k$ (see", "Algebra, Definition \\ref{algebra-definition-geometrically-regular}).", "\\end{enumerate}" ], "refs": [ "algebra-definition-geometrically-regular" ], "proofs": [ { "contents": [ "It is clear that (1) implies (2).", "Assume (2). This in particular implies that $\\mathcal{O}_{X, x}$", "is a regular local ring. Let $k \\subset k'$ be a finite purely inseparable", "field extension. Consider the ring $\\mathcal{O}_{X, x} \\otimes_k k'$.", "By Algebra, Lemma \\ref{algebra-lemma-p-ring-map}", "its spectrum is the same as the spectrum of $\\mathcal{O}_{X, x}$.", "Hence it is a local ring also", "(Algebra, Lemma \\ref{algebra-lemma-characterize-local-ring}).", "Therefore there is a unique point $x' \\in X_{k'}$ lying over $x$", "and $\\mathcal{O}_{X_{k'}, x'} \\cong \\mathcal{O}_{X, x} \\otimes_k k'$.", "By assumption this is a regular ring. Hence we deduce (3)", "from the definition of a geometrically regular ring.", "\\medskip\\noindent", "Assume (3). Let $k \\subset k'$ be a field extension. Since", "$\\Spec(k') \\to \\Spec(k)$ is surjective, also", "$X_{k'} \\to X$ is surjective", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-surjective}).", "Let $x' \\in X_{k'}$ be any point lying over $x$.", "The local ring $\\mathcal{O}_{X_{k'}, x'}$", "is a localization of the ring $\\mathcal{O}_{X, x} \\otimes_k k'$.", "Hence it is regular by assumption and (1) is proved." ], "refs": [ "algebra-lemma-p-ring-map", "algebra-lemma-characterize-local-ring", "morphisms-lemma-base-change-surjective" ], "ref_ids": [ 582, 397, 5165 ] } ], "ref_ids": [ 1555 ] }, { "id": 10959, "type": "theorem", "label": "varieties-lemma-geometrically-regular", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-regular", "contents": [ "Let $k$ be a field.", "Let $X$ be a locally Noetherian scheme over $k$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically regular,", "\\item $X_{k'}$ is a regular scheme for every finitely generated field", "extension $k \\subset k'$,", "\\item $X_{k'}$ is a regular scheme for every finite purely inseparable", "field extension $k \\subset k'$,", "\\item for every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$", "is geometrically regular (see", "Algebra, Definition \\ref{algebra-definition-geometrically-regular}), and", "\\item there exists an affine open covering $X = \\bigcup U_i$ such that", "each $\\mathcal{O}_X(U_i)$ is geometrically regular over $k$.", "\\end{enumerate}" ], "refs": [ "algebra-definition-geometrically-regular" ], "proofs": [ { "contents": [ "Assume (1). Then for every finitely generated field extension", "$k \\subset k'$ and", "every point $x' \\in X_{k'}$ the local ring of $X_{k'}$ at $x'$", "is regular. By Properties, Lemma \\ref{properties-lemma-characterize-regular}", "this means that $X_{k'}$ is regular. Hence (2).", "\\medskip\\noindent", "It is clear that (2) implies (3).", "\\medskip\\noindent", "Assume (3) and let $U \\subset X$ be an affine open subscheme.", "Then $U_{k'}$ is a regular scheme for any finite purely inseparable", "extension $k \\subset k'$ (including $k = k'$). This means that", "$k' \\otimes_k \\mathcal{O}(U)$ is a regular ring for all", "finite purely inseparable extensions $k \\subset k'$. Hence", "$\\mathcal{O}(U)$ is a geometrically regular $k$-algebra", "and we see that (4) holds.", "\\medskip\\noindent", "It is clear that (4) implies (5). Let $X = \\bigcup U_i$ be an affine", "open covering as in (5). For any field extension $k \\subset k'$ the base", "change $X_{k'}$ is gotten by gluing the spectra of the", "rings $\\mathcal{O}_X(U_i) \\otimes_k k'$ (see", "Schemes, Section \\ref{schemes-section-fibre-products}).", "Hence $X_{k'}$ is regular. So (1) holds." ], "refs": [ "properties-lemma-characterize-regular" ], "ref_ids": [ 2975 ] } ], "ref_ids": [ 1555 ] }, { "id": 10960, "type": "theorem", "label": "varieties-lemma-geometrically-regular-upstairs", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-regular-upstairs", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme over $k$.", "Let $k'/k$ be a finitely generated field extension.", "Let $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying over $x$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically regular at $x$,", "\\item $X_{k'}$ is geometrically regular at $x'$.", "\\end{enumerate}", "In particular, $X$ is geometrically regular over $k$ if and only if", "$X_{k'}$ is geometrically regular over $k'$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2). Assume (2).", "Let $k \\subset k''$ be a finite purely inseparable field extension", "and let $x'' \\in X_{k''}$ be a point lying over $x$ (actually it is", "unique). We can find a common, finitely generated, field extension", "$k \\subset k'''$ (i.e.\\ with both $k' \\subset k'''$ and $k'' \\subset k'''$)", "and a point $x''' \\in X_{k'''}$ lying over both $x'$ and $x''$.", "Consider the map of local rings", "$$", "\\mathcal{O}_{X_{k''}, x''} \\longrightarrow \\mathcal{O}_{X_{k'''}, x''''}.", "$$", "This is a flat local ring homomorphism of Noetherian local rings", "and hence faithfully flat.", "By (2) we see that the local ring on the right is regular.", "Thus by Algebra, Lemma \\ref{algebra-lemma-flat-under-regular}", "we conclude that $\\mathcal{O}_{X_{k''}, x''}$ is regular.", "By Lemma \\ref{lemma-geometrically-regular-at-point} we see that $X$", "is geometrically regular at $x$." ], "refs": [ "algebra-lemma-flat-under-regular", "varieties-lemma-geometrically-regular-at-point" ], "ref_ids": [ 981, 10958 ] } ], "ref_ids": [] }, { "id": 10961, "type": "theorem", "label": "varieties-lemma-flat-under-geometrically-regular", "categories": [ "varieties" ], "title": "varieties-lemma-flat-under-geometrically-regular", "contents": [ "Let $k$ be a field.", "Let $f : X \\to Y$ be a morphism of locally Noetherian schemes over $k$.", "Let $x \\in X$ be a point and set $y = f(x)$.", "If $X$ is geometrically regular at $x$ and", "$f$ is flat at $x$ then $Y$ is geometrically regular at $y$.", "In particular, if $X$ is geometrically regular over $k$ and", "$f$ is flat and surjective, then $Y$ is geometrically regular over $k$." ], "refs": [], "proofs": [ { "contents": [ "Let $k'$ be finite purely inseparable extension of $k$.", "Let $f' : X_{k'} \\to Y_{k'}$ be the base change of $f$.", "Let $x' \\in X_{k'}$ be the unique point lying over $x$.", "If we show that $Y_{k'}$ is regular at $y' = f'(x')$, then", "$Y$ is geometrically regular over $k$ at $y'$, see", "Lemma \\ref{lemma-geometrically-regular}.", "By", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-module-flat}", "the morphism $X_{k'} \\to Y_{k'}$ is flat at $x'$.", "Hence the ring map", "$$", "\\mathcal{O}_{Y_{k'}, y'}", "\\longrightarrow", "\\mathcal{O}_{X_{k'}, x'}", "$$", "is a flat local homomorphism of local Noetherian rings with", "right hand side regular by assumption. Hence the left hand side", "is a regular local ring by", "Algebra, Lemma \\ref{algebra-lemma-flat-under-regular}." ], "refs": [ "varieties-lemma-geometrically-regular", "morphisms-lemma-base-change-module-flat", "algebra-lemma-flat-under-regular" ], "ref_ids": [ 10959, 5264, 981 ] } ], "ref_ids": [] }, { "id": 10962, "type": "theorem", "label": "varieties-lemma-geometrically-regular-smooth", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-regular-smooth", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme locally of finite type over $k$.", "Let $x \\in X$.", "Then $X$ is geometrically regular at $x$ if and only if $X \\to \\Spec(k)$", "is smooth at $x$ (Morphisms, Definition \\ref{morphisms-definition-smooth})." ], "refs": [ "morphisms-definition-smooth" ], "proofs": [ { "contents": [ "The question is local around $x$,", "hence we may assume that $X = \\Spec(A)$", "for some finite type $k$-algebra.", "Let $x$ correspond to the prime $\\mathfrak p$.", "\\medskip\\noindent", "If $A$ is smooth over $k$ at $\\mathfrak p$, then we may localize $A$", "and assume that $A$ is smooth over $k$. In this case $k' \\otimes_k A$", "is smooth over $k'$ for all extension fields $k'/k$, and each of", "these Noetherian rings is regular by", "Algebra, Lemma \\ref{algebra-lemma-characterize-smooth-over-field}.", "\\medskip\\noindent", "Assume $X$ is geometrically regular at $x$.", "Consider the residue field $K := \\kappa(x) = \\kappa(\\mathfrak p)$ of $x$.", "It is a finitely generated extension of $k$.", "By Algebra, Lemma \\ref{algebra-lemma-make-separable}", "there exists a finite purely inseparable", "extension $k \\subset k'$ such that the compositum", "$k'K$ is a separable field extension of $k'$.", "Let $\\mathfrak p' \\subset A' = k' \\otimes_k A$ be a prime ideal", "lying over $\\mathfrak p$. It is the unique prime lying over $\\mathfrak p$, see", "Algebra, Lemma \\ref{algebra-lemma-p-ring-map}.", "Hence the residue field $K' := \\kappa(\\mathfrak p')$", "is the compositum $k'K$. By assumption the local ring", "$(A')_{\\mathfrak p'}$ is regular. Hence by", "Algebra, Lemma \\ref{algebra-lemma-separable-smooth}", "we see that $k' \\to A'$ is smooth at $\\mathfrak p'$.", "This in turn implies that $k \\to A$ is smooth at $\\mathfrak p$ by", "Algebra, Lemma \\ref{algebra-lemma-smooth-field-change-local}.", "The lemma is proved." ], "refs": [ "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-make-separable", "algebra-lemma-p-ring-map", "algebra-lemma-separable-smooth", "algebra-lemma-smooth-field-change-local" ], "ref_ids": [ 1223, 573, 582, 1225, 1202 ] } ], "ref_ids": [ 5564 ] }, { "id": 10963, "type": "theorem", "label": "varieties-lemma-CM-base-change", "categories": [ "varieties" ], "title": "varieties-lemma-CM-base-change", "contents": [ "Let $X$ be a locally Noetherian scheme over the field $k$.", "Let $k \\subset k'$ be a finitely generated field extension.", "Let $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying", "over $x$. Then we have", "$$", "\\mathcal{O}_{X, x}\\text{ is Cohen-Macaulay}", "\\Leftrightarrow", "\\mathcal{O}_{X_{k'}, x'}\\text{ is Cohen-Macaulay}", "$$", "If $X$ is locally of finite type over $k$, the same holds for any", "field extension $k \\subset k'$." ], "refs": [], "proofs": [ { "contents": [ "The first case of the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-CM-geometrically-CM}.", "The second case of the lemma is equivalent to", "Algebra, Lemma \\ref{algebra-lemma-extend-field-CM-locus}." ], "refs": [ "algebra-lemma-CM-geometrically-CM", "algebra-lemma-extend-field-CM-locus" ], "ref_ids": [ 1388, 1126 ] } ], "ref_ids": [] }, { "id": 10964, "type": "theorem", "label": "varieties-lemma-locally-finite-type-Jacobson", "categories": [ "varieties" ], "title": "varieties-lemma-locally-finite-type-Jacobson", "contents": [ "Let $X$ be a scheme which is locally of finite type over $k$.", "Then", "\\begin{enumerate}", "\\item for any closed point $x \\in X$ the extension $k \\subset \\kappa(x)$", "is algebraic, and", "\\item $X$ is a Jacobson scheme", "(Properties, Definition \\ref{properties-definition-jacobson}).", "\\end{enumerate}" ], "refs": [ "properties-definition-jacobson" ], "proofs": [ { "contents": [ "A scheme is Jacobson if and only if it has an affine open covering", "by Jacobson schemes, see", "Properties, Lemma \\ref{properties-lemma-locally-jacobson}.", "The property on residue fields at closed points is also local on $X$.", "Hence we may assume that $X$ is affine. In this case the result", "is a consequence of the Hilbert Nullstellensatz, see", "Algebra, Theorem \\ref{algebra-theorem-nullstellensatz}.", "It also follows from a combination of", "Morphisms, Lemmas \\ref{morphisms-lemma-jacobson-finite-type-points},", "\\ref{morphisms-lemma-Jacobson-universally-Jacobson}, and", "\\ref{morphisms-lemma-ubiquity-Jacobson-schemes}." ], "refs": [ "properties-lemma-locally-jacobson", "algebra-theorem-nullstellensatz", "morphisms-lemma-jacobson-finite-type-points", "morphisms-lemma-Jacobson-universally-Jacobson", "morphisms-lemma-ubiquity-Jacobson-schemes" ], "ref_ids": [ 2964, 316, 5211, 5212, 5213 ] } ], "ref_ids": [ 3072 ] }, { "id": 10965, "type": "theorem", "label": "varieties-lemma-make-Jacobson", "categories": [ "varieties" ], "title": "varieties-lemma-make-Jacobson", "contents": [ "Let $X$ be a scheme over a field $k$.", "For any field extension $k \\subset K$ whose cardinality is large enough", "we have", "\\begin{enumerate}", "\\item for any closed point $x \\in X_K$ the extension $K \\subset \\kappa(x)$", "is algebraic, and", "\\item $X_K$ is a Jacobson scheme", "(Properties, Definition \\ref{properties-definition-jacobson}).", "\\end{enumerate}" ], "refs": [ "properties-definition-jacobson" ], "proofs": [ { "contents": [ "Choose an affine open covering $X = \\bigcup U_i$.", "By", "Algebra, Lemma \\ref{algebra-lemma-base-change-Jacobson}", "and", "Properties, Lemma \\ref{properties-lemma-affine-jacobson}", "there exist cardinals $\\kappa_i$ such that $U_{i, K}$ has", "the desired properties over $K$ if $\\#(K) \\geq \\kappa_i$.", "Set $\\kappa = \\max\\{\\kappa_i\\}$. Then if the cardinality of", "$K$ is larger than $\\kappa$ we see that each $U_{i, K}$ satisfies", "the conclusions of the lemma. Hence $X_K$ is Jacobson by", "Properties, Lemma \\ref{properties-lemma-locally-jacobson}.", "The statement on residue fields at closed points of $X_K$", "follows from the corresponding", "statements for residue fields of closed points of the $U_{i, K}$." ], "refs": [ "algebra-lemma-base-change-Jacobson", "properties-lemma-affine-jacobson", "properties-lemma-locally-jacobson" ], "ref_ids": [ 474, 2963, 2964 ] } ], "ref_ids": [ 3072 ] }, { "id": 10966, "type": "theorem", "label": "varieties-lemma-ample-after-field-extension", "categories": [ "varieties" ], "title": "varieties-lemma-ample-after-field-extension", "contents": [ "Let $k$ be a field. Let $X$ be a scheme over $k$.", "If there exists an ample invertible sheaf on $X_K$ for some", "field extension $k \\subset K$, then $X$ has an ample invertible", "sheaf." ], "refs": [], "proofs": [ { "contents": [ "Let $k \\subset K$ be a field extension such that $X_K$ has", "an ample invertible sheaf $\\mathcal{L}$.", "The morphism $X_K \\to X$ is surjective. Hence $X$ is quasi-compact", "as the image of a quasi-compact scheme (Properties, Definition", "\\ref{properties-definition-ample}). Since $X_K$ is quasi-separated", "(by Properties, Lemma", "\\ref{properties-lemma-affine-s-opens-cover-quasi-separated})", "we see that $X$ is quasi-separated: If $U, V \\subset X$ are", "affine open, then $(U \\cap V)_K = U_K \\cap V_K$ is quasi-compact", "and $(U \\cap V)_K \\to U \\cap V$ is surjective. Thus", "Schemes, Lemma \\ref{schemes-lemma-characterize-quasi-separated} applies.", "\\medskip\\noindent", "Write $K = \\colim A_i$ as the colimit of the subalgebras of $K$", "which are of finite type over $k$. Denote", "$X_i = X \\times_{\\Spec(k)} \\Spec(A_i)$.", "Since $X_K = \\lim X_i$ we find an $i$ and an invertible sheaf'", "$\\mathcal{L}_i$ on $X_i$ whose pullback to $X_K$ is $\\mathcal{L}$", "(Limits, Lemma \\ref{limits-lemma-descend-invertible-modules};", "here and below we use that $X$ is quasi-compact and quasi-separated as", "just shown). By Limits, Lemma \\ref{limits-lemma-limit-ample}", "we may assume $\\mathcal{L}_i$ is ample after possibly increasing $i$.", "Fix such an $i$ and let $\\mathfrak m \\subset A_i$ be a maximal", "ideal. By the Hilbert Nullstellensatz", "(Algebra, Theorem \\ref{algebra-theorem-nullstellensatz})", "the residue field $k' = A_i/\\mathfrak m$ is a finite", "extension of $k$. Hence $X_{k'} \\subset X_i$ is a closed subscheme", "hence has an ample invertible sheaf", "(Properties, Lemma \\ref{properties-lemma-ample-on-closed}).", "Since $X_{k'} \\to X$ is finite locally free we conclude", "that $X$ has an ample invertible sheaf by", "Divisors, Proposition \\ref{divisors-proposition-push-down-ample}." ], "refs": [ "properties-definition-ample", "properties-lemma-affine-s-opens-cover-quasi-separated", "schemes-lemma-characterize-quasi-separated", "limits-lemma-descend-invertible-modules", "limits-lemma-limit-ample", "algebra-theorem-nullstellensatz", "properties-lemma-ample-on-closed", "divisors-proposition-push-down-ample" ], "ref_ids": [ 3088, 3045, 7709, 15079, 15045, 316, 3041, 8081 ] } ], "ref_ids": [] }, { "id": 10967, "type": "theorem", "label": "varieties-lemma-quasi-affine-after-field-extension", "categories": [ "varieties" ], "title": "varieties-lemma-quasi-affine-after-field-extension", "contents": [ "Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is quasi-affine", "for some field extension $k \\subset K$, then $X$ is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "Let $k \\subset K$ be a field extension such that $X_K$ is quasi-affine.", "The morphism $X_K \\to X$ is surjective. Hence $X$ is quasi-compact", "as the image of a quasi-compact scheme (Properties, Definition", "\\ref{properties-definition-quasi-affine}). Since $X_K$ is quasi-separated", "(as an open subscheme of an affine scheme)", "we see that $X$ is quasi-separated: If $U, V \\subset X$ are", "affine open, then $(U \\cap V)_K = U_K \\cap V_K$ is quasi-compact", "and $(U \\cap V)_K \\to U \\cap V$ is surjective. Thus", "Schemes, Lemma \\ref{schemes-lemma-characterize-quasi-separated} applies.", "\\medskip\\noindent", "Write $K = \\colim A_i$ as the colimit of the subalgebras of $K$", "which are of finite type over $k$. Denote", "$X_i = X \\times_{\\Spec(k)} \\Spec(A_i)$.", "Since $X_K = \\lim X_i$ we find an $i$ such that $X_i$ is quasi-affine", "(Limits, Lemma \\ref{limits-lemma-limit-quasi-affine};", "here we use that $X$ is quasi-compact and quasi-separated as", "just shown). By the Hilbert Nullstellensatz", "(Algebra, Theorem \\ref{algebra-theorem-nullstellensatz})", "the residue field $k' = A_i/\\mathfrak m$ is a finite", "extension of $k$. Hence $X_{k'} \\subset X_i$ is a closed", "subscheme hence is quasi-affine (Properties, Lemma", "\\ref{properties-lemma-quasi-affine-locally-closed}).", "Since $X_{k'} \\to X$ is finite locally free we conclude by", "Divisors, Lemma \\ref{divisors-lemma-push-down-quasi-affine}." ], "refs": [ "properties-definition-quasi-affine", "schemes-lemma-characterize-quasi-separated", "limits-lemma-limit-quasi-affine", "algebra-theorem-nullstellensatz", "properties-lemma-quasi-affine-locally-closed", "divisors-lemma-push-down-quasi-affine" ], "ref_ids": [ 3083, 7709, 15042, 316, 3054, 7971 ] } ], "ref_ids": [] }, { "id": 10968, "type": "theorem", "label": "varieties-lemma-quasi-projective-after-field-extension", "categories": [ "varieties" ], "title": "varieties-lemma-quasi-projective-after-field-extension", "contents": [ "Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is quasi-projective", "over $K$ for some field extension $k \\subset K$, then $X$ is quasi-projective", "over $k$." ], "refs": [], "proofs": [ { "contents": [ "By definition a morphism of schemes $g : Y \\to T$ is quasi-projective", "if it is locally of finite type, quasi-compact, and there exists", "a $g$-ample invertible sheaf on $Y$.", "Let $k \\subset K$ be a field extension such that $X_K$ is quasi-projective", "over $K$. Let $\\Spec(A) \\subset X$ be an affine open. Then $U_K$ is an", "affine open subscheme of $X_K$, hence $A_K$ is a $K$-algebra of finite type.", "Then $A$ is a $k$-algebra of finite type by", "Algebra, Lemma \\ref{algebra-lemma-finite-type-descends}.", "Hence $X \\to \\Spec(k)$ is locally of finite type.", "Since $X_K \\to \\Spec(K)$ is quasi-compact, we see that $X_K$ is", "quasi-compact, hence $X$ is quasi-compact, hence $X \\to \\Spec(k)$", "is of finite type. By Morphisms, Lemma", "\\ref{morphisms-lemma-finite-type-over-affine-ample-very-ample}", "we see that $X_K$ has an ample invertible sheaf.", "Then $X$ has an ample invertible sheaf by", "Lemma \\ref{lemma-ample-after-field-extension}.", "Hence $X \\to \\Spec(k)$ is quasi-projective by Morphisms, Lemma", "\\ref{morphisms-lemma-finite-type-over-affine-ample-very-ample}." ], "refs": [ "algebra-lemma-finite-type-descends", "morphisms-lemma-finite-type-over-affine-ample-very-ample", "varieties-lemma-ample-after-field-extension", "morphisms-lemma-finite-type-over-affine-ample-very-ample" ], "ref_ids": [ 1079, 5394, 10966, 5394 ] } ], "ref_ids": [] }, { "id": 10969, "type": "theorem", "label": "varieties-lemma-proper-after-field-extension", "categories": [ "varieties" ], "title": "varieties-lemma-proper-after-field-extension", "contents": [ "Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is proper", "over $K$ for some field extension $k \\subset K$, then $X$ is proper", "over $k$." ], "refs": [], "proofs": [ { "contents": [ "Let $k \\subset K$ be a field extension such that $X_K$ is proper over $K$.", "Recall that this implies $X_K$ is separated and quasi-compact", "(Morphisms, Definition \\ref{morphisms-definition-proper}).", "The morphism $X_K \\to X$ is surjective. Hence $X$ is quasi-compact", "as the image of a quasi-compact scheme (Properties, Definition", "\\ref{properties-definition-ample}). Since $X_K$ is separated", "we see that $X$ is quasi-separated: If $U, V \\subset X$ are", "affine open, then $(U \\cap V)_K = U_K \\cap V_K$ is quasi-compact", "and $(U \\cap V)_K \\to U \\cap V$ is surjective. Thus", "Schemes, Lemma \\ref{schemes-lemma-characterize-quasi-separated} applies.", "\\medskip\\noindent", "Write $K = \\colim A_i$ as the colimit of the subalgebras of $K$", "which are of finite type over $k$. Denote", "$X_i = X \\times_{\\Spec(k)} \\Spec(A_i)$.", "By Limits, Lemma \\ref{limits-lemma-eventually-proper}", "there exists an $i$ such that $X_i \\to \\Spec(A_i)$ is proper.", "Here we use that $X$ is quasi-compact and quasi-separated as", "just shown. Choose a maximal ideal $\\mathfrak m \\subset A_i$.", "By the Hilbert Nullstellensatz", "(Algebra, Theorem \\ref{algebra-theorem-nullstellensatz})", "the residue field $k' = A_i/\\mathfrak m$ is a finite", "extension of $k$. The base change $X_{k'} \\to \\Spec(k')$", "is proper (Morphisms, Lemma \\ref{morphisms-lemma-base-change-proper}).", "Since $k \\subset k'$ is finite both $X_{k'} \\to X$ and the composition", "$X_{k'} \\to \\Spec(k)$", "are proper as well (Morphisms, Lemmas \\ref{morphisms-lemma-finite-proper},", "\\ref{morphisms-lemma-base-change-proper}, and", "\\ref{morphisms-lemma-composition-proper}).", "The first implies that $X$ is separated over $k$ as $X_{k'}$", "is separated", "(Morphisms, Lemma \\ref{morphisms-lemma-image-universally-closed-separated}).", "The second implies that $X \\to \\Spec(k)$ is proper", "by Morphisms, Lemma \\ref{morphisms-lemma-image-proper-is-proper}." ], "refs": [ "morphisms-definition-proper", "properties-definition-ample", "schemes-lemma-characterize-quasi-separated", "limits-lemma-eventually-proper", "algebra-theorem-nullstellensatz", "morphisms-lemma-base-change-proper", "morphisms-lemma-finite-proper", "morphisms-lemma-base-change-proper", "morphisms-lemma-composition-proper", "morphisms-lemma-image-universally-closed-separated", "morphisms-lemma-image-proper-is-proper" ], "ref_ids": [ 5571, 3088, 7709, 15089, 316, 5409, 5445, 5409, 5408, 5415, 5413 ] } ], "ref_ids": [] }, { "id": 10970, "type": "theorem", "label": "varieties-lemma-projective-after-field-extension", "categories": [ "varieties" ], "title": "varieties-lemma-projective-after-field-extension", "contents": [ "Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is projective", "over $K$ for some field extension $k \\subset K$, then $X$ is projective", "over $k$." ], "refs": [], "proofs": [ { "contents": [ "A scheme over $k$ is projective over $k$ if and only if it is", "quasi-projective and proper over $k$. See", "Morphisms, Lemma \\ref{morphisms-lemma-projective-is-quasi-projective-proper}.", "Thus the lemma follows from", "Lemmas \\ref{lemma-quasi-projective-after-field-extension} and", "\\ref{lemma-proper-after-field-extension}." ], "refs": [ "morphisms-lemma-projective-is-quasi-projective-proper", "varieties-lemma-quasi-projective-after-field-extension", "varieties-lemma-proper-after-field-extension" ], "ref_ids": [ 5430, 10968, 10969 ] } ], "ref_ids": [] }, { "id": 10971, "type": "theorem", "label": "varieties-lemma-tangent-space", "categories": [ "varieties" ], "title": "varieties-lemma-tangent-space", "contents": [ "The set of dotted arrows making (\\ref{equation-tangent-space}) commute", "has a canonical $\\kappa(x)$-vector space structure." ], "refs": [], "proofs": [ { "contents": [ "Set $\\kappa = \\kappa(x)$. Observe that we have a pushout in the", "category of schemes", "$$", "\\Spec(\\kappa[\\epsilon]) \\amalg_{\\Spec(\\kappa)} \\Spec(\\kappa[\\epsilon])", "= \\Spec(\\kappa[\\epsilon_1, \\epsilon_2])", "$$", "where $\\kappa[\\epsilon_1, \\epsilon_2]$ is the $\\kappa$-algebra with", "basis $1, \\epsilon_1, \\epsilon_2$ and", "$\\epsilon_1^2 = \\epsilon_1\\epsilon_2 = \\epsilon_2^2 = 0$.", "This follows immediately from the corresponding result for", "rings and the description of morphisms from spectra of local rings", "to schemes in", "Schemes, Lemma \\ref{schemes-lemma-morphism-from-spec-local-ring}.", "Given two arrows", "$\\theta_1, \\theta_2 : \\Spec(\\kappa[\\epsilon]) \\to X$", "we can consider the morphism", "$$", "\\theta_1 + \\theta_2 :", "\\Spec(\\kappa[\\epsilon]) \\to", "\\Spec(\\kappa[\\epsilon_1, \\epsilon_2])", "\\xrightarrow{\\theta_1, \\theta_2} X", "$$", "where the first arrow is given by $\\epsilon_i \\mapsto \\epsilon$.", "On the other hand, given $\\lambda \\in \\kappa$ there is a self map", "of $\\Spec(\\kappa[\\epsilon])$ corresponding to the $\\kappa$-algebra", "endomorphism of $\\kappa[\\epsilon]$ which sends $\\epsilon$ to $\\lambda \\epsilon$.", "Precomposing $\\theta : \\Spec(\\kappa[\\epsilon]) \\to X$", "by this selfmap gives $\\lambda \\theta$. The reader can verify", "the axioms of a vector space by verifying the existence", "of suitable commutative diagrams of schemes. We omit the details.", "(An alternative proof would be to express everything in terms of local", "rings and then verify the vector space axioms on the level of ring maps.)" ], "refs": [ "schemes-lemma-morphism-from-spec-local-ring" ], "ref_ids": [ 7683 ] } ], "ref_ids": [] }, { "id": 10972, "type": "theorem", "label": "varieties-lemma-tangent-space-cotangent-space", "categories": [ "varieties" ], "title": "varieties-lemma-tangent-space-cotangent-space", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$.", "There is a canonical isomorphism", "$$", "T_{X/S, x} = \\Hom_{\\mathcal{O}_{X, x}}(\\Omega_{X/S, x}, \\kappa(x))", "$$", "of vector spaces over $\\kappa(x)$." ], "refs": [], "proofs": [ { "contents": [ "Set $\\kappa = \\kappa(x)$.", "Given $\\theta \\in T_{X/S, x}$ we obtain a map", "$$", "\\theta^*\\Omega_{X/S} \\to", "\\Omega_{\\Spec(\\kappa[\\epsilon])/\\Spec(\\kappa(s))} \\to", "\\Omega_{\\Spec(\\kappa[\\epsilon])/\\Spec(\\kappa)}", "$$", "Taking sections we obtain an $\\mathcal{O}_{X, x}$-linear map", "$\\xi_\\theta : \\Omega_{X/S, x} \\to \\kappa \\text{d}\\epsilon$, i.e.,", "an element of the right hand side of the", "formula of the lemma. To show that $\\theta \\mapsto \\xi_\\theta$ is", "an isomorphism we can replace $S$ by $s$ and $X$ by the", "scheme theoretic fibre $X_s$. Indeed, both sides of the", "formula only depend on the scheme theoretic fibre;", "this is clear for $T_{X/S, x}$ and for the RHS see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-differentials}.", "We may also replace $X$ by the spectrum of $\\mathcal{O}_{X, x}$", "as this does not change $T_{X/S, x}$", "(Schemes, Lemma \\ref{schemes-lemma-morphism-from-spec-local-ring})", "nor $\\Omega_{X/S, x}$", "(Modules, Lemma \\ref{modules-lemma-stalk-module-differentials}).", "\\medskip\\noindent", "Let $(A, \\mathfrak m, \\kappa)$ be a local ring over a field $k$.", "To finish the proof we have to show that any $A$-linear map", "$\\xi : \\Omega_{A/k} \\to \\kappa$ comes from a unique $k$-algebra", "map $\\varphi : A \\to \\kappa[\\epsilon]$ agreeing with the canonical", "map $c : A \\to \\kappa$ modulo $\\epsilon$. Write", "$\\varphi(a) = c(a) + D(a) \\epsilon$", "the reader sees that $a \\mapsto D(a)$ is a $k$-derivation.", "Using the universal property of $\\Omega_{A/k}$ we see that each", "$D$ corresponds to a unique $\\xi$ and vice versa. This finishes the proof." ], "refs": [ "morphisms-lemma-base-change-differentials", "schemes-lemma-morphism-from-spec-local-ring", "modules-lemma-stalk-module-differentials" ], "ref_ids": [ 5314, 7683, 13314 ] } ], "ref_ids": [] }, { "id": 10973, "type": "theorem", "label": "varieties-lemma-tangent-space-rational-point", "categories": [ "varieties" ], "title": "varieties-lemma-tangent-space-rational-point", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point and let $s = f(x) \\in S$.", "Assume that $\\kappa(x) = \\kappa(s)$. Then there are canonical isomorphisms", "$$", "\\mathfrak m_x/(\\mathfrak m_x^2 + \\mathfrak m_s\\mathcal{O}_{X, x})", "=", "\\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)", "$$", "and", "$$", "T_{X/S, x} =", "\\Hom_{\\kappa(x)}(", "\\mathfrak m_x/(\\mathfrak m_x^2 + \\mathfrak m_s\\mathcal{O}_{X, x}),", "\\kappa(x))", "$$", "This works more generally if $\\kappa(x)/\\kappa(s)$ is a separable", "algebraic extension." ], "refs": [], "proofs": [ { "contents": [ "The second isomorphism follows from the first by", "Lemma \\ref{lemma-tangent-space-cotangent-space}.", "For the first, we can replace $S$ by $s$ and $X$ by $X_s$, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-differentials}.", "We may also replace $X$ by the spectrum of $\\mathcal{O}_{X, x}$, see", "Modules, Lemma \\ref{modules-lemma-stalk-module-differentials}.", "Thus we have to show the following algebra fact: let", "$(A, \\mathfrak m, \\kappa)$ be a local ring over a field $k$", "such that $\\kappa/k$ is separable algebraic. Then the canonical map", "$$", "\\mathfrak m/\\mathfrak m^2", "\\longrightarrow", "\\Omega_{A/k} \\otimes \\kappa", "$$", "is an isomorphism. Observe that", "$\\mathfrak m/\\mathfrak m^2 = H_1(\\NL_{\\kappa/A})$. By", "Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL}", "it suffices to show that $\\Omega_{\\kappa/k} = 0$ and", "$H_1(\\NL_{\\kappa/k}) = 0$. Since $\\kappa$ is the union of", "its finite separable extensions in $k$ it suffices to prove", "this when $\\kappa$ is a finite separable extension of $k$", "(Algebra, Lemma \\ref{algebra-lemma-colimits-NL}).", "In this case the ring map $k \\to \\kappa$ is \\'etale", "and hence $\\NL_{\\kappa/k} = 0$ (more or less by definition, see", "Algebra, Section \\ref{algebra-section-etale})." ], "refs": [ "varieties-lemma-tangent-space-cotangent-space", "morphisms-lemma-base-change-differentials", "modules-lemma-stalk-module-differentials", "algebra-lemma-exact-sequence-NL", "algebra-lemma-colimits-NL" ], "ref_ids": [ 10972, 5314, 13314, 1153, 1157 ] } ], "ref_ids": [] }, { "id": 10974, "type": "theorem", "label": "varieties-lemma-map-tangent-spaces", "categories": [ "varieties" ], "title": "varieties-lemma-map-tangent-spaces", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes over a base scheme $S$.", "Let $x \\in X$ be a point. Set $y = f(x)$. If $\\kappa(y) = \\kappa(x)$,", "then $f$ induces a natural linear map", "$$", "\\text{d}f : T_{X/S, x} \\longrightarrow T_{Y/S, y}", "$$", "which is dual to the linear map", "$\\Omega_{Y/S, y} \\otimes \\kappa(y) \\to \\Omega_{X/S, x}$", "via the identifications of Lemma \\ref{lemma-tangent-space-cotangent-space}." ], "refs": [ "varieties-lemma-tangent-space-cotangent-space" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10972 ] }, { "id": 10975, "type": "theorem", "label": "varieties-lemma-tangent-space-product", "categories": [ "varieties" ], "title": "varieties-lemma-tangent-space-product", "contents": [ "Let $X$, $Y$ be schemes over a base $S$. Let $x \\in X$ and $y \\in Y$ with", "the same image point $s \\in S$ such that $\\kappa(s) = \\kappa(x)$ and", "$\\kappa(s) = \\kappa(y)$. There is a canonical isomorphism", "$$", "T_{X \\times_S Y/S, (x, y)} = T_{X/S, x} \\oplus T_{Y/S, y}", "$$", "The map from left to right is induced by the maps on tangent spaces coming", "from the projections $X \\times_S Y \\to X$ and $X \\times_S Y \\to Y$.", "The map from right to left is induced by the maps", "$1 \\times y : X_s \\to X_s \\times_s Y_s$ and", "$x \\times 1 : Y_s \\to X_s \\times_s Y_s$ via the identification", "(\\ref{equation-tangent-space-fibre}) of", "tangent spaces with tangent spaces of fibres." ], "refs": [], "proofs": [ { "contents": [ "The direct sum decomposition follows from", "Morphisms, Lemma \\ref{morphisms-lemma-differential-product}", "via Lemma \\ref{lemma-tangent-space-rational-point}. Compatibility", "with the maps comes from Lemma \\ref{lemma-map-tangent-spaces}." ], "refs": [ "morphisms-lemma-differential-product", "varieties-lemma-tangent-space-rational-point", "varieties-lemma-map-tangent-spaces" ], "ref_ids": [ 5315, 10973, 10974 ] } ], "ref_ids": [] }, { "id": 10976, "type": "theorem", "label": "varieties-lemma-injective-tangent-spaces-unramified", "categories": [ "varieties" ], "title": "varieties-lemma-injective-tangent-spaces-unramified", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes locally of finite type over a", "base scheme $S$. Let $x \\in X$ be a point. Set $y = f(x)$ and assume", "that $\\kappa(y) = \\kappa(x)$. Then the following are equivalent", "\\begin{enumerate}", "\\item $\\text{d}f : T_{X/S, x} \\longrightarrow T_{Y/S, y}$ is injective, and", "\\item $f$ is unramified at $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The morphism $f$ is locally of finite type by", "Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type}.", "The map $\\text{d}f$ is injective, if and only if", "$\\Omega_{Y/S, y} \\otimes \\kappa(y) \\to \\Omega_{X/S, x} \\otimes \\kappa(x)$", "is surjective (Lemma \\ref{lemma-map-tangent-spaces}).", "The exact sequence $f^*\\Omega_{Y/S} \\to \\Omega_{X/S} \\to \\Omega_{X/Y} \\to 0$", "(Morphisms, Lemma \\ref{morphisms-lemma-triangle-differentials})", "then shows that this happens if and only if", "$\\Omega_{X/Y, x} \\otimes \\kappa(x) = 0$.", "Hence the result follows from", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-at-point}." ], "refs": [ "morphisms-lemma-permanence-finite-type", "varieties-lemma-map-tangent-spaces", "morphisms-lemma-triangle-differentials", "morphisms-lemma-unramified-at-point" ], "ref_ids": [ 5204, 10974, 5313, 5355 ] } ], "ref_ids": [] }, { "id": 10977, "type": "theorem", "label": "varieties-lemma-quasi-finite-in-codim-1", "categories": [ "varieties" ], "title": "varieties-lemma-quasi-finite-in-codim-1", "contents": [ "Let $f : X \\to Y$ be locally of finite type. Let $y \\in Y$ be a point", "such that $\\mathcal{O}_{Y, y}$ is Noetherian of dimension $\\leq 1$.", "Assume in addition one of the following conditions is satisfied", "\\begin{enumerate}", "\\item for every generic point $\\eta$ of an irreducible component", "of $X$ the field extension $\\kappa(\\eta) \\supset \\kappa(f(\\eta))$", "is finite (or algebraic),", "\\item for every generic point $\\eta$ of an irreducible component", "of $X$ such that $f(\\eta) \\leadsto y$ the field extension", "$\\kappa(\\eta) \\supset \\kappa(f(\\eta))$ is finite (or algebraic),", "\\item $f$ is quasi-finite at every generic point of an", "irreducible component of $X$,", "\\item $Y$ is locally Noetherian and $f$", "is quasi-finite at a dense set of points of $X$,", "\\item add more here.", "\\end{enumerate}", "Then $f$ is quasi-finite at every point of $X$ lying over $y$." ], "refs": [], "proofs": [ { "contents": [ "Condition (4) implies $X$ is locally Noetherian", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}).", "The set of points at which morphism is quasi-finite is open", "(Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-points-open}).", "A dense open of a locally Noetherian scheme contains all generic", "point of irreducible components, hence (4) implies (3).", "Condition (3) implies condition", "(1) by Morphisms, Lemma \\ref{morphisms-lemma-residue-field-quasi-finite}.", "Condition (1) implies condition (2).", "Thus it suffices to prove the lemma in case (2) holds.", "\\medskip\\noindent", "Assume (2) holds. Recall that $\\Spec(\\mathcal{O}_{Y, y})$", "is the set of points of $Y$ specializing to $y$, see", "Schemes, Lemma \\ref{schemes-lemma-specialize-points}.", "Combined with", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-quasi-finite}", "this shows we may replace $Y$ by $\\Spec(\\mathcal{O}_{Y, y})$.", "Thus we may assume $Y = \\Spec(B)$ where $B$ is a Noetherian", "local ring of dimension $\\leq 1$ and $y$ is the closed point.", "\\medskip\\noindent", "Let $X = \\bigcup X_i$ be the irreducible components of $X$ viewed", "as reduced closed subschemes. If we can show each fibre $X_{i, y}$", "is a discrete space, then $X_y = \\bigcup X_{i, y}$ is discrete as", "well and we conclude that $X \\to Y$ is quasi-finite at all points", "of $X_y$ by Morphisms, Lemma", "\\ref{morphisms-lemma-quasi-finite-at-point-characterize}.", "Thus we may assume $X$ is an integral scheme.", "\\medskip\\noindent", "If $X \\to Y$ maps the generic point $\\eta$ of $X$ to $y$, then $X$", "is the spectrum of a finite extension of $\\kappa(y)$ and the", "result is true. Assume that $X$ maps $\\eta$ to a point corresponding", "to a minimal prime $\\mathfrak q$ of $B$ different from $\\mathfrak m_B$.", "We obtain a factorization $X \\to \\Spec(B/\\mathfrak q) \\to \\Spec(B)$.", "Let $x \\in X$ be a point lying over $y$.", "By the dimension formula", "(Morphisms, Lemma \\ref{morphisms-lemma-dimension-formula})", "we have", "$$", "\\dim(\\mathcal{O}_{X, x}) \\leq \\dim(B/\\mathfrak q) +", "\\text{trdeg}_{\\kappa(\\mathfrak q)}(R(X)) - \\text{trdeg}_{\\kappa(y)} \\kappa(x)", "$$", "We know that $\\dim(B/\\mathfrak q) = 1$, that the generic point of $X$", "is not equal to $x$ and specializes to $x$ and that $R(X)$ is algebraic", "over $\\kappa(\\mathfrak q)$. Thus we get", "$$", "1 \\leq 1 - \\text{trdeg}_{\\kappa(y)} \\kappa(x)", "$$", "Hence every point $x$ of $X_y$ is closed in $X_y$ by", "Morphisms, Lemma", "\\ref{morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre}", "and hence $X \\to Y$ is quasi-finite at every point $x$ of $X_y$ by", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}", "(which also implies that $X_y$ is a discrete topological space)." ], "refs": [ "morphisms-lemma-finite-type-noetherian", "morphisms-lemma-quasi-finite-points-open", "morphisms-lemma-residue-field-quasi-finite", "schemes-lemma-specialize-points", "morphisms-lemma-base-change-quasi-finite", "morphisms-lemma-quasi-finite-at-point-characterize", "morphisms-lemma-dimension-formula", "morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre", "morphisms-lemma-quasi-finite-at-point-characterize" ], "ref_ids": [ 5202, 5521, 5225, 7684, 5233, 5226, 5493, 5222, 5226 ] } ], "ref_ids": [] }, { "id": 10978, "type": "theorem", "label": "varieties-lemma-finite-in-codim-1", "categories": [ "varieties" ], "title": "varieties-lemma-finite-in-codim-1", "contents": [ "Let $f : X \\to Y$ be a proper morphism. Let $y \\in Y$ be a point", "such that $\\mathcal{O}_{Y, y}$ is Noetherian of dimension $\\leq 1$.", "Assume in addition one of the following conditions is satisfied", "\\begin{enumerate}", "\\item for every generic point $\\eta$ of an irreducible component", "of $X$ the field extension $\\kappa(\\eta) \\supset \\kappa(f(\\eta))$", "is finite (or algebraic),", "\\item for every generic point $\\eta$ of an irreducible component", "of $X$ such that $f(\\eta) \\leadsto y$ the field extension", "$\\kappa(\\eta) \\supset \\kappa(f(\\eta))$ is finite (or algebraic),", "\\item $f$ is quasi-finite at every generic point of $X$,", "\\item $Y$ is locally Noetherian and $f$", "is quasi-finite at a dense set of points of $X$,", "\\item add more here.", "\\end{enumerate}", "Then there exists an open neighbourhood $V \\subset Y$ of $y$ such that", "$f^{-1}(V) \\to V$ is finite." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-quasi-finite-in-codim-1} the morphism $f$ is", "quasi-finite at every point of the fibre $X_y$. Hence", "$X_y$ is a discrete topological space", "(Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}).", "As $f$ is proper the fibre $X_y$ is quasi-compact, i.e., finite.", "Thus we can apply Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-finite-fibre-finite-in-neighbourhood}", "to conclude." ], "refs": [ "varieties-lemma-quasi-finite-in-codim-1", "morphisms-lemma-quasi-finite-at-point-characterize", "coherent-lemma-proper-finite-fibre-finite-in-neighbourhood" ], "ref_ids": [ 10977, 5226, 3366 ] } ], "ref_ids": [] }, { "id": 10979, "type": "theorem", "label": "varieties-lemma-modification-normal-iso-over-codimension-1", "categories": [ "varieties" ], "title": "varieties-lemma-modification-normal-iso-over-codimension-1", "contents": [ "Let $X$ be a Noetherian scheme. Let $f : Y \\to X$ be a birational proper", "morphism of schemes with $Y$ reduced. Let $U \\subset X$ be the", "maximal open over which $f$ is an isomorphism. Then $U$ contains", "\\begin{enumerate}", "\\item every point of codimension $0$ in $X$,", "\\item every $x \\in X$ of codimension $1$ on $X$ such that", "$\\mathcal{O}_{X, x}$ is a discrete valuation ring,", "\\item every $x \\in X$ such that the fibre of $Y \\to X$ over $x$ is", "finite and such that $\\mathcal{O}_{X, x}$ is normal, and", "\\item every $x \\in X$ such that $f$ is quasi-finite at some", "$y \\in Y$ lying over $x$ and $\\mathcal{O}_{X, x}$ is normal.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from Morphisms, Lemma", "\\ref{morphisms-lemma-birational-isomorphism-over-dense-open}.", "Part (2) follows from part (3) and Lemma \\ref{lemma-finite-in-codim-1}", "(and the fact that finite morphisms have finite fibres).", "\\medskip\\noindent", "Part (3) follows from part (4) and", "Morphisms, Lemma \\ref{morphisms-lemma-finite-fibre}", "but we will also give a direct proof.", "Let $x \\in X$ be as in (3). By", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-finite-fibre-finite-in-neighbourhood}", "we may assume $f$ is finite. We may assume $X$ affine.", "This reduces us to", "the case of a finite birational morphism of Noetherian affine schemes", "$Y \\to X$ and $x \\in X$ such that $\\mathcal{O}_{X, x}$ is a", "normal domain. Since $\\mathcal{O}_{X, x}$ is a domain and $X$", "is Noetherian, we may replace $X$ by an affine open of $x$ which", "is integral. Then, since $Y \\to X$ is birational and $Y$ is reduced", "we see that $Y$ is integral. Writing $X = \\Spec(A)$ and $Y = \\Spec(B)$", "we see that $A \\subset B$ is a finite inclusion of domains having the same", "field of fractions. If $\\mathfrak p \\subset A$ is the prime corresponding", "to $x$, then $A_\\mathfrak p$ being normal implies that", "$A_\\mathfrak p \\subset B_\\mathfrak p$ is an equality.", "Since $B$ is a finite $A$-module, we see there exists an", "$a \\in A$, $a \\not \\in \\mathfrak p$ such that $A_a \\to B_a$", "is an isomorphism.", "\\medskip\\noindent", "Let $x \\in X$ and $y \\in Y$ be as in (4). After replacing $X$", "by an affine open neighbourhood we may assume $X = \\Spec(A)$", "and $A \\subset \\mathcal{O}_{X, x}$, see", "Properties, Lemma \\ref{properties-lemma-ring-affine-open-injective-local-ring}.", "Then $A$ is a domain and hence $X$ is integral.", "Since $f$ is birational and $Y$ is reduced", "it follows that $Y$ is integral too. Consider the ring map", "$\\mathcal{O}_{X, x} \\to \\mathcal{O}_{Y, y}$. This is a ring map", "which is essentially of finite type, the residue field extension", "is finite, and $\\dim(\\mathcal{O}_{Y, y}/\\mathfrak m_x\\mathcal{O}_{Y, y}) = 0$", "(to see this trace through the definitions of quasi-finite", "maps in", "Morphisms, Definition \\ref{morphisms-definition-quasi-finite} and", "Algebra, Definition \\ref{algebra-definition-quasi-finite}). By", "Algebra, Lemma \\ref{algebra-lemma-essentially-finite-type-fibre-dim-zero}", "$\\mathcal{O}_{Y, y}$ is the localization of a finite", "$\\mathcal{O}_{X, x}$-algebra $B$. Of course we may replace", "$B$ by the image of $B$ in $\\mathcal{O}_{Y, y}$ and assume", "that $B$ is a domain with the same fraction field as $\\mathcal{O}_{Y, y}$.", "Then $\\mathcal{O}_{X, x} \\subset B$", "have the same fraction field as $f$ is birational. Since", "$\\mathcal{O}_{X, x}$ is normal, we conclude that", "$\\mathcal{O}_{X, x} = B$ (because finite implies integral),", "in particular, we see that $\\mathcal{O}_{X, x} = \\mathcal{O}_{Y, y}$. By", "Morphisms, Lemma \\ref{morphisms-lemma-morphism-defined-local-ring}", "after shrinking $X$ we may assume there is a section", "$X \\to Y$ of $f$ mapping $x$ to $y$ and inducing the given", "isomorphism on local rings. Since $X \\to Y$ is closed", "(by Schemes, Lemma \\ref{schemes-lemma-section-immersion})", "necessarily maps the generic point of $X$ to the generic point of $Y$", "it follows that the image of $X \\to Y$ is $Y$.", "Then $Y = X$ and we've proved what we wanted to show." ], "refs": [ "morphisms-lemma-birational-isomorphism-over-dense-open", "varieties-lemma-finite-in-codim-1", "morphisms-lemma-finite-fibre", "coherent-lemma-proper-finite-fibre-finite-in-neighbourhood", "properties-lemma-ring-affine-open-injective-local-ring", "morphisms-definition-quasi-finite", "algebra-definition-quasi-finite", "algebra-lemma-essentially-finite-type-fibre-dim-zero", "morphisms-lemma-morphism-defined-local-ring" ], "ref_ids": [ 5490, 10978, 5227, 3366, 3065, 5553, 1522, 1069, 5418 ] } ], "ref_ids": [] }, { "id": 10980, "type": "theorem", "label": "varieties-lemma-noether-normalization", "categories": [ "varieties" ], "title": "varieties-lemma-noether-normalization", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ with", "image $s \\in S$. Let $V \\subset S$ be an affine open neighbourhood", "of $s$. If $f$ is locally of finite type and $\\dim_x(X_s) = d$,", "then there exists an affine open $U \\subset X$ with", "$x \\in U$ and $f(U) \\subset V$ and a factorization", "$$", "U \\xrightarrow{\\pi} \\mathbf{A}^d_V \\to V", "$$", "of $f|_U : U \\to V$ such that $\\pi$ is quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Algebra, Lemma \\ref{algebra-lemma-quasi-finite-over-polynomial-algebra}." ], "refs": [ "algebra-lemma-quasi-finite-over-polynomial-algebra" ], "ref_ids": [ 1071 ] } ], "ref_ids": [] }, { "id": 10981, "type": "theorem", "label": "varieties-lemma-noether-normalization-affine", "categories": [ "varieties" ], "title": "varieties-lemma-noether-normalization-affine", "contents": [ "Let $f : X \\to S$ be a finite type morphism of affine schemes.", "Let $s \\in S$. If $\\dim(X_s) = d$, then there exists a factorization", "$$", "X \\xrightarrow{\\pi} \\mathbf{A}^d_S \\to S", "$$", "of $f$ such that the morphism $\\pi_s : X_s \\to \\mathbf{A}^d_{\\kappa(s)}$", "of fibres over $s$ is finite." ], "refs": [], "proofs": [ { "contents": [ "Write $S = \\Spec(A)$ and $X = \\Spec(B)$ and let $A \\to B$ be the ring", "map corresponding to $f$. Let $\\mathfrak p \\subset A$ be the prime ideal", "corresponding to $s$. We can choose a surjection", "$A[x_1, \\ldots, x_r] \\to B$. By", "Algebra, Lemma \\ref{algebra-lemma-Noether-normalization}", "there exist elements $y_1, \\ldots, y_d \\in A$ in the $\\mathbf{Z}$-subalgebra", "of $A$ generated by $x_1, \\ldots, x_r$ such that the $A$-algebra homomorphism", "$A[t_1, \\ldots, t_d] \\to B$ sending $t_i$ to $y_i$ induces a finite", "$\\kappa(\\mathfrak p)$-algebra homomorphism", "$\\kappa(\\mathfrak p)[t_1, \\ldots, t_d] \\to B \\otimes_A \\kappa(\\mathfrak p)$.", "This proves the lemma." ], "refs": [ "algebra-lemma-Noether-normalization" ], "ref_ids": [ 1001 ] } ], "ref_ids": [] }, { "id": 10982, "type": "theorem", "label": "varieties-lemma-geometric-structure-unramified", "categories": [ "varieties" ], "title": "varieties-lemma-geometric-structure-unramified", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$.", "Let $V = \\Spec(A)$ be an affine open neighbourhood of $f(x)$ in $S$.", "If $f$ is unramified at $x$, then there exist exists an affine open", "$U \\subset X$ with $x \\in U$ and $f(U) \\subset V$", "such that we have a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] & U \\ar[l] \\ar[rd] \\ar[r]^-j &", "\\Spec(A[t]_{g'}/(g)) \\ar[d] \\ar[r] &", "\\Spec(A[t]) = \\mathbf{A}^1_V \\ar[ld] \\\\", "Y & & V \\ar[ll]", "}", "$$", "where $j$ is an immersion, $g \\in A[t]$ is a monic polynomial, and", "$g'$ is the derivative of $g$ with respect to $t$. If $f$ is \\'etale", "at $x$, then we may choose the diagram such that $j$ is an open immersion." ], "refs": [], "proofs": [ { "contents": [ "The unramified case is a translation of", "Algebra, Proposition \\ref{algebra-proposition-unramified-locally-standard}.", "In the \\'etale case this is a translation of", "Algebra, Proposition \\ref{algebra-proposition-etale-locally-standard}", "or equivalently it follows from", "Morphisms, Lemma \\ref{morphisms-lemma-etale-locally-standard-etale}", "although the statements differ slightly." ], "refs": [ "algebra-proposition-unramified-locally-standard", "algebra-proposition-etale-locally-standard", "morphisms-lemma-etale-locally-standard-etale" ], "ref_ids": [ 1428, 1427, 5371 ] } ], "ref_ids": [] }, { "id": 10983, "type": "theorem", "label": "varieties-lemma-unramfied-over-affine", "categories": [ "varieties" ], "title": "varieties-lemma-unramfied-over-affine", "contents": [ "Let $f : X \\to S$ be a finite type morphism of affine schemes.", "Let $x \\in X$ with image $s \\in S$. Let", "$$", "r =", "\\dim_{\\kappa(x)} \\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) =", "\\dim_{\\kappa(x)} \\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) =", "\\dim_{\\kappa(x)} T_{X/S, x}", "$$", "Then there exists a factorization", "$$", "X \\xrightarrow{\\pi} \\mathbf{A}^r_S \\to S", "$$", "of $f$ such that $\\pi$ is unramified at $x$." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms, Lemma \\ref{morphisms-lemma-finite-type-differentials}", "the first dimension is finite.", "The first equality follows as the restriction of", "$\\Omega_{X/S}$ to the fibre is the module of differentials", "from Morphisms, Lemma", "\\ref{morphisms-lemma-base-change-differentials}.", "The last equality follows from Lemma \\ref{lemma-tangent-space-cotangent-space}.", "Thus we see that the statement makes sense.", "\\medskip\\noindent", "To prove the lemma write $S = \\Spec(A)$ and $X = \\Spec(B)$ and let", "$A \\to B$ be the ring map corresponding to $f$. Let $\\mathfrak q \\subset B$", "be the prime ideal corresponding to $x$. Choose a surjection of $A$-algebras", "$A[x_1, \\ldots, x_t] \\to B$. Since $\\Omega_{B/A}$ is generated by", "$\\text{d}x_1, \\ldots, \\text{d}x_t$ we see that their images in", "$\\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)$ generate", "this as a $\\kappa(x)$-vector space. After renumbering we may assume", "that $\\text{d}x_1, \\ldots, \\text{d}x_r$ map to a basis of", "$\\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)$.", "We claim that $P = A[x_1, \\ldots, x_r] \\to B$ is unramified at $\\mathfrak q$.", "To see this it suffices to show that $\\Omega_{B/P, \\mathfrak q} = 0$", "(Algebra, Lemma \\ref{algebra-lemma-unramified}).", "Note that $\\Omega_{B/P}$ is the quotient of $\\Omega_{B/A}$ by the", "submodule generated by $\\text{d}x_1, \\ldots, \\text{d}x_r$.", "Hence", "$\\Omega_{B/P, \\mathfrak q} \\otimes_{B_\\mathfrak q} \\kappa(\\mathfrak q) = 0$", "by our choice of $x_1, \\ldots, x_r$.", "By Nakayama's lemma, more precisely Algebra, Lemma \\ref{algebra-lemma-NAK}", "part (2) which applies as $\\Omega_{B/P}$ is finite (see reference above),", "we conclude that $\\Omega_{B/P, \\mathfrak q} = 0$." ], "refs": [ "morphisms-lemma-finite-type-differentials", "morphisms-lemma-base-change-differentials", "varieties-lemma-tangent-space-cotangent-space", "algebra-lemma-unramified", "algebra-lemma-NAK" ], "ref_ids": [ 5316, 5314, 10972, 1266, 401 ] } ], "ref_ids": [] }, { "id": 10984, "type": "theorem", "label": "varieties-lemma-immersion-into-affine", "categories": [ "varieties" ], "title": "varieties-lemma-immersion-into-affine", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ with image $s \\in S$. Let $V \\subset S$ be", "an affine open neighbourhood of $s$. If $f$ is locally of", "finite type and", "$$", "r =", "\\dim_{\\kappa(x)} \\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) =", "\\dim_{\\kappa(x)} \\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) =", "\\dim_{\\kappa(x)} T_{X/S, x}", "$$", "then there exist", "\\begin{enumerate}", "\\item an affine open $U \\subset X$ with $x \\in U$ and $f(U) \\subset V$ and a", "factorization", "$$", "U \\xrightarrow{j} \\mathbf{A}^{r + 1}_V \\to V", "$$", "of $f|_U$ such that $j$ is an immersion, or", "\\item an affine open $U \\subset X$ with $x \\in U$ and $f(U) \\subset V$ and a", "factorization", "$$", "U \\xrightarrow{j} D \\to V", "$$", "of $f|_U$ such that $j$ is a closed immersion and $D \\to V$", "is smooth of relative dimension $r$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Pick any affine open $U \\subset X$ with $x \\in U$ and $f(U) \\subset V$.", "Apply Lemma \\ref{lemma-unramfied-over-affine} to $U \\to V$ to", "get $U \\to \\mathbf{A}^r_V \\to V$ as in the statement of that lemma.", "By Lemma \\ref{lemma-geometric-structure-unramified}", "we get a factorization", "$$", "U \\xrightarrow{j} D \\xrightarrow{j'} \\mathbf{A}^{r + 1}_V", "\\xrightarrow{p} \\mathbf{A}^r_V \\to V", "$$", "where $j$ and $j'$ are immersions, $p$ is the projection, and", "$p \\circ j'$ is standard \\'etale. Thus we see in particular that", "(1) and (2) hold." ], "refs": [ "varieties-lemma-unramfied-over-affine", "varieties-lemma-geometric-structure-unramified" ], "ref_ids": [ 10983, 10982 ] } ], "ref_ids": [] }, { "id": 10985, "type": "theorem", "label": "varieties-lemma-dimension-fibre-in-codim-1", "categories": [ "varieties" ], "title": "varieties-lemma-dimension-fibre-in-codim-1", "contents": [ "Let $f : X \\to Y$ be locally of finite type. Let $x \\in X$ be a point", "with image $y \\in Y$ such that $\\mathcal{O}_{Y, y}$ is Noetherian of", "dimension $\\leq 1$. Let $d \\geq 0$ be an integer such that for every", "generic point $\\eta$ of an irreducible component of $X$ which contains", "$x$, we have $\\dim_\\eta(X_{f(\\eta)}) = d$. Then $\\dim_x(X_y) = d$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\Spec(\\mathcal{O}_{Y, y})$", "is the set of points of $Y$ specializing to $y$, see", "Schemes, Lemma \\ref{schemes-lemma-specialize-points}.", "Thus we may replace $Y$ by $\\Spec(\\mathcal{O}_{Y, y})$", "and assume $Y = \\Spec(B)$ where $B$ is a Noetherian", "local ring of dimension $\\leq 1$ and $y$ is the closed point.", "We may also replace $X$ by an affine neighbourhood of $x$.", "\\medskip\\noindent", "Let $X = \\bigcup X_i$ be the irreducible components of $X$ viewed", "as reduced closed subschemes. If we can show each fibre $X_{i, y}$", "has dimension $d$, then $X_y = \\bigcup X_{i, y}$ has dimension $d$ as", "well. Thus we may assume $X$ is an integral scheme.", "\\medskip\\noindent", "If $X \\to Y$ maps the generic point $\\eta$ of $X$ to $y$, then $X$", "is a scheme over $\\kappa(y)$ and the result is true by assumption.", "Assume that $X$ maps $\\eta$ to a point $\\xi \\in Y$ corresponding", "to a minimal prime $\\mathfrak q$ of $B$ different from $\\mathfrak m_B$.", "We obtain a factorization $X \\to \\Spec(B/\\mathfrak q) \\to \\Spec(B)$.", "By the dimension formula", "(Morphisms, Lemma \\ref{morphisms-lemma-dimension-formula})", "we have", "$$", "\\dim(\\mathcal{O}_{X, x}) + \\text{trdeg}_{\\kappa(y)} \\kappa(x) \\leq", "\\dim(B/\\mathfrak q) + \\text{trdeg}_{\\kappa(\\mathfrak q)}(R(X))", "$$", "We have $\\dim(B/\\mathfrak q) = 1$. We have", "$\\text{trdeg}_{\\kappa(\\mathfrak q)}(R(X)) = d$ by", "our assumption that $\\dim_\\eta(X_\\xi) = d$, see", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}.", "Since $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X_s, x}$ has a kernel", "(as $\\eta \\mapsto \\xi \\not = y$) and since $\\mathcal{O}_{X, x}$", "is a Noetherian domain we see that", "$\\dim(\\mathcal{O}_{X, x}) > \\dim(\\mathcal{O}_{X_y, x})$.", "We conclude that", "$$", "\\dim_x(X_s) =", "\\dim(\\mathcal{O}_{X_s, x}) + \\text{trdeg}_{\\kappa(y)} \\kappa(x) \\leq d", "$$", "(Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}).", "On the other hand, we have", "$\\dim_x(X_s) \\geq \\dim_\\eta(X_{f(\\eta)}) = d$ by", "Morphisms, Lemma \\ref{morphisms-lemma-openness-bounded-dimension-fibres}." ], "refs": [ "schemes-lemma-specialize-points", "morphisms-lemma-dimension-formula", "morphisms-lemma-dimension-fibre-at-a-point", "morphisms-lemma-dimension-fibre-at-a-point", "morphisms-lemma-openness-bounded-dimension-fibres" ], "ref_ids": [ 7684, 5493, 5277, 5277, 5280 ] } ], "ref_ids": [] }, { "id": 10986, "type": "theorem", "label": "varieties-lemma-dominate-valuation-ring-dimension-fibres", "categories": [ "varieties" ], "title": "varieties-lemma-dominate-valuation-ring-dimension-fibres", "contents": [ "Let $f : X \\to \\Spec(R)$ be a morphism from an irreducible", "scheme to the spectrum of a valuation ring. If $f$ is locally", "of finite type and surjective, then the special fibre is", "equidimensional of dimension equal to the dimension of the generic fibre." ], "refs": [], "proofs": [ { "contents": [ "We may replace $X$ by its reduction because this does not change the", "dimension of $X$ or of the special fibre. Then $X$ is integral and", "the lemma follows from Algebra, Lemma", "\\ref{algebra-lemma-finite-type-domain-over-valuation-ring-dim-fibres}." ], "refs": [ "algebra-lemma-finite-type-domain-over-valuation-ring-dim-fibres" ], "ref_ids": [ 1078 ] } ], "ref_ids": [] }, { "id": 10987, "type": "theorem", "label": "varieties-lemma-dimension-fibre-in-higher-codimension", "categories": [ "varieties" ], "title": "varieties-lemma-dimension-fibre-in-higher-codimension", "contents": [ "Let $f : X \\to Y$ be locally of finite type. Let $x \\in X$ be a point", "with image $y \\in Y$ such that $\\mathcal{O}_{Y, y}$ is Noetherian. Let", "$d \\geq 0$ be an integer such that for every generic point $\\eta$ of an", "irreducible component of $X$ which contains $x$, we have", "$f(\\eta) \\not = y$ and $\\dim_\\eta(X_{f(\\eta)}) = d$. Then", "$\\dim_x(X_y) \\leq d + \\dim(\\mathcal{O}_{Y, y}) - 1$." ], "refs": [], "proofs": [ { "contents": [ "Exactly as in the proof of Lemma \\ref{lemma-dimension-fibre-in-codim-1}", "we reduce to the case $X = \\Spec(A)$ with $A$ a domain and $Y = \\Spec(B)$", "where $B$ is a Noetherian local ring whose maximal ideal corresponds to $y$.", "After replacing $B$ by $B/\\Ker(B \\to A)$ we may assume that $B$", "is a domain and that $B \\subset A$.", "Then we use the dimension formula", "(Morphisms, Lemma \\ref{morphisms-lemma-dimension-formula}) to get", "$$", "\\dim(\\mathcal{O}_{X, x}) + \\text{trdeg}_{\\kappa(y)} \\kappa(x) \\leq", "\\dim(B) + \\text{trdeg}_B(A)", "$$", "We have $\\text{trdeg}_B(A) = d$ by", "our assumption that $\\dim_\\eta(X_\\xi) = d$, see", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}.", "Since $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X_s, x}$ has a kernel", "(as $f(\\eta) \\not = y$) and since $\\mathcal{O}_{X, x}$", "is a Noetherian domain we see that", "$\\dim(\\mathcal{O}_{X, x}) > \\dim(\\mathcal{O}_{X_y, x})$.", "We conclude that", "$$", "\\dim_x(X_s) =", "\\dim(\\mathcal{O}_{X_s, x}) + \\text{trdeg}_{\\kappa(y)} \\kappa(x)", "< \\dim(B) + d", "$$", "(equality by Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point})", "which proves what we want." ], "refs": [ "varieties-lemma-dimension-fibre-in-codim-1", "morphisms-lemma-dimension-formula", "morphisms-lemma-dimension-fibre-at-a-point", "morphisms-lemma-dimension-fibre-at-a-point" ], "ref_ids": [ 10985, 5493, 5277, 5277 ] } ], "ref_ids": [] }, { "id": 10988, "type": "theorem", "label": "varieties-lemma-algebraic-scheme-dim-0", "categories": [ "varieties" ], "title": "varieties-lemma-algebraic-scheme-dim-0", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme of", "dimension $0$. Then $X$ is a disjoint union of spectra of local Artinian", "$k$-algebras $A$ with $\\dim_k(A) < \\infty$. If $X$ is an algebraic $k$-scheme", "of dimension $0$, then in addition $X$ is affine and the morphism", "$X \\to \\Spec(k)$ is finite." ], "refs": [], "proofs": [ { "contents": [ "Let $X$ be a locally algebraic $k$-scheme of dimension $0$.", "Let $U = \\Spec(A) \\subset X$ be an affine open subscheme.", "Since $\\dim(X) = 0$ we see that $\\dim(A) = 0$.", "By Noether normalization, see", "Algebra, Lemma \\ref{algebra-lemma-Noether-normalization}", "we see that there exists a finite injection $k \\to A$, i.e.,", "$\\dim_k(A) < \\infty$. Hence $A$ is Artinian, see", "Algebra, Lemma \\ref{algebra-lemma-finite-dimensional-algebra}.", "This implies that $A = A_1 \\times \\ldots \\times A_r$ is a product", "of finitely many Artinian local rings, see", "Algebra, Lemma \\ref{algebra-lemma-artinian-finite-length}.", "Of course $\\dim_k(A_i) < \\infty$ for each $i$ as the sum of", "these dimensions equals $\\dim_k(A)$.", "\\medskip\\noindent", "The arguments above show that $X$ has an open covering whose members are", "finite discrete topological spaces. Hence $X$ is a discrete topological space.", "It follows that $X$ is isomorphic to the disjoint union of its connected", "components each of which is a singleton. Since a singleton scheme is affine", "we conclude (by the results of the paragraph above) that each of these", "singletons is the spectrum of a local Artinian $k$-algebra $A$ with", "$\\dim_k(A) < \\infty$.", "\\medskip\\noindent", "Finally, if $X$ is an algebraic $k$-scheme of dimension $0$, then", "$X$ is quasi-compact hence is a finite disjoint union", "$X = \\Spec(A_1) \\amalg \\ldots \\amalg \\Spec(A_r)$", "hence affine (see", "Schemes, Lemma \\ref{schemes-lemma-disjoint-union-affines})", "and we have seen the finiteness of $X \\to \\Spec(k)$ in the", "first paragraph of the proof." ], "refs": [ "algebra-lemma-Noether-normalization", "algebra-lemma-finite-dimensional-algebra", "algebra-lemma-artinian-finite-length", "schemes-lemma-disjoint-union-affines" ], "ref_ids": [ 1001, 642, 646, 7659 ] } ], "ref_ids": [] }, { "id": 10989, "type": "theorem", "label": "varieties-lemma-dimension-locally-algebraic", "categories": [ "varieties" ], "title": "varieties-lemma-dimension-locally-algebraic", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.", "\\begin{enumerate}", "\\item", "\\label{item-catenary}", "The topological space of $X$ is catenary", "(Topology, Definition \\ref{topology-definition-catenary}).", "\\item", "\\label{item-dimension-at-closed-point}", "For $x \\in X$ closed, we have $\\dim_x(X) = \\dim(\\mathcal{O}_{X, x})$.", "\\item", "\\label{item-dimension-irreducible}", "For $X$ irreducible we have $\\dim(X) = \\dim(U)$ for any", "nonempty open $U \\subset X$ and $\\dim(X) = \\dim_x(X)$", "for any $x \\in X$.", "\\item", "\\label{item-irreducible-maximal-chains}", "For $X$ irreducible any chain of irreducible closed subsets can be", "extended to a maximal chain and all maximal chains of irreducible", "closed subsets have length equal to $\\dim(X)$.", "\\item", "\\label{item-dimension-irreducibles-passing-through}", "For $x \\in X$ we have", "$\\dim_x(X) = \\max \\dim(Z) = \\min \\dim(\\mathcal{O}_{X, x'})$", "where the maximum is over irreducible components", "$Z \\subset X$ containing $x$ and the minimum is over", "specializations $x \\leadsto x'$ with $x'$ closed in $X$.", "\\item", "\\label{item-dimension-irreducible-trdeg}", "If $X$ is irreducible with generic point $x$, then", "$\\dim(X) = \\text{trdeg}_k(\\kappa(x))$.", "\\item", "\\label{item-immediate-specialization}", "If $x \\leadsto x'$ is an immediate specialization", "of points of $X$, then we have", "$\\text{trdeg}_k(\\kappa(x)) = \\text{trdeg}_k(\\kappa(x')) + 1$.", "\\item", "\\label{item-dimension-sup-trdeg}", "The dimension of $X$ is the supremum of the numbers", "$\\text{trdeg}_k(\\kappa(x))$ where $x$ runs over the", "generic points of the irreducible components of $X$.", "\\item", "\\label{item-specialization}", "If $x \\leadsto x'$ is a nontrivial specialization of points of $X$, then", "\\begin{enumerate}", "\\item $\\dim_x(X) \\leq \\dim_{x'}(X)$,", "\\item $\\dim(\\mathcal{O}_{X, x}) < \\dim(\\mathcal{O}_{X, x'})$,", "\\item $\\text{trdeg}_k(\\kappa(x)) > \\text{trdeg}_k(\\kappa(x'))$, and", "\\item any maximal chain of nontrivial specializations", "$x = x_0 \\leadsto x_1 \\leadsto \\ldots \\leadsto x_n = x$ has", "length $n = \\text{trdeg}_k(\\kappa(x)) - \\text{trdeg}_k(\\kappa(x'))$.", "\\end{enumerate}", "\\item", "\\label{item-dimension-formula}", "For $x \\in X$ we have", "$\\dim_x(X) = \\text{trdeg}_k(\\kappa(x)) + \\dim(\\mathcal{O}_{X, x})$.", "\\item", "\\label{item-immediate-specialization-local-ring}", "If $x \\leadsto x'$ is an immediate specialization", "of points of $X$ and $X$ is irreducible or equidimensional, then", "$\\dim(\\mathcal{O}_{X, x'}) = \\dim(\\mathcal{O}_{X, x}) + 1$.", "\\end{enumerate}" ], "refs": [ "topology-definition-catenary" ], "proofs": [ { "contents": [ "Instead on relying on the more general results proved earlier", "we will reduce the statements to the corresponding statements", "for finite type $k$-algebras and cite results from the chapter", "on commutative algebra.", "\\medskip\\noindent", "Proof of (\\ref{item-catenary}). This is local on $X$ by", "Topology, Lemma \\ref{topology-lemma-catenary}. Thus we may", "assume $X = \\Spec(A)$ where $A$ is a finite type $k$-algebra.", "We have to show that $A$ is catenary", "(Algebra, Lemma \\ref{algebra-lemma-catenary}).", "We can reduce to $k[x_1, \\ldots, x_n]$ using", "Algebra, Lemma \\ref{algebra-lemma-quotient-catenary} and then apply", "Algebra, Lemma \\ref{algebra-lemma-dimension-height-polynomial-ring}.", "Alternatively, this holds because $k$ is Cohen-Macaulay (trivially) and", "Cohen-Macaulay rings are universally catenary", "(Algebra, Lemma \\ref{algebra-lemma-CM-ring-catenary}).", "\\medskip\\noindent", "Proof of (\\ref{item-dimension-at-closed-point}). Choose an affine", "neighbourhood $U = \\Spec(A)$ of $x$. Then $\\dim_x(X) = \\dim_x(U)$.", "Hence we reduce to the affine case, which is", "Algebra, Lemma \\ref{algebra-lemma-dimension-closed-point-finite-type-field}.", "\\medskip\\noindent", "Proof of (\\ref{item-dimension-irreducible}). It suffices to show that", "any two nonempty affine opens $U, U' \\subset X$ have the same dimension", "(any finite chain of irreducible subsets meets an affine open).", "Pick a closed point $x$ of $X$ with $x \\in U \\cap U'$. This is possible", "because $X$ is irreducible, hence $U \\cap U'$ is nonempty, hence there is", "such a closed point because $X$ is Jacobson by", "Lemma \\ref{lemma-locally-finite-type-Jacobson}. Then", "$\\dim(U) = \\dim(\\mathcal{O}_{X, x}) = \\dim(U')$ by", "Algebra, Lemma \\ref{algebra-lemma-dimension-spell-it-out}", "(strictly speaking you have to replace $X$ by its reduction before", "applying the lemma).", "\\medskip\\noindent", "Proof of (\\ref{item-irreducible-maximal-chains}). Given a chain", "of irreducible closed subsets we can find an affine open $U \\subset X$", "which meets the smallest one. Thus the statement follows from", "Algebra, Lemma \\ref{algebra-lemma-dimension-spell-it-out} ", "and $\\dim(U) = \\dim(X)$ which we have seen in", "(\\ref{item-dimension-irreducible}).", "\\medskip\\noindent", "Proof of (\\ref{item-dimension-irreducibles-passing-through}).", "Choose an affine neighbourhood $U = \\Spec(A)$ of $x$. Then", "$\\dim_x(X) = \\dim_x(U)$. The rule $Z \\mapsto Z \\cap U$", "is a bijection between irreducible components of $X$ passing through", "$x$ and irreducible components of $U$ passing through $x$.", "Also, $\\dim(Z \\cap U) = \\dim(Z)$ for such $Z$ by", "(\\ref{item-dimension-irreducible}).", "Hence the statement follows from", "Algebra, Lemma \\ref{algebra-lemma-dimension-at-a-point-finite-type-over-field}.", "\\medskip\\noindent", "Proof of (\\ref{item-dimension-irreducible-trdeg}). By", "(\\ref{item-dimension-irreducible}) this reduces to the case where", "$X = \\Spec(A)$ is affine. In this case it follows from", "Algebra, Lemma \\ref{algebra-lemma-dimension-prime-polynomial-ring}", "applied to $A_{red}$.", "\\medskip\\noindent", "Proof of (\\ref{item-immediate-specialization}).", "Let $Z = \\overline{\\{x\\}} \\supset Z' = \\overline{\\{x'\\}}$.", "Then it follows from (\\ref{item-irreducible-maximal-chains}) that", "$Z \\supset Z'$ is the start of a maximal chain of", "irreducible closed subschemes in $Z$", "and consequently $\\dim(Z) = \\dim(Z') + 1$.", "We conclude by (\\ref{item-dimension-irreducible-trdeg}).", "\\medskip\\noindent", "Proof of (\\ref{item-dimension-sup-trdeg}). A simple topological argument", "shows that $\\dim(X) = \\sup \\dim(Z)$ where the supremum is over the", "irreducible components of $X$ (hint: use", "Topology, Lemma \\ref{topology-lemma-irreducible}).", "Thus this follows from (\\ref{item-dimension-irreducible-trdeg}).", "\\medskip\\noindent", "Proof of (\\ref{item-specialization}). Part (a) follows from the", "fact that any open $U \\subset X$ containing $x'$ also contains $x$.", "Part (b) follows because $\\mathcal{O}_{X, x}$ is a localization of", "$\\mathcal{O}_{X, x'}$ hence any chain of primes in $\\mathcal{O}_{X, x}$", "corresponds to a chain of primes in $\\mathcal{O}_{X, x'}$ which can", "be extended by adding $\\mathfrak m_{x'}$ at the end.", "Both (c) and (d) follow formally from (\\ref{item-immediate-specialization}).", "\\medskip\\noindent", "Proof of (\\ref{item-dimension-formula}). Choose an affine", "neighbourhood $U = \\Spec(A)$ of $x$. Then $\\dim_x(X) = \\dim_x(U)$.", "Hence we reduce to the affine case, which is", "Algebra, Lemma \\ref{algebra-lemma-dimension-at-a-point-finite-type-field}.", "\\medskip\\noindent", "Proof of (\\ref{item-immediate-specialization-local-ring}).", "If $X$ is equidimensional", "(Topology, Definition \\ref{topology-definition-equidimensional})", "then $\\dim(X)$ is equal to the dimension of every irreducible", "component of $X$, whence $\\dim_x(X) = \\dim(X) = \\dim_{x'}(X)$", "by (\\ref{item-dimension-irreducibles-passing-through}).", "Thus this follows from (\\ref{item-immediate-specialization})." ], "refs": [ "topology-lemma-catenary", "algebra-lemma-catenary", "algebra-lemma-quotient-catenary", "algebra-lemma-dimension-height-polynomial-ring", "algebra-lemma-CM-ring-catenary", "algebra-lemma-dimension-closed-point-finite-type-field", "varieties-lemma-locally-finite-type-Jacobson", "algebra-lemma-dimension-spell-it-out", "algebra-lemma-dimension-spell-it-out", "algebra-lemma-dimension-at-a-point-finite-type-over-field", "algebra-lemma-dimension-prime-polynomial-ring", "topology-lemma-irreducible", "algebra-lemma-dimension-at-a-point-finite-type-field", "topology-definition-equidimensional" ], "ref_ids": [ 8226, 931, 935, 993, 937, 996, 10964, 994, 994, 995, 1005, 8213, 1007, 8357 ] } ], "ref_ids": [ 8359 ] }, { "id": 10990, "type": "theorem", "label": "varieties-lemma-dimension-fibres-locally-algebraic", "categories": [ "varieties" ], "title": "varieties-lemma-dimension-fibres-locally-algebraic", "contents": [ "Let $k$ be a field. Let $f : X \\to Y$ be a morphism of locally algebraic", "$k$-schemes.", "\\begin{enumerate}", "\\item For $y \\in Y$, the fibre $X_y$ is a locally", "algebraic scheme over $\\kappa(y)$ hence all the results of", "Lemma \\ref{lemma-dimension-locally-algebraic} apply.", "\\item Assume $X$ is irreducible. Set $Z = \\overline{f(X)}$ and", "$d = \\dim(X) - \\dim(Z)$. Then", "\\begin{enumerate}", "\\item $\\dim_x(X_{f(x)}) \\geq d$ for all $x \\in X$,", "\\item the set of $x \\in X$ with $\\dim_x(X_{f(x)}) = d$ is dense open,", "\\item if $\\dim(\\mathcal{O}_{Z, f(x)}) \\geq 1$, then", "$\\dim_x(X_{f(x)}) \\leq d + \\dim(\\mathcal{O}_{Z, f(x)}) - 1$,", "\\item if $\\dim(\\mathcal{O}_{Z, f(x)}) = 1$, then $\\dim_x(X_{f(x)}) = d$,", "\\end{enumerate}", "\\item For $x \\in X$ with $y = f(x)$ we have", "$\\dim_x(X_y) \\geq \\dim_x(X) - \\dim_y(Y)$.", "\\end{enumerate}" ], "refs": [ "varieties-lemma-dimension-locally-algebraic" ], "proofs": [ { "contents": [ "The morphism $f$ is locally of finite type by", "Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type}.", "Hence the base change $X_y \\to \\Spec(\\kappa(y))$ is locally", "of finite type. This proves (1).", "In the rest of the proof we will freely use the results", "of Lemma \\ref{lemma-dimension-locally-algebraic} for $X$, $Y$, and", "the fibres of $f$.", "\\medskip\\noindent", "Proof of (2). Let $\\eta \\in X$ be the generic point and set", "$\\xi = f(\\eta)$. Then $Z = \\overline{\\{\\xi\\}}$. Hence", "$$", "d = \\dim(X) - \\dim(Z) =", "\\text{trdeg}_k \\kappa(\\eta) - \\text{trdeg}_k \\kappa(\\xi) =", "\\text{trdeg}_{\\kappa(\\xi)} \\kappa(\\eta) = \\dim_\\eta(X_\\xi)", "$$", "Thus parts (2)(a) and (2)(b) follow from", "Morphisms, Lemma \\ref{morphisms-lemma-openness-bounded-dimension-fibres}.", "Parts (2)(c) and (2)(d) follow from", "Lemmas \\ref{lemma-dimension-fibre-in-higher-codimension} and", "\\ref{lemma-dimension-fibre-in-codim-1}.", "\\medskip\\noindent", "Proof of (3). Let $x \\in X$. Let $X' \\subset X$ be a irreducible component", "of $X$ passing through $x$ of dimension $\\dim_x(X)$. Then (2) implies that", "$\\dim_x(X_y) \\geq \\dim(X') - \\dim(Z')$ where $Z' \\subset Y$", "is the closure of the image of $X'$. This proves (3)." ], "refs": [ "morphisms-lemma-permanence-finite-type", "varieties-lemma-dimension-locally-algebraic", "morphisms-lemma-openness-bounded-dimension-fibres", "varieties-lemma-dimension-fibre-in-higher-codimension", "varieties-lemma-dimension-fibre-in-codim-1" ], "ref_ids": [ 5204, 10989, 5280, 10987, 10985 ] } ], "ref_ids": [ 10989 ] }, { "id": 10991, "type": "theorem", "label": "varieties-lemma-dimension-product-locally-algebraic", "categories": [ "varieties" ], "title": "varieties-lemma-dimension-product-locally-algebraic", "contents": [ "\\begin{slogan}", "The dimension of the product is the sum of the dimensions.", "\\end{slogan}", "Let $k$ be a field. Let $X$, $Y$ be locally algebraic $k$-schemes.", "\\begin{enumerate}", "\\item For $z \\in X \\times Y$ lying over $(x, y)$ we have", "$\\dim_z(X \\times Y) = \\dim_x(X) + \\dim_y(Y)$.", "\\item We have $\\dim(X \\times Y) = \\dim(X) + \\dim(Y)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Consider the factorization", "$$", "X \\times Y \\longrightarrow Y \\longrightarrow \\Spec(k)", "$$", "of the structure morphism. The first morphism $p : X \\times Y \\to Y$", "is flat as a base change of the flat morphism $X \\to \\Spec(k)$", "by Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat}.", "Moreover, we have $\\dim_z(p^{-1}(y)) = \\dim_x(X)$ by", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}.", "Hence $\\dim_z(X \\times Y) = \\dim_x(X) + \\dim_y(Y)$ by", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point-additive}.", "Part (2) is a direct consequence of (1)." ], "refs": [ "morphisms-lemma-base-change-flat", "morphisms-lemma-dimension-fibre-after-base-change", "morphisms-lemma-dimension-fibre-at-a-point-additive" ], "ref_ids": [ 5265, 5279, 5278 ] } ], "ref_ids": [] }, { "id": 10992, "type": "theorem", "label": "varieties-lemma-complete-local-ring-structure-as-algebra", "categories": [ "varieties" ], "title": "varieties-lemma-complete-local-ring-structure-as-algebra", "contents": [ "Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$.", "Let $x \\in X$ be a point with residue field $\\kappa$.", "There is an isomorphism", "\\begin{equation}", "\\label{equation-complete-local-ring}", "\\kappa[[x_1, \\ldots, x_n]]/I \\longrightarrow \\mathcal{O}_{X, x}^\\wedge", "\\end{equation}", "inducing the identity on residue fields.", "In general we cannot choose (\\ref{equation-complete-local-ring})", "to be a $k$-algebra isomorphism. However, if the extension $\\kappa/k$", "is separable, then we can choose", "(\\ref{equation-complete-local-ring}) to be an isomorphism of $k$-algebras." ], "refs": [], "proofs": [ { "contents": [ "The existence of the isomorphism is an immediate consequence of", "the Cohen structure theorem\\footnote{Note that if $\\kappa$ has", "characteristic $p$, then the theorem just says we get a surjection", "$\\Lambda[[x_1, \\ldots, x_n]] \\to \\mathcal{O}_{X, x}^\\wedge$ where", "$\\Lambda$ is a Cohen ring for $\\kappa$. But of course in this", "case the map factors through $\\Lambda/p\\Lambda[[x_1, \\ldots, x_n]]$", "and $\\Lambda/p\\Lambda = \\kappa$.}", "(Algebra, Theorem \\ref{algebra-theorem-cohen-structure-theorem}).", "\\medskip\\noindent", "Let $p$ be an odd prime number, let $k = \\mathbf{F}_p(t)$, and", "$A = k[x, y]/(y^2 + x^p - t)$. Then the completion $A^\\wedge$ of $A$", "in the maximal ideal", "$\\mathfrak m = (y)$ is isomorphic to $k(t^{1/p})[[z]]$ as a ring", "but not as a $k$-algebra. The reason is that $A^\\wedge$", "does not contain an element whose $p$th power is $t$ (as the reader", "can see by computing modulo $y^2$).", "This also shows that any isomorphism (\\ref{equation-complete-local-ring})", "cannot be a $k$-algebra isomorphism.", "\\medskip\\noindent", "If $\\kappa/k$ is separable, then there is a $k$-algebra", "homomorphism $\\kappa \\to \\mathcal{O}_{X, x}^\\wedge$", "inducing the identity on residue fields by", "More on Algebra, Lemma \\ref{more-algebra-lemma-lift-residue-field}.", "Let $f_1, \\ldots, f_n \\in \\mathfrak m_x$ be generators.", "Consider the map", "$$", "\\kappa[[x_1, \\ldots, x_n]] \\longrightarrow \\mathcal{O}_{X, x}^\\wedge,\\quad", "x_i \\longmapsto f_i", "$$", "Since both sides are $(x_1, \\ldots, x_n)$-adically complete", "(the right hand side by", "Algebra, Lemmas \\ref{algebra-lemma-hathat-finitely-generated})", "this map is surjective by", "Algebra, Lemma \\ref{algebra-lemma-completion-generalities}", "as it is surjective modulo $(x_1, \\ldots, x_n)$ by", "construction." ], "refs": [ "algebra-theorem-cohen-structure-theorem", "more-algebra-lemma-lift-residue-field", "algebra-lemma-hathat-finitely-generated", "algebra-lemma-completion-generalities" ], "ref_ids": [ 327, 10023, 859, 858 ] } ], "ref_ids": [] }, { "id": 10993, "type": "theorem", "label": "varieties-lemma-base-change-complete-local-ring", "categories": [ "varieties" ], "title": "varieties-lemma-base-change-complete-local-ring", "contents": [ "Let $K/k$ be an extension of fields. Let $X$ be a locally algebraic", "$k$-scheme. Set $Y = X_K$. Let $y \\in Y$ be a point with image $x \\in X$.", "Assume that $\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y})$", "and that $\\kappa(x)/k$ is separable.", "Choose an isomorphism", "$$", "\\kappa(x)[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m) \\longrightarrow", "\\mathcal{O}_{X, x}^\\wedge", "$$", "of $k$-algebras as in (\\ref{equation-complete-local-ring}).", "Then we have an isomorphism", "$$", "\\kappa(y)[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m) \\longrightarrow", "\\mathcal{O}_{Y, y}^\\wedge", "$$", "of $K$-algebras as in (\\ref{equation-complete-local-ring}). Here we use", "$\\kappa(x) \\to \\kappa(y)$ to view $g_j$ as a power series", "over $\\kappa(y)$." ], "refs": [], "proofs": [ { "contents": [ "The local ring map $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{Y, y}$", "induces a local ring map", "$\\mathcal{O}_{X, x}^\\wedge \\to \\mathcal{O}_{Y, y}^\\wedge$.", "The induced map", "$$", "\\kappa(x) \\to \\kappa(x)[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m)", "\\to \\mathcal{O}_{X, x}^\\wedge \\to \\mathcal{O}_{Y, y}^\\wedge", "$$", "composed with the projection to $\\kappa(y)$ is the canonical", "homomorphism $\\kappa(x) \\to \\kappa(y)$.", "By Lemma \\ref{lemma-change-fields-flat} the residue field", "$\\kappa(y)$ is a localization of $\\kappa(x) \\otimes_k K$", "at the kernel $\\mathfrak p_0$ of $\\kappa(x) \\otimes_k K \\to \\kappa(y)$.", "On the other hand, by Lemma \\ref{lemma-change-fields-algebraic-unramified}", "the local ring $(\\kappa(x) \\otimes_k K)_{\\mathfrak p_0}$", "is equal to $\\kappa(y)$. Hence the map", "$$", "\\kappa(x) \\otimes_k K \\to \\mathcal{O}_{Y, y}^\\wedge", "$$", "factors canonically through $\\kappa(y)$. We obtain a commutative", "diagram", "$$", "\\xymatrix{", "\\kappa(y) \\ar[rr] & & \\mathcal{O}_{Y, y}^\\wedge \\\\", "\\kappa(x) \\ar[r] \\ar[u] &", "\\kappa(x)[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m) \\ar[r] &", "\\mathcal{O}_{X, x}^\\wedge \\ar[u]", "}", "$$", "Let $f_i \\in \\mathfrak m_x^\\wedge \\subset \\mathcal{O}_{X, x}^\\wedge$", "be the image of $x_i$. Observe that", "$\\mathfrak m_x^\\wedge = (f_1, \\ldots, f_n)$ as the map is surjective.", "Consider the map", "$$", "\\kappa(y)[[x_1, \\ldots, x_n]] \\longrightarrow \\mathcal{O}_{Y, y}^\\wedge,\\quad", "x_i \\longmapsto f_i", "$$", "where here $f_i$ really means the image of $f_i$ in $\\mathfrak m_y^\\wedge$.", "Since $\\mathfrak m_x \\mathcal{O}_{Y, y} = \\mathfrak m_y$", "by Lemma \\ref{lemma-change-fields-algebraic-unramified}", "we see that the right hand side is complete with respect to", "$(x_1, \\ldots, x_n)$ (use Algebra, Lemma", "\\ref{algebra-lemma-hathat-finitely-generated} to see that", "it is a complete local ring).", "Since both sides are $(x_1, \\ldots, x_n)$-adically complete", "our map is surjective by", "Algebra, Lemma \\ref{algebra-lemma-completion-generalities}", "as it is surjective modulo $(x_1, \\ldots, x_n)$.", "Of course the power series $g_1, \\ldots, g_m$", "are mapped to zero under this map, as they already map to zero", "in $\\mathcal{O}_{X, x}^\\wedge$. Thus we have the commutative diagram", "$$", "\\xymatrix{", "\\kappa(y)[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m) \\ar[r] &", "\\mathcal{O}_{Y, y}^\\wedge \\\\", "\\kappa(x)[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m) \\ar[r] \\ar[u] &", "\\mathcal{O}_{X, x}^\\wedge \\ar[u]", "}", "$$", "We still need to show that the top horizontal arrow is an isomorphism.", "We already know that it is surjective. We know that", "$\\mathcal{O}_{X, x} \\to \\mathcal{O}_{Y, y}$ is flat", "(Lemma \\ref{lemma-change-fields-flat}), which implies that", "$\\mathcal{O}_{X, x}^\\wedge \\to \\mathcal{O}_{Y, y}^\\wedge$ is flat", "(More on Algebra, Lemma \\ref{more-algebra-lemma-flat-completion}).", "Thus we may apply Algebra, Lemma \\ref{algebra-lemma-mod-injective}", "with $R = \\kappa(x)[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m)$,", "with $S = \\kappa(y)[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m)$,", "with $M = \\mathcal{O}_{Y, y}^\\wedge$, and with $N = S$", "to conclude that the map is injective." ], "refs": [ "varieties-lemma-change-fields-flat", "varieties-lemma-change-fields-algebraic-unramified", "varieties-lemma-change-fields-algebraic-unramified", "algebra-lemma-hathat-finitely-generated", "algebra-lemma-completion-generalities", "varieties-lemma-change-fields-flat", "more-algebra-lemma-flat-completion", "algebra-lemma-mod-injective" ], "ref_ids": [ 10903, 10905, 10905, 859, 858, 10903, 10049, 883 ] } ], "ref_ids": [] }, { "id": 10994, "type": "theorem", "label": "varieties-lemma-globally-generated-base-change", "categories": [ "varieties" ], "title": "varieties-lemma-globally-generated-base-change", "contents": [ "Let $X \\to \\Spec(A)$ be a morphism of schemes. Let $A \\subset A'$", "be a faithfully flat ring map. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is globally generated", "if and only if the base change $\\mathcal{F}_{A'}$ is globally generated." ], "refs": [], "proofs": [ { "contents": [ "More precisely, set $X_{A'} = X \\times_{\\Spec(A)} \\Spec(A')$.", "Let $\\mathcal{F}_{A'} = p^*\\mathcal{F}$ where $p : X_{A'} \\to X$", "is the projection. By", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}", "we have", "$H^0(X_{k'}, \\mathcal{F}_{A'}) = H^0(X, \\mathcal{F}) \\otimes_A A'$.", "Thus if $s_i$, $i \\in I$ are generators for $H^0(X, \\mathcal{F})$", "as an $A$-module, then their images in $H^0(X_{A'}, \\mathcal{F}_{A'})$", "are generators for $H^0(X_{A'}, \\mathcal{F}_{A'})$ as an $A'$-module.", "Thus we have to show that the map", "$\\alpha : \\bigoplus_{i \\in I} \\mathcal{O}_X \\to \\mathcal{F}$,", "$(f_i) \\mapsto \\sum f_i s_i$", "is surjective if and only if $p^*\\alpha$ is surjective.", "This we may check over an affine open $U = \\Spec(B)$ of $X$.", "Then $\\mathcal{F}|_U$ corresponds to a $B$-module $M$", "and $s_i|_U$ to elements $x_i \\in M$. Thus we have to show", "that $\\bigoplus_{i \\in I} B \\to M$ is surjective if and only", "if the base change $\\bigoplus_{i \\in I} B \\otimes_A A' \\to M \\otimes_A A'$", "is surjective. This is true because $A \\to A'$ is faithfully flat." ], "refs": [ "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 3298 ] } ], "ref_ids": [] }, { "id": 10995, "type": "theorem", "label": "varieties-lemma-very-ample-vanish-at-point", "categories": [ "varieties" ], "title": "varieties-lemma-very-ample-vanish-at-point", "contents": [ "Let $k$ be an infinite field. Let $X$ be a scheme of finite type over $k$.", "Let $\\mathcal{L}$ be a very ample invertible sheaf on $X$.", "Let $n \\geq 0$ and $x, x_1, \\ldots, x_n \\in X$ be points with", "$x$ a $k$-rational point, i.e., $\\kappa(x) = k$, and", "$x \\not = x_i$ for $i = 1, \\ldots, n$.", "Then there exists an $s \\in H^0(X, \\mathcal{L})$ which vanishes at", "$x$ but not at $x_i$." ], "refs": [], "proofs": [ { "contents": [ "If $n = 0$ the result is trivial, hence we assume $n > 0$.", "By definition of a very ample invertible sheaf, the lemma immediately", "reduces to the case where $X = \\mathbf{P}^r_k$ for some $r > 0$", "and $\\mathcal{L} = \\mathcal{O}_X(1)$. Write", "$\\mathbf{P}^r_k = \\text{Proj}(k[T_0, \\ldots, T_r])$.", "Set $V = H^0(X, \\mathcal{L}) = kT_0 \\oplus \\ldots \\oplus kT_r$.", "Since $x$ is a $k$-rational point, we see that the set", "$s \\in V$ which vanish at $x$ is a codimension $1$ subspace", "$W \\subset V$ and that $W$ generates the homogeneous prime", "ideal corresponding to $x$. Since $x_i \\not = x$ the corresponding", "homogeneous prime $\\mathfrak p_i \\subset k[T_0, \\ldots, T_r]$ does", "not contain $W$. Since $k$ is infinite, we then see that", "$W \\not = \\bigcup W \\cap \\mathfrak q_i$ and the proof is complete." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10996, "type": "theorem", "label": "varieties-lemma-generated-by-dim-plus-1-sections", "categories": [ "varieties" ], "title": "varieties-lemma-generated-by-dim-plus-1-sections", "contents": [ "Let $k$ be an infinite field. Let $X$ be an algebraic $k$-scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $V \\to \\Gamma(X, \\mathcal{L})$ be a linear map of $k$-vector spaces", "whose image generates $\\mathcal{L}$. Then there exists a subspace", "$W \\subset V$ with $\\dim_k(W) \\leq \\dim(X) + 1$ which generates $\\mathcal{L}$." ], "refs": [], "proofs": [ { "contents": [ "Throughout the proof we will use that for every $x \\in X$", "the linear map", "$$", "\\psi_x : V \\to \\Gamma(X, \\mathcal{L}) \\to \\mathcal{L}_x \\to", "\\mathcal{L}_x \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)", "$$", "is nonzero. The proof is by induction on $\\dim(X)$.", "\\medskip\\noindent", "The base case is $\\dim(X) = 0$. In this case $X$ has finitely many points", "$X = \\{x_1, \\ldots, x_n\\}$ (see for example", "Lemma \\ref{lemma-algebraic-scheme-dim-0}). Since $k$ is infinite", "there exists a vector $v \\in V$ such that $\\psi_{x_i}(v) \\not = 0$", "for all $i$. Then $W = k\\cdot v$ does the job.", "\\medskip\\noindent", "Assume $\\dim(X) > 0$. Let $X_i \\subset X$ be the irreducible components", "of dimension equal to $\\dim(X)$. Since $X$ is Noetherian there are only", "finitely many of these. For each $i$ pick a point $x_i \\in X_i$.", "As above choose $v \\in V$ such that $\\psi_{x_i}(v) \\not = 0$", "for all $i$. Let $Z \\subset X$ be the zero scheme of the image", "of $v$ in $\\Gamma(X, \\mathcal{L})$, see", "Divisors, Definition \\ref{divisors-definition-zero-scheme-s}.", "By construction $\\dim(Z) < \\dim(X)$. By induction we can find", "$W \\subset V$ with $\\dim(W) \\leq \\dim(X)$ such that $W$ generates", "$\\mathcal{L}|_Z$. Then $W + k\\cdot v$ generates $\\mathcal{L}$." ], "refs": [ "varieties-lemma-algebraic-scheme-dim-0", "divisors-definition-zero-scheme-s" ], "ref_ids": [ 10988, 8094 ] } ], "ref_ids": [] }, { "id": 10997, "type": "theorem", "label": "varieties-lemma-separate-points-tangent-vectors", "categories": [ "varieties" ], "title": "varieties-lemma-separate-points-tangent-vectors", "contents": [ "Let $k$ be an algebraically closed field.", "Let $X$ be a proper $k$-scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $V \\subset H^0(X, \\mathcal{L})$ be a $k$-subvector space. If", "\\begin{enumerate}", "\\item for every pair of distinct closed points $x, y \\in X$", "there is a section $s \\in V$ which vanishes at $x$ but not at $y$, and", "\\item for every closed point $x \\in X$ and nonzero tangent vector", "$\\theta \\in T_{X/k, x}$ there exist a section $s \\in V$", "which vanishes at $x$ but whose pullback by $\\theta$ is nonzero,", "\\end{enumerate}", "then $\\mathcal{L}$ is very ample and the canonical morphism", "$\\varphi_{\\mathcal{L}, V} : X \\to \\mathbf{P}(V)$", "is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Condition (1) implies in particular that the elements of $V$ generate", "$\\mathcal{L}$ over $X$. Hence we get a canonical morphism", "$$", "\\varphi = \\varphi_{\\mathcal{L}, V} :", "X", "\\longrightarrow", "\\mathbf{P}(V)", "$$", "by Constructions, Example \\ref{constructions-example-projective-space}.", "The morphism $\\varphi$ is proper by", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}.", "By (1) the map $\\varphi$ is injective on closed points", "(computation omitted).", "In particular, the fibre over any closed point of $\\mathbf{P}(V)$", "is a singleton (small detail omitted). Thus we see that", "$\\varphi$ is finite, for example use Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-finite-fibre-finite-in-neighbourhood}.", "To finish the proof it suffices to show that the map", "$$", "\\varphi^\\sharp :", "\\mathcal{O}_{\\mathbf{P}(V)}", "\\longrightarrow", "\\varphi_*\\mathcal{O}_X", "$$", "is surjective. This we may check on stalks at closed points.", "Let $x \\in X$ be a closed point with image the closed", "point $p = \\varphi(x) \\in \\mathbf{P}(V)$. Since", "$\\varphi^{-1}(\\{p\\}) = \\{x\\}$ by (1) and since $\\varphi$", "is proper (hence closed), we see that $\\varphi^{-1}(U)$", "runs through a fundamental system of open neighbourhoods", "of $x$ as $U$ runs through a fundamental system of open neighbourhoods of $p$.", "We conclude that on stalks at $p$ we obtain the map", "$$", "\\varphi^\\sharp_x :", "\\mathcal{O}_{\\mathbf{P}(V), p}", "\\longrightarrow", "\\mathcal{O}_{X, x}", "$$", "In particular, $\\mathcal{O}_{X, x}$ is a finite", "$\\mathcal{O}_{\\mathbf{P}(V), p}$-module.", "Moreover, the residue fields of $x$ and $p$ are equal to $k$", "(as $k$ is algebraically closed -- use the Hilbert Nullstellensatz).", "Finally, condition (2) implies that the map", "$$", "T_{X/k, x} \\longrightarrow T_{\\mathbf{P}(V)/k, p}", "$$", "is injective since any nonzero $\\theta$ in the kernel of this map", "couldn't possibly satisfy the conclusion of (2).", "In terms of the map of local rings above this means that", "$$", "\\mathfrak m_p/\\mathfrak m_p^2 \\longrightarrow", "\\mathfrak m_x/\\mathfrak m_x^2", "$$", "is surjective, see Lemma \\ref{lemma-tangent-space-rational-point}.", "Now the proof is finished by applying", "Algebra, Lemma \\ref{algebra-lemma-when-surjective-local}." ], "refs": [ "morphisms-lemma-image-proper-scheme-closed", "coherent-lemma-proper-finite-fibre-finite-in-neighbourhood", "varieties-lemma-tangent-space-rational-point", "algebra-lemma-when-surjective-local" ], "ref_ids": [ 5411, 3366, 10973, 402 ] } ], "ref_ids": [] }, { "id": 10998, "type": "theorem", "label": "varieties-lemma-variant-separate-points-tangent-vectors", "categories": [ "varieties" ], "title": "varieties-lemma-variant-separate-points-tangent-vectors", "contents": [ "Let $k$ be an algebraically closed field.", "Let $X$ be a proper $k$-scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Suppose that for every closed subscheme $Z \\subset X$", "of dimension $0$ and degree $2$ over $k$ the map", "$$", "H^0(X, \\mathcal{L}) \\longrightarrow H^0(Z, \\mathcal{L}|_Z)", "$$", "is surjective. Then $\\mathcal{L}$ is very ample on $X$ over $k$." ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of", "Lemma \\ref{lemma-separate-points-tangent-vectors}.", "Namely, given distinct closed points", "$x, y \\in X$ taking $Z = x \\cup y$ (viewed as closed", "subscheme) we get condition (1) of the lemma.", "And given a nonzero tangent vector $\\theta \\in T_{X/k, x}$", "the morphism $\\theta : \\Spec(k[\\epsilon]) \\to X$", "is a closed immersion. Setting $Z = \\Im(\\theta)$", "we obtain condition (2) of the lemma." ], "refs": [ "varieties-lemma-separate-points-tangent-vectors" ], "ref_ids": [ 10997 ] } ], "ref_ids": [] }, { "id": 10999, "type": "theorem", "label": "varieties-lemma-closure-of-product", "categories": [ "varieties" ], "title": "varieties-lemma-closure-of-product", "contents": [ "Let $k$ be a field.", "Let $X$, $Y$ be schemes over $k$, and let", "$A \\subset X$, $B \\subset Y$ be subsets.", "Set", "$$", "AB =", "\\{z \\in X \\times_k Y \\mid \\text{pr}_X(z) \\in A, \\ \\text{pr}_Y(z) \\in B\\}", "\\subset X \\times_k Y", "$$", "Then set theoretically we have", "$$", "\\overline{A} \\times_k \\overline{B} = \\overline{AB}", "$$" ], "refs": [], "proofs": [ { "contents": [ "The inclusion $\\overline{AB} \\subset \\overline{A} \\times_k \\overline{B}$", "is immediate.", "We may replace $X$ and $Y$ by the reduced closed subschemes $\\overline{A}$", "and $\\overline{B}$.", "Let $W \\subset X \\times_k Y$ be a nonempty open subset. By", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open}", "the subset $U = \\text{pr}_X(W)$ is nonempty open in $X$.", "Hence $A \\cap U$ is nonempty. Pick $a \\in A \\cap U$.", "Denote $Y_{\\kappa(a)} = \\{a\\} \\times_k Y$", "the fibre of $\\text{pr}_X : X \\times_k Y \\to X$ over $a$. By", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open}", "again the morphism $Y_a \\to Y$ is open as", "$\\Spec(\\kappa(a)) \\to \\Spec(k)$ is universally open.", "Hence the nonempty open", "subset $W_a = W \\times_{X \\times_k Y} Y_a$", "maps to a nonempty open subset of $Y$.", "We conclude there exists a $b \\in B$ in the image.", "Hence $AB \\cap W \\not = \\emptyset$ as desired." ], "refs": [ "morphisms-lemma-scheme-over-field-universally-open", "morphisms-lemma-scheme-over-field-universally-open" ], "ref_ids": [ 5254, 5254 ] } ], "ref_ids": [] }, { "id": 11000, "type": "theorem", "label": "varieties-lemma-closure-image-product-map", "categories": [ "varieties" ], "title": "varieties-lemma-closure-image-product-map", "contents": [ "Let $k$ be a field.", "Let $f : A \\to X$, $g : B \\to Y$ be morphisms of schemes over $k$.", "Then set theoretically we have", "$$", "\\overline{f(A)} \\times_k \\overline{g(B)} =", "\\overline{(f \\times g)(A \\times_k B)}", "$$" ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-closure-of-product}", "as the image of $f \\times g$ is $f(A)g(B)$", "in the notation of that lemma." ], "refs": [ "varieties-lemma-closure-of-product" ], "ref_ids": [ 10999 ] } ], "ref_ids": [] }, { "id": 11001, "type": "theorem", "label": "varieties-lemma-scheme-theoretic-image-product-map", "categories": [ "varieties" ], "title": "varieties-lemma-scheme-theoretic-image-product-map", "contents": [ "Let $k$ be a field.", "Let $f : A \\to X$, $g : B \\to Y$ be quasi-compact morphisms of schemes", "over $k$. Let $Z \\subset X$ be the scheme theoretic image of $f$, see", "Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-image}.", "Similarly, let $Z' \\subset Y$ be the scheme theoretic image of $g$.", "Then $Z \\times_k Z'$ is the scheme theoretic image of $f \\times g$." ], "refs": [ "morphisms-definition-scheme-theoretic-image" ], "proofs": [ { "contents": [ "Recall that $Z$ is the smallest closed subscheme of $X$ through which", "$f$ factors. Similarly for $Z'$. Let $W \\subset X \\times_k Y$ be the", "scheme theoretic image of $f \\times g$. As $f \\times g$ factors through", "$Z \\times_k Z'$ we see that $W \\subset Z \\times_k Z'$.", "\\medskip\\noindent", "To prove the other inclusion let $U \\subset X$ and $V \\subset Y$ be", "affine opens. By", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}", "the scheme $Z \\cap U$ is the scheme theoretic image of", "$f|_{f^{-1}(U)} : f^{-1}(U) \\to U$, and similarly for", "$Z' \\cap V$ and $W \\cap U \\times_k V$. Hence we may assume", "$X$ and $Y$ affine. As $f$ and $g$ are quasi-compact this implies", "that $A = \\bigcup U_i$ is a finite union of affines and", "$B = \\bigcup V_j$ is a finite union of affines.", "Then we may replace $A$ by $\\coprod U_i$ and $B$ by", "$\\coprod V_j$, i.e., we may assume that $A$ and $B$ are affine as well.", "In this case $Z$ is cut out by", "$\\Ker(\\Gamma(X, \\mathcal{O}_X) \\to \\Gamma(A, \\mathcal{O}_A))$", "and similarly for $Z'$ and $W$. Hence the result follows from", "the equality", "$$", "\\Gamma(A \\times_k B, \\mathcal{O}_{A \\times_k B})", "=", "\\Gamma(A, \\mathcal{O}_A) \\otimes_k \\Gamma(B, \\mathcal{O}_B)", "$$", "which holds as $A$ and $B$ are affine. Details omitted." ], "refs": [ "morphisms-lemma-quasi-compact-scheme-theoretic-image" ], "ref_ids": [ 5146 ] } ], "ref_ids": [ 5539 ] }, { "id": 11002, "type": "theorem", "label": "varieties-lemma-char-zero-differentials-free-smooth", "categories": [ "varieties" ], "title": "varieties-lemma-char-zero-differentials-free-smooth", "contents": [ "Let $k$ be a field. Let $X$ be a scheme over $k$.", "Assume", "\\begin{enumerate}", "\\item $X$ is locally of finite type over $k$,", "\\item $\\Omega_{X/k}$ is locally free, and", "\\item $k$ has characteristic zero.", "\\end{enumerate}", "Then the structure morphism $X \\to \\Spec(k)$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Algebra, Lemma \\ref{algebra-lemma-characteristic-zero-local-smooth}." ], "refs": [ "algebra-lemma-characteristic-zero-local-smooth" ], "ref_ids": [ 1227 ] } ], "ref_ids": [] }, { "id": 11003, "type": "theorem", "label": "varieties-lemma-char-p-differentials-free-smooth", "categories": [ "varieties" ], "title": "varieties-lemma-char-p-differentials-free-smooth", "contents": [ "Let $k$ be a field. Let $X$ be a scheme over $k$.", "Assume", "\\begin{enumerate}", "\\item $X$ is locally of finite type over $k$,", "\\item $\\Omega_{X/k}$ is locally free,", "\\item $X$ is reduced, and", "\\item $k$ is perfect.", "\\end{enumerate}", "Then the structure morphism $X \\to \\Spec(k)$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$ be a point. As $X$ is locally Noetherian (see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian})", "there are finitely many irreducible components", "$X_1, \\ldots, X_n$ passing through $x$ (see", "Properties, Lemma \\ref{properties-lemma-Noetherian-topology} and", "Topology, Lemma \\ref{topology-lemma-Noetherian}).", "Let $\\eta_i \\in X_i$ be the generic point. As $X$ is reduced we have", "$\\mathcal{O}_{X, \\eta_i} = \\kappa(\\eta_i)$, see", "Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring}.", "Moreover, $\\kappa(\\eta_i)$ is a finitely generated field extension", "of the perfect field $k$ hence separably generated over $k$ (see", "Algebra, Section \\ref{algebra-section-separability}).", "It follows that $\\Omega_{X/k, \\eta_i} = \\Omega_{\\kappa(\\eta_i)/k}$", "is free of rank the transcendence degree of $\\kappa(\\eta_i)$ over $k$. By", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}", "we conclude that $\\dim_{\\eta_i}(X_i) = \\text{rank}_{\\eta_i}(\\Omega_{X/k})$.", "Since $x \\in X_1 \\cap \\ldots \\cap X_n$ we see that", "$$", "\\text{rank}_x(\\Omega_{X/k}) = \\text{rank}_{\\eta_i}(\\Omega_{X/k}) = \\dim(X_i).", "$$", "Therefore $\\dim_x(X) = \\text{rank}_x(\\Omega_{X/k})$, see", "Algebra, Lemma \\ref{algebra-lemma-dimension-at-a-point-finite-type-over-field}.", "It follows that $X \\to \\Spec(k)$ is smooth at $x$ for example by", "Algebra, Lemma \\ref{algebra-lemma-characterize-smooth-over-field}." ], "refs": [ "morphisms-lemma-finite-type-noetherian", "properties-lemma-Noetherian-topology", "topology-lemma-Noetherian", "algebra-lemma-minimal-prime-reduced-ring", "morphisms-lemma-dimension-fibre-at-a-point", "algebra-lemma-dimension-at-a-point-finite-type-over-field", "algebra-lemma-characterize-smooth-over-field" ], "ref_ids": [ 5202, 2954, 8220, 418, 5277, 995, 1223 ] } ], "ref_ids": [] }, { "id": 11004, "type": "theorem", "label": "varieties-lemma-smooth-regular", "categories": [ "varieties" ], "title": "varieties-lemma-smooth-regular", "contents": [ "\\begin{slogan}", "Smooth over a field implies regular", "\\end{slogan}", "Let $X \\to \\Spec(k)$ be a smooth morphism where $k$ is a field.", "Then $X$ is a regular scheme." ], "refs": [], "proofs": [ { "contents": [ "(See also", "Lemma \\ref{lemma-geometrically-regular-smooth}.)", "By", "Algebra, Lemma \\ref{algebra-lemma-characterize-smooth-over-field}", "every local ring $\\mathcal{O}_{X, x}$ is regular.", "And because $X$ is locally of finite type over $k$ it is locally", "Noetherian. Hence $X$ is regular by", "Properties, Lemma \\ref{properties-lemma-characterize-regular}." ], "refs": [ "varieties-lemma-geometrically-regular-smooth", "algebra-lemma-characterize-smooth-over-field", "properties-lemma-characterize-regular" ], "ref_ids": [ 10962, 1223, 2975 ] } ], "ref_ids": [] }, { "id": 11005, "type": "theorem", "label": "varieties-lemma-smooth-geometrically-normal", "categories": [ "varieties" ], "title": "varieties-lemma-smooth-geometrically-normal", "contents": [ "Let $X \\to \\Spec(k)$ be a smooth morphism where $k$ is a field.", "Then $X$ is geometrically regular, geometrically normal, and", "geometrically reduced over $k$." ], "refs": [], "proofs": [ { "contents": [ "(See also", "Lemma \\ref{lemma-geometrically-regular-smooth}.)", "Let $k'$ be a finite purely inseparable extension of $k$.", "It suffices to prove that $X_{k'}$ is regular, normal, reduced, see", "Lemmas \\ref{lemma-geometrically-regular},", "\\ref{lemma-geometrically-normal}, and", "\\ref{lemma-check-only-finite-inseparable-extensions}.", "By", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-smooth}", "the morphism $X_{k'} \\to \\Spec(k')$ is smooth too.", "Hence it suffices to show that a scheme $X$ smooth over a field is regular,", "normal, and reduced. We see that $X$ is regular by", "Lemma \\ref{lemma-smooth-regular}.", "Hence", "Properties, Lemma \\ref{properties-lemma-regular-normal}", "guarantees that $X$ is normal." ], "refs": [ "varieties-lemma-geometrically-regular-smooth", "varieties-lemma-geometrically-regular", "varieties-lemma-geometrically-normal", "varieties-lemma-check-only-finite-inseparable-extensions", "morphisms-lemma-base-change-smooth", "varieties-lemma-smooth-regular", "properties-lemma-regular-normal" ], "ref_ids": [ 10962, 10959, 10952, 10909, 5327, 11004, 2977 ] } ], "ref_ids": [] }, { "id": 11006, "type": "theorem", "label": "varieties-lemma-affine-space-over-field", "categories": [ "varieties" ], "title": "varieties-lemma-affine-space-over-field", "contents": [ "Let $k$ be a field. Let $d \\geq 0$. Let $W \\subset \\mathbf{A}^d_k$", "be nonempty open. Then there exists a closed point $w \\in W$ such that", "$k \\subset \\kappa(w)$ is finite separable." ], "refs": [], "proofs": [ { "contents": [ "After possible shrinking $W$ we may assume that", "$W = \\mathbf{A}^d_k \\setminus V(f)$ for some $f \\in k[x_1, \\ldots, x_d]$.", "If the lemma is wrong then $f(a_1, \\ldots, a_d) = 0$ for all", "$(a_1, \\ldots, a_d) \\in (k^{sep})^d$. This is absurd as $k^{sep}$", "is an infinite field." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11007, "type": "theorem", "label": "varieties-lemma-smooth-separable-closed-points-dense", "categories": [ "varieties" ], "title": "varieties-lemma-smooth-separable-closed-points-dense", "contents": [ "Let $k$ be a field. If $X$ is smooth over $\\Spec(k)$ then", "the set", "$$", "\\{x \\in X\\text{ closed such that }k \\subset \\kappa(x)", "\\text{ is finite separable}\\}", "$$", "is dense in $X$." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that given a nonempty smooth $X$ over $k$", "there exists at least one closed point whose residue field is", "finite separable over $k$. To see this, choose a diagram", "$$", "\\xymatrix{", "X & U \\ar[l] \\ar[r]^-\\pi & \\mathbf{A}^d_k", "}", "$$", "with $\\pi$ \\'etale, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-etale-over-affine-space}.", "The morphism $\\pi : U \\to \\mathbf{A}^d_k$ is open, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-open}.", "By", "Lemma \\ref{lemma-affine-space-over-field}", "we may choose a closed point $w \\in \\pi(U)$ whose residue field is", "finite separable over $k$. Pick any $x \\in U$ with $\\pi(x) = w$. By", "Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}", "the field extension $\\kappa(w) \\subset \\kappa(x)$ is finite separable.", "Hence $k \\subset \\kappa(x)$ is finite separable. The point $x$ is a", "closed point of $X$ by", "Morphisms, Lemma", "\\ref{morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre}." ], "refs": [ "morphisms-lemma-smooth-etale-over-affine-space", "morphisms-lemma-etale-open", "varieties-lemma-affine-space-over-field", "morphisms-lemma-etale-over-field", "morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre" ], "ref_ids": [ 5377, 5370, 11006, 5364, 5222 ] } ], "ref_ids": [] }, { "id": 11008, "type": "theorem", "label": "varieties-lemma-geometrically-reduced-dense-smooth-open", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-reduced-dense-smooth-open", "contents": [ "Let $X$ be a scheme over a field $k$.", "If $X$ is locally of finite type and geometrically reduced over $k$", "then $X$ contains a dense open which is smooth over $k$." ], "refs": [], "proofs": [ { "contents": [ "The problem is local on $X$, hence we may assume $X$ is quasi-compact.", "Let $X = X_1 \\cup \\ldots \\cup X_n$ be the irreducible components of $X$.", "Then $Z = \\bigcup_{i \\not = j} X_i \\cap X_j$ is nowhere dense in $X$.", "Hence we may replace $X$ by $X \\setminus Z$. As $X \\setminus Z$ is a", "disjoint union of irreducible schemes, this reduces us to the case", "where $X$ is irreducible. As $X$ is irreducible and reduced, it is", "integral, see", "Properties, Lemma \\ref{properties-lemma-characterize-integral}.", "Let $\\eta \\in X$ be its generic point.", "Then the function field $K = k(X) = \\kappa(\\eta)$ is geometrically", "reduced over $k$, hence separable over $k$, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-separable-field-extensions}.", "Let $U = \\Spec(A) \\subset X$ be any nonempty affine open", "so that $K = A_{(0)}$ is the fraction field of $A$. Apply", "Algebra, Lemma \\ref{algebra-lemma-separable-smooth}", "to conclude that $A$ is smooth at $(0)$ over $k$.", "By definition this means that some principal localization", "of $A$ is smooth over $k$ and we win." ], "refs": [ "properties-lemma-characterize-integral", "algebra-lemma-characterize-separable-field-extensions", "algebra-lemma-separable-smooth" ], "ref_ids": [ 2947, 569, 1225 ] } ], "ref_ids": [] }, { "id": 11009, "type": "theorem", "label": "varieties-lemma-dense-smooth-open-variety-over-perfect-field", "categories": [ "varieties" ], "title": "varieties-lemma-dense-smooth-open-variety-over-perfect-field", "contents": [ "Let $k$ be a perfect field. Let $X$ be a locally algebraic", "reduced $k$-scheme, for example a variety over $k$. Then we have", "$$", "\\{x \\in X \\mid X \\to \\Spec(k)\\text{ is smooth at }x\\} =", "\\{x \\in X \\mid \\mathcal{O}_{X, x}\\text{ is regular}\\}", "$$", "and this is a dense open subscheme of $X$." ], "refs": [], "proofs": [ { "contents": [ "The equality of the two sets follows immediately from", "Algebra, Lemma \\ref{algebra-lemma-separable-smooth} and the definitions", "(see Algebra, Definition \\ref{algebra-definition-perfect} for the definition", "of a perfect field). The set is open because the set of points where", "a morphism of schemes is smooth is open, see", "Morphisms, Definition \\ref{morphisms-definition-smooth}.", "Finally, we give two arguments to see that it is dense:", "(1) The generic points of $X$ are in the set as the local rings at", "generic points are fields (Algebra, Lemma", "\\ref{algebra-lemma-minimal-prime-reduced-ring}) hence regular.", "(2) We use that $X$ is geometrically reduced by", "Lemma \\ref{lemma-perfect-reduced} and hence", "Lemma \\ref{lemma-geometrically-reduced-dense-smooth-open} applies." ], "refs": [ "algebra-lemma-separable-smooth", "algebra-definition-perfect", "morphisms-definition-smooth", "algebra-lemma-minimal-prime-reduced-ring", "varieties-lemma-perfect-reduced", "varieties-lemma-geometrically-reduced-dense-smooth-open" ], "ref_ids": [ 1225, 1462, 5564, 418, 10907, 11008 ] } ], "ref_ids": [] }, { "id": 11010, "type": "theorem", "label": "varieties-lemma-flat-under-smooth", "categories": [ "varieties" ], "title": "varieties-lemma-flat-under-smooth", "contents": [ "Let $k$ be a field. Let $f : X \\to Y$ be a morphism of schemes locally", "of finite type over $k$. Let $x \\in X$ be a point and set $y = f(x)$.", "If $X \\to \\Spec(k)$ is smooth at $x$ and $f$ is flat at $x$", "then $Y \\to \\Spec(k)$ is smooth at $y$. In particular, if $X$ is", "smooth over $k$ and $f$ is flat and surjective, then $Y$ is smooth over $k$." ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that $Y$ is geometrically regular at $y$, see", "Lemma \\ref{lemma-geometrically-regular-smooth}.", "This follows from", "Lemma \\ref{lemma-flat-under-geometrically-regular}", "(and", "Lemma \\ref{lemma-geometrically-regular-smooth}", "applied to $(X, x)$)." ], "refs": [ "varieties-lemma-geometrically-regular-smooth", "varieties-lemma-flat-under-geometrically-regular", "varieties-lemma-geometrically-regular-smooth" ], "ref_ids": [ 10962, 10961, 10962 ] } ], "ref_ids": [] }, { "id": 11011, "type": "theorem", "label": "varieties-lemma-variety-with-smooth-rational-point", "categories": [ "varieties" ], "title": "varieties-lemma-variety-with-smooth-rational-point", "contents": [ "Let $k$ be a field. Let $X$ be a variety over $k$ which has", "a $k$-rational point $x$ such that $X$ is smooth at $x$.", "Then $X$ is geometrically integral over $k$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be the smooth locus of $X$. By assumption $U$ is nonempty", "and hence dense and scheme theoretically dense. Then", "$U_{\\overline{k}} \\subset X_{\\overline{k}}$ is dense and", "scheme theoretically dense as well (some details omitted).", "Thus it suffices to show that $U$ is geometrically integral.", "Because $U$ has a $k$-rational point it is geometrically connected", "by Lemma \\ref{lemma-geometrically-connected-if-connected-and-point}.", "On the other hand, $U_{\\overline{k}}$ is reduced and normal", "(Lemma \\ref{lemma-smooth-geometrically-normal}.", "Since a connected normal Noetherian scheme", "is integral (Properties, Lemma \\ref{properties-lemma-normal-Noetherian})", "the proof is complete." ], "refs": [ "varieties-lemma-geometrically-connected-if-connected-and-point", "varieties-lemma-smooth-geometrically-normal", "properties-lemma-normal-Noetherian" ], "ref_ids": [ 10926, 11005, 2970 ] } ], "ref_ids": [] }, { "id": 11012, "type": "theorem", "label": "varieties-lemma-regular-functions-proper-variety", "categories": [ "varieties" ], "title": "varieties-lemma-regular-functions-proper-variety", "contents": [ "Let $X$ be a proper variety over $k$. Then", "\\begin{enumerate}", "\\item $K = H^0(X, \\mathcal{O}_X)$ is a field which is", "a finite extension of the field $k$,", "\\item if $X$ is geometrically reduced, then $K/k$ is separable,", "\\item if $X$ is geometrically irreducible, then $K/k$", "is purely inseparable,", "\\item if $X$ is geometrically integral, then $K = k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-proper-geometrically-reduced-global-sections}." ], "refs": [ "varieties-lemma-proper-geometrically-reduced-global-sections" ], "ref_ids": [ 10948 ] } ], "ref_ids": [] }, { "id": 11013, "type": "theorem", "label": "varieties-lemma-normalization-locally-algebraic", "categories": [ "varieties" ], "title": "varieties-lemma-normalization-locally-algebraic", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic scheme over $k$.", "Let $\\nu : X^\\nu \\to X$ be the normalization morphism, see", "Morphisms, Definition \\ref{morphisms-definition-normalization}.", "Then", "\\begin{enumerate}", "\\item $\\nu$ is finite, dominant, and $X^\\nu$ is a disjoint", "union of normal irreducible locally algebraic schemes over $k$,", "\\item $\\nu$ factors as $X^\\nu \\to X_{red} \\to X$ and the first", "morphism is the normalization morphism of $X_{red}$,", "\\item if $X$ is a reduced algebraic scheme, then $\\nu$ is", "birational,", "\\item if $X$ is a variety, then $X^\\nu$ is a variety and", "$\\nu$ is a finite birational morphism of varieties.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-normalization" ], "proofs": [ { "contents": [ "Since $X$ is locally of finite type over a field, we see that", "$X$ is locally Noetherian", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian})", "hence every quasi-compact open has finitely many irreducible", "components (Properties, Lemma", "\\ref{properties-lemma-Noetherian-irreducible-components}).", "Thus Morphisms, Definition \\ref{morphisms-definition-normalization} applies.", "The normalization $X^\\nu$ is always a disjoint union of normal integral", "schemes and the normalization morphism $\\nu$ is always dominant, see", "Morphisms, Lemma \\ref{morphisms-lemma-normalization-normal}.", "Since $X$ is universally Nagata", "(Morphisms, Lemma \\ref{morphisms-lemma-ubiquity-nagata})", "we see that $\\nu$ is finite", "(Morphisms, Lemma \\ref{morphisms-lemma-nagata-normalization}).", "Hence $X^\\nu$ is locally algebraic too.", "At this point we have proved (1).", "\\medskip\\noindent", "Part (2) is Morphisms, Lemma \\ref{morphisms-lemma-normalization-reduced}.", "\\medskip\\noindent", "Part (3) is Morphisms, Lemma \\ref{morphisms-lemma-normalization-birational}.", "\\medskip\\noindent", "Part (4) follows from (1), (2), (3), and the fact that $X^\\nu$ is", "separated as a scheme finite over a separated scheme." ], "refs": [ "morphisms-lemma-finite-type-noetherian", "properties-lemma-Noetherian-irreducible-components", "morphisms-definition-normalization", "morphisms-lemma-normalization-normal", "morphisms-lemma-ubiquity-nagata", "morphisms-lemma-nagata-normalization", "morphisms-lemma-normalization-reduced", "morphisms-lemma-normalization-birational" ], "ref_ids": [ 5202, 2956, 5592, 5515, 5219, 5520, 5512, 5517 ] } ], "ref_ids": [ 5592 ] }, { "id": 11014, "type": "theorem", "label": "varieties-lemma-relative-normalization-finite", "categories": [ "varieties" ], "title": "varieties-lemma-relative-normalization-finite", "contents": [ "Let $k$ be a field. Let $f : Y \\to X$ be a quasi-compact", "morphism of locally algebraic schemes over $k$. Let $X'$", "be the normalization of $X$ in $Y$. If $Y$ is reduced, then", "$X' \\to X$ is finite." ], "refs": [], "proofs": [ { "contents": [ "Since $Y$ is quasi-separated (by", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian-quasi-separated} and", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian})", "the morphism $f$ is quasi-separated", "(Schemes, Lemma \\ref{schemes-lemma-compose-after-separated}).", "Hence Morphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}", "applies. The result follows from Morphisms, Lemma", "\\ref{morphisms-lemma-nagata-normalization-finite-general}.", "This uses that locally algebraic schemes are locally Noetherian", "(hence have locally finitely many irreducible components)", "and that locally algebraic schemes are Nagata", "(Morphisms, Lemma \\ref{morphisms-lemma-ubiquity-nagata}).", "Some small details omitted." ], "refs": [ "properties-lemma-locally-Noetherian-quasi-separated", "morphisms-lemma-finite-type-noetherian", "schemes-lemma-compose-after-separated", "morphisms-definition-normalization-X-in-Y", "morphisms-lemma-nagata-normalization-finite-general", "morphisms-lemma-ubiquity-nagata" ], "ref_ids": [ 2953, 5202, 7715, 5591, 5509, 5219 ] } ], "ref_ids": [] }, { "id": 11015, "type": "theorem", "label": "varieties-lemma-finite-extension-geometrically-normal", "categories": [ "varieties" ], "title": "varieties-lemma-finite-extension-geometrically-normal", "contents": [ "Let $k$ be a field. Let $X$ be an algebraic $k$-scheme.", "Then there exists a finite purely inseparable extension $k \\subset k'$", "such that the normalization $Y$ of $X_{k'}$ is geometrically normal over $k'$." ], "refs": [], "proofs": [ { "contents": [ "Let $K = k^{perf}$ be the perfect closure. Let $Y_K$ be the", "normalization of $X_K$, see Lemma \\ref{lemma-normalization-locally-algebraic}.", "By Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "there exists a finite sub extension $K/k'/k$ and a morphism", "$\\nu : Y \\to X_{k'}$ of finite presentation whose base change to $K$", "is the normalization morphism $\\nu_K : Y_K \\to X_K$.", "Observe that $Y$ is geometrically normal over $k'$", "(Lemma \\ref{lemma-geometrically-normal}).", "After increasing $k'$ we may assume $Y \\to X_{k'}$ is finite", "(Limits, Lemma \\ref{limits-lemma-descend-finite-finite-presentation}).", "Since $\\nu_K : Y_K \\to X_K$ is the normalization morphism,", "it induces a birational morphism $Y_K \\to (X_K)_{red}$.", "Hence there is a dense open $V_K \\subset X_K$ such that", "$\\nu_K^{-1}(V_K) \\to V_K$ is a closed immersion", "(inducing an isomorphism of $\\nu_K^{-1}(V_K)$ with $V_{K, red}$, see", "for example Morphisms, Lemma", "\\ref{morphisms-lemma-birational-isomorphism-over-dense-open}).", "After increasing $k'$ we find $V_K$ is the base change of a dense open", "$V \\subset Y$ and the morphism $\\nu^{-1}(V) \\to V$ is a closed immersion", "(Limits, Lemmas \\ref{limits-lemma-descend-opens} and", "\\ref{limits-lemma-descend-closed-immersion-finite-presentation}).", "It follows readily from this that $\\nu$ is the normalization", "morphism and the proof is complete." ], "refs": [ "varieties-lemma-normalization-locally-algebraic", "limits-lemma-descend-finite-presentation", "varieties-lemma-geometrically-normal", "limits-lemma-descend-finite-finite-presentation", "morphisms-lemma-birational-isomorphism-over-dense-open", "limits-lemma-descend-opens", "limits-lemma-descend-closed-immersion-finite-presentation" ], "ref_ids": [ 11013, 15077, 10952, 15058, 5490, 15041, 15060 ] } ], "ref_ids": [] }, { "id": 11016, "type": "theorem", "label": "varieties-lemma-normalization-and-change-of-fields", "categories": [ "varieties" ], "title": "varieties-lemma-normalization-and-change-of-fields", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.", "Let $K/k$ be an extension of fields. Let $\\nu : X^\\nu \\to X$", "be the normalization of $X$ and let $Y^\\nu \\to X_K$ be the", "normalization of the base change. Then the canonical morphism", "$$", "Y^\\nu \\longrightarrow X^\\nu \\times_{\\Spec(k)} \\Spec(K)", "$$", "is an isomorphism if $K/k$ is separable and a universal homeomorphism", "in general." ], "refs": [], "proofs": [ { "contents": [ "Set $Y = X_K$. Let $X^{(0)}$, resp.\\ $Y^{(0)}$ be the set of generic points", "of irreducible components of $X$, resp.\\ $Y$. Then the projection morphism", "$\\pi : Y \\to X$ satisfies $\\pi(Y^{(0)}) = X^{(0)}$. This is true because", "$\\pi$ is surjective, open, and generizing, see", "Morphisms, Lemmas \\ref{morphisms-lemma-scheme-over-field-universally-open} and", "\\ref{morphisms-lemma-open-generizing}.", "If we view $X^{(0)}$, resp.\\ $Y^{(0)}$ as (reduced) schemes, then", "$X^\\nu$, resp.\\ $Y^\\nu$ is the normalization of $X$, resp.\\ $Y$ in", "$X^{(0)}$, resp.\\ $Y^{(0})$.", "Thus Morphisms, Lemma \\ref{morphisms-lemma-functoriality-normalization}", "gives a canonical morphism $Y^\\nu \\to X^\\nu$ over $Y \\to X$ which in", "turn gives the canonical morphism of the lemma by the universal", "property of the fibre product.", "\\medskip\\noindent", "To prove this morphism has the properties stated in the lemma we may", "assume $X = \\Spec(A)$ is affine. Let $Q(A_{red})$ be the", "total ring of fractions of $A_{red}$. Then $X^\\nu$ is the spectrum of", "the integral closure $A'$ of $A$ in $Q(A_{red})$, see", "Morphisms, Lemmas \\ref{morphisms-lemma-normalization-reduced} and", "\\ref{morphisms-lemma-description-normalization}.", "Similarly, $Y^\\nu$ is the spectrum of the integral closure $B'$ of", "$A \\otimes_k K$ in $Q((A \\otimes_k K)_{red})$. There is a canonical", "map $Q(A_{red}) \\to Q((A \\otimes_k K)_{red})$, a canonical map", "$A' \\to B'$, and the morphism of the lemma corresponds to the induced map", "$$", "A' \\otimes_k K \\longrightarrow B'", "$$", "of $K$-algebras. The kernel consists of nilpotent", "elements as the kernel of $Q(A_{red}) \\otimes_k K \\to Q((A \\otimes_k K)_{red})$", "is the set of nilpotent elements.", "\\medskip\\noindent", "If $K/k$ is separable, then $A' \\otimes_k K$ is normal by", "Lemma \\ref{lemma-base-change-normal-by-separable}. In particular", "it is reduced, whence $Q((A \\otimes_k K)_{red}) = Q(A' \\otimes_k K)$", "and $B' = A' \\otimes_k K$ by", "Algebra, Lemma \\ref{algebra-lemma-characterize-reduced-ring-normal}.", "\\medskip\\noindent", "Assume $K/k$ is not separable. Then the characteristic of $k$ is $p > 0$.", "We will show that for every $b \\in B'$ there is a power $q$ of $p$", "such that $b^q$ is in the image of $A' \\otimes_k K$. This will prove", "that the displayed map is a universal homeomorphism by", "Algebra, Lemma \\ref{algebra-lemma-p-ring-map}.", "For a given $b$ there is a subfield $F \\subset K$", "with $F/k$ finitely generated such that $b$ is contained in", "$Q((A \\otimes_k F)_{red})$ and is integral over $A \\otimes_k F$.", "Choose a monic polynomial $P = T^d + \\alpha_1 T^{d - 1} + \\ldots + \\alpha _d$", "with $P(b) = 0$ and $\\alpha_i \\in A \\otimes_k F$.", "Choose a transcendence basis $t_1, \\ldots, t_r$ for $F$ over $k$.", "Let $F/F'/k(t_1, \\ldots, t_r)$ be the maximal separable", "subextension (Fields, Lemma \\ref{fields-lemma-separable-first}).", "Since $F/F'$ is finite purely inseparable, there is a $q$ such", "that $\\lambda^q \\in F'$ for all $\\lambda \\in F$. Then $b^q$ is", "in $Q((A \\otimes_k F')_{red})$ and satisfies the polynomial", "$T^d + \\alpha_1^q T^{d - 1} + \\ldots + \\alpha _d^q$ with", "$\\alpha_i^q \\in A \\otimes_k F'$. By the separable case", "we see that $b^q \\in A' \\otimes_k F'$ and the proof is complete." ], "refs": [ "morphisms-lemma-scheme-over-field-universally-open", "morphisms-lemma-open-generizing", "morphisms-lemma-functoriality-normalization", "morphisms-lemma-normalization-reduced", "morphisms-lemma-description-normalization", "varieties-lemma-base-change-normal-by-separable", "algebra-lemma-characterize-reduced-ring-normal", "algebra-lemma-p-ring-map", "fields-lemma-separable-first" ], "ref_ids": [ 5254, 5255, 5500, 5512, 5513, 10955, 515, 582, 4482 ] } ], "ref_ids": [] }, { "id": 11017, "type": "theorem", "label": "varieties-lemma-geometrically-normal-in-codim-1", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-normal-in-codim-1", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.", "Let $\\nu : X^\\nu \\to X$ be the normalization of $X$.", "Let $x \\in X$ be a point such that (a) $\\mathcal{O}_{X, x}$", "is reduced, (b) $\\dim(\\mathcal{O}_{X, x}) = 1$, and (c)", "for every $x' \\in X^\\nu$ with $\\nu(x') = x$ the extension", "$\\kappa(x')/k$ is separable. Then $X$ is geometrically reduced at $x$", "and $X^\\nu$ is geometrically regular at $x'$ with $\\nu(x') = x$." ], "refs": [], "proofs": [ { "contents": [ "We will use the results of Lemma \\ref{lemma-normalization-locally-algebraic}", "without further mention. Let $x' \\in X^\\nu$ be a point over $x$.", "By dimension theory (Section \\ref{section-algebraic-schemes}) we have", "$\\dim(\\mathcal{O}_{X^\\nu, x'}) = 1$. Since $X^\\nu$ is normal, we", "see that $\\mathcal{O}_{X^\\nu, x'}$ is a discrete valuation ring", "(Properties, Lemma \\ref{properties-lemma-criterion-normal}).", "Thus $\\mathcal{O}_{X^\\nu, x'}$ is a regular local $k$-algebra", "whose residue field is separable over $k$. Hence", "$k \\to \\mathcal{O}_{X^\\nu, x'}$ is formally smooth", "in the $\\mathfrak m_{x'}$-adic topology, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-regular-implies-fs}.", "Then $\\mathcal{O}_{X^\\nu, x'}$ is geometrically regular", "over $k$ by", "More on Algebra, Theorem \\ref{more-algebra-theorem-regular-fs}.", "Thus $X^\\nu$ is geometrically regular at $x'$ by", "Lemma \\ref{lemma-geometrically-regular-at-point}.", "\\medskip\\noindent", "Since $\\mathcal{O}_{X, x}$ is reduced, the family of maps", "$\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X^\\nu, x'}$ is injective.", "Since $\\mathcal{O}_{X^\\nu, x'}$ is a geometrically reduced", "$k$-algebra, it follows immediately that $\\mathcal{O}_{X, x}$", "is a geometrically reduced $k$-algebra. Hence $X$ is geometrically", "reduced at $x$ by Lemma \\ref{lemma-geometrically-reduced-at-point}." ], "refs": [ "varieties-lemma-normalization-locally-algebraic", "properties-lemma-criterion-normal", "more-algebra-lemma-regular-implies-fs", "more-algebra-theorem-regular-fs", "varieties-lemma-geometrically-regular-at-point", "varieties-lemma-geometrically-reduced-at-point" ], "ref_ids": [ 11013, 2989, 10025, 9802, 10958, 10906 ] } ], "ref_ids": [] }, { "id": 11018, "type": "theorem", "label": "varieties-lemma-open-in-normal-proper", "categories": [ "varieties" ], "title": "varieties-lemma-open-in-normal-proper", "contents": [ "Let $k$ be an algebraically closed field.", "Let $\\overline{X}$ be a proper variety over $k$.", "Let $X \\subset \\overline{X}$ be an open subscheme.", "Assume $X$ is normal.", "Then $\\mathcal{O}^*(X)/k^*$ is a finitely generated abelian group." ], "refs": [], "proofs": [ { "contents": [ "Since the statement only concerns $X$, we may replace $\\overline{X}$", "by a different proper variety over $k$. Let $\\nu : \\overline{X}^\\nu \\to", "\\overline{X}$ be the normalization morphism. By", "Lemma \\ref{lemma-normalization-locally-algebraic}", "we have that $\\nu$ is finite and $\\overline{X}^\\nu$ is a", "variety. Since $X$ is normal, we see that $\\nu^{-1}(X) \\to X$ is an", "isomorphism (tiny detail omitted). Finally, we see that", "$\\overline{X}^\\nu$ is proper over $k$ as a finite morphism is proper", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-proper})", "and compositions of proper morphisms", "are proper (Morphisms, Lemma \\ref{morphisms-lemma-composition-proper}).", "Thus we may and do assume $\\overline{X}$ is normal.", "\\medskip\\noindent", "We will use without further mention that for any affine open $U$ of", "$\\overline{X}$ the ring $\\mathcal{O}(U)$ is a finitely generated", "$k$-algebra, which is Noetherian, a domain and normal, see", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-permanence},", "Properties, Definition \\ref{properties-definition-integral},", "Properties, Lemmas \\ref{properties-lemma-locally-Noetherian} and", "\\ref{properties-lemma-locally-normal},", "Morphisms, Lemma \\ref{morphisms-lemma-locally-finite-type-characterize}.", "\\medskip\\noindent", "Let $\\xi_1, \\ldots, \\xi_r$ be the generic points of the complement of $X$", "in $\\overline{X}$. There are finitely many since $\\overline{X}$ has a", "Noetherian underlying topological space (see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian},", "Properties, Lemma \\ref{properties-lemma-Noetherian-topology}, and", "Topology, Lemma \\ref{topology-lemma-Noetherian}).", "For each $i$ the local ring $\\mathcal{O}_i = \\mathcal{O}_{X, \\xi_i}$", "is a normal Noetherian local domain (as a localization of a", "Noetherian normal domain). Let $J \\subset \\{1, \\ldots, r\\}$ be the set of", "indices $i$ such that $\\dim(\\mathcal{O}_i) = 1$. For $j \\in J$ the", "local ring $\\mathcal{O}_j$ is a discrete valuation ring, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-dvr}.", "Hence we obtain a valuation", "$$", "v_j : k(\\overline{X})^* \\longrightarrow \\mathbf{Z}", "$$", "with the property that $v_j(f) \\geq 0 \\Leftrightarrow f \\in \\mathcal{O}_j$.", "\\medskip\\noindent", "Think of $\\mathcal{O}(X)$ as a sub $k$-algebra of $k(X) = k(\\overline{X})$.", "We claim that the kernel of the map", "$$", "\\mathcal{O}(X)^* \\longrightarrow", "\\prod\\nolimits_{j \\in J} \\mathbf{Z},", "\\quad", "f \\longmapsto \\prod v_j(f)", "$$", "is $k^*$. It is clear that this claim proves the lemma.", "Namely, suppose that $f \\in \\mathcal{O}(X)$ is an element of the kernel.", "Let $U = \\Spec(B) \\subset \\overline{X}$ be any affine open.", "Then $B$ is a Noetherian normal domain.", "For every height one prime $\\mathfrak q \\subset B$ with corresponding", "point $\\xi \\in X$ we see that either $\\xi = \\xi_j$ for some $j \\in J$", "or that $\\xi \\in X$. The reason is that", "$\\text{codim}(\\overline{\\{\\xi\\}}, \\overline{X}) = 1$ by", "Properties, Lemma \\ref{properties-lemma-codimension-local-ring}", "and hence if $\\xi \\in \\overline{X} \\setminus X$ it must be a", "generic point of $\\overline{X} \\setminus X$, hence equal to some", "$\\xi_j$, $j \\in J$.", "We conclude that $f \\in \\mathcal{O}_{X, \\xi} = B_{\\mathfrak q}$", "in either case as $f$ is in the kernel of the map. Thus", "$f \\in \\bigcap_{\\text{ht}(\\mathfrak q) = 1} B_{\\mathfrak q} = B$, see", "Algebra, Lemma", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}.", "In other words, we see that", "$f \\in \\Gamma(\\overline{X}, \\mathcal{O}_{\\overline{X}})$.", "But since $k$ is algebraically closed we conclude that", "$f \\in k$ by", "Lemma \\ref{lemma-regular-functions-proper-variety}." ], "refs": [ "varieties-lemma-normalization-locally-algebraic", "morphisms-lemma-finite-proper", "morphisms-lemma-composition-proper", "algebra-lemma-Noetherian-permanence", "properties-definition-integral", "properties-lemma-locally-Noetherian", "properties-lemma-locally-normal", "morphisms-lemma-locally-finite-type-characterize", "morphisms-lemma-finite-type-noetherian", "properties-lemma-Noetherian-topology", "topology-lemma-Noetherian", "algebra-lemma-characterize-dvr", "properties-lemma-codimension-local-ring", "algebra-lemma-normal-domain-intersection-localizations-height-1", "varieties-lemma-regular-functions-proper-variety" ], "ref_ids": [ 11013, 5445, 5408, 448, 3068, 2951, 2966, 5198, 5202, 2954, 8220, 1023, 2979, 1313, 11012 ] } ], "ref_ids": [] }, { "id": 11019, "type": "theorem", "label": "varieties-lemma-units-integral-finite-type-algebraically-closed", "categories": [ "varieties" ], "title": "varieties-lemma-units-integral-finite-type-algebraically-closed", "contents": [ "Let $k$ be an algebraically closed field.", "Let $X$ be an integral scheme locally of finite type over $k$.", "Then $\\mathcal{O}^*(X)/k^*$ is a finitely generated abelian group." ], "refs": [], "proofs": [ { "contents": [ "As $X$ is integral the restriction mapping", "$\\mathcal{O}(X) \\to \\mathcal{O}(U)$ is injective for any", "nonempty open subscheme $U \\subset X$. Hence we may assume", "that $X$ is affine. Choose a closed immersion", "$X \\to \\mathbf{A}^n_k$", "and denote $\\overline{X}$ the closure of $X$ in $\\mathbf{P}^n_k$", "via the usual immersion $\\mathbf{A}^n_k \\to \\mathbf{P}^n_k$.", "Thus we may assume that $X$ is an affine open of a projective", "variety $\\overline{X}$.", "\\medskip\\noindent", "Let $\\nu : \\overline{X}^\\nu \\to \\overline{X}$ be the normalization", "morphism, see", "Morphisms, Definition \\ref{morphisms-definition-normalization}.", "We know that $\\nu$ is finite, dominant, and that $\\overline{X}^\\nu$", "is a normal irreducible scheme, see", "Morphisms, Lemmas \\ref{morphisms-lemma-normalization-normal},", "\\ref{morphisms-lemma-Japanese-normalization}, and", "\\ref{morphisms-lemma-ubiquity-nagata}.", "It follows that $\\overline{X}^\\nu$ is a proper variety,", "because $\\overline{X} \\to \\Spec(k)$ is proper as a composition", "of a finite and a proper morphism (see results in", "Morphisms, Sections \\ref{morphisms-section-proper} and", "\\ref{morphisms-section-integral}).", "It also follows that $\\nu$ is a surjective morphism, because", "the image of $\\nu$ is closed and contains the generic point of $\\overline{X}$.", "Hence setting $X^\\nu = \\nu^{-1}(X)$ we see that it suffices to prove the", "result for $X^\\nu$. In other words, we may assume that $X$ is a nonempty", "open of a normal proper variety $\\overline{X}$. This case is handled by", "Lemma \\ref{lemma-open-in-normal-proper}." ], "refs": [ "morphisms-definition-normalization", "morphisms-lemma-normalization-normal", "morphisms-lemma-Japanese-normalization", "morphisms-lemma-ubiquity-nagata", "varieties-lemma-open-in-normal-proper" ], "ref_ids": [ 5592, 5515, 5519, 5219, 11018 ] } ], "ref_ids": [] }, { "id": 11020, "type": "theorem", "label": "varieties-lemma-units-general-algebraically-closed", "categories": [ "varieties" ], "title": "varieties-lemma-units-general-algebraically-closed", "contents": [ "Let $k$ be an algebraically closed field.", "Let $X$ be a connected reduced scheme which is locally of finite type", "over $k$ with finitely many irreducible components.", "Then $\\mathcal{O}^*(X)/k^*$ is a finitely generated abelian group." ], "refs": [], "proofs": [ { "contents": [ "Let $X = \\bigcup X_i$ be the irreducible components. By", "Lemma \\ref{lemma-units-integral-finite-type-algebraically-closed}", "we see that $\\mathcal{O}(X_i)^*/k^*$ is a finitely generated", "abelian group. Let $f \\in \\mathcal{O}(X)^*$ be in the kernel", "of the map", "$$", "\\mathcal{O}(X)^* \\longrightarrow \\prod \\mathcal{O}(X_i)^*/k^*.", "$$", "Then for each $i$ there exists an element $\\lambda_i \\in k$", "such that $f|_{X_i} = \\lambda_i$.", "By restricting to $X_i \\cap X_j$ we conclude that", "$\\lambda_i = \\lambda_j$ if $X_i \\cap X_j \\not = \\emptyset$.", "Since $X$ is connected we conclude that all $\\lambda_i$ agree", "and hence that $f \\in k^*$. This proves that", "$$", "\\mathcal{O}(X)^*/k^* \\subset \\prod \\mathcal{O}(X_i)^*/k^*", "$$", "and the lemma follows as on the right we have a product of finitely", "many finitely generated abelian groups." ], "refs": [ "varieties-lemma-units-integral-finite-type-algebraically-closed" ], "ref_ids": [ 11019 ] } ], "ref_ids": [] }, { "id": 11021, "type": "theorem", "label": "varieties-lemma-integral-closure-ground-field", "categories": [ "varieties" ], "title": "varieties-lemma-integral-closure-ground-field", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme over $k$ which is connected and reduced.", "Then the integral closure of $k$ in $\\Gamma(X, \\mathcal{O}_X)$", "is a field." ], "refs": [], "proofs": [ { "contents": [ "Let $k' \\subset \\Gamma(X, \\mathcal{O}_X)$ be the integral closure of", "$k$. Then $X \\to \\Spec(k)$ factors through $\\Spec(k')$, see", "Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}.", "As $X$ is reduced we see that $k'$ has no nonzero nilpotent elements.", "As $k \\to k'$ is integral we see that every prime ideal of $k'$ is", "both a maximal ideal and a minimal prime,", "and $\\Spec(k')$ is totally disconnected, see", "Algebra, Lemmas \\ref{algebra-lemma-integral-no-inclusion} and", "\\ref{algebra-lemma-ring-with-only-minimal-primes}.", "As $X$ is connected the morphism $X \\to \\Spec(k')$ is", "constant, say with image the point corresponding to", "$\\mathfrak p \\subset k'$. Then any $f \\in k'$, $f \\not \\in \\mathfrak p$", "maps to an invertible element of $\\mathcal{O}_X$. By definition", "of $k'$ this then forces $f$ to be a unit of $k'$. Hence we see", "that $k'$ is local with maximal ideal $\\mathfrak p$, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-local-ring}.", "Since we've already seen that $k'$ is reduced this implies that", "$k'$ is a field, see", "Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring}." ], "refs": [ "schemes-lemma-morphism-into-affine", "algebra-lemma-integral-no-inclusion", "algebra-lemma-ring-with-only-minimal-primes", "algebra-lemma-characterize-local-ring", "algebra-lemma-minimal-prime-reduced-ring" ], "ref_ids": [ 7655, 498, 426, 397, 418 ] } ], "ref_ids": [] }, { "id": 11022, "type": "theorem", "label": "varieties-lemma-units-variety", "categories": [ "varieties" ], "title": "varieties-lemma-units-variety", "contents": [ "Let $k$ be a field.", "Let $X$ be a variety over $k$.", "The group $\\mathcal{O}(X)^*/k^*$ is a finitely generated abelian group", "provided at least one of the following conditions holds:", "\\begin{enumerate}", "\\item $k$ is integrally closed in $\\Gamma(X, \\mathcal{O}_X)$,", "\\item $k$ is algebraically closed in $k(X)$,", "\\item $X$ is geometrically integral over $k$, or", "\\item $k$ is the ``intersection'' of the field extensions", "$k \\subset \\kappa(x)$ where $x$ runs over the closed points of $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We see that (1) is enough by", "Proposition \\ref{proposition-units-general}.", "We omit the verification that each of (2), (3), (4) implies (1)." ], "refs": [ "varieties-proposition-units-general" ], "ref_ids": [ 11136 ] } ], "ref_ids": [] }, { "id": 11023, "type": "theorem", "label": "varieties-lemma-kunneth", "categories": [ "varieties" ], "title": "varieties-lemma-kunneth", "contents": [ "Let $k$ be a field. Let $X$ and $Y$ be schemes over $k$ and", "let $\\mathcal{F}$, resp.\\ $\\mathcal{G}$ be a quasi-coherent", "$\\mathcal{O}_X$-module, resp.\\ $\\mathcal{O}_Y$-module.", "Then we have a canonical isomorphism", "$$", "H^n(X \\times_{\\Spec(k)} Y, \\text{pr}_1^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X \\times_{\\Spec(k)} Y}} \\text{pr}_2^*\\mathcal{G}) =", "\\bigoplus\\nolimits_{p + q = n}", "H^p(X, \\mathcal{F}) \\otimes_k H^q(Y, \\mathcal{G})", "$$", "provided $X$ and $Y$ are quasi-compact and have affine", "diagonal\\footnote{The case where $X$ and $Y$ are quasi-separated", "will be discussed in Lemma \\ref{lemma-kunneth-general} below.}", "(for example if $X$ and $Y$ are separated)." ], "refs": [ "varieties-lemma-kunneth-general" ], "proofs": [ { "contents": [ "In this proof unadorned products and tensor products are over $k$. As maps", "$$", "H^p(X, \\mathcal{F}) \\otimes H^q(Y, \\mathcal{G})", "\\longrightarrow", "H^n(X \\times Y, \\text{pr}_1^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X \\times Y}} \\text{pr}_2^*\\mathcal{G})", "$$", "we use functoriality of cohomology to get maps", "$H^p(X, \\mathcal{F}) \\to H^p(X \\times Y, \\text{pr}_1^*\\mathcal{F})$ and", "$H^p(Y, \\mathcal{G}) \\to H^p(X \\times Y, \\text{pr}_2^*\\mathcal{G})$", "and then we use the cup product", "$$", "\\cup :", "H^p(X \\times Y, \\text{pr}_1^*\\mathcal{F})", "\\otimes H^q(X \\times Y, \\text{pr}_2^*\\mathcal{G})", "\\longrightarrow", "H^n(X \\times Y, \\text{pr}_1^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X \\times Y}} \\text{pr}_2^*\\mathcal{G})", "$$", "The result is true when $X$ and $Y$ are affine by the vanishing of", "higher cohomology groups on affines (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero})", "and the definitions (of pullbacks of quasi-coherent modules and tensor", "products of quasi-coherent modules).", "\\medskip\\noindent", "Choose finite affine open coverings", "$\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ and", "$\\mathcal{V} : Y = \\bigcup_{j \\in J} V_j$.", "This determines an affine open covering", "$\\mathcal{W} : X \\times Y = \\bigcup_{(i, j) \\in I \\times J} U_i \\times V_j$.", "Note that $\\mathcal{W}$ is a refinement of", "$\\text{pr}_1^{-1}\\mathcal{U}$ and of $\\text{pr}_2^{-1}\\mathcal{V}$.", "Thus by Cohomology, Lemma \\ref{cohomology-lemma-functoriality-cech}", "we obtain maps", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W}, \\text{pr}_1^*\\mathcal{F})", "\\quad\\text{and}\\quad", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G}) \\to", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W}, \\text{pr}_2^*\\mathcal{G})", "$$", "compatible with pullback maps on cohomology. In Cohomology, Equation", "(\\ref{cohomology-equation-needs-signs})", "we have constructed a map of complexes", "$$", "\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W}, \\text{pr}_1^*\\mathcal{F})", "\\otimes", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W}, \\text{pr}_2^*\\mathcal{G}))", "\\longrightarrow", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W},", "\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}}", "\\text{pr}_2^*\\mathcal{G})", "$$", "defining the cup product on cohomology. Combining the above we", "obtain a map of complexes", "\\begin{equation}", "\\label{equation-kunneth-on-cech}", "\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\otimes", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G}))", "\\longrightarrow", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W},", "\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}}", "\\text{pr}_2^*\\mathcal{G})", "\\end{equation}", "We warn the reader that this map is not an isomorphism of", "complexes. Recall that we may compute the cohomologies of our", "quasi-coherent sheaves using our coverings", "(Cohomology of Schemes, Lemmas", "\\ref{coherent-lemma-affine-diagonal} and", "\\ref{coherent-lemma-cech-cohomology-quasi-coherent}).", "Thus on cohomology (\\ref{equation-kunneth-on-cech}) reproduces the map of", "the lemma.", "\\medskip\\noindent", "Consider a short exact", "sequence $0 \\to \\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{F}'' \\to 0$", "of quasi-coherent modules. Since the construction of", "(\\ref{equation-kunneth-on-cech}) is functorial in $\\mathcal{F}$ and since the", "formation of the relevant {\\v C}ech complexes is exact in", "the variable $\\mathcal{F}$ (because we are taking sections over", "affine opens) we find a map between short exact sequence of", "complexes", "$$", "\\xymatrix{", "\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "\\otimes", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G})) \\ar[r] \\ar[d] &", "\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}')", "\\otimes", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G})) \\ar[r] \\ar[d] &", "\\text{Tot}(", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}'')", "\\otimes", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G})) \\ar[d] \\\\", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W},", "\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}}", "\\text{pr}_2^*\\mathcal{G}) \\ar[r] &", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W},", "\\text{pr}_1^*\\mathcal{F}' \\otimes_{\\mathcal{O}_{X \\times Y}}", "\\text{pr}_2^*\\mathcal{G}) \\ar[r] &", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{W},", "\\text{pr}_1^*\\mathcal{F}'' \\otimes_{\\mathcal{O}_{X \\times Y}}", "\\text{pr}_2^*\\mathcal{G})", "}", "$$", "(we have dropped the outer zeros).", "Looking at long exact cohomology sequences we find that if the result of", "the lemma holds for $2$-out-of-$3$ of", "$\\mathcal{F}, \\mathcal{F}', \\mathcal{F}''$, then it holds for", "the third.", "\\medskip\\noindent", "Observe that $X$ has finite cohomological dimension for", "quasi-coherent modules, see Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-vanishing-nr-affines}.", "Using induction on", "$d(\\mathcal{F}) = \\max \\{d \\mid H^d(X, \\mathcal{F}) \\not = 0\\}$", "we will reduce to the case $d(\\mathcal{F}) = 0$.", "Assume $d(\\mathcal{F}) > 0$.", "By Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-affine-diagonal-universal-delta-functor}", "we have seen that", "there exists an embedding $\\mathcal{F} \\to \\mathcal{F}'$", "such that $H^p(X, \\mathcal{F}') = 0$ for all $p \\geq 1$.", "Setting $\\mathcal{F}'' = \\Coker(\\mathcal{F} \\to \\mathcal{F}')$", "we see that $d(\\mathcal{F}'') < d(\\mathcal{F})$.", "Then we can apply the result from the previous paragraph", "to see that it suffices to prove the lemma for $\\mathcal{F}'$", "and $\\mathcal{F}''$ thereby proving the induction step.", "\\medskip\\noindent", "Arguing in the same fashion for $\\mathcal{G}$ we find that we", "may assume that both $\\mathcal{F}$ and $\\mathcal{G}$", "have nonzero cohomology only in degree $0$.", "Let $V \\subset Y$ be an affine open. Consider the affine open covering", "$\\mathcal{U}_V : X \\times V = \\bigcup_{i \\in I} U_i \\times V$.", "It is immediate that", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\otimes \\mathcal{G}(V) =", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}_V,", "\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}}", "\\text{pr}_2^*\\mathcal{G})", "$$", "(equality of complexes). We conclude that", "$$", "R\\text{pr}_{2, *}(\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}}", "\\text{pr}_2^*\\mathcal{G})", "\\cong", "\\Gamma(X, \\mathcal{F}) \\otimes_k \\mathcal{G} \\cong", "\\bigoplus\\nolimits_{\\alpha \\in A} \\mathcal{G}", "$$", "on $Y$. Here $A$ is a basis for the $k$-vector space $\\Gamma(X, \\mathcal{F})$.", "Cohomology on $Y$ commutes with direct sums", "(Cohomology, Lemma \\ref{cohomology-lemma-quasi-separated-cohomology-colimit}).", "Using the Leray spectral sequence for $\\text{pr}_2$", "(via Cohomology, Lemma \\ref{cohomology-lemma-apply-Leray})", "we conclude that", "$H^n(X \\times Y, \\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}}", "\\text{pr}_2^*\\mathcal{G})$", "is zero for $n > 0$ and isomorphic to", "$H^0(X, \\mathcal{F}) \\otimes H^0(Y, \\mathcal{G})$ for $n = 0$.", "This finishes the proof (except that we should check that the", "isomorphism is indeed given by cup product in degree $0$; we omit", "the verification)." ], "refs": [ "coherent-lemma-quasi-coherent-affine-cohomology-zero", "cohomology-lemma-functoriality-cech", "coherent-lemma-affine-diagonal", "coherent-lemma-cech-cohomology-quasi-coherent", "coherent-lemma-vanishing-nr-affines", "coherent-lemma-affine-diagonal-universal-delta-functor", "cohomology-lemma-quasi-separated-cohomology-colimit", "cohomology-lemma-apply-Leray" ], "ref_ids": [ 3282, 2075, 3285, 3286, 3292, 3293, 2082, 2071 ] } ], "ref_ids": [ 11024 ] }, { "id": 11024, "type": "theorem", "label": "varieties-lemma-kunneth-general", "categories": [ "varieties" ], "title": "varieties-lemma-kunneth-general", "contents": [ "Let $k$ be a field. Let $X$ and $Y$ be schemes over $k$ and", "let $\\mathcal{F}$, resp.\\ $\\mathcal{G}$ be a quasi-coherent", "$\\mathcal{O}_X$-module, resp.\\ $\\mathcal{O}_Y$-module.", "Then we have a canonical isomorphism", "$$", "H^n(X \\times_{\\Spec(k)} Y, \\text{pr}_1^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X \\times_{\\Spec(k)} Y}} \\text{pr}_2^*\\mathcal{G}) =", "\\bigoplus\\nolimits_{p + q = n}", "H^p(X, \\mathcal{F}) \\otimes_k H^q(Y, \\mathcal{G})", "$$", "provided $X$ and $Y$ are quasi-compact and quasi-separated." ], "refs": [], "proofs": [ { "contents": [ "If $X$ and $Y$ are separated or more generally have affine diagonal, then", "please see Lemma \\ref{lemma-kunneth} for ``better'' proof", "(the feature it has over this proof is that it identifies the", "maps as pullbacks followed by cup products).", "Let $X'$, resp.\\ $Y'$ be the infinitesimal thickening of $X$, resp.\\ $Y$", "whose structure sheaf is", "$\\mathcal{O}_{X'} = \\mathcal{O}_X \\oplus \\mathcal{F}$,", "resp.\\ $\\mathcal{O}_{Y'} = \\mathcal{O}_Y \\oplus \\mathcal{G}$ where", "$\\mathcal{F}$, resp.\\ $\\mathcal{G}$ is an ideal of square zero.", "Then", "$$", "\\mathcal{O}_{X' \\times Y'} =", "\\mathcal{O}_{X \\times Y}", "\\oplus \\text{pr}_1^*\\mathcal{F}", "\\oplus \\text{pr}_2^*\\mathcal{G}", "\\oplus \\text{pr}_1^*\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X \\times Y}} \\text{pr}_2^*\\mathcal{G}", "$$", "as sheaves on $X \\times Y$. In this way we see that it suffices to", "prove that", "$$", "H^n(X \\times Y, \\mathcal{O}_{X \\times Y}) =", "\\bigoplus\\nolimits_{p + q = n}", "H^p(X, \\mathcal{O}_X) \\otimes_k H^q(Y, \\mathcal{O}_Y)", "$$", "for any pair of quasi-compact and quasi-separated schemes over $k$.", "Some details omitted.", "\\medskip\\noindent", "To prove this statement we use cohomology and base change in the form", "of Cohomology of Schemes, Lemma \\ref{coherent-lemma-hypercoverings}.", "This lemma tells us there exists a bounded below complex of $k$-vector spaces,", "i.e., a complex $\\mathcal{K}^\\bullet$ of quasi-coherent modules on $\\Spec(k)$,", "which universally computes the cohomology of $Y$ over $\\Spec(k)$.", "In particular, we see that", "$$", "R\\text{pr}_{1, *}(\\mathcal{O}_{X \\times Y}) \\cong", "(X \\to \\Spec(k))^*\\mathcal{K}^\\bullet", "$$", "in $D(\\mathcal{O}_X)$. Up to homotopy the complex $\\mathcal{K}^\\bullet$", "is isomorphic to $\\bigoplus_{q \\geq 0} H^q(Y, \\mathcal{O}_Y)[-q]$ because", "this is true for every complex of vector spaces over a field.", "We conclude that", "$$", "R\\text{pr}_{1, *}(\\mathcal{O}_{X \\times Y}) \\cong", "\\bigoplus\\nolimits_{q \\geq 0}", "H^q(Y, \\mathcal{O}_Y)[-q] \\otimes_k \\mathcal{O}_X", "$$", "in $D(\\mathcal{O}_X)$. Then we have", "\\begin{align*}", "R\\Gamma(X \\times Y, \\mathcal{O}_{X \\times Y})", "& =", "R\\Gamma(X, R\\text{pr}_{1, *}(\\mathcal{O}_{X \\times Y})) \\\\", "& =", "R\\Gamma(X, \\bigoplus\\nolimits_{q \\geq 0}", "H^q(Y, \\mathcal{O}_Y)[-q] \\otimes_k \\mathcal{O}_X) \\\\", "& =", "\\bigoplus\\nolimits_{q \\geq 0}", "R\\Gamma(X, H^q(Y, \\mathcal{O}_Y) \\otimes \\mathcal{O}_X)[-q] \\\\", "& =", "\\bigoplus\\nolimits_{q \\geq 0}", "R\\Gamma(X, \\mathcal{O}_X) \\otimes_k H^q(Y, \\mathcal{O}_Y)[-q] \\\\", "& =", "\\bigoplus\\nolimits_{p, q \\geq 0}", "H^p(X, \\mathcal{O}_X)[-p] \\otimes_k H^q(Y, \\mathcal{O}_Y)[-q]", "\\end{align*}", "as desired. The first equality by Leray for $\\text{pr}_1$", "(Cohomology, Lemma \\ref{cohomology-lemma-before-Leray}).", "The second by our decomposition of the total direct image given above.", "The third because cohomology always commutes with finite direct sums", "(and cohomology of $Y$ vanishes in sufficiently large degree by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-vanishing-nr-affines-quasi-separated}).", "The fourth because cohomology on $X$ commutes with infinite", "direct sums by Cohomology, Lemma", "\\ref{cohomology-lemma-quasi-separated-cohomology-colimit}.", "The final equality by our remark on the derived category", "of a field above." ], "refs": [ "varieties-lemma-kunneth", "coherent-lemma-hypercoverings", "cohomology-lemma-before-Leray", "coherent-lemma-vanishing-nr-affines-quasi-separated", "cohomology-lemma-quasi-separated-cohomology-colimit" ], "ref_ids": [ 11023, 3303, 2068, 3294, 2082 ] } ], "ref_ids": [] }, { "id": 11025, "type": "theorem", "label": "varieties-lemma-change-rings-pic-pre", "categories": [ "varieties" ], "title": "varieties-lemma-change-rings-pic-pre", "contents": [ "Let $A \\to B$ be a faithfully flat ring map. Let $X$ be a quasi-compact and", "quasi-separated scheme over $A$. Let $\\mathcal{L}$ be an invertible", "$\\mathcal{O}_X$-module whose pullback to $X_B$ is trivial. Then", "$H^0(X, \\mathcal{L})$ and $H^0(X, \\mathcal{L}^{\\otimes -1})$ are invertible", "$H^0(X, \\mathcal{O}_X)$-modules and the", "multiplication map induces an isomorphism", "$$", "H^0(X, \\mathcal{L}) \\otimes_{H^0(X, \\mathcal{O}_X)}", "H^0(X, \\mathcal{L}^{\\otimes -1}) \\longrightarrow", "H^0(X, \\mathcal{O}_X)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Denote $\\mathcal{L}_B$ the pullback of $\\mathcal{L}$ to $X_B$.", "Choose an isomorphism $\\mathcal{L}_B \\to \\mathcal{O}_{X_B}$.", "Set $R = H^0(X, \\mathcal{O}_X)$, $M = H^0(X, \\mathcal{L})$ and think of", "$M$ as an $R$-module. For every quasi-coherent $\\mathcal{O}_X$-module", "$\\mathcal{F}$ with pullback $\\mathcal{F}_B$ on $X_B$ there is a", "canonical isomorphism", "$H^0(X_B, \\mathcal{F}_B) = H^0(X, \\mathcal{F}) \\otimes_A B$, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}.", "Thus we have", "$$", "M \\otimes_R (R \\otimes_A B) =", "M \\otimes_A B = H^0(X_B, \\mathcal{L}_B) \\cong", "H^0(X_B, \\mathcal{O}_{X_B}) = R \\otimes_A B", "$$", "Since $R \\to R \\otimes_A B$ is faithfully flat (as the base change", "of the faithfully flat map $A \\to B$), we conclude", "that $M$ is an invertible $R$-module by", "Algebra, Proposition \\ref{algebra-proposition-ffdescent-finite-projectivity}.", "Similarly $N = H^0(X, \\mathcal{L}^{\\otimes -1})$ is an invertible $R$-module.", "To see that the statement on tensor products is true, use that it is true", "after pulling back to $X_B$ and faithful flatness of $R \\to R \\otimes_A B$.", "Some details omitted." ], "refs": [ "coherent-lemma-flat-base-change-cohomology", "algebra-proposition-ffdescent-finite-projectivity" ], "ref_ids": [ 3298, 1413 ] } ], "ref_ids": [] }, { "id": 11026, "type": "theorem", "label": "varieties-lemma-change-rings-pic", "categories": [ "varieties" ], "title": "varieties-lemma-change-rings-pic", "contents": [ "Let $A \\to B$ be a faithfully flat ring map.", "Let $X$ be a scheme over $A$ such that", "\\begin{enumerate}", "\\item $X$ is quasi-compact and quasi-separated, and", "\\item $R = H^0(X, \\mathcal{O}_X)$ is a semi-local ring.", "\\end{enumerate}", "Then the pullback map $\\Pic(X) \\to \\Pic(X_B)$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module", "whose pullback $\\mathcal{L}'$ to $X_B$ is trivial.", "Set $M = H^0(X, \\mathcal{L})$ and $N = H^0(X, \\mathcal{L}^{\\otimes - 1})$.", "By Lemma \\ref{lemma-change-rings-pic-pre} the $R$-modules $M$ and $N$ are", "invertible. Since $R$ is semi-local $M \\cong R$ and $N \\cong R$, see", "Algebra, Lemma \\ref{algebra-lemma-locally-free-semi-local-free}.", "Choose generators $s \\in M$ and $t \\in N$. Then", "$st \\in R = H^0(X, \\mathcal{O}_X)$ is a unit by the last part", "of Lemma \\ref{lemma-change-rings-pic-pre}.", "We conclude that $s$ and $t$ define trivializations of $\\mathcal{L}$ and", "$\\mathcal{L}^{\\otimes -1}$ over $X$." ], "refs": [ "varieties-lemma-change-rings-pic-pre", "algebra-lemma-locally-free-semi-local-free", "varieties-lemma-change-rings-pic-pre" ], "ref_ids": [ 11025, 799, 11025 ] } ], "ref_ids": [] }, { "id": 11027, "type": "theorem", "label": "varieties-lemma-change-fields-pic", "categories": [ "varieties" ], "title": "varieties-lemma-change-fields-pic", "contents": [ "Let $k'/k$ be a field extension.", "Let $X$ be a scheme over $k$ such that", "\\begin{enumerate}", "\\item $X$ is quasi-compact and quasi-separated, and", "\\item $R = H^0(X, \\mathcal{O}_X)$ is semi-local, e.g., if $\\dim_k R < \\infty$.", "\\end{enumerate}", "Then the pullback map $\\Pic(X) \\to \\Pic(X_{k'})$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-change-rings-pic}.", "If $\\dim_k R < \\infty$, then", "$R$ is Artinian and hence semi-local (Algebra, Lemmas", "\\ref{algebra-lemma-finite-dimensional-algebra} and", "\\ref{algebra-lemma-artinian-finite-nr-max})." ], "refs": [ "varieties-lemma-change-rings-pic", "algebra-lemma-finite-dimensional-algebra", "algebra-lemma-artinian-finite-nr-max" ], "ref_ids": [ 11026, 642, 643 ] } ], "ref_ids": [] }, { "id": 11028, "type": "theorem", "label": "varieties-lemma-rational-equivalence-for-Pic", "categories": [ "varieties" ], "title": "varieties-lemma-rational-equivalence-for-Pic", "contents": [ "Let $k$ be a field. Let $X$ be a normal variety over $k$.", "Let $U \\subset \\mathbf{A}^n_k$ be an open subscheme with", "$k$-rational points $p, q \\in U(k)$. For every invertible", "module $\\mathcal{L}$ on $X \\times_{\\Spec(k)} U$ the restrictions", "$\\mathcal{L}|_{X \\times p}$ and $\\mathcal{L}|_{X \\times q}$", "are isomorphic." ], "refs": [], "proofs": [ { "contents": [ "The fibres of $X \\times_{\\Spec(k)} U \\to X$ are open subschemes", "of affine $n$-space over fields. Hence these fibres have", "trivial Picard groups by", "Divisors, Lemma \\ref{divisors-lemma-open-subscheme-UFD}.", "Applying Divisors, Lemma \\ref{divisors-lemma-in-image-pullback}", "we see that $\\mathcal{L}$ is the pullback of an invertible", "module $\\mathcal{N}$ on $X$." ], "refs": [ "divisors-lemma-open-subscheme-UFD", "divisors-lemma-in-image-pullback" ], "ref_ids": [ 8033, 8030 ] } ], "ref_ids": [] }, { "id": 11029, "type": "theorem", "label": "varieties-lemma-euler-characteristic-additive", "categories": [ "varieties" ], "title": "varieties-lemma-euler-characteristic-additive", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$.", "Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "be a short exact sequence of coherent modules on $X$. Then", "$$", "\\chi(X, \\mathcal{F}_2) = \\chi(X, \\mathcal{F}_1) + \\chi(X, \\mathcal{F}_3)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Consider the long exact sequence of cohomology", "$$", "0 \\to H^0(X, \\mathcal{F}_1) \\to H^0(X, \\mathcal{F}_2) \\to", "H^0(X, \\mathcal{F}_3) \\to H^1(X, \\mathcal{F}_1) \\to \\ldots", "$$", "associated to the short exact sequence of the lemma. The rank-nullity theorem", "in linear algebra shows that", "$$", "0 = \\dim H^0(X, \\mathcal{F}_1) - \\dim H^0(X, \\mathcal{F}_2)", "+ \\dim H^0(X, \\mathcal{F}_3) - \\dim H^1(X, \\mathcal{F}_1) + \\ldots", "$$", "This immediately implies the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11030, "type": "theorem", "label": "varieties-lemma-chi-tensor-finite", "categories": [ "varieties" ], "title": "varieties-lemma-chi-tensor-finite", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{F}$", "be a coherent sheaf with $\\dim(\\text{Supp}(\\mathcal{F})) \\leq 0$.", "Then", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is generated by global sections,", "\\item $H^i(X, \\mathcal{F}) = 0$ for $i > 0$,", "\\item $\\chi(X, \\mathcal{F}) = \\dim_k\\Gamma(X, \\mathcal{F})$, and", "\\item", "$\\chi(X, \\mathcal{F} \\otimes \\mathcal{E}) = n\\chi(X, \\mathcal{F})$", "for every locally free module $\\mathcal{E}$ of rank $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-support-closed}", "we see that $\\mathcal{F} = i_*\\mathcal{G}$ where $i : Z \\to X$ is the inclusion", "of the scheme theoretic support of $\\mathcal{F}$ and where $\\mathcal{G}$", "is a coherent $\\mathcal{O}_Z$-module. Since the dimension of $Z$ is", "$0$, we see $Z$ is affine (Properties, Lemma", "\\ref{properties-lemma-locally-Noetherian-dimension-0}).", "Hence $\\mathcal{G}$ is globally generated and the higher", "cohomology groups of $\\mathcal{G}$ are zero", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}).", "Hence $\\mathcal{F} = i_*\\mathcal{G}$ is globally generated.", "Since the cohomologies of $\\mathcal{F}$ and $\\mathcal{G}$ agree", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology})", "we conclude that the higher cohomology groups of $\\mathcal{F}$ are zero", "which gives the first formula. By the projection formula", "(Cohomology, Lemma \\ref{cohomology-lemma-projection-formula}) we have", "$$", "i_*(\\mathcal{G} \\otimes i^*\\mathcal{E}) = \\mathcal{F} \\otimes \\mathcal{E}.", "$$", "Since $Z$ has dimension $0$ the locally free sheaf $i^*\\mathcal{E}$", "is isomorphic to $\\mathcal{O}_Z^{\\oplus n}$ and arguing as above", "we see that the second formula holds." ], "refs": [ "coherent-lemma-coherent-support-closed", "properties-lemma-locally-Noetherian-dimension-0", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "coherent-lemma-relative-affine-cohomology", "cohomology-lemma-projection-formula" ], "ref_ids": [ 3314, 2981, 3282, 3284, 2243 ] } ], "ref_ids": [] }, { "id": 11031, "type": "theorem", "label": "varieties-lemma-euler-characteristic-extend-base-field", "categories": [ "varieties" ], "title": "varieties-lemma-euler-characteristic-extend-base-field", "contents": [ "Let $k \\subset k'$ be an extension of fields. Let $X$ be a proper scheme", "over $k$. Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Let $\\mathcal{F}'$ be the pullback of $\\mathcal{F}$ to $X_{k'}$.", "Then $\\chi(X, \\mathcal{F}) = \\chi(X', \\mathcal{F}')$." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$$", "H^i(X_{k'}, \\mathcal{F}') = H^i(X, \\mathcal{F}) \\otimes_k k'", "$$", "by flat base change, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}." ], "refs": [ "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 3298 ] } ], "ref_ids": [] }, { "id": 11032, "type": "theorem", "label": "varieties-lemma-euler-characteristic-morphism", "categories": [ "varieties" ], "title": "varieties-lemma-euler-characteristic-morphism", "contents": [ "Let $k$ be a field. Let $f : Y \\to X$ be a morphism of proper schemes over", "$k$. Let $\\mathcal{G}$ be a coherent $\\mathcal{O}_Y$-module. Then", "$$", "\\chi(Y, \\mathcal{G}) = \\sum (-1)^i \\chi(X, R^if_*\\mathcal{G})", "$$" ], "refs": [], "proofs": [ { "contents": [ "The formula makes sense: the sheaves $R^if_*\\mathcal{G}$ are coherent", "and only a finite number of them are nonzero, see", "Cohomology of Schemes,", "Proposition \\ref{coherent-proposition-proper-pushforward-coherent} and", "Lemma \\ref{coherent-lemma-quasi-coherence-higher-direct-images}.", "By Cohomology, Lemma \\ref{cohomology-lemma-Leray} there is a spectral", "sequence with", "$$", "E_2^{p, q} = H^p(X, R^qf_*\\mathcal{G})", "$$", "converging to $H^{p + q}(Y, \\mathcal{G})$. By finiteness of cohomology", "on $X$ we see that only a finite number of $E_2^{p, q}$ are nonzero", "and each $E_2^{p, q}$ is a finite dimensional vector space. It follows", "that the same is true for $E_r^{p, q}$ for $r \\geq 2$ and that", "$$", "\\sum (-1)^{p + q} \\dim_k E_r^{p, q}", "$$", "is independent of $r$. Since for $r$ large enough we have", "$E_r^{p, q} = E_\\infty^{p, q}$ and since convergence means there", "is a filtration on $H^n(Y, \\mathcal{G})$ whose graded pieces are", "$E_\\infty^{p, q}$ with $p + q = n$ (this is the meaning of convergence", "of the spectral sequence), we conclude.", "Compare also with the more general", "Homology, Lemma \\ref{homology-lemma-biregular-ss-relation-in-K0}." ], "refs": [ "coherent-proposition-proper-pushforward-coherent", "coherent-lemma-quasi-coherence-higher-direct-images", "cohomology-lemma-Leray", "homology-lemma-biregular-ss-relation-in-K0" ], "ref_ids": [ 3401, 3295, 2070, 12102 ] } ], "ref_ids": [] }, { "id": 11033, "type": "theorem", "label": "varieties-lemma-projective-space-smooth", "categories": [ "varieties" ], "title": "varieties-lemma-projective-space-smooth", "contents": [ "\\begin{slogan}", "Projective space is smooth.", "\\end{slogan}", "Let $k$ be a field and $n \\geq 0$. Then $\\mathbf{P}^n_k$ is a", "smooth projective variety of dimension $n$ over $k$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11034, "type": "theorem", "label": "varieties-lemma-intersection-in-affine-space", "categories": [ "varieties" ], "title": "varieties-lemma-intersection-in-affine-space", "contents": [ "Let $k$ be a field and $n \\geq 0$. Let $X, Y \\subset \\mathbf{A}^n_k$", "be closed subsets. Assume that $X$ and $Y$ are equidimensional,", "$\\dim(X) = r$ and $\\dim(Y) = s$.", "Then every irreducible component of $X \\cap Y$ has dimension $\\geq r + s - n$." ], "refs": [], "proofs": [ { "contents": [ "Consider the closed subscheme $X \\times Y \\subset \\mathbf{A}^{2n}_k$", "where we use coordinates $x_1, \\ldots, x_n, y_1, \\ldots, y_n$. Then", "$X \\cap Y = X \\times Y \\cap V(x_1 - y_1, \\ldots, x_n - y_n)$.", "Let $t \\in X \\cap Y \\subset X \\times Y$ be a closed point.", "By Lemma \\ref{lemma-dimension-product-locally-algebraic}", "we have $\\dim_t(X \\times Y) = \\dim(X) + \\dim(Y)$.", "Thus $\\dim(\\mathcal{O}_{X \\times Y, t}) = r + s$ by", "Lemma \\ref{lemma-dimension-locally-algebraic}.", "By Algebra, Lemma \\ref{algebra-lemma-one-equation}", "we conclude that", "$$", "\\dim(\\mathcal{O}_{X \\cap Y, t}) =", "\\dim(\\mathcal{O}_{X \\times Y, t}/(x_1 - y_1, \\ldots, x_n - y_n)) \\geq", "r + s - n", "$$", "This implies the result by Lemma \\ref{lemma-dimension-locally-algebraic}." ], "refs": [ "varieties-lemma-dimension-product-locally-algebraic", "varieties-lemma-dimension-locally-algebraic", "varieties-lemma-dimension-locally-algebraic" ], "ref_ids": [ 10991, 10989, 10989 ] } ], "ref_ids": [] }, { "id": 11035, "type": "theorem", "label": "varieties-lemma-intersection-in-projective-space", "categories": [ "varieties" ], "title": "varieties-lemma-intersection-in-projective-space", "contents": [ "Let $k$ be a field and $n \\geq 0$. Let $X, Y \\subset \\mathbf{P}^n_k$", "be nonempty closed subsets. If $\\dim(X) = r$ and $\\dim(Y) = s$ and", "$r + s \\geq n$, then $X \\cap Y$ is nonempty and", "$\\dim(X \\cap Y) \\geq r + s - n$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\mathbf{A}^n = \\Spec(k[x_0, \\ldots, x_n])$ and", "$\\mathbf{P}^n = \\text{Proj}(k[T_0, \\ldots, T_n])$.", "Consider the morphism", "$\\pi : \\mathbf{A}^{n + 1} \\setminus \\{0\\} \\to \\mathbf{P}^n$", "which sends $(x_0, \\ldots, x_n)$ to the point $[x_0 : \\ldots : x_n]$.", "More precisely, it is the morphism associated to the pair", "$(\\mathcal{O}_{\\mathbf{A}^{n + 1} \\setminus \\{0\\}}, (x_0, \\ldots, x_n))$,", "see Constructions, Lemma \\ref{constructions-lemma-projective-space}.", "Over the standard affine open $D_+(T_i)$ we get the morphism", "associated to the ring map", "$$", "k\\left[\\frac{T_0}{T_i}, \\ldots, \\frac{T_n}{T_i}\\right]", "\\longrightarrow", "k\\left[T_0, \\ldots, T_n, \\frac{1}{T_i}\\right] \\cong", "k\\left[\\frac{T_0}{T_i}, \\ldots, \\frac{T_n}{T_i}\\right]", "\\left[T_i, \\frac{1}{T_i}\\right]", "$$", "which is surjective and smooth of relative dimension $1$", "with irreducible fibres (details omitted).", "Hence $\\pi^{-1}(X)$ and $\\pi^{-1}(Y)$ are nonempty closed subsets of", "dimension $r + 1$ and $s + 1$. Choose an irreducible component", "$V \\subset \\pi^{-1}(X)$ of dimension $r + 1$ and an", "irreducible component $W \\subset \\pi^{-1}(Y)$ of dimension $s + 1$.", "Observe that this implies $V$ and $W$ contain every fibre of $\\pi$", "they meet (since $\\pi$ has irreducible fibres of dimension $1$", "and since Lemma \\ref{lemma-dimension-fibres-locally-algebraic}", "says the fibres of $V \\to \\pi(V)$ and $W \\to \\pi(W)$ have dimension $\\geq 1$).", "Let $\\overline{V}$ and $\\overline{W}$ be the closure of", "$V$ and $W$ in $\\mathbf{A}^{n + 1}$. Since $0 \\in \\mathbf{A}^{n + 1}$", "is in the closure of every fibre of $\\pi$ we see that", "$0 \\in \\overline{V} \\cap \\overline{W}$. By", "Lemma \\ref{lemma-intersection-in-affine-space}", "we have $\\dim(\\overline{V} \\cap \\overline{W}) \\geq r + s - n + 1$.", "Arguing as above using Lemma \\ref{lemma-dimension-fibres-locally-algebraic}", "again, we conclude that $\\pi(V \\cap W) \\subset X \\cap Y$", "has dimension at least $r + s - n$ as desired." ], "refs": [ "constructions-lemma-projective-space", "varieties-lemma-dimension-fibres-locally-algebraic", "varieties-lemma-intersection-in-affine-space", "varieties-lemma-dimension-fibres-locally-algebraic" ], "ref_ids": [ 12621, 10990, 11034, 10990 ] } ], "ref_ids": [] }, { "id": 11036, "type": "theorem", "label": "varieties-lemma-equation-codim-1-in-projective-space", "categories": [ "varieties" ], "title": "varieties-lemma-equation-codim-1-in-projective-space", "contents": [ "Let $k$ be a field. Let $Z \\subset \\mathbf{P}^n_k$ be a closed subscheme", "which has no embedded points such that every irreducible component", "of $Z$ has dimension $n - 1$. Then the ideal $I(Z) \\subset k[T_0, \\ldots, T_n]$", "corresponding to $Z$ is principal." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Divisors, Lemma \\ref{divisors-lemma-equation-codim-1-in-projective-space}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11037, "type": "theorem", "label": "varieties-lemma-hyperplane", "categories": [ "varieties" ], "title": "varieties-lemma-hyperplane", "contents": [ "Let $k$ be a field. Let $n \\geq 1$.", "Let $i : H \\to \\mathbf{P}^n_k$ be a hyperplane.", "Then there exists an isomorphism", "$$", "\\varphi : \\mathbf{P}^{n - 1}_k \\longrightarrow H", "$$", "such that $i^*\\mathcal{O}(1)$ pulls back to $\\mathcal{O}(1)$." ], "refs": [], "proofs": [ { "contents": [ "We have $\\mathbf{P}^n_k = \\text{Proj}(k[T_0, \\ldots, T_n])$.", "The section $s$ corresponds to a homogeneous form in $T_0, \\ldots, T_n$", "of degree $1$, see", "Cohomology of Schemes, Section", "\\ref{coherent-section-cohomology-projective-space}.", "Say $s = \\sum a_i T_i$.", "Constructions, Lemma \\ref{constructions-lemma-closed-in-projective-space}", "gives that", "$H = \\text{Proj}(k[T_0, \\ldots, T_n]/I)$ for the graded ideal $I$", "defined by setting $I_d$ equal to the kernel of the map", "$\\Gamma(\\mathbf{P}^n_k, \\mathcal{O}(d)) \\to \\Gamma(H, i^*\\mathcal{O}(d))$.", "By our construction of $Z(s)$ in Divisors, ", "Definition \\ref{divisors-definition-zero-scheme-s}", "we see that on $D_{+}(T_j)$ the ideal of $H$ is generated by", "$\\sum a_i T_i/T_j$ in the polynomial ring", "$k[T_0/T_j, \\ldots, T_n/T_j]$. Thus it is clear that $I$ is the ideal", "generated by $\\sum a_i T_i$. Note that", "$$", "k[T_0, \\ldots, T_n]/I = k[T_0, \\ldots, T_n]/(\\sum a_i T_i) \\cong", "k[S_0, \\ldots, S_{n - 1}]", "$$", "as graded rings. For example, if $a_n \\not = 0$, then mapping", "$S_i$ equal to the class of $T_i$ works. We obtain the desired isomorphism", "by functoriality of $\\text{Proj}$.", "Equality of twists of structure sheaves follows for example from", "Constructions, Lemma", "\\ref{constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj}." ], "refs": [ "constructions-lemma-closed-in-projective-space", "divisors-definition-zero-scheme-s", "constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj" ], "ref_ids": [ 12625, 8094, 12612 ] } ], "ref_ids": [] }, { "id": 11038, "type": "theorem", "label": "varieties-lemma-exact-sequence-induction", "categories": [ "varieties" ], "title": "varieties-lemma-exact-sequence-induction", "contents": [ "Let $k$ be an infinite field. Let $n \\geq 1$.", "Let $\\mathcal{F}$ be a coherent module on $\\mathbf{P}^n_k$.", "Then there exist a nonzero section", "$s \\in \\Gamma(\\mathbf{P}^n_k, \\mathcal{O}(1))$", "and a short exact sequence", "$$", "0 \\to \\mathcal{F}(-1) \\to \\mathcal{F} \\to i_*\\mathcal{G} \\to 0", "$$", "where $i : H \\to \\mathbf{P}^n_k$ is the hyperplane $H$ associated to $s$", "and $\\mathcal{G} = i^*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "The map $\\mathcal{F}(-1) \\to \\mathcal{F}$ comes from", "Constructions, Equation (\\ref{constructions-equation-multiply-on-sheaf})", "with $n = 1$, $m = -1$ and the section $s$ of $\\mathcal{O}(1)$.", "Let's work out what this map looks like if we restrict it to", "$D_{+}(T_0)$. Write $D_{+}(T_0) = \\Spec(k[x_1, \\ldots, x_n])$", "with $x_i = T_i/T_0$. Identify $\\mathcal{O}(1)|_{D_{+}(T_0)}$ with", "$\\mathcal{O}$ using the section $T_0$. Hence if", "$s = \\sum a_iT_i$ then $s|_{D_{+}(T_0)} = a_0 + \\sum a_ix_i$", "with the identification chosen above. Furthermore, suppose", "$\\mathcal{F}|_{D_{+}(T_0)}$ corresponds to the finite", "$k[x_1, \\ldots, x_n]$-module $M$. Via the identification", "$\\mathcal{F}(-1) = \\mathcal{F} \\otimes \\mathcal{O}(-1)$", "and our chosen trivialization of $\\mathcal{O}(1)$ we see that", "$\\mathcal{F}(-1)$ corresponds to $M$ as well. Thus", "restricting $\\mathcal{F}(-1) \\to \\mathcal{F}$ to $D_{+}(T_0)$ gives", "the map", "$$", "M \\xrightarrow{a_0 + \\sum a_ix_i} M", "$$", "To see that the arrow is injective, it suffices to pick", "$a_0 + \\sum a_ix_i$ outside any of the associated primes of $M$, see", "Algebra, Lemma \\ref{algebra-lemma-ass-zero-divisors}. By", "Algebra, Lemma \\ref{algebra-lemma-finite-ass}", "the set $\\text{Ass}(M)$ of associated primes of $M$ is finite.", "Note that for $\\mathfrak p \\in \\text{Ass}(M)$ the intersection", "$\\mathfrak p \\cap \\{a_0 + \\sum a_i x_i\\}$ is a proper $k$-subvector space.", "We conclude that there is a finite family of proper sub vector spaces", "$V_1, \\ldots, V_m \\subset \\Gamma(\\mathbf{P}^n_k, \\mathcal{O}(1))$", "such that if we take $s$ outside of $\\bigcup V_i$, then multiplication", "by $s$ is injective over $D_{+}(T_0)$.", "Similarly for the restriction to $D_{+}(T_j)$ for $j = 1, \\ldots, n$.", "Since $k$ is infinite, a finite union of proper sub vector spaces", "is never equal to the whole space, hence we may choose $s$ such", "that the map is injective.", "The cokernel of $\\mathcal{F}(-1) \\to \\mathcal{F}$ is annihilated", "by $\\Im(s : \\mathcal{O}(-1) \\to \\mathcal{O})$ which is", "the ideal sheaf of $H$ by", "Divisors, Definition \\ref{divisors-definition-zero-scheme-s}.", "Hence we obtain $\\mathcal{G}$ on $H$ using", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-i-star-equivalence}." ], "refs": [ "algebra-lemma-ass-zero-divisors", "algebra-lemma-finite-ass", "divisors-definition-zero-scheme-s", "coherent-lemma-i-star-equivalence" ], "ref_ids": [ 704, 701, 8094, 3315 ] } ], "ref_ids": [] }, { "id": 11039, "type": "theorem", "label": "varieties-lemma-m-regular-extend-base-field", "categories": [ "varieties" ], "title": "varieties-lemma-m-regular-extend-base-field", "contents": [ "Let $k \\subset k'$ be an extension of fields. Let $n \\geq 0$.", "Let $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$.", "Let $\\mathcal{F}'$ be the pullback of $\\mathcal{F}$ to $\\mathbf{P}^n_{k'}$.", "Then $\\mathcal{F}$ is $m$-regular if and only if $\\mathcal{F}'$ is", "$m$-regular." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$$", "H^i(\\mathbf{P}^n_{k'}, \\mathcal{F}') =", "H^i(\\mathbf{P}^n_k, \\mathcal{F}) \\otimes_k k'", "$$", "by flat base change, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}." ], "refs": [ "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 3298 ] } ], "ref_ids": [] }, { "id": 11040, "type": "theorem", "label": "varieties-lemma-m-regular", "categories": [ "varieties" ], "title": "varieties-lemma-m-regular", "contents": [ "In the situation of Lemma \\ref{lemma-exact-sequence-induction},", "if $\\mathcal{F}$ is $m$-regular, then $\\mathcal{G}$ is $m$-regular", "on $H \\cong \\mathbf{P}^{n - 1}_k$." ], "refs": [ "varieties-lemma-exact-sequence-induction" ], "proofs": [ { "contents": [ "Recall that $H^i(\\mathbf{P}^n_k, i_*\\mathcal{G}) = H^i(H, \\mathcal{G})$ by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology}.", "Hence we see that for $i \\geq 1$ we get", "$$", "H^i(\\mathbf{P}^n_k, \\mathcal{F}(m - i)) \\to", "H^i(H, \\mathcal{G}(m - i)) \\to", "H^{i + 1}(\\mathbf{P}^n_k, \\mathcal{F}(m - 1 - i))", "$$", "by Remark \\ref{remark-exact-sequence-induction-cohomology}.", "The lemma follows." ], "refs": [ "coherent-lemma-relative-affine-cohomology", "varieties-remark-exact-sequence-induction-cohomology" ], "ref_ids": [ 3284, 11167 ] } ], "ref_ids": [ 11038 ] }, { "id": 11041, "type": "theorem", "label": "varieties-lemma-m-regular-up", "categories": [ "varieties" ], "title": "varieties-lemma-m-regular-up", "contents": [ "Let $k$ be a field. Let $n \\geq 0$.", "Let $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$.", "If $\\mathcal{F}$ is $m$-regular, then $\\mathcal{F}$ is", "$(m + 1)$-regular." ], "refs": [], "proofs": [ { "contents": [ "We prove this by induction on $n$. If $n = 0$ every sheaf is $m$-regular", "for all $m$ and there is nothing to prove. By", "Lemma \\ref{lemma-m-regular-extend-base-field} we may replace $k$", "by an infinite overfield and assume $k$ is infinite.", "Thus we may apply Lemma \\ref{lemma-exact-sequence-induction}.", "By Lemma \\ref{lemma-m-regular} we know that $\\mathcal{G}$ is", "$m$-regular. By induction on $n$ we see that $\\mathcal{G}$ is", "$(m + 1)$-regular. Considering the long exact cohomology sequence", "associated to the sequence", "$$", "0 \\to \\mathcal{F}(m - i) \\to \\mathcal{F}(m + 1 - i)", "\\to i_*\\mathcal{G}(m + 1 - i) \\to 0", "$$", "as in Remark \\ref{remark-exact-sequence-induction-cohomology}", "the reader easily deduces for $i \\geq 1$ the vanishing of", "$H^i(\\mathbf{P}^n_k, \\mathcal{F}(m + 1 - i))$ from the (known) vanishing of", "$H^i(\\mathbf{P}^n_k, \\mathcal{F}(m - i))$ and", "$H^i(\\mathbf{P}^n_k, \\mathcal{G}(m + 1 - i))$." ], "refs": [ "varieties-lemma-m-regular-extend-base-field", "varieties-lemma-exact-sequence-induction", "varieties-lemma-m-regular", "varieties-remark-exact-sequence-induction-cohomology" ], "ref_ids": [ 11039, 11038, 11040, 11167 ] } ], "ref_ids": [] }, { "id": 11042, "type": "theorem", "label": "varieties-lemma-m-regular-multiply", "categories": [ "varieties" ], "title": "varieties-lemma-m-regular-multiply", "contents": [ "Let $k$ be a field. Let $n \\geq 0$.", "Let $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$.", "If $\\mathcal{F}$ is $m$-regular, then the multiplication map", "$$", "H^0(\\mathbf{P}^n_k, \\mathcal{F}(m)) \\otimes_k", "H^0(\\mathbf{P}^n_k, \\mathcal{O}(1))", "\\longrightarrow", "H^0(\\mathbf{P}^n_k, \\mathcal{F}(m + 1))", "$$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $k \\subset k'$ be an extension of fields. Let $\\mathcal{F}'$", "be as in Lemma \\ref{lemma-m-regular-extend-base-field}. By", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}", "the base change of the linear map of the lemma to $k'$ is the", "same linear map for the sheaf $\\mathcal{F}'$. Since $k \\to k'$", "is faithfully flat it suffices to prove the lemma over $k'$, i.e.,", "we may assume $k$ is infinite.", "\\medskip\\noindent", "Assume $k$ is infinite. We prove the lemma by induction on $n$.", "The case $n = 0$ is trivial as $\\mathcal{O}(1) \\cong \\mathcal{O}$", "is generated by $T_0$. For $n > 0$ apply", "Lemma \\ref{lemma-exact-sequence-induction}", "and tensor the sequence by $\\mathcal{O}(m + 1)$ to get", "$$", "0 \\to \\mathcal{F}(m) \\xrightarrow{s} \\mathcal{F}(m + 1) \\to", "i_*\\mathcal{G}(m + 1) \\to 0", "$$", "see Remark \\ref{remark-exact-sequence-induction-cohomology}.", "Let $t \\in H^0(\\mathbf{P}^n_k, \\mathcal{F}(m + 1))$.", "By induction the image $\\overline{t} \\in H^0(H, \\mathcal{G}(m + 1))$", "is the image of $\\sum g_i \\otimes \\overline{s}_i$ with", "$\\overline{s}_i \\in \\Gamma(H, \\mathcal{O}(1))$ and", "$g_i \\in H^0(H, \\mathcal{G}(m))$. Since $\\mathcal{F}$ is $m$-regular", "we have $H^1(\\mathbf{P}^n_k, \\mathcal{F}(m - 1)) = 0$, hence long", "exact cohomology sequence associated to the short exact sequence", "$$", "0 \\to \\mathcal{F}(m - 1) \\xrightarrow{s} \\mathcal{F}(m) \\to", "i_*\\mathcal{G}(m) \\to 0", "$$", "shows we can lift $g_i$ to $f_i \\in H^0(\\mathbf{P}^n_k, \\mathcal{F}(m))$.", "We can also lift $\\overline{s}_i$ to", "$s_i \\in H^0(\\mathbf{P}^n_k, \\mathcal{O}(1))$ (see proof of", "Lemma \\ref{lemma-hyperplane} for example). After substracting the", "image of $\\sum f_i \\otimes s_i$ from $t$ we see that we may assume", "$\\overline{t} = 0$. But this exactly means that $t$ is the", "image of $f \\otimes s$ for some $f \\in H^0(\\mathbf{P}^n_k, \\mathcal{F}(m))$", "as desired." ], "refs": [ "varieties-lemma-m-regular-extend-base-field", "coherent-lemma-flat-base-change-cohomology", "varieties-lemma-exact-sequence-induction", "varieties-remark-exact-sequence-induction-cohomology", "varieties-lemma-hyperplane" ], "ref_ids": [ 11039, 3298, 11038, 11167, 11037 ] } ], "ref_ids": [] }, { "id": 11043, "type": "theorem", "label": "varieties-lemma-m-regular-globally-generated", "categories": [ "varieties" ], "title": "varieties-lemma-m-regular-globally-generated", "contents": [ "Let $k$ be a field. Let $n \\geq 0$.", "Let $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$.", "If $\\mathcal{F}$ is $m$-regular, then $\\mathcal{F}(m)$ is", "globally generated." ], "refs": [], "proofs": [ { "contents": [ "For all $d \\gg 0$ the sheaf $\\mathcal{F}(d)$ is globally generated.", "This follows for example from the first part of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-projective}.", "Pick $d \\geq m$ such that $\\mathcal{F}(d)$ is globally generated.", "Choose a basis $f_1, \\ldots, f_r \\in H^0(\\mathbf{P}^n_k, \\mathcal{F})$.", "By Lemma \\ref{lemma-m-regular-multiply} every element", "$f \\in H^0(\\mathbf{P}^n_k, \\mathcal{F}(d))$ can be written as", "$f = \\sum P_if_i$ for some $P_i \\in k[T_0, \\ldots, T_n]$ homogeneous", "of degree $d - m$. Since the sections $f$ generate $\\mathcal{F}(d)$", "it follows that the sections $f_i$ generate $\\mathcal{F}(m)$." ], "refs": [ "coherent-lemma-coherent-projective", "varieties-lemma-m-regular-multiply" ], "ref_ids": [ 3338, 11042 ] } ], "ref_ids": [] }, { "id": 11044, "type": "theorem", "label": "varieties-lemma-hilbert-polynomial", "categories": [ "varieties" ], "title": "varieties-lemma-hilbert-polynomial", "contents": [ "Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent sheaf", "on $\\mathbf{P}^n_k$. The function", "$$", "d \\longmapsto \\chi(\\mathbf{P}^n_k, \\mathcal{F}(d))", "$$", "is a polynomial." ], "refs": [], "proofs": [ { "contents": [ "We prove this by induction on $n$. If $n = 0$, then", "$\\mathbf{P}^n_k = \\Spec(k)$ and $\\mathcal{F}(d) = \\mathcal{F}$.", "Hence in this case the function is constant, i.e., a polynomial", "of degree $0$. Assume $n > 0$. By", "Lemma \\ref{lemma-euler-characteristic-extend-base-field}", "we may assume $k$ is infinite. Apply", "Lemma \\ref{lemma-exact-sequence-induction}.", "Applying Lemma \\ref{lemma-euler-characteristic-additive}", "to the twisted sequences", "$0 \\to \\mathcal{F}(d - 1) \\to \\mathcal{F}(d) \\to i_*\\mathcal{G}(d) \\to 0$", "we obtain", "$$", "\\chi(\\mathbf{P}^n_k, \\mathcal{F}(d)) -", "\\chi(\\mathbf{P}^n_k, \\mathcal{F}(d - 1)) =", "\\chi(H, \\mathcal{G}(d))", "$$", "See Remark \\ref{remark-exact-sequence-induction-cohomology}.", "Since $H \\cong \\mathbf{P}^{n - 1}_k$", "by induction the right hand side is a polynomial.", "The lemma is finished by noting that any function", "$f : \\mathbf{Z} \\to \\mathbf{Z}$ with the property that the map", "$d \\mapsto f(d) - f(d - 1)$ is a polynomial, is itself a polynomial.", "We omit the proof of this fact (hint: compare with", "Algebra, Lemma \\ref{algebra-lemma-numerical-polynomial})." ], "refs": [ "varieties-lemma-euler-characteristic-extend-base-field", "varieties-lemma-exact-sequence-induction", "varieties-lemma-euler-characteristic-additive", "varieties-remark-exact-sequence-induction-cohomology", "algebra-lemma-numerical-polynomial" ], "ref_ids": [ 11031, 11038, 11029, 11167, 670 ] } ], "ref_ids": [] }, { "id": 11045, "type": "theorem", "label": "varieties-lemma-hilbert-polynomial-H0", "categories": [ "varieties" ], "title": "varieties-lemma-hilbert-polynomial-H0", "contents": [ "Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent sheaf", "on $\\mathbf{P}^n_k$ with Hilbert polynomial $P \\in \\mathbf{Q}[t]$.", "Then", "$$", "P(d) = \\dim_k H^0(\\mathbf{P}^n_k, \\mathcal{F}(d))", "$$", "for all $d \\gg 0$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the vanishing of cohomology of high enough twists", "of $\\mathcal{F}$. See", "Cohomology of Schemes,", "Lemma \\ref{coherent-lemma-coherent-projective}." ], "refs": [ "coherent-lemma-coherent-projective" ], "ref_ids": [ 3338 ] } ], "ref_ids": [] }, { "id": 11046, "type": "theorem", "label": "varieties-lemma-bound-quotients-free", "categories": [ "varieties" ], "title": "varieties-lemma-bound-quotients-free", "contents": [ "Let $k$ be a field. Let $n \\geq 0$. Let $r \\geq 1$. Let $P \\in \\mathbf{Q}[t]$.", "There exists an integer $m$ depending on $n$, $r$, and $P$", "with the following property: if", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{O}^{\\oplus r} \\to \\mathcal{F} \\to 0", "$$", "is a short exact sequence of coherent sheaves on $\\mathbf{P}^n_k$", "and $\\mathcal{F}$ has Hilbert polynomial $P$, then", "$\\mathcal{K}$ is $m$-regular." ], "refs": [], "proofs": [ { "contents": [ "We prove this by induction on $n$. If $n = 0$, then", "$\\mathbf{P}^n_k = \\Spec(k)$ and any coherent module is $0$-regular", "and any surjective map is surjective on global sections.", "Assume $n > 0$. Consider an exact sequence as in the lemma.", "Let $P' \\in \\mathbf{Q}[t]$ be the polynomial", "$P'(t) = P(t) - P(t - 1)$. Let $m'$ be the integer", "which works for $n - 1$, $r$, and $P'$.", "By Lemmas \\ref{lemma-m-regular-extend-base-field} and", "\\ref{lemma-euler-characteristic-extend-base-field}", "we may replace $k$ by a field extension, hence we may assume", "$k$ is infinite. Apply ", "Lemma \\ref{lemma-exact-sequence-induction}", "to the coherent sheaf $\\mathcal{F}$.", "The Hilbert polynomial of $\\mathcal{F}' = i^*\\mathcal{F}$", "is $P'$ (see proof of Lemma \\ref{lemma-hilbert-polynomial}).", "Since $i^*$ is right exact we see that $\\mathcal{F}'$ is", "a quotient of $\\mathcal{O}_H^{\\oplus r} = i^*\\mathcal{O}^{\\oplus r}$.", "Thus the induction hypothesis applies to $\\mathcal{F}'$ on", "$H \\cong \\mathbf{P}^{n - 1}_k$ (Lemma \\ref{lemma-hyperplane}).", "Note that the map $\\mathcal{K}(-1) \\to \\mathcal{K}$ is injective", "as $\\mathcal{K} \\subset \\mathcal{O}^{\\oplus r}$ and has", "cokernel $i_*\\mathcal{H}$ where $\\mathcal{H} = i^*\\mathcal{K}$.", "By the snake lemma (Homology, Lemma \\ref{homology-lemma-snake})", "we obtain a commutative diagram with exact columns and rows", "$$", "\\xymatrix{", "& 0 \\ar[d] & 0 \\ar[d] & 0 \\ar[d] \\\\", "0 \\ar[r] &", "\\mathcal{K}(-1) \\ar[r] \\ar[d] &", "\\mathcal{O}^{\\oplus r}(-1) \\ar[r] \\ar[d] &", "\\mathcal{F}(-1) \\ar[d] \\ar[r] & 0 \\\\", "0 \\ar[r] &", "\\mathcal{K} \\ar[r] \\ar[d] &", "\\mathcal{O}^{\\oplus r} \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[d] \\ar[r] & 0\\\\", "0 \\ar[r] &", "i_*\\mathcal{H} \\ar[r] \\ar[d] &", "i_*\\mathcal{O}_H^{\\oplus r} \\ar[r] \\ar[d] &", "i_*\\mathcal{F}' \\ar[r] \\ar[d] & 0 \\\\", "& 0 & 0 & 0", "}", "$$", "Thus the induction hypothesis applies to the exact sequence", "$0 \\to \\mathcal{H} \\to \\mathcal{O}_H^{\\oplus r} \\to \\mathcal{F}' \\to 0$", "on $H \\cong \\mathbf{P}^{n - 1}_k$ (Lemma \\ref{lemma-hyperplane})", "and $\\mathcal{H}$ is $m'$-regular. Recall that this implies that", "$\\mathcal{H}$ is $d$-regular for all $d \\geq m'$", "(Lemma \\ref{lemma-m-regular-up}).", "\\medskip\\noindent", "Let $i \\geq 2$ and $d \\geq m'$. It follows from the long exact", "cohomology sequence associated to the left column of the diagram", "above and the vanishing of $H^{i - 1}(H, \\mathcal{H}(d))$", "that the map", "$$", "H^i(\\mathbf{P}^n_k, \\mathcal{K}(d - 1))", "\\longrightarrow", "H^i(\\mathbf{P}^n_k, \\mathcal{K}(d))", "$$", "is injective. As these groups are zero for $d \\gg 0$", "(Cohomology of Schemes,", "Lemma \\ref{coherent-lemma-coherent-projective})", "we conclude $H^i(\\mathbf{P}^n_k, \\mathcal{K}(d))$ are zero", "for all $d \\geq m'$ and $i \\geq 2$.", "\\medskip\\noindent", "We still have to control $H^1$. First we observe that all the maps", "$$", "H^1(\\mathbf{P}^n_k, \\mathcal{K}(m' - 1)) \\to", "H^1(\\mathbf{P}^n_k, \\mathcal{K}(m')) \\to", "H^1(\\mathbf{P}^n_k, \\mathcal{K}(m' + 1)) \\to \\ldots", "$$", "are surjective by the vanishing of $H^1(H, \\mathcal{H}(d))$ for $d \\geq m'$.", "Suppose $d > m'$ is such that", "$$", "H^1(\\mathbf{P}^n_k, \\mathcal{K}(d - 1))", "\\longrightarrow", "H^1(\\mathbf{P}^n_k, \\mathcal{K}(d))", "$$", "is injective. Then", "$H^0(\\mathbf{P}^n_k, \\mathcal{K}(d)) \\to H^0(H, \\mathcal{H}(d))$", "is surjective. Consider the commutative diagram", "$$", "\\xymatrix{", "H^0(\\mathbf{P}^n_k, \\mathcal{K}(d)) \\otimes_k", "H^0(\\mathbf{P}^n_k, \\mathcal{O}(1))", "\\ar[r] \\ar[d] &", "H^0(\\mathbf{P}^n_k, \\mathcal{K}(d + 1)) \\ar[d] \\\\", "H^0(H, \\mathcal{H}(d)) \\otimes_k", "H^0(H, \\mathcal{O}_H(1))", "\\ar[r] &", "H^0(H, \\mathcal{H}(d + 1))", "}", "$$", "By Lemma \\ref{lemma-m-regular-multiply}", "we see that the bottom horizontal arrow is surjective.", "Hence the right vertical arrow is surjective. We conclude that", "$$", "H^1(\\mathbf{P}^n_k, \\mathcal{K}(d))", "\\longrightarrow", "H^1(\\mathbf{P}^n_k, \\mathcal{K}(d + 1))", "$$", "is injective. By induction we see that", "$$", "H^1(\\mathbf{P}^n_k, \\mathcal{K}(d - 1)) \\to", "H^1(\\mathbf{P}^n_k, \\mathcal{K}(d)) \\to", "H^1(\\mathbf{P}^n_k, \\mathcal{K}(d + 1)) \\to \\ldots", "$$", "are all injective and we conclude that", "$H^1(\\mathbf{P}^n_k, \\mathcal{K}(d - 1)) = 0$", "because of the eventual vanishing of these groups. Thus the dimensions", "of the groups $H^1(\\mathbf{P}^n_k, \\mathcal{K}(d))$ for $d \\geq m'$", "are strictly decreasing until they become zero. It follows that the", "regularity of $\\mathcal{K}$ is bounded by", "$m' + \\dim_k H^1(\\mathbf{P}^n_k, \\mathcal{K}(m'))$.", "On the other hand, by the vanishing of the higher cohomology groups", "we have", "$$", "\\dim_k H^1(\\mathbf{P}^n_k, \\mathcal{K}(m')) = ", "- \\chi(\\mathbf{P}^n_k, \\mathcal{K}(m')) +", "\\dim_k H^0(\\mathbf{P}^n_k, \\mathcal{K}(m'))", "$$", "Note that the $H^0$ has dimension bounded by the dimension of", "$H^0(\\mathbf{P}^n_k, \\mathcal{O}^{\\oplus r}(m'))$ which is", "at most $r{n + m' \\choose n}$ if $m' > 0$ and zero if not.", "Finally, the term $\\chi(\\mathbf{P}^n_k, \\mathcal{K}(m'))$ is equal", "to $r{n + m' \\choose n} - P(m')$. This gives a bound of the", "desired type finishing the proof of the lemma." ], "refs": [ "varieties-lemma-m-regular-extend-base-field", "varieties-lemma-euler-characteristic-extend-base-field", "varieties-lemma-exact-sequence-induction", "varieties-lemma-hilbert-polynomial", "varieties-lemma-hyperplane", "homology-lemma-snake", "varieties-lemma-hyperplane", "varieties-lemma-m-regular-up", "coherent-lemma-coherent-projective", "varieties-lemma-m-regular-multiply" ], "ref_ids": [ 11039, 11031, 11038, 11044, 11037, 12027, 11037, 11041, 3338, 11042 ] } ], "ref_ids": [] }, { "id": 11047, "type": "theorem", "label": "varieties-lemma-frobenius-endomorphism-identity", "categories": [ "varieties" ], "title": "varieties-lemma-frobenius-endomorphism-identity", "contents": [ "Let $p > 0$ be a prime number.", "Let $f : X \\to Y$ be a morphism of schemes in characteristic $p$.", "Then the diagram", "$$", "\\xymatrix{", "X \\ar[d]_f \\ar[r]_{F_X} & X \\ar[d]^f \\\\", "Y \\ar[r]^{F_Y} & Y", "}", "$$", "commutes." ], "refs": [], "proofs": [ { "contents": [ "This follows from the following trivial algebraic fact: if $\\varphi : A \\to B$", "is a homomorphism of rings of characteristic $p$, then", "$\\varphi(a^p) = \\varphi(a)^p$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11048, "type": "theorem", "label": "varieties-lemma-frobenius", "categories": [ "varieties" ], "title": "varieties-lemma-frobenius", "contents": [ "Let $p > 0$ be a prime number. Let $X$ be a scheme in characteristic $p$.", "Then the absolute frobenius $F_X : X \\to X$", "is a universal homeomorphism, is integral, and", "induces purely inseparable residue field extensions." ], "refs": [], "proofs": [ { "contents": [ "This follows from the corresponding results for the frobenius endomorphism", "$F_A : A \\to A$ of a ring $A$ of characteristic $p > 0$. See the", "discussion in Algebra, Section \\ref{algebra-section-universal-homeomorphism},", "for example Lemma \\ref{algebra-lemma-p-ring-map}." ], "refs": [ "algebra-lemma-p-ring-map" ], "ref_ids": [ 582 ] } ], "ref_ids": [] }, { "id": 11049, "type": "theorem", "label": "varieties-lemma-relative-frobenius-endomorphism-identity", "categories": [ "varieties" ], "title": "varieties-lemma-relative-frobenius-endomorphism-identity", "contents": [ "Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$.", "Let $f : X \\to Y$ be a morphism of schemes over $S$ .", "Then the diagram", "$$", "\\xymatrix{", "X \\ar[d]_f \\ar[r]_{F_{X/S}} & X^{(p)} \\ar[d]^{f^{(p)}} \\\\", "Y \\ar[r]^{F_{Y/S}} & Y^{(p)}", "}", "$$", "commutes." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-frobenius-endomorphism-identity}", "and the definitions." ], "refs": [ "varieties-lemma-frobenius-endomorphism-identity" ], "ref_ids": [ 11047 ] } ], "ref_ids": [] }, { "id": 11050, "type": "theorem", "label": "varieties-lemma-relative-frobenius", "categories": [ "varieties" ], "title": "varieties-lemma-relative-frobenius", "contents": [ "Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$.", "Let $X$ be a scheme over $S$.", "Then the relative frobenius $F_{X/S} : X \\to X^{(p)}$", "is a universal homeomorphism, is integral, and", "induces purely inseparable residue field extensions." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-frobenius} the morphisms $F_X : X \\to X$ and the base change", "$h : X^{(p)} \\to X$ of $F_S$ are universal homeomorphisms.", "Since $h \\circ F_{X/S} = F_X$ we conclude that $F_{X/S}$ is a universal", "homeomorphism. By Morphisms, Lemmas", "\\ref{morphisms-lemma-universal-homeomorphism} and", "\\ref{morphisms-lemma-universally-injective}", "we conclude that $F_{X/S}$ has the other properties as well." ], "refs": [ "varieties-lemma-frobenius", "morphisms-lemma-universal-homeomorphism", "morphisms-lemma-universally-injective" ], "ref_ids": [ 11048, 5454, 5167 ] } ], "ref_ids": [] }, { "id": 11051, "type": "theorem", "label": "varieties-lemma-relative-frobenius-omega", "categories": [ "varieties" ], "title": "varieties-lemma-relative-frobenius-omega", "contents": [ "Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$.", "Let $X$ be a scheme over $S$. Then $\\Omega_{X/S} = \\Omega_{X/X^{(p)}}$." ], "refs": [], "proofs": [ { "contents": [ "This translates into the following algebra fact.", "Let $A \\to B$ be a homomorphism of rings of characteristic $p$.", "Set $B' = B \\otimes_{A, F_A} A$ and consider the ring map", "$F_{B/A} : B' \\to B$, $b \\otimes a \\mapsto b^pa$.", "Then our assertion is that $\\Omega_{B/A} = \\Omega_{B/B'}$.", "This is true because $\\text{d}(b^pa) = 0$ if", "$\\text{d} : B \\to \\Omega_{B/A}$ is the universal derivation", "and hence $\\text{d}$ is a $B'$-derivation." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11052, "type": "theorem", "label": "varieties-lemma-relative-frobenius-finite", "categories": [ "varieties" ], "title": "varieties-lemma-relative-frobenius-finite", "contents": [ "Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$.", "Let $X$ be a scheme over $S$. If $X \\to S$ is locally of finite type,", "then $F_{X/S}$ is finite." ], "refs": [], "proofs": [ { "contents": [ "This translates into the following algebra fact.", "Let $A \\to B$ be a finite type homomorphism of rings of characteristic $p$.", "Set $B' = B \\otimes_{A, F_A} A$ and consider the ring map", "$F_{B/A} : B' \\to B$, $b \\otimes a \\mapsto b^pa$.", "Then our assertion is that $F_{B/A}$ is finite.", "Namely, if $x_1, \\ldots, x_n \\in B$ are generators over $A$,", "then $x_i$ is integral over $B'$ because $x_i^p = F_{B/A}(x_i \\otimes 1)$.", "Hence $F_{B/A} : B' \\to B$ is finite by", "Algebra, Lemma \\ref{algebra-lemma-characterize-finite-in-terms-of-integral}." ], "refs": [ "algebra-lemma-characterize-finite-in-terms-of-integral" ], "ref_ids": [ 484 ] } ], "ref_ids": [] }, { "id": 11053, "type": "theorem", "label": "varieties-lemma-geometrically-reduced-p", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-reduced-p", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$.", "Then $X$ is geometrically reduced if and only if $X^{(p)}$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "Consider the absolute frobenius $F_k : k \\to k$. Then $F_k(k) = k^p$", "in other words, $F_k : k \\to k$ is isomorphic to the embedding of", "$k$ into $k^{1/p}$. Thus the lemma follows from", "Lemma \\ref{lemma-geometrically-reduced}." ], "refs": [ "varieties-lemma-geometrically-reduced" ], "ref_ids": [ 10908 ] } ], "ref_ids": [] }, { "id": 11054, "type": "theorem", "label": "varieties-lemma-inseparable-deg-p-smooth", "categories": [ "varieties" ], "title": "varieties-lemma-inseparable-deg-p-smooth", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $X$ be a variety over $k$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X^{(p)}$ is reduced,", "\\item $X$ is geometrically reduced,", "\\item there is a nonempty open $U \\subset X$ smooth over $k$.", "\\end{enumerate}", "In this case $X^{(p)}$ is a variety over $k$ and $F_{X/k} : X \\to X^{(p)}$", "is a finite dominant morphism of degree $p^{\\dim(X)}$." ], "refs": [], "proofs": [ { "contents": [ "We have seen the equivalence of (1) and (2) in", "Lemma \\ref{lemma-geometrically-reduced-p}.", "We have seen that (2) implies (3) in", "Lemma \\ref{lemma-geometrically-reduced-dense-smooth-open}.", "If (3) holds, then $U$ is geometrically reduced", "(see for example Lemma \\ref{lemma-geometrically-regular-smooth})", "and hence $X$ is geometrically reduced by", "Lemma \\ref{lemma-generic-points-geometrically-reduced}.", "In this way we see that (1), (2), and (3) are equivalent.", "\\medskip\\noindent", "Assume (1), (2), and (3) hold. Since $F_{X/k}$ is a homeomorphism", "(Lemma \\ref{lemma-relative-frobenius}) we see that $X^{(p)}$ is a variety.", "Then $F_{X/k}$ is finite by Lemma \\ref{lemma-relative-frobenius-finite}.", "It is dominant as it is surjective. To compute the degree", "(Morphisms, Definition \\ref{morphisms-definition-degree})", "it suffices to compute the degree of $F_{U/k} : U \\to U^{(p)}$", "(as $F_{U/k} = F_{X/k}|_U$ by", "Lemma \\ref{lemma-relative-frobenius-endomorphism-identity}).", "After shrinking $U$ a bit we may assume there exists an", "\\'etale morphism $h : U \\to \\mathbf{A}^n_k$, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-etale-over-affine-space}.", "Of course $n = \\dim(U)$ because", "$\\mathbf{A}^n_k \\to \\Spec(k)$ is smooth of relative dimension $n$,", "the \\'etale morphism $h$ is smooth of relative dimension $0$, and", "$U \\to \\Spec(k)$ is smooth of relative dimension $\\dim(U)$", "and relative dimensions add up correctly", "(Morphisms, Lemma \\ref{morphisms-lemma-composition-relative-dimension-d}).", "Observe that $h$ is a generically finite dominant morphism", "of varieties, and hence $\\deg(h)$ is defined.", "By Lemma \\ref{lemma-relative-frobenius-endomorphism-identity}", "we have a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_{F_{X/k}} \\ar[d]_h & & X^{(p)} \\ar[d]^{h^{(p)}} \\\\", "\\mathbf{A}^n_k \\ar[rr]^{F_{\\mathbf{A}^n_k/k}} & &", "(\\mathbf{A}^n_k)^{(p)}", "}", "$$", "Since $h^{(p)}$ is a base change of $h$ it is \\'etale as well", "and it follows that $h^{(p)}$ is a generically finite dominant", "morphism of varieties as well. The degree of $h^{(p)}$ is the", "degree of the extension $k(X^{(p)})/k((\\mathbf{A}^n_k)^{(p)})$", "which is the same as the degree of the extension $k(X)/k(\\mathbf{A}^n_k)$", "because $h^{(p)}$ is the base change of $h$ (small detail omitted).", "By multiplicativity of degrees", "(Morphisms, Lemma \\ref{morphisms-lemma-degree-composition})", "it suffices to show that the degree of $F_{\\mathbf{A}^n_k/k}$", "is $p^n$. To see this observe that", "$(\\mathbf{A}^n_k)^{(p)} = \\mathbf{A}^n_k$ and that", "$F_{\\mathbf{A}^n_k/k}$ is given by the map sending the", "coordinates to their $p$th powers." ], "refs": [ "varieties-lemma-geometrically-reduced-p", "varieties-lemma-geometrically-reduced-dense-smooth-open", "varieties-lemma-geometrically-regular-smooth", "varieties-lemma-generic-points-geometrically-reduced", "varieties-lemma-relative-frobenius", "varieties-lemma-relative-frobenius-finite", "morphisms-definition-degree", "varieties-lemma-relative-frobenius-endomorphism-identity", "morphisms-lemma-smooth-etale-over-affine-space", "morphisms-lemma-composition-relative-dimension-d", "varieties-lemma-relative-frobenius-endomorphism-identity", "morphisms-lemma-degree-composition" ], "ref_ids": [ 11053, 11008, 10962, 10912, 11050, 11052, 5587, 11049, 5377, 5285, 11049, 5492 ] } ], "ref_ids": [] }, { "id": 11055, "type": "theorem", "label": "varieties-lemma-glue-valuation-ring", "categories": [ "varieties" ], "title": "varieties-lemma-glue-valuation-ring", "contents": [ "In Situation \\ref{situation-glue} assume that $B$ is a valuation ring.", "Then for every unit $u$ of $A$ either $u \\in R$ or $u^{-1} \\in R$." ], "refs": [], "proofs": [ { "contents": [ "Namely, if the image $c$ of $u$ in $K$ is in $B$, then $u \\in R$.", "Otherwise, $c^{-1} \\in B$", "(Algebra, Lemma \\ref{algebra-lemma-valuation-ring-x-or-x-inverse})", "and $u^{-1} \\in R$." ], "refs": [ "algebra-lemma-valuation-ring-x-or-x-inverse" ], "ref_ids": [ 609 ] } ], "ref_ids": [] }, { "id": 11056, "type": "theorem", "label": "varieties-lemma-glue-separated", "categories": [ "varieties" ], "title": "varieties-lemma-glue-separated", "contents": [ "In Situation \\ref{situation-glue} assume $A$ is a Noetherian", "ring of dimension $1$. The following are equivalent", "\\begin{enumerate}", "\\item $A \\otimes B \\to K$ is not surjective,", "\\item there exists a discrete valuation ring $\\mathcal{O} \\subset K$", "containing both $A$ and $B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (2) implies (1). On the other hand, if $A \\otimes B \\to K$", "is not surjective, then the image $C \\subset K$ is not a field hence", "$C$ has a nonzero maximal ideal $\\mathfrak m$. Choose a valuation ring", "$\\mathcal{O} \\subset K$ dominating $C_\\mathfrak m$. By", "Algebra, Lemma \\ref{algebra-lemma-krull-akizuki} applied to", "$A \\subset \\mathcal{O}$ the ring $\\mathcal{O}$ is Noetherian.", "Hence $\\mathcal{O}$ is a discrete valuation ring by", "Algebra, Lemma \\ref{algebra-lemma-valuation-ring-Noetherian-discrete}." ], "refs": [ "algebra-lemma-krull-akizuki", "algebra-lemma-valuation-ring-Noetherian-discrete" ], "ref_ids": [ 1027, 623 ] } ], "ref_ids": [] }, { "id": 11057, "type": "theorem", "label": "varieties-lemma-semi-local", "categories": [ "varieties" ], "title": "varieties-lemma-semi-local", "contents": [ "In Situation \\ref{situation-glue} assume", "\\begin{enumerate}", "\\item $A$ is a Noetherian semi-local domain of dimension $1$,", "\\item $B$ is a discrete valuation ring,", "\\end{enumerate}", "Then we have the following two possibilities", "\\begin{enumerate}", "\\item[(a)] If $A^*$ is not contained in $R$, then", "$\\Spec(A) \\to \\Spec(R)$ and $\\Spec(B) \\to \\Spec(R)$", "are open immersions covering $\\Spec(R)$ and $K = A \\otimes_R B$.", "\\item[(b)] If $A^*$ is contained in $R$, then $B$ dominates one of", "the local rings of $A$ at a maximal ideal and $A \\otimes B \\to K$", "is not surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assumption (a) implies there is a unit $u$ of $A$ whose image in $K$ lies in", "the maximal ideal of $B$. Then $u$ is a nonzerodivisor of $R$ and for", "every $a \\in A$ there exists an $n$ such that $u^n a \\in R$. It follows", "that $A = R_u$.", "\\medskip\\noindent", "Let $\\mathfrak m_A$ be the Jacobson radical of $A$. Let $x \\in \\mathfrak m_A$", "be a nonzero element. Since $\\dim(A) = 1$ we see that $K = A_x$.", "After replacing $x$ by $x^n u^m$ for some $n \\geq 1$ and", "$m \\in \\mathbf{Z}$ we may assume $x$ maps to a unit of $B$.", "We see that for every $b \\in B$ we have that $x^nb$", "in the image of $R$ for some $n$. Thus $B = R_x$.", "\\medskip\\noindent", "Let $z \\in R$. If $z \\not \\in \\mathfrak m_A$ and $z$ does not map to", "an element of $\\mathfrak m_B$, then $z$ is invertible.", "Thus $x + u$ is invertible in $R$. Hence $\\Spec(R) = D(x) \\cup D(u)$.", "We have seen above that $D(u) = \\Spec(A)$ and $D(x) = \\Spec(B)$.", "\\medskip\\noindent", "Case (b). If $x \\in \\mathfrak m_A$, then $1 + x$ is a unit and", "hence $1 + x \\in R$, i.e, $x \\in R$. Thus we see that", "$\\mathfrak m_A \\subset R \\subset A$. In fact, in this case", "$A$ is integral over $R$. Namely, write", "$A/\\mathfrak m_A = \\kappa_1 \\times \\ldots \\times \\kappa_n$", "as a product of fields. Say $x = (c_1, \\ldots, c_r, 0, \\ldots, 0)$", "is an element with $c_i \\not = 0$. Then", "$$", "x^2 - x(c_1, \\ldots, c_r, 1, \\ldots, 1) = 0", "$$", "Since $R$ contains all units we see that $A/\\mathfrak m_A$ is", "integral over the image of $R$ in it, and hence $A$ is integral over $R$.", "It follows that $R \\subset A \\subset B$ as $B$ is integrally closed.", "Moreover, if $x \\in \\mathfrak m_A$ is nonzero, then", "$K = A_x = \\bigcup x^{-n}A = \\bigcup x^{-n}R$.", "Hence $x^{-1} \\not \\in B$, i.e., $x \\in \\mathfrak m_B$.", "We conclude $\\mathfrak m_A \\subset \\mathfrak m_B$.", "Thus $A \\cap \\mathfrak m_B$ is a maximal ideal of $A$ thereby", "finishing the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11058, "type": "theorem", "label": "varieties-lemma-semi-local-dimension-one-conductor", "categories": [ "varieties" ], "title": "varieties-lemma-semi-local-dimension-one-conductor", "contents": [ "Let $B$ be a semi-local Noetherian domain of dimension $1$.", "Let $B'$ be the integral closure of $B$ in its fraction field.", "Then $B'$ is a semi-local Dedekind domain.", "Let $x$ be a nonzero element of the Jacobson radical of $B'$.", "Then for every $y \\in B'$ there exists an $n$ such that", "$x^n y \\in B$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak m_B$ be the Jacobson radical of $B$.", "The structure of $B'$ results from", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-Dedekind}.", "Given $x, y \\in B'$ as in the statement of the lemma consider", "the subring $B \\subset A \\subset B'$ generated by $x$ and $y$.", "Then $A$ is finite over $B$ (Algebra, Lemma", "\\ref{algebra-lemma-characterize-finite-in-terms-of-integral}).", "Since the fraction fields of $B$ and $A$ are the same we see", "that the finite module $A/B$ is supported on the set of", "closed points of $B$. Thus $\\mathfrak m_B^n A \\subset B$ for", "a suitable $n$. Moreover, $\\Spec(B') \\to \\Spec(A)$ is", "surjective (Algebra, Lemma \\ref{algebra-lemma-integral-overring-surjective}),", "hence $A$ is semi-local as well. It also follows that", "$x$ is in the Jacobson radical $\\mathfrak m_A$ of $A$. Note that", "$\\mathfrak m_A = \\sqrt{\\mathfrak m_B A}$.", "Thus $x^m y \\in \\mathfrak m_B A$ for some $m$.", "Then $x^{nm} y \\in B$." ], "refs": [ "algebra-lemma-integral-closure-Dedekind", "algebra-lemma-characterize-finite-in-terms-of-integral", "algebra-lemma-integral-overring-surjective" ], "ref_ids": [ 1042, 484, 495 ] } ], "ref_ids": [] }, { "id": 11059, "type": "theorem", "label": "varieties-lemma-semi-local-both-side", "categories": [ "varieties" ], "title": "varieties-lemma-semi-local-both-side", "contents": [ "In Situation \\ref{situation-glue} assume", "\\begin{enumerate}", "\\item $A$ is a Noetherian semi-local domain of dimension $1$,", "\\item $B$ is a Noetherian semi-local domain of dimension $1$,", "\\item $A \\otimes B \\to K$ is surjective.", "\\end{enumerate}", "Then $\\Spec(A) \\to \\Spec(R)$ and $\\Spec(B) \\to \\Spec(R)$", "are open immersions covering $\\Spec(R)$ and $K = A \\otimes_R B$." ], "refs": [], "proofs": [ { "contents": [ "Special case: $B$ is integrally closed in $K$. This means that", "$B$ is a Dedekind domain", "(Algebra, Lemma \\ref{algebra-lemma-characterize-Dedekind})", "whence all of its localizations at maximal ideals are discrete valuation rings.", "Let $\\mathfrak m_1, \\ldots, \\mathfrak m_r$ be the maximal ideals of", "$B$. We set", "$$", "R_1 = A \\times_K B_{\\mathfrak m_1}", "$$", "Observing that $A \\otimes_{R_1} B_{\\mathfrak m_1} \\to K$ is surjective", "we conclude from Lemma \\ref{lemma-semi-local} that $A$ and", "$B_{\\mathfrak m_1}$ define open subschemes covering $\\Spec(R_1)$ and that", "$K = A \\otimes_{R_1} B_{\\mathfrak m_1}$. In particular $R_1$ is a", "semi-local Noetherian ring of dimension $1$. By induction we define", "$$", "R_{i + 1} = R_i \\times_K B_{\\mathfrak m_{i + 1}}", "$$", "for $i = 1, \\ldots, r - 1$. Observe that $R = R_n$ because", "$B = B_{\\mathfrak m_1} \\cap \\ldots \\cap B_{\\mathfrak m_r}$ (see", "Algebra, Lemma", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}).", "It follows from the inductive procedure that $R \\to A$ defines an", "open immersion $\\Spec(A) \\to \\Spec(R)$. On the other hand, the", "maximal ideals $\\mathfrak n_i$ of $R$ not in this open correspond", "to the maximal ideals $\\mathfrak m_i$ of $B$ and in fact", "the ring map $R \\to B$ defines an isomorphisms", "$R_{\\mathfrak n_i} \\to B_{\\mathfrak m_i}$ (details omitted; hint:", "in each step we added exactly one maximal ideal to $\\Spec(R_i)$).", "It follows that $\\Spec(B) \\to \\Spec(R)$ is an open immersion", "as desired.", "\\medskip\\noindent", "General case. Let $B' \\subset K$ be the integral closure of $B$.", "See Lemma \\ref{lemma-semi-local-dimension-one-conductor}.", "Then the special case applies to $R' = A \\times_K B'$.", "Pick $x \\in R'$ which is not contained in the maximal", "ideals of $A$ and is contained in the maximal ideals of $B'$", "(see Algebra, Lemma \\ref{algebra-lemma-chinese-remainder}).", "By Lemma \\ref{lemma-semi-local-dimension-one-conductor}", "there exists an integer $n$ such that $x^n \\in R = A \\times_K B$.", "Replace $x$ by $x^n$ so $x \\in R$. For every $y \\in R'$ there exists", "an integer $n$ such that $x^n y \\in R$. On the other hand,", "it is clear that $R'_x = A$. Thus $R_x = A$.", "Exchanging the roles of $A$ and $B$ we also find an $y \\in R$", "such that $B = R_y$. Note that inverting both $x$ and $y$", "leaves no primes except $(0)$. Thus $K = R_{xy} = R_x \\otimes_R R_y$.", "This finishes the proof." ], "refs": [ "algebra-lemma-characterize-Dedekind", "varieties-lemma-semi-local", "algebra-lemma-normal-domain-intersection-localizations-height-1", "varieties-lemma-semi-local-dimension-one-conductor", "algebra-lemma-chinese-remainder", "varieties-lemma-semi-local-dimension-one-conductor" ], "ref_ids": [ 1041, 11057, 1313, 11058, 380, 11058 ] } ], "ref_ids": [] }, { "id": 11060, "type": "theorem", "label": "varieties-lemma-glue-a-bunch-of-local-rings", "categories": [ "varieties" ], "title": "varieties-lemma-glue-a-bunch-of-local-rings", "contents": [ "Let $K$ be a field. Let $A_1, \\ldots, A_r \\subset K$ be Noetherian", "semi-local rings of dimension $1$ with fraction field $K$. If", "$A_i \\otimes A_j \\to K$ is surjective for all $i \\not = j$, then", "there exists a Noetherian semi-local domain $A \\subset K$", "of dimension $1$ contained in $A_1, \\ldots, A_r$ such that", "\\begin{enumerate}", "\\item $A \\to A_i$ induces an open immersion $j_i : \\Spec(A_i) \\to \\Spec(A)$,", "\\item $\\Spec(A)$ is the union of the opens $j_i(\\Spec(A_i))$,", "\\item each closed point of $\\Spec(A)$ lies in exactly one of these", "opens.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Namely, we can take $A = A_1 \\cap \\ldots \\cap A_r$. First we note that (3),", "once (1) and (2) have been proven, follows from the assumption that", "$A_i \\otimes A_j \\to K$ is surjective since if", "$\\mathfrak m \\in j_i(\\Spec(A_i)) \\cap j_j(\\Spec(A_j))$, then", "$A_i \\otimes A_j \\to K$ ends up in $A_\\mathfrak m$.", "To prove (1) and (2) we argue by induction on $r$.", "If $r > 1$ by induction we have the results (1) and (2) for", "$B = A_2 \\cap \\ldots \\cap A_r$. Then we apply", "Lemma \\ref{lemma-semi-local-both-side} to see they hold for", "$A = A_1 \\cap B$." ], "refs": [ "varieties-lemma-semi-local-both-side" ], "ref_ids": [ 11059 ] } ], "ref_ids": [] }, { "id": 11061, "type": "theorem", "label": "varieties-lemma-create-globally-generated", "categories": [ "varieties" ], "title": "varieties-lemma-create-globally-generated", "contents": [ "Let $A$ be a domain with fraction field $K$. Let $B_1, \\ldots, B_r \\subset K$", "be Noetherian $1$-dimensional semi-local rings whose fraction", "fields are $K$. If $A \\otimes B_i \\to K$ are surjective for $i = 1, \\ldots, r$,", "then there exists an $x \\in A$ such that $x^{-1}$ is in the Jacobson radical of", "$B_i$ for $i = 1, \\ldots, r$." ], "refs": [], "proofs": [ { "contents": [ "Let $B_i'$ be the integral closure of $B_i$ in $K$. Suppose we find a", "nonzero $x \\in A$ such that $x^{-1}$ is in the Jacobson radical of $B'_i$ for", "$i = 1, \\ldots, r$. Then by", "Lemma \\ref{lemma-semi-local-dimension-one-conductor},", "after replacing $x$ by a power we get $x^{-1} \\in B_i$.", "Since $\\Spec(B'_i) \\to \\Spec(B_i)$ is surjective we", "see that $x^{-1}$ is then also in the Jacobson radical of $B_i$.", "Thus we may assume that each $B_i$ is a semi-local Dedekind domain.", "\\medskip\\noindent", "If $B_i$ is not local, then remove $B_i$ from the list and", "add back the finite collection of local rings $(B_i)_\\mathfrak m$.", "Thus we may assume that $B_i$ is a discrete valuation ring for", "$i = 1, \\ldots, r$.", "\\medskip\\noindent", "Let $v_i : K \\to \\mathbf{Z}$, $i = 1, \\ldots, r$", "be the corresponding discrete valuations (see", "Algebra, Lemma \\ref{algebra-lemma-characterize-Dedekind}).", "We are looking for a nonzero $x \\in A$ with $v_i(x) < 0$ for", "$i = 1, \\ldots, r$. We will prove this by induction on $r$.", "\\medskip\\noindent", "If $r = 1$ and the result is wrong, then $A \\subset B$ and the map", "$A \\otimes B \\to K$ is not surjective, contradiction.", "\\medskip\\noindent", "If $r > 1$, then by induction we can find a nonzero $x \\in A$ such that", "$v_i(x) < 0$ for $i = 1, \\ldots, r - 1$. If $v_r(x) < 0$ then we are", "done, so we may assume $v_r(x) \\geq 0$. By the base case we can find", "$y \\in A$ nonzero such that $v_r(y) < 0$. After replacing $x$ by a", "power we may assume that $v_i(x) < v_i(y)$ for $i = 1, \\ldots, r - 1$.", "Then $x + y$ is the element we are looking for." ], "refs": [ "varieties-lemma-semi-local-dimension-one-conductor", "algebra-lemma-characterize-Dedekind" ], "ref_ids": [ 11058, 1041 ] } ], "ref_ids": [] }, { "id": 11062, "type": "theorem", "label": "varieties-lemma-power-equal", "categories": [ "varieties" ], "title": "varieties-lemma-power-equal", "contents": [ "Let $A$ be a Noetherian local ring of dimension $1$.", "Let $L = \\prod A_\\mathfrak p$ where the product is over", "the minimal primes of $A$. Let $a_1, a_2 \\in \\mathfrak m_A$", "map to the same element of $L$. Then $a_1^n = a_2^n$ for", "some $n > 0$." ], "refs": [], "proofs": [ { "contents": [ "Write $a_1 = a_2 + x$. Then $x$ maps to zero in $L$. Hence", "$x$ is a nilpotent element of $A$ because $\\bigcap \\mathfrak p$", "is the radical of $(0)$ and the annihilator $I$ of $x$ contains", "a power of the maximal ideal because $\\mathfrak p \\not \\in V(I)$", "for all minimal primes. Say $x^k = 0$ and $\\mathfrak m^n \\subset I$.", "Then", "$$", "a_1^{k + n} = a_2^{k + n} + {n + k \\choose 1} a_2^{n + k - 1} x +", "{n + k \\choose 2} a_2^{n + k - 2} x^2 + \\ldots +", "{n + k \\choose k - 1} a_2^{n + 1} x^{k - 1} = a_2^{n + k}", "$$", "because $a_2 \\in \\mathfrak m_A$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11063, "type": "theorem", "label": "varieties-lemma-power-works", "categories": [ "varieties" ], "title": "varieties-lemma-power-works", "contents": [ "Let $A$ be a Noetherian local ring of dimension $1$.", "Let $L = \\prod A_\\mathfrak p$ and $I = \\bigcap \\mathfrak p$", "where the product and intersection are over", "the minimal primes of $A$. Let $f \\in L$ be an element", "of the form $f = i + a$ where $a \\in \\mathfrak m_A$ and", "$i \\in IL$. Then some power of $f$ is in the image of $A \\to L$." ], "refs": [], "proofs": [ { "contents": [ "Since $A$ is Noetherian we have $I^t = 0$ for some $t > 0$.", "Suppose that we know that $f = a + i$ with $i \\in I^kL$.", "Then $f^n = a^n + na^{n - 1}i \\bmod I^{k + 1}L$.", "Hence it suffices to show that $na^{n - 1}i$ is in", "the image of $I^k \\to I^kL$ for some $n \\gg 0$.", "To see this, pick a $g \\in A$ such that $\\mathfrak m_A = \\sqrt{(g)}$", "(Algebra, Lemma \\ref{algebra-lemma-height-1}). Then $L = A_g$ for example by", "Algebra, Proposition \\ref{algebra-proposition-dimension-zero-ring}.", "On the other hand, there is an $n$ such that $a^n \\in (g)$.", "Hence we can clear denominators for elements of $L$", "by multiplying by a high power of $a$." ], "refs": [ "algebra-lemma-height-1", "algebra-proposition-dimension-zero-ring" ], "ref_ids": [ 682, 1410 ] } ], "ref_ids": [] }, { "id": 11064, "type": "theorem", "label": "varieties-lemma-good-intersection", "categories": [ "varieties" ], "title": "varieties-lemma-good-intersection", "contents": [ "Let $A$ be a Noetherian local ring of dimension $1$.", "Let $L = \\prod A_\\mathfrak p$ where the product is over", "the minimal primes of $A$. Let $K \\to L$ be an integral ring map.", "Then there exist $a \\in \\mathfrak m_A$ and $x \\in K$", "which map to the same element of $L$ such that $\\mathfrak m_A = \\sqrt{(a)}$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-power-works} we may replace $A$ by", "$A/(\\bigcap \\mathfrak p)$ and assume that $A$ is a reduced ring", "(some details omitted).", "We may also replace $K$ by the image of $K \\to L$.", "Then $K$ is a reduced ring. The map $\\Spec(L) \\to \\Spec(K)$ is", "surjective and closed (details omitted). Hence $\\Spec(K)$ is a finite", "discrete space. It follows that $K$ is a finite product of fields.", "\\medskip\\noindent", "Let $\\mathfrak p_j$, $j = 1, \\ldots, m$ be the minimal primes of $A$.", "Set $L_j$ be the fraction field of $A_j$ so that", "$L = \\prod_{j = 1, \\ldots, m} L_j$.", "Let $A_j$ be the normalization of $A/\\mathfrak p_j$. Then", "$A_j$ is a semi-local Dedekind domain with at least", "one maximal ideal, see", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-Dedekind}.", "Let $n$ be the sum of the numbers of maximal ideals", "in $A_1, \\ldots, A_m$. For such a maximal ideal $\\mathfrak m \\subset A_j$", "we consider the function", "$$", "v_{\\mathfrak m} : L \\to L_j \\to \\mathbf{Z} \\cup \\{\\infty\\}", "$$", "where the second arrow is the discrete valuation corresponding", "to the discrete valuation ring $(A_j)_{\\mathfrak m}$ extended", "by mapping $0$ to $\\infty$. In this way we obtain $n$ functions", "$v_1, \\ldots, v_n : L \\to \\mathbf{Z} \\cup \\{\\infty\\}$.", "We will find an element $x \\in K$ such that $v_i(x) < 0$", "for all $i = 1, \\ldots, n$.", "\\medskip\\noindent", "First we claim that for each $i$ there exists an element $x \\in K$", "with $v_i(x) < 0$. Namely, suppose that $v_i$ corresponds to", "$\\mathfrak m \\subset A_j$. If $v_i(x) \\geq 0$ for all $x \\in K$,", "then $K$ maps into $(A_j)_{\\mathfrak m}$ inside the fraction field", "$L_j$ of $A_j$.", "The image of $K$ in $L_j$ is a field over $L_j$ is", "algebraic by Algebra, Lemma \\ref{algebra-lemma-integral-under-field}.", "Combined we get a contradiction with Algebra, Lemma", "\\ref{algebra-lemma-valuation-ring-cap-field-finite}.", "\\medskip\\noindent", "Suppose we have found an element $x \\in K$ such that", "$v_1(x) < 0, \\ldots, v_r(x) < 0$ for some $r < n$. If $v_{r + 1}(x) < 0$,", "then $x$ works for $r + 1$. If not, then choose some $y \\in K$ with", "$v_{r + 1}(y) < 0$ as is possible by the result of the previous", "paragraph. After replacing $x$ by $x^n$ for some $n > 0$,", "we may assume $v_i(x) < v_i(y)$ for $i = 1, \\ldots, r$. Then", "$v_j(x + y) = v_j(x) < 0$ for $j = 1, \\ldots, r$ by properties", "of valuations and similarly $v_{r + 1}(x + y) = v_{r + 1}(y) < 0$.", "Arguing by induction, we find", "$x \\in K$ with $v_i(x) < 0$ for $i = 1, \\ldots, n$.", "\\medskip\\noindent", "In particular, the element $x \\in K$ has nonzero projection", "in each factor of $K$ (recall that $K$ is a finite product of", "fields and if some component of $x$ was zero, then one", "of the values $v_i(x)$ would be $\\infty$). Hence $x$ is", "invertible and $x^{-1} \\in K$ is an element with", "$\\infty > v_i(x^{-1}) > 0$ for all $i$. It follows from", "Lemma \\ref{lemma-semi-local-dimension-one-conductor} that", "for some $e < 0$ the element $x^e \\in K$ maps to an element of", "$\\mathfrak m_A/\\mathfrak p_j \\subset A/\\mathfrak p_j$ for all", "$j = 1, \\ldots, m$. Observe that the cokernel of the map", "$\\mathfrak m_A \\to \\prod \\mathfrak m_A/\\mathfrak p_j$ is", "annihilated by a power of $\\mathfrak m_A$. Hence after replacing", "$e$ by a more negative $e$, we find an element $a \\in \\mathfrak m_A$", "whose image in $\\mathfrak m_A/\\mathfrak p_j$ is equal to", "the image of $x^e$. The pair $(a, x^e)$ satisfies the", "conclusions of the lemma." ], "refs": [ "varieties-lemma-power-works", "algebra-lemma-integral-closure-Dedekind", "algebra-lemma-integral-under-field", "algebra-lemma-valuation-ring-cap-field-finite", "varieties-lemma-semi-local-dimension-one-conductor" ], "ref_ids": [ 11063, 1042, 496, 613, 11058 ] } ], "ref_ids": [] }, { "id": 11065, "type": "theorem", "label": "varieties-lemma-localization-semi-local", "categories": [ "varieties" ], "title": "varieties-lemma-localization-semi-local", "contents": [ "Let $A$ be a ring. Let $\\mathfrak p_1, \\ldots, \\mathfrak p_r$", "be a finite set of a primes of $A$. Let", "$S = A \\setminus \\bigcup \\mathfrak p_i$. Then $S$ is a multiplicative", "system and $S^{-1}A$ is a semi-local ring whose maximal ideals", "correspond to the maximal elements of the set $\\{\\mathfrak p_i\\}$." ], "refs": [], "proofs": [ { "contents": [ "If $a, b \\in A$ and $a, b \\in S$, then $a, b \\not \\in \\mathfrak p_i$", "hence $ab \\not \\in \\mathfrak p_i$, hence $ab \\in S$. Also $1 \\in S$.", "Thus $S$ is a multiplicative subset of $A$. By the description of", "$\\Spec(S^{-1}A)$ in", "Algebra, Lemma \\ref{algebra-lemma-spec-localization}", "and by", "Algebra, Lemma \\ref{algebra-lemma-silly}", "we see that the primes of $S^{-1}A$ correspond to the primes of", "$A$ contained in one of the $\\mathfrak p_i$.", "Hence the maximal ideals of $S^{-1}A$ correspond one-to-one with the", "maximal (w.r.t.\\ inclusion) elements of the set", "$\\{\\mathfrak p_1, \\ldots, \\mathfrak p_r\\}$." ], "refs": [ "algebra-lemma-spec-localization", "algebra-lemma-silly" ], "ref_ids": [ 391, 378 ] } ], "ref_ids": [] }, { "id": 11066, "type": "theorem", "label": "varieties-lemma-affine", "categories": [ "varieties" ], "title": "varieties-lemma-affine", "contents": [ "Let $X$ be a scheme all of whose local rings are Noetherian of dimension", "$\\leq 1$. Let $U \\subset X$ be a retrocompact open. Denote", "$j : U \\to X$ the inclusion morphism. Then $R^pj_*\\mathcal{F} = 0$, $p > 0$", "for every quasi-coherent $\\mathcal{O}_U$-module $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "We may check the vanishing of $R^pj_*\\mathcal{F}$ at stalks.", "Formation of $R^qj_*$ commutes with flat base change, see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology}.", "Thus we may assume that $X$ is the spectrum of a Noetherian local", "ring of dimension $\\leq 1$. In this case $X$ has a closed point", "$x$ and finitely many other points $x_1, \\ldots, x_n$ which specialize", "to $x$ but not each other (see", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-irreducible-components}).", "If $x \\in U$, then $U = X$ and the result is clear. If not, then", "$U = \\{x_1, \\ldots, x_r\\}$ for some $r$ after possibly renumbering", "the points. Then $U$ is affine", "(Schemes, Lemma \\ref{schemes-lemma-scheme-finite-discrete-affine}).", "Thus the result follows from Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-relative-affine-vanishing}." ], "refs": [ "coherent-lemma-flat-base-change-cohomology", "algebra-lemma-Noetherian-irreducible-components", "schemes-lemma-scheme-finite-discrete-affine", "coherent-lemma-relative-affine-vanishing" ], "ref_ids": [ 3298, 453, 7678, 3283 ] } ], "ref_ids": [] }, { "id": 11067, "type": "theorem", "label": "varieties-lemma-open-in-affine-curve-affine", "categories": [ "varieties" ], "title": "varieties-lemma-open-in-affine-curve-affine", "contents": [ "Let $X$ be an affine scheme all of whose local rings are Noetherian", "of dimension $\\leq 1$. Then any quasi-compact open $U \\subset X$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Denote $j : U \\to X$ the inclusion morphism. Let $\\mathcal{F}$", "be a quasi-coherent $\\mathcal{O}_U$-module.", "By Lemma \\ref{lemma-affine} the higher direct images", "$R^pj_*\\mathcal{F}$ are zero. The $\\mathcal{O}_X$-module $j_*\\mathcal{F}$", "is quasi-coherent", "(Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}).", "Hence it has vanishing higher cohomology groups by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "By the Leray spectral sequence", "Cohomology, Lemma \\ref{cohomology-lemma-apply-Leray}", "we have $H^p(U, \\mathcal{F}) = 0$ for all $p > 0$.", "Thus $U$ is affine, for example by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-compact-h1-zero-covering}." ], "refs": [ "varieties-lemma-affine", "schemes-lemma-push-forward-quasi-coherent", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "cohomology-lemma-apply-Leray", "coherent-lemma-quasi-compact-h1-zero-covering" ], "ref_ids": [ 11066, 7730, 3282, 2071, 3287 ] } ], "ref_ids": [] }, { "id": 11068, "type": "theorem", "label": "varieties-lemma-complement-codim-1-closed-points", "categories": [ "varieties" ], "title": "varieties-lemma-complement-codim-1-closed-points", "contents": [ "Let $X$ be a scheme. Let $U \\subset X$ be an open. Assume", "\\begin{enumerate}", "\\item $U$ is a retrocompact open of $X$,", "\\item $X \\setminus U$ is discrete, and", "\\item for $x \\in X \\setminus U$ the local ring", "$\\mathcal{O}_{X, x}$ is Noetherian of dimension $\\leq 1$.", "\\end{enumerate}", "Then (1) there exists an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$", "and a section $s$ such that $U = X_s$ and (2) the map", "$\\Pic(X) \\to \\Pic(U)$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $X \\setminus U = \\{x_i; i \\in I\\}$.", "Choose affine opens $U_i \\subset X$ with $x_i \\in X$ and", "$x_j \\not \\in U_i$ for $j \\not = i$. This is possible by condition (2).", "Say $U_i = \\Spec(A_i)$. Let $\\mathfrak m_i \\subset A_i$ be the maximal", "ideal corresponding to $x_i$. By our assumption on the local rings", "there are only a finite number of prime ideals", "$\\mathfrak q \\subset \\mathfrak m_i$,", "$\\mathfrak q \\not = \\mathfrak m_i$ (see", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-irreducible-components}).", "Thus by prime avoidance (Algebra, Lemma", "\\ref{algebra-lemma-silly}) we can find $f_i \\in \\mathfrak m_i$", "not contained in any of those primes. Then", "$V(f_i) = \\{\\mathfrak m_i\\} \\amalg Z_i$ for some closed subset", "$Z_i \\subset U_i$ because $Z_i$ is a retrocompact open subset of", "$V(f_i)$ closed under specialization, see", "Algebra, Lemma \\ref{algebra-lemma-constructible-stable-specialization-closed}.", "After shrinking $U_i$ we may assume $V(f_i) = \\{x_i\\}$. Then", "$$", "\\mathcal{U} : X = U \\cup \\bigcup U_i", "$$", "is an open covering of $X$. Consider the $2$-cocycle with values", "in $\\mathcal{O}_X^*$ given by $f_i$ on $U \\cap U_i$ and by", "$f_i/f_j$ on $U_i \\cap U_j$. This defines a line bundle", "$\\mathcal{L}$ such that the section $s$ defined by $1$ on $U$", "and $f_i$ on $U_i$ is as in the statement of the lemma.", "\\medskip\\noindent", "Let $\\mathcal{N}$ be an invertible $\\mathcal{O}_U$-module.", "Let $N_i$ be the invertible $(A_i)_{f_i}$ module such that", "$\\mathcal{N}|_{U \\cap U_i}$ is equal to $\\tilde N_i$.", "Observe that $(A_{\\mathfrak m_i})_{f_i}$ is an Artinian ring", "(as a dimension zero Noetherian ring, see", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-dimension-0}).", "Thus it is a product of local rings", "(Algebra, Lemma \\ref{algebra-lemma-artinian-finite-length}) and", "hence has trivial Picard group. Thus, after shrinking $U_i$", "(i.e., after replacing $A_i$ by $(A_i)_g$ for some $g \\in A_i$,", "$g \\not \\in \\mathfrak m_i$)", "we can assume that $N_i = (A_i)_{f_i}$, i.e., that", "$\\mathcal{N}|_{U \\cap U_i}$ is trivial. In this case it is", "clear how to extend $\\mathcal{N}$ to an invertible sheaf over $X$", "(by extending it by a trivial invertible module over each $U_i$)." ], "refs": [ "algebra-lemma-Noetherian-irreducible-components", "algebra-lemma-silly", "algebra-lemma-constructible-stable-specialization-closed", "algebra-lemma-Noetherian-dimension-0", "algebra-lemma-artinian-finite-length" ], "ref_ids": [ 453, 378, 553, 680, 646 ] } ], "ref_ids": [] }, { "id": 11069, "type": "theorem", "label": "varieties-lemma-find-globally-generated", "categories": [ "varieties" ], "title": "varieties-lemma-find-globally-generated", "contents": [ "Let $X$ be an integral separated scheme. Let $U \\subset X$ be a nonempty", "affine open such that $X \\setminus U$ is a finite set of points", "$x_1, \\ldots, x_r$ with $\\mathcal{O}_{X, x_i}$ Noetherian of dimension $1$.", "Then there exists a globally generated invertible $\\mathcal{O}_X$-module", "$\\mathcal{L}$ and a section $s$ such that $U = X_s$." ], "refs": [], "proofs": [ { "contents": [ "Say $U = \\Spec(A)$ and let $K$ be the function field of $X$.", "Write $B_i = \\mathcal{O}_{X, x_i}$ and $\\mathfrak m_i = \\mathfrak m_{x_i}$.", "Since $x_i \\not \\in U$ we see that the open", "$U \\times_X \\Spec(B_i)$ of $\\Spec(B_i)$ has only one point, i.e.,", "$U \\times_X \\Spec(B_i) = \\Spec(K)$.", "Since $X$ is separated, we find that $\\Spec(K)$ is a closed subscheme", "of $U \\times \\Spec(B_i)$, i.e., the map $A \\otimes B_i \\to K$ is a surjection.", "By Lemma \\ref{lemma-create-globally-generated} we can find a nonzero", "$f \\in A$ such that $f^{-1} \\in \\mathfrak m_i$ for $i = 1, \\ldots, r$.", "Pick opens $x_i \\in U_i \\subset X$ such that $f^{-1} \\in \\mathcal{O}(U_i)$.", "Then", "$$", "\\mathcal{U} : X = U \\cup \\bigcup U_i", "$$", "is an open covering of $X$. Consider the $2$-cocycle with values", "in $\\mathcal{O}_X^*$ given by $f$ on $U \\cap U_i$ and by", "$1$ on $U_i \\cap U_j$. This defines a line bundle $\\mathcal{L}$", "with two sections:", "\\begin{enumerate}", "\\item a section $s$ defined by $1$ on $U$", "and $f^{-1}$ on $U_i$ is as in the statement of the lemma, and", "\\item a section $t$ defined by $f$ on $U$ and", "$1$ on $U_i$.", "\\end{enumerate}", "Note that $X_t \\supset U_1 \\cup \\ldots \\cup U_r$.", "Hence $s, t$ generate $\\mathcal{L}$ and the lemma is proved." ], "refs": [ "varieties-lemma-create-globally-generated" ], "ref_ids": [ 11061 ] } ], "ref_ids": [] }, { "id": 11070, "type": "theorem", "label": "varieties-lemma-enough-globally-generated-ample", "categories": [ "varieties" ], "title": "varieties-lemma-enough-globally-generated-ample", "contents": [ "Let $X$ be a quasi-compact scheme. If for every $x \\in X$", "there exists a pair $(\\mathcal{L}, s)$ consisting of a globally generated", "invertible sheaf $\\mathcal{L}$ and a global section $s$ such that", "$x \\in X_s$ and $X_s$ is affine, then $X$ has an ample invertible", "sheaf." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is quasi-compact we can find a finite collection", "$(\\mathcal{L}_i, s_i)$, $i = 1, \\ldots, n$", "of pairs such that $X_{s_i}$ is affine and $X = \\bigcup X_{s_i}$.", "Again because $X$ is quasi-compact we can find, for each $i$, a finite", "collection of sections $t_{i, j}$, $j = 1, \\ldots, m_i$", "such that $X = \\bigcup X_{t_{i, j}}$. Set $t_{i, 0} = s_i$.", "Consider the invertible sheaf", "$$", "\\mathcal{L} = \\mathcal{L}_1 ", "\\otimes_{\\mathcal{O}_X} \\ldots", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}_n", "$$", "and the global sections", "$$", "\\tau_J = t_{1, j_1} \\otimes \\ldots \\otimes t_{n, j_n}", "$$", "By Properties, Lemma \\ref{properties-lemma-affine-cap-s-open}", "the open $X_{\\tau_J}$ is affine as soon as $j_i = 0$ for some $i$.", "It is a simple matter to see that these opens cover $X$.", "Hence $\\mathcal{L}$ is ample by definition." ], "refs": [ "properties-lemma-affine-cap-s-open" ], "ref_ids": [ 3042 ] } ], "ref_ids": [] }, { "id": 11071, "type": "theorem", "label": "varieties-lemma-dim-1-noetherian-integral-separated-has-ample", "categories": [ "varieties" ], "title": "varieties-lemma-dim-1-noetherian-integral-separated-has-ample", "contents": [ "Let $X$ be a Noetherian integral separated scheme of dimension $1$.", "Then $X$ has an ample invertible sheaf." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine open covering $X = U_1 \\cup \\ldots \\cup U_n$.", "Since $X$ is Noetherian, each of the sets $X \\setminus U_i$ is finite.", "Thus by Lemma \\ref{lemma-find-globally-generated}", "we can find a pair $(\\mathcal{L}_i, s_i)$", "consisting of a globally generated invertible sheaf $\\mathcal{L}_i$", "and a global section $s_i$ such that $U_i = X_{s_i}$.", "We conclude that $X$ has an ample invertible sheaf by", "Lemma \\ref{lemma-enough-globally-generated-ample}." ], "refs": [ "varieties-lemma-find-globally-generated", "varieties-lemma-enough-globally-generated-ample" ], "ref_ids": [ 11069, 11070 ] } ], "ref_ids": [] }, { "id": 11072, "type": "theorem", "label": "varieties-lemma-surjective-pic-birational-finite", "categories": [ "varieties" ], "title": "varieties-lemma-surjective-pic-birational-finite", "contents": [ "Let $f : X \\to Y$ be a finite morphism of schemes. Assume there", "exists an open $V \\subset Y$ such that $f^{-1}(V) \\to V$ is an", "isomorphism and $Y \\setminus V$ is a discrete space. Then every", "invertible $\\mathcal{O}_X$-module is the pullback of an invertible", "$\\mathcal{O}_Y$-module." ], "refs": [], "proofs": [ { "contents": [ "We will use that $\\Pic(X) = H^1(X, \\mathcal{O}_X^*)$, see", "Cohomology, Lemma \\ref{cohomology-lemma-h1-invertible}.", "Consider the Leray spectral sequence for the abelian sheaf $\\mathcal{O}_X^*$", "and $f$, see Cohomology, Lemma \\ref{cohomology-lemma-Leray}.", "Consider the induced map", "$$", "H^1(X, \\mathcal{O}_X^*) \\longrightarrow H^0(Y, R^1f_*\\mathcal{O}_X^*)", "$$", "Divisors, Lemma \\ref{divisors-lemma-finite-trivialize-invertible-upstairs}", "says exactly that this map is zero. Hence Leray gives", "$H^1(X, \\mathcal{O}_X^*) = H^1(Y, f_*\\mathcal{O}_X^*)$.", "Next we consider the map", "$$", "f^\\sharp : \\mathcal{O}_Y^* \\longrightarrow f_*\\mathcal{O}_X^*", "$$", "By assumption the kernel and cokernel of this map are supported", "on the closed subset $T = Y \\setminus V$ of $Y$.", "Since $T$ is a discrete topological space by assumption", "the higher cohomology groups of any abelian sheaf on $Y$ supported", "on $T$ is zero (follows from", "Cohomology, Lemma \\ref{cohomology-lemma-cohomology-and-closed-immersions},", "Modules, Lemma \\ref{modules-lemma-i-star-exact}, and", "the fact that $H^i(T, \\mathcal{F}) = 0$ for any $i > 0$", "and any abelian sheaf $\\mathcal{F}$ on $T$).", "Breaking the displayed map into short exact sequences", "$$", "0 \\to \\Ker(f^\\sharp) \\to \\mathcal{O}_Y^* \\to \\Im(f^\\sharp) \\to 0,\\quad", "0 \\to \\Im(f^\\sharp) \\to f_*\\mathcal{O}_X^* \\to \\Coker(f^\\sharp) \\to 0", "$$", "we first conclude that $H^1(Y, \\mathcal{O}_Y^*) \\to H^1(Y, \\Im(f^\\sharp))$", "is surjective and then that", "$H^1(Y, \\Im(f^\\sharp)) \\to H^1(Y, f_*\\mathcal{O}_X^*)$ is surjective.", "Combining all the above we find that $H^1(Y, \\mathcal{O}_Y^*) \\to", "H^1(X, \\mathcal{O}_X^*)$ is surjective as desired." ], "refs": [ "cohomology-lemma-h1-invertible", "cohomology-lemma-Leray", "divisors-lemma-finite-trivialize-invertible-upstairs", "cohomology-lemma-cohomology-and-closed-immersions", "modules-lemma-i-star-exact" ], "ref_ids": [ 2036, 2070, 7963, 2084, 13232 ] } ], "ref_ids": [] }, { "id": 11073, "type": "theorem", "label": "varieties-lemma-glue-invertible-sheaves", "categories": [ "varieties" ], "title": "varieties-lemma-glue-invertible-sheaves", "contents": [ "Let $X$ be a scheme. Let $Z_1, \\ldots, Z_n \\subset X$ be closed", "subschemes. Let $\\mathcal{L}_i$ be an invertible sheaf on $Z_i$.", "Assume that", "\\begin{enumerate}", "\\item $X$ is reduced,", "\\item $X = \\bigcup Z_i$ set theoretically, and", "\\item $Z_i \\cap Z_j$ is a discrete topological space for $i \\not = j$.", "\\end{enumerate}", "Then there exists an invertible sheaf $\\mathcal{L}$ on $X$ whose restriction", "to $Z_i$ is $\\mathcal{L}_i$. Moreover, if we are given sections", "$s_i \\in \\Gamma(Z_i, \\mathcal{L}_i)$ which are nonvanishing at the", "points of $Z_i \\cap Z_j$, then we can choose $\\mathcal{L}$ such", "that there exists a $s \\in \\Gamma(X, \\mathcal{L})$ with", "$s|_{Z_i} = s_i$ for all $i$." ], "refs": [], "proofs": [ { "contents": [ "The existence of $\\mathcal{L}$ can be deduced from", "Lemma \\ref{lemma-surjective-pic-birational-finite}", "but we will also give a direct proof and we will use", "the direct proof to see the statement about sections is true.", "Set $T = \\bigcup_{i \\not = j} Z_i \\cap Z_j$. As $X$ is reduced we have", "$$", "X \\setminus T = \\bigcup (Z_i \\setminus T)", "$$", "as schemes. Assumption (3) implies $T$ is a discrete subset of $X$.", "Thus for each $t \\in T$ we can find an open $U_t \\subset X$", "with $t \\in U_t$ but $t' \\not \\in U_t$ for $t' \\in T$, $t' \\not = t$.", "By shrinking $U_t$ if necessary, we may assume that there exist isomorphisms", "$\\varphi_{t, i} : \\mathcal{L}_i|_{U_t \\cap Z_i} \\to", "\\mathcal{O}_{U_t \\cap Z_i}$. Furthermore, for each $i$ choose an open covering", "$$", "Z_i \\setminus T = \\bigcup\\nolimits_j U_{ij}", "$$", "such that there exist isomorphisms", "$\\varphi_{i, j} : \\mathcal{L}_i|_{U_{ij}} \\cong \\mathcal{O}_{U_{ij}}$.", "Observe that", "$$", "\\mathcal{U} : X = \\bigcup U_t \\cup \\bigcup U_{ij}", "$$", "is an open covering of $X$. We claim that we can use the isomorphisms", "$\\varphi_{t, i}$ and $\\varphi_{i, j}$ to define a $2$-cocycle with values", "in $\\mathcal{O}_X^*$ for this covering that defines $\\mathcal{L}$ as", "in the statement of the lemma.", "\\medskip\\noindent", "Namely, if $i \\not = i'$, then $U_{i, j} \\cap U_{i', j'} = \\emptyset$", "and there is nothing to do. For $U_{i, j} \\cap U_{i, j'}$ we have", "$\\mathcal{O}_X(U_{i, j} \\cap U_{i, j'}) =", "\\mathcal{O}_{Z_i}(U_{i, j} \\cap U_{i, j'})$ by the first remark of the proof.", "Thus the transition function for $\\mathcal{L}_i$ (more precisely", "$\\varphi_{i, j} \\circ \\varphi_{i, j'}^{-1}$) defines the value of our", "cocycle on this intersection.", "For $U_t \\cap U_{i, j}$ we can do the same thing.", "Finally, for $t \\not = t'$ we have", "$$", "U_t \\cap U_{t'} = \\coprod (U_t \\cap U_{t'}) \\cap Z_i", "$$", "and moreover the intersection $U_t \\cap U_{t'} \\cap Z_i$ is contained", "in $Z_i \\setminus T$. Hence by the same reasoning as before we see that", "$$", "\\mathcal{O}_X(U_t \\cap U_{t'}) =", "\\prod \\mathcal{O}_{Z_i}(U_t \\cap U_{t'} \\cap Z_i)", "$$", "and we can use the transition functions for $\\mathcal{L}_i$ (more precisely", "$\\varphi_{t, i} \\circ \\varphi_{t', i}^{-1}$) to define the value of", "our cocycle on $U_t \\cap U_{t'}$. This finishes the proof of existence", "of $\\mathcal{L}$.", "\\medskip\\noindent", "Given sections $s_i$ as in the last assertion of the lemma, in the argument", "above, we choose $U_t$ such that $s_i|_{U_t \\cap Z_i}$ is nonvanishing and", "we choose $\\varphi_{t, i}$ such that $\\varphi_{t, i}(s_i|_{U_t \\cap Z_i}) = 1$.", "Then using $1$ over $U_t$ and $\\varphi_{i, j}(s_i|_{U_{i, j}})$ over", "$U_{i, j}$ will define a section of $\\mathcal{L}$ which restricts", "to $s_i$ over $Z_i$." ], "refs": [ "varieties-lemma-surjective-pic-birational-finite" ], "ref_ids": [ 11072 ] } ], "ref_ids": [] }, { "id": 11074, "type": "theorem", "label": "varieties-lemma-dim-1-noetherian-reduced-separated-has-ample", "categories": [ "varieties" ], "title": "varieties-lemma-dim-1-noetherian-reduced-separated-has-ample", "contents": [ "Let $X$ be a Noetherian reduced separated scheme of dimension $1$.", "Then $X$ has an ample invertible sheaf." ], "refs": [], "proofs": [ { "contents": [ "Let $Z_i$, $i = 1, \\ldots, n$ be the irreducible components of $X$.", "We view these as reduced closed subschemes of $X$.", "By Lemma \\ref{lemma-dim-1-noetherian-integral-separated-has-ample}", "there exist ample invertible sheaves $\\mathcal{L}_i$ on $Z_i$.", "Set $T = \\bigcup_{i \\not = j} Z_i \\cap Z_j$. As $X$ is Noetherian", "of dimension $1$, the set $T$ is finite and consists of closed", "points of $X$. For each $i$ we may, possibly after replacing", "$\\mathcal{L}_i$ by a power, choose $s_i \\in \\Gamma(Z_i, \\mathcal{L}_i)$", "such that $(Z_i)_{s_i}$ is affine and contains $T \\cap Z_i$, see", "Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-principal-affine}.", "\\medskip\\noindent", "By Lemma \\ref{lemma-glue-invertible-sheaves} we can find an invertible sheaf", "$\\mathcal{L}$ on $X$ and $s \\in \\Gamma(X, \\mathcal{L})$", "such that $(\\mathcal{L}, s)|_{Z_i} = (\\mathcal{L}_i, s_i)$.", "Observe that $X_s$ contains $T$ and is set theoretically equal", "to the affine closed subschemes $(Z_i)_{s_i}$. Thus it is affine by", "Limits, Lemma \\ref{limits-lemma-affines-glued-in-closed-affine}.", "To finish the proof, it suffices to find for every $x \\in X$, $x \\not \\in T$", "an integer $m > 0$ and a section $t \\in \\Gamma(X, \\mathcal{L}^{\\otimes m})$", "such that $X_t$ is affine and $x \\in X_t$. Since $x \\not \\in T$", "we see that $x \\in Z_i$ for some unique $i$, say $i = 1$.", "Let $Z \\subset X$ be the reduced closed subscheme whose underlying", "topological space is $Z_2 \\cup \\ldots \\cup Z_n$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the ideal", "sheaf of $Z$. Denote that $\\mathcal{I}_1 \\subset \\mathcal{O}_{Z_1}$", "the inverse image of this ideal sheaf under the inclusion", "morphism $Z_1 \\to X$. Observe that", "$$", "\\Gamma(X, \\mathcal{I}\\mathcal{L}^{\\otimes m}) =", "\\Gamma(Z_1, \\mathcal{I}_1 \\mathcal{L}_1^{\\otimes m})", "$$", "see Remark \\ref{remark-conductor}. Thus it suffices to find $m > 0$", "and $t \\in \\Gamma(Z_1, \\mathcal{I}_1 \\mathcal{L}_1^{\\otimes m})$", "with $x \\in (Z_1)_t$ affine. Since $\\mathcal{L}_1$ is ample", "and since $x$ is not in $Z_1 \\cap T = V(\\mathcal{I}_1)$", "we can find a section", "$t_1 \\in \\Gamma(Z_1, \\mathcal{I}_1 \\mathcal{L}_1^{\\otimes m_1})$", "with $x \\in (Z_1)_{t_1}$, see Properties, Proposition", "\\ref{properties-proposition-characterize-ample}.", "Since $\\mathcal{L}_1$ is ample we can find a section", "$t_2 \\in \\Gamma(Z_1, \\mathcal{L}_1^{\\otimes m_2})$", "with $x \\in (Z_1)_{t_2}$ and $(Z_1)_{t_2}$ affine, see", "Properties, Definition \\ref{properties-definition-ample}.", "Set $m = m_1 + m_2$ and $t = t_1 t_2$. Then", "$t \\in \\Gamma(Z_1, \\mathcal{I}_1 \\mathcal{L}_1^{\\otimes m})$", "with $x \\in (Z_1)_t$ by construction and", "$(Z_1)_t$ is affine by Properties, Lemma", "\\ref{properties-lemma-affine-cap-s-open}." ], "refs": [ "varieties-lemma-dim-1-noetherian-integral-separated-has-ample", "properties-lemma-ample-finite-set-in-principal-affine", "varieties-lemma-glue-invertible-sheaves", "limits-lemma-affines-glued-in-closed-affine", "varieties-remark-conductor", "properties-proposition-characterize-ample", "properties-definition-ample", "properties-lemma-affine-cap-s-open" ], "ref_ids": [ 11071, 3063, 11073, 15083, 11169, 3067, 3088, 3042 ] } ], "ref_ids": [] }, { "id": 11075, "type": "theorem", "label": "varieties-lemma-lift-line-bundle-from-reduction-dimension-1", "categories": [ "varieties" ], "title": "varieties-lemma-lift-line-bundle-from-reduction-dimension-1", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes.", "If the underlying topological space of $X$ is Noetherian and", "$\\dim(X) \\leq 1$, then $\\Pic(X) \\to \\Pic(Z)$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Consider the short exact sequence", "$$", "0 \\to (1 + \\mathcal{I}) \\cap \\mathcal{O}_X^* \\to", "\\mathcal{O}^*_X \\to i_*\\mathcal{O}^*_Z \\to 0", "$$", "of sheaves of abelian groups on $X$ where $\\mathcal{I}$", "is the quasi-coherent sheaf of ideals corresponding to $Z$.", "Since $\\dim(X) \\leq 1$ we see that $H^2(X, \\mathcal{F}) = 0$", "for any abelian sheaf $\\mathcal{F}$, see", "Cohomology, Proposition \\ref{cohomology-proposition-vanishing-Noetherian}.", "Hence the map $H^1(X, \\mathcal{O}^*_X) \\to H^1(X, i_*\\mathcal{O}_Z^*)$", "is surjective. By", "Cohomology, Lemma \\ref{cohomology-lemma-cohomology-and-closed-immersions}", "we have $H^1(X, i_*\\mathcal{O}_Z^*) = H^1(Z, \\mathcal{O}_Z^*)$.", "This proves the lemma by", "Cohomology, Lemma \\ref{cohomology-lemma-h1-invertible}." ], "refs": [ "cohomology-proposition-vanishing-Noetherian", "cohomology-lemma-cohomology-and-closed-immersions", "cohomology-lemma-h1-invertible" ], "ref_ids": [ 2246, 2084, 2036 ] } ], "ref_ids": [] }, { "id": 11076, "type": "theorem", "label": "varieties-lemma-surjection-on-pic-quasi-finite", "categories": [ "varieties" ], "title": "varieties-lemma-surjection-on-pic-quasi-finite", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume", "$Y$ is Noetherian of dimension $\\leq 1$, $f$ is finite, and", "there exists a dense open $V \\subset Y$ such that", "$f^{-1}(V) \\to V$ is a closed immersion. Then every invertible", "$\\mathcal{O}_X$-module is the pullback of an invertible $\\mathcal{O}_Y$-module." ], "refs": [], "proofs": [ { "contents": [ "We factor $f$ as $X \\to Z \\to Y$ where $Z$ is the scheme theoretic", "image of $f$. Then $X \\to Z$ is an isomorphism over $V \\cap Z$", "and Lemma \\ref{lemma-surjective-pic-birational-finite} applies.", "On the other hand,", "Lemma \\ref{lemma-lift-line-bundle-from-reduction-dimension-1}", "applies to $Z \\to Y$. Some details omitted." ], "refs": [ "varieties-lemma-surjective-pic-birational-finite", "varieties-lemma-lift-line-bundle-from-reduction-dimension-1" ], "ref_ids": [ 11072, 11075 ] } ], "ref_ids": [] }, { "id": 11077, "type": "theorem", "label": "varieties-lemma-pre-pre-delta-invariant", "categories": [ "varieties" ], "title": "varieties-lemma-pre-pre-delta-invariant", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian $1$-dimensional local ring.", "Let $f \\in \\mathfrak m$. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathfrak m = \\sqrt{(f)}$,", "\\item $f$ is not contained in any minimal prime of $A$, and", "\\item $A_f = \\prod_{\\mathfrak p\\text{ minimal}} A_\\mathfrak p$ as $A$-algebras.", "\\end{enumerate}", "Such an $f \\in \\mathfrak m$ exists. If $\\text{depth}(A) = 1$ (for example", "$A$ is reduced), then (1) -- (3) are also equivalent to", "\\begin{enumerate}", "\\item[(4)] $f$ is a nonzerodivisor,", "\\item[(5)] $A_f$ is the total ring of fractions of $A$.", "\\end{enumerate}", "If $A$ is reduced, then (1) -- (5) are also equivalent to", "\\begin{enumerate}", "\\item[(6)] $A_f$ is the product of the residue fields at the minimal", "primes of $A$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The spectrum of $A$ has finitely many primes", "$\\mathfrak p_1, \\ldots, \\mathfrak p_n$ besides $\\mathfrak m$", "and these are all minimal, see", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-irreducible-components}.", "Then the equivalence of (1) and (2) follows from", "Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}.", "Clearly, (3) implies (2). Conversely, if (2) is true,", "then the spectrum of $A_f$ is the subset", "$\\{\\mathfrak p_1, \\ldots, \\mathfrak p_n\\}$ of $\\Spec(A)$", "with induced topology, see", "Algebra, Lemma \\ref{algebra-lemma-spec-localization}.", "This is a finite discrete topological space.", "Hence $A_f = \\prod_{\\mathfrak p\\text{ minimal}} A_\\mathfrak p$ by", "Algebra, Proposition \\ref{algebra-proposition-dimension-zero-ring}.", "The existence of an $f$ is asserted in", "Algebra, Lemma \\ref{algebra-lemma-height-1}.", "\\medskip\\noindent", "Assume $A$ has depth $1$. (This is the maximum by", "Algebra, Lemma \\ref{algebra-lemma-bound-depth} and holds if $A$ is reduced by", "Algebra, Lemma \\ref{algebra-lemma-criterion-reduced}.)", "Then $\\mathfrak m$ is not an associated prime of $A$.", "Every minimal prime of $A$ is an associated prime", "(Algebra, Proposition", "\\ref{algebra-proposition-minimal-primes-associated-primes}).", "Hence the set of nonzerodivisors of $A$ is exactly the set of elements", "not contained in any of the minimal primes by", "Algebra, Lemma \\ref{algebra-lemma-ass-zero-divisors}.", "Thus (4) is equivalent to (2).", "Part (5) is equivalent to (3) by", "Algebra, Lemma \\ref{algebra-lemma-total-ring-fractions-no-embedded-points}.", "\\medskip\\noindent", "Then $A_\\mathfrak p$ is a field for", "$\\mathfrak p \\subset A$ minimal, see", "Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring}.", "Hence (3) is equivalent ot (6)." ], "refs": [ "algebra-lemma-Noetherian-irreducible-components", "algebra-lemma-Zariski-topology", "algebra-lemma-spec-localization", "algebra-proposition-dimension-zero-ring", "algebra-lemma-height-1", "algebra-lemma-bound-depth", "algebra-lemma-criterion-reduced", "algebra-proposition-minimal-primes-associated-primes", "algebra-lemma-ass-zero-divisors", "algebra-lemma-total-ring-fractions-no-embedded-points", "algebra-lemma-minimal-prime-reduced-ring" ], "ref_ids": [ 453, 389, 391, 1410, 682, 770, 1310, 1412, 704, 421, 418 ] } ], "ref_ids": [] }, { "id": 11078, "type": "theorem", "label": "varieties-lemma-pre-delta-invariant", "categories": [ "varieties" ], "title": "varieties-lemma-pre-delta-invariant", "contents": [ "Let $(A, \\mathfrak m)$ be a reduced Nagata $1$-dimensional local ring.", "Let $A'$ be the integral closure of $A$ in the total ring of fractions", "of $A$. Then $A'$ is a normal Nagata ring, $A \\to A'$ is finite, and", "$A'/A$ has finite length as an $A$-module." ], "refs": [], "proofs": [ { "contents": [ "The total ring of fractions is essentially of finite type over $A$", "hence $A \\to A'$ is finite because $A$ is Nagata, see Algebra, Lemma", "\\ref{algebra-lemma-nagata-in-reduced-finite-type-finite-integral-closure}.", "The ring $A'$ is normal for example by", "Algebra, Lemma \\ref{algebra-lemma-characterize-reduced-ring-normal} and", "\\ref{algebra-lemma-Noetherian-irreducible-components}.", "The ring $A'$ is Nagata for example by", "Algebra, Lemma \\ref{algebra-lemma-quasi-finite-over-nagata}.", "Choose $f \\in \\mathfrak m$ as in Lemma \\ref{lemma-pre-pre-delta-invariant}.", "As $A' \\subset A_f$ it is clear that $A_f = A'_f$. Hence the support of the", "finite $A$-module $A'/A$ is contained in $\\{\\mathfrak m\\}$.", "It follows that it has finite length by", "Algebra, Lemma \\ref{algebra-lemma-support-point}." ], "refs": [ "algebra-lemma-nagata-in-reduced-finite-type-finite-integral-closure", "algebra-lemma-characterize-reduced-ring-normal", "algebra-lemma-Noetherian-irreducible-components", "algebra-lemma-quasi-finite-over-nagata", "varieties-lemma-pre-pre-delta-invariant", "algebra-lemma-support-point" ], "ref_ids": [ 1347, 515, 453, 1350, 11077, 693 ] } ], "ref_ids": [] }, { "id": 11079, "type": "theorem", "label": "varieties-lemma-delta-invariant-is-zero", "categories": [ "varieties" ], "title": "varieties-lemma-delta-invariant-is-zero", "contents": [ "Let $A$ be a reduced Nagata local ring of dimension $1$.", "The $\\delta$-invariant of $A$ is $0$ if and only if", "$A$ is a discrete valuation ring." ], "refs": [], "proofs": [ { "contents": [ "If $A$ is a discrete valuation ring, then $A$ is normal and", "the ring $A'$ is equal to $A$. Conversely, if the", "$\\delta$-invariant of $A$ is $0$, then $A$ is integrally", "closed in its total ring of fractions which implies that", "$A$ is normal", "(Algebra, Lemma \\ref{algebra-lemma-characterize-reduced-ring-normal})", "and this forces $A$ to be a discrete valuation ring by", "Algebra, Lemma \\ref{algebra-lemma-characterize-dvr}." ], "refs": [ "algebra-lemma-characterize-reduced-ring-normal", "algebra-lemma-characterize-dvr" ], "ref_ids": [ 515, 1023 ] } ], "ref_ids": [] }, { "id": 11080, "type": "theorem", "label": "varieties-lemma-normalization-same-after-completion", "categories": [ "varieties" ], "title": "varieties-lemma-normalization-same-after-completion", "contents": [ "Let $A$ be a reduced Nagata local ring of dimension $1$.", "Let $A \\to A'$ be as in Lemma \\ref{lemma-pre-delta-invariant}.", "Let $A^h$, $A^{sh}$, resp.\\ $A^\\wedge$", "be the henselization, strict henselization, reps.\\ completion of $A$.", "Then $A^h$, $A^{sh}$, resp. $A^\\wedge$ is a reduced Nagata local", "ring of dimension $1$ and", "$A' \\otimes_A A^h$, $A' \\otimes_A A^{sh}$, resp. $A' \\otimes_A A^\\wedge$", "is the integral closure of $A^h$, $A^{sh}$, resp.\\ $A^\\wedge$", "in its total ring of fractions." ], "refs": [ "varieties-lemma-pre-delta-invariant" ], "proofs": [ { "contents": [ "Observe that $A^\\wedge$ is reduced, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-completion-reduced}.", "The rings $A^h$ and $A^{sh}$ are reduced by", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-reduced}.", "The dimensions of $A$, $A^h$, $A^{sh}$, and $A^\\wedge$ are the same", "by More on Algebra, Lemmas", "\\ref{more-algebra-lemma-completion-dimension} and", "\\ref{more-algebra-lemma-henselization-dimension}.", "\\medskip\\noindent", "Recall that a Noetherian local ring is Nagata if and only if the", "formal fibres of $A$ are geometrically reduced, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-Nagata-local-ring}.", "This property is inherited by $A^h$ and $A^{sh}$, see", "the material in More on Algebra, Section", "\\ref{more-algebra-section-properties-formal-fibres}", "and especially Lemma \\ref{more-algebra-lemma-henselization-P-ring}.", "The completion is Nagata by", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-complete-local-Nagata}.", "\\medskip\\noindent", "Now we come to the statement on integral closures. Before continuing", "let us pick $f \\in \\mathfrak m$ as in", "Lemma \\ref{lemma-pre-pre-delta-invariant}.", "Then the image of $f$ in $A^h$, $A^{sh}$, and $A^\\wedge$", "clearly is an element satisfying properties (1) -- (6)", "in that ring.", "\\medskip\\noindent", "Since $A \\to A'$ is finite we see that", "$A' \\otimes_A A^h$ and $A' \\otimes_A A^{sh}$", "is the product of henselian local rings finite over $A^h$ and", "$A^{sh}$, see Algebra, Lemma \\ref{algebra-lemma-finite-over-henselian}.", "Each of these local rings is the henselization of $A'$ at a", "maximal ideal $\\mathfrak m' \\subset A'$ lying over $\\mathfrak m$, see", "Algebra, Lemma \\ref{algebra-lemma-quasi-finite-henselization} or", "\\ref{algebra-lemma-quasi-finite-strict-henselization}.", "Hence these local rings are normal domains by", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-normal}.", "It follows that $A' \\otimes_A A^h$ and $A' \\otimes_A A^{sh}$", "are normal rings. Since $A^h \\to A' \\otimes_A A^h$ and", "$A^{sh} \\to A' \\otimes_A A^{sh}$ are finite (hence integral) and since", "$A' \\otimes_A A^h \\subset (A^h)_f = Q(A^h)$ and", "$A' \\otimes_A A^{sh} \\subset (A^{sh})_f = Q(A^{sh})$", "we conclude that $A' \\otimes_A A^h$ and $A' \\otimes_A A^{sh}$ are", "the desired integral closures.", "\\medskip\\noindent", "For the completion we argue in entirely the same manner. First,", "by Algebra, Lemma \\ref{algebra-lemma-completion-finite-extension} we have", "$$", "A' \\otimes_A A^\\wedge = (A')^\\wedge =", "\\prod\\nolimits (A'_{\\mathfrak m'})^\\wedge", "$$", "The local rings $A'_{\\mathfrak m'}$ are normal and have dimension $1$", "(by Algebra, Lemma \\ref{algebra-lemma-finite-in-codim-1} for example or", "the discussion in", "Algebra, Section \\ref{algebra-section-homomorphism-dimension}).", "Thus $A'_{\\mathfrak m'}$ is a discrete valuation ring, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-dvr}.", "Hence $(A'_{\\mathfrak m'})^\\wedge$ is a discrete valuation ring", "by More on Algebra, Lemma \\ref{more-algebra-lemma-completion-dvr}.", "It follows that $A' \\otimes_A A^\\wedge$ is a normal ring and", "we can conclude in exactly the same manner as before." ], "refs": [ "more-algebra-lemma-completion-reduced", "more-algebra-lemma-henselization-reduced", "more-algebra-lemma-completion-dimension", "more-algebra-lemma-henselization-dimension", "more-algebra-lemma-Nagata-local-ring", "more-algebra-lemma-henselization-P-ring", "algebra-lemma-Noetherian-complete-local-Nagata", "varieties-lemma-pre-pre-delta-invariant", "algebra-lemma-finite-over-henselian", "algebra-lemma-quasi-finite-henselization", "algebra-lemma-quasi-finite-strict-henselization", "more-algebra-lemma-henselization-normal", "algebra-lemma-completion-finite-extension", "algebra-lemma-finite-in-codim-1", "algebra-lemma-characterize-dvr", "more-algebra-lemma-completion-dvr" ], "ref_ids": [ 10047, 10058, 10042, 10061, 10107, 10100, 1353, 11077, 1277, 1300, 1306, 10060, 876, 991, 1023, 10046 ] } ], "ref_ids": [ 11078 ] }, { "id": 11081, "type": "theorem", "label": "varieties-lemma-delta-same-after-completion", "categories": [ "varieties" ], "title": "varieties-lemma-delta-same-after-completion", "contents": [ "Let $A$ be a reduced Nagata local ring of dimension $1$.", "The $\\delta$-invariant of $A$ is the same as the", "$\\delta$-invariant of the henselization, strict henselization,", "or the completion of $A$." ], "refs": [], "proofs": [ { "contents": [ "Let us do this in case of the completion $B = A^\\wedge$;", "the other cases are proved in exactly the same manner.", "Let $A'$, resp.\\ $B'$ be the integral closure of $A$, resp.\\ $B$", "in its total ring of fractions.", "Then $B' = A' \\otimes_A B$ by", "Lemma \\ref{lemma-normalization-same-after-completion}.", "Hence $B'/B = A'/A \\otimes_A B$.", "The equality now follows from", "Algebra, Lemma \\ref{algebra-lemma-pullback-module}", "and the fact that $B \\otimes_A \\kappa_A = \\kappa_B$." ], "refs": [ "varieties-lemma-normalization-same-after-completion", "algebra-lemma-pullback-module" ], "ref_ids": [ 11080, 640 ] } ], "ref_ids": [] }, { "id": 11082, "type": "theorem", "label": "varieties-lemma-delta-invariant-and-change-of-fields", "categories": [ "varieties" ], "title": "varieties-lemma-delta-invariant-and-change-of-fields", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.", "Let $K/k$ be a field extension and set $Y = X_K$.", "Let $y \\in Y$ with image $x \\in X$.", "Assume $X$ is geometrically reduced at $x$ and", "$\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y}) = 1$.", "Then", "$$", "\\delta\\text{-invariant of }X\\text{ at }x \\leq", "\\delta\\text{-invariant of }Y\\text{ at }y", "$$" ], "refs": [], "proofs": [ { "contents": [ "Set $A = \\mathcal{O}_{X, x}$ and $B = \\mathcal{O}_{Y, y}$.", "By Lemma \\ref{lemma-geometrically-reduced-at-point}", "we see that $A$ is geometrically reduced.", "Hence $B$ is a localization of $A \\otimes_k K$.", "Let $A \\to A'$ be as in Lemma \\ref{lemma-pre-delta-invariant}.", "Then", "$$", "B' = B \\otimes_{(A \\otimes_k K)} (A' \\otimes_k K)", "$$", "is finite over $B$ and $B \\to B'$ induces", "an isomorphism on total rings of fractions. Namely, pick $f \\in \\mathfrak m_A$", "satisfying (1) -- (6) of Lemma \\ref{lemma-pre-pre-delta-invariant};", "since $\\dim(B) = 1$ we see that $f \\in \\mathfrak m_B$", "playes the same role for $B$ and we see that $B_f = B'_f$ because $A_f = A'_f$.", "Let $B''$ be the integral closure of $B$ in its", "total ring of fractions as in Lemma \\ref{lemma-pre-delta-invariant}.", "Then $B' \\subset B''$. Thus the $\\delta$-invariant of $Y$ at $y$ is", "$\\text{length}_B(B''/B)$ and", "\\begin{align*}", "\\text{length}_B(B''/B)", "& \\geq", "\\text{length}_B(B'/B) \\\\", "& =", "\\text{length}_B((A'/A) \\otimes_A B) \\\\", "& =", "\\text{length}_B(B/\\mathfrak m_A B) \\text{length}_A(A'/A)", "\\end{align*}", "by Algebra, Lemma \\ref{algebra-lemma-pullback-module}", "since $A \\to B$ is flat (as a localization of $A \\to A \\otimes_k K$).", "Since $\\text{length}_A(A'/A)$ is the $\\delta$-invariant of $X$ at $x$", "and since", "$\\text{length}_B(B/\\mathfrak m_A B) \\geq 1$ the lemma is proved." ], "refs": [ "varieties-lemma-geometrically-reduced-at-point", "varieties-lemma-pre-delta-invariant", "varieties-lemma-pre-pre-delta-invariant", "varieties-lemma-pre-delta-invariant", "algebra-lemma-pullback-module" ], "ref_ids": [ 10906, 11078, 11077, 11078, 640 ] } ], "ref_ids": [] }, { "id": 11083, "type": "theorem", "label": "varieties-lemma-delta-invariant-and-change-of-fields-better", "categories": [ "varieties" ], "title": "varieties-lemma-delta-invariant-and-change-of-fields-better", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.", "Let $K/k$ be a field extension and set $Y = X_K$.", "Let $y \\in Y$ with image $x \\in X$.", "Assume assumptions (a), (b), (c) of", "Lemma \\ref{lemma-geometrically-normal-in-codim-1}", "hold for $x \\in X$ and that $\\dim(\\mathcal{O}_{Y, y}) = 1$.", "Then the $\\delta$-invariant", "of $X$ at $x$ is $\\delta$-invariant of $Y$ at $y$." ], "refs": [ "varieties-lemma-geometrically-normal-in-codim-1" ], "proofs": [ { "contents": [ "Set $A = \\mathcal{O}_{X, x}$ and $B = \\mathcal{O}_{Y, y}$.", "By Lemma \\ref{lemma-geometrically-normal-in-codim-1}", "we see that $A$ is geometrically reduced.", "Hence $B$ is a localization of $A \\otimes_k K$.", "Let $A \\to A'$ be as in Lemma \\ref{lemma-pre-delta-invariant}.", "By Lemma \\ref{lemma-geometrically-normal-in-codim-1}", "we see that $A' \\otimes_k K$ is normal.", "Hence", "$$", "B' = B \\otimes_{(A \\otimes_k K)} (A' \\otimes_k K)", "$$", "is normal, finite over $B$, and $B \\to B'$ induces", "an isomorphism on total rings of fractions. Namely, pick $f \\in \\mathfrak m_A$", "satisfying (1) -- (6) of Lemma \\ref{lemma-pre-pre-delta-invariant};", "since $\\dim(B) = 1$ we see that $f \\in \\mathfrak m_B$", "playes the same role for $B$ and we see that $B_f = B'_f$ because $A_f = A'_f$.", "It follows that $B \\to B'$ is as in Lemma \\ref{lemma-pre-delta-invariant}", "for $B$. Thus we have to show that", "$\\text{length}_A(A'/A) =", "\\text{length}_B(B'/B) = \\text{length}_B((A'/A) \\otimes_A B)$.", "Since $A \\to B$ is flat (as a localization of $A \\to A \\otimes_k K$)", "and since $\\mathfrak m_B = \\mathfrak m_A B$ (because", "$B/\\mathfrak m_A B$ is zero dimensional by the remarks above and", "a localization of $K \\otimes_k \\kappa(x)$ which is reduced as", "$\\kappa(x)$ is separable over $k$) we conclude by", "Algebra, Lemma \\ref{algebra-lemma-pullback-module}." ], "refs": [ "varieties-lemma-geometrically-normal-in-codim-1", "varieties-lemma-pre-delta-invariant", "varieties-lemma-geometrically-normal-in-codim-1", "varieties-lemma-pre-pre-delta-invariant", "varieties-lemma-pre-delta-invariant", "algebra-lemma-pullback-module" ], "ref_ids": [ 11017, 11078, 11017, 11077, 11078, 640 ] } ], "ref_ids": [ 11017 ] }, { "id": 11084, "type": "theorem", "label": "varieties-lemma-number-of-branches", "categories": [ "varieties" ], "title": "varieties-lemma-number-of-branches", "contents": [ "Let $X$ be a scheme. Assume every quasi-compact open of $X$ has", "finitely many irreducible components. Let $\\nu : X^\\nu \\to X$", "be the normalization of $X$. Let $x \\in X$.", "\\begin{enumerate}", "\\item The number of branches of $X$ at $x$ is the number of", "inverse images of $x$ in $X^\\nu$.", "\\item The number of geometric branches of $X$ at $x$ is", "$\\sum_{\\nu(x^\\nu) = x} [\\kappa(x^\\nu) : \\kappa(x)]_s$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First note that the assumption on $X$ exactly means that the", "normalization is defined, see", "Morphisms, Definition \\ref{morphisms-definition-normalization}.", "Then the stalk $A' = (\\nu_*\\mathcal{O}_{X^\\nu})_x$ is the", "integral closure of $A = \\mathcal{O}_{X, x}$ in the total", "ring of fractions of $A_{red}$, see", "Morphisms, Lemma \\ref{morphisms-lemma-stalk-normalization}.", "Since $\\nu$ is an integral morphism, we see that", "the points of $X^\\nu$ lying over $x$ correspond", "to the primes of $A'$ lying over the maximal ideal $\\mathfrak m$", "of $A$. As $A \\to A'$ is integral, this is the same thing as", "the maximal ideals of $A'$", "(Algebra, Lemmas \\ref{algebra-lemma-integral-no-inclusion} and", "\\ref{algebra-lemma-integral-going-up}).", "Thus the lemma now follows", "from its algebraic counterpart:", "More on Algebra, Lemma \\ref{more-algebra-lemma-number-of-branches-1}." ], "refs": [ "morphisms-definition-normalization", "morphisms-lemma-stalk-normalization", "algebra-lemma-integral-no-inclusion", "algebra-lemma-integral-going-up", "more-algebra-lemma-number-of-branches-1" ], "ref_ids": [ 5592, 5514, 498, 500, 10469 ] } ], "ref_ids": [] }, { "id": 11085, "type": "theorem", "label": "varieties-lemma-geometric-branches-and-change-of-fields", "categories": [ "varieties" ], "title": "varieties-lemma-geometric-branches-and-change-of-fields", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.", "Let $K/k$ be an extension of fields. Let $y \\in X_K$ be a", "point with image $x$ in $X$. Then the number of", "geometric branches of $X$ at $x$ is the number of geometric", "branches of $X_K$ at $y$." ], "refs": [], "proofs": [ { "contents": [ "Write $Y = X_K$ and let $X^\\nu$, resp.\\ $Y^\\nu$ be the normalization", "of $X$, resp.\\ $Y$. Consider the commutative diagram", "$$", "\\xymatrix{", "Y^\\nu \\ar[r] \\ar[d] & X^\\nu_K \\ar[r] \\ar[d]_{\\nu_K} & X^\\nu \\ar[d]_\\nu \\\\", "Y \\ar@{=}[r] & Y \\ar[r] & X", "}", "$$", "By Lemma \\ref{lemma-normalization-and-change-of-fields} we see that", "the left top horizontal arrow is a universal homeomorphism.", "Hence it induces purely inseparable residue field extensions, see", "Morphisms, Lemmas \\ref{morphisms-lemma-universal-homeomorphism} and", "\\ref{morphisms-lemma-universally-injective}.", "Thus the number of geometric branches of $Y$ at $y$ is", "$\\sum_{\\nu_K(y') = y} [\\kappa(y') : \\kappa(y)]_s$", "by Lemma \\ref{lemma-number-of-branches}. Similarly", "$\\sum_{\\nu(x') = x} [\\kappa(x') : \\kappa(x)]_s$ is the number of", "geometric branches of $X$ at $x$. Using Schemes, Lemma", "\\ref{schemes-lemma-points-fibre-product}", "our statement follows from the following algebra fact:", "given a field extension $l/\\kappa$ and", "an algebraic field extension $m/\\kappa$, then", "$$", "\\sum\\nolimits_{m \\otimes_\\kappa l \\to m'} [m' : l']_s = [m : \\kappa]_s", "$$", "where the sum is over the quotient fields of $m \\otimes_\\kappa l$.", "One can prove this in an elementary way, or one can use", "Lemma \\ref{lemma-separably-closed-field-connected-components}", "applied to", "$$", "\\Spec(m \\otimes_\\kappa l) \\times_{\\Spec(l)} \\Spec(\\overline{l}) =", "\\Spec(m) \\otimes_{\\Spec(\\kappa)} \\Spec(\\overline{l})", "\\longrightarrow", "\\Spec(m) \\times_{\\Spec(\\kappa)} \\Spec(\\overline{\\kappa})", "$$", "because one can interpret $[m : \\kappa]_s$ as the number of", "connected components of the right hand side and the sum", "$\\sum_{m \\otimes_\\kappa l \\to m'} [m' : l']_s$", "as the number of connected components of the left hand side." ], "refs": [ "varieties-lemma-normalization-and-change-of-fields", "morphisms-lemma-universal-homeomorphism", "morphisms-lemma-universally-injective", "varieties-lemma-number-of-branches", "schemes-lemma-points-fibre-product", "varieties-lemma-separably-closed-field-connected-components" ], "ref_ids": [ 11016, 5454, 5167, 11084, 7693, 10918 ] } ], "ref_ids": [] }, { "id": 11086, "type": "theorem", "label": "varieties-lemma-geometrically-unibranch-and-change-of-fields", "categories": [ "varieties" ], "title": "varieties-lemma-geometrically-unibranch-and-change-of-fields", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.", "Let $K/k$ be an extension of fields. Let $y \\in X_K$ be a", "point with image $x$ in $X$. Then $X$ is geometrically unibranch", "at $x$ if and only if $X_K$ is geometrically unibranch at $y$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from", "Lemma \\ref{lemma-geometric-branches-and-change-of-fields}", "and More on Algebra, Lemma \\ref{more-algebra-lemma-number-of-branches-1}." ], "refs": [ "varieties-lemma-geometric-branches-and-change-of-fields", "more-algebra-lemma-number-of-branches-1" ], "ref_ids": [ 11085, 10469 ] } ], "ref_ids": [] }, { "id": 11087, "type": "theorem", "label": "varieties-lemma-delta-number-branches-inequality-sh", "categories": [ "varieties" ], "title": "varieties-lemma-delta-number-branches-inequality-sh", "contents": [ "Let $(A, \\mathfrak m)$ be a strictly henselian", "$1$-dimensional reduced Nagata local ring. Then", "$$", "\\delta\\text{-invariant of }A \\geq \\text{number of geometric branches of }A - 1", "$$", "If equality holds, then $A$ is a wedge of $n \\geq 1$ strictly henselian", "discrete valuation rings." ], "refs": [], "proofs": [ { "contents": [ "The number of geometric branches is equal to the number of branches of $A$", "(immediate from", "More on Algebra, Definition \\ref{more-algebra-definition-number-of-branches}).", "Let $A \\to A'$ be as in Lemma \\ref{lemma-pre-delta-invariant}.", "Observe that the number of branches of $A$ is the number", "of maximal ideals of $A'$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-number-of-branches-1}.", "There is a surjection", "$$", "A'/A \\longrightarrow", "\\left(\\prod\\nolimits_{\\mathfrak m'} \\kappa(\\mathfrak m')\\right)/", "\\kappa(\\mathfrak m)", "$$", "Since $\\dim_{\\kappa(\\mathfrak m)} \\prod \\kappa(\\mathfrak m')$", "is $\\geq$ the number of branches, the inequality is obvious.", "\\medskip\\noindent", "If equality holds, then $\\kappa(\\mathfrak m') = \\kappa(\\mathfrak m)$", "for all $\\mathfrak m' \\subset A'$ and the displayed arrow above", "is an isomorphism. Since $A$ is henselian and", "$A \\to A'$ is finite, we see that $A'$ is a product of local", "henselian rings, see Algebra, Lemma \\ref{algebra-lemma-finite-over-henselian}.", "The factors are the local rings $A'_{\\mathfrak m'}$ and as", "$A'$ is normal, these factors are discrete valuation rings", "(Algebra, Lemma \\ref{algebra-lemma-characterize-dvr}).", "Since the displayed arrow is an isomorphism we see that", "$A$ is indeed the wedge of these local rings." ], "refs": [ "more-algebra-definition-number-of-branches", "varieties-lemma-pre-delta-invariant", "more-algebra-lemma-number-of-branches-1", "algebra-lemma-finite-over-henselian", "algebra-lemma-characterize-dvr" ], "ref_ids": [ 10638, 11078, 10469, 1277, 1023 ] } ], "ref_ids": [] }, { "id": 11088, "type": "theorem", "label": "varieties-lemma-delta-number-branches-inequality", "categories": [ "varieties" ], "title": "varieties-lemma-delta-number-branches-inequality", "contents": [ "Let $(A, \\mathfrak m)$ be a $1$-dimensional reduced Nagata local ring. Then", "$$", "\\delta\\text{-invariant of }A \\geq \\text{number of geometric branches of }A - 1", "$$" ], "refs": [], "proofs": [ { "contents": [ "We may replace $A$ by the strict henselization of $A$ without", "changing the $\\delta$-invariant", "(Lemma \\ref{lemma-delta-same-after-completion}) and", "without changing the number of geometric branches of $A$", "(this is immediate from the definition, see", "More on Algebra, Definition \\ref{more-algebra-definition-number-of-branches}).", "Thus we may assume $A$ is strictly henselian and we may", "apply Lemma \\ref{lemma-delta-number-branches-inequality-sh}." ], "refs": [ "varieties-lemma-delta-same-after-completion", "more-algebra-definition-number-of-branches", "varieties-lemma-delta-number-branches-inequality-sh" ], "ref_ids": [ 11081, 10638, 11087 ] } ], "ref_ids": [] }, { "id": 11089, "type": "theorem", "label": "varieties-lemma-normalize-noetherian-dim-1", "categories": [ "varieties" ], "title": "varieties-lemma-normalize-noetherian-dim-1", "contents": [ "Let $X$ be a locally Noetherian scheme of dimension $1$.", "Let $\\nu : X^\\nu \\to X$ be the normalization. Then", "\\begin{enumerate}", "\\item $\\nu$ is integral, surjective, and induces a bijection", "on irreducible components,", "\\item there is a factorization $X^\\nu \\to X_{red} \\to X$", "and the morphism $X^\\nu \\to X_{red}$ is the normalization", "of $X_{red}$,", "\\item $X^\\nu \\to X_{red}$ is birational,", "\\item for every closed point $x \\in X$ the stalk", "$(\\nu_*\\mathcal{O}_{X^\\nu})_x$ is the integral closure", "of $\\mathcal{O}_{X, x}$ in the total ring of fractions", "of $(\\mathcal{O}_{X, x})_{red} = \\mathcal{O}_{X_{red}, x}$,", "\\item the fibres of $\\nu$ are finite and the residue", "field extensions are finite,", "\\item $X^\\nu$ is a disjoint union of integral normal Noetherian", "schemes and each affine open is the spectrum of a finite", "product of Dedekind domains.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Many of the results are in fact general properties of the normalization", "morphism, see", "Morphisms, Lemmas \\ref{morphisms-lemma-normalization-reduced},", "\\ref{morphisms-lemma-stalk-normalization},", "\\ref{morphisms-lemma-normalization-normal}, and", "\\ref{morphisms-lemma-normalization-birational}.", "What is not clear is that the fibres are finite,", "that the induced residue field extensions are finite, and", "that $X^\\nu$ locally looks like the spectrum of a", "Dedekind domain (and hence is Noetherian).", "To see this we may assume that $X = \\Spec(A)$ is", "affine, Noetherian, dimension $1$, and that $A$ is reduced.", "Then we may use the description in", "Morphisms, Lemma \\ref{morphisms-lemma-description-normalization}", "to reduce to the case where $A$ is a Noetherian domain", "of dimension $1$. In this case the desired properties", "follow from Krull-Akizuki in the form stated in", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-Dedekind}." ], "refs": [ "morphisms-lemma-normalization-reduced", "morphisms-lemma-stalk-normalization", "morphisms-lemma-normalization-normal", "morphisms-lemma-normalization-birational", "morphisms-lemma-description-normalization", "algebra-lemma-integral-closure-Dedekind" ], "ref_ids": [ 5512, 5514, 5515, 5517, 5513, 1042 ] } ], "ref_ids": [] }, { "id": 11090, "type": "theorem", "label": "varieties-lemma-prepare-delta-invariant", "categories": [ "varieties" ], "title": "varieties-lemma-prepare-delta-invariant", "contents": [ "Let $X$ be a reduced Nagata scheme of dimension $1$. Let $\\nu : X^\\nu \\to X$", "be the normalization. Let $x \\in X$ denote a closed point. Then", "\\begin{enumerate}", "\\item $\\nu : X^\\nu \\to X$ is finite, surjective, and birational,", "\\item $\\mathcal{O}_X \\subset \\nu_*\\mathcal{O}_{X^\\nu}$ and", "$\\nu_*\\mathcal{O}_{X^\\nu}/\\mathcal{O}_X$ is a direct sum of", "skyscraper sheaves $\\mathcal{Q}_x$ in the singular points $x$ of $X$,", "\\item $A' = (\\nu_*\\mathcal{O}_{X^\\nu})_x$ is the integral closure", "of $A = \\mathcal{O}_{X, x}$ in its total ring of fractions,", "\\item $\\mathcal{Q}_x = A'/A$ has finite length equal to the", "$\\delta$-invariant of $X$ at $x$,", "\\item $A'$ is a semi-local ring which is a finite product of", "Dedekind domains,", "\\item $A^\\wedge$ is a reduced Noetherian complete", "local ring of dimension $1$,", "\\item $(A')^\\wedge$ is the integral closure of $A^\\wedge$", "in its total ring of fractions,", "\\item $(A')^\\wedge$ is a finite product of", "complete discrete valuation rings, and", "\\item $A'/A \\cong (A')^\\wedge/A^\\wedge$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may and will use all the results of", "Lemma \\ref{lemma-normalize-noetherian-dim-1}.", "Finiteness of $\\nu$ follows from", "Morphisms, Lemma \\ref{morphisms-lemma-nagata-normalization}.", "Since $X$ is reduced, Nagata, of dimension $1$, we see that the", "regular locus is a dense open $U \\subset X$ by", "More on Algebra, Proposition \\ref{more-algebra-proposition-ubiquity-J-2}.", "Since a regular scheme is normal, this shows that $\\nu$", "is an isomorphism over $U$.", "Since $\\dim(X) \\leq 1$ this implies that $\\nu$ is not an isomorphism", "over a discrete set of closed points $x \\in X$.", "In particular we see that we have a short exact sequence", "$$", "0 \\to \\mathcal{O}_X \\to \\nu_*\\mathcal{O}_{X^\\nu} \\to", "\\bigoplus\\nolimits_{x \\in X \\setminus U} \\mathcal{Q}_x \\to 0", "$$", "As we have the description of the stalks of $\\nu_*\\mathcal{O}_{X^\\nu}$", "by Lemma \\ref{lemma-normalize-noetherian-dim-1}, we conclude that", "$Q_x = A'/A$ indeed has length equal to the $\\delta$-invariant of $X$ at $x$.", "Note that $Q_x \\not = 0$ exactly when $x$ is a singular point for", "example by Lemma \\ref{lemma-delta-invariant-is-zero}.", "The description of $A'$ as a product of semi-local Dedekind domains follows", "from Lemma \\ref{lemma-normalize-noetherian-dim-1} as well.", "The relationship between $A$, $A'$, and $(A')^\\wedge$", "we have see in Lemma \\ref{lemma-normalization-same-after-completion}", "(and its proof)." ], "refs": [ "varieties-lemma-normalize-noetherian-dim-1", "morphisms-lemma-nagata-normalization", "more-algebra-proposition-ubiquity-J-2", "varieties-lemma-normalize-noetherian-dim-1", "varieties-lemma-delta-invariant-is-zero", "varieties-lemma-normalize-noetherian-dim-1", "varieties-lemma-normalization-same-after-completion" ], "ref_ids": [ 11089, 5520, 10578, 11089, 11079, 11089, 11080 ] } ], "ref_ids": [] }, { "id": 11091, "type": "theorem", "label": "varieties-lemma-characterize-open-immersion", "categories": [ "varieties" ], "title": "varieties-lemma-characterize-open-immersion", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $X^0$ denote the set", "of generic points of irreducible components of $X$. If", "\\begin{enumerate}", "\\item $f$ is separated,", "\\item there is an open covering $X = \\bigcup U_i$ such that", "$f|_{U_i} : U_i \\to Y$ is an open immersion, and", "\\item if $\\xi, \\xi' \\in X^0$, $\\xi \\not = \\xi'$, then $f(\\xi) \\not = f(\\xi')$,", "\\end{enumerate}", "then $f$ is an open immersion." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $y = f(x) = f(x')$. Pick a specialization $y_0 \\leadsto y$", "where $y_0$ is a generic point of an irreducible component of $Y$.", "Since $f$ is locally on the source an isomorphism we can pick specializations", "$x_0 \\leadsto x$ and $x'_0 \\leadsto x'$ mapping to $y_0 \\leadsto y$.", "Note that $x_0, x'_0 \\in X^0$. Hence $x_0 = x'_0$ by assumption (3).", "As $f$ is separated we conclude that $x = x'$. Thus $f$ is an open immersion." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11092, "type": "theorem", "label": "varieties-lemma-local-isomorphism", "categories": [ "varieties" ], "title": "varieties-lemma-local-isomorphism", "contents": [ "Let $X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point with", "image $s \\in S$. If", "\\begin{enumerate}", "\\item $\\mathcal{O}_{X, x} = \\mathcal{O}_{S, s}$,", "\\item $X$ is reduced,", "\\item $X \\to S$ is of finite type, and", "\\item $S$ has finitely many irreducible components,", "\\end{enumerate}", "then there exists an open neighbourhood $U$", "of $x$ such that $f|_U$ is an open immersion." ], "refs": [], "proofs": [ { "contents": [ "We may remove the (finitely many) irreducible components of $S$", "which do not contain $s$. We may replace $S$ by an affine open", "neighbourhood of $s$. We may replace $X$ by an affine open neighbourhood", "of $x$. Say $S = \\Spec(A)$ and $X = \\Spec(B)$. Let $\\mathfrak q \\subset B$,", "resp.\\ $\\mathfrak p \\subset A$ be the prime ideal corresponding to $x$,", "resp.\\ $s$. As $A$ is a reduced and all of the minimal primes of", "$A$ are contained in $\\mathfrak p$ we see that $A \\subset A_\\mathfrak p$.", "As $X \\to S$ is of finite type, $B$ is of finite type over $A$.", "Let $b_1, \\ldots, b_n \\in B$ be elements which generate $B$ over $A$", "Since $A_\\mathfrak p = B_\\mathfrak q$ we can find", "$f \\in A$, $f \\not \\in \\mathfrak p$", "and $a_i \\in A$ such that $b_i$ and $a_i/f$ have the same image", "in $B_\\mathfrak q$. Thus we can find $g \\in B$, $g \\not \\in \\mathfrak q$", "such that $g(fb_i - a_i) = 0$ in $B$. It follows that the image of", "$A_f \\to B_{fg}$ contains the images of $b_1, \\ldots, b_n$, in", "particular also the image of $g$.", "Choose $n \\geq 0$ and $f' \\in A$ such that $f'/f^n$ maps", "to the image of $g$ in $B_{fg}$. Since $A_\\mathfrak p = B_\\mathfrak q$", "we see that $f' \\not \\in \\mathfrak p$.", "We conclude that $A_{ff'} \\to B_{fg}$ is surjective.", "Finally, as $A_{ff'} \\subset A_\\mathfrak p = B_\\mathfrak q$ (see above)", "the map $A_{ff'} \\to B_{fg}$ is injective, hence an isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11093, "type": "theorem", "label": "varieties-lemma-points-in-affine", "categories": [ "varieties" ], "title": "varieties-lemma-points-in-affine", "contents": [ "Let $f : T \\to X$ be a morphism of schemes. Let $X^0$, resp.\\ $T^0$", "denote the sets of generic points of irreducible components.", "Let $t_1, \\ldots, t_m \\in T$ be a finite set of points", "with images $x_j = f(t_j)$. If", "\\begin{enumerate}", "\\item $T$ is affine,", "\\item $X$ is quasi-separated,", "\\item $X^0$ is finite", "\\item $f(T^0) \\subset X^0$ and $f : T^0 \\to X^0$ is injective, and", "\\item $\\mathcal{O}_{X, x_j} = \\mathcal{O}_{T, t_j}$,", "\\end{enumerate}", "then there exists an affine open of $X$ containing $x_1, \\ldots, x_r$." ], "refs": [], "proofs": [ { "contents": [ "Using Limits, Proposition \\ref{limits-proposition-affine}", "there is an immediate reduction to the case where $X$ and $T$ are reduced.", "Details omitted.", "\\medskip\\noindent", "Assume $X$ and $T$ are reduced. We may write $T = \\lim_{i \\in I} T_i$", "as a directed limit of schemes of finite presentation over $X$", "with affine transition morphisms, see", "Limits, Lemma \\ref{limits-lemma-relative-approximation}.", "Pick $i \\in I$ such that $T_i$ is affine, see", "Limits, Lemma \\ref{limits-lemma-limit-affine}.", "Say $T_i = \\Spec(R_i)$ and $T = \\Spec(R)$.", "Let $R' \\subset R$ be the image of $R_i \\to R$.", "Then $T' = \\Spec(R')$ is affine, reduced, of finite type over $X$,", "and $T \\to T'$ dominant. For $j = 1, \\ldots, r$ let $t'_j \\in T'$", "be the image of $t_j$. Consider the local ring maps", "$$", "\\mathcal{O}_{X, x_j} \\to", "\\mathcal{O}_{T', t'_j} \\to", "\\mathcal{O}_{T, t_j}", "$$", "Denote $(T')^0$ the set of generic points of irreducible", "components of $T'$. Let $\\xi \\leadsto t'_j$ be a specialization with", "$\\xi \\in (T')^0$. As $T \\to T'$ is dominant we can choose $\\eta \\in T^0$ ", "mapping to $\\xi$ (warning: a priori we do not know that $\\eta$ specializes", "to $t_j$). Assumption (3) applied to $\\eta$ tells us that the image $\\theta$", "of $\\xi$ in $X$ corresponds to a minimal prime of $\\mathcal{O}_{X, x_j}$.", "Lifting $\\xi$ via the isomorphism of (5) we obtain a specialization", "$\\eta' \\leadsto t_j$ with $\\eta' \\in T^0$ mapping to $\\theta \\leadsto x_j$.", "The injectivity of (4) shows that $\\eta = \\eta'$. Thus", "every minimal prime of $\\mathcal{O}_{T', t'_j}$ lies below", "a minimal prime of $\\mathcal{O}_{T, t_j}$. We conclude that", "$\\mathcal{O}_{T', t'_j} \\to \\mathcal{O}_{T, t_j}$ is injective,", "hence both maps above are isomorphisms.", "\\medskip\\noindent", "By Lemma \\ref{lemma-local-isomorphism} there exists an open", "$U \\subset T'$ containing all the points $t'_j$ such that", "$U \\to X$ is a local isomorphism as in", "Lemma \\ref{lemma-characterize-open-immersion}.", "By that lemma we see that $U \\to X$ is an open immersion.", "Finally, by", "Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-affine}", "we can find an open $W \\subset U \\subset T'$", "containing all the $t'_j$. The image of $W$ in $X$ is the", "desired affine open." ], "refs": [ "limits-proposition-affine", "limits-lemma-relative-approximation", "limits-lemma-limit-affine", "varieties-lemma-local-isomorphism", "varieties-lemma-characterize-open-immersion", "properties-lemma-ample-finite-set-in-affine" ], "ref_ids": [ 15129, 15055, 15043, 11092, 11091, 3062 ] } ], "ref_ids": [] }, { "id": 11094, "type": "theorem", "label": "varieties-lemma-finite-set-codim-1-points-in-affine", "categories": [ "varieties" ], "title": "varieties-lemma-finite-set-codim-1-points-in-affine", "contents": [ "Let $X$ be an integral separated scheme. Let $x_1, \\ldots, x_r \\in X$", "be a finite set of points such that $\\mathcal{O}_{X, x_i}$", "is Noetherian of dimension $\\leq 1$. Then there exists an affine", "open subscheme of $X$ containing all of $x_1, \\ldots, x_r$." ], "refs": [], "proofs": [ { "contents": [ "Let $K$ be the field of rational functions of $X$.", "Set $A_i = \\mathcal{O}_{X, x_i}$. Then $A_i \\subset K$ and $K$", "is the fraction field of $A_i$. Since $X$ is separated, and", "$x_i \\not = x_j$ there cannot be a valuation ring $\\mathcal{O} \\subset K$", "dominating both $A_i$ and $A_j$. Namely, considering the", "diagram", "$$", "\\xymatrix{", "\\Spec(\\mathcal{O}) \\ar[r] \\ar[d] & \\Spec(A_1) \\ar[d] \\\\", "\\Spec(A_2) \\ar[r] & X", "}", "$$", "and applying the valuative criterion of separatedness", "(Schemes, Lemma \\ref{schemes-lemma-separated-implies-valuative})", "we would get $x_i = x_j$. Thus we see by", "Lemma \\ref{lemma-glue-separated}", "that $A_i \\otimes A_j \\to K$ is surjective for all $i \\not = j$.", "By Lemma \\ref{lemma-glue-a-bunch-of-local-rings}", "we see that $A = A_1 \\cap \\ldots \\cap A_r$ is a Noetherian", "semi-local ring with exactly $r$ maximal ideals", "$\\mathfrak m_1, \\ldots, \\mathfrak m_r$ such that $A_i = A_{\\mathfrak m_i}$.", "Moreover,", "$$", "\\Spec(A) = \\Spec(A_1) \\cup \\ldots \\cup \\Spec(A_r)", "$$", "is an open covering and the intersection of any two pieces of this", "covering is $\\Spec(K)$. Thus the given morphisms $\\Spec(A_i) \\to X$", "glue to a morphism of schemes", "$$", "\\Spec(A) \\longrightarrow X", "$$", "mapping $\\mathfrak m_i$ to $x_i$ and inducing isomorphisms of local rings.", "Thus the result follows from Lemma \\ref{lemma-points-in-affine}." ], "refs": [ "schemes-lemma-separated-implies-valuative", "varieties-lemma-glue-separated", "varieties-lemma-glue-a-bunch-of-local-rings", "varieties-lemma-points-in-affine" ], "ref_ids": [ 7719, 11056, 11060, 11093 ] } ], "ref_ids": [] }, { "id": 11095, "type": "theorem", "label": "varieties-lemma-extra-silly", "categories": [ "varieties" ], "title": "varieties-lemma-extra-silly", "contents": [ "Let $A$ be a ring, $I \\subset A$ an ideal,", "$\\mathfrak p_1, \\ldots, \\mathfrak p_r$ primes of $A$, and", "$\\overline{f} \\in A/I$ an element. If $I \\not \\subset \\mathfrak p_i$", "for all $i$, then there exists an $f \\in A$, $f \\not \\in \\mathfrak p_i$", "which maps to $\\overline{f}$ in $A/I$." ], "refs": [], "proofs": [ { "contents": [ "We may assume there are no inclusion relations among the $\\mathfrak p_i$", "(by removing the smaller primes). First pick any $f \\in A$ lifting", "$\\overline{f}$. Let $S$ be the set $s \\in \\{1, \\ldots, r\\}$ such", "that $f \\in \\mathfrak p_s$. If $S$ is empty we are done. If not,", "consider the ideal $J = I \\prod_{i \\not \\in S} \\mathfrak p_i$.", "Note that $J$ is not contained in $\\mathfrak p_s$ for $s \\in S$", "because there are no inclusions among the $\\mathfrak p_i$ and because", "$I$ is not contained in any $\\mathfrak p_i$.", "Hence we can choose $g \\in J$, $g \\not \\in \\mathfrak p_s$ for", "$s \\in S$ by Algebra, Lemma \\ref{algebra-lemma-silly}.", "Then $f + g$ is a solution to the problem posed by the lemma." ], "refs": [ "algebra-lemma-silly" ], "ref_ids": [ 378 ] } ], "ref_ids": [] }, { "id": 11096, "type": "theorem", "label": "varieties-lemma-finite-set-codim-1-points-in-affine-per-component", "categories": [ "varieties" ], "title": "varieties-lemma-finite-set-codim-1-points-in-affine-per-component", "contents": [ "Let $X$ be a scheme. Let $T \\subset X$ be finite set of points. Assume", "\\begin{enumerate}", "\\item $X$ has finitely many irreducible components $Z_1, \\ldots, Z_t$, and", "\\item $Z_i \\cap T$ is contained in an affine open of the reduced", "induced subscheme corresponding to $Z_i$.", "\\end{enumerate}", "Then there exists an affine open subscheme of $X$ containing $T$." ], "refs": [], "proofs": [ { "contents": [ "Using Limits, Proposition \\ref{limits-proposition-affine}", "there is an immediate reduction to the case where $X$", "is reduced. Details omitted. In the rest of the proof we", "endow every closed subset of $X$ with the induced reduced closed", "subscheme structure.", "\\medskip\\noindent", "We argue by induction that we can find an affine open", "$U \\subset Z_1 \\cup \\ldots \\cup Z_r$ containing", "$T \\cap (Z_1 \\cup \\ldots \\cup Z_r)$. For $r = 1$ this holds by assumption.", "Say $r > 1$ and let $U \\subset Z_1 \\cup \\ldots \\cup Z_{r - 1}$", "be an affine open containing $T \\cap (Z_1 \\cup \\ldots \\cup Z_{r - 1})$.", "Let $V \\subset X_r$ be an affine open containing $T \\cap Z_r$ (exists by", "assumption). Then $U \\cap V$ contains", "$T \\cap ( Z_1 \\cup \\ldots \\cup Z_{r - 1} ) \\cap Z_r$.", "Hence", "$$", "\\Delta = (U \\cap Z_r) \\setminus (U \\cap V)", "$$", "does not contain any element of $T$. Note that $\\Delta$ is a closed", "subset of $U$. By prime avoidance (Algebra, Lemma \\ref{algebra-lemma-silly}),", "we can find a standard open $U'$ of $U$ containing $T \\cap U$ and avoiding", "$\\Delta$, i.e., $U' \\cap Z_r \\subset U \\cap V$.", "After replacing $U$ by $U'$ we may assume that $U \\cap V$ is", "closed in $U$.", "\\medskip\\noindent", "Using that by the same arguments as above also the set", "$\\Delta' = (U \\cap (Z_1 \\cup \\ldots \\cup Z_{r - 1})) \\setminus (U \\cap V)$", "does not contain any element of $T$ we find a $h \\in \\mathcal{O}(V)$", "such that $D(h) \\subset V$ contains $T \\cap V$ and such that", "$U \\cap D(h) \\subset U \\cap V$. Using that $U \\cap V$ is closed in $U$", "we can use Lemma \\ref{lemma-extra-silly}", "to find an element $g \\in \\mathcal{O}(U)$ whose restriction to $U \\cap V$", "equals the restriction of $h$ to $U \\cap V$ and such that", "$T \\cap U \\subset D(g)$. Then we can replace $U$ by $D(g)$", "and $V$ by $D(h)$ to reach the situation where $U \\cap V$ is", "closed in both $U$ and $V$. In this case the scheme $U \\cup V$ is", "affine by Limits, Lemma \\ref{limits-lemma-affines-glued-in-closed-affine}.", "This proves the induction step and thereby the lemma." ], "refs": [ "limits-proposition-affine", "algebra-lemma-silly", "varieties-lemma-extra-silly", "limits-lemma-affines-glued-in-closed-affine" ], "ref_ids": [ 15129, 378, 11095, 15083 ] } ], "ref_ids": [] }, { "id": 11097, "type": "theorem", "label": "varieties-lemma-proper-minus-point", "categories": [ "varieties" ], "title": "varieties-lemma-proper-minus-point", "contents": [ "Let $X$ be an irreducible scheme of dimension $> 0$ over a field $k$.", "Let $x \\in X$ be a closed point. The open subscheme $X \\setminus \\{x\\}$", "is not proper over $k$." ], "refs": [], "proofs": [ { "contents": [ "Namely, choose a specialization $x' \\leadsto x$ with $x' \\not = x$", "(for example take $x'$ to be the generic point). By", "Schemes, Lemma \\ref{schemes-lemma-points-specialize}", "there exists a morphism $a : \\Spec(A) \\to X$", "where $A$ is a valuation ring with fraction field $K$", "such that the generic point of $\\Spec(A)$", "maps to $x'$ and the closed point of $\\Spec(A)$ maps to $x$.", "The morphism $\\Spec(K) \\to X \\setminus \\{x\\}$ does", "not extend to a morphism $b : \\Spec(A) \\to X \\setminus \\{x\\}$", "since by the uniqueness in Schemes, Lemma", "\\ref{schemes-lemma-separated-implies-valuative}", "we would have $a = b$ as morphisms into $X$ which is absurd.", "Hence the valuative criterion", "(Schemes, Proposition", "\\ref{schemes-proposition-characterize-universally-closed})", "shows that $X \\setminus \\{x\\} \\to \\Spec(k)$", "is not universally closed, hence not proper." ], "refs": [ "schemes-lemma-points-specialize", "schemes-lemma-separated-implies-valuative", "schemes-proposition-characterize-universally-closed" ], "ref_ids": [ 7704, 7719, 7733 ] } ], "ref_ids": [] }, { "id": 11098, "type": "theorem", "label": "varieties-lemma-dim-1-quasi-projective", "categories": [ "varieties" ], "title": "varieties-lemma-dim-1-quasi-projective", "contents": [ "Let $X$ be a separated finite type scheme over a field $k$.", "If $\\dim(X) \\leq 1$ then $X$ is H-quasi-projective over $k$." ], "refs": [], "proofs": [ { "contents": [ "By Proposition \\ref{proposition-dim-1-noetherian-separated-has-ample}", "the scheme $X$ has an ample invertible sheaf $\\mathcal{L}$.", "By Morphisms, Lemma \\ref{morphisms-lemma-quasi-projective-finite-type-over-S}", "we see that $X$ is isomorphic to a locally", "closed subscheme of $\\mathbf{P}^n_k$ over $\\Spec(k)$. This is", "the definition of being H-quasi-projective over $k$, see", "Morphisms, Definition \\ref{morphisms-definition-quasi-projective}." ], "refs": [ "varieties-proposition-dim-1-noetherian-separated-has-ample", "morphisms-lemma-quasi-projective-finite-type-over-S", "morphisms-definition-quasi-projective" ], "ref_ids": [ 11138, 5393, 5570 ] } ], "ref_ids": [] }, { "id": 11099, "type": "theorem", "label": "varieties-lemma-dim-1-proper-projective", "categories": [ "varieties" ], "title": "varieties-lemma-dim-1-proper-projective", "contents": [ "Let $X$ be a proper scheme over a field $k$.", "If $\\dim(X) \\leq 1$ then $X$ is H-projective over $k$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-dim-1-quasi-projective} we see that $X$ is a", "locally closed subscheme of $\\mathbf{P}^n_k$ for some field $k$.", "Since $X$ is proper over $k$ it follows that $X$ is a closed subscheme", "of $\\mathbf{P}^n_k$", "(Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed})." ], "refs": [ "varieties-lemma-dim-1-quasi-projective", "morphisms-lemma-image-proper-scheme-closed" ], "ref_ids": [ 11098, 5411 ] } ], "ref_ids": [] }, { "id": 11100, "type": "theorem", "label": "varieties-lemma-dim-1-projective-completion", "categories": [ "varieties" ], "title": "varieties-lemma-dim-1-projective-completion", "contents": [ "Let $X$ be a separated scheme of finite type over $k$.", "If $\\dim(X) \\leq 1$, then there exists an open immersion", "$j : X \\to \\overline{X}$ with the following properties", "\\begin{enumerate}", "\\item $\\overline{X}$ is H-projective over $k$, i.e., $\\overline{X}$", "is a closed subscheme of $\\mathbf{P}^d_k$ for some $d$,", "\\item $j(X) \\subset \\overline{X}$ is dense and scheme", "theoretically dense,", "\\item $\\overline{X} \\setminus X = \\{x_1, \\ldots, x_n\\}$", "for some closed points $x_i \\in \\overline{X}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-dim-1-quasi-projective} we may assume $X$ is a", "locally closed subscheme of $\\mathbf{P}^d_k$ for some $d$. Let", "$\\overline{X} \\subset \\mathbf{P}^d_k$ be the scheme theoretic image", "of $X \\to \\mathbf{P}^d_k$, see Morphisms, Definition", "\\ref{morphisms-definition-scheme-theoretic-image}.", "The description in", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-immersion}", "gives properties (1) and (2).", "Then $\\dim(X) = 1 \\Rightarrow \\dim(\\overline{X}) = 1$ for example by", "looking at generic points, see", "Lemma \\ref{lemma-dimension-locally-algebraic}.", "As $\\overline{X}$ is Noetherian, it then", "follows that $\\overline{X} \\setminus X = \\{x_1, \\ldots, x_n\\}$", "is a finite set of closed points." ], "refs": [ "varieties-lemma-dim-1-quasi-projective", "morphisms-definition-scheme-theoretic-image", "morphisms-lemma-quasi-compact-immersion", "varieties-lemma-dimension-locally-algebraic" ], "ref_ids": [ 11098, 5539, 5154, 10989 ] } ], "ref_ids": [] }, { "id": 11101, "type": "theorem", "label": "varieties-lemma-reduced-dim-1-projective-completion", "categories": [ "varieties" ], "title": "varieties-lemma-reduced-dim-1-projective-completion", "contents": [ "Let $X$ be a separated scheme of finite type over $k$.", "If $X$ is reduced and $\\dim(X) \\leq 1$, then there exists", "an open immersion $j : X \\to \\overline{X}$ such that", "\\begin{enumerate}", "\\item $\\overline{X}$ is H-projective over $k$, i.e., $\\overline{X}$", "is a closed subscheme of $\\mathbf{P}^d_k$ for some $d$,", "\\item $j(X) \\subset \\overline{X}$ is dense and scheme", "theoretically dense,", "\\item $\\overline{X} \\setminus X = \\{x_1, \\ldots, x_n\\}$", "for some closed points $x_i \\in \\overline{X}$,", "\\item the local rings $\\mathcal{O}_{\\overline{X}, x_i}$", "are discrete valuation rings for $i = 1, \\ldots, n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $j : X \\to \\overline{X}$ be as in", "Lemma \\ref{lemma-dim-1-projective-completion}.", "Consider the normalization $X'$ of $\\overline{X}$ in", "$X$. By Lemma \\ref{lemma-relative-normalization-finite}", "the morphism $X' \\to \\overline{X}$ is finite.", "By Morphisms, Lemma \\ref{morphisms-lemma-finite-projective}", "$X' \\to \\overline{X}$ is projective. By Morphisms, Lemma", "\\ref{morphisms-lemma-projective-over-quasi-projective-is-H-projective}", "we see that $X' \\to \\overline{X}$ is H-projective.", "By Morphisms, Lemma \\ref{morphisms-lemma-H-projective-composition}", "we see that $X' \\to \\Spec(k)$ is H-projective.", "Let $\\{x'_1, \\ldots, x'_m\\} \\subset X'$ be the inverse", "image of $\\{x_1, \\ldots, x_n\\} = \\overline{X} \\setminus X$.", "Then $\\dim(\\mathcal{O}_{X', x'_i}) = 1$ for all $1 \\leq i \\leq m$.", "Hence the local rings $\\mathcal{O}_{X', x'}$", "are discrete valuation rings by", "Morphisms, Lemma", "\\ref{morphisms-lemma-relative-normalization-normal-codim-1}.", "Then $X \\to X'$ and $\\{x'_1, \\ldots, x'_m\\}$ is as desired." ], "refs": [ "varieties-lemma-dim-1-projective-completion", "varieties-lemma-relative-normalization-finite", "morphisms-lemma-finite-projective", "morphisms-lemma-projective-over-quasi-projective-is-H-projective", "morphisms-lemma-H-projective-composition", "morphisms-lemma-relative-normalization-normal-codim-1" ], "ref_ids": [ 11100, 11014, 5450, 5433, 5424, 5511 ] } ], "ref_ids": [] }, { "id": 11102, "type": "theorem", "label": "varieties-lemma-curve-affine-projective", "categories": [ "varieties" ], "title": "varieties-lemma-curve-affine-projective", "contents": [ "Let $X$ be a curve over $k$. Then either $X$ is an affine scheme or $X$", "is H-projective over $k$." ], "refs": [], "proofs": [ { "contents": [ "Choose $X \\to \\overline{X}$ as in", "Lemma \\ref{lemma-dim-1-projective-completion}.", "By Lemma \\ref{lemma-find-globally-generated}", "we can find a globally generated invertible sheaf $\\mathcal{L}$", "on $\\overline{X}$ and a section $s \\in \\Gamma(\\overline{X}, \\mathcal{L})$", "such that $X = \\overline{X}_s$.", "Choose a basis $s = s_0, s_1, \\ldots, s_m$ of the finite dimensional", "$k$-vector space $\\Gamma(\\overline{X}, \\mathcal{L})$", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-over-affine-cohomology-finite}).", "We obtain a corresponding morphism", "$$", "f : \\overline{X} \\longrightarrow \\mathbf{P}^m_k", "$$", "such that the inverse image of $D_{+}(T_0)$ is $X$, see", "Constructions, Lemma \\ref{constructions-lemma-projective-space}.", "In particular, $f$ is non-constant, i.e., $\\Im(f)$ has more", "than one point. A topological argument shows that $f$ maps the generic", "point $\\eta$ of $\\overline{X}$ to a nonclosed point of $\\mathbf{P}^n_k$.", "Hence if $y \\in \\mathbf{P}^n_k$ is a closed point, then $f^{-1}(\\{y\\})$", "is a closed set of $\\overline{X}$ not containing $\\eta$, hence finite.", "By Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-finite-fibre-finite-in-neighbourhood}\\footnote{One", "can avoid using this lemma which relies on the theorem of formal", "functions. Namely, $\\overline{X}$ is projective hence it suffices to show", "a proper morphism $f : X \\to Y$ with finite fibres between quasi-projective", "schemes over $k$ is finite. To do this, one chooses an affine open of $X$", "containing the fibre of $f$ over a point $y$ using that any finite set of", "points of a quasi-projective scheme over $k$ is contained in an affine.", "Shrinking $Y$ to a small affine neighbourhood of $y$ one reduces to the", "case of a proper morphism between affines. Such a morphism is finite by", "Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed}.}", "we conclude that $f$ is finite. Hence $X = f^{-1}(D_{+}(T_0))$", "is affine." ], "refs": [ "varieties-lemma-dim-1-projective-completion", "varieties-lemma-find-globally-generated", "coherent-lemma-proper-over-affine-cohomology-finite", "constructions-lemma-projective-space", "coherent-lemma-proper-finite-fibre-finite-in-neighbourhood", "morphisms-lemma-integral-universally-closed" ], "ref_ids": [ 11100, 11069, 3355, 12621, 3366, 5441 ] } ], "ref_ids": [] }, { "id": 11103, "type": "theorem", "label": "varieties-lemma-dim-1-nonproper-affine", "categories": [ "varieties" ], "title": "varieties-lemma-dim-1-nonproper-affine", "contents": [ "Let $X$ be a separated scheme of finite type over $k$.", "If $\\dim(X) \\leq 1$ and no irreducible component of $X$", "is proper of dimension $1$, then $X$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Let $X = \\bigcup X_i$ be the decomposition of $X$ into irreducible components.", "We think of $X_i$ as an integral scheme (using the reduced induced scheme", "structure, see Schemes, Definition", "\\ref{schemes-definition-reduced-induced-scheme}).", "In particular $X_i$ is a singleton (hence affine) or a curve", "hence affine by Lemma \\ref{lemma-curve-affine-projective}.", "Then $\\coprod X_i \\to X$ is finite surjective and $\\coprod X_i$ is affine.", "Thus we see that $X$ is affine by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-image-affine-finite-morphism-affine-Noetherian}." ], "refs": [ "schemes-definition-reduced-induced-scheme", "varieties-lemma-curve-affine-projective", "coherent-lemma-image-affine-finite-morphism-affine-Noetherian" ], "ref_ids": [ 7745, 11102, 3337 ] } ], "ref_ids": [] }, { "id": 11104, "type": "theorem", "label": "varieties-lemma-degree-base-change", "categories": [ "varieties" ], "title": "varieties-lemma-degree-base-change", "contents": [ "Let $k \\subset k'$ be an extension of fields. Let $X$ be a proper scheme of", "dimension $\\leq 1$ over $k$. Let $\\mathcal{E}$ be a locally free", "$\\mathcal{O}_X$-module of constant rank $n$. Then the degree of", "$\\mathcal{E}/X/k$ is equal to the degree of", "$\\mathcal{E}_{k'}/X_{k'}/k'$." ], "refs": [], "proofs": [ { "contents": [ "More precisely, set $X_{k'} = X \\times_{\\Spec(k)} \\Spec(k')$.", "Let $\\mathcal{E}_{k'} = p^*\\mathcal{E}$ where $p : X_{k'} \\to X$", "is the projection. By", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}", "we have", "$H^i(X_{k'}, \\mathcal{E}_{k'}) = H^i(X, \\mathcal{E}) \\otimes_k k'$", "and", "$H^i(X_{k'}, \\mathcal{O}_{X_{k'}}) = H^i(X, \\mathcal{O}_X) \\otimes_k k'$.", "Hence we see that the Euler characteristics are unchanged, hence the", "degree is unchanged." ], "refs": [ "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 3298 ] } ], "ref_ids": [] }, { "id": 11105, "type": "theorem", "label": "varieties-lemma-degree-additive", "categories": [ "varieties" ], "title": "varieties-lemma-degree-additive", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$", "over $k$. Let $0 \\to \\mathcal{E}_1 \\to \\mathcal{E}_2 \\to \\mathcal{E}_3 \\to 0$", "be a short exact sequence of locally free $\\mathcal{O}_X$-modules", "each of finite constant rank. Then", "$$", "\\deg(\\mathcal{E}_2) = \\deg(\\mathcal{E}_1) + \\deg(\\mathcal{E}_3)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from additivity of Euler characteristics", "(Lemma \\ref{lemma-euler-characteristic-additive})", "and additivity of ranks." ], "refs": [ "varieties-lemma-euler-characteristic-additive" ], "ref_ids": [ 11029 ] } ], "ref_ids": [] }, { "id": 11106, "type": "theorem", "label": "varieties-lemma-degree-birational-pullback", "categories": [ "varieties" ], "title": "varieties-lemma-degree-birational-pullback", "contents": [ "Let $k$ be a field. Let $f : X' \\to X$ be a birational morphism of", "proper schemes of dimension $\\leq 1$ over $k$. Then", "$$", "\\deg(f^*\\mathcal{E}) = \\deg(\\mathcal{E})", "$$", "for every finite locally free sheaf of constant rank. More generally", "it suffices if $f$ induces a bijection between irreducible components", "of dimension $1$ and isomorphisms of local rings at the corresponding", "generic points." ], "refs": [], "proofs": [ { "contents": [ "The morphism $f$ is proper", "(Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed})", "and has fibres of dimension $\\leq 0$. Hence $f$ is finite", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-finite-fibre-finite-in-neighbourhood}).", "Thus", "$$", "Rf_*f^*\\mathcal{E} = f_*f^*\\mathcal{E} =", "\\mathcal{E} \\otimes_{\\mathcal{O}_X} f_*\\mathcal{O}_{X'}", "$$", "Since $f$ induces an isomorphism on local rings at generic points of", "all irreducible components of dimension $1$ we see that the kernel", "and cokernel", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{O}_X \\to f_*\\mathcal{O}_{X'}", "\\to \\mathcal{Q} \\to 0", "$$", "have supports of dimension $\\leq 0$. Note that tensoring this with", "$\\mathcal{E}$ is still an exact sequence as $\\mathcal{E}$ is locally free.", "We obtain", "\\begin{align*}", "\\chi(X, \\mathcal{E}) - \\chi(X', f^*\\mathcal{E})", "& =", "\\chi(X, \\mathcal{E}) - \\chi(X, f_*f^*\\mathcal{E}) \\\\", "& =", "\\chi(X, \\mathcal{E}) - \\chi(X, \\mathcal{E} \\otimes f_*\\mathcal{O}_{X'}) \\\\", "& =", "\\chi(X, \\mathcal{K} \\otimes \\mathcal{E}) -", "\\chi(X, \\mathcal{Q} \\otimes \\mathcal{E}) \\\\", "& =", "n\\chi(X, \\mathcal{K}) -", "n\\chi(X, \\mathcal{Q}) \\\\", "& =", "n\\chi(X, \\mathcal{O}_X) - n\\chi(X, f_*\\mathcal{O}_{X'}) \\\\", "& =", "n\\chi(X, \\mathcal{O}_X) - n\\chi(X', \\mathcal{O}_{X'})", "\\end{align*}", "which proves what we want. The first equality as $f$ is finite, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology}.", "The second equality by projection formula, see", "Cohomology, Lemma \\ref{cohomology-lemma-projection-formula}.", "The third by additivity of Euler characteristics, see", "Lemma \\ref{lemma-euler-characteristic-additive}.", "The fourth by Lemma \\ref{lemma-chi-tensor-finite}." ], "refs": [ "morphisms-lemma-image-proper-scheme-closed", "coherent-lemma-proper-finite-fibre-finite-in-neighbourhood", "coherent-lemma-relative-affine-cohomology", "cohomology-lemma-projection-formula", "varieties-lemma-euler-characteristic-additive", "varieties-lemma-chi-tensor-finite" ], "ref_ids": [ 5411, 3366, 3284, 2243, 11029, 11030 ] } ], "ref_ids": [] }, { "id": 11107, "type": "theorem", "label": "varieties-lemma-degree-on-proper-curve", "categories": [ "varieties" ], "title": "varieties-lemma-degree-on-proper-curve", "contents": [ "Let $k$ be a field. Let $X$ be a proper curve over $k$ with generic point", "$\\xi$. Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $n$", "and let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Then", "$$", "\\chi(X, \\mathcal{E} \\otimes \\mathcal{F}) =", "r \\deg(\\mathcal{E}) + n \\chi(X, \\mathcal{F})", "$$", "where $r = \\dim_{\\kappa(\\xi)} \\mathcal{F}_\\xi$ is the rank of $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{P}$ be the property of coherent sheaves $\\mathcal{F}$", "on $X$ expressing that the formula of the lemma holds. We claim that", "the assumptions (1) and (2) of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-property}", "hold for $\\mathcal{P}$. Namely, (1) holds because the Euler characteristic", "and the rank $r$ are additive in short exact sequences of coherent sheaves.", "And (2) holds too: If $Z = X$ then we may take", "$\\mathcal{G} = \\mathcal{O}_X$ and $\\mathcal{P}(\\mathcal{O}_X)$", "is true by the definition of degree. If $i : Z \\to X$ is the inclusion", "of a closed point we may take $\\mathcal{G} = i_*\\mathcal{O}_Z$", "and $\\mathcal{P}$ holds by Lemma \\ref{lemma-chi-tensor-finite}", "and the fact that $r = 0$ in this case." ], "refs": [ "coherent-lemma-property", "varieties-lemma-chi-tensor-finite" ], "ref_ids": [ 3332, 11030 ] } ], "ref_ids": [] }, { "id": 11108, "type": "theorem", "label": "varieties-lemma-degree-in-terms-of-components", "categories": [ "varieties" ], "title": "varieties-lemma-degree-in-terms-of-components", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$.", "Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $n$.", "Then", "$$", "\\deg(\\mathcal{E}) = \\sum m_i \\deg(\\mathcal{E}|_{C_i})", "$$", "where $C_i \\subset X$, $i = 1, \\ldots, t$ are the irreducible components", "of dimension $1$ with reduced induced scheme structure and $m_i$ is the", "multiplicity of $C_i$ in $X$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the statement makes sense because $C_i \\to \\Spec(k)$", "is proper of dimension $1$ (Morphisms, Lemmas", "\\ref{morphisms-lemma-closed-immersion-proper} and", "\\ref{morphisms-lemma-composition-proper}). Consider the open subscheme", "$U_i = X \\setminus (\\bigcup_{j \\not = i} C_j)$ and let $X_i \\subset X$", "be the scheme theoretic closure of $U_i$. Note that $X_i \\cap U_i = U_i$", "(scheme theoretically) and that $X_i \\cap U_j = \\emptyset$", "(set theoretically) for $i \\not = j$; this follows from the description", "of scheme theoretic closure in", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-immersion}.", "Thus we may apply Lemma \\ref{lemma-degree-birational-pullback} to the morphism", "$X' = \\bigcup X_i \\to X$. Since it is clear that $C_i \\subset X_i$", "(scheme theoretically) and that the multiplicity of $C_i$ in $X_i$ is equal", "to the multiplicity of $C_i$ in $X$, we see that we reduce to the case", "discussed in the following paragraph.", "\\medskip\\noindent", "Assume $X$ is irreducible with generic point $\\xi$. Let", "$C = X_{red}$ have multiplicity $m$.", "We have to show that $\\deg(\\mathcal{E}) = m \\deg(\\mathcal{E}|_C)$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the ideal defining the closed", "subscheme $C$. Let $e \\geq 0$ be minimal such that $\\mathcal{I}^{e + 1} = 0$", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-power-ideal-kills-sheaf}).", "We argue by induction on $e$. If $e = 0$, then $X = C$ and the result", "is immediate. Otherwise we set $\\mathcal{F} = \\mathcal{I}^e$ viewed as", "a coherent $\\mathcal{O}_C$-module (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-i-star-equivalence}).", "Let $X' \\subset X$ be the closed subscheme cut out by the", "coherent ideal $\\mathcal{I}^e$ and let $m'$ be the multiplicity", "of $C$ in $X'$. Taking stalks at $\\xi$ of the short exact sequence", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{O}_X \\to \\mathcal{O}_{X'} \\to 0", "$$", "we find (use Algebra, Lemmas \\ref{algebra-lemma-length-additive},", "\\ref{algebra-lemma-dimension-is-length}, and", "\\ref{algebra-lemma-length-independent}) that", "$$", "m = \\text{length}_{\\mathcal{O}_{X, \\xi}} \\mathcal{O}_{X, \\xi}", "= \\dim_{\\kappa(\\xi)} \\mathcal{F}_\\xi +", "\\text{length}_{\\mathcal{O}_{X', \\xi}} \\mathcal{O}_{X', \\xi}", "= r + m'", "$$", "where $r$ is the rank of $\\mathcal{F}$ as a coherent sheaf on $C$.", "Tensoring with $\\mathcal{E}$ we obtain a short exact sequence", "$$", "0 \\to \\mathcal{E}|_C \\otimes \\mathcal{F} \\to \\mathcal{E} \\to", "\\mathcal{E} \\otimes \\mathcal{O}_{X'} \\to 0", "$$", "By induction we have", "$\\chi(\\mathcal{E} \\otimes \\mathcal{O}_{X'}) = m' \\deg(\\mathcal{E}|_C)$.", "By Lemma \\ref{lemma-degree-on-proper-curve} we have", "$\\chi(\\mathcal{E}|_C \\otimes \\mathcal{F}) =", "r \\deg(\\mathcal{E}|_C) + n \\chi(\\mathcal{F})$.", "Putting everything together we obtain the result." ], "refs": [ "morphisms-lemma-closed-immersion-proper", "morphisms-lemma-composition-proper", "morphisms-lemma-quasi-compact-immersion", "varieties-lemma-degree-birational-pullback", "coherent-lemma-power-ideal-kills-sheaf", "coherent-lemma-i-star-equivalence", "algebra-lemma-length-additive", "algebra-lemma-dimension-is-length", "algebra-lemma-length-independent", "varieties-lemma-degree-on-proper-curve" ], "ref_ids": [ 5410, 5408, 5154, 11106, 3320, 3315, 631, 634, 633, 11107 ] } ], "ref_ids": [] }, { "id": 11109, "type": "theorem", "label": "varieties-lemma-degree-tensor-product", "categories": [ "varieties" ], "title": "varieties-lemma-degree-tensor-product", "contents": [ "Let $k$ be a field, let $X$ be a proper scheme of dimension $\\leq 1$", "over $k$, and let $\\mathcal{E}$, $\\mathcal{V}$ be locally free", "$\\mathcal{O}_X$-modules of constant finite rank. Then", "$$", "\\deg(\\mathcal{E} \\otimes \\mathcal{V}) =", "\\text{rank}(\\mathcal{E}) \\deg(\\mathcal{V}) +", "\\text{rank}(\\mathcal{V}) \\deg(\\mathcal{E})", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-degree-in-terms-of-components} and elementary", "arithmetic, we reduce to the case of a proper curve.", "This case follows from Lemma \\ref{lemma-degree-on-proper-curve}." ], "refs": [ "varieties-lemma-degree-in-terms-of-components", "varieties-lemma-degree-on-proper-curve" ], "ref_ids": [ 11108, 11107 ] } ], "ref_ids": [] }, { "id": 11110, "type": "theorem", "label": "varieties-lemma-degree-and-det", "categories": [ "varieties" ], "title": "varieties-lemma-degree-and-det", "contents": [ "Let $k$ be a field, let $X$ be a proper scheme of dimension $\\leq 1$", "over $k$, and let $\\mathcal{E}$ be a locally free", "$\\mathcal{O}_X$-module of rank $n$. Then", "$$", "\\deg(\\mathcal{E}) = \\deg(\\wedge^n(\\mathcal{E})) = \\deg(\\det(\\mathcal{E}))", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-degree-in-terms-of-components} and elementary", "arithmetic, we reduce to the case of a proper curve.", "Then there exists a modification $f : X' \\to X$ such that", "$f^*\\mathcal{E}$ has a filtration whose successive", "quotients are invertible modules, see", "Divisors, Lemma \\ref{divisors-lemma-filter-after-modification}.", "By Lemma \\ref{lemma-degree-birational-pullback} we may work on $X'$.", "Thus we may assume we have a filtration", "$$", "0 = \\mathcal{E}_0 \\subset \\mathcal{E}_1 \\subset \\mathcal{E}_2 \\subset", "\\ldots \\subset \\mathcal{E}_n = \\mathcal{E}", "$$", "by locally free $\\mathcal{O}_X$-modules", "with $\\mathcal{L}_i = \\mathcal{E}_i/\\mathcal{E}_{i - 1}$ is invertible.", "By Modules, Lemma \\ref{modules-lemma-det-ses} and induction we find", "$\\det(\\mathcal{E}) = \\mathcal{L}_1 \\otimes \\ldots \\otimes \\mathcal{L}_n$.", "Thus the equality follows from Lemma \\ref{lemma-degree-tensor-product}", "and additivity (Lemma \\ref{lemma-degree-additive})." ], "refs": [ "varieties-lemma-degree-in-terms-of-components", "divisors-lemma-filter-after-modification", "varieties-lemma-degree-birational-pullback", "modules-lemma-det-ses", "varieties-lemma-degree-tensor-product", "varieties-lemma-degree-additive" ], "ref_ids": [ 11108, 8079, 11106, 13306, 11109, 11105 ] } ], "ref_ids": [] }, { "id": 11111, "type": "theorem", "label": "varieties-lemma-degree-effective-Cartier-divisor", "categories": [ "varieties" ], "title": "varieties-lemma-degree-effective-Cartier-divisor", "contents": [ "Let $k$ be a field, let $X$ be a proper scheme of dimension $\\leq 1$", "over $k$. Let $D$ be an effective Cartier divisor on $X$.", "Then $D$ is finite over $\\Spec(k)$ of degree", "$\\deg(D) = \\dim_k \\Gamma(D, \\mathcal{O}_D)$. For a locally free sheaf", "$\\mathcal{E}$ of rank $n$ we have", "$$", "\\deg(\\mathcal{E}(D)) = n\\deg(D) + \\deg(\\mathcal{E})", "$$", "where $\\mathcal{E}(D) = \\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(D)$." ], "refs": [], "proofs": [ { "contents": [ "Since $D$ is nowhere dense in $X$ (Divisors, Lemma", "\\ref{divisors-lemma-complement-effective-Cartier-divisor})", "we see that $\\dim(D) \\leq 0$. Hence $D$ is finite over $k$", "by Lemma \\ref{lemma-algebraic-scheme-dim-0}. Since $k$ is a field,", "the morphism $D \\to \\Spec(k)$ is finite locally free and hence has", "a degree", "(Morphisms, Definition \\ref{morphisms-definition-finite-locally-free}),", "which is clearly equal to $\\dim_k \\Gamma(D, \\mathcal{O}_D)$", "as stated in the lemma. By", "Divisors, Definition", "\\ref{divisors-definition-invertible-sheaf-effective-Cartier-divisor}", "there is a short exact sequence", "$$", "0 \\to \\mathcal{O}_X \\to \\mathcal{O}_X(D) \\to i_*i^*\\mathcal{O}_X(D) \\to 0", "$$", "where $i : D \\to X$ is the closed immersion. Tensoring with $\\mathcal{E}$", "we obtain a short exact sequence", "$$", "0 \\to \\mathcal{E} \\to \\mathcal{E}(D) \\to i_*i^*\\mathcal{E}(D) \\to 0", "$$", "The equation of the lemma follows from additivity of", "the Euler characteristic (Lemma \\ref{lemma-euler-characteristic-additive})", "and Lemma \\ref{lemma-chi-tensor-finite}." ], "refs": [ "divisors-lemma-complement-effective-Cartier-divisor", "varieties-lemma-algebraic-scheme-dim-0", "morphisms-definition-finite-locally-free", "divisors-definition-invertible-sheaf-effective-Cartier-divisor", "varieties-lemma-euler-characteristic-additive", "varieties-lemma-chi-tensor-finite" ], "ref_ids": [ 7929, 10988, 5578, 8092, 11029, 11030 ] } ], "ref_ids": [] }, { "id": 11112, "type": "theorem", "label": "varieties-lemma-divisible", "categories": [ "varieties" ], "title": "varieties-lemma-divisible", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$", "which is reduced and connected.", "Let $\\kappa = H^0(X, \\mathcal{O}_X)$. Then $\\kappa/k$ is a", "finite extension of fields and $w = [\\kappa : k]$ divides", "\\begin{enumerate}", "\\item $\\deg(\\mathcal{E})$ for all locally free $\\mathcal{O}_X$-modules", "$\\mathcal{E}$,", "\\item $[\\kappa(x) : k]$ for all closed points $x \\in X$, and", "\\item $\\deg(D)$ for all closed subschemes $D \\subset X$", "of dimension zero.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See Lemma \\ref{lemma-proper-geometrically-reduced-global-sections}", "for the assertions about $\\kappa$.", "For every quasi-coherent $\\mathcal{O}_X$-module, the", "$k$-vector spaces $H^i(X, \\mathcal{F})$ are $\\kappa$-vector spaces.", "The divisibilities easily follow from this statement and the", "definitions." ], "refs": [ "varieties-lemma-proper-geometrically-reduced-global-sections" ], "ref_ids": [ 10948 ] } ], "ref_ids": [] }, { "id": 11113, "type": "theorem", "label": "varieties-lemma-degree-pullback-map-proper-curves", "categories": [ "varieties" ], "title": "varieties-lemma-degree-pullback-map-proper-curves", "contents": [ "Let $k$ be a field. Let $f : X \\to Y$ be a nonconstant morphism of", "proper curves over $k$. Let $\\mathcal{E}$ be a locally free", "$\\mathcal{O}_Y$-module. Then", "$$", "\\deg(f^*\\mathcal{E}) = \\deg(X/Y) \\deg(\\mathcal{E})", "$$" ], "refs": [], "proofs": [ { "contents": [ "The degree of $X$ over $Y$ is defined in", "Morphisms, Definition \\ref{morphisms-definition-degree}.", "Thus $f_*\\mathcal{O}_X$ is a coherent $\\mathcal{O}_Y$-module", "of rank $\\deg(X/Y)$, i.e.,", "$\\deg(X/Y) = \\dim_{\\kappa(\\xi)} (f_*\\mathcal{O}_X)_\\xi$ where $\\xi$", "is the generic point of $Y$. Thus we obtain", "\\begin{align*}", "\\chi(X, f^*\\mathcal{E})", "& =", "\\chi(Y, f_*f^*\\mathcal{E}) \\\\", "& =", "\\chi(Y, \\mathcal{E} \\otimes f_*\\mathcal{O}_X) \\\\", "& =", "\\deg(X/Y) \\deg(\\mathcal{E}) + n \\chi(Y, f_*\\mathcal{O}_X) \\\\", "& =", "\\deg(X/Y) \\deg(\\mathcal{E}) + n \\chi(X, \\mathcal{O}_X)", "\\end{align*}", "as desired. The first equality as $f$ is finite, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology}.", "The second equality by projection formula, see", "Cohomology, Lemma \\ref{cohomology-lemma-projection-formula}.", "The third equality by Lemma \\ref{lemma-degree-on-proper-curve}." ], "refs": [ "morphisms-definition-degree", "coherent-lemma-relative-affine-cohomology", "cohomology-lemma-projection-formula", "varieties-lemma-degree-on-proper-curve" ], "ref_ids": [ 5587, 3284, 2243, 11107 ] } ], "ref_ids": [] }, { "id": 11114, "type": "theorem", "label": "varieties-lemma-check-invertible-sheaf-trivial", "categories": [ "varieties" ], "title": "varieties-lemma-check-invertible-sheaf-trivial", "contents": [ "Let $k$ be a field. Let $X$ be a proper curve over $k$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item If $\\mathcal{L}$ has a nonzero section, then", "$\\deg(\\mathcal{L}) \\geq 0$.", "\\item If $\\mathcal{L}$ has a nonzero section $s$ which vanishes", "at a point, then $\\deg(\\mathcal{L}) > 0$.", "\\item If $\\mathcal{L}$ and $\\mathcal{L}^{-1}$ have nonzero sections, then", "$\\mathcal{L} \\cong \\mathcal{O}_X$.", "\\item If $\\deg(\\mathcal{L}) \\leq 0$ and $\\mathcal{L}$ has a nonzero", "section, then $\\mathcal{L} \\cong \\mathcal{O}_X$.", "\\item If $\\mathcal{N} \\to \\mathcal{L}$ is a nonzero map of invertible", "$\\mathcal{O}_X$-modules, then $\\deg(\\mathcal{L}) \\geq \\deg(\\mathcal{N})$", "and if equality holds then it is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $s$ be a nonzero section of $\\mathcal{L}$. Since $X$ is a curve, we", "see that $s$ is a regular section. Hence there is an effective", "Cartier divisor $D \\subset X$ and an isomorphism", "$\\mathcal{L} \\to \\mathcal{O}_X(D)$ mapping $s$ the canonical", "section $1$ of $\\mathcal{O}_X(D)$, see", "Divisors, Lemma \\ref{divisors-lemma-characterize-OD}.", "Then $\\deg(\\mathcal{L}) = \\deg(D)$ by", "Lemma \\ref{lemma-degree-effective-Cartier-divisor}.", "As $\\deg(D) \\geq 0$ and $= 0$", "if and only if $D = \\emptyset$, this proves (1) and (2).", "In case (3) we see that $\\deg(\\mathcal{L}) = 0$ and", "$D = \\emptyset$. Similarly for (4). To see (5) apply", "(1) and (4) to the invertible sheaf", "$$", "\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}^{\\otimes -1} =", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{N}, \\mathcal{L})", "$$", "which has degree $\\deg(\\mathcal{L}) - \\deg(\\mathcal{N})$", "by Lemma \\ref{lemma-degree-tensor-product}." ], "refs": [ "divisors-lemma-characterize-OD", "varieties-lemma-degree-effective-Cartier-divisor", "varieties-lemma-degree-tensor-product" ], "ref_ids": [ 7944, 11111, 11109 ] } ], "ref_ids": [] }, { "id": 11115, "type": "theorem", "label": "varieties-lemma-no-sections-dual-nef", "categories": [ "varieties" ], "title": "varieties-lemma-no-sections-dual-nef", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$", "which is reduced, connected, and equidimensional of dimension $1$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "If $\\deg(\\mathcal{L}|_C) \\leq 0$ for all irreducible components $C$", "of $X$, then either $H^0(X, \\mathcal{L}) = 0$ or", "$\\mathcal{L} \\cong \\mathcal{O}_X$." ], "refs": [], "proofs": [ { "contents": [ "Let $s \\in H^0(X, \\mathcal{L})$ be nonzero. Since $X$ is reduced there exists", "an irreducible component $C$ of $X$ with $s|_C \\not = 0$.", "But if $s|_C$ is nonzero, then", "$s$ is nonwhere vanishing on $C$ by", "Lemma \\ref{lemma-check-invertible-sheaf-trivial}.", "This in turn implies $s$ is nowhere vanishing on", "every irreducible component of $X$ meeting $C$.", "Since $X$ is connected, we conclude that $s$", "vanishes nowhere and the lemma follows." ], "refs": [ "varieties-lemma-check-invertible-sheaf-trivial" ], "ref_ids": [ 11114 ] } ], "ref_ids": [] }, { "id": 11116, "type": "theorem", "label": "varieties-lemma-ample-curve", "categories": [ "varieties" ], "title": "varieties-lemma-ample-curve", "contents": [ "Let $k$ be a field. Let $X$ be a proper curve over $k$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Then $\\mathcal{L}$ is ample if and only if $\\deg(\\mathcal{L}) > 0$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{L}$ is ample, then there exists an $n > 0$ and a section", "$s \\in H^0(X, \\mathcal{L}^{\\otimes n})$ with $X_s$ affine. Since", "$X$ isn't affine (otherwise by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}", "$X$ would be finite), we see that $s$ vanishes at some point.", "Hence $\\deg(\\mathcal{L}^{\\otimes n}) > 0$ by", "Lemma \\ref{lemma-check-invertible-sheaf-trivial}.", "By Lemma \\ref{lemma-degree-tensor-product}", "we conclude that $\\deg(\\mathcal{L}) = 1/n\\deg(\\mathcal{L}^{\\otimes n}) > 0$.", "\\medskip\\noindent", "Assume $\\deg(\\mathcal{L}) > 0$. Then", "$$", "\\dim_k H^0(X, \\mathcal{L}^{\\otimes n}) \\geq \\chi(X, \\mathcal{L}^n)", "= n\\deg(\\mathcal{L}) + \\chi(X, \\mathcal{O}_X)", "$$", "grows linearly with $n$. Hence for any finite collection of closed", "points $x_1, \\ldots, x_t$ of $X$, we can find an $n$ such that", "$\\dim_k H^0(X, \\mathcal{L}^{\\otimes n}) > \\sum \\dim_k \\kappa(x_i)$.", "(Recall that by Hilbert Nullstellensatz, the extension fields", "$k \\subset \\kappa(x_i)$ are finite, see for example", "Morphisms, Lemma \\ref{morphisms-lemma-closed-point-fibre-locally-finite-type}).", "Hence we can find a nonzero $s \\in H^0(X, \\mathcal{L}^{\\otimes n})$", "vanishing in $x_1, \\ldots, x_t$. In particular, if we choose", "$x_1, \\ldots, x_t$ such that $X \\setminus \\{x_1, \\ldots, x_t\\}$", "is affine, then $X_s$ is affine too", "(for example by Properties, Lemma \\ref{properties-lemma-affine-cap-s-open}", "although if we choose our finite set such that", "$\\mathcal{L}|_{X \\setminus \\{x_1, \\ldots, x_t\\}}$ is trivial, then", "it is immediate). The conclusion is that we can find an $n > 0$", "and a nonzero section $s \\in H^0(X, \\mathcal{L}^{\\otimes n})$ such", "that $X_s$ is affine.", "\\medskip\\noindent", "We will show that for every quasi-coherent sheaf of ideals $\\mathcal{I}$", "there exists an $m > 0$ such that", "$H^1(X, \\mathcal{I} \\otimes \\mathcal{L}^{\\otimes m})$ is zero.", "This will finish the proof by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-vanshing-gives-ample}.", "To see this we consider the maps", "$$", "\\mathcal{I} \\xrightarrow{s}", "\\mathcal{I} \\otimes \\mathcal{L}^{\\otimes n} \\xrightarrow{s}", "\\mathcal{I} \\otimes \\mathcal{L}^{\\otimes 2n} \\xrightarrow{s} \\ldots", "$$", "Since $\\mathcal{I}$ is torsion free, these maps are injective and", "isomorphisms over $X_s$, hence the cokernels have vanishing $H^1$", "(by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-coherent-support-dimension-0} for example).", "We conclude that the maps of vector spaces", "$$", "H^1(X, \\mathcal{I}) \\to", "H^1(X, \\mathcal{I} \\otimes \\mathcal{L}^{\\otimes n}) \\to", "H^1(X, \\mathcal{I} \\otimes \\mathcal{L}^{\\otimes 2n}) \\to \\ldots", "$$", "are surjective. On the other hand, the dimension of $H^1(X, \\mathcal{I})$", "is finite, and every element maps to zero eventually by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-section-affine-open-kills-classes}.", "Thus for some $e > 0$ we see that", "$H^1(X, \\mathcal{I} \\otimes \\mathcal{L}^{\\otimes en})$ is zero.", "This finishes the proof." ], "refs": [ "morphisms-lemma-finite-proper", "varieties-lemma-check-invertible-sheaf-trivial", "varieties-lemma-degree-tensor-product", "morphisms-lemma-closed-point-fibre-locally-finite-type", "properties-lemma-affine-cap-s-open", "coherent-lemma-vanshing-gives-ample", "coherent-lemma-coherent-support-dimension-0" ], "ref_ids": [ 5445, 11114, 11109, 5223, 3042, 3346, 3317 ] } ], "ref_ids": [] }, { "id": 11117, "type": "theorem", "label": "varieties-lemma-ampleness-in-terms-of-degrees-components", "categories": [ "varieties" ], "title": "varieties-lemma-ampleness-in-terms-of-degrees-components", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$", "over $k$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $C_i \\subset X$, $i = 1, \\ldots, t$ be the irreducible components", "of dimension $1$. The following are equivalent:", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is ample, and", "\\item $\\deg(\\mathcal{L}|_{C_i}) > 0$ for $i = 1, \\ldots, t$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $x_1, \\ldots, x_r \\in X$ be the isolated closed points.", "Think of $x_i = \\Spec(\\kappa(x_i))$ as a scheme.", "Consider the morphism of schemes", "$$", "f : C_1 \\amalg \\ldots \\amalg C_t \\amalg x_1 \\amalg \\ldots \\amalg x_r", "\\longrightarrow X", "$$", "This is a finite surjective morphism of schemes proper over $k$", "(details omitted). Thus $\\mathcal{L}$ is ample if and only if", "$f^*\\mathcal{L}$ is ample (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-surjective-finite-morphism-ample}).", "Thus we conclude by Lemma \\ref{lemma-ample-curve}." ], "refs": [ "coherent-lemma-surjective-finite-morphism-ample", "varieties-lemma-ample-curve" ], "ref_ids": [ 3347, 11116 ] } ], "ref_ids": [] }, { "id": 11118, "type": "theorem", "label": "varieties-lemma-regular-point-on-curve", "categories": [ "varieties" ], "title": "varieties-lemma-regular-point-on-curve", "contents": [ "Let $k$ be a field. Let $X$ be a curve over $k$. Let $x \\in X$ be a closed", "point. We think of $x$ as a (reduced) closed subscheme of $X$ with sheaf", "of ideals $\\mathcal{I}$. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{O}_{X, x}$ is regular,", "\\item $\\mathcal{O}_{X, x}$ is normal,", "\\item $\\mathcal{O}_{X, x}$ is a discrete valuation ring,", "\\item $\\mathcal{I}$ is an invertible $\\mathcal{O}_X$-module,", "\\item $x$ is an effective Cartier divisor on $X$.", "\\end{enumerate}", "If $k$ is perfect, these are also equivalent to", "\\begin{enumerate}", "\\item[(6)] $X \\to \\Spec(k)$ is smooth at $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is a curve, the local ring $\\mathcal{O}_{X, x}$ is a Noetherian", "local domain of dimension $1$ (Lemma \\ref{lemma-dimension-locally-algebraic}).", "Parts (4) and (5) are equivalent by definition and are equivalent to", "$\\mathcal{I}_x = \\mathfrak m_x \\subset \\mathcal{O}_{X, x}$ having one generator", "(Divisors, Lemma \\ref{divisors-lemma-effective-Cartier-in-points}).", "The equivalence of (1), (2), (3), (4), and (5) therefore follows from", "Algebra, Lemma \\ref{algebra-lemma-characterize-dvr}. The final statement", "follows from Lemma \\ref{lemma-dense-smooth-open-variety-over-perfect-field}." ], "refs": [ "varieties-lemma-dimension-locally-algebraic", "divisors-lemma-effective-Cartier-in-points", "algebra-lemma-characterize-dvr", "varieties-lemma-dense-smooth-open-variety-over-perfect-field" ], "ref_ids": [ 10989, 7946, 1023, 11009 ] } ], "ref_ids": [] }, { "id": 11119, "type": "theorem", "label": "varieties-lemma-general-degree-g-line-bundle", "categories": [ "varieties" ], "title": "varieties-lemma-general-degree-g-line-bundle", "contents": [ "Let $k$ be an algebraically closed field. Let $X$ be a proper curve over $k$.", "Then there exist", "\\begin{enumerate}", "\\item an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ with", "$\\dim_k H^0(X, \\mathcal{L}) = 1$ and $H^1(X, \\mathcal{L}) = 0$, and", "\\item an invertible $\\mathcal{O}_X$-module $\\mathcal{N}$ with", "$\\dim_k H^0(X, \\mathcal{N}) = 0$ and $H^1(X, \\mathcal{N}) = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a closed immersion $i : X \\to \\mathbf{P}^n_k$", "(Lemma \\ref{lemma-dim-1-proper-projective}).", "Setting $\\mathcal{L} = i^*\\mathcal{O}_{\\mathbf{P}^n}(d)$", "for $d \\gg 0$ we see that there exists an invertible sheaf", "$\\mathcal{L}$ with $H^0(X, \\mathcal{L}) \\not = 0$ and", "$H^1(X, \\mathcal{L}) = 0$ (see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-vanshing-gives-ample}", "for vanishing and the references therein for nonvanishing).", "We will finish the proof of (1) by descending induction on", "$t = \\dim_k H^0(X, \\mathcal{L})$. The base case $t = 1$ is trivial.", "Assume $t > 1$.", "\\medskip\\noindent", "Let $U \\subset X$ be the nonempty open subset of nonsingular points", "studied in Lemma \\ref{lemma-dense-smooth-open-variety-over-perfect-field}.", "Let $s \\in H^0(X, \\mathcal{L})$ be nonzero. There exists a closed", "point $x \\in U$ such that $s$ does not vanish in $x$. Let $\\mathcal{I}$", "be the ideal sheaf of $i : x \\to X$ as in", "Lemma \\ref{lemma-regular-point-on-curve}. Look at the", "short exact sequence", "$$", "0 \\to \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L} \\to", "\\mathcal{L} \\to i_*i^*\\mathcal{L} \\to 0", "$$", "Observe that $H^0(X, i_*i^*\\mathcal{L}) = H^0(x, i^*\\mathcal{L})$", "has dimension $1$ as $x$ is a $k$-rational point ($k$ is algebraically", "closed). Since $s$ does not vanish at $x$ we conclude that", "$$", "H^0(X, \\mathcal{L}) \\longrightarrow H^0(X, i_*i^*\\mathcal{L})", "$$", "is surjective. Hence", "$\\dim_k H^0(X, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}) = t - 1$.", "Finally, the long exact sequence of cohomology also shows that", "$H^1(X, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}) = 0$", "thereby finishing the proof of the induction step.", "\\medskip\\noindent", "To get an invertible sheaf as in (2) take an invertible sheaf", "$\\mathcal{L}$ as in (1) and do the argument in the previous paragraph", "one more time." ], "refs": [ "varieties-lemma-dim-1-proper-projective", "coherent-lemma-vanshing-gives-ample", "varieties-lemma-dense-smooth-open-variety-over-perfect-field", "varieties-lemma-regular-point-on-curve" ], "ref_ids": [ 11099, 3346, 11009, 11118 ] } ], "ref_ids": [] }, { "id": 11120, "type": "theorem", "label": "varieties-lemma-vanishing-degree-2g-and-1-line-bundle", "categories": [ "varieties" ], "title": "varieties-lemma-vanishing-degree-2g-and-1-line-bundle", "contents": [ "Let $k$ be an algebraically closed field. Let $X$ be a proper curve over $k$.", "Set $g = \\dim_k H^1(X, \\mathcal{O}_X)$. For every invertible", "$\\mathcal{O}_X$-module $\\mathcal{L}$ with $\\deg(\\mathcal{L}) \\geq 2g - 1$", "we have $H^1(X, \\mathcal{L}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{N}$ be the invertible module we found in", "Lemma \\ref{lemma-general-degree-g-line-bundle} part (2).", "The degree of $\\mathcal{N}$ is", "$\\chi(X, \\mathcal{N}) - \\chi(X, \\mathcal{O}_X) = 0 - (1 - g) = g - 1$.", "Hence the degree of $\\mathcal{L} \\otimes \\mathcal{N}^{\\otimes - 1}$", "is $\\deg(\\mathcal{L}) - (g - 1) \\geq g$.", "Hence", "$\\chi(X, \\mathcal{L} \\otimes \\mathcal{N}^{\\otimes -1}) \\geq g + 1 - g = 1$.", "Thus there is a nonzero global section $s$ whose zero scheme is an", "effective Cartier divisor $D$ of degree $\\deg(\\mathcal{L}) - (g - 1)$.", "This gives a short exact sequence", "$$", "0 \\to \\mathcal{N} \\xrightarrow{s} \\mathcal{L} \\to i_*(\\mathcal{L}|_D) \\to 0", "$$", "where $i : D \\to X$ is the inclusion morphism. We conclude that", "$H^0(X, \\mathcal{L})$ maps isomorphically to $H^0(D, \\mathcal{L}|_D)$", "which has dimension $\\deg(\\mathcal{L}) - (g - 1)$. The result follows", "from the definition of degree." ], "refs": [ "varieties-lemma-general-degree-g-line-bundle" ], "ref_ids": [ 11119 ] } ], "ref_ids": [] }, { "id": 11121, "type": "theorem", "label": "varieties-lemma-numerical-polynomial-from-euler", "categories": [ "varieties" ], "title": "varieties-lemma-numerical-polynomial-from-euler", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{F}$", "be a coherent $\\mathcal{O}_X$-module. Let", "$\\mathcal{L}_1, \\ldots, \\mathcal{L}_r$ be invertible $\\mathcal{O}_X$-modules.", "The map", "$$", "(n_1, \\ldots, n_r) \\longmapsto", "\\chi(X, \\mathcal{F} \\otimes", "\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r})", "$$", "is a numerical polynomial in $n_1, \\ldots, n_r$ of total degree at", "most the dimension of the support of $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "We prove this by induction on $\\dim(\\text{Supp}(\\mathcal{F}))$.", "If this number is zero, then the function is constant with value", "$\\dim_k \\Gamma(X, \\mathcal{F})$ by Lemma \\ref{lemma-chi-tensor-finite}.", "Assume $\\dim(\\text{Supp}(\\mathcal{F})) > 0$.", "\\medskip\\noindent", "If $\\mathcal{F}$ has embedded associated points, then we can consider", "the short exact sequence", "$0 \\to \\mathcal{K} \\to \\mathcal{F} \\to \\mathcal{F}' \\to 0$", "constructed in Divisors, Lemma \\ref{divisors-lemma-remove-embedded-points}.", "Since the dimension of the support of $\\mathcal{K}$ is strictly less,", "the result holds for $\\mathcal{K}$ by induction hypothesis and with", "strictly smaller total degree.", "By additivity of the Euler characteristic", "(Lemma \\ref{lemma-euler-characteristic-additive})", "it suffices to prove the result for $\\mathcal{F}'$. Thus we may assume", "$\\mathcal{F}$ does not have embedded associated points.", "\\medskip\\noindent", "If $i : Z \\to X$ is a closed immersion and $\\mathcal{F} = i_*\\mathcal{G}$,", "then we see that the result for $X$, $\\mathcal{F}$,", "$\\mathcal{L}_1, \\ldots, \\mathcal{L}_r$ is equivalent to the result", "for $Z$, $\\mathcal{G}$, $i^*\\mathcal{L}_1, \\ldots, i^*\\mathcal{L}_r$", "(since the cohomologies agree, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology}).", "Applying Divisors, Lemma \\ref{divisors-lemma-no-embedded-points-endos}", "we may assume that $X$ has no embedded components and", "$X = \\text{Supp}(\\mathcal{F})$.", "\\medskip\\noindent", "Pick a regular meromorphic section $s$ of $\\mathcal{L}_1$, see", "Divisors, Lemma \\ref{divisors-lemma-regular-meromorphic-section-exists}.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the ideal of", "denominators of $s$ and consider the maps", "$$", "\\mathcal{I}\\mathcal{F} \\to \\mathcal{F},\\quad", "\\mathcal{I}\\mathcal{F} \\to \\mathcal{F} \\otimes \\mathcal{L}_1", "$$", "of Divisors, Lemma \\ref{divisors-lemma-make-maps-regular-section}.", "These are injective and have cokernels $\\mathcal{Q}$, $\\mathcal{Q}'$", "supported on nowhere dense closed subschemes of $X = \\text{Supp}(\\mathcal{F})$.", "Tensoring with the invertible module", "$\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes \\mathcal{L}_1^{\\otimes n_1}$", "is exact, hence using additivity again", "we see that", "\\begin{align*}", "&\\chi(X, \\mathcal{F} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}) -", "\\chi(X, \\mathcal{F} \\otimes \\mathcal{L}_1^{\\otimes n_1 + 1}", "\\otimes \\ldots \\otimes \\mathcal{L}_r^{\\otimes n_r}) \\\\", "& =", "\\chi(\\mathcal{Q} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}) -", "\\chi(\\mathcal{Q}' \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r})", "\\end{align*}", "Thus we see that the function $P(n_1, \\ldots, n_r)$ of the lemma has", "the property that", "$$", "P(n_1 + 1, n_2, \\ldots, n_r) - P(n_1, \\ldots, n_r)", "$$", "is a numerical polynomial of total degree $<$ the dimension", "of the support of $\\mathcal{F}$. Of course by symmetry the same", "thing is true for", "$$", "P(n_1, \\ldots, n_{i - 1}, n_i + 1, n_{i + 1}, \\ldots, n_r)", "- P(n_1, \\ldots, n_r)", "$$", "for any $i \\in \\{1, \\ldots, r\\}$. A simple arithmetic argument shows", "that $P$ is a numerical polynomial of total degree at most", "$\\dim(\\text{Supp}(\\mathcal{F}))$." ], "refs": [ "varieties-lemma-chi-tensor-finite", "divisors-lemma-remove-embedded-points", "varieties-lemma-euler-characteristic-additive", "coherent-lemma-relative-affine-cohomology", "divisors-lemma-no-embedded-points-endos" ], "ref_ids": [ 11030, 7870, 11029, 3284, 7871 ] } ], "ref_ids": [] }, { "id": 11122, "type": "theorem", "label": "varieties-lemma-numerical-polynomial-leading-term", "categories": [ "varieties" ], "title": "varieties-lemma-numerical-polynomial-leading-term", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let", "$\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let", "$\\mathcal{L}_1, \\ldots, \\mathcal{L}_r$ be invertible $\\mathcal{O}_X$-modules.", "Let $d = \\dim(\\text{Supp}(\\mathcal{F}))$.", "Let $Z_i \\subset X$ be the irreducible components", "of $\\text{Supp}(\\mathcal{F})$ of dimension $d$. Let $\\xi_i \\in Z_i$", "be the generic point and set", "$m_i = \\text{length}_{\\mathcal{O}_{X, \\xi_i}}(\\mathcal{F}_{\\xi_i})$.", "Then", "$$", "\\chi(X, \\mathcal{F} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}) -", "\\sum\\nolimits_i", "m_i\\ \\chi(Z_i, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}|_{Z_i})", "$$", "is a numerical polynomial in $n_1, \\ldots, n_r$ of total degree $< d$." ], "refs": [], "proofs": [ { "contents": [ "Consider pairs $(\\xi , Z)$ where $Z \\subset X$ is an integral", "closed subscheme of dimension $d$ and $\\xi$ is its generic point.", "Then the finite $\\mathcal{O}_{X, \\xi}$-module $\\mathcal{F}_\\xi$", "has support contained in $\\{\\xi\\}$ hence the length", "$m_Z = \\text{length}_{\\mathcal{O}_{X, \\xi}}(\\mathcal{F}_\\xi)$", "is finite (Algebra, Lemma \\ref{algebra-lemma-support-point})", "and zero unless $Z = Z_i$ for some $i$. Thus the expression", "of the lemma can be written as", "$$", "E(\\mathcal{F}) =", "\\chi(X, \\mathcal{F} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}) -", "\\sum\\nolimits", "m_Z\\ \\chi(Z, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}|_Z)", "$$", "where the sum is over integral closed subschemes $Z \\subset X$", "of dimension $d$. The assignment $\\mathcal{F} \\mapsto E(\\mathcal{F})$", "is additive in short exact sequences", "$0 \\to \\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{F}'' \\to 0$", "of coherent $\\mathcal{O}_X$-modules whose support has dimension", "$\\leq d$. This follows from additivity of Euler characteristics", "(Lemma \\ref{lemma-euler-characteristic-additive})", "and additivity of lengths", "(Algebra, Lemma \\ref{algebra-lemma-length-additive}).", "Let us apply Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-filter}", "to find a filtration", "$$", "0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset", "\\ldots \\subset \\mathcal{F}_m = \\mathcal{F}", "$$", "by coherent subsheaves such that for each $j = 1, \\ldots, m$", "there exists an integral closed subscheme $V_j \\subset X$", "and a sheaf of ideals $\\mathcal{I}_j \\subset \\mathcal{O}_{V_j}$", "such that", "$$", "\\mathcal{F}_j/\\mathcal{F}_{j - 1}", "\\cong (V_j \\to X)_* \\mathcal{I}_j", "$$", "By the additivity we remarked upon above it suffices to", "prove the result for each of the subquotients", "$\\mathcal{F}_j/\\mathcal{F}_{j - 1}$. Thus it suffices to prove", "the result when $\\mathcal{F} = (V \\to X)_*\\mathcal{I}$ where", "$V \\subset X$ is an integral closed subscheme of dimension $\\leq d$.", "If $\\dim(V) < d$ and more generally for $\\mathcal{F}$", "whose support has dimension $< d$, then the first term", "in $E(\\mathcal{F})$ has total degree $< d$ by", "Lemma \\ref{lemma-numerical-polynomial-from-euler}", "and the second term is zero. If $\\dim(V) = d$, then we can use the", "short exact sequence", "$$", "0 \\to (V \\to X)_*\\mathcal{I} \\to (V \\to X)_*\\mathcal{O}_V", "\\to (V \\to X)_*(\\mathcal{O}_V/\\mathcal{I}) \\to 0", "$$", "The result holds for the middle sheaf because", "the only $Z$ occurring in the sum is $Z = V$", "with $m_Z = 1$ and because", "$$", "H^i(X, ((V \\to X)_*\\mathcal{O}_V) \\otimes ", " \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}) =", "H^i(V, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}|_V)", "$$", "by the projection formula", "(Cohomology, Section \\ref{cohomology-section-projection-formula}) and", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-relative-affine-cohomology};", "so in this case we actually have $E(\\mathcal{F}) = 0$.", "The result holds for the sheaf on the right because its support", "has dimension $< d$. Thus the result holds for the sheaf on the", "left and the lemma is proved." ], "refs": [ "algebra-lemma-support-point", "varieties-lemma-euler-characteristic-additive", "algebra-lemma-length-additive", "coherent-lemma-coherent-filter", "varieties-lemma-numerical-polynomial-from-euler", "coherent-lemma-relative-affine-cohomology" ], "ref_ids": [ 693, 11029, 631, 3329, 11121, 3284 ] } ], "ref_ids": [] }, { "id": 11123, "type": "theorem", "label": "varieties-lemma-intersection-number-integer", "categories": [ "varieties" ], "title": "varieties-lemma-intersection-number-integer", "contents": [ "In the situation of Definition \\ref{definition-intersection-number}", "the intersection number", "$(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$", "is an integer." ], "refs": [ "varieties-definition-intersection-number" ], "proofs": [ { "contents": [ "Any numerical polynomial of degree $e$ in $n_1, \\ldots, n_d$ can be", "written uniquely as a $\\mathbf{Z}$-linear combination of the functions", "${n_1 \\choose k_1}{n_2 \\choose k_2} \\ldots {n_d \\choose k_d}$ with", "$k_1 + \\ldots + k_d \\leq e$. Apply this with $e = d$.", "Left as an exercise." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 11162 ] }, { "id": 11124, "type": "theorem", "label": "varieties-lemma-intersection-number-additive", "categories": [ "varieties" ], "title": "varieties-lemma-intersection-number-additive", "contents": [ "In the situation of Definition \\ref{definition-intersection-number}", "the intersection number", "$(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$", "is additive: if $\\mathcal{L}_i = \\mathcal{L}_i' \\otimes \\mathcal{L}_i''$,", "then we have", "$$", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_i \\cdots \\mathcal{L}_d \\cdot Z) =", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_i' \\cdots \\mathcal{L}_d \\cdot Z) +", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_i'' \\cdots \\mathcal{L}_d \\cdot Z)", "$$" ], "refs": [ "varieties-definition-intersection-number" ], "proofs": [ { "contents": [ "This is true because by Lemma \\ref{lemma-numerical-polynomial-from-euler}", "the function", "$$", "(n_1, \\ldots, n_{i - 1}, n_i', n_i'', n_{i + 1}, \\ldots, n_d)", "\\mapsto", "\\chi(Z, \\mathcal{L}_1^{\\otimes n_1} \\otimes", "\\ldots \\otimes (\\mathcal{L}_i')^{\\otimes n_i'} \\otimes", "(\\mathcal{L}_i'')^{\\otimes n_i''} \\otimes \\ldots \\otimes", "\\mathcal{L}_d^{\\otimes n_d}|_Z)", "$$", "is a numerical polynomial of total degree at most $d$ in $d + 1$ variables." ], "refs": [ "varieties-lemma-numerical-polynomial-from-euler" ], "ref_ids": [ 11121 ] } ], "ref_ids": [ 11162 ] }, { "id": 11125, "type": "theorem", "label": "varieties-lemma-intersection-number-in-terms-of-components", "categories": [ "varieties" ], "title": "varieties-lemma-intersection-number-in-terms-of-components", "contents": [ "In the situation of Definition \\ref{definition-intersection-number}", "let $Z_i \\subset Z$ be the irreducible components of dimension $d$. Let", "$m_i = \\text{length}_{\\mathcal{O}_{X, \\xi_i}}(\\mathcal{O}_{Z, \\xi_i})$", "where $\\xi_i \\in Z_i$ is the generic point. Then", "$$", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) =", "\\sum m_i(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z_i)", "$$" ], "refs": [ "varieties-definition-intersection-number" ], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-numerical-polynomial-leading-term}", "and the definitions." ], "refs": [ "varieties-lemma-numerical-polynomial-leading-term" ], "ref_ids": [ 11122 ] } ], "ref_ids": [ 11162 ] }, { "id": 11126, "type": "theorem", "label": "varieties-lemma-intersection-number-and-pullback", "categories": [ "varieties" ], "title": "varieties-lemma-intersection-number-and-pullback", "contents": [ "Let $k$ be a field. Let $f : Y \\to X$ be a morphism of proper schemes over $k$.", "Let $Z \\subset Y$ be an integral closed subscheme of dimension $d$ and let", "$\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible $\\mathcal{O}_X$-modules.", "Then", "$$", "(f^*\\mathcal{L}_1 \\cdots f^*\\mathcal{L}_d \\cdot Z) =", "\\deg(f|_Z : Z \\to f(Z)) (\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot f(Z))", "$$", "where $\\deg(Z \\to f(Z))$ is as in", "Morphisms, Definition \\ref{morphisms-definition-degree}", "or $0$ if $\\dim(f(Z)) < d$." ], "refs": [ "morphisms-definition-degree" ], "proofs": [ { "contents": [ "The left hand side is computed using the coefficient of $n_1 \\ldots n_d$", "in the function", "$$", "\\chi(Y, \\mathcal{O}_Z \\otimes f^*\\mathcal{L}_1^{\\otimes n_1} \\otimes", "\\ldots \\otimes f^*\\mathcal{L}_d^{\\otimes n_d}) =", "\\sum (-1)^i", "\\chi(X, R^if_*\\mathcal{O}_Z \\otimes", "\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_d^{\\otimes n_d})", "$$", "The equality follows from Lemma \\ref{lemma-euler-characteristic-morphism}", "and the projection formula", "(Cohomology, Lemma \\ref{cohomology-lemma-projection-formula}).", "If $f(Z)$ has dimension $< d$, then the right hand side", "is a polynomial of total degree $< d$ by", "Lemma \\ref{lemma-numerical-polynomial-from-euler}", "and the result is true. Assume $\\dim(f(Z)) = d$. Let", "$\\xi \\in f(Z)$ be the generic point. By", "dimension theory (see Lemmas \\ref{lemma-dimension-locally-algebraic} and", "\\ref{lemma-dimension-fibres-locally-algebraic})", "the generic point of $Z$ is the unique point of $Z$ mapping to $\\xi$.", "Then $f : Z \\to f(Z)$ is finite over a nonempty open of $f(Z)$, see", "Morphisms, Lemma \\ref{morphisms-lemma-generically-finite}.", "Thus $\\deg(f : Z \\to f(Z))$ is defined and in fact it is equal", "to the length of the stalk of $f_*\\mathcal{O}_Z$ at $\\xi$", "over $\\mathcal{O}_{X, \\xi}$. Moreover, the stalk of", "$R^if_*\\mathcal{O}_X$ at $\\xi$ is zero for $i > 0$ because", "we just saw that $f|_Z$ is finite in a neighbourhood of $\\xi$", "(so that Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-finite-pushforward-coherent} gives the vanishing).", "Thus the terms $\\chi(X, R^if_*\\mathcal{O}_Z \\otimes", "\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_d^{\\otimes n_d})$ with $i > 0$ have total", "degree $< d$ and", "$$", "\\chi(X, f_*\\mathcal{O}_Z \\otimes", "\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_d^{\\otimes n_d})", "=", "\\deg(f : Z \\to f(Z)) \\chi(f(Z),", "\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_d^{\\otimes n_d}|_{f(Z)})", "$$", "modulo a polynomial of total degree $< d$ by", "Lemma \\ref{lemma-numerical-polynomial-leading-term}.", "The desired result follows." ], "refs": [ "varieties-lemma-euler-characteristic-morphism", "cohomology-lemma-projection-formula", "varieties-lemma-numerical-polynomial-from-euler", "varieties-lemma-dimension-locally-algebraic", "varieties-lemma-dimension-fibres-locally-algebraic", "morphisms-lemma-generically-finite", "coherent-lemma-finite-pushforward-coherent", "varieties-lemma-numerical-polynomial-leading-term" ], "ref_ids": [ 11032, 2243, 11121, 10989, 10990, 5487, 3316, 11122 ] } ], "ref_ids": [ 5587 ] }, { "id": 11127, "type": "theorem", "label": "varieties-lemma-numerical-intersection-effective-Cartier-divisor", "categories": [ "varieties" ], "title": "varieties-lemma-numerical-intersection-effective-Cartier-divisor", "contents": [ "Let $k$ be a field. Let $X$ be proper over $k$. Let $Z \\subset X$ be", "a closed subscheme of dimension $d$. Let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$", "be invertible $\\mathcal{O}_X$-modules. Assume there exists an", "effective Cartier divisor $D \\subset Z$ such that", "$\\mathcal{L}_1|_Z \\cong \\mathcal{O}_Z(D)$. Then", "$$", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) =", "(\\mathcal{L}_2 \\cdots \\mathcal{L}_d \\cdot D)", "$$" ], "refs": [], "proofs": [ { "contents": [ "We may replace $X$ by $Z$ and $\\mathcal{L}_i$ by $\\mathcal{L}_i|_Z$.", "Thus we may assume $X = Z$ and $\\mathcal{L}_1 = \\mathcal{O}_X(D)$.", "Then $\\mathcal{L}_1^{-1}$ is the ideal sheaf of $D$ and we can", "consider the short exact sequence", "$$", "0 \\to \\mathcal{L}_1^{\\otimes -1} \\to \\mathcal{O}_X \\to \\mathcal{O}_D \\to 0", "$$", "Set", "$P(n_1, \\ldots, n_d) =", "\\chi(X, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_d^{\\otimes n_d})$", "and", "$Q(n_1, \\ldots, n_d) =", "\\chi(D, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_d^{\\otimes n_d}|_D)$.", "We conclude from additivity that", "$$", "P(n_1, \\ldots, n_d) - P(n_1 - 1, n_2, \\ldots, n_d) =", "Q(n_1, \\ldots, n_d)", "$$", "Because the total degree of $P$ is at most $d$, we see that", "the coefficient of $n_1 \\ldots n_d$ in $P$ is equal to the coefficient", "of $n_2 \\ldots n_d$ in $Q$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11128, "type": "theorem", "label": "varieties-lemma-ample-positive", "categories": [ "varieties" ], "title": "varieties-lemma-ample-positive", "contents": [ "Let $k$ be a field. Let $X$ be proper over $k$. Let $Z \\subset X$ be", "a closed subscheme of dimension $d$. If $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$", "are ample, then $(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$ is positive." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by induction on $d$. The case $d = 0$", "follows from Lemma \\ref{lemma-chi-tensor-finite}. Assume $d > 0$.", "By Lemma \\ref{lemma-intersection-number-in-terms-of-components}", "we may assume that $Z$ is an integral closed subscheme.", "In fact, we may replace $X$ by $Z$ and $\\mathcal{L}_i$", "by $\\mathcal{L}_i|_Z$ to reduce to the case $Z = X$ is a", "proper variety of dimension $d$.", "By Lemma \\ref{lemma-intersection-number-additive}", "we may replace $\\mathcal{L}_1$ by a positive tensor power.", "Thus we may assume there exists a nonzero section", "$s \\in \\Gamma(X, \\mathcal{L}_1)$", "such that $X_s$ is affine (here we use the definition of", "ample invertible sheaf, see", "Properties, Definition \\ref{properties-definition-ample}).", "Observe that $X$ is not affine because proper and affine", "implies finite (Morphisms, Lemma \\ref{morphisms-lemma-finite-proper})", "which contradicts $d > 0$. It follows that $s$ has a nonempty vanishing", "scheme $Z(s) \\subset X$. Since $X$ is a variety, $s$ is a regular section", "of $\\mathcal{L}_1$, so $Z(s)$ is an effective Cartier divisor,", "thus $Z(s)$ has codimension $1$ in $X$, and", "hence $Z(s)$ has dimension $d - 1$ (here we use material from", "Divisors, Sections \\ref{divisors-section-effective-Cartier-divisors} and", "\\ref{divisors-section-Noetherian-effective-Cartier} and from dimension theory", "as in Lemma \\ref{lemma-dimension-locally-algebraic}).", "By Lemma \\ref{lemma-numerical-intersection-effective-Cartier-divisor}", "we have", "$$", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot X) =", "(\\mathcal{L}_2 \\cdots \\mathcal{L}_d \\cdot Z(s))", "$$", "By induction the right hand side is positive and the proof is complete." ], "refs": [ "varieties-lemma-chi-tensor-finite", "varieties-lemma-intersection-number-in-terms-of-components", "varieties-lemma-intersection-number-additive", "properties-definition-ample", "morphisms-lemma-finite-proper", "varieties-lemma-dimension-locally-algebraic", "varieties-lemma-numerical-intersection-effective-Cartier-divisor" ], "ref_ids": [ 11030, 11125, 11124, 3088, 5445, 10989, 11127 ] } ], "ref_ids": [] }, { "id": 11129, "type": "theorem", "label": "varieties-lemma-degree-finite-morphism-in-terms-degrees", "categories": [ "varieties" ], "title": "varieties-lemma-degree-finite-morphism-in-terms-degrees", "contents": [ "Let $k$ be a field. Let $f : Y \\to X$ be a finite", "dominant morphism of proper varieties over $k$. Let $\\mathcal{L}$", "be an ample invertible $\\mathcal{O}_X$-module.", "Then", "$$", "\\deg_{f^*\\mathcal{L}}(Y) = \\deg(f) \\deg_\\mathcal{L}(X)", "$$", "where $\\deg(f)$ is as in", "Morphisms, Definition \\ref{morphisms-definition-degree}." ], "refs": [ "morphisms-definition-degree" ], "proofs": [ { "contents": [ "The statement makes sense because $f^*\\mathcal{L}$ is ample by", "Morphisms, Lemma \\ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}.", "Having said this the result is a special case of", "Lemma \\ref{lemma-intersection-number-and-pullback}." ], "refs": [ "morphisms-lemma-pullback-ample-tensor-relatively-ample", "varieties-lemma-intersection-number-and-pullback" ], "ref_ids": [ 5383, 11126 ] } ], "ref_ids": [ 5587 ] }, { "id": 11130, "type": "theorem", "label": "varieties-lemma-intersection-numbers-and-degrees-on-curves", "categories": [ "varieties" ], "title": "varieties-lemma-intersection-numbers-and-degrees-on-curves", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$.", "Let $Z \\subset X$ be a closed subscheme of dimension $\\leq 1$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Then", "$$", "(\\mathcal{L} \\cdot Z) = \\deg(\\mathcal{L}|_Z)", "$$", "where $\\deg(\\mathcal{L}|_Z)$ is as in", "Definition \\ref{definition-degree-invertible-sheaf}.", "If $\\mathcal{L}$ is ample, then", "$\\deg_\\mathcal{L}(Z) = \\deg(\\mathcal{L}|_Z)$." ], "refs": [ "varieties-definition-degree-invertible-sheaf" ], "proofs": [ { "contents": [ "This follows from the fact that the function", "$n \\mapsto \\chi(Z, \\mathcal{L}|_Z^{\\otimes n})$ has degree $1$", "and hence the leading coefficient is the difference of consecutive values." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 11161 ] }, { "id": 11131, "type": "theorem", "label": "varieties-lemma-generate-over-complement", "categories": [ "varieties" ], "title": "varieties-lemma-generate-over-complement", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$", "be an ample invertible $\\mathcal{O}_X$-module. Let $Z \\subset X$ be a", "closed subscheme. Then there exists an integer $n_0$ such that for all", "$n \\geq n_0$ the kernel $V_n$ of", "$\\Gamma(X, \\mathcal{L}^{\\otimes n}) \\to \\Gamma(Z, \\mathcal{L}^{\\otimes n}|_Z)$", "generates $\\mathcal{L}^{\\otimes n}|_{X \\setminus Z}$ and", "the canonical morphism", "$$", "X \\setminus Z \\longrightarrow \\mathbf{P}(V_n)", "$$", "is an immersion of schemes over $k$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent ideal sheaf of", "$Z$. Observe that via the inclusion", "$\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n} \\subset", "\\mathcal{L}^{\\otimes n}$ we have", "$V_n = \\Gamma(X, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n})$.", "Choose $n_1$ such that for $n \\geq n_1$ the sheaf", "$\\mathcal{I} \\otimes \\mathcal{L}^{\\otimes n}$ is globally generated, see", "Properties, Proposition \\ref{properties-proposition-characterize-ample}.", "It follows that $V_n$ gererates $\\mathcal{L}^{\\otimes n}|_{X \\setminus Z}$", "for $n \\geq n_1$.", "\\medskip\\noindent", "For $n \\geq n_1$ denote", "$\\psi_n : V_n \\to", "\\Gamma(X \\setminus Z, \\mathcal{L}^{\\otimes n}|_{X \\setminus Z})$", "the restriction map. We get a canonical morphism", "$$", "\\varphi = \\varphi_{\\mathcal{L}^{\\otimes n}|_{X \\setminus Z}, \\psi_n} :", "X \\setminus Z", "\\longrightarrow", "\\mathbf{P}(V_n)", "$$", "by Constructions, Example \\ref{constructions-example-projective-space}.", "Choose $n_2$ such that for all $n \\geq n_2$ the invertible sheaf", "$\\mathcal{L}^{\\otimes n}$ is very ample on $X$.", "We claim that $n_0 = n_1 + n_2$ works.", "\\medskip\\noindent", "Proof of the claim. Say $n \\geq n_0$ and write $n = n_1 + n'$.", "For $x \\in X \\setminus Z$ we can choose $s_1 \\in V_1$ not", "vanishing at $x$. Set $V' = \\Gamma(X, \\mathcal{L}^{\\otimes n'})$.", "By our choice of $n$ and $n'$ we see that the corresponding morphism", "$\\varphi' : X \\to \\mathbf{P}(V')$ is a closed immersion. Thus if we choose", "$s' \\in \\Gamma(X, \\mathcal{L}^{\\otimes n'})$ not vanishing at $x$,", "then $X_{s'} = (\\varphi')^{-1}(D_+(s')$ (see", "Constructions, Lemma \\ref{constructions-lemma-invertible-map-into-proj})", "is affine and $X_{s'} \\to D_+(s')$ is a closed immersion.", "Then $s = s_1 \\otimes s' \\in V_n$ does not vanish at $x$.", "If $D_+(s) \\subset \\mathbf{P}(V_n)$ denotes the", "corresponding open affine space of our projective space, then", "$\\varphi^{-1}(D_+(s)) = X_s \\subset X \\setminus Z$ (see reference above).", "The open $X_s = X_{s'} \\cap X_{s_1}$ is affine, see", "Properties, Lemma \\ref{properties-lemma-affine-cap-s-open}.", "Consider the ring map", "$$", "\\text{Sym}(V)_{(s)} \\longrightarrow \\mathcal{O}_X(X_s)", "$$", "defining the morphism $X_s \\to D_+(s)$. Because $X_{s'} \\to D_+(s')$", "is a closed immersion, the images of the elements", "$$", "\\frac{s_1 \\otimes t'}{s_1 \\otimes s'}", "$$", "where $t' \\in V'$ generate the image of", "$\\mathcal{O}_X(X_{s'}) \\to \\mathcal{O}_X(X_s)$.", "Since $X_s \\to X_{s'}$ is an open immersion,", "this implies that $X_s \\to D_+(s)$ is an immersion of affine schemes", "(see below). Thus $\\varphi_n$ is an immersion by", "Morphisms, Lemma \\ref{morphisms-lemma-check-immersion}.", "\\medskip\\noindent", "Let $a : A' \\to A$ and $c : B \\to A$ be ring maps such that", "$\\Spec(a)$ is an immersion and $\\Im(a) \\subset \\Im(c)$.", "Set $B' = A' \\times_A B$ with projections $b : B' \\to B$", "and $c' : B' \\to A'$. By assumption $c'$", "is surjective and hence $\\Spec(c')$ is a closed immersion.", "Whence $\\Spec(c') \\circ \\Spec(a)$ is an immersion", "(Schemes, Lemma \\ref{schemes-lemma-composition-immersion}).", "Then $\\Spec(c)$ has to be an immersion because it factors", "the immersion $\\Spec(c') \\circ \\Spec(a) = \\Spec(b) \\circ \\Spec(c)$, see", "Morphisms, Lemma \\ref{morphisms-lemma-immersion-permanence}." ], "refs": [ "properties-proposition-characterize-ample", "constructions-lemma-invertible-map-into-proj", "properties-lemma-affine-cap-s-open", "morphisms-lemma-check-immersion", "schemes-lemma-composition-immersion", "morphisms-lemma-immersion-permanence" ], "ref_ids": [ 3067, 12628, 3042, 5135, 7732, 5132 ] } ], "ref_ids": [] }, { "id": 11132, "type": "theorem", "label": "varieties-lemma-bertini", "categories": [ "varieties" ], "title": "varieties-lemma-bertini", "contents": [ "In Situation \\ref{situation-family-divisors} assume", "\\begin{enumerate}", "\\item $X$ is smooth over $k$,", "\\item the image of $\\psi : V \\to \\Gamma(X, \\mathcal{L})$", "generates $\\mathcal{L}$,", "\\item the corresponding morphism", "$\\varphi_{\\mathcal{L}, \\psi} : X \\to \\mathbf{P}(V)$ is", "an immersion.", "\\end{enumerate}", "Then for general $v \\in V \\otimes_k k'$", "the scheme $H_v$ is smooth over $k'$." ], "refs": [], "proofs": [ { "contents": [ "(We observe that $X$ is separated and finite type as a locally", "closed subscheme of a projective space.)", "Let us use the notation introduced above the statement of the lemma.", "We consider the projections", "$$", "\\xymatrix{", "\\mathbf{A}^r_k \\times_k X \\ar[d] &", "H_{univ} \\ar[l] \\ar[ld]^p \\ar[r] \\ar[rd]_q &", "\\mathbf{A}^r_k \\times_k X \\ar[d] \\\\", "X & &", "\\mathbf{A}^r_k", "}", "$$", "Let $\\Sigma \\subset H_{univ}$ be the singular locus of the morphsm", "$q : H_{univ} \\to \\mathbf{A}^r_k$, i.e., the set of points where", "$q$ is not smooth. Then $\\Sigma$ is closed because the smooth locus", "of a morphism is open by definition. Since the fibre of a", "smooth morphism is smooth, it suffices to prove $q(\\Sigma)$", "is contained in a proper closed subset of $\\mathbf{A}^r_k$.", "Since $\\Sigma$ (with reduced induced scheme structure) is a", "finite type scheme over $k$ it suffices to prove $\\dim(\\Sigma) < r$", "This follows from Lemma \\ref{lemma-dimension-fibres-locally-algebraic}.", "Since dimensions aren't changed by replacing $k$ by a bigger field", "(Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}),", "we may and do assume $k$ is algebraically closed.", "By dimension theory (Lemma \\ref{lemma-dimension-fibres-locally-algebraic}),", "it suffices to prove that for $x \\in X \\setminus Z$ closed we have", "$p^{-1}(\\{x\\}) \\cap \\Sigma$ has dimension $< r - \\dim(X')$", "where $X'$ is the unique irreducible component of $X$ containing $x$.", "As $X$ is smooth over $k$ and $x$ is a closed point we have", "$\\dim(X') = \\dim \\mathfrak m_x/\\mathfrak m_x^2$", "(Morphisms, Lemma \\ref{morphisms-lemma-smooth-omega-finite-locally-free}", "and Algebra, Lemma \\ref{algebra-lemma-rank-omega}).", "Thus we win if", "$$", "\\dim p^{-1}(x) \\cap \\Sigma < r - \\dim \\mathfrak m_x/\\mathfrak m_x^2", "$$", "for all $x \\in X$ closed.", "\\medskip\\noindent", "Since $V$ globally generated $\\mathcal{L}$, for every irreducible component", "$X'$ of $X$ there is a nonempty Zariski open of $\\mathbf{A}^r$ such that the", "fibres of $q$ over this open do not contain $X'$. (For example, if $x' \\in X'$", "is a closed point, then we can take the open corresponding to those", "vectors $v \\in V$ such that $\\psi(v)$ does not vanish at $x'$. This", "open will be the complement of a hyperplane in $\\mathbf{A}^r_k$.)", "Let $U \\subset \\mathbf{A}^r$ be the (nonempty) intersection of these opens.", "Then the fibres of $q^{-1}(U) \\to U$ are effective Cartier divisors", "on the fibres of $U \\times_k X \\to U$ (because a nonvanishing section", "of an invertible module on an integral scheme is a regular section). Hence the", "morphism $q^{-1}(U) \\to U$ is flat by", "Divisors, Lemma \\ref{divisors-lemma-fibre-Cartier}.", "Thus for $x \\in X$ closed and $v \\in V = \\mathbf{A}^r_k(k)$,", "if $(x, v) \\in H_{univ}$, i.e., if $x \\in H_v$", "then $q$ is smooth at $(x, v)$ if and only if the fibre", "$H_v$ is smooth at $x$, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-at-point}.", "\\medskip\\noindent", "Consider the image $\\psi(v)_x$ in the stalk $\\mathcal{L}_x$", "of the section corresponding to $v \\in V$. We have", "$$", "x \\in H_v \\Leftrightarrow \\psi(v)_x \\in \\mathfrak m_x\\mathcal{L}_x", "$$", "If this is true, then we have", "$$", "H_v\\text{ singular at }x \\Leftrightarrow", "\\psi(v)_x \\in \\mathfrak m_x^2\\mathcal{L}_x", "$$", "Namely, $\\psi(v)_x$ is not contained in $\\mathfrak m_x^2\\mathcal{L}_x$", "$\\Leftrightarrow$", "the local equation for $H_v \\subset X$ at $x$ is not contained", "in $\\mathfrak m_x^2$", "$\\Leftrightarrow$", "$\\mathcal{O}_{H_v, x}$ is regular", "(Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM})", "$\\Leftrightarrow$", "$H_v$ is smooth at $x$ over $k$", "(Algebra, Lemma \\ref{algebra-lemma-separable-smooth}).", "We conclude that the closed points of $p^{-1}(x) \\cap \\Sigma$ correspond", "to those $v \\in V$ such that $\\psi(v)_x \\in \\mathfrak m_x^2\\mathcal{L}_x$.", "However, as $\\varphi_{\\mathcal{L}, \\psi}$ is an immersion the", "map", "$$", "V \\longrightarrow \\mathcal{L}_x/\\mathfrak m_x^2\\mathcal{L}_x", "$$", "is surjective (small detail omitted). By the above, the closed points", "of the locus $p^{-1}(x) \\cap \\Sigma$ viewed as a subspace of $V$", "is the kernel of this map and hence has dimension", "$r - \\dim \\mathfrak m_x/\\mathfrak m_x^2 - 1$ as desired." ], "refs": [ "varieties-lemma-dimension-fibres-locally-algebraic", "morphisms-lemma-dimension-fibre-after-base-change", "varieties-lemma-dimension-fibres-locally-algebraic", "morphisms-lemma-smooth-omega-finite-locally-free", "algebra-lemma-rank-omega", "divisors-lemma-fibre-Cartier", "morphisms-lemma-smooth-at-point", "algebra-lemma-regular-ring-CM", "algebra-lemma-separable-smooth" ], "ref_ids": [ 10990, 5279, 10990, 5334, 1221, 7978, 5335, 941, 1225 ] } ], "ref_ids": [] }, { "id": 11133, "type": "theorem", "label": "varieties-lemma-vanishin-h0-negative", "categories": [ "varieties" ], "title": "varieties-lemma-vanishin-h0-negative", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$", "be an ample invertible $\\mathcal{O}_X$-module. Let $\\mathcal{F}$ be a", "coherent $\\mathcal{O}_X$-module. If $\\text{Ass}(\\mathcal{F})$ does not", "contain any closed points, then", "$\\Gamma(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) = 0$", "for $n \\ll 0$." ], "refs": [], "proofs": [ { "contents": [ "For a coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "let $\\mathcal{P}(\\mathcal{F})$ be the property: there", "exists an $n_0 \\in \\mathbf{Z}$ such that for $n \\leq n_0$", "every section $s$ of", "$\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}$", "has support consisting only of closed points.", "Since $\\text{Ass}(\\mathcal{F}) =", "\\text{Ass}(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n})$", "we see that it suffices to prove $\\mathcal{P}$", "holds for all coherent modules on $X$.", "To do this we will prove that conditions (1), (2), and (3) of", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-property-higher-rank-cohomological}", "are satisfied.", "\\medskip\\noindent", "To see condition (1) suppose that", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0", "$$", "is a short exact sequence of coherent $\\mathcal{O}_X$-modules", "such that we have $\\mathcal{P}$ for $\\mathcal{F}_i$, $i = 1, 2$.", "Let $n_1, n_2$ be the cutoffs we find.", "Let $\\mathcal{F}'_2 \\subset \\mathcal{F}_2$ be the maximal", "coherent submodule whose support is a finite set of closed", "points. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the annihilator", "of $\\mathcal{F}'_2$. Since $\\mathcal{L}$ is ample, we can find an $e > 0$", "such that $\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes e}$", "is globally generated. Set $n_0 = \\min(n_2, n_1 - e)$. Let $n \\leq n_0$", "and let $t$ be a global section of", "$\\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}$.", "The image of $t$ in $\\mathcal{F}_2 \\otimes \\mathcal{L}^{\\otimes n}$", "falls into $\\mathcal{F}'_2 \\otimes \\mathcal{L}^{\\otimes n}$ because", "$n \\leq n_2$. Hence for any", "$s \\in \\Gamma(X, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes e})$", "the product $t \\otimes s$ lies in", "$\\mathcal{F}_1 \\otimes \\mathcal{L}^{\\otimes n + e}$.", "Thus $t \\otimes s$ has support contained in", "the finite set of closed points in $\\text{Ass}(\\mathcal{F}_1)$", "because $n + e \\leq n_1$. Since by our choice of $e$", "we may choose $s$ invertible in any point", "not in the support of $\\mathcal{F}'_2$", "we conclude that the support of $t$ is contained in the union of the ", "finite set of closed points in $\\text{Ass}(\\mathcal{F}_1)$ and", "the finite set of closed points in $\\text{Ass}(\\mathcal{F}_2)$.", "This finishes the proof of condition (1).", "\\medskip\\noindent", "Condition (2) is immediate.", "\\medskip\\noindent", "For condition (3) we choose $\\mathcal{G} = \\mathcal{O}_Z$.", "In this case, if $Z$ is a closed point of $X$, then there is", "nothing the show. If $\\dim(Z) > 0$, then we will show that", "$\\Gamma(Z, \\mathcal{L}^{\\otimes n}|_Z) = 0$", "for $n < 0$. Namely, let $s$ be a nonzero section of a negative", "power of $\\mathcal{L}|_Z$. Choose a nonzero section $t$ of a positive", "power of $\\mathcal{L}|_Z$ (this is possible as $\\mathcal{L}$ is ample, see", "Properties, Proposition \\ref{properties-proposition-characterize-ample}).", "Then $s^{\\deg(t)} \\otimes t^{\\deg(s)}$", "is a nonzero global section of $\\mathcal{O}_Z$", "(because $Z$ is integral) and hence", "a unit (Lemma \\ref{lemma-proper-geometrically-reduced-global-sections}).", "This implies that $t$ is a trivializing section", "of a positive power of $\\mathcal{L}$.", "Thus the function $n \\mapsto \\dim_k \\Gamma(X, \\mathcal{L}^{\\otimes n})$", "is bounded on an infinite set of positive integers which contradicts", "asymptotic Riemann-Roch", "(Proposition \\ref{proposition-asymptotic-riemann-roch})", "since $\\dim(Z) > 0$." ], "refs": [ "coherent-lemma-property-higher-rank-cohomological", "properties-proposition-characterize-ample", "varieties-lemma-proper-geometrically-reduced-global-sections", "varieties-proposition-asymptotic-riemann-roch" ], "ref_ids": [ 3334, 3067, 10948, 11140 ] } ], "ref_ids": [] }, { "id": 11134, "type": "theorem", "label": "varieties-lemma-vanishin-h1-negative", "categories": [ "varieties" ], "title": "varieties-lemma-vanishin-h1-negative", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$", "be an ample invertible $\\mathcal{O}_X$-module. Let $\\mathcal{F}$ be a", "coherent $\\mathcal{O}_X$-module. Assume that", "for $x \\in X$ closed we have $\\text{depth}(\\mathcal{F}_x) \\geq 2$.", "Then", "$H^1(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes m}) = 0$", "for $m \\ll 0$." ], "refs": [], "proofs": [ { "contents": [ "Choose a closed immersion $i : X \\to \\mathbf{P}^n_k$ such that", "$i^*\\mathcal{O}(1) \\cong \\mathcal{L}^{\\otimes e}$ for some $e > 0$", "(see Morphisms, Lemma", "\\ref{morphisms-lemma-finite-type-over-affine-ample-very-ample}).", "Then it suffices to prove the lemma for", "$$", "\\mathcal{G} =", "i_*(\\mathcal{F} \\oplus \\mathcal{F} \\otimes \\mathcal{L} \\oplus \\ldots", "\\oplus \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes e - 1})", "\\quad\\text{and}\\quad \\mathcal{O}(1)", "$$", "on $\\mathbf{P}^n_k$. Namely, we have", "$$", "H^1(\\mathbf{P}^n_k, \\mathcal{G}(m)) =", "\\bigoplus\\nolimits_{j = 0, \\ldots, e - 1}", "H^1(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes j + me})", "$$", "by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology}.", "Also, if $y \\in \\mathbf{P}^n_k$ is a closed point then", "$\\text{depth}(\\mathcal{G}_y) = \\infty$ if $y \\not \\in i(X)$", "and $\\text{depth}(\\mathcal{G}_y) = \\text{depth}(\\mathcal{F}_x)$", "if $y = i(x)$ because in this case", "$\\mathcal{G}_y \\cong \\mathcal{F}_x^{\\oplus e}$ as a module over", "$\\mathcal{O}_{\\mathbf{P}^n_k, x}$ and we can use for example", "Algebra, Lemma \\ref{algebra-lemma-depth-goes-down-finite}", "to get the equality.", "\\medskip\\noindent", "Assume $X = \\mathbf{P}^n_k$ and $\\mathcal{L} = \\mathcal{O}(1)$ and", "$k$ is infinite. Choose $s \\in H^0(\\mathbf{P}^1_k, \\mathcal{O}(1))$", "which determines an exact sequence", "$$", "0 \\to \\mathcal{F}(-1) \\xrightarrow{s} \\mathcal{F} \\to \\mathcal{G} \\to 0", "$$", "as in Lemma \\ref{lemma-exact-sequence-induction}. Since the map", "$\\mathcal{F}(-1) \\to \\mathcal{F}$ is affine locally given by", "multiplying by a nonzerodivisor on $\\mathcal{F}$", "we see that for $x \\in \\mathbf{P}^n_k$ closed we have", "$\\text{depth}(\\mathcal{G}_x) \\geq 1$, see", "Algebra, Lemma \\ref{algebra-lemma-depth-drops-by-one}.", "Hence by Lemma \\ref{lemma-vanishin-h0-negative}", "we have $H^0(\\mathcal{G}(m)) = 0$ for $m \\ll 0$.", "Looking at the long exact sequence of cohomology after twisting", "(see Remark \\ref{remark-exact-sequence-induction-cohomology})", "we find that the sequence of numbers", "$$", "\\dim H^1(\\mathbf{P}^n_k, \\mathcal{F}(m))", "$$", "stabilizes for $m \\leq m_0$ for some integer $m_0$.", "Let $N$ be the common dimension of these", "spaces for $m \\leq m_0$. We have to show $N = 0$.", "\\medskip\\noindent", "For $d > 0$ and $m \\leq m_0$ consider the bilinear map", "$$", "H^0(\\mathbf{P}^n_k, \\mathcal{O}(d)) \\times", "H^1(\\mathbf{P}^n_k, \\mathcal{F}(m - d))", "\\longrightarrow", "H^1(\\mathbf{P}^n_k, \\mathcal{F}(m))", "$$", "By linear algebra, there is a codimension $\\leq N^2$ subspace", "$V_m \\subset H^0(\\mathbf{P}^n_k, \\mathcal{O}(d))$ such that", "multiplication by $s' \\in V_m$ annihilates", "$H^1(\\mathbf{P}^n_k, \\mathcal{F}(m - d))$.", "Observe that for $m' < m \\leq m_0$ the diagram", "$$", "\\xymatrix{", "H^0(\\mathbf{P}^n_k, \\mathcal{O}(d)) \\times", "H^1(\\mathbf{P}^n_k, \\mathcal{F}(m' - d)) \\ar[r] \\ar[d]^{1 \\times s^{m' - m}} &", "H^1(\\mathbf{P}^n_k, \\mathcal{F}(m')) \\ar[d]^{s^{m' - m}}\\\\", "H^0(\\mathbf{P}^n_k, \\mathcal{O}(d)) \\times", "H^1(\\mathbf{P}^n_k, \\mathcal{F}(m - d)) \\ar[r] &", "H^1(\\mathbf{P}^n_k, \\mathcal{F}(m))", "}", "$$", "commutes with isomorphisms going vertically. Thus $V_m = V$ is", "independent of $m \\leq m_0$. For $x \\in \\text{Ass}(\\mathcal{F})$", "set $Z = \\overline{\\{x\\}}$. For $d$ large enough the linear map", "$$", "H^0(\\mathbf{P}^n_k, \\mathcal{O}(d)) \\to H^0(Z, \\mathcal{O}(d)|_Z)", "$$", "has rank $> N^2$ because $\\dim(Z) \\geq 1$ (for example this follows", "from asymptotic Riemann-Roch and ampleness $\\mathcal{O}(1)$; details", "omitted). Hence we can find $s' \\in V$ such that $s'$ does not vanish", "in any associated point of $\\mathcal{F}$ (use that the set", "of associated points is finite). Then we obtain", "$$", "0 \\to \\mathcal{F}(-d) \\xrightarrow{s'} \\mathcal{F} \\to \\mathcal{G}' \\to 0", "$$", "and as before we conclude as before that multiplication by $s'$", "on $H^1(\\mathbf{P}^n_k, \\mathcal{F}(m - d))$ is injective", "for $m \\ll 0$. This contradicts the choice of $s'$ unless", "$N = 0$ as desired.", "\\medskip\\noindent", "We still have to treat the case where $k$ is finite.", "In this case let $K/k$ be any infinite algebraic field extension.", "Denote $\\mathcal{F}_K$ and $\\mathcal{L}_K$ the pullbacks", "of $\\mathcal{F}$ and $\\mathcal{L}$ to $X_K = \\Spec(K) \\times_{\\Spec(k)} X$.", "We have", "$$", "H^1(X_K, \\mathcal{F}_K \\otimes \\mathcal{L}_K^{\\otimes m}) =", "H^1(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes m}) \\otimes_k K", "$$", "by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology}.", "On the other hand, a closed point $x_K$ of $X_K$ maps to a closed point", "$x$ of $X$ because $K/k$ is an algebraic extension. The", "ring map $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X_K, x_K}$", "is flat (Lemma \\ref{lemma-change-fields-flat}). Hence we have", "$$", "\\text{depth}(\\mathcal{F}_{x_K}) =", "\\text{depth}(\\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}}", "\\mathcal{O}_{X_K, x_K}) \\geq", "\\text{depth}(\\mathcal{F}_x)", "$$", "by Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck-module}", "(in fact equality holds here but we don't need it).", "Therefore the result over $k$ follows", "from the result over the infinite field $K$ and the proof is complete." ], "refs": [ "morphisms-lemma-finite-type-over-affine-ample-very-ample", "coherent-lemma-relative-affine-cohomology", "algebra-lemma-depth-goes-down-finite", "varieties-lemma-exact-sequence-induction", "algebra-lemma-depth-drops-by-one", "varieties-lemma-vanishin-h0-negative", "varieties-remark-exact-sequence-induction-cohomology", "coherent-lemma-flat-base-change-cohomology", "varieties-lemma-change-fields-flat", "algebra-lemma-apply-grothendieck-module" ], "ref_ids": [ 5394, 3284, 778, 11038, 774, 11133, 11167, 3298, 10903, 1360 ] } ], "ref_ids": [] }, { "id": 11135, "type": "theorem", "label": "varieties-lemma-connectedness-ample-divisor", "categories": [ "varieties" ], "title": "varieties-lemma-connectedness-ample-divisor", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$", "be an ample invertible $\\mathcal{O}_X$-module. Let", "$s \\in \\Gamma(X, \\mathcal{L})$. Assume", "\\begin{enumerate}", "\\item $s$ is a regular section", "(Divisors, Definition \\ref{divisors-definition-regular-section}),", "\\item for every closed point $x \\in X$ we have", "$\\text{depth}(\\mathcal{O}_{X, x}) \\geq 2$, and", "\\item $X$ is connected.", "\\end{enumerate}", "Then the zero scheme $Z(s)$ of $s$ is connected." ], "refs": [], "proofs": [ { "contents": [ "Since $s$ is a regular section, so is", "$s^n \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ for all $n > 1$.", "Moreover, the inclusion morphism $Z(s) \\to Z(s^n)$ is a bijection", "on underlying topological spaces. Hence if $Z(s)$ is disconnected,", "so is $Z(s^n)$. Now consider the canonical short exact sequence", "$$", "0 \\to \\mathcal{L}^{\\otimes -n} \\xrightarrow{s^n}", "\\mathcal{O}_X \\to \\mathcal{O}_{Z(s^n)} \\to 0", "$$", "Consider the $k$-algebra $R_n = \\Gamma(X, \\mathcal{O}_{Z(s^n)})$.", "If $Z(s)$ is disconnected, i.e., $Z(s^n)$ is disconnected,", "then either $R_n$ is zero in case $Z(s^n) = \\emptyset$ or", "$R_n$ contains a nontrivial idempotent in case $Z(s^n) = U \\amalg V$ with", "$U, V \\subset Z(s^n)$ open and nonempty (the reader may wish to consult", "Lemma \\ref{lemma-proper-geometrically-reduced-global-sections}).", "Thus the map $\\Gamma(X, \\mathcal{O}_X) \\to R_n$ cannot be an isomorphism.", "It follows that either $H^0(X, \\mathcal{L}^{\\otimes -n})$ or", "$H^1(X, \\mathcal{L}^{\\otimes -n})$ is nonzero for infinitely", "many positive $n$. This contradicts Lemma \\ref{lemma-vanishin-h0-negative} or", "\\ref{lemma-vanishin-h1-negative}", "and the proof is complete." ], "refs": [ "varieties-lemma-proper-geometrically-reduced-global-sections", "varieties-lemma-vanishin-h0-negative", "varieties-lemma-vanishin-h1-negative" ], "ref_ids": [ 10948, 11133, 11134 ] } ], "ref_ids": [] }, { "id": 11136, "type": "theorem", "label": "varieties-proposition-units-general", "categories": [ "varieties" ], "title": "varieties-proposition-units-general", "contents": [ "Let $k$ be a field. Let $X$ be a scheme over $k$. Assume that $X$ is", "locally of finite type over $k$, connected, reduced, and has finitely many", "irreducible components. Then $\\mathcal{O}(X)^*/k^*$ is a finitely generated", "abelian group if in addition to the conditions above at least", "one of the following conditions is satisfied:", "\\begin{enumerate}", "\\item the integral closure of $k$ in $\\Gamma(X, \\mathcal{O}_X)$ is $k$,", "\\item $X$ has a $k$-rational point, or", "\\item $X$ is geometrically integral.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\overline{k}$ be an algebraic closure of $k$.", "Let $Y$ be a connected component of $(X_{\\overline{k}})_{red}$.", "Note that the canonical morphism $p : Y \\to X$ is open (by", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open})", "and closed (by", "Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed}).", "Hence $p(Y) = X$ as $X$ was assumed connected. In particular, as", "$X$ is reduced this implies $\\mathcal{O}(X) \\subset \\mathcal{O}(Y)$. By", "Lemma \\ref{lemma-galois-action-irreducible-components-locally-finite-type}", "we see that $Y$ has finitely many irreducible components.", "Thus", "Lemma \\ref{lemma-units-general-algebraically-closed}", "applies to $Y$. This implies that if $\\mathcal{O}(X)^*/k^*$ is", "not a finitely generated abelian group, then there exist elements", "$f \\in \\mathcal{O}(X)$, $f \\not \\in k$ which map to an element of", "$\\overline{k}$ via the map $\\mathcal{O}(X) \\to \\mathcal{O}(Y)$.", "In this case $f$ is algebraic over $k$, hence", "integral over $k$. Thus, if condition (1) holds, then this cannot happen.", "To finish the proof we show that conditions (2) and (3) imply (1).", "\\medskip\\noindent", "Let $k \\subset k' \\subset \\Gamma(X, \\mathcal{O}_X)$ be the integral", "closure of $k$ in $\\Gamma(X, \\mathcal{O}_X)$. By", "Lemma \\ref{lemma-integral-closure-ground-field}", "we see that $k'$ is a field.", "If $e : \\Spec(k) \\to X$ is a $k$-rational point, then", "$e^\\sharp : \\Gamma(X, \\mathcal{O}_X) \\to k$ is a section to the", "inclusion map $k \\to \\Gamma(X, \\mathcal{O}_X)$. In particular the", "restriction of $e^\\sharp$ to $k'$ is a field map $k' \\to k$ over $k$,", "which clearly shows that (2) implies (1).", "\\medskip\\noindent", "If the integral closure $k'$ of $k$ in $\\Gamma(X, \\mathcal{O}_X)$", "is not trivial, then we see that $X$ is either not geometrically connected", "(if $k \\subset k'$ is not purely inseparable) or that $X$ is not", "geometrically reduced (if $k \\subset k'$ is nontrivial purely inseparable).", "Details omitted. Hence (3) implies (1)." ], "refs": [ "morphisms-lemma-scheme-over-field-universally-open", "morphisms-lemma-integral-universally-closed", "varieties-lemma-galois-action-irreducible-components-locally-finite-type", "varieties-lemma-units-general-algebraically-closed", "varieties-lemma-integral-closure-ground-field" ], "ref_ids": [ 5254, 5441, 10944, 11020, 11021 ] } ], "ref_ids": [] }, { "id": 11137, "type": "theorem", "label": "varieties-proposition-unique-base-field", "categories": [ "varieties" ], "title": "varieties-proposition-unique-base-field", "contents": [ "Let $X$ be a scheme. Let $a : X \\to \\Spec(k_1)$ and", "$b : X \\to \\Spec(k_2)$ be morphisms from $X$ to spectra of fields.", "Assume $a, b$ are locally of finite type, and", "$X$ is reduced, and connected. Then we have", "$k_1' = k_2'$, where $k_i' \\subset \\Gamma(X, \\mathcal{O}_X)$ is", "the integral closure of $k_i$ in $\\Gamma(X, \\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "First, assume the lemma holds in case $X$ is quasi-compact (we will", "do the quasi-compact case below).", "As $X$ is locally of finite type over a field, it is locally Noetherian, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}.", "In particular this means that it is locally connected,", "connected components of open subsets are open, and", "intersections of quasi-compact opens are quasi-compact, see", "Properties, Lemma \\ref{properties-lemma-Noetherian-topology},", "Topology, Lemma \\ref{topology-lemma-locally-connected},", "Topology, Section \\ref{topology-section-noetherian}, and", "Topology, Lemma \\ref{topology-lemma-constructible-Noetherian-space}.", "Pick an open covering $X = \\bigcup_{i \\in I} U_i$", "such that each $U_i$ is quasi-compact and connected.", "For each $i$ let $K_i \\subset \\mathcal{O}_X(U_i)$ be the integral", "closure of $k_1$ and of $k_2$.", "For each pair $i, j \\in I$ we decompose", "$$", "U_i \\cap U_j = \\coprod U_{i, j, l}", "$$", "into its finitely many connected components. Write", "$K_{i, j, l} \\subset \\mathcal{O}(U_{i, j, l})$", "for the integral closure of $k_1$ and of $k_2$. By", "Lemma \\ref{lemma-integral-closure-ground-field}", "the rings $K_i$ and $K_{i, j, l}$ are fields.", "Now we claim that $k_1'$ and $k_2'$ both equal the kernel of the map", "$$", "\\prod K_i \\longrightarrow \\prod K_{i, j, l}, \\quad", "(x_i)_i \\longmapsto x_i|_{U_{i, j, l}} - x_j|_{U_{i, j, l}}", "$$", "which proves what we want.", "Namely, it is clear that $k_1'$ is contained in this kernel.", "On the other hand, suppose that $(x_i)_i$ is in the kernel.", "By the sheaf condition $(x_i)_i$ corresponds to $f \\in \\mathcal{O}(X)$.", "Pick some $i_0 \\in I$ and let $P(T) \\in k_1[T]$ be a monic polynomial", "with $P(x_{i_0}) = 0$. Then we claim that $P(f) = 0$ which proves", "that $f \\in k_1$. To prove this we have to show that $P(x_i) = 0$", "for all $i$. Pick $i \\in I$. As $X$ is connected there exists a", "sequence $i_0, i_1, \\ldots, i_n = i \\in I$ such that", "$U_{i_t} \\cap U_{i_{t + 1}} \\not = \\emptyset$. Now this means that", "for each $t$ there exists an $l_t$ such that $x_{i_t}$ and", "$x_{i_{t + 1}}$ map to the same element of the field $K_{i, j, l}$.", "Hence if $P(x_{i_t}) = 0$, then $P(x_{i_{t + 1}}) = 0$. By", "induction, starting with $P(x_{i_0}) = 0$ we deduce that", "$P(x_i) = 0$ as desired.", "\\medskip\\noindent", "To finish the proof of the lemma we prove the lemma under the", "additional hypothesis that $X$ is quasi-compact. By", "Lemma \\ref{lemma-integral-closure-ground-field}", "after replacing $k_i$ by $k_i'$", "we may assume that $k_i$ is integrally closed in $\\Gamma(X, \\mathcal{O}_X)$.", "This implies that $\\mathcal{O}(X)^*/k_i^*$ is a finitely generated", "abelian group, see", "Proposition \\ref{proposition-units-general}.", "Let $k_{12} = k_1 \\cap k_2$ as a subring of $\\mathcal{O}(X)$.", "Note that $k_{12}$ is a field. Since", "$$", "k_1^*/k_{12}^* \\longrightarrow \\mathcal{O}(X)^*/k_2^*", "$$", "we see that $k_1^*/k_{12}^*$ is a finitely generated abelian group as well.", "Hence there exist $\\alpha_1, \\ldots, \\alpha_n \\in k_1^*$ such that", "every element $\\lambda \\in k_1$ has the form", "$$", "\\lambda = c \\alpha_1^{e_1} \\ldots \\alpha_n^{e_n}", "$$", "for some $e_i \\in \\mathbf{Z}$ and $c \\in k_{12}$.", "In particular, the ring map", "$$", "k_{12}[x_1, \\ldots, x_n, \\frac{1}{x_1 \\ldots x_n}] \\longrightarrow k_1, \\quad", "x_i \\longmapsto \\alpha_i", "$$", "is surjective. By the Hilbert Nullstellensatz,", "Algebra, Theorem \\ref{algebra-theorem-nullstellensatz}", "we conclude that $k_1$ is a finite extension of $k_{12}$.", "In the same way we conclude that $k_2$ is a finite extension of $k_{12}$.", "In particular both $k_1$ and $k_2$ are contained in the integral closure", "$k_{12}'$ of $k_{12}$ in $\\Gamma(X, \\mathcal{O}_X)$. But since $k_{12}'$", "is a field by", "Lemma \\ref{lemma-integral-closure-ground-field}", "and since we chose $k_i$ to be integrally closed in $\\Gamma(X, \\mathcal{O}_X)$", "we conclude that $k_1 = k_{12} = k_2$ as desired." ], "refs": [ "morphisms-lemma-finite-type-noetherian", "properties-lemma-Noetherian-topology", "topology-lemma-locally-connected", "topology-lemma-constructible-Noetherian-space", "varieties-lemma-integral-closure-ground-field", "varieties-lemma-integral-closure-ground-field", "varieties-proposition-units-general", "algebra-theorem-nullstellensatz", "varieties-lemma-integral-closure-ground-field" ], "ref_ids": [ 5202, 2954, 8211, 8267, 11021, 11021, 11136, 316, 11021 ] } ], "ref_ids": [] }, { "id": 11138, "type": "theorem", "label": "varieties-proposition-dim-1-noetherian-separated-has-ample", "categories": [ "varieties" ], "title": "varieties-proposition-dim-1-noetherian-separated-has-ample", "contents": [ "Let $X$ be a Noetherian separated scheme of dimension $1$.", "Then $X$ has an ample invertible sheaf." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset X$ be the reduction of $X$. By", "Lemma \\ref{lemma-dim-1-noetherian-reduced-separated-has-ample}", "the scheme $Z$ has an ample invertible sheaf.", "Thus by Lemma \\ref{lemma-lift-line-bundle-from-reduction-dimension-1}", "there exists an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$", "on $X$ whose restriction to $Z$ is ample.", "Then $\\mathcal{L}$ is ample by an application of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-ample-on-reduction}." ], "refs": [ "varieties-lemma-dim-1-noetherian-reduced-separated-has-ample", "varieties-lemma-lift-line-bundle-from-reduction-dimension-1", "coherent-lemma-ample-on-reduction" ], "ref_ids": [ 11074, 11075, 3350 ] } ], "ref_ids": [] }, { "id": 11139, "type": "theorem", "label": "varieties-proposition-finite-set-of-points-of-codim-1-in-affine", "categories": [ "varieties" ], "title": "varieties-proposition-finite-set-of-points-of-codim-1-in-affine", "contents": [ "Let $X$ be a separated scheme such that every quasi-compact open", "has a finite number of irreducible components. Let $x_1, \\ldots, x_r \\in X$", "be points such that $\\mathcal{O}_{X, x_i}$ is Noetherian", "of dimension $\\leq 1$. Then there exists an affine open subscheme", "of $X$ containing all of $x_1, \\ldots, x_r$." ], "refs": [], "proofs": [ { "contents": [ "We can replace $X$ by a quasi-compact open containing $x_1, \\ldots, x_r$", "hence we may assume that $X$ has finitely many irreducible components.", "By Lemma \\ref{lemma-finite-set-codim-1-points-in-affine-per-component}", "we reduce to the case where $X$ is integral. This case is", "Lemma \\ref{lemma-finite-set-codim-1-points-in-affine}." ], "refs": [ "varieties-lemma-finite-set-codim-1-points-in-affine-per-component", "varieties-lemma-finite-set-codim-1-points-in-affine" ], "ref_ids": [ 11096, 11094 ] } ], "ref_ids": [] }, { "id": 11140, "type": "theorem", "label": "varieties-proposition-asymptotic-riemann-roch", "categories": [ "varieties" ], "title": "varieties-proposition-asymptotic-riemann-roch", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $d$.", "Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.", "Then", "$$", "\\dim_k \\Gamma(X, \\mathcal{L}^{\\otimes n}) \\sim c n^d + l.o.t.", "$$", "where $c = \\deg_\\mathcal{L}(X)/d!$ is a positive constant." ], "refs": [], "proofs": [ { "contents": [ "This follows from the definitions,", "Lemma \\ref{lemma-ample-positive}, and the vanishing", "of higher cohomology in", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-vanshing-gives-ample}." ], "refs": [ "varieties-lemma-ample-positive", "coherent-lemma-vanshing-gives-ample" ], "ref_ids": [ 11128, 3346 ] } ], "ref_ids": [] }, { "id": 11171, "type": "theorem", "label": "cotangent-theorem-quillen-spectral-sequence", "categories": [ "cotangent" ], "title": "cotangent-theorem-quillen-spectral-sequence", "contents": [ "Let $A \\to B$ be a surjective ring map. Consider the sheaf", "$\\Omega = \\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B}$ of", "$\\underline{B}$-modules on $\\mathcal{C}_{B/A}$, see", "Section \\ref{section-compute-L-pi-shriek}.", "Then there is a spectral sequence with $E_1$-page", "$$", "E_1^{p, q} =", "H_{- p - q}(\\mathcal{C}_{B/A}, \\text{Sym}^p_{\\underline{B}}(\\Omega))", "\\Rightarrow \\text{Tor}^A_{- p - q}(B, B)", "$$", "with $d_r$ of bidegree $(r, -r + 1)$.", "Moreover, $H_i(\\mathcal{C}_{B/A}, \\text{Sym}^k_{\\underline{B}}(\\Omega)) = 0$", "for $i < k$." ], "refs": [], "proofs": [ { "contents": [ "Let $I \\subset A$ be the kernel of $A \\to B$. Let", "$\\mathcal{J} \\subset \\mathcal{O}$", "be the kernel of $\\mathcal{O} \\to \\underline{B}$. Then", "$I\\mathcal{O} \\subset \\mathcal{J}$. Set", "$\\mathcal{K} = \\mathcal{J}/I\\mathcal{O}$ and", "$\\overline{\\mathcal{O}} = \\mathcal{O}/I\\mathcal{O}$.", "\\medskip\\noindent", "For every object $U = (P \\to B)$ of $\\mathcal{C}_{B/A}$", "we can choose an isomorphism $P = A[E]$ such that the map", "$P \\to B$ maps each $e \\in E$ to zero. Then", "$J = \\mathcal{J}(U) \\subset P = \\mathcal{O}(U)$", "is equal to $J = IP + (e; e \\in E)$. Moreover", "$\\overline{\\mathcal{O}}(U) = B[E]$ and $K = \\mathcal{K}(U) = (e; e \\in E)$", "is the ideal generated by the variables in the polynomial ring $B[E]$.", "In particular it is clear that", "$$", "K/K^2 \\xrightarrow{\\text{d}} \\Omega_{P/A} \\otimes_P B", "$$", "is a bijection. In other words, $\\Omega = \\mathcal{K}/\\mathcal{K}^2$", "and $\\text{Sym}_B^k(\\Omega) = \\mathcal{K}^k/\\mathcal{K}^{k + 1}$.", "Note that $\\pi_!(\\Omega) = \\Omega_{B/A} = 0$ (Lemma \\ref{lemma-identify-H0})", "as $A \\to B$ is surjective", "(Algebra, Lemma \\ref{algebra-lemma-trivial-differential-surjective}).", "By Lemma \\ref{lemma-vanishing-symmetric-powers} we conclude that", "$$", "H_i(\\mathcal{C}_{B/A}, \\mathcal{K}^k/\\mathcal{K}^{k + 1}) =", "H_i(\\mathcal{C}_{B/A}, \\text{Sym}^k_{\\underline{B}}(\\Omega)) = 0", "$$", "for $i < k$. This proves the final statement of the theorem.", "\\medskip\\noindent", "The approach to the theorem is to note that", "$$", "B \\otimes_A^\\mathbf{L} B = L\\pi_!(\\mathcal{O}) \\otimes_A^\\mathbf{L} B =", "L\\pi_!(\\mathcal{O} \\otimes_{\\underline{A}}^\\mathbf{L} \\underline{B}) =", "L\\pi_!(\\overline{\\mathcal{O}})", "$$", "The first equality by Lemma \\ref{lemma-apply-O-B-comparison},", "the second equality by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-change-of-rings}, and", "the third equality as $\\mathcal{O}$ is flat over $\\underline{A}$.", "The sheaf $\\overline{\\mathcal{O}}$ has a filtration", "$$", "\\ldots \\subset", "\\mathcal{K}^3 \\subset", "\\mathcal{K}^2 \\subset", "\\mathcal{K} \\subset", "\\overline{\\mathcal{O}}", "$$", "This induces a filtration $F$ on a complex $C$ representing", "$L\\pi_!(\\overline{\\mathcal{O}})$ with $F^pC$ representing", "$L\\pi_!(\\mathcal{K}^p)$ (construction of $C$ and $F$ omitted).", "Consider the spectral sequence of", "Homology, Section \\ref{homology-section-filtered-complex}", "associated to $(C, F)$. It has $E_1$-page", "$$", "E_1^{p, q} = H_{- p - q}(\\mathcal{C}_{B/A}, \\mathcal{K}^p/\\mathcal{K}^{p + 1})", "\\quad\\Rightarrow\\quad", "H_{- p - q}(\\mathcal{C}_{B/A}, \\overline{\\mathcal{O}}) = ", "\\text{Tor}_{- p - q}^A(B, B)", "$$", "and differentials $E_r^{p, q} \\to E_r^{p + r, q - r + 1}$. To show convergence", "we will show that for every $k$ there exists a $c$ such that", "$H_i(\\mathcal{C}_{B/A}, \\mathcal{K}^n) = 0$", "for $i < k$ and $n > c$\\footnote{A posteriori", "the ``correct'' vanishing $H_i(\\mathcal{C}_{B/A}, \\mathcal{K}^n) = 0$ for", "$i < n$ can be concluded.}.", "\\medskip\\noindent", "Given $k \\geq 0$ set $c = k^2$. We claim that", "$$", "H_i(\\mathcal{C}_{B/A}, \\mathcal{K}^{n + c}) \\to", "H_i(\\mathcal{C}_{B/A}, \\mathcal{K}^n)", "$$", "is zero for $i < k$ and all $n \\geq 0$. Note that", "$\\mathcal{K}^n/\\mathcal{K}^{n + c}$ has a finite filtration whose successive", "quotients $\\mathcal{K}^m/\\mathcal{K}^{m + 1}$, $n \\leq m < n + c$", "have $H_i(\\mathcal{C}_{B/A}, \\mathcal{K}^m/\\mathcal{K}^{m + 1}) = 0$", "for $i < n$ (see above). Hence the claim implies", "$H_i(\\mathcal{C}_{B/A}, \\mathcal{K}^{n + c}) = 0$ for $i < k$ and all", "$n \\geq k$ which is what we need to show.", "\\medskip\\noindent", "Proof of the claim. Recall that for any $\\mathcal{O}$-module $\\mathcal{F}$", "the map $\\mathcal{F} \\to \\mathcal{F} \\otimes_\\mathcal{O}^\\mathbf{L} B$", "induces an isomorphism on applying $L\\pi_!$, see", "Lemma \\ref{lemma-O-homology-B-homology}.", "Consider the map", "$$", "\\mathcal{K}^{n + k} \\otimes_\\mathcal{O}^\\mathbf{L} B", "\\longrightarrow", "\\mathcal{K}^n \\otimes_\\mathcal{O}^\\mathbf{L} B", "$$", "We claim that this map induces the zero map on cohomology sheaves", "in degrees $0, -1, \\ldots, - k + 1$. If this second claim holds, then", "the $k$-fold composition", "$$", "\\mathcal{K}^{n + c} \\otimes_\\mathcal{O}^\\mathbf{L} B", "\\longrightarrow", "\\mathcal{K}^n \\otimes_\\mathcal{O}^\\mathbf{L} B", "$$", "factors through $\\tau_{\\leq -k}\\mathcal{K}^n \\otimes_\\mathcal{O}^\\mathbf{L} B$", "hence induces zero on $H_i(\\mathcal{C}_{B/A}, -) = L_i\\pi_!( - )$", "for $i < k$, see", "Derived Categories, Lemma \\ref{derived-lemma-trick-vanishing-composition}.", "By the remark above this means the same thing is true for", "$H_i(\\mathcal{C}_{B/A}, \\mathcal{K}^{n + c}) \\to", "H_i(\\mathcal{C}_{B/A}, \\mathcal{K}^n)$", "which proves the (first) claim.", "\\medskip\\noindent", "Proof of the second claim. The statement is local, hence we may work", "over an object $U = (P \\to B)$ as above. We have to show", "the maps", "$$", "\\text{Tor}_i^P(B, K^{n + k}) \\to \\text{Tor}_i^P(B, K^n)", "$$", "are zero for $i < k$. There is a spectral sequence", "$$", "\\text{Tor}_a^P(P/IP, \\text{Tor}_b^{P/IP}(B, K^n))", "\\Rightarrow", "\\text{Tor}_{a + b}^P(B, K^n),", "$$", "see More on Algebra, Example \\ref{more-algebra-example-tor-change-rings}.", "Thus it suffices to prove the maps", "$$", "\\text{Tor}_i^{P/IP}(B, K^{n + 1}) \\to \\text{Tor}_i^{P/IP}(B, K^n)", "$$", "are zero for all $i$. This is Lemma \\ref{lemma-map-tors-zero}." ], "refs": [ "cotangent-lemma-identify-H0", "algebra-lemma-trivial-differential-surjective", "cotangent-lemma-vanishing-symmetric-powers", "cotangent-lemma-apply-O-B-comparison", "sites-cohomology-lemma-change-of-rings", "cotangent-lemma-O-homology-B-homology", "derived-lemma-trick-vanishing-composition", "cotangent-lemma-map-tors-zero" ], "ref_ids": [ 11177, 1131, 11205, 11184, 4347, 11183, 1817, 11206 ] } ], "ref_ids": [] }, { "id": 11172, "type": "theorem", "label": "cotangent-lemma-colimit-cotangent-complex", "categories": [ "cotangent" ], "title": "cotangent-lemma-colimit-cotangent-complex", "contents": [ "Let $A_i \\to B_i$ be a system of ring maps over a directed index", "set $I$. Then $\\colim L_{B_i/A_i} = L_{\\colim B_i/\\colim A_i}$." ], "refs": [], "proofs": [ { "contents": [ "This is true because the forgetful functor", "$V : A\\textit{-Alg} \\to \\textit{Sets}$ and its adjoint", "$U : \\textit{Sets} \\to A\\textit{-Alg}$ commute with filtered colimits.", "Moreover, the functor $B/A \\mapsto \\Omega_{B/A}$ does as well", "(Algebra, Lemma \\ref{algebra-lemma-colimit-differentials})." ], "refs": [ "algebra-lemma-colimit-differentials" ], "ref_ids": [ 1130 ] } ], "ref_ids": [] }, { "id": 11173, "type": "theorem", "label": "cotangent-lemma-identify-pi-shriek", "categories": [ "cotangent" ], "title": "cotangent-lemma-identify-pi-shriek", "contents": [ "With notation as above let $P_\\bullet$ be a simplicial $A$-algebra", "endowed with an augmentation $\\epsilon : P_\\bullet \\to B$.", "Assume each $P_n$ is a polynomial algebra over $A$ and $\\epsilon$", "is a trivial Kan fibration on underlying simplicial sets. Then", "$$", "L\\pi_!(\\mathcal{F}) = \\mathcal{F}(P_\\bullet, \\epsilon)", "$$", "in $D(\\textit{Ab})$, resp.\\ $D(B)$ functorially in $\\mathcal{F}$ in", "$\\textit{Ab}(\\mathcal{C})$, resp.\\ $\\textit{Mod}(\\underline{B})$." ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion of Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution} to prove this.", "Given an object $U = (Q, \\beta)$ of $\\mathcal{C}$ we have to show that", "$$", "S_\\bullet = \\Mor_\\mathcal{C}((Q, \\beta), (P_\\bullet, \\epsilon))", "$$", "is homotopy equivalent to a singleton.", "Write $Q = A[E]$ for some set $E$ (this is possible by our choice of", "the category $\\mathcal{C}$). We see that", "$$", "S_\\bullet = \\Mor_{\\textit{Sets}}((E, \\beta|_E), (P_\\bullet, \\epsilon))", "$$", "Let $*$ be the constant simplicial set on a singleton. For $b \\in B$", "let $F_{b, \\bullet}$ be the simplicial set defined by the cartesian", "diagram", "$$", "\\xymatrix{", "F_{b, \\bullet} \\ar[r] \\ar[d] & P_\\bullet \\ar[d]_\\epsilon \\\\", "{*} \\ar[r]^b & B", "}", "$$", "With this notation $S_\\bullet = \\prod_{e \\in E} F_{\\beta(e), \\bullet}$.", "Since we assumed $\\epsilon$ is a trivial Kan fibration we see that", "$F_{b, \\bullet} \\to *$ is a trivial Kan fibration", "(Simplicial, Lemma \\ref{simplicial-lemma-trivial-kan-base-change}).", "Thus $S_\\bullet \\to *$ is a trivial Kan fibration", "(Simplicial, Lemma \\ref{simplicial-lemma-product-trivial-kan}).", "Therefore $S_\\bullet$ is homotopy equivalent to $*$", "(Simplicial, Lemma \\ref{simplicial-lemma-trivial-kan-homotopy})." ], "refs": [ "sites-cohomology-lemma-compute-by-cosimplicial-resolution", "simplicial-lemma-trivial-kan-base-change", "simplicial-lemma-product-trivial-kan", "simplicial-lemma-trivial-kan-homotopy" ], "ref_ids": [ 4348, 14887, 14890, 14892 ] } ], "ref_ids": [] }, { "id": 11174, "type": "theorem", "label": "cotangent-lemma-pi-shriek-standard", "categories": [ "cotangent" ], "title": "cotangent-lemma-pi-shriek-standard", "contents": [ "Let $A \\to B$ be a ring map. Let $\\epsilon : P_\\bullet \\to B$", "be the standard resolution of $B$ over $A$. Let $\\pi$ be as in", "(\\ref{equation-pi}). Then", "$$", "L\\pi_!(\\mathcal{F}) = \\mathcal{F}(P_\\bullet, \\epsilon)", "$$", "in $D(\\textit{Ab})$, resp.\\ $D(B)$ functorially in $\\mathcal{F}$ in", "$\\textit{Ab}(\\mathcal{C})$, resp.\\ $\\textit{Mod}(\\underline{B})$." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "We will apply Lemma \\ref{lemma-identify-pi-shriek}.", "Since the terms $P_n$ are polynomial algebras we see the first", "assumption of that lemma is satisfied. The second assumption is proved", "as follows. By", "Simplicial, Lemma \\ref{simplicial-lemma-standard-simplicial-homotopy}", "the map $\\epsilon$ is a homotopy equivalence of underlying", "simplicial sets. By", "Simplicial, Lemma \\ref{simplicial-lemma-homotopy-equivalence}", "this implies $\\epsilon$ induces a quasi-isomorphism of associated", "complexes of abelian groups. By", "Simplicial, Lemma \\ref{simplicial-lemma-qis-simplicial-abelian-groups}", "this implies that $\\epsilon$ is a trivial Kan fibration of underlying", "simplicial sets." ], "refs": [ "cotangent-lemma-identify-pi-shriek", "simplicial-lemma-standard-simplicial-homotopy", "simplicial-lemma-homotopy-equivalence", "simplicial-lemma-qis-simplicial-abelian-groups" ], "ref_ids": [ 11173, 14909, 14900, 14899 ] } ], "ref_ids": [] }, { "id": 11175, "type": "theorem", "label": "cotangent-lemma-compute-cotangent-complex", "categories": [ "cotangent" ], "title": "cotangent-lemma-compute-cotangent-complex", "contents": [ "Let $A \\to B$ be a ring map. Let $\\pi$ and $i$ be as in (\\ref{equation-pi}).", "There is a canonical isomorphism", "$$", "L_{B/A} = L\\pi_!(Li^*\\Omega_{\\mathcal{O}/A}) =", "L\\pi_!(i^*\\Omega_{\\mathcal{O}/A}) =", "L\\pi_!(\\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B})", "$$", "in $D(B)$." ], "refs": [], "proofs": [ { "contents": [ "For an object $\\alpha : P \\to B$ of the category $\\mathcal{C}$", "the module $\\Omega_{P/A}$ is a free $P$-module. Thus", "$\\Omega_{\\mathcal{O}/A}$ is a flat $\\mathcal{O}$-module. Hence", "$Li^*\\Omega_{\\mathcal{O}/A} = i^*\\Omega_{\\mathcal{O}/A}$ is the sheaf", "of $\\underline{B}$-modules which associates to $\\alpha : P \\to A$ the", "$B$-module $\\Omega_{P/A} \\otimes_{P, \\alpha} B$.", "By Lemma \\ref{lemma-pi-shriek-standard}", "we see that the right hand side is computed by", "the value of this sheaf on the standard resolution which is our", "definition of the left hand side", "(Definition \\ref{definition-cotangent-complex-ring-map})." ], "refs": [ "cotangent-lemma-pi-shriek-standard", "cotangent-definition-cotangent-complex-ring-map" ], "ref_ids": [ 11174, 11245 ] } ], "ref_ids": [] }, { "id": 11176, "type": "theorem", "label": "cotangent-lemma-pi-lower-shriek-constant-sheaf", "categories": [ "cotangent" ], "title": "cotangent-lemma-pi-lower-shriek-constant-sheaf", "contents": [ "If $A \\to B$ is a ring map, then $L\\pi_!(\\pi^{-1}M) = M$", "with $\\pi$ as in (\\ref{equation-pi})." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-identify-pi-shriek} which tells us", "$L\\pi_!(\\pi^{-1}M)$ is computed by $(\\pi^{-1}M)(P_\\bullet, \\epsilon)$", "which is the constant simplicial object on $M$." ], "refs": [ "cotangent-lemma-identify-pi-shriek" ], "ref_ids": [ 11173 ] } ], "ref_ids": [] }, { "id": 11177, "type": "theorem", "label": "cotangent-lemma-identify-H0", "categories": [ "cotangent" ], "title": "cotangent-lemma-identify-H0", "contents": [ "If $A \\to B$ is a ring map, then $H^0(L_{B/A}) = \\Omega_{B/A}$." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by a direct calculation.", "We will use the identification of Lemma \\ref{lemma-compute-cotangent-complex}.", "There is clearly a map from $\\Omega_{\\mathcal{O}/A} \\otimes \\underline{B}$", "to the constant sheaf with value $\\Omega_{B/A}$. Thus this map induces", "a map", "$$", "H^0(L_{B/A}) = H^0(L\\pi_!(\\Omega_{\\mathcal{O}/A} \\otimes \\underline{B}))", "= \\pi_!(\\Omega_{\\mathcal{O}/A} \\otimes \\underline{B}) \\to \\Omega_{B/A}", "$$", "By choosing an object $P \\to B$ of $\\mathcal{C}_{B/A}$ with $P \\to B$", "surjective we see that this map is surjective (by", "Algebra, Lemma \\ref{algebra-lemma-differential-surjective}).", "To show that it is injective, suppose that $P \\to B$ is an object", "of $\\mathcal{C}_{B/A}$ and that $\\xi \\in \\Omega_{P/A} \\otimes_P B$", "is an element which maps to zero in $\\Omega_{B/A}$.", "We first choose factorization $P \\to P' \\to B$ such that $P' \\to B$", "is surjective and $P'$ is a polynomial algebra over $A$.", "We may replace $P$ by $P'$. If $B = P/I$, then the kernel", "$\\Omega_{P/A} \\otimes_P B \\to \\Omega_{B/A}$ is the image of", "$I/I^2$ (Algebra, Lemma \\ref{algebra-lemma-differential-seq}).", "Say $\\xi$ is the image of $f \\in I$.", "Then we consider the two maps $a, b : P' = P[x] \\to P$, the first of which", "maps $x$ to $0$ and the second of which maps $x$ to $f$ (in both", "cases $P[x] \\to B$ maps $x$ to zero). We see that $\\xi$ and $0$", "are the image of $\\text{d}x \\otimes 1$ in $\\Omega_{P'/A} \\otimes_{P'} B$.", "Thus $\\xi$ and $0$ have the same image in the colimit (see", "Cohomology on Sites, Example \\ref{sites-cohomology-example-category-to-point})", "$\\pi_!(\\Omega_{\\mathcal{O}/A} \\otimes \\underline{B})$ as desired." ], "refs": [ "cotangent-lemma-compute-cotangent-complex", "algebra-lemma-differential-surjective", "algebra-lemma-differential-seq" ], "ref_ids": [ 11175, 1132, 1135 ] } ], "ref_ids": [] }, { "id": 11178, "type": "theorem", "label": "cotangent-lemma-pi-lower-shriek-polynomial-algebra", "categories": [ "cotangent" ], "title": "cotangent-lemma-pi-lower-shriek-polynomial-algebra", "contents": [ "If $B$ is a polynomial algebra over the ring $A$, then", "with $\\pi$ as in (\\ref{equation-pi}) we have that", "$\\pi_!$ is exact and $\\pi_!\\mathcal{F} = \\mathcal{F}(B \\to B)$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-identify-pi-shriek} which tells us", "the constant simplicial algebra on $B$ can be used to compute $L\\pi_!$." ], "refs": [ "cotangent-lemma-identify-pi-shriek" ], "ref_ids": [ 11173 ] } ], "ref_ids": [] }, { "id": 11179, "type": "theorem", "label": "cotangent-lemma-cotangent-complex-polynomial-algebra", "categories": [ "cotangent" ], "title": "cotangent-lemma-cotangent-complex-polynomial-algebra", "contents": [ "If $B$ is a polynomial algebra over the ring $A$, then", "$L_{B/A}$ is quasi-isomorphic to $\\Omega_{B/A}[0]$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from", "Lemmas \\ref{lemma-compute-cotangent-complex} and", "\\ref{lemma-pi-lower-shriek-polynomial-algebra}." ], "refs": [ "cotangent-lemma-compute-cotangent-complex", "cotangent-lemma-pi-lower-shriek-polynomial-algebra" ], "ref_ids": [ 11175, 11178 ] } ], "ref_ids": [] }, { "id": 11180, "type": "theorem", "label": "cotangent-lemma-polynomial", "categories": [ "cotangent" ], "title": "cotangent-lemma-polynomial", "contents": [ "Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map.", "Let $\\mathcal{A}$ be the category of $A$-algebra maps $C \\to B$. Let", "$n \\geq 0$ and let $P_\\bullet$ be a simplicial object of $\\mathcal{A}$", "such that", "\\begin{enumerate}", "\\item $P_\\bullet \\to B$ is a trivial Kan fibration of simplicial sets,", "\\item $P_k$ is finite type over $A$ for $k \\leq n$,", "\\item $P_\\bullet = \\text{cosk}_n \\text{sk}_n P_\\bullet$ as simplicial", "objects of $\\mathcal{A}$.", "\\end{enumerate}", "Then $P_{n + 1}$ is a finite type $A$-algebra." ], "refs": [], "proofs": [ { "contents": [ "Although the proof we give of this lemma is straightforward, it is a bit", "messy. To clarify the idea we explain what happens for low $n$ before giving", "the proof in general. For example, if $n = 0$, then (3) means that", "$P_1 = P_0 \\times_B P_0$. Since the ring map $P_0 \\to B$ is surjective, this", "is of finite type over $A$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-fibre-product-finite-type}.", "\\medskip\\noindent", "If $n = 1$, then (3) means that", "$$", "P_2 = \\{(f_0, f_1, f_2) \\in P_1^3 \\mid", "d_0f_0 = d_0f_1,\\ d_1f_0 = d_0f_2,\\ d_1f_1 = d_1f_2 \\}", "$$", "where the equalities take place in $P_0$. Observe that the triple", "$$", "(d_0f_0, d_1f_0, d_1f_1) = (d_0f_1, d_0f_2, d_1f_2)", "$$", "is an element of the fibre product $P_0 \\times_B P_0 \\times_B P_0$ over $B$", "because the maps $d_i : P_1 \\to P_0$ are morphisms over $B$. Thus we get", "a map", "$$", "\\psi : P_2 \\longrightarrow P_0 \\times_B P_0 \\times_B P_0", "$$", "The fibre of $\\psi$ over an element", "$(g_0, g_1, g_2) \\in P_0 \\times_B P_0 \\times_B P_0$", "is the set of triples $(f_0, f_1, f_2)$ of $1$-simplices", "with $(d_0, d_1)(f_0) = (g_0, g_1)$, $(d_0, d_1)(f_1) = (g_0, g_2)$,", "and $(d_0, d_1)(f_2) = (g_1, g_2)$. As $P_\\bullet \\to B$ is a trivial", "Kan fibration the map $(d_0, d_1) : P_1 \\to P_0 \\times_B P_0$ is", "surjective. Thus we see that $P_2$ fits into the cartesian diagram", "$$", "\\xymatrix{", "P_2 \\ar[d] \\ar[r] & P_1^3 \\ar[d] \\\\", "P_0 \\times_B P_0 \\times_B P_0 \\ar[r] & (P_0 \\times_B P_0)^3", "}", "$$", "By More on Algebra, Lemma \\ref{more-algebra-lemma-formal-consequence}", "we conclude. The general case is similar, but requires a bit more notation.", "\\medskip\\noindent", "The case $n > 1$. By Simplicial, Lemma \\ref{simplicial-lemma-cosk-above-object}", "the condition $P_\\bullet = \\text{cosk}_n \\text{sk}_n P_\\bullet$", "implies the same thing is true in the category of simplicial", "$A$-algebras and hence in the category of sets (as the forgetful", "functor from $A$-algebras to sets commutes with limits). Thus", "$$", "P_{n + 1} =", "\\Mor(\\Delta[n + 1], P_\\bullet) =", "\\Mor(\\text{sk}_n \\Delta[n + 1], \\text{sk}_n P_\\bullet)", "$$", "by Simplicial, Lemma \\ref{simplicial-lemma-simplex-map} and", "Equation (\\ref{simplicial-equation-cosk}). We will prove by induction", "on $1 \\leq k < m \\leq n + 1$ that the ring", "$$", "Q_{k, m} = \\Mor(\\text{sk}_k \\Delta[m], \\text{sk}_k P_\\bullet)", "$$", "is of finite type over $A$. The case $k = 1$, $1 < m \\leq n + 1$", "is entirely similar to the discussion above in the case $n = 1$.", "Namely, there is a cartesian diagram", "$$", "\\xymatrix{", "Q_{1, m} \\ar[d] \\ar[r] & P_1^N \\ar[d] \\\\", "P_0 \\times_B \\ldots \\times_B P_0 \\ar[r] & (P_0 \\times_B P_0)^N", "}", "$$", "where $N = {m + 1 \\choose 2}$. We conclude as before.", "\\medskip\\noindent", "Let $1 \\leq k_0 \\leq n$ and assume $Q_{k, m}$ is of finite type", "over $A$ for all $1 \\leq k \\leq k_0$ and $k < m \\leq n + 1$.", "For $k_0 + 1 < m \\leq n + 1$ we claim there is a cartesian square", "$$", "\\xymatrix{", "Q_{k_0 + 1, m} \\ar[d] \\ar[r] & P_{k_0 + 1}^N \\ar[d] \\\\", "Q_{k_0, m} \\ar[r] & Q_{k_0, k_0 + 1}^N", "}", "$$", "where $N$ is the number of nondegenerate $(k_0 + 1)$-simplices", "of $\\Delta[m]$. Namely, to see this is true, think of an element of", "$Q_{k_0 + 1, m}$ as a function $f$ from the $(k_0 + 1)$-skeleton", "of $\\Delta[m]$ to $P_\\bullet$. We can restrict $f$ to the $k_0$-skeleton", "which gives the left vertical map of the diagram. We can also restrict", "to each nondegenerate $(k_0 + 1)$-simplex which gives the top horizontal", "arrow. Moreover, to give such an $f$ is the same thing as giving its", "restriction to $k_0$-skeleton and to each nondegenerate", "$(k_0 + 1)$-face, provided these agree on the overlap, and this", "is exactly the content of the diagram. Moreover, the fact that", "$P_\\bullet \\to B$ is a trivial Kan fibration implies that", "the map", "$$", "P_{k_0} \\to Q_{k_0, k_0 + 1} = \\Mor(\\partial \\Delta[k_0 + 1], P_\\bullet)", "$$", "is surjective as every map $\\partial \\Delta[k_0 + 1] \\to B$ can be extended", "to $\\Delta[k_0 + 1] \\to B$ for $k_0 \\geq 1$ (small argument about constant", "simplicial sets omitted). Since by induction hypothesis the rings", "$Q_{k_0, m}$, $Q_{k_0, k_0 + 1}$ are finite type $A$-algebras, so is", "$Q_{k_0 + 1, m}$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-formal-consequence}", "once more." ], "refs": [ "more-algebra-lemma-fibre-product-finite-type", "more-algebra-lemma-formal-consequence", "simplicial-lemma-cosk-above-object", "simplicial-lemma-simplex-map", "more-algebra-lemma-formal-consequence" ], "ref_ids": [ 9814, 9815, 14843, 14817, 9815 ] } ], "ref_ids": [] }, { "id": 11181, "type": "theorem", "label": "cotangent-lemma-pi-shriek-finite", "categories": [ "cotangent" ], "title": "cotangent-lemma-pi-shriek-finite", "contents": [ "Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map.", "Let $\\pi$, $\\underline{B}$ be as in (\\ref{equation-pi}).", "If $\\mathcal{F}$ is an $\\underline{B}$-module such that", "$\\mathcal{F}(P, \\alpha)$ is a finite $B$-module for all", "$\\alpha : P = A[x_1, \\ldots, x_n] \\to B$, then the cohomology modules", "of $L\\pi_!(\\mathcal{F})$ are finite $B$-modules." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-identify-pi-shriek} and", "Proposition \\ref{proposition-polynomial}", "we can compute $L\\pi_!(\\mathcal{F})$ by a complex", "constructed out of the values of $\\mathcal{F}$ on finite type", "polynomial algebras." ], "refs": [ "cotangent-lemma-identify-pi-shriek", "cotangent-proposition-polynomial" ], "ref_ids": [ 11173, 11241 ] } ], "ref_ids": [] }, { "id": 11182, "type": "theorem", "label": "cotangent-lemma-cotangent-finite", "categories": [ "cotangent" ], "title": "cotangent-lemma-cotangent-finite", "contents": [ "Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map.", "Then $H^n(L_{B/A})$ is a finite $B$-module for all $n \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemmas \\ref{lemma-compute-cotangent-complex} and", "\\ref{lemma-pi-shriek-finite}." ], "refs": [ "cotangent-lemma-compute-cotangent-complex", "cotangent-lemma-pi-shriek-finite" ], "ref_ids": [ 11175, 11181 ] } ], "ref_ids": [] }, { "id": 11183, "type": "theorem", "label": "cotangent-lemma-O-homology-B-homology", "categories": [ "cotangent" ], "title": "cotangent-lemma-O-homology-B-homology", "contents": [ "Let $A \\to B$ be a ring map. Let $\\pi$, $\\mathcal{O}$, $\\underline{B}$", "be as in (\\ref{equation-pi}). For any $\\mathcal{O}$-module $\\mathcal{F}$", "we have", "$$", "L\\pi_!(\\mathcal{F}) = L\\pi_!(Li^*\\mathcal{F}) =", "L\\pi_!(\\mathcal{F} \\otimes_\\mathcal{O}^\\mathbf{L} \\underline{B})", "$$", "in $D(\\textit{Ab})$." ], "refs": [], "proofs": [ { "contents": [ "It suffices to verify the assumptions of Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-O-homology-qis}", "hold for $\\mathcal{O} \\to \\underline{B}$ on $\\mathcal{C}_{B/A}$.", "We will use the results of Remark \\ref{remark-resolution} without", "further mention. Choose a resolution $P_\\bullet$ of $B$ over $A$ to get a", "suitable cosimplicial object $U_\\bullet$ of $\\mathcal{C}_{B/A}$.", "Since $P_\\bullet \\to B$ induces a quasi-isomorphism on associated", "complexes of abelian groups we see that $L\\pi_!\\mathcal{O} = B$.", "On the other hand $L\\pi_!\\underline{B}$ is computed by", "$\\underline{B}(U_\\bullet) = B$. This verifies the second assumption of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-O-homology-qis}", "and we are done with the proof." ], "refs": [ "sites-cohomology-lemma-O-homology-qis", "cotangent-remark-resolution", "sites-cohomology-lemma-O-homology-qis" ], "ref_ids": [ 4352, 11256, 4352 ] } ], "ref_ids": [] }, { "id": 11184, "type": "theorem", "label": "cotangent-lemma-apply-O-B-comparison", "categories": [ "cotangent" ], "title": "cotangent-lemma-apply-O-B-comparison", "contents": [ "Let $A \\to B$ be a ring map. Let $\\pi$, $\\mathcal{O}$, $\\underline{B}$", "be as in (\\ref{equation-pi}). We have", "$$", "L\\pi_!(\\mathcal{O}) = L\\pi_!(\\underline{B}) = B", "\\quad\\text{and}\\quad", "L_{B/A} = L\\pi_!(\\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B}) =", "L\\pi_!(\\Omega_{\\mathcal{O}/A})", "$$", "in $D(\\textit{Ab})$." ], "refs": [], "proofs": [ { "contents": [ "This is just an application of Lemma \\ref{lemma-O-homology-B-homology}", "(and the first equality on the right is", "Lemma \\ref{lemma-compute-cotangent-complex})." ], "refs": [ "cotangent-lemma-O-homology-B-homology", "cotangent-lemma-compute-cotangent-complex" ], "ref_ids": [ 11183, 11175 ] } ], "ref_ids": [] }, { "id": 11185, "type": "theorem", "label": "cotangent-lemma-special-case-triangle", "categories": [ "cotangent" ], "title": "cotangent-lemma-special-case-triangle", "contents": [ "Let $A \\to B \\to C$ be ring maps. If $B$ is a polynomial algebra over", "$A$, then there is a distinguished triangle ", "$L_{B/A} \\otimes_B^\\mathbf{L} C \\to L_{C/A} \\to L_{C/B} \\to", "L_{B/A} \\otimes_B^\\mathbf{L} C[1]$ in $D(C)$." ], "refs": [], "proofs": [ { "contents": [ "We will use the observations of Remark \\ref{remark-resolution}", "without further mention. Choose a resolution $\\epsilon : P_\\bullet \\to C$", "of $C$ over $B$ (for example the standard resolution). Since $B$ is a", "polynomial algebra over $A$ we see that $P_\\bullet$ is also a resolution of", "$C$ over $A$. Hence $L_{C/A}$ is computed by", "$\\Omega_{P_\\bullet/A} \\otimes_{P_\\bullet, \\epsilon} C$", "and $L_{C/B}$ is computed by", "$\\Omega_{P_\\bullet/B} \\otimes_{P_\\bullet, \\epsilon} C$.", "Since for each $n$ we have the short exact sequence", "$0 \\to \\Omega_{B/A} \\otimes_B P_n \\to \\Omega_{P_n/A} \\to \\Omega_{P_n/B}$", "(Algebra, Lemma \\ref{algebra-lemma-ses-formally-smooth})", "and since $L_{B/A} = \\Omega_{B/A}[0]$", "(Lemma \\ref{lemma-cotangent-complex-polynomial-algebra})", "we obtain the result." ], "refs": [ "cotangent-remark-resolution", "algebra-lemma-ses-formally-smooth", "cotangent-lemma-cotangent-complex-polynomial-algebra" ], "ref_ids": [ 11256, 1209, 11179 ] } ], "ref_ids": [] }, { "id": 11186, "type": "theorem", "label": "cotangent-lemma-flat-base-change", "categories": [ "cotangent" ], "title": "cotangent-lemma-flat-base-change", "contents": [ "Assume (\\ref{equation-commutative-square}) induces a quasi-isomorphism", "$B \\otimes_A^\\mathbf{L} A' = B'$. Then, with notation as in", "(\\ref{equation-double-square}) and", "$\\mathcal{F}' \\in \\textit{Ab}(\\mathcal{C}')$,", "we have $L\\pi_!(g^{-1}\\mathcal{F}') = L\\pi'_!(\\mathcal{F}')$." ], "refs": [], "proofs": [ { "contents": [ "We use the results of Remark \\ref{remark-resolution} without", "further mention. We will apply Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-get-it-now}. Let $P_\\bullet \\to B$ be a resolution.", "If we can show that $u(P_\\bullet) = P_\\bullet \\otimes_A A' \\to B'$", "is a quasi-isomorphism, then we are done. The complex of $A$-modules", "$s(P_\\bullet)$ associated to $P_\\bullet$ (viewed as a simplicial $A$-module)", "is a free $A$-module resolution of $B$. Namely, $P_n$ is a free $A$-module and", "$s(P_\\bullet) \\to B$ is a quasi-isomorphism. Thus $B \\otimes_A^\\mathbf{L} A'$", "is computed by $s(P_\\bullet) \\otimes_A A' = s(P_\\bullet \\otimes_A A')$.", "Therefore the assumption of the lemma signifies that", "$\\epsilon' : P_\\bullet \\otimes_A A' \\to B'$ is a quasi-isomorphism." ], "refs": [ "cotangent-remark-resolution", "sites-cohomology-lemma-get-it-now" ], "ref_ids": [ 11256, 4349 ] } ], "ref_ids": [] }, { "id": 11187, "type": "theorem", "label": "cotangent-lemma-flat-base-change-cotangent-complex", "categories": [ "cotangent" ], "title": "cotangent-lemma-flat-base-change-cotangent-complex", "contents": [ "If (\\ref{equation-commutative-square}) induces a quasi-isomorphism", "$B \\otimes_A^\\mathbf{L} A' = B'$, then the functoriality map", "induces an isomorphism", "$$", "L_{B/A} \\otimes_B^\\mathbf{L} B' \\longrightarrow L_{B'/A'}", "$$" ], "refs": [], "proofs": [ { "contents": [ "We will use the notation introduced in Equation (\\ref{equation-double-square}).", "We have", "$$", "L_{B/A} \\otimes_B^\\mathbf{L} B' =", "L\\pi_!(\\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B})", "\\otimes_B^\\mathbf{L} B' =", "L\\pi_!(Lh^*(\\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B}))", "$$", "the first equality by Lemma \\ref{lemma-compute-cotangent-complex}", "and the second by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-change-of-rings}.", "Since $\\Omega_{\\mathcal{O}/A}$ is a flat $\\mathcal{O}$-module,", "we see that $\\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B}$", "is a flat $\\underline{B}$-module. Thus", "$Lh^*(\\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B}) =", "\\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B'}$", "which is equal to", "$g^{-1}(\\Omega_{\\mathcal{O}'/A'} \\otimes_{\\mathcal{O}'} \\underline{B'})$", "by inspection.", "we conclude by Lemma \\ref{lemma-flat-base-change}", "and the fact that $L_{B'/A'}$ is computed by", "$L\\pi'_!(\\Omega_{\\mathcal{O}'/A'} \\otimes_{\\mathcal{O}'} \\underline{B'})$." ], "refs": [ "cotangent-lemma-compute-cotangent-complex", "sites-cohomology-lemma-change-of-rings", "cotangent-lemma-flat-base-change" ], "ref_ids": [ 11175, 4347, 11186 ] } ], "ref_ids": [] }, { "id": 11188, "type": "theorem", "label": "cotangent-lemma-cotangent-complex-product", "categories": [ "cotangent" ], "title": "cotangent-lemma-cotangent-complex-product", "contents": [ "Let $A \\to B$ and $A \\to C$ be ring maps.", "Then the map $L_{B \\times C/A} \\to L_{B/A} \\oplus L_{C/A}$ is", "an isomorphism in $D(B \\times C)$." ], "refs": [], "proofs": [ { "contents": [ "Although this lemma can be deduced from the fundamental triangle", "we will give a direct and elementary proof of this now.", "Factor the ring map $A \\to B \\times C$ as $A \\to A[x] \\to B \\times C$", "where $x \\mapsto (1, 0)$. By Lemma \\ref{lemma-special-case-triangle}", "we have a distinguished triangle", "$$", "L_{A[x]/A} \\otimes_{A[x]}^\\mathbf{L} (B \\times C) \\to L_{B \\times C/A} \\to", "L_{B \\times C/A[x]} \\to L_{A[x]/A} \\otimes_{A[x]}^\\mathbf{L} (B \\times C)[1]", "$$", "in $D(B \\times C)$. Similarly we have the distinguished triangles", "$$", "\\begin{matrix}", "L_{A[x]/A} \\otimes_{A[x]}^\\mathbf{L} B \\to L_{B/A} \\to L_{B/A[x]}", "\\to L_{A[x]/A} \\otimes_{A[x]}^\\mathbf{L} B[1] \\\\", "L_{A[x]/A} \\otimes_{A[x]}^\\mathbf{L} C \\to L_{C/A} \\to L_{C/A[x]}", "\\to L_{A[x]/A} \\otimes_{A[x]}^\\mathbf{L} C[1]", "\\end{matrix}", "$$", "Thus it suffices to prove the result for $B \\times C$ over $A[x]$.", "Note that $A[x] \\to A[x, x^{-1}]$ is flat, that", "$(B \\times C) \\otimes_{A[x]} A[x, x^{-1}] = B \\otimes_{A[x]} A[x, x^{-1}]$,", "and that $C \\otimes_{A[x]} A[x, x^{-1}] = 0$.", "Thus by base change (Lemma \\ref{lemma-flat-base-change-cotangent-complex})", "the map $L_{B \\times C/A[x]} \\to L_{B/A[x]} \\oplus L_{C/A[x]}$", "becomes an isomorphism after inverting $x$.", "In the same way one shows that the map becomes an isomorphism after", "inverting $x - 1$. This proves the lemma." ], "refs": [ "cotangent-lemma-special-case-triangle", "cotangent-lemma-flat-base-change-cotangent-complex" ], "ref_ids": [ 11185, 11187 ] } ], "ref_ids": [] }, { "id": 11189, "type": "theorem", "label": "cotangent-lemma-triangle-ses", "categories": [ "cotangent" ], "title": "cotangent-lemma-triangle-ses", "contents": [ "With notation as in (\\ref{equation-three-maps}) set", "$$", "\\begin{matrix}", "\\Omega_1 = \\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B}", "\\text{ on }\\mathcal{C}_{B/A} \\\\", "\\Omega_2 = \\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{C}", "\\text{ on }\\mathcal{C}_{C/A} \\\\", "\\Omega_3 = \\Omega_{\\mathcal{O}/B} \\otimes_\\mathcal{O} \\underline{C}", "\\text{ on }\\mathcal{C}_{C/B}", "\\end{matrix}", "$$", "Then we have a canonical short exact sequence of sheaves", "of $\\underline{C}$-modules", "$$", "0 \\to g_1^{-1}\\Omega_1 \\otimes_{\\underline{B}} \\underline{C} \\to", "g_2^{-1}\\Omega_2 \\to", "g_3^{-1}\\Omega_3 \\to 0", "$$", "on $\\mathcal{C}_{C/B/A}$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $g_i^{-1}$ is gotten by simply precomposing with $u_i$.", "Given an object $U = (P \\to B, Q \\to C)$ we have a split", "short exact sequence", "$$", "0 \\to \\Omega_{P/A} \\otimes Q \\to \\Omega_{Q/A} \\to \\Omega_{Q/P} \\to 0", "$$", "for example by Algebra, Lemma \\ref{algebra-lemma-ses-formally-smooth}.", "Tensoring with $C$ over $Q$ we obtain a short exact sequence", "$$", "0 \\to \\Omega_{P/A} \\otimes C \\to \\Omega_{Q/A} \\otimes C \\to", "\\Omega_{Q/P} \\otimes C \\to 0", "$$", "We have $\\Omega_{P/A} \\otimes C = \\Omega_{P/A} \\otimes B \\otimes C$", "whence this is the value of", "$g_1^{-1}\\Omega_1 \\otimes_{\\underline{B}} \\underline{C}$", "on $U$. The module $\\Omega_{Q/A} \\otimes C$ is the value of", "$g_2^{-1}\\Omega_2$ on $U$.", "We have $\\Omega_{Q/P} \\otimes C = \\Omega_{Q \\otimes_P B/B} \\otimes C$", "by Algebra, Lemma \\ref{algebra-lemma-differentials-base-change}", "hence this is the value of", "$g_3^{-1}\\Omega_3$ on $U$. Thus the short exact sequence of the", "lemma comes from assigning to $U$ the last displayed short exact", "sequence." ], "refs": [ "algebra-lemma-ses-formally-smooth", "algebra-lemma-differentials-base-change" ], "ref_ids": [ 1209, 1138 ] } ], "ref_ids": [] }, { "id": 11190, "type": "theorem", "label": "cotangent-lemma-polynomial-on-top", "categories": [ "cotangent" ], "title": "cotangent-lemma-polynomial-on-top", "contents": [ "With notation as in (\\ref{equation-three-maps})", "suppose that $C$ is a polynomial algebra over $B$. Then", "$L\\pi_!(g_3^{-1}\\mathcal{F}) = L\\pi_{3, !}\\mathcal{F} = \\pi_{3, !}\\mathcal{F}$", "for any abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}_{C/B}$" ], "refs": [], "proofs": [ { "contents": [ "Write $C = B[E]$ for some set $E$. Choose a resolution", "$P_\\bullet \\to B$ of $B$ over $A$. For every $n$ consider", "the object $U_n = (P_n \\to B, P_n[E] \\to C)$ of $\\mathcal{C}_{C/B/A}$.", "Then $U_\\bullet$ is a cosimplicial object of $\\mathcal{C}_{C/B/A}$.", "Note that $u_3(U_\\bullet)$ is the constant cosimplicial", "object of $\\mathcal{C}_{C/B}$ with value $(C \\to C)$.", "We will prove that the object $U_\\bullet$ of $\\mathcal{C}_{C/B/A}$", "satisfies the hypotheses of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}.", "This implies the lemma as it shows that $L\\pi_!(g_3^{-1}\\mathcal{F})$", "is computed by the constant simplicial abelian group", "$\\mathcal{F}(C \\to C)$ which is the value of", "$L\\pi_{3, !}\\mathcal{F} = \\pi_{3, !}\\mathcal{F}$ by", "Lemma \\ref{lemma-pi-lower-shriek-polynomial-algebra}.", "\\medskip\\noindent", "Let $U = (\\beta : P \\to B, \\gamma : Q \\to C)$ be an object of", "$\\mathcal{C}_{C/B/A}$. We may write $P = A[S]$ and $Q = A[S \\amalg T]$", "by the definition of our category $\\mathcal{C}_{C/B/A}$. We have to show that", "$$", "\\Mor_{\\mathcal{C}_{C/B/A}}(U_\\bullet, U)", "$$", "is homotopy equivalent to a singleton simplicial set $*$. Observe that this", "simplicial set is the product", "$$", "\\prod\\nolimits_{s \\in S} F_s \\times \\prod\\nolimits_{t \\in T} F'_t", "$$", "where $F_s$ is the corresponding simplicial set for", "$U_s = (A[\\{s\\}] \\to B, A[\\{s\\}] \\to C)$", "and $F'_t$ is the corresponding simplicial set for", "$U_t = (A \\to B, A[\\{t\\}] \\to C)$. Namely, the object $U$", "is the product $\\prod U_s \\times \\prod U_t$ in $\\mathcal{C}_{C/B/A}$.", "It suffices each $F_s$ and $F'_t$ is homotopy equivalent to $*$, see", "Simplicial, Lemma \\ref{simplicial-lemma-products-homotopy}.", "The case of $F_s$ follows as $P_\\bullet \\to B$ is a trivial Kan", "fibration (as a resolution) and $F_s$ is the fibre of this map over", "$\\beta(s)$. (Use Simplicial, Lemmas", "\\ref{simplicial-lemma-trivial-kan-base-change} and", "\\ref{simplicial-lemma-trivial-kan-homotopy}).", "The case of $F'_t$ is more interesting. Here we are saying that", "the fibre of", "$$", "P_\\bullet[E] \\longrightarrow C = B[E]", "$$", "over $\\gamma(t) \\in C$ is homotopy equivalent to a point. In fact we", "will show this map is a trivial Kan fibration. Namely,", "$P_\\bullet \\to B$ is a trivial can fibration. For any ring $R$", "we have", "$$", "R[E] =", "\\colim_{\\Sigma \\subset \\text{Map}(E, \\mathbf{Z}_{\\geq 0})\\text{ finite}}", "\\prod\\nolimits_{I \\in \\Sigma} R", "$$", "(filtered colimit). Thus the displayed map of simplicial sets is a", "filtered colimit of trivial Kan fibrations, whence a trivial Kan fibration", "by Simplicial, Lemma \\ref{simplicial-lemma-filtered-colimit-trivial-kan}." ], "refs": [ "sites-cohomology-lemma-compute-by-cosimplicial-resolution", "cotangent-lemma-pi-lower-shriek-polynomial-algebra", "simplicial-lemma-products-homotopy", "simplicial-lemma-trivial-kan-base-change", "simplicial-lemma-trivial-kan-homotopy", "simplicial-lemma-filtered-colimit-trivial-kan" ], "ref_ids": [ 4348, 11178, 14875, 14887, 14892, 14891 ] } ], "ref_ids": [] }, { "id": 11191, "type": "theorem", "label": "cotangent-lemma-triangle-compute-lower-shriek", "categories": [ "cotangent" ], "title": "cotangent-lemma-triangle-compute-lower-shriek", "contents": [ "With notation as in (\\ref{equation-three-maps}) we have", "$Lg_{i, !} \\circ g_i^{-1} = \\text{id}$ for $i = 1, 2, 3$", "and hence also $L\\pi_! \\circ g_i^{-1} = L\\pi_{i, !}$ for", "$i = 1, 2, 3$." ], "refs": [], "proofs": [ { "contents": [ "Proof for $i = 1$. We claim the functor $\\mathcal{C}_{C/B/A}$", "is a fibred category over $\\mathcal{C}_{B/A}$", "Namely, suppose given $(P \\to B, Q \\to C)$", "and a morphism $(P' \\to B) \\to (P \\to B)$ of $\\mathcal{C}_{B/A}$.", "Recall that this means we have an $A$-algebra homomorphism", "$P \\to P'$ compatible with maps to $B$. Then we set $Q' = Q \\otimes_P P'$", "with induced map to $C$ and the morphism", "$$", "(P' \\to B, Q' \\to C) \\longrightarrow (P \\to B, Q \\to C)", "$$", "in $\\mathcal{C}_{C/B/A}$ (note reversal arrows again) is strongly cartesian", "in $\\mathcal{C}_{C/B/A}$ over $\\mathcal{C}_{B/A}$. Moreover, observe", "that the fibre category of $u_1$ over $P \\to B$ is the category", "$\\mathcal{C}_{C/P}$. Let $\\mathcal{F}$ be an abelian sheaf on", "$\\mathcal{C}_{B/A}$. Since we have a fibred category we may apply", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-compute-left-derived-pi-shriek}.", "Thus $L_ng_{1, !}g_1^{-1}\\mathcal{F}$ is the (pre)sheaf", "which assigns to $U \\in \\Ob(\\mathcal{C}_{B/A})$ the $n$th homology of", "$g_1^{-1}\\mathcal{F}$ restricted to the fibre category over $U$.", "Since these restrictions are constant the desired result follows from", "Lemma \\ref{lemma-pi-lower-shriek-constant-sheaf}", "via our identifications of fibre categories above.", "\\medskip\\noindent", "The case $i = 2$.", "We claim $\\mathcal{C}_{C/B/A}$ is a fibred category over $\\mathcal{C}_{C/A}$", "is a fibred category. Namely, suppose given $(P \\to B, Q \\to C)$", "and a morphism $(Q' \\to C) \\to (Q \\to C)$ of $\\mathcal{C}_{C/A}$.", "Recall that this means we have a $B$-algebra homomorphism", "$Q \\to Q'$ compatible with maps to $C$. Then", "$$", "(P \\to B, Q' \\to C) \\longrightarrow (P \\to B, Q \\to C)", "$$", "is strongly cartesian in $\\mathcal{C}_{C/B/A}$ over $\\mathcal{C}_{C/A}$.", "Note that the fibre category of $u_2$ over $Q \\to C$ has an final", "(beware reversal arrows) object, namely, $(A \\to B, Q \\to C)$. Let", "$\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}_{C/A}$.", "Since we have a fibred category we may apply", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-compute-left-derived-pi-shriek}.", "Thus $L_ng_{2, !}g_2^{-1}\\mathcal{F}$ is the (pre)sheaf", "which assigns to $U \\in \\Ob(\\mathcal{C}_{C/A})$ the $n$th homology of", "$g_1^{-1}\\mathcal{F}$ restricted to the fibre category over $U$.", "Since these restrictions are constant the desired result follows from", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-initial-final}", "because the fibre categories all have final objects.", "\\medskip\\noindent", "The case $i = 3$. In this case we will apply", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-compute-left-derived-g-shriek}", "to $u = u_3 : \\mathcal{C}_{C/B/A} \\to \\mathcal{C}_{C/B}$", "and $\\mathcal{F}' = g_3^{-1}\\mathcal{F}$ for some abelian sheaf", "$\\mathcal{F}$ on $\\mathcal{C}_{C/B}$.", "Suppose $U = (\\overline{Q} \\to C)$ is an object of $\\mathcal{C}_{C/B}$.", "Then $\\mathcal{I}_U = \\mathcal{C}_{\\overline{Q}/B/A}$ (again beware", "of reversal of arrows). The sheaf $\\mathcal{F}'_U$ is given by the", "rule $(P \\to B, Q \\to \\overline{Q}) \\mapsto \\mathcal{F}(Q \\otimes_P B \\to C)$.", "In other words, this sheaf is the pullback of a sheaf", "on $\\mathcal{C}_{\\overline{Q}/C}$ via the morphism", "$\\Sh(\\mathcal{C}_{\\overline{Q}/B/A}) \\to \\Sh(\\mathcal{C}_{\\overline{Q}/B})$.", "Thus Lemma \\ref{lemma-polynomial-on-top} shows that", "$H_n(\\mathcal{I}_U, \\mathcal{F}'_U) = 0$ for $n > 0$", "and equal to $\\mathcal{F}(\\overline{Q} \\to C)$ for $n = 0$.", "The aforementioned Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-compute-left-derived-g-shriek}", "implies that $Lg_{3, !}(g_3^{-1}\\mathcal{F}) = \\mathcal{F}$ and", "the proof is done." ], "refs": [ "sites-cohomology-lemma-compute-left-derived-pi-shriek", "cotangent-lemma-pi-lower-shriek-constant-sheaf", "sites-cohomology-lemma-compute-left-derived-pi-shriek", "sites-cohomology-lemma-initial-final", "sites-cohomology-lemma-compute-left-derived-g-shriek", "cotangent-lemma-polynomial-on-top", "sites-cohomology-lemma-compute-left-derived-g-shriek" ], "ref_ids": [ 4354, 11176, 4354, 4346, 4355, 11190, 4355 ] } ], "ref_ids": [] }, { "id": 11192, "type": "theorem", "label": "cotangent-lemma-localize-at-bottom", "categories": [ "cotangent" ], "title": "cotangent-lemma-localize-at-bottom", "contents": [ "Let $A \\to A' \\to B$ be ring maps such that $B = B \\otimes_A^\\mathbf{L} A'$.", "Then $L_{B/A} = L_{B/A'}$ in $D(B)$." ], "refs": [], "proofs": [ { "contents": [ "According to the discussion above (i.e., using", "Lemma \\ref{lemma-flat-base-change})", "and Lemma \\ref{lemma-compute-cotangent-complex}", "we have to show that the sheaf given", "by the rule $(P \\to B) \\mapsto \\Omega_{P/A} \\otimes_P B$ on $\\mathcal{C}_{B/A}$", "is the pullback of the sheaf given by the rule", "$(P \\to B) \\mapsto \\Omega_{P/A'} \\otimes_P B$.", "The pullback functor $g^{-1}$ is given by precomposing with the", "functor $u : \\mathcal{C}_{B/A} \\to \\mathcal{C}_{B/A'}$,", "$(P \\to B) \\mapsto (P \\otimes_A A' \\to B)$.", "Thus we have to show that", "$$", "\\Omega_{P/A} \\otimes_P B =", "\\Omega_{P \\otimes_A A'/A'} \\otimes_{(P \\otimes_A A')} B", "$$", "By Algebra, Lemma \\ref{algebra-lemma-differentials-base-change}", "the right hand side is equal to", "$$", "(\\Omega_{P/A} \\otimes_A A') \\otimes_{(P \\otimes_A A')} B", "$$", "Since $P$ is a polynomial algebra over $A$ the module", "$\\Omega_{P/A}$ is free and the equality is obvious." ], "refs": [ "cotangent-lemma-flat-base-change", "cotangent-lemma-compute-cotangent-complex", "algebra-lemma-differentials-base-change" ], "ref_ids": [ 11186, 11175, 1138 ] } ], "ref_ids": [] }, { "id": 11193, "type": "theorem", "label": "cotangent-lemma-derived-diagonal", "categories": [ "cotangent" ], "title": "cotangent-lemma-derived-diagonal", "contents": [ "Let $A \\to B$ be a ring map such that $B = B \\otimes_A^\\mathbf{L} B$.", "Then $L_{B/A} = 0$ in $D(B)$." ], "refs": [], "proofs": [ { "contents": [ "This is true because $L_{B/A} = L_{B/B} = 0$ by", "Lemmas \\ref{lemma-localize-at-bottom} and", "\\ref{lemma-cotangent-complex-polynomial-algebra}." ], "refs": [ "cotangent-lemma-localize-at-bottom", "cotangent-lemma-cotangent-complex-polynomial-algebra" ], "ref_ids": [ 11192, 11179 ] } ], "ref_ids": [] }, { "id": 11194, "type": "theorem", "label": "cotangent-lemma-bootstrap", "categories": [ "cotangent" ], "title": "cotangent-lemma-bootstrap", "contents": [ "Let $A \\to B$ be a ring map such that $\\text{Tor}^A_i(B, B) = 0$ for $i > 0$", "and such that $L_{B/B \\otimes_A B} = 0$.", "Then $L_{B/A} = 0$ in $D(B)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-flat-base-change-cotangent-complex} we see that", "$L_{B/A} \\otimes_B^\\mathbf{L} (B \\otimes_A B) = L_{B \\otimes_A B/B}$.", "Now we use the distinguished triangle (\\ref{equation-triangle})", "$$", "L_{B \\otimes_A B/B} \\otimes^\\mathbf{L}_{(B \\otimes_A B)} B \\to", "L_{B/B} \\to L_{B/B \\otimes_A B} \\to", "L_{B \\otimes_A B/B} \\otimes^\\mathbf{L}_{(B \\otimes_A B)} B[1]", "$$", "associated to the ring maps $B \\to B \\otimes_A B \\to B$ and the vanishing of", "$L_{B/B}$ (Lemma \\ref{lemma-cotangent-complex-polynomial-algebra}) and", "$L_{B/B \\otimes_A B}$ (assumed) to see that", "$$", "0 =", "L_{B \\otimes_A B/B} \\otimes^\\mathbf{L}_{(B \\otimes_A B)} B =", "L_{B/A} \\otimes_B^\\mathbf{L} (B \\otimes_A B)", "\\otimes^\\mathbf{L}_{(B \\otimes_A B)} B = L_{B/A}", "$$", "as desired." ], "refs": [ "cotangent-lemma-flat-base-change-cotangent-complex", "cotangent-lemma-cotangent-complex-polynomial-algebra" ], "ref_ids": [ 11187, 11179 ] } ], "ref_ids": [] }, { "id": 11195, "type": "theorem", "label": "cotangent-lemma-when-zero", "categories": [ "cotangent" ], "title": "cotangent-lemma-when-zero", "contents": [ "The cotangent complex $L_{B/A}$ is zero in each of the following cases:", "\\begin{enumerate}", "\\item $A \\to B$ and $B \\otimes_A B \\to B$ are flat, i.e., $A \\to B$", "is weakly \\'etale", "(More on Algebra, Definition \\ref{more-algebra-definition-weakly-etale}),", "\\item $A \\to B$ is a flat epimorphism of rings,", "\\item $B = S^{-1}A$ for some multiplicative subset $S \\subset A$,", "\\item $A \\to B$ is unramified and flat,", "\\item $A \\to B$ is \\'etale,", "\\item $A \\to B$ is a filtered colimit of ring maps for which", "the cotangent complex vanishes,", "\\item $B$ is a henselization of a local ring of $A$,", "\\item $B$ is a strict henselization of a local ring of $A$, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [ "more-algebra-definition-weakly-etale" ], "proofs": [ { "contents": [ "In case (1) we may apply", "Lemma \\ref{lemma-derived-diagonal}", "to the surjective flat ring map $B \\otimes_A B \\to B$", "to conclude that $L_{B/B \\otimes_A B} = 0$ and then we use", "Lemma \\ref{lemma-bootstrap}", "to conclude. The cases (2) -- (5) are each special cases of (1).", "Part (6) follows from Lemma \\ref{lemma-colimit-cotangent-complex}.", "Parts (7) and (8) follows from the fact that (strict) henselizations", "are filtered colimits of \\'etale ring extensions of $A$, see", "Algebra, Lemmas \\ref{algebra-lemma-henselization-different} and", "\\ref{algebra-lemma-strict-henselization-different}." ], "refs": [ "cotangent-lemma-derived-diagonal", "cotangent-lemma-bootstrap", "cotangent-lemma-colimit-cotangent-complex", "algebra-lemma-henselization-different", "algebra-lemma-strict-henselization-different" ], "ref_ids": [ 11193, 11194, 11172, 1298, 1304 ] } ], "ref_ids": [ 10635 ] }, { "id": 11196, "type": "theorem", "label": "cotangent-lemma-localize-on-top", "categories": [ "cotangent" ], "title": "cotangent-lemma-localize-on-top", "contents": [ "Let $A \\to B \\to C$ be ring maps such that $L_{C/B} = 0$.", "Then $L_{C/A} = L_{B/A} \\otimes_B^\\mathbf{L} C$." ], "refs": [], "proofs": [ { "contents": [ "This is a trivial consequence of ", "the distinguished triangle (\\ref{equation-triangle})." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11197, "type": "theorem", "label": "cotangent-lemma-localize", "categories": [ "cotangent" ], "title": "cotangent-lemma-localize", "contents": [ "Let $A \\to B$ be ring maps and $S \\subset A$, $T \\subset B$ multiplicative", "subsets such that $S$ maps into $T$.", "Then $L_{T^{-1}B/S^{-1}A} = L_{B/A} \\otimes_B T^{-1}B$", "in $D(T^{-1}B)$." ], "refs": [], "proofs": [ { "contents": [ "Lemma \\ref{lemma-localize-on-top} shows that", "$L_{T^{-1}B/A} = L_{B/A} \\otimes_B T^{-1}B$", "and Lemma \\ref{lemma-localize-at-bottom}", "shows that $L_{T^{-1}B/A} = L_{T^{-1}B/S^{-1}A}$." ], "refs": [ "cotangent-lemma-localize-on-top", "cotangent-lemma-localize-at-bottom" ], "ref_ids": [ 11196, 11192 ] } ], "ref_ids": [] }, { "id": 11198, "type": "theorem", "label": "cotangent-lemma-cotangent-complex-henselization", "categories": [ "cotangent" ], "title": "cotangent-lemma-cotangent-complex-henselization", "contents": [ "Let $A \\to B$ be a local ring homomorphism of local rings.", "Let $A^h \\to B^h$, resp.\\ $A^{sh} \\to B^{sh}$ be the induced", "maps of henselizations, resp.\\ strict henselizations.", "Then", "$$", "L_{B^h/A^h} = L_{B^h/A} = L_{B/A} \\otimes_B^\\mathbf{L} B^h", "\\quad\\text{resp.}\\quad", "L_{B^{sh}/A^{sh}} = L_{B^{sh}/A} = L_{B/A} \\otimes_B^\\mathbf{L} B^{sh}", "$$", "in $D(B^h)$, resp.\\ $D(B^{sh})$." ], "refs": [], "proofs": [ { "contents": [ "The complexes $L_{A^h/A}$, $L_{A^{sh}/A}$, $L_{B^h/B}$, and", "$L_{B^{sh}/B}$ are all zero by Lemma \\ref{lemma-when-zero}.", "Using the fundamental distinguished triangle (\\ref{equation-triangle})", "for $A \\to B \\to B^h$ we obtain", "$L_{B^h/A} = L_{B/A} \\otimes_B^\\mathbf{L} B^h$.", "Using the fundamental triangle for $A \\to A^h \\to B^h$", "we obtain $L_{B^h/A^h} = L_{B^h/A}$.", "Similarly for strict henselizations." ], "refs": [ "cotangent-lemma-when-zero" ], "ref_ids": [ 11195 ] } ], "ref_ids": [] }, { "id": 11199, "type": "theorem", "label": "cotangent-lemma-when-projective", "categories": [ "cotangent" ], "title": "cotangent-lemma-when-projective", "contents": [ "If $A \\to B$ is a smooth ring map, then $L_{B/A} = \\Omega_{B/A}[0]$." ], "refs": [], "proofs": [ { "contents": [ "We have the agreement in cohomological degree $0$ by", "Lemma \\ref{lemma-identify-H0}.", "Thus it suffices to prove the other cohomology groups", "are zero. It suffices to prove this locally on $\\Spec(B)$ as", "$L_{B_g/A} = (L_{B/A})_g$ for $g \\in B$ by Lemma \\ref{lemma-localize-on-top}.", "Thus we may assume that $A \\to B$ is standard smooth", "(Algebra, Lemma \\ref{algebra-lemma-smooth-syntomic}), i.e.,", "that we can factor $A \\to B$ as", "$A \\to A[x_1, \\ldots, x_n] \\to B$ with $A[x_1, \\ldots, x_n] \\to B$", "\\'etale. In this case Lemmas \\ref{lemma-when-zero} and", "Lemma \\ref{lemma-localize-on-top} show that", "$L_{B/A} = L_{A[x_1, \\ldots, x_n]/A} \\otimes B$", "whence the conclusion by", "Lemma \\ref{lemma-cotangent-complex-polynomial-algebra}." ], "refs": [ "cotangent-lemma-identify-H0", "cotangent-lemma-localize-on-top", "algebra-lemma-smooth-syntomic", "cotangent-lemma-when-zero", "cotangent-lemma-localize-on-top", "cotangent-lemma-cotangent-complex-polynomial-algebra" ], "ref_ids": [ 11177, 11196, 1195, 11195, 11196, 11179 ] } ], "ref_ids": [] }, { "id": 11200, "type": "theorem", "label": "cotangent-lemma-frobenius-homotopy", "categories": [ "cotangent" ], "title": "cotangent-lemma-frobenius-homotopy", "contents": [ "Let $A \\to B$ be a ring map with $p = 0$ in $A$. Let $P_\\bullet$ be the", "standard resolution of $B$ over $A$. The map $P_\\bullet \\to P_\\bullet$", "induced by the diagram", "$$", "\\xymatrix{", "B \\ar[r]_{F_B} & B \\\\", "A \\ar[u] \\ar[r]^{F_A} & A \\ar[u]", "}", "$$", "discussed in Section \\ref{section-functoriality} is homotopic to the Frobenius", "endomorphism $P_\\bullet \\to P_\\bullet$ given by Frobenius on each $P_n$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{A}$ be the category of $\\mathbf{F}_p$-algebra maps", "$A \\to B$. Let $\\mathcal{S}$ be the category of pairs $(A, E)$", "where $A$ is an $\\mathbf{F}_p$-algebra and $E$ is a set. Consider the", "adjoint functors", "$$", "V : \\mathcal{A} \\to \\mathcal{S}, \\quad (A \\to B) \\mapsto (A, B)", "$$", "and", "$$", "U : \\mathcal{S} \\to \\mathcal{A}, \\quad (A, E) \\mapsto (A \\to A[E])", "$$", "Let $X$ be the simplicial object in ", "in the category of functors from $\\mathcal{A}$ to $\\mathcal{A}$", "constructed in Simplicial, Section \\ref{simplicial-section-standard}.", "It is clear that $P_\\bullet = X(A \\to B)$ because if we fix", "$A$ then.", "\\medskip\\noindent", "Set $Y = U \\circ V$. Recall that $X$ is constructed from $Y$", "and certain maps and has terms $X_n = Y \\circ \\ldots \\circ Y$", "with $n + 1$ terms; the construction is given in", "Simplicial, Example \\ref{simplicial-example-godement} and please see", "proof of Simplicial, Lemma \\ref{simplicial-lemma-standard-simplicial}", "for details.", "\\medskip\\noindent", "Let $f : \\text{id}_\\mathcal{A} \\to \\text{id}_\\mathcal{A}$", "be the Frobenius endomorphism of the identity functor.", "In other words, we set $f_{A \\to B} = (F_A, F_B) : (A \\to B) \\to (A \\to B)$.", "Then our two maps on $X(A \\to B)$ are given by the natural transformations", "$f \\star 1_X$ and $1_X \\star f$. Details omitted.", "Thus we conclude by Simplicial, Lemma", "\\ref{simplicial-lemma-godement-before-after}." ], "refs": [ "simplicial-lemma-standard-simplicial", "simplicial-lemma-godement-before-after" ], "ref_ids": [ 14908, 14907 ] } ], "ref_ids": [] }, { "id": 11201, "type": "theorem", "label": "cotangent-lemma-frobenius-acts-as-zero", "categories": [ "cotangent" ], "title": "cotangent-lemma-frobenius-acts-as-zero", "contents": [ "Let $p$ be a prime number. Let $A \\to B$ be a ring homomorphism", "and assume that $p = 0$ in $A$. The map $L_{B/A} \\to L_{B/A}$", "of Section \\ref{section-functoriality} induced by the", "Frobenius maps $F_A$ and $F_B$ is homotopic to zero." ], "refs": [], "proofs": [ { "contents": [ "Let $P_\\bullet$ be the standard resolution of $B$ over $A$.", "By Lemma \\ref{lemma-frobenius-homotopy} the map $P_\\bullet \\to P_\\bullet$", "induced by $F_A$ and $F_B$ is homotopic to the map", "$F_{P_\\bullet} : P_\\bullet \\to P_\\bullet$ given by", "Frobenius on each term. Hence we obtain what we want as clearly", "$F_{P_\\bullet}$ induces the zero zero map $\\Omega_{P_n/A} \\to \\Omega_{P_n/A}$", "(since the derivative of a $p$th power is zero)." ], "refs": [ "cotangent-lemma-frobenius-homotopy" ], "ref_ids": [ 11200 ] } ], "ref_ids": [] }, { "id": 11202, "type": "theorem", "label": "cotangent-lemma-perfect-zero", "categories": [ "cotangent" ], "title": "cotangent-lemma-perfect-zero", "contents": [ "Let $p$ be a prime number. Let $A \\to B$ be a ring homomorphism", "and assume that $p = 0$ in $A$. If $A$ and $B$ are perfect, then", "$L_{B/A}$ is zero in $D(B)$." ], "refs": [], "proofs": [ { "contents": [ "The map $(F_A, F_B) : (A \\to B) \\to (A \\to B)$ is an isomorphism", "hence induces an isomorphism on $L_{B/A}$ and on the other hand", "induces zero on $L_{B/A}$ by Lemma \\ref{lemma-frobenius-acts-as-zero}." ], "refs": [ "cotangent-lemma-frobenius-acts-as-zero" ], "ref_ids": [ 11201 ] } ], "ref_ids": [] }, { "id": 11203, "type": "theorem", "label": "cotangent-lemma-surjection", "categories": [ "cotangent" ], "title": "cotangent-lemma-surjection", "contents": [ "\\begin{slogan}", "The cohomology of the cotangent complex of a surjective ring map is trivial in", "degree zero; it is the kernel modulo its square in degree $-1$.", "\\end{slogan}", "Let $A \\to B$ be a surjective ring map with kernel $I$.", "Then $H^0(L_{B/A}) = 0$ and $H^{-1}(L_{B/A}) = I/I^2$.", "This isomorphism comes from the map (\\ref{equation-comparison-map})", "for the object $(A \\to B)$ of $\\mathcal{C}_{B/A}$." ], "refs": [], "proofs": [ { "contents": [ "We will show below (using the surjectivity of $A \\to B$)", "that there exists a short exact sequence", "$$", "0 \\to \\pi^{-1}(I/I^2) \\to \\mathcal{J}/\\mathcal{J}^2 \\to \\Omega \\to 0", "$$", "of sheaves on $\\mathcal{C}_{B/A}$. Taking $L\\pi_!$ and", "the associated long exact sequence of homology, and using the", "vanishing of $H_1(\\mathcal{C}_{B/A}, \\mathcal{J}/\\mathcal{J}^2)$ and", "$H_0(\\mathcal{C}_{B/A}, \\mathcal{J}/\\mathcal{J}^2)$", "shown in Remark \\ref{remark-make-map} we obtain what we want using", "Lemma \\ref{lemma-pi-lower-shriek-constant-sheaf}.", "\\medskip\\noindent", "What is left is to verify the local statement mentioned above.", "For every object $U = (P \\to B)$ of $\\mathcal{C}_{B/A}$", "we can choose an isomorphism $P = A[E]$ such that the map", "$P \\to B$ maps each $e \\in E$ to zero. Then", "$J = \\mathcal{J}(U) \\subset P = \\mathcal{O}(U)$", "is equal to $J = IP + (e; e \\in E)$. The value on $U$ of the short sequence", "of sheaves above is the sequence", "$$", "0 \\to I/I^2 \\to J/J^2 \\to \\Omega_{P/A} \\otimes_P B \\to 0", "$$", "Verification omitted (hint: the only tricky point is that", "$IP \\cap J^2 = IJ$; which follows for example from", "More on Algebra, Lemma \\ref{more-algebra-lemma-conormal-sequence-H1-regular})." ], "refs": [ "cotangent-remark-make-map", "cotangent-lemma-pi-lower-shriek-constant-sheaf", "more-algebra-lemma-conormal-sequence-H1-regular" ], "ref_ids": [ 11260, 11176, 9980 ] } ], "ref_ids": [] }, { "id": 11204, "type": "theorem", "label": "cotangent-lemma-relation-with-naive-cotangent-complex", "categories": [ "cotangent" ], "title": "cotangent-lemma-relation-with-naive-cotangent-complex", "contents": [ "Let $A \\to B$ be a ring map. Then $\\tau_{\\geq -1}L_{B/A}$", "is canonically quasi-isomorphic to the naive cotangent complex." ], "refs": [], "proofs": [ { "contents": [ "Consider $P = A[B] \\to B$ with kernel $I$. The naive cotangent", "complex $\\NL_{B/A}$ of $B$ over $A$ is the complex", "$I/I^2 \\to \\Omega_{P/A} \\otimes_P B$,", "see Algebra, Definition \\ref{algebra-definition-naive-cotangent-complex}.", "Observe that in (\\ref{equation-comparison-map}) we have already", "constructed a canonical map", "$$", "c : \\NL_{B/A} \\longrightarrow \\tau_{\\geq -1}L_{B/A}", "$$", "Consider the distinguished triangle (\\ref{equation-triangle})", "$$", "L_{P/A} \\otimes_P^\\mathbf{L} B \\to L_{B/A} \\to L_{B/P} \\to ", "(L_{P/A} \\otimes_P^\\mathbf{L} B)[1]", "$$", "associated to the ring maps $A \\to A[B] \\to B$. We know that", "$L_{P/A} = \\Omega_{P/A}[0] = \\NL_{P/A}$ in $D(P)$", "(Lemma \\ref{lemma-cotangent-complex-polynomial-algebra}", "and", "Algebra, Lemma \\ref{algebra-lemma-NL-polynomial-algebra})", "and that", "$\\tau_{\\geq -1}L_{B/P} = I/I^2[1] = \\NL_{B/P}$ in $D(B)$", "(Lemma \\ref{lemma-surjection} and", "Algebra, Lemma \\ref{algebra-lemma-NL-surjection}).", "To show $c$ is a quasi-isomorphism it suffices by", "Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL}", "and the long exact cohomology sequence associated to the", "distinguished triangle", "to show that the maps $L_{P/A} \\to L_{B/A} \\to L_{B/P}$ are compatible", "on cohomology groups with the corresponding maps", "$\\NL_{P/A} \\to \\NL_{B/A} \\to \\NL_{B/P}$", "of the naive cotangent complex. We omit the verification." ], "refs": [ "algebra-definition-naive-cotangent-complex", "cotangent-lemma-cotangent-complex-polynomial-algebra", "algebra-lemma-NL-polynomial-algebra", "cotangent-lemma-surjection", "algebra-lemma-NL-surjection", "algebra-lemma-exact-sequence-NL" ], "ref_ids": [ 1529, 11179, 1152, 11203, 1154, 1153 ] } ], "ref_ids": [] }, { "id": 11205, "type": "theorem", "label": "cotangent-lemma-vanishing-symmetric-powers", "categories": [ "cotangent" ], "title": "cotangent-lemma-vanishing-symmetric-powers", "contents": [ "Notation and assumptions as in", "Cohomology on Sites, Example \\ref{sites-cohomology-example-category-to-point}.", "Assume $\\mathcal{C}$ has a cosimplicial object as in", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}.", "Let $\\mathcal{F}$ be a flat $\\underline{B}$-module such that", "$H_0(\\mathcal{C}, \\mathcal{F}) = 0$.", "Then $H_l(\\mathcal{C}, \\text{Sym}_{\\underline{B}}^k(\\mathcal{F})) = 0$", "for $l < k$." ], "refs": [ "sites-cohomology-lemma-compute-by-cosimplicial-resolution" ], "proofs": [ { "contents": [ "We drop the subscript ${}_{\\underline{B}}$ from tensor products, wedge powers,", "and symmetric powers. We will prove the lemma by induction on $k$.", "The cases $k = 0, 1$ follow from the assumptions. If $k > 1$ consider", "the exact complex", "$$", "\\ldots \\to", "\\wedge^2\\mathcal{F} \\otimes \\text{Sym}^{k - 2}\\mathcal{F} \\to", "\\mathcal{F} \\otimes \\text{Sym}^{k - 1}\\mathcal{F} \\to", "\\text{Sym}^k\\mathcal{F} \\to 0", "$$", "with differentials as in the Koszul complex. If we think of this as a", "resolution of $\\text{Sym}^k\\mathcal{F}$, then this gives a first quadrant", "spectral sequence", "$$", "E_1^{p, q} =", "H_p(\\mathcal{C},", "\\wedge^{q + 1}\\mathcal{F} \\otimes \\text{Sym}^{k - q - 1}\\mathcal{F})", "\\Rightarrow", "H_{p + q}(\\mathcal{C}, \\text{Sym}^k(\\mathcal{F}))", "$$", "By Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-eilenberg-zilber}", "we have", "$$", "L\\pi_!(\\wedge^{q + 1}\\mathcal{F} \\otimes \\text{Sym}^{k - q - 1}\\mathcal{F}) =", "L\\pi_!(\\wedge^{q + 1}\\mathcal{F}) \\otimes_B^\\mathbf{L}", "L\\pi_!(\\text{Sym}^{k - q - 1}\\mathcal{F}))", "$$", "It follows (from the construction of derived tensor products) that", "the induction hypothesis combined with the vanishing", "of $H_0(\\mathcal{C}, \\wedge^{q + 1}(\\mathcal{F})) = 0$ will prove what we want.", "This is true because $\\wedge^{q + 1}(\\mathcal{F})$ is a quotient", "of $\\mathcal{F}^{\\otimes q + 1}$ and", "$H_0(\\mathcal{C}, \\mathcal{F}^{\\otimes q + 1})$", "is a quotient of $H_0(\\mathcal{C}, \\mathcal{F})^{\\otimes q + 1}$", "which is zero." ], "refs": [ "sites-cohomology-lemma-eilenberg-zilber" ], "ref_ids": [ 4351 ] } ], "ref_ids": [ 4348 ] }, { "id": 11206, "type": "theorem", "label": "cotangent-lemma-map-tors-zero", "categories": [ "cotangent" ], "title": "cotangent-lemma-map-tors-zero", "contents": [ "Let $A$ be a ring. Let $P = A[E]$ be a polynomial ring.", "Set $I = (e; e \\in E) \\subset P$. The maps", "$\\text{Tor}_i^P(A, I^{n + 1}) \\to \\text{Tor}_i^P(A, I^n)$", "are zero for all $i$ and $n$." ], "refs": [], "proofs": [ { "contents": [ "Denote $x_e \\in P$ the variable corresponding to $e \\in E$.", "A free resolution of $A$ over $P$ is given by the Koszul complex", "$K_\\bullet$ on the $x_e$. Here $K_i$ has basis given by wedges", "$e_1 \\wedge \\ldots \\wedge e_i$, $e_1, \\ldots, e_i \\in E$ and $d(e) = x_e$.", "Thus $K_\\bullet \\otimes_P I^n = I^nK_\\bullet$ computes", "$\\text{Tor}_i^P(A, I^n)$. Observe that everything is graded", "with $\\deg(x_e) = 1$, $\\deg(e) = 1$, and $\\deg(a) = 0$ for $a \\in A$.", "Suppose $\\xi \\in I^{n + 1}K_i$ is a cocycle homogeneous of degree $m$.", "Note that $m \\geq i + 1 + n$. Then $\\xi = \\text{d}\\eta$ for some", "$\\eta \\in K_{i + 1}$ as $K_\\bullet$ is exact in degrees $ > 0$.", "(The case $i = 0$ is left to the reader.)", "Now $\\deg(\\eta) = m \\geq i + 1 + n$. Hence writing $\\eta$", "in terms of the basis we see the coordinates are in $I^n$.", "Thus $\\xi$ maps to zero in the homology of $I^nK_\\bullet$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11207, "type": "theorem", "label": "cotangent-lemma-polynomial-ring-unique", "categories": [ "cotangent" ], "title": "cotangent-lemma-polynomial-ring-unique", "contents": [ "Let $P = A[S]$ be a polynomial ring over $A$. Let $M$ be a $(P, P)$-bimodule", "over $A$. Given $m_s \\in M$ for $s \\in S$, there exists a unique", "$A$-biderivation $\\lambda : P \\to M$ mapping $s$ to $m_s$ for $s \\in S$." ], "refs": [], "proofs": [ { "contents": [ "We set", "$$", "\\lambda(s_1 \\ldots s_t) =", "\\sum s_1 \\ldots s_{i - 1} m_{s_i} s_{i + 1} \\ldots s_t", "$$", "in $M$. Extending by $A$-linearity we obtain a biderivation." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11208, "type": "theorem", "label": "cotangent-lemma-compare-higher", "categories": [ "cotangent" ], "title": "cotangent-lemma-compare-higher", "contents": [ "In the situation above denote $L$ the complex", "(\\ref{equation-lichtenbaum-schlessinger}).", "There is a canonical map $L_{B/A} \\to L$ in $D(A)$ which", "induces an isomorphism $\\tau_{\\geq -2}L_{B/A} \\to L$ in $D(B)$." ], "refs": [], "proofs": [ { "contents": [ "Let $P_\\bullet \\to B$ be a resolution of $B$ over $A$", "(Remark \\ref{remark-resolution}). We will identify $L_{B/A}$ with", "$\\Omega_{P_\\bullet/A} \\otimes B$. To construct the map we", "make some choices.", "\\medskip\\noindent", "Choose an $A$-algebra map $\\psi : P_0 \\to P$ compatible with the", "given maps $P_0 \\to B$ and $P \\to B$.", "\\medskip\\noindent", "Write $P_1 = A[S]$ for some set $S$. For $s \\in S$ we may write", "$$", "\\psi(d_0(s) - d_1(s)) = \\sum p_{s, t} f_t", "$$", "for some $p_{s, t} \\in P$. Think of $F = \\bigoplus_{t \\in T} P$", "as a $(P_1, P_1)$-bimodule via the maps $(\\psi \\circ d_0, \\psi \\circ d_1)$.", "By Lemma \\ref{lemma-polynomial-ring-unique} we obtain a unique", "$A$-biderivation $\\lambda : P_1 \\to F$ mapping $s$ to the vector with", "coordinates $p_{s, t}$. By construction the composition", "$$", "P_1 \\longrightarrow F \\longrightarrow P", "$$", "sends $f \\in P_1$ to $\\psi(d_0(f) - d_1(f))$ because the map", "$f \\mapsto \\psi(d_0(f) - d_1(f))$ is an $A$-biderivation agreeing with", "the composition on generators.", "\\medskip\\noindent", "For $g \\in P_2$ we claim that $\\lambda(d_0(g) - d_1(g) + d_2(g))$", "is an element of $Rel$. Namely, by the last remark of the previous", "paragraph the image of $\\lambda(d_0(g) - d_1(g) + d_2(g))$ in $P$ is", "$$", "\\psi((d_0 - d_1)(d_0(g) - d_1(g) + d_2(g)))", "$$", "which is zero by Simplicial, Section \\ref{simplicial-section-complexes}).", "\\medskip\\noindent", "The choice of $\\psi$ determines a map", "$$", "\\text{d}\\psi \\otimes 1 :", "\\Omega_{P_0/A} \\otimes B", "\\longrightarrow", "\\Omega_{P/A} \\otimes B", "$$", "Composing $\\lambda$ with the map $F \\to F \\otimes B$ gives a", "usual $A$-derivation as the two $P_1$-module structures on", "$F \\otimes B$ agree. Thus $\\lambda$ determines a map", "$$", "\\overline{\\lambda} :", "\\Omega_{P_1/A} \\otimes B", "\\longrightarrow", "F \\otimes B", "$$", "Finally, We obtain a $B$-linear map", "$$", "q :", "\\Omega_{P_2/A} \\otimes B", "\\longrightarrow", "Rel/TrivRel", "$$", "by mapping $\\text{d}g$ to the class of $\\lambda(d_0(g) - d_1(g) + d_2(g))$", "in the quotient.", "\\medskip\\noindent", "The diagram", "$$", "\\xymatrix{", "\\Omega_{P_3/A} \\otimes B \\ar[r] \\ar[d] &", "\\Omega_{P_2/A} \\otimes B \\ar[r] \\ar[d]_q &", "\\Omega_{P_1/A} \\otimes B \\ar[r] \\ar[d]_{\\overline{\\lambda}} &", "\\Omega_{P_0/A} \\otimes B \\ar[d]_{\\text{d}\\psi \\otimes 1} \\\\", "0 \\ar[r] &", "Rel/TrivRel \\ar[r] &", "F \\otimes B \\ar[r] &", "\\Omega_{P/A} \\otimes B", "}", "$$", "commutes (calculation omitted) and we obtain the map of the lemma.", "By Remark \\ref{remark-explicit-comparison-map} and", "Lemma \\ref{lemma-relation-with-naive-cotangent-complex} we see that this map", "induces isomorphisms $H_1(L_{B/A}) \\to H_1(L)$ and $H_0(L_{B/A}) \\to H_0(L)$.", "\\medskip\\noindent", "It remains to see that our map $L_{B/A} \\to L$ induces an isomorphism", "$H_2(L_{B/A}) \\to H_2(L)$. Choose a resolution of $B$ over $A$ with", "$P_0 = P = A[u_i]$ and then $P_1$ and $P_2$ as in", "Example \\ref{example-resolution-length-two}.", "In Remark \\ref{remark-elucidate-degree-two} we have constructed an exact", "sequence", "$$", "\\wedge^2_B(J_0/J_0^2) \\to \\text{Tor}_2^{P_0}(B, B) \\to H^{-2}(L_{B/A}) \\to 0", "$$", "where $P_0 = P$ and $J_0 = \\Ker(P \\to B) = I$.", "Calculating the Tor group using the short exact sequences", "$0 \\to I \\to P \\to B \\to 0$ and $0 \\to Rel \\to F \\to I \\to 0$", "we find that", "$\\text{Tor}_2^P(B, B) = \\Ker(Rel \\otimes B \\to F \\otimes B)$.", "The image of the map $\\wedge^2_B(I/I^2) \\to \\text{Tor}_2^P(B, B)$", "under this identification is exactly the image of $TrivRel \\otimes B$.", "Thus we see that $H_2(L_{B/A}) \\cong H_2(L)$.", "\\medskip\\noindent", "Finally, we have to check that our map $L_{B/A} \\to L$ actually induces", "this isomorphism. We will use the notation and results discussed in", "Example \\ref{example-resolution-length-two} and", "Remarks \\ref{remark-elucidate-degree-two} and \\ref{remark-surjection}", "without further mention. Pick an element $\\xi$ of", "$\\text{Tor}_2^{P_0}(B, B) = \\Ker(I \\otimes_P I \\to I^2)$.", "Write $\\xi = \\sum h_{t', t}f_{t'} \\otimes f_t$ for some", "$h_{t', t} \\in P$. Tracing through the exact sequences above we", "find that $\\xi$ corresponds to the image in $Rel \\otimes B$", "of the element $r \\in Rel \\subset F = \\bigoplus_{t \\in T} P$ with", "$t$th coordinate $r_t = \\sum_{t' \\in T} h_{t', t}f_{t'}$.", "On the other hand, $\\xi$ corresponds to the element of", "$H_2(L_{B/A}) = H_2(\\Omega)$ which is the image", "via $\\text{d} : H_2(\\mathcal{J}/\\mathcal{J}^2) \\to H_2(\\Omega)$", "of the boundary of $\\xi$ under the $2$-extension", "$$", "0 \\to", "\\text{Tor}_2^\\mathcal{O}(\\underline{B}, \\underline{B})", "\\to ", "\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{J} \\to \\mathcal{J}", "\\to", "\\mathcal{J}/\\mathcal{J}^2 \\to 0", "$$", "We compute the successive transgressions of our element. First we have", "$$", "\\xi = (d_0 - d_1)(- \\sum s_0(h_{t', t} f_{t'}) \\otimes x_t)", "$$", "and next we have", "$$", "\\sum s_0(h_{t', t} f_{t'}) x_t = d_0(v_r) - d_1(v_r) + d_2(v_r)", "$$", "by our choice of the variables $v$ in", "Example \\ref{example-resolution-length-two}.", "We may choose our map $\\lambda$ above such that", "$\\lambda(u_i) = 0$ and $\\lambda(x_t) = - e_t$ where $e_t \\in F$", "denotes the basis vector corresponding to $t \\in T$.", "Hence the construction of our map $q$ above sends $\\text{d}v_r$ to", "$$", "\\lambda(\\sum s_0(h_{t', t} f_{t'}) x_t) =", "\\sum\\nolimits_t \\left(\\sum\\nolimits_{t'} h_{t', t}f_{t'}\\right) e_t", "$$", "matching the image of $\\xi$ in $Rel \\otimes B$ (the two minus signs", "we found above cancel out). This agreement finishes the proof." ], "refs": [ "cotangent-remark-resolution", "cotangent-lemma-polynomial-ring-unique", "cotangent-remark-explicit-comparison-map", "cotangent-lemma-relation-with-naive-cotangent-complex", "cotangent-remark-elucidate-degree-two", "cotangent-remark-elucidate-degree-two", "cotangent-remark-surjection" ], "ref_ids": [ 11256, 11207, 11261, 11204, 11265, 11265, 11262 ] } ], "ref_ids": [] }, { "id": 11209, "type": "theorem", "label": "cotangent-lemma-special-case", "categories": [ "cotangent" ], "title": "cotangent-lemma-special-case", "contents": [ "Let $A = \\mathbf{Z}[x_1, \\ldots, x_n] \\to B = \\mathbf{Z}$", "be the ring map which sends $x_i$ to $0$ for $i = 1, \\ldots, n$.", "Let $I = (x_1, \\ldots, x_n) \\subset A$. Then $L_{B/A}$ is quasi-isomorphic to", "$I/I^2[1]$." ], "refs": [], "proofs": [ { "contents": [ "There are several ways to prove this. For example one can explicitly construct", "a resolution of $B$ over $A$ and compute. We will use (\\ref{equation-triangle}).", "Namely, consider the distinguished triangle", "$$", "L_{\\mathbf{Z}[x_1, \\ldots, x_n]/\\mathbf{Z}}", "\\otimes_{\\mathbf{Z}[x_1, \\ldots, x_n]} \\mathbf{Z} \\to", "L_{\\mathbf{Z}/\\mathbf{Z}} \\to", "L_{\\mathbf{Z}/\\mathbf{Z}[x_1, \\ldots, x_n]}\\to", "L_{\\mathbf{Z}[x_1, \\ldots, x_n]/\\mathbf{Z}}", "\\otimes_{\\mathbf{Z}[x_1, \\ldots, x_n]} \\mathbf{Z}[1]", "$$", "The complex $L_{\\mathbf{Z}[x_1, \\ldots, x_n]/\\mathbf{Z}}$", "is quasi-isomorphic to $\\Omega_{\\mathbf{Z}[x_1, \\ldots, x_n]/\\mathbf{Z}}$ by", "Lemma \\ref{lemma-cotangent-complex-polynomial-algebra}.", "The complex $L_{\\mathbf{Z}/\\mathbf{Z}}$ is zero in $D(\\mathbf{Z})$ by", "Lemma \\ref{lemma-when-zero}.", "Thus we see that $L_{B/A}$ has only one nonzero cohomology group", "which is as described in the lemma by Lemma \\ref{lemma-surjection}." ], "refs": [ "cotangent-lemma-cotangent-complex-polynomial-algebra", "cotangent-lemma-when-zero", "cotangent-lemma-surjection" ], "ref_ids": [ 11179, 11195, 11203 ] } ], "ref_ids": [] }, { "id": 11210, "type": "theorem", "label": "cotangent-lemma-mod-regular-sequence", "categories": [ "cotangent" ], "title": "cotangent-lemma-mod-regular-sequence", "contents": [ "Let $A \\to B$ be a surjective ring map whose kernel $I$ is generated", "by a Koszul-regular sequence (for example a regular sequence).", "Then $L_{B/A}$ is quasi-isomorphic to $I/I^2[1]$." ], "refs": [], "proofs": [ { "contents": [ "Let $f_1, \\ldots, f_r \\in I$ be a Koszul regular sequence generating $I$.", "Consider the ring map $\\mathbf{Z}[x_1, \\ldots, x_r] \\to A$ sending", "$x_i$ to $f_i$. Since $x_1, \\ldots, x_r$ is a regular sequence in", "$\\mathbf{Z}[x_1, \\ldots, x_r]$ we see that the Koszul complex", "on $x_1, \\ldots, x_r$ is a free resolution of", "$\\mathbf{Z} = \\mathbf{Z}[x_1, \\ldots, x_r]/(x_1, \\ldots, x_r)$", "over $\\mathbf{Z}[x_1, \\ldots, x_r]$", "(see More on Algebra, Lemma \\ref{more-algebra-lemma-regular-koszul-regular}).", "Thus the assumption that $f_1, \\ldots, f_r$ is Koszul regular", "exactly means that", "$B = A \\otimes_{\\mathbf{Z}[x_1, \\ldots, x_r]}^\\mathbf{L} \\mathbf{Z}$.", "Hence", "$L_{B/A} = L_{\\mathbf{Z}/\\mathbf{Z}[x_1, \\ldots, x_r]}", "\\otimes_\\mathbf{Z}^\\mathbf{L} B$ by", "Lemmas \\ref{lemma-flat-base-change-cotangent-complex} and", "\\ref{lemma-special-case}." ], "refs": [ "more-algebra-lemma-regular-koszul-regular", "cotangent-lemma-flat-base-change-cotangent-complex", "cotangent-lemma-special-case" ], "ref_ids": [ 9973, 11187, 11209 ] } ], "ref_ids": [] }, { "id": 11211, "type": "theorem", "label": "cotangent-lemma-mod-Koszul-regular-ideal", "categories": [ "cotangent" ], "title": "cotangent-lemma-mod-Koszul-regular-ideal", "contents": [ "Let $A \\to B$ be a surjective ring map whose kernel $I$ is Koszul.", "Then $L_{B/A}$ is quasi-isomorphic to $I/I^2[1]$." ], "refs": [], "proofs": [ { "contents": [ "Locally on $\\Spec(A)$ the ideal $I$ is generated by a Koszul regular", "sequence, see More on Algebra, Definition", "\\ref{more-algebra-definition-regular-ideal}.", "Hence this follows from", "Lemma \\ref{lemma-flat-base-change-cotangent-complex}." ], "refs": [ "more-algebra-definition-regular-ideal", "cotangent-lemma-flat-base-change-cotangent-complex" ], "ref_ids": [ 10608, 11187 ] } ], "ref_ids": [] }, { "id": 11212, "type": "theorem", "label": "cotangent-lemma-tensor-product-tor-independent", "categories": [ "cotangent" ], "title": "cotangent-lemma-tensor-product-tor-independent", "contents": [ "If $A$ and $B$ are Tor independent $R$-algebras, then the object $E$", "in (\\ref{equation-tensor-product}) is zero. In this case we have", "$$", "L_{A \\otimes_R B/R} =", "L_{A/R} \\otimes_A^\\mathbf{L} (A \\otimes_R B) \\oplus", "L_{B/R} \\otimes_B^\\mathbf{L} (A \\otimes_R B)", "$$", "which is represented by the complex", "$L_{A/R} \\otimes_R B \\oplus L_{B/R} \\otimes_R A $", "of $A \\otimes_R B$-modules." ], "refs": [], "proofs": [ { "contents": [ "The first two statements are immediate from", "Lemma \\ref{lemma-flat-base-change-cotangent-complex}.", "The last statement follows as $L_{A/R}$ is a complex", "of free $A$-modules, hence $L_{A/R} \\otimes_A^\\mathbf{L} (A \\otimes_R B)$", "is represented by", "$L_{A/R} \\otimes_A (A \\otimes_R B) = L_{A/R} \\otimes_R B$" ], "refs": [ "cotangent-lemma-flat-base-change-cotangent-complex" ], "ref_ids": [ 11187 ] } ], "ref_ids": [] }, { "id": 11213, "type": "theorem", "label": "cotangent-lemma-tensor-product", "categories": [ "cotangent" ], "title": "cotangent-lemma-tensor-product", "contents": [ "Let $R$ be a ring and let $A$, $B$ be $R$-algebras. The object $E$", "in (\\ref{equation-tensor-product}) satisfies", "$$", "H^i(E) =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & i \\geq -1 \\\\", "\\text{Tor}_1^R(A, B) & \\text{if} & i = -2", "\\end{matrix}", "\\right.", "$$" ], "refs": [], "proofs": [ { "contents": [ "We use the description of $E$ as the cone on", "$L_{B/R} \\otimes_B^\\mathbf{L} (A \\otimes_R B) \\to L_{A \\otimes_R B/A}$.", "By Lemma \\ref{lemma-compare-higher} the canonical truncations", "$\\tau_{\\geq -2}L_{B/R}$ and $\\tau_{\\geq -2}L_{A \\otimes_R B/A}$", "are computed by the Lichtenbaum-Schlessinger complex", "(\\ref{equation-lichtenbaum-schlessinger}).", "These isomorphisms are compatible with functoriality", "(Remark \\ref{remark-functoriality-lichtenbaum-schlessinger}).", "Thus in this proof we work with the Lichtenbaum-Schlessinger complexes.", "\\medskip\\noindent", "Choose a polynomial algebra $P$ over $R$ and a surjection $P \\to B$.", "Choose generators $f_t \\in P$, $t \\in T$ of the kernel of this surjection.", "Let $Rel \\subset F = \\bigoplus_{t \\in T} P$ be the kernel of the map", "$F \\to P$ which maps the basis vector corresponding to $t$ to $f_t$.", "Set $P_A = A \\otimes_R P$ and $F_A = A \\otimes_R F = P_A \\otimes_P F$.", "Let $Rel_A$ be the kernel of the map $F_A \\to P_A$. Using the exact sequence", "$$", "0 \\to Rel \\to F \\to P \\to B \\to 0", "$$", "and standard short exact sequences for Tor we obtain an exact sequence", "$$", "A \\otimes_R Rel \\to Rel_A \\to \\text{Tor}_1^R(A, B) \\to 0", "$$", "Note that $P_A \\to A \\otimes_R B$ is a surjection whose kernel is generated", "by the elements $1 \\otimes f_t$ in $P_A$. Denote $TrivRel_A \\subset Rel_A$", "the $P_A$-submodule generated by the elements", "$(\\ldots, 1 \\otimes f_{t'}, 0, \\ldots,", "0, - 1 \\otimes f_t \\otimes 1, 0, \\ldots)$.", "Since $TrivRel \\otimes_R A \\to TrivRel_A$ is surjective, we find a", "canonical exact sequence", "$$", "A \\otimes_R (Rel/TrivRel) \\to Rel_A/TrivRel_A \\to \\text{Tor}_1^R(A, B) \\to 0", "$$", "The map of Lichtenbaum-Schlessinger complexes is given by the diagram", "$$", "\\xymatrix{", "Rel_A/TrivRel_A \\ar[r] &", "F_A \\otimes_{P_A} (A \\otimes_R B) \\ar[r] &", "\\Omega_{P_A/A \\otimes_R B} \\otimes_{P_A} (A \\otimes_R B) \\\\", "Rel/TrivRel \\ar[r] \\ar[u]_{-2} &", "F \\otimes_P B \\ar[r] \\ar[u]_{-1} &", "\\Omega_{P/A} \\otimes_P B \\ar[u]_0", "}", "$$", "Note that vertical maps $-1$ and $-0$ induce an isomorphism after applying", "the functor $A \\otimes_R - = P_A \\otimes_P -$ to the source and the vertical", "map $-2$ gives exactly the map whose cokernel is the desired Tor module", "as we saw above." ], "refs": [ "cotangent-lemma-compare-higher", "cotangent-remark-functoriality-lichtenbaum-schlessinger" ], "ref_ids": [ 11208, 11266 ] } ], "ref_ids": [] }, { "id": 11214, "type": "theorem", "label": "cotangent-lemma-find-obstruction", "categories": [ "cotangent" ], "title": "cotangent-lemma-find-obstruction", "contents": [ "In the situation above we have", "\\begin{enumerate}", "\\item There is a canonical element $\\xi \\in \\Ext^2_B(L_{B/A}, N)$", "whose vanishing is a sufficient and necessary condition for the existence", "of a solution to (\\ref{equation-to-solve}).", "\\item If there exists a solution, then the set of", "isomorphism classes of solutions is principal homogeneous under", "$\\Ext^1_B(L_{B/A}, N)$.", "\\item Given a solution $B'$, the set of automorphisms of $B'$", "fitting into (\\ref{equation-to-solve}) is canonically isomorphic", "to $\\Ext^0_B(L_{B/A}, N)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Via the identifications $\\NL_{B/A} = \\tau_{\\geq -1}L_{B/A}$", "(Lemma \\ref{lemma-relation-with-naive-cotangent-complex}) and", "$H^0(L_{B/A}) = \\Omega_{B/A}$", "(Lemma \\ref{lemma-identify-H0})", "we have seen parts (2) and (3) in", "Deformation Theory, Lemmas \\ref{defos-lemma-huge-diagram} and", "\\ref{defos-lemma-choices}.", "\\medskip\\noindent", "Proof of (1). We will use the results of", "Deformation Theory, Lemma \\ref{defos-lemma-extensions-of-rings}", "without further mention.", "Let $\\alpha \\in \\Ext^1_A(\\NL_{A/\\mathbf{Z}}, I)$", "be the element corresponding to the isomorphism class of $A'$.", "The existence of $B'$ corresponds to an element", "$\\beta \\in \\Ext_B^1(\\NL_{B/\\mathbf{Z}}, N)$", "which maps to the image of $\\alpha$ in", "$\\Ext^1_A(\\NL_{A/\\mathbf{Z}}, N)$. Note that", "$$", "\\Ext^1_A(\\NL_{A/\\mathbf{Z}}, N) =", "\\Ext^1_A(L_{A/\\mathbf{Z}}, N) =", "\\Ext^1_B(L_{A/\\mathbf{Z}} \\otimes_A^\\mathbf{L} B, N)", "$$", "and", "$$", "\\Ext^1_B(\\NL_{B/\\mathbf{Z}}, N) =", "\\Ext^1_B(L_{B/\\mathbf{Z}}, N)", "$$", "by Lemma \\ref{lemma-relation-with-naive-cotangent-complex}.", "Since the distinguished triangle (\\ref{equation-triangle})", "for $\\mathbf{Z} \\to A \\to B$ gives rise to a long exact sequence", "$$", "\\ldots \\to", "\\Ext^1_B(L_{B/\\mathbf{Z}}, N) \\to", "\\Ext^1_B(L_{A/\\mathbf{Z}} \\otimes_A^\\mathbf{L} B, N) \\to", "\\Ext^2_B(L_{B/A}, N) \\to \\ldots", "$$", "we obtain the result with $\\xi$ the image of $\\alpha$." ], "refs": [ "cotangent-lemma-relation-with-naive-cotangent-complex", "cotangent-lemma-identify-H0", "defos-lemma-huge-diagram", "defos-lemma-choices", "defos-lemma-extensions-of-rings", "cotangent-lemma-relation-with-naive-cotangent-complex" ], "ref_ids": [ 11204, 11177, 13369, 13371, 13372, 11204 ] } ], "ref_ids": [] }, { "id": 11215, "type": "theorem", "label": "cotangent-lemma-pullback-cotangent-morphism-topoi", "categories": [ "cotangent" ], "title": "cotangent-lemma-pullback-cotangent-morphism-topoi", "contents": [ "Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi.", "Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings", "on $\\mathcal{C}$. Then", "$f^{-1}L_{\\mathcal{B}/\\mathcal{A}} = L_{f^{-1}\\mathcal{B}/f^{-1}\\mathcal{A}}$." ], "refs": [], "proofs": [ { "contents": [ "The diagram", "$$", "\\xymatrix{", "\\mathcal{A}\\textit{-Alg} \\ar[d]_{f^{-1}} \\ar[r] &", "\\Sh(\\mathcal{C}) \\ar@<1ex>[l] \\ar[d]^{f^{-1}} \\\\", "f^{-1}\\mathcal{A}\\textit{-Alg} \\ar[r] & \\Sh(\\mathcal{D}) \\ar@<1ex>[l]", "}", "$$", "commutes." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11216, "type": "theorem", "label": "cotangent-lemma-compute-L-morphism-sheaves-rings", "categories": [ "cotangent" ], "title": "cotangent-lemma-compute-L-morphism-sheaves-rings", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a", "homomorphism of sheaves of rings on $\\mathcal{C}$. Then", "$H^i(L_{\\mathcal{B}/\\mathcal{A}})$ is the sheaf associated to the", "presheaf $U \\mapsto H^i(L_{\\mathcal{B}(U)/\\mathcal{A}(U)})$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{C}'$ be the site we get by endowing $\\mathcal{C}$ with the", "chaotic topology (presheaves are sheaves). There is a morphism of topoi", "$f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$ where $f_*$ is the inclusion", "of sheaves into presheaves and $f^{-1}$ is sheafification.", "By Lemma \\ref{lemma-pullback-cotangent-morphism-topoi}", "it suffices to prove the result for $\\mathcal{C}'$, i.e.,", "in case $\\mathcal{C}$ has the chaotic topology.", "\\medskip\\noindent", "If $\\mathcal{C}$ carries the chaotic topology, then", "$L_{\\mathcal{B}/\\mathcal{A}}(U)$ is equal to", "$L_{\\mathcal{B}(U)/\\mathcal{A}(U)}$ because", "$$", "\\xymatrix{", "\\mathcal{A}\\textit{-Alg} \\ar[d]_{\\text{sections over }U} \\ar[r] &", "\\Sh(\\mathcal{C}) \\ar@<1ex>[l] \\ar[d]^{\\text{sections over }U} \\\\", "\\mathcal{A}(U)\\textit{-Alg} \\ar[r] & \\textit{Sets} \\ar@<1ex>[l]", "}", "$$", "commutes." ], "refs": [ "cotangent-lemma-pullback-cotangent-morphism-topoi" ], "ref_ids": [ 11215 ] } ], "ref_ids": [] }, { "id": 11217, "type": "theorem", "label": "cotangent-lemma-H0-L-morphism-sheaves-rings", "categories": [ "cotangent" ], "title": "cotangent-lemma-H0-L-morphism-sheaves-rings", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a", "homomorphism of sheaves of rings on $\\mathcal{C}$. Then", "$H^0(L_{\\mathcal{B}/\\mathcal{A}}) = \\Omega_{\\mathcal{B}/\\mathcal{A}}$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemmas \\ref{lemma-compute-L-morphism-sheaves-rings}", "and \\ref{lemma-identify-H0} and", "Modules on Sites, Lemma \\ref{sites-modules-lemma-differentials-sheafify}." ], "refs": [ "cotangent-lemma-compute-L-morphism-sheaves-rings", "cotangent-lemma-identify-H0", "sites-modules-lemma-differentials-sheafify" ], "ref_ids": [ 11216, 11177, 14229 ] } ], "ref_ids": [] }, { "id": 11218, "type": "theorem", "label": "cotangent-lemma-compute-L-product-sheaves-rings", "categories": [ "cotangent" ], "title": "cotangent-lemma-compute-L-product-sheaves-rings", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$", "and $\\mathcal{A} \\to \\mathcal{B}'$ be homomorphisms of sheaves of rings", "on $\\mathcal{C}$. Then", "$$", "L_{\\mathcal{B} \\times \\mathcal{B}'/\\mathcal{A}}", "\\longrightarrow", "L_{\\mathcal{B}/\\mathcal{A}} \\oplus L_{\\mathcal{B}'/\\mathcal{A}}", "$$", "is an isomorphism in $D(\\mathcal{B} \\times \\mathcal{B}')$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-compute-L-morphism-sheaves-rings}", "it suffices to prove this for ring maps.", "In the case of rings this is", "Lemma \\ref{lemma-cotangent-complex-product}." ], "refs": [ "cotangent-lemma-compute-L-morphism-sheaves-rings", "cotangent-lemma-cotangent-complex-product" ], "ref_ids": [ 11216, 11188 ] } ], "ref_ids": [] }, { "id": 11219, "type": "theorem", "label": "cotangent-lemma-triangle-sheaves-rings", "categories": [ "cotangent" ], "title": "cotangent-lemma-triangle-sheaves-rings", "contents": [ "Let $\\mathcal{D}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B} \\to \\mathcal{C}$", "be homomorphisms of sheaves of rings on $\\mathcal{D}$.", "There is a canonical distinguished triangle", "$$", "L_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B}^\\mathbf{L} \\mathcal{C}", "\\to L_{\\mathcal{C}/\\mathcal{A}} \\to L_{\\mathcal{C}/\\mathcal{B}} \\to", "L_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B}^\\mathbf{L} \\mathcal{C}[1]", "$$", "in $D(\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "We will use the method described in", "Remarks \\ref{remark-triangle} and \\ref{remark-explicit-map}", "to construct the triangle; we will freely use the results mentioned there.", "As in those remarks we first construct the triangle in case", "$\\mathcal{B} \\to \\mathcal{C}$ is an injective map of sheaves of rings.", "In this case we set", "\\begin{enumerate}", "\\item $\\mathcal{P}_\\bullet$ is the standard resolution of $\\mathcal{B}$", "over $\\mathcal{A}$,", "\\item $\\mathcal{Q}_\\bullet$ is the standard resolution of $\\mathcal{C}$", "over $\\mathcal{A}$,", "\\item $\\mathcal{R}_\\bullet$ is the standard resolution of $\\mathcal{C}$", "over $\\mathcal{B}$,", "\\item $\\mathcal{S}_\\bullet$ is the standard resolution of $\\mathcal{B}$", "over $\\mathcal{B}$,", "\\item $\\overline{\\mathcal{Q}}_\\bullet =", "\\mathcal{Q}_\\bullet \\otimes_{\\mathcal{P}_\\bullet} \\mathcal{B}$, and", "\\item $\\overline{\\mathcal{R}}_\\bullet =", "\\mathcal{R}_\\bullet \\otimes_{\\mathcal{S}_\\bullet} \\mathcal{B}$.", "\\end{enumerate}", "The distinguished triangle is the distinguished triangle associated", "to the short exact sequence", "of simplicial $\\mathcal{C}$-modules", "$$", "0 \\to", "\\Omega_{\\mathcal{P}_\\bullet/\\mathcal{A}}", "\\otimes_{\\mathcal{P}_\\bullet} \\mathcal{C} \\to", "\\Omega_{\\mathcal{Q}_\\bullet/\\mathcal{A}}", "\\otimes_{\\mathcal{Q}_\\bullet} \\mathcal{C} \\to", "\\Omega_{\\overline{\\mathcal{Q}}_\\bullet/\\mathcal{B}}", "\\otimes_{\\overline{\\mathcal{Q}}_\\bullet} \\mathcal{C} \\to 0", "$$", "The first two terms are equal to the first two terms of the triangle", "of the statement of the lemma. The identification of the last term with", "$L_{\\mathcal{C}/\\mathcal{B}}$ uses the quasi-isomorphisms of complexes", "$$", "L_{\\mathcal{C}/\\mathcal{B}} =", "\\Omega_{\\mathcal{R}_\\bullet/\\mathcal{B}}", "\\otimes_{\\mathcal{R}_\\bullet} \\mathcal{C}", "\\longrightarrow", "\\Omega_{\\overline{\\mathcal{R}}_\\bullet/\\mathcal{B}}", "\\otimes_{\\overline{\\mathcal{R}}_\\bullet} \\mathcal{C}", "\\longleftarrow", "\\Omega_{\\overline{\\mathcal{Q}}_\\bullet/\\mathcal{B}}", "\\otimes_{\\overline{\\mathcal{Q}}_\\bullet} \\mathcal{C}", "$$", "All the constructions used above can first be done on the level", "of presheaves and then sheafified. Hence to prove sequences are exact,", "or that map are quasi-isomorphisms it suffices to prove the corresponding", "statement for the ring maps", "$\\mathcal{A}(U) \\to \\mathcal{B}(U) \\to \\mathcal{C}(U)$", "which are known. This finishes the proof in the case that", "$\\mathcal{B} \\to \\mathcal{C}$ is injective.", "\\medskip\\noindent", "In general, we reduce to the case where $\\mathcal{B} \\to \\mathcal{C}$ is", "injective by replacing $\\mathcal{C}$ by $\\mathcal{B} \\times \\mathcal{C}$ if", "necessary. This is possible by the argument given in", "Remark \\ref{remark-triangle} by", "Lemma \\ref{lemma-compute-L-product-sheaves-rings}." ], "refs": [ "cotangent-remark-triangle", "cotangent-remark-explicit-map", "cotangent-remark-triangle", "cotangent-lemma-compute-L-product-sheaves-rings" ], "ref_ids": [ 11258, 11259, 11258, 11218 ] } ], "ref_ids": [] }, { "id": 11220, "type": "theorem", "label": "cotangent-lemma-stalk-cotangent-complex", "categories": [ "cotangent" ], "title": "cotangent-lemma-stalk-cotangent-complex", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a", "homomorphism of sheaves of rings on $\\mathcal{C}$. If $p$ is a point", "of $\\mathcal{C}$, then", "$(L_{\\mathcal{B}/\\mathcal{A}})_p = L_{\\mathcal{B}_p/\\mathcal{A}_p}$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-pullback-cotangent-morphism-topoi}." ], "refs": [ "cotangent-lemma-pullback-cotangent-morphism-topoi" ], "ref_ids": [ 11215 ] } ], "ref_ids": [] }, { "id": 11221, "type": "theorem", "label": "cotangent-lemma-compare-cotangent-complex-with-naive", "categories": [ "cotangent" ], "title": "cotangent-lemma-compare-cotangent-complex-with-naive", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a", "homomorphism of sheaves of rings on $\\mathcal{C}$.", "There is a canonical map", "$L_{\\mathcal{B}/\\mathcal{A}} \\to \\NL_{\\mathcal{B}/\\mathcal{A}}$", "which identifies the naive cotangent complex with the truncation", "$\\tau_{\\geq -1}L_{\\mathcal{B}/\\mathcal{A}}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{P}_\\bullet$ be the standard resolution of $\\mathcal{B}$", "over $\\mathcal{A}$.", "Let $\\mathcal{I} = \\Ker(\\mathcal{A}[\\mathcal{B}] \\to \\mathcal{B})$.", "Recall that $\\mathcal{P}_0 = \\mathcal{A}[\\mathcal{B}]$. The map of the", "lemma is given by the commutative diagram", "$$", "\\xymatrix{", "L_{\\mathcal{B}/\\mathcal{A}} \\ar[d] & \\ldots \\ar[r] &", "\\Omega_{\\mathcal{P}_2/\\mathcal{A}} \\otimes_{\\mathcal{P}_2} \\mathcal{B}", "\\ar[r] \\ar[d] &", "\\Omega_{\\mathcal{P}_1/\\mathcal{A}} \\otimes_{\\mathcal{P}_1} \\mathcal{B}", "\\ar[r] \\ar[d] &", "\\Omega_{\\mathcal{P}_0/\\mathcal{A}} \\otimes_{\\mathcal{P}_0} \\mathcal{B}", "\\ar[d] \\\\", "\\NL_{\\mathcal{B}/\\mathcal{A}} & \\ldots \\ar[r] &", "0 \\ar[r] & ", "\\mathcal{I}/\\mathcal{I}^2 \\ar[r] &", "\\Omega_{\\mathcal{P}_0/\\mathcal{A}} \\otimes_{\\mathcal{P}_0} \\mathcal{B}", "}", "$$", "We construct the downward arrow with target $\\mathcal{I}/\\mathcal{I}^2$", "by sending a local section $\\text{d}f \\otimes b$ to the class of", "$(d_0(f) - d_1(f))b$ in $\\mathcal{I}/\\mathcal{I}^2$.", "Here $d_i : \\mathcal{P}_1 \\to \\mathcal{P}_0$,", "$i = 0, 1$ are the two face maps of the simplicial structure.", "This makes sense as $d_0 - d_1$ maps $\\mathcal{P}_1$ into", "$\\mathcal{I} = \\Ker(\\mathcal{P}_0 \\to \\mathcal{B})$.", "We omit the verification that this rule is well defined.", "Our map is compatible with the differential", "$\\Omega_{\\mathcal{P}_1/\\mathcal{A}} \\otimes_{\\mathcal{P}_1} \\mathcal{B}", "\\to \\Omega_{\\mathcal{P}_0/\\mathcal{A}} \\otimes_{\\mathcal{P}_0} \\mathcal{B}$", "as this differential maps a local section $\\text{d}f \\otimes b$ to", "$\\text{d}(d_0(f) - d_1(f)) \\otimes b$. Moreover, the differential", "$\\Omega_{\\mathcal{P}_2/\\mathcal{A}} \\otimes_{\\mathcal{P}_2} \\mathcal{B}", "\\to \\Omega_{\\mathcal{P}_1/\\mathcal{A}} \\otimes_{\\mathcal{P}_1} \\mathcal{B}$", "maps a local section $\\text{d}f \\otimes b$ to", "$\\text{d}(d_0(f) - d_1(f) + d_2(f)) \\otimes b$", "which are annihilated by our downward arrow. Hence a map of complexes.", "\\medskip\\noindent", "To see that our map induces an isomorphism on the cohomology sheaves", "$H^0$ and $H^{-1}$ we argue as follows. Let $\\mathcal{C}'$ be the site", "with the same underlying category as $\\mathcal{C}$ but endowed with the", "chaotic topology. Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$ be", "the morphism of topoi whose pullback functor is sheafification.", "Let $\\mathcal{A}' \\to \\mathcal{B}'$ be the given map, but thought of", "as a map of sheaves of rings on $\\mathcal{C}'$. The construction above", "gives a map $L_{\\mathcal{B}'/\\mathcal{A}'} \\to \\NL_{\\mathcal{B}'/\\mathcal{A}'}$", "on $\\mathcal{C}'$ whose value over any object $U$ of $\\mathcal{C}'$", "is just the map", "$$", "L_{\\mathcal{B}(U)/\\mathcal{A}(U)} \\to \\NL_{\\mathcal{B}(U)/\\mathcal{A}(U)}", "$$", "of Remark \\ref{remark-explicit-comparison-map} which induces an isomorphism", "on $H^0$ and $H^{-1}$. Since", "$f^{-1}L_{\\mathcal{B}'/\\mathcal{A}'} = L_{\\mathcal{B}/\\mathcal{A}}$", "(Lemma \\ref{lemma-pullback-cotangent-morphism-topoi})", "and", "$f^{-1}\\NL_{\\mathcal{B}'/\\mathcal{A}'} = \\NL_{\\mathcal{B}/\\mathcal{A}}$", "(Modules on Sites, Lemma \\ref{sites-modules-lemma-pullback-NL})", "the lemma is proved." ], "refs": [ "cotangent-remark-explicit-comparison-map", "cotangent-lemma-pullback-cotangent-morphism-topoi", "sites-modules-lemma-pullback-NL" ], "ref_ids": [ 11261, 11215, 14241 ] } ], "ref_ids": [] }, { "id": 11222, "type": "theorem", "label": "cotangent-lemma-H0-L-morphism-ringed-spaces", "categories": [ "cotangent" ], "title": "cotangent-lemma-H0-L-morphism-ringed-spaces", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$ be a morphism of", "ringed spaces. Then $H^0(L_{X/S}) = \\Omega_{X/S}$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-H0-L-morphism-sheaves-rings}." ], "refs": [ "cotangent-lemma-H0-L-morphism-sheaves-rings" ], "ref_ids": [ 11217 ] } ], "ref_ids": [] }, { "id": 11223, "type": "theorem", "label": "cotangent-lemma-triangle-ringed-spaces", "categories": [ "cotangent" ], "title": "cotangent-lemma-triangle-ringed-spaces", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces.", "Then there is a canonical distinguished triangle", "$$", "Lf^* L_{Y/Z} \\to L_{X/Z} \\to L_{X/Y} \\to Lf^*L_{Y/Z}[1]", "$$", "in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Set $h = g \\circ f$ so that $h^{-1}\\mathcal{O}_Z = f^{-1}g^{-1}\\mathcal{O}_Z$.", "By Lemma \\ref{lemma-pullback-cotangent-morphism-topoi} we have", "$f^{-1}L_{Y/Z} = L_{f^{-1}\\mathcal{O}_Y/h^{-1}\\mathcal{O}_Z}$", "and this is a complex of flat $f^{-1}\\mathcal{O}_Y$-modules.", "Hence the distinguished triangle above is an example of the", "distinguished triangle of", "Lemma \\ref{lemma-triangle-sheaves-rings}", "with $\\mathcal{A} = h^{-1}\\mathcal{O}_Z$, $\\mathcal{B} = f^{-1}\\mathcal{O}_Y$,", "and $\\mathcal{C} = \\mathcal{O}_X$." ], "refs": [ "cotangent-lemma-pullback-cotangent-morphism-topoi", "cotangent-lemma-triangle-sheaves-rings" ], "ref_ids": [ 11215, 11219 ] } ], "ref_ids": [] }, { "id": 11224, "type": "theorem", "label": "cotangent-lemma-compare-cotangent-complex-with-naive-ringed-spaces", "categories": [ "cotangent" ], "title": "cotangent-lemma-compare-cotangent-complex-with-naive-ringed-spaces", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of", "ringed spaces. There is a canonical map $L_{X/Y} \\to \\NL_{X/Y}$ which", "identifies the naive cotangent complex with the truncation", "$\\tau_{\\geq -1}L_{X/Y}$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-compare-cotangent-complex-with-naive}." ], "refs": [ "cotangent-lemma-compare-cotangent-complex-with-naive" ], "ref_ids": [ 11221 ] } ], "ref_ids": [] }, { "id": 11225, "type": "theorem", "label": "cotangent-lemma-find-obstruction-ringed-spaces", "categories": [ "cotangent" ], "title": "cotangent-lemma-find-obstruction-ringed-spaces", "contents": [ "In the situation above we have", "\\begin{enumerate}", "\\item There is a canonical element", "$\\xi \\in \\Ext^2_{\\mathcal{O}_X}(L_{X/S}, \\mathcal{G})$", "whose vanishing is a sufficient and necessary condition for the existence", "of a solution to (\\ref{equation-to-solve-ringed-spaces}).", "\\item If there exists a solution, then the set of", "isomorphism classes of solutions is principal homogeneous under", "$\\Ext^1_{\\mathcal{O}_X}(L_{X/S}, \\mathcal{G})$.", "\\item Given a solution $X'$, the set of automorphisms of $X'$", "fitting into (\\ref{equation-to-solve-ringed-spaces}) is canonically isomorphic", "to $\\Ext^0_{\\mathcal{O}_X}(L_{X/S}, \\mathcal{G})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Via the identifications $\\NL_{X/S} = \\tau_{\\geq -1}L_{X/S}$", "(Lemma \\ref{lemma-compare-cotangent-complex-with-naive-ringed-spaces})", "and", "$H^0(L_{X/S}) = \\Omega_{X/S}$", "(Lemma \\ref{lemma-H0-L-morphism-ringed-spaces})", "we have seen parts (2) and (3) in", "Deformation Theory, Lemmas \\ref{defos-lemma-huge-diagram-ringed-spaces} and", "\\ref{defos-lemma-choices-ringed-spaces}.", "\\medskip\\noindent", "Proof of (1). We will use the results of", "Deformation Theory, Lemma \\ref{defos-lemma-extensions-of-ringed-spaces}", "without further mention.", "Let $\\alpha \\in \\Ext^1_{\\mathcal{O}_S}(\\NL_{S/\\mathbf{Z}}, \\mathcal{J})$", "be the element corresponding to the isomorphism class of $S'$.", "The existence of $X'$ corresponds to an element", "$\\beta \\in \\Ext_{\\mathcal{O}_X}^1(\\NL_{X/\\mathbf{Z}}, \\mathcal{G})$", "which maps to the image of $\\alpha$ in", "$\\Ext^1_{\\mathcal{O}_X}(Lf^*\\NL_{S/\\mathbf{Z}}, \\mathcal{G})$.", "Note that", "$$", "\\Ext^1_{\\mathcal{O}_X}(Lf^*\\NL_{S/\\mathbf{Z}}, \\mathcal{G}) =", "\\Ext^1_{\\mathcal{O}_X}(Lf^*L_{S/\\mathbf{Z}}, \\mathcal{G})", "$$", "and", "$$", "\\Ext^1_{\\mathcal{O}_X}(\\NL_{X/\\mathbf{Z}}, \\mathcal{G}) =", "\\Ext^1_{\\mathcal{O}_X}(L_{X/\\mathbf{Z}}, \\mathcal{G})", "$$", "by Lemma \\ref{lemma-compare-cotangent-complex-with-naive-ringed-spaces}.", "The distinguished triangle of Lemma \\ref{lemma-triangle-ringed-spaces}", "for $X \\to S \\to (*, \\mathbf{Z})$ gives rise to a long exact sequence", "$$", "\\ldots \\to", "\\Ext^1_{\\mathcal{O}_X}(L_{X/\\mathbf{Z}}, \\mathcal{G}) \\to", "\\Ext^1_{\\mathcal{O}_X}(Lf^*L_{S/\\mathbf{Z}}, \\mathcal{G}) \\to", "\\Ext^2_{\\mathcal{O}_X}(L_{X/S}, \\mathcal{G}) \\to \\ldots", "$$", "We obtain the result with $\\xi$ the image of $\\alpha$." ], "refs": [ "cotangent-lemma-compare-cotangent-complex-with-naive-ringed-spaces", "cotangent-lemma-H0-L-morphism-ringed-spaces", "defos-lemma-huge-diagram-ringed-spaces", "defos-lemma-choices-ringed-spaces", "defos-lemma-extensions-of-ringed-spaces", "cotangent-lemma-compare-cotangent-complex-with-naive-ringed-spaces", "cotangent-lemma-triangle-ringed-spaces" ], "ref_ids": [ 11224, 11222, 13387, 13389, 13390, 11224, 11223 ] } ], "ref_ids": [] }, { "id": 11226, "type": "theorem", "label": "cotangent-lemma-H0-L-morphism-ringed-topoi", "categories": [ "cotangent" ], "title": "cotangent-lemma-H0-L-morphism-ringed-topoi", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to", "(\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$ be a morphism of", "ringed topoi. Then $H^0(L_f) = \\Omega_f$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-H0-L-morphism-sheaves-rings}." ], "refs": [ "cotangent-lemma-H0-L-morphism-sheaves-rings" ], "ref_ids": [ 11217 ] } ], "ref_ids": [] }, { "id": 11227, "type": "theorem", "label": "cotangent-lemma-triangle-ringed-topoi", "categories": [ "cotangent" ], "title": "cotangent-lemma-triangle-ringed-topoi", "contents": [ "Let $f : (\\Sh(\\mathcal{C}_1), \\mathcal{O}_1) \\to", "(\\Sh(\\mathcal{C}_2), \\mathcal{O}_2)$ and", "$g : (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\to", "(\\Sh(\\mathcal{C}_3), \\mathcal{O}_3)$ be morphisms of ringed topoi.", "Then there is a canonical distinguished triangle", "$$", "Lf^* L_g \\to L_{g \\circ f} \\to L_f \\to Lf^*L_g[1]", "$$", "in $D(\\mathcal{O}_1)$." ], "refs": [], "proofs": [ { "contents": [ "Set $h = g \\circ f$ so that $h^{-1}\\mathcal{O}_3 = f^{-1}g^{-1}\\mathcal{O}_3$.", "By Lemma \\ref{lemma-pullback-cotangent-morphism-topoi} we have", "$f^{-1}L_g = L_{f^{-1}\\mathcal{O}_2/h^{-1}\\mathcal{O}_3}$", "and this is a complex of flat $f^{-1}\\mathcal{O}_2$-modules.", "Hence the distinguished triangle above is an example of the", "distinguished triangle of", "Lemma \\ref{lemma-triangle-sheaves-rings}", "with $\\mathcal{A} = h^{-1}\\mathcal{O}_3$, $\\mathcal{B} = f^{-1}\\mathcal{O}_2$,", "and $\\mathcal{C} = \\mathcal{O}_1$." ], "refs": [ "cotangent-lemma-pullback-cotangent-morphism-topoi", "cotangent-lemma-triangle-sheaves-rings" ], "ref_ids": [ 11215, 11219 ] } ], "ref_ids": [] }, { "id": 11228, "type": "theorem", "label": "cotangent-lemma-compare-cotangent-complex-with-naive-ringed-topoi", "categories": [ "cotangent" ], "title": "cotangent-lemma-compare-cotangent-complex-with-naive-ringed-topoi", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to", "(\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$ be a morphism of", "ringed topoi. There is a canonical map $L_f \\to \\NL_f$ which", "identifies the naive cotangent complex with the truncation", "$\\tau_{\\geq -1}L_f$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Lemma \\ref{lemma-compare-cotangent-complex-with-naive}." ], "refs": [ "cotangent-lemma-compare-cotangent-complex-with-naive" ], "ref_ids": [ 11221 ] } ], "ref_ids": [] }, { "id": 11229, "type": "theorem", "label": "cotangent-lemma-find-obstruction-ringed-topoi", "categories": [ "cotangent" ], "title": "cotangent-lemma-find-obstruction-ringed-topoi", "contents": [ "In the situation above we have", "\\begin{enumerate}", "\\item There is a canonical element", "$\\xi \\in \\Ext^2_\\mathcal{O}(L_f, \\mathcal{G})$", "whose vanishing is a sufficient and necessary condition for the existence", "of a solution to (\\ref{equation-to-solve-ringed-topoi}).", "\\item If there exists a solution, then the set of", "isomorphism classes of solutions is principal homogeneous under", "$\\Ext^1_\\mathcal{O}(L_f, \\mathcal{G})$.", "\\item Given a solution $X'$, the set of automorphisms of $X'$", "fitting into (\\ref{equation-to-solve-ringed-topoi}) is canonically isomorphic", "to $\\Ext^0_\\mathcal{O}(L_f, \\mathcal{G})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Via the identifications $\\NL_f = \\tau_{\\geq -1}L_f$", "(Lemma \\ref{lemma-compare-cotangent-complex-with-naive-ringed-topoi}) and", "$H^0(L_{X/S}) = \\Omega_{X/S}$", "(Lemma \\ref{lemma-H0-L-morphism-ringed-topoi})", "we have seen parts (2) and (3) in", "Deformation Theory, Lemmas \\ref{defos-lemma-huge-diagram-ringed-topoi} and", "\\ref{defos-lemma-choices-ringed-topoi}.", "\\medskip\\noindent", "Proof of (1). We will use the results of", "Deformation Theory, Lemma \\ref{defos-lemma-extensions-of-ringed-topoi}", "without further mention. Denote", "$$", "p : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(*), \\mathbf{Z})", "\\quad\\text{and}\\quad", "q : (\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B}) \\to (\\Sh(*), \\mathbf{Z}).", "$$", "Let $\\alpha \\in \\Ext^1_{\\mathcal{O}_\\mathcal{B}}(\\NL_q, \\mathcal{J})$", "be the element corresponding to the isomorphism class of", "$\\mathcal{O}_{\\mathcal{B}'}$. The existence of $\\mathcal{O}'$", "corresponds to an element", "$\\beta \\in \\Ext_\\mathcal{O}^1(\\NL_p, \\mathcal{G})$", "which maps to the image of $\\alpha$ in", "$\\Ext^1_{\\mathcal{O}_X}(Lf^*\\NL_q, \\mathcal{G})$.", "Note that", "$$", "\\Ext^1_{\\mathcal{O}_X}(Lf^*\\NL_q, \\mathcal{G}) =", "\\Ext^1_{\\mathcal{O}_X}(Lf^*L_q, \\mathcal{G})", "$$", "and", "$$", "\\Ext^1_{\\mathcal{O}_X}(\\NL_p, \\mathcal{G}) =", "\\Ext^1_{\\mathcal{O}_X}(L_p, \\mathcal{G})", "$$", "by Lemma \\ref{lemma-compare-cotangent-complex-with-naive-ringed-topoi}.", "The distinguished triangle of Lemma \\ref{lemma-triangle-ringed-topoi}", "for $p = q \\circ f$ gives rise to a long exact sequence", "$$", "\\ldots \\to", "\\Ext^1_{\\mathcal{O}_X}(L_p, \\mathcal{G}) \\to", "\\Ext^1_{\\mathcal{O}_X}(Lf^*L_q, \\mathcal{G}) \\to", "\\Ext^2_{\\mathcal{O}_X}(L_f, \\mathcal{G}) \\to \\ldots", "$$", "We obtain the result with $\\xi$ the image of $\\alpha$." ], "refs": [ "cotangent-lemma-compare-cotangent-complex-with-naive-ringed-topoi", "cotangent-lemma-H0-L-morphism-ringed-topoi", "defos-lemma-huge-diagram-ringed-topoi", "defos-lemma-choices-ringed-topoi", "defos-lemma-extensions-of-ringed-topoi", "cotangent-lemma-compare-cotangent-complex-with-naive-ringed-topoi", "cotangent-lemma-triangle-ringed-topoi" ], "ref_ids": [ 11228, 11226, 13407, 13409, 13410, 11228, 11227 ] } ], "ref_ids": [] }, { "id": 11230, "type": "theorem", "label": "cotangent-lemma-morphism-affine-schemes", "categories": [ "cotangent" ], "title": "cotangent-lemma-morphism-affine-schemes", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $U = \\Spec(A) \\subset X$", "and $V = \\Spec(B) \\subset Y$ be affine opens such that $f(U) \\subset V$.", "There is a canonical map", "$$", "\\widetilde{L_{B/A}} \\longrightarrow L_{X/Y}|_U", "$$", "of complexes which is an isomorphism in $D(\\mathcal{O}_U)$.", "This map is compatible with restricting to smaller affine opens", "of $X$ and $Y$." ], "refs": [], "proofs": [ { "contents": [ "By Remark \\ref{remark-map-sections-over-U}", "there is a canonical map of complexes", "$L_{\\mathcal{O}_X(U)/f^{-1}\\mathcal{O}_Y(U)} \\to L_{X/Y}(U)$", "of $B = \\mathcal{O}_X(U)$-modules, which is compatible", "with further restrictions. Using the canonical map", "$A \\to f^{-1}\\mathcal{O}_Y(U)$ we obtain a canonical map", "$L_{B/A} \\to L_{\\mathcal{O}_X(U)/f^{-1}\\mathcal{O}_Y(U)}$", "of complexes of $B$-modules.", "Using the universal property of the $\\widetilde{\\ }$", "functor (see Schemes, Lemma \\ref{schemes-lemma-compare-constructions})", "we obtain a map as in the statement of the lemma.", "We may check this map is an isomorphism on cohomology sheaves", "by checking it induces isomorphisms on stalks.", "This follows immediately from", "Lemmas \\ref{lemma-stalk-cotangent-complex} and \\ref{lemma-localize}", "(and the description of the stalks of", "$\\mathcal{O}_X$ and $f^{-1}\\mathcal{O}_Y$", "at a point $\\mathfrak p \\in \\Spec(B)$ as $B_\\mathfrak p$ and", "$A_\\mathfrak q$ where $\\mathfrak q = A \\cap \\mathfrak p$; references", "used are Schemes, Lemma \\ref{schemes-lemma-spec-sheaves}", "and", "Sheaves, Lemma \\ref{sheaves-lemma-stalk-pullback})." ], "refs": [ "cotangent-remark-map-sections-over-U", "schemes-lemma-compare-constructions", "cotangent-lemma-stalk-cotangent-complex", "cotangent-lemma-localize", "schemes-lemma-spec-sheaves", "sheaves-lemma-stalk-pullback" ], "ref_ids": [ 11267, 7660, 11220, 11197, 7651, 14507 ] } ], "ref_ids": [] }, { "id": 11231, "type": "theorem", "label": "cotangent-lemma-scheme-over-ring", "categories": [ "cotangent" ], "title": "cotangent-lemma-scheme-over-ring", "contents": [ "Let $\\Lambda$ be a ring. Let $X$ be a scheme over $\\Lambda$.", "Then", "$$", "L_{X/\\Spec(\\Lambda)} = L_{\\mathcal{O}_X/\\underline{\\Lambda}}", "$$", "where $\\underline{\\Lambda}$ is the constant sheaf with value", "$\\Lambda$ on $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $p : X \\to \\Spec(\\Lambda)$ be the structure morphism.", "Let $q : \\Spec(\\Lambda) \\to (*, \\Lambda)$ be the obvious morphism.", "By the distinguished triangle of Lemma \\ref{lemma-triangle-ringed-spaces}", "it suffices to show that $L_q = 0$. To see this it suffices to", "show for $\\mathfrak p \\in \\Spec(\\Lambda)$ that", "$$", "(L_q)_\\mathfrak p =", "L_{\\mathcal{O}_{\\Spec(\\Lambda), \\mathfrak p}/\\Lambda} =", "L_{\\Lambda_\\mathfrak p/\\Lambda}", "$$", "(Lemma \\ref{lemma-stalk-cotangent-complex})", "is zero which follows from Lemma \\ref{lemma-when-zero}." ], "refs": [ "cotangent-lemma-triangle-ringed-spaces", "cotangent-lemma-stalk-cotangent-complex", "cotangent-lemma-when-zero" ], "ref_ids": [ 11223, 11220, 11195 ] } ], "ref_ids": [] }, { "id": 11232, "type": "theorem", "label": "cotangent-lemma-category-fibred", "categories": [ "cotangent" ], "title": "cotangent-lemma-category-fibred", "contents": [ "In the situation above the category", "$\\mathcal{C}_{X/\\Lambda}$ is fibred over $X_{Zar}$." ], "refs": [], "proofs": [ { "contents": [ "Given an object $U \\to \\mathbf{A}$ of $\\mathcal{C}_{X/\\Lambda}$ and a morphism", "$U' \\to U$ of $X_{Zar}$ consider the object $U' \\to \\mathbf{A}$ of", "$\\mathcal{C}_{X/\\Lambda}$ where $U' \\to \\mathbf{A}$ is the composition of", "$U \\to \\mathbf{A}$ and $U' \\to U$. The morphism", "$(U' \\to \\mathbf{A}) \\to (U \\to \\mathbf{A})$ of", "$\\mathcal{C}_{X/\\Lambda}$ is strongly cartesian over $X_{Zar}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11233, "type": "theorem", "label": "cotangent-lemma-cotangent-morphism-schemes", "categories": [ "cotangent" ], "title": "cotangent-lemma-cotangent-morphism-schemes", "contents": [ "In the situation above there is a canonical isomorphism", "$$", "L_{X/\\Lambda} = ", "L\\pi_!(Li^*\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}) =", "L\\pi_!(i^*\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}) =", "L\\pi_!(\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}", "\\otimes_\\mathcal{O} \\underline{\\mathcal{O}}_X)", "$$", "in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "We first observe that for any object $(U \\to \\mathbf{A})$ of", "$\\mathcal{C}_{X/\\Lambda}$", "the value of the sheaf $\\mathcal{O}$ is a polynomial algebra over $\\Lambda$.", "Hence $\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}$ is a flat $\\mathcal{O}$-module", "and we conclude the second and third equalities of the statement of the", "lemma hold.", "\\medskip\\noindent", "By Remark \\ref{remark-compute-L-pi-shriek} the object", "$L\\pi_!(\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}", "\\otimes_\\mathcal{O} \\underline{\\mathcal{O}}_X)$", "is computed as the sheafification of the complex of presheaves", "$$", "U \\mapsto", "\\left(\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}", "\\otimes_\\mathcal{O} \\underline{\\mathcal{O}}_X\\right)(\\mathbf{A}_{\\bullet, U})", "=", "\\Omega_{P_{\\bullet, U}/\\Lambda} \\otimes_{P_{\\bullet, U}} \\mathcal{O}_X(U) =", "L_{\\mathcal{O}_X(U)/\\Lambda}", "$$", "using notation as in Remark \\ref{remark-compute-L-pi-shriek}.", "Now Remark \\ref{remark-map-sections-over-U} shows that", "$L\\pi_!(\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}", "\\otimes_\\mathcal{O} \\underline{\\mathcal{O}}_X)$", "computes the cotangent complex of the map of rings", "$\\underline{\\Lambda} \\to \\mathcal{O}_X$ on $X$.", "This is what we want by Lemma \\ref{lemma-scheme-over-ring}." ], "refs": [ "cotangent-remark-compute-L-pi-shriek", "cotangent-remark-compute-L-pi-shriek", "cotangent-remark-map-sections-over-U", "cotangent-lemma-scheme-over-ring" ], "ref_ids": [ 11268, 11268, 11267, 11231 ] } ], "ref_ids": [] }, { "id": 11234, "type": "theorem", "label": "cotangent-lemma-etale-localization", "categories": [ "cotangent" ], "title": "cotangent-lemma-etale-localization", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_p \\ar[r]_g & V \\ar[d]^q \\\\", "X \\ar[r]^f & Y", "}", "$$", "of algebraic spaces over $S$ with $p$ and $q$ \\'etale.", "Then there is a canonical identification", "$L_{X/Y}|_{U_\\etale} = L_{U/V}$ in $D(\\mathcal{O}_U)$." ], "refs": [], "proofs": [ { "contents": [ "Formation of the cotangent complex commutes with pullback", "(Lemma \\ref{lemma-pullback-cotangent-morphism-topoi}) and", "we have $p_{small}^{-1}\\mathcal{O}_X = \\mathcal{O}_U$ and", "$g_{small}^{-1}\\mathcal{O}_{V_\\etale} =", "p_{small}^{-1}f_{small}^{-1}\\mathcal{O}_{Y_\\etale}$", "because $q_{small}^{-1}\\mathcal{O}_{Y_\\etale} =", "\\mathcal{O}_{V_\\etale}$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-etale-exact-pullback}).", "Tracing through the definitions we conclude that", "$L_{X/Y}|_{U_\\etale} = L_{U/V}$." ], "refs": [ "cotangent-lemma-pullback-cotangent-morphism-topoi", "spaces-properties-lemma-etale-exact-pullback" ], "ref_ids": [ 11215, 11897 ] } ], "ref_ids": [] }, { "id": 11235, "type": "theorem", "label": "cotangent-lemma-compare-spaces-schemes", "categories": [ "cotangent" ], "title": "cotangent-lemma-compare-spaces-schemes", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $X$ and $Y$ representable by schemes $X_0$ and $Y_0$.", "Then there is a canonical identification", "$L_{X/Y} = \\epsilon^*L_{X_0/Y_0}$ in $D(\\mathcal{O}_X)$", "where $\\epsilon$ is as in Derived Categories of Spaces, Section", "\\ref{spaces-perfect-section-derived-quasi-coherent-etale}", "and $L_{X_0/Y_0}$ is as in", " Definition \\ref{definition-cotangent-morphism-schemes}." ], "refs": [ "cotangent-definition-cotangent-morphism-schemes" ], "proofs": [ { "contents": [ "Let $f_0 : X_0 \\to Y_0$ be the morphism of schemes corresponding to $f$.", "There is a canonical map", "$\\epsilon^{-1}f_0^{-1}\\mathcal{O}_{Y_0} \\to f_{small}^{-1}\\mathcal{O}_Y$", "compatible with", "$\\epsilon^\\sharp : \\epsilon^{-1}\\mathcal{O}_{X_0} \\to \\mathcal{O}_X$", "because there is a commutative diagram", "$$", "\\xymatrix{", "X_{0, Zar} \\ar[d]_{f_0} & X_\\etale \\ar[l]^\\epsilon \\ar[d]^f \\\\", "Y_{0, Zar} & Y_\\etale \\ar[l]_\\epsilon", "}", "$$", "see Derived Categories of Spaces, Remark", "\\ref{spaces-perfect-remark-match-total-direct-images}.", "Thus we obtain a canonical map", "$$", "\\epsilon^{-1}L_{X_0/Y_0} =", "\\epsilon^{-1}L_{\\mathcal{O}_{X_0}/f_0^{-1}\\mathcal{O}_{Y_0}} =", "L_{\\epsilon^{-1}\\mathcal{O}_{X_0}/\\epsilon^{-1}f_0^{-1}\\mathcal{O}_{Y_0}}", "\\longrightarrow", "L_{\\mathcal{O}_X/f^{-1}_{small}\\mathcal{O}_Y} = L_{X/Y}", "$$", "by the functoriality discussed in Section \\ref{section-cotangent-complex}", "and Lemma \\ref{lemma-pullback-cotangent-morphism-topoi}.", "To see that the induced map $\\epsilon^*L_{X_0/Y_0} \\to L_{X/Y}$ is an", "isomorphism we may check on stalks at geometric points", "(Properties of Spaces, Theorem", "\\ref{spaces-properties-theorem-exactness-stalks}).", "We will use Lemma \\ref{lemma-stalk-cotangent-complex}", "to compute the stalks. Let $\\overline{x} : \\Spec(k) \\to X_0$", "be a geometric point lying over $x \\in X_0$, with", "$\\overline{y} = f \\circ \\overline{x}$ lying over $y \\in Y_0$. Then", "$$", "L_{X/Y, \\overline{x}} =", "L_{\\mathcal{O}_{X, \\overline{x}}/\\mathcal{O}_{Y, \\overline{y}}}", "$$", "and", "$$", "(\\epsilon^*L_{X_0/Y_0})_{\\overline{x}} =", "L_{X_0/Y_0, x} \\otimes_{\\mathcal{O}_{X_0, x}}", "\\mathcal{O}_{X, \\overline{x}} =", "L_{\\mathcal{O}_{X_0, x}/\\mathcal{O}_{Y_0, y}}", "\\otimes_{\\mathcal{O}_{X_0, x}} \\mathcal{O}_{X, \\overline{x}}", "$$", "Some details omitted (hint: use that the stalk of a pullback", "is the stalk at the image point, see", "Sites, Lemma \\ref{sites-lemma-point-morphism-sites},", "as well as the corresponding result for modules, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-pullback-stalk}).", "Observe that $\\mathcal{O}_{X, \\overline{x}}$ is the strict", "henselization of $\\mathcal{O}_{X_0, x}$ and similarly", "for $\\mathcal{O}_{Y, \\overline{y}}$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring}).", "Thus the result follows from", "Lemma \\ref{lemma-cotangent-complex-henselization}." ], "refs": [ "spaces-perfect-remark-match-total-direct-images", "cotangent-lemma-pullback-cotangent-morphism-topoi", "spaces-properties-theorem-exactness-stalks", "cotangent-lemma-stalk-cotangent-complex", "sites-lemma-point-morphism-sites", "sites-modules-lemma-pullback-stalk", "spaces-properties-lemma-describe-etale-local-ring", "cotangent-lemma-cotangent-complex-henselization" ], "ref_ids": [ 2768, 11215, 11813, 11220, 8603, 14245, 11884, 11198 ] } ], "ref_ids": [ 11253 ] }, { "id": 11236, "type": "theorem", "label": "cotangent-lemma-space-over-ring", "categories": [ "cotangent" ], "title": "cotangent-lemma-space-over-ring", "contents": [ "Let $\\Lambda$ be a ring. Let $X$ be an algebraic space over $\\Lambda$.", "Then", "$$", "L_{X/\\Spec(\\Lambda)} = L_{\\mathcal{O}_X/\\underline{\\Lambda}}", "$$", "where $\\underline{\\Lambda}$ is the constant sheaf with value", "$\\Lambda$ on $X_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "Let $p : X \\to \\Spec(\\Lambda)$ be the structure morphism.", "Let $q : \\Spec(\\Lambda)_\\etale \\to (*, \\Lambda)$", "be the obvious morphism. By the distinguished triangle of", "Lemma \\ref{lemma-triangle-ringed-topoi}", "it suffices to show that $L_q = 0$. To see this it suffices to", "show", "(Properties of Spaces, Theorem", "\\ref{spaces-properties-theorem-exactness-stalks})", "for a geometric point $\\overline{t} : \\Spec(k) \\to \\Spec(\\Lambda)$ that", "$$", "(L_q)_{\\overline{t}} =", "L_{\\mathcal{O}_{\\Spec(\\Lambda)_\\etale, \\overline{t}}/\\Lambda}", "$$", "(Lemma \\ref{lemma-stalk-cotangent-complex})", "is zero. Since $\\mathcal{O}_{\\Spec(\\Lambda)_\\etale, \\overline{t}}$", "is a strict henselization of a local ring of $\\Lambda$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring})", "this follows from Lemma \\ref{lemma-when-zero}." ], "refs": [ "cotangent-lemma-triangle-ringed-topoi", "spaces-properties-theorem-exactness-stalks", "cotangent-lemma-stalk-cotangent-complex", "spaces-properties-lemma-describe-etale-local-ring", "cotangent-lemma-when-zero" ], "ref_ids": [ 11227, 11813, 11220, 11884, 11195 ] } ], "ref_ids": [] }, { "id": 11237, "type": "theorem", "label": "cotangent-lemma-category-fibred-space", "categories": [ "cotangent" ], "title": "cotangent-lemma-category-fibred-space", "contents": [ "In the situation above the category", "$\\mathcal{C}_{X/\\Lambda}$ is fibred over $X_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "Given an object $U \\to \\mathbf{A}$ of $\\mathcal{C}_{X/\\Lambda}$ and a morphism", "$U' \\to U$ of $X_\\etale$ consider the object $U' \\to \\mathbf{A}$ of", "$\\mathcal{C}_{X/\\Lambda}$ where $U' \\to \\mathbf{A}$ is the composition of", "$U \\to \\mathbf{A}$", "and $U' \\to U$. The morphism $(U' \\to \\mathbf{A}) \\to (U \\to \\mathbf{A})$ of", "$\\mathcal{C}_{X/\\Lambda}$ is strongly cartesian over $X_\\etale$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11238, "type": "theorem", "label": "cotangent-lemma-cotangent-morphism-spaces", "categories": [ "cotangent" ], "title": "cotangent-lemma-cotangent-morphism-spaces", "contents": [ "In the situation above there is a canonical isomorphism", "$$", "L_{X/\\Lambda} = ", "L\\pi_!(Li^*\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}) =", "L\\pi_!(i^*\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}) =", "L\\pi_!(\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}", "\\otimes_\\mathcal{O} \\underline{\\mathcal{O}}_X)", "$$", "in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "We first observe that for any object $(U \\to \\mathbf{A})$ of", "$\\mathcal{C}_{X/\\Lambda}$", "the value of the sheaf $\\mathcal{O}$ is a polynomial algebra over $\\Lambda$.", "Hence $\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}$ is a flat $\\mathcal{O}$-module", "and we conclude the second and third equalities of the statement of the", "lemma hold.", "\\medskip\\noindent", "By Remark \\ref{remark-compute-L-pi-shriek-spaces} the object", "$L\\pi_!(\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}", "\\otimes_\\mathcal{O} \\underline{\\mathcal{O}}_X)$", "is computed as the sheafification of the complex of presheaves", "$$", "U \\mapsto", "\\left(\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}", "\\otimes_\\mathcal{O} \\underline{\\mathcal{O}}_X\\right)(\\mathbf{A}_{\\bullet, U})", "=", "\\Omega_{P_{\\bullet, U}/\\Lambda} \\otimes_{P_{\\bullet, U}} \\mathcal{O}_X(U) =", "L_{\\mathcal{O}_X(U)/\\Lambda}", "$$", "using notation as in Remark \\ref{remark-compute-L-pi-shriek-spaces}.", "Now Remark \\ref{remark-map-sections-over-U} shows that", "$L\\pi_!(\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}", "\\otimes_\\mathcal{O} \\underline{\\mathcal{O}}_X)$", "computes the cotangent complex of the map of rings", "$\\underline{\\Lambda} \\to \\mathcal{O}_X$ on $X_\\etale$.", "This is what we want by Lemma \\ref{lemma-space-over-ring}." ], "refs": [ "cotangent-remark-compute-L-pi-shriek-spaces", "cotangent-remark-compute-L-pi-shriek-spaces", "cotangent-remark-map-sections-over-U", "cotangent-lemma-space-over-ring" ], "ref_ids": [ 11269, 11269, 11267, 11236 ] } ], "ref_ids": [] }, { "id": 11239, "type": "theorem", "label": "cotangent-lemma-fibre-product-tor-independent", "categories": [ "cotangent" ], "title": "cotangent-lemma-fibre-product-tor-independent", "contents": [ "In the situation above, if $X$ and $Y$ are Tor independent over $B$, then", "the object $E$ in (\\ref{equation-fibre-product}) is zero. In this case we", "have", "$$", "L_{X \\times_B Y/B} = Lp^*L_{X/B} \\oplus Lq^*L_{Y/B}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $W$ and a surjective \\'etale morphism $W \\to B$.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X \\times_B W$.", "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y \\times_B W$.", "Then $U \\times_W V \\to X \\times_B Y$ is surjective \\'etale too.", "Hence it suffices to prove that the restriction of $E$ to $U \\times_W V$", "is zero. By Lemma \\ref{lemma-compare-spaces-schemes} and", "Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-tor-independent}", "this reduces us to the case of schemes.", "Taking suitable affine opens we reduce to the case of affine schemes.", "Using ", "Lemma \\ref{lemma-morphism-affine-schemes}", "we reduce to the case of a tensor product of rings, i.e., to", "Lemma \\ref{lemma-tensor-product-tor-independent}." ], "refs": [ "cotangent-lemma-compare-spaces-schemes", "spaces-perfect-lemma-tor-independent", "cotangent-lemma-morphism-affine-schemes", "cotangent-lemma-tensor-product-tor-independent" ], "ref_ids": [ 11235, 2719, 11230, 11212 ] } ], "ref_ids": [] }, { "id": 11240, "type": "theorem", "label": "cotangent-lemma-fibre-product", "categories": [ "cotangent" ], "title": "cotangent-lemma-fibre-product", "contents": [ "Let $S$ be a scheme. Let $X \\to B$ and $Y \\to B$ be morphisms of algebraic", "spaces over $S$. The object $E$ in (\\ref{equation-fibre-product}) satisfies", "$H^i(E) = 0$ for $i = 0, -1$ and for a geometric point", "$(\\overline{x}, \\overline{y}) : \\Spec(k) \\to X \\times_B Y$ we have", "$$", "H^{-2}(E)_{(\\overline{x}, \\overline{y})} =", "\\text{Tor}_1^R(A, B) \\otimes_{A \\otimes_R B} C", "$$", "where $R = \\mathcal{O}_{B, \\overline{b}}$, $A = \\mathcal{O}_{X, \\overline{x}}$,", "$B = \\mathcal{O}_{Y, \\overline{y}}$, and", "$C = \\mathcal{O}_{X \\times_B Y, (\\overline{x}, \\overline{y})}$." ], "refs": [], "proofs": [ { "contents": [ "The formation of the cotangent complex commutes with taking stalks", "and pullbacks, see", "Lemmas \\ref{lemma-stalk-cotangent-complex} and", "\\ref{lemma-pullback-cotangent-morphism-topoi}.", "Note that $C$ is a henselization of $A \\otimes_R B$.", "$L_{C/R} = L_{A \\otimes_R B/R} \\otimes_{A \\otimes_R B} C$", "by the results of Section \\ref{section-localization}.", "Thus the stalk of $E$ at our geometric point is the cone of the", "map $L_{A/R} \\otimes C \\to L_{A \\otimes_R B/R} \\otimes C$.", "Therefore the results of the lemma follow from", "the case of rings, i.e., Lemma \\ref{lemma-tensor-product}." ], "refs": [ "cotangent-lemma-stalk-cotangent-complex", "cotangent-lemma-pullback-cotangent-morphism-topoi", "cotangent-lemma-tensor-product" ], "ref_ids": [ 11220, 11215, 11213 ] } ], "ref_ids": [] }, { "id": 11241, "type": "theorem", "label": "cotangent-proposition-polynomial", "categories": [ "cotangent" ], "title": "cotangent-proposition-polynomial", "contents": [ "Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map.", "There exists a simplicial $A$-algebra $P_\\bullet$ with an augmentation", "$\\epsilon : P_\\bullet \\to B$ such that each $P_n$ is a polynomial algebra", "of finite type over $A$ and such that $\\epsilon$ is a trivial", "Kan fibration of simplicial sets." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{A}$ be the category of $A$-algebra maps $C \\to B$.", "In this proof our simplicial objects and skeleton and coskeleton", "functors will be taken in this category.", "\\medskip\\noindent", "Choose a polynomial algebra $P_0$ of finite type over $A$ and a surjection", "$P_0 \\to B$. As a first approximation we take", "$P_\\bullet = \\text{cosk}_0(P_0)$. In other words, $P_\\bullet$ is the simplicial", "$A$-algebra with terms $P_n = P_0 \\times_A \\ldots \\times_A P_0$.", "(In the final paragraph of the proof this simplicial object will", "be denoted $P^0_\\bullet$.) By", "Simplicial, Lemma \\ref{simplicial-lemma-cosk-minus-one-equivalence}", "the map $P_\\bullet \\to B$ is a trivial Kan fibration of simplicial sets.", "Also, observe that $P_\\bullet = \\text{cosk}_0 \\text{sk}_0 P_\\bullet$.", "\\medskip\\noindent", "Suppose for some $n \\geq 0$ we have constructed $P_\\bullet$", "(in the final paragraph of the proof this will be $P^n_\\bullet$)", "such that", "\\begin{enumerate}", "\\item[(a)] $P_\\bullet \\to B$ is a trivial Kan fibration of simplicial sets,", "\\item[(b)] $P_k$ is a finitely generated polynomial algebra for", "$0 \\leq k \\leq n$, and", "\\item[(c)] $P_\\bullet = \\text{cosk}_n \\text{sk}_n P_\\bullet$", "\\end{enumerate}", "By Lemma \\ref{lemma-polynomial}", "we can find a finitely generated polynomial algebra $Q$ over $A$", "and a surjection $Q \\to P_{n + 1}$. Since $P_n$ is a polynomial algebra", "the $A$-algebra maps $s_i : P_n \\to P_{n + 1}$ lift to maps", "$s'_i : P_n \\to Q$. Set $d'_j : Q \\to P_n$ equal to the composition of", "$Q \\to P_{n + 1}$ and $d_j : P_{n + 1} \\to P_n$.", "We obtain a truncated simplicial object $P'_\\bullet$ of $\\mathcal{A}$", "by setting $P'_k = P_k$ for $k \\leq n$ and $P'_{n + 1} = Q$ and morphisms", "$d'_i = d_i$ and $s'_i = s_i$ in degrees $k \\leq n - 1$ and using the", "morphisms $d'_j$ and $s'_i$ in degree $n$. Extend this to a full simplicial", "object $P'_\\bullet$ of $\\mathcal{A}$ using $\\text{cosk}_{n + 1}$. By", "functoriality of the coskeleton functors there is a morphism", "$P'_\\bullet \\to P_\\bullet$ of simplicial objects extending the", "given morphism of $(n + 1)$-truncated simplicial objects.", "(This morphism will be denoted $P^{n + 1}_\\bullet \\to P^n_\\bullet$", "in the final paragraph of the proof.)", "\\medskip\\noindent", "Note that conditions (b) and (c) are satisfied for $P'_\\bullet$ with $n$", "replaced by $n + 1$. We claim the map $P'_\\bullet \\to P_\\bullet$ satisfies", "assumptions (1), (2), (3), and (4) of", "Simplicial, Lemmas \\ref{simplicial-lemma-section}", "with $n + 1$ instead of $n$. Conditions (1) and (2) hold by construction.", "By Simplicial, Lemma \\ref{simplicial-lemma-cosk-above-object}", "we see that we have", "$P_\\bullet = \\text{cosk}_{n + 1}\\text{sk}_{n + 1}P_\\bullet$", "and", "$P'_\\bullet = \\text{cosk}_{n + 1}\\text{sk}_{n + 1}P'_\\bullet$", "not only in $\\mathcal{A}$ but also in the category of $A$-algebras,", "whence in the category of sets (as the forgetful functor from $A$-algebras", "to sets commutes with all limits). This proves (3) and (4). Thus the lemma", "applies and $P'_\\bullet \\to P_\\bullet$ is a trivial Kan fibration. By", "Simplicial, Lemma \\ref{simplicial-lemma-trivial-kan-composition}", "we conclude that $P'_\\bullet \\to B$ is a trivial Kan fibration and (a)", "holds as well.", "\\medskip\\noindent", "To finish the proof we take the inverse limit $P_\\bullet = \\lim P^n_\\bullet$", "of the sequence of simplicial algebras", "$$", "\\ldots \\to P^2_\\bullet \\to P^1_\\bullet \\to P^0_\\bullet", "$$", "constructed above. The map $P_\\bullet \\to B$ is a trivial Kan fibration by", "Simplicial, Lemma \\ref{simplicial-lemma-limit-trivial-kan}.", "However, the construction above stabilizes in each degree", "to a fixed finitely generated polynomial algebra as desired." ], "refs": [ "simplicial-lemma-cosk-minus-one-equivalence", "cotangent-lemma-polynomial", "simplicial-lemma-cosk-above-object", "simplicial-lemma-trivial-kan-composition", "simplicial-lemma-limit-trivial-kan" ], "ref_ids": [ 14903, 11180, 14843, 14888, 14889 ] } ], "ref_ids": [] }, { "id": 11242, "type": "theorem", "label": "cotangent-proposition-triangle", "categories": [ "cotangent" ], "title": "cotangent-proposition-triangle", "contents": [ "Let $A \\to B \\to C$ be ring maps. There is a canonical distinguished", "triangle", "$$", "L_{B/A} \\otimes_B^\\mathbf{L} C \\to L_{C/A} \\to L_{C/B} \\to", "L_{B/A} \\otimes_B^\\mathbf{L} C[1]", "$$", "in $D(C)$." ], "refs": [], "proofs": [ { "contents": [ "Consider the short exact sequence of sheaves of", "Lemma \\ref{lemma-triangle-ses}", "and apply the derived functor $L\\pi_!$ to obtain a distinguished", "triangle", "$$", "L\\pi_!(g_1^{-1}\\Omega_1 \\otimes_{\\underline{B}} \\underline{C}) \\to", "L\\pi_!(g_2^{-1}\\Omega_2) \\to", "L\\pi_!(g_3^{-1}\\Omega_3) \\to", "L\\pi_!(g_1^{-1}\\Omega_1 \\otimes_{\\underline{B}} \\underline{C})[1]", "$$", "in $D(C)$. Using Lemmas \\ref{lemma-triangle-compute-lower-shriek} and", "\\ref{lemma-compute-cotangent-complex}", "we see that the second and third terms agree with $L_{C/A}$ and $L_{C/B}$", "and the first one equals", "$$", "L\\pi_{1, !}(\\Omega_1 \\otimes_{\\underline{B}} \\underline{C}) =", "L\\pi_{1, !}(\\Omega_1) \\otimes_B^\\mathbf{L} C =", "L_{B/A} \\otimes_B^\\mathbf{L} C", "$$", "The first equality by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-change-of-rings}", "(and flatness of $\\Omega_1$ as a sheaf of modules over $\\underline{B}$)", "and the second by Lemma \\ref{lemma-compute-cotangent-complex}." ], "refs": [ "cotangent-lemma-triangle-ses", "cotangent-lemma-triangle-compute-lower-shriek", "cotangent-lemma-compute-cotangent-complex", "sites-cohomology-lemma-change-of-rings", "cotangent-lemma-compute-cotangent-complex" ], "ref_ids": [ 11189, 11191, 11175, 4347, 11175 ] } ], "ref_ids": [] }, { "id": 11243, "type": "theorem", "label": "cotangent-proposition-cotangent-complex-local-complete-intersection", "categories": [ "cotangent" ], "title": "cotangent-proposition-cotangent-complex-local-complete-intersection", "contents": [ "Let $A \\to B$ be a local complete intersection map.", "Then $L_{B/A}$ is a perfect complex with tor amplitude in $[-1, 0]$." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjection $P = A[x_1, \\ldots, x_n] \\to B$ with kernel $J$.", "By Lemma \\ref{lemma-relation-with-naive-cotangent-complex}", "we see that $J/J^2 \\to \\bigoplus B\\text{d}x_i$", "is quasi-isomorphic to $\\tau_{\\geq -1}L_{B/A}$.", "Note that $J/J^2$ is finite projective", "(More on Algebra, Lemma", "\\ref{more-algebra-lemma-quasi-regular-ideal-finite-projective}),", "hence $\\tau_{\\geq -1}L_{B/A}$ is a perfect complex with", "tor amplitude in $[-1, 0]$.", "Thus it suffices to show that $H^i(L_{B/A}) = 0$ for $i \\not \\in [-1, 0]$.", "This follows from (\\ref{equation-triangle})", "$$", "L_{P/A} \\otimes_P^\\mathbf{L} B \\to L_{B/A} \\to L_{B/P} \\to", "L_{P/A} \\otimes_P^\\mathbf{L} B[1]", "$$", "and Lemma \\ref{lemma-mod-Koszul-regular-ideal}", "to see that $H^i(L_{B/P})$ is zero unless $i \\in \\{-1, 0\\}$.", "(We also use Lemma \\ref{lemma-cotangent-complex-polynomial-algebra}", "for the term on the left.)" ], "refs": [ "cotangent-lemma-relation-with-naive-cotangent-complex", "more-algebra-lemma-quasi-regular-ideal-finite-projective", "cotangent-lemma-mod-Koszul-regular-ideal", "cotangent-lemma-cotangent-complex-polynomial-algebra" ], "ref_ids": [ 11204, 9996, 11211, 11179 ] } ], "ref_ids": [] }, { "id": 11270, "type": "theorem", "label": "spaces-cohomology-theorem-formal-functions", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-theorem-formal-functions", "contents": [ "In Situation \\ref{situation-formal-functions}. Fix $p \\geq 0$.", "The system of maps", "$$", "H^p(X, \\mathcal{F})/I^nH^p(X, \\mathcal{F})", "\\longrightarrow", "H^p(X, \\mathcal{F}/I^n\\mathcal{F})", "$$", "define an isomorphism of limits", "$$", "H^p(X, \\mathcal{F})^\\wedge", "\\longrightarrow", "\\lim_n H^p(X, \\mathcal{F}/I^n\\mathcal{F})", "$$", "where the left hand side is the completion of the $A$-module", "$H^p(X, \\mathcal{F})$ with respect to the ideal $I$, see", "Algebra, Section \\ref{algebra-section-completion}.", "Moreover, this is in fact a homeomorphism for the limit topologies." ], "refs": [], "proofs": [ { "contents": [ "In fact, this follows immediately from", "Lemma \\ref{lemma-ML-cohomology-powers-ideal}. We spell out the details.", "Set $M = H^p(X, \\mathcal{F})$ and $M_n = H^p(X, \\mathcal{F}/I^n\\mathcal{F})$.", "Denote $N_n = \\Im(M \\to M_n)$.", "By the description of the limit in Homology, Section", "\\ref{homology-section-inverse-systems} we have", "$$", "\\lim_n M_n", "=", "\\{(x_n) \\in \\prod M_n \\mid \\varphi_i(x_n) = x_{n - 1}, \\ n = 2, 3, \\ldots\\}", "$$", "Pick an element $x = (x_n) \\in \\lim_n M_n$.", "By Lemma \\ref{lemma-ML-cohomology-powers-ideal} part (3)", "we have $x_n \\in N_n$ for all $n$ since by", "definition $x_n$ is the image of some $x_{n + m} \\in M_{n + m}$ for", "all $m$. By Lemma \\ref{lemma-ML-cohomology-powers-ideal} part (1)", "we see that there exists a factorization", "$$", "M \\to N_n \\to M/I^{n - c_1}M", "$$", "of the reduction map. Denote $y_n \\in M/I^{n - c_1}M$ the image of $x_n$", "for $n \\geq c_1$. Since for $n' \\geq n$ the composition", "$M \\to M_{n'} \\to M_n$ is the given map $M \\to M_n$ we see that", "$y_{n'}$ maps to $y_n$ under the canonical map", "$M/I^{n' - c_1}M \\to M/I^{n - c_1}M$. Hence $y = (y_{n + c_1})$", "defines an element of $\\lim_n M/I^nM$.", "We omit the verification that $y$ maps to $x$ under the", "map", "$$", "M^\\wedge = \\lim_n M/I^nM \\longrightarrow \\lim_n M_n", "$$", "of the lemma. We also omit the verification on topologies." ], "refs": [ "spaces-cohomology-lemma-ML-cohomology-powers-ideal", "spaces-cohomology-lemma-ML-cohomology-powers-ideal", "spaces-cohomology-lemma-ML-cohomology-powers-ideal" ], "ref_ids": [ 11338, 11338, 11338 ] } ], "ref_ids": [] }, { "id": 11271, "type": "theorem", "label": "spaces-cohomology-lemma-higher-direct-image", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-higher-direct-image", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. If $f$ is quasi-compact and quasi-separated, then $R^if_*$", "transforms quasi-coherent $\\mathcal{O}_X$-modules into", "quasi-coherent $\\mathcal{O}_Y$-modules." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\to Y$ be an \\'etale morphism where $V$ is an affine scheme. Set", "$U = V \\times_Y X$ and denote $f' : U \\to V$ the induced morphism.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. By", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-pushforward-etale-base-change-modules}", "we have", "$R^if'_*(\\mathcal{F}|_U) = (R^if_*\\mathcal{F})|_V$.", "Since the property of being a quasi-coherent module is local in the", "\\'etale topology on $Y$ (see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-characterize-quasi-coherent})", "we may replace $Y$ by $V$, i.e., we may assume $Y$ is an affine scheme.", "\\medskip\\noindent", "Assume $Y$ is affine. Since $f$ is quasi-compact we see that $X$", "is quasi-compact. Thus we may choose an affine scheme $U$ and a surjective", "\\'etale morphism $g : U \\to X$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-quasi-compact-affine-cover}.", "Picture", "$$", "\\xymatrix{", "U \\ar[r]_g \\ar[rd]_{f \\circ g} & X \\ar[d]^f \\\\", "& Y", "}", "$$", "The morphism $g : U \\to X$ is representable, separated", "and quasi-compact because $X$ is quasi-separated. Hence the lemma", "holds for $g$ (by the discussion above the lemma).", "It also holds for $f \\circ g : U \\to Y$ (as this is a morphism", "of affine schemes).", "\\medskip\\noindent", "In the situation described in the previous paragraph we will show by", "induction on $n$ that $IH_n$: for any quasi-coherent sheaf $\\mathcal{F}$", "on $X$ the sheaves $R^if\\mathcal{F}$", "are quasi-coherent for $i \\leq n$.", "The case $n = 0$ follows from", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-pushforward}.", "Assume $IH_n$. In the rest of the proof we show that $IH_{n + 1}$ holds.", "\\medskip\\noindent", "Let $\\mathcal{H}$ be a quasi-coherent $\\mathcal{O}_U$-module.", "Consider the Leray spectral sequence", "$$", "E_2^{p, q} = R^pf_* R^qg_* \\mathcal{H}", "\\Rightarrow", "R^{p + q}(f \\circ g)_*\\mathcal{H}", "$$", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-relative-Leray}.", "As $R^qg_*\\mathcal{H}$ is quasi-coherent by $IH_n$ all the sheaves", "$R^pf_*R^qg_*\\mathcal{H}$ are quasi-coherent for $p \\leq n$.", "The sheaves $R^{p + q}(f \\circ g)_*\\mathcal{H}$ are all", "quasi-coherent (in fact zero for $p + q > 0$ but we do not need this).", "Looking in degrees $\\leq n + 1$ the only module which we do not", "yet know is quasi-coherent is $E_2^{n + 1, 0} = R^{n + 1}f_*g_*\\mathcal{H}$.", "Moreover, the differentials", "$d_r^{n + 1, 0} : E_r^{n + 1, 0} \\to E_r^{n + 1 + r, 1 - r}$", "are zero as the target is zero. Using that $\\QCoh(\\mathcal{O}_X)$", "is a weak Serre subcategory of $\\textit{Mod}(\\mathcal{O}_X)$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-properties-quasi-coherent}) it", "follows that $R^{n + 1}f_*g_*\\mathcal{H}$", "is quasi-coherent (details omitted).", "\\medskip\\noindent", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Set $\\mathcal{H} = g^*\\mathcal{F}$. The adjunction mapping", "$\\mathcal{F} \\to g_*g^*\\mathcal{F} = g_*\\mathcal{H}$ is injective", "as $U \\to X$ is surjective \\'etale. Consider the exact sequence", "$$", "0 \\to \\mathcal{F} \\to g_*\\mathcal{H} \\to \\mathcal{G} \\to 0", "$$", "where $\\mathcal{G}$ is the cokernel of the first map and in particular", "quasi-coherent. Applying the long exact cohomology sequence we obtain", "$$", "R^nf_*g_*\\mathcal{H} \\to", "R^nf_*\\mathcal{G} \\to", "R^{n + 1}f_*\\mathcal{F} \\to", "R^{n + 1}f_*g_*\\mathcal{H} \\to", "R^{n + 1}f_*\\mathcal{G}", "$$", "The cokernel of the first arrow is quasi-coherent and", "we have seen above that $R^{n + 1}f_*g_*\\mathcal{H}$ is quasi-coherent.", "Thus $R^{n + 1}f_*\\mathcal{F}$ has a $2$-step filtration", "where the first step is quasi-coherent and the second a submodule of", "a quasi-coherent sheaf. Since $\\mathcal{F}$ is an arbitrary quasi-coherent", "$\\mathcal{O}_X$-module, this result also holds for $\\mathcal{G}$.", "Thus we can choose an exact sequence", "$0 \\to \\mathcal{A} \\to R^{n + 1}f_*\\mathcal{G} \\to \\mathcal{B}$", "with $\\mathcal{A}$, $\\mathcal{B}$ quasi-coherent $\\mathcal{O}_Y$-modules.", "Then the kernel $\\mathcal{K}$ of", "$R^{n + 1}f_*g_*\\mathcal{H} \\to R^{n + 1}f_*\\mathcal{G}", "\\to \\mathcal{B}$ is quasi-coherent, whereupon we obtain a map", "$\\mathcal{K} \\to \\mathcal{A}$ whose kernel $\\mathcal{K}'$ is", "quasi-coherent too. Hence $R^{n + 1}f_*\\mathcal{F}$ sits in an exact", "sequence", "$$", "R^nf_*g_*\\mathcal{H} \\to", "R^nf_*\\mathcal{G} \\to", "R^{n + 1}f_*\\mathcal{F} \\to \\mathcal{K}' \\to 0", "$$", "with all modules quasi-coherent except for possibly $R^{n + 1}f_*\\mathcal{F}$.", "We conclude that $R^{n + 1}f_*\\mathcal{F}$ is quasi-coherent, i.e.,", "$IH_{n + 1}$ holds as desired." ], "refs": [ "spaces-properties-lemma-pushforward-etale-base-change-modules", "spaces-properties-lemma-characterize-quasi-coherent", "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-morphisms-lemma-pushforward", "sites-cohomology-lemma-relative-Leray", "spaces-properties-lemma-properties-quasi-coherent" ], "ref_ids": [ 11898, 11911, 11832, 4760, 4222, 11912 ] } ], "ref_ids": [] }, { "id": 11272, "type": "theorem", "label": "spaces-cohomology-lemma-quasi-coherence-higher-direct-images-application", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-quasi-coherence-higher-direct-images-application", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-separated and quasi-compact", "morphism of algebraic spaces over $S$. For any quasi-coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ and any affine object $V$ of", "$Y_\\etale$ we have", "$$", "H^q(V \\times_Y X, \\mathcal{F}) = H^0(V, R^qf_*\\mathcal{F})", "$$", "for all $q \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Since formation of $Rf_*$ commutes with \\'etale localization", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-pushforward-etale-base-change-modules})", "we may replace $Y$ by $V$ and assume $Y = V$ is affine.", "Consider the Leray spectral sequence", "$E_2^{p, q} = H^p(Y, R^qf_*\\mathcal{F})$", "converging to $H^{p + q}(X, \\mathcal{F})$, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-Leray}.", "By Lemma \\ref{lemma-higher-direct-image}", "we see that the sheaves $R^qf_*\\mathcal{F}$ are quasi-coherent. By", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}", "we see that $E_2^{p, q} = 0$ when $p > 0$.", "Hence the spectral sequence degenerates at $E_2$ and we win." ], "refs": [ "spaces-properties-lemma-pushforward-etale-base-change-modules", "sites-cohomology-lemma-Leray", "spaces-cohomology-lemma-higher-direct-image", "coherent-lemma-quasi-coherent-affine-cohomology-zero" ], "ref_ids": [ 11898, 4220, 11271, 3282 ] } ], "ref_ids": [] }, { "id": 11273, "type": "theorem", "label": "spaces-cohomology-lemma-finite-higher-direct-image-zero", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-finite-higher-direct-image-zero", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an integral (for example finite)", "morphism of algebraic spaces. Then", "$f_* : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$", "is an exact functor and $R^pf_* = 0$ for $p > 0$." ], "refs": [], "proofs": [ { "contents": [ "By Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-pushforward-etale-base-change}", "we may compute the higher direct images on an \\'etale cover of $Y$.", "Hence we may assume $Y$ is a scheme. This implies that", "$X$ is a scheme (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-integral-local}).", "In this case we may apply", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-what-integral}.", "For the finite case the reader may wish to consult the less technical", "\\'Etale Cohomology, Proposition", "\\ref{etale-cohomology-proposition-finite-higher-direct-image-zero}." ], "refs": [ "spaces-properties-lemma-pushforward-etale-base-change", "spaces-morphisms-lemma-integral-local", "etale-cohomology-lemma-what-integral", "etale-cohomology-proposition-finite-higher-direct-image-zero" ], "ref_ids": [ 11867, 4940, 6454, 6703 ] } ], "ref_ids": [] }, { "id": 11274, "type": "theorem", "label": "spaces-cohomology-lemma-stalk-push-finite", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-stalk-push-finite", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a finite morphism of algebraic", "spaces over $S$. Let $\\overline{y}$ be a geometric point of $Y$ with", "lifts $\\overline{x}_1, \\ldots, \\overline{x}_n$ in $X$. Then", "$$", "(f_*\\mathcal{F})_{\\overline{y}} =", "\\bigoplus\\nolimits_{i = 1, \\ldots, n}", "\\mathcal{F}_{\\overline{x}_i}", "$$", "for any sheaf $\\mathcal{F}$ on $X_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "Choose an \\'etale neighbourhood $(V, \\overline{v})$ of $\\overline{y}$.", "Then the stalk $(f_*\\mathcal{F})_{\\overline{y}}$", "is the stalk of $f_*\\mathcal{F}|_V$ at $\\overline{v}$.", "By Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-pushforward-etale-base-change}", "we may replace $Y$ by $V$ and $X$ by $X \\times_Y V$.", "Then $Z \\to X$ is a finite morphism of schemes and the result is", "\\'Etale Cohomology, Proposition", "\\ref{etale-cohomology-proposition-finite-higher-direct-image-zero}." ], "refs": [ "spaces-properties-lemma-pushforward-etale-base-change", "etale-cohomology-proposition-finite-higher-direct-image-zero" ], "ref_ids": [ 11867, 6703 ] } ], "ref_ids": [] }, { "id": 11275, "type": "theorem", "label": "spaces-cohomology-lemma-finite-rings", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-finite-rings", "contents": [ "Let $S$ be a scheme. Let $\\pi : X \\to Y$ be a finite morphism of algebraic", "spaces over $S$. Let $\\mathcal{A}$ be a sheaf of rings on $X_\\etale$.", "Let $\\mathcal{B}$ be a sheaf of rings on $Y_\\etale$.", "Let $\\varphi : \\mathcal{B} \\to \\pi_*\\mathcal{A}$", "be a homomorphism of sheaves of rings so that we obtain a", "morphism of ringed topoi", "$$", "f = (\\pi, \\varphi) :", "(\\Sh(X_\\etale), \\mathcal{A})", "\\longrightarrow", "(\\Sh(Y_\\etale), \\mathcal{B}).", "$$", "For a sheaf of $\\mathcal{A}$-modules $\\mathcal{F}$ and a", "sheaf of $\\mathcal{B}$-modules $\\mathcal{G}$ the canonical map", "$$", "\\mathcal{G} \\otimes_\\mathcal{B} f_*\\mathcal{F}", "\\longrightarrow", "f_*(f^*\\mathcal{G} \\otimes_\\mathcal{A} \\mathcal{F}).", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The map is the map adjoint to the map", "$$", "f^*\\mathcal{G} \\otimes_\\mathcal{A}", "f^* f_*\\mathcal{F} =", "f^*(\\mathcal{G} \\otimes_\\mathcal{B} f_*\\mathcal{F})", "\\longrightarrow", "f^*\\mathcal{G} \\otimes_\\mathcal{A} \\mathcal{F}", "$$", "coming from $\\text{id} : f^*\\mathcal{G} \\to f^*\\mathcal{G}$", "and the adjunction map $f^* f_*\\mathcal{F} \\to \\mathcal{F}$.", "To see this map is an isomorphism, we may check on stalks", "(Properties of Spaces, Theorem", "\\ref{spaces-properties-theorem-exactness-stalks}).", "Let $\\overline{y}$ be a geometric point of $Y$ and", "let $\\overline{x}_1, \\ldots, \\overline{x}_n$ be the geometric", "points of $X$ lying over $\\overline{y}$.", "Working out what our maps does on stalks, we see that we", "have to show", "$$", "\\mathcal{G}_{\\overline{y}}", "\\otimes_{\\mathcal{B}_{\\overline{y}}}", "\\left(", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} \\mathcal{F}_{\\overline{x}_i}", "\\right) =", "\\bigoplus\\nolimits_{i = 1, \\ldots, n}", "(\\mathcal{G}_{\\overline{y}}", "\\otimes_{\\mathcal{B}_{\\overline{x}}}", "\\mathcal{A}_{\\overline{x}_i}) \\otimes_{\\mathcal{A}_{\\overline{x}_i}}", "\\mathcal{F}_{\\overline{x}_i}", "$$", "which holds true. Here we have used that", "taking tensor products commutes with taking stalks, the", "behaviour of stalks under pullback", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-stalk-pullback}, and", "the behaviour of stalks under pushforward along a closed immersion", "Lemma \\ref{lemma-stalk-push-finite}." ], "refs": [ "spaces-properties-theorem-exactness-stalks", "spaces-properties-lemma-stalk-pullback", "spaces-cohomology-lemma-stalk-push-finite" ], "ref_ids": [ 11813, 11875, 11274 ] } ], "ref_ids": [] }, { "id": 11276, "type": "theorem", "label": "spaces-cohomology-lemma-projection-formula-finite", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-projection-formula-finite", "contents": [ "With $S$, $X$, $Y$, $\\pi$, $\\mathcal{A}$, $\\mathcal{B}$, $\\varphi$, and $f$", "as in Lemma \\ref{lemma-finite-rings} we have", "$$", "K \\otimes_\\mathcal{B}^\\mathbf{L} Rf_*M =", "Rf_*(Lf^*K \\otimes_\\mathcal{A}^\\mathbf{L} M)", "$$", "in $D(\\mathcal{B})$ for any $K \\in D(\\mathcal{B})$ and", "$M \\in D(\\mathcal{A})$." ], "refs": [ "spaces-cohomology-lemma-finite-rings" ], "proofs": [ { "contents": [ "Since $f_*$ is exact (Lemma \\ref{lemma-finite-higher-direct-image-zero})", "the functor $Rf_*$ is computed by applying $f_*$ to any representative complex.", "Choose a complex $\\mathcal{K}^\\bullet$ of $\\mathcal{B}$-modules", "representing $K$ which is K-flat with flat terms, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-K-flat-resolution}.", "Then $f^*\\mathcal{K}^\\bullet$ is K-flat with flat terms, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-pullback-K-flat}.", "Choose any complex $\\mathcal{M}^\\bullet$ of $\\mathcal{A}$-modules", "representing $M$. Then", "we have to show", "$$", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{B} f_*\\mathcal{M}^\\bullet)", "=", "f_*\\text{Tot}(f^*\\mathcal{K}^\\bullet \\otimes_\\mathcal{A} \\mathcal{M}^\\bullet)", "$$", "because by our choices these complexes represent the right and left hand", "side of the formula in the lemma.", "Since $f_*$ commutes with direct sums", "(for example by the description of the stalks in", "Lemma \\ref{lemma-stalk-push-finite}),", "this reduces to the equalities", "$$", "\\mathcal{K}^n \\otimes_\\mathcal{B} f_*\\mathcal{M}^m", "=", "f_*(f^*\\mathcal{K}^n \\otimes_\\mathcal{A} \\mathcal{M}^m)", "$$", "which are true by Lemma \\ref{lemma-finite-rings}." ], "refs": [ "spaces-cohomology-lemma-finite-higher-direct-image-zero", "sites-cohomology-lemma-K-flat-resolution", "sites-cohomology-lemma-pullback-K-flat", "spaces-cohomology-lemma-stalk-push-finite", "spaces-cohomology-lemma-finite-rings" ], "ref_ids": [ 11273, 4236, 4241, 11274, 11275 ] } ], "ref_ids": [ 11275 ] }, { "id": 11277, "type": "theorem", "label": "spaces-cohomology-lemma-colimits", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-colimits", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "If $X$ is quasi-compact and quasi-separated, then", "$$", "\\colim_i H^p(X, \\mathcal{F}_i)", "\\longrightarrow", "H^p(X, \\colim_i \\mathcal{F}_i)", "$$", "is an isomorphism", "for every filtered diagram of abelian sheaves on $X_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-colim-works-over-collection}.", "Namely, let $\\mathcal{B} \\subset \\Ob(X_{spaces, \\etale})$", "be the set of quasi-compact and quasi-separated spaces \\'etale over $X$.", "Note that if $U \\in \\mathcal{B}$ then, because $U$ is quasi-compact,", "the collection of finite coverings $\\{U_i \\to U\\}$ with $U_i \\in \\mathcal{B}$", "is cofinal in the set of coverings of $U$ in $X_{spaces, \\etale}$. By", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence}", "the set $\\mathcal{B}$ satisfies all the assumptions of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-colim-works-over-collection}.", "Since $X \\in \\mathcal{B}$ we win." ], "refs": [ "sites-cohomology-lemma-colim-works-over-collection", "spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence", "sites-cohomology-lemma-colim-works-over-collection" ], "ref_ids": [ 4224, 4744, 4224 ] } ], "ref_ids": [] }, { "id": 11278, "type": "theorem", "label": "spaces-cohomology-lemma-colimit-cohomology", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-colimit-cohomology", "contents": [ "\\begin{slogan}", "Higher direct images of qcqs morphisms commute with filtered colimits", "of sheaves.", "\\end{slogan}", "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and quasi-separated", "morphism of algebraic spaces over $S$. Let $\\mathcal{F} = \\colim \\mathcal{F}_i$", "be a filtered colimit of abelian sheaves on $X_\\etale$.", "Then for any $p \\geq 0$ we have", "$$", "R^pf_*\\mathcal{F} = \\colim R^pf_*\\mathcal{F}_i.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Recall that $R^pf_*\\mathcal{F}$ is the sheaf on $Y_{spaces, \\etale}$", "associated to $V \\mapsto H^p(V \\times_Y X, \\mathcal{F})$, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-higher-direct-images}", "and Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-functoriality-etale-site}.", "Recall that the colimit is the sheaf associated to the presheaf colimit.", "Hence we can apply Lemma \\ref{lemma-colimits}", "to $H^p(V \\times_Y X, -)$ where $V$ is affine to conclude (because", "when $V$ is affine, then $V \\times_Y X$ is quasi-compact and quasi-separated).", "Strictly speaking this also uses Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-alternative} to see that there exist", "enough affine objects." ], "refs": [ "sites-cohomology-lemma-higher-direct-images", "spaces-properties-lemma-functoriality-etale-site", "spaces-cohomology-lemma-colimits", "spaces-properties-lemma-alternative" ], "ref_ids": [ 4189, 11864, 11277, 11863 ] } ], "ref_ids": [] }, { "id": 11279, "type": "theorem", "label": "spaces-cohomology-lemma-finite-presentation-quasi-compact-colimit", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-finite-presentation-quasi-compact-colimit", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Let $I$ be a directed set and", "let $(\\mathcal{F}_i, \\varphi_{ii'})$ be a system over $I$", "of quasi-coherent $\\mathcal{O}_X$-modules. Let $\\mathcal{G}$ be an", "$\\mathcal{O}_X$-module of finite presentation. Then we have", "$$", "\\colim_i \\Hom_X(\\mathcal{G}, \\mathcal{F}_i)", "=", "\\Hom_X(\\mathcal{G}, \\colim_i \\mathcal{F}_i).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $U$ and a surjective \\'etale morphism", "$U \\to X$. Set $R = U \\times_X U$. Note that $R$ is a quasi-compact", "(as $X$ is quasi-separated and $U$ quasi-compact) and separated (as", "$U$ is separated) scheme. Hence we have", "$$", "\\colim_i \\Hom_U(\\mathcal{G}|_U, \\mathcal{F}_i|_U)", "=", "\\Hom_U(\\mathcal{G}|_U, \\colim_i \\mathcal{F}_i|_U).", "$$", "by Modules, Lemma \\ref{modules-lemma-finite-presentation-quasi-compact-colimit}", "(and the material on restriction to", "schemes \\'etale over $X$, see", "Properties of Spaces, Sections \\ref{spaces-properties-section-quasi-coherent}", "and \\ref{spaces-properties-section-properties-modules}). Similarly for $R$.", "Since $\\QCoh(\\mathcal{O}_X) = \\QCoh(U, R, s, t, c)$ (see", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-quasi-coherent})", "the result follows formally." ], "refs": [ "modules-lemma-finite-presentation-quasi-compact-colimit", "spaces-properties-proposition-quasi-coherent" ], "ref_ids": [ 13252, 11920 ] } ], "ref_ids": [] }, { "id": 11280, "type": "theorem", "label": "spaces-cohomology-lemma-product-is-tensor-product", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-product-is-tensor-product", "contents": [ "Let $S$ be a scheme. Let $f_i : U_i \\to X$ be \\'etale morphisms", "of algebraic spaces over $S$. Then there are isomorphisms", "$$", "f_{1, !}\\underline{\\mathbf{Z}} \\otimes_{\\mathbf{Z}}", "f_{2, !}\\underline{\\mathbf{Z}}", "\\longrightarrow", "f_{12, !}\\underline{\\mathbf{Z}}", "$$", "where $f_{12} : U_1 \\times_X U_2 \\to X$ is the structure morphism", "and", "$$", "(f_1 \\amalg f_2)_! \\underline{\\mathbf{Z}}", "\\longrightarrow", "f_{1, !}\\underline{\\mathbf{Z}} \\oplus", "f_{2, !}\\underline{\\mathbf{Z}}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Once we have defined the map it will be an isomorphism by our description", "of stalks above. To define the map it suffices to work on the level of", "presheaves. Thus we have to define a map", "$$", "\\left(\\bigoplus\\nolimits_{\\varphi_1 \\in \\Mor_X(V, U_1)} \\mathbf{Z}\\right)", "\\otimes_{\\mathbf{Z}}", "\\left(\\bigoplus\\nolimits_{\\varphi_2 \\in \\Mor_X(V, U_2)} \\mathbf{Z}\\right)", "\\longrightarrow", "\\bigoplus\\nolimits_{\\varphi \\in \\Mor_X(V, U_1 \\times_X U_2)}", "\\mathbf{Z}", "$$", "We map the element $1_{\\varphi_1} \\otimes 1_{\\varphi_2}$ to the element", "$1_{\\varphi_1 \\times \\varphi_2}$ with obvious notation. We omit the proof", "of the second equality." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11281, "type": "theorem", "label": "spaces-cohomology-lemma-alternating-cech-to-cohomology", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-alternating-cech-to-cohomology", "contents": [ "Let $S$ be a scheme. Let $f : U \\to X$ be a surjective \\'etale morphism", "of algebraic spaces over $S$. Let $\\mathcal{F}$ be an object of", "$\\textit{Ab}(X_\\etale)$. There exists a canonical map", "$$", "\\check{\\mathcal{C}}^\\bullet_{alt}(f, \\mathcal{F})", "\\longrightarrow", "R\\Gamma(X, \\mathcal{F})", "$$", "in $D(\\textit{Ab})$. Moreover, there is a spectral sequence with $E_1$-page", "$$", "E_1^{p, q} =", "\\Ext_{\\textit{Ab}(X_\\etale)}^q(K^p, \\mathcal{F})", "$$", "converging to $H^{p + q}(X, \\mathcal{F})$ where", "$K^p = \\wedge^{p + 1}f_!\\underline{\\mathbf{Z}}$." ], "refs": [], "proofs": [ { "contents": [ "Recall that we have the quasi-isomorphism", "$K^\\bullet \\to \\underline{\\mathbf{Z}}[0]$, see", "(\\ref{equation-quasi-isomorphism}).", "Choose an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$", "in $\\textit{Ab}(X_\\etale)$. Consider the double complex", "$\\Hom(K^\\bullet, \\mathcal{I}^\\bullet)$ with terms", "$\\Hom(K^p, \\mathcal{I}^q)$. The differential", "$d_1^{p, q} : A^{p, q} \\to A^{p + 1, q}$", "is the one coming from the differential $K^{p + 1} \\to K^p$", "and the differential $d_2^{p, q} : A^{p, q} \\to A^{p, q + 1}$ is the", "one coming from the differential", "$\\mathcal{I}^q \\to \\mathcal{I}^{q + 1}$.", "Denote $\\text{Tot}(\\Hom(K^\\bullet, \\mathcal{I}^\\bullet))$", "the associated total complex, see", "Homology, Section \\ref{homology-section-double-complexes}.", "We will use the two spectral", "sequences $({}'E_r, {}'d_r)$ and $({}''E_r, {}''d_r)$", "associated to this double complex, see", "Homology, Section \\ref{homology-section-double-complex}.", "\\medskip\\noindent", "Because $K^\\bullet$ is a resolution of $\\underline{\\mathbf{Z}}$", "we see that the complexes", "$$", "\\Hom(K^\\bullet, \\mathcal{I}^q) :", "\\Hom(K^0, \\mathcal{I}^q) \\to", "\\Hom(K^1, \\mathcal{I}^q) \\to", "\\Hom(K^2, \\mathcal{I}^q) \\to \\ldots", "$$", "are acyclic in positive degrees and have $H^0$ equal to", "$\\Gamma(X, \\mathcal{I}^q)$. Hence by", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}", "the natural map", "$$", "\\mathcal{I}^\\bullet(X) \\longrightarrow", "\\text{Tot}(\\Hom(K^\\bullet, \\mathcal{I}^\\bullet))", "$$", "is a quasi-isomorphism of complexes of abelian groups. In particular", "we conclude that", "$H^n(\\text{Tot}(\\Hom(K^\\bullet, \\mathcal{I}^\\bullet))) = H^n(X, \\mathcal{F})$.", "\\medskip\\noindent", "The map $\\check{\\mathcal{C}}^\\bullet_{alt}(f, \\mathcal{F}) \\to", "R\\Gamma(X, \\mathcal{F})$ of the lemma is the composition of", "$\\check{\\mathcal{C}}^\\bullet_{alt}(f, \\mathcal{F}) \\to", "\\text{Tot}(\\Hom(K^\\bullet, \\mathcal{I}^\\bullet))$", "with the inverse of the displayed quasi-isomorphism.", "\\medskip\\noindent", "Finally, consider the spectral sequence $({}'E_r, {}'d_r)$.", "We have", "$$", "E_1^{p, q} = q\\text{th cohomology of }", "\\Hom(K^p, \\mathcal{I}^0) \\to", "\\Hom(K^p, \\mathcal{I}^1) \\to", "\\Hom(K^p, \\mathcal{I}^2) \\to \\ldots", "$$", "This proves the lemma." ], "refs": [ "homology-lemma-double-complex-gives-resolution" ], "ref_ids": [ 12106 ] } ], "ref_ids": [] }, { "id": 11282, "type": "theorem", "label": "spaces-cohomology-lemma-compute", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-compute", "contents": [ "Let $S$ be a scheme. Let $f : U \\to X$ be a surjective, \\'etale, and separated", "morphism of algebraic spaces over $S$. For $p \\geq 0$ set", "$$", "W_p = U \\times_X \\ldots \\times_X U \\setminus \\text{all diagonals}", "$$", "where the fibre product has $p + 1$ factors.", "There is a free action of $S_{p + 1}$ on $W_p$ over $X$ and", "$$", "\\Hom(K^p, \\mathcal{F}) = S_{p + 1}\\text{-anti-invariant elements of }", "\\mathcal{F}(W_p)", "$$", "functorially in $\\mathcal{F}$ where", "$K^p = \\wedge^{p + 1}f_!\\underline{\\mathbf{Z}}$." ], "refs": [], "proofs": [ { "contents": [ "Because $U \\to X$ is separated the diagonal $U \\to U \\times_X U$ is a", "closed immersion. Since $U \\to X$ is \\'etale the diagonal", "$U \\to U \\times_X U$ is an open immersion, see", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-etale-unramified} and", "\\ref{spaces-morphisms-lemma-diagonal-unramified-morphism}.", "Hence $W_p$ is an open and closed subspace of", "$U^{p + 1} = U \\times_X \\ldots \\times_X U$. The action of $S_{p + 1}$", "on $W_p$ is free as we've thrown out the fixed points of the action.", "By", "Lemma \\ref{lemma-product-is-tensor-product}", "we see that", "$$", "(f_!\\underline{\\mathbf{Z}})^{\\otimes p + 1} =", "f^{p + 1}_!\\underline{\\mathbf{Z}} = (W_p \\to X)_!\\underline{\\mathbf{Z}}", "\\oplus Rest", "$$", "where $f^{p + 1} : U^{p + 1} \\to X$ is the structure morphism.", "Looking at stalks over a geometric point $\\overline{x}$ of $X$", "we see that", "$$", "\\left(", "\\bigoplus\\nolimits_{\\overline{u} \\mapsto \\overline{x}} \\mathbf{Z}", "\\right)^{\\otimes p + 1}", "\\longrightarrow", "(W_p \\to X)_!\\underline{\\mathbf{Z}}_{\\overline{x}}", "$$", "is the quotient whose kernel is generated by all tensors", "$1_{\\overline{u}_0} \\otimes \\ldots \\otimes 1_{\\overline{u}_p}$", "where $\\overline{u}_i = \\overline{u}_j$ for some $i \\not = j$.", "Thus the quotient map", "$$", "(f_!\\underline{\\mathbf{Z}})^{\\otimes p + 1}", "\\longrightarrow", "\\wedge^{p + 1}f_!\\underline{\\mathbf{Z}}", "$$", "factors through $(W_p \\to X)_!\\underline{\\mathbf{Z}}$, i.e., we get", "$$", "(f_!\\underline{\\mathbf{Z}})^{\\otimes p + 1}", "\\longrightarrow", "(W_p \\to X)_!\\underline{\\mathbf{Z}}", "\\longrightarrow", "\\wedge^{p + 1}f_!\\underline{\\mathbf{Z}}", "$$", "This already proves that $\\Hom(K^p, \\mathcal{F})$ is (functorially) a", "subgroup of", "$$", "\\Hom((W_p \\to X)_!\\underline{\\mathbf{Z}}, \\mathcal{F}) = \\mathcal{F}(W_p)", "$$", "To identify it with the $S_{p + 1}$-anti-invariants we have to prove that", "the surjection $(W_p \\to X)_!\\underline{\\mathbf{Z}}", "\\to \\wedge^{p + 1}f_!\\underline{\\mathbf{Z}}$ is the maximal", "$S_{p + 1}$-anti-invariant quotient. In other words, we have to show that", "$\\wedge^{p + 1}f_!\\underline{\\mathbf{Z}}$ is the quotient of", "$(W_p \\to X)_!\\underline{\\mathbf{Z}}$ by the subsheaf generated by", "the local sections $s - \\text{sign}(\\sigma)\\sigma(s)$ where $s$ is", "a local section of $(W_p \\to X)_!\\underline{\\mathbf{Z}}$.", "This can be checked on the stalks, where it is clear." ], "refs": [ "spaces-morphisms-lemma-etale-unramified", "spaces-morphisms-lemma-diagonal-unramified-morphism", "spaces-cohomology-lemma-product-is-tensor-product" ], "ref_ids": [ 4913, 4902, 11280 ] } ], "ref_ids": [] }, { "id": 11283, "type": "theorem", "label": "spaces-cohomology-lemma-twist", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-twist", "contents": [ "Let $S$ be a scheme. Let $W$ be an algebraic space over $S$.", "Let $G$ be a finite group acting freely on $W$.", "Let $U = W/G$, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quotient}.", "Let $\\chi : G \\to \\{+1, -1\\}$ be a character.", "Then there exists a rank 1 locally free sheaf of $\\mathbf{Z}$-modules", "$\\underline{\\mathbf{Z}}(\\chi)$ on $U_\\etale$ such that for every", "abelian sheaf $\\mathcal{F}$ on $U_\\etale$ we have", "$$", "H^0(W, \\mathcal{F}|_W)^\\chi =", "H^0(U, \\mathcal{F} \\otimes_{\\mathbf{Z}} \\underline{\\mathbf{Z}}(\\chi))", "$$" ], "refs": [ "spaces-properties-lemma-quotient" ], "proofs": [ { "contents": [ "The quotient morphism $q : W \\to U$ is a $G$-torsor, i.e., there exists", "a surjective \\'etale morphism $U' \\to U$ such that", "$W \\times_U U' = \\coprod_{g \\in G} U'$ as spaces with $G$-action over $U'$.", "(Namely, $U' = W$ works.) Hence $q_*\\underline{\\mathbf{Z}}$ is a finite", "locally free $\\mathbf{Z}$-module with an action of $G$. For any", "geometric point $\\overline{u}$ of $U$, then we get $G$-equivariant", "isomorphisms", "$$", "(q_*\\underline{\\mathbf{Z}})_{\\overline{u}}", "= \\bigoplus\\nolimits_{\\overline{w} \\mapsto \\overline{u}} \\mathbf{Z}", "= \\bigoplus\\nolimits_{g \\in G} \\mathbf{Z} = \\mathbf{Z}[G]", "$$", "where the second $=$ uses a geometric point", "$\\overline{w}_0$ lying over $\\overline{u}$ and", "maps the summand corresponding to $g \\in G$ to the summand", "corresponding to $g(\\overline{w}_0)$. We have", "$$", "H^0(W, \\mathcal{F}|_W) =", "H^0(U, \\mathcal{F} \\otimes_\\mathbf{Z} q_*\\underline{\\mathbf{Z}})", "$$", "because", "$q_*\\mathcal{F}|_W = \\mathcal{F} \\otimes_\\mathbf{Z} q_*\\underline{\\mathbf{Z}}$", "as one can check by restricting to $U'$. Let", "$$", "\\underline{\\mathbf{Z}}(\\chi) =", "(q_*\\underline{\\mathbf{Z}})^\\chi \\subset", "q_*\\underline{\\mathbf{Z}}", "$$", "be the subsheaf of sections that transform according to $\\chi$. For", "any geometric point $\\overline{u}$ of $U$ we have", "$$", "\\underline{\\mathbf{Z}}(\\chi)_{\\overline{u}} =", "\\mathbf{Z} \\cdot \\sum\\nolimits_g \\chi(g) g", "\\subset", "\\mathbf{Z}[G] = (q_*\\underline{\\mathbf{Z}})_{\\overline{u}}", "$$", "It follows that $\\underline{\\mathbf{Z}}(\\chi)$ is locally free of", "rank 1 (more precisely, this should be checked after restricting to $U'$).", "Note that for any $\\mathbf{Z}$-module $M$ the $\\chi$-semi-invariants", "of $M[G]$ are the elements of the form $m \\cdot \\sum\\nolimits_g \\chi(g) g$.", "Thus we see that for any abelian sheaf $\\mathcal{F}$ on $U$ we have", "$$", "\\left(\\mathcal{F} \\otimes_\\mathbf{Z} q_*\\underline{\\mathbf{Z}}\\right)^\\chi", "=", "\\mathcal{F} \\otimes_\\mathbf{Z} \\underline{\\mathbf{Z}}(\\chi)", "$$", "because we have equality at all stalks. The result of the lemma follows by", "taking global sections." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 11916 ] }, { "id": 11284, "type": "theorem", "label": "spaces-cohomology-lemma-alternating-spectral-sequence", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-alternating-spectral-sequence", "contents": [ "Let $S$ be a scheme. Let $f : U \\to X$ be a surjective, \\'etale, and", "separated morphism of algebraic spaces over $S$. For $p \\geq 0$ set", "$$", "W_p = U \\times_X \\ldots \\times_X U \\setminus \\text{all diagonals}", "$$", "(with $p + 1$ factors) as in Lemma \\ref{lemma-compute}.", "Let $\\chi_p : S_{p + 1} \\to \\{+1, -1\\}$ be the sign character.", "Let $U_p = W_p/S_{p + 1}$ and $\\underline{\\mathbf{Z}}(\\chi_p)$ be as in", "Lemma \\ref{lemma-twist}.", "Then the spectral sequence of", "Lemma \\ref{lemma-alternating-cech-to-cohomology}", "has $E_1$-page", "$$", "E_1^{p, q} =", "H^q(U_p, \\mathcal{F}|_{U_p} \\otimes_\\mathbf{Z} \\underline{\\mathbf{Z}}(\\chi_p))", "$$", "and converges to $H^{p + q}(X, \\mathcal{F})$." ], "refs": [ "spaces-cohomology-lemma-compute", "spaces-cohomology-lemma-twist", "spaces-cohomology-lemma-alternating-cech-to-cohomology" ], "proofs": [ { "contents": [ "Note that since the action of $S_{p + 1}$ on $W_p$ is over $X$ we do", "obtain a morphism $U_p \\to X$. Since $W_p \\to X$ is \\'etale and since", "$W_p \\to U_p$ is surjective \\'etale, it follows", "that also $U_p \\to X$ is \\'etale, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-etale-local}.", "Therefore an injective object of", "$\\textit{Ab}(X_\\etale)$ restricts to an injective object of", "$\\textit{Ab}(U_{p, \\etale})$, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-of-open}.", "Moreover, the functor", "$\\mathcal{G} \\mapsto", "\\mathcal{G} \\otimes_\\mathbf{Z} \\underline{\\mathbf{Z}}(\\chi_p))$", "is an auto-equivalence of $\\textit{Ab}(U_p)$, whence transforms injective", "objects into injective objects and is exact (because", "$\\underline{\\mathbf{Z}}(\\chi_p)$ is an invertible", "$\\underline{\\mathbf{Z}}$-module). Thus given an injective resolution", "$\\mathcal{F} \\to \\mathcal{I}^\\bullet$ in $\\textit{Ab}(X_\\etale)$", "the complex", "$$", "\\Gamma(U_p,", "\\mathcal{I}^0|_{U_p} \\otimes_\\mathbf{Z} \\underline{\\mathbf{Z}}(\\chi_p))", "\\to", "\\Gamma(U_p,", "\\mathcal{I}^1|_{U_p} \\otimes_\\mathbf{Z} \\underline{\\mathbf{Z}}(\\chi_p))", "\\to", "\\Gamma(U_p,", "\\mathcal{I}^2|_{U_p} \\otimes_\\mathbf{Z} \\underline{\\mathbf{Z}}(\\chi_p))", "\\to \\ldots", "$$", "computes", "$H^*(U_p,", "\\mathcal{F}|_{U_p} \\otimes_\\mathbf{Z} \\underline{\\mathbf{Z}}(\\chi_p))$.", "On the other hand, by", "Lemma \\ref{lemma-twist}", "it is equal to the complex of $S_{p + 1}$-anti-invariants in", "$$", "\\Gamma(W_p, \\mathcal{I}^0) \\to", "\\Gamma(W_p, \\mathcal{I}^1) \\to", "\\Gamma(W_p, \\mathcal{I}^2) \\to \\ldots", "$$", "which by", "Lemma \\ref{lemma-compute}", "is equal to the complex", "$$", "\\Hom(K^p, \\mathcal{I}^0) \\to", "\\Hom(K^p, \\mathcal{I}^1) \\to", "\\Hom(K^p, \\mathcal{I}^2) \\to \\ldots", "$$", "which computes", "$\\Ext^*_{\\textit{Ab}(X_\\etale)}(K^p, \\mathcal{F})$.", "Putting everything together we win." ], "refs": [ "spaces-morphisms-lemma-etale-local", "sites-cohomology-lemma-cohomology-of-open", "spaces-cohomology-lemma-twist", "spaces-cohomology-lemma-compute" ], "ref_ids": [ 4905, 4186, 11283, 11282 ] } ], "ref_ids": [ 11282, 11283, 11281 ] }, { "id": 11285, "type": "theorem", "label": "spaces-cohomology-lemma-quasi-coherent-twist", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-quasi-coherent-twist", "contents": [ "With $S$, $W$, $G$, $U$, $\\chi$ as in", "Lemma \\ref{lemma-twist}.", "If $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_U$-module,", "then so is $\\mathcal{F} \\otimes_{\\mathbf{Z}} \\underline{\\mathbf{Z}}(\\chi)$." ], "refs": [ "spaces-cohomology-lemma-twist" ], "proofs": [ { "contents": [ "The $\\mathcal{O}_U$-module structure is clear. To check that", "$\\mathcal{F} \\otimes_{\\mathbf{Z}} \\underline{\\mathbf{Z}}(\\chi)$", "is quasi-coherent it suffices to check \\'etale locally.", "Hence the lemma follows as $\\underline{\\mathbf{Z}}(\\chi)$", "is finite locally free as a $\\underline{\\mathbf{Z}}$-module." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 11283 ] }, { "id": 11286, "type": "theorem", "label": "spaces-cohomology-lemma-vanishing-quasi-separated", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-vanishing-quasi-separated", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Assume $X$ is quasi-compact and quasi-separated.", "Then we can choose", "\\begin{enumerate}", "\\item an affine scheme $U$,", "\\item a surjective \\'etale morphism $f : U \\to X$,", "\\item an integer $d$ bounding the degrees of the fibres of $U \\to X$,", "\\item for every $p = 0, 1, \\ldots, d$ a surjective \\'etale morphism", "$V_p \\to U_p$ from an affine scheme $V_p$ where $U_p$ is as in", "Lemma \\ref{lemma-alternating-spectral-sequence}, and", "\\item an integer $d_p$ bounding the degree of the fibres of $V_p \\to U_p$.", "\\end{enumerate}", "Moreover, whenever we have (1) -- (5), then for any quasi-coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ we have $H^q(X, \\mathcal{F}) = 0$ for", "$q \\geq \\max(d_p + p)$." ], "refs": [ "spaces-cohomology-lemma-alternating-spectral-sequence" ], "proofs": [ { "contents": [ "Since $X$ is quasi-compact we can find a surjective \\'etale morphism", "$U \\to X$ with $U$ affine, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}.", "By", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-bounded-fibres}", "the fibres of $f$ are universally bounded, hence we can find $d$.", "We have $U_p = W_p/S_{p + 1}$ and $W_p \\subset U \\times_X \\ldots \\times_X U$", "is open and closed. Since $X$ is quasi-separated the schemes", "$W_p$ are quasi-compact, hence $U_p$ is quasi-compact.", "Since $U$ is separated, the schemes $W_p$ are separated, hence", "$U_p$ is separated by (the absolute version of)", "Spaces, Lemma \\ref{spaces-lemma-quotient-finite-separated}.", "By", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}", "we can find the morphisms $V_p \\to W_p$.", "By", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-bounded-fibres}", "we can find the integers $d_p$.", "\\medskip\\noindent", "At this point the proof uses the spectral sequence", "$$", "E_1^{p, q} =", "H^q(U_p, \\mathcal{F}|_{U_p} \\otimes_\\mathbf{Z} \\underline{\\mathbf{Z}}(\\chi_p))", "\\Rightarrow", "H^{p + q}(X, \\mathcal{F})", "$$", "see", "Lemma \\ref{lemma-alternating-spectral-sequence}.", "By definition of the integer $d$ we see that $U_p = 0$ for $p \\geq d$.", "By Proposition \\ref{proposition-vanishing}", "and", "Lemma \\ref{lemma-quasi-coherent-twist}", "we see that", "$H^q(U_p,", "\\mathcal{F}|_{U_p} \\otimes_\\mathbf{Z} \\underline{\\mathbf{Z}}(\\chi_p))$", "is zero for $q \\geq d_p$ for $p = 0, \\ldots, d$.", "Whence the lemma." ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "decent-spaces-lemma-bounded-fibres", "spaces-lemma-quotient-finite-separated", "spaces-properties-lemma-quasi-compact-affine-cover", "decent-spaces-lemma-bounded-fibres", "spaces-cohomology-lemma-alternating-spectral-sequence", "spaces-cohomology-proposition-vanishing", "spaces-cohomology-lemma-quasi-coherent-twist" ], "ref_ids": [ 11832, 9466, 8165, 11832, 9466, 11284, 11345, 11285 ] } ], "ref_ids": [ 11284 ] }, { "id": 11287, "type": "theorem", "label": "spaces-cohomology-lemma-vanishing-higher-direct-images", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-vanishing-higher-direct-images", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a", "morphism of algebraic spaces over $S$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is quasi-compact and quasi-separated, and", "\\item $Y$ is quasi-compact.", "\\end{enumerate}", "Then there exists an integer $n(X \\to Y)$ such that", "for any algebraic space $Y'$, any morphism $Y' \\to Y$", "and any quasi-coherent sheaf $\\mathcal{F}'$ on $X' = Y' \\times_Y X$", "the higher direct images $R^if'_*\\mathcal{F}'$ are zero for", "$i \\geq n(X \\to Y)$." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\to Y$ be a surjective \\'etale morphism where $V$ is an affine", "scheme, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}.", "Suppose we prove the result for the base change $f_V : V \\times_Y X \\to V$.", "Then the result holds for $f$ with $n(X \\to Y) = n(X_V \\to V)$.", "Namely, if $Y' \\to Y$ and $\\mathcal{F}'$ are as in the lemma, then", "$R^if'_*\\mathcal{F}'|_{V \\times_Y Y'}$ is equal to", "$R^if'_{V, *}\\mathcal{F}'|_{X'_V}$ where", "$f'_V : X'_V = V \\times_Y Y' \\times_Y X \\to V \\times_Y Y' = Y'_V$, see", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-pushforward-etale-base-change-modules}.", "Thus we may assume that $Y$ is an affine scheme.", "\\medskip\\noindent", "Moreover, to prove the vanishing for all $Y' \\to Y$ and", "$\\mathcal{F}'$ it suffices to do so when $Y'$ is an affine scheme.", "In this case, $R^if'_*\\mathcal{F}'$ is quasi-coherent by", "Lemma \\ref{lemma-higher-direct-image}.", "Hence it suffices to prove that $H^i(X', \\mathcal{F}') = 0$, because", "$H^i(X', \\mathcal{F}') = H^0(Y', R^if'_*\\mathcal{F}')$ by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-apply-Leray}", "and the vanishing of higher cohomology of quasi-coherent sheaves", "on affine algebraic spaces", "(Proposition \\ref{proposition-vanishing}).", "\\medskip\\noindent", "Choose $U \\to X$, $d$, $V_p \\to U_p$ and $d_p$ as in", "Lemma \\ref{lemma-vanishing-quasi-separated}.", "For any affine scheme $Y'$ and morphism $Y' \\to Y$ denote", "$X' = Y' \\times_Y X$, $U' = Y' \\times_Y U$, $V'_p = Y' \\times_Y V_p$.", "Then $U' \\to X'$, $d' = d$, $V'_p \\to U'_p$ and $d'_p = d$", "is a collection of choices as in", "Lemma \\ref{lemma-vanishing-quasi-separated}", "for the algebraic space $X'$ (details omitted).", "Hence we see that $H^i(X', \\mathcal{F}') = 0$ for $i \\geq \\max(p + d_p)$", "and we win." ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-properties-lemma-pushforward-etale-base-change-modules", "spaces-cohomology-lemma-higher-direct-image", "sites-cohomology-lemma-apply-Leray", "spaces-cohomology-proposition-vanishing", "spaces-cohomology-lemma-vanishing-quasi-separated", "spaces-cohomology-lemma-vanishing-quasi-separated" ], "ref_ids": [ 11832, 11898, 11271, 4221, 11345, 11286, 11286 ] } ], "ref_ids": [] }, { "id": 11288, "type": "theorem", "label": "spaces-cohomology-lemma-affine-vanishing-higher-direct-images", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-affine-vanishing-higher-direct-images", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine", "morphism of algebraic spaces over $S$. Then", "$R^if_*\\mathcal{F} = 0$ for $i > 0$ and any quasi-coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Recall that an affine morphism of algebraic spaces is representable.", "Hence this follows from (\\ref{equation-representable-higher-direct-image}) and", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-vanishing}." ], "refs": [ "coherent-lemma-relative-affine-vanishing" ], "ref_ids": [ 3283 ] } ], "ref_ids": [] }, { "id": 11289, "type": "theorem", "label": "spaces-cohomology-lemma-relative-affine-cohomology", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-relative-affine-cohomology", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine", "morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then $H^i(X, \\mathcal{F}) = H^i(Y, f_*\\mathcal{F})$ for all $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-affine-vanishing-higher-direct-images}", "and the Leray spectral sequence. See", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-apply-Leray}." ], "refs": [ "spaces-cohomology-lemma-affine-vanishing-higher-direct-images", "sites-cohomology-lemma-apply-Leray" ], "ref_ids": [ 11288, 4221 ] } ], "ref_ids": [] }, { "id": 11290, "type": "theorem", "label": "spaces-cohomology-lemma-sections-with-support-acyclic", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-sections-with-support-acyclic", "contents": [ "Let $S$ be a scheme.", "Let $i : Z \\to X$ be a closed immersion of algebraic spaces over $S$.", "Let $\\mathcal{I}$ be an injective abelian sheaf on $X_\\etale$.", "Then $\\mathcal{H}_Z(\\mathcal{I})$ is an injective abelian sheaf", "on $Z_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "Observe that for any abelian sheaf $\\mathcal{G}$ on $Z_\\etale$", "we have", "$$", "\\Hom_Z(\\mathcal{G}, \\mathcal{H}_Z(\\mathcal{F})) =", "\\Hom_X(i_*\\mathcal{G}, \\mathcal{F})", "$$", "because after all any section of $i_*\\mathcal{G}$ has support in $Z$.", "Since $i_*$ is exact (Lemma \\ref{lemma-finite-higher-direct-image-zero})", "and as $\\mathcal{I}$ is injective on $X_\\etale$ we conclude that", "$\\mathcal{H}_Z(\\mathcal{I})$ is injective on $Z_\\etale$." ], "refs": [ "spaces-cohomology-lemma-finite-higher-direct-image-zero" ], "ref_ids": [ 11273 ] } ], "ref_ids": [] }, { "id": 11291, "type": "theorem", "label": "spaces-cohomology-lemma-cohomology-with-support-sheaf-on-support", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-cohomology-with-support-sheaf-on-support", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of", "algebraic spaces over $S$. Let $\\mathcal{G}$ be an injective abelian", "sheaf on $Z_\\etale$. Then $\\mathcal{H}^p_Z(i_*\\mathcal{G}) = 0$ for $p > 0$." ], "refs": [], "proofs": [ { "contents": [ "This is true because the functor $i_*$ is exact", "(Lemma \\ref{lemma-finite-higher-direct-image-zero}) and transforms", "injective abelian sheaves into injective abelian sheaves", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-pushforward-injective-flat})." ], "refs": [ "spaces-cohomology-lemma-finite-higher-direct-image-zero", "sites-cohomology-lemma-pushforward-injective-flat" ], "ref_ids": [ 11273, 4218 ] } ], "ref_ids": [] }, { "id": 11292, "type": "theorem", "label": "spaces-cohomology-lemma-etale-localization-sheaf-with-support", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-etale-localization-sheaf-with-support", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an \\'etale morphism of", "algebraic spaces over $S$. Let $Z \\subset Y$ be a closed subspace", "such that $f^{-1}(Z) \\to Z$ is an isomorphism of algebraic spaces.", "Let $\\mathcal{F}$ be an abelian sheaf on $X$. Then", "$$", "\\mathcal{H}^q_Z(\\mathcal{F}) = \\mathcal{H}^q_{f^{-1}(Z)}(f^{-1}\\mathcal{F})", "$$", "as abelian sheaves on $Z = f^{-1}(Z)$ and we", "have $H^q_Z(Y, \\mathcal{F}) = H^q_{f^{-1}(Z)}(X, f^{-1}\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Because $f$ is \\'etale an injective resolution of $\\mathcal{F}$", "pulls back to an injective resolution of $f^{-1}\\mathcal{F}$.", "Hence it suffices to check the equality for $\\mathcal{H}_Z(-)$", "which follows from the definitions. The proof for cohomology with", "supports is the same. Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11293, "type": "theorem", "label": "spaces-cohomology-lemma-complexes-with-support-on-closed", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-complexes-with-support-on-closed", "contents": [ "Let $S$ be a scheme.", "Let $i : Z \\to X$ be a closed immersion of algebraic spaces over $S$.", "The map $Ri_* = i_* : D(Z_\\etale) \\to D(X_\\etale)$", "induces an equivalence $D(Z_\\etale) \\to D_{|Z|}(X_\\etale)$ with quasi-inverse", "$$", "i^{-1}|_{D_Z(X_\\etale)} = R\\mathcal{H}_Z|_{D_{|Z|}(X_\\etale)}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Recall that $i^{-1}$ and $i_*$ is an adjoint pair of", "exact functors such that $i^{-1}i_*$ is isomorphic to the identify", "functor on abelian sheaves. See", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-stalk-pullback} and", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-closed-immersion-push-pull}.", "Thus $i_* : D(Z_\\etale) \\to D_Z(X_\\etale)$ is fully faithful and", "$i^{-1}$ determines", "a left inverse. On the other hand, suppose that $K$ is an object of", "$D_Z(X_\\etale)$ and consider the adjunction map", "$K \\to i_*i^{-1}K$.", "Using exactness of $i_*$ and $i^{-1}$", "this induces the adjunction maps", "$H^n(K) \\to i_*i^{-1}H^n(K)$ on cohomology sheaves.", "Since these cohomology", "sheaves are supported on $Z$ we see these adjunction maps are isomorphisms", "and we conclude that $D(Z_\\etale) \\to D_Z(X_\\etale)$ is an equivalence.", "\\medskip\\noindent", "To finish the proof we have to show that $R\\mathcal{H}_Z(K) = i^{-1}K$", "if $K$ is an object of $D_Z(X_\\etale)$. To do this we can use that", "$K = i_*i^{-1}K$", "as we've just proved this is the case. Then we", "can choose a K-injective representative $\\mathcal{I}^\\bullet$ for", "$i^{-1}K$.", "Since $i_*$ is the right adjoint to the exact functor", "$i^{-1}$, the", "complex $i_*\\mathcal{I}^\\bullet$ is K-injective", "(Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}).", "We see that $R\\mathcal{H}_Z(K)$ is computed by", "$\\mathcal{H}_Z(i_*\\mathcal{I}^\\bullet) = \\mathcal{I}^\\bullet$", "as desired." ], "refs": [ "spaces-properties-lemma-stalk-pullback", "spaces-morphisms-lemma-closed-immersion-push-pull", "derived-lemma-adjoint-preserve-K-injectives" ], "ref_ids": [ 11875, 4768, 1915 ] } ], "ref_ids": [] }, { "id": 11294, "type": "theorem", "label": "spaces-cohomology-lemma-vanishing-above-dimension", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-vanishing-above-dimension", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. Assume $\\dim(X) \\leq d$ for some integer $d$.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf $\\mathcal{F}$ on $X$.", "\\begin{enumerate}", "\\item $H^q(X, \\mathcal{F}) = 0$ for $q > d$,", "\\item $H^d(X, \\mathcal{F}) \\to H^d(U, \\mathcal{F})$ is surjective", "for any quasi-compact open $U \\subset X$,", "\\item $H^q_Z(X, \\mathcal{F}) = 0$ for $q > d$ for any closed subspace", "$Z \\subset X$ whose complement is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-dimension-decent-invariant-under-etale}", "every algebraic space $Y$ \\'etale over $X$ has dimension $\\leq d$.", "If $Y$ is quasi-separated, the dimension of $Y$ is equal to the", "Krull dimension of $|Y|$ by ", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-dimension-decent-space}.", "Also, if $Y$ is a scheme, then \\'etale cohomology of $\\mathcal{F}$", "over $Y$, resp.\\ \\'etale cohomology of $\\mathcal{F}$ with support in a", "closed subscheme, agrees with usual cohomology of $\\mathcal{F}$,", "resp.\\ usual cohomology with support in the closed subscheme.", "See", "Descent, Proposition \\ref{descent-proposition-same-cohomology-quasi-coherent}", "and", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-cohomology-with-support-quasi-coherent}.", "We will use these facts without further mention.", "\\medskip\\noindent", "By Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated}", "there exist an integer $n$ and open subspaces", "$$", "\\emptyset = U_{n + 1} \\subset", "U_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X", "$$", "with the following property: setting $T_p = U_p \\setminus U_{p + 1}$", "(with reduced induced subspace structure) there exists a quasi-compact", "separated scheme $V_p$ and a surjective \\'etale morphism $f_p : V_p \\to U_p$", "such that $f_p^{-1}(T_p) \\to T_p$ is an isomorphism.", "\\medskip\\noindent", "As $U_n = V_n$ is a scheme, our initial remarks imply the cohomology of", "$\\mathcal{F}$ over $U_n$ vanishes in degrees $> d$ by", "Cohomology, Proposition", "\\ref{cohomology-proposition-cohomological-dimension-spectral}.", "Suppose we have shown, by induction, that", "$H^q(U_{p + 1}, \\mathcal{F}|_{U_{p + 1}}) = 0$ for $q > d$.", "It suffices to show $H_{T_p}^q(U_p, \\mathcal{F})$ for", "$q > d$ is zero in order to conclude the vanishing of cohomology", "of $\\mathcal{F}$ over $U_p$ in degrees $> d$.", "However, we have", "$$", "H^q_{T_p}(U_p, \\mathcal{F}) = H^q_{f_p^{-1}(T_p)}(V_p, \\mathcal{F})", "$$", "by Lemma \\ref{lemma-etale-localization-sheaf-with-support}", "and as $V_p$ is a scheme we obtain the desired vanishing from", "Cohomology, Proposition", "\\ref{cohomology-proposition-cohomological-dimension-spectral}.", "In this way we conclude that (1) is true.", "\\medskip\\noindent", "To prove (2) let $U \\subset X$ be a quasi-compact open subspace.", "Consider the open subspace $U' = U \\cup U_n$. Let $Z = U' \\setminus U$.", "Then $g : U_n \\to U'$ is an \\'etale morphism such that", "$g^{-1}(Z) \\to Z$ is an isomorphism. Hence by", "Lemma \\ref{lemma-etale-localization-sheaf-with-support}", "we have $H^q_Z(U', \\mathcal{F}) = H^q_Z(U_n, \\mathcal{F})$", "which vanishes in degree $> d$ because $U_n$ is a scheme", "and we can apply", "Cohomology, Proposition", "\\ref{cohomology-proposition-cohomological-dimension-spectral}.", "We conclude that $H^d(U', \\mathcal{F}) \\to H^d(U, \\mathcal{F})$", "is surjective. Assume, by induction, that we have reduced", "our problem to the case where $U$ contains $U_{p + 1}$.", "Then we set $U' = U \\cup U_p$, set $Z = U' \\setminus U$, and", "we argue using the morphism $f_p : V_p \\to U'$ which is \\'etale", "and has the property that $f_p^{-1}(Z) \\to Z$ is an isomorphism.", "In other words, we again see that", "$$", "H^q_Z(U', \\mathcal{F}) = H^q_{f_p^{-1}(Z)}(V_p, \\mathcal{F})", "$$", "and we again see this vanishes in degrees $> d$.", "We conclude that $H^d(U', \\mathcal{F}) \\to H^d(U, \\mathcal{F})$", "is surjective. Eventually we reach the stage where $U_1 = X \\subset U$", "which finishes the proof.", "\\medskip\\noindent", "A formal argument shows that (2) implies (3)." ], "refs": [ "spaces-properties-lemma-dimension-decent-invariant-under-etale", "decent-spaces-lemma-dimension-decent-space", "descent-proposition-same-cohomology-quasi-coherent", "etale-cohomology-lemma-cohomology-with-support-quasi-coherent", "decent-spaces-lemma-filter-quasi-compact-quasi-separated", "cohomology-proposition-cohomological-dimension-spectral", "spaces-cohomology-lemma-etale-localization-sheaf-with-support", "cohomology-proposition-cohomological-dimension-spectral", "spaces-cohomology-lemma-etale-localization-sheaf-with-support", "cohomology-proposition-cohomological-dimension-spectral" ], "ref_ids": [ 11887, 9496, 14754, 6570, 9480, 2247, 11292, 2247, 11292, 2247 ] } ], "ref_ids": [] }, { "id": 11295, "type": "theorem", "label": "spaces-cohomology-lemma-affine-base-change", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-affine-base-change", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic", "spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "In this case $f_*\\mathcal{F} \\cong Rf_*\\mathcal{F}$ is a quasi-coherent", "sheaf, and for every diagram (\\ref{equation-base-change-diagram})", "we have", "$$", "g^*f_*\\mathcal{F} = f'_*(g')^*\\mathcal{F}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "By the discussion surrounding", "(\\ref{equation-representable-higher-direct-image})", "this reduces to the case of an affine morphism of schemes which", "is treated in Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-affine-base-change}." ], "refs": [ "coherent-lemma-affine-base-change" ], "ref_ids": [ 3297 ] } ], "ref_ids": [] }, { "id": 11296, "type": "theorem", "label": "spaces-cohomology-lemma-flat-base-change-cohomology", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-flat-base-change-cohomology", "contents": [ "Let $S$ be a scheme. Consider a cartesian diagram of algebraic spaces", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module", "with pullback $\\mathcal{F}' = (g')^*\\mathcal{F}$.", "Assume that $g$ is flat and that $f$ is quasi-compact and quasi-separated.", "For any $i \\geq 0$", "\\begin{enumerate}", "\\item the base change map of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-base-change-map-flat-case}", "is an isomorphism", "$$", "g^*R^if_*\\mathcal{F} \\longrightarrow R^if'_*\\mathcal{F}',", "$$", "\\item if $Y = \\Spec(A)$ and $Y' = \\Spec(B)$, then", "$H^i(X, \\mathcal{F}) \\otimes_A B = H^i(X', \\mathcal{F}')$.", "\\end{enumerate}" ], "refs": [ "sites-cohomology-lemma-base-change-map-flat-case" ], "proofs": [ { "contents": [ "The morphism $g'$ is flat by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-flat}.", "Note that flatness of $g$ and $g'$ is equivalent to flatness", "of the morphisms of small \\'etale ringed sites, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-flat-morphism-sites}.", "Hence we can apply", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-base-change-map-flat-case}", "to obtain a base change map", "$$", "g^*R^pf_*\\mathcal{F} \\longrightarrow R^pf'_*\\mathcal{F}'", "$$", "To prove this map is an isomorphism we can work locally in the \\'etale", "topology on $Y'$. Thus we may assume that $Y$ and $Y'$ are affine", "schemes. Say $Y = \\Spec(A)$ and $Y' = \\Spec(B)$.", "In this case we are really trying to show that the map", "$$", "H^p(X, \\mathcal{F}) \\otimes_A B \\longrightarrow H^p(X_B, \\mathcal{F}_B)", "$$", "is an isomorphism where $X_B = \\Spec(B) \\times_{\\Spec(A)} X$ and", "$\\mathcal{F}_B$ is the pullback of $\\mathcal{F}$ to $X_B$.", "In other words, it suffices to prove (2).", "\\medskip\\noindent", "Fix $A \\to B$ a flat ring map and let $X$ be a quasi-compact and", "quasi-separated algebraic space over $A$. Note that $g' : X_B \\to X$", "is affine as a base change of $\\Spec(B) \\to \\Spec(A)$. Hence", "the higher direct images $R^i(g')_*\\mathcal{F}_B$ are zero by", "Lemma \\ref{lemma-affine-vanishing-higher-direct-images}.", "Thus $H^p(X_B, \\mathcal{F}_B) = H^p(X, g'_*\\mathcal{F}_B)$, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-apply-Leray}.", "Moreover, we have", "$$", "g'_*\\mathcal{F}_B = \\mathcal{F} \\otimes_{\\underline{A}} \\underline{B}", "$$", "where $\\underline{A}$, $\\underline{B}$ denotes the constant sheaf of", "rings with value $A$, $B$. Namely, it is clear that there is a map", "from right to left. For any affine scheme $U$ \\'etale over $X$ we have", "\\begin{align*}", "g'_*\\mathcal{F}_B(U) & = \\mathcal{F}_B(\\Spec(B) \\times_{\\Spec(A)} U) \\\\", "& =", "\\Gamma(\\Spec(B) \\times_{\\Spec(A)} U,", "(\\Spec(B) \\times_{\\Spec(A)} U \\to U)^*\\mathcal{F}|_U) \\\\", "& =", "B \\otimes_A \\mathcal{F}(U)", "\\end{align*}", "hence the map is an isomorphism. Write $B = \\colim M_i$ as a filtered", "colimit of finite free $A$-modules $M_i$ using Lazard's theorem, see", "Algebra, Theorem \\ref{algebra-theorem-lazard}.", "We deduce that", "\\begin{align*}", "H^p(X, g'_*\\mathcal{F}_B) &", "= H^p(X, \\mathcal{F} \\otimes_{\\underline{A}} \\underline{B}) \\\\", "& = H^p(X, \\colim_i \\mathcal{F} \\otimes_{\\underline{A}} \\underline{M_i}) \\\\", "& = \\colim_i H^p(X, \\mathcal{F} \\otimes_{\\underline{A}} \\underline{M_i}) \\\\", "& = \\colim_i H^p(X, \\mathcal{F}) \\otimes_A M_i \\\\", "& = H^p(X, \\mathcal{F}) \\otimes_A \\colim_i M_i \\\\", "& = H^p(X, \\mathcal{F}) \\otimes_A B", "\\end{align*}", "The first equality because", "$g'_*\\mathcal{F}_B = \\mathcal{F} \\otimes_{\\underline{A}} \\underline{B}$", "as seen above.", "The second because $\\otimes$ commutes with colimits.", "The third equality because cohomology on $X$ commutes with", "colimits (see", "Lemma \\ref{lemma-colimits}).", "The fourth equality because $M_i$ is finite free (i.e., because cohomology", "commutes with finite direct sums).", "The fifth because $\\otimes$ commutes with colimits.", "The sixth by choice of our system." ], "refs": [ "spaces-morphisms-lemma-base-change-flat", "spaces-morphisms-lemma-flat-morphism-sites", "sites-cohomology-lemma-base-change-map-flat-case", "spaces-cohomology-lemma-affine-vanishing-higher-direct-images", "sites-cohomology-lemma-apply-Leray", "algebra-theorem-lazard", "spaces-cohomology-lemma-colimits" ], "ref_ids": [ 4853, 4858, 4223, 11288, 4221, 318, 11277 ] } ], "ref_ids": [ 4223 ] }, { "id": 11297, "type": "theorem", "label": "spaces-cohomology-lemma-coherent-Noetherian", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-coherent-Noetherian", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a locally Noetherian algebraic space over $S$.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is coherent,", "\\item $\\mathcal{F}$ is a quasi-coherent, finite type $\\mathcal{O}_X$-module,", "\\item $\\mathcal{F}$ is a finitely presented $\\mathcal{O}_X$-module,", "\\item for any \\'etale morphism $\\varphi : U \\to X$ where $U$ is a scheme", "the pullback $\\varphi^*\\mathcal{F}$ is a coherent module on $U$, and", "\\item there exists a surjective \\'etale morphism $\\varphi : U \\to X$", "where $U$ is a scheme such that the pullback $\\varphi^*\\mathcal{F}$ is", "a coherent module on $U$.", "\\end{enumerate}", "In particular $\\mathcal{O}_X$ is coherent, any invertible", "$\\mathcal{O}_X$-module is coherent, and more generally any", "finite locally free $\\mathcal{O}_X$-module is coherent." ], "refs": [], "proofs": [ { "contents": [ "To be sure, if $X$ is a locally Noetherian algebraic space and", "$U \\to X$ is an \\'etale morphism, then $U$ is locally Noetherian, see", "Properties of Spaces, Section \\ref{spaces-properties-section-types-properties}.", "The lemma then follows from the points (1) -- (5) made in", "Properties of Spaces, Section \\ref{spaces-properties-section-properties-modules}", "and the corresponding result for coherent modules on locally", "Noetherian schemes, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-Noetherian}." ], "refs": [ "coherent-lemma-coherent-Noetherian" ], "ref_ids": [ 3308 ] } ], "ref_ids": [] }, { "id": 11298, "type": "theorem", "label": "spaces-cohomology-lemma-coherent-abelian-Noetherian", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-coherent-abelian-Noetherian", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.", "The category of coherent $\\mathcal{O}_X$-modules is abelian. More precisely,", "the kernel and cokernel of a map of coherent $\\mathcal{O}_X$-modules are", "coherent. Any extension of coherent sheaves is coherent." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $f : U \\to X$.", "Pullback $f^*$ is an exact functor as it equals a restriction functor, see", "Properties of Spaces, Equation", "(\\ref{spaces-properties-equation-restrict-modules}).", "By", "Lemma \\ref{lemma-coherent-Noetherian} we can check whether an", "$\\mathcal{O}_X$-module $\\mathcal{F}$ is", "coherent by checking whether $f^*\\mathcal{F}$ is coherent. Hence the", "lemma follows from the case of schemes which is", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-abelian-Noetherian}." ], "refs": [ "spaces-cohomology-lemma-coherent-Noetherian", "coherent-lemma-coherent-abelian-Noetherian" ], "ref_ids": [ 11297, 3309 ] } ], "ref_ids": [] }, { "id": 11299, "type": "theorem", "label": "spaces-cohomology-lemma-coherent-Noetherian-quasi-coherent-sub-quotient", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-coherent-Noetherian-quasi-coherent-sub-quotient", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a locally Noetherian algebraic space over $S$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Any quasi-coherent submodule of $\\mathcal{F}$ is coherent.", "Any quasi-coherent quotient module of $\\mathcal{F}$ is coherent." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $f : U \\to X$.", "Pullback $f^*$ is an exact functor as it equals a restriction functor, see", "Properties of Spaces, Equation", "(\\ref{spaces-properties-equation-restrict-modules}).", "By", "Lemma \\ref{lemma-coherent-Noetherian} we can check whether an", "$\\mathcal{O}_X$-module $\\mathcal{G}$ is", "coherent by checking whether $f^*\\mathcal{H}$ is coherent. Hence the", "lemma follows from the case of schemes which is", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient}." ], "refs": [ "spaces-cohomology-lemma-coherent-Noetherian", "coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient" ], "ref_ids": [ 11297, 3310 ] } ], "ref_ids": [] }, { "id": 11300, "type": "theorem", "label": "spaces-cohomology-lemma-tensor-hom-coherent", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-tensor-hom-coherent", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a locally Noetherian algebraic space over $S$,.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.", "The $\\mathcal{O}_X$-modules $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$", "and $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ are", "coherent." ], "refs": [], "proofs": [ { "contents": [ "Via Lemma \\ref{lemma-coherent-Noetherian} this follows from the result", "for schemes, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-tensor-hom-coherent}." ], "refs": [ "spaces-cohomology-lemma-coherent-Noetherian", "coherent-lemma-tensor-hom-coherent" ], "ref_ids": [ 11297, 3311 ] } ], "ref_ids": [] }, { "id": 11301, "type": "theorem", "label": "spaces-cohomology-lemma-local-isomorphism", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-local-isomorphism", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.", "Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a homomorphism", "of $\\mathcal{O}_X$-modules. Let $\\overline{x}$ be a geometric point of $X$", "lying over $x \\in |X|$.", "\\begin{enumerate}", "\\item If $\\mathcal{F}_{\\overline{x}} = 0$ then there exists an open", "neighbourhood $X' \\subset X$ of $x$ such that $\\mathcal{F}|_{X'} = 0$.", "\\item If $\\varphi_{\\overline{x}} : \\mathcal{G}_{\\overline{x}} \\to", "\\mathcal{F}_{\\overline{x}}$ is injective, then there exists an open", "neighbourhood $X' \\subset X$ of $x$ such that $\\varphi|_{X'}$ is injective.", "\\item If $\\varphi_{\\overline{x}} : \\mathcal{G}_{\\overline{x}} \\to", "\\mathcal{F}_{\\overline{x}}$ is surjective, then there exists an open", "neighbourhood $X' \\subset X$ of $x$ such that $\\varphi|_{X'}$ is surjective.", "\\item If $\\varphi_{\\overline{x}} : \\mathcal{G}_{\\overline{x}} \\to", "\\mathcal{F}_{\\overline{x}}$ is bijective, then there exists an open", "neighbourhood $X' \\subset X$ of $x$ such that $\\varphi|_{X'}$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi : U \\to X$ be an \\'etale morphism where $U$ is a scheme and", "let $u \\in U$ be a point mapping to $x$. By", "Properties of Spaces, Lemmas", "\\ref{spaces-properties-lemma-stalk-quasi-coherent} and", "\\ref{spaces-properties-lemma-describe-etale-local-ring}", "as well as", "More on Algebra, Lemma \\ref{more-algebra-lemma-dumb-properties-henselization}", "we see that $\\varphi_{\\overline{x}}$ is injective, surjective, or bijective", "if and only if $\\varphi_u : \\varphi^*\\mathcal{F}_u \\to \\varphi^*\\mathcal{G}_u$", "has the corresponding property. Thus we can apply the schemes version of", "this lemma to see that (after possibly shrinking $U$) the map", "$\\varphi^*\\mathcal{F} \\to \\varphi^*\\mathcal{G}$ is injective, surjective,", "or an isomorphism. Let $X' \\subset X$ be the open subspace corresponding", "to $|\\varphi|(|U|) \\subset |X|$, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-open-subspaces}.", "Since $\\{U \\to X'\\}$ is a covering for the \\'etale topology, we conclude", "that $\\varphi|_{X'}$ is injective, surjective, or an isomorphism as desired.", "Finally, observe that (1) follows from (2) by looking at the map", "$\\mathcal{F} \\to 0$." ], "refs": [ "spaces-properties-lemma-stalk-quasi-coherent", "spaces-properties-lemma-describe-etale-local-ring", "more-algebra-lemma-dumb-properties-henselization", "spaces-properties-lemma-open-subspaces" ], "ref_ids": [ 11909, 11884, 10055, 11823 ] } ], "ref_ids": [] }, { "id": 11302, "type": "theorem", "label": "spaces-cohomology-lemma-coherent-support-closed", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-coherent-support-closed", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $i : Z \\to X$", "be the scheme theoretic support of $\\mathcal{F}$ and $\\mathcal{G}$", "the quasi-coherent $\\mathcal{O}_Z$-module such that", "$i_*\\mathcal{G} = \\mathcal{F}$, see", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-scheme-theoretic-support}.", "Then $\\mathcal{G}$ is a coherent $\\mathcal{O}_Z$-module." ], "refs": [ "spaces-morphisms-definition-scheme-theoretic-support" ], "proofs": [ { "contents": [ "The statement of the lemma makes sense as a coherent module is in", "particular of finite type. Moreover, as $Z \\to X$ is a closed immersion", "it is locally of finite type and hence $Z$ is locally Noetherian, see", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-immersion-locally-finite-type} and", "\\ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}.", "Finally, as $\\mathcal{G}$ is of finite type it is a coherent", "$\\mathcal{O}_Z$-module by", "Lemma \\ref{lemma-coherent-Noetherian}" ], "refs": [ "spaces-morphisms-lemma-immersion-locally-finite-type", "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "spaces-cohomology-lemma-coherent-Noetherian" ], "ref_ids": [ 4819, 4817, 11297 ] } ], "ref_ids": [ 4993 ] }, { "id": 11303, "type": "theorem", "label": "spaces-cohomology-lemma-i-star-equivalence", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-i-star-equivalence", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of locally", "Noetherian algebraic spaces over $S$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent sheaf of ideals", "cutting out $Z$. The functor $i_*$ induces an equivalence between the", "category of coherent $\\mathcal{O}_X$-modules annihilated by $\\mathcal{I}$", "and the category of coherent $\\mathcal{O}_Z$-modules." ], "refs": [], "proofs": [ { "contents": [ "The functor is fully faithful by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-i-star-equivalence}.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module", "annihilated by $\\mathcal{I}$. By", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-i-star-equivalence}", "we can write $\\mathcal{F} = i_*\\mathcal{G}$ for some quasi-coherent", "sheaf $\\mathcal{G}$ on $Z$. To check that $\\mathcal{G}$ is coherent", "we can work \\'etale locally (Lemma \\ref{lemma-coherent-Noetherian}).", "Choosing an \\'etale covering by a scheme we conclude that", "$\\mathcal{G}$ is coherent by the case of schemes", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-i-star-equivalence}).", "Hence the functor is fully faithful and the proof is done." ], "refs": [ "spaces-morphisms-lemma-i-star-equivalence", "spaces-morphisms-lemma-i-star-equivalence", "spaces-cohomology-lemma-coherent-Noetherian", "coherent-lemma-i-star-equivalence" ], "ref_ids": [ 4771, 4771, 11297, 3315 ] } ], "ref_ids": [] }, { "id": 11304, "type": "theorem", "label": "spaces-cohomology-lemma-finite-pushforward-coherent", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-finite-pushforward-coherent", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a finite morphism of algebraic", "spaces over $S$ with $Y$ locally Noetherian. Let $\\mathcal{F}$ be a", "coherent $\\mathcal{O}_X$-module. Assume $f$ is finite and $Y$ locally", "Noetherian. Then $R^pf_*\\mathcal{F} = 0$ for $p > 0$ and", "$f_*\\mathcal{F}$ is coherent." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Then $V \\times_Y X \\to V$ is a finite morphism of locally Noetherian", "schemes. By (\\ref{equation-representable-higher-direct-image}) we reduce", "to the case of schemes which is", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-finite-pushforward-coherent}." ], "refs": [ "coherent-lemma-finite-pushforward-coherent" ], "ref_ids": [ 3316 ] } ], "ref_ids": [] }, { "id": 11305, "type": "theorem", "label": "spaces-cohomology-lemma-acc-coherent", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-acc-coherent", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "The ascending chain condition holds for quasi-coherent submodules", "of $\\mathcal{F}$. In other words, given any sequence", "$$", "\\mathcal{F}_1 \\subset \\mathcal{F}_2 \\subset \\ldots \\subset \\mathcal{F}", "$$", "of quasi-coherent submodules, then", "$\\mathcal{F}_n = \\mathcal{F}_{n + 1} = \\ldots $ for some $n \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $U$ and a surjective \\'etale morphism $U \\to X$", "(see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "Then $U$ is a Noetherian scheme (by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}).", "If $\\mathcal{F}_n|_U = \\mathcal{F}_{n + 1}|_U = \\ldots$", "then $\\mathcal{F}_n = \\mathcal{F}_{n + 1} = \\ldots$.", "Hence the result follows from the case of schemes, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-acc-coherent}." ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "coherent-lemma-acc-coherent" ], "ref_ids": [ 11832, 4817, 3319 ] } ], "ref_ids": [] }, { "id": 11306, "type": "theorem", "label": "spaces-cohomology-lemma-power-ideal-kills-sheaf", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-power-ideal-kills-sheaf", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$. Let", "$\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals", "corresponding to a closed subspace $Z \\subset X$. Then there is some", "$n \\geq 0$ such that $\\mathcal{I}^n\\mathcal{F} = 0$ if and only if", "$\\text{Supp}(\\mathcal{F}) \\subset Z$ (set theoretically)." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $U$ and a surjective \\'etale morphism $U \\to X$", "(see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "Then $U$ is a Noetherian scheme (by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}).", "Note that $\\mathcal{I}^n\\mathcal{F}|_U = 0$ if and only if", "$\\mathcal{I}^n\\mathcal{F} = 0$ and similarly for the condition on", "the support. Hence the result follows from the case of schemes, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-power-ideal-kills-sheaf}." ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "coherent-lemma-power-ideal-kills-sheaf" ], "ref_ids": [ 11832, 4817, 3320 ] } ], "ref_ids": [] }, { "id": 11307, "type": "theorem", "label": "spaces-cohomology-lemma-Artin-Rees", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-Artin-Rees", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$. Let", "$\\mathcal{G} \\subset \\mathcal{F}$ be a quasi-coherent subsheaf.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of", "ideals. Then there exists a $c \\geq 0$ such that for all $n \\geq c$ we", "have", "$$", "\\mathcal{I}^{n - c}(\\mathcal{I}^c\\mathcal{F} \\cap \\mathcal{G})", "=", "\\mathcal{I}^n\\mathcal{F} \\cap \\mathcal{G}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $U$ and a surjective \\'etale morphism $U \\to X$", "(see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "Then $U$ is a Noetherian scheme (by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}).", "The equality of the lemma holds if and only if it holds after", "restricting to $U$. Hence the result follows from the case of schemes, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-Artin-Rees}." ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "coherent-lemma-Artin-Rees" ], "ref_ids": [ 11832, 4817, 3321 ] } ], "ref_ids": [] }, { "id": 11308, "type": "theorem", "label": "spaces-cohomology-lemma-homs-over-open", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-homs-over-open", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\mathcal{G}$ be a coherent $\\mathcal{O}_X$-module.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of", "ideals. Denote $Z \\subset X$ the corresponding closed subspace and", "set $U = X \\setminus Z$. There is a canonical isomorphism", "$$", "\\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n\\mathcal{G}, \\mathcal{F})", "\\longrightarrow", "\\Hom_{\\mathcal{O}_U}(\\mathcal{G}|_U, \\mathcal{F}|_U).", "$$", "In particular we have an isomorphism", "$$", "\\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n, \\mathcal{F})", "\\longrightarrow", "\\Gamma(U, \\mathcal{F}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $W$ be an affine scheme and let $W \\to X$ be a surjective \\'etale", "morphism (see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}).", "Set $R = W \\times_X W$. Then $W$ and $R$ are Noetherian schemes, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}.", "Hence the result hold for the restrictions of $\\mathcal{F}$, $\\mathcal{G}$,", "and $\\mathcal{I}$, $U$, $Z$ to $W$ and $R$ by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-homs-over-open}.", "It follows formally that the result holds over $X$." ], "refs": [ "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "coherent-lemma-homs-over-open" ], "ref_ids": [ 11832, 4817, 3322 ] } ], "ref_ids": [] }, { "id": 11309, "type": "theorem", "label": "spaces-cohomology-lemma-prepare-filter-support", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-prepare-filter-support", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$. Suppose that", "$\\text{Supp}(\\mathcal{F}) = Z \\cup Z'$ with $Z$, $Z'$ closed.", "Then there exists a short exact sequence of coherent sheaves", "$$", "0 \\to \\mathcal{G}' \\to \\mathcal{F} \\to \\mathcal{G} \\to 0", "$$", "with $\\text{Supp}(\\mathcal{G}') \\subset Z'$ and", "$\\text{Supp}(\\mathcal{G}) \\subset Z$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the sheaf of ideals", "defining the reduced induced closed subspace structure on $Z$, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-reduced-closed-subspace}.", "Consider the subsheaves", "$\\mathcal{G}'_n = \\mathcal{I}^n\\mathcal{F}$ and the", "quotients $\\mathcal{G}_n = \\mathcal{F}/\\mathcal{I}^n\\mathcal{F}$.", "For each $n$ we have a short exact sequence", "$$", "0 \\to \\mathcal{G}'_n \\to \\mathcal{F} \\to \\mathcal{G}_n \\to 0", "$$", "For every geometric point $\\overline{x}$ of $Z' \\setminus Z$ we have", "$\\mathcal{I}_{\\overline{x}} = \\mathcal{O}_{X, \\overline{x}}$", "and hence $\\mathcal{G}_{n, \\overline{x}} = 0$. Thus we see that", "$\\text{Supp}(\\mathcal{G}_n) \\subset Z$. Note that $X \\setminus Z'$", "is a Noetherian algebraic space. Hence by", "Lemma \\ref{lemma-power-ideal-kills-sheaf}", "there exists an $n$ such that $\\mathcal{G}'_n|_{X \\setminus Z'} =", "\\mathcal{I}^n\\mathcal{F}|_{X \\setminus Z'} = 0$.", "For such an $n$ we see that $\\text{Supp}(\\mathcal{G}'_n) \\subset Z'$.", "Thus setting $\\mathcal{G}' = \\mathcal{G}'_n$ and $\\mathcal{G} = \\mathcal{G}_n$", "works." ], "refs": [ "spaces-properties-lemma-reduced-closed-subspace", "spaces-cohomology-lemma-power-ideal-kills-sheaf" ], "ref_ids": [ 11846, 11306 ] } ], "ref_ids": [] }, { "id": 11310, "type": "theorem", "label": "spaces-cohomology-lemma-prepare-filter-irreducible", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-prepare-filter-irreducible", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$. Assume that the scheme", "theoretic support of $\\mathcal{F}$ is a reduced $Z \\subset X$ with", "$|Z|$ irreducible. Then there exist an integer $r > 0$, a nonzero", "sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$, and an injective", "map of coherent sheaves", "$$", "i_*\\left(\\mathcal{I}^{\\oplus r}\\right) \\to \\mathcal{F}", "$$", "whose cokernel is supported on a proper closed subspace of $Z$." ], "refs": [], "proofs": [ { "contents": [ "By assumption there exists a coherent $\\mathcal{O}_Z$-module", "$\\mathcal{G}$ with support $Z$ and $\\mathcal{F} \\cong i_*\\mathcal{G}$, see", "Lemma \\ref{lemma-coherent-support-closed}. Hence it suffices to prove the", "lemma for the case $Z = X$ and $i = \\text{id}$.", "\\medskip\\noindent", "By Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}", "there exists a dense open subspace $U \\subset X$ which is a scheme. Note that", "$U$ is a Noetherian integral scheme. After shrinking $U$ we may assume", "that $\\mathcal{F}|_U \\cong \\mathcal{O}_U^{\\oplus r}$ (for example by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-prepare-filter-irreducible}", "or by a direct algebra argument). Let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be a quasi-coherent sheaf of ideals whose associated closed subspace", "is the complement of $U$ in $X$ (see for example", "Properties of Spaces, Section \\ref{spaces-properties-section-reduced}). ", "By Lemma \\ref{lemma-homs-over-open} there exists an $n \\geq 0$ and a", "morphism $\\mathcal{I}^n(\\mathcal{O}_X^{\\oplus r}) \\to \\mathcal{F}$", "which recovers our isomorphism over $U$. Since", "$\\mathcal{I}^n(\\mathcal{O}_X^{\\oplus r}) = (\\mathcal{I}^n)^{\\oplus r}$", "we get a map as in the lemma. It is injective: namely, if $\\sigma$ is", "a nonzero section of $\\mathcal{I}^{\\oplus r}$ over a scheme $W$ \\'etale", "over $X$, then because $X$ hence $W$ is reduced the support of $\\sigma$", "contains a nonempty open of $W$. But the kernel of", "$(\\mathcal{I}^n)^{\\oplus r} \\to \\mathcal{F}$ is zero", "over a dense open, hence $\\sigma$ cannot be a section of the kernel." ], "refs": [ "spaces-cohomology-lemma-coherent-support-closed", "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme", "coherent-lemma-prepare-filter-irreducible", "spaces-cohomology-lemma-homs-over-open" ], "ref_ids": [ 11302, 11917, 3328, 11308 ] } ], "ref_ids": [] }, { "id": 11311, "type": "theorem", "label": "spaces-cohomology-lemma-coherent-filter", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-coherent-filter", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$. There exists a filtration", "$$", "0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset", "\\ldots \\subset \\mathcal{F}_m = \\mathcal{F}", "$$", "by coherent subsheaves such that for each $j = 1, \\ldots, m$", "there exists a reduced closed subspace $Z_j \\subset X$ with $|Z_j|$", "irreducible and a sheaf of ideals $\\mathcal{I}_j \\subset \\mathcal{O}_{Z_j}$", "such that", "$$", "\\mathcal{F}_j/\\mathcal{F}_{j - 1}", "\\cong (Z_j \\to X)_* \\mathcal{I}_j", "$$" ], "refs": [], "proofs": [ { "contents": [ "Consider the collection", "$$", "\\mathcal{T} =", "\\left\\{", "\\begin{matrix}", "T \\subset |X|", "\\text{ closed such that there exists a coherent sheaf }", "\\mathcal{F} \\\\", "\\text{ with }", "\\text{Supp}(\\mathcal{F}) = T", "\\text{ for which the lemma is wrong}", "\\end{matrix}", "\\right\\}", "$$", "We are trying to show that $\\mathcal{T}$ is empty. If not, then", "because $|X|$ is Noetherian (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-Noetherian-topology})", "we can choose a minimal element $T \\in \\mathcal{T}$. This means that", "there exists a coherent sheaf $\\mathcal{F}$ on $X$ whose support is $T$", "and for which the lemma does not hold. Clearly $T \\not = \\emptyset$ since", "the only sheaf whose support is empty is the zero sheaf for which the", "lemma does hold (with $m = 0$).", "\\medskip\\noindent", "If $T$ is not irreducible, then we can write $T = Z_1 \\cup Z_2$", "with $Z_1, Z_2$ closed and strictly smaller than $T$.", "Then we can apply Lemma \\ref{lemma-prepare-filter-support}", "to get a short exact sequence of coherent sheaves", "$$", "0 \\to", "\\mathcal{G}_1 \\to", "\\mathcal{F} \\to", "\\mathcal{G}_2 \\to 0", "$$", "with $\\text{Supp}(\\mathcal{G}_i) \\subset Z_i$. By minimality of", "$T$ each of $\\mathcal{G}_i$ has a filtration as in the statement", "of the lemma. By considering the induced filtration on $\\mathcal{F}$", "we arrive at a contradiction. Hence we conclude", "that $T$ is irreducible.", "\\medskip\\noindent", "Suppose $T$ is irreducible. Let $\\mathcal{J}$ be the sheaf of ideals", "defining the reduced induced closed subspace structure on $T$,", "see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-reduced-closed-subspace}.", "By Lemma \\ref{lemma-power-ideal-kills-sheaf} we see there exists", "an $n \\geq 0$ such that $\\mathcal{J}^n\\mathcal{F} = 0$. Hence we obtain", "a filtration", "$$", "0 = \\mathcal{I}^n\\mathcal{F} \\subset \\mathcal{I}^{n - 1}\\mathcal{F}", "\\subset \\ldots \\subset \\mathcal{I}\\mathcal{F} \\subset \\mathcal{F}", "$$", "each of whose successive subquotients is annihilated by $\\mathcal{J}$.", "Hence if each of these subquotients has a filtration as in the statement", "of the lemma then also $\\mathcal{F}$ does. In other words we may", "assume that $\\mathcal{J}$ does annihilate $\\mathcal{F}$.", "\\medskip\\noindent", "Assume $T$ is irreducible and $\\mathcal{J}\\mathcal{F} = 0$ where", "$\\mathcal{J}$ is as above. Then the scheme theoretic support of", "$\\mathcal{F}$ is $T$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-i-star-equivalence}.", "Hence we can apply Lemma \\ref{lemma-prepare-filter-irreducible}.", "This gives a short exact sequence", "$$", "0 \\to", "i_*(\\mathcal{I}^{\\oplus r}) \\to", "\\mathcal{F} \\to", "\\mathcal{Q} \\to 0", "$$", "where the support of $\\mathcal{Q}$ is a proper closed subset of $T$.", "Hence we see that $\\mathcal{Q}$ has a filtration of the desired type", "by minimality of $T$. But then clearly $\\mathcal{F}$ does too, which is", "our final contradiction." ], "refs": [ "spaces-properties-lemma-Noetherian-topology", "spaces-cohomology-lemma-prepare-filter-support", "spaces-properties-lemma-reduced-closed-subspace", "spaces-cohomology-lemma-power-ideal-kills-sheaf", "spaces-morphisms-lemma-i-star-equivalence", "spaces-cohomology-lemma-prepare-filter-irreducible" ], "ref_ids": [ 11891, 11309, 11846, 11306, 4771, 11310 ] } ], "ref_ids": [] }, { "id": 11312, "type": "theorem", "label": "spaces-cohomology-lemma-property-initial", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-property-initial", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $\\mathcal{P}$ be a property of coherent sheaves on $X$. Assume", "\\begin{enumerate}", "\\item For any short exact sequence of coherent sheaves", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0", "$$", "if $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$", "then so does $\\mathcal{F}$.", "\\item For every reduced closed subspace $Z \\subset X$ with $|Z|$ irreducible", "and every quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$", "we have $\\mathcal{P}$ for $i_*\\mathcal{I}$.", "\\end{enumerate}", "Then property $\\mathcal{P}$ holds for every coherent sheaf on $X$." ], "refs": [], "proofs": [ { "contents": [ "First note that if $\\mathcal{F}$ is a coherent sheaf with a filtration", "$$", "0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset", "\\ldots \\subset \\mathcal{F}_m = \\mathcal{F}", "$$", "by coherent subsheaves such that each of $\\mathcal{F}_i/\\mathcal{F}_{i - 1}$", "has property $\\mathcal{P}$, then so does $\\mathcal{F}$.", "This follows from the property (1) for $\\mathcal{P}$.", "On the other hand, by Lemma \\ref{lemma-coherent-filter}", "we can filter any $\\mathcal{F}$", "with successive subquotients as in (2).", "Hence the lemma follows." ], "refs": [ "spaces-cohomology-lemma-coherent-filter" ], "ref_ids": [ 11311 ] } ], "ref_ids": [] }, { "id": 11313, "type": "theorem", "label": "spaces-cohomology-lemma-property-higher-rank-cohomological", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-property-higher-rank-cohomological", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $\\mathcal{P}$ be a property of coherent sheaves on $X$. Assume", "\\begin{enumerate}", "\\item For any short exact sequence of coherent sheaves", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0", "$$", "if $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$", "then so does $\\mathcal{F}$.", "\\item If $\\mathcal{P}$ holds for $\\mathcal{F}^{\\oplus r}$ for", "some $r \\geq 1$, then it holds for $\\mathcal{F}$.", "\\item For every reduced closed subspace $i : Z \\to X$ with", "$|Z|$ irreducible there exists a coherent sheaf $\\mathcal{G}$ on $Z$", "such that", "\\begin{enumerate}", "\\item $\\text{Supp}(\\mathcal{G}) = Z$,", "\\item for every nonzero quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_Z$ there exists a quasi-coherent", "subsheaf $\\mathcal{G}' \\subset \\mathcal{I}\\mathcal{G}$ such that", "$\\text{Supp}(\\mathcal{G}/\\mathcal{G}')$ is proper closed in $|Z|$", "and such that $\\mathcal{P}$ holds for $i_*\\mathcal{G}'$.", "\\end{enumerate}", "\\end{enumerate}", "Then property $\\mathcal{P}$ holds for every coherent sheaf on $X$." ], "refs": [], "proofs": [ { "contents": [ "Consider the collection", "$$", "\\mathcal{T} =", "\\left\\{", "\\begin{matrix}", "T \\subset |X|", "\\text{ nonempty closed such that there exists a coherent sheaf } \\\\", "\\mathcal{F}", "\\text{ with }", "\\text{Supp}(\\mathcal{F}) = T", "\\text{ for which the lemma is wrong}", "\\end{matrix}", "\\right\\}", "$$", "We are trying to show that $\\mathcal{T}$ is empty. If not, then", "because $|X|$ is Noetherian (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-Noetherian-topology})", "we can choose a minimal element $T \\in \\mathcal{T}$. This means that", "there exists a coherent sheaf $\\mathcal{F}$ on $X$ whose support is $T$", "and for which the lemma does not hold.", "\\medskip\\noindent", "If $T$ is not irreducible, then we can write $T = Z_1 \\cup Z_2$", "with $Z_1, Z_2$ closed and strictly smaller than $T$.", "Then we can apply Lemma \\ref{lemma-prepare-filter-support}", "to get a short exact sequence of coherent sheaves", "$$", "0 \\to", "\\mathcal{G}_1 \\to", "\\mathcal{F} \\to", "\\mathcal{G}_2 \\to 0", "$$", "with $\\text{Supp}(\\mathcal{G}_i) \\subset Z_i$. By minimality of", "$T$ each of $\\mathcal{G}_i$ has $\\mathcal{P}$. Hence $\\mathcal{F}$", "has property $\\mathcal{P}$ by (1), a contradiction.", "\\medskip\\noindent", "Suppose $T$ is irreducible. Let $\\mathcal{J}$ be the sheaf of ideals", "defining the reduced induced closed subspace structure on $T$,", "see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-reduced-closed-subspace}.", "By Lemma \\ref{lemma-power-ideal-kills-sheaf} we see there exists", "an $n \\geq 0$ such that $\\mathcal{J}^n\\mathcal{F} = 0$. Hence we obtain", "a filtration", "$$", "0 = \\mathcal{J}^n\\mathcal{F} \\subset \\mathcal{J}^{n - 1}\\mathcal{F}", "\\subset \\ldots \\subset \\mathcal{J}\\mathcal{F} \\subset \\mathcal{F}", "$$", "each of whose successive subquotients is annihilated by $\\mathcal{J}$.", "Hence if each of these subquotients has a filtration as in the statement", "of the lemma then also $\\mathcal{F}$ does by (1). In other words we may", "assume that $\\mathcal{J}$ does annihilate $\\mathcal{F}$.", "\\medskip\\noindent", "Assume $T$ is irreducible and $\\mathcal{J}\\mathcal{F} = 0$ where", "$\\mathcal{J}$ is as above. Denote $i : Z \\to X$ the closed subspace", "corresponding to $\\mathcal{J}$. Then $\\mathcal{F} = i_*\\mathcal{H}$", "for some coherent $\\mathcal{O}_Z$-module $\\mathcal{H}$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-i-star-equivalence}", "and Lemma \\ref{lemma-coherent-support-closed}.", "Let $\\mathcal{G}$ be the coherent sheaf on $Z$ satisfying", "(3)(a) and (3)(b). We apply Lemma \\ref{lemma-prepare-filter-irreducible}", "to get injective maps", "$$", "\\mathcal{I}_1^{\\oplus r_1} \\to \\mathcal{H}", "\\quad\\text{and}\\quad", "\\mathcal{I}_2^{\\oplus r_2} \\to \\mathcal{G}", "$$", "where the support of the cokernels are proper closed in $Z$. Hence we find", "an nonempty open $V \\subset Z$ such that", "$$", "\\mathcal{H}^{\\oplus r_2}_V \\cong \\mathcal{G}^{\\oplus r_1}_V", "$$", "Let $\\mathcal{I} \\subset \\mathcal{O}_Z$ be a quasi-coherent ideal sheaf", "cutting out $Z \\setminus V$ we obtain", "(Lemma \\ref{lemma-homs-over-open})", "a map", "$$", "\\mathcal{I}^n\\mathcal{G}^{\\oplus r_1} \\longrightarrow \\mathcal{H}^{\\oplus r_2}", "$$", "which is an isomorphism over $V$. The kernel is supported on $Z \\setminus V$", "hence annihilated by some power of $\\mathcal{I}$, see", "Lemma \\ref{lemma-power-ideal-kills-sheaf}. Thus after increasing", "$n$ we may assume the displayed map is injective, see", "Lemma \\ref{lemma-Artin-Rees}. Applying (3)(b) we find", "$\\mathcal{G}' \\subset \\mathcal{I}^n\\mathcal{G}$ such that", "$$", "(i_*\\mathcal{G}')^{\\oplus r_1} \\longrightarrow", "i_*\\mathcal{H}^{\\oplus r_2} = \\mathcal{F}^{\\oplus r_2}", "$$", "is injective with cokernel supported in a proper closed subset of $Z$", "and such that property $\\mathcal{P}$ holds for $i_*\\mathcal{G}'$.", "By (1) property $\\mathcal{P}$ holds for $(i_*\\mathcal{G}')^{\\oplus r_1}$.", "By (1) and minimality of $T = |Z|$ property $\\mathcal{P}$ holds for", "$\\mathcal{F}^{\\oplus r_2}$. And finally by (2) property $\\mathcal{P}$", "holds for $\\mathcal{F}$ which is the desired contradiction." ], "refs": [ "spaces-properties-lemma-Noetherian-topology", "spaces-cohomology-lemma-prepare-filter-support", "spaces-properties-lemma-reduced-closed-subspace", "spaces-cohomology-lemma-power-ideal-kills-sheaf", "spaces-morphisms-lemma-i-star-equivalence", "spaces-cohomology-lemma-coherent-support-closed", "spaces-cohomology-lemma-prepare-filter-irreducible", "spaces-cohomology-lemma-homs-over-open", "spaces-cohomology-lemma-power-ideal-kills-sheaf", "spaces-cohomology-lemma-Artin-Rees" ], "ref_ids": [ 11891, 11309, 11846, 11306, 4771, 11302, 11310, 11308, 11306, 11307 ] } ], "ref_ids": [] }, { "id": 11314, "type": "theorem", "label": "spaces-cohomology-lemma-property-higher-rank-cohomological-variant", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-property-higher-rank-cohomological-variant", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $\\mathcal{P}$ be a property of coherent sheaves on $X$. Assume", "\\begin{enumerate}", "\\item For any short exact sequence of coherent sheaves on $X$", "if two out of three have property $\\mathcal{P}$ so does the third.", "\\item If $\\mathcal{P}$ holds for $\\mathcal{F}^{\\oplus r}$ for", "some $r \\geq 1$, then it holds for $\\mathcal{F}$.", "\\item For every reduced closed subspace $i : Z \\to X$ with", "$|Z|$ irreducible there exists a coherent sheaf $\\mathcal{G}$ on $X$", "whose scheme theoretic support is $Z$ such that $\\mathcal{P}$ holds for", "$\\mathcal{G}$.", "\\end{enumerate}", "Then property $\\mathcal{P}$ holds for every coherent sheaf on $X$." ], "refs": [], "proofs": [ { "contents": [ "We will show that conditions (1) and (2) of", "Lemma \\ref{lemma-property-initial} hold. This is clear for condition (1).", "To show that (2) holds, let", "$$", "\\mathcal{T} =", "\\left\\{", "\\begin{matrix}", "i : Z \\to X \\text{ reduced closed subspace with }|Z|\\text{ irreducible such}\\\\", "\\text{ that }i_*\\mathcal{I}\\text{ does not have }\\mathcal{P}", "\\text{ for some quasi-coherent }\\mathcal{I} \\subset \\mathcal{O}_Z", "\\end{matrix}", "\\right\\}", "$$", "If $\\mathcal{T}$ is nonempty, then since $X$ is Noetherian, we can", "find an $i : Z \\to X$ which is minimal in $\\mathcal{T}$. We will show", "that this leads to a contradiction.", "\\medskip\\noindent", "Let $\\mathcal{G}$ be the sheaf whose scheme theoretic support is $Z$ whose", "existence is assumed in assumption (3). Let", "$\\varphi : i_*\\mathcal{I}^{\\oplus r} \\to \\mathcal{G}$ be as in", "Lemma \\ref{lemma-prepare-filter-irreducible}. Let", "$$", "0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset", "\\ldots \\subset \\mathcal{F}_m = \\Coker(\\varphi)", "$$", "be a filtration as in Lemma \\ref{lemma-coherent-filter}. By minimality", "of $Z$ and assumption (1) we see that $\\Coker(\\varphi)$ has", "property $\\mathcal{P}$. As $\\varphi$ is injective we conclude using", "assumption (1) once more that $i_*\\mathcal{I}^{\\oplus r}$ has property", "$\\mathcal{P}$. Using assumption (2) we conclude that $i_*\\mathcal{I}$", "has property $\\mathcal{P}$.", "\\medskip\\noindent", "Finally, if $\\mathcal{J} \\subset \\mathcal{O}_Z$ is a second quasi-coherent", "sheaf of ideals, set $\\mathcal{K} = \\mathcal{I} \\cap \\mathcal{J}$", "and consider the short exact sequences", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{I} \\to \\mathcal{I}/\\mathcal{K} \\to 0", "\\quad", "\\text{and}", "\\quad", "0 \\to \\mathcal{K} \\to \\mathcal{J} \\to \\mathcal{J}/\\mathcal{K} \\to 0", "$$", "Arguing as above, using the minimality of $Z$, we see that", "$i_*\\mathcal{I}/\\mathcal{K}$ and $i_*\\mathcal{J}/\\mathcal{K}$", "satisfy $\\mathcal{P}$. Hence by assumption (1) we conclude that", "$i_*\\mathcal{K}$ and then $i_*\\mathcal{J}$ satisfy $\\mathcal{P}$.", "In other words, $Z$ is not an element of $\\mathcal{T}$ which is", "the desired contradiction." ], "refs": [ "spaces-cohomology-lemma-property-initial", "spaces-cohomology-lemma-prepare-filter-irreducible", "spaces-cohomology-lemma-coherent-filter" ], "ref_ids": [ 11312, 11310, 11311 ] } ], "ref_ids": [] }, { "id": 11315, "type": "theorem", "label": "spaces-cohomology-lemma-directed-colimit-coherent", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-directed-colimit-coherent", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Every quasi-coherent $\\mathcal{O}_X$-module is the filtered colimit", "of its coherent submodules." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "If $\\mathcal{G}, \\mathcal{H} \\subset \\mathcal{F}$ are coherent", "$\\mathcal{O}_X$-submodules then the image of", "$\\mathcal{G} \\oplus \\mathcal{H} \\to \\mathcal{F}$ is another", "coherent $\\mathcal{O}_X$-submodule which contains both of them", "(see Lemmas \\ref{lemma-coherent-abelian-Noetherian} and", "\\ref{lemma-coherent-Noetherian-quasi-coherent-sub-quotient}).", "In this way we see that the system is directed.", "Hence it now suffices to show that $\\mathcal{F}$ can be written as", "a filtered colimit of coherent modules, as then we can take the", "images of these modules in $\\mathcal{F}$ to conclude there are", "enough of them.", "\\medskip\\noindent", "Let $U$ be an affine scheme and $U \\to X$ a surjective \\'etale morphism.", "Set $R = U \\times_X U$ so that $X = U/R$ as usual. By", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-quasi-coherent}", "we see that $\\QCoh(\\mathcal{O}_X) = \\QCoh(U, R, s, t, c)$.", "Hence we reduce to showing the corresponding thing for", "$\\QCoh(U, R, s, t, c)$. Thus the result follows from", "the more general Groupoids, Lemma \\ref{groupoids-lemma-colimit-coherent}." ], "refs": [ "spaces-cohomology-lemma-coherent-abelian-Noetherian", "spaces-cohomology-lemma-coherent-Noetherian-quasi-coherent-sub-quotient", "spaces-properties-proposition-quasi-coherent", "groupoids-lemma-colimit-coherent" ], "ref_ids": [ 11298, 11299, 11920, 9632 ] } ], "ref_ids": [] }, { "id": 11316, "type": "theorem", "label": "spaces-cohomology-lemma-direct-colimit-finite-presentation", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-direct-colimit-finite-presentation", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic", "spaces over $S$ with $Y$ Noetherian. Then every quasi-coherent", "$\\mathcal{O}_X$-module is a filtered colimit of finitely presented", "$\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Write $f_*\\mathcal{F} = \\colim \\mathcal{H}_i$ with $\\mathcal{H}_i$", "a coherent $\\mathcal{O}_Y$-module, see", "Lemma \\ref{lemma-directed-colimit-coherent}.", "By Lemma \\ref{lemma-coherent-Noetherian} the modules $\\mathcal{H}_i$", "are $\\mathcal{O}_Y$-modules of finite presentation. Hence", "$f^*\\mathcal{H}_i$ is an $\\mathcal{O}_X$-module of finite presentation, see", "Properties of Spaces, Section", "\\ref{spaces-properties-section-properties-modules}.", "We claim the map", "$$", "\\colim f^*\\mathcal{H}_i = f^*f_*\\mathcal{F} \\to \\mathcal{F}", "$$", "is surjective as $f$ is assumed affine, Namely, choose", "a scheme $V$ and a surjective \\'etale morphism $V \\to Y$. Set", "$U = X \\times_Y V$. Then $U$ is a scheme, $f' : U \\to V$ is affine, and", "$U \\to X$ is surjective \\'etale. By", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-pushforward-etale-base-change-modules}", "we see that $f'_*(\\mathcal{F}|_U) = f_*\\mathcal{F}|_V$ and similarly", "for pullbacks. Thus the restriction of $f^*f_*\\mathcal{F} \\to \\mathcal{F}$", "to $U$ is the map", "$$", "f^*f_*\\mathcal{F}|_U = (f')^*(f_*\\mathcal{F})|_V) =", "(f')^*f'_*(\\mathcal{F}|_U) \\to \\mathcal{F}|_U", "$$", "which is surjective as $f'$ is an affine morphism of schemes.", "Hence the claim holds.", "\\medskip\\noindent", "We conclude that every quasi-coherent module on $X$ is a quotient of a", "filtered colimit of finitely presented modules. In particular, we see that", "$\\mathcal{F}$ is a cokernel of a map", "$$", "\\colim_{j \\in J} \\mathcal{G}_j \\longrightarrow \\colim_{i \\in I} \\mathcal{H}_i", "$$", "with $\\mathcal{G}_j$ and $\\mathcal{H}_i$ finitely presented. Note", "that for every $j \\in I$ there exist $i \\in I$ and a morphism", "$\\alpha : \\mathcal{G}_j \\to \\mathcal{H}_i$ such that", "$$", "\\xymatrix{", "\\mathcal{G}_j \\ar[r]_\\alpha \\ar[d] & \\mathcal{H}_i \\ar[d] \\\\", "\\colim_{j \\in J} \\mathcal{G}_j \\ar[r] &", "\\colim_{i \\in I} \\mathcal{H}_i", "}", "$$", "commutes, see", "Lemma \\ref{lemma-finite-presentation-quasi-compact-colimit}.", "In this situation $\\Coker(\\alpha)$ is a finitely presented", "$\\mathcal{O}_X$-module which comes endowed with a map", "$\\Coker(\\alpha) \\to \\mathcal{F}$. Consider the set $K$ of", "triples $(i, j, \\alpha)$ as above. We say that", "$(i, j, \\alpha) \\leq (i', j', \\alpha')$ if and only if", "$i \\leq i'$, $j \\leq j'$, and the diagram", "$$", "\\xymatrix{", "\\mathcal{G}_j \\ar[r]_\\alpha \\ar[d] & \\mathcal{H}_i \\ar[d] \\\\", "\\mathcal{G}_{j'} \\ar[r]^{\\alpha'} &", "\\mathcal{H}_{i'}", "}", "$$", "commutes. It follows from the above that $K$ is", "a directed partially ordered set,", "$$", "\\mathcal{F} = \\colim_{(i, j, \\alpha) \\in K} \\Coker(\\alpha),", "$$", "and we win." ], "refs": [ "spaces-cohomology-lemma-directed-colimit-coherent", "spaces-cohomology-lemma-coherent-Noetherian", "spaces-properties-lemma-pushforward-etale-base-change-modules", "spaces-cohomology-lemma-finite-presentation-quasi-compact-colimit" ], "ref_ids": [ 11315, 11297, 11898, 11279 ] } ], "ref_ids": [] }, { "id": 11317, "type": "theorem", "label": "spaces-cohomology-lemma-vanishing-compute", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-vanishing-compute", "contents": [ "In Situation \\ref{situation-vanishing} for an $A$-module $M$ we have", "$p_*(M \\otimes_A \\mathcal{O}_X) = \\widetilde{M}$ and", "$\\Gamma(X, M \\otimes_A \\mathcal{O}_X) = M$." ], "refs": [], "proofs": [ { "contents": [ "The equality $p_*(M \\otimes_A \\mathcal{O}_X) = \\widetilde{M}$ follows", "from the equality $\\Gamma(X, M \\otimes_A \\mathcal{O}_X) = M$ as", "$p_*(M \\otimes_A \\mathcal{O}_X)$ is a quasi-coherent module on", "$\\Spec(A)$ by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-pushforward}.", "Observe that $\\Gamma(X, \\bigoplus_{i \\in I} \\mathcal{O}_X) =", "\\bigoplus_{i \\in I} A$ by Lemma \\ref{lemma-colimits}. Hence the", "lemma holds for free modules. Choose a short exact sequence", "$F_1 \\to F_0 \\to M$ where $F_0, F_1$ are free $A$-modules. Since", "$H^1(X, -)$ is zero the global sections functor is right exact.", "Moreover the pullback $p^*$ is right exact as well. Hence we see", "that", "$$", "\\Gamma(X, F_1 \\otimes_A \\mathcal{O}_X) \\to", "\\Gamma(X, F_0 \\otimes_A \\mathcal{O}_X) \\to", "\\Gamma(X, M \\otimes_A \\mathcal{O}_X) \\to 0", "$$", "is exact. The result follows." ], "refs": [ "spaces-morphisms-lemma-pushforward", "spaces-cohomology-lemma-colimits" ], "ref_ids": [ 4760, 11277 ] } ], "ref_ids": [] }, { "id": 11318, "type": "theorem", "label": "spaces-cohomology-lemma-vanishing-base-change", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-vanishing-base-change", "contents": [ "In Situation \\ref{situation-vanishing}.", "\\begin{enumerate}", "\\item Given an affine morphism $X' \\to X$ of algebraic spaces, we have", "$H^1(X', \\mathcal{F}') = 0$ for every quasi-coherent", "$\\mathcal{O}_{X'}$-module $\\mathcal{F}'$.", "\\item Given an $A$-algebra $A'$ setting $X' = X \\times_{\\Spec(A)} \\Spec(A')$", "the morphism $X' \\to X$ is affine and $\\Gamma(X', \\mathcal{O}_{X'}) = A'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from Lemma \\ref{lemma-affine-vanishing-higher-direct-images}", "and the Leray spectral sequence (Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-Leray}). Let $A \\to A'$ be as in (2).", "Then $X' \\to X$ is affine because affine morphisms are preserved under", "base change (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-affine}) and the", "fact that a morphism of affine schemes is affine. The equality", "$\\Gamma(X', \\mathcal{O}_{X'}) = A'$ follows as", "$(X' \\to X)_*\\mathcal{O}_{X'} = A' \\otimes_A \\mathcal{O}_X$", "by Lemma \\ref{lemma-affine-base-change} and thus", "$$", "\\Gamma(X', \\mathcal{O}_{X'}) =", "\\Gamma(X, (X' \\to X)_*\\mathcal{O}_{X'}) =", "\\Gamma(X, A' \\otimes_A \\mathcal{O}_X) = A'", "$$", "by Lemma \\ref{lemma-vanishing-compute}." ], "refs": [ "spaces-cohomology-lemma-affine-vanishing-higher-direct-images", "sites-cohomology-lemma-Leray", "spaces-morphisms-lemma-base-change-affine", "spaces-cohomology-lemma-affine-base-change", "spaces-cohomology-lemma-vanishing-compute" ], "ref_ids": [ 11288, 4220, 4800, 11295, 11317 ] } ], "ref_ids": [] }, { "id": 11319, "type": "theorem", "label": "spaces-cohomology-lemma-vanishing-separate-closed", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-vanishing-separate-closed", "contents": [ "In Situation \\ref{situation-vanishing}. Let $Z_0, Z_1 \\subset |X|$", "be disjoint closed subsets. Then there exists an $a \\in A$ such that", "$Z_0 \\subset V(a)$ and $Z_1 \\subset V(a - 1)$." ], "refs": [], "proofs": [ { "contents": [ "We may and do endow $Z_0$, $Z_1$ with the reduced induced subspace structure", "(Properties of Spaces, Definition", "\\ref{spaces-properties-definition-reduced-induced-space}) and we denote", "$i_0 : Z_0 \\to X$ and $i_1 : Z_1 \\to X$ the corresponding closed immersions.", "Since $Z_0 \\cap Z_1 = \\emptyset$ we see that the canonical map of", "quasi-coherent $\\mathcal{O}_X$-modules", "$$", "\\mathcal{O}_X", "\\longrightarrow", "i_{0, *}\\mathcal{O}_{Z_0} \\oplus i_{1, *}\\mathcal{O}_{Z_1}", "$$", "is surjective (look at stalks at geometric points). Since $H^1(X, -)$ is", "zero on the kernel of this map the induced map of global sections is", "surjective. Thus we can find $a \\in A$ which maps to the global section", "$(0, 1)$ of the right hand side." ], "refs": [ "spaces-properties-definition-reduced-induced-space" ], "ref_ids": [ 11932 ] } ], "ref_ids": [] }, { "id": 11320, "type": "theorem", "label": "spaces-cohomology-lemma-vanishing-injective", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-vanishing-injective", "contents": [ "In Situation \\ref{situation-vanishing} the morphism $p : X \\to \\Spec(A)$ is", "universally injective." ], "refs": [], "proofs": [ { "contents": [ "Let $A \\to k$ be a ring homomorphism where $k$ is a field. It suffices to", "show that $\\Spec(k) \\times_{\\Spec(A)} X$ has at most one point (see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-injective-local}).", "Using Lemma \\ref{lemma-vanishing-base-change} we may assume that $A$", "is a field and we have to show that $|X|$ has at most one point.", "\\medskip\\noindent", "Let's think of $X$ as an algebraic space over $\\Spec(k)$ and let's", "use the notation $X(K)$ to denote $K$-valued points of $X$", "for any extension $k \\subset K$, see", "Morphisms of Spaces, Section \\ref{spaces-morphisms-section-points-fields}.", "If $k \\subset K$ is an algebraically closed field extension", "of large transcendence degree, then we see that $X(K) \\to |X|$", "is surjective, see Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-large-enough}. Hence, after replacing $k$", "by $K$, we see that it suffices to prove that $X(k)$ is a singleton", "(in the case $A = k)$.", "\\medskip\\noindent", "Let $x, x' \\in X(k)$. By Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-algebraic-residue-field-extension-closed-point}", "we see that $x$ and $x'$ are closed points of $|X|$. Hence $x$ and $x'$", "map to distinct points of $\\Spec(k)$ if $x \\not = x'$ by", "Lemma \\ref{lemma-vanishing-separate-closed}. We conclude that", "$x = x'$ as desired." ], "refs": [ "spaces-morphisms-lemma-universally-injective-local", "spaces-cohomology-lemma-vanishing-base-change", "spaces-morphisms-lemma-large-enough", "decent-spaces-lemma-algebraic-residue-field-extension-closed-point", "spaces-cohomology-lemma-vanishing-separate-closed" ], "ref_ids": [ 4795, 11318, 4821, 9508, 11319 ] } ], "ref_ids": [] }, { "id": 11321, "type": "theorem", "label": "spaces-cohomology-lemma-vanishing-separated", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-vanishing-separated", "contents": [ "In Situation \\ref{situation-vanishing} the morphism $p : X \\to \\Spec(A)$ is", "separated." ], "refs": [], "proofs": [ { "contents": [ "By Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-there-is-a-scheme-integral-over}", "we can find a scheme $Y$ and a surjective integral morphism", "$Y \\to X$. Since an integral morphism is affine, we can apply", "Lemma \\ref{lemma-vanishing-base-change}", "to see that $H^1(Y, \\mathcal{G}) = 0$ for every", "quasi-coherent $\\mathcal{O}_Y$-module $\\mathcal{G}$.", "Since $Y \\to X$ is quasi-compact and $X$ is quasi-compact,", "we see that $Y$ is quasi-compact.", "Since $Y$ is a scheme, we may apply", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-compact-h1-zero-covering}", "to see that $Y$ is affine. Hence $Y$ is separated.", "Note that an integral morphism is affine and universally closed, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-integral-universally-closed}.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-image-universally-closed-separated}", "we see that $X$ is a separated algebraic space." ], "refs": [ "decent-spaces-lemma-there-is-a-scheme-integral-over", "spaces-cohomology-lemma-vanishing-base-change", "coherent-lemma-quasi-compact-h1-zero-covering", "spaces-morphisms-lemma-integral-universally-closed", "spaces-morphisms-lemma-image-universally-closed-separated" ], "ref_ids": [ 9483, 11318, 3287, 4944, 4750 ] } ], "ref_ids": [] }, { "id": 11322, "type": "theorem", "label": "spaces-cohomology-lemma-Noetherian-h1-zero", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-Noetherian-h1-zero", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Assume that for every coherent $\\mathcal{O}_X$-module", "$\\mathcal{F}$ we have $H^1(X, \\mathcal{F}) = 0$.", "Then $X$ is an affine scheme." ], "refs": [], "proofs": [ { "contents": [ "The assumption implies that $H^1(X, \\mathcal{F}) = 0$ for every quasi-coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ by", "Lemmas \\ref{lemma-directed-colimit-coherent} and \\ref{lemma-colimits}.", "Then $X$ is affine by", "Proposition \\ref{proposition-vanishing-affine}." ], "refs": [ "spaces-cohomology-lemma-directed-colimit-coherent", "spaces-cohomology-lemma-colimits", "spaces-cohomology-proposition-vanishing-affine" ], "ref_ids": [ 11315, 11277, 11346 ] } ], "ref_ids": [] }, { "id": 11323, "type": "theorem", "label": "spaces-cohomology-lemma-Noetherian-h1-zero-invertible", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-Noetherian-h1-zero-invertible", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Assume that for every coherent $\\mathcal{O}_X$-module", "$\\mathcal{F}$ there exists an $n \\geq 1$ such that", "$H^1(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) = 0$.", "Then $X$ is a scheme and $\\mathcal{L}$ is ample on $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $s \\in H^0(X, \\mathcal{L}^{\\otimes d})$ be a global section.", "Let $U \\subset X$ be the open subspace over which $s$ is a generator", "of $\\mathcal{L}^{\\otimes d}$. In particular we have", "$\\mathcal{L}^{\\otimes d}|_U \\cong \\mathcal{O}_U$.", "We claim that $U$ is affine.", "\\medskip\\noindent", "Proof of the claim. We will show that $H^1(U, \\mathcal{F}) = 0$", "for every quasi-coherent $\\mathcal{O}_U$-module $\\mathcal{F}$.", "This will prove the claim by Proposition \\ref{proposition-vanishing-affine}.", "Denote $j : U \\to X$ the inclusion morphism.", "Since \\'etale locally the morphism $j$ is affine", "(by Morphisms, Lemma \\ref{morphisms-lemma-affine-s-open})", "we see that $j$ is affine (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-affine-local}).", "Hence we have", "$$", "H^1(U, \\mathcal{F}) = H^1(X, j_*\\mathcal{F})", "$$", "by Lemma \\ref{lemma-affine-vanishing-higher-direct-images}", "(and Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-apply-Leray}).", "Write $j_*\\mathcal{F} = \\colim \\mathcal{F}_i$ as a filtered colimit", "of coherent $\\mathcal{O}_X$-modules, see", "Lemma \\ref{lemma-directed-colimit-coherent}. Then", "$$", "H^1(X, j_*\\mathcal{F}) = \\colim H^1(X, \\mathcal{F}_i)", "$$", "by Lemma \\ref{lemma-colimits}.", "Thus it suffices to show that $H^1(X, \\mathcal{F}_i)$ maps", "to zero in $H^1(U, j^*\\mathcal{F}_i)$. By assumption there exists", "an $n \\geq 1$ such that", "$$", "H^1(X,", "\\mathcal{F}_i \\otimes_{\\mathcal{O}_X}", "(\\mathcal{O}_X \\oplus \\mathcal{L} \\oplus \\ldots", "\\oplus \\mathcal{L}^{\\otimes d - 1})", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) = 0", "$$", "Hence there exists an $a \\geq 0$ such that", "$H^1(X, \\mathcal{F}_i \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes ad}) = 0$.", "On the other hand, the map", "$$", "s^a : \\mathcal{F}_i \\longrightarrow", "\\mathcal{F}_i \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes ad}", "$$", "is an isomorphism after restriction to $U$. Contemplating the", "commutative diagram", "$$", "\\xymatrix{", "H^1(X, \\mathcal{F}_i) \\ar[r] \\ar[d]_{s^a} & H^1(U, j^*\\mathcal{F}_i)", "\\ar[d]^{\\cong} \\\\", "H^1(X, \\mathcal{F}_i \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes ad}) \\ar[r] &", "H^1(U,", "j^*(\\mathcal{F}_i \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes ad}))", "}", "$$", "we conclude that the map $H^1(X, \\mathcal{F}_i) \\to H^1(U, j^*\\mathcal{F}_i)$", "is zero and the claim holds.", "\\medskip\\noindent", "Let $x \\in |X|$ be a closed point. By Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-decent-space-closed-point}", "we can represent $x$ by a closed immersion $i : \\Spec(k) \\to X$", "(this also uses that a quasi-separated algebraic space is", "decent, see Decent Spaces, Section", "\\ref{decent-spaces-section-reasonable-decent}).", "Thus $\\mathcal{O}_X \\to i_*\\mathcal{O}_{\\Spec(k)}$ is surjective.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the kernel and choose", "$d \\geq 1$ such that", "$H^1(X, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) = 0$.", "Then", "$$", "H^0(X, \\mathcal{L}^{\\otimes d}) \\to", "H^0(X,", "i_*\\mathcal{O}_{\\Spec(k)} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) =", "H^0(\\Spec(k), i^*\\mathcal{L}^{\\otimes d}) \\cong k", "$$", "is surjective by the long exact cohomology sequence. Hence", "there exists an $s \\in H^0(X, \\mathcal{L}^{\\otimes d})$", "such that $x \\in U$ where $U$ is the open subspace corresponding to $s$", "as above. Thus $x$ is in the schematic locus", "(see Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme})", "of $X$ by our claim.", "\\medskip\\noindent", "To conclude that $X$ is a scheme, it suffices to show that", "any open subset of $|X|$ which contains all the closed points", "is equal to $|X|$. This follows from the fact that $|X|$", "is a Noetherian topological space, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-Noetherian-sober}.", "Finally, if $X$ is a scheme, then we can apply", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-compact-h1-zero-invertible}", "to conclude that $\\mathcal{L}$ is ample." ], "refs": [ "spaces-cohomology-proposition-vanishing-affine", "morphisms-lemma-affine-s-open", "spaces-morphisms-lemma-affine-local", "spaces-cohomology-lemma-affine-vanishing-higher-direct-images", "sites-cohomology-lemma-apply-Leray", "spaces-cohomology-lemma-directed-colimit-coherent", "spaces-cohomology-lemma-colimits", "decent-spaces-lemma-decent-space-closed-point", "spaces-properties-lemma-subscheme", "spaces-properties-lemma-Noetherian-sober", "coherent-lemma-quasi-compact-h1-zero-invertible" ], "ref_ids": [ 11346, 5178, 4798, 11288, 4221, 11315, 11277, 9510, 11848, 11892, 3289 ] } ], "ref_ids": [] }, { "id": 11324, "type": "theorem", "label": "spaces-cohomology-lemma-finite-morphism-Noetherian", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-finite-morphism-Noetherian", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is finite, surjective and $X$ locally Noetherian.", "Let $i : Z \\to X$ be a closed immersion. Denote $i' : Z' \\to Y$", "the inverse image of $Z$", "(Morphisms of Spaces, Section \\ref{spaces-morphisms-section-closed-immersions})", "and $f' : Z' \\to Z$", "the induced morphism. Then $\\mathcal{G} = f'_*\\mathcal{O}_{Z'}$", "is a coherent $\\mathcal{O}_Z$-module whose support is $Z$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $f'$ is the base change of $f$ and hence is finite", "and surjective by Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-surjective} and", "\\ref{spaces-morphisms-lemma-base-change-integral}.", "Note that $Y$, $Z$, and $Z'$ are locally Noetherian by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}", "(and the fact that closed immersions and finite morphisms", "are of finite type). By Lemma \\ref{lemma-finite-pushforward-coherent}", "we see that $\\mathcal{G}$ is a coherent $\\mathcal{O}_Z$-module.", "The support of $\\mathcal{G}$ is closed in $|Z|$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-support-finite-type}.", "Hence if the support of $\\mathcal{G}$ is not equal to $|Z|$,", "then after replacing $X$ by an open subspace we may assume", "$\\mathcal{G} = 0$ but $Z \\not = \\emptyset$.", "This would mean that $f'_*\\mathcal{O}_{Z'} = 0$. In particular the section", "$1 \\in \\Gamma(Z', \\mathcal{O}_{Z'}) = \\Gamma(Z, f'_*\\mathcal{O}_{Z'})$", "would be zero which would imply $Z' = \\emptyset$ is the empty", "algebraic space. This is impossible as $Z' \\to Z$ is surjective." ], "refs": [ "spaces-morphisms-lemma-base-change-surjective", "spaces-morphisms-lemma-base-change-integral", "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "spaces-cohomology-lemma-finite-pushforward-coherent", "spaces-morphisms-lemma-support-finite-type" ], "ref_ids": [ 4727, 4942, 4817, 11304, 4777 ] } ], "ref_ids": [] }, { "id": 11325, "type": "theorem", "label": "spaces-cohomology-lemma-affine-morphism-projection-ideal", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-affine-morphism-projection-ideal", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces", "over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $Y$.", "Let $\\mathcal{I}$ be a quasi-coherent sheaf of ideals on $X$.", "If $f$ is affine then", "$\\mathcal{I}f_*\\mathcal{F} = f_*(f^{-1}\\mathcal{I}\\mathcal{F})$", "(with notation as explained in the proof)." ], "refs": [], "proofs": [ { "contents": [ "The notation means the following. Since $f^{-1}$ is an exact functor", "we see that $f^{-1}\\mathcal{I}$ is a sheaf", "of ideals of $f^{-1}\\mathcal{O}_X$. Via the map", "$f^\\sharp : f^{-1}\\mathcal{O}_X \\to \\mathcal{O}_Y$", "on $Y_\\etale$ this acts on $\\mathcal{F}$.", "Then $f^{-1}\\mathcal{I}\\mathcal{F}$ is the subsheaf", "generated by sums of local sections of the form $as$ where $a$", "is a local section of $f^{-1}\\mathcal{I}$ and $s$ is a local section", "of $\\mathcal{F}$. It is a quasi-coherent $\\mathcal{O}_Y$-submodule", "of $\\mathcal{F}$ because it is also the image of a natural map", "$f^*\\mathcal{I} \\otimes_{\\mathcal{O}_Y} \\mathcal{F} \\to \\mathcal{F}$.", "\\medskip\\noindent", "Having said this the proof is straightforward. Namely, the question is", "\\'etale local on $X$ and hence we may assume $X$ is an affine scheme.", "In this case the result is a consequence of the corresponding result", "for schemes, see Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-affine-morphism-projection-ideal}." ], "refs": [ "coherent-lemma-affine-morphism-projection-ideal" ], "ref_ids": [ 3336 ] } ], "ref_ids": [] }, { "id": 11326, "type": "theorem", "label": "spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic", "spaces over $S$. Assume", "\\begin{enumerate}", "\\item $f$ finite,", "\\item $f$ surjective,", "\\item $Y$ affine, and", "\\item $X$ Noetherian.", "\\end{enumerate}", "Then $X$ is affine." ], "refs": [], "proofs": [ { "contents": [ "We will prove that under the assumptions of the lemma for any coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ we have $H^1(X, \\mathcal{F}) = 0$.", "This implies that $H^1(X, \\mathcal{F}) = 0$ for every quasi-coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ by", "Lemmas \\ref{lemma-directed-colimit-coherent} and \\ref{lemma-colimits}.", "Then it follows that $X$ is affine from", "Proposition \\ref{proposition-vanishing-affine}.", "\\medskip\\noindent", "Let $\\mathcal{P}$ be the property of coherent sheaves", "$\\mathcal{F}$ on $X$ defined by the rule", "$$", "\\mathcal{P}(\\mathcal{F}) \\Leftrightarrow H^1(X, \\mathcal{F}) = 0.", "$$", "We are going to apply Lemma \\ref{lemma-property-higher-rank-cohomological}.", "Thus we have to verify (1), (2) and (3) of that lemma for $\\mathcal{P}$.", "Property (1) follows from the long exact cohomology sequence associated", "to a short exact sequence of sheaves. Property (2) follows since", "$H^1(X, -)$ is an additive functor. To see (3) let $i : Z \\to X$ be", "a reduced closed subspace with $|Z|$ irreducible. Let $i' : Z' \\to Y$", "and $f' : Z' \\to Z$ be as in Lemma \\ref{lemma-finite-morphism-Noetherian}", "and set $\\mathcal{G} = f'_*\\mathcal{O}_{Z'}$. We claim that", "$\\mathcal{G}$ satisfies properties (3)(a) and (3)(b) of", "Lemma \\ref{lemma-property-higher-rank-cohomological}", "which will finish the proof. Property (3)(a) we have seen in", "Lemma \\ref{lemma-finite-morphism-Noetherian}. To see (3)(b) let", "$\\mathcal{I}$ be a nonzero quasi-coherent sheaf of ideals on $Z$.", "Denote $\\mathcal{I}' \\subset \\mathcal{O}_{Z'}$ the quasi-coherent", "ideal $(f')^{-1}\\mathcal{I} \\mathcal{O}_{Z'}$, i.e., the", "image of $(f')^*\\mathcal{I} \\to \\mathcal{O}_{Z'}$.", "By Lemma \\ref{lemma-affine-morphism-projection-ideal} we have", "$f_*\\mathcal{I}' = \\mathcal{I} \\mathcal{G}$.", "We claim the common value", "$\\mathcal{G}' = \\mathcal{I} \\mathcal{G} = f'_*\\mathcal{I}'$", "satisfies the condition expressed in (3)(b).", "First, it is clear that the support of $\\mathcal{G}/\\mathcal{G}'$", "is contained in the support of $\\mathcal{O}_Z/\\mathcal{I}$", "which is a proper subspace of $|Z|$ as $\\mathcal{I}$ is a", "nonzero ideal sheaf on the reduced and irreducible algebraic space $Z$.", "The morphism $f'$ is affine, hence $R^1f'_*\\mathcal{I}' = 0$ by", "Lemma \\ref{lemma-affine-vanishing-higher-direct-images}.", "As $Z'$ is affine (as a closed subscheme of an affine scheme)", "we have $H^1(Z', \\mathcal{I}') = 0$. Hence the Leray", "spectral sequence (in the form", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-apply-Leray})", "implies that $H^1(Z, f'_*\\mathcal{I}') = 0$.", "Since $i : Z \\to X$ is affine we conclude that", "$R^1i_*f'_*\\mathcal{I}' = 0$ hence $H^1(X, i_*f'_*\\mathcal{I}') = 0$", "by Leray again. In other words, we have $H^1(X, i_*\\mathcal{G}') = 0$", "as desired." ], "refs": [ "spaces-cohomology-lemma-directed-colimit-coherent", "spaces-cohomology-lemma-colimits", "spaces-cohomology-proposition-vanishing-affine", "spaces-cohomology-lemma-property-higher-rank-cohomological", "spaces-cohomology-lemma-finite-morphism-Noetherian", "spaces-cohomology-lemma-property-higher-rank-cohomological", "spaces-cohomology-lemma-finite-morphism-Noetherian", "spaces-cohomology-lemma-affine-morphism-projection-ideal", "spaces-cohomology-lemma-affine-vanishing-higher-direct-images", "sites-cohomology-lemma-apply-Leray" ], "ref_ids": [ 11315, 11277, 11346, 11313, 11324, 11313, 11324, 11325, 11288, 4221 ] } ], "ref_ids": [] }, { "id": 11327, "type": "theorem", "label": "spaces-cohomology-lemma-weak-chow", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-weak-chow", "contents": [ "Let $A$ be a ring. Let $X$ be an algebraic space over $\\Spec(A)$", "whose structure morphism $X \\to \\Spec(A)$ is separated of finite type.", "Then there exists a proper surjective morphism $X' \\to X$", "where $X'$ is a scheme which is H-quasi-projective over $\\Spec(A)$." ], "refs": [], "proofs": [ { "contents": [ "Let $W$ be an affine scheme and let $f : W \\to X$ be a surjective", "\\'etale morphism. There exists an integer $d$ such that all geometric", "fibres of f have $\\leq d$ points (because $X$ is a separated algebraic", "hence reasonable, see", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-bounded-fibres}).", "Picking $d$ minimal we get a nonempty open $U \\subset X$ such that", "$f^{-1}(U) \\to U$ is finite \\'etale of degree $d$, see", "Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-quasi-compact-reasonable-stratification}.", "Let", "$$", "V \\subset W \\times_X W \\times_X \\ldots \\times_X W", "$$", "($d$ factors in the fibre product) be the complement of all the diagonals.", "Because $W \\to X$ is separated the diagonal $W \\to W \\times_X W$ is a", "closed immersion. Since $W \\to X$ is \\'etale the diagonal", "$W \\to W \\times_X W$ is an open immersion, see", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-etale-unramified} and", "\\ref{spaces-morphisms-lemma-diagonal-unramified-morphism}.", "Hence the diagonals are open and closed subschemes", "of the quasi-compact scheme $W \\times_X \\ldots \\times_X W$.", "In particular we conclude $V$ is a quasi-compact scheme.", "Choose an open immersion $W \\subset Y$ with $Y$ H-projective over", "$A$ (this is possible as $W$ is affine and of finite type over $A$;", "for example we can use", "Morphisms, Lemmas", "\\ref{morphisms-lemma-quasi-affine-finite-type-over-S} and", "\\ref{morphisms-lemma-H-quasi-projective-open-H-projective}).", "Let", "$$", "Z \\subset Y \\times_A Y \\times_A \\ldots \\times_A Y", "$$", "be the scheme theoretic image of the composition", "$V \\to W \\times_X \\ldots \\times_X W \\to Y \\times_A \\ldots \\times_A Y$.", "Observe that this morphism is quasi-compact since $V$ is quasi-compact", "and $Y \\times_A \\ldots \\times_A Y$ is separated.", "Note that $V \\to Z$ is an open immersion as", "$V \\to Y \\times_A \\ldots \\times_A Y$ is an immersion, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-immersion}.", "The projection morphisms give $d$ morphisms $g_i : Z \\to Y$.", "These morphisms $g_i$ are projective as $Y$ is projective over $A$, see", "material in Morphisms, Section \\ref{morphisms-section-projective}.", "We set", "$$", "X' = \\bigcup g_i^{-1}(W) \\subset Z", "$$", "There is a morphism $X' \\to X$ whose restriction to $g_i^{-1}(W)$ is", "the composition $g_i^{-1}(W) \\to W \\to X$.", "Namely, these morphisms agree over $V$ hence agree over", "$g_i^{-1}(W) \\cap g_j^{-1}(W)$ by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-equality-of-morphisms}.", "Claim: the morphism $X' \\to X$ is proper.", "\\medskip\\noindent", "If the claim holds, then the lemma follows by induction on $d$.", "Namely, by construction $X'$ is H-quasi-projective over $\\Spec(A)$.", "The image of $X' \\to X$ contains the open $U$ as $V$ surjects onto $U$.", "Denote $T$ the reduced induced algebraic space structure on $X \\setminus U$.", "Then $T \\times_X W$ is a closed subscheme of $W$, hence affine.", "Moreover, the morphism $T \\times_X W \\to T$ is \\'etale and every geometric", "fibre has $< d$ points. By induction hypothesis there exists a proper", "surjective morphism $T' \\to T$ where $T'$ is a scheme H-quasi-projective", "over $\\Spec(A)$. Since $T$ is a closed subspace of $X$ we see that", "$T' \\to X$ is a proper morphism. Thus the lemma follows by taking the", "proper surjective morphism $X' \\amalg T' \\to X$.", "\\medskip\\noindent", "Proof of the claim. By construction the morphism $X' \\to X$ is separated", "and of finite type. We will check conditions (1) -- (4) of", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-refined-valuative-criterion-universally-closed}", "for the morphisms $V \\to X'$ and $X' \\to X$.", "Conditions (1) and (2) we have seen above.", "Condition (3) holds as $X' \\to X$ is separated (as a morphism whose", "source is a separated algebraic space). Thus it suffices to check", "liftability to $X'$ for diagrams", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & V \\ar[d] \\\\", "\\Spec(R) \\ar[r] & X", "}", "$$", "where $R$ is a valuation ring with fraction field $K$.", "Note that the top horizontal map is given by $d$ pairwise distinct", "$K$-valued points $w_1, \\ldots, w_d$ of $W$. In fact, this", "is a complete set of inverse images of the point $x \\in X(K)$", "coming from the diagram. Since $W \\to X$ is surjective,", "we can, after possibly replacing $R$ by an extension of valuation rings,", "lift the morphism $\\Spec(R) \\to X$ to a morphism $w : \\Spec(R) \\to W$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-lift-valuation-ring-through-flat-morphism}.", "Since $w_1, \\ldots, w_d$ is a complete collection of inverse images of", "$x$ we see that $w|_{\\Spec(K)}$ is equal to one of them, say $w_i$.", "Thus we see that we get a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & Z \\ar[d]_{g_i}\\\\", "\\Spec(R) \\ar[r]^w & Y", "}", "$$", "By the valuative criterion of properness for the projective", "morphism $g_i$ we can lift $w$ to $z : \\Spec(R) \\to Z$, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-projective-proper}", "and", "Schemes, Proposition \\ref{schemes-proposition-characterize-universally-closed}.", "The image of $z$ is in $g_i^{-1}(W) \\subset X'$ and the proof is complete." ], "refs": [ "decent-spaces-lemma-bounded-fibres", "decent-spaces-lemma-quasi-compact-reasonable-stratification", "spaces-morphisms-lemma-etale-unramified", "spaces-morphisms-lemma-diagonal-unramified-morphism", "morphisms-lemma-quasi-affine-finite-type-over-S", "morphisms-lemma-H-quasi-projective-open-H-projective", "morphisms-lemma-quasi-compact-immersion", "spaces-morphisms-lemma-equality-of-morphisms", "spaces-morphisms-lemma-refined-valuative-criterion-universally-closed", "spaces-morphisms-lemma-lift-valuation-ring-through-flat-morphism", "morphisms-lemma-locally-projective-proper", "schemes-proposition-characterize-universally-closed" ], "ref_ids": [ 9466, 9475, 4913, 4902, 5392, 5428, 5154, 4791, 4934, 4933, 5422, 7733 ] } ], "ref_ids": [] }, { "id": 11328, "type": "theorem", "label": "spaces-cohomology-lemma-check-separated-dvr", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-check-separated-dvr", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. Assume", "\\begin{enumerate}", "\\item $Y$ is locally Noetherian,", "\\item $f$ is locally of finite type and quasi-separated,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y", "}", "$$", "where $A$ is a discrete valuation ring and $K$ its fraction field,", "there is at most one dotted arrow making the diagram commute.", "\\end{enumerate}", "Then $f$ is separated." ], "refs": [], "proofs": [ { "contents": [ "To prove $f$ is separated, we may work \\'etale locally on $Y$", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-separated-local}).", "Choose an affine scheme $U$ and an \\'etale morphism $U \\to X \\times_Y X$.", "Set $V = X \\times_{\\Delta, X \\times_Y X} U$ which is quasi-compact because", "$f$ is quasi-separated. Consider a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & V \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & U", "}", "$$", "We can interpret the composition $\\Spec(A) \\to U \\to X \\times_Y X$", "as a pair of morphisms $a, b : \\Spec(A) \\to X$ agreeing as morphisms", "into $Y$ and equal when restricted to $\\Spec(K)$. Hence our assumption", "(3) guarantees $a = b$ and we find the dotted arrow in the diagram.", "By Limits, Lemma \\ref{limits-lemma-Noetherian-dvr-valuative-proper}", "we conclude that $V \\to U$ is proper. In other words, $\\Delta$ is proper.", "Since $\\Delta$ is a monomorphism, we find that $\\Delta$ is a", "closed immersion (\\'Etale Morphisms, Lemma", "\\ref{etale-lemma-characterize-closed-immersion}) as desired." ], "refs": [ "spaces-morphisms-lemma-separated-local", "limits-lemma-Noetherian-dvr-valuative-proper", "etale-lemma-characterize-closed-immersion" ], "ref_ids": [ 4722, 15099, 10702 ] } ], "ref_ids": [] }, { "id": 11329, "type": "theorem", "label": "spaces-cohomology-lemma-check-proper-dvr", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-check-proper-dvr", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. Assume", "\\begin{enumerate}", "\\item $Y$ is locally Noetherian,", "\\item $f$ is of finite type and quasi-separated,", "\\item for every commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y", "}", "$$", "where $A$ is a discrete valuation ring and $K$ its fraction field,", "there is a unique dotted arrow making the diagram commute.", "\\end{enumerate}", "Then $f$ is proper." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove $f$ is universally closed because $f$ is separated by", "Lemma \\ref{lemma-check-separated-dvr}.", "To do this we may work \\'etale locally on $Y$", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-local}).", "Hence we may assume $Y = \\Spec(A)$ is a Noetherian affine scheme.", "Choose $X' \\to X$ as in the weak form of Chow's lemma", "(Lemma \\ref{lemma-weak-chow}). We claim that $X' \\to \\Spec(A)$", "is universally closed. The claim implies the lemma by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-image-proper-is-proper}.", "To prove this, according to", "Limits, Lemma \\ref{limits-lemma-check-universally-closed-Noetherian}", "it suffices to prove that in every solid commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X' \\ar[r] & X \\ar[d] \\\\", "\\Spec(A) \\ar[rr] \\ar@{-->}[ru]^a \\ar@{-->}[rru]_b & & Y", "}", "$$", "where $A$ is a dvr with fraction field $K$ we can find the", "dotted arrow $a$. By assumption we can find the dotted arrow $b$.", "Then the morphism $X' \\times_{X, b} \\Spec(A) \\to \\Spec(A)$", "is a proper morphism of schemes and by the valuative criterion", "for morphisms of schemes we can lift $b$ to the desired morphism $a$." ], "refs": [ "spaces-cohomology-lemma-check-separated-dvr", "spaces-morphisms-lemma-universally-closed-local", "spaces-cohomology-lemma-weak-chow", "spaces-morphisms-lemma-image-proper-is-proper", "limits-lemma-check-universally-closed-Noetherian" ], "ref_ids": [ 11328, 4748, 11327, 4921, 15100 ] } ], "ref_ids": [] }, { "id": 11330, "type": "theorem", "label": "spaces-cohomology-lemma-kill-by-twisting", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-kill-by-twisting", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[r]_i \\ar[rd]_f & \\mathbf{P}^n_Y \\ar[d] \\\\", "& Y", "}", "$$", "of algebraic spaces over $S$. Assume $i$ is a closed immersion", "and $Y$ Noetherian. Set $\\mathcal{L} = i^*\\mathcal{O}_{\\mathbf{P}^n_Y}(1)$.", "Let $\\mathcal{F}$ be a coherent module on $X$.", "Then there exists an integer $d_0$ such that for all $d \\geq d_0$ we have", "$R^pf_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) = 0$", "for all $p > 0$." ], "refs": [], "proofs": [ { "contents": [ "Checking whether $R^pf_*(\\mathcal{F} \\otimes \\mathcal{L}^{\\otimes d})$", "is zero can be done \\'etale locally on $Y$, see", "Equation (\\ref{equation-representable-higher-direct-image}).", "Hence we may assume $Y$ is the spectrum of a Noetherian ring. In this case", "$X$ is a scheme and the result follows from", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-kill-by-twisting}." ], "refs": [ "coherent-lemma-kill-by-twisting" ], "ref_ids": [ 3344 ] } ], "ref_ids": [] }, { "id": 11331, "type": "theorem", "label": "spaces-cohomology-lemma-proper-pushforward-coherent", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-proper-pushforward-coherent", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism", "of algebraic spaces over $S$ with $Y$ locally Noetherian.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Then $R^if_*\\mathcal{F}$ is a coherent $\\mathcal{O}_Y$-module", "for all $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "We first remark that $X$ is a locally Noetherian algebraic space", "by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}.", "Hence the statement of the lemma makes sense. Moreover, computing", "$R^if_*\\mathcal{F}$ commutes with \\'etale localization on $Y$", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-pushforward-etale-base-change-modules})", "and checking whether $R^if_*\\mathcal{F}$ coherent can be done", "\\'etale locally on $Y$ (Lemma \\ref{lemma-coherent-Noetherian}).", "Hence we may assume that $Y = \\Spec(A)$ is a Noetherian affine scheme.", "\\medskip\\noindent", "Assume $Y = \\Spec(A)$ is an affine scheme. Note that $f$ is locally", "of finite presentation", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation}).", "Thus it is of finite presentation, hence $X$ is Noetherian", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-noetherian}).", "Thus Lemma \\ref{lemma-property-higher-rank-cohomological-variant}", "applies to the category of coherent modules of $X$.", "For a coherent sheaf $\\mathcal{F}$ on $X$ we say $\\mathcal{P}$ holds", "if and only if $R^if_*\\mathcal{F}$ is a coherent module on $\\Spec(A)$.", "We will show that conditions (1), (2), and (3) of", "Lemma \\ref{lemma-property-higher-rank-cohomological-variant} hold", "for this property thereby finishing the proof of the lemma.", "\\medskip\\noindent", "Verification of condition (1). Let", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0", "$$", "be a short exact sequence of coherent sheaves on $X$.", "Consider the long exact sequence of higher direct images", "$$", "R^{p - 1}f_*\\mathcal{F}_3 \\to", "R^pf_*\\mathcal{F}_1 \\to", "R^pf_*\\mathcal{F}_2 \\to", "R^pf_*\\mathcal{F}_3 \\to", "R^{p + 1}f_*\\mathcal{F}_1", "$$", "Then it is clear that if 2-out-of-3 of the sheaves $\\mathcal{F}_i$", "have property $\\mathcal{P}$, then the higher direct images of the", "third are sandwiched in this exact complex between two coherent", "sheaves. Hence these higher direct images are also coherent by", "Lemmas \\ref{lemma-coherent-abelian-Noetherian} and", "\\ref{lemma-coherent-Noetherian-quasi-coherent-sub-quotient}.", "Hence property $\\mathcal{P}$ holds for the third as well.", "\\medskip\\noindent", "Verification of condition (2). This follows immediately from the fact", "that $R^if_*(\\mathcal{F}_1 \\oplus \\mathcal{F}_2) =", "R^if_*\\mathcal{F}_1 \\oplus R^if_*\\mathcal{F}_2$ and that a summand", "of a coherent module is coherent (see lemmas cited above).", "\\medskip\\noindent", "Verification of condition (3). Let $i : Z \\to X$ be a closed immersion", "with $Z$ reduced and $|Z|$ irreducible. Set $g = f \\circ i : Z \\to \\Spec(A)$.", "Let $\\mathcal{G}$ be a coherent module on $Z$ whose scheme theoretic support", "is equal to $Z$ such that $R^pg_*\\mathcal{G}$ is coherent for all $p$.", "Then $\\mathcal{F} = i_*\\mathcal{G}$ is a coherent module on", "$X$ whose support scheme theoretic support is $Z$ such that", "$R^pf_*\\mathcal{F} = R^pg_*\\mathcal{G}$. To see this use", "the Leray spectral sequence", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-relative-Leray})", "and the fact that $R^qi_*\\mathcal{G} = 0$ for $q > 0$ by", "Lemma \\ref{lemma-affine-vanishing-higher-direct-images}", "and the fact that a closed immersion is affine.", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-closed-immersion-affine}).", "Thus we reduce to finding a coherent sheaf $\\mathcal{G}$ on $Z$", "with support equal to $Z$ such that $R^pg_*\\mathcal{G}$ is coherent", "for all $p$.", "\\medskip\\noindent", "We apply Lemma \\ref{lemma-weak-chow} to the morphism $Z \\to \\Spec(A)$.", "Thus we get a diagram", "$$", "\\xymatrix{", "Z \\ar[rd]_g & Z' \\ar[d]^-{g'} \\ar[l]^\\pi \\ar[r]_i & \\mathbf{P}^n_A \\ar[dl] \\\\", "& \\Spec(A) &", "}", "$$", "with $\\pi : Z' \\to Z$ proper surjective and $i$ an immersion.", "Since $Z \\to \\Spec(A)$ is proper we conclude that $g'$ is proper", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-composition-proper}).", "Hence $i$ is a closed immersion", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-universally-closed-permanence} and", "\\ref{spaces-morphisms-lemma-immersion-when-closed}).", "It follows that the morphism", "$i' = (i, \\pi) : \\mathbf{P}^n_A \\times_{\\Spec(A)} Z' = \\mathbf{P}^n_Z$ is", "a closed immersion", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-semi-diagonal}).", "Set", "$$", "\\mathcal{L} =", "i^*\\mathcal{O}_{\\mathbf{P}^n_A}(1) =", "(i')^*\\mathcal{O}_{\\mathbf{P}^n_Z}(1)", "$$", "We may apply Lemma \\ref{lemma-kill-by-twisting}", "to $\\mathcal{L}$ and $\\pi$ as well as $\\mathcal{L}$ and $g'$. ", "Hence for all $d \\gg 0$ we have", "$R^p\\pi_*\\mathcal{L}^{\\otimes d} = 0$ for all $p > 0$ and", "$R^p(g')_*\\mathcal{L}^{\\otimes d} = 0$ for all $p > 0$.", "Set $\\mathcal{G} = \\pi_*\\mathcal{L}^{\\otimes d}$.", "By the Leray spectral sequence", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-relative-Leray})", "we have", "$$", "E_2^{p, q} = R^pg_* R^q\\pi_*\\mathcal{L}^{\\otimes d}", "\\Rightarrow", "R^{p + q}(g')_*\\mathcal{L}^{\\otimes d}", "$$", "and by choice of $d$ the only nonzero terms in $E_2^{p, q}$ are", "those with $q = 0$ and the only nonzero terms of", "$R^{p + q}(g')_*\\mathcal{L}^{\\otimes d}$ are those with $p = q = 0$.", "This implies that $R^pg_*\\mathcal{G} = 0$ for $p > 0$ and", "that $g_*\\mathcal{G} = (g')_*\\mathcal{L}^{\\otimes n}$.", "Applying", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-locally-projective-pushforward}", "we see that $g_*\\mathcal{G} = (g')_*\\mathcal{L}^{\\otimes d}$ is", "coherent.", "\\medskip\\noindent", "We still have to check that the support of $\\mathcal{G}$ is $Z$.", "This follows from the fact that $\\mathcal{L}^{\\otimes d}$ has", "lots of global sections. We spell it out here.", "Note that $\\mathcal{L}^{\\otimes d}$ is globally generated for all $d \\geq 0$", "because the same is true for $\\mathcal{O}_{\\mathbf{P}^n}(d)$.", "Pick a point $z \\in Z'$ mapping to the generic point $\\xi$ of $Z$", "which we can do as $\\pi$ is surjective.", "(Observe that $Z$ does indeed have a generic point as $|Z|$ is irreducible", "and $Z$ is Noetherian, hence quasi-separated, hence $|Z|$ is a sober", "topological space by", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-separated-sober}.)", "Pick $s \\in \\Gamma(Z', \\mathcal{L}^{\\otimes d})$ which does not vanish at $z$.", "Since $\\Gamma(Z, \\mathcal{G}) = \\Gamma(Z', \\mathcal{L}^{\\otimes d})$", "we may think of $s$ as a global section of $\\mathcal{G}$.", "Choose a geometric point $\\overline{z}$ of $Z'$ lying over $z$", "and denote $\\overline{\\xi} = g' \\circ \\overline{z}$", "the corresponding geometric point of $Z$. The adjunction map", "$$", "(g')^*\\mathcal{G} =", "(g')^*g'_*\\mathcal{L}^{\\otimes d} \\longrightarrow \\mathcal{L}^{\\otimes d}", "$$", "induces a map of stalks", "$\\mathcal{G}_{\\overline{\\xi}} \\to \\mathcal{L}_{\\overline{z}}$, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-stalk-pullback-quasi-coherent}.", "Moreover the adjunction map sends the pullback of $s$ (viewed as a section", "of $\\mathcal{G}$) to $s$ (viewed as a section of $\\mathcal{L}^{\\otimes d}$).", "Thus the image of $s$ in the vector space which is the source of the arrow", "$$", "\\mathcal{G}_{\\overline{\\xi}} \\otimes \\kappa(\\overline{\\xi})", "\\longrightarrow", "\\mathcal{L}^{\\otimes d}_{\\overline{z}} \\otimes \\kappa(\\overline{z})", "$$", "isn't zero since by choice of $s$ the image in the target of the arrow", "is nonzero. Hence $\\xi$ is in the support of $\\mathcal{G}$", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-support-finite-type}).", "Since $|Z|$ is irreducible and $Z$ is reduced we conclude that", "the scheme theoretic support of $\\mathcal{G}$ is all of $Z$ as desired." ], "refs": [ "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "spaces-properties-lemma-pushforward-etale-base-change-modules", "spaces-cohomology-lemma-coherent-Noetherian", "spaces-morphisms-lemma-noetherian-finite-type-finite-presentation", "spaces-morphisms-lemma-finite-presentation-noetherian", "spaces-cohomology-lemma-property-higher-rank-cohomological-variant", "spaces-cohomology-lemma-property-higher-rank-cohomological-variant", "spaces-cohomology-lemma-coherent-abelian-Noetherian", "spaces-cohomology-lemma-coherent-Noetherian-quasi-coherent-sub-quotient", "sites-cohomology-lemma-relative-Leray", "spaces-cohomology-lemma-affine-vanishing-higher-direct-images", "spaces-morphisms-lemma-closed-immersion-affine", "spaces-cohomology-lemma-weak-chow", "spaces-morphisms-lemma-composition-proper", "spaces-morphisms-lemma-universally-closed-permanence", "spaces-morphisms-lemma-immersion-when-closed", "spaces-morphisms-lemma-semi-diagonal", "spaces-cohomology-lemma-kill-by-twisting", "sites-cohomology-lemma-relative-Leray", "coherent-lemma-locally-projective-pushforward", "spaces-properties-lemma-quasi-separated-sober", "spaces-properties-lemma-stalk-pullback-quasi-coherent", "spaces-morphisms-lemma-support-finite-type" ], "ref_ids": [ 4817, 11898, 11297, 4844, 4843, 11314, 11314, 11298, 11299, 4222, 11288, 4801, 11327, 4918, 4920, 4763, 4716, 11330, 4222, 3345, 11852, 11910, 4777 ] } ], "ref_ids": [] }, { "id": 11332, "type": "theorem", "label": "spaces-cohomology-lemma-proper-over-affine-cohomology-finite", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-proper-over-affine-cohomology-finite", "contents": [ "Let $A$ be a Noetherian ring.", "Let $f : X \\to \\Spec(A)$ be a proper morphism of algebraic spaces.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Then $H^i(X, \\mathcal{F})$ is finite $A$-module for all $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "This is just the affine case of Lemma \\ref{lemma-proper-pushforward-coherent}.", "Namely, by Lemma \\ref{lemma-higher-direct-image} we know that", "$R^if_*\\mathcal{F}$ is a quasi-coherent sheaf. Hence it is the quasi-coherent", "sheaf associated to the $A$-module", "$\\Gamma(\\Spec(A), R^if_*\\mathcal{F}) = H^i(X, \\mathcal{F})$.", "The equality holds by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-apply-Leray}", "and vanishing of higher cohomology groups of quasi-coherent modules", "on affine schemes (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}).", "By Lemma \\ref{lemma-coherent-Noetherian} we see $R^if_*\\mathcal{F}$ is", "a coherent sheaf if and only if $H^i(X, \\mathcal{F})$", "is an $A$-module of finite type. Hence", "Lemma \\ref{lemma-proper-pushforward-coherent} gives us the conclusion." ], "refs": [ "spaces-cohomology-lemma-proper-pushforward-coherent", "spaces-cohomology-lemma-higher-direct-image", "sites-cohomology-lemma-apply-Leray", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "spaces-cohomology-lemma-coherent-Noetherian", "spaces-cohomology-lemma-proper-pushforward-coherent" ], "ref_ids": [ 11331, 11271, 4221, 3282, 11297, 11331 ] } ], "ref_ids": [] }, { "id": 11333, "type": "theorem", "label": "spaces-cohomology-lemma-graded-finiteness", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-graded-finiteness", "contents": [ "Let $A$ be a Noetherian ring.", "Let $B$ be a finitely generated graded $A$-algebra.", "Let $f : X \\to \\Spec(A)$ be a proper morphism of algebraic spaces.", "Set $\\mathcal{B} = f^*\\widetilde B$.", "Let $\\mathcal{F}$ be a quasi-coherent", "graded $\\mathcal{B}$-module of finite type.", "For every $p \\geq 0$ the graded $B$-module $H^p(X, \\mathcal{F})$", "is a finite $B$-module." ], "refs": [], "proofs": [ { "contents": [ "To prove this we consider the fibre product diagram", "$$", "\\xymatrix{", "X' = \\Spec(B) \\times_{\\Spec(A)} X", "\\ar[r]_-\\pi \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "\\Spec(B) \\ar[r] &", "\\Spec(A)", "}", "$$", "Note that $f'$ is a proper morphism, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-proper}.", "Also, $B$ is a finitely generated $A$-algebra, and hence", "Noetherian (Algebra, Lemma \\ref{algebra-lemma-Noetherian-permanence}).", "This implies that $X'$ is a Noetherian algebraic space", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-noetherian}).", "Note that $X'$ is the relative spectrum of the quasi-coherent", "$\\mathcal{O}_X$-algebra $\\mathcal{B}$ by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-affine-equivalence-algebras}.", "Since $\\mathcal{F}$ is a quasi-coherent $\\mathcal{B}$-module", "we see that there is a unique quasi-coherent", "$\\mathcal{O}_{X'}$-module $\\mathcal{F}'$ such that", "$\\pi_*\\mathcal{F}' = \\mathcal{F}$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-affine-equivalence-modules}.", "Since $\\mathcal{F}$ is finite type as a $\\mathcal{B}$-module we", "conclude that $\\mathcal{F}'$ is a finite type", "$\\mathcal{O}_{X'}$-module (details omitted). In other words,", "$\\mathcal{F}'$ is a coherent $\\mathcal{O}_{X'}$-module", "(Lemma \\ref{lemma-coherent-Noetherian}).", "Since the morphism $\\pi : X' \\to X$ is affine we have", "$$", "H^p(X, \\mathcal{F}) = H^p(X', \\mathcal{F}')", "$$", "by", "Lemma \\ref{lemma-affine-vanishing-higher-direct-images}", "and", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-apply-Leray}.", "Thus the lemma follows from", "Lemma \\ref{lemma-proper-over-affine-cohomology-finite}." ], "refs": [ "spaces-morphisms-lemma-base-change-proper", "algebra-lemma-Noetherian-permanence", "spaces-morphisms-lemma-finite-presentation-noetherian", "spaces-morphisms-lemma-affine-equivalence-algebras", "spaces-morphisms-lemma-affine-equivalence-modules", "spaces-cohomology-lemma-coherent-Noetherian", "spaces-cohomology-lemma-affine-vanishing-higher-direct-images", "sites-cohomology-lemma-apply-Leray", "spaces-cohomology-lemma-proper-over-affine-cohomology-finite" ], "ref_ids": [ 4917, 448, 4843, 4802, 4803, 11297, 11288, 4221, 11332 ] } ], "ref_ids": [] }, { "id": 11334, "type": "theorem", "label": "spaces-cohomology-lemma-vanshing-gives-ample", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-vanshing-gives-ample", "contents": [ "Let $R$ be a Noetherian ring. Let $X$ be a proper algebraic space", "over $R$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is a scheme and $\\mathcal{L}$ is ample on $X$,", "\\item for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ there exists", "an $n_0 \\geq 0$ such that", "$H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}) = 0$ for all $n \\geq n_0$", "and $p > 0$, and", "\\item for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ there exists", "an $n \\geq 1$ such that", "$H^1(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}) = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) follows from", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-vanshing-gives-ample}.", "The implication (2) $\\Rightarrow$ (3) is trivial.", "The implication (3) $\\Rightarrow$ (1) is", "Lemma \\ref{lemma-Noetherian-h1-zero-invertible}." ], "refs": [ "coherent-lemma-vanshing-gives-ample", "spaces-cohomology-lemma-Noetherian-h1-zero-invertible" ], "ref_ids": [ 3346, 11323 ] } ], "ref_ids": [] }, { "id": 11335, "type": "theorem", "label": "spaces-cohomology-lemma-surjective-finite-morphism-ample", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-surjective-finite-morphism-ample", "contents": [ "Let $R$ be a Noetherian ring. Let $f : Y \\to X$ be a morphism of", "algebraic spaces proper over $R$. Let $\\mathcal{L}$ be an", "invertible $\\mathcal{O}_X$-module. Assume $f$ is finite and surjective.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is a scheme and $\\mathcal{L}$ is ample, and", "\\item $Y$ is a scheme and $f^*\\mathcal{L}$ is ample.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Then $Y$ is a scheme as a finite morphism is representable", "(by schemes), see Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-integral-local}. Hence (2)", "follows from Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-surjective-finite-morphism-ample}.", "\\medskip\\noindent", "Assume (2). Let $P$ be the following property on", "coherent $\\mathcal{O}_X$-modules $\\mathcal{F}$: there exists an $n_0$", "such that $H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}) = 0$", "for all $n \\geq n_0$ and $p > 0$. We will prove that $P$ holds", "for any coherent $\\mathcal{O}_X$-module $\\mathcal{F}$, which implies", "$\\mathcal{L}$ is ample by Lemma \\ref{lemma-vanshing-gives-ample}.", "We are going to apply Lemma \\ref{lemma-property-higher-rank-cohomological}.", "Thus we have to verify (1), (2) and (3) of that lemma for $P$.", "Property (1) follows from the long exact cohomology sequence associated", "to a short exact sequence of sheaves and the fact that tensoring with", "an invertible sheaf is an exact functor. Property (2) follows since", "$H^p(X, -)$ is an additive functor.", "\\medskip\\noindent", "To see (3) let $i : Z \\to X$ be a reduced closed subspace with $|Z|$", "irreducible. Let $i' : Z' \\to Y$ and $f' : Z' \\to Z$ be as in", "Lemma \\ref{lemma-finite-morphism-Noetherian} and set", "$\\mathcal{G} = f'_*\\mathcal{O}_{Z'}$. We claim that", "$\\mathcal{G}$ satisfies properties (3)(a) and (3)(b) of", "Lemma \\ref{lemma-property-higher-rank-cohomological}", "which will finish the proof. Property (3)(a) we have seen in", "Lemma \\ref{lemma-finite-morphism-Noetherian}. To see (3)(b) let", "$\\mathcal{I}$ be a nonzero quasi-coherent sheaf of ideals on $Z$.", "Denote $\\mathcal{I}' \\subset \\mathcal{O}_{Z'}$ the quasi-coherent", "ideal $(f')^{-1}\\mathcal{I} \\mathcal{O}_{Z'}$, i.e., the", "image of $(f')^*\\mathcal{I} \\to \\mathcal{O}_{Z'}$.", "By Lemma \\ref{lemma-affine-morphism-projection-ideal} we have", "$f_*\\mathcal{I}' = \\mathcal{I} \\mathcal{G}$. We claim the common value", "$\\mathcal{G}' = \\mathcal{I} \\mathcal{G} = f'_*\\mathcal{I}'$", "satisfies the condition expressed in (3)(b).", "First, it is clear that the support of $\\mathcal{G}/\\mathcal{G}'$", "is contained in the support of $\\mathcal{O}_Z/\\mathcal{I}$", "which is a proper subspace of $|Z|$ as $\\mathcal{I}$ is a", "nonzero ideal sheaf on the reduced and irreducible algebraic space $Z$.", "Recall that $f'_*$, $i_*$, and $i'_*$ transform coherent modules into", "coherent modules, see Lemmas \\ref{lemma-finite-pushforward-coherent} and", "\\ref{lemma-i-star-equivalence}.", "As $Y$ is a scheme and $\\mathcal{L}$ is ample", "we see from Lemma \\ref{lemma-vanshing-gives-ample}", "that there exists an $n_0$ such that", "$$", "H^p(Y, i'_*\\mathcal{I}' \\otimes_{\\mathcal{O}_Y} f^*\\mathcal{L}^{\\otimes n}) = 0", "$$", "for $n \\geq n_0$ and $p > 0$. Now we get", "\\begin{align*}", "H^p(X, i_*\\mathcal{G}' \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n})", "& =", "H^p(Z, \\mathcal{G'} \\otimes_{\\mathcal{O}_Z} i^*\\mathcal{L}^{\\otimes n}) \\\\", "& =", "H^p(Z,", "f'_*\\mathcal{I}' \\otimes_{\\mathcal{O}_Z} i^*\\mathcal{L}^{\\otimes n})) \\\\", "& =", "H^p(Z, f'_*(\\mathcal{I}' \\otimes_{\\mathcal{O}_{Z'}}", "(f')^*i^*\\mathcal{L}^{\\otimes n})) \\\\", "& =", "H^p(Z, f'_*(\\mathcal{I}' \\otimes_{\\mathcal{O}_{Z'}}", "(i')^*f^*\\mathcal{L}^{\\otimes n})) \\\\", "& =", "H^p(Z',", "\\mathcal{I}' \\otimes_{\\mathcal{O}_{Z'}} (i')^*f^*\\mathcal{L}^{\\otimes n})) \\\\", "& =", "H^p(Y, i'_*\\mathcal{I}' \\otimes_{\\mathcal{O}_Y} f^*\\mathcal{L}^{\\otimes n}) = 0", "\\end{align*}", "Here we have used the projection formula and the Leray spectral sequence", "(see Cohomology on Sites, Sections", "\\ref{sites-cohomology-section-projection-formula} and", "\\ref{sites-cohomology-section-leray})", "and Lemma \\ref{lemma-finite-higher-direct-image-zero}.", "This verifies property (3)(b) of", "Lemma \\ref{lemma-property-higher-rank-cohomological} as desired." ], "refs": [ "spaces-morphisms-lemma-integral-local", "coherent-lemma-surjective-finite-morphism-ample", "spaces-cohomology-lemma-vanshing-gives-ample", "spaces-cohomology-lemma-property-higher-rank-cohomological", "spaces-cohomology-lemma-finite-morphism-Noetherian", "spaces-cohomology-lemma-property-higher-rank-cohomological", "spaces-cohomology-lemma-finite-morphism-Noetherian", "spaces-cohomology-lemma-affine-morphism-projection-ideal", "spaces-cohomology-lemma-finite-pushforward-coherent", "spaces-cohomology-lemma-i-star-equivalence", "spaces-cohomology-lemma-vanshing-gives-ample", "spaces-cohomology-lemma-finite-higher-direct-image-zero", "spaces-cohomology-lemma-property-higher-rank-cohomological" ], "ref_ids": [ 4940, 3347, 11334, 11313, 11324, 11313, 11324, 11325, 11304, 11303, 11334, 11273, 11313 ] } ], "ref_ids": [] }, { "id": 11336, "type": "theorem", "label": "spaces-cohomology-lemma-cohomology-powers-ideal-times-F", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-cohomology-powers-ideal-times-F", "contents": [ "In Situation \\ref{situation-formal-functions}.", "Set $B = \\bigoplus_{n \\geq 0} I^n$.", "Then for every $p \\geq 0$ the graded $B$-module", "$\\bigoplus_{n \\geq 0} H^p(X, I^n\\mathcal{F})$ is", "a finite $B$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{B} = \\bigoplus I^n\\mathcal{O}_X = f^*\\widetilde{B}$.", "Then $\\bigoplus I^n\\mathcal{F}$ is a finite type", "graded $\\mathcal{B}$-module. Hence the result follows", "from Lemma \\ref{lemma-graded-finiteness}." ], "refs": [ "spaces-cohomology-lemma-graded-finiteness" ], "ref_ids": [ 11333 ] } ], "ref_ids": [] }, { "id": 11337, "type": "theorem", "label": "spaces-cohomology-lemma-cohomology-powers-ideal-application", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-cohomology-powers-ideal-application", "contents": [ "In Situation \\ref{situation-formal-functions}.", "For every $p \\geq 0$ there exists an integer $c \\geq 0$ such that", "\\begin{enumerate}", "\\item the multiplication map", "$I^{n - c} \\otimes H^p(X, I^c\\mathcal{F}) \\to H^p(X, I^n\\mathcal{F})$", "is surjective for all $n \\geq c$, and", "\\item the image of $H^p(X, I^{n + m}\\mathcal{F}) \\to H^p(X, I^n\\mathcal{F})$", "is contained in the submodule $I^{m - c} H^p(X, I^n\\mathcal{F})$", "for all $n \\geq 0$, $m \\geq c$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-cohomology-powers-ideal-times-F}", "we can find $d_1, \\ldots, d_t \\geq 0$, and", "$x_i \\in H^p(X, I^{d_i}\\mathcal{F})$ such that", "$\\bigoplus_{n \\geq 0} H^p(X, I^n\\mathcal{F})$ is generated", "by $x_1, \\ldots, x_t$ over $B = \\bigoplus_{n \\geq 0} I^n$.", "Take $c = \\max\\{d_i\\}$. It is clear that (1) holds.", "For (2) let $b = \\max(0, n - c)$.", "Consider the commutative diagram of $A$-modules", "$$", "\\xymatrix{", "I^{n + m - c - b} \\otimes I^b \\otimes", "H^p(X, I^c\\mathcal{F}) \\ar[r] \\ar[d] &", "I^{n + m - c} \\otimes H^p(X, I^c\\mathcal{F}) \\ar[r] &", "H^p(X, I^{n + m}\\mathcal{F}) \\ar[d] \\\\", "I^{n + m - c - b} \\otimes H^p(X, I^n\\mathcal{F}) \\ar[rr] & &", "H^p(X, I^n\\mathcal{F})", "}", "$$", "By part (1) of the lemma the composition of the horizontal arrows", "is surjective if $n + m \\geq c$. On the other hand, it is clear", "that $n + m - c - b \\geq m - c$. Hence part (2)." ], "refs": [ "spaces-cohomology-lemma-cohomology-powers-ideal-times-F" ], "ref_ids": [ 11336 ] } ], "ref_ids": [] }, { "id": 11338, "type": "theorem", "label": "spaces-cohomology-lemma-ML-cohomology-powers-ideal", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-ML-cohomology-powers-ideal", "contents": [ "In Situation \\ref{situation-formal-functions}.", "Fix $p \\geq 0$.", "\\begin{enumerate}", "\\item There exists a $c_1 \\geq 0$ such that for all $n \\geq c_1$", "we have", "$$", "\\Ker(", "H^p(X, \\mathcal{F}) \\to H^p(X, \\mathcal{F}/I^n\\mathcal{F})", ")", "\\subset", "I^{n - c_1}H^p(X, \\mathcal{F}).", "$$", "\\item The inverse system", "$$", "\\left(H^p(X, \\mathcal{F}/I^n\\mathcal{F})\\right)_{n \\in \\mathbf{N}}", "$$", "satisfies the Mittag-Leffler condition (see", "Homology, Definition \\ref{homology-definition-Mittag-Leffler}).", "\\item In fact for any $p$ and $n$ there exists a $c_2(n) \\geq n$", "such that", "$$", "\\Im(H^p(X, \\mathcal{F}/I^k\\mathcal{F})", "\\to H^p(X, \\mathcal{F}/I^n\\mathcal{F}))", "=", "\\Im(H^p(X, \\mathcal{F})", "\\to H^p(X, \\mathcal{F}/I^n\\mathcal{F}))", "$$", "for all $k \\geq c_2(n)$.", "\\end{enumerate}" ], "refs": [ "homology-definition-Mittag-Leffler" ], "proofs": [ { "contents": [ "Let $c_1 = \\max\\{c_p, c_{p + 1}\\}$, where $c_p, c_{p +1}$ are the integers", "found in Lemma \\ref{lemma-cohomology-powers-ideal-application} for", "$H^p$ and $H^{p + 1}$. We will use this constant in the proofs of", "(1), (2) and (3).", "\\medskip\\noindent", "Let us prove part (1). Consider the short exact sequence", "$$", "0 \\to I^n\\mathcal{F} \\to \\mathcal{F} \\to \\mathcal{F}/I^n\\mathcal{F} \\to 0", "$$", "From the long exact cohomology sequence we see that", "$$", "\\Ker(", "H^p(X, \\mathcal{F}) \\to H^p(X, \\mathcal{F}/I^n\\mathcal{F})", ")", "=", "\\Im(", "H^p(X, I^n\\mathcal{F}) \\to H^p(X, \\mathcal{F})", ")", "$$", "Hence by our choice of $c_1$ we see that this is contained in", "$I^{n - c_1}H^p(X, \\mathcal{F})$ for $n \\geq c_1$.", "\\medskip\\noindent", "Note that part (3) implies part (2) by definition of the Mittag-Leffler", "condition.", "\\medskip\\noindent", "Let us prove part (3).", "Fix an $n$ throughout the rest of the proof.", "Consider the commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "I^n\\mathcal{F} \\ar[r] &", "\\mathcal{F} \\ar[r] &", "\\mathcal{F}/I^n\\mathcal{F} \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "I^{n + m}\\mathcal{F} \\ar[r] \\ar[u] &", "\\mathcal{F} \\ar[r] \\ar[u] &", "\\mathcal{F}/I^{n + m}\\mathcal{F} \\ar[r] \\ar[u] &", "0", "}", "$$", "This gives rise to the following commutative diagram", "$$", "\\xymatrix{", "H^p(X, I^n\\mathcal{F}) \\ar[r] &", "H^p(X, \\mathcal{F}) \\ar[r] &", "H^p(X, \\mathcal{F}/I^n\\mathcal{F}) \\ar[r]_\\delta &", "H^{p + 1}(X, I^n\\mathcal{F}) \\\\", "H^p(X, I^{n + m}\\mathcal{F}) \\ar[r] \\ar[u] &", "H^p(X, \\mathcal{F}) \\ar[r] \\ar[u]^1 &", "H^p(X, \\mathcal{F}/I^{n + m}\\mathcal{F}) \\ar[r] \\ar[u] &", "H^{p + 1}(X, I^{n + m}\\mathcal{F}) \\ar[u]^a", "}", "$$", "If $m \\geq c_1$ we see that the image of $a$ is", "contained in $I^{m - c_1} H^{p + 1}(X, I^n\\mathcal{F})$.", "By the Artin-Rees lemma (see Algebra, Lemma \\ref{algebra-lemma-map-AR})", "there exists an integer $c_3(n)$ such that", "$$", "I^N H^{p + 1}(X, I^n\\mathcal{F}) \\cap \\Im(\\delta)", "\\subset", "\\delta\\left(I^{N - c_3(n)}H^p(X, \\mathcal{F}/I^n\\mathcal{F})\\right)", "$$", "for all $N \\geq c_3(n)$. As $H^p(X, \\mathcal{F}/I^n\\mathcal{F})$", "is annihilated by $I^n$, we see that if $m \\geq c_3(n) + c_1 + n$,", "then", "$$", "\\Im(H^p(X, \\mathcal{F}/I^{n + m}\\mathcal{F})", "\\to H^p(X, \\mathcal{F}/I^n\\mathcal{F}))", "=", "\\Im(H^p(X, \\mathcal{F})", "\\to H^p(X, \\mathcal{F}/I^n\\mathcal{F}))", "$$", "In other words, part (3) holds with $c_2(n) = c_3(n) + c_1 + n$." ], "refs": [ "spaces-cohomology-lemma-cohomology-powers-ideal-application", "algebra-lemma-map-AR" ], "ref_ids": [ 11337, 626 ] } ], "ref_ids": [ 12188 ] }, { "id": 11339, "type": "theorem", "label": "spaces-cohomology-lemma-spell-out-theorem-formal-functions", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-spell-out-theorem-formal-functions", "contents": [ "Let $A$ be a ring. Let $I \\subset A$ be an ideal. Assume $A$ is", "Noetherian and complete with respect to $I$.", "Let $f : X \\to \\Spec(A)$ be a proper morphism of algebraic spaces.", "Let $\\mathcal{F}$ be a coherent sheaf on $X$. Then", "$$", "H^p(X, \\mathcal{F}) = \\lim_n H^p(X, \\mathcal{F}/I^n\\mathcal{F})", "$$", "for all $p \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of the theorem on formal functions", "(Theorem \\ref{theorem-formal-functions}) in the", "case of a complete Noetherian base ring. Namely, in this case the", "$A$-module $H^p(X, \\mathcal{F})$ is finite", "(Lemma \\ref{lemma-proper-over-affine-cohomology-finite}) hence", "$I$-adically complete (Algebra, Lemma \\ref{algebra-lemma-completion-tensor})", "and we see that completion on the left hand side is not necessary." ], "refs": [ "spaces-cohomology-theorem-formal-functions", "spaces-cohomology-lemma-proper-over-affine-cohomology-finite", "algebra-lemma-completion-tensor" ], "ref_ids": [ 11270, 11332, 869 ] } ], "ref_ids": [] }, { "id": 11340, "type": "theorem", "label": "spaces-cohomology-lemma-formal-functions-stalk", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-formal-functions-stalk", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ and let $\\mathcal{F}$ be a quasi-coherent sheaf on $Y$. Assume", "\\begin{enumerate}", "\\item $Y$ locally Noetherian,", "\\item $f$ proper, and", "\\item $\\mathcal{F}$ coherent.", "\\end{enumerate}", "Let $\\overline{y}$ be a geometric point of $Y$.", "Consider the ``infinitesimal neighbourhoods''", "$$", "\\xymatrix{", "X_n =", "\\Spec(\\mathcal{O}_{Y, \\overline{y}}/\\mathfrak m_{\\overline{y}}^n) \\times_Y X", "\\ar[r]_-{i_n} \\ar[d]_{f_n} &", "X \\ar[d]^f \\\\", "\\Spec(\\mathcal{O}_{Y, \\overline{y}}/\\mathfrak m_{\\overline{y}}^n)", "\\ar[r]^-{c_n} & Y", "}", "$$", "of the fibre $X_1 = X_{\\overline{y}}$ and set", "$\\mathcal{F}_n = i_n^*\\mathcal{F}$. Then we have", "$$", "\\left(R^pf_*\\mathcal{F}\\right)_{\\overline{y}}^\\wedge", "\\cong", "\\lim_n H^p(X_n, \\mathcal{F}_n)", "$$", "as $\\mathcal{O}_{Y, \\overline{y}}^\\wedge$-modules." ], "refs": [], "proofs": [ { "contents": [ "This is just a reformulation of a special case of the theorem", "on formal functions, Theorem \\ref{theorem-formal-functions}.", "Let us spell it out. Note that $\\mathcal{O}_{Y, \\overline{y}}$", "is a Noetherian local ring, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-Noetherian-local-ring-Noetherian}.", "Consider the canonical morphism", "$c : \\Spec(\\mathcal{O}_{Y, \\overline{y}}) \\to Y$.", "This is a flat morphism as it identifies local rings.", "Denote $f' : X' \\to \\Spec(\\mathcal{O}_{Y, \\overline{y}})$", "the base change of $f$ to this local ring. We see that", "$c^*R^pf_*\\mathcal{F} = R^pf'_*\\mathcal{F}'$ by", "Lemma \\ref{lemma-flat-base-change-cohomology}.", "Moreover, we have canonical identifications $X_n = X'_n$", "for all $n \\geq 1$.", "\\medskip\\noindent", "Hence we may assume that $Y = \\Spec(A)$ is the spectrum of", "a strictly henselian Noetherian local ring $A$ with maximal ideal", "$\\mathfrak m$ and that $\\overline{y} \\to Y$ is equal to", "$\\Spec(A/\\mathfrak m) \\to Y$. It follows that", "$$", "\\left(R^pf_*\\mathcal{F}\\right)_{\\overline{y}} =", "\\Gamma(Y, R^pf_*\\mathcal{F}) =", "H^p(X, \\mathcal{F})", "$$", "because $(Y, \\overline{y})$ is an initial object in the category", "of \\'etale neighbourhoods of $\\overline{y}$.", "The morphisms $c_n$ are each closed immersions.", "Hence their base changes $i_n$ are closed immersions as well.", "Note that $i_{n, *}\\mathcal{F}_n = i_{n, *}i_n^*\\mathcal{F}", "= \\mathcal{F}/\\mathfrak m^n\\mathcal{F}$. By the Leray spectral sequence", "for $i_n$, and Lemma \\ref{lemma-finite-pushforward-coherent} we see that", "$$", "H^p(X_n, \\mathcal{F}_n) =", "H^p(X, i_{n, *}\\mathcal{F}) =", "H^p(X, \\mathcal{F}/\\mathfrak m^n\\mathcal{F})", "$$", "Hence we may indeed apply the theorem on formal functions to compute", "the limit in the statement of the lemma and we win." ], "refs": [ "spaces-cohomology-theorem-formal-functions", "spaces-properties-lemma-Noetherian-local-ring-Noetherian", "spaces-cohomology-lemma-flat-base-change-cohomology", "spaces-cohomology-lemma-finite-pushforward-coherent" ], "ref_ids": [ 11270, 11893, 11296, 11304 ] } ], "ref_ids": [] }, { "id": 11341, "type": "theorem", "label": "spaces-cohomology-lemma-higher-direct-images-zero-finite-fibre", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-higher-direct-images-zero-finite-fibre", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $\\overline{y}$ be a geometric point of $Y$.", "Assume", "\\begin{enumerate}", "\\item $Y$ locally Noetherian,", "\\item $f$ is proper, and", "\\item $X_{\\overline{y}}$ has discrete underlying topological space.", "\\end{enumerate}", "Then for any coherent sheaf $\\mathcal{F}$ on $X$ we have", "$(R^pf_*\\mathcal{F})_{\\overline{y}} = 0$ for all $p > 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\kappa(\\overline{y})$ be the residue field of the local", "ring of $\\mathcal{O}_{Y, \\overline{y}}$. As in", "Lemma \\ref{lemma-formal-functions-stalk}", "we set $X_{\\overline{y}} = X_1 = \\Spec(\\kappa(\\overline{y})) \\times_Y X$.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-quasi-finite-at-point}", "the morphism $f : X \\to Y$ is quasi-finite at each of", "the points of the fibre of $X \\to Y$ over $\\overline{y}$.", "It follows that $X_{\\overline{y}} \\to \\overline{y}$ is separated and", "quasi-finite. Hence $X_{\\overline{y}}$ is a scheme", "by Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}.", "Since it is quasi-compact its underlying topological space is a finite", "discrete space. Then it is an affine scheme by", "Schemes, Lemma \\ref{schemes-lemma-scheme-finite-discrete-affine}.", "By Lemma \\ref{lemma-image-affine-finite-morphism-affine-Noetherian}", "it follows that the algebraic spaces $X_n$ are affine schemes as well.", "Moreover, the underlying topological of each $X_n$ is the same", "as that of $X_1$. Hence it follows that $H^p(X_n, \\mathcal{F}_n) = 0$", "for all $p > 0$. Hence we see that", "$(R^pf_*\\mathcal{F})_{\\overline{y}}^\\wedge = 0$", "by Lemma \\ref{lemma-formal-functions-stalk}.", "Note that $R^pf_*\\mathcal{F}$ is coherent by", "Lemma \\ref{lemma-proper-pushforward-coherent} and", "hence $R^pf_*\\mathcal{F}_{\\overline{y}}$ is a finite", "$\\mathcal{O}_{Y, \\overline{y}}$-module.", "By Algebra, Lemma \\ref{algebra-lemma-completion-tensor}", "this implies that $(R^pf_*\\mathcal{F})_{\\overline{y}} = 0$." ], "refs": [ "spaces-cohomology-lemma-formal-functions-stalk", "spaces-morphisms-lemma-quasi-finite-at-point", "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "schemes-lemma-scheme-finite-discrete-affine", "spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian", "spaces-cohomology-lemma-formal-functions-stalk", "spaces-cohomology-lemma-proper-pushforward-coherent", "algebra-lemma-completion-tensor" ], "ref_ids": [ 11340, 4877, 4983, 7678, 11326, 11340, 11331, 869 ] } ], "ref_ids": [] }, { "id": 11342, "type": "theorem", "label": "spaces-cohomology-lemma-higher-direct-images-zero-above-dimension-fibre", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-higher-direct-images-zero-above-dimension-fibre", "contents": [ "\\begin{slogan}", "For proper maps, stalks of higher direct images are trivial in degrees", "larger than the dimension of the fibre.", "\\end{slogan}", "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\overline{y}$ be a geometric point of $Y$.", "Assume", "\\begin{enumerate}", "\\item $Y$ locally Noetherian,", "\\item $f$ is proper, and", "\\item $\\dim(X_{\\overline{y}}) = d$.", "\\end{enumerate}", "Then for any coherent sheaf $\\mathcal{F}$ on $X$ we have", "$(R^pf_*\\mathcal{F})_{\\overline{y}} = 0$ for all $p > d$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\kappa(\\overline{y})$ be the residue field of the local", "ring of $\\mathcal{O}_{Y, \\overline{y}}$. As in", "Lemma \\ref{lemma-formal-functions-stalk}", "we set $X_{\\overline{y}} = X_1 = \\Spec(\\kappa(\\overline{y})) \\times_Y X$.", "Moreover, the underlying topological space of each infinitesimal", "neighbourhood $X_n$ is the same as that of $X_{\\overline{y}}$.", "Hence $H^p(X_n, \\mathcal{F}_n) = 0$ for all $p > d$ by", "Lemma \\ref{lemma-vanishing-above-dimension}.", "Hence we see that $(R^pf_*\\mathcal{F})_{\\overline{y}}^\\wedge = 0$", "by Lemma \\ref{lemma-formal-functions-stalk} for $p > d$.", "Note that $R^pf_*\\mathcal{F}$ is coherent by", "Lemma \\ref{lemma-proper-pushforward-coherent} and", "hence $R^pf_*\\mathcal{F}_{\\overline{y}}$ is a finite", "$\\mathcal{O}_{Y, \\overline{y}}$-module.", "By Algebra, Lemma \\ref{algebra-lemma-completion-tensor}", "this implies that $(R^pf_*\\mathcal{F})_{\\overline{y}} = 0$." ], "refs": [ "spaces-cohomology-lemma-formal-functions-stalk", "spaces-cohomology-lemma-vanishing-above-dimension", "spaces-cohomology-lemma-formal-functions-stalk", "spaces-cohomology-lemma-proper-pushforward-coherent", "algebra-lemma-completion-tensor" ], "ref_ids": [ 11340, 11294, 11340, 11331, 869 ] } ], "ref_ids": [] }, { "id": 11343, "type": "theorem", "label": "spaces-cohomology-lemma-characterize-finite", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-characterize-finite", "contents": [ "(For a more general version see", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-characterize-finite}).", "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume $Y$ is locally Noetherian.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is finite, and", "\\item $f$ is proper and $|X_k|$ is a discrete space", "for every morphism $\\Spec(k) \\to Y$ where $k$ is a field.", "\\end{enumerate}" ], "refs": [ "spaces-more-morphisms-lemma-characterize-finite" ], "proofs": [ { "contents": [ "A finite morphism is proper according to", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-proper}.", "A finite morphism is quasi-finite according to", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-quasi-finite}.", "A quasi-finite morphism has discrete fibres $X_k$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-locally-quasi-finite}.", "Hence a finite morphism is proper and has discrete fibres $X_k$.", "\\medskip\\noindent", "Assume $f$ is proper with discrete fibres $X_k$.", "We want to show $f$ is finite.", "In fact it suffices to prove $f$ is affine.", "Namely, if $f$ is affine, then it follows that", "$f$ is integral by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-integral-universally-closed}", "whereupon it follows from", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-integral}", "that $f$ is finite.", "\\medskip\\noindent", "To show that $f$ is affine we may assume that $Y$ is affine, and our", "goal is to show that $X$ is affine too.", "Since $f$ is proper we see that $X$ is separated and quasi-compact.", "We will show that for any coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ we have $H^1(X, \\mathcal{F}) = 0$.", "This implies that $H^1(X, \\mathcal{F}) = 0$ for every quasi-coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ by", "Lemmas \\ref{lemma-directed-colimit-coherent} and \\ref{lemma-colimits}.", "Then it follows that $X$ is affine from", "Proposition \\ref{proposition-vanishing-affine}. By", "Lemma \\ref{lemma-higher-direct-images-zero-finite-fibre}", "we conclude that the stalks of $R^1f_*\\mathcal{F}$", "are zero for all geometric points of $Y$.", "In other words, $R^1f_*\\mathcal{F} = 0$. Hence we see from", "the Leray Spectral Sequence for $f$ that", "$H^1(X , \\mathcal{F}) = H^1(Y, f_*\\mathcal{F})$.", "Since $Y$ is affine, and $f_*\\mathcal{F}$ is quasi-coherent", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-pushforward})", "we conclude $H^1(Y, f_*\\mathcal{F}) = 0$", "from", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "Hence $H^1(X, \\mathcal{F}) = 0$ as desired." ], "refs": [ "spaces-morphisms-lemma-finite-proper", "spaces-morphisms-lemma-finite-quasi-finite", "spaces-morphisms-lemma-locally-quasi-finite", "spaces-morphisms-lemma-integral-universally-closed", "spaces-morphisms-lemma-finite-integral", "spaces-cohomology-lemma-directed-colimit-coherent", "spaces-cohomology-lemma-colimits", "spaces-cohomology-proposition-vanishing-affine", "spaces-cohomology-lemma-higher-direct-images-zero-finite-fibre", "spaces-morphisms-lemma-pushforward", "coherent-lemma-quasi-coherent-affine-cohomology-zero" ], "ref_ids": [ 4946, 4945, 4833, 4944, 4943, 11315, 11277, 11346, 11341, 4760, 3282 ] } ], "ref_ids": [ 173 ] }, { "id": 11344, "type": "theorem", "label": "spaces-cohomology-lemma-proper-finite-fibre-finite-in-neighbourhood", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-lemma-proper-finite-fibre-finite-in-neighbourhood", "contents": [ "(For a more general version see", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood}).", "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $\\overline{y}$ be a geometric point of $Y$.", "Assume", "\\begin{enumerate}", "\\item $Y$ is locally Noetherian,", "\\item $f$ is proper, and", "\\item $|X_{\\overline{y}}|$ is finite.", "\\end{enumerate}", "Then there exists an open neighbourhood $V \\subset Y$ of $\\overline{y}$", "such that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite." ], "refs": [ "spaces-more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood" ], "proofs": [ { "contents": [ "The morphism $f$ is quasi-finite at all the geometric points of $X$", "lying over $\\overline{y}$ by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-quasi-finite-at-point}.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part} the", "set of points at which $f$ is quasi-finite is an open subspace $U \\subset X$.", "Let $Z = X \\setminus U$. Then $\\overline{y} \\not \\in f(Z)$. Since $f$", "is proper the set $f(Z) \\subset Y$ is closed. Choose any open neighbourhood", "$V \\subset Y$ of $\\overline{y}$ with $Z \\cap V = \\emptyset$. Then", "$f^{-1}(V) \\to V$ is locally quasi-finite and proper.", "Hence $f^{-1}(V) \\to V$ has discrete fibres $X_k$", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-quasi-finite})", "which are quasi-compact hence finite.", "Thus $f^{-1}(V) \\to V$", "is finite by Lemma \\ref{lemma-characterize-finite}." ], "refs": [ "spaces-morphisms-lemma-quasi-finite-at-point", "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part", "spaces-morphisms-lemma-locally-quasi-finite", "spaces-cohomology-lemma-characterize-finite" ], "ref_ids": [ 4877, 4876, 4833, 11343 ] } ], "ref_ids": [ 174 ] }, { "id": 11345, "type": "theorem", "label": "spaces-cohomology-proposition-vanishing", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-proposition-vanishing", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Assume $X$ is quasi-compact and separated.", "Let $U$ be an affine scheme, and let", "$f : U \\to X$ be a surjective \\'etale morphism.", "Let $d$ be an upper bound for the size of the fibres of", "$|U| \\to |X|$. Then for any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "we have $H^q(X, \\mathcal{F}) = 0$ for $q \\geq d$." ], "refs": [], "proofs": [ { "contents": [ "We will use the spectral sequence of", "Lemma \\ref{lemma-alternating-spectral-sequence}.", "The lemma applies since $f$ is separated as $U$ is separated, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-compose-after-separated}.", "Since $X$ is separated the scheme $U \\times_X \\ldots \\times_X U$ is a closed", "subscheme of", "$U \\times_{\\Spec(\\mathbf{Z})} \\ldots \\times_{\\Spec(\\mathbf{Z})} U$", "hence is affine. Thus $W_p$ is affine. Hence $U_p = W_p/S_{p + 1}$ is an", "affine scheme by", "Groupoids, Proposition \\ref{groupoids-proposition-finite-flat-equivalence}.", "The discussion in", "Section \\ref{section-higher-direct-image}", "shows that cohomology of quasi-coherent sheaves on $W_p$ (as an algebraic", "space) agrees with the cohomology of the corresponding quasi-coherent", "sheaf on the underlying affine scheme, hence vanishes in positive degrees by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "By", "Lemma \\ref{lemma-quasi-coherent-twist}", "the sheaves", "$\\mathcal{F}|_{U_p} \\otimes_\\mathbf{Z} \\underline{\\mathbf{Z}}(\\chi_p)$", "are quasi-coherent. Hence", "$H^q(W_p,", "\\mathcal{F}|_{U_p} \\otimes_\\mathbf{Z} \\underline{\\mathbf{Z}}(\\chi_p))$", "is zero when $q > 0$. By our definition of the integer $d$ we see that", "$W_p = \\emptyset$ for $p \\geq d$. Hence also", "$H^0(W_p,", "\\mathcal{F}|_{U_p} \\otimes_\\mathbf{Z} \\underline{\\mathbf{Z}}(\\chi_p))$", "is zero when $p \\geq d$.", "This proves the proposition." ], "refs": [ "spaces-cohomology-lemma-alternating-spectral-sequence", "spaces-morphisms-lemma-compose-after-separated", "groupoids-proposition-finite-flat-equivalence", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "spaces-cohomology-lemma-quasi-coherent-twist" ], "ref_ids": [ 11284, 4720, 9669, 3282, 11285 ] } ], "ref_ids": [] }, { "id": 11346, "type": "theorem", "label": "spaces-cohomology-proposition-vanishing-affine", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-proposition-vanishing-affine", "contents": [ "\\begin{slogan}", "Serre's criterion for affineness in the setting of algebraic spaces.", "\\end{slogan}", "A quasi-compact and quasi-separated algebraic space is affine", "if and only if all higher cohomology groups of quasi-coherent sheaves", "vanish. More precisely, any algebraic space as in", "Situation \\ref{situation-vanishing} is an affine scheme." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $U = \\Spec(B)$ and a surjective \\'etale", "morphism $\\varphi : U \\to X$. Set $R = U \\times_X U$. As $p$ is separated", "(Lemma \\ref{lemma-vanishing-separated}) we see that $R$ is a", "closed subscheme of $U \\times_{\\Spec(A)} U = \\Spec(B \\otimes_A B)$.", "Hence $R = \\Spec(C)$ is affine too and the ring map", "$$", "B \\otimes_A B \\longrightarrow C", "$$", "is surjective. Let us denote the two maps $s, t : B \\to C$ as usual. Pick", "$g_1, \\ldots, g_m \\in B$ such that $s(g_1), \\ldots, s(g_m)$ generate $C$", "over $t : B \\to C$ (which is possible as $t : B \\to C$ is of finite", "presentation and the displayed map is surjective). Then $g_1, \\ldots, g_m$", "give global sections of $\\varphi_*\\mathcal{O}_U$ and the map", "$$", "\\mathcal{O}_X[z_1, \\ldots, z_n] \\longrightarrow \\varphi_*\\mathcal{O}_U,", "\\quad", "z_j \\longmapsto g_j", "$$", "is surjective: you can check this by restricting to $U$.", "Namely, $\\varphi^*\\varphi_*\\mathcal{O}_U = t_*\\mathcal{O}_R$", "(by Lemma \\ref{lemma-flat-base-change-cohomology})", "hence you get exactly the condition that $s(g_i)$ generate $C$", "over $t : B \\to C$. By the vanishing of $H^1$ of the kernel we see that", "$$", "\\Gamma(X, \\mathcal{O}_X[x_1, \\ldots, x_n]) =", "A[x_1, \\ldots, x_n] \\longrightarrow", "\\Gamma(X, \\varphi_*\\mathcal{O}_U) = \\Gamma(U, \\mathcal{O}_U) = B", "$$", "is surjective. Thus we conclude that $B$ is a finite type $A$-algebra.", "Hence $X \\to \\Spec(A)$ is of finite type and separated.", "By Lemma \\ref{lemma-vanishing-injective}", "and", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-locally-quasi-finite}", "it is also locally quasi-finite. Hence $X \\to \\Spec(A)$ is representable by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable}", "and $X$ is a scheme. Finally $X$ is affine, hence equal to $\\Spec(A)$,", "by an application of Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-compact-h1-zero-covering}." ], "refs": [ "spaces-cohomology-lemma-vanishing-separated", "spaces-cohomology-lemma-flat-base-change-cohomology", "spaces-cohomology-lemma-vanishing-injective", "spaces-morphisms-lemma-locally-quasi-finite", "spaces-morphisms-lemma-locally-quasi-finite-separated-representable", "coherent-lemma-quasi-compact-h1-zero-covering" ], "ref_ids": [ 11321, 11296, 11320, 4833, 4972, 3287 ] } ], "ref_ids": [] }, { "id": 11350, "type": "theorem", "label": "artin-theorem-contractions", "categories": [ "artin" ], "title": "artin-theorem-contractions", "contents": [ "\\begin{reference}", "\\cite[Theorem 3.1]{ArtinII}", "\\end{reference}", "Let $S$ be a locally Noetherian scheme such that $\\mathcal{O}_{S, s}$", "is a G-ring for all finite type points $s \\in S$. Let $X'$ be an algebraic", "space locally of finite type over $S$. Let $T' \\subset |X'|$ be a closed", "subset. Let $W$ be a locally Noetherian formal algebraic space over $S$", "with $W_{red}$ locally of finite type over $S$. Finally, we let", "$$", "g : X'_{/T'} \\longrightarrow W", "$$", "be a formal modification, see Algebraization of Formal Spaces, Definition", "\\ref{restricted-definition-formal-modification}. If $X'$ and $W$ are", "separated\\footnote{See Remark \\ref{remark-separated-needed}.} over $S$, then", "there exists a proper morphism $f : X' \\to X$ of algebraic spaces over $S$,", "a closed subset $T \\subset |X|$, and an isomorphism $a : X_{/T} \\to W$", "of formal algebraic spaces such that", "\\begin{enumerate}", "\\item $T'$ is the inverse image of $T$ by $|f| : |X'| \\to |X|$,", "\\item $f : X' \\to X$ maps $X' \\setminus T'$ isomorphically to", "$X \\setminus T$, and", "\\item $g = a \\circ f_{/T}$ where $f_{/T} : X'_{/T'} \\to X_{/T}$", "is the induced morphism.", "\\end{enumerate}", "In other words, $(f : X' \\to X, T, a)$ is a solution as defined earlier in", "this section." ], "refs": [ "restricted-definition-formal-modification", "artin-remark-separated-needed" ], "proofs": [ { "contents": [ "Let $F$ be the functor constructed using $X'$, $T'$, $W$, $g$ in this section.", "By Lemma \\ref{lemma-functor-is-solution} it suffices to show that", "$F$ corresponds to an algebraic space $X$ locally of finite type over $S$.", "In order to do this, we will apply", "Proposition \\ref{proposition-spaces-diagonal-representable-noetherian}.", "Namely, by Lemma \\ref{lemma-diagonal-contractions}", "the diagonal of $F$ is representable by closed immersions", "and by", "Lemmas \\ref{lemma-sheaf}, \\ref{lemma-limit-preserving},", "\\ref{lemma-rs}, \\ref{lemma-finite-dim},", "\\ref{lemma-formal-object-effective}, and \\ref{lemma-openness-versality}", "we have axioms [0], [1], [2], [3], [4], and [5]." ], "refs": [ "artin-lemma-functor-is-solution", "artin-proposition-spaces-diagonal-representable-noetherian", "artin-lemma-diagonal-contractions", "artin-lemma-sheaf", "artin-lemma-limit-preserving", "artin-lemma-rs", "artin-lemma-finite-dim", "artin-lemma-formal-object-effective", "artin-lemma-openness-versality" ], "ref_ids": [ 11404, 11416, 11406, 11407, 11408, 11409, 11410, 11411, 11413 ] } ], "ref_ids": [ 2447, 11441 ] }, { "id": 11351, "type": "theorem", "label": "artin-lemma-predeformation-category", "categories": [ "artin" ], "title": "artin-lemma-predeformation-category", "contents": [ "The functor $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ defined above", "is a predeformation category." ], "refs": [], "proofs": [ { "contents": [ "We have to show that $\\mathcal{F}$ is (a) cofibred in groupoids over", "$\\mathcal{C}_\\Lambda$ and (b) that $\\mathcal{F}(k)$ is a category equivalent", "to a category with a single object and a single morphism.", "\\medskip\\noindent", "Proof of (a). The fibre categories of $\\mathcal{F}$", "over $\\mathcal{C}_\\Lambda$ are groupoids as the fibre categories", "of $\\mathcal{X}$ are groupoids. Let $A \\to A'$ be a morphism of", "$\\mathcal{C}_\\Lambda$ and let $x_0 \\to x$ be an object of $\\mathcal{F}(A)$.", "Because $\\mathcal{X}$ is fibred in groupoids, we can find a morphism", "$x' \\to x$ lying over $\\Spec(A') \\to \\Spec(A)$. Since the composition", "$A \\to A' \\to k$ is equal the given map $A \\to k$ we see (by uniqueness", "of pullbacks up to isomorphism) that the pullback via $\\Spec(k) \\to \\Spec(A')$", "of $x'$ is $x_0$, i.e., that there exists a morphism $x_0 \\to x'$", "lying over $\\Spec(k) \\to \\Spec(A')$ compatible with", "$x_0 \\to x$ and $x' \\to x$. This proves that $\\mathcal{F}$ has", "pushforwards. We conclude by (the dual of)", "Categories, Lemma \\ref{categories-lemma-fibred-groupoids}.", "\\medskip\\noindent", "Proof of (b). If $A = k$, then $\\Spec(k) = \\Spec(A)$ and since $\\mathcal{X}$", "is fibred in groupoids over $(\\Sch/S)_{fppf}$ we see that given any object", "$x_0 \\to x$ in $\\mathcal{F}(k)$ the morphism $x_0 \\to x$ is an isomorphism.", "Hence every object of $\\mathcal{F}(k)$ is isomorphic to $x_0 \\to x_0$.", "Clearly the only self morphism of $x_0 \\to x_0$ in $\\mathcal{F}$ is", "the identity." ], "refs": [ "categories-lemma-fibred-groupoids" ], "ref_ids": [ 12294 ] } ], "ref_ids": [] }, { "id": 11352, "type": "theorem", "label": "artin-lemma-formally-smooth-on-deformation-categories", "categories": [ "artin" ], "title": "artin-lemma-formally-smooth-on-deformation-categories", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$", "be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "Assume either", "\\begin{enumerate}", "\\item $F$ is formally smooth on objects (Criteria for Representability,", "Section \\ref{criteria-section-formally-smooth}),", "\\item $F$ is representable by algebraic spaces and formally smooth, or", "\\item $F$ is representable by algebraic spaces and smooth.", "\\end{enumerate}", "Then for every finite type field $k$ over $S$ and object", "$x_0$ of $\\mathcal{X}$ over $k$ the functor (\\ref{equation-functoriality})", "is smooth in the sense of", "Formal Deformation Theory, Definition", "\\ref{formal-defos-definition-smooth-morphism}." ], "refs": [ "formal-defos-definition-smooth-morphism" ], "proofs": [ { "contents": [ "Case (1) is a matter of unwinding the definitions.", "Assumption (2) implies (1) by", "Criteria for Representability, Lemma", "\\ref{criteria-lemma-representable-by-spaces-formally-smooth}.", "Assumption (3) implies (2) by", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-smooth-formally-smooth}", "and the principle of", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-representable-transformations-property-implication}." ], "refs": [ "criteria-lemma-representable-by-spaces-formally-smooth", "spaces-more-morphisms-lemma-smooth-formally-smooth", "algebraic-lemma-representable-transformations-property-implication" ], "ref_ids": [ 3107, 110, 8459 ] } ], "ref_ids": [ 3520 ] }, { "id": 11353, "type": "theorem", "label": "artin-lemma-fibre-product-deformation-categories", "categories": [ "artin" ], "title": "artin-lemma-fibre-product-deformation-categories", "contents": [ "Let $S$ be a locally Noetherian scheme. Let", "$$", "\\xymatrix{", "\\mathcal{W} \\ar[d] \\ar[r] & \\mathcal{Z} \\ar[d] \\\\", "\\mathcal{X} \\ar[r] & \\mathcal{Y}", "}", "$$", "be a $2$-fibre product of categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$. Let $k$ be a finite type field over $S$ and", "$w_0$ an object of $\\mathcal{W}$ over $k$. Let $x_0, z_0, y_0$ be", "the images of $w_0$ under the morphisms in the diagram. Then", "$$", "\\xymatrix{", "\\mathcal{F}_{\\mathcal{W}, k, w_0} \\ar[d] \\ar[r] &", "\\mathcal{F}_{\\mathcal{Z}, k, z_0} \\ar[d] \\\\", "\\mathcal{F}_{\\mathcal{X}, k, x_0} \\ar[r] & \\mathcal{F}_{\\mathcal{Y}, k, y_0}", "}", "$$", "is a fibre product of predeformation categories." ], "refs": [], "proofs": [ { "contents": [ "This is a matter of unwinding the definitions. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11354, "type": "theorem", "label": "artin-lemma-pushout", "categories": [ "artin" ], "title": "artin-lemma-pushout", "contents": [ "\\begin{slogan}", "Algebraic stacks satisfy the (strong) Rim-Schlessinger condition", "\\end{slogan}", "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d] & X' \\ar[d] \\\\", "Y \\ar[r] & Y'", "}", "$$", "be a pushout in the category of schemes over $S$ where $X \\to X'$", "is a thickening and $X \\to Y$ is affine, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.", "Let $\\mathcal{Z}$ be an algebraic stack over $S$.", "Then the functor of fibre categories", "$$", "\\mathcal{Z}_{Y'}", "\\longrightarrow", "\\mathcal{Z}_Y \\times_{\\mathcal{Z}_X} \\mathcal{Z}_{X'}", "$$", "is an equivalence of categories." ], "refs": [ "more-morphisms-lemma-pushout-along-thickening" ], "proofs": [ { "contents": [ "Let $y'$ be an object of left hand side. The sheaf", "$\\mathit{Isom}(y', y')$ on the category of schemes over $Y'$", "is representable by an algebraic space $I$ over $Y'$, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-representable-diagonal}.", "We conclude that the functor of the lemma is fully faithful as", "$Y'$ is the pushout in the category of algebraic spaces as", "well as the category of schemes, see", "Pushouts of Spaces, Lemma", "\\ref{spaces-pushouts-lemma-pushout-along-thickening-schemes}.", "\\medskip\\noindent", "Let $(y, x', f)$ be an object of the right hand side. Here $f : y|_X \\to x'|_X$", "is an isomorphism. To finish the proof we have to construct an object $y'$ of", "$\\mathcal{Z}_{Y'}$ whose restrictions to $Y$ and $X'$ agree with $y$ and $x'$", "in a manner compatible with $\\varphi$. In fact, it suffices to construct $y'$", "fppf locally on $Y'$, see", "Stacks, Lemma \\ref{stacks-lemma-characterize-essentially-surjective-when-ff}.", "Choose a representable algebraic stack", "$\\mathcal{W}$ and a surjective smooth morphism $\\mathcal{W} \\to \\mathcal{Z}$.", "Then", "$$", "(\\Sch/Y)_{fppf} \\times_{y, \\mathcal{Z}} \\mathcal{W}", "\\quad\\text{and}\\quad", "(\\Sch/X')_{fppf} \\times_{x', \\mathcal{Z}} \\mathcal{W}", "$$", "are algebraic stacks representable by algebraic spaces $V$ and $U'$", "smooth over $Y$ and $X'$. The isomorphism $f$ induces an isomorphism", "$\\varphi : V \\times_Y X \\to U' \\times_{X'} X$ over $X$. By", "Pushouts of Spaces, Lemmas", "\\ref{spaces-pushouts-lemma-pushout-along-thickening} and", "\\ref{spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat}", "we see that the pushout $V' = V \\amalg_{V \\times_Y X} U'$ is", "an algebraic space smooth over $Y'$ whose base change to", "$Y$ and $X'$ recovers $V$ and $U'$ in a manner compatible with $\\varphi$.", "\\medskip\\noindent", "Let $W$ be the algebraic space representing $\\mathcal{W}$.", "The projections $V \\to W$ and $U' \\to W$ agree as morphisms", "over $V \\times_Y X \\cong U' \\times_{X'} X$ hence the universal", "property of the pushout determines a morphism of algebraic spaces", "$V' \\to W$. Choose a scheme $Y_1'$ and a surjective \\'etale morphism", "$Y_1' \\to V'$. Set $Y_1 = Y \\times_{Y'} Y_1'$,", "$X_1' = X' \\times_{Y'} Y_1'$, $X_1 = X \\times_{Y'} Y_1'$.", "The composition", "$$", "(\\Sch/Y_1') \\to (\\Sch/V') \\to (\\Sch/W) = \\mathcal{W} \\to \\mathcal{Z}", "$$", "corresponds by the $2$-Yoneda lemma to an object $y_1'$ of $\\mathcal{Z}$", "over $Y_1'$ whose restriction to $Y_1$ and $X_1'$ agrees with $y|_{Y_1}$", "and $x'|_{X_1'}$ in a manner compatible with $f|_{X_1}$. Thus we have", "constructed our desired object smooth locally over $Y'$ and we win." ], "refs": [ "algebraic-lemma-representable-diagonal", "spaces-pushouts-lemma-pushout-along-thickening-schemes", "stacks-lemma-characterize-essentially-surjective-when-ff", "spaces-pushouts-lemma-pushout-along-thickening", "spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat" ], "ref_ids": [ 8461, 10858, 8945, 10859, 10864 ] } ], "ref_ids": [ 13762 ] }, { "id": 11355, "type": "theorem", "label": "artin-lemma-algebraic-stack-RS", "categories": [ "artin" ], "title": "artin-lemma-algebraic-stack-RS", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack over a locally Noetherian base", "$S$. Then $\\mathcal{X}$ satisfies (RS)." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions and Lemma \\ref{lemma-pushout}." ], "refs": [ "artin-lemma-pushout" ], "ref_ids": [ 11354 ] } ], "ref_ids": [] }, { "id": 11356, "type": "theorem", "label": "artin-lemma-fibre-product-RS", "categories": [ "artin" ], "title": "artin-lemma-fibre-product-RS", "contents": [ "Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and", "$q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. If $\\mathcal{X}$, $\\mathcal{Y}$,", "and $\\mathcal{Z}$ satisfy (RS), then so", "does $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$." ], "refs": [], "proofs": [ { "contents": [ "This is formal. Let ", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d] & X' \\ar[d] \\\\", "Y \\ar[r] & Y' = Y \\amalg_X X'", "}", "$$", "be a diagram as in Definition \\ref{definition-RS}. We have to show that", "$$", "(\\mathcal{X} \\times_{\\mathcal{Y}} \\mathcal{Z})_{Y'}", "\\longrightarrow", "(\\mathcal{X} \\times_{\\mathcal{Y}} \\mathcal{Z})_Y", "\\times_{(\\mathcal{X} \\times_{\\mathcal{Y}} \\mathcal{Z})_X}", "(\\mathcal{X} \\times_{\\mathcal{Y}} \\mathcal{Z})_{X'}", "$$", "is an equivalence. Using the definition of the $2$-fibre product", "this becomes", "\\begin{equation}", "\\label{equation-RS-fibre-product}", "\\mathcal{X}_{Y'} \\times_{\\mathcal{Y}_{Y'}} \\mathcal{Z}_{Y'}", "\\longrightarrow", "(\\mathcal{X}_Y \\times_{\\mathcal{Y}_Y} \\mathcal{Z}_Y)", "\\times_{(\\mathcal{X}_X \\times_{\\mathcal{Y}_X} \\mathcal{Z}_X)}", "(\\mathcal{X}_{X'} \\times_{\\mathcal{Y}_{X'}} \\mathcal{Z}_{X'}).", "\\end{equation}", "We are given that each of the functors", "$$", "\\mathcal{X}_{Y'} \\to \\mathcal{X}_Y \\times_{\\mathcal{Y}_Y} \\mathcal{Z}_Y,", "\\quad", "\\mathcal{Y}_{Y'} \\to \\mathcal{X}_X \\times_{\\mathcal{Y}_X} \\mathcal{Z}_X,", "\\quad", "\\mathcal{Z}_{Y'} \\to", "\\mathcal{X}_{X'} \\times_{\\mathcal{Y}_{X'}} \\mathcal{Z}_{X'}", "$$", "are equivalences. An object of the right hand side of", "(\\ref{equation-RS-fibre-product}) is a system", "$$", "((x_Y, z_Y, \\phi_Y), (x_{X'}, z_{X'}, \\phi_{X'}), (\\alpha, \\beta)).", "$$", "Then $(x_Y, x_{Y'}, \\alpha)$ is isomorphic to the image of an object", "$x_{Y'}$ in $\\mathcal{X}_{Y'}$ and $(z_Y, z_{Y'}, \\beta)$ is isomorphic", "to the image of an object $z_{Y'}$ of $\\mathcal{Z}_{Y'}$. The pair of", "morphisms $(\\phi_Y, \\phi_{X'})$ corresponds to a morphism $\\psi$", "between the images of $x_{Y'}$ and $z_{Y'}$ in $\\mathcal{Y}_{Y'}$.", "Then $(x_{Y'}, z_{Y'}, \\psi)$ is an object of the left hand side of", "(\\ref{equation-RS-fibre-product}) mapping to the given object of the", "right hand side. This proves that (\\ref{equation-RS-fibre-product}) is", "essentially surjective. We omit the proof that it is fully faithful." ], "refs": [ "artin-definition-RS" ], "ref_ids": [ 11417 ] } ], "ref_ids": [] }, { "id": 11357, "type": "theorem", "label": "artin-lemma-deformation-category", "categories": [ "artin" ], "title": "artin-lemma-deformation-category", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$ satisfying (RS). For any field", "$k$ of finite type over $S$ and any object $x_0$ of $\\mathcal{X}$ lying", "over $k$ the predeformation category", "$p : \\mathcal{F}_{\\mathcal{X}, k, x_0} \\to \\mathcal{C}_\\Lambda$", "(\\ref{equation-predeformation-category}) is a deformation category, see", "Formal Deformation Theory, Definition", "\\ref{formal-defos-definition-deformation-category}." ], "refs": [ "formal-defos-definition-deformation-category" ], "proofs": [ { "contents": [ "Set $\\mathcal{F} = \\mathcal{F}_{\\mathcal{X}, k, x_0}$.", "Let $f_1 : A_1 \\to A$ and $f_2 : A_2 \\to A$ be ring maps in", "$\\mathcal{C}_\\Lambda$ with $f_2$ surjective. We have to show that", "the functor", "$$", "\\mathcal{F}(A_1 \\times_A A_2)", "\\longrightarrow", "\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)", "$$", "is an equivalence, see", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-RS-2-categorical}.", "Set $X = \\Spec(A)$, $X' = \\Spec(A_2)$, $Y = \\Spec(A_1)$ and", "$Y' = \\Spec(A_1 \\times_A A_2)$. Note that $Y' = Y \\amalg_X X'$ in the", "category of schemes, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.", "We know that in the diagram of functors of fibre categories", "$$", "\\xymatrix{", "\\mathcal{X}_{Y'} \\ar[r] \\ar[d] &", "\\mathcal{X}_Y \\times_{\\mathcal{X}_X} \\mathcal{X}_{X'} \\ar[d] \\\\", "\\mathcal{X}_{\\Spec(k)} \\ar@{=}[r] & \\mathcal{X}_{\\Spec(k)}", "}", "$$", "the top horizontal arrow is an equivalence by", "Definition \\ref{definition-RS}.", "Since $\\mathcal{F}(B)$ is the category of objects of $\\mathcal{X}_{\\Spec(B)}$", "with an identification with $x_0$ over $k$ we win." ], "refs": [ "formal-defos-lemma-RS-2-categorical", "more-morphisms-lemma-pushout-along-thickening", "artin-definition-RS" ], "ref_ids": [ 3468, 13762, 11417 ] } ], "ref_ids": [ 3530 ] }, { "id": 11358, "type": "theorem", "label": "artin-lemma-change-of-field", "categories": [ "artin" ], "title": "artin-lemma-change-of-field", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $k$ be a", "field of finite type over $S$ and let $l/k$ be a finite extension.", "Let $x_0$ be an object of $\\mathcal{F}$ lying over $\\Spec(k)$.", "Denote $x_{l, 0}$ the restriction of $x_0$ to $\\Spec(l)$.", "Then there is a canonical functor", "$$", "(\\mathcal{F}_{\\mathcal{X}, k , x_0})_{l/k}", "\\longrightarrow", "\\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}}", "$$", "of categories cofibred in groupoids over $\\mathcal{C}_{\\Lambda, l}$.", "If $\\mathcal{X}$ satisfies (RS), then this functor is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Consider a factorization", "$$", "\\Spec(l) \\to \\Spec(B) \\to S", "$$", "as in (\\ref{equation-factor}). By definition we have", "$$", "(\\mathcal{F}_{\\mathcal{X}, k , x_0})_{l/k}(B) =", "\\mathcal{F}_{\\mathcal{X}, k, x_0}(B \\times_l k)", "$$", "see Formal Deformation Theory, Situation", "\\ref{formal-defos-situation-change-of-fields}. Thus an object of this", "is a morphism $x_0 \\to x$ of $\\mathcal{X}$ lying over the morphism", "$\\Spec(k) \\to \\Spec(B \\times_l k)$. Choosing pullback functor for $\\mathcal{X}$", "we can associate to $x_0 \\to x$ the morphism $x_{l, 0} \\to x_B$", "where $x_B$ is the restriction of $x$ to $\\Spec(B)$ (via the morphism", "$\\Spec(B) \\to \\Spec(B \\times_l k)$ coming from $B \\times_l k \\subset B$).", "This construction is functorial in $B$ and compatible with morphisms.", "\\medskip\\noindent", "Next, assume $\\mathcal{X}$ satisfies (RS). Consider the diagrams", "$$", "\\vcenter{", "\\xymatrix{", "l & B \\ar[l] \\\\", "k \\ar[u] & B \\times_l k \\ar[l] \\ar[u]", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "\\Spec(l) \\ar[d] \\ar[r] & \\Spec(B) \\ar[d] \\\\", "\\Spec(k) \\ar[r] & \\Spec(B \\times_l k)", "}", "}", "$$", "The diagram on the left is a fibre product of rings. The diagram on the", "right is a pushout in the category of schemes, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.", "These schemes are all of finite type over $S$ (see remarks following", "Definition \\ref{definition-RS}). Hence (RS) kicks in to give an equivalence", "of fibre categories", "$$", "\\mathcal{X}_{\\Spec(B \\times_l k)}", "\\longrightarrow", "\\mathcal{X}_{\\Spec(k)}", "\\times_{\\mathcal{X}_{\\Spec(l)}}", "\\mathcal{X}_{\\Spec(B)}", "$$", "This implies that the functor defined above gives an equivalence of", "fibre categories. Hence the functor is an equivalence on categories", "cofibred in groupoids by (the dual of)", "Categories, Lemma \\ref{categories-lemma-equivalence-fibred-categories}." ], "refs": [ "more-morphisms-lemma-pushout-along-thickening", "artin-definition-RS", "categories-lemma-equivalence-fibred-categories" ], "ref_ids": [ 13762, 11417, 12297 ] } ], "ref_ids": [] }, { "id": 11359, "type": "theorem", "label": "artin-lemma-finite-dimension", "categories": [ "artin" ], "title": "artin-lemma-finite-dimension", "contents": [ "Let $S$ be a locally Noetherian scheme. Assume", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is an algebraic stack,", "\\item $U$ is a scheme locally of finite type over $S$, and", "\\item $(\\Sch/U)_{fppf} \\to \\mathcal{X}$ is a smooth surjective", "morphism.", "\\end{enumerate}", "Then, for any $\\mathcal{F} = \\mathcal{F}_{\\mathcal{X}, k, x_0}$ as in", "Section \\ref{section-predeformation-categories}", "the tangent space $T\\mathcal{F}$ and infinitesimal automorphism space", "$\\text{Inf}(\\mathcal{F})$ have finite dimension over $k$" ], "refs": [], "proofs": [ { "contents": [ "Let us write $\\mathcal{U} = (\\Sch/U)_{fppf}$. By our definition", "of algebraic stacks the $1$-morphism $\\mathcal{U} \\to \\mathcal{X}$", "is representable by algebraic spaces. Hence in particular the", "2-fibre product", "$$", "\\mathcal{U}_{x_0} = (\\Sch/\\Spec(k))_{fppf} \\times_\\mathcal{X} \\mathcal{U}", "$$", "is representable by an algebraic space $U_{x_0}$ over $\\Spec(k)$. Then", "$U_{x_0} \\to \\Spec(k)$ is smooth and surjective (in particular $U_{x_0}$", "is nonempty). By Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-smooth-separable-closed-points-dense}", "we can find a finite extension $l \\supset k$ and a point", "$\\Spec(l) \\to U_{x_0}$ over $k$. We have", "$$", "(\\mathcal{F}_{\\mathcal{X}, k , x_0})_{l/k} =", "\\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}}", "$$", "by Lemma \\ref{lemma-change-of-field} and the fact that $\\mathcal{X}$", "satisfies (RS). Thus we see that", "$$", "T\\mathcal{F} \\otimes_k l \\cong T\\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}}", "\\quad\\text{and}\\quad", "\\text{Inf}(\\mathcal{F}) \\otimes_k l \\cong", "\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}})", "$$", "by", "Formal Deformation Theory, Lemmas", "\\ref{formal-defos-lemma-tangent-space-change-of-field} and", "\\ref{formal-defos-lemma-inf-aut-change-of-field}", "(these are applicable by", "Lemmas \\ref{lemma-algebraic-stack-RS} and", "\\ref{lemma-deformation-category} and", "Remark \\ref{remark-deformation-category-implies}).", "Hence it suffices to prove that $T\\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}}$", "and $\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}})$", "have finite dimension over $l$. Note that $x_{l, 0}$ comes from a point", "$u_0$ of $\\mathcal{U}$ over $l$.", "\\medskip\\noindent", "We interrupt the flow of the argument to show that the lemma for", "infinitesimal automorphisms follows from the lemma for tangent spaces.", "Namely, let", "$\\mathcal{R} = \\mathcal{U} \\times_\\mathcal{X} \\mathcal{U}$.", "Let $r_0$ be the $l$-valued point $(u_0, u_0, \\text{id}_{x_0})$ of", "$\\mathcal{R}$. Combining", "Lemma \\ref{lemma-fibre-product-deformation-categories} and", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-deformation-functor-diagonal}", "we see that", "$$", "\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}})", "\\subset", "T\\mathcal{F}_{\\mathcal{R}, l, r_0}", "$$", "Note that $\\mathcal{R}$ is an algebraic stack, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-2-fibre-product-general}.", "Also, $\\mathcal{R}$ is representable by an algebraic space $R$", "smooth over $U$ (via either projection, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-stack-presentation}).", "Hence, choose an scheme $U'$ and a surjective \\'etale morphism", "$U' \\to R$ we see that $U'$ is smooth over $U$, hence locally of", "finite type over $S$. As $(\\Sch/U')_{fppf} \\to \\mathcal{R}$ is", "surjective and smooth, we have reduced the question to the case", "of tangent spaces.", "\\medskip\\noindent", "The functor (\\ref{equation-functoriality})", "$$", "\\mathcal{F}_{\\mathcal{U}, l, u_0}", "\\longrightarrow", "\\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}}", "$$", "is smooth by Lemma \\ref{lemma-formally-smooth-on-deformation-categories}.", "The induced map on tangent spaces", "$$", "T\\mathcal{F}_{\\mathcal{U}, l, u_0}", "\\longrightarrow", "T\\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}}", "$$", "is $l$-linear (by", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-k-linear-differential})", "and surjective (as smooth maps of predeformation categories induce", "surjective maps on tangent spaces by", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-smooth-morphism-essentially-surjective}).", "Hence it suffices to prove that the tangent space of the deformation", "space associated to the representable algebraic stack $\\mathcal{U}$", "at the point $u_0$ is finite dimensional. Let $\\Spec(R) \\subset U$ be", "an affine open such that $u_0 : \\Spec(l) \\to U$ factors through $\\Spec(R)$", "and such that $\\Spec(R) \\to S$ factors through $\\Spec(\\Lambda) \\subset S$.", "Let $\\mathfrak m_R \\subset R$ be the kernel of the $\\Lambda$-algebra map", "$\\varphi_0 : R \\to l$ corresponding to $u_0$. Note that $R$, being of finite", "type over the Noetherian ring $\\Lambda$, is a Noetherian ring. Hence", "$\\mathfrak m_R = (f_1, \\ldots, f_n)$ is a finitely generated ideal.", "We have", "$$", "T\\mathcal{F}_{\\mathcal{U}, l, u_0}", "=", "\\{\\varphi : R \\to l[\\epsilon] \\mid", "\\varphi \\text{ is a } \\Lambda\\text{-algebra map and }", "\\varphi \\bmod \\epsilon = \\varphi_0\\}", "$$", "An element of the right hand side is determined by its values on", "$f_1, \\ldots, f_n$ hence the dimension is at most $n$ and we win.", "Some details omitted." ], "refs": [ "spaces-over-fields-lemma-smooth-separable-closed-points-dense", "artin-lemma-change-of-field", "formal-defos-lemma-tangent-space-change-of-field", "formal-defos-lemma-inf-aut-change-of-field", "artin-lemma-algebraic-stack-RS", "artin-lemma-deformation-category", "artin-remark-deformation-category-implies", "artin-lemma-fibre-product-deformation-categories", "formal-defos-lemma-deformation-functor-diagonal", "algebraic-lemma-2-fibre-product-general", "algebraic-lemma-stack-presentation", "artin-lemma-formally-smooth-on-deformation-categories", "formal-defos-lemma-k-linear-differential", "formal-defos-lemma-smooth-morphism-essentially-surjective" ], "ref_ids": [ 12873, 11358, 3506, 3507, 11355, 11357, 11427, 11353, 3493, 8466, 8474, 11352, 3453, 3434 ] } ], "ref_ids": [] }, { "id": 11360, "type": "theorem", "label": "artin-lemma-fibre-product-tangent-spaces", "categories": [ "artin" ], "title": "artin-lemma-fibre-product-tangent-spaces", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$", "and $q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$,", "$\\mathcal{Y}$, $\\mathcal{Z}$ satisfy (RS).", "Let $k$ be a field of finite type over $S$ and let $w_0$ be an object of", "$\\mathcal{W} = \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ over $k$.", "Denote $x_0, y_0, z_0$ the objects of $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$", "you get from $w_0$. Then there is a $6$-term exact sequence", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\text{Inf}(\\mathcal{F}_{\\mathcal{W}, k, w_0}) \\ar[r] &", "\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0}) \\oplus", "\\text{Inf}(\\mathcal{F}_{\\mathcal{Z}, k, z_0}) \\ar[r] &", "\\text{Inf}(\\mathcal{F}_{\\mathcal{Y}, k, y_0}) \\ar[lld] \\\\", " &", "T\\mathcal{F}_{\\mathcal{W}, k, w_0} \\ar[r] &", "T\\mathcal{F}_{\\mathcal{X}, k, x_0} \\oplus", "T\\mathcal{F}_{\\mathcal{Z}, k, z_0} \\ar[r] &", "T\\mathcal{F}_{\\mathcal{Y}, k, y_0}", "}", "$$", "of $k$-vector spaces." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-fibre-product-RS} we see that $\\mathcal{W}$", "satisfies (RS) and hence the lemma makes sense. To see the lemma", "is true, apply Lemmas \\ref{lemma-fibre-product-deformation-categories} and", "\\ref{lemma-deformation-category}", "and Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-deformation-categories-fiber-product-morphisms}." ], "refs": [ "artin-lemma-fibre-product-RS", "artin-lemma-fibre-product-deformation-categories", "artin-lemma-deformation-category", "formal-defos-lemma-deformation-categories-fiber-product-morphisms" ], "ref_ids": [ 11356, 11353, 11357, 3482 ] } ], "ref_ids": [] }, { "id": 11361, "type": "theorem", "label": "artin-lemma-smooth-lift-formal", "categories": [ "artin" ], "title": "artin-lemma-smooth-lift-formal", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$", "be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "Let $\\eta = (R, \\eta_n, g_n)$ be a formal object of $\\mathcal{Y}$", "and let $\\xi_1$ be an object of $\\mathcal{X}$ with $F(\\xi_1) \\cong \\eta_1$.", "If $F$ is formally smooth on objects (see", "Criteria for Representability, Section \\ref{criteria-section-formally-smooth}),", "then there exists a formal object $\\xi = (R, \\xi_n, f_n)$ of $\\mathcal{X}$", "such that $F(\\xi) \\cong \\eta$." ], "refs": [], "proofs": [ { "contents": [ "Note that each of the morphisms", "$\\Spec(R/\\mathfrak m^n) \\to \\Spec(R/\\mathfrak m^{n + 1})$ is a first order", "thickening of affine schemes over $S$. Hence the assumption on $F$ means", "that we can successively lift $\\xi_1$ to objects $\\xi_2, \\xi_3, \\ldots$", "of $\\mathcal{X}$ endowed with compatible isomorphisms", "$\\eta_n|_{\\Spec(R/\\mathfrak m^{n - 1})} \\cong \\eta_{n - 1}$", "and $F(\\eta_n) \\cong \\xi_n$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11362, "type": "theorem", "label": "artin-lemma-effective", "categories": [ "artin" ], "title": "artin-lemma-effective", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be an algebraic", "stack over $S$. The functor (\\ref{equation-approximation}) is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Case I: $\\mathcal{X}$ is representable (by a scheme). Say", "$\\mathcal{X} = (\\Sch/X)_{fppf}$ for some scheme $X$ over $S$.", "Unwinding the definitions we have to prove the following: Given", "a Noetherian complete local $S$-algebra $R$ with $R/\\mathfrak m$ of", "finite type over $S$ we have", "$$", "\\Mor_S(\\Spec(R), X) \\longrightarrow \\lim \\Mor_S(\\Spec(R/\\mathfrak m^n), X)", "$$", "is bijective. This follows from Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-map-into-scheme}.", "\\medskip\\noindent", "Case II. $\\mathcal{X}$ is representable by an algebraic space. Say", "$\\mathcal{X}$ is representable by $X$. Again we have to show that", "$$", "\\Mor_S(\\Spec(R), X) \\longrightarrow \\lim \\Mor_S(\\Spec(R/\\mathfrak m^n), X)", "$$", "is bijective for $R$ as above. This is Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-map-into-algebraic-space}.", "\\medskip\\noindent", "Case III: General case of an algebraic stack. A general remark is that", "the left and right hand side of (\\ref{equation-approximation}) are", "categories fibred in groupoids over the category of affine schemes", "over $S$ which are spectra of Noetherian complete local rings", "with residue field of finite type over $S$. We will also see in the", "proof below that they form stacks for a certain topology on this", "category.", "\\medskip\\noindent", "We first prove fully faithfulness. Let $R$ be a Noetherian complete", "local $S$-algebra with $k = R/\\mathfrak m$ of finite type over $S$.", "Let $x, x'$ be objects of $\\mathcal{X}$ over $R$. As $\\mathcal{X}$ is", "an algebraic stack $\\mathit{Isom}(x, x')$ is representable by an", "algebraic space $I$ over $\\Spec(R)$, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-representable-diagonal}.", "Applying Case II to $I$ over $\\Spec(R)$ implies immediately that", "(\\ref{equation-approximation}) is fully faithful on fibre categories over", "$\\Spec(R)$. Hence the functor is fully faithful by", "Categories, Lemma \\ref{categories-lemma-equivalence-fibred-categories}.", "\\medskip\\noindent", "Essential surjectivity. Let $\\xi = (R, \\xi_n, f_n)$ be a formal object of", "$\\mathcal{X}$. Choose a scheme $U$ over $S$ and a surjective smooth morphism", "$f : (\\Sch/U)_{fppf} \\to \\mathcal{X}$. For every $n$ consider the fibre product", "$$", "(\\Sch/\\Spec(R/\\mathfrak m^n))_{fppf}", "\\times_{\\xi_n, \\mathcal{X}, f}", "(\\Sch/U)_{fppf}", "$$", "By assumption this is representable by an algebraic space $V_n$ surjective and", "smooth over $\\Spec(R/\\mathfrak m^n)$. The morphisms", "$f_n : \\xi_n \\to \\xi_{n + 1}$ induce cartesian squares", "$$", "\\xymatrix{", "V_{n + 1} \\ar[d] & V_n \\ar[d] \\ar[l] \\\\", "\\Spec(R/\\mathfrak m^{n + 1}) & \\Spec(R/\\mathfrak m^n) \\ar[l]", "}", "$$", "of algebraic spaces. By Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-smooth-separable-closed-points-dense}", "we can find a finite separable extension $k \\subset k'$ and a point", "$v'_1 : \\Spec(k') \\to V_1$ over $k$. Let $R \\subset R'$ be the finite \\'etale", "extension whose residue field extension is $k \\subset k'$ (exists and", "is unique by", "Algebra, Lemmas \\ref{algebra-lemma-henselian-cat-finite-etale} and", "\\ref{algebra-lemma-complete-henselian}).", "By the infinitesimal lifting criterion of smoothness (see", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-smooth-formally-smooth})", "applied to $V_n \\to \\Spec(R/\\mathfrak m^n)$ for $n = 2, 3, 4, \\ldots$", "we can successively find morphisms", "$v'_n : \\Spec(R'/(\\mathfrak m')^n) \\to V_n$ over $\\Spec(R/\\mathfrak m^n)$", "fitting into commutative diagrams", "$$", "\\xymatrix{", "\\Spec(R'/(\\mathfrak m')^{n + 1}) \\ar[d]_{v'_{n + 1}} &", "\\Spec(R'/(\\mathfrak m')^n) \\ar[d]^{v'_n} \\ar[l] \\\\", "V_{n + 1} & V_n \\ar[l]", "}", "$$", "Composing with the projection morphisms $V_n \\to U$ we obtain a compatible", "system of morphisms $u'_n : \\Spec(R'/(\\mathfrak m')^n) \\to U$.", "By Case I the family $(u'_n)$ comes from a unique", "morphism $u' : \\Spec(R') \\to U$. Denote $x'$ the object of $\\mathcal{X}$", "over $\\Spec(R')$ we get by applying the $1$-morphism $f$ to $u'$.", "By construction, there exists a morphism of formal objects", "$$", "(\\ref{equation-approximation})(x') =", "(R', x'|_{\\Spec(R'/(\\mathfrak m')^n)}, \\ldots)", "\\longrightarrow", "(R, \\xi_n, f_n)", "$$", "lying over $\\Spec(R') \\to \\Spec(R)$. Note that $R' \\otimes_R R'$ is a finite", "product of spectra of Noetherian complete local rings to which our current", "discussion applies. Denote $p_0, p_1 : \\Spec(R' \\otimes_R R') \\to \\Spec(R')$", "the two projections. By the fully faithfulness shown above there exists", "a canonical isomorphism $\\varphi : p_0^*x' \\to p_1^*x'$ because we have", "such isomorphisms over", "$\\Spec((R' \\otimes_R R')/\\mathfrak m^n(R' \\otimes_R R'))$.", "We omit the proof that the isomorphism $\\varphi$ satisfies the cocycle", "condition (see Stacks, Definition \\ref{stacks-definition-descent-data}).", "Since $\\{\\Spec(R') \\to \\Spec(R)\\}$ is an fppf covering we conclude", "that $x'$ descends to an object $x$ of $\\mathcal{X}$ over $\\Spec(R)$.", "We omit the proof that $x_n$ is the restriction of $x$ to", "$\\Spec(R/\\mathfrak m^n)$." ], "refs": [ "formal-spaces-lemma-map-into-scheme", "formal-spaces-lemma-map-into-algebraic-space", "algebraic-lemma-representable-diagonal", "categories-lemma-equivalence-fibred-categories", "spaces-over-fields-lemma-smooth-separable-closed-points-dense", "algebra-lemma-henselian-cat-finite-etale", "algebra-lemma-complete-henselian", "spaces-more-morphisms-lemma-smooth-formally-smooth", "stacks-definition-descent-data" ], "ref_ids": [ 3963, 3964, 8461, 12297, 12873, 1280, 1282, 110, 8993 ] } ], "ref_ids": [] }, { "id": 11363, "type": "theorem", "label": "artin-lemma-fibre-product-effective", "categories": [ "artin" ], "title": "artin-lemma-fibre-product-effective", "contents": [ "Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and", "$q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. If the functor", "(\\ref{equation-approximation}) is an equivalence for ", "$\\mathcal{X}$, $\\mathcal{Y}$, and $\\mathcal{Z}$, then it is ", "an equivalence for $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$." ], "refs": [], "proofs": [ { "contents": [ "The left and the right hand side of (\\ref{equation-approximation})", "for $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ are simply the $2$-fibre", "products of the left and the right hand side of (\\ref{equation-approximation})", "for $\\mathcal{X}$, $\\mathcal{Z}$ over $\\mathcal{Y}$.", "Hence the result follows as taking $2$-fibre products is compatible", "with equivalences of categories, see", "Categories, Lemma \\ref{categories-lemma-equivalence-2-fibre-product}." ], "refs": [ "categories-lemma-equivalence-2-fibre-product" ], "ref_ids": [ 12272 ] } ], "ref_ids": [] }, { "id": 11364, "type": "theorem", "label": "artin-lemma-approximate", "categories": [ "artin" ], "title": "artin-lemma-approximate", "contents": [ "Let $S$ be a locally Noetherian scheme. Let", "$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category", "fibred in groupoids. Let $x$ be an object of", "$\\mathcal{X}$ lying over $\\Spec(R)$ where $R$ is a Noetherian complete", "local ring with residue field $k$ of finite type over $S$. Let $s \\in S$", "be the image of $\\Spec(k) \\to S$. Assume that (a) $\\mathcal{O}_{S, s}$ is", "a G-ring and (b) $p$ is limit preserving on objects. Then for every", "integer $N \\geq 1$ there exist", "\\begin{enumerate}", "\\item a finite type $S$-algebra $A$,", "\\item a maximal ideal $\\mathfrak m_A \\subset A$,", "\\item an object $x_A$ of $\\mathcal{X}$ over $\\Spec(A)$,", "\\item an $S$-isomorphism $R/\\mathfrak m_R^N \\cong A/\\mathfrak m_A^N$,", "\\item an isomorphism", "$x|_{\\Spec(R/\\mathfrak m_R^N)} \\cong x_A|_{\\Spec(A/\\mathfrak m_A^N)}$", "compatible with (4), and", "\\item an isomorphism", "$\\text{Gr}_{\\mathfrak m_R}(R) \\cong \\text{Gr}_{\\mathfrak m_A}(A)$", "of graded $k$-algebras.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose an affine open $\\Spec(\\Lambda) \\subset S$ such that $k$ is a finite", "$\\Lambda$-algebra, see", "Morphisms, Lemma \\ref{morphisms-lemma-point-finite-type}.", "We may and do replace $S$ by $\\Spec(\\Lambda)$.", "\\medskip\\noindent", "We may write $R$ as a directed colimit $R = \\colim C_j$ where each", "$C_j$ is a finite type $\\Lambda$-algebra (see", "Algebra, Lemma \\ref{algebra-lemma-ring-colimit-fp}).", "By assumption (b) the object $x$ is isomorphic to the restriction of", "an object over one of the $C_j$. Hence we may choose a finite type", "$\\Lambda$-algebra $C$, a $\\Lambda$-algebra map $C \\to R$, and an object", "$x_C$ of $\\mathcal{X}$ over $\\Spec(C)$ such that $x = x_C|_{\\Spec(R)}$.", "The choice of $C$ is a bookkeeping device and could be avoided.", "For later use, let us write $C = \\Lambda[y_1, \\ldots, y_u]/(f_1, \\ldots, f_v)$", "and we denote $\\overline{a}_i \\in R$ the image of $y_i$ under the", "map $C \\to R$. Set $\\mathfrak m_C = C \\cap \\mathfrak m_R$.", "\\medskip\\noindent", "Choose a $\\Lambda$-algebra surjection $\\Lambda[x_1, \\ldots, x_s] \\to k$", "and denote $\\mathfrak m'$ the kernel.", "By the universal property of polynomial rings we may lift this", "to a $\\Lambda$-algebra map $\\Lambda[x_1, \\ldots, x_s] \\to R$.", "We add some variables (i.e., we increase $s$ a bit) mapping to generators", "of $\\mathfrak m_R$. Having done this we see that", "$\\Lambda[x_1, \\ldots, x_s] \\to R/\\mathfrak m_R^2$ is surjective.", "Then we see that", "\\begin{equation}", "\\label{equation-surjection}", "P = \\Lambda[x_1, \\ldots, x_s]_{\\mathfrak m'}^\\wedge \\longrightarrow R", "\\end{equation}", "is a surjective map of Noetherian complete local rings, see for example", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-surjective-cotangent-space}.", "\\medskip\\noindent", "Choose lifts $a_i \\in P$ of $\\overline{a}_i$ we found above.", "Choose generators $b_1, \\ldots, b_r \\in P$ for the kernel of", "(\\ref{equation-surjection}).", "Choose $c_{ji} \\in P$ such that", "$$", "f_j(a_1, \\ldots, a_u) = \\sum c_{ji} b_i", "$$", "in $P$ which is possible by the choices made so far. Choose generators", "$$", "k_1, \\ldots, k_t \\in", "\\Ker(P^{\\oplus r} \\xrightarrow{(b_1, \\ldots, b_r)} P)", "$$", "and write $k_i = (k_{i1}, \\ldots, k_{ir})$ and $K = (k_{ij})$", "so that", "$$", "P^{\\oplus t} \\xrightarrow{K}", "P^{\\oplus r} \\xrightarrow{(b_1, \\ldots, b_r)}", "P \\to R \\to 0", "$$", "is an exact sequence of $P$-modules. In particular we have", "$\\sum k_{ij} b_j = 0$. After possibly increasing $N$ we may", "assume $N - 1$ works in the Artin-Rees lemma for the first two maps of this", "exact sequence (see More on Algebra, Section", "\\ref{more-algebra-section-artin-rees} for terminology).", "\\medskip\\noindent", "By assumption $\\mathcal{O}_{S, s} = \\Lambda_{\\Lambda \\cap \\mathfrak m'}$ is", "a G-ring. Hence by More on Algebra, Proposition", "\\ref{more-algebra-proposition-finite-type-over-G-ring}", "the ring $\\Lambda[x_1, \\ldots, x_s]_{\\mathfrak m'}$ is a $G$-ring.", "Hence by Smoothing Ring Maps, Theorem", "\\ref{smoothing-theorem-approximation-property-variant}", "there exist an \\'etale ring map", "$$", "\\Lambda[x_1, \\ldots, x_s]_{\\mathfrak m'} \\to B,", "$$", "a maximal ideal $\\mathfrak m_B$ of $B$ lying over $\\mathfrak m'$, and", "elements $a'_i, b'_i, c'_{ij}, k'_{ij} \\in B'$ such that", "\\begin{enumerate}", "\\item $\\kappa(\\mathfrak m') = \\kappa(\\mathfrak m_B)$ which implies", "that $\\Lambda[x_1, \\ldots, x_s]_{\\mathfrak m'} \\subset B_{\\mathfrak m_B}", "\\subset P$ and $P$ is identified with the completion of $B$ at", "$\\mathfrak m_B$, see remark preceding Smoothing Ring Maps, Theorem", "\\ref{smoothing-theorem-approximation-property-variant},", "\\item $a_i - a'_i, b_i - b'_i, c_{ij} - c'_{ij}, k_{ij} - k'_{ij} \\in", "(\\mathfrak m')^N P$, and", "\\item $f_j(a'_1, \\ldots, a'_u) = \\sum c'_{ji} b'_i$ and $\\sum k'_{ij}b'_j = 0$.", "\\end{enumerate}", "Set $A = B/(b'_1, \\ldots, b'_r)$ and denote $\\mathfrak m_A$ the", "image of $\\mathfrak m_B$ in $A$. (Note that $A$ is essentially of finite", "type over $\\Lambda$; at the end of the proof we will show how to obtain", "an $A$ which is of finite type over $\\Lambda$.) There is a ring map", "$C \\to A$ sending $y_i \\mapsto a'_i$ because the $a'_i$ satisfy", "the desired equations modulo $(b'_1, \\ldots, b'_r)$.", "Note that $A/\\mathfrak m_A^N = R/\\mathfrak m_R^N$ as quotients of", "$P = B^\\wedge$ by property (2) above. Set $x_A = x_C|_{\\Spec(A)}$.", "Since the maps", "$$", "C \\to A \\to A/\\mathfrak m_A^N \\cong R/\\mathfrak m_R^N", "\\quad\\text{and}\\quad", "C \\to R \\to R/\\mathfrak m_R^N", "$$", "are equal we see that $x_A$ and $x$ agree modulo $\\mathfrak m_R^N$", "via the isomorphism $A/\\mathfrak m_A^N = R/\\mathfrak m_R^N$. At this", "point we have shown properties (1) -- (5) of the statement of the lemma.", "To see (6) note that", "$$", "P^{\\oplus t} \\xrightarrow{K}", "P^{\\oplus r} \\xrightarrow{(b_1, \\ldots, b_r)}", "P", "\\quad\\text{and}\\quad", "P^{\\oplus t} \\xrightarrow{K'}", "P^{\\oplus r} \\xrightarrow{(b'_1, \\ldots, b'_r)}", "P", "$$", "are two complexes of $P$-modules which are congruent modulo", "$(\\mathfrak m')^N$ with the first one being exact. By our choice of $N$", "above we see from", "More on Algebra, Lemma \\ref{more-algebra-lemma-approximate-complex-graded}", "that $R = P/(b_1, \\ldots, b_r)$ and", "$P/(b'_1, \\ldots, b'_r) = B^\\wedge/(b'_1, \\ldots, b'_r) = A^\\wedge$", "have isomorphic associated graded algebras, which is what we wanted to show.", "\\medskip\\noindent", "This last paragraph of the proof serves to clean up the issue that $A$ is", "essentially of finite type over $S$ and not yet of finite type.", "The construction above gives $A = B/(b'_1, \\ldots, b'_r)$ and", "$\\mathfrak m_A \\subset A$ with $B$ \\'etale over", "$\\Lambda[x_1, \\ldots, x_s]_{\\mathfrak m'}$. Hence $A$ is of finite", "type over the Noetherian ring $\\Lambda[x_1, \\ldots, x_s]_{\\mathfrak m'}$.", "Thus we can write $A = (A_0)_{\\mathfrak m'}$ for some finite type", "$\\Lambda[x_1, \\ldots, x_n]$ algebra $A_0$. Then", "$A = \\colim (A_0)_f$ where", "$f \\in \\Lambda[x_1, \\ldots, x_n] \\setminus \\mathfrak m'$, see", "Algebra, Lemma \\ref{algebra-lemma-localization-colimit}.", "Because $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ is limit preserving on", "objects, we see that", "$x_A$ comes from some object $x_{(A_0)_f}$ over $\\Spec((A_0)_f)$ for", "an $f$ as above. After replacing $A$ by $(A_0)_f$ and $x_A$ by", "$x_{(A_0)_f}$ and $\\mathfrak m_A$ by $(A_0)_f \\cap \\mathfrak m_A$", "the proof is finished." ], "refs": [ "morphisms-lemma-point-finite-type", "algebra-lemma-ring-colimit-fp", "formal-defos-lemma-surjective-cotangent-space", "more-algebra-proposition-finite-type-over-G-ring", "smoothing-theorem-approximation-property-variant", "smoothing-theorem-approximation-property-variant", "more-algebra-lemma-approximate-complex-graded", "algebra-lemma-localization-colimit" ], "ref_ids": [ 5205, 1091, 3421, 10581, 5607, 5607, 9812, 348 ] } ], "ref_ids": [] }, { "id": 11365, "type": "theorem", "label": "artin-lemma-fibre-product-limit-preserving", "categories": [ "artin" ], "title": "artin-lemma-fibre-product-limit-preserving", "contents": [ "Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and", "$q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$.", "\\begin{enumerate}", "\\item If $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ and", "$\\mathcal{Z} \\to (\\Sch/S)_{fppf}$ are limit preserving on objects and", "$\\mathcal{Y}$ is limit preserving, then", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to (\\Sch/S)_{fppf}$ is", "limit preserving on objects.", "\\item If $\\mathcal{X}$, $\\mathcal{Y}$,", "and $\\mathcal{Z}$ are limit preserving, then so", "is $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is formal. Proof of (1). Let $T = \\lim_{i \\in I} T_i$ be the directed", "limit of affine schemes $T_i$ over $S$. We will prove that the functor", "$\\colim \\mathcal{X}_{T_i} \\to \\mathcal{X}_T$ is essentially surjective.", "Recall that an object of the fibre product over $T$ is a quadruple", "$(T, x, z, \\alpha)$ where $x$ is an object of $\\mathcal{X}$ lying over $T$,", "$z$ is an object of $\\mathcal{Z}$ lying over $T$, and", "$\\alpha : p(x) \\to q(z)$ is a morphism in the fibre category of", "$\\mathcal{Y}$ over $T$. By assumption on $\\mathcal{X}$ and $\\mathcal{Z}$", "we can find an $i$ and objects $x_i$ and $z_i$ over $T_i$ such that", "$x_i|_T \\cong T$ and $z_i|_T \\cong z$. Then $\\alpha$ corresponds to", "an isomorphism $p(x_i)|_T \\to q(z_i)|_T$ which comes from an isomorphism", "$\\alpha_{i'} : p(x_i)|_{T_{i'}} \\to q(z_i)|_{T_{i'}}$ by our assumption on", "$\\mathcal{Y}$. After replacing $i$ by $i'$, $x_i$ by $x_i|_{T_{i'}}$, and", "$z_i$ by $z_i|_{T_{i'}}$ we see that $(T_i, x_i, z_i, \\alpha_i)$", "is an object of the fibre product over $T_i$ which restricts to", "an object isomorphic to $(T, x, z, \\alpha)$ over $T$ as desired.", "\\medskip\\noindent", "We omit the arguments showing that $\\colim \\mathcal{X}_{T_i} \\to \\mathcal{X}_T$", "is fully faithful in (2)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11366, "type": "theorem", "label": "artin-lemma-limit-preserving-algebraic-space", "categories": [ "artin" ], "title": "artin-lemma-limit-preserving-algebraic-space", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{X}$ is a stack in setoids and", "$\\mathcal{X} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects,", "\\item $\\mathcal{X}$ is a stack in setoids and limit preserving,", "\\item $\\mathcal{X}$ is representable by an algebraic space", "locally of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Under each of the three assumptions $\\mathcal{X}$ is representable", "by an algebraic space $X$ over $S$, see Algebraic Stacks, Proposition", "\\ref{algebraic-proposition-algebraic-stack-no-automorphisms}.", "It is clear that (1) and (2) are equivalent as a functor between", "setoids is an equivalence if and only if it is surjective on isomorphism", "classes. Finally, (1) and (3) are equivalent by", "Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-characterize-locally-finite-presentation}." ], "refs": [ "algebraic-proposition-algebraic-stack-no-automorphisms", "spaces-limits-proposition-characterize-locally-finite-presentation" ], "ref_ids": [ 8480, 4655 ] } ], "ref_ids": [] }, { "id": 11367, "type": "theorem", "label": "artin-lemma-diagonal", "categories": [ "artin" ], "title": "artin-lemma-diagonal", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred", "in groupoids over $(\\Sch/S)_{fppf}$. Assume", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is", "representable by algebraic spaces and $\\mathcal{X}$ is limit preserving.", "Then $\\Delta$ is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "We apply Criteria for Representability, Lemma", "\\ref{criteria-lemma-check-property-limit-preserving}.", "Let $V$ be an affine scheme $V$ locally of finite presentation over $S$", "and let $\\theta$ be an object of $\\mathcal{X} \\times \\mathcal{X}$", "over $V$. Let $F_\\theta$ be an algebraic space representing", "$\\mathcal{X} \\times_{\\Delta, \\mathcal{X} \\times \\mathcal{X}, \\theta}", "(\\Sch/V)_{fppf}$ and let $f_\\theta : F_\\theta \\to V$ be the canonical morphism", "(see Algebraic Stacks, Section", "\\ref{algebraic-section-morphisms-representable-by-algebraic-spaces}).", "It suffices to show that", "$F_\\theta \\to V$ has the corresponding properties. By", "Lemmas \\ref{lemma-fibre-product-limit-preserving} and", "\\ref{lemma-limit-preserving-algebraic-space}", "we see that $F_\\theta \\to S$ is locally of finite presentation.", "It follows that $F_\\theta \\to V$ is locally of finite type", "by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-permanence-finite-type}." ], "refs": [ "criteria-lemma-check-property-limit-preserving", "artin-lemma-fibre-product-limit-preserving", "artin-lemma-limit-preserving-algebraic-space", "spaces-morphisms-lemma-permanence-finite-type" ], "ref_ids": [ 3104, 11365, 11366, 4818 ] } ], "ref_ids": [] }, { "id": 11368, "type": "theorem", "label": "artin-lemma-versality-matches", "categories": [ "artin" ], "title": "artin-lemma-versality-matches", "contents": [ "With notation as in Definition \\ref{definition-versal}.", "Let $R = \\mathcal{O}_{U, u_0}^\\wedge$.", "Let $\\xi$ be the formal object of $\\mathcal{X}$", "over $R$ associated to $x|_{\\Spec(R)}$, see (\\ref{equation-approximation}).", "Then", "$$", "x\\text{ is versal at }u_0", "\\Leftrightarrow", "\\xi\\text{ is versal}", "$$" ], "refs": [ "artin-definition-versal" ], "proofs": [ { "contents": [ "Observe that $\\mathcal{O}_{U, u_0}$ is a Noetherian local $S$-algebra", "with residue field $k$. Hence $R = \\mathcal{O}_{U, u_0}^\\wedge$ is an object of", "$\\mathcal{C}_\\Lambda^\\wedge$, see Formal Deformation Theory, Definition", "\\ref{formal-defos-definition-completion-CLambda}.", "Recall that $\\xi$ is versal if", "$\\underline{\\xi} : \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to", "\\mathcal{F}_{\\mathcal{X}, k, x_0}$", "is smooth and $x$ is versal at $u_0$ if", "$\\hat x : \\mathcal{F}_{(\\Sch/U)_{fppf}, k, u_0}", "\\to \\mathcal{F}_{\\mathcal{X}, k, x_0}$ is smooth.", "There is an identification of predeformation categories", "$$", "\\underline{R}|_{\\mathcal{C}_\\Lambda}", "=", "\\mathcal{F}_{(\\Sch/U)_{fppf}, k, u_0},", "$$", "see Formal Deformation Theory, Remark", "\\ref{formal-defos-remark-formal-objects-yoneda} for notation.", "Namely, given an Artinian local $S$-algebra $A$ with residue field", "identified with $k$ we have", "$$", "\\Mor_{\\mathcal{C}_\\Lambda^\\wedge}(R, A) =", "\\{\\varphi \\in \\Mor_S(\\Spec(A), U) \\mid \\varphi|_{\\Spec(k)} = u_0\\}", "$$", "Unwinding the definitions the reader verifies that the resulting map", "$$", "\\underline{R}|_{\\mathcal{C}_\\Lambda} =", "\\mathcal{F}_{(\\Sch/U)_{fppf}, k, u_0}", "\\xrightarrow{\\hat x}", "\\mathcal{F}_{\\mathcal{X}, k, x_0},", "$$", "is equal to $\\underline{\\xi}$ and we see that the lemma is true." ], "refs": [ "formal-defos-definition-completion-CLambda", "formal-defos-remark-formal-objects-yoneda" ], "ref_ids": [ 3514, 3553 ] } ], "ref_ids": [ 11422 ] }, { "id": 11369, "type": "theorem", "label": "artin-lemma-versal-implies-smooth", "categories": [ "artin" ], "title": "artin-lemma-versal-implies-smooth", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $f : U \\to V$", "be a morphism of schemes locally of finite type over $S$.", "Let $u_0 \\in U$ be a finite type point. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is smooth at $u_0$,", "\\item $f$ viewed as an object of $(\\Sch/V)_{fppf}$ over $U$ is", "versal at $u_0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a restatement of More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-lifting-along-artinian-at-point}." ], "refs": [ "more-morphisms-lemma-lifting-along-artinian-at-point" ], "ref_ids": [ 13741 ] } ], "ref_ids": [] }, { "id": 11370, "type": "theorem", "label": "artin-lemma-versal-change-of-field", "categories": [ "artin" ], "title": "artin-lemma-versal-change-of-field", "contents": [ "Let $S$, $\\mathcal{X}$, $U$, $x$, $u_0$ be as in", "Definition \\ref{definition-versal}. Let $l$ be a field and let", "$u_{l, 0} : \\Spec(l) \\to U$ be a morphism with image $u_0$ such that", "$l/k = \\kappa(u_0)$ is finite. Set $x_{l, 0} = x_0|_{\\Spec(l)}$.", "If $\\mathcal{X}$ satisfies (RS) and $x$ is versal at $u_0$, then", "$$", "\\mathcal{F}_{(\\Sch/U)_{fppf}, l, u_{l, 0}}", "\\longrightarrow", "\\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}}", "$$", "is smooth." ], "refs": [ "artin-definition-versal" ], "proofs": [ { "contents": [ "Note that $(\\Sch/U)_{fppf}$ satisfies (RS) by", "Lemma \\ref{lemma-algebraic-stack-RS}.", "Hence the functor of the lemma is the functor", "$$", "(\\mathcal{F}_{(\\Sch/U)_{fppf}, k , u_0})_{l/k}", "\\longrightarrow", "(\\mathcal{F}_{\\mathcal{X}, k , x_0})_{l/k}", "$$", "associated to $\\hat x$, see Lemma \\ref{lemma-change-of-field}.", "Hence the lemma follows from", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-change-of-fields-smooth}." ], "refs": [ "artin-lemma-algebraic-stack-RS", "artin-lemma-change-of-field", "formal-defos-lemma-change-of-fields-smooth" ], "ref_ids": [ 11355, 11358, 3508 ] } ], "ref_ids": [ 11422 ] }, { "id": 11371, "type": "theorem", "label": "artin-lemma-base-change-versal", "categories": [ "artin" ], "title": "artin-lemma-base-change-versal", "contents": [ "Let $S$, $\\mathcal{X}$, $U$, $x$, $u_0$ be as in", "Definition \\ref{definition-versal}. Assume", "\\begin{enumerate}", "\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$", "is representable by algebraic spaces,", "\\item $\\Delta$ is locally of finite type", "(for example if $\\mathcal{X}$ is limit preserving), and", "\\item $\\mathcal{X}$ has (RS).", "\\end{enumerate}", "Let $V$ be a scheme locally of finite type over $S$", "and let $y$ be an object of $\\mathcal{X}$ over $V$.", "Form the $2$-fibre product", "$$", "\\xymatrix{", "\\mathcal{Z} \\ar[r] \\ar[d] & (\\Sch/U)_{fppf} \\ar[d]^x \\\\", "(\\Sch/V)_{fppf} \\ar[r]^y & \\mathcal{X}", "}", "$$", "Let $Z$ be the algebraic space representing $\\mathcal{Z}$", "and let $z_0 \\in |Z|$ be a finite type point lying over $u_0$.", "If $x$ is versal at $u_0$, then", "the morphism $Z \\to V$ is smooth at $z_0$." ], "refs": [ "artin-definition-versal" ], "proofs": [ { "contents": [ "(The parenthetical remark in the statement holds by", "Lemma \\ref{lemma-diagonal}.)", "Observe that $Z$ exists by assumption (1) and Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-representable-diagonal}. By assumption (2) we see that", "$Z \\to V \\times_S U$ is locally of finite type.", "Choose a scheme $W$, a closed point $w_0 \\in W$, and", "an \\'etale morphism $W \\to Z$ mapping $w_0$ to $z_0$, see", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-finite-type-point}.", "Then $W$ is locally of finite type over $S$ and", "$w_0$ is a finite type point of $W$.", "Let $l = \\kappa(z_0)$. Denote $z_{l, 0}$, $v_{l, 0}$,", "$u_{l, 0}$, and $x_{l, 0}$ the objects of", "$\\mathcal{Z}$, $(\\Sch/V)_{fppf}$, $(\\Sch/U)_{fppf}$,", "and $\\mathcal{X}$ over $\\Spec(l)$ obtained by pullback to $\\Spec(l) = w_0$.", "Consider", "$$", "\\xymatrix{", "\\mathcal{F}_{(\\Sch/W)_{fppf}, l, w_0} \\ar[r] &", "\\mathcal{F}_{\\mathcal{Z}, l, z_{l, 0}} \\ar[d] \\ar[r] &", "\\mathcal{F}_{(\\Sch/U)_{fppf}, l, u_{l, 0}} \\ar[d] \\\\", "& \\mathcal{F}_{(\\Sch/V)_{fppf}, l, v_{l, 0}} \\ar[r] &", "\\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}}", "}", "$$", "By Lemma \\ref{lemma-fibre-product-deformation-categories}", "the square is a fibre product of predeformation categories.", "By Lemma \\ref{lemma-versal-change-of-field}", "we see that the right vertical arrow is smooth.", "By Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-smooth-properties}", "the left vertical arrow is smooth.", "By Lemma \\ref{lemma-formally-smooth-on-deformation-categories}", "we see that the left horizontal arrow is smooth.", "We conclude that the map", "$$", "\\mathcal{F}_{(\\Sch/W)_{fppf}, l, w_0} \\to", "\\mathcal{F}_{(\\Sch/V)_{fppf}, l, v_{l, 0}}", "$$", "is smooth by Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-smooth-properties}.", "Thus we conclude that $W \\to V$ is smooth at $w_0$ by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-lifting-along-artinian-at-point}.", "This exactly means that $Z \\to V$ is smooth at $z_0$", "and the proof is complete." ], "refs": [ "artin-lemma-diagonal", "algebraic-lemma-representable-diagonal", "spaces-morphisms-definition-finite-type-point", "artin-lemma-fibre-product-deformation-categories", "artin-lemma-versal-change-of-field", "formal-defos-lemma-smooth-properties", "artin-lemma-formally-smooth-on-deformation-categories", "formal-defos-lemma-smooth-properties", "more-morphisms-lemma-lifting-along-artinian-at-point" ], "ref_ids": [ 11367, 8461, 5004, 11353, 11370, 3433, 11352, 3433, 13741 ] } ], "ref_ids": [ 11422 ] }, { "id": 11372, "type": "theorem", "label": "artin-lemma-approximate-versal", "categories": [ "artin" ], "title": "artin-lemma-approximate-versal", "contents": [ "Let $S$ be a locally Noetherian scheme. Let", "$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.", "Let $\\xi = (R, \\xi_n, f_n)$ be a formal object of $\\mathcal{X}$ with", "$\\xi_1$ lying over $\\Spec(k) \\to S$ with image $s \\in S$. Assume", "\\begin{enumerate}", "\\item $\\xi$ is versal,", "\\item $\\xi$ is effective,", "\\item $\\mathcal{O}_{S, s}$ is a G-ring, and", "\\item $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects.", "\\end{enumerate}", "Then there exist a morphism of finite type $U \\to S$, a finite type", "point $u_0 \\in U$ with residue field $k$, and an object $x$ of $\\mathcal{X}$", "over $U$ such that $x$ is versal at $u_0$ and such that", "$x|_{\\Spec(\\mathcal{O}_{U, u_0}/\\mathfrak m_{u_0}^n)} \\cong \\xi_n$." ], "refs": [], "proofs": [ { "contents": [ "Choose an object $x_R$ of $\\mathcal{X}$ lying over $\\Spec(R)$ whose associated", "formal object is $\\xi$. Let $N = 2$ and apply Lemma \\ref{lemma-approximate}.", "We obtain $A, \\mathfrak m_A, x_A, \\ldots$.", "Let $\\eta = (A^\\wedge, \\eta_n, g_n)$ be the formal object associated to", "$x_A|_{\\Spec(A^\\wedge)}$. We have a diagram", "$$", "\\vcenter{", "\\xymatrix{", "& \\eta \\ar[d] \\\\", "\\xi \\ar[r] \\ar@{..>}[ru] & \\xi_2 = \\eta_2", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "& A^\\wedge \\ar[d] \\\\", "R \\ar[r] \\ar@{..>}[ru] & R/\\mathfrak m_R^2 = A/\\mathfrak m_A^2", "}", "}", "$$", "The versality of $\\xi$ means exactly that we can find the", "dotted arrows in the diagrams, because we can successively find", "morphisms $\\xi \\to \\eta_3$, $\\xi \\to \\eta_4$, and so on by", "Formal Deformation Theory, Remark \\ref{formal-defos-remark-versal-object}.", "The corresponding ring map $R \\to A^\\wedge$ is surjective by", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-surjective-cotangent-space}.", "On the other hand, we have", "$\\dim_k \\mathfrak m_R^n/\\mathfrak m_R^{n + 1} =", "\\dim_k \\mathfrak m_A^n/\\mathfrak m_A^{n + 1}$ for all $n$ by construction.", "Hence $R/\\mathfrak m_R^n$ and $A/\\mathfrak m_A^n$ have the same (finite)", "length as $\\Lambda$-modules by additivity of length and", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-length}.", "It follows that $R/\\mathfrak m_R^n \\to A/\\mathfrak m_A^n$ is an isomorphism", "for all $n$, hence $R \\to A^\\wedge$ is an isomorphism. Thus $\\eta$ is", "isomorphic to a versal object, hence versal itself. By", "Lemma \\ref{lemma-versality-matches}", "we conclude that $x_A$ is versal at the point $u_0$ of", "$U = \\Spec(A)$ corresponding to $\\mathfrak m_A$." ], "refs": [ "artin-lemma-approximate", "formal-defos-remark-versal-object", "formal-defos-lemma-surjective-cotangent-space", "formal-defos-lemma-length", "artin-lemma-versality-matches" ], "ref_ids": [ 11364, 3559, 3421, 3415, 11368 ] } ], "ref_ids": [] }, { "id": 11373, "type": "theorem", "label": "artin-lemma-versal-smooth", "categories": [ "artin" ], "title": "artin-lemma-versal-smooth", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $U$ be a scheme locally", "of finite type over $S$. Let $x$ be an object of $\\mathcal{X}$ over $U$.", "Assume that $x$ is versal at every finite type point of $U$ and that", "$\\mathcal{X}$ satisfies (RS). Then $x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$", "satisfies (\\ref{equation-smooth})." ], "refs": [], "proofs": [ { "contents": [ "Let $\\Spec(l) \\to U$ be a morphism with $l$ of finite type over $S$.", "Then the image $u_0 \\in U$ is a finite type point of $U$ and", "$\\kappa(u_0) \\subset l$ is a finite extension, see discussion in", "Morphisms, Section \\ref{morphisms-section-points-finite-type}.", "Hence we see that", "$\\mathcal{F}_{(\\Sch/U)_{fppf}, l, u_{l, 0}} \\to", "\\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}}$", "is smooth by Lemma \\ref{lemma-versal-change-of-field}." ], "refs": [ "artin-lemma-versal-change-of-field" ], "ref_ids": [ 11370 ] } ], "ref_ids": [] }, { "id": 11374, "type": "theorem", "label": "artin-lemma-composition-smooth", "categories": [ "artin" ], "title": "artin-lemma-composition-smooth", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$", "and $g : \\mathcal{Y} \\to \\mathcal{Z}$ be composable $1$-morphisms of", "categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $f$ and $g$", "satisfy (\\ref{equation-smooth}) so does $g \\circ f$." ], "refs": [], "proofs": [ { "contents": [ "This follows formally from Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-smooth-properties}." ], "refs": [ "formal-defos-lemma-smooth-properties" ], "ref_ids": [ 3433 ] } ], "ref_ids": [] }, { "id": 11375, "type": "theorem", "label": "artin-lemma-base-change-smooth", "categories": [ "artin" ], "title": "artin-lemma-base-change-smooth", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$", "and $\\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of", "categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $f$", "satisfies (\\ref{equation-smooth}) so does the projection", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Lemma \\ref{lemma-fibre-product-deformation-categories}", "and", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-smooth-properties}." ], "refs": [ "artin-lemma-fibre-product-deformation-categories", "formal-defos-lemma-smooth-properties" ], "ref_ids": [ 11353, 3433 ] } ], "ref_ids": [] }, { "id": 11376, "type": "theorem", "label": "artin-lemma-smooth-smooth", "categories": [ "artin" ], "title": "artin-lemma-smooth-smooth", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$", "be a $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "If $f$ is formally smooth on objects, then $f$ satisfies", "(\\ref{equation-smooth}). If $f$ is representable by algebraic spaces", "and smooth, then $f$ satisfies (\\ref{equation-smooth})." ], "refs": [], "proofs": [ { "contents": [ "A reformulation of Lemma \\ref{lemma-formally-smooth-on-deformation-categories}." ], "refs": [ "artin-lemma-formally-smooth-on-deformation-categories" ], "ref_ids": [ 11352 ] } ], "ref_ids": [] }, { "id": 11377, "type": "theorem", "label": "artin-lemma-implies-smooth", "categories": [ "artin" ], "title": "artin-lemma-implies-smooth", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$", "be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "Assume", "\\begin{enumerate}", "\\item $f$ is representable by algebraic spaces,", "\\item $f$ satisfies (\\ref{equation-smooth}),", "\\item $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects, and", "\\item $\\mathcal{Y}$ is limit preserving.", "\\end{enumerate}", "Then $f$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "The key ingredient of the proof is More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-lifting-along-artinian-at-point}", "which (almost) says that a morphism of schemes of finite type over $S$", "satisfying (\\ref{equation-smooth}) is a smooth morphism. The other", "arguments of the proof are essentially bookkeeping.", "\\medskip\\noindent", "Let $V$ be a scheme over $S$ and let $y$ be an object of $\\mathcal{Y}$ over", "$V$. Let $Z$ be an algebraic space representing the $2$-fibre product", "$\\mathcal{Z} = \\mathcal{X} \\times_{f, \\mathcal{X}, y} (\\Sch/V)_{fppf}$.", "We have to show that the projection morphism $Z \\to V$ is smooth, see", "Algebraic Stacks, Definition", "\\ref{algebraic-definition-relative-representable-property}.", "In fact, it suffices to do this when $V$ is an affine scheme", "locally of finite presentation over $S$, see", "Criteria for Representability, Lemma", "\\ref{criteria-lemma-check-property-limit-preserving}.", "Then $(\\Sch/V)_{fppf}$ is limit preserving by", "Lemma \\ref{lemma-limit-preserving-algebraic-space}.", "Hence $Z \\to S$ is locally of finite presentation by", "Lemmas \\ref{lemma-fibre-product-limit-preserving} and", "\\ref{lemma-limit-preserving-algebraic-space}.", "Choose a scheme $W$ and a surjective \\'etale morphism $W \\to Z$.", "Then $W$ is locally of finite presentation over $S$.", "\\medskip\\noindent", "Since $f$ satisfies (\\ref{equation-smooth}) we see that so does", "$\\mathcal{Z} \\to (\\Sch/V)_{fppf}$, see Lemma \\ref{lemma-base-change-smooth}.", "Next, we see that $(\\Sch/W)_{fppf} \\to \\mathcal{Z}$ satisfies", "(\\ref{equation-smooth}) by Lemma \\ref{lemma-smooth-smooth}.", "Thus the composition", "$$", "(\\Sch/W)_{fppf} \\to \\mathcal{Z} \\to (\\Sch/V)_{fppf}", "$$", "satisfies (\\ref{equation-smooth}) by Lemma \\ref{lemma-composition-smooth}.", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-lifting-along-artinian-at-point}", "shows that the composition $W \\to Z \\to V$ is smooth at every finite type", "point $w_0$ of $W$. Since the smooth locus is open we conclude", "that $W \\to V$ is a smooth morphism of schemes by", "Morphisms, Lemma \\ref{morphisms-lemma-enough-finite-type-points}.", "Thus we conclude that $Z \\to V$ is a smooth morphism", "of algebraic spaces by definition." ], "refs": [ "more-morphisms-lemma-lifting-along-artinian-at-point", "algebraic-definition-relative-representable-property", "criteria-lemma-check-property-limit-preserving", "artin-lemma-limit-preserving-algebraic-space", "artin-lemma-fibre-product-limit-preserving", "artin-lemma-limit-preserving-algebraic-space", "artin-lemma-base-change-smooth", "artin-lemma-smooth-smooth", "artin-lemma-composition-smooth", "more-morphisms-lemma-lifting-along-artinian-at-point", "morphisms-lemma-enough-finite-type-points" ], "ref_ids": [ 13741, 8483, 3104, 11366, 11365, 11366, 11375, 11376, 11374, 13741, 5210 ] } ], "ref_ids": [] }, { "id": 11378, "type": "theorem", "label": "artin-lemma-get-smooth", "categories": [ "artin" ], "title": "artin-lemma-get-smooth", "contents": [ "Let $S$ be a locally Noetherian scheme. Let", "$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.", "Let $k$ be a finite type field over $S$ and let $x_0$ be an object of", "$\\mathcal{X}$ over $\\Spec(k)$ with image $s \\in S$. Assume", "\\begin{enumerate}", "\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is", "representable by algebraic spaces,", "\\item $\\mathcal{X}$ satisfies axioms [1], [2], [3] (see", "Section \\ref{section-axioms}),", "\\item every formal object of $\\mathcal{X}$ is effective,", "\\item openness of versality holds for $\\mathcal{X}$, and", "\\item $\\mathcal{O}_{S, s}$ is a G-ring.", "\\end{enumerate}", "Then there exist a morphism of finite type $U \\to S$ and an object", "$x$ of $\\mathcal{X}$ over $U$ such that", "$$", "x : (\\Sch/U)_{fppf} \\longrightarrow \\mathcal{X}", "$$", "is smooth and such that there exists a finite type point $u_0 \\in U$", "whose residue field is $k$ and such that $x|_{u_0} \\cong x_0$." ], "refs": [], "proofs": [ { "contents": [ "By axiom [2], Lemma \\ref{lemma-deformation-category}, and", "Remark \\ref{remark-deformation-category-implies}", "we see that $\\mathcal{F}_{\\mathcal{X}, k, x_0}$ satisfies (S1) and (S2).", "Since also the tangent space has finite dimension by axiom [3]", "we deduce from Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-versal-object-existence}", "that $\\mathcal{F}_{\\mathcal{X}, k, x_0}$ has a versal formal object $\\xi$.", "Assumption (3) says $\\xi$ is effective. By axiom [1] and", "Lemma \\ref{lemma-approximate-versal}", "there exists a morphism of finite type $U \\to S$, an object $x$ of", "$\\mathcal{X}$ over $U$, and a finite type point $u_0$ of $U$ with residue", "field $k$ such that $x$ is versal at $u_0$ and such that", "$x|_{\\Spec(k)} \\cong x_0$. By openness of versality we may shrink", "$U$ and assume that $x$ is versal at every finite type point of $U$.", "We claim that", "$$", "x : (\\Sch/U)_{fppf} \\longrightarrow \\mathcal{X}", "$$", "is smooth which proves the lemma. Namely, by Lemma \\ref{lemma-versal-smooth}", "$x$ satisfies (\\ref{equation-smooth})", "whereupon Lemma \\ref{lemma-implies-smooth}", "finishes the proof." ], "refs": [ "artin-lemma-deformation-category", "artin-remark-deformation-category-implies", "formal-defos-lemma-versal-object-existence", "artin-lemma-approximate-versal", "artin-lemma-versal-smooth", "artin-lemma-implies-smooth" ], "ref_ids": [ 11357, 11427, 3458, 11372, 11373, 11377 ] } ], "ref_ids": [] }, { "id": 11379, "type": "theorem", "label": "artin-lemma-monomorphism", "categories": [ "artin" ], "title": "artin-lemma-monomorphism", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $a : F \\to G$ be a transformation", "of functors $(\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Assume that", "\\begin{enumerate}", "\\item $a$ is injective,", "\\item $F$ satisfies axioms [0], [1], [2], [4], and [5],", "\\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$,", "\\item $G$ is an algebraic space locally of finite type over $S$,", "\\end{enumerate}", "Then $F$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-dimension} the functor $G$ satisfies [3].", "As $F \\to G$ is injective, we conclude that $F$ also satisfies [3].", "Moreover, as $F \\to G$ is injective, we see that given schemes", "$U$, $V$ and morphisms $U \\to F$ and $V \\to F$, then", "$U \\times_F V = U \\times_G V$. Hence $\\Delta : F \\to F \\times F$ is", "representable (by schemes) as this holds for $G$ by assumption.", "Thus Proposition \\ref{proposition-spaces-diagonal-representable}", "applies\\footnote{The set", "theoretic condition [-1] holds for $F$ as it holds for $G$. Details", "omitted.}." ], "refs": [ "artin-lemma-finite-dimension", "artin-proposition-spaces-diagonal-representable" ], "ref_ids": [ 11359, 11414 ] } ], "ref_ids": [] }, { "id": 11380, "type": "theorem", "label": "artin-lemma-diagonal-representable", "categories": [ "artin" ], "title": "artin-lemma-diagonal-representable", "contents": [ "Let $S$ be a locally Noetherian scheme. Let", "$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.", "Assume that", "\\begin{enumerate}", "\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$", "is representable by algebraic spaces,", "\\item $\\mathcal{X}$ satisfies axioms [-1], [0], [1], [2], [3] (see", "Section \\ref{section-axioms}),", "\\item every formal object of $\\mathcal{X}$ is effective,", "\\item $\\mathcal{X}$ satisfies openness of versality, and", "\\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$.", "\\end{enumerate}", "Then $\\mathcal{X}$ is an algebraic stack." ], "refs": [], "proofs": [ { "contents": [ "Lemma \\ref{lemma-get-smooth} applies to $\\mathcal{X}$. Using this we", "choose, for every finite type field $k$ over $S$ and every", "isomorphism class of object $x_0 \\in \\Ob(\\mathcal{X}_{\\Spec(k)})$,", "an affine scheme $U_{k, x_0}$ of finite type over $S$ and a smooth morphism", "$(\\Sch/U_{k, x_0})_{fppf} \\to \\mathcal{X}$ such that there exists a finite", "type point $u_{k, x_0} \\in U_{k, x_0}$ with residue field $k$ such that $x_0$", "is the image of $u_{k, x_0}$. Then", "$$", "(\\Sch/U)_{fppf} \\to \\mathcal{X},", "\\quad\\text{with}\\quad", "U = \\coprod\\nolimits_{k, x_0} U_{k, x_0}", "$$", "is smooth\\footnote{Set theoretical remark: This coproduct is (isomorphic to)", "an object of $(\\Sch/S)_{fppf}$ as we have a bound on the index set", "by axiom [-1], see Sets, Lemma \\ref{sets-lemma-what-is-in-it}.}.", "To finish the proof it suffices to show this map is surjective,", "see Criteria for Representability, Lemma \\ref{criteria-lemma-stacks-etale}", "(this is where we use axiom [0]). By Criteria for Representability, Lemma", "\\ref{criteria-lemma-check-property-limit-preserving}", "it suffices to show that", "$(\\Sch/U)_{fppf} \\times_\\mathcal{X} (\\Sch/V)_{fppf} \\to (\\Sch/V)_{fppf}$", "is surjective for those $y : (\\Sch/V)_{fppf} \\to \\mathcal{X}$ where", "$V$ is an affine scheme locally of finite presentation", "over $S$. By assumption (1) the fibre product", "$(\\Sch/U)_{fppf} \\times_\\mathcal{X} (\\Sch/V)_{fppf}$ is representable", "by an algebraic space $W$. Then $W \\to V$ is smooth, hence the image is", "open. Hence it suffices to show that the image of $W \\to V$ contains all", "finite type points of $V$, see", "Morphisms, Lemma \\ref{morphisms-lemma-enough-finite-type-points}.", "Let $v_0 \\in V$ be a finite type point. Then $k = \\kappa(v_0)$ is", "a finite type field over $S$. Denote $x_0 = y|_{\\Spec(k)}$", "the pullback of $y$ by $v_0$. Then $(u_{k, x_0}, v_0)$ will give", "a morphism $\\Spec(k) \\to W$ whose composition with $W \\to V$", "is $v_0$ and we win." ], "refs": [ "artin-lemma-get-smooth", "sets-lemma-what-is-in-it", "criteria-lemma-stacks-etale", "criteria-lemma-check-property-limit-preserving", "morphisms-lemma-enough-finite-type-points" ], "ref_ids": [ 11378, 8795, 3141, 3104, 5210 ] } ], "ref_ids": [] }, { "id": 11381, "type": "theorem", "label": "artin-lemma-algebraic-stack-RS-star", "categories": [ "artin" ], "title": "artin-lemma-algebraic-stack-RS-star", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack over a base $S$.", "Then $\\mathcal{X}$ satisfies (RS*)." ], "refs": [], "proofs": [ { "contents": [ "This is implied by Lemma \\ref{lemma-pushout}, see", "remarks following Definition \\ref{definition-RS-star}." ], "refs": [ "artin-lemma-pushout", "artin-definition-RS-star" ], "ref_ids": [ 11354, 11424 ] } ], "ref_ids": [] }, { "id": 11382, "type": "theorem", "label": "artin-lemma-fibre-product-RS-star", "categories": [ "artin" ], "title": "artin-lemma-fibre-product-RS-star", "contents": [ "Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and", "$q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. If $\\mathcal{X}$, $\\mathcal{Y}$,", "and $\\mathcal{Z}$ satisfy (RS*), then so", "does $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Lemma \\ref{lemma-fibre-product-RS}." ], "refs": [ "artin-lemma-fibre-product-RS" ], "ref_ids": [ 11356 ] } ], "ref_ids": [] }, { "id": 11383, "type": "theorem", "label": "artin-lemma-single-point", "categories": [ "artin" ], "title": "artin-lemma-single-point", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$ having (RS*).", "Let $x$ be an object of $\\mathcal{X}$ over an affine scheme $U$", "of finite type over $S$. Let $u \\in U$ be a finite type point such that", "$x$ is not versal at $u$. Then there exists a morphism $x \\to y$", "of $\\mathcal{X}$ lying over $U \\to T$ satisfying", "\\begin{enumerate}", "\\item the morphism $U \\to T$ is a first order thickening,", "\\item we have a short exact sequence", "$$", "0 \\to \\kappa(u) \\to \\mathcal{O}_T \\to \\mathcal{O}_U \\to 0", "$$", "\\item there does {\\bf not} exist a pair $(W, \\alpha)$", "consisting of an open neighbourhood $W \\subset T$ of $u$", "and a morphism $\\beta : y|_W \\to x$ such that the composition", "$$", "x|_{U \\cap W} \\xrightarrow{\\text{restriction of }x \\to y}", "y|_W \\xrightarrow{\\beta} x", "$$", "is the canonical morphism $x|_{U \\cap W} \\to x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $R = \\mathcal{O}_{U, u}^\\wedge$. Let $k = \\kappa(u)$", "be the residue field of $R$. Let $\\xi$ be the formal object", "of $\\mathcal{X}$ over $R$ associated to $x$. Since $x$ is not", "versal at $u$, we see that $\\xi$ is not versal, see", "Lemma \\ref{lemma-versality-matches}. By the discussion following", "Definition \\ref{definition-versal-formal-object}", "this means we can find", "morphisms $\\xi_1 \\to x_A \\to x_B$ of $\\mathcal{X}$ lying over", "closed immersions $\\Spec(k) \\to \\Spec(A) \\to \\Spec(B)$", "where $A, B$ are Artinian local rings with residue field $k$,", "an $n \\geq 1$ and a commutative diagram", "$$", "\\vcenter{", "\\xymatrix{", "& x_A \\ar[ld] \\\\", "\\xi_n & \\xi_1 \\ar[u] \\ar[l]", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "& \\Spec(A) \\ar[ld] \\\\", "\\Spec(R/\\mathfrak m^n) & \\Spec(k) \\ar[u] \\ar[l]", "}", "}", "$$", "such that there does {\\bf not} exist an $m \\geq n$ and a commutative diagram", "$$", "\\vcenter{", "\\xymatrix{", "& & x_B \\ar[lldd] \\\\", "& & x_A \\ar[ld] \\ar[u] \\\\", "\\xi_m & \\xi_n \\ar[l] & \\xi_1 \\ar[u] \\ar[l]", "}", "}", "\\quad\\text{lying over}", "\\vcenter{", "\\xymatrix{", "& & \\Spec(B) \\ar[lldd] \\\\", "& & \\Spec(A) \\ar[ld] \\ar[u] \\\\", "\\Spec(R/\\mathfrak m^m) &", "\\Spec(R/\\mathfrak m^n) \\ar[l] &", "\\Spec(k) \\ar[u] \\ar[l]", "}", "}", "$$", "We may moreover assume that $B \\to A$ is a small", "extension, i.e., that the kernel $I$ of the surjection $B \\to A$", "is isomorphic to $k$ as an $A$-module.", "This follows from Formal Deformation Theory, Remark", "\\ref{formal-defos-remark-versal-object}.", "Then we simply define", "$$", "T = U \\amalg_{\\Spec(A)} \\Spec(B)", "$$", "By property (RS*) we find $y$ over $T$ whose restriction to", "$\\Spec(B)$ is $x_B$ and whose restriction to $U$ is $x$", "(this gives the arrow $x \\to y$ lying over $U \\to T$).", "To finish the proof we verify conditions (1), (2), and (3).", "\\medskip\\noindent", "By the construction of the pushout we have a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "I \\ar[r] &", "B \\ar[r] &", "A \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "I \\ar[r] \\ar[u] &", "\\Gamma(T, \\mathcal{O}_T) \\ar[r] \\ar[u] &", "\\Gamma(U, \\mathcal{O}_U) \\ar[r] \\ar[u] &", "0", "}", "$$", "with exact rows. This immediately proves (1) and (2).", "To finish the proof we will argue by contradiction.", "Assume we have a pair $(W, \\beta)$ as in (3).", "Since $\\Spec(B) \\to T$ factors through $W$ we get the morphism", "$$", "x_B \\to y|_W \\xrightarrow{\\beta} x", "$$", "Since $B$ is Artinian local with residue field $k = \\kappa(u)$", "we see that $x_B \\to x$ lies over a morphism $\\Spec(B) \\to U$", "which factors through $\\Spec(\\mathcal{O}_{U, u}/\\mathfrak m_u^m)$", "for some $m \\geq n$. In other words, $x_B \\to x$ factors", "through $\\xi_m$ giving a map $x_B \\to \\xi_m$.", "The compatibility condition on the morphism $\\alpha$", "in condition (3) translates into the condition that", "$$", "\\xymatrix{", "x_B \\ar[d] & x_A \\ar[d] \\ar[l] \\\\", "\\xi_m & \\xi_n \\ar[l]", "}", "$$", "is commutative. This gives the contradiction we were looking for." ], "refs": [ "artin-lemma-versality-matches", "artin-definition-versal-formal-object", "formal-defos-remark-versal-object" ], "ref_ids": [ 11368, 11421, 3559 ] } ], "ref_ids": [] }, { "id": 11384, "type": "theorem", "label": "artin-lemma-generalization-versality", "categories": [ "artin" ], "title": "artin-lemma-generalization-versality", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred", "in groupoids over $(\\Sch/S)_{fppf}$. Assume", "\\begin{enumerate}", "\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is", "representable by algebraic spaces,", "\\item $\\mathcal{X}$ has (RS*),", "\\item $\\mathcal{X}$ is limit preserving.", "\\end{enumerate}", "Let $x$ be an object of $\\mathcal{X}$ over a scheme $U$ of finite type over", "$S$. Let $u \\leadsto u_0$ be a specialization of finite type points of $U$", "such that $x$ is versal at $u_0$. Then $x$ is versal at $u$." ], "refs": [], "proofs": [ { "contents": [ "After shrinking $U$ we may assume $U$ is affine and $U$ maps into an", "affine open $\\Spec(\\Lambda)$ of $S$. If $x$ is not versal at $u$ then", "we may pick $x \\to y$ lying over $U \\to T$ as in", "Lemma \\ref{lemma-single-point}. Write $U = \\Spec(R_0)$ and $T = \\Spec(R)$.", "The morphism $U \\to T$ corresponds to a surjective ring map", "$R \\to R_0$ whose kernel is an ideal of square zero.", "By assumption (3) we get that $y$ comes from an object $x'$ over", "$U' = \\Spec(R')$ for some finite type $\\Lambda$-subalgebra", "$R' \\subset R$. After increasing $R'$ we may and do assume that", "$R' \\to R_0$ is surjective, so that $U \\subset U'$ is a first order thickening.", "Thus we now have", "$$", "x \\to y \\to x'", "\\text{ lying over }", "U \\to T \\to U'", "$$", "By assumption (1) there is an algebraic space $Z$ over $S$ representing", "$$", "(\\Sch/U)_{fppf} \\times_{x, \\mathcal{X}, x'} (\\Sch/U')_{fppf}", "$$", "see Algebraic Stacks, Lemma \\ref{algebraic-lemma-representable-diagonal}.", "By construction of $2$-fibre products, a $V$-valued point of $Z$", "corresponds to a triple $(a, a', \\alpha)$ consisting of morphisms", "$a : V \\to U$, $a' : V \\to U'$ and a morphism $\\alpha : a^*x \\to (a')^*x'$.", "We obtain a commutative diagram", "$$", "\\xymatrix{", "U \\ar[rd] \\ar[rdd] \\ar[rrd] \\\\", "& Z \\ar[r]_{p'} \\ar[d]^p & U' \\ar[d] \\\\", "& U \\ar[r] & S", "}", "$$", "The morphism $i : U \\to Z$ comes the isomorphism $x \\to x'|_U$.", "Let $z_0 = i(u_0) \\in Z$. By Lemma \\ref{lemma-base-change-versal}", "we see that $Z \\to U'$ is smooth at $z_0$. After replacing $U$ by an", "affine open neighbourhood of $u_0$, replacing $U'$ by the corresponding", "open, and replacing $Z$ by the intersection of the inverse images", "of these opens by $p$ and $p'$, we reach the situation where", "$Z \\to U'$ is smooth along $i(U)$. Since $u \\leadsto u_0$ the point", "$u$ is in this open. Condition (3) of Lemma \\ref{lemma-single-point}", "is clearly preserved by shrinking $U$ (all of the schemes $U$, $T$, $U'$", "have the same underlying topological space).", "Since $U \\to U'$ is a first order thickening of affine schemes,", "we can choose a morphism $i' : U' \\to Z$", "such that $p' \\circ i' = \\text{id}_{U'}$ and", "whose restriction to $U$ is $i$", "(More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-smooth-formally-smooth}).", "Pulling back the universal morphism $p^*x \\to (p')^*x'$ by $i'$", "we obtain a morphism", "$$", "x' \\to x", "$$", "lying over $p \\circ i' : U' \\to U$ such that the composition", "$$", "x \\to x' \\to x", "$$", "is the identity. Recall that we have $y \\to x'$ lying over", "the morphism $T \\to U'$. Composing we get a morphism", "$y \\to x$ whose existence contradicts condition", "(3) of Lemma \\ref{lemma-single-point}.", "This contradiction finishes the proof." ], "refs": [ "artin-lemma-single-point", "algebraic-lemma-representable-diagonal", "artin-lemma-base-change-versal", "artin-lemma-single-point", "spaces-more-morphisms-lemma-smooth-formally-smooth", "artin-lemma-single-point" ], "ref_ids": [ 11383, 8461, 11371, 11383, 110, 11383 ] } ], "ref_ids": [] }, { "id": 11385, "type": "theorem", "label": "artin-lemma-infinite-sequence-pre", "categories": [ "artin" ], "title": "artin-lemma-infinite-sequence-pre", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$ having (RS*).", "Let $x$ be an object of", "$\\mathcal{X}$ over an affine scheme $U$ of finite type over $S$.", "Let $u_n \\in U$, $n \\geq 1$ be finite type points such that", "(a) there are no specializations $u_n \\leadsto u_m$ for $n \\not = m$, and", "(b) $x$ is not versal at $u_n$ for all $n$. Then there exist morphisms", "$$", "x \\to x_1 \\to x_2 \\to \\ldots", "\\quad\\text{in }\\mathcal{X}\\text{ lying over }\\quad", "U \\to U_1 \\to U_2 \\to \\ldots", "$$", "over $S$ such that", "\\begin{enumerate}", "\\item for each $n$ the morphism $U \\to U_n$ is a first order", "thickening,", "\\item for each $n$ we have a short exact sequence", "$$", "0 \\to \\kappa(u_n) \\to \\mathcal{O}_{U_n} \\to \\mathcal{O}_{U_{n - 1}} \\to 0", "$$", "with $U_0 = U$ for $n = 1$,", "\\item for each $n$ there does {\\bf not} exist a pair $(W, \\alpha)$", "consisting of an open neighbourhood $W \\subset U_n$ of $u_n$", "and a morphism $\\alpha : x_n|_W \\to x$", "such that the composition", "$$", "x|_{U \\cap W} \\xrightarrow{\\text{restriction of }x \\to x_n}", "x_n|_W \\xrightarrow{\\alpha} x", "$$", "is the canonical morphism $x|_{U \\cap W} \\to x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since there are no specializations among the points $u_n$ (and in", "particular the $u_n$ are pairwise distinct), for every $n$", "we can find an open $U' \\subset U$", "such that $u_n \\in U'$ and $u_i \\not \\in U'$ for $i = 1, \\ldots, n - 1$.", "By Lemma \\ref{lemma-single-point} for each $n \\geq 1$ we can find", "$$", "x \\to y_n", "\\quad\\text{in }\\mathcal{X}\\text{ lying over}\\quad", "U \\to T_n", "$$", "such that", "\\begin{enumerate}", "\\item the morphism $U \\to T_n$ is a first order thickening,", "\\item we have a short exact sequence", "$$", "0 \\to \\kappa(u_n) \\to \\mathcal{O}_{T_n} \\to \\mathcal{O}_U \\to 0", "$$", "\\item there does {\\bf not} exist a pair $(W, \\alpha)$", "consisting of an open neighbourhood $W \\subset T_n$ of $u_n$", "and a morphism $\\beta : y_n|_W \\to x$ such that the composition", "$$", "x|_{U \\cap W} \\xrightarrow{\\text{restriction of }x \\to y_n}", "y_n|_W \\xrightarrow{\\beta} x", "$$", "is the canonical morphism $x|_{U \\cap W} \\to x$.", "\\end{enumerate}", "Thus we can define inductively", "$$", "U_1 = T_1, \\quad", "U_{n + 1} = U_n \\amalg_U T_{n + 1}", "$$", "Setting $x_1 = y_1$ and using (RS*) we find inductively", "$x_{n + 1}$ over $U_{n + 1}$ restricting to", "$x_n$ over $U_n$ and $y_{n + 1}$ over $T_{n + 1}$.", "Property (1) for $U \\to U_n$ follows from the construction", "of the pushout in More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-pushout-along-thickening}.", "Property (2) for $U_n$ similarly follows from", "property (2) for $T_n$ by the construction of the pushout.", "After shrinking to an open neighbourhood $U'$ of $u_n$", "as discussed above, property (3) for $(U_n, x_n)$ follows from property (3)", "for $(T_n, y_n)$ simply because the corresponding open subschemes", "of $T_n$ and $U_n$ are isomorphic. Some details omitted." ], "refs": [ "artin-lemma-single-point", "more-morphisms-lemma-pushout-along-thickening" ], "ref_ids": [ 11383, 13762 ] } ], "ref_ids": [] }, { "id": 11386, "type": "theorem", "label": "artin-lemma-SGE-implies-openness-versality", "categories": [ "artin" ], "title": "artin-lemma-SGE-implies-openness-versality", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred", "in groupoids over $(\\Sch/S)_{fppf}$. Assume", "\\begin{enumerate}", "\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is", "representable by algebraic spaces,", "\\item $\\mathcal{X}$ has (RS*),", "\\item $\\mathcal{X}$ is limit preserving,", "\\item systems $(\\xi_n)$ as in Remark \\ref{remark-strong-effectiveness}", "where $\\Ker(R_m \\to R_n)$ is an ideal of square zero for all $m \\geq n$", "are effective.", "\\end{enumerate}", "Then $\\mathcal{X}$ satisfies openness of versality." ], "refs": [ "artin-remark-strong-effectiveness" ], "proofs": [ { "contents": [ "Choose a scheme $U$ locally of finite type over $S$,", "a finite type point $u_0$ of $U$, and an object $x$ of $\\mathcal{X}$", "over $U$ such that $x$ is versal at $u_0$. After shrinking", "$U$ we may assume $U$ is affine and $U$ maps into an affine open", "$\\Spec(\\Lambda)$ of $S$. Let $E \\subset U$ be the set of finite type", "points $u$ such that $x$ is not versal at $u$. By", "Lemma \\ref{lemma-generalization-versality}", "if $u \\in E$ then $u_0$ is not a specialization of $u$.", "If openness of versality does not hold, then $u_0$ is in the closure", "$\\overline{E}$ of $E$. By", "Properties, Lemma \\ref{properties-lemma-countable-dense-subset}", "we may choose a countable subset $E' \\subset E$ with the same closure", "as $E$. By Properties, Lemma \\ref{properties-lemma-maximal-points}", "we may assume there are no specializations among the points of $E'$.", "Observe that $E'$ has to be (countably) infinite as $u_0$", "isn't the specialization of any point of $E'$ as pointed out above.", "Thus we can write $E' = \\{u_1, u_2, u_3, \\ldots\\}$, there", "are no specializations among the $u_i$, and $u_0$ is in the closure", "of $E'$.", "\\medskip\\noindent", "Choose $x \\to x_1 \\to x_2 \\to \\ldots$ lying over", "$U \\to U_1 \\to U_2 \\to \\ldots$ as in Lemma \\ref{lemma-infinite-sequence-pre}.", "Write $U_n = \\Spec(R_n)$ and $U = \\Spec(R_0)$.", "Set $R = \\lim R_n$. Observe that $R \\to R_0$ is surjective", "with kernel an ideal of square zero. By assumption (4)", "we get $\\xi$ over $\\Spec(R)$ whose base change to $R_n$ is $x_n$.", "By assumption (3) we get that $\\xi$ comes from an object $\\xi'$ over", "$U' = \\Spec(R')$ for some finite type $\\Lambda$-subalgebra", "$R' \\subset R$. After increasing $R'$ we may and do assume that", "$R' \\to R_0$ is surjective, so that $U \\subset U'$ is a first order thickening.", "Thus we now have", "$$", "x \\to x_1 \\to x_2 \\to \\ldots \\to \\xi'", "\\text{ lying over }", "U \\to U_1 \\to U_2 \\to \\ldots \\to U'", "$$", "By assumption (1) there is an algebraic space $Z$ over $S$ representing", "$$", "(\\Sch/U)_{fppf} \\times_{x, \\mathcal{X}, \\xi'} (\\Sch/U')_{fppf}", "$$", "see Algebraic Stacks, Lemma \\ref{algebraic-lemma-representable-diagonal}.", "By construction of $2$-fibre products, a $T$-valued point of $Z$", "corresponds to a triple $(a, a', \\alpha)$ consisting of morphisms", "$a : T \\to U$, $a' : T \\to U'$ and a morphism $\\alpha : a^*x \\to (a')^*\\xi'$.", "We obtain a commutative diagram", "$$", "\\xymatrix{", "U \\ar[rd] \\ar[rdd] \\ar[rrd] \\\\", "& Z \\ar[r]_{p'} \\ar[d]^p & U' \\ar[d] \\\\", "& U \\ar[r] & S", "}", "$$", "The morphism $i : U \\to Z$ comes the isomorphism $x \\to \\xi'|_U$.", "Let $z_0 = i(u_0) \\in Z$. By Lemma \\ref{lemma-base-change-versal}", "we see that $Z \\to U'$ is smooth at $z_0$. After replacing $U$ by an", "affine open neighbourhood of $u_0$, replacing $U'$ by the corresponding", "open, and replacing $Z$ by the intersection of the inverse images", "of these opens by $p$ and $p'$, we reach the situation where", "$Z \\to U'$ is smooth along $i(U)$. Note that this", "also involves replacing $u_n$ by a subsequence, namely", "by those indices such that $u_n$ is in the open. Moreover, condition", "(3) of Lemma \\ref{lemma-infinite-sequence-pre}", "is clearly preserved by shrinking $U$", "(all of the schemes $U$, $U_n$, $U'$ have the same underlying", "topological space).", "Since $U \\to U'$ is a first order thickening of affine schemes,", "we can choose a morphism $i' : U' \\to Z$", "such that $p' \\circ i' = \\text{id}_{U'}$ and", "whose restriction to $U$ is $i$", "(More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-smooth-formally-smooth}).", "Pulling back the universal morphism", "$p^*x \\to (p')^*\\xi'$ by $i'$ we obtain a morphism", "$$", "\\xi' \\to x", "$$", "lying over $p \\circ i' : U' \\to U$ such that the composition", "$$", "x \\to \\xi' \\to x", "$$", "is the identity. Recall that we have $x_1 \\to \\xi'$ lying over", "the morphism $U_1 \\to U'$. Composing we get a morphism", "$x_1 \\to x$ whose existence contradicts condition", "(3) of Lemma \\ref{lemma-infinite-sequence-pre}.", "This contradiction finishes the proof." ], "refs": [ "artin-lemma-generalization-versality", "properties-lemma-countable-dense-subset", "properties-lemma-maximal-points", "artin-lemma-infinite-sequence-pre", "algebraic-lemma-representable-diagonal", "artin-lemma-base-change-versal", "artin-lemma-infinite-sequence-pre", "spaces-more-morphisms-lemma-smooth-formally-smooth", "artin-lemma-infinite-sequence-pre" ], "ref_ids": [ 11384, 2962, 2961, 11385, 8461, 11371, 11385, 110, 11385 ] } ], "ref_ids": [ 11429 ] }, { "id": 11387, "type": "theorem", "label": "artin-lemma-functoriality", "categories": [ "artin" ], "title": "artin-lemma-functoriality", "contents": [ "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism", "of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let", "$$", "\\xymatrix{", "B' \\ar[r] & B \\\\", "A' \\ar[u] \\ar[r] & A \\ar[u]", "}", "$$", "be a commutative diagram of $S$-algebras. Let $x$ be an object of $\\mathcal{X}$", "over $\\Spec(A)$, let $y$ be an object of $\\mathcal{Y}$ over $\\Spec(B)$,", "and let $\\phi : f(x)|_{\\Spec(B)} \\to y$ be a morphism of $\\mathcal{Y}$", "over $\\Spec(B)$. Then there is a canonical functor", "$$", "\\textit{Lift}(x, A') \\longrightarrow \\textit{Lift}(y, B')", "$$", "of categories of lifts induced by $f$ and $\\phi$. The construction is", "compatible with compositions of $1$-morphisms of categories fibred in", "groupoids in an obvious manner." ], "refs": [], "proofs": [ { "contents": [ "This lemma proves itself." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11388, "type": "theorem", "label": "artin-lemma-properties-lift-RS-star", "categories": [ "artin" ], "title": "artin-lemma-properties-lift-RS-star", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$ satisfies", "condition (RS*). Let $A$ be an $S$-algebra and let $x$ be an object of", "$\\mathcal{X}$ over $\\Spec(A)$.", "\\begin{enumerate}", "\\item There exists an $A$-linear functor", "$\\text{Inf}_x : \\text{Mod}_A \\to \\text{Mod}_A$", "such that given a deformation situation $(x, A' \\to A)$ and a lift $x'$", "there is an isomorphism $\\text{Inf}_x(I) \\to \\text{Inf}(x'/x)$ where", "$I = \\Ker(A' \\to A)$.", "\\item There exists an $A$-linear functor", "$T_x : \\text{Mod}_A \\to \\text{Mod}_A$", "such that", "\\begin{enumerate}", "\\item given $M$ in $\\text{Mod}_A$ there is a bijection", "$T_x(M) \\to \\text{Lift}(x, A[M])$,", "\\item given a deformation situation $(x, A' \\to A)$ there is an action", "$$", "T_x(I) \\times \\text{Lift}(x, A') \\to \\text{Lift}(x, A')", "$$", "where $I = \\Ker(A' \\to A)$. It is simply transitive if", "$\\text{Lift}(x, A') \\not = \\emptyset$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We define $\\text{Inf}_x$ as the functor", "$$", "\\text{Mod}_A \\longrightarrow \\textit{Sets},\\quad", "M \\longrightarrow", "\\text{Inf}(x'_M/x) = \\text{Lift}(\\text{id}_x, A[M])", "$$", "mapping $M$ to the group of infinitesimal automorphisms", "of the trivial deformation $x'_M$ of $x$ to $\\Spec(A[M])$", "or equivalently the group of lifts of $\\text{id}_x$ in", "$\\mathit{Aut}_\\mathcal{X}(x'_M)$.", "We define $T_x$ as the functor", "$$", "\\text{Mod}_A \\longrightarrow \\textit{Sets},\\quad", "M \\longrightarrow \\text{Lift}(x, A[M])", "$$", "of isomorphism classes of infintesimal deformations of $x$ to", "$\\Spec(A[M])$. We apply Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-linear-functor}", "to $\\text{Inf}_x$ and $T_x$. This lemma is applicable, since", "(RS*) tells us that", "$$", "\\textit{Lift}(x, A[M \\times N]) =", "\\textit{Lift}(x, A[M]) \\times \\textit{Lift}(x, A[N])", "$$", "as categories (and trivial deformations match up too).", "\\medskip\\noindent", "Let $(x, A' \\to A)$ be a deformation situation. Consider the ring map", "$g : A' \\times_A A' \\to A[I]$ defined by the", "rule $g(a_1, a_2) = \\overline{a_1} \\oplus a_2 - a_1$.", "There is an isomorphism", "$$", "A' \\times_A A' \\longrightarrow A' \\times_A A[I]", "$$", "given by $(a_1, a_2) \\mapsto (a_1, g(a_1, a_2))$.", "This isomorphism commutes with the projections to $A'$ on the first", "factor, and hence with the projections to $A$. Thus applying (RS*)", "twice we find equivalences of categories", "\\begin{align*}", "\\textit{Lift}(x, A') \\times \\textit{Lift}(x, A')", "& =", "\\textit{Lift}(x, A' \\times_A A') \\\\", "& =", "\\textit{Lift}(x, A' \\times_A A[I]) \\\\", "& =", "\\textit{Lift}(x, A') \\times \\textit{Lift}(x, A[I])", "\\end{align*}", "Using these maps and projection onto the last factor of the last product", "we see that we obtain ``difference maps''", "$$", "\\text{Inf}(x'/x) \\times \\text{Inf}(x'/x)", "\\longrightarrow", "\\text{Inf}_x(I)", "\\quad\\text{and}\\quad", "\\text{Lift}(x, A') \\times \\text{Lift}(x, A')", "\\longrightarrow", "T_x(I)", "$$", "These difference maps satisfy the transitivity rule", "``$(x'_1 - x'_2) + (x'_2 - x'_3) = x'_1 - x'_3$'' because", "$$", "\\xymatrix{", "A' \\times_A A' \\times_A A'", "\\ar[rrrrr]_-{(a_1, a_2, a_3) \\mapsto (g(a_1, a_2), g(a_2, a_3))}", "\\ar[rrrrrd]_{(a_1, a_2, a_3) \\mapsto g(a_1, a_3)} & & & & &", "A[I] \\times_A A[I] = A[I \\times I] \\ar[d]^{+} \\\\", "& & & & & A[I]", "}", "$$", "is commutative. Inverting the string of equivalences above we obtain", "an action which is free and transitive provided $\\text{Inf}(x'/x)$,", "resp.\\ $\\text{Lift}(x, A')$ is nonempty. Note that $\\text{Inf}(x'/x)$", "is always nonempty as it is a group." ], "refs": [ "formal-defos-lemma-linear-functor" ], "ref_ids": [ 3445 ] } ], "ref_ids": [] }, { "id": 11389, "type": "theorem", "label": "artin-lemma-ses-inf-and-T", "categories": [ "artin" ], "title": "artin-lemma-ses-inf-and-T", "contents": [ "Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$", "and $q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$,", "$\\mathcal{Y}$, $\\mathcal{Z}$ satisfy (RS*).", "Let $A$ be an $S$-algebra and let $w$ be an object of", "$\\mathcal{W} = \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ over $A$.", "Denote $x, y, z$ the objects of $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$", "you get from $w$. For any $A$-module $M$ there is a $6$-term exact sequence", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\text{Inf}_w(M) \\ar[r] &", "\\text{Inf}_x(M) \\oplus \\text{Inf}_z(M) \\ar[r] &", "\\text{Inf}_y(M) \\ar[lld] \\\\", " &", "T_w(M) \\ar[r] &", "T_x(M) \\oplus T_z(M) \\ar[r] &", "T_y(M)", "}", "$$", "of $A$-modules." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-fibre-product-RS-star} we see that $\\mathcal{W}$", "satisfies (RS*) and hence $T_w(M)$ and $\\text{Inf}_w(M)$ are defined.", "The horizontal arrows are defined using the functoriality of", "Lemma \\ref{lemma-functoriality}.", "\\medskip\\noindent", "Definition of the ``boundary'' map $\\delta : \\text{Inf}_y(M) \\to T_w(M)$.", "Choose isomorphisms $p(x) \\to y$ and $y \\to q(z)$ such that", "$w = (x, z, p(x) \\to y \\to q(z))$ in the description of", "the $2$-fibre product of", "Categories, Lemma \\ref{categories-lemma-2-product-fibred-categories}", "and more precisely", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.", "Let $x', y', z', w'$ denote the trivial deformation of", "$x, y, z, w$ over $A[M]$. By pullback we get isomorphisms", "$y' \\to p(x')$ and $q(z') \\to y'$. An element $\\alpha \\in \\text{Inf}_y(M)$", "is the same thing as an automorphism $\\alpha : y' \\to y'$", "over $A[M]$ which restricts to the identity on $y$ over $A$.", "Thus setting", "$$", "\\delta(\\alpha) =", "(x', z', p(x') \\to y' \\xrightarrow{\\alpha} y' \\to q(z'))", "$$", "we obtain an object of $T_w(M)$. This is a map of $A$-modules", "by Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-morphism-linear-functors}.", "\\medskip\\noindent", "The rest of the proof is exactly the same as the proof of", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-deformation-categories-fiber-product-morphisms}." ], "refs": [ "artin-lemma-fibre-product-RS-star", "artin-lemma-functoriality", "categories-lemma-2-product-fibred-categories", "categories-lemma-2-product-categories-over-C", "formal-defos-lemma-morphism-linear-functors", "formal-defos-lemma-deformation-categories-fiber-product-morphisms" ], "ref_ids": [ 11382, 11387, 12296, 12280, 3446, 3482 ] } ], "ref_ids": [] }, { "id": 11390, "type": "theorem", "label": "artin-lemma-get-openness-obstruction-theory", "categories": [ "artin" ], "title": "artin-lemma-get-openness-obstruction-theory", "contents": [ "\\begin{reference}", "This is \\cite[Theorem 4.4]{Hall-coherent}", "\\end{reference}", "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred", "in groupoids over $(\\Sch/S)_{fppf}$. Assume", "\\begin{enumerate}", "\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is", "representable by algebraic spaces,", "\\item $\\mathcal{X}$ has (RS*),", "\\item $\\mathcal{X}$ is limit preserving,", "\\item there exists an obstruction theory\\footnote{Analyzing the proof", "the reader sees that in fact it suffices to check", "the functoriality (ii) of obstruction classes in", "Definition \\ref{definition-obstruction-theory}", "for maps $(y, B' \\to B) \\to (x, A' \\to A)$", "with $B = A$ and $y = x$.},", "\\item for an object $x$ of $\\mathcal{X}$ over $\\Spec(A)$", "and $A$-modules $M_n$, $n \\geq 1$ we have", "\\begin{enumerate}", "\\item $T_x(\\prod M_n) = \\prod T_x(M_n)$,", "\\item $\\mathcal{O}_x(\\prod M_n) \\to \\prod \\mathcal{O}_x(M_n)$", "is injective.", "\\end{enumerate}", "\\end{enumerate}", "Then $\\mathcal{X}$ satisfies openness of versality." ], "refs": [ "artin-definition-obstruction-theory" ], "proofs": [ { "contents": [ "We prove this by verifying condition (4) of", "Lemma \\ref{lemma-SGE-implies-openness-versality}.", "Let $(\\xi_n)$ and $(R_n)$ be as in Remark \\ref{remark-strong-effectiveness}", "such that $\\Ker(R_m \\to R_n)$ is an ideal of square zero", "for all $m \\geq n$. Set $A = R_1$ and $x = \\xi_1$.", "Denote $M_n = \\Ker(R_n \\to R_1)$.", "Then $M_n$ is an $A$-module. Set $R = \\lim R_n$.", "Let", "$$", "\\tilde R = \\{(r_1, r_2, r_3 \\ldots) \\in \\prod R_n", "\\text{ such that all have the same image in }A\\}", "$$", "Then $\\tilde R \\to A$ is surjective with kernel $M = \\prod M_n$.", "There is a map $R \\to \\tilde R$ and a map", "$\\tilde R \\to A[M]$, $(r_1, r_2, r_3, \\ldots) \\mapsto", "(r_1, r_2 - r_1, r_3 - r_2, \\ldots)$.", "Together these give a short exact sequence", "$$", "(x, R \\to A) \\to (x, \\tilde R \\to A) \\to (x, A[M])", "$$", "of deformation situations, see", "Remark \\ref{remark-short-exact-sequence-thickenings}.", "The associated sequence of kernels", "$0 \\to \\lim M_n \\to M \\to M \\to 0$", "is the canonical sequence computing the limit", "of the system of modules $(M_n)$.", "\\medskip\\noindent", "Let $o_x(\\tilde R) \\in \\mathcal{O}_x(M)$ be the obstruction element.", "Since we have the lifts $\\xi_n$ we see that $o_x(\\tilde R)$", "maps to zero in $\\mathcal{O}_x(M_n)$. By assumption (5)(b)", "we see that $o_x(\\tilde R) = 0$. Choose a lift $\\tilde \\xi$", "of $x$ to $\\Spec(\\tilde R)$. Let $\\tilde \\xi_n$ be the", "restriction of $\\tilde \\xi$ to $\\Spec(R_n)$. There exists", "elements $t_n \\in T_x(M_n)$ such that", "$t_n \\cdot \\tilde \\xi_n = \\xi_n$ by", "Lemma \\ref{lemma-properties-lift-RS-star} part (2)(b).", "By assumption (5)(a) we can find $t \\in T_x(M)$", "mapping to $t_n$ in $T_x(M_n)$. After replacing", "$\\tilde \\xi$ by $t \\cdot \\tilde \\xi$ we find that", "$\\tilde \\xi$ restricts to $\\xi_n$ over $\\Spec(R_n)$ for all $n$.", "In particular, since $\\xi_{n + 1}$ restricts to $\\xi_n$", "over $\\Spec(R_n)$, the restriction $\\overline{\\xi}$ of $\\tilde \\xi$", "to $\\Spec(A[M])$ has the property that it restricts to", "the trivial deformation over $\\Spec(A[M_n])$ for all $n$.", "Hence by assumption (5)(a) we find that $\\overline{\\xi}$", "is the trivial deformation of $x$. By axiom (RS*)", "applied to $R = \\tilde R \\times_{A[M]} A$", "this implies that $\\tilde \\xi$ is the pullback", "of a deformation $\\xi$ of $x$ over $R$. This finishes the proof." ], "refs": [ "artin-lemma-SGE-implies-openness-versality", "artin-remark-strong-effectiveness", "artin-remark-short-exact-sequence-thickenings", "artin-lemma-properties-lift-RS-star" ], "ref_ids": [ 11386, 11429, 11433, 11388 ] } ], "ref_ids": [ 11425 ] }, { "id": 11391, "type": "theorem", "label": "artin-lemma-compute-ext-into-field", "categories": [ "artin" ], "title": "artin-lemma-compute-ext-into-field", "contents": [ "Let $A \\to k$ be a ring map with $k$ a field. Let $E \\in D^-(A)$.", "Then $\\Ext^i_A(E, k) = \\Hom_k(H^{-i}(E \\otimes^\\mathbf{L} k), k)$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Replace $E$ by a bounded above complex of free $A$-modules", "and compute both sides." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11392, "type": "theorem", "label": "artin-lemma-construct-essential-surjection", "categories": [ "artin" ], "title": "artin-lemma-construct-essential-surjection", "contents": [ "Let $\\Lambda \\to A \\to k$ be finite type ring maps of Noetherian rings with", "$k = \\kappa(\\mathfrak p)$ for some prime $\\mathfrak p$ of $A$. Let", "$\\xi : E \\to \\NL_{A/\\Lambda}$ be morphism of $D^{-}(A)$ such that", "$H^{-1}(\\xi \\otimes^{\\mathbf{L}} k)$ is not surjective.", "Then there exists a surjection $A' \\to A$ of $\\Lambda$-algebras", "such that", "\\begin{enumerate}", "\\item[(a)] $I = \\Ker(A' \\to A)$ has square zero and is isomorphic to $k$", "as an $A$-module,", "\\item[(b)] $\\Omega_{A'/\\Lambda} \\otimes k = \\Omega_{A/\\Lambda} \\otimes k$, and", "\\item[(c)] $E \\to \\NL_{A/A'}$ is zero.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $f \\in A$, $f \\not \\in \\mathfrak p$. Suppose that $A'' \\to A_f$", "satisfies (a), (b), (c) for the induced map", "$E \\otimes_A A_f \\to \\NL_{A_f/\\Lambda}$, see", "Algebra, Lemma \\ref{algebra-lemma-localize-NL}.", "Then we can set $A' = A'' \\times_{A_f} A$ and get a solution.", "Namely, it is clear that $A' \\to A$ satisfies (a) because", "$\\Ker(A' \\to A) = \\Ker(A'' \\to A) = I$. Pick", "$f'' \\in A''$ lifting $f$. Then the localization of $A'$ at", "$(f'', f)$ is isomorphic to $A''$", "(for example by", "More on Algebra, Lemma \\ref{more-algebra-lemma-diagram-localize}).", "Thus (b) and (c) are clear for $A'$ too.", "In this way we see that we may replace $A$ by the localization", "$A_f$ (finitely many times).", "In particular (after such a replacement) we may assume that $\\mathfrak p$", "is a maximal ideal of $A$, see", "Morphisms, Lemma \\ref{morphisms-lemma-point-finite-type}.", "\\medskip\\noindent", "Choose a presentation $A = \\Lambda[x_1, \\ldots, x_n]/J$. Then", "$\\NL_{A/\\Lambda}$ is (canonically) homotopy equivalent to", "$$", "J/J^2", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} A\\text{d}x_i,", "$$", "see Algebra, Lemma \\ref{algebra-lemma-NL-homotopy}. After localizing", "if necessary (using Nakayama's lemma) we can choose generators", "$f_1, \\ldots, f_m$ of $J$ such that $f_j \\otimes 1$ form a basis for", "$J/J^2 \\otimes_A k$. Moreover, after renumbering, we can assume that the", "images of $\\text{d}f_1, \\ldots, \\text{d}f_r$ form a", "basis for the image of $J/J^2 \\otimes k \\to \\bigoplus k\\text{d}x_i$", "and that $\\text{d}f_{r + 1}, \\ldots, \\text{d}f_m$ map to zero in", "$\\bigoplus k\\text{d}x_i$. With these choices the space", "$$", "H^{-1}(\\NL_{A/\\Lambda} \\otimes^{\\mathbf{L}}_A k) =", "H^{-1}(\\NL_{A/\\Lambda} \\otimes_A k)", "$$", "has basis $f_{r + 1} \\otimes 1, \\ldots, f_m \\otimes 1$. Changing basis", "once again we may assume that the image of $H^{-1}(\\xi \\otimes^{\\mathbf{L}} k)$", "is contained in the $k$-span of", "$f_{r + 1} \\otimes 1, \\ldots, f_{m - 1} \\otimes 1$.", "Set", "$$", "A' = \\Lambda[x_1, \\ldots, x_n]/(f_1, \\ldots, f_{m - 1}, \\mathfrak pf_m)", "$$", "By construction $A' \\to A$ satisfies (a). Since $\\text{d}f_m$ maps", "to zero in $\\bigoplus k\\text{d}x_i$ we see that (b) holds. Finally, by", "construction the induced map $E \\to \\NL_{A/A'} = I[1]$ induces the zero map", "$H^{-1}(E \\otimes_A^\\mathbf{L} k) \\to I \\otimes_A k$. By", "Lemma \\ref{lemma-compute-ext-into-field}", "we see that the composition is zero." ], "refs": [ "algebra-lemma-localize-NL", "more-algebra-lemma-diagram-localize", "morphisms-lemma-point-finite-type", "algebra-lemma-NL-homotopy", "artin-lemma-compute-ext-into-field" ], "ref_ids": [ 1161, 9816, 5205, 1151, 11391 ] } ], "ref_ids": [] }, { "id": 11393, "type": "theorem", "label": "artin-lemma-characterize-versal", "categories": [ "artin" ], "title": "artin-lemma-characterize-versal", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$ satisfying (RS*).", "Let $U = \\Spec(A)$ be an", "affine scheme of finite type over $S$ which maps into an affine open", "$\\Spec(\\Lambda)$. Let $x$ be an object of $\\mathcal{X}$ over $U$.", "Let $\\xi : E \\to \\NL_{A/\\Lambda}$ be a morphism of $D^{-}(A)$. Assume", "\\begin{enumerate}", "\\item[(i)] for every deformation situation $(x, A' \\to A)$ we have:", "$x$ lifts to $\\Spec(A')$ if and only if", "$E \\to \\NL_{A/\\Lambda} \\to \\NL_{A/A'}$ is zero, and", "\\item[(ii)] there is an isomorphism of functors", "$T_x(-) \\to \\Ext^0_A(E, -)$", "such that $E \\to \\NL_{A/\\Lambda} \\to \\Omega^1_{A/\\Lambda}$", "corresponds to the canonical element (see", "Remark \\ref{remark-canonical-element}).", "\\end{enumerate}", "Let $u_0 \\in U$ be a finite type point with residue field", "$k = \\kappa(u_0)$. Consider the following statements", "\\begin{enumerate}", "\\item $x$ is versal at $u_0$, and", "\\item $\\xi : E \\to \\NL_{A/\\Lambda}$ induces a surjection", "$H^{-1}(E \\otimes_A^{\\mathbf{L}} k) \\to", "H^{-1}(\\NL_{A/\\Lambda} \\otimes_A^{\\mathbf{L}} k)$", "and an injection", "$H^0(E \\otimes_A^{\\mathbf{L}} k) \\to", "H^0(\\NL_{A/\\Lambda} \\otimes_A^{\\mathbf{L}} k)$.", "\\end{enumerate}", "Then we always have (2) $\\Rightarrow$ (1) and we have (1) $\\Rightarrow$ (2)", "if $u_0$ is a closed point." ], "refs": [ "artin-remark-canonical-element" ], "proofs": [ { "contents": [ "Let $\\mathfrak p = \\Ker(A \\to k)$ be the prime corresponding to $u_0$.", "\\medskip\\noindent", "Assume that $x$ versal at $u_0$ and that $u_0$ is a closed point of $U$.", "If $H^{-1}(\\xi \\otimes_A^{\\mathbf{L}} k)$ is not surjective, then", "let $A' \\to A$ be an extension with kernel $I$ as in", "Lemma \\ref{lemma-construct-essential-surjection}.", "Because $u_0$ is a closed point, we see that $I$ is a finite $A$-module,", "hence that $A'$ is a finite type $\\Lambda$-algebra (this fails if", "$u_0$ is not closed). In particular $A'$ is Noetherian.", "By property (c) for $A'$ and (i) for $\\xi$ we see that $x$ lifts to", "an object $x'$ over $A'$.", "Let $\\mathfrak p' \\subset A'$ be kernel of the surjective map to $k$.", "By Artin-Rees (Algebra, Lemma \\ref{algebra-lemma-Artin-Rees})", "there exists an $n > 1$ such that $(\\mathfrak p')^n \\cap I = 0$.", "Then we see that", "$$", "B' = A'/(\\mathfrak p')^n \\longrightarrow A/\\mathfrak p^n = B", "$$", "is a small, essential extension of local Artinian rings, see", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-essential-surjection}.", "On the other hand, as $x$ is versal at $u_0$ and as $x'|_{\\Spec(B')}$", "is a lift of $x|_{\\Spec(B)}$, there exists an integer", "$m \\geq n$ and a map $q : A/\\mathfrak p^m \\to B'$", "such that the composition", "$A/\\mathfrak p^m \\to B' \\to B$ is the quotient map.", "Since the maximal ideal of $B'$ has $n$th power equal to zero, this", "$q$ factors through $B$ which contradicts the fact that $B' \\to B$ is an", "essential surjection. This contradiction shows that", "$H^{-1}(\\xi \\otimes_A^{\\mathbf{L}} k)$", "is surjective.", "\\medskip\\noindent", "Assume that $x$ versal at $u_0$. By Lemma \\ref{lemma-compute-ext-into-field}", "the map $H^0(\\xi \\otimes_A^{\\mathbf{L}} k)$ is dual to the map", "$\\Ext^0_A(\\NL_{A/\\Lambda}, k) \\to \\text{Ext}^0_A(E, k)$. Note that", "$$", "\\Ext^0_A(\\NL_{A/\\Lambda}, k) = \\text{Der}_\\Lambda(A, k)", "\\quad\\text{and}\\quad", "T_x(k) = \\Ext^0_A(E, k)", "$$", "Condition (ii) assures us the map", "$\\Ext^0_A(\\NL_{A/\\Lambda}, k) \\to \\text{Ext}^0_A(E, k)$", "sends a tangent vector $\\theta$ to $U$ at $u_0$ to the corresponding", "infinitesimal deformation of $x_0$, see Remark \\ref{remark-canonical-element}.", "Hence if $x$ is versal, then this map is surjective, see", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-versal-criterion}.", "Hence $H^0(\\xi \\otimes_A^{\\mathbf{L}} k)$ is injective.", "This finishes the proof of (1) $\\Rightarrow$ (2) in case $u_0$ is a", "closed point.", "\\medskip\\noindent", "For the rest of the proof assume $H^{-1}(E \\otimes_A^\\mathbf{L} k) \\to", "H^{-1}(\\NL_{A/\\Lambda} \\otimes_A^\\mathbf{L} k)$", "is surjective and", "$H^0(E \\otimes_A^\\mathbf{L} k) \\to", "H^0(\\NL_{A/\\Lambda} \\otimes_A^\\mathbf{L} k)$", "injective. Set $R = A_\\mathfrak p^\\wedge$ and let $\\eta$ be the", "formal object over $R$ associated to $x|_{\\Spec(R)}$.", "The map $d\\underline{\\eta}$ on tangent spaces is surjective", "because it is identified with the dual of the injective map", "$H^0(E \\otimes_A^{\\mathbf{L}} k) \\to", "H^0(\\NL_{A/\\Lambda} \\otimes_A^{\\mathbf{L}} k)$", "(see previous paragraph). According to", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-versal-criterion}", "it suffices to prove the following:", "Let $C' \\to C$ be a small extension of finite type Artinian local", "$\\Lambda$-algebras with residue field $k$. Let $R \\to C$ be a", "$\\Lambda$-algebra map compatible with identifications of residue fields.", "Let $y = x|_{\\Spec(C)}$ and let $y'$ be a lift of $y$ to $C'$.", "To show: we can lift the $\\Lambda$-algebra map $R \\to C$ to $R \\to C'$.", "\\medskip\\noindent", "Observe that it suffices to lift the $\\Lambda$-algebra map $A \\to C$.", "Let $I = \\Ker(C' \\to C)$. Note that $I$ is a $1$-dimensional $k$-vector", "space. The obstruction $ob$ to lifting $A \\to C$ is an element of", "$\\Ext^1_A(\\NL_{A/\\Lambda}, I)$, see Example \\ref{example-key}.", "By Lemma \\ref{lemma-compute-ext-into-field} and our assumption the map", "$\\xi$ induces an injection", "$$", "\\Ext^1_A(\\NL_{A/\\Lambda}, I)", "\\longrightarrow", "\\Ext^1_A(E, I)", "$$", "By the construction of $ob$ and (i) the image of $ob$ in $\\Ext^1_A(E, I)$", "is the obstruction to lifting $x$ to $A \\times_C C'$. By (RS*) the fact that", "$y/C$ lifts to $y'/C'$ implies that $x$ lifts to $A \\times_C C'$. Hence", "$ob = 0$ and we are done." ], "refs": [ "artin-lemma-construct-essential-surjection", "algebra-lemma-Artin-Rees", "formal-defos-lemma-essential-surjection", "artin-lemma-compute-ext-into-field", "artin-remark-canonical-element", "formal-defos-lemma-versal-criterion", "formal-defos-lemma-versal-criterion", "artin-lemma-compute-ext-into-field" ], "ref_ids": [ 11392, 625, 3420, 11391, 11435, 3456, 3456, 11391 ] } ], "ref_ids": [ 11435 ] }, { "id": 11394, "type": "theorem", "label": "artin-lemma-openness", "categories": [ "artin" ], "title": "artin-lemma-openness", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$ satisfying (RS*).", "Let $U = \\Spec(A)$ be an affine scheme of finite type over $S$ which maps", "into an affine open $\\Spec(\\Lambda)$. Let $x$ be an object of $\\mathcal{X}$", "over $U$. Let $\\xi : E \\to \\NL_{A/\\Lambda}$ be a morphism of $D^{-}(A)$.", "Assume", "\\begin{enumerate}", "\\item[(i)] for every deformation situation $(x, A' \\to A)$ we have:", "$x$ lifts to $\\Spec(A')$ if and only if", "$E \\to \\NL_{A/\\Lambda} \\to \\NL_{A/A'}$ is zero,", "\\item[(ii)] there is an isomorphism of functors", "$T_x(-) \\to \\Ext^0_A(E, -)$", "such that $E \\to \\NL_{A/\\Lambda} \\to \\Omega^1_{A/\\Lambda}$", "corresponds to the canonical element (see", "Remark \\ref{remark-canonical-element}),", "\\item[(iii)] the cohomology groups of $E$ are finite $A$-modules.", "\\end{enumerate}", "If $x$ is versal at a closed point $u_0 \\in U$,", "then there exists an open neighbourhood $u_0 \\in U' \\subset U$", "such that $x$ is versal at every finite type point of $U'$." ], "refs": [ "artin-remark-canonical-element" ], "proofs": [ { "contents": [ "Let $C$ be the cone of $\\xi$ so that we have a distinguished triangle", "$$", "E \\to \\NL_{A/\\Lambda} \\to C \\to E[1]", "$$", "in $D^{-}(A)$. By Lemma \\ref{lemma-characterize-versal}", "the assumption that $x$ is versal at $u_0$ implies that", "$H^{-1}(C \\otimes^\\mathbf{L} k) = 0$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-cut-complex-in-two}", "there exists an $f \\in A$ not contained in the prime corresponding to", "$u_0$ such that $H^{-1}(C \\otimes^\\mathbf{L}_A M) = 0$ for", "any $A_f$-module $M$. Using", "Lemma \\ref{lemma-characterize-versal}", "again we see that we have versality for all finite type points of", "the open $D(f) \\subset U$." ], "refs": [ "artin-lemma-characterize-versal", "more-algebra-lemma-cut-complex-in-two", "artin-lemma-characterize-versal" ], "ref_ids": [ 11393, 10238, 11393 ] } ], "ref_ids": [ 11435 ] }, { "id": 11395, "type": "theorem", "label": "artin-lemma-naive-obstruction-theory-qis", "categories": [ "artin" ], "title": "artin-lemma-naive-obstruction-theory-qis", "contents": [ "Let $S$ and $\\mathcal{X}$ be as in", "Definition \\ref{definition-naive-obstruction-theory}", "and let $\\mathcal{X}$ be endowed with a naive obstruction theory.", "Let $A \\to B$ and $y \\to x$ be as in (\\ref{item-functoriality}).", "Let $k$ be a $B$-algebra which is a field. Then the functoriality", "map $E_x \\to E_y$ induces bijections", "$$", "H^i(E_x \\otimes_A^{\\mathbf{L}} k) \\to H^i(E_y \\otimes_B^{\\mathbf{L}} k)", "$$", "for $i = 0, 1$." ], "refs": [ "artin-definition-naive-obstruction-theory" ], "proofs": [ { "contents": [ "Let $z = x|_{\\Spec(k)}$. Then (RS*) implies that", "$$", "\\textit{Lift}(x, A[k]) = \\textit{Lift}(z, k[k])", "\\quad\\text{and}\\quad", "\\textit{Lift}(y, B[k]) = \\textit{Lift}(z, k[k])", "$$", "because $A[k] = A \\times_k k[k]$ and $B[k] = B \\times_k k[k]$.", "Hence the properties of a naive obstruction theory imply that the", "functoriality map $E_x \\to E_y$ induces bijections", "$\\Ext^i_A(E_x, k) \\to \\text{Ext}^i_B(E_y, k)$", "for $i = -1, 0$. By Lemma \\ref{lemma-compute-ext-into-field} our maps", "$H^i(E_x \\otimes_A^{\\mathbf{L}} k) \\to H^i(E_y \\otimes_B^{\\mathbf{L}} k)$,", "$i = 0, 1$ induce isomorphisms on dual vector spaces hence are isomorphisms." ], "refs": [ "artin-lemma-compute-ext-into-field" ], "ref_ids": [ 11391 ] } ], "ref_ids": [ 11426 ] }, { "id": 11396, "type": "theorem", "label": "artin-lemma-naive-obstruction-theory-gives-openness", "categories": [ "artin" ], "title": "artin-lemma-naive-obstruction-theory-gives-openness", "contents": [ "Let $S$ be a locally Noetherian scheme. Let", "$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}^{opp}$ be a category fibred in groupoids.", "Assume that $\\mathcal{X}$ satisfies (RS*)", "and that $\\mathcal{X}$ has a naive obstruction theory.", "Then openness of versality holds for $\\mathcal{X}$ provided the", "complexes $E_x$ of Definition \\ref{definition-naive-obstruction-theory}", "have finitely generated cohomology groups for pairs $(A, x)$ where", "$A$ is of finite type over $S$." ], "refs": [ "artin-definition-naive-obstruction-theory" ], "proofs": [ { "contents": [ "Let $U$ be a scheme locally of finite type over $S$, let $x$ be an object of", "$\\mathcal{X}$ over $U$, and let $u_0$ be a finite type point of $U$ such that", "$x$ is versal at $u_0$. We may first shrink $U$ to an affine scheme such", "that $u_0$ is a closed point and such that $U \\to S$ maps into an affine", "open $\\Spec(\\Lambda)$. Say $U = \\Spec(A)$. Let", "$\\xi_x : E_x \\to \\NL_{A/\\Lambda}$ be the obstruction map.", "At this point we may apply Lemma \\ref{lemma-openness} to conclude." ], "refs": [ "artin-lemma-openness" ], "ref_ids": [ 11394 ] } ], "ref_ids": [ 11426 ] }, { "id": 11397, "type": "theorem", "label": "artin-lemma-dual-obstruction", "categories": [ "artin" ], "title": "artin-lemma-dual-obstruction", "contents": [ "In Situation \\ref{situation-dual}. Assume furthermore that", "\\begin{enumerate}", "\\item[(iv)] given a short exact sequence of deformation situations", "as in Remark \\ref{remark-short-exact-sequence-thickenings} and", "a lift $x'_2 \\in \\text{Lift}(x, A_2')$ then", "$o_x(A_3') \\in H^2(K^\\bullet \\otimes_A^\\mathbf{L} I_3)$", "equals $\\partial\\theta$ where", "$\\theta \\in H^1(K^\\bullet \\otimes_A^\\mathbf{L} I_1)$", "is the element corresponding to $x'_2|_{\\Spec(A_1')}$ via", "$A_1' = A[I_1]$ and the given map", "$T_x(-) \\to H^1(K^\\bullet \\otimes_A^\\mathbf{L} -)$.", "\\end{enumerate}", "In this case there exists an element", "$\\xi \\in H^1(K^\\bullet \\otimes_A^\\mathbf{L} \\NL_{A/\\Lambda})$", "such that", "\\begin{enumerate}", "\\item for every deformation situation $(x, A' \\to A)$ we have", "$\\xi_{A'} = o_x(A')$, and", "\\item $\\xi_{can}$ matches the canonical element of", "Remark \\ref{remark-canonical-element} via the given transformation", "$T_x(-) \\to H^1(K^\\bullet \\otimes_A^\\mathbf{L} -)$.", "\\end{enumerate}" ], "refs": [ "artin-remark-short-exact-sequence-thickenings", "artin-remark-canonical-element" ], "proofs": [ { "contents": [ "Choose a $\\alpha : \\Lambda[x_1, \\ldots, x_n] \\to A$ with kernel $J$.", "Write $P = \\Lambda[x_1, \\ldots, x_n]$. In the rest of this proof we work with", "$$", "\\NL(\\alpha) = (J/J^2 \\longrightarrow \\bigoplus A \\text{d}x_i)", "$$", "which is permissible by", "Algebra, Lemma \\ref{algebra-lemma-NL-homotopy}", "and", "More on Algebra, Lemma \\ref{more-algebra-lemma-derived-tor-homotopy}.", "Consider the element", "$o_x(P/J^2) \\in H^2(K^\\bullet \\otimes_A^\\mathbf{L} J/J^2)$ and consider", "the quotient", "$$", "C = (P/J^2 \\times \\bigoplus A \\text{d}x_i)/(J/J^2)", "$$", "where $J/J^2$ is embedded diagonally. Note that $C \\to A$ is a surjection", "with kernel $\\bigoplus A\\text{d}x_i$. Moreover there is a section", "$A \\to C$ to $C \\to A$ given by mapping the class of $f \\in P$ to the class", "of $(f, \\text{d}f)$ in the pushout. For later use, denote $x_C$ the", "pullback of $x$ along the corresponding morphism $\\Spec(C) \\to \\Spec(A)$.", "Thus we see that $o_x(C) = 0$.", "We conclude that $o_x(P/J^2)$ maps to zero in", "$H^2(K^\\bullet \\otimes_A^\\mathbf{L} \\bigoplus A\\text{d}x_i)$.", "It follows that there exists some element", "$\\xi \\in H^1(K^\\bullet \\otimes_A^\\mathbf{L} \\NL(\\alpha))$", "mapping to $o_x(P/J^2)$.", "\\medskip\\noindent", "Note that for any deformation situation $(x, A' \\to A)$ there exists", "a $\\Lambda$-algebra map $P/J^2 \\to A'$ compatible with the augmentations", "to $A$. Hence the", "element $\\xi$ satisfies the first property of the lemma by construction", "and property (ii) of Situation \\ref{situation-dual}.", "\\medskip\\noindent", "Note that our choice of $\\xi$ was well defined up to the choice of an", "element of $H^1(K^\\bullet \\otimes_A^\\mathbf{L} \\bigoplus A\\text{d}x_i)$.", "We will show that after modifying $\\xi$ by an element of the aforementioned", "group we can arrange it so that the second assertion of the lemma is true.", "Let $C' \\subset C$ be the image of $P/J^2$ under the", "$\\Lambda$-algebra map $P/J^2 \\to C$ (inclusion of first factor).", "Observe that", "$\\Ker(C' \\to A) = \\Im(J/J^2 \\to \\bigoplus A\\text{d}x_i)$.", "Set $\\overline{C} = A[\\Omega_{A/\\Lambda}]$. The map", "$P/J^2 \\times \\bigoplus A \\text{d}x_i \\to \\overline{C}$,", "$(f, \\sum f_i \\text{d}x_i) \\mapsto (f \\bmod J, \\sum f_i \\text{d}x_i)$", "factors through a surjective map $C \\to \\overline{C}$. Then", "$$", "(x, \\overline{C} \\to A) \\to (x, C \\to A) \\to (x, C' \\to A)", "$$", "is a short exact sequence of deformation situations. The", "associated splitting $\\overline{C} = A[\\Omega_{A/\\Lambda}]$ (from", "Remark \\ref{remark-short-exact-sequence-thickenings}) equals the given", "splitting above. Moreover, the section $A \\to C$ composed with the map", "$C \\to \\overline{C}$", "is the map $(1, \\text{d}) : A \\to A[\\Omega_{A/\\Lambda}]$ of", "Remark \\ref{remark-canonical-element}.", "Thus $x_C$ restricts to the canonical element $x_{can}$ of", "$T_x(\\Omega_{A/\\Lambda}) = \\text{Lift}(x, A[\\Omega_{A/\\Lambda}])$.", "By condition (iv) we conclude that $o_x(P/J^2)$ maps to $\\partial x_{can}$", "in", "$$", "H^1(K^\\bullet \\otimes_A^\\mathbf{L} \\Im(J/J^2 \\to \\bigoplus A\\text{d}x_i))", "$$", "By construction $\\xi$ maps to $o_x(P/J^2)$. It follows that", "$x_{can}$ and $\\xi_{can}$ map to the same element in the", "displayed group which means (by the long exact cohomology sequence)", "that they differ by an element of", "$H^1(K^\\bullet \\otimes_A^\\mathbf{L} \\bigoplus A\\text{d}x_i)$", "as desired." ], "refs": [ "algebra-lemma-NL-homotopy", "more-algebra-lemma-derived-tor-homotopy", "artin-remark-short-exact-sequence-thickenings", "artin-remark-canonical-element" ], "ref_ids": [ 1151, 10121, 11433, 11435 ] } ], "ref_ids": [ 11433, 11435 ] }, { "id": 11398, "type": "theorem", "label": "artin-lemma-dual-openness", "categories": [ "artin" ], "title": "artin-lemma-dual-openness", "contents": [ "In Situation \\ref{situation-dual} assume that (iv) of", "Lemma \\ref{lemma-dual-obstruction} holds and that $K^\\bullet$ is a", "perfect object of $D(A)$. In this case, if $x$ is versal at a closed", "point $u_0 \\in U$ then there exists an open neighbourhood", "$u_0 \\in U' \\subset U$ such that $x$ is versal at every finite type", "point of $U'$." ], "refs": [ "artin-lemma-dual-obstruction" ], "proofs": [ { "contents": [ "We may assume that $K^\\bullet$ is a finite complex of finite projective", "$A$-modules. Thus the derived tensor product with $K^\\bullet$ is the", "same as simply tensoring with $K^\\bullet$. Let", "$E^\\bullet$ be the dual perfect complex to $K^\\bullet$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-dual-perfect-complex}.", "(So $E^n = \\Hom_A(K^{-n}, A)$ with differentials the transpose of the", "differentials of $K^\\bullet$.) Let $E \\in D^{-}(A)$ denote the", "object represented by the complex $E^\\bullet[-1]$.", "Let $\\xi \\in H^1(\\text{Tot}(K^\\bullet \\otimes_A \\NL_{A/\\Lambda}))$", "be the element constructed in Lemma \\ref{lemma-dual-obstruction}", "and denote $\\xi : E = E^\\bullet[-1] \\to \\NL_{A/\\Lambda}$ the corresponding", "map (loc.cit.). We claim that the pair $(E, \\xi)$ satisfies all the", "assumptions of Lemma \\ref{lemma-openness} which finishes the proof.", "\\medskip\\noindent", "Namely, assumption (i) of Lemma \\ref{lemma-openness} follows from conclusion", "(1) of Lemma \\ref{lemma-dual-obstruction}", "and the fact that $H^2(K^\\bullet \\otimes_A^\\mathbf{L} -) =", "\\Ext^1(E, -)$ by loc.cit. Assumption (ii) of", "Lemma \\ref{lemma-openness} follows from conclusion (2) of", "Lemma \\ref{lemma-dual-obstruction}", "and the fact that $H^1(K^\\bullet \\otimes_A^\\mathbf{L} -) =", "\\Ext^0(E, -)$ by loc.cit. Assumption (iii) of Lemma \\ref{lemma-openness}", "is clear." ], "refs": [ "more-algebra-lemma-dual-perfect-complex", "artin-lemma-dual-obstruction", "artin-lemma-openness", "artin-lemma-openness", "artin-lemma-dual-obstruction", "artin-lemma-openness", "artin-lemma-dual-obstruction", "artin-lemma-openness" ], "ref_ids": [ 10224, 11397, 11394, 11394, 11397, 11394, 11397, 11394 ] } ], "ref_ids": [ 11397 ] }, { "id": 11399, "type": "theorem", "label": "artin-lemma-canonical-extension", "categories": [ "artin" ], "title": "artin-lemma-canonical-extension", "contents": [ "Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.", "Restricting along the inclusion functor", "$(\\textit{Noetherian}/S)_\\tau \\to (\\Sch/S)_\\tau$", "defines an equivalence of categories between", "\\begin{enumerate}", "\\item the category of limit preserving sheaves on", "$(\\Sch/S)_\\tau$ and", "\\item the category of limit preserving sheaves on", "$(\\textit{Noetherian}/S)_\\tau$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $F : (\\textit{Noetherian}/S)_\\tau^{opp} \\to \\textit{Sets}$", "be a functor which is both limit preserving and a sheaf.", "By Topologies, Lemmas", "\\ref{topologies-lemma-extend} and \\ref{topologies-lemma-extend-sheaf-general}", "there exists a unique functor", "$F' : (\\Sch/S)_\\tau^{opp} \\to \\textit{Sets}$", "which is limit preserving, a sheaf, and restricts to $F$.", "In fact, the construction of $F'$ in", "Topologies, Lemma \\ref{topologies-lemma-extend}", "is functorial in $F$ and this construction is a quasi-inverse", "to restriction. Some details omitted." ], "refs": [ "topologies-lemma-extend", "topologies-lemma-extend-sheaf-general", "topologies-lemma-extend" ], "ref_ids": [ 12517, 12519, 12517 ] } ], "ref_ids": [] }, { "id": 11400, "type": "theorem", "label": "artin-lemma-representable-limit-preserving", "categories": [ "artin" ], "title": "artin-lemma-representable-limit-preserving", "contents": [ "Let $X$ be an object of $(\\textit{Noetherian}/S)_\\tau$. If the functor", "of points $h_X : (\\textit{Noetherian}/S)_\\tau^{opp} \\to \\textit{Sets}$", "is limit preserving, then $X$ is locally of finite presentation over $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset X$ be an affine open subscheme which maps into an affine", "open $U \\subset S$. We may write $V = \\lim V_i$ as a directed limit of affine", "schemes $V_i$ of finite presentation over $U$, see", "Algebra, Lemma \\ref{algebra-lemma-ring-colimit-fp}.", "By assumption, the arrow $V \\to X$ factors as $V \\to V_i \\to X$", "for some $i$. After increasing $i$ we may assume $V_i \\to X$", "factors through $V$ as the inverse image of $V \\subset X$ in $V_i$", "eventually becomes equal to $V_i$ by", "Limits, Lemma \\ref{limits-lemma-descend-opens}.", "Then the identity morphism $V \\to V$ factors through $V_i$ for some $i$", "in the category of schemes over $U$. Thus $V \\to U$ is of finite presentation;", "the corresponding algebra fact is that if $B$ is an $A$-algebra", "such that $\\text{id} : B \\to B$ factors through a finitely presented", "$A$-algebra, then $B$ is of finite presentation over $A$ (nice exercise).", "Hence $X$ is locally of finite presentation over $S$." ], "refs": [ "algebra-lemma-ring-colimit-fp", "limits-lemma-descend-opens" ], "ref_ids": [ 1091, 15041 ] } ], "ref_ids": [] }, { "id": 11401, "type": "theorem", "label": "artin-lemma-representable", "categories": [ "artin" ], "title": "artin-lemma-representable", "contents": [ "Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.", "Let $F', G' : (\\Sch/S)_\\tau^{opp} \\to \\textit{Sets}$ be limit preserving", "and sheaves. Let $a' : F' \\to G'$ be a transformation of functors.", "Denote $a : F \\to G$ the restriction of $a' : F' \\to G'$ to", "$(\\textit{Noetherian}/S)_\\tau$. The following are equivalent", "\\begin{enumerate}", "\\item $a'$ is representable (as a transformation of functors, see", "Categories, Definition \\ref{categories-definition-representable-morphism}), and", "\\item for every object $V$ of $(\\textit{Noetherian}/S)_\\tau$", "and every map $V \\to G$ the fibre product", "$F \\times_G V : (\\textit{Noetherian}/S)_\\tau^{opp} \\to \\textit{Sets}$", "is a representable functor, and", "\\item same as in (2) but only for $V$ affine finite type over $S$", "mapping into an affine open of $S$.", "\\end{enumerate}" ], "refs": [ "categories-definition-representable-morphism" ], "proofs": [ { "contents": [ "Assume (1). By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-locally-finite-presentation-permanence}", "the transformation $a'$ is limit preserving\\footnote{This", "makes sense even if $\\tau \\not = fppf$ as the underlying", "category of $(\\Sch/S)_\\tau$ equals the underlying category", "of $(\\Sch/S)_{fppf}$ and the statement doesn't refer to the topology.}.", "Take $\\xi : V \\to G$ as in (2). Denote $V' = V$ but viewed as an", "object of $(\\Sch/S)_\\tau$. Since $G$ is the restriction of", "$G'$ to $(\\textit{Noetherian}/S)_\\tau$ we see that", "$\\xi \\in G(V)$ corresponds to $\\xi' \\in G'(V')$.", "By assumption $V' \\times_{\\xi', G'} F'$ is representable", "by a scheme $U'$. The morphism of schemes $U' \\to V'$ corresponding to", "the projection $V' \\times_{\\xi', G'} F' \\to V'$ is locally of finite", "presentation by", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-base-change-locally-finite-presentation} and", "Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}.", "Hence $U'$ is a locally Noetherian scheme and therefore $U'$ is", "isomorphic to an object $U$ of $(\\textit{Noetherian}/S)_\\tau$.", "Then $U$ represents $F \\times_G V$ as desired.", "\\medskip\\noindent", "The implication (2) $\\Rightarrow$ (3) is immediate. Assume (3).", "We will prove (1). Let $T$ be an object of $(\\Sch/S)_\\tau$", "and let $T \\to G'$ be a morphism. We have to show", "the functor $F' \\times_{G'} T$ is representable by a scheme $X$", "over $T$. Let $\\mathcal{B}$ be the set of affine opens of", "$T$ which map into an affine open of $S$. This is a basis", "for the topology of $T$. Below we will show that for $W \\in \\mathcal{B}$", "the fibre product $F' \\times_{G'} W$ is representable", "by a scheme $X_W$ over $W$. If $W_1 \\subset W_2$ in $\\mathcal{B}$, then", "we obtain an isomorphism $X_{W_1} \\to X_{W_2} \\times_{W_2} W_1$ because both", "$X_{W_1}$ and $X_{W_2} \\times_{W_2} W_1$ represent the functor", "$F' \\times_{G'} W_1$.", "These isomorphisms are canonical and satisfy the cocycle condition", "mentioned in Constructions, Lemma", "\\ref{constructions-lemma-relative-glueing}.", "Hence we can glue the schemes $X_W$ to a scheme $X$ over $T$.", "Compatibility of the glueing maps with the maps", "$X_W \\to F'$ provide us with a map $X \\to F'$.", "The resulting map $X \\to F' \\times_{G'} T$ is an", "isomorphism as we may check this locally on $T$ (as source", "and target of this arrow are sheaves for the Zariski topology).", "\\medskip\\noindent", "Let $W$ be an affine scheme which maps into an affine open $U \\subset S$.", "Let $W \\to G'$ be a map.", "Still assuming (3) we have to show that $F' \\times_{G'} W$", "is representable by a scheme.", "We may write $W = \\lim V'_i$ as a directed limit of affine", "schemes $V'_i$ of finite presentation over $U$, see", "Algebra, Lemma \\ref{algebra-lemma-ring-colimit-fp}.", "Since $V'_i$ is of finite type over an Noetherian scheme,", "we see that $V'_i$ is a Noetherian scheme.", "Denote $V_i = V'_i$ but viewed as an object of", "$(\\textit{Noetherian}/S)_\\tau$. As $G'$", "is limit preserving can choose an $i$ and a map", "$V'_i \\to G'$ such that $W \\to G'$ is the composition", "$W \\to V'_i \\to G'$. Since $G$ is the restriction of $G'$", "to $(\\textit{Noetherian}/S)_\\tau$ the morphism $V'_i \\to G'$", "is the same thing as a morphism $V_i \\to G$ (see above).", "By assumption (3) the functor $F \\times_G V_i$ is representable by an object", "$X_i$ of $(\\textit{Noetherian}/S)_\\tau$.", "The functor $F \\times_G V_i$ is limit preserving", "as it is the restriction of $F' \\times_{G'} V'_i$", "and this functor is limit preserving by", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-fibre-product-locally-finite-presentation},", "the assumption that $F'$ and $G'$ are limit preserving, and", "Limits, Remark \\ref{limits-remark-limit-preserving} which", "tells us that the functor of points of $V'_i$ is limit preserving.", "By Lemma \\ref{lemma-representable-limit-preserving}", "we conclude that $X_i$ is locally of finite presentation over $S$.", "Denote $X'_i = X_i$ but viewed as an object of", "$(\\Sch/S)_\\tau$. Then we see that $F' \\times_{G'} V'_i$", "and the functors of points $h_{X'_i}$ are both extensions", "of $h_{X_i} : (\\textit{Noetherian}/S)_\\tau^{opp} \\to \\textit{Sets}$", "to limit preserving sheaves on $(\\Sch/S)_\\tau$.", "By the equivalence of categories of Lemma \\ref{lemma-canonical-extension}", "we deduce that $X'_i$ represents $F' \\times_{G'} V'_i$.", "Then finally", "$$", "F' \\times_{G'} W = F' \\times_{G'} V'_i \\times_{V'_i} W =", "X'_i \\times_{V'_i} W", "$$", "is representable as desired." ], "refs": [ "spaces-limits-lemma-locally-finite-presentation-permanence", "spaces-limits-lemma-base-change-locally-finite-presentation", "limits-proposition-characterize-locally-finite-presentation", "constructions-lemma-relative-glueing", "algebra-lemma-ring-colimit-fp", "spaces-limits-lemma-fibre-product-locally-finite-presentation", "limits-remark-limit-preserving", "artin-lemma-representable-limit-preserving", "artin-lemma-canonical-extension" ], "ref_ids": [ 4558, 4559, 15127, 12581, 1091, 4560, 15130, 11400, 11399 ] } ], "ref_ids": [ 12348 ] }, { "id": 11402, "type": "theorem", "label": "artin-lemma-functor", "categories": [ "artin" ], "title": "artin-lemma-functor", "contents": [ "In Situation \\ref{situation-contractions} the rule $F$ that sends", "a locally Noetherian scheme $V$ over $S$ to the set of triples", "$(Z, u', \\hat x)$ satisfying the compatibility condition and which sends a", "a morphism $\\varphi : V_2 \\to V_1$ of locally Noetherian schemes over $S$", "to the map", "$$", "F(\\varphi) : F(V_1) \\longrightarrow F(V_2)", "$$", "sending an element $(Z_1, u'_1, \\hat x_1)$ of $F(V_1)$ to", "$(Z_2, u'_2, \\hat x_2)$ in $F(V_2)$ given by", "\\begin{enumerate}", "\\item $Z_2 \\subset V_2$ is the inverse image of $Z_1$ by $\\varphi$,", "\\item $u'_2$ is the composition of $u'_1$ and", "$\\varphi|_{V_2 \\setminus Z_2} : V_2 \\setminus Z_2 \\to V_1 \\setminus Z_1$,", "\\item $\\hat x_2$ is the composition of $\\hat x_1$ and", "$\\varphi_{/Z_2} : V_{2, /Z_2} \\to V_{1, /Z_1}$", "\\end{enumerate}", "is a contravariant functor." ], "refs": [], "proofs": [ { "contents": [ "To see the compatibility condition between $u'_2$ and $\\hat x_2$, let", "$V'_1 \\to V_1$, $\\hat x'_1$, and $x'_1$ witness the compatibility between", "$u'_1$ and $\\hat x_1$. Set $V'_2 = V_2 \\times_{V_1} V'_1$, set", "$\\hat x'_2$ equal to the composition of $\\hat x'_1$ and", "$V'_{2, /Z_2} \\to V'_{1, /Z_1}$, and set $x'_2$", "equal to the composition of $x'_1$ and $V'_2 \\to V'_1$.", "Then $V'_2 \\to V_2$, $\\hat x'_2$, and $x'_2$ witness the compatibility between", "$u'_2$ and $\\hat x_2$. We omit the detailed verification." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11403, "type": "theorem", "label": "artin-lemma-solution", "categories": [ "artin" ], "title": "artin-lemma-solution", "contents": [ "In Situation \\ref{situation-contractions} if there exists a solution", "$(f : X' \\to X, T, a)$ then there is a functorial bijection", "$F(V) = \\Mor_S(V, X)$ on the category of", "locally Noetherian schemes $V$ over $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $V$ be a locally Noetherian scheme over $S$.", "Let $x : V \\to X$ be a morphism over $S$.", "Then we get an element $(Z, u', \\hat x)$ in $F(V)$ as follows", "\\begin{enumerate}", "\\item $Z \\subset V$ is the inverse image of $T$ by $x$,", "\\item $u' : V \\setminus Z \\to U' = U$ is the restriction of", "$x$ to $V \\setminus Z$,", "\\item $\\hat x : V_{/Z} \\to W$ is the composition of", "$x_{/Z} : V_{/Z} \\to X_{/T}$ with the isomorphism $a : X_{/T} \\to W$.", "\\end{enumerate}", "This triple satisfies the compatibility condition because we", "can take $V' = V \\times_{x, X} X'$, we can take $\\hat x'$", "the completion of the projection $x' : V' \\to X'$.", "\\medskip\\noindent", "Conversely, suppose given an element $(Z, u', \\hat x)$ of $F(V)$. We claim", "there is a unique morphism $x : V \\to X$ compatible with $u'$ and $\\hat x$.", "Namely, let $V' \\to V$, $\\hat x'$, and $x'$ witness the", "compatibility between $u'$ and $\\hat x$. Then", "Algebraization of Formal Spaces, Proposition", "\\ref{restricted-proposition-glue-modification}", "is exactly the result we need to find", "a unique morphism $x : V \\to X$ agreeing with", "$\\hat x$ over $V_{/Z}$ and with $x'$ over $V'$ (and a fortiori", "agreeing with $u'$ over $V \\setminus Z$).", "\\medskip\\noindent", "We omit the verification that the two constructions above define inverse", "bijections between their respective domains." ], "refs": [ "restricted-proposition-glue-modification" ], "ref_ids": [ 2433 ] } ], "ref_ids": [] }, { "id": 11404, "type": "theorem", "label": "artin-lemma-functor-is-solution", "categories": [ "artin" ], "title": "artin-lemma-functor-is-solution", "contents": [ "In Situation \\ref{situation-contractions} if there exists an", "algebraic space $X$ locally of finite type over $S$ and a", "functorial bijection $F(V) = \\Mor_S(V, X)$ on the category of", "locally Noetherian schemes $V$ over $S$, then $X$ is a solution." ], "refs": [], "proofs": [ { "contents": [ "We have to construct a proper morphism $f : X' \\to X$, a closed subset", "$T \\subset |X|$, and an isomorphism $a : X_{/T} \\to W$ with properties", "(1), (2), (3) listed just below Situation \\ref{situation-contractions}.", "\\medskip\\noindent", "The discussion in this proof is a bit pedantic because we want", "to carefully match the underlying categories. In this paragraph", "we explain how the adventurous reader can proceed less timidly.", "Namely, the reader may extend our definition of the functor $F$", "to all locally Noetherian algebraic spaces over $S$.", "Doing so the reader may then conclude that $F$ and $X$ agree as", "functors on the category of these algebraic spaces, i.e.,", "$X$ represents $F$. Then one considers the universal object", "$(T, u', \\hat x)$ in $F(X)$. Then the reader will find that", "for the triple $X'' \\to X$, $\\hat x'$, $x'$ witnessing the compatibility", "between $u'$ and $\\hat x$ the morphism $x' : X'' \\to X'$ is an isomorphism", "and this will produce $f : X' \\to X$ by inverting $x'$. Finally, we already", "have $T \\subset |X|$ and the reader may show that $\\hat x$ is an isomorphism", "which can served as the last ingredient namely $a$.", "\\medskip\\noindent", "Denote $h_X(-) = \\Mor_S(-, X)$ the functor of", "points of $X$ restricted to the category", "$(\\textit{Noetherian}/S)_\\etale$ of Section \\ref{section-noetherian}.", "By Limits of Spaces, Remark \\ref{spaces-limits-remark-limit-preserving}", "the algebraic spaces $X$ and $X'$ are limit preserving. Hence so", "are the restrictions $h_X$ and $h_{X'}$.", "To construct $f$ it therefore suffices to construct a", "transformation $h_{X'} \\to h_X = F$, see Lemma \\ref{lemma-canonical-extension}.", "Thus let $V \\to S$ be an object of $(\\textit{Noetherian}/S)_\\etale$", "and let $\\tilde x : V \\to X'$ be in $h_{X'}(V)$.", "Then we get an element $(Z, u', \\hat x)$ in $F(V)$ as follows", "\\begin{enumerate}", "\\item $Z \\subset V$ is the inverse image of $T'$ by $\\tilde x$,", "\\item $u' : V \\setminus Z \\to U'$ is the restriction of", "$\\tilde x$ to $V \\setminus Z$,", "\\item $\\hat x : V_{/Z} \\to W$ is the composition of", "$x_{/Z} : V_{/Z} \\to X'_{/T'}$ with $g : X'_{/T'} \\to W$.", "\\end{enumerate}", "This triple satisfies the compatibility condition: first we always", "obtain $V' \\to V$ and $\\hat x' : V'_{/Z'} \\to X'_{/T'}$ for free", "(see discussion preceding Lemma \\ref{lemma-functor}).", "Then we just define $x' : V' \\to X'$ to be the composition", "of $V' \\to V$ and the morphism $\\tilde x : V \\to X'$.", "We omit the verification that this works.", "\\medskip\\noindent", "If $\\xi : V \\to X$ is an \\'etale morphism where $V$ is a scheme, then", "we obtain $\\xi = (Z, u', \\hat x) \\in F(V) = h_X(V) = X(V)$.", "Of course, if $\\varphi : V' \\to V$ is a further \\'etale morphism of schemes,", "then $(Z, u', \\hat x)$ pulled back to $F(V')$ corresponds to", "$\\xi \\circ \\varphi$.", "The closed subset $T \\subset |X|$ is just defined as the closed", "subset such that $\\xi : V \\to X$ for $\\xi = (Z, u', \\hat x)$", "pulls $T$ back to $Z$", "\\medskip\\noindent", "Consider Noetherian schemes $V$ over $S$ and a morphism", "$\\xi : V \\to X$ corresponding to $(Z, u', \\hat x)$ as above.", "Then we see that $\\xi(V)$ is set theoretically contained in $T$", "if and only if $V = Z$ (as topological spaces). Hence we see that", "$X_{/T}$ agrees with $W$ as a functor. This produces the isomorphism", "$a : X_{/T} \\to W$. (We've omitted a small detail here which is", "that for the locally Noetherian formal algebraic spaces $X_{/T}$ and", "$W$ it suffices to check one gets an isomorphism after evaluating", "on locally Noetherian schemes over $S$.)", "\\medskip\\noindent", "We omit the proof of conditions (1), (2), and (3)." ], "refs": [ "spaces-limits-remark-limit-preserving", "artin-lemma-canonical-extension", "artin-lemma-functor" ], "ref_ids": [ 4663, 11399, 11402 ] } ], "ref_ids": [] }, { "id": 11405, "type": "theorem", "label": "artin-lemma-closed-immersion", "categories": [ "artin" ], "title": "artin-lemma-closed-immersion", "contents": [ "In Situation \\ref{situation-contractions} assume given a closed", "subset $Z \\subset S$ such that", "\\begin{enumerate}", "\\item the inverse image of $Z$ in $X'$ is $T'$,", "\\item $U' \\to S \\setminus Z$ is a closed immersion,", "\\item $W \\to S_{/Z}$ is a closed immersion.", "\\end{enumerate}", "Then there exists a solution $(f : X' \\to X, T, a)$", "and moreover $X \\to S$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Suppose we have a closed subscheme $X \\subset S$ such that", "$X \\cap (S \\setminus Z) = U'$ and $X_{/Z} = W$. Then", "$X$ represents the functor $F$ (some details omitted) and hence ", "is a solution. To find $X$ is clearly a local question on $S$.", "In this way we reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $S = \\Spec(A)$ is affine. Let $I \\subset A$ be the", "radical ideal cutting out $Z$.", "Write $I = (f_1, \\ldots, f_r)$. By assumption we are given", "\\begin{enumerate}", "\\item the closed immersion $U' \\to S \\setminus Z$ determines", "ideals $J_i \\subset A[1/f_i]$ such that $J_i$ and $J_j$", "generate the same ideal in $A[1/f_if_j]$,", "\\item the closed immersion $W \\to S_{/Z}$ is the map", "$\\text{Spf}(A^\\wedge/J') \\to \\text{Spf}(A^\\wedge)$ for some", "ideal $J' \\subset A^\\wedge$ in the $I$-adic completion $A^\\wedge$ of $A$.", "\\end{enumerate}", "To finish the proof we need to find an ideal $J \\subset A$", "such that $J_i = J[1/f_i]$ and $J' = JA^\\wedge$. By", "More on Algebra, Proposition \\ref{more-algebra-proposition-equivalence}", "it suffices to show that $J_i$ and $J'$ generate the same ideal", "in $A^\\wedge[1/f_i]$ for all $i$.", "\\medskip\\noindent", "Recall that $A' = H^0(X', \\mathcal{O})$ is a finite $A$-algebra", "whose formation commutes with flat base change", "(Cohomology of Spaces, Lemmas", "\\ref{spaces-cohomology-lemma-proper-over-affine-cohomology-finite} and", "\\ref{spaces-cohomology-lemma-flat-base-change-cohomology}). Denote", "$J'' = \\Ker(A \\to A')$\\footnote{Contrary to what the reader", "may expect, the ideals $J$ and $J''$ won't agreee in general.}.", "We have $J_i = J''A[1/f_i]$ as follows", "from base change to the spectrum of $A[1/f_i]$.", "Observe that we have a commutative diagram", "$$", "\\xymatrix{", "X' \\ar[d] &", "X'_{/T'} \\times_{S_{/Z}} \\text{Spf}(A^\\wedge) \\ar[l] \\ar[d] &", "X'_{/T'} \\times_W \\text{Spf}(A^\\wedge/J') \\ar@{=}[l] \\ar[d] \\\\", "\\Spec(A) &", "\\text{Spf}(A^\\wedge) \\ar[l] &", "\\text{Spf}(A^\\wedge/J') \\ar[l]", "}", "$$", "The middle vertical arrow is the completion of the left vertical", "arrow along the obvious closed subsets. By the theorem on formal", "functions we have", "$$", "(A')^\\wedge = \\Gamma(X' \\times_S \\Spec(A^\\wedge), \\mathcal{O}) =", "\\lim H^0(X' \\times_S \\Spec(A/I^n), \\mathcal{O})", "$$", "See Cohomology of Spaces, Theorem", "\\ref{spaces-cohomology-theorem-formal-functions}.", "From the diagram we conclude that $J'$ maps to zero in $(A')^\\wedge$.", "Hence $J' \\subset J'' A^\\wedge$. Consider the arrows", "$$", "X'_{/T'} \\to", "\\text{Spf}(A^\\wedge/J''A^\\wedge) \\to", "\\text{Spf}(A^\\wedge/J') = W", "$$", "We know the composition $g$ is a formal modification", "(in particular rig-\\'etale and rig-surjective) and the second", "arrow is a closed immersion (in particular an adic monomorphism).", "Hence $X'_{/T'} \\to \\text{Spf}(A^\\wedge/J''A^\\wedge)$ is", "rig-surjective and rig-\\'etale, see", "Algebraization of Formal Spaces, Lemmas", "\\ref{restricted-lemma-rig-surjective-alternative-permanence} and", "\\ref{restricted-lemma-rig-etale-alternative-permanence}.", "Applying Algebraization of Formal Spaces, Lemmas", "\\ref{restricted-lemma-rig-etale-descent} and", "\\ref{restricted-lemma-permanence-rig-surjective}", "we conclude that $\\text{Spf}(A^\\wedge/J''A^\\wedge) \\to W$", "is rig-\\'etale and rig-surjective.", "By Algebraization of Formal Spaces, Lemma", "\\ref{restricted-lemma-closed-immersion-rig-smooth-rig-surjective}", "we conclude that $I^n J'' A^\\wedge \\subset J'$ for some $n > 0$.", "It follows that $J'' A^\\wedge[1/f_i] = J' A^\\wedge[1/f_i]$ and", "we deduce $J_i A^\\wedge[1/f_i] = J' A^\\wedge[1/f_i]$ for all", "$i$ as desired." ], "refs": [ "more-algebra-proposition-equivalence", "spaces-cohomology-lemma-proper-over-affine-cohomology-finite", "spaces-cohomology-lemma-flat-base-change-cohomology", "spaces-cohomology-theorem-formal-functions", "restricted-lemma-rig-surjective-alternative-permanence", "restricted-lemma-rig-etale-alternative-permanence", "restricted-lemma-rig-etale-descent", "restricted-lemma-permanence-rig-surjective", "restricted-lemma-closed-immersion-rig-smooth-rig-surjective" ], "ref_ids": [ 10587, 11332, 11296, 11270, 2387, 2383, 2396, 2388, 2395 ] } ], "ref_ids": [] }, { "id": 11406, "type": "theorem", "label": "artin-lemma-diagonal-contractions", "categories": [ "artin" ], "title": "artin-lemma-diagonal-contractions", "contents": [ "In Situation \\ref{situation-contractions} assume $X' \\to S$", "and $W \\to S$ are separated. Then the diagonal $\\Delta : F \\to F \\times F$", "is representable by closed immersions." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-closed-immersion}", "with the discussion in Remark \\ref{remark-diagonal}." ], "refs": [ "artin-lemma-closed-immersion", "artin-remark-diagonal" ], "ref_ids": [ 11405, 11440 ] } ], "ref_ids": [] }, { "id": 11407, "type": "theorem", "label": "artin-lemma-sheaf", "categories": [ "artin" ], "title": "artin-lemma-sheaf", "contents": [ "In Situation \\ref{situation-contractions} the functor", "$F$ satisfies the sheaf property for all \\'etale coverings", "of locally Noetherian schemes over $S$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: morphisms may be defined \\'etale locally." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11408, "type": "theorem", "label": "artin-lemma-limit-preserving", "categories": [ "artin" ], "title": "artin-lemma-limit-preserving", "contents": [ "In Situation \\ref{situation-contractions} the functor $F$ is limit preserving:", "for any directed limit $V = \\lim V_\\lambda$ of Noetherian affine schemes", "over $S$ we have $F(V) = \\colim F(V_\\lambda)$." ], "refs": [], "proofs": [ { "contents": [ "This is an absurdly long proof. Much of it consists of standard", "arguments on limits and \\'etale localization. We urge the reader to skip ahead", "to the last part of the proof where something interesting happens.", "\\medskip\\noindent", "Let $V = \\lim_{\\lambda \\in \\Lambda} V_i$ be a directed limit of schemes", "over $S$ with $V$ and $V_\\lambda$ Noetherian and with affine transition", "morphisms. See Limits, Section \\ref{limits-section-limits} for material on", "limits of schemes. We will prove that $\\colim F(V_\\lambda) \\to F(V)$", "is bijective.", "\\medskip\\noindent", "Proof of injectivity: notation.", "Let $\\lambda \\in \\Lambda$ and", "$\\xi_{\\lambda, 1}, \\xi_{\\lambda, 2} \\in F(V_\\lambda)$ ", "be elements which restrict to the same element of $F(V)$.", "Write $\\xi_{\\lambda, 1} = (Z_{\\lambda, 1}, u'_{\\lambda, 1},", "\\hat x_{\\lambda, 1})$ and", "$\\xi_{\\lambda, 2} = (Z_{\\lambda, 2}, u'_{\\lambda, 2}, \\hat x_{\\lambda, 2})$.", "\\medskip\\noindent", "Proof of injectivity: agreement of $Z_{\\lambda, i}$.", "Since $Z_{\\lambda, 1}$ and $Z_{\\lambda, 2}$ restrict to the same closed", "subset of $V$, we may after increasing $i$ assume", "$Z_{\\lambda, 1} = Z_{\\lambda, 2}$, see", "Limits, Lemma \\ref{limits-lemma-inverse-limit-top} and", "Topology, Lemma \\ref{topology-lemma-describe-limits}.", "Let us denote the common value $Z_\\lambda \\subset V_\\lambda$, for", "$\\mu \\geq \\lambda$", "denote $Z_\\mu \\subset V_\\mu$ the inverse image in $V_\\mu$ and", "and denote $Z$ the inverse image in $V$. We will use below that", "$Z = \\lim_{\\mu \\geq \\lambda} Z_\\mu$ as schemes if we view $Z$", "and $Z_\\mu$ as reduced closed subschemes.", "\\medskip\\noindent", "Proof of injectivity: agreement of $u'_{\\lambda, i}$.", "Since $U'$ is locally of finite type over $S$ and since", "the restrictions of $u'_{\\lambda, 1}$ and $u'_{\\lambda, 2}$", "to $V \\setminus Z$ are the same, we may after increasing $\\lambda$ assume", "$u'_{\\lambda, 1} = u'_{\\lambda, 2}$, see Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}.", "Let us denote the common value $u'_\\lambda$ and denote", "$u'$ the restriction to $V \\setminus Z$.", "\\medskip\\noindent", "Proof of injectivity: restatement.", "At this point we have", "$\\xi_{\\lambda, 1} = (Z_\\lambda, u'_\\lambda, \\hat x_{\\lambda, 1})$ and", "$\\xi_{\\lambda, 2} = (Z_\\lambda, u'_\\lambda, \\hat x_{\\lambda, 2})$.", "The main problem we face in this part of the proof", "is to show that the morphisms", "$\\hat x_{\\lambda, 1}$ and $\\hat x_{\\lambda, 2}$ become the same after", "increasing $\\lambda$.", "\\medskip\\noindent", "Proof of injectivity: agreement of $\\hat x_{\\lambda, i}|_{Z_\\lambda}$.", "Consider the morpisms", "$\\hat x_{\\lambda, 1}|_{Z_\\lambda}, \\hat x_{\\lambda, 2}|_{Z_\\lambda} :", "Z_\\lambda \\to W_{red}$.", "These morphisms restrict to the same morphism $Z \\to W_{red}$.", "Since $W_{red}$ is a scheme locally of finite type over $S$", "we see using Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}", "that after replacing $\\lambda$ by a bigger index", "we may assume", "$\\hat x_{\\lambda, 1}|_{Z_\\lambda} = \\hat x_{\\lambda, 2}|_{Z_\\lambda} :", "Z_\\lambda \\to W_{red}$.", "\\medskip\\noindent", "Proof of injectivity: end.", "Next, we are going to apply the discussion in", "Remark \\ref{remark-diagonal} to $V_\\lambda$ and the two elements", "$\\xi_{\\lambda, 1}, \\xi_{\\lambda, 2} \\in F(V_\\lambda)$.", "This gives us", "\\begin{enumerate}", "\\item $e_\\lambda : E_\\lambda' \\to V_\\lambda$", "separated and locally of finite type,", "\\item $e_\\lambda^{-1}(V_\\lambda \\setminus Z_\\lambda) \\to", "V_\\lambda \\setminus Z_\\lambda$ is an isomorphism,", "\\item a monomorphism $E_{W, \\lambda} \\to V_{\\lambda, /Z_\\lambda}$", "which is the equalizer of $\\hat x_{\\lambda, 1}$ and $\\hat x_{\\lambda, 2}$,", "\\item a formal modification $E'_{\\lambda, /Z_\\lambda} \\to E_{W, \\lambda}$", "\\end{enumerate}", "Assertion (2) holds by assertion (2) in Remark \\ref{remark-diagonal}", "and the preparatory work we did above getting", "$u'_{\\lambda, 1} = u'_{\\lambda, 2} = u'_\\lambda$.", "Since $Z_\\lambda = (V_{\\lambda, /Z_\\lambda})_{red}$ factors through", "$E_{W, \\lambda}$ because", "$\\hat x_{\\lambda, 1}|_{Z_\\lambda} = \\hat x_{\\lambda, 2}|_{Z_\\lambda}$", "we see from", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-monomorphism-iso-over-red}", "that $E_{W, \\lambda} \\to V_{\\lambda, /Z_\\lambda}$ is a closed immersion.", "Then we see from assertion (4) in Remark \\ref{remark-diagonal}", "and Lemma \\ref{lemma-closed-immersion} applied to the triple", "$E_\\lambda'$, $e_\\lambda^{-1}(Z_\\lambda)$,", "$E'_{\\lambda, /Z_\\lambda} \\to E_{W, \\lambda}$ over $V_\\lambda$ that", "there exists a closed immersion $E_\\lambda \\to V_\\lambda$", "which is a solution for this triple.", "Next we use assertion (5) in Remark \\ref{remark-diagonal}", "which combined with Lemma \\ref{lemma-solution}", "says that $E_\\lambda$ is the ``equalizer'' of $\\xi_{\\lambda, 1}$", "and $\\xi_{\\lambda, 2}$. In particular, we see that $V \\to V_\\lambda$", "factors through $E_\\lambda$. Then using Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}", "once more we find $\\mu \\geq \\lambda$ such that $V_\\mu \\to V_\\lambda$", "factors through $E_\\lambda$ and hence the pullbacks of", "$\\xi_{\\lambda, 1}$ and $\\xi_{\\lambda, 2}$ to $V_\\mu$ are the same", "as desired.", "\\medskip\\noindent", "Proof of surjectivity: statement.", "Let $\\xi = (Z, u', \\hat x)$ be an element of $F(V)$.", "We have to find a $\\lambda \\in \\Lambda$ and an element", "$\\xi_\\lambda \\in F(V_\\lambda)$ restricting to $\\xi$.", "\\medskip\\noindent", "Proof of surjectivity: the question is \\'etale local.", "By the unicity proved in the previous part of the proof and by the", "sheaf property of $F$ in Lemma \\ref{lemma-sheaf}, the problem", "is local on $V$ in the \\'etale topology. More precisely, let $v \\in V$.", "We claim it suffices to find an \\'etale morphism", "$(\\tilde V, \\tilde v) \\to (V, v)$ and some", "$\\lambda$, some an \\'etale morphism $\\tilde V_\\lambda \\to V_\\lambda$,", "and some element $\\tilde \\xi_\\lambda \\in F(\\tilde V_\\lambda)$ such that", "$\\tilde V = \\tilde V_\\lambda \\times_{V_\\lambda} V$", "and $\\xi|_U = \\tilde \\xi_\\lambda|_U$. We omit a detailed proof of", "this claim\\footnote{To prove this", "one assembles a collection of the morphisms $\\tilde V \\to V$", "into a finite \\'etale covering and shows that the corresponding morphisms", "$\\tilde V_\\lambda \\to V_\\lambda$ form an \\'etale covering as well (after", "increasing $\\lambda$). Next one uses the injectivity to see that", "the elements $\\tilde \\xi_\\lambda$ glue (after increasing $\\lambda$)", "and one uses the sheaf property for $F$ to descend these", "elements to an element of $F(V_\\lambda)$.}.", "\\medskip\\noindent", "Proof of surjectivity: rephrasing the problem.", "Recall that any \\'etale morphism $(\\tilde V, \\tilde v) \\to (V, v)$", "with $\\tilde V$ affine is the base change of an \\'etale morphism", "$\\tilde V_\\lambda \\to V_\\lambda$ with $\\tilde V_\\lambda$ affine", "for some $\\lambda$, see for example", "Topologies, Lemma \\ref{topologies-lemma-limit-fppf-topology}.", "Given $\\tilde V_\\lambda$ we have", "$\\tilde V = \\lim_{\\mu \\geq \\lambda} \\tilde V_\\lambda \\times_{V_\\lambda} V_\\mu$.", "Hence given $(\\tilde V, \\tilde v) \\to (V, v)$ \\'etale with $\\tilde V$ affine,", "we may replace $(V, v)$ by $(\\tilde V, \\tilde v)$ and $\\xi$ by", "the restriction of $\\xi$ to $\\tilde V$.", "\\medskip\\noindent", "Proof of surjectivity: reduce to base being affine. In particular,", "suppose $\\tilde S \\subset S$ is an affine open subscheme such", "that $v \\in V$ maps to a point of $\\tilde S$. Then we may according", "to the previous paragraph, replace $V$ by $\\tilde V = \\tilde S \\times_S V$.", "Of course, if we do this, it suffices to solve the problem", "for the functor $F$ restricted to the category of locally Noetherian", "schemes over $\\tilde S$. This functor is of course the functor", "associated to the whole situation base changed to $\\tilde S$.", "Thus we may and do assume $S = \\Spec(R)$ is a Noetherian affine scheme", "for the rest of the proof.", "\\medskip\\noindent", "Proof of surjectivity: easy case.", "If $v \\in V \\setminus Z$, then we can take $\\tilde V = V \\setminus Z$.", "This descends to an open subscheme $\\tilde V_\\lambda \\subset V_\\lambda$", "for some $\\lambda$ by Limits, Lemma", "\\ref{limits-lemma-descend-opens}.", "Next, after increasing $\\lambda$ we may assume", "there is a morphism $u'_\\lambda : \\tilde V_\\lambda \\to U'$", "restricting to $u'$. Taking", "$\\tilde \\xi_\\lambda = (\\emptyset, u'_\\lambda, \\emptyset)$", "gives the desired element of $F(\\tilde V_\\lambda)$.", "\\medskip\\noindent", "Proof of surjectivity: hard case and reduction to affine $W$.", "The most difficult case comes from considering $v \\in Z \\subset V$.", "We claim that we can reduce this to the case where $W$ is an", "affine formal scheme; we urge the reader to skip this argument\\footnote{Artin's", "approach to the proof of this lemma is to work around this and", "consequently he can avoid proving the injectivity first. Namely, Artin", "consistently works with a finite affine \\'etale coverings of all spaces", "in sight keeping track of the maps between them during the proof.", "In hindsight that might be preferable to what we do here.}.", "Namely, we can choose an \\'etale morphism", "$\\tilde W \\to W$ where $\\tilde W$ is an affine formal algebraic space", "such that the image of $v$ by $\\hat x : V_{/Z} \\to W$ is", "in the image of $\\tilde W \\to W$ (on reductions).", "Then the morphisms", "$$", "p : \\tilde W \\times_{W, g} X'_{/T'} \\longrightarrow X'_{/T'}", "$$", "and", "$$", "q : \\tilde W \\times_{W, \\hat x} V_{/Z} \\to V_{/Z}", "$$", "are \\'etale morphisms of locally Noetherian formal algebraic spaces.", "By (an easy case of) Algebraization of Formal Spaces, Theorem", "\\ref{restricted-theorem-dilatations-general}", "there exists a morphism $\\tilde X' \\to X'$ of algebraic spaces", "which is locally of finite type, is an isomorphism over $U'$, and", "such that $\\tilde X'_{/T'} \\to X'_{/T'}$ is isomorphic to $p$.", "By Algebraization of Formal Spaces, Lemma \\ref{restricted-lemma-output-etale}", "the morphism $\\tilde X' \\to X'$ is \\'etale. Denote", "$\\tilde T' \\subset |\\tilde X'|$ the inverse image of $T'$.", "Denote $\\tilde U' \\subset \\tilde X'$ the complementary open subspace.", "Denote $\\tilde g' : \\tilde X'_{/\\tilde T'} \\to \\tilde W$", "the formal modification which is the base change of $g$ by", "$\\tilde W \\to W$. Then we see that", "$$", "\\tilde X',\\ \\tilde T',\\ \\tilde U',\\ \\tilde W,", "\\ \\tilde g : \\tilde X'_{/\\tilde T'} \\to \\tilde W", "$$", "is another example of Situation \\ref{situation-contractions}.", "Denote $\\tilde F$ the functor constructed from this triple.", "There is a transformation of functors", "$$", "\\tilde F \\longrightarrow F", "$$", "constructed using the morphisms $\\tilde X' \\to X'$ and", "$\\tilde W \\to W$ in the obvious manner; details omitted.", "\\medskip\\noindent", "Proof of surjectivity: hard case and reduction to affine $W$, part 2.", "By the same theorem as used above, there exists a morphism", "$\\tilde V \\to V$ of algebraic spaces which is locally of finite type,", "is an isomorphism over $V \\setminus Z$ and such that", "$\\tilde V_{/Z} \\to V_{/Z}$ is isomorphic to $q$.", "Denote $\\tilde Z \\subset \\tilde V$ the inverse image of $Z$.", "By Algebraization of Formal Spaces, Lemmas", "\\ref{restricted-lemma-output-etale} and \\ref{restricted-lemma-output-separated}", "the morphism $\\tilde V \\to V$ is \\'etale and separated.", "In particular $\\tilde V$ is a (locally Noetherian) scheme, see for example", "Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}.", "We have the morphism $u'$ which we may view as a morphism", "$$", "\\tilde u' : \\tilde V \\setminus \\tilde Z \\longrightarrow \\tilde U'", "$$", "where $\\tilde U' \\subset \\tilde X'$ is the open mapping isomorphically", "to $U'$. We have a morphism", "$$", "\\tilde {\\hat x} :", "\\tilde V_{/\\tilde Z} = \\tilde W \\times_{W, \\hat x} V_{/Z}", "\\longrightarrow", "\\tilde W", "$$", "Namely, here we just use the projection. Thus we have the triple", "$$", "\\tilde \\xi = (\\tilde Z, \\tilde u', \\tilde {\\hat x}) \\in \\tilde F(\\tilde V)", "$$", "We omit proving the compatibility condition; hints: if $V' \\to V$, $\\hat x'$,", "and $x'$ witness the compatibility between $u'$ and $\\hat x$, then one", "sets $\\tilde V' = V' \\times_V \\tilde V$ which comes with morphsms", "$\\tilde{\\hat x}'$ and $\\tilde x'$ and show this works.", "The image of $\\tilde \\xi$ under the transformation $\\tilde F \\to F$", "is the restriction of $\\xi$ to $\\tilde V$.", "\\medskip\\noindent", "Proof of surjectivity: hard case and reduction to affine $W$, part 3.", "By our choice of $\\tilde W \\to W$, there is an affine open", "$\\tilde V_{open} \\subset \\tilde V$ (we're running out of notation)", "whose image in $V$ contains our chosen point $v \\in V$.", "Now by the case studied in the next paragraph and the remarks made", "earlier, we can descend $\\tilde \\xi|_{\\tilde V_{open}}$", "to some element $\\tilde \\xi_\\lambda$ of $\\tilde F$ over", "$\\tilde V_{\\lambda, open}$", "for some \\'etale morphism $\\tilde V_{\\lambda, open} \\to V_\\lambda$", "whose base change to $V$ is $\\tilde V_{open}$.", "Applying the transformation of functors $\\tilde F \\to F$", "we obtain the element of $F(\\tilde V_{\\lambda, open})$", "we were looking for. This reduces us to the case discussed", "in the next paragraph.", "\\medskip\\noindent", "Proof of surjectivity: the case of an affine $W$. We have $v \\in Z \\subset V$", "and $W$ is an affine formal algebraic space. Recall that", "$$", "\\xi = (Z, u', \\hat x) \\in F(V)", "$$", "We may still replace $V$ by an \\'etale neighbourhood of $v$.", "In particular we may and do assume $V$ and $V_\\lambda$ are affine.", "\\medskip\\noindent", "Proof of surjectivity: descending $Z$. We can find a $\\lambda$", "and a closed subscheme $Z_\\lambda \\subset V_\\lambda$ such that", "$Z$ is the base change of $Z_\\lambda$ to $V$. See", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}.", "Warning: we don't know (and in general it won't be true)", "that $Z_\\lambda$ is a reduced closed subscheme of $V_\\lambda$.", "For $\\mu \\geq \\lambda$ denote $Z_\\mu \\subset V_\\mu$ the scheme theoretic", "inverse image in $V_\\mu$. We will use below that", "$Z = \\lim_{\\mu \\geq \\lambda} Z_\\mu$ as schemes.", "\\medskip\\noindent", "Proof of surjectivity: descending $u'$.", "Since $U'$ is locally of finite type over $S$", "we may assume after increasing $\\lambda$", "that there exists a morphism", "$u'_\\lambda : V_\\lambda \\setminus Z_\\lambda \\to U'$", "whose restriction to $V \\setminus Z$ is $u'$.", "See Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}.", "For $\\mu \\geq \\lambda$ we will denote $u'_\\mu$ the restriction", "of $u'_\\lambda$ to $V_\\mu \\setminus Z_\\mu$.", "\\medskip\\noindent", "Proof of surjectivity: descending a witness.", "Let $V' \\to V$, $\\hat x'$, and $x'$ witness the compatibility between", "$u'$ and $\\hat x$. Using the same references as above we may assume", "(after increasing $\\lambda$) that there exists a morphism", "$V'_\\lambda \\to V_\\lambda$ of finite type whose base change to $V$", "is $V' \\to V$. After increasing $\\lambda$ we may assume", "$V'_\\lambda \\to V_\\lambda$ is proper", "(Limits, Lemma \\ref{limits-lemma-eventually-proper}).", "Next, we may assume $V'_\\lambda \\to V_\\lambda$ is an isomorphism", "over $V_\\lambda \\setminus Z_\\lambda$", "(Limits, Lemma \\ref{limits-lemma-descend-isomorphism}).", "Next, we may assume there is a morphism $x'_\\lambda : V'_\\lambda \\to X'$", "whose restriction to $V'$ is $x'$.", "Increasing $\\lambda$ again we may assume $x'_\\lambda$", "agrees with $u'_\\lambda$ over $V_\\lambda \\setminus Z_\\lambda$.", "For $\\mu \\geq \\lambda$ we denote", "$V'_\\mu$ and $x'_\\mu$ the base change of $V'_\\lambda$ and the", "restriction of $x'_\\lambda$.", "\\medskip\\noindent", "Proof of surjectivity: algebra.", "Write $W = \\text{Spf}(B)$, $V = \\Spec(A)$, and", "for $\\mu \\geq \\lambda$ write $V_\\mu = \\Spec(A_\\mu)$.", "Denote $I_\\mu \\subset A_\\mu$ and $I \\subset A$", "the ideals cutting out $Z_\\mu$ and $Z$.", "Then $I_\\lambda A_\\mu = I_\\mu$ and $I_\\lambda A = I$.", "The morphism $\\hat x$ determines and is determined by a", "continuous ring map", "$$", "(\\hat x)^\\sharp : B \\longrightarrow A^\\wedge", "$$", "where $A^\\wedge$ is the $I$-adic completion of $A$.", "To finish the proof we have to show that this map descends to a map into", "$A_\\mu^\\wedge$ for some sufficiently large $\\mu$ where $A_\\mu^\\wedge$", "is the $I_\\mu$-adic completion of $A_\\mu$.", "This is a nontrivial fact; Artin writes in his paper", "\\cite{ArtinII}: ``Since the data (3.5) involve $I$-adic completions,", "which do not commute with direct limits, the verification is somewhat", "delicate. It is an algebraic analogue of a convergence proof in analysis.''", "\\medskip\\noindent", "Proof of surjectivity: algebra, more rings.", "Let us denote", "$$", "C_\\mu = \\Gamma(V'_\\mu, \\mathcal{O})", "\\quad\\text{and}\\quad", "C = \\Gamma(V', \\mathcal{O})", "$$", "Observe that $A \\to C$ and $A_\\mu \\to C_\\mu$ are finite ring maps as", "$V' \\to V$ and $V'_\\mu \\to V_\\mu$ are proper morphisms, see", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-over-affine-cohomology-finite}.", "Since $V = \\lim V_\\mu$ and $V' = \\lim V'_\\mu$", "we have", "$$", "A = \\colim A_\\mu", "\\quad\\text{and}\\quad", "C = \\colim C_\\mu", "$$", "by Limits, Lemma \\ref{limits-lemma-descend-section}\\footnote{We don't", "know that $C_\\mu = C_\\lambda \\otimes_{A_\\lambda} A_\\mu$ as the", "various morphisms aren't flat.}. For an element", "$a \\in I$, resp.\\ $a \\in I_\\mu$ the maps $A_a \\to C_a$,", "resp.\\ $(A_\\mu)_a \\to (C_\\mu)_a$ are isomorphisms by flat base change", "(Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-flat-base-change-cohomology}).", "Hence the kernel and cokernel of $A \\to C$ is supported", "on $V(I)$ and similarly for $A_\\mu \\to C_\\mu$.", "We conclude the kernel and cokernel of $A \\to C$", "are annihilated by a power of $I$ and the kernel and cokernel of", "$A_\\mu \\to C_\\mu$ are annihilated by a power of $I_\\mu$, see", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-power-ideal-kills-module}.", "\\medskip\\noindent", "Proof of surjectivity: algebra, more ring maps.", "Denote $Z_n \\subset V$ the $n$th infinitesimal", "neighbourhood of $Z$ and denote $Z_{\\mu, n} \\subset V_\\mu$", "the $n$th infinitesimal neighbourhoof of $Z_\\mu$.", "By the theorem on formal functions", "(Cohomology of Spaces, Theorem", "\\ref{spaces-cohomology-theorem-formal-functions})", "we have", "$$", "C^\\wedge = \\lim_n H^0(V' \\times_V Z_n, \\mathcal{O})", "\\quad\\text{and}\\quad", "C_\\mu^\\wedge =", "\\lim_n H^0(V'_\\mu \\times_{V_\\mu} Z_{\\mu, n}, \\mathcal{O})", "$$", "where $C^\\wedge$ and $C_\\mu^\\wedge$ are the completion with", "respect to $I$ and $I_\\mu$.", "Combining the completion of the morphism", "$x'_\\mu : V'_\\mu \\to X'$ with the morphism $g : X'_{/T'} \\to W$ we obtain", "$$", "g \\circ x'_{\\mu, /Z_\\mu} :", "V'_{\\mu, /Z_\\mu} = \\colim V_\\mu' \\times_{V_\\mu} Z_{\\mu, n}", "\\longrightarrow", "W", "$$", "and hence by the description of the completion", "$C_\\mu^\\wedge$ above we obtain a continuous ring homomorphism", "$$", "(g \\circ x'_{\\mu, /Z_\\mu})^\\sharp : B \\longrightarrow C_\\mu^\\wedge", "$$", "The fact that $V' \\to V$, $\\hat x'$, $x'$ witnesses the compatibility", "between $u'$ and $\\hat x$ implies the", "commutativity of the following diagram", "$$", "\\xymatrix{", "C_\\mu^\\wedge \\ar[r] &", "C^\\wedge \\\\", "B \\ar[u]^{(g \\circ x'_{\\mu, /Z_\\mu})^\\sharp} \\ar[r]^{(\\hat x)^\\sharp} &", "A^\\wedge \\ar[u]", "}", "$$", "\\medskip\\noindent", "Proof of surjectivity: more algebra arguments. Recall that the finite", "$A$-modules $\\Ker(A \\to C)$ and $\\Coker(A \\to C)$ are annihilated", "by a power of $I$ and similarly the finite $A_\\mu$-modules", "$\\Ker(A_\\mu \\to C_\\mu)$ and $\\Coker(A_\\mu \\to C_\\mu)$ are annihilated", "by a power of $I_\\mu$. This implies that these modules are", "equal to their completions. Since $I$-adic completion on the category of", "finite $A$-modules is exact (see", "Algebra, Section \\ref{algebra-section-completion-noetherian})", "it follows that we have", "$$", "\\Coker(A^\\wedge \\to C^\\wedge) = \\Coker(A \\to C)", "$$", "and similarly for kernels and for the maps $A_\\mu \\to C_\\mu$.", "Of course we also have", "$$", "\\Ker(A \\to C) = \\colim \\Ker(A_\\mu \\to C_\\mu)", "\\quad\\text{and}\\quad", "\\Coker(A \\to C) = \\colim \\Coker(A_\\mu \\to C_\\mu)", "$$", "Recall that $S = \\Spec(R)$ is affine. All of the ring maps", "above are $R$-algebra homomorphisms as all of the morphisms", "are morphisms over $S$. By", "Algebraization of Formal Spaces, Lemma", "\\ref{restricted-lemma-Noetherian-finite-type-red}", "we see that $B$ is topologically of finite type over $R$.", "Say $B$ is topologically generated by $b_1, \\ldots, b_n$.", "Pick some $\\mu$ (for example $\\lambda$) and consider", "the elements", "$$", "\\text{images of }", "(g \\circ x'_{\\mu, /Z_\\mu})^\\sharp(b_1)", ", \\ldots,", "(g \\circ x'_{\\mu, /Z_\\mu})^\\sharp(b_n)", "\\text{ in }\\Coker(A_\\mu \\to C_\\mu)", "$$", "The image of these elements in $\\Coker(\\alpha)$ are zero", "by the commutativity of the square above. Since", "$\\Coker(A \\to C) = \\colim \\Coker(A_\\mu \\to C_\\mu)$ and these", "cokernels are equal to their completions", "we see that after increasing $\\mu$ we may assume these", "images are all zero. This means that the continuous", "homomorphism $(g \\circ x'_{\\mu, /Z_\\mu})^\\sharp$ has image contained", "in $\\Im(A_\\mu \\to C_\\mu)$.", "Choose elements $a_{\\mu, j} \\in (A_\\mu)^\\wedge$ mapping to", "$(g \\circ x'_{\\mu, /Z_\\mu})^\\sharp(b_1)$ in $(C_\\mu)^\\wedge$.", "Then $a_{\\mu, j} \\in A_\\mu^\\wedge$ and $(\\hat x)^\\sharp(b_j) \\in A^\\wedge$", "map to the same element of $C^\\wedge$ by the commutativity of the", "square above. Since", "$\\Ker(A \\to C) = \\colim \\Ker(A_\\mu \\to C_\\mu)$ and these kernels", "are equal to their completions, we may after increasing", "$\\mu$ adjust our choices of $a_{\\mu, j}$ such that", "the image of $a_{\\mu, j}$ in $A^\\wedge$ is equal to $(\\hat x)^\\sharp(b_j)$.", "\\medskip\\noindent", "Proof of surjectivity: final algebra arguments.", "Let $\\mathfrak b \\subset B$ be the ideal of topologically nilpotent", "elements. Let $J \\subset R[x_1, \\ldots, x_n]$ be the ideal", "consisting of those $h(x_1, \\ldots, x_n)$ such that", "$h(b_1, \\ldots, b_n) \\in \\mathfrak b$. Then we get a continuous", "surjection of topological $R$-algebras", "$$", "\\Phi : R[x_1, \\ldots, x_n]^\\wedge \\longrightarrow B,\\quad", "x_j \\longmapsto b_j", "$$", "where the completion on the left hand side is with respect to $J$.", "Since $R[x_1, \\ldots, x_n]$ is Noetherian we can choose", "generators $h_1, \\ldots, h_m$ for $J$. By the commutativity", "of the square above we see that $h_j(a_{\\mu, 1}, \\ldots, a_{\\mu, n})$ is", "an element of $A_\\mu^\\wedge$ whose image in $A^\\wedge$ is contained", "in $IA^\\wedge$. Namely, the ring map $(\\hat x)^\\sharp$ is continuous", "and $IA^\\wedge$ is the ideal of topological nilpotent elements", "of $A^\\wedge$ because $A^\\wedge/IA^\\wedge = A/I$ is reduced.", "(See Algebra, Section \\ref{algebra-section-completion-noetherian}", "for results on completion in Noetherian rings.)", "Since $A/I = \\colim A_\\mu/I_\\mu$ we conclude that after increasing", "$\\mu$ we may assume $h_j(a_{\\mu, 1}, \\ldots, a_{\\mu, n})$ is in", "$I_\\mu A_\\mu^\\wedge$. In particular the elements", "$h_j(a_{\\mu, 1}, \\ldots, a_{\\mu, n})$ of $A_\\mu^\\wedge$", "are topologically nilpotent in $A_\\mu^\\wedge$.", "Thus we obtain a continuous $R$-algebra homomorphism", "$$", "\\Psi : R[x_1, \\ldots, x_n]^\\wedge \\longrightarrow A_\\mu^\\wedge,\\quad", "x_j \\longmapsto a_{\\mu, j}", "$$", "In order to conclude what we want, we need to see if $\\Ker(\\Phi)$ is", "annihilated by $\\Psi$. This may not be true, but we can achieve", "this after increasing $\\mu$. Indeed, since", "$R[x_1, \\ldots, x_n]^\\wedge$ is Noetherian,", "we can choose generators $g_1, \\ldots, g_l$ of the ideal", "$\\Ker(\\Phi)$. Then we see that", "$$", "\\Psi(g_1), \\ldots, \\Psi(g_l) \\in", "\\Ker(A_\\mu^\\wedge \\to C_\\mu^\\wedge) = \\Ker(A_\\mu \\to C_\\mu)", "$$", "map to zero in $\\Ker(A \\to C) = \\colim \\Ker(A_\\mu \\to C_\\mu)$.", "Hence increasing $\\mu$ as before we get the desired result.", "\\medskip\\noindent", "Proof of surjectivity: mopping up. The continuous ring homomorphism", "$B \\to (A_\\mu)^\\wedge$ constructed above determines a morphism", "$\\hat x_\\mu : V_{\\mu, /Z_\\mu} \\to W$.", "The compatibility of $\\hat x_\\mu$ and $u'_\\mu$ follows", "from the fact that the ring map $B \\to (A_\\mu)^\\wedge$", "is by construction compatible with the ring map $A_\\mu \\to C_\\mu$.", "In fact, the compatibility will be witnessed by the proper morphism", "$V'_\\mu \\to V_\\mu$ and the morphisms", "$x'_\\mu$ and $\\hat x'_\\mu = x'_{\\mu, /Z_\\mu}$ we used in", "the construction. This finishes the proof." ], "refs": [ "limits-lemma-inverse-limit-top", "topology-lemma-describe-limits", "limits-proposition-characterize-locally-finite-presentation", "limits-proposition-characterize-locally-finite-presentation", "artin-remark-diagonal", "artin-remark-diagonal", "formal-spaces-lemma-monomorphism-iso-over-red", "artin-remark-diagonal", "artin-lemma-closed-immersion", "artin-remark-diagonal", "artin-lemma-solution", "limits-proposition-characterize-locally-finite-presentation", "artin-lemma-sheaf", "topologies-lemma-limit-fppf-topology", "limits-lemma-descend-opens", "restricted-theorem-dilatations-general", "restricted-lemma-output-etale", "restricted-lemma-output-etale", "restricted-lemma-output-separated", "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "limits-lemma-descend-finite-presentation", "limits-proposition-characterize-locally-finite-presentation", "limits-lemma-eventually-proper", "limits-lemma-descend-isomorphism", "spaces-cohomology-lemma-proper-over-affine-cohomology-finite", "spaces-cohomology-lemma-flat-base-change-cohomology", "algebra-lemma-Noetherian-power-ideal-kills-module", "spaces-cohomology-theorem-formal-functions", "restricted-lemma-Noetherian-finite-type-red" ], "ref_ids": [ 15033, 8249, 15127, 15127, 11440, 11440, 3947, 11440, 11405, 11440, 11403, 15127, 11407, 12518, 15041, 2292, 2426, 2426, 2424, 4983, 15077, 15127, 15089, 15066, 11332, 11296, 694, 11270, 2332 ] } ], "ref_ids": [] }, { "id": 11409, "type": "theorem", "label": "artin-lemma-rs", "categories": [ "artin" ], "title": "artin-lemma-rs", "contents": [ "In Situation \\ref{situation-contractions} the functor $F$ satisfies", "the Rim-Schlessinger condition (RS)." ], "refs": [], "proofs": [ { "contents": [ "Recall that the condition only involves the evaluation $F(V)$ of the functor", "$F$ on schemes $V$ over $S$ which are spectra of Artinian local rings", "and the restriction maps $F(V_2) \\to F(V_1)$ for morphisms $V_1 \\to V_2$", "of schemes over $S$ which are spectra of Artinian local rings.", "Thus let $V/S$ be the spetruim of an Artinian local ring.", "If $\\xi = (Z, u', \\hat x) \\in F(V)$ then either $Z = \\emptyset$", "or $Z = V$ (set theoretically). In the first case we see that", "$\\hat x$ is a morphism from the empty formal algebraic space", "into $W$. In the second case we see that $u'$ is a morphism from", "the empty scheme into $X'$ and we see that $\\hat x : V \\to W$", "is a morphism into $W$. We conclude that", "$$", "F(V) = U'(V) \\amalg W(V)", "$$", "and moreover for $V_1 \\to V_2$ as above the induced map", "$F(V_2) \\to F(V_1)$ is compatible with this decomposition.", "Hence it suffices to prove that both $U'$ and $W$ satisfy the", "Rim-Schlessinger condition. For $U'$ this follows from", "Lemma \\ref{lemma-algebraic-stack-RS}.", "To see that it is true for $W$, we write $W = \\colim W_n$ as in", "Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-structure-locally-noetherian}.", "Say $V = \\Spec(A)$ with $(A, \\mathfrak m)$ an Artinian local ring.", "Pick $n \\geq 1$ such that $\\mathfrak m^n = 0$. Then we have", "$W(V) = W_n(V)$. Hence we see that the Rim-Schlessinger condition", "for $W$ follows from the Rim-Schlessinger condition for $W_n$ for", "all $n$ (which in turn follows from", "Lemma \\ref{lemma-algebraic-stack-RS})." ], "refs": [ "artin-lemma-algebraic-stack-RS", "formal-spaces-lemma-structure-locally-noetherian", "artin-lemma-algebraic-stack-RS" ], "ref_ids": [ 11355, 3921, 11355 ] } ], "ref_ids": [] }, { "id": 11410, "type": "theorem", "label": "artin-lemma-finite-dim", "categories": [ "artin" ], "title": "artin-lemma-finite-dim", "contents": [ "In Situation \\ref{situation-contractions} the tangent spaces of", "the functor $F$ are finite dimensional." ], "refs": [], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-rs} we have seen that", "$F(V) = U'(V) \\amalg W(V)$ if $V$ is the spectrum of an Artinian", "local ring. The tangent spaces are computed entirely from evaluations", "of $F$ on such schemes over $S$.", "Hence it suffices to prove that the tangent spaces", "of the functors $U'$ and $W$ are finite dimensional.", "For $U'$ this follows from", "Lemma \\ref{lemma-finite-dimension}.", "Write $W = \\colim W_n$ as in the proof of Lemma \\ref{lemma-rs}.", "Then we see that the tangent spaces of $W$ are equal to the", "tangent spaces of $W_2$, as to get at the tangent space", "we only need to evaluate $W$ on spectra of Artinian local", "rings $(A, \\mathfrak m)$ with $\\mathfrak m^2 = 0$.", "Then again we see that the tangent spaces of $W_2$ have", "finite dimension by", "Lemma \\ref{lemma-finite-dimension}." ], "refs": [ "artin-lemma-rs", "artin-lemma-finite-dimension", "artin-lemma-rs", "artin-lemma-finite-dimension" ], "ref_ids": [ 11409, 11359, 11409, 11359 ] } ], "ref_ids": [] }, { "id": 11411, "type": "theorem", "label": "artin-lemma-formal-object-effective", "categories": [ "artin" ], "title": "artin-lemma-formal-object-effective", "contents": [ "In Situation \\ref{situation-contractions} assume $X' \\to S$ is separated.", "Then every formal object for $F$ is effective." ], "refs": [], "proofs": [ { "contents": [ "A formal object $\\xi = (R, \\xi_n)$ of $F$ consists of a Noetherian", "complete local $S$-algebra $R$ whose residue field is of finite type", "over $S$, together with elements $\\xi_n \\in F(\\Spec(R/\\mathfrak m^n))$", "for all $n$ such that $\\xi_{n + 1}|_{\\Spec(R/\\mathfrak m^n)} = \\xi_n$.", "By the discussion in the proof of Lemma \\ref{lemma-rs}", "we see that either $\\xi$ is a formal object of $U'$ or a formal", "object of $W$. In the first case we see that $\\xi$ is effective", "by Lemma \\ref{lemma-effective}. The second case is the interesting case.", "Set $V = \\Spec(R)$. We will construct an element", "$(Z, u', \\hat x) \\in F(V)$ whose image in $F(\\Spec(R/\\mathfrak m^n))$", "is $\\xi_n$ for all $n \\geq 1$.", "\\medskip\\noindent", "We may view the collection of elements $\\xi_n$ as a morphism", "$$", "\\xi : \\text{Spf}(R) \\longrightarrow W", "$$", "of locally Noetherian formal algebraic spaces over $S$. Observe that $\\xi$", "is {\\it not} an adic morphism in general. To fix this, let $I \\subset R$", "be the ideal corresponding to the formal closed subspace", "$$", "\\text{Spf}(R) \\times_{\\xi, W} W_{red} \\subset \\text{Spf}(R)", "$$", "Note that $I \\subset \\mathfrak m_R$. Set $Z = V(I) \\subset V = \\Spec(R)$.", "Since $R$ is $\\mathfrak m_R$-adically complete it is a fortiori", "$I$-adically complete (Algebra, Lemma \\ref{algebra-lemma-complete-by-sub}).", "Moreover, we claim that for each $n \\geq 1$ the morphism", "$$", "\\xi|_{\\text{Spf}(R/I^n)} :", "\\text{Spf}(R/I^n)", "\\longrightarrow", "W", "$$", "actually comes from a morphism", "$$", "\\xi'_n : \\Spec(R/I^n) \\longrightarrow W", "$$", "Namely, this follows from writing $W = \\colim W_n$ as in the", "proof of Lemma \\ref{lemma-rs}, noticing that $\\xi|_{\\text{Spf}(R/I^n)}$", "maps into $W_n$, and applying Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-map-into-algebraic-space}", "to algebraize this to a morphism $\\Spec(R/I^n) \\to W_n$", "as desired. Let us denote $\\text{Spf}'(R) = V_{/Z}$ the formal spectrum", "of $R$ endowed with the $I$-adic topology -- equivalently the formal", "completion of $V$ along $Z$. Using the morphisms", "$\\xi'_n$ we obtain an adic morphism", "$$", "\\hat x = (\\xi'_n) : \\text{Spf}'(R) \\longrightarrow W", "$$", "of locally Noetherian formal algebraic spaces over $S$.", "Consider the base change", "$$", "\\text{Spf}'(R) \\times_{\\hat x, W, g} X'_{/T'} \\longrightarrow \\text{Spf}'(R)", "$$", "This is a formal modification by", "Algebraization of Formal Spaces, Lemma", "\\ref{restricted-lemma-base-change-formal-modification}.", "Hence by the main theorem on dilatations", "(Algebraization of Formal Spaces, Theorem \\ref{restricted-theorem-dilatations})", "we obtain a proper morphism", "$$", "V' \\longrightarrow V = \\Spec(R)", "$$", "which is an isomorphism over $\\Spec(R) \\setminus V(I)$ and", "whose completion recovers the formal modification above, in other words", "$$", "V' \\times_{\\Spec(R)} \\Spec(R/I^n) =", "\\Spec(R/I^n) \\times_{\\xi'_n, W, g} X'_{/T'}", "$$", "This in particular tells us we have a compatible system of morphisms", "$$", "V' \\times_{\\Spec(R)} \\Spec(R/I^n) \\longrightarrow X' \\times_S \\Spec(R/I^n)", "$$", "Hence by Grothendieck's algebraization theorem (in the form of", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-algebraize-morphism})", "we obtain a morphism", "$$", "x' : V' \\to X'", "$$", "over $S$ recovering the morphisms displayed above. Then finally", "setting $u' : V \\setminus Z \\to X'$ the restriction of $x'$ to", "$V \\setminus Z \\subset V'$ gives the third component of our", "desired element $(Z, u', \\hat x) \\in F(V)$." ], "refs": [ "artin-lemma-rs", "artin-lemma-effective", "algebra-lemma-complete-by-sub", "artin-lemma-rs", "formal-spaces-lemma-map-into-algebraic-space", "restricted-lemma-base-change-formal-modification", "restricted-theorem-dilatations", "spaces-more-morphisms-lemma-algebraize-morphism" ], "ref_ids": [ 11409, 11362, 864, 11409, 3964, 2411, 2293, 211 ] } ], "ref_ids": [] }, { "id": 11412, "type": "theorem", "label": "artin-lemma-openness-smoothness", "categories": [ "artin" ], "title": "artin-lemma-openness-smoothness", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $V$ be a scheme locally", "of finite type over $S$. Let $Z \\subset V$ be closed. Let $W$ be", "a locally Noetherian formal algebraic space over $S$ such that", "$W_{red}$ is locally of finite type over $S$. Let $g : V_{/Z} \\to W$", "be an adic morphism of formal algebraic spaces over $S$. Let $v \\in V$", "be a closed point such that $g$ is versal at $v$ (as in", "Section \\ref{section-axioms-functors}).", "Then after replacing $V$ by an open neighbourhood of $v$ the", "morphism $g$ is smooth (see proof)." ], "refs": [], "proofs": [ { "contents": [ "Since $g$ is adic it is representable by algebraic spaces (Formal Spaces,", "Section \\ref{formal-spaces-section-adic}).", "Thus by saying $g$ is smooth we mean that $g$ should be smooth", "in the sense of", "Bootstrap, Definition \\ref{bootstrap-definition-property-transformation}.", "\\medskip\\noindent", "Write $W = \\colim W_n$ as in Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-structure-locally-noetherian}.", "Set $V_n = V_{/Z} \\times_{\\hat x, W} W_n$.", "Then $V_n$ is a closed subscheme with underlying set $Z$.", "Smoothness of $V \\to W$ is equivalent to the smoothness", "of all the morphisms $V_n \\to W_n$ (this holds because any morphism", "$T \\to W$ with $T$ a quasi-compact scheme factors through", "$W_n$ for some $n$). We know that the morphism $V_n \\to W_n$", "is smooth at $v$ by", "Lemma \\ref{lemma-base-change-versal}\\footnote{The lemma applies since", "the diagonal of $W$ is representable by algebraic spaces and", "locally of finite type, see Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-diagonal-morphism-formal-algebraic-spaces}", "and we have seen that $W$ has (RS) in the proof of Lemma \\ref{lemma-rs}.}.", "Of course this means that given any $n$ we can shrink $V$", "such that $V_n \\to W_n$ is smooth. The problem is to find", "an open which works for all $n$ at the same time.", "\\medskip\\noindent", "The question is local on $V$, hence we may assume $S = \\Spec(R)$ and", "$V = \\Spec(A)$ are affine.", "\\medskip\\noindent", "In this paragraph we reduce to the case where $W$ is an affine formal", "algebraic space. Choose an affine formal scheme $W'$ and an \\'etale morphism", "$W' \\to W$ such that the image of $v$ in $W_{red}$ is in the", "image of $W'_{red} \\to W_{red}$. Then $V_{/Z} \\times_{g, W} W' \\to V_{/Z}$", "is an adic \\'etale morphism of formal algebraic spaces over $S$", "and $V_{/Z} \\times_{g, W} W'$ is an affine fromal algebraic space.", "By Algebraization of Formal Spaces,", "Lemma \\ref{restricted-lemma-algebraize-rig-etale-affine}", "there exists an \\'etale morphism $\\varphi : V' \\to V$ of affine schemes", "such that the completion of $V'$ along $Z' = \\varphi^{-1}(Z)$", "is isomorphic to $V_{/Z} \\times_{g, W} W'$ over $V_{/Z}$.", "Observe that $v$ is the image of some $v' \\in V'$.", "Since smoothness is preserved under base change we see that", "$V'_n \\to W'_n$ is smooth for all $n$. In the next paragraph", "we show that after replacing $V'$ by an open neighbourhood of $v'$", "the morphisms $V'_n \\to W'_n$ are smooth for all $n$.", "Then, after we replace $V$ by the open image of $V' \\to V$,", "we obtain that $V_n \\to W_n$ is smooth by \\'etale descent of smoothness.", "Some details omitted.", "\\medskip\\noindent", "Assume $S = \\Spec(R)$, $V = \\Spec(A)$, $Z = V(I)$, and $W = \\text{Spf}(B)$.", "Let $v$ correspond to the maximal ideal $I \\subset \\mathfrak m \\subset A$.", "We are given an adic continuous $R$-algebra homomorphism", "$$", "B \\longrightarrow A^\\wedge", "$$", "Let $\\mathfrak b \\subset B$ be the ideal of topologically nilpotent", "elements (this is the maximal ideal of definition of the Noetherian adic", "topological ring $B$). Observe that $\\mathfrak b A^\\wedge$ and", "$IA^\\wedge$ are both ideals of definition of the Noetherian adic", "ring $A^\\wedge$. Also, $\\mathfrak m A^\\wedge$ is a maximal ideal", "of $A^\\wedge$ containing both $\\mathfrak b A^\\wedge$ and $IA^\\wedge$.", "We are given that", "$$", "B_n = B/\\mathfrak b^n \\to A^\\wedge/\\mathfrak b^n A^\\wedge = A_n", "$$", "is smooth at $\\mathfrak m$ for all $n$. By the discussion above", "we may and do assume that $B_1 \\to A_1$ is a smooth ring map.", "Denote $\\mathfrak m_1 \\subset A_1$ the maximal ideal corresponing", "to $\\mathfrak m$. Since smoothness implies flatness, we see that:", "for all $n \\geq 1$ the map", "$$", "\\mathfrak b^n/\\mathfrak b^{n + 1} \\otimes_{B_1} (A_1)_{\\mathfrak m_1}", "\\longrightarrow", "\\left(\\mathfrak b^nA^\\wedge/\\mathfrak b^{n + 1}A^\\wedge\\right)_{\\mathfrak m_1}", "$$", "is an isomorphism (see", "Algebra, Lemma \\ref{algebra-lemma-what-does-it-mean-again}).", "Consider the Rees algebra", "$$", "B' = \\bigoplus\\nolimits_{n \\geq 0} \\mathfrak b^n/\\mathfrak b^{n + 1}", "$$", "which is a finite type graded algebra over the Noetherian ring $B_1$ and", "the Rees algebra", "$$", "A' = \\bigoplus\\nolimits_{n \\geq 0}", "\\mathfrak b^nA^\\wedge/\\mathfrak b^{n + 1}A^\\wedge", "$$", "which is a a finite type graded algebra over the Noetherian ring $A_1$.", "Consider the homomorphism of graded $A_1$-algebras", "$$", "\\Psi : B' \\otimes_{B_1} A_1 \\longrightarrow A'", "$$", "By the above this map is an isomorphism after localizing at", "the maximal ideal $\\mathfrak m_1$ of $A_1$.", "Hence $\\Ker(\\Psi)$, resp.\\ $\\Coker(\\Psi)$ is a finite module", "over $B' \\otimes_{B_1} A_1$, resp.\\ $A'$ whose localization", "at $\\mathfrak m_1$ is zero. It follows that after replacing", "$A_1$ (and correspondingly $A$) by a principal localization", "we may assume $\\Psi$ is an isomorphism. (This is the key step of the proof.)", "Then working backwards we see that $B_n \\to A_n$ is flat, see", "Algebra, Lemma \\ref{algebra-lemma-what-does-it-mean-again}.", "Hence $A_n \\to B_n$ is smooth (as a flat ring map with smooth", "fibres, see Algebra, Lemma \\ref{algebra-lemma-flat-fibre-smooth})", "and the proof is complete." ], "refs": [ "bootstrap-definition-property-transformation", "formal-spaces-lemma-structure-locally-noetherian", "artin-lemma-base-change-versal", "formal-spaces-lemma-diagonal-morphism-formal-algebraic-spaces", "artin-lemma-rs", "restricted-lemma-algebraize-rig-etale-affine", "algebra-lemma-what-does-it-mean-again", "algebra-lemma-what-does-it-mean-again", "algebra-lemma-flat-fibre-smooth" ], "ref_ids": [ 2638, 3921, 11371, 3893, 11409, 2412, 891, 891, 1200 ] } ], "ref_ids": [] }, { "id": 11413, "type": "theorem", "label": "artin-lemma-openness-versality", "categories": [ "artin" ], "title": "artin-lemma-openness-versality", "contents": [ "In Situation \\ref{situation-contractions} the functor", "$F$ satisfies openness of versality." ], "refs": [], "proofs": [ { "contents": [ "We have to show the following. Given a scheme $V$ locally of finite type over", "$S$, given $\\xi \\in F(V)$, and given a finite type point $v_0 \\in V$ such that", "$\\xi$ is versal at $v_0$, after replacing $V$ by an open neighbourhood", "of $v_0$ we have that $\\xi$ is versal at every finite type point of $V$.", "Write $\\xi = (Z, u', \\hat x)$.", "\\medskip\\noindent", "First case: $v_0 \\not \\in Z$. Then we can first replace $V$ by", "$V \\setminus Z$. Hence we see that $\\xi = (\\emptyset, u', \\emptyset)$", "and the morphism $u' : V \\to X'$ is versal at $v_0$.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-lifting-along-artinian-at-point}", "this means that $u' : V \\to X'$ is smooth at $v_0$.", "Since the set of a points where a morphism is smooth is open,", "we can after shrinking $V$ assume $u'$ is smooth.", "Then the same lemma tells us that $\\xi$ is versal at every", "point as desired.", "\\medskip\\noindent", "Second case: $v_0 \\in Z$. Write $W = \\colim W_n$ as in", "Formal Spaces, Lemma \\ref{formal-spaces-lemma-structure-locally-noetherian}.", "By Lemma \\ref{lemma-openness-smoothness} we may assume $\\hat x : V_{/Z} \\to W$", "is a smooth morphism of formal algebraic spaces. It follows immediately", "that $\\xi = (Z, u', \\hat x)$ is versal at all finite type points of $Z$.", "Let $V' \\to V$, $\\hat x'$, and $x'$ witness the compatibility between $u'$", "and $\\hat x$. We see that $\\hat x' : V'_{/Z} \\to X'_{/T'}$ is smooth as a", "base change of $\\hat x$. Since $\\hat x'$ is the completion of", "$x' : V' \\to X'$ this implies that $x' : V' \\to X'$ is smooth at all", "points of $(V' \\to V)^{-1}(Z) = |x'|^{-1}(T') \\subset |V'|$", "by the already used More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-lifting-along-artinian-at-point}.", "Since the set of smooth points of a morphism is open, we see that", "the closed set of points $B \\subset |V'|$ where $x'$ is not smooth", "does not meet $(V' \\to V)^{-1}(Z)$. Since $V' \\to V$ is proper and", "hence closed, we see that $(V' \\to V)(B) \\subset V$ is a closed", "subset not meeting $Z$. Hence after shrinking $V$ we may assume", "$B = \\emptyset$, i.e., $x'$ is smooth. By the discussion in the previous", "paragraph this exactly means that $\\xi$ is versal at all finite type", "points of $V$ not contained in $Z$ and the proof is complete." ], "refs": [ "spaces-more-morphisms-lemma-lifting-along-artinian-at-point", "formal-spaces-lemma-structure-locally-noetherian", "artin-lemma-openness-smoothness", "spaces-more-morphisms-lemma-lifting-along-artinian-at-point" ], "ref_ids": [ 119, 3921, 11412, 119 ] } ], "ref_ids": [] }, { "id": 11414, "type": "theorem", "label": "artin-proposition-spaces-diagonal-representable", "categories": [ "artin" ], "title": "artin-proposition-spaces-diagonal-representable", "contents": [ "Let $S$ be a locally Noetherian scheme. Let", "$F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor. Assume that", "\\begin{enumerate}", "\\item $\\Delta : F \\to F \\times F$ is representable by algebraic spaces,", "\\item $F$ satisfies axioms [-1], [0], [1], [2], [3], [4], [5]", "(see Section \\ref{section-axioms-functors}), and", "\\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$.", "\\end{enumerate}", "Then $F$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "Lemma \\ref{lemma-get-smooth} applies to $F$. Using this we", "choose, for every finite type field $k$ over $S$ and $x_0 \\in F(\\Spec(k))$,", "an affine scheme $U_{k, x_0}$ of finite type over $S$ and a smooth morphism", "$U_{k, x_0} \\to F$ such that there exists a finite type point", "$u_{k, x_0} \\in U_{k, x_0}$ with residue field $k$ such that $x_0$", "is the image of $u_{k, x_0}$. Then", "$$", "U = \\coprod\\nolimits_{k, x_0} U_{k, x_0} \\longrightarrow F", "$$", "is smooth\\footnote{Set theoretical remark: This coproduct is (isomorphic)", "to an object of $(\\Sch/S)_{fppf}$ as we have a bound on the index set", "by axiom [-1], see Sets, Lemma \\ref{sets-lemma-what-is-in-it}.}.", "To finish the proof it suffices to show this map is surjective,", "see Bootstrap, Lemma \\ref{bootstrap-lemma-spaces-etale-smooth-cover}", "(this is where we use axiom [0]). By Criteria for Representability, Lemma", "\\ref{criteria-lemma-check-property-limit-preserving}", "it suffices to show that $U \\times_F V \\to V$ is surjective for those", "$V \\to F$ where $V$ is an affine scheme locally of finite presentation", "over $S$. Since $U \\times_F V \\to V$ is smooth the image is open. Hence", "it suffices to show that the image of $U \\times_F V \\to V$ contains all", "finite type points of $V$, see", "Morphisms, Lemma \\ref{morphisms-lemma-enough-finite-type-points}.", "Let $v_0 \\in V$ be a finite type point. Then $k = \\kappa(v_0)$ is", "a finite type field over $S$. Denote $x_0$ the composition", "$\\Spec(k) \\xrightarrow{v_0} V \\to F$. Then", "$(u_{k, x_0}, v_0) : \\Spec(k) \\to U \\times_F V$ is a point mapping to", "$v_0$ and we win." ], "refs": [ "artin-lemma-get-smooth", "sets-lemma-what-is-in-it", "bootstrap-lemma-spaces-etale-smooth-cover", "criteria-lemma-check-property-limit-preserving", "morphisms-lemma-enough-finite-type-points" ], "ref_ids": [ 11378, 8795, 2635, 3104, 5210 ] } ], "ref_ids": [] }, { "id": 11415, "type": "theorem", "label": "artin-proposition-second-diagonal-representable", "categories": [ "artin" ], "title": "artin-proposition-second-diagonal-representable", "contents": [ "Let $S$ be a locally Noetherian scheme. Let", "$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.", "Assume that", "\\begin{enumerate}", "\\item $\\Delta_\\Delta : \\mathcal{X} \\to", "\\mathcal{X} \\times_{\\mathcal{X} \\times \\mathcal{X}} \\mathcal{X}$", "is representable by algebraic spaces,", "\\item $\\mathcal{X}$ satisfies axioms [-1], [0], [1], [2], [3], [4], and [5]", "(see Section \\ref{section-axioms}),", "\\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$.", "\\end{enumerate}", "Then $\\mathcal{X}$ is an algebraic stack." ], "refs": [], "proofs": [ { "contents": [ "We first prove that $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$", "is representable by algebraic spaces. To do this it suffices to show", "that", "$$", "\\mathcal{Y} =", "\\mathcal{X} \\times_{\\Delta, \\mathcal{X} \\times \\mathcal{X}, y} (\\Sch/V)_{fppf}", "$$", "is representable by an algebraic space for any affine scheme $V$ locally", "of finite presentation over $S$ and object $y$ of", "$\\mathcal{X} \\times \\mathcal{X}$ over $V$, see", "Criteria for Representability, Lemma", "\\ref{criteria-lemma-check-representable-limit-preserving}\\footnote{The", "set theoretic condition in Criteria for Representability, Lemma", "\\ref{criteria-lemma-check-representable-limit-preserving}", "will hold: the size of the algebraic space $Y$ representing $\\mathcal{Y}$ is", "suitably bounded. Namely, $Y \\to S$ will be locally of finite type and $Y$", "will satisfy axiom [-1]. Details omitted.}.", "Observe that $\\mathcal{Y}$ is fibred in setoids", "(Stacks, Lemma \\ref{stacks-lemma-isom-as-2-fibre-product})", "and let $Y : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$,", "$T \\mapsto \\Ob(\\mathcal{Y}_T)/\\cong$ be the functor of isomorphism", "classes. We will apply", "Proposition \\ref{proposition-spaces-diagonal-representable}", "to see that $Y$ is an algebraic space.", "\\medskip\\noindent", "Note that", "$\\Delta_\\mathcal{Y} : \\mathcal{Y} \\to \\mathcal{Y} \\times \\mathcal{Y}$", "(and hence also $Y \\to Y \\times Y$)", "is representable by algebraic spaces by condition (1) and", "Criteria for Representability, Lemma \\ref{criteria-lemma-second-diagonal}.", "Observe that $Y$ is a sheaf for the \\'etale topology by", "Stacks, Lemmas \\ref{stacks-lemma-stack-in-setoids-characterize} and", "\\ref{stacks-lemma-2-fibre-product-gives-stack-in-setoids}, i.e.,", "axiom [0] holds. Also $Y$ is limit preserving by", "Lemma \\ref{lemma-fibre-product-limit-preserving}, i.e., we have [1].", "Note that $Y$ has (RS), i.e., axiom [2] holds, by", "Lemmas \\ref{lemma-algebraic-stack-RS} and", "\\ref{lemma-fibre-product-RS}. Axiom [3] for $Y$ follows", "from Lemmas \\ref{lemma-finite-dimension} and", "\\ref{lemma-fibre-product-tangent-spaces}.", "Axiom [4] follows from Lemmas \\ref{lemma-effective} and", "\\ref{lemma-fibre-product-effective}.", "Axiom [5] for $Y$ follows directly from openness of versality", "for $\\Delta_\\mathcal{X}$ which is part of axiom [5] for $\\mathcal{X}$.", "Thus all the assumptions of", "Proposition \\ref{proposition-spaces-diagonal-representable}", "are satisfied and $Y$ is an algebraic space.", "\\medskip\\noindent", "At this point it follows from Lemma \\ref{lemma-diagonal-representable}", "that $\\mathcal{X}$ is an algebraic stack." ], "refs": [ "criteria-lemma-check-representable-limit-preserving", "criteria-lemma-check-representable-limit-preserving", "stacks-lemma-isom-as-2-fibre-product", "artin-proposition-spaces-diagonal-representable", "criteria-lemma-second-diagonal", "stacks-lemma-stack-in-setoids-characterize", "stacks-lemma-2-fibre-product-gives-stack-in-setoids", "artin-lemma-fibre-product-limit-preserving", "artin-lemma-algebraic-stack-RS", "artin-lemma-fibre-product-RS", "artin-lemma-finite-dimension", "artin-lemma-fibre-product-tangent-spaces", "artin-lemma-effective", "artin-lemma-fibre-product-effective", "artin-proposition-spaces-diagonal-representable", "artin-lemma-diagonal-representable" ], "ref_ids": [ 3103, 3103, 8936, 11414, 3098, 8951, 8954, 11365, 11355, 11356, 11359, 11360, 11362, 11363, 11414, 11380 ] } ], "ref_ids": [] }, { "id": 11416, "type": "theorem", "label": "artin-proposition-spaces-diagonal-representable-noetherian", "categories": [ "artin" ], "title": "artin-proposition-spaces-diagonal-representable-noetherian", "contents": [ "Let $S$ be a locally Noetherian scheme. Let", "$F : (\\textit{Noetherian}/S)_\\etale^{opp} \\to \\textit{Sets}$", "be a functor. Assume that", "\\begin{enumerate}", "\\item $\\Delta : F \\to F \\times F$ is representable", "(as a transformation of functors, see", "Categories, Definition \\ref{categories-definition-representable-morphism}),", "\\item $F$ satisfies axioms [-1], [0], [1], [2], [3], [4], [5]", "(see above), and", "\\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$.", "\\end{enumerate}", "Then there exists a unique algebraic space", "$F' : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$", "whose restriction to $(\\textit{Noetherian}/S)_\\etale$ is $F$", "(see proof for elucidation)." ], "refs": [ "categories-definition-representable-morphism" ], "proofs": [ { "contents": [ "Recall that the sites $(\\Sch/S)_{fppf}$ and $(\\Sch/S)_\\etale$ have the same", "underlying category, see discussion in Section \\ref{section-noetherian}.", "Similarly the sites $(\\textit{Noetherian}/S)_\\etale$ and", "$(\\textit{Noetherian}/S)_{fppf}$ have the same underlying categories.", "By axioms [0] and [1] the functor $F$ is a sheaf and", "limit preserving.", "Let $F' : (\\Sch/S)_\\etale^{opp} \\to \\textit{Sets}$", "be the unique extension of $F$ which is a sheaf (for the \\'etale topology)", "and which is limit preserving, see", "Lemma \\ref{lemma-canonical-extension}.", "Then $F'$ satisfies axioms [0] and [1] as given in", "Section \\ref{section-axioms-functors}.", "By Lemma \\ref{lemma-representable} we see that", "$\\Delta' : F' \\to F' \\times F'$ is representable (by schemes).", "On the other hand, it is immediately clear that", "$F'$ satisfies axioms [-1], [2], [3], [4], [5] of", "Section \\ref{section-axioms-functors}", "as each of these involves only evaluating $F'$ at objects", "of $(\\textit{Noetherian}/S)_\\etale$ and we've assumed the", "corresponding conditions for $F$.", "Whence $F'$ is an algebraic space by", "Proposition \\ref{proposition-spaces-diagonal-representable}." ], "refs": [ "artin-lemma-canonical-extension", "artin-lemma-representable", "artin-proposition-spaces-diagonal-representable" ], "ref_ids": [ 11399, 11401, 11414 ] } ], "ref_ids": [ 12348 ] }, { "id": 11442, "type": "theorem", "label": "obsolete-theorem-coherent-algebraic-general", "categories": [ "obsolete" ], "title": "obsolete-theorem-coherent-algebraic-general", "contents": [ "Let $S$ be a scheme. Let $f : X \\to B$ be morphism of algebraic spaces", "over $S$. Assume that $f$ is of finite presentation and separated. Then", "$\\textit{Coh}_{X/B}$ is an algebraic stack over $S$." ], "refs": [], "proofs": [ { "contents": [ "This theorem is a copy of Quot, Theorem", "\\ref{quot-theorem-coherent-algebraic-general}.", "The reason we have this copy here is that with the", "material in this section we get a second proof (as discussed", "at the beginning of this section). Namely,", "we argue exactly as in the proof of", "Quot, Theorem \\ref{quot-theorem-coherent-algebraic}", "except that we substitute", "Lemma \\ref{lemma-coherent-defo-thy-general} for", "Quot, Lemma \\ref{quot-lemma-coherent-defo-thy}." ], "refs": [ "quot-theorem-coherent-algebraic-general", "quot-theorem-coherent-algebraic", "obsolete-lemma-coherent-defo-thy-general", "quot-lemma-coherent-defo-thy" ], "ref_ids": [ 3147, 3146, 11501, 3168 ] } ], "ref_ids": [] }, { "id": 11443, "type": "theorem", "label": "obsolete-theorem-equivalence-sheaves-point", "categories": [ "obsolete" ], "title": "obsolete-theorem-equivalence-sheaves-point", "contents": [ "Let $S = \\Spec(K)$ with $K$ a field.", "Let $\\overline{s}$ be a geometric point of $S$.", "Let $G = \\text{Gal}_{\\kappa(s)}$ denote the absolute Galois group.", "Then there is an equivalence of categories", "$\\Sh(S_\\etale) \\to G\\textit{-Sets}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{s}}$." ], "refs": [], "proofs": [ { "contents": [ "This is a duplicate of \\'Etale Cohomology, Theorem", "\\ref{etale-cohomology-theorem-equivalence-sheaves-point}." ], "refs": [ "etale-cohomology-theorem-equivalence-sheaves-point" ], "ref_ids": [ 6386 ] } ], "ref_ids": [] }, { "id": 11444, "type": "theorem", "label": "obsolete-lemma-finite-presentation-module-independent", "categories": [ "obsolete" ], "title": "obsolete-lemma-finite-presentation-module-independent", "contents": [ "Let $M$ be an $R$-module of finite presentation.", "For any surjection $\\alpha : R^{\\oplus n} \\to M$ the", "kernel of $\\alpha$ is a finite $R$-module." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Algebra, Lemma \\ref{algebra-lemma-extension}." ], "refs": [ "algebra-lemma-extension" ], "ref_ids": [ 330 ] } ], "ref_ids": [] }, { "id": 11445, "type": "theorem", "label": "obsolete-lemma-p-ring-map", "categories": [ "obsolete" ], "title": "obsolete-lemma-p-ring-map", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. If", "\\begin{enumerate}", "\\item for any $x \\in S$ there exists $n > 0$ such that", "$x^n$ is in the image of $\\varphi$, and", "\\item for any $x \\in \\Ker(\\varphi)$ there exists $n > 0$", "such that $x^n = 0$,", "\\end{enumerate}", "then $\\varphi$ induces a homeomorphism on spectra. Given a", "prime number $p$ such that", "\\begin{enumerate}", "\\item[(a)] $S$ is generated as an $R$-algebra by elements $x$ such", "that there exists an $n > 0$ with $x^{p^n} \\in \\varphi(R)$ and", "$p^nx \\in \\varphi(R)$, and", "\\item[(b)] the kernel of $\\varphi$ is generated by nilpotent elements,", "\\end{enumerate}", "then (1) and (2) hold, and for any ring map $R \\to R'$", "the ring map $R' \\to R' \\otimes_R S$ also satisfies (a), (b), (1), and (2)", "and in particular induces a homeomorphism on spectra." ], "refs": [], "proofs": [ { "contents": [ "This is a combination of", "Algebra, Lemmas \\ref{algebra-lemma-powers} and", "\\ref{algebra-lemma-p-ring-map}." ], "refs": [ "algebra-lemma-powers", "algebra-lemma-p-ring-map" ], "ref_ids": [ 578, 582 ] } ], "ref_ids": [] }, { "id": 11446, "type": "theorem", "label": "obsolete-lemma-lift-elements-ideal", "categories": [ "obsolete" ], "title": "obsolete-lemma-lift-elements-ideal", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak p \\subset R$ be a prime.", "Let $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p$.", "Assume $S_{\\mathfrak q}$ is essentially of finite type over $R_\\mathfrak p$.", "Assume given", "\\begin{enumerate}", "\\item an integer $n \\geq 0$,", "\\item a prime $\\mathfrak a \\subset \\kappa(\\mathfrak p)[x_1, \\ldots, x_n]$,", "\\item a surjective $\\kappa(\\mathfrak p)$-homomorphism", "$$", "\\psi : (\\kappa(\\mathfrak p)[x_1, \\ldots, x_n])_{\\mathfrak a}", "\\longrightarrow", "S_{\\mathfrak q}/\\mathfrak p S_{\\mathfrak q},", "$$", "and", "\\item elements $\\overline{f}_1, \\ldots, \\overline{f}_e$ in $\\Ker(\\psi)$.", "\\end{enumerate}", "Then there exist", "\\begin{enumerate}", "\\item an integer $m \\geq 0$,", "\\item and element $g \\in S$, $g \\not\\in \\mathfrak q$,", "\\item a map", "$$", "\\Psi :", "R[x_1, \\ldots, x_n, x_{n + 1}, \\ldots, x_{n + m}]", "\\longrightarrow", "S_g,", "$$", "and", "\\item elements $f_1, \\ldots, f_e, f_{e + 1}, \\ldots, f_{e + m}$", "of $\\Ker(\\Psi)$", "\\end{enumerate}", "such that", "\\begin{enumerate}", "\\item the following diagram commutes", "$$", "\\xymatrix{", "R[x_1, \\ldots, x_{n + m}] \\ar[d]_\\Psi", "\\ar[rr]_-{x_{n + j} \\mapsto 0} & &", "(\\kappa(\\mathfrak p)[x_1, \\ldots, x_n])_{\\mathfrak a} \\ar[d]^\\psi \\\\", "S_g \\ar[rr] & &", "S_{\\mathfrak q}/\\mathfrak p S_{\\mathfrak q}", "},", "$$", "\\item the element $f_i$, $i \\leq n$ maps to a unit times", "$\\overline{f}_i$ in the local ring", "$$", "(\\kappa(\\mathfrak p)[x_1, \\ldots, x_{n + m}])_{", "(\\mathfrak a, x_{n + 1}, \\ldots, x_{n + m})},", "$$", "\\item the element $f_{e + j}$ maps to", "a unit times $x_{n + j}$ in the same local ring, and", "\\item the induced map $R[x_1, \\ldots, x_{n + m}]_{\\mathfrak b}", "\\to S_{\\mathfrak q}$ is surjective, where", "$\\mathfrak b = \\Psi^{-1}(\\mathfrak qS_g)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We claim that it suffices to prove the lemma in case $R$", "and $S$ are local with maximal ideals $\\mathfrak p$ and $\\mathfrak q$.", "Namely, suppose we have constructed", "$$", "\\Psi' : R_{\\mathfrak p}[x_1, \\ldots, x_{n + m}]", "\\longrightarrow", "S_{\\mathfrak q}", "$$", "and $f_1', \\ldots, f_{e + m}' \\in R_{\\mathfrak p}[x_1, \\ldots, x_{n + m}]$", "with all the required properties. Then there exists an element", "$f \\in R$, $f \\not \\in \\mathfrak p$ such that each", "$ff_k'$ comes from an element $f_k \\in R[x_1, \\ldots, x_{n + m}]$.", "Moreover, for a suitable $g \\in S$, $g \\not \\in \\mathfrak q$", "the elements $\\Psi'(x_i)$ are the image of elements", "$y_i \\in S_g$. Let $\\Psi$ be the $R$-algebra map defined", "by the rule $\\Psi(x_i) = y_i$. Since $\\Psi(f_i)$ is zero", "in the localization $S_{\\mathfrak q}$ we may after possibly", "replacing $g$ assume that $\\Psi(f_i) = 0$. This proves the claim.", "\\medskip\\noindent", "Thus we may assume $R$ and $S$ are local", "with maximal ideals $\\mathfrak p$ and $\\mathfrak q$.", "Pick $y_1, \\ldots, y_n \\in S$ such that", "$y_i \\bmod \\mathfrak pS = \\psi(x_i)$.", "Let $y_{n + 1}, \\ldots, y_{n + m} \\in S$ be elements which generate", "an $R$-subalgebra of which $S$ is the localization.", "These exist by the assumption that $S$ is essentially of", "finite type over $R$. Since $\\psi$ is surjective we", "may write $y_{n + j} \\bmod \\mathfrak pS = \\psi(h_j)$ for", "some $h_j \\in \\kappa(\\mathfrak p)[x_1, \\ldots, x_n]_{\\mathfrak a}$.", "Write $h_j = g_j/d$, $g_j \\in \\kappa(\\mathfrak p)[x_1, \\ldots, x_n]$", "for some common denominator $d \\in \\kappa(\\mathfrak p)[x_1, \\ldots, x_n]$,", "$d \\not \\in \\mathfrak a$. Choose lifts $G_j, D \\in R[x_1, \\ldots, x_n]$", "of $g_j$ and $d$. Set", "$y_{n + j}' = D(y_1, \\ldots, y_n) y_{n + j} - G_j(y_1, \\ldots, y_n)$.", "By construction $y_{n + j}' \\in \\mathfrak p S$.", "It is clear that $y_1, \\ldots, y_n, y_n', \\ldots, y_{n + m}'$", "generate an $R$-subalgebra of $S$ whose localization is $S$.", "We define", "$$", "\\Psi : R[x_1, \\ldots, x_{n + m}] \\to S", "$$", "to be the map that sends $x_i$ to $y_i$ for $i = 1, \\ldots, n$", "and $x_{n + j}$ to $y'_{n + j}$ for $j = 1, \\ldots, m$. Properties", "(1) and (4) are clear by construction. Moreover the ideal", "$\\mathfrak b$ maps onto the ideal", "$(\\mathfrak a, x_{n + 1}, \\ldots, x_{n + m})$", "in the polynomial ring $\\kappa(\\mathfrak p)[x_1, \\ldots, x_{n + m}]$.", "\\medskip\\noindent", "Denote $J = \\Ker(\\Psi)$. We have a short exact sequence", "$$", "0 \\to J_{\\mathfrak b}", "\\to R[x_1, \\ldots, x_{n + m}]_{\\mathfrak b}", "\\to S_{\\mathfrak q}", "\\to 0.", "$$", "The surjectivity comes from our choice of", "$y_1, \\ldots, y_n, y_n', \\ldots, y_{n + m}'$ above.", "This implies that", "$$", "J_{\\mathfrak b}/ \\mathfrak pJ_{\\mathfrak b}", "\\to \\kappa(\\mathfrak p)[x_1, \\ldots, x_{n + m}]_{", "(\\mathfrak a, x_{n + 1}, \\ldots, x_{n + m})}", "\\to S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}", "\\to 0", "$$", "is exact. By construction $x_i$ maps to $\\psi(x_i)$ and", "$x_{n + j}$ maps to zero under the last map.", "Thus it is easy to choose $f_i$ as in", "(2) and (3) of the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11447, "type": "theorem", "label": "obsolete-lemma-spec-localization-first", "categories": [ "obsolete" ], "title": "obsolete-lemma-spec-localization-first", "contents": [ "Let $S$ be a multiplicative set of $A$. Then the map", "$$", "f: \\Spec(S^{-1}A)\\longrightarrow \\Spec(A)", "$$", "induced by the canonical ring map", "$A \\to S^{-1}A$ is a homeomorphism onto its image and", "$\\Im(f) = \\{ \\mathfrak p \\in \\Spec(A) : \\mathfrak p\\cap S = \\emptyset \\}$." ], "refs": [], "proofs": [ { "contents": [ "This is a duplicate of Algebra, Lemma \\ref{algebra-lemma-spec-localization}." ], "refs": [ "algebra-lemma-spec-localization" ], "ref_ids": [ 391 ] } ], "ref_ids": [] }, { "id": 11448, "type": "theorem", "label": "obsolete-lemma-finite-type-flat-over-integral-algebra", "categories": [ "obsolete" ], "title": "obsolete-lemma-finite-type-flat-over-integral-algebra", "contents": [ "Let $A \\to B$ be a finite type, flat ring map with $A$ an integral", "domain. Then $B$ is a finitely presented $A$-algebra." ], "refs": [], "proofs": [ { "contents": [ "Special case of More on Flatness, Proposition", "\\ref{flat-proposition-flat-finite-type-finite-presentation-domain}." ], "refs": [ "flat-proposition-flat-finite-type-finite-presentation-domain" ], "ref_ids": [ 6201 ] } ], "ref_ids": [] }, { "id": 11449, "type": "theorem", "label": "obsolete-lemma-helper-finite-type-flat-finite-presentation", "categories": [ "obsolete" ], "title": "obsolete-lemma-helper-finite-type-flat-finite-presentation", "contents": [ "Let $R$ be a domain with fraction field $K$.", "Let $S = R[x_1, \\ldots, x_n]$ be a polynomial ring over $R$.", "Let $M$ be a finite $S$-module. Assume that $M$ is flat over $R$.", "If for every subring $R \\subset R' \\subset K$, $R \\not = R'$", "the module $M \\otimes_R R'$ is finitely presented", "over $S \\otimes_R R'$, then $M$ is finitely presented over $S$." ], "refs": [], "proofs": [ { "contents": [ "This lemma is true because $M$ is finitely presented even without the", "assumption that $M \\otimes_R R'$ is finitely presented for every $R'$", "as in the statement of the lemma. This follows from More on Flatness,", "Proposition \\ref{flat-proposition-flat-finite-type-finite-presentation-domain}.", "Originally this lemma had an erroneous proof (thanks to Ofer Gabber", "for finding the gap) and was used in an alternative proof of", "the proposition cited. To reinstate this lemma, we need a correct argument", "in case $R$ is a local normal domain using only", "results from the chapters on commutative algebra; please email", "\\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}", "if you have an argument." ], "refs": [ "flat-proposition-flat-finite-type-finite-presentation-domain" ], "ref_ids": [ 6201 ] } ], "ref_ids": [] }, { "id": 11450, "type": "theorem", "label": "obsolete-lemma-relative-effective-cartier-algebra", "categories": [ "obsolete" ], "title": "obsolete-lemma-relative-effective-cartier-algebra", "contents": [ "Let $A \\to B$ be a ring map. Let $f \\in B$. Assume that", "\\begin{enumerate}", "\\item $A \\to B$ is flat,", "\\item $f$ is a nonzerodivisor, and", "\\item $A \\to B/fB$ is flat.", "\\end{enumerate}", "Then for every ideal $I \\subset A$ the map", "$f : B/IB \\to B/IB$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Note that $IB = I \\otimes_A B$ and $I(B/fB) = I \\otimes_A B/fB$", "by the flatness of $B$ and $B/fB$ over $A$.", "In particular $IB/fIB \\cong I \\otimes_A B/fB$ maps injectively", "into $B/fB$. Hence the result follows from the snake lemma applied", "to the diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "I \\otimes_A B \\ar[r] \\ar[d]^f &", "B \\ar[r] \\ar[d]^f &", "B/IB \\ar[r] \\ar[d]^f &", "0 \\\\", "0 \\ar[r] &", "I \\otimes_A B \\ar[r] &", "B \\ar[r] &", "B/IB \\ar[r] &", "0", "}", "$$", "with exact rows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11451, "type": "theorem", "label": "obsolete-lemma-faithfully-flat-injective", "categories": [ "obsolete" ], "title": "obsolete-lemma-faithfully-flat-injective", "contents": [ "If $R \\to S$ is a faithfully flat ring map then for every $R$-module", "$M$ the map $M \\to S \\otimes_R M$, $x \\mapsto 1 \\otimes x$ is injective." ], "refs": [], "proofs": [ { "contents": [ "This lemma is a duplicate of", "Algebra, Lemma \\ref{algebra-lemma-faithfully-flat-universally-injective}." ], "refs": [ "algebra-lemma-faithfully-flat-universally-injective" ], "ref_ids": [ 814 ] } ], "ref_ids": [] }, { "id": 11452, "type": "theorem", "label": "obsolete-lemma-not-domain", "categories": [ "obsolete" ], "title": "obsolete-lemma-not-domain", "contents": [ "Let $(R, \\mathfrak m)$ be a reduced Noetherian local ring of dimension $1$", "and let $x \\in \\mathfrak m$ be a nonzerodivisor. Let", "$\\mathfrak q_1, \\ldots, \\mathfrak q_r$ be the minimal primes of $R$.", "Then", "$$", "\\text{length}_R(R/(x)) = \\sum\\nolimits_i \\text{ord}_{R/\\mathfrak q_i}(x)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Special (very easy) case of", "Chow Homology, Lemma \\ref{chow-lemma-additivity-divisors-restricted}." ], "refs": [ "chow-lemma-additivity-divisors-restricted" ], "ref_ids": [ 5653 ] } ], "ref_ids": [] }, { "id": 11453, "type": "theorem", "label": "obsolete-lemma-bound-primes", "categories": [ "obsolete" ], "title": "obsolete-lemma-bound-primes", "contents": [ "Let $A$ be a Noetherian local normal domain of dimension $2$.", "For $f \\in \\mathfrak m$ nonzero denote", "$\\text{div}(f) = \\sum n_i (\\mathfrak p_i)$", "the divisor associated to $f$ on the punctured spectrum of $A$.", "We set $|f| = \\sum n_i$. There exist integers $N$ and $M$", "such that $|f + g| \\leq M$ for all $g \\in \\mathfrak m^N$." ], "refs": [], "proofs": [ { "contents": [ "Pick $h \\in \\mathfrak m$ such that $f, h$ is a regular sequence in $A$", "(this follows from Algebra, Lemmas \\ref{algebra-lemma-criterion-normal} and", "\\ref{algebra-lemma-depth-drops-by-one}).", "We will prove the lemma with $M = \\text{length}_A(A/(f, h))$ and with", "$N$ any integer such that $\\mathfrak m^N \\subset (f, h)$. Such", "an integer $N$ exists because $\\sqrt{(f, h)} = \\mathfrak m$. Note that", "$M = \\text{length}_A(A/(f + g, h))$ for all $g \\in \\mathfrak m^N$", "because $(f, h) = (f + g, h)$. This moreover implies that $f + g, h$", "is a regular sequence in $A$ too, see", "Algebra, Lemma \\ref{algebra-lemma-reformulate-CM}.", "Now suppose that $\\text{div}(f + g ) = \\sum m_j (\\mathfrak q_j)$.", "Then consider the map", "$$", "c : A/(f + g) \\longrightarrow \\prod A/\\mathfrak q_j^{(m_j)}", "$$", "where $\\mathfrak q_j^{(m_j)}$ is the symbolic power, see", "Algebra, Section \\ref{algebra-section-symbolic-power}.", "Since $A$ is normal, we see that $A_{\\mathfrak q_i}$ is", "a discrete valuation ring and hence", "$$", "A_{\\mathfrak q_i}/(f + g) =", "A_{\\mathfrak q_i}/\\mathfrak q_i^{m_i} A_{\\mathfrak q_i} =", "(A/\\mathfrak q_i^{(m_i)})_{\\mathfrak q_i}", "$$", "Since $V(f + g, h) = \\{\\mathfrak m\\}$ this implies that $c$ becomes", "an isomorphism on inverting $h$ (small detail omitted). Since $h$ is a", "nonzerodivisor on $A/(f + g)$ we see that the length of $A/(f + g, h)$", "equals the Herbrand quotient $e_A(A/(f + g), 0, h)$", "as defined in Chow Homology, Section", "\\ref{chow-section-periodic-complexes}.", "Similarly the length of $A/(h, \\mathfrak q_j^{(m_j)})$ equals", "$e_A(A/\\mathfrak q_j^{(m_j)}, 0, h)$. Then we have", "\\begin{align*}", "M & = \\text{length}_A(A/(f + g, h) \\\\", "& =", "e_A(A/(f + g), 0, h) \\\\", "& =", "\\sum\\nolimits_i e_A(A/\\mathfrak q_j^{(m_j)}, 0, h) \\\\", "& =", "\\sum\\nolimits_i \\sum\\nolimits_{m = 0, \\ldots, m_j - 1}", "e_A(\\mathfrak q_j^{(m)}/\\mathfrak q_j^{(m + 1)}, 0, h)", "\\end{align*}", "The equalities follow from Chow Homology, Lemmas", "\\ref{chow-lemma-additivity-periodic-length} and", "\\ref{chow-lemma-finite-periodic-length}", "using in particular that", "the cokernel of $c$ has finite length as discussed above.", "It is straightforward to prove that", "$e_A(\\mathfrak q^{(m)}/\\mathfrak q^{(m + 1)}, 0, h)$", "is at least $1$ by Nakayama's lemma. This finishes the proof of the lemma." ], "refs": [ "algebra-lemma-criterion-normal", "algebra-lemma-depth-drops-by-one", "algebra-lemma-reformulate-CM", "chow-lemma-additivity-periodic-length", "chow-lemma-finite-periodic-length" ], "ref_ids": [ 1311, 774, 923, 5649, 5650 ] } ], "ref_ids": [] }, { "id": 11454, "type": "theorem", "label": "obsolete-lemma-flat-over-gorenstein-gorenstein-fibre", "categories": [ "obsolete" ], "title": "obsolete-lemma-flat-over-gorenstein-gorenstein-fibre", "contents": [ "Let $A \\to B$ be a flat local homomorphism of Noetherian local rings.", "If $A$ and $B/\\mathfrak m_A B$ are Gorenstein, then $B$ is Gorenstein." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-flat-under-gorenstein}." ], "refs": [ "dualizing-lemma-flat-under-gorenstein" ], "ref_ids": [ 2885 ] } ], "ref_ids": [] }, { "id": 11455, "type": "theorem", "label": "obsolete-lemma-kill-local", "categories": [ "obsolete" ], "title": "obsolete-lemma-kill-local", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module.", "Let $s$ be an integer. Assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex,", "\\item if $\\mathfrak p \\not \\in V(I)$ and", "$V(\\mathfrak p) \\cap V(I) \\not = \\{\\mathfrak m\\}$, then", "$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > s$.", "\\end{enumerate}", "Then there exists an $n > 0$ and an ideal $J \\subset A$", "with $V(J) \\cap V(I) = \\{\\mathfrak m\\}$ such that $JI^n$ annihilates", "$H^i_\\mathfrak m(M)$ for $i \\leq s$." ], "refs": [], "proofs": [ { "contents": [ "According to", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-sitting-in-degrees}", "we have to show this for the finite $A$-module", "$E^i = \\text{Ext}^{-i}_A(M, \\omega_A^\\bullet)$", "for $i \\leq s$. The support $Z$ of $E^0 \\oplus \\ldots \\oplus E^s$", "is closed in $\\Spec(A)$ and does not contain any prime as in (2).", "Hence it is contained in $V(JI^n)$ for some $J$ as in", "the statement of the lemma." ], "refs": [ "local-cohomology-lemma-sitting-in-degrees" ], "ref_ids": [ 9737 ] } ], "ref_ids": [] }, { "id": 11456, "type": "theorem", "label": "obsolete-lemma-algebraize-local-cohomology-bis-bis", "categories": [ "obsolete" ], "title": "obsolete-lemma-algebraize-local-cohomology-bis-bis", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module.", "Let $s$ and $d$ be integers. Assume", "\\begin{enumerate}", "\\item[(a)] $A$ has a dualizing complex,", "\\item[(b)] $\\text{cd}(A, I) \\leq d$,", "\\item[(c)] if $\\mathfrak p \\not \\in V(I)$ then", "$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) > s$ or", "$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > d + s$.", "\\end{enumerate}", "Then the assumptions of", "Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-algebraize-local-cohomology-bis} hold", "for $A, I, \\mathfrak m, M$ and", "$H^i_\\mathfrak m(M) \\to \\lim H^i_\\mathfrak m(M/I^nM)$", "is an isomorphism for $i \\leq s$ and these modules are", "annihilated by a power of $I$." ], "refs": [ "algebraization-lemma-algebraize-local-cohomology-bis" ], "proofs": [ { "contents": [ "The assumptions of Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-algebraize-local-cohomology-bis}", "by the more general Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-bootstrap-bis-bis}.", "Then the conclusion of Algebraic and Formal Geometry, Lemma", "\\ref{algebraization-lemma-algebraize-local-cohomology-bis}", "gives the second statement." ], "refs": [ "algebraization-lemma-algebraize-local-cohomology-bis", "algebraization-lemma-bootstrap-bis-bis", "algebraization-lemma-algebraize-local-cohomology-bis" ], "ref_ids": [ 12724, 12725, 12724 ] } ], "ref_ids": [ 12724 ] }, { "id": 11457, "type": "theorem", "label": "obsolete-lemma-combine-one", "categories": [ "obsolete" ], "title": "obsolete-lemma-combine-one", "contents": [ "In Algebraic and Formal Geometry, Situation", "\\ref{algebraization-situation-bootstrap}", "we have $H^s_\\mathfrak a(M) = \\lim H^s_\\mathfrak a(M/I^nM)$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from Algebraic and Formal Geometry, Theorem", "\\ref{algebraization-theorem-final-bootstrap}.", "The original version of this lemma, which had additional", "assumptions, was superseded by the this theorem." ], "refs": [ "algebraization-theorem-final-bootstrap" ], "ref_ids": [ 12673 ] } ], "ref_ids": [] }, { "id": 11458, "type": "theorem", "label": "obsolete-lemma-change-equation-multiply", "categories": [ "obsolete" ], "title": "obsolete-lemma-change-equation-multiply", "contents": [ "Let $R$ be a ring and let $\\varphi : R[x] \\to S$ be", "a ring map. Let $t \\in S$. If $t$ is integral over", "$R[x]$, then there exists an $\\ell \\geq 0$ such that", "for every $a \\in R$ the element $\\varphi(a)^\\ell t$", "is integral over $\\varphi_a : R[y] \\to S$, defined by", "$y \\mapsto \\varphi(ax)$ and $r \\mapsto \\varphi(r)$", "for $r\\in R$." ], "refs": [], "proofs": [ { "contents": [ "Say $t^d + \\sum_{i < d} \\varphi(f_i)t^i = 0$", "with $f_i \\in R[x]$. Let $\\ell$ be the maximum degree", "in $x$ of all the $f_i$. Multiply the equation", "by $\\varphi(a)^\\ell$ to get", "$\\varphi(a)^\\ell t^d + \\sum_{i < d} \\varphi(a^\\ell f_i)t^i = 0$.", "Note that each $\\varphi(a^\\ell f_i)$ is in the image of", "$\\varphi_a$. The result follows from", "Algebra, Lemma \\ref{algebra-lemma-make-integral-trivial}." ], "refs": [ "algebra-lemma-make-integral-trivial" ], "ref_ids": [ 1057 ] } ], "ref_ids": [] }, { "id": 11459, "type": "theorem", "label": "obsolete-lemma-make-integral-less-trivial", "categories": [ "obsolete" ], "title": "obsolete-lemma-make-integral-less-trivial", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "Suppose $t \\in S$ satisfies the", "relation $\\varphi(a_0) + \\varphi(a_1)t + \\ldots + \\varphi(a_n) t^n = 0$.", "Set $u_n = \\varphi(a_n)$, $u_{n-1} = u_n t + \\varphi(a_{n-1})$,", "and so on till $u_1 = u_2 t + \\varphi(a_1)$.", "Then all of $u_n, u_{n-1}, \\ldots, u_1$ and", "$u_nt, u_{n-1}t, \\ldots, u_1t$ are integral over $R$,", "and the ideals $(\\varphi(a_0), \\ldots, \\varphi(a_n))$ and", "$(u_n, \\ldots, u_1)$ of $S$ are equal." ], "refs": [], "proofs": [ { "contents": [ "We prove this by induction on $n$. As $u_n = \\varphi(a_n)$ we", "conclude from", "Algebra, Lemma \\ref{algebra-lemma-make-integral-trivial}", "that $u_nt$ is integral over $R$. Of course", "$u_n = \\varphi(a_n)$ is integral over $R$. Then", "$u_{n - 1} = u_n t + \\varphi(a_{n - 1})$ is integral over $R$ (see", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-is-ring})", "and we have", "$$", "\\varphi(a_0) + \\varphi(a_1)t + \\ldots + \\varphi(a_{n - 1})t^{n - 1} +", "u_{n - 1}t^{n - 1} = 0.", "$$", "Hence by the induction hypothesis applied to the map", "$S' \\to S$ where $S'$ is the integral closure of $R$ in $S$", "and the displayed equation we see that", "$u_{n-1}, \\ldots, u_1$ and $u_{n-1}t, \\ldots, u_1t$", "are all in $S'$ too. The statement on the ideals is immediate from the", "shape of the elements and the fact that $u_1t + \\varphi(a_0) = 0$." ], "refs": [ "algebra-lemma-make-integral-trivial", "algebra-lemma-integral-closure-is-ring" ], "ref_ids": [ 1057, 486 ] } ], "ref_ids": [] }, { "id": 11460, "type": "theorem", "label": "obsolete-lemma-make-integral-not-in-ideal", "categories": [ "obsolete" ], "title": "obsolete-lemma-make-integral-not-in-ideal", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "Suppose $t \\in S$ satisfies the", "relation $\\varphi(a_0) + \\varphi(a_1)t + \\ldots + \\varphi(a_n) t^n = 0$.", "Let $J \\subset S$ be an ideal such that for at", "least one $i$ we have $\\varphi(a_i) \\not \\in J$.", "Then there exists a $u \\in S$, $u \\not\\in J$ such", "that both $u$ and $ut$ are integral over $R$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from Lemma \\ref{lemma-make-integral-less-trivial}", "since one of the elements $u_i$ will not be in $J$." ], "refs": [ "obsolete-lemma-make-integral-less-trivial" ], "ref_ids": [ 11459 ] } ], "ref_ids": [] }, { "id": 11461, "type": "theorem", "label": "obsolete-lemma-P1", "categories": [ "obsolete" ], "title": "obsolete-lemma-P1", "contents": [ "Let $R$ be a ring.", "Let $F(X, Y) \\in R[X, Y]$ be homogeneous of degree", "$d$. Assume that for every prime $\\mathfrak p$ of $R$", "at least one coefficient of $F$ is not in $\\mathfrak p$.", "Let $S = R[X, Y]/(F)$ as a graded ring.", "Then for all $n \\geq d$ the $R$-module $S_n$", "is finite locally free of rank $d$." ], "refs": [], "proofs": [ { "contents": [ "The $R$-module $S_n$ has a presentation", "$$", "R[X, Y]_{n-d} \\to R[X, Y]_n \\to S_n \\to 0.", "$$", "Thus by Algebra, Lemma \\ref{algebra-lemma-cokernel-flat}", "it is enough to show that multiplication", "by $F$ induces an injective map", "$\\kappa(\\mathfrak p)[X, Y]", "\\to \\kappa(\\mathfrak p)[X, Y]$", "for all primes $\\mathfrak p$.", "This is clear from the assumption that", "$F$ does not map to the zero polynomial mod $\\mathfrak p$.", "The assertion on ranks is clear from this as well." ], "refs": [ "algebra-lemma-cokernel-flat" ], "ref_ids": [ 804 ] } ], "ref_ids": [] }, { "id": 11462, "type": "theorem", "label": "obsolete-lemma-rel-prime-pols", "categories": [ "obsolete" ], "title": "obsolete-lemma-rel-prime-pols", "contents": [ "Let $k$ be a field. Let $F, G \\in k[X, Y]$ be homogeneous", "of degrees $d, e$. Assume $F, G$ relatively prime.", "Then multiplication by $G$ is injective on $S = k[X, Y]/(F)$." ], "refs": [], "proofs": [ { "contents": [ "This is one way to define ``relatively prime''. If you have another", "definition, then you can show it is equivalent to this one." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11463, "type": "theorem", "label": "obsolete-lemma-P1-localize", "categories": [ "obsolete" ], "title": "obsolete-lemma-P1-localize", "contents": [ "Let $R$ be a ring. Let $F(X, Y) \\in R[X, Y]$ be homogeneous of degree", "$d$. Let $S = R[X, Y]/(F)$ as a graded ring.", "Let $\\mathfrak p \\subset R$ be a prime such that", "some coefficient of $F$ is not in $\\mathfrak p$.", "There exists an $f \\in R$ $f \\not\\in \\mathfrak p$,", "an integer $e$, and a $G \\in R[X, Y]_e$", "such that multiplication by $G$ induces isomorphisms", "$(S_n)_f \\to (S_{n + e})_f$ for all $n \\geq d$." ], "refs": [], "proofs": [ { "contents": [ "During the course of the proof we may replace $R$ by $R_f$", "for $f\\in R$, $f\\not\\in \\mathfrak p$ (finitely often).", "As a first step we do such a replacement such that", "some coefficient of $F$ is invertible in $R$.", "In particular the modules $S_n$ are now locally", "free of rank $d$ for $n \\geq d$ by Lemma \\ref{lemma-P1}.", "Pick any $G \\in R[X, Y]_e$ such that the image of", "$G$ in $\\kappa(\\mathfrak p)[X, Y]$ is relatively", "prime to the image of $F(X, Y)$ (this is possible for some $e$).", "Apply Algebra, Lemma \\ref{algebra-lemma-cokernel-flat} to the map", "induced by multiplication by $G$ from $S_d \\to S_{d + e}$.", "By our choice of $G$ and Lemma \\ref{lemma-rel-prime-pols}", "we see", "$S_d \\otimes \\kappa(\\mathfrak p) \\to S_{d + e} \\otimes \\kappa(\\mathfrak p)$", "is bijective. Thus, after replacing $R$ by $R_f$ for a suitable", "$f$ we may assume that $G : S_d \\to S_{d + e}$", "is bijective. This in turn implies that the image", "of $G$ in $\\kappa(\\mathfrak p')[X, Y]$ is relatively", "prime to the image of $F$ for all primes $\\mathfrak p'$", "of $R$. And then by Algebra, Lemma \\ref{algebra-lemma-cokernel-flat}", "again we see that all the maps", "$G : S_d \\to S_{d + e}$, $n \\geq d$ are isomorphisms." ], "refs": [ "obsolete-lemma-P1", "algebra-lemma-cokernel-flat", "obsolete-lemma-rel-prime-pols", "algebra-lemma-cokernel-flat" ], "ref_ids": [ 11461, 804, 11462, 804 ] } ], "ref_ids": [] }, { "id": 11464, "type": "theorem", "label": "obsolete-lemma-finite-after-localization", "categories": [ "obsolete" ], "title": "obsolete-lemma-finite-after-localization", "contents": [ "Let $R$ be a ring, let $f \\in R$.", "Suppose we have $S$, $S'$ and the solid arrows", "forming the following commutative diagram of rings", "$$", "\\xymatrix{", "& S'' \\ar@{-->}[rd] \\ar@{-->}[dd] &", "\\\\", "R \\ar[rr] \\ar@{-->}[ru] \\ar[d] & & S \\ar[d]", "\\\\", "R_f \\ar[r] & S' \\ar[r] & S_f", "}", "$$", "Assume that $R_f \\to S'$ is finite. Then we can find", "a finite ring map $R \\to S''$ and dotted arrows as", "in the diagram such that $S' = (S'')_f$." ], "refs": [], "proofs": [ { "contents": [ "Namely, suppose that $S'$ is generated by", "$x_i$ over $R_f$, $i = 1, \\ldots, w$. Let $P_i(t) \\in R_f[t]$", "be a monic polynomial such that $P_i(x_i) = 0$.", "Say $P_i$ has degree $d_i > 0$. Write", "$P_i(t) = t^{d_i} + \\sum_{j < d_i} (a_{ij}/f^n) t^j$", "for some uniform $n$. Also write", "the image of $x_i$ in $S_f$ as $g_i / f^n$", "for suitable $g_i \\in S$. Then we know", "that the element", "$\\xi_i = f^{nd_i} g_i^{d_i} + \\sum_{j < d_i} f^{n(d_i - j)} a_{ij} g_i^j$", "of $S$ is killed by a power of $f$.", "Hence upon increasing $n$ to $n'$, which replaces", "$g_i$ by $f^{n' - n}g_i$ we may assume $\\xi_i = 0$.", "Then $S'$ is generated by the elements", "$f^n x_i$, each of which is a zero of the", "monic polynomial $Q_i(t) = t^{d_i} +", "\\sum_{j < d_i} f^{n(d_i - j)} a_{ij} t^j$", "with coefficients in $R$. Also, by construction", "$Q_i(f^ng_i) = 0$ in $S$. Thus we get a finite $R$-algebra", "$S'' = R[z_1, \\ldots, z_w]/(Q_1(z_1), \\ldots, Q_w(z_w))$", "which fits into a commutative diagram as above.", "The map $\\alpha : S'' \\to S$ maps $z_i$ to $f^ng_i$ and", "the map $\\beta : S'' \\to S'$ maps $z_i$ to $f^nx_i$.", "It may not yet be the case that $\\beta$ induces an", "isomorphism $(S'')_f \\cong S'$.", "For the moment we only know that this map", "is surjective. The problem is that there could be", "elements $h/f^n \\in (S'')_f$ which map to zero", "in $S'$ but are not zero. In this case $\\beta(h)$", "is an element of $S$ such that $f^N \\beta(h) = 0$", "for some $N$. Thus $f^N h$ is an element ot the ideal", "$J = \\{h \\in S'' \\mid \\alpha(h) = 0 \\text{ and }", "\\beta(h) = 0\\}$ of $S''$. OK, and it is easy to see that", "$S''/J$ does the job." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11465, "type": "theorem", "label": "obsolete-lemma-formally-smooth-smooth", "categories": [ "obsolete" ], "title": "obsolete-lemma-formally-smooth-smooth", "contents": [ "Let $R$ be a ring. Let $S$ be a $R$-algebra.", "If $S$ is of finite presentation and formally smooth over $R$", "then $S$ is smooth over $R$." ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Proposition \\ref{algebra-proposition-smooth-formally-smooth}." ], "refs": [ "algebra-proposition-smooth-formally-smooth" ], "ref_ids": [ 1426 ] } ], "ref_ids": [] }, { "id": 11466, "type": "theorem", "label": "obsolete-lemma-get-morphism-general", "categories": [ "obsolete" ], "title": "obsolete-lemma-get-morphism-general", "contents": [ "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.", "Let $t$ be the minimal number of generators for $I$.", "Let $C$ be a Noetherian $I$-adically complete $A$-algebra.", "There exists an integer $d \\geq 0$ depending only on", "$I \\subset A \\to C$ with the following property: given", "\\begin{enumerate}", "\\item $c \\geq 0$ and $B$ in", "Algebraization of Formal Spaces, Equation (\\ref{restricted-equation-C-prime})", "such that for $a \\in I^c$", "multiplication by $a$ on $\\NL_{B/A}^\\wedge$ is zero in $D(B)$,", "\\item an integer $n > 2t\\max(c, d)$,", "\\item an $A/I^n$-algebra map $\\psi_n : B/I^nB \\to C/I^nC$,", "\\end{enumerate}", "there exists a map $\\varphi : B \\to C$ of $A$-algebras such", "that $\\psi_n \\bmod I^{m - c} = \\varphi \\bmod I^{m - c}$", "with $m = \\lfloor \\frac{n}{t} \\rfloor$." ], "refs": [], "proofs": [ { "contents": [ "This lemma has been obsoleted by the stronger", "Algebraization of Formal Spaces, Lemma", "\\ref{restricted-lemma-get-morphism-general-better}.", "In fact, we will deduce the lemma from it.", "\\medskip\\noindent", "Let $I \\subset A \\to C$ be given as in the statement above.", "Denote $d(\\text{Gr}_I(C))$ and $q(\\text{Gr}_I(C))$ the integers found in ", "Local Cohomology, Section \\ref{local-cohomology-section-uniform}.", "Observe that $t$ is an upper bound for the minimal number of generators", "of $IC$ and hence we have $d(\\text{Gr}_I(C)) + 1 \\leq t$, see discussion in", "Local Cohomology, Section \\ref{local-cohomology-section-uniform}.", "We may and do assume $t \\geq 1$ since otherwise the lemma does", "not say anything. We claim that the lemma is true with", "$$", "d = q(\\text{Gr}_I(C))", "$$", "Namely, suppose that $c$, $B$, $n$, $\\psi_n$ are as in the statement above.", "Then we see that", "$$", "n > 2t\\max(c, d) \\Rightarrow n \\geq 2tc + 1 \\Rightarrow", "n \\geq 2(d(\\text{Gr}_I(C)) + 1)c + 1", "$$", "On the other hand, we have", "$$", "n > 2t\\max(c, d) \\Rightarrow n > t(c + d) \\Rightarrow", "n \\geq q(C) + tc \\geq q(\\text{Gr}_I(C)) + (d(\\text{Gr}_I(C)) + 1)c", "$$", "Hence the assumptions of", "Algebraization of Formal Spaces, Lemma", "\\ref{restricted-lemma-get-morphism-general-better}", "are satisfied and we obtain an $A$-algebra homomorphism", "$\\varphi : B \\to C$ which is congruent with $\\psi_n$", "module $I^{n - (d(\\text{Gr}_I(C)) + 1)c}C$.", "Since", "\\begin{align*}", "n - (d(\\text{Gr}_I(C)) + 1)c", "& = \\frac{n}{t} + \\frac{(t - 1)n}{t} - (d(\\text{Gr}_I(C)) + 1)c \\\\", "& \\geq \\frac{n}{t} + \\frac{(d(\\text{Gr}_I(C))n}{t} - (d(\\text{Gr}_I(C)) + 1)c \\\\", "& > \\frac{n}{t} + \\frac{d(\\text{Gr}_I(C))2tc}{t} - (d(\\text{Gr}_I(C)) + 1)c \\\\", "& = \\frac{n}{t} + 2d(\\text{Gr}_I(C))c - (d(\\text{Gr}_I(C)) + 1)c \\\\", "& = \\frac{n}{t} + d(\\text{Gr}_I(C))c - c \\\\", "& \\geq m - c", "\\end{align*}", "we see that we have the congruence of", "$\\varphi$ and $\\psi_n$ module $I^{m - c}C$ as desired." ], "refs": [ "restricted-lemma-get-morphism-general-better", "restricted-lemma-get-morphism-general-better" ], "ref_ids": [ 2306, 2306 ] } ], "ref_ids": [] }, { "id": 11467, "type": "theorem", "label": "obsolete-lemma-pullback-K-flat", "categories": [ "obsolete" ], "title": "obsolete-lemma-pullback-K-flat", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$ be a ringed topos.", "For any complex of $\\mathcal{O}_\\mathcal{C}$-modules $\\mathcal{G}^\\bullet$", "there exists a quasi-isomorphism $\\mathcal{K}^\\bullet \\to \\mathcal{G}^\\bullet$", "such that $f^*\\mathcal{K}^\\bullet$ is a K-flat complex of", "$\\mathcal{O}_\\mathcal{D}$-modules for any morphism", "$f : (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\to", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$ of ringed topoi." ], "refs": [], "proofs": [ { "contents": [ "This follows from Cohomology on Sites, Lemmas", "\\ref{sites-cohomology-lemma-K-flat-resolution} and", "\\ref{sites-cohomology-lemma-pullback-K-flat}." ], "refs": [ "sites-cohomology-lemma-K-flat-resolution", "sites-cohomology-lemma-pullback-K-flat" ], "ref_ids": [ 4236, 4241 ] } ], "ref_ids": [] }, { "id": 11468, "type": "theorem", "label": "obsolete-lemma-Rlim-of-system", "categories": [ "obsolete" ], "title": "obsolete-lemma-Rlim-of-system", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(K_n)$", "be an inverse system of objects of $D(\\mathcal{O})$.", "Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset.", "Let $d \\in \\mathbf{N}$. Assume", "\\begin{enumerate}", "\\item $K_n$ is an object of $D^+(\\mathcal{O})$ for all $n$,", "\\item for $q \\in \\mathbf{Z}$ there exists", "$n(q)$ such that $H^q(K_{n + 1}) \\to H^q(K_n)$ is an isomorphism for", "$n \\geq n(q)$,", "\\item every object of $\\mathcal{C}$ has a covering whose members are", "elements of $\\mathcal{B}$,", "\\item for every $U \\in \\mathcal{B}$ we have $H^p(U, H^q(K_n)) = 0$", "for $p > d$ and all $q$.", "\\end{enumerate}", "Then we have $H^m(R\\lim K_n) = \\lim H^m(K_n)$ for all $m \\in \\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "Set $K = R\\lim K_n$. Let $U \\in \\mathcal{B}$. For each $n$ there is a spectral", "sequence", "$$", "H^p(U, H^q(K_n)) \\Rightarrow H^{p + q}(U, K_n)", "$$", "which converges as $K_n$ is bounded below, see", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "If we fix $m \\in \\mathbf{Z}$, then we see from our assumption (4)", "that only $H^p(U, H^q(K_n))$ contribute to $H^m(U, K_n)$", "for $0 \\leq p \\leq d$ and $m - d \\leq q \\leq m$. By assumption (2)", "this implies that $H^m(U, K_{n + 1}) \\to H^m(U, K_n)$ is an isomorphism", "as soon as $n \\geq \\max{n(m), \\ldots, n(m - d)}$. The functor $R\\Gamma(U, -)$", "commutes with derived limits by", "Injectives, Lemma \\ref{injectives-lemma-RF-commutes-with-Rlim}.", "Thus we have", "$$", "H^m(U, K) = H^m(R\\lim R\\Gamma(U, K_n))", "$$", "On the other hand we have just seen that the complexes $R\\Gamma(U, K_n)$", "have eventually constant cohomology groups. Thus by", "More on Algebra, Remark \\ref{more-algebra-remark-compare-derived-limit}", "we find that $H^m(U, K)$ is equal to $H^m(U, K_n)$ for", "all $n \\gg 0$ for some bound independent of $U \\in \\mathcal{B}$.", "Pick such an $n$. Finally, recall that $H^m(K)$ is the sheafification of", "the presheaf $U \\mapsto H^m(U, K)$ and $H^m(K_n)$ is the sheafification", "of the presheaf $U \\mapsto H^m(U, K_n)$. On the elements", "of $\\mathcal{B}$ these presheaves have the same values. Therefore assumption", "(3) guarantees that the sheafifications are the same too.", "The lemma follows." ], "refs": [ "derived-lemma-two-ss-complex-functor", "injectives-lemma-RF-commutes-with-Rlim", "more-algebra-remark-compare-derived-limit" ], "ref_ids": [ 1871, 7796, 10658 ] } ], "ref_ids": [] }, { "id": 11469, "type": "theorem", "label": "obsolete-lemma-trivialities-cohomological-descent-abelian", "categories": [ "obsolete" ], "title": "obsolete-lemma-trivialities-cohomological-descent-abelian", "contents": [ "In Simplicial Spaces, Situation", "\\ref{spaces-simplicial-situation-simplicial-site}", "let $a_0$ be an augmentation towards a site $\\mathcal{D}$ as in", "Simplicial Spaces, Remark \\ref{spaces-simplicial-remark-augmentation-site}.", "Suppose given strictly full weak Serre subcategories", "$$", "\\mathcal{A} \\subset \\textit{Ab}(\\mathcal{D}),\\quad", "\\mathcal{A}_n \\subset \\textit{Ab}(\\mathcal{C}_n)", "$$", "Then", "\\begin{enumerate}", "\\item[(1)]", "the collection of abelian sheaves $\\mathcal{F}$ on $\\mathcal{C}_{total}$", "whose restriction to $\\mathcal{C}_n$ is in $\\mathcal{A}_n$ for all $n$", "is a strictly full weak Serre subcategory", "$\\mathcal{A}_{total} \\subset \\textit{Ab}(\\mathcal{C}_{total})$.", "\\end{enumerate}", "If $a_n^{-1}$ sends $\\mathcal{A}$ into $\\mathcal{A}_n$", "for all $n$, then", "\\begin{enumerate}", "\\item[(2)] $a^{-1}$ sends $\\mathcal{A}$ into $\\mathcal{A}_{total}$ and", "\\item[(3)] $a^{-1}$ sends $D_\\mathcal{A}(\\mathcal{D})$ into", "$D_{\\mathcal{A}_{total}}(\\mathcal{C}_{total})$.", "\\end{enumerate}", "If $R^qa_{n, *}$ sends $\\mathcal{A}_n$ into $\\mathcal{A}$", "for all $n, q$, then", "\\begin{enumerate}", "\\item[(4)] $R^qa_*$ sends $\\mathcal{A}_{total}$ into $\\mathcal{A}$ for all $q$,", "and", "\\item[(5)] $Ra_*$ sends $D_{\\mathcal{A}_{total}}^+(\\mathcal{C}_{total})$", "into $D_\\mathcal{A}^+(\\mathcal{D})$.", "\\end{enumerate}" ], "refs": [ "spaces-simplicial-remark-augmentation-site" ], "proofs": [ { "contents": [ "The only interesting assertions are (4) and (5).", "Part (4) follows from the spectral sequence in", "Simplicial Spaces, Lemma", "\\ref{spaces-simplicial-lemma-augmentation-spectral-sequence}", "and Homology, Lemma \\ref{homology-lemma-biregular-ss-converges}.", "Then part (5) follows by considering the spectral sequence", "associated to the canonical filtration on an object", "$K$ of $D_{\\mathcal{A}_{total}}^+(\\mathcal{C}_{total})$ given by truncations.", "We omit the details." ], "refs": [ "spaces-simplicial-lemma-augmentation-spectral-sequence", "homology-lemma-biregular-ss-converges" ], "ref_ids": [ 9043, 12101 ] } ], "ref_ids": [ 9152 ] }, { "id": 11470, "type": "theorem", "label": "obsolete-lemma-glue-f-upper-shriek", "categories": [ "obsolete" ], "title": "obsolete-lemma-glue-f-upper-shriek", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes.", "There exists a unique functor", "$f^! : \\textit{Ab}(Y_\\etale) \\to \\textit{Ab}(X_\\etale)$ such that", "\\begin{enumerate}", "\\item for any open $j : U \\to X$ with $f \\circ j$ separated there", "is a canonical isomorphism $j^! \\circ f^! = (f \\circ j)^!$, and", "\\item these isomorphisms for $U \\subset U' \\subset X$ are compatible", "with the isomorphisms in More \\'Etale Cohomology, Lemma", "\\ref{more-etale-lemma-upper-shriek-restriction}.", "\\end{enumerate}" ], "refs": [ "more-etale-lemma-upper-shriek-restriction" ], "proofs": [ { "contents": [ "Immediate consequence of More \\'Etale Cohomology, Lemmas", "\\ref{more-etale-lemma-lqf-f-upper-shriek} and", "\\ref{more-etale-lemma-upper-shriek-restriction}." ], "refs": [ "more-etale-lemma-lqf-f-upper-shriek", "more-etale-lemma-upper-shriek-restriction" ], "ref_ids": [ 8828, 8830 ] } ], "ref_ids": [ 8830 ] }, { "id": 11471, "type": "theorem", "label": "obsolete-lemma-lqf-f-upper-shriek-stalk", "categories": [ "obsolete" ], "title": "obsolete-lemma-lqf-f-upper-shriek-stalk", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes which is locally quasi-finite.", "For an abelian group $A$ and a geometric point", "$\\overline{y} : \\Spec(k) \\to Y$ we have", "$f^!(\\overline{y}_*A) = \\prod\\nolimits_{f(\\overline{x}) = \\overline{y}}", "\\overline{x}_*A$." ], "refs": [], "proofs": [ { "contents": [ "Follows from the corresponding statement in", "More \\'Etale Cohomology, Lemma \\ref{more-etale-lemma-lqf-f-upper-shriek}." ], "refs": [ "more-etale-lemma-lqf-f-upper-shriek" ], "ref_ids": [ 8828 ] } ], "ref_ids": [] }, { "id": 11472, "type": "theorem", "label": "obsolete-lemma-lqf-f-shriek-composition", "categories": [ "obsolete" ], "title": "obsolete-lemma-lqf-f-shriek-composition", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be composable locally", "quasi-finite morphisms of schemes. Then $g_! \\circ f_! = (g \\circ f)_!$", "and $f^! \\circ g^! = (g \\circ f)^!$." ], "refs": [], "proofs": [ { "contents": [ "Combination of More \\'Etale Cohomology, Lemmas", "\\ref{more-etale-lemma-lqf-shriek-composition} and", "\\ref{more-etale-lemma-upper-shriek-restriction}." ], "refs": [ "more-etale-lemma-lqf-shriek-composition", "more-etale-lemma-upper-shriek-restriction" ], "ref_ids": [ 8827, 8830 ] } ], "ref_ids": [] }, { "id": 11473, "type": "theorem", "label": "obsolete-lemma-P-not-preserved-base-change", "categories": [ "obsolete" ], "title": "obsolete-lemma-P-not-preserved-base-change", "contents": [ "Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded algebras.", "Let $N$ be a differential graded $(A, B)$-bimodule with property", "(P). Let $M$ be a differential graded $A$-module with property (P).", "Then $Q = M \\otimes_A N$ is a differential graded $B$-module which represents", "$M \\otimes_A^\\mathbf{L} N$ in $D(B)$ and which has a filtration", "$$", "0 = F_{-1}Q \\subset F_0Q \\subset F_1Q \\subset \\ldots \\subset Q", "$$", "by differential graded submodules such that $Q = \\bigcup F_pQ$,", "the inclusions $F_iQ \\to F_{i + 1}Q$ are admissible monomorphisms,", "the quotients $F_{i + 1}Q/F_iQ$ are isomorphic as differential", "graded $B$-modules to a direct sum of $(A \\otimes_R B)[k]$." ], "refs": [], "proofs": [ { "contents": [ "Choose filtrations $F_\\bullet$ on $M$ and $N$. Then consider the filtration", "on $Q = M \\otimes_A N$ given by", "$$", "F_n(Q) = \\sum\\nolimits_{i + j = n}", "F_i(M) \\otimes_A F_j(N)", "$$", "This is clearly a differential graded $B$-submodule. We see that", "$$", "F_n(Q)/F_{n - 1}(Q) =", "\\bigoplus\\nolimits_{i + j = n}", "F_i(M)/F_{i - 1}(M) \\otimes_A F_j(N)/F_{j - 1}(N)", "$$", "for example because the filtration of $M$ is split in the category", "of graded $A$-modules. Since by assumption the quotients on the right", "hand side are isomorphic to direct sums of shifts of $A$ and", "$A \\otimes_R B$ and since", "$A \\otimes_A (A \\otimes_R B) = A \\otimes_R B$,", "we conclude that the left hand side is a direct sum of shifts", "of $A \\otimes_R B$ as a differential graded $B$-module.", "(Warning: $Q$ does not have a structure of $(A, B)$-bimodule.)", "This proves the first statement of the lemma.", "The second statement is immediate from", "the definition of the functor in", "Differential Graded Algebra, Lemma \\ref{dga-lemma-derived-bc}." ], "refs": [ "dga-lemma-derived-bc" ], "ref_ids": [ 13105 ] } ], "ref_ids": [] }, { "id": 11474, "type": "theorem", "label": "obsolete-lemma-equiv", "categories": [ "obsolete" ], "title": "obsolete-lemma-equiv", "contents": [ "Assumptions and notation as in", "Simplicial, Lemma \\ref{simplicial-lemma-section}.", "There exists a section $g : U \\to V$ to the morphism $f$ and", "the composition $g \\circ f$ is homotopy equivalent to the identity", "on $V$. In particular, the morphism $f$ is a homotopy equivalence." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Simplicial, Lemmas \\ref{simplicial-lemma-section} and", "\\ref{simplicial-lemma-trivial-kan-homotopy}." ], "refs": [ "simplicial-lemma-trivial-kan-homotopy" ], "ref_ids": [ 14892 ] } ], "ref_ids": [] }, { "id": 11475, "type": "theorem", "label": "obsolete-lemma-cosk-hom-deltak", "categories": [ "obsolete" ], "title": "obsolete-lemma-cosk-hom-deltak", "contents": [ "Let $\\mathcal{C}$ be a category with finite coproducts", "and finite limits. Let $X$ be an object of $\\mathcal{C}$.", "Let $k \\geq 0$. The canonical map", "$$", "\\Hom(\\Delta[k], X)", "\\longrightarrow", "\\text{cosk}_1 \\text{sk}_1 \\Hom(\\Delta[k], X)", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "For any simplicial object $V$ we have", "\\begin{eqnarray*}", "\\Mor(V, \\text{cosk}_1 \\text{sk}_1 \\Hom(\\Delta[k], X))", "& = &", "\\Mor(\\text{sk}_1 V, \\text{sk}_1 \\Hom(\\Delta[k], X)) \\\\", "& = &", "\\Mor(i_{1!} \\text{sk}_1 V, \\Hom(\\Delta[k], X)) \\\\", "& = &", "\\Mor(i_{1!} \\text{sk}_1 V \\times \\Delta[k], X)", "\\end{eqnarray*}", "The first equality by the adjointness of $\\text{sk}$ and $\\text{cosk}$,", "the second equality by the adjointness of $i_{1!}$ and $\\text{sk}_1$, and", "the first equality by", "Simplicial, Definition \\ref{simplicial-definition-hom-from-simplicial-set}", "where the last $X$ denotes the constant simplicial object with value $X$.", "By Simplicial, Lemma \\ref{simplicial-lemma-augmentation-howto} an element", "in this set depends only on the terms of degree $0$ and $1$", "of $i_{1!} \\text{sk}_1 V \\times \\Delta[k]$. These", "agree with the degree $0$ and $1$ terms of", "$V \\times \\Delta[k]$, see", "Simplicial, Lemma \\ref{simplicial-lemma-recovering-U-for-real}.", "Thus the set above is equal to", "$\\Mor(V \\times \\Delta[k], X) = \\Mor(V, \\Hom(\\Delta[k], X))$." ], "refs": [ "simplicial-definition-hom-from-simplicial-set", "simplicial-lemma-augmentation-howto", "simplicial-lemma-recovering-U-for-real" ], "ref_ids": [ 14923, 14845, 14849 ] } ], "ref_ids": [] }, { "id": 11476, "type": "theorem", "label": "obsolete-lemma-cosk0-hom-deltak", "categories": [ "obsolete" ], "title": "obsolete-lemma-cosk0-hom-deltak", "contents": [ "Let $\\mathcal{C}$ be a category. Let $X$ be an object of $\\mathcal{C}$", "such that the self products $X \\times \\ldots \\times X$ exist.", "Let $k \\geq 0$ and let $C[k]$ be as in", "Simplicial, Example \\ref{simplicial-example-simplex-cosimplicial-set}.", "With notation as in", "Simplicial, Lemma \\ref{simplicial-lemma-morphism-into-product}", "the canonical map", "$$", "\\Hom(C[k], X)_1", "\\longrightarrow", "(\\text{cosk}_0 \\text{sk}_0 \\Hom(C[k], X))_1", "$$", "is identified with the map", "$$", "\\prod\\nolimits_{\\alpha : [k] \\to [1]} X", "\\longrightarrow", "X \\times X", "$$", "which is the projection onto the factors where $\\alpha$", "is a constant map." ], "refs": [ "simplicial-lemma-morphism-into-product" ], "proofs": [ { "contents": [ "This is shown in the proof of", "Hypercoverings, Lemma \\ref{hypercovering-lemma-covering}." ], "refs": [ "hypercovering-lemma-covering" ], "ref_ids": [ 8405 ] } ], "ref_ids": [ 14822 ] }, { "id": 11477, "type": "theorem", "label": "obsolete-lemma-stein-projective", "categories": [ "obsolete" ], "title": "obsolete-lemma-stein-projective", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring.", "Let $X \\subset \\mathbf{P}^n_R$ be a closed subscheme.", "Assume that $R = \\Gamma(X, \\mathcal{O}_X)$. Then the special fibre", "$X_k$ is geometrically connected." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "More on Morphisms, Theorem", "\\ref{more-morphisms-theorem-stein-factorization-general}." ], "refs": [ "more-morphisms-theorem-stein-factorization-general" ], "ref_ids": [ 13675 ] } ], "ref_ids": [] }, { "id": 11478, "type": "theorem", "label": "obsolete-lemma-property-irreducible-higher-rank", "categories": [ "obsolete" ], "title": "obsolete-lemma-property-irreducible-higher-rank", "contents": [ "Let $X$ be a Noetherian scheme.", "Let $Z_0 \\subset X$ be an irreducible closed subset with generic point $\\xi$.", "Let $\\mathcal{P}$ be a property of coherent sheaves on $X$ such that", "\\begin{enumerate}", "\\item For any short exact sequence of coherent sheaves if two", "out of three of them have property $\\mathcal{P}$ then so does the", "third.", "\\item If $\\mathcal{P}$ holds for a direct sum of coherent sheaves", "then it holds for both.", "\\item For every integral closed subscheme $Z \\subset Z_0 \\subset X$,", "$Z \\not = Z_0$ and every quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_Z$ we have", "$\\mathcal{P}$ for $(Z \\to X)_*\\mathcal{I}$.", "\\item There exists some coherent sheaf $\\mathcal{G}$ on $X$ such that", "\\begin{enumerate}", "\\item $\\text{Supp}(\\mathcal{G}) = Z_0$,", "\\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$, and", "\\item property $\\mathcal{P}$ holds for $\\mathcal{G}$.", "\\end{enumerate}", "\\end{enumerate}", "Then property $\\mathcal{P}$ holds for every coherent sheaf", "$\\mathcal{F}$ on $X$ whose support is contained in $Z_0$." ], "refs": [], "proofs": [ { "contents": [ "The proof is a variant on the proof of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-property-irreducible}.", "In exactly the same manner as in that proof we see that", "any coherent sheaf whose support is strictly contained in $Z_0$", "has property $\\mathcal{P}$.", "\\medskip\\noindent", "Consider a coherent sheaf $\\mathcal{G}$ as in (3).", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-prepare-filter-irreducible}", "there exists a sheaf of ideals $\\mathcal{I}$ on $Z_0$ and", "a short exact sequence", "$$", "0 \\to", "\\left((Z_0 \\to X)_*\\mathcal{I}\\right)^{\\oplus r} \\to", "\\mathcal{G} \\to", "\\mathcal{Q} \\to 0", "$$", "where the support of $\\mathcal{Q}$ is strictly contained in $Z_0$.", "In particular $r > 0$ and $\\mathcal{I}$ is nonzero", "because the support of $\\mathcal{G}$ is equal to $Z$.", "Since $\\mathcal{Q}$ has property $\\mathcal{P}$ we conclude that", "also $\\left((Z_0 \\to X)_*\\mathcal{I}\\right)^{\\oplus r}$", "has property $\\mathcal{P}$.", "By (2) we deduce property $\\mathcal{P}$ for", "$(Z_0 \\to X)_*\\mathcal{I}$. Slotting this into the proof of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-property-irreducible}", "at the appropriate point gives the lemma.", "Some details omitted." ], "refs": [ "coherent-lemma-property-irreducible", "coherent-lemma-prepare-filter-irreducible", "coherent-lemma-property-irreducible" ], "ref_ids": [ 3331, 3328, 3331 ] } ], "ref_ids": [] }, { "id": 11479, "type": "theorem", "label": "obsolete-lemma-property-higher-rank", "categories": [ "obsolete" ], "title": "obsolete-lemma-property-higher-rank", "contents": [ "Let $X$ be a Noetherian scheme.", "Let $\\mathcal{P}$ be a property of coherent sheaves on $X$ such that", "\\begin{enumerate}", "\\item For any short exact sequence of coherent sheaves if two", "out of three of them have property $\\mathcal{P}$ then so does the", "third.", "\\item If $\\mathcal{P}$ holds for a direct sum of coherent sheaves", "then it holds for both.", "\\item For every integral closed subscheme $Z \\subset X$", "with generic point $\\xi$ there exists", "some coherent sheaf $\\mathcal{G}$ such that", "\\begin{enumerate}", "\\item $\\text{Supp}(\\mathcal{G}) = Z$,", "\\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$, and", "\\item property $\\mathcal{P}$ holds for $\\mathcal{G}$.", "\\end{enumerate}", "\\end{enumerate}", "Then property $\\mathcal{P}$ holds for every coherent sheaf", "on $X$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-property-irreducible-higher-rank}", "in exactly the same way that", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-property} follows from", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-property-irreducible}." ], "refs": [ "obsolete-lemma-property-irreducible-higher-rank", "coherent-lemma-property", "coherent-lemma-property-irreducible" ], "ref_ids": [ 11478, 3332, 3331 ] } ], "ref_ids": [] }, { "id": 11480, "type": "theorem", "label": "obsolete-lemma-section-maps-back-into", "categories": [ "obsolete" ], "title": "obsolete-lemma-section-maps-back-into", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{L})$ be a section.", "Let $\\mathcal{F}' \\subset \\mathcal{F}$ be quasi-coherent", "$\\mathcal{O}_X$-modules. Assume that", "\\begin{enumerate}", "\\item $X$ is quasi-compact,", "\\item $\\mathcal{F}$ is of finite type, and", "\\item $\\mathcal{F}'|_{X_s} = \\mathcal{F}|_{X_s}$.", "\\end{enumerate}", "Then there exists an $n \\geq 0$ such that", "multiplication by $s^n$ on $\\mathcal{F}$ factors", "through $\\mathcal{F}'$." ], "refs": [], "proofs": [ { "contents": [ "In other words we claim that", "$s^n\\mathcal{F} \\subset", "\\mathcal{F}' \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}$", "for some $n \\geq 0$. In other words, we claim that the quotient map", "$\\mathcal{F} \\to \\mathcal{F}/\\mathcal{F}'$ becomes", "zero after multiplying by a power of $s$.", "This follows from Properties, Lemma", "\\ref{properties-lemma-section-maps-backwards}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11481, "type": "theorem", "label": "obsolete-lemma-bound-degree-in-nbhd-generic-point", "categories": [ "obsolete" ], "title": "obsolete-lemma-bound-degree-in-nbhd-generic-point", "contents": [ "Let $f : X \\to Y$ be a morphism schemes. Assume", "\\begin{enumerate}", "\\item $X$ and $Y$ are integral schemes,", "\\item $f$ is locally of finite type and dominant,", "\\item $f$ is either quasi-compact or separated,", "\\item $f$ is generically finite, i.e., one of (1) -- (5) of", "Morphisms, Lemma \\ref{morphisms-lemma-finite-degree} holds.", "\\end{enumerate}", "Then there is a nonempty open $V \\subset Y$ such that", "$f^{-1}(V) \\to V$ is finite locally free of degree $\\deg(X/Y)$.", "In particular, the degrees of the fibres of $f^{-1}(V) \\to V$", "are bounded by $\\deg(X/Y)$." ], "refs": [ "morphisms-lemma-finite-degree" ], "proofs": [ { "contents": [ "We may choose $V$ such that $f^{-1}(V) \\to V$ is finite.", "Then we may shrink $V$ and assume that $f^{-1}(V) \\to V$", "is flat and of finite presentation by generic flatness", "(Morphisms, Proposition \\ref{morphisms-proposition-generic-flatness}).", "Then the morphism is finite locally free by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}.", "Since $V$ is irreducible the morphism has a fixed degree.", "The final statement follows from this and", "Morphisms, Lemma \\ref{morphisms-lemma-finite-locally-free-universally-bounded}." ], "refs": [ "morphisms-proposition-generic-flatness", "morphisms-lemma-finite-flat", "morphisms-lemma-finite-locally-free-universally-bounded" ], "ref_ids": [ 5533, 5471, 5525 ] } ], "ref_ids": [ 5491 ] }, { "id": 11482, "type": "theorem", "label": "obsolete-lemma-factors-through-quotient", "categories": [ "obsolete" ], "title": "obsolete-lemma-factors-through-quotient", "contents": [ "Let $S = \\Spec(R)$ be an affine scheme. Let $X$ be an algebraic space over", "$S$. Let $q_i : \\mathcal{F} \\to \\mathcal{Q}_i$, $i = 1, 2$", "be surjective maps of quasi-coherent $\\mathcal{O}_X$-modules.", "Assume $\\mathcal{Q}_1$ flat over $S$. Let $T \\to S$ be a quasi-compact", "morphism of schemes such that there exists a factorization", "$$", "\\xymatrix{", "& \\mathcal{F}_T \\ar[rd]^{q_{2, T}} \\ar[ld]_{q_{1, T}} \\\\", "\\mathcal{Q}_{1, T} & & \\mathcal{Q}_{2, T} \\ar@{..>}[ll]", "}", "$$", "Then exists a closed subscheme $Z \\subset S$ such that", "(a) $T \\to S$ factors through $Z$ and (b)", "$q_{1, Z}$ factors through $q_{2, Z}$.", "If $\\Ker(q_2)$ is a finite type $\\mathcal{O}_X$-module and $X$", "quasi-compact, then we can take $Z \\to S$ of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Apply Flatness on Spaces, Lemma \\ref{spaces-flat-lemma-F-zero-somewhat-closed}", "to the map $\\Ker(q_2) \\to \\mathcal{Q}_1$." ], "refs": [ "spaces-flat-lemma-F-zero-somewhat-closed" ], "ref_ids": [ 7178 ] } ], "ref_ids": [] }, { "id": 11483, "type": "theorem", "label": "obsolete-lemma-separated-fpqc", "categories": [ "obsolete" ], "title": "obsolete-lemma-separated-fpqc", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a fpqc covering of schemes over $S$.", "Then the map", "$$", "\\Mor_S(T, X)", "\\longrightarrow", "\\prod\\nolimits_{i \\in I} \\Mor_S(T_i, X)", "$$", "is injective." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-sheaf-fpqc}." ], "refs": [ "spaces-properties-proposition-sheaf-fpqc" ], "ref_ids": [ 11919 ] } ], "ref_ids": [] }, { "id": 11484, "type": "theorem", "label": "obsolete-lemma-sheaf-fpqc-open-covering", "categories": [ "obsolete" ], "title": "obsolete-lemma-sheaf-fpqc-open-covering", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $X = \\bigcup_{j \\in J} X_j$ be a Zariski covering, see", "Spaces, Definition \\ref{spaces-definition-Zariski-open-covering}.", "If each $X_j$ satisfies the sheaf property for the fpqc topology", "then $X$ satisfies the sheaf property for the fpqc topology." ], "refs": [ "spaces-definition-Zariski-open-covering" ], "proofs": [ { "contents": [ "This is true because all algebraic spaces satisfy the sheaf property", "for the fpqc topology, see", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-sheaf-fpqc}." ], "refs": [ "spaces-properties-proposition-sheaf-fpqc" ], "ref_ids": [ 11919 ] } ], "ref_ids": [ 8179 ] }, { "id": 11485, "type": "theorem", "label": "obsolete-lemma-sheaf-fpqc-quasi-separated", "categories": [ "obsolete" ], "title": "obsolete-lemma-sheaf-fpqc-quasi-separated", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "If $X$ is Zariski locally quasi-separated over $S$, then $X$ satisfies", "the sheaf condition for the fpqc topology." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of the general", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-sheaf-fpqc}." ], "refs": [ "spaces-properties-proposition-sheaf-fpqc" ], "ref_ids": [ 11919 ] } ], "ref_ids": [] }, { "id": 11486, "type": "theorem", "label": "obsolete-lemma-reasonable-kolmogorov", "categories": [ "obsolete" ], "title": "obsolete-lemma-reasonable-kolmogorov", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a reasonable algebraic space over $S$.", "Then $|X|$ is Kolmogorov (see", "Topology, Definition \\ref{topology-definition-generic-point})." ], "refs": [ "topology-definition-generic-point" ], "proofs": [ { "contents": [ "Follows from the definitions and", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-kolmogorov}." ], "refs": [ "decent-spaces-lemma-kolmogorov" ], "ref_ids": [ 9495 ] } ], "ref_ids": [ 8354 ] }, { "id": 11487, "type": "theorem", "label": "obsolete-lemma-scheme-very-reasonable", "categories": [ "obsolete" ], "title": "obsolete-lemma-scheme-very-reasonable", "contents": [ "A scheme is very reasonable." ], "refs": [], "proofs": [ { "contents": [ "This is true because the identity map is a quasi-compact, surjective", "\\'etale morphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11488, "type": "theorem", "label": "obsolete-lemma-very-reasonable-Zariski-local", "categories": [ "obsolete" ], "title": "obsolete-lemma-very-reasonable-Zariski-local", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "If there exists a Zariski open covering $X = \\bigcup X_i$ such that", "each $X_i$ is very reasonable, then $X$ is very reasonable." ], "refs": [], "proofs": [ { "contents": [ "This is case $(\\epsilon)$ of", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-properties-local}." ], "refs": [ "decent-spaces-lemma-properties-local" ], "ref_ids": [ 9467 ] } ], "ref_ids": [] }, { "id": 11489, "type": "theorem", "label": "obsolete-lemma-quasi-separated-very-reasonable", "categories": [ "obsolete" ], "title": "obsolete-lemma-quasi-separated-very-reasonable", "contents": [ "An algebraic space which is Zariski locally quasi-separated is very reasonable.", "In particular any quasi-separated algebraic space is very reasonable." ], "refs": [], "proofs": [ { "contents": [ "This is one of the implications of", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-bounded-fibres}." ], "refs": [ "decent-spaces-lemma-bounded-fibres" ], "ref_ids": [ 9466 ] } ], "ref_ids": [] }, { "id": 11490, "type": "theorem", "label": "obsolete-lemma-representable-very-reasonable", "categories": [ "obsolete" ], "title": "obsolete-lemma-representable-very-reasonable", "contents": [ "Let $S$ be a scheme.", "Let $X$, $Y$ be algebraic spaces over $S$.", "Let $Y \\to X$ be a representable morphism.", "If $X$ is very reasonable, so is $Y$." ], "refs": [], "proofs": [ { "contents": [ "This is case $(\\epsilon)$ of", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-representable-properties}." ], "refs": [ "decent-spaces-lemma-representable-properties" ], "ref_ids": [ 9468 ] } ], "ref_ids": [] }, { "id": 11491, "type": "theorem", "label": "obsolete-lemma-very-reasonable-quasi-compact-pieces", "categories": [ "obsolete" ], "title": "obsolete-lemma-very-reasonable-quasi-compact-pieces", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a very reasonable algebraic space over $S$.", "There exists a set of schemes", "$U_i$ and morphisms $U_i \\to X$ such that", "\\begin{enumerate}", "\\item each $U_i$ is a quasi-compact scheme,", "\\item each $U_i \\to X$ is \\'etale,", "\\item both projections $U_i \\times_X U_i \\to U_i$ are quasi-compact, and", "\\item the morphism $\\coprod U_i \\to X$ is surjective (and \\'etale).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Decent Spaces, Definition \\ref{decent-spaces-definition-very-reasonable}", "says that there exist $U_i \\to X$ such that (2), (3) and (4) hold.", "Fix $i$, and set $R_i = U_i \\times_X U_i$, and denote $s, t : R_i \\to U_i$", "the projections.", "For any affine open $W \\subset U_i$ the open $W' = t(s^{-1}(W)) \\subset U_i$", "is a quasi-compact $R_i$-invariant open (see", "Groupoids, Lemma \\ref{groupoids-lemma-constructing-invariant-opens}).", "Hence $W'$ is a quasi-compact scheme, $W' \\to X$ is \\'etale, and", "$W' \\times_X W' = s^{-1}(W') = t^{-1}(W')$ so both projections", "$W' \\times_X W' \\to W'$ are quasi-compact. This means the family of", "$W' \\to X$, where $W \\subset U_i$ runs through the members of affine", "open coverings of the $U_i$ gives what we want." ], "refs": [ "decent-spaces-definition-very-reasonable", "groupoids-lemma-constructing-invariant-opens" ], "ref_ids": [ 9562, 9645 ] } ], "ref_ids": [] }, { "id": 11492, "type": "theorem", "label": "obsolete-lemma-vanishing-surjective", "categories": [ "obsolete" ], "title": "obsolete-lemma-vanishing-surjective", "contents": [ "In Cohomology of Spaces, Situation \\ref{spaces-cohomology-situation-vanishing}", "the morphism $p : X \\to \\Spec(A)$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "This lemma was originally used in the proof of", "Cohomology of Spaces, Proposition", "\\ref{spaces-cohomology-proposition-vanishing-affine}", "but now is a consequence of it." ], "refs": [ "spaces-cohomology-proposition-vanishing-affine" ], "ref_ids": [ 11346 ] } ], "ref_ids": [] }, { "id": 11493, "type": "theorem", "label": "obsolete-lemma-vanishing-universally-closed", "categories": [ "obsolete" ], "title": "obsolete-lemma-vanishing-universally-closed", "contents": [ "In Cohomology of Spaces, Situation \\ref{spaces-cohomology-situation-vanishing}", "the morphism $p : X \\to \\Spec(A)$ is universally closed." ], "refs": [], "proofs": [ { "contents": [ "This lemma was originally used in the proof of", "Cohomology of Spaces, Proposition", "\\ref{spaces-cohomology-proposition-vanishing-affine}", "but now is a consequence of it." ], "refs": [ "spaces-cohomology-proposition-vanishing-affine" ], "ref_ids": [ 11346 ] } ], "ref_ids": [] }, { "id": 11494, "type": "theorem", "label": "obsolete-lemma-infinite-sequence", "categories": [ "obsolete" ], "title": "obsolete-lemma-infinite-sequence", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$ having (RS*).", "Let $x$ be an object of", "$\\mathcal{X}$ over an affine scheme $U$ of finite type over $S$.", "Let $u_n \\in U$, $n \\geq 1$ be pairwise distinct finite type points", "such that $x$ is not versal at $u_n$ for all $n$. After replacing", "$u_n$ by a subsequence, there exist morphisms", "$$", "x \\to x_1 \\to x_2 \\to \\ldots", "\\quad\\text{in }\\mathcal{X}\\text{ lying over }\\quad", "U \\to U_1 \\to U_2 \\to \\ldots", "$$", "over $S$ such that", "\\begin{enumerate}", "\\item for each $n$ the morphism $U \\to U_n$ is a first order", "thickening,", "\\item for each $n$ we have a short exact sequence", "$$", "0 \\to \\kappa(u_n) \\to \\mathcal{O}_{U_n} \\to \\mathcal{O}_{U_{n - 1}} \\to 0", "$$", "with $U_0 = U$ for $n = 1$,", "\\item for each $n$ there does {\\bf not} exist a pair $(W, \\alpha)$", "consisting of an open neighbourhood $W \\subset U_n$ of $u_n$", "and a morphism $\\alpha : x_n|_W \\to x$", "such that the composition", "$$", "x|_{U \\cap W} \\xrightarrow{\\text{restriction of }x \\to x_n}", "x_n|_W \\xrightarrow{\\alpha} x", "$$", "is the canonical morphism $x|_{U \\cap W} \\to x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma was originally used in the proof of a criterion for", "openness of versality", "(Artin's Axioms, Lemma \\ref{artin-lemma-SGE-implies-openness-versality}) but it", "got replaced by Artin's Axioms, Lemma \\ref{artin-lemma-infinite-sequence-pre}", "from which it readily follows. Namely,", "after replacing $u_n$, $n \\geq 1$ by a subsequence we may and do", "assume that there are no specializations among these points, see", "Properties, Lemma \\ref{properties-lemma-thin-infinite-sequence}.", "Then we can apply", "Artin's Axioms, Lemma \\ref{artin-lemma-infinite-sequence-pre}", "to finish the proof." ], "refs": [ "artin-lemma-SGE-implies-openness-versality", "artin-lemma-infinite-sequence-pre", "properties-lemma-thin-infinite-sequence", "artin-lemma-infinite-sequence-pre" ], "ref_ids": [ 11386, 11385, 2960, 11385 ] } ], "ref_ids": [] }, { "id": 11495, "type": "theorem", "label": "obsolete-lemma-category-fibred", "categories": [ "obsolete" ], "title": "obsolete-lemma-category-fibred", "contents": [ "Let $X \\to Y$ be a morphism of schemes.", "\\begin{enumerate}", "\\item The category $\\mathcal{C}_{X/Y}$ is fibred over $X_{Zar}$.", "\\item The category $\\mathcal{C}_{X/Y}$ is fibred over $Y_{Zar}$.", "\\item The category $\\mathcal{C}_{X/Y}$ is fibred over the", "category of pairs $(U, V)$ where $U \\subset X$, $V \\subset Y$ are", "open and $f(U) \\subset V$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Ad (1). Given an object $U \\to A$ of $\\mathcal{C}_{X/Y}$ and a morphism", "$U' \\to U$ of $X_{Zar}$ consider the object $i' : U' \\to A$ of", "$\\mathcal{C}_{X/Y}$ where $i'$ is the composition of $i$ and $U' \\to U$.", "The morphism $(U' \\to A) \\to (U \\to A)$ of $\\mathcal{C}_{X/Y}$", "is strongly cartesian over $X_{Zar}$.", "\\medskip\\noindent", "Ad (2). Given an object $U \\to A/V$ and $V' \\to V$ we can set", "$U' = U \\cap f^{-1}(V')$ and $A' = V' \\times_V A$ to obtain a strongly", "cartesian morphism $(U' \\to A') \\to (U \\to A)$ over $V' \\to V$.", "\\medskip\\noindent", "Ad (3). Denote $(X/Y)_{Zar}$ the category in (3). Given $U \\to A/V$", "and a morphism $(U', V') \\to (U, V)$ in $(X/Y)_{Zar}$ we can consider", "$A' = V' \\times_V A$. Then the morphism $(U' \\to A'/V') \\to (U \\to A/V)$", "is strongly cartesian in $\\mathcal{C}_{X/Y}$ over $(X/Y)_{Zar}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11496, "type": "theorem", "label": "obsolete-lemma-equivalence", "categories": [ "obsolete" ], "title": "obsolete-lemma-equivalence", "contents": [ "Notation and assumptions as above. Let $p : X_A \\to X$ denote the projection.", "Given $A'$ denote $p' : X_{A'} \\to X$ the projection. The functor $p'_*$", "induces an equivalence of categories between", "\\begin{enumerate}", "\\item the category $\\textit{Lift}(\\mathcal{F}, A')$, and", "\\item the category $\\textit{Lift}(p_*\\mathcal{F}, A')$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "FIXME." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11497, "type": "theorem", "label": "obsolete-lemma-second-equivalence", "categories": [ "obsolete" ], "title": "obsolete-lemma-second-equivalence", "contents": [ "Notation and assumptions as above.", "The functor $\\pi_!$ induces an equivalence of categories between", "\\begin{enumerate}", "\\item the category", "$\\textit{Lift}_\\mathcal{O}(i_*\\underline{\\mathcal{G}}, A')$, and", "\\item the category $\\textit{Lift}(\\mathcal{G}, A')$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "FIXME." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11498, "type": "theorem", "label": "obsolete-lemma-second-equivalence-obs", "categories": [ "obsolete" ], "title": "obsolete-lemma-second-equivalence-obs", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-second-equivalence}.", "Consider the object", "$$", "L = L(\\Lambda, X, A, \\mathcal{G}) = L\\pi_!(Li^*(i_*(\\underline{\\mathcal{G}})))", "$$", "of $D(\\mathcal{O}_X \\otimes_\\Lambda A)$. Given a surjection $A' \\to A$ of", "$\\Lambda$-algebras with square zero kernel $I$ we have", "\\begin{enumerate}", "\\item The category $\\textit{Lift}(\\mathcal{G}, A')$ is nonempty", "if and only if a certain class", "$\\xi \\in \\Ext^2_{\\mathcal{O}_X \\otimes A}(L, \\mathcal{G} \\otimes_A I)$", "is zero.", "\\item If $\\textit{Lift}(\\mathcal{G}, A')$ is nonempty, then", "$\\text{Lift}(\\mathcal{G}, A')$ is principal homogeneous under", "$\\Ext^1_{\\mathcal{O}_X \\otimes A}(L, \\mathcal{G} \\otimes_A I)$.", "\\item Given a lift $\\mathcal{G}'$, the set of automorphisms of", "$\\mathcal{G}'$ which pull back to $\\text{id}_\\mathcal{G}$ is canonically", "isomorphic to", "$\\Ext^0_{\\mathcal{O}_X \\otimes A}(L, \\mathcal{G} \\otimes_A I)$.", "\\end{enumerate}" ], "refs": [ "obsolete-lemma-second-equivalence" ], "proofs": [ { "contents": [ "FIXME." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 11497 ] }, { "id": 11499, "type": "theorem", "label": "obsolete-lemma-pseudo-coherent", "categories": [ "obsolete" ], "title": "obsolete-lemma-pseudo-coherent", "contents": [ "In the situation of Proposition \\ref{proposition-conclusion}, if", "$X \\to \\Spec(\\Lambda)$ is locally of finite type and $\\Lambda$ is Noetherian,", "then $L$ is pseudo-coherent." ], "refs": [ "obsolete-proposition-conclusion" ], "proofs": [ { "contents": [ "FIXME." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 11524 ] }, { "id": 11500, "type": "theorem", "label": "obsolete-lemma-ob-is-obstruction", "categories": [ "obsolete" ], "title": "obsolete-lemma-ob-is-obstruction", "contents": [ "In the situation of Remark \\ref{remark-construction-ob} assume that", "$\\mathcal{F}$ is flat over $U$. Then the vanishing of the class", "$\\xi_{U'}$ is a necessary and sufficient condition for the existence of a", "$\\mathcal{O}_{X \\times_B U'}$-module $\\mathcal{F}'$ flat over $U'$", "with $i^*\\mathcal{F}' \\cong \\mathcal{F}$." ], "refs": [ "obsolete-remark-construction-ob" ], "proofs": [ { "contents": [ "[Proof (sketch)]", "We will use the criterion of", "Deformation Theory, Lemma \\ref{defos-lemma-inf-obs-ext-rel-ringed-topoi}.", "We will abbreviate $\\mathcal{O} = \\mathcal{O}_{X \\times_B U}$ and", "$\\mathcal{O}' = \\mathcal{O}_{X \\times_B U'}$.", "Consider the short exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_{U'} \\to \\mathcal{O}_U \\to 0.", "$$", "Let $\\mathcal{J} \\subset \\mathcal{O}'$ be the quasi-coherent", "sheaf of ideals cutting out $X \\times_B U$. By the above we obtain an exact", "sequence", "$$", "\\text{Tor}_1^{\\mathcal{O}_B}(\\mathcal{O}_X, \\mathcal{O}_U) \\to", "q^*\\mathcal{I} \\to \\mathcal{J} \\to 0", "$$", "where the $\\text{Tor}_1^{\\mathcal{O}_B}(\\mathcal{O}_X, \\mathcal{O}_U)$", "is an abbreviation for", "$$", "\\text{Tor}_1^{h^{-1}\\mathcal{O}_B}(p^{-1}\\mathcal{O}_X, q^{-1}\\mathcal{O}_U)", "\\otimes_{(p^{-1}\\mathcal{O}_X\\otimes_{h^{-1}\\mathcal{O}_B}q^{-1}\\mathcal{O}_U)}", "\\mathcal{O}.", "$$", "Tensoring with $\\mathcal{F}$ we obtain the exact sequence", "$$", "\\mathcal{F} \\otimes_\\mathcal{O}", "\\text{Tor}_1^{\\mathcal{O}_B}(\\mathcal{O}_X, \\mathcal{O}_U) \\to", "\\mathcal{F} \\otimes_\\mathcal{O}", "q^*\\mathcal{I} \\to", "\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{J} \\to 0", "$$", "(Note that the roles of the letters $\\mathcal{I}$ and $\\mathcal{J}$", "are reversed relative to the notation in", "Deformation Theory, Lemma \\ref{defos-lemma-inf-obs-ext-rel-ringed-topoi}.)", "Condition (1) of the lemma is that the last map above is an", "isomorphism, i.e., that the first map is zero.", "The vanishing of this map may be checked on stalks at geometric points ", "$\\overline{z} = (\\overline{x}, \\overline{u}) : \\Spec(k) \\to X \\times_B U$.", "Set $R = \\mathcal{O}_{B, \\overline{b}}$, $A = \\mathcal{O}_{X, \\overline{x}}$,", "$B = \\mathcal{O}_{U, \\overline{u}}$, and", "$C = \\mathcal{O}_{\\overline{z}}$.", "By Cotangent, Lemma \\ref{cotangent-lemma-fibre-product}", "and the defining triangle for $E(\\mathcal{F})$ we see that", "$$", "H^{-2}(E(\\mathcal{F}))_{\\overline{z}} =", "\\mathcal{F}_{\\overline{z}} \\otimes \\text{Tor}_1^R(A, B)", "$$", "The map $\\xi_{U'}$ therefore induces a map", "$$", "\\mathcal{F}_{\\overline{z}} \\otimes \\text{Tor}_1^R(A, B)", "\\longrightarrow", "\\mathcal{F}_{\\overline{z}} \\otimes_B \\mathcal{I}_{\\overline{u}}", "$$", "We claim this map is the same as the stalk of the map described above", "(proof omitted; this is a purely ring theoretic statement).", "Thus we see that condition (1) of ", "Deformation Theory, Lemma \\ref{defos-lemma-inf-obs-ext-rel-ringed-topoi}", "is equivalent to the vanishing", "$H^{-2}(\\xi_{U'}) :", "H^{-2}(E(\\mathcal{F})) \\to \\mathcal{F} \\otimes \\mathcal{I}$.", "\\medskip\\noindent", "To finish the proof we show that, assuming that condition (1) is satisfied,", "condition (2) is equivalent to the vanishing of $\\xi_{U'}$. In the rest", "of the proof we write $\\mathcal{F} \\otimes \\mathcal{I}$ to denote", "$\\mathcal{F} \\otimes_\\mathcal{O} q^*\\mathcal{I} =", "\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{J}$. A consideration", "of the spectral sequence", "$$", "\\Ext^i(H^{-j}(E(\\mathcal{F})), \\mathcal{F} \\otimes \\mathcal{I})", "\\Rightarrow", "\\Ext^{i + j}(E(\\mathcal{F}), \\mathcal{F} \\otimes \\mathcal{I})", "$$", "using that $H^0(E(\\mathcal{F})) = \\mathcal{F}$ and", "$H^{-1}(E(\\mathcal{F})) = 0$", "shows that there is an exact sequence", "$$", "0 \\to", "\\Ext^2(\\mathcal{F}, \\mathcal{F} \\otimes \\mathcal{I}) \\to", "\\Ext^2(E(\\mathcal{F}), \\mathcal{F} \\otimes \\mathcal{I}) \\to", "\\Hom(H^{-2}(E(\\mathcal{F})), \\mathcal{F} \\otimes \\mathcal{I})", "$$", "Thus our element $\\xi_{U'}$ is an element of", "$\\Ext^2(\\mathcal{F}, \\mathcal{F} \\otimes \\mathcal{I})$.", "The proof is finished by showing this element agrees with the", "element of", "Deformation Theory, Lemma \\ref{defos-lemma-inf-obs-ext-rel-ringed-topoi}", "a verification we omit." ], "refs": [ "defos-lemma-inf-obs-ext-rel-ringed-topoi", "defos-lemma-inf-obs-ext-rel-ringed-topoi", "cotangent-lemma-fibre-product", "defos-lemma-inf-obs-ext-rel-ringed-topoi", "defos-lemma-inf-obs-ext-rel-ringed-topoi" ], "ref_ids": [ 13403, 13403, 11240, 13403, 13403 ] } ], "ref_ids": [ 11561 ] }, { "id": 11501, "type": "theorem", "label": "obsolete-lemma-coherent-defo-thy-general", "categories": [ "obsolete" ], "title": "obsolete-lemma-coherent-defo-thy-general", "contents": [ "In Quot, Situation \\ref{quot-situation-coherent} assume that", "$S$ is a locally Noetherian scheme and $S = B$.", "Let $\\mathcal{X} = \\textit{Coh}_{X/B}$.", "Then we have openness of versality for $\\mathcal{X}$ (see", "Artin's Axioms, Definition \\ref{artin-definition-openness-versality})." ], "refs": [ "artin-definition-openness-versality" ], "proofs": [ { "contents": [ "[Proof (sketch)]", "Let $U \\to S$ be of finite type morphism of schemes, $x$ an object of", "$\\mathcal{X}$ over $U$ and $u_0 \\in U$ a finite type point such that", "$x$ is versal at $u_0$. After shrinking $U$ we may assume that $u_0$", "is a closed point (Morphisms, Lemma \\ref{morphisms-lemma-point-finite-type})", "and $U = \\Spec(A)$ with $U \\to S$ mapping into an", "affine open $\\Spec(\\Lambda)$ of $S$. We will use", "Artin's Axioms, Lemma \\ref{artin-lemma-dual-openness} to prove the lemma.", "Let $\\mathcal{F}$ be the coherent module on $X_A = \\Spec(A) \\times_S X$", "flat over $A$ corresponding to the given object $x$.", "\\medskip\\noindent", "Choose $E(\\mathcal{F})$ and $e_\\mathcal{F}$ as in", "Remark \\ref{remark-construction-E}.", "The description of the cohomology sheaves of $E(\\mathcal{F})$ shows", "that", "$$", "\\Ext^1(E(\\mathcal{F}), \\mathcal{F} \\otimes_A M) =", "\\Ext^1(\\mathcal{F}, \\mathcal{F} \\otimes_A M)", "$$", "for any $A$-module $M$. Using this and using", "Deformation Theory, Lemma \\ref{defos-lemma-inf-ext-rel-ringed-topoi}", "we have an isomorphism of functors", "$$", "T_x(M) = \\Ext^1_{X_A}(E(\\mathcal{F}), \\mathcal{F} \\otimes_A M)", "$$", "By Lemma \\ref{lemma-ob-is-obstruction} given any surjection $A' \\to A$", "of $\\Lambda$-algebras with square zero kernel $I$ we have an obstruction class", "$$", "\\xi_{A'} \\in \\Ext^2_{X_A}(E(\\mathcal{F}), \\mathcal{F} \\otimes_A I)", "$$", "Apply Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-compute-ext}", "to the computation of the Ext groups", "$\\Ext^i_{X_A}(E(\\mathcal{F}), \\mathcal{F} \\otimes_A M)$", "for $i \\leq m$ with $m = 2$. We omit the verification that", "$E(\\mathcal{F})$ is in $D^-_{\\textit{Coh}}$; hint: use", "Cotangent, Lemma \\ref{cotangent-lemma-cotangent-finite}.", "We find a perfect object $K \\in D(A)$", "and functorial isomorphisms", "$$", "H^i(K \\otimes_A^\\mathbf{L} M)", "\\longrightarrow", "\\Ext^i_{X_A}(E(\\mathcal{F}), \\mathcal{F} \\otimes_A M)", "$$", "for $i \\leq m$ compatible with boundary maps. This object $K$, together", "with the displayed identifications above gives us a datum as in", "Artin's Axioms, Situation \\ref{artin-situation-dual}.", "Finally, condition (iv) of", "Artin's Axioms, Lemma \\ref{artin-lemma-dual-obstruction}", "holds by a variant of", "Deformation Theory, Lemma \\ref{defos-lemma-verify-iv-ringed-topoi}", "whose formulation and proof we omit.", "Thus Artin's Axioms, Lemma \\ref{artin-lemma-dual-openness}", "applies and the lemma is proved." ], "refs": [ "morphisms-lemma-point-finite-type", "artin-lemma-dual-openness", "obsolete-remark-construction-E", "defos-lemma-inf-ext-rel-ringed-topoi", "obsolete-lemma-ob-is-obstruction", "spaces-perfect-lemma-compute-ext", "cotangent-lemma-cotangent-finite", "artin-lemma-dual-obstruction", "defos-lemma-verify-iv-ringed-topoi", "artin-lemma-dual-openness" ], "ref_ids": [ 5205, 11398, 11560, 13402, 11500, 2733, 11182, 11397, 13406, 11398 ] } ], "ref_ids": [ 11423 ] }, { "id": 11502, "type": "theorem", "label": "obsolete-lemma-henselian", "categories": [ "obsolete" ], "title": "obsolete-lemma-henselian", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "The category of", "Algebraization of Formal Spaces, Equation", "(\\ref{restricted-equation-modification})", "for $A$ is equivalent to the category", "Algebraization of Formal Spaces, Equation", "(\\ref{restricted-equation-modification})", "for the henselization $A^h$ of $A$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Algebraization of Formal Spaces, Lemma", "\\ref{restricted-lemma-equivalence-to-completion}." ], "refs": [ "restricted-lemma-equivalence-to-completion" ], "ref_ids": [ 2430 ] } ], "ref_ids": [] }, { "id": 11503, "type": "theorem", "label": "obsolete-lemma-double-dual-rational", "categories": [ "obsolete" ], "title": "obsolete-lemma-double-dual-rational", "contents": [ "In Resolution of Surfaces, Situation \\ref{resolve-situation-rational}.", "Let $M$ be a finite reflexive $A$-module. Let $M \\otimes_A \\mathcal{O}_X$", "denote the pullback of the associated $\\mathcal{O}_S$-module. Then", "$M \\otimes_A \\mathcal{O}_X$ maps onto its double dual." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F} = (M \\otimes_A \\mathcal{O}_X)^{**}$ be the double dual and", "let $\\mathcal{F}' \\subset \\mathcal{F}$ be the image of the evaluation map", "$M \\otimes_A \\mathcal{O}_X \\to \\mathcal{F}$. Then we have a short exact", "sequence", "$$", "0 \\to \\mathcal{F}' \\to \\mathcal{F} \\to \\mathcal{Q} \\to 0", "$$", "Since $X$ is normal, the local rings $\\mathcal{O}_{X, x}$ are discrete", "valuation rings for points of codimension $1$ (see", "Properties, Lemma \\ref{properties-lemma-criterion-normal}).", "Hence $\\mathcal{Q}_x = 0$ for such points by", "More on Algebra, Lemma \\ref{more-algebra-lemma-cokernel-map-double-dual-dvr}.", "Thus $\\mathcal{Q}$ is supported in finitely many closed points and is", "globally generated by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-support-dimension-0}.", "We obtain the exact sequence", "$$", "0 \\to H^0(X, \\mathcal{F}') \\to H^0(X, \\mathcal{F}) \\to H^0(X, \\mathcal{Q}) \\to 0", "$$", "because $\\mathcal{F}'$ is generated by global sections", "(Resolution of Surfaces, Lemma \\ref{resolve-lemma-globally-generated}).", "Since $X \\to \\Spec(A)$ is an isomorphism over the complement of the", "closed point, and since $M$ is reflexive, we see that the maps", "$$", "M \\to H^0(X, \\mathcal{F}') \\to H^0(X, \\mathcal{F})", "$$", "induce isomorphisms after localization at any nonmaximal prime of $A$.", "Hence these maps are isomorphisms by More on Algebra, Lemma", "\\ref{more-algebra-lemma-check-isomorphism-via-depth-and-ass}", "and the fact that reflexive modules over normal rings have property $(S_2)$", "(More on Algebra, Lemma \\ref{more-algebra-lemma-reflexive-over-normal}).", "Thus we conclude that $\\mathcal{Q} = 0$ as desired." ], "refs": [ "properties-lemma-criterion-normal", "more-algebra-lemma-cokernel-map-double-dual-dvr", "coherent-lemma-coherent-support-dimension-0", "resolve-lemma-globally-generated", "more-algebra-lemma-check-isomorphism-via-depth-and-ass", "more-algebra-lemma-reflexive-over-normal" ], "ref_ids": [ 2989, 9924, 3317, 11670, 9933, 9937 ] } ], "ref_ids": [] }, { "id": 11504, "type": "theorem", "label": "obsolete-lemma-good-blowing-up", "categories": [ "obsolete" ], "title": "obsolete-lemma-good-blowing-up", "contents": [ "Let $b : X' \\to X$ be the blowing up of a smooth projective", "scheme over a field $k$ in a smooth closed subscheme $Z \\subset X$.", "Picture", "$$", "\\xymatrix{", "E \\ar[r]_j \\ar[d]_\\pi & X' \\ar[d]^b \\\\", "Z \\ar[r]^i & X", "}", "$$", "Assume there exists an element of $K_0(X)$ whose restriction to", "$Z$ is equal to the class of $\\mathcal{C}_{Z/X}$ in $K_0(Z)$.", "Then $[Lb^*\\mathcal{O}_Z] = [\\mathcal{O}_E] \\cdot \\alpha''$", "in $K_0(X')$ for some $\\alpha'' \\in K_0(X')$." ], "refs": [], "proofs": [ { "contents": [ "The schemes $X$, $X'$, $E$, $Z$ are smooth and projective over", "$k$ and hence we have $K'_0(X) = K_0(X) = K_0(\\textit{Vect}(X)) =", "K_0(D^b_{\\textit{Coh}}(X)))$", "and similarly for the other $3$. See", "Derived Categories of Schemes, Lemmas \\ref{perfect-lemma-Noetherian-Kprime},", "\\ref{perfect-lemma-Kprime-K}, and \\ref{perfect-lemma-K-is-old-K}.", "We will switch between these versions at will in this proof.", "Consider the short exact sequence", "$$", "0 \\to \\mathcal{F} \\to \\pi^*\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{E/X'} \\to 0", "$$", "of finite locally free $\\mathcal{O}_E$-modules defining $\\mathcal{F}$.", "Observe that $\\mathcal{C}_{E/X'} = \\mathcal{O}_{X'}(-E)|_E$", "is the restriction of the invertible $\\mathcal{O}_X$-module", "$\\mathcal{O}_{X'}(-E)$.", "Let $\\alpha \\in K_0(X)$ be an element such that", "$i^*\\alpha = [\\mathcal{C}_{Z/X}]$ in $K_0(Z)$.", "Let $\\alpha' = b^*\\alpha - [\\mathcal{O}_{X'}(-E)]$.", "Then $j^*\\alpha' = [\\mathcal{F}]$. We deduce that", "$j^*\\lambda^i(\\alpha') = [\\wedge^i(\\mathcal{F})]$ by", "Weil Cohomology Theories, Lemma \\ref{weil-lemma-lambda-operations}.", "This means that $[\\mathcal{O}_E] \\cdot \\alpha' = [\\wedge^i\\mathcal{F}]$", "in $K_0(X)$, see", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-projection-formula}.", "Let $r$ be the maximum codimension of an irreducible component of $Z$", "in $X$. A computation which we omit shows that", "$H^{-i}(Lb^*\\mathcal{O}_Z) = \\wedge^i\\mathcal{F}$", "for $i \\geq 0, 1, \\ldots, r - 1$ and zero in other degrees.", "It follows that in $K_0(X)$ we have", "\\begin{align*}", "[Lb^*\\mathcal{O}_Z] & =", "\\sum\\nolimits_{i = 0, \\ldots, r - 1} (-1)^i[\\wedge^i\\mathcal{F}] \\\\", "& =", "\\sum\\nolimits_{i = 0, \\ldots, r - 1} (-1)^i[\\mathcal{O}_E] \\lambda^i(\\alpha') \\\\", "& =", "[\\mathcal{O}_E] \\left(\\sum\\nolimits_{i = 0, \\ldots, r - 1}", "(-1)^i \\lambda^i(\\alpha')\\right)", "\\end{align*}", "This proves the lemma with", "$\\alpha'' = \\sum_{i = 0, \\ldots, r - 1} (-1)^i \\lambda^i(\\alpha')$." ], "refs": [ "perfect-lemma-Noetherian-Kprime", "perfect-lemma-Kprime-K", "perfect-lemma-K-is-old-K", "weil-lemma-lambda-operations", "perfect-lemma-projection-formula" ], "ref_ids": [ 7097, 7099, 7100, 5092, 7101 ] } ], "ref_ids": [] }, { "id": 11505, "type": "theorem", "label": "obsolete-lemma-gysin-factors-principal", "categories": [ "obsolete" ], "title": "obsolete-lemma-gysin-factors-principal", "contents": [ "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $X$ be integral and $n = \\dim_\\delta(X)$.", "Let $a \\in \\Gamma(X, \\mathcal{O}_X)$ be a nonzero function.", "Let $i : D = Z(a) \\to X$ be the closed immersion of the zero scheme of $a$.", "Let $f \\in R(X)^*$.", "In this case $i^*\\text{div}_X(f) = 0$ in $A_{n - 2}(D)$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Chow Homology, Lemma \\ref{chow-lemma-gysin-factors-general}." ], "refs": [ "chow-lemma-gysin-factors-general" ], "ref_ids": [ 5720 ] } ], "ref_ids": [] }, { "id": 11506, "type": "theorem", "label": "obsolete-lemma-push-pull-effective-Cartier", "categories": [ "obsolete" ], "title": "obsolete-lemma-push-pull-effective-Cartier", "contents": [ "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $f : X \\to Y$ be a proper morphism.", "Let $D \\subset Y$ be an effective Cartier divisor.", "Assume $X$, $Y$ integral, $n = \\dim_\\delta(X) = \\dim_\\delta(Y)$ and", "$f$ dominant. Then", "$$", "f_*[f^{-1}(D)]_{n - 1} = [R(X) : R(Y)] [D]_{n - 1}.", "$$", "In particular if $f$ is birational then $f_*[f^{-1}(D)]_{n - 1} = [D]_{n - 1}$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Chow Homology, Lemma \\ref{chow-lemma-equal-c1-as-cycles}", "and the fact that $D$ is the zero", "scheme of the canonical section $1_D$ of $\\mathcal{O}_X(D)$." ], "refs": [ "chow-lemma-equal-c1-as-cycles" ], "ref_ids": [ 5710 ] } ], "ref_ids": [] }, { "id": 11507, "type": "theorem", "label": "obsolete-lemma-blowing-up-denominators", "categories": [ "obsolete" ], "title": "obsolete-lemma-blowing-up-denominators", "contents": [ "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Assume $X$ integral with $\\dim_\\delta(X) = n$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s$ be a nonzero meromorphic section of $\\mathcal{L}$.", "Let $U \\subset X$ be the maximal open subscheme such that", "$s$ corresponds to a section of $\\mathcal{L}$ over $U$.", "There exists a projective morphism", "$$", "\\pi : X' \\longrightarrow X", "$$", "such that", "\\begin{enumerate}", "\\item $X'$ is integral,", "\\item $\\pi|_{\\pi^{-1}(U)} : \\pi^{-1}(U) \\to U$ is an isomorphism,", "\\item there exist effective Cartier divisors $D, E \\subset X'$", "such that", "$$", "\\pi^*\\mathcal{L} = \\mathcal{O}_{X'}(D - E),", "$$", "\\item the meromorphic section $s$ corresponds, via the isomorphism above,", "to the meromorphic section $1_D \\otimes (1_E)^{-1}$ (see", "Divisors, Definition", "\\ref{divisors-definition-invertible-sheaf-effective-Cartier-divisor}),", "\\item we have", "$$", "\\pi_*([D]_{n - 1} - [E]_{n - 1}) = \\text{div}_\\mathcal{L}(s)", "$$", "in $Z_{n - 1}(X)$.", "\\end{enumerate}" ], "refs": [ "divisors-definition-invertible-sheaf-effective-Cartier-divisor" ], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent ideal sheaf", "of denominators of $s$, see Divisors, Definition", "\\ref{divisors-definition-regular-meromorphic-ideal-denominators}.", "By Divisors, Lemma \\ref{divisors-lemma-blowing-up-denominators}", "we get (2), (3), and (4).", "By Divisors, Lemma \\ref{divisors-lemma-blow-up-integral-scheme}", "we get (1). By Divisors, Lemma \\ref{divisors-lemma-blowing-up-projective}", "the morphism $\\pi$ is projective.", "We still have to prove (5).", "By Chow Homology, Lemma \\ref{chow-lemma-equal-c1-as-cycles} we have", "$$", "\\pi_*(\\text{div}_{\\mathcal{L}'}(s')) = \\text{div}_\\mathcal{L}(s).", "$$", "Hence it suffices to show that", "$\\text{div}_{\\mathcal{L}'}(s') = [D]_{n - 1} - [E]_{n - 1}$.", "This follows from the equality", "$s' = 1_D \\otimes 1_E^{-1}$ and additivity, see", "Divisors, Lemma \\ref{divisors-lemma-c1-additive}." ], "refs": [ "divisors-definition-regular-meromorphic-ideal-denominators", "divisors-lemma-blowing-up-denominators", "divisors-lemma-blow-up-integral-scheme", "divisors-lemma-blowing-up-projective", "chow-lemma-equal-c1-as-cycles", "divisors-lemma-c1-additive" ], "ref_ids": [ 8105, 8075, 8059, 8063, 5710, 8027 ] } ], "ref_ids": [ 8092 ] }, { "id": 11508, "type": "theorem", "label": "obsolete-lemma-two-divisors", "categories": [ "obsolete" ], "title": "obsolete-lemma-two-divisors", "contents": [ "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Assume $X$ integral and $\\dim_\\delta(X) = n$.", "Let $D_1, D_2$ be two effective Cartier divisors in $X$.", "Let $Z$ be an open and closed subscheme of the scheme $D_1 \\cap D_2$.", "Assume $\\dim_\\delta(D_1 \\cap D_2 \\setminus Z) \\leq n - 2$.", "Then there exists a morphism", "$b : X' \\to X$, and Cartier divisors", "$D_1', D_2', E$ on $X'$ with the following properties", "\\begin{enumerate}", "\\item $X'$ is integral,", "\\item $b$ is projective,", "\\item $b$ is the blowup of $X$ in the closed subscheme $Z$,", "\\item $E = b^{-1}(Z)$,", "\\item $b^{-1}(D_1) = D'_1 + E$, and $b^{-1}D_2 = D_2' + E$,", "\\item $\\dim_\\delta(D'_1 \\cap D'_2) \\leq n - 2$, and if", "$Z = D_1 \\cap D_2$ then $D'_1 \\cap D'_2 = \\emptyset$,", "\\item for every integral closed subscheme $W'$", "with $\\dim_\\delta(W') = n - 1$ we have", "\\begin{enumerate}", "\\item if $\\epsilon_{W'}(D'_1, E) > 0$, then setting", "$W = b(W')$ we have", "$\\dim_\\delta(W) = n - 1$ and", "$$", "\\epsilon_{W'}(D'_1, E) < \\epsilon_W(D_1, D_2),", "$$", "\\item if $\\epsilon_{W'}(D'_2, E) > 0$, then setting", "$W = b(W')$ we have", "$\\dim_\\delta(W) = n - 1$ and", "$$", "\\epsilon_{W'}(D'_2, E) < \\epsilon_W(D_1, D_2),", "$$", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that the quasi-coherent ideal sheaf", "$\\mathcal{I} = \\mathcal{I}_{D_1} + \\mathcal{I}_{D_2}$", "defines the scheme theoretic intersection $D_1 \\cap D_2 \\subset X$.", "Since $Z$ is a union of connected components of $D_1 \\cap D_2$", "we see that for every $z \\in Z$ the kernel of", "$\\mathcal{O}_{X, z} \\to \\mathcal{O}_{Z, z}$ is equal to $\\mathcal{I}_z$.", "Let $b : X' \\to X$ be the blowup of $X$ in $Z$. (So Zariski locally", "around $Z$ it is the blowup of $X$ in $\\mathcal{I}$.)", "Denote $E = b^{-1}(Z)$ the corresponding effective Cartier divisor, see", "Divisors,", "Lemma \\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}.", "Since $Z \\subset D_1$ we have $E \\subset f^{-1}(D_1)$ and hence", "$D_1 = D_1' + E$ for some effective Cartier divisor $D'_1 \\subset X'$,", "see Divisors, Lemma \\ref{divisors-lemma-difference-effective-Cartier-divisors}.", "Similarly $D_2 = D_2' + E$. This takes care of assertions (1) -- (5).", "\\medskip\\noindent", "Note that if $W'$ is as in (7) (a) or (7) (b), then the image $W$", "of $W'$ is contained in $D_1 \\cap D_2$. If $W$ is not contained in", "$Z$, then $b$ is an isomorphism at the generic point of $W$ and", "we see that $\\dim_\\delta(W) = \\dim_\\delta(W') = n - 1$ which", "contradicts the assumption that", "$\\dim_\\delta(D_1 \\cap D_2 \\setminus Z) \\leq n - 2$.", "Hence $W \\subset Z$. This means that", "to prove (6) and (7) we may work locally around $Z$ on $X$.", "\\medskip\\noindent", "Thus we may assume that $X = \\Spec(A)$ with", "$A$ a Noetherian domain, and $D_1 = \\Spec(A/a)$,", "$D_2 = \\Spec(A/b)$ and $Z = D_1 \\cap D_2$.", "Set $I = (a, b)$. Since $A$ is a domain and $a, b \\not = 0$ we can", "cover the blowup by two patches, namely", "$U = \\Spec(A[s]/(as - b))$ and $V = \\Spec(A[t]/(bt -a))$.", "These patches are glued using the isomorphism", "$A[s, s^{-1}]/(as - b) \\cong A[t, t^{-1}]/(bt - a)$", "which maps $s$ to $t^{-1}$.", "The effective Cartier divisor $E$ is described by", "$\\Spec(A[s]/(as - b, a)) \\subset U$ and", "$\\Spec(A[t]/(bt - a, b)) \\subset V$.", "The closed subscheme $D'_1$ corresponds to", "$\\Spec(A[t]/(bt - a, t)) \\subset U$.", "The closed subscheme $D'_2$ corresponds to", "$\\Spec(A[s]/(as -b, s)) \\subset V$.", "Since ``$ts = 1$'' we see that $D'_1 \\cap D'_2 = \\emptyset$.", "\\medskip\\noindent", "Suppose we have a prime $\\mathfrak q \\subset A[s]/(as - b)$", "of height one with $s, a \\in \\mathfrak q$.", "Let $\\mathfrak p \\subset A$ be the corresponding prime of $A$.", "Observe that $a, b \\in \\mathfrak p$.", "By the dimension formula we see that $\\dim(A_{\\mathfrak p}) = 1$", "as well. The final assertion to be shown is that", "$$", "\\text{ord}_{A_{\\mathfrak p}}(a)", "\\text{ord}_{A_{\\mathfrak p}}(b)", ">", "\\text{ord}_{B_{\\mathfrak q}}(a)", "\\text{ord}_{B_{\\mathfrak q}}(s)", "$$", "where $B = A[s]/(as - b)$. By", "Algebra, Lemma \\ref{algebra-lemma-quasi-finite-extension-dim-1}", "we have $\\text{ord}_{A_{\\mathfrak p}}(x) \\geq \\text{ord}_{B_{\\mathfrak q}}(x)$", "for $x = a, b$. Since $\\text{ord}_{B_{\\mathfrak q}}(s) > 0$ we win", "by additivity of the $\\text{ord}$ function and the fact that", "$as = b$." ], "refs": [ "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "divisors-lemma-difference-effective-Cartier-divisors", "algebra-lemma-quasi-finite-extension-dim-1" ], "ref_ids": [ 8054, 7932, 1068 ] } ], "ref_ids": [] }, { "id": 11509, "type": "theorem", "label": "obsolete-lemma-sum-divisors-associated-Weil", "categories": [ "obsolete" ], "title": "obsolete-lemma-sum-divisors-associated-Weil", "contents": [ "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Assume $X$ integral and $\\dim_\\delta(X) = n$.", "Let $\\{D_i\\}_{i \\in I}$ be a locally finite collection", "of effective Cartier divisors on $X$.", "Suppose given $n_i \\geq 0$ for $i \\in I$.", "Then", "$$", "[D]_{n - 1} = \\sum\\nolimits_i n_i[D_i]_{n - 1}", "$$", "in $Z_{n - 1}(X)$." ], "refs": [], "proofs": [ { "contents": [ "Since we are proving an equality of cycles we may work locally on $X$.", "Hence this reduces to a finite sum, and by induction to a sum of", "two effective Cartier divisors $D = D_1 + D_2$.", "By Chow Homology, Lemma \\ref{chow-lemma-compute-c1} we see that", "$D_1 = \\text{div}_{\\mathcal{O}_X(D_1)}(1_{D_1})$ where", "$1_{D_1}$ denotes the canonical section of $\\mathcal{O}_X(D_1)$.", "Of course we have the same statement for $D_2$ and $D$.", "Since $1_D = 1_{D_1} \\otimes 1_{D_2}$ via the identification", "$\\mathcal{O}_X(D) = \\mathcal{O}_X(D_1) \\otimes \\mathcal{O}_X(D_2)$", "we win by Divisors, Lemma \\ref{divisors-lemma-c1-additive}." ], "refs": [ "chow-lemma-compute-c1", "divisors-lemma-c1-additive" ], "ref_ids": [ 5703, 8027 ] } ], "ref_ids": [] }, { "id": 11510, "type": "theorem", "label": "obsolete-lemma-blowing-up-intersections", "categories": [ "obsolete" ], "title": "obsolete-lemma-blowing-up-intersections", "contents": [ "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Assume $X$ integral and $\\dim_\\delta(X) = d$.", "Let $\\{D_i\\}_{i \\in I}$ be a locally finite collection of", "effective Cartier divisors on $X$.", "Assume that for all $\\{i, j, k\\} \\subset I$, $\\#\\{i, j, k\\} = 3$", "we have $D_i \\cap D_j \\cap D_k = \\emptyset$.", "Then there exist", "\\begin{enumerate}", "\\item an open subscheme $U \\subset X$ with", "$\\dim_\\delta(X \\setminus U) \\leq d - 3$,", "\\item a morphism $b : U' \\to U$, and", "\\item effective Cartier divisors $\\{D'_j\\}_{j \\in J}$ on $U'$", "\\end{enumerate}", "with the following properties:", "\\begin{enumerate}", "\\item $b$ is proper morphism $b : U' \\to U$,", "\\item $U'$ is integral,", "\\item $b$ is an isomorphism over the complement of the union of the pairwise", "intersections of the $D_i|_U$,", "\\item $\\{D'_j\\}_{j \\in J}$ is a locally finite collection of effective", "Cartier divisors on $U'$,", "\\item $\\dim_\\delta(D'_j \\cap D'_{j'}) \\leq d - 2$ if $j \\not = j'$, and", "\\item $b^{-1}(D_i|_U) = \\sum n_{ij} D'_j$ for certain $n_{ij} \\geq 0$.", "\\end{enumerate}", "Moreover, if $X$ is quasi-compact, then we may assume $U = X$ in the above." ], "refs": [], "proofs": [ { "contents": [ "Let us first prove this in the quasi-compact case, since it is perhaps", "the most interesting case. In this case we produce inductively a sequence", "of blowups", "$$", "X = X_0 \\xleftarrow{b_0} X_1 \\xleftarrow{b_1} X_2 \\leftarrow \\ldots", "$$", "and finite sets of effective Cartier divisors $\\{D_{n, i}\\}_{i \\in I_n}$.", "At each stage these will have the property that any triple", "intersection $D_{n, i} \\cap D_{n, j} \\cap D_{n, k}$ is empty.", "Moreover, for each $n \\geq 0$ we will have", "$I_{n + 1} = I_n \\amalg P(I_n)$ where $P(I_n)$ denotes", "the set of pairs of elements of $I_n$. Finally, we will have", "$$", "b_n^{-1}(D_{n, i}) = D_{n + 1, i} +", "\\sum\\nolimits_{i' \\in I_n, i' \\not = i} D_{n + 1, \\{i, i'\\}}", "$$", "We conclude that for each $n \\geq 0$ we have", "$(b_0 \\circ \\ldots \\circ b_n)^{-1}(D_i)$ is a nonnegative", "integer combination of the divisors $D_{n + 1, j}$, $j \\in I_{n + 1}$.", "\\medskip\\noindent", "To start the induction we set $X_0 = X$ and", "$I_0 = I$ and $D_{0, i} = D_i$.", "\\medskip\\noindent", "Given $(X_n, \\{D_{n, i}\\}_{i \\in I_n})$ let $X_{n + 1}$ be the", "blowup of $X_n$ in the closed subscheme", "$Z_n = \\bigcup_{\\{i, i'\\} \\in P(I_n)} D_{n, i} \\cap D_{n, i'}$.", "Note that the closed subschemes $D_{n, i} \\cap D_{n, i'}$ are pairwise", "disjoint by our assumption on triple intersections.", "In other words we may write", "$Z_n = \\coprod_{\\{i, i'\\} \\in P(I_n)} D_{n, i} \\cap D_{n, i'}$.", "Moreover, in a Zariski neighbourhood of $D_{n, i} \\cap D_{n, i'}$ the", "morphism $b_n$ is equal to the blowup of the scheme $X_n$", "in the closed subscheme $D_{n, i} \\cap D_{n, i'}$, and the results", "of Lemma \\ref{lemma-two-divisors} apply.", "Hence setting $D_{n + 1, \\{i, i'\\}} = b_n^{-1}(D_i \\cap D_{i'})$", "we get an effective Cartier divisor.", "The Cartier divisors $D_{n + 1, \\{i, i'\\}}$ are pairwise disjoint.", "Clearly we have", "$b_n^{-1}(D_{n, i}) \\supset D_{n + 1, \\{i, i'\\}}$ for", "every $i' \\in I_n$, $i' \\not = i$. Hence, applying", "Divisors, Lemma \\ref{divisors-lemma-difference-effective-Cartier-divisors}", "we see that indeed $b^{-1}(D_{n, i}) = D_{n + 1, i} +", "\\sum\\nolimits_{i' \\in I_n, i' \\not = i} D_{n + 1, \\{i, i'\\}}$", "for some effective Cartier divisor $D_{n + 1, i}$ on $X_{n + 1}$.", "In a neighbourhood of $D_{n + 1, \\{i, i'\\}}$ these divisors", "$D_{n + 1, i}$ play the role of the primed divisors of", "Lemma \\ref{lemma-two-divisors}. In particular we conclude that", "$D_{n + 1, i} \\cap D_{n + 1, i'} = \\emptyset$ if $i \\not = i'$,", "$i, i' \\in I_n$ by part (6) of Lemma \\ref{lemma-two-divisors}.", "This already implies that triple intersections", "of the divisors $D_{n + 1, i}$ are zero.", "\\medskip\\noindent", "OK, and at this point we can use the quasi-compactness of $X$", "to conclude that the invariant", "\\begin{equation}", "\\label{equation-invariant}", "\\epsilon(X, \\{D_i\\}_{i \\in I}) =", "\\max\\{\\epsilon_Z(D_i, D_{i'}) \\mid", "Z \\subset X,", "\\dim_\\delta(Z) = d - 1,", "\\{i, i'\\} \\in P(I)\\}", "\\end{equation}", "is finite, since after all each $D_i$ has at most finitely many irreducible", "components. We claim that for some $n$ the invariant", "$\\epsilon(X_n, \\{D_{n, i}\\}_{i \\in I_n})$ is zero. Namely, if not then", "by Lemma \\ref{lemma-two-divisors} we have a strictly decreasing sequence", "$$", "\\epsilon(X, \\{D_i\\}_{i \\in I})", "=", "\\epsilon(X_0, \\{D_{0, i}\\}_{i \\in I_0})", ">", "\\epsilon(X_1, \\{D_{1, i}\\}_{i \\in I_1})", ">", "\\ldots", "$$", "of positive integers which is a contradiction. Take $n$ with", "invariant $\\epsilon(X_n, \\{D_{n, i}\\}_{i \\in I_n})$ equal to zero.", "This means that there is no integral closed subscheme $Z \\subset X_n$", "and no pair of indices $i, i' \\in I_n$", "such that $\\epsilon_Z(D_{n, i}, D_{n, i'}) > 0$.", "In other words, $\\dim_\\delta(D_{n, i}, D_{n, i'}) \\leq d - 2$ for", "all pairs $\\{i, i'\\} \\in P(I_n)$ as desired.", "\\medskip\\noindent", "Next, we come to the general case where we no longer assume that", "the scheme $X$ is quasi-compact. The problem with the idea from", "the first part of the proof is that we may get and infinite sequence", "of blowups with centers dominating a fixed point of $X$. In order to", "avoid this we cut out suitable closed subsets of codimension $\\geq 3$", "at each stage. Namely, we will construct by induction", "a sequence of morphisms having the following shape", "$$", "\\xymatrix{", "X = X_0 \\\\", "U_0 \\ar[u]^{j_0} & X_1 \\ar[l]_{b_0} \\\\", " & U_1 \\ar[u]^{j_1} & X_2 \\ar[l]_{b_1} \\\\", " & & U_2 \\ar[u]^{j_2} & X_3 \\ar[l]_{b_2}", "}", "$$", "Each of the morphisms $j_n : U_n \\to X_n$ will be an open immersion.", "Each of the morphisms $b_n : X_{n + 1} \\to U_n$ will be a proper birational", "morphism of integral schemes. As in the quasi-compact case we will have", "effective Cartier divisors $\\{D_{n, i}\\}_{i \\in I_n}$ on $X_n$.", "At each stage these will have the property that any triple", "intersection $D_{n, i} \\cap D_{n, j} \\cap D_{n, k}$ is empty.", "Moreover, for each $n \\geq 0$ we will have", "$I_{n + 1} = I_n \\amalg P(I_n)$ where $P(I_n)$ denotes", "the set of pairs of elements of $I_n$.", "Finally, we will arrange it so that", "$$", "b_n^{-1}(D_{n, i}|_{U_n}) = D_{n + 1, i} +", "\\sum\\nolimits_{i' \\in I_n, i' \\not = i} D_{n + 1, \\{i, i'\\}}", "$$", "\\medskip\\noindent", "We start the induction by setting $X_0 = X$,", "$I_0 = I$ and $D_{0, i} = D_i$.", "\\medskip\\noindent", "Given $(X_n, \\{D_{n, i}\\})$ we construct the open subscheme", "$U_n$ as follows. For each pair $\\{i, i'\\} \\in P(I_n)$ consider", "the closed subscheme $D_{n, i} \\cap D_{n, i'}$. This has ``good''", "irreducible components which have $\\delta$-dimension $d - 2$ and", "``bad'' irreducible components which have $\\delta$-dimension $d - 1$.", "Let us set", "$$", "\\text{Bad}(i, i')", "=", "\\bigcup\\nolimits_{W \\subset D_{n, i} \\cap D_{n, i'}", "\\text{ irred.\\ comp. with }\\dim_\\delta(W) = d - 1} W", "$$", "and similarly", "$$", "\\text{Good}(i, i')", "=", "\\bigcup\\nolimits_{W \\subset D_{n, i} \\cap D_{n, i'}", "\\text{ irred.\\ comp. with }\\dim_\\delta(W) = d - 2} W.", "$$", "Then $D_{n, i} \\cap D_{n, i'} = \\text{Bad}(i, i') \\cup \\text{Good}(i, i')$", "and moreover we have", "$\\dim_\\delta(\\text{Bad}(i, i') \\cap \\text{Good}(i, i')) \\leq d - 3$.", "Here is our choice of $U_n$:", "$$", "U_n", "=", "X_n", "\\setminus", "\\bigcup\\nolimits_{\\{i, i'\\} \\in P(I_n)}", "\\text{Bad}(i, i') \\cap \\text{Good}(i, i').", "$$", "By our condition on triple intersections of the divisors $D_{n, i}$", "we see that the union is actually a disjoint union. Moreover,", "we see that (as a scheme)", "$$", "D_{n, i}|_{U_n} \\cap D_{n, i'}|_{U_n}", "=", "Z_{n, i, i'} \\amalg G_{n, i, i'}", "$$", "where $Z_{n, i, i'}$ is $\\delta$-equidimensional of dimension $d - 1$", "and $G_{n, i, i'}$ is $\\delta$-equidimensional of dimension $d - 2$.", "(So topologically $Z_{n, i, i'}$ is the union of the bad components", "but throw out intersections with good components.) Finally we set", "$$", "Z_n =", "\\bigcup\\nolimits_{\\{i, i'\\} \\in P(I_n)} Z_{n, i, i'} =", "\\coprod\\nolimits_{\\{i, i'\\} \\in P(I_n)} Z_{n, i, i'},", "$$", "and we let $b_n : X_{n + 1} \\to X_n$ be the blowup in $Z_n$.", "Note that Lemma \\ref{lemma-two-divisors}", "applies to the morphism $b_n : X_{n + 1} \\to X_n$ locally around", "each of the loci $D_{n, i}|_{U_n} \\cap D_{n, i'}|_{U_n}$. Hence,", "exactly as in the first part of the proof we obtain effective", "Cartier divisors $D_{n + 1, \\{i, i'\\}}$ for $\\{i, i'\\} \\in P(I_n)$", "and effective Cartier divisors $D_{n + 1, i}$ for $i \\in I_n$", "such that", "$b_n^{-1}(D_{n, i}|_{U_n}) = D_{n + 1, i} +", "\\sum\\nolimits_{i' \\in I_n, i' \\not = i} D_{n + 1, \\{i, i'\\}}$.", "For each $n$ denote $\\pi_n : X_n \\to X$ the morphism obtained", "as the composition $j_0 \\circ \\ldots \\circ j_{n - 1} \\circ b_{n - 1}$.", "\\medskip\\noindent", "{\\bf Claim:} given any quasi-compact open $V \\subset X$", "for all sufficiently large $n$ the maps", "$$", "\\pi_n^{-1}(V) \\leftarrow \\pi_{n + 1}^{-1}(V) \\leftarrow \\ldots", "$$", "are all isomorphisms. Namely, if the map", "$\\pi_n^{-1}(V) \\leftarrow \\pi_{n + 1}^{-1}(V)$ is not an isomorphism,", "then $Z_{n, i, i'} \\cap \\pi_n^{-1}(V) \\not = \\emptyset$ for some", "$\\{i, i'\\} \\in P(I_n)$. Hence there exists an irreducible component", "$W \\subset D_{n, i} \\cap D_{n, i'}$ with $\\dim_\\delta(W) = d - 1$.", "In particular we see that $\\epsilon_W(D_{n, i}, D_{n, i'}) > 0$.", "Applying Lemma \\ref{lemma-two-divisors} repeatedly we see that", "$$", "\\epsilon_W(D_{n, i}, D_{n, i'})", "<", "\\epsilon(V, \\{D_i|_V\\}) - n", "$$", "with $\\epsilon(V, \\{D_i|_V\\})$ as in (\\ref{equation-invariant}).", "Since $V$ is quasi-compact, we have", "$\\epsilon(V, \\{D_i|_V\\}) < \\infty$ and taking $n > \\epsilon(V, \\{D_i|_V\\})$", "we see the result.", "\\medskip\\noindent", "Note that by construction the difference $X_n \\setminus U_n$", "has $\\dim_\\delta(X_n \\setminus U_n) \\leq d - 3$.", "Let $T_n = \\pi_n(X_n \\setminus U_n)$ be its image in $X$.", "Traversing in the diagram of maps above using each $b_n$ is closed", "it follows that $T_0 \\cup \\ldots \\cup T_n$ is a closed subset of $X$", "for each $n$. Any $t \\in T_n$ satisfies $\\delta(t) \\leq d - 3$", "by construction. Hence $\\overline{T_n} \\subset X$ is a closed subset", "with $\\dim_\\delta(T_n) \\leq d - 3$. By the claim above we see", "that for any quasi-compact open $V \\subset X$ we have", "$T_n \\cap V \\not = \\emptyset$ for at most finitely many $n$.", "Hence $\\{\\overline{T_n}\\}_{n \\geq 0}$ is a locally finite", "collection of closed subsets, and we may set", "$U = X \\setminus \\bigcup \\overline{T_n}$. This will be", "$U$ as in the lemma.", "\\medskip\\noindent", "Note that $U_n \\cap \\pi_n^{-1}(U) = \\pi_n^{-1}(U)$ by construction", "of $U$. Hence all the morphisms", "$$", "b_n : \\pi_{n + 1}^{-1}(U) \\longrightarrow \\pi_n^{-1}(U)", "$$", "are proper. Moreover, by the claim they eventually become isomorphisms", "over each quasi-compact open of $X$. Hence we can define", "$$", "U' = \\lim_n \\pi_n^{-1}(U).", "$$", "The induced morphism $b : U' \\to U$ is proper since this is local", "on $U$, and over each compact open the limit stabilizes. Similarly", "we set $J = \\bigcup_{n \\geq 0} I_n$ using the inclusions", "$I_n \\to I_{n + 1}$ from the construction. For $j \\in J$ choose", "an $n_0$ such that $j$ corresponds to $i \\in I_{n_0}$ and define", "$D'_j = \\lim_{n \\geq n_0} D_{n, i}$. Again this makes sense", "as locally over $X$ the morphisms stabilize.", "The other claims of the lemma are verified as in the case", "of a quasi-compact $X$." ], "refs": [ "obsolete-lemma-two-divisors", "divisors-lemma-difference-effective-Cartier-divisors", "obsolete-lemma-two-divisors", "obsolete-lemma-two-divisors", "obsolete-lemma-two-divisors", "obsolete-lemma-two-divisors", "obsolete-lemma-two-divisors" ], "ref_ids": [ 11508, 7932, 11508, 11508, 11508, 11508, 11508 ] } ], "ref_ids": [] }, { "id": 11511, "type": "theorem", "label": "obsolete-lemma-improved-additivity", "categories": [ "obsolete" ], "title": "obsolete-lemma-improved-additivity", "contents": [ "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\{i_j : D_j \\to X \\}_{j \\in J}$ be a locally finite collection", "of effective Cartier divisors on $X$. Let $n_j > 0$, $j\\in J$.", "Set $D = \\sum_{j \\in J} n_j D_j$, and denote $i : D \\to X$ the", "inclusion morphism. Let $\\alpha \\in Z_{k + 1}(X)$. Then", "$$", "p : \\coprod\\nolimits_{j \\in J} D_j \\longrightarrow D", "$$", "is proper and", "$$", "i^*\\alpha = p_*\\left(\\sum n_j i_j^*\\alpha\\right)", "$$", "in $\\CH_k(D)$." ], "refs": [], "proofs": [ { "contents": [ "The proof of this lemma is made a bit longer than expected", "by a subtlety concerning infinite sums of rational equivalences.", "In the quasi-compact case the family $D_j$ is finite and the result", "is altogether easy and a straightforward consequence of", "Chow Homology, Lemma \\ref{chow-lemma-compute-c1} and", "Divisors, Lemma \\ref{divisors-lemma-c1-additive} and the definitions.", "\\medskip\\noindent", "The morphism $p$ is proper since the family $\\{D_j\\}_{j \\in J}$", "is locally finite. Write $\\alpha = \\sum_{a \\in A} m_a [W_a]$", "with $W_a \\subset X$ an integral closed subscheme of", "$\\delta$-dimension $k + 1$.", "Denote $i_a : W_a \\to X$ the closed immersion.", "We assume that $m_a \\not = 0$ for all $a \\in A$ such that", "$\\{W_a\\}_{a \\in A}$ is locally finite on $X$.", "\\medskip\\noindent", "Observe that by", "Chow Homology, Definition \\ref{chow-definition-gysin-homomorphism}", "the class $i^*\\alpha$ is the class of a cycle", "$\\sum m_a\\beta_a$ for certain $\\beta_a \\in Z_k(W_a \\cap D)$.", "Namely, if $W_a \\not \\subset D$ then $\\beta_a = [D \\cap W_a]_k$", "and if $W_a \\subset D$, then $\\beta_a$ is a cycle", "representing $c_1(\\mathcal{O}_X(D)) \\cap [W_a]$.", "\\medskip\\noindent", "For each $a \\in A$ write $J = J_{a, 1} \\amalg J_{a, 2} \\amalg J_{a, 3}$", "where", "\\begin{enumerate}", "\\item $j \\in J_{a, 1}$ if and only if $W_a \\cap D_j = \\emptyset$,", "\\item $j \\in J_{a, 2}$ if and only if", "$W_a \\not = W_a \\cap D_1 \\not = \\emptyset$, and", "\\item $j \\in J_{a, 3}$ if and only if $W_a \\subset D_j$.", "\\end{enumerate}", "Since the family $\\{D_j\\}$ is locally finite we see that", "$J_{a, 3}$ is a finite set. For every $a \\in A$ and $j \\in J$", "we choose a cycle $\\beta_{a, j} \\in Z_k(W_a \\cap D_j)$ as follows", "\\begin{enumerate}", "\\item if $j \\in J_{a, 1}$ we set $\\beta_{a, j} = 0$,", "\\item if $j \\in J_{a, 2}$ we set $\\beta_{a, j} = [D_j \\cap W_a]_k$, and", "\\item if $j \\in J_{a, 3}$ we choose $\\beta_{a, j} \\in Z_k(W_a)$", "representing $c_1(i_a^*\\mathcal{O}_X(D_j)) \\cap [W_j]$.", "\\end{enumerate}", "We claim that", "$$", "\\beta_a \\sim_{rat}", "\\sum\\nolimits_{j \\in J} n_j \\beta_{a, j}", "$$", "in $\\CH_k(W_a \\cap D)$.", "\\medskip\\noindent", "Case I: $W_a \\not \\subset D$. In this case $J_{a, 3} = \\emptyset$.", "Thus it suffices to show that", "$[D \\cap W_a]_k = \\sum n_j [D_j \\cap W_a]_k$ as cycles.", "This is Lemma \\ref{lemma-sum-divisors-associated-Weil}.", "\\medskip\\noindent", "Case II: $W_a \\subset D$. In this case $\\beta_a$ is a cycle representing", "$c_1(i_a^*\\mathcal{O}_X(D)) \\cap [W_a]$.", "Write $D = D_{a, 1} + D_{a, 2} + D_{a, 3}$ with", "$D_{a, s} = \\sum_{j \\in J_{a, s}} n_jD_j$.", "By Divisors, Lemma \\ref{divisors-lemma-c1-additive} we have", "\\begin{eqnarray*}", "c_1(i_a^*\\mathcal{O}_X(D)) \\cap [W_a] & = &", "c_1(i_a^*\\mathcal{O}_X(D_{a, 1})) \\cap [W_a] +", "c_1(i_a^*\\mathcal{O}_X(D_{a, 2})) \\cap [W_a] \\\\", "& &", " + c_1(i_a^*\\mathcal{O}_X(D_{a, 3})) \\cap [W_a].", "\\end{eqnarray*}", "It is clear that the first term of the sum is zero.", "Since $J_{a, 3}$ is finite we see that the last term agrees", "with $\\sum\\nolimits_{j \\in J_{a, 3}} n_jc_1(i_a^*\\mathcal{L}_j) \\cap [W_a]$,", "see Divisors, Lemma \\ref{divisors-lemma-c1-additive}.", "This is represented by $\\sum_{j \\in J_{a, 3}} n_j \\beta_{a, j}$.", "Finally, by Case I we see that the middle term is represented by the cycle", "$\\sum\\nolimits_{j \\in J_{a, 2}} n_j[D_j \\cap W_a]_k =", "\\sum_{j \\in J_{a, 2}} n_j\\beta_{a, j}$.", "Whence the claim in this case.", "\\medskip\\noindent", "At this point we are ready to finish the proof of the lemma.", "Namely, we have $i^*D \\sim_{rat} \\sum m_a\\beta_a$ by our", "choice of $\\beta_a$. For each $a$ we have", "$\\beta_a \\sim_{rat} \\sum_j \\beta_{a, j}$ with the rational", "equivalence taking place on $D \\cap W_a$.", "Since the collection of closed subschemes $D \\cap W_a$", "is locally finite on $D$, we see that also", "$\\sum m_a \\beta_a \\sim_{rat} \\sum_{a, j} m_a\\beta_{a, j}$", "on $D$! (See", "Chow Homology, Remark \\ref{chow-remark-infinite-sums-rational-equivalences}.)", "Ok, and now it is clear that $\\sum_a m_a\\beta_{a, j}$ (viewed", "as a cycle on $D_j$) represents $i_j^*\\alpha$ and hence", "$\\sum_{a, j} m_a\\beta_{a, j}$ represents $p_* \\sum_j i_j^*\\alpha$", "and we win." ], "refs": [ "chow-lemma-compute-c1", "divisors-lemma-c1-additive", "chow-definition-gysin-homomorphism", "obsolete-lemma-sum-divisors-associated-Weil", "divisors-lemma-c1-additive", "divisors-lemma-c1-additive", "chow-remark-infinite-sums-rational-equivalences" ], "ref_ids": [ 5703, 8027, 5915, 11509, 8027, 8027, 5931 ] } ], "ref_ids": [] }, { "id": 11512, "type": "theorem", "label": "obsolete-lemma-commutativity-effective-Cartier-proper-intersection", "categories": [ "obsolete" ], "title": "obsolete-lemma-commutativity-effective-Cartier-proper-intersection", "contents": [ "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Assume $X$ integral and $\\dim_\\delta(X) = n$.", "Let $D$, $D'$ be effective Cartier divisors on $X$.", "Assume $\\dim_\\delta(D \\cap D') = n - 2$. Let $i : D \\to X$,", "resp.\\ $i' : D' \\to X$ be the corresponding closed immersions.", "Then", "\\begin{enumerate}", "\\item there exists a cycle $\\alpha \\in Z_{n - 2}(D \\cap D')$", "whose pushforward to $D$ represents", "$i^*[D']_{n - 1} \\in \\CH_{n - 2}(D)$", "and whose pushforward to $D'$ represents", "$(i')^*[D]_{n - 1} \\in \\CH_{n - 2}(D')$, and", "\\item we have", "$$", "D \\cdot [D']_{n - 1}", "=", "D' \\cdot [D]_{n - 1}", "$$", "in $\\CH_{n - 2}(X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) is a trivial consequence of part (1).", "Let us write $[D]_{n - 1} = \\sum n_a[Z_a]$ and", "$[D']_{n - 1} = \\sum m_b[Z_b]$ with $Z_a$ the irreducible", "components of $D$ and $[Z_b]$ the irreducible", "components of $D'$. According to", "Chow Homology, Definition \\ref{chow-definition-gysin-homomorphism},", "we have $i^*D' = \\sum m_b i^*[Z_b]$ and $(i')^*D = \\sum n_a(i')^*[Z_a]$.", "By assumption, none of the irreducible components $Z_b$", "is contained in $D$, and hence $i^*[Z_b] = [Z_b\\cap D]_{n - 2}$", "by definition. Similarly $(i')^*[Z_a] = [Z_a \\cap D']_{n - 2}$.", "Hence we are trying to prove the equality of cycles", "$$", "\\sum n_a[Z_a \\cap D']_{n - 2} = \\sum m_b[Z_b \\cap D]_{n - 2}", "$$", "which are indeed supported on $D \\cap D'$.", "Let $W \\subset X$ be an integral closed subscheme", "with $\\dim_\\delta(W) = n - 2$. Let $\\xi \\in W$ be its generic point.", "Set $R = \\mathcal{O}_{X, \\xi}$. It is a Noetherian local domain.", "Note that $\\dim(R) = 2$. Let $f \\in R$, resp.\\ $f' \\in R$", "be an element defining the ideal of $D$, resp.\\ $D'$.", "By assumption $\\dim(R/(f, f')) = 0$. Let", "$\\mathfrak q'_1, \\ldots, \\mathfrak q'_t \\subset R$ be the minimal", "primes over $(f')$, let $\\mathfrak q_1, \\ldots, \\mathfrak q_s \\subset R$", "be the minimal primes over $(f)$.", "The equality above comes down to the equality", "$$", "\\sum_{i = 1, \\ldots, s}", "\\text{length}_{R_{\\mathfrak q_i}}(R_{\\mathfrak q_i}/(f))", "\\text{ord}_{R/\\mathfrak q_i}(f')", "=", "\\sum_{j = 1, \\ldots, t}", "\\text{length}_{R_{\\mathfrak q'_j}}(R_{\\mathfrak q'_j}/(f'))", "\\text{ord}_{R/\\mathfrak q'_j}(f).", "$$", "By Chow Homology, Lemma \\ref{chow-lemma-length-multiplication} ", "applied with $M = R/(f)$ the left hand side of", "this equation is equal to", "$$", "\\text{length}_R(R/(f, f'))", "-", "\\text{length}_R(\\Ker(f' : R/(f) \\to R/(f)))", "$$", "OK, and now we note that", "$\\Ker(f' : R/(f) \\to R/(f))$ is canonically isomorphic", "to $((f) \\cap (f'))/(ff')$ via the map $x \\bmod (f) \\mapsto", "f'x \\bmod (ff')$. Hence the left hand side is", "$$", "\\text{length}_R(R/(f, f'))", "-", "\\text{length}_R((f) \\cap (f')/(ff'))", "$$", "Since this is symmetric in $f$ and $f'$ we win." ], "refs": [ "chow-definition-gysin-homomorphism", "chow-lemma-length-multiplication" ], "ref_ids": [ 5915, 5652 ] } ], "ref_ids": [] }, { "id": 11513, "type": "theorem", "label": "obsolete-lemma-commutativity-effective-Cartier-proper-intersection-infinite", "categories": [ "obsolete" ], "title": "obsolete-lemma-commutativity-effective-Cartier-proper-intersection-infinite", "contents": [ "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Assume $X$ integral and $\\dim_\\delta(X) = n$.", "Let $\\{D_j\\}_{j \\in J}$ be a locally finite collection of", "effective Cartier divisors on $X$. Let $n_j, m_j \\geq 0$ be", "collections of nonnegative integers. Set", "$D = \\sum n_j D_j$ and $D' = \\sum m_j D_j$.", "Assume that $\\dim_\\delta(D_j \\cap D_{j'}) = n - 2$ for every", "$j \\not = j'$. Then $D \\cdot [D']_{n - 1} = D' \\cdot [D]_{n - 1}$ in", "$\\CH_{n - 2}(X)$." ], "refs": [], "proofs": [ { "contents": [ "This lemma is a trivial consequence of", "Lemmas \\ref{lemma-sum-divisors-associated-Weil} and", "\\ref{lemma-commutativity-effective-Cartier-proper-intersection}", "in case the sums are finite, e.g., if $X$ is quasi-compact.", "Hence we suggest the reader skip the proof.", "\\medskip\\noindent", "Here is the proof in the general case.", "Let $i_j : D_j \\to X$ be the closed immersions", "Let $p : \\coprod D_j \\to X$ denote coproduct of the morphisms $i_j$.", "Let $\\{Z_a\\}_{a \\in A}$ be the collection of irreducible components of", "$\\bigcup D_j$. For each $j$ we write", "$$", "[D_j]_{n - 1} = \\sum d_{j, a}[Z_a].", "$$", "By Lemma \\ref{lemma-sum-divisors-associated-Weil} we have", "$$", "[D]_{n - 1} = \\sum n_j d_{j, a} [Z_a],", "\\quad", "[D']_{n - 1} = \\sum m_j d_{j, a} [Z_a].", "$$", "By Lemma \\ref{lemma-improved-additivity}", "we have", "$$", "D \\cdot [D']_{n - 1} = p_*\\left(\\sum n_j i_j^*[D']_{n - 1} \\right),", "\\quad", "D' \\cdot [D]_{n - 1} = p_*\\left(\\sum m_{j'} i_{j'}^*[D]_{n - 1} \\right).", "$$", "As in the definition of the Gysin homomorphisms (see", "Chow Homology, Definition \\ref{chow-definition-gysin-homomorphism})", "we choose cycles $\\beta_{a, j}$ on $D_j \\cap Z_a$ representing", "$i_j^*[Z_a]$. (Note that in fact $\\beta_{a, j} = [D_j \\cap Z_a]_{n - 2}$", "if $Z_a$ is not contained in $D_j$, i.e., there is no choice in that case.)", "Now since $p$ is a closed immersion when restricted to each of the $D_j$", "we can (and we will) view $\\beta_{a, j}$ as a cycle on $X$.", "Plugging in the formulas for $[D]_{n - 1}$ and $[D']_{n - 1}$ obtained", "above we see that", "$$", "D \\cdot [D']_{n - 1} =", "\\sum\\nolimits_{j, j', a} n_j m_{j'} d_{j', a} \\beta_{a, j},", "\\quad", "D' \\cdot [D]_{n - 1} =", "\\sum\\nolimits_{j, j', a} m_{j'} n_j d_{j, a} \\beta_{a, j'}.", "$$", "Moreover, with the same conventions we also have", "$$", "D_j \\cdot [D_{j'}]_{n - 1} = \\sum d_{j', a} \\beta_{a, j}.", "$$", "In these terms", "Lemma \\ref{lemma-commutativity-effective-Cartier-proper-intersection}", "(see also its proof)", "says that for $j \\not = j'$ the cycles", "$\\sum d_{j', a} \\beta_{a, j}$ and $\\sum d_{j, a} \\beta_{a, j'}$", "are equal as cycles! Hence we see that", "\\begin{eqnarray*}", "D \\cdot [D']_{n - 1}", "& = &", "\\sum\\nolimits_{j, j', a} n_j m_{j'} d_{j', a} \\beta_{a, j} \\\\", "& = &", "\\sum\\nolimits_{j \\not = j'} n_j m_{j'}", "\\left(\\sum\\nolimits_a d_{j', a} \\beta_{a, j}\\right) +", "\\sum\\nolimits_{j, a} n_j m_j d_{j, a} \\beta_{a, j} \\\\", "& = &", "\\sum\\nolimits_{j \\not = j'} n_j m_{j'}", "\\left(\\sum\\nolimits_a d_{j, a} \\beta_{a, j'}\\right) +", "\\sum\\nolimits_{j, a} n_j m_j d_{j, a} \\beta_{a, j} \\\\", "& = &", "\\sum\\nolimits_{j, j', a} m_{j'} n_j d_{j, a} \\beta_{a, j'} \\\\", "& = &", "D' \\cdot [D]_{n - 1}", "\\end{eqnarray*}", "and we win." ], "refs": [ "obsolete-lemma-sum-divisors-associated-Weil", "obsolete-lemma-commutativity-effective-Cartier-proper-intersection", "obsolete-lemma-sum-divisors-associated-Weil", "obsolete-lemma-improved-additivity", "chow-definition-gysin-homomorphism", "obsolete-lemma-commutativity-effective-Cartier-proper-intersection" ], "ref_ids": [ 11509, 11512, 11509, 11511, 5915, 11512 ] } ], "ref_ids": [] }, { "id": 11514, "type": "theorem", "label": "obsolete-lemma-commutativity-effective-Cartier", "categories": [ "obsolete" ], "title": "obsolete-lemma-commutativity-effective-Cartier", "contents": [ "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Assume $X$ integral and $\\dim_\\delta(X) = n$.", "Let $D$, $D'$ be effective Cartier divisors on $X$.", "Then", "$$", "D \\cdot [D']_{n - 1} = D' \\cdot [D]_{n - 1}", "$$", "in $\\CH_{n - 2}(X)$." ], "refs": [], "proofs": [ { "contents": [ "[First proof of Lemma \\ref{lemma-commutativity-effective-Cartier}]", "First, let us prove this in case $X$ is quasi-compact.", "In this case, apply", "Lemma \\ref{lemma-blowing-up-intersections} to $X$ and the", "two element set $\\{D, D'\\}$ of effective Cartier divisors.", "Thus we get a proper morphism $b : X' \\to X$,", "a finite collection of effective Cartier", "divisors $D'_j \\subset X'$ intersecting pairwise in codimension $\\geq 2$,", "with $b^{-1}(D) = \\sum n_j D'_j$, and $b^{-1}(D') = \\sum m_j D'_j$.", "Note that $b_*[b^{-1}(D)]_{n - 1} = [D]_{n - 1}$ in $Z_{n - 1}(X)$", "and similarly for $D'$,", "see Lemma \\ref{lemma-push-pull-effective-Cartier}.", "Hence, by Chow Homology, Lemma \\ref{chow-lemma-pushforward-cap-c1} we have", "$$", "D \\cdot [D']_{n - 1} = b_*\\left(b^{-1}(D) \\cdot [b^{-1}(D')]_{n - 1}\\right)", "$$", "in $\\CH_{n - 2}(X)$ and similarly for the other term. Hence the", "lemma follows from the equality", "$b^{-1}(D) \\cdot [b^{-1}(D')]_{n - 1} = b^{-1}(D') \\cdot [b^{-1}(D)]_{n - 1}$", "in $\\CH_{n - 2}(X')$ of Lemma", "\\ref{lemma-commutativity-effective-Cartier-proper-intersection-infinite}.", "\\medskip\\noindent", "Note that in the proof above, each referenced lemma works also", "in the general case (when $X$ is not assumed quasi-compact). The", "only minor change in the general case is that the morphism", "$b : U' \\to U$ we get from applying", "Lemma \\ref{lemma-blowing-up-intersections}", "has as its target", "an open $U \\subset X$ whose complement has codimension $\\geq 3$.", "Hence by Chow Homology, Lemma \\ref{chow-lemma-restrict-to-open} we see that", "$\\CH_{n - 2}(U) = \\CH_{n - 2}(X)$", "and after replacing $X$ by $U$ the rest of the proof goes through", "unchanged." ], "refs": [ "obsolete-lemma-commutativity-effective-Cartier", "obsolete-lemma-blowing-up-intersections", "obsolete-lemma-push-pull-effective-Cartier", "chow-lemma-pushforward-cap-c1", "obsolete-lemma-commutativity-effective-Cartier-proper-intersection-infinite", "obsolete-lemma-blowing-up-intersections", "chow-lemma-restrict-to-open" ], "ref_ids": [ 11514, 11510, 11506, 5711, 11513, 11510, 5690 ] } ], "ref_ids": [] }, { "id": 11515, "type": "theorem", "label": "obsolete-lemma-dualizing-components", "categories": [ "obsolete" ], "title": "obsolete-lemma-dualizing-components", "contents": [ "In Semistable Reduction, Situation \\ref{models-situation-regular-model}", "the dualizing module of $C_i$ over $k$ is", "$$", "\\omega_{C_i} = \\omega_X(C_i)|_{C_i}", "$$", "where $\\omega_X$ is as above." ], "refs": [], "proofs": [ { "contents": [ "Let $t : C_i \\to X$ be the closed immersion. Since $t$ is", "the inclusion of an effective Cartier divisor we conclude from", "Duality for Schemes, Lemmas", "\\ref{duality-lemma-twisted-inverse-image-closed} and", "\\ref{duality-lemma-sheaf-with-exact-support-effective-Cartier}", "that we have $t^!(\\mathcal{L}) = \\mathcal{L}(C_i)|_{C_i}$", "for every invertible $\\mathcal{O}_X$-module $\\mathcal{L}$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "C_i \\ar[r]_t \\ar[d]_g & X \\ar[d]^f \\\\", "\\Spec(k) \\ar[r]^s & \\Spec(R)", "}", "$$", "Observe that $C_i$ is a Gorenstein curve", "(Semistable Reduction, Lemma \\ref{models-lemma-gorenstein}) with invertible", "dualizing module $\\omega_{C_i}$ characterized by the property", "$\\omega_{C_i}[0] = g^!\\mathcal{O}_{\\Spec(k)}$. See", "Algebraic Curves, Lemma \\ref{curves-lemma-duality-dim-1}, its proof, and", "Algebraic Curves, Lemmas \\ref{curves-lemma-duality-dim-1-CM} and", "\\ref{curves-lemma-rr}.", "On the other hand, $s^!(R[1]) = k$ and hence", "$$", "\\omega_{C_i}[0] =", "g^! s^!(R[1]) = t^!f^!(R[1]) = t^!\\omega_X", "$$", "Combining the above we obtain the statement of the lemma." ], "refs": [ "duality-lemma-twisted-inverse-image-closed", "duality-lemma-sheaf-with-exact-support-effective-Cartier", "models-lemma-gorenstein", "curves-lemma-duality-dim-1", "curves-lemma-duality-dim-1-CM", "curves-lemma-rr" ], "ref_ids": [ 13525, 13544, 9236, 6250, 6251, 6256 ] } ], "ref_ids": [] }, { "id": 11516, "type": "theorem", "label": "obsolete-lemma-directed-colimit-finite-type", "categories": [ "obsolete" ], "title": "obsolete-lemma-directed-colimit-finite-type", "contents": [ "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then $\\mathcal{F}$ is the directed colimit of its finite type", "quasi-coherent submodules." ], "refs": [], "proofs": [ { "contents": [ "This is a duplicate of Properties, Lemma", "\\ref{properties-lemma-quasi-coherent-colimit-finite-type}." ], "refs": [ "properties-lemma-quasi-coherent-colimit-finite-type" ], "ref_ids": [ 3020 ] } ], "ref_ids": [] }, { "id": 11517, "type": "theorem", "label": "obsolete-lemma-points-monomorphism", "categories": [ "obsolete" ], "title": "obsolete-lemma-points-monomorphism", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The map $\\{\\Spec(k) \\to X \\text{ monomorphism}\\} \\to |X|$ is injective." ], "refs": [], "proofs": [ { "contents": [ "This is a duplicate of", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-monomorphism}." ], "refs": [ "spaces-properties-lemma-points-monomorphism" ], "ref_ids": [ 11826 ] } ], "ref_ids": [] }, { "id": 11518, "type": "theorem", "label": "obsolete-lemma-locally-ringed-space-direct-summand-free", "categories": [ "obsolete" ], "title": "obsolete-lemma-locally-ringed-space-direct-summand-free", "contents": [ "Let $X$ be a locally ringed space. A direct summand of a finite free", "$\\mathcal{O}_X$-module is finite locally free." ], "refs": [], "proofs": [ { "contents": [ "This is a duplicate of Modules, Lemma", "\\ref{modules-lemma-direct-summand-of-locally-free-is-locally-free}." ], "refs": [ "modules-lemma-direct-summand-of-locally-free-is-locally-free" ], "ref_ids": [ 13266 ] } ], "ref_ids": [] }, { "id": 11519, "type": "theorem", "label": "obsolete-lemma-characterize-injective", "categories": [ "obsolete" ], "title": "obsolete-lemma-characterize-injective", "contents": [ "Let $R$ be a ring. Let $E$ be an $R$-module. The following are equivalent", "\\begin{enumerate}", "\\item $E$ is an injective $R$-module, and", "\\item given an ideal $I \\subset R$ and a module map $\\varphi : I \\to E$", "there exists an extension of $\\varphi$ to an $R$-module map $R \\to E$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is Baer's criterion, see", "Injectives, Lemma \\ref{injectives-lemma-criterion-baer}." ], "refs": [ "injectives-lemma-criterion-baer" ], "ref_ids": [ 7771 ] } ], "ref_ids": [] }, { "id": 11520, "type": "theorem", "label": "obsolete-lemma-periodic-length", "categories": [ "obsolete" ], "title": "obsolete-lemma-periodic-length", "contents": [ "Let $R$ be a local ring.", "\\begin{enumerate}", "\\item If $(M, N, \\varphi, \\psi)$ is a $2$-periodic complex", "such that $M$, $N$ have finite length. Then", "$e_R(M, N, \\varphi, \\psi) = \\text{length}_R(M) - \\text{length}_R(N)$.", "\\item If $(M, \\varphi, \\psi)$ is a $(2, 1)$-periodic complex", "such that $M$ has finite length. Then", "$e_R(M, \\varphi, \\psi) = 0$.", "\\item Suppose that we have a short exact sequence of", "$2$-periodic complexes", "$$", "0 \\to (M_1, N_1, \\varphi_1, \\psi_1)", "\\to (M_2, N_2, \\varphi_2, \\psi_2)", "\\to (M_3, N_3, \\varphi_3, \\psi_3)", "\\to 0", "$$", "If two out of three have cohomology modules of finite length so does", "the third and we have", "$$", "e_R(M_2, N_2, \\varphi_2, \\psi_2) =", "e_R(M_1, N_1, \\varphi_1, \\psi_1) +", "e_R(M_3, N_3, \\varphi_3, \\psi_3).", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from Chow Homology, Lemmas", "\\ref{chow-lemma-additivity-periodic-length} and", "\\ref{chow-lemma-finite-periodic-length}." ], "refs": [ "chow-lemma-additivity-periodic-length", "chow-lemma-finite-periodic-length" ], "ref_ids": [ 5649, 5650 ] } ], "ref_ids": [] }, { "id": 11521, "type": "theorem", "label": "obsolete-lemma-examples-have-RS", "categories": [ "obsolete" ], "title": "obsolete-lemma-examples-have-RS", "contents": [ "Deformation Problems, Examples", "\\ref{examples-defos-example-finite-projective-modules},", "\\ref{examples-defos-example-representations},", "\\ref{examples-defos-example-continuous-representations}, and", "\\ref{examples-defos-example-graded-algebras}", "satisfy the Rim-Schlessinger condition (RS)." ], "refs": [], "proofs": [ { "contents": [ "This follows from Deformation Problems, Lemmas", "\\ref{examples-defos-lemma-finite-projective-modules-RS},", "\\ref{examples-defos-lemma-representations-RS},", "\\ref{examples-defos-lemma-continuous-representations-RS}, and", "\\ref{examples-defos-lemma-graded-algebras-RS}." ], "refs": [ "examples-defos-lemma-finite-projective-modules-RS", "examples-defos-lemma-representations-RS", "examples-defos-lemma-continuous-representations-RS", "examples-defos-lemma-graded-algebras-RS" ], "ref_ids": [ 8724, 8726, 8729, 8732 ] } ], "ref_ids": [] }, { "id": 11522, "type": "theorem", "label": "obsolete-lemma-tangent-and-inf", "categories": [ "obsolete" ], "title": "obsolete-lemma-tangent-and-inf", "contents": [ "We have the following canonical $k$-vector space identifications:", "\\begin{enumerate}", "\\item In Deformation Problems, Example", "\\ref{examples-defos-example-finite-projective-modules}", "if $x_0 = (k, V)$, then $T_{x_0}\\mathcal{F} = (0)$", "and $\\text{Inf}_{x_0}(\\mathcal{F}) = \\text{End}_k(V)$", "are finite dimensional.", "\\item In Deformation Problems, Example", "\\ref{examples-defos-example-representations}", "if $x_0 = (k, V, \\rho_0)$, then", "$T_{x_0}\\mathcal{F} = \\Ext^1_{k[\\Gamma]}(V, V) = H^1(\\Gamma, \\text{End}_k(V))$", "and $\\text{Inf}_{x_0}(\\mathcal{F}) = H^0(\\Gamma, \\text{End}_k(V))$", "are finite dimensional if $\\Gamma$ is finitely generated.", "\\item In Deformation Problems, Example", "\\ref{examples-defos-example-continuous-representations}", "if $x_0 = (k, V, \\rho_0)$, then", "$T_{x_0}\\mathcal{F} = H^1_{cont}(\\Gamma, \\text{End}_k(V))$", "and", "$\\text{Inf}_{x_0}(\\mathcal{F}) = H^0_{cont}(\\Gamma, \\text{End}_k(V))$", "are finite dimensional if $\\Gamma$ is topologically finitely generated.", "\\item In Deformation Problems, Example", "\\ref{examples-defos-example-graded-algebras}", "if $x_0 = (k, P)$, then", "$T_{x_0}\\mathcal{F}$ and $\\text{Inf}_{x_0}(\\mathcal{F}) = \\text{Der}_k(P, P)$", "are finite dimensional if $P$ is finitely generated over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from Deformation Problems, Lemmas", "\\ref{examples-defos-lemma-finite-projective-modules-TI},", "\\ref{examples-defos-lemma-representations-TI},", "\\ref{examples-defos-lemma-continuous-representations-TI}, and", "\\ref{examples-defos-lemma-graded-algebras-TI}." ], "refs": [ "examples-defos-lemma-finite-projective-modules-TI", "examples-defos-lemma-representations-TI", "examples-defos-lemma-continuous-representations-TI", "examples-defos-lemma-graded-algebras-TI" ], "ref_ids": [ 8725, 8727, 8730, 8733 ] } ], "ref_ids": [] }, { "id": 11523, "type": "theorem", "label": "obsolete-proposition-lqf-shriek", "categories": [ "obsolete" ], "title": "obsolete-proposition-lqf-shriek", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism. There exist", "adjoint functors $f_! : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$", "and $f^! : \\textit{Ab}(Y_\\etale) \\to \\textit{Ab}(X_\\etale)$", "with the following properties", "\\begin{enumerate}", "\\item the functor $f^!$ is the one constructed in More \\'Etale Cohomology,", "Lemma \\ref{more-etale-lemma-lqf-f-upper-shriek},", "\\item for any open $j : U \\to X$ with $f \\circ j$ separated", "there is a canonical isomorphism $f_! \\circ j_! = (f \\circ j)_!$, and", "\\item these isomorphisms for $U \\subset U' \\subset X$ are compatible", "with the isomorphisms in More \\'Etale Cohomology,", "Lemma \\ref{more-etale-lemma-f-shriek-composition}.", "\\end{enumerate}" ], "refs": [ "more-etale-lemma-lqf-f-upper-shriek", "more-etale-lemma-f-shriek-composition" ], "proofs": [ { "contents": [ "See More \\'Etale Cohomology, Sections", "\\ref{more-etale-section-finite-support} and", "\\ref{more-etale-section-duality-locally-quasi-finite}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8828, 8816 ] }, { "id": 11524, "type": "theorem", "label": "obsolete-proposition-conclusion", "categories": [ "obsolete" ], "title": "obsolete-proposition-conclusion", "contents": [ "With $\\Lambda$, $X$, $A$, $\\mathcal{F}$ as above. There exists a canonical", "object $L = L(\\Lambda, X, A, \\mathcal{F})$ of $D(X_A)$ such that given a", "surjection $A' \\to A$ of $\\Lambda$-algebras with square zero kernel $I$ we", "have", "\\begin{enumerate}", "\\item The category $\\textit{Lift}(\\mathcal{F}, A')$ is nonempty", "if and only if a certain class $\\xi \\in \\Ext^2_{X_A}(L,", "\\mathcal{F} \\otimes_A I)$ is zero.", "\\item If $\\textit{Lift}(\\mathcal{F}, A')$ is nonempty, then", "$\\text{Lift}(\\mathcal{F}, A')$ is principal homogeneous under", "$\\Ext^1_{X_A}(L, \\mathcal{F} \\otimes_A I)$.", "\\item Given a lift $\\mathcal{F}'$, the set of automorphisms of", "$\\mathcal{F}'$ which pull back to $\\text{id}_\\mathcal{F}$ is canonically", "isomorphic to", "$\\Ext^0_{X_A}(L, \\mathcal{F} \\otimes_A I)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "FIXME." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11565, "type": "theorem", "label": "stacks-sheaves-lemma-1-morphisms-presheaves", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-1-morphisms-presheaves", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Y} \\to \\mathcal{Z}$", "be $1$-morphisms of categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$. Then $(g \\circ f)^p = f^p \\circ g^p$ and", "there is a canonical isomorphism", "${}_p(g \\circ f) \\to {}_pg \\circ {}_pf$", "compatible with adjointness of $(f^p, {}_pf)$, $(g^p, {}_pg)$, and", "$((g \\circ f)^p, {}_p(g \\circ f))$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{H}$ be a presheaf on $\\mathcal{Z}$. Then", "$(g \\circ f)^p\\mathcal{H} = f^p (g^p\\mathcal{H})$ is given", "by the equalities", "$$", "(g \\circ f)^p\\mathcal{H}(x) = \\mathcal{H}((g \\circ f)(x))", "= \\mathcal{H}(g(f(x))) = f^p (g^p\\mathcal{H})(x).", "$$", "We omit the verification that this is compatible with restriction maps.", "\\medskip\\noindent", "Next, we define the transformation ${}_p(g \\circ f) \\to {}_pg \\circ {}_pf$.", "Let $\\mathcal{F}$ be a presheaf on $\\mathcal{X}$.", "If $z$ is an object of $\\mathcal{Z}$ then we get a", "category $\\mathcal{J}$ of quadruples", "$(x, f(x) \\to y, y, g(y) \\to z)$ and a category $\\mathcal{I}$", "of pairs $(x, g(f(x)) \\to z)$. There is a canonical functor", "$\\mathcal{J} \\to \\mathcal{I}$ sending the object", "$(x, \\alpha : f(x) \\to y, y, \\beta : g(y) \\to z)$ to", "$(x, \\beta \\circ f(\\alpha) : g(f(x)) \\to z)$. This gives the arrow in", "\\begin{align*}", "({}_p(g \\circ f)\\mathcal{F})(z) & =", "\\lim_{g(f(x)) \\to z} \\mathcal{F}(x) \\\\", "& = \\lim_\\mathcal{I} \\mathcal{F} \\\\", "& \\to \\lim_\\mathcal{J} \\mathcal{F} \\\\", "& = \\lim_{g(y) \\to z}", "\\Big(\\lim_{f(x) \\to y} \\mathcal{F}(x)\\Big) \\\\", "& =", "({}_pg \\circ {}_pf\\mathcal{F})(x)", "\\end{align*}", "by", "Categories, Lemma \\ref{categories-lemma-functorial-limit}.", "We omit the verification that this is compatible with restriction maps.", "An alternative to this direct construction is to define", "${}_p(g \\circ f) \\cong {}_pg \\circ {}_pf$", "as the unique map compatible with the adjointness properties. This also", "has the advantage that one does not need to prove the compatibility.", "\\medskip\\noindent", "Compatibility with adjointness of $(f^p, {}_pf)$, $(g^p, {}_pg)$, and", "$((g \\circ f)^p, {}_p(g \\circ f))$ means that given presheaves", "$\\mathcal{H}$ and $\\mathcal{F}$ as above we have a commutative diagram", "$$", "\\xymatrix{", "\\Mor_{\\textit{PSh}(\\mathcal{X})}(f^pg^p\\mathcal{H}, \\mathcal{F})", "\\ar@{=}[r] \\ar@{=}[d] &", "\\Mor_{\\textit{PSh}(\\mathcal{Y})}(g^p\\mathcal{H}, {}_pf\\mathcal{F})", "\\ar@{=}[r] &", "\\Mor_{\\textit{PSh}(\\mathcal{Y})}(\\mathcal{H}, {}_pg{}_pf\\mathcal{F})", "\\\\", "\\Mor_{\\textit{PSh}(\\mathcal{X})}((g \\circ f)^p\\mathcal{G}, \\mathcal{F})", "\\ar@{=}[rr] & &", "\\Mor_{\\textit{PSh}(\\mathcal{Y})}(\\mathcal{G}, {}_p(g \\circ f)\\mathcal{F})", "\\ar[u]", "}", "$$", "Proof omitted." ], "refs": [ "categories-lemma-functorial-limit" ], "ref_ids": [ 12211 ] } ], "ref_ids": [] }, { "id": 11566, "type": "theorem", "label": "stacks-sheaves-lemma-2-morphisms-presheaves", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-2-morphisms-presheaves", "contents": [ "Let $f, g : \\mathcal{X} \\to \\mathcal{Y}$ be $1$-morphisms of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $t : f \\to g$", "be a $2$-morphism of categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$. Assigned to $t$ there are canonical", "isomorphisms of functors", "$$", "t^p : g^p \\longrightarrow f^p", "\\quad\\text{and}\\quad", "{}_pt : {}_pf \\longrightarrow {}_pg", "$$", "which compatible with adjointness of $(f^p, {}_pf)$ and", "$(g^p, {}_pg)$ and with", "vertical and horizontal composition of $2$-morphisms." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{G}$ be a presheaf on $\\mathcal{Y}$. Then", "$t^p : g^p\\mathcal{G} \\to f^p\\mathcal{G}$ is given by the family", "of maps", "$$", "g^p\\mathcal{G}(x) = \\mathcal{G}(g(x))", "\\xrightarrow{\\mathcal{G}(t_x)}", "\\mathcal{G}(f(x)) = f^p\\mathcal{G}(x)", "$$", "parametrized by $x \\in \\Ob(\\mathcal{X})$. This makes sense as", "$t_x : f(x) \\to g(x)$ and $\\mathcal{G}$ is a contravariant functor.", "We omit the verification that this is compatible with restriction", "mappings.", "\\medskip\\noindent", "To define the transformation ${}_pt$ for $y \\in \\Ob(\\mathcal{Y})$", "define ${}_y^f\\mathcal{I}$, resp.\\ ${}_y^g\\mathcal{I}$ to be the category", "of pairs $(x, \\psi : f(x) \\to y)$, resp.\\ $(x, \\psi : g(x) \\to y)$, see", "Sites, Section \\ref{sites-section-more-functoriality-PSh}.", "Note that $t$ defines a functor", "${}_yt : {}_y^g\\mathcal{I} \\to {}_y^f\\mathcal{I}$", "given by the rule", "$$", "(x, g(x) \\to y) \\longmapsto (x, f(x) \\xrightarrow{t_x} g(x) \\to y).", "$$", "Note that for $\\mathcal{F}$ a presheaf on $\\mathcal{X}$ the composition", "of ${}_yt$ with $\\mathcal{F} : {}_y^f\\mathcal{I}^{opp} \\to \\textit{Sets}$,", "$(x, f(x) \\to y) \\mapsto \\mathcal{F}(x)$ is equal to", "$\\mathcal{F} : {}_y^g\\mathcal{I}^{opp} \\to \\textit{Sets}$. Hence by", "Categories, Lemma \\ref{categories-lemma-functorial-limit}", "we get for every $y \\in \\Ob(\\mathcal{Y})$ a canonical map", "$$", "({}_pf\\mathcal{F})(y) = \\lim_{{}_y^f\\mathcal{I}} \\mathcal{F}", "\\longrightarrow", "\\lim_{{}_y^g\\mathcal{I}} \\mathcal{F} = ({}_pg\\mathcal{F})(y)", "$$", "We omit the verification that this is compatible with restriction", "mappings. An alternative to this direct construction is to define", "${}_pt$ as the unique map compatible with the adjointness properties", "of the pairs $(f^p, {}_pf)$ and $(g^p, {}_pg)$ (see below). This also", "has the advantage that one does not need to prove the compatibility.", "\\medskip\\noindent", "Compatibility with adjointness of $(f^p, {}_pf)$ and $(g^p, {}_pg)$ means", "that given presheaves $\\mathcal{G}$ and $\\mathcal{F}$ as above we have", "a commutative diagram", "$$", "\\xymatrix{", "\\Mor_{\\textit{PSh}(\\mathcal{X})}(f^p\\mathcal{G}, \\mathcal{F})", "\\ar@{=}[r] \\ar[d]_{- \\circ t^p} &", "\\Mor_{\\textit{PSh}(\\mathcal{Y})}(\\mathcal{G}, {}_pf\\mathcal{F})", "\\ar[d]^{{}_pt \\circ -} \\\\", "\\Mor_{\\textit{PSh}(\\mathcal{X})}(g^p\\mathcal{G}, \\mathcal{F})", "\\ar@{=}[r] &", "\\Mor_{\\textit{PSh}(\\mathcal{Y})}(\\mathcal{G}, {}_pg\\mathcal{F})", "}", "$$", "Proof omitted. Hint: Work through the proof of", "Sites, Lemma \\ref{sites-lemma-adjoints-pu}", "and observe the compatibility from the explicit description of the", "horizontal and vertical maps in the diagram.", "\\medskip\\noindent", "We omit the verification that this is compatible with vertical and horizontal", "compositions. Hint: The proof of this for $t^p$ is straightforward and", "one can conclude that this holds for the ${}_pt$ maps using compatibility", "with adjointness." ], "refs": [ "categories-lemma-functorial-limit", "sites-lemma-adjoints-pu" ], "ref_ids": [ 12211, 8537 ] } ], "ref_ids": [] }, { "id": 11567, "type": "theorem", "label": "stacks-sheaves-lemma-functoriality-sheaves", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-functoriality-sheaves", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. Let", "$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.", "The functors ${}_pf$ and $f^p$ of (\\ref{equation-pushforward-pullback})", "transform $\\tau$ sheaves into $\\tau$ sheaves and define a morphism", "of topoi", "$f : \\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from", "Stacks, Lemma \\ref{stacks-lemma-topology-inherited-functorial}." ], "refs": [ "stacks-lemma-topology-inherited-functorial" ], "ref_ids": [ 8970 ] } ], "ref_ids": [] }, { "id": 11568, "type": "theorem", "label": "stacks-sheaves-lemma-base-change", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-base-change", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\ar[r]_{g'} \\ar[d]_{f'} &", "\\mathcal{X} \\ar[d]^f \\\\", "\\mathcal{Y}' \\ar[r]^g & \\mathcal{Y}", "}", "$$", "be a $2$-cartesian diagram of categories fibred in groupoids over $S$.", "Then we have a canonical isomorphism", "$$", "g^{-1}f_*\\mathcal{F} \\longrightarrow f'_*(g')^{-1}\\mathcal{F}", "$$", "functorial in the presheaf $\\mathcal{F}$ on $\\mathcal{X}$." ], "refs": [], "proofs": [ { "contents": [ "Given an object $y'$ of $\\mathcal{Y}'$ over $V$", "there is an equivalence", "$$", "(\\Sch/V)_{fppf} \\times_{g(y'), \\mathcal{Y}} \\mathcal{X}", "=", "(\\Sch/V)_{fppf} \\times_{y', \\mathcal{Y}'}", "(\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X})", "$$", "Hence by (\\ref{equation-pushforward}) a bijection", "$g^{-1}f_*\\mathcal{F}(y') \\to f'_*(g')^{-1}\\mathcal{F}(y')$.", "We omit the verification that this is compatible with restriction", "mappings." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11569, "type": "theorem", "label": "stacks-sheaves-lemma-representable", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-representable", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. The following are", "equivalent", "\\begin{enumerate}", "\\item $f$ is representable, and", "\\item for every $y \\in \\Ob(\\mathcal{Y})$ the functor", "$\\mathcal{X}^{opp} \\to \\textit{Sets}$,", "$x \\mapsto \\Mor_\\mathcal{Y}(f(x), y)$", "is representable.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "According to the discussion in", "Algebraic Stacks, Section \\ref{algebraic-section-representable-morphism}", "we see that $f$ is representable if and only if", "for every $y \\in \\Ob(\\mathcal{Y})$", "lying over $U$ the $2$-fibre product", "$(\\Sch/U)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}$", "is representable, i.e., of the form $(\\Sch/V_y)_{fppf}$ for some", "scheme $V_y$ over $U$. Objects in this $2$-fibre products are triples", "$(h : V \\to U, x, \\alpha : f(x) \\to h^*y)$ where $\\alpha$ lies", "over $\\text{id}_V$. Dropping the $h$ from the notation we see that this", "is equivalent to the data of an object $x$ of $\\mathcal{X}$ and a", "morphism $f(x) \\to y$. Hence the $2$-fibre product is", "representable by $V_y$ and $f(x_y) \\to y$ where $x_y$ is an object", "of $\\mathcal{X}$ over $V_y$ if and only if the functor in (2) is representable", "by $x_y$ with universal object a map $f(x_y) \\to y$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11570, "type": "theorem", "label": "stacks-sheaves-lemma-representable-pushforward", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-representable-pushforward", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a representable $1$-morphism of", "categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let", "$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.", "Then the functor $u : \\mathcal{Y}_\\tau \\to \\mathcal{X}_\\tau$ is continuous", "and defines a morphism of sites $\\mathcal{X}_\\tau \\to \\mathcal{Y}_\\tau$", "which induces the same morphism of topoi", "$\\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$", "as the morphism $f$ constructed in", "Lemma \\ref{lemma-functoriality-sheaves}.", "Moreover, $f_*\\mathcal{F}(y) = \\mathcal{F}(u(y))$ for any presheaf", "$\\mathcal{F}$ on $\\mathcal{X}$." ], "refs": [ "stacks-sheaves-lemma-functoriality-sheaves" ], "proofs": [ { "contents": [ "Let $\\{y_i \\to y\\}$ be a $\\tau$-covering in $\\mathcal{Y}$. By definition", "this simply means that $\\{q(y_i) \\to q(y)\\}$ is a $\\tau$-covering of", "schemes. By the final remark above the lemma we see that", "$\\{p(u(y_i)) \\to p(u(y))\\}$ is the base change of the $\\tau$-covering", "$\\{q(y_i) \\to q(y)\\}$ by $p(u(y)) \\to q(y)$, hence is itself a", "$\\tau$-covering by the axioms of a site. Hence $\\{u(y_i) \\to u(y)\\}$", "is a $\\tau$-covering of $\\mathcal{X}$. This proves that $u$ is", "continuous.", "\\medskip\\noindent", "Let's use the notation $u_p, u_s, u^p, u^s$ of", "Sites, Sections \\ref{sites-section-functoriality-PSh} and", "\\ref{sites-section-continuous-functors}.", "If we can show the final assertion of the lemma, then we see that", "$f_* = u^p = u^s$ (by continuity of $u$ seen above) and hence by adjointness", "$f^{-1} = u_s$ which will prove $u_s$ is exact, hence that $u$ determines", "a morphism of sites, and the equality will be clear as well.", "To see that $f_*\\mathcal{F}(y) = \\mathcal{F}(u(y))$ note that by", "definition", "$$", "f_*\\mathcal{F}(y) = ({}_pf\\mathcal{F})(y) =", "\\lim_{f(x) \\to y} \\mathcal{F}(x).", "$$", "Since $u(y)$ is a final object in the category the limit is taken", "over we conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 11567 ] }, { "id": 11571, "type": "theorem", "label": "stacks-sheaves-lemma-functoriality-structure-sheaf", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-functoriality-structure-sheaf", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. Let", "$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.", "There is a canonical identification", "$f^{-1}\\mathcal{O}_\\mathcal{X} = \\mathcal{O}_\\mathcal{Y}$", "which turns", "$f : \\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$", "into a morphism of ringed topoi." ], "refs": [], "proofs": [ { "contents": [ "Denote $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ and", "$q : \\mathcal{Y} \\to (\\Sch/S)_{fppf}$ the structural functors.", "Then $q = p \\circ f$, hence $q^{-1} = f^{-1} \\circ p^{-1}$ by", "Lemma \\ref{lemma-1-morphisms-presheaves}.", "The result follows." ], "refs": [ "stacks-sheaves-lemma-1-morphisms-presheaves" ], "ref_ids": [ 11565 ] } ], "ref_ids": [] }, { "id": 11572, "type": "theorem", "label": "stacks-sheaves-lemma-compare-with-scheme", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-compare-with-scheme", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred", "in groupoids over $(\\Sch/S)$. Assume $\\mathcal{X}$ is representable", "by a scheme $X$. For $\\tau \\in \\{Zar,\\linebreak[0] \\etale,\\linebreak[0]", "smooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$", "there is a canonical equivalence", "$$", "(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X}) =", "((\\Sch/X)_\\tau, \\mathcal{O}_X)", "$$", "of ringed sites." ], "refs": [], "proofs": [ { "contents": [ "This follows by choosing an equivalence", "$(\\Sch/X)_\\tau \\to \\mathcal{X}$ of categories fibred in groupoids", "over $(\\Sch/S)_{fppf}$ and using the functoriality of", "the construction $\\mathcal{X} \\leadsto \\mathcal{X}_\\tau$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11573, "type": "theorem", "label": "stacks-sheaves-lemma-compare-with-morphism-of-schemes", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-compare-with-morphism-of-schemes", "contents": [ "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism", "of categories fibred in groupoids over $S$.", "Assume $\\mathcal{X}$, $\\mathcal{Y}$ are representable by schemes", "$X$, $Y$. Let $f : X \\to Y$ be the morphism of schemes corresponding", "to $f$. For $\\tau \\in \\{Zar,\\linebreak[0] \\etale,\\linebreak[0]", "smooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$", "the morphism of ringed topoi", "$f : (\\Sh(\\mathcal{X}_\\tau), \\mathcal{O}_\\mathcal{X}) \\to", "(\\Sh(\\mathcal{Y}_\\tau), \\mathcal{O}_\\mathcal{Y})$", "agrees with the morphism of ringed topoi", "$f : (\\Sh((\\Sch/X)_\\tau), \\mathcal{O}_X) \\to ", "(\\Sh((\\Sch/Y)_\\tau), \\mathcal{O}_Y)$ via the identifications of", "Lemma \\ref{lemma-compare-with-scheme}." ], "refs": [ "stacks-sheaves-lemma-compare-with-scheme" ], "proofs": [ { "contents": [ "Follows by unwinding the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 11572 ] }, { "id": 11574, "type": "theorem", "label": "stacks-sheaves-lemma-localizing", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-localizing", "contents": [ "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred", "in groupoids. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.", "Let $x \\in \\Ob(\\mathcal{X})$ lying over $U = p(x)$.", "The functor $p$ induces an equivalence of sites", "$\\mathcal{X}_\\tau/x \\to (\\Sch/U)_\\tau$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Stacks, Lemma \\ref{stacks-lemma-localizing}." ], "refs": [ "stacks-lemma-localizing" ], "ref_ids": [ 8971 ] } ], "ref_ids": [] }, { "id": 11575, "type": "theorem", "label": "stacks-sheaves-lemma-comparison", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-comparison", "contents": [ "Let $\\mathcal{F}$ be an \\'etale sheaf on $\\mathcal{X} \\to (\\Sch/S)_{fppf}$.", "\\begin{enumerate}", "\\item If $\\varphi : x \\to y$ and $\\psi : y \\to z$", "are morphisms of $\\mathcal{X}$ lying over $a : U \\to V$ and", "$b : V \\to W$, then the composition", "$$", "a_{small}^{-1}(b_{small}^{-1} (\\mathcal{F}|_{W_\\etale}))", "\\xrightarrow{a_{small}^{-1}c_\\psi}", "a_{small}^{-1}(\\mathcal{F}|_{V_\\etale})", "\\xrightarrow{c_\\varphi}", "\\mathcal{F}|_{U_\\etale}", "$$", "is equal to $c_{\\psi \\circ \\varphi}$ via the identification", "$$", "(b \\circ a)_{small}^{-1}(\\mathcal{F}|_{W_\\etale}) =", "a_{small}^{-1}(b_{small}^{-1} (\\mathcal{F}|_{W_\\etale})).", "$$", "\\item If $\\varphi : x \\to y$ lies over an \\'etale morphism of schemes", "$a : U \\to V$, then (\\ref{equation-comparison}) is an isomorphism.", "\\item Suppose $f : \\mathcal{Y} \\to \\mathcal{X}$ is a $1$-morphism of", "categories fibred in groupoids over $(\\Sch/S)_{fppf}$ and $y$ is", "an object of $\\mathcal{Y}$ lying over the scheme $U$ with image", "$x = f(y)$. Then there is a canonical identification", "$f^{-1}\\mathcal{F}|_{U_\\etale} = \\mathcal{F}|_{U_\\etale}$.", "\\item Moreover, given $\\psi : y' \\to y$ in $\\mathcal{Y}$ lying over", "$a : U' \\to U$ the comparison map", "$c_\\psi : a_{small}^{-1}(F^{-1}\\mathcal{F}|_{U_\\etale}) \\to", "F^{-1}\\mathcal{F}|_{U'_\\etale}$ is equal to the", "comparison map $c_{f(\\psi)} : a_{small}^{-1}\\mathcal{F}|_{U_\\etale}", "\\to \\mathcal{F}|_{U'_\\etale}$ via the identifications in (3).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The verification of these properties is omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11576, "type": "theorem", "label": "stacks-sheaves-lemma-localizing-structure-sheaf", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-localizing-structure-sheaf", "contents": [ "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred", "in groupoids. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.", "Let $x \\in \\Ob(\\mathcal{X})$ lying over $U = p(x)$.", "The equivalence of", "Lemma \\ref{lemma-localizing}", "extends to an equivalence of ringed sites", "$(\\mathcal{X}_\\tau/x, \\mathcal{O}_\\mathcal{X}|_x) \\to", "((\\Sch/U)_\\tau, \\mathcal{O})$." ], "refs": [ "stacks-sheaves-lemma-localizing" ], "proofs": [ { "contents": [ "This is immediate from the construction of the structure sheaves." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 11574 ] }, { "id": 11577, "type": "theorem", "label": "stacks-sheaves-lemma-enough-points", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-enough-points", "contents": [ "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred", "in groupoids. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.", "The site $\\mathcal{X}_\\tau$ has enough points." ], "refs": [], "proofs": [ { "contents": [ "By", "Sites, Lemma \\ref{sites-lemma-enough-points-local}", "we have to show that there exists a family of objects $x$ of $\\mathcal{X}$", "such that $\\mathcal{X}_\\tau/x$ has enough points and such that the sheaves", "$h_x^\\#$ cover the final object of the category of sheaves.", "By", "Lemma \\ref{lemma-localizing}", "and", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-points-fppf}", "we see that $\\mathcal{X}_\\tau/x$ has enough points for every object", "$x$ and we win." ], "refs": [ "sites-lemma-enough-points-local", "stacks-sheaves-lemma-localizing", "etale-cohomology-lemma-points-fppf" ], "ref_ids": [ 8612, 11574, 6427 ] } ], "ref_ids": [] }, { "id": 11578, "type": "theorem", "label": "stacks-sheaves-lemma-compare", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-compare", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category", "fibred in groupoids. Assume $\\mathcal{X}$ is representable by an algebraic", "space $F$. Then there exists a continuous and cocontinuous functor", "$", "F_\\etale \\to \\mathcal{X}_\\etale", "$", "which induces a morphism of ringed sites", "$$", "\\pi_F :", "(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})", "\\longrightarrow", "(F_\\etale, \\mathcal{O}_F)", "$$", "and a morphism of ringed topoi", "$$", "i_F :", "(\\Sh(F_\\etale), \\mathcal{O}_F)", "\\longrightarrow", "(\\Sh(\\mathcal{X}_\\etale), \\mathcal{O}_\\mathcal{X})", "$$", "such that $\\pi_F \\circ i_F = \\text{id}$. Moreover $\\pi_{F, *} = i_F^{-1}$." ], "refs": [], "proofs": [ { "contents": [ "Choose an equivalence $j : \\mathcal{S}_F \\to \\mathcal{X}$, see", "Algebraic Stacks, Sections \\ref{algebraic-section-split} and", "\\ref{algebraic-section-representable-by-algebraic-spaces}.", "An object of $F_\\etale$ is a scheme $U$ together with an", "\\'etale morphism $\\varphi : U \\to F$. Then $\\varphi$ is an object", "of $\\mathcal{S}_F$ over $U$. Hence $j(\\varphi)$ is an object of", "$\\mathcal{X}$ over $U$. In this way $j$ induces a functor", "$u : F_\\etale \\to \\mathcal{X}$. It is clear that", "$u$ is continuous and cocontinuous for the \\'etale topology on", "$\\mathcal{X}$. Since $j$ is an equivalence, the functor $u$ is fully", "faithful. Also, fibre products and equalizers exist in $F_\\etale$", "and $u$ commutes with them because these are computed on the level", "of underlying schemes in $F_\\etale$. Thus", "Sites, Lemmas \\ref{sites-lemma-when-shriek},", "\\ref{sites-lemma-preserve-equalizers}, and", "\\ref{sites-lemma-back-and-forth}", "apply. In particular $u$ defines a morphism of topoi", "$i_F : \\Sh(F_\\etale) \\to \\Sh(\\mathcal{X}_\\etale)$", "and there exists a left adjoint $i_{F, !}$ of $i_F^{-1}$ which commutes", "with fibre products and equalizers.", "\\medskip\\noindent", "We claim that $i_{F, !}$ is exact. If this is true, then we can define", "$\\pi_F$ by the rules $\\pi_F^{-1} = i_{F, !}$ and $\\pi_{F, *} = i_F^{-1}$", "and everything is clear. To prove the claim, note that we already know", "that $i_{F, !}$", "is right exact and preserves fibre products. Hence it suffices to show", "that $i_{F, !}* = *$ where $*$ indicates the final object in the category", "of sheaves of sets. Let $U$ be a scheme and let", "$\\varphi : U \\to F$ be surjective and \\'etale. Set $R = U \\times_F U$.", "Then", "$$", "\\xymatrix{", "h_R \\ar@<1ex>[r] \\ar@<-1ex>[r] & h_U \\ar[r] & {*}", "}", "$$", "is a coequalizer diagram in $\\Sh(F_\\etale)$. Using the", "right exactness of $i_{F, !}$, using $i_{F, !} = (u_p\\ )^\\#$, and using", "Sites, Lemma \\ref{sites-lemma-pullback-representable-presheaf}", "we see that", "$$", "\\xymatrix{", "h_{u(R)} \\ar@<1ex>[r] \\ar@<-1ex>[r] & h_{u(U)} \\ar[r] & i_{F, !}{*}", "}", "$$", "is a coequalizer diagram in $\\Sh(F_\\etale)$. Using that", "$j$ is an equivalence and that $F = U/R$ it follows that", "the coequalizer in $\\Sh(\\mathcal{X}_\\etale)$ of the", "two maps $h_{u(R)} \\to h_{u(U)}$ is $*$. We omit the proof that", "these morphisms are compatible with structure sheaves." ], "refs": [ "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers", "sites-lemma-back-and-forth", "sites-lemma-pullback-representable-presheaf" ], "ref_ids": [ 8545, 8546, 8547, 8501 ] } ], "ref_ids": [] }, { "id": 11579, "type": "theorem", "label": "stacks-sheaves-lemma-compare-morphism", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-compare-morphism", "contents": [ "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism", "of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume", "$\\mathcal{X}$, $\\mathcal{Y}$ are representable by algebraic spaces $F$, $G$.", "Denote $f : F \\to G$ the induced morphism of algebraic spaces, and", "$f_{small} : F_\\etale \\to G_\\etale$", "the corresponding morphism of ringed topoi. Then", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{X}_\\etale), \\mathcal{O}_\\mathcal{X})", "\\ar[d]_{\\pi_F} \\ar[rr]_f & &", "(\\Sh(\\mathcal{Y}_\\etale), \\mathcal{O}_\\mathcal{Y}) \\ar[d]^{\\pi_G} \\\\", "(\\Sh(F_\\etale), \\mathcal{O}_F) \\ar[rr]^{f_{small}} & &", "(\\Sh(G_\\etale), \\mathcal{O}_G)", "}", "$$", "is a commutative diagram of ringed topoi." ], "refs": [], "proofs": [ { "contents": [ "This is similar to", "Topologies, Lemma \\ref{topologies-lemma-morphism-big-small-etale} (3)", "but there is a small snag due to the fact that $F \\to G$ may not be", "representable by schemes. In particular we don't get a commutative diagram", "of ringed sites, but only a commutative diagram of ringed topoi.", "\\medskip\\noindent", "Before we start the proof proper, we choose equivalences", "$j : \\mathcal{S}_F \\to \\mathcal{X}$ and", "$j' : \\mathcal{S}_G \\to \\mathcal{Y}$ which induce functors", "$u : F_\\etale \\to \\mathcal{X}$ and", "$u' : G_\\etale \\to \\mathcal{Y}$ as in the proof of", "Lemma \\ref{lemma-compare}. Because of the 2-functoriality of", "sheaves on categories fibred in groupoids over $\\Sch_{fppf}$", "(see discussion in Section \\ref{section-presheaves})", "we may assume that $\\mathcal{X} = \\mathcal{S}_F$ and", "$\\mathcal{Y} = \\mathcal{S}_G$ and that $f : \\mathcal{S}_F \\to \\mathcal{S}_G$", "is the functor associated to the morphism $f : F \\to G$. Correspondingly", "we will omit $u$ and $u'$ from the notation, i.e., given an object", "$U \\to F$ of $F_\\etale$ we denote $U/F$", "the corresponding object of $\\mathcal{X}$. Similarly for $G$.", "\\medskip\\noindent", "Let $\\mathcal{G}$ be a sheaf on $\\mathcal{X}_\\etale$.", "To prove (2) we compute $\\pi_{G, *}f_*\\mathcal{G}$ and", "$f_{small, *}\\pi_{F, *}\\mathcal{G}$. To do this let $V \\to G$ be an object", "of $G_\\etale$. Then", "$$", "\\pi_{G, *}f_*\\mathcal{G}(V) = f_*\\mathcal{G}(V/G) =", "\\Gamma\\Big(", "(\\Sch/V)_{fppf} \\times_{\\mathcal{Y}} \\mathcal{X},", "\\ \\text{pr}^{-1}\\mathcal{G}\\Big)", "$$", "see (\\ref{equation-pushforward}). The fibre product in the formula is", "$$", "(\\Sch/V)_{fppf} \\times_{\\mathcal{Y}} \\mathcal{X} =", "(\\Sch/V)_{fppf} \\times_{\\mathcal{S}_G} \\mathcal{S}_F =", "\\mathcal{S}_{V \\times_G F}", "$$", "i.e., it is the split category fibred in groupoids associated to the", "algebraic space $V \\times_G F$. And $\\text{pr}^{-1}\\mathcal{G}$ is a", "sheaf on $\\mathcal{S}_{V \\times_G F}$ for the \\'etale topology.", "\\medskip\\noindent", "In particular, if $V \\times_G F$ is representable, i.e., if it is a scheme,", "then $\\pi_{G, *}f_*\\mathcal{G}(V) = \\mathcal{G}(V \\times_G F/F)$ and", "also", "$$", "f_{small, *}\\pi_{F, *}\\mathcal{G}(V) =", "\\pi_{F, *}\\mathcal{G}(V \\times_G F) =", "\\mathcal{G}(V \\times_G F/F)", "$$", "which proves the desired equality in this special case.", "\\medskip\\noindent", "In general, choose a scheme $U$ and a surjective \\'etale morphism", "$U \\to V \\times_G F$. Set $R = U \\times_{V \\times_G F} U$. Then", "$U/V \\times_G F$ and $R/V \\times_G F$ are objects of the fibre", "product category above. Since $\\text{pr}^{-1}\\mathcal{G}$ is a", "sheaf for the \\'etale topology on $\\mathcal{S}_{V \\times_G F}$", "the diagram", "$$", "\\xymatrix{", "\\Gamma\\Big(", "(\\Sch/V)_{fppf} \\times_{\\mathcal{Y}} \\mathcal{X},", "\\ \\text{pr}^{-1}\\mathcal{G}\\Big)", "\\ar[r] &", "\\text{pr}^{-1}\\mathcal{G}(U/V \\times_G F) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\text{pr}^{-1}\\mathcal{G}(R/V \\times_G F)", "}", "$$", "is an equalizer diagram. Note that", "$\\text{pr}^{-1}\\mathcal{G}(U/V \\times_G F) = \\mathcal{G}(U/F)$", "and $\\text{pr}^{-1}\\mathcal{G}(R/V \\times_G F) = \\mathcal{G}(R/F)$", "by the definition of pullbacks. Moreover, by the material in", "Properties of Spaces, Section \\ref{spaces-properties-section-etale-site}", "(especially,", "Properties of Spaces,", "Remark \\ref{spaces-properties-remark-explain-equivalence} and", "Lemma \\ref{spaces-properties-lemma-functoriality-etale-site})", "we see that there is an equalizer diagram", "$$", "\\xymatrix{", "f_{small, *}\\pi_{F, *}\\mathcal{G}(V)", "\\ar[r] &", "\\pi_{F, *}\\mathcal{G}(U/F) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\pi_{F, *}\\mathcal{G}(R/F)", "}", "$$", "Since we also have $\\pi_{F, *}\\mathcal{G}(U/F) = \\mathcal{G}(U/F)$", "and $\\pi_{F, *}\\mathcal{G}(U/F) = \\mathcal{G}(U/F)$", "we obtain a canonical identification", "$f_{small, *}\\pi_{F, *}\\mathcal{G}(V) = \\pi_{G, *}f_*\\mathcal{G}(V)$.", "We omit the proof that this is compatible with restriction mappings", "and that it is functorial in $\\mathcal{G}$." ], "refs": [ "topologies-lemma-morphism-big-small-etale", "stacks-sheaves-lemma-compare", "spaces-properties-remark-explain-equivalence", "spaces-properties-lemma-functoriality-etale-site" ], "ref_ids": [ 12455, 11578, 11953, 11864 ] } ], "ref_ids": [] }, { "id": 11580, "type": "theorem", "label": "stacks-sheaves-lemma-pullback-quasi-coherent", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-pullback-quasi-coherent", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$.", "The pullback functor", "$f^* = f^{-1} : \\textit{Mod}(\\mathcal{O}_\\mathcal{Y}) \\to", "\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$", "preserves quasi-coherent sheaves." ], "refs": [], "proofs": [ { "contents": [ "This is a general fact, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-local-pullback}." ], "refs": [ "sites-modules-lemma-local-pullback" ], "ref_ids": [ 14186 ] } ], "ref_ids": [] }, { "id": 11581, "type": "theorem", "label": "stacks-sheaves-lemma-characterize-quasi-coherent", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-characterize-quasi-coherent", "contents": [ "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category", "fibred in groupoids. Let $\\mathcal{F}$", "be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules. Then $\\mathcal{F}$", "is quasi-coherent if and only if $x^*\\mathcal{F}$ is a quasi-coherent", "sheaf on $(\\Sch/U)_{fppf}$ for every object $x$ of", "$\\mathcal{X}$ with $U = p(x)$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-pullback-quasi-coherent}", "the condition is necessary. Conversely, since $x^*\\mathcal{F}$", "is just the restriction to $\\mathcal{X}_{fppf}/x$ we see that it", "is sufficient directly from the definition of a quasi-coherent sheaf", "(and the fact that the notion of being quasi-coherent is an intrinsic", "property of sheaves of modules, see", "Modules on Sites, Section \\ref{sites-modules-section-intrinsic})." ], "refs": [ "stacks-sheaves-lemma-pullback-quasi-coherent" ], "ref_ids": [ 11580 ] } ], "ref_ids": [] }, { "id": 11582, "type": "theorem", "label": "stacks-sheaves-lemma-characterize-quasi-coherent-bis", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-characterize-quasi-coherent-bis", "contents": [ "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category", "fibred in groupoids. Let $\\mathcal{F}$ be a presheaf of", "modules on $\\mathcal{X}$. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is an object of", "$\\textit{Mod}(\\mathcal{X}_{Zar}, \\mathcal{O}_\\mathcal{X})$", "and $\\mathcal{F}$ is a quasi-coherent module on", "$(\\mathcal{X}_{Zar}, \\mathcal{O}_\\mathcal{X})$ in the sense of", "Modules on Sites, Definition \\ref{sites-modules-definition-site-local},", "\\item $\\mathcal{F}$ is an object of", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "and $\\mathcal{F}$ is a quasi-coherent module on", "$(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ in the sense of", "Modules on Sites, Definition \\ref{sites-modules-definition-site-local}, and", "\\item $\\mathcal{F}$ is a quasi-coherent module on $\\mathcal{X}$", "in the sense of Definition \\ref{definition-quasi-coherent}.", "\\end{enumerate}" ], "refs": [ "sites-modules-definition-site-local", "sites-modules-definition-site-local", "stacks-sheaves-definition-quasi-coherent" ], "proofs": [ { "contents": [ "Assume either (1), (2), or (3) holds.", "Let $x$ be an object of $\\mathcal{X}$ lying over the scheme $U$.", "Recall that $x^*\\mathcal{F} = x^{-1}\\mathcal{F}$ is just the", "restriction to $\\mathcal{X}/x = (\\Sch/U)_\\tau$ where", "$\\tau = fppf$, $\\tau = \\etale$, or $\\tau = Zar$, see", "Section \\ref{section-restriction}.", "By the definition of quasi-coherent modules on a ringed site", "this restriction is quasi-coherent provided $\\mathcal{F}$ is.", "By Descent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent}", "we see that $x^*\\mathcal{F}$ is the sheaf associated to", "a quasi-coherent $\\mathcal{O}_U$-module and is therefore", "a quasi-coherent module in the fppf, \\'etale, and Zariski", "topology; here we also use", "Descent, Lemma \\ref{descent-lemma-sheaf-condition-holds} and", "Definition \\ref{descent-definition-structure-sheaf}.", "Since this holds for every object $x$ of $\\mathcal{X}$,", "we see that $\\mathcal{F}$ is a sheaf in any of the three topologies.", "Moreover, we find that $\\mathcal{F}$ is quasi-coherent in any", "of the three topologies directly from the definition of being", "quasi-coherent and the fact that $x$ is an arbitrary object of $\\mathcal{X}$." ], "refs": [ "descent-proposition-equivalence-quasi-coherent", "descent-lemma-sheaf-condition-holds", "descent-definition-structure-sheaf" ], "ref_ids": [ 14755, 14621, 14766 ] } ], "ref_ids": [ 14289, 14289, 11629 ] }, { "id": 11583, "type": "theorem", "label": "stacks-sheaves-lemma-quasi-coherent", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-quasi-coherent", "contents": [ "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category", "fibred in groupoids. Let $\\mathcal{F}$", "be a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules. Then $\\mathcal{F}$", "is quasi-coherent if and only if the following two conditions hold", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is locally quasi-coherent, and", "\\item for any morphism $\\varphi : x \\to y$ of $\\mathcal{X}$ lying over", "$f : U \\to V$ the comparison map", "$c_\\varphi : f_{small}^*\\mathcal{F}|_{V_\\etale} \\to", "\\mathcal{F}|_{U_\\etale}$ of", "(\\ref{equation-comparison-modules}) is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{F}$ is quasi-coherent. Then $\\mathcal{F}$ is a sheaf", "for the fppf topology, hence a sheaf for the \\'etale topology. Moreover,", "any pullback of $\\mathcal{F}$ to a ringed topos is quasi-coherent, hence", "the restrictions $x^*\\mathcal{F}|_{U_\\etale}$ are quasi-coherent.", "This proves $\\mathcal{F}$ is locally quasi-coherent.", "Let $y$ be an object of $\\mathcal{X}$ with $V = p(y)$.", "We have seen that $\\mathcal{X}/y = (\\Sch/V)_{fppf}$. By", "Descent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent}", "it follows that $y^*\\mathcal{F}$ is the quasi-coherent module", "associated to a (usual) quasi-coherent module $\\mathcal{F}_V$ on", "the scheme $V$. Hence certainly the comparison maps", "(\\ref{equation-comparison-modules}) are isomorphisms.", "\\medskip\\noindent", "Conversely, suppose that $\\mathcal{F}$ satisfies (1) and (2).", "Let $y$ be an object of $\\mathcal{X}$ with $V = p(y)$. Denote", "$\\mathcal{F}_V$ the quasi-coherent module on", "the scheme $V$ corresponding to the restriction", "$y^*\\mathcal{F}|_{V_\\etale}$ which is quasi-coherent by", "assumption (1), see", "Descent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent}.", "Condition (2) now signifies that the restrictions", "$x^*\\mathcal{F}|_{U_\\etale}$ for $x$ over $y$ are each", "isomorphic to the (\\'etale sheaf associated to the) pullback of $\\mathcal{F}_V$", "via the corresponding morphism of schemes $U \\to V$.", "Hence $y^*\\mathcal{F}$ is the sheaf on $(\\Sch/V)_{fppf}$", "associated to $\\mathcal{F}_V$. Hence it is quasi-coherent (by", "Descent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent}", "again) and we see that $\\mathcal{F}$ is quasi-coherent on $\\mathcal{X}$ by", "Lemma \\ref{lemma-characterize-quasi-coherent}." ], "refs": [ "descent-proposition-equivalence-quasi-coherent", "descent-proposition-equivalence-quasi-coherent", "descent-proposition-equivalence-quasi-coherent", "stacks-sheaves-lemma-characterize-quasi-coherent" ], "ref_ids": [ 14755, 14755, 14755, 11581 ] } ], "ref_ids": [] }, { "id": 11584, "type": "theorem", "label": "stacks-sheaves-lemma-pullback-lqc", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-pullback-lqc", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. The pullback functor", "$f^* = f^{-1} :", "\\textit{Mod}(\\mathcal{Y}_\\etale, \\mathcal{O}_\\mathcal{Y})", "\\to", "\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "preserves locally quasi-coherent sheaves." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{G}$ be locally quasi-coherent on $\\mathcal{Y}$.", "Choose an object $x$ of $\\mathcal{X}$ lying over the scheme $U$.", "The restriction $x^*f^*\\mathcal{G}|_{U_\\etale}$ equals", "$(f \\circ x)^*\\mathcal{G}|_{U_\\etale}$", "hence is a quasi-coherent sheaf by assumption on $\\mathcal{G}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11585, "type": "theorem", "label": "stacks-sheaves-lemma-lqc-colimits", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-lqc-colimits", "contents": [ "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in", "groupoids.", "\\begin{enumerate}", "\\item The category $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$", "has colimits and they agree with colimits in the category", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.", "\\item The category $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$", "is abelian with kernels and cokernels computed in", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$,", "in other words the inclusion functor is exact.", "\\item Given a short exact sequence", "$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$ of", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "if two out of three are locally quasi-coherent so is the third.", "\\item Given $\\mathcal{F}, \\mathcal{G}$ in", "$\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$", "the tensor product $\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{G}$", "in $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "is an object of $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$.", "\\item Given $\\mathcal{F}, \\mathcal{G}$ in", "$\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$", "with $\\mathcal{F}$ locally of finite presentation on", "$\\mathcal{X}_\\etale$ the sheaf", "$\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$", "in $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "is an object of $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Each of these statements follows from the corresponding statement of", "Descent, Lemma \\ref{descent-lemma-equivalence-quasi-coherent-limits}.", "For example, suppose that", "$\\mathcal{I} \\to \\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$,", "$i \\mapsto \\mathcal{F}_i$ is a diagram.", "Consider the object $\\mathcal{F} = \\colim_i \\mathcal{F}_i$ of", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.", "For any object $x$ of $\\mathcal{X}$ with $U = p(x)$ the pullback functor", "$x^*$ commutes with all colimits as it is a left adjoint. Hence", "$x^*\\mathcal{F} = \\colim_i x^*\\mathcal{F}_i$. Similarly we have", "$x^*\\mathcal{F}|_{U_\\etale} =", "\\colim_i x^*\\mathcal{F}_i|_{U_\\etale}$.", "Now by assumption each $x^*\\mathcal{F}_i|_{U_\\etale}$", "is quasi-coherent, hence the colimit is quasi-coherent by the", "aforementioned", "Descent, Lemma \\ref{descent-lemma-equivalence-quasi-coherent-limits}.", "This proves (1).", "\\medskip\\noindent", "It follows from (1) that cokernels exist in", "$\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$ and agree with the cokernels computed", "in $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of", "$\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$ and let", "$\\mathcal{K} = \\Ker(\\varphi)$ computed in", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.", "If we can show that $\\mathcal{K}$ is a locally quasi-coherent module,", "then the proof of (2) is complete. To see this, note that kernels", "are computed in the category of presheaves (no sheafification necessary).", "Hence $\\mathcal{K}|_{U_\\etale}$ is the kernel of the map", "$\\mathcal{F}|_{U_\\etale} \\to \\mathcal{G}|_{U_\\etale}$,", "i.e., is the kernel of a map of quasi-coherent sheaves on $U_\\etale$", "whence quasi-coherent by", "Descent, Lemma \\ref{descent-lemma-equivalence-quasi-coherent-limits}.", "This proves (2).", "\\medskip\\noindent", "Parts (3), (4), and (5) follow in exactly the same way. Details omitted." ], "refs": [ "descent-lemma-equivalence-quasi-coherent-limits", "descent-lemma-equivalence-quasi-coherent-limits", "descent-lemma-equivalence-quasi-coherent-limits" ], "ref_ids": [ 14627, 14627, 14627 ] } ], "ref_ids": [] }, { "id": 11586, "type": "theorem", "label": "stacks-sheaves-lemma-qc-colimits", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-qc-colimits", "contents": [ "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category", "fibred in groupoids.", "\\begin{enumerate}", "\\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$", "has colimits and they agree with colimits in the categories", "$\\textit{Mod}(\\mathcal{X}_{Zar}, \\mathcal{O}_\\mathcal{X})$,", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$,", "$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$, and", "$\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$.", "\\item Given $\\mathcal{F}, \\mathcal{G}$ in", "$\\QCoh(\\mathcal{O}_\\mathcal{X})$", "the tensor product $\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{G}$", "in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$", "is an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$.", "\\item Given $\\mathcal{F}, \\mathcal{G}$ in", "$\\QCoh(\\mathcal{O}_\\mathcal{X})$", "with $\\mathcal{F}$ locally of finite presentation on", "$\\mathcal{X}_{fppf}$ the sheaf", "$\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$", "in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$", "is an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\to \\QCoh(\\mathcal{O}_\\mathcal{X})$,", "$i \\mapsto \\mathcal{F}_i$ be a diagram.", "Viewing $\\mathcal{F}_i$ as quasi-coherent modules", "in the Zariski topology (Lemma \\ref{lemma-characterize-quasi-coherent-bis}), we", "may consider the object $\\mathcal{F} = \\colim_i \\mathcal{F}_i$ of", "$\\textit{Mod}(\\mathcal{X}_{Zar}, \\mathcal{O}_\\mathcal{X})$.", "For any object $x$ of $\\mathcal{X}$ with $U = p(x)$ the restriction functor", "$x^*$ (Section \\ref{section-restriction})", "commutes with all colimits as it is a left adjoint. Hence", "$x^*\\mathcal{F} = \\colim_i x^*\\mathcal{F}_i$ in", "$\\textit{Mod}((\\Sch/U)_{Zar}, \\mathcal{O})$.", "Observe that $x_i^*\\mathcal{F}_i$ is a quasi-coherent object", "(because restrictions of quasi-coherent modules are quasi-coherent).", "Thus by the equivalence in", "Descent, Proposition \\ref{descent-proposition-equivalence-quasi-coherent}", "and by the compatibility with colimits in", "Descent, Lemma \\ref{descent-lemma-equivalence-quasi-coherent-limits}", "we conclude that $x^*\\mathcal{F}$ is quasi-coherent.", "Thus $\\mathcal{F}$ is quasi-coherent, see", "Lemma \\ref{lemma-characterize-quasi-coherent-bis}.", "Thus we see that $\\QCoh(\\mathcal{O}_\\mathcal{X})$", "has colimits and they agree with colimits in the category", "$\\textit{Mod}(\\mathcal{X}_{Zar}, \\mathcal{O}_\\mathcal{X})$.", "Since the other categories listed are full subcategories", "of $\\textit{Mod}(\\mathcal{X}_{Zar}, \\mathcal{O}_\\mathcal{X})$", "we conclude part (1) holds.", "\\medskip\\noindent", "Parts (2) and (3) are proved in the same way.", "Details omitted." ], "refs": [ "stacks-sheaves-lemma-characterize-quasi-coherent-bis", "descent-proposition-equivalence-quasi-coherent", "descent-lemma-equivalence-quasi-coherent-limits", "stacks-sheaves-lemma-characterize-quasi-coherent-bis" ], "ref_ids": [ 11582, 14755, 14627, 11582 ] } ], "ref_ids": [] }, { "id": 11587, "type": "theorem", "label": "stacks-sheaves-lemma-stackification", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-stackification", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. If", "$f$ induces an equivalence of stackifications, then the morphism", "of topoi", "$f : \\Sh(\\mathcal{X}_{fppf}) \\to \\Sh(\\mathcal{Y}_{fppf})$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "We may assume $\\mathcal{Y}$ is the stackification of $\\mathcal{X}$.", "We claim that $f : \\mathcal{X} \\to \\mathcal{Y}$ is a special cocontinuous", "functor, see", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}", "which will prove the lemma. By", "Stacks, Lemma \\ref{stacks-lemma-topology-inherited-functorial}", "the functor $f$ is continuous and cocontinuous. By", "Stacks, Lemma \\ref{stacks-lemma-stackify}", "we see that conditions (3), (4), and (5) of", "Sites, Lemma \\ref{sites-lemma-equivalence}", "hold." ], "refs": [ "sites-definition-special-cocontinuous-functor", "stacks-lemma-topology-inherited-functorial", "stacks-lemma-stackify", "sites-lemma-equivalence" ], "ref_ids": [ 8672, 8970, 8961, 8578 ] } ], "ref_ids": [] }, { "id": 11588, "type": "theorem", "label": "stacks-sheaves-lemma-stackification-quasi-coherent", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-stackification-quasi-coherent", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. If", "$f$ induces an equivalence of stackifications, then $f^*$", "induces equivalences", "$\\textit{Mod}(\\mathcal{O}_\\mathcal{X}) \\to", "\\textit{Mod}(\\mathcal{O}_\\mathcal{Y})$", "and", "$\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to", "\\QCoh(\\mathcal{O}_\\mathcal{Y})$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $\\mathcal{Y}$ is the stackification of $\\mathcal{X}$.", "The first assertion is clear from", "Lemma \\ref{lemma-stackification}", "and", "$\\mathcal{O}_\\mathcal{X} = f^{-1}\\mathcal{O}_\\mathcal{Y}$.", "Pullback of quasi-coherent sheaves are quasi-coherent, see", "Lemma \\ref{lemma-pullback-quasi-coherent}.", "Hence it suffices to show that if $f^*\\mathcal{G}$ is", "quasi-coherent, then $\\mathcal{G}$ is.", "To see this, let $y$ be an object of $\\mathcal{Y}$.", "Translating the condition that $\\mathcal{Y}$ is the stackification", "of $\\mathcal{X}$ we see there exists an fppf covering $\\{y_i \\to y\\}$", "in $\\mathcal{Y}$ such that $y_i \\cong f(x_i)$ for some", "$x_i$ object of $\\mathcal{X}$. Say $x_i$ and $y_i$ lie over the scheme $U_i$.", "Then $f^*\\mathcal{G}$ being quasi-coherent, means that $x_i^*f^*\\mathcal{G}$", "is quasi-coherent. Since $x_i^*f^*\\mathcal{G}$ is isomorphic to", "$y_i^*\\mathcal{G}$ (as sheaves on $(\\Sch/U_i)_{fppf}$ we", "see that $y_i^*\\mathcal{G}$ is quasi-coherent.", "It follows from", "Modules on Sites, Lemma \\ref{sites-modules-lemma-local-final-object}", "that the restriction of $\\mathcal{G}$ to $\\mathcal{Y}/y$ is", "quasi-coherent. Hence $\\mathcal{G}$ is quasi-coherent by", "Lemma \\ref{lemma-characterize-quasi-coherent}." ], "refs": [ "stacks-sheaves-lemma-stackification", "stacks-sheaves-lemma-pullback-quasi-coherent", "sites-modules-lemma-local-final-object", "stacks-sheaves-lemma-characterize-quasi-coherent" ], "ref_ids": [ 11587, 11580, 14185, 11581 ] } ], "ref_ids": [] }, { "id": 11589, "type": "theorem", "label": "stacks-sheaves-lemma-map-from-quasi-coherent", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-map-from-quasi-coherent", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.", "Assume $s, t$ are flat and locally of finite presentation.", "Let $\\mathcal{X} = [U/R]$ be the quotient stack. Denote", "$\\pi : \\mathcal{S}_U \\to \\mathcal{X}$ the quotient map.", "Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_\\mathcal{X}$-module, and let $\\mathcal{H}$ be any object", "of $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.", "The map", "$$", "\\Hom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{H})", "\\longrightarrow", "\\Hom_{\\mathcal{O}_U}(x^*\\mathcal{F}|_{U_\\etale},", "x^*\\mathcal{H}|_{U_\\etale}),", "\\quad", "\\phi \\longmapsto x^*\\phi|_{U_\\etale}", "$$", "is injective and its image consists of exactly those", "$\\varphi : x^*\\mathcal{F}|_{U_\\etale} \\to", "x^*\\mathcal{H}|_{U_\\etale}$ which give rise to a commutative", "diagram", "$$", "\\xymatrix{", "s_{small}^*(x^*\\mathcal{F}|_{U_\\etale})", "\\ar[r] \\ar[d]^{s_{small}^*\\varphi} &", "(x \\circ s)^*\\mathcal{F}|_{R_\\etale} =", "(x \\circ t)^*\\mathcal{F}|_{R_\\etale} &", "t_{small}^*(x^*\\mathcal{F}|_{U_\\etale})", "\\ar[l] \\ar[d]_{t_{small}^*\\varphi} \\\\", "s_{small}^*(x^*\\mathcal{H}|_{U_\\etale})", "\\ar[r] &", "(x \\circ s)^*\\mathcal{H}|_{R_\\etale} =", "(x \\circ t)^*\\mathcal{H}|_{R_\\etale} &", "t_{small}^*(x^*\\mathcal{H}|_{U_\\etale})", "\\ar[l]", "}", "$$", "of modules on $R_\\etale$", "where the horizontal arrows are the comparison maps", "(\\ref{equation-comparison-algebraic-spaces-modules})." ], "refs": [], "proofs": [ { "contents": [ "According to", "Lemma \\ref{lemma-stackification-quasi-coherent}", "the stackification map $[U/_{\\!p}R] \\to [U/R]$ (see", "Groupoids in Spaces, Definition", "\\ref{spaces-groupoids-definition-quotient-stack})", "induces an equivalence of categories of quasi-coherent sheaves", "and of fppf $\\mathcal{O}$-modules.", "Thus it suffices to prove the lemma with $\\mathcal{X} = [U/_{\\!p}R]$.", "By Proposition \\ref{proposition-quasi-coherent}", "and its proof there exists a quasi-coherent module", "$(\\mathcal{G}, \\alpha)$ on $(U, R, s, t, c)$ such that", "$\\mathcal{F}$ is given by the rule", "$\\mathcal{F}(T, u) = \\Gamma(T, u^*\\mathcal{G})$.", "In particular $x^*\\mathcal{F}|_{U_\\etale} = \\mathcal{G}$", "and it is clear that the map of the statement of the", "lemma is injective. Moreover, given a map", "$\\varphi : \\mathcal{G} \\to x^*\\mathcal{H}|_{U_\\etale}$", "and given any object", "$y = (T, u)$ of $[U/_{\\!p}R]$ we can consider the map", "$$", "\\mathcal{F}(y) = \\Gamma(T, u^*\\mathcal{G})", "\\xrightarrow{u_{small}^*\\varphi}", "\\Gamma(T, u_{small}^*x^*\\mathcal{H}|_{U_\\etale})", "\\rightarrow", "\\Gamma(T, y^*\\mathcal{H}|_{T_\\etale}) = \\mathcal{H}(y)", "$$", "where the second arrow is the comparison map", "(\\ref{equation-comparison-modules}) for the sheaf $\\mathcal{H}$.", "This assignment is compatible with the restriction mappings of the", "sheaves $\\mathcal{F}$ and $\\mathcal{G}$ for morphisms of", "$[U/_{\\!p}R]$ if the cocycle condition of", "the lemma is satisfied. Proof omitted. Hint: the restriction maps", "of $\\mathcal{F}$ are made explicit in terms of $(\\mathcal{G}, \\alpha)$", "in the proof of", "Proposition \\ref{proposition-quasi-coherent}." ], "refs": [ "stacks-sheaves-lemma-stackification-quasi-coherent", "spaces-groupoids-definition-quotient-stack", "stacks-sheaves-proposition-quasi-coherent", "stacks-sheaves-proposition-quasi-coherent" ], "ref_ids": [ 11588, 9354, 11617, 11617 ] } ], "ref_ids": [] }, { "id": 11590, "type": "theorem", "label": "stacks-sheaves-lemma-quasi-coherent-algebraic-stack", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-quasi-coherent-algebraic-stack", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack over $S$.", "\\begin{enumerate}", "\\item If $[U/R] \\to \\mathcal{X}$ is a presentation of $\\mathcal{X}$", "then there is a canonical equivalence", "$\\QCoh(\\mathcal{O}_\\mathcal{X}) \\cong", "\\QCoh(U, R, s, t, c)$.", "\\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$ is abelian.", "\\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$", "has colimits and they agree with colimits in the category", "$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.", "\\item Given $\\mathcal{F}, \\mathcal{G}$ in", "$\\QCoh(\\mathcal{O}_\\mathcal{X})$", "the tensor product $\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{G}$", "in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$", "is an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$.", "\\item Given $\\mathcal{F}, \\mathcal{G}$ in", "$\\QCoh(\\mathcal{O}_\\mathcal{X})$", "with $\\mathcal{F}$ locally of finite presentation on", "$\\mathcal{X}_{fppf}$ the sheaf", "$\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$", "in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$", "is an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Properties (3), (4), and (5) were proven in", "Lemma \\ref{lemma-qc-colimits}.", "Part (1) is", "Proposition \\ref{proposition-quasi-coherent}.", "Part (2) follows from", "Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-abelian}", "as discussed above." ], "refs": [ "stacks-sheaves-lemma-qc-colimits", "stacks-sheaves-proposition-quasi-coherent", "spaces-groupoids-lemma-abelian" ], "ref_ids": [ 11586, 11617, 9305 ] } ], "ref_ids": [] }, { "id": 11591, "type": "theorem", "label": "stacks-sheaves-lemma-cohomology-restriction", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-cohomology-restriction", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred in groupoids", "over $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zariski, \\etale, smooth,", "syntomic, fppf\\}$. Let $x \\in \\Ob(\\mathcal{X})$ be an object lying", "over the scheme $U$. Let $\\mathcal{F}$ be", "an object of $\\textit{Ab}(\\mathcal{X}_\\tau)$ or", "$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$. Then", "$$", "H^p_\\tau(x, \\mathcal{F}) = H^p((\\Sch/U)_\\tau, x^{-1}\\mathcal{F})", "$$", "and if $\\tau = \\etale$, then we also have", "$$", "H^p_\\etale(x, \\mathcal{F}) =", "H^p(U_\\etale, \\mathcal{F}|_{U_\\etale}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-of-open}", "and the equivalence of ", "Lemma \\ref{lemma-localizing-structure-sheaf}.", "The second statement follows from the first combined with", "\\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-compare-cohomology-big-small}." ], "refs": [ "sites-cohomology-lemma-cohomology-of-open", "stacks-sheaves-lemma-localizing-structure-sheaf", "etale-cohomology-lemma-compare-cohomology-big-small" ], "ref_ids": [ 4186, 11576, 6411 ] } ], "ref_ids": [] }, { "id": 11592, "type": "theorem", "label": "stacks-sheaves-lemma-pushforward-injective", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-pushforward-injective", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. Let", "$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.", "\\begin{enumerate}", "\\item $f_*\\mathcal{I}$ is injective in $\\textit{Ab}(\\mathcal{Y}_\\tau)$", "for $\\mathcal{I}$ injective in $\\textit{Ab}(\\mathcal{X}_\\tau)$, and", "\\item $f_*\\mathcal{I}$ is injective in", "$\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$", "for $\\mathcal{I}$ injective in", "$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows formally from the fact that $f^{-1}$ is an exact", "left adjoint of $f_*$, see", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}." ], "refs": [ "homology-lemma-adjoint-preserve-injectives" ], "ref_ids": [ 12116 ] } ], "ref_ids": [] }, { "id": 11593, "type": "theorem", "label": "stacks-sheaves-lemma-fibre-products", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-fibre-products", "contents": [ "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$", "be a category fibred in groupoids.", "\\begin{enumerate}", "\\item The category $\\mathcal{X}$ has fibre products.", "\\item If the $\\mathit{Isom}$-presheaves of $\\mathcal{X}$", "are representable by algebraic spaces, then $\\mathcal{X}$ has equalizers.", "\\item If $\\mathcal{X}$ is an algebraic stack (or more generally", "a quotient stack), then $\\mathcal{X}$ has equalizers.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows", "Categories, Lemma \\ref{categories-lemma-fibred-groupoids-fibre-product-goes-up}", "as $(\\Sch/S)_{fppf}$ has fibre products.", "\\medskip\\noindent", "Let $a, b : x \\to y$ be morphisms of $\\mathcal{X}$.", "Set $U = p(x)$ and $V = p(y)$. The category of schemes has equalizers", "hence we can let $W \\to U$ be the equalizer of $p(a)$ and $p(b)$.", "Denote $c : z \\to x$ a morphism of $\\mathcal{X}$ lying over $W \\to U$.", "The equalizer of $a$ and $b$, if it exists, is the equalizer of $a \\circ c$", "and $b \\circ c$. Thus we may assume that $p(a) = p(b) = f : U \\to V$.", "As $\\mathcal{X}$ is fibred in groupoids, there exists a unique automorphism", "$i : x \\to x$ in the fibre category of $\\mathcal{X}$ over $U$ such that", "$a \\circ i = b$. Again the equalizer of $a$ and $b$ is the equalizer", "of $\\text{id}_x$ and $i$. Recall that the $\\mathit{Isom}_\\mathcal{X}(x)$", "is the presheaf on $(\\Sch/U)_{fppf}$ which to", "$T/U$ associates the set of automorphisms of $x|_T$ in the fibre category", "of $\\mathcal{X}$ over $T$, see", "Stacks, Definition \\ref{stacks-definition-mor-presheaf}.", "If $\\mathit{Isom}_\\mathcal{X}(x)$ is representable by an algebraic space", "$G \\to U$, then we see that $\\text{id}_x$ and $i$ define morphisms", "$e, i : U \\to G$ over $U$. Set $M = U \\times_{e, G, i} U$, which by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-section-immersion}", "is a scheme. Then it is clear that $x|_M \\to x$ is the equalizer of", "the maps $\\text{id}_x$ and $i$ in $\\mathcal{X}$.", "This proves (2).", "\\medskip\\noindent", "If $\\mathcal{X} = [U/R]$ for some groupoid in algebraic spaces", "$(U, R, s, t, c)$ over $S$, then the hypothesis of (2) holds by", "Bootstrap, Lemma \\ref{bootstrap-lemma-quotient-stack-isom}.", "If $\\mathcal{X}$ is an algebraic stack, then we can choose a", "presentation $[U/R] \\cong \\mathcal{X}$ by", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-stack-presentation}." ], "refs": [ "categories-lemma-fibred-groupoids-fibre-product-goes-up", "stacks-definition-mor-presheaf", "bootstrap-lemma-quotient-stack-isom", "algebraic-lemma-stack-presentation" ], "ref_ids": [ 12303, 8992, 2629, 8474 ] } ], "ref_ids": [] }, { "id": 11594, "type": "theorem", "label": "stacks-sheaves-lemma-fibre-products-morphism", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-fibre-products-morphism", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$.", "\\begin{enumerate}", "\\item The functor $f$ transforms fibre products into fibre products.", "\\item If $f$ is faithful, then $f$ transforms equalizers into equalizers.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By", "Categories, Lemma \\ref{categories-lemma-fibred-groupoids-fibre-product-goes-up}", "we see that a fibre product in $\\mathcal{X}$ is any commutative square lying", "over a fibre product diagram in $(\\Sch/S)_{fppf}$. Similarly for", "$\\mathcal{Y}$. Hence (1) is clear.", "\\medskip\\noindent", "Let $x \\to x'$ be the equalizer of two morphisms $a, b : x' \\to x''$", "in $\\mathcal{X}$. We will show that $f(x) \\to f(x')$ is the equalizer", "of $f(a)$ and $f(b)$. Let $y \\to f(x)$ be a morphism of $\\mathcal{Y}$", "equalizing $f(a)$ and $f(b)$. Say $x, x', x''$ lie over the schemes", "$U, U', U''$ and $y$ lies over $V$. Denote $h : V \\to U'$ the image", "of $y \\to f(x)$ in the category of schemes. The morphism", "$y \\to f(x)$ is isomorphic to $f(h^*x') \\to f(x')$ by the axioms of", "fibred categories. Hence, as $f$ is faithful, we see that", "$h^*x' \\to x'$ equalizes $a$ and $b$. Thus we obtain a unique morphism", "$h^*x' \\to x$ whose image $y = f(h^*x') \\to f(x)$ is the desired morphism", "in $\\mathcal{Y}$." ], "refs": [ "categories-lemma-fibred-groupoids-fibre-product-goes-up" ], "ref_ids": [ 12303 ] } ], "ref_ids": [] }, { "id": 11595, "type": "theorem", "label": "stacks-sheaves-lemma-fibre-products-preserve-properties", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-fibre-products-preserve-properties", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Z} \\to \\mathcal{Y}$", "be faithful $1$-morphisms of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$.", "\\begin{enumerate}", "\\item the functor $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Y}$", "is faithful, and", "\\item if $\\mathcal{X}, \\mathcal{Z}$ have equalizers, so does", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We think of objects in $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ as", "quadruples $(U, x, z, \\alpha)$ where $\\alpha : f(x) \\to g(z)$ is an", "isomorphism over $U$, see", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.", "A morphism $(U, x, z, \\alpha) \\to (U', x', z', \\alpha')$ is a", "pair of morphisms $a : x \\to x'$ and $b : z \\to z'$ compatible", "with $\\alpha$ and $\\alpha'$. Thus it is clear that if $f$ and", "$g$ are faithful, so is the functor", "$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Y}$.", "Now, suppose that", "$(a, b), (a', b') : (U, x, z, \\alpha) \\to (U', x', z', \\alpha')$", "are two morphisms of the $2$-fibre product. Then consider the equalizer", "$x'' \\to x$ of $a$ and $a'$ and the equalizer $z'' \\to z$ of $b$ and $b'$.", "Since $f$ commutes with equalizers (by", "Lemma \\ref{lemma-fibre-products-morphism})", "we see that $f(x'') \\to f(x)$ is the equalizer of $f(a)$ and $f(a')$.", "Similarly, $g(z'') \\to g(z)$ is the equalizer of $g(b)$ and $g(b')$.", "Picture", "$$", "\\xymatrix{", "f(x'') \\ar[r] \\ar@{..>}[d]_{\\alpha''}&", "f(x) \\ar[d]_\\alpha", "\\ar@<0.5ex>[r]^{f(a)}", "\\ar@<-0.5ex>[r]_{f(a')}", " &", "f(x') \\ar[d]^{\\alpha'} \\\\", "g(z'') \\ar[r] &", "g(z)", "\\ar@<0.5ex>[r]^{g(b)}", "\\ar@<-0.5ex>[r]_{g(b')}", " &", "g(z')", "}", "$$", "It is clear that the dotted arrow exists and is an isomorphism.", "However, it is not a priori the case that the image of $\\alpha''$", "in the category of schemes is the identity of its source. On the other", "hand, the existence of $\\alpha''$ means that we can assume that $x''$", "and $z''$ are defined over the same scheme and that the morphisms", "$x'' \\to x$ and $z'' \\to z$ have the same image in the category of schemes.", "Redoing the diagram above we see that the dotted arrow now does", "project to an identity morphism and we win. Some details omitted." ], "refs": [ "categories-lemma-2-product-categories-over-C", "stacks-sheaves-lemma-fibre-products-morphism" ], "ref_ids": [ 12280, 11594 ] } ], "ref_ids": [] }, { "id": 11596, "type": "theorem", "label": "stacks-sheaves-lemma-pullback-injective", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-pullback-injective", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. Let", "$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.", "The functor", "$f^{-1} : \\textit{Ab}(\\mathcal{Y}_\\tau) \\to \\textit{Ab}(\\mathcal{X}_\\tau)$", "has a left adjoint", "$f_! : \\textit{Ab}(\\mathcal{X}_\\tau) \\to \\textit{Ab}(\\mathcal{Y}_\\tau)$.", "If $f$ is faithful and $\\mathcal{X}$ has equalizers, then", "\\begin{enumerate}", "\\item $f_!$ is exact, and", "\\item $f^{-1}\\mathcal{I}$ is injective in $\\textit{Ab}(\\mathcal{X}_\\tau)$", "for $\\mathcal{I}$ injective in $\\textit{Ab}(\\mathcal{Y}_\\tau)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By", "Stacks, Lemma \\ref{stacks-lemma-topology-inherited-functorial}", "the functor $f$ is continuous and cocontinuous. Hence by", "Modules on Sites, Lemma \\ref{sites-modules-lemma-g-shriek-adjoint}", "the functor", "$f^{-1} : \\textit{Ab}(\\mathcal{Y}_\\tau) \\to \\textit{Ab}(\\mathcal{X}_\\tau)$", "has a left adjoint", "$f_! : \\textit{Ab}(\\mathcal{X}_\\tau) \\to \\textit{Ab}(\\mathcal{Y}_\\tau)$.", "To see (1) we apply", "Modules on Sites, Lemma \\ref{sites-modules-lemma-exactness-lower-shriek}", "and to see that the hypotheses of that lemma are satisfied use", "Lemmas \\ref{lemma-fibre-products} and", "\\ref{lemma-fibre-products-morphism}", "above. Part (2) follows from this formally, see", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}." ], "refs": [ "stacks-lemma-topology-inherited-functorial", "sites-modules-lemma-g-shriek-adjoint", "sites-modules-lemma-exactness-lower-shriek", "stacks-sheaves-lemma-fibre-products", "stacks-sheaves-lemma-fibre-products-morphism", "homology-lemma-adjoint-preserve-injectives" ], "ref_ids": [ 8970, 14164, 14165, 11593, 11594, 12116 ] } ], "ref_ids": [] }, { "id": 11597, "type": "theorem", "label": "stacks-sheaves-lemma-pullback-injective-modules", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-pullback-injective-modules", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. Let", "$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.", "The functor", "$f^* : \\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y}) \\to", "\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$", "has a left adjoint", "$f_! : \\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X}) \\to", "\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$ which", "agrees with the functor $f_!$ of Lemma \\ref{lemma-pullback-injective}", "on underlying abelian sheaves.", "If $f$ is faithful and $\\mathcal{X}$ has equalizers, then", "\\begin{enumerate}", "\\item $f_!$ is exact, and", "\\item $f^{-1}\\mathcal{I}$ is injective in", "$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$", "for $\\mathcal{I}$ injective in", "$\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{X})$.", "\\end{enumerate}" ], "refs": [ "stacks-sheaves-lemma-pullback-injective" ], "proofs": [ { "contents": [ "Recall that $f$ is a continuous and cocontinuous functor of sites", "and that $f^{-1}\\mathcal{O}_\\mathcal{Y} = \\mathcal{O}_\\mathcal{X}$. Hence", "Modules on Sites, Lemma \\ref{sites-modules-lemma-lower-shriek-modules}", "implies $f^*$ has a left adjoint $f_!^{Mod}$.", "Let $x$ be an object of $\\mathcal{X}$ lying over the scheme $U$.", "Then $f$ induces an equivalence of ringed sites", "$$", "\\mathcal{X}/x \\longrightarrow \\mathcal{Y}/f(x)", "$$", "as both sides are equivalent to $(\\Sch/U)_\\tau$, see", "Lemma \\ref{lemma-localizing-structure-sheaf}.", "Modules on Sites, Remark \\ref{sites-modules-remark-when-shriek-equal}", "shows that $f_!$ agrees with the functor on abelian sheaves.", "\\medskip\\noindent", "Assume now that $\\mathcal{X}$ has equalizers and that $f$ is faithful.", "Lemma \\ref{lemma-pullback-injective}", "tells us that $f_!$ is exact. Finally,", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives}", "implies the statement on pullbacks of injective modules." ], "refs": [ "sites-modules-lemma-lower-shriek-modules", "stacks-sheaves-lemma-localizing-structure-sheaf", "sites-modules-remark-when-shriek-equal", "stacks-sheaves-lemma-pullback-injective", "homology-lemma-adjoint-preserve-injectives" ], "ref_ids": [ 14262, 11576, 14312, 11596, 12116 ] } ], "ref_ids": [ 11596 ] }, { "id": 11598, "type": "theorem", "label": "stacks-sheaves-lemma-generalities", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-generalities", "contents": [ "Generalities on {\\v C}ech complexes.", "\\begin{enumerate}", "\\item If", "$$", "\\xymatrix{", "\\mathcal{V} \\ar[d]_g \\ar[r]_h & \\mathcal{U} \\ar[d]^f \\\\", "\\mathcal{Y} \\ar[r]^e & \\mathcal{X}", "}", "$$", "is $2$-commutative diagram of categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$, then there is a morphism of {\\v C}ech complexes", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U} \\to \\mathcal{X}, \\mathcal{F})", "\\longrightarrow", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V} \\to \\mathcal{Y}, e^{-1}\\mathcal{F})", "$$", "\\item if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism,", "\\item if $f, f' : \\mathcal{U} \\to \\mathcal{X}$ are $2$-isomorphic, then", "the associated {\\v C}ech complexes are isomorphic,", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In the situation of (1) let $t : f \\circ h \\to e \\circ g$ be a $2$-morphism.", "The map on complexes is given in degree $n$ by", "pullback along the $1$-morphisms", "$\\mathcal{V}_n \\to \\mathcal{U}_n$ given by the rule", "$$", "(v_0, \\ldots, v_n, y, \\beta_0, \\ldots, \\beta_n)", "\\longmapsto", "(h(v_0), \\ldots, h(v_n), e(y),", "e(\\beta_0) \\circ t_{v_0}, \\ldots, e(\\beta_n) \\circ t_{v_n}).", "$$", "For (2), note that pullback on global sections is an isomorphism", "for any presheaf of sets when the pullback is along an equivalence", "of categories. Part (3) follows on combining (1) and (2)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11599, "type": "theorem", "label": "stacks-sheaves-lemma-homotopy", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-homotopy", "contents": [ "If there exists a $1$-morphism $s : \\mathcal{X} \\to \\mathcal{U}$", "such that $f \\circ s$ is $2$-isomorphic to $\\text{id}_\\mathcal{X}$", "then the extended {\\v C}ech complex is homotopic to zero." ], "refs": [], "proofs": [ { "contents": [ "Set $\\mathcal{U}' = \\mathcal{U} \\times_\\mathcal{X} \\mathcal{X}$", "equal to the fibre product as described in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.", "Set $f' : \\mathcal{U}' \\to \\mathcal{X}$ equal to the second projection.", "Then $\\mathcal{U} \\to \\mathcal{U}'$, $u \\mapsto (u, f(x), 1)$", "is an equivalence over $\\mathcal{X}$, hence we may replace", "$(\\mathcal{U}, f)$ by $(\\mathcal{U}', f')$ by", "Lemma \\ref{lemma-generalities}.", "The advantage of this is that now $f'$ has a section $s'$ such", "that $f' \\circ s' = \\text{id}_\\mathcal{X}$ on the nose. Namely, if", "$t : s \\circ f \\to \\text{id}_\\mathcal{X}$ is a $2$-isomorphism", "then we can set $s'(x) = (s(x), x, t_x)$. Thus we may assume that", "$f \\circ s = \\text{id}_\\mathcal{X}$.", "\\medskip\\noindent", "In the case that $f \\circ s = \\text{id}_\\mathcal{X}$ the result follows", "from general principles. We give the homotopy explicitly. Namely,", "for $n \\geq 0$ define $s_n : \\mathcal{U}_n \\to \\mathcal{U}_{n + 1}$", "to be the $1$-morphism defined by the rule on objects", "$$", "(u_0, \\ldots, u_n, x, \\alpha_0, \\ldots, \\alpha_n)", "\\longmapsto", "(u_0, \\ldots, u_n, s(x), x,", "\\alpha_0, \\ldots, \\alpha_n, \\text{id}_x).", "$$", "Define", "$$", "h^{n + 1} :", "\\Gamma(\\mathcal{U}_{n + 1}, f_{n + 1}^{-1}\\mathcal{F})", "\\longrightarrow", "\\Gamma(\\mathcal{U}_n, f_n^{-1}\\mathcal{F})", "$$", "as pullback along $s_n$. We also set $s_{-1} = s$ and", "$h^0 : \\Gamma(\\mathcal{U}_0, f_0^{-1}\\mathcal{F}) \\to", "\\Gamma(\\mathcal{X}, \\mathcal{F})$ equal to pullback along $s_{-1}$.", "Then the family of maps $\\{h^n\\}_{n \\geq 0}$ is a homotopy between", "$1$ and $0$ on the extended {\\v C}ech complex." ], "refs": [ "categories-lemma-2-product-categories-over-C", "stacks-sheaves-lemma-generalities" ], "ref_ids": [ 12280, 11598 ] } ], "ref_ids": [] }, { "id": 11600, "type": "theorem", "label": "stacks-sheaves-lemma-generalities-sheafified", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-generalities-sheafified", "contents": [ "Generalities on relative {\\v C}ech complexes.", "\\begin{enumerate}", "\\item If", "$$", "\\xymatrix{", "\\mathcal{V} \\ar[d]_g \\ar[r]_h & \\mathcal{U} \\ar[d]^f \\\\", "\\mathcal{Y} \\ar[r]^e & \\mathcal{X}", "}", "$$", "is $2$-commutative diagram of categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$, then there is a morphism", "$e^{-1}\\mathcal{K}^\\bullet(f, \\mathcal{F}) \\to", "\\mathcal{K}^\\bullet(g, e^{-1}\\mathcal{F})$.", "\\item if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism,", "\\item if $f, f' : \\mathcal{U} \\to \\mathcal{X}$ are $2$-isomorphic, then", "the associated relative {\\v C}ech complexes are isomorphic,", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Literally the same as the proof of", "Lemma \\ref{lemma-generalities}", "using the pullback maps of", "Remark \\ref{remark-cech-complex-presheaves}." ], "refs": [ "stacks-sheaves-lemma-generalities", "stacks-sheaves-remark-cech-complex-presheaves" ], "ref_ids": [ 11598, 11633 ] } ], "ref_ids": [] }, { "id": 11601, "type": "theorem", "label": "stacks-sheaves-lemma-homotopy-sheafified", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-homotopy-sheafified", "contents": [ "If there exists a $1$-morphism $s : \\mathcal{X} \\to \\mathcal{U}$", "such that $f \\circ s$ is $2$-isomorphic to $\\text{id}_\\mathcal{X}$", "then the extended relative {\\v C}ech complex is homotopic to zero." ], "refs": [], "proofs": [ { "contents": [ "Literally the same as the proof of", "Lemma \\ref{lemma-homotopy}." ], "refs": [ "stacks-sheaves-lemma-homotopy" ], "ref_ids": [ 11599 ] } ], "ref_ids": [] }, { "id": 11602, "type": "theorem", "label": "stacks-sheaves-lemma-base-change-cech-complex", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-base-change-cech-complex", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{V} \\ar[d]_g \\ar[r]_h & \\mathcal{U} \\ar[d]^f \\\\", "\\mathcal{Y} \\ar[r]^e & \\mathcal{X}", "}", "$$", "be a $2$-fibre product of categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$ and let $\\mathcal{F}$ be an abelian presheaf", "on $\\mathcal{X}$. Then the map", "$e^{-1}\\mathcal{K}^\\bullet(f, \\mathcal{F}) \\to", "\\mathcal{K}^\\bullet(g, e^{-1}\\mathcal{F})$", "of", "Lemma \\ref{lemma-generalities-sheafified}", "is an isomorphism of complexes of abelian presheaves." ], "refs": [ "stacks-sheaves-lemma-generalities-sheafified" ], "proofs": [ { "contents": [ "Let $y$ be an object of $\\mathcal{Y}$ lying over the scheme $T$.", "Set $x = e(y)$. We are going to show that the map induces an isomorphism", "on sections over $y$. Note that", "$$", "\\Gamma(y, e^{-1}\\mathcal{K}^\\bullet(f, \\mathcal{F})) =", "\\Gamma(x, \\mathcal{K}^\\bullet(f, \\mathcal{F})) =", "\\check{\\mathcal{C}}^\\bullet(", "(\\Sch/T)_{fppf} \\times_{x, \\mathcal{X}} \\mathcal{U} \\to", "(\\Sch/T)_{fppf}, x^{-1}\\mathcal{F})", "$$", "by", "Remark \\ref{remark-cech-complex-sections}. On the other hand,", "$$", "\\Gamma(y, \\mathcal{K}^\\bullet(g, e^{-1}\\mathcal{F})) =", "\\check{\\mathcal{C}}^\\bullet(", "(\\Sch/T)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{V} \\to", "(\\Sch/T)_{fppf}, y^{-1}e^{-1}\\mathcal{F})", "$$", "also by", "Remark \\ref{remark-cech-complex-sections}.", "Note that $y^{-1}e^{-1}\\mathcal{F} = x^{-1}\\mathcal{F}$", "and since the diagram is $2$-cartesian the $1$-morphism", "$$", "(\\Sch/T)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{V} \\to", "(\\Sch/T)_{fppf} \\times_{x, \\mathcal{X}} \\mathcal{U}", "$$", "is an equivalence. Hence the map on sections over $y$ is an", "isomorphism by", "Lemma \\ref{lemma-generalities}." ], "refs": [ "stacks-sheaves-remark-cech-complex-sections", "stacks-sheaves-remark-cech-complex-sections", "stacks-sheaves-lemma-generalities" ], "ref_ids": [ 11634, 11634, 11598 ] } ], "ref_ids": [ 11600 ] }, { "id": 11603, "type": "theorem", "label": "stacks-sheaves-lemma-check-exactness-covering", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-check-exactness-covering", "contents": [ "Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred", "in groupoids over $(\\Sch/S)_{fppf}$. Let", "$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.", "Let", "$$", "\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}", "$$", "be a complex in $\\textit{Ab}(\\mathcal{X}_\\tau)$. Assume that", "\\begin{enumerate}", "\\item for every object $x$ of $\\mathcal{X}$ there exists a covering", "$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic", "to $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$, and", "\\item $f^{-1}\\mathcal{F} \\to f^{-1}\\mathcal{G} \\to f^{-1}\\mathcal{H}$ is exact.", "\\end{enumerate}", "Then the sequence $\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "Let $x$ be an object of $\\mathcal{X}$ lying over the scheme $T$.", "Consider the sequence", "$x^{-1}\\mathcal{F} \\to x^{-1}\\mathcal{G} \\to x^{-1}\\mathcal{H}$", "of abelian sheaves on $(\\Sch/T)_\\tau$. It suffices to show", "this sequence is exact. By assumption there exists a $\\tau$-covering", "$\\{T_i \\to T\\}$ such that $x|_{T_i}$ is isomorphic to $f(u_i)$ for", "some object $u_i$ of $\\mathcal{U}$ over $T_i$ and moreover the sequence", "$u_i^{-1}f^{-1}\\mathcal{F} \\to u_i^{-1}f^{-1}\\mathcal{G} \\to", "u_i^{-1}f^{-1}\\mathcal{H}$ of abelian sheaves on $(\\Sch/T_i)_\\tau$", "is exact. Since", "$u_i^{-1}f^{-1}\\mathcal{F} = x^{-1}\\mathcal{F}|_{(\\Sch/T_i)_\\tau}$", "we conclude that the sequence", "$x^{-1}\\mathcal{F} \\to x^{-1}\\mathcal{G} \\to x^{-1}\\mathcal{H}$", "become exact after localizing at each of the members of a covering,", "hence the sequence is exact." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11604, "type": "theorem", "label": "stacks-sheaves-lemma-cech-to-cohomology", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-cech-to-cohomology", "contents": [ "Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred", "in groupoids over $(\\Sch/S)_{fppf}$. Let", "$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{X}_\\tau$,", "\\item for every object $x$ of $\\mathcal{X}$ there exists a covering", "$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic", "to $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,", "\\item the category $\\mathcal{U}$ has equalizers, and", "\\item the functor $f$ is faithful.", "\\end{enumerate}", "Then there is a first quadrant spectral sequence of abelian groups", "$$", "E_1^{p, q} = H^q((\\mathcal{U}_p)_\\tau, f_p^{-1}\\mathcal{F})", "\\Rightarrow", "H^{p + q}(\\mathcal{X}_\\tau, \\mathcal{F})", "$$", "converging to the cohomology of $\\mathcal{F}$ in the $\\tau$-topology." ], "refs": [], "proofs": [ { "contents": [ "Before we start the proof we make some remarks. By", "Lemma \\ref{lemma-fibre-products-preserve-properties}", "(and induction) all of the categories fibred in groupoids $\\mathcal{U}_p$", "have equalizers and all of the morphisms $f_p : \\mathcal{U}_p \\to \\mathcal{X}$", "are faithful. Let $\\mathcal{I}$ be an injective object", "of $\\textit{Ab}(\\mathcal{X}_\\tau)$. By", "Lemma \\ref{lemma-pullback-injective}", "we see $f_p^{-1}\\mathcal{I}$ is an injective object of", "$\\textit{Ab}((\\mathcal{U}_p)_\\tau)$.", "Hence $f_{p, *}f_p^{-1}\\mathcal{I}$ is an injective object of", "$\\textit{Ab}(\\mathcal{X}_\\tau)$ by", "Lemma \\ref{lemma-pushforward-injective}.", "Hence", "Proposition \\ref{proposition-exactness-cech-complex}", "shows that the extended relative {\\v C}ech complex", "$$", "\\ldots \\to 0 \\to", "\\mathcal{I} \\to", "f_{0, *}f_0^{-1}\\mathcal{I} \\to", "f_{1, *}f_1^{-1}\\mathcal{I} \\to", "f_{2, *}f_2^{-1}\\mathcal{I} \\to \\ldots", "$$", "is an exact complex in $\\textit{Ab}(\\mathcal{X}_\\tau)$ all of whose", "terms are injective. Taking global sections of this complex is exact", "and we see that the {\\v C}ech complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U} \\to \\mathcal{X}, \\mathcal{I})$", "is quasi-isomorphic to $\\Gamma(\\mathcal{X}_\\tau, \\mathcal{I})[0]$.", "\\medskip\\noindent", "With these preliminaries out of the way consider the two spectral sequences", "associated to the double complex (see", "Homology, Section \\ref{homology-section-double-complex})", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U} \\to \\mathcal{X}, \\mathcal{I}^\\bullet)", "$$", "where $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ is an injective resolution", "in $\\textit{Ab}(\\mathcal{X}_\\tau)$.", "The discussion above shows that", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution}", "applies which shows that", "$\\Gamma(\\mathcal{X}_\\tau, \\mathcal{I}^\\bullet)$", "is quasi-isomorphic to the total complex associated to the double complex.", "By our remarks above the complex $f_p^{-1}\\mathcal{I}^\\bullet$ is an", "injective resolution of $f_p^{-1}\\mathcal{F}$. Hence the other spectral", "sequence is as indicated in the lemma." ], "refs": [ "stacks-sheaves-lemma-fibre-products-preserve-properties", "stacks-sheaves-lemma-pullback-injective", "stacks-sheaves-lemma-pushforward-injective", "stacks-sheaves-proposition-exactness-cech-complex", "homology-lemma-double-complex-gives-resolution" ], "ref_ids": [ 11595, 11596, 11592, 11619, 12106 ] } ], "ref_ids": [] }, { "id": 11605, "type": "theorem", "label": "stacks-sheaves-lemma-cech-to-cohomology-modules", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-cech-to-cohomology-modules", "contents": [ "Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred", "in groupoids over $(\\Sch/S)_{fppf}$. Let", "$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is an object of", "$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$,", "\\item for every object $x$ of $\\mathcal{X}$ there exists a covering", "$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic", "to $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,", "\\item the category $\\mathcal{U}$ has equalizers, and", "\\item the functor $f$ is faithful.", "\\end{enumerate}", "Then there is a first quadrant spectral sequence of", "$\\Gamma(\\mathcal{O}_\\mathcal{X})$-modules", "$$", "E_1^{p, q} = H^q((\\mathcal{U}_p)_\\tau, f_p^*\\mathcal{F})", "\\Rightarrow", "H^{p + q}(\\mathcal{X}_\\tau, \\mathcal{F})", "$$", "converging to the cohomology of $\\mathcal{F}$ in the $\\tau$-topology." ], "refs": [], "proofs": [ { "contents": [ "The proof of this lemma is identical to the proof of", "Lemma \\ref{lemma-cech-to-cohomology}", "except that it uses an injective resolution in", "$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$", "and it uses", "Lemma \\ref{lemma-pullback-injective-modules}", "instead of", "Lemma \\ref{lemma-pullback-injective}." ], "refs": [ "stacks-sheaves-lemma-cech-to-cohomology", "stacks-sheaves-lemma-pullback-injective-modules", "stacks-sheaves-lemma-pullback-injective" ], "ref_ids": [ 11604, 11597, 11596 ] } ], "ref_ids": [] }, { "id": 11606, "type": "theorem", "label": "stacks-sheaves-lemma-surjective-flat-locally-finite-presentation", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-surjective-flat-locally-finite-presentation", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of", "categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "\\begin{enumerate}", "\\item Assume that $f$ is representable by algebraic spaces, surjective,", "flat, and locally of finite presentation. Then for any object $y$ of", "$\\mathcal{Y}$ there exists an fppf covering $\\{y_i \\to y\\}$ and objects", "$x_i$ of $\\mathcal{X}$ such that $f(x_i) \\cong y_i$ in $\\mathcal{Y}$.", "\\item Assume that $f$ is representable by algebraic spaces, surjective,", "and smooth. Then for any object $y$ of", "$\\mathcal{Y}$ there exists an \\'etale covering $\\{y_i \\to y\\}$ and objects", "$x_i$ of $\\mathcal{X}$ such that $f(x_i) \\cong y_i$ in $\\mathcal{Y}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Suppose that $y$ lies over the scheme $V$.", "We may think of $y$ as a morphism $(\\Sch/V)_{fppf} \\to \\mathcal{Y}$.", "By definition the $2$-fibre product", "$\\mathcal{X} \\times_\\mathcal{Y} (\\Sch/V)_{fppf}$", "is representable by an algebraic space $W$ and the morphism", "$W \\to V$ is surjective, flat, and locally of finite presentation.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to W$.", "Then $U \\to V$ is also surjective, flat, and locally of finite presentation", "(see Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-etale-flat},", "\\ref{spaces-morphisms-lemma-etale-locally-finite-presentation},", "\\ref{spaces-morphisms-lemma-composition-surjective},", "\\ref{spaces-morphisms-lemma-composition-finite-presentation}, and", "\\ref{spaces-morphisms-lemma-composition-flat}).", "Hence $\\{U \\to V\\}$ is an fppf covering. Denote $x$ the object of", "$\\mathcal{X}$ over $U$ corresponding to the $1$-morphism", "$(\\Sch/U)_{fppf} \\to \\mathcal{X}$. Then $\\{f(x) \\to y\\}$ is", "the desired fppf covering of $\\mathcal{Y}$.", "\\medskip\\noindent", "Proof of (2). Suppose that $y$ lies over the scheme $V$.", "We may think of $y$ as a morphism $(\\Sch/V)_{fppf} \\to \\mathcal{Y}$.", "By definition the $2$-fibre product", "$\\mathcal{X} \\times_\\mathcal{Y} (\\Sch/V)_{fppf}$", "is representable by an algebraic space $W$ and the morphism", "$W \\to V$ is surjective and smooth.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to W$.", "Then $U \\to V$ is also surjective and smooth", "(see Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-etale-smooth},", "\\ref{spaces-morphisms-lemma-composition-surjective}, and", "\\ref{spaces-morphisms-lemma-composition-smooth}).", "Hence $\\{U \\to V\\}$ is a smooth covering. By", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-etale-dominates-smooth}", "there exists an \\'etale covering $\\{V_i \\to V\\}$ such that", "each $V_i \\to V$ factors through $U$. Denote $x_i$ the object of", "$\\mathcal{X}$ over $V_i$ corresponding to the $1$-morphism", "$$", "(\\Sch/V_i)_{fppf} \\to (\\Sch/U)_{fppf} \\to \\mathcal{X}.", "$$", "Then $\\{f(x_i) \\to y\\}$ is", "the desired \\'etale covering of $\\mathcal{Y}$." ], "refs": [ "spaces-morphisms-lemma-etale-flat", "spaces-morphisms-lemma-etale-locally-finite-presentation", "spaces-morphisms-lemma-composition-surjective", "spaces-morphisms-lemma-composition-finite-presentation", "spaces-morphisms-lemma-composition-flat", "spaces-morphisms-lemma-etale-smooth", "spaces-morphisms-lemma-composition-surjective", "spaces-morphisms-lemma-composition-smooth", "more-morphisms-lemma-etale-dominates-smooth" ], "ref_ids": [ 4910, 4911, 4726, 4839, 4852, 4909, 4726, 4886, 13880 ] } ], "ref_ids": [] }, { "id": 11607, "type": "theorem", "label": "stacks-sheaves-lemma-cech-to-cohomology-relative", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-cech-to-cohomology-relative", "contents": [ "Let $f : \\mathcal{U} \\to \\mathcal{X}$ and", "$g : \\mathcal{X} \\to \\mathcal{Y}$", "be composable $1$-morphisms of categories fibred", "in groupoids over $(\\Sch/S)_{fppf}$. Let", "$\\tau \\in \\{Zar, \\etale, smooth, syntomic, \\linebreak[0] fppf\\}$.", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{X}_\\tau$,", "\\item for every object $x$ of $\\mathcal{X}$ there exists a covering", "$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic", "to $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,", "\\item the category $\\mathcal{U}$ has equalizers, and", "\\item the functor $f$ is faithful.", "\\end{enumerate}", "Then there is a first quadrant spectral sequence of abelian sheaves", "on $\\mathcal{Y}_\\tau$", "$$", "E_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}", "\\Rightarrow", "R^{p + q}g_*\\mathcal{F}", "$$", "where all higher direct images are computed in the $\\tau$-topology." ], "refs": [], "proofs": [ { "contents": [ "Note that the assumptions on $f : \\mathcal{U} \\to \\mathcal{X}$", "and $\\mathcal{F}$ are identical to those in", "Lemma \\ref{lemma-cech-to-cohomology}.", "Hence the preliminary remarks made in the proof of that lemma", "hold here also. These remarks imply in particular that", "$$", "0 \\to g_*\\mathcal{I} \\to", "(g \\circ f_0)_*f_0^{-1}\\mathcal{I} \\to", "(g \\circ f_1)_*f_1^{-1}\\mathcal{I} \\to \\ldots", "$$", "is exact if $\\mathcal{I}$ is an injective object of", "$\\textit{Ab}(\\mathcal{X}_\\tau)$.", "Having said this, consider the two spectral sequences of", "Homology, Section \\ref{homology-section-double-complex}", "associated to the double complex $\\mathcal{C}^{\\bullet, \\bullet}$ with terms", "$$", "\\mathcal{C}^{p, q} = (g \\circ f_p)_*\\mathcal{I}^q", "$$", "where $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ is an injective resolution", "in $\\textit{Ab}(\\mathcal{X}_\\tau)$. The first spectral sequence implies, via", "Homology, Lemma \\ref{homology-lemma-double-complex-gives-resolution},", "that $g_*\\mathcal{I}^\\bullet$ is quasi-isomorphic to the total complex", "associated to $\\mathcal{C}^{\\bullet, \\bullet}$.", "Since $f_p^{-1}\\mathcal{I}^\\bullet$ is an injective resolution of", "$f_p^{-1}\\mathcal{F}$ (see", "Lemma \\ref{lemma-pullback-injective})", "the second spectral sequence has terms", "$E_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}$ as in the statement", "of the lemma." ], "refs": [ "stacks-sheaves-lemma-cech-to-cohomology", "homology-lemma-double-complex-gives-resolution", "stacks-sheaves-lemma-pullback-injective" ], "ref_ids": [ 11604, 12106, 11596 ] } ], "ref_ids": [] }, { "id": 11608, "type": "theorem", "label": "stacks-sheaves-lemma-cech-to-cohomology-relative-modules", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-cech-to-cohomology-relative-modules", "contents": [ "Let $f : \\mathcal{U} \\to \\mathcal{X}$ and", "$g : \\mathcal{X} \\to \\mathcal{Y}$", "be composable $1$-morphisms of categories fibred", "in groupoids over $(\\Sch/S)_{fppf}$. Let", "$\\tau \\in \\{Zar, \\etale, smooth, syntomic, \\linebreak[0] fppf\\}$.", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is an object of", "$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$,", "\\item for every object $x$ of $\\mathcal{X}$ there exists a covering", "$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic", "to $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,", "\\item the category $\\mathcal{U}$ has equalizers, and", "\\item the functor $f$ is faithful.", "\\end{enumerate}", "Then there is a first quadrant spectral sequence in", "$\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$", "$$", "E_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}", "\\Rightarrow", "R^{p + q}g_*\\mathcal{F}", "$$", "where all higher direct images are computed in the $\\tau$-topology." ], "refs": [], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Lemma \\ref{lemma-cech-to-cohomology-relative}", "except that it uses an injective resolution in", "$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$", "and it uses", "Lemma \\ref{lemma-pullback-injective-modules}", "instead of", "Lemma \\ref{lemma-pullback-injective}." ], "refs": [ "stacks-sheaves-lemma-cech-to-cohomology-relative", "stacks-sheaves-lemma-pullback-injective-modules", "stacks-sheaves-lemma-pullback-injective" ], "ref_ids": [ 11607, 11597, 11596 ] } ], "ref_ids": [] }, { "id": 11609, "type": "theorem", "label": "stacks-sheaves-lemma-pushforward-restriction", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-pushforward-restriction", "contents": [ "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a", "$1$-morphism of algebraic stacks\\footnote{This result should hold", "for any $1$-morphism of categories fibred in groupoids over", "$(\\Sch/S)_{fppf}$.} over $S$.", "Let $\\tau \\in \\{Zariski,\\linebreak[0] \\etale,\\linebreak[0]", "smooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$.", "Let $\\mathcal{F}$ be", "an object of $\\textit{Ab}(\\mathcal{X}_\\tau)$ or", "$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$.", "Then the sheaf $R^if_*\\mathcal{F}$ is the sheaf associated to the", "presheaf", "$$", "y \\longmapsto", "H^i_\\tau\\Big((\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X},", "\\ \\text{pr}^{-1}\\mathcal{F}\\Big)", "$$", "Here $y$ is a typical object of $\\mathcal{Y}$ lying over the scheme $V$." ], "refs": [], "proofs": [ { "contents": [ "Choose an injective resolution $\\mathcal{F}[0] \\to \\mathcal{I}^\\bullet$.", "By the formula for pushforward (\\ref{equation-pushforward}) we see that", "$R^if_*\\mathcal{F}$ is the sheaf associated to the presheaf which associates", "to $y$ the cohomology of the complex", "$$", "\\begin{matrix}", "\\Gamma\\Big((\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X},", "\\ \\text{pr}^{-1}\\mathcal{I}^{i - 1}\\Big) \\\\", "\\downarrow \\\\", "\\Gamma\\Big((\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X},", "\\ \\text{pr}^{-1}\\mathcal{I}^i\\Big) \\\\", "\\downarrow \\\\", "\\Gamma\\Big((\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X},", "\\ \\text{pr}^{-1}\\mathcal{I}^{i + 1}\\Big)", "\\end{matrix}", "$$", "Since $\\text{pr}^{-1}$ is exact, it suffices to show that", "$\\text{pr}^{-1}$ preserves injectives. This follows from", "Lemmas \\ref{lemma-pullback-injective} and", "\\ref{lemma-pullback-injective-modules}", "as well as the fact that $\\text{pr}$ is a representable morphism of", "algebraic stacks (so that $\\text{pr}$ is faithful by", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}", "and that", "$(\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}$", "has equalizers by ", "Lemma \\ref{lemma-fibre-products})." ], "refs": [ "stacks-sheaves-lemma-pullback-injective", "stacks-sheaves-lemma-pullback-injective-modules", "algebraic-lemma-characterize-representable-by-algebraic-spaces", "stacks-sheaves-lemma-fibre-products" ], "ref_ids": [ 11596, 11597, 8469, 11593 ] } ], "ref_ids": [] }, { "id": 11610, "type": "theorem", "label": "stacks-sheaves-lemma-base-change-higher-direct-images", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-base-change-higher-direct-images", "contents": [ "Let $S$ be a scheme. Let", "$\\tau \\in \\{Zariski,\\linebreak[0] \\etale,\\linebreak[0]", "smooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$. Let", "$$", "\\xymatrix{", "\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\ar[r]_{g'} \\ar[d]_{f'} &", "\\mathcal{X} \\ar[d]^f \\\\", "\\mathcal{Y}' \\ar[r]^g & \\mathcal{Y}", "}", "$$", "be a $2$-cartesian diagram of algebraic stacks over $S$. Then the base change", "map is an isomorphism", "$$", "g^{-1}Rf_*\\mathcal{F} \\longrightarrow Rf'_*(g')^{-1}\\mathcal{F}", "$$", "functorial for $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{X}_\\tau)$", "or $\\mathcal{F}$ in $\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$." ], "refs": [], "proofs": [ { "contents": [ "The isomorphism $g^{-1}f_*\\mathcal{F} = f'_*(g')^{-1}\\mathcal{F}$ is", "Lemma \\ref{lemma-base-change} (and it holds for arbitrary presheaves).", "For the derived direct images, there is a base change map because the", "morphisms $g$ and $g'$ are flat, see", "Cohomology on Sites, Section \\ref{sites-cohomology-section-base-change-map}.", "To see that this map is a quasi-isomorphism we can use that for", "an object $y'$ of $\\mathcal{Y}'$ over a scheme $V$ there is an equivalence", "$$", "(\\Sch/V)_{fppf} \\times_{g(y'), \\mathcal{Y}} \\mathcal{X}", "=", "(\\Sch/V)_{fppf} \\times_{y', \\mathcal{Y}'}", "(\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X})", "$$", "We conclude that the induced map", "$g^{-1}R^if_*\\mathcal{F} \\to R^if'_*(g')^{-1}\\mathcal{F}$", "is an isomorphism by", "Lemma \\ref{lemma-pushforward-restriction}." ], "refs": [ "stacks-sheaves-lemma-base-change", "stacks-sheaves-lemma-pushforward-restriction" ], "ref_ids": [ 11568, 11609 ] } ], "ref_ids": [] }, { "id": 11611, "type": "theorem", "label": "stacks-sheaves-lemma-compare-injectives", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-compare-injectives", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$", "representable by the algebraic space $F$.", "\\begin{enumerate}", "\\item If $\\mathcal{I}$ injective in $\\textit{Ab}(\\mathcal{X}_\\etale)$, then", "$\\mathcal{I}|_{F_\\etale}$ is injective in $\\textit{Ab}(F_\\etale)$,", "\\item If $\\mathcal{I}^\\bullet$ is a K-injective complex in", "$\\textit{Ab}(\\mathcal{X}_\\etale)$, then $\\mathcal{I}^\\bullet|_{F_\\etale}$", "is a K-injective complex in $\\textit{Ab}(F_\\etale)$.", "\\end{enumerate}", "The same does not hold for modules." ], "refs": [], "proofs": [ { "contents": [ "This follows formally from the fact that the restriction functor", "$\\pi_{F, *} = i_F^{-1}$ (see Lemma \\ref{lemma-compare})", "is right adjoint to the exact functor $\\pi_F^{-1}$, see", "Homology, Lemma \\ref{homology-lemma-adjoint-preserve-injectives} and", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}.", "To see that the lemma does not hold for modules, we refer the", "reader to \\'Etale Cohomology, Lemma", "\\ref{etale-cohomology-lemma-compare-injectives}." ], "refs": [ "stacks-sheaves-lemma-compare", "homology-lemma-adjoint-preserve-injectives", "derived-lemma-adjoint-preserve-K-injectives", "etale-cohomology-lemma-compare-injectives" ], "ref_ids": [ 11578, 12116, 1915, 6653 ] } ], "ref_ids": [] }, { "id": 11612, "type": "theorem", "label": "stacks-sheaves-lemma-compare-morphism-cohomology", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-compare-morphism-cohomology", "contents": [ "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism", "of algebraic stacks over $S$. Assume $\\mathcal{X}$, $\\mathcal{Y}$ are", "representable by algebraic spaces $F$, $G$. Denote $f : F \\to G$ the", "induced morphism of algebraic spaces.", "\\begin{enumerate}", "\\item For any $\\mathcal{F} \\in \\textit{Ab}(\\mathcal{X}_\\etale)$", "we have", "$$", "(Rf_*\\mathcal{F})|_{G_\\etale} =", "Rf_{small, *}(\\mathcal{F}|_{F_\\etale})", "$$", "in $D(G_\\etale)$.", "\\item For any object $\\mathcal{F}$ of", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$", "we have", "$$", "(Rf_*\\mathcal{F})|_{G_\\etale} =", "Rf_{small, *}(\\mathcal{F}|_{F_\\etale})", "$$", "in $D(\\mathcal{O}_G)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows immediately from", "Lemma \\ref{lemma-compare-injectives}", "and (\\ref{equation-compare-big-small})", "on choosing an injective resolution of $\\mathcal{F}$.", "\\medskip\\noindent", "Part (2) can be proved as follows. In Lemma \\ref{lemma-compare-morphism}", "we have seen that $\\pi_G \\circ f = f_{small} \\circ \\pi_F$ as morphisms", "of ringed sites. Hence we obtain", "$R\\pi_{G, *} \\circ Rf_* = Rf_{small, *} \\circ R\\pi_{F, *}$", "by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-derived-pushforward-composition}.", "Since the restriction functors $\\pi_{F, *}$ and $\\pi_{G, *}$", "are exact, we conclude." ], "refs": [ "stacks-sheaves-lemma-compare-injectives", "stacks-sheaves-lemma-compare-morphism", "sites-cohomology-lemma-derived-pushforward-composition" ], "ref_ids": [ 11611, 11579, 4250 ] } ], "ref_ids": [] }, { "id": 11613, "type": "theorem", "label": "stacks-sheaves-lemma-compare-representable-morphism-cohomology", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-compare-representable-morphism-cohomology", "contents": [ "Let $S$ be a scheme. Consider a $2$-fibre product square", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[r]_{g'} \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\", "\\mathcal{Y}' \\ar[r]^g & \\mathcal{Y}", "}", "$$", "of algebraic stacks over $S$. Assume that $f$ is representable by algebraic", "spaces and that $\\mathcal{Y}'$ is representable by an algebraic space $G'$.", "Then $\\mathcal{X}'$ is representable by an algebraic space $F'$ and", "denoting $f' : F' \\to G'$ the induced morphism of algebraic spaces", "we have", "$$", "g^{-1}(Rf_*\\mathcal{F})|_{G'_\\etale} =", "Rf'_{small, *}((g')^{-1}\\mathcal{F}|_{F'_\\etale})", "$$", "for any $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{X}_\\etale)$", "or in", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$" ], "refs": [], "proofs": [ { "contents": [ "Follows formally on combining", "Lemmas \\ref{lemma-base-change-higher-direct-images} and", "\\ref{lemma-compare-morphism-cohomology}." ], "refs": [ "stacks-sheaves-lemma-base-change-higher-direct-images", "stacks-sheaves-lemma-compare-morphism-cohomology" ], "ref_ids": [ 11610, 11612 ] } ], "ref_ids": [] }, { "id": 11614, "type": "theorem", "label": "stacks-sheaves-lemma-lqc-flat-base-change-fppf-sheaf", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-lqc-flat-base-change-fppf-sheaf", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$.", "Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules.", "Assume", "\\begin{enumerate}", "\\item[(a)] $\\mathcal{F}$ is locally quasi-coherent, and", "\\item[(b)] for any morphism $\\varphi : x \\to y$ of $\\mathcal{X}$ which lies", "over a morphism of schemes $f : U \\to V$ which is flat and", "locally of finite presentation the comparison map", "$c_\\varphi : f_{small}^*\\mathcal{F}|_{V_\\etale} \\to", "\\mathcal{F}|_{U_\\etale}$ of", "(\\ref{equation-comparison-modules}) is an isomorphism.", "\\end{enumerate}", "Then $\\mathcal{F}$ is a sheaf for the fppf topology." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{x_i \\to x\\}$ be an fppf covering of $\\mathcal{X}$ lying over the", "fppf covering $\\{f_i : U_i \\to U\\}$ of schemes over $S$.", "By assumption the restriction $\\mathcal{G} = \\mathcal{F}|_{U_\\etale}$", "is quasi-coherent and the comparison maps", "$f_{i, small}^*\\mathcal{G} \\to \\mathcal{F}|_{U_{i, \\etale}}$", "are isomorphisms. Hence the sheaf condition for $\\mathcal{F}$", "and the covering $\\{x_i \\to x\\}$ is equivalent to the sheaf condition", "for $\\mathcal{G}^a$ on $(\\Sch/U)_{fppf}$ and the covering $\\{U_i \\to U\\}$", "which holds by", "Descent, Lemma \\ref{descent-lemma-sheaf-condition-holds}." ], "refs": [ "descent-lemma-sheaf-condition-holds" ], "ref_ids": [ 14621 ] } ], "ref_ids": [] }, { "id": 11615, "type": "theorem", "label": "stacks-sheaves-lemma-compare-fppf-etale", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-compare-fppf-etale", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$.", "Let $\\mathcal{F}$ be a presheaf $\\mathcal{O}_\\mathcal{X}$-module such that", "\\begin{enumerate}", "\\item[(a)] $\\mathcal{F}$ is locally quasi-coherent, and", "\\item[(b)] for any morphism $\\varphi : x \\to y$ of $\\mathcal{X}$ which lies", "over a morphism of schemes $f : U \\to V$ which is flat and", "locally of finite presentation, the comparison map", "$c_\\varphi : f_{small}^*\\mathcal{F}|_{V_\\etale} \\to", "\\mathcal{F}|_{U_\\etale}$ of", "(\\ref{equation-comparison-modules}) is an isomorphism.", "\\end{enumerate}", "Then $\\mathcal{F}$ is an $\\mathcal{O}_\\mathcal{X}$-module and", "we have the following", "\\begin{enumerate}", "\\item If $\\epsilon : \\mathcal{X}_{fppf} \\to \\mathcal{X}_\\etale$", "is the comparison morphism, then", "$R\\epsilon_*\\mathcal{F} = \\epsilon_*\\mathcal{F}$.", "\\item The cohomology groups $H^p_{fppf}(\\mathcal{X}, \\mathcal{F})$ are equal", "to the cohomology groups computed in the \\'etale topology on $\\mathcal{X}$.", "Similarly for the cohomology groups $H^p_{fppf}(x, \\mathcal{F})$ and the", "derived versions $R\\Gamma(\\mathcal{X}, \\mathcal{F})$ and", "$R\\Gamma(x, \\mathcal{F})$.", "\\item If $f : \\mathcal{X} \\to \\mathcal{Y}$ is a $1$-morphism of", "categories fibred in groupoids over $(\\Sch/S)_{fppf}$ then", "$R^if_*\\mathcal{F}$ is equal to the fppf-sheafification of the", "higher direct image computed in the \\'etale cohomology.", "Similarly for derived pullback.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The assertion that $\\mathcal{F}$ is an $\\mathcal{O}_\\mathcal{X}$-module", "follows from", "Lemma \\ref{lemma-lqc-flat-base-change-fppf-sheaf}.", "Note that $\\epsilon$ is a morphism of sites given by the identity", "functor on $\\mathcal{X}$. The sheaf $R^p\\epsilon_*\\mathcal{F}$ is therefore", "the sheaf associated to the presheaf", "$x \\mapsto H^p_{fppf}(x, \\mathcal{F})$, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-higher-direct-images}.", "To prove (1) it suffices to show that", "$H^p_{fppf}(x, \\mathcal{F}) = 0$ for $p > 0$", "whenever $x$ lies over an affine scheme $U$. By", "Lemma \\ref{lemma-cohomology-restriction}", "we have", "$H^p_{fppf}(x, \\mathcal{F}) = H^p((\\Sch/U)_{fppf}, x^{-1}\\mathcal{F})$.", "Combining", "Descent, Lemma \\ref{descent-lemma-quasi-coherent-and-flat-base-change}", "with Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}", "we see that these cohomology groups are zero.", "\\medskip\\noindent", "We have seen above that $\\epsilon_*\\mathcal{F}$ and $\\mathcal{F}$ are the", "sheaves on $\\mathcal{X}_\\etale$ and $\\mathcal{X}_{fppf}$", "corresponding to the same presheaf on $\\mathcal{X}$ (and this is true more", "generally for any sheaf in the fppf topology on $\\mathcal{X}$).", "We often abusively identify $\\mathcal{F}$ and $\\epsilon_*\\mathcal{F}$", "and this is the sense in which parts (2) and (3) of the lemma should be", "understood. Thus part (2) follows formally from (1) and the Leray spectral", "sequence, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-apply-Leray}.", "\\medskip\\noindent", "Finally we prove (3). The sheaf $R^if_*\\mathcal{F}$", "(resp.\\ $Rf_{\\etale, *}\\mathcal{F}$)", "is the sheaf associated to the presheaf", "$$", "y \\longmapsto", "H^i_\\tau\\Big((\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X},", "\\ \\text{pr}^{-1}\\mathcal{F}\\Big)", "$$", "where $\\tau$ is $fppf$ (resp.\\ $\\etale$), see", "Lemma \\ref{lemma-pushforward-restriction}.", "Note that $\\text{pr}^{-1}\\mathcal{F}$ satisfies properties (a) and (b)", "also (by Lemmas \\ref{lemma-pullback-lqc} and \\ref{lemma-comparison}),", "hence these two presheaves are equal by (2).", "This immediately implies (3)." ], "refs": [ "stacks-sheaves-lemma-lqc-flat-base-change-fppf-sheaf", "sites-cohomology-lemma-higher-direct-images", "stacks-sheaves-lemma-cohomology-restriction", "descent-lemma-quasi-coherent-and-flat-base-change", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "sites-cohomology-lemma-apply-Leray", "stacks-sheaves-lemma-pushforward-restriction", "stacks-sheaves-lemma-pullback-lqc", "stacks-sheaves-lemma-comparison" ], "ref_ids": [ 11614, 4189, 11591, 14631, 3282, 4221, 11609, 11584, 11575 ] } ], "ref_ids": [] }, { "id": 11616, "type": "theorem", "label": "stacks-sheaves-lemma-cohomology-on-subcategory", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-lemma-cohomology-on-subcategory", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$.", "Let $\\tau = \\etale$ (resp.\\ $\\tau = fppf$). Let", "$\\mathcal{X}' \\subset \\mathcal{X}$ be a full subcategory with the", "following properties", "\\begin{enumerate}", "\\item if $x \\to x'$ is a morphism of $\\mathcal{X}$ which lies over a", "smooth (resp.\\ flat and locally finitely presented) morphism of", "schemes and $x' \\in \\Ob(\\mathcal{X}')$, then $x \\in \\Ob(\\mathcal{X}')$, and", "\\item there exists an object $x \\in \\Ob(\\mathcal{X}')$ lying over", "a scheme $U$ such that the associated $1$-morphism", "$x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$ is smooth and surjective.", "\\end{enumerate}", "We get a site $\\mathcal{X}'_\\tau$ by declaring a covering of $\\mathcal{X}'$", "to be any family of morphisms $\\{x_i \\to x\\}$ in $\\mathcal{X}'$ which is a", "covering in $\\mathcal{X}_\\tau$. Then the inclusion functor", "$\\mathcal{X}' \\to \\mathcal{X}_\\tau$ is fully faithful, cocontinuous, and", "continuous, whence defines a morphism of topoi", "$$", "g : \\Sh(\\mathcal{X}'_\\tau) \\longrightarrow \\Sh(\\mathcal{X}_\\tau)", "$$", "and $H^p(\\mathcal{X}'_\\tau, g^{-1}\\mathcal{F}) =", "H^p(\\mathcal{X}_\\tau, \\mathcal{F})$ for all $p \\geq 0$ and all", "$\\mathcal{F} \\in \\textit{Ab}(\\mathcal{X}_\\tau)$." ], "refs": [], "proofs": [ { "contents": [ "Note that assumption (1) implies that if $\\{x_i \\to x\\}$ is a covering", "of $\\mathcal{X}_\\tau$ and $x \\in \\Ob(\\mathcal{X}')$, then we have", "$x_i \\in \\Ob(\\mathcal{X}')$. Hence we see that $\\mathcal{X}' \\to \\mathcal{X}$", "is continuous and cocontinuous as the coverings of objects of", "$\\mathcal{X}'_\\tau$ agree with their coverings seen as objects of", "$\\mathcal{X}_\\tau$. We obtain the morphism $g$ and the functor", "$g^{-1}$ is identified with the restriction functor, see", "Sites, Lemma \\ref{sites-lemma-when-shriek}.", "\\medskip\\noindent", "In particular, if $\\{x_i \\to x\\}$ is a covering in $\\mathcal{X}'_\\tau$,", "then for any abelian sheaf $\\mathcal{F}$ on $\\mathcal{X}$ then", "$$", "\\check H^p(\\{x_i \\to x\\}, g^{-1}\\mathcal{F}) =", "\\check H^p(\\{x_i \\to x\\}, \\mathcal{F})", "$$", "Thus if $\\mathcal{I}$ is an injective abelian sheaf on $\\mathcal{X}_\\tau$", "then we see that the higher {\\v C}ech cohomology groups are zero", "(Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-injective-trivial-cech}).", "Hence $H^p(x, g^{-1}\\mathcal{I}) = 0$ for all objects $x$", "of $\\mathcal{X}'$", "(Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-cech-vanish-collection}).", "In other words injective abelian sheaves on $\\mathcal{X}_\\tau$", "are right acyclic for the functor $H^0(x, g^{-1}-)$.", "It follows that $H^p(x, g^{-1}\\mathcal{F}) = H^p(x, \\mathcal{F})$", "for all $\\mathcal{F} \\in \\textit{Ab}(\\mathcal{X})$ and all", "$x \\in \\Ob(\\mathcal{X}')$.", "\\medskip\\noindent", "Choose an object $x \\in \\mathcal{X}'$ lying over a scheme $U$", "as in assumption (2). In particular $\\mathcal{X}/x \\to \\mathcal{X}$", "is a morphism of algebraic stacks which representable by algebraic spaces,", "surjective, and smooth. (Note that $\\mathcal{X}/x$ is equivalent to", "$(\\Sch/U)_{fppf}$, see Lemma \\ref{lemma-localizing}.)", "The map of sheaves", "$$", "h_x \\longrightarrow *", "$$", "in $\\Sh(\\mathcal{X}_\\tau)$ is surjective. Namely, for any object $x'$", "of $\\mathcal{X}$ there exists a $\\tau$-covering $\\{x'_i \\to x'\\}$", "such that there exist morphisms $x'_i \\to x$, see", "Lemma \\ref{lemma-surjective-flat-locally-finite-presentation}.", "Since $g$ is exact, the map of sheaves", "$$", "g^{-1}h_x \\longrightarrow * = g^{-1}*", "$$", "in $\\Sh(\\mathcal{X}'_\\tau)$ is surjective also. Let $h_{x, n}$ be", "the $(n + 1)$-fold product $h_x \\times \\ldots \\times h_x$.", "Then we have spectral sequences", "\\begin{equation}", "\\label{equation-spectral-sequence-one}", "E_1^{p, q} = H^q(h_{x, p}, \\mathcal{F}) \\Rightarrow", "H^{p + q}(\\mathcal{X}_\\tau, \\mathcal{F})", "\\end{equation}", "and", "\\begin{equation}", "\\label{equation-spectral-sequence-two}", "E_1^{p, q} = H^q(g^{-1}h_{x, p}, g^{-1}\\mathcal{F}) \\Rightarrow", "H^{p + q}(\\mathcal{X}'_\\tau, g^{-1}\\mathcal{F})", "\\end{equation}", "see Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-cech-to-cohomology-sheaf-sets}.", "\\medskip\\noindent", "Case I: $\\mathcal{X}$ has a final object $x$ which is also an object of", "$\\mathcal{X}'$. This case follows immediately from the discussion", "in the second paragraph above.", "\\medskip\\noindent", "Case II: $\\mathcal{X}$ is representable by an algebraic space $F$.", "In this case the sheaves $h_{x, n}$ are representable by an", "object $x_n$ in $\\mathcal{X}$. (Namely, if $\\mathcal{S}_F = \\mathcal{X}$", "and $x : U \\to F$ is the given object, then $h_{x, n}$ is representable", "by the object $U \\times_F \\ldots \\times_F U \\to F$ of $\\mathcal{S}_F$.)", "It follows that $H^q(h_{x, p}, \\mathcal{F}) = H^q(x_p, \\mathcal{F})$.", "The morphisms $x_n \\to x$ lie over smooth morphisms of schemes, hence", "$x_n \\in \\mathcal{X}'$ for all $n$. Hence", "$H^q(g^{-1}h_{x, p}, g^{-1}\\mathcal{F}) = H^q(x_p, g^{-1}\\mathcal{F})$.", "Thus in the two spectral sequences", "(\\ref{equation-spectral-sequence-one}) and", "(\\ref{equation-spectral-sequence-two}) above the $E_1^{p, q}$ terms agree", "by the discussion in the second paragraph. The lemma follows in Case II", "as well.", "\\medskip\\noindent", "Case III: $\\mathcal{X}$ is an algebraic stack. We claim that in this case", "the cohomology groups $H^q(h_{x, p}, \\mathcal{F})$ and", "$H^q(g^{-1}h_{x, n}, g^{-1}\\mathcal{F})$ agree by Case II above.", "Once we have proved this the result will follow as before.", "\\medskip\\noindent", "Namely, consider the category $\\mathcal{X}/h_{x, n}$, see", "Sites, Lemma \\ref{sites-lemma-localize-topos-site}.", "Since $h_{x, n}$ is the $(n + 1)$-fold product of $h_x$ an", "object of this category is an $(n + 2)$-tuple", "$(y, s_0, \\ldots, s_n)$ where $y$ is an object of $\\mathcal{X}$ and each", "$s_i : y \\to x$ is a morphism of $\\mathcal{X}$.", "This is a category over $(\\Sch/S)_{fppf}$. There is an equivalence", "$$", "\\mathcal{X}/h_{x, n}", "\\longrightarrow", "(\\Sch/U)_{fppf} \\times_\\mathcal{X} \\ldots \\times_\\mathcal{X} (\\Sch/U)_{fppf}", "=: \\mathcal{U}_n", "$$", "over $(\\Sch/S)_{fppf}$. Namely, if $x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$ also", "denotes the $1$-morphism associated with $x$ and", "$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ the structure functor,", "then we can think of $(y, s_0, \\ldots, s_n)$ as", "$(y, f_0, \\ldots, f_n, \\alpha_0, \\ldots, \\alpha_n)$", "where $y$ is an object of $\\mathcal{X}$, $f_i : p(y) \\to p(x)$ is a", "morphism of schemes, and $\\alpha_i : y \\to x(f_i)$ an isomorphism.", "The category of $2n+3$-tuples", "$(y, f_0, \\ldots, f_n, \\alpha_0, \\ldots, \\alpha_n)$", "is an incarnation of the $(n + 1)$-fold fibred product $\\mathcal{U}_n$", "of algebraic stacks displayed above, as we discussed in", "Section \\ref{section-cech}.", "By Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-cohomology-on-sheaf-sets}", "we have", "$$", "H^p(\\mathcal{U}_n, \\mathcal{F}|_{\\mathcal{U}_n}) =", "H^p(\\mathcal{X}/h_{x, n}, \\mathcal{F}|_{\\mathcal{X}/h_{x, n}}) =", "H^p(h_{x, n}, \\mathcal{F}).", "$$", "Finally, we discuss the ``primed'' analogue of this. Namely,", "$\\mathcal{X}'/h_{x, n}$ corresponds, via the equivalence above", "to the full subcategory $\\mathcal{U}'_n \\subset \\mathcal{U}_n$", "consisting of those tuples", "$(y, f_0, \\ldots, f_n, \\alpha_0, \\ldots, \\alpha_n)$", "with $y \\in \\mathcal{X}'$. Hence certainly property (1) of the", "statement of the lemma holds", "for the inclusion $\\mathcal{U}'_n \\subset \\mathcal{U}_n$.", "To see property (2) choose an object $\\xi = (y, s_0, \\ldots, s_n)$ which", "lies over a scheme $W$ such that $(\\Sch/W)_{fppf} \\to \\mathcal{U}_n$", "is smooth and surjective (this is possible as $\\mathcal{U}_n$ is", "an algebraic stack). Then", "$(\\Sch/W)_{fppf} \\to \\mathcal{U}_n \\to (\\Sch/U)_{fppf}$", "is smooth as a composition of base changes of the morphism", "$x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$, see", "Algebraic Stacks, Lemmas", "\\ref{algebraic-lemma-base-change-representable-transformations-property} and", "\\ref{algebraic-lemma-composition-representable-transformations-property}.", "Thus axiom (1) for $\\mathcal{X}$ implies that $y$ is an object of", "$\\mathcal{X}'$ whence $\\xi$ is an object of $\\mathcal{U}'_n$.", "Using again", "$$", "H^p(\\mathcal{U}'_n, \\mathcal{F}|_{\\mathcal{U}'_n}) =", "H^p(\\mathcal{X}'/h_{x, n}, \\mathcal{F}|_{\\mathcal{X}'/h_{x, n}}) =", "H^p(g^{-1}h_{x, n}, g^{-1}\\mathcal{F}).", "$$", "we now can use Case II for ", "$\\mathcal{U}'_n \\subset \\mathcal{U}_n$", "to conclude." ], "refs": [ "sites-lemma-when-shriek", "sites-cohomology-lemma-injective-trivial-cech", "sites-cohomology-lemma-cech-vanish-collection", "stacks-sheaves-lemma-localizing", "stacks-sheaves-lemma-surjective-flat-locally-finite-presentation", "sites-cohomology-lemma-cech-to-cohomology-sheaf-sets", "sites-lemma-localize-topos-site", "sites-cohomology-lemma-cohomology-on-sheaf-sets", "algebraic-lemma-base-change-representable-transformations-property", "algebraic-lemma-composition-representable-transformations-property" ], "ref_ids": [ 8545, 4198, 4205, 11574, 11606, 4214, 8585, 4215, 8456, 8455 ] } ], "ref_ids": [] }, { "id": 11617, "type": "theorem", "label": "stacks-sheaves-proposition-quasi-coherent", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-proposition-quasi-coherent", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.", "Let $\\mathcal{X} = [U/R]$ be the quotient stack.", "The category of quasi-coherent modules on $\\mathcal{X}$", "is equivalent to the category of quasi-coherent modules", "on $(U, R, s, t, c)$." ], "refs": [], "proofs": [ { "contents": [ "Denote $\\QCoh(U, R, s, t, c)$ the category of quasi-coherent modules", "on the groupoid $(U, R, s, t, c)$. We will construct quasi-inverse functors", "$$", "\\QCoh(\\mathcal{O}_\\mathcal{X})", "\\longleftrightarrow", "\\QCoh(U, R, s, t, c).", "$$", "According to", "Lemma \\ref{lemma-stackification-quasi-coherent}", "the stackification map $[U/_{\\!p}R] \\to [U/R]$ (see", "Groupoids in Spaces, Definition", "\\ref{spaces-groupoids-definition-quotient-stack})", "induces an equivalence of categories of quasi-coherent sheaves.", "Thus it suffices to prove the lemma with $\\mathcal{X} = [U/_{\\!p}R]$.", "\\medskip\\noindent", "Recall that an object $x = (T, u)$ of $\\mathcal{X} = [U/_{\\!p}R]$", "is given by a scheme $T$ and a morphism $u : T \\to U$. A morphism", "$(T, u) \\to (T', u')$ is given by a pair $(f, r)$ where $f : T \\to T'$", "and $r : T \\to R$ with $s \\circ r = u$ and $t \\circ r = u' \\circ f$.", "Let us call a {\\it special morphism} any morphism of the form", "$(f, e \\circ u' \\circ f) : (T, u' \\circ f) \\to (T', u')$.", "The category of $(T, u)$ with special morphisms is just the", "category of schemes over $U$.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $\\mathcal{X}$.", "Then we obtain for every $x = (T, u)$ a quasi-coherent sheaf", "$\\mathcal{F}_{(T, u)} = x^*\\mathcal{F}|_{T_\\etale}$ on $T$.", "Moreover, for any morphism $(f, r) : x = (T, u) \\to (T', u') = x'$", "we obtain a comparison isomorphism", "$$", "c_{(f, r)} :", "f_{small}^*\\mathcal{F}_{(T', u')}", "\\longrightarrow", "\\mathcal{F}_{(T, u)}", "$$", "see Lemma \\ref{lemma-quasi-coherent}. Moreover, these isomorphisms are", "compatible with compositions, see", "Lemma \\ref{lemma-comparison}.", "If $U$, $R$ are schemes, then we can", "construct the quasi-coherent sheaf on the groupoid as follows: First", "the object $(U, \\text{id})$ corresponds to a quasi-coherent sheaf", "$\\mathcal{F}_{(U, \\text{id})}$ on $U$. Next, the isomorphism", "$\\alpha : t_{small}^*\\mathcal{F}_{(U, \\text{id})} \\to", "s_{small}^*\\mathcal{F}_{(U, \\text{id})}$", "comes from", "\\begin{enumerate}", "\\item the morphism $(R, \\text{id}_R) : (R, s) \\to (R, t)$", "in the category $[U/_{\\!p}R]$ which produces an isomorphism", "$\\mathcal{F}_{(R, t)} \\to \\mathcal{F}_{(R, s)}$,", "\\item the special morphism $(R, s) \\to (U, \\text{id})$ which produces an", "isomorphism", "$s_{small}^*\\mathcal{F}_{(U, \\text{id})} \\to \\mathcal{F}_{(R, s)}$, and", "\\item the special morphism $(R, t) \\to (U, \\text{id})$ which produces an", "isomorphism $t_{small}^*\\mathcal{F}_{(U, \\text{id})} \\to \\mathcal{F}_{(R, t)}$.", "\\end{enumerate}", "The cocycle condition for $\\alpha$ follows from the condition", "that $(U, R, s, t, c)$ is groupoid, i.e., that composition is", "associative (details omitted).", "\\medskip\\noindent", "To do this in general, i.e., when $U$ and $R$ are algebraic spaces,", "it suffices to explain how to associate to an algebraic space $(W, u)$ over", "$U$ a quasi-coherent sheaf $\\mathcal{F}_{(W, u)}$ and to construct the", "comparison maps for morphisms between these. We set", "$\\mathcal{F}_{(W, u)} = x^*\\mathcal{F}|_{W_\\etale}$", "where $x$ is the $1$-morphism", "$\\mathcal{S}_W \\to \\mathcal{S}_U \\to [U/_{\\!p}R]$ and the comparison", "maps are explained in (\\ref{equation-comparison-algebraic-spaces-modules}).", "\\medskip\\noindent", "Conversely, suppose that $(\\mathcal{G}, \\alpha)$ is a quasi-coherent", "module on $(U, R, s, t, c)$. We are going to define a presheaf of modules", "$\\mathcal{F}$ on $\\mathcal{X}$ as follows. Given an object", "$(T, u)$ of $[U/_{\\!p}R]$ we set", "$$", "\\mathcal{F}(T, u) : = \\Gamma(T, u_{small}^*\\mathcal{G}).", "$$", "Given a morphism $(f, r) : (T, u) \\to (T', u')$ we get a map", "\\begin{align*}", "\\mathcal{F}(T', u') & = \\Gamma(T', (u')_{small}^*\\mathcal{G}) \\\\", "& \\to \\Gamma(T, f_{small}^*(u')_{small}^*\\mathcal{G}) =", "\\Gamma(T, (u' \\circ f)_{small}^*\\mathcal{G}) \\\\", "& = \\Gamma(T, (t \\circ r)_{small}^*\\mathcal{G}) =", "\\Gamma(T, r_{small}^*t_{small}^*\\mathcal{G}) \\\\", "& \\to \\Gamma(T, r_{small}^*s_{small}^*\\mathcal{G}) =", "\\Gamma(T, (s \\circ r)_{small}^*\\mathcal{G}) \\\\", "& = \\Gamma(T, u_{small}^*\\mathcal{G}) \\\\", "& = \\mathcal{F}(T, u)", "\\end{align*}", "where the first arrow is pullback along $f$ and the second arrow is", "$\\alpha$. Note that if $(T, r)$ is a special morphism, then this", "map is just pullback along $f$ as $e_{small}^*\\alpha = \\text{id}$ by", "the axioms of a sheaf of quasi-coherent modules on a groupoid.", "The cocycle condition implies that $\\mathcal{F}$ is a presheaf", "of modules (details omitted). It is immediate from the definition that", "$\\mathcal{F}$ is quasi-coherent when pulled back to", "$(\\Sch/T)_{fppf}$ (by the simple description of the", "restriction maps of $\\mathcal{F}$ in case of a special morphism).", "\\medskip\\noindent", "We omit the verification that the functors constructed above are", "quasi-inverse to each other." ], "refs": [ "stacks-sheaves-lemma-stackification-quasi-coherent", "spaces-groupoids-definition-quotient-stack", "stacks-sheaves-lemma-quasi-coherent", "stacks-sheaves-lemma-comparison" ], "ref_ids": [ 11588, 9354, 11583, 11575 ] } ], "ref_ids": [] }, { "id": 11618, "type": "theorem", "label": "stacks-sheaves-proposition-coherator", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-proposition-coherator", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack over $S$.", "\\begin{enumerate}", "\\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$ is a Grothendieck", "abelian category. Consequently, $\\QCoh(\\mathcal{O}_\\mathcal{X})$", "has enough injectives and all limits.", "\\item The inclusion functor", "$\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to", "\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ has a right adjoint\\footnote{This", "functor is sometimes called the {\\it coherator}.}", "$$", "Q :", "\\textit{Mod}(\\mathcal{O}_\\mathcal{X})", "\\longrightarrow", "\\QCoh(\\mathcal{O}_\\mathcal{X})", "$$", "such that for every quasi-coherent sheaf $\\mathcal{F}$ the adjunction mapping", "$Q(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This proof is a repeat of the proof in the case of schemes, see", "Properties, Proposition \\ref{properties-proposition-coherator}", "and the case of algebraic spaces, see", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-coherator}.", "We advise the reader to read either of those proofs first.", "\\medskip\\noindent", "Part (1) means $\\QCoh(\\mathcal{O}_\\mathcal{X})$ (a) has all colimits,", "(b) filtered colimits are exact, and (c) has a generator, see", "Injectives, Section \\ref{injectives-section-grothendieck-conditions}.", "By Lemma \\ref{lemma-quasi-coherent-algebraic-stack}", "colimits in $\\QCoh(\\mathcal{O}_X)$ exist and agree", "with colimits in $\\textit{Mod}(\\mathcal{O}_X)$. By", "Modules on Sites, Lemma \\ref{sites-modules-lemma-limits-colimits}", "filtered colimits are exact. Hence (a) and (b) hold.", "\\medskip\\noindent", "Choose a presentation $\\mathcal{X} = [U/R]$ so that $(U, R, s, t, c)$", "is a smooth groupoid in algebraic spaces and in particular $s$ and $t$", "are flat morphisms of algebraic spaces. By", "Lemma \\ref{lemma-quasi-coherent-algebraic-stack}", "above we have", "$\\QCoh(\\mathcal{O}_\\mathcal{X}) = \\QCoh(U, R, s, t, c)$.", "By Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-set-generators}", "there exists a set $T$ and a family $(\\mathcal{F}_t)_{t \\in T}$ of", "quasi-coherent sheaves on $\\mathcal{X}$ such that every quasi-coherent", "sheaf on $\\mathcal{X}$ is the directed colimit of its subsheaves", "which are isomorphic to one of the $\\mathcal{F}_t$.", "Thus $\\bigoplus_t \\mathcal{F}_t$ is", "a generator of $\\QCoh(\\mathcal{O}_X)$ and we conclude that (c) holds.", "The assertions on limits and injectives hold in any", "Grothendieck abelian category, see", "Injectives, Theorem", "\\ref{injectives-theorem-injective-embedding-grothendieck} and", "Lemma \\ref{injectives-lemma-grothendieck-products}.", "\\medskip\\noindent", "Proof of (2). To construct $Q$ we use the following general procedure.", "Given an object $\\mathcal{F}$ of $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$", "we consider the functor", "$$", "\\QCoh(\\mathcal{O}_\\mathcal{X})^{opp}", "\\longrightarrow", "\\textit{Sets},", "\\quad", "\\mathcal{G}", "\\longmapsto", "\\Hom_\\mathcal{X}(\\mathcal{G}, \\mathcal{F})", "$$", "This functor transforms colimits into limits,", "hence is representable, see", "Injectives, Lemma \\ref{injectives-lemma-grothendieck-brown}.", "Thus there exists a quasi-coherent sheaf $Q(\\mathcal{F})$", "and a functorial isomorphism", "$\\Hom_\\mathcal{X}(\\mathcal{G}, \\mathcal{F}) =", "\\Hom_\\mathcal{X}(\\mathcal{G}, Q(\\mathcal{F}))$", "for $\\mathcal{G}$ in $\\QCoh(\\mathcal{O}_\\mathcal{X})$.", "By the Yoneda lemma", "(Categories, Lemma \\ref{categories-lemma-yoneda})", "the construction $\\mathcal{F} \\leadsto Q(\\mathcal{F})$ is", "functorial in $\\mathcal{F}$. By construction $Q$ is a right", "adjoint to the inclusion functor.", "The fact that $Q(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism", "when $\\mathcal{F}$ is quasi-coherent is a formal consequence of the fact", "that the inclusion functor", "$\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to", "\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$", "is fully faithful." ], "refs": [ "properties-proposition-coherator", "spaces-properties-proposition-coherator", "stacks-sheaves-lemma-quasi-coherent-algebraic-stack", "sites-modules-lemma-limits-colimits", "stacks-sheaves-lemma-quasi-coherent-algebraic-stack", "spaces-groupoids-lemma-set-generators", "injectives-theorem-injective-embedding-grothendieck", "injectives-lemma-grothendieck-products", "injectives-lemma-grothendieck-brown", "categories-lemma-yoneda" ], "ref_ids": [ 3066, 11921, 11590, 14158, 11590, 9307, 7767, 7794, 7793, 12203 ] } ], "ref_ids": [] }, { "id": 11619, "type": "theorem", "label": "stacks-sheaves-proposition-exactness-cech-complex", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-proposition-exactness-cech-complex", "contents": [ "Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred", "in groupoids over $(\\Sch/S)_{fppf}$. Let", "$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.", "If", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{X}_\\tau$, and", "\\item for every object $x$ of $\\mathcal{X}$ there exists a covering", "$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic", "to $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,", "\\end{enumerate}", "then the extended relative {\\v C}ech complex", "$$", "\\ldots \\to 0 \\to", "\\mathcal{F} \\to", "f_{0, *}f_0^{-1}\\mathcal{F} \\to", "f_{1, *}f_1^{-1}\\mathcal{F} \\to", "f_{2, *}f_2^{-1}\\mathcal{F} \\to \\ldots", "$$", "is exact in $\\textit{Ab}(\\mathcal{X}_\\tau)$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-check-exactness-covering}", "it suffices to check exactness after pulling back to $\\mathcal{U}$.", "By", "Lemma \\ref{lemma-base-change-cech-complex}", "the pullback of the extended relative {\\v C}ech complex is isomorphic", "to the extend relative {\\v C}ech complex for the morphism", "$\\mathcal{U} \\times_\\mathcal{X} \\mathcal{U} \\to \\mathcal{U}$", "and an abelian sheaf on $\\mathcal{U}_\\tau$. Since there is a section", "$\\Delta_{\\mathcal{U}/\\mathcal{X}} : \\mathcal{U} \\to", "\\mathcal{U} \\times_\\mathcal{X} \\mathcal{U}$ exactness follows from", "Lemma \\ref{lemma-homotopy-sheafified}." ], "refs": [ "stacks-sheaves-lemma-check-exactness-covering", "stacks-sheaves-lemma-base-change-cech-complex", "stacks-sheaves-lemma-homotopy-sheafified" ], "ref_ids": [ 11603, 11602, 11601 ] } ], "ref_ids": [] }, { "id": 11620, "type": "theorem", "label": "stacks-sheaves-proposition-smooth-covering-compute-cohomology", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-proposition-smooth-covering-compute-cohomology", "contents": [ "Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of algebraic stacks.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be an abelian \\'etale sheaf on $\\mathcal{X}$.", "Assume that $f$ is representable by algebraic spaces, surjective, and smooth.", "Then there is a spectral sequence", "$$", "E_1^{p, q} = H^q_\\etale(\\mathcal{U}_p, f_p^{-1}\\mathcal{F})", "\\Rightarrow", "H^{p + q}_\\etale(\\mathcal{X}, \\mathcal{F})", "$$", "\\item Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{X}$.", "Assume that $f$ is representable by algebraic spaces, surjective, flat,", "and locally of finite presentation. Then there is", "a spectral sequence", "$$", "E_1^{p, q} = H^q_{fppf}(\\mathcal{U}_p, f_p^{-1}\\mathcal{F})", "\\Rightarrow", "H^{p + q}_{fppf}(\\mathcal{X}, \\mathcal{F})", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To see this we will check the hypotheses (1) -- (4) of", "Lemma \\ref{lemma-cech-to-cohomology}.", "The $1$-morphism $f$ is faithful by", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}.", "This proves (4).", "Hypothesis (3) follows from the fact that $\\mathcal{U}$ is an algebraic", "stack, see", "Lemma \\ref{lemma-fibre-products}.", "To see (2) apply", "Lemma \\ref{lemma-surjective-flat-locally-finite-presentation}.", "Condition (1) is satisfied by fiat." ], "refs": [ "stacks-sheaves-lemma-cech-to-cohomology", "algebraic-lemma-characterize-representable-by-algebraic-spaces", "stacks-sheaves-lemma-fibre-products", "stacks-sheaves-lemma-surjective-flat-locally-finite-presentation" ], "ref_ids": [ 11604, 8469, 11593, 11606 ] } ], "ref_ids": [] }, { "id": 11621, "type": "theorem", "label": "stacks-sheaves-proposition-smooth-covering-compute-direct-image", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-proposition-smooth-covering-compute-direct-image", "contents": [ "Let $f : \\mathcal{U} \\to \\mathcal{X}$ and $g : \\mathcal{X} \\to \\mathcal{Y}$", "be composable $1$-morphisms of algebraic stacks.", "\\begin{enumerate}", "\\item Assume that $f$ is representable by algebraic spaces, surjective and", "smooth.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is in $\\textit{Ab}(\\mathcal{X}_\\etale)$", "then there is a spectral sequence", "$$", "E_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}", "\\Rightarrow", "R^{p + q}g_*\\mathcal{F}", "$$", "in $\\textit{Ab}(\\mathcal{Y}_\\etale)$ with higher direct images", "computed in the \\'etale topology.", "\\item If $\\mathcal{F}$ is in", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ then", "there is a spectral sequence", "$$", "E_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}", "\\Rightarrow", "R^{p + q}g_*\\mathcal{F}", "$$", "in $\\textit{Mod}(\\mathcal{Y}_\\etale, \\mathcal{O}_\\mathcal{Y})$.", "\\end{enumerate}", "\\item Assume that $f$ is representable by algebraic spaces, surjective,", "flat, and locally of finite presentation.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is in $\\textit{Ab}(\\mathcal{X})$ then there is", "a spectral sequence", "$$", "E_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}", "\\Rightarrow", "R^{p + q}g_*\\mathcal{F}", "$$", "in $\\textit{Ab}(\\mathcal{Y})$ with higher direct images", "computed in the fppf topology.", "\\item If $\\mathcal{F}$ is in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ then", "there is a spectral sequence", "$$", "E_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}", "\\Rightarrow", "R^{p + q}g_*\\mathcal{F}", "$$", "in $\\textit{Mod}(\\mathcal{O}_\\mathcal{Y})$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To see this we will check the hypotheses (1) -- (4) of", "Lemma \\ref{lemma-cech-to-cohomology-relative} and", "Lemma \\ref{lemma-cech-to-cohomology-relative-modules}.", "The $1$-morphism $f$ is faithful by", "Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}.", "This proves (4).", "Hypothesis (3) follows from the fact that $\\mathcal{U}$ is an algebraic", "stack, see", "Lemma \\ref{lemma-fibre-products}.", "To see (2) apply", "Lemma \\ref{lemma-surjective-flat-locally-finite-presentation}.", "Condition (1) is satisfied by fiat in all four cases." ], "refs": [ "stacks-sheaves-lemma-cech-to-cohomology-relative", "stacks-sheaves-lemma-cech-to-cohomology-relative-modules", "algebraic-lemma-characterize-representable-by-algebraic-spaces", "stacks-sheaves-lemma-fibre-products", "stacks-sheaves-lemma-surjective-flat-locally-finite-presentation" ], "ref_ids": [ 11607, 11608, 8469, 11593, 11606 ] } ], "ref_ids": [] }, { "id": 11635, "type": "theorem", "label": "resolve-theorem-resolve", "categories": [ "resolve" ], "title": "resolve-theorem-resolve", "contents": [ "\\begin{reference}", "\\cite[Theorem on page 151]{Lipman}", "\\end{reference}", "Let $Y$ be a two dimensional integral Noetherian scheme. The following are", "equivalent", "\\begin{enumerate}", "\\item there exists an alteration $X \\to Y$ with $X$ regular,", "\\item there exists a resolution of singularities of $Y$,", "\\item $Y$ has a resolution of singularities by normalized blowups,", "\\item the normalization $Y^\\nu \\to Y$ is finite, $Y^\\nu$ has", "finitely many singular points $y_1, \\ldots, y_m$, and for each", "$y_i$ the completion of $\\mathcal{O}_{Y^\\nu, y_i}$ is normal.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implications (3) $\\Rightarrow$ (2) $\\Rightarrow$ (1) are immediate.", "\\medskip\\noindent", "Let $X \\to Y$ be an alteration with $X$ regular. Then $Y^\\nu \\to Y$", "is finite by Lemma \\ref{lemma-regular-alteration-implies}.", "Consider the factorization $f : X \\to Y^\\nu$ from ", "Morphisms, Lemma \\ref{morphisms-lemma-normalization-normal}.", "The morphism $f$ is finite over an open $V \\subset Y^\\nu$ containing", "every point of codimension $\\leq 1$ in $Y^\\nu$", "by Varieties, Lemma \\ref{varieties-lemma-finite-in-codim-1}.", "Then $f$ is flat over $V$ by", "Algebra, Lemma \\ref{algebra-lemma-CM-over-regular-flat}", "and the fact that a normal local ring", "of dimension $\\leq 2$ is Cohen-Macaulay by Serre's criterion", "(Algebra, Lemma \\ref{algebra-lemma-criterion-normal}).", "Then $V$ is regular by Algebra, Lemma \\ref{algebra-lemma-descent-regular}.", "As $Y^\\nu$ is Noetherian we conclude that", "$Y^\\nu \\setminus V = \\{y_1, \\ldots, y_m\\}$ is finite.", "By Lemma \\ref{lemma-regular-alteration-implies-local}", "the completion of $\\mathcal{O}_{Y^\\nu, y_i}$ is normal.", "In this way we see that (1) $\\Rightarrow$ (4).", "\\medskip\\noindent", "Assume (4). We have to prove (3). We may immediately replace", "$Y$ by its normalization. Let $y_1, \\ldots, y_m \\in Y$ be the", "singular points. Applying", "Lemmas \\ref{lemma-resolve-complete} and", "\\ref{lemma-existence-implies-existence-by-normalized-blowing-ups}", "we find there exists a finite sequence of normalized blowups", "$$", "Y_{i, n_i} \\to Y_{i, n_i - 1} \\to \\ldots \\to \\Spec(\\mathcal{O}^\\wedge_{Y, y_i})", "$$", "such that $Y_{i, n_i}$ is regular. By", "Lemma \\ref{lemma-normalized-blowup-completion}", "there is a corresponding sequence of normalized blowing ups", "$$", "X_{i, n_i} \\to \\ldots \\to X_{i, 1} \\to \\Spec(\\mathcal{O}_{Y, y_i})", "$$", "Then $X_{i, n_i}$ is a regular scheme by", "Lemma \\ref{lemma-port-regularity-to-completion}.", "By Lemma \\ref{lemma-equivalence-sequence-normalized-blowups}", "we can fit these normalized blowing ups", "into a corresponding sequence", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to Y", "$$", "and of course $X_n$ is regular too (look at the local rings).", "This completes the proof." ], "refs": [ "resolve-lemma-regular-alteration-implies", "morphisms-lemma-normalization-normal", "varieties-lemma-finite-in-codim-1", "algebra-lemma-CM-over-regular-flat", "algebra-lemma-criterion-normal", "algebra-lemma-descent-regular", "resolve-lemma-regular-alteration-implies-local", "resolve-lemma-resolve-complete", "resolve-lemma-existence-implies-existence-by-normalized-blowing-ups", "resolve-lemma-normalized-blowup-completion", "resolve-lemma-port-regularity-to-completion", "resolve-lemma-equivalence-sequence-normalized-blowups" ], "ref_ids": [ 11688, 5515, 10978, 1107, 1311, 1373, 11690, 11692, 11691, 11685, 11680, 11655 ] } ], "ref_ids": [] }, { "id": 11636, "type": "theorem", "label": "resolve-lemma-trace-well-defined", "categories": [ "resolve" ], "title": "resolve-lemma-trace-well-defined", "contents": [ "Let $\\varphi : R[x]/(x^p - a) \\to R[y]/(y^p - b)$ be an $R$-algebra", "homomorphism. Then $\\text{Tr}_x = \\text{Tr}_y \\circ \\varphi$." ], "refs": [], "proofs": [ { "contents": [ "Say $\\varphi(x) = \\lambda_0 + \\lambda_1 y + \\ldots + \\lambda_{p - 1}y^{p - 1}$", "with $\\lambda_i \\in R$. The condition that mapping $x$ to", "$\\lambda_0 + \\lambda_1 y + \\ldots + \\lambda_{p - 1}y^{p - 1}$", "induces an $R$-algebra homomorphism $R[x]/(x^p - a) \\to R[y]/(y^p - b)$", "is equivalent to the condition that", "$$", "a = \\lambda_0^p + \\lambda_1^p b + \\ldots + \\lambda_{p - 1}^pb^{p - 1}", "$$", "in the ring $R$. Consider the polynomial ring", "$$", "R_{univ} = \\mathbf{F}_p[b, \\lambda_0, \\ldots, \\lambda_{p - 1}]", "$$", "with the element", "$a = \\lambda_0^p + \\lambda_1^p b + \\ldots + \\lambda_{p - 1}^pb^{p - 1}$", "Consider the universal algebra map", "$\\varphi_{univ} : R_{univ}[x]/(x^p - a) \\to R_{univ}[y]/(y^p - b)$", "given by mapping $x$ to", "$\\lambda_0 + \\lambda_1 y + \\ldots + \\lambda_{p - 1}y^{p - 1}$.", "We obtain a canonical map", "$$", "R_{univ} \\longrightarrow R", "$$", "sending $b, \\lambda_i$ to $b, \\lambda_i$. By construction we get a", "commutative diagram", "$$", "\\xymatrix{", "R_{univ}[x]/(x^p - a) \\ar[r] \\ar[d]_{\\varphi_{univ}} &", "R[x]/(x^p - a) \\ar[d]^\\varphi \\\\", "R_{univ}[y]/(y^p - b) \\ar[r] & R[y]/(y^p - b)", "}", "$$", "and the horizontal arrows are compatible with the trace maps. Hence it", "suffices to prove the lemma for the map $\\varphi_{univ}$. Thus we may", "assume $R = \\mathbf{F}_p[b, \\lambda_0, \\ldots, \\lambda_{p - 1}]$", "is a polynomial ring. We will check the lemma holds in this case", "by evaluating", "$\\text{Tr}_y(\\varphi(x)^i\\text{d}\\varphi(x))$ for $i = 0 , \\ldots, p - 1$.", "\\medskip\\noindent", "The case $0 \\leq i \\leq p - 2$. Expand", "$$", "(\\lambda_0 + \\lambda_1 y + \\ldots + \\lambda_{p - 1}y^{p - 1})^i", "(\\lambda_1 + 2 \\lambda_2 y + \\ldots + (p - 1)\\lambda_{p - 1}y^{p - 2})", "$$", "in the ring $R[y]/(y^p - b)$. We have to show that the coefficient", "of $y^{p - 1}$ is zero. For this it suffices to show that", "the expression above as a polynomial in $y$ has vanishing", "coefficients in front of the powers $y^{pk - 1}$.", "Then we write our polynomial as", "$$", "\\frac{\\text{d}}{(i + 1)\\text{d}y}", "(\\lambda_0 + \\lambda_1 y + \\ldots + \\lambda_{p - 1}y^{p - 1})^{i + 1}", "$$", "and indeed the coefficients of $y^{kp - 1}$ are all zero.", "\\medskip\\noindent", "The case $i = p - 1$. Expand", "$$", "(\\lambda_0 + \\lambda_1 y + \\ldots + \\lambda_{p - 1}y^{p - 1})^{p - 1}", "(\\lambda_1 + 2 \\lambda_2 y + \\ldots + (p - 1)\\lambda_{p - 1}y^{p - 2})", "$$", "in the ring $R[y]/(y^p - b)$. To finish the proof we have to show that", "the coefficient of $y^{p - 1}$ times $\\text{d}b$ is $\\text{d}a$.", "Here we use that $R$ is $S/pS$ where", "$S = \\mathbf{Z}[b, \\lambda_0, \\ldots, \\lambda_{p - 1}]$.", "Then the above, as a polynomial in $y$, is equal to", "$$", "\\frac{\\text{d}}{p\\text{d}y}", "(\\lambda_0 + \\lambda_1 y + \\ldots + \\lambda_{p - 1}y^{p - 1})^p", "$$", "Since $\\frac{\\text{d}}{\\text{d}y}(y^{pk}) = pk y^{pk - 1}$", "it suffices to understand the coefficients of $y^{pk}$ in the polynomial", "$(\\lambda_0 + \\lambda_1 y + \\ldots + \\lambda_{p - 1}y^{p - 1})^p$", "modulo $p$. The sum of these terms gives", "$$", "\\lambda_0^p + \\lambda_1^py^p + \\ldots + \\lambda_{p - 1}^py^{p(p - 1)}", "\\bmod p", "$$", "Whence we see that we obtain after applying the operator", "$\\frac{\\text{d}}{p\\text{d}y}$ and after reducing modulo $y^p - b$", "the value", "$$", "\\lambda_1^p + 2\\lambda_2^pb + \\ldots + (p - 1)\\lambda_{p - 1}b^{p - 2}", "$$", "for the coefficient of $y^{p - 1}$ we wanted to compute. Now because", "$a = \\lambda_0^p + \\lambda_1^p b + \\ldots + \\lambda_{p - 1}^pb^{p - 1}$", "in $R$ we obtain that", "$$", "\\text{d}a = (\\lambda_1^p + 2 \\lambda_2^p b + \\ldots +", "(p - 1) \\lambda_{p - 1}^p b^{p - 2}) \\text{d}b", "$$", "in $R$. This proves that the coefficient of $y^{p - 1}$ is as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11637, "type": "theorem", "label": "resolve-lemma-trace-higher", "categories": [ "resolve" ], "title": "resolve-lemma-trace-higher", "contents": [ "Let $\\mathbf{F}_p \\subset \\Lambda \\subset R \\subset S$ be ring extensions", "and assume that $S$ is isomorphic to $R[x]/(x^p - a)$ for some $a \\in R$.", "Then there are canonical $R$-linear maps", "$$", "\\text{Tr} :", "\\Omega^{t + 1}_{S/\\Lambda}", "\\longrightarrow", "\\Omega_{R/\\Lambda}^{t + 1}", "$$", "for $t \\geq 0$ such that", "$$", "\\eta_1 \\wedge \\ldots \\wedge \\eta_t \\wedge x^i\\text{d}x", "\\longmapsto", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & 0 \\leq i \\leq p - 2, \\\\", "\\eta_1 \\wedge \\ldots \\wedge \\eta_t \\wedge \\text{d}a & \\text{if} & i = p - 1", "\\end{matrix}", "\\right.", "$$", "for $\\eta_i \\in \\Omega_{R/\\Lambda}$ and such that $\\text{Tr}$ annihilates the", "image of", "$S \\otimes_R \\Omega_{R/\\Lambda}^{t + 1} \\to \\Omega_{S/\\Lambda}^{t + 1}$." ], "refs": [], "proofs": [ { "contents": [ "For $t = 0$ we use the composition", "$$", "\\Omega_{S/\\Lambda} \\to \\Omega_{S/R} \\to \\Omega_R \\to \\Omega_{R/\\Lambda}", "$$", "where the second map is Lemma \\ref{lemma-trace-well-defined}.", "There is an exact sequence", "$$", "H_1(L_{S/R}) \\xrightarrow{\\delta} \\Omega_{R/\\Lambda} \\otimes_R S \\to", "\\Omega_{S/\\Lambda} \\to \\Omega_{S/R} \\to 0", "$$", "(Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL}).", "The module $\\Omega_{S/R}$ is free over $S$ with basis $\\text{d}x$", "and the module $H_1(L_{S/R})$ is free over $S$ with basis $x^p - a$", "which $\\delta$ maps to $-\\text{d}a \\otimes 1$ in", "$\\Omega_{R/\\Lambda} \\otimes_R S$. In particular, if we set", "$$", "M = \\Coker(R \\to \\Omega_{R/\\Lambda}, 1 \\mapsto -\\text{d}a)", "$$", "then we see that $\\Coker(\\delta) = M \\otimes_R S$. We obtain a", "canonical map", "$$", "\\Omega^{t + 1}_{S/\\Lambda} \\to", "\\wedge_S^t(\\Coker(\\delta)) \\otimes_S \\Omega_{S/R} =", "\\wedge^t_R(M) \\otimes_R \\Omega_{S/R}", "$$", "Now, since the image of the map", "$\\text{Tr} : \\Omega_{S/R} \\to \\Omega_{R/\\Lambda}$", "of Lemma \\ref{lemma-trace-well-defined} is contained in $R\\text{d}a$ we", "see that wedging with an element in the image annihilates $\\text{d}a$.", "Hence there is a canonical map", "$$", "\\wedge^t_R(M) \\otimes_R \\Omega_{S/R} \\to \\Omega_{R/\\Lambda}^{t + 1}", "$$", "mapping", "$\\overline{\\eta}_1 \\wedge \\ldots \\wedge \\overline{\\eta}_t \\wedge \\omega$", "to $\\eta_1 \\wedge \\ldots \\wedge \\eta_t \\wedge \\text{Tr}(\\omega)$." ], "refs": [ "resolve-lemma-trace-well-defined", "algebra-lemma-exact-sequence-NL", "resolve-lemma-trace-well-defined" ], "ref_ids": [ 11636, 1153, 11636 ] } ], "ref_ids": [] }, { "id": 11638, "type": "theorem", "label": "resolve-lemma-trace-extends", "categories": [ "resolve" ], "title": "resolve-lemma-trace-extends", "contents": [ "Let $S$ be a scheme over $\\mathbf{F}_p$. Let $f : Y \\to X$ be a finite morphism", "of Noetherian normal integral schemes over $S$. Assume", "\\begin{enumerate}", "\\item the extension of function fields is purely inseparable of degree $p$, and", "\\item $\\Omega_{X/S}$ is a coherent $\\mathcal{O}_X$-module (for example", "if $X$ is of finite type over $S$).", "\\end{enumerate}", "For $i \\geq 1$ there is a canonical map", "$$", "\\text{Tr} : f_*\\Omega^i_{Y/S} \\longrightarrow (\\Omega_{X/S}^i)^{**}", "$$", "whose stalk in the generic point of $X$ recovers the trace map of", "Lemma \\ref{lemma-trace-higher}." ], "refs": [ "resolve-lemma-trace-higher" ], "proofs": [ { "contents": [ "The exact sequence $f^*\\Omega_{X/S} \\to \\Omega_{Y/S} \\to \\Omega_{Y/X} \\to 0$", "shows that $\\Omega_{Y/S}$ and hence $f_*\\Omega_{Y/S}$ are coherent modules", "as well. Thus it suffices to prove the trace map in the generic point", "extends to stalks at $x \\in X$ with $\\dim(\\mathcal{O}_{X, x}) = 1$, see", "Divisors, Lemma \\ref{divisors-lemma-describe-reflexive-hull}.", "Thus we reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $X = \\Spec(A)$ and $Y = \\Spec(B)$ with $A$ a discrete valuation", "ring and $B$ finite over $A$. Since the induced extension $K \\subset L$", "of fraction fields is purely inseparable, we see that $B$ is local too.", "Hence $B$ is a discrete valuation ring too. Then either", "\\begin{enumerate}", "\\item $B/A$ has ramification index $p$ and hence $B = A[x]/(x^p - a)$", "where $a \\in A$ is a uniformizer, or", "\\item $\\mathfrak m_B = \\mathfrak m_A B$ and the residue field", "$B/\\mathfrak m_A B$ is purely inseparable of degree $p$ over", "$\\kappa_A = A/\\mathfrak m_A$.", "Choose any $x \\in B$ whose residue class is not in $\\kappa_A$", "and then we'll have $B = A[x]/(x^p - a)$ where $a \\in A$ is", "a unit.", "\\end{enumerate}", "Let $\\Spec(\\Lambda) \\subset S$ be an affine open such that", "$X$ maps into $\\Spec(\\Lambda)$. Then we can apply", "Lemma \\ref{lemma-trace-higher}", "to see that the trace map extends to", "$\\Omega^i_{B/\\Lambda} \\to \\Omega^i_{A/\\Lambda}$", "for all $i \\geq 1$." ], "refs": [ "divisors-lemma-describe-reflexive-hull", "resolve-lemma-trace-higher" ], "ref_ids": [ 7925, 11637 ] } ], "ref_ids": [ 11637 ] }, { "id": 11639, "type": "theorem", "label": "resolve-lemma-blowup", "categories": [ "resolve" ], "title": "resolve-lemma-blowup", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$.", "Let $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$", "wotj exceptional divisor $E$. There is a closed immersion", "$$", "r : X \\longrightarrow \\mathbf{P}^1_S", "$$", "over $S$ such that", "\\begin{enumerate}", "\\item $r|_E : E \\to \\mathbf{P}^1_\\kappa$ is an isomorphism,", "\\item $\\mathcal{O}_X(E) = \\mathcal{O}_X(-1) =", "r^*\\mathcal{O}_{\\mathbf{P}^1}(-1)$, and", "\\item $\\mathcal{C}_{E/X} = (r|_E)^*\\mathcal{O}_{\\mathbf{P}^1}(1)$ and", "$\\mathcal{N}_{E/X} = (r|_E)^*\\mathcal{O}_{\\mathbf{P}^1}(-1)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As $A$ is regular of dimension $2$ we can write $\\mathfrak m = (x, y)$.", "Then $x$ and $y$ placed in degree $1$ generate the Rees algebra", "$\\bigoplus_{n \\geq 0} \\mathfrak m^n$ over $A$. Recall that", "$X = \\text{Proj}(\\bigoplus_{n \\geq 0} \\mathfrak m^n)$, see", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-affine}.", "Thus the surjection", "$$", "A[T_0, T_1] \\longrightarrow \\bigoplus\\nolimits_{n \\geq 0} \\mathfrak m^n,", "\\quad", "T_0 \\mapsto x,\\ T_1 \\mapsto y", "$$", "of graded $A$-algebras induces a closed immersion", "$r : X \\to \\mathbf{P}^1_S = \\text{Proj}(A[T_0, T_1])$", "such that $\\mathcal{O}_X(1) = r^*\\mathcal{O}_{\\mathbf{P}^1_S}(1)$, see", "Constructions, Lemma", "\\ref{constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj}.", "This proves (2) because $\\mathcal{O}_X(E) = \\mathcal{O}_X(-1)$", "by Divisors, Lemma", "\\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}.", "\\medskip\\noindent", "To prove (1) note that", "$$", "\\left(\\bigoplus\\nolimits_{n \\geq 0} \\mathfrak m^n\\right) \\otimes_A \\kappa =", "\\bigoplus\\nolimits_{n \\geq 0} \\mathfrak m^n/\\mathfrak m^{n + 1} \\cong", "\\kappa[\\overline{x}, \\overline{y}]", "$$", "a polynomial algebra, see Algebra, Lemma \\ref{algebra-lemma-regular-graded}.", "This proves that the fibre of $X \\to S$ over $\\Spec(\\kappa)$ is equal to", "$\\text{Proj}(\\kappa[\\overline{x}, \\overline{y}]) = \\mathbf{P}^1_\\kappa$, see", "Constructions, Lemma \\ref{constructions-lemma-base-change-map-proj}.", "Recall that $E$ is the closed subscheme of $X$ defined by", "$\\mathfrak m\\mathcal{O}_X$, i.e., $E = X_\\kappa$.", "By our choice of the morphism $r$ we see that $r|_E$ in fact", "produces the identification of $E = X_\\kappa$ with the special", "fibre of $\\mathbf{P}^1_S \\to S$.", "\\medskip\\noindent", "Part (3) follows from (1) and (2) and Divisors, Lemma", "\\ref{divisors-lemma-conormal-effective-Cartier-divisor}." ], "refs": [ "divisors-lemma-blowing-up-affine", "constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj", "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "algebra-lemma-regular-graded", "constructions-lemma-base-change-map-proj", "divisors-lemma-conormal-effective-Cartier-divisor" ], "ref_ids": [ 8052, 12612, 8054, 939, 12613, 7938 ] } ], "ref_ids": [] }, { "id": 11640, "type": "theorem", "label": "resolve-lemma-blowup-regular", "categories": [ "resolve" ], "title": "resolve-lemma-blowup-regular", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$.", "Let $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$.", "Then $X$ is an irreducible regular scheme." ], "refs": [], "proofs": [ { "contents": [ "Observe that $X$ is integral by", "Divisors, Lemma \\ref{divisors-lemma-blow-up-integral-scheme}", "and", "Algebra, Lemma \\ref{algebra-lemma-regular-domain}.", "To see $X$ is regular it suffices to check that $\\mathcal{O}_{X, x}$", "is regular for closed points $x \\in X$, see", "Properties, Lemma \\ref{properties-lemma-characterize-regular}.", "Let $x \\in X$ be a closed point. Since $f$ is proper $x$ maps to", "$\\mathfrak m$, i.e., $x$ is a point of the exceptional divisor $E$.", "Then $E$ is an effective Cartier divisor and $E \\cong \\mathbf{P}^1_\\kappa$.", "Thus if $g \\in \\mathfrak m_x \\subset \\mathcal{O}_{X, x}$ is a local", "equation for $E$, then", "$\\mathcal{O}_{X, x}/(g) \\cong \\mathcal{O}_{\\mathbf{P}^1_\\kappa, x}$.", "Since $\\mathbf{P}^1_\\kappa$ is covered by two affine opens which are the", "spectrum of a polynomial ring over $\\kappa$, we see that", "$\\mathcal{O}_{\\mathbf{P}^1_\\kappa, x}$ is regular by", "Algebra, Lemma \\ref{algebra-lemma-dim-affine-space}.", "We conclude by", "Algebra, Lemma \\ref{algebra-lemma-regular-mod-x}." ], "refs": [ "divisors-lemma-blow-up-integral-scheme", "algebra-lemma-regular-domain", "properties-lemma-characterize-regular", "algebra-lemma-dim-affine-space", "algebra-lemma-regular-mod-x" ], "ref_ids": [ 8059, 940, 2975, 992, 945 ] } ], "ref_ids": [] }, { "id": 11641, "type": "theorem", "label": "resolve-lemma-blowup-pic", "categories": [ "resolve" ], "title": "resolve-lemma-blowup-pic", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$.", "Let $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$.", "Then $\\Pic(X) = \\mathbf{Z}$ generated by $\\mathcal{O}_X(E)$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $E = \\mathbf{P}^1_\\kappa$ has Picard group $\\mathbf{Z}$", "with generator $\\mathcal{O}(1)$, see", "Divisors, Lemma \\ref{divisors-lemma-Pic-projective-space-UFD}.", "By Lemma \\ref{lemma-blowup} the invertible $\\mathcal{O}_X$-module", "$\\mathcal{O}_X(E)$ restricts to $\\mathcal{O}(-1)$. Hence", "$\\mathcal{O}_X(E)$ generates an infinite cyclic group in $\\Pic(X)$.", "Since $A$ is regular it is a UFD, see More on Algebra, ", "Lemma \\ref{more-algebra-lemma-regular-local-UFD}.", "Then the punctured spectrum $U = S \\setminus \\{\\mathfrak m\\} = X \\setminus E$", "has trivial Picard group, see", "Divisors, Lemma \\ref{divisors-lemma-open-subscheme-UFD}.", "Hence for every invertible $\\mathcal{O}_X$-module $\\mathcal{L}$", "there is an isomorphism $s : \\mathcal{O}_U \\to \\mathcal{L}|_U$.", "Then $s$ is a regular meromorphic section of $\\mathcal{L}$", "and we see that $\\text{div}_\\mathcal{L}(s) = nE$ for some", "$n \\in \\mathbf{Z}$", "(Divisors, Definition \\ref{divisors-definition-divisor-invertible-sheaf}).", "By Divisors, Lemma \\ref{divisors-lemma-normal-c1-injective}", "(and the fact that $X$ is normal by Lemma \\ref{lemma-blowup-regular})", "we conclude that $\\mathcal{L} = \\mathcal{O}_X(nE)$." ], "refs": [ "divisors-lemma-Pic-projective-space-UFD", "resolve-lemma-blowup", "more-algebra-lemma-regular-local-UFD", "divisors-lemma-open-subscheme-UFD", "divisors-definition-divisor-invertible-sheaf", "divisors-lemma-normal-c1-injective", "resolve-lemma-blowup-regular" ], "ref_ids": [ 8034, 11639, 10544, 8033, 8111, 8028, 11640 ] } ], "ref_ids": [] }, { "id": 11642, "type": "theorem", "label": "resolve-lemma-cohomology-of-blowup", "categories": [ "resolve" ], "title": "resolve-lemma-cohomology-of-blowup", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$.", "Let $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item $H^p(X, \\mathcal{F}) = 0$ for $p \\not \\in \\{0, 1\\}$,", "\\item $H^1(X, \\mathcal{O}_X(n)) = 0$ for $n \\geq -1$,", "\\item $H^1(X, \\mathcal{F}) = 0$ if $\\mathcal{F}$ or $\\mathcal{F}(1)$", "is globally generated,", "\\item $H^0(X, \\mathcal{O}_X(n)) = \\mathfrak m^{\\max(0, n)}$,", "\\item $\\text{length}_A H^1(X, \\mathcal{O}_X(n)) = -n(-n - 1)/2$", "if $n < 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $\\mathfrak m = (x, y)$, then $X$ is covered by the spectra", "of the affine blowup algebras $A[\\frac{\\mathfrak m}{x}]$ and", "$A[\\frac{\\mathfrak m}{y}]$ because $x$ and $y$ placed in degree $1$", "generate the Rees algebra $\\bigoplus \\mathfrak m^n$ over $A$.", "See Divisors, Lemma \\ref{divisors-lemma-blowing-up-affine} and", "Constructions, Lemma \\ref{constructions-lemma-proj-quasi-compact}.", "Since $X$ is separated by", "Constructions, Lemma \\ref{constructions-lemma-proj-separated}", "we see that cohomology of quasi-coherent sheaves vanishes in", "degrees $\\geq 2$ by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-vanishing-nr-affines}.", "\\medskip\\noindent", "Let $i : E \\to X$ be the exceptional divisor, see", "Divisors, Definition \\ref{divisors-definition-blow-up}.", "Recall that $\\mathcal{O}_X(-E) = \\mathcal{O}_X(1)$ is", "$f$-relatively ample, see", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}.", "Hence we know that $H^1(X, \\mathcal{O}_X(-nE)) = 0$ for some $n > 0$,", "see Cohomology of Schemes, Lemma \\ref{coherent-lemma-kill-by-twisting}.", "Consider the filtration", "$$", "\\mathcal{O}_X(-nE) \\subset \\mathcal{O}_X(-(n - 1)E) \\subset", "\\ldots \\subset \\mathcal{O}_X(-E) \\subset \\mathcal{O}_X \\subset \\mathcal{O}_X(E)", "$$", "The successive quotients are the sheaves", "$$", "\\mathcal{O}_X(-t E)/\\mathcal{O}_X(-(t + 1)E) =", "\\mathcal{O}_X(t)/\\mathcal{I}(t) =", "i_*\\mathcal{O}_E(t)", "$$", "where $\\mathcal{I} = \\mathcal{O}_X(-E)$ is the ideal sheaf of $E$.", "By Lemma \\ref{lemma-blowup} we have $E = \\mathbf{P}^1_\\kappa$ and", "$\\mathcal{O}_E(1)$ indeed corresponds to the usual Serre twist of", "the structure sheaf on $\\mathbf{P}^1$. Hence the cohomology", "of $\\mathcal{O}_E(t)$ vanishes in degree $1$ for $t \\geq -1$, see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cohomology-projective-space-over-ring}.", "Since this is equal to $H^1(X, i_*\\mathcal{O}_E(t))$ (by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology})", "we find that $H^1(X, \\mathcal{O}_X(-(t + 1)E)) \\to H^1(X, \\mathcal{O}_X(-tE))$", "is surjective for $t \\geq -1$. Hence", "$$", "0 = H^1(X, \\mathcal{O}_X(-nE))", "\\longrightarrow", "H^1(X, \\mathcal{O}_X(-tE)) = H^1(X, \\mathcal{O}_X(t))", "$$", "is surjective for $t \\geq -1$ which proves (2).", "\\medskip\\noindent", "Let $\\mathcal{F}$ be globally generated. This means there exists", "a short exact sequence", "$$", "0 \\to \\mathcal{G} \\to \\bigoplus\\nolimits_{i \\in I} \\mathcal{O}_X", "\\to \\mathcal{F} \\to 0", "$$", "Note that $H^1(X, \\bigoplus_{i \\in I} \\mathcal{O}_X) =", "\\bigoplus_{i \\in I} H^1(X, \\mathcal{O}_X)$ by", "Cohomology, Lemma \\ref{cohomology-lemma-quasi-separated-cohomology-colimit}.", "By part (2) we have $H^1(X, \\mathcal{O}_X) = 0$.", "If $\\mathcal{F}(1)$ is globally generated, then we can find a", "surjection $\\bigoplus_{i \\in I} \\mathcal{O}_X(-1) \\to \\mathcal{F}$", "and argue in a similar fashion.", "In other words, part (3) follows from part (2).", "\\medskip\\noindent", "For part (4) we note that for all $n$ large enough we have", "$\\Gamma(X, \\mathcal{O}_X(n)) = \\mathfrak m^n$, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-recover-tail-graded-module}.", "If $n \\geq 0$, then we can use the short exact sequence", "$$", "0 \\to \\mathcal{O}_X(n) \\to \\mathcal{O}_X(n - 1) \\to", "i_*\\mathcal{O}_E(n - 1) \\to 0", "$$", "and the vanishing of $H^1$ for the sheaf on the left to get a commutative", "diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathfrak m^{\\max(0, n)} \\ar[r] \\ar[d] &", "\\mathfrak m^{\\max(0, n - 1)} \\ar[r] \\ar[d] &", "\\mathfrak m^{\\max(0, n)}/\\mathfrak m^{\\max(0, n - 1)} \\ar[r] \\ar[d] & 0\\\\", "0 \\ar[r] &", "\\Gamma(X, \\mathcal{O}_X(n)) \\ar[r] &", "\\Gamma(X, \\mathcal{O}_X(n - 1)) \\ar[r] &", "\\Gamma(E, \\mathcal{O}_E(n - 1)) \\ar[r] & 0", "}", "$$", "with exact rows. In fact, the rows are exact also for $n < 0$", "because in this case the groups on the right are zero.", "In the proof of Lemma \\ref{lemma-blowup}", "we have seen that the right vertical arrow is an isomorphism", "(details omitted). Hence if the left vertical arrow is an isomorphism, so", "is the middle one. In this way we see that (4) holds by", "descending induction on $n$.", "\\medskip\\noindent", "Finally, we prove (5) by descending induction on $n$ and the sequences", "$$", "0 \\to \\mathcal{O}_X(n) \\to \\mathcal{O}_X(n - 1) \\to", "i_*\\mathcal{O}_E(n - 1) \\to 0", "$$", "Namely, for $n \\geq -1$ we already know $H^1(X, \\mathcal{O}_X(n)) = 0$.", "Since", "$$", "H^1(X, i_*\\mathcal{O}_E(-2)) =", "H^1(E, \\mathcal{O}_E(-2)) =", "H^1(\\mathbf{P}^1_\\kappa, \\mathcal{O}(-2)) \\cong \\kappa", "$$", "by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cohomology-projective-space-over-ring}", "which has length $1$ as an $A$-module, we conclude from the long exact", "cohomology sequence that (5) holds for $n = -2$. And so on and so forth." ], "refs": [ "divisors-lemma-blowing-up-affine", "constructions-lemma-proj-quasi-compact", "constructions-lemma-proj-separated", "coherent-lemma-vanishing-nr-affines", "divisors-definition-blow-up", "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "coherent-lemma-kill-by-twisting", "resolve-lemma-blowup", "coherent-lemma-cohomology-projective-space-over-ring", "coherent-lemma-relative-affine-cohomology", "cohomology-lemma-quasi-separated-cohomology-colimit", "coherent-lemma-recover-tail-graded-module", "resolve-lemma-blowup", "coherent-lemma-cohomology-projective-space-over-ring" ], "ref_ids": [ 8052, 12598, 12597, 3292, 8112, 8054, 3344, 11639, 3304, 3284, 2082, 3340, 11639, 3304 ] } ], "ref_ids": [] }, { "id": 11643, "type": "theorem", "label": "resolve-lemma-blowup-improve", "categories": [ "resolve" ], "title": "resolve-lemma-blowup-improve", "contents": [ "Let $(A, \\mathfrak m)$ be a regular local ring of dimension $2$.", "Let $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$.", "Let $\\mathfrak m^n \\subset I \\subset \\mathfrak m$ be an ideal.", "Let $d \\geq 0$ be the largest integer such that", "$$", "I \\mathcal{O}_X \\subset \\mathcal{O}_X(-dE)", "$$", "where $E$ is the exceptional divisor. Set", "$\\mathcal{I}' = I\\mathcal{O}_X(dE) \\subset \\mathcal{O}_X$.", "Then $d > 0$, the sheaf", "$\\mathcal{O}_X/\\mathcal{I}'$ is supported in finitely many", "closed points $x_1, \\ldots, x_r$ of $X$, and", "\\begin{align*}", "\\text{length}_A(A/I)", "& >", "\\text{length}_A \\Gamma(X, \\mathcal{O}_X/\\mathcal{I}') \\\\", "& \\geq", "\\sum\\nolimits_{i = 1, \\ldots, r}", "\\text{length}_{\\mathcal{O}_{X, x_i}}", "(\\mathcal{O}_{X, x_i}/\\mathcal{I}'_{x_i})", "\\end{align*}" ], "refs": [], "proofs": [ { "contents": [ "Since $I \\subset \\mathfrak m$ we see that every element of $I$", "vanishes on $E$. Thus we see that $d \\geq 1$. On the other hand, since", "$\\mathfrak m^n \\subset I$ we see that $d \\leq n$. Consider the", "short exact sequence", "$$", "0 \\to I\\mathcal{O}_X \\to \\mathcal{O}_X \\to \\mathcal{O}_X/I\\mathcal{O}_X \\to 0", "$$", "Since $I\\mathcal{O}_X$ is globally generated, we see that", "$H^1(X, I\\mathcal{O}_X) = 0$ by Lemma \\ref{lemma-cohomology-of-blowup}.", "Hence we obtain a surjection", "$A/I \\to \\Gamma(X, \\mathcal{O}_X/I\\mathcal{O}_X)$. Consider the short exact", "sequence", "$$", "0 \\to", "\\mathcal{O}_X(-dE)/I\\mathcal{O}_X \\to", "\\mathcal{O}_X/I\\mathcal{O}_X \\to", "\\mathcal{O}_X/\\mathcal{O}_X(-dE) \\to 0", "$$", "By Divisors, Lemma \\ref{divisors-lemma-codim-1-part}", "we see that $\\mathcal{O}_X(-dE)/I\\mathcal{O}_X$ is supported in finitely many", "closed points of $X$. In particular, this coherent sheaf has vanishing higher", "cohomology groups (detail omitted). Thus in the following diagram", "$$", "\\xymatrix{", "& & A/I \\ar[d] \\\\", "0 \\ar[r] &", "\\Gamma(X, \\mathcal{O}_X(-dE)/I\\mathcal{O}_X) \\ar[r] &", "\\Gamma(X, \\mathcal{O}_X/I\\mathcal{O}_X) \\ar[r] &", "\\Gamma(X, \\mathcal{O}_X/\\mathcal{O}_X(-dE)) \\ar[r] & 0", "}", "$$", "the bottom row is exact and the vertical arrow surjective. We have", "$$", "\\text{length}_A \\Gamma(X, \\mathcal{O}_X(-dE)/I\\mathcal{O}_X) <", "\\text{length}_A(A/I)", "$$", "since $\\Gamma(X, \\mathcal{O}_X/\\mathcal{O}_X(-dE))$ is nonzero.", "Namely, the image of $1 \\in \\Gamma(X, \\mathcal{O}_X)$", "is nonzero as $d > 0$.", "\\medskip\\noindent", "To finish the proof we translate the results above into the statements", "of the lemma. Since", "$\\mathcal{O}_X(dE)$ is invertible we have", "$$", "\\mathcal{O}_X/\\mathcal{I}' =", "\\mathcal{O}_X(-dE)/I\\mathcal{O}_X \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(dE).", "$$", "Thus $\\mathcal{O}_X/\\mathcal{I}'$ and $\\mathcal{O}_X(-dE)/I\\mathcal{O}_X$", "are supported in the same set of finitely many", "closed points, say $x_1, \\ldots, x_r \\in E \\subset X$.", "Moreover we obtain", "$$", "\\Gamma(X, \\mathcal{O}_X(-dE)/I\\mathcal{O}_X) =", "\\bigoplus \\mathcal{O}_X(-dE)_{x_i}/I\\mathcal{O}_{X, x_i}", "\\cong", "\\bigoplus \\mathcal{O}_{X, x_i}/\\mathcal{I}'_{x_i} =", "\\Gamma(X, \\mathcal{O}_X/\\mathcal{I}')", "$$", "because an invertible module over a local ring is trivial.", "Thus we obtain the strict inequality. We also get the second because", "$$", "\\text{length}_A(\\mathcal{O}_{X, x_i}/\\mathcal{I}'_{x_i}) \\geq", "\\text{length}_{\\mathcal{O}_{X, x_i}}(\\mathcal{O}_{X, x_i}/\\mathcal{I}'_{x_i})", "$$", "as is immediate from the definition of length." ], "refs": [ "resolve-lemma-cohomology-of-blowup", "divisors-lemma-codim-1-part" ], "ref_ids": [ 11642, 7952 ] } ], "ref_ids": [] }, { "id": 11644, "type": "theorem", "label": "resolve-lemma-differentials-of-blowup", "categories": [ "resolve" ], "title": "resolve-lemma-differentials-of-blowup", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$.", "Let $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$.", "Then $\\Omega_{X/S} = i_*\\Omega_{E/\\kappa}$, where $i : E \\to X$", "is the immersion of the exceptional divisor." ], "refs": [], "proofs": [ { "contents": [ "Writing $\\mathbf{P}^1 = \\mathbf{P}^1_S$, let", "$r : X \\to \\mathbf{P}^1$ be as in Lemma \\ref{lemma-blowup}.", "Then we have an exact sequence", "$$", "\\mathcal{C}_{X/\\mathbf{P}^1} \\to r^*\\Omega_{\\mathbf{P}^1/S} \\to", "\\Omega_{X/S} \\to 0", "$$", "see Morphisms, Lemma \\ref{morphisms-lemma-differentials-relative-immersion}.", "Since $\\Omega_{\\mathbf{P}^1/S}|_E = \\Omega_{E/\\kappa}$ by", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-differentials}", "it suffices to see that the first arrow defines a surjection", "onto the kernel of the canonical map", "$r^*\\Omega_{\\mathbf{P}^1/S} \\to i_*\\Omega_{E/\\kappa}$.", "This we can do locally. With notation as in the proof of", "Lemma \\ref{lemma-blowup} on an affine open of $X$ the morphism $f$", "corresponds to the ring map", "$$", "A \\to A[t]/(xt - y)", "$$", "where $x, y \\in \\mathfrak m$ are generators. Thus", "$\\text{d}(xt - y) = x\\text{d}t$ and $y\\text{d}t = t \\cdot x \\text{d}t$", "which proves what we want." ], "refs": [ "resolve-lemma-blowup", "morphisms-lemma-differentials-relative-immersion", "morphisms-lemma-base-change-differentials", "resolve-lemma-blowup" ], "ref_ids": [ 11639, 5319, 5314, 11639 ] } ], "ref_ids": [] }, { "id": 11645, "type": "theorem", "label": "resolve-lemma-make-ideal-principal", "categories": [ "resolve" ], "title": "resolve-lemma-make-ideal-principal", "contents": [ "Let $X$ be a Noetherian scheme. Let $T \\subset X$ be a finite set of", "closed points $x$ such that $\\mathcal{O}_{X, x}$ is", "regular of dimension $2$ for $x \\in T$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent", "sheaf of ideals such that $\\mathcal{O}_X/\\mathcal{I}$ is supported", "on $T$.", "Then there exists a sequence", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "where $X_{i + 1} \\to X_i$ is the blowing up of $X_i$ at a closed", "point $x_i$ lying above a point of $T$ such that", "$\\mathcal{I}\\mathcal{O}_{X_n}$ is an invertible ideal sheaf." ], "refs": [], "proofs": [ { "contents": [ "Say $T = \\{x_1, \\ldots, x_r\\}$. Set", "$$", "n_i = \\text{length}_{\\mathcal{O}_{X, x_i}}(\\mathcal{O}_{X, x_i}/I_i)", "$$", "This is finite as $\\mathcal{O}_X/\\mathcal{I}$ is supported on $T$", "and hence $\\mathcal{O}_{X, x_i}/I_i$ has support equal to", "$\\{\\mathfrak m_{x_i}\\}$ (see Algebra, Lemma \\ref{algebra-lemma-support-point}).", "We are going to use induction on $\\sum n_i$. If $n_i = 0$ for all", "$i$, then $\\mathcal{I} = \\mathcal{O}_X$ and we are done.", "\\medskip\\noindent", "Suppose $n_i > 0$. Let $X' \\to X$ be the blowing up of $X$ in $x_i$", "(see discussion above the lemma).", "Since $\\Spec(\\mathcal{O}_{X, x_i}) \\to X$ is flat we see that", "$X' \\times_X \\Spec(\\mathcal{O}_{X, x_i})$ is the blowup of", "the ring $\\mathcal{O}_{X, x_i}$ in the maximal ideal, see", "Divisors, Lemma", "\\ref{divisors-lemma-flat-base-change-blowing-up}.", "Hence the square in the commutative diagram", "$$", "\\xymatrix{", "\\text{Proj}(\\bigoplus\\nolimits_{d \\geq 0} \\mathfrak m_{x_i}^d) \\ar[r] \\ar[d] &", "X' \\ar[d] \\\\", "\\Spec(\\mathcal{O}_{X, x_i}) \\ar[r] & X", "}", "$$", "is cartesian. Let $E \\subset X'$ and", "$E' \\subset \\text{Proj}(\\bigoplus\\nolimits_{d \\geq 0} \\mathfrak m_{x_i}^d)$", "be the exceptional divisors. Let $d \\geq 1$ be the integer found in", "Lemma \\ref{lemma-blowup-improve} for the ideal", "$\\mathcal{I}_i \\subset \\mathcal{O}_{X, x_i}$.", "Since the horizontal arrows in the diagram are flat, since", "$E' \\to E$ is surjective, and since $E'$ is the pullback of $E$, we see that", "$$", "\\mathcal{I}\\mathcal{O}_{X'} \\subset \\mathcal{O}_{X'}(-dE)", "$$", "(some details omitted).", "Set $\\mathcal{I}' = \\mathcal{I}\\mathcal{O}_{X'}(dE) \\subset \\mathcal{O}_{X'}$.", "Then we see that $\\mathcal{O}_{X'}/\\mathcal{I}'$ is supported in finitely", "many closed points $T' \\subset |X'|$ because this holds over", "$X \\setminus \\{x_i\\}$ and for the pullback to", "$\\text{Proj}(\\bigoplus\\nolimits_{d \\geq 0} \\mathfrak m_{x_i}^d)$.", "The final assertion of Lemma \\ref{lemma-blowup-improve}", "tells us that the sum of the lengths of the stalks", "$\\mathcal{O}_{X', x'}/\\mathcal{I}'\\mathcal{O}_{X', x'}$", "for $x'$ lying over $x_i$ is $< n_i$. Hence the sum of the lengths", "has decreased.", "\\medskip\\noindent", "By induction hypothesis, there exists a sequence", "$$", "X'_n \\to \\ldots \\to X'_1 \\to X'", "$$", "of blowups at closed points lying over $T'$ such that", "$\\mathcal{I}'\\mathcal{O}_{X'_n}$ is invertible. Since", "$\\mathcal{I}'\\mathcal{O}_{X'}(-dE) = \\mathcal{I}\\mathcal{O}_{X'}$, we see", "that $\\mathcal{I}\\mathcal{O}_{X'_n} =", "\\mathcal{I}'\\mathcal{O}_{X'_n}(-d(f')^{-1}E)$", "where $f' : X'_n \\to X'$ is the composition.", "Note that $(f')^{-1}E$ is an effective Cartier divisor by", "Divisors, Lemma \\ref{divisors-lemma-blow-up-pullback-effective-Cartier}.", "Thus we are done by", "Divisors, Lemma \\ref{divisors-lemma-sum-effective-Cartier-divisors}." ], "refs": [ "algebra-lemma-support-point", "divisors-lemma-flat-base-change-blowing-up", "resolve-lemma-blowup-improve", "resolve-lemma-blowup-improve", "divisors-lemma-blow-up-pullback-effective-Cartier", "divisors-lemma-sum-effective-Cartier-divisors" ], "ref_ids": [ 693, 8053, 11643, 11643, 8061, 7931 ] } ], "ref_ids": [] }, { "id": 11646, "type": "theorem", "label": "resolve-lemma-dominate-by-blowing-up-in-points", "categories": [ "resolve" ], "title": "resolve-lemma-dominate-by-blowing-up-in-points", "contents": [ "Let $X$ be a Noetherian scheme. Let $T \\subset X$ be a finite set of", "closed points $x$ such that $\\mathcal{O}_{X, x}$ is a regular local", "ring of dimension $2$. Let $f : Y \\to X$ be a proper morphism of", "schemes which is an isomorphism over $U = X \\setminus T$.", "Then there exists a sequence", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "where $X_{i + 1} \\to X_i$ is the blowing up of $X_i$ at a closed", "point $x_i$ lying above a point of $T$ and a factorization $X_n \\to Y \\to X$", "of the composition." ], "refs": [], "proofs": [ { "contents": [ "By More on Flatness, Lemma \\ref{flat-lemma-dominate-modification-by-blowup} ", "there exists a $U$-admissible blowup $X' \\to X$ which dominates", "$Y \\to X$. Hence we may assume there exists an ideal sheaf", "$\\mathcal{I} \\subset \\mathcal{O}_X$ such that", "$\\mathcal{O}_X/\\mathcal{I}$ is supported on $T$ and such that", "$Y$ is the blowing up of $X$ in $\\mathcal{I}$.", "By Lemma \\ref{lemma-make-ideal-principal} ", "there exists a sequence", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "where $X_{i + 1} \\to X_i$ is the blowing up of $X_i$ at a closed", "point $x_i$ lying above a point of $T$ such that", "$\\mathcal{I}\\mathcal{O}_{X_n}$ is an invertible ideal sheaf.", "By the universal property of blowing up", "(Divisors, Lemma", "\\ref{divisors-lemma-universal-property-blowing-up})", "we find the desired factorization." ], "refs": [ "flat-lemma-dominate-modification-by-blowup", "resolve-lemma-make-ideal-principal", "divisors-lemma-universal-property-blowing-up" ], "ref_ids": [ 6126, 11645, 8055 ] } ], "ref_ids": [] }, { "id": 11647, "type": "theorem", "label": "resolve-lemma-extend-rational-map-blowing-up", "categories": [ "resolve" ], "title": "resolve-lemma-extend-rational-map-blowing-up", "contents": [ "Let $S$ be a scheme. Let $X$ be a scheme over $S$ which is", "regular and has dimension $2$. Let $Y$ be a proper", "scheme over $S$. Given an $S$-rational map $f : U \\to Y$ from", "$X$ to $Y$ there exists a sequence", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "and an $S$-morphism $f_n : X_n \\to Y$ such that $X_{i + 1} \\to X_i$", "blowing up of $X_i$ at a closed point not lying over $U$", "and $f_n$ and $f$ agree." ], "refs": [], "proofs": [ { "contents": [ "We may assume $U$ contains every point of codimension $1$, see", "Morphisms, Lemma \\ref{morphisms-lemma-extend-across}.", "Hence the complement $T \\subset X$ of $U$ is a finite set", "of closed points whose local rings are regular of dimension $2$.", "Applying", "Divisors, Lemma \\ref{divisors-lemma-extend-rational-map-after-modification}", "we find a proper morphism $p : X' \\to X$ which is an isomorphism", "over $U$ and a morphism $f' : X' \\to Y$ agreeing with $f$ over $U$.", "Apply Lemma \\ref{lemma-dominate-by-blowing-up-in-points}", "to the morphism $p : X' \\to X$. The composition $X_n \\to X' \\to Y$ is", "the desired morphism." ], "refs": [ "morphisms-lemma-extend-across", "divisors-lemma-extend-rational-map-after-modification", "resolve-lemma-dominate-by-blowing-up-in-points" ], "ref_ids": [ 5419, 8080, 11646 ] } ], "ref_ids": [] }, { "id": 11648, "type": "theorem", "label": "resolve-lemma-Nagata-normalized-blowup", "categories": [ "resolve" ], "title": "resolve-lemma-Nagata-normalized-blowup", "contents": [ "In Definition \\ref{definition-normalized-blowup} if $X$ is Nagata,", "then the normalized blowing up of $X$ at $x$ is", "normal, Nagata, and proper over $X$." ], "refs": [ "resolve-definition-normalized-blowup" ], "proofs": [ { "contents": [ "The blowup morphism $X' \\to X$ is proper", "(as $X$ is locally Noetherian we may apply", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-projective}).", "Thus $X'$ is Nagata", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-type-nagata}).", "Therefore the normalization $X'' \\to X'$ is finite", "(Morphisms, Lemma \\ref{morphisms-lemma-nagata-normalization})", "and we conclude that $X'' \\to X$ is proper as well", "(Morphisms, Lemmas \\ref{morphisms-lemma-finite-proper} and", "\\ref{morphisms-lemma-composition-proper}).", "It follows that the normalized blowing up", "is a normal (Morphisms, Lemma", "\\ref{morphisms-lemma-normalization-normal})", "Nagata algebraic space." ], "refs": [ "divisors-lemma-blowing-up-projective", "morphisms-lemma-finite-type-nagata", "morphisms-lemma-nagata-normalization", "morphisms-lemma-finite-proper", "morphisms-lemma-composition-proper", "morphisms-lemma-normalization-normal" ], "ref_ids": [ 8063, 5218, 5520, 5445, 5408, 5515 ] } ], "ref_ids": [ 11712 ] }, { "id": 11649, "type": "theorem", "label": "resolve-lemma-dominate-by-normalized-blowing-up", "categories": [ "resolve" ], "title": "resolve-lemma-dominate-by-normalized-blowing-up", "contents": [ "Let $X$ be a scheme which is Noetherian, Nagata, and has dimension $2$.", "Let $f : Y \\to X$ be a proper birational morphism.", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "X_n \\ar[r] \\ar[d] &", "X_{n - 1} \\ar[r] &", "\\ldots \\ar[r] &", "X_1 \\ar[r] &", "X_0 \\ar[d] \\\\", "Y \\ar[rrrr] & & & & X", "}", "$$", "where $X_0 \\to X$ is the normalization and", "where $X_{i + 1} \\to X_i$ is the normalized blowing up of $X_i$ at a closed", "point." ], "refs": [], "proofs": [ { "contents": [ "We will use the results of Morphisms, Sections", "\\ref{morphisms-section-nagata},", "\\ref{morphisms-section-dimension-formula}, and", "\\ref{morphisms-section-normalization} without further mention.", "We may replace $Y$ by its normalization. Let $X_0 \\to X$", "be the normalization. The morphism $Y \\to X$ factors through $X_0$.", "Thus we may assume that both $X$ and $Y$ are normal.", "\\medskip\\noindent", "Assume $X$ and $Y$ are normal. The morphism $f : Y \\to X$ is an isomorphism", "over an open which contains every point of codimension $0$ and $1$ in $Y$ and", "every point of $Y$ over which the fibre is finite, see Varieties, Lemma", "\\ref{varieties-lemma-modification-normal-iso-over-codimension-1}.", "Hence there is a finite set of closed points $T \\subset X$", "such that $f$ is an isomorphism over $X \\setminus T$. For each $x \\in T$", "the fibre $Y_x$ is a proper geometrically connected scheme of dimension $1$", "over $\\kappa(x)$, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-geometrically-connected-fibres-towards-normal}.", "Thus", "$$", "BadCurves(f) = \\{C \\subset Y\\text{ closed} \\mid", "\\dim(C) = 1, f(C) = \\text{a point}\\}", "$$", "is a finite set. We will prove the lemma by induction on the number", "of elements of $BadCurves(f)$. The base case is the case where $BadCurves(f)$", "is empty, and in that case $f$ is an isomorphism.", "\\medskip\\noindent", "Fix $x \\in T$. Let $X' \\to X$ be the normalized blowup of $X$ at $x$ and let", "$Y'$ be the normalization of $Y \\times_X X'$. Picture", "$$", "\\xymatrix{", "Y' \\ar[r]_{f'} \\ar[d] & X' \\ar[d] \\\\", "Y \\ar[r]^f & X", "}", "$$", "Let $x' \\in X'$ be a closed point lying over $x$ such that", "the fibre $Y'_{x'}$ has dimension $\\geq 1$. Let $C' \\subset Y'$", "be an irreducible component of $Y'_{x'}$, i.e., $C' \\in BadCurves(f')$.", "Since $Y' \\to Y \\times_X X'$ is finite we see that $C'$ must map", "to an irreducible component $C \\subset Y_x$.", "If is clear that $C \\in BadCurves(f)$.", "Since $Y' \\to Y$ is birational and hence an isomorphism over points of", "codimension $1$ in $Y$, we see that we obtain an injective map", "$$", "BadCurves(f') \\longrightarrow BadCurves(f)", "$$", "Thus it suffices to show that after a finite number of these", "normalized blowups we get rid at of at least one of the bad", "curves, i.e., the displayed map is not surjective.", "\\medskip\\noindent", "We will get rid of a bad curve using an argument due to Zariski.", "Pick $C \\in BadCurves(f)$ lying over our $x$. Denote $\\mathcal{O}_{Y, C}$", "the local ring of $Y$ at the generic point of $C$. Choose an element", "$u \\in \\mathcal{O}_{X, C}$ whose image in the residue field", "$R(C)$ is transcendental over $\\kappa(x)$ (we can do this because", "$R(C)$ has transcendence degree $1$ over $\\kappa(x)$ by", "Varieties, Lemma \\ref{varieties-lemma-dimension-locally-algebraic}).", "We can write $u = a/b$ with $a, b \\in \\mathcal{O}_{X, x}$ as", "$\\mathcal{O}_{Y, C}$ and $\\mathcal{O}_{X, x}$ have the same", "fraction fields. By our choice of $u$ it must be the case that", "$a, b \\in \\mathfrak m_x$. Hence", "$$", "N_{u, a, b} = \\min", "\\{\\text{ord}_{\\mathcal{O}_{Y, C}}(a), \\text{ord}_{\\mathcal{O}_{Y, C}}(b)\\} > 0", "$$", "Thus we can do descending induction on this integer.", "Let $X' \\to X$ be the normalized blowing up of $x$", "and let $Y'$ be the normalization of $X' \\times_X Y$ as above. We will", "show that if $C$ is the image of some bad curve $C' \\subset Y'$", "lying over $x' \\in X'$, then", "there exists a choice of $a', b' \\mathcal{O}_{X', x'}$", "such that $N_{u, a', b'} < N_{u, a, b}$. This will finish the proof.", "Namely, since $X' \\to X$ factors through the blowing up, we see that", "there exists a nonzero element $d \\in \\mathfrak m_{x'}$ such that", "$a = a' d$ and $b = b' d$ (namely, take $d$ to be the local equation", "for the exceptional divisor of the blowup). Since $Y' \\to Y$", "is an isomorphism over an open containing the generic point of $C$", "(seen above) we see that $\\mathcal{O}_{Y', C'} = \\mathcal{O}_{Y, C}$.", "Hence", "$$", "\\text{ord}_{\\mathcal{O}_{Y, C}}(a) =", "\\text{ord}_{\\mathcal{O}_{Y', C'}}(a' d) =", "\\text{ord}_{\\mathcal{O}_{Y', C'}}(a') +", "\\text{ord}_{\\mathcal{O}_{Y', C'}}(d) >", "\\text{ord}_{\\mathcal{O}_{Y', C'}}(a')", "$$", "Similarly for $b$ and the proof is complete." ], "refs": [ "varieties-lemma-modification-normal-iso-over-codimension-1", "more-morphisms-lemma-geometrically-connected-fibres-towards-normal", "varieties-lemma-dimension-locally-algebraic" ], "ref_ids": [ 10979, 13945, 10989 ] } ], "ref_ids": [] }, { "id": 11650, "type": "theorem", "label": "resolve-lemma-extend-rational-map-normalized-blowing-up", "categories": [ "resolve" ], "title": "resolve-lemma-extend-rational-map-normalized-blowing-up", "contents": [ "Let $S$ be a scheme. Let $X$ be a scheme over $S$ which is", "Noetherian, Nagata, and has dimension $2$. Let $Y$ be a proper", "scheme over $S$. Given an $S$-rational map $f : U \\to Y$ from", "$X$ to $Y$ there exists a sequence", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 \\to X", "$$", "and an $S$-morphism $f_n : X_n \\to Y$ such that $X_0 \\to X$ is the", "normalization, $X_{i + 1} \\to X_i$ is the normalized blowing up of", "$X_i$ at a closed point, and $f_n$ and $f$ agree." ], "refs": [], "proofs": [ { "contents": [ "Applying", "Divisors, Lemma \\ref{divisors-lemma-extend-rational-map-after-modification}", "we find a proper morphism $p : X' \\to X$ which is an isomorphism", "over $U$ and a morphism $f' : X' \\to Y$ agreeing with $f$ over $U$.", "Apply Lemma \\ref{lemma-dominate-by-normalized-blowing-up}", "to the morphism $p : X' \\to X$. The composition $X_n \\to X' \\to Y$ is", "the desired morphism." ], "refs": [ "divisors-lemma-extend-rational-map-after-modification", "resolve-lemma-dominate-by-normalized-blowing-up" ], "ref_ids": [ 8080, 11649 ] } ], "ref_ids": [] }, { "id": 11651, "type": "theorem", "label": "resolve-lemma-equivalence", "categories": [ "resolve" ], "title": "resolve-lemma-equivalence", "contents": [ "The functor $F$ (\\ref{equation-equivalence}) is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "For $n = 1$ this is Limits, Lemma \\ref{limits-lemma-modifications}.", "For $n > 1$ the lemma can be proved in exactly the same way or it", "can be deduced from it. For example, suppose that", "$g_i : Y_i \\to S_i$ are objects of $FP_{S_i, s_i}$.", "Then by the case $n = 1$ we can find $f'_i : X'_i \\to S$", "of finite presentation", "which are isomorphisms over $S \\setminus \\{s_i\\}$ and whose", "base change to $S_i$ is $g_i$. Then we can set", "$$", "f : X = X'_1 \\times_S \\ldots \\times_S X'_n \\to S", "$$", "This is an object of $FP_{S, \\{s_1, \\ldots, s_n\\}}$", "whose base change by $S_i \\to S$ recovers $g_i$. Thus the functor", "is essentially surjective. We omit the proof of", "fully faithfulness." ], "refs": [ "limits-lemma-modifications" ], "ref_ids": [ 15117 ] } ], "ref_ids": [] }, { "id": 11652, "type": "theorem", "label": "resolve-lemma-equivalence-properties", "categories": [ "resolve" ], "title": "resolve-lemma-equivalence-properties", "contents": [ "Let $S, s_i, S_i$ be as in (\\ref{equation-equivalence}).", "If $f : X \\to S$ corresponds to $g_i : Y_i \\to S_i$ under $F$,", "then $f$ is separated, proper, finite, if and only if $g_i$ is so", "for $i = 1, \\ldots, n$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Limits, Lemma", "\\ref{limits-lemma-modifications-properties}." ], "refs": [ "limits-lemma-modifications-properties" ], "ref_ids": [ 15118 ] } ], "ref_ids": [] }, { "id": 11653, "type": "theorem", "label": "resolve-lemma-equivalence-fibre", "categories": [ "resolve" ], "title": "resolve-lemma-equivalence-fibre", "contents": [ "Let $S, s_i, S_i$ be as in (\\ref{equation-equivalence}).", "If $f : X \\to S$ corresponds to $g_i : Y_i \\to S_i$ under $F$,", "then $X_{s_i} \\cong (Y_i)_{s_i}$ as schemes over $\\kappa(s_i)$." ], "refs": [], "proofs": [ { "contents": [ "This is clear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11654, "type": "theorem", "label": "resolve-lemma-equivalence-sequence-blowups", "categories": [ "resolve" ], "title": "resolve-lemma-equivalence-sequence-blowups", "contents": [ "Let $S, s_i, S_i$ be as in (\\ref{equation-equivalence})", "and assume $f : X \\to S$ corresponds to $g_i : Y_i \\to S_i$ under $F$.", "Then there exists a factorization", "$$", "X = Z_m \\to Z_{m - 1} \\to \\ldots \\to Z_1 \\to Z_0 = S", "$$", "of $f$ where $Z_{j + 1} \\to Z_j$ is the blowing up of $Z_j$ at a closed", "point $z_j$ lying over $\\{s_1, \\ldots, s_n\\}$ if and only if for each", "$i$ there exists a factorization", "$$", "Y_i = Z_{i, m_i} \\to Z_{i, m_i - 1} \\to \\ldots \\to Z_{i, 1} \\to Z_{i, 0} = S_i", "$$", "of $g_i$ where $Z_{i, j + 1} \\to Z_{i, j}$ is the blowing up of $Z_{i, j}$", "at a closed point $z_{i, j}$ lying over $s_i$." ], "refs": [], "proofs": [ { "contents": [ "Let's start with a sequence of blowups", "$Z_m \\to Z_{m - 1} \\to \\ldots \\to Z_1 \\to Z_0 = S$.", "The first morphism $Z_1 \\to S$ is given", "by blowing up one of the $s_i$, say $s_1$. Applying $F$", "to $Z_1 \\to S$ we find a blowup $Z_{1, 1} \\to S_1$ at $s_1$", "is the blowing up at $s_1$ and otherwise $Z_{i, 0} = S_i$ for $i > 1$.", "In the next step, we either blow up one of the $s_i$, $i \\geq 2$", "on $Z_1$ or we pick a closed point $z_1$ of the fibre of $Z_1 \\to S$", "over $s_1$. In the first case it is clear what to do and in", "the second case we use that $(Z_1)_{s_1} \\cong (Z_{1, 1})_{s_1}$", "(Lemma \\ref{lemma-equivalence-fibre})", "to get a closed point $z_{1, 1} \\in Z_{1, 1}$ corresponding to $z_1$.", "Then we set $Z_{1, 2} \\to Z_{1, 1}$ equal to the blowing up", "in $z_{1, 1}$. Continuing in this manner we construct the factorizations", "of each $g_i$.", "\\medskip\\noindent", "Conversely, given sequences of blowups", "$Z_{i, m_i} \\to Z_{i, m_i - 1} \\to \\ldots \\to Z_{i, 1} \\to Z_{i, 0} = S_i$", "we construct the sequence of blowing ups of $S$ in exactly the same manner." ], "refs": [ "resolve-lemma-equivalence-fibre" ], "ref_ids": [ 11653 ] } ], "ref_ids": [] }, { "id": 11655, "type": "theorem", "label": "resolve-lemma-equivalence-sequence-normalized-blowups", "categories": [ "resolve" ], "title": "resolve-lemma-equivalence-sequence-normalized-blowups", "contents": [ "Let $S, s_i, S_i$ be as in (\\ref{equation-equivalence})", "and assume $f : X \\to S$ corresponds to $g_i : Y_i \\to S_i$ under $F$.", "Assume every quasi-compact open of $S$ has finitely many irreducible", "components. Then there exists a factorization", "$$", "X = Z_m \\to Z_{m - 1} \\to \\ldots \\to Z_1 \\to Z_0 = S", "$$", "of $f$ where $Z_{j + 1} \\to Z_j$ is the normalized blowing up of $Z_j$", "at a closed point $z_j$ lying over $\\{x_1, \\ldots, x_n\\}$ if and only if", "for each $i$ there exists a factorization", "$$", "Y_i = Z_{i, m_i} \\to Z_{i, m_i - 1} \\to \\ldots \\to Z_{i, 1} \\to Z_{i, 0} = S_i", "$$", "of $g_i$ where $Z_{i, j + 1} \\to Z_{i, j}$ is the normalized blowing up of", "$Z_{i, j}$ at a closed point $z_{i, j}$ lying over $s_i$." ], "refs": [], "proofs": [ { "contents": [ "The assumption on $S$ is used to assure us (successively) that", "the schemes we are normalizing have locally finitely many irreducible", "components so that the statement makes sense. Having said this the", "lemma follows by the exact same argument as used to prove", "Lemma \\ref{lemma-equivalence-sequence-blowups}." ], "refs": [ "resolve-lemma-equivalence-sequence-blowups" ], "ref_ids": [ 11654 ] } ], "ref_ids": [] }, { "id": 11656, "type": "theorem", "label": "resolve-lemma-dominate-by-scheme-modification", "categories": [ "resolve" ], "title": "resolve-lemma-dominate-by-scheme-modification", "contents": [ "In Situation \\ref{situation-vanishing} there exists a $U$-admissible", "blowup $X' \\to S$ which dominates $X$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "More on Flatness, Lemma \\ref{flat-lemma-dominate-modification-by-blowup}." ], "refs": [ "flat-lemma-dominate-modification-by-blowup" ], "ref_ids": [ 6126 ] } ], "ref_ids": [] }, { "id": 11657, "type": "theorem", "label": "resolve-lemma-nice-meromorphic-function", "categories": [ "resolve" ], "title": "resolve-lemma-nice-meromorphic-function", "contents": [ "In Situation \\ref{situation-vanishing} there exists a nonzero", "$f \\in \\mathfrak m$ such that for every $i = 1, \\ldots, r$ there exist", "\\begin{enumerate}", "\\item a closed point $x_i \\in C_i$ with $x_i \\not \\in C_j$ for $j \\not = i$,", "\\item a factorization $f = g_i f_i$ of $f$ in $\\mathcal{O}_{X, x_i}$", "such that $g_i \\in \\mathfrak m_{x_i}$ maps to a nonzero element", "of $\\mathcal{O}_{C_i, x_i}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use the observations made following Situation \\ref{situation-vanishing}", "without further mention. Pick a closed point $x_i \\in C_i$ which is not in", "$C_j$ for $j \\not = i$. Pick $g_i \\in \\mathfrak m_{x_i}$ which maps to a", "nonzero element of $\\mathcal{O}_{C_i, x_i}$. Since the fraction field of $A$", "is the fraction field of $\\mathcal{O}_{X_i, x_i}$ we can write", "$g_i = a_i/b_i$ for some $a_i, b_i \\in A$. Take $f = \\prod a_i$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11658, "type": "theorem", "label": "resolve-lemma-nontrivial-normal-bundle", "categories": [ "resolve" ], "title": "resolve-lemma-nontrivial-normal-bundle", "contents": [ "In Situation \\ref{situation-vanishing} assume $X$ is normal.", "Let $Z \\subset X$ be a nonempty effective Cartier divisor such that", "$Z \\subset X_s$ set theoretically.", "Then the conormal sheaf of $Z$ is not trivial.", "More precisely, there exists an $i$ such that $C_i \\subset Z$", "and $\\deg(\\mathcal{C}_{Z/X}|_{C_i}) > 0$." ], "refs": [], "proofs": [ { "contents": [ "We will use the observations made following Situation \\ref{situation-vanishing}", "without further mention. Let $f$ be a function as in", "Lemma \\ref{lemma-nice-meromorphic-function}.", "Let $\\xi_i \\in C_i$ be the generic point. Let", "$\\mathcal{O}_i$ be the local ring of $X$ at $\\xi_i$. Then $\\mathcal{O}_i$", "is a discrete valuation ring. Let $e_i$ be the valuation of", "$f$ in $\\mathcal{O}_i$, so $e_i > 0$. Let $h_i \\in \\mathcal{O}_i$ be a local", "equation for $Z$ and let $d_i$ be its valuation. Then $d_i \\geq 0$.", "Choose and fix $i$ with $d_i/e_i$ maximal (then $d_i > 0$ as", "$Z$ is not empty). Replace $f$ by $f^{d_i}$ and $Z$ by $e_iZ$.", "This is permissible, by the relation", "$\\mathcal{O}_X(e_i Z) = \\mathcal{O}_X(Z)^{\\otimes e_i}$,", "the relation between the conormal sheaf and $\\mathcal{O}_X(Z)$", "(see Divisors, Lemmas", "\\ref{divisors-lemma-invertible-sheaf-sum-effective-Cartier-divisors}", "and \\ref{divisors-lemma-conormal-effective-Cartier-divisor}, and", "since the degree gets multiplied by $e_i$, see", "Varieties, Lemma \\ref{varieties-lemma-degree-tensor-product}.", "Let $\\mathcal{I}$ be the ideal sheaf of $Z$ so that", "$\\mathcal{C}_{Z/X} = \\mathcal{I}|_Z$. Consider the image $\\overline{f}$", "of $f$ in $\\Gamma(Z, \\mathcal{O}_Z)$. By our choices above we see", "that $\\overline{f}$ vanishes in the generic points of irreducible", "components of $Z$ (these are all generic points of $C_j$ as $Z$ is", "contained in the special fibre). On the other hand, $Z$ is $(S_1)$ by", "Divisors, Lemma \\ref{divisors-lemma-normal-effective-Cartier-divisor-S1}.", "Thus the scheme $Z$ has no embedded associated points and", "we conclude that $\\overline{f} = 0$ (Divisors, Lemmas", "\\ref{divisors-lemma-S1-no-embedded} and", "\\ref{divisors-lemma-restriction-injective-open-contains-weakly-ass}).", "Hence $f$ is a global section of $\\mathcal{I}$", "which generates $\\mathcal{I}_{\\xi_i}$ by construction.", "Thus the image $s_i$ of $f$ in $\\Gamma(C_i, \\mathcal{I}|_{C_i})$ is nonzero.", "However, our choice of $f$ guarantees that $s_i$ has a zero at $x_i$.", "Hence the degree of $\\mathcal{I}|_{C_i}$ is $> 0$ by", "Varieties, Lemma \\ref{varieties-lemma-check-invertible-sheaf-trivial}." ], "refs": [ "resolve-lemma-nice-meromorphic-function", "divisors-lemma-invertible-sheaf-sum-effective-Cartier-divisors", "divisors-lemma-conormal-effective-Cartier-divisor", "varieties-lemma-degree-tensor-product", "divisors-lemma-normal-effective-Cartier-divisor-S1", "divisors-lemma-S1-no-embedded", "divisors-lemma-restriction-injective-open-contains-weakly-ass", "varieties-lemma-check-invertible-sheaf-trivial" ], "ref_ids": [ 11657, 7940, 7938, 11109, 7950, 7867, 7876, 11114 ] } ], "ref_ids": [] }, { "id": 11659, "type": "theorem", "label": "resolve-lemma-H1-injective", "categories": [ "resolve" ], "title": "resolve-lemma-H1-injective", "contents": [ "In Situation \\ref{situation-vanishing} assume $X$ is normal", "and $A$ Nagata. The map", "$$", "H^1(X, \\mathcal{O}_X) \\longrightarrow H^1(f^{-1}(U), \\mathcal{O}_X)", "$$", "is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $0 \\to \\mathcal{O}_X \\to \\mathcal{E} \\to \\mathcal{O}_X \\to 0$ be the", "extension corresponding to a nontrivial element $\\xi$ of", "$H^1(X, \\mathcal{O}_X)$", "(Cohomology, Lemma \\ref{cohomology-lemma-h1-extensions}).", "Let $\\pi : P = \\mathbf{P}(\\mathcal{E}) \\to X$", "be the projective bundle associated to $\\mathcal{E}$.", "The surjection $\\mathcal{E} \\to \\mathcal{O}_X$", "defines a section $\\sigma : X \\to P$ whose conormal sheaf is", "isomorphic to $\\mathcal{O}_X$ (Divisors, Lemma", "\\ref{divisors-lemma-conormal-sheaf-section-projective-bundle}).", "If the restriction of $\\xi$ to $f^{-1}(U)$ is trivial, then we get", "a map $\\mathcal{E}|_{f^{-1}(U)} \\to \\mathcal{O}_{f^{-1}(U)}$ splitting", "the injection $\\mathcal{O}_X \\to \\mathcal{E}$. This defines a second", "section $\\sigma' : f^{-1}(U) \\to P$ disjoint from $\\sigma$. Since $\\xi$", "is nontrivial we conclude that $\\sigma'$ cannot extend to all of $X$", "and be disjoint from $\\sigma$. Let $X' \\subset P$ be the", "scheme theoretic image of $\\sigma'$ (Morphisms,", "Definition \\ref{morphisms-definition-scheme-theoretic-image}).", "Picture", "$$", "\\xymatrix{", "& X' \\ar[rd]_g \\ar[r] & P \\ar[d]_\\pi \\\\", "f^{-1}(U) \\ar[ru]_{\\sigma'} \\ar[rr] & & X \\ar@/_/[u]_\\sigma", "}", "$$", "The morphism $P \\setminus \\sigma(X) \\to X$ is affine.", "If $X' \\cap \\sigma(X) = \\emptyset$, then $X' \\to X$ is both affine", "and proper, hence finite", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}),", "hence an isomorphism (as $X$ is normal, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-birational-over-normal}).", "This is impossible as mentioned above.", "\\medskip\\noindent", "Let $X^\\nu$ be the normalization of $X'$.", "Since $A$ is Nagata, we see that $X^\\nu \\to X'$ is finite", "(Morphisms, Lemmas \\ref{morphisms-lemma-nagata-normalization} and", "\\ref{morphisms-lemma-ubiquity-nagata}). Let $Z \\subset X^\\nu$ be the", "pullback of the effective Cartier divisor $\\sigma(X) \\subset P$.", "By the above we see that $Z$ is not empty and is contained", "in the closed fibre of $X^\\nu \\to S$.", "Since $P \\to X$ is smooth, we see that $\\sigma(X)$ is an effective", "Cartier divisor", "(Divisors, Lemma \\ref{divisors-lemma-section-smooth-regular-immersion}).", "Hence $Z \\subset X^\\nu$ is an effective Cartier divisor too.", "Since the conormal sheaf of $\\sigma(X)$ in $P$ is $\\mathcal{O}_X$, the", "conormal sheaf of $Z$ in $X^\\nu$ (which is a priori invertible)", "is $\\mathcal{O}_Z$ by", "Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial-flat}.", "This is impossible by", "Lemma \\ref{lemma-nontrivial-normal-bundle}", "and the proof is complete." ], "refs": [ "cohomology-lemma-h1-extensions", "morphisms-definition-scheme-theoretic-image", "morphisms-lemma-finite-proper", "morphisms-lemma-finite-birational-over-normal", "morphisms-lemma-nagata-normalization", "morphisms-lemma-ubiquity-nagata", "morphisms-lemma-conormal-functorial-flat", "resolve-lemma-nontrivial-normal-bundle" ], "ref_ids": [ 2035, 5539, 5445, 5518, 5520, 5219, 5305, 11658 ] } ], "ref_ids": [] }, { "id": 11660, "type": "theorem", "label": "resolve-lemma-R1-injective", "categories": [ "resolve" ], "title": "resolve-lemma-R1-injective", "contents": [ "In Situation \\ref{situation-vanishing} assume $X$ is normal and $A$ Nagata.", "Then", "$$", "\\Hom_{D(A)}(\\kappa[-1], Rf_*\\mathcal{O}_X)", "$$", "is zero. This uses $D(A) = D_\\QCoh(\\mathcal{O}_S)$ to think of", "$Rf_*\\mathcal{O}_X$ as an object of $D(A)$." ], "refs": [], "proofs": [ { "contents": [ "By adjointness of $Rf_*$ and $Lf^*$ such a map is the same thing", "as a map $\\alpha : Lf^*\\kappa[-1] \\to \\mathcal{O}_X$. Note that", "$$", "H^i(Lf^*\\kappa[-1]) =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & i > 1 \\\\", "\\mathcal{O}_{X_s} & \\text{if} & i = 1 \\\\", "\\text{some }\\mathcal{O}_{X_s}\\text{-module} & \\text{if} & i \\leq 0", "\\end{matrix}", "\\right.", "$$", "Since $\\Hom(H^0(Lf^*\\kappa[-1]), \\mathcal{O}_X) = 0$ as $\\mathcal{O}_X$", "is torsion free, the spectral sequence for $\\Ext$", "(Cohomology on Sites, Example", "\\ref{sites-cohomology-example-hom-complex-into-sheaf})", "implies that", "$\\Hom_{D(\\mathcal{O}_X)}(Lf^*\\kappa[-1], \\mathcal{O}_X)$ is equal to", "$\\Ext^1_{\\mathcal{O}_X}(\\mathcal{O}_{X_s}, \\mathcal{O}_X)$.", "We conclude that", "$\\alpha : Lf^*\\kappa[-1] \\to \\mathcal{O}_X$ is given by an extension", "$$", "0 \\to \\mathcal{O}_X \\to \\mathcal{E} \\to \\mathcal{O}_{X_s} \\to 0", "$$", "By Lemma \\ref{lemma-H1-injective} the pullback of this extension", "via the surjection $\\mathcal{O}_X \\to \\mathcal{O}_{X_s}$ is zero", "(since this pullback is clearly split over $f^{-1}(U)$).", "Thus $1 \\in \\mathcal{O}_{X_s}$ lifts to a global section $s$ of", "$\\mathcal{E}$. Multiplying $s$ by the ideal sheaf $\\mathcal{I}$", "of $X_s$ we obtain an $\\mathcal{O}_X$-module map", "$c_s : \\mathcal{I} \\to \\mathcal{O}_X$. Applying $f_*$ we obtain", "an $A$-linear map $f_*c_s : \\mathfrak m \\to A$. Since $A$ is", "a Noetherian normal local domain this map is given by multiplication", "by an element $a \\in A$. Changing $s$ into $s - a$ we find that", "$s$ is annihilated by $\\mathcal{I}$ and the extension is trivial", "as desired." ], "refs": [ "resolve-lemma-H1-injective" ], "ref_ids": [ 11659 ] } ], "ref_ids": [] }, { "id": 11661, "type": "theorem", "label": "resolve-lemma-exact-sequence", "categories": [ "resolve" ], "title": "resolve-lemma-exact-sequence", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian normal local domain", "of dimension $2$. Consider a commutative diagram", "$$", "\\xymatrix{", "X' \\ar[rd]_{f'} \\ar[rr]_g & & X \\ar[ld]^f \\\\", "& \\Spec(A)", "}", "$$", "where $f$ and $f'$ are modifications as in Situation \\ref{situation-vanishing}", "and $X$ normal. Then we have a short exact sequence", "$$", "0 \\to H^1(X, \\mathcal{O}_X) \\to H^1(X', \\mathcal{O}_{X'}) \\to", "H^0(X, R^1g_*\\mathcal{O}_{X'}) \\to 0", "$$", "Also $\\dim(\\text{Supp}(R^1g_*\\mathcal{O}_{X'})) = 0$", "and $R^1g_*\\mathcal{O}_{X'}$ is generated by global sections." ], "refs": [], "proofs": [ { "contents": [ "We will use the observations made following Situation \\ref{situation-vanishing}", "without further mention. As $X$ is normal and $g$ is dominant and", "birational, we have $g_*\\mathcal{O}_{X'} = \\mathcal{O}_X$, see for", "example More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-geometrically-connected-fibres-towards-normal}.", "Since the fibres of $g$ have dimension $\\leq 1$, we have", "$R^pg_*\\mathcal{O}_{X'} = 0$ for $p > 1$, see for example", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-higher-direct-images-zero-above-dimension-fibre}.", "The support of $R^1g_*\\mathcal{O}_{X'}$ is contained in the set of points of", "$X$ where the fibres of $g'$ have dimension $\\geq 1$. Thus", "it is contained in the set of images of those", "irreducible components $C' \\subset X'_s$ which map to points of $X_s$", "which is a finite set of closed points", "(recall that $X'_s \\to X_s$ is a morphism of proper $1$-dimensional", "schemes over $\\kappa$). Then $R^1g_*\\mathcal{O}_{X'}$ is globally", "generated by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-support-dimension-0}.", "Using the morphism $f : X \\to S$ and the references above we find that", "$H^p(X, \\mathcal{F}) = 0$ for $p > 1$ for any coherent $\\mathcal{O}_X$-module", "$\\mathcal{F}$. Hence the short exact sequence of the lemma is a consequence", "of the Leray spectral sequence for $g$ and $\\mathcal{O}_{X'}$, see", "Cohomology, Lemma \\ref{cohomology-lemma-Leray}." ], "refs": [ "more-morphisms-lemma-geometrically-connected-fibres-towards-normal", "coherent-lemma-higher-direct-images-zero-above-dimension-fibre", "coherent-lemma-coherent-support-dimension-0", "cohomology-lemma-Leray" ], "ref_ids": [ 13945, 3364, 3317, 2070 ] } ], "ref_ids": [] }, { "id": 11662, "type": "theorem", "label": "resolve-lemma-bound-a-torsion", "categories": [ "resolve" ], "title": "resolve-lemma-bound-a-torsion", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain", "of dimension $2$. Let $a \\in A$ be nonzero. There exists an integer $N$ such", "that for every modification $f : X \\to \\Spec(A)$ with $X$ normal the", "$A$-module", "$$", "M_{X, a} = \\Coker(A \\longrightarrow H^0(Z, \\mathcal{O}_Z))", "$$", "where $Z \\subset X$ is cut out by $a$ has length bounded by $N$." ], "refs": [], "proofs": [ { "contents": [ "By the short exact sequence", "$", "0 \\to \\mathcal{O}_X \\xrightarrow{a} \\mathcal{O}_X \\to \\mathcal{O}_Z \\to 0", "$", "we see that", "\\begin{equation}", "\\label{equation-a-torsion}", "M_{X, a} = H^1(X, \\mathcal{O}_X)[a]", "\\end{equation}", "Here $N[a] = \\{n \\in N \\mid an = 0\\}$ for an $A$-module $N$. Thus", "if $a$ divides $b$, then $M_{X, a} \\subset M_{X, b}$.", "Suppose that for some $c \\in A$ the modules $M_{X, c}$", "have bounded length. Then for every $X$ we have an exact sequence", "$$", "0 \\to M_{X, c} \\to M_{X, c^2} \\to M_{X, c}", "$$", "where the second arrow is given by multiplication by $c$. Hence we see that", "$M_{X, c^2}$ has bounded length as well. Thus it suffices to find a $c \\in A$", "for which the lemma is true such that $a$ divides $c^n$ for some $n > 0$.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-divides-radical}", "we may assume $A/(a)$ is a reduced ring.", "\\medskip\\noindent", "Assume that $A/(a)$ is reduced. Let $A/(a) \\subset B$ be the normalization", "of $A/(a)$ in its quotient ring. Because $A$ is Nagata, we see that", "$\\Coker(A \\to B)$ is finite. We claim the length of this finite", "module is a bound. To see this, consider $f : X \\to \\Spec(A)$ as in the lemma", "and let $Z' \\subset Z$ be the scheme theoretic closure of $Z \\cap f^{-1}(U)$.", "Then $Z' \\to \\Spec(A/(a))$ is finite for example by Varieties, Lemma", "\\ref{varieties-lemma-finite-in-codim-1}.", "Hence $Z' = \\Spec(B')$ with $A/(a) \\subset B' \\subset B$.", "On the other hand, we claim the map", "$$", "H^0(Z, \\mathcal{O}_Z) \\to H^0(Z', \\mathcal{O}_{Z'})", "$$", "is injective. Namely, if $s \\in H^0(Z, \\mathcal{O}_Z)$", "is in the kernel, then", "the restriction of $s$ to $f^{-1}(U) \\cap Z$ is zero.", "Hence the image of $s$ in $H^1(X, \\mathcal{O}_X)$ vanishes in", "$H^1(f^{-1}(U), \\mathcal{O}_X)$. By Lemma \\ref{lemma-H1-injective}", "we see that $s$ comes from an element $\\tilde s$ of $A$. But by", "assumption $\\tilde s$ maps to zero in $B'$ which implies that $s = 0$.", "Putting everything together we see that", "$M_{X, a}$ is a subquotient of $B'/A$, namely not every element", "of $B'$ extends to a global section of $\\mathcal{O}_Z$, but in", "any case the length of $M_{X, a}$ is bounded by the length of $B/A$." ], "refs": [ "more-algebra-lemma-divides-radical", "varieties-lemma-finite-in-codim-1", "resolve-lemma-H1-injective" ], "ref_ids": [ 10568, 10978, 11659 ] } ], "ref_ids": [] }, { "id": 11663, "type": "theorem", "label": "resolve-lemma-rational-propagates", "categories": [ "resolve" ], "title": "resolve-lemma-rational-propagates", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain of", "dimension $2$ which defines a rational singularity. Let $A \\subset B$", "be a local extension of domains with the same fraction field", "which is essentially of finite type such", "that $\\dim(B) = 2$ and $B$ normal. Then $B$ defines a rational singularity." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite type $A$-algebra $C$ such that $B = C_\\mathfrak q$", "for some prime $\\mathfrak q \\subset C$. After replacing", "$C$ by the image of $C$ in $B$ we may assume that $C$ is a domain", "with fraction field equal to the fraction field of $A$.", "Then we can choose a closed immersion $\\Spec(C) \\to \\mathbf{A}^n_A$", "and take the closure in $\\mathbf{P}^n_A$ to conclude that $B$", "is isomorphic to $\\mathcal{O}_{X, x}$ for some closed point $x \\in X$", "of a projective modification $X \\to \\Spec(A)$.", "(Morphisms, Lemma \\ref{morphisms-lemma-dimension-formula},", "shows that $\\kappa(x)$ is finite over $\\kappa$ and then", "Morphisms, Lemma", "\\ref{morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre}", "shows that $x$ is a closed point.)", "Let $\\nu : X^\\nu \\to X$ be the normalization.", "Since $A$ is Nagata the morphism $\\nu$ is finite (Morphisms, Lemma", "\\ref{morphisms-lemma-nagata-normalization}).", "Thus $X^\\nu$ is projective over $A$ by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-category-projective}.", "Since $B = \\mathcal{O}_{X, x}$ is normal, we see that", "$\\mathcal{O}_{X, x} = (\\nu_*\\mathcal{O}_{X^\\nu})_x$.", "Hence there is a unique point $x^\\nu \\in X^\\nu$ lying over $x$", "and $\\mathcal{O}_{X^\\nu, x^\\nu} = \\mathcal{O}_{X, x}$.", "Thus we may assume $X$ is normal and projective over $A$.", "Let $Y \\to \\Spec(\\mathcal{O}_{X, x}) = \\Spec(B)$", "be a modification with $Y$ normal.", "We have to show that $H^1(Y, \\mathcal{O}_Y) = 0$. By", "Limits, Lemma \\ref{limits-lemma-modifications}", "we can find a morphism of schemes $g : X' \\to X$ which is an isomorphism", "over $X \\setminus \\{x\\}$ such that $X' \\times_X \\Spec(\\mathcal{O}_{X, x})$", "is isomorphic to $Y$. Then $g$ is a modification as it is proper by", "Limits, Lemma \\ref{limits-lemma-modifications-properties}.", "The local ring of $X'$ at a point of $x'$ is either isomorphic", "to the local ring of $X$ at $g(x')$ if $g(x') \\not = x$ and", "if $g(x') = x$, then the local ring of $X'$ at $x'$ is isomorphic", "to the local ring of $Y$ at the corresponding point. Hence we see", "that $X'$ is normal as both $X$ and $Y$ are normal.", "Thus $H^1(X', \\mathcal{O}_{X'}) = 0$ by our assumption on $A$.", "By Lemma \\ref{lemma-exact-sequence} we have $R^1g_*\\mathcal{O}_{X'} = 0$.", "Clearly this means that $H^1(Y, \\mathcal{O}_Y) = 0$ as desired." ], "refs": [ "morphisms-lemma-dimension-formula", "morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre", "morphisms-lemma-nagata-normalization", "more-morphisms-lemma-category-projective", "limits-lemma-modifications", "limits-lemma-modifications-properties", "resolve-lemma-exact-sequence" ], "ref_ids": [ 5493, 5222, 5520, 13932, 15117, 15118, 11661 ] } ], "ref_ids": [] }, { "id": 11664, "type": "theorem", "label": "resolve-lemma-reduce-to-rational", "categories": [ "resolve" ], "title": "resolve-lemma-reduce-to-rational", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain", "of dimension $2$. If reduction to rational singularities is possible", "for $A$, then there exists a finite sequence of normalized blowups", "$$", "X = X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = \\Spec(A)", "$$", "in closed points such that for any closed point $x \\in X$", "the local ring $\\mathcal{O}_{X, x}$ defines a rational singularity.", "In particular $X \\to \\Spec(A)$ is a modification and $X$", "is a normal scheme projective over $A$." ], "refs": [], "proofs": [ { "contents": [ "We choose a modification $X \\to \\Spec(A)$ with $X$ normal", "which maximizes the length of $H^1(X, \\mathcal{O}_X)$.", "By Lemma \\ref{lemma-exact-sequence}", "for any further modification $g : X' \\to X$ with $X'$ normal", "we have $R^1g_*\\mathcal{O}_{X'} = 0$ and", "$H^1(X, \\mathcal{O}_X) = H^1(X', \\mathcal{O}_{X'})$.", "\\medskip\\noindent", "Let $x \\in X$ be a closed point. We will show that $\\mathcal{O}_{X, x}$", "defines a rational singularity. Let $Y \\to \\Spec(\\mathcal{O}_{X, x})$", "be a modification with $Y$ normal. We have to show that", "$H^1(Y, \\mathcal{O}_Y) = 0$. By", "Limits, Lemma \\ref{limits-lemma-modifications}", "we can find a morphism of schemes $g : X' \\to X$ which is an isomorphism", "over $X \\setminus \\{x\\}$ such that $X' \\times_X \\Spec(\\mathcal{O}_{X, x})$", "is isomorphic to $Y$. Then $g$ is a modification as it is proper by", "Limits, Lemma \\ref{limits-lemma-modifications-properties}.", "The local ring of $X'$ at a point of $x'$ is either isomorphic", "to the local ring of $X$ at $g(x')$ if $g(x') \\not = x$ and", "if $g(x') = x$, then the local ring of $X'$ at $x'$ is isomorphic", "to the local ring of $Y$ at the corresponding point. Hence we see", "that $X'$ is normal as both $X$ and $Y$ are normal. By maximality", "we have $R^1g_*\\mathcal{O}_{X'} = 0$ (see first paragraph). Clearly", "this means that $H^1(Y, \\mathcal{O}_Y) = 0$ as desired.", "\\medskip\\noindent", "The conclusion is that we've found one normal modification $X$", "of $\\Spec(A)$ such that the local rings of $X$ at closed points all define", "rational singularities. Then we choose a sequence of normalized", "blowups $X_n \\to \\ldots \\to X_1 \\to \\Spec(A)$ such that $X_n$", "dominates $X$, see Lemma \\ref{lemma-dominate-by-normalized-blowing-up}.", "For a closed point $x' \\in X_n$ mapping to $x \\in X$ we can apply", "Lemma \\ref{lemma-rational-propagates} to the ring map", "$\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X_n, x'}$", "to see that $\\mathcal{O}_{X_n, x'}$ defines a rational singularity." ], "refs": [ "resolve-lemma-exact-sequence", "limits-lemma-modifications", "limits-lemma-modifications-properties", "resolve-lemma-dominate-by-normalized-blowing-up", "resolve-lemma-rational-propagates" ], "ref_ids": [ 11661, 15117, 15118, 11649, 11663 ] } ], "ref_ids": [] }, { "id": 11665, "type": "theorem", "label": "resolve-lemma-go-up-separable", "categories": [ "resolve" ], "title": "resolve-lemma-go-up-separable", "contents": [ "Let $A \\to B$ be a finite injective local ring map of local normal", "Nagata domains of dimension $2$. Assume that the induced extension of", "fraction fields is separable. If reduction to rational singularities", "is possible for $A$ then it is possible for $B$." ], "refs": [], "proofs": [ { "contents": [ "Let $n$ be the degree of the fraction field extension $K \\subset L$.", "Let $\\text{Trace}_{L/K} : L \\to K$ be the trace. Since the extension is finite", "separable the trace pairing $(h, g) \\mapsto \\text{Trace}_{L/K}(fg)$", "is a nondegenerate bilinear form on $L$ over $K$. See", "Fields, Lemma \\ref{fields-lemma-separable-trace-pairing}.", "Pick $b_1, \\ldots, b_n \\in B$ which form a basis of $L$ over $K$.", "By the above $d = \\det(\\text{Trace}_{L/K}(b_ib_j)) \\in A$ is nonzero.", "\\medskip\\noindent", "Let $Y \\to \\Spec(B)$ be a modification with $Y$ normal. We can find", "a $U$-admissible blowup $X'$ of $\\Spec(A)$ such that the strict transform", "$Y'$ of $Y$ is finite over $X'$, see More on Flatness, Lemma", "\\ref{flat-lemma-finite-after-blowing-up}. Picture", "$$", "\\xymatrix{", "Y' \\ar[d] \\ar[r] & Y \\ar[r] & \\Spec(B) \\ar[d] \\\\", "X' \\ar[rr] & & \\Spec(A)", "}", "$$", "After replacing $X'$ and $Y'$ by their normalizations we may assume that", "$X'$ and $Y'$ are normal modifications of $\\Spec(A)$ and $\\Spec(B)$.", "In this way we reduce to the case where there exists a commutative diagram", "$$", "\\xymatrix{", "Y \\ar[d]_\\pi \\ar[r]_-g & \\Spec(B) \\ar[d] \\\\", "X \\ar[r]^-f & \\Spec(A)", "}", "$$", "with $X$ and $Y$ normal modifications of $\\Spec(A)$ and $\\Spec(B)$ and", "$\\pi$ finite.", "\\medskip\\noindent", "The trace map on $L$ over $K$ extends to a map of $\\mathcal{O}_X$-modules", "$\\text{Trace} : \\pi_*\\mathcal{O}_Y \\to \\mathcal{O}_X$. Consider the map", "$$", "\\Phi : \\pi_*\\mathcal{O}_Y \\longrightarrow \\mathcal{O}_X^{\\oplus n},\\quad", "s \\longmapsto (\\text{Trace}(b_1s), \\ldots, \\text{Trace}(b_ns))", "$$", "This map is injective (because it is injective in the generic point)", "and there is a map", "$$", "\\mathcal{O}_X^{\\oplus n} \\longrightarrow \\pi_*\\mathcal{O}_Y,\\quad", "(s_1, \\ldots, s_n) \\longmapsto \\sum b_i s_i", "$$", "whose composition with $\\Phi$ has matrix $\\text{Trace}(b_ib_j)$.", "Hence the cokernel of $\\Phi$ is annihilated by $d$. Thus we see that", "we have an exact sequence", "$$", "H^0(X, \\Coker(\\Phi)) \\to H^1(Y, \\mathcal{O}_Y) \\to", "H^1(X, \\mathcal{O}_X)^{\\oplus n}", "$$", "Since the right hand side is bounded by assumption, it suffices to show", "that the $d$-torsion in $H^1(Y, \\mathcal{O}_Y)$ is bounded.", "This is the content of Lemma \\ref{lemma-bound-a-torsion} and", "(\\ref{equation-a-torsion})." ], "refs": [ "fields-lemma-separable-trace-pairing", "flat-lemma-finite-after-blowing-up", "resolve-lemma-bound-a-torsion" ], "ref_ids": [ 4503, 6124, 11662 ] } ], "ref_ids": [] }, { "id": 11666, "type": "theorem", "label": "resolve-lemma-regular-rational", "categories": [ "resolve" ], "title": "resolve-lemma-regular-rational", "contents": [ "Let $A$ be a Nagata regular local ring of dimension $2$. Then $A$ defines", "a rational singularity." ], "refs": [], "proofs": [ { "contents": [ "(The assumption that $A$ be Nagata is not necessary for this proof,", "but we've only defined the notion of rational singularity in the", "case of Nagata $2$-dimensional normal local domains.)", "Let $X \\to \\Spec(A)$ be a modification with $X$ normal. By", "Lemma \\ref{lemma-dominate-by-blowing-up-in-points}", "we can dominate $X$ by a scheme $X_n$ which is the last in a sequence", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = \\Spec(A)", "$$", "of blowing ups in closed points. By Lemma \\ref{lemma-blowup-regular}", "the schemes $X_i$ are regular, in particular", "normal (Algebra, Lemma \\ref{algebra-lemma-regular-normal}).", "By Lemma \\ref{lemma-exact-sequence} we have", "$H^1(X, \\mathcal{O}_X) \\subset H^1(X_n, \\mathcal{O}_{X_n})$.", "Thus it suffices to prove $H^1(X_n, \\mathcal{O}_{X_n}) = 0$.", "Using Lemma \\ref{lemma-exact-sequence} again, we", "see that it suffices to prove $R^1(X_i \\to X_{i - 1})_*\\mathcal{O}_{X_i} = 0$", "for $i = 1, \\ldots, n$. This follows from", "Lemma \\ref{lemma-cohomology-of-blowup}." ], "refs": [ "resolve-lemma-dominate-by-blowing-up-in-points", "resolve-lemma-blowup-regular", "algebra-lemma-regular-normal", "resolve-lemma-exact-sequence", "resolve-lemma-exact-sequence", "resolve-lemma-cohomology-of-blowup" ], "ref_ids": [ 11646, 11640, 1312, 11661, 11661, 11642 ] } ], "ref_ids": [] }, { "id": 11667, "type": "theorem", "label": "resolve-lemma-bound-dualizing-implies-bound", "categories": [ "resolve" ], "title": "resolve-lemma-bound-dualizing-implies-bound", "contents": [ "Let $A$ be a local normal Nagata domain of dimension $2$ which has a", "dualizing complex $\\omega_A^\\bullet$. If there exists a nonzero $d \\in A$", "such that for all normal modifications $X \\to \\Spec(A)$ the cokernel of the", "trace map", "$$", "\\Gamma(X, \\omega_X) \\to \\omega_A", "$$", "is annihilated by $d$, then reduction to rational singularities", "is possible for $A$." ], "refs": [], "proofs": [ { "contents": [ "For $X \\to \\Spec(A)$ as in the statement we have to bound", "$H^1(X, \\mathcal{O}_X)$. Let $\\omega_X$ be the dualizing module", "of $X$ as in the statement of Grauert-Riemenschneider", "(Proposition \\ref{proposition-Grauert-Riemenschneider}).", "The trace map is the map $Rf_*\\omega_X \\to \\omega_A$ described", "in Duality for Schemes, Section \\ref{duality-section-trace}.", "By Grauert-Riemenschneider we have $Rf_*\\omega_X = f_*\\omega_X$", "thus the trace map indeed produces a map $\\Gamma(X, \\omega_X) \\to \\omega_A$.", "By duality we have $Rf_*\\omega_X = R\\Hom_A(Rf_*\\mathcal{O}_X, \\omega_A)$", "(this uses that $\\omega_X[2]$ is the dualizing complex on $X$", "normalized relative to $\\omega_A[2]$,", "see Duality for Schemes, Lemma \\ref{duality-lemma-duality-bootstrap}", "or more directly Section \\ref{duality-section-duality} or even more directly", "Example \\ref{duality-example-iso-on-RSheafHom-noetherian}).", "The distinguished triangle", "$$", "A \\to Rf_*\\mathcal{O}_X \\to R^1f_*\\mathcal{O}_X[-1] \\to A[1]", "$$", "is transformed by $R\\Hom_A(-, \\omega_A)$ into the short exact sequence", "$$", "0 \\to f_*\\omega_X \\to \\omega_A \\to", "\\Ext_A^2(R^1f_*\\mathcal{O}_X, \\omega_A) \\to 0", "$$", "(and $\\Ext_A^i(R^1f_*\\mathcal{O}_X, \\omega_A) = 0$ for $i \\not = 2$;", "this will follow from the discussion below as well).", "Since $R^1f_*\\mathcal{O}_X$ is supported in $\\{\\mathfrak m\\}$, the", "local duality theorem tells us that", "$$", "\\Ext_A^2(R^1f_*\\mathcal{O}_X, \\omega_A) =", "\\Ext_A^0(R^1f_*\\mathcal{O}_X, \\omega_A[2]) =", "\\Hom_A(R^1f_*\\mathcal{O}_X, E)", "$$", "is the Matlis dual of $R^1f_*\\mathcal{O}_X$ (and the other", "ext groups are zero), see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-special-case-local-duality}.", "By the equivalence of categories inherent in Matlis duality", "(Dualizing Complexes, Proposition \\ref{dualizing-proposition-matlis}),", "if $R^1f_*\\mathcal{O}_X$ is not annihilated by $d$,", "then neither is the $\\Ext^2$ above. Hence we see that", "$H^1(X, \\mathcal{O}_X)$ is annihilated by $d$. Thus the required", "boundedness follows from Lemma \\ref{lemma-bound-a-torsion} and", "(\\ref{equation-a-torsion})." ], "refs": [ "resolve-proposition-Grauert-Riemenschneider", "duality-lemma-duality-bootstrap", "dualizing-lemma-special-case-local-duality", "dualizing-proposition-matlis", "resolve-lemma-bound-a-torsion" ], "ref_ids": [ 11711, 13575, 2873, 2924, 11662 ] } ], "ref_ids": [] }, { "id": 11668, "type": "theorem", "label": "resolve-lemma-compare-differentials-dualizing", "categories": [ "resolve" ], "title": "resolve-lemma-compare-differentials-dualizing", "contents": [ "Let $p$ be a prime number.", "Let $A$ be a regular local ring of dimension $2$ and characteristic $p$.", "Let $A_0 \\subset A$ be a subring such that $\\Omega_{A/A_0}$ is free", "of rank $r < \\infty$. Set $\\omega_A = \\Omega^r_{A/A_0}$. If $X \\to \\Spec(A)$", "is the result of a sequence of blowups in closed points, then", "there exists a map", "$$", "\\varphi_X : (\\Omega^r_{X/\\Spec(A_0)})^{**} \\longrightarrow \\omega_X", "$$", "extending the given identification in the generic point." ], "refs": [], "proofs": [ { "contents": [ "Observe that $A$ is Gorenstein (Dualizing Complexes,", "Lemma \\ref{dualizing-lemma-regular-gorenstein})", "and hence the invertible module $\\omega_A$ does indeed serve", "as a dualizing module. Moreover, any $X$ as in the lemma", "has an invertible dualizing module $\\omega_X$ as $X$ is regular", "(hence Gorenstein) and proper over $A$, see", "Remark \\ref{remark-dualizing-setup} and", "Lemma \\ref{lemma-blowup-regular}.", "Suppose we have constructed the map", "$\\varphi_X : (\\Omega^r_{X/A_0})^{**} \\to \\omega_X$", "and suppose that $b : X' \\to X$ is a blowup in a closed point.", "Set $\\Omega^r_X = (\\Omega^r_{X/A_0})^{**}$ and", "$\\Omega^r_{X'} = (\\Omega^r_{X'/A_0})^{**}$. Since $\\omega_{X'} = b^!(\\omega_X)$", "a map $\\Omega^r_{X'} \\to \\omega_{X'}$ is the same thing as a map", "$Rb_*(\\Omega^r_{X'}) \\to \\omega_X$. See discussion in", "Remark \\ref{remark-dualizing-setup} and", "Duality for Schemes, Section \\ref{duality-section-duality}.", "Thus in turn it suffices to produce a map", "$$", "Rb_*(\\Omega^r_{X'}) \\longrightarrow \\Omega^r_X", "$$", "The sheaves $\\Omega^r_{X'}$ and $\\Omega^r_X$ are invertible, see", "Divisors, Lemma \\ref{divisors-lemma-reflexive-over-regular-dim-2}.", "Consider the exact sequence", "$$", "b^*\\Omega_{X/A_0} \\to \\Omega_{X'/A_0} \\to \\Omega_{X'/X} \\to 0", "$$", "A local calculation shows that $\\Omega_{X'/X}$ is isomorphic", "to an invertible module on the exceptional divisor $E$, see", "Lemma \\ref{lemma-differentials-of-blowup}. It follows that", "either", "$$", "\\Omega^r_{X'} \\cong (b^*\\Omega^r_X)(E)", "\\quad\\text{or}\\quad", "\\Omega^r_{X'} \\cong b^*\\Omega^r_X", "$$", "see Divisors, Lemma \\ref{divisors-lemma-wedge-product-ses}.", "(The second possibility never happens in characteristic zero, but", "can happen in characteristic $p$.) In both cases we see that", "$R^1b_*(\\Omega^r_{X'}) = 0$ and $b_*(\\Omega^r_{X'}) = \\Omega^r_X$ by", "Lemma \\ref{lemma-cohomology-of-blowup}." ], "refs": [ "dualizing-lemma-regular-gorenstein", "resolve-remark-dualizing-setup", "resolve-lemma-blowup-regular", "resolve-remark-dualizing-setup", "divisors-lemma-reflexive-over-regular-dim-2", "resolve-lemma-differentials-of-blowup", "divisors-lemma-wedge-product-ses", "resolve-lemma-cohomology-of-blowup" ], "ref_ids": [ 2880, 11717, 11640, 11717, 7926, 11644, 7957, 11642 ] } ], "ref_ids": [] }, { "id": 11669, "type": "theorem", "label": "resolve-lemma-go-up-degree-p", "categories": [ "resolve" ], "title": "resolve-lemma-go-up-degree-p", "contents": [ "Let $p$ be a prime number. Let $A$ be a complete regular local ring of", "dimension $2$ and characteristic $p$. Let $L/K$ be a degree $p$ inseparable", "extension of the fraction field $K$ of $A$. Let $B \\subset L$ be the integral", "closure of $A$. Then reduction to rational singularities is possible for $B$." ], "refs": [], "proofs": [ { "contents": [ "We have $A = k[[x, y]]$. Write $L = K[x]/(x^p - f)$ for some $f \\in A$", "and denote $g \\in B$ the congruence class of $x$, i.e., the element such", "that $g^p = f$. By Algebra, Lemma \\ref{algebra-lemma-derivative-zero-pth-power}", "we see that $\\text{d}f$ is nonzero in $\\Omega_{K/\\mathbf{F}_p}$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-power-series-ring-subfields}", "there exists a subfield $k^p \\subset k' \\subset k$ with", "$p^e = [k : k'] < \\infty$ such that $\\text{d}f$ is nonzero", "in $\\Omega_{K/K_0}$ where $K_0$ is the fraction field of", "$A_0 = k'[[x^p, y^p]] \\subset A$. Then", "$$", "\\Omega_{A/A_0} =", "A \\otimes_k \\Omega_{k/k'} \\oplus A \\text{d}x \\oplus A \\text{d}y", "$$", "is finite free of rank $e + 2$. Set $\\omega_A = \\Omega^{e + 2}_{A/A_0}$.", "Consider the canonical map", "$$", "\\text{Tr} :", "\\Omega^{e + 2}_{B/A_0}", "\\longrightarrow", "\\Omega^{e + 2}_{A/A_0} = \\omega_A", "$$", "of Lemma \\ref{lemma-trace-extends}. By duality this determines a map", "$$", "c : \\Omega^{e + 2}_{B/A_0} \\to \\omega_B = \\Hom_A(B, \\omega_A)", "$$", "Claim: the cokernel of $c$ is annihilated by a nonzero element of $B$.", "\\medskip\\noindent", "Since $\\text{d}f$ is nonzero in $\\Omega_{A/A_0}$ we can find", "$\\eta_1, \\ldots, \\eta_{e + 1} \\in \\Omega_{A/A_0}$ such that", "$\\theta = \\eta_1 \\wedge \\ldots \\wedge \\eta_{e + 1} \\wedge \\text{d}f$ is", "nonzero in $\\omega_A = \\Omega^{e + 2}_{A/A_0}$. To prove the claim we", "will construct elements $\\omega_i$ of $\\Omega^{e + 2}_{B/A_0}$,", "$i = 0, \\ldots, p - 1$ which are mapped to", "$\\varphi_i \\in \\omega_B = \\Hom_A(B, \\omega_A)$", "with $\\varphi_i(g^j) = \\delta_{ij}\\theta$ for $j = 0, \\ldots, p - 1$.", "Since $\\{1, g, \\ldots, g^{p - 1}\\}$ is a basis for $L/K$ this", "proves the claim. We set", "$\\eta = \\eta_1 \\wedge \\ldots \\wedge \\eta_{e + 1}$", "so that $\\theta = \\eta \\wedge \\text{d}f$.", "Set $\\omega_i = \\eta \\wedge g^{p - 1 - i}\\text{d}g$. Then", "by construction we have", "$$", "\\varphi_i(g^j) = \\text{Tr}(g^j \\eta \\wedge g^{p - 1 - i}\\text{d}g) =", "\\text{Tr}(\\eta \\wedge g^{p - 1 - i + j}\\text{d}g) = \\delta_{ij} \\theta", "$$", "by the explicit description of the trace map in Lemma \\ref{lemma-trace-higher}.", "\\medskip\\noindent", "Let $Y \\to \\Spec(B)$ be a normal modification. Exactly as in the proof of", "Lemma \\ref{lemma-go-up-separable} we can reduce to the case where $Y$", "is finite over a modification $X$ of $\\Spec(A)$. By", "Lemma \\ref{lemma-dominate-by-blowing-up-in-points} we may", "even assume $X \\to \\Spec(A)$ is the result of a sequence of", "blowing ups in closed points. Picture:", "$$", "\\xymatrix{", "Y \\ar[d]_\\pi \\ar[r]_-g & \\Spec(B) \\ar[d] \\\\", "X \\ar[r]^-f & \\Spec(A)", "}", "$$", "We may apply Lemma \\ref{lemma-trace-extends} to $\\pi$ and", "we obtain the first arrow in", "$$", "\\pi_*(\\Omega^{e + 2}_{Y/A_0})", "\\xrightarrow{\\text{Tr}}", "(\\Omega^{e + 2}_{X/A_0})^{**}", "\\xrightarrow{\\varphi_X}", "\\omega_X", "$$", "and the second arrow is from", "Lemma \\ref{lemma-compare-differentials-dualizing}", "(because $f$ is a sequence of blowups in closed points).", "By duality for the finite morphism $\\pi$ this corresponds to a map", "$$", "c_Y : \\Omega^{e + 2}_{Y/A_0} \\longrightarrow \\omega_Y", "$$", "extending the map $c$ above. Hence we see that the image of", "$\\Gamma(Y, \\omega_Y) \\to \\omega_B$ contains the image of $c$.", "By our claim we see that the cokernel is annihilated by", "a fixed nonzero element of $B$. We conclude by", "Lemma \\ref{lemma-bound-dualizing-implies-bound}." ], "refs": [ "algebra-lemma-derivative-zero-pth-power", "more-algebra-lemma-power-series-ring-subfields", "resolve-lemma-trace-extends", "resolve-lemma-trace-higher", "resolve-lemma-go-up-separable", "resolve-lemma-dominate-by-blowing-up-in-points", "resolve-lemma-trace-extends", "resolve-lemma-compare-differentials-dualizing", "resolve-lemma-bound-dualizing-implies-bound" ], "ref_ids": [ 1315, 10071, 11638, 11637, 11665, 11646, 11638, 11668, 11667 ] } ], "ref_ids": [] }, { "id": 11670, "type": "theorem", "label": "resolve-lemma-globally-generated", "categories": [ "resolve" ], "title": "resolve-lemma-globally-generated", "contents": [ "In Situation \\ref{situation-rational}.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then", "\\begin{enumerate}", "\\item $H^p(X, \\mathcal{F}) = 0$ for $p \\not \\in \\{0, 1\\}$, and", "\\item $H^1(X, \\mathcal{F}) = 0$ if $\\mathcal{F}$ is globally generated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-higher-direct-images-zero-above-dimension-fibre}.", "If $\\mathcal{F}$ is globally generated, then there is a surjection", "$\\bigoplus_{i \\in I} \\mathcal{O}_X \\to \\mathcal{F}$. By part (1)", "and the long exact sequence of cohomology this", "induces a surjection on $H^1$. Since $H^1(X, \\mathcal{O}_X) = 0$", "as $S$ has a rational singularity, and since $H^1(X, -)$ commutes", "with direct sums", "(Cohomology, Lemma \\ref{cohomology-lemma-quasi-separated-cohomology-colimit})", "we conclude." ], "refs": [ "coherent-lemma-higher-direct-images-zero-above-dimension-fibre", "cohomology-lemma-quasi-separated-cohomology-colimit" ], "ref_ids": [ 3364, 2082 ] } ], "ref_ids": [] }, { "id": 11671, "type": "theorem", "label": "resolve-lemma-sections-powers-I-rational", "categories": [ "resolve" ], "title": "resolve-lemma-sections-powers-I-rational", "contents": [ "In Situation \\ref{situation-rational} assume", "$E = X_s$ is an effective Cartier divisor.", "Let $\\mathcal{I}$ be the ideal sheaf of $E$. Then", "$H^0(X, \\mathcal{I}^n) = \\mathfrak m^n$ and", "$H^1(X, \\mathcal{I}^n) = 0$." ], "refs": [], "proofs": [ { "contents": [ "We have $H^0(X, \\mathcal{O}_X) = A$, see discussion following", "Situation \\ref{situation-vanishing}. Then", "$\\mathfrak m \\subset H^0(X, \\mathcal{I}) \\subset H^0(X, \\mathcal{O}_X)$.", "The second inclusion is not an equality as $X_s \\not = \\emptyset$.", "Thus $H^0(X, \\mathcal{I}) = \\mathfrak m$.", "As $\\mathcal{I}^n = \\mathfrak m^n\\mathcal{O}_X$ our", "Lemma \\ref{lemma-globally-generated} shows that $H^1(X, \\mathcal{I}^n) = 0$.", "\\medskip\\noindent", "Choose generators $x_1, \\ldots, x_{\\mu + 1}$ of $\\mathfrak m$. These define", "global sections of $\\mathcal{I}$ which generate it. Hence", "a short exact sequence", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{O}_X^{\\oplus \\mu + 1} \\to \\mathcal{I} \\to 0", "$$", "Then $\\mathcal{F}$ is a finite locally free $\\mathcal{O}_X$-module", "of rank $\\mu$ and $\\mathcal{F} \\otimes \\mathcal{I}$ is globally", "generated by Constructions, Lemma", "\\ref{constructions-lemma-globally-generated-omega-twist-1}.", "Hence $\\mathcal{F} \\otimes \\mathcal{I}^n$", "is globally generated for all $n \\geq 1$. Thus for $n \\geq 2$ we can", "consider the exact sequence", "$$", "0 \\to \\mathcal{F} \\otimes \\mathcal{I}^{n - 1} \\to", "(\\mathcal{I}^{n - 1})^{\\oplus \\mu + 1} \\to", "\\mathcal{I}^n \\to 0", "$$", "Applying the long exact sequence of cohomology using that", "$H^1(X, \\mathcal{F} \\otimes \\mathcal{I}^{n - 1}) = 0$ by", "Lemma \\ref{lemma-globally-generated}", "we obtain that every", "element of $H^0(X, \\mathcal{I}^n)$ is of the form $\\sum x_i a_i$", "for some $a_i \\in H^0(X, \\mathcal{I}^{n - 1})$. This shows that", "$H^0(X, \\mathcal{I}^n) = \\mathfrak m^n$ by induction." ], "refs": [ "resolve-lemma-globally-generated", "constructions-lemma-globally-generated-omega-twist-1", "resolve-lemma-globally-generated" ], "ref_ids": [ 11670, 12627, 11670 ] } ], "ref_ids": [] }, { "id": 11672, "type": "theorem", "label": "resolve-lemma-blow-up-normal-rational", "categories": [ "resolve" ], "title": "resolve-lemma-blow-up-normal-rational", "contents": [ "In Situation \\ref{situation-rational}", "the blowup of $\\Spec(A)$ in $\\mathfrak m$ is normal." ], "refs": [], "proofs": [ { "contents": [ "Let $X' \\to \\Spec(A)$ be the blowup, in other words", "$$", "X' = \\text{Proj}(A \\oplus \\mathfrak m \\oplus \\mathfrak m^2 \\oplus \\ldots).", "$$", "is the Proj of the Rees algebra. This in particular shows that", "$X'$ is integral and that $X' \\to \\Spec(A)$ is a projective", "modification. Let $X$ be the normalization of $X'$.", "Since $A$ is Nagata, we see that $\\nu : X \\to X'$ is finite", "(Morphisms, Lemma \\ref{morphisms-lemma-nagata-normalization}).", "Let $E' \\subset X'$ be the exceptional divisor and let $E \\subset X$", "be the inverse image. Let $\\mathcal{I}' \\subset \\mathcal{O}_{X'}$", "and $\\mathcal{I} \\subset \\mathcal{O}_X$ be their ideal sheaves.", "Recall that $\\mathcal{I}' = \\mathcal{O}_{X'}(1)$", "(Divisors, Lemma \\ref{divisors-lemma-blowing-up-projective}).", "Observe that $\\mathcal{I} = \\nu^*\\mathcal{I}'$ and that $E$ is an", "effective Cartier divisor (Divisors, Lemma", "\\ref{divisors-lemma-pullback-effective-Cartier-defined}).", "We are trying to show that $\\nu$ is an isomorphism. As $\\nu$ is finite,", "it suffices to show that $\\mathcal{O}_{X'} \\to \\nu_*\\mathcal{O}_X$", "is an isomorphism. If not, then we can find an $n \\geq 0$ such that", "$$", "H^0(X', (\\mathcal{I}')^n) \\not =", "H^0(X', (\\nu_*\\mathcal{O}_X) \\otimes (\\mathcal{I}')^n)", "$$", "for example because we can recover quasi-coherent $\\mathcal{O}_{X'}$-modules", "from their associated graded modules, see", "Properties, Lemma \\ref{properties-lemma-ample-quasi-coherent}.", "By the projection formula we have", "$$", "H^0(X', (\\nu_*\\mathcal{O}_X) \\otimes (\\mathcal{I}')^n) =", "H^0(X, \\nu^*(\\mathcal{I}')^n) =", "H^0(X, \\mathcal{I}^n) = \\mathfrak m^n", "$$", "the last equality by Lemma \\ref{lemma-sections-powers-I-rational}.", "On the other hand, there is clearly an injection", "$\\mathfrak m^n \\to H^0(X', (\\mathcal{I}')^n)$. Since", "$H^0(X', (\\mathcal{I}')^n)$ is torsion free we conclude equality holds", "for all $n$, hence $X = X'$." ], "refs": [ "morphisms-lemma-nagata-normalization", "divisors-lemma-blowing-up-projective", "divisors-lemma-pullback-effective-Cartier-defined", "properties-lemma-ample-quasi-coherent", "resolve-lemma-sections-powers-I-rational" ], "ref_ids": [ 5520, 8063, 7936, 3057, 11671 ] } ], "ref_ids": [] }, { "id": 11673, "type": "theorem", "label": "resolve-lemma-cohomology-blow-up-rational", "categories": [ "resolve" ], "title": "resolve-lemma-cohomology-blow-up-rational", "contents": [ "In Situation \\ref{situation-rational}.", "Let $X$ be the blowup of $\\Spec(A)$ in $\\mathfrak m$. Let $E \\subset X$", "be the exceptional divisor. With $\\mathcal{O}_X(1) = \\mathcal{I}$ as", "usual and $\\mathcal{O}_E(1) = \\mathcal{O}_X(1)|_E$ we have", "\\begin{enumerate}", "\\item $E$ is a proper Cohen-Macaulay curve over $\\kappa$.", "\\item $\\mathcal{O}_E(1)$ is very ample", "\\item $\\deg(\\mathcal{O}_E(1)) \\geq 1$ and equality holds only if", "$A$ is a regular local ring,", "\\item $H^1(E, \\mathcal{O}_E(n)) = 0$ for $n \\geq 0$, and", "\\item $H^0(E, \\mathcal{O}_E(n)) = \\mathfrak m^n/\\mathfrak m^{n + 1}$", "for $n \\geq 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathcal{O}_X(1)$ is very ample by construction, we see that", "its restriction to the special fibre $E$ is very ample as well.", "By Lemma \\ref{lemma-blow-up-normal-rational} the scheme $X$ is normal.", "Then $E$ is Cohen-Macaulay by", "Divisors, Lemma \\ref{divisors-lemma-normal-effective-Cartier-divisor-S1}.", "Lemma \\ref{lemma-sections-powers-I-rational} applies and we obtain", "(4) and (5) from the exact sequences", "$$", "0 \\to \\mathcal{I}^{n + 1} \\to \\mathcal{I}^n \\to i_*\\mathcal{O}_E(n) \\to 0", "$$", "and the long exact cohomology sequence. In particular, we see that", "$$", "\\deg(\\mathcal{O}_E(1)) = \\chi(E, \\mathcal{O}_E(1)) - \\chi(E, \\mathcal{O}_E) =", "\\dim(\\mathfrak m/\\mathfrak m^2) - 1", "$$", "by Varieties, Definition \\ref{varieties-definition-degree-invertible-sheaf}.", "Thus (3) follows as well." ], "refs": [ "resolve-lemma-blow-up-normal-rational", "divisors-lemma-normal-effective-Cartier-divisor-S1", "resolve-lemma-sections-powers-I-rational", "varieties-definition-degree-invertible-sheaf" ], "ref_ids": [ 11672, 7950, 11671, 11161 ] } ], "ref_ids": [] }, { "id": 11674, "type": "theorem", "label": "resolve-lemma-dualizing-rational", "categories": [ "resolve" ], "title": "resolve-lemma-dualizing-rational", "contents": [ "In Situation \\ref{situation-rational} assume $A$ has a", "dualizing complex $\\omega_A^\\bullet$. With $\\omega_X$ the dualizing", "module of $X$, the trace map $H^0(X, \\omega_X) \\to \\omega_A$ is an", "isomorphism and consequently there is a canonical map", "$f^*\\omega_A \\to \\omega_X$." ], "refs": [], "proofs": [ { "contents": [ "By Grauert-Riemenschneider", "(Proposition \\ref{proposition-Grauert-Riemenschneider}) we see that", "$Rf_*\\omega_X = f_*\\omega_X$. By duality we have a short exact", "sequence", "$$", "0 \\to f_*\\omega_X \\to \\omega_A \\to", "\\Ext^2_A(R^1f_*\\mathcal{O}_X, \\omega_A) \\to 0", "$$", "(for example see proof of Lemma \\ref{lemma-bound-dualizing-implies-bound})", "and since $A$ defines a rational singularity we obtain", "$f_*\\omega_X = \\omega_A$." ], "refs": [ "resolve-proposition-Grauert-Riemenschneider", "resolve-lemma-bound-dualizing-implies-bound" ], "ref_ids": [ 11711, 11667 ] } ], "ref_ids": [] }, { "id": 11675, "type": "theorem", "label": "resolve-lemma-dualizing-blow-up-rational", "categories": [ "resolve" ], "title": "resolve-lemma-dualizing-blow-up-rational", "contents": [ "In Situation \\ref{situation-rational} assume $A$ has a", "dualizing complex $\\omega_A^\\bullet$ and is not regular.", "Let $X$ be the blowup of $\\Spec(A)$ in $\\mathfrak m$ with", "exceptional divisor $E \\subset X$. Let $\\omega_X$", "be the dualizing module of $X$. Then", "\\begin{enumerate}", "\\item $\\omega_E = \\omega_X|_E \\otimes \\mathcal{O}_E(-1)$,", "\\item $H^1(X, \\omega_X(n)) = 0$ for $n \\geq 0$,", "\\item the map $f^*\\omega_A \\to \\omega_X$ of", "Lemma \\ref{lemma-dualizing-rational} is surjective.", "\\end{enumerate}" ], "refs": [ "resolve-lemma-dualizing-rational" ], "proofs": [ { "contents": [ "We will use the results of Lemma \\ref{lemma-cohomology-blow-up-rational}", "without further mention. Observe that", "$\\omega_E = \\omega_X|_E \\otimes \\mathcal{O}_E(-1)$", "by Duality for Schemes, Lemmas", "\\ref{duality-lemma-sheaf-with-exact-support-effective-Cartier} and", "\\ref{duality-lemma-twisted-inverse-image-closed}. Thus", "$\\omega_X|_E = \\omega_E(1)$. Consider the short exact sequences", "$$", "0 \\to \\omega_X(n + 1) \\to \\omega_X(n) \\to i_*\\omega_E(n + 1) \\to 0", "$$", "By Algebraic Curves, Lemma \\ref{curves-lemma-vanishing-twist}", "we see that $H^1(E, \\omega_E(n + 1)) = 0$ for $n \\geq 0$.", "Thus we see that the maps", "$$", "\\ldots \\to H^1(X, \\omega_X(2)) \\to H^1(X, \\omega_X(1)) \\to H^1(X, \\omega_X)", "$$", "are surjective. Since $H^1(X, \\omega_X(n))$ is zero for $n \\gg 0$", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-kill-by-twisting})", "we conclude that (2) holds.", "\\medskip\\noindent", "By Algebraic Curves, Lemma", "\\ref{curves-lemma-tensor-omega-with-globally-generated-invertible}", "we see that $\\omega_X|_E = \\omega_E \\otimes \\mathcal{O}_E(1)$", "is globally generated. Since we seen above that", "$H^1(X, \\omega_X(1)) = 0$ the map $H^0(X, \\omega_X) \\to H^0(E, \\omega_X|_E)$", "is surjective. We conclude that $\\omega_X$ is globally generated", "hence (3) holds because $\\Gamma(X, \\omega_X) = \\omega_A$ is used", "in Lemma \\ref{lemma-dualizing-rational} to define the map." ], "refs": [ "resolve-lemma-cohomology-blow-up-rational", "duality-lemma-sheaf-with-exact-support-effective-Cartier", "duality-lemma-twisted-inverse-image-closed", "curves-lemma-vanishing-twist", "coherent-lemma-kill-by-twisting", "curves-lemma-tensor-omega-with-globally-generated-invertible", "resolve-lemma-dualizing-rational" ], "ref_ids": [ 11673, 13544, 13525, 6259, 3344, 6262, 11674 ] } ], "ref_ids": [ 11674 ] }, { "id": 11676, "type": "theorem", "label": "resolve-lemma-rational-to-gorenstein", "categories": [ "resolve" ], "title": "resolve-lemma-rational-to-gorenstein", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain of", "dimension $2$ which defines a rational singularity. Assume $A$ has", "a dualizing complex. Then there exists a finite sequence of blowups in", "singular closed points", "$$", "X = X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = \\Spec(A)", "$$", "such that $X_i$ is normal for each $i$ and such that", "the dualizing sheaf $\\omega_X$ of $X$ is an invertible", "$\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "The dualizing module $\\omega_A$ is a finite $A$-module whose stalk at", "the generic point is invertible. Namely, $\\omega_A \\otimes_A K$", "is a dualizing module for the fraction field $K$ of $A$, hence has", "rank $1$. Thus there exists a blowup $b : Y \\to \\Spec(A)$ such that", "the strict transform of $\\omega_A$ with respect to $b$ is an invertible", "$\\mathcal{O}_Y$-module, see", "Divisors, Lemma \\ref{divisors-lemma-blowup-fitting-ideal}.", "By Lemma \\ref{lemma-dominate-by-normalized-blowing-up}", "we can choose a sequence of normalized blowups", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to \\Spec(A)", "$$", "such that $X_n$ dominates $Y$. By Lemma \\ref{lemma-blow-up-normal-rational}", "and arguing by induction each $X_i \\to X_{i - 1}$ is simply a blowing up.", "\\medskip\\noindent", "We claim that $\\omega_{X_n}$ is invertible. Since $\\omega_{X_n}$", "is a coherent $\\mathcal{O}_{X_n}$-module, it suffices to see its stalks", "are invertible modules. If $x \\in X_n$ is a regular point, then this is", "clear from the fact that regular schemes are", "Gorenstein (Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-regular-gorenstein}). If $x$ is a singular point of", "$X_n$, then each of the images $x_i \\in X_i$ of $x$ is a singular point", "(because the blowup of a regular point is regular by", "Lemma \\ref{lemma-blowup-regular}).", "Consider the canonical map $f_n^*\\omega_A \\to \\omega_{X_n}$ of", "Lemma \\ref{lemma-dualizing-rational}. For each $i$ the morphism", "$X_{i + 1} \\to X_i$ is either a blowup of $x_i$ or an isomorphism", "at $x_i$. Since $x_i$ is always a singular point, it follows from", "Lemma \\ref{lemma-dualizing-blow-up-rational}", "and induction that the maps $f_i^*\\omega_A \\to \\omega_{X_i}$", "is always surjective on stalks at $x_i$. Hence", "$$", "(f_n^*\\omega_A)_x \\longrightarrow \\omega_{X_n, x}", "$$", "is surjective. On the other hand, by our choice of $b$ the quotient", "of $f_n^*\\omega_A$ by its torsion submodule is an invertible module", "$\\mathcal{L}$. Moreover, the dualizing module is torsion free", "(Duality for Schemes, Lemma \\ref{duality-lemma-dualizing-module}).", "It follows that $\\mathcal{L}_x \\cong \\omega_{X_n, x}$ and the proof is", "complete." ], "refs": [ "divisors-lemma-blowup-fitting-ideal", "resolve-lemma-dominate-by-normalized-blowing-up", "resolve-lemma-blow-up-normal-rational", "dualizing-lemma-regular-gorenstein", "resolve-lemma-blowup-regular", "resolve-lemma-dualizing-rational", "resolve-lemma-dualizing-blow-up-rational", "duality-lemma-dualizing-module" ], "ref_ids": [ 8078, 11649, 11672, 2880, 11640, 11674, 11675, 13583 ] } ], "ref_ids": [] }, { "id": 11677, "type": "theorem", "label": "resolve-lemma-sequence-blowups", "categories": [ "resolve" ], "title": "resolve-lemma-sequence-blowups", "contents": [ "Let $X$ be a locally Noetherian scheme. Let", "$$", "(X, p) = (X_0, p_0) \\leftarrow (X_1, p_1) \\leftarrow (X_2, p_2) \\leftarrow", "(X_3, p_3) \\leftarrow \\ldots", "$$", "be a sequence of blowups such that", "\\begin{enumerate}", "\\item $p_i$ is closed, maps to $p_{i - 1}$, and", "$\\kappa(p_i) = \\kappa(p_{i - 1})$,", "\\item there exists an $x_1 \\in \\mathfrak m_p$ whose image", "in $\\mathfrak m_{p_i}$, $i > 0$ defines the exceptional divisor", "$E_i \\subset X_i$.", "\\end{enumerate}", "Then the sequence is obtained from a nonsingular arc $a : T \\to X$", "as above." ], "refs": [], "proofs": [ { "contents": [ "Let us write $\\mathcal{O}_n = \\mathcal{O}_{X_n, p_n}$", "and $\\mathcal{O} = \\mathcal{O}_{X, p}$. Denote", "$\\mathfrak m \\subset \\mathcal{O}$ and $\\mathfrak m_n \\subset \\mathcal{O}_n$", "the maximal ideals.", "\\medskip\\noindent", "We claim that $x_1^t \\not \\in \\mathfrak m_n^{t + 1}$.", "Namely, if this were the case, then in the local ring", "$\\mathcal{O}_{n + 1}$ the element $x_1^t$ would be in the ideal of", "$(t + 1)E_{n + 1}$.", "This contradicts the assumption that $x_1$ defines $E_{n + 1}$.", "\\medskip\\noindent", "For every $n$ choose generators $y_{n, 1}, \\ldots, y_{n, t_n}$", "for $\\mathfrak m_n$. As", "$\\mathfrak m_n \\mathcal{O}_{n + 1} = x_1\\mathcal{O}_{n + 1}$", "by assumption (2), we can write $y_{n, i} = a_{n, i} x_1$", "for some $a_{n, i} \\in \\mathcal{O}_{n + 1}$. Since", "the map $\\mathcal{O}_n \\to \\mathcal{O}_{n + 1}$ defines", "an isomorphism on residue fields by (1) we can choose", "$c_{n, i} \\in \\mathcal{O}_n$ having the same residue class as", "$a_{n, i}$. Then we see that", "$$", "\\mathfrak m_n = (x_1, z_{n, 1}, \\ldots, z_{n, t_n}),", "\\quad z_{n, i} = y_{n, i} - c_{n, i} x_1", "$$", "and the elements $z_{n, i}$ map to elements of $\\mathfrak m_{n + 1}^2$", "in $\\mathcal{O}_{n + 1}$.", "\\medskip\\noindent", "Let us consider", "$$", "J_n = \\Ker(\\mathcal{O} \\to \\mathcal{O}_n/\\mathfrak m_n^{n + 1})", "$$", "We claim that $\\mathcal{O}/J_n$ has length $n + 1$ and that", "$\\mathcal{O}/(x_1) + J_n$ equals the residue field. For $n = 0$", "this is immediate. Assume the statement holds for $n$.", "Let $f \\in J_n$. Then in $\\mathcal{O}_n$ we have", "$$", "f = a x_1^{n + 1} + x_1^n A_1(z_{n, i}) +", "x_1^{n - 1} A_2(z_{n, i}) + \\ldots + A_{n + 1}(z_{n, i})", "$$", "for some $a \\in \\mathcal{O}_n$ and some $A_i$ homogeneous of degree $i$", "with coefficients in $\\mathcal{O}_n$. Since $\\mathcal{O} \\to \\mathcal{O}_n$", "identifies residue fields, we may choose $a \\in \\mathcal{O}$", "(argue as in the construction of $z_{n, i}$ above).", "Taking the image in", "$\\mathcal{O}_{n + 1}$ we see that $f$ and $a x_1^{n + 1}$", "have the same image modulo $\\mathfrak m_{n + 1}^{n + 2}$.", "Since $x_n^{n + 1} \\not \\in \\mathfrak m_{n + 1}^{n + 2}$", "it follows that $J_n/J_{n + 1}$ has length $1$ and the claim is true.", "\\medskip\\noindent", "Consider $R = \\lim \\mathcal{O}/J_n$. This is a quotient of", "the $\\mathfrak m$-adic completion of $\\mathcal{O}$ hence it is", "a complete Noetherian local ring. On the other hand, it is", "not finite length and $x_1$ generates the maximal ideal.", "Thus $R$ is a complete discrete valuation ring.", "The map $\\mathcal{O} \\to R$ lifts to a local homomorphism", "$\\mathcal{O}_n \\to R$ for every $n$. There are two ways to show this:", "(1) for every $n$ one can use a similar procedure", "to construct $\\mathcal{O}_n \\to R_n$ and then one can", "show that $\\mathcal{O} \\to \\mathcal{O}_n \\to R_n$ factors", "through an isomorphism $R \\to R_n$, or (2) one can use", "Divisors, Lemma \\ref{divisors-lemma-characterize-affine-blowup}", "to show that $\\mathcal{O}_n$ is a localization of a repeated", "affine blowup algebra to explicitly construct a map $\\mathcal{O}_n \\to R$.", "Having said this it is clear that our sequence of blowups", "comes from the nonsingular arc $a : T = \\Spec(R) \\to X$." ], "refs": [ "divisors-lemma-characterize-affine-blowup" ], "ref_ids": [ 8056 ] } ], "ref_ids": [] }, { "id": 11678, "type": "theorem", "label": "resolve-lemma-sequence-blowups-along-arc-becomes-nonsingular", "categories": [ "resolve" ], "title": "resolve-lemma-sequence-blowups-along-arc-becomes-nonsingular", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local domain of", "dimension $2$. Let $A \\to R$ be a surjection onto a", "complete discrete valuation ring.", "This defines a nonsingular arc $a : T = \\Spec(R) \\to \\Spec(A)$. Let", "$$", "\\Spec(A) = X_0 \\leftarrow X_1 \\leftarrow X_2 \\leftarrow X_3 \\leftarrow \\ldots", "$$", "be the sequence of blowing ups constructed from $a$.", "If $A_\\mathfrak p$ is a regular local ring where", "$\\mathfrak p = \\Ker(A \\to R)$, then", "for some $i$ the scheme $X_i$ is regular at $x_i$." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1 \\in \\mathfrak m$ map to a uniformizer of $R$.", "Observe that $\\kappa(\\mathfrak p) = K$ is the", "fraction field of $R$. Write $\\mathfrak p = (x_2, \\ldots, x_r)$", "with $r$ minimal. If $r = 2$, then $\\mathfrak m = (x_1, x_2)$", "and $A$ is regular and the lemma is true. Assume $r > 2$.", "After renumbering if necessary,", "we may assume that $x_2$ maps to a uniformizer of $A_\\mathfrak p$.", "Then $\\mathfrak p/\\mathfrak p^2 + (x_2)$ is annihilated by a power", "of $x_1$. For $i > 2$ we can find $n_i \\geq 0$ and $a_i \\in A$", "such that", "$$", "x_1^{n_i} x_i - a_i x_2 = \\sum\\nolimits_{2 \\leq j \\leq k} a_{jk} x_jx_k", "$$", "for some $a_{jk} \\in A$. If $n_i = 0$ for some $i$, then we can remove", "$x_i$ from the list of generators of $\\mathfrak p$ and we win by", "induction on $r$. If for some $i$ the element $a_i$ is a unit, then", "we can remove $x_2$ from the list of generators of $\\mathfrak p$", "and we win in the same manner. Thus either", "$a_i \\in \\mathfrak p$ or $a_i = u_i x_1^{m_1} \\bmod \\mathfrak p$", "for some $m_1 > 0$ and unit $u_i \\in A$. Thus we have either", "$$", "x_1^{n_i} x_i = \\sum\\nolimits_{2 \\leq j \\leq k} a_{jk} x_jx_k", "\\quad\\text{or}\\quad", "x_1^{n_i} x_i - u_i x_1^{m_i} x_2 =", "\\sum\\nolimits_{2 \\leq j \\leq k} a_{jk} x_jx_k", "$$", "We will prove that after blowing up the integers $n_i$, $m_i$", "decrease which will finish the proof.", "\\medskip\\noindent", "Let us see what happens with these equations on the affine blowup", "algebra $A' = A[\\mathfrak m/x_1]$. As $\\mathfrak m = (x_1, \\ldots, x_r)$", "we see that $A'$ is generated over $R$ by $y_i = x_i/x_1$ for $i \\geq 2$.", "Clearly $A \\to R$ extends to $A' \\to R$ with kernel", "$(y_2, \\ldots, y_r)$. Then we see that either", "$$", "x_1^{n_i - 1} y_i = \\sum\\nolimits_{2 \\leq j \\leq k} a_{jk} y_jy_k", "\\quad\\text{or}\\quad", "x_1^{n_i - 1} y_i - u_i x_1^{m_1 - 1} y_2 =", "\\sum\\nolimits_{2 \\leq j \\leq k} a_{jk} y_jy_k", "$$", "and the proof is complete." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11679, "type": "theorem", "label": "resolve-lemma-iso-completions", "categories": [ "resolve" ], "title": "resolve-lemma-iso-completions", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a local ring with finitely generated", "maximal ideal $\\mathfrak m$. Let $X$ be a scheme over $A$.", "Let $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$ where", "$A^\\wedge$ is the $\\mathfrak m$-adic completion of $A$.", "For a point $q \\in Y$ with image $p \\in X$ lying", "over the closed point of $\\Spec(A)$ the", "local ring map $\\mathcal{O}_{X, p} \\to \\mathcal{O}_{Y, q}$", "induces an isomorphism on completions." ], "refs": [], "proofs": [ { "contents": [ "We may assume $X$ is affine. Then we may write $X = \\Spec(B)$.", "Let $\\mathfrak q \\subset B' = B \\otimes_A A^\\wedge$ be the", "prime corresponding to $q$ and let $\\mathfrak p \\subset B$", "be the prime ideal corresponding to $p$.", "By Algebra, Lemma \\ref{algebra-lemma-hathat-finitely-generated}", "we have", "$$", "B'/(\\mathfrak m^\\wedge)^n B' =", "A^\\wedge/(\\mathfrak m^\\wedge)^n \\otimes_A B =", "A/\\mathfrak m^n \\otimes_A B = B/\\mathfrak m^n B", "$$", "for all $n$. Since $\\mathfrak m B \\subset \\mathfrak p$ and", "$\\mathfrak m^\\wedge B' \\subset \\mathfrak q$ we see that", "$B/\\mathfrak p^n$ and $B'/\\mathfrak q^n$ are both", "quotients of the ring displayed above by the $n$th power", "of the same prime ideal. The lemma follows." ], "refs": [ "algebra-lemma-hathat-finitely-generated" ], "ref_ids": [ 859 ] } ], "ref_ids": [] }, { "id": 11680, "type": "theorem", "label": "resolve-lemma-port-regularity-to-completion", "categories": [ "resolve" ], "title": "resolve-lemma-port-regularity-to-completion", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Let $X \\to \\Spec(A)$ be a morphism which is locally of finite type.", "Set $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$. Let $y \\in Y$ with", "image $x \\in X$. Then", "\\begin{enumerate}", "\\item if $\\mathcal{O}_{Y, y}$ is regular, then $\\mathcal{O}_{X, x}$", "is regular,", "\\item if $y$ is in the closed fibre, then $\\mathcal{O}_{Y, y}$ is regular", "$\\Leftrightarrow \\mathcal{O}_{X, x}$ is regular, and", "\\item If $X$ is proper over $A$, then $X$ is regular", "if and only if $Y$ is regular.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $A \\to A^\\wedge$ is faithfully flat", "(Algebra, Lemma \\ref{algebra-lemma-completion-faithfully-flat}),", "we see that $Y \\to X$ is flat. Hence (1) by", "Algebra, Lemma \\ref{algebra-lemma-descent-regular}.", "Lemma \\ref{lemma-iso-completions} shows the morphism $Y \\to X$", "induces an isomorphism on complete local rings at points", "of the special fibres. Thus (2) by", "More on Algebra, Lemma \\ref{more-algebra-lemma-completion-regular}.", "If $X$ is proper over $A$, then $Y$ is proper over $A^\\wedge$", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-proper})", "and we see every closed point of $X$ and $Y$ lies in the closed fibre.", "Thus we see that $Y$ is a regular scheme if and only if $X$ is so by", "Properties, Lemma \\ref{properties-lemma-characterize-regular}." ], "refs": [ "algebra-lemma-completion-faithfully-flat", "algebra-lemma-descent-regular", "resolve-lemma-iso-completions", "more-algebra-lemma-completion-regular", "morphisms-lemma-base-change-proper", "properties-lemma-characterize-regular" ], "ref_ids": [ 871, 1373, 11679, 10045, 5409, 2975 ] } ], "ref_ids": [] }, { "id": 11681, "type": "theorem", "label": "resolve-lemma-descend-admissible-blowup", "categories": [ "resolve" ], "title": "resolve-lemma-descend-admissible-blowup", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring with completion $A^\\wedge$.", "Let $U \\subset \\Spec(A)$ and $U^\\wedge \\subset \\Spec(A^\\wedge)$ be the", "punctured spectra. If $Y \\to \\Spec(A^\\wedge)$ is a $U^\\wedge$-admissible", "blowup, then there exists a $U$-admissible blowup $X \\to \\Spec(A)$", "such that $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$." ], "refs": [], "proofs": [ { "contents": [ "By definition there exists an ideal $J \\subset A^\\wedge$ such that", "$V(J) = \\{\\mathfrak m A^\\wedge\\}$ and such that $Y$ is the blowup", "of $S^\\wedge$ in the closed subscheme defined by $J$, see", "Divisors, Definition \\ref{divisors-definition-admissible-blowup}.", "Since $A^\\wedge$ is Noetherian this implies", "$\\mathfrak m^n A^\\wedge \\subset J$ for some $n$.", "Since $A^\\wedge/\\mathfrak m^n A^\\wedge = A/\\mathfrak m^n$", "we find an ideal $\\mathfrak m^n \\subset I \\subset A$", "such that $J = I A^\\wedge$. Let $X \\to S$ be the blowup in $I$.", "Since $A \\to A^\\wedge$ is flat", "we conclude that the base change of $X$ is $Y$ by", "Divisors, Lemma \\ref{divisors-lemma-flat-base-change-blowing-up}." ], "refs": [ "divisors-definition-admissible-blowup", "divisors-lemma-flat-base-change-blowing-up" ], "ref_ids": [ 8114, 8053 ] } ], "ref_ids": [] }, { "id": 11682, "type": "theorem", "label": "resolve-lemma-blowup-still-good", "categories": [ "resolve" ], "title": "resolve-lemma-blowup-still-good", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Nagata local normal domain of", "dimension $2$. Assume $A$ defines a rational singularity and that", "the completion $A^\\wedge$ of $A$ is normal. Then", "\\begin{enumerate}", "\\item $A^\\wedge$ defines a rational singularity, and", "\\item if $X \\to \\Spec(A)$ is the blowing up in $\\mathfrak m$, then", "for a closed point $x \\in X$ the completion $\\mathcal{O}_{X, x}$ is normal.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $Y \\to \\Spec(A^\\wedge)$ be a modification with $Y$ normal.", "We have to show that $H^1(Y, \\mathcal{O}_Y) = 0$. By Varieties, Lemma", "\\ref{varieties-lemma-modification-normal-iso-over-codimension-1}", "$Y \\to \\Spec(A^\\wedge)$ is an isomorphism over the punctured", "spectrum $U^\\wedge = \\Spec(A^\\wedge) \\setminus \\{\\mathfrak m^\\wedge\\}$.", "By Lemma \\ref{lemma-dominate-by-scheme-modification}", "there exists a $U^\\wedge$-admissible blowup $Y' \\to \\Spec(A^\\wedge)$", "dominating $Y$. By Lemma \\ref{lemma-descend-admissible-blowup}", "we find there exists a $U$-admissible blowup $X \\to \\Spec(A)$", "whose base change to $A^\\wedge$ dominates $Y$.", "Since $A$ is Nagata, we can replace $X$ by its normalization", "after which $X \\to \\Spec(A)$ is a normal modification (but", "possibly no longer a $U$-admissible blowup).", "Then $H^1(X, \\mathcal{O}_X) = 0$ as $A$ defines a rational", "singularity. It follows that", "$H^1(X \\times_{\\Spec(A)} \\Spec(A^\\wedge),", "\\mathcal{O}_{X \\times_{\\Spec(A)} \\Spec(A^\\wedge)}) = 0$", "by flat base change (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology}", "and flatness of $A \\to A^\\wedge$ by", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}).", "We find that $H^1(Y, \\mathcal{O}_Y) = 0$ by", "Lemma \\ref{lemma-exact-sequence}.", "\\medskip\\noindent", "Finally, let $X \\to \\Spec(A)$ be the blowing up of $\\Spec(A)$", "in $\\mathfrak m$. Then $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$", "is the blowing up of $\\Spec(A^\\wedge)$ in $\\mathfrak m^\\wedge$.", "By Lemma \\ref{lemma-blow-up-normal-rational} we see that both $Y$", "and $X$ are normal. On the other hand, $A^\\wedge$ is excellent", "(More on Algebra, Proposition", "\\ref{more-algebra-proposition-ubiquity-excellent})", "hence every affine open in $Y$ is the spectrum of an", "excellent normal domain", "(More on Algebra, Lemma \\ref{more-algebra-lemma-finite-type-over-excellent}).", "Thus for $y \\in Y$ the ring map", "$\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{Y, y}^\\wedge$", "is regular and by", "More on Algebra, Lemma \\ref{more-algebra-lemma-normal-goes-up}", "we find that $\\mathcal{O}_{Y, y}^\\wedge$ is normal.", "If $x \\in X$ is a closed point of the special fibre,", "then there is a unique closed point $y \\in Y$ lying over $x$.", "Since $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{Y, y}$ induces", "an isomorphism on completions (Lemma \\ref{lemma-iso-completions})", "we conclude." ], "refs": [ "varieties-lemma-modification-normal-iso-over-codimension-1", "resolve-lemma-dominate-by-scheme-modification", "resolve-lemma-descend-admissible-blowup", "coherent-lemma-flat-base-change-cohomology", "algebra-lemma-completion-flat", "resolve-lemma-exact-sequence", "resolve-lemma-blow-up-normal-rational", "more-algebra-proposition-ubiquity-excellent", "more-algebra-lemma-finite-type-over-excellent", "more-algebra-lemma-normal-goes-up", "resolve-lemma-iso-completions" ], "ref_ids": [ 10979, 11656, 11681, 3298, 870, 11661, 11672, 10584, 10106, 10041, 11679 ] } ], "ref_ids": [] }, { "id": 11683, "type": "theorem", "label": "resolve-lemma-formally-unramified", "categories": [ "resolve" ], "title": "resolve-lemma-formally-unramified", "contents": [ "Let $(A, \\mathfrak m)$ be a local Noetherian ring. Let", "$X$ be a scheme over $A$. Assume", "\\begin{enumerate}", "\\item $A$ is analytically unramified", "(Algebra, Definition \\ref{algebra-definition-analytically-unramified}),", "\\item $X$ is locally of finite type over $A$, and", "\\item $X \\to \\Spec(A)$ is \\'etale at the generic points of irreducible", "components of $X$.", "\\end{enumerate}", "Then the normalization of $X$ is finite over $X$." ], "refs": [ "algebra-definition-analytically-unramified" ], "proofs": [ { "contents": [ "Since $A$ is analytically unramified it is reduced", "by Algebra, Lemma \\ref{algebra-lemma-analytically-unramified-easy}.", "Since the normalization of $X$ depends only on the reduction", "of $X$, we may replace $X$ by its reduction $X_{red}$; note", "that $X_{red} \\to X$ is an isomorphism over the open $U$ where", "$X \\to \\Spec(A)$ is \\'etale because $U$ is reduced", "(Descent, Lemma \\ref{descent-lemma-reduced-local-smooth})", "hence condition (3) remains true after this replacement.", "In addition we may and do assume that $X = \\Spec(B)$ is affine.", "\\medskip\\noindent", "The map", "$$", "K = \\prod\\nolimits_{\\mathfrak p \\subset A\\text{ minimal}} \\kappa(\\mathfrak p)", "\\longrightarrow", "K^\\wedge = \\prod\\nolimits_{\\mathfrak p^\\wedge \\subset A^\\wedge\\text{ minimal}}", "\\kappa(\\mathfrak p^\\wedge)", "$$", "is injective because $A \\to A^\\wedge$ is faithfully flat", "(Algebra, Lemma \\ref{algebra-lemma-completion-faithfully-flat})", "hence induces a surjective map between sets of minimal primes", "(by going down for flat ring maps, see", "Algebra, Section \\ref{algebra-section-going-up}).", "Both sides are finite products of fields as our rings are Noetherian.", "Let $L = \\prod_{\\mathfrak q \\subset B\\text{ minimal}} \\kappa(\\mathfrak q)$.", "Our assumption (3) implies that $L = B \\otimes_A K$ and that", "$K \\to L$ is a finite \\'etale ring map (this is true", "because $A \\to B$ is generically finite, for example use", "Algebra, Lemma \\ref{algebra-lemma-generically-finite}", "or the more detailed results in Morphisms, Section", "\\ref{morphisms-section-generically-finite}).", "Since $B$ is reduced we see that $B \\subset L$.", "This implies that", "$$", "C = B \\otimes_A A^\\wedge \\subset", "L \\otimes_A A^\\wedge = L \\otimes_K K^\\wedge = M", "$$", "Then $M$ is the total ring of fractions of $C$ and", "is a finite product of fields as a finite separable", "algebra over $K^\\wedge$. It follows that $C$ is reduced", "and that its normalization $C'$ is the integral closure of", "$C$ in $M$. The normalization $B'$ of $B$ is the integral", "closure of $B$ in $L$. By flatness of $A \\to A^\\wedge$", "we obtain an injective map $B' \\otimes_A A^\\wedge \\to M$ whose", "image is contained in $C'$. Picture", "$$", "B' \\otimes_A A^\\wedge \\longrightarrow C'", "$$", "As $A^\\wedge$ is Nagata (by", "Algebra, Lemma \\ref{algebra-lemma-Noetherian-complete-local-Nagata}),", "we see that $C'$ is finite over", "$C = B \\otimes_A A^\\wedge$ (see", "Algebra, Lemmas", "\\ref{algebra-lemma-Noetherian-complete-local-Nagata} and", "\\ref{algebra-lemma-nagata-in-reduced-finite-type-finite-integral-closure}).", "As $C$ is Noetherian, we conclude that", "$B' \\otimes_A A^\\wedge$ is finite over $C = B \\otimes_A A^\\wedge$.", "Therefore by faithfully flat descent", "(Algebra, Lemma \\ref{algebra-lemma-descend-properties-modules})", "we see that $B'$ is finite over $B$ which is what we had to show." ], "refs": [ "algebra-lemma-analytically-unramified-easy", "descent-lemma-reduced-local-smooth", "algebra-lemma-completion-faithfully-flat", "algebra-lemma-generically-finite", "algebra-lemma-Noetherian-complete-local-Nagata", "algebra-lemma-Noetherian-complete-local-Nagata", "algebra-lemma-nagata-in-reduced-finite-type-finite-integral-closure", "algebra-lemma-descend-properties-modules" ], "ref_ids": [ 1354, 14653, 871, 1056, 1353, 1353, 1347, 819 ] } ], "ref_ids": [ 1553 ] }, { "id": 11684, "type": "theorem", "label": "resolve-lemma-normalization-completion", "categories": [ "resolve" ], "title": "resolve-lemma-normalization-completion", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Let $X \\to \\Spec(A)$ be a morphism which is locally of finite type.", "Set $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$.", "If the complement of the special fibre in $Y$ is normal, then", "the normalization $X^\\nu \\to X$ is finite and the base change", "of $X^\\nu$ to $\\Spec(A^\\wedge)$ recovers the normalization of $Y$." ], "refs": [], "proofs": [ { "contents": [ "There is an immediate reduction to the case where $X = \\Spec(B)$", "is affine with $B$ a finite type $A$-algebra. Set $C = B \\otimes_A A^\\wedge$", "so that $Y = \\Spec(C)$. Since", "$A \\to A^\\wedge$ is faithfully flat, for any prime $\\mathfrak q \\subset B$", "there exists a prime $\\mathfrak r \\subset C$ lying over $\\mathfrak q$.", "Then $B_\\mathfrak q \\to C_\\mathfrak r$ is faithfully flat. Hence if", "$\\mathfrak q$ does not lie over $\\mathfrak m$, then $C_\\mathfrak r$", "is normal by assumption on $Y$ and we conclude that $B_\\mathfrak q$", "is normal by Algebra, Lemma \\ref{algebra-lemma-descent-normal}.", "In this way we see that $X$ is normal away from the special fibre.", "\\medskip\\noindent", "Recall that the complete Noetherian local ring $A^\\wedge$ is Nagata", "(Algebra, Lemma \\ref{algebra-lemma-Noetherian-complete-local-Nagata}).", "Hence the normalization $Y^\\nu \\to Y$ is finite", "(Morphisms, Lemma \\ref{morphisms-lemma-nagata-normalization})", "and an isomorphism away from the special fibre. Say $Y^\\nu = \\Spec(C')$.", "Then $C \\to C'$ is finite and an isomorphism away from $V(\\mathfrak m C)$.", "Since $B \\to C$ is flat and induces an isomorphism", "$B/\\mathfrak m B \\to C/\\mathfrak m C$ there exists a finite", "ring map $B \\to B'$ whose base change to $C$ recovers $C \\to C'$.", "See More on Algebra, Lemma", "\\ref{more-algebra-lemma-application-formal-glueing} and", "Remark \\ref{more-algebra-remark-formal-glueing-algebras}.", "Thus we find a finite morphism $X' \\to X$ which is an isomorphism", "away from the special fibre and whose base change recovers $Y^\\nu \\to Y$.", "By the discussion in the first paragraph we see that $X'$ is normal at", "points not on the special fibre. For a point $x \\in X'$ on the special", "fibre we have a corresponding point $y \\in Y^\\nu$ and a flat map", "$\\mathcal{O}_{X', x} \\to \\mathcal{O}_{Y^\\nu, y}$.", "Since $\\mathcal{O}_{Y^\\nu, y}$ is normal, so is $\\mathcal{O}_{X', x}$, see", "Algebra, Lemma \\ref{algebra-lemma-descent-normal}.", "Thus $X'$ is normal and it follows that it is the normalization of $X$." ], "refs": [ "algebra-lemma-descent-normal", "algebra-lemma-Noetherian-complete-local-Nagata", "morphisms-lemma-nagata-normalization", "more-algebra-lemma-application-formal-glueing", "more-algebra-remark-formal-glueing-algebras", "algebra-lemma-descent-normal" ], "ref_ids": [ 1372, 1353, 5520, 10352, 10663, 1372 ] } ], "ref_ids": [] }, { "id": 11685, "type": "theorem", "label": "resolve-lemma-normalized-blowup-completion", "categories": [ "resolve" ], "title": "resolve-lemma-normalized-blowup-completion", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local domain whose completion", "$A^\\wedge$ is normal. Then given any sequence", "$$", "Y_n \\to Y_{n - 1} \\to \\ldots \\to Y_1 \\to \\Spec(A^\\wedge)", "$$", "of normalized blowups, there exists a sequence of (proper) normalized blowups", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to \\Spec(A)", "$$", "whose base change to $A^\\wedge$ recovers the given sequence." ], "refs": [], "proofs": [ { "contents": [ "Given the sequence $Y_n \\to \\ldots \\to Y_1 \\to Y_0 = \\Spec(A^\\wedge)$ we", "inductively construct $X_n \\to \\ldots \\to X_1 \\to X_0 = \\Spec(A)$.", "The base case is $i = 0$. Given $X_i$ whose base change is $Y_i$,", "let $Y'_i \\to Y_i$ be the blowing up in the closed point $y_i \\in Y_i$", "such that $Y_{i + 1}$ is the normalization of $Y_i$.", "Since the closed fibres of $Y_i$ and $X_i$ are isomorphic, the point", "$y_i$ corresponds to a closed point $x_i$ on the special fibre of $X_i$.", "Let $X'_i \\to X_i$ be the blowup of $X_i$ in $x_i$. Then the base change", "of $X'_i$ to $\\Spec(A^\\wedge)$ is isomorphic to $Y'_i$. ", "By Lemma \\ref{lemma-normalization-completion}", "the normalization $X_{i + 1} \\to X'_i$ is finite and its base change", "to $\\Spec(A^\\wedge)$ is isomorphic to $Y_{i + 1}$." ], "refs": [ "resolve-lemma-normalization-completion" ], "ref_ids": [ 11684 ] } ], "ref_ids": [] }, { "id": 11686, "type": "theorem", "label": "resolve-lemma-issquare", "categories": [ "resolve" ], "title": "resolve-lemma-issquare", "contents": [ "Let $\\kappa$ be a field. Let $I \\subset \\kappa[x, y]$ be an ideal. Let", "$$", "a + b x + c y + d x^2 + exy + f y^2 \\in I^2", "$$", "for some $a, b, c, d, e, f \\in k$ not all zero. If the colength", "of $I$ in $\\kappa[x, y]$ is $> 1$, then", "$a + b x + c y + d x^2 + exy + f y^2 = j(g + hx + iy)^2$", "for some $j, g, h, i \\in \\kappa$." ], "refs": [], "proofs": [ { "contents": [ "Consider the partial derivatives $b + 2dx + ey$ and", "$c + ex + 2fy$. By the Leibniz rules these are contained in $I$.", "If one of these is nonzero, then after a linear change of coordinates,", "i.e., of the form $x \\mapsto \\alpha + \\beta x + \\gamma y$ and", "$y \\mapsto \\delta + \\epsilon x + \\zeta y$, we may assume", "that $x \\in I$. Then we see that $I = (x)$ or $I = (x, F)$ with", "$F$ a monic polynomial of degree $\\geq 2$ in $y$.", "In the first case the statement is clear. In the second case", "observe that we can write any element in $I^2$ in the form", "$$", "A(x, y) x^2 + B(y) x F + C(y) F^2", "$$", "for some $A(x, y) \\in \\kappa[x, y]$ and $B, C \\in \\kappa[y]$.", "Thus", "$$", "a + b x + c y + d x^2 + exy + f y^2 = A(x, y) x^2 + B(y) x F + C(y) F^2", "$$", "and by degree reasons we see that $B = C = 0$ and $A$ is a constant.", "\\medskip\\noindent", "To finish the proof we need to deal with the case that both", "partial derivatives are zero. This can only happen in characteristic $2$", "and then we get", "$$", "a + d x^2 + f y^2 \\in I^2", "$$", "We may assume $f$ is nonzero (if not, then switch the roles of $x$ and $y$).", "After dividing by $f$ we obtain the case where the characteristic of", "$\\kappa$ is $2$ and", "$$", "a + d x^2 + y^2 \\in I^2", "$$", "If $a$ and $d$ are squares in $\\kappa$, then we are done. If not,", "then there exists a derivation $\\theta : \\kappa \\to \\kappa$ with", "$\\theta(a) \\not = 0$ or $\\theta(d) \\not = 0$, see", "Algebra, Lemma \\ref{algebra-lemma-derivative-zero-pth-power}.", "We can extend this to a derivation of $\\kappa[x, y]$ by setting", "$\\theta(x) = \\theta(y) = 0$. Then we find that", "$$", "\\theta(a) + \\theta(d) x^2 \\in I", "$$", "The case $\\theta(d) = 0$ is absurd. Thus we may assume", "that $\\alpha + x^2 \\in I$ for some $\\alpha \\in \\kappa$.", "Combining with the above we find that $a + \\alpha d + y^2 \\in I$.", "Hence", "$$", "J = (\\alpha + x^2, a + \\alpha d + y^2) \\subset I", "$$", "with codimension at most $2$. Observe that", "$J/J^2$ is free over $\\kappa[x, y]/J$ with basis", "$\\alpha + x^2$ and $a + \\alpha d + y^2$.", "Thus $a + d x^2 + y^2 =", "1 \\cdot (a + \\alpha d + y^2) + d \\cdot (\\alpha + x^2) \\in I^2$", "implies that the inclusion $J \\subset I$ is strict.", "Thus we find a nonzero element of the form $g + hx + iy + jxy$ in $I$.", "If $j = 0$, then $I$ contains a linear form and we can", "conclude as in the first paragraph. Thus $j \\not = 0$", "and $\\dim_\\kappa(I/J) = 1$ (otherwise we could find", "an element as above in $I$ with $j = 0$).", "We conclude that $I$ has the form", "$(\\alpha + x^2, \\beta + y^2, g + hx + iy + jxy)$", "with $j \\not = 0$ and has colength $3$.", "In this case $a + dx^2 + y^2 \\in I^2$ is impossible.", "This can be shown by a direct computation, but we prefer to argue", "as follows. Namely, to prove this statement we may assume that", "$\\kappa$ is algebraically closed. Then we can do a coordinate", "change $x \\mapsto \\sqrt{\\alpha} + x$ and $y \\mapsto \\sqrt{\\beta} + y$", "and assume that $I = (x^2, y^2, g' + h'x + i'y + jxy)$ with the same $j$.", "Then $g' = h' = i' = 0$ otherwise the colength of $I$ is not $3$.", "Thus we get $I = (x^2, y^2, xy)$ and the result is clear." ], "refs": [ "algebra-lemma-derivative-zero-pth-power" ], "ref_ids": [ 1315 ] } ], "ref_ids": [] }, { "id": 11687, "type": "theorem", "label": "resolve-lemma-resolve-rational-double-points", "categories": [ "resolve" ], "title": "resolve-lemma-resolve-rational-double-points", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata", "domain of dimension $2$ which defines a rational singularity,", "whose completion is normal, and which is Gorenstein.", "Then there exists a finite sequence of blowups in", "singular closed points", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = \\Spec(A)", "$$", "such that $X_n$ is regular and such that each intervening", "schemes $X_i$ is normal with finitely many singular points", "of the same type." ], "refs": [], "proofs": [ { "contents": [ "This is exactly what was proved in the discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11688, "type": "theorem", "label": "resolve-lemma-regular-alteration-implies", "categories": [ "resolve" ], "title": "resolve-lemma-regular-alteration-implies", "contents": [ "Let $Y$ be a Noetherian integral scheme. Assume there exists an alteration", "$f : X \\to Y$ with $X$ regular. Then the normalization $Y^\\nu \\to Y$", "is finite and $Y$ has a dense open which is regular." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove this when $Y = \\Spec(A)$ where $A$ is a Noetherian domain.", "Let $B$ be the integral closure of $A$ in its fraction field.", "Set $C = \\Gamma(X, \\mathcal{O}_X)$. By", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-over-affine-cohomology-finite}", "we see that $C$ is a finite $A$-module. As $X$ is normal", "(Properties, Lemma \\ref{properties-lemma-regular-normal})", "we see that $C$ is normal domain", "(Properties, Lemma \\ref{properties-lemma-normal-integral-sections}).", "Thus $B \\subset C$ and we conclude that $B$ is finite over $A$", "as $A$ is Noetherian.", "\\medskip\\noindent", "There exists a nonempty open $V \\subset Y$ such that $f^{-1}V \\to V$", "is finite, see Morphisms, Definition \\ref{morphisms-definition-alteration}.", "After shrinking $V$ we may assume that $f^{-1}V \\to V$ is flat", "(Morphisms, Proposition \\ref{morphisms-proposition-generic-flatness}).", "Thus $f^{-1}V \\to V$ is faithfully flat. Then $V$ is regular by", "Algebra, Lemma \\ref{algebra-lemma-descent-regular}." ], "refs": [ "coherent-lemma-proper-over-affine-cohomology-finite", "properties-lemma-regular-normal", "properties-lemma-normal-integral-sections", "morphisms-definition-alteration", "morphisms-proposition-generic-flatness", "algebra-lemma-descent-regular" ], "ref_ids": [ 3355, 2977, 2972, 5589, 5533, 1373 ] } ], "ref_ids": [] }, { "id": 11689, "type": "theorem", "label": "resolve-lemma-algebra-helper", "categories": [ "resolve" ], "title": "resolve-lemma-algebra-helper", "contents": [ "Let $(A, \\mathfrak m)$ be a local Noetherian ring. Let $B \\subset C$", "be finite $A$-algebras. Assume that (a) $B$ is a normal ring, and", "(b) the $\\mathfrak m$-adic completion $C^\\wedge$ is a normal ring.", "Then $B^\\wedge$ is a normal ring." ], "refs": [], "proofs": [ { "contents": [ "Consider the commutative diagram", "$$", "\\xymatrix{", "B \\ar[r] \\ar[d] & C \\ar[d] \\\\", "B^\\wedge \\ar[r] & C^\\wedge", "}", "$$", "Recall that $\\mathfrak m$-adic completion on the category of", "finite $A$-modules is exact because it is given by tensoring with", "the flat $A$-algebra $A^\\wedge$", "(Algebra, Lemma \\ref{algebra-lemma-completion-flat}).", "We will use Serre's criterion", "(Algebra, Lemma \\ref{algebra-lemma-criterion-normal})", "to prove that the Noetherian ring $B^\\wedge$ is normal.", "Let $\\mathfrak q \\subset B^\\wedge$ be a prime lying over", "$\\mathfrak p \\subset B$. If $\\dim(B_\\mathfrak p) \\geq 2$, then", "$\\text{depth}(B_\\mathfrak p) \\geq 2$ and since", "$B_\\mathfrak p \\to B^\\wedge_\\mathfrak q$ is flat we find", "that $\\text{depth}(B^\\wedge_\\mathfrak q) \\geq 2$", "(Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck}).", "If $\\dim(B_\\mathfrak p) \\leq 1$, then $B_\\mathfrak p$ is", "either a discrete valuation ring or a field.", "In that case $C_\\mathfrak p$ is faithfully flat over $B_\\mathfrak p$", "(because it is finite and torsion free).", "Hence $B^\\wedge_\\mathfrak p \\to C^\\wedge_\\mathfrak p$ is", "faithfully flat and the same holds after localizing at $\\mathfrak q$.", "As $C^\\wedge$ and hence any localization is $(S_2)$ we conclude that", "$B^\\wedge_\\mathfrak p$ is $(S_2)$ by", "Algebra, Lemma \\ref{algebra-lemma-descent-Sk}.", "All in all we find that", "$(S_2)$ holds for $B^\\wedge$. To prove that $B^\\wedge$ is", "$(R_1)$ we only have to consider primes $\\mathfrak q \\subset B^\\wedge$", "with $\\dim(B^\\wedge_\\mathfrak q) \\leq 1$. Since", "$\\dim(B^\\wedge_\\mathfrak q) = \\dim(B_\\mathfrak p) +", "\\dim(B^\\wedge_\\mathfrak q/\\mathfrak p B^\\wedge_\\mathfrak q)$ by", "Algebra, Lemma \\ref{algebra-lemma-dimension-base-fibre-total}", "we find that $\\dim(B_\\mathfrak p) \\leq 1$ and", "we see that $B^\\wedge_\\mathfrak q \\to C^\\wedge_\\mathfrak q$", "is faithfully flat as before. We conclude using", "Algebra, Lemma \\ref{algebra-lemma-descent-Rk}." ], "refs": [ "algebra-lemma-completion-flat", "algebra-lemma-criterion-normal", "algebra-lemma-apply-grothendieck", "algebra-lemma-descent-Sk", "algebra-lemma-dimension-base-fibre-total", "algebra-lemma-descent-Rk" ], "ref_ids": [ 870, 1311, 1361, 1374, 986, 1375 ] } ], "ref_ids": [] }, { "id": 11690, "type": "theorem", "label": "resolve-lemma-regular-alteration-implies-local", "categories": [ "resolve" ], "title": "resolve-lemma-regular-alteration-implies-local", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a local Noetherian domain.", "Assume there exists an alteration $f : X \\to \\Spec(A)$", "with $X$ regular. Then", "\\begin{enumerate}", "\\item there exists a nonzero $f \\in A$ such that $A_f$ is regular,", "\\item the integral closure $B$ of $A$ in its fraction field is finite over $A$,", "\\item the $\\mathfrak m$-adic completion of $B$ is a normal ring, i.e., the", "completions of $B$ at its maximal ideals are normal domains, and", "\\item the generic formal fibre of $A$ is regular.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1) and (2) follow from Lemma \\ref{lemma-regular-alteration-implies}.", "We have to redo part of the proof of that lemma in order to set up notation", "for the proof of (3). Set $C = \\Gamma(X, \\mathcal{O}_X)$. By", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-over-affine-cohomology-finite}", "we see that $C$ is a finite $A$-module. As $X$ is normal", "(Properties, Lemma \\ref{properties-lemma-regular-normal})", "we see that $C$ is normal domain", "(Properties, Lemma \\ref{properties-lemma-normal-integral-sections}).", "Thus $B \\subset C$ and we conclude that $B$ is finite over $A$", "as $A$ is Noetherian. By Lemma \\ref{lemma-algebra-helper}", "in order to prove (3) it suffices to show", "that the $\\mathfrak m$-adic completion $C^\\wedge$ is normal.", "\\medskip\\noindent", "By Algebra, Lemma \\ref{algebra-lemma-completion-finite-extension}", "the completion $C^\\wedge$ is the product of the completions of", "$C$ at the prime ideals of $C$ lying over $\\mathfrak m$.", "There are finitely many of these and these are the maximal", "ideals $\\mathfrak m_1, \\ldots, \\mathfrak m_r$ of $C$.", "(The corresponding result for $B$ explains the final statement of the lemma.)", "Thus replacing $A$ by $C_{\\mathfrak m_i}$ and $X$ by", "$X_i = X \\times_{\\Spec(C)} \\Spec(C_{\\mathfrak m_i})$", "we reduce to the case discussed in the next paragraph.", "(Note that $\\Gamma(X_i, \\mathcal{O}) = C_{\\mathfrak m_i}$ by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}.)", "\\medskip\\noindent", "Here $A$ is a Noetherian local normal domain and $f : X \\to \\Spec(A)$ is a", "regular alteration with $\\Gamma(X, \\mathcal{O}_X) = A$.", "We have to show that the completion $A^\\wedge$", "of $A$ is a normal domain. By", "Lemma \\ref{lemma-port-regularity-to-completion}", "$Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$ is regular.", "Since $\\Gamma(Y, \\mathcal{O}_Y) = A^\\wedge$ by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology},", "we conclude that $A^\\wedge$ is normal as before.", "Namely, $Y$ is normal by", "Properties, Lemma \\ref{properties-lemma-regular-normal}.", "It is connected because $\\Gamma(Y, \\mathcal{O}_Y) = A^\\wedge$ is local.", "Hence $Y$ is normal and integral (as connected and normal", "implies integral for Noetherian schemes). Thus", "$\\Gamma(Y, \\mathcal{O}_Y) = A^\\wedge$", "is a normal domain by", "Properties, Lemma \\ref{properties-lemma-normal-integral-sections}.", "This proves (3).", "\\medskip\\noindent", "Proof of (4). Let $\\eta \\in \\Spec(A)$ denote the generic point", "and denote by a subscript $\\eta$ the base change to $\\eta$.", "Since $f$ is an alteration, the scheme $X_\\eta$ is finite and", "faithfully flat over $\\eta$. Since $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$", "is regular by Lemma \\ref{lemma-port-regularity-to-completion}", "we see that $Y_\\eta$ is regular (as a limit of opens in $Y$).", "Then $Y_\\eta \\to \\Spec(A^\\wedge \\otimes_A \\kappa(\\eta))$ is finite", "faithfully flat onto the generic formal fibre. We conclude by", "Algebra, Lemma \\ref{algebra-lemma-descent-regular}." ], "refs": [ "resolve-lemma-regular-alteration-implies", "coherent-lemma-proper-over-affine-cohomology-finite", "properties-lemma-regular-normal", "properties-lemma-normal-integral-sections", "resolve-lemma-algebra-helper", "algebra-lemma-completion-finite-extension", "coherent-lemma-flat-base-change-cohomology", "resolve-lemma-port-regularity-to-completion", "coherent-lemma-flat-base-change-cohomology", "properties-lemma-regular-normal", "properties-lemma-normal-integral-sections", "resolve-lemma-port-regularity-to-completion", "algebra-lemma-descent-regular" ], "ref_ids": [ 11688, 3355, 2977, 2972, 11689, 876, 3298, 11680, 3298, 2977, 2972, 11680, 1373 ] } ], "ref_ids": [] }, { "id": 11691, "type": "theorem", "label": "resolve-lemma-existence-implies-existence-by-normalized-blowing-ups", "categories": [ "resolve" ], "title": "resolve-lemma-existence-implies-existence-by-normalized-blowing-ups", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Assume $A$ is normal and has dimension $2$.", "If $\\Spec(A)$ has a resolution of singularities,", "then $\\Spec(A)$ has a resolution by normalized blowups." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-regular-alteration-implies-local}", "the completion $A^\\wedge$ of $A$ is normal.", "By Lemma \\ref{lemma-port-regularity-to-completion} we see", "that $\\Spec(A^\\wedge)$ has a resolution.", "By Lemma \\ref{lemma-normalized-blowup-completion}", "any sequence $Y_n \\to Y_{n - 1} \\to \\ldots \\to \\Spec(A^\\wedge)$", "of normalized blowups of comes from a sequence of normalized", "blowups $X_n \\to \\ldots \\to \\Spec(A)$. Moreover if $Y_n$ is", "regular, then $X_n$ is regular by", "Lemma \\ref{lemma-port-regularity-to-completion}.", "Thus it suffices to prove the lemma in case $A$ is complete.", "\\medskip\\noindent", "Assume in addition $A$ is a complete. We will use that $A$ is Nagata", "(Algebra, Proposition \\ref{algebra-proposition-ubiquity-nagata}),", "excellent (More on Algebra, Proposition", "\\ref{more-algebra-proposition-ubiquity-excellent}),", "and has a dualizing complex", "(Dualizing Complexes, Lemma \\ref{dualizing-lemma-ubiquity-dualizing}).", "Moreover, the same is true for any ring essentially of finite type over $A$.", "If $B$ is a excellent local normal domain, then the completion", "$B^\\wedge$ is normal (as $B \\to B^\\wedge$ is regular and", "More on Algebra, Lemma \\ref{more-algebra-lemma-normal-goes-up} applies).", "We will use this without further mention in the rest of the proof.", "\\medskip\\noindent", "Let $X \\to \\Spec(A)$ be a resolution of singularities.", "Choose a sequence of normalized blowing ups", "$$", "Y_n \\to Y_{n - 1} \\to \\ldots \\to Y_1 \\to \\Spec(A)", "$$", "dominating $X$ (Lemma \\ref{lemma-dominate-by-normalized-blowing-up}).", "The morphism $Y_n \\to X$ is an isomorphism away from", "finitely many points of $X$.", "Hence we can apply Lemma \\ref{lemma-dominate-by-blowing-up-in-points}", "to find a sequence of blowing ups", "$$", "X_m \\to X_{m - 1} \\to \\ldots \\to X", "$$", "in closed points such that $X_m$ dominates $Y_n$. Diagram", "$$", "\\xymatrix{", "& Y_n \\ar[rd] \\ar[rr] & & \\Spec(A) \\\\", "X_m \\ar[rr] \\ar[ru] & & X \\ar[ru]", "}", "$$", "To prove the lemma it suffices to show that a finite number of normalized", "blowups of $Y_n$ produce a regular scheme. By our diagram above we see that", "$Y_n$ has a resolution (namely $X_m$). As $Y_n$ is a normal surface", "this implies that $Y_n$ has at most finitely many singularities", "$y_1, \\ldots, y_t$ (because $X_m \\to Y_n$ is an isomorphism away from", "the fibres of dimension $1$, see Varieties, Lemma", "\\ref{varieties-lemma-modification-normal-iso-over-codimension-1}).", "\\medskip\\noindent", "Let $x_a \\in X$ be the image of $y_a$. Then $\\mathcal{O}_{X, x_a}$", "is regular and hence defines a rational singularity", "(Lemma \\ref{lemma-regular-rational}).", "Apply Lemma \\ref{lemma-rational-propagates} to", "$\\mathcal{O}_{X, x_a} \\to \\mathcal{O}_{Y_n, y_a}$", "to see that $\\mathcal{O}_{Y_n, y_a}$ defines a", "rational singularity. By Lemma \\ref{lemma-rational-to-gorenstein}", "there exists a finite sequence of blowups in singular closed points", "$$", "Y_{a, n_a} \\to Y_{a, n_a - 1} \\to \\ldots \\to \\Spec(\\mathcal{O}_{Y_n, y_a})", "$$", "such that $Y_{a, n_a}$ is Gorenstein, i.e., has an", "invertible dualizing module. By (the essentially trivial)", "Lemma \\ref{lemma-equivalence-sequence-blowups}", "with $n' = \\sum n_a$ these sequences correspond to a sequence of", "blowups", "$$", "Y_{n + n'} \\to Y_{n + n' - 1} \\to \\ldots \\to Y_n", "$$", "such that $Y_{n + n'}$ is normal and", "the local rings of $Y_{n + n'}$ are Gorenstein. Using the references", "given above we can dominate $Y_{n + n'}$ by a sequence of blowups", "$X_{m + m'} \\to \\ldots \\to X_m$ dominating $Y_{n + n'}$ as in the following", "$$", "\\xymatrix{", " & Y_{n + n'} \\ar[rr] & & Y_n \\ar[rd] \\ar[rr] & & \\Spec(A) \\\\", "X_{m + m'} \\ar[ru] \\ar[rr] & & X_m \\ar[rr] \\ar[ru] & & X \\ar[ru]", "}", "$$", "Thus again $Y_{n + n'}$ has a finite number of singular points", "$y'_1, \\ldots, y'_s$, but this time the singularities are", "rational double points, more precisely, the local rings", "$\\mathcal{O}_{Y_{n + n'}, y'_b}$ are as in", "Lemma \\ref{lemma-resolve-rational-double-points}.", "Arguing exactly as above we conclude that the lemma is true." ], "refs": [ "resolve-lemma-regular-alteration-implies-local", "resolve-lemma-port-regularity-to-completion", "resolve-lemma-normalized-blowup-completion", "resolve-lemma-port-regularity-to-completion", "algebra-proposition-ubiquity-nagata", "more-algebra-proposition-ubiquity-excellent", "dualizing-lemma-ubiquity-dualizing", "more-algebra-lemma-normal-goes-up", "resolve-lemma-dominate-by-normalized-blowing-up", "resolve-lemma-dominate-by-blowing-up-in-points", "varieties-lemma-modification-normal-iso-over-codimension-1", "resolve-lemma-regular-rational", "resolve-lemma-rational-propagates", "resolve-lemma-rational-to-gorenstein", "resolve-lemma-equivalence-sequence-blowups", "resolve-lemma-resolve-rational-double-points" ], "ref_ids": [ 11690, 11680, 11685, 11680, 1431, 10584, 2890, 10041, 11649, 11646, 10979, 11666, 11663, 11676, 11654, 11687 ] } ], "ref_ids": [] }, { "id": 11692, "type": "theorem", "label": "resolve-lemma-resolve-complete", "categories": [ "resolve" ], "title": "resolve-lemma-resolve-complete", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian complete local ring.", "Assume $A$ is a normal domain of dimension $2$. Then $\\Spec(A)$ has a", "resolution of singularities." ], "refs": [], "proofs": [ { "contents": [ "A Noetherian complete local ring is J-2", "(More on Algebra, Proposition \\ref{more-algebra-proposition-ubiquity-J-2}),", "Nagata (Algebra, Proposition \\ref{algebra-proposition-ubiquity-nagata}),", "excellent (More on Algebra, Proposition", "\\ref{more-algebra-proposition-ubiquity-excellent}),", "and has a dualizing complex", "(Dualizing Complexes, Lemma \\ref{dualizing-lemma-ubiquity-dualizing}).", "Moreover, the same is true for any ring essentially of finite type over $A$.", "If $B$ is a excellent local normal domain, then the completion", "$B^\\wedge$ is normal (as $B \\to B^\\wedge$ is regular and", "More on Algebra, Lemma \\ref{more-algebra-lemma-normal-goes-up} applies).", "In other words, the local rings which we encounter in the rest of the proof", "will have the required ``excellency'' properties required of them.", "\\medskip\\noindent", "Choose $A_0 \\subset A$ with $A_0$ a regular complete local ring", "and $A_0 \\to A$ finite, see Algebra, Lemma", "\\ref{algebra-lemma-complete-local-Noetherian-domain-finite-over-regular}.", "This induces a finite extension of fraction fields $K_0 \\subset K$.", "We will argue by induction on $[K : K_0]$. The base case is", "when the degree is $1$ in which case $A_0 = A$ and the result is true.", "\\medskip\\noindent", "Suppose there is an intermediate field $K_0 \\subset L \\subset K$,", "$K_0 \\not = L \\not = K$. Let $B \\subset A$ be the integral closure", "of $A_0$ in $L$. By induction we choose a resolution of singularities", "$Y \\to \\Spec(B)$. Let $X$ be the normalization", "of $Y \\times_{\\Spec(B)} \\Spec(A)$. Picture:", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d] & \\Spec(A) \\ar[d] \\\\", "Y \\ar[r] & \\Spec(B)", "}", "$$", "Since $A$ is J-2 the regular locus of $X$ is open. Since $X$", "is a normal surface we conclude that $X$ has at worst finitely", "many singular points $x_1, \\ldots, x_n$ which are closed points with", "$\\dim(\\mathcal{O}_{X, x_i}) = 2$.", "For each $i$ let $y_i \\in Y$ be the image.", "Since", "$\\mathcal{O}_{Y, y_i}^\\wedge \\to \\mathcal{O}_{X, x_i}^\\wedge$", "is finite of smaller degree than before we conclude by", "induction hypothesis that $\\mathcal{O}_{X, x_i}^\\wedge$", "has resolution of singularities. By", "Lemma \\ref{lemma-existence-implies-existence-by-normalized-blowing-ups}", "there is a sequence", "$$", "Z^\\wedge_{i, n_i} \\to \\ldots \\to Z^\\wedge_{i, 1} \\to", "\\Spec(\\mathcal{O}_{X, x_i}^\\wedge)", "$$", "of normalized blowups with $Z^\\wedge_{i, n_i}$ regular.", "By Lemma \\ref{lemma-normalized-blowup-completion}", "there is a corresponding sequence of normalized blowing ups", "$$", "Z_{i, n_i} \\to \\ldots \\to Z_{i, 1} \\to \\Spec(\\mathcal{O}_{X, x_i})", "$$", "Then $Z_{i, n_i}$ is a regular scheme by", "Lemma \\ref{lemma-port-regularity-to-completion}.", "By Lemma \\ref{lemma-equivalence-sequence-normalized-blowups}", "we can fit these normalized blowing ups", "into a corresponding sequence", "$$", "Z_n \\to Z_{n - 1} \\to \\ldots \\to Z_1 \\to X", "$$", "and of course $Z_n$ is regular too (look at the local rings).", "This proves the induction step.", "\\medskip\\noindent", "Assume there is no intermediate field $K_0 \\subset L \\subset K$", "with $K_0 \\not = L \\not = K$. Then either $K/K_0$ is separable", "or the characteristic to $K$ is $p$ and $[K : K_0] = p$.", "Then either Lemma \\ref{lemma-go-up-separable} or \\ref{lemma-go-up-degree-p}", "implies that reduction to rational singularities is possible.", "By Lemma \\ref{lemma-reduce-to-rational} we conclude that there exists a", "normal modification $X \\to \\Spec(A)$ such that for", "every singular point $x$ of $X$ the local ring $\\mathcal{O}_{X, x}$", "defines a rational singularity. Since $A$ is J-2 we find that $X$ has", "finitely many singular points $x_1, \\ldots, x_n$.", "By Lemma \\ref{lemma-rational-to-gorenstein}", "there exists a finite sequence of blowups in singular closed points", "$$", "X_{i, n_i} \\to X_{i, n_i - 1} \\to \\ldots \\to \\Spec(\\mathcal{O}_{X, x_i})", "$$", "such that $X_{i, n_i}$ is Gorenstein, i.e., has an", "invertible dualizing module. By (the essentially trivial)", "Lemma \\ref{lemma-equivalence-sequence-blowups}", "with $n = \\sum n_a$ these sequences correspond to a sequence of", "blowups", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X", "$$", "such that $X_n$ is normal and the local rings of $X_n$ are Gorenstein.", "Again $X_n$ has a finite number of singular points", "$x'_1, \\ldots, x'_s$, but this time the singularities are", "rational double points, more precisely, the local rings", "$\\mathcal{O}_{X_n, x'_i}$ are as in", "Lemma \\ref{lemma-resolve-rational-double-points}.", "Arguing exactly as above we conclude that the lemma is true." ], "refs": [ "more-algebra-proposition-ubiquity-J-2", "algebra-proposition-ubiquity-nagata", "more-algebra-proposition-ubiquity-excellent", "dualizing-lemma-ubiquity-dualizing", "more-algebra-lemma-normal-goes-up", "algebra-lemma-complete-local-Noetherian-domain-finite-over-regular", "resolve-lemma-existence-implies-existence-by-normalized-blowing-ups", "resolve-lemma-normalized-blowup-completion", "resolve-lemma-port-regularity-to-completion", "resolve-lemma-equivalence-sequence-normalized-blowups", "resolve-lemma-go-up-separable", "resolve-lemma-go-up-degree-p", "resolve-lemma-reduce-to-rational", "resolve-lemma-rational-to-gorenstein", "resolve-lemma-equivalence-sequence-blowups", "resolve-lemma-resolve-rational-double-points" ], "ref_ids": [ 10578, 1431, 10584, 2890, 10041, 1332, 11691, 11685, 11680, 11655, 11665, 11669, 11664, 11676, 11654, 11687 ] } ], "ref_ids": [] }, { "id": 11693, "type": "theorem", "label": "resolve-lemma-resolve-curve", "categories": [ "resolve" ], "title": "resolve-lemma-resolve-curve", "contents": [ "Let $Y$ be a one dimensional integral Noetherian scheme.", "The following are equivalent", "\\begin{enumerate}", "\\item there exists an alteration $X \\to Y$ with $X$ regular,", "\\item there exists a resolution of singularities of $Y$,", "\\item there exists a finite sequence", "$Y_n \\to Y_{n - 1} \\to \\ldots \\to Y_1 \\to Y$ of blowups", "in closed points with $Y_n$ regular, and", "\\item the normalization $Y^\\nu \\to Y$ is finite.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implications (3) $\\Rightarrow$ (2) $\\Rightarrow$ (1) are immediate.", "The implication (1) $\\Rightarrow$ (4) follows from", "Lemma \\ref{lemma-regular-alteration-implies}.", "Observe that a normal one dimensional scheme is regular hence", "the implication (4) $\\Rightarrow$ (2) is clear as well.", "Thus it remains to show that the equivalent conditions (1), (2), and", "(4) imply (3).", "\\medskip\\noindent", "Let $f : X \\to Y$ be a resolution of singularities. Since the dimension", "of $Y$ is one we see that $f$ is finite by", "Varieties, Lemma \\ref{varieties-lemma-finite-in-codim-1}.", "We will construct factorizations", "$$", "X \\to \\ldots \\to Y_2 \\to Y_1 \\to Y", "$$", "where $Y_i \\to Y_{i - 1}$ is a blowing up of a closed point and not", "an isomorphism as long as $Y_{i - 1}$ is not regular.", "Each of these morphisms will be finite (by the same", "reason as above) and we will get a corresponding system", "$$", "f_*\\mathcal{O}_X \\supset \\ldots \\supset", "f_{2, *}\\mathcal{O}_{Y_2} \\supset", "f_{1, *}\\mathcal{O}_{Y_1} \\supset \\mathcal{O}_Y", "$$", "where $f_i : Y_i \\to Y$ is the structure morphism.", "Since $Y$ is Noetherian, this increasing sequence of coherent submodules", "must stabilize", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-acc-coherent})", "which proves that for some $n$ the scheme $Y_n$ is regular", "as desired. To construct $Y_i$ given $Y_{i - 1}$ we pick a singular", "closed point $y_{i - 1} \\in Y_{i - 1}$ and we let $Y_i \\to Y_{i - 1}$", "be the corresponding blowup. Since $X$ is regular of dimension $1$", "(and hence the local rings at closed points are discrete valuation", "rings and in particular PIDs), the ideal sheaf", "$\\mathfrak m_{y_{i - 1}} \\cdot \\mathcal{O}_X$ is invertible.", "By the universal property of blowing up (Divisors, Lemma", "\\ref{divisors-lemma-universal-property-blowing-up})", "this gives us a factorization", "$X \\to Y_i$.", "Finally, $Y_i \\to Y_{i - 1}$ is not an isomorphism as", "$\\mathfrak m_{y_{i - 1}}$ is not an invertible ideal." ], "refs": [ "resolve-lemma-regular-alteration-implies", "varieties-lemma-finite-in-codim-1", "coherent-lemma-acc-coherent", "divisors-lemma-universal-property-blowing-up" ], "ref_ids": [ 11688, 10978, 3319, 8055 ] } ], "ref_ids": [] }, { "id": 11694, "type": "theorem", "label": "resolve-lemma-blowup-curve", "categories": [ "resolve" ], "title": "resolve-lemma-blowup-curve", "contents": [ "Let $X$ be a Noetherian scheme. Let $Y \\subset X$ be an integral closed", "subscheme of dimension $1$ satisfying the equivalent conditions of", "Lemma \\ref{lemma-resolve-curve}. Then there exists a finite sequence", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X", "$$", "of blowups in closed points such that the strict transform of $Y$", "in $X_n$ is a regular curve." ], "refs": [ "resolve-lemma-resolve-curve" ], "proofs": [ { "contents": [ "Let $Y_n \\to Y_{n - 1} \\to \\ldots \\to Y_1 \\to Y$ be the sequence of", "blowups given to us by Lemma \\ref{lemma-resolve-curve}. Let", "$X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X$ be the corresponding", "sequence of blowups of $X$. This works because the strict transform", "is the blowup by Divisors, Lemma \\ref{divisors-lemma-strict-transform}." ], "refs": [ "resolve-lemma-resolve-curve", "divisors-lemma-strict-transform" ], "ref_ids": [ 11693, 8065 ] } ], "ref_ids": [ 11693 ] }, { "id": 11695, "type": "theorem", "label": "resolve-lemma-blowup-nonsingular-curves-meeting-at-point", "categories": [ "resolve" ], "title": "resolve-lemma-blowup-nonsingular-curves-meeting-at-point", "contents": [ "In the situation above let $X' \\to X$ be the blowing up of $X$ in $p$.", "Let $Y', Z' \\subset X'$ be the strict transforms of $Y, Z$.", "If $\\mathcal{O}_{Y, p}$ is regular, then", "\\begin{enumerate}", "\\item $Y' \\to Y$ is an isomorphism,", "\\item $Y'$ meets the exceptional fibre $E \\subset X'$ in one point", "$q$ and $m_q(Y \\cap E) = 1$,", "\\item if $q \\in Z'$ too, then $m_q(Y \\cap Z') < m_p(Y \\cap Z)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathcal{O}_{X, p} \\to \\mathcal{O}_{Y, p}$ is surjective and", "$\\mathcal{O}_{Y, p}$ is a discrete valuation ring, we can pick an", "element $x_1 \\in \\mathfrak m_p$ mapping to a uniformizer in", "$\\mathcal{O}_{Y, p}$. Choose an affine open $U = \\Spec(A)$ containing", "$p$ such that $x_1 \\in A$. Let $\\mathfrak m \\subset A$ be the", "maximal ideal corresponding to $p$. Let $I, J \\subset A$", "be the ideals defining $Y, Z$ in $\\Spec(A)$. After shrinking $U$", "we may assume that $\\mathfrak m = I + (x_1)$, in other words,", "that $V(x_1) \\cap U \\cap Y = \\{p\\}$ scheme theoretically.", "We conclude that $p$ is an effective Cartier divisor on $Y$ and", "since $Y'$ is the blowing up of $Y$ in $p$", "(Divisors, Lemma \\ref{divisors-lemma-strict-transform})", "we see that $Y' \\to Y$ is an isomorphism by", "Divisors, Lemma \\ref{divisors-lemma-blow-up-effective-Cartier-divisor}.", "The relationship $\\mathfrak m = I + (x_1)$ implies that", "$\\mathfrak m^n \\subset I + (x_1^n)$ hence we can define a map", "$$", "\\psi : A[\\textstyle{\\frac{\\mathfrak m}{x_1}}] \\longrightarrow A/I", "$$", "by sending $y/x_1^n \\in A[\\frac{\\mathfrak m}{x_1}]$", "to the class of $a$ in $A/I$ where $a$", "is chosen such that $y \\equiv ax_1^n \\bmod I$.", "Then $\\psi$ corresponds to the morphism of $Y \\cap U$ into $X'$", "over $U$ given by $Y' \\cong Y$. Since the image of $x_1$", "in $A[\\frac{\\mathfrak m}{x_1}]$ cuts out the exceptional divisor", "we conclude that $m_q(Y', E) = 1$. Finally, since", "$J \\subset \\mathfrak m$ implies that the ideal", "$J' \\subset A[\\frac{\\mathfrak m}{x_1}]$ certainly", "contains the elements $f/x_1$ for $f \\in J$.", "Thus if we choose $f \\in J$ whose image $\\overline{f}$ in $A/I$ has", "minimal valuation equal to $m_p(Y \\cap Z)$, then we see that", "$\\psi(f/x_1) = \\overline{f}/x_1$ in $A/I$ has valuation one less", "proving the last part of the lemma." ], "refs": [ "divisors-lemma-strict-transform", "divisors-lemma-blow-up-effective-Cartier-divisor" ], "ref_ids": [ 8065, 8057 ] } ], "ref_ids": [] }, { "id": 11696, "type": "theorem", "label": "resolve-lemma-blowup-curves", "categories": [ "resolve" ], "title": "resolve-lemma-blowup-curves", "contents": [ "Let $X$ be a Noetherian scheme. Let $Y_i \\subset X$, $i = 1, \\ldots, n$", "be an integral closed subschemes of dimension $1$ each satisfying the", "equivalent conditions of Lemma \\ref{lemma-resolve-curve}. Then there", "exists a finite sequence", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X", "$$", "of blowups in closed points such that the strict transform $Y'_i \\subset X_n$", "of $Y_i$ in $X_n$ are pairwise disjoint regular curves." ], "refs": [ "resolve-lemma-resolve-curve" ], "proofs": [ { "contents": [ "It follows from Lemma \\ref{lemma-blowup-curve} that we may assume $Y_i$", "is a regular curve for $i = 1, \\ldots, n$. For every $i \\not = j$", "and $p \\in Y_i \\cap Y_j$ we have the invariant", "$m_p(Y_i \\cap Y_j)$ (\\ref{equation-multiplicity}).", "If the maximum of these numbers is $> 1$, then we can decrease", "it (Lemma \\ref{lemma-blowup-nonsingular-curves-meeting-at-point})", "by blowing up in all the points $p$ where the maximum is attained.", "If the maximum is $1$ then we can separate the curves using the", "same lemma by blowing up in all these points $p$." ], "refs": [ "resolve-lemma-blowup-curve", "resolve-lemma-blowup-nonsingular-curves-meeting-at-point" ], "ref_ids": [ 11694, 11695 ] } ], "ref_ids": [ 11693 ] }, { "id": 11697, "type": "theorem", "label": "resolve-lemma-turn-into-effective-Cartier", "categories": [ "resolve" ], "title": "resolve-lemma-turn-into-effective-Cartier", "contents": [ "Let $X$ be a regular scheme of dimension $2$. Let $Z \\subset X$", "be a proper closed subscheme. There exists a sequence", "$$", "X_n \\to \\ldots \\to X_1 \\to X", "$$", "of blowing ups in closed points such that the inverse image $Z_n$ of $Z$", "in $X_n$ is an effective Cartier divisor." ], "refs": [], "proofs": [ { "contents": [ "Let $D \\subset Z$ be the largest effective Cartier divisor contained in $Z$.", "Then $\\mathcal{I}_Z \\subset \\mathcal{I}_D$ and the quotient is supported", "in closed points by Divisors, Lemma \\ref{divisors-lemma-codim-1-part}.", "Thus we can write $\\mathcal{I}_Z = \\mathcal{I}_{Z'} \\mathcal{I}_D$", "where $Z' \\subset X$ is a closed subscheme which set theoretically", "consists of finitely many closed points. Applying", "Lemma \\ref{lemma-make-ideal-principal}", "we find a sequence of blowups as in the statement of our lemma", "such that $\\mathcal{I}_{Z'}\\mathcal{O}_{X_n}$ is invertible.", "This proves the lemma." ], "refs": [ "divisors-lemma-codim-1-part", "resolve-lemma-make-ideal-principal" ], "ref_ids": [ 7952, 11645 ] } ], "ref_ids": [] }, { "id": 11698, "type": "theorem", "label": "resolve-lemma-embedded-resolution", "categories": [ "resolve" ], "title": "resolve-lemma-embedded-resolution", "contents": [ "Let $X$ be a regular scheme of dimension $2$. Let $Z \\subset X$", "be a proper closed subscheme such that every irreducible component", "$Y \\subset Z$ of dimension $1$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-resolve-curve}. Then there exists a sequence", "$$", "X_n \\to \\ldots \\to X_1 \\to X", "$$", "of blowups in closed points such that the inverse image $Z_n$ of $Z$", "in $X_n$ is an effective Cartier divisor supported on a strict normal crossings", "divisor." ], "refs": [ "resolve-lemma-resolve-curve" ], "proofs": [ { "contents": [ "Let $X' \\to X$ be a blowup in a closed point $p$. Then the inverse image", "$Z' \\subset X'$ of $Z$ is supported on the strict transform of $Z$ and", "the exceptional divisor. The exceptional divisor is a regular curve", "(Lemma \\ref{lemma-blowup}) and the strict transform $Y'$ of each irreducible", "component $Y$ is either equal to $Y$ or the blowup of $Y$ at $p$.", "Thus in this process we do not produce additional singular components", "of dimension $1$. Thus it follows from", "Lemmas \\ref{lemma-turn-into-effective-Cartier} and \\ref{lemma-blowup-curves}", "that we may assume $Z$ is an effective Cartier divisor and", "that all irreducible components $Y$ of $Z$ are regular.", "(Of course we cannot assume the irreducible components are", "pairwise disjoint because in each blowup of a point of $Z$", "we add a new irreducible component to $Z$, namely the exceptional divisor.)", "\\medskip\\noindent", "Assume $Z$ is an effective Cartier divisor whose irreducible components", "$Y_i$ are regular. For every $i \\not = j$", "and $p \\in Y_i \\cap Y_j$ we have the invariant", "$m_p(Y_i \\cap Y_j)$ (\\ref{equation-multiplicity}).", "If the maximum of these numbers is $> 1$, then we can decrease", "it (Lemma \\ref{lemma-blowup-nonsingular-curves-meeting-at-point})", "by blowing up in all the points $p$ where the maximum is attained", "(note that the ``new'' invariants $m_{q_i}(Y'_i \\cap E)$ are always $1$).", "If the maximum is $1$ then, if $p \\in Y_1 \\cap \\ldots \\cap Y_r$", "for some $r > 2$ and not any of the others (for example), then after", "blowing up $p$ we see that $Y'_1, \\ldots, Y'_r$ do not meet in points", "above $p$ and $m_{q_i}(Y'_i, E) = 1$ where $Y'_i \\cap E = \\{q_i\\}$.", "Thus continuing to blowup points where more than $3$", "of the components of $Z$ meet, we reach the situation where", "for every closed point $p \\in X$ there is either", "(a) no curves $Y_i$ passing through $p$,", "(b) exactly one curve $Y_i$ passing through $p$ and $\\mathcal{O}_{Y_i, p}$", "is regular, or (c) exactly two curves $Y_i$, $Y_j$ passing through", "$p$, the local rings $\\mathcal{O}_{Y_i, p}$, $\\mathcal{O}_{Y_j, p}$", "are regular and $m_p(Y_i \\cap Y_j) = 1$.", "This means that $\\sum Y_i$ is a strict normal crossings", "divisor on the regular surface $X$, see", "\\'Etale Morphisms, Lemma", "\\ref{etale-lemma-strict-normal-crossings}." ], "refs": [ "resolve-lemma-blowup", "resolve-lemma-turn-into-effective-Cartier", "resolve-lemma-blowup-curves", "resolve-lemma-blowup-nonsingular-curves-meeting-at-point", "etale-lemma-strict-normal-crossings" ], "ref_ids": [ 11639, 11697, 11696, 11695, 10721 ] } ], "ref_ids": [ 11693 ] }, { "id": 11699, "type": "theorem", "label": "resolve-lemma-factor-through-contraction", "categories": [ "resolve" ], "title": "resolve-lemma-factor-through-contraction", "contents": [ "Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an", "exceptional curve of the first kind. If a contraction $X \\to X'$", "of $E$ exists, then it has the following universal property:", "for every morphism $\\varphi : X \\to Y$ such that $\\varphi(E)$", "is a point, there is a unique factorization", "$X \\to X' \\to Y$ of $\\varphi$." ], "refs": [], "proofs": [ { "contents": [ "Let $b : X \\to X'$ be a contraction of $E$. As a topological space", "$X'$ is the quotient of $X$ by the relation identifying all points", "of $E$ to one point. Namely, $b$ is proper", "(Divisors, Lemma \\ref{divisors-lemma-blowing-up-projective} and", "Morphisms, Lemma \\ref{morphisms-lemma-locally-projective-proper})", "and surjective, hence defines a submersive map of topological", "spaces (Topology, Lemma", "\\ref{topology-lemma-closed-morphism-quotient-topology}).", "On the other hand, the canonical map", "$\\mathcal{O}_{X'} \\to b_*\\mathcal{O}_X$ is an isomorphism. Namely,", "this is clear over the complement of the image point $x \\in X'$ of $E$", "and on stalks at $x$ the map is an isomorphism by part (4) of", "Lemma \\ref{lemma-cohomology-of-blowup}.", "Thus the pair $(X', \\mathcal{O}_{X'})$ is constructed", "from $X$ by taking the quotient as a topological space", "and endowing this with $b_*\\mathcal{O}_X$ as structure sheaf.", "\\medskip\\noindent", "Given $\\varphi$ we can let $\\varphi' : X' \\to Y$ be the unique map of", "topological spaces such that $\\varphi = \\varphi' \\circ b$.", "Then the map", "$$", "\\varphi^\\sharp : \\varphi^{-1}\\mathcal{O}_Y =", "b^{-1}((\\varphi')^{-1}\\mathcal{O}_Y) \\to \\mathcal{O}_X", "$$", "is adjoint to a map", "$$", "(\\varphi')^\\sharp :", "(\\varphi')^{-1}\\mathcal{O}_Y \\to b_*\\mathcal{O}_X = \\mathcal{O}_{X'}", "$$", "Then $(\\varphi', (\\varphi')^\\sharp)$ is a morphism of ringed spaces", "from $X'$ to $Y$ such that we get the desired factorization. Since", "$\\varphi$ is a morphism of locally ringed spaces, it follows that", "$\\varphi'$ is too. Namely, the only thing to check is that the map", "$\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X', x}$ is local, where $y \\in Y$", "is the image of $E$ under $\\varphi$. This is true because an element", "$f \\in \\mathfrak m_y$ pulls back to a function on $X$ which is zero", "in every point of $E$ hence the pull back of $f$ to $X'$ is a function", "defined on a neighbourhood of $x$ in $X'$ with the same property.", "Then it is clear that this function must vanish at $x$ as desired." ], "refs": [ "divisors-lemma-blowing-up-projective", "morphisms-lemma-locally-projective-proper", "topology-lemma-closed-morphism-quotient-topology", "resolve-lemma-cohomology-of-blowup" ], "ref_ids": [ 8063, 5422, 8204, 11642 ] } ], "ref_ids": [] }, { "id": 11700, "type": "theorem", "label": "resolve-lemma-contraction-unique", "categories": [ "resolve" ], "title": "resolve-lemma-contraction-unique", "contents": [ "Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an", "exceptional curve of the first kind. If there exists a contraction", "of $E$, then it is unique up to unique isomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the universal property of", "Lemma \\ref{lemma-factor-through-contraction}." ], "refs": [ "resolve-lemma-factor-through-contraction" ], "ref_ids": [ 11699 ] } ], "ref_ids": [] }, { "id": 11701, "type": "theorem", "label": "resolve-lemma-exceptional-first-kind-local", "categories": [ "resolve" ], "title": "resolve-lemma-exceptional-first-kind-local", "contents": [ "Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an", "exceptional curve of the first kind. Let $E_n = nE$ and", "denote $\\mathcal{O}_n$ its structure sheaf. Then", "$$", "A = \\lim H^0(E_n, \\mathcal{O}_n)", "$$", "is a complete local Noetherian regular local ring of dimension $2$", "and $\\Ker(A \\to H^0(E_n, \\mathcal{O}_n))$ is the $n$th power of", "its maximal ideal." ], "refs": [], "proofs": [ { "contents": [ "Recall that there exists an isomorphism $\\mathbf{P}^1_k \\to E$", "such that the normal sheaf of $E$ in $X$ pulls back to $\\mathcal{O}(-1)$.", "Then $H^0(E, \\mathcal{O}_E) = k$.", "We will denote $\\mathcal{O}_n(iE)$ the restriction of the invertible", "sheaf $\\mathcal{O}_X(iE)$ to $E_n$ for all $n \\geq 1$ and $i \\in \\mathbf{Z}$.", "Recall that $\\mathcal{O}_X(-nE)$ is the ideal sheaf of $E_n$. Hence", "for $d \\geq 0$ we obtain a short exact sequence", "$$", "0 \\to \\mathcal{O}_E(-(d + n)E) \\to", "\\mathcal{O}_{n + 1}(-dE) \\to", "\\mathcal{O}_n(-dE) \\to 0", "$$", "Since $\\mathcal{O}_E(-(d + n)E) = \\mathcal{O}_{\\mathbf{P}^1_k}(d + n)$", "the first cohomology group vanishes for all $d \\geq 0$ and $n \\geq 1$.", "We conclude that the transition maps of the system", "$H^0(E_n, \\mathcal{O}_n(-dE))$ are surjective. For $d = 0$", "we get an inverse system of surjections of rings such that the", "kernel of each transition map is a nilpotent ideal.", "Hence $A = \\lim H^0(E_n, \\mathcal{O}_n)$ is a local ring", "with residue field $k$ and maximal ideal", "$$", "\\lim \\Ker(H^0(E_n, \\mathcal{O}_n) \\to H^0(E, \\mathcal{O}_E)) =", "\\lim H^0(E_n, \\mathcal{O}_n(-E))", "$$", "Pick $x, y$ in this kernel mapping to a $k$-basis of", "$H^0(E, \\mathcal{O}_E(-E)) = H^0(\\mathbf{P}^1_k, \\mathcal{O}(1))$.", "Then $x^d, x^{d - 1}y, \\ldots, y^d$ are", "elements of $\\lim H^0(E_n, \\mathcal{O}_n(-dE))$ which map to a basis", "of $H^0(E, \\mathcal{O}_E(-dE)) = H^0(\\mathbf{P}^1_k, \\mathcal{O}(d))$.", "In this way we see that $A$ is separated and complete with respect", "to the linear topology defined by the kernels", "$$", "I_n = \\Ker(A \\longrightarrow H^0(E_n, \\mathcal{O}_n))", "$$", "We have $x, y \\in I_1$, $I_d I_{d'} \\subset I_{d + d'}$", "and $I_d/I_{d + 1}$ is a free $k$-module on $x^d, x^{d - 1}y, \\ldots, y^d$.", "We will show that $I_d = (x, y)^d$. Namely, if $z_e \\in I_e$ with", "$e \\geq d$, then we can write", "$$", "z_e = a_{e, 0} x^d + a_{e, 1} x^{d - 1}y + \\ldots + a_{e, d}y^d + z_{e + 1}", "$$", "where $a_{e, j} \\in (x, y)^{e - d}$ and $z_{e + 1} \\in I_{e + 1}$", "by our description of $I_d/I_{d + 1}$. Thus starting with some", "$z = z_d \\in I_d$ we can do this inductively", "$$", "z = \\sum\\nolimits_{e \\geq d} \\sum\\nolimits_j a_{e, j} x^{d - j} y^j", "$$", "with some $a_{e, j} \\in (x, y)^{e - d}$. Then $a_j = \\sum_{e \\geq d} a_{e, j}$", "exists (by completeness and the fact that $a_{e, j} \\in I_{e - d}$)", "and we have $z = \\sum a_{e, j} x^{d - j} y^j$.", "Hence $I_d = (x, y)^d$.", "Thus $A$ is $(x, y)$-adically complete. Then $A$ is", "Noetherian by Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian}.", "It is clear that the dimension is $2$ by the description", "of $(x, y)^d/(x, y)^{d + 1}$ and", "Algebra, Proposition \\ref{algebra-proposition-dimension}.", "Since the maximal ideal", "is generated by two elements it is regular." ], "refs": [ "algebra-lemma-completion-Noetherian", "algebra-proposition-dimension" ], "ref_ids": [ 873, 1411 ] } ], "ref_ids": [] }, { "id": 11702, "type": "theorem", "label": "resolve-lemma-contraction", "categories": [ "resolve" ], "title": "resolve-lemma-contraction", "contents": [ "Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an", "exceptional curve of the first kind. If there exists a morphism", "$f : X \\to Y$ such that", "\\begin{enumerate}", "\\item $Y$ is Noetherian,", "\\item $f$ is proper,", "\\item $f$ maps $E$ to a point $y$ of $Y$,", "\\item $f$ is quasi-finite at every point not in $E$,", "\\end{enumerate}", "Then there exists a contraction of $E$ and it is the Stein", "factorization of $f$." ], "refs": [], "proofs": [ { "contents": [ "We apply More on Morphisms, Theorem", "\\ref{more-morphisms-theorem-stein-factorization-Noetherian}", "to get a Stein factorization $X \\to X' \\to Y$.", "Then $X \\to X'$ satisfies all the hypotheses of", "the lemma (some details omitted).", "Thus after replacing $Y$ by $X'$ we may in addition", "assume that $f_*\\mathcal{O}_X = \\mathcal{O}_Y$ and", "that the fibres of $f$ are geometrically connected.", "\\medskip\\noindent", "Assume that $f_*\\mathcal{O}_X = \\mathcal{O}_Y$ and", "that the fibres of $f$ are geometrically connected.", "Note that $y \\in Y$ is a closed point as $f$ is closed and $E$ is closed.", "The restriction $f^{-1}(Y \\setminus \\{y\\}) \\to Y \\setminus \\{y\\}$", "of $f$ is a finite morphism", "(More on Morphisms, Lemma \\ref{more-morphisms-lemma-characterize-finite}).", "Hence this restriction is an isomorphism since", "$f_*\\mathcal{O}_X = \\mathcal{O}_Y$ since finite morphisms are affine.", "To prove that $\\mathcal{O}_{Y, y}$ is regular of dimension", "$2$ we consider the isomorphism", "$$", "\\mathcal{O}_{Y, y}^\\wedge \\longrightarrow", "\\lim H^0(X \\times_Y \\Spec(\\mathcal{O}_{Y, y}/\\mathfrak m_y^n), \\mathcal{O})", "$$", "of Cohomology of Schemes, Lemma \\ref{coherent-lemma-formal-functions-stalk}.", "Let $E_n = nE$ as in Lemma \\ref{lemma-exceptional-first-kind-local}.", "Observe that", "$$", "E_n \\subset X \\times_Y \\Spec(\\mathcal{O}_{Y, y}/\\mathfrak m_y^n)", "$$", "because $E \\subset X_y = X \\times_Y \\Spec(\\kappa(y))$.", "On the other hand, since $E = f^{-1}(\\{y\\})$ set theoretically", "(because the fibres of $f$ are geometrically connected), we see that", "the scheme theoretic fibre $X_y$ is scheme theoretically contained in", "$E_n$ for some $n > 0$. Namely, apply", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-power-ideal-kills-sheaf}", "to the coherent $\\mathcal{O}_X$-module $\\mathcal{F} = \\mathcal{O}_{X_y}$", "and the ideal sheaf $\\mathcal{I}$ of $E$ and use that", "$\\mathcal{I}^n$ is the ideal sheaf of $E_n$. This shows that", "$$", "X \\times_Y \\Spec(\\mathcal{O}_{Y, y}/\\mathfrak m_y^m) \\subset E_{nm}", "$$", "Thus the inverse limit displayed above is equal to", "$\\lim H^0(E_n, \\mathcal{O}_n)$", "which is a regular two dimensional local ring by", "Lemma \\ref{lemma-exceptional-first-kind-local}.", "Hence $\\mathcal{O}_{Y, y}$ is a two dimensional regular local", "ring because its completion is so", "(More on Algebra, Lemma \\ref{more-algebra-lemma-completion-regular} and", "\\ref{more-algebra-lemma-completion-dimension}).", "\\medskip\\noindent", "We still have to prove that $f : X \\to Y$ is the blowup $b : Y' \\to Y$", "of $Y$ at $y$. We encourage the reader to find her own proof.", "First, we note that Lemma \\ref{lemma-exceptional-first-kind-local}", "also implies that $X_y = E$ scheme theoretically.", "Since the ideal sheaf of $E$ is invertible, this shows", "that $f^{-1}\\mathfrak m_y \\cdot \\mathcal{O}_X$ is invertible.", "Hence we obtain a factorization", "$$", "X \\to Y' \\to Y", "$$", "of the morphism $f$ by the universal property of blowing up, see", "Divisors, Lemma \\ref{divisors-lemma-universal-property-blowing-up}.", "Recall that the exceptional fibre of $E' \\subset Y'$ is an exceptional", "curve of the first kind by Lemma \\ref{lemma-blowup}.", "Let $g : E \\to E'$ be the induced morphism.", "Because for both $E'$ and $E$ the conormal sheaf is generated", "by (pullbacks of) $a$ and $b$, we see that the canonical map", "$g^*\\mathcal{C}_{E'/Y'} \\to \\mathcal{C}_{E/X}$", "(Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial})", "is surjective. Since both are invertible, this map is an isomorphism.", "Since $\\mathcal{C}_{E/X}$ has positive degree, it follows that $g$", "cannot be a constant morphism.", "Hence $g$ has finite fibres. Hence $g$ is a finite morphism", "(same reference as above). However, since $Y'$ is regular", "(and hence normal) at all points of $E'$ and since $X \\to Y'$", "is birational and an isomorphism away from $E'$, we conclude", "that $X \\to Y'$ is an isomorphism by", "Varieties, Lemma", "\\ref{varieties-lemma-modification-normal-iso-over-codimension-1}." ], "refs": [ "more-morphisms-theorem-stein-factorization-Noetherian", "more-morphisms-lemma-characterize-finite", "coherent-lemma-formal-functions-stalk", "resolve-lemma-exceptional-first-kind-local", "coherent-lemma-power-ideal-kills-sheaf", "resolve-lemma-exceptional-first-kind-local", "more-algebra-lemma-completion-regular", "more-algebra-lemma-completion-dimension", "resolve-lemma-exceptional-first-kind-local", "divisors-lemma-universal-property-blowing-up", "resolve-lemma-blowup", "morphisms-lemma-conormal-functorial", "varieties-lemma-modification-normal-iso-over-codimension-1" ], "ref_ids": [ 13674, 13903, 3362, 11701, 3320, 11701, 10045, 10042, 11701, 8055, 11639, 5304, 10979 ] } ], "ref_ids": [] }, { "id": 11703, "type": "theorem", "label": "resolve-lemma-pic-blowup", "categories": [ "resolve" ], "title": "resolve-lemma-pic-blowup", "contents": [ "Let $b : X \\to X'$ be the contraction of an", "exceptional curve of the first kind $E \\subset X$.", "Then there is a short exact sequence", "$$", "0 \\to \\Pic(X') \\to \\Pic(X) \\to \\mathbf{Z} \\to 0", "$$", "where the first map is pullback by $b$ and the second map sends", "$\\mathcal{L}$ to the degree of $\\mathcal{L}$ on the exceptional", "curve $E$. The sequence is split by the map", "$n \\mapsto \\mathcal{O}_X(-nE)$." ], "refs": [], "proofs": [ { "contents": [ "Since $E = \\mathbf{P}^1_k$ we see that the Picard group of $E$", "is $\\mathbf{Z}$, see Divisors, Lemma", "\\ref{divisors-lemma-Pic-projective-space-UFD}.", "Hence we can think of the last map as $\\mathcal{L} \\mapsto \\mathcal{L}|_E$.", "The degree of the restriction of $\\mathcal{O}_X(E)$ to $E$ is $-1$", "by definition of exceptional curves of the first kind. Combining these", "remarks we see that it", "suffices to show that $\\Pic(X') \\to \\Pic(X)$ is injective", "with image the invertible sheaves restricting to $\\mathcal{O}_E$ on $E$.", "\\medskip\\noindent", "Given an invertible $\\mathcal{O}_{X'}$-module", "$\\mathcal{L}'$ we claim the map $\\mathcal{L}' \\to b_*b^*\\mathcal{L}'$", "is an isomorphism. This is clear everywhere except possibly at the image", "point $x \\in X'$ of $E$. To check it is an isomorphism on stalks", "at $x$ we may replace $X'$ by an open neighbourhood at $x$ and", "assume $\\mathcal{L}'$ is $\\mathcal{O}_{X'}$. Then we have to", "show that the map $\\mathcal{O}_{X'} \\to b_*\\mathcal{O}_X$", "is an isomorphism. This follows from Lemma \\ref{lemma-cohomology-of-blowup}", "part (4).", "\\medskip\\noindent", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module with", "$\\mathcal{L}|_E = \\mathcal{O}_E$. Then we claim ", "(1) $b_*\\mathcal{L}$ is invertible and", "(2) $b^*b_*\\mathcal{L} \\to \\mathcal{L}$ is an isomorphism.", "Statements (1) and (2) are clear over $X' \\setminus \\{x\\}$.", "Thus it suffices to prove (1) and (2) after base change", "to $\\Spec(\\mathcal{O}_{X', x})$.", "Computing $b_*$ commutes with flat base change", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology})", "and similarly for $b^*$ and formation of the adjunction map.", "But if $X'$ is the spectrum of a regular local ring", "then $\\mathcal{L}$ is trivial by the description of", "the Picard group in Lemma \\ref{lemma-blowup-pic}. Thus", "the claim is proved.", "\\medskip\\noindent", "Combining the claims proved in the previous two paragraphs we", "see that the map $\\mathcal{L} \\mapsto b_*\\mathcal{L}$", "is an inverse to the map", "$$", "\\Pic(X') \\longrightarrow \\Ker(\\Pic(X) \\to \\Pic(E))", "$$", "and the lemma is proved." ], "refs": [ "divisors-lemma-Pic-projective-space-UFD", "resolve-lemma-cohomology-of-blowup", "coherent-lemma-flat-base-change-cohomology", "resolve-lemma-blowup-pic" ], "ref_ids": [ 8034, 11642, 3298, 11641 ] } ], "ref_ids": [] }, { "id": 11704, "type": "theorem", "label": "resolve-lemma-lift-sections-and-h1", "categories": [ "resolve" ], "title": "resolve-lemma-lift-sections-and-h1", "contents": [ "Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an", "exceptional curve of the first kind. Let $\\mathcal{L}$ be", "an invertible $\\mathcal{O}_X$-module.", "Let $n$ be the integer such that $\\mathcal{L}|_E$ has degree $n$", "viewed as an invertible module on $\\mathbf{P}^1$. Then", "\\begin{enumerate}", "\\item If $H^1(X, \\mathcal{L}) = 0$ and $n \\geq 0$, then", "$H^1(X, \\mathcal{L}(iE)) = 0$ for $0 \\leq i \\leq n + 1$.", "\\item If $n \\leq 0$, then", "$H^1(X, \\mathcal{L}) \\subset H^1(X, \\mathcal{L}(E))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\mathcal{L}|_E = \\mathcal{O}(n)$ by", "Divisors, Lemma \\ref{divisors-lemma-Pic-projective-space-UFD}.", "Use induction, the long exact cohomology sequence associated to the", "short exact sequence", "$$", "0 \\to \\mathcal{L} \\to \\mathcal{L}(E) \\to \\mathcal{L}(E)|_E \\to 0,", "$$", "and use the fact that $H^1(\\mathbf{P}^1, \\mathcal{O}(d)) = 0$ for", "$d \\geq -1$ and $H^0(\\mathbf{P}^1, \\mathcal{O}(d)) = 0$ for", "$d \\leq -1$. Some details omitted." ], "refs": [ "divisors-lemma-Pic-projective-space-UFD" ], "ref_ids": [ 8034 ] } ], "ref_ids": [] }, { "id": 11705, "type": "theorem", "label": "resolve-lemma-contract-ample", "categories": [ "resolve" ], "title": "resolve-lemma-contract-ample", "contents": [ "Let $S = \\Spec(R)$ be an affine Noetherian scheme.", "Let $X \\to S$ be a proper morphism. Let $\\mathcal{L}$ be an", "ample invertible sheaf on $X$. Let $E \\subset X$ be an", "exceptional curve of the first kind. Then", "\\begin{enumerate}", "\\item there exists a contraction $b : X \\to X'$ of $E$,", "\\item $X'$ is proper over $S$, and", "\\item the invertible $\\mathcal{O}_{X'}$-module $\\mathcal{L}'$", "is ample with $\\mathcal{L}'$ as in Remark \\ref{remark-pic-blowup}.", "\\end{enumerate}" ], "refs": [ "resolve-remark-pic-blowup" ], "proofs": [ { "contents": [ "Let $n$ be the degree of $\\mathcal{L}|_E$ as in", "Lemma \\ref{lemma-lift-sections-and-h1}.", "Observe that $n > 0$ as $\\mathcal{L}$ is ample on $E$", "(Varieties, Lemma \\ref{varieties-lemma-ample-curve} and", "Properties, Lemma \\ref{properties-lemma-ample-on-closed}).", "After replacing $\\mathcal{L}$ by a power we may assume", "$H^i(X, \\mathcal{L}^{\\otimes e}) = 0$ for all $i > 0$ and $e > 0$, see", "Cohomology of Schemes,", "Lemma \\ref{coherent-lemma-vanshing-gives-ample}.", "Finally, after replacing $\\mathcal{L}$ by another power we may assume", "there exist global sections $t_0, \\ldots, t_n$ of $\\mathcal{L}$", "which define a closed immersion $\\psi : X \\to \\mathbf{P}^n_S$, see", "Morphisms, Lemma", "\\ref{morphisms-lemma-finite-type-over-affine-ample-very-ample}.", "\\medskip\\noindent", "Set $\\mathcal{M} = \\mathcal{L}(nE)$. Then $\\mathcal{M}|_E \\cong \\mathcal{O}_E$.", "Since we have the short exact sequence", "$$", "0 \\to \\mathcal{M}(-E) \\to \\mathcal{M} \\to \\mathcal{O}_E \\to 0", "$$", "and since $H^1(X, \\mathcal{M}(-E))$ is zero", "(by Lemma \\ref{lemma-lift-sections-and-h1} and the fact that $n > 0$)", "we can pick a section $s_{n + 1}$ of $\\mathcal{M}$ which generates", "$\\mathcal{M}|_E$.", "Finally, denote $s_0, \\ldots, s_n$ the sections of $\\mathcal{M}$", "we get from the sections $t_0, \\ldots, t_n$ of $\\mathcal{L}$", "chosen above via $\\mathcal{L} \\subset \\mathcal{L}(nE) = \\mathcal{M}$.", "Combined the sections $s_0, \\ldots, s_n, s_{n + 1}$", "generate $\\mathcal{M}$ in every point of $X$ and therefore define", "a morphism", "$$", "\\varphi : X \\longrightarrow \\mathbf{P}^{n + 1}_S", "$$", "over $S$, see Constructions, Lemma \\ref{constructions-lemma-projective-space}.", "\\medskip\\noindent", "Below we will check the conditions of Lemma \\ref{lemma-contraction}.", "Once this is done we see that the Stein factorization", "$X \\to X' \\to \\mathbf{P}^{n + 1}_S$ of $\\varphi$ is the desired contraction", "which proves (1).", "Moreover, the morphism $X' \\to \\mathbf{P}^{n + 1}_S$ is finite", "hence $X'$ is proper over $S$", "(Morphisms, Lemmas \\ref{morphisms-lemma-finite-proper} and", "\\ref{morphisms-lemma-composition-proper}). This proves (2).", "Observe that $X'$ has an ample invertible sheaf. Namely the pullback", "$\\mathcal{M}'$ of $\\mathcal{O}_{\\mathbf{P}^{n + 1}_S}(1)$ is ample by", "Morphisms, Lemma \\ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}.", "Observe that $\\mathcal{M}'$ pulls back to $\\mathcal{M}$ on $X$", "(by Constructions, Lemma \\ref{constructions-lemma-projective-space}).", "Finally, $\\mathcal{M} = \\mathcal{L}(nE)$. Since in the arguments above", "we have replaced the original $\\mathcal{L}$ by a positive power", "we conclude that the invertible $\\mathcal{O}_{X'}$-module $\\mathcal{L}'$", "mentioned in (3) of the lemma is ample on $X'$ by", "Properties, Lemma \\ref{properties-lemma-ample-power-ample}.", "\\medskip\\noindent", "Easy observations: $\\mathbf{P}^{n + 1}_S$ is Noetherian and $\\varphi$ is proper.", "Details omitted.", "\\medskip\\noindent", "Next, we observe that any point of $U = X \\setminus E$ is mapped", "to the open subscheme $W$ of $\\mathbf{P}^{n + 1}_S$ where one of the", "first $n + 1$ homogeneous coordinates is nonzero. On the other hand,", "any point of $E$ is mapped to a point where the first $n + 1$ homogeneous", "coordinates are all zero, in particular into the complement of $W$.", "Moreover, it is clear that there is a factorization", "$$", "U = \\varphi^{-1}(W) \\xrightarrow{\\varphi|_U} W \\xrightarrow{pr} \\mathbf{P}^n_S", "$$", "of $\\psi|_U$ where $pr$ is the projection using the first", "$n + 1$ coordinates and $\\psi : X \\to \\mathbf{P}^n_S$ is the embedding chosen", "above. It follows that $\\varphi|_U : U \\to W$ is quasi-finite.", "\\medskip\\noindent", "Finally, we consider the map $\\varphi|_E : E \\to \\mathbf{P}^{n + 1}_S$.", "Observe that for any point $x \\in E$ the image $\\varphi(x)$", "has its first $n + 1$ coordinates equal to zero, i.e., the morphism", "$\\varphi|_E$ factors through the closed subscheme", "$\\mathbf{P}^0_S \\cong S$. The morphism $E \\to S = \\Spec(R)$", "factors as $E \\to \\Spec(H^0(E, \\mathcal{O}_E)) \\to \\Spec(R)$", "by Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}.", "Since by assumption $H^0(E, \\mathcal{O}_E)$ is a field we conclude", "that $E$ maps to a point in $S \\subset \\mathbf{P}^{n + 1}_S$", "which finishes the proof." ], "refs": [ "resolve-lemma-lift-sections-and-h1", "varieties-lemma-ample-curve", "properties-lemma-ample-on-closed", "coherent-lemma-vanshing-gives-ample", "morphisms-lemma-finite-type-over-affine-ample-very-ample", "resolve-lemma-lift-sections-and-h1", "constructions-lemma-projective-space", "resolve-lemma-contraction", "morphisms-lemma-finite-proper", "morphisms-lemma-composition-proper", "morphisms-lemma-pullback-ample-tensor-relatively-ample", "constructions-lemma-projective-space", "properties-lemma-ample-power-ample", "schemes-lemma-morphism-into-affine" ], "ref_ids": [ 11704, 11116, 3041, 3346, 5394, 11704, 12621, 11702, 5445, 5408, 5383, 12621, 3040, 7655 ] } ], "ref_ids": [ 11718 ] }, { "id": 11706, "type": "theorem", "label": "resolve-lemma-contract-when-quasi-projective", "categories": [ "resolve" ], "title": "resolve-lemma-contract-when-quasi-projective", "contents": [ "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of finite type.", "Let $E \\subset X$ be an exceptional curve of the first kind which is in a", "fibre of $f$.", "\\begin{enumerate}", "\\item If $X$ is projective over $S$, then there exists a contraction", "$X \\to X'$ of $E$ and $X'$ is projective over $S$.", "\\item If $X$ is quasi-projective over $S$, then there exists a contraction", "$X \\to X'$ of $E$ and $X'$ is quasi-projective over $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Both cases follow from Lemma \\ref{lemma-contract-ample}", "using standard results on ample invertible modules and", "(quasi-)projective morphisms.", "\\medskip\\noindent", "Proof of (2). Projectivity of $f$ means that $f$ is proper and there exists", "an $f$-ample invertible module $\\mathcal{L}$, see", "Morphisms, Lemma \\ref{morphisms-lemma-projective-is-quasi-projective-proper}", "and Definition \\ref{morphisms-definition-quasi-projective}.", "Let $U \\subset S$ be an affine open containing the image of $E$.", "By Lemma \\ref{lemma-contract-ample} there exists a contraction", "$c : f^{-1}(U) \\to V'$ of $E$ and an ample invertible module", "$\\mathcal{N}'$ on $V'$ whose pullback to $f^{-1}(U)$ is equal to", "$\\mathcal{L}(nE)|_{f^{-1}(U)}$. Let $v \\in V'$ be the closed point", "such that $c$ is the blowing up of $v$.", "Then we can glue $V'$ and $X \\setminus E$ along", "$f^{-1}(U) \\setminus E = V' \\setminus \\{v\\}$", "to get a scheme $X'$ over $S$. The morphisms $c$ and", "$\\text{id}_{X \\setminus E}$ glue to a morphism $b : X \\to X'$", "which is the contraction of $E$. The inverse image of $U$ in $X'$", "is proper over $U$. On the other hand, the restriction of $X' \\to S$", "to the complement of the image of $v$ in $S$ is isomorphic to the", "restriction of $X \\to S$ to that open. Hence $X' \\to S$ is proper", "(as being proper is local on the base by", "Morphisms, Lemma \\ref{morphisms-lemma-proper-local-on-the-base}).", "Finally, $\\mathcal{N}'$ and $\\mathcal{L}|_{X \\setminus E}$ restrict to", "isomorphic invertible modules over $f^{-1}(U) \\setminus E = V' \\setminus \\{v\\}$", "and hence glue to an invertible module $\\mathcal{L}'$ over $X'$.", "The restriction of $\\mathcal{L}'$ to the inverse image of $U$", "in $X'$ is ample because this is true for $\\mathcal{N}'$.", "For affine opens of $S$ avoiding the image of $v$, we see that", "the same is true because it holds for $\\mathcal{L}$.", "Thus $\\mathcal{L}'$ is $(X' \\to S)$-relatively ample by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-relatively-ample}", "and (2) is proved.", "\\medskip\\noindent", "Proof of (3). We can write $X$ as an open subscheme of a scheme", "$\\overline{X}$ projective over $S$ by Morphisms, Lemma", "\\ref{morphisms-lemma-quasi-projective-open-projective}.", "By (2) there is a contraction $b : \\overline{X} \\to \\overline{X}'$", "and $\\overline{X}'$ is projective over $S$. Then we let", "$X' \\subset \\overline{X}$ be the image of $X \\to \\overline{X}'$;", "this is an open as $b$ is an isomorphism away from $E$.", "Then $X \\to X'$ is the desired contraction. Note that", "$X'$ is quasi-projective over $S$ as it has an", "$S$-relatively ample invertible module", "by the construction in the proof of part (2)." ], "refs": [ "resolve-lemma-contract-ample", "morphisms-lemma-projective-is-quasi-projective-proper", "morphisms-definition-quasi-projective", "resolve-lemma-contract-ample", "morphisms-lemma-proper-local-on-the-base", "morphisms-lemma-characterize-relatively-ample", "morphisms-lemma-quasi-projective-open-projective" ], "ref_ids": [ 11705, 5430, 5570, 11705, 5407, 5380, 5429 ] } ], "ref_ids": [] }, { "id": 11707, "type": "theorem", "label": "resolve-lemma-regular-dim-2-quasi-projective", "categories": [ "resolve" ], "title": "resolve-lemma-regular-dim-2-quasi-projective", "contents": [ "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a", "separated morphism of finite type with $X$ regular of dimension $2$.", "Then $X$ is quasi-projective over $S$." ], "refs": [], "proofs": [ { "contents": [ "By Chow's lemma", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-chow-Noetherian})", "there exists a proper morphism $\\pi : X' \\to X$ which is an isomorphism", "over a dense open $U \\subset X$ such that $X' \\to S$ is H-quasi-projective.", "By Lemma \\ref{lemma-extend-rational-map-blowing-up}", "there exists a sequence of blowups in closed points", "$$", "X_n \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "and an $S$-morphism $X_n \\to X'$ extending the rational map $U \\to X'$.", "Observe that $X_n \\to X$ is projective by", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-projective} and", "Morphisms, Lemma \\ref{morphisms-lemma-composition-projective}.", "This implies that $X_n \\to X'$ is projective by", "Morphisms, Lemma \\ref{morphisms-lemma-projective-permanence}.", "Hence $X_n \\to S$ is quasi-projective by", "Morphisms, Lemma \\ref{morphisms-lemma-composition-quasi-projective}", "(and the fact that a projective morphism is quasi-projective, see", "Morphisms, Lemma \\ref{morphisms-lemma-projective-quasi-projective}).", "By Lemma \\ref{lemma-contract-when-quasi-projective}", "(and uniqueness of contractions Lemma \\ref{lemma-contraction-unique})", "we conclude that $X_{n - 1}, \\ldots, X_0 = X$ are quasi-projective over $S$", "as desired." ], "refs": [ "coherent-lemma-chow-Noetherian", "resolve-lemma-extend-rational-map-blowing-up", "divisors-lemma-blowing-up-projective", "morphisms-lemma-composition-projective", "morphisms-lemma-projective-permanence", "morphisms-lemma-composition-quasi-projective", "morphisms-lemma-projective-quasi-projective", "resolve-lemma-contract-when-quasi-projective", "resolve-lemma-contraction-unique" ], "ref_ids": [ 3354, 11647, 8063, 5431, 5432, 5400, 5427, 11706, 11700 ] } ], "ref_ids": [] }, { "id": 11708, "type": "theorem", "label": "resolve-lemma-regular-dim-2-projective", "categories": [ "resolve" ], "title": "resolve-lemma-regular-dim-2-projective", "contents": [ "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a", "proper morphism with $X$ regular of dimension $2$.", "Then $X$ is projective over $S$." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-regular-dim-2-quasi-projective} and", "Morphisms, Lemma \\ref{morphisms-lemma-projective-is-quasi-projective-proper}." ], "refs": [ "resolve-lemma-regular-dim-2-quasi-projective", "morphisms-lemma-projective-is-quasi-projective-proper" ], "ref_ids": [ 11707, 5430 ] } ], "ref_ids": [] }, { "id": 11709, "type": "theorem", "label": "resolve-lemma-proper-birational-regular-surfaces", "categories": [ "resolve" ], "title": "resolve-lemma-proper-birational-regular-surfaces", "contents": [ "Let $f : X \\to Y$ be a proper birational morphism between", "integral Noetherian schemes regular of dimension $2$.", "Then $f$ is a sequence of blowups in closed points." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset Y$ be the maximal open over which $f$ is an isomorphism.", "Then $V$ contains all codimension $1$ points of $V$ (Varieties,", "Lemma \\ref{varieties-lemma-modification-normal-iso-over-codimension-1}).", "Let $y \\in Y$ be a closed point not contained in $V$.", "Then we want to show that $f$ factors through the blowup $b : Y' \\to Y$", "of $Y$ at $y$. Namely, if this is true, then at least one (and in fact", "exactly one) component of the fibre $f^{-1}(y)$ will map isomorphically", "onto the exceptional curve in $Y'$ and the number of curves", "in fibres of $X \\to Y'$ will be strictly less that the number of curves", "in fibres of $X \\to Y$, so we conclude by induction. Some details omitted.", "\\medskip\\noindent", "By Lemma \\ref{lemma-extend-rational-map-blowing-up}", "we know that there exists a sequence of blowing ups", "$$", "X' = X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "in closed points lying over the fibre $f^{-1}(y)$", "and a morphism $X' \\to Y'$ such that", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r] & X \\ar[d]^f \\\\", "Y' \\ar[r] & Y", "}", "$$", "is commutative. We want to show that the morphism $X' \\to Y'$", "factors through $X$ and hence we can use induction on $n$ to", "reduce to the case where $X' \\to X$ is the blowup of $X$", "in a closed point $x \\in X$ mapping to $y$.", "\\medskip\\noindent", "Let $E \\subset X'$ be the exceptional fibre of the blowing up", "$X' \\to X$. If $E$ maps to a point in $Y'$, then we obtain the", "desired factorization by Lemma \\ref{lemma-factor-through-contraction}.", "We will prove that", "if this is not the case we obtain a contradiction. Namely,", "if $f'(E)$ is not a point, then", "$E' = f'(E)$ must be the exceptional curve in $Y'$.", "Picture", "$$", "\\xymatrix{", "E \\ar[r] \\ar[d]_g & X' \\ar[d]_{f'} \\ar[r] & X \\ar[d]^f \\\\", "E' \\ar[r] & Y' \\ar[r] & Y", "}", "$$", "Arguing as before $f'$ is an isomorphism in an open neighbourhood", "of the generic point of $E'$. Hence $g : E \\to E'$ is a finite birational", "morphism. Then the inverse of $g$ (a rational map) is everywhere defined", "by Morphisms, Lemma \\ref{morphisms-lemma-extend-across} and $g$ is", "an isomorphism. Consider the map", "$$", "g^*\\mathcal{C}_{E'/Y'} \\longrightarrow \\mathcal{C}_{E/X'}", "$$", "of Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial}.", "Since the source and target are invertible modules of degree $1$", "on $E = E' = \\mathbf{P}^1_\\kappa$ and since the map is", "nonzero (as $f'$ is an isomorphism in the generic point of $E$)", "we conclude it is an isomorphism. By", "Morphisms, Lemma \\ref{morphisms-lemma-two-immersions}", "we conclude that $\\Omega_{X'/Y'}|_E = 0$.", "This means that $f'$ is unramified at every point of $E$", "(Morphisms, Lemma \\ref{morphisms-lemma-unramified-at-point}).", "Hence $f'$ is quasi-finite at every point of $E$", "(Morphisms, Lemma \\ref{morphisms-lemma-unramified-quasi-finite}).", "Hence the maximal open $V' \\subset Y'$ over which $f'$ is an", "isomorphism contains $E'$ by Varieties, Lemma", "\\ref{varieties-lemma-modification-normal-iso-over-codimension-1}.", "This in turn implies that the inverse image of $y$ in", "$X'$ is $E'$. Hence the inverse image of $y$ in $X$ is $x$.", "Hence $x \\in X$ is in the maximal open over which", "$f$ is an isomorphism by Varieties, Lemma", "\\ref{varieties-lemma-modification-normal-iso-over-codimension-1}.", "This is a contradiction as we assumed that $y$ is not", "in this open." ], "refs": [ "varieties-lemma-modification-normal-iso-over-codimension-1", "resolve-lemma-extend-rational-map-blowing-up", "resolve-lemma-factor-through-contraction", "morphisms-lemma-extend-across", "morphisms-lemma-conormal-functorial", "morphisms-lemma-two-immersions", "morphisms-lemma-unramified-at-point", "morphisms-lemma-unramified-quasi-finite", "varieties-lemma-modification-normal-iso-over-codimension-1", "varieties-lemma-modification-normal-iso-over-codimension-1" ], "ref_ids": [ 10979, 11647, 11699, 5419, 5304, 5321, 5355, 5351, 10979, 10979 ] } ], "ref_ids": [] }, { "id": 11710, "type": "theorem", "label": "resolve-lemma-birational-regular-surfaces", "categories": [ "resolve" ], "title": "resolve-lemma-birational-regular-surfaces", "contents": [ "Let $S$ be a Noetherian scheme. Let $X$ and $Y$ be proper", "integral schemes over $S$ which are regular of dimension $2$.", "Then $X$ and $Y$ are $S$-birational if and only if there", "exists a diagram of $S$-morphisms", "$$", "X = X_0 \\leftarrow X_1 \\leftarrow \\ldots \\leftarrow X_n = Y_m", "\\to \\ldots \\to Y_1 \\to Y_0 = Y", "$$", "where each morphism is a blowup in a closed point." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be open and let $f : U \\to Y$ be the given", "$S$-rational map (which is invertible as an $S$-rational map).", "By Lemma \\ref{lemma-extend-rational-map-blowing-up}", "we can factor $f$ as $X_n \\to \\ldots \\to X_1 \\to X_0 = X$", "and $f_n : X_n \\to Y$. Since $X_n$ is proper over $S$ and", "$Y$ separated over $S$ the morphism $f_n$ is proper.", "Clearly $f_n$ is birational. Hence $f_n$ is a composition", "of contractions by Lemma \\ref{lemma-proper-birational-regular-surfaces}.", "We omit the proof of the converse." ], "refs": [ "resolve-lemma-extend-rational-map-blowing-up", "resolve-lemma-proper-birational-regular-surfaces" ], "ref_ids": [ 11647, 11709 ] } ], "ref_ids": [] }, { "id": 11711, "type": "theorem", "label": "resolve-proposition-Grauert-Riemenschneider", "categories": [ "resolve" ], "title": "resolve-proposition-Grauert-Riemenschneider", "contents": [ "In Situation \\ref{situation-vanishing} assume", "\\begin{enumerate}", "\\item $X$ is a normal scheme,", "\\item $A$ is Nagata and has a dualizing complex $\\omega_A^\\bullet$.", "\\end{enumerate}", "Let $\\omega_X$ be the dualizing module of $X$", "(Remark \\ref{remark-dualizing-setup}). Then $R^1f_*\\omega_X = 0$." ], "refs": [ "resolve-remark-dualizing-setup" ], "proofs": [ { "contents": [ "In this proof we will use the identification $D(A) = D_\\QCoh(\\mathcal{O}_S)$", "to identify quasi-coherent $\\mathcal{O}_S$-modules with $A$-modules.", "Moreover, we may assume that $\\omega_A^\\bullet$ is normalized, see", "Dualizing Complexes, Section \\ref{dualizing-section-dualizing-local}.", "Since $X$ is a Noetherian normal $2$-dimensional scheme", "it is Cohen-Macaulay (Properties, Lemma", "\\ref{properties-lemma-normal-dimension-2-Cohen-Macaulay}).", "Thus $\\omega_X^\\bullet = \\omega_X[2]$ (Duality for Schemes, Lemma", "\\ref{duality-lemma-dualizing-module-CM-scheme} and the", "normalization in Duality for Schemes, Example", "\\ref{duality-example-proper-over-local}).", "If the proposition is false, then we can find a nonzero map", "$R^1f_*\\omega_X \\to \\kappa$. In other words we obtain a nonzero map", "$\\alpha : Rf_*\\omega_X^\\bullet \\to \\kappa[1]$.", "Applying $R\\Hom_A(-, \\omega_A^\\bullet)$ we get a nonzero map", "$$", "\\beta : \\kappa[-1] \\longrightarrow Rf_*\\mathcal{O}_X", "$$", "which is impossible by Lemma \\ref{lemma-R1-injective}.", "To see that $R\\Hom_A(-, \\omega_A^\\bullet)$ does what we said, first", "note that", "$$", "R\\Hom_A(\\kappa[1], \\omega_A^\\bullet) =", "R\\Hom_A(\\kappa, \\omega_A^\\bullet)[-1] =", "\\kappa[-1]", "$$", "as $\\omega_A^\\bullet$ is normalized and we have", "$$", "R\\Hom_A(Rf_*\\omega_X^\\bullet, \\omega_A^\\bullet) =", "Rf_*R\\SheafHom_{\\mathcal{O}_X}(\\omega_X^\\bullet, \\omega_X^\\bullet) =", "Rf_*\\mathcal{O}_X", "$$", "The first equality by", "Duality for Schemes, Example \\ref{duality-example-iso-on-RSheafHom-noetherian}", "and the fact that $\\omega_X^\\bullet = f^!\\omega_A^\\bullet$", "by construction, and the second equality because $\\omega_X^\\bullet$", "is a dualizing complex for $X$ (which goes back to", "Duality for Schemes, Lemma \\ref{duality-lemma-shriek-dualizing})." ], "refs": [ "properties-lemma-normal-dimension-2-Cohen-Macaulay", "duality-lemma-dualizing-module-CM-scheme", "resolve-lemma-R1-injective", "duality-lemma-shriek-dualizing" ], "ref_ids": [ 2991, 13586, 11660, 13560 ] } ], "ref_ids": [ 11717 ] }, { "id": 11783, "type": "theorem", "label": "spaces-duality-lemma-equivalent-definitions", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-equivalent-definitions", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.", "Let $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. The following are equivalent", "\\begin{enumerate}", "\\item For every \\'etale morphism $U \\to X$ where $U$ is a scheme", "the restriction $K|_U$ is a dualizing complex for $U$ (as discussed above).", "\\item There exists a surjective \\'etale morphism $U \\to X$ where $U$ is a", "scheme such that $K|_U$ is a dualizing complex for $U$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $U \\to X$ is surjective \\'etale where $U$ is a scheme.", "Let $V \\to X$ be an \\'etale morphism where $V$ is a scheme.", "Then", "$$", "U \\leftarrow U \\times_X V \\rightarrow V", "$$", "are \\'etale morphisms of schemes with the arrow to $V$ surjective.", "Hence we can use Duality for Schemes, Lemma \\ref{duality-lemma-descent-ascent}", "to see that if $K|_U$ is a dualizing complex for $U$, then", "$K|_V$ is a dualizing complex for $V$." ], "refs": [ "duality-lemma-descent-ascent" ], "ref_ids": [ 13605 ] } ], "ref_ids": [] }, { "id": 11784, "type": "theorem", "label": "spaces-duality-lemma-affine-duality", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-affine-duality", "contents": [ "Let $A$ be a Noetherian ring and let $X = \\Spec(A)$. Let", "$\\mathcal{O}_\\etale$ be the structure sheaf of $X$ on the", "small \\'etale site of $X$. Let $K, L$ be objects of $D(A)$.", "If $K \\in D_{\\textit{Coh}}(A)$ and $L$ has finite injective", "dimension, then", "$$", "\\epsilon^*\\widetilde{R\\Hom_A(K, L)} =", "R\\SheafHom_{\\mathcal{O}_\\etale}(\\epsilon^*\\widetilde{K},", "\\epsilon^*\\widetilde{L})", "$$", "in $D(\\mathcal{O}_\\etale)$ where", "$\\epsilon : (X_\\etale, \\mathcal{O}_\\etale) \\to (X, \\mathcal{O}_X)$", "is as in Derived Categories of Spaces, Section", "\\ref{spaces-perfect-section-derived-quasi-coherent-etale}." ], "refs": [], "proofs": [ { "contents": [ "By Duality for Schemes, Lemma \\ref{duality-lemma-affine-duality}", "we have a canonical isomorphism", "$$", "\\widetilde{R\\Hom_A(K, L)} =", "R\\SheafHom_{\\mathcal{O}_X}(\\widetilde{K}, \\widetilde{L})", "$$", "in $D(\\mathcal{O}_X)$. There is a canonical map", "$$", "\\epsilon^*R\\Hom_{\\mathcal{O}_X}(\\widetilde{K}, \\widetilde{L})", "\\longrightarrow", "R\\SheafHom_{\\mathcal{O}_\\etale}(\\epsilon^*\\widetilde{K},", "\\epsilon^*\\widetilde{L})", "$$", "in $D(\\mathcal{O}_\\etale)$, see Cohomology on Sites, Remark", "\\ref{sites-cohomology-remark-prepare-fancy-base-change}.", "We will show the left and right hand side of this arrow", "have isomorphic cohomology sheaves, but we will omit the", "verification that the isomorphism is given by this arrow.", "\\medskip\\noindent", "We may assume that $L$ is given by a finite complex $I^\\bullet$", "of injective $A$-modules. By induction on the length of $I^\\bullet$", "and compatibility of the constructions with distinguished triangles,", "we reduce to the case that $L = I[0]$ where $I$ is an injective $A$-module.", "Recall that the cohomology sheaves of", "$R\\SheafHom_{\\mathcal{O}_\\etale}(\\epsilon^*\\widetilde{K},", "\\epsilon^*\\widetilde{L}))$", "are the sheafifications of the presheaf sending $U$ \\'etale", "over $X$ to the $i$th ext group between the restrictions of", "$\\epsilon^*\\widetilde{K}$ and $\\epsilon^*\\widetilde{L}$", "to $U_\\etale$. See", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-section-RHom-over-U}.", "If $U = \\Spec(B)$ is affine, then this ext group", "is equal to $\\text{Ext}^i_B(K \\otimes_A B, L \\otimes_A B)$", "by the equivalence of", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site} and", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded}", "(this also uses the compatibilities detailed in", "Derived Categories of Spaces, Remark", "\\ref{spaces-perfect-remark-match-total-direct-images}).", "Since $A \\to B$ is \\'etale, we see that", "$I \\otimes_A B$ is an injective $B$-module", "by Dualizing Complexes, Lemma \\ref{dualizing-lemma-injective-goes-up}.", "Hence we see that", "\\begin{align*}", "\\Ext^n_B(K \\otimes_A B, I \\otimes_A B)", "& =", "\\Hom_B(H^{-n}(K \\otimes_A B), I \\otimes_A B) \\\\", "& =", "\\Hom_{A_f}(H^{-n}(K) \\otimes_A B, I \\otimes_A B) \\\\", "& =", "\\Hom_A(H^{-n}(K), I) \\otimes_A B \\\\", "& =", "\\text{Ext}^n_A(K, I) \\otimes_A B", "\\end{align*}", "The penultimate equality because $H^{-n}(K)$ is a finite $A$-module, see", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-pseudo-coherence-and-base-change-ext}.", "Therefore the cohomology sheaves of the left and right hand", "side of the equality in the lemma are the same." ], "refs": [ "duality-lemma-affine-duality", "sites-cohomology-remark-prepare-fancy-base-change", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "perfect-lemma-affine-compare-bounded", "spaces-perfect-remark-match-total-direct-images", "dualizing-lemma-injective-goes-up", "more-algebra-lemma-pseudo-coherence-and-base-change-ext" ], "ref_ids": [ 13497, 4431, 2644, 6941, 2768, 2917, 10165 ] } ], "ref_ids": [] }, { "id": 11785, "type": "theorem", "label": "spaces-duality-lemma-dualizing-spaces", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-dualizing-spaces", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.", "Let $K$ be a dualizing complex on $X$.", "Then $K$ is an object of $D_{\\textit{Coh}}(\\mathcal{O}_X)$", "and $D = R\\SheafHom_{\\mathcal{O}_X}(-, K)$ induces an anti-equivalence", "$$", "D :", "D_{\\textit{Coh}}(\\mathcal{O}_X)", "\\longrightarrow", "D_{\\textit{Coh}}(\\mathcal{O}_X)", "$$", "which comes equipped with a canonical isomorphism", "$\\text{id} \\to D \\circ D$. If $X$ is quasi-compact, then", "$D$ exchanges $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and", "$D^-_{\\textit{Coh}}(\\mathcal{O}_X)$ and induces an equivalence", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be an \\'etale morphism with $U$ affine. Say $U = \\Spec(A)$ and", "let $\\omega_A^\\bullet$ be a dualizing complex for $A$ corresponding to $K|_U$", "as in Lemma \\ref{lemma-equivalent-definitions} and", "Duality for Schemes, Lemma \\ref{duality-lemma-equivalent-definitions}.", "By Lemma \\ref{lemma-affine-duality} the diagram", "$$", "\\xymatrix{", "D_{\\textit{Coh}}(A) \\ar[r] \\ar[d]_{R\\Hom_A(-, \\omega_A^\\bullet)} &", "D_{\\textit{Coh}}(\\mathcal{O}_\\etale)", "\\ar[d]^{R\\SheafHom_{\\mathcal{O}_\\etale}(-, K|_U)} \\\\", "D_{\\textit{Coh}}(A) \\ar[r] &", "D(\\mathcal{O}_\\etale)", "}", "$$", "commutes where $\\mathcal{O}_\\etale$ is the structure sheaf of the", "small \\'etale site of $U$. Since formation of $R\\SheafHom$ commutes", "with restriction, we conclude that $D$ sends", "$D_{\\textit{Coh}}(\\mathcal{O}_X)$ into", "$D_{\\textit{Coh}}(\\mathcal{O}_X)$. Moreover, the canonical map", "$$", "L \\longrightarrow", "R\\SheafHom_{\\mathcal{O}_X}(R\\SheafHom_{\\mathcal{O}_X}(L, K), K)", "$$", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-internal-hom-evaluate})", "is an isomorphism for all $L$ in $D_{\\textit{Coh}}(\\mathcal{O}_X)$", "because this is true over all $U$ as above by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-dualizing}.", "The statement on boundedness properties of the functor $D$", "in the quasi-compact case also follows from the corresponding", "statements of Dualizing Complexes, Lemma \\ref{dualizing-lemma-dualizing}." ], "refs": [ "spaces-duality-lemma-equivalent-definitions", "duality-lemma-equivalent-definitions", "spaces-duality-lemma-affine-duality", "sites-cohomology-lemma-internal-hom-evaluate", "dualizing-lemma-dualizing", "dualizing-lemma-dualizing" ], "ref_ids": [ 11783, 13496, 11784, 4331, 2848, 2848 ] } ], "ref_ids": [] }, { "id": 11786, "type": "theorem", "label": "spaces-duality-lemma-dualizing-unique-spaces", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-dualizing-unique-spaces", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a locally Noetherian algebraic space over $S$.", "If $K$ and $K'$ are dualizing complexes on $X$, then $K'$", "is isomorphic to $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$", "for some invertible object $L$ of $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Set", "$$", "L = R\\SheafHom_{\\mathcal{O}_X}(K, K')", "$$", "This is an invertible object of $D(\\mathcal{O}_X)$, because affine locally", "this is true. Use Lemma \\ref{lemma-affine-duality} and", "Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-dualizing-unique} and its proof.", "The evaluation map $L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K \\to K'$", "is an isomorphism for the same reason." ], "refs": [ "spaces-duality-lemma-affine-duality", "dualizing-lemma-dualizing-unique" ], "ref_ids": [ 11784, 2850 ] } ], "ref_ids": [] }, { "id": 11787, "type": "theorem", "label": "spaces-duality-lemma-dimension-function-scheme", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-dimension-function-scheme", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian", "quasi-separated algebraic space over $S$.", "Let $\\omega_X^\\bullet$ be a dualizing complex on $X$. Then $X$ the function", "$|X| \\to \\mathbf{Z}$ defined by", "$$", "x \\longmapsto \\delta(x)\\text{ such that }", "\\omega_{X, \\overline{x}}^\\bullet[-\\delta(x)]", "\\text{ is a normalized dualizing complex over }", "\\mathcal{O}_{X, \\overline{x}}", "$$", "is a dimension function on $|X|$." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $U \\to X$ be a surjective \\'etale morphism.", "Let $\\omega_U^\\bullet$ be the dualizing complex on $U$ associated", "to $\\omega_X^\\bullet|_U$.", "If $u \\in U$ maps to $x \\in |X|$, then $\\mathcal{O}_{X, \\overline{x}}$", "is the strict henselization of $\\mathcal{O}_{U, u}$. By", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-flat-unramified}", "we see that if $\\omega^\\bullet$ is a normalized dualizing complex", "for $\\mathcal{O}_{U, u}$, then", "$\\omega^\\bullet \\otimes_{\\mathcal{O}_{U, u}} \\mathcal{O}_{X, \\overline{x}}$", "is a normalized dualizing complex for $\\mathcal{O}_{X, \\overline{x}}$.", "Hence we see that the dimension function $U \\to \\mathbf{Z}$ of", "Duality for Schemes, Lemma \\ref{duality-lemma-dimension-function-scheme}", "for the scheme $U$ and the complex", "$\\omega_U^\\bullet$ is equal to the composition of $U \\to |X|$ with $\\delta$.", "Using the specializations in $|X|$ lift to specializations in $U$", "and that nontrivial specializations in $U$ map to", "nontrivial specializations in $X$", "(Decent Spaces, Lemmas \\ref{decent-spaces-lemma-decent-specialization} and", "\\ref{decent-spaces-lemma-decent-no-specializations-map-to-same-point})", "an easy topological argument shows that $\\delta$ is a dimension function", "on $|X|$." ], "refs": [ "dualizing-lemma-flat-unramified", "duality-lemma-dimension-function-scheme", "decent-spaces-lemma-decent-specialization", "decent-spaces-lemma-decent-no-specializations-map-to-same-point" ], "ref_ids": [ 2887, 13501, 9494, 9493 ] } ], "ref_ids": [] }, { "id": 11788, "type": "theorem", "label": "spaces-duality-lemma-twisted-inverse-image", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-twisted-inverse-image", "contents": [ "\\begin{reference}", "This is almost the same as \\cite[Example 4.2]{Neeman-Grothendieck}.", "\\end{reference}", "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism between quasi-separated and quasi-compact", "algebraic spaces over $S$. The functor $Rf_* : D_\\QCoh(X) \\to D_\\QCoh(Y)$", "has a right adjoint." ], "refs": [], "proofs": [ { "contents": [ "We will prove a right adjoint exists by verifying the hypotheses of", "Derived Categories, Proposition \\ref{derived-proposition-brown}.", "First off, the category $D_\\QCoh(\\mathcal{O}_X)$ has direct sums, see", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-quasi-coherence-direct-sums}.", "The category $D_\\QCoh(\\mathcal{O}_X)$ is compactly generated by", "Derived Categories of Spaces, Theorem", "\\ref{spaces-perfect-theorem-bondal-van-den-Bergh}.", "Since $X$ and $Y$ are quasi-compact and quasi-separated, so is $f$, see", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-compose-after-separated} and", "\\ref{spaces-morphisms-lemma-quasi-compact-permanence}.", "Hence the functor $Rf_*$ commutes with direct sums, see", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-quasi-coherence-pushforward-direct-sums}.", "This finishes the proof." ], "refs": [ "derived-proposition-brown", "spaces-perfect-lemma-quasi-coherence-direct-sums", "spaces-perfect-theorem-bondal-van-den-Bergh", "spaces-morphisms-lemma-compose-after-separated", "spaces-morphisms-lemma-quasi-compact-permanence", "spaces-perfect-lemma-quasi-coherence-pushforward-direct-sums" ], "ref_ids": [ 1966, 2646, 2640, 4720, 4743, 2653 ] } ], "ref_ids": [] }, { "id": 11789, "type": "theorem", "label": "spaces-duality-lemma-twisted-inverse-image-bounded-below", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-twisted-inverse-image-bounded-below", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-twisted-inverse-image}.", "Let $a : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$ be the right", "adjoint to $Rf_*$. Then $a$ maps", "$D^+_\\QCoh(\\mathcal{O}_Y)$ into $D^+_\\QCoh(\\mathcal{O}_X)$.", "In fact, there exists an integer $N$ such that", "$H^i(K) = 0$ for $i \\leq c$ implies $H^i(a(K)) = 0$ for $i \\leq c - N$." ], "refs": [ "spaces-duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "By Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-quasi-coherence-direct-image}", "the functor $Rf_*$ has finite cohomological dimension. In other words,", "there exist an integer $N$ such that", "$H^i(Rf_*L) = 0$ for $i \\geq N + c$ if $H^i(L) = 0$ for $i \\geq c$.", "Say $K \\in D^+_\\QCoh(\\mathcal{O}_Y)$ has $H^i(K) = 0$ for $i \\leq c$.", "Then", "$$", "\\Hom_{D(\\mathcal{O}_X)}(\\tau_{\\leq c - N}a(K), a(K)) =", "\\Hom_{D(\\mathcal{O}_Y)}(Rf_*\\tau_{\\leq c - N}a(K), K) = 0", "$$", "by what we said above. Clearly, this implies that", "$H^i(a(K)) = 0$ for $i \\leq c - N$." ], "refs": [ "spaces-perfect-lemma-quasi-coherence-direct-image" ], "ref_ids": [ 2652 ] } ], "ref_ids": [ 11788 ] }, { "id": 11790, "type": "theorem", "label": "spaces-duality-lemma-iso-on-RSheafHom", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-iso-on-RSheafHom", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated", "algebraic spaces over $S$.", "Let $a$ be the right adjoint to", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$.", "Let $L \\in D_\\QCoh(\\mathcal{O}_X)$ and", "$K \\in D_\\QCoh(\\mathcal{O}_Y)$.", "Then the map (\\ref{equation-sheafy-trace})", "$$", "Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K))", "\\longrightarrow", "R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K)", "$$", "becomes an isomorphism after applying the functor", "$DQ_Y : D(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_Y)$", "discussed in Derived Categories of Spaces, Section", "\\ref{spaces-perfect-section-better-coherator}." ], "refs": [], "proofs": [ { "contents": [ "The statement makes sense as $DQ_Y$ exists by", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-better-coherator}.", "Since $DQ_Y$ is the right adjoint to the inclusion", "functor $D_\\QCoh(\\mathcal{O}_Y) \\to D(\\mathcal{O}_Y)$", "to prove the lemma we have to show that for any", "$M \\in D_\\QCoh(\\mathcal{O}_Y)$", "the map (\\ref{equation-sheafy-trace}) induces an bijection", "$$", "\\Hom_Y(M, Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K)))", "\\longrightarrow", "\\Hom_Y(M, R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K))", "$$", "To see this we use the following string of equalities", "\\begin{align*}", "\\Hom_Y(M, Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K)))", "& =", "\\Hom_X(Lf^*M, R\\SheafHom_{\\mathcal{O}_X}(L, a(K))) \\\\", "& =", "\\Hom_X(Lf^*M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L, a(K)) \\\\", "& =", "\\Hom_Y(Rf_*(Lf^*M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L), K) \\\\", "& =", "\\Hom_Y(M \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L, K) \\\\", "& =", "\\Hom_Y(M, R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K))", "\\end{align*}", "The first equality holds by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-adjoint}.", "The second equality by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-internal-hom}.", "The third equality by construction of $a$.", "The fourth equality by Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-cohomology-base-change} (this is the important step).", "The fifth by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-internal-hom}." ], "refs": [ "spaces-perfect-lemma-better-coherator", "sites-cohomology-lemma-adjoint", "sites-cohomology-lemma-internal-hom", "spaces-perfect-lemma-cohomology-base-change", "sites-cohomology-lemma-internal-hom" ], "ref_ids": [ 2715, 4249, 4328, 2718, 4328 ] } ], "ref_ids": [] }, { "id": 11791, "type": "theorem", "label": "spaces-duality-lemma-iso-global-hom", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-iso-global-hom", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of quasi-separated and quasi-compact", "algebraic spaces over $S$.", "For all $L \\in D_\\QCoh(\\mathcal{O}_X)$ and $K \\in D_\\QCoh(\\mathcal{O}_Y)$", "(\\ref{equation-sheafy-trace}) induces an isomorphism", "$R\\Hom_X(L, a(K)) \\to R\\Hom_Y(Rf_*L, K)$ of global derived homs." ], "refs": [], "proofs": [ { "contents": [ "By construction (Cohomology on Sites, Section", "\\ref{sites-cohomology-section-global-RHom}) the complexes", "$$", "R\\Hom_X(L, a(K)) =", "R\\Gamma(X, R\\SheafHom_{\\mathcal{O}_X}(L, a(K))) =", "R\\Gamma(Y, Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K)))", "$$", "and", "$$", "R\\Hom_Y(Rf_*L, K) = R\\Gamma(Y, R\\SheafHom_{\\mathcal{O}_X}(Rf_*L, a(K)))", "$$", "Thus the lemma is a consequence of Lemma \\ref{lemma-iso-on-RSheafHom}.", "Namely, a map $E \\to E'$ in $D(\\mathcal{O}_Y)$ which induces", "an isomorphism $DQ_Y(E) \\to DQ_Y(E')$ induces a quasi-isomorphism", "$R\\Gamma(Y, E) \\to R\\Gamma(Y, E')$. Indeed we have", "$H^i(Y, E) = \\Ext^i_Y(\\mathcal{O}_Y, E) = \\Hom(\\mathcal{O}_Y[-i], E) =", "\\Hom(\\mathcal{O}_Y[-i], DQ_Y(E))$ because $\\mathcal{O}_Y[-i]$", "is in $D_\\QCoh(\\mathcal{O}_Y)$ and $DQ_Y$ is the right adjoint", "to the inclusion functor $D_\\QCoh(\\mathcal{O}_Y) \\to D(\\mathcal{O}_Y)$." ], "refs": [ "spaces-duality-lemma-iso-on-RSheafHom" ], "ref_ids": [ 11790 ] } ], "ref_ids": [] }, { "id": 11792, "type": "theorem", "label": "spaces-duality-lemma-flat-precompose-pus", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-flat-precompose-pus", "contents": [ "In diagram (\\ref{equation-base-change}) the map", "$a \\circ Rg_* \\leftarrow Rg'_* \\circ a'$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The base change map $Lg^* \\circ Rf_* K \\to Rf'_* \\circ L(g')^*K$", "is an isomorphism for every $K$ in $D_\\QCoh(\\mathcal{O}_X)$ by", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-compare-base-change}", "(this uses the assumption of Tor independence).", "Thus the corresponding transformation between adjoint functors", "is an isomorphism as well." ], "refs": [ "spaces-perfect-lemma-compare-base-change" ], "ref_ids": [ 2720 ] } ], "ref_ids": [] }, { "id": 11793, "type": "theorem", "label": "spaces-duality-lemma-compose-base-change-maps", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-compose-base-change-maps", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "X' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^l \\ar[d]_{g'} & Y \\ar[d]^g \\\\", "Z' \\ar[r]^m & Z", "}", "$$", "of quasi-compact and quasi-separated algebraic spaces over $S$ where", "both diagrams are cartesian and where $f$ and $l$", "as well as $g$ and $m$ are Tor independent.", "Then the maps (\\ref{equation-base-change-map})", "for the two squares compose to give the base", "change map for the outer rectangle (see proof for a precise statement)." ], "refs": [], "proofs": [ { "contents": [ "It follows from the assumptions that $g \\circ f$ and $m$ are Tor", "independent (details omitted), hence the statement makes sense.", "In this proof we write $k^*$ in place of $Lk^*$ and $f_*$ instead", "of $Rf_*$. Let $a$, $b$, and $c$ be the right adjoints of", "Lemma \\ref{lemma-twisted-inverse-image}", "for $f$, $g$, and $g \\circ f$ and similarly for the primed versions.", "The arrow corresponding to the top square is the composition", "$$", "\\gamma_{top} :", "k^* \\circ a \\to k^* \\circ a \\circ l_* \\circ l^*", "\\xleftarrow{\\xi_{top}} k^* \\circ k_* \\circ a' \\circ l^* \\to a' \\circ l^*", "$$", "where $\\xi_{top} : k_* \\circ a' \\to a \\circ l_*$", "is an isomorphism (hence can be inverted)", "and is the arrow ``dual'' to the base change map", "$l^* \\circ f_* \\to f'_* \\circ k^*$. The outer arrows come", "from the canonical maps $1 \\to l_* \\circ l^*$ and $k^* \\circ k_* \\to 1$.", "Similarly for the second square we have", "$$", "\\gamma_{bot} :", "l^* \\circ b \\to l^* \\circ b \\circ m_* \\circ m^*", "\\xleftarrow{\\xi_{bot}} l^* \\circ l_* \\circ b' \\circ m^* \\to b' \\circ m^*", "$$", "For the outer rectangle we get", "$$", "\\gamma_{rect} :", "k^* \\circ c \\to k^* \\circ c \\circ m_* \\circ m^*", "\\xleftarrow{\\xi_{rect}} k^* \\circ k_* \\circ c' \\circ m^* \\to c' \\circ m^*", "$$", "We have $(g \\circ f)_* = g_* \\circ f_*$ and hence", "$c = a \\circ b$ and similarly $c' = a' \\circ b'$.", "The statement of the lemma is that $\\gamma_{rect}$", "is equal to the composition", "$$", "k^* \\circ c = k^* \\circ a \\circ b \\xrightarrow{\\gamma_{top}}", "a' \\circ l^* \\circ b \\xrightarrow{\\gamma_{bot}}", "a' \\circ b' \\circ m^* = c' \\circ m^*", "$$", "To see this we contemplate the following diagram:", "$$", "\\xymatrix{", "& & k^* \\circ a \\circ b \\ar[d] \\ar[lldd] \\\\", "& & k^* \\circ a \\circ l_* \\circ l^* \\circ b \\ar[ld] \\\\", "k^* \\circ a \\circ b \\circ m_* \\circ m^* \\ar[r] &", "k^* \\circ a \\circ l_* \\circ l^* \\circ b \\circ m_* \\circ m^* &", "k^* \\circ k_* \\circ a' \\circ l^* \\circ b \\ar[u]_{\\xi_{top}} \\ar[d] \\ar[ld] \\\\", "& k^*\\circ k_* \\circ a' \\circ l^* \\circ b \\circ m_* \\circ m^*", "\\ar[u]_{\\xi_{top}} \\ar[rd] &", "a' \\circ l^* \\circ b \\ar[d] \\\\", "k^* \\circ k_* \\circ a' \\circ b' \\circ m^* \\ar[uu]_{\\xi_{rect}} \\ar[ddrr] &", "k^*\\circ k_* \\circ a' \\circ l^* \\circ l_* \\circ b' \\circ m^*", "\\ar[u]_{\\xi_{bot}} \\ar[l] \\ar[dr] &", "a' \\circ l^* \\circ b \\circ m_* \\circ m^* \\\\", "& & a' \\circ l^* \\circ l_* \\circ b' \\circ m^* \\ar[u]_{\\xi_{bot}} \\ar[d] \\\\", "& & a' \\circ b' \\circ m^*", "}", "$$", "Going down the right hand side we have the composition and going", "down the left hand side we have $\\gamma_{rect}$.", "All the quadrilaterals on the right hand side of this diagram commute", "by Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats}", "or more simply the discussion preceding", "Categories, Definition \\ref{categories-definition-horizontal-composition}.", "Hence we see that it suffices to show the diagram", "$$", "\\xymatrix{", "a \\circ l_* \\circ l^* \\circ b \\circ m_* &", "a \\circ b \\circ m_* \\ar[l] \\\\", "k_* \\circ a' \\circ l^* \\circ b \\circ m_* \\ar[u]_{\\xi_{top}} & \\\\", "k_* \\circ a' \\circ l^* \\circ l_* \\circ b' \\ar[u]_{\\xi_{bot}} \\ar[r] &", "k_* \\circ a' \\circ b' \\ar[uu]_{\\xi_{rect}}", "}", "$$", "becomes commutative if we invert the arrows $\\xi_{top}$, $\\xi_{bot}$,", "and $\\xi_{rect}$ (note that this is different from asking the", "diagram to be commutative). However, the diagram", "$$", "\\xymatrix{", "& a \\circ l_* \\circ l^* \\circ b \\circ m_* \\\\", "a \\circ l_* \\circ l^* \\circ l_* \\circ b'", "\\ar[ru]^{\\xi_{bot}} & &", "k_* \\circ a' \\circ l^* \\circ b \\circ m_* \\ar[ul]_{\\xi_{top}} \\\\", "& k_* \\circ a' \\circ l^* \\circ l_* \\circ b'", "\\ar[ul]^{\\xi_{top}} \\ar[ur]_{\\xi_{bot}}", "}", "$$", "commutes by Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats}.", "Since the diagrams", "$$", "\\vcenter{", "\\xymatrix{", "a \\circ l_* \\circ l^* \\circ b \\circ m_* & a \\circ b \\circ m \\ar[l] \\\\", "a \\circ l_* \\circ l^* \\circ l_* \\circ b' \\ar[u] &", "a \\circ l_* \\circ b' \\ar[l] \\ar[u]", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "a \\circ l_* \\circ l^* \\circ l_* \\circ b' \\ar[r] & a \\circ l_* \\circ b' \\\\", "k_* \\circ a' \\circ l^* \\circ l_* \\circ b' \\ar[u] \\ar[r] &", "k_* \\circ a' \\circ b' \\ar[u]", "}", "}", "$$", "commute (see references cited) and since the composition of", "$l_* \\to l_* \\circ l^* \\circ l_* \\to l_*$ is the identity,", "we find that it suffices to prove that", "$$", "k \\circ a' \\circ b' \\xrightarrow{\\xi_{bot}} a \\circ l_* \\circ b", "\\xrightarrow{\\xi_{top}} a \\circ b \\circ m_*", "$$", "is equal to $\\xi_{rect}$ (via the identifications $a \\circ b = c$", "and $a' \\circ b' = c'$). This is the statement dual to", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-compose-base-change}", "and the proof is complete." ], "refs": [ "spaces-duality-lemma-twisted-inverse-image", "categories-lemma-properties-2-cat-cats", "categories-definition-horizontal-composition", "categories-lemma-properties-2-cat-cats", "sites-cohomology-remark-compose-base-change" ], "ref_ids": [ 11788, 12269, 12377, 12269, 4425 ] } ], "ref_ids": [] }, { "id": 11794, "type": "theorem", "label": "spaces-duality-lemma-compose-base-change-maps-horizontal", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-compose-base-change-maps-horizontal", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "X'' \\ar[r]_{g'} \\ar[d]_{f''} & X' \\ar[r]_g \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y'' \\ar[r]^{h'} & Y' \\ar[r]^h & Y", "}", "$$", "of quasi-compact and quasi-separated algebraic spaces over $S$ where", "both diagrams are cartesian and where $f$ and $h$", "as well as $f'$ and $h'$ are Tor independent.", "Then the maps (\\ref{equation-base-change-map})", "for the two squares compose to give the base", "change map for the outer rectangle (see proof for a precise statement)." ], "refs": [], "proofs": [ { "contents": [ "It follows from the assumptions that $f$ and $h \\circ h'$ are Tor", "independent (details omitted), hence the statement makes sense.", "In this proof we write $g^*$ in place of $Lg^*$ and $f_*$ instead", "of $Rf_*$. Let $a$, $a'$, and $a''$ be the right adjoints of", "Lemma \\ref{lemma-twisted-inverse-image}", "for $f$, $f'$, and $f''$. The arrow corresponding to the right", "square is the composition", "$$", "\\gamma_{right} :", "g^* \\circ a \\to g^* \\circ a \\circ h_* \\circ h^*", "\\xleftarrow{\\xi_{right}} g^* \\circ g_* \\circ a' \\circ h^* \\to a' \\circ h^*", "$$", "where $\\xi_{right} : g_* \\circ a' \\to a \\circ h_*$", "is an isomorphism (hence can be inverted)", "and is the arrow ``dual'' to the base change map", "$h^* \\circ f_* \\to f'_* \\circ g^*$. The outer arrows come", "from the canonical maps $1 \\to h_* \\circ h^*$ and $g^* \\circ g_* \\to 1$.", "Similarly for the left square we have", "$$", "\\gamma_{left} :", "(g')^* \\circ a' \\to (g')^* \\circ a' \\circ (h')_* \\circ (h')^*", "\\xleftarrow{\\xi_{left}}", "(g')^* \\circ (g')_* \\circ a'' \\circ (h')^* \\to a'' \\circ (h')^*", "$$", "For the outer rectangle we get", "$$", "\\gamma_{rect} :", "k^* \\circ a \\to", "k^* \\circ a \\circ m_* \\circ m^* \\xleftarrow{\\xi_{rect}}", "k^* \\circ k_* \\circ a'' \\circ m^* \\to", "a'' \\circ m^*", "$$", "where $k = g \\circ g'$ and $m = h \\circ h'$.", "We have $k^* = (g')^* \\circ g^*$ and $m^* = (h')^* \\circ h^*$.", "The statement of the lemma is that $\\gamma_{rect}$", "is equal to the composition", "$$", "k^* \\circ a =", "(g')^* \\circ g^* \\circ a \\xrightarrow{\\gamma_{right}}", "(g')^* \\circ a' \\circ h^* \\xrightarrow{\\gamma_{left}}", "a'' \\circ (h')^* \\circ h^* = a'' \\circ m^*", "$$", "To see this we contemplate the following diagram", "$$", "\\xymatrix{", "& (g')^* \\circ g^* \\circ a \\ar[d] \\ar[ddl] \\\\", "& (g')^* \\circ g^* \\circ a \\circ h_* \\circ h^* \\ar[ld] \\\\", "(g')^* \\circ g^* \\circ a \\circ h_* \\circ (h')_* \\circ (h')^* \\circ h^* &", "(g')^* \\circ g^* \\circ g_* \\circ a' \\circ h^*", "\\ar[u]_{\\xi_{right}} \\ar[d] \\ar[ld] \\\\", "(g')^* \\circ g^* \\circ g_* \\circ a' \\circ (h')_* \\circ (h')^* \\circ h^*", "\\ar[u]_{\\xi_{right}} \\ar[dr] &", "(g')^* \\circ a' \\circ h^* \\ar[d] \\\\", "(g')^* \\circ g^* \\circ g_* \\circ (g')_* \\circ a'' \\circ (h')^* \\circ h^*", "\\ar[u]_{\\xi_{left}} \\ar[ddr] \\ar[dr] &", "(g')^* \\circ a' \\circ (h')_* \\circ (h')^* \\circ h^* \\\\", "& (g')^*\\circ (g')_* \\circ a'' \\circ (h')^* \\circ h^*", "\\ar[u]_{\\xi_{left}} \\ar[d] \\\\", "& a'' \\circ (h')^* \\circ h^*", "}", "$$", "Going down the right hand side we have the composition and going", "down the left hand side we have $\\gamma_{rect}$.", "All the quadrilaterals on the right hand side of this diagram commute", "by Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats}", "or more simply the discussion preceding", "Categories, Definition \\ref{categories-definition-horizontal-composition}.", "Hence we see that it suffices to show that", "$$", "g_* \\circ (g')_* \\circ a'' \\xrightarrow{\\xi_{left}}", "g_* \\circ a' \\circ (h')_* \\xrightarrow{\\xi_{right}}", "a \\circ h_* \\circ (h')_*", "$$", "is equal to $\\xi_{rect}$. This is the statement dual to", "Cohomology, Remark \\ref{cohomology-remark-compose-base-change-horizontal}", "and the proof is complete." ], "refs": [ "spaces-duality-lemma-twisted-inverse-image", "categories-lemma-properties-2-cat-cats", "categories-definition-horizontal-composition", "cohomology-remark-compose-base-change-horizontal" ], "ref_ids": [ 11788, 12269, 12377, 2271 ] } ], "ref_ids": [] }, { "id": 11795, "type": "theorem", "label": "spaces-duality-lemma-more-base-change", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-more-base-change", "contents": [ "In diagram (\\ref{equation-base-change}) assume in addition", "$g : Y' \\to Y$ is a morphism of affine schemes and $f : X \\to Y$ is proper.", "Then the base change map (\\ref{equation-base-change-map}) induces an", "isomorphism", "$$", "L(g')^*a(K) \\longrightarrow a'(Lg^*K)", "$$", "in the following cases", "\\begin{enumerate}", "\\item for all $K \\in D_\\QCoh(\\mathcal{O}_X)$ if $f$", "is flat of finite presentation,", "\\item for all $K \\in D_\\QCoh(\\mathcal{O}_X)$ if $f$", "is perfect and $Y$ Noetherian,", "\\item for $K \\in D_\\QCoh^+(\\mathcal{O}_X)$ if $g$ has finite Tor dimension", "and $Y$ Noetherian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $Y = \\Spec(A)$ and $Y' = \\Spec(A')$. As a base change of an affine", "morphism, the morphism $g'$ is affine. Let $M$ be a perfect generator", "for $D_\\QCoh(\\mathcal{O}_X)$, see Derived Categories of Spaces, Theorem", "\\ref{spaces-perfect-theorem-bondal-van-den-Bergh}. Then $L(g')^*M$ is a", "generator for $D_\\QCoh(\\mathcal{O}_{X'})$, see", "Derived Categories of Spaces, Remark", "\\ref{spaces-perfect-remark-pullback-generator}.", "Hence it suffices to show that (\\ref{equation-base-change-map})", "induces an isomorphism", "\\begin{equation}", "\\label{equation-iso}", "R\\Hom_{X'}(L(g')^*M, L(g')^*a(K))", "\\longrightarrow", "R\\Hom_{X'}(L(g')^*M, a'(Lg^*K))", "\\end{equation}", "of global hom complexes, see", "Cohomology on Sites, Section \\ref{sites-cohomology-section-global-RHom},", "as this will imply the cone of $L(g')^*a(K) \\to a'(Lg^*K)$", "is zero.", "The structure of the proof is as follows: we will first show that", "these Hom complexes are isomorphic and in the last part of the proof", "we will show that the isomorphism is induced by (\\ref{equation-iso}).", "\\medskip\\noindent", "The left hand side. Because $M$ is perfect, the canonical map", "$$", "R\\Hom_X(M, a(K)) \\otimes^\\mathbf{L}_A A'", "\\longrightarrow", "R\\Hom_{X'}(L(g')^*M, L(g')^*a(K))", "$$", "is an isomorphism by Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-affine-morphism-and-hom-out-of-perfect}.", "We can combine this with the isomorphism", "$R\\Hom_Y(Rf_*M, K) = R\\Hom_X(M, a(K))$", "of Lemma \\ref{lemma-iso-global-hom}", "to get that the left hand side equals", "$R\\Hom_Y(Rf_*M, K) \\otimes^\\mathbf{L}_A A'$.", "\\medskip\\noindent", "The right hand side. Here we first use the isomorphism", "$$", "R\\Hom_{X'}(L(g')^*M, a'(Lg^*K)) = R\\Hom_{Y'}(Rf'_*L(g')^*M, Lg^*K)", "$$", "of Lemma \\ref{lemma-iso-global-hom}. Since $f$ and $g$ are", "Tor independent the base change", "map $Lg^*Rf_*M \\to Rf'_*L(g')^*M$ is an isomorphism by", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-compare-base-change}.", "Hence we may rewrite this as $R\\Hom_{Y'}(Lg^*Rf_*M, Lg^*K)$.", "Since $Y$, $Y'$ are affine and $K$, $Rf_*M$ are in $D_\\QCoh(\\mathcal{O}_Y)$", "(Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-quasi-coherence-direct-image})", "we have a canonical map", "$$", "\\beta :", "R\\Hom_Y(Rf_*M, K) \\otimes^\\mathbf{L}_A A'", "\\longrightarrow", "R\\Hom_{Y'}(Lg^*Rf_*M, Lg^*K)", "$$", "in $D(A')$. This is the arrow", "More on Algebra, Equation (\\ref{more-algebra-equation-base-change-RHom})", "where we have used Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-affine-compare-bounded} and", "\\ref{perfect-lemma-quasi-coherence-internal-hom}", "to translate back and forth into algebra.", "\\begin{enumerate}", "\\item If $f$ is flat and of finite presentation, the complex $Rf_*M$", "is perfect on $Y$ by Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-flat-proper-perfect-direct-image-general}", "and $\\beta$ is an isomorphism by", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom} part (1).", "\\item If $f$ is perfect and $Y$ Noetherian, the complex $Rf_*M$", "is perfect on $Y$ by More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-perfect-proper-perfect-direct-image}", "and $\\beta$ is an isomorphism as before.", "\\item If $g$ has finite tor dimension and $Y$ is Noetherian,", "the complex $Rf_*M$ is pseudo-coherent on $Y$", "(Derived Categories of Spaces, Lemmas", "\\ref{spaces-perfect-lemma-direct-image-coherent} and", "\\ref{spaces-perfect-lemma-identify-pseudo-coherent-noetherian})", "and $\\beta$ is an isomorphism by", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom} part (4).", "\\end{enumerate}", "We conclude that we obtain the same answer as in the previous paragraph.", "\\medskip\\noindent", "In the rest of the proof we show that the identifications of", "the left and right hand side of (\\ref{equation-iso})", "given in the second and third paragraph are in fact given by", "(\\ref{equation-iso}). To make our formulas manageable", "we will use $(-, -)_X = R\\Hom_X(-, -)$, use $- \\otimes A'$", "in stead of $- \\otimes_A^\\mathbf{L} A'$, and we will abbreviate", "$g^* = Lg^*$ and $f_* = Rf_*$. Consider the following", "commutative diagram", "$$", "\\xymatrix{", "((g')^*M, (g')^*a(K))_{X'} \\ar[d] &", "(M, a(K))_X \\otimes A' \\ar[l]^-\\alpha \\ar[d] &", "(f_*M, K)_Y \\otimes A' \\ar@{=}[l] \\ar[d] \\\\", "((g')^*M, (g')^*a(g_*g^*K))_{X'} &", "(M, a(g_*g^*K))_X \\otimes A' \\ar[l]^-\\alpha &", "(f_*M, g_*g^*K)_Y \\otimes A' \\ar@{=}[l] \\ar@/_4pc/[dd]_{\\mu'} \\\\", "((g')^*M, (g')^*g'_*a'(g^*K))_{X'} \\ar[u] \\ar[d] &", "(M, g'_*a'(g^*K))_X \\otimes A' \\ar[u] \\ar[l]^-\\alpha \\ar[ld]^\\mu &", "(f_*M, K) \\otimes A' \\ar[d]^\\beta \\\\", "((g')^*M, a'(g^*K))_{X'} &", "(f'_*(g')^*M, g^*K)_{Y'} \\ar@{=}[l] \\ar[r] &", "(g^*f_*M, g^*K)_{Y'}", "}", "$$", "The arrows labeled $\\alpha$ are the maps from", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-affine-morphism-and-hom-out-of-perfect}", "for the diagram with corners $X', X, Y', Y$.", "The upper part of the diagram is commutative as the horizontal arrows are", "functorial in the entries.", "The middle vertical arrows come from the invertible transformation", "$g'_* \\circ a' \\to a \\circ g_*$ of Lemma \\ref{lemma-flat-precompose-pus}", "and therefore the middle square is commutative.", "Going down the left hand side is (\\ref{equation-iso}).", "The upper horizontal arrows provide the identifications used in the", "second paragraph of the proof.", "The lower horizontal arrows including $\\beta$ provide the identifications", "used in the third paragraph of the proof. Given $E \\in D(A)$,", "$E' \\in D(A')$, and $c : E \\to E'$ in $D(A)$ we will denote", "$\\mu_c : E \\otimes A' \\to E'$ the map induced by $c$", "and the adjointness of restriction and base change;", "if $c$ is clear we write $\\mu = \\mu_c$, i.e., we", "drop $c$ from the notation. The map $\\mu$ in the diagram is of this", "form with $c$ given by the identification", "$(M, g'_*a(g^*K))_X = ((g')^*M, a'(g^*K))_{X'}$", "; the triangle involving $\\mu$ is commutative by", "Derived Categories of Spaces, Remark", "\\ref{spaces-perfect-remark-multiplication-map}.", "\\medskip\\noindent", "Observe that", "$$", "\\xymatrix{", "(M, a(g_*g^*K))_X &", "(f_*M, g_* g^*K)_Y \\ar@{=}[l] &", "(g^*f_*M, g^*K)_{Y'} \\ar@{=}[l] \\\\", "(M, g'_* a'(g^*K))_X \\ar[u] &", "((g')^*M, a'(g^*K))_{X'} \\ar@{=}[l] &", "(f'_*(g')^*M, g^*K)_{Y'} \\ar@{=}[l] \\ar[u]", "}", "$$", "is commutative by the very definition of the transformation", "$g'_* \\circ a' \\to a \\circ g_*$. Letting $\\mu'$ be as above", "corresponding to the identification", "$(f_*M, g_*g^*K)_X = (g^*f_*M, g^*K)_{Y'}$, then the", "hexagon commutes as well. Thus it suffices to show that", "$\\beta$ is equal to the composition of", "$(f_*M, K)_Y \\otimes A' \\to (f_*M, g_*g^*K)_X \\otimes A'$", "and $\\mu'$. To do this, it suffices to prove the two induced maps", "$(f_*M, K)_Y \\to (g^*f_*M, g^*K)_{Y'}$ are the same.", "In other words, it suffices to show the diagram", "$$", "\\xymatrix{", "R\\Hom_A(E, K) \\ar[rr]_{\\text{induced by }\\beta} \\ar[rd] & &", "R\\Hom_{A'}(E \\otimes_A^\\mathbf{L} A', K \\otimes_A^\\mathbf{L} A') \\\\", "& R\\Hom_A(E, K \\otimes_A^\\mathbf{L} A') \\ar[ru]", "}", "$$", "commutes for all $E, K \\in D(A)$. Since this is how $\\beta$ is constructed in", "More on Algebra, Section \\ref{more-algebra-section-base-change-RHom}", "the proof is complete." ], "refs": [ "spaces-perfect-theorem-bondal-van-den-Bergh", "spaces-perfect-remark-pullback-generator", "spaces-perfect-lemma-affine-morphism-and-hom-out-of-perfect", "spaces-duality-lemma-iso-global-hom", "spaces-duality-lemma-iso-global-hom", "spaces-perfect-lemma-compare-base-change", "spaces-perfect-lemma-quasi-coherence-direct-image", "perfect-lemma-affine-compare-bounded", "perfect-lemma-quasi-coherence-internal-hom", "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "more-algebra-lemma-base-change-RHom", "spaces-more-morphisms-lemma-perfect-proper-perfect-direct-image", "spaces-perfect-lemma-direct-image-coherent", "spaces-perfect-lemma-identify-pseudo-coherent-noetherian", "more-algebra-lemma-base-change-RHom", "spaces-perfect-lemma-affine-morphism-and-hom-out-of-perfect", "spaces-duality-lemma-flat-precompose-pus", "spaces-perfect-remark-multiplication-map" ], "ref_ids": [ 2640, 2771, 2721, 11791, 11791, 2720, 2652, 6941, 6981, 2738, 10418, 234, 2665, 2697, 10418, 2721, 11792, 2777 ] } ], "ref_ids": [] }, { "id": 11796, "type": "theorem", "label": "spaces-duality-lemma-trace-map-and-base-change", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-trace-map-and-base-change", "contents": [ "Suppose we have a diagram (\\ref{equation-base-change}).", "Then the maps", "$1 \\star \\text{Tr}_f : Lg^* \\circ Rf_* \\circ a \\to Lg^*$ and", "$\\text{Tr}_{f'} \\star 1 : Rf'_* \\circ a' \\circ Lg^* \\to Lg^*$", "agree via the base change maps", "$\\beta : Lg^* \\circ Rf_* \\to Rf'_* \\circ L(g')^*$", "(Cohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change})", "and $\\alpha : L(g')^* \\circ a \\to a' \\circ Lg^*$", "(\\ref{equation-base-change-map}).", "More precisely, the diagram", "$$", "\\xymatrix{", "Lg^* \\circ Rf_* \\circ a", "\\ar[d]_{\\beta \\star 1} \\ar[r]_-{1 \\star \\text{Tr}_f} &", "Lg^* \\\\", "Rf'_* \\circ L(g')^* \\circ a \\ar[r]^{1 \\star \\alpha} &", "Rf'_* \\circ a' \\circ Lg^* \\ar[u]_{\\text{Tr}_{f'} \\star 1}", "}", "$$", "of transformations of functors commutes." ], "refs": [ "sites-cohomology-remark-base-change" ], "proofs": [ { "contents": [ "In this proof we write $f_*$ for $Rf_*$ and $g^*$ for $Lg^*$ and we", "drop $\\star$ products with identities as one can figure out which ones", "to add as long as the source and target of the transformation is known.", "Recall that $\\beta : g^* \\circ f_* \\to f'_* \\circ (g')^*$ is an isomorphism", "and that $\\alpha$ is defined using", "the isomorphism $\\beta^\\vee : g'_* \\circ a' \\to a \\circ g_*$", "which is the adjoint of $\\beta$, see Lemma \\ref{lemma-flat-precompose-pus}", "and its proof. First we note that the top horizontal arrow", "of the diagram in the lemma is equal to the composition", "$$", "g^* \\circ f_* \\circ a \\to", "g^* \\circ f_* \\circ a \\circ g_* \\circ g^* \\to", "g^* \\circ g_* \\circ g^* \\to g^*", "$$", "where the first arrow is the unit for $(g^*, g_*)$, the second arrow", "is $\\text{Tr}_f$, and the third arrow is the counit for $(g^*, g_*)$.", "This is a simple consequence of the fact that the composition", "$g^* \\to g^* \\circ g_* \\circ g^* \\to g^*$ of unit and counit is the identity.", "Consider the diagram", "$$", "\\xymatrix{", "& g^* \\circ f_* \\circ a \\ar[ld]_\\beta \\ar[d] \\ar[r]_{\\text{Tr}_f} & g^* \\\\", "f'_* \\circ (g')^* \\circ a \\ar[dr] &", "g^* \\circ f_* \\circ a \\circ g_* \\circ g^* \\ar[d]_\\beta \\ar[ru] &", "g^* \\circ f_* \\circ g'_* \\circ a' \\circ g^* \\ar[l]_{\\beta^\\vee} \\ar[d]_\\beta &", "f'_* \\circ a' \\circ g^* \\ar[lu]_{\\text{Tr}_{f'}} \\\\", "& f'_* \\circ (g')^* \\circ a \\circ g_* \\circ g^* &", "f'_* \\circ (g')^* \\circ g'_* \\circ a' \\circ g^* \\ar[ru] \\ar[l]_{\\beta^\\vee}", "}", "$$", "In this diagram the two squares commute ", "Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats}", "or more simply the discussion preceding", "Categories, Definition \\ref{categories-definition-horizontal-composition}.", "The triangle commutes by the discussion above. By", "Categories, Lemma", "\\ref{categories-lemma-transformation-between-functors-and-adjoints}", "the square", "$$", "\\xymatrix{", "g^* \\circ f_* \\circ g'_* \\circ a' \\ar[d]_{\\beta^\\vee} \\ar[r]_-\\beta &", "f'_* \\circ (g')^* \\circ g'_* \\circ a' \\ar[d] \\\\", "g^* \\circ f_* \\circ a \\circ g_* \\ar[r] &", "\\text{id}", "}", "$$", "commutes which implies the pentagon in the big diagram commutes.", "Since $\\beta$ and $\\beta^\\vee$ are isomorphisms, and since going on", "the outside of the big diagram equals", "$\\text{Tr}_f \\circ \\alpha \\circ \\beta$ by definition this proves the lemma." ], "refs": [ "spaces-duality-lemma-flat-precompose-pus", "categories-lemma-properties-2-cat-cats", "categories-definition-horizontal-composition", "categories-lemma-transformation-between-functors-and-adjoints" ], "ref_ids": [ 11792, 12269, 12377, 12251 ] } ], "ref_ids": [ 4424 ] }, { "id": 11797, "type": "theorem", "label": "spaces-duality-lemma-unit-and-base-change", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-unit-and-base-change", "contents": [ "Suppose we have a diagram (\\ref{equation-base-change}). Then the maps", "$1 \\star \\eta_f : L(g')^* \\to L(g')^* \\circ a \\circ Rf_*$ and", "$\\eta_{f'} \\star 1 : L(g')^* \\to a' \\circ Rf'_* \\circ L(g')^*$", "agree via the base change maps", "$\\beta : Lg^* \\circ Rf_* \\to Rf'_* \\circ L(g')^*$", "(Cohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change})", "and $\\alpha : L(g')^* \\circ a \\to a' \\circ Lg^*$", "(\\ref{equation-base-change-map}).", "More precisely, the diagram", "$$", "\\xymatrix{", "L(g')^* \\ar[r]_-{1 \\star \\eta_f} \\ar[d]_{\\eta_{f'} \\star 1} &", "L(g')^* \\circ a \\circ Rf_* \\ar[d]^\\alpha \\\\", "a' \\circ Rf'_* \\circ L(g')^* &", "a' \\circ Lg^* \\circ Rf_* \\ar[l]_-\\beta", "}", "$$", "of transformations of functors commutes." ], "refs": [ "sites-cohomology-remark-base-change" ], "proofs": [ { "contents": [ "This proof is dual to the proof of Lemma \\ref{lemma-trace-map-and-base-change}.", "In this proof we write $f_*$ for $Rf_*$ and $g^*$ for $Lg^*$ and we", "drop $\\star$ products with identities as one can figure out which ones", "to add as long as the source and target of the transformation is known.", "Recall that $\\beta : g^* \\circ f_* \\to f'_* \\circ (g')^*$ is an isomorphism", "and that $\\alpha$ is defined using", "the isomorphism $\\beta^\\vee : g'_* \\circ a' \\to a \\circ g_*$", "which is the adjoint of $\\beta$, see Lemma \\ref{lemma-flat-precompose-pus}", "and its proof. First we note that the left vertical arrow", "of the diagram in the lemma is equal to the composition", "$$", "(g')^* \\to (g')^* \\circ g'_* \\circ (g')^* \\to", "(g')^* \\circ g'_* \\circ a' \\circ f'_* \\circ (g')^* \\to", "a' \\circ f'_* \\circ (g')^*", "$$", "where the first arrow is the unit for $((g')^*, g'_*)$, the second arrow", "is $\\eta_{f'}$, and the third arrow is the counit for $((g')^*, g'_*)$.", "This is a simple consequence of the fact that the composition", "$(g')^* \\to (g')^* \\circ (g')_* \\circ (g')^* \\to (g')^*$", "of unit and counit is the identity. Consider the diagram", "$$", "\\xymatrix{", "& (g')^* \\circ a \\circ f_* \\ar[r] &", "(g')^* \\circ a \\circ g_* \\circ g^* \\circ f_*", "\\ar[ld]_\\beta \\\\", "(g')^* \\ar[ru]^{\\eta_f} \\ar[dd]_{\\eta_{f'}} \\ar[rd] &", "(g')^* \\circ a \\circ g_* \\circ f'_* \\circ (g')^* &", "(g')^* \\circ g'_* \\circ a' \\circ g^* \\circ f_*", "\\ar[u]_{\\beta^\\vee} \\ar[ld]_\\beta \\ar[d] \\\\", "& (g')^* \\circ g'_* \\circ a' \\circ f'_* \\circ (g')^*", "\\ar[ld] \\ar[u]_{\\beta^\\vee} &", "a' \\circ g^* \\circ f_* \\ar[lld]^\\beta \\\\", "a' \\circ f'_* \\circ (g')^*", "}", "$$", "In this diagram the two squares commute ", "Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats}", "or more simply the discussion preceding", "Categories, Definition \\ref{categories-definition-horizontal-composition}.", "The triangle commutes by the discussion above. By the dual of", "Categories, Lemma", "\\ref{categories-lemma-transformation-between-functors-and-adjoints}", "the square", "$$", "\\xymatrix{", "\\text{id} \\ar[r] \\ar[d] &", "g'_* \\circ a' \\circ g^* \\circ f_* \\ar[d]^\\beta \\\\", "g'_* \\circ a' \\circ g^* \\circ f_* \\ar[r]^{\\beta^\\vee} &", "a \\circ g_* \\circ f'_* \\circ (g')^*", "}", "$$", "commutes which implies the pentagon in the big diagram commutes.", "Since $\\beta$ and $\\beta^\\vee$ are isomorphisms, and since going on", "the outside of the big diagram equals", "$\\beta \\circ \\alpha \\circ \\eta_f$ by definition this proves the lemma." ], "refs": [ "spaces-duality-lemma-trace-map-and-base-change", "spaces-duality-lemma-flat-precompose-pus", "categories-lemma-properties-2-cat-cats", "categories-definition-horizontal-composition", "categories-lemma-transformation-between-functors-and-adjoints" ], "ref_ids": [ 11796, 11792, 12269, 12377, 12251 ] } ], "ref_ids": [ 4424 ] }, { "id": 11798, "type": "theorem", "label": "spaces-duality-lemma-compare-with-pullback-perfect", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-compare-with-pullback-perfect", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated", "algebraic spaces over $S$. The map", "$Lf^*K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} a(L) \\to", "a(K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} L)$", "defined above for $K, L \\in D_\\QCoh(\\mathcal{O}_Y)$", "is an isomorphism if $K$ is perfect. In particular,", "(\\ref{equation-compare-with-pullback}) is an isomorphism if $K$ is perfect." ], "refs": [], "proofs": [ { "contents": [ "Let $K^\\vee$ be the ``dual'' to $K$, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-dual-perfect-complex}.", "For $M \\in D_\\QCoh(\\mathcal{O}_X)$ we have", "\\begin{align*}", "\\Hom_{D(\\mathcal{O}_Y)}(Rf_*M, K \\otimes^\\mathbf{L}_{\\mathcal{O}_Y} L)", "& =", "\\Hom_{D(\\mathcal{O}_Y)}(", "Rf_*M \\otimes^\\mathbf{L}_{\\mathcal{O}_Y} K^\\vee, L) \\\\", "& =", "\\Hom_{D(\\mathcal{O}_X)}(", "M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} Lf^*K^\\vee, a(L)) \\\\", "& =", "\\Hom_{D(\\mathcal{O}_X)}(M,", "Lf^*K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} a(L))", "\\end{align*}", "Second equality by the definition of $a$ and the projection formula", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-projection-formula})", "or the more general Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-cohomology-base-change}.", "Hence the result by the Yoneda lemma." ], "refs": [ "sites-cohomology-lemma-dual-perfect-complex", "sites-cohomology-lemma-projection-formula", "spaces-perfect-lemma-cohomology-base-change" ], "ref_ids": [ 4390, 4396, 2718 ] } ], "ref_ids": [] }, { "id": 11799, "type": "theorem", "label": "spaces-duality-lemma-restriction-compare-with-pullback", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-restriction-compare-with-pullback", "contents": [ "Suppose we have a diagram (\\ref{equation-base-change}).", "Let $K \\in D_\\QCoh(\\mathcal{O}_Y)$. The diagram", "$$", "\\xymatrix{", "L(g')^*(Lf^*K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} a(\\mathcal{O}_Y))", "\\ar[r] \\ar[d] & L(g')^*a(K) \\ar[d] \\\\", "L(f')^*Lg^*K \\otimes_{\\mathcal{O}_{X'}}^\\mathbf{L} a'(\\mathcal{O}_{Y'})", "\\ar[r] & a'(Lg^*K)", "}", "$$", "commutes where the horizontal arrows are the maps", "(\\ref{equation-compare-with-pullback}) for $K$ and $Lg^*K$", "and the vertical maps are constructed using", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change} and", "(\\ref{equation-base-change-map})." ], "refs": [ "sites-cohomology-remark-base-change" ], "proofs": [ { "contents": [ "In this proof we will write $f_*$ for $Rf_*$ and $f^*$ for $Lf^*$, etc,", "and we will write $\\otimes$ for $\\otimes^\\mathbf{L}_{\\mathcal{O}_X}$, etc.", "Let us write (\\ref{equation-compare-with-pullback}) as the composition", "\\begin{align*}", "f^*K \\otimes a(\\mathcal{O}_Y)", "& \\to", "a(f_*(f^*K \\otimes a(\\mathcal{O}_Y))) \\\\", "& \\leftarrow", "a(K \\otimes f_*a(\\mathcal{O}_K)) \\\\", "& \\to", "a(K \\otimes \\mathcal{O}_Y) \\\\", "& \\to", "a(K)", "\\end{align*}", "Here the first arrow is the unit $\\eta_f$, the second arrow is $a$", "applied to Cohomology on Sites, Equation", "(\\ref{sites-cohomology-equation-projection-formula-map}) which is an", "isomorphism by Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-cohomology-base-change}, the third arrow is", "$a$ applied to $\\text{id}_K \\otimes \\text{Tr}_f$, and the fourth", "arrow is $a$ applied to the isomorphism $K \\otimes \\mathcal{O}_Y = K$.", "The proof of the lemma consists in showing that each of these", "maps gives rise to a commutative square as in the statement of the lemma.", "For $\\eta_f$ and $\\text{Tr}_f$ this is", "Lemmas \\ref{lemma-unit-and-base-change} and", "\\ref{lemma-trace-map-and-base-change}.", "For the arrow using Cohomology on Sites, Equation", "(\\ref{sites-cohomology-equation-projection-formula-map})", "this is Cohomology on Sites, Remark", "\\ref{sites-cohomology-remark-compatible-with-diagram}.", "For the multiplication map it is clear. This finishes the proof." ], "refs": [ "spaces-perfect-lemma-cohomology-base-change", "spaces-duality-lemma-unit-and-base-change", "spaces-duality-lemma-trace-map-and-base-change", "sites-cohomology-remark-compatible-with-diagram" ], "ref_ids": [ 2718, 11797, 11796, 4440 ] } ], "ref_ids": [ 4424 ] }, { "id": 11800, "type": "theorem", "label": "spaces-duality-lemma-proper-flat", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-proper-flat", "contents": [ "Let $S$ be a scheme.", "Let $Y$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $f : X \\to Y$ be a morphism of algebraic spaces which is proper, flat, and", "of finite presentation.", "Let $a$ be the right adjoint for", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of", "Lemma \\ref{lemma-twisted-inverse-image}. Then $a$ commutes with direct sums." ], "refs": [ "spaces-duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "Let $P$ be a perfect object of $D(\\mathcal{O}_X)$. By", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-flat-proper-perfect-direct-image-general}", "the complex $Rf_*P$ is perfect on $Y$.", "Let $K_i$ be a family of objects of $D_\\QCoh(\\mathcal{O}_Y)$.", "Then", "\\begin{align*}", "\\Hom_{D(\\mathcal{O}_X)}(P, a(\\bigoplus K_i))", "& =", "\\Hom_{D(\\mathcal{O}_Y)}(Rf_*P, \\bigoplus K_i) \\\\", "& =", "\\bigoplus \\Hom_{D(\\mathcal{O}_Y)}(Rf_*P, K_i) \\\\", "& =", "\\bigoplus \\Hom_{D(\\mathcal{O}_X)}(P, a(K_i))", "\\end{align*}", "because a perfect object is compact (Derived Categories of Spaces,", "Proposition \\ref{spaces-perfect-proposition-compact-is-perfect}).", "Since $D_\\QCoh(\\mathcal{O}_X)$ has a perfect generator", "(Derived Categories of Spaces, Theorem", "\\ref{spaces-perfect-theorem-bondal-van-den-Bergh})", "we conclude that the map $\\bigoplus a(K_i) \\to a(\\bigoplus K_i)$", "is an isomorphism, i.e., $a$ commutes with direct sums." ], "refs": [ "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "spaces-perfect-proposition-compact-is-perfect", "spaces-perfect-theorem-bondal-van-den-Bergh" ], "ref_ids": [ 2738, 2758, 2640 ] } ], "ref_ids": [ 11788 ] }, { "id": 11801, "type": "theorem", "label": "spaces-duality-lemma-compare-with-pullback-flat-proper", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-compare-with-pullback-flat-proper", "contents": [ "Let $S$ be a scheme.", "Let $Y$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $f : X \\to Y$ be a morphism of algebraic spaces which is proper, flat, and", "of finite presentation.", "The map (\\ref{equation-compare-with-pullback}) is an isomorphism", "for every object $K$ of $D_\\QCoh(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-proper-flat} we know that $a$ commutes", "with direct sums. Hence the collection of objects of", "$D_\\QCoh(\\mathcal{O}_Y)$ for which (\\ref{equation-compare-with-pullback})", "is an isomorphism is a strictly full, saturated, triangulated", "subcategory of $D_\\QCoh(\\mathcal{O}_Y)$ which is moreover", "preserved under taking direct sums. Since $D_\\QCoh(\\mathcal{O}_Y)$", "is a module category (Derived Categories of Spaces, Theorem", "\\ref{spaces-perfect-theorem-DQCoh-is-Ddga}) generated by a single", "perfect object (Derived Categories of Spaces, Theorem", "\\ref{spaces-perfect-theorem-bondal-van-den-Bergh})", "we can argue as in", "More on Algebra, Remark \\ref{more-algebra-remark-P-resolution}", "to see that it suffices to prove (\\ref{equation-compare-with-pullback})", "is an isomorphism for a single perfect object.", "However, the result holds for perfect objects, see", "Lemma \\ref{lemma-compare-with-pullback-perfect}." ], "refs": [ "spaces-duality-lemma-proper-flat", "spaces-perfect-theorem-DQCoh-is-Ddga", "spaces-perfect-theorem-bondal-van-den-Bergh", "more-algebra-remark-P-resolution", "spaces-duality-lemma-compare-with-pullback-perfect" ], "ref_ids": [ 11800, 2641, 2640, 10653, 11798 ] } ], "ref_ids": [] }, { "id": 11802, "type": "theorem", "label": "spaces-duality-lemma-properties-relative-dualizing", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-properties-relative-dualizing", "contents": [ "Let $Y$ be an affine scheme. Let $f : X \\to Y$ be a morphism of", "algebraic spaces which is proper, flat, and of finite presentation.", "Let $a$ be the right adjoint for", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of", "Lemma \\ref{lemma-twisted-inverse-image}.", "Then", "\\begin{enumerate}", "\\item $a(\\mathcal{O}_Y)$ is a $Y$-perfect object of $D(\\mathcal{O}_X)$,", "\\item $Rf_*a(\\mathcal{O}_Y)$ has vanishing cohomology sheaves", "in positive degrees,", "\\item $\\mathcal{O}_X \\to", "R\\SheafHom_{\\mathcal{O}_X}(a(\\mathcal{O}_Y), a(\\mathcal{O}_Y))$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [ "spaces-duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "For a perfect object $E$ of $D(\\mathcal{O}_X)$ we have", "\\begin{align*}", "Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_{X/Y}^\\bullet)", "& =", "Rf_*R\\SheafHom_{\\mathcal{O}_X}(E^\\vee, \\omega_{X/Y}^\\bullet) \\\\", "& =", "R\\SheafHom_{\\mathcal{O}_Y}(Rf_*E^\\vee, \\mathcal{O}_Y) \\\\", "& =", "(Rf_*E^\\vee)^\\vee", "\\end{align*}", "For the first equality, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-dual-perfect-complex}.", "For the second equality, see Lemma \\ref{lemma-iso-on-RSheafHom},", "Remark \\ref{remark-iso-on-RSheafHom}, and ", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-flat-proper-perfect-direct-image-general}.", "The third equality is the definition of the dual. In particular", "these references also show that the outcome is a perfect object", "of $D(\\mathcal{O}_Y)$. We conclude that $\\omega_{X/Y}^\\bullet$", "is $Y$-perfect by More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-characterize-relatively-perfect}.", "This proves (1).", "\\medskip\\noindent", "Let $M$ be an object of $D_\\QCoh(\\mathcal{O}_Y)$. Then", "\\begin{align*}", "\\Hom_Y(M, Rf_*a(\\mathcal{O}_Y)) & =", "\\Hom_X(Lf^*M, a(\\mathcal{O}_Y)) \\\\", "& =", "\\Hom_Y(Rf_*Lf^*M, \\mathcal{O}_Y) \\\\", "& =", "\\Hom_Y(M \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*\\mathcal{O}_Y, \\mathcal{O}_Y)", "\\end{align*}", "The first equality holds by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-adjoint}.", "The second equality by construction of $a$.", "The third equality by Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-cohomology-base-change}.", "Recall $Rf_*\\mathcal{O}_X$ is perfect of tor amplitude in $[0, N]$", "for some $N$, see", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-flat-proper-perfect-direct-image-general}.", "Thus we can represent $Rf_*\\mathcal{O}_X$ by a complex of", "finite projective modules sitting in degrees $[0, N]$", "(using More on Algebra, Lemma \\ref{more-algebra-lemma-perfect}", "and the fact that $Y$ is affine).", "Hence if $M = \\mathcal{O}_Y[-i]$ for some $i > 0$, then the last", "group is zero. Since $Y$ is affine we conclude that", "$H^i(Rf_*a(\\mathcal{O}_Y)) = 0$ for $i > 0$.", "This proves (2).", "\\medskip\\noindent", "Let $E$ be a perfect object of $D_\\QCoh(\\mathcal{O}_X)$. Then", "we have", "\\begin{align*}", "\\Hom_X(E, R\\SheafHom_{\\mathcal{O}_X}(a(\\mathcal{O}_Y), a(\\mathcal{O}_Y))", "& =", "\\Hom_X(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} a(\\mathcal{O}_Y),", "a(\\mathcal{O}_Y)) \\\\", "& =", "\\Hom_Y(Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} a(\\mathcal{O}_Y)),", "\\mathcal{O}_Y) \\\\", "& =", "\\Hom_Y(Rf_*(R\\SheafHom_{\\mathcal{O}_X}(E^\\vee, a(\\mathcal{O}_Y))),", "\\mathcal{O}_Y) \\\\", "& =", "\\Hom_Y(R\\SheafHom_{\\mathcal{O}_Y}(Rf_*E^\\vee, \\mathcal{O}_Y),", "\\mathcal{O}_Y) \\\\", "& =", "R\\Gamma(Y, Rf_*E^\\vee) \\\\", "& =", "\\Hom_X(E, \\mathcal{O}_X)", "\\end{align*}", "The first equality holds by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-internal-hom}.", "The second equality is the definition of $a$.", "The third equality comes from the construction of the dual perfect", "complex $E^\\vee$, see Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-dual-perfect-complex}.", "The fourth equality follows from the equality", "$Rf_*R\\SheafHom_{\\mathcal{O}_X}(E^\\vee, \\omega_{X/Y}^\\bullet) =", "R\\SheafHom_{\\mathcal{O}_Y}(Rf_*E^\\vee, \\mathcal{O}_Y)$", "shown in the first paragraph of the proof.", "The fifth equality holds by double duality for perfect complexes", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-dual-perfect-complex})", "and the fact that $Rf_*E$ is perfect by", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-flat-proper-perfect-direct-image-general}", "The last equality is Leray for $f$.", "This string of equalities essentially shows (3)", "holds by the Yoneda lemma. Namely, the object", "$R\\SheafHom(a(\\mathcal{O}_Y), a(\\mathcal{O}_Y))$", "is in $D_\\QCoh(\\mathcal{O}_X)$ by Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-quasi-coherence-internal-hom}.", "Taking $E = \\mathcal{O}_X$ in the above we get a map", "$\\alpha : \\mathcal{O}_X \\to", "R\\SheafHom_{\\mathcal{O}_X}(a(\\mathcal{O}_Y), a(\\mathcal{O}_Y))$", "corresponding to", "$\\text{id}_{\\mathcal{O}_X} \\in \\Hom_X(\\mathcal{O}_X, \\mathcal{O}_X)$.", "Since all the isomorphisms above are functorial in $E$ we", "see that the cone on $\\alpha$ is an object $C$ of $D_\\QCoh(\\mathcal{O}_X)$", "such that $\\Hom(E, C) = 0$ for all perfect $E$.", "Since the perfect objects generate", "(Derived Categories of Spaces, Theorem", "\\ref{spaces-perfect-theorem-bondal-van-den-Bergh})", "we conclude that $\\alpha$ is an isomorphism." ], "refs": [ "sites-cohomology-lemma-dual-perfect-complex", "spaces-duality-lemma-iso-on-RSheafHom", "spaces-duality-remark-iso-on-RSheafHom", "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "spaces-more-morphisms-lemma-characterize-relatively-perfect", "sites-cohomology-lemma-adjoint", "spaces-perfect-lemma-cohomology-base-change", "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "more-algebra-lemma-perfect", "sites-cohomology-lemma-internal-hom", "sites-cohomology-lemma-dual-perfect-complex", "sites-cohomology-lemma-dual-perfect-complex", "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "spaces-perfect-lemma-quasi-coherence-internal-hom", "spaces-perfect-theorem-bondal-van-den-Bergh" ], "ref_ids": [ 4390, 11790, 11811, 2738, 268, 4249, 2718, 2738, 10212, 4328, 4390, 4390, 2738, 2700, 2640 ] } ], "ref_ids": [ 11788 ] }, { "id": 11803, "type": "theorem", "label": "spaces-duality-lemma-relative-dualizing-RHom", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-relative-dualizing-RHom", "contents": [ "Let $S$ be a scheme. Let $X \\to Y$ be a proper, flat morphism of", "algebraic spaces which is of finite presentation.", "If $(\\omega_{X/Y}^\\bullet, \\tau)$ is a relative dualizing complex,", "then $\\mathcal{O}_X \\to", "R\\SheafHom_{\\mathcal{O}_X}(\\omega_{X/Y}^\\bullet, \\omega_{X/Y}^\\bullet)$", "is an isomorphism and $Rf_*\\omega_{X/Y}^\\bullet$ has vanishing cohomology", "sheaves in positive degrees." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove this after base change to an affine scheme \\'etale", "over $Y$ in which case it follows from", "Lemma \\ref{lemma-properties-relative-dualizing}." ], "refs": [ "spaces-duality-lemma-properties-relative-dualizing" ], "ref_ids": [ 11802 ] } ], "ref_ids": [] }, { "id": 11804, "type": "theorem", "label": "spaces-duality-lemma-uniqueness-relative-dualizing", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-uniqueness-relative-dualizing", "contents": [ "Let $S$ be a scheme. Let $X \\to Y$ be a proper, flat morphism of", "algebraic spaces which is of finite presentation.", "If $(\\omega_j^\\bullet, \\tau_j)$, $j = 1, 2$", "are two relative dualizing complexes on $X/Y$,", "then there is a unique isomorphism", "$(\\omega_1^\\bullet, \\tau_1) \\to (\\omega_2^\\bullet, \\tau_2)$." ], "refs": [], "proofs": [ { "contents": [ "Consider $g : Y' \\to Y$ \\'etale with $Y'$ an affine scheme", "and denote $X' = Y' \\times_Y X$ the base change.", "By Definition \\ref{definition-relative-dualizing-proper-flat}", "and the discussion following, there is a unique isomorphism", "$\\iota : (\\omega_1^\\bullet|_{X'}, \\tau_1|_{Y'}) \\to", "(\\omega_2^\\bullet|_{X'}, \\tau_2|_{Y'})$. If $Y'' \\to Y'$", "is a further \\'etale morphism of affines and $X'' = Y'' \\times_Y X$,", "then $\\iota|_{X''}$ is the unique isomorphism", "$(\\omega_1^\\bullet|_{X''}, \\tau_1|_{Y''}) \\to", "(\\omega_2^\\bullet|_{X''}, \\tau_2|_{Y''})$ (by uniqueness).", "Also we have", "$$", "\\text{Ext}^p_{X'}(\\omega_1^\\bullet|_{X'}, \\omega_2^\\bullet|_{X'}) = 0,", "\\quad p < 0", "$$", "because", "$\\mathcal{O}_{X'} \\cong", "R\\SheafHom_{\\mathcal{O}_{X'}}(\\omega_1^\\bullet|_{X'}, \\omega_1^\\bullet|_{X'})", "\\cong", "R\\SheafHom_{\\mathcal{O}_{X'}}(\\omega_1^\\bullet|_{X'}, \\omega_2^\\bullet|_{X'})$", "by Lemma \\ref{lemma-relative-dualizing-RHom}.", "\\medskip\\noindent", "Choose a \\'etale hypercovering $b : V \\to Y$ such that each", "$V_n = \\coprod_{i \\in I_n} Y_{n, i}$ with $Y_{n, i}$ affine.", "This is possible by Hypercoverings, Lemma", "\\ref{hypercovering-lemma-hypercovering-object} and", "Remark \\ref{hypercovering-remark-take-unions-hypercovering-X}", "(to replace the hypercovering produced", "in the lemma by the one having disjoint unions in each degree).", "Denote $X_{n, i} = Y_{n, i} \\times_Y X$ and $U_n = V_n \\times_Y X$", "so that we obtain an \\'etale hypercovering", "$a : U \\to X$ (Hypercoverings, Lemma", "\\ref{hypercovering-lemma-hypercovering-morphism-sites})", "with $U_n = \\coprod X_{n, i}$.", "The assumptions of Simplicial Spaces, Lemma", "\\ref{spaces-simplicial-lemma-fppf-neg-ext-zero-hom}", "are satisfied for $a : U \\to X$ and the complexes", "$\\omega_1^\\bullet$ and $\\omega_2^\\bullet$.", "Hence we obtain a unique morphism", "$\\iota : \\omega_1^\\bullet \\to \\omega_2^\\bullet$", "whose restriction to $X_{0, i}$ is the unique", "isomorphism $(\\omega_1^\\bullet|_{X_{0, i}}, \\tau_1|_{Y_{0, i}}) \\to", "(\\omega_2^\\bullet|_{X_{0, i}}, \\tau_2|_{Y_{0, i}})$", "We still have to see that the diagram", "$$", "\\xymatrix{", "Rf_*\\omega_1^\\bullet \\ar[rd]_{\\tau_1} \\ar[rr]_{Rf_*\\iota} & &", "Rf_*\\omega_1^\\bullet \\ar[ld]^{\\tau_2} \\\\", "& \\mathcal{O}_Y", "}", "$$", "is commutative. However, we know that $Rf_*\\omega_1^\\bullet$ and", "$Rf_*\\omega_2^\\bullet$ have vanishing cohomology sheaves in positive", "degrees (Lemma \\ref{lemma-relative-dualizing-RHom})", "thus this commutativity may be proved after", "restricting to the affines $Y_{0, i}$ where it holds by construction." ], "refs": [ "spaces-duality-definition-relative-dualizing-proper-flat", "spaces-duality-lemma-relative-dualizing-RHom", "hypercovering-lemma-hypercovering-object", "hypercovering-remark-take-unions-hypercovering-X", "hypercovering-lemma-hypercovering-morphism-sites", "spaces-simplicial-lemma-fppf-neg-ext-zero-hom", "spaces-duality-lemma-relative-dualizing-RHom" ], "ref_ids": [ 11810, 11803, 8415, 8434, 8417, 9139, 11803 ] } ], "ref_ids": [] }, { "id": 11805, "type": "theorem", "label": "spaces-duality-lemma-covering-enough", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-covering-enough", "contents": [ "Let $S$ be a scheme. Let $X \\to Y$ be a proper, flat morphism of", "algebraic spaces which is of finite presentation.", "Let $(\\omega^\\bullet, \\tau)$ be a pair consisting", "of a $Y$-perfect object of $D(\\mathcal{O}_X)$ and a map", "$\\tau : Rf_*\\omega^\\bullet \\to \\mathcal{O}_Y$.", "Assume we have cartesian diagrams", "$$", "\\xymatrix{", "X_i \\ar[r]_{g_i'} \\ar[d]_{f_i} & X \\ar[d]^f \\\\", "Y_i \\ar[r]^{g_i} & Y", "}", "$$", "with $Y_i$ affine such that $\\{g_i : Y_i \\to Y\\}$ is an \\'etale covering", "and isomorphisms of pairs $(\\omega^\\bullet|_{X_i}, \\tau|_{Y_i})", "\\to (a_i(\\mathcal{O}_{Y_i}), \\text{Tr}_{f_i, \\mathcal{O}_{Y_i}})$", "as in Definition \\ref{definition-relative-dualizing-proper-flat}.", "Then $(\\omega^\\bullet, \\tau)$ is a relative dualizing complex for $X$ over $Y$." ], "refs": [ "spaces-duality-definition-relative-dualizing-proper-flat" ], "proofs": [ { "contents": [ "Let $g : Y' \\to Y$ and $X', f', g', a'$ be as in", "Definition \\ref{definition-relative-dualizing-proper-flat}.", "Set $((\\omega')^\\bullet, \\tau') = (L(g')^*\\omega^\\bullet, Lg^*\\tau)$.", "We can find a finite \\'etale covering", "$\\{Y'_j \\to Y'\\}$ by affines which refines $\\{Y_i \\times_Y Y' \\to Y'\\}$", "(Topologies, Lemma \\ref{topologies-lemma-etale-affine}).", "Thus for each $j$ there is an $i_j$ and a morphism", "$k_j : Y'_j \\to Y_{i_j}$ over $Y$. Consider the fibre products", "$$", "\\xymatrix{", "X'_j \\ar[r]_{h_j'} \\ar[d]_{f'_j} &", "X' \\ar[d]^{f'} \\\\", "Y'_j \\ar[r]^{h_j} & Y'", "}", "$$", "Denote $k'_j : X'_j \\to X_{i_j}$ the induced morphism (base change", "of $k_j$ by $f_{i_j}$). Restricting the given isomorphisms to $Y'_j$", "via the morphism $k'_j$ we get isomorphisms of pairs", "$((\\omega')^\\bullet|_{X'_j}, \\tau'|_{Y'_j})", "\\to (a_j(\\mathcal{O}_{Y'_j}), \\text{Tr}_{f'_j, \\mathcal{O}_{Y'_j}})$.", "After replacing $f : X \\to Y$ by $f' : X' \\to Y'$ we reduce to the", "problem solved in the next paragraph.", "\\medskip\\noindent", "Assume $Y$ is affine. Problem: show $(\\omega^\\bullet, \\tau)$ is", "isomorphic to $(\\omega_{X/Y}^\\bullet, \\text{Tr}) =", "(a(\\mathcal{O}_Y), \\text{Tr}_{f, \\mathcal{O}_Y})$.", "We may assume our covering $\\{Y_i \\to Y\\}$ is given by a single", "surjective \\'etale morphism $\\{g : Y' \\to Y\\}$ of affines.", "Namely, we can first replace $\\{g_i: Y_i \\to Y\\}$ by a finite", "subcovering, and then we can set $g = \\coprod g_i : Y' = \\coprod Y_i \\to Y$;", "some details omitted. Set $X' = Y' \\times_Y X$ with maps", "$f', g'$ as in Definition \\ref{definition-relative-dualizing-proper-flat}.", "Then all we're given is that we have an isomorphism", "$$", "(\\omega^\\bullet|_{X'}, \\tau|_{Y'}) \\to", "(a'(\\mathcal{O}_{Y'}), \\text{Tr}_{f', \\mathcal{O}_{Y'}})", "$$", "Since $(\\omega_{X/Y}^\\bullet, \\text{Tr})$ is a relative dualizing complex", "(see discussion following", "Definition \\ref{definition-relative-dualizing-proper-flat})", "there is a unique isomorphism", "$$", "(\\omega_{X/Y}^\\bullet|_{X'}, \\text{Tr}|_{Y'}) \\to", "(a'(\\mathcal{O}_{Y'}), \\text{Tr}_{f', \\mathcal{O}_{Y'}})", "$$", "Uniqueness by Lemma \\ref{lemma-uniqueness-relative-dualizing} for example.", "Combining the displayed isomorphisms we find an isomorphism", "$$", "\\alpha :", "(\\omega^\\bullet|_{X'}, \\tau|_{Y'}) \\to", "(\\omega_{X/Y}^\\bullet|_{X'}, \\text{Tr}|_{Y'})", "$$", "Set $Y'' = Y' \\times_Y Y'$ and $X'' = Y'' \\times_Y X$ the two", "pullbacks of $\\alpha$ to $X''$ have to be the same by uniqueness again.", "Since we have vanishing negative self exts for", "$\\omega_{X'/Y'}^\\bullet$ over $X'$ (Lemma \\ref{lemma-relative-dualizing-RHom})", "and since this remains true after pulling back by any projection", "$Y' \\times_Y \\ldots \\times_Y Y' \\to Y'$ (small detail omitted -- compare", "with the proof of Lemma \\ref{lemma-uniqueness-relative-dualizing}),", "we find that $\\alpha$ descends to an isomorphism", "$\\omega^\\bullet \\to \\omega_{X/Y}^\\bullet$", "over $X$ by Simplicial Spaces, Lemma", "\\ref{spaces-simplicial-lemma-fppf-neg-ext-zero-hom}." ], "refs": [ "spaces-duality-definition-relative-dualizing-proper-flat", "topologies-lemma-etale-affine", "spaces-duality-definition-relative-dualizing-proper-flat", "spaces-duality-definition-relative-dualizing-proper-flat", "spaces-duality-lemma-uniqueness-relative-dualizing", "spaces-duality-lemma-relative-dualizing-RHom", "spaces-duality-lemma-uniqueness-relative-dualizing", "spaces-simplicial-lemma-fppf-neg-ext-zero-hom" ], "ref_ids": [ 11810, 12447, 11810, 11810, 11804, 11803, 11804, 9139 ] } ], "ref_ids": [ 11810 ] }, { "id": 11806, "type": "theorem", "label": "spaces-duality-lemma-existence-relative-dualizing", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-existence-relative-dualizing", "contents": [ "Let $S$ be a scheme. Let $X \\to Y$ be a proper, flat morphism of", "algebraic spaces which is of finite presentation.", "There exists a relative dualizing complex $(\\omega_{X/Y}^\\bullet, \\tau)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a \\'etale hypercovering $b : V \\to Y$ such that each", "$V_n = \\coprod_{i \\in I_n} Y_{n, i}$ with $Y_{n, i}$ affine.", "This is possible by Hypercoverings, Lemma", "\\ref{hypercovering-lemma-hypercovering-object} and", "Remark \\ref{hypercovering-remark-take-unions-hypercovering-X}", "(to replace the hypercovering produced", "in the lemma by the one having disjoint unions in each degree).", "Denote $X_{n, i} = Y_{n, i} \\times_Y X$ and $U_n = V_n \\times_Y X$", "so that we obtain an \\'etale hypercovering", "$a : U \\to X$ (Hypercoverings, Lemma", "\\ref{hypercovering-lemma-hypercovering-morphism-sites})", "with $U_n = \\coprod X_{n, i}$.", "For each $n, i$ there exists a relative dualizing complex", "$(\\omega_{n, i}^\\bullet, \\tau_{n, i})$ on $X_{n, i}/Y_{n, i}$.", "See discussion following", "Definition \\ref{definition-relative-dualizing-proper-flat}.", "For $\\varphi : [m] \\to [n]$ and $i \\in I_n$ consider the morphisms", "$g_{\\varphi, i} : Y_{n, i} \\to Y_{m, \\alpha(\\varphi)}$ and", "$g'_{\\varphi, i} : X_{n, i} \\to X_{m, \\alpha(\\varphi)}$", "which are part of the structure of the given hypercoverings", "(Hypercoverings, Section \\ref{hypercovering-section-hypercovering-sites}).", "Then we have a unique isomorphisms", "$$", "\\iota_{n, i, \\varphi} :", "(L(g'_{n, i})^*\\omega_{n, i}^\\bullet, Lg_{n, i}^*\\tau_{n, i})", "\\longrightarrow", "(\\omega_{m, \\alpha(\\varphi)(i)}^\\bullet, \\tau_{m, \\alpha(\\varphi)(i)})", "$$", "of pairs, see discussion following", "Definition \\ref{definition-relative-dualizing-proper-flat}.", "Observe that $\\omega_{n, i}^\\bullet$ has vanishing negative", "self exts on $X_{n, i}$ by Lemma \\ref{lemma-relative-dualizing-RHom}.", "Denote $(\\omega_n^\\bullet, \\tau_n)$ the pair on $U_n/V_n$ constructed using", "the pairs $(\\omega_{n, i}^\\bullet, \\tau_{n, i})$ for $i \\in I_n$.", "For $\\varphi : [m] \\to [n]$ and $i \\in I_n$ consider the morphisms", "$g_\\varphi : V_n \\to V_m$ and $g'_\\varphi : U_n \\to U_m$ which are part", "of the structure of the simplicial algebraic spaces $V$ and $U$.", "Then we have unique isomorphisms", "$$", "\\iota_\\varphi :", "(L(g'_\\varphi)^*\\omega_n^\\bullet, Lg_\\varphi^*\\tau_n)", "\\longrightarrow", "(\\omega_m^\\bullet, \\tau_m)", "$$", "of pairs constructed from the isomorphisms on the pieces.", "The uniqueness guarantees that these isomorphisms satisfy", "the transitivity condition as formulated in", "Simplicial Spaces, Definition", "\\ref{spaces-simplicial-definition-cartesian-derived-modules}.", "The assumptions of Simplicial Spaces, Lemma", "\\ref{spaces-simplicial-lemma-fppf-glue-neg-ext-zero}", "are satisfied for $a : U \\to X$, the complexes $\\omega_n^\\bullet$", "and the isomorphisms $\\iota_\\varphi$\\footnote{This lemma uses only", "$\\omega_0^\\bullet$ and the two maps $\\delta_1^1, \\delta_0^1 : [1] \\to [0]$.", "The reader can skip the first few lines of the proof of the", "referenced lemma", "because here we actually are already given a simplicial system of", "the derived category of modules.}.", "Thus we obtain an object $\\omega^\\bullet$ of $D_\\QCoh(\\mathcal{O}_X)$", "together with an isomorphism", "$\\iota_0 : \\omega^\\bullet|_{U_0} \\to \\omega_0^\\bullet$", "compatible with the two isomorphisms", "$\\iota_{\\delta^1_0}$ and $\\iota_{\\delta^1_1}$.", "Finally, we apply", "Simplicial Spaces, Lemma", "\\ref{spaces-simplicial-lemma-fppf-neg-ext-zero-hom}", "to find a unique morphism", "$$", "\\tau : Rf_*\\omega^\\bullet \\longrightarrow \\mathcal{O}_Y", "$$", "whose restriction to $V_0$ agrees with $\\tau_0$; some details omitted --", "compare with the end of the proof of", "Lemma \\ref{lemma-uniqueness-relative-dualizing}", "for example to see why we have", "the required vanishing of negative exts.", "By Lemma \\ref{lemma-covering-enough}", "the pair $(\\omega^\\bullet, \\tau)$ is a relative dualizing complex", "and the proof is complete." ], "refs": [ "hypercovering-lemma-hypercovering-object", "hypercovering-remark-take-unions-hypercovering-X", "hypercovering-lemma-hypercovering-morphism-sites", "spaces-duality-definition-relative-dualizing-proper-flat", "spaces-duality-definition-relative-dualizing-proper-flat", "spaces-duality-lemma-relative-dualizing-RHom", "spaces-simplicial-definition-cartesian-derived-modules", "spaces-simplicial-lemma-fppf-glue-neg-ext-zero", "spaces-simplicial-lemma-fppf-neg-ext-zero-hom", "spaces-duality-lemma-uniqueness-relative-dualizing", "spaces-duality-lemma-covering-enough" ], "ref_ids": [ 8415, 8434, 8417, 11810, 11810, 11803, 9149, 9140, 9139, 11804, 11805 ] } ], "ref_ids": [] }, { "id": 11807, "type": "theorem", "label": "spaces-duality-lemma-base-change-relative-dualizing", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-base-change-relative-dualizing", "contents": [ "Let $S$ be a scheme. Consider a cartesian square", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "of algebraic spaces over $S$. Assume $X \\to Y$ is proper, flat, and", "of finite presentation. Let $(\\omega_{X/Y}^\\bullet, \\tau)$ be a", "relative dualizing complex for $f$. Then", "$(L(g')^*\\omega_{X/Y}^\\bullet, Lg^*\\tau)$ is a relative dualizing", "complex for $f'$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $L(g')^*\\omega_{X/Y}^\\bullet$ is $Y'$-perfect by", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-base-change-relatively-perfect}.", "The other condition of", "Definition \\ref{definition-relative-dualizing-proper-flat}", "holds by transitivity of fibre products." ], "refs": [ "spaces-more-morphisms-lemma-base-change-relatively-perfect", "spaces-duality-definition-relative-dualizing-proper-flat" ], "ref_ids": [ 262, 11810 ] } ], "ref_ids": [] }, { "id": 11808, "type": "theorem", "label": "spaces-duality-lemma-compare", "categories": [ "spaces-duality" ], "title": "spaces-duality-lemma-compare", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "quasi-compact and quasi-separated algebraic spaces over $S$.", "Assume $X$ and $Y$ are representable and let $f_0 : X_0 \\to Y_0$ be a", "morphism of schemes representing $f$ (awkward but temporary notation).", "Let $a : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$", "be the right adjoint of $Rf_*$ from Lemma \\ref{lemma-twisted-inverse-image}.", "Let $a_0 : D_\\QCoh(\\mathcal{O}_{Y_0}) \\to D_\\QCoh(\\mathcal{O}_{X_0})$", "be the right adjoint of $Rf_*$ from", "Duality for Schemes, Lemma \\ref{duality-lemma-twisted-inverse-image}.", "Then ", "$$", "\\xymatrix{", "D_\\QCoh(\\mathcal{O}_{X_0})", "\\ar@{=}[rrrrrr]_{\\text{Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}}}", "& & & & & &", "D_\\QCoh(\\mathcal{O}_X) \\\\", "D_\\QCoh(\\mathcal{O}_{Y_0}) \\ar[u]^{a_0}", "\\ar@{=}[rrrrrr]^{\\text{Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}}}", "& & & & & &", "D_\\QCoh(\\mathcal{O}_Y) \\ar[u]_a", "}", "$$", "is commutative." ], "refs": [ "spaces-duality-lemma-twisted-inverse-image", "duality-lemma-twisted-inverse-image", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site" ], "proofs": [ { "contents": [ "Follows from uniqueness of adjoints and the compatibilities of", "Derived Categories of Spaces, Remark", "\\ref{spaces-perfect-remark-match-total-direct-images}." ], "refs": [ "spaces-perfect-remark-match-total-direct-images" ], "ref_ids": [ 2768 ] } ], "ref_ids": [ 11788, 13503, 2644, 2644 ] }, { "id": 11813, "type": "theorem", "label": "spaces-properties-theorem-exactness-stalks", "categories": [ "spaces-properties" ], "title": "spaces-properties-theorem-exactness-stalks", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "A map $a : \\mathcal{F} \\to \\mathcal{G}$ of sheaves of sets is injective", "(resp.\\ surjective) if and only if the map on stalks", "$a_{\\overline{x}} : \\mathcal{F}_{\\overline{x}} \\to \\mathcal{G}_{\\overline{x}}$", "is injective (resp.\\ surjective) for all geometric points of $X$.", "A sequence of abelian sheaves on $X_\\etale$ is exact", "if and only if it is exact on all stalks at geometric points of $S$." ], "refs": [], "proofs": [ { "contents": [ "We know the theorem is true if $X$ is a scheme, see", "\\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-exactness-stalks}.", "Choose a surjective \\'etale morphism $f : U \\to X$ where $U$ is a scheme.", "Since $\\{U \\to X\\}$ is a covering (in $X_{spaces, \\etale}$) we can check", "whether a map of sheaves is injective, or surjective by restricting", "to $U$. Now if $\\overline{u} : \\Spec(k) \\to U$ is a geometric", "point of $U$, then", "$(\\mathcal{F}|_U)_{\\overline{u}} = \\mathcal{F}_{\\overline{x}}$", "where $\\overline{x} = f \\circ \\overline{u}$. (This is clear from the", "colimits defining the stalks at $\\overline{u}$ and $\\overline{x}$, but", "it also follows from", "Lemma \\ref{lemma-stalk-pullback}.)", "Hence the result for $U$ implies the result for $X$ and we win." ], "refs": [ "etale-cohomology-theorem-exactness-stalks", "spaces-properties-lemma-stalk-pullback" ], "ref_ids": [ 6376, 11875 ] } ], "ref_ids": [] }, { "id": 11814, "type": "theorem", "label": "spaces-properties-theorem-fully-faithful", "categories": [ "spaces-properties" ], "title": "spaces-properties-theorem-fully-faithful", "contents": [ "Let $X$, $Y$ be algebraic spaces over $\\Spec(\\mathbf{Z})$.", "Let", "$$", "(g, g^\\sharp) :", "(\\Sh(X_\\etale), \\mathcal{O}_X)", "\\longrightarrow", "(\\Sh(Y_\\etale), \\mathcal{O}_Y)", "$$", "be a morphism of locally ringed topoi. Then there exists a", "unique morphism of algebraic spaces $f : X \\to Y$ such that", "$(g, g^\\sharp)$ is isomorphic to $(f_{small}, f^\\sharp)$.", "In other words, the construction", "$$", "\\textit{Spaces}/\\Spec(\\mathbf{Z})", "\\longrightarrow \\textit{Locally ringed topoi},", "\\quad", "X \\longrightarrow (X_\\etale, \\mathcal{O}_X)", "$$", "is fully faithful (morphisms up to $2$-isomorphisms on the right hand side)." ], "refs": [], "proofs": [ { "contents": [ "The uniqueness we have seen in", "Lemma \\ref{lemma-faithful}.", "Thus it suffices to prove existence.", "In this proof we will freely use the identifications of", "Equation (\\ref{equation-localize-at-scheme-ringed})", "as well as the result of", "Lemma \\ref{lemma-relocalize-morphism-at-schemes}.", "\\medskip\\noindent", "Let $U \\in \\Ob(X_\\etale)$, let", "$V \\in \\Ob(Y_\\etale)$", "and let $s \\in g^{-1}h_V(U)$ be a section. We may think of", "$s$ as a map of sheaves $s : h_U \\to g^{-1}h_V$. By", "Modules on Sites,", "Lemma \\ref{sites-modules-lemma-relocalize-morphism-ringed-topoi}", "we obtain a commutative diagram of morphisms of ringed topoi", "$$", "\\xymatrix{", "(\\Sh(X_\\etale/U), \\mathcal{O}_U)", "\\ar[rr]_-{(j, j^\\sharp)} \\ar[d]_{(g_s, g_s^\\sharp)} & &", "(\\Sh(X_\\etale), \\mathcal{O}_X) \\ar[d]^{(g, g^\\sharp)} \\\\", "(\\Sh(V_\\etale), \\mathcal{O}_V) \\ar[rr] & &", "(\\Sh(Y_\\etale), \\mathcal{O}_Y).", "}", "$$", "By", "\\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-fully-faithful}", "we obtain a unique morphism of schemes $f_s : U \\to V$ such that", "$(g_s, g_s^\\sharp)$ is $2$-isomorphic to $(f_{s, small}, f_s^\\sharp)$.", "The construction $(U, V, s) \\leadsto f_s$ just explained satisfies", "the following functoriality property: Suppose given morphisms", "$a : U' \\to U$ in $X_\\etale$ and $b : V' \\to V$ in $Y_\\etale$", "and a map $s' : h_{U'} \\to g^{-1}h_{V'}$ such that the diagram", "$$", "\\xymatrix{", "h_{U'} \\ar[d]_a \\ar[r]_{s'} & g^{-1}h_{V'} \\ar[d]^{g^{-1}b} \\\\", "h_U \\ar[r]^s & g^{-1}h_V", "}", "$$", "commutes. Then the diagram", "$$", "\\xymatrix{", "U' \\ar[r]_-{f_{s'}} \\ar[d]_a & u(V') \\ar[d]^{u(b)} \\\\", "U \\ar[r]^-{f_s} & u(V)", "}", "$$", "of schemes commutes. The reason this is true is that the same condition", "holds for the morphisms $(g_s, g_s^\\sharp)$ constructed in", "Modules on Sites,", "Lemma \\ref{sites-modules-lemma-relocalize-morphism-ringed-topoi}", "and the uniqueness in", "\\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-fully-faithful}.", "\\medskip\\noindent", "The problem is to glue the morphisms $f_s$ to a morphism of algebraic", "spaces. To do this first choose a scheme $V$ and a surjective \\'etale", "morphism $V \\to Y$. This means that $h_V \\to *$ is surjective and hence", "$g^{-1}h_V \\to *$ is surjective too. This means there exists a scheme $U$", "and a surjective \\'etale morphism $U \\to X$ and a morphism", "$s : h_U \\to g^{-1}h_V$. Next, set $R = V \\times_Y V$ and", "$R' = U \\times_X U$. Then we get", "$g^{-1}h_R = g^{-1}h_V \\times g^{-1}h_V$ as $g^{-1}$ is exact.", "Thus $s$ induces a morphism $s \\times s : h_{R'} \\to g^{-1}h_R$.", "Applying the constructions above we see that we get a", "commutative diagram of morphisms of schemes", "$$", "\\xymatrix{", "R' \\ar@<1ex>[d] \\ar@<-1ex>[d] \\ar[rr]_{f_{s \\times s}} & &", "R \\ar@<1ex>[d] \\ar@<-1ex>[d] \\\\", "U \\ar[rr]^{f_s} & &", "V", "}", "$$", "Since we have $X = U/R'$ and $Y = V/R$ (see", "Spaces, Lemma \\ref{spaces-lemma-space-presentation})", "we conclude that this diagram", "defines a morphism of algebraic spaces $f : X \\to Y$ fitting", "into an obvious commutative diagram.", "Now we still have to show that $(f_{small}, f^\\sharp)$ is", "$2$-isomorphic to $(g, g^\\sharp)$.", "Let $t_V : f_{s, small}^{-1} \\to g_s^{-1}$ and", "$t_R : f_{s \\times s, small}^{-1} \\to g_{s \\times s}^{-1}$ be", "the $2$-isomorphisms which are given to us by the construction above.", "Let $\\mathcal{G}$ be a sheaf on $Y_\\etale$. Then we see that", "$t_V$ defines an isomorphism", "$$", "f_{small}^{-1}\\mathcal{G}|_{U_\\etale}", "=", "f_{s, small}^{-1}\\mathcal{G}|_{V_\\etale}", "\\xrightarrow{t_V}", "g_s^{-1}\\mathcal{G}|_{V_\\etale}", "=", "g^{-1}\\mathcal{G}|_{U_\\etale}.", "$$", "Moreover, this isomorphism pulled back to $R'$ via either projection", "$R' \\to U$ is the isomorphism", "$$", "f_{small}^{-1}\\mathcal{G}|_{R'_\\etale}", "=", "f_{s \\times s, small}^{-1}\\mathcal{G}|_{R_\\etale}", "\\xrightarrow{t_R}", "g_{s \\times s}^{-1}\\mathcal{G}|_{R_\\etale}", "=", "g^{-1}\\mathcal{G}|_{R'_\\etale}.", "$$", "Since $\\{U \\to X\\}$ is a covering in the site $X_{spaces, \\etale}$", "this means the first displayed isomorphism descends to an isomorphism", "$t : f_{small}^{-1}\\mathcal{G} \\to g^{-1}\\mathcal{G}$", "of sheaves (small detail omitted). The isomorphism is functorial", "in $\\mathcal{G}$ since $t_V$ and $t_R$ are transformations of functors.", "Finally, $t$ is compatible with $f^\\sharp$ and $g^\\sharp$ as", "$t_V$ and $t_R$ are (some details omitted).", "This finishes the proof of the theorem." ], "refs": [ "spaces-properties-lemma-faithful", "spaces-properties-lemma-relocalize-morphism-at-schemes", "sites-modules-lemma-relocalize-morphism-ringed-topoi", "etale-cohomology-theorem-fully-faithful", "sites-modules-lemma-relocalize-morphism-ringed-topoi", "etale-cohomology-theorem-fully-faithful", "spaces-lemma-space-presentation" ], "ref_ids": [ 11905, 11901, 14182, 6381, 14182, 6381, 8149 ] } ], "ref_ids": [] }, { "id": 11815, "type": "theorem", "label": "spaces-properties-lemma-trivial-implications", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-trivial-implications", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "We have the following implications among the separation axioms", "of Definition \\ref{definition-separated}:", "\\begin{enumerate}", "\\item separated implies all the others,", "\\item quasi-separated implies Zariski locally quasi-separated.", "\\end{enumerate}" ], "refs": [ "spaces-properties-definition-separated" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 11922 ] }, { "id": 11816, "type": "theorem", "label": "spaces-properties-lemma-characterize-quasi-separated", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-characterize-quasi-separated", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is a quasi-separated algebraic space,", "\\item for $U \\to X$, $V \\to X$ with $U$, $V$ quasi-compact schemes", "the fibre product $U \\times_X V$ is quasi-compact,", "\\item for $U \\to X$, $V \\to X$ with $U$, $V$ affine", "the fibre product $U \\times_X V$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Using Spaces, Lemma", "\\ref{spaces-lemma-category-of-spaces-over-smaller-base-scheme}", "we see that we may assume $S = \\Spec(\\mathbf{Z})$.", "Since $U \\times_X V = X \\times_{X \\times X} (U \\times V)$", "and since $U \\times V$ is quasi-compact if $U$ and $V$ are so,", "we see that (1) implies (2). It is clear that (2) implies (3).", "Assume (3). Choose a scheme $W$ and a surjective \\'etale morphism", "$W \\to X$. Then $W \\times W \\to X \\times X$ is surjective \\'etale.", "Hence it suffices to show that", "$$", "j : W \\times_X W = X \\times_{(X \\times X)} (W \\times W) \\to W \\times W", "$$", "is quasi-compact, see Spaces, Lemma", "\\ref{spaces-lemma-descent-representable-transformations-property}.", "If $U \\subset W$ and $V \\subset W$ are affine opens, then", "$j^{-1}(U \\times V) = U \\times_X V$ is quasi-compact by assumption.", "Since the affine opens $U \\times V$ form an affine open covering of", "$W \\times W$ (Schemes, Lemma \\ref{schemes-lemma-affine-covering-fibre-product})", "we conclude by", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-affine}." ], "refs": [ "spaces-lemma-category-of-spaces-over-smaller-base-scheme", "spaces-lemma-descent-representable-transformations-property", "schemes-lemma-affine-covering-fibre-product", "schemes-lemma-quasi-compact-affine" ], "ref_ids": [ 8170, 8134, 7692, 7697 ] } ], "ref_ids": [] }, { "id": 11817, "type": "theorem", "label": "spaces-properties-lemma-characterize-separated", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-characterize-separated", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is a separated algebraic space,", "\\item for $U \\to X$, $V \\to X$ with $U$, $V$ affine", "the fibre product $U \\times_X V$ is affine and", "$$", "\\mathcal{O}(U) \\otimes_\\mathbf{Z} \\mathcal{O}(V)", "\\longrightarrow", "\\mathcal{O}(U \\times_X V)", "$$", "is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Using Spaces, Lemma", "\\ref{spaces-lemma-category-of-spaces-over-smaller-base-scheme}", "we see that we may assume $S = \\Spec(\\mathbf{Z})$.", "Since $U \\times_X V = X \\times_{X \\times X} (U \\times V)$", "and since $U \\times V$ is affine if $U$ and $V$ are so,", "we see that (1) implies (2).", "Assume (2). Choose a scheme $W$ and a surjective \\'etale morphism", "$W \\to X$. Then $W \\times W \\to X \\times X$ is surjective \\'etale.", "Hence it suffices to show that", "$$", "j : W \\times_X W = X \\times_{(X \\times X)} (W \\times W) \\to W \\times W", "$$", "is a closed immersion, see Spaces, Lemma", "\\ref{spaces-lemma-descent-representable-transformations-property}.", "If $U \\subset W$ and $V \\subset W$ are affine opens, then", "$j^{-1}(U \\times V) = U \\times_X V$ is affine by assumption", "and the map $U \\times_X V \\to U \\times V$ is a closed immersion", "because the corresponding ring map is surjective.", "Since the affine opens $U \\times V$ form an affine open covering", "of $W \\times W$", "(Schemes, Lemma \\ref{schemes-lemma-affine-covering-fibre-product})", "we conclude by", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion}." ], "refs": [ "spaces-lemma-category-of-spaces-over-smaller-base-scheme", "spaces-lemma-descent-representable-transformations-property", "schemes-lemma-affine-covering-fibre-product", "morphisms-lemma-closed-immersion" ], "ref_ids": [ 8170, 8134, 7692, 5125 ] } ], "ref_ids": [] }, { "id": 11818, "type": "theorem", "label": "spaces-properties-lemma-scheme-points", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-scheme-points", "contents": [ "Let $S$ be a scheme. Let $X$ be a scheme over $S$.", "The points of $X$ as a scheme are in canonical 1-1 correspondence", "with the points of $X$ as an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "This is Schemes, Lemma \\ref{schemes-lemma-characterize-points}." ], "refs": [ "schemes-lemma-characterize-points" ], "ref_ids": [ 7685 ] } ], "ref_ids": [] }, { "id": 11819, "type": "theorem", "label": "spaces-properties-lemma-points-cartesian", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-points-cartesian", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "Z \\times_Y X \\ar[r] \\ar[d] & X \\ar[d] \\\\", "Z \\ar[r] & Y", "}", "$$", "be a cartesian diagram of algebraic spaces over $S$. Then the map of sets", "of points", "$$", "|Z \\times_Y X|", "\\longrightarrow", "|Z| \\times_{|Y|} |X|", "$$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Namely, suppose given fields $K$, $L$ and morphisms", "$\\Spec(K) \\to X$, $\\Spec(L) \\to Z$, then the", "assumption that they agree as elements of $|Y|$ means that", "there is a common extension $K \\subset M$ and $L \\subset M$", "such that", "$\\Spec(M) \\to \\Spec(K) \\to X \\to Y$ and", "$\\Spec(M) \\to \\Spec(L) \\to Z \\to Y$ agree.", "And this is exactly the condition which says you get a", "morphism $\\Spec(M) \\to Z \\times_Y X$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11820, "type": "theorem", "label": "spaces-properties-lemma-characterize-surjective", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-characterize-surjective", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $f : T \\to X$ be a morphism from a scheme to $X$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f : T \\to X$ is surjective (according to", "Spaces, Definition \\ref{spaces-definition-relative-representable-property}),", "and", "\\item $|f| : |T| \\to |X|$ is surjective.", "\\end{enumerate}" ], "refs": [ "spaces-definition-relative-representable-property" ], "proofs": [ { "contents": [ "Assume (1). Let $x : \\Spec(K) \\to X$ be a morphism", "from the spectrum of a field into $X$. By assumption the morphism of", "schemes $\\Spec(K) \\times_X T \\to \\Spec(K)$ is surjective.", "Hence there exists a field extension $K \\subset K'$ and a morphism", "$\\Spec(K') \\to \\Spec(K) \\times_X T$ such that the left", "square in the diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] &", "\\Spec(K) \\times_X T \\ar[d] \\ar[r] &", "T \\ar[d] \\\\", "\\Spec(K) \\ar@{=}[r] &", "\\Spec(K) \\ar[r]^-x & X", "}", "$$", "is commutative. This shows that $|f| : |T| \\to |X|$ is surjective.", "\\medskip\\noindent", "Assume (2). Let $Z \\to X$ be a morphism where $Z$ is", "a scheme. We have to show that the morphism of schemes $Z \\times_X T \\to T$", "is surjective, i.e., that $|Z \\times_X T| \\to |Z|$ is surjective.", "This follows from (2) and", "Lemma \\ref{lemma-points-cartesian}." ], "refs": [ "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 11819 ] } ], "ref_ids": [ 8173 ] }, { "id": 11821, "type": "theorem", "label": "spaces-properties-lemma-points-presentation", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-points-presentation", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $X = U/R$ be a presentation of $X$, see", "Spaces, Definition \\ref{spaces-definition-presentation}.", "Then the image of $|R| \\to |U| \\times |U|$ is an equivalence relation", "and $|X|$ is the quotient of $|U|$ by this equivalence relation." ], "refs": [ "spaces-definition-presentation" ], "proofs": [ { "contents": [ "The assumption means that $U$ is a scheme, $p : U \\to X$ is a surjective,", "\\'etale morphism, $R = U \\times_X U$ is a scheme and defines an \\'etale", "equivalence relation on $U$ such that $X = U/R$ as sheaves. By", "Lemma \\ref{lemma-characterize-surjective}", "we see that $|U| \\to |X|$ is surjective. By", "Lemma \\ref{lemma-points-cartesian}", "the map", "$$", "|R| \\longrightarrow |U| \\times_{|X|} |U|", "$$", "is surjective. Hence the image of $|R| \\to |U| \\times |U|$ is", "exactly the set of pairs $(u_1, u_2) \\in |U| \\times |U|$", "such that $u_1$ and $u_2$ have the same image in $|X|$.", "Combining these two statements we get the result of the lemma." ], "refs": [ "spaces-properties-lemma-characterize-surjective", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 11820, 11819 ] } ], "ref_ids": [ 8177 ] }, { "id": 11822, "type": "theorem", "label": "spaces-properties-lemma-topology-points", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-topology-points", "contents": [ "Let $S$ be a scheme. There exists a unique topology on the sets of points", "of algebraic spaces over $S$ with the following properties:", "\\begin{enumerate}", "\\item if $X$ is a scheme over $S$, then the topology on $|X|$ is the usual one", "(via the identification of Lemma \\ref{lemma-scheme-points}),", "\\item for every morphism of algebraic spaces $X \\to Y$ over $S$", "the map $|X| \\to |Y|$ is continuous, and", "\\item for every \\'etale morphism $U \\to X$ with $U$ a scheme", "the map of topological spaces $|U| \\to |X|$ is continuous and open.", "\\end{enumerate}" ], "refs": [ "spaces-properties-lemma-scheme-points" ], "proofs": [ { "contents": [ "Let $X$ be an algebraic space over $S$. Let $p : U \\to X$ be a", "surjective \\'etale morphism where $U$ is a scheme over $S$.", "We define $W \\subset |X|$ is open if and only if $|p|^{-1}(W)$", "is an open subset of $|U|$. This is a topology on $|X|$", "(it is the quotient topology on $|X|$, see", "Topology, Lemma \\ref{topology-lemma-quotient}).", "\\medskip\\noindent", "Let us prove that the topology is independent of the choice of", "the presentation. To do this it suffices to show that if $U'$ is a scheme,", "and $U' \\to X$ is an \\'etale morphism, then the map $|U'| \\to |X|$", "(with topology on $|X|$ defined using $U \\to X$ as above)", "is open and continuous; which in addition will prove that (3) holds.", "Set $U'' = U \\times_X U'$, so that we have the commutative diagram", "$$", "\\xymatrix{", "U'' \\ar[r] \\ar[d] & U' \\ar[d] \\\\", "U \\ar[r] & X", "}", "$$", "As $U \\to X$ and $U' \\to X$ are \\'etale we see that", "both $U'' \\to U$ and $U'' \\to U'$ are \\'etale morphisms of schemes.", "Moreover, $U'' \\to U'$ is surjective. Hence", "we get a commutative diagram of maps of sets", "$$", "\\xymatrix{", "|U''| \\ar[r] \\ar[d] & |U'| \\ar[d] \\\\", "|U| \\ar[r] & |X|", "}", "$$", "The lower horizontal arrow is surjective (see", "Lemma \\ref{lemma-characterize-surjective}", "or", "Lemma \\ref{lemma-points-presentation})", "and continuous by definition of the topology on $|X|$.", "The top horizontal arrow is surjective, continuous, and open by", "Morphisms, Lemma \\ref{morphisms-lemma-etale-open}.", "The left vertical arrow is continuous and open (by", "Morphisms, Lemma \\ref{morphisms-lemma-etale-open}", "again.) Hence it follows formally that the right vertical", "arrow is continuous and open.", "\\medskip\\noindent", "To finish the proof we prove (2).", "Let $a : X \\to Y$ be a morphism of algebraic spaces. According to", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}", "we can find a diagram", "$$", "\\xymatrix{", "U \\ar[d]_p \\ar[r]_\\alpha & V \\ar[d]^q \\\\", "X \\ar[r]^a & Y", "}", "$$", "where $U$ and $V$ are schemes, and $p$ and $q$ are surjective and \\'etale.", "This gives rise to the diagram", "$$", "\\xymatrix{", "|U| \\ar[d]_p \\ar[r]_\\alpha & |V| \\ar[d]^q \\\\", "|X| \\ar[r]^a & |Y|", "}", "$$", "where all but the lower horizontal arrows are known to be continuous and", "the two vertical arrows are surjective and open. It follows that the", "lower horizontal arrow is continuous as desired." ], "refs": [ "topology-lemma-quotient", "spaces-properties-lemma-characterize-surjective", "spaces-properties-lemma-points-presentation", "morphisms-lemma-etale-open", "morphisms-lemma-etale-open", "spaces-lemma-lift-morphism-presentations" ], "ref_ids": [ 8202, 11820, 11821, 5370, 5370, 8159 ] } ], "ref_ids": [ 11818 ] }, { "id": 11823, "type": "theorem", "label": "spaces-properties-lemma-open-subspaces", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-open-subspaces", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item The rule $X' \\mapsto |X'|$ defines an inclusion preserving", "bijection between open subspaces $X'$ (see", "Spaces, Definition \\ref{spaces-definition-immersion})", "of $X$, and opens of the topological space $|X|$.", "\\item A family $\\{X_i \\subset X\\}_{i \\in I}$ of open subspaces of $X$", "is a Zariski covering (see", "Spaces, Definition \\ref{spaces-definition-Zariski-open-covering})", "if and only if $|X| = \\bigcup |X_i|$.", "\\end{enumerate}", "In other words, the small Zariski site $X_{Zar}$ of $X$ is canonically", "identified with a site associated to the topological space $|X|$ (see", "Sites, Example \\ref{sites-example-site-topological})." ], "refs": [ "spaces-definition-immersion", "spaces-definition-Zariski-open-covering" ], "proofs": [ { "contents": [ "In order to prove (1) let us construct the inverse of the rule.", "Namely, suppose that $W \\subset |X|$ is open. Choose a presentation", "$X = U/R$ corresponding to the surjective \\'etale map", "$p : U \\to X$ and \\'etale maps $s, t : R \\to U$.", "By construction we see that $|p|^{-1}(W)$ is an", "open of $U$. Denote $W' \\subset U$ the corresponding open subscheme.", "It is clear that $R' = s^{-1}(W') = t^{-1}(W')$ is a Zariski open", "of $R$ which defines an \\'etale equivalence relation on $W'$.", "By Spaces, Lemma \\ref{spaces-lemma-finding-opens} the morphism", "$X' = W'/R' \\to X$ is an open immersion. Hence $X'$ is an algebraic space", "by Spaces, Lemma \\ref{spaces-lemma-representable-over-space}.", "By construction $|X'| = W$, i.e., $X'$ is a subspace of $X$", "corresponding to $W$. Thus (1) is proved.", "\\medskip\\noindent", "To prove (2), note that if $\\{X_i \\subset X\\}_{i \\in I}$ is a collection", "of open subspaces, then it is a Zariski covering if and only if the", "$U = \\bigcup U \\times_X X_i$ is an open covering. This follows from", "the definition of a Zariski covering and the fact that the morphism", "$U \\to X$ is surjective as a map of presheaves on $(\\Sch/S)_{fppf}$.", "On the other hand, we see that $|X| = \\bigcup |X_i|$ if and only if", "$U = \\bigcup U \\times_X X_i$ by Lemma \\ref{lemma-points-presentation}", "(and the fact that the projections $U \\times_X X_i \\to X_i$ are surjective", "and \\'etale). Thus the equivalence of (2) follows." ], "refs": [ "spaces-lemma-finding-opens", "spaces-lemma-representable-over-space", "spaces-properties-lemma-points-presentation" ], "ref_ids": [ 8151, 8156, 11821 ] } ], "ref_ids": [ 8178, 8179 ] }, { "id": 11824, "type": "theorem", "label": "spaces-properties-lemma-factor-through-open-subspace", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-factor-through-open-subspace", "contents": [ "Let $S$ be a scheme.", "Let $X$, $Y$ be algebraic spaces over $S$.", "Let $X' \\subset X$ be an open subspace.", "Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$.", "Then $f$ factors through $X'$ if and only if $|f| : |Y| \\to |X|$", "factors through $|X'| \\subset |X|$." ], "refs": [], "proofs": [ { "contents": [ "By Spaces, Lemma \\ref{spaces-lemma-base-change-immersions}", "we see that $Y' = Y \\times_X X' \\to Y$ is an open immersion.", "If $|f|(|Y|) \\subset |X'|$, then clearly $|Y'| = |Y|$. Hence $Y' = Y$ by", "Lemma \\ref{lemma-open-subspaces}." ], "refs": [ "spaces-lemma-base-change-immersions", "spaces-properties-lemma-open-subspaces" ], "ref_ids": [ 8161, 11823 ] } ], "ref_ids": [] }, { "id": 11825, "type": "theorem", "label": "spaces-properties-lemma-etale-image-open", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-etale-image-open", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic spaces over $S$.", "Let $U$ be a scheme and let $f : U \\to X$ be an \\'etale morphism.", "Let $X' \\subset X$ be the open subspace corresponding to", "the open $|f|(|U|) \\subset |X|$ via", "Lemma \\ref{lemma-open-subspaces}.", "Then $f$ factors through a surjective \\'etale morphism $f' : U \\to X'$.", "Moreover, if $R = U \\times_X U$, then $R = U \\times_{X'} U$ and $X'$ has", "the presentation $X' = U/R$." ], "refs": [ "spaces-properties-lemma-open-subspaces" ], "proofs": [ { "contents": [ "The existence of the factorization follows from", "Lemma \\ref{lemma-factor-through-open-subspace}.", "The morphism $f'$ is surjective according to", "Lemma \\ref{lemma-characterize-surjective}.", "To see $f'$ is \\'etale, suppose that $T \\to X'$ is a morphism", "where $T$ is a scheme. Then $T \\times_X U = T \\times_{X'} U$", "as $X' \\to X$ is a monomorphism of sheaves. Thus the projection", "$T \\times_{X'} U \\to T$ is \\'etale as we assumed $f$ \\'etale.", "We have $U \\times_X U = U \\times_{X'} U$ as $X' \\to X$ is a monomorphism.", "Then $X' = U/R$ follows from", "Spaces, Lemma \\ref{spaces-lemma-space-presentation}." ], "refs": [ "spaces-properties-lemma-factor-through-open-subspace", "spaces-properties-lemma-characterize-surjective", "spaces-lemma-space-presentation" ], "ref_ids": [ 11824, 11820, 8149 ] } ], "ref_ids": [ 11823 ] }, { "id": 11826, "type": "theorem", "label": "spaces-properties-lemma-points-monomorphism", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-points-monomorphism", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Consider the map", "$$", "\\{\\Spec(k) \\to X \\text{ monomorphism where }k\\text{ is a field}\\}", "\\longrightarrow", "|X|", "$$", "This map is injective." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $\\varphi_i : \\Spec(k_i) \\to X$ are monomorphisms", "for $i = 1, 2$. If $\\varphi_1$ and $\\varphi_2$ define the same point", "of $|X|$, then we see that the scheme", "$$", "Y = \\Spec(k_1) \\times_{\\varphi_1, X, \\varphi_2} \\Spec(k_2)", "$$", "is nonempty. Since the base change of a monomorphism is a monomorphism", "this means that the projection morphisms $Y \\to \\Spec(k_i)$", "are monomorphisms. Hence $\\Spec(k_1) = Y = \\Spec(k_2)$", "as schemes over $X$, see", "Schemes, Lemma \\ref{schemes-lemma-mono-towards-spec-field}.", "We conclude that $\\varphi_1 = \\varphi_2$, which proves the lemma." ], "refs": [ "schemes-lemma-mono-towards-spec-field" ], "ref_ids": [ 7729 ] } ], "ref_ids": [] }, { "id": 11827, "type": "theorem", "label": "spaces-properties-lemma-quasi-compact-space", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-quasi-compact-space", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Then $X$ is quasi-compact if and only if $|X|$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and an \\'etale surjective morphism $U \\to X$.", "We will use Lemma \\ref{lemma-characterize-surjective}.", "If $U$ is quasi-compact, then since $|U| \\to |X|$ is surjective", "we conclude that $|X|$ is quasi-compact.", "If $|X|$ is quasi-compact, then since $|U| \\to |X|$ is open", "we see that there exists a quasi-compact open $U' \\subset U$", "such that $|U'| \\to |X|$ is surjective (and still \\'etale).", "Hence we win." ], "refs": [ "spaces-properties-lemma-characterize-surjective" ], "ref_ids": [ 11820 ] } ], "ref_ids": [] }, { "id": 11828, "type": "theorem", "label": "spaces-properties-lemma-finite-disjoint-quasi-compact", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-finite-disjoint-quasi-compact", "contents": [ "A finite disjoint union of quasi-compact algebraic spaces is", "a quasi-compact algebraic space." ], "refs": [], "proofs": [ { "contents": [ "This is clear from", "Lemma \\ref{lemma-quasi-compact-space}", "and the corresponding topological fact." ], "refs": [ "spaces-properties-lemma-quasi-compact-space" ], "ref_ids": [ 11827 ] } ], "ref_ids": [] }, { "id": 11829, "type": "theorem", "label": "spaces-properties-lemma-space-locally-quasi-compact", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-space-locally-quasi-compact", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Every point of $|X|$ has a fundamental system of open", "quasi-compact neighbourhoods.", "In particular $|X|$ is locally quasi-compact in the sense of", "Topology, Definition \\ref{topology-definition-locally-quasi-compact}." ], "refs": [ "topology-definition-locally-quasi-compact" ], "proofs": [ { "contents": [ "This follows formally from the fact that there exists a scheme", "$U$ and a surjective, open, continuous map", "$U \\to |X|$ of topological spaces. To be a bit more precise, if", "$u \\in U$ maps to $x \\in |X|$, then the images of the affine", "neighbourhoods of $u$ will give a fundamental system of quasi-compact", "open neighbourhoods of $x$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8361 ] }, { "id": 11830, "type": "theorem", "label": "spaces-properties-lemma-cover-by-union-affines", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-cover-by-union-affines", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "There exists a surjective \\'etale morphism $U \\to X$ where", "$U$ is a disjoint union of affine schemes.", "We may in addition assume each of these affines", "maps into an affine open of $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\to X$ be a surjective \\'etale morphism.", "Let $V = \\bigcup_{i \\in I} V_i$ be a Zariski open covering", "such that each $V_i$ maps into an affine open of $S$.", "Then set $U = \\coprod_{i \\in I} V_i$ with induced morphism", "$U \\to V \\to X$. This is \\'etale and surjective as a composition", "of \\'etale and surjective representable", "transformations of functors (via the general principle", "Spaces, Lemma", "\\ref{spaces-lemma-composition-representable-transformations-property}", "and", "Morphisms, Lemmas \\ref{morphisms-lemma-composition-surjective} and", "\\ref{morphisms-lemma-composition-etale})." ], "refs": [ "spaces-lemma-composition-representable-transformations-property", "morphisms-lemma-composition-surjective", "morphisms-lemma-composition-etale" ], "ref_ids": [ 8132, 5163, 5360 ] } ], "ref_ids": [] }, { "id": 11831, "type": "theorem", "label": "spaces-properties-lemma-union-of-quasi-compact", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-union-of-quasi-compact", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "There exists a Zariski covering $X = \\bigcup X_i$", "such that each algebraic space $X_i$ has a surjective", "\\'etale covering by an affine scheme. We may in addition assume", "each $X_i$ maps into an affine open of $S$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-cover-by-union-affines} we can find a surjective", "\\'etale morphism $U = \\coprod U_i \\to X$, with $U_i$ affine and mapping", "into an affine open of $S$. Let $X_i \\subset X$ be the open subspace", "of $X$ such that $U_i \\to X$ factors through an \\'etale surjective morphism", "$U_i \\to X_i$, see", "Lemma \\ref{lemma-etale-image-open}.", "Since $U = \\bigcup U_i$ we see that $X = \\bigcup X_i$.", "As $U_i \\to X_i$ is surjective it follows that $X_i \\to S$ maps into", "an affine open of $S$." ], "refs": [ "spaces-properties-lemma-cover-by-union-affines", "spaces-properties-lemma-etale-image-open" ], "ref_ids": [ 11830, 11825 ] } ], "ref_ids": [] }, { "id": 11832, "type": "theorem", "label": "spaces-properties-lemma-quasi-compact-affine-cover", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-quasi-compact-affine-cover", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Then $X$ is quasi-compact if and only if", "there exists an \\'etale surjective morphism $U \\to X$", "with $U$ an affine scheme." ], "refs": [], "proofs": [ { "contents": [ "If there exists an \\'etale surjective morphism $U \\to X$ with $U$", "affine then $X$ is quasi-compact by Definition \\ref{definition-quasi-compact}.", "Conversely, if $X$ is quasi-compact, then $|X|$ is quasi-compact.", "Let $U = \\coprod_{i \\in I} U_i$ be a disjoint union of affine schemes", "with an \\'etale and surjective map $\\varphi : U \\to X$", "(Lemma \\ref{lemma-cover-by-union-affines}).", "Then $|X| = \\bigcup \\varphi(|U_i|)$ and", "by quasi-compactness there is a finite subset $i_1, \\ldots, i_n$", "such that $|X| = \\bigcup \\varphi(|U_{i_j}|)$. Hence", "$U_{i_1} \\cup \\ldots \\cup U_{i_n}$ is an affine scheme with a", "finite surjective morphism towards $X$." ], "refs": [ "spaces-properties-definition-quasi-compact", "spaces-properties-lemma-cover-by-union-affines" ], "ref_ids": [ 11925, 11830 ] } ], "ref_ids": [] }, { "id": 11833, "type": "theorem", "label": "spaces-properties-lemma-separated-cover", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-separated-cover", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $U$ be a separated scheme and $U \\to X$ \\'etale.", "Then $U \\to X$ is separated, and $R = U \\times_X U$ is a separated scheme." ], "refs": [], "proofs": [ { "contents": [ "Let $X' \\subset X$ be the open subscheme such that $U \\to X$ factors", "through an \\'etale surjection $U \\to X'$, see", "Lemma \\ref{lemma-etale-image-open}.", "If $U \\to X'$ is separated, then so is $U \\to X$, see", "Spaces, Lemma", "\\ref{spaces-lemma-composition-representable-transformations-property}", "(as the open immersion $X' \\to X$ is separated by", "Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}", "and", "Schemes, Lemma \\ref{schemes-lemma-immersions-monomorphisms}).", "Moreover, since $U \\times_{X'} U = U \\times_X U$ it suffices", "to prove the result after replacing $X$ by $X'$, i.e., we may", "assume $U \\to X$ surjective.", "Consider the commutative diagram", "$$", "\\xymatrix{", "R = U \\times_X U \\ar[r] \\ar[d] & U \\ar[d] \\\\", "U \\ar[r] & X", "}", "$$", "In the proof of", "Spaces, Lemma \\ref{spaces-lemma-properties-diagonal}", "we have seen that $j : R \\to U \\times_S U$ is separated.", "The morphism of schemes $U \\to S$ is separated as $U$ is a separated", "scheme, see", "Schemes, Lemma \\ref{schemes-lemma-compose-after-separated}.", "Hence $U \\times_S U \\to U$ is separated as a base change, see", "Schemes, Lemma \\ref{schemes-lemma-separated-permanence}.", "Hence the scheme $U \\times_S U$ is separated (by the same lemma).", "Since $j$ is separated we see in the same way that $R$ is separated.", "Hence $R \\to U$ is a separated morphism (by", "Schemes, Lemma \\ref{schemes-lemma-compose-after-separated}", "again). Thus by", "Spaces, Lemma \\ref{spaces-lemma-representable-morphisms-spaces-property}", "and the diagram above we conclude that $U \\to X$ is separated." ], "refs": [ "spaces-properties-lemma-etale-image-open", "spaces-lemma-composition-representable-transformations-property", "spaces-lemma-representable-transformations-property-implication", "schemes-lemma-immersions-monomorphisms", "spaces-lemma-properties-diagonal", "schemes-lemma-compose-after-separated", "schemes-lemma-separated-permanence", "schemes-lemma-compose-after-separated", "spaces-lemma-representable-morphisms-spaces-property" ], "ref_ids": [ 11825, 8132, 8136, 7727, 8163, 7715, 7714, 7715, 8157 ] } ], "ref_ids": [] }, { "id": 11834, "type": "theorem", "label": "spaces-properties-lemma-quasi-separated", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-quasi-separated", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "If there exists a quasi-separated scheme $U$ and a surjective", "\\'etale morphism $U \\to X$ such that either of the projections", "$U \\times_X U \\to U$ is quasi-compact, then $X$ is quasi-separated." ], "refs": [], "proofs": [ { "contents": [ "We may think of $X$ as an algebraic space over $\\mathbf{Z}$.", "Consider the cartesian diagram", "$$", "\\xymatrix{", "U \\times_X U \\ar[r] \\ar[d]_j & X \\ar[d]^\\Delta \\\\", "U \\times U \\ar[r] & X \\times X", "}", "$$", "Since $U$ is quasi-separated the projection $U \\times U \\to U$ is", "quasi-separated (as a base change of a quasi-separated morphism", "of schemes, see Schemes, Lemma \\ref{schemes-lemma-separated-permanence}).", "Hence the assumption in the lemma implies $j$ is quasi-compact by", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}.", "By Spaces, Lemma \\ref{spaces-lemma-representable-morphisms-spaces-property}", "we see that $\\Delta$ is quasi-compact as desired." ], "refs": [ "schemes-lemma-separated-permanence", "schemes-lemma-quasi-compact-permanence", "spaces-lemma-representable-morphisms-spaces-property" ], "ref_ids": [ 7714, 7716, 8157 ] } ], "ref_ids": [] }, { "id": 11835, "type": "theorem", "label": "spaces-properties-lemma-quasi-separated-quasi-compact-pieces", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-quasi-separated-quasi-compact-pieces", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is Zariski locally quasi-separated over $S$,", "\\item $X$ is Zariski locally quasi-separated,", "\\item there exists a Zariski open covering $X = \\bigcup X_i$", "such that for each $i$ there exists an affine scheme", "$U_i$ and a quasi-compact surjective \\'etale", "morphism $U_i \\to X_i$, and", "\\item there exists a Zariski open covering $X = \\bigcup X_i$", "such that for each $i$ there exists an affine scheme", "$U_i$ which maps into an affine open of $S$ and a quasi-compact", "surjective \\'etale morphism $U_i \\to X_i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $U_i \\to X_i \\subset X$ are as in (3). To prove (4)", "choose for each $i$ a finite affine open covering $U_i =", "U_{i1} \\cup \\ldots \\cup U_{in_i}$ such that each $U_{ij}$ maps", "into an affine open of $S$. The compositions", "$U_{ij} \\to U_i \\to X_i$ are \\'etale and quasi-compact (see", "Spaces, Lemma", "\\ref{spaces-lemma-composition-representable-transformations-property}).", "Let $X_{ij} \\subset X_i$ be the open subspace corresponding to", "the image of $|U_{ij}| \\to |X_i|$, see", "Lemma \\ref{lemma-etale-image-open}.", "Note that $U_{ij} \\to X_{ij}$ is quasi-compact as $X_{ij} \\subset X_i$", "is a monomorphism and as $U_{ij} \\to X$ is quasi-compact.", "Then $X = \\bigcup X_{ij}$ is a covering as in (4).", "The implication (4) $\\Rightarrow$ (3) is immediate.", "\\medskip\\noindent", "Assume (4). To show that $X$ is Zariski locally quasi-separated over $S$", "it suffices to show that $X_i$ is quasi-separated over $S$.", "Hence we may assume there exists an affine scheme $U$ mapping into", "an affine open of $S$ and a quasi-compact surjective \\'etale", "morphism $U \\to X$. Consider the fibre product square", "$$", "\\xymatrix{", "U \\times_X U \\ar[r] \\ar[d] & U \\times_S U \\ar[d] \\\\", "X \\ar[r]^-{\\Delta_{X/S}} & X \\times_S X", "}", "$$", "The right vertical arrow is surjective \\'etale (see", "Spaces, Lemma", "\\ref{spaces-lemma-product-representable-transformations-property})", "and $U \\times_S U$ is affine (as $U$ maps into an affine open of $S$, see", "Schemes, Section \\ref{schemes-section-fibre-products}),", "and $U \\times_X U$ is quasi-compact", "because the projection $U \\times_X U \\to U$ is quasi-compact as a", "base change of $U \\to X$. It follows from", "Spaces, Lemma \\ref{spaces-lemma-representable-morphisms-spaces-property}", "that $\\Delta_{X/S}$ is quasi-compact as desired.", "\\medskip\\noindent", "Assume (1). To prove (3) there is an immediate reduction to the case", "where $X$ is quasi-separated over $S$. By", "Lemma \\ref{lemma-union-of-quasi-compact}", "we can find a Zariski open covering $X = \\bigcup X_i$ such that each", "$X_i$ maps into an affine open of $S$, and such that there exist affine", "schemes $U_i$ and surjective \\'etale morphisms $U_i \\to X_i$.", "Since $U_i \\to S$ maps into an affine open of $S$ we see that", "$U_i \\times_S U_i$ is affine, see", "Schemes, Section \\ref{schemes-section-fibre-products}.", "As $X$ is quasi-separated over $S$, the morphisms", "$$", "R_i = U_i \\times_{X_i} U_i = U_i \\times_X U_i", "\\longrightarrow", "U_i \\times_S U_i", "$$", "as base changes of $\\Delta_{X/S}$ are quasi-compact. Hence we conclude", "that $R_i$ is a quasi-compact scheme. This in turn implies that each", "projection $R_i \\to U_i$ is quasi-compact. Hence, applying", "Spaces, Lemma \\ref{spaces-lemma-representable-morphisms-spaces-property}", "to the covering $U_i \\to X_i$ and the morphism $U_i \\to X_i$", "we conclude that the morphisms $U_i \\to X_i$ are quasi-compact as desired.", "\\medskip\\noindent", "At this point we see that (1), (3), and (4) are equivalent. Since (3) does", "not refer to the base scheme we conclude that these are also equivalent", "with (2)." ], "refs": [ "spaces-lemma-composition-representable-transformations-property", "spaces-properties-lemma-etale-image-open", "spaces-lemma-product-representable-transformations-property", "spaces-lemma-representable-morphisms-spaces-property", "spaces-properties-lemma-union-of-quasi-compact", "spaces-lemma-representable-morphisms-spaces-property" ], "ref_ids": [ 8132, 11825, 8135, 8157, 11831, 8157 ] } ], "ref_ids": [] }, { "id": 11836, "type": "theorem", "label": "spaces-properties-lemma-finite-fibres-presentation", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-finite-fibres-presentation", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $U$ be a scheme. Let $\\varphi : U \\to X$ be an \\'etale morphism such that", "the projections $R = U \\times_X U \\to U$ are quasi-compact; for example if", "$\\varphi$ is quasi-compact. Then the fibres of", "$$", "|U| \\to |X|", "\\quad\\text{and}\\quad", "|R| \\to |X|", "$$", "are finite." ], "refs": [], "proofs": [ { "contents": [ "Denote $R = U \\times_X U$, and $s, t : R \\to U$ the projections.", "Let $u \\in U$ be a point, and let $x \\in |X|$ be its image.", "The fibre of $|U| \\to |X|$ over $x$ is equal to", "$s(t^{-1}(\\{u\\}))$ by", "Lemma \\ref{lemma-points-cartesian},", "and the fibre of $|R| \\to |X|$ over $x$ is $t^{-1}(s(t^{-1}(\\{u\\})))$.", "Since $t : R \\to U$ is \\'etale and quasi-compact, it has finite fibres", "(as its fibres are disjoint unions of spectra of fields by", "Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}", "and quasi-compact). Hence we win." ], "refs": [ "spaces-properties-lemma-points-cartesian", "morphisms-lemma-etale-over-field" ], "ref_ids": [ 11819, 5364 ] } ], "ref_ids": [] }, { "id": 11837, "type": "theorem", "label": "spaces-properties-lemma-type-property", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-type-property", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{P}$ be a property of schemes which is local in the \\'etale", "topology, see", "Descent, Definition \\ref{descent-definition-property-local}.", "The following are equivalent", "\\begin{enumerate}", "\\item for some scheme $U$ and surjective \\'etale morphism $U \\to X$", "the scheme $U$ has property $\\mathcal{P}$, and", "\\item for every scheme $U$ and every \\'etale morphism $U \\to X$", "the scheme $U$ has property $\\mathcal{P}$.", "\\end{enumerate}", "If $X$ is representable this is equivalent to $\\mathcal{P}(X)$." ], "refs": [ "descent-definition-property-local" ], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is immediate.", "For the converse, choose a surjective \\'etale morphism $U \\to X$", "with $U$ a scheme that has $\\mathcal{P}$ and let $V$ be an \\'etale", "$X$-scheme. Then $U \\times_X V \\rightarrow V$ is an \\'etale surjection", "of schemes, so $V$ inherits $\\mathcal{P}$ from $U \\times_X V$, which in", "turn inherits $\\mathcal{P}$ from $U$ (see discussion following", "Descent, Definition \\ref{descent-definition-property-local}).", "The last claim is clear from (1) and", "Descent, Definition \\ref{descent-definition-property-local}." ], "refs": [ "descent-definition-property-local", "descent-definition-property-local" ], "ref_ids": [ 14768, 14768 ] } ], "ref_ids": [ 14768 ] }, { "id": 11838, "type": "theorem", "label": "spaces-properties-lemma-local-source-target-at-point", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-local-source-target-at-point", "contents": [ "Let $\\mathcal{P}$ be a property of germs of schemes which is \\'etale local, see", "Descent, Definition \\ref{descent-definition-local-at-point}.", "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$ be a point of $X$.", "Consider \\'etale morphisms $a : U \\to X$ where $U$ is a scheme.", "The following are equivalent", "\\begin{enumerate}", "\\item for any $U \\to X$ as above and $u \\in U$ with $a(u) = x$ we have", "$\\mathcal{P}(U, u)$, and", "\\item for some $U \\to X$ as above and $u \\in U$ with $a(u) = x$ we have", "$\\mathcal{P}(U, u)$.", "\\end{enumerate}", "If $X$ is representable, then this is equivalent to $\\mathcal{P}(X, x)$." ], "refs": [ "descent-definition-local-at-point" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 14771 ] }, { "id": 11839, "type": "theorem", "label": "spaces-properties-lemma-locally-constructible", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-locally-constructible", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $E \\subset |X|$ be a subset. The following are equivalent", "\\begin{enumerate}", "\\item for every \\'etale morphism $U \\to X$ where $U$ is a", "scheme the inverse image of $E$ in $U$ is a locally constructible", "subset of $U$,", "\\item for every \\'etale morphism $U \\to X$ where $U$ is an", "affine scheme the inverse image of $E$ in $U$ is a constructible", "subset of $U$,", "\\item for some surjective \\'etale morphism $U \\to X$ where $U$ is a", "scheme the inverse image of $E$ in $U$ is a locally constructible", "subset of $U$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Properties, Lemma \\ref{properties-lemma-locally-constructible}", "we see that (1) and (2) are equivalent. It is immediate that (1)", "implies (3). Thus we assume we have a surjective \\'etale morphism", "$\\varphi : U \\to X$ where $U$ is a scheme such that $\\varphi^{-1}(E)$", "is locally constructible. Let $\\varphi' : U' \\to X$ be another", "\\'etale morphism where $U'$ is a scheme. Then we have", "$$", "E'' = \\text{pr}_1^{-1}(\\varphi^{-1}(E)) = \\text{pr}_2^{-1}((\\varphi')^{-1}(E))", "$$", "where $\\text{pr}_1 : U \\times_X U' \\to U$ and", "$\\text{pr}_2 : U \\times_X U' \\to U'$ are the projections.", "By Morphisms, Lemma \\ref{morphisms-lemma-inverse-image-constructible}", "we see that $E''$ is locally constructible in $U \\times_X U'$.", "Let $W' \\subset U'$ be an affine open. Since $\\text{pr}_2$ is", "\\'etale and hence open, we can choose a quasi-compact open", "$W'' \\subset U \\times_X U'$ with $\\text{pr}_2(W'') = W'$.", "Then $\\text{pr}_2|_{W''} : W'' \\to W'$ is quasi-compact.", "We have $W' \\cap (\\varphi')^{-1}(E) = \\text{pr}_2(E'' \\cap W'')$", "as $\\varphi$ is surjective, see Lemma \\ref{lemma-points-cartesian}.", "Thus $W' \\cap (\\varphi')^{-1}(E) = \\text{pr}_2(E'' \\cap W'')$", "is locally constructible by", "Morphisms, Theorem \\ref{morphisms-theorem-chevalley} as desired." ], "refs": [ "properties-lemma-locally-constructible", "morphisms-lemma-inverse-image-constructible", "spaces-properties-lemma-points-cartesian", "morphisms-theorem-chevalley" ], "ref_ids": [ 2938, 5249, 11819, 5123 ] } ], "ref_ids": [] }, { "id": 11840, "type": "theorem", "label": "spaces-properties-lemma-pre-dimension-local-ring", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-pre-dimension-local-ring", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$ be a point.", "Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$.", "The following are equivalent", "\\begin{enumerate}", "\\item for some scheme $U$ and \\'etale morphism $a : U \\to X$ and point", "$u \\in U$ with $a(u) = x$ we have $\\dim(\\mathcal{O}_{U, u}) = d$,", "\\item for any scheme $U$, any \\'etale morphism $a : U \\to X$, and any point", "$u \\in U$ with $a(u) = x$ we have $\\dim(\\mathcal{O}_{U, u}) = d$.", "\\end{enumerate}", "If $X$ is a scheme, this is equivalent to $\\dim(\\mathcal{O}_{X, x}) = d$." ], "refs": [], "proofs": [ { "contents": [ "Combine", "Lemma \\ref{lemma-local-source-target-at-point} and", "Descent, Lemma \\ref{descent-lemma-dimension-local-ring-local}." ], "refs": [ "spaces-properties-lemma-local-source-target-at-point", "descent-lemma-dimension-local-ring-local" ], "ref_ids": [ 11838, 14661 ] } ], "ref_ids": [] }, { "id": 11841, "type": "theorem", "label": "spaces-properties-lemma-dimension", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-dimension", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The following quantities are equal:", "\\begin{enumerate}", "\\item The dimension of $X$.", "\\item The supremum of the dimensions of the local rings of $X$.", "\\item The supremum of $\\dim_x(X)$ for $x \\in |X|$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The numbers in (1) and (3) are equal by Definition \\ref{definition-dimension}.", "Let $U \\to X$ be a surjective \\'etale morphism from a scheme $U$.", "The supremum of $\\dim_x(X)$ for $x \\in |X|$ is the same as the", "supremum of $\\dim_u(U)$ for points $u$ of $U$ by definition.", "This is the same as the supremum of $\\dim(\\mathcal{O}_{U, u})$ by", "Properties, Lemma \\ref{properties-lemma-dimension}. This in turn", "is the same as (2) by definition." ], "refs": [ "spaces-properties-definition-dimension", "properties-lemma-dimension" ], "ref_ids": [ 11930, 2978 ] } ], "ref_ids": [] }, { "id": 11842, "type": "theorem", "label": "spaces-properties-lemma-codimension-0-points", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-codimension-0-points", "contents": [ "Let $S$ be a scheme and let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$. Consider \\'etale morphisms $a : U \\to X$ where", "$U$ is a scheme. The following are equivalent", "\\begin{enumerate}", "\\item $x$ is a point of codimension $0$ on $X$,", "\\item for some $U \\to X$ as above and $u \\in U$ with $a(u) = x$,", "the point $u$ is the generic point of an irreducible component of $U$, and", "\\item for any $U \\to X$ as above and any $u \\in U$ mapping to $x$,", "the point $u$ is the generic point of an irreducible component of $U$.", "\\end{enumerate}", "If $X$ is representable, this is equivalent to $x$ being a generic", "point of an irreducible component of $|X|$." ], "refs": [], "proofs": [ { "contents": [ "Observe that a point $u$ of a scheme $U$ is a generic point of an", "irreducible component of $U$ if and only if $\\dim(\\mathcal{O}_{U, u}) = 0$", "(Properties, Lemma \\ref{properties-lemma-generic-point}).", "Hence this follows from the definition of the codimension of a", "point on $X$ (Definition \\ref{definition-dimension-local-ring})." ], "refs": [ "properties-lemma-generic-point", "spaces-properties-definition-dimension-local-ring" ], "ref_ids": [ 2980, 11931 ] } ], "ref_ids": [] }, { "id": 11843, "type": "theorem", "label": "spaces-properties-lemma-codimension-0-points-dense", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-codimension-0-points-dense", "contents": [ "Let $S$ be a scheme and let $X$ be an algebraic space over $S$.", "The set of codimension $0$ points of $X$ is dense in $|X|$." ], "refs": [], "proofs": [ { "contents": [ "If $U$ is a scheme, then the set of generic points of irreducible", "components is dense in $U$ (holds for any quasi-sober topological space).", "Thus if $U \\to X$ is a surjective \\'etale morphism, then the set", "of codimension $0$ points of $X$ is the image of a dense subset of", "$|U|$ (Lemma \\ref{lemma-codimension-0-points}). Since $|X|$ has the", "quotient topology for $|U| \\to |X|$ we conclude." ], "refs": [ "spaces-properties-lemma-codimension-0-points" ], "ref_ids": [ 11842 ] } ], "ref_ids": [] }, { "id": 11844, "type": "theorem", "label": "spaces-properties-lemma-subspace-induced-topology", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-subspace-induced-topology", "contents": [ "Let $S$ be a scheme. Let $Z \\to X$ be an immersion of algebraic spaces.", "Then $|Z| \\to |X|$ is a homeomorphism of $|Z|$ onto a locally closed subset", "of $|X|$." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme and $U \\to X$ a surjective \\'etale morphism.", "Then $Z \\times_X U \\to U$ is an immersion of schemes, hence gives a", "homeomorphism of $|Z \\times_X U|$ with a locally closed subset $T'$", "of $|U|$. By Lemma \\ref{lemma-points-cartesian} the subset", "$T'$ is the inverse image of the image $T$ of $|Z| \\to |X|$.", "The map $|Z| \\to |X|$ is injective because the transformation of", "functors $Z \\to X$ is injective, see", "Spaces, Section \\ref{spaces-section-Zariski}. By", "Topology, Lemma \\ref{topology-lemma-open-morphism-quotient-topology}", "we see that $T$ is locally closed in $|X|$. Moreover, the continuous", "map $|Z| \\to T$ is a homeomorphism as the map $|Z \\times_X U| \\to T'$", "is a homeomorphism and $|Z \\times_Y U| \\to |Z|$ is submersive." ], "refs": [ "spaces-properties-lemma-points-cartesian", "topology-lemma-open-morphism-quotient-topology" ], "ref_ids": [ 11819, 8203 ] } ], "ref_ids": [] }, { "id": 11845, "type": "theorem", "label": "spaces-properties-lemma-subspaces-presentation", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-subspaces-presentation", "contents": [ "Let $S$ be a scheme. Let $j : R \\to U \\times_S U$ be an \\'etale equivalence", "relation. Let $X = U/R$ be the associated algebraic space", "(Spaces, Theorem \\ref{spaces-theorem-presentation}). There is a", "canonical bijection", "$$", "R\\text{-invariant locally closed subschemes }Z'\\text{ of }U", "\\leftrightarrow", "\\text{locally closed subspaces }Z\\text{ of }X", "$$", "Moreover, if $Z \\to X$ is closed (resp.\\ open) if and only if", "$Z' \\to U$ is closed (resp.\\ open)." ], "refs": [ "spaces-theorem-presentation" ], "proofs": [ { "contents": [ "Denote $\\varphi : U \\to X$ the canonical map. The bijection sends", "$Z \\to X$ to $Z' = Z \\times_X U \\to U$. It is immediate from the definition", "that $Z' \\to U$ is an immersion, resp.\\ closed immersion, resp.\\ open", "immersion if $Z \\to X$ is so. It is also clear that $Z'$ is $R$-invariant", "(see Groupoids, Definition \\ref{groupoids-definition-invariant-open}).", "\\medskip\\noindent", "Conversely, assume that $Z' \\to U$ is an immersion which is $R$-invariant.", "Let $R'$ be the restriction of $R$ to $Z'$, see", "Groupoids, Definition \\ref{groupoids-definition-restrict-groupoid}.", "Since $R' = R \\times_{s, U} Z' = Z' \\times_{U, t} R$ in this case", "we see that $R'$ is an \\'etale equivalence relation on $Z'$. By", "Spaces, Theorem \\ref{spaces-theorem-presentation} we see", "$Z = Z'/R'$ is an algebraic space. By construction we have", "$U \\times_X Z = Z'$, so $U \\times_X Z \\to Z$ is an immersion.", "Note that the property ``immersion'' is preserved under base change", "and fppf local on the base (see Spaces, Section \\ref{spaces-section-lists}).", "Moreover, immersions are separated and locally quasi-finite (see", "Schemes, Lemma \\ref{schemes-lemma-immersions-monomorphisms}", "and", "Morphisms, Lemma \\ref{morphisms-lemma-immersion-locally-quasi-finite}).", "Hence by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend}", "immersions satisfy descent for fppf covering. This means all the hypotheses of", "Spaces,", "Lemma \\ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale}", "are satisfied for $Z \\to X$, $\\mathcal{P}=$``immersion'',", "and the \\'etale surjective morphism $U \\to X$. We conclude that $Z \\to X$", "is representable and an immersion, which is the", "definition of a subspace (see", "Spaces, Definition \\ref{spaces-definition-immersion}).", "\\medskip\\noindent", "It is clear that these constructions are inverse to each other and we win." ], "refs": [ "groupoids-definition-invariant-open", "groupoids-definition-restrict-groupoid", "spaces-theorem-presentation", "schemes-lemma-immersions-monomorphisms", "morphisms-lemma-immersion-locally-quasi-finite", "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "spaces-lemma-morphism-sheaves-with-P-effective-descent-etale", "spaces-definition-immersion" ], "ref_ids": [ 9685, 9684, 8124, 7727, 5236, 13949, 8158, 8178 ] } ], "ref_ids": [ 8124 ] }, { "id": 11846, "type": "theorem", "label": "spaces-properties-lemma-reduced-closed-subspace", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-reduced-closed-subspace", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $T \\subset |X|$ be a closed subset.", "There exists a unique closed subspace $Z \\subset X$ with", "the following properties: (a) we have $|Z| = T$, and (b) $Z$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be a surjective \\'etale morphism, where $U$ is a scheme.", "Set $R = U \\times_X U$, so that $X = U/R$, see", "Spaces, Lemma \\ref{spaces-lemma-space-presentation}.", "As usual we denote $s, t : R \\to U$ the two projection morphisms.", "By Lemma \\ref{lemma-points-presentation}", "we see that $T$ corresponds to a closed subset $T' \\subset |U|$ such", "that $s^{-1}(T') = t^{-1}(T')$.", "Let $Z' \\subset U$ be the reduced induced scheme structure on $T'$.", "In this case the fibre products", "$Z' \\times_{U, t} R$ and $Z' \\times_{U, s} R$ are closed subschemes", "of $R$", "(Schemes, Lemma \\ref{schemes-lemma-base-change-immersion})", "which are \\'etale over $Z'$", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-etale}),", "and hence reduced", "(because being reduced is local in the \\'etale topology, see", "Remark \\ref{remark-list-properties-local-etale-topology}).", "Since they have the same underlying topological space (see above)", "we conclude that $Z' \\times_{U, t} R = Z' \\times_{U, s} R$.", "Thus we can apply Lemma \\ref{lemma-subspaces-presentation}", "to obtain a closed subspace $Z \\subset X$ whose pullback to $U$ is $Z'$.", "By construction $|Z| = T$ and $Z$ is reduced. This proves existence.", "We omit the proof of uniqueness." ], "refs": [ "spaces-lemma-space-presentation", "spaces-properties-lemma-points-presentation", "schemes-lemma-base-change-immersion", "morphisms-lemma-base-change-etale", "spaces-properties-remark-list-properties-local-etale-topology", "spaces-properties-lemma-subspaces-presentation" ], "ref_ids": [ 8149, 11821, 7695, 5361, 11950, 11845 ] } ], "ref_ids": [] }, { "id": 11847, "type": "theorem", "label": "spaces-properties-lemma-map-into-reduction", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-map-into-reduction", "contents": [ "Let $S$ be a scheme.", "Let $X$, $Y$ be algebraic spaces over $S$.", "Let $Z \\subset X$ be a closed subspace.", "Assume $Y$ is reduced.", "A morphism $f : Y \\to X$ factors through $Z$ if and only if", "$f(|Y|) \\subset |Z|$." ], "refs": [], "proofs": [ { "contents": [ "Assume $f(|Y|) \\subset |Z|$. Choose a diagram", "$$", "\\xymatrix{", "V \\ar[d]_b \\ar[r]_h & U \\ar[d]^a \\\\", "Y \\ar[r]^f & X", "}", "$$", "where $U$, $V$ are schemes, and the vertical arrows are surjective and", "\\'etale. The scheme $V$ is reduced, see", "Lemma \\ref{lemma-type-property}.", "Hence $h$ factors through $a^{-1}(Z)$ by", "Schemes, Lemma \\ref{schemes-lemma-map-into-reduction}.", "So $a \\circ h$ factors through $Z$.", "As $Z \\subset X$ is a subsheaf, and $V \\to Y$ is a surjection of sheaves", "on $(\\Sch/S)_{fppf}$ we conclude that $X \\to Y$ factors", "through $Z$." ], "refs": [ "spaces-properties-lemma-type-property", "schemes-lemma-map-into-reduction" ], "ref_ids": [ 11837, 7682 ] } ], "ref_ids": [] }, { "id": 11848, "type": "theorem", "label": "spaces-properties-lemma-subscheme", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-subscheme", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "There exists a largest open subspace $X' \\subset X$ which is a scheme." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be an \\'etale surjective morphism, where $U$ is a scheme.", "Let $R = U \\times_X U$. The open subspaces of $X$ correspond $1 - 1$", "with open subschemes of $U$ which are $R$-invariant. Hence there is a", "set of them. Let $X_i$, $i \\in I$ be the set of open subspaces", "of $X$ which are schemes, i.e., are representable. Consider the", "open subspace $X' \\subset X$ whose underlying set of points is", "the open $\\bigcup |X_i|$ of $|X|$. By", "Lemma \\ref{lemma-characterize-surjective}", "we see that", "$$", "\\coprod X_i \\longrightarrow X'", "$$", "is a surjective map of sheaves on $(\\Sch/S)_{fppf}$.", "But since each $X_i \\to X'$ is representable by open immersions", "we see that in fact the map is surjective in the Zariski", "topology. Namely, if $T \\to X'$ is a morphism from a scheme", "into $X'$, then $X_i \\times_X' T$ is an open subscheme of $T$.", "Hence we can apply", "Schemes, Lemma \\ref{schemes-lemma-glue-functors}", "to see that $X'$ is a scheme." ], "refs": [ "spaces-properties-lemma-characterize-surjective", "schemes-lemma-glue-functors" ], "ref_ids": [ 11820, 7688 ] } ], "ref_ids": [] }, { "id": 11849, "type": "theorem", "label": "spaces-properties-lemma-quasi-separated-finite-etale-cover-dense-open-scheme", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-quasi-separated-finite-etale-cover-dense-open-scheme", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "If there exists a finite, \\'etale, surjective morphism", "$U \\to X$ where $U$ is a quasi-separated scheme, then", "there exists a dense open subspace $X'$ of $X$ which is a scheme.", "More precisely, every point $x \\in |X|$ of codimension $0$ in $X$", "is contained in $X'$." ], "refs": [], "proofs": [ { "contents": [ "Let $X' \\subset X$ be the maximal open subspace which is a scheme", "(Lemma \\ref{lemma-subscheme}).", "Let $x \\in |X|$ be a point of codimension $0$ on $X$.", "By Lemma \\ref{lemma-codimension-0-points-dense}", "it suffices to show $x \\in X'$.", "Let $U \\to X$ be as in the statement of the lemma.", "Write $R = U \\times_X U$ and denote $s, t : R \\to U$ the projections as usual.", "Note that $s, t$ are surjective, finite and \\'etale.", "By Lemma \\ref{lemma-finite-fibres-presentation}", "the fibre of $|U| \\to |X|$ over $x$ is finite, say", "$\\{\\eta_1, \\ldots, \\eta_n\\}$. By Lemma \\ref{lemma-codimension-0-points}", "each $\\eta_i$ is the generic point of an irreducible component of $U$.", "By Properties, Lemma \\ref{properties-lemma-maximal-points-affine}", "we can find an affine open $W \\subset U$ containing", "$\\{\\eta_1, \\ldots, \\eta_n\\}$", "(this is where we use that $U$ is quasi-separated). By", "Groupoids, Lemma \\ref{groupoids-lemma-find-invariant-affine}", "we may assume that $W$ is $R$-invariant.", "Since $W \\subset U$ is an $R$-invariant affine open, the restriction", "$R_W$ of $R$ to $W$ equals $R_W = s^{-1}(W) = t^{-1}(W)$ (see", "Groupoids, Definition \\ref{groupoids-definition-invariant-open}", "and discussion following it). In particular the maps $R_W \\to W$ are", "finite \\'etale also. It follows that $R_W$ is affine.", "Thus we see that $W/R_W$ is a scheme, by", "Groupoids, Proposition \\ref{groupoids-proposition-finite-flat-equivalence}.", "On the other hand, $W/R_W$ is an open subspace of $X$ by", "Spaces, Lemma \\ref{spaces-lemma-finding-opens} and it contains", "$x$ by construction." ], "refs": [ "spaces-properties-lemma-subscheme", "spaces-properties-lemma-codimension-0-points-dense", "spaces-properties-lemma-finite-fibres-presentation", "spaces-properties-lemma-codimension-0-points", "properties-lemma-maximal-points-affine", "groupoids-lemma-find-invariant-affine", "groupoids-definition-invariant-open", "groupoids-proposition-finite-flat-equivalence", "spaces-lemma-finding-opens" ], "ref_ids": [ 11848, 11843, 11836, 11842, 3059, 9664, 9685, 9669, 8151 ] } ], "ref_ids": [] }, { "id": 11850, "type": "theorem", "label": "spaces-properties-lemma-quotient-scheme", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-quotient-scheme", "contents": [ "Let $S$ be a scheme. Let $G \\to S$ be a group scheme. Let $X \\to S$ be", "a morphism of schemes. Let $a : G \\times_S X \\to X$ be an action. Assume that", "\\begin{enumerate}", "\\item $G \\to S$ is finite locally free,", "\\item the action $a$ is free,", "\\item $X \\to S$ is affine, or quasi-affine, or projective, or", "quasi-projective, or $X$ is isomorphic to an open subscheme of an", "affine scheme, or $X$ is isomorphic to an open subscheme of $\\text{Proj}(A)$", "for some graded ring $A$, or $G \\to S$ is radicial.", "\\end{enumerate}", "Then the fppf quotient sheaf $X/G$ is a scheme and $X \\to X/G$", "is an fppf $G$-torsor." ], "refs": [], "proofs": [ { "contents": [ "We first show that $X/G$ is a scheme. Since the action is free the morphism", "$j = (a, \\text{pr}) : G \\times_S X \\to X \\times_S X$", "is a monomorphism and hence an equivalence relation, see", "Groupoids, Lemma \\ref{groupoids-lemma-free-action}. The maps", "$s, t : G \\times_S X \\to X$", "are finite locally free as we've assumed that $G \\to S$ is finite locally", "free. To conclude it now suffices to prove the last assumption of", "Proposition \\ref{proposition-finite-flat-equivalence-global} holds.", "Since the action of $G$ is over $S$ it suffices to prove that", "any finite set of points in a fibre of $X \\to S$ is contained in an", "affine open of $X$. If $X$ is isomorphic to an open subscheme of an", "affine scheme or isomorphic to an open subscheme of $\\text{Proj}(A)$", "for some graded ring $A$ this follows from", "Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-affine}.", "If $X \\to S$ is affine, or quasi-affine, or projective, or", "quasi-projective, we may replace $S$ by an affine open and we", "get back to the case we just dealt with. If $G \\to S$ is radicial,", "then the orbits of points on $X$ under the action of $G$ are singletons", "and the condition trivially holds. Some details omitted.", "\\medskip\\noindent", "To see that $X \\to X/G$ is an fppf $G$-torsor", "(Groupoids, Definition \\ref{groupoids-definition-principal-homogeneous-space})", "we have to show that $G \\times_S X \\to X \\times_{X/G} X$", "is an isomorphism and that $X \\to X/G$ fppf locally has sections.", "The second part is clear from the fact that $X \\to X/G$ is surjective", "as a map of fppf sheaves (by construction). The first part follows from", "the isomorphism $R = U \\times_M U$ in the conclusion of", "Proposition \\ref{proposition-finite-flat-equivalence-global}", "(note that $R = G \\times_S X$ in our case)." ], "refs": [ "groupoids-lemma-free-action", "spaces-properties-proposition-finite-flat-equivalence-global", "properties-lemma-ample-finite-set-in-affine", "groupoids-definition-principal-homogeneous-space", "spaces-properties-proposition-finite-flat-equivalence-global" ], "ref_ids": [ 9614, 11918, 3062, 9679, 11918 ] } ], "ref_ids": [] }, { "id": 11851, "type": "theorem", "label": "spaces-properties-lemma-quotient-separated", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-quotient-separated", "contents": [ "Notation and assumptions as in", "Proposition \\ref{proposition-finite-flat-equivalence-global}. Then", "\\begin{enumerate}", "\\item if $U$ is quasi-separated over $S$, then $U/R$ is quasi-separated", "over $S$,", "\\item if $U$ is quasi-separated, then $U/R$ is quasi-separated,", "\\item if $U$ is separated over $S$, then $U/R$ is separated over $S$,", "\\item if $U$ is separated, then $U/R$ is separated, and", "\\item add more here.", "\\end{enumerate}", "Similar results hold in the setting of Lemma \\ref{lemma-quotient-scheme}." ], "refs": [ "spaces-properties-proposition-finite-flat-equivalence-global", "spaces-properties-lemma-quotient-scheme" ], "proofs": [ { "contents": [ "Since $M$ represents the quotient sheaf we have a cartesian diagram", "$$", "\\xymatrix{", "R \\ar[r]_-j \\ar[d] & U \\times_S U \\ar[d] \\\\", "M \\ar[r] & M \\times_S M", "}", "$$", "of schemes. Since $U \\times_S U \\to M \\times_S M$ is surjective finite locally", "free, to show that $M \\to M \\times_S M$ is quasi-compact, resp.\\ a closed", "immersion, it suffices to show that $j : R \\to U \\times_S U$ is", "quasi-compact, resp.\\ a closed immersion, see", "Descent, Lemmas \\ref{descent-lemma-descending-property-quasi-compact} and", "\\ref{descent-lemma-descending-property-closed-immersion}.", "Since $j : R \\to U \\times_S U$ is a morphism over $U$ and since", "$R$ is finite over $U$, we see that $j$ is quasi-compact as soon", "as the projection $U \\times_S U \\to U$ is quasi-separated", "(Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}).", "Since $j$ is a monomorphism and locally of finite type, we see that", "$j$ is a closed immersion as soon as it is proper", "(\\'Etale Morphisms, Lemma \\ref{etale-lemma-characterize-closed-immersion})", "which will be the case as soon as the projection", "$U \\times_S U \\to U$ is separated", "(Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}).", "This proves (1) and (3). To prove (2) and (4) we replace $S$ by", "$\\Spec(\\mathbf{Z})$, see Definition \\ref{definition-separated}.", "Since Lemma \\ref{lemma-quotient-scheme} is proved through an application of", "Proposition \\ref{proposition-finite-flat-equivalence-global}", "the final statement is clear too." ], "refs": [ "descent-lemma-descending-property-quasi-compact", "descent-lemma-descending-property-closed-immersion", "schemes-lemma-quasi-compact-permanence", "etale-lemma-characterize-closed-immersion", "morphisms-lemma-image-proper-scheme-closed", "spaces-properties-definition-separated", "spaces-properties-lemma-quotient-scheme", "spaces-properties-proposition-finite-flat-equivalence-global" ], "ref_ids": [ 14666, 14684, 7716, 10702, 5411, 11922, 11850, 11918 ] } ], "ref_ids": [ 11918, 11850 ] }, { "id": 11852, "type": "theorem", "label": "spaces-properties-lemma-quasi-separated-sober", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-quasi-separated-sober", "contents": [ "Let $S$ be a scheme. Let $X$ be a Zariski locally quasi-separated", "algebraic space over $S$. Then the topological space $|X|$ is sober (see", "Topology, Definition \\ref{topology-definition-generic-point})." ], "refs": [ "topology-definition-generic-point" ], "proofs": [ { "contents": [ "Combining", "Topology, Lemma \\ref{topology-lemma-sober-local}", "and", "Lemma \\ref{lemma-quasi-separated-quasi-compact-pieces}", "we see that we may assume that there exists an affine scheme $U$", "and a surjective, quasi-compact, \\'etale morphism $U \\to X$.", "Set $R = U \\times_X U$ with projection maps $s, t : R \\to U$. Applying", "Lemma \\ref{lemma-finite-fibres-presentation}", "we see that the fibres of $s, t$ are finite. It follows all the assumptions of", "Topology, Lemma \\ref{topology-lemma-quotient-kolmogorov}", "are met, and we conclude that $|X|$ is Kolmogorov\\footnote{", "Actually we use here also", "Schemes, Lemma \\ref{schemes-lemma-scheme-sober} (soberness schemes),", "Morphisms, Lemmas \\ref{morphisms-lemma-etale-flat}", "and \\ref{morphisms-lemma-generalizations-lift-flat} (generalizations", "lift along \\'etale morphisms),", "Lemma \\ref{lemma-points-presentation} (points on an algebraic space in", "terms of a presentation), and", "Lemma \\ref{lemma-topology-points} (openness quotient map).}.", "\\medskip\\noindent", "It remains to show that every irreducible closed subset", "$T \\subset |X|$ has a generic point. By", "Lemma \\ref{lemma-reduced-closed-subspace}", "there exists a closed subspace $Z \\subset X$ with $|Z| = |T|$.", "Note that $U \\times_X Z \\to Z$ is a quasi-compact, surjective, \\'etale", "morphism from an affine scheme to $Z$, hence $Z$ is Zariski locally", "quasi-separated by", "Lemma \\ref{lemma-quasi-separated-quasi-compact-pieces}.", "By", "Proposition \\ref{proposition-locally-quasi-separated-open-dense-scheme}", "we see that there exists an open dense subspace $Z' \\subset Z$", "which is a scheme. This means that $|Z'| \\subset T$ is open dense.", "Hence the topological space $|Z'|$ is irreducible, which means that", "$Z'$ is an irreducible scheme. By", "Schemes, Lemma \\ref{schemes-lemma-scheme-sober}", "we conclude that $|Z'|$ is the closure of a single point", "$\\eta \\in |Z'| \\subset T$ and hence also $T = \\overline{\\{\\eta\\}}$, and we win." ], "refs": [ "topology-lemma-sober-local", "spaces-properties-lemma-quasi-separated-quasi-compact-pieces", "spaces-properties-lemma-finite-fibres-presentation", "topology-lemma-quotient-kolmogorov", "schemes-lemma-scheme-sober", "morphisms-lemma-etale-flat", "morphisms-lemma-generalizations-lift-flat", "spaces-properties-lemma-points-presentation", "spaces-properties-lemma-topology-points", "spaces-properties-lemma-reduced-closed-subspace", "spaces-properties-lemma-quasi-separated-quasi-compact-pieces", "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme", "schemes-lemma-scheme-sober" ], "ref_ids": [ 8216, 11835, 11836, 8288, 7672, 5369, 5266, 11821, 11822, 11846, 11835, 11917, 7672 ] } ], "ref_ids": [ 8354 ] }, { "id": 11853, "type": "theorem", "label": "spaces-properties-lemma-quasi-compact-quasi-separated-spectral", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-quasi-compact-quasi-separated-spectral", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$. The topological space $|X|$ is a spectral space." ], "refs": [], "proofs": [ { "contents": [ "By Topology, Definition \\ref{topology-definition-spectral-space}", "we have to check that $|X|$ is sober, quasi-compact, has a basis", "of quasi-compact opens, and the intersection of any two", "quasi-compact opens is quasi-compact. By", "Lemma \\ref{lemma-quasi-separated-sober} we see that $|X|$ is sober.", "By Lemma \\ref{lemma-quasi-compact-space} we see that $|X|$ is quasi-compact.", "By Lemma \\ref{lemma-quasi-compact-affine-cover} there exists an affine scheme", "$U$ and a surjective \\'etale morphism $f : U \\to X$.", "Since $|f| : |U| \\to |X|$ is open and continuous and since $|U|$ has", "a basis of quasi-compact opens, we conclude that $|X|$ has a basis", "of quasi-compact opens. Finally, suppose that", "$A, B \\subset |X|$ are quasi-compact open. Then $A = |X'|$ and $B = |X''|$", "for some open subspaces $X', X'' \\subset X$ (Lemma \\ref{lemma-open-subspaces})", "and we can choose affine schemes $V$ and $W$ and surjective", "\\'etale morphisms $V \\to X'$ and $W \\to X''$", "(Lemma \\ref{lemma-quasi-compact-affine-cover}).", "Then $A \\cap B$ is the image of", "$|V \\times_X W| \\to |X|$ (Lemma \\ref{lemma-points-cartesian}).", "Since $V \\times_X W$ is quasi-compact as $X$ is quasi-separated", "(Lemma \\ref{lemma-characterize-quasi-separated})", "we conclude that $A \\cap B$ is quasi-compact and the proof is finished." ], "refs": [ "topology-definition-spectral-space", "spaces-properties-lemma-quasi-separated-sober", "spaces-properties-lemma-quasi-compact-space", "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-properties-lemma-open-subspaces", "spaces-properties-lemma-quasi-compact-affine-cover", "spaces-properties-lemma-points-cartesian", "spaces-properties-lemma-characterize-quasi-separated" ], "ref_ids": [ 8370, 11852, 11827, 11832, 11823, 11832, 11819, 11816 ] } ], "ref_ids": [] }, { "id": 11854, "type": "theorem", "label": "spaces-properties-lemma-point-like-spaces", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-point-like-spaces", "contents": [ "Let $S$ be a scheme. Let $k$ be a field.", "Let $X$ be an algebraic space over $S$ and assume that there exists", "a surjective \\'etale morphism $\\Spec(k) \\to X$.", "If $X$ is quasi-separated, then $X \\cong \\Spec(k')$", "where $k' \\subset k$ is a finite separable extension." ], "refs": [], "proofs": [ { "contents": [ "Set $R = \\Spec(k) \\times_X \\Spec(k)$, so that we have a", "fibre product diagram", "$$", "\\xymatrix{", "R \\ar[r]_-s \\ar[d]_-t & \\Spec(k) \\ar[d] \\\\", "\\Spec(k) \\ar[r] & X", "}", "$$", "By", "Spaces, Lemma \\ref{spaces-lemma-space-presentation}", "we know $X = \\Spec(k)/R$ is the quotient sheaf.", "Because $\\Spec(k) \\to X$ is \\'etale, the morphisms $s$ and $t$ are", "\\'etale. Hence $R = \\coprod_{i \\in I} \\Spec(k_i)$ is a disjoint", "union of spectra of fields, and both $s$ and $t$", "induce finite separable field extensions $s, t : k \\subset k_i$, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}.", "Because", "$$", "R = \\Spec(k) \\times_X \\Spec(k)", "= (\\Spec(k) \\times_S \\Spec(k)) \\times_{X \\times_S X, \\Delta} X", "$$", "and since $\\Delta$ is quasi-compact by assumption we conclude that", "$R \\to \\Spec(k) \\times_S \\Spec(k)$ is quasi-compact.", "Hence $R$ is quasi-compact as $\\Spec(k) \\times_S \\Spec(k)$ is", "affine. We conclude that $I$ is finite. This implies", "that $s$ and $t$ are finite locally free morphisms. Hence by", "Groupoids, Proposition \\ref{groupoids-proposition-finite-flat-equivalence}", "we conclude that $\\Spec(k)/R$ is", "represented by $\\Spec(k')$, with $k' \\subset k$ finite locally free", "where", "$$", "k' = \\{x \\in k \\mid s_i(x) = t_i(x)\\text{ for all }i \\in I\\}", "$$", "It is easy to see that $k'$ is a field." ], "refs": [ "spaces-lemma-space-presentation", "morphisms-lemma-etale-over-field", "groupoids-proposition-finite-flat-equivalence" ], "ref_ids": [ 8149, 5364, 9669 ] } ], "ref_ids": [] }, { "id": 11855, "type": "theorem", "label": "spaces-properties-lemma-etale-over-space", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-etale-over-space", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $U$, $U'$ be schemes over $S$.", "\\begin{enumerate}", "\\item If $U \\to U'$ is an \\'etale morphism of schemes, and", "if $U' \\to X$ is an \\'etale morphism from $U'$ to $X$, then the", "composition $U \\to X$ is an \\'etale morphism from $U$ to $X$.", "\\item If $\\varphi : U \\to X$ and $\\varphi' : U' \\to X$ are", "\\'etale morphisms towards $X$, and if $\\chi : U \\to U'$ is a", "morphism of schemes such that $\\varphi = \\varphi' \\circ \\chi$,", "then $\\chi$ is an \\'etale morphism of schemes.", "\\item If $\\chi : U \\to U'$ is a surjective \\'etale morphism", "of schemes and $\\varphi' : U' \\to X$ is a morphism such that", "$\\varphi = \\varphi' \\circ \\chi$ is \\'etale, then $\\varphi'$", "is \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that our definition of an \\'etale morphism from a scheme into an", "algebraic space comes from", "Spaces, Definition", "\\ref{spaces-definition-relative-representable-property}", "via the fact that any morphism from a scheme into an algebraic space", "is representable.", "\\medskip\\noindent", "Part (1) of the lemma follows from this, the fact that", "\\'etale morphisms are preserved under composition", "(Morphisms, Lemma", "\\ref{morphisms-lemma-composition-etale})", "and", "Spaces, Lemmas", "\\ref{spaces-lemma-composition-representable-transformations-property} and", "\\ref{spaces-lemma-morphism-schemes-gives-representable-transformation-property}", "(which are formal).", "\\medskip\\noindent", "To prove part (2) choose a scheme $W$ over $S$ and a", "surjective \\'etale morphism $W \\to X$. Consider the base change", "$\\chi_W : W \\times_X U \\to W \\times_X U'$ of $\\chi$.", "As $W \\times_X U$ and $W \\times_X U'$ are \\'etale over $W$, we conclude that", "$\\chi_W$ is \\'etale, by", "Morphisms, Lemma \\ref{morphisms-lemma-etale-permanence-two}.", "On the other hand, in the commutative diagram", "$$", "\\xymatrix{", "W \\times_X U \\ar[r] \\ar[d] & W \\times_X U' \\ar[d] \\\\", "U \\ar[r] & U'", "}", "$$", "the two vertical arrows are \\'etale and surjective.", "Hence by", "Descent, Lemma \\ref{descent-lemma-syntomic-smooth-etale-permanence}", "we conclude that $U \\to U'$ is \\'etale.", "\\medskip\\noindent", "To prove part (2) choose a scheme $W$ over $S$ and a morphism $W \\to X$.", "As above we consider the diagram", "$$", "\\xymatrix{", "W \\times_X U \\ar[r] \\ar[d] & W \\times_X U' \\ar[d] \\ar[r] & W \\ar[d] \\\\", "U \\ar[r] & U' \\ar[r] & X", "}", "$$", "Now we know that $W \\times_X U \\to W \\times_X U'$ is surjective \\'etale", "(as a base change of $U \\to U'$)", "and that $W \\times_X U \\to W$ is \\'etale. Thus $W \\times_X U' \\to W$", "is \\'etale by Descent, Lemma", "\\ref{descent-lemma-syntomic-smooth-etale-permanence}. By definition", "this means that $\\varphi'$ is \\'etale." ], "refs": [ "spaces-definition-relative-representable-property", "morphisms-lemma-composition-etale", "spaces-lemma-composition-representable-transformations-property", "spaces-lemma-morphism-schemes-gives-representable-transformation-property", "morphisms-lemma-etale-permanence-two", "descent-lemma-syntomic-smooth-etale-permanence", "descent-lemma-syntomic-smooth-etale-permanence" ], "ref_ids": [ 8173, 5360, 8132, 8131, 5376, 14643, 14643 ] } ], "ref_ids": [] }, { "id": 11856, "type": "theorem", "label": "spaces-properties-lemma-etale-local", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-etale-local", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is \\'etale,", "\\item there exists a surjective \\'etale morphism $\\varphi : U \\to X$,", "where $U$ is a scheme, such that the composition $f \\circ \\varphi$ is", "\\'etale (as a morphism of algebraic spaces),", "\\item there exists a surjective \\'etale morphism $\\psi : V \\to Y$,", "where $V$ is a scheme, such that the base change $V \\times_X Y \\to V$", "is \\'etale (as a morphism of algebraic spaces),", "\\item there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes, the vertical arrows are \\'etale, and the", "left vertical arrow is surjective such that the horizontal arrow is \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us prove that (4) implies (1). Assume a diagram as in (4) given.", "Let $W \\to X$ be an \\'etale morphism with $W$ a scheme. Then we see", "that $W \\times_X U \\to U$ is \\'etale. Hence $W \\times_X U \\to V$ is \\'etale", "as the composition of the \\'etale morphisms of schemes $W \\times_X U \\to U$", "and $U \\to V$. Therefore $W \\times_X U \\to Y$ is \\'etale by", "Lemma \\ref{lemma-etale-over-space} (1). Since also", "the projection $W \\times_X U \\to W$ is surjective and \\'etale, we conclude", "from Lemma \\ref{lemma-etale-over-space} (3) that $W \\to Y$ is \\'etale.", "\\medskip\\noindent", "Let us prove that (1) implies (4). Assume (1). Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U \\to X$ and $V \\to Y$ are surjective and \\'etale, see", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}.", "By assumption the morphism $U \\to Y$ is \\'etale,", "and hence $U \\to V$ is \\'etale by Lemma \\ref{lemma-etale-over-space} (2).", "\\medskip\\noindent", "We omit the proof that (2) and (3) are also equivalent to (1)." ], "refs": [ "spaces-properties-lemma-etale-over-space", "spaces-properties-lemma-etale-over-space", "spaces-lemma-lift-morphism-presentations", "spaces-properties-lemma-etale-over-space" ], "ref_ids": [ 11855, 11855, 8159, 11855 ] } ], "ref_ids": [] }, { "id": 11857, "type": "theorem", "label": "spaces-properties-lemma-composition-etale", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-composition-etale", "contents": [ "The composition of two \\'etale morphisms of algebraic spaces", "is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11858, "type": "theorem", "label": "spaces-properties-lemma-base-change-etale", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-base-change-etale", "contents": [ "The base change of an \\'etale morphism of algebraic spaces", "by any morphism of algebraic spaces is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Let $X \\to Y$ be an \\'etale morphism of algebraic spaces over $S$.", "Let $Z \\to Y$ be a morphism of algebraic spaces.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "Choose a scheme $W$ and a surjective \\'etale morphism $W \\to Z$.", "Then $U \\to Y$ is \\'etale, hence in the diagram", "$$", "\\xymatrix{", "W \\times_Y U \\ar[d] \\ar[r] & W \\ar[d] \\\\", "Z \\times_Y X \\ar[r] & Z", "}", "$$", "the top horizontal arrow is \\'etale.", "Moreover, the left vertical arrow is surjective", "and \\'etale (verification omitted). Hence we conclude that the lower", "horizontal arrow is \\'etale by Lemma \\ref{lemma-etale-local}." ], "refs": [ "spaces-properties-lemma-etale-local" ], "ref_ids": [ 11856 ] } ], "ref_ids": [] }, { "id": 11859, "type": "theorem", "label": "spaces-properties-lemma-etale-permanence", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-etale-permanence", "contents": [ "Let $S$ be a scheme. Let $X, Y, Z$ be algebraic spaces.", "Let $g : X \\to Z$, $h : Y \\to Z$ be \\'etale morphisms and let", "$f : X \\to Y$ be a morphism such that $h \\circ f = g$.", "Then $f$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_\\chi & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U \\to X$ and $V \\to Y$ are surjective and \\'etale, see", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}.", "By assumption the morphisms $\\varphi : U \\to X \\to Z$ and", "$\\psi : V \\to Y \\to Z$ are \\'etale. Moreover, $\\psi \\circ \\chi = \\varphi$", "by our assumption on $f, g, h$.", "Hence $U \\to V$ is \\'etale by Lemma \\ref{lemma-etale-over-space}", "part (2)." ], "refs": [ "spaces-lemma-lift-morphism-presentations", "spaces-properties-lemma-etale-over-space" ], "ref_ids": [ 8159, 11855 ] } ], "ref_ids": [] }, { "id": 11860, "type": "theorem", "label": "spaces-properties-lemma-etale-open", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-etale-open", "contents": [ "Let $S$ be a scheme.", "If $X \\to Y$ is an \\'etale morphism of algebraic spaces over $S$,", "then the associated map $|X| \\to |Y|$ of topological spaces", "is open." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the diagram in", "Lemma \\ref{lemma-etale-local}", "and", "Lemma \\ref{lemma-topology-points}." ], "refs": [ "spaces-properties-lemma-etale-local", "spaces-properties-lemma-topology-points" ], "ref_ids": [ 11856, 11822 ] } ], "ref_ids": [] }, { "id": 11861, "type": "theorem", "label": "spaces-properties-lemma-etale-over-field-scheme", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-etale-over-field-scheme", "contents": [ "Let $S$ be a scheme. Let $X \\to \\Spec(k)$", "be \\'etale morphism over $S$, where $k$ is a field.", "Then $X$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be an affine scheme, and let $U \\to X$ be an \\'etale morphism. By", "Definition \\ref{definition-etale}", "we see that $U \\to \\Spec(k)$ is an \\'etale", "morphism. Hence $U = \\coprod_{i = 1, \\ldots, n} \\Spec(k_i)$", "is a finite disjoint union of spectra of finite separable extensions", "$k_i$ of $k$, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}.", "The $R = U \\times_X U \\to U \\times_{\\Spec(k)} U$ is a monomorphism", "and $U \\times_{\\Spec(k)} U$ is also a finite disjoint union of", "spectra of finite separable extensions of $k$. Hence by", "Schemes, Lemma \\ref{schemes-lemma-mono-towards-spec-field}", "we see that $R$ is similarly a finite disjoint union of", "spectra of finite separable extensions of $k$.", "This $U$ and $R$ are affine and", "both projections $R \\to U$ are finite locally free.", "Hence $U/R$ is a scheme by", "Groupoids, Proposition \\ref{groupoids-proposition-finite-flat-equivalence}.", "By", "Spaces, Lemma \\ref{spaces-lemma-finding-opens}", "it is also an open subspace of $X$. By", "Lemma \\ref{lemma-subscheme}", "we conclude that $X$ is a scheme." ], "refs": [ "spaces-properties-definition-etale", "morphisms-lemma-etale-over-field", "schemes-lemma-mono-towards-spec-field", "groupoids-proposition-finite-flat-equivalence", "spaces-lemma-finding-opens", "spaces-properties-lemma-subscheme" ], "ref_ids": [ 11933, 5364, 7729, 9669, 8151, 11848 ] } ], "ref_ids": [] }, { "id": 11862, "type": "theorem", "label": "spaces-properties-lemma-compare-etale-sites", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-compare-etale-sites", "contents": [ "The functor", "$$", "X_\\etale \\longrightarrow X_{spaces, \\etale}, \\quad", "U/X \\longmapsto U/X", "$$", "is a special cocontinuous functor", "(Sites, Definition \\ref{sites-definition-special-cocontinuous-functor})", "and hence induces an equivalence of topoi", "$\\Sh(X_\\etale) \\to \\Sh(X_{spaces, \\etale})$." ], "refs": [ "sites-definition-special-cocontinuous-functor" ], "proofs": [ { "contents": [ "We have to show that the functor satisfies the assumptions (1) -- (5) of", "Sites, Lemma \\ref{sites-lemma-equivalence}.", "It is clear that the functor is continuous and cocontinuous, which", "proves assumptions (1) and (2).", "Assumptions (3) and (4) hold simply because the functor is fully faithful.", "Assumption (5) holds, because an algebraic space by definition has", "a covering by a scheme." ], "refs": [ "sites-lemma-equivalence" ], "ref_ids": [ 8578 ] } ], "ref_ids": [ 8672 ] }, { "id": 11863, "type": "theorem", "label": "spaces-properties-lemma-alternative", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-alternative", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $X_{affine, \\etale}$ denote the full subcategory of", "$X_\\etale$ whose objects are those", "$U/X \\in \\Ob(X_\\etale)$ with $U$ affine.", "A covering of $X_{affine, \\etale}$ will be a", "standard \\'etale covering, see", "Topologies, Definition \\ref{topologies-definition-standard-etale}.", "Then restriction", "$$", "\\mathcal{F} \\longmapsto \\mathcal{F}|_{X_{affine, \\etale}}", "$$", "defines an equivalence of topoi", "$\\Sh(X_\\etale) \\cong \\Sh(X_{affine, \\etale})$." ], "refs": [ "topologies-definition-standard-etale" ], "proofs": [ { "contents": [ "This you can show directly from the definitions, and is a good exercise.", "But it also follows immediately from", "Sites, Lemma \\ref{sites-lemma-equivalence}", "by checking that the inclusion functor", "$X_{affine, \\etale} \\to X_\\etale$ is a special cocontinuous", "functor as in", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}." ], "refs": [ "sites-lemma-equivalence", "sites-definition-special-cocontinuous-functor" ], "ref_ids": [ 8578, 8672 ] } ], "ref_ids": [ 12527 ] }, { "id": 11864, "type": "theorem", "label": "spaces-properties-lemma-functoriality-etale-site", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-functoriality-etale-site", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item The continuous functor", "$$", "Y_{spaces, \\etale} \\longrightarrow X_{spaces, \\etale}, \\quad", "V \\longmapsto X \\times_Y V", "$$", "induces a morphism of sites", "$$", "f_{spaces, \\etale} :", "X_{spaces, \\etale}", "\\to", "Y_{spaces, \\etale}.", "$$", "\\item The rule $f \\mapsto f_{spaces, \\etale}$ is compatible with", "compositions, in other words $(f \\circ g)_{spaces, \\etale}", "= f_{spaces, \\etale} \\circ g_{spaces, \\etale}$ (see", "Sites, Definition \\ref{sites-definition-composition-morphisms-sites}).", "\\item The morphism of topoi associated to $f_{spaces, \\etale}$", "induces, via Lemma \\ref{lemma-compare-etale-sites}, a morphism of topoi", "$f_{small} : \\Sh(X_\\etale) \\to \\Sh(Y_\\etale)$", "whose construction is compatible with compositions.", "\\item If $f$ is a representable morphism of algebraic spaces,", "then $f_{small}$ comes from a morphism of sites", "$X_\\etale \\to Y_\\etale$,", "corresponding to the continuous functor $V \\mapsto X \\times_Y V$.", "\\end{enumerate}" ], "refs": [ "sites-definition-composition-morphisms-sites", "spaces-properties-lemma-compare-etale-sites" ], "proofs": [ { "contents": [ "Let us show that the functor described in (1) satisfies the assumptions", "of Sites, Proposition \\ref{sites-proposition-get-morphism}.", "Thus we have to show that", "$Y_{spaces, \\etale}$ has a final object (namely $Y$) and that", "the functor transforms this into a final object in $X_{spaces, \\etale}$", "(namely $X$). This is clear as $X \\times_Y Y = X$ in any category.", "Next, we have to show that $Y_{spaces, \\etale}$ has fibre products.", "This is true since the category of algebraic spaces has fibre products,", "and since $V \\times_Y V'$ is \\'etale over $Y$ if $V$ and $V'$ are \\'etale", "over $Y$ (see Lemmas \\ref{lemma-composition-etale} and", "\\ref{lemma-base-change-etale} above).", "OK, so the proposition applies and we see that we get a morphism", "of sites as described in (1).", "\\medskip\\noindent", "Part (2) you get by unwinding the definitions.", "Part (3) is clear by using the equivalences for $X$ and $Y$", "from Lemma \\ref{lemma-compare-etale-sites} above.", "Part (4) follows, because if $f$ is representable, then the", "functors above fit into a commutative diagram", "$$", "\\xymatrix{", "X_\\etale \\ar[r] &", "X_{spaces, \\etale} \\\\", "Y_\\etale \\ar[r] \\ar[u] &", "Y_{spaces, \\etale} \\ar[u]", "}", "$$", "of categories." ], "refs": [ "sites-proposition-get-morphism", "spaces-properties-lemma-composition-etale", "spaces-properties-lemma-base-change-etale", "spaces-properties-lemma-compare-etale-sites" ], "ref_ids": [ 8641, 11857, 11858, 11862 ] } ], "ref_ids": [ 8666, 11862 ] }, { "id": 11865, "type": "theorem", "label": "spaces-properties-lemma-f-map", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-f-map", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$ and", "let $\\mathcal{G}$ be a sheaf of sets on $Y_\\etale$.", "There are canonical bijections between the following three sets:", "\\begin{enumerate}", "\\item The set of maps $\\mathcal{G} \\to f_{small, *}\\mathcal{F}$.", "\\item The set of maps $f_{small}^{-1}\\mathcal{G} \\to \\mathcal{F}$.", "\\item The set of $f$-maps $\\varphi : \\mathcal{G} \\to \\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that (1) and (2) are the same because the functors $f_{small, *}$", "and $f_{small}^{-1}$ are a pair of adjoint functors.", "Suppose that $\\alpha : f_{small}^{-1}\\mathcal{G} \\to \\mathcal{F}$", "is a map of sheaves on $Y_\\etale$. Let a diagram", "$$", "\\xymatrix{", "U \\ar[d]_g \\ar[r]_{j_U} & X \\ar[d]^f \\\\", "V \\ar[r]^{j_V} & Y", "}", "$$", "as in Definition \\ref{definition-f-map} be given.", "By the commutativity of the diagram we also get a map", "$g_{small}^{-1}(j_V)^{-1}\\mathcal{G} \\to (j_U)^{-1}\\mathcal{F}$", "(compare Sites, Section \\ref{sites-section-localize} for the", "description of the localization functors). Hence we certainly", "get a map", "$\\varphi_{(V, U, g)} :", "\\mathcal{G}(V) = (j_V)^{-1}\\mathcal{G}(V)", "\\to", "(j_U)^{-1}\\mathcal{F}(U) = \\mathcal{F}(U)$.", "We omit the verification that this rule is compatible with", "further restrictions and defines an $f$-map from $\\mathcal{G}$ to", "$\\mathcal{F}$.", "\\medskip\\noindent", "Conversely, suppose that we are given an $f$-map", "$\\varphi = (\\varphi_{(U, V, g)})$.", "Let $\\mathcal{G}'$ (resp.\\ $\\mathcal{F}'$) denote the extension of", "$\\mathcal{G}$ (resp.\\ $\\mathcal{F}$) to $Y_{spaces, \\etale}$", "(resp.\\ $X_{spaces, \\etale}$), see", "Lemma \\ref{lemma-compare-etale-sites}.", "Then we have to construct a map of sheaves", "$$", "\\mathcal{G}' \\longrightarrow (f_{spaces, \\etale})_*\\mathcal{F}'", "$$", "To do this, let $V \\to Y$ be an \\'etale morphism of algebraic spaces.", "We have to construct a map of sets", "$$", "\\mathcal{G}'(V) \\to \\mathcal{F}'(X \\times_Y V)", "$$", "Choose an \\'etale surjective morphism $V' \\to V$ with $V'$ a scheme,", "and after that choose an \\'etale surjective morphism", "$U' \\to X \\times_U V'$ with $U'$ a scheme. We get a morphism of", "schemes $g' : U' \\to V'$ and also a morphism of schemes", "$$", "g'' : U' \\times_{X \\times_Y V} U' \\longrightarrow V' \\times_V V'", "$$", "Consider the following diagram", "$$", "\\xymatrix{", "\\mathcal{F}'(X \\times_Y V) \\ar[r] &", "\\mathcal{F}(U') \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\mathcal{F}(U' \\times_{X \\times_Y V} U') \\\\", "\\mathcal{G}'(X \\times_Y V) \\ar[r] \\ar@{..>}[u] &", "\\mathcal{G}(V') \\ar@<1ex>[r] \\ar@<-1ex>[r] \\ar[u]_{\\varphi_{(U', V', g')}} &", "\\mathcal{G}(V' \\times_V V') \\ar[u]_{\\varphi_{(U'', V'', g'')}}", "}", "$$", "The compatibility of the maps $\\varphi_{...}$", "with restriction shows that the two right squares commute.", "The definition of coverings in $X_{spaces, \\etale}$ shows that", "the horizontal rows are equalizer diagrams. Hence we get", "the dotted arrow. We leave it to the reader to show that these", "arrows are compatible with the restriction mappings." ], "refs": [ "spaces-properties-definition-f-map", "spaces-properties-lemma-compare-etale-sites" ], "ref_ids": [ 11937, 11862 ] } ], "ref_ids": [] }, { "id": 11866, "type": "theorem", "label": "spaces-properties-lemma-etale-morphism-topoi", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-etale-morphism-topoi", "contents": [ "Let $S$ be a scheme, and let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is \\'etale. In this case there is a functor", "$$", "j : X_\\etale \\to Y_\\etale, \\quad", "(\\varphi : U \\to X) \\mapsto (f \\circ \\varphi : U \\to Y)", "$$", "which is cocontinuous. The morphism of topoi $f_{small}$ is the", "morphism of topoi associated to $j$, see", "Sites, Lemma \\ref{sites-lemma-cocontinuous-morphism-topoi}.", "Moreover, $j$ is continuous as well, hence", "Sites, Lemma \\ref{sites-lemma-when-shriek}", "applies. In particular $f_{small}^{-1}\\mathcal{G}(U) = \\mathcal{G}(jU)$", "for all sheaves $\\mathcal{G}$ on $Y_\\etale$." ], "refs": [ "sites-lemma-cocontinuous-morphism-topoi", "sites-lemma-when-shriek" ], "proofs": [ { "contents": [ "Note that by our very definition of an \\'etale morphism of algebraic spaces", "(Definition \\ref{definition-etale}) it is", "indeed the case that the rule given defines a functor $j$ as indicated.", "It is clear that $j$ is cocontinuous and continuous, simply because a covering", "$\\{U_i \\to U\\}$ of $j(\\varphi : U \\to X)$ in $Y_\\etale$ is the", "same thing as a covering of $(\\varphi : U \\to X)$ in $X_\\etale$. It", "remains to show that $j$ induces the same morphism of topoi as $f_{small}$.", "To see this we consider the diagram", "$$", "\\xymatrix{", "X_\\etale \\ar[r] \\ar[d]^j &", "X_{spaces, \\etale} \\ar@/_/[d]_{j_{spaces}} \\\\", "Y_\\etale \\ar[r] &", "Y_{spaces, \\etale} \\ar@/_/[u]_{v : V \\mapsto X \\times_Y V}", "}", "$$", "of categories. Here the functor $j_{spaces}$ is the obvious extension of $j$", "to the category $X_{spaces, \\etale}$. Thus the inner square is", "commutative. In fact $j_{spaces}$ can be identified with the", "localization functor", "$j_X : Y_{spaces, \\etale}/X \\to Y_{spaces, \\etale}$", "discussed in", "Sites, Section \\ref{sites-section-localize}.", "Hence, by", "Sites, Lemma \\ref{sites-lemma-localize-given-products}", "the cocontinuous functor $j_{spaces}$ and the functor $v$ of the diagram", "induce the same morphism of topoi. By", "Sites, Lemma \\ref{sites-lemma-composition-cocontinuous}", "the commutativity of the inner square (consisting of cocontinuous functors", "between sites) gives a commutative diagram of associated morphisms of topoi.", "Hence, by the construction of $f_{small}$ in", "Lemma \\ref{lemma-functoriality-etale-site} we win." ], "refs": [ "spaces-properties-definition-etale", "sites-lemma-localize-given-products", "sites-lemma-composition-cocontinuous", "spaces-properties-lemma-functoriality-etale-site" ], "ref_ids": [ 11933, 8567, 8544, 11864 ] } ], "ref_ids": [ 8543, 8545 ] }, { "id": 11867, "type": "theorem", "label": "spaces-properties-lemma-pushforward-etale-base-change", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-pushforward-etale-base-change", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X' \\ar[r] \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "be a cartesian square of algebraic spaces over $S$. Let", "$\\mathcal{F}$ be a sheaf on $X_\\etale$. If $g$ is \\'etale, then", "\\begin{enumerate}", "\\item $f'_{small, *}(\\mathcal{F}|_{X'}) = (f_{small, *}\\mathcal{F})|_{Y'}$", "in $\\Sh(Y'_\\etale)$\\footnote{Also", "$(f')_{small}^{-1}(\\mathcal{G}|_{Y'}) = (f_{small}^{-1}\\mathcal{G})|_{X'}$", "because of commutativity of the diagram and (\\ref{equation-restrict})}, and", "\\item if $\\mathcal{F}$ is an abelian sheaf, then", "$R^if'_{small, *}(\\mathcal{F}|_{X'}) = (R^if_{small, *}\\mathcal{F})|_{Y'}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the following diagram of functors", "$$", "\\xymatrix{", "X'_{spaces, \\etale} \\ar[r]_j &", "X_{spaces, \\etale} \\\\", "Y'_{spaces, \\etale} \\ar[r]^j \\ar[u]^{V' \\mapsto V' \\times_{Y'} X'} &", "Y_{spaces, \\etale} \\ar[u]_{V \\mapsto V \\times_Y X}", "}", "$$", "The horizontal arrows are localizations and the vertical arrows induce", "morphisms of sites. Hence the last statement of", "Sites, Lemma \\ref{sites-lemma-localize-morphism}", "gives (1). To see (2) apply (1) to an injective resolution of $\\mathcal{F}$", "and use that restriction is exact and preserves injectives (see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-of-open})." ], "refs": [ "sites-lemma-localize-morphism", "sites-cohomology-lemma-cohomology-of-open" ], "ref_ids": [ 8571, 4186 ] } ], "ref_ids": [] }, { "id": 11868, "type": "theorem", "label": "spaces-properties-lemma-characterize-sheaf-small-etale", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-characterize-sheaf-small-etale", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "A sheaf $\\mathcal{F}$ on $X_\\etale$ is given by the following data:", "\\begin{enumerate}", "\\item for every $U \\in \\Ob(X_\\etale)$ a sheaf", "$\\mathcal{F}_U$ on $U_\\etale$,", "\\item for every $f : U' \\to U$ in $X_\\etale$ an isomorphism", "$c_f : f_{small}^{-1}\\mathcal{F}_U \\to \\mathcal{F}_{U'}$.", "\\end{enumerate}", "These data are subject to the condition that given any $f : U' \\to U$", "and $g : U'' \\to U'$ in $X_\\etale$ the composition", "$g_{small}^{-1}c_f \\circ c_g$ is equal to $c_{f \\circ g}$." ], "refs": [], "proofs": [ { "contents": [ "Given a sheaf $\\mathcal{F}$ on $X_\\etale$ and an object", "$\\varphi : U \\to X$ of", "$X_\\etale$ we set $\\mathcal{F}_U = \\varphi_{small}^{-1}\\mathcal{F}$.", "If $\\varphi' : U' \\to X$ is a second object of $X_\\etale$, and", "$f : U' \\to U$ is a morphism between them, then", "the isomorphism $c_f$ comes from the fact that", "$f_{small}^{-1} \\circ \\varphi_{small}^{-1} = (\\varphi')^{-1}_{small}$, see", "Lemma \\ref{lemma-functoriality-etale-site}. The condition on the", "transitivity of the isomorphisms $c_f$ follows from the functoriality", "of the small \\'etale sites also; verification omitted.", "\\medskip\\noindent", "Conversely, suppose we are given a collection of data $(\\mathcal{F}_U, c_f)$", "as in the lemma. In this case we simply define $\\mathcal{F}$ by the rule", "$U \\mapsto \\mathcal{F}_U(U)$. Details omitted." ], "refs": [ "spaces-properties-lemma-functoriality-etale-site" ], "ref_ids": [ 11864 ] } ], "ref_ids": [] }, { "id": 11869, "type": "theorem", "label": "spaces-properties-lemma-descent-sheaf", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-descent-sheaf", "contents": [ "With $S$, $\\varphi : U \\to X$, and $(U, R, s, t, c, e, i)$ as above.", "For any sheaf $\\mathcal{F}$ on $X_\\etale$ the", "sheaf\\footnote{In this lemma", "and its proof we write simply $\\varphi^{-1}$ instead of $\\varphi_{small}^{-1}$", "and similarly for all the other pullbacks.}", "$\\mathcal{G} = \\varphi^{-1}\\mathcal{F}$ comes equipped with a canonical", "isomorphism", "$$", "\\alpha :", "t^{-1}\\mathcal{G}", "\\longrightarrow", "s^{-1}\\mathcal{G}", "$$", "such that the diagram", "$$", "\\xymatrix{", "& \\text{pr}_1^{-1}t^{-1}\\mathcal{G} \\ar[r]_-{\\text{pr}_1^{-1}\\alpha} &", "\\text{pr}_1^{-1}s^{-1}\\mathcal{G} \\ar@{=}[rd] & \\\\", "\\text{pr}_0^{-1}s^{-1}\\mathcal{G} \\ar@{=}[ru] & & &", "c^{-1}s^{-1}\\mathcal{G} \\\\", "&", "\\text{pr}_0^{-1}t^{-1}\\mathcal{G} \\ar[lu]^{\\text{pr}_0^{-1}\\alpha} \\ar@{=}[r] &", "c^{-1}t^{-1}\\mathcal{G} \\ar[ru]_{c^{-1}\\alpha}", "}", "$$", "is a commutative. The functor $\\mathcal{F} \\mapsto (\\mathcal{G}, \\alpha)$", "defines an equivalence of categories between sheaves on", "$X_\\etale$ and pairs $(\\mathcal{G}, \\alpha)$ as above." ], "refs": [], "proofs": [ { "contents": [ "[First proof of Lemma \\ref{lemma-descent-sheaf}]", "Let $\\mathcal{C} = X_{spaces, \\etale}$. By", "Lemma \\ref{lemma-etale-morphism-topoi}", "and its proof we have $U_{spaces, \\etale} = \\mathcal{C}/U$", "and the pullback functor $\\varphi^{-1}$ is just the restriction functor.", "Moreover, $\\{U \\to X\\}$ is a covering of the site $\\mathcal{C}$ and", "$R = U \\times_X U$. The isomorphism $\\alpha$ is just the canonical", "identification", "$$", "\\left(\\mathcal{F}|_{\\mathcal{C}/U}\\right)|_{\\mathcal{C}/U \\times_X U}", "=", "\\left(\\mathcal{F}|_{\\mathcal{C}/U}\\right)|_{\\mathcal{C}/U \\times_X U}", "$$", "and the commutativity of the diagram is the cocycle condition for glueing", "data. Hence this lemma is a special case of glueing of sheaves, see", "Sites, Section \\ref{sites-section-glueing-sheaves}." ], "refs": [ "spaces-properties-lemma-descent-sheaf", "spaces-properties-lemma-etale-morphism-topoi" ], "ref_ids": [ 11869, 11866 ] } ], "ref_ids": [] }, { "id": 11870, "type": "theorem", "label": "spaces-properties-lemma-cofinal-etale", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-cofinal-etale", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\overline{x}$ be a geometric point of $X$.", "The category of \\'etale neighborhoods is cofiltered. More precisely:", "\\begin{enumerate}", "\\item Let $(U_i, \\overline{u}_i)_{i = 1, 2}$ be two \\'etale neighborhoods of", "$\\overline{x}$ in $X$. Then there exists a third \\'etale neighborhood", "$(U, \\overline{u})$ and morphisms", "$(U, \\overline{u}) \\to (U_i, \\overline{u}_i)$, $i = 1, 2$.", "\\item Let $h_1, h_2: (U, \\overline{u}) \\to (U', \\overline{u}')$ be two", "morphisms between \\'etale neighborhoods of $\\overline{s}$. Then there exist an", "\\'etale neighborhood $(U'', \\overline{u}'')$ and a morphism", "$h : (U'', \\overline{u}'') \\to (U, \\overline{u})$", "which equalizes $h_1$ and $h_2$, i.e., such that", "$h_1 \\circ h = h_2 \\circ h$.", "\\end{enumerate}", "Moreover, given any \\'etale neighbourhood", "$(U, \\overline{u}) \\to (X, \\overline{x})$", "there exists a morphism of \\'etale neighbourhoods", "$(U', \\overline{u}') \\to (U, \\overline{u})$", "where $U'$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "For part (1), consider the fibre product $U = U_1 \\times_X U_2$.", "It is \\'etale over both $U_1$ and $U_2$ because \\'etale morphisms are", "preserved under base change and composition, see", "Lemmas \\ref{lemma-base-change-etale} and \\ref{lemma-composition-etale}.", "The map $\\overline{u} \\to U$ defined by $(\\overline{u}_1, \\overline{u}_2)$", "gives it the structure of an \\'etale neighborhood mapping to both", "$U_1$ and $U_2$.", "\\medskip\\noindent", "For part (2), define $U''$ as the fibre product", "$$", "\\xymatrix{", "U'' \\ar[r] \\ar[d] & U \\ar[d]^{(h_1, h_2)} \\\\", "U' \\ar[r]^-\\Delta & U' \\times_X U'.", "}", "$$", "Since $\\overline{u}$ and $\\overline{u}'$ agree over $X$ with $\\overline{x}$,", "we see that $\\overline{u}'' = (\\overline{u}, \\overline{u}')$ is a geometric", "point of $U''$. In particular $U'' \\not = \\emptyset$.", "Moreover, since $U'$ is \\'etale over $X$, so is the fibre product", "$U'\\times_X U'$ (as seen above in the case of $U_1 \\times_X U_2$).", "Hence the vertical arrow $(h_1, h_2)$ is \\'etale by", "Lemma \\ref{lemma-etale-permanence}.", "Therefore $U''$ is \\'etale over $U'$ by base change, and hence also", "\\'etale over $X$ (because compositions of \\'etale morphisms are \\'etale).", "Thus $(U'', \\overline{u}'')$ is a solution to the problem posed by (2).", "\\medskip\\noindent", "To see the final assertion, choose any surjective \\'etale morphism", "$U' \\to U$ where $U'$ is a scheme. Then", "$U' \\times_U \\overline{u}$ is a scheme surjective and \\'etale over", "$\\overline{u} = \\Spec(k)$ with $k$ algebraically closed.", "It follows (see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field})", "that $U' \\times_U \\overline{u} \\to \\overline{u}$ has a section", "which gives us the desired $\\overline{u}'$." ], "refs": [ "spaces-properties-lemma-base-change-etale", "spaces-properties-lemma-composition-etale", "spaces-properties-lemma-etale-permanence", "morphisms-lemma-etale-over-field" ], "ref_ids": [ 11858, 11857, 11859, 5364 ] } ], "ref_ids": [] }, { "id": 11871, "type": "theorem", "label": "spaces-properties-lemma-geometric-lift-to-usual", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-geometric-lift-to-usual", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\overline{x} : \\Spec(k) \\to X$ be a geometric point of $X$", "lying over $x \\in |X|$. Let $\\varphi : U \\to X$ be an \\'etale morphism", "of algebraic spaces and let $u \\in |U|$ with $\\varphi(u) = x$.", "Then there exists a geometric point", "$\\overline{u} : \\Spec(k) \\to U$ lying over $u$ with", "$\\overline{x} = \\varphi \\circ \\overline{u}$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $U'$ with $u' \\in U'$ and an \\'etale morphism", "$U' \\to U$ which maps $u'$ to $u$. If we can prove the lemma for", "$(U', u') \\to (X, x)$ then the lemma follows. Hence we may assume that", "$U$ is a scheme, in particular that $U \\to X$ is representable.", "Then look at the cartesian diagram", "$$", "\\xymatrix{", "\\Spec(k) \\times_{\\overline{x}, X, \\varphi} U", "\\ar[d]_{\\text{pr}_1} \\ar[r]_-{\\text{pr}_2} & U", "\\ar[d]^\\varphi \\\\", "\\Spec(k) \\ar[r]^-{\\overline{x}} & X", "}", "$$", "The projection $\\text{pr}_1$ is the base change of an \\'etale morphisms", "so it is \\'etale, see", "Lemma \\ref{lemma-base-change-etale}.", "Therefore, the scheme $\\Spec(k) \\times_{\\overline{x}, X, \\varphi} U$", "is a disjoint union of finite separable extensions of $k$, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}.", "But $k$ is algebraically closed, so all these extensions are trivial,", "so $\\Spec(k) \\times_{\\overline{x}, X, \\varphi} U$", "is a disjoint union of copies of $\\Spec(k)$ and each of", "these corresponds to a geometric point $\\overline{u}$ with", "$\\varphi \\circ \\overline{u} = \\overline{x}$. By", "Lemma \\ref{lemma-points-cartesian}", "the map", "$$", "|\\Spec(k) \\times_{\\overline{x}, X, \\varphi} U|", "\\longrightarrow", "|\\Spec(k)| \\times_{|X|} |U|", "$$", "is surjective, hence we can pick $\\overline{u}$ to lie over $u$." ], "refs": [ "spaces-properties-lemma-base-change-etale", "morphisms-lemma-etale-over-field", "spaces-properties-lemma-points-cartesian" ], "ref_ids": [ 11858, 5364, 11819 ] } ], "ref_ids": [] }, { "id": 11872, "type": "theorem", "label": "spaces-properties-lemma-geometric-lift-to-cover", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-geometric-lift-to-cover", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\overline{x}$ be a geometric point of $X$.", "Let $(U, \\overline{u})$ an \\'etale neighborhood of $\\overline{x}$.", "Let $\\{\\varphi_i : U_i \\to U\\}_{i \\in I}$ be an \\'etale covering in", "$X_{spaces, \\etale}$.", "Then there exist $i \\in I$ and $\\overline{u}_i : \\overline{x} \\to U_i$", "such that $\\varphi_i : (U_i, \\overline{u}_i) \\to (U, \\overline{u})$", "is a morphism of \\'etale neighborhoods." ], "refs": [], "proofs": [ { "contents": [ "Let $u \\in |U|$ be the image of $\\overline{u}$.", "As $|U| = \\bigcup_{i \\in I} \\varphi_i(|U_i|)$ there exists an", "$i$ and a point $u_i \\in U_i$ mapping to $x$. Apply", "Lemma \\ref{lemma-geometric-lift-to-usual}", "to $(U_i, u_i) \\to (U, u)$ and $\\overline{u}$ to", "get the desired geometric point." ], "refs": [ "spaces-properties-lemma-geometric-lift-to-usual" ], "ref_ids": [ 11871 ] } ], "ref_ids": [] }, { "id": 11873, "type": "theorem", "label": "spaces-properties-lemma-stalk-gives-point", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-stalk-gives-point", "contents": [ "\\begin{slogan}", "A geometric point of an algebraic space gives a point of its \\'etale topos.", "\\end{slogan}", "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $\\overline{x}$ be a geometric point of $X$.", "Consider the functor", "$$", "u : X_\\etale \\longrightarrow \\textit{Sets}, \\quad", "U \\longmapsto |U_{\\overline{x}}|", "$$", "Then $u$ defines a point $p$ of the site $X_\\etale$", "(Sites, Definition \\ref{sites-definition-point})", "and its associated stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_p$", "(Sites, Equation \\ref{sites-equation-stalk})", "is the functor $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$", "defined above." ], "refs": [ "sites-definition-point" ], "proofs": [ { "contents": [ "In the proof of", "Lemma \\ref{lemma-geometric-lift-to-cover}", "we have seen that the scheme $U_{\\overline{x}}$ is a disjoint union of", "schemes isomorphic to $\\overline{x}$. Thus we can also think of", "$|U_{\\overline{x}}|$ as the set of geometric points of $U$ lying over", "$\\overline{x}$, i.e., as the collection of morphisms", "$\\overline{u} : \\overline{x} \\to U$ fitting into the diagram of", "Definition \\ref{definition-geometric-point}.", "From this it follows that $u(X)$ is a singleton, and that", "$u(U \\times_V W) = u(U) \\times_{u(V)} u(W)$", "whenever $U \\to V$ and $W \\to V$ are morphisms in $X_\\etale$.", "And, given a covering $\\{U_i \\to U\\}_{i \\in I}$ in $X_\\etale$ we see", "that $\\coprod u(U_i) \\to u(U)$ is surjective by", "Lemma \\ref{lemma-geometric-lift-to-cover}.", "Hence", "Sites, Proposition \\ref{sites-proposition-point-limits}", "applies, so $p$ is a point of the site $X_\\etale$.", "Finally, the our functor $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{s}}$", "is given by exactly the same colimit as the functor", "$\\mathcal{F} \\mapsto \\mathcal{F}_p$ associated to $p$ in", "Sites, Equation \\ref{sites-equation-stalk}", "which proves the final assertion." ], "refs": [ "spaces-properties-lemma-geometric-lift-to-cover", "spaces-properties-definition-geometric-point", "spaces-properties-lemma-geometric-lift-to-cover", "sites-proposition-point-limits" ], "ref_ids": [ 11872, 11938, 11872, 8642 ] } ], "ref_ids": [ 8675 ] }, { "id": 11874, "type": "theorem", "label": "spaces-properties-lemma-stalk-exact", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-stalk-exact", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $\\overline{x}$ be a geometric point of $X$.", "\\begin{enumerate}", "\\item The stalk functor", "$\\textit{PAb}(X_\\etale) \\to \\textit{Ab}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$", "is exact.", "\\item We have $(\\mathcal{F}^\\#)_{\\overline{x}} = \\mathcal{F}_{\\overline{x}}$", "for any presheaf of sets $\\mathcal{F}$ on $X_\\etale$.", "\\item The functor", "$\\textit{Ab}(X_\\etale) \\to \\textit{Ab}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$ is exact.", "\\item Similarly the functors", "$\\textit{PSh}(X_\\etale) \\to \\textit{Sets}$ and", "$\\Sh(X_\\etale) \\to \\textit{Sets}$ given by the stalk functor", "$\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$ are exact (see", "Categories, Definition \\ref{categories-definition-exact})", "and commute with arbitrary colimits.", "\\end{enumerate}" ], "refs": [ "categories-definition-exact" ], "proofs": [ { "contents": [ "This result follows from the general material in", "Modules on Sites, Section \\ref{sites-modules-section-stalks}.", "This is true because $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$", "comes from a point of the small \\'etale site of $X$, see", "Lemma \\ref{lemma-stalk-gives-point}. See the proof of", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-stalk-exact}", "for a direct proof of some of these statements in the setting of", "the small \\'etale site of a scheme." ], "refs": [ "spaces-properties-lemma-stalk-gives-point", "etale-cohomology-lemma-stalk-exact" ], "ref_ids": [ 11873, 6425 ] } ], "ref_ids": [ 12370 ] }, { "id": 11875, "type": "theorem", "label": "spaces-properties-lemma-stalk-pullback", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-stalk-pullback", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item The functor", "$f_{small}^{-1} :", "\\textit{Ab}(Y_\\etale)", "\\to", "\\textit{Ab}(X_\\etale)$", "is exact.", "\\item The functor", "$f_{small}^{-1} :", "\\Sh(Y_\\etale)", "\\to", "\\Sh(X_\\etale)$", "is exact, i.e., it commutes with finite limits and colimits, see", "Categories, Definition \\ref{categories-definition-exact}.", "\\item For any \\'etale morphism $V \\to Y$ of algebraic spaces", "we have $f_{small}^{-1}h_V = h_{X \\times_Y V}$.", "\\item Let $\\overline{x} \\to X$ be a geometric point.", "Let $\\mathcal{G}$ be a sheaf on $Y_\\etale$.", "Then there is a canonical identification", "$$", "(f_{small}^{-1}\\mathcal{G})_{\\overline{x}} = \\mathcal{G}_{\\overline{y}}.", "$$", "where $\\overline{y} = f \\circ \\overline{x}$.", "\\end{enumerate}" ], "refs": [ "categories-definition-exact" ], "proofs": [ { "contents": [ "Recall that $f_{small}$ is defined via $f_{spaces, small}$ in", "Lemma \\ref{lemma-functoriality-etale-site}.", "Parts (1), (2) and (3) are general consequences of the fact that", "$f_{spaces, \\etale} :", "X_{spaces, \\etale}", "\\to", "Y_{spaces, \\etale}$", "is a morphism of sites, see", "Sites, Definition \\ref{sites-definition-morphism-sites}", "for (2),", "Modules on Sites, Lemma \\ref{sites-modules-lemma-flat-pullback-exact}", "for (1), and", "Sites, Lemma \\ref{sites-lemma-pullback-representable-sheaf}", "for (3).", "\\medskip\\noindent", "Proof of (4). This statement is a special case of", "Sites, Lemma \\ref{sites-lemma-point-morphism-sites}", "via", "Lemma \\ref{lemma-stalk-gives-point}.", "We also provide a direct proof. Note that by", "Lemma \\ref{lemma-stalk-exact}.", "taking stalks commutes with sheafification.", "Let $\\mathcal{G}'$ be the sheaf on $Y_{spaces, \\etale}$ whose restriction", "to $Y_\\etale$ is $\\mathcal{G}$.", "Recall that $f_{spaces, \\etale}^{-1}\\mathcal{G}'$ is the sheaf", "associated to the presheaf", "$$", "U \\longrightarrow \\colim_{U \\to X \\times_Y V} \\mathcal{G}'(V),", "$$", "see", "Sites, Sections \\ref{sites-section-continuous-functors} and", "\\ref{sites-section-functoriality-PSh}.", "Thus we have", "\\begin{align*}", "(f_{spaces, \\etale}^{-1}\\mathcal{G}')_{\\overline{x}}", "& =", "\\colim_{(U, \\overline{u})} f_{spaces, \\etale}^{-1}\\mathcal{G}'(U)", "\\\\", "& = \\colim_{(U, \\overline{u})}", "\\colim_{a : U \\to X \\times_Y V} \\mathcal{G}'(V) \\\\", "& = \\colim_{(V, \\overline{v})} \\mathcal{G}'(V) \\\\", "& = \\mathcal{G}'_{\\overline{y}}", "\\end{align*}", "in the third equality the pair $(U, \\overline{u})$ and the map", "$a : U \\to X \\times_Y V$ corresponds to the pair $(V, a \\circ \\overline{u})$.", "Since the stalk of $\\mathcal{G}'$", "(resp.\\ $f_{spaces, \\etale}^{-1}\\mathcal{G}'$)", "agrees with the stalk of $\\mathcal{G}$ (resp.\\ $f_{small}^{-1}\\mathcal{G}$),", "see", "Equation (\\ref{equation-stalk-spaces-etale})", "the result follows." ], "refs": [ "spaces-properties-lemma-functoriality-etale-site", "sites-definition-morphism-sites", "sites-modules-lemma-flat-pullback-exact", "sites-lemma-pullback-representable-sheaf", "sites-lemma-point-morphism-sites", "spaces-properties-lemma-stalk-gives-point", "spaces-properties-lemma-stalk-exact" ], "ref_ids": [ 11864, 8665, 14223, 8524, 8603, 11873, 11874 ] } ], "ref_ids": [ 12370 ] }, { "id": 11876, "type": "theorem", "label": "spaces-properties-lemma-points-small-etale-site", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-points-small-etale-site", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $p : \\Sh(pt) \\to \\Sh(X_\\etale)$", "be a point of the small \\'etale topos of $X$.", "Then there exists a geometric point $\\overline{x}$ of $X$", "such that the stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_p$", "is isomorphic to the stalk functor", "$\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$." ], "refs": [], "proofs": [ { "contents": [ "By", "Sites, Lemma \\ref{sites-lemma-point-site-topos}", "there is a one to one correspondence between points of the site and points", "of the associated topos. Hence we may assume that $p$ is given by", "a functor $u : X_\\etale \\to \\textit{Sets}$ which defines a point", "of the site $X_\\etale$.", "Let $U \\in \\Ob(X_\\etale)$ be an object whose structure morphism", "$j : U \\to X$ is surjective. Note that $h_U$ is a sheaf", "which surjects onto the final sheaf. Since taking stalks is exact", "we see that $(h_U)_p = u(U)$ is not empty (use", "Sites, Lemma \\ref{sites-lemma-points-recover}).", "Pick $x \\in u(U)$. By", "Sites, Lemma \\ref{sites-lemma-point-localize}", "we obtain a point $q : \\Sh(pt) \\to \\Sh(U_\\etale)$", "such that $p = j_{small} \\circ q$, so that", "$\\mathcal{F}_p = (\\mathcal{F}|_U)_q$ functorially.", "By", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-points-small-etale-site}", "there is a geometric point $\\overline{u}$ of $U$ and a functorial", "isomorphism $\\mathcal{G}_q = \\mathcal{G}_{\\overline{u}}$", "for $\\mathcal{G} \\in \\Sh(U_\\etale)$. Set", "$\\overline{x} = j \\circ \\overline{u}$. Then we see that", "$\\mathcal{F}_{\\overline{x}} \\cong (\\mathcal{F}|_U)_{\\overline{u}}$", "functorially in $\\mathcal{F}$ on $X_\\etale$ by", "Lemma \\ref{lemma-stalk-pullback}", "and we win." ], "refs": [ "sites-lemma-point-site-topos", "sites-lemma-points-recover", "sites-lemma-point-localize", "etale-cohomology-lemma-points-small-etale-site", "spaces-properties-lemma-stalk-pullback" ], "ref_ids": [ 8596, 8593, 8605, 6426, 11875 ] } ], "ref_ids": [] }, { "id": 11877, "type": "theorem", "label": "spaces-properties-lemma-support-subsheaf-final", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-support-subsheaf-final", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a subsheaf of the final object of the \\'etale", "topos of $X$ (see", "Sites, Example \\ref{sites-example-singleton-sheaf}).", "Then there exists a unique open", "$W \\subset X$ such that $\\mathcal{F} = h_W$." ], "refs": [], "proofs": [ { "contents": [ "The condition means that $\\mathcal{F}(U)$ is a singleton or", "empty for all $\\varphi : U \\to X$ in $\\Ob(X_{spaces, \\etale})$.", "In particular local sections always glue. If", "$\\mathcal{F}(U) \\not = \\emptyset$, then", "$\\mathcal{F}(\\varphi(U)) \\not = \\emptyset$ because", "$\\varphi(U) \\subset X$ is an open subspace", "(Lemma \\ref{lemma-etale-open})", "and", "$\\{\\varphi : U \\to \\varphi(U)\\}$ is a covering in $X_{spaces, \\etale}$.", "Take", "$W = \\bigcup_{\\varphi : U \\to S, \\mathcal{F}(U) \\not = \\emptyset} \\varphi(U)$", "to conclude." ], "refs": [ "spaces-properties-lemma-etale-open" ], "ref_ids": [ 11860 ] } ], "ref_ids": [] }, { "id": 11878, "type": "theorem", "label": "spaces-properties-lemma-zero-over-image", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-zero-over-image", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be an abelian sheaf on $X_{spaces, \\etale}$.", "Let $\\sigma \\in \\mathcal{F}(U)$ be a local section.", "There exists an open subspace $W \\subset U$ such that", "\\begin{enumerate}", "\\item $W \\subset U$ is the largest open subspace of $U$ such", "that $\\sigma|_W = 0$,", "\\item for every $\\varphi : V \\to U$ in $X_\\etale$ we have", "$$", "\\sigma|_V = 0 \\Leftrightarrow \\varphi(V) \\subset W,", "$$", "\\item for every geometric point $\\overline{u}$ of $U$ we have", "$$", "(U, \\overline{u}, \\sigma) = 0\\text{ in }\\mathcal{F}_{\\overline{s}}", "\\Leftrightarrow", "\\overline{u} \\in W", "$$", "where $\\overline{s} = (U \\to S) \\circ \\overline{u}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathcal{F}$ is a sheaf in the \\'etale topology the restriction of", "$\\mathcal{F}$ to $U_{Zar}$ is a sheaf on $U$ in the Zariski topology.", "Hence there exists a Zariski open $W$ having property (1), see", "Modules, Lemma \\ref{modules-lemma-support-section-closed}. Let", "$\\varphi : V \\to U$ be an arrow of $X_\\etale$. Note that", "$\\varphi(V) \\subset U$ is an open subspace", "(Lemma \\ref{lemma-etale-open})", "and that $\\{V \\to \\varphi(V)\\}$ is an \\'etale covering. Hence if", "$\\sigma|_V = 0$, then by the sheaf condition for $\\mathcal{F}$ we", "see that $\\sigma|_{\\varphi(V)} = 0$. This proves (2).", "To prove (3) we have to show that if $(U, \\overline{u}, \\sigma)$", "defines the zero element of $\\mathcal{F}_{\\overline{s}}$, then", "$\\overline{u} \\in W$. This is true because the assumption means", "there exists a morphism of \\'etale neighbourhoods", "$(V, \\overline{v}) \\to (U, \\overline{u})$ such that", "$\\sigma|_V = 0$. Hence by (2) we see that $V \\to U$ maps into $W$, and", "hence $\\overline{u} \\in W$." ], "refs": [ "spaces-properties-lemma-etale-open" ], "ref_ids": [ 11860 ] } ], "ref_ids": [] }, { "id": 11879, "type": "theorem", "label": "spaces-properties-lemma-support-section-closed", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-support-section-closed", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$.", "Let $U \\in \\Ob(X_\\etale)$ and $\\sigma \\in \\mathcal{F}(U)$.", "\\begin{enumerate}", "\\item The support of $\\sigma$ is closed in $|X|$.", "\\item The support of $\\sigma + \\sigma'$ is contained in the union of", "the supports of $\\sigma, \\sigma' \\in \\mathcal{F}(X)$.", "\\item If $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a map of", "abelian sheaves on $X_\\etale$, then the support of $\\varphi(\\sigma)$ is", "contained in the support of $\\sigma \\in \\mathcal{F}(U)$.", "\\item The support of $\\mathcal{F}$ is the union of the images of the", "supports of all local sections of $\\mathcal{F}$.", "\\item If $\\mathcal{F} \\to \\mathcal{G}$ is surjective then the support", "of $\\mathcal{G}$ is a subset of the support of $\\mathcal{F}$.", "\\item If $\\mathcal{F} \\to \\mathcal{G}$ is injective then the support", "of $\\mathcal{F}$ is a subset of the support of $\\mathcal{G}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) holds by definition.", "Parts (2) and (3) hold because they holds for the restriction of", "$\\mathcal{F}$ and $\\mathcal{G}$ to $U_{Zar}$, see", "Modules, Lemma \\ref{modules-lemma-support-section-closed}.", "Part (4) is a direct consequence of", "Lemma \\ref{lemma-zero-over-image} part (3).", "Parts (5) and (6) follow from the other parts." ], "refs": [ "spaces-properties-lemma-zero-over-image" ], "ref_ids": [ 11878 ] } ], "ref_ids": [] }, { "id": 11880, "type": "theorem", "label": "spaces-properties-lemma-support-sheaf-rings-closed", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-support-sheaf-rings-closed", "contents": [ "The support of a sheaf of rings on the small \\'etale site of an", "algebraic space is closed." ], "refs": [], "proofs": [ { "contents": [ "This is true because (according to our conventions)", "a ring is $0$ if and only if", "$1 = 0$, and hence the support of a sheaf of rings", "is the support of the unit section." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11881, "type": "theorem", "label": "spaces-properties-lemma-sheaf-condition-holds", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-sheaf-condition-holds", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The rule $U \\mapsto \\Gamma(U, \\mathcal{O}_U)$ defines", "a sheaf of rings on $X_\\etale$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definition of a covering and", "Descent, Lemma \\ref{descent-lemma-sheaf-condition-holds}." ], "refs": [ "descent-lemma-sheaf-condition-holds" ], "ref_ids": [ 14621 ] } ], "ref_ids": [] }, { "id": 11882, "type": "theorem", "label": "spaces-properties-lemma-morphism-ringed-topoi", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-morphism-ringed-topoi", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Then there is a canonical map", "$f^\\sharp : f_{small}^{-1}\\mathcal{O}_Y \\to \\mathcal{O}_X$ such that", "$$", "(f_{small}, f^\\sharp) :", "(\\Sh(X_\\etale), \\mathcal{O}_X)", "\\longrightarrow", "(\\Sh(Y_\\etale), \\mathcal{O}_Y)", "$$", "is a morphism of ringed topoi. Furthermore,", "\\begin{enumerate}", "\\item The construction $f \\mapsto (f_{small}, f^\\sharp)$ is compatible with", "compositions.", "\\item If $f$ is a morphism of schemes, then $f^\\sharp$ is the map described in", "Descent, Remark \\ref{descent-remark-change-topologies-ringed}.", "\\end{enumerate}" ], "refs": [ "descent-remark-change-topologies-ringed" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-f-map} it suffices to give an $f$-map from", "$\\mathcal{O}_Y$ to $\\mathcal{O}_X$. In other words, for every", "commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_g \\ar[r] & X \\ar[d]^f \\\\", "V \\ar[r] & Y", "}", "$$", "where $U \\in X_\\etale$, $V \\in Y_\\etale$ we have to give a", "map of rings", "$", "(f^\\sharp)_{(U, V, g)} :", "\\Gamma(V, \\mathcal{O}_V)", "\\to", "\\Gamma(U, \\mathcal{O}_U).", "$", "Of course we just take $(f^\\sharp)_{(U, V, g)} = g^\\sharp$.", "It is clear that this is compatible with restriction mappings", "and hence indeed gives an $f$-map.", "We omit checking compatibility with compositions and agreement with the", "construction in", "Descent, Remark \\ref{descent-remark-change-topologies-ringed}." ], "refs": [ "spaces-properties-lemma-f-map", "descent-remark-change-topologies-ringed" ], "ref_ids": [ 11865, 14792 ] } ], "ref_ids": [ 14792 ] }, { "id": 11883, "type": "theorem", "label": "spaces-properties-lemma-reduced-space", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-reduced-space", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is reduced,", "\\item for every $x \\in |X|$ the local ring of $X$ at $x$ is", "reduced (Remark \\ref{remark-list-properties-local-ring-local-etale-topology}).", "\\end{enumerate}", "In this case $\\Gamma(X, \\mathcal{O}_X)$ is a reduced ring and", "if $f \\in \\Gamma(X, \\mathcal{O}_X)$ has $X = V(f)$, then $f = 0$." ], "refs": [ "spaces-properties-remark-list-properties-local-ring-local-etale-topology" ], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows from", "Properties, Lemma \\ref{properties-lemma-characterize-reduced}", "applied to affine schemes \\'etale over $X$. The final statements", "follow the cited lemma and fact that $\\Gamma(X, \\mathcal{O}_X)$ is", "a subring of $\\Gamma(U, \\mathcal{O}_U)$ for some", "reduced scheme $U$ \\'etale over $X$." ], "refs": [ "properties-lemma-characterize-reduced" ], "ref_ids": [ 2945 ] } ], "ref_ids": [ 11951 ] }, { "id": 11884, "type": "theorem", "label": "spaces-properties-lemma-describe-etale-local-ring", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-describe-etale-local-ring", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $\\overline{x}$ be a geometric point of $X$.", "Let $(U, \\overline{u})$ be an \\'etale neighbourhood of $\\overline{x}$", "where $U$ is a scheme. Then we have", "$$", "\\mathcal{O}_{X, \\overline{x}} =", "\\mathcal{O}_{U, \\overline{u}} =", "\\mathcal{O}_{U, u}^{sh}", "$$", "where the left hand side is the stalk of the structure sheaf of $X$,", "and the right hand side is the strict henselization of the local ring", "of $U$ at the point $u$ at which $\\overline{u}$ is centered." ], "refs": [], "proofs": [ { "contents": [ "We know that the structure sheaf $\\mathcal{O}_U$ on", "$U_\\etale$ is the restriction of the structure sheaf of $X$.", "Hence the first equality follows from", "Lemma \\ref{lemma-stalk-pullback} part (4).", "The second equality is explained in", "\\'Etale Cohomology,", "Lemma \\ref{etale-cohomology-lemma-describe-etale-local-ring}." ], "refs": [ "spaces-properties-lemma-stalk-pullback", "etale-cohomology-lemma-describe-etale-local-ring" ], "ref_ids": [ 11875, 6433 ] } ], "ref_ids": [] }, { "id": 11885, "type": "theorem", "label": "spaces-properties-lemma-etale-site-locally-ringed", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-etale-site-locally-ringed", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "The small \\'etale site $X_\\etale$ endowed with its", "structure sheaf $\\mathcal{O}_X$ is a locally ringed site, see", "Modules on Sites, Definition \\ref{sites-modules-definition-locally-ringed}." ], "refs": [ "sites-modules-definition-locally-ringed" ], "proofs": [ { "contents": [ "This follows because the stalks", "$\\mathcal{O}_{X, \\overline{x}}$ are", "local, and because $S_\\etale$ has enough points, see", "Lemmas \\ref{lemma-describe-etale-local-ring} and", "Theorem \\ref{theorem-exactness-stalks}.", "See", "Modules on Sites, Lemma \\ref{sites-modules-lemma-locally-ringed-stalk} and", "\\ref{sites-modules-lemma-ringed-stalk-not-zero}", "for the fact that this implies the small \\'etale site is locally ringed." ], "refs": [ "spaces-properties-lemma-describe-etale-local-ring", "spaces-properties-theorem-exactness-stalks", "sites-modules-lemma-locally-ringed-stalk", "sites-modules-lemma-ringed-stalk-not-zero" ], "ref_ids": [ 11884, 11813, 14254, 14255 ] } ], "ref_ids": [ 14302 ] }, { "id": 11886, "type": "theorem", "label": "spaces-properties-lemma-dimension-local-ring", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-dimension-local-ring", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$ be a point. Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$.", "The following are equivalent", "\\begin{enumerate}", "\\item the dimension of the local ring of $X$ at $x$", "(Definition \\ref{definition-dimension-local-ring}) is $d$,", "\\item $\\dim(\\mathcal{O}_{X, \\overline{x}}) = d$ for some geometric", "point $\\overline{x}$ lying over $x$, and", "\\item $\\dim(\\mathcal{O}_{X, \\overline{x}}) = d$ for any geometric", "point $\\overline{x}$ lying over $x$.", "\\end{enumerate}" ], "refs": [ "spaces-properties-definition-dimension-local-ring" ], "proofs": [ { "contents": [ "The equivalence of (2) and (3) follows from the fact that the", "isomorphism type of $\\mathcal{O}_{X, \\overline{x}}$ only depends", "on $x \\in |X|$, see Remark \\ref{remark-map-stalks}.", "Using Lemma \\ref{lemma-describe-etale-local-ring}", "the equivalence of (1) and (2)$+$(3) comes down to the", "following statement: Given any local ring $R$ we have", "$\\dim(R) = \\dim(R^{sh})$. This is", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-dimension}." ], "refs": [ "spaces-properties-remark-map-stalks", "spaces-properties-lemma-describe-etale-local-ring", "more-algebra-lemma-henselization-dimension" ], "ref_ids": [ 11955, 11884, 10061 ] } ], "ref_ids": [ 11931 ] }, { "id": 11887, "type": "theorem", "label": "spaces-properties-lemma-dimension-decent-invariant-under-etale", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-dimension-decent-invariant-under-etale", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an \\'etale morphism", "of algebraic spaces over $S$. Let $x \\in X$. Then", "(1) $\\dim_x(X) = \\dim_{f(x)}(Y)$ and (2) the dimension of", "the local ring of $X$ at $x$ equals the dimension of", "the local ring of $Y$ at $f(x)$. If $f$ is surjective, then", "(3) $\\dim(X) = \\dim(Y)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a point $u \\in U$ and an \\'etale morphism", "$U \\to X$ which maps $u$ to $x$. Then the composition $U \\to Y$", "is also \\'etale and maps $u$ to $f(x)$. Thus the statements (1) and (2)", "follow as the relevant integers are defined in terms of the behaviour", "of the scheme $U$ at $u$. See", "Definition \\ref{definition-dimension-at-point} for (1). Part (3) is", "an immediate consequence of (1), see Definition \\ref{definition-dimension}." ], "refs": [ "spaces-properties-definition-dimension-at-point", "spaces-properties-definition-dimension" ], "ref_ids": [ 11929, 11930 ] } ], "ref_ids": [] }, { "id": 11888, "type": "theorem", "label": "spaces-properties-lemma-reduced-local-ring", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-reduced-local-ring", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$ be a point. The following are equivalent", "\\begin{enumerate}", "\\item the local ring of $X$ at $x$ is reduced", "(Remark \\ref{remark-list-properties-local-ring-local-etale-topology}),", "\\item $\\mathcal{O}_{X, \\overline{x}}$ is reduced for some geometric", "point $\\overline{x}$ lying over $x$, and", "\\item $\\mathcal{O}_{X, \\overline{x}}$ is reduced for any geometric", "point $\\overline{x}$ lying over $x$.", "\\end{enumerate}" ], "refs": [ "spaces-properties-remark-list-properties-local-ring-local-etale-topology" ], "proofs": [ { "contents": [ "The equivalence of (2) and (3) follows from the fact that the", "isomorphism type of $\\mathcal{O}_{X, \\overline{x}}$ only depends", "on $x \\in |X|$, see Remark \\ref{remark-map-stalks}.", "Using Lemma \\ref{lemma-describe-etale-local-ring}", "the equivalence of (1) and (2)$+$(3) comes down to the", "following statement: a local ring is reduced if and only if", "its strict henselization is reduced. This is", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-reduced}." ], "refs": [ "spaces-properties-remark-map-stalks", "spaces-properties-lemma-describe-etale-local-ring", "more-algebra-lemma-henselization-reduced" ], "ref_ids": [ 11955, 11884, 10058 ] } ], "ref_ids": [ 11951 ] }, { "id": 11889, "type": "theorem", "label": "spaces-properties-lemma-irreducible-local-ring", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-irreducible-local-ring", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$ be a point.", "The following are equivalent", "\\begin{enumerate}", "\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and", "$u \\in U$ with $a(u) = x$ the local ring $\\mathcal{O}_{U, u}$ has a", "unique minimal prime,", "\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and", "$u \\in U$ with $a(u) = x$ there is a unique irreducible component of $U$", "through $u$,", "\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and", "$u \\in U$ with $a(u) = x$ the local ring $\\mathcal{O}_{U, u}$", "is unibranch,", "\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and", "$u \\in U$ with $a(u) = x$ the local ring $\\mathcal{O}_{U, u}$", "is geometrically unibranch,", "\\item $\\mathcal{O}_{X, \\overline{x}}$ has a unique minimal prime", "for any geometric point $\\overline{x}$ lying over $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows from the fact that irreducible", "components of $U$ passing through $u$ are in $1$-$1$ correspondence with", "minimal primes of the local ring of $U$ at $u$. Let $a : U \\to X$ and", "$u \\in U$ be as in (1). Then $\\mathcal{O}_{X, \\overline{x}}$ is", "the strict henselization of $\\mathcal{O}_{U, u}$ by", "Lemma \\ref{lemma-describe-etale-local-ring}.", "In particular (4) and (5) are equivalent by", "More on Algebra, Lemma \\ref{more-algebra-lemma-geometrically-unibranch}.", "The equivalence of (2), (3), and (4) follows from", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-nr-branches}." ], "refs": [ "spaces-properties-lemma-describe-etale-local-ring", "more-algebra-lemma-geometrically-unibranch", "more-morphisms-lemma-nr-branches" ], "ref_ids": [ 11884, 10468, 13872 ] } ], "ref_ids": [] }, { "id": 11890, "type": "theorem", "label": "spaces-properties-lemma-nr-branches-local-ring", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-nr-branches-local-ring", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$ be a point. Let $n \\in \\{1, 2, \\ldots\\}$ be an integer.", "The following are equivalent", "\\begin{enumerate}", "\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and", "$u \\in U$ with $a(u) = x$ the number of minimal primes", "of the local ring $\\mathcal{O}_{U, u}$ is $\\leq n$", "and for at least one choice of $U, a, u$ it is $n$,", "\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and", "$u \\in U$ with $a(u) = x$ the number irreducible components of", "$U$ passing through $u$ is $\\leq n$ and for at least one choice", "of $U, a, u$ it is $n$,", "\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and $u \\in U$", "with $a(u) = x$ the number of branches of $U$ at $u$ is $\\leq n$", "and for at least one choice of $U, a, u$ it is $n$,", "\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and $u \\in U$", "with $a(u) = x$ the number of geometric branches of $U$ at $u$ is $n$, and", "\\item the number of minimal prime ideals of", "$\\mathcal{O}_{X, \\overline{x}}$ is $n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows from the fact that irreducible", "components of $U$ passing through $u$ are in $1$-$1$ correspondence with", "minimal primes of the local ring of $U$ at $u$. Let $a : U \\to X$ and", "$u \\in U$ be as in (1). Then $\\mathcal{O}_{X, \\overline{x}}$ is the", "strict henselization of $\\mathcal{O}_{U, u}$ by", "Lemma \\ref{lemma-describe-etale-local-ring}. Recall that", "the (geometric) number of branches of $U$ at $u$ is the number", "of minimal prime ideals of the (strict) henselization of $\\mathcal{O}_{U, u}$.", "In particular (4) and (5) are equivalent.", "The equivalence of (2), (3), and (4) follows from", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-nr-branches}." ], "refs": [ "spaces-properties-lemma-describe-etale-local-ring", "more-morphisms-lemma-nr-branches" ], "ref_ids": [ 11884, 13872 ] } ], "ref_ids": [] }, { "id": 11891, "type": "theorem", "label": "spaces-properties-lemma-Noetherian-topology", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-Noetherian-topology", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item If $X$ is locally Noetherian then $|X|$ is a locally Noetherian", "topological space.", "\\item If $X$ is quasi-compact and locally Noetherian, then $|X|$", "is a Noetherian topological space.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $X$ is locally Noetherian.", "Choose a scheme $U$ and a surjective \\'etale morphism", "$U \\to X$. As $X$ is locally Noetherian we see that $U$ is locally", "Noetherian. By", "Properties, Lemma \\ref{properties-lemma-Noetherian-topology}", "this means that $|U|$ is a locally Noetherian topological space.", "Since $|U| \\to |X|$ is open and surjective we conclude that", "$|X|$ is locally Noetherian by", "Topology, Lemma \\ref{topology-lemma-image-Noetherian}.", "This proves (1). If $X$ is quasi-compact and locally Noetherian,", "then $|X|$ is quasi-compact and locally Noetherian. Hence $|X|$", "is Noetherian by", "Topology,", "Lemma \\ref{topology-lemma-quasi-compact-locally-Noetherian-Noetherian}." ], "refs": [ "properties-lemma-Noetherian-topology", "topology-lemma-image-Noetherian", "topology-lemma-quasi-compact-locally-Noetherian-Noetherian" ], "ref_ids": [ 2954, 8221, 8240 ] } ], "ref_ids": [] }, { "id": 11892, "type": "theorem", "label": "spaces-properties-lemma-Noetherian-sober", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-Noetherian-sober", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "If $X$ is Noetherian, then $|X|$ is a sober Noetherian topological space." ], "refs": [], "proofs": [ { "contents": [ "A quasi-separated algebraic space has an underlying sober topological", "space, see", "Lemma \\ref{lemma-quasi-separated-sober}.", "It is Noetherian by", "Lemma \\ref{lemma-Noetherian-topology}." ], "refs": [ "spaces-properties-lemma-quasi-separated-sober", "spaces-properties-lemma-Noetherian-topology" ], "ref_ids": [ 11852, 11891 ] } ], "ref_ids": [] }, { "id": 11893, "type": "theorem", "label": "spaces-properties-lemma-Noetherian-local-ring-Noetherian", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-Noetherian-local-ring-Noetherian", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $\\overline{x}$ be a geometric point of $X$. Then", "$\\mathcal{O}_{X, \\overline{x}}$ is a Noetherian local ring." ], "refs": [], "proofs": [ { "contents": [ "Choose an \\'etale neighbourhood $(U, \\overline{u})$ of $\\overline{x}$", "where $U$ is a scheme. Then $\\mathcal{O}_{X, \\overline{x}}$ is the", "strict henselization of the local ring of $U$ at $u$, see", "Lemma \\ref{lemma-describe-etale-local-ring}.", "By our definition of Noetherian spaces the scheme $U$ is Noetherian.", "Hence we conclude by", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-noetherian}." ], "refs": [ "spaces-properties-lemma-describe-etale-local-ring", "more-algebra-lemma-henselization-noetherian" ], "ref_ids": [ 11884, 10057 ] } ], "ref_ids": [] }, { "id": 11894, "type": "theorem", "label": "spaces-properties-lemma-regular", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-regular", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a locally Noetherian algebraic space over $S$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is regular, and", "\\item every \\'etale local ring $\\mathcal{O}_{X, \\overline{x}}$ is", "regular.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $U \\to X$ be a surjective \\'etale morphism.", "By assumption $U$ is locally Noetherian. Moreover, every \\'etale local", "ring $\\mathcal{O}_{X, \\overline{x}}$ is the strict henselization of", "a local ring on $U$ and conversely, see", "Lemma \\ref{lemma-describe-etale-local-ring}.", "Thus by", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-regular}", "we see that (2) is equivalent to every local ring of $U$ being", "regular, i.e., $U$ being a regular scheme (see", "Properties, Lemma \\ref{properties-lemma-characterize-regular}).", "This equivalent to (1) by", "Definition \\ref{definition-type-property}." ], "refs": [ "spaces-properties-lemma-describe-etale-local-ring", "more-algebra-lemma-henselization-regular", "properties-lemma-characterize-regular", "spaces-properties-definition-type-property" ], "ref_ids": [ 11884, 10064, 2975, 11926 ] } ], "ref_ids": [] }, { "id": 11895, "type": "theorem", "label": "spaces-properties-lemma-regular-at-x", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-regular-at-x", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$ be a point. The following are equivalent", "\\begin{enumerate}", "\\item $X$ is regular at $x$, and", "\\item the \\'etale local ring $\\mathcal{O}_{X, \\overline{x}}$ is", "regular for any (equivalently some) geometric point $\\overline{x}$", "lying over $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme, $u \\in U$ a point, and let $a : U \\to X$ be an", "\\'etale morphism mapping $u$ to $x$. For any geometric point", "$\\overline{x}$ of $X$ lying over $x$, the \\'etale local", "ring $\\mathcal{O}_{X, \\overline{x}}$ is the strict henselization of", "a local ring on $U$ at $u$, see", "Lemma \\ref{lemma-describe-etale-local-ring}.", "Thus we conclude by", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-regular}." ], "refs": [ "spaces-properties-lemma-describe-etale-local-ring", "more-algebra-lemma-henselization-regular" ], "ref_ids": [ 11884, 10064 ] } ], "ref_ids": [] }, { "id": 11896, "type": "theorem", "label": "spaces-properties-lemma-regular-normal", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-regular-normal", "contents": [ "A regular algebraic space is normal." ], "refs": [], "proofs": [ { "contents": [ "This follows from the definitions and the case of schemes", "See Properties, Lemma \\ref{properties-lemma-regular-normal}." ], "refs": [ "properties-lemma-regular-normal" ], "ref_ids": [ 2977 ] } ], "ref_ids": [] }, { "id": 11897, "type": "theorem", "label": "spaces-properties-lemma-etale-exact-pullback", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-etale-exact-pullback", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be an \\'etale morphism of algebraic spaces over $S$.", "Then $f^{-1}\\mathcal{O}_Y = \\mathcal{O}_X$, and", "$f^*\\mathcal{G} = f_{small}^{-1}\\mathcal{G}$ for any sheaf of", "$\\mathcal{O}_Y$-modules $\\mathcal{G}$. In particular,", "$f^* : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_Y)$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "By the description of inverse image in Lemma \\ref{lemma-etale-morphism-topoi}", "and the definition of the structure sheaves it is clear that", "$f_{small}^{-1}\\mathcal{O}_Y = \\mathcal{O}_X$. Since the pullback", "$$", "f^*\\mathcal{G} =", "f_{small}^{-1}\\mathcal{G} \\otimes_{f_{small}^{-1}\\mathcal{O}_Y}", "\\mathcal{O}_X", "$$", "by definition we conclude that $f^*\\mathcal{G} = f_{small}^{-1}\\mathcal{G}$.", "The exactness is clear because $f_{small}^{-1}$ is exact, as $f_{small}$", "is a morphism of topoi." ], "refs": [ "spaces-properties-lemma-etale-morphism-topoi" ], "ref_ids": [ 11866 ] } ], "ref_ids": [] }, { "id": 11898, "type": "theorem", "label": "spaces-properties-lemma-pushforward-etale-base-change-modules", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-pushforward-etale-base-change-modules", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X' \\ar[r] \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "be a cartesian square of algebraic spaces over $S$. Let", "$\\mathcal{F} \\in \\textit{Mod}(\\mathcal{O}_X)$. If $g$ is \\'etale, then", "$f'_*(\\mathcal{F}|_{X'}) = (f_*\\mathcal{F})|_{Y'}$\\footnote{Also", "$(f')^*(\\mathcal{G}|_{Y'}) = (f^*\\mathcal{G})|_{X'}$", "by commutativity of the diagram and (\\ref{equation-restrict-modules})} and", "$R^if'_*(\\mathcal{F}|_{X'}) = (R^if_*\\mathcal{F})|_{Y'}$ in", "$\\textit{Mod}(\\mathcal{O}_{Y'})$." ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of", "Lemma \\ref{lemma-pushforward-etale-base-change}", "in the case of modules." ], "refs": [ "spaces-properties-lemma-pushforward-etale-base-change" ], "ref_ids": [ 11867 ] } ], "ref_ids": [] }, { "id": 11899, "type": "theorem", "label": "spaces-properties-lemma-characterize-module-small-etale", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-characterize-module-small-etale", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "A sheaf $\\mathcal{F}$ of $\\mathcal{O}_X$-modules is given by the following", "data:", "\\begin{enumerate}", "\\item for every $U \\in \\Ob(X_\\etale)$ a sheaf", "$\\mathcal{F}_U$ of $\\mathcal{O}_U$-modules on $U_\\etale$,", "\\item for every $f : U' \\to U$ in $X_\\etale$ an isomorphism", "$c_f : f_{small}^*\\mathcal{F}_U \\to \\mathcal{F}_{U'}$.", "\\end{enumerate}", "These data are subject to the condition that given any $f : U' \\to U$", "and $g : U'' \\to U'$ in $X_\\etale$ the composition", "$g_{small}^{-1}c_f \\circ c_g$ is equal to $c_{f \\circ g}$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-etale-exact-pullback}", "and \\ref{lemma-characterize-sheaf-small-etale}, and use the fact that", "any morphism between objects of $X_\\etale$ is an \\'etale morphism", "of schemes." ], "refs": [ "spaces-properties-lemma-etale-exact-pullback", "spaces-properties-lemma-characterize-sheaf-small-etale" ], "ref_ids": [ 11897, 11868 ] } ], "ref_ids": [] }, { "id": 11900, "type": "theorem", "label": "spaces-properties-lemma-relocalize-morphism", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-relocalize-morphism", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "U \\ar[d]_p \\ar[r]_g & V \\ar[d]^q \\\\", "X \\ar[r]^f & Y", "}", "$$", "be a commutative diagram of algebraic spaces over $S$ with $p$ and $q$ \\'etale.", "Via the identifications", "(\\ref{equation-localize-ringed}) for $U \\to X$ and $V \\to Y$", "the morphism of ringed topoi", "$$", "(g_{spaces, \\etale}, g^\\sharp) :", "(\\Sh(U_{spaces, \\etale}), \\mathcal{O}_U)", "\\longrightarrow", "(\\Sh(V_{spaces, \\etale}), \\mathcal{O}_V)", "$$", "is $2$-isomorphic to the morphism $(f_{spaces, \\etale, c}, f_c^\\sharp)$", "constructed in", "Modules on Sites,", "Lemma \\ref{sites-modules-lemma-relocalize-morphism-ringed-sites}", "starting with the morphism of ringed sites", "$(f_{spaces, \\etale}, f^\\sharp)$ and", "the map $c : U \\to V \\times_Y X$ corresponding to $g$." ], "refs": [ "sites-modules-lemma-relocalize-morphism-ringed-sites" ], "proofs": [ { "contents": [ "The morphism $(f_{spaces, \\etale, c}, f_c^\\sharp)$ is defined as a", "composition $f' \\circ j$", "of a localization and a base change map. Similarly $g$ is a composition", "$U \\to V \\times_Y X \\to V$. Hence it suffices to prove", "the lemma in the following two cases: (1) $f = \\text{id}$, and", "(2) $U = X \\times_Y V$. In case (1) the morphism $g : U \\to V$ is", "\\'etale, see", "Lemma \\ref{lemma-etale-permanence}.", "Hence $(g_{spaces, \\etale}, g^\\sharp)$ is a localization morphism", "by the discussion surrounding", "Equations (\\ref{equation-localize}) and", "(\\ref{equation-localize-ringed})", "which is exactly the content of the lemma in this case.", "In case (2) the morphism $g_{spaces, \\etale}$", "comes from the morphism of ringed sites given by the functor", "$V_{spaces, \\etale} \\to U_{spaces, \\etale}$,", "$V'/V \\mapsto V' \\times_V U/U$", "which is also what the morphism $f'$ is defined by, see", "Sites, Lemma \\ref{sites-lemma-localize-morphism}.", "We omit the verification that $(f')^\\sharp = g^\\sharp$", "in this case (both are the restriction of $f^\\sharp$", "to $U_{spaces, \\etale}$)." ], "refs": [ "spaces-properties-lemma-etale-permanence", "sites-lemma-localize-morphism" ], "ref_ids": [ 11859, 8571 ] } ], "ref_ids": [ 14175 ] }, { "id": 11901, "type": "theorem", "label": "spaces-properties-lemma-relocalize-morphism-at-schemes", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-relocalize-morphism-at-schemes", "contents": [ "Same notation and assumptions as in", "Lemma \\ref{lemma-relocalize-morphism}", "except that we also assume $U$ and $V$ are schemes.", "Via the identifications", "(\\ref{equation-localize-at-scheme-ringed})", "for $U \\to X$ and $V \\to Y$ the morphism of ringed topoi", "$$", "(g_{small}, g^\\sharp) :", "(\\Sh(U_\\etale), \\mathcal{O}_U)", "\\longrightarrow", "(\\Sh(V_\\etale), \\mathcal{O}_V)", "$$", "is $2$-isomorphic to the morphism $(f_{small, s}, f_s^\\sharp)$", "constructed in", "Modules on Sites,", "Lemma \\ref{sites-modules-lemma-relocalize-morphism-ringed-topoi}", "starting with $(f_{small}, f^\\sharp)$ and", "the map $s : h_U \\to f_{small}^{-1}h_V$ corresponding to $g$." ], "refs": [ "spaces-properties-lemma-relocalize-morphism", "sites-modules-lemma-relocalize-morphism-ringed-topoi" ], "proofs": [ { "contents": [ "Note that $(g_{small}, g^\\sharp)$ is $2$-isomorphic as a", "morphism of ringed topoi to the morphism of ringed topoi", "associated to the morphism of ringed sites", "$(g_{spaces, \\etale}, g^\\sharp)$. Hence we conclude by", "Lemma \\ref{lemma-relocalize-morphism}", "and", "Modules on Sites,", "Lemma \\ref{sites-modules-lemma-relocalize-morphism-compare}." ], "refs": [ "spaces-properties-lemma-relocalize-morphism", "sites-modules-lemma-relocalize-morphism-compare" ], "ref_ids": [ 11900, 14183 ] } ], "ref_ids": [ 11900, 14182 ] }, { "id": 11902, "type": "theorem", "label": "spaces-properties-lemma-sheaf-gives-space", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-sheaf-gives-space", "contents": [ "Let $S$ be a scheme and let $Y$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a sheaf of sets on $Y_\\etale$.", "Provided a set theoretic condition is satisfied (see proof)", "the functor $X$ associated to $\\mathcal{F}$ above is an algebraic space", "and there is an \\'etale morphism $f : X \\to Y$ of algebraic spaces", "such that $\\mathcal{F} = f_{small, *}*$ where $*$ is the final object", "of the category $\\Sh(X_\\etale)$ (constant sheaf with value a singleton)." ], "refs": [], "proofs": [ { "contents": [ "Let us prove that $X$ is a sheaf for the fppf topology. Namely, suppose", "that $\\{g_i : T_i \\to T\\}$ is a covering of $(\\Sch/S)_{fppf}$ and", "$(y_i, s_i) \\in X(T_i)$ satisfy the glueing condition, i.e., the", "restriction of $(y_i, s_i)$ and $(y_j, s_j)$ to $T_i \\times_T T_j$ agree.", "Then since $Y$ is a sheaf for the fppf topology, we see that", "the $y_i$ give rise to a unique morphism $y : T \\to Y$ such that", "$y_i = y \\circ g_i$. Then we see that", "$y_{i, small}^{-1}\\mathcal{F} = g_{i, small}^{-1}y_{small}^{-1}\\mathcal{F}$.", "Hence the sections $s_i$ glue uniquely to a section of", "$y_{small}^{-1}\\mathcal{F}$ by", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-describe-pullback}.", "\\medskip\\noindent", "The construction that sends $\\mathcal{F} \\in \\Ob(\\Sh(Y_\\etale))$", "to $X \\in \\Ob((\\Sch/S)_{fppf})$ preserves finite limits and", "all colimits since each of the functors $y_{small}^{-1}$ have", "this property. Of course, if $V \\in \\Ob(Y_\\etale)$, then", "the construction sends the representable sheaf $h_V$ on $Y_\\etale$", "to the representable functor represented by $V$.", "\\medskip\\noindent", "By Sites, Lemma \\ref{sites-lemma-sheaf-coequalizer-representable}", "we can find a set $I$, for each $i \\in I$ an object $V_i$ of $Y_\\etale$", "and a surjective map of sheaves", "$$", "\\coprod h_{V_i} \\longrightarrow \\mathcal{F}", "$$", "on $Y_\\etale$. The set theoretic condition we need is that the index", "set $I$ is not too large\\footnote{It suffices if the supremum of the", "cardinalities of the stalks of $\\mathcal{F}$ at geometric points of $Y$", "is bounded by the size of some object of $(\\Sch/S)_{fppf}$.}.", "Then $V = \\coprod V_i$ is an object of $(\\Sch/S)_{fppf}$ and", "therefore an object of $Y_\\etale$ and we have a surjective map", "$h_V \\to \\mathcal{F}$.", "\\medskip\\noindent", "Observe that the product of $h_V$ with itself in $\\Sh(Y_\\etale)$", "is $h_{V \\times_Y V}$. Consider the fibre product", "$$", "h_V \\times_\\mathcal{F} h_V \\subset h_{V \\times_Y V}", "$$", "There is an open subscheme $R$ of $V \\times_Y V$ such that", "$h_V \\times_\\mathcal{F} h_V = h_R$, see", "Lemma \\ref{lemma-support-subsheaf-final} (small detail omitted).", "By the Yoneda lemma we obtain two morphisms $s, t : R \\to V$ in $Y_\\etale$", "and we find a coequalizer diagram", "$$", "\\xymatrix{", "h_R \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "h_V \\ar[r] &", "\\mathcal{F}", "}", "$$", "in $\\Sh(Y_\\etale)$. Of course the morphisms $s, t$ are \\'etale", "and define an \\'etale equivalence relation $(t, s) : R \\to V \\times_S V$.", "\\medskip\\noindent", "By the discussion in the preceding two paragraphs we find a", "coequalizer diagram", "$$", "\\xymatrix{", "R \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "V \\ar[r] &", "X", "}", "$$", "in $(\\Sch/S)_{fppf}$. Thus $X = V/R$ is an algebraic space by", "Spaces, Theorem \\ref{spaces-theorem-presentation}.", "The other statements follow readily from this; details omitted." ], "refs": [ "etale-cohomology-lemma-describe-pullback", "sites-lemma-sheaf-coequalizer-representable", "spaces-properties-lemma-support-subsheaf-final", "spaces-theorem-presentation" ], "ref_ids": [ 6438, 8520, 11877, 8124 ] } ], "ref_ids": [] }, { "id": 11903, "type": "theorem", "label": "spaces-properties-lemma-morphism-locally-ringed", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-morphism-locally-ringed", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The morphism of ringed topoi $(f_{small}, f^\\sharp)$", "associated to $f$ is a morphism of locally ringed topoi, see", "Modules on Sites,", "Definition \\ref{sites-modules-definition-morphism-locally-ringed-topoi}." ], "refs": [ "sites-modules-definition-morphism-locally-ringed-topoi" ], "proofs": [ { "contents": [ "Note that the assertion makes sense since we have seen that", "$(X_\\etale, \\mathcal{O}_{X_\\etale})$ and", "$(Y_\\etale, \\mathcal{O}_{Y_\\etale})$", "are locally ringed sites, see", "Lemma \\ref{lemma-etale-site-locally-ringed}.", "Moreover, we know that $X_\\etale$ has enough points, see", "Theorem \\ref{theorem-exactness-stalks}.", "Hence it suffices to prove that $(f_{small}, f^\\sharp)$", "satisfies condition (3) of", "Modules on Sites,", "Lemma \\ref{sites-modules-lemma-locally-ringed-morphism}.", "To see this take a point $p$ of $X_\\etale$. By", "Lemma \\ref{lemma-points-small-etale-site}", "$p$ corresponds to a geometric point $\\overline{x}$ of $X$.", "By", "Lemma \\ref{lemma-stalk-pullback}", "the point $q = f_{small} \\circ p$ corresponds to the", "geometric point $\\overline{y} = f \\circ \\overline{x}$ of $Y$.", "Hence the assertion we have to prove is that the induced map", "of \\'etale local rings", "$$", "\\mathcal{O}_{Y, \\overline{y}} \\longrightarrow \\mathcal{O}_{X, \\overline{x}}", "$$", "is a local ring map. You can prove this directly, but instead we deduce it", "from the corresponding result for schemes. To do this choose a commutative", "diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "where $U$ and $V$ are schemes, and the vertical arrows are surjective", "\\'etale (see", "Spaces, Lemma \\ref{spaces-lemma-lift-morphism-presentations}).", "Choose a lift $\\overline{u} : \\overline{x} \\to U$ (possible by", "Lemma \\ref{lemma-geometric-lift-to-cover}).", "Set $\\overline{v} = \\psi \\circ \\overline{u}$. We obtain a commutative", "diagram of \\'etale local rings", "$$", "\\xymatrix{", "\\mathcal{O}_{U, \\overline{u}} &", "\\mathcal{O}_{V, \\overline{v}} \\ar[l] \\\\", "\\mathcal{O}_{X, \\overline{x}} \\ar[u] &", "\\mathcal{O}_{Y, \\overline{y}}. \\ar[l] \\ar[u]", "}", "$$", "By", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-morphism-locally-ringed}", "the top horizontal arrow is a local ring map. Finally by", "Lemma \\ref{lemma-describe-etale-local-ring}", "the vertical arrows are isomorphisms. Hence we win." ], "refs": [ "spaces-properties-lemma-etale-site-locally-ringed", "spaces-properties-theorem-exactness-stalks", "sites-modules-lemma-locally-ringed-morphism", "spaces-properties-lemma-points-small-etale-site", "spaces-properties-lemma-stalk-pullback", "spaces-lemma-lift-morphism-presentations", "spaces-properties-lemma-geometric-lift-to-cover", "etale-cohomology-lemma-morphism-locally-ringed", "spaces-properties-lemma-describe-etale-local-ring" ], "ref_ids": [ 11885, 11813, 14258, 11876, 11875, 8159, 11872, 6442, 11884 ] } ], "ref_ids": [ 14304 ] }, { "id": 11904, "type": "theorem", "label": "spaces-properties-lemma-2-morphism", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-2-morphism", "contents": [ "Let $S$ be a scheme.", "Let $X$, $Y$ be algebraic spaces over $S$.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $t$ be a $2$-morphism from $(f_{small}, f^\\sharp)$ to itself, see", "Modules on Sites,", "Definition \\ref{sites-modules-definition-2-morphism-ringed-topoi}.", "Then $t = \\text{id}$." ], "refs": [ "sites-modules-definition-2-morphism-ringed-topoi" ], "proofs": [ { "contents": [ "Let $X'$, resp.\\ $Y'$ be $X$ viewed as an algebraic space over", "$\\Spec(\\mathbf{Z})$, see", "Spaces, Definition \\ref{spaces-definition-base-change}.", "It is clear from the construction that $(X_{small}, \\mathcal{O})$", "is equal to $(X'_{small}, \\mathcal{O})$ and similarly for $Y$.", "Hence we may work with $X'$ and $Y'$. In other words we may", "assume that $S = \\Spec(\\mathbf{Z})$.", "\\medskip\\noindent", "Assume $S = \\Spec(\\mathbf{Z})$, $f : X \\to Y$ and $t$ are as in", "the lemma. This means that $t : f^{-1}_{small} \\to f^{-1}_{small}$", "is a transformation of functors such that the diagram", "$$", "\\xymatrix{", "f_{small}^{-1}\\mathcal{O}_Y", "\\ar[rd]_{f^\\sharp} & &", "f_{small}^{-1}\\mathcal{O}_Y \\ar[ll]^t \\ar[ld]^{f^\\sharp} \\\\", "& \\mathcal{O}_X", "}", "$$", "is commutative. Suppose $V \\to Y$ is \\'etale with $V$ affine.", "Write $V = \\Spec(B)$. Choose generators $b_j \\in B$, $j \\in J$", "for $B$ as a $\\mathbf{Z}$-algebra. Set", "$T = \\Spec(\\mathbf{Z}[\\{x_j\\}_{j \\in J}])$.", "In the following we will use that", "$\\Mor_{\\Sch}(U, T) = \\prod_{j \\in J} \\Gamma(U, \\mathcal{O}_U)$", "for any scheme $U$ without further mention.", "The surjective ring map $\\mathbf{Z}[x_j] \\to B$, $x_j \\mapsto b_j$", "corresponds to a closed immersion $V \\to T$.", "We obtain a monomorphism", "$$", "i : V \\longrightarrow T_Y = T \\times Y", "$$", "of algebraic spaces over $Y$. In terms of sheaves on $Y_\\etale$", "the morphism $i$ induces an injection", "$h_i : h_V \\to \\prod_{j \\in J} \\mathcal{O}_Y$ of sheaves.", "The base change $i' : X \\times_Y V \\to T_X$ of $i$ to $X$", "is a monomorphism too", "(Spaces,", "Lemma \\ref{spaces-lemma-base-change-representable-transformations-property}).", "Hence $i' : X \\times_Y V \\to T_X$ is a monomorphism, which", "in turn means that", "$h_{i'} : h_{X \\times_Y V} \\to \\prod_{j \\in J} \\mathcal{O}_X$", "is an injection of sheaves.", "Via the identification $f_{small}^{-1}h_V = h_{X \\times_Y V}$ of", "Lemma \\ref{lemma-stalk-pullback}", "the map $h_{i'}$ is equal to", "$$", "\\xymatrix{", "f_{small}^{-1}h_V \\ar[r]^-{f^{-1}h_i} &", "\\prod_{j \\in J} f_{small}^{-1}\\mathcal{O}_Y", "\\ar[r]^{\\prod f^\\sharp} &", "\\prod_{j \\in J} \\mathcal{O}_X", "}", "$$", "(verification omitted). This means that the map", "$t : f_{small}^{-1}h_V \\to f_{small}^{-1}h_V$", "fits into the commutative diagram", "$$", "\\xymatrix{", "f_{small}^{-1}h_V \\ar[r]^-{f^{-1}h_i} \\ar[d]^t &", "\\prod_{j \\in J} f_{small}^{-1}\\mathcal{O}_Y", "\\ar[r]^-{\\prod f^\\sharp} \\ar[d]^{\\prod t} &", "\\prod_{j \\in J} \\mathcal{O}_X \\ar[d]^{\\text{id}}\\\\", "f_{small}^{-1}h_V \\ar[r]^-{f^{-1}h_i} &", "\\prod_{j \\in J} f_{small}^{-1}\\mathcal{O}_Y", "\\ar[r]^-{\\prod f^\\sharp} &", "\\prod_{j \\in J} \\mathcal{O}_X", "}", "$$", "The commutativity of the right square holds by our assumption on $t$", "explained above.", "Since the composition of the horizontal arrows is injective", "by the discussion above we conclude that the left vertical arrow", "is the identity map as well. Any sheaf of sets on", "$Y_\\etale$ admits a surjection from a (huge) coproduct of sheaves", "of the form $h_V$ with $V$ affine (combine", "Lemma \\ref{lemma-alternative}", "with", "Sites, Lemma \\ref{sites-lemma-sheaf-coequalizer-representable}).", "Thus we conclude that $t : f_{small}^{-1} \\to f_{small}^{-1}$", "is the identity transformation as desired." ], "refs": [ "spaces-definition-base-change", "spaces-lemma-base-change-representable-transformations-property", "spaces-properties-lemma-stalk-pullback", "spaces-properties-lemma-alternative", "sites-lemma-sheaf-coequalizer-representable" ], "ref_ids": [ 8183, 8133, 11875, 11863, 8520 ] } ], "ref_ids": [ 14281 ] }, { "id": 11905, "type": "theorem", "label": "spaces-properties-lemma-faithful", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-faithful", "contents": [ "Let $S$ be a scheme.", "Let $X$, $Y$ be algebraic spaces over $S$.", "Any two morphisms $a, b : X \\to Y$ of algebraic spaces over $S$", "for which there exists a $2$-isomorphism", "$(a_{small}, a^\\sharp) \\cong (b_{small}, b^\\sharp)$", "in the $2$-category of ringed topoi are equal." ], "refs": [], "proofs": [ { "contents": [ "Let $t : a_{small}^{-1} \\to b_{small}^{-1}$ be the $2$-isomorphism.", "We may equivalently think of $t$ as a transformation", "$t : a_{spaces, \\etale}^{-1} \\to b_{spaces, \\etale}^{-1}$", "since there is not difference between sheaves on $X_\\etale$", "and sheaves on $X_{spaces, \\etale}$.", "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_p \\ar[r]_\\alpha & V \\ar[d]^q \\\\", "X \\ar[r]^a & Y", "}", "$$", "where $U$ and $V$ are schemes, and $p$ and $q$ are surjective \\'etale.", "Consider the diagram", "$$", "\\xymatrix{", "h_U \\ar[r]_-\\alpha \\ar@{=}[d] & a_{spaces, \\etale}^{-1}h_V \\ar[d]^t \\\\", "h_U \\ar@{..>}[r] & b_{spaces, \\etale}^{-1}h_V", "}", "$$", "Since the sheaf $b_{spaces, \\etale}^{-1}h_V$ is isomorphic to", "$h_{V \\times_{Y, b} X}$ we see that the dotted arrow comes from a", "morphism of schemes", "$\\beta : U \\to V$ fitting into a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_p \\ar[r]_\\beta & V \\ar[d]^q \\\\", "X \\ar[r]^b & Y", "}", "$$", "We claim that there exists a sequence of $2$-isomorphisms", "\\begin{align*}", "(\\alpha_{small}, \\alpha^\\sharp)", "& \\cong", "(\\alpha_{spaces, \\etale}, \\alpha^\\sharp) \\\\", "& \\cong", "(a_{spaces, \\etale, c}, a_c^\\sharp) \\\\", "& \\cong", "(b_{spaces, \\etale, d}, b_d^\\sharp) \\\\", "& \\cong", "(\\beta_{spaces, \\etale}, \\beta^\\sharp) \\\\", "& \\cong", "(\\beta_{small}, \\beta^\\sharp)", "\\end{align*}", "The first and the last $2$-isomorphisms come from the identifications", "between sheaves on $U_{spaces, \\etale}$ and sheaves on", "$U_\\etale$ and similarly for $V$. The second and fourth", "$2$-isomorphisms are those of", "Lemma \\ref{lemma-relocalize-morphism}", "with $c : U \\to X \\times_{a, Y} V$ induced by $\\alpha$ and", "$d : U \\to X \\times_{b, Y} V$ induced by $\\beta$.", "The middle $2$-isomorphism comes from the transformation $t$.", "Namely, the functor $a_{spaces, \\etale, c}^{-1}$ corresponds", "to the functor", "$$", "(\\mathcal{H} \\to h_V) \\longmapsto", "(a_{spaces, \\etale}^{-1}\\mathcal{H}", "\\times_{a_{spaces, \\etale}^{-1}h_V, \\alpha}", "h_U \\to", "h_U)", "$$", "and similarly for $b_{spaces, \\etale, d}^{-1}$, see", "Sites, Lemma \\ref{sites-lemma-relocalize-morphism}.", "This uses the identification of sheaves on $Y_{spaces, \\etale}/V$", "as arrows $(\\mathcal{H} \\to h_V)$ in $\\Sh(Y_{spaces, \\etale})$", "and similarly for $U/X$, see", "Sites, Lemma \\ref{sites-lemma-essential-image-j-shriek}.", "Via this identification the structure sheaf $\\mathcal{O}_V$ corresponds to the", "pair $(\\mathcal{O}_Y \\times h_V \\to h_V)$ and similarly", "for $\\mathcal{O}_U$, see", "Modules on Sites,", "Lemma \\ref{sites-modules-lemma-localize-compare}.", "Since $t$ switches $\\alpha$ and $\\beta$ we see that $t$ induces an isomorphism", "$$", "t :", "a_{spaces, \\etale}^{-1}\\mathcal{H}", "\\times_{a_{spaces, \\etale}^{-1}h_V, \\alpha}", "h_U", "\\longrightarrow", "b_{spaces, \\etale}^{-1}\\mathcal{H}", "\\times_{b_{spaces, \\etale}^{-1}h_V, \\beta}", "h_U", "$$", "over $h_U$ functorially in $(\\mathcal{H} \\to h_V)$. Also, $t$ is compatible", "with $a_c^\\sharp$ and $b_d^\\sharp$ as $t$ is", "compatible with $a^\\sharp$ and $b^\\sharp$ by our description", "of the structure sheaves $\\mathcal{O}_U$ and $\\mathcal{O}_V$", "above. Hence, the morphisms of ringed topoi", "$(\\alpha_{small}, \\alpha^\\sharp)$ and $(\\beta_{small}, \\beta^\\sharp)$", "are $2$-isomorphic. By", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-faithful}", "we conclude $\\alpha = \\beta$! Since $p : U \\to X$ is a surjection", "of sheaves it follows that $a = b$." ], "refs": [ "spaces-properties-lemma-relocalize-morphism", "sites-lemma-relocalize-morphism", "sites-lemma-essential-image-j-shriek", "sites-modules-lemma-localize-compare", "etale-cohomology-lemma-faithful" ], "ref_ids": [ 11900, 8573, 8555, 14177, 6444 ] } ], "ref_ids": [] }, { "id": 11906, "type": "theorem", "label": "spaces-properties-lemma-isomorphism-ringed-topoi", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-isomorphism-ringed-topoi", "contents": [ "Let $X$, $Y$ be algebraic spaces over $\\mathbf{Z}$. If", "$$", "(g, g^\\sharp) :", "(\\Sh(X_\\etale), \\mathcal{O}_X)", "\\longrightarrow", "(\\Sh(Y_\\etale), \\mathcal{O}_Y)", "$$", "is an isomorphism of ringed topoi, then there exists a unique", "morphism $f : X \\to Y$ of algebraic spaces such that", "$(g, g^\\sharp)$ is isomorphic to $(f_{small}, f^\\sharp)$", "and moreover $f$ is an isomorphism of algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "By", "Theorem \\ref{theorem-fully-faithful}", "it suffices to show that $(g, g^\\sharp)$ is a morphism of", "locally ringed topoi. By", "Modules on Sites, Lemma \\ref{sites-modules-lemma-locally-ringed-morphism}", "(and since the site $X_\\etale$ has enough points)", "it suffices to check that the map", "$\\mathcal{O}_{Y, q} \\to \\mathcal{O}_{X, p}$ induced by $g^\\sharp$", "is a local ring map where $q = f \\circ p$ and $p$ is any point of", "$X_\\etale$. As it is an isomorphism this is clear." ], "refs": [ "spaces-properties-theorem-fully-faithful", "sites-modules-lemma-locally-ringed-morphism" ], "ref_ids": [ 11814, 14258 ] } ], "ref_ids": [] }, { "id": 11907, "type": "theorem", "label": "spaces-properties-lemma-pullback-quasi-coherent", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-pullback-quasi-coherent", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The pullback functor", "$f^* : \\textit{Mod}(\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X)$", "preserves quasi-coherent sheaves." ], "refs": [], "proofs": [ { "contents": [ "This is a general fact, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-local-pullback}." ], "refs": [ "sites-modules-lemma-local-pullback" ], "ref_ids": [ 14186 ] } ], "ref_ids": [] }, { "id": 11908, "type": "theorem", "label": "spaces-properties-lemma-characterize-quasi-coherent-small-etale", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-characterize-quasi-coherent-small-etale", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "A quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "is given by the following data:", "\\begin{enumerate}", "\\item for every $U \\in \\Ob(X_\\etale)$ a quasi-coherent", "$\\mathcal{O}_U$-module $\\mathcal{F}_U$ on $U_\\etale$,", "\\item for every $f : U' \\to U$ in $X_\\etale$ an isomorphism", "$c_f : f_{small}^*\\mathcal{F}_U \\to \\mathcal{F}_{U'}$.", "\\end{enumerate}", "These data are subject to the condition that given any $f : U' \\to U$", "and $g : U'' \\to U'$ in $X_\\etale$ the composition", "$g_{small}^{-1}c_f \\circ c_g$ is equal to $c_{f \\circ g}$." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-pullback-quasi-coherent} and", "\\ref{lemma-characterize-module-small-etale}." ], "refs": [ "spaces-properties-lemma-pullback-quasi-coherent", "spaces-properties-lemma-characterize-module-small-etale" ], "ref_ids": [ 11907, 11899 ] } ], "ref_ids": [] }, { "id": 11909, "type": "theorem", "label": "spaces-properties-lemma-stalk-quasi-coherent", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-stalk-quasi-coherent", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $x \\in |X|$ be a point and let $\\overline{x}$ be a geometric", "point lying over $x$. Finally, let", "$\\varphi : (U, \\overline{u}) \\to (X, \\overline{x})$", "be an \\'etale neighbourhood where $U$ is a scheme.", "Then", "$$", "(\\varphi^*\\mathcal{F})_u \\otimes_{\\mathcal{O}_{U, u}}", "\\mathcal{O}_{X, \\overline{x}} =", "\\mathcal{F}_{\\overline{x}}", "$$", "where $u \\in U$ is the image of $\\overline{u}$." ], "refs": [], "proofs": [ { "contents": [ "Note that $\\mathcal{O}_{X, \\overline{x}} = \\mathcal{O}_{U, u}^{sh}$ by", "Lemma \\ref{lemma-describe-etale-local-ring}", "hence the tensor product makes sense. Moreover, from", "Definition \\ref{definition-stalk}", "it is clear that", "$$", "\\mathcal{F}_{\\overline{u}} = \\colim (\\varphi^*\\mathcal{F})_u", "$$", "where the colimit is over $\\varphi : (U, \\overline{u}) \\to (X, \\overline{x})$", "as in the lemma. Hence there is a canonical map from left to right in", "the statement of the lemma. We have a similar colimit description for", "$\\mathcal{O}_{X, \\overline{x}}$", "and by", "Lemma \\ref{lemma-characterize-quasi-coherent-small-etale}", "we have", "$$", "((\\varphi')^*\\mathcal{F})_{u'} =", "(\\varphi^*\\mathcal{F})_u \\otimes_{\\mathcal{O}_{U, u}} \\mathcal{O}_{U', u'}", "$$", "whenever $(U', \\overline{u}') \\to (U, \\overline{u})$ is a morphism of", "\\'etale neighbourhoods. To complete the proof we use that", "$\\otimes$ commutes with colimits." ], "refs": [ "spaces-properties-lemma-describe-etale-local-ring", "spaces-properties-definition-stalk", "spaces-properties-lemma-characterize-quasi-coherent-small-etale" ], "ref_ids": [ 11884, 11940, 11908 ] } ], "ref_ids": [] }, { "id": 11910, "type": "theorem", "label": "spaces-properties-lemma-stalk-pullback-quasi-coherent", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-stalk-pullback-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.", "Let $\\overline{x}$ be a geometric point of $X$ and let", "$\\overline{y} = f \\circ \\overline{x}$ be the image in $Y$.", "Then there is a canonical isomorphism", "$$", "(f^*\\mathcal{G})_{\\overline{x}} =", "\\mathcal{G}_{\\overline{y}} \\otimes_{\\mathcal{O}_{Y, \\overline{y}}}", "\\mathcal{O}_{X, \\overline{x}}", "$$", "of the stalk of the pullback with the tensor product of the stalk", "with the local ring of $X$ at $\\overline{x}$." ], "refs": [], "proofs": [ { "contents": [ "Since $f^*\\mathcal{G} =", "f_{small}^{-1}\\mathcal{G} \\otimes_{f_{small}^{-1}\\mathcal{O}_Y} \\mathcal{O}_X$", "this follows from the description of stalks of pullbacks in", "Lemma \\ref{lemma-stalk-pullback}", "and the fact that taking stalks commutes with tensor products.", "A more direct way to see this is as follows.", "Choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_p \\ar[r]_\\alpha & V \\ar[d]^q \\\\", "X \\ar[r]^a & Y", "}", "$$", "where $U$ and $V$ are schemes, and $p$ and $q$ are surjective \\'etale.", "By", "Lemma \\ref{lemma-geometric-lift-to-usual}", "we can choose a geometric point $\\overline{u}$ of $U$ such that", "$\\overline{x} = p \\circ \\overline{u}$. Set", "$\\overline{v} = \\alpha \\circ \\overline{u}$.", "Then we see that", "\\begin{align*}", "(f^*\\mathcal{G})_{\\overline{x}} & =", "(p^*f^*\\mathcal{G})_u \\otimes_{\\mathcal{O}_{U, u}}", "\\mathcal{O}_{X, \\overline{x}} \\\\", "& = (\\alpha^*q^*\\mathcal{G})_u \\otimes_{\\mathcal{O}_{U, u}}", "\\mathcal{O}_{X, \\overline{x}} \\\\", "& = (q^*\\mathcal{G})_v \\otimes_{\\mathcal{O}_{V, v}}", "\\mathcal{O}_{U, u} \\otimes_{\\mathcal{O}_{U, u}}", "\\mathcal{O}_{X, \\overline{x}} \\\\", "& = (q^*\\mathcal{G})_v \\otimes_{\\mathcal{O}_{V, v}}", "\\mathcal{O}_{X, \\overline{x}} \\\\", "& = (q^*\\mathcal{G})_v \\otimes_{\\mathcal{O}_{V, v}}", "\\mathcal{O}_{Y, \\overline{y}} \\otimes_{\\mathcal{O}_{Y, \\overline{y}}}", "\\mathcal{O}_{X, \\overline{x}} \\\\", "& = \\mathcal{G}_{\\overline{y}} \\otimes_{\\mathcal{O}_{Y, \\overline{y}}}", "\\mathcal{O}_{X, \\overline{x}}", "\\end{align*}", "Here we have used", "Lemma \\ref{lemma-stalk-quasi-coherent} (twice)", "and the corresponding result for pullbacks of quasi-coherent sheaves", "on schemes, see", "Sheaves, Lemma \\ref{sheaves-lemma-stalk-pullback-modules}." ], "refs": [ "spaces-properties-lemma-stalk-pullback", "spaces-properties-lemma-geometric-lift-to-usual", "spaces-properties-lemma-stalk-quasi-coherent", "sheaves-lemma-stalk-pullback-modules" ], "ref_ids": [ 11875, 11871, 11909, 14523 ] } ], "ref_ids": [] }, { "id": 11911, "type": "theorem", "label": "spaces-properties-lemma-characterize-quasi-coherent", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-characterize-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module,", "\\item there exists an \\'etale morphism $f : Y \\to X$ of", "algebraic spaces over $S$ with $|f| : |Y| \\to |X|$ surjective", "such that $f^*\\mathcal{F}$ is quasi-coherent on $Y$,", "\\item there exists a scheme $U$ and a surjective \\'etale morphism", "$\\varphi : U \\to X$ such that $\\varphi^*\\mathcal{F}$ is a quasi-coherent", "$\\mathcal{O}_U$-module, and", "\\item for every affine scheme $U$ and \\'etale morphism $\\varphi : U \\to X$ the", "restriction $\\varphi^*\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_U$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2) by considering $\\text{id}_X$.", "Assume $f : Y \\to X$ is as in (2), and let $V \\to Y$ be a surjective", "\\'etale morphism from a scheme towards $Y$. Then the composition $V \\to X$ is", "surjective \\'etale as well", "and by Lemma \\ref{lemma-pullback-quasi-coherent} the pullback of $\\mathcal{F}$", "to $V$ is quasi-coherent as well. Hence we see that (2) implies (3).", "\\medskip\\noindent", "Let $U \\to X$ be as in (3). Let us use the abuse of notation introduced", "in Equation (\\ref{equation-restrict-modules}).", "As $\\mathcal{F}|_{U_\\etale}$ is quasi-coherent there exists an", "\\'etale covering $\\{U_i \\to U\\}$ such that", "$\\mathcal{F}|_{U_{i, \\etale}}$ has a global presentation, see", "Modules on Sites, Definition \\ref{sites-modules-definition-global} and", "Lemma \\ref{sites-modules-lemma-local-final-object}.", "Let $V \\to X$ be an object of $X_\\etale$. Since $U \\to X$ is", "surjective and \\'etale, the family of maps $\\{U_i \\times_X V \\to V\\}$ is an", "\\'etale covering", "of $V$. Via the morphisms $U_i \\times_X V \\to U_i$ we can restrict the", "global presentations of $\\mathcal{F}|_{U_{i, \\etale}}$ to get a global", "presentation of $\\mathcal{F}|_{(U_i \\times_X V)_\\etale}$", "Hence the sheaf $\\mathcal{F}$ on $X_\\etale$ satisfies the condition of", "Modules on Sites, Definition \\ref{sites-modules-definition-site-local}", "and hence is quasi-coherent.", "\\medskip\\noindent", "The equivalence of (3) and (4) comes from the fact that any scheme has", "an affine open covering." ], "refs": [ "spaces-properties-lemma-pullback-quasi-coherent", "sites-modules-definition-global", "sites-modules-lemma-local-final-object", "sites-modules-definition-site-local" ], "ref_ids": [ 11907, 14286, 14185, 14289 ] } ], "ref_ids": [] }, { "id": 11912, "type": "theorem", "label": "spaces-properties-lemma-properties-quasi-coherent", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-properties-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The category $\\QCoh(\\mathcal{O}_X)$ of quasi-coherent sheaves on $X$", "has the following properties:", "\\begin{enumerate}", "\\item Any direct sum of quasi-coherent sheaves is quasi-coherent.", "\\item Any colimit of quasi-coherent sheaves is quasi-coherent.", "\\item The kernel and cokernel of a morphism of quasi-coherent sheaves", "is quasi-coherent.", "\\item Given a short exact sequence of $\\mathcal{O}_X$-modules", "$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "if two out of three are quasi-coherent so is the third.", "\\item Given two quasi-coherent $\\mathcal{O}_X$-modules", "the tensor product is quasi-coherent.", "\\item Given two quasi-coherent $\\mathcal{O}_X$-modules", "$\\mathcal{F}$, $\\mathcal{G}$ such that $\\mathcal{F}$", "is of finite presentation (see", "Section \\ref{section-properties-modules}),", "then the internal hom", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$", "is quasi-coherent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that we have the corresponding result for quasi-coherent modules", "on schemes, see", "Schemes, Section \\ref{schemes-section-quasi-coherent}.", "We will reduce the lemma to this case by \\'etale localization.", "Choose a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$.", "In order to formulate this proof correctly, we temporarily go back", "to making the (pedantic) distinction between a quasi-coherent sheaf", "$\\mathcal{G}$ on the scheme $U$ and the associated quasi-coherent sheaf", "$\\mathcal{G}^a$ (see", "Descent, Definition \\ref{descent-definition-structure-sheaf})", "on $U_\\etale$", "We have a commutative diagram", "$$", "\\xymatrix{", "\\QCoh(\\mathcal{O}_X) \\ar[r] \\ar[d] &", "\\QCoh(\\mathcal{O}_U) \\ar[d] \\\\", "\\textit{Mod}(\\mathcal{O}_X) \\ar[r] &", "\\textit{Mod}(\\mathcal{O}_U)", "}", "$$", "The bottom horizontal arrow is the restriction functor", "(\\ref{equation-restrict-modules})", "$\\mathcal{G} \\mapsto \\mathcal{G}|_{U_\\etale}$.", "This functor has both a left adjoint and a right adjoint, see", "Modules on Sites, Section \\ref{sites-modules-section-localize},", "hence commutes with all limits and colimits.", "Moreover, we know that an object of $\\textit{Mod}(\\mathcal{O}_X)$ is in", "$\\QCoh(\\mathcal{O}_X)$ if and only if its restriction to $U$ is in", "$\\QCoh(\\mathcal{O}_U)$, see", "Lemma \\ref{lemma-characterize-quasi-coherent}.", "Let $\\mathcal{F}_i$ be a family of", "quasi-coherent $\\mathcal{O}_X$-modules. Then $\\bigoplus \\mathcal{F}_i$", "is an $\\mathcal{O}_X$-module whose restriction to $U$ is the direct sum", "of the restrictions. Let $\\mathcal{G}_i$ be a quasi-coherent sheaf", "on $U$ with $\\mathcal{F}_i|_{U_\\etale} = \\mathcal{G}_i^a$.", "Combining the above with", "Descent, Lemma \\ref{descent-lemma-equivalence-quasi-coherent-limits}", "we see that", "$$", "\\Big(\\bigoplus \\mathcal{F}_i\\Big)|_{U_\\etale} =", "\\bigoplus \\mathcal{F}_i|_{U_\\etale} =", "\\bigoplus \\mathcal{G}_i^a =", "\\Big(\\bigoplus \\mathcal{G}_i\\Big)^a", "$$", "hence $\\bigoplus \\mathcal{F}_i$ is quasi-coherent and (1) follows.", "The other statements are proved just so (using the same references)." ], "refs": [ "descent-definition-structure-sheaf", "spaces-properties-lemma-characterize-quasi-coherent", "descent-lemma-equivalence-quasi-coherent-limits" ], "ref_ids": [ 14766, 11911, 14627 ] } ], "ref_ids": [] }, { "id": 11913, "type": "theorem", "label": "spaces-properties-lemma-locally-projective", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-locally-projective", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item for some scheme $U$ and surjective \\'etale morphism", "$U \\to X$ the restriction $\\mathcal{F}|_U$ is locally projective", "on $U$, and", "\\item for any scheme $U$ and any \\'etale morphism", "$U \\to X$ the restriction $\\mathcal{F}|_U$ is locally projective", "on $U$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $U \\to X$ be as in (1) and let $V \\to X$ be \\'etale where", "$V$ is a scheme. Then $\\{U \\times_X V \\to V\\}$ is an fppf covering", "of schemes. Hence if $\\mathcal{F}|_U$ is locally projective, then", "$\\mathcal{F}|_{U \\times_X V}$ is locally projective (see", "Properties, Lemma \\ref{properties-lemma-locally-projective-pullback})", "and hence $\\mathcal{F}|_V$ is locally projective, see", "Descent, Lemma \\ref{descent-lemma-locally-projective-descends}." ], "refs": [ "properties-lemma-locally-projective-pullback", "descent-lemma-locally-projective-descends" ], "ref_ids": [ 3017, 14618 ] } ], "ref_ids": [] }, { "id": 11914, "type": "theorem", "label": "spaces-properties-lemma-locally-projective-pullback", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-locally-projective-pullback", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.", "If $\\mathcal{G}$ is locally projective on $Y$, then $f^*\\mathcal{G}$", "is locally projective on $X$." ], "refs": [], "proofs": [ { "contents": [ "Choose a surjective \\'etale morphism $V \\to Y$ with $V$ a scheme.", "Choose a surjective \\'etale morphism $U \\to V \\times_Y X$ with", "$U$ a scheme. Denote $\\psi : U \\to V$ the induced morphism.", "Then", "$$", "f^*\\mathcal{G}|_U = \\psi^*(\\mathcal{G}|_V)", "$$", "Hence the lemma follows from the definition and the result in the", "case of schemes, see", "Properties, Lemma \\ref{properties-lemma-locally-projective-pullback}." ], "refs": [ "properties-lemma-locally-projective-pullback" ], "ref_ids": [ 3017 ] } ], "ref_ids": [] }, { "id": 11915, "type": "theorem", "label": "spaces-properties-lemma-morphism-to-affine-scheme", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-morphism-to-affine-scheme", "contents": [ "Let $X$ be an algebraic space over $\\mathbf{Z}$.", "Let $T$ be an affine scheme.", "The map", "$$", "\\Mor(X, T)", "\\longrightarrow", "\\Hom(\\Gamma(T, \\mathcal{O}_T), \\Gamma(X, \\mathcal{O}_X))", "$$", "which maps $f$ to $f^\\sharp$ (on global sections) is bijective." ], "refs": [], "proofs": [ { "contents": [ "We construct the inverse of the map.", "Let $\\varphi : \\Gamma(T, \\mathcal{O}_T) \\to \\Gamma(X, \\mathcal{O}_X)$", "be a ring map. Choose a presentation $X = U/R$, see", "Spaces, Definition \\ref{spaces-definition-presentation}.", "By", "Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}", "the composition", "$$", "\\Gamma(T, \\mathcal{O}_T) \\to \\Gamma(X, \\mathcal{O}_X) \\to", "\\Gamma(U, \\mathcal{O}_U)", "$$", "corresponds to a unique morphism of schemes $g : U \\to T$. By the same lemma", "the two compositions $R \\to U \\to T$ are equal. Hence we obtain a morphism", "$f : X = U/R \\to T$ such that $U \\to X \\to T$ equals $g$. By construction", "the diagram", "$$", "\\xymatrix{", "\\Gamma(U, \\mathcal{O}_U) & \\Gamma(X, \\mathcal{O}_X) \\ar[l]^{f^\\sharp} \\\\", "& \\Gamma(T, \\mathcal{O}_T) \\ar[lu]^{g^\\sharp} \\ar[u]^\\varphi", "}", "$$", "commutes. Hence $f^\\sharp$ equals $\\varphi$ because $U \\to X$ is an", "\\'etale covering and $\\mathcal{O}_X$ is a sheaf on $X_\\etale$.", "The uniqueness of $f$ follows from the uniqueness of $g$." ], "refs": [ "spaces-definition-presentation", "schemes-lemma-morphism-into-affine" ], "ref_ids": [ 8177, 7655 ] } ], "ref_ids": [] }, { "id": 11916, "type": "theorem", "label": "spaces-properties-lemma-quotient", "categories": [ "spaces-properties" ], "title": "spaces-properties-lemma-quotient", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $G$ be an abstract group with a free action on $X$.", "Then the quotient sheaf $X/G$ is an algebraic space." ], "refs": [], "proofs": [ { "contents": [ "The statement means that the sheaf $F$ associated to the presheaf", "$$", "T \\longmapsto X(T)/G", "$$", "is an algebraic space. To see this we will construct a presentation.", "Namely, choose a scheme $U$ and a surjective \\'etale morphism", "$\\varphi : U \\to X$. Set $V = \\coprod_{g \\in G} U$ and set", "$\\psi : V \\to X$ equal to $a(g) \\circ \\varphi$ on the component corresponding", "to $g \\in G$. Let $G$ act on $V$ by permuting the components, i.e.,", "$g_0 \\in G$ maps the component corresponding to $g$ to the component", "corresponding to $g_0g$ via the identity morphism of $U$.", "Then $\\psi$ is a $G$-equivariant morphism, i.e., we reduce to the", "case dealt with in the next paragraph.", "\\medskip\\noindent", "Assume that there exists a $G$-action on $U$ and that $U \\to X$ is surjective,", "\\'etale and $G$-equivariant. In this case there is an induced", "action of $G$ on $R = U \\times_X U$ compatible with the projection", "mappings $t, s : R \\to U$. Now we claim that", "$$", "X/G = U/\\coprod\\nolimits_{g \\in G} R", "$$", "where the map", "$$", "j : \\coprod\\nolimits_{g \\in G} R", "\\longrightarrow", "U \\times_S U", "$$", "is given by $(r, g) \\mapsto (t(r), g(s(r)))$. Note that $j$ is a monomorphism:", "If $(t(r), g(s(r))) = (t(r'), g'(s(r')))$, then", "$t(r) = t(r')$, hence $r$ and $r'$ have the same image in $X$ under", "both $s$ and $t$, hence $g = g'$ (as $G$ acts freely on $X$), hence", "$s(r) = s(r')$, hence $r = r'$ (as $R$ is an equivalence relation on $U$).", "Moreover $j$ is an equivalence relation (details omitted).", "Both projections $\\coprod\\nolimits_{g \\in G} R \\to U$ are \\'etale, as", "$s$ and $t$ are \\'etale. Thus $j$ is an \\'etale equivalence relation", "and $U/\\coprod\\nolimits_{g \\in G} R$ is an algebraic space by", "Spaces, Theorem \\ref{spaces-theorem-presentation}.", "There is a map", "$$", "U/\\coprod\\nolimits_{g \\in G} R \\longrightarrow X/G", "$$", "induced by the map $U \\to X$. We omit the proof that it is an", "isomorphism of sheaves." ], "refs": [ "spaces-theorem-presentation" ], "ref_ids": [ 8124 ] } ], "ref_ids": [] }, { "id": 11917, "type": "theorem", "label": "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme", "categories": [ "spaces-properties" ], "title": "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is", "Zariski locally quasi-separated (for example if $X$ is quasi-separated), then", "there exists a dense open subspace $X'$ of $X$ which is a scheme.", "More precisely, every point $x \\in |X|$ of codimension $0$ on $X$", "is contained in $X'$." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$ by Lemma \\ref{lemma-subscheme}.", "Thus by Lemma \\ref{lemma-quasi-separated-quasi-compact-pieces}", "we may assume that there exists an affine scheme $U$ and a", "surjective, quasi-compact, \\'etale morphism $U \\to X$.", "Moreover $U \\to X$ is separated (Lemma \\ref{lemma-separated-cover}).", "Set $R = U \\times_X U$ and denote $s, t : R \\to U$ the projections", "as usual. Then $s, t$ are surjective, quasi-compact, separated, and", "\\'etale. Hence $s, t$ are also quasi-finite and have finite fibres", "(Morphisms,", "Lemmas \\ref{morphisms-lemma-etale-locally-quasi-finite},", "\\ref{morphisms-lemma-quasi-finite-locally-quasi-compact}, and", "\\ref{morphisms-lemma-quasi-finite}).", "By Morphisms, Lemma \\ref{morphisms-lemma-generically-finite}", "for every $\\eta \\in U$ which is the generic point of an", "irreducible component of $U$, there exists an open neighbourhood", "$V \\subset U$ of $\\eta$ such that $s^{-1}(V) \\to V$ is finite. By", "Descent, Lemma \\ref{descent-lemma-descending-property-finite}", "being finite is fpqc (and in particular \\'etale) local on the target.", "Hence we may apply", "More on Groupoids, Lemma \\ref{more-groupoids-lemma-property-invariant}", "which says that the largest open $W \\subset U$ over which $s$ is", "finite is $R$-invariant. By the above $W$ contains every", "generic point of an irreducible component of $U$.", "The restriction $R_W$ of $R$ to $W$ equals $R_W = s^{-1}(W) = t^{-1}(W)$", "(see Groupoids, Definition \\ref{groupoids-definition-invariant-open}", "and discussion following it).", "By construction $s_W, t_W : R_W \\to W$ are finite \\'etale.", "Consider the open subspace $X' = W/R_W \\subset X$ (see", "Spaces, Lemma \\ref{spaces-lemma-finding-opens}).", "By construction the inclusion map $X' \\to X$", "induces a bijection on points of codimension $0$.", "This reduces us to", "Lemma \\ref{lemma-quasi-separated-finite-etale-cover-dense-open-scheme}." ], "refs": [ "spaces-properties-lemma-subscheme", "spaces-properties-lemma-quasi-separated-quasi-compact-pieces", "spaces-properties-lemma-separated-cover", "morphisms-lemma-etale-locally-quasi-finite", "morphisms-lemma-quasi-finite-locally-quasi-compact", "morphisms-lemma-quasi-finite", "morphisms-lemma-generically-finite", "descent-lemma-descending-property-finite", "more-groupoids-lemma-property-invariant", "groupoids-definition-invariant-open", "spaces-lemma-finding-opens", "spaces-properties-lemma-quasi-separated-finite-etale-cover-dense-open-scheme" ], "ref_ids": [ 11848, 11835, 11833, 5363, 5229, 5230, 5487, 14688, 2460, 9685, 8151, 11849 ] } ], "ref_ids": [] }, { "id": 11918, "type": "theorem", "label": "spaces-properties-proposition-finite-flat-equivalence-global", "categories": [ "spaces-properties" ], "title": "spaces-properties-proposition-finite-flat-equivalence-global", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Assume", "\\begin{enumerate}", "\\item $s, t : R \\to U$ finite locally free,", "\\item $j = (t, s)$ is an equivalence, and", "\\item for a dense set of points $u \\in U$ the $R$-equivalence class", "$t(s^{-1}(\\{u\\}))$ is contained in an affine open of $U$.", "\\end{enumerate}", "Then there exists a finite locally free morphism $U \\to M$", "of schemes over $S$ such that $R = U \\times_M U$ and such that $M$", "represents the quotient sheaf $U/R$ in the fppf topology." ], "refs": [], "proofs": [ { "contents": [ "By assumption (3) and", "Groupoids, Lemma \\ref{groupoids-lemma-find-invariant-affine}", "we can find an open covering $U = \\bigcup U_i$ such that each $U_i$", "is an $R$-invariant affine open of $U$. Set $R_i = R|_{U_i}$.", "Consider the fppf sheaves $F = U/R$ and $F_i = U_i/R_i$.", "By Spaces, Lemma \\ref{spaces-lemma-finding-opens} the morphisms", "$F_i \\to F$ are representable and open immersions.", "By Groupoids, Proposition \\ref{groupoids-proposition-finite-flat-equivalence}", "the sheaves $F_i$ are representable by affine schemes.", "If $T$ is a scheme and $T \\to F$ is a morphism, then $V_i = F_i \\times_F T$", "is open in $T$ and we claim that $T = \\bigcup V_i$. Namely,", "fppf locally on $T$ we can lift $T \\to F$ to a morphism", "$f : T \\to U$ and in that case $f^{-1}(U_i) \\subset V_i$.", "Hence we conclude that $F$ is representable by a scheme, see", "Schemes, Lemma \\ref{schemes-lemma-glue-functors}." ], "refs": [ "groupoids-lemma-find-invariant-affine", "spaces-lemma-finding-opens", "groupoids-proposition-finite-flat-equivalence", "schemes-lemma-glue-functors" ], "ref_ids": [ 9664, 8151, 9669, 7688 ] } ], "ref_ids": [] }, { "id": 11919, "type": "theorem", "label": "spaces-properties-proposition-sheaf-fpqc", "categories": [ "spaces-properties" ], "title": "spaces-properties-proposition-sheaf-fpqc", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then", "$X$ satisfies the sheaf property for the fpqc topology." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is a sheaf for the Zariski topology it suffices to show", "the following. Given a surjective flat morphism of affines", "$f : T' \\to T$ we have:", "$X(T)$ is the equalizer of the two maps $X(T') \\to X(T' \\times_T T')$.", "See Topologies, Lemma \\ref{topologies-lemma-sheaf-property-fpqc}", "(there is a little argument omitted here because the lemma cited", "is formulated for functors defined on the category of all schemes).", "\\medskip\\noindent", "Let $a, b : T \\to X$ be two morphisms such that $a \\circ f = b \\circ f$.", "We have to show $a = b$. Consider the fibre product", "$$", "E = X \\times_{\\Delta_{X/S}, X \\times_S X, (a, b)} T.", "$$", "By Spaces, Lemma \\ref{spaces-lemma-properties-diagonal}", "the morphism $\\Delta_{X/S}$ is a representable monomorphism. Hence", "$E \\to T$ is a monomorphism of schemes. Our assumption that", "$a \\circ f = b \\circ f$ implies that $T' \\to T$ factors (uniquely) through $E$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "T' \\times_T E \\ar[r] \\ar[d] & E \\ar[d] \\\\", "T' \\ar[r] \\ar@/^5ex/[u] \\ar[ru] & T", "}", "$$", "Since the projection $T' \\times_T E \\to T'$ is a monomorphism", "with a section we conclude it is an isomorphism. Hence we conclude that", "$E \\to T$ is an isomorphism by", "Descent, Lemma \\ref{descent-lemma-descending-property-isomorphism}.", "This means $a = b$ as desired.", "\\medskip\\noindent", "Next, let $c : T' \\to X$ be a morphism such that the two compositions", "$T' \\times_T T' \\to T' \\to X$ are the same. We have to find a morphism", "$a : T \\to X$ whose composition with $T' \\to T$ is $c$. Choose an", "affine scheme $U$ and an \\'etale morphism $U \\to X$ such that the image", "of $|U| \\to |X|$ contains the image of $|c| : |T'| \\to |X|$.", "This is possible by Lemmas \\ref{lemma-topology-points} and", "\\ref{lemma-cover-by-union-affines}, the fact that a finite disjoint union of", "affines is affine, and the fact that $|T'|$ is quasi-compact", "(small argument omitted). Since $U \\to X$ is separated", "(Lemma \\ref{lemma-separated-cover}), we see that", "$$", "V = U \\times_{X, c} T' \\longrightarrow T'", "$$", "is a surjective, \\'etale, separated morphism of schemes", "(to see that it is surjective use Lemma \\ref{lemma-points-cartesian}", "and our choice of $U \\to X$). The fact that", "$c \\circ \\text{pr}_0 = c \\circ \\text{pr}_1$ means that we obtain a", "descent datum on $V/T'/T$", "(Descent, Definition \\ref{descent-definition-descent-datum})", "because", "\\begin{align*}", "V \\times_{T'} (T' \\times_T T')", "& =", "U \\times_{X, c \\circ \\text{pr}_0} (T' \\times_T T') \\\\", "& =", "(T' \\times_T T') \\times_{c \\circ \\text{pr}_1, X} U \\\\", "& =", "(T' \\times_T T') \\times_{T'} V", "\\end{align*}", "The morphism $V \\to T'$ is ind-quasi-affine by", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-separated-ind-quasi-affine}", "(because \\'etale morphisms are locally quasi-finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-locally-quasi-finite}).", "By More on Groupoids, Lemma \\ref{more-groupoids-lemma-ind-quasi-affine}", "the descent datum is effective. Say $W \\to T$ is a morphism", "such that there is an isomorphism $\\alpha : T' \\times_T W \\to V$", "compatible with the given descent datum on $V$ and the canonical descent", "datum on $T' \\times_T W$. Then $W \\to T$ is surjective and \\'etale", "(Descent, Lemmas \\ref{descent-lemma-descending-property-surjective} and", "\\ref{descent-lemma-descending-property-etale}).", "Consider the composition", "$$", "b' : T' \\times_T W \\longrightarrow V = U \\times_{X, c} T' \\longrightarrow U", "$$", "The two compositions", "$b' \\circ (\\text{pr}_0, 1), ", "b' \\circ (\\text{pr}_1, 1) :", "(T' \\times_T T') \\times_T W \\to T' \\times_T W \\to U$", "agree by our choice of $\\alpha$ and the corresponding property of $c$", "(computation omitted). Hence $b'$ descends to a morphism $b : W \\to U$ by", "Descent, Lemma \\ref{descent-lemma-fpqc-universal-effective-epimorphisms}.", "The diagram", "$$", "\\xymatrix{", "T' \\times_T W \\ar[r] \\ar[d] & W \\ar[r]_b & U \\ar[d] \\\\", "T' \\ar[rr]^c & & X", "}", "$$", "is commutative. What this means is that we have proved the existence", "of $a$ \\'etale locally on $T$, i.e., we have an $a' : W \\to X$.", "However, since we have proved uniqueness", "in the first paragraph, we find that this \\'etale local solution", "satisfies the glueing condition, i.e., we have", "$\\text{pr}_0^*a' = \\text{pr}_1^*a'$ as elements of $X(W \\times_T W)$.", "Since $X$ is an \\'etale sheaf we find a unique $a \\in X(T)$ restricting", "to $a'$ on $W$." ], "refs": [ "topologies-lemma-sheaf-property-fpqc", "spaces-lemma-properties-diagonal", "descent-lemma-descending-property-isomorphism", "spaces-properties-lemma-topology-points", "spaces-properties-lemma-cover-by-union-affines", "spaces-properties-lemma-separated-cover", "spaces-properties-lemma-points-cartesian", "descent-definition-descent-datum", "more-morphisms-lemma-etale-separated-ind-quasi-affine", "morphisms-lemma-etale-locally-quasi-finite", "more-groupoids-lemma-ind-quasi-affine", "descent-lemma-descending-property-surjective", "descent-lemma-descending-property-etale", "descent-lemma-fpqc-universal-effective-epimorphisms" ], "ref_ids": [ 12502, 8163, 14682, 11822, 11830, 11833, 11819, 14776, 14045, 5363, 2505, 14672, 14694, 14638 ] } ], "ref_ids": [] }, { "id": 11920, "type": "theorem", "label": "spaces-properties-proposition-quasi-coherent", "categories": [ "spaces-properties" ], "title": "spaces-properties-proposition-quasi-coherent", "contents": [ "With $S$, $\\varphi : U \\to X$, and $(U, R, s, t, c)$ as above.", "For any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the", "sheaf $\\varphi^*\\mathcal{F}$ comes equipped with a canonical", "isomorphism", "$$", "\\alpha : t^*\\varphi^*\\mathcal{F} \\longrightarrow s^*\\varphi^*\\mathcal{F}", "$$", "which satisfies the conditions of", "Groupoids, Definition \\ref{groupoids-definition-groupoid-module}", "and therefore defines a quasi-coherent sheaf on $(U, R, s, t, c)$.", "The functor $\\mathcal{F} \\mapsto (\\varphi^*\\mathcal{F}, \\alpha)$", "defines an equivalence of categories", "$$", "\\begin{matrix}", "\\text{Quasi-coherent} \\\\", "\\mathcal{O}_X\\text{-modules}", "\\end{matrix}", "\\longleftrightarrow", "\\begin{matrix}", "\\text{Quasi-coherent modules}\\\\", "\\text{on }(U, R, s, t, c)", "\\end{matrix}", "$$" ], "refs": [ "groupoids-definition-groupoid-module" ], "proofs": [ { "contents": [ "In the statement of the proposition, and in this proof we think of a", "quasi-coherent sheaf on a scheme as a quasi-coherent sheaf on the small", "\\'etale site of that scheme. This is permissible by the results of", "Descent, Section \\ref{descent-section-quasi-coherent-sheaves}.", "\\medskip\\noindent", "The existence of $\\alpha$ comes from the fact that", "$\\varphi \\circ t = \\varphi \\circ s$ and that pullback is", "functorial in the morphism, see discussion surrounding", "Equation (\\ref{equation-push-pull}). In exactly the same way, i.e., by", "functoriality of pullback, we see that the isomorphism $\\alpha$ satisfies", "condition (1) of", "Groupoids, Definition \\ref{groupoids-definition-groupoid-module}.", "To see condition (2) of the definition it suffices to see that $\\alpha$", "is an isomorphism which is clear. The construction", "$\\mathcal{F} \\mapsto (\\varphi^*\\mathcal{F}, \\alpha)$", "is clearly functorial in the quasi-coherent sheaf $\\mathcal{F}$.", "Hence we obtain the functor from left to right in the displayed", "formula of the lemma.", "\\medskip\\noindent", "Conversely, suppose that $(\\mathcal{F}, \\alpha)$ is a quasi-coherent", "sheaf on $(U, R, s, t, c)$. Let $V \\to X$ be an object of $X_\\etale$.", "In this case the morphism $V' = U \\times_X V \\to V$ is a surjective \\'etale", "morphism of schemes, and hence $\\{V' \\to V\\}$ is an \\'etale", "covering of $V$. Moreover, the quasi-coherent sheaf $\\mathcal{F}$", "pulls back to a quasi-coherent sheaf $\\mathcal{F}'$ on $V'$.", "Since $R = U \\times_X U$ with $t = \\text{pr}_0$ and $s = \\text{pr}_0$", "we see that $V' \\times_V V' = R \\times_X V$ with projection maps", "$V' \\times_V V' \\to V'$ equal to the pullbacks of $t$ and $s$. Hence", "$\\alpha$ pulls back to an isomorphism", "$\\alpha' : \\text{pr}_0^*\\mathcal{F}' \\to \\text{pr}_1^*\\mathcal{F}'$, and", "the pair $(\\mathcal{F}', \\alpha')$ is a descend datum for quasi-coherent", "sheaves with respect to $\\{V' \\to V\\}$. By", "Descent, Proposition", "\\ref{descent-proposition-fpqc-descent-quasi-coherent}", "this descent datum is effective, and we obtain a quasi-coherent", "$\\mathcal{O}_V$-module $\\mathcal{F}_V$ on $V_\\etale$.", "To see that this gives a quasi-coherent sheaf on $X_\\etale$ we have", "to show (by", "Lemma \\ref{lemma-characterize-quasi-coherent-small-etale})", "that for any morphism $f : V_1 \\to V_2$ in $X_\\etale$", "there is a canonical isomorphism", "$c_f : \\mathcal{F}_{V_1} \\to \\mathcal{F}_{V_2}$", "compatible with compositions of morphisms. We omit the verification.", "We also omit the verification that this defines a functor from the", "category on the right to the category on the left which is inverse", "to the functor described above." ], "refs": [ "groupoids-definition-groupoid-module", "descent-proposition-fpqc-descent-quasi-coherent", "spaces-properties-lemma-characterize-quasi-coherent-small-etale" ], "ref_ids": [ 9682, 14753, 11908 ] } ], "ref_ids": [ 9682 ] }, { "id": 11921, "type": "theorem", "label": "spaces-properties-proposition-coherator", "categories": [ "spaces-properties" ], "title": "spaces-properties-proposition-coherator", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item The category $\\QCoh(\\mathcal{O}_X)$ is a Grothendieck", "abelian category. Consequently, $\\QCoh(\\mathcal{O}_X)$", "has enough injectives and all limits.", "\\item The inclusion functor", "$\\QCoh(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X)$", "has a right adjoint\\footnote{This functor is sometimes called", "the {\\it coherator}.}", "$$", "Q : \\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\QCoh(\\mathcal{O}_X)", "$$", "such that for every quasi-coherent sheaf $\\mathcal{F}$ the adjunction mapping", "$Q(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This proof is a repeat of the proof in the case of schemes, see", "Properties, Proposition \\ref{properties-proposition-coherator}.", "We advise the reader to read that proof first.", "\\medskip\\noindent", "Part (1) means $\\QCoh(\\mathcal{O}_X)$ (a) has all colimits,", "(b) filtered colimits are exact, and (c) has a generator, see", "Injectives, Section \\ref{injectives-section-grothendieck-conditions}.", "By Lemma \\ref{lemma-properties-quasi-coherent}", "colimits in $\\QCoh(\\mathcal{O}_X)$ exist and agree", "with colimits in $\\textit{Mod}(\\mathcal{O}_X)$. By", "Modules on Sites, Lemma \\ref{sites-modules-lemma-limits-colimits}", "filtered colimits are exact. Hence (a) and (b) hold.", "\\medskip\\noindent", "To construct a generator, choose a presentation $X = U/R$ so that", "$(U, R, s, t, c)$ is an", "\\'etale groupoid scheme and in particular $s$ and $t$ are flat morphisms", "of schemes. Pick a cardinal $\\kappa$ as in", "Groupoids, Lemma \\ref{groupoids-lemma-colimit-kappa}.", "Pick a collection $(\\mathcal{E}_t, \\alpha_t)_{t \\in T}$ of", "$\\kappa$-generated quasi-coherent modules on", "$(U, R, s, t, c)$ as in", "Groupoids, Lemma \\ref{groupoids-lemma-set-of-iso-classes}.", "Let $\\mathcal{F}_t$ be the quasi-coherent module on $X$ which", "corresponds to the quasi-coherent module $(\\mathcal{E}_t, \\alpha_t)$ via", "the equivalence of categories of", "Proposition \\ref{proposition-quasi-coherent}.", "Then we see that every quasi-coherent module $\\mathcal{H}$ is the", "directed colimit of its quasi-coherent submodules which are isomorphic", "to one of the $\\mathcal{F}_t$. Thus $\\bigoplus_t \\mathcal{F}_t$ is", "a generator of $\\QCoh(\\mathcal{O}_X)$ and we conclude that (c) holds.", "The assertions on limits and injectives hold in any", "Grothendieck abelian category, see", "Injectives, Theorem", "\\ref{injectives-theorem-injective-embedding-grothendieck} and", "Lemma \\ref{injectives-lemma-grothendieck-products}.", "\\medskip\\noindent", "Proof of (2). To construct $Q$ we use the following general procedure.", "Given an object $\\mathcal{F}$ of $\\textit{Mod}(\\mathcal{O}_X)$", "we consider the functor", "$$", "\\QCoh(\\mathcal{O}_X)^{opp} \\longrightarrow \\textit{Sets},\\quad", "\\mathcal{G} \\longmapsto \\Hom_X(\\mathcal{G}, \\mathcal{F})", "$$", "This functor transforms colimits into limits,", "hence is representable, see", "Injectives, Lemma \\ref{injectives-lemma-grothendieck-brown}.", "Thus there exists a quasi-coherent sheaf $Q(\\mathcal{F})$", "and a functorial isomorphism", "$\\Hom_X(\\mathcal{G}, \\mathcal{F}) = \\Hom_X(\\mathcal{G}, Q(\\mathcal{F}))$", "for $\\mathcal{G}$ in $\\QCoh(\\mathcal{O}_X)$. By the Yoneda lemma", "(Categories, Lemma \\ref{categories-lemma-yoneda})", "the construction $\\mathcal{F} \\leadsto Q(\\mathcal{F})$ is", "functorial in $\\mathcal{F}$. By construction $Q$ is a right", "adjoint to the inclusion functor.", "The fact that $Q(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism", "when $\\mathcal{F}$ is quasi-coherent is a formal consequence of the fact", "that the inclusion functor", "$\\QCoh(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X)$", "is fully faithful." ], "refs": [ "properties-proposition-coherator", "spaces-properties-lemma-properties-quasi-coherent", "sites-modules-lemma-limits-colimits", "groupoids-lemma-colimit-kappa", "groupoids-lemma-set-of-iso-classes", "spaces-properties-proposition-quasi-coherent", "injectives-theorem-injective-embedding-grothendieck", "injectives-lemma-grothendieck-products", "injectives-lemma-grothendieck-brown", "categories-lemma-yoneda" ], "ref_ids": [ 3066, 11912, 14158, 9635, 9634, 11920, 7767, 7794, 7793, 12203 ] } ], "ref_ids": [] }, { "id": 11956, "type": "theorem", "label": "intersection-theorem-multiplicity-with-koszul", "categories": [ "intersection" ], "title": "intersection-theorem-multiplicity-with-koszul", "contents": [ "\\begin{reference}", "\\cite[Theorem 1 in part B of Chapter IV]{Serre_algebre_locale}", "\\end{reference}", "Let $A$ be a Noetherian local ring. Let $I = (f_1, \\ldots, f_r) \\subset A$", "be an ideal of definition. Let $M$ be a finite $A$-module. Then", "$$", "e_I(M, r) = \\sum", "(-1)^i\\text{length}_A H_i(K_\\bullet(f_1, \\ldots, f_r) \\otimes_A M)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let us change the Koszul complex $K_\\bullet(f_1, \\ldots, f_r)$ into a cochain", "complex $K^\\bullet$ by setting $K^n = K_{-n}(f_1, \\ldots, f_r)$.", "Then $K^\\bullet$ is sitting in degrees $-r, \\ldots, 0$ and", "$H^i(K^\\bullet \\otimes_A M) = H_{-i}(K_\\bullet(f_1, \\ldots, f_r) \\otimes_A M)$.", "The statement of the theorem makes sense as the modules", "$H^i(K^\\bullet \\otimes M)$ are annihilated by $f_1, \\ldots, f_r$", "(More on Algebra, Lemma \\ref{more-algebra-lemma-homotopy-koszul})", "hence have finite length.", "Define a filtration on the complex $K^\\bullet$ by setting", "$$", "F^p(K^n \\otimes_A M) =", "I^{\\max(0, p + n)}(K^n \\otimes_A M),\\quad p \\in \\mathbf{Z}", "$$", "Since $f_i I^p \\subset I^{p + 1}$ this is a filtration by subcomplexes.", "Thus we have a filtered complex and we obtain a spectral sequence, see", "Homology, Section \\ref{homology-section-filtered-complex}.", "We have", "$$", "E_0 = \\bigoplus\\nolimits_{p, q} E_0^{p, q} =", "\\bigoplus\\nolimits_{p, q} \\text{gr}^p(K^{p + q} \\otimes_A M) =", "\\text{Gr}_I(K^\\bullet \\otimes_A M)", "$$", "Since $K^n$ is finite free we have", "$$", "\\text{Gr}_I(K^\\bullet \\otimes_A M) =", "\\text{Gr}_I(K^\\bullet) \\otimes_{\\text{Gr}_I(A)} \\text{Gr}_I(M)", "$$", "Note that $\\text{Gr}_I(K^\\bullet)$ is the Koszul", "complex over $\\text{Gr}_I(A)$ on the elements", "$\\overline{f}_1, \\ldots, \\overline{f}_r \\in I/I^2$.", "A simple calculation (omitted)", "shows that the differential $d_0$ on $E_0$", "agrees with the differential coming from the Koszul complex.", "Since $\\text{Gr}_I(M)$ is a finite $\\text{Gr}_I(A)$-module", "and since $\\text{Gr}_I(A)$ is Noetherian (as a quotient", "of $A/I[x_1, \\ldots, x_r]$ with $x_i \\mapsto \\overline{f}_i$), the", "cohomology module $E_1 = \\bigoplus E_1^{p, q}$", "is a finite $\\text{Gr}_I(A)$-module. However, as above", "$E_1$ is annihilated by $\\overline{f}_1, \\ldots, \\overline{f}_r$.", "We conclude $E_1$ has finite length.", "In particular we find that $\\text{Gr}^p_F(K^\\bullet \\otimes M)$ is", "acyclic for $p \\gg 0$.", "\\medskip\\noindent", "Next, we check that the spectral sequence above converges", "using Homology, Lemma \\ref{homology-lemma-filtered-complex-ss-converges}.", "The required equalities follow easily from the Artin-Rees lemma", "in the form stated in Algebra, Lemma \\ref{algebra-lemma-map-AR}.", "Thus we see that", "\\begin{align*}", "\\sum (-1)^i\\text{length}_A(H^i(K^\\bullet \\otimes_A M))", "& =", "\\sum (-1)^{p + q} \\text{length}_A(E_\\infty^{p, q}) \\\\", "& =", "\\sum (-1)^{p + q} \\text{length}_A(E_1^{p, q})", "\\end{align*}", "because as we've seen above the length of $E_1$ is finite", "(of course this uses additivity of lengths). Pick $t$ so", "large that $\\text{Gr}^p_F(K^\\bullet \\otimes M)$", "is acyclic for $p \\geq t$ (see above). Using", "additivity again we see that", "$$", "\\sum (-1)^{p + q} \\text{length}_A(E_1^{p, q}) =", "\\sum\\nolimits_n \\sum\\nolimits_{p \\leq t}", "(-1)^n \\text{length}_A(\\text{gr}^p(K^n \\otimes_A M))", "$$", "This is equal to", "$$", "\\sum\\nolimits_{n = -r, \\ldots, 0} (-1)^n{r \\choose |n|} \\chi_{I, M}(t + n)", "$$", "by our choice of filtration above and the definition of $\\chi_{I, M}$ in", "Algebra, Section \\ref{algebra-section-Noetherian-local}.", "The lemma follows from Lemma \\ref{lemma-leading-coefficient}", "and the definition of $e_I(M, r)$." ], "refs": [ "more-algebra-lemma-homotopy-koszul", "homology-lemma-filtered-complex-ss-converges", "algebra-lemma-map-AR", "intersection-lemma-leading-coefficient" ], "ref_ids": [ 9960, 12100, 626, 11972 ] } ], "ref_ids": [] }, { "id": 11957, "type": "theorem", "label": "intersection-theorem-well-defined", "categories": [ "intersection" ], "title": "intersection-theorem-well-defined", "contents": [ "Let $X$ be a nonsingular projective variety. Let $\\alpha$, resp.\\ $\\beta$", "be an $r$, resp.\\ $s$ cycle on $X$. Assume that $\\alpha$ and $\\beta$", "intersect properly so that $\\alpha \\cdot \\beta$ is defined. Finally,", "assume that $\\alpha \\sim_{rat} 0$. Then $\\alpha \\cdot \\beta \\sim_{rat} 0$." ], "refs": [], "proofs": [ { "contents": [ "Pick a closed immersion $X \\subset \\mathbf{P}^N$.", "By linearity it suffices to prove the result when $\\beta = [Z]$ for some", "$s$-dimensional closed subvariety $Z \\subset X$ which intersects $\\alpha$", "properly. The condition $\\alpha \\sim_{rat} 0$ means there", "are finitely many $(r + 1)$-dimensional closed subvarieties", "$W_i \\subset X \\times \\mathbf{P}^1$ such that", "$$", "\\alpha = \\sum [W_{i, a_i}]_r - [W_{i, b_i}]_r", "$$", "for some pairs of points $a_i, b_i$ of $\\mathbf{P}^1$.", "Let $W_{i, a_i}^t$ and $W_{i, b_i}^t$ be the irreducible components", "of $W_{i, a_i}$ and $W_{i, b_i}$.", "We will use induction on the maximum $d$ of the integers", "$$", "\\dim(Z \\cap W_{i, a_i}^t),\\quad \\dim(Z \\cap W_{i, b_i}^t)", "$$", "The main problem in the rest of the proof is that although we know that $Z$", "intersects $\\alpha$ properly, it may not be the case that $Z$ intersects the", "``intermediate'' varieties $W_{i, a_i}^t$", "and $W_{i, b_i}^t$ properly, i.e., it may happen that $d > r + s - \\dim(X)$.", "\\medskip\\noindent", "Base case: $d = r + s - \\dim(X)$. In this case all the intersections of", "$Z$ with the $W_{i, a_i}^t$ and $W_{i, b_i}^t$ are proper and the", "desired result follows from Lemma \\ref{lemma-well-defined-special-case},", "because it applies to show that", "$[Z] \\cdot [W_{i, a_i}]_r \\sim_{rat} [Z] \\cdot [W_{i, b_i}]_r$ for", "each $i$.", "\\medskip\\noindent", "Induction step: $d > r + s - \\dim(X)$. Apply", "Lemma \\ref{lemma-moving} to $Z \\subset X$ and", "the family of subvarieties $\\{W_{i, a_i}^t, W_{i, b_i}^t\\}$. Then we find a", "closed subvariety $C \\subset \\mathbf{P}^N$ intersecting $X$", "properly such that", "$$", "C \\cdot X = [Z] + \\sum m_j [Z_j]", "$$", "and such that", "$$", "\\dim(Z_j \\cap W_{i, a_i}^t) \\leq \\dim(Z \\cap W_{i, a_i}^t),\\quad", "\\dim(Z_j \\cap W_{i, b_i}^t) \\leq \\dim(Z \\cap W_{i, b_i}^t)", "$$", "with strict inequality if the right hand side is $> r + s - \\dim(X)$.", "This implies two things: (a) the induction hypothesis applies to", "each $Z_j$, and (b) $C \\cdot X$ and $\\alpha$ intersect properly (because", "$\\alpha$ is a linear combination of those $[W_{i, a_i}^t]$ and", "$[W_{i, a_i}^t]$ which intersect $Z$ properly).", "Next, pick $C' \\subset \\mathbf{P}^N \\times \\mathbf{P}^1$", "as in Lemma \\ref{lemma-move} with respect to $C$, $X$, and", "$W_{i, a_i}^t$, $W_{i, b_i}^t$.", "Write $C' \\cdot X \\times \\mathbf{P}^1 = \\sum n_k [E_k]$ for", "some subvarieties $E_k \\subset X \\times \\mathbf{P}^1$ of", "dimension $s + 1$. Note that $n_k > 0$ for all $k$ by", "Proposition \\ref{proposition-positivity}.", "By Lemma \\ref{lemma-transfer} we have", "$$", "[Z] + \\sum m_j [Z_j] = \\sum n_k[E_{k, 0}]_s", "$$", "Since $E_{k, 0} \\subset C \\cap X$ we see that $[E_{k, 0}]_s$ and $\\alpha$", "intersect properly. On the other hand, the cycle", "$$", "\\gamma = \\sum n_k[E_{k, \\infty}]_s", "$$", "is supported on $C'_\\infty \\cap X$ and hence", "properly intersects each $W_{i, a_i}^t$, $W_{i, b_i}^t$.", "Thus by the base case and linearity, we see that", "$$", "\\gamma \\cdot \\alpha \\sim_{rat} 0", "$$", "As we have seen that $E_{k, 0}$ and $E_{k, \\infty}$", "intersect $\\alpha$ properly Lemma \\ref{lemma-well-defined-special-case}", "applied to $E_k \\subset X \\times \\mathbf{P}^1$ and $\\alpha$ gives", "$$", "[E_{k, 0}] \\cdot \\alpha \\sim_{rat} [E_{k, \\infty}] \\cdot \\alpha", "$$", "Putting everything together we have", "\\begin{align*}", "[Z] \\cdot \\alpha", "& =", "(\\sum n_k[E_{k, 0}]_r - \\sum m_j[Z_j]) \\cdot \\alpha \\\\", "& \\sim_{rat}", "\\sum n_k [E_{k, 0}] \\cdot \\alpha \\quad (\\text{by induction hypothesis})\\\\", "& \\sim_{rat}", "\\sum n_k [E_{k, \\infty}] \\cdot \\alpha \\quad (\\text{by the lemma})\\\\", "& =", "\\gamma \\cdot \\alpha \\\\", "& \\sim_{rat}", "0 \\quad (\\text{by base case})", "\\end{align*}", "This finishes the proof." ], "refs": [ "intersection-lemma-well-defined-special-case", "intersection-lemma-moving", "intersection-lemma-move", "intersection-proposition-positivity", "intersection-lemma-transfer", "intersection-lemma-well-defined-special-case" ], "ref_ids": [ 12000, 11997, 11998, 12002, 11989, 12000 ] } ], "ref_ids": [] }, { "id": 11958, "type": "theorem", "label": "intersection-lemma-push-coherent", "categories": [ "intersection" ], "title": "intersection-lemma-push-coherent", "contents": [ "\\begin{reference}", "See \\cite[Chapter V]{Serre_algebre_locale}.", "\\end{reference}", "Suppose that $f : X \\to Y$ is a proper morphism of varieties.", "Let $\\mathcal{F}$ be a coherent sheaf with", "$\\dim(\\text{Supp}(\\mathcal{F})) \\leq k$, then", "$f_*[\\mathcal{F}]_k = [f_*\\mathcal{F}]_k$. In particular, if", "$Z \\subset X$ is a closed subscheme of dimension $\\leq k$, then", "$f_*[Z]_k = [f_*\\mathcal{O}_Z]_k$." ], "refs": [], "proofs": [ { "contents": [ "See Chow Homology, Lemma \\ref{chow-lemma-cycle-push-sheaf}." ], "refs": [ "chow-lemma-cycle-push-sheaf" ], "ref_ids": [ 5676 ] } ], "ref_ids": [] }, { "id": 11959, "type": "theorem", "label": "intersection-lemma-compose-pushforward", "categories": [ "intersection" ], "title": "intersection-lemma-compose-pushforward", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms of", "varieties. Then $g_* \\circ f_* = (g \\circ f)_*$ as maps $Z_k(X) \\to Z_k(Z)$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Chow Homology, Lemma \\ref{chow-lemma-compose-pushforward}." ], "refs": [ "chow-lemma-compose-pushforward" ], "ref_ids": [ 5674 ] } ], "ref_ids": [] }, { "id": 11960, "type": "theorem", "label": "intersection-lemma-pullback", "categories": [ "intersection" ], "title": "intersection-lemma-pullback", "contents": [ "Let $f : X \\to Y$ be a flat morphism of varieties. Set $r = \\dim(X) - \\dim(Y)$.", "Then $f^*[\\mathcal{F}]_k = [f^*\\mathcal{F}]_{k + r}$", "if $\\mathcal{F}$ is a coherent sheaf on $Y$ and the dimension of the", "support of $\\mathcal{F}$ is at most $k$." ], "refs": [], "proofs": [ { "contents": [ "See Chow Homology, Lemma \\ref{chow-lemma-pullback-coherent}." ], "refs": [ "chow-lemma-pullback-coherent" ], "ref_ids": [ 5681 ] } ], "ref_ids": [] }, { "id": 11961, "type": "theorem", "label": "intersection-lemma-compose-flat-pullback", "categories": [ "intersection" ], "title": "intersection-lemma-compose-flat-pullback", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be flat morphisms of", "varieties. Then $g \\circ f$ is flat and $f^* \\circ g^* = (g \\circ f)^*$", "as maps $Z_k(Z) \\to Z_{k + \\dim(X) - \\dim(Z)}(X)$." ], "refs": [], "proofs": [ { "contents": [ "Special case of Chow Homology, Lemma \\ref{chow-lemma-compose-flat-pullback}." ], "refs": [ "chow-lemma-compose-flat-pullback" ], "ref_ids": [ 5680 ] } ], "ref_ids": [] }, { "id": 11962, "type": "theorem", "label": "intersection-lemma-rational-equivalence", "categories": [ "intersection" ], "title": "intersection-lemma-rational-equivalence", "contents": [ "Let $X$ be a variety. Let $W \\subset X$ be a subvariety", "of dimension $k + 1$. Let $f \\in \\mathbf{C}(W)^*$ be a nonzero rational", "function on $W$. Then $\\text{div}_W(f)$ is rationally equivalent to zero on", "$X$. Conversely, these principal divisors generate the abelian group of", "cycles rationally equivalent to zero on $X$." ], "refs": [], "proofs": [ { "contents": [ "The first assertion follows from", "Chow Homology, Lemma \\ref{chow-lemma-rational-function}.", "More precisely, let $W' \\subset X \\times \\mathbf{P}^1$ be the closure", "of the graph of $f$. Then $\\text{div}_W(f) = [W'_0]_k - [W'_\\infty]$", "in $Z_k(W) \\subset Z_k(X)$, see part (6) of", "Chow Homology, Lemma \\ref{chow-lemma-rational-function}.", "\\medskip\\noindent", "For the second, let $W' \\subset X \\times \\mathbf{P}^1$ be a closed", "subvariety of dimension $k + 1$ which dominates $\\mathbf{P}^1$.", "We will show that $[W'_0]_k - [W'_\\infty]_k$ is a principal divisor", "which will finish the proof. Let $W \\subset X$ be the image of $W'$", "under the projection to $X$. Then $W' \\to W$ is proper and generically", "finite\\footnote{If $W' \\to W$ is birational, then the result follows", "from Chow Homology, Lemma \\ref{chow-lemma-rational-function}.", "Our task is to show that even if $W' \\to W$", "has degree $> 1$ the basic rational equivalence", "$[W'_0]_k \\sim_{rat} [W'_\\infty]_k$ comes from a principal divisor", "on a subvariety of $X$.}. Let $f$ denote the projection $W' \\to \\mathbf{P}^1$", "viewed as an element of $\\mathbf{C}(W')^*$. Let", "$g = \\text{Nm}(f) \\in \\mathbf{C}(W)^*$ be the norm. By", "Chow Homology, Lemma \\ref{chow-lemma-proper-pushforward-alteration}", "we have", "$$", "\\text{div}_W(g) = \\text{pr}_{X, *}\\text{div}_{W'}(f)", "$$", "Since $\\text{div}_{W'}(f) = [W'_0]_k - [W'_\\infty]_k$ ", "by Chow Homology, Lemma \\ref{chow-lemma-rational-function}", "the proof is complete." ], "refs": [ "chow-lemma-rational-function", "chow-lemma-rational-function", "chow-lemma-rational-function", "chow-lemma-proper-pushforward-alteration", "chow-lemma-rational-function" ], "ref_ids": [ 5688, 5688, 5688, 5687, 5688 ] } ], "ref_ids": [] }, { "id": 11963, "type": "theorem", "label": "intersection-lemma-dimension-product-varieties", "categories": [ "intersection" ], "title": "intersection-lemma-dimension-product-varieties", "contents": [ "Let $X$ and $Y$ be varieties. Then $X \\times Y$ is a variety and", "$\\dim(X \\times Y) = \\dim(X) + \\dim(Y)$." ], "refs": [], "proofs": [ { "contents": [ "The scheme $X \\times Y = X \\times_{\\Spec(\\mathbf{C})} Y$ is a variety by", "Varieties, Lemma \\ref{varieties-lemma-product-varieties}.", "The statement on dimension is", "Varieties, Lemma \\ref{varieties-lemma-dimension-product-locally-algebraic}." ], "refs": [ "varieties-lemma-product-varieties", "varieties-lemma-dimension-product-locally-algebraic" ], "ref_ids": [ 10901, 10991 ] } ], "ref_ids": [] }, { "id": 11964, "type": "theorem", "label": "intersection-lemma-pullback-by-regular-immersion", "categories": [ "intersection" ], "title": "intersection-lemma-pullback-by-regular-immersion", "contents": [ "Let $f : X \\to Y$ be a morphism of varieties.", "\\begin{enumerate}", "\\item If $Z \\subset Y$ is a subvariety dimension $d$ and $f$ is a regular", "immersion of codimension $c$, then every irreducible component", "of $f^{-1}(Z)$ has dimension $\\geq d - c$.", "\\item If $Z \\subset Y$ is a subvariety of dimension $d$ and", "$f$ is a local complete intersection morphism of relative dimension $r$,", "then every irreducible component of $f^{-1}(Z)$ has dimension $\\geq d + r$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). We may work locally, hence we may assume that", "$Y = \\Spec(A)$ and $X = V(f_1, \\ldots, f_c)$ where $f_1, \\ldots, f_c$", "is a regular sequence in $A$. If $Z = \\Spec(A/\\mathfrak p)$, then", "we see that $f^{-1}(Z) = \\Spec(A/\\mathfrak p + (f_1, \\ldots, f_c))$.", "If $V$ is an irreducible component of $f^{-1}(Z)$, then we can", "choose a closed point $v \\in V$ not contained in any other irreducible", "component of $f^{-1}(Z)$. Then", "$$", "\\dim(Z) = \\dim \\mathcal{O}_{Z, v}", "\\quad\\text{and}\\quad", "\\dim(V) = \\dim \\mathcal{O}_{V, v} = \\dim \\mathcal{O}_{Z, v}/(f_1, \\ldots, f_c)", "$$", "The first equality for example by", "Algebra, Lemma \\ref{algebra-lemma-dimension-prime-polynomial-ring}", "and the second equality by our choice of closed point.", "The result now follows from the fact that dividing by one element", "in the maximal ideal decreases the dimension by at most $1$, see", "Algebra, Lemma \\ref{algebra-lemma-one-equation}.", "\\medskip\\noindent", "Proof of (2). Choose a factorization as in the definition of a", "local complete intersection and apply (1). Some details omitted." ], "refs": [ "algebra-lemma-dimension-prime-polynomial-ring" ], "ref_ids": [ 1005 ] } ], "ref_ids": [] }, { "id": 11965, "type": "theorem", "label": "intersection-lemma-diagonal-regular-immersion", "categories": [ "intersection" ], "title": "intersection-lemma-diagonal-regular-immersion", "contents": [ "Let $X$ be a nonsingular variety. Then the diagonal", "$\\Delta : X \\to X \\times X$ is a regular immersion of codimension $\\dim(X)$." ], "refs": [], "proofs": [ { "contents": [ "In fact, any closed immersion between nonsingular projective", "varieties is a regular immersion, see Divisors,", "Lemma \\ref{divisors-lemma-immersion-smooth-into-smooth-regular-immersion}." ], "refs": [ "divisors-lemma-immersion-smooth-into-smooth-regular-immersion" ], "ref_ids": [ 8007 ] } ], "ref_ids": [] }, { "id": 11966, "type": "theorem", "label": "intersection-lemma-intersect-in-smooth", "categories": [ "intersection" ], "title": "intersection-lemma-intersect-in-smooth", "contents": [ "Let $X$ be a nonsingular variety and let $W,V \\subset X$", "be closed subvarieties with $\\dim(W) = s$ and $\\dim(V) = r$. Then every", "irreducible component $Z$ of $V \\cap W$ has dimension $\\geq r + s - \\dim(X)$." ], "refs": [], "proofs": [ { "contents": [ "Since $V \\cap W = \\Delta^{-1}(V \\times W)$ (scheme theoretically)", "we conclude by Lemmas \\ref{lemma-diagonal-regular-immersion} and", "\\ref{lemma-pullback-by-regular-immersion}." ], "refs": [ "intersection-lemma-diagonal-regular-immersion", "intersection-lemma-pullback-by-regular-immersion" ], "ref_ids": [ 11965, 11964 ] } ], "ref_ids": [] }, { "id": 11967, "type": "theorem", "label": "intersection-lemma-tensor-coherent", "categories": [ "intersection" ], "title": "intersection-lemma-tensor-coherent", "contents": [ "Let $X$ be a locally Noetherian scheme.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ and $\\mathcal{G}$ are coherent $\\mathcal{O}_X$-modules,", "then $\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is too.", "\\item If $L$ and $K$ are in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, then", "so is $L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us explain how to prove (1) in a more elementary way and part (2)", "using previously developed general theory.", "\\medskip\\noindent", "Proof of (1). Since formation of $\\text{Tor}$ commutes with localization", "we may assume $X$ is affine. Hence $X = \\Spec(A)$ for some Noetherian", "ring $A$ and $\\mathcal{F}$, $\\mathcal{G}$ correspond to finite $A$-modules", "$M$ and $N$ (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-coherent-Noetherian}).", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-tensor-product} we may", "compute the $\\text{Tor}$'s by first computing the $\\text{Tor}$'s", "of $M$ and $N$ over $A$, and then taking the associated $\\mathcal{O}_X$-module.", "Since the modules $\\text{Tor}_p^A(M, N)$ are finite by", "Algebra, Lemma \\ref{algebra-lemma-tor-noetherian}", "we conclude.", "\\medskip\\noindent", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-identify-pseudo-coherent-noetherian}", "the assumption is equivalent to asking $L$ and $K$ to be", "(locally) pseudo-coherent. Then $L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K$", "is pseudo-coherent by", "Cohomology, Lemma \\ref{cohomology-lemma-tensor-pseudo-coherent}." ], "refs": [ "coherent-lemma-coherent-Noetherian", "perfect-lemma-quasi-coherence-tensor-product", "algebra-lemma-tor-noetherian", "perfect-lemma-identify-pseudo-coherent-noetherian", "cohomology-lemma-tensor-pseudo-coherent" ], "ref_ids": [ 3308, 6945, 785, 6976, 2208 ] } ], "ref_ids": [] }, { "id": 11968, "type": "theorem", "label": "intersection-lemma-compute-tor-nonsingular", "categories": [ "intersection" ], "title": "intersection-lemma-compute-tor-nonsingular", "contents": [ "Let $X$ be a nonsingular variety.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.", "The $\\mathcal{O}_X$-module", "$\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$", "is coherent, has stalk at $x$ equal to", "$\\text{Tor}_p^{\\mathcal{O}_{X, x}}(\\mathcal{F}_x, \\mathcal{G}_x)$,", "is supported on", "$\\text{Supp}(\\mathcal{F}) \\cap \\text{Supp}(\\mathcal{G})$, and", "is nonzero only for $p \\in \\{0, \\ldots, \\dim(X)\\}$." ], "refs": [], "proofs": [ { "contents": [ "The result on stalks was discussed above and it implies the support", "condition. The $\\text{Tor}$'s are coherent by", "Lemma \\ref{lemma-tensor-coherent}. The vanishing of negative", "$\\text{Tor}$'s is immediate from the construction. The", "vanishing of $\\text{Tor}_p$ for $p > \\dim(X)$ can be seen as follows:", "the local rings $\\mathcal{O}_{X, x}$ are regular", "(as $X$ is nonsingular) of dimension $\\leq \\dim(X)$", "(Algebra, Lemma \\ref{algebra-lemma-dimension-prime-polynomial-ring}),", "hence $\\mathcal{O}_{X, x}$ has finite global dimension $\\leq \\dim(X)$", "(Algebra, Lemma \\ref{algebra-lemma-finite-gl-dim-finite-dim-regular})", "which implies that $\\text{Tor}$-groups of modules vanish beyond the dimension", "(More on Algebra, Lemma \\ref{more-algebra-lemma-finite-gl-dim-tor-dimension})." ], "refs": [ "intersection-lemma-tensor-coherent", "algebra-lemma-dimension-prime-polynomial-ring", "algebra-lemma-finite-gl-dim-finite-dim-regular", "more-algebra-lemma-finite-gl-dim-tor-dimension" ], "ref_ids": [ 11967, 1005, 980, 10186 ] } ], "ref_ids": [] }, { "id": 11969, "type": "theorem", "label": "intersection-lemma-transversal", "categories": [ "intersection" ], "title": "intersection-lemma-transversal", "contents": [ "Let $X$ be a nonsingular variety. Let $V, W \\subset X$ be", "closed subvarieties which intersect properly. Let $Z$ be an irreducible", "component of $V \\cap W$ and assume that the multiplicity", "(in the sense of Section \\ref{section-cycle-of-closed}) of $Z$", "in the closed subscheme $V \\cap W$ is $1$.", "Then $e(X, V \\cdot W, Z) = 1$ and $V$ and $W$ are smooth", "in a general point of $Z$." ], "refs": [], "proofs": [ { "contents": [ "Let $(A, \\mathfrak m, \\kappa) =", "(\\mathcal{O}_{X, \\xi}, \\mathfrak m_\\xi, \\kappa(\\xi))$ where $\\xi \\in Z$", "is the generic point. Then $\\dim(A) = \\dim(X) - \\dim(Z)$, see", "Varieties, Lemma \\ref{varieties-lemma-dimension-locally-algebraic}.", "Let $I, J \\subset A$ cut out the trace of $V$ and $W$", "in $\\Spec(A)$. Set $\\overline{I} = I + \\mathfrak m^2/\\mathfrak m^2$.", "Then $\\dim_\\kappa \\overline{I} \\leq \\dim(X) - \\dim(V)$ with equality", "if and only if $A/I$ is regular (this follows from the lemma cited", "above and the definition of regular rings, see", "Algebra, Definition \\ref{algebra-definition-regular-local}", "and the discussion preceding it). Similarly for $\\overline{J}$.", "If the multiplicity is $1$, then", "$\\text{length}_A(A/I + J) = 1$, hence $I + J = \\mathfrak m$, hence", "$\\overline{I} + \\overline{J} = \\mathfrak m/\\mathfrak m^2$.", "Then we get equality everywhere (because the intersection is", "proper). Hence we find $f_1, \\ldots, f_a \\in I$ and $g_1, \\ldots g_b \\in J$", "such that $\\overline{f}_1, \\ldots, \\overline{g}_b$ is a basis", "for $\\mathfrak m/\\mathfrak m^2$. Then $f_1, \\ldots, g_b$ is a", "regular system of parameters and a regular sequence", "(Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}).", "The same lemma shows $A/(f_1, \\ldots, f_a)$ is a regular local ring", "of dimension $\\dim(X) - \\dim(V)$, hence $A/(f_1, \\ldots, f_a) \\to A/I$", "is an isomorphism (if the kernel is nonzero, then the dimension", "of $A/I$ is strictly less, see", "Algebra, Lemmas \\ref{algebra-lemma-regular-domain} and", "\\ref{algebra-lemma-one-equation}).", "We conclude $I = (f_1, \\ldots, f_a)$ and $J = (g_1, \\ldots, g_b)$", "by symmetry. Thus the Koszul complex $K_\\bullet(A, f_1, \\ldots, f_a)$", "on $f_1, \\ldots, f_a$ is a resolution of $A/I$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-regular-koszul-regular}.", "Hence", "\\begin{align*}", "\\text{Tor}_p^A(A/I, A/J)", "& =", "H_p(K_\\bullet(A, f_1, \\ldots, f_a) \\otimes_A A/J) \\\\", "& =", "H_p(K_\\bullet(A/J, f_1 \\bmod J, \\ldots, f_a \\bmod J))", "\\end{align*}", "Since we've seen above that $f_1 \\bmod J, \\ldots, f_a \\bmod J$ is", "a regular system of parameters in the regular local ring $A/J$", "we conclude that there is only one cohomology group, namely", "$H_0 = A/(I + J) = \\kappa$. This finishes the proof." ], "refs": [ "varieties-lemma-dimension-locally-algebraic", "algebra-definition-regular-local", "algebra-lemma-regular-ring-CM", "algebra-lemma-regular-domain", "more-algebra-lemma-regular-koszul-regular" ], "ref_ids": [ 10989, 1480, 941, 940, 9973 ] } ], "ref_ids": [] }, { "id": 11970, "type": "theorem", "label": "intersection-lemma-multiplicity-ses", "categories": [ "intersection" ], "title": "intersection-lemma-multiplicity-ses", "contents": [ "Let $A$ be a Noetherian local ring. Let $I \\subset A$ be an ideal of", "definition. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence", "of finite $A$-modules. Let $d \\geq \\dim(\\text{Supp}(M))$. Then", "$$", "e_I(M, d) = e_I(M', d) + e_I(M'', d)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions and", "Algebra, Lemma \\ref{algebra-lemma-hilbert-ses-chi}." ], "refs": [ "algebra-lemma-hilbert-ses-chi" ], "ref_ids": [ 678 ] } ], "ref_ids": [] }, { "id": 11971, "type": "theorem", "label": "intersection-lemma-multiplicity-as-a-sum", "categories": [ "intersection" ], "title": "intersection-lemma-multiplicity-as-a-sum", "contents": [ "Let $A$ be a Noetherian local ring. Let $I \\subset A$ be an ideal of", "definition. Let $M$ be a finite $A$-module. Let $d \\geq \\dim(\\text{Supp}(M))$.", "Then", "$$", "e_I(M, d) =", "\\sum \\text{length}_{A_\\mathfrak p}(M_\\mathfrak p) e_I(A/\\mathfrak p, d)", "$$", "where the sum is over primes $\\mathfrak p \\subset A$ with", "$\\dim(A/\\mathfrak p) = d$." ], "refs": [], "proofs": [ { "contents": [ "Both the left and side and the right hand side are additive in short", "exact sequences of modules of dimension $\\leq d$, see", "Lemma \\ref{lemma-multiplicity-ses} and", "Algebra, Lemma \\ref{algebra-lemma-length-additive}.", "Hence by Algebra, Lemma \\ref{algebra-lemma-filter-Noetherian-module}", "it suffices to prove this when $M = A/\\mathfrak q$ for some", "prime $\\mathfrak q$ of $A$ with $\\dim(A/\\mathfrak q) \\leq d$.", "This case is obvious." ], "refs": [ "intersection-lemma-multiplicity-ses", "algebra-lemma-length-additive", "algebra-lemma-filter-Noetherian-module" ], "ref_ids": [ 11970, 631, 691 ] } ], "ref_ids": [] }, { "id": 11972, "type": "theorem", "label": "intersection-lemma-leading-coefficient", "categories": [ "intersection" ], "title": "intersection-lemma-leading-coefficient", "contents": [ "Let $P$ be a polynomial of degree $r$ with leading coefficient $a$.", "Then", "$$", "r! a = \\sum\\nolimits_{i = 0, \\ldots, r} (-1)^i{r \\choose i} P(t - i)", "$$", "for any $t$." ], "refs": [], "proofs": [ { "contents": [ "Let us write $\\Delta$ the operator which to a polynomial $P$ associates", "the polynomial $\\Delta(P) = P(t) - P(t - 1)$. We claim that", "$$", "\\Delta^r(P) = \\sum\\nolimits_{i = 0, \\ldots, r} (-1)^i {r \\choose i} P(t - i)", "$$", "This is true for $r = 0, 1$ by inspection. Assume it is true for $r$.", "Then we compute", "\\begin{align*}", "\\Delta^{r + 1}(P)", "& =", "\\sum\\nolimits_{i = 0, \\ldots, r} (-1)^i {r \\choose i} \\Delta(P)(t - i) \\\\", "& =", "\\sum\\nolimits_{n = -r, \\ldots, 0} (-1)^i {r \\choose i}", "(P(t - i) - P(t - i - 1))", "\\end{align*}", "Thus the claim follows from the equality", "$$", "{r + 1 \\choose i} = {r \\choose i} + {r \\choose i - 1}", "$$", "The lemma follows from the fact that $\\Delta(P)$ is of degree $r - 1$", "with leading coefficient $ra$ if the degree of $P$ is $r$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11973, "type": "theorem", "label": "intersection-lemma-intersection-multiplicity-CM", "categories": [ "intersection" ], "title": "intersection-lemma-intersection-multiplicity-CM", "contents": [ "Let $X$ be a nonsingular variety and $W, V \\subset X$ closed", "subvarieties which intersect properly. Let $Z$ be an irreducible component", "of $V \\cap W$ with generic point $\\xi$. Assume that $\\mathcal{O}_{W, \\xi}$", "and $\\mathcal{O}_{V, \\xi}$ are Cohen-Macaulay. Then", "$$", "e(X, V \\cdot W, Z) =", "\\text{length}_{\\mathcal{O}_{X, \\xi}}(\\mathcal{O}_{V \\cap W, \\xi})", "$$", "where $V \\cap W$ is the scheme theoretic intersection.", "In particular, if both $V$ and $W$ are Cohen-Macaulay, then", "$V \\cdot W = [V \\cap W]_{\\dim(V) + \\dim(W) - \\dim(X)}$." ], "refs": [], "proofs": [ { "contents": [ "Set $A = \\mathcal{O}_{X, \\xi}$, $B = \\mathcal{O}_{V, \\xi}$, and", "$C = \\mathcal{O}_{W, \\xi}$. By Auslander-Buchsbaum", "(Algebra, Proposition \\ref{algebra-proposition-Auslander-Buchsbaum})", "we can find a finite free resolution $F_\\bullet \\to B$ of", "length", "$$", "\\text{depth}(A) - \\text{depth}(B) =", "\\dim(A) - \\dim(B) = \\dim(C)", "$$", "First equality as $A$ and $B$ are Cohen-Macaulay and the second", "as $V$ and $W$ intersect properly. Then $F_\\bullet \\otimes_A C$ is a", "complex of finite free modules representing $B \\otimes_A^\\mathbf{L} C$", "hence has cohomology modules with support in $\\{\\mathfrak m_A\\}$.", "By the Acyclicity lemma (Algebra, Lemma \\ref{algebra-lemma-acyclic})", "which applies as $C$ is Cohen-Macaulay", "we conclude that $F_\\bullet \\otimes_A C$ has nonzero", "cohomology only in degree $0$. This finishes the proof." ], "refs": [ "algebra-proposition-Auslander-Buchsbaum", "algebra-lemma-acyclic" ], "ref_ids": [ 1423, 913 ] } ], "ref_ids": [] }, { "id": 11974, "type": "theorem", "label": "intersection-lemma-one-ideal-ci", "categories": [ "intersection" ], "title": "intersection-lemma-one-ideal-ci", "contents": [ "Let $A$ be a Noetherian local ring. Let $I = (f_1, \\ldots, f_r)$ be an ideal", "generated by a regular sequence. Let $M$ be a finite $A$-module. Assume that", "$\\dim(\\text{Supp}(M/IM)) = 0$. Then", "$$", "e_I(M, r) = \\sum (-1)^i\\text{length}_A(\\text{Tor}_i^A(A/I, M))", "$$", "Here $e_I(M, r)$ is as in Remark \\ref{remark-trivial-generalization}." ], "refs": [ "intersection-remark-trivial-generalization" ], "proofs": [ { "contents": [ "Since $f_1, \\ldots, f_r$ is a regular sequence the Koszul complex", "$K_\\bullet(f_1, \\ldots, f_r)$ is a resolution of $A/I$ over $A$, see", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-noetherian-finite-all-equivalent}.", "Thus the right hand side is equal to", "$$", "\\sum (-1)^i\\text{length}_A H_i(K_\\bullet(f_1, \\ldots, f_r) \\otimes_A M)", "$$", "Now the result follows immediately from", "Theorem \\ref{theorem-multiplicity-with-koszul} if $I$ is an ideal", "of definition. In general, we replace $A$ by $\\overline{A} = A/\\text{Ann}(M)$", "and $f_1, \\ldots, f_r$ by $\\overline{f}_1, \\ldots, \\overline{f}_r$", "which is allowed because", "$$", "K_\\bullet(f_1, \\ldots, f_r) \\otimes_A M =", "K_\\bullet(\\overline{f}_1, \\ldots, \\overline{f}_r) \\otimes_{\\overline{A}} M", "$$", "Since $e_I(M, r) = e_{\\overline{I}}(M, r)$ where", "$\\overline{I} = (\\overline{f}_1, \\ldots, \\overline{f}_r) \\subset \\overline{A}$", "is an ideal of definition the result follows from", "Theorem \\ref{theorem-multiplicity-with-koszul} in this case as well." ], "refs": [ "more-algebra-lemma-noetherian-finite-all-equivalent", "intersection-theorem-multiplicity-with-koszul", "intersection-theorem-multiplicity-with-koszul" ], "ref_ids": [ 9978, 11956, 11956 ] } ], "ref_ids": [ 12005 ] }, { "id": 11975, "type": "theorem", "label": "intersection-lemma-multiplicity-with-lci", "categories": [ "intersection" ], "title": "intersection-lemma-multiplicity-with-lci", "contents": [ "Let $X$ be a nonsingular variety. Let $W,V \\subset X$ be", "closed subvarieties which intersect properly. Let $Z$ be an irreducible", "component of $V \\cap W$ with generic point $\\xi$.", "Suppose the ideal of $V$ in $\\mathcal{O}_{X, \\xi}$ is cut out by", "a regular sequence $f_1, \\ldots, f_c \\in \\mathcal{O}_{X, \\xi}$.", "Then $e(X, V\\cdot W, Z)$ is equal to $c!$ times the leading coefficient in", "the Hilbert polynomial", "$$", "t \\mapsto \\text{length}_{\\mathcal{O}_{X, \\xi}}", "\\mathcal{O}_{W, \\xi}/(f_1, \\ldots, f_c)^t,\\quad t \\gg 0.", "$$", "In particular, this coefficient is $> 0$." ], "refs": [], "proofs": [ { "contents": [ "The equality", "$$", "e(X, V\\cdot W, Z) = e_{(f_1, \\ldots, f_c)}(\\mathcal{O}_{W, \\xi}, c)", "$$", "follows from the more general Lemma \\ref{lemma-one-ideal-ci}.", "To see that $e_{(f_1, \\ldots, f_c)}(\\mathcal{O}_{W, \\xi}, c)$ is", "$> 0$ or equivalently that $e_{(f_1, \\ldots, f_c)}(\\mathcal{O}_{W, \\xi}, c)$", "is the leading coefficient of the Hilbert polynomial", "it suffices to show that the", "dimension of $\\mathcal{O}_{W, \\xi}$ is $c$, because the degree of the", "Hilbert polynomial is equal to the dimension by", "Algebra, Proposition \\ref{algebra-proposition-dimension}.", "Say $\\dim(V) = r$, $\\dim(W) = s$, and $\\dim(X) = n$. Then", "$\\dim(Z) = r + s - n$ as the intersection is proper. Thus", "the transcendence degree of $\\kappa(\\xi)$ over $\\mathbf{C}$ is", "$r + s - n$, see Algebra, Lemma", "\\ref{algebra-lemma-dimension-prime-polynomial-ring}.", "We have $r + c = n$ because $V$ is cut out by a regular sequence", "in a neighbourhood of $\\xi$, see", "Divisors, Lemma \\ref{divisors-lemma-Noetherian-scheme-regular-ideal}", "and then Lemma \\ref{lemma-pullback-by-regular-immersion}", "applies (for example). Thus", "$$", "\\dim(\\mathcal{O}_{W, \\xi}) = s - (r + s - n) = s - ((n - c) + s - n) = c", "$$", "the first equality by Algebra, Lemma", "\\ref{algebra-lemma-dimension-at-a-point-finite-type-field}." ], "refs": [ "intersection-lemma-one-ideal-ci", "algebra-proposition-dimension", "algebra-lemma-dimension-prime-polynomial-ring", "divisors-lemma-Noetherian-scheme-regular-ideal", "intersection-lemma-pullback-by-regular-immersion", "algebra-lemma-dimension-at-a-point-finite-type-field" ], "ref_ids": [ 11974, 1411, 1005, 7988, 11964, 1007 ] } ], "ref_ids": [] }, { "id": 11976, "type": "theorem", "label": "intersection-lemma-multiplicity-with-effective-Cartier-divisor", "categories": [ "intersection" ], "title": "intersection-lemma-multiplicity-with-effective-Cartier-divisor", "contents": [ "In Lemma \\ref{lemma-multiplicity-with-lci} assume that $c = 1$, i.e., $V$", "is an effective Cartier divisor. Then", "$$", "e(X, V \\cdot W, Z) =", "\\text{length}_{\\mathcal{O}_{X, \\xi}}", "(\\mathcal{O}_{W, \\xi}/f_1\\mathcal{O}_{W, \\xi}).", "$$" ], "refs": [ "intersection-lemma-multiplicity-with-lci" ], "proofs": [ { "contents": [ "In this case the image of $f_1$ in $\\mathcal{O}_{W, \\xi}$ is nonzero by", "properness of intersection, hence a nonzerodivisor divisor. Moreover,", "$\\mathcal{O}_{W, \\xi}$ is a Noetherian local domain of dimension $1$.", "Thus", "$$", "\\text{length}_{\\mathcal{O}_{X, \\xi}}", "(\\mathcal{O}_{W, \\xi}/f_1^t\\mathcal{O}_{W, \\xi}) =", "t \\text{length}_{\\mathcal{O}_{X, \\xi}}", "(\\mathcal{O}_{W, \\xi}/f_1\\mathcal{O}_{W, \\xi})", "$$", "for all $t \\geq 1$, see Algebra, Lemma \\ref{algebra-lemma-ord-additive}.", "This proves the lemma." ], "refs": [ "algebra-lemma-ord-additive" ], "ref_ids": [ 1043 ] } ], "ref_ids": [ 11975 ] }, { "id": 11977, "type": "theorem", "label": "intersection-lemma-multiplicity-lci-CM", "categories": [ "intersection" ], "title": "intersection-lemma-multiplicity-lci-CM", "contents": [ "In Lemma \\ref{lemma-multiplicity-with-lci} assume that", "the local ring $\\mathcal{O}_{W, \\xi}$ is Cohen-Macaulay. Then we", "have", "$$", "e(X, V \\cdot W, Z) =", "\\text{length}_{\\mathcal{O}_{X, \\xi}} (\\mathcal{O}_{W, \\xi}/", "f_1\\mathcal{O}_{W, \\xi} + \\ldots + f_c\\mathcal{O}_{W, \\xi}).", "$$" ], "refs": [ "intersection-lemma-multiplicity-with-lci" ], "proofs": [ { "contents": [ "This follows immediately from Lemma \\ref{lemma-intersection-multiplicity-CM}.", "Alternatively, we can deduce it from Lemma \\ref{lemma-multiplicity-with-lci}.", "Namely, by Algebra, Lemma \\ref{algebra-lemma-reformulate-CM}", "we see that $f_1, \\ldots, f_c$ is a regular sequence in", "$\\mathcal{O}_{W, \\xi}$. Then", "Algebra, Lemma \\ref{algebra-lemma-regular-quasi-regular} shows that", "$f_1, \\ldots, f_c$ is a quasi-regular sequence.", "This easily implies the length of", "$\\mathcal{O}_{W, \\xi}/(f_1, \\ldots, f_c)^t$ is", "$$", "{c + t \\choose c}", "\\text{length}_{\\mathcal{O}_{X, \\xi}} (\\mathcal{O}_{W, \\xi}/", "f_1\\mathcal{O}_{W, \\xi} + \\ldots + f_c\\mathcal{O}_{W, \\xi}).", "$$", "Looking at the leading coefficient we conclude." ], "refs": [ "intersection-lemma-intersection-multiplicity-CM", "intersection-lemma-multiplicity-with-lci", "algebra-lemma-reformulate-CM", "algebra-lemma-regular-quasi-regular" ], "ref_ids": [ 11973, 11975, 923, 746 ] } ], "ref_ids": [ 11975 ] }, { "id": 11978, "type": "theorem", "label": "intersection-lemma-rational-equivalence-and-intersection", "categories": [ "intersection" ], "title": "intersection-lemma-rational-equivalence-and-intersection", "contents": [ "Let $X$ be a nonsingular variety. Let $a, b \\in \\mathbf{P}^1$", "be distinct closed points. Let $k \\geq 0$.", "\\begin{enumerate}", "\\item If $W \\subset X \\times \\mathbf{P}^1$ is a closed subvariety", "of dimension $k + 1$ which intersects $X \\times a$ properly, then", "\\begin{enumerate}", "\\item $[W_a]_k = W \\cdot X \\times a$ as cycles on $X \\times \\mathbf{P}^1$, and", "\\item $[W_a]_k = \\text{pr}_{X, *}(W \\cdot X \\times a)$ as cycles on $X$.", "\\end{enumerate}", "\\item Let $\\alpha$ be a $(k + 1)$-cycle on $X \\times \\mathbf{P}^1$", "which intersects $X \\times a$ and $X \\times b$ properly. Then", "$pr_{X,*}( \\alpha \\cdot X \\times a - \\alpha \\cdot X \\times b)$", "is rationally equivalent to zero.", "\\item Conversely, any $k$-cycle which is", "rationally equivalent to $0$ is of this form.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First we observe that $X \\times a$ is an effective Cartier divisor in", "$X \\times \\mathbf{P}^1$ and that $W_a$ is the scheme theoretic intersection", "of $W$ with $X \\times a$. Hence the equality in (1)(a) is", "immediate from the definitions and the calculation of intersection", "multiplicity in case of a Cartier divisor given in", "Lemma \\ref{lemma-multiplicity-with-effective-Cartier-divisor}.", "Part (1)(b) holds because $W_a \\to X \\times \\mathbf{P}^1 \\to X$ maps", "isomorphically onto its image which is how we viewed $W_a$", "as a closed subscheme of $X$ in Section \\ref{section-rational-equivalence}.", "Parts (2) and (3) are formal consequences of part (1) and the definitions." ], "refs": [ "intersection-lemma-multiplicity-with-effective-Cartier-divisor" ], "ref_ids": [ 11976 ] } ], "ref_ids": [] }, { "id": 11979, "type": "theorem", "label": "intersection-lemma-transversal-subschemes", "categories": [ "intersection" ], "title": "intersection-lemma-transversal-subschemes", "contents": [ "Let $X$ be a nonsingular variety. Let $r, s \\geq 0$ and let", "$Y, Z \\subset X$ be closed subschemes with $\\dim(Y) \\leq r$ and", "$\\dim(Z) \\leq s$. Assume $[Y]_r = \\sum n_i[Y_i]$ and", "$[Z]_s = \\sum m_j[Z_j]$ intersect properly.", "Let $T$ be an irreducible component of $Y_{i_0} \\cap Z_{j_0}$", "for some $i_0$ and $j_0$ and assume that the multiplicity", "(in the sense of Section \\ref{section-cycle-of-closed}) of $T$", "in the closed subscheme $Y \\cap Z$ is $1$.", "Then", "\\begin{enumerate}", "\\item the coefficient of $T$ in $[Y]_r \\cdot [Z]_s$ is $1$,", "\\item $Y$ and $Z$ are nonsingular at the generic point of $Z$,", "\\item $n_{i_0} = 1$, $m_{j_0} = 1$, and", "\\item $T$ is not contained in $Y_i$ or $Z_j$ for $i \\not = i_0$ and", "$j \\not = j_0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Set $n = \\dim(X)$, $a = n - r$, $b = n - s$. Observe that", "$\\dim(T) = r + s - n = n - a - b$ by the assumption that the", "intersections are transversal. Let $(A, \\mathfrak m, \\kappa) =", "(\\mathcal{O}_{X, \\xi}, \\mathfrak m_\\xi, \\kappa(\\xi))$ where $\\xi \\in T$", "is the generic point. Then $\\dim(A) = a + b$, see", "Varieties, Lemma \\ref{varieties-lemma-dimension-locally-algebraic}.", "Let $I_0, I, J_0, J \\subset A$ cut out the trace of", "$Y_{i_0}$, $Y$, $Z_{j_0}$, $Z$ in $\\Spec(A)$.", "Then $\\dim(A/I) = \\dim(A/I_0) = b$ and $\\dim(A/J) = \\dim(A/J_0) = a$", "by the same reference. Set $\\overline{I} = I + \\mathfrak m^2/\\mathfrak m^2$.", "Then $I \\subset I_0 \\subset \\mathfrak m$ and", "$J \\subset J_0 \\subset \\mathfrak m$ and $I + J = \\mathfrak m$.", "By Lemma \\ref{lemma-transversal} and its proof we see that", "$I_0 = (f_1, \\ldots, f_a)$ and $J_0 = (g_1, \\ldots, g_b)$", "where $f_1, \\ldots, g_b$ is a regular system of parameters", "for the regular local ring $A$. Since $I + J = \\mathfrak m$, the map", "$$", "I \\oplus J \\to", "\\mathfrak m/\\mathfrak m^2 =", "\\kappa f_1", "\\oplus \\ldots \\oplus", "\\kappa f_a", "\\oplus", "\\kappa g_1", "\\oplus \\ldots \\oplus", "\\kappa g_b", "$$", "is surjective. We conclude that we can find", "$f_1', \\ldots, f_a' \\in I$ and $g'_1, \\ldots, g_b' \\in J$", "whose residue classes in $\\mathfrak m/\\mathfrak m^2$ are equal to the", "residue classes of $f_1, \\ldots, f_a$ and $g_1, \\ldots, g_b$.", "Then $f'_1, \\ldots, g'_b$ is a regular system of parameters of $A$.", "By Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM} we find that", "$A/(f'_1, \\ldots, f'_a)$ is a regular local ring of dimension $b$.", "Thus any nontrivial quotient of $A/(f'_1, \\ldots, f'_a)$", "has strictly smaller dimension", "(Algebra, Lemmas \\ref{algebra-lemma-regular-domain} and", "\\ref{algebra-lemma-one-equation}). Hence $I = (f'_1, \\ldots, f'_a) = I_0$.", "By symmetry $J = J_0$. This proves (2), (3), and (4).", "Finally, the coefficient of $T$ in $[Y]_r \\cdot [Z]_s$", "is the coefficient of $T$ in $Y_{i_0} \\cdot Z_{j_0}$ which is", "$1$ by Lemma \\ref{lemma-transversal}." ], "refs": [ "varieties-lemma-dimension-locally-algebraic", "intersection-lemma-transversal", "algebra-lemma-regular-ring-CM", "algebra-lemma-regular-domain", "intersection-lemma-transversal" ], "ref_ids": [ 10989, 11969, 941, 940, 11969 ] } ], "ref_ids": [] }, { "id": 11980, "type": "theorem", "label": "intersection-lemma-exterior-product-rational-equivalence", "categories": [ "intersection" ], "title": "intersection-lemma-exterior-product-rational-equivalence", "contents": [ "Let $X$ and $Y$ be varieties.", "Let $\\alpha \\in Z_r(X)$ and $\\beta \\in Z_s(Y)$.", "If $\\alpha \\sim_{rat} 0$ or $\\beta \\sim_{rat} 0$, then", "$\\alpha \\times \\beta \\sim_{rat} 0$." ], "refs": [], "proofs": [ { "contents": [ "By linearity and symmetry in $X$ and $Y$, it suffices to prove this when", "$\\alpha = [V]$ for some subvariety $V \\subset X$ of dimension $s$ and", "$\\beta = [W_a]_s - [W_b]_s$ for some closed subvariety", "$W \\subset Y \\times \\mathbf{P}^1$ of dimension $s + 1$ which", "intersects $Y \\times a$ and $Y \\times b$ properly. In this case", "the lemma follows if we can prove", "$$", "[(V \\times W)_a]_{r + s} = [V] \\times [W_a]_s", "$$", "and similarly with $a$ replaced by $b$. Namely, then we see that", "$\\alpha \\times \\beta = [(V \\times W)_a]_{r + s} - [(V \\times W)_b]_{r + s}$", "as desired. To see the displayed equality we note the equality", "$$", "V \\times W_a = (V \\times W)_a", "$$", "of schemes. The projection $V \\times W_a \\to W_a$ induces a bijection", "of irreducible components (see for example", "Varieties, Lemma \\ref{varieties-lemma-bijection-irreducible-components}).", "Let $W' \\subset W_a$ be an irreducible component with generic point $\\zeta$.", "Then $V \\times W'$ is the corresponding irreducible component of", "$V \\times W_a$ (see Lemma \\ref{lemma-dimension-product-varieties}).", "Let $\\xi$ be the generic point of $V \\times W'$. We have to show that", "$$", "\\text{length}_{\\mathcal{O}_{Y, \\zeta}}(\\mathcal{O}_{W_a, \\zeta}) =", "\\text{length}_{\\mathcal{O}_{X \\times Y, \\xi}}(", "\\mathcal{O}_{V \\times W_a, \\xi})", "$$", "In this formula we may replace", "$\\mathcal{O}_{Y, \\zeta}$ by $\\mathcal{O}_{W_a, \\zeta}$ and", "we may replace", "$\\mathcal{O}_{X \\times Y, \\zeta}$ by $\\mathcal{O}_{V \\times W_a, \\zeta}$", "(see Algebra, Lemma \\ref{algebra-lemma-length-independent}).", "As $\\mathcal{O}_{W_a, \\zeta} \\to \\mathcal{O}_{V \\times W_a, \\xi}$ is flat,", "by Algebra, Lemma \\ref{algebra-lemma-pullback-module} it suffices", "to show that", "$$", "\\text{length}_{\\mathcal{O}_{V \\times W_a, \\xi}}(", "\\mathcal{O}_{V \\times W_a, \\xi}/", "\\mathfrak m_\\zeta\\mathcal{O}_{V \\times W_a, \\xi}) = 1", "$$", "This is true because the quotient on the right is the local ring", "$\\mathcal{O}_{V \\times W', \\xi}$ of a variety at a generic point", "hence equal to $\\kappa(\\xi)$." ], "refs": [ "varieties-lemma-bijection-irreducible-components", "intersection-lemma-dimension-product-varieties", "algebra-lemma-length-independent", "algebra-lemma-pullback-module" ], "ref_ids": [ 10934, 11963, 633, 640 ] } ], "ref_ids": [] }, { "id": 11981, "type": "theorem", "label": "intersection-lemma-exterior-product", "categories": [ "intersection" ], "title": "intersection-lemma-exterior-product", "contents": [ "Let $X$ and $Y$ be nonsingular varieties.", "Let $\\alpha \\in Z_r(X)$ and $\\beta \\in Z_s(Y)$.", "Then", "\\begin{enumerate}", "\\item $\\text{pr}_Y^*(\\beta) = [X] \\times \\beta$ and", "$\\text{pr}_X^*(\\alpha) = \\alpha \\times [Y]$,", "\\item $\\alpha \\times [Y]$ and $[X]\\times \\beta$", "intersect properly on $X\\times Y$, and", "\\item we have", "$\\alpha \\times \\beta =", "(\\alpha \\times [Y])\\cdot ([X]\\times\\beta) =", "pr_Y^*(\\alpha) \\cdot pr_X^*(\\beta)$", "in $Z_{r + s}(X \\times Y)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By linearity we may assume $\\alpha = [V]$ and $\\beta = [W]$.", "Then (1) says that $\\text{pr}_Y^{-1}(W) = X \\times W$ and", "$\\text{pr}_X^{-1}(V) = V \\times Y$. This is clear.", "Part (2) holds because $X \\times W \\cap V \\times Y = V \\times W$ and", "$\\dim(V \\times W) = r + s$ by Lemma \\ref{lemma-dimension-product-varieties}.", "\\medskip\\noindent", "Proof of (3).", "Let $\\xi$ be the generic point of $V \\times W$.", "Since the projections $X \\times W \\to W$ is smooth as a base change of", "$X \\to \\Spec(\\mathbf{C})$, we see that $X \\times W$ is nonsingular", "at every point lying over the generic point of $W$, in particular at $\\xi$.", "Similarly for $V \\times Y$. Hence $\\mathcal{O}_{X \\times W, \\xi}$", "and $\\mathcal{O}_{V \\times Y, \\xi}$ are Cohen-Macaulay local rings", "and Lemma \\ref{lemma-intersection-multiplicity-CM} applies.", "Since $V \\times Y \\cap X \\times W = V \\times W$ scheme theoretically", "the proof is complete." ], "refs": [ "intersection-lemma-dimension-product-varieties", "intersection-lemma-intersection-multiplicity-CM" ], "ref_ids": [ 11963, 11973 ] } ], "ref_ids": [] }, { "id": 11982, "type": "theorem", "label": "intersection-lemma-tor-and-diagonal", "categories": [ "intersection" ], "title": "intersection-lemma-tor-and-diagonal", "contents": [ "Let $X$ be a nonsingular variety.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ and $\\mathcal{G}$ are coherent $\\mathcal{O}_X$-modules,", "then there are canonical isomorphisms", "$$", "\\text{Tor}_i^{\\mathcal{O}_{X \\times X}}(\\mathcal{O}_\\Delta,", "\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times X}}", "\\text{pr}_2^*\\mathcal{G})", "=", "\\Delta_*\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})", "$$", "\\item If $K$ and $M$ are in $D_\\QCoh(\\mathcal{O}_X)$, then", "there is a canonical isomorphism", "$$", "L\\Delta^* \\left(", "L\\text{pr}_1^*K \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L} L\\text{pr}_2^*M", "\\right)", "= K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M", "$$", "in $D_\\QCoh(\\mathcal{O}_X)$ and a canonical isomorphism", "$$", "\\mathcal{O}_\\Delta \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L}", "L\\text{pr}_1^*K \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L} L\\text{pr}_2^*M", "= \\Delta_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)", "$$", "in $D_\\QCoh(X \\times X)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us explain how to prove (1) in a more elementary way and part (2) using", "more general theory. As (2) implies (1) the reader can skip the proof", "of (1).", "\\medskip\\noindent", "Proof of (1). Choose an affine open $\\Spec(A) \\subset X$.", "Then $A$ is a Noetherian $\\mathbf{C}$-algebra and", "$\\mathcal{F}$, $\\mathcal{G}$ correspond to finite $A$-modules", "$M$ and $N$ (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-coherent-Noetherian}).", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-tensor-product} we may", "compute $\\text{Tor}_i$ over $\\mathcal{O}_X$", "by first computing the $\\text{Tor}$'s", "of $M$ and $N$ over $A$, and then taking the associated $\\mathcal{O}_X$-module.", "For the $\\text{Tor}_i$ over $\\mathcal{O}_{X \\times X}$ we compute", "the tor of $A$ and $M \\otimes_\\mathbf{C} N$ over $A \\otimes_\\mathbf{C} A$", "and then take the associated $\\mathcal{O}_{X \\times X}$-module.", "Hence on this affine patch we have to prove that", "$$", "\\text{Tor}_i^{A \\otimes_\\mathbf{C} A}(A, M \\otimes_\\mathbf{C} N) =", "\\text{Tor}_i^A(M, N)", "$$", "To see this choose resolutions $F_\\bullet \\to M$ and $G_\\bullet \\to M$", "by finite free $A$-modules", "(Algebra, Lemma \\ref{algebra-lemma-resolution-by-finite-free}).", "Note that $\\text{Tot}(F_\\bullet \\otimes_\\mathbf{C} G_\\bullet)$", "is a resolution of $M \\otimes_\\mathbf{C} N$ as it computes", "Tor groups over $\\mathbf{C}$! Of course the terms of", "$F_\\bullet \\otimes_\\mathbf{C} G_\\bullet$ are finite free", "$A \\otimes_\\mathbf{C} A$-modules. Hence the left hand side", "of the displayed equation is the module", "$$", "H_i(A \\otimes_{A \\otimes_\\mathbf{C} A}", "\\text{Tot}(F_\\bullet \\otimes_\\mathbf{C} G_\\bullet))", "$$", "and the right hand side is the module", "$$", "H_i(\\text{Tot}(F_\\bullet \\otimes_A G_\\bullet))", "$$", "Since $A \\otimes_{A \\otimes_\\mathbf{C} A} (F_p \\otimes_\\mathbf{C} G_q)", "= F_p \\otimes_A G_q$ we see that these modules are equal.", "This defines an isomorphism over the affine open $\\Spec(A) \\times \\Spec(A)$", "(which is good enough for the application to equality of intersection numbers).", "We omit the proof that these isomorphisms glue.", "\\medskip\\noindent", "Proof of (2). The second statement follows from the first by the", "projection formula as stated in", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-cohomology-base-change}.", "To see the first, represent $K$ and $M$ by K-flat complexes", "$\\mathcal{K}^\\bullet$ and $\\mathcal{M}^\\bullet$.", "Since pullback and tensor product preserve K-flat complexes", "(Cohomology, Lemmas \\ref{cohomology-lemma-tensor-product-K-flat} and", "\\ref{cohomology-lemma-pullback-K-flat})", "we see that it suffices to show", "$$", "\\Delta^*\\text{Tot}(", "\\text{pr}_1^*\\mathcal{K}^\\bullet", "\\otimes_{\\mathcal{O}_{X \\times X}} \\text{pr}_2^*\\mathcal{M}^\\bullet)", "=", "\\text{Tot}(", "\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{M}^\\bullet)", "$$", "Thus it suffices to see that there are canonical isomorphisms", "$$", "\\Delta^*(\\text{pr}_1^*\\mathcal{K}", "\\otimes_{\\mathcal{O}_{X \\times X}} \\text{pr}_2^*\\mathcal{M})", "\\longrightarrow", "\\mathcal{K} \\otimes_{\\mathcal{O}_X} \\mathcal{M}", "$$", "whenever $\\mathcal{K}$ and $\\mathcal{M}$ are $\\mathcal{O}_X$-modules", "(not necessarily quasi-coherent or flat).", "We omit the details." ], "refs": [ "coherent-lemma-coherent-Noetherian", "perfect-lemma-quasi-coherence-tensor-product", "algebra-lemma-resolution-by-finite-free", "perfect-lemma-cohomology-base-change", "cohomology-lemma-tensor-product-K-flat", "cohomology-lemma-pullback-K-flat" ], "ref_ids": [ 3308, 6945, 761, 7025, 2105, 2108 ] } ], "ref_ids": [] }, { "id": 11983, "type": "theorem", "label": "intersection-lemma-reduction-diagonal", "categories": [ "intersection" ], "title": "intersection-lemma-reduction-diagonal", "contents": [ "Let $X$ be a nonsingular variety. Let $\\alpha$, resp.\\ $\\beta$", "be an $r$-cycle, resp.\\ $s$-cycle on $X$. Assume $\\alpha$ and $\\beta$", "intersect properly. Then", "\\begin{enumerate}", "\\item $\\alpha \\times \\beta$ and $[\\Delta]$ intersect properly", "\\item we have $\\Delta_*(\\alpha \\cdot \\beta) = [\\Delta] \\cdot \\alpha\\times\\beta$", "as cycles on $X \\times X$,", "\\item if $X$ is proper, then", "$\\text{pr}_{1, *}([\\Delta] \\cdot \\alpha\\times\\beta) = \\alpha\\cdot\\beta$,", "where $pr_1 : X\\times X \\to X$ is the projection.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By linearity it suffices to prove this when $\\alpha = [V]$ and $\\beta = [W]$", "for some closed subvarieties $V \\subset X$ and $W \\subset Y$ which intersect", "properly. Recall that $V \\times W$ is a closed subvariety of dimension $r + s$.", "Observe that scheme theoretically we have", "$V \\cap W = \\Delta^{-1}(V \\times W)$ as well as", "$\\Delta(V \\cap W) = \\Delta \\cap V \\times W$.", "This proves (1).", "\\medskip\\noindent", "Proof of (2). Let $Z \\subset V \\cap W$ be an irreducible component", "with generic point $\\xi$. We have to show that the coefficient of", "$Z$ in $\\alpha \\cdot \\beta$ is the same as the coefficient of", "$\\Delta(Z)$ in $[\\Delta] \\cdot \\alpha \\times \\beta$. The first is given", "by the integer", "$$", "\\sum (-1)^i", "\\text{length}_{\\mathcal{O}_{X, \\xi}}", "\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{O}_V, \\mathcal{O}_W)_\\xi", "$$", "and the second by the integer", "$$", "\\sum (-1)^i", "\\text{length}_{\\mathcal{O}_{X \\times Y, \\Delta(\\xi)}}", "\\text{Tor}_i^{\\mathcal{O}_{X \\times Y}}(", "\\mathcal{O}_\\Delta, \\mathcal{O}_{V \\times W})_{\\Delta(\\xi)}", "$$", "However, by Lemma \\ref{lemma-tor-and-diagonal} we have", "$$", "\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{O}_V, \\mathcal{O}_W)_\\xi \\cong", "\\text{Tor}_i^{\\mathcal{O}_{X \\times Y}}(", "\\mathcal{O}_\\Delta, \\mathcal{O}_{V \\times W})_{\\Delta(\\xi)}", "$$", "as $\\mathcal{O}_{X \\times X, \\Delta(\\xi)}$-modules. Thus equality", "of lengths (by Algebra, Lemma \\ref{algebra-lemma-length-independent}", "to be precise).", "\\medskip\\noindent", "Part (2) implies (3) because", "$\\text{pr}_{1, *} \\circ \\Delta_* = \\text{id}$ by", "Lemma \\ref{lemma-compose-pushforward}." ], "refs": [ "intersection-lemma-tor-and-diagonal", "algebra-lemma-length-independent", "intersection-lemma-compose-pushforward" ], "ref_ids": [ 11982, 633, 11959 ] } ], "ref_ids": [] }, { "id": 11984, "type": "theorem", "label": "intersection-lemma-tor-sheaf", "categories": [ "intersection" ], "title": "intersection-lemma-tor-sheaf", "contents": [ "\\begin{reference}", "\\cite[Chapter V]{Serre_algebre_locale}", "\\end{reference}", "Let $X$ be a nonsingular variety. Let $\\mathcal{F}$ and", "$\\mathcal{G}$ be coherent sheaves on $X$ with", "$\\dim(\\text{Supp}(\\mathcal{F})) \\leq r$,", "$\\dim(\\text{Supp}(\\mathcal{G})) \\leq s$, and", "$\\dim(\\text{Supp}(\\mathcal{F}) \\cap \\text{Supp}(\\mathcal{G}) )", "\\leq r + s - \\dim X$. In this case $[\\mathcal{F}]_r$ and $[\\mathcal{G}]_s$", "intersect properly and", "$$", "[\\mathcal{F}]_r \\cdot [\\mathcal{G}]_s =", "\\sum (-1)^p", "[\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})]_{r + s - \\dim(X)}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "The statement that $[\\mathcal{F}]_r$ and $[\\mathcal{G}]_s$ intersect properly", "is immediate. Since we are proving an equality of cycles we may work", "locally on $X$. (Observe that the formation of the intersection", "product of cycles, the formation of $\\text{Tor}$-sheaves, and", "forming the cycle associated to a coherent sheaf, each commute with", "restriction to open subschemes.) Thus we may and do assume that $X$ is affine.", "\\medskip\\noindent", "Denote", "$$", "RHS(\\mathcal{F}, \\mathcal{G}) = [\\mathcal{F}]_r \\cdot [\\mathcal{G}]_s", "\\quad\\text{and}\\quad", "LHS(\\mathcal{F}, \\mathcal{G}) = \\sum (-1)^p", "[\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})]_{r + s - \\dim(X)}", "$$", "Consider a short exact sequence", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0", "$$", "of coherent sheaves on $X$ with", "$\\text{Supp}(\\mathcal{F}_i) \\subset \\text{Supp}(\\mathcal{F})$,", "then both $LHS(\\mathcal{F}_i, \\mathcal{G})$ and", "$RHS(\\mathcal{F}_i, \\mathcal{G})$ are defined for $i = 1, 2, 3$", "and we have", "$$", "RHS(\\mathcal{F}_2, \\mathcal{G}) =", "RHS(\\mathcal{F}_1, \\mathcal{G}) + RHS(\\mathcal{F}_3, \\mathcal{G})", "$$", "and similarly for LHS. Namely, the support condition guarantees that", "everything is defined, the short exact sequence and additivity of lengths", "gives", "$$", "[\\mathcal{F}_2]_r = [\\mathcal{F}_1]_r + [\\mathcal{F}_3]_r", "$$", "(Chow Homology, Lemma \\ref{chow-lemma-additivity-sheaf-cycle})", "which implies additivity for RHS. The long exact sequence of $\\text{Tor}$s", "$$", "\\ldots \\to \\text{Tor}_1(\\mathcal{F}_3, \\mathcal{G}) \\to", "\\text{Tor}_0(\\mathcal{F}_1, \\mathcal{G}) \\to", "\\text{Tor}_0(\\mathcal{F}_2, \\mathcal{G}) \\to", "\\text{Tor}_0(\\mathcal{F}_3, \\mathcal{G}) \\to 0", "$$", "and additivity of lengths as before implies additivity for LHS.", "\\medskip\\noindent", "By Algebra, Lemma \\ref{algebra-lemma-filter-Noetherian-module}", "and the fact that $X$ is affine, we can find a filtration of $\\mathcal{F}$", "whose graded pieces are structure sheaves of closed subvarieties of", "$\\text{Supp}(\\mathcal{F})$. The additivity shown in the previous paragraph,", "implies that it suffices to prove $LHS = RHS$ with", "$\\mathcal{F}$ replaced by $\\mathcal{O}_V$ where", "$V \\subset \\text{Supp}(\\mathcal{F})$.", "By symmetry we can do the same for $\\mathcal{G}$.", "This reduces us to proving that", "$$", "LHS(\\mathcal{O}_V, \\mathcal{O}_W) = RHS(\\mathcal{O}_V, \\mathcal{O}_W)", "$$", "where $W \\subset \\text{Supp}(\\mathcal{G})$ is a closed subvariety.", "If $\\dim(V) = r$ and $\\dim(W) = s$, then this equality is the", "{\\bf definition} of $V \\cdot W$. On the other hand, if", "$\\dim(V) < r$ or $\\dim(W) < s$, i.e., $[V]_r = 0$ or $[W]_s = 0$,", "then we have to prove that $RHS(\\mathcal{O}_V, \\mathcal{O}_W) = 0$", "\\footnote{The reader can see that this is not a triviality by", "taking $r = s = 1$ and $X$ a nonsingular surface and $V = W$", "a closed point $x$ of $X$. In this case there are $3$ nonzero", "$\\text{Tor}$s of lengths $1, 2, 1$ at $x$.}.", "\\medskip\\noindent", "Let $Z \\subset V \\cap W$ be an irreducible component of dimension", "$r + s - \\dim(X)$. This is the maximal dimension of a component", "and it suffices to show that the coefficient of $Z$ in $RHS$ is zero.", "Let $\\xi \\in Z$ be the generic point. Write $A = \\mathcal{O}_{X, \\xi}$,", "$B = \\mathcal{O}_{X \\times X, \\Delta(\\xi)}$, and", "$C = \\mathcal{O}_{V \\times W, \\Delta(\\xi)}$.", "By Lemma \\ref{lemma-tor-and-diagonal} we have", "$$", "\\text{coeff of }Z\\text{ in }", "RHS(\\mathcal{O}_V, \\mathcal{O}_W) = ", "\\sum (-1)^i", "\\text{length}_B \\text{Tor}_i^B(A, C)", "$$", "Since $\\dim(V) < r$ or $\\dim(W) < s$ we have $\\dim(V \\times W) < r + s$", "which implies $\\dim(C) < \\dim(X)$ (small detail omitted). Moreover, the", "kernel $I$ of $B \\to A$ is generated by a regular sequence of", "length $\\dim(X)$ (Lemma \\ref{lemma-diagonal-regular-immersion}).", "Hence vanishing by Lemma \\ref{lemma-one-ideal-ci} because", "the Hilbert function of $C$ with respect to $I$ has degree $\\dim(C) < n$", "by Algebra, Proposition \\ref{algebra-proposition-dimension}." ], "refs": [ "chow-lemma-additivity-sheaf-cycle", "algebra-lemma-filter-Noetherian-module", "intersection-lemma-tor-and-diagonal", "intersection-lemma-diagonal-regular-immersion", "intersection-lemma-one-ideal-ci", "algebra-proposition-dimension" ], "ref_ids": [ 5671, 691, 11982, 11965, 11974, 1411 ] } ], "ref_ids": [] }, { "id": 11985, "type": "theorem", "label": "intersection-lemma-associative", "categories": [ "intersection" ], "title": "intersection-lemma-associative", "contents": [ "Let $X$ be a nonsingular variety. Let $U, V, W$ be closed", "subvarieties. Assume that $U, V, W$ intersect properly pairwise", "and that $\\dim(U \\cap V \\cap W) \\leq \\dim(U) + \\dim(V) + \\dim(W) - 2\\dim(X)$.", "Then", "$$", "U \\cdot (V \\cdot W) = (U \\cdot V) \\cdot W", "$$", "as cycles on $X$." ], "refs": [], "proofs": [ { "contents": [ "We are going to use Lemma \\ref{lemma-tor-sheaf} without further mention.", "This implies that", "\\begin{align*}", "V \\cdot W", "& =", "\\sum (-1)^i [\\text{Tor}_i(\\mathcal{O}_V, \\mathcal{O}_W)]_{b + c - n} \\\\", "U \\cdot (V \\cdot W)", "& =", "\\sum (-1)^{i + j}", "[", "\\text{Tor}_j(\\mathcal{O}_U, \\text{Tor}_i(\\mathcal{O}_V, \\mathcal{O}_W))", "]_{a + b + c - 2n} \\\\", "U \\cdot V", "& =", "\\sum (-1)^i [\\text{Tor}_i(\\mathcal{O}_U, \\mathcal{O}_V)]_{a + b - n} \\\\", "(U \\cdot V) \\cdot W", "& =", "\\sum (-1)^{i + j}", "[", "\\text{Tor}_j(\\text{Tor}_i(\\mathcal{O}_U, \\mathcal{O}_V), \\mathcal{O}_W))", "]_{a + b + c - 2n}", "\\end{align*}", "where $\\dim(U) = a$, $\\dim(V) = b$, $\\dim(W) = c$, $\\dim(X) = n$.", "The assumptions in the lemma guarantee that the coherent sheaves", "in the formulae above satisfy the required bounds on dimensions", "of supports in order to make sense of these. Now consider the object", "$$", "K =", "\\mathcal{O}_U \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{O}_V", "\\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{O}_W", "$$", "of the derived category $D_{\\textit{Coh}}(\\mathcal{O}_X)$.", "We claim that the expressions obtained above for", "$U \\cdot (V \\cdot W)$ and $(U \\cdot V) \\cdot W$", "are equal to", "$$", "\\sum (-1)^k [H^k(K)]_{a + b + c - 2n}", "$$", "This will prove the lemma. By symmetry it suffices to prove one", "of these equalities. To do this we represent $\\mathcal{O}_U$", "and $\\mathcal{O}_V \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_W$", "by K-flat complexes $M^\\bullet$ and $L^\\bullet$ and use the", "spectral sequence associated to the double complex", "$K^\\bullet \\otimes_{\\mathcal{O}_X} L^\\bullet$ in", "Homology, Section \\ref{homology-section-double-complex}.", "This is a spectral sequence with $E_2$ page", "$$", "E_2^{p, q} =", "\\text{Tor}_{-p}(\\mathcal{O}_U, \\text{Tor}_{-q}(\\mathcal{O}_V, \\mathcal{O}_W))", "$$", "converging to $H^{p + q}(K)$ (details omitted; compare with", "More on Algebra, Example \\ref{more-algebra-example-tor}).", "Since lengths are additive in short", "exact sequences we see that the result is true." ], "refs": [ "intersection-lemma-tor-sheaf" ], "ref_ids": [ 11984 ] } ], "ref_ids": [] }, { "id": 11986, "type": "theorem", "label": "intersection-lemma-flat-pull-back-and-intersections-sheaves", "categories": [ "intersection" ], "title": "intersection-lemma-flat-pull-back-and-intersections-sheaves", "contents": [ "Let $f : X \\to Y$ be a flat morphism of nonsingular varieties. Set", "$e = \\dim(X) - \\dim(Y)$. Let $\\mathcal{F}$ and $\\mathcal{G}$ be coherent", "sheaves on $Y$ with $\\dim(\\text{Supp}(\\mathcal{F})) \\leq r$,", "$\\dim(\\text{Supp}(\\mathcal{G})) \\leq s$, and", "$\\dim(\\text{Supp}(\\mathcal{F}) \\cap \\text{Supp}(\\mathcal{G}) )", "\\leq r + s - \\dim(Y)$. In this case the cycles", "$[f^*\\mathcal{F}]_{r + e}$ and $[f^*\\mathcal{G}]_{s + e}$", "intersect properly and", "$$", "f^*([\\mathcal{F}]_r \\cdot [\\mathcal{G}]_s) =", "[f^*\\mathcal{F}]_{r + e} \\cdot [f^*\\mathcal{G}]_{s + e}", "$$" ], "refs": [], "proofs": [ { "contents": [ "The statement that $[f^*\\mathcal{F}]_{r + e}$ and $[f^*\\mathcal{G}]_{s + e}$", "intersect properly is immediate from the assumption that $f$ has", "relative dimension $e$. By", "Lemmas \\ref{lemma-tor-sheaf} and \\ref{lemma-pullback}", "it suffices to show that", "$$", "f^*\\text{Tor}_i^{\\mathcal{O}_Y}(\\mathcal{F}, \\mathcal{G}) =", "\\text{Tor}_i^{\\mathcal{O}_X}(f^*\\mathcal{F}, f^*\\mathcal{G})", "$$", "as $\\mathcal{O}_X$-modules. This follows from", "Cohomology, Lemma \\ref{cohomology-lemma-pullback-tensor-product}", "and the fact that $f^*$ is exact, so $Lf^*\\mathcal{F} = f^*\\mathcal{F}$", "and similarly for $\\mathcal{G}$." ], "refs": [ "intersection-lemma-tor-sheaf", "intersection-lemma-pullback", "cohomology-lemma-pullback-tensor-product" ], "ref_ids": [ 11984, 11960, 2118 ] } ], "ref_ids": [] }, { "id": 11987, "type": "theorem", "label": "intersection-lemma-flat-pullback-and-intersections", "categories": [ "intersection" ], "title": "intersection-lemma-flat-pullback-and-intersections", "contents": [ "Let $f : X \\to Y$ be a flat morphism of nonsingular varieties.", "Let $\\alpha$ be a $r$-cycle on $Y$ and $\\beta$ an $s$-cycle on $Y$.", "Assume that $\\alpha$ and $\\beta$ intersect properly. Then $f^*\\alpha$", "and $f^*\\beta$ intersect properly and", "$f^*( \\alpha \\cdot \\beta ) = f^*\\alpha \\cdot f^*\\beta$." ], "refs": [], "proofs": [ { "contents": [ "By linearity we may assume that $\\alpha = [V]$ and $\\beta = [W]$", "for some closed subvarieties $V, W \\subset Y$ of dimension $r, s$.", "Say $f$ has relative dimension $e$. Then the lemma is a special case of", "Lemma \\ref{lemma-flat-pull-back-and-intersections-sheaves}", "because $[V] = [\\mathcal{O}_V]_r$, $[W] = [\\mathcal{O}_W]_r$,", "$f^*[V] = [f^{-1}(V)]_{r + e} = [f^*\\mathcal{O}_V]_{r + e}$, and", "$f^*[W] = [f^{-1}(W)]_{s + e} = [f^*\\mathcal{O}_W]_{s + e}$." ], "refs": [ "intersection-lemma-flat-pull-back-and-intersections-sheaves" ], "ref_ids": [ 11986 ] } ], "ref_ids": [] }, { "id": 11988, "type": "theorem", "label": "intersection-lemma-projection-formula-flat", "categories": [ "intersection" ], "title": "intersection-lemma-projection-formula-flat", "contents": [ "\\begin{reference}", "See \\cite[Chapter V, C), Section 7, formula (10)]{Serre_algebre_locale}", "for a more general formula.", "\\end{reference}", "Let $f : X \\to Y$ be a flat proper morphism of nonsingular varieties.", "Set $e = \\dim(X) - \\dim(Y)$. Let $\\alpha$ be an $r$-cycle on $X$ and let", "$\\beta$ be a $s$-cycle on $Y$. Assume that $\\alpha$ and $f^*(\\beta)$ intersect", "properly. Then $f_*(\\alpha)$ and $\\beta$ intersect properly and", "$$", "f_*(\\alpha) \\cdot \\beta = f_*( \\alpha \\cdot f^*\\beta)", "$$" ], "refs": [], "proofs": [ { "contents": [ "By linearity we reduce to the case where $\\alpha = [V]$ and", "$\\beta = [W]$ for some closed subvariety $V \\subset X$ and", "$W \\subset Y$ of dimension $r$ and $s$. Then $f^{-1}(W)$ has", "pure dimension $s + e$. We assume the cycles", "$[V]$ and $f^*[W]$ intersect properly. We will use without", "further mention the fact that $V \\cap f^{-1}(W) \\to f(V) \\cap W$", "is surjective.", "\\medskip\\noindent", "Let $a$ be the dimension of the generic fibre of $V \\to f(V)$.", "If $a > 0$, then $f_*[V] = 0$. In particular $f_*\\alpha$ and $\\beta$", "intersect properly. To finish this case we have to show that", "$f_*([V] \\cdot f^*[W]) = 0$. However, since every fibre of", "$V \\to f(V)$ has dimension $\\geq a$ (see", "Morphisms, Lemma \\ref{morphisms-lemma-openness-bounded-dimension-fibres})", "we conclude that every irreducible component $Z$ of $V \\cap f^{-1}(W)$", "has fibres of dimension $\\geq a$ over $f(Z)$. This certainly", "implies what we want.", "\\medskip\\noindent", "Assume that $V \\to f(V)$ is generically finite. Let $Z \\subset f(V) \\cap W$", "be an irreducible component. Let $Z_i \\subset V \\cap f^{-1}(W)$,", "$i = 1, \\ldots, t$ be the irreducible components of $V \\cap f^{-1}(W)$", "dominating $Z$. By assumption each $Z_i$ has dimension", "$r + s + e - \\dim(X) = r + s - \\dim(Y)$. Hence", "$\\dim(Z) \\leq r + s - \\dim(Y)$. Thus we see that $f(V)$ and $W$", "intersect properly, $\\dim(Z) = r + s - \\dim(Y)$, and each", "$Z_i \\to Z$ is generically finite. In particular, it follows that", "$V \\to f(V)$ has finite fibre over the generic point $\\xi$ of $Z$.", "Thus $V \\to Y$ is finite in an open neighbourhood of $\\xi$, see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-finite-fibre-finite-in-neighbourhood}.", "Using a very general projection formula for derived tensor products, we get", "$$", "Rf_*(\\mathcal{O}_V \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*\\mathcal{O}_W) =", "Rf_*\\mathcal{O}_V \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} \\mathcal{O}_W", "$$", "see Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-cohomology-base-change}.", "Since $f$ is flat, we see that $Lf^*\\mathcal{O}_W = f^*\\mathcal{O}_W$.", "Since $f|_V$ is finite in an open neighbourhood of $\\xi$ we have", "$$", "(Rf_*\\mathcal{F})_\\xi = (f_*\\mathcal{F})_\\xi", "$$", "for any coherent sheaf on $X$ whose support is contained in $V$", "(see Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-higher-direct-images-zero-finite-fibre}). Thus", "we conclude that", "\\begin{equation}", "\\label{equation-stalks}", "\\left(", "f_*\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{O}_V, f^*\\mathcal{O}_W)", "\\right)_\\xi =", "\\left(\\text{Tor}_i^{\\mathcal{O}_Y}(f_*\\mathcal{O}_V, \\mathcal{O}_W)\\right)_\\xi", "\\end{equation}", "for all $i$. Since $f^*[W] = [f^*\\mathcal{O}_W]_{s + e}$ by", "Lemma \\ref{lemma-pullback} we have", "$$", "[V] \\cdot f^*[W] =", "\\sum (-1)^i", "[\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{O}_V,", "f^*\\mathcal{O}_W)]_{r + s - \\dim(Y)}", "$$", "by Lemma \\ref{lemma-tor-sheaf}. Applying", "Lemma \\ref{lemma-push-coherent}", "we find", "$$", "f_*([V] \\cdot f^*[W]) =", "\\sum (-1)^i", "[f_*\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{O}_V,", "f^*\\mathcal{O}_W)]_{r + s - \\dim(Y)}", "$$", "Since $f_*[V] = [f_*\\mathcal{O}_V]_r$ by", "Lemma \\ref{lemma-push-coherent} we have", "$$", "[f_*V] \\cdot [W] =", "\\sum (-1)^i", "[\\text{Tor}_i^{\\mathcal{O}_X}(f_*\\mathcal{O}_V,", "\\mathcal{O}_W)]_{r + s - \\dim(Y)}", "$$", "again by Lemma \\ref{lemma-tor-sheaf}.", "Comparing the formula for $f_*([V] \\cdot f^*[W])$ with", "the formula for $f_*[V] \\cdot [W]$ and looking at the", "coefficient of $Z$ by taking lengths of stalks at $\\xi$, we see that", "(\\ref{equation-stalks}) finishes the proof." ], "refs": [ "morphisms-lemma-openness-bounded-dimension-fibres", "coherent-lemma-proper-finite-fibre-finite-in-neighbourhood", "perfect-lemma-cohomology-base-change", "coherent-lemma-higher-direct-images-zero-finite-fibre", "intersection-lemma-pullback", "intersection-lemma-tor-sheaf", "intersection-lemma-push-coherent", "intersection-lemma-push-coherent", "intersection-lemma-tor-sheaf" ], "ref_ids": [ 5280, 3366, 7025, 3363, 11960, 11984, 11958, 11958, 11984 ] } ], "ref_ids": [] }, { "id": 11989, "type": "theorem", "label": "intersection-lemma-transfer", "categories": [ "intersection" ], "title": "intersection-lemma-transfer", "contents": [ "Let $X \\to P$ be a closed immersion of nonsingular varieties.", "Let $C' \\subset P \\times \\mathbf{P}^1$ be a closed subvariety of dimension", "$r + 1$. Assume", "\\begin{enumerate}", "\\item the fibre $C = C'_0$ has dimension $r$, i.e., $C' \\to \\mathbf{P}^1$", "is dominant,", "\\item $C'$ intersects $X \\times \\mathbf{P}^1$ properly,", "\\item $[C]_r$ intersects $X$ properly.", "\\end{enumerate}", "Then setting $\\alpha = [C]_r \\cdot X$ viewed as cycle on $X$ and", "$\\beta = C' \\cdot X \\times \\mathbf{P}^1$ viewed as cycle on", "$X \\times \\mathbf{P}^1$, we have", "$$", "\\alpha = \\text{pr}_{X, *}(\\beta \\cdot X \\times 0)", "$$", "as cycles on $X$ where $\\text{pr}_X : X \\times \\mathbf{P}^1 \\to X$ is the", "projection." ], "refs": [], "proofs": [ { "contents": [ "Let $\\text{pr} : P \\times \\mathbf{P}^1 \\to P$ be the projection.", "Since we are proving an equality of cycles it suffices to think of", "$\\alpha$, resp.\\ $\\beta$ as a cycle on $P$, resp.\\ $P \\times \\mathbf{P}^1$", "and prove the result for pushing forward by $\\text{pr}$.", "Because $\\text{pr}^*X = X \\times \\mathbf{P}^1$ and", "$\\text{pr}$ defines an isomorphism of $C'_0$ onto $C$", "the projection formula (Lemma \\ref{lemma-projection-formula-flat})", "gives", "$$", "\\text{pr}_*([C'_0]_r \\cdot X \\times \\mathbf{P}^1) = [C]_r \\cdot X = \\alpha", "$$", "On the other hand, we have $[C'_0]_r = C' \\cdot P \\times 0$", "as cycles on $P \\times \\mathbf{P}^1$", "by Lemma \\ref{lemma-rational-equivalence-and-intersection}.", "Hence", "$$", "[C'_0]_r \\cdot X \\times \\mathbf{P}^1 =", "(C' \\cdot P \\times 0) \\cdot X \\times \\mathbf{P}^1 =", "(C' \\cdot X \\times \\mathbf{P}^1) \\cdot P \\times 0", "$$", "by associativity (Lemma \\ref{lemma-associative}) and commutativity of the", "intersection product. It remains to show that the intersection product of", "$C' \\cdot X \\times \\mathbf{P}^1$ with $P \\times 0$ on", "$P \\times \\mathbf{P}^1$ is equal (as a cycle) to the intersection product of", "$\\beta$ with $X \\times 0$ on $X \\times \\mathbf{P}^1$. Write", "$C' \\cdot X \\times \\mathbf{P}^1 = \\sum n_k[E_k]$ and hence", "$\\beta = \\sum n_k[E_k]$ for some subvarieties", "$E_k \\subset X \\times \\mathbf{P}^1 \\subset P \\times \\mathbf{P}^1$.", "Then both intersections are equal to $\\sum m_k[E_{k, 0}]$ by", "Lemma \\ref{lemma-rational-equivalence-and-intersection} applied twice.", "This finishes the proof." ], "refs": [ "intersection-lemma-projection-formula-flat", "intersection-lemma-rational-equivalence-and-intersection", "intersection-lemma-associative", "intersection-lemma-rational-equivalence-and-intersection" ], "ref_ids": [ 11988, 11978, 11985, 11978 ] } ], "ref_ids": [] }, { "id": 11990, "type": "theorem", "label": "intersection-lemma-projection-generically-finite", "categories": [ "intersection" ], "title": "intersection-lemma-projection-generically-finite", "contents": [ "Let $V$ be a vector space of dimension $n + 1$.", "Let $X \\subset \\mathbf{P}(V)$ be a closed subscheme.", "If $X \\not = \\mathbf{P}(V)$, then there is a nonempty Zariski open", "$U \\subset \\mathbf{P}(V)$", "such that for all closed points $p \\in U$ the restriction", "of the projection $r_p$ defines a finite morphism", "$r_p|_X : X \\to \\mathbf{P}(W_p)$." ], "refs": [], "proofs": [ { "contents": [ "We claim the lemma holds with $U = \\mathbf{P}(V) \\setminus X$. For a closed", "point $p$ of $U$ we indeed obtain a morphism $r_p|_X : X \\to \\mathbf{P}(W_p)$.", "This morphism is proper because $X$ is a proper scheme", "(Morphisms, Lemmas \\ref{morphisms-lemma-locally-projective-proper} and", "\\ref{morphisms-lemma-image-proper-scheme-closed}). On the other hand, the", "fibres of $r_p$ are affine lines as can be seen by a direct calculation.", "Hence the fibres of $r_p|X$ are proper and affine, whence finite", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}).", "Finally, a proper morphism with finite fibres is finite", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-characterize-finite})." ], "refs": [ "morphisms-lemma-locally-projective-proper", "morphisms-lemma-image-proper-scheme-closed", "morphisms-lemma-finite-proper", "coherent-lemma-characterize-finite" ], "ref_ids": [ 5422, 5411, 5445, 3365 ] } ], "ref_ids": [] }, { "id": 11991, "type": "theorem", "label": "intersection-lemma-projection-generically-immersion", "categories": [ "intersection" ], "title": "intersection-lemma-projection-generically-immersion", "contents": [ "Let $V$ be a vector space of dimension $n + 1$.", "Let $X \\subset \\mathbf{P}(V)$ be a closed subvariety.", "Let $x \\in X$ be a nonsingular point.", "\\begin{enumerate}", "\\item If $\\dim(X) < n - 1$, then there is a nonempty Zariski open", "$U \\subset \\mathbf{P}(V)$ such that for all closed points $p \\in U$ the", "morphism $r_p|_X : X \\to r_p(X)$ is an", "isomorphism over an open neighbourhood of $r_p(x)$.", "\\item If $\\dim(X) = n - 1$, then there is a nonempty Zariski open", "$U \\subset \\mathbf{P}(V)$ such that for all closed points $p \\in U$ the", "morphism $r_p|_X : X \\to \\mathbf{P}(W_p)$ is \\'etale at $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Note that if $x, y \\in X$ have the same image under", "$r_p$ then $p$ is on the line $\\overline{xy}$.", "Consider the finite type scheme", "$$", "T = \\{(y, p) \\mid", "y \\in X \\setminus \\{x\\},\\ p \\in \\mathbf{P}(V),\\ p \\in \\overline{xy}\\}", "$$", "and the morphisms $T \\to X$ and $T \\to \\mathbf{P}(V)$ given by", "$(y, p) \\mapsto y$ and $(y, p) \\mapsto p$.", "Since each fibre of $T \\to X$ is a line, we see that", "the dimension of $T$ is $\\dim(X) + 1 < \\dim(\\mathbf{P}(V))$.", "Hence $T \\to \\mathbf{P}(V)$ is not surjective. On the other hand,", "consider the finite type scheme", "$$", "T' = \\{p \\mid", "p \\in \\mathbf{P}(V) \\setminus \\{x\\},", "\\ \\overline{xp}\\text{ tangent to }X\\text{ at }x\\}", "$$", "Then the dimension of $T'$ is $\\dim(X) < \\dim(\\mathbf{P}(V))$.", "Thus the morphism $T' \\to \\mathbf{P}(V)$ is not surjective either.", "Let $U \\subset \\mathbf{P}(V) \\setminus X$ be nonempty open and disjoint", "from these images; such a $U$ exists because the images of $T$ and $T'$", "in $\\mathbf{P}(V)$ are constructible by", "Morphisms, Lemma \\ref{morphisms-lemma-chevalley}.", "Then for $p \\in U$ closed the projection", "$r_p|_X : X \\to \\mathbf{P}(W_p)$ is injective on the", "tangent space at $x$ and $r_p^{-1}(\\{r_p(x)\\}) = \\{x\\}$.", "This means that $r_p$ is unramified at $x$", "(Varieties, Lemma \\ref{varieties-lemma-injective-tangent-spaces-unramified}),", "finite by Lemma \\ref{lemma-projection-generically-finite},", "and $r_p^{-1}(\\{r_p(x)\\}) = \\{x\\}$ thus \\'Etale Morphisms, Lemma", "\\ref{etale-lemma-finite-unramified-one-point} applies and", "there is an open neighbourhood $R$ of $r_p(x)$", "in $\\mathbf{P}(W_p)$ such that $(r_p|_X)^{-1}(R) \\to R$ is a", "closed immersion which proves (1).", "\\medskip\\noindent", "Proof of (2). In this case we still conclude that the morphism", "$T' \\to \\mathbf{P}(V)$ is not surjective.", "Arguing as above we conclude that for $U \\subset \\mathbf{P}(V)$", "avoiding $X$ and the image of $T'$, the projection", "$r_p|_X : X \\to \\mathbf{P}(W_p)$ is \\'etale at $x$ and finite." ], "refs": [ "morphisms-lemma-chevalley", "varieties-lemma-injective-tangent-spaces-unramified", "intersection-lemma-projection-generically-finite", "etale-lemma-finite-unramified-one-point" ], "ref_ids": [ 5250, 10976, 11990, 10703 ] } ], "ref_ids": [] }, { "id": 11992, "type": "theorem", "label": "intersection-lemma-projection-injective", "categories": [ "intersection" ], "title": "intersection-lemma-projection-injective", "contents": [ "Let $V$ be a vector space of dimension $n + 1$.", "Let $Y, Z \\subset \\mathbf{P}(V)$ be closed subvarieties.", "There is a nonempty Zariski open $U \\subset \\mathbf{P}(V)$", "such that for all closed points $p \\in U$ we have", "$$", "Y \\cap r_p^{-1}(r_p(Z)) = (Y \\cap Z) \\cup E", "$$", "with $E \\subset Y$ closed and", "$\\dim(E) \\leq \\dim(Y) + \\dim(Z) + 1 - n$." ], "refs": [], "proofs": [ { "contents": [ "Set $Y' = Y \\setminus Y \\cap Z$.", "Let $y \\in Y'$, $z \\in Z$ be closed points with $r_p(y) = r_p(z)$.", "Then $p$ is on the line $\\overline{yz}$ passing through $y$ and $z$.", "Consider the finite type scheme", "$$", "T = \\{(y, z, p) \\mid y \\in Y', z \\in Z, p \\in \\overline{yz}\\}", "$$", "and the morphism $T \\to \\mathbf{P}(V)$ given by $(y, z, p) \\mapsto p$.", "Observe that $T$ is irreducible and that $\\dim(T) = \\dim(Y) + \\dim(Z) + 1$.", "Hence the general fibre of $T \\to \\mathbf{P}(V)$ has dimension at most", "$\\dim(Y) + \\dim(Z) + 1 - n$, more precisely, there exists a nonempty", "open $U \\subset \\mathbf{P}(V) \\setminus (Y \\cup Z)$ over", "which the fibre has dimension at most $\\dim(Y) + \\dim(Z) + 1 - n$", "(Varieties, Lemma \\ref{varieties-lemma-dimension-fibres-locally-algebraic}).", "Let $p \\in U$ be a closed point and let $F \\subset T$ be the fibre", "of $T \\to \\mathbf{P}(V)$ over $p$. Then", "$$", "(Y \\cap r_p^{-1}(r_p(Z))) \\setminus (Y \\cap Z)", "$$", "is the image of $F \\to Y$, $(y, z, p) \\mapsto y$. Again by", "Varieties, Lemma \\ref{varieties-lemma-dimension-fibres-locally-algebraic}", "the closure of the image of $F \\to Y$ has dimension at most", "$\\dim(Y) + \\dim(Z) + 1 - n$." ], "refs": [ "varieties-lemma-dimension-fibres-locally-algebraic", "varieties-lemma-dimension-fibres-locally-algebraic" ], "ref_ids": [ 10990, 10990 ] } ], "ref_ids": [] }, { "id": 11993, "type": "theorem", "label": "intersection-lemma-find-lines", "categories": [ "intersection" ], "title": "intersection-lemma-find-lines", "contents": [ "Let $V$ be a vector space. Let $B \\subset \\mathbf{P}(V)$", "be a closed subvariety of codimension $\\geq 2$.", "Let $p \\in \\mathbf{P}(V)$ be a closed point, $p \\not \\in B$.", "Then there exists a line $\\ell \\subset \\mathbf{P}(V)$", "with $\\ell \\cap B = \\emptyset$. Moreover, these lines", "sweep out an open subset of $\\mathbf{P}(V)$." ], "refs": [], "proofs": [ { "contents": [ "Consider the image of $B$ under the projection", "$r_p : \\mathbf{P}(V) \\to \\mathbf{P}(W_p)$.", "Since $\\dim(W_p) = \\dim(V) - 1$, we see that $r_p(B)$", "has codimension $\\geq 1$ in $\\mathbf{P}(W_p)$.", "For any $q \\in \\mathbf{P}(V)$ with $r_p(q) \\not \\in r_p(B)$", "we see that the line $\\ell = \\overline{pq}$ connecting $p$ and $q$ works." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11994, "type": "theorem", "label": "intersection-lemma-doubly-transitive", "categories": [ "intersection" ], "title": "intersection-lemma-doubly-transitive", "contents": [ "Let $V$ be a vector space. Let $G = \\text{PGL}(V)$.", "Then $G \\times \\mathbf{P}(V) \\to \\mathbf{P}(V)$ is", "doubly transitive." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This follows from the fact that $\\text{GL}(V)$ acts doubly", "transitive on pairs of linearly independent vectors." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11995, "type": "theorem", "label": "intersection-lemma-determinant", "categories": [ "intersection" ], "title": "intersection-lemma-determinant", "contents": [ "Let $k$ be a field. Let $n \\geq 1$ be an integer and let", "$x_{ij}, 1 \\leq i, j \\leq n$ be variables. Then", "$$", "\\det", "\\left(", "\\begin{matrix}", "x_{11} & x_{12} & \\ldots & x_{1n} \\\\", "x_{21} & \\ldots & \\ldots & \\ldots \\\\", "\\ldots & \\ldots & \\ldots & \\ldots \\\\", "x_{n1} & \\ldots & \\ldots & x_{nn}", "\\end{matrix}", "\\right)", "$$", "is an irreducible element of the polynomial ring $k[x_{ij}]$." ], "refs": [], "proofs": [ { "contents": [ "Let $V$ be an $n$ dimensional vector space. Translating into geometry", "the lemma signifies that the variety $C$ of non-invertible linear maps", "$V \\to V$ is irreducible. Let $W$ be a vector space of dimension $n - 1$.", "By elementary linear algebra, the morphism", "$$", "\\Hom(W, V) \\times \\Hom(V, W) \\longrightarrow \\Hom(V, V),\\quad", "(\\psi, \\varphi) \\longmapsto \\psi \\circ \\varphi", "$$", "has image $C$. Since the source is irreducible, so is the image." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11996, "type": "theorem", "label": "intersection-lemma-make-family", "categories": [ "intersection" ], "title": "intersection-lemma-make-family", "contents": [ "With notation as above. Let $X, Y$ be closed subvarieties of $\\mathbf{P}(V)$", "which intersect properly such that $X \\not = \\mathbf{P}(V)$ and", "$X \\cap Y \\not = \\emptyset$. For a general line $\\ell \\subset \\mathbf{P}$", "with $[\\text{id}_V] \\in \\ell$ we have", "\\begin{enumerate}", "\\item $X \\subset U_g$ for all $[g] \\in \\ell$,", "\\item $g(X)$ intersects $Y$ properly for all $[g] \\in \\ell$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $B \\subset \\mathbf{P}$ be the set of ``bad'' points, i.e., those", "points $[g]$ that violate either (1) or (2). Note that", "$[\\text{id}_V] \\not \\in B$ by assumption. Moreover, $B$ is closed.", "Hence it suffices to prove that $\\dim(B) \\leq \\dim(\\mathbf{P}) - 2$", "(Lemma \\ref{lemma-find-lines}).", "\\medskip\\noindent", "First, consider the open $G = \\text{PGL}(V) \\subset \\mathbf{P}$", "consisting of points $[g]$ such that $g : V \\to V$ is invertible.", "Since $G$ acts doubly transitively on $\\mathbf{P}(V)$", "(Lemma \\ref{lemma-doubly-transitive})", "we see that", "$$", "T = \\{(x, y, [g]) \\mid x \\in X, y \\in Y, [g] \\in G, r_g(x) = y\\}", "$$", "is a locally trivial fibration over $X \\times Y$ with fibre equal", "to the stabilizer of a point in $G$. Hence $T$ is a variety.", "Observe that the fibre of $T \\to G$ over $[g]$ is $r_g(X) \\cap Y$.", "The morphism $T \\to G$ is surjective, because any translate of $X$", "intersects $Y$ (note that by the assumption that $X$ and $Y$ intersect", "properly and that $X \\cap Y \\not = \\emptyset$ we see that", "$\\dim(X) + \\dim(Y) \\geq \\dim(\\mathbf{P}(V))$ and then", "Varieties, Lemma \\ref{varieties-lemma-intersection-in-projective-space}", "implies all translates of $X$ intersect $Y$).", "Since the dimensions of fibres of a dominant morphism of varieties do", "not jump in codimension $1$", "(Varieties, Lemma \\ref{varieties-lemma-dimension-fibres-locally-algebraic})", "we conclude that $B \\cap G$ has codimension $\\geq 2$.", "\\medskip\\noindent", "Next we look at the complement $Z = \\mathbf{P} \\setminus G$.", "This is an irreducible variety because the determinant is an", "irreducible polynomial (Lemma \\ref{lemma-determinant}).", "Thus it suffices to prove that $B$ does not contain the", "generic point of $Z$. For a general point $[g] \\in Z$", "the cokernel $V \\to \\Coker(g)$ has dimension $1$, hence", "$U(g)$ is the complement of a point. Since $X \\not = \\mathbf{P}(V)$", "we see that for a general $[g] \\in Z$ we have $X \\subset U(g)$.", "Moreover, the morphism $r_g|_X : X \\to r_g(X)$ is finite, hence", "$\\dim(r_g(X)) = \\dim(X)$.", "On the other hand, for such a $g$ the image of $r_g$ is the", "closed subspace $H = \\mathbf{P}(\\Im(g)) \\subset \\mathbf{P}(V)$", "which has codimension $1$.", "For general point of $Z$ we see that $H \\cap Y$ has dimension $1$", "less than $Y$ (compare with", "Varieties, Lemma \\ref{varieties-lemma-exact-sequence-induction}).", "Thus we see that we have to show that $r_g(X)$ and $H \\cap Y$", "intersect properly in $H$. For a fixed choice of $H$, we can", "by postcomposing $g$ by an automorphism, move $r_g(X)$ by", "an arbitrary automorphism of $H = \\mathbf{P}(\\Im(g))$.", "Thus we can argue as above to conclude that the intersection", "of $H \\cap Y$ with $r_g(X)$ is proper for general $g$ with given", "$H = \\mathbf{P}(\\Im(g))$. Some details omitted." ], "refs": [ "intersection-lemma-find-lines", "intersection-lemma-doubly-transitive", "varieties-lemma-intersection-in-projective-space", "varieties-lemma-dimension-fibres-locally-algebraic", "intersection-lemma-determinant", "varieties-lemma-exact-sequence-induction" ], "ref_ids": [ 11993, 11994, 11035, 10990, 11995, 11038 ] } ], "ref_ids": [] }, { "id": 11997, "type": "theorem", "label": "intersection-lemma-moving", "categories": [ "intersection" ], "title": "intersection-lemma-moving", "contents": [ "\\begin{reference}", "See \\cite{Roberts}.", "\\end{reference}", "Let $X \\subset \\mathbf{P}^N$ be a nonsingular closed subvariety.", "Let $n = \\dim(X)$ and $0 \\leq d, d' < n$. Let $Z \\subset X$ be a closed", "subvariety of dimension $d$ and $T_i \\subset X$, $i \\in I$ be a finite", "collection of closed subvarieties of dimension $d'$. Then there exists", "a subvariety $C \\subset \\mathbf{P}^N$ such that $C$ intersects $X$", "properly and such that", "$$", "C \\cdot X = Z + \\sum\\nolimits_{j \\in J} m_j Z_j", "$$", "where $Z_j \\subset X$ are irreducible of dimension $d$, distinct from $Z$, and", "$$", "\\dim(Z_j \\cap T_i) \\leq \\dim(Z \\cap T_i)", "$$", "with strict inequality if $Z$ does not intersect $T_i$ properly in $X$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\mathbf{P}^N = \\mathbf{P}(V_N)$ so $\\dim(V_N) = N + 1$ and set", "$X_N = X$. We are going to choose a sequence of projections from points", "\\begin{align*}", "& r_N : ", "\\mathbf{P}(V_N) \\setminus \\{p_N\\} \\to \\mathbf{P}(V_{N - 1}), \\\\", "& r_{N - 1} :", "\\mathbf{P}(V_{N - 1}) \\setminus \\{p_{N - 1}\\} \\to \\mathbf{P}(V_{N - 2}), \\\\", "& \\ldots,\\\\", "& r_{n + 1} :", "\\mathbf{P}(V_{n + 1}) \\setminus \\{p_{n + 1}\\} \\to \\mathbf{P}(V_n)", "\\end{align*}", "as in Section \\ref{section-projection}. At each step we will choose", "$p_N, p_{N - 1}, \\ldots, p_{n + 1}$ in a suitable Zariski open set.", "Pick a closed point $x \\in Z \\subset X$. For every $i$ pick", "closed points $x_{it} \\in T_i \\cap Z$, at least one in each irreducible", "component of $T_i \\cap Z$. Taking the composition we obtain", "a morphism", "$$", "\\pi = (r_{n + 1} \\circ \\ldots \\circ r_N)|_X :", "X \\longrightarrow \\mathbf{P}(V_n)", "$$", "which has the following properties", "\\begin{enumerate}", "\\item $\\pi$ is finite,", "\\item $\\pi$ is \\'etale at $x$ and all $x_{it}$,", "\\item $\\pi|_Z : Z \\to \\pi(Z)$ is an isomorphism", "over an open neighbourhood of $\\pi(x_{it})$,", "\\item $T_i \\cap \\pi^{-1}(\\pi(Z)) = (T_i \\cap Z) \\cup E_i$ with", "$E_i \\subset T_i$ closed and", "$\\dim(E_i) \\leq d + d' + 1 - (n + 1) = d + d' - n$.", "\\end{enumerate}", "It follows in a straightforward manner from", "Lemmas \\ref{lemma-projection-generically-finite},", "\\ref{lemma-projection-generically-immersion}, and", "\\ref{lemma-projection-injective} and induction that we can do this;", "observe that the last projection is from $\\mathbf{P}(V_{n + 1})$ and that", "$\\dim(V_{n + 1}) = n + 2$ which explains the inequality in (4).", "\\medskip\\noindent", "Let $C \\subset \\mathbf{P}(V_N)$ be the scheme theoretic closure of", "$(r_{n + 1} \\circ \\ldots \\circ r_N)^{-1}(\\pi(Z))$. Because $\\pi$", "is \\'etale at the point $x$ of $Z$, we see that the closed subscheme", "$C \\cap X$ contains $Z$ with multiplicity $1$ (local calculation omitted).", "Hence by Lemma \\ref{lemma-transversal-subschemes} we conclude that", "$$", "C \\cdot X = [Z] + \\sum m_j[Z_j]", "$$", "for some subvarieties $Z_j \\subset X$ of dimension $d$. Note that", "$$", "C \\cap X = \\pi^{-1}(\\pi(Z))", "$$", "set theoretically. Hence", "$T_i \\cap Z_j \\subset T_i \\cap \\pi^{-1}(\\pi(Z)) \\subset T_i \\cap Z \\cup E_i$.", "For any irreducible component of $T_i \\cap Z$ contained in $E_i$ we", "have the desired dimension bound. Finally, let $V$ be an irreducible", "component of $T_i \\cap Z_j$ which is contained in $T_i \\cap Z$. To finish", "the proof it suffices to show that $V$ does not contain any of the", "points $x_{it}$, because then $\\dim(V) < \\dim(Z \\cap T_i)$.", "To show this it suffices to show that $x_{it} \\not \\in Z_j$", "for all $i, t, j$.", "\\medskip\\noindent", "Set $Z' = \\pi(Z)$ and $Z'' = \\pi^{-1}(Z')$, scheme theoretically. By", "condition (3) we can find an open $U \\subset \\mathbf{P}(V_n)$ containing", "$\\pi(x_{it})$ such that $\\pi^{-1}(U) \\cap Z \\to U \\cap Z'$ is an isomorphism.", "In particular, $Z \\to Z'$ is a local isomorphism at $x_{it}$.", "On the other hand, $Z'' \\to Z'$ is \\'etale at $x_{it}$ by condition (2).", "Hence the closed immersion $Z \\to Z''$ is \\'etale at $x_{it}$", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-permanence}).", "Thus $Z = Z''$ in a Zariski neighbourhood of $x_{it}$ which proves", "the assertion." ], "refs": [ "intersection-lemma-projection-generically-finite", "intersection-lemma-projection-generically-immersion", "intersection-lemma-projection-injective", "intersection-lemma-transversal-subschemes", "morphisms-lemma-etale-permanence" ], "ref_ids": [ 11990, 11991, 11992, 11979, 5375 ] } ], "ref_ids": [] }, { "id": 11998, "type": "theorem", "label": "intersection-lemma-move", "categories": [ "intersection" ], "title": "intersection-lemma-move", "contents": [ "Let $C \\subset \\mathbf{P}^N$ be a closed subvariety.", "Let $X \\subset \\mathbf{P}^N$ be subvariety and let $T_i \\subset X$", "be a finite collection of closed subvarieties.", "Assume that $C$ and $X$ intersect properly.", "Then there exists a closed subvariety", "$C' \\subset \\mathbf{P}^N \\times \\mathbf{P}^1$ such that", "\\begin{enumerate}", "\\item $C' \\to \\mathbf{P}^1$ is dominant,", "\\item $C'_0 = C$ scheme theoretically,", "\\item $C'$ and $X \\times \\mathbf{P}^1$ intersect properly,", "\\item $C'_\\infty$ properly intersects each of the given $T_i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $C \\cap X = \\emptyset$, then we take the constant family", "$C' = C \\times \\mathbf{P}^1$. Thus we may and do assume", "$C \\cap X \\not = \\emptyset$.", "\\medskip\\noindent", "Write $\\mathbf{P}^N = \\mathbf{P}(V)$ so $\\dim(V) = N + 1$. Let", "$E = \\text{End}(V)$. Let $E^\\vee = \\Hom(E, \\mathbf{C})$. Set", "$\\mathbf{P} = \\mathbf{P}(E^\\vee)$ as in Lemma \\ref{lemma-make-family}.", "Choose a general line $\\ell \\subset \\mathbf{P}$ passing through $\\text{id}_V$.", "Set $C' \\subset \\ell \\times \\mathbf{P}(V)$ equal to the", "closed subscheme having fibre $r_g(C)$ over $[g] \\in \\ell$.", "More precisely, $C'$ is the image of", "$$", "\\ell \\times C \\subset \\mathbf{P} \\times \\mathbf{P}(V)", "$$", "under the morphism (\\ref{equation-r-psi}). By Lemma \\ref{lemma-make-family}", "this makes sense, i.e., $\\ell \\times C \\subset U(\\psi)$. The morphism", "$\\ell \\times C \\to C'$ is finite and $C'_{[g]} = r_g(C)$ set theoretically", "for all $[g] \\in \\ell$. Parts (1) and (2) are clear with", "$0 = [\\text{id}_V] \\in \\ell$. Part (3) follows from the fact", "that $r_g(C)$ and $X$ intersect properly for all $[g] \\in \\ell$.", "Part (4) follows from the fact that a general point $\\infty = [g] \\in \\ell$", "is a general point of $\\mathbf{P}$ and for such as point", "$r_g(C) \\cap T$ is proper for any closed subvariety $T$ of $\\mathbf{P}(V)$.", "Details omitted." ], "refs": [ "intersection-lemma-make-family", "intersection-lemma-make-family" ], "ref_ids": [ 11996, 11996 ] } ], "ref_ids": [] }, { "id": 11999, "type": "theorem", "label": "intersection-lemma-moving-move", "categories": [ "intersection" ], "title": "intersection-lemma-moving-move", "contents": [ "Let $X$ be a nonsingular projective variety. Let $\\alpha$ be an", "$r$-cycle and $\\beta$ be an $s$-cycle on $X$. Then there exists", "an $r$-cycle $\\alpha'$ such that $\\alpha' \\sim_{rat} \\alpha$ and", "such that $\\alpha'$ and $\\beta$ intersect properly." ], "refs": [], "proofs": [ { "contents": [ "Write $\\beta = \\sum n_i[T_i]$ for some subvarieties $T_i \\subset X$", "of dimension $s$. By linearity we may assume that $\\alpha = [Z]$ for", "some irreducible closed subvariety $Z \\subset X$ of dimension $r$.", "We will prove the lemma by induction on the maximum $e$ of the integers", "$$", "\\dim(Z \\cap T_i)", "$$", "The base case is $e = r + s - \\dim(X)$. In this case $Z$ intersects", "$\\beta$ properly and the lemma is trivial.", "\\medskip\\noindent", "Induction step. Assume that $e > r + s - \\dim(X)$. Choose an embedding", "$X \\subset \\mathbf{P}^N$ and apply Lemma \\ref{lemma-moving} to find a", "closed subvariety $C \\subset \\mathbf{P}^N$ such that", "$C \\cdot X = [Z] + \\sum m_j[Z_j]$ and such that the induction", "hypothesis applies to each $Z_j$. Next, apply Lemma \\ref{lemma-move}", "to $C$, $X$, $T_i$ to find $C' \\subset \\mathbf{P}^N \\times \\mathbf{P}^1$.", "Let $\\gamma = C' \\cdot X \\times \\mathbf{P}^1$ viewed as a cycle", "on $X \\times \\mathbf{P}^1$. By Lemma \\ref{lemma-transfer} we have", "$$", "[Z] + \\sum m_j[Z_j] = \\text{pr}_{X, *}(\\gamma \\cdot X \\times 0)", "$$", "On the other hand the cycle", "$\\gamma_\\infty = \\text{pr}_{X, *}(\\gamma \\cdot X \\times \\infty)$", "is supported on $C'_\\infty \\cap X$ hence intersects $\\beta$ transversally.", "Thus we see that $[Z] \\sim_{rat} - \\sum m_j[Z_j] + \\gamma_\\infty$", "by Lemma \\ref{lemma-rational-equivalence-and-intersection}. Since by", "induction each $[Z_j]$ is rationally equivalent to a cycle which properly", "intersects $\\beta$ this finishes the proof." ], "refs": [ "intersection-lemma-moving", "intersection-lemma-move", "intersection-lemma-transfer", "intersection-lemma-rational-equivalence-and-intersection" ], "ref_ids": [ 11997, 11998, 11989, 11978 ] } ], "ref_ids": [] }, { "id": 12000, "type": "theorem", "label": "intersection-lemma-well-defined-special-case", "categories": [ "intersection" ], "title": "intersection-lemma-well-defined-special-case", "contents": [ "Let $X$ be a nonsingular variety. Let", "$W \\subset X \\times \\mathbf{P}^1$ be an $(s + 1)$-dimensional subvariety", "dominating $\\mathbf{P}^1$. Let $W_a$, resp.\\ $W_b$ be the fibre of", "$W \\to \\mathbf{P}^1$ over $a$, resp.\\ $b$. Let $V$ be a $r$-dimensional", "subvariety of $X$ such that $V$ intersects both $W_a$ and", "$W_b$ properly. Then $[V] \\cdot [W_a]_r \\sim_{rat} [V] \\cdot [W_b]_r$." ], "refs": [], "proofs": [ { "contents": [ "We have $[W_a]_r = \\text{pr}_{X,*}(W \\cdot X \\times a)$ and similarly for", "$[W_b]_r$, see Lemma \\ref{lemma-rational-equivalence-and-intersection}.", "Thus we reduce to showing", "$$", "V \\cdot \\text{pr}_{X,*}( W \\cdot X \\times a) \\sim_{rat} V \\cdot", "\\text{pr}_{X,*}( W \\cdot X\\times b).", "$$", "Applying the projection formula", "Lemma \\ref{lemma-projection-formula-flat} we get", "$$", "V \\cdot \\text{pr}_{X,*}( W \\cdot X\\times a) =", "\\text{pr}_{X,*}(V \\times \\mathbf{P}^1 \\cdot (W \\cdot X\\times a))", "$$", "and similarly for $b$. Thus we reduce to showing", "$$", "\\text{pr}_{X,*}(V \\times \\mathbf{P}^1 \\cdot (W \\cdot X\\times a))", "\\sim_{rat}", "\\text{pr}_{X,*}(V \\times \\mathbf{P}^1 \\cdot (W \\cdot X\\times b))", "$$", "If $V \\times \\mathbf{P}^1$ intersects $W$ properly, then", "associativity for the intersection multiplicities", "(Lemma \\ref{lemma-associative})", "gives $V \\times \\mathbf{P}^1 \\cdot (W \\cdot X\\times a) =", "(V \\times \\mathbf{P}^1 \\cdot W) \\cdot X \\times a$", "and similarly for $b$. Thus we reduce to showing", "$$", "\\text{pr}_{X,*}((V \\times \\mathbf{P}^1 \\cdot W) \\cdot X\\times a)", "\\sim_{rat}", "\\text{pr}_{X,*}((V \\times \\mathbf{P}^1 \\cdot W) \\cdot X\\times b)", "$$", "which is true by Lemma \\ref{lemma-rational-equivalence-and-intersection}.", "\\medskip\\noindent", "The argument above does not quite work. The obstruction is that", "we do not know that $V \\times \\mathbf{P}^1$ and $W$ intersect properly.", "We only know that $V$ and $W_a$ and $V$ and $W_b$ intersect properly.", "Let $Z_i$, $i \\in I$ be the irreducible components of", "$V \\times \\mathbf{P}^1 \\cap W$. Then we know that", "$\\dim(Z_i) \\geq r + 1 + s + 1 - n - 1 = r + s + 1 - n$ where $n = \\dim(X)$, see", "Lemma \\ref{lemma-intersect-in-smooth}. Since we have assumed", "that $V$ and $W_a$ intersect properly, we see that", "$\\dim(Z_{i, a}) = r + s - n$ or $Z_{i, a} = \\emptyset$.", "On the other hand, if $Z_{i, a} \\not = \\emptyset$, then", "$\\dim(Z_{i, a}) \\geq \\dim(Z_i) - 1 = r + s - n$.", "It follows that $\\dim(Z_i) = r + s + 1 - n$ if $Z_i$ meets $X \\times a$", "and in this case $Z_i \\to \\mathbf{P}^1$ is surjective.", "Thus we may write $I = I' \\amalg I''$ where $I'$ is the set of $i \\in I$", "such that $Z_i \\to \\mathbf{P}^1$ is surjective and $I''$ is the set of", "$i \\in I$ such that $Z_i$ lies over a closed point $t_i \\in \\mathbf{P}^1$", "with $t_i \\not = a$ and $t_i \\not = b$. Consider the cycle", "$$", "\\gamma = \\sum\\nolimits_{i \\in I'} e_i [Z_i]", "$$", "where we take", "$$", "e_i = \\sum\\nolimits_p (-1)^p", "\\text{length}_{\\mathcal{O}_{X \\times \\mathbf{P}^1, Z_i}}", "\\text{Tor}_p^{\\mathcal{O}_{X \\times \\mathbf{P}^1, Z_i}}(", "\\mathcal{O}_{V \\times \\mathbf{P}^1, Z_i}, \\mathcal{O}_{W, Z_i})", "$$", "We will show that $\\gamma$ can be used as a replacement for", "the intersection product of $V \\times \\mathbf{P}^1$ and $W$.", "\\medskip\\noindent", "We will show this using associativity of intersection products in exactly", "the same way as above. Let $U = \\mathbf{P}^1 \\setminus \\{t_i, i \\in I''\\}$.", "Note that $X \\times a$ and $X \\times b$ are contained in $X \\times U$.", "The subvarieties", "$$", "V \\times U,\\quad W_U,\\quad X \\times a\\quad\\text{of}\\quad X \\times U", "$$", "intersect transversally pairwise by our choice of $U$ and moreover", "$\\dim(V \\times U \\cap W_U \\cap X \\times a) = \\dim(V \\cap W_a)$ has", "the expected dimension. Thus we see that", "$$", "V \\times U \\cdot (W_U \\cdot X \\times a) =", "(V \\times U \\cdot W_U) \\cdot X \\times a", "$$", "as cycles on $X \\times U$ by Lemma \\ref{lemma-associative}.", "By construction $\\gamma$ restricts to the cycle $V \\times U \\cdot W_U$", "on $X \\times U$. Trivially,", "$V \\times \\mathbf{P}^1 \\cdot (W \\times X \\times a)$ restricts", "to $V \\times U \\cdot (W_U \\cdot X \\times a)$ on $X \\times U$.", "Hence", "$$", "V \\times \\mathbf{P}^1 \\cdot (W \\cdot X \\times a) =", "\\gamma \\cdot X \\times a", "$$", "as cycles on $X \\times \\mathbf{P}^1$ (because both sides", "are contained in $X \\times U$ and are equal after restricting", "to $X \\times U$ by what was said before). Since we have the same for $b$", "we conclude", "\\begin{align*}", "V \\cdot [W_a]", "& =", "\\text{pr}_{X,*}(V \\times \\mathbf{P}^1 \\cdot (W \\cdot X\\times a)) \\\\", "& =", "\\text{pr}_{X, *}(\\gamma \\cdot X \\times a) \\\\", "& \\sim_{rat} ", "\\text{pr}_{X, *}(\\gamma \\cdot X \\times b) \\\\", "& =", "\\text{pr}_{X,*}(V \\times \\mathbf{P}^1 \\cdot (W \\cdot X\\times b)) \\\\", "& =", "V \\cdot [W_b]", "\\end{align*}", "The first and the last equality by the first paragraph of the proof,", "the second and penultimate equalities were shown in this paragraph, and", "the middle equivalence is", "Lemma \\ref{lemma-rational-equivalence-and-intersection}." ], "refs": [ "intersection-lemma-rational-equivalence-and-intersection", "intersection-lemma-projection-formula-flat", "intersection-lemma-associative", "intersection-lemma-rational-equivalence-and-intersection", "intersection-lemma-intersect-in-smooth", "intersection-lemma-associative", "intersection-lemma-rational-equivalence-and-intersection" ], "ref_ids": [ 11978, 11988, 11985, 11978, 11966, 11985, 11978 ] } ], "ref_ids": [] }, { "id": 12001, "type": "theorem", "label": "intersection-lemma-pullback-and-intersection-product", "categories": [ "intersection" ], "title": "intersection-lemma-pullback-and-intersection-product", "contents": [ "Let $f : X \\to Y$ be a morphism of nonsingular projective varieties.", "The pullback map on chow groups satisfies:", "\\begin{enumerate}", "\\item $f^* : \\CH^*(Y) \\to \\CH^*(X)$ is a ring map,", "\\item $(g \\circ f)^* = f^* \\circ g^*$ for a composable pair $f, g$,", "\\item the projection formula holds: $f_*(\\alpha) \\cdot \\beta =", "f_*( \\alpha \\cdot f^*\\beta)$, and", "\\item if $f$ is flat then it agrees with the previous definition.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "All of these follow readily from the results above.", "\\medskip\\noindent", "For (1) it suffices to show that", "$\\text{pr}_{X,*}( \\Gamma_f \\cdot \\alpha \\cdot \\beta) =", "\\text{pr}_{X,*}(\\Gamma_f \\cdot \\alpha) \\cdot", "\\text{pr}_{X,*}(\\Gamma_f \\cdot \\beta)$", "for cycles $\\alpha$, $\\beta$ on $X \\times Y$. If $\\alpha$ is a cycle on", "$X \\times Y$ which intersects $\\Gamma_f$ properly, then it is easy", "to see that", "$$", "\\Gamma_f \\cdot \\alpha =", "\\Gamma_f \\cdot \\text{pr}_X^*(\\text{pr}_{X,*}(\\Gamma_f \\cdot \\alpha))", "$$", "as cycles because $\\Gamma_f$ is a graph. Thus we get the first", "equality in", "\\begin{align*}", "\\text{pr}_{X,*}(\\Gamma_f \\cdot \\alpha \\cdot \\beta)", "& =", "\\text{pr}_{X,*}(", "\\Gamma_f \\cdot", "\\text{pr}_X^*(\\text{pr}_{X,*}(\\Gamma_f \\cdot \\alpha)) \\cdot \\beta) \\\\", "& =", "\\text{pr}_{X,*}(\\text{pr}_X^*(\\text{pr}_{X,*}(\\Gamma_f \\cdot \\alpha))", "\\cdot (\\Gamma_f \\cdot \\beta)) \\\\", "& =", "\\text{pr}_{X,*}(\\Gamma_f \\cdot \\alpha) \\cdot", "\\text{pr}_{X,*}(\\Gamma_f \\cdot \\beta)", "\\end{align*}", "the last step by the projection formula in the flat case", "(Lemma \\ref{lemma-projection-formula-flat}).", "\\medskip\\noindent", "If $g : Y \\to Z$ then property (2) follows formally from the observation that", "$$", "\\Gamma =", "\\text{pr}_{X \\times Y}^*\\Gamma_f \\cdot", "\\text{pr}_{Y \\times Z}^*\\Gamma_g", "$$", "in $Z_*(X \\times Y \\times Z)$ where $\\Gamma = \\{(x, f(x), g(f(x))\\}$", "and maps isomorphically to $\\Gamma_{g \\circ f}$ in $X \\times Z$.", "The equality follows from the scheme theoretic equality and", "Lemma \\ref{lemma-transversal}.", "\\medskip\\noindent", "For (3) we use the projection formula for flat maps twice", "\\begin{align*}", "f_*(\\alpha \\cdot pr_{X, *}(\\Gamma_f \\cdot pr_Y^*(\\beta)))", "& =", "f_*(pr_{X, *}(pr_X^*\\alpha \\cdot \\Gamma_f \\cdot pr_Y^*(\\beta))) \\\\", "& =", "pr_{Y, *}(pr_X^*\\alpha \\cdot \\Gamma_f \\cdot pr_Y^*(\\beta))) \\\\", "& =", "pt_{Y, *}(pr_X^*\\alpha \\cdot \\Gamma_f) \\cdot \\beta \\\\", "& =", "f_*(\\alpha) \\cdot \\beta", "\\end{align*}", "where in the last equality we use the remark on graphs made above.", "This proves (3).", "\\medskip\\noindent", "Property (4) rests on identifying the intersection product", "$\\Gamma_f \\cdot pr_Y^*\\alpha$ in the case $f$ is flat. Namely, in this", "case if $V \\subset Y$ is a closed subvariety, then every generic point", "$\\xi$ of the scheme $f^{-1}(V) \\cong \\Gamma_f \\cap pr_Y^{-1}(V)$", "lies over the generic point of $V$. Hence the local ring of", "$pr_Y^{-1}(V) = X \\times V$ at $\\xi$ is Cohen-Macaulay. Since", "$\\Gamma_f \\subset X \\times Y$ is a regular immersion (as a morphism of", "smooth projective varieties) we find that", "$$", "\\Gamma_f \\cdot pr_Y^*[V] = [\\Gamma_f \\cap pr_Y^{-1}(V)]_d", "$$", "with $d$ the dimension of $\\Gamma_f \\cap pr_Y^{-1}(V)$, see", "Lemma \\ref{lemma-multiplicity-lci-CM}. Since $\\Gamma_f \\cap pr_Y^{-1}(V)$", "maps isomorphically to $f^{-1}(V)$ we conclude." ], "refs": [ "intersection-lemma-projection-formula-flat", "intersection-lemma-transversal", "intersection-lemma-multiplicity-lci-CM" ], "ref_ids": [ 11988, 11969, 11977 ] } ], "ref_ids": [] }, { "id": 12002, "type": "theorem", "label": "intersection-proposition-positivity", "categories": [ "intersection" ], "title": "intersection-proposition-positivity", "contents": [ "\\begin{reference}", "This is one of the main results of \\cite{Serre_algebre_locale}.", "\\end{reference}", "Let $X$ be a nonsingular variety. Let $V \\subset X$ and", "$W \\subset Y$ be closed subvarieties which intersect properly.", "Let $Z \\subset V \\cap W$ be an irreducible component.", "Then $e(X, V \\cdot W, Z) > 0$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-reduction-diagonal} we have", "$$", "e(X, V \\cdot W, Z) = e(X \\times X, \\Delta \\cdot V \\times W, \\Delta(Z))", "$$", "Since $\\Delta : X \\to X \\times X$ is a regular immersion", "(see Lemma \\ref{lemma-diagonal-regular-immersion}), we see that", "$e(X \\times X, \\Delta \\cdot V \\times W, \\Delta(Z))$ is a positive", "integer by Lemma \\ref{lemma-multiplicity-with-lci}." ], "refs": [ "intersection-lemma-reduction-diagonal", "intersection-lemma-diagonal-regular-immersion", "intersection-lemma-multiplicity-with-lci" ], "ref_ids": [ 11983, 11965, 11975 ] } ], "ref_ids": [] }, { "id": 12008, "type": "theorem", "label": "homology-lemma-preadditive-zero", "categories": [ "homology" ], "title": "homology-lemma-preadditive-zero", "contents": [ "Let $\\mathcal{A}$ be a preadditive category.", "Let $x$ be an object of $\\mathcal{A}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $x$ is an initial object,", "\\item $x$ is a final object, and", "\\item $\\text{id}_x = 0$ in $\\Mor_\\mathcal{A}(x, x)$.", "\\end{enumerate}", "Furthermore, if such an object $0$ exists, then a morphism", "$\\alpha : x \\to y$ factors through $0$ if and only if $\\alpha = 0$." ], "refs": [], "proofs": [ { "contents": [ "First assume that $x$ is either (1) initial or (2) final.", "In both cases, it follows that $\\Mor(x,x)$ is a trivial abelian group", "containing $\\text{id}_x$, thus $\\text{id}_x = 0$ in", "$\\Mor(x, x)$, which shows that each of (1) and (2) implies (3).", "\\medskip\\noindent", "Now assume that $\\text{id}_x = 0$ in $\\Mor(x,x)$. Let $y$", "be an arbitrary object of $\\mathcal{A}$ and let $f \\in \\Mor(x ,y)$.", "Denote $C : \\Mor(x,x) \\times \\Mor(x,y) \\to \\Mor(x,y)$ the composition map.", "Then $f = C(0, f)$ and since $C$ is bilinear we have $C(0, f) = 0$.", "Thus $f = 0$. Hence $x$ is initial in $\\mathcal{A}$.", "A similar argument for $f \\in \\Mor(y, x)$ can be used to show that", "$x$ is also final. Thus (3) implies both (1) and (2)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12009, "type": "theorem", "label": "homology-lemma-preadditive-direct-sum", "categories": [ "homology" ], "title": "homology-lemma-preadditive-direct-sum", "contents": [ "Let $\\mathcal{A}$ be a preadditive category.", "Let $x, y \\in \\Ob(\\mathcal{A})$.", "If the product $x \\times y$ exists, then so does", "the coproduct $x \\amalg y$.", "If the coproduct $x \\amalg y$ exists, then so does", "the product $x \\times y$. In this case", "also $x \\amalg y \\cong x \\times y$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $z = x \\times y$ with projections", "$p : z \\to x$ and $q : z \\to y$. Denote $i : x \\to z$", "the morphism corresponding to $(1, 0)$. Denote $j : y \\to z$", "the morphism corresponding to $(0, 1)$. Thus we have the", "commutative diagram", "$$", "\\xymatrix{", "x \\ar[rr]^1 \\ar[rd]^i & & x \\\\", "& z \\ar[ru]^p \\ar[rd]^q & \\\\", "y \\ar[rr]^1 \\ar[ru]^j & & y", "}", "$$", "where the diagonal compositions are zero. It follows that", "$i \\circ p + j \\circ q : z \\to z$ is the identity since", "it is a morphism which upon composing with $p$ gives $p$", "and upon composing with $q$ gives $q$.", "Suppose given morphisms $a : x \\to w$ and $b : y \\to w$.", "Then we can form the map $a \\circ p + b \\circ q : z \\to w$.", "In this way we get a bijection $\\Mor(z, w)", "= \\Mor(x, w) \\times \\Mor(y, w)$ which", "show that $z = x \\amalg y$.", "\\medskip\\noindent", "We leave it to the reader to construct the morphisms", "$p, q$ given a coproduct $x \\amalg y$ instead of a", "product." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12010, "type": "theorem", "label": "homology-lemma-additive-additive", "categories": [ "homology" ], "title": "homology-lemma-additive-additive", "contents": [ "Let $\\mathcal{A}$, $\\mathcal{B}$ be preadditive categories.", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor.", "Then $F$ transforms direct sums to direct sums and zero to zero." ], "refs": [], "proofs": [ { "contents": [ "Suppose $F$ is additive. A direct sum $z$", "of $x$ and $y$ is characterized by having morphisms", "$i : x \\to z$, $j : y \\to z$, $p : z \\to x$ and", "$q : z \\to y$ such that $p \\circ i = \\text{id}_x$,", "$q \\circ j = \\text{id}_y$, $p \\circ j = 0$, $q \\circ i = 0$", "and $i \\circ p + j \\circ q = \\text{id}_z$, according", "to Remark \\ref{remark-direct-sum}. Clearly $F(x), F(y), F(z)$", "and the morphisms $F(i), F(j), F(p), F(q)$ satisfy exactly the", "same relations (by additivity) and we see that $F(z)$ is", "a direct sum of $F(x)$ and $F(y)$.", "Hence, $F$ transforms direct sums to direct sums.", "\\medskip\\noindent", "To see that $F$ transforms zero to zero, use the", "characterization (3) of the zero object in", "Lemma \\ref{lemma-preadditive-zero}." ], "refs": [ "homology-remark-direct-sum", "homology-lemma-preadditive-zero" ], "ref_ids": [ 12189, 12008 ] } ], "ref_ids": [] }, { "id": 12011, "type": "theorem", "label": "homology-lemma-additive-cat-biproduct-kernel", "categories": [ "homology" ], "title": "homology-lemma-additive-cat-biproduct-kernel", "contents": [ "Let $\\mathcal{C}$ be a preadditive category.", "Let $x \\oplus y$ with morphisms $i, j, p, q$ as in", "Lemma \\ref{lemma-preadditive-direct-sum}", "be a direct sum in $\\mathcal{C}$. Then $i : x \\to x \\oplus y$", "is a kernel of $q : x \\oplus y \\rightarrow y$. Dually, $p$ is", "a cokernel for $j$." ], "refs": [ "homology-lemma-preadditive-direct-sum" ], "proofs": [ { "contents": [ "Let $f : z' \\to x \\oplus y$ be a morphism such that $q \\circ f = 0$.", "We have to show that there exists a unique morphism $g : z' \\to x$", "such that $f = i \\circ g$. Since $i \\circ p + j \\circ q$ is the identity on", "$x \\oplus y$ we see that", "$$", "f = (i \\circ p + j \\circ q) \\circ f = i \\circ p \\circ f", "$$", "and hence $g = p \\circ f$ works. Uniqueness holds because $p \\circ i$", "is the identity on $x$. The proof of the second statement is dual." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12009 ] }, { "id": 12012, "type": "theorem", "label": "homology-lemma-kernel-mono", "categories": [ "homology" ], "title": "homology-lemma-kernel-mono", "contents": [ "Let $\\mathcal{C}$ be a preadditive category.", "Let $f : x \\to y$ be a morphism in $\\mathcal{C}$.", "\\begin{enumerate}", "\\item If a kernel of $f$ exists, then", "this kernel is a monomorphism.", "\\item If a cokernel of $f$ exists, then", "this cokernel is an epimorphism.", "\\item If a kernel and coimage of $f$ exist, then", "the coimage is an epimorphism.", "\\item If a cokernel and image of $f$ exist, then", "the image is a monomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows easily from the uniqueness required in the", "definition of a kernel. The proof of (2) is dual.", "Part (3) follows from (2), since the coimage is a cokernel.", "Similarly, (4) follows from (1)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12013, "type": "theorem", "label": "homology-lemma-coim-im-map", "categories": [ "homology" ], "title": "homology-lemma-coim-im-map", "contents": [ "Let $f : x \\to y$ be a morphism in a preadditive category", "such that the kernel, cokernel, image and coimage all exist.", "Then $f$ can be factored uniquely as", "$x \\to \\Coim(f) \\to \\Im(f) \\to y$." ], "refs": [], "proofs": [ { "contents": [ "There is a canonical morphism $\\Coim(f) \\to y$", "because $\\Ker(f) \\to x \\to y$ is zero.", "The composition $\\Coim(f) \\to y \\to \\Coker(f)$", "is zero, because it is the unique morphism which gives", "rise to the morphism $x \\to y \\to \\Coker(f)$ which", "is zero", "(the uniqueness follows from", "Lemma \\ref{lemma-kernel-mono} (3)).", "Hence $\\Coim(f) \\to y$ factors uniquely through", "$\\Im(f) \\to y$, which gives us the desired map." ], "refs": [ "homology-lemma-kernel-mono" ], "ref_ids": [ 12012 ] } ], "ref_ids": [] }, { "id": 12014, "type": "theorem", "label": "homology-lemma-karoubian", "categories": [ "homology" ], "title": "homology-lemma-karoubian", "contents": [ "Let $\\mathcal{C}$ be a preadditive category. The following", "are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{C}$ is Karoubian,", "\\item every idempotent endomorphism of an object of $\\mathcal{C}$ has a", "cokernel, and", "\\item given an idempotent endomorphism $p : z \\to z$ of $\\mathcal{C}$", "there exists a direct sum decomposition $z = x \\oplus y$ such", "that $p$ corresponds to the projection onto $y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and let $p : z \\to z$ be as in (3).", "Let $x = \\Ker(p)$ and $y = \\Ker(1 - p)$. There are maps", "$x \\to z$ and $y \\to z$. Since $(1 - p)p = 0$ we see that $p : z \\to z$", "factors through $y$, hence we obtain a morphism $z \\to y$. Similarly", "we obtain a morphism $z \\to x$. We omit the verification that these", "four morphisms induce an isomorphism $x = y \\oplus z$ as in", "Remark \\ref{remark-direct-sum}.", "Thus (1) $\\Rightarrow$ (3). The implication (2) $\\Rightarrow$ (3)", "is dual. Finally, condition (3) implies (1) and (2) by", "Lemma \\ref{lemma-additive-cat-biproduct-kernel}." ], "refs": [ "homology-remark-direct-sum", "homology-lemma-additive-cat-biproduct-kernel" ], "ref_ids": [ 12189, 12011 ] } ], "ref_ids": [] }, { "id": 12015, "type": "theorem", "label": "homology-lemma-projectors-have-images", "categories": [ "homology" ], "title": "homology-lemma-projectors-have-images", "contents": [ "Let $\\mathcal{D}$ be a preadditive category.", "\\begin{enumerate}", "\\item If $\\mathcal{D}$ has countable products and kernels of maps which", "have a right inverse, then $\\mathcal{D}$ is Karoubian.", "\\item If $\\mathcal{D}$ has countable coproducts and cokernels of", "maps which have a left inverse, then $\\mathcal{D}$ is Karoubian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X$ be an object of $\\mathcal{D}$ and let $e : X \\to X$ be an idempotent.", "The functor", "$$", "W \\longmapsto \\Ker(", "\\Mor_\\mathcal{D}(W, X)", "\\xrightarrow{e}", "\\Mor_\\mathcal{D}(W, X)", ")", "$$", "if representable if and only if $e$ has a kernel. Note that for any", "abelian group $A$ and idempotent endomorphism $e : A \\to A$ we have", "$$", "\\Ker(e : A \\to A)", "= \\Ker(\\Phi :", "\\prod\\nolimits_{n \\in \\mathbf{N}} A", "\\to", "\\prod\\nolimits_{n \\in \\mathbf{N}} A", ")", "$$", "where", "$$", "\\Phi(a_1, a_2, a_3, \\ldots) = (ea_1 + (1 - e)a_2, ea_2 + (1 - e)a_3, \\ldots)", "$$", "Moreover, $\\Phi$ has the right inverse", "$$", "\\Psi(a_1, a_2, a_3, \\ldots) =", "(a_1, (1 - e)a_1 + ea_2, (1 - e)a_2 + ea_3, \\ldots).", "$$", "Hence (1) holds. The proof of (2) is dual (using the dual definition", "of a Karoubian category, namely condition (2) of", "Lemma \\ref{lemma-karoubian})." ], "refs": [ "homology-lemma-karoubian" ], "ref_ids": [ 12014 ] } ], "ref_ids": [] }, { "id": 12016, "type": "theorem", "label": "homology-lemma-abelian-opposite", "categories": [ "homology" ], "title": "homology-lemma-abelian-opposite", "contents": [ "Let $\\mathcal{A}$ be a preadditive category.", "The additions on sets of morphisms make", "$\\mathcal{A}^{opp}$ into a preadditive category.", "Furthermore, $\\mathcal{A}$ is additive if and only if $\\mathcal{A}^{opp}$", "is additive, and", "$\\mathcal{A}$ is abelian if and only if $\\mathcal{A}^{opp}$ is abelian." ], "refs": [], "proofs": [ { "contents": [ "The first statement is straightforward.", "To see that $\\mathcal{A}$ is additive if and only if $\\mathcal{A}^{opp}$", "is additive, recall that additivity can be characterized by", "the existence of a zero object and direct sums, which are both", "preserved when passing to the opposite category.", "Finally, to see that", "$\\mathcal{A}$ is abelian if and only if $\\mathcal{A}^{opp}$ is abelian,", "observes that kernels, cokernels, images and coimages in", "$\\mathcal{A}^{opp}$ correspond to", "cokernels, kernels, coimages and images in $\\mathcal{A}$,", "respectively." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12017, "type": "theorem", "label": "homology-lemma-characterize-injective", "categories": [ "homology" ], "title": "homology-lemma-characterize-injective", "contents": [ "Let $f : x \\to y$ be a morphism in an abelian category $\\mathcal{A}$. Then", "\\begin{enumerate}", "\\item $f$ is injective if and only if $f$ is a monomorphism, and", "\\item $f$ is surjective if and only if $f$ is an epimorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Recall that $\\Ker(f)$ is an object representing the", "functor sending $z$ to", "$\\Ker(\\Mor_\\mathcal{A}(z, x) \\to \\Mor_\\mathcal{A}(z, y))$.", "Thus $\\Ker(f)$ is $0$ if and only if", "$\\Mor_\\mathcal{A}(z, x) \\to \\Mor_\\mathcal{A}(z, y)$", "is injective for all $z$ if and only if $f$ is a monomorphism.", "The proof of (2) is similar." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12018, "type": "theorem", "label": "homology-lemma-colimit-abelian-category", "categories": [ "homology" ], "title": "homology-lemma-colimit-abelian-category", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "All finite limits and finite colimits exist in $\\mathcal{A}$." ], "refs": [], "proofs": [ { "contents": [ "To show that finite limits exist it suffices to show", "that finite products and equalizers exist, see", "Categories, Lemma \\ref{categories-lemma-finite-limits-exist}.", "Finite products exist", "by definition and the equalizer of $a, b : x \\to y$ is", "the kernel of $a - b$. The argument for finite colimits", "is similar but dual to this." ], "refs": [ "categories-lemma-finite-limits-exist" ], "ref_ids": [ 12224 ] } ], "ref_ids": [] }, { "id": 12019, "type": "theorem", "label": "homology-lemma-check-exactness", "categories": [ "homology" ], "title": "homology-lemma-check-exactness", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a complex of $\\mathcal{A}$.", "\\begin{enumerate}", "\\item $M_1 \\to M_2 \\to M_3 \\to 0$ is exact if and only if", "$$", "0 \\to \\Hom_\\mathcal{A}(M_3, N) \\to", "\\Hom_\\mathcal{A}(M_2, N) \\to \\Hom_\\mathcal{A}(M_1, N)", "$$", "is an exact sequence of abelian groups for all objects $N$ of", "$\\mathcal{A}$, and", "\\item $0 \\to M_1 \\to M_2 \\to M_3$ is exact if and only if", "$$", "0 \\to \\Hom_\\mathcal{A}(N, M_1) \\to \\Hom_\\mathcal{A}(N, M_2) \\to", "\\Hom_\\mathcal{A}(N, M_1)", "$$", "is an exact sequence of abelian groups for all objects $N$ of $\\mathcal{A}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: See", "Algebra, Lemma \\ref{algebra-lemma-hom-exact}." ], "refs": [ "algebra-lemma-hom-exact" ], "ref_ids": [ 352 ] } ], "ref_ids": [] }, { "id": 12020, "type": "theorem", "label": "homology-lemma-ses-split", "categories": [ "homology" ], "title": "homology-lemma-ses-split", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $0 \\to A \\to B \\to C \\to 0$", "be a short exact sequence.", "\\begin{enumerate}", "\\item Given a morphism $s : C \\to B$ left inverse to", "$B \\to C$, there exists a unique $\\pi : B \\to A$", "such that $(s, \\pi)$ splits the short exact sequence", "as in Definition \\ref{definition-ses-split}.", "\\item Given a morphism $\\pi : B \\to A$ right inverse to", "$A \\to B$, there exists a unique $s : C \\to B$", "such that $(s, \\pi)$ splits the short exact sequence", "as in Definition \\ref{definition-ses-split}.", "\\end{enumerate}" ], "refs": [ "homology-definition-ses-split", "homology-definition-ses-split" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12140, 12140 ] }, { "id": 12021, "type": "theorem", "label": "homology-lemma-characterize-cartesian", "categories": [ "homology" ], "title": "homology-lemma-characterize-cartesian", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let", "$$", "\\xymatrix{", "w\\ar[r]^f\\ar[d]_g", "& y\\ar[d]^h\\\\", "x\\ar[r]^k", "& z", "}", "$$", "be a commutative diagram. ", "\\begin{enumerate}", "\\item The diagram is cartesian if and only if ", "$$", "0 \\to w \\xrightarrow{(g, f)} x \\oplus y \\xrightarrow{(k, -h)} z", "$$", "is exact.", "\\item The diagram is cocartesian if and only if ", "$$", "w \\xrightarrow{(g, -f)} x \\oplus y \\xrightarrow{(k, h)} z \\to 0", "$$", "is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $u = (g, f) : w \\to x \\oplus y$ and $v = (k, -h) : x \\oplus y \\to z$. ", "Let $p : x \\oplus y \\to x$ and $q : x \\oplus y \\to y$ be the canonical ", "projections. Let $i : \\Ker(v) \\to x \\oplus y$ be the canonical ", "injection. By Example \\ref{example-fibre-product-pushouts}, the diagram is ", "cartesian if and only if there exists an isomorphism ", "$r : \\Ker(v) \\to w$ with $f \\circ r = q \\circ i$ and ", "$g \\circ r = p \\circ i$. The sequence ", "$0 \\to w \\overset{u} \\to x \\oplus y \\overset{v} \\to z$ is exact if and ", "only if there exists an isomorphism $r : \\Ker(v) \\to w$ with ", "$u \\circ r = i$. But given $r : \\Ker(v) \\to w$, we have ", "$f \\circ r = q \\circ i$ and $g \\circ r = p \\circ i$ if and ", "only if $q \\circ u \\circ r= f \\circ r = q \\circ i$ and ", "$p \\circ u \\circ r = g \\circ r = p \\circ i$, hence if and only if", "$u \\circ r = i$. This proves (1), and then (2) follows by duality." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12022, "type": "theorem", "label": "homology-lemma-cartesian-kernel", "categories": [ "homology" ], "title": "homology-lemma-cartesian-kernel", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let", "$$", "\\xymatrix{", "w\\ar[r]^f\\ar[d]_g", "& y\\ar[d]^h\\\\", "x\\ar[r]^k", "& z", "}", "$$", "be a commutative diagram.", "\\begin{enumerate}", "\\item If the diagram is cartesian, then the morphism ", "$\\Ker(f)\\to\\Ker(k)$ induced by $g$ is an isomorphism.", "\\item If the diagram is cocartesian, then the morphism ", "$\\Coker(f)\\to\\Coker(k)$ induced by $h$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose the diagram is cartesian. Let ", "$e:\\Ker(f)\\to\\Ker(k)$ be induced by $g$. Let ", "$i:\\Ker(f)\\to w$ and $j:\\Ker(k)\\to x$ be the canonical ", "injections. There exists $t:\\Ker(k)\\to w$ with $f\\circ t=0$ ", "and $g\\circ t=j$. Hence, there exists $u:\\Ker(k)\\to\\Ker(f)$ ", "with $i\\circ u=t$. It follows ", "$g\\circ i\\circ u\\circ e=g\\circ t\\circ e=j\\circ e=g\\circ i$ and ", "$f\\circ i\\circ u\\circ e=0=f\\circ i$, hence $i\\circ u\\circ e=i$. Since ", "$i$ is a monomorphism this implies $u\\circ e=\\text{id}_{\\Ker(f)}$.", "Furthermore, we have $j\\circ e\\circ u=g\\circ i\\circ u=g\\circ t=j$. ", "Since $j$ is a monomorphism this implies $e\\circ u=\\text{id}_{\\Ker(k)}$.", "This proves (1). Now, (2) follows by duality." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12023, "type": "theorem", "label": "homology-lemma-cartesian-cocartesian", "categories": [ "homology" ], "title": "homology-lemma-cartesian-cocartesian", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let", "$$", "\\xymatrix{", "w\\ar[r]^f\\ar[d]_g", "& y\\ar[d]^h\\\\", "x\\ar[r]^k", "& z", "}", "$$", "be a commutative diagram.", "\\begin{enumerate}", "\\item If the diagram is cartesian and $k$ is an epimorphism, ", "then the diagram is cocartesian and $f$ is an epimorphism.", "\\item If the diagram is cocartesian and $g$ is a monomorphism, ", "then the diagram is cartesian and $h$ is a monomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose the diagram is cartesian and $k$ is an epimorphism. ", "Let $u = (g, f) : w \\to x \\oplus y$ and let $v = (k, -h) : x \\oplus y \\to z$. ", "As $k$ is an epimorphism, $v$ is an epimorphism, too. Therefore ", "and by Lemma \\ref{lemma-characterize-cartesian}, the sequence ", "$0\\to w\\overset{u}\\to x\\oplus y\\overset{v}\\to z\\to 0$ is exact. Thus, the ", "diagram is cocartesian by Lemma \\ref{lemma-characterize-cartesian}. Finally, ", "$f$ is an epimorphism by Lemma \\ref{lemma-cartesian-kernel} and ", "Lemma \\ref{lemma-characterize-injective}. This proves (1), and (2) ", "follows by duality." ], "refs": [ "homology-lemma-characterize-cartesian", "homology-lemma-characterize-cartesian", "homology-lemma-cartesian-kernel", "homology-lemma-characterize-injective" ], "ref_ids": [ 12021, 12021, 12022, 12017 ] } ], "ref_ids": [] }, { "id": 12024, "type": "theorem", "label": "homology-lemma-epimorphism-universal-abelian-category", "categories": [ "homology" ], "title": "homology-lemma-epimorphism-universal-abelian-category", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item If $x \\to y$ is surjective, then for every $z \\to y$ the", "projection $x \\times_y z \\to z$ is surjective.", "\\item If $x \\to y$ is injective, then for every $x \\to z$ the", "morphism $z \\to z \\amalg_x y$ is injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Immediately from Lemma \\ref{lemma-characterize-injective} and", "Lemma \\ref{lemma-cartesian-cocartesian}." ], "refs": [ "homology-lemma-characterize-injective", "homology-lemma-cartesian-cocartesian" ], "ref_ids": [ 12017, 12023 ] } ], "ref_ids": [] }, { "id": 12025, "type": "theorem", "label": "homology-lemma-check-exactness-fibre-product", "categories": [ "homology" ], "title": "homology-lemma-check-exactness-fibre-product", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $f:x\\to y$ and $g:y\\to z$ ", "be morphisms with $g\\circ f=0$. Then, the following statements are equivalent:", "\\begin{enumerate}", "\\item The sequence $x\\overset{f}\\to y\\overset{g}\\to z$ is exact.", "\\item For every $h:w\\to y$ with $g\\circ h=0$ there exist an object $v$, ", "an epimorphism $k:v\\to w$ and a morphism $l:v\\to x$ with $h\\circ k=f\\circ l$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $i:\\Ker(g)\\to y$ be the canonical injection. Let ", "$p:x\\to\\Coim(f)$ be the canonical projection. Let ", "$j:\\Im(f)\\to\\Ker(g)$ be the canonical injection.", "\\medskip\\noindent", "Suppose (1) holds. Let $h:w\\to y$ with $g\\circ h=0$. There exists ", "$c:w\\to\\Ker(g)$ with $i\\circ c=h$. ", "Let $v=x\\times_{\\Ker(g)}w$ with canonical projections ", "$k:v\\to w$ and $l:v\\to x$, so that $c\\circ k=j\\circ p\\circ l$. ", "Then, $h\\circ k=i\\circ c\\circ k=i\\circ j\\circ p\\circ l=f\\circ l$. ", "As $j\\circ p$ is an epimorphism by hypothesis, $k$ is an ", "epimorphism by Lemma \\ref{lemma-cartesian-cocartesian}. This implies (2).", "\\medskip\\noindent", "Suppose (2) holds. Then, $g\\circ i=0$. So, there are an object ", "$w$, an epimorphism $k:w\\to\\Ker(g)$ and a morphism ", "$l:w\\to x$ with $f\\circ l=i\\circ k$. It follows ", "$i\\circ j\\circ p\\circ l=f\\circ l=i\\circ k$. Since $i$ is a ", "monomorphism we see that $j\\circ p\\circ l=k$ is an epimorphism. ", "So, $j$ is an epimorphisms and thus an isomorphism. This implies (1)." ], "refs": [ "homology-lemma-cartesian-cocartesian" ], "ref_ids": [ 12023 ] } ], "ref_ids": [] }, { "id": 12026, "type": "theorem", "label": "homology-lemma-exact-kernel-sequence", "categories": [ "homology" ], "title": "homology-lemma-exact-kernel-sequence", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let", "$$", "\\xymatrix{", "x \\ar[r]^f \\ar[d]^\\alpha &", "y \\ar[r]^g \\ar[d]^\\beta &", "z \\ar[d]^\\gamma\\\\", "u \\ar[r]^k & v \\ar[r]^l & w", "}", "$$", "be a commutative diagram.", "\\begin{enumerate}", "\\item If the first row is exact and $k$ is a monomorphism, then the induced ", "sequence $\\Ker(\\alpha) \\to \\Ker(\\beta) \\to \\Ker(\\gamma)$ ", "is exact.", "\\item If the second row is exact and $g$ is an epimorphism, then the induced ", "sequence", "$\\Coker(\\alpha) \\to \\Coker(\\beta) \\to \\Coker(\\gamma)$ ", "is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose the first row is exact and $k$ is a monomorphism. Let ", "$a:\\Ker(\\alpha)\\to\\Ker(\\beta)$ and ", "$b:\\Ker(\\beta)\\to\\Ker(\\gamma)$ be the induced morphisms. ", "Let $h:\\Ker(\\alpha)\\to x$, $i:\\Ker(\\beta)\\to y$ and ", "$j:\\Ker(\\gamma)\\to z$ be the canonical injections. As $j$ is ", "a monomorphism we have $b\\circ a=0$. Let $c:s\\to\\Ker(\\beta)$ ", "with $b\\circ c=0$. Then, $g\\circ i\\circ c=j\\circ b\\circ c=0$. By ", "Lemma \\ref{lemma-check-exactness-fibre-product} there are an object $t$, an ", "epimorphism $d:t\\to s$ and a morphism $e:t\\to x$ with ", "$i\\circ c\\circ d=f\\circ e$. Then, ", "$k\\circ \\alpha\\circ e=\\beta\\circ f\\circ e=\\beta\\circ i\\circ c\\circ d=0$. ", "As $k$ is a monomorphism we get $\\alpha\\circ e=0$. So, there exists ", "$m:t\\to\\Ker(\\alpha)$ with $h\\circ m=e$. It follows ", "$i\\circ a\\circ m=f\\circ h\\circ m=f\\circ e=i\\circ c\\circ d$. ", "As $i$ is a monomorphism we get $a\\circ m=c\\circ d$. Thus, ", "Lemma \\ref{lemma-check-exactness-fibre-product} implies (1), and then ", "(2) follows by duality." ], "refs": [ "homology-lemma-check-exactness-fibre-product", "homology-lemma-check-exactness-fibre-product" ], "ref_ids": [ 12025, 12025 ] } ], "ref_ids": [] }, { "id": 12027, "type": "theorem", "label": "homology-lemma-snake", "categories": [ "homology" ], "title": "homology-lemma-snake", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let ", "$$", "\\xymatrix{", "& x \\ar[r]^f \\ar[d]^\\alpha &", "y \\ar[r]^g \\ar[d]^\\beta &", "z \\ar[r] \\ar[d]^\\gamma &", "0 \\\\", "0 \\ar[r] & u \\ar[r]^k & v \\ar[r]^l & w", "}", "$$", "be a commutative diagram with exact rows.", "\\begin{enumerate}", "\\item There exists a unique morphism ", "$\\delta : \\Ker(\\gamma) \\rightarrow \\Coker(\\alpha)$ ", "such that the diagram", "$$", "\\xymatrix{", "y \\ar[d]_\\beta &", "y \\times_z \\Ker(\\gamma) \\ar[l]_{\\pi'} \\ar[r]^{\\pi} &", "\\Ker(\\gamma) \\ar[d]^\\delta \\\\", "v \\ar[r]^{\\iota'} & \\Coker(\\alpha) \\amalg_u v &", "\\Coker(\\alpha) \\ar[l]_\\iota", "}", "$$", "commutes, where $\\pi$ and $\\pi'$ are the canonical projections ", "and $\\iota$ and $\\iota'$ are the canonical coprojections.", "\\item The induced sequence ", "$$", "\\Ker(\\alpha)\\overset{f'} \\to \\Ker(\\beta) ", "\\overset{g'}\\to \\Ker(\\gamma)\\overset{\\delta}\\to", "\\Coker(\\alpha) \\overset{k'}\\to \\Coker(\\beta) ", "\\overset{l'}\\to \\Coker(\\gamma)", "$$", "is exact. If $f$ is injective then so is $f'$, and if $l$ is ", "surjective then so is $l'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As $\\pi$ is an epimorphism and $\\iota$ is a monomorphism by ", "Lemma \\ref{lemma-cartesian-cocartesian}, uniqueness of $\\delta$ is clear.", "Let $p=y\\times_z\\Ker(\\gamma)$ and $q=\\Coker(\\alpha)\\amalg_uv$. ", "Let $h:\\Ker(\\beta)\\to y$, $i:\\Ker(\\gamma)\\to z$ and ", "$j:\\Ker(\\pi)\\rightarrow p$ be the canonical injections. ", "Let $p:u\\to\\Coker(\\alpha)$ be the canonical projection. ", "Keeping in mind Lemma \\ref{lemma-cartesian-cocartesian} we get a commutative ", "diagram with exact rows ", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Ker(\\pi) \\ar[r]^j &", "p \\ar[r]^{\\pi} \\ar[d]_{\\pi'} &", "\\Ker(\\gamma) \\ar[d]_i \\ar[r] & 0 \\\\", "& x \\ar[r]^f \\ar[d]_\\alpha & y \\ar[r]^g \\ar[d]_\\beta &", "z \\ar[d]_\\gamma \\ar[r] & 0 \\\\", "0 \\ar[r] & u \\ar[r]^k \\ar[d]_p &", "v \\ar[r]^l \\ar[d]_{\\iota'} & w & \\\\", "0 \\ar[r] & \\Coker(\\alpha) \\ar[r]^\\iota & q & &", "}", "$$", "As $l \\circ \\beta \\circ \\pi' = \\gamma \\circ i \\circ \\pi = 0$ and as the third ", "row of the diagram above is exact, there is an $a:p\\to u$ ", "with $k \\circ a = \\beta \\circ \\pi'$. As the upper right quadrangle of the ", "diagram above is cartesian, Lemma \\ref{lemma-cartesian-kernel} yields an ", "epimorphism $b : x \\to \\Ker(\\pi)$ with $\\pi' \\circ j \\circ b = f$. ", "It follows ", "$k \\circ a \\circ j \\circ b = \\beta \\circ \\pi' \\circ j \\circ b =", "\\beta \\circ f = k \\circ \\alpha$.", "As $k$ is a monomorphism this implies $a \\circ j \\circ b = \\alpha$. It follows ", "$p \\circ a \\circ j \\circ b = p \\circ \\alpha = 0$. As $b$ is an epimorphism this ", "implies $p\\circ a\\circ j=0$. Therefore, as the top row of the diagram ", "above is exact, there exists", "$\\delta : \\Ker(\\gamma) \\to \\Coker(\\alpha)$ with", "$\\delta \\circ \\pi = p \\circ a$. It follows ", "$\\iota \\circ \\delta \\circ \\pi = \\iota \\circ p \\circ a =", "\\iota' \\circ k \\circ a = \\iota' \\circ \\beta \\circ \\pi'$", "as desired.", "\\medskip\\noindent", "As the upper right quadrangle in the diagram above is cartesian there ", "is a $c : \\Ker(\\beta) \\to p$ with $\\pi' \\circ c = h$ and $\\pi \\circ c = g'$. ", "It follows ", "$\\iota \\circ \\delta \\circ g' = \\iota \\circ \\delta \\circ \\pi \\circ c =", "\\iota' \\circ \\beta \\circ \\pi' \\circ c = \\iota' \\circ \\beta \\circ h = 0$. ", "As $\\iota$ is a monomorphism this implies $\\delta \\circ g' = 0$.", "\\medskip\\noindent", "Next, let $d : r \\to \\Ker(\\gamma)$ with $\\delta \\circ d = 0$. Applying", "Lemma \\ref{lemma-check-exactness-fibre-product} to the exact sequence ", "$p \\overset{\\pi}\\to \\Ker(\\gamma) \\to 0$ and $d$ yields an object $s$, ", "an epimorphism $m : s \\to r$ and a morphism $n : s \\to p$ with ", "$\\pi \\circ n = d \\circ m$. As $p \\circ a \\circ n = \\delta \\circ d \\circ m = 0$, ", "applying Lemma \\ref{lemma-check-exactness-fibre-product} to the exact sequence ", "$x \\overset{\\alpha}\\to u \\overset{p}\\to \\Coker(\\alpha)$ and ", "$a \\circ n$ yields an object $t$, an epimorphism $\\varepsilon : t \\to s$ and ", "a morphism $\\zeta : t \\to x$ with", "$a \\circ n \\circ \\varepsilon = \\alpha \\circ \\zeta$. ", "It holds ", "$\\beta \\circ \\pi' \\circ n \\circ \\varepsilon =", "k \\circ \\alpha \\circ \\zeta = \\beta \\circ f \\circ \\zeta$. ", "Let $\\eta = \\pi' \\circ n \\circ \\varepsilon - f \\circ \\zeta : t \\to y$. Then, ", "$\\beta \\circ \\eta = 0$. It follows that there is a ", "$\\vartheta : t \\to \\Ker(\\beta)$ with $\\eta = h \\circ \\vartheta$. It holds ", "$i \\circ g' \\circ \\vartheta = g \\circ h \\circ \\vartheta =", "g \\circ \\pi' \\circ n \\circ \\varepsilon - g \\circ f \\circ \\zeta =", "i \\circ \\pi \\circ n \\circ \\varepsilon = i \\circ d \\circ m \\circ \\varepsilon$. ", "As $i$ is a monomorphism we get", "$g' \\circ \\vartheta = d \\circ m \\circ \\varepsilon$. ", "Thus, as $m \\circ \\varepsilon$ is an epimorphism,", "Lemma \\ref{lemma-check-exactness-fibre-product} implies that ", "$\\Ker(\\beta) \\overset{g'}\\to \\Ker(\\gamma) \\overset{\\delta}\\to \\Coker(\\alpha)$ ", "is exact. Then, the claim follows by Lemma \\ref{lemma-exact-kernel-sequence}", "and duality." ], "refs": [ "homology-lemma-cartesian-cocartesian", "homology-lemma-cartesian-cocartesian", "homology-lemma-cartesian-kernel", "homology-lemma-check-exactness-fibre-product", "homology-lemma-check-exactness-fibre-product", "homology-lemma-check-exactness-fibre-product", "homology-lemma-exact-kernel-sequence" ], "ref_ids": [ 12023, 12023, 12022, 12025, 12025, 12025, 12026 ] } ], "ref_ids": [] }, { "id": 12028, "type": "theorem", "label": "homology-lemma-snake-natural", "categories": [ "homology" ], "title": "homology-lemma-snake-natural", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let ", "$$", "\\xymatrix{", "& & & x\\ar[ld]\\ar[rr]\\ar[dd]^(.4)\\alpha", "& & y\\ar[ld]\\ar[rr]\\ar[dd]^(.4)\\beta", "& & z\\ar[ld]\\ar[rr]\\ar[dd]^(.4)\\gamma", "& & 0\\\\", "& & x'\\ar[rr]\\ar[dd]^(.4){\\alpha'}", "& & y'\\ar[rr]\\ar[dd]^(.4){\\beta'}", "& & z'\\ar[rr]\\ar[dd]^(.4){\\gamma'}", "& & 0", "& \\\\", "& 0\\ar[rr]", "& & u\\ar[ld]\\ar[rr]", "& & v\\ar[ld]\\ar[rr]", "& & w\\ar[ld]", "& & \\\\", "0\\ar[rr]", "& & u'\\ar[rr]", "& & v'\\ar[rr]", "& & w'", "& & &", "}", "$$", "be a commutative diagram with exact rows. Then, the induced diagram", "$$", "\\xymatrix@C=15pt{", "\\Ker(\\alpha) \\ar[r] \\ar[d] &", "\\Ker(\\beta) \\ar[r] \\ar[d] &", "\\Ker(\\gamma) \\ar[r]^(.45){\\delta} \\ar[d] &", "\\Coker(\\alpha) \\ar[r] \\ar[d] &", "\\Coker(\\beta) \\ar[r] \\ar[d] &", "\\Coker(\\gamma) \\ar[d] \\\\", "\\Ker(\\alpha') \\ar[r] &", "\\Ker(\\beta') \\ar[r] &", "\\Ker(\\gamma') \\ar[r]^(.45){\\delta'} &", "\\Coker(\\alpha') \\ar[r] &", "\\Coker(\\beta') \\ar[r] &", "\\Coker(\\gamma')", "}", "$$", "commutes." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12029, "type": "theorem", "label": "homology-lemma-four-lemma", "categories": [ "homology" ], "title": "homology-lemma-four-lemma", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let", "$$", "\\xymatrix{", "w \\ar[r] \\ar[d]^\\alpha & x \\ar[r] \\ar[d]^\\beta & y \\ar[r] \\ar[d]^\\gamma &", "z \\ar[d]^\\delta \\\\", "w' \\ar[r] & x' \\ar[r] & y' \\ar[r] & z'", "}", "$$", "be a commutative diagram with exact rows.", "\\begin{enumerate}", "\\item If $\\alpha, \\gamma$ are surjective and $\\delta$ is injective, then", "$\\beta$ is surjective.", "\\item If $\\beta, \\delta$ are injective and $\\alpha$ is surjective, then", "$\\gamma$ is injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $\\alpha, \\gamma$ are surjective and $\\delta$ is injective.", "We may replace $w'$ by $\\Im(w' \\to x')$, i.e., we may assume", "that $w' \\to x'$ is injective.", "We may replace $z$ by $\\Im(y \\to z)$, i.e., we may assume that", "$y \\to z$ is surjective. Then we may apply", "Lemma \\ref{lemma-snake}", "to", "$$", "\\xymatrix{", "& \\Ker(y \\to z) \\ar[r] \\ar[d] & y \\ar[r] \\ar[d] & z \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] & \\Ker(y' \\to z') \\ar[r] & y' \\ar[r] & z'", "}", "$$", "to conclude that $\\Ker(y \\to z) \\to \\Ker(y' \\to z')$ is", "surjective. Finally, we apply", "Lemma \\ref{lemma-snake}", "to", "$$", "\\xymatrix{", "& w \\ar[r] \\ar[d] & x \\ar[r] \\ar[d] & \\Ker(y \\to z) \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] & w' \\ar[r] & x' \\ar[r] & \\Ker(y' \\to z')", "}", "$$", "to conclude that $x \\to x'$ is surjective. This proves (1). The proof", "of (2) is dual to this." ], "refs": [ "homology-lemma-snake", "homology-lemma-snake" ], "ref_ids": [ 12027, 12027 ] } ], "ref_ids": [] }, { "id": 12030, "type": "theorem", "label": "homology-lemma-five-lemma", "categories": [ "homology" ], "title": "homology-lemma-five-lemma", "contents": [ "\\begin{reference}", "\\cite[Lemma 4.5 page 16]{Eilenberg-Steenrod}", "\\end{reference}", "Let $\\mathcal{A}$ be an abelian category. Let", "$$", "\\xymatrix{", "v \\ar[r] \\ar[d]^\\alpha &", "w \\ar[r] \\ar[d]^\\beta &", "x \\ar[r] \\ar[d]^\\gamma &", "y \\ar[r] \\ar[d]^\\delta &", "z \\ar[d]^\\epsilon \\\\", "v' \\ar[r] & w' \\ar[r] & x' \\ar[r] & y' \\ar[r] & z'", "}", "$$", "be a commutative diagram with exact rows. If $\\beta, \\delta$", "are isomorphisms, $\\epsilon$ is injective, and $\\alpha$ is surjective", "then $\\gamma$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Lemma \\ref{lemma-four-lemma}." ], "refs": [ "homology-lemma-four-lemma" ], "ref_ids": [ 12029 ] } ], "ref_ids": [] }, { "id": 12031, "type": "theorem", "label": "homology-lemma-baer-sum", "categories": [ "homology" ], "title": "homology-lemma-baer-sum", "contents": [ "The construction $(E_1, E_2) \\mapsto E_1 + E_2$", "above defines a commutative group", "law on $\\Ext_\\mathcal{A}(B, A)$ which is", "functorial in both variables." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12032, "type": "theorem", "label": "homology-lemma-six-term-sequence-ext", "categories": [ "homology" ], "title": "homology-lemma-six-term-sequence-ext", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a short exact sequence", "in $\\mathcal{A}$.", "\\begin{enumerate}", "\\item There is a canonical six term exact sequence of abelian groups", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Hom_\\mathcal{A}(M_3, N) \\ar[r] &", "\\Hom_\\mathcal{A}(M_2, N) \\ar[r] &", "\\Hom_\\mathcal{A}(M_1, N) \\ar[lld] \\\\", "& \\Ext_\\mathcal{A}(M_3, N) \\ar[r] &", "\\Ext_\\mathcal{A}(M_2, N) \\ar[r] &", "\\Ext_\\mathcal{A}(M_1, N)", "}", "$$", "for all objects $N$ of $\\mathcal{A}$, and", "\\item there is a canonical six term exact sequence of abelian groups", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Hom_\\mathcal{A}(N, M_1) \\ar[r] &", "\\Hom_\\mathcal{A}(N, M_2) \\ar[r] &", "\\Hom_\\mathcal{A}(N, M_3) \\ar[lld] \\\\", "& \\Ext_\\mathcal{A}(N, M_1) \\ar[r] &", "\\Ext_\\mathcal{A}(N, M_2) \\ar[r] &", "\\Ext_\\mathcal{A}(N, M_3)", "}", "$$", "for all objects $N$ of $\\mathcal{A}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: The boundary maps are defined using either the pushout", "or pullback of the given short exact sequence." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12033, "type": "theorem", "label": "homology-lemma-additive-functor", "categories": [ "homology" ], "title": "homology-lemma-additive-functor", "contents": [ "Let $\\mathcal{A}$ and $\\mathcal{B}$ be additive categories.", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.", "The following are equivalent", "\\begin{enumerate}", "\\item $F$ is additive,", "\\item $F(A) \\oplus F(B) \\to F(A \\oplus B)$ is an isomorphism for", "all $A, B \\in \\mathcal{A}$, and", "\\item $F(A \\oplus B) \\to F(A) \\oplus F(B)$ is an isomorphism for", "all $A, B \\in \\mathcal{A}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Additive functors commute with direct sums by", "Lemma \\ref{lemma-additive-additive} hence (1)", "implies (2) and (3). On the other hand (2) and (3)", "are equivalent because the composition", "$F(A) \\oplus F(B) \\to F(A \\oplus B) \\to F(A) \\oplus F(B)$", "is the identity map. Assume (2) and (3) hold.", "Let $f, g : A \\to B$ be maps. Then $f + g$ is equal to", "the composition", "$$", "A \\to A \\oplus A \\xrightarrow{\\text{diag}(f, g)} B \\oplus B \\to B", "$$", "Apply the functor $F$ and consider the following diagram", "$$", "\\xymatrix{", "F(A) \\ar[r] \\ar[rd] &", "F(A \\oplus A) \\ar[rr]_{F(\\text{diag}(f, g))} & &", "F(B \\oplus B) \\ar[r] \\ar[d] &", "F(B) \\\\", "&", "F(A) \\oplus F(A) \\ar[u] \\ar[rr]^{\\text{diag}(F(f), F(g))} & &", "F(B) \\oplus F(B) \\ar[ru]", "}", "$$", "We claim this is commutative. For the middle square we can verify it", "separately for each of the for induced maps $F(A) \\to F(B)$", "where it follows from the fact that $F$ is a functor (in other words", "this square commutes even if $F$ does not satisfy any properties", "beyond being a functor). For the triangle on the left, we use that", "$F(A \\oplus A) \\to F(A) \\oplus F(A)$ is an isomorphism", "to see that it suffice to check after composition with", "this map and this check is trivial. Dually for the other triangle.", "Thus going around the bottom is equal to $F(f + g)$ and we conclude." ], "refs": [ "homology-lemma-additive-additive" ], "ref_ids": [ 12010 ] } ], "ref_ids": [] }, { "id": 12034, "type": "theorem", "label": "homology-lemma-exact-functor", "categories": [ "homology" ], "title": "homology-lemma-exact-functor", "contents": [ "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.", "\\begin{enumerate}", "\\item If $F$ is either left or right exact, then it is additive.", "\\item $F$ is left exact if and only if", "for every short exact sequence", "$0 \\to A \\to B \\to C \\to 0$", "the sequence $0 \\to F(A) \\to F(B) \\to F(C)$", "is exact.", "\\item $F$ is right exact if and only if for every short exact sequence", "$0 \\to A \\to B \\to C \\to 0$", "the sequence $F(A) \\to F(B) \\to F(C) \\to 0$", "is exact.", "\\item $F$ is exact if and only if for every short exact sequence", "$0 \\to A \\to B \\to C \\to 0$", "the sequence $0 \\to F(A) \\to F(B) \\to F(C) \\to 0$", "is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $F$ is left exact, i.e., $F$ commutes with finite limits, then", "$F$ sends products to products, hence $F$ preserved direct sums,", "hence $F$ is additive by Lemma \\ref{lemma-additive-functor}.", "On the other hand, suppose that for every short exact sequence", "$0 \\to A \\to B \\to C \\to 0$ the sequence $0 \\to F(A) \\to F(B) \\to F(C)$", "is exact. Let $A, B$ be two objects. Then we have a short", "exact sequence", "$$", "0 \\to A \\to A \\oplus B \\to B \\to 0", "$$", "see for example Lemma \\ref{lemma-additive-cat-biproduct-kernel}.", "By assumption, the lower row in the commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "F(A) \\ar[d] \\ar[r] &", "F(A) \\oplus F(B) \\ar[r] \\ar[d] &", "F(B) \\ar[d] \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "F(A) \\ar[r] &", "F(A \\oplus B) \\ar[r] &", "F(B)", "}", "$$", "is exact. Hence by the snake lemma (Lemma \\ref{lemma-snake})", "we conclude that $F(A) \\oplus F(B) \\to F(A \\oplus B)$ is an", "isomorphism. Hence $F$ is additive in this case as well.", "Thus for the rest of the proof we may assume $F$ is additive.", "\\medskip\\noindent", "Denote $f : B \\to C$ a map from $B$ to $C$.", "Exactness of $0 \\to A \\to B \\to C$ just means that", "$A = \\Ker(f)$. Clearly the kernel of $f$ is", "the equalizer of the two maps $f$ and $0$ from $B$ to $C$.", "Hence if $F$ commutes with limits, then $F(\\Ker(f))", "= \\Ker(F(f))$ which exactly means that", "$0 \\to F(A) \\to F(B) \\to F(C)$ is exact.", "\\medskip\\noindent", "Conversely, suppose that $F$ is additive and", "transforms any short exact sequence $0 \\to A \\to B \\to C \\to 0$ into", "an exact sequence $0 \\to F(A) \\to F(B) \\to F(C)$.", "Because it is additive it commutes with direct sums", "and hence finite products in $\\mathcal{A}$. To show", "it commutes with finite limits it therefore", "suffices to show that it commutes with", "equalizers. But equalizers in an abelian category", "are the same as the kernel of the difference map,", "hence it suffices to show that $F$ commutes with", "taking kernels. Let $f : A \\to B$ be a morphism.", "Factor $f$ as $A \\to I \\to B$ with $f' : A \\to I$ surjective", "and $i : I \\to B$ injective. (This is possible by the", "definition of an abelian category.) Then it is", "clear that $\\Ker(f) = \\Ker(f')$. Also", "$0 \\to \\Ker(f') \\to A \\to I \\to 0$", "and", "$0 \\to I \\to B \\to B/I \\to 0$", "are short exact. By the condition imposed on $F$", "we see that", "$0 \\to F(\\Ker(f')) \\to F(A) \\to F(I)$", "and", "$0 \\to F(I) \\to F(B) \\to F(B/I)$", "are exact. Hence it is also the case that", "$F(\\Ker(f'))$ is the kernel of the map", "$F(A) \\to F(B)$, and we win.", "\\medskip\\noindent", "The proof of (3) is similar to the proof of (2).", "Statement (4) is a combination of (2) and (3)." ], "refs": [ "homology-lemma-additive-functor", "homology-lemma-additive-cat-biproduct-kernel", "homology-lemma-snake" ], "ref_ids": [ 12033, 12011, 12027 ] } ], "ref_ids": [] }, { "id": 12035, "type": "theorem", "label": "homology-lemma-exact-functor-ext", "categories": [ "homology" ], "title": "homology-lemma-exact-functor-ext", "contents": [ "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor.", "For every pair of objects $A, B$ of $\\mathcal{A}$ the", "functor $F$ induces an abelian group homomorphism", "$$", "\\Ext_\\mathcal{A}(B, A)", "\\longrightarrow", "\\Ext_\\mathcal{B}(F(B), F(A))", "$$", "which maps the extension $E$ to $F(E)$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12036, "type": "theorem", "label": "homology-lemma-adjoint-get-abelian", "categories": [ "homology" ], "title": "homology-lemma-adjoint-get-abelian", "contents": [ "Let $a : \\mathcal{A} \\to \\mathcal{B}$ and $b : \\mathcal{B} \\to \\mathcal{A}$", "be functors. Assume that", "\\begin{enumerate}", "\\item $\\mathcal{A}$, $\\mathcal{B}$ are additive categories,", "$a$, $b$ are additive functors, and $a$ is right adjoint to $b$,", "\\item $\\mathcal{B}$ is abelian and $b$ is left exact, and", "\\item $ba \\cong \\text{id}_\\mathcal{A}$.", "\\end{enumerate}", "Then $\\mathcal{A}$ is abelian." ], "refs": [], "proofs": [ { "contents": [ "As $\\mathcal{B}$ is abelian we see that all finite limits and colimits", "exist in $\\mathcal{B}$ by Lemma \\ref{lemma-colimit-abelian-category}.", "Since $b$ is a left adjoint we see that $b$ is also right exact", "and hence exact, see", "Categories, Lemma \\ref{categories-lemma-exact-adjoint}.", "Let $\\varphi : B_1 \\to B_2$ be a morphism of $\\mathcal{B}$.", "In particular, if $K = \\Ker(B_1 \\to B_2)$, then $K$ is", "the equalizer of $0$ and $\\varphi$ and hence", "$bK$ is the equalizer of $0$ and $b\\varphi$, hence", "$bK$ is the kernel of $b\\varphi$. Similarly, if", "$Q = \\Coker(B_1 \\to B_2)$, then $Q$ is", "the coequalizer of $0$ and $\\varphi$ and hence", "$bQ$ is the coequalizer of $0$ and $b\\varphi$, hence", "$bQ$ is the cokernel of $b\\varphi$. Thus we see that every morphism", "of the form $b\\varphi$ in $\\mathcal{A}$ has a kernel and a cokernel.", "However, since $ba \\cong \\text{id}$ we see that every morphism of", "$\\mathcal{A}$ is of this form, and we conclude that kernels and", "cokernels exist in $\\mathcal{A}$. In fact, the argument shows that", "if $\\psi : A_1 \\to A_2$ is a morphism then", "$$", "\\Ker(\\psi) = b\\Ker(a\\psi),", "\\quad\\text{and}\\quad", "\\Coker(\\psi) = b\\Coker(a\\psi).", "$$", "Now we still have to show that $\\Coim(\\psi)= \\Im(\\psi)$.", "We do this as follows.", "First note that since $\\mathcal{A}$ has kernels and cokernels it", "has all finite limits and colimits (see proof of", "Lemma \\ref{lemma-colimit-abelian-category}).", "Hence we see by Categories, Lemma \\ref{categories-lemma-exact-adjoint}", "that $a$ is left exact and", "hence transforms kernels (=equalizers) into kernels.", "\\begin{align*}", "\\Coim(\\psi)", "& =", "\\Coker(\\Ker(\\psi) \\to A_1)", "& \\text{by definition} \\\\", "& =", "b\\Coker(a(\\Ker(\\psi) \\to A_1))", "& \\text{by formula above} \\\\", "& =", "b\\Coker(\\Ker(a\\psi) \\to aA_1))", "& a\\text{ preserves kernels} \\\\", "& =", "b\\Coim(a\\psi)", "& \\text{by definition} \\\\", "& =", "b\\Im(a\\psi)", "& \\mathcal{B}\\text{ is abelian} \\\\", "& =", "b\\Ker(aA_2 \\to \\Coker(a\\psi))", "& \\text{by definition} \\\\", "& =", "\\Ker(baA_2 \\to b\\Coker(a\\psi))", "& b\\text{ preserves kernels} \\\\", "& =", "\\Ker(A_2 \\to b\\Coker(a\\psi))", "& ba = \\text{id}_\\mathcal{A} \\\\", "& =", "\\Ker(A_2 \\to \\Coker(\\psi))", "& \\text{by formula above} \\\\", "& =", "\\Im(\\psi)", "& \\text{by definition}", "\\end{align*}", "Thus the lemma holds." ], "refs": [ "homology-lemma-colimit-abelian-category", "categories-lemma-exact-adjoint", "homology-lemma-colimit-abelian-category", "categories-lemma-exact-adjoint" ], "ref_ids": [ 12018, 12250, 12018, 12250 ] } ], "ref_ids": [] }, { "id": 12037, "type": "theorem", "label": "homology-lemma-localization-preadditive", "categories": [ "homology" ], "title": "homology-lemma-localization-preadditive", "contents": [ "Let $\\mathcal{C}$ be a preadditive category.", "Let $S$ be a left or right multiplicative system.", "There exists a canonical preadditive structure on", "$S^{-1}\\mathcal{C}$ such that the localization functor", "$Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ is additive." ], "refs": [], "proofs": [ { "contents": [ "We will prove this in the case $S$ is a left multiplicative system.", "The case where $S$ is a right multiplicative system is dual.", "Suppose that $X, Y$ are objects of $\\mathcal{C}$ and that", "$\\alpha, \\beta : X \\to Y$ are morphisms in $S^{-1}\\mathcal{C}$. According to", "Categories, Lemma \\ref{categories-lemma-morphisms-left-localization}", "we may represent these by pairs $s^{-1}f, s^{-1}g$ with common denominator", "$s$. In this case we define $\\alpha + \\beta$ to be the equivalence class of", "$s^{-1}(f + g)$. In the rest of the proof we show that this is well defined", "and that composition is bilinear. Once this is done it is clear that", "$Q$ is an additive functor.", "\\medskip\\noindent", "Let us show construction above is well defined.", "An abstract way of saying this is that filtered colimits of", "abelian groups agree with filtered colimits of sets and to use", "Categories,", "Equation (\\ref{categories-equation-left-localization-morphisms-colimit}).", "We can work this out in a bit more detail as follows.", "Say $s : Y \\to Y_1$ and $f, g : X \\to Y_1$. Suppose we have a second", "representation of $\\alpha, \\beta$ as $(s')^{-1}f', (s')^{-1}g'$ with", "$s' : Y \\to Y_2$ and $f', g' : X \\to Y_2$. By", "Categories, Remark \\ref{categories-remark-left-localization-morphisms-colimit}", "we can find a morphism $s_3 : Y \\to Y_3$ and morphisms", "$a_1 : Y_1 \\to Y_3$, $a_2 : Y_2 \\to Y_3$ such that", "$a_1 \\circ s = s_3 = a_2 \\circ s'$ and also", "$a_1 \\circ f = a_2 \\circ f'$ and $a_1 \\circ g = a_2 \\circ g'$.", "Hence we see that $s^{-1}(f + g)$ is equivalent to", "\\begin{align*}", "s_3^{-1}(a_1 \\circ (f + g)) & =", "s_3^{-1}(a_1 \\circ f + a_1 \\circ g) \\\\", "& = s_3^{-1}(a_2 \\circ f' + a_2 \\circ g') \\\\", "& = s_3^{-1}(a_2 \\circ (f' + g'))", "\\end{align*}", "which is equivalent to $(s')^{-1}(f' + g')$.", "\\medskip\\noindent", "Fix $s : Y \\to Y'$ and $f, g : X \\to Y'$ with", "$\\alpha = s^{-1}f$ and $\\beta = s^{-1}g$ as morphisms $X \\to Y$", "in $S^{-1}\\mathcal{C}$.", "To show that composition is bilinear first consider the case of a", "morphism $\\gamma : Y \\to Z$ in $S^{-1}\\mathcal{C}$. Say $\\gamma = t^{-1}h$", "for some $h : Y \\to Z'$ and $t : Z \\to Z'$ in $S$. Using LMS2 we", "choose morphisms $a : Y' \\to Z''$ and $t' : Z' \\to Z''$ in $S$ such", "that $a \\circ s = t' \\circ h$. Picture", "$$", "\\xymatrix{", "& & Z \\ar[d]^t \\\\", "& Y \\ar[r]^h \\ar[d]^s & Z' \\ar[d]^{t'} \\\\", "X \\ar[r]^{f, g} & Y' \\ar[r]^a & Z''", "}", "$$", "Then", "$\\gamma \\circ \\alpha = (t' \\circ t)^{-1}(a \\circ f)$ and", "$\\gamma \\circ \\beta = (t' \\circ t)^{-1}(a \\circ g)$.", "Hence we see that $\\gamma \\circ (\\alpha + \\beta)$ is represented", "by $(t' \\circ t)^{-1}(a \\circ (f + g)) =", "(t' \\circ t)^{-1}(a \\circ f + a \\circ g)$ which represents", "$\\gamma \\circ \\alpha + \\gamma \\circ \\beta$.", "\\medskip\\noindent", "Finally, assume that $\\delta : W \\to X$ is another morphism of", "$S^{-1}\\mathcal{C}$. Say $\\delta = r^{-1}i$ for some", "$i : W \\to X'$ and $r : X \\to X'$ in $S$. We claim that we can find", "a morphism $s : Y' \\to Y''$ in $S$ and morphisms $a'', b'' : X' \\to Y''$", "such that the following diagram commutes", "$$", "\\xymatrix{", "& & & Y \\ar[d]^s \\\\", "& X \\ar[rr]^{f, g, f + g} \\ar[d]^s & & Y' \\ar[d]^{s'} \\\\", "W \\ar[r]^i & X' \\ar[rr]^{a'', b'', a'' + b''} & & Y''", "}", "$$", "Namely, using LMS2 we can first choose", "$s_1 : Y' \\to Y_1$, $s_2 : Y' \\to Y_2$ in $S$ and", "$a : X' \\to Y_1$, $b : X' \\to Y_2$ such that", "$a \\circ s = s_1 \\circ f$ and $b \\circ s = s_2 \\circ f$.", "Then using that the category $Y'/S$ is filtered (see", "Categories, Remark \\ref{categories-remark-left-localization-morphisms-colimit}),", "we can", "find a $s' : Y' \\to Y''$ and morphisms $a' : Y_1 \\to Y''$, $b' : Y_2 \\to Y''$", "such that $s' = a' \\circ s_1$ and $s' = b' \\circ s_2$. Setting", "$a'' = a' \\circ a$ and $b'' = b' \\circ b$ works.", "At this point we see that the compositions", "$\\alpha \\circ \\delta$ and $\\beta \\circ \\delta$ are represented by", "$(s' \\circ s)^{-1}a''$ and $(s' \\circ s)^{-1}b''$.", "Hence $\\alpha \\circ \\delta + \\beta \\circ \\delta$ is represented", "by $(s' \\circ s)^{-1}(a'' + b'')$ which by the diagram again", "is a representative of $(\\alpha + \\beta) \\circ \\delta$." ], "refs": [ "categories-lemma-morphisms-left-localization", "categories-remark-left-localization-morphisms-colimit", "categories-remark-left-localization-morphisms-colimit" ], "ref_ids": [ 12256, 12424, 12424 ] } ], "ref_ids": [] }, { "id": 12038, "type": "theorem", "label": "homology-lemma-localization-additive", "categories": [ "homology" ], "title": "homology-lemma-localization-additive", "contents": [ "Let $\\mathcal{C}$ be an additive category.", "Let $S$ be a left or right multiplicative system.", "Then $S^{-1}\\mathcal{C}$ is an additive category and the localization functor", "$Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ is additive." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-localization-preadditive}", "we see that $S^{-1}\\mathcal{C}$ is preadditive and that $Q$ is additive.", "Recall that the functor $Q$ commutes with finite colimits", "(resp.\\ finite limits), see", "Categories, Lemmas \\ref{categories-lemma-left-localization-limits} and", "\\ref{categories-lemma-right-localization-limits}.", "We conclude that $S^{-1}\\mathcal{C}$ has a zero object and", "direct sums, see", "Lemmas \\ref{lemma-preadditive-zero} and", "\\ref{lemma-preadditive-direct-sum}." ], "refs": [ "homology-lemma-localization-preadditive", "categories-lemma-left-localization-limits", "categories-lemma-right-localization-limits", "homology-lemma-preadditive-zero", "homology-lemma-preadditive-direct-sum" ], "ref_ids": [ 12037, 12259, 12265, 12008, 12009 ] } ], "ref_ids": [] }, { "id": 12039, "type": "theorem", "label": "homology-lemma-kernel-localization", "categories": [ "homology" ], "title": "homology-lemma-kernel-localization", "contents": [ "Let $\\mathcal{C}$ be an additive category. Let $S$ be a multiplicative", "system. Let $X$ be an object", "of $\\mathcal{C}$. The following are equivalent", "\\begin{enumerate}", "\\item $Q(X) = 0$ in $S^{-1}\\mathcal{C}$,", "\\item there exists $Y \\in \\Ob(\\mathcal{C})$ such that", "$0 : X \\to Y$ is an element of $S$, and", "\\item there exists $Z \\in \\Ob(\\mathcal{C})$ such that", "$0 : Z \\to X$ is an element of $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (2) holds we see that $0 = Q(0) : Q(X) \\to Q(Y)$ is an isomorphism.", "In the additive category $S^{-1}\\mathcal{C}$ this implies that $Q(X) = 0$.", "Hence (2) $\\Rightarrow$ (1). Similarly, (3) $\\Rightarrow$ (1).", "Suppose that $Q(X) = 0$. This implies that the morphism", "$f : 0 \\to X$ is transformed into an isomorphism in $S^{-1}\\mathcal{C}$.", "Hence by", "Categories, Lemma \\ref{categories-lemma-what-gets-inverted}", "there exists a morphism $g : Z \\to 0$ such that $fg \\in S$. This proves", "(1) $\\Rightarrow$ (3). Similarly, (1) $\\Rightarrow$ (2)." ], "refs": [ "categories-lemma-what-gets-inverted" ], "ref_ids": [ 12268 ] } ], "ref_ids": [] }, { "id": 12040, "type": "theorem", "label": "homology-lemma-localization-abelian", "categories": [ "homology" ], "title": "homology-lemma-localization-abelian", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item If $S$ is a left multiplicative system, then", "the category $S^{-1}\\mathcal{A}$ has cokernels and the functor", "$Q : \\mathcal{A} \\to S^{-1}\\mathcal{A}$ commutes with them.", "\\item If $S$ is a right multiplicative system, then", "the category $S^{-1}\\mathcal{A}$ has kernels and the functor", "$Q : \\mathcal{A} \\to S^{-1}\\mathcal{A}$ commutes with them.", "\\item If $S$ is a multiplicative system, then the category", "$S^{-1}\\mathcal{A}$ is abelian and the functor", "$Q : \\mathcal{A} \\to S^{-1}\\mathcal{A}$ is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $S$ is a left multiplicative system. Let $a : X \\to Y$ be a morphism", "of $S^{-1}\\mathcal{A}$. Then $a = s^{-1}f$ for some $s : Y \\to Y'$", "in $S$ and $f : X \\to Y'$. Since $Q(s)$ is an isomorphism we see that", "the existence of $\\Coker(a : X \\to Y)$ is equivalent to the existence", "of $\\Coker(Q(f) : X \\to Y')$. Since $\\Coker(Q(f))$ is the", "coequalizer of $0$ and $Q(f)$ we see that $\\Coker(Q(f))$ is", "represented by $Q(\\Coker(f))$ by", "Categories, Lemma \\ref{categories-lemma-left-localization-limits}.", "This proves (1).", "\\medskip\\noindent", "Part (2) is dual to part (1).", "\\medskip\\noindent", "If $S$ is a multiplicative system, then $S$ is both a left and a right", "multiplicative system. Thus we see that $S^{-1}\\mathcal{A}$ has", "kernels and cokernels and $Q$ commutes with kernels and cokernels.", "To finish the proof of (3) we have to show that $\\Coim = \\Im$ in", "$S^{-1}\\mathcal{A}$. Again using that any arrow in $S^{-1}\\mathcal{A}$", "is isomorphic to an arrow $Q(f)$ we see that the result follows", "from the result for $\\mathcal{A}$." ], "refs": [ "categories-lemma-left-localization-limits" ], "ref_ids": [ 12259 ] } ], "ref_ids": [] }, { "id": 12041, "type": "theorem", "label": "homology-lemma-ses-artinian", "categories": [ "homology" ], "title": "homology-lemma-ses-artinian", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $0 \\to A_1 \\to A_2 \\to A_3 \\to 0$", "be a short exact sequence of $\\mathcal{A}$. Then $A_2$ is Artinian", "if and only if $A_1$ and $A_3$ are Artinian." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12042, "type": "theorem", "label": "homology-lemma-ses-noetherian", "categories": [ "homology" ], "title": "homology-lemma-ses-noetherian", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $0 \\to A_1 \\to A_2 \\to A_3 \\to 0$", "be a short exact sequence of $\\mathcal{A}$. Then $A_2$ is Noetherian", "if and only if $A_1$ and $A_3$ are Noetherian." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12043, "type": "theorem", "label": "homology-lemma-finite-length", "categories": [ "homology" ], "title": "homology-lemma-finite-length", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $A$ be an object", "of $\\mathcal{A}$. The following are equivalent", "\\begin{enumerate}", "\\item $A$ is Artinian and Noetherian, and", "\\item there exists a filtration", "$0 \\subset A_1 \\subset A_2 \\subset \\ldots \\subset A_n = A$", "by subobjects such that $A_i/A_{i - 1}$ is simple for $i = 1, \\ldots, n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). If $A$ is zero, then (2) holds. If $A$ is not zero, then", "there exists a smallest nonzero object $A_1 \\subset A$ by the Artinian", "property. Of course $A_1$ is simple. If $A_1 = A$, then we are done.", "If not, then we can find $A_1 \\subset A_2 \\subset A$ minimal", "with $A_2 \\not = A_1$. Then $A_2/A_1$ is simple. Continuing in this way, we", "can find a sequence $0 \\subset A_1 \\subset A_2 \\subset \\ldots $", "of subobjects of $A$ such that $A_i/A_{i - 1}$ is simple. Since $A$", "is Noetherian, we conclude that the process stops. Hence (2) follows.", "\\medskip\\noindent", "Assume (2). We will prove (1) by induction on $n$. If $n = 1$, then", "$A$ is simple and clearly Noetherian and Artinian. If the result holds", "for $n - 1$, then we use the short exact sequence", "$0 \\to A_{n - 1} \\to A_n \\to A_n/A_{n - 1} \\to 0$", "and Lemmas \\ref{lemma-ses-artinian} and \\ref{lemma-ses-noetherian}", "to conclude for $n$." ], "refs": [ "homology-lemma-ses-artinian", "homology-lemma-ses-noetherian" ], "ref_ids": [ 12041, 12042 ] } ], "ref_ids": [] }, { "id": 12044, "type": "theorem", "label": "homology-lemma-jordan-holder", "categories": [ "homology" ], "title": "homology-lemma-jordan-holder", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $A$ be an object", "of $\\mathcal{A}$ satisfying the equivalent conditions of", "Lemma \\ref{lemma-finite-length}. Given two filtrations", "$$", "0 \\subset A_1 \\subset A_2 \\subset \\ldots \\subset A_n = A", "\\quad\\text{and}\\quad", "0 \\subset B_1 \\subset B_2 \\subset \\ldots \\subset B_m = A", "$$", "with $S_i = A_i/A_{i - 1}$ and $T_j = B_j/B_{j - 1}$ simple objects we have", "$n = m$ and there exists a permutation $\\sigma$ of $\\{1, \\ldots, n\\}$", "such that $S_i \\cong T_{\\sigma(i)}$ for all $i \\in \\{1, \\ldots, n\\}$." ], "refs": [ "homology-lemma-finite-length" ], "proofs": [ { "contents": [ "Let $j$ be the smallest index such that $A_1 \\subset B_j$.", "Then the map $S_1 = A_1 \\to B_j/B_{j - 1} = T_j$ is an isomorphism.", "Moreover, the object $A/A_1 = A_n/A_1 = B_m/A_1$", "has the two filtrations", "$$", "0 \\subset A_2/A_1 \\subset A_3/A_1 \\subset \\ldots \\subset A_n/A_1", "$$", "and", "$$", "0 \\subset (B_1 + A_1)/A_1 \\subset \\ldots \\subset", "(B_{j - 1} + A_1)/A_1 = B_j/A_1 \\subset B_{j + 1}/A_1", "\\subset \\ldots \\subset B_m/A_1", "$$", "We conclude by induction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12043 ] }, { "id": 12045, "type": "theorem", "label": "homology-lemma-characterize-serre-subcategory", "categories": [ "homology" ], "title": "homology-lemma-characterize-serre-subcategory", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $\\mathcal{C}$ be a subcategory of $\\mathcal{A}$.", "Then $\\mathcal{C}$ is a Serre subcategory if and only if", "the following conditions are satisfied:", "\\begin{enumerate}", "\\item $0 \\in \\Ob(\\mathcal{C})$,", "\\item $\\mathcal{C}$ is a strictly full subcategory of $\\mathcal{A}$,", "\\item any subobject or quotient of an object of $\\mathcal{C}$ is an object", "of $\\mathcal{C}$,", "\\item if $A \\in \\Ob(\\mathcal{A})$ is an extension of objects of $\\mathcal{C}$", "then also $A \\in \\Ob(\\mathcal{C})$.", "\\end{enumerate}", "Moreover, a Serre subcategory is an abelian category and", "the inclusion functor is exact." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12046, "type": "theorem", "label": "homology-lemma-characterize-weak-serre-subcategory", "categories": [ "homology" ], "title": "homology-lemma-characterize-weak-serre-subcategory", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $\\mathcal{C}$ be a subcategory of $\\mathcal{A}$.", "Then $\\mathcal{C}$ is a weak Serre subcategory if and only if", "the following conditions are satisfied:", "\\begin{enumerate}", "\\item $0 \\in \\Ob(\\mathcal{C})$,", "\\item $\\mathcal{C}$ is a strictly full subcategory of $\\mathcal{A}$,", "\\item kernels and cokernels in $\\mathcal{A}$ of morphisms", "between objects of $\\mathcal{C}$ are in $\\mathcal{C}$,", "\\item if $A \\in \\Ob(\\mathcal{A})$ is an extension of objects of $\\mathcal{C}$", "then also $A \\in \\Ob(\\mathcal{C})$.", "\\end{enumerate}", "Moreover, a weak Serre subcategory is an abelian category and", "the inclusion functor is exact." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12047, "type": "theorem", "label": "homology-lemma-kernel-exact-functor", "categories": [ "homology" ], "title": "homology-lemma-kernel-exact-functor", "contents": [ "Let $\\mathcal{A}$, $\\mathcal{B}$ be abelian categories.", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor.", "Then the full subcategory of objects $C$ of $\\mathcal{A}$", "such that $F(C) = 0$ forms a Serre subcategory of $\\mathcal{A}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12048, "type": "theorem", "label": "homology-lemma-serre-subcategory-is-kernel", "categories": [ "homology" ], "title": "homology-lemma-serre-subcategory-is-kernel", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $\\mathcal{C} \\subset \\mathcal{A}$ be a Serre subcategory.", "There exists an abelian category $\\mathcal{A}/\\mathcal{C}$", "and an exact functor", "$$", "F : \\mathcal{A} \\longrightarrow \\mathcal{A}/\\mathcal{C}", "$$", "which is essentially surjective and whose kernel is $\\mathcal{C}$.", "The category $\\mathcal{A}/\\mathcal{C}$ and the functor $F$ are", "characterized by the following universal property: For any exact", "functor $G : \\mathcal{A} \\to \\mathcal{B}$ such that", "$\\mathcal{C} \\subset \\Ker(G)$ there exists a factorization", "$G = H \\circ F$ for a unique exact functor", "$H : \\mathcal{A}/\\mathcal{C} \\to \\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the set of arrows of $\\mathcal{A}$ defined by", "the following formula", "$$", "S = \\{f \\in \\text{Arrows}(\\mathcal{A}) \\mid", "\\Ker(f), \\Coker(f) \\in \\Ob(\\mathcal{C}) \\}.", "$$", "We claim that $S$ is a multiplicative system. To prove this we have", "to check MS1, MS2, MS3, see", "Categories, Definition \\ref{categories-definition-multiplicative-system}.", "\\medskip\\noindent", "It is clear that identities are elements of $S$. Suppose that", "$f : A \\to B$ and $g : B \\to C$ are elements of $S$.", "There are exact sequences", "$$", "\\begin{matrix}", "0 \\to \\Ker(f) \\to \\Ker(gf) \\to \\Ker(g) \\\\", "\\Coker(f) \\to \\Coker(gf) \\to \\Coker(g) \\to 0", "\\end{matrix}", "$$", "Hence it follows that $gf \\in S$. This proves MS1. (In fact, a similar", "argument will show that $S$ is a saturated multiplicative system, see", "Categories, Definition", "\\ref{categories-definition-saturated-multiplicative-system}.)", "\\medskip\\noindent", "Consider a solid diagram", "$$", "\\xymatrix{", "A \\ar[d]_t \\ar[r]_g & B \\ar@{..>}[d]^s \\\\", "C \\ar@{..>}[r]^f & C \\amalg_A B", "}", "$$", "with $t \\in S$. Set", "$W = C \\amalg_A B = \\Coker((t, -g) : A \\to C \\oplus B)$.", "Then $\\Ker(t) \\to \\Ker(s)$ is surjective and", "$\\Coker(t) \\to \\Coker(s)$ is an isomorphism. Hence", "$s$ is an element of $S$. This proves LMS2 and the proof of RMS2 is dual.", "\\medskip\\noindent", "Finally, consider morphisms $f, g : B \\to C$ and a morphism $s : A \\to B$", "in $S$ such that $f \\circ s = g \\circ s$. This means that", "$(f - g) \\circ s = 0$. In turn this means that", "$I = \\Im(f - g) \\subset C$ is a quotient of $\\Coker(s)$", "hence an object of $\\mathcal{C}$. Thus $t : C \\to C' = C/I$ is an", "element of $S$ such that $t \\circ (f - g) = 0$, i.e., such that", "$t \\circ f = t \\circ g$. This proves LMS3 and the proof of", "RMS3 is dual.", "\\medskip\\noindent", "Having proved that $S$ is a multiplicative system we set", "$\\mathcal{A}/\\mathcal{C} = S^{-1}\\mathcal{A}$, and we set", "$F$ equal to the localization functor $Q$. By", "Lemma \\ref{lemma-localization-abelian}", "the category $\\mathcal{A}/\\mathcal{C}$ is abelian and $F$ is exact.", "If $X$ is in the kernel of $F = Q$, then by", "Lemma \\ref{lemma-kernel-localization}", "we see that $0 : X \\to Z$ is an element of $S$ and hence", "$X$ is an object of $\\mathcal{C}$, i.e., the kernel of", "$F$ is $\\mathcal{C}$.", "Finally, if $G$ is as in the statement of the lemma, then $G$ turns", "every element of $S$ into an isomorphism. Hence we obtain the", "functor $H : \\mathcal{A}/\\mathcal{C} \\to \\mathcal{B}$ from", "the universal property of localization, see", "Categories, Lemma \\ref{categories-lemma-properties-left-localization}." ], "refs": [ "categories-definition-multiplicative-system", "categories-definition-saturated-multiplicative-system", "homology-lemma-localization-abelian", "homology-lemma-kernel-localization", "categories-lemma-properties-left-localization" ], "ref_ids": [ 12373, 12376, 12040, 12039, 12258 ] } ], "ref_ids": [] }, { "id": 12049, "type": "theorem", "label": "homology-lemma-quotient-by-kernel-exact-functor", "categories": [ "homology" ], "title": "homology-lemma-quotient-by-kernel-exact-functor", "contents": [ "Let $\\mathcal{A}$, $\\mathcal{B}$ be abelian categories.", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor.", "Let $\\mathcal{C} \\subset \\mathcal{A}$ be a Serre subcategory", "contained in the kernel of $F$.", "Then $\\mathcal{C} = \\Ker(F)$ if and only if the induced functor", "$\\overline{F} : \\mathcal{A}/\\mathcal{C} \\to \\mathcal{B}$", "(Lemma \\ref{lemma-serre-subcategory-is-kernel}) is faithful." ], "refs": [ "homology-lemma-serre-subcategory-is-kernel" ], "proofs": [ { "contents": [ "We will use the results of Lemma \\ref{lemma-serre-subcategory-is-kernel}", "without further mention.", "The ``only if'' direction is true because the kernel of $\\overline{F}$ is zero", "by construction. Namely, if $f : X \\to Y$ is a morphism in", "$\\mathcal{A}/\\mathcal{C}$ such that $\\overline{F}(f) = 0$, then", "$\\overline{F}(\\Im(f)) = \\Im(\\overline{F}(f)) = 0$, hence $\\Im(f) = 0$ by the", "assumption on the kernel of $F$. Thus $f = 0$.", "\\medskip\\noindent", "For the ``if'' direction, let $X$ be an object of $\\mathcal{A}$ such that $F(X)", "= 0$. Then $\\overline{F}(\\text{id}_X) = \\text{id}_{\\overline{F}(X)} = 0$, thus", "$\\text{id}_X = 0$ in $\\mathcal{A}/\\mathcal{C}$ by faithfulness of", "$\\overline{F}$. Hence $X = 0$ in $\\mathcal{A}/\\mathcal{C}$, that is $X \\in", "\\Ob(\\mathcal{C})$." ], "refs": [ "homology-lemma-serre-subcategory-is-kernel" ], "ref_ids": [ 12048 ] } ], "ref_ids": [ 12048 ] }, { "id": 12050, "type": "theorem", "label": "homology-lemma-exact-functor-K-groups", "categories": [ "homology" ], "title": "homology-lemma-exact-functor-K-groups", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor between", "abelian categories. Then $F$ induces a homomorphism of $K$-groups", "$K_0(F) : K_0(\\mathcal{A}) \\to K_0(\\mathcal{B})$ by simply setting", "$K_0(F)([A]) = [F(A)]$." ], "refs": [], "proofs": [ { "contents": [ "Proves itself." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12051, "type": "theorem", "label": "homology-lemma-serre-subcategory-K-groups", "categories": [ "homology" ], "title": "homology-lemma-serre-subcategory-K-groups", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $\\mathcal{C} \\subset \\mathcal{A}$ be a Serre subcategory and", "set $\\mathcal{B} = \\mathcal{A}/\\mathcal{C}$.", "\\begin{enumerate}", "\\item The exact functors $\\mathcal{C} \\to \\mathcal{A}$ and", "$\\mathcal{A} \\to \\mathcal{B}$ induce an exact sequence", "$$", "K_0(\\mathcal{C}) \\to", "K_0(\\mathcal{A}) \\to", "K_0(\\mathcal{B}) \\to", "0", "$$", "of $K$-groups, and", "\\item the kernel of $K_0(\\mathcal{C}) \\to K_0(\\mathcal{A})$ is equal", "to the collection of elements of the form", "$$", "[H^0(M, \\varphi, \\psi)] - [H^1(M, \\varphi, \\psi)]", "$$", "where $(M, \\varphi, \\psi)$ is a complex as in (\\ref{equation-cyclic-complex})", "with the property that it becomes exact in $\\mathcal{B}$; in other words", "that $H^0(M, \\varphi, \\psi)$ and $H^1(M, \\varphi, \\psi)$ are", "objects of $\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). It is clear that $K_0(\\mathcal{A}) \\to K_0(\\mathcal{B})$", "is surjective and that the composition $K_0(\\mathcal{C}) \\to", "K_0(\\mathcal{A}) \\to K_0(\\mathcal{B})$ is zero. Let $x \\in K_0(\\mathcal{A})$", "be an element mapping to zero in $K_0(\\mathcal{B})$. We can write", "$x = [A] - [A']$ with $A, A'$ in $\\mathcal{A}$ (fun exercise).", "Denote $B, B'$ the corresponding objects of $\\mathcal{B}$. The fact that", "$x$ maps to zero in $K_0(\\mathcal{B})$", "means that there exists a finite set $I = I^+ \\amalg I^{-}$,", "for each $i \\in I$ a short exact sequence", "$$", "0 \\to B_i \\to B'_i \\to B''_i \\to 0", "$$", "in $\\mathcal{B}$ such that we have", "$$", "[B] - [B'] = \\sum\\nolimits_{i \\in I^{+}} ([B'_i] - [B_i] - [B''_i])", "-", "\\sum\\nolimits_{i \\in I^{-}} ([B'_i] - [B_i] - [B''_i])", "$$", "in the free abelian group on isomorphism classes of objects of $\\mathcal{B}$.", "We can rewrite this as", "$$", "[B]", "+ \\sum\\nolimits_{i \\in I^{+}} ([B_i] + [B''_i])", "+ \\sum\\nolimits_{i \\in I^{-}} [B'_i]", "=", "[B']", "+ \\sum\\nolimits_{i \\in I^{-}} ([B_i] + [B''_i])", "+ \\sum\\nolimits_{i \\in I^{+}} [B'_i].", "$$", "Since the right and left hand side should contain the same isomorphism classes", "of objects of $\\mathcal{B}$ counted with multiplicity, this means there should", "be a bijection", "$$", "\\tau :", "\\{B\\} \\amalg \\{B_i, B''_i; i \\in I^+\\} \\amalg \\{B'_i; i \\in I^-\\}", "\\longrightarrow", "\\{B'\\} \\amalg \\{B_i, B''_i; i \\in I^-\\} \\amalg \\{B'_i; i \\in I^+\\}", "$$", "such that $N$ and $\\tau(N)$ are isomorphic in $\\mathcal{B}$.", "The proof of Lemmas \\ref{lemma-serre-subcategory-is-kernel} and", "\\ref{lemma-localization-abelian} show that we choose for $i \\in I$", "a short exact sequence", "$$", "0 \\to A_i \\to A'_i \\to A''_i \\to 0", "$$", "in $\\mathcal{A}$ such that $B_i, B'_i, B''_i$ are isomorphic to the", "images of $A_i, A'_i, A''_i$ in $\\mathcal{B}$. This implies that the", "corresponding bijection", "$$", "\\tau :", "\\{A\\} \\amalg \\{A_i, A''_i; i \\in I^+\\} \\amalg \\{A'_i; i \\in I^-\\}", "\\longrightarrow", "\\{A'\\} \\amalg \\{A_i, A''_i; i \\in I^-\\} \\amalg \\{A'_i; i \\in I^+\\}", "$$", "satisfies the property that $M$ and $\\tau(M)$ are objects of $\\mathcal{A}$", "which become isomorphic in $\\mathcal{B}$. This means $[M] - [\\tau(M)]$", "is in the image of $K_0(\\mathcal{C}) \\to K_0(\\mathcal{A})$. Namely,", "the isomorphism in $\\mathcal{B}$ is given by a diagram", "$M \\leftarrow M' \\rightarrow \\tau(M)$ in $\\mathcal{A}$ where both", "$M' \\to M$ and $M' \\to \\tau(M)$ have kernel and cokernel in $\\mathcal{C}$.", "Working backwards we conclude that $x = [A] - [A']$ is in the image", "of $K_0(\\mathcal{C}) \\to K_0(\\mathcal{A})$ and the proof of part (1)", "is complete.", "\\medskip\\noindent", "Proof of (2). The proof is similar to the proof of (1) but slightly", "more bookkeeping is involved. First we remark that any class of the type", "$[H^0(M, \\varphi, \\psi)] - [H^1(M, \\varphi, \\psi)]$ is zero", "in $K_0(\\mathcal{A})$ by the following calculation", "\\begin{align*}", "0 & = [M] - [M] \\\\", "& = [\\Ker(\\varphi)] + [\\Im(\\varphi)]", "- [\\Ker(\\psi)] - [\\Im(\\psi)] \\\\", "& =", "[\\Ker(\\varphi)/\\Im(\\psi)] -", "[\\Ker(\\psi)/\\Im(\\varphi)] \\\\", "& = [H^1(M, \\varphi, \\psi)] - [H^0(M, \\varphi, \\psi)]", "\\end{align*}", "as desired. Hence it suffices to show that any element in the kernel", "of $K_0(\\mathcal{C}) \\to K_0(\\mathcal{A})$ is of this form.", "\\medskip\\noindent", "Any element $x$ in $K_0(\\mathcal{C})$ can be represented as the", "difference $x = [P] - [Q]$ of two objects of $\\mathcal{C}$ (fun exercise).", "Suppose that this element maps to zero in $K_0(\\mathcal{A})$.", "This means that there exist", "\\begin{enumerate}", "\\item a finite set $I = I^{+} \\amalg I^{-}$,", "\\item for $i \\in I$ a short exact sequence $0 \\to A_i \\to B_i \\to C_i \\to 0$", "in $\\mathcal{A}$", "\\end{enumerate}", "such that", "$$", "[P] - [Q] =", "\\sum\\nolimits_{i \\in I^{+}} ([B_i] - [A_i] - [C_i])", "-", "\\sum\\nolimits_{i \\in I^{-}} ([B_i] - [A_i] - [C_i])", "$$", "in the free abelian group on the objects of $\\mathcal{A}$.", "We can rewrite this as", "$$", "[P]", "+ \\sum\\nolimits_{i \\in I^{+}} ([A_i] + [C_i])", "+ \\sum\\nolimits_{i \\in I^{-}} [B_i]", "=", "[Q]", "+ \\sum\\nolimits_{i \\in I^{-}} ([A_i] + [C_i])", "+ \\sum\\nolimits_{i \\in I^{+}} [B_i].", "$$", "Since the right and left hand side should contain the same objects", "of $\\mathcal{A}$ counted with multiplicity, this means there should be", "a bijection $\\tau$ between the terms which occur above. Set", "$$", "T^{+} =", "\\{p\\}\\ \\amalg\\ \\{a, c\\} \\times I^{+}\\ \\amalg\\ \\{b\\} \\times I^{-}", "$$", "and", "$$", "T^{-} =", "\\{q\\}\\ \\amalg\\ \\{a, c\\} \\times I^{-}\\ \\amalg\\ \\{b\\} \\times I^{+}.", "$$", "Set $T = T^{+} \\amalg T^{-} = \\{p, q\\} \\amalg \\{a, b, c\\} \\times I$.", "For $t \\in T$ define", "$$", "O(t)", "=", "\\left\\{", "\\begin{matrix}", "P & \\text{if} & t = p \\\\", "Q & \\text{if} & t = q \\\\", "A_i & \\text{if} & t = (a, i) \\\\", "B_i & \\text{if} & t = (b, i) \\\\", "C_i & \\text{if} & t = (c, i)", "\\end{matrix}", "\\right.", "$$", "Hence we can view $\\tau : T^{+} \\to T^{-}$ as a bijection", "such that $O(t) = O(\\tau(t))$ for all $t \\in T^{+}$.", "Let $t^{-}_0 = \\tau(p)$ and let $t^{+}_0 \\in T^{+}$ be the", "unique element such that $\\tau(t^{+}_0) = q$.", "Consider the object", "$$", "M^{+} = \\bigoplus\\nolimits_{t \\in T^{+}} O(t)", "$$", "By using $\\tau$ we see that it is equal to the object", "$$", "M^{-} = \\bigoplus\\nolimits_{t \\in T^{-}} O(t)", "$$", "Consider the map", "$$", "\\varphi : M^{+} \\longrightarrow M^{-}", "$$", "which on the summand $O(t) = A_i$ corresponding to $t = (a, i)$, $i \\in I^{+}$", "uses the map $A_i \\to B_i$ into the summand $O((b, i)) = B_i$ of $M^{-}$", "and on the summand $O(t) = B_i$ corresponding to $(b, i)$, $i \\in I^{-}$", "uses the map $B_i \\to C_i$ into the summand $O((c, i)) = C_i$ of $M^{-}$.", "The map is zero on the summands corresponding to $p$", "and $(c, i)$, $i \\in I^{+}$.", "Similarly, consider the map", "$$", "\\psi : M^{-} \\longrightarrow M^{+}", "$$", "which on the summand $O(t) = A_i$ corresponding to $t = (a, i)$, $i \\in I^{-}$", "uses the map $A_i \\to B_i$ into the summand $O((b, i)) = B_i$ of $M^{+}$", "and on the summand $O(t) = B_i$ corresponding to $(b, i)$, $i \\in I^{+}$", "uses the map $B_i \\to C_i$ into the summand $O((c, i)) = C_i$ of $M^{+}$.", "The map is zero on the summands corresponding to $q$ and", "$(c, i)$, $i \\in I^{-}$.", "\\medskip\\noindent", "Note that the kernel of $\\varphi$ is equal to the direct sum of the", "summand $P$ and the summands $O((c, i)) = C_i$, $i \\in I^{+}$ and", "the subobjects $A_i$ inside the summands $O((b, i)) = B_i$, $i \\in I^{-}$.", "The image of $\\psi$ is equal to the direct sum of the", "summands $O((c, i)) = C_i$, $i \\in I^{+}$ and", "the subobjects $A_i$ inside the summands $O((b, i)) = B_i$, $i \\in I^{-}$.", "In other words we see that", "$$", "P \\cong \\Ker(\\varphi)/\\Im(\\psi).", "$$", "In exactly the same way we see that", "$$", "Q \\cong \\Ker(\\psi)/\\Im(\\varphi).", "$$", "Since as we remarked above the existence of the bijection", "$\\tau$ shows that $M^{+} = M^{-}$ we see that the lemma follows." ], "refs": [ "homology-lemma-serre-subcategory-is-kernel", "homology-lemma-localization-abelian" ], "ref_ids": [ 12048, 12040 ] } ], "ref_ids": [] }, { "id": 12052, "type": "theorem", "label": "homology-lemma-efface-implies-universal", "categories": [ "homology" ], "title": "homology-lemma-efface-implies-universal", "contents": [ "Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories.", "Let $F = (F^n, \\delta_F)$ be a $\\delta$-functor", "from $\\mathcal{A}$ to $\\mathcal{B}$.", "Suppose that for every $n > 0$ and any $A \\in \\Ob(\\mathcal{A})$", "there exists an injective morphism $u : A \\to B$ (depending on $A$ and $n$)", "such that $F^n(u) : F^n(A) \\to F^n(B)$ is zero. Then $F$ is a universal", "$\\delta$-functor." ], "refs": [], "proofs": [ { "contents": [ "Let $G = (G^n, \\delta_G)$ be a $\\delta$-functor", "from $\\mathcal{A}$ to $\\mathcal{B}$ and let $t : F^0 \\to G^0$", "be a morphism of functors. We have to show there exists", "a unique morphism of $\\delta$-functors $\\{t^n\\}_{n \\geq 0} : F \\to G$", "such that $t = t^0$. We construct $t^n$ by induction on $n$.", "For $n = 0$ we set $t^0 = t$.", "Suppose we have already constructed a unique sequence of", "transformation of functors $t^i$ for $i \\leq n$ compatible with", "the maps $\\delta$ in degrees $\\leq n$.", "\\medskip\\noindent", "Let $A \\in \\Ob(\\mathcal{A})$. By assumption we may choose", "a embedding $u : A \\to B$ such that $F^{n + 1}(u) = 0$.", "Let $C = B/u(A)$. The long exact cohomology sequence for", "the short exact sequence $0 \\to A \\to B \\to C \\to 0$ and the", "$\\delta$-functor $F$ gives that", "$F^{n + 1}(A) = \\Coker(F^n(B) \\to F^n(C))$ by our choice of $u$.", "Since we have already defined $t^n$ we can set", "$$", "t^{n + 1}_A : F^{n + 1}(A) \\to G^{n + 1}(A)", "$$", "equal to the unique map such that", "$$", "\\xymatrix{", "\\Coker(F^n(B) \\to F^n(C)) \\ar[r]_{t^n}", "\\ar[d]_{\\delta_{F, A \\to B \\to C}} &", "\\Coker(G^n(B) \\to G^n(C))", "\\ar[d]^{\\delta_{G, A \\to B \\to C}} \\\\", "F^{n + 1}(A) \\ar[r]^{t^{n + 1}_A} &", "G^{n + 1}(A)", "}", "$$", "commutes. This is clearly uniquely determined by the requirements", "imposed. We omit the verification that this defines a transformation", "of functors." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12053, "type": "theorem", "label": "homology-lemma-uniqueness-universal-delta-functor", "categories": [ "homology" ], "title": "homology-lemma-uniqueness-universal-delta-functor", "contents": [ "Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories.", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.", "If there exists a universal $\\delta$-functor", "$(F^n, \\delta_F)$ from $\\mathcal{A}$ to $\\mathcal{B}$", "with $F^0 = F$, then it is determined up to unique isomorphism", "of $\\delta$-functors." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12054, "type": "theorem", "label": "homology-lemma-compose-homotopy", "categories": [ "homology" ], "title": "homology-lemma-compose-homotopy", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let $f, g : B_\\bullet \\to C_\\bullet$ be morphisms", "of chain complexes. Suppose given morphisms of chain", "complexes $a : A_\\bullet \\to B_\\bullet$, and", "$c : C_\\bullet \\to D_\\bullet$.", "If $\\{h_i : B_i \\to C_{i + 1}\\}$ defines a homotopy", "between $f$ and $g$, then $\\{c_{i + 1} \\circ h_i \\circ a_i\\}$", "defines a homotopy between $c \\circ f \\circ a$ and", "$c \\circ g \\circ a$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12055, "type": "theorem", "label": "homology-lemma-cat-chain-abelian", "categories": [ "homology" ], "title": "homology-lemma-cat-chain-abelian", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item The category of chain complexes in $\\mathcal{A}$ is", "abelian.", "\\item A morphism of complexes", "$f : A_\\bullet \\to B_\\bullet$ is injective", "if and only if each $f_n : A_n \\to B_n$ is injective.", "\\item A morphism of complexes", "$f : A_\\bullet \\to B_\\bullet$ is surjective", "if and only if each $f_n : A_n \\to B_n$ is surjective.", "\\item A sequence of chain complexes", "$$", "A_\\bullet \\xrightarrow{f} B_\\bullet \\xrightarrow{g} C_\\bullet", "$$", "is exact at $B_\\bullet$ if and only if each sequence", "$$", "A_i \\xrightarrow{f_i} B_i \\xrightarrow{g_i} C_i", "$$", "is exact at $B_i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12056, "type": "theorem", "label": "homology-lemma-map-homology-homotopy", "categories": [ "homology" ], "title": "homology-lemma-map-homology-homotopy", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item If the maps $f, g : A_\\bullet \\to B_\\bullet$ are", "homotopic, then the induced maps $H_i(f)$ and $H_i(g)$", "are equal.", "\\item If the map $f : A_\\bullet \\to B_\\bullet$ is a homotopy", "equivalence, then $f$ is a quasi-isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12057, "type": "theorem", "label": "homology-lemma-long-exact-sequence-chain", "categories": [ "homology" ], "title": "homology-lemma-long-exact-sequence-chain", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Suppose that", "$$", "0 \\to", "A_\\bullet \\to", "B_\\bullet \\to", "C_\\bullet \\to", "0", "$$", "is a short exact sequence of chain complexes of $\\mathcal{A}$.", "Then there is a canonical long exact homology sequence", "$$", "\\xymatrix{", "\\ldots & \\ldots & \\ldots \\ar[lld] \\\\", "H_i(A_\\bullet) \\ar[r] & H_i(B_\\bullet) \\ar[r] & H_i(C_\\bullet) \\ar[lld] \\\\", "H_{i - 1}(A_\\bullet) \\ar[r] &", "H_{i - 1}(B_\\bullet) \\ar[r] &", "H_{i - 1}(C_\\bullet) \\ar[lld] \\\\", "\\ldots & \\ldots & \\ldots \\\\", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted. The maps come from the Snake Lemma \\ref{lemma-snake}", "applied to the diagrams", "$$", "\\xymatrix{", "&", "A_i/\\Im(d_{A, i + 1}) \\ar[r] \\ar[d]^{d_{A, i}} &", "B_i/\\Im(d_{B, i + 1}) \\ar[r] \\ar[d]^{d_{B, i}} &", "C_i/\\Im(d_{C, i + 1}) \\ar[r] \\ar[d]^{d_{C, i}} &", "0 \\\\", "0 \\ar[r] &", "\\Ker(d_{A, i - 1}) \\ar[r] &", "\\Ker(d_{B, i - 1}) \\ar[r] &", "\\Ker(d_{C, i - 1}) &", "}", "$$" ], "refs": [ "homology-lemma-snake" ], "ref_ids": [ 12027 ] } ], "ref_ids": [] }, { "id": 12058, "type": "theorem", "label": "homology-lemma-compose-homotopy-cochain", "categories": [ "homology" ], "title": "homology-lemma-compose-homotopy-cochain", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let $f, g : B^\\bullet \\to C^\\bullet$ be morphisms", "of cochain complexes. Suppose given morphisms of cochain", "complexes $a : A^\\bullet \\to B^\\bullet$, and", "$c : C^\\bullet \\to D^\\bullet$.", "If $\\{h^i : B^i \\to C^{i - 1}\\}$ defines a homotopy", "between $f$ and $g$, then $\\{c^{i - 1} \\circ h^i \\circ a^i\\}$", "defines a homotopy between $c \\circ f \\circ a$ and", "$c \\circ g \\circ a$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12059, "type": "theorem", "label": "homology-lemma-cat-cochain-abelian", "categories": [ "homology" ], "title": "homology-lemma-cat-cochain-abelian", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item The category of cochain complexes in $\\mathcal{A}$ is", "abelian.", "\\item A morphism of cochain complexes", "$f : A^\\bullet \\to B^\\bullet$ is injective", "if and only if each $f^n : A^n \\to B^n$ is injective.", "\\item A morphism of cochain complexes", "$f : A^\\bullet \\to B^\\bullet$ is surjective", "if and only if each $f^n : A^n \\to B^n$ is surjective.", "\\item A sequence of cochain complexes", "$$", "A^\\bullet \\xrightarrow{f} B^\\bullet \\xrightarrow{g} C^\\bullet", "$$", "is exact at $B^\\bullet$ if and only if each sequence", "$$", "A^i \\xrightarrow{f^i} B^i \\xrightarrow{g^i} C^i", "$$", "is exact at $B^i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12060, "type": "theorem", "label": "homology-lemma-map-cohomology-homotopy-cochain", "categories": [ "homology" ], "title": "homology-lemma-map-cohomology-homotopy-cochain", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item If the maps $f, g : A^\\bullet \\to B^\\bullet$ are", "homotopic, then the induced maps $H^i(f)$ and $H^i(g)$", "are equal.", "\\item If $f : A^\\bullet \\to B^\\bullet$ is a homotopy equivalence,", "then $f$ is a quasi-isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12061, "type": "theorem", "label": "homology-lemma-long-exact-sequence-cochain", "categories": [ "homology" ], "title": "homology-lemma-long-exact-sequence-cochain", "contents": [ "\\begin{slogan}", "Short exact sequences of complexes give rise to long exact sequences", "of (co)homology.", "\\end{slogan}", "Let $\\mathcal{A}$ be an abelian category.", "Suppose that", "$$", "0 \\to", "A^\\bullet \\to", "B^\\bullet \\to", "C^\\bullet \\to", "0", "$$", "is a short exact sequence of chain complexes of $\\mathcal{A}$.", "Then there is a canonical long exact cohomology sequence", "$$", "\\xymatrix{", "\\ldots & \\ldots & \\ldots \\ar[lld] \\\\", "H^i(A^\\bullet) \\ar[r] &", "H^i(B^\\bullet) \\ar[r] &", "H^i(C^\\bullet) \\ar[lld] \\\\", "H^{i + 1}(A^\\bullet) \\ar[r] &", "H^{i + 1}(B^\\bullet) \\ar[r] &", "H^{i + 1}(C^\\bullet) \\ar[lld] \\\\", "\\ldots & \\ldots & \\ldots \\\\", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted. The maps come from the Snake Lemma \\ref{lemma-snake}", "applied to the diagrams", "$$", "\\xymatrix{", "&", "A^i/\\Im(d_A^{i - 1}) \\ar[r] \\ar[d]^{d_A^i} &", "B^i/\\Im(d_B^{i - 1}) \\ar[r] \\ar[d]^{d_B^i} &", "C^i/\\Im(d_C^{i - 1}) \\ar[r] \\ar[d]^{d_C^i} &", "0 \\\\", "0 \\ar[r] &", "\\Ker(d_A^{i + 1}) \\ar[r] &", "\\Ker(d_B^{i + 1}) \\ar[r] &", "\\Ker(d_C^{i + 1}) &", "}", "$$" ], "refs": [ "homology-lemma-snake" ], "ref_ids": [ 12027 ] } ], "ref_ids": [] }, { "id": 12062, "type": "theorem", "label": "homology-lemma-homotopy-shift", "categories": [ "homology" ], "title": "homology-lemma-homotopy-shift", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Suppose that $A_\\bullet$ and $B_\\bullet$ are", "chain complexes. Given any morphism of chain", "complexes $a : A_\\bullet \\to B_\\bullet$ there", "is a bijection between the set of homotopies", "from $a$ to $a$ and", "$\\Mor_{\\text{Ch}(\\mathcal{A})}(A_\\bullet, B[1]_\\bullet)$.", "More generally, the set of homotopies between", "$a$ and $b$ is either empty or a principal homogeneous", "space under the group", "$\\Mor_{\\text{Ch}(\\mathcal{A})}(A_\\bullet, B[1]_\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "See above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12063, "type": "theorem", "label": "homology-lemma-ses-termwise-split", "categories": [ "homology" ], "title": "homology-lemma-ses-termwise-split", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let", "$$", "0 \\to A_\\bullet \\to B_\\bullet \\to C_\\bullet \\to 0", "$$", "be a short exact sequence of complexes.", "Suppose that $\\{s_n : C_n \\to B_n\\}$ is a family", "of morphisms which split the short exact sequences", "$0 \\to A_n \\to B_n \\to C_n \\to 0$. Let", "$\\pi_n : B_n \\to A_n$ be the associated", "projections, see Lemma \\ref{lemma-ses-split}.", "Then the family of morphisms", "$$", "\\pi_{n - 1} \\circ d_{B, n} \\circ s_n", ":", "C_n \\to A_{n - 1}", "$$", "define a morphism of complexes $\\delta(s) : C_\\bullet \\to A[-1]_\\bullet$." ], "refs": [ "homology-lemma-ses-split" ], "proofs": [ { "contents": [ "Denote $i : A_\\bullet \\to B_\\bullet$ and $q : B_\\bullet \\to C_\\bullet$", "the maps of complexes in the short exact sequence. Then", "$i_{n - 1} \\circ \\pi_{n - 1} \\circ d_{B, n} \\circ s_n =", "d_{B, n} \\circ s_n - s_{n - 1} \\circ d_{C, n}$. Hence", "$i_{n - 2} \\circ d_{A, n - 1} \\circ \\pi_{n - 1} \\circ d_{B, n} \\circ s_n =", "d_{B, n - 1} \\circ (d_{B, n} \\circ s_n - s_{n - 1} \\circ d_{C, n}) =", "- d_{B, n - 1} \\circ s_{n - 1} \\circ d_{C, n}$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12020 ] }, { "id": 12064, "type": "theorem", "label": "homology-lemma-ses-termwise-split-long", "categories": [ "homology" ], "title": "homology-lemma-ses-termwise-split-long", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-ses-termwise-split} above.", "The morphism of complexes $\\delta(s) : C_\\bullet \\to A[-1]_\\bullet$", "induces the maps", "$$", "H_i(\\delta(s)) :", "H_i(C_\\bullet) \\longrightarrow H_i(A[-1]_\\bullet) = H_{i - 1}(A_\\bullet)", "$$", "which occur in the long exact homology sequence associated", "to the short exact sequence of chain complexes by", "Lemma \\ref{lemma-long-exact-sequence-chain}." ], "refs": [ "homology-lemma-ses-termwise-split", "homology-lemma-long-exact-sequence-chain" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12063, 12057 ] }, { "id": 12065, "type": "theorem", "label": "homology-lemma-ses-termwise-split-homotopy", "categories": [ "homology" ], "title": "homology-lemma-ses-termwise-split-homotopy", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-ses-termwise-split} above.", "Suppose $\\{s'_n : C_n \\to B_n\\}$ is a second choice of splittings.", "Write $s'_n = s_n + i_n \\circ h_n$ for some unique", "morphisms $h_n : C_n \\to A_n$. The family of maps", "$\\{h_n : C_n \\to A[-1]_{n + 1}\\}$ is a homotopy between", "the associated morphisms", "$\\delta(s), \\delta(s') : C_\\bullet \\to A[-1]_\\bullet$." ], "refs": [ "homology-lemma-ses-termwise-split" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12063 ] }, { "id": 12066, "type": "theorem", "label": "homology-lemma-homotopy-shift-cochain", "categories": [ "homology" ], "title": "homology-lemma-homotopy-shift-cochain", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Suppose that $A^\\bullet$ and $B^\\bullet$ are", "cochain complexes. Given any morphism of cochain", "complexes $a : A^\\bullet \\to B^\\bullet$ there", "is a bijection between the set of homotopies", "from $a$ to $a$ and", "$\\Mor_{\\text{CoCh}(\\mathcal{A})}(A^\\bullet, B[-1]^\\bullet)$.", "More generally, the set of homotopies between", "$a$ and $b$ is either empty or a principal homogeneous", "space under the group", "$\\Mor_{\\text{CoCh}(\\mathcal{A})}(A^\\bullet, B[-1]^\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "See above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12067, "type": "theorem", "label": "homology-lemma-ses-termwise-split-cochain", "categories": [ "homology" ], "title": "homology-lemma-ses-termwise-split-cochain", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let", "$$", "0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0", "$$", "be a complex (!) of complexes.", "Suppose that we are given splittings $B^n = A^n \\oplus C^n$", "compatible with the maps in the displayed sequence.", "Let $s^n : C^n \\to B^n$ and $\\pi^n : B^n \\to A^n$ be the", "corresponding maps. Then the family of morphisms", "$$", "\\pi^{n + 1} \\circ d_B^n \\circ s^n", ":", "C^n \\to A^{n + 1}", "$$", "define a morphism of complexes $\\delta : C^\\bullet \\to A[1]^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "Denote $i : A^\\bullet \\to B^\\bullet$ and $q : B^\\bullet \\to C^\\bullet$", "the maps of complexes in the short exact sequence. Then", "$i^{n + 1} \\circ \\pi^{n + 1} \\circ d_B^n \\circ s^n =", "d_B^n \\circ s^n - s^{n + 1} \\circ d_C^n$. Hence", "$i^{n + 2} \\circ d_A^{n + 1} \\circ \\pi^{n + 1} \\circ d_B^n \\circ s^n =", "d_B^{n + 1} \\circ (d_B^n \\circ s^n - s^{n + 1} \\circ d_C^n) =", "- d_B^{n + 1} \\circ s^{n + 1} \\circ d_C^n$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12068, "type": "theorem", "label": "homology-lemma-ses-termwise-split-long-cochain", "categories": [ "homology" ], "title": "homology-lemma-ses-termwise-split-long-cochain", "contents": [ "Notation and assumptions as in", "Lemma \\ref{lemma-ses-termwise-split-cochain} above.", "Assume in addition that $\\mathcal{A}$ is abelian.", "The morphism of complexes $\\delta : C^\\bullet \\to A[1]^\\bullet$", "induces the maps", "$$", "H^i(\\delta) :", "H^i(C^\\bullet) \\longrightarrow H^i(A[1]^\\bullet) = H^{i + 1}(A^\\bullet)", "$$", "which occur in the long exact homology sequence associated", "to the short exact sequence of cochain complexes by", "Lemma \\ref{lemma-long-exact-sequence-cochain}." ], "refs": [ "homology-lemma-ses-termwise-split-cochain", "homology-lemma-long-exact-sequence-cochain" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12067, 12061 ] }, { "id": 12069, "type": "theorem", "label": "homology-lemma-ses-termwise-split-homotopy-cochain", "categories": [ "homology" ], "title": "homology-lemma-ses-termwise-split-homotopy-cochain", "contents": [ "Notation and assumptions as in", "Lemma \\ref{lemma-ses-termwise-split-cochain}.", "Let $\\alpha : A^\\bullet \\to B^\\bullet$,", "$\\beta : B^\\bullet \\to C^\\bullet$ be the given", "morphisms of complexes.", "Suppose $(s')^n : C^n \\to B^n$ and $(\\pi')^n : B^n \\to A^n$", "is a second choice of splittings.", "Write $(s')^n = s^n + \\alpha^n \\circ h^n$ and", "$(\\pi')^n = \\pi^n + g^n \\circ \\beta^n$ for some unique", "morphisms $h^n : C^n \\to A^n$ and $g^n : C^n \\to A^n$. Then", "\\begin{enumerate}", "\\item $g^n = - h^n$, and", "\\item the family of maps $\\{g^n : C^n \\to A[1]^{n - 1}\\}$ is a homotopy", "between $\\delta, \\delta' : C^\\bullet \\to A[1]^\\bullet$, more precisely", "$(\\delta')^n = \\delta^n + g^{n + 1} \\circ d_C^n + d_{A[1]}^{n - 1} \\circ g^n$.", "\\end{enumerate}" ], "refs": [ "homology-lemma-ses-termwise-split-cochain" ], "proofs": [ { "contents": [ "As $(s')^n$ and $(\\pi')^n$ are splittings we have $(\\pi')^n \\circ (s')^n = 0$.", "Hence", "$$", "0 = ( \\pi^n + g^n \\circ \\beta^n ) \\circ ( s^n + \\alpha^n \\circ h^n ) =", "g^n \\circ \\beta^n \\circ s^n + \\pi^n \\circ \\alpha^n \\circ h^n =", "g^n + h^n", "$$", "which proves (1). We compute $(\\delta')^n$ as follows", "$$", "( \\pi^{n + 1} + g^{n + 1} \\circ \\beta^{n + 1} )", "\\circ d_B^n \\circ", "( s^n + \\alpha^n \\circ h^n )", "= \\delta^n + g^{n + 1} \\circ d_C^n + d_A^n \\circ h^n", "$$", "Since $h^n = -g^n$ and since $d_{A[1]}^{n - 1} = -d_A^n$ we conclude that (2)", "holds." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12067 ] }, { "id": 12070, "type": "theorem", "label": "homology-lemma-graded", "categories": [ "homology" ], "title": "homology-lemma-graded", "contents": [ "Let $\\mathcal{A}$ be an abelian category. The category of graded objects", "$\\text{Gr}(\\mathcal{A})$ is abelian." ], "refs": [], "proofs": [ { "contents": [ "Let $f : A = (A^i) \\to B = (B^i)$ be a morphism of graded objects", "of $\\mathcal{A}$ given by", "morphisms $f^i : A^i \\to B^i$ of $\\mathcal{A}$.", "Then we have $\\Ker(f) = (\\Ker(f^i))$ and $\\Coker(f) = (\\Coker(f^i))$", "in the category $\\text{Gr}(\\mathcal{A})$.", "Since we have $\\Im = \\Coim$ in $\\mathcal{A}$", "we see the same thing holds in $\\text{Gr}(\\mathcal{A})$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12071, "type": "theorem", "label": "homology-lemma-additive-dual", "categories": [ "homology" ], "title": "homology-lemma-additive-dual", "contents": [ "Let $\\mathcal{A}$ be an additive monoidal category.", "If $Y_i$, $i = 1, 2$ are left duals of $X_i$, $i = 1, 2$, then", "$Y_1 \\oplus Y_2$ is a left dual of $X_1 \\oplus X_2$." ], "refs": [], "proofs": [ { "contents": [ "Follows from uniqueness of adjoints and", "Categories, Remark \\ref{categories-remark-left-dual-adjoint}." ], "refs": [ "categories-remark-left-dual-adjoint" ], "ref_ids": [ 12430 ] } ], "ref_ids": [] }, { "id": 12072, "type": "theorem", "label": "homology-lemma-Karoubian-dual", "categories": [ "homology" ], "title": "homology-lemma-Karoubian-dual", "contents": [ "In a Karoubian additive monoidal category every summand", "of an object which has a left dual has a left dual." ], "refs": [], "proofs": [ { "contents": [ "We will use Categories, Lemma \\ref{categories-lemma-left-dual}", "without further mention.", "Let $X$ be an object which has a left dual $Y$. We have", "$$", "\\Hom(X, X) = \\Hom(\\mathbf{1}, X \\otimes Y) = \\Hom(Y, Y)", "$$", "If $a : X \\to X$ corresponds to $b : Y \\to Y$ then $b$ is the unique", "endomorphism of $Y$ such that precomposing by $a$ on", "$$", "\\Hom(Z' \\otimes X, Z) = \\Hom(Z', Z \\otimes Y)", "$$", "is the same as postcomposing by $1 \\otimes b$.", "Hence the bijection $\\Hom(X, X) \\to \\Hom(Y, Y)$, $a \\mapsto b$", "is an isomorphism of the opposite of the algebra $\\Hom(X, X)$ with", "the algebra $\\Hom(Y, Y)$. In particular, if $X = X_1 \\oplus X_2$, then", "the corresponding projectors $e_1, e_2$ are mapped to idempotents", "in $\\Hom(Y, Y)$. If $Y = Y_1 \\oplus Y_2$ is the corresponding direct", "sum decomposition of $Y$ (Section \\ref{section-karoubian})", "then we see that under the bijection", "$\\Hom(Z' \\otimes X, Z) = \\Hom(Z', Z \\otimes Y)$", "we have $\\Hom(Z' \\otimes X_i, Z) = \\Hom(Z', Z \\otimes Y_i)$", "functorially as subgroups for $i = 1, 2$.", "It follows that $Y_i$ is the left dual of", "$X_i$ by the discussion in", "Categories, Remark \\ref{categories-remark-left-dual-adjoint}." ], "refs": [ "categories-lemma-left-dual", "categories-remark-left-dual-adjoint" ], "ref_ids": [ 12325, 12430 ] } ], "ref_ids": [] }, { "id": 12073, "type": "theorem", "label": "homology-lemma-left-dual-graded-vector-spaces", "categories": [ "homology" ], "title": "homology-lemma-left-dual-graded-vector-spaces", "contents": [ "Let $F$ be a field. Let $\\mathcal{C}$ be the category of graded", "$F$-vector spaces viewed as a monoidal category as in", "Example \\ref{example-graded-vector-spaces}. If $V$ in $\\mathcal{C}$", "has a left dual $W$, then $\\sum_n \\dim_F V^n < \\infty$", "and the map $\\epsilon$ defines nondegenerate pairings", "$W^{-n} \\times V^n \\to F$." ], "refs": [], "proofs": [ { "contents": [ "As unit we take", "By Categories, Definition \\ref{categories-definition-dual} we have", "maps", "$$", "\\eta : \\mathbf{1} \\to V \\otimes W\\quad", "\\epsilon : W \\otimes V \\to \\mathbf{1}", "$$", "Since $\\mathbf{1} = F$ placed in degree $0$, we", "may think of $\\epsilon$ as a sequence of pairings", "$W^{-n} \\times V^n \\to F$ as in the statement of the lemma.", "Choose bases $\\{e_{n, i}\\}_{i \\in I_n}$ for $V^n$ for all $n$.", "Write", "$$", "\\eta(1) = \\sum e_{n, i} \\otimes w_{-n, i}", "$$", "for some elements $w_{-n, i} \\in W^{-n}$ almost all of which are zero!", "The condition that $(\\epsilon \\otimes 1) \\circ (1 \\otimes \\eta)$ is the", "identity on $W$ means that", "$$", "\\sum\\nolimits_{n, i} \\epsilon(w, e_{n, i})w_{-n, i} = w", "$$", "Thus we see that $W$ is generated as a graded vector space", "by the finitely many nonzero vectors $w_{-n, i}$.", "The condition that $(1 \\otimes \\epsilon) \\circ (\\eta \\otimes 1)$", "is the identity of $V$ means that", "$$", "\\sum\\nolimits_{n, i} e_{n, i}\\ \\epsilon(w_{-n, i}, v) = v", "$$", "In particular, setting $v = e_{n, i}$ we conclude that", "$\\epsilon(w_{-n, i}, e_{n, i'}) = \\delta_{ii'}$. Thus", "we find that the statement of the lemma holds and that", "$\\{w_{-n, i}\\}_{i \\in I_n}$ is the dual basis for $W^{-n}$ to", "the chosen basis for $V^n$." ], "refs": [ "categories-definition-dual" ], "ref_ids": [ 12407 ] } ], "ref_ids": [] }, { "id": 12074, "type": "theorem", "label": "homology-lemma-filtered", "categories": [ "homology" ], "title": "homology-lemma-filtered", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "The category of filtered objects $\\text{Fil}(\\mathcal{A})$", "has the following properties:", "\\begin{enumerate}", "\\item It is an additive category.", "\\item It has a zero object.", "\\item It has kernels and cokernels, images and coimages.", "\\item In general it is not an abelian category.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that $\\text{Fil}(\\mathcal{A})$ is additive with direct", "sum given by $(A, F) \\oplus (B, F) = (A \\oplus B, F)$ where", "$F^p(A \\oplus B) = F^pA \\oplus F^pB$.", "The kernel of a morphism $f : (A, F) \\to (B, F)$ of filtered", "objects is the injection $\\Ker(f) \\subset A$ where $\\Ker(f)$", "is endowed with the induced filtration.", "The cokernel of a morphism $f : A \\to B$ of filtered", "objects is the surjection $B \\to \\Coker(f)$ where $\\Coker(f)$", "is endowed with the quotient filtration. Since all kernels and cokernels", "exist, so do all coimages and images. See", "Example \\ref{example-not-abelian}", "for the last statement." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12075, "type": "theorem", "label": "homology-lemma-characterize-strict-general", "categories": [ "homology" ], "title": "homology-lemma-characterize-strict-general", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $f : A \\to B$ be a morphism of filtered objects of $\\mathcal{A}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is strict,", "\\item the morphism $\\Coim(f) \\to \\Im(f)$ of", "Lemma \\ref{lemma-coim-im-map}", "is an isomorphism.", "\\end{enumerate}" ], "refs": [ "homology-lemma-coim-im-map" ], "proofs": [ { "contents": [ "Note that $\\Coim(f) \\to \\Im(f)$ is an isomorphism of", "objects of $\\mathcal{A}$, and that part (2) signifies that it is", "an isomorphism of filtered objects.", "By the description of kernels and cokernels in the proof of", "Lemma \\ref{lemma-filtered}", "we see that the filtration on $\\Coim(f)$ is the", "quotient filtration coming from $A \\to \\Coim(f)$.", "Similarly, the filtration on $\\Im(f)$ is the induced", "filtration coming from the injection $\\Im(f) \\to B$.", "The definition of strict is exactly that the quotient filtration", "is the induced filtration." ], "refs": [ "homology-lemma-filtered" ], "ref_ids": [ 12074 ] } ], "ref_ids": [ 12013 ] }, { "id": 12076, "type": "theorem", "label": "homology-lemma-add-summand-strict-monomorphism", "categories": [ "homology" ], "title": "homology-lemma-add-summand-strict-monomorphism", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $f : A \\to B$ be a strict monomorphism of filtered objects.", "Let $g : A \\to C$ be a morphism of filtered objects.", "Then $f \\oplus g : A \\to B \\oplus C$ is a strict monomorphism." ], "refs": [], "proofs": [ { "contents": [ "Clear from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12077, "type": "theorem", "label": "homology-lemma-add-summand-strict-epimorphism", "categories": [ "homology" ], "title": "homology-lemma-add-summand-strict-epimorphism", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $f : B \\to A$ be a strict epimorphism of filtered objects.", "Let $g : C \\to A$ be a morphism of filtered objects.", "Then $f \\oplus g : B \\oplus C \\to A$ is a strict epimorphism." ], "refs": [], "proofs": [ { "contents": [ "Clear from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12078, "type": "theorem", "label": "homology-lemma-induced-and-quotient-strict", "categories": [ "homology" ], "title": "homology-lemma-induced-and-quotient-strict", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $(A, F)$, $(B, F)$ be filtered objects.", "Let $u : A \\to B$ be a morphism of filtered objects.", "If $u$ is injective then $u$ is strict if and only if the filtration", "on $A$ is the induced filtration.", "If $u$ is surjective then $u$ is strict if and only if the filtration", "on $B$ is the quotient filtration." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12079, "type": "theorem", "label": "homology-lemma-composition-strict", "categories": [ "homology" ], "title": "homology-lemma-composition-strict", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $f : A \\to B$, $g : B \\to C$", "be strict morphisms of filtered objects.", "\\begin{enumerate}", "\\item In general the composition $g \\circ f$ is not strict.", "\\item If $g$ is injective, then $g \\circ f$ is strict.", "\\item If $f$ is surjective, then $g \\circ f$ is strict.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $B$ a vector space over a field $k$ with basis $e_1, e_2$, with the", "filtration $F^nB = B$ for $n < 0$, with $F^0B = ke_1$, and $F^nB = 0$ for", "$n > 0$. Now take $A = k(e_1 + e_2)$ and $C = B/ke_2$ with filtrations", "induced by $B$, i.e., such that $A \\to B$ and $B \\to C$ are strict", "(Lemma \\ref{lemma-induced-and-quotient-strict}).", "Then $F^n(A) = A$ for $n < 0$ and $F^n(A) = 0$ for $n \\geq 0$. ", "Also $F^n(C) = C$ for $n \\leq 0$ and $F^n(C) = 0$ for $n > 0$.", "So the (nonzero) composition $A \\to C$ is not strict.", "\\medskip\\noindent", "Assume $g$ is injective. Then", "\\begin{align*}", "g(f(F^pA)) & = g(f(A) \\cap F^pB) \\\\", "& = g(f(A)) \\cap g(F^p(B)) \\\\", "& = (g \\circ f)(A) \\cap (g(B) \\cap F^pC) \\\\", "& = (g \\circ f)(A) \\cap F^pC.", "\\end{align*}", "The first equality as $f$ is strict, the second because $g$ is injective,", "the third because $g$ is strict, and the fourth because", "$(g \\circ f)(A) \\subset g(B)$.", "\\medskip\\noindent", "Assume $f$ is surjective. Then", "\\begin{align*}", "(g \\circ f)^{-1}(F^iC) & = f^{-1}(F^iB + \\Ker(g)) \\\\", "& = f^{-1}(F^iB) + f^{-1}(\\Ker(g)) \\\\", "& = F^iA + \\Ker(f) + \\Ker(g \\circ f) \\\\", "& = F^iA + \\Ker(g \\circ f)", "\\end{align*}", "The first equality because $g$ is strict, the second because $f$ is", "surjective, the third because $f$ is strict, and the last because", "$\\Ker(f) \\subset \\Ker(g \\circ f)$." ], "refs": [ "homology-lemma-induced-and-quotient-strict" ], "ref_ids": [ 12078 ] } ], "ref_ids": [] }, { "id": 12080, "type": "theorem", "label": "homology-lemma-filtration-subobject", "categories": [ "homology" ], "title": "homology-lemma-filtration-subobject", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $(A, F)$ be a filtered object of $\\mathcal{A}$.", "Let $X \\subset Y \\subset A$ be subobjects of $A$.", "On the object", "$$", "Y/X = \\Ker(A/X \\to A/Y)", "$$", "the quotient filtration coming from the induced filtration on $Y$ and the", "induced filtration coming from the quotient filtration on $A/X$ agree.", "Any of the morphisms $X \\to Y$, $X \\to A$, $Y \\to A$, $Y \\to A/X$,", "$Y \\to Y/X$, $Y/X \\to A/X$ are strict (with induced/quotient filtrations)." ], "refs": [], "proofs": [ { "contents": [ "The quotient filtration $Y/X$ is given by", "$F^p(Y/X) = F^pY/(X \\cap F^pY) = F^pY/F^pX$", "because $F^pY = Y \\cap F^pA$ and $F^pX = X \\cap F^pA$.", "The induced filtration from the injection $Y/X \\to A/X$ is given by", "\\begin{align*}", "F^p(Y/X) & = Y/X \\cap F^p(A/X) \\\\", "& = Y/X \\cap (F^pA + X)/X \\\\", "& = (Y \\cap F^pA)/(X \\cap F^pA) \\\\", "& = F^pY/F^pX.", "\\end{align*}", "Hence the first statement of the lemma.", "The proof of the other cases is similar." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12081, "type": "theorem", "label": "homology-lemma-pushout-filtered", "categories": [ "homology" ], "title": "homology-lemma-pushout-filtered", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A, B, C \\in \\text{Fil}(\\mathcal{A})$.", "Let $f : A \\to B$ and $g : A \\to C$ be morphisms.", "Then there exists a pushout", "$$", "\\xymatrix{", "A \\ar[r]_f \\ar[d]_g & B \\ar[d]^{g'} \\\\", "C \\ar[r]^{f'} & C \\amalg_A B", "}", "$$", "in $\\text{Fil}(\\mathcal{A})$. If $f$ is strict, so is $f'$." ], "refs": [], "proofs": [ { "contents": [ "Set $C \\amalg_A B$ equal to $\\Coker((1, -1) : A \\to C \\oplus B)$", "in $\\text{Fil}(\\mathcal{A})$. This cokernel exists, by", "Lemma \\ref{lemma-filtered}.", "It is a pushout, see", "Example \\ref{example-fibre-product-pushouts}.", "Note that $F^p(C \\amalg_A B)$ is the image of $F^pC \\oplus F^pB$.", "Hence", "$$", "(f')^{-1}(F^p(C \\amalg_A B)) = g(f^{-1}(F^pB))) + F^pC", "$$", "Whence the last statement." ], "refs": [ "homology-lemma-filtered" ], "ref_ids": [ 12074 ] } ], "ref_ids": [] }, { "id": 12082, "type": "theorem", "label": "homology-lemma-fibre-product-filtered", "categories": [ "homology" ], "title": "homology-lemma-fibre-product-filtered", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A, B, C \\in \\text{Fil}(\\mathcal{A})$.", "Let $f : B \\to A$ and $g : C \\to A$ be morphisms.", "Then there exists a fibre product", "$$", "\\xymatrix{", "B \\times_A C \\ar[r]_{g'} \\ar[d]_{f'} & B \\ar[d]^f \\\\", "C \\ar[r]^g & A", "}", "$$", "in $\\text{Fil}(\\mathcal{A})$. If $f$ is strict, so is $f'$." ], "refs": [], "proofs": [ { "contents": [ "This lemma is dual to", "Lemma \\ref{lemma-pushout-filtered}." ], "refs": [ "homology-lemma-pushout-filtered" ], "ref_ids": [ 12081 ] } ], "ref_ids": [] }, { "id": 12083, "type": "theorem", "label": "homology-lemma-ses-gr", "categories": [ "homology" ], "title": "homology-lemma-ses-gr", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item Let $A$ be a filtered object and $X \\subset A$. Then for each $p$", "the sequence", "$$", "0 \\to \\text{gr}^p(X) \\to \\text{gr}^p(A) \\to \\text{gr}^p(A/X) \\to 0", "$$", "is exact (with induced filtration on $X$ and quotient filtration on $A/X$).", "\\item Let $f : A \\to B$ be a morphism of filtered objects of $\\mathcal{A}$.", "Then for each $p$ the sequences", "$$", "0 \\to \\text{gr}^p(\\Ker(f)) \\to \\text{gr}^p(A) \\to", "\\text{gr}^p(\\Coim(f)) \\to 0", "$$", "and", "$$", "0 \\to \\text{gr}^p(\\Im(f)) \\to \\text{gr}^p(B) \\to", "\\text{gr}^p(\\Coker(f)) \\to 0", "$$", "are exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have $F^{p + 1}X = X \\cap F^{p + 1}A$, hence map", "$\\text{gr}^p(X) \\to \\text{gr}^p(A)$ is injective. Dually the map", "$\\text{gr}^p(A) \\to \\text{gr}^p(A/X)$ is surjective.", "The kernel of $F^pA/F^{p + 1}A \\to A/X + F^{p + 1}A$", "is clearly $F^{p + 1}A + X \\cap F^pA/F^{p + 1}A = F^pX/F^{p + 1}X$", "hence exactness in the middle.", "The two short exact sequence of (2) are special cases of the", "short exact sequence of (1)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12084, "type": "theorem", "label": "homology-lemma-characterize-strict", "categories": [ "homology" ], "title": "homology-lemma-characterize-strict", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $f : A \\to B$ be a morphism of finite", "filtered objects of $\\mathcal{A}$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is strict,", "\\item the morphism $\\Coim(f) \\to \\Im(f)$ is an isomorphism,", "\\item $\\text{gr}(\\Coim(f)) \\to \\text{gr}(\\Im(f))$ is an", "isomorphism,", "\\item the sequence", "$\\text{gr}(\\Ker(f)) \\to \\text{gr}(A) \\to \\text{gr}(B)$", "is exact,", "\\item the sequence $\\text{gr}(A) \\to \\text{gr}(B) \\to", "\\text{gr}(\\Coker(f))$ is exact, and", "\\item the sequence", "$$", "0 \\to", "\\text{gr}(\\Ker(f)) \\to", "\\text{gr}(A) \\to", "\\text{gr}(B) \\to", "\\text{gr}(\\Coker(f)) \\to 0", "$$", "is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is", "Lemma \\ref{lemma-characterize-strict-general}.", "By", "Lemma \\ref{lemma-ses-gr}", "we see that (4), (5), (6) imply (3) and that (3) implies (4), (5), (6).", "Hence it suffices to show that (3) implies (2).", "Thus we have to show that if $f : A \\to B$ is an injective and surjective", "map of finite filtered objects which induces and isomorphism", "$\\text{gr}(A) \\to \\text{gr}(B)$, then $f$ induces an isomorphism of", "filtered objects. In other words, we have to show that", "$f(F^pA) = F^pB$ for all $p$.", "As the filtrations are finite we may prove this by descending induction", "on $p$. Suppose that $f(F^{p + 1}A) = F^{p + 1}B$.", "Then commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "F^{p + 1}A \\ar[r] \\ar[d]^f &", "F^pA \\ar[r] \\ar[d]^f &", "\\text{gr}^p(A) \\ar[r] \\ar[d]^{\\text{gr}^p(f)} &", "0 \\\\", "0 \\ar[r] &", "F^{p + 1}B \\ar[r] &", "F^pB \\ar[r] &", "\\text{gr}^p(B) \\ar[r] &", "0", "}", "$$", "and the five lemma imply that $f(F^pA) = F^pB$." ], "refs": [ "homology-lemma-characterize-strict-general", "homology-lemma-ses-gr" ], "ref_ids": [ 12075, 12083 ] } ], "ref_ids": [] }, { "id": 12085, "type": "theorem", "label": "homology-lemma-filtered-complex", "categories": [ "homology" ], "title": "homology-lemma-filtered-complex", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $A \\to B \\to C$ be a complex", "of filtered objects of $\\mathcal{A}$. Assume $\\alpha : A \\to B$ and", "$\\beta : B \\to C$ are strict morphisms of filtered objects. Then", "$\\text{gr}(\\Ker(\\beta)/\\Im(\\alpha)) =", "\\Ker(\\text{gr}(\\beta))/\\Im(\\text{gr}(\\alpha)))$." ], "refs": [], "proofs": [ { "contents": [ "This follows formally from", "Lemma \\ref{lemma-ses-gr}", "and the fact that", "$\\Coim(\\alpha) \\cong \\Im(\\alpha)$ and", "$\\Coim(\\beta) \\cong \\Im(\\beta)$ by", "Lemma \\ref{lemma-characterize-strict-general}." ], "refs": [ "homology-lemma-ses-gr", "homology-lemma-characterize-strict-general" ], "ref_ids": [ 12083, 12075 ] } ], "ref_ids": [] }, { "id": 12086, "type": "theorem", "label": "homology-lemma-filtered-acyclic", "categories": [ "homology" ], "title": "homology-lemma-filtered-acyclic", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A \\to B \\to C$ be a complex of filtered objects of $\\mathcal{A}$.", "Assume $A, B, C$ have finite filtrations and that", "$\\text{gr}(A) \\to \\text{gr}(B) \\to \\text{gr}(C)$ is exact.", "Then", "\\begin{enumerate}", "\\item for each $p \\in \\mathbf{Z}$ the sequence", "$\\text{gr}^p(A) \\to \\text{gr}^p(B) \\to \\text{gr}^p(C)$ is exact,", "\\item for each $p \\in \\mathbf{Z}$ the sequence", "$F^p(A) \\to F^p(B) \\to F^p(C)$ is exact,", "\\item for each $p \\in \\mathbf{Z}$ the sequence", "$A/F^p(A) \\to B/F^p(B) \\to C/F^p(C)$ is exact,", "\\item the maps $A \\to B$ and $B \\to C$ are strict, and", "\\item $A \\to B \\to C$ is exact (as a sequence in $\\mathcal{A}$).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is immediate from the definitions.", "We will prove (3) by induction on the length of the filtrations.", "If each of $A$, $B$, $C$ has only one", "nonzero graded part, then (3) holds as $\\text{gr}(A) = A$, etc.", "Let $n$ be the largest integer such that at least one of", "$F^nA, F^nB, F^nC$ is nonzero. Set $A' = A/F^nA$, $B' = B/F^nB$,", "$C' = C/F^nC$ with induced filtrations. Note that", "$\\text{gr}(A) = F^nA \\oplus \\text{gr}(A')$", "and similarly for $B$ and $C$. The induction hypothesis", "applies to $A' \\to B' \\to C'$, which implies that", "$A/F^p(A) \\to B/F^p(B) \\to C/F^p(C)$ is exact for $p \\geq n$.", "To conclude the same for $p = n + 1$, i.e., to prove that $A \\to B \\to C$", "is exact we use the commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] & F^nA \\ar[r] \\ar[d] & A \\ar[r] \\ar[d] & A' \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] & F^nB \\ar[r] \\ar[d] & B \\ar[r] \\ar[d] & B' \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] & F^nC \\ar[r] & C \\ar[r] & C' \\ar[r] & 0", "}", "$$", "whose rows are short exact sequences of objects of $\\mathcal{A}$.", "The proof of (2) is dual. Of course (5) follows from (2).", "\\medskip\\noindent", "To prove (4) denote $f : A \\to B$ and $g : B \\to C$ the given morphisms.", "We know that $f(F^p(A)) = \\Ker(F^p(B) \\to F^p(C))$ by (2) and", "$f(A) = \\Ker(g)$ by (5). Hence", "$f(F^p(A)) = \\Ker(F^p(B) \\to F^p(C)) =", "\\Ker(g) \\cap F^p(B) = f(A) \\cap F^p(B)$ which proves that", "$f$ is strict. The proof that $g$ is strict is dual to this." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12087, "type": "theorem", "label": "homology-lemma-derived-exact-couple", "categories": [ "homology" ], "title": "homology-lemma-derived-exact-couple", "contents": [ "Let $(A, E, \\alpha, f, g)$ be an exact couple in an abelian category", "$\\mathcal{A}$. Set", "\\begin{enumerate}", "\\item $d = g \\circ f : E \\to E$ so that $d \\circ d = 0$,", "\\item $E' = \\Ker(d)/\\Im(d)$,", "\\item $A' = \\Im(\\alpha)$,", "\\item $\\alpha' : A' \\to A'$ induced by $\\alpha$,", "\\item $f' : E' \\to A'$ induced by $f$,", "\\item $g' : A' \\to E'$ induced by ``$g \\circ \\alpha^{-1}$''.", "\\end{enumerate}", "Then we have", "\\begin{enumerate}", "\\item $\\Ker(d) = f^{-1}(\\Ker(g)) = f^{-1}(\\Im(\\alpha))$,", "\\item $\\Im(d) = g(\\Im(f)) = g(\\Ker(\\alpha))$,", "\\item $(A', E', \\alpha', f', g')$ is an exact couple.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12088, "type": "theorem", "label": "homology-lemma-spectral-sequence-associated-exact-couple", "categories": [ "homology" ], "title": "homology-lemma-spectral-sequence-associated-exact-couple", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $(A, E, \\alpha, f, g)$ be an exact couple.", "Let $(E_r, d_r)_{r \\geq 1}$ be the spectral sequence", "associated to the exact couple.", "In this case we have", "$$", "0 = B_1 \\subset \\ldots \\subset", "B_{r + 1} = g(\\Ker(\\alpha^r))", "\\subset \\ldots \\subset", "Z_{r + 1} = f^{-1}(\\Im(\\alpha^r))", "\\subset \\ldots \\subset Z_1 = E", "$$", "and the map $d_{r + 1} : E_{r + 1} \\to E_{r + 1}$", "is described by the following rule:", "For any (test) object $T$ of $\\mathcal{A}$ and any elements", "$x : T \\to Z_{r + 1}$ and $y : T \\to A$ such that", "$f \\circ x = \\alpha^r \\circ y$ we have", "$$", "d_{r + 1} \\circ \\overline{x} = \\overline{g \\circ y}", "$$", "where $\\overline{x} : T \\to E_{r + 1}$ is the", "induced morphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12089, "type": "theorem", "label": "homology-lemma-differential-objects-abelian", "categories": [ "homology" ], "title": "homology-lemma-differential-objects-abelian", "contents": [ "\\begin{slogan}", "The category of differential objects of an abelian category is itself", "an abelian category.", "\\end{slogan}", "Let $\\mathcal{A}$ be an abelian category.", "The category of differential objects of $\\mathcal{A}$ is abelian." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12090, "type": "theorem", "label": "homology-lemma-differential-objects-ses", "categories": [ "homology" ], "title": "homology-lemma-differential-objects-ses", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $0 \\to (A, d) \\to (B, d) \\to (C, d) \\to 0$ be a short exact sequence", "of differential objects. Then we get an exact homology sequence", "$$", "\\ldots \\to H(C, d) \\to H(A, d) \\to H(B, d) \\to H(C, d) \\to \\ldots", "$$" ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-long-exact-sequence-cochain}", "to the short exact sequence of complexes", "$$", "\\begin{matrix}", "0 & \\to & A & \\to & B & \\to & C & \\to & 0 \\\\", "& & \\downarrow & & \\downarrow & & \\downarrow \\\\", "0 & \\to & A & \\to & B & \\to & C & \\to & 0 \\\\", "& & \\downarrow & & \\downarrow & & \\downarrow \\\\", "0 & \\to & A & \\to & B & \\to & C & \\to & 0", "\\end{matrix}", "$$", "where the vertical arrows are $d$." ], "refs": [ "homology-lemma-long-exact-sequence-cochain" ], "ref_ids": [ 12061 ] } ], "ref_ids": [] }, { "id": 12091, "type": "theorem", "label": "homology-lemma-spectral-sequence-filtered-differential", "categories": [ "homology" ], "title": "homology-lemma-spectral-sequence-filtered-differential", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a", "filtered differential object of $\\mathcal{A}$. There is a", "spectral sequence $(E_r, d_r)_{r \\geq 0}$ in $\\text{Gr}(\\mathcal{A})$", "associated to $(K, F, d)$ such that $d_r : E_r \\to E_r[r]$", "for all $r$ and such that the graded pieces", "$E_r^p$ and maps $d_r^p : E_r^p \\to E_r^{p + r}$", "are as given above. Furthermore, $E_0^p = \\text{gr}^p K$,", "$d_0^p = \\text{gr}^p(d)$, and $E_1^p = H(\\text{gr}^pK, d)$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{A}$ has countable direct sums and if countable direct", "sums are exact, then this follows from the discussion above.", "In general, we proceed as follows; we strongly suggest the reader", "skip this proof. Consider the object $A = (F^{p + 1}K)$ of", "$\\text{Gr}(\\mathcal{A})$, i.e., we put $F^{p + 1}K$ in degree $p$", "(the funny shift in numbering to get numbering correct later on).", "We endow it with a differential $d$ by using $d$ on each component.", "Then $(A, d)$ is a differential object of $\\text{Gr}(\\mathcal{A})$.", "Consider the map", "$$", "\\alpha : A \\to A[-1]", "$$", "which is given in degree $p$ by the inclusions $F^{p + 1}A \\to F^pA$.", "This is clearly an injective morphism of differential objects", "$\\alpha : (A, d) \\to (A, d)[-1]$. Hence, we can apply", "Remark \\ref{remark-differential-object-selfmap}", "with $S = \\text{id}$ and $T = [1]$.", "The corresponding spectral sequence $(E_r, d_r)_{r \\geq 0}$", "in $\\text{Gr}(\\mathcal{A})$ is the spectral sequence we are looking", "for. Let us unwind the definitions a bit.", "First of all we have $E_r = (E_r^p)$ is an object of $\\text{Gr}(\\mathcal{A})$.", "Then, since $T^rS = [r]$ we have $d_r : E_r \\to E_r[r]$ which means that", "$d_r^p : E_r^p \\to E_r^{p + r}$.", "\\medskip\\noindent", "To see that the description of the graded pieces hold, we argue", "as above. Namely, first we have $E_0 = \\Coker(\\alpha : A \\to A[-1])$", "and by our choice of numbering above this gives", "$E_0^p = \\text{gr}^pK$. The first differential is given by", "$d_0^p = \\text{gr}^pd : E_0^p \\to E_0^p$.", "Next, the description of the boundaries $B_r$ and the cocycles $Z_r$", "in Remark \\ref{remark-differential-object-selfmap}", "translates into a straightforward manner into the formulae", "for $Z_r^p$ and $B_r^p$ given above." ], "refs": [ "homology-remark-differential-object-selfmap", "homology-remark-differential-object-selfmap" ], "ref_ids": [ 12197, 12197 ] } ], "ref_ids": [] }, { "id": 12092, "type": "theorem", "label": "homology-lemma-spectral-sequence-filtered-differential-d1", "categories": [ "homology" ], "title": "homology-lemma-spectral-sequence-filtered-differential-d1", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered", "differential object of $\\mathcal{A}$. The spectral sequence", "$(E_r, d_r)_{r \\geq 0}$ associated to $(K, F, d)$ has", "$$", "d_1^p :", "E_1^p = H(\\text{gr}^pK)", "\\longrightarrow", "H(\\text{gr}^{p + 1}K) = E_1^{p + 1}", "$$", "equal to the boundary map in homology associated to the short", "exact sequence of differential objects", "$$", "0 \\to \\text{gr}^{p + 1}K \\to F^pK/F^{p + 2}K \\to \\text{gr}^pK \\to 0.", "$$" ], "refs": [], "proofs": [ { "contents": [ "This is clear from the formula for the differential $d_1^p$", "given just above Lemma \\ref{lemma-spectral-sequence-filtered-differential}." ], "refs": [ "homology-lemma-spectral-sequence-filtered-differential" ], "ref_ids": [ 12091 ] } ], "ref_ids": [] }, { "id": 12093, "type": "theorem", "label": "homology-lemma-compute-filtered-cohomology", "categories": [ "homology" ], "title": "homology-lemma-compute-filtered-cohomology", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered", "differential object of $\\mathcal{A}$. If $Z_\\infty^p$ and $B_\\infty^p$", "exist (see proof), then", "\\begin{enumerate}", "\\item the limit $E_\\infty$ exists and is graded having", "$E_\\infty^p = Z_\\infty^p/B_\\infty^p$ in degree $p$, and", "\\item the associated graded $\\text{gr}(H(K))$ of the cohomology of $K$", "is a graded subquotient of the graded limit object $E_\\infty$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The objects $Z_\\infty$, $B_\\infty$, and the limit", "$E_\\infty = Z_\\infty/B_\\infty$ of", "Definition \\ref{definition-limit-spectral-sequence}", "are objects of $\\text{Gr}(\\mathcal{A})$ by our construction of", "the spectral sequence in the proof of", "Lemma \\ref{lemma-spectral-sequence-filtered-differential}.", "Since $Z_r = \\bigoplus Z_r^p$ and $B_r = \\bigoplus B_r^p$, if we assume that", "$$", "Z_\\infty^p = \\bigcap\\nolimits_r Z_r^p =", "\\frac{\\bigcap_r (F^pK \\cap d^{-1}(F^{p + r}K) + F^{p + 1}K)}{F^{p + 1}K}", "$$", "and", "$$", "B_\\infty^p = \\bigcup\\nolimits_r B_r^p =", "\\frac{\\bigcup_r (F^pK \\cap d(F^{p - r + 1}K) + F^{p + 1}K)}{F^{p + 1}K}.", "$$", "exist, then $Z_\\infty$ and $B_\\infty$ exist with degree $p$ parts", "$Z_\\infty^p$ and $B_\\infty^p$ (follows from an elementary argument", "about unions and intersections of graded subobjects). Thus", "$$", "E_\\infty^p =", "\\frac{\\bigcap_r (F^pK \\cap d^{-1}(F^{p + r}K) + F^{p + 1}K)}", "{\\bigcup_r (F^pK \\cap d(F^{p - r + 1}K) + F^{p + 1}K)}.", "$$", "where the top and bottom exist. We have", "\\begin{equation}", "\\label{equation-on-top}", "\\Ker(d) \\cap F^pK + F^{p + 1}K", "\\subset", "\\bigcap\\nolimits_r \\left(F^pK \\cap d^{-1}(F^{p + r}K) + F^{p + 1}K\\right)", "\\end{equation}", "and", "\\begin{equation}", "\\label{equation-at-bottom}", "\\bigcup\\nolimits_r \\left(F^pK \\cap d(F^{p - r + 1}K) + F^{p + 1}K\\right)", "\\subset", "\\Im(d) \\cap F^pK + F^{p + 1}K.", "\\end{equation}", "Thus a subquotient of $E_\\infty^p$ is", "$$", "\\frac{\\Ker(d) \\cap F^pK + F^{p + 1}K}{\\Im(d) \\cap F^pK + F^{p + 1}K} =", "\\frac{\\Ker(d) \\cap F^pK}{\\Im(d) \\cap F^pK + \\Ker(d) \\cap F^{p + 1}K}", "$$", "Comparing with the formula given for $\\text{gr}^pH(K)$ in the discussion", "following", "Definition \\ref{definition-filtration-cohomology-filtered-differential}", "we conclude." ], "refs": [ "homology-definition-limit-spectral-sequence", "homology-lemma-spectral-sequence-filtered-differential", "homology-definition-filtration-cohomology-filtered-differential" ], "ref_ids": [ 12168, 12091, 12175 ] } ], "ref_ids": [] }, { "id": 12094, "type": "theorem", "label": "homology-lemma-filtered-differential-ss-converges", "categories": [ "homology" ], "title": "homology-lemma-filtered-differential-ss-converges", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $(K, F, d)$ be a filtered differential object of $\\mathcal{A}$.", "The associated spectral sequence", "\\begin{enumerate}", "\\item weakly converges to $H(K)$ if and only if for every", "$p \\in \\mathbf{Z}$ we have equality in equations", "(\\ref{equation-at-bottom}) and (\\ref{equation-on-top}),", "\\item abuts to $H(K)$ if and only if it weakly converges to $H(K)$ and", "$\\bigcap_p (\\Ker(d) \\cap F^pK + \\Im(d)) = \\Im(d)$", "and $\\bigcup_p (\\Ker(d) \\cap F^pK + \\Im(d)) = \\Ker(d)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Immediate from the discussions above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12095, "type": "theorem", "label": "homology-lemma-spectral-sequence-filtered-complex", "categories": [ "homology" ], "title": "homology-lemma-spectral-sequence-filtered-complex", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a", "filtered complex of $\\mathcal{A}$. There is a spectral sequence", "$(E_r, d_r)_{r \\geq 0}$ in the category of bigraded objects of $\\mathcal{A}$", "associated to $(K^\\bullet, F)$ such that $d_r$ has bidegree $(r, - r + 1)$", "and such that $E_r$ has bigraded pieces $E_r^{p, q}$ and maps", "$d_r^{p, q} : E_r^{p, q} \\to E_r^{p + r, q - r + 1}$ as given above.", "Furthermore, we have $E_0^{p, q} = \\text{gr}^p(K^{p + q})$,", "$d_0^{p, q} = \\text{gr}^p(d^{p + q})$,", "and $E_1^{p, q} = H^{p + q}(\\text{gr}^p(K^\\bullet))$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{A}$ has countable direct sums and if countable direct", "sums are exact, then this follows from the discussion above.", "In general, we proceed as follows; we strongly suggest the reader", "skip this proof. Consider the bigraded object $A = (F^{p + 1}K^{p + 1 + q})$", "of $\\mathcal{A}$, i.e., we put $F^{p + 1}K^{p + 1 + q}$ in degree $(p, q)$", "(the funny shift in numbering to get numbering correct later on).", "We endow it with a differential $d : A \\to A[0, 1]$ by using $d$", "on each component. Then $(A, d)$ is a differential bigraded object.", "Consider the map", "$$", "\\alpha : A \\to A[-1, 1]", "$$", "which is given in degree $(p, q)$ by the inclusion", "$F^{p + 1}K^{p + 1 + q} \\to F^pK^{p + 1 + q}$.", "This is an injective morphism of differential objects", "$\\alpha : (A, d) \\to (A, d)[-1, 1]$. Hence, we can apply", "Remark \\ref{remark-differential-object-selfmap}", "with $S = [0, 1]$ and $T = [1, -1]$.", "The corresponding spectral sequence $(E_r, d_r)_{r \\geq 0}$", "of bigraded objects is the spectral sequence we are looking", "for. Let us unwind the definitions a bit. First of all we have", "$E_r = (E_r^{p, q})$. Then, since $T^rS = [r, -r + 1]$", "we have $d_r : E_r \\to E_r[r, -r + 1]$ which means that", "$d_r^{p, q} : E_r^{p, q} \\to E_r^{p + r, q - r + 1}$.", "\\medskip\\noindent", "To see that the description of the graded pieces hold, we argue", "as above. Namely, first we have", "$$", "E_0 = \\Coker(\\alpha : A \\to A[-1, 1])[0, -1] =", "\\Coker(\\alpha[0, -1] : A[0, -1] \\to A[-1, 0])", "$$", "and by our choice of numbering above this gives", "$$", "E_0^{p, q} = \\Coker(F^{p + 1}K^{p + q} \\to F^pK^{p + q}) = \\text{gr}^pK^{p + q}", "$$", "The first differential is given by", "$d_0^{p, q} = \\text{gr}^pd^{p + q} : E_0^{p, q} \\to E_0^{p, q + 1}$.", "Next, the description of the boundaries $B_r$ and the cocycles $Z_r$", "in Remark \\ref{remark-differential-object-selfmap}", "translates into a straightforward manner into the formulae", "for $Z_r^{p, q}$ and $B_r^{p, q}$ given above." ], "refs": [ "homology-remark-differential-object-selfmap", "homology-remark-differential-object-selfmap" ], "ref_ids": [ 12197, 12197 ] } ], "ref_ids": [] }, { "id": 12096, "type": "theorem", "label": "homology-lemma-spectral-sequence-filtered-complex-d1", "categories": [ "homology" ], "title": "homology-lemma-spectral-sequence-filtered-complex-d1", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $(K^\\bullet, F)$ be a filtered complex of $\\mathcal{A}$.", "Assume $\\mathcal{A}$ has countable direct sums.", "Let $(E_r, d_r)_{r \\geq 0}$ be the spectral sequence", "associated to $(K^\\bullet, F)$.", "\\begin{enumerate}", "\\item The map", "$$", "d_1^{p, q} :", "E_1^{p, q} = H^{p + q}(\\text{gr}^p(K^\\bullet))", "\\longrightarrow", "E_1^{p + 1, q} = H^{p + q + 1}(\\text{gr}^{p + 1}(K^\\bullet))", "$$", "is equal to the boundary map in cohomology associated to the short", "exact sequence of complexes", "$$", "0 \\to \\text{gr}^{p + 1}(K^\\bullet) \\to", "F^pK^\\bullet/F^{p + 2}K^\\bullet \\to \\text{gr}^p(K^\\bullet) \\to 0.", "$$", "\\item Assume that $d(F^pK) \\subset F^{p + 1}K$ for all $p \\in \\mathbf{Z}$.", "Then $d$ induces the zero differential on $\\text{gr}^p(K^\\bullet)$", "and hence", "$E_1^{p, q} = \\text{gr}^p(K^\\bullet)^{p + q}$.", "Furthermore, in this case", "$$", "d_1^{p, q} :", "E_1^{p, q} = \\text{gr}^p(K^\\bullet)^{p + q}", "\\longrightarrow", "E_1^{p + 1, q} = \\text{gr}^{p + 1}(K^\\bullet)^{p + q + 1}", "$$", "is the morphism induced by $d$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is clear from the formula given for the differential", "$d_1^{p, q}$ just above Lemma \\ref{lemma-spectral-sequence-filtered-complex}." ], "refs": [ "homology-lemma-spectral-sequence-filtered-complex" ], "ref_ids": [ 12095 ] } ], "ref_ids": [] }, { "id": 12097, "type": "theorem", "label": "homology-lemma-spectral-sequence-filtered-complex-functorial", "categories": [ "homology" ], "title": "homology-lemma-spectral-sequence-filtered-complex-functorial", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $\\alpha : (K^\\bullet, F) \\to (L^\\bullet, F)$ be a morphism of", "filtered complexes of $\\mathcal{A}$. Let $(E_r(K), d_r)_{r \\geq 0}$,", "resp.\\ $(E_r(L), d_r)_{r \\geq 0}$ be the spectral sequence associated", "to $(K^\\bullet, F)$, resp.\\ $(L^\\bullet, F)$.", "The morphism $\\alpha$ induces a canonical morphism of spectral", "sequences $\\{\\alpha_r : E_r(K) \\to E_r(L)\\}_{r \\geq 0}$ compatible", "with the bigradings." ], "refs": [], "proofs": [ { "contents": [ "Obvious from the explicit representation of the terms of the", "spectral sequences." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12098, "type": "theorem", "label": "homology-lemma-compute-cohomology-filtered-complex", "categories": [ "homology" ], "title": "homology-lemma-compute-cohomology-filtered-complex", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a filtered", "complex of $\\mathcal{A}$. If $Z_\\infty^{p, q}$ and $B_\\infty^{p, q}$ exist", "(see proof), then", "\\begin{enumerate}", "\\item the limit $E_\\infty$ exists and is a bigraded object having", "$E_\\infty^{p, q} = Z_\\infty^{p, q}/B_\\infty^{p, q}$ in bidegree $(p, q)$,", "\\item the $p$th graded part $\\text{gr}^pH^n(K^\\bullet)$ of the", "$n$th cohomology object of $K^\\bullet$ is a subquotient of", "$E_\\infty^{p, n - p}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The objects $Z_\\infty$, $B_\\infty$, and the limit", "$E_\\infty = Z_\\infty/B_\\infty$ of", "Definition \\ref{definition-limit-spectral-sequence}", "are bigraded objects of $\\mathcal{A}$ by our construction of the", "spectral sequence in Lemma \\ref{lemma-spectral-sequence-filtered-complex}.", "Since $Z_r = \\bigoplus Z_r^{p, q}$ and $B_r = \\bigoplus B_r^{p, q}$,", "if we assume that", "$$", "Z_\\infty^{p, q} = \\bigcap\\nolimits_r Z_r^{p, q} =", "\\bigcap\\nolimits_r", "\\frac{F^pK^{p + q} \\cap d^{-1}(F^{p + r}K^{p + q + 1}) + F^{p + 1}K^{p + q}}", "{F^{p + 1}K^{p + q}}", "$$", "and", "$$", "B_\\infty^{p, q} = \\bigcup\\nolimits_r B_r^{p, q} =", "\\bigcup\\nolimits_r", "\\frac{F^pK^{p + q} \\cap d(F^{p - r + 1}K^{p + q - 1}) + F^{p + 1}K^{p + q}}", "{F^{p + 1}K^{p + q}}", "$$", "exist, then $Z_\\infty$ and $B_\\infty$ exist with bidegree $(p, q)$", "parts $Z_\\infty^{p, q}$ and $B_\\infty^{p, q}$ (follows from an elementary", "argument about unions and intersections of bigraded objects). Thus", "$$", "E_\\infty^{p, q} =", "\\frac{\\bigcap_r (F^pK^{p + q} \\cap d^{-1}(F^{p + r}K^{p + q + 1})", "+ F^{p + 1}K^{p + q})}", "{\\bigcup_r (F^pK^{p + q} \\cap d(F^{p - r + 1}K^{p + q - 1})", "+ F^{p + 1}K^{p + q})}.", "$$", "where the top and the bottom exist. With $n = p + q$ we have", "\\begin{equation}", "\\label{equation-on-top-bigraded}", "\\Ker(d) \\cap F^pK^{n} + F^{p + 1}K^{n}", "\\subset", "\\bigcap\\nolimits_r", "\\left(", "F^pK^{n} \\cap d^{-1}(F^{p + r}K^{n + 1}) + F^{p + 1}K^{n}", "\\right)", "\\end{equation}", "and", "\\begin{equation}", "\\label{equation-at-bottom-bigraded}", "\\bigcup\\nolimits_r", "\\left(", "F^pK^{n} \\cap d(F^{p - r + 1}K^{n - 1}) + F^{p + 1}K^{n}", "\\right)", "\\subset", "\\Im(d) \\cap F^pK^{n} + F^{p + 1}K^{n}.", "\\end{equation}", "Thus a subquotient of $E_\\infty^{p, q}$ is", "$$", "\\frac{\\Ker(d) \\cap F^pK^{n} + F^{p + 1}K^n}", "{\\Im(d) \\cap F^pK^{n} + F^{p + 1}K^{n}} =", "\\frac{\\Ker(d) \\cap F^pK^n}{\\Im(d) \\cap F^pK^n + \\Ker(d) \\cap F^{p + 1}K^n}", "$$", "Comparing with (\\ref{equation-graded-cohomology}) we conclude." ], "refs": [ "homology-definition-limit-spectral-sequence", "homology-lemma-spectral-sequence-filtered-complex" ], "ref_ids": [ 12168, 12095 ] } ], "ref_ids": [] }, { "id": 12099, "type": "theorem", "label": "homology-lemma-relate-boundedness", "categories": [ "homology" ], "title": "homology-lemma-relate-boundedness", "contents": [ "In the situation of Definition \\ref{definition-bounded-ss}.", "Let $Z_r^{p, q}, B_r^{p, q} \\subset E_{r_0}^{p, q}$ be the", "$(p, q)$-graded parts of $Z_r, B_r$ defined as in", "Section \\ref{section-spectral-sequence}.", "\\begin{enumerate}", "\\item The spectral sequence is regular if and only if for all $p, q$", "there exists an $r = r(p, q)$ such that", "$Z_r^{p, q} = Z_{r + 1}^{p, q} = \\ldots$", "\\item The spectral sequence is coregular if and only if for all $p, q$", "there exists an $r = r(p, q)$ such that", "$B_r^{p, q} = B_{r + 1}^{p, q} = \\ldots$", "\\item The spectral sequence is bounded if and only if it is both", "bounded below and bounded above.", "\\item If the spectral sequence is bounded below, then it is regular.", "\\item If the spectral sequence is bounded above, then it is coregular.", "\\end{enumerate}" ], "refs": [ "homology-definition-bounded-ss" ], "proofs": [ { "contents": [ "Omitted. Hint: If $E_r^{p, q} = 0$, then we have $E_{r'}^{p, q} = 0$", "for all $r' \\geq r$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12179 ] }, { "id": 12100, "type": "theorem", "label": "homology-lemma-filtered-complex-ss-converges", "categories": [ "homology" ], "title": "homology-lemma-filtered-complex-ss-converges", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a filtered", "complex of $\\mathcal{A}$. The associated spectral sequence", "\\begin{enumerate}", "\\item weakly converges to $H^*(K^\\bullet)$ if and only if for every", "$p, q \\in \\mathbf{Z}$ we have equality in equations", "(\\ref{equation-at-bottom-bigraded}) and (\\ref{equation-on-top-bigraded}),", "\\item abuts to $H^*(K)$ if and only if it weakly converges to $H^*(K^\\bullet)$", "and we have", "$\\bigcap_p (\\Ker(d) \\cap F^pK^n + \\Im(d) \\cap K^n) = \\Im(d) \\cap K^n$", "and", "$\\bigcup_p (\\Ker(d) \\cap F^pK^n + \\Im(d) \\cap K^n) = \\Ker(d) \\cap K^n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Immediate from the discussions above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12101, "type": "theorem", "label": "homology-lemma-biregular-ss-converges", "categories": [ "homology" ], "title": "homology-lemma-biregular-ss-converges", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a", "filtered complex of $\\mathcal{A}$. Assume that the filtration on each $K^n$", "is finite (see Definition \\ref{definition-filtered}). Then", "\\begin{enumerate}", "\\item the spectral sequence associated to $(K^\\bullet, F)$ is bounded,", "\\item the filtration on each $H^n(K^\\bullet)$ is finite,", "\\item the spectral sequence associated to $(K^\\bullet, F)$ converges", "to $H^*(K^\\bullet)$,", "\\item if $\\mathcal{C} \\subset \\mathcal{A}$ is a weak Serre subcategory", "and for some $r$ we have $E_r^{p, q} \\in \\mathcal{C}$ for all", "$p, q \\in \\mathbf{Z}$, then $H^n(K^\\bullet)$ is in $\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [ "homology-definition-filtered" ], "proofs": [ { "contents": [ "Part (1) follows as $E_0^{p, n - p} = \\text{gr}^p K^n$.", "Part (2) is clear from Equation (\\ref{equation-filtration-cohomology}).", "We will use Lemma \\ref{lemma-filtered-complex-ss-converges} to prove", "that the spectral sequence weakly converges. Fix $p, n \\in \\mathbf{Z}$.", "Looking at the right hand side of (\\ref{equation-on-top-bigraded})", "we see that we get $F^pK^n \\cap \\Ker(d) + F^{p + 1}K^n$ because", "$F^{p + r}K^n = 0$ for $r \\gg 0$. Thus (\\ref{equation-on-top-bigraded})", "is an equality. Look at the left hand side of (\\ref{equation-on-top-bigraded}).", "The expression is equal to the right hand side since", "$F^{p - r + 1}K^{n - 1} = K^{n - 1}$ for $r \\gg 0$.", "Thus (\\ref{equation-on-top-bigraded}) is an equality. Since the filtration", "on $H^n(K^\\bullet)$ is finite by (2) we see that we have abutment.", "To prove we have convergence we have to show the spectral sequence is", "regular which follows as it is bounded", "(Lemma \\ref{lemma-relate-boundedness}) and we have", "to show that $H^n(K^\\bullet) = \\lim_p H^n(K^\\bullet)/F^pH^n(K^\\bullet)$", "which follows from the fact that the filtration on $H^*(K^\\bullet)$", "is finite proved in part (2).", "\\medskip\\noindent", "Proof of (4). Assume that for some $r \\geq 0$ we have", "$E_r^{p, q} \\in \\mathcal{C}$ for some weak Serre subcategory", "$\\mathcal{C}$ of $\\mathcal{A}$. Then $E_{r + 1}^{p, q}$ is", "in $\\mathcal{C}$ as well, see", "Lemma \\ref{lemma-characterize-weak-serre-subcategory}.", "By boundedness proved above (which implies that the spectral sequence", "is both regular and coregular, see Lemma \\ref{lemma-relate-boundedness})", "we can find an $r' \\geq r$ such that $E_\\infty^{p, q} = E_{r'}^{p, q}$", "for all $p, q$ with $p + q = n$. Thus $H^n(K^\\bullet)$ is an object", "of $\\mathcal{A}$ which has a finite filtration whose graded pieces", "are in $\\mathcal{C}$. This implies that $H^n(K^\\bullet)$ is in $\\mathcal{C}$", "by Lemma \\ref{lemma-characterize-weak-serre-subcategory}." ], "refs": [ "homology-lemma-filtered-complex-ss-converges", "homology-lemma-relate-boundedness", "homology-lemma-characterize-weak-serre-subcategory", "homology-lemma-relate-boundedness", "homology-lemma-characterize-weak-serre-subcategory" ], "ref_ids": [ 12100, 12099, 12046, 12099, 12046 ] } ], "ref_ids": [ 12165 ] }, { "id": 12102, "type": "theorem", "label": "homology-lemma-biregular-ss-relation-in-K0", "categories": [ "homology" ], "title": "homology-lemma-biregular-ss-relation-in-K0", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a", "filtered complex of $\\mathcal{A}$. Assume that the filtration on each $K^n$", "is finite (see Definition \\ref{definition-filtered}) and that for some", "$r$ we have only a finite number of nonzero $E_r^{p, q}$. Then", "only a finite number of $H^n(K^\\bullet)$ are nonzero and we have", "$$", "\\sum (-1)^n[H^n(K^\\bullet)] = \\sum (-1)^{p + q} [E_r^{p, q}]", "$$", "in $K_0(\\mathcal{A}')$ where $\\mathcal{A}'$ is the smallest weak", "Serre subcategory of $\\mathcal{A}$ containing the objects", "$E_r^{p, q}$." ], "refs": [ "homology-definition-filtered" ], "proofs": [ { "contents": [ "Denote $E_r^{even}$ and $E_r^{odd}$ the even and odd part of $E_r$", "defined as the direct sum of the $(p, q)$ components with $p + q$ even", "and odd. The differential $d_r$ defines maps", "$\\varphi : E_r^{even} \\to E_r^{odd}$ and $\\psi : E_r^{odd} \\to E_r^{even}$", "whose compositions either way give zero.", "Then we see that", "\\begin{align*}", "[E_r^{even}] - [E_r^{odd}] & =", "[\\Ker(\\varphi)] + [\\Im(\\varphi)] - [\\Ker(\\psi)] - [\\Im(\\psi)] \\\\", "& =", "[\\Ker(\\varphi)/\\Im(\\psi)] - [\\Ker(\\psi)/\\Im(\\varphi)] \\\\", "& =", "[E_{r + 1}^{even}] - [E_{r + 1}^{odd}]", "\\end{align*}", "Note that all the intervening objects are in the smallest Serre", "subcategory containing the objects $E_r^{p, q}$.", "Continuing in this manner we see that we can increase $r$ at will.", "Since there are only a finite number of pairs $(p, q)$ for which", "$E_r^{p, q}$ is nonzero, a property which is inherited by", "$E_{r + 1}, E_{r + 2}, \\ldots$, we see that we may assume that $d_r = 0$.", "At this stage we see that $H^n(K^\\bullet)$ has a finite filtration", "(Lemma \\ref{lemma-biregular-ss-converges}) whose graded pieces", "are exactly the $E_r^{p, n - p}$ and the result is clear." ], "refs": [ "homology-lemma-biregular-ss-converges" ], "ref_ids": [ 12101 ] } ], "ref_ids": [ 12165 ] }, { "id": 12103, "type": "theorem", "label": "homology-lemma-ss-converges-trivial", "categories": [ "homology" ], "title": "homology-lemma-ss-converges-trivial", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a", "filtered complex of $\\mathcal{A}$. Assume", "\\begin{enumerate}", "\\item for every $n$ there exist $p_0(n)$ such that", "$H^n(F^pK^\\bullet) = 0$ for $p \\geq p_0(n)$,", "\\item for every $n$ there exist $p_1(n)$ such that", "$H^n(F^pK^\\bullet) \\to H^n(K^\\bullet)$ is an isomorphism", "for $p \\leq p_1(n)$.", "\\end{enumerate}", "Then", "\\begin{enumerate}", "\\item the spectral sequence associated to $(K^\\bullet, F)$ is bounded,", "\\item the filtration on each $H^n(K^\\bullet)$ is finite,", "\\item the spectral sequence associated to $(K^\\bullet, F)$ converges", "to $H^*(K^\\bullet)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Fix $n$. Using the long exact cohomology sequence associated to", "the short exact sequence of complexes", "$$", "0 \\to F^{p + 1}K^\\bullet \\to F^pK^\\bullet \\to \\text{gr}^pK^\\bullet \\to 0", "$$", "we find that $E_1^{p, n - p} = 0$ for $p \\geq \\max(p_0(n), p_0(n + 1))$ and", "$p < \\min(p_1(n), p_1(n + 1))$. Hence the spectral sequence is bounded", "(Definition \\ref{definition-bounded-ss}). This proves (1).", "\\medskip\\noindent", "It is clear from the assumptions and", "Definition \\ref{definition-filtration-cohomology-filtered-complex}", "that the filtration on $H^n(K^\\bullet)$ is finite. This proves (2).", "\\medskip\\noindent", "Next we prove that the spectral sequence weakly converges to", "$H^*(K^\\bullet)$ using", "Lemma \\ref{lemma-filtered-complex-ss-converges}.", "Let us show that we have equality in (\\ref{equation-on-top-bigraded}).", "Namely, for $p + r > p_0(n + 1)$ the map", "$$", "d : F^pK^{n} \\cap d^{-1}(F^{p + r}K^{n + 1}) \\to F^{p + r}K^{n + 1}", "$$", "ends up in the image of $d : F^{p + r}K^n \\to F^{p + r}K^{n + 1}$", "because the complex $F^{p + r}K^\\bullet$ is exact in degree $n + 1$.", "We conclude that $F^pK^{n} \\cap d^{-1}(F^{p + r}K^{n + 1}) =", "d(F^{p + r}K^n) + \\Ker(d) \\cap F^pK^n$. Hence for such $r$ we have", "$$", "\\Ker(d) \\cap F^pK^{n} + F^{p + 1}K^{n} =", "F^pK^{n} \\cap d^{-1}(F^{p + r}K^{n + 1}) + F^{p + 1}K^{n}", "$$", "which proves the desired equality. To show that we have equality in", "(\\ref{equation-at-bottom-bigraded}) we use that for $p - r + 1 < p_1(n - 1)$", "we have", "$$", "d(F^{p - r + 1}K^{n - 1}) = \\Im(d) \\cap F^{p - r + 1}K^n", "$$", "because the map $F^{p - r + 1}K^\\bullet \\to K^\\bullet$ induces an", "isomorphism on cohomology in degree $n - 1$. This shows that", "we have", "$$", "F^pK^{n} \\cap d(F^{p - r + 1}K^{n - 1}) + F^{p + 1}K^{n} =", "\\Im(d) \\cap F^pK^{n} + F^{p + 1}K^{n}", "$$", "for such $r$ which proves the desired equality.", "\\medskip\\noindent", "To see that the spectral sequence abuts to $H^*(K^\\bullet)$ using", "Lemma \\ref{lemma-filtered-complex-ss-converges} we have to show that", "$\\bigcap_p (\\Ker(d) \\cap F^pK^n + \\Im(d) \\cap K^n) = \\Im(d) \\cap K^n$", "and", "$\\bigcup_p (\\Ker(d) \\cap F^pK^n + \\Im(d) \\cap K^n) = \\Ker(d) \\cap K^n$.", "For $p \\geq p_0(n)$ we have", "$\\Ker(d) \\cap F^pK^n + \\Im(d) \\cap K^n = \\Im(d) \\cap K^n$", "and for $p \\leq p_1(n)$ we have", "$\\Ker(d) \\cap F^pK^n + \\Im(d) \\cap K^n = \\Ker(d) \\cap K^n$.", "Combining weak convergence, abutment, and boundedness we see", "that (2) and (3) are true." ], "refs": [ "homology-definition-bounded-ss", "homology-definition-filtration-cohomology-filtered-complex", "homology-lemma-filtered-complex-ss-converges", "homology-lemma-filtered-complex-ss-converges" ], "ref_ids": [ 12179, 12178, 12100, 12100 ] } ], "ref_ids": [] }, { "id": 12104, "type": "theorem", "label": "homology-lemma-ss-double-complex", "categories": [ "homology" ], "title": "homology-lemma-ss-double-complex", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $K^{\\bullet, \\bullet}$ be a double complex.", "The spectral sequences associated to $K^{\\bullet, \\bullet}$", "have the following terms:", "\\begin{enumerate}", "\\item ${}'E_0^{p, q} = K^{p, q}$ with", "${}'d_0^{p, q} = (-1)^p d_2^{p, q} : K^{p, q} \\to K^{p, q + 1}$,", "\\item ${}''E_0^{p, q} = K^{q, p}$ with", "${}''d_0^{p, q} = d_1^{q, p} : K^{q, p} \\to K^{q + 1, p}$,", "\\item ${}'E_1^{p, q} = H^q(K^{p, \\bullet})$ with", "${}'d_1^{p, q} = H^q(d_1^{p, \\bullet})$,", "\\item ${}''E_1^{p, q} = H^q(K^{\\bullet, p})$ with", "${}''d_1^{p, q} = (-1)^q H^q(d_2^{\\bullet, p})$,", "\\item ${}'E_2^{p, q} = H^p_I(H^q_{II}(K^{\\bullet, \\bullet}))$,", "\\item ${}''E_2^{p, q} = H^p_{II}(H^q_I(K^{\\bullet, \\bullet}))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12105, "type": "theorem", "label": "homology-lemma-first-quadrant-ss", "categories": [ "homology" ], "title": "homology-lemma-first-quadrant-ss", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $K^{\\bullet, \\bullet}$", "be a double complex. Assume that for every $n \\in \\mathbf{Z}$ there are", "only finitely many nonzero $K^{p, q}$ with $p + q = n$. Then", "\\begin{enumerate}", "\\item the two spectral sequences associated to $K^{\\bullet, \\bullet}$", "are bounded,", "\\item the filtrations $F_I$, $F_{II}$ on each $H^n(K^\\bullet)$ are finite,", "\\item the spectral sequences $({}'E_r, {}'d_r)_{r \\geq 0}$ and", "$({}''E_r, {}''d_r)_{r \\geq 0}$ converge to", "$H^*(\\text{Tot}(K^{\\bullet, \\bullet}))$,", "\\item if $\\mathcal{C} \\subset \\mathcal{A}$ is a weak Serre subcategory", "and for some $r$ we have ${}'E_r^{p, q} \\in \\mathcal{C}$ for all", "$p, q \\in \\mathbf{Z}$, then $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$", "is in $\\mathcal{C}$. Similarly for $({}''E_r, {}''d_r)_{r \\geq 0}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-biregular-ss-converges}." ], "refs": [ "homology-lemma-biregular-ss-converges" ], "ref_ids": [ 12101 ] } ], "ref_ids": [] }, { "id": 12106, "type": "theorem", "label": "homology-lemma-double-complex-gives-resolution", "categories": [ "homology" ], "title": "homology-lemma-double-complex-gives-resolution", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $K^\\bullet$ be a complex.", "Let $A^{\\bullet, \\bullet}$ be a double complex.", "Let $\\alpha^p : K^p \\to A^{p, 0}$ be morphisms.", "Assume that", "\\begin{enumerate}", "\\item For every $n \\in \\mathbf{Z}$ there are only finitely many nonzero", "$A^{p, q}$ with $p + q = n$.", "\\item We have $A^{p, q} = 0$ if $q < 0$.", "\\item The morphisms $\\alpha^p$ give rise to a morphism", "of complexes $\\alpha : K^\\bullet \\to A^{\\bullet, 0}$.", "\\item The complex $A^{p, \\bullet}$ is exact in all degrees", "$q \\not = 0$ and the morphism $K^p \\to A^{p, 0}$ induces", "an isomorphism $K^p \\to \\Ker(d_2^{p, 0})$.", "\\end{enumerate}", "Then $\\alpha$ induces a quasi-isomorphism", "$$", "K^\\bullet \\longrightarrow \\text{Tot}(A^{\\bullet, \\bullet})", "$$", "of complexes.", "Moreover, there is a variant of this lemma involving the second", "variable $q$ instead of $p$." ], "refs": [], "proofs": [ { "contents": [ "The map is simply the map given by the morphisms", "$K^n \\to A^{n, 0} \\to \\text{Tot}^n(A^{\\bullet, \\bullet})$,", "which are easily seen to define a morphism of complexes.", "Consider the spectral sequence $({}'E_r, {}'d_r)_{r \\geq 0}$", "associated to the double complex $A^{\\bullet, \\bullet}$.", "By Lemma \\ref{lemma-first-quadrant-ss} this spectral sequence converges", "and the induced filtration on $H^n(\\text{Tot}(A^{\\bullet, \\bullet}))$", "is finite for each $n$.", "By Lemma \\ref{lemma-ss-double-complex} and assumption (4) we have", "${}'E_1^{p, q} = 0$ unless $q = 0$ and $'E_1^{p, 0} = K^p$", "with differential ${}'d_1^{p, 0}$ identified with $d_K^p$.", "Hence ${}'E_2^{p, 0} = H^p(K^\\bullet)$ and zero otherwise.", "This clearly implies $d_2^{p, q} = d_3^{p, q} = \\ldots = 0$", "for degree reasons. Hence we conclude that", "$H^n(\\text{Tot}(A^{\\bullet, \\bullet})) = H^n(K^\\bullet)$.", "We omit the verification that this identification is given by the", "morphism of complexes $K^\\bullet \\to \\text{Tot}(A^{\\bullet, \\bullet})$", "introduced above." ], "refs": [ "homology-lemma-first-quadrant-ss", "homology-lemma-ss-double-complex" ], "ref_ids": [ 12105, 12104 ] } ], "ref_ids": [] }, { "id": 12107, "type": "theorem", "label": "homology-lemma-homotopy-complex-complexes", "categories": [ "homology" ], "title": "homology-lemma-homotopy-complex-complexes", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $M^\\bullet$ be a complex of $\\mathcal{A}$. Let", "$$", "a :", "M^\\bullet[0]", "\\longrightarrow", "\\left(A^{0, \\bullet} \\to A^{1, \\bullet} \\to A^{2, \\bullet} \\to \\ldots \\right)", "$$", "be a homotopy equivalence in the category of complexes of complexes", "of $\\mathcal{A}$. Then the map", "$\\alpha : M^\\bullet \\to \\text{Tot}(A^{\\bullet, \\bullet})$", "induced by $M^\\bullet \\to A^{0, \\bullet}$ is a homotopy equivalence." ], "refs": [], "proofs": [ { "contents": [ "The statement makes sense as a complex of complexes is the same thing", "as a double complex. The assumption means there is a map", "$$", "b :", "\\left(A^{0, \\bullet} \\to A^{1, \\bullet} \\to A^{2, \\bullet} \\to \\ldots \\right)", "\\longrightarrow", "M^\\bullet[0]", "$$", "such that $a \\circ b$ and $b \\circ a$ are homotopic to the identity", "in the category of complexes of complexes. This means that $b \\circ a$", "is the identity of $M^\\bullet[0]$ (because there is only one term in", "degree $0$). Also, observe that $b$ is given by a map", "$b^0 : A^{0, \\bullet} \\to M^\\bullet$ and zero in all other degrees.", "Thus $b$ induces a map", "$\\beta : \\text{Tot}(A^{\\bullet, \\bullet}) \\to M^\\bullet$", "and $\\beta \\circ \\alpha$ is the identity on $M^\\bullet$.", "Finally, we have to show that the map", "$\\alpha \\circ \\beta$ is homotopic to the identity.", "For this we choose maps of complexes", "$h^n : A^{n, \\bullet} \\to A^{n - 1, \\bullet}$", "such that $a \\circ b - \\text{id} = d_1 \\circ h + h \\circ d_1$", "which exist by assumption. Here $d_1 : A^{n, \\bullet} \\to A^{n + 1, \\bullet}$", "are the differentials of the complex of complexes. We will also", "denote $d_2$ the differentials of the complexes $A^{n, \\bullet}$", "for all $n$. Let $h^{n, m} : A^{n, m} \\to A^{n - 1, m}$ be the components of", "$h^n$. Then we can consider", "$$", "h' : \\text{Tot}(A^{\\bullet, \\bullet})^k =", "\\bigoplus\\nolimits_{n + m = k} A^{n, m}", "\\to", "\\bigoplus\\nolimits_{n + m = k - 1} A^{n, m} =", "\\text{Tot}(A^{\\bullet, \\bullet})^{k - 1}", "$$", "given by $h^{n, m}$ on the summand $A^{n, m}$. Then we compute", "that the map", "$$", "d_{\\text{Tot}(A^{\\bullet, \\bullet})} \\circ h' +", "h' \\circ d_{\\text{Tot}(A^{\\bullet, \\bullet})}", "$$", "restricted to the summand $A^{n, m}$ is equal to", "$$", "d_1^{n - 1, m} \\circ h^{n, m} +", "(-1)^{n - 1} d_2^{n - 1, m} \\circ h^{n, m} +", "h^{n + 1, m} \\circ d_1^{n, m} + h^{n, m + 1} \\circ (-1)^nd_2^{n, m}", "$$", "Since $h^n$ is a map of complexes, the terms", "$(-1)^{n - 1} d_2^{n - 1, m} \\circ h^{n, m}$ and", "$h^{n, m + 1} \\circ (-1)^nd_2^{n, m}$ cancel.", "The other two terms give", "$(\\alpha \\circ \\beta)|_{A^{n, m}} - \\text{id}_{A^{n, m}}$", "because $a \\circ b - \\text{id} = d_1 \\circ h + h \\circ d_1$.", "This finishes the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12108, "type": "theorem", "label": "homology-lemma-right-resolution-gives-qis", "categories": [ "homology" ], "title": "homology-lemma-right-resolution-gives-qis", "contents": [ "Let $M^\\bullet$ be a complex of abelian groups. Let", "$$", "0 \\to M^\\bullet \\to A_0^\\bullet \\to A_1^\\bullet \\to A_2^\\bullet \\to \\ldots", "$$", "be an exact complex of complexes of abelian groups. Set", "$A^{p, q} = A_p^q$ to obtain a double complex.", "Then the map $M^\\bullet \\to \\text{Tot}(A^{\\bullet, \\bullet})$", "induced by $M^\\bullet \\to A_0^\\bullet$ is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "If there exists a $t \\in \\mathbf{Z}$ such that $A_0^q = 0$ for $q < t$, then", "this follows immediately from", "Lemma \\ref{lemma-double-complex-gives-resolution}", "(with $p$ and $q$ swapped as in the final statement of that lemma).", "OK, but for every $t \\in \\mathbf{Z}$ we have a complex", "$$", "0 \\to", "\\sigma_{\\geq t}M^\\bullet \\to", "\\sigma_{\\geq t}A_0^\\bullet \\to", "\\sigma_{\\geq t}A_1^\\bullet \\to", "\\sigma_{\\geq t}A_2^\\bullet \\to \\ldots", "$$", "of stupid truncations. Denote $A(t)^{\\bullet, \\bullet}$ the corresponding", "double complex. Every element $\\xi$ of $H^n(\\text{Tot}(A^{\\bullet, \\bullet}))$", "is the image of an element of $H^n(\\text{Tot}(A(t)^{\\bullet, \\bullet}))$", "for some $t$ (look at explicit representatives of cohomology classes).", "Hence $\\xi$ is in the image of $H^n(\\sigma_{\\geq t}M^\\bullet)$.", "Thus the map $H^n(M^\\bullet) \\to H^n(\\text{Tot}(A^{\\bullet, \\bullet}))$", "is surjective. It is injective because for all $t$ the map", "$H^n(\\sigma_{\\geq t}M^\\bullet) \\to H^n(\\text{Tot}(A(t)^{\\bullet, \\bullet}))$", "is injective and similar arguments." ], "refs": [ "homology-lemma-double-complex-gives-resolution" ], "ref_ids": [ 12106 ] } ], "ref_ids": [] }, { "id": 12109, "type": "theorem", "label": "homology-lemma-good-resolution-gives-qis", "categories": [ "homology" ], "title": "homology-lemma-good-resolution-gives-qis", "contents": [ "Let $M^\\bullet$ be a complex of abelian groups. Let", "$$", "\\ldots \\to A_2^\\bullet \\to A_1^\\bullet \\to A_0^\\bullet \\to M^\\bullet \\to 0", "$$", "be an exact complex of complexes of abelian groups such that for all", "$p \\in \\mathbf{Z}$ the complexes", "$$", "\\ldots \\to \\Ker(d_{A_2^\\bullet}^p) \\to \\Ker(d_{A_1^\\bullet}^p)", "\\to \\Ker(d_{A_0^\\bullet}^p) \\to \\Ker(d_{M^\\bullet}^p) \\to 0", "$$", "are exact as well. Set $A^{p, q} = A_{-p}^q$ to obtain a double", "complex. Then $\\text{Tot}(A^{\\bullet, \\bullet}) \\to M^\\bullet$", "induced by $A_0^\\bullet \\to M^\\bullet$ is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Using the short exact sequences", "$0 \\to \\Ker(d^p_{A_n^\\bullet}) \\to A_n^p \\to \\Im(d^p_{A_n^\\bullet}) \\to 0$", "and the assumptions we see that", "$$", "\\ldots \\to \\Im(d_{A_2^\\bullet}^p) \\to \\Im(d_{A_1^\\bullet}^p)", "\\to \\Im(d_{A_0^\\bullet}^p) \\to \\Im(d_{M^\\bullet}^p) \\to 0", "$$", "is exact for all $p \\in \\mathbf{Z}$. Repeating with the exact sequences", "$0 \\to \\Im(d^{p - 1}_{A_n^\\bullet}) \\to \\Ker(d^p_{A_n^\\bullet})", "\\to H^p(A_n^\\bullet) \\to 0$ we find that", "$$", "\\ldots \\to H^p(A_2^\\bullet) \\to H^p(A_1^\\bullet)", "\\to H^p(A_0^\\bullet) \\to H^p(M^\\bullet) \\to 0", "$$", "is exact for all $p \\in \\mathbf{Z}$.", "\\medskip\\noindent", "Write $T^\\bullet = \\text{Tot}(A^{\\bullet, \\bullet})$. We will show that", "$H^0(T^\\bullet) \\to H^0(M^\\bullet)$ is an isomorphism. The same argument", "works for other degrees. Let $x \\in \\Ker(\\text{d}_{T^\\bullet}^0)$ represent", "an element $\\xi \\in H^0(T^\\bullet)$.", "Write $x = \\sum_{i = n, \\ldots, 0} x_i$ with $x_i \\in A_i^i$.", "Assume $n > 0$. Then $x_n$ is in the kernel of $d_{A_n^\\bullet}^n$", "and maps to zero in $H^n(A_{n - 1}^\\bullet)$ because it maps", "to an element which is the boundary of $x_{n - 1}$ up to sign.", "By the first paragraph of the proof, we find that", "$x_n \\bmod \\Im(d^{n - 1}_{A_n^\\bullet})$", "is in the image of $H^n(A_{n + 1}^\\bullet) \\to H^n(A_n^\\bullet)$.", "Thus we can modify $x$ by a boundary and reach the situation", "where $x_n$ is a boundary. Modifying $x$ once more we see that", "we may assume $x_n = 0$. By induction we see that every cohomology", "class $\\xi$ is represented by a cocycle $x = x_0$.", "Finally, the condition on exactness of kernels tells us", "two such cocycles $x_0$ and $x_0'$ are cohomologous if", "and only if their image in $H^0(M^\\bullet)$ are the same." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12110, "type": "theorem", "label": "homology-lemma-good-right-resolution-gives-qis", "categories": [ "homology" ], "title": "homology-lemma-good-right-resolution-gives-qis", "contents": [ "Let $M^\\bullet$ be a complex of abelian groups. Let", "$$", "0 \\to M^\\bullet \\to A_0^\\bullet \\to A_1^\\bullet \\to A_2^\\bullet \\to \\ldots", "$$", "be an exact complex of complexes of abelian groups", "such that for all $p \\in \\mathbf{Z}$ the complexes", "$$", "0 \\to", "\\Coker(d_{M^\\bullet}^p) \\to", "\\Coker(d_{A_0^\\bullet}^p) \\to", "\\Coker(d_{A_1^\\bullet}^p) \\to", "\\Coker(d_{A_2^\\bullet}^p) \\to \\ldots", "$$", "are exact as well. Set $A^{p, q} = A_p^q$ to obtain a double", "complex. Let $\\text{Tot}_\\pi(A^{\\bullet, \\bullet})$ be the", "product total complex associated to the double complex", "(see proof). Then the map", "$M^\\bullet \\to \\text{Tot}_\\pi(A^{\\bullet, \\bullet})$", "induced by $M^\\bullet \\to A_0^\\bullet$ is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Abbreviating $T^\\bullet = \\text{Tot}_\\pi(A^{\\bullet, \\bullet})$", "we define", "$$", "T^n = \\prod\\nolimits_{p + q = n} A^{p, q} =", "\\prod\\nolimits_{p + q = n} A_p^q", "\\quad\\text{with}\\quad", "\\text{d}_{T^\\bullet}^n =", "\\prod\\nolimits_{n = p + q} (f_p^q + (-1)^pd_{A_p^\\bullet}^q)", "$$", "where $f_p^\\bullet : A_p^\\bullet \\to A_{p + 1}^\\bullet$", "are the maps of complexes in the lemma.", "\\medskip\\noindent", "We will show that $H^0(M^\\bullet) \\to H^0(T^\\bullet)$ is an isomorphism.", "The same argument works for other degrees.", "Let $x \\in \\Ker(\\text{d}_{T^\\bullet}^0)$ represent $\\xi \\in H^0(T^\\bullet)$.", "Write $x = (x_i)$ with $x_i \\in A_i^{-i}$.", "Note that $x_0$ maps to zero in $\\Coker(A_1^{-1} \\to A_1^0)$.", "Hence we see that $x_0 = m_0 + d_{A_0^\\bullet}^{-1}(y)$ for some", "$m_0 \\in M^0$ and $y \\in A_0^{-1}$.", "Then $d_{M^\\bullet}(m_0) = 0$ because $\\text{d}_{A_0^\\bullet}(x_0) = 0$", "as $\\text{d}_{T^\\bullet}(x) = 0$.", "Thus, replacing $\\xi$ by something in the image of", "$H^0(M^\\bullet) \\to H^0(T^\\bullet)$ we may assume that $x_0$", "is in $\\Im(d^{-1}_{A_0^\\bullet})$.", "\\medskip\\noindent", "Assume $x_0 \\in \\Im(d^{-1}_{A_0^\\bullet})$. We claim that in this", "case $\\xi = 0$. To prove this we find, by induction on $n$ elements", "$y_0, y_1, \\ldots, y_n$ with $y_i \\in A_i^{-i - 1}$ such that", "$x_0 = \\text{d}_{A_0}^{-1}(y_0)$ and", "$x_j = f_{j - 1}^{-j}(y_{j - 1}) + (-1)^j d^{-j - 1}_{A_{-j}^\\bullet}(y_j)$", "for $j = 1, \\ldots, n$. This is clear for $n = 0$. Proof of induction step:", "suppose we have found $y_0, \\ldots, y_{n - 1}$. Then", "$w_n = x_n - f_{n - 1}^{-n}(y_{n - 1})$ is in the kernel of", "$d^{-n}_{A_n^\\bullet}$ and maps to zero in $H^n(A_{n + 1}^\\bullet)$", "(because it maps to an element which is a boundary the boundary", "of $x_{n + 1}$ up to sign). Exactly as in the proof of", "Lemma \\ref{lemma-good-resolution-gives-qis}", "the assumptions of the lemma imply that", "$$", "0 \\to", "H^p(M^\\bullet) \\to", "H^p(A_0^\\bullet) \\to", "H^p(A_1^\\bullet) \\to", "H^p(A_2^\\bullet) \\to \\ldots", "$$", "is exact for all $p \\in \\mathbf{Z}$. Thus after changing $y_{n - 1}$", "by an element in $\\Ker(d^{n - 1}_{A_{n - 1}^\\bullet})$ we may assume", "that $w_n$ maps to zero in $H^{-n}(A_n^\\bullet)$. This means we", "can find $y_n$ as desired. Observe that this procedure does not", "change $y_0, \\ldots, y_{n - 2}$. Hence continuing ad infinitum", "we find an element $y = (y_i)$ in $T^{n - 1}$ with $d_{T^\\bullet}(y) = \\xi$.", "This shows that $H^0(M^\\bullet) \\to H^0(T^\\bullet)$ is surjective.", "\\medskip\\noindent", "Suppose that $m_0 \\in \\Ker(d^0_{M^\\bullet})$ maps to zero in $H^0(T^\\bullet)$.", "Say it maps to the differential applied to $y = (y_i) \\in T^{-1}$ .", "Then $y_0 \\in A_0^{-1}$ maps to zero in $\\Coker(d^{-2}_{A_1^\\bullet})$.", "By assumption this means that $y_0 \\bmod \\Im(d^{-2}_{A_0^\\bullet})$", "is the image of some $z \\in M^{-1}$. It follows that", "$m_0 = d^{-1}_{M^\\bullet}(z)$. This proves injectivity and the proof is", "complete." ], "refs": [ "homology-lemma-good-resolution-gives-qis" ], "ref_ids": [ 12109 ] } ], "ref_ids": [] }, { "id": 12111, "type": "theorem", "label": "homology-lemma-resolution-gives-qis", "categories": [ "homology" ], "title": "homology-lemma-resolution-gives-qis", "contents": [ "Let $M^\\bullet$ be a complex of abelian groups. Let", "$$", "\\ldots \\to A_2^\\bullet \\to A_1^\\bullet \\to A_0^\\bullet \\to M^\\bullet \\to 0", "$$", "be an exact complex of complexes of abelian groups. Set $A^{p, q} = A_{-p}^q$", "to obtain a double complex. Let $\\text{Tot}_\\pi(A^{\\bullet, \\bullet})$", "be the product total complex associated to the double complex (see proof).", "Then the map $\\text{Tot}_\\pi(A^{\\bullet, \\bullet}) \\to M^\\bullet$", "induced by $A_0^\\bullet \\to M^\\bullet$ is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Abbreviating $T^\\bullet = \\text{Tot}_\\pi(A^{\\bullet, \\bullet})$", "we define", "$$", "T^n = \\prod\\nolimits_{p + q = n} A^{p, q} =", "\\prod\\nolimits_{p + q = n} A_{-p}^q", "\\quad\\text{with}\\quad", "\\text{d}_{T^\\bullet}^n =", "\\prod\\nolimits_{n = p + q} (f_{-p}^q + (-1)^pd_{A_{-p}^\\bullet}^q)", "$$", "where $f_p^\\bullet : A_p^\\bullet \\to A_{p - 1}^\\bullet$", "are the maps of complexes in the lemma.", "We will show that $T^\\bullet$ is acyclic when", "$M^\\bullet$ is the zero complex. This will suffice by", "the following trick. Set $B_n^\\bullet = A_{n + 1}^\\bullet$", "and $B_0^\\bullet = M^\\bullet$. Then we have an exact sequence", "$$", "\\ldots \\to B_2^\\bullet \\to B_1^\\bullet \\to B_0^\\bullet \\to 0 \\to 0", "$$", "as in the lemma. Let $S^\\bullet = \\text{Tot}_\\pi(B^{\\bullet, \\bullet})$.", "Then there is an obvious short exact sequence of complexes", "$$", "0 \\to M^\\bullet \\to S^\\bullet \\to T^\\bullet[1] \\to 0", "$$", "and we conclude by the long exact cohomology sequence. Some details omitted.", "\\medskip\\noindent", "Assume $M^\\bullet = 0$. We will show $H^0(T^\\bullet) = 0$. The same argument", "works for other degrees. Let $x =(x_n) \\in \\Ker(d_{T^\\bullet})$", "map to $\\xi \\in H^0(T^\\bullet)$ with $x_n \\in A^{-n, n} = A_n^n$.", "Since $M^0 = 0$ we find that $x_0 = f_1^0(y_0)$ for some $y_0 \\in A_1^0$.", "Then $x_1 - d^0_{A_1^\\bullet}(y_0) = f_2^1(y_1)$", "because it is mapped to zero by $f_1^1$ as $x$ is a cocycle.", "for some $y_1 \\in A_2^1$. Continuing, using induction, we find", "$y = (y_i) \\in T^{-1}$ with $d_{T^\\bullet}(y) = x$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12112, "type": "theorem", "label": "homology-lemma-characterize-injectives", "categories": [ "homology" ], "title": "homology-lemma-characterize-injectives", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $I$ be an object of $\\mathcal{A}$.", "The following are equivalent:", "\\begin{enumerate}", "\\item The object $I$ is injective.", "\\item The functor $B \\mapsto \\Hom_\\mathcal{A}(B, I)$", "is exact.", "\\item Any short exact sequence", "$$", "0 \\to I \\to A \\to B \\to 0", "$$", "in $\\mathcal{A}$ is split.", "\\item We have $\\Ext_\\mathcal{A}(B, I) = 0$ for", "all $B \\in \\Ob(\\mathcal{A})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12113, "type": "theorem", "label": "homology-lemma-product-injectives", "categories": [ "homology" ], "title": "homology-lemma-product-injectives", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Suppose $I_\\omega$, $\\omega \\in \\Omega$ is a set of injective", "objects of $\\mathcal{A}$. If $\\prod_{\\omega \\in \\Omega} I_\\omega$", "exists then it is injective." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12114, "type": "theorem", "label": "homology-lemma-characterize-projectives", "categories": [ "homology" ], "title": "homology-lemma-characterize-projectives", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $P$ be an object of $\\mathcal{A}$.", "The following are equivalent:", "\\begin{enumerate}", "\\item The object $P$ is projective.", "\\item The functor $B \\mapsto \\Hom_\\mathcal{A}(P, B)$", "is exact.", "\\item Any short exact sequence", "$$", "0 \\to A \\to B \\to P \\to 0", "$$", "in $\\mathcal{A}$ is split.", "\\item We have $\\Ext_\\mathcal{A}(P, A) = 0$ for", "all $A \\in \\Ob(\\mathcal{A})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12115, "type": "theorem", "label": "homology-lemma-coproduct-projectives", "categories": [ "homology" ], "title": "homology-lemma-coproduct-projectives", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Suppose $P_\\omega$, $\\omega \\in \\Omega$ is a set of projective", "objects of $\\mathcal{A}$. If $\\coprod_{\\omega \\in \\Omega} P_\\omega$", "exists then it is projective." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12116, "type": "theorem", "label": "homology-lemma-adjoint-preserve-injectives", "categories": [ "homology" ], "title": "homology-lemma-adjoint-preserve-injectives", "contents": [ "\\begin{slogan}", "A functor with an exact left adjoint preserves injectives", "\\end{slogan}", "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.", "Let $u : \\mathcal{A} \\to \\mathcal{B}$ and", "$v : \\mathcal{B} \\to \\mathcal{A}$ be additive functors. Assume", "\\begin{enumerate}", "\\item $u$ is right adjoint to $v$, and", "\\item $v$ transforms injective maps into injective maps.", "\\end{enumerate}", "Then $u$ transforms injectives into injectives." ], "refs": [], "proofs": [ { "contents": [ "Let $I$ be an injective object of $\\mathcal{A}$.", "Let $\\varphi : N \\to M$ be an injective map in $\\mathcal{B}$ and let", "$\\alpha : N \\to uI$ be a morphism.", "By adjointness we get a morphism $\\alpha : vN \\to I$ and", "by assumption $v\\varphi : vN \\to vM$ is injective.", "Hence as $I$ is an injective object we get a morphism", "$\\beta : vM \\to I$ extending $\\alpha$. By adjointness", "again this corresponds to a morphism $\\beta : M \\to uI$ as", "desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12117, "type": "theorem", "label": "homology-lemma-adjoint-enough-injectives", "categories": [ "homology" ], "title": "homology-lemma-adjoint-enough-injectives", "contents": [ "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.", "Let $u : \\mathcal{A} \\to \\mathcal{B}$ and", "$v : \\mathcal{B} \\to \\mathcal{A}$ be additive functors.", "Assume", "\\begin{enumerate}", "\\item $u$ is right adjoint to $v$,", "\\item $v$ transforms injective maps into injective maps,", "\\item $\\mathcal{A}$ has enough injectives, and", "\\item $vB = 0$ implies $B = 0$ for any $B \\in \\Ob(\\mathcal{B})$.", "\\end{enumerate}", "Then $\\mathcal{B}$ has enough injectives." ], "refs": [], "proofs": [ { "contents": [ "Pick $B \\in \\Ob(\\mathcal{B})$.", "Pick an injection $vB \\to I$ for $I$", "an injective object of $\\mathcal{A}$.", "According to Lemma \\ref{lemma-adjoint-preserve-injectives}", "and the assumptions the corresponding map", "$B \\to uI$ is the injection of $B$ into an injective object." ], "refs": [ "homology-lemma-adjoint-preserve-injectives" ], "ref_ids": [ 12116 ] } ], "ref_ids": [] }, { "id": 12118, "type": "theorem", "label": "homology-lemma-adjoint-functorial-injectives", "categories": [ "homology" ], "title": "homology-lemma-adjoint-functorial-injectives", "contents": [ "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.", "Let $u : \\mathcal{A} \\to \\mathcal{B}$ and", "$v : \\mathcal{B} \\to \\mathcal{A}$ be additive functors.", "Assume", "\\begin{enumerate}", "\\item $u$ is right adjoint to $v$,", "\\item $v$ transforms injective maps into injective maps,", "\\item $\\mathcal{A}$ has enough injectives,", "\\item $vB = 0$ implies $B = 0$ for any $B \\in \\Ob(\\mathcal{B})$, and", "\\item $\\mathcal{A}$ has functorial injective hulls.", "\\end{enumerate}", "Then $\\mathcal{B}$ has functorial injective hulls." ], "refs": [], "proofs": [ { "contents": [ "Let $A \\mapsto (A \\to J(A))$ be a functorial", "injective hull on $\\mathcal{A}$. Then", "$B \\mapsto (B \\to uJ(vB))$ is a functorial", "injective hull on $\\mathcal{B}$. Compare with the", "proof of Lemma \\ref{lemma-adjoint-enough-injectives}." ], "refs": [ "homology-lemma-adjoint-enough-injectives" ], "ref_ids": [ 12117 ] } ], "ref_ids": [] }, { "id": 12119, "type": "theorem", "label": "homology-lemma-partially-defined-adjoint", "categories": [ "homology" ], "title": "homology-lemma-partially-defined-adjoint", "contents": [ "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.", "Let $u : \\mathcal{A} \\to \\mathcal{B}$ be a functor.", "If there exists a subset $\\mathcal{P} \\subset \\Ob(\\mathcal{B})$", "such that", "\\begin{enumerate}", "\\item every object of $\\mathcal{B}$ is a quotient of an element", "of $\\mathcal{P}$, and", "\\item for every $P \\in \\mathcal{P}$ there exists an object", "$Q$ of $\\mathcal{A}$ such that", "$\\Hom_\\mathcal{A}(Q, A) = \\Hom_\\mathcal{B}(P, u(A))$ functorially", "in $A$,", "\\end{enumerate}", "then there exists a left adjoint $v$ of $u$." ], "refs": [], "proofs": [ { "contents": [ "By the Yoneda lemma (Categories, Lemma \\ref{categories-lemma-yoneda})", "the object $Q$ of $\\mathcal{A}$ corresponding to $P$ is defined up to", "unique isomorphism by the formula", "$\\Hom_\\mathcal{A}(Q, A) = \\Hom_\\mathcal{B}(P, u(A))$. Let us write", "$Q = v(P)$. Denote $i_P : P \\to u(v(P))$ the map corresponding to", "$\\text{id}_{v(P)}$ in $\\Hom_\\mathcal{A}(v(P), v(P))$. Functoriality", "in (2) implies that the bijection is given by", "$$", "\\Hom_\\mathcal{A}(v(P), A) \\to \\Hom_\\mathcal{B}(P, u(A)),\\quad", "\\varphi \\mapsto u(\\varphi) \\circ i_P", "$$", "For any pair of elements $P_1, P_2 \\in \\mathcal{P}$ there is a canonical map", "$$", "\\Hom_\\mathcal{B}(P_2, P_1)", "\\to", "\\Hom_\\mathcal{A}(v(P_2), v(P_1)),\\quad", "\\varphi \\mapsto v(\\varphi)", "$$", "which is characterized by the rule", "$u(v(\\varphi)) \\circ i_{P_2} = i_{P_1} \\circ \\varphi$ in", "$\\Hom_\\mathcal{B}(P_2, u(v(P_1)))$.", "Note that $\\varphi \\mapsto v(\\varphi)$ is", "compatible with composition; this can be seen directly", "from the characterization. Hence $P \\mapsto v(P)$ is a functor from", "the full subcategory of $\\mathcal{B}$ whose objects are the elements", "of $\\mathcal{P}$.", "\\medskip\\noindent", "Given an arbitrary object $B$ of $\\mathcal{B}$ choose an exact sequence", "$$", "P_2 \\to P_1 \\to B \\to 0", "$$", "which is possible by assumption (1). Define $v(B)$ to be the object of", "$\\mathcal{A}$ fitting into the exact sequence", "$$", "v(P_2) \\to v(P_1) \\to v(B) \\to 0", "$$", "Then", "\\begin{align*}", "\\Hom_\\mathcal{A}(v(B), A)", "& =", "\\Ker(\\Hom_\\mathcal{A}(v(P_1), A) \\to \\Hom_\\mathcal{A}(v(P_2), A)) \\\\", "& =", "\\Ker(\\Hom_\\mathcal{B}(P_1, u(A)) \\to \\Hom_\\mathcal{B}(P_2, u(A))) \\\\", "& =", "\\Hom_\\mathcal{B}(B, u(A))", "\\end{align*}", "Hence we see that we may take $\\mathcal{P} = \\Ob(\\mathcal{B})$, i.e., we", "see that $v$ is everywhere defined." ], "refs": [ "categories-lemma-yoneda" ], "ref_ids": [ 12203 ] } ], "ref_ids": [] }, { "id": 12120, "type": "theorem", "label": "homology-lemma-essentially-constant-into-karoubian", "categories": [ "homology" ], "title": "homology-lemma-essentially-constant-into-karoubian", "contents": [ "Let $\\mathcal{I}$ be a category, let $\\mathcal{A}$ be a pre-additive", "Karoubian category, and let $M : \\mathcal{I} \\to \\mathcal{A}$ be a diagram.", "\\begin{enumerate}", "\\item Assume $\\mathcal{I}$ is filtered. The following are equivalent", "\\begin{enumerate}", "\\item $M$ is essentially constant,", "\\item $X = \\colim M$ exists and there exists a cofinal filtered subcategory", "$\\mathcal{I}' \\subset \\mathcal{I}$ and for $i' \\in \\Ob(\\mathcal{I}')$", "a direct sum decomposition $M_{i'} = X_{i'} \\oplus Z_{i'}$ such that", "$X_{i'}$ maps isomorphically to $X$ and $Z_{i'}$ to zero in $M_{i''}$", "for some $i' \\to i''$ in $\\mathcal{I}'$.", "\\end{enumerate}", "\\item Assume $\\mathcal{I}$ is cofiltered. The following are equivalent", "\\begin{enumerate}", "\\item $M$ is essentially constant,", "\\item $X = \\lim M$ exists and there exists an initial cofiltered subcategory", "$\\mathcal{I}' \\subset \\mathcal{I}$ and for $i' \\in \\Ob(\\mathcal{I}')$", "a direct sum decomposition $M_{i'} = X_{i'} \\oplus Z_{i'}$", "such that $X$ maps isomorphically to $X_{i'}$ and $M_{i''} \\to Z_{i'}$", "is zero for some $i'' \\to i'$ in $\\mathcal{I}'$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1)(a), i.e., $\\mathcal{I}$ is filtered and $M$ is essentially", "constant. Let $X = \\colim M_i$. Choose $i$ and $X \\to M_i$ as in", "Categories, Definition", "\\ref{categories-definition-essentially-constant-diagram}.", "Let $\\mathcal{I}'$ be the full subcategory consisting of objects", "which are the target of a morphism with source $i$.", "Suppose $i' \\in \\Ob(\\mathcal{I}')$ and choose a morphism $i \\to i'$.", "Then $X \\to M_i \\to M_{i'}$ composed with $M_{i'} \\to X$ is the identity", "on $X$. As $\\mathcal{A}$ is Karoubian, we find a direct summand", "decomposition $M_{i'} = X_{i'} \\oplus Z_{i'}$, where", "$Z_{i'} = \\Ker(M_{i'} \\to X)$ and $X_{i'}$ maps isomorphically to $X$.", "Pick $i \\to k$ and $i' \\to k$ such that $M_{i'} \\to X \\to M_i \\to M_k$", "equals $M_{i'} \\to M_k$ as in Categories, Definition", "\\ref{categories-definition-essentially-constant-diagram}.", "Then we see that $M_{i'} \\to M_k$ annihilates $Z_{i'}$.", "Thus (1)(b) holds.", "\\medskip\\noindent", "Assume (1)(b), i.e., $\\mathcal{I}$ is filtered and we have", "$\\mathcal{I}' \\subset \\mathcal{I}$ and for $i' \\in \\Ob(\\mathcal{I}')$", "a direct sum decomposition $M_{i'} = X_{i'} \\oplus Z_{i'}$", "as stated in the lemma. To see that $M$ is essentially constant", "we can replace $\\mathcal{I}$ by $\\mathcal{I}'$, see", "Categories, Lemma \\ref{categories-lemma-cofinal-essentially-constant}.", "Pick any $i \\in \\Ob(\\mathcal{I})$", "and denote $X \\to M_i$ the inverse of the isomorphism $X_i \\to X$", "followed by the inclusion map $X_i \\to M_i$. If $j$ is", "a second object, then choose $j \\to k$ such that $Z_j \\to M_k$ is", "zero. Since $\\mathcal{I}$ is filtered we may also assume there is", "a morphism $i \\to k$ (after possibly increasing $k$). Then", "$M_j \\to X \\to M_i \\to M_k$ and $M_j \\to M_k$ both annihilate $Z_j$.", "Thus after postcomposing by a morphism $M_k \\to M_l$ which annihilates", "the summand $Z_k$, we find that $M_j \\to X \\to M_i \\to M_l$ and", "$M_j \\to M_l$ are equal, i.e., $M$ is essentially constant.", "\\medskip\\noindent", "The proof of (2) is dual." ], "refs": [ "categories-definition-essentially-constant-diagram", "categories-definition-essentially-constant-diagram", "categories-lemma-cofinal-essentially-constant" ], "ref_ids": [ 12368, 12368, 12242 ] } ], "ref_ids": [] }, { "id": 12121, "type": "theorem", "label": "homology-lemma-direct-sum-from-product-colimit", "categories": [ "homology" ], "title": "homology-lemma-direct-sum-from-product-colimit", "contents": [ "Let $\\mathcal{I}$ be a category. Let $\\mathcal{A}$ be an additive, Karoubian", "category. Let $F : \\mathcal{I} \\to \\mathcal{A}$ and", "$G : \\mathcal{I} \\to \\mathcal{A}$ be functors. The following are equivalent", "\\begin{enumerate}", "\\item $\\colim_\\mathcal{I} F \\oplus G$ exists, and", "\\item $\\colim_\\mathcal{I} F$ and $\\colim_\\mathcal{I} G$ exist.", "\\end{enumerate}", "In this case $\\colim_\\mathcal{I} F \\oplus G =", "\\colim_\\mathcal{I} F \\oplus \\colim_\\mathcal{I} G$." ], "refs": [], "proofs": [ { "contents": [ "Assume (1) holds. Set $W = \\colim_\\mathcal{I} F \\oplus G$.", "Note that the projection onto $F$ defines natural transformation", "$F \\oplus G \\to F \\oplus G$ which is idempotent. Hence we obtain", "an idempotent endomorphism $W \\to W$ by", "Categories, Lemma \\ref{categories-lemma-functorial-colimit}.", "Since $\\mathcal{A}$ is Karoubian we get a corresponding direct", "sum decomposition $W = X \\oplus Y$, see Lemma \\ref{lemma-karoubian}.", "A straightforward argument (omitted) shows that", "$X = \\colim_\\mathcal{I} F$ and $Y = \\colim_\\mathcal{I} G$.", "Thus (2) holds. We omit the proof that (2) implies (1)." ], "refs": [ "categories-lemma-functorial-colimit", "homology-lemma-karoubian" ], "ref_ids": [ 12210, 12014 ] } ], "ref_ids": [] }, { "id": 12122, "type": "theorem", "label": "homology-lemma-direct-sum-from-product-essentially-constant", "categories": [ "homology" ], "title": "homology-lemma-direct-sum-from-product-essentially-constant", "contents": [ "Let $\\mathcal{I}$ be a filtered category. Let $\\mathcal{A}$", "be an additive, Karoubian category. Let $F : \\mathcal{I} \\to \\mathcal{A}$ and", "$G : \\mathcal{I} \\to \\mathcal{A}$ be functors. The following are equivalent", "\\begin{enumerate}", "\\item $F \\oplus G : \\mathcal{I} \\to \\mathcal{A}$", "is essentially constant, and", "\\item $F$ and $G$ are essentially constant.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) holds. In particular $W = \\colim_\\mathcal{I} F \\oplus G$ exists", "and hence by Lemma \\ref{lemma-direct-sum-from-product-colimit}", "we have $W = X \\oplus Y$ with $X = \\colim_\\mathcal{I} F$ and", "$Y = \\colim_\\mathcal{I} G$. A straightforward argument (omitted)", "using for example the characterization of", "Categories, Lemma \\ref{categories-lemma-characterize-essentially-constant-ind}", "shows that $F$ is essentially constant with value $X$ and $G$ is essentially", "constant with value $Y$. Thus (2) holds. The proof that (2) implies (1)", "is omitted." ], "refs": [ "homology-lemma-direct-sum-from-product-colimit", "categories-lemma-characterize-essentially-constant-ind" ], "ref_ids": [ 12121, 12240 ] } ], "ref_ids": [] }, { "id": 12123, "type": "theorem", "label": "homology-lemma-inverse-systems-abelian", "categories": [ "homology" ], "title": "homology-lemma-inverse-systems-abelian", "contents": [ "Let $\\mathcal{C}$ be a category.", "\\begin{enumerate}", "\\item If $\\mathcal{C}$ is an additive category, then the category", "of inverse systems with values in $\\mathcal{C}$ is an additive category.", "\\item If $\\mathcal{C}$ is an abelian category, then the category", "of inverse systems with values in $\\mathcal{C}$ is an abelian category.", "A sequence $(K_i) \\to (L_i) \\to (M_i)$ of inverse systems", "is exact if and only if each $K_i \\to L_i \\to N_i$ is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12124, "type": "theorem", "label": "homology-lemma-Mittag-Leffler", "categories": [ "homology" ], "title": "homology-lemma-Mittag-Leffler", "contents": [ "Let", "$$", "0 \\to (A_i) \\to (B_i) \\to (C_i) \\to 0", "$$", "be a short exact sequence of inverse systems of abelian groups.", "\\begin{enumerate}", "\\item In any case the sequence", "$$", "0 \\to \\lim_i A_i \\to \\lim_i B_i \\to \\lim_i C_i", "$$", "is exact.", "\\item If $(B_i)$ is ML, then also $(C_i)$ is ML.", "\\item If $(A_i)$ is ML, then", "$$", "0 \\to \\lim_i A_i \\to \\lim_i B_i \\to \\lim_i C_i \\to 0", "$$", "is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Nice exercise. See", "Algebra, Lemma \\ref{algebra-lemma-Mittag-Leffler} for part (3)." ], "refs": [ "algebra-lemma-Mittag-Leffler" ], "ref_ids": [ 827 ] } ], "ref_ids": [] }, { "id": 12125, "type": "theorem", "label": "homology-lemma-apply-Mittag-Leffler", "categories": [ "homology" ], "title": "homology-lemma-apply-Mittag-Leffler", "contents": [ "Let", "$$", "(A_i) \\to (B_i) \\to (C_i) \\to (D_i)", "$$", "be an exact sequence of inverse systems of abelian groups. If the", "system $(A_i)$ is ML, then the sequence", "$$", "\\lim_i B_i \\to \\lim_i C_i \\to \\lim_i D_i", "$$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "Let $Z_i = \\Ker(C_i \\to D_i)$ and $I_i = \\Im(A_i \\to B_i)$.", "Then $\\lim Z_i = \\Ker(\\lim C_i \\to \\lim D_i)$ and", "we get a short exact sequence of systems", "$$", "0 \\to (I_i) \\to (B_i) \\to (Z_i) \\to 0", "$$", "Moreover, by", "Lemma \\ref{lemma-Mittag-Leffler}", "we see that $(I_i)$ has (ML), thus another application of", "Lemma \\ref{lemma-Mittag-Leffler}", "shows that $\\lim B_i \\to \\lim Z_i$ is surjective which", "proves the lemma." ], "refs": [ "homology-lemma-Mittag-Leffler", "homology-lemma-Mittag-Leffler" ], "ref_ids": [ 12124, 12124 ] } ], "ref_ids": [] }, { "id": 12126, "type": "theorem", "label": "homology-lemma-essentially-constant", "categories": [ "homology" ], "title": "homology-lemma-essentially-constant", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $(A_i)$ be an inverse system in $\\mathcal{A}$ with limit $A = \\lim A_i$.", "Then $(A_i)$ is essentially constant (see", "Categories, Definition", "\\ref{categories-definition-essentially-constant-diagram})", "if and only if there exists an $i$ and for all $j \\geq i$ a direct sum", "decomposition $A_j = A \\oplus Z_j$ such that", "(a) the maps $A_{j'} \\to A_j$ are compatible with the direct sum", "decompositions, (b) for all $j$ there exists some $j' \\geq j$ such that", "$Z_{j'} \\to Z_j$ is zero." ], "refs": [ "categories-definition-essentially-constant-diagram" ], "proofs": [ { "contents": [ "Assume $(A_i)$ is essentially constant. Then there exists an $i$ and", "a morphism $A_i \\to A$ such that $A \\to A_i \\to A$ is the identity and", "for all $j \\geq i$ there exists", "a $j' \\geq j$ such that $A_{j'} \\to A_j$ factors as", "$A_{j'} \\to A_i \\to A \\to A_j$ (the last map comes from $A = \\lim A_i$).", "Hence setting $Z_j = \\Ker(A_j \\to A)$ for all $j \\geq i$ works.", "Proof of the converse is omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12368 ] }, { "id": 12127, "type": "theorem", "label": "homology-lemma-exact-sequence-ML", "categories": [ "homology" ], "title": "homology-lemma-exact-sequence-ML", "contents": [ "Let", "$$", "0 \\to (A_i) \\to (B_i) \\to (C_i) \\to 0", "$$", "be an exact sequence of inverse systems of abelian groups.", "If $(A_i)$ has ML and $(C_i)$ is essentially constant, then $(B_i)$ has ML." ], "refs": [], "proofs": [ { "contents": [ "After renumbering we may assume that $C_i = C \\oplus Z_i$ compatible with", "transition maps and that for all $i$ there exists an $i' \\geq i$ such that", "$Z_{i'} \\to Z_i$ is zero, see", "Lemma \\ref{lemma-essentially-constant}.", "Pick $i$. Let $c \\geq i$ by an integer such that", "$\\Im(A_c \\to A) = \\Im(A_{i'} \\to A_i)$", "for all $i' \\geq c$. Let $c' \\geq c$ be an integer such that", "$Z_{c'} \\to Z_c$ is zero. For $i' \\geq c'$ consider the maps", "$$", "\\xymatrix{", "0 \\ar[r] & A_{i'} \\ar[d] \\ar[r] & B_{i'} \\ar[d] \\ar[r] &", "C \\oplus Z_{i'} \\ar[d] \\ar[r] & 0 \\\\", "0 \\ar[r] & A_{c'} \\ar[d] \\ar[r] & B_{c'} \\ar[d] \\ar[r] &", "C \\oplus Z_{c'} \\ar[d]\\ar[r] & 0 \\\\", "0 \\ar[r] & A_c \\ar[d] \\ar[r] & B_c \\ar[d] \\ar[r] &", "C \\oplus Z_c \\ar[d]\\ar[r] & 0 \\\\", "0 \\ar[r] & A_i \\ar[r] & B_i \\ar[r] & C \\oplus Z_i \\ar[r] & 0", "}", "$$", "Because $Z_{c'} \\to Z_c$ is zero the image $\\Im(B_{c'} \\to B_c)$", "is an extension $C$ by a subgroup $A' \\subset A_c$ which", "contains the image of $A_{c'} \\to A_c$. Hence $\\Im(B_{c'} \\to B_i)$", "is an extension of $C$ by the image of $A'$ which is the image of", "$A_c \\to A_i$ by our choice of $c$. In exactly the same way one shows", "that $\\Im(B_{i'} \\to B_i)$ is an extension of $C$ by", "the image of $A_c \\to A_i$. Hence", "$\\Im(B_{c'} \\to B_i) = \\Im(B_{i'} \\to B_i)$ and we win." ], "refs": [ "homology-lemma-essentially-constant" ], "ref_ids": [ 12126 ] } ], "ref_ids": [] }, { "id": 12128, "type": "theorem", "label": "homology-lemma-apply-Mittag-Leffler-again", "categories": [ "homology" ], "title": "homology-lemma-apply-Mittag-Leffler-again", "contents": [ "Let", "$$", "(A^{-2}_i \\to A^{-1}_i \\to A^0_i \\to A^1_i)", "$$", "be an inverse system of complexes of abelian groups and denote", "$A^{-2} \\to A^{-1} \\to A^0 \\to A^1$ its limit. Denote", "$(H_i^{-1})$, $(H_i^0)$ the inverse systems of cohomologies, and", "denote $H^{-1}$, $H^0$ the cohomologies of $A^{-2} \\to A^{-1} \\to A^0 \\to A^1$.", "If $(A^{-2}_i)$ and $(A^{-1}_i)$ are ML and", "$(H^{-1}_i)$ is essentially constant, then", "$H^0 = \\lim H_i^0$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z^j_i = \\Ker(A^j_i \\to A^{j + 1}_i)$ and", "$I^j_i = \\Im(A^{j - 1}_i \\to A^j_i)$.", "Note that $\\lim Z^0_i = \\Ker(\\lim A^0_i \\to \\lim A^1_i)$ as", "taking kernels commutes with limits.", "The systems $(I^{-1}_i)$ and $(I^0_i)$ have ML as quotients of", "the systems $(A^{-2}_i)$ and $(A^{-1}_i)$, see", "Lemma \\ref{lemma-Mittag-Leffler}.", "Thus an exact sequence", "$$", "0 \\to (I^{-1}_i) \\to (Z^{-1}_i) \\to (H^{-1}_i) \\to 0", "$$", "of inverse systems where $(I^{-1}_i)$ has ML", "and where $(H^{-1}_i)$ is essentially constant by assumption.", "Hence $(Z^{-1}_i)$ has ML by", "Lemma \\ref{lemma-exact-sequence-ML}.", "The exact sequence", "$$", "0 \\to (Z^{-1}_i) \\to (A^{-1}_i) \\to (I^0_i) \\to 0", "$$", "and an application of", "Lemma \\ref{lemma-Mittag-Leffler}", "shows that $\\lim A^{-1}_i \\to \\lim I^0_i$ is surjective.", "Finally, the exact sequence", "$$", "0 \\to (I^0_i) \\to (Z^0_i) \\to (H^0_i) \\to 0", "$$", "and", "Lemma \\ref{lemma-Mittag-Leffler}", "show that $\\lim I^0_i \\to \\lim Z^0_i \\to \\lim H^0_i \\to 0$", "is exact. Putting everything together we win." ], "refs": [ "homology-lemma-Mittag-Leffler", "homology-lemma-exact-sequence-ML", "homology-lemma-Mittag-Leffler", "homology-lemma-Mittag-Leffler" ], "ref_ids": [ 12124, 12127, 12124, 12124 ] } ], "ref_ids": [] }, { "id": 12129, "type": "theorem", "label": "homology-lemma-ML-over-ordinals", "categories": [ "homology" ], "title": "homology-lemma-ML-over-ordinals", "contents": [ "Let $\\alpha$ be an ordinal. Let $K_\\beta^\\bullet$, $\\beta < \\alpha$", "be an inverse system of complexes of abelian groups over $\\alpha$. If", "for all $\\beta < \\alpha$ the complex $K_\\beta^\\bullet$ is acyclic and", "the map", "$$", "K^n_\\beta \\longrightarrow \\lim_{\\gamma < \\beta} K^n_\\gamma", "$$", "is surjective, then the complex", "$\\lim_{\\beta < \\alpha} K_\\beta^\\bullet$ is acyclic." ], "refs": [], "proofs": [ { "contents": [ "By transfinite induction we prove this holds for every ordinal", "$\\alpha$ and every system as in the lemma. In particular, whilst", "proving the result for $\\alpha$ we may assume the complexes", "$\\lim_{\\gamma < \\beta} K^n_\\gamma$ are acyclic.", "\\medskip\\noindent", "Let $x \\in \\lim_{\\beta < \\alpha} K^0_\\alpha$ with $\\text{d}(x) = 0$.", "We will find a $y \\in K^{-1}_\\alpha$ with $\\text{d}(y) = x$.", "Write $x = (x_\\beta)$ where $x_\\beta \\in K_\\beta^0$ is the", "image of $x$ for $\\beta < \\alpha$. We will construct $y = (y_\\beta)$", "by transfinite induction.", "\\medskip\\noindent", "For $\\beta = 0$ let $y_0 \\in K_0^{-1}$", "be any element with $\\text{d}(y_0) = x_0$.", "\\medskip\\noindent", "For $\\beta = \\gamma + 1$ a successor, we have to find an element $y_\\beta$", "which maps both to $y_\\gamma$ by the transition map", "$f : K^\\bullet_\\beta \\to K^\\bullet_\\gamma$ and to $x_\\beta$ under the", "differential. As a first approximation we choose $y'_\\beta$ with", "$\\text{d}(y'_\\beta) = x_\\beta$. Then the difference $y_\\gamma - f(y'_\\beta)$", "is in the kernel of the differential, hence equal to $\\text{d}(z_\\gamma)$", "for some $z_\\gamma \\in K^{-2}_\\gamma$.", "By assumption, the map $f^{-2} : K^{-2}_\\beta \\to K^{-2}_\\gamma$", "is surjective. Hence we write $z_\\gamma = f(z_\\beta)$", "and change $y'_\\beta$ into $y_\\beta = y'_\\beta + \\text{d}(z_\\beta)$", "which works.", "\\medskip\\noindent", "If $\\beta$ is a limit ordinal, then we have the element", "$(y_\\gamma)_{\\gamma < \\beta}$ in $\\lim_{\\gamma < \\beta} K^{-1}_\\gamma$", "whose differential is the image of $x_\\beta$. Thus we can argue in exactly", "the same manner as above using the termwise surjective map of complexes", "$f : K_\\beta^\\bullet \\to \\lim_{\\gamma < \\beta} K_\\gamma^\\bullet$", "and the fact (see first paragraph of proof) that we may assume", "$\\lim_{\\gamma < \\beta} K_\\gamma^\\bullet$ is acyclic by induction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12130, "type": "theorem", "label": "homology-lemma-product-abelian-groups-exact", "categories": [ "homology" ], "title": "homology-lemma-product-abelian-groups-exact", "contents": [ "Let $I$ be a set. For $i \\in I$ let $L_i \\to M_i \\to N_i$ be a complex", "of abelian groups. Let $H_i = \\Ker(M_i \\to N_i)/\\Im(L_i \\to M_i)$", "be the cohomology. Then", "$$", "\\prod L_i \\to \\prod M_i \\to \\prod N_i", "$$", "is a complex of abelian groups with homology $\\prod H_i$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12200, "type": "theorem", "label": "categories-theorem-adjoint-functor", "categories": [ "categories" ], "title": "categories-theorem-adjoint-functor", "contents": [ "Let $G : \\mathcal{C} \\to \\mathcal{D}$ be a functor of big categories.", "Assume $\\mathcal{C}$ has limits, $G$ commutes with them, and for", "every object $y$ of $\\mathcal{D}$ there exists a set of pairs", "$(x_i, f_i)_{i \\in I}$ with $x_i \\in \\Ob(\\mathcal{C})$,", "$f_i \\in \\Mor_\\mathcal{D}(y, G(x_i))$ such that for any", "pair $(x, f)$ with $x \\in \\Ob(\\mathcal{C})$,", "$f \\in \\Mor_\\mathcal{C}(y, G(x))$ there is an $i$ and a morphism", "$h : x_i \\to x$ such that $f = G(h) \\circ f_i$.", "Then $G$ has a left adjoint $F$." ], "refs": [], "proofs": [ { "contents": [ "The assumptions imply that for every object $y$ of $\\mathcal{D}$", "the functor $x \\mapsto \\Mor_\\mathcal{D}(y, G(x))$ satisfies the", "assumptions of Lemma \\ref{lemma-a-version-of-brown}.", "Thus it is representable by an object, let's call it $F(y)$.", "An application of Yoneda's lemma (Lemma \\ref{lemma-yoneda})", "turns the rule $y \\mapsto F(y)$ into a functor which", "by construction is an adjoint to $G$. We omit the details." ], "refs": [ "categories-lemma-a-version-of-brown", "categories-lemma-yoneda" ], "ref_ids": [ 12253, 12203 ] } ], "ref_ids": [] }, { "id": 12201, "type": "theorem", "label": "categories-lemma-construct-quasi-inverse", "categories": [ "categories" ], "title": "categories-lemma-construct-quasi-inverse", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a fully faithful functor.", "Suppose for every $X \\in \\Ob(\\mathcal{B})$ given an", "object $j(X)$ of $\\mathcal{A}$ and an isomorphism $i_X : X \\to F(j(X))$.", "Then there is a unique functor $j : \\mathcal{B} \\to \\mathcal{A}$", "such that $j$ extends the rule on objects, and the isomorphisms", "$i_X$ define an isomorphism of functors", "$\\text{id}_\\mathcal{B} \\to F \\circ j$. Moreover, $j$ and $F$", "are quasi-inverse equivalences of categories." ], "refs": [], "proofs": [ { "contents": [ "This lemma proves itself." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12202, "type": "theorem", "label": "categories-lemma-equivalence-categories", "categories": [ "categories" ], "title": "categories-lemma-equivalence-categories", "contents": [ "A functor is an equivalence of categories if and only if it is both fully", "faithful and essentially surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be essentially surjective and fully", "faithful. As by convention all categories are small and as $F$ is essentially", "surjective we can, using the axiom of choice, choose for every", "$X \\in \\Ob(\\mathcal{B})$ an object $j(X)$ of $\\mathcal{A}$ and an", "isomorphism $i_X : X \\to F(j(X))$. Then we apply", "Lemma \\ref{lemma-construct-quasi-inverse}", "using that $F$ is fully faithful." ], "refs": [ "categories-lemma-construct-quasi-inverse" ], "ref_ids": [ 12201 ] } ], "ref_ids": [] }, { "id": 12203, "type": "theorem", "label": "categories-lemma-yoneda", "categories": [ "categories" ], "title": "categories-lemma-yoneda", "contents": [ "\\begin{reference}", "Appeared in some form in \\cite{Yoneda-homology}. Used by Grothendieck in a", "generalized form in \\cite{Gr-II}.", "\\end{reference}", "Let $U, V \\in \\Ob(\\mathcal{C})$.", "Given any morphism of functors $s : h_U \\to h_V$", "there is a unique morphism $\\phi : U \\to V$", "such that $h(\\phi) = s$. In other words the", "functor $h$ is fully faithful. More generally,", "given any contravariant functor $F$ and any object", "$U$ of $\\mathcal{C}$ we have a natural bijection", "$$", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(h_U, F) \\longrightarrow F(U),", "\\quad", "s \\longmapsto s_U(\\text{id}_U).", "$$" ], "refs": [], "proofs": [ { "contents": [ "For the first statement, just take", "$\\phi = s_U(\\text{id}_U) \\in \\Mor_\\mathcal{C}(U, V)$.", "For the second statement, given $\\xi \\in F(U)$ define", "$s$ by $s_V : h_U(V) \\to F(V)$ by sending the element $f : V \\to U$", "of $h_U(V) = \\Mor_\\mathcal{C}(V, U)$ to $F(f)(\\xi)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12204, "type": "theorem", "label": "categories-lemma-composition-representable", "categories": [ "categories" ], "title": "categories-lemma-composition-representable", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $f : x \\to y$, and $g : y \\to z$ be representable.", "Then $g \\circ f : x \\to z$ is representable." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12205, "type": "theorem", "label": "categories-lemma-base-change-representable", "categories": [ "categories" ], "title": "categories-lemma-base-change-representable", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $f : x \\to y$ be representable.", "Let $y' \\to y$ be a morphism of $\\mathcal{C}$.", "Then the morphism $x' := x \\times_y y' \\to y'$ is representable also." ], "refs": [], "proofs": [ { "contents": [ "Let $z \\to y'$ be a morphism. The fibre product", "$x' \\times_{y'} z$ is supposed to represent the", "functor", "\\begin{eqnarray*}", "w & \\mapsto & h_{x'}(w)\\times_{h_{y'}(w)} h_z(w) \\\\", "& = & (h_x(w) \\times_{h_y(w)} h_{y'}(w)) \\times_{h_{y'}(w)} h_z(w) \\\\", "& = & h_x(w) \\times_{h_y(w)} h_z(w)", "\\end{eqnarray*}", "which is representable by assumption." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12206, "type": "theorem", "label": "categories-lemma-fibre-product-presheaves", "categories": [ "categories" ], "title": "categories-lemma-fibre-product-presheaves", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $F, G, H : \\mathcal{C}^{opp} \\to \\textit{Sets}$", "be functors. Let $a : F \\to G$ and $b : H \\to G$ be", "transformations of functors. Then the fibre product", "$F \\times_{a, G, b} H$ in the category", "$\\textit{PSh}(\\mathcal{C})$", "exists and is given by the formula", "$$", "(F \\times_{a, G, b} H)(X) =", "F(X) \\times_{a_X, G(X), b_X} H(X)", "$$", "for any object $X$ of $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12207, "type": "theorem", "label": "categories-lemma-representable-over-representable", "categories": [ "categories" ], "title": "categories-lemma-representable-over-representable", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $a : F \\to G$ be a morphism of contravariant functors", "from $\\mathcal{C}$ to $\\textit{Sets}$. If $a$ is representable,", "and $G$ is a representable functor, then $F$ is representable." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12208, "type": "theorem", "label": "categories-lemma-representable-diagonal", "categories": [ "categories" ], "title": "categories-lemma-representable-diagonal", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $F : \\mathcal{C}^{opp} \\to \\textit{Sets}$ be a functor.", "Assume $\\mathcal{C}$ has products of pairs of objects and fibre products.", "The following are equivalent:", "\\begin{enumerate}", "\\item the diagonal $\\Delta : F \\to F \\times F$ is representable,", "\\item for every $U$ in $\\mathcal{C}$,", "and any $\\xi \\in F(U)$ the map $\\xi : h_U \\to F$ is representable,", "\\item for every pair $U, V$ in $\\mathcal{C}$", "and any $\\xi \\in F(U)$, $\\xi' \\in F(V)$ the fibre product", "$h_U \\times_{\\xi, F, \\xi'} h_V$ is representable.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will continue to use the Yoneda lemma to identify $F(U)$", "with transformations $h_U \\to F$ of functors.", "\\medskip\\noindent", "Equivalence of (2) and (3). Let $U, \\xi, V, \\xi'$ be as in (3).", "Both (2) and (3) tell us exactly that $h_U \\times_{\\xi, F, \\xi'} h_V$", "is representable; the only difference is that the statement", "(3) is symmetric in $U$ and $V$ whereas (2) is not.", "\\medskip\\noindent", "Assume condition (1). Let $U, \\xi, V, \\xi'$", "be as in (3). Note that $h_U \\times h_V = h_{U \\times V}$ is representable.", "Denote $\\eta : h_{U \\times V} \\to F \\times F$ the map", "corresponding to the product $\\xi \\times \\xi' : h_U \\times h_V \\to F \\times F$.", "Then the fibre product $F \\times_{\\Delta, F \\times F, \\eta} h_{U \\times V}$", "is representable by assumption. This means there exists", "$W \\in \\Ob(\\mathcal{C})$, morphisms", "$W \\to U$, $W \\to V$ and $h_W \\to F$ such that", "$$", "\\xymatrix{", "h_W \\ar[d] \\ar[r] & h_U \\times h_V \\ar[d]^{\\xi \\times \\xi'} \\\\", "F \\ar[r] & F \\times F", "}", "$$", "is cartesian. Using the explicit description of fibre products", "in Lemma \\ref{lemma-fibre-product-presheaves} the reader sees that this", "implies that $h_W = h_U \\times_{\\xi, F, \\xi'} h_V$ as desired.", "\\medskip\\noindent", "Assume the equivalent conditions (2) and (3). Let $U$ be an object", "of $\\mathcal{C}$ and let $(\\xi, \\xi') \\in (F \\times F)(U)$.", "By (3) the fibre product $h_U \\times_{\\xi, F, \\xi'} h_U$ is", "representable. Choose an object $W$ and an isomorphism", "$h_W \\to h_U \\times_{\\xi, F, \\xi'} h_U$. The two projections", "$\\text{pr}_i : h_U \\times_{\\xi, F, \\xi'} h_U \\to h_U$", "correspond to morphisms $p_i : W \\to U$ by Yoneda. Consider", "$W' = W \\times_{(p_1, p_2), U \\times U} U$. It is formal", "to show that $W'$ represents $F \\times_{\\Delta, F \\times F} h_U$", "because", "$$", "h_{W'} = h_W \\times_{h_U \\times h_U} h_U", "= (h_U \\times_{\\xi, F, \\xi'} h_U) \\times_{h_U \\times h_U} h_U", "= F \\times_{F \\times F} h_U.", "$$", "Thus $\\Delta$ is representable and this finishes the proof." ], "refs": [ "categories-lemma-fibre-product-presheaves" ], "ref_ids": [ 12206 ] } ], "ref_ids": [] }, { "id": 12209, "type": "theorem", "label": "categories-lemma-characterize-mono-epi", "categories": [ "categories" ], "title": "categories-lemma-characterize-mono-epi", "contents": [ "Let $\\mathcal{C}$ be a category, and let $f : X \\to Y$ be", "a morphism of $\\mathcal{C}$. Then", "\\begin{enumerate}", "\\item $f$ is a monomorphism if and only if $X$ is the fibre", "product $X \\times_Y X$, and", "\\item $f$ is an epimorphism if and only if $Y$ is the pushout", "$Y \\amalg_X Y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12210, "type": "theorem", "label": "categories-lemma-functorial-colimit", "categories": [ "categories" ], "title": "categories-lemma-functorial-colimit", "contents": [ "Suppose that $M : \\mathcal{I} \\to \\mathcal{C}$,", "and $N : \\mathcal{J} \\to \\mathcal{C}$ are diagrams", "whose colimits exist. Suppose", "$H : \\mathcal{I} \\to \\mathcal{J}$ is", "a functor, and suppose $t : M \\to N \\circ H$", "is a transformation of functors.", "Then there is a unique morphism", "$$", "\\theta :", "\\colim_\\mathcal{I} M", "\\longrightarrow", "\\colim_\\mathcal{J} N", "$$", "such that all the diagrams", "$$", "\\xymatrix{", "M_i \\ar[d]_{t_i} \\ar[r]", "&", "\\colim_\\mathcal{I} M \\ar[d]^{\\theta}", "\\\\", "N_{H(i)} \\ar[r]", "&", "\\colim_\\mathcal{J} N", "}", "$$", "commute." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12211, "type": "theorem", "label": "categories-lemma-functorial-limit", "categories": [ "categories" ], "title": "categories-lemma-functorial-limit", "contents": [ "Suppose that $M : \\mathcal{I} \\to \\mathcal{C}$,", "and $N : \\mathcal{J} \\to \\mathcal{C}$ are diagrams", "whose limits exist. Suppose $H : \\mathcal{I} \\to \\mathcal{J}$ is", "a functor, and suppose $t : N \\circ H \\to M$", "is a transformation of functors.", "Then there is a unique morphism", "$$", "\\theta :", "\\lim_\\mathcal{J} N", "\\longrightarrow", "\\lim_\\mathcal{I} M", "$$", "such that all the diagrams", "$$", "\\xymatrix{", "\\lim_\\mathcal{J} N \\ar[d]^{\\theta} \\ar[r]", "&", "N_{H(i)} \\ar[d]_{t_i}", "\\\\", "\\lim_\\mathcal{I} M \\ar[r]", "&", "M_i", "}", "$$", "commute." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12212, "type": "theorem", "label": "categories-lemma-colimits-commute", "categories": [ "categories" ], "title": "categories-lemma-colimits-commute", "contents": [ "Let $\\mathcal{I}$, $\\mathcal{J}$ be index categories.", "Let $M : \\mathcal{I} \\times \\mathcal{J} \\to \\mathcal{C}$ be a functor.", "We have", "$$", "\\colim_i \\colim_j M_{i, j}", "=", "\\colim_{i, j} M_{i, j}", "=", "\\colim_j \\colim_i M_{i, j}", "$$", "provided all the indicated colimits exist. Similar for limits." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12213, "type": "theorem", "label": "categories-lemma-limits-products-equalizers", "categories": [ "categories" ], "title": "categories-lemma-limits-products-equalizers", "contents": [ "Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram.", "Write $I = \\Ob(\\mathcal{I})$ and $A = \\text{Arrows}(\\mathcal{I})$.", "Denote $s, t : A \\to I$ the source and target maps.", "Suppose that $\\prod_{i \\in I} M_i$ and $\\prod_{a \\in A} M_{t(a)}$", "exist. Suppose that the equalizer of", "$$", "\\xymatrix{", "\\prod_{i \\in I} M_i", "\\ar@<1ex>[r]^\\phi \\ar@<-1ex>[r]_\\psi", "&", "\\prod_{a \\in A} M_{t(a)}", "}", "$$", "exists, where the morphisms are determined by their components", "as follows: $p_a \\circ \\psi = M(a) \\circ p_{s(a)}$", "and $p_a \\circ \\phi = p_{t(a)}$. Then this equalizer is the", "limit of the diagram." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12214, "type": "theorem", "label": "categories-lemma-colimits-coproducts-coequalizers", "categories": [ "categories" ], "title": "categories-lemma-colimits-coproducts-coequalizers", "contents": [ "\\begin{slogan}", "If all coproducts and coequalizers exist, all colimits exist.", "\\end{slogan}", "Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram.", "Write $I = \\Ob(\\mathcal{I})$ and $A = \\text{Arrows}(\\mathcal{I})$.", "Denote $s, t : A \\to I$ the source and target maps.", "Suppose that $\\coprod_{i \\in I} M_i$ and $\\coprod_{a \\in A} M_{s(a)}$", "exist. Suppose that the coequalizer of", "$$", "\\xymatrix{", "\\coprod_{a \\in A} M_{s(a)}", "\\ar@<1ex>[r]^\\phi \\ar@<-1ex>[r]_\\psi", "&", "\\coprod_{i \\in I} M_i", "}", "$$", "exists, where the morphisms are determined by their components", "as follows: The component $M_{s(a)}$ maps via $\\psi$", "to the component $M_{t(a)}$ via the morphism $a$.", "The component $M_{s(a)}$ maps via $\\phi$ to the component", "$M_{s(a)}$ by the identity morphism. Then this coequalizer is the", "colimit of the diagram." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12215, "type": "theorem", "label": "categories-lemma-connected-limit-over-X", "categories": [ "categories" ], "title": "categories-lemma-connected-limit-over-X", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $X$ be an object of $\\mathcal{C}$.", "Let $M : \\mathcal{I} \\to \\mathcal{C}/X$ be a diagram", "in the category of objects over $X$.", "If the index category $\\mathcal{I}$ is connected", "and the limit of $M$ exists in $\\mathcal{C}/X$,", "then the limit of the composition", "$\\mathcal{I} \\to \\mathcal{C}/X \\to \\mathcal{C}$", "exists and is the same." ], "refs": [], "proofs": [ { "contents": [ "Let $M \\to X$ be an object representing the limit in $\\mathcal{C}/X$.", "Consider the functor", "$$", "W \\longmapsto \\lim_i \\Mor_\\mathcal{C}(W, M_i).", "$$", "Let $(\\varphi_i)$ be an element of the set on the right.", "Since each $M_i$ comes equipped with a morphism $s_i : M_i \\to X$ we", "get morphisms $f_i = s_i \\circ \\varphi_i : W \\to X$. But as $\\mathcal{I}$", "is connected we see that all $f_i$ are equal. Since $\\mathcal{I}$", "is nonempty there is at least one $f_i$.", "Hence this common value $W \\to X$ defines the structure of an", "object of $W$ in $\\mathcal{C}/X$ and $(\\varphi_i)$ defines is an", "element of $\\lim_i \\Mor_{\\mathcal{C}/X}(W, M_i)$.", "Thus we obtain a unique morphism $\\phi : W \\to M$ such that", "$\\varphi_i$ is the composition of $\\phi$ with $M \\to M_i$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12216, "type": "theorem", "label": "categories-lemma-connected-colimit-under-X", "categories": [ "categories" ], "title": "categories-lemma-connected-colimit-under-X", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $X$ be an object of $\\mathcal{C}$.", "Let $M : \\mathcal{I} \\to X/\\mathcal{C}$ be a diagram", "in the category of objects under $X$.", "If the index category $\\mathcal{I}$ is connected", "and the colimit of $M$ exists in $X/\\mathcal{C}$,", "then the colimit of the composition", "$\\mathcal{I} \\to X/\\mathcal{C} \\to \\mathcal{C}$", "exists and is the same." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This lemma is dual to Lemma \\ref{lemma-connected-limit-over-X}." ], "refs": [ "categories-lemma-connected-limit-over-X" ], "ref_ids": [ 12215 ] } ], "ref_ids": [] }, { "id": 12217, "type": "theorem", "label": "categories-lemma-cofinal", "categories": [ "categories" ], "title": "categories-lemma-cofinal", "contents": [ "Let $H : \\mathcal{I} \\to \\mathcal{J}$ be a functor of categories. Assume", "$\\mathcal{I}$ is cofinal in $\\mathcal{J}$. Then for every diagram", "$M : \\mathcal{J} \\to \\mathcal{C}$ we have a canonical isomorphism", "$$", "\\colim_\\mathcal{I} M \\circ H", "=", "\\colim_\\mathcal{J} M", "$$", "if either side exists." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12218, "type": "theorem", "label": "categories-lemma-initial", "categories": [ "categories" ], "title": "categories-lemma-initial", "contents": [ "Let $H : \\mathcal{I} \\to \\mathcal{J}$ be a functor of categories.", "Assume $\\mathcal{I}$ is initial in $\\mathcal{J}$.", "Then for every diagram $M : \\mathcal{J} \\to \\mathcal{C}$ we", "have a canonical isomorphism", "$$", "\\lim_\\mathcal{I} M \\circ H = \\lim_\\mathcal{J} M", "$$", "if either side exists." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12219, "type": "theorem", "label": "categories-lemma-colimit-constant-connected-fibers", "categories": [ "categories" ], "title": "categories-lemma-colimit-constant-connected-fibers", "contents": [ "Let $F : \\mathcal{I} \\to \\mathcal{I}'$ be a functor.", "Assume", "\\begin{enumerate}", "\\item the fibre categories (see", "Definition \\ref{definition-fibre-category})", "of $\\mathcal{I}$ over $\\mathcal{I}'$ are all connected, and", "\\item for every morphism $\\alpha' : x' \\to y'$ in $\\mathcal{I}'$ there", "exist a morphism $\\alpha : x \\to y$ in $\\mathcal{I}$ such that", "$F(\\alpha) = \\alpha'$.", "\\end{enumerate}", "Then for every diagram $M : \\mathcal{I}' \\to \\mathcal{C}$", "the colimit $\\colim_\\mathcal{I} M \\circ F$ exists if and only", "if $\\colim_{\\mathcal{I}'} M$ exists and if so these colimits", "agree." ], "refs": [ "categories-definition-fibre-category" ], "proofs": [ { "contents": [ "One can prove this by showing that $\\mathcal{I}$ is cofinal in", "$\\mathcal{I}'$ and applying Lemma \\ref{lemma-cofinal}.", "But we can also prove it directly as follows.", "It suffices to show that for any object $T$ of $\\mathcal{C}$ we have", "$$", "\\lim_{\\mathcal{I}^{opp}} \\Mor_\\mathcal{C}(M_{F(i)}, T)", "=", "\\lim_{(\\mathcal{I}')^{opp}} \\Mor_\\mathcal{C}(M_{i'}, T)", "$$", "If $(g_{i'})_{i' \\in \\Ob(\\mathcal{I}')}$ is an element of", "the right hand side, then setting $f_i = g_{F(i)}$ we obtain an", "element $(f_i)_{i \\in \\Ob(\\mathcal{I})}$ of the left hand side.", "Conversely, let $(f_i)_{i \\in \\Ob(\\mathcal{I})}$ be an element of the", "left hand side. Note that on each (connected)", "fibre category $\\mathcal{I}_{i'}$ the functor $M \\circ F$", "is constant with value $M_{i'}$. Hence the morphisms", "$f_i$ for $i \\in \\Ob(\\mathcal{I})$ with $F(i) = i'$", "are all the same and determine a well defined morphism", "$g_{i'} : M_{i'} \\to T$. By assumption (2) the collection", "$(g_{i'})_{i' \\in \\Ob(\\mathcal{I}')}$ defines an element", "of the right hand side." ], "refs": [ "categories-lemma-cofinal" ], "ref_ids": [ 12217 ] } ], "ref_ids": [ 12386 ] }, { "id": 12220, "type": "theorem", "label": "categories-lemma-product-with-connected", "categories": [ "categories" ], "title": "categories-lemma-product-with-connected", "contents": [ "Let $\\mathcal{I}$ and $\\mathcal{J}$ be a categories and denote", "$p : \\mathcal{I} \\times \\mathcal{J} \\to \\mathcal{J}$ the projection.", "If $\\mathcal{I}$ is connected, then for a diagram", "$M : \\mathcal{J} \\to \\mathcal{C}$ the colimit $\\colim_\\mathcal{J} M$ exists", "if and only if $\\colim_{\\mathcal{I} \\times \\mathcal{J}} M \\circ p$ exists and", "if so these colimits are equal." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-colimit-constant-connected-fibers}." ], "refs": [ "categories-lemma-colimit-constant-connected-fibers" ], "ref_ids": [ 12219 ] } ], "ref_ids": [] }, { "id": 12221, "type": "theorem", "label": "categories-lemma-finite-diagram-category", "categories": [ "categories" ], "title": "categories-lemma-finite-diagram-category", "contents": [ "Let $\\mathcal{I}$ be a category with", "\\begin{enumerate}", "\\item $\\Ob(\\mathcal{I})$ is finite, and", "\\item there exist finitely many morphisms", "$f_1, \\ldots, f_m \\in \\text{Arrows}(\\mathcal{I})$ such", "that every morphism of $\\mathcal{I}$ is a composition", "$f_{j_1} \\circ f_{j_2} \\circ \\ldots \\circ f_{j_k}$.", "\\end{enumerate}", "Then there exists a functor $F : \\mathcal{J} \\to \\mathcal{I}$", "such that", "\\begin{enumerate}", "\\item[(a)] $\\mathcal{J}$ is a finite category, and", "\\item[(b)] for any diagram $M : \\mathcal{I} \\to \\mathcal{C}$ the", "(co)limit of $M$ over $\\mathcal{I}$ exists if and only if", "the (co)limit of $M \\circ F$ over $\\mathcal{J}$ exists and in this case", "the (co)limits are canonically isomorphic.", "\\end{enumerate}", "Moreover, $\\mathcal{J}$ is connected (resp.\\ nonempty) if and only if", "$\\mathcal{I}$ is so." ], "refs": [], "proofs": [ { "contents": [ "Say $\\Ob(\\mathcal{I}) = \\{x_1, \\ldots, x_n\\}$.", "Denote $s, t : \\{1, \\ldots, m\\} \\to \\{1, \\ldots, n\\}$ the functions", "such that $f_j : x_{s(j)} \\to x_{t(j)}$.", "We set $\\Ob(\\mathcal{J}) = \\{y_1, \\ldots, y_n, z_1, \\ldots, z_n\\}$", "Besides the identity morphisms we introduce morphisms", "$g_j : y_{s(j)} \\to z_{t(j)}$, $j = 1, \\ldots, m$ and morphisms", "$h_i : y_i \\to z_i$, $i = 1, \\ldots, n$. Since all of the nonidentity", "morphisms in $\\mathcal{J}$ go from a $y$ to a $z$ there are no", "compositions to define and no associativities to check.", "Set $F(y_i) = F(z_i) = x_i$. Set $F(g_j) = f_j$ and $F(h_i) = \\text{id}_{x_i}$.", "It is clear that $F$ is a functor.", "It is clear that $\\mathcal{J}$ is finite.", "It is clear that $\\mathcal{J}$ is connected, resp.\\ nonempty", "if and only if $\\mathcal{I}$ is so.", "\\medskip\\noindent", "Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram.", "Consider an object $W$ of $\\mathcal{C}$ and morphisms", "$q_i : W \\to M(x_i)$ as in", "Definition \\ref{definition-limit}.", "Then by taking $q_i : W \\to M(F(y_i)) = M(F(z_i)) = M(x_i)$ we obtain", "a family of maps as in", "Definition \\ref{definition-limit}", "for the diagram $M \\circ F$.", "Conversely, suppose we are given maps", "$qy_i : W \\to M(F(y_i))$ and $qz_i : W \\to M(F(z_i))$", "as in", "Definition \\ref{definition-limit}", "for the diagram $M \\circ F$. Since", "$$", "M(F(h_i)) = \\text{id} : M(F(y_i)) = M(x_i) \\longrightarrow M(x_i) = M(F(z_i))", "$$", "we conclude that $qy_i = qz_i$ for all $i$. Set $q_i$ equal to this common", "value. The compatibility of", "$q_{s(j)} = qy_{s(j)}$ and $q_{t(j)} = qz_{t(j)}$ with the morphism", "$M(f_j)$ guarantees that the family $q_i$ is compatible with all morphisms", "in $\\mathcal{I}$ as by assumption every such morphism is a composition", "of the morphisms $f_j$. Thus we have found a canonical bijection", "$$", "\\lim_{B \\in \\Ob(\\mathcal{J})} \\Mor_\\mathcal{C}(W, M(F(B)))", "=", "\\lim_{A \\in \\Ob(\\mathcal{I})} \\Mor_\\mathcal{C}(W, M(A))", "$$", "which implies the statement on limits in the lemma. The statement on colimits", "is proved in the same way (proof omitted)." ], "refs": [ "categories-definition-limit", "categories-definition-limit", "categories-definition-limit" ], "ref_ids": [ 12356, 12356, 12356 ] } ], "ref_ids": [] }, { "id": 12222, "type": "theorem", "label": "categories-lemma-fibre-products-equalizers-exist", "categories": [ "categories" ], "title": "categories-lemma-fibre-products-equalizers-exist", "contents": [ "Let $\\mathcal{C}$ be a category.", "The following are equivalent:", "\\begin{enumerate}", "\\item Connected finite limits exist in $\\mathcal{C}$.", "\\item Equalizers and fibre products exist in $\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since equalizers and fibre products are finite connected", "limits we see that (1) implies (2). For the converse, let $\\mathcal{I}$", "be a finite connected diagram category. Let", "$F : \\mathcal{J} \\to \\mathcal{I}$", "be the functor of diagram categories constructed in the proof of", "Lemma \\ref{lemma-finite-diagram-category}.", "Then we see that we may replace $\\mathcal{I}$ by $\\mathcal{J}$.", "The result is that we may assume that", "$\\Ob(\\mathcal{I}) = \\{x_1, \\ldots, x_n\\} \\amalg \\{y_1, \\ldots, y_m\\}$", "with $n, m \\geq 1$ such that all nonidentity morphisms in $\\mathcal{I}$", "are morphisms $f : x_i \\to y_j$ for some $i$ and $j$.", "\\medskip\\noindent", "Suppose that $n > 1$. Since $\\mathcal{I}$ is connected there", "exist indices $i_1, i_2$ and $j_0$ and morphisms $a : x_{i_1} \\to y_{j_0}$", "and $b : x_{i_2} \\to y_{j_0}$. Consider the category", "$$", "\\mathcal{I}' =", "\\{x\\} \\amalg \\{x_1, \\ldots, \\hat x_{i_1}, \\ldots, \\hat x_{i_2}, \\ldots x_n\\}", "\\amalg \\{y_1, \\ldots, y_m\\}", "$$", "with", "$$", "\\Mor_{\\mathcal{I}'}(x, y_j) = \\Mor_\\mathcal{I}(x_{i_1}, y_j)", "\\amalg \\Mor_\\mathcal{I}(x_{i_2}, y_j)", "$$", "and all other morphism sets the same as in $\\mathcal{I}$. For any functor", "$M : \\mathcal{I} \\to \\mathcal{C}$ we can construct a functor", "$M' : \\mathcal{I}' \\to \\mathcal{C}$ by setting", "$$", "M'(x) = M(x_{i_1}) \\times_{M(a), M(y_{j_0}), M(b)} M(x_{i_2})", "$$", "and for a morphism $f' : x \\to y_j$ corresponding to, say,", "$f : x_{i_1} \\to y_j$ we set $M'(f) = M(f) \\circ \\text{pr}_1$.", "Then the functor $M$ has a limit if and only if the functor $M'$ has", "a limit (proof omitted). Hence by induction we reduce to the case $n = 1$.", "\\medskip\\noindent", "If $n = 1$, then the limit of any $M : \\mathcal{I} \\to \\mathcal{C}$ is", "the successive equalizer of pairs of maps $x_1 \\to y_j$ hence", "exists by assumption." ], "refs": [ "categories-lemma-finite-diagram-category" ], "ref_ids": [ 12221 ] } ], "ref_ids": [] }, { "id": 12223, "type": "theorem", "label": "categories-lemma-almost-finite-limits-exist", "categories": [ "categories" ], "title": "categories-lemma-almost-finite-limits-exist", "contents": [ "Let $\\mathcal{C}$ be a category.", "The following are equivalent:", "\\begin{enumerate}", "\\item Nonempty finite limits exist in $\\mathcal{C}$.", "\\item Products of pairs and equalizers exist in $\\mathcal{C}$.", "\\item Products of pairs and fibre products exist in $\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since products of pairs, fibre products, and equalizers are limits with", "nonempty index categories we see that (1) implies both (2) and (3).", "Assume (2). Then finite nonempty products and equalizers exist. Hence by", "Lemma \\ref{lemma-limits-products-equalizers}", "we see that finite nonempty limits exist, i.e., (1) holds. Assume (3).", "If $a, b : A \\to B$ are morphisms of $\\mathcal{C}$, then the", "equalizer of $a, b$ is", "$$", "(A \\times_{a, B, b} A)\\times_{(pr_1, pr_2), A \\times A, \\Delta} A.", "$$", "Thus (3) implies (2), and the lemma is proved." ], "refs": [ "categories-lemma-limits-products-equalizers" ], "ref_ids": [ 12213 ] } ], "ref_ids": [] }, { "id": 12224, "type": "theorem", "label": "categories-lemma-finite-limits-exist", "categories": [ "categories" ], "title": "categories-lemma-finite-limits-exist", "contents": [ "Let $\\mathcal{C}$ be a category.", "The following are equivalent:", "\\begin{enumerate}", "\\item Finite limits exist in $\\mathcal{C}$.", "\\item Finite products and equalizers exist.", "\\item The category has a final object and fibred products exist.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since products of pairs, fibre products, equalizers, and final objects", "are limits over finite index categories we see that (1) implies both (2)", "and (3). By", "Lemma \\ref{lemma-limits-products-equalizers}", "above we see that (2) implies (1). Assume (3).", "Note that the product $A \\times B$ is the fibre product over the", "final object. If $a, b : A \\to B$ are morphisms of $\\mathcal{C}$, then the", "equalizer of $a, b$ is", "$$", "(A \\times_{a, B, b} A)\\times_{(pr_1, pr_2), A \\times A, \\Delta} A.", "$$", "Thus (3) implies (2) and the lemma is proved." ], "refs": [ "categories-lemma-limits-products-equalizers" ], "ref_ids": [ 12213 ] } ], "ref_ids": [] }, { "id": 12225, "type": "theorem", "label": "categories-lemma-push-outs-coequalizers-exist", "categories": [ "categories" ], "title": "categories-lemma-push-outs-coequalizers-exist", "contents": [ "Let $\\mathcal{C}$ be a category.", "The following are equivalent:", "\\begin{enumerate}", "\\item Connected finite colimits exist in $\\mathcal{C}$.", "\\item Coequalizers and pushouts exist in $\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is dual to", "Lemma \\ref{lemma-fibre-products-equalizers-exist}." ], "refs": [ "categories-lemma-fibre-products-equalizers-exist" ], "ref_ids": [ 12222 ] } ], "ref_ids": [] }, { "id": 12226, "type": "theorem", "label": "categories-lemma-almost-finite-colimits-exist", "categories": [ "categories" ], "title": "categories-lemma-almost-finite-colimits-exist", "contents": [ "Let $\\mathcal{C}$ be a category.", "The following are equivalent:", "\\begin{enumerate}", "\\item Nonempty finite colimits exist in $\\mathcal{C}$.", "\\item Coproducts of pairs and coequalizers exist in $\\mathcal{C}$.", "\\item Coproducts of pairs and pushouts exist in $\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is the dual of", "Lemma \\ref{lemma-almost-finite-limits-exist}." ], "refs": [ "categories-lemma-almost-finite-limits-exist" ], "ref_ids": [ 12223 ] } ], "ref_ids": [] }, { "id": 12227, "type": "theorem", "label": "categories-lemma-colimits-exist", "categories": [ "categories" ], "title": "categories-lemma-colimits-exist", "contents": [ "Let $\\mathcal{C}$ be a category.", "The following are equivalent:", "\\begin{enumerate}", "\\item Finite colimits exist in $\\mathcal{C}$,", "\\item Finite coproducts and coequalizers exist in $\\mathcal{C}$, and", "\\item The category has an initial object and pushouts exist.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is dual to Lemma \\ref{lemma-finite-limits-exist}." ], "refs": [ "categories-lemma-finite-limits-exist" ], "ref_ids": [ 12224 ] } ], "ref_ids": [] }, { "id": 12228, "type": "theorem", "label": "categories-lemma-directed-commutes", "categories": [ "categories" ], "title": "categories-lemma-directed-commutes", "contents": [ "Let $\\mathcal{I}$ and $\\mathcal{J}$ be index categories.", "Assume that $\\mathcal{I}$ is filtered and $\\mathcal{J}$ is finite.", "Let $M : \\mathcal{I} \\times \\mathcal{J} \\to \\textit{Sets}$,", "$(i, j) \\mapsto M_{i, j}$ be a diagram of diagrams of sets.", "In this case", "$$", "\\colim_i \\lim_j M_{i, j}", "=", "\\lim_j \\colim_i M_{i, j}.", "$$", "In particular, colimits over $\\mathcal{I}$ commute with finite products,", "fibre products, and equalizers of sets." ], "refs": [], "proofs": [ { "contents": [ "Omitted. In fact, it is a fun exercise to prove that a category is", "filtered if and only if colimits over the category commute with finite", "limits (into the category of sets)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12229, "type": "theorem", "label": "categories-lemma-cofinal-in-filtered", "categories": [ "categories" ], "title": "categories-lemma-cofinal-in-filtered", "contents": [ "Let $\\mathcal{I}$ be a category. Let $\\mathcal{J}$ be a full subcategory.", "Assume that $\\mathcal{I}$ is filtered. Assume also that for any object", "$i$ of $\\mathcal{I}$, there exists a morphism $i \\to j$", "to some object $j$ of $\\mathcal{J}$. Then $\\mathcal{J}$", "is filtered and cofinal in $\\mathcal{I}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Pleasant exercise of the notions involved." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12230, "type": "theorem", "label": "categories-lemma-preserve-products", "categories": [ "categories" ], "title": "categories-lemma-preserve-products", "contents": [ "Let $\\mathcal{I}$ be an index category, i.e., a category. Assume", "that for every pair of objects $x, y$ of $\\mathcal{I}$", "there exists an object $z$ and morphisms $x \\to z$ and $y \\to z$.", "Then", "\\begin{enumerate}", "\\item If $M$ and $N$ are diagrams of sets over $\\mathcal{I}$,", "then $\\colim (M_i \\times N_i) \\to \\colim M_i \\times \\colim N_i$", "is surjective,", "\\item in general colimits of diagrams of sets over $\\mathcal{I}$", "do not commute with finite nonempty products.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $(\\overline{m}, \\overline{n})$", "be an element of $\\colim M_i \\times \\colim N_i$.", "Then we can find $m \\in M_x$ and $n \\in N_y$ for some", "$x, y \\in \\Ob(\\mathcal{I})$ such that $m$ mapsto", "$\\overline{m}$ and $n$ mapsto $\\overline{n}$. See", "Section \\ref{section-limit-sets}.", "Choose $a : x \\to z$ and $b : y \\to z$", "in $\\mathcal{I}$. Then $(M(a)(m), N(b)(n))$ is an element of", "$(M \\times N)_z$ whose image in $\\colim (M_i \\times N_i)$", "maps to $(\\overline{m}, \\overline{n})$ as desired.", "\\medskip\\noindent", "Proof of (2). Let $G$ be a non-trivial group and", "let $\\mathcal{I}$ be the one-object category with endomorphism monoid $G$.", "Then $\\mathcal{I}$ trivially satisfies the condition stated in the lemma.", "Now let $G$ act on itself by translation and view the $G$-set $G$", "as a set-valued $\\mathcal{I}$-diagram. Then", "$$", "\\colim_\\mathcal{I} G \\times \\colim_\\mathcal{I} G \\cong G/G \\times G/G", "$$", "is not isomorphic to", "$$", "\\colim_\\mathcal{I} (G \\times G) \\cong (G \\times G)/G", "$$", "This example indicates that you cannot just drop the additional", "condition Lemma \\ref{lemma-directed-commutes}", "even if you only care about finite products." ], "refs": [ "categories-lemma-directed-commutes" ], "ref_ids": [ 12228 ] } ], "ref_ids": [] }, { "id": 12231, "type": "theorem", "label": "categories-lemma-colimits-abelian-as-sets", "categories": [ "categories" ], "title": "categories-lemma-colimits-abelian-as-sets", "contents": [ "Let $\\mathcal{I}$ be an index category, i.e., a category. Assume", "that for every pair of objects $x, y$ of $\\mathcal{I}$", "there exists an object $z$ and morphisms $x \\to z$ and $y \\to z$.", "Let $M : \\mathcal{I} \\to \\textit{Ab}$ be a diagram of abelian", "groups over $\\mathcal{I}$. Then the colimit of $M$ in the category", "of sets surjects onto the colimit of $M$ in the category of", "abelian groups." ], "refs": [], "proofs": [ { "contents": [ "Recall that the colimit in the category of sets is the quotient of", "the disjoint union $\\coprod M_i$ by relation, see", "Section \\ref{section-limit-sets}.", "Similarly, the colimit in the category of abelian groups is a quotient", "of the direct sum $\\bigoplus M_i$.", "The assumption of the lemma means that given $i, j \\in \\Ob(\\mathcal{I})$", "and $m \\in M_i$ and $n \\in M_j$, then we can find an object", "$k$ and morphisms $a : i \\to k$ and $b : j \\to k$.", "Thus $m + n$ is represented in the colimit by the element", "$M(a)(m) + M(b)(n)$ of $M_k$. Thus the $\\coprod M_i$", "surjects onto the colimit." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12232, "type": "theorem", "label": "categories-lemma-split-into-connected", "categories": [ "categories" ], "title": "categories-lemma-split-into-connected", "contents": [ "Let $\\mathcal{I}$ be an index category, i.e., a category. Assume", "that for every solid diagram", "$$", "\\xymatrix{", "x \\ar[d] \\ar[r] & y \\ar@{..>}[d] \\\\", "z \\ar@{..>}[r] & w", "}", "$$", "in $\\mathcal{I}$ there exists an object $w$ and dotted arrows", "making the diagram commute. Then $\\mathcal{I}$ is either empty", "or a nonempty disjoint union of connected categories having", "the same property." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{I}$ is the empty category, then the lemma is true.", "Otherwise, we define a relation on objects of $\\mathcal{I}$ by", "saying that $x \\sim y$ if there exists a $z$ and", "morphisms $x \\to z$ and $y \\to z$. This is an equivalence", "relation by the assumption of the lemma. Hence $\\Ob(\\mathcal{I})$", "is a disjoint union of equivalence classes. Let $\\mathcal{I}_j$", "be the full subcategories corresponding to these equivalence classes.", "Then $\\mathcal{I} = \\coprod \\mathcal{I}_j$ with $\\mathcal{I}_j$", "nonempty as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12233, "type": "theorem", "label": "categories-lemma-preserve-injective-maps", "categories": [ "categories" ], "title": "categories-lemma-preserve-injective-maps", "contents": [ "Let $\\mathcal{I}$ be an index category, i.e., a category. Assume", "that for every solid diagram", "$$", "\\xymatrix{", "x \\ar[d] \\ar[r] & y \\ar@{..>}[d] \\\\", "z \\ar@{..>}[r] & w", "}", "$$", "in $\\mathcal{I}$ there exists an object $w$ and dotted arrows", "making the diagram commute. Then", "\\begin{enumerate}", "\\item an injective morphism $M \\to N$ of diagrams of sets over", "$\\mathcal{I}$ gives rise to an injective map $\\colim M_i \\to \\colim N_i$", "of sets,", "\\item in general the same is not the case for diagrams of abelian", "groups and their colimits.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{I}$ is the empty category, then the lemma is true.", "Thus we may assume $\\mathcal{I}$ is nonempty. In this case", "we can write $\\mathcal{I} = \\coprod \\mathcal{I}_j$ where each", "$\\mathcal{I}_j$ is nonempty and satisfies the same property, see", "Lemma \\ref{lemma-split-into-connected}. Since", "$\\colim_\\mathcal{I} M = \\coprod_j \\colim_{\\mathcal{I}_j} M|_{\\mathcal{I}_j}$", "this reduces the proof of (1) to the connected case.", "\\medskip\\noindent", "Assume $\\mathcal{I}$ is connected and $M \\to N$ is injective, i.e.,", "all the maps $M_i \\to N_i$ are injective.", "We identify $M_i$ with the image of $M_i \\to N_i$, i.e.,", "we will think of $M_i$ as a subset of $N_i$.", "We will use the description of the colimits given in", "Section \\ref{section-limit-sets} without further mention.", "Let $s, s' \\in \\colim M_i$ map to the same element of $\\colim N_i$.", "Say $s$ comes from an element $m$ of $M_i$ and $s'$ comes from an", "element $m'$ of $M_{i'}$. Then we can find a sequence", "$i = i_0, i_1, \\ldots, i_n = i'$ of objects of $\\mathcal{I}$", "and morphisms", "$$", "\\xymatrix{", "&", "i_1 \\ar[ld] \\ar[rd] & &", "i_3 \\ar[ld] & &", "i_{2n-1} \\ar[rd] & \\\\", "i = i_0 & &", "i_2 & &", "\\ldots & &", "i_{2n} = i'", "}", "$$", "and elements $n_{i_j} \\in N_{i_j}$ mapping to each other under", "the maps $N_{i_{2k-1}} \\to N_{i_{2k-2}}$ and $N_{i_{2k-1}}", "\\to N_{i_{2k}}$ induced from the maps in $\\mathcal{I}$ above", "with $n_{i_0} = m$ and $n_{i_{2n}} = m'$. We will prove by induction", "on $n$ that this implies $s = s'$. The base case $n = 0$ is trivial.", "Assume $n \\geq 1$. Using the assumption on $\\mathcal{I}$", "we find a commutative diagram", "$$", "\\xymatrix{", "& i_1 \\ar[ld] \\ar[rd] \\\\", "i_0 \\ar[rd] & & i_2 \\ar[ld] \\\\", "& w", "}", "$$", "We conclude that $m$ and $n_{i_2}$ map to the same element of $N_w$", "because both are the image of the element $n_{i_1}$.", "In particular, this element is an element $m'' \\in M_w$ which", "gives rise to the same element as $s$ in $\\colim M_i$.", "Then we find the chain", "$$", "\\xymatrix{", "&", "i_3 \\ar[ld] \\ar[rd] & &", "i_5 \\ar[ld] & &", "i_{2n-1} \\ar[rd] & \\\\", "w & &", "i_4 & &", "\\ldots & &", "i_{2n} = i'", "}", "$$", "and the elements $n_{i_j}$ for $j \\geq 3$ which has a smaller length", "than the chain we started with. This proves the induction step and the", "proof of (1) is complete.", "\\medskip\\noindent", "Let $G$ be a group and let $\\mathcal{I}$ be the one-object category with", "endomorphism monoid $G$. Then $\\mathcal{I}$ satisfies the condition stated", "in the lemma because given $g_1, g_2 \\in G$ we can find $h_1, h_2 \\in G$", "with $h_1 g_1 = h_2 g_2$. An diagram $M$ over $\\mathcal{I}$ in", "$\\textit{Ab}$ is the same thing as an abelian group $M$ with $G$-action", "and $\\colim_\\mathcal{I} M$ is the coinvariants $M_G$ of $M$.", "Take $G$ the group of order $2$ acting trivially on $M = \\mathbf{Z}/2\\mathbf{Z}$", "mapping into the first summand of", "$N = \\mathbf{Z}/2\\mathbf{Z} \\times \\mathbf{Z}/2\\mathbf{Z}$", "where the nontrivial element of $G$ acts by", "$(x, y) \\mapsto (x + y, y)$. Then $M_G \\to N_G$ is zero." ], "refs": [ "categories-lemma-split-into-connected" ], "ref_ids": [ 12232 ] } ], "ref_ids": [] }, { "id": 12234, "type": "theorem", "label": "categories-lemma-split-into-directed", "categories": [ "categories" ], "title": "categories-lemma-split-into-directed", "contents": [ "Let $\\mathcal{I}$ be an index category, i.e., a category.", "Assume", "\\begin{enumerate}", "\\item for every pair of morphisms $a : w \\to x$ and $b : w \\to y$", "in $\\mathcal{I}$ there exists an object $z$ and morphisms $c : x \\to z$", "and $d : y \\to z$ such that $c \\circ a = d \\circ b$, and", "\\item for every pair of morphisms $a, b : x \\to y$ there exists", "a morphism $c : y \\to z$ such that $c \\circ a = c \\circ b$.", "\\end{enumerate}", "Then $\\mathcal{I}$ is a (possibly empty) union", "of disjoint filtered index categories $\\mathcal{I}_j$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{I}$ is the empty category, then the lemma is true.", "Otherwise, we define a relation on objects of $\\mathcal{I}$ by", "saying that $x \\sim y$ if there exists a $z$ and", "morphisms $x \\to z$ and $y \\to z$. This is an equivalence", "relation by the first assumption of the lemma. Hence $\\Ob(\\mathcal{I})$", "is a disjoint union of equivalence classes. Let $\\mathcal{I}_j$", "be the full subcategories corresponding to these equivalence classes.", "The rest is clear from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12235, "type": "theorem", "label": "categories-lemma-almost-directed-commutes-equalizers", "categories": [ "categories" ], "title": "categories-lemma-almost-directed-commutes-equalizers", "contents": [ "Let $\\mathcal{I}$ be an index category satisfying the hypotheses of", "Lemma \\ref{lemma-split-into-directed} above. Then colimits over $\\mathcal{I}$", "commute with fibre products and equalizers in sets (and more generally", "with finite connected limits)." ], "refs": [ "categories-lemma-split-into-directed" ], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-split-into-directed}", "we may write $\\mathcal{I} = \\coprod \\mathcal{I}_j$ with each $\\mathcal{I}_j$", "filtered. By", "Lemma \\ref{lemma-directed-commutes}", "we see that colimits of $\\mathcal{I}_j$ commute with equalizers and", "fibred products. Thus it suffices to show that equalizers and fibre products", "commute with coproducts in the category of sets (including empty coproducts).", "In other words, given a set $J$ and sets $A_j, B_j, C_j$ and set maps", "$A_j \\to B_j$, $C_j \\to B_j$ for $j \\in J$ we have to show that", "$$", "(\\coprod\\nolimits_{j \\in J} A_j)", "\\times_{(\\coprod\\nolimits_{j \\in J} B_j)}", "(\\coprod\\nolimits_{j \\in J} C_j)", "=", "\\coprod\\nolimits_{j \\in J} A_j \\times_{B_j} C_j", "$$", "and given $a_j, a'_j : A_j \\to B_j$ that", "$$", "\\text{Equalizer}(", "\\coprod\\nolimits_{j \\in J} a_j,", "\\coprod\\nolimits_{j \\in J} a'_j)", "=", "\\coprod\\nolimits_{j \\in J}", "\\text{Equalizer}(a_j, a'_j)", "$$", "This is true even if $J = \\emptyset$. Details omitted." ], "refs": [ "categories-lemma-split-into-directed", "categories-lemma-directed-commutes" ], "ref_ids": [ 12234, 12228 ] } ], "ref_ids": [ 12234 ] }, { "id": 12236, "type": "theorem", "label": "categories-lemma-directed-category-system", "categories": [ "categories" ], "title": "categories-lemma-directed-category-system", "contents": [ "Let $\\mathcal{I}$ be a filtered index category.", "There exists a directed set $I$", "and a system $(x_i, \\varphi_{ii'})$ over $I$ in $\\mathcal{I}$", "with the following properties:", "\\begin{enumerate}", "\\item For every category $\\mathcal{C}$ and every diagram", "$M : \\mathcal{I} \\to \\mathcal{C}$ with values in $\\mathcal{C}$,", "denote $(M(x_i), M(\\varphi_{ii'}))$", "the corresponding system over $I$. If", "$\\colim_{i \\in I} M(x_i)$ exists then so does", "$\\colim_\\mathcal{I} M$ and the transformation", "$$", "\\theta :", "\\colim_{i \\in I} M(x_i)", "\\longrightarrow", "\\colim_\\mathcal{I} M", "$$", "of Lemma \\ref{lemma-functorial-colimit} is an isomorphism.", "\\item For every category $\\mathcal{C}$ and every diagram", "$M : \\mathcal{I}^{opp} \\to \\mathcal{C}$ in $\\mathcal{C}$, denote", "$(M(x_i), M(\\varphi_{ii'}))$ the corresponding inverse system", "over $I$. If $\\lim_{i \\in I} M(x_i)$ exists then so does", "$\\lim_\\mathcal{I} M$ and the transformation", "$$", "\\theta :", "\\lim_{\\mathcal{I}^{opp}} M", "\\longrightarrow", "\\lim_{i \\in I} M(x_i)", "$$", "of Lemma \\ref{lemma-functorial-limit} is an isomorphism.", "\\end{enumerate}" ], "refs": [ "categories-lemma-functorial-colimit", "categories-lemma-functorial-limit" ], "proofs": [ { "contents": [ "As explained in the text following", "Definition \\ref{definition-system-over-poset}, we may view", "preordered sets as categories and systems as functors.", "Throughout the proof, we will freely shift between these two points of view.", "We prove the first statement by constructing a category", "$\\mathcal{I}_0$, corresponding to a directed set\\footnote{In fact,", "our construction will produce a directed partially ordered set.}, and a cofinal", "functor $M_0 : \\mathcal{I}_0 \\to \\mathcal{I}$. Then, by", "Lemma \\ref{lemma-cofinal}, the colimit of a diagram", "$M : \\mathcal{I} \\to \\mathcal{C}$ coincides with the", "colimit of the diagram $M \\circ M_0 | \\mathcal{I}_0 \\to \\mathcal{C}$,", "from which the statement follows. The second statement is dual to the", "first and may be proved by interpreting a limit in $\\mathcal{C}$ as", "a colimit in $\\mathcal{C}^{opp}$. We omit the details.", "\\medskip\\noindent", "A category $\\mathcal{F}$ is called {\\em finitely generated} if", "there exists a finite set $F$ of arrows in $\\mathcal{F}$, such that", "each arrow in $\\mathcal{F}$ may be obtained by composing", "arrows from $F$. In particular, this implies that $\\mathcal{F}$ has", "finitely many objects. We start the proof by reducing to the case", "when $\\mathcal{I}$ has the property that every finitely generated", "subcategory of $\\mathcal{I}$ may be extended to a finitely", "generated subcategory with a unique final object.", "\\medskip\\noindent", "Let $\\omega$ denote the directed set of finite ordinals, which", "we view as a filtered category. It is easy to verify that the", "product category $\\mathcal{I}\\times \\omega$ is also filtered,", "and the projection", "$\\Pi : \\mathcal{I} \\times \\omega \\to \\mathcal{I}$", "is cofinal.", "\\medskip\\noindent", "Now let $\\mathcal{F}$ be any finitely generated", "subcategory of $\\mathcal{I}\\times \\omega$.", "By using the axioms of a filtered category and a simple induction", "argument on a finite set of generators of $\\mathcal{F}$,", "we may construct a cocone $(\\{f_i\\}, i_\\infty)$ in $\\mathcal{I}$", "for the diagram $\\mathcal{F} \\to \\mathcal{I}$. That is, a morphism", "$f_i : i \\to i_\\infty$ for every object $i$ in $\\mathcal{F}$", "such that for each arrow $f : i \\to i'$ in $\\mathcal{F}$", "we have $f_i = f\\circ f_{i'}$. We can also choose $i_\\infty$ such", "that there are no arrows from $i_\\infty$ to an object in $\\mathcal{F}$.", "This is possible since", "we may always post-compose the arrows $f_i$ with an arrow", "which is the identity on the $\\mathcal{I}$-component and", "strictly increasing on the $\\omega$-component.", "Now let $\\mathcal{F}^+$ denote the category consisting of all", "objects and arrows in $\\mathcal{F}$", "together with the object $i_\\infty$, the identity", "arrow $\\text{id}_{i_\\infty}$ and the arrows $f_i$.", "Since there are no arrows from $i_\\infty$ in $\\mathcal{F}^+$", "to any object of $\\mathcal{F}$, the arrow set in $\\mathcal{F}^+$", "is closed under composition, so $\\mathcal{F}^+$ is indeed", "a category. By construction, it is a finitely", "generated subcategory of $\\mathcal{I}$ which has $i_\\infty$ as", "unique final object. Since, by Lemma \\ref{lemma-cofinal},", "the colimit of any diagram $M : \\mathcal{I} \\to \\mathcal{C}$", "coincides with the colimit of $M\\circ\\Pi$ , this gives the desired", "reduction.", "\\medskip\\noindent", "The set of all finitely generated subcategories of $\\mathcal{I}$", "with a unique final object is naturally ordered by inclusion.", "We take $\\mathcal{I}_0$ to be the category corresponding", "to this set. We also have a functor", "$M_0 : \\mathcal{I}_0 \\to \\mathcal{I}$, which takes an", "arrow $\\mathcal{F} \\subset \\mathcal{F'}$ in", "$\\mathcal{I}_0$ to the unique map from the final object of", "$\\mathcal{F}$ to the final object of $\\mathcal{F}'$.", "Given any two finitely generated subcategories of", "$\\mathcal{I}$, the category generated by these two categories is", "also finitely generated. By our assumption on $\\mathcal{I}$, it is", "also contained in a finitely generated subcategory of $\\mathcal{I}$", "with a unique final object. This shows that $\\mathcal{I}_0$ is directed.", "\\medskip\\noindent", "Finally, we verify that $M_0$ is cofinal. Since any", "object of $\\mathcal{I}$ is the final object in the subcategory", "consisting of only that object and its identity arrow, the functor", "$M_0$ is surjective on objects. In particular, Condition (1) of", "Definition \\ref{definition-cofinal} is satisfied. Given", "an object $i$ of $\\mathcal{I}$, $\\mathcal{F}_1, \\mathcal{F}_2$ in", "$\\mathcal{I}_0$ and maps $\\varphi_1 : i \\to M_0(\\mathcal{F}_1)$", "and $\\varphi_2 : i \\to M_0(\\mathcal{F}_2)$ in", "$\\mathcal{I}$, we can take $\\mathcal{F}_{12}$ to be a finitely", "generated category with a unique final object containing", "$\\mathcal{F}_1$, $\\mathcal{F}_2$ and the morphisms $\\varphi_1, \\varphi_2$.", "The resulting diagram commutes", "$$", "\\xymatrix{", "& M_0(\\mathcal{F}_{12}) & \\\\", "M_0(\\mathcal{F}_{1}) \\ar[ru] & & M_0(\\mathcal{F}_{2}) \\ar[lu] \\\\", "& i \\ar[lu] \\ar[ru]", "}", "$$", "since it lives in the category $\\mathcal{F}_{12}$ and", "$M_0(\\mathcal{F}_{12})$ is final in", "this category. Hence also Condition (2) is satisfied, which concludes", "the proof." ], "refs": [ "categories-definition-system-over-poset", "categories-lemma-cofinal", "categories-lemma-cofinal", "categories-definition-cofinal" ], "ref_ids": [ 12366, 12217, 12217, 12361 ] } ], "ref_ids": [ 12210, 12211 ] }, { "id": 12237, "type": "theorem", "label": "categories-lemma-nonempty-limit", "categories": [ "categories" ], "title": "categories-lemma-nonempty-limit", "contents": [ "If $S : \\mathcal{I} \\to \\textit{Sets}$ is a cofiltered diagram of sets", "and all the $S_i$ are finite nonempty, then $\\lim_i S_i$ is nonempty.", "In other words, the limit of a directed inverse system of finite nonempty sets", "is nonempty." ], "refs": [], "proofs": [ { "contents": [ "The two statements are equivalent by", "Lemma \\ref{lemma-directed-category-system}.", "Let $I$ be a directed set and let $(S_i)_{i \\in I}$", "be an inverse system of finite nonempty sets over $I$.", "Let us say that a {\\it subsystem} $T$ is a family $T = (T_i)_{i \\in I}$", "of nonempty subsets $T_i \\subset S_i$ such that $T_{i'}$ is mapped", "into $T_i$ by the transition map $S_{i'} \\to S_i$ for all $i' \\geq i$.", "Denote $\\mathcal{T}$ the set of subsystems. We order $\\mathcal{T}$", "by inclusion. Suppose $T_\\alpha$, $\\alpha \\in A$ is a totally ordered family", "of elements of $\\mathcal{T}$. Say $T_\\alpha = (T_{\\alpha, i})_{i \\in I}$.", "Then we can find a lower bound $T = (T_i)_{i \\in I}$ by setting", "$T_i = \\bigcap_{\\alpha \\in A} T_{\\alpha, i}$ which is manifestly a", "finite nonempty subset of $S_i$ as all the $T_{\\alpha, i}$ are nonempty", "and as the $T_\\alpha$ form a totally ordered family. Thus we may", "apply Zorn's lemma to see that $\\mathcal{T}$ has minimal elements.", "\\medskip\\noindent", "Let's analyze what a minimal element $T \\in \\mathcal{T}$ looks like.", "First observe that the maps $T_{i'} \\to T_i$ are all surjective.", "Namely, as $I$ is a directed set and $T_i$ is finite,", "the intersection $T'_i = \\bigcap_{i' \\geq i} \\Im(T_{i'} \\to T_i)$", "is nonempty. Thus $T' = (T'_i)$ is a subsystem contained in $T$ and", "by minimality $T' = T$. Finally, we claim that $T_i$ is a singleton", "for each $i$. Namely, if $x \\in T_i$, then we can define", "$T'_{i'} = (T_{i'} \\to T_i)^{-1}(\\{x\\})$ for $i' \\geq i$ and", "$T'_j = T_j$ if $j \\not \\geq i$. This is another subsystem as we've seen", "above that the transition maps of the subsystem $T$ are surjective.", "By minimality we see that $T = T'$ which indeed implies that $T_i$", "is a singleton. This holds for every $i \\in I$, hence we see that", "$T_i = \\{x_i\\}$ for some $x_i \\in S_i$ with $x_{i'} \\mapsto x_i$", "under the map $S_{i'} \\to S_i$ for every $i' \\geq i$. In other words,", "$(x_i) \\in \\lim S_i$ and the lemma is proved." ], "refs": [ "categories-lemma-directed-category-system" ], "ref_ids": [ 12236 ] } ], "ref_ids": [] }, { "id": 12238, "type": "theorem", "label": "categories-lemma-essentially-constant-is-limit-colimit", "categories": [ "categories" ], "title": "categories-lemma-essentially-constant-is-limit-colimit", "contents": [ "Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram.", "If $\\mathcal{I}$ is filtered and $M$ is essentially", "constant as an ind-object, then $X = \\colim M_i$ exists and $M$", "is essentially constant with value $X$.", "If $\\mathcal{I}$ is cofiltered and $M$ is essentially", "constant as a pro-object, then $X = \\lim M_i$ exists and $M$ is", "essentially constant with value $X$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. This is a good excercise in the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12239, "type": "theorem", "label": "categories-lemma-image-essentially-constant", "categories": [ "categories" ], "title": "categories-lemma-image-essentially-constant", "contents": [ "Let $\\mathcal{C}$ be a category. Let $M : \\mathcal{I} \\to \\mathcal{C}$", "be a diagram with filtered (resp.\\ cofiltered) index category $\\mathcal{I}$.", "Let $F : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "If $M$ is essentially constant as an ind-object (resp.\\ pro-object),", "then so is $F \\circ M : \\mathcal{I} \\to \\mathcal{D}$." ], "refs": [], "proofs": [ { "contents": [ "If $X$ is a value for $M$, then it follows immediately from the", "definition that $F(X)$ is a value for $F \\circ M$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12240, "type": "theorem", "label": "categories-lemma-characterize-essentially-constant-ind", "categories": [ "categories" ], "title": "categories-lemma-characterize-essentially-constant-ind", "contents": [ "Let $\\mathcal{C}$ be a category. Let $M : \\mathcal{I} \\to \\mathcal{C}$", "be a diagram with filtered index category $\\mathcal{I}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $M$ is an essentially constant ind-object, and", "\\item $X = \\colim_i M_i$ exists and for any $W$ in $\\mathcal{C}$", "the map", "$$", "\\colim_i \\Mor_\\mathcal{C}(W, M_i) \\longrightarrow", "\\Mor_\\mathcal{C}(W, X)", "$$", "is bijective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (2) holds. Then $\\text{id}_X \\in \\Mor_\\mathcal{C}(X, X)$", "comes from a morphism $X \\to M_i$ for some $i$, i.e., $X \\to M_i \\to X$", "is the identity. Then both maps", "$$", "\\Mor_\\mathcal{C}(W, X)", "\\longrightarrow", "\\colim_i \\Mor_\\mathcal{C}(W, M_i)", "\\longrightarrow", "\\Mor_\\mathcal{C}(W, X)", "$$", "are bijective for all $W$ where the first one is induced by the morphism", "$X \\to M_i$ we found above, and the composition is the identity. This means", "that the composition", "$$", "\\colim_i \\Mor_\\mathcal{C}(W, M_i)", "\\longrightarrow", "\\Mor_\\mathcal{C}(W, X)", "\\longrightarrow", "\\colim_i \\Mor_\\mathcal{C}(W, M_i)", "$$", "is the identity too. Setting $W = M_j$ and starting with $\\text{id}_{M_j}$", "in the colimit, we see that $M_j \\to X \\to M_i \\to M_k$ is equal to", "$M_j \\to M_k$ for some $k$ large enough. This proves (1) holds.", "The proof of (1) $\\Rightarrow$ (2) is omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12241, "type": "theorem", "label": "categories-lemma-characterize-essentially-constant-pro", "categories": [ "categories" ], "title": "categories-lemma-characterize-essentially-constant-pro", "contents": [ "Let $\\mathcal{C}$ be a category. Let $M : \\mathcal{I} \\to \\mathcal{C}$", "be a diagram with cofiltered index category $\\mathcal{I}$.", "The following are equivalent", "\\begin{enumerate}", "\\item $M$ is an essentially constant pro-object, and", "\\item $X = \\lim_i M_i$ exists and for any $W$ in $\\mathcal{C}$", "the map", "$$", "\\colim_{i \\in \\mathcal{I}^{opp}} \\Mor_\\mathcal{C}(M_i, W)", "\\longrightarrow", "\\Mor_\\mathcal{C}(X, W)", "$$", "is bijective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (2) holds. Then $\\text{id}_X \\in \\Mor_\\mathcal{C}(X, X)$", "comes from a morphism $M_i \\to X$ for some $i$, i.e., $X \\to M_i \\to X$", "is the identity. Then both maps", "$$", "\\Mor_\\mathcal{C}(X, W)", "\\longrightarrow", "\\colim_i \\Mor_\\mathcal{C}(M_i, W)", "\\longrightarrow", "\\Mor_\\mathcal{C}(X, W)", "$$", "are bijective for all $W$ where the first one is induced by the morphism", "$M_i \\to X$ we found above, and the composition is the identity. This means", "that the composition", "$$", "\\colim_i \\Mor_\\mathcal{C}(M_i, W)", "\\longrightarrow", "\\Mor_\\mathcal{C}(X, W)", "\\longrightarrow", "\\colim_i \\Mor_\\mathcal{C}(M_i, W)", "$$", "is the identity too. Setting $W = M_j$ and starting with $\\text{id}_{M_j}$", "in the colimit, we see that $M_k \\to M_i \\to X \\to M_j$ is equal to", "$M_k \\to M_j$ for some $k$ large enough. This proves (1) holds.", "The proof of (1) $\\Rightarrow$ (2) is omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12242, "type": "theorem", "label": "categories-lemma-cofinal-essentially-constant", "categories": [ "categories" ], "title": "categories-lemma-cofinal-essentially-constant", "contents": [ "Let $\\mathcal{C}$ be a category. Let $H : \\mathcal{I} \\to \\mathcal{J}$", "be a functor of filtered index categories. If $H$ is cofinal, then", "any diagram $M : \\mathcal{J} \\to \\mathcal{C}$ is essentially constant", "if and only if $M \\circ H$ is essentially constant." ], "refs": [], "proofs": [ { "contents": [ "This follows formally from", "Lemmas \\ref{lemma-characterize-essentially-constant-ind} and", "\\ref{lemma-cofinal}." ], "refs": [ "categories-lemma-characterize-essentially-constant-ind", "categories-lemma-cofinal" ], "ref_ids": [ 12240, 12217 ] } ], "ref_ids": [] }, { "id": 12243, "type": "theorem", "label": "categories-lemma-essentially-constant-over-product", "categories": [ "categories" ], "title": "categories-lemma-essentially-constant-over-product", "contents": [ "Let $\\mathcal{I}$ and $\\mathcal{J}$ be filtered categories and denote", "$p : \\mathcal{I} \\times \\mathcal{J} \\to \\mathcal{J}$ the projection.", "Then $\\mathcal{I} \\times \\mathcal{J}$ is filtered and a diagram", "$M : \\mathcal{J} \\to \\mathcal{C}$ is essentially constant if and only", "if $M \\circ p : \\mathcal{I} \\times \\mathcal{J} \\to \\mathcal{C}$", "is essentially constant." ], "refs": [], "proofs": [ { "contents": [ "We omit the verification that $\\mathcal{I} \\times \\mathcal{J}$ is", "filtered. The equivalence follows from", "Lemma \\ref{lemma-cofinal-essentially-constant}", "because $p$ is cofinal (verification omitted)." ], "refs": [ "categories-lemma-cofinal-essentially-constant" ], "ref_ids": [ 12242 ] } ], "ref_ids": [] }, { "id": 12244, "type": "theorem", "label": "categories-lemma-initial-essentially-constant", "categories": [ "categories" ], "title": "categories-lemma-initial-essentially-constant", "contents": [ "Let $\\mathcal{C}$ be a category. Let $H : \\mathcal{I} \\to \\mathcal{J}$", "be a functor of cofiltered index categories. If $H$ is initial, then", "any diagram $M : \\mathcal{J} \\to \\mathcal{C}$ is essentially constant", "if and only if $M \\circ H$ is essentially constant." ], "refs": [], "proofs": [ { "contents": [ "This follows formally from", "Lemmas \\ref{lemma-characterize-essentially-constant-pro},", "\\ref{lemma-initial}, \\ref{lemma-cofinal}, and", "the fact that if $\\mathcal{I}$ is initial in $\\mathcal{J}$,", "then $\\mathcal{I}^{opp}$ is cofinal in $\\mathcal{J}^{opp}$." ], "refs": [ "categories-lemma-characterize-essentially-constant-pro", "categories-lemma-initial", "categories-lemma-cofinal" ], "ref_ids": [ 12241, 12218, 12217 ] } ], "ref_ids": [] }, { "id": 12245, "type": "theorem", "label": "categories-lemma-characterize-left-exact", "categories": [ "categories" ], "title": "categories-lemma-characterize-left-exact", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.", "Suppose all finite limits exist in $\\mathcal{A}$,", "see Lemma \\ref{lemma-finite-limits-exist}.", "The following are equivalent:", "\\begin{enumerate}", "\\item $F$ is left exact,", "\\item $F$ commutes with finite products and equalizers, and", "\\item $F$ transforms a final object of $\\mathcal{A}$", "into a final object of $\\mathcal{B}$, and commutes with fibre products.", "\\end{enumerate}" ], "refs": [ "categories-lemma-finite-limits-exist" ], "proofs": [ { "contents": [ "Lemma \\ref{lemma-limits-products-equalizers} shows that (2) implies (1).", "Suppose (3) holds. The fibre product over the final object is the product.", "If $a, b : A \\to B$ are morphisms of $\\mathcal{A}$, then the", "equalizer of $a, b$ is", "$$", "(A \\times_{a, B, b} A)\\times_{(pr_1, pr_2), A \\times A, \\Delta} A.", "$$", "Thus (3) implies (2). Finally (1) implies (3) because", "the empty limit is a final object, and fibre products are limits." ], "refs": [ "categories-lemma-limits-products-equalizers" ], "ref_ids": [ 12213 ] } ], "ref_ids": [ 12224 ] }, { "id": 12246, "type": "theorem", "label": "categories-lemma-adjoint-exists", "categories": [ "categories" ], "title": "categories-lemma-adjoint-exists", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor between categories.", "If for each $y \\in \\Ob(\\mathcal{D})$ the functor", "$x \\mapsto \\Mor_\\mathcal{D}(u(x), y)$ is representable, then", "$u$ has a right adjoint." ], "refs": [], "proofs": [ { "contents": [ "For each $y$ choose an object $v(y)$ and an isomorphism", "$\\Mor_\\mathcal{C}(-, v(y)) \\to \\Mor_\\mathcal{D}(u(-), y)$", "of functors. By Yoneda's lemma (Lemma \\ref{lemma-yoneda})", "for any morphism $g : y \\to y'$ the transformation of functors", "$$", "\\Mor_\\mathcal{C}(-, v(y)) \\to \\Mor_\\mathcal{D}(u(-), y) \\to", "\\Mor_\\mathcal{D}(u(-), y') \\to \\Mor_\\mathcal{C}(-, v(y'))", "$$", "corresponds to a unique morphism $v(g) : v(y) \\to v(y')$.", "We omit the verification that $v$ is a functor and that", "it is right adjoint to $u$." ], "refs": [ "categories-lemma-yoneda" ], "ref_ids": [ 12203 ] } ], "ref_ids": [] }, { "id": 12247, "type": "theorem", "label": "categories-lemma-left-adjoint-composed-fully-faithful", "categories": [ "categories" ], "title": "categories-lemma-left-adjoint-composed-fully-faithful", "contents": [ "\\begin{reference}", "Bhargav Bhatt, private communication.", "\\end{reference}", "Let $u$ be a left adjoint to $v$ as in Definition \\ref{definition-adjoint}.", "\\begin{enumerate}", "\\item If $v \\circ u$ is fully faithful, then $u$ is fully faithful.", "\\item If $u \\circ v$ is fully faithful, then $v$ is fully faithful.", "\\end{enumerate}" ], "refs": [ "categories-definition-adjoint" ], "proofs": [ { "contents": [ "Proof of (2). Assume $u \\circ v$ is fully faithful.", "Say we have $X$, $Y$ in $\\mathcal{D}$.", "Then the natural composite map", "$$", "\\Mor(X,Y) \\to \\Mor(v(X),v(Y)) \\to \\Mor(u(v(X)), u(v(Y)))", "$$", "is a bijection, so $v$ is at least faithful. To show full faithfulness,", "we must show that the second map above is injective.", "But the adjunction between $u$ and $v$ says that", "$$", "\\Mor(v(X), v(Y)) \\to \\Mor(u(v(X)), u(v(Y))) \\to \\Mor(u(v(X)), Y)", "$$", "is a bijection, where the first map is natural one and", "the second map comes from the counit $u(v(Y)) \\to Y$ of the adjunction.", "So this says that", "$\\Mor(v(X), v(Y)) \\to \\Mor(u(v(X)), u(v(Y)))$", "is also injective, as wanted. The proof of (1) is dual to this." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12371 ] }, { "id": 12248, "type": "theorem", "label": "categories-lemma-adjoint-fully-faithful", "categories": [ "categories" ], "title": "categories-lemma-adjoint-fully-faithful", "contents": [ "Let $u$ be a left adjoint to $v$ as in Definition \\ref{definition-adjoint}.", "Then", "\\begin{enumerate}", "\\item $u$ is fully faithful $\\Leftrightarrow$ $\\text{id} \\cong v \\circ u$", "$\\Leftrightarrow$ $\\eta : \\textit{id} \\to v \\circ u$ is an isomorphism,", "\\item $v$ is fully faithful $\\Leftrightarrow$", "$u \\circ v \\cong \\text{id}$ $\\Leftrightarrow$", "$\\epsilon : u \\circ v \\to \\text{id}$ is an isomorphism.", "\\end{enumerate}" ], "refs": [ "categories-definition-adjoint" ], "proofs": [ { "contents": [ "Proof of (1).", "Assume $u$ is fully faithful. We will show $\\eta_X : X \\to v(u(X))$", "is an isomorphism. Let $X' \\to v(u(X))$ be any morphism.", "By adjointness this corresponds to a morphism $u(X') \\to u(X)$. By fully", "faithfulness of $u$ this corresponds to a unique morphism $X' \\to X$.", "Thus we see that post-composing by $\\eta_X$ defines a bijection", "$\\Mor(X', X) \\to \\Mor(X', v(u(X)))$. Hence $\\eta_X$ is an isomorphism.", "If there exists an isomorphism $\\text{id} \\cong v \\circ u$ of functors,", "then $v \\circ u$ is fully faithful. By", "Lemma \\ref{lemma-left-adjoint-composed-fully-faithful} we see", "that $u$ is fully faithful. By the above this implies $\\eta$", "is an isomorphism. Thus all $3$ conditions are equivalent (and these", "conditions are also equivalent to $v \\circ u$ being fully faithful).", "\\medskip\\noindent", "Part (2) is dual to part (1)." ], "refs": [ "categories-lemma-left-adjoint-composed-fully-faithful" ], "ref_ids": [ 12247 ] } ], "ref_ids": [ 12371 ] }, { "id": 12249, "type": "theorem", "label": "categories-lemma-adjoint-exact", "categories": [ "categories" ], "title": "categories-lemma-adjoint-exact", "contents": [ "Let $u$ be a left adjoint to $v$ as in Definition \\ref{definition-adjoint}.", "\\begin{enumerate}", "\\item Suppose that $M : \\mathcal{I} \\to \\mathcal{C}$ is a diagram,", "and suppose that $\\colim_\\mathcal{I} M$ exists in", "$\\mathcal{C}$. Then $u(\\colim_\\mathcal{I} M) =", "\\colim_\\mathcal{I} u \\circ M$. In other words,", "$u$ commutes with (representable) colimits.", "\\item Suppose that $M : \\mathcal{I} \\to \\mathcal{D}$ is a diagram,", "and suppose that $\\lim_\\mathcal{I} M$ exists in", "$\\mathcal{D}$. Then $v(\\lim_\\mathcal{I} M) =", "\\lim_\\mathcal{I} v \\circ M$. In other words $v$ commutes", "with representable limits.", "\\end{enumerate}" ], "refs": [ "categories-definition-adjoint" ], "proofs": [ { "contents": [ "A morphism from a colimit into an object is the same as a compatible", "system of morphisms from the constituents of the limit into the", "object, see Remark \\ref{remark-limit-colim}. So", "$$", "\\begin{matrix}", "\\Mor_\\mathcal{D}(u(\\colim_{i \\in \\mathcal{I}} M_i), Y) &", "= & \\Mor_\\mathcal{C}(\\colim_{i \\in \\mathcal{I}} M_i, v(Y)) \\\\", "& = &", "\\lim_{i \\in \\mathcal{I}^{opp}}", "\\Mor_\\mathcal{C}(M_i, v(Y)) \\\\", "& = &", "\\lim_{i \\in \\mathcal{I}^{opp}}", "\\Mor_\\mathcal{D}(u(M_i), Y)", "\\end{matrix}", "$$", "proves that $u(\\colim_{i \\in \\mathcal{I}} M_i)$ is", "the colimit we are looking for.", "A similar argument works for the other statement." ], "refs": [ "categories-remark-limit-colim" ], "ref_ids": [ 12415 ] } ], "ref_ids": [ 12371 ] }, { "id": 12250, "type": "theorem", "label": "categories-lemma-exact-adjoint", "categories": [ "categories" ], "title": "categories-lemma-exact-adjoint", "contents": [ "Let $u$ be a left adjoint of $v$ as in Definition \\ref{definition-adjoint}.", "\\begin{enumerate}", "\\item If $\\mathcal{C}$ has finite colimits, then $u$ is right exact.", "\\item If $\\mathcal{D}$ has finite limits, then $v$ is left exact.", "\\end{enumerate}" ], "refs": [ "categories-definition-adjoint" ], "proofs": [ { "contents": [ "Obvious from the definitions and Lemma \\ref{lemma-adjoint-exact}." ], "refs": [ "categories-lemma-adjoint-exact" ], "ref_ids": [ 12249 ] } ], "ref_ids": [ 12371 ] }, { "id": 12251, "type": "theorem", "label": "categories-lemma-transformation-between-functors-and-adjoints", "categories": [ "categories" ], "title": "categories-lemma-transformation-between-functors-and-adjoints", "contents": [ "Let $u_1, u_2 : \\mathcal{C} \\to \\mathcal{D}$ be functors with right", "adjoints $v_1, v_2 : \\mathcal{D} \\to \\mathcal{C}$. Let $\\beta : u_2 \\to u_1$", "be a transformation of functors. Let $\\beta^\\vee : v_1 \\to v_2$ be", "the corresponding transformation of adjoint functors. Then", "$$", "\\xymatrix{", "u_2 \\circ v_1 \\ar[r]_\\beta \\ar[d]_{\\beta^\\vee} &", "u_1 \\circ v_1 \\ar[d] \\\\", "u_2 \\circ v_2 \\ar[r] & \\text{id}", "}", "$$", "is commutative where the unlabeled arrows are the counit transformations." ], "refs": [], "proofs": [ { "contents": [ "This is true because $\\beta^\\vee_D : v_1D \\to v_2D$ is the unique", "morphism such that the induced maps $\\Mor(C, v_1D) \\to \\Mor(C, v_2D)$", "is the map $\\Mor(u_1C, D) \\to \\Mor(u_2C, D)$ induced by", "$\\beta_C : u_2C \\to u_1C$. Namely, this means the map", "$$", "\\Mor(u_1 v_1 D, D') \\to \\Mor(u_2 v_1 D, D')", "$$", "induced by $\\beta_{v_1 D}$ is the same as the map", "$$", "\\Mor(v_1 D, v_1 D') \\to \\Mor(v_1 D, v_2 D')", "$$", "induced by $\\beta^\\vee_{D'}$. Taking $D' = D$ we find that the counit", "$u_1 v_1 D \\to D$ precomposed by $\\beta_{v_1D}$ corresponds to $\\beta^\\vee_D$", "under adjunction. This exactly means that the diagram commutes when", "evaluated on $D$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12252, "type": "theorem", "label": "categories-lemma-compose-counits", "categories": [ "categories" ], "title": "categories-lemma-compose-counits", "contents": [ "Let $\\mathcal{A}$, $\\mathcal{B}$, and $\\mathcal{C}$ be categories.", "Let $v : \\mathcal{A} \\to \\mathcal{B}$ and", "$v' : \\mathcal{B} \\to \\mathcal{C}$ be functors", "with left adjoints $u$ and $u'$ respectively. Then", "\\begin{enumerate}", "\\item The functor $v'' = v' \\circ v$ has a left adjoint equal to", "$u'' = u \\circ u'$.", "\\item Given $X$ in $\\mathcal{A}$ we have", "\\begin{equation}", "\\label{equation-compose-counits}", "\\epsilon_X^v \\circ u(\\epsilon^{v'}_{v(X)}) = \\epsilon^{v''}_X :", "u''(v''(X)) \\to X", "\\end{equation}", "Where $\\epsilon$ is the counit of the adjunctions.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us unwind the formula in (2) because this will also immediately", "prove (1). First, the counit of the adjunctions for the pairs", "$(u, v)$ and $(u', v')$ are maps", "$\\epsilon_X^v : u(v(X)) \\to X$ and", "$\\epsilon_Y^{v'} : u'(v'(Y)) \\to Y$, see discussion following", "Definition \\ref{definition-adjoint}.", "With $u''$ and $v''$ as in (1) we unwind everything", "$$", "u''(v''(X)) = u(u'(v'(v(X)))) \\xrightarrow{u(\\epsilon_{v(X)}^{v'})}", "u(v(X)) \\xrightarrow{\\epsilon_X^v} X", "$$", "to get the map on the left hand side of (\\ref{equation-compose-counits}).", "Let us denote this by $\\epsilon_X^{v''}$ for now.", "To see that this is the counit of an adjoint pair", "$(u'', v'')$ we have to show that given $Z$ in $\\mathcal{C}$", "the rule that sends a morphism $\\beta : Z \\to v''(X)$", "to $\\alpha = \\epsilon_X^{v''} \\circ u''(\\beta) : u''(Z) \\to X$", "is a bijection on sets of morphisms.", "This is true because, this is the composition of the", "rule sending $\\beta$ to $\\epsilon_{v(X)}^{v'} \\circ u'(\\beta)$", "which is a bijection by assumption on $(u', v')$ and then", "sending this to", "$\\epsilon_X^v \\circ u(\\epsilon_{v(X)}^{v'} \\circ u'(\\beta))$", "which is a bijection by assumption on $(u, v)$." ], "refs": [ "categories-definition-adjoint" ], "ref_ids": [ 12371 ] } ], "ref_ids": [] }, { "id": 12253, "type": "theorem", "label": "categories-lemma-a-version-of-brown", "categories": [ "categories" ], "title": "categories-lemma-a-version-of-brown", "contents": [ "Let $\\mathcal{C}$ be a big\\footnote{See Remark \\ref{remark-big-categories}.}", "category which has limits. Let $F : \\mathcal{C} \\to \\textit{Sets}$ be a", "functor. Assume that", "\\begin{enumerate}", "\\item $F$ commutes with limits,", "\\item there exists a family $\\{x_i\\}_{i \\in I}$ of objects of $\\mathcal{C}$", "and for each $i \\in I$ an element $f_i \\in F(x_i)$", "such that for $y \\in \\Ob(\\mathcal{C})$ and $g \\in F(y)$", "there exists an $i$ and a morphism $\\varphi : x_i \\to y$", "with $F(\\varphi)(f_i) = g$.", "\\end{enumerate}", "Then $F$ is representable, i.e., there exists an object $x$", "of $\\mathcal{C}$ such that", "$$", "F(y) = \\Mor_\\mathcal{C}(x, y)", "$$", "functorially in $y$." ], "refs": [ "categories-remark-big-categories" ], "proofs": [ { "contents": [ "Let $\\mathcal{I}$ be the category whose objects are the pairs $(x_i, f_i)$", "and whose morphisms $(x_i, f_i) \\to (x_{i'}, f_{i'})$ are maps", "$\\varphi : x_i \\to x_{i'}$ in $\\mathcal{C}$", "such that $F(\\varphi)(f_i) = f_{i'}$. Set", "$$", "x = \\lim_{(x_i, f_i) \\in \\mathcal{I}} x_i", "$$", "(this will not be the $x$ we are looking for, see below).", "The limit exists by assumption. As $F$ commutes with limits", "we have", "$$", "F(x) = \\lim_{(x_i, f_i) \\in \\mathcal{I}} F(x_i).", "$$", "Hence there is a universal element $f \\in F(x)$ which maps to $f_i \\in F(x_i)$", "under $F$ applied to the projection map $x \\to x_i$.", "Using $f$ we obtain a transformation of functors", "$$", "\\xi : \\Mor_\\mathcal{C}(x, - ) \\longrightarrow F(-)", "$$", "see Section \\ref{section-opposite}. Let $y$ be an arbitrary object of", "$\\mathcal{C}$ and let $g \\in F(y)$. Choose $x_i \\to y$ such that $f_i$", "maps to $g$ which is possible by assumption. Then $F$ applied to the maps", "$$", "x \\longrightarrow x_i \\longrightarrow y", "$$", "(the first being the projection map of the limit defining $x$)", "sends $f$ to $g$. Hence the transformation $\\xi$ is surjective.", "\\medskip\\noindent", "In order to find the object representing $F$ we let $e : x' \\to x$ be the", "equalizer of all self maps $\\varphi : x \\to x$ with $F(\\varphi)(f) = f$.", "Since $F$ commutes with limits, it commutes with equalizers, and", "we see there exists an $f' \\in F(x')$ mapping to $f$ in $F(x)$.", "Since $\\xi$ is surjective and since $f'$ maps to $f$ we see that", "also $\\xi' : \\Mor_\\mathcal{C}(x', -) \\to F(-)$ is surjective.", "Finally, suppose that $a, b : x' \\to y$ are two maps such that", "$F(a)(f') = F(b)(f')$. We have to show $a = b$. Consider the equalizer", "$e' : x'' \\to x'$. Again we find $f'' \\in F(x'')$ mapping to $f'$.", "Choose a map $\\psi : x \\to x''$ such that $F(\\psi)(f) = f''$.", "Then we see that $e \\circ e' \\circ \\psi : x \\to x$ is a morphism", "with $F(e \\circ e' \\circ \\psi)(f) = f$. Hence", "$e \\circ e' \\circ \\psi \\circ e = e$. Since $e$ is a monomorphism,", "this implies that $e'$ is an epimorphism, thus $a = b$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12410 ] }, { "id": 12254, "type": "theorem", "label": "categories-lemma-extend-functor-by-colim", "categories": [ "categories" ], "title": "categories-lemma-extend-functor-by-colim", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be big categories having filtered", "colimits. Let $\\mathcal{C}' \\subset \\mathcal{C}$ be a small full subcategory", "consisting of categorically compact objects of $\\mathcal{C}$ such that every", "object of $\\mathcal{C}$ is a filtered colimit of objects of $\\mathcal{C}'$.", "Then every functor $F' : \\mathcal{C}' \\to \\mathcal{D}$ has a unique", "extension $F : \\mathcal{C} \\to \\mathcal{D}$ commuting with filtered colimits." ], "refs": [], "proofs": [ { "contents": [ "For every object $X$ of $\\mathcal{C}$ we may write $X$ as a filtered", "colimit $X = \\colim X_i$ with $X_i \\in \\Ob(\\mathcal{C}')$. Then we set", "$$", "F(X) = \\colim F'(X_i)", "$$", "in $\\mathcal{D}$. We will show below that this construction does not", "depend on the choice of the colimit presentation of $X$.", "\\medskip\\noindent", "Suppose given a morphism $\\alpha : X \\to Y$ of $\\mathcal{C}$", "and $X = \\colim_{i \\in I} X_i$ and $Y = \\colim_{j \\in J} Y_i$", "are written as filtered colimit of objects in $\\mathcal{C}'$.", "For each $i \\in I$ since $X_i$ is a categorically compact object of", "$\\mathcal{C}$ we can find a $j \\in J$ and a commutative diagram", "$$", "\\xymatrix{", "X_i \\ar[r] \\ar[d] & X \\ar[d]^\\alpha \\\\", "Y_j \\ar[r] & Y", "}", "$$", "Then we obtain a morphism $F'(X_i) \\to F'(Y_j) \\to F(Y)$ where the second", "morphism is the coprojection into $F(Y) = \\colim F'(Y_j)$. The arrow", "$\\beta_i : F'(X_i) \\to F(Y)$ does not depend on the choice of $j$.", "For $i \\leq i'$ the composition", "$$", "F'(X_i) \\to F'(X_{i'}) \\xrightarrow{\\beta_{i'}} F(Y)", "$$", "is equal to $\\beta_i$. Thus we obtain a well defined arrow", "$$", "F(\\alpha) : F(X) = \\colim F(X_i) \\to F(Y)", "$$", "by the universal property of the colimit. If $\\alpha' : Y \\to Z$ is a", "second morphism of $\\mathcal{C}$ and $Z = \\colim Z_k$ is also", "written as filtered colimit of objects in $\\mathcal{C}'$, then", "it is a pleasant exercise to show that the induced", "morphisms $F(\\alpha) : F(X) \\to F(Y)$ and $F(\\alpha') : F(Y) \\to F(Z)$", "compose to the morphism $F(\\alpha' \\circ \\alpha)$. Details omitted.", "\\medskip\\noindent", "In particular, if we are given two presentations", "$X = \\colim X_i$ and $X = \\colim X'_{i'}$ as filtered", "colimits of systems in $\\mathcal{C}'$, then we get mutually inverse", "arrows $\\colim F'(X_i) \\to \\colim F'(X'_{i'})$ and", "$\\colim F'(X'_{i'}) \\to \\colim F'(X_i)$. In other words, the", "value $F(X)$ is well defined independent of the choice of the", "presentation of $X$ as a filtered colimit of objects of $\\mathcal{C}'$.", "Together with the functoriality of $F$ discussed in the previous", "paragraph, we find that $F$ is a functor. Also, it is clear that", "$F(X) = F'(X)$ if $X \\in \\Ob(\\mathcal{C}')$.", "\\medskip\\noindent", "The uniqueness statement in the lemma is clear, provided", "we show that $F$ commutes with filtered colimits (because this", "statement doesn't make sense otherwise). To show this, suppose that", "$X = \\colim_{\\lambda \\in \\Lambda} X_\\lambda$", "is a filtered colimit of $\\mathcal{C}$. Since $F$ is a functor", "we certainly get a map", "$$", "\\colim_\\lambda F(X_\\lambda) \\longrightarrow F(X)", "$$", "On the other hand, write $X = \\colim X_i$", "as a filtered colimit of objects of $\\mathcal{C}'$. ", "As above, for each $i \\in I$ we can choose a $\\lambda \\in \\Lambda$", "and a commutative diagram", "$$", "\\xymatrix{", "X_i \\ar[rr] \\ar[rd] & & X_\\lambda \\ar[ld] \\\\", "& X", "}", "$$", "As above this determines a well defined morphism", "$F'(X_i) \\to \\colim_\\lambda F(X_\\lambda)$ compatible", "with transition morphisms and hence a morphism", "$$", "F(X) = \\colim_i F'(X_i) \\longrightarrow \\colim_\\lambda F(X_\\lambda)", "$$", "This morphism is inverse to the morphism above (details omitted)", "and proves that $F(X) = \\colim_\\lambda F(X_\\lambda)$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12255, "type": "theorem", "label": "categories-lemma-left-localization", "categories": [ "categories" ], "title": "categories-lemma-left-localization", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative", "system.", "\\begin{enumerate}", "\\item The relation on pairs defined above is an equivalence relation.", "\\item The composition rule given above is well defined on equivalence", "classes.", "\\item Composition is associative (and the identity morphisms satisfy", "the identity axioms), and hence $S^{-1}\\mathcal{C}$ is a category.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let us say two pairs $p_1 = (f_1 : X \\to Y_1, s_1 : Y \\to Y_1)$", "and $p_2 = (f_2 : X \\to Y_2, s_2 : Y \\to Y_2)$ are elementary equivalent", "if there exists a morphism $a : Y_1 \\to Y_2$ of $\\mathcal{C}$ such that", "$a \\circ f_1 = f_2$ and $a \\circ s_1 = s_2$. Diagram:", "$$", "\\xymatrix{", "X \\ar@{=}[d] \\ar[r]_{f_1} & Y_1 \\ar[d]^a & Y \\ar[l]^{s_1} \\ar@{=}[d] \\\\", "X \\ar[r]^{f_2} & Y_2 & Y \\ar[l]_{s_2}", "}", "$$", "Let us denote this property by saying $p_1Ep_2$.", "Note that $pEp$ and $aEb, bEc \\Rightarrow aEc$.", "(Despite its name, $E$ is not an equivalence", "relation.)", "Part (1) claims that the relation", "$p \\sim p' \\Leftrightarrow \\exists q: pEq \\wedge p'Eq$", "(where $q$ is supposed to be a pair satisfying the", "same conditions as $p$ and $p'$)", "is an equivalence relation. A simple formal argument, using the properties", "of $E$ above, shows that it suffices to prove", "$p_3Ep_1, p_3Ep_2 \\Rightarrow p_1 \\sim p_2$.", "Thus suppose that we are given a commutative diagram", "$$", "\\xymatrix{", " & Y_1 & \\\\", "X \\ar[ru]^{f_1} \\ar[r]^{f_3} \\ar[rd]_{f_2} &", "Y_3 \\ar[u]_{a_{31}} \\ar[d]^{a_{32}} &", "Y \\ar[lu]_{s_1} \\ar[l]_{s_3} \\ar[ld]^{s_2} \\\\", "& Y_2 &", "}", "$$", "with $s_i \\in S$.", "First we apply LMS2 to get a commutative diagram", "$$", "\\xymatrix{", "Y \\ar[d]_{s_1} \\ar[r]_{s_2} & Y_2 \\ar@{..>}[d]^{a_{24}} \\\\", "Y_1 \\ar@{..>}[r]^{a_{14}} & Y_4", "}", "$$", "with $a_{24} \\in S$. Then, we have", "$$", "a_{14} \\circ a_{31} \\circ s_3 =", "a_{14} \\circ s_1 =", "a_{24} \\circ s_2 =", "a_{24} \\circ a_{32} \\circ s_3.", "$$", "Hence, by LMS3, there exists a", "morphism $s_{44} : Y_4 \\to Y'_4$ such that $s_{44} \\in S$ and", "$s_{44} \\circ a_{14} \\circ a_{31}", "= s_{44} \\circ a_{24} \\circ a_{32}$.", "Hence, after replacing $Y_4$, $a_{14}$ and $a_{24}$ by $Y'_4$,", "$s_{44} \\circ a_{14}$ and $s_{44} \\circ a_{24}$, we may assume", "that $a_{14} \\circ a_{31} = a_{24} \\circ a_{32}$ (and", "we still have $a_{24} \\in S$ and", "$a_{14} \\circ s_1 = a_{24} \\circ s_2$). Set", "$$", "f_4 =", "a_{14} \\circ f_1 =", "a_{14} \\circ a_{31} \\circ f_3 =", "a_{24} \\circ a_{32} \\circ f_3 =", "a_{24} \\circ f_2", "$$", "and", "$s_4 = a_{14} \\circ s_1 = a_{24} \\circ s_2$. Then, the diagram", "$$", "\\xymatrix{", "X \\ar@{=}[d] \\ar[r]_{f_1} & Y_1 \\ar[d]^{a_{14}} & Y \\ar[l]^{s_1} \\ar@{=}[d] \\\\", "X \\ar[r]^{f_4} & Y_4 & Y \\ar[l]_{s_4}", "}", "$$", "commutes, and we have $s_4 \\in S$ (by LMS1). Thus, $p_1 E p_4$,", "where $p_4 = (f_4, s_4)$. Similarly, $p_2 E p_4$. Combining these,", "we find $p_1 \\sim p_2$.", "\\medskip\\noindent", "Proof of (2). Let $p = (f : X \\to Y', s : Y \\to Y')$ and", "$q = (g : Y \\to Z', t : Z \\to Z')$ be pairs as in the definition of composition", "above. To compose we choose a diagram", "$$", "\\xymatrix{", "Y \\ar[d]_s \\ar[r]_g & Z' \\ar[d]^{u_2} \\\\", "Y' \\ar[r]^{h_2} & Z_2", "}", "$$", "with $u_2 \\in S$. We first show that the equivalence class of the pair", "$r_2 = (h_2 \\circ f : X \\to Z_2, u_2 \\circ t : Z \\to Z_2)$", "is independent of the choice of $(Z_2, h_2, u_2)$. Namely, suppose", "that $(Z_3, h_3, u_3)$ is another choice with corresponding composition", "$r_3 = (h_3 \\circ f : X \\to Z_3, u_3 \\circ t : Z \\to Z_3)$.", "Then by LMS2 we can choose a diagram", "$$", "\\xymatrix{", "Z' \\ar[d]_{u_2} \\ar[r]_{u_3} & Z_3 \\ar[d]^{u_{34}} \\\\", "Z_2 \\ar[r]^{h_{24}} & Z_4", "}", "$$", "with $u_{34} \\in S$. We have $h_2 \\circ s = u_2 \\circ g$", "and similarly $h_3 \\circ s = u_3 \\circ g$. Now,", "$$", "u_{34} \\circ h_3 \\circ s", "= u_{34} \\circ u_3 \\circ g", "= h_{24} \\circ u_2 \\circ g", "= h_{24} \\circ h_2 \\circ s.", "$$", "Hence, LMS3 shows that there", "exists a $Z'_4$ and an $s_{44} : Z_4 \\to Z'_4$ such that", "$s_{44} \\circ u_{34} \\circ h_3 = s_{44} \\circ h_{24} \\circ h_2$.", "Replacing $Z_4$, $h_{24}$ and $u_{34}$ by $Z'_4$,", "$s_{44} \\circ h_{24}$ and $s_{44} \\circ u_{34}$, we may", "assume that $u_{34} \\circ h_3 = h_{24} \\circ h_2$.", "Meanwhile, the relations $u_{34} \\circ u_3 = h_{24} \\circ u_2$", "and $u_{34} \\in S$ continue to hold. We can now set", "$h_4 = u_{34} \\circ h_3 = h_{24} \\circ h_2$ and", "$u_4 = u_{34} \\circ u_3 = h_{24} \\circ u_2$. Then, we have a", "commutative diagram", "$$", "\\xymatrix{", "X \\ar@{=}[d] \\ar[r]_{h_2\\circ f} &", "Z_2 \\ar[d]^{h_{24}} &", "Z \\ar[l]^{u_2 \\circ t} \\ar@{=}[d] \\\\", "X \\ar@{=}[d] \\ar[r]^{h_4\\circ f} &", "Z_4 &", "Z \\ar@{=}[d] \\ar[l]_{u_4 \\circ t} \\\\", "X \\ar[r]^{h_3 \\circ f} &", "Z_3 \\ar[u]^{u_{34}} &", "Z \\ar[l]_{u_3 \\circ t}", "}", "$$", "Hence we obtain a pair", "$r_4 =", "(h_4 \\circ f : X \\to Z_4, u_4 \\circ t : Z \\to Z_4)$", "and the above diagram shows that we have $r_2Er_4$ and", "$r_3Er_4$, whence $r_2 \\sim r_3$, as desired. Thus it now", "makes sense to define $p \\circ q$ as the equivalence class of", "all possible pairs $r$ obtained as above.", "\\medskip\\noindent", "To finish the proof of (2) we have to show that given pairs", "$p_1, p_2, q$ such that $p_1Ep_2$ then $p_1 \\circ q = p_2 \\circ q$ and", "$q \\circ p_1 = q \\circ p_2$ whenever the compositions make sense.", "To do this, write $p_1 = (f_1 : X \\to Y_1, s_1 : Y \\to Y_1)$ and", "$p_2 = (f_2 : X \\to Y_2, s_2 : Y \\to Y_2)$ and let", "$a : Y_1 \\to Y_2$ be a morphism of $\\mathcal{C}$ such that", "$f_2 = a \\circ f_1$ and $s_2 = a \\circ s_1$.", "First assume that $q = (g : Y \\to Z', t : Z \\to Z')$.", "In this case choose a commutative diagram as the one on the left", "$$", "\\vcenter{", "\\xymatrix{", "Y \\ar[d]_{s_2} \\ar[r]^g & Z' \\ar[d]^u \\\\", "Y_2 \\ar[r]^h & Z''", "}", "}", "\\quad", "\\Rightarrow", "\\quad", "\\vcenter{", "\\xymatrix{", "Y \\ar[d]_{s_1} \\ar[r]^g & Z' \\ar[d]^u \\\\", "Y_1 \\ar[r]^{h \\circ a} & Z''", "}", "}", "$$", "(with $u \\in S$),", "which implies the diagram on the right is commutative as well.", "Using these diagrams we see that both compositions $q \\circ p_1$", "and $q \\circ p_2$ are the equivalence class of", "$(h \\circ a \\circ f_1 : X \\to Z'', u \\circ t : Z \\to Z'')$.", "Thus $q \\circ p_1 = q \\circ p_2$.", "The proof of the other case, in which we have to show", "$p_1 \\circ q = p_2 \\circ q$, is omitted. (It is similar to the", "case we did.)", "\\medskip\\noindent", "Proof of (3). We have to prove associativity of composition.", "Consider a solid diagram", "$$", "\\xymatrix{", "& & & Z \\ar[d] \\\\", "& & Y \\ar[d] \\ar[r] & Z' \\ar@{..>}[d] \\\\", "& X \\ar[d] \\ar[r] & Y' \\ar@{..>}[d] \\ar@{..>}[r] & Z'' \\ar@{..>}[d] \\\\", "W \\ar[r] & X' \\ar@{..>}[r] & Y'' \\ar@{..>}[r] & Z'''", "}", "$$", "(whose vertical arrows belong to $S$)", "which gives rise to three composable pairs.", "Using LMS2 we can choose the dotted arrows making the squares commutative", "and such that the vertical arrows are in $S$.", "Then it is clear that the composition of the three pairs", "is the equivalence class of the pair", "$(W \\to Z''', Z \\to Z''')$ gotten by composing the", "horizontal arrows on the bottom row and the vertical arrows", "on the right column.", "\\medskip\\noindent", "We leave it to the reader to check the identity axioms." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12256, "type": "theorem", "label": "categories-lemma-morphisms-left-localization", "categories": [ "categories" ], "title": "categories-lemma-morphisms-left-localization", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative", "system of morphisms of $\\mathcal{C}$. Given any finite collection", "$g_i : X_i \\to Y$ of morphisms of $S^{-1}\\mathcal{C}$", "(indexed by $i$),", "we can find an element $s : Y \\to Y'$ of $S$ and", "a family of morphisms $f_i : X_i \\to Y'$ of $\\mathcal{C}$ such that", "each $g_i$ is the equivalence class of the pair", "$(f_i : X_i \\to Y', s : Y \\to Y')$." ], "refs": [], "proofs": [ { "contents": [ "For each $i$ choose a representative $(X_i \\to Y_i, s_i : Y \\to Y_i)$", "of $g_i$.", "The lemma follows if we can find a morphism $s : Y \\to Y'$ in $S$ such that", "for each $i$ there is a morphism $a_i : Y_i \\to Y'$ with", "$a_i \\circ s_i = s$. If we have two indices $i = 1, 2$, then we can", "do this by completing the square", "$$", "\\xymatrix{", "Y \\ar[d]_{s_1} \\ar[r]_{s_2} & Y_2 \\ar[d]^{t_2} \\\\", "Y_1 \\ar[r]^{a_1} & Y'", "}", "$$", "with $t_2 \\in S$ as is possible by", "Definition \\ref{definition-multiplicative-system}.", "Then $s = t_2 \\circ s_2 \\in S$ works.", "If we have $n > 2$ morphisms, then we use the above trick to reduce", "to the case of $n - 1$ morphisms, and we win by induction." ], "refs": [ "categories-definition-multiplicative-system" ], "ref_ids": [ 12373 ] } ], "ref_ids": [] }, { "id": 12257, "type": "theorem", "label": "categories-lemma-equality-morphisms-left-localization", "categories": [ "categories" ], "title": "categories-lemma-equality-morphisms-left-localization", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative", "system of morphisms of $\\mathcal{C}$. Let $A, B : X \\to Y$ be", "morphisms of $S^{-1}\\mathcal{C}$ which are the equivalence", "classes of $(f : X \\to Y', s : Y \\to Y')$ and", "$(g : X \\to Y', s : Y \\to Y')$. Then", "$A = B$ if and only if there exists a morphism", "$a : Y' \\to Y''$ with $a \\circ s \\in S$ and", "such that $a \\circ f = a \\circ g$." ], "refs": [], "proofs": [ { "contents": [ "The equality of $A$ and $B$ means that there exists a commutative diagram", "$$", "\\xymatrix{", " & Y' \\ar[d]^u & \\\\", "X \\ar[ru]^f \\ar[r]^h \\ar[rd]_g &", "Z &", "Y \\ar[lu]_s \\ar[l]_t \\ar[ld]^s \\\\", "& Y' \\ar[u]_v &", "}", "$$", "with $t \\in S$. In particular $u \\circ s = v \\circ s$. Hence by LMS3 there", "exists a $s' : Z \\to Y''$ in $S$ such that $s' \\circ u = s' \\circ v$.", "Setting $a$ equal to this common value does the job." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12258, "type": "theorem", "label": "categories-lemma-properties-left-localization", "categories": [ "categories" ], "title": "categories-lemma-properties-left-localization", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative", "system of morphisms of $\\mathcal{C}$.", "\\begin{enumerate}", "\\item The rules $X \\mapsto X$ and", "$(f : X \\to Y) \\mapsto (f : X \\to Y, \\text{id}_Y : Y \\to Y)$", "define a functor $Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$.", "\\item For any $s \\in S$ the morphism $Q(s)$ is an isomorphism in", "$S^{-1}\\mathcal{C}$.", "\\item If $G : \\mathcal{C} \\to \\mathcal{D}$ is any functor such that", "$G(s)$ is invertible for every $s \\in S$, then there exists a", "unique functor $H : S^{-1}\\mathcal{C} \\to \\mathcal{D}$", "such that $H \\circ Q = G$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1) and (2) are clear. (In (2), the inverse of $Q(s)$ is", "the equivalence class of the pair $(\\text{id}_Y, s)$.)", "To see (3) just set $H(X) = G(X)$", "and set $H((f : X \\to Y', s : Y \\to Y')) = G(s)^{-1} \\circ G(f)$.", "Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12259, "type": "theorem", "label": "categories-lemma-left-localization-limits", "categories": [ "categories" ], "title": "categories-lemma-left-localization-limits", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative", "system of morphisms of $\\mathcal{C}$. The localization functor", "$Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ commutes with finite colimits." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I}$ be a finite category and let", "$\\mathcal{I} \\to \\mathcal{C}$, $i \\mapsto X_i$", "be a functor whose colimit exists. Then using", "(\\ref{equation-left-localization-morphisms-colimit}),", "the fact that $Y/S$ is filtered, and", "Lemma \\ref{lemma-directed-commutes} we have", "\\begin{align*}", "\\Mor_{S^{-1}\\mathcal{C}}(Q(\\colim X_i), Q(Y))", "& =", "\\colim_{(s : Y \\to Y') \\in Y/S} \\Mor_\\mathcal{C}(\\colim X_i, Y') \\\\", "& =", "\\colim_{(s : Y \\to Y') \\in Y/S} \\lim_i \\Mor_\\mathcal{C}(X_i, Y') \\\\", "& =", "\\lim_i \\colim_{(s : Y \\to Y') \\in Y/S} \\Mor_\\mathcal{C}(X_i, Y') \\\\", "& =", "\\lim_i \\Mor_{S^{-1}\\mathcal{C}}(Q(X_i), Q(Y))", "\\end{align*}", "and this isomorphism commutes with the projections", "from both sides to the set", "$\\Mor_{S^{-1}\\mathcal{C}}(Q(X_j), Q(Y))$ for each", "$j \\in \\Ob(\\mathcal{I})$. Thus, $Q(\\colim X_i)$ satisfies", "the universal property for the colimit of the functor", "$i \\mapsto Q(X_i)$; hence, it is this colimit, as desired." ], "refs": [ "categories-lemma-directed-commutes" ], "ref_ids": [ 12228 ] } ], "ref_ids": [] }, { "id": 12260, "type": "theorem", "label": "categories-lemma-left-localization-diagram", "categories": [ "categories" ], "title": "categories-lemma-left-localization-diagram", "contents": [ "Let $\\mathcal{C}$ be a category. Let $S$ be a left multiplicative", "system. If $f : X \\to Y$, $f' : X' \\to Y'$ are two morphisms of", "$\\mathcal{C}$ and if", "$$", "\\xymatrix{", "Q(X) \\ar[d]_{Q(f)} \\ar[r]_a & Q(X') \\ar[d]^{Q(f')} \\\\", "Q(Y) \\ar[r]^b & Q(Y')", "}", "$$", "is a commutative diagram in $S^{-1}\\mathcal{C}$, then there exists", "a morphism $f'' : X'' \\to Y''$ in $\\mathcal{C}$ and a commutative", "diagram", "$$", "\\xymatrix{", "X \\ar[d]_f \\ar[r]_g & X'' \\ar[d]^{f''} & X' \\ar[d]^{f'} \\ar[l]^s \\\\", "Y \\ar[r]^h & Y'' & Y' \\ar[l]_t", "}", "$$", "in $\\mathcal{C}$ with $s, t \\in S$ and $a = s^{-1}g$, $b = t^{-1}h$." ], "refs": [], "proofs": [ { "contents": [ "We choose maps and objects in the following way:", "First write $a = s^{-1}g$ for some $s : X' \\to X''$ in $S$ and", "$g : X \\to X''$. By LMS2 we can find $t : Y' \\to Y''$ in $S$ and", "$f'' : X'' \\to Y''$ such that", "$$", "\\xymatrix{", "X \\ar[d]_f \\ar[r]_g & X'' \\ar[d]^{f''} & X' \\ar[d]^{f'} \\ar[l]^s \\\\", "Y & Y'' & Y' \\ar[l]_t", "}", "$$", "commutes. Now in this diagram we are going to repeatedly change our", "choice of", "$$", "X'' \\xrightarrow{f''} Y'' \\xleftarrow{t} Y'", "$$", "by postcomposing both $t$ and $f''$ by a morphism $d : Y'' \\to Y'''$", "with the property that $d \\circ t \\in S$. According to", "Remark \\ref{remark-left-localization-morphisms-colimit}", "we may after such a replacement assume that there exists a morphism", "$h : Y \\to Y''$ such that $b = t^{-1}h$ holds\\footnote{Here is a", "more down-to-earth way to see this:", "Write $b = q^{-1}i$ for some $q : Y' \\to Z$ in $S$ and some", "$i : Y \\to Z$. By LMS2 we can find $r : Y'' \\to Y'''$ in $S$ and", "$j : Z \\to Y'''$ such that $j \\circ q = r \\circ t$. Now, set", "$d = r$ and $h = j \\circ i$.}. At this point we have everything", "as in the lemma except that we don't know that the left square of the", "diagram commutes.", "But the definition of composition in $S^{-1} \\mathcal{C}$ shows that", "$b \\circ Q\\left(f\\right)$ is the equivalence class of the pair", "$(h \\circ f : X \\to Y'', t : Y' \\to Y'')$ (since $b$ is the", "equivalence class of the pair $(h : Y \\to Y'', t : Y' \\to Y'')$,", "while $Q\\left(f\\right)$ is the equivalence class of the pair", "$(f : X \\to Y, \\text{id} : Y \\to Y)$), while", "$Q\\left(f'\\right) \\circ a$ is the equivalence class of the pair", "$(f'' \\circ g : X \\to Y'', t : Y' \\to Y'')$ (since $a$ is the", "equivalence class of the pair $(g : X \\to X'', s : X' \\to X'')$,", "while $Q\\left(f'\\right)$ is the equivalence class of the pair", "$(f' : X' \\to Y', \\text{id} : Y' \\to Y')$).", "Since we know that", "$b \\circ Q\\left(f\\right) = Q\\left(f'\\right) \\circ a$, we thus", "conclude that the equivalence classes of the pairs", "$(h \\circ f : X \\to Y'', t : Y' \\to Y'')$ and", "$(f'' \\circ g : X \\to Y'', t : Y' \\to Y'')$ are equal.", "Hence using", "Lemma \\ref{lemma-equality-morphisms-left-localization}", "we can find a morphism $d : Y'' \\to Y'''$ such that", "$d \\circ t \\in S$ and $d \\circ h \\circ f = d \\circ f'' \\circ g$.", "Hence we make one more replacement of the kind described", "above and we win." ], "refs": [ "categories-remark-left-localization-morphisms-colimit", "categories-lemma-equality-morphisms-left-localization" ], "ref_ids": [ 12424, 12257 ] } ], "ref_ids": [] }, { "id": 12261, "type": "theorem", "label": "categories-lemma-right-localization", "categories": [ "categories" ], "title": "categories-lemma-right-localization", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative", "system.", "\\begin{enumerate}", "\\item The relation on pairs defined above is an equivalence relation.", "\\item The composition rule given above is well defined on equivalence", "classes.", "\\item Composition is associative (and the identity morphisms satisfy", "the identity axioms), and hence $S^{-1}\\mathcal{C}$ is a category.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma is dual to", "Lemma \\ref{lemma-left-localization}.", "It follows formally from that lemma by replacing", "$\\mathcal{C}$ by its opposite category in which", "$S$ is a left multiplicative system." ], "refs": [ "categories-lemma-left-localization" ], "ref_ids": [ 12255 ] } ], "ref_ids": [] }, { "id": 12262, "type": "theorem", "label": "categories-lemma-morphisms-right-localization", "categories": [ "categories" ], "title": "categories-lemma-morphisms-right-localization", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative", "system of morphisms of $\\mathcal{C}$. Given any finite collection", "$g_i : X \\to Y_i$ of morphisms of $S^{-1}\\mathcal{C}$", "(indexed by $i$),", "we can find an element $s : X' \\to X$ of $S$ and a family", "of morphisms $f_i : X' \\to Y_i$ of $\\mathcal{C}$ such that", "$g_i$ is the equivalence class of the pair", "$(f_i : X' \\to Y_i, s : X' \\to X)$." ], "refs": [], "proofs": [ { "contents": [ "This lemma is the dual of", "Lemma \\ref{lemma-morphisms-left-localization}", "and follows formally from that lemma by replacing all", "categories in sight by their opposites." ], "refs": [ "categories-lemma-morphisms-left-localization" ], "ref_ids": [ 12256 ] } ], "ref_ids": [] }, { "id": 12263, "type": "theorem", "label": "categories-lemma-equality-morphisms-right-localization", "categories": [ "categories" ], "title": "categories-lemma-equality-morphisms-right-localization", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative", "system of morphisms of $\\mathcal{C}$. Let $A, B : X \\to Y$ be", "morphisms of $S^{-1}\\mathcal{C}$ which are the equivalence", "classes of $(f : X' \\to Y, s : X' \\to X)$ and", "$(g : X' \\to Y, s : X' \\to X)$. Then", "$A = B$ if and only if there exists a morphism", "$a : X'' \\to X'$ with $s \\circ a \\in S$ and", "such that $f \\circ a = g \\circ a$." ], "refs": [], "proofs": [ { "contents": [ "This is dual to", "Lemma \\ref{lemma-equality-morphisms-left-localization}." ], "refs": [ "categories-lemma-equality-morphisms-left-localization" ], "ref_ids": [ 12257 ] } ], "ref_ids": [] }, { "id": 12264, "type": "theorem", "label": "categories-lemma-properties-right-localization", "categories": [ "categories" ], "title": "categories-lemma-properties-right-localization", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative", "system of morphisms of $\\mathcal{C}$.", "\\begin{enumerate}", "\\item The rules $X \\mapsto X$ and", "$(f : X \\to Y) \\mapsto (f : X \\to Y, \\text{id}_X : X \\to X)$", "define a functor $Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$.", "\\item For any $s \\in S$ the morphism $Q(s)$ is an isomorphism in", "$S^{-1}\\mathcal{C}$.", "\\item If $G : \\mathcal{C} \\to \\mathcal{D}$ is any functor such that", "$G(s)$ is invertible for every $s \\in S$, then there exists a", "unique functor $H : S^{-1}\\mathcal{C} \\to \\mathcal{D}$", "such that $H \\circ Q = G$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This lemma is the dual of", "Lemma \\ref{lemma-properties-left-localization}", "and follows formally from that lemma by replacing all", "categories in sight by their opposites." ], "refs": [ "categories-lemma-properties-left-localization" ], "ref_ids": [ 12258 ] } ], "ref_ids": [] }, { "id": 12265, "type": "theorem", "label": "categories-lemma-right-localization-limits", "categories": [ "categories" ], "title": "categories-lemma-right-localization-limits", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative", "system of morphisms of $\\mathcal{C}$. The localization functor", "$Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ commutes with finite limits." ], "refs": [], "proofs": [ { "contents": [ "This is dual to Lemma \\ref{lemma-left-localization-limits}." ], "refs": [ "categories-lemma-left-localization-limits" ], "ref_ids": [ 12259 ] } ], "ref_ids": [] }, { "id": 12266, "type": "theorem", "label": "categories-lemma-right-localization-diagram", "categories": [ "categories" ], "title": "categories-lemma-right-localization-diagram", "contents": [ "Let $\\mathcal{C}$ be a category. Let $S$ be a right multiplicative", "system. If $f : X \\to Y$, $f' : X' \\to Y'$ are two morphisms of", "$\\mathcal{C}$ and if", "$$", "\\xymatrix{", "Q(X) \\ar[d]_{Q(f)} \\ar[r]_a & Q(X') \\ar[d]^{Q(f')} \\\\", "Q(Y) \\ar[r]^b & Q(Y')", "}", "$$", "is a commutative diagram in $S^{-1}\\mathcal{C}$, then there exists", "a morphism $f'' : X'' \\to Y''$ in $\\mathcal{C}$ and a commutative", "diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & X'' \\ar[l]^s \\ar[d]^{f''} \\ar[r]_g & X' \\ar[d]^{f'} \\\\", "Y & Y'' \\ar[l]_t \\ar[r]^h & Y'", "}", "$$", "in $\\mathcal{C}$ with $s, t \\in S$ and $a = gs^{-1}$, $b = ht^{-1}$." ], "refs": [], "proofs": [ { "contents": [ "This lemma is dual to", "Lemma \\ref{lemma-left-localization-diagram}." ], "refs": [ "categories-lemma-left-localization-diagram" ], "ref_ids": [ 12260 ] } ], "ref_ids": [] }, { "id": 12267, "type": "theorem", "label": "categories-lemma-multiplicative-system", "categories": [ "categories" ], "title": "categories-lemma-multiplicative-system", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a multiplicative system.", "The category of left fractions and the category of right fractions", "$S^{-1}\\mathcal{C}$ are canonically isomorphic." ], "refs": [], "proofs": [ { "contents": [ "Denote $\\mathcal{C}_{left}$, $\\mathcal{C}_{right}$ the two categories", "of fractions. By the universal properties of", "Lemmas \\ref{lemma-properties-left-localization} and", "\\ref{lemma-properties-right-localization}", "we obtain functors $\\mathcal{C}_{left} \\to \\mathcal{C}_{right}$", "and $\\mathcal{C}_{right} \\to \\mathcal{C}_{left}$.", "By the uniqueness statement in the universal properties, these", "functors are each other's inverse." ], "refs": [ "categories-lemma-properties-left-localization", "categories-lemma-properties-right-localization" ], "ref_ids": [ 12258, 12264 ] } ], "ref_ids": [] }, { "id": 12268, "type": "theorem", "label": "categories-lemma-what-gets-inverted", "categories": [ "categories" ], "title": "categories-lemma-what-gets-inverted", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a multiplicative system.", "Denote $Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ the localization functor.", "The set", "$$", "\\hat S = \\{f \\in \\text{Arrows}(\\mathcal{C}) \\mid", "Q(f) \\text{ is an isomorphism}\\}", "$$", "is equal to", "$$", "S' = \\{f \\in \\text{Arrows}(\\mathcal{C}) \\mid", "\\text{there exist }g, h\\text{ such that }gf, fh \\in S\\}", "$$", "and is the smallest saturated multiplicative system containing $S$.", "In particular, if $S$ is saturated, then $\\hat S = S$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $S \\subset S' \\subset \\hat S$ because elements of", "$S'$ map to morphisms in $S^{-1}\\mathcal{C}$ which have both left", "and right inverses. Note that $S'$ satisfies MS4, and that", "$\\hat S$ satisfies MS1. Next, we prove that $S' = \\hat S$.", "\\medskip\\noindent", "Let $f \\in \\hat S$. Let $s^{-1}g = ht^{-1}$ be the inverse morphism", "in $S^{-1}\\mathcal{C}$. (We may use both left fractions and right", "fractions to describe morphisms in $S^{-1}\\mathcal{C}$, see", "Lemma \\ref{lemma-multiplicative-system}.)", "The relation $\\text{id}_X = s^{-1}gf$ in $S^{-1}\\mathcal{C}$ means", "there exists a commutative diagram", "$$", "\\xymatrix{", " & X' \\ar[d]^u & \\\\", "X \\ar[ru]^{gf} \\ar[r]^{f'} \\ar[rd]_{\\text{id}_X} &", "X'' &", "X \\ar[lu]_s \\ar[l]_{s'} \\ar[ld]^{\\text{id}_X} \\\\", "& X \\ar[u]_v &", "}", "$$", "for some morphisms $f', u, v$ and $s' \\in S$. Hence $ugf = s' \\in S$.", "Similarly, using that $\\text{id}_Y = fht^{-1}$ one proves that", "$fhw \\in S$ for some $w$. We conclude that $f \\in S'$. Thus", "$S' = \\hat S$. Provided we prove that $S' = \\hat S$ is a", "multiplicative system it is now clear that this implies that $S' = \\hat S$", "is the smallest saturated system containing $S$.", "\\medskip\\noindent", "Our remarks above take care of MS1 and MS4, so to finish the proof of the", "lemma we have to show that LMS2, RMS2, LMS3, RMS3 hold for $\\hat S$.", "Let us check that LMS2 holds for $\\hat S$. Suppose we have a solid diagram", "$$", "\\xymatrix{", "X \\ar[d]_t \\ar[r]_g & Y \\ar@{..>}[d]^s \\\\", "Z \\ar@{..>}[r]^f & W", "}", "$$", "with $t \\in \\hat S$. Pick a morphism $a : Z \\to Z'$ such that", "$at \\in S$. Then we can use LMS2 for $S$ to find a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_t \\ar[r]_g & Y \\ar[dd]^s \\\\", "Z \\ar[d]_a \\\\", "Z' \\ar[r]^{f'} & W", "}", "$$", "and setting $f = f' \\circ a$ we win. The proof of RMS2 is dual to this.", "Finally, suppose given a pair of morphisms $f, g : X \\to Y$ and", "$t \\in \\hat S$ with target $X$ such that $ft = gt$.", "Then we pick a morphism $b$ such that $tb \\in S$. Then", "$ftb = gtb$ which implies by LMS3 for $S$ that there exists an $s \\in S$", "with source $Y$ such that $sf = sg$ as desired. The proof of", "RMS3 is dual to this." ], "refs": [ "categories-lemma-multiplicative-system" ], "ref_ids": [ 12267 ] } ], "ref_ids": [] }, { "id": 12269, "type": "theorem", "label": "categories-lemma-properties-2-cat-cats", "categories": [ "categories" ], "title": "categories-lemma-properties-2-cat-cats", "contents": [ "The horizontal and vertical compositions have the following", "properties", "\\begin{enumerate}", "\\item $\\circ$ and $\\star$ are associative,", "\\item the identity transformations $\\text{id}_F$", "are units for $\\circ$,", "\\item the identity transformations of the identity functors", "$\\text{id}_{\\text{id}_\\mathcal{A}}$", "are units for $\\star$ and $\\circ$, and", "\\item given a diagram", "$$", "\\xymatrix{", "\\mathcal{A}", "\\rruppertwocell^F{t}", "\\ar[rr]_(.3){F'}", "\\rrlowertwocell_{F''}{t'}", "& &", "\\mathcal{B}", "\\rruppertwocell^G{s}", "\\ar[rr]_(.3){G'}", "\\rrlowertwocell_{G''}{s'}", "& &", "\\mathcal{C}", "}", "$$", "we have $ (s' \\circ s) \\star (t' \\circ t) = (s' \\star t') \\circ (s \\star t)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The last statement turns using our previous notation into the following", "equation", "$$", "s'_{F''}", "\\circ", "{}_{G'}t'", "\\circ", "s_{F'}", "\\circ", "{}_Gt", "=", "(s' \\circ s)_{F''}", "\\circ", "{}_G(t' \\circ t).", "$$", "According to our result above applied to the middle composition", "we may rewrite the left hand side as", "$", "s'_{F''}", "\\circ", "s_{F''}", "\\circ", "{}_Gt'", "\\circ", "{}_Gt", "$", "which is easily shown to be equal to the right hand side." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12270, "type": "theorem", "label": "categories-lemma-2-fibre-product-categories", "categories": [ "categories" ], "title": "categories-lemma-2-fibre-product-categories", "contents": [ "In the $(2, 1)$-category of categories $2$-fibre products exist and", "are given by the construction of", "Example \\ref{example-2-fibre-product-categories}." ], "refs": [], "proofs": [ { "contents": [ "Let us check the universal property:", "let $\\mathcal{W}$ be a category, let", "$a : \\mathcal{W} \\to \\mathcal{A}$ and", "$b : \\mathcal{W} \\to \\mathcal{B}$ be functors, and", "let $t : F \\circ a \\to G \\circ b$ be an isomorphism of functors.", "\\medskip\\noindent", "Consider the functor", "$\\gamma : \\mathcal{W} \\to \\mathcal{A} \\times_\\mathcal{C}\\mathcal{B}$", "given by $W \\mapsto (a(W), b(W), t_W)$.", "(Check this is a functor omitted.)", "Moreover, consider $\\alpha : a \\to p \\circ \\gamma$ and", "$\\beta : b \\to q \\circ \\gamma$ obtained from the identities", "$p \\circ \\gamma = a$ and $q \\circ \\gamma = b$. Then it is", "clear that $(\\gamma, \\alpha, \\beta)$ is a morphism", "from $(W, a, b, t)$ to", "$(\\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}, p, q, \\psi)$.", "\\medskip\\noindent", "Let", "$(k, \\alpha', \\beta') :", "(W, a, b, t) \\to (\\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}, p, q, \\psi)$", "be a second such morphism. For an object $W$ of $\\mathcal{W}$ let us write", "$k(W) = (a_k(W), b_k(W), t_{k, W})$. Hence $p(k(W)) = a_k(W)$ and so on.", "The map $\\alpha'$ corresponds to functorial maps", "$\\alpha' : a(W) \\to a_k(W)$. Since we are working in the", "$(2, 1)$-category of categories, in fact each of the maps", "$a(W) \\to a_k(W)$ is an isomorphism. We can use these", "(and their counterparts $b(W) \\to b_k(W)$) to get isomorphisms", "$$", "\\delta_W :", "\\gamma(W) = (a(W), b(W), t_W)", "\\longrightarrow", "(a_k(W), b_k(W), t_{k, W}) = k(W).", "$$", "It is straightforward to show that $\\delta$ defines a", "$2$-isomorphism between $\\gamma$ and $k$ in the $2$-category", "of $2$-commutative diagrams as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12271, "type": "theorem", "label": "categories-lemma-functoriality-2-fibre-product", "categories": [ "categories" ], "title": "categories-lemma-functoriality-2-fibre-product", "contents": [ "Let", "$$", "\\xymatrix{", "& \\mathcal{Y} \\ar[d]_I \\ar[rd]^K & \\\\", "\\mathcal{X} \\ar[r]^H \\ar[rd]^L &", "\\mathcal{Z} \\ar[rd]^M & \\mathcal{B} \\ar[d]^G \\\\", "& \\mathcal{A} \\ar[r]^F & \\mathcal{C}", "}", "$$", "be a $2$-commutative diagram of categories.", "A choice of isomorphisms", "$\\alpha : G \\circ K \\to M \\circ I$ and", "$\\beta : M \\circ H \\to F \\circ L$", "determines a morphism", "$$", "\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y}", "\\longrightarrow", "\\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}", "$$", "of $2$-fibre products associated to this situation." ], "refs": [], "proofs": [ { "contents": [ "Just use the functor", "$$", "(X, Y, \\phi) \\longmapsto (L(X), K(Y),", "\\alpha^{-1}_Y \\circ M(\\phi) \\circ \\beta^{-1}_X)", "$$", "on objects and", "$$", "(a, b) \\longmapsto (L(a), K(b))", "$$", "on morphisms." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12272, "type": "theorem", "label": "categories-lemma-equivalence-2-fibre-product", "categories": [ "categories" ], "title": "categories-lemma-equivalence-2-fibre-product", "contents": [ "Assumptions as in Lemma \\ref{lemma-functoriality-2-fibre-product}.", "\\begin{enumerate}", "\\item If $K$ and $L$ are faithful", "then the morphism", "$\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to", "\\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}$", "is faithful.", "\\item If $K$ and $L$ are fully faithful and $M$ is faithful", "then the morphism", "$\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to", "\\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}$", "is fully faithful.", "\\item If $K$ and $L$ are equivalences and $M$ is fully faithful", "then the morphism", "$\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to", "\\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}$", "is an equivalence.", "\\end{enumerate}" ], "refs": [ "categories-lemma-functoriality-2-fibre-product" ], "proofs": [ { "contents": [ "Let $(X, Y, \\phi)$ and $(X', Y', \\phi')$ be objects of", "$\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y}$.", "Set $Z = H(X)$ and identify it with $I(Y)$ via $\\phi$.", "Also, identify $M(Z)$ with $F(L(X))$ via $\\alpha_X$ and", "identify $M(Z)$ with $G(K(Y))$ via $\\beta_Y$. Similarly for", "$Z' = H(X')$ and $M(Z')$.", "The map on morphisms is the map", "$$", "\\xymatrix{", "\\Mor_\\mathcal{X}(X, X')", "\\times_{\\Mor_\\mathcal{Z}(Z, Z')}", "\\Mor_\\mathcal{Y}(Y, Y')", "\\ar[d] \\\\", "\\Mor_\\mathcal{A}(L(X), L(X'))", "\\times_{\\Mor_\\mathcal{C}(M(Z), M(Z'))}", "\\Mor_\\mathcal{B}(K(Y), K(Y'))", "}", "$$", "Hence parts (1) and (2) follow. Moreover, if $K$ and $L$", "are equivalences and $M$ is fully faithful, then any object", "$(A, B, \\phi)$ is in the essential image for the following reasons:", "Pick $X$, $Y$ such that $L(X) \\cong A$ and $K(Y) \\cong B$.", "Then the fully faithfulness of $M$ guarantees that we can", "find an isomorphism $H(X) \\cong I(Y)$. Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12271 ] }, { "id": 12273, "type": "theorem", "label": "categories-lemma-associativity-2-fibre-product", "categories": [ "categories" ], "title": "categories-lemma-associativity-2-fibre-product", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{A} \\ar[rd] & & \\mathcal{C} \\ar[ld] \\ar[rd] & & \\mathcal{E} \\ar[ld] \\\\", "& \\mathcal{B} & & \\mathcal{D}", "}", "$$", "be a diagram of categories and functors.", "Then there is a canonical isomorphism", "$$", "(\\mathcal{A} \\times_\\mathcal{B} \\mathcal{C}) \\times_\\mathcal{D} \\mathcal{E}", "\\cong", "\\mathcal{A} \\times_\\mathcal{B} (\\mathcal{C} \\times_\\mathcal{D} \\mathcal{E})", "$$", "of categories." ], "refs": [], "proofs": [ { "contents": [ "Just use the functor", "$$", "((A, C, \\phi), E, \\psi)", "\\longmapsto", "(A, (C, E, \\psi), \\phi)", "$$", "if you know what I mean." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12274, "type": "theorem", "label": "categories-lemma-triple-2-fibre-product-pr02", "categories": [ "categories" ], "title": "categories-lemma-triple-2-fibre-product-pr02", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{A} \\ar[rd] & & \\mathcal{C} \\ar[ld] \\ar[rd] & & \\mathcal{E} \\ar[ld] \\\\", "& \\mathcal{B} \\ar[rd]_F & & \\mathcal{D} \\ar[ld]^G \\\\", "& & \\mathcal{F} &", "}", "$$", "be a commutative diagram of categories and functors.", "Then there is a canonical functor", "$$", "\\text{pr}_{02} :", "\\mathcal{A} \\times_\\mathcal{B} \\mathcal{C} \\times_\\mathcal{D} \\mathcal{E}", "\\longrightarrow", "\\mathcal{A} \\times_\\mathcal{F} \\mathcal{E}", "$$", "of categories." ], "refs": [], "proofs": [ { "contents": [ "If we write", "$\\mathcal{A} \\times_\\mathcal{B} \\mathcal{C}", "\\times_\\mathcal{D} \\mathcal{E}$", "as", "$(\\mathcal{A} \\times_\\mathcal{B} \\mathcal{C})", "\\times_\\mathcal{D} \\mathcal{E}$", "then we can just use the functor", "$$", "((A, C, \\phi), E, \\psi)", "\\longmapsto", "(A, E, G(\\psi) \\circ F(\\phi))", "$$", "if you know what I mean." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12275, "type": "theorem", "label": "categories-lemma-2-fibre-product-erase-factor", "categories": [ "categories" ], "title": "categories-lemma-2-fibre-product-erase-factor", "contents": [ "Let", "$$", "\\mathcal{A} \\to", "\\mathcal{B} \\leftarrow \\mathcal{C} \\leftarrow \\mathcal{D}", "$$", "be a diagram of categories and functors.", "Then there is a canonical isomorphism", "$$", "\\mathcal{A} \\times_\\mathcal{B} \\mathcal{C} \\times_\\mathcal{C} \\mathcal{D}", "\\cong", "\\mathcal{A} \\times_\\mathcal{B} \\mathcal{D}", "$$", "of categories." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12276, "type": "theorem", "label": "categories-lemma-diagonal-1", "categories": [ "categories" ], "title": "categories-lemma-diagonal-1", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{C}_3 \\ar[r] \\ar[d] & \\mathcal{S} \\ar[d]^\\Delta \\\\", "\\mathcal{C}_1 \\times \\mathcal{C}_2 \\ar[r]^{G_1 \\times G_2} &", "\\mathcal{S} \\times \\mathcal{S}", "}", "$$", "be a $2$-fibre product of categories.", "Then there is a canonical isomorphism", "$\\mathcal{C}_3 \\cong", "\\mathcal{C}_1 \\times_{G_1, \\mathcal{S}, G_2} \\mathcal{C}_2$." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $\\mathcal{C}_3$ is the category", "$(\\mathcal{C}_1 \\times \\mathcal{C}_2)\\times_{\\mathcal{S} \\times \\mathcal{S}}", "\\mathcal{S}$ constructed in Example \\ref{example-2-fibre-product-categories}.", "Hence an object is a triple", "$((X_1, X_2), S, \\phi)$ where", "$\\phi = (\\phi_1, \\phi_2) : (G_1(X_1), G_2(X_2)) \\to (S, S)$", "is an isomorphism. Thus we can associate to this the triple", "$(X_1, X_2, \\phi_2^{-1} \\circ \\phi_1)$.", "Conversely, if $(X_1, X_2, \\psi)$ is an object of", "$\\mathcal{C}_1 \\times_{G_1, \\mathcal{S}, G_2} \\mathcal{C}_2$,", "then we can associate to this the triple", "$((X_1, X_2), G_2(X_2), (\\psi, \\text{id}_{G_2(X_2)}))$.", "We claim these constructions given mutually inverse functors.", "We omit describing how to deal with morphisms", "and showing they are mutually inverse." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12277, "type": "theorem", "label": "categories-lemma-diagonal-2", "categories": [ "categories" ], "title": "categories-lemma-diagonal-2", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{C}' \\ar[r] \\ar[d] & \\mathcal{S} \\ar[d]^\\Delta \\\\", "\\mathcal{C} \\ar[r]^{G_1 \\times G_2} &", "\\mathcal{S} \\times \\mathcal{S}", "}", "$$", "be a $2$-fibre product of categories.", "Then there is a canonical isomorphism", "$$", "\\mathcal{C}' \\cong", "(\\mathcal{C} \\times_{G_1, \\mathcal{S}, G_2} \\mathcal{C})", "\\times_{(p, q), \\mathcal{C} \\times \\mathcal{C}, \\Delta}", "\\mathcal{C}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "An object of the right hand side is given by", "$((C_1, C_2, \\phi), C_3, \\psi)$ where", "$\\phi : G_1(C_1) \\to G_2(C_2)$ is an isomorphism", "and $\\psi = (\\psi_1, \\psi_2) : (C_1, C_2) \\to (C_3, C_3)$ is", "an isomorphism. Hence we can associate to this the triple", "$(C_3, G_1(C_1), (G_1(\\psi_1^{-1}), \\phi^{-1} \\circ G_2(\\psi_2^{-1})))$", "which is an object of $\\mathcal{C}'$.", "Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12278, "type": "theorem", "label": "categories-lemma-fibre-product-after-map", "categories": [ "categories" ], "title": "categories-lemma-fibre-product-after-map", "contents": [ "Let $\\mathcal{A} \\to \\mathcal{C}$, $\\mathcal{B} \\to \\mathcal{C}$", "and $\\mathcal{C} \\to \\mathcal{D}$ be functors between categories.", "Then the diagram", "$$", "\\xymatrix{", "\\mathcal{A} \\times_\\mathcal{C} \\mathcal{B} \\ar[d] \\ar[r] &", "\\mathcal{A} \\times_\\mathcal{D} \\mathcal{B} \\ar[d] \\\\", "\\mathcal{C} \\ar[r]^-{\\Delta_{\\mathcal{C}/\\mathcal{D}}} \\ar[r] &", "\\mathcal{C} \\times_\\mathcal{D} \\mathcal{C}", "}", "$$", "is a $2$-fibre product diagram." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12279, "type": "theorem", "label": "categories-lemma-base-change-diagonal", "categories": [ "categories" ], "title": "categories-lemma-base-change-diagonal", "contents": [ "Let", "$$", "\\xymatrix{", "\\mathcal{U} \\ar[d] \\ar[r] & \\mathcal{V} \\ar[d] \\\\", "\\mathcal{X} \\ar[r] & \\mathcal{Y}", "}", "$$", "be a $2$-fibre product of categories. Then the diagram", "$$", "\\xymatrix{", "\\mathcal{U} \\ar[d] \\ar[r] &", "\\mathcal{U} \\times_\\mathcal{V} \\mathcal{U} \\ar[d] \\\\", "\\mathcal{X} \\ar[r] &", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}", "}", "$$", "is $2$-cartesian." ], "refs": [], "proofs": [ { "contents": [ "This is a purely $2$-category theoretic statement, valid in any", "$(2, 1)$-category with $2$-fibre products. Explicitly, it follows", "from the following chain of equivalences:", "\\begin{align*}", "\\mathcal{X} \\times_{(\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X})}", "(\\mathcal{U} \\times_\\mathcal{V} \\mathcal{U})", "& =", "\\mathcal{X} \\times_{(\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X})}", "((\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{V})", "\\times_\\mathcal{V} (\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{V})) \\\\", "& =", "\\mathcal{X} \\times_{(\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X})}", "(\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}", "\\times_\\mathcal{Y} \\mathcal{V}) \\\\", "& =", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{V} = \\mathcal{U}", "\\end{align*}", "see", "Lemmas \\ref{lemma-associativity-2-fibre-product} and", "\\ref{lemma-2-fibre-product-erase-factor}." ], "refs": [ "categories-lemma-associativity-2-fibre-product", "categories-lemma-2-fibre-product-erase-factor" ], "ref_ids": [ 12273, 12275 ] } ], "ref_ids": [] }, { "id": 12280, "type": "theorem", "label": "categories-lemma-2-product-categories-over-C", "categories": [ "categories" ], "title": "categories-lemma-2-product-categories-over-C", "contents": [ "Let $\\mathcal{C}$ be a category.", "The $(2, 1)$-category of categories", "over $\\mathcal{C}$ has 2-fibre products.", "Suppose that", "$F : \\mathcal{X} \\to \\mathcal{S}$ and", "$G : \\mathcal{Y} \\to \\mathcal{S}$ are morphisms of categories over", "$\\mathcal{C}$.", "An explicit 2-fibre product", "$\\mathcal{X} \\times_\\mathcal{S}\\mathcal{Y}$ is given by the following", "description", "\\begin{enumerate}", "\\item an object of $\\mathcal{X} \\times_\\mathcal{S} \\mathcal{Y}$ is a quadruple", "$(U, x, y, f)$, where $U \\in \\Ob(\\mathcal{C})$,", "$x\\in \\Ob(\\mathcal{X}_U)$, $y\\in \\Ob(\\mathcal{Y}_U)$,", "and $f : F(x) \\to G(y)$ is an isomorphism in $\\mathcal{S}_U$,", "\\item a morphism $(U, x, y, f) \\to (U', x', y', f')$ is given by a pair", "$(a, b)$, where $a : x \\to x'$ is a morphism in $\\mathcal{X}$, and", "$b : y \\to y'$ is a", "morphism in $\\mathcal{Y}$ such that", "\\begin{enumerate}", "\\item $a$ and $b$ induce the same morphism $U \\to U'$, and", "\\item the diagram", "$$", "\\xymatrix{", "F(x) \\ar[r]^f \\ar[d]^{F(a)} & G(y) \\ar[d]^{G(b)} \\\\", "F(x') \\ar[r]^{f'} & G(y')", "}", "$$", "is commutative.", "\\end{enumerate}", "\\end{enumerate}", "The functors $p : \\mathcal{X} \\times_\\mathcal{S}\\mathcal{Y} \\to \\mathcal{X}$", "and $q : \\mathcal{X} \\times_\\mathcal{S}\\mathcal{Y} \\to \\mathcal{Y}$ are the", "forgetful functors in this case. The transformation $\\psi : F \\circ p \\to", "G \\circ q$ is given on the object $\\xi = (U, x, y, f)$ by", "$\\psi_\\xi = f : F(p(\\xi)) = F(x) \\to G(y) = G(q(\\xi))$." ], "refs": [], "proofs": [ { "contents": [ "Let us check the universal property: let", "$p_\\mathcal{W} : \\mathcal{W}\\to \\mathcal{C}$", "be a category over $\\mathcal{C}$, let $X : \\mathcal{W} \\to \\mathcal{X}$ and", "$Y : \\mathcal{W} \\to \\mathcal{Y}$ be functors over $\\mathcal{C}$, and let", "$t : F \\circ X \\to G \\circ Y$ be an isomorphism of functors over $\\mathcal{C}$.", "The desired functor", "$\\gamma : \\mathcal{W} \\to \\mathcal{X} \\times_\\mathcal{S} \\mathcal{Y}$", "is given by $W \\mapsto (p_\\mathcal{W}(W), X(W), Y(W), t_W)$.", "Details omitted; compare with Lemma \\ref{lemma-2-fibre-product-categories}." ], "refs": [ "categories-lemma-2-fibre-product-categories" ], "ref_ids": [ 12270 ] } ], "ref_ids": [] }, { "id": 12281, "type": "theorem", "label": "categories-lemma-fibre-2-fibre-product-categories-over-C", "categories": [ "categories" ], "title": "categories-lemma-fibre-2-fibre-product-categories-over-C", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $f : \\mathcal{X} \\to \\mathcal{S}$ and", "$g : \\mathcal{Y} \\to \\mathcal{S}$ be morphisms of categories over", "$\\mathcal{C}$. For any object $U$ of $\\mathcal{C}$ we have", "the following identity of", "fibre categories", "$$", "\\left(\\mathcal{X} \\times_\\mathcal{S}\\mathcal{Y}\\right)_U", "=", "\\mathcal{X}_U \\times_{\\mathcal{S}_U} \\mathcal{Y}_U", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12282, "type": "theorem", "label": "categories-lemma-composition-cartesian", "categories": [ "categories" ], "title": "categories-lemma-composition-cartesian", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$.", "\\begin{enumerate}", "\\item The composition of two strongly cartesian morphisms", "is strongly cartesian.", "\\item Any isomorphism of $\\mathcal{S}$ is strongly cartesian.", "\\item Any strongly cartesian morphism $\\varphi$ such that $p(\\varphi)$", "is an isomorphism, is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $\\varphi : y \\to x$ and $\\psi : z \\to y$ be", "strongly cartesian. Let $t$ be an arbitrary object of $\\mathcal{S}$.", "Then we have", "\\begin{align*}", "& \\Mor_\\mathcal{S}(t, z) \\\\", "& =", "\\Mor_\\mathcal{S}(t, y)", "\\times_{\\Mor_\\mathcal{C}(p(t), p(y))}", "\\Mor_\\mathcal{C}(p(t), p(z)) \\\\", "& =", "\\Mor_\\mathcal{S}(t, x)", "\\times_{\\Mor_\\mathcal{C}(p(t), p(x))}", "\\Mor_\\mathcal{C}(p(t), p(y))", "\\times_{\\Mor_\\mathcal{C}(p(t), p(y))}", "\\Mor_\\mathcal{C}(p(t), p(z)) \\\\", "& =", "\\Mor_\\mathcal{S}(t, x)", "\\times_{\\Mor_\\mathcal{C}(p(t), p(x))}", "\\Mor_\\mathcal{C}(p(t), p(z))", "\\end{align*}", "hence $z \\to x$ is strongly cartesian.", "\\medskip\\noindent", "Proof of (2). Let $y \\to x$ be an isomorphism. Then $p(y) \\to p(x)$", "is an isomorphism too. Hence", "$\\Mor_\\mathcal{C}(p(z), p(y)) \\to", "\\Mor_\\mathcal{C}(p(z), p(x))$", "is a bijection. Hence", "$\\Mor_\\mathcal{S}(z, x)", "\\times_{\\Mor_\\mathcal{C}(p(z), p(x))}", "\\Mor_\\mathcal{C}(p(z), p(y))$ is bijective to", "$\\Mor_\\mathcal{S}(z, x)$.", "Hence the displayed map of", "Definition \\ref{definition-cartesian-over-C}", "is a bijection as $y \\to x$ is an isomorphism, and we conclude that", "$y \\to x$ is strongly cartesian.", "\\medskip\\noindent", "Proof of (3). Assume $\\varphi : y \\to x$ is strongly cartesian with", "$p(\\varphi) : p(y) \\to p(x)$ an isomorphism. Applying the definition with", "$z = x$ shows that $(\\text{id}_x, p(\\varphi)^{-1})$ comes from a unique", "morphism $\\chi : x \\to y$. We omit the verification that $\\chi$ is the", "inverse of $\\varphi$." ], "refs": [ "categories-definition-cartesian-over-C" ], "ref_ids": [ 12387 ] } ], "ref_ids": [] }, { "id": 12283, "type": "theorem", "label": "categories-lemma-cartesian-over-cartesian", "categories": [ "categories" ], "title": "categories-lemma-cartesian-over-cartesian", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{C}$", "be composable functors between categories. Let $x \\to y$ be a morphism of", "$\\mathcal{A}$. If $x \\to y$ is strongly $\\mathcal{B}$-cartesian", "and $F(x) \\to F(y)$ is strongly $\\mathcal{C}$-cartesian, then", "$x \\to y$ is strongly $\\mathcal{C}$-cartesian." ], "refs": [], "proofs": [ { "contents": [ "This follows directly from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12284, "type": "theorem", "label": "categories-lemma-strongly-cartesian-fibre-product", "categories": [ "categories" ], "title": "categories-lemma-strongly-cartesian-fibre-product", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$.", "Let $x \\to y$ and $z \\to y$ be morphisms of $\\mathcal{S}$.", "Assume", "\\begin{enumerate}", "\\item $x \\to y$ is strongly cartesian,", "\\item $p(x) \\times_{p(y)} p(z)$ exists, and", "\\item there exists a strongly cartesian morphism $a : w \\to z$ in", "$\\mathcal{S}$ with $p(w) = p(x) \\times_{p(y)} p(z)$ and", "$p(a) = \\text{pr}_2 : p(x) \\times_{p(y)} p(z) \\to p(z)$.", "\\end{enumerate}", "Then the fibre product $x \\times_y z$ exists and is isomorphic to $w$." ], "refs": [], "proofs": [ { "contents": [ "Since $x \\to y$ is strongly cartesian there exists a unique morphism", "$b : w \\to x$ such that $p(b) = \\text{pr}_1$. To see that $w$ is the", "fibre product we compute", "\\begin{align*}", "& \\Mor_\\mathcal{S}(t, w) \\\\", "& = \\Mor_\\mathcal{S}(t, z)", "\\times_{\\Mor_\\mathcal{C}(p(t), p(z))}", "\\Mor_\\mathcal{C}(p(t), p(w)) \\\\", "& = \\Mor_\\mathcal{S}(t, z)", "\\times_{\\Mor_\\mathcal{C}(p(t), p(z))}", "(\\Mor_\\mathcal{C}(p(t), p(x))", "\\times_{\\Mor_\\mathcal{C}(p(t), p(y))}", "\\Mor_\\mathcal{C}(p(t), p(z))) \\\\", "& = \\Mor_\\mathcal{S}(t, z)", "\\times_{\\Mor_\\mathcal{C}(p(t), p(y))}", "\\Mor_\\mathcal{C}(p(t), p(x)) \\\\", "& = \\Mor_\\mathcal{S}(t, z)", "\\times_{\\Mor_\\mathcal{S}(t, y)}", "\\Mor_\\mathcal{S}(t, y)", "\\times_{\\Mor_\\mathcal{C}(p(t), p(y))}", "\\Mor_\\mathcal{C}(p(t), p(x)) \\\\", "& = \\Mor_\\mathcal{S}(t, z)", "\\times_{\\Mor_\\mathcal{S}(t, y)}", "\\Mor_\\mathcal{S}(t, x)", "\\end{align*}", "as desired. The first equality holds because $a : w \\to z$ is strongly", "cartesian and the last equality holds because $x \\to y$ is strongly", "cartesian." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12285, "type": "theorem", "label": "categories-lemma-fibred", "categories": [ "categories" ], "title": "categories-lemma-fibred", "contents": [ "Assume $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category.", "Assume given a choice of pullbacks for $p : \\mathcal{S} \\to \\mathcal{C}$.", "\\begin{enumerate}", "\\item For any pair of composable morphisms $f : V \\to U$,", "$g : W \\to V$ there is a unique isomorphism", "$$", "\\alpha_{g, f} :", "(f \\circ g)^\\ast", "\\longrightarrow", "g^\\ast \\circ f^\\ast", "$$", "as functors $\\mathcal{S}_U \\to \\mathcal{S}_W$", "such that for every $y\\in \\Ob(\\mathcal{S}_U)$ the following", "diagram commutes", "$$", "\\xymatrix{", "g^\\ast f^\\ast y \\ar[r]", "&", "f^\\ast y \\ar[d] \\\\", "(f \\circ g)^\\ast y \\ar[r]", "\\ar[u]^{(\\alpha_{g, f})_y}", "&", "y", "}", "$$", "\\item If $f = \\text{id}_U$, then there is a canonical isomorphism", "$\\alpha_U : \\text{id} \\to (\\text{id}_U)^*$ as functors", "$\\mathcal{S}_U \\to \\mathcal{S}_U$.", "\\item The quadruple", "$(U \\mapsto \\mathcal{S}_U, f \\mapsto f^*, \\alpha_{g, f}, \\alpha_U)$", "defines a pseudo functor from $\\mathcal{C}^{opp}$ to", "the $(2, 1)$-category of categories, see", "Definition \\ref{definition-functor-into-2-category}.", "\\end{enumerate}" ], "refs": [ "categories-definition-functor-into-2-category" ], "proofs": [ { "contents": [ "In fact, it is clear that the commutative diagram of", "part (1) uniquely determines the morphism", "$(\\alpha_{g, f})_y$ in the fibre category", "$\\mathcal{S}_W$. It is an isomorphism since both", "the morphism $(f \\circ g)^*y \\to y$", "and the composition $g^*f^*y \\to f^*y \\to y$ are strongly", "cartesian morphisms lifting $f \\circ g$ (see discussion", "following Definition \\ref{definition-cartesian-over-C} and", "Lemma \\ref{lemma-composition-cartesian}). In the same way,", "since $\\text{id}_x : x \\to x$ is clearly strongly cartesian", "over $\\text{id}_U$ (with $U = p(x)$) we see that there exists", "an isomorphism $(\\alpha_U)_x : x \\to (\\text{id}_U)^*x$.", "(Of course we could have assumed beforehand that $f^*x = x$", "whenever $f$ is an identity morphism, but it is better for", "the sake of generality not to assume this.)", "We omit the verification that $\\alpha_{g, f}$ and", "$\\alpha_U$ so obtained are transformations of functors.", "We also omit the verification of (3)." ], "refs": [ "categories-definition-cartesian-over-C", "categories-lemma-composition-cartesian" ], "ref_ids": [ 12387, 12282 ] } ], "ref_ids": [ 12381 ] }, { "id": 12286, "type": "theorem", "label": "categories-lemma-fibred-equivalent", "categories": [ "categories" ], "title": "categories-lemma-fibred-equivalent", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{S}_1$, $\\mathcal{S}_2$ be categories over $\\mathcal{C}$.", "Suppose that $\\mathcal{S}_1$ and $\\mathcal{S}_2$ are equivalent", "as categories over $\\mathcal{C}$.", "Then $\\mathcal{S}_1$ is fibred over $\\mathcal{C}$ if and only if", "$\\mathcal{S}_2$ is fibred over $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Denote $p_i : \\mathcal{S}_i \\to \\mathcal{C}$ the given functors.", "Let $F : \\mathcal{S}_1 \\to \\mathcal{S}_2$,", "$G : \\mathcal{S}_2 \\to \\mathcal{S}_1$ be functors over $\\mathcal{C}$, and let", "$i : F \\circ G \\to \\text{id}_{\\mathcal{S}_2}$,", "$j : G \\circ F \\to \\text{id}_{\\mathcal{S}_1}$ be isomorphisms of", "functors over $\\mathcal{C}$.", "We claim that in this case $F$ maps strongly cartesian morphisms", "to strongly cartesian morphisms. Namely, suppose that", "$\\varphi : y \\to x$ is strongly cartesian in $\\mathcal{S}_1$.", "Set $f : V \\to U$ equal to $p_1(\\varphi)$. Suppose that", "$z' \\in \\Ob(\\mathcal{S}_2)$, with $W = p_2(z')$, and we are given", "$g : W \\to V$ and $\\psi' : z' \\to F(x)$ such that", "$p_2(\\psi') = f \\circ g$. Then", "$$", "\\psi = j \\circ G(\\psi') : G(z') \\to G(F(x)) \\to x", "$$", "is a morphism in $\\mathcal{S}_1$ with $p_1(\\psi) = f \\circ g$.", "Hence by assumption there exists a unique morphism $\\xi : G(z') \\to y$", "lying over $g$ such that $\\psi = \\varphi \\circ \\xi$. This in turn gives a", "morphism", "$$", "\\xi' = F(\\xi) \\circ i^{-1} : z' \\to F(G(z')) \\to F(y)", "$$", "lying over $g$ with $\\psi' = F(\\varphi) \\circ \\xi'$. We omit the verification", "that $\\xi'$ is unique." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12287, "type": "theorem", "label": "categories-lemma-2-product-fibred-categories-over-C", "categories": [ "categories" ], "title": "categories-lemma-2-product-fibred-categories-over-C", "contents": [ "Let $\\mathcal{C}$ be a category.", "The $(2, 1)$-category of fibred categories", "over $\\mathcal{C}$ has 2-fibre products, and", "they are described as in", "Lemma \\ref{lemma-2-product-categories-over-C}." ], "refs": [ "categories-lemma-2-product-categories-over-C" ], "proofs": [ { "contents": [ "Basically what one has to show here is that given", "$F : \\mathcal{X} \\to \\mathcal{S}$ and", "$G : \\mathcal{Y} \\to \\mathcal{S}$ morphisms of fibred", "categories over $\\mathcal{C}$, then the category", "$\\mathcal{X} \\times_\\mathcal{S} \\mathcal{Y}$", "described in Lemma \\ref{lemma-2-product-categories-over-C} is fibred.", "Let us show that $\\mathcal{X} \\times_\\mathcal{S} \\mathcal{Y}$", "has plenty of strongly cartesian morphisms.", "Namely, suppose we have $(U, x, y, \\phi)$ an object of", "$\\mathcal{X} \\times_\\mathcal{S} \\mathcal{Y}$.", "And suppose $f : V \\to U$ is a morphism in $\\mathcal{C}$.", "Choose strongly cartesian morphisms $a : f^*x \\to x$ in $\\mathcal{X}$", "lying over $f$ and $b : f^*y \\to y$ in $\\mathcal{Y}$ lying over $f$.", "By assumption $F(a)$ and $G(b)$ are strongly cartesian.", "Since $\\phi : F(x) \\to G(y)$ is an isomorphism, by the uniqueness", "of strongly cartesian morphisms we find a unique isomorphism", "$f^*\\phi : F(f^*x) \\to G(f^*y)$ such that", "$G(b) \\circ f^*\\phi = \\phi \\circ F(a)$. In other words", "$(a, b) : (V, f^*x, f^*y, f^*\\phi) \\to (U, x, y, \\phi)$", "is a morphism in $\\mathcal{X} \\times_\\mathcal{S} \\mathcal{Y}$.", "We omit the verification that this is a strongly cartesian morphism", "(and that these are in fact the only strongly cartesian morphisms)." ], "refs": [ "categories-lemma-2-product-categories-over-C" ], "ref_ids": [ 12280 ] } ], "ref_ids": [ 12280 ] }, { "id": 12288, "type": "theorem", "label": "categories-lemma-cute", "categories": [ "categories" ], "title": "categories-lemma-cute", "contents": [ "Let $\\mathcal{C}$ be a category. Let $U \\in \\Ob(\\mathcal{C})$.", "If $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category", "and $p$ factors through $p' : \\mathcal{S} \\to \\mathcal{C}/U$", "then $p' : \\mathcal{S} \\to \\mathcal{C}/U$ is a fibred category." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $\\varphi : x' \\to x$ is strongly cartesian with respect to $p$.", "We claim that $\\varphi$ is strongly cartesian with respect to $p'$ also.", "Set $g = p'(\\varphi)$, so that $g : V'/U \\to V/U$", "for some morphisms $f : V \\to U$ and $f' : V' \\to U$.", "Let $z \\in \\Ob(\\mathcal{S})$. Set $p'(z) = (W \\to U)$.", "To show that $\\varphi$ is strongly cartesian for $p'$ we have to show", "$$", "\\Mor_\\mathcal{S}(z, x')", "\\longrightarrow", "\\Mor_\\mathcal{S}(z, x)", "\\times_{\\Mor_{\\mathcal{C}/U}(W/U, V/U)}", "\\Mor_{\\mathcal{C}/U}(W/U, V'/U),", "$$", "given by $\\psi' \\longmapsto (\\varphi \\circ \\psi', p'(\\psi'))$", "is bijective. Suppose given an element $(\\psi, h)$ of the", "right hand side, then in particular $g \\circ h = p(\\psi)$,", "and by the condition that $\\varphi$ is strongly cartesian we", "get a unique morphism $\\psi' : z \\to x'$ with $\\psi = \\varphi \\circ \\psi'$", "and $p(\\psi') = h$. OK, and now $p'(\\psi') : W/U \\to V/U$", "is a morphism whose corresponding map $W \\to V$ is $h$, hence", "equal to $h$ as a morphism in $\\mathcal{C}/U$. Thus $\\psi'$ is", "a unique morphism $z \\to x'$ which maps to the given pair $(\\psi, h)$.", "This proves the claim.", "\\medskip\\noindent", "Finally, suppose given $g : V'/U \\to V/U$ and $x$ with $p'(x) = V/U$.", "Since $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category we", "see there exists a strongly cartesian morphism $\\varphi : x' \\to x$", "with $p(\\varphi) = g$. By the same argument as above it follows", "that $p'(\\varphi) = g : V'/U \\to V/U$. And as seen above the morphism", "$\\varphi$ is strongly cartesian. Thus the conditions of", "Definition \\ref{definition-fibred-category} are satisfied and we win." ], "refs": [ "categories-definition-fibred-category" ], "ref_ids": [ 12388 ] } ], "ref_ids": [] }, { "id": 12289, "type": "theorem", "label": "categories-lemma-fibred-over-fibred", "categories": [ "categories" ], "title": "categories-lemma-fibred-over-fibred", "contents": [ "Let $\\mathcal{A} \\to \\mathcal{B} \\to \\mathcal{C}$ be functors between", "categories. If $\\mathcal{A}$ is fibred over $\\mathcal{B}$ and", "$\\mathcal{B}$ is fibred over $\\mathcal{C}$, then $\\mathcal{A}$", "is fibred over $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the definitions and", "Lemma \\ref{lemma-cartesian-over-cartesian}." ], "refs": [ "categories-lemma-cartesian-over-cartesian" ], "ref_ids": [ 12283 ] } ], "ref_ids": [] }, { "id": 12290, "type": "theorem", "label": "categories-lemma-fibred-category-representable-goes-up", "categories": [ "categories" ], "title": "categories-lemma-fibred-category-representable-goes-up", "contents": [ "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category.", "Let $x \\to y$ and $z \\to y$ be morphisms of $\\mathcal{S}$", "with $x \\to y$ strongly cartesian. If $p(x) \\times_{p(y)} p(z)$ exists,", "then $x \\times_y z$ exists, $p(x \\times_y z) = p(x) \\times_{p(y)} p(z)$,", "and $x \\times_y z \\to z$ is strongly cartesian." ], "refs": [], "proofs": [ { "contents": [ "Pick a strongly cartesian morphism", "$\\text{pr}_2^*z \\to z$ lying over", "$\\text{pr}_2 : p(x) \\times_{p(y)} p(z) \\to p(z)$. Then", "$\\text{pr}_2^*z = x \\times_y z$ by", "Lemma \\ref{lemma-strongly-cartesian-fibre-product}." ], "refs": [ "categories-lemma-strongly-cartesian-fibre-product" ], "ref_ids": [ 12284 ] } ], "ref_ids": [] }, { "id": 12291, "type": "theorem", "label": "categories-lemma-ameliorate-morphism-fibred-categories", "categories": [ "categories" ], "title": "categories-lemma-ameliorate-morphism-fibred-categories", "contents": [ "Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{X} \\to \\mathcal{Y}$", "be a $1$-morphism of fibred categories over $\\mathcal{C}$.", "There exist $1$-morphisms of fibred categories over $\\mathcal{C}$", "$$", "\\xymatrix{", "\\mathcal{X} \\ar@<1ex>[r]^u &", "\\mathcal{X}' \\ar[r]^v \\ar@<1ex>[l]^w & \\mathcal{Y}", "}", "$$", "such that $F = v \\circ u$ and such that", "\\begin{enumerate}", "\\item $u : \\mathcal{X} \\to \\mathcal{X}'$ is fully faithful,", "\\item $w$ is left adjoint to $u$, and", "\\item $v : \\mathcal{X}' \\to \\mathcal{Y}$ is a fibred category.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $p : \\mathcal{X} \\to \\mathcal{C}$ and $q : \\mathcal{Y} \\to \\mathcal{C}$", "the structure functors. We construct $\\mathcal{X}'$ explicitly as follows.", "An object of $\\mathcal{X}'$ is a quadruple $(U, x, y, f)$ where", "$x \\in \\Ob(\\mathcal{X}_U)$, $y \\in \\Ob(\\mathcal{Y}_U)$", "and $f : y \\to F(x)$ is a morphism in $\\mathcal{Y}_U$.", "A morphism $(a, b) : (U, x, y, f) \\to (U', x', y', f')$ is given", "by $a : x \\to x'$ and $b : y \\to y'$ with $p(a) = q(b) : U \\to U'$", "and such that $f' \\circ b = F(a) \\circ f$.", "\\medskip\\noindent", "Let us make a choice of pullbacks for both $p$ and $q$ and let us", "use the same notation to indicate them.", "Let $(U, x, y, f)$ be an object and let $h : V \\to U$ be a morphism.", "Consider the morphism $c : (V, h^*x, h^*y, h^*f) \\to (U, x, y, f)$", "coming from the given strongly cartesian maps $h^*x \\to x$ and $h^*y \\to y$.", "We claim $c$ is strongly cartesian in $\\mathcal{X}'$ over $\\mathcal{C}$.", "Namely, suppose we are given an object $(W, x', y', f')$ of $\\mathcal{X}'$,", "a morphism $(a, b) : (W, x', y', f') \\to (U, x, y, f)$ lying over", "$W \\to U$, and a factorization $W \\to V \\to U$ of $W \\to U$ through $h$.", "As $h^*x \\to x$ and $h^*y \\to y$ are strongly cartesian we obtain morphisms", "$a' : x' \\to h^*x$ and $b' : y' \\to h^*y$ lying over the given morphism", "$W \\to V$. Consider the diagram", "$$", "\\xymatrix{", "y' \\ar[d]_{f'} \\ar[r] & h^*y \\ar[r] \\ar[d]_{h^*f} & y \\ar[d]_f \\\\", "F(x') \\ar[r] & F(h^*x) \\ar[r] & F(x)", "}", "$$", "The outer rectangle and the right square commute.", "Since $F$ is a $1$-morphism of fibred categories the morphism", "$F(h^*x) \\to F(x)$ is strongly cartesian.", "Hence the left square commutes by the universal property", "of strongly cartesian morphisms. This proves that $\\mathcal{X}'$", "is fibred over $\\mathcal{C}$.", "\\medskip\\noindent", "The functor $u : \\mathcal{X} \\to \\mathcal{X}'$ is given by", "$x \\mapsto (p(x), x, F(x), \\text{id})$. This is fully faithful.", "The functor $\\mathcal{X}' \\to \\mathcal{Y}$ is given by", "$(U, x, y, f) \\mapsto y$. The functor $w : \\mathcal{X}' \\to \\mathcal{X}$", "is given by $(U, x, y, f) \\mapsto x$. Each of these functors is", "a $1$-morphism of fibred categories over $\\mathcal{C}$ by our", "description of strongly cartesian morphisms of $\\mathcal{X}'$ over", "$\\mathcal{C}$. Adjointness of $w$ and $u$ means that", "$$", "\\Mor_\\mathcal{X}(x, x') =", "\\Mor_{\\mathcal{X}'}((U, x, y, f), (p(x'), x', F(x'), \\text{id})),", "$$", "which follows immediately from the definitions.", "\\medskip\\noindent", "Finally, we have to show that $\\mathcal{X}' \\to \\mathcal{Y}$ is a fibred", "category. Let $c : y' \\to y$ be a morphism in $\\mathcal{Y}$", "and let $(U, x, y, f)$ be an object of $\\mathcal{X}'$ lying over $y$.", "Set $V = q(y')$ and let $h = q(c) : V \\to U$. Let $a : h^*x \\to x$", "and $b : h^*y \\to y$ be the strongly cartesian morphisms covering $h$.", "Since $F$ is a $1$-morphism of fibred categories we may identify", "$h^*F(x) = F(h^*x)$ with strongly cartesian morphism", "$F(a) : F(h^*x) \\to F(x)$. By the universal property", "of $b : h^*y \\to y$ there is a morphism $c' : y' \\to h^*y$ in", "$\\mathcal{Y}_V$ such that $c = b \\circ c'$. We claim that", "$$", "(a, c) : (V, h^*x, y', h^*f \\circ c') \\longrightarrow (U, x, y, f)", "$$", "is strongly cartesian in $\\mathcal{X}'$ over $\\mathcal{Y}$. To see this", "let $(W, x_1, y_1, f_1)$ be an object of $\\mathcal{X}'$, let", "$(a_1, b_1) : (W, x_1, y_1, f_1) \\to (U, x, y, f)$ be a morphism", "and let $b_1 = c \\circ b_1'$ for some morphism $b_1' : y_1 \\to y'$.", "Then", "$$", "(a_1', b_1') : (W, x_1, y_1, f_1) \\longrightarrow (V, h^*x, y', h^*f \\circ c')", "$$", "(where $a_1' : x_1 \\to h^*x$ is the unique morphism lying over the", "given morphism $q(b_1') : W \\to V$ such that $a_1 = a \\circ a_1'$)", "is the desired morphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12292, "type": "theorem", "label": "categories-lemma-inertia-fibred-category", "categories": [ "categories" ], "title": "categories-lemma-inertia-fibred-category", "contents": [ "Let $\\mathcal{C}$ be a category. Let", "$p : \\mathcal{S} \\to \\mathcal{C}$ and", "$p' : \\mathcal{S}' \\to \\mathcal{C}$ be fibred categories.", "Let $F : \\mathcal{S} \\to \\mathcal{S}'$ be a $1$-morphism of", "fibred categories over $\\mathcal{C}$. Consider the category", "$\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ over $\\mathcal{C}$ whose", "\\begin{enumerate}", "\\item objects are pairs $(x, \\alpha)$ where $x \\in \\Ob(\\mathcal{S})$", "and $\\alpha : x \\to x$ is an automorphism with $F(\\alpha) = \\text{id}$,", "\\item morphisms $(x, \\alpha) \\to (y, \\beta)$ are given by morphisms", "$\\phi : x \\to y$ such that", "$$", "\\xymatrix{", "x\\ar[r]_\\phi\\ar[d]_\\alpha &", "y\\ar[d]^{\\beta} \\\\", "x\\ar[r]^\\phi &", "y \\\\", "}", "$$", "commutes, and", "\\item the functor $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'} \\to \\mathcal{C}$", "is given by $(x, \\alpha) \\mapsto p(x)$.", "\\end{enumerate}", "Then", "\\begin{enumerate}", "\\item there is an equivalence", "$$", "\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'} \\longrightarrow", "\\mathcal{S}", "\\times_{\\Delta, (\\mathcal{S} \\times_{\\mathcal{S}'} \\mathcal{S}), \\Delta}", "\\mathcal{S}", "$$", "in the $(2, 1)$-category of categories over $\\mathcal{C}$, and", "\\item $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ is a fibred category over", "$\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that (2) follows from (1) by", "Lemmas \\ref{lemma-2-product-fibred-categories-over-C} and", "\\ref{lemma-fibred-equivalent}. Thus it suffices to prove (1).", "We will use without further mention the construction of the $2$-fibre product", "from", "Lemma \\ref{lemma-2-product-fibred-categories-over-C}.", "In particular an object of", "$\\mathcal{S}", "\\times_{\\Delta, (\\mathcal{S} \\times_{\\mathcal{S}'} \\mathcal{S}), \\Delta}", "\\mathcal{S}$", "is a triple $(x, y, (\\iota, \\kappa))$ where $x$ and $y$ are objects of", "$\\mathcal{S}$, and", "$(\\iota, \\kappa) : (x, x, \\text{id}_{F(x)}) \\to (y, y, \\text{id}_{F(y)})$", "is an isomorphism in $\\mathcal{S} \\times_{\\mathcal{S}'} \\mathcal{S}$.", "This just means that $\\iota, \\kappa : x \\to y$ are isomorphisms and that", "$F(\\iota) = F(\\kappa)$. Consider the functor", "$$", "I_{\\mathcal{S}/\\mathcal{S}'}", "\\longrightarrow", "\\mathcal{S}", "\\times_{\\Delta, (\\mathcal{S} \\times_{\\mathcal{S}'} \\mathcal{S}), \\Delta}", "\\mathcal{S}", "$$", "which to an object $(x, \\alpha)$ of the left hand side assigns the object", "$(x, x, (\\alpha, \\text{id}_x))$ of the right hand side", "and to a morphism $\\phi$ of the left hand side", "assigns the morphism $(\\phi, \\phi)$ of the right hand side.", "We claim that a quasi-inverse to that morphism is given by the", "functor", "$$", "\\mathcal{S}", "\\times_{\\Delta, (\\mathcal{S} \\times_{\\mathcal{S}'} \\mathcal{S}), \\Delta}", "\\mathcal{S}", "\\longrightarrow", "I_{\\mathcal{S}/\\mathcal{S}'}", "$$", "which to an object $(x, y, (\\iota, \\kappa))$ of the left hand side", "assigns the object $(x, \\kappa^{-1} \\circ \\iota)$ of the right hand side", "and to a morphism", "$(\\phi, \\phi') : (x, y, (\\iota, \\kappa)) \\to (z, w, (\\lambda, \\mu))$", "of the left hand side assigns the morphism $\\phi$.", "Indeed, the endo-functor of $I_{\\mathcal{S}/\\mathcal{S}'}$ induced", "by composing the two functors above is the identity on the nose, and", "the endo-functor induced on", "$\\mathcal{S}", "\\times_{\\Delta, (\\mathcal{S} \\times_{\\mathcal{S}'} \\mathcal{S}), \\Delta}", "\\mathcal{S}$", "is isomorphic to", "the identity via the natural isomorphism", "$$", "(\\text{id}_x, \\kappa) :", "(x, x, (\\kappa^{-1} \\circ \\iota, \\text{id}_x))", "\\longrightarrow", "(x, y, (\\iota, \\kappa)).", "$$", "Some details omitted." ], "refs": [ "categories-lemma-2-product-fibred-categories-over-C", "categories-lemma-fibred-equivalent", "categories-lemma-2-product-fibred-categories-over-C" ], "ref_ids": [ 12287, 12286, 12287 ] } ], "ref_ids": [] }, { "id": 12293, "type": "theorem", "label": "categories-lemma-relative-inertia-as-fibre-product", "categories": [ "categories" ], "title": "categories-lemma-relative-inertia-as-fibre-product", "contents": [ "Let $F : \\mathcal{S} \\to \\mathcal{S}'$ be a $1$-morphism of categories", "fibred over a category $\\mathcal{C}$. Then the diagram", "$$", "\\xymatrix{", "\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}", "\\ar[d]_{F \\circ (\\ref{equation-inertia-structure-map})}", "\\ar[rr]_{(\\ref{equation-comparison})} & &", "\\mathcal{I}_\\mathcal{S} \\ar[d]^{(\\ref{equation-functorial})} \\\\", "\\mathcal{S}' \\ar[rr]^e & &", "\\mathcal{I}_{\\mathcal{S}'}", "}", "$$", "is a $2$-fibre product." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12294, "type": "theorem", "label": "categories-lemma-fibred-groupoids", "categories": [ "categories" ], "title": "categories-lemma-fibred-groupoids", "contents": [ "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a functor.", "The following are equivalent", "\\begin{enumerate}", "\\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a category", "fibred in groupoids, and", "\\item all fibre categories are groupoids and", "$\\mathcal{S}$ is a fibred category over $\\mathcal{C}$.", "\\end{enumerate}", "Moreover, in this case every morphism of $\\mathcal{S}$ is", "strongly cartesian. In addition, given $f^\\ast x \\to x$", "lying over $f$ for all $f: V \\to U = p(x)$ the data", "$(U \\mapsto \\mathcal{S}_U, f \\mapsto f^*, \\alpha_{f, g}, \\alpha_U)$", "constructed in Lemma \\ref{lemma-fibred}", "defines a pseudo functor from $\\mathcal{C}^{opp}$ in to", "the $(2, 1)$-category of groupoids." ], "refs": [ "categories-lemma-fibred" ], "proofs": [ { "contents": [ "Assume $p : \\mathcal{S} \\to \\mathcal{C}$ is fibred in groupoids.", "To show all fibre categories $\\mathcal{S}_U$ for", "$U \\in \\Ob(\\mathcal{C})$", "are groupoids, we must exhibit for every $f : y \\to x$ in $\\mathcal{S}_U$ an", "inverse morphism. The diagram on the left (in $\\mathcal{S}_U$) is mapped by", "$p$ to the diagram on the right:", "$$", "\\xymatrix{", "y \\ar[r]^f & x & U \\ar[r]^{\\text{id}_U} & U \\\\", "x \\ar@{-->}[u] \\ar[ru]_{\\text{id}_x} & &", "U \\ar@{-->}[u]\\ar[ru]_{\\text{id}_U} & \\\\", "}", "$$", "Since only $\\text{i}d_U$ makes the diagram on the right commute, there is a", "unique $g : x \\to y$ making the diagram on the left commute, so", "$fg = \\text{id}_x$. By a similar argument there is a unique $h : y \\to x$ so", "that $gh = \\text{id}_y$. Then $fgh = f : y \\to x$. We have $fg = \\text{id}_x$,", "so $h = f$. Condition (2) of Definition \\ref{definition-fibred-groupoids} says", "exactly that every morphism of $\\mathcal{S}$ is strongly cartesian. Hence", "condition (1) of Definition \\ref{definition-fibred-groupoids} implies that", "$\\mathcal{S}$ is a fibred category over $\\mathcal{C}$.", "\\medskip\\noindent", "Conversely, assume all fibre categories are groupoids and", "$\\mathcal{S}$ is a fibred category over $\\mathcal{C}$.", "We have to check conditions (1) and (2) of", "Definition \\ref{definition-fibred-groupoids}.", "The first condition follows trivially. Let $\\phi : y \\to x$,", "$\\psi : z \\to x$ and $f : p(z) \\to p(y)$ such that", "$p(\\phi) \\circ f = p(\\psi)$ be as in condition (2) of", "Definition \\ref{definition-fibred-groupoids}.", "Write $U = p(x)$, $V = p(y)$, $W = p(z)$, $p(\\phi) = g : V \\to U$,", "$p(\\psi) = h : W \\to U$. Choose a strongly cartesian $g^*x \\to x$", "lying over $g$. Then we get a morphism $i : y \\to g^*x$ in", "$\\mathcal{S}_V$, which is therefore an isomorphism. We", "also get a morphism $j : z \\to g^*x$ corresponding to", "the pair $(\\psi, f)$ as $g^*x \\to x$ is strongly cartesian.", "Then one checks that $\\chi = i^{-1} \\circ j$ is a solution.", "\\medskip\\noindent", "We have seen in the proof of (1) $\\Rightarrow$ (2) that", "every morphism of $\\mathcal{S}$ is strongly cartesian.", "The final statement follows directly from Lemma \\ref{lemma-fibred}." ], "refs": [ "categories-definition-fibred-groupoids", "categories-definition-fibred-groupoids", "categories-definition-fibred-groupoids", "categories-definition-fibred-groupoids", "categories-lemma-fibred" ], "ref_ids": [ 12392, 12392, 12392, 12392, 12285 ] } ], "ref_ids": [ 12285 ] }, { "id": 12295, "type": "theorem", "label": "categories-lemma-fibred-gives-fibred-groupoids", "categories": [ "categories" ], "title": "categories-lemma-fibred-gives-fibred-groupoids", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category.", "Let $\\mathcal{S}'$ be the subcategory of $\\mathcal{S}$ defined", "as follows", "\\begin{enumerate}", "\\item $\\Ob(\\mathcal{S}') = \\Ob(\\mathcal{S})$, and", "\\item for $x, y \\in \\Ob(\\mathcal{S}')$ the set of morphisms between $x$", "and $y$ in $\\mathcal{S}'$ is the set of strongly cartesian morphisms between", "$x$ and $y$ in $\\mathcal{S}$.", "\\end{enumerate}", "Let $p' : \\mathcal{S}' \\to \\mathcal{C}$ be the restriction of $p$", "to $\\mathcal{S}'$. Then $p' : \\mathcal{S}' \\to \\mathcal{C}$ is fibred", "in groupoids." ], "refs": [], "proofs": [ { "contents": [ "Note that the construction makes sense since by", "Lemma \\ref{lemma-composition-cartesian}", "the identity morphism of any object of $\\mathcal{S}$ is strongly cartesian,", "and the composition of strongly cartesian morphisms is strongly cartesian.", "The first lifting property of", "Definition \\ref{definition-fibred-groupoids}", "follows from the condition that in a fibred category", "given any morphism $f : V \\to U$ and $x$ lying over $U$ there exists", "a strongly cartesian morphism $\\varphi : y \\to x$ lying over $f$.", "Let us check the second lifting property of", "Definition \\ref{definition-fibred-groupoids}", "for the category $p' : \\mathcal{S}' \\to \\mathcal{C}$ over $\\mathcal{C}$.", "To do this we argue as in the discussion following", "Definition \\ref{definition-fibred-groupoids}.", "Thus in Diagram \\ref{equation-fibred-groupoids} the", "morphisms $\\phi$, $\\psi$ and $\\gamma$ are strongly cartesian morphisms", "of $\\mathcal{S}$.", "Hence $\\gamma$ and $\\phi \\circ \\psi$ are strongly cartesian morphisms", "of $\\mathcal{S}$ lying over the same arrow of $\\mathcal{C}$ and", "having the same target in $\\mathcal{S}$. By the discussion following", "Definition \\ref{definition-cartesian-over-C}", "this means these two arrows are isomorphic as desired (here we use also", "that any isomorphism in $\\mathcal{S}$ is strongly cartesian, by", "Lemma \\ref{lemma-composition-cartesian} again)." ], "refs": [ "categories-lemma-composition-cartesian", "categories-definition-fibred-groupoids", "categories-definition-fibred-groupoids", "categories-definition-fibred-groupoids", "categories-definition-cartesian-over-C", "categories-lemma-composition-cartesian" ], "ref_ids": [ 12282, 12392, 12392, 12392, 12387, 12282 ] } ], "ref_ids": [] }, { "id": 12296, "type": "theorem", "label": "categories-lemma-2-product-fibred-categories", "categories": [ "categories" ], "title": "categories-lemma-2-product-fibred-categories", "contents": [ "Let $\\mathcal{C}$ be a category.", "The $2$-category of categories fibred in groupoids", "over $\\mathcal{C}$ has 2-fibre products, and they are described as in", "Lemma \\ref{lemma-2-product-categories-over-C}." ], "refs": [ "categories-lemma-2-product-categories-over-C" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-2-product-fibred-categories-over-C}", "the fibre product as described in", "Lemma \\ref{lemma-2-product-categories-over-C} is a fibred category.", "Hence it suffices to prove that the fibre categories are", "groupoids, see Lemma \\ref{lemma-fibred-groupoids}.", "By Lemma \\ref{lemma-fibre-2-fibre-product-categories-over-C}", "it is enough to show that the $2$-fibre product of groupoids", "is a groupoid, which is clear (from the construction in", "Lemma \\ref{lemma-2-fibre-product-categories} for example)." ], "refs": [ "categories-lemma-2-product-fibred-categories-over-C", "categories-lemma-2-product-categories-over-C", "categories-lemma-fibred-groupoids", "categories-lemma-fibre-2-fibre-product-categories-over-C", "categories-lemma-2-fibre-product-categories" ], "ref_ids": [ 12287, 12280, 12294, 12281, 12270 ] } ], "ref_ids": [ 12280 ] }, { "id": 12297, "type": "theorem", "label": "categories-lemma-equivalence-fibred-categories", "categories": [ "categories" ], "title": "categories-lemma-equivalence-fibred-categories", "contents": [ "Let $p : \\mathcal{S}\\to \\mathcal{C}$ and", "$p' : \\mathcal{S'}\\to \\mathcal{C}$ be categories fibred in groupoids, and", "suppose that $G : \\mathcal{S}\\to \\mathcal {S}'$ is a functor over", "$\\mathcal{C}$.", "\\begin{enumerate}", "\\item Then $G$ is faithful (resp.\\ fully faithful, resp.\\ an equivalence)", "if and only if for each $U\\in\\Ob(\\mathcal{C})$ the induced functor", "$G_U : \\mathcal{S}_U\\to \\mathcal{S}'_U$ is faithful", "(resp.\\ fully faithful, resp.\\ an equivalence).", "\\item If $G$ is an equivalence, then $G$ is an equivalence in the", "$2$-category of categories fibred in groupoids over $\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $x, y$ be objects of $\\mathcal{S}$ lying over the same object $U$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "\\Mor_\\mathcal{S}(x, y) \\ar[rd]_p \\ar[rr]_G & &", "\\Mor_{\\mathcal{S}'}(G(x), G(y)) \\ar[ld]^{p'} \\\\", "& \\Mor_\\mathcal{C}(U, U) &", "}", "$$", "From this diagram it is clear that if $G$ is faithful (resp.\\ fully faithful)", "then so is each $G_U$.", "\\medskip\\noindent", "Suppose $G$ is an equivalence. For every object", "$x'$ of $\\mathcal{S}'$ there exists an object $x$ of $\\mathcal{S}$", "such that $G(x)$ is isomorphic to $x'$. Suppose that $x'$ lies", "over $U'$ and $x$ lies over $U$. Then there is an isomorphism", "$f : U' \\to U$ in $\\mathcal{C}$, namely, $p'$ applied to the", "isomorphism $x' \\to G(x)$. By the axioms of a category fibred", "in groupoids there exists an arrow $f^*x \\to x$ of $\\mathcal{S}$", "lying over $f$. Hence there exists an isomorphism", "$\\alpha : x' \\to G(f^*x)$ such that $p'(\\alpha) = \\text{id}_{U'}$", "(this time by the axioms for $\\mathcal{S}'$). All in all we conclude", "that for every object $x'$ of $\\mathcal{S}'$ we can choose", "a pair $(o_{x'}, \\alpha_{x'})$ consisting of an object", "$o_{x'}$ of $\\mathcal{S}$ and an isomorphism $\\alpha_{x'} : x' \\to G(o_{x'})$", "with $p'(\\alpha_{x'}) = \\text{id}_{p'(x')}$.", "From this point on we proceed as usual (see proof of", "Lemma \\ref{lemma-equivalence-categories}) to produce an inverse", "functor $F : \\mathcal{S}' \\to \\mathcal{S}$, by taking", "$x' \\mapsto o_{x'}$ and $\\varphi' : x' \\to y'$ to the unique", "arrow $\\varphi_{\\varphi'} : o_{x'} \\to o_{y'}$ with", "$\\alpha_{y'}^{-1} \\circ G(\\varphi_{\\varphi'}) \\circ \\alpha_{x'} = \\varphi'$.", "With these choices $F$ is a functor over $\\mathcal{C}$.", "We omit the verification that $G \\circ F$ and $F \\circ G$ are", "$2$-isomorphic to the respective identity functors", "(in the $2$-category of categories fibred in groupoids over $\\mathcal{C}$).", "\\medskip\\noindent", "Suppose that $G_U$ is faithful (resp.\\ fully faithful)", "for all $U\\in\\Ob(\\mathcal C)$. To", "show that $G$ is faithful (resp.\\ fully faithful)", "we have to show for any objects", "$x, y\\in\\Ob(\\mathcal{S})$ that $G$ induces an", "injection (resp.\\ bijection) between", "$\\Mor_\\mathcal{S}(x, y)$ and", "$\\Mor_{\\mathcal{S}'}(G(x), G(y))$.", "Set $U = p(x)$ and $V = p(y)$.", "It suffices to prove that $G$", "induces an injection (resp.\\ bijection) between morphism", "$x \\to y$ lying over $f$ to morphisms $G(x) \\to G(y)$ lying over $f$", "for any morphism $f : U \\to V$.", "Now fix $f : U \\to V$. Denote $f^*y \\to y$ a pullback.", "Then also $G(f^*y) \\to G(y)$ is a pullback.", "The set of morphisms from $x$ to $y$ lying over $f$", "is bijective to the set of morphisms between", "$x$ and $f^*y$ lying over $\\text{id}_U$. (By the second axiom", "of a category fibred in groupoids.) Similarly", "the set of morphisms from $G(x)$ to $G(y)$ lying over $f$", "is bijective to the set of morphisms between", "$G(x)$ and $G(f^*y)$ lying over $\\text{id}_U$.", "Hence the fact that $G_U$ is faithful (resp.\\ fully faithful)", "gives the desired result.", "\\medskip\\noindent", "Finally suppose for all $G_U$ is an equivalence for all $U$, so it is", "fully faithful and essentially surjective. We have seen this implies $G$ is", "fully faithful, and thus to prove it is an equivalence we have to prove that", "it is essentially surjective. This is clear, for if $z'\\in", "\\Ob(\\mathcal{S}')$ then $z'\\in \\Ob(\\mathcal{S}'_U)$ where", "$U = p'(z')$. Since $G_U$ is essentially surjective we know that", "$z'$ is isomorphic, in $\\mathcal{S}'_U$, to an object of the form", "$G_U(z)$ for some $z\\in \\Ob(\\mathcal{S}_U)$. But morphisms", "in $\\mathcal{S}'_U$ are morphisms in $\\mathcal{S}'$ and hence $z'$ is", "isomorphic to $G(z)$ in $\\mathcal{S}'$." ], "refs": [ "categories-lemma-equivalence-categories" ], "ref_ids": [ 12202 ] } ], "ref_ids": [] }, { "id": 12298, "type": "theorem", "label": "categories-lemma-fully-faithful-diagonal-equivalence", "categories": [ "categories" ], "title": "categories-lemma-fully-faithful-diagonal-equivalence", "contents": [ "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S}\\to \\mathcal{C}$ and", "$p' : \\mathcal{S'}\\to \\mathcal{C}$ be categories fibred in groupoids.", "Let $G : \\mathcal{S}\\to \\mathcal {S}'$ be a functor over $\\mathcal{C}$.", "Then $G$ is fully faithful if and only if the diagonal", "$$", "\\Delta_G :", "\\mathcal{S}", "\\longrightarrow", "\\mathcal{S} \\times_{G, \\mathcal{S}', G} \\mathcal{S}", "$$", "is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-equivalence-fibred-categories}", "it suffices to look at fibre categories over an object $U$ of $\\mathcal{C}$.", "An object of the right hand side is a triple $(x, x', \\alpha)$ where", "$\\alpha : G(x) \\to G(x')$ is a morphism in $\\mathcal{S}'_U$.", "The functor $\\Delta_G$ maps the object $x$ of $\\mathcal{S}_U$", "to the triple $(x, x, \\text{id}_{G(x)})$. Note that $(x, x', \\alpha)$", "is in the essential image of $\\Delta_G$ if and only if $\\alpha = G(\\beta)$", "for some morphism $\\beta : x \\to x'$ in $\\mathcal{S}_U$ (details omitted).", "Hence in order for $\\Delta_G$ to be an equivalence, every $\\alpha$ has to", "be the image of a morphism $\\beta : x \\to x'$, and also every two", "distinct morphisms $\\beta, \\beta' : x \\to x'$ have to give distinct", "morphisms $G(\\beta), G(\\beta')$. This proves the lemma." ], "refs": [ "categories-lemma-equivalence-fibred-categories" ], "ref_ids": [ 12297 ] } ], "ref_ids": [] }, { "id": 12299, "type": "theorem", "label": "categories-lemma-morphisms-equivalent-fibred-groupoids", "categories": [ "categories" ], "title": "categories-lemma-morphisms-equivalent-fibred-groupoids", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{S}_i$, $i = 1, 2, 3, 4$ be categories fibred in", "groupoids over $\\mathcal{C}$.", "Suppose that $\\varphi : \\mathcal{S}_1 \\to \\mathcal{S}_2$ and", "$\\psi : \\mathcal{S}_3 \\to \\mathcal{S}_4$ are equivalences", "over $\\mathcal{C}$. Then", "$$", "\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{S}_2, \\mathcal{S}_3)", "\\longrightarrow", "\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{S}_1, \\mathcal{S}_4),", "\\quad \\alpha \\longmapsto \\psi \\circ \\alpha \\circ \\varphi", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "This is a generality and holds in any $2$-category." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12300, "type": "theorem", "label": "categories-lemma-inertia-fibred-groupoids", "categories": [ "categories" ], "title": "categories-lemma-inertia-fibred-groupoids", "contents": [ "Let $\\mathcal{C}$ be a category.", "If $p : \\mathcal{S} \\to \\mathcal{C}$ is fibred in groupoids, then", "so is the inertia fibred category $\\mathcal{I}_\\mathcal{S} \\to \\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Clear from the construction in", "Lemma \\ref{lemma-inertia-fibred-category}", "or by using (from the same lemma) that", "$I_\\mathcal{S} \\to \\mathcal{S}", "\\times_{\\Delta, \\mathcal{S} \\times_\\mathcal{C} \\mathcal{S}, \\Delta}\\mathcal{S}$", "is an equivalence and appealing to", "Lemma \\ref{lemma-2-product-fibred-categories}." ], "refs": [ "categories-lemma-inertia-fibred-category", "categories-lemma-2-product-fibred-categories" ], "ref_ids": [ 12292, 12296 ] } ], "ref_ids": [] }, { "id": 12301, "type": "theorem", "label": "categories-lemma-cute-groupoids", "categories": [ "categories" ], "title": "categories-lemma-cute-groupoids", "contents": [ "Let $\\mathcal{C}$ be a category. Let $U \\in \\Ob(\\mathcal{C})$.", "If $p : \\mathcal{S} \\to \\mathcal{C}$ is a category fibred in groupoids", "and $p$ factors through $p' : \\mathcal{S} \\to \\mathcal{C}/U$", "then $p' : \\mathcal{S} \\to \\mathcal{C}/U$ is fibred in groupoids." ], "refs": [], "proofs": [ { "contents": [ "We have already seen in Lemma \\ref{lemma-cute} that $p'$ is a fibred", "category. Hence it suffices to prove the fibre categories are groupoids,", "see Lemma \\ref{lemma-fibred-groupoids}.", "For $V \\in \\Ob(\\mathcal{C})$ we have", "$$", "\\mathcal{S}_V = \\coprod\\nolimits_{f : V \\to U} \\mathcal{S}_{(f : V \\to U)}", "$$", "where the left hand side is the fibre category of $p$ and the right hand side", "is the disjoint union of the fibre categories of $p'$.", "Hence the result." ], "refs": [ "categories-lemma-cute", "categories-lemma-fibred-groupoids" ], "ref_ids": [ 12288, 12294 ] } ], "ref_ids": [] }, { "id": 12302, "type": "theorem", "label": "categories-lemma-fibred-in-groupoids-over-fibred-in-groupoids", "categories": [ "categories" ], "title": "categories-lemma-fibred-in-groupoids-over-fibred-in-groupoids", "contents": [ "Let $\\mathcal{A} \\to \\mathcal{B} \\to \\mathcal{C}$ be functors between", "categories. If $\\mathcal{A}$ is fibred in groupoids over $\\mathcal{B}$", "and $\\mathcal{B}$ is fibred in groupoids over $\\mathcal{C}$, then", "$\\mathcal{A}$ is fibred in groupoids over $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "One can prove this directly from the definition. However, we will argue", "using the criterion of Lemma \\ref{lemma-fibred-groupoids}.", "By Lemma \\ref{lemma-fibred-over-fibred} we see that $\\mathcal{A}$", "is fibred over $\\mathcal{C}$. To finish the proof we show that the fibre", "category $\\mathcal{A}_U$ is a groupoid for $U$ in $\\mathcal{C}$.", "Namely, if $x \\to y$ is a morphism of $\\mathcal{A}_U$, then its", "image in $\\mathcal{B}$ is an isomorphism as $\\mathcal{B}_U$ is", "a groupoid. But then $x \\to y$ is an isomorphism, for example by", "Lemma \\ref{lemma-composition-cartesian} and the fact that every", "morphism of $\\mathcal{A}$ is strongly $\\mathcal{B}$-cartesian", "(see Lemma \\ref{lemma-fibred-groupoids})." ], "refs": [ "categories-lemma-fibred-groupoids", "categories-lemma-fibred-over-fibred", "categories-lemma-composition-cartesian", "categories-lemma-fibred-groupoids" ], "ref_ids": [ 12294, 12289, 12282, 12294 ] } ], "ref_ids": [] }, { "id": 12303, "type": "theorem", "label": "categories-lemma-fibred-groupoids-fibre-product-goes-up", "categories": [ "categories" ], "title": "categories-lemma-fibred-groupoids-fibre-product-goes-up", "contents": [ "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids.", "Let $x \\to y$ and $z \\to y$ be morphisms of $\\mathcal{S}$.", "If $p(x) \\times_{p(y)} p(z)$ exists, then", "$x \\times_y z$ exists and $p(x \\times_y z) = p(x) \\times_{p(y)} p(z)$." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Lemma \\ref{lemma-fibred-category-representable-goes-up}." ], "refs": [ "categories-lemma-fibred-category-representable-goes-up" ], "ref_ids": [ 12290 ] } ], "ref_ids": [] }, { "id": 12304, "type": "theorem", "label": "categories-lemma-ameliorate-morphism-categories-fibred-groupoids", "categories": [ "categories" ], "title": "categories-lemma-ameliorate-morphism-categories-fibred-groupoids", "contents": [ "Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{X} \\to \\mathcal{Y}$", "be a $1$-morphism of categories fibred in groupoids over $\\mathcal{C}$.", "There exists a factorization $\\mathcal{X} \\to \\mathcal{X}' \\to \\mathcal{Y}$", "by $1$-morphisms of categories fibred in groupoids over $\\mathcal{C}$ such", "that $\\mathcal{X} \\to \\mathcal{X}'$ is an equivalence over $\\mathcal{C}$", "and such that $\\mathcal{X}'$ is a category fibred in groupoids over", "$\\mathcal{Y}$." ], "refs": [], "proofs": [ { "contents": [ "Denote $p : \\mathcal{X} \\to \\mathcal{C}$ and $q : \\mathcal{Y} \\to \\mathcal{C}$", "the structure functors. We construct $\\mathcal{X}'$ explicitly as follows.", "An object of $\\mathcal{X}'$ is a quadruple $(U, x, y, f)$ where", "$x \\in \\Ob(\\mathcal{X}_U)$, $y \\in \\Ob(\\mathcal{Y}_U)$", "and $f : F(x) \\to y$ is an isomorphism in $\\mathcal{Y}_U$.", "A morphism $(a, b) : (U, x, y, f) \\to (U', x', y', f')$ is given", "by $a : x \\to x'$ and $b : y \\to y'$ with $p(a) = q(b)$ and", "such that $f' \\circ F(a) = b \\circ f$. In other words", "$\\mathcal{X}' = \\mathcal{X} \\times_{F, \\mathcal{Y}, \\text{id}} \\mathcal{Y}$", "with the construction of the $2$-fibre product from", "Lemma \\ref{lemma-2-product-categories-over-C}.", "By", "Lemma \\ref{lemma-2-product-fibred-categories}", "we see that $\\mathcal{X}'$ is a category fibred in groupoids over", "$\\mathcal{C}$ and that $\\mathcal{X}' \\to \\mathcal{Y}$ is a morphism of", "categories over $\\mathcal{C}$. As functor $\\mathcal{X} \\to \\mathcal{X}'$ we take", "$x \\mapsto (p(x), x, F(x), \\text{id}_{F(x)})$ on objects and", "$(a : x \\to x') \\mapsto (a, F(a))$ on morphisms. It is clear that", "the composition $\\mathcal{X} \\to \\mathcal{X}' \\to \\mathcal{Y}$", "equals $F$. We omit the verification that", "$\\mathcal{X} \\to \\mathcal{X}'$ is an equivalence of fibred categories over", "$\\mathcal{C}$.", "\\medskip\\noindent", "Finally, we have to show that $\\mathcal{X}' \\to \\mathcal{Y}$ is a category", "fibred in groupoids. Let $b : y' \\to y$ be a morphism in $\\mathcal{Y}$", "and let $(U, x, y, f)$ be an object of $\\mathcal{X}'$ lying over $y$.", "Because $\\mathcal{X}$ is fibred in groupoids over $\\mathcal{C}$ we", "can find a morphism $a : x' \\to x$ lying over $U' = q(y') \\to q(y) = U$.", "Since $\\mathcal{Y}$ is fibred in groupoids over $\\mathcal{C}$ and since", "both $F(x') \\to F(x)$ and $y' \\to y$ lie over the same morphism $U' \\to U$", "we can find $f' : F(x') \\to y'$ lying over $\\text{id}_{U'}$ such that", "$f \\circ F(a) = b \\circ f'$. Hence we obtain", "$(a, b) : (U', x', y', f') \\to (U, x, y, f)$.", "This verifies the first condition (1) of", "Definition \\ref{definition-fibred-groupoids}.", "To see (2) let", "$(a, b) : (U', x', y', f') \\to (U, x, y, f)$ and", "$(a', b') : (U'', x'', y'', f'') \\to (U, x, y, f)$ be morphisms of", "$\\mathcal{X}'$ and let $b'' : y' \\to y''$ be a morphism of $\\mathcal{Y}$", "such that $b' \\circ b'' = b$. We have to show that there exists", "a unique morphism $a'' : x' \\to x''$ such that", "$f'' \\circ F(a'') = b'' \\circ f'$ and such that", "$(a', b') \\circ (a'', b'') = (a, b)$. Because $\\mathcal{X}$ is fibred", "in groupoids we know there exists a unique morphism", "$a'' : x' \\to x''$ such that $a' \\circ a'' = a$ and $p(a'') = q(b'')$.", "Because $\\mathcal{Y}$ is fibred in groupoids we see that", "$F(a'')$ is the unique morphism $F(x') \\to F(x'')$ such that", "$F(a') \\circ F(a'') = F(a)$ and $q(F(a'')) = q(b'')$. The relation", "$f'' \\circ F(a'') = b'' \\circ f'$ follows from this and the given", "relations $f \\circ F(a) = b \\circ f'$ and $f \\circ F(a') = b' \\circ f''$." ], "refs": [ "categories-lemma-2-product-categories-over-C", "categories-lemma-2-product-fibred-categories", "categories-definition-fibred-groupoids" ], "ref_ids": [ 12280, 12296, 12392 ] } ], "ref_ids": [] }, { "id": 12305, "type": "theorem", "label": "categories-lemma-amelioration-unique", "categories": [ "categories" ], "title": "categories-lemma-amelioration-unique", "contents": [ "Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{X} \\to \\mathcal{Y}$", "be a $1$-morphism of categories fibred in groupoids over $\\mathcal{C}$.", "Assume we have a $2$-commutative diagram", "$$", "\\xymatrix{", "\\mathcal{X}' \\ar[rd]_f &", "\\mathcal{X} \\ar[l]^a \\ar[d]^F \\ar[r]_b &", "\\mathcal{X}'' \\ar[ld]^g \\\\", "& \\mathcal{Y}", "}", "$$", "where $a$ and $b$ are equivalences of categories over $\\mathcal{C}$", "and $f$ and $g$ are categories fibred in groupoids. Then there exists", "an equivalence $h : \\mathcal{X}'' \\to \\mathcal{X}'$ of categories over", "$\\mathcal{Y}$ such that $h \\circ b$ is $2$-isomorphic to $a$ as $1$-morphisms", "of categories over $\\mathcal{C}$. If the diagram above actually commutes, then", "we can arrange it so that $h \\circ b$ is $2$-isomorphic to $a$ as", "$1$-morphisms of categories over $\\mathcal{Y}$." ], "refs": [], "proofs": [ { "contents": [ "We will show that both $\\mathcal{X}'$ and $\\mathcal{X}''$ over $\\mathcal{Y}$", "are equivalent to the category fibred in groupoids", "$\\mathcal{X} \\times_{F, \\mathcal{Y}, \\text{id}} \\mathcal{Y}$", "over $\\mathcal{Y}$, see proof of", "Lemma \\ref{lemma-ameliorate-morphism-categories-fibred-groupoids}.", "Choose a quasi-inverse $b^{-1} : \\mathcal{X}'' \\to \\mathcal{X}$ in the", "$2$-category of categories over $\\mathcal{C}$.", "Since the right triangle of the diagram is $2$-commutative we see that", "$$", "\\xymatrix{", "\\mathcal{X} \\ar[d]_F & \\mathcal{X}'' \\ar[l]^{b^{-1}} \\ar[d]^g \\\\", "\\mathcal{Y} & \\mathcal{Y} \\ar[l]", "}", "$$", "is $2$-commutative. Hence we obtain a $1$-morphism", "$c : \\mathcal{X}'' \\to", "\\mathcal{X} \\times_{F, \\mathcal{Y}, \\text{id}} \\mathcal{Y}$", "by the universal property of the $2$-fibre product. Moreover $c$", "is a morphism of categories over $\\mathcal{Y}$ (!) and an equivalence", "(by the assumption that $b$ is an equivalence, see", "Lemma \\ref{lemma-equivalence-2-fibre-product}).", "Hence $c$ is an equivalence in the $2$-category of categories fibred", "in groupoids over $\\mathcal{Y}$ by", "Lemma \\ref{lemma-equivalence-fibred-categories}.", "\\medskip\\noindent", "We still have to construct a $2$-isomorphism between $c \\circ b$ and", "the functor $d : \\mathcal{X} \\to", "\\mathcal{X} \\times_{F, \\mathcal{Y}, \\text{id}} \\mathcal{Y}$,", "$x \\mapsto (p(x), x, F(x), \\text{id}_{F(x)})$", "constructed in the proof of", "Lemma \\ref{lemma-ameliorate-morphism-categories-fibred-groupoids}.", "Let $\\alpha : F \\to g \\circ b$ and $\\beta : b^{-1} \\circ b \\to \\text{id}$", "be $2$-isomorphisms between $1$-morphisms of categories over $\\mathcal{C}$.", "Note that $c \\circ b$ is given by the rule", "$$", "x \\mapsto (p(x), b^{-1}(b(x)), g(b(x)), \\alpha_x \\circ F(\\beta_x))", "$$", "on objects. Then we see that", "$$", "(\\beta_x, \\alpha_x) :", "(p(x), x, F(x), \\text{id}_{F(x)})", "\\longrightarrow", "(p(x), b^{-1}(b(x)), g(b(x)), \\alpha_x \\circ F(\\beta_x))", "$$", "is a functorial isomorphism which gives our $2$-morphism", "$d \\to b \\circ c$. Finally, if the diagram commutes then", "$\\alpha_x$ is the identity for all $x$ and we see that this", "$2$-morphism is a $2$-morphism in the $2$-category of categories", "over $\\mathcal{Y}$." ], "refs": [ "categories-lemma-ameliorate-morphism-categories-fibred-groupoids", "categories-lemma-equivalence-2-fibre-product", "categories-lemma-equivalence-fibred-categories", "categories-lemma-ameliorate-morphism-categories-fibred-groupoids" ], "ref_ids": [ 12304, 12272, 12297, 12304 ] } ], "ref_ids": [] }, { "id": 12306, "type": "theorem", "label": "categories-lemma-when-split", "categories": [ "categories" ], "title": "categories-lemma-when-split", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{S}$ be a fibred category over $\\mathcal{C}$.", "Then $\\mathcal{S}$ is split if and only if for some choice", "of pullbacks (see Definition \\ref{definition-pullback-functor-fibred-category})", "the pullback functors", "$(f \\circ g)^*$ and $g^* \\circ f^*$ are equal." ], "refs": [ "categories-definition-pullback-functor-fibred-category" ], "proofs": [ { "contents": [ "This is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12389 ] }, { "id": 12307, "type": "theorem", "label": "categories-lemma-fibred-strict", "categories": [ "categories" ], "title": "categories-lemma-fibred-strict", "contents": [ "Let $ p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category.", "There exists a contravariant functor $F : \\mathcal{C} \\to \\textit{Cat}$", "such that $\\mathcal{S}$ is equivalent to $\\mathcal{S}_F$", "in the $2$-category of fibred categories over $\\mathcal{C}$. In other", "words, every fibred category is equivalent to a split one." ], "refs": [], "proofs": [ { "contents": [ "Let us make a choice of pullbacks (see", "Definition \\ref{definition-pullback-functor-fibred-category}).", "By Lemma \\ref{lemma-fibred} we get pullback functors $f^*$ for", "every morphism $f$ of $\\mathcal{C}$.", "\\medskip\\noindent", "We construct a new category $\\mathcal{S}'$ as follows.", "The objects of $\\mathcal{S}'$ are pairs $(x, f)$", "consisting of a morphism $f : V \\to U$ of $\\mathcal{C}$", "and an object $x$ of $\\mathcal{S}$ over $U$, i.e.,", "$x\\in \\Ob(\\mathcal{S}_U)$. The functor", "$p' : \\mathcal{S}' \\to \\mathcal{C}$ will map the pair $(x, f)$ to the source", "of the morphism $f$, in other words $p'(x, f : V\\to U) = V$. A morphism", "$\\varphi : (x_1, f_1: V_1 \\to U_1) \\to (x_2, f_2 : V_2 \\to U_2)$ is given by a", "pair $(\\varphi, g)$ consisting of a morphism $g : V_1 \\to V_2$ and a morphism", "$\\varphi : f_1^\\ast x_1 \\to f_2^\\ast x_2$ with $p(\\varphi) = g$. It is no", "problem to define the composition law: $(\\varphi, g) \\circ (\\psi, h) =", "(\\varphi \\circ \\psi, g\\circ h)$ for any pair of composable morphisms.", "There is a natural functor $\\mathcal{S} \\to \\mathcal{S}'$ which simply maps", "$x$ over $U$ to the pair $(x, \\text{id}_U)$.", "\\medskip\\noindent", "At this point we need to check that $p'$ makes $\\mathcal{S}'$ into a", "fibred category over $\\mathcal{C}$, and we need to check that", "$\\mathcal{S} \\to \\mathcal{S}'$ is an equivalence of categories over", "$\\mathcal{C}$ which maps strongly cartesian morphisms to strongly", "cartesian morphisms. We omit the verifications.", "\\medskip\\noindent", "Finally, we can define pullback functors on $\\mathcal{S}'$", "by setting $g^\\ast(x, f) = (x, f \\circ g)$ on objects if", "$g : V' \\to V$ and $f : V \\to U$. On morphisms", "$(\\varphi, \\text{id}_V) : (x_1, f_1) \\to (x_2, f_2)$", "between morphisms in $\\mathcal{S}'_V$ we set $g^\\ast(\\varphi, \\text{id}_V) =", "(g^\\ast\\varphi, \\text{id}_{V'})$ where we use the unique identifications", "$g^\\ast f_i^\\ast x_i = (f_i \\circ g)^\\ast x_i$ from Lemma", "\\ref{lemma-fibred} to think of $g^\\ast\\varphi$ as a morphism from", "$(f_1 \\circ g)^\\ast x_1$ to $(f_2 \\circ g)^\\ast x_2$. Clearly, these pullback", "functors $g^\\ast$ have the property that", "$g_1^\\ast \\circ g_2^\\ast = (g_2\\circ g_1)^\\ast$, in other words $\\mathcal{S}'$", "is split as desired." ], "refs": [ "categories-definition-pullback-functor-fibred-category", "categories-lemma-fibred", "categories-lemma-fibred" ], "ref_ids": [ 12389, 12285, 12285 ] } ], "ref_ids": [] }, { "id": 12308, "type": "theorem", "label": "categories-lemma-fibred-groupoids-strict", "categories": [ "categories" ], "title": "categories-lemma-fibred-groupoids-strict", "contents": [ "Let $ p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids.", "There exists a contravariant functor $F : \\mathcal{C} \\to \\textit{Groupoids}$", "such that $\\mathcal{S}$ is equivalent to $\\mathcal{S}_F$ over $\\mathcal{C}$.", "In other words, every category fibred in groupoids is equivalent to a split one." ], "refs": [], "proofs": [ { "contents": [ "Make a choice of pullbacks (see", "Definition \\ref{definition-pullback-functor-fibred-category}).", "By Lemmas \\ref{lemma-fibred} and \\ref{lemma-fibred-groupoids}", "we get pullback functors $f^*$ for", "every morphism $f$ of $\\mathcal{C}$.", "\\medskip\\noindent", "We construct a new category $\\mathcal{S}'$ as follows.", "The objects of $\\mathcal{S}'$ are pairs $(x, f)$", "consisting of a morphism $f : V \\to U$ of $\\mathcal{C}$", "and an object $x$ of $\\mathcal{S}$ over $U$, i.e.,", "$x\\in \\Ob(\\mathcal{S}_U)$. The functor", "$p' : \\mathcal{S}' \\to \\mathcal{C}$ will map the pair $(x, f)$ to the source", "of the morphism $f$, in other words $p'(x, f : V\\to U) = V$. A morphism", "$\\varphi : (x_1, f_1: V_1 \\to U_1) \\to (x_2, f_2 : V_2 \\to U_2)$ is given by a", "pair $(\\varphi, g)$ consisting of a morphism $g : V_1 \\to V_2$ and a morphism", "$\\varphi : f_1^\\ast x_1 \\to f_2^\\ast x_2$ with $p(\\varphi) = g$. It is no", "problem to define the composition law: $(\\varphi, g) \\circ (\\psi, h) =", "(\\varphi \\circ \\psi, g\\circ h)$ for any pair of composable morphisms.", "There is a natural functor $\\mathcal{S} \\to \\mathcal{S}'$ which simply maps", "$x$ over $U$ to the pair $(x, \\text{id}_U)$.", "\\medskip\\noindent", "At this point we need to check that $p'$ makes $\\mathcal{S}'$ into a category", "fibred in groupoids over $\\mathcal{C}$, and we need to check that", "$\\mathcal{S} \\to \\mathcal{S}'$ is an equivalence of categories over", "$\\mathcal{C}$. We omit the verifications.", "\\medskip\\noindent", "Finally, we can define pullback functors on $\\mathcal{S}'$", "by setting $g^\\ast(x, f) = (x, f \\circ g)$ on objects if", "$g : V' \\to V$ and $f : V \\to U$. On morphisms", "$(\\varphi, \\text{id}_V) : (x_1, f_1) \\to (x_2, f_2)$", "between morphisms in $\\mathcal{S}'_V$ we set $g^\\ast(\\varphi, \\text{id}_V) =", "(g^\\ast\\varphi, \\text{id}_{V'})$ where we use the unique identifications", "$g^\\ast f_i^\\ast x_i = (f_i \\circ g)^\\ast x_i$ from Lemma", "\\ref{lemma-fibred-groupoids} to think of $g^\\ast\\varphi$ as a morphism from", "$(f_1 \\circ g)^\\ast x_1$ to $(f_2 \\circ g)^\\ast x_2$. Clearly, these pullback", "functors $g^\\ast$ have the property that", "$g_1^\\ast \\circ g_2^\\ast = (g_2\\circ g_1)^\\ast$, in other words $\\mathcal{S}'$", "is split as desired." ], "refs": [ "categories-definition-pullback-functor-fibred-category", "categories-lemma-fibred", "categories-lemma-fibred-groupoids", "categories-lemma-fibred-groupoids" ], "ref_ids": [ 12389, 12285, 12294, 12294 ] } ], "ref_ids": [] }, { "id": 12309, "type": "theorem", "label": "categories-lemma-2-product-categories-fibred-sets", "categories": [ "categories" ], "title": "categories-lemma-2-product-categories-fibred-sets", "contents": [ "Let $\\mathcal{C}$ be a category.", "The 2-category of categories fibred in sets over $\\mathcal{C}$", "has 2-fibre products. More precisely, the 2-fibre product described in", "Lemma \\ref{lemma-2-product-categories-over-C}", "returns a category fibred in sets if one starts out with such." ], "refs": [ "categories-lemma-2-product-categories-over-C" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12280 ] }, { "id": 12310, "type": "theorem", "label": "categories-lemma-2-category-fibred-sets", "categories": [ "categories" ], "title": "categories-lemma-2-category-fibred-sets", "contents": [ "\\begin{slogan}", "Categories fibred in sets are precisely presheaves.", "\\end{slogan}", "Let $\\mathcal{C}$ be a category.", "The only $2$-morphisms between categories fibred in sets are identities.", "In other words, the $2$-category of categories fibred in sets is a category.", "Moreover, there is an equivalence of categories", "$$", "\\left\\{", "\\begin{matrix}", "\\text{the category of presheaves}\\\\", "\\text{of sets over }\\mathcal{C}", "\\end{matrix}", "\\right\\}", "\\leftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{the category of categories}\\\\", "\\text{fibred in sets over }\\mathcal{C}", "\\end{matrix}", "\\right\\}", "$$", "The functor from left to right is the construction", "$F \\to \\mathcal{S}_F$ discussed in", "Example \\ref{example-presheaf}.", "The functor from right to left assigns to $p : \\mathcal{S} \\to \\mathcal{C}$", "the presheaf of objects $U \\mapsto \\Ob(\\mathcal{S}_U)$." ], "refs": [], "proofs": [ { "contents": [ "The first assertion is clear, as the only morphisms in the fibre", "categories are identities.", "\\medskip\\noindent", "Suppose that $p :", "\\mathcal{S} \\to \\mathcal{C}$ is fibred in sets. Let $f : V \\to U$", "be a morphism in $\\mathcal{C}$ and let $x \\in \\Ob(\\mathcal{S}_U)$.", "Then there is exactly one choice for the object $f^\\ast x$. Thus we see that", "$(f \\circ g)^\\ast x = g^\\ast(f^\\ast x)$ for $f, g$ as in Lemma", "\\ref{lemma-fibred-groupoids}. It follows that we may think of the", "assignments $U \\mapsto \\Ob(\\mathcal{S}_U)$ and $f \\mapsto f^\\ast$", "as a presheaf on $\\mathcal{C}$." ], "refs": [ "categories-lemma-fibred-groupoids" ], "ref_ids": [ 12294 ] } ], "ref_ids": [] }, { "id": 12311, "type": "theorem", "label": "categories-lemma-2-product-categories-fibred-setoids", "categories": [ "categories" ], "title": "categories-lemma-2-product-categories-fibred-setoids", "contents": [ "Let $\\mathcal{C}$ be a category.", "The 2-category of categories fibred in setoids over $\\mathcal{C}$", "has 2-fibre products. More precisely, the 2-fibre product described in", "Lemma \\ref{lemma-2-product-categories-over-C} returns a category fibred in", "setoids if one starts out with such." ], "refs": [ "categories-lemma-2-product-categories-over-C" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12280 ] }, { "id": 12312, "type": "theorem", "label": "categories-lemma-setoid-fibres", "categories": [ "categories" ], "title": "categories-lemma-setoid-fibres", "contents": [ "Let $\\mathcal{C}$ be a category. Let $\\mathcal{S}$ be a category", "over $\\mathcal{C}$.", "\\begin{enumerate}", "\\item If $\\mathcal{S} \\to \\mathcal{S}'$ is an equivalence", "over $\\mathcal{C}$ with $\\mathcal{S}'$ fibred in sets over $\\mathcal{C}$,", "then", "\\begin{enumerate}", "\\item $\\mathcal{S}$ is fibred in setoids over $\\mathcal{C}$, and", "\\item for each $U \\in \\Ob(\\mathcal{C})$ the map", "$\\Ob(\\mathcal{S}_U) \\to \\Ob(\\mathcal{S}'_U)$", "identifies the target as the set of isomorphism classes of the source.", "\\end{enumerate}", "\\item If $p : \\mathcal{S} \\to \\mathcal{C}$ is a category fibred in setoids,", "then there exists a category fibred in sets", "$p' : \\mathcal{S}' \\to \\mathcal{C}$ and an equivalence", "$\\text{can} : \\mathcal{S} \\to \\mathcal{S}'$ over $\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us prove (2).", "An object of the category $\\mathcal{S}'$ will be a pair $(U, \\xi)$, where", "$U \\in \\Ob(\\mathcal{C})$ and $\\xi$ is an isomorphism class of objects", "of $\\mathcal{S}_U$. A morphism $(U, \\xi) \\to (V , \\psi)$ is given by a", "morphism $x \\to y$, where $x \\in \\xi$ and $y \\in \\psi$. Here we identify", "two morphisms $x \\to y$ and $x' \\to y'$ if they induce the same morphism", "$U \\to V$, and if for some choices of isomorphisms $x \\to x'$ in", "$\\mathcal{S}_U$ and $y \\to y'$ in $\\mathcal{S}_V$ the compositions", "$x \\to x' \\to y'$ and $x \\to y \\to y'$ agree. By construction there are", "surjective maps on objects and morphisms from $\\mathcal{S} \\to", "\\mathcal{S}'$. We define composition of morphisms in $\\mathcal{S}'$ to", "be the unique law that turns $\\mathcal{S} \\to \\mathcal{S}'$ into a functor.", "Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12313, "type": "theorem", "label": "categories-lemma-2-category-fibred-setoids", "categories": [ "categories" ], "title": "categories-lemma-2-category-fibred-setoids", "contents": [ "Let $\\mathcal{C}$ be a category. The construction of", "Lemma \\ref{lemma-setoid-fibres}", "part (2) gives a functor", "$$", "F :", "\\left\\{", "\\begin{matrix}", "\\text{the 2-category of categories}\\\\", "\\text{fibred in setoids over }\\mathcal{C}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{the category of categories}\\\\", "\\text{fibred in sets over }\\mathcal{C}", "\\end{matrix}", "\\right\\}", "$$", "(see", "Definition \\ref{definition-functor-into-2-category}).", "This functor is an equivalence in the following sense:", "\\begin{enumerate}", "\\item for any two 1-morphisms $f, g : \\mathcal{S}_1 \\to \\mathcal{S}_2$", "with $F(f) = F(g)$ there exists a unique 2-isomorphism $f \\to g$,", "\\item for any morphism $h : F(\\mathcal{S}_1) \\to F(\\mathcal{S}_2)$", "there exists a 1-morphism $f : \\mathcal{S}_1 \\to \\mathcal{S}_2$", "with $F(f) = h$, and", "\\item any category fibred in sets $\\mathcal{S}$ is equal to $F(\\mathcal{S})$.", "\\end{enumerate}", "In particular, defining $F_i \\in \\textit{PSh}(\\mathcal{C})$ by the", "rule $F_i(U) = \\Ob(\\mathcal{S}_{i, U})/\\cong$, we have", "$$", "\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{S}_1, \\mathcal{S}_2)", "\\Big/", "2\\text{-isomorphism}", "=", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(F_1, F_2)", "$$", "More precisely, given any map $\\phi : F_1 \\to F_2$ there exists a", "$1$-morphism $f : \\mathcal{S}_1 \\to \\mathcal{S}_2$ which induces", "$\\phi$ on isomorphism classes of objects and", "which is unique up to unique $2$-isomorphism." ], "refs": [ "categories-lemma-setoid-fibres", "categories-definition-functor-into-2-category" ], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-2-category-fibred-sets}", "the target of $F$ is a category hence the assertion makes sense.", "The construction of", "Lemma \\ref{lemma-setoid-fibres} part (2)", "assigns to $\\mathcal{S}$ the category fibred in sets whose value over", "$U$ is the set of isomorphism classes in $\\mathcal{S}_U$. Hence it", "is clear that it defines a functor as indicated.", "Let $f, g : \\mathcal{S}_1 \\to \\mathcal{S}_2$", "with $F(f) = F(g)$ be as in (1). For each object $U$ of $\\mathcal{C}$", "and each object $x$ of $\\mathcal{S}_{1, U}$ we see that $f(x) \\cong g(x)$", "by assumption. As $\\mathcal{S}_2$ is fibred in setoids there exists", "a unique isomorphism $t_x : f(x) \\to g(x)$ in $\\mathcal{S}_{2, U}$.", "Clearly the rule $x \\mapsto t_x$ gives the desired $2$-isomorphism", "$f \\to g$. We omit the proofs of (2) and (3).", "To see the final assertion use", "Lemma \\ref{lemma-2-category-fibred-sets}", "to see that the right hand side is equal to", "$\\Mor_{\\textit{Cat}/\\mathcal{C}}(F(\\mathcal{S}_1), F(\\mathcal{S}_2))$", "and apply (1) and (2) above." ], "refs": [ "categories-lemma-2-category-fibred-sets", "categories-lemma-setoid-fibres", "categories-lemma-2-category-fibred-sets" ], "ref_ids": [ 12310, 12312, 12310 ] } ], "ref_ids": [ 12312, 12381 ] }, { "id": 12314, "type": "theorem", "label": "categories-lemma-characterize-fibred-setoids-inertia", "categories": [ "categories" ], "title": "categories-lemma-characterize-fibred-setoids-inertia", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids.", "The following are equivalent:", "\\begin{enumerate}", "\\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a category fibred in setoids, and", "\\item the canonical $1$-morphism $\\mathcal{I}_\\mathcal{S} \\to \\mathcal{S}$,", "see (\\ref{equation-inertia-structure-map}), is an equivalence (of categories", "over $\\mathcal{C}$).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (2). The category $\\mathcal{I}_\\mathcal{S}$ has objects", "$(x, \\alpha)$ where $x \\in \\mathcal{S}$, say with $p(x) = U$, and", "$\\alpha : x \\to x$ is a morphism in $\\mathcal{S}_U$. Hence if", "$\\mathcal{I}_\\mathcal{S} \\to \\mathcal{S}$ is an equivalence over $\\mathcal{C}$", "then every pair of objects $(x, \\alpha)$, $(x, \\alpha')$ are isomorphic", "in the fibre category of $\\mathcal{I}_\\mathcal{S}$ over $U$.", "Looking at the definition of morphisms in $\\mathcal{I}_\\mathcal{S}$", "we conclude that $\\alpha$, $\\alpha'$ are conjugate in the group", "of automorphisms of $x$. Hence taking $\\alpha' = \\text{id}_x$ we conclude", "that every automorphism of $x$ is equal to the identity.", "Since $\\mathcal{S} \\to \\mathcal{C}$ is fibred in groupoids this", "implies that $\\mathcal{S} \\to \\mathcal{C}$ is fibred in setoids.", "We omit the proof of (1) $\\Rightarrow$ (2)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12315, "type": "theorem", "label": "categories-lemma-category-fibred-setoids-presheaves-products", "categories": [ "categories" ], "title": "categories-lemma-category-fibred-setoids-presheaves-products", "contents": [ "Let $\\mathcal{C}$ be a category.", "The construction of", "Lemma \\ref{lemma-2-category-fibred-setoids}", "which associates to a category fibred in setoids a presheaf is", "compatible with products, in the sense that the presheaf associated", "to a $2$-fibre product $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$", "is the fibre product of the presheaves associated to", "$\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$." ], "refs": [ "categories-lemma-2-category-fibred-setoids" ], "proofs": [ { "contents": [ "Let $U \\in \\Ob(\\mathcal{C})$. The lemma just says that", "$$", "\\Ob((\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z})_U)/\\!\\cong", "\\quad \\text{equals} \\quad", "\\Ob(\\mathcal{X}_U)/\\!\\cong", "\\ \\times_{\\Ob(\\mathcal{Y}_U)/\\!\\cong}", "\\ \\Ob(\\mathcal{Z}_U)/\\!\\cong", "$$", "the proof of which we omit. (But note that this would not be true", "in general if the category $\\mathcal{Y}_U$ is not a setoid.)" ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12313 ] }, { "id": 12316, "type": "theorem", "label": "categories-lemma-characterize-representable-fibred-category", "categories": [ "categories" ], "title": "categories-lemma-characterize-representable-fibred-category", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids.", "\\begin{enumerate}", "\\item $\\mathcal{S}$ is representable if and only if", "the following conditions are satisfied:", "\\begin{enumerate}", "\\item $\\mathcal{S}$ is fibred in setoids, and", "\\item the presheaf $U \\mapsto \\Ob(\\mathcal{S}_U)/\\cong$ is", "representable.", "\\end{enumerate}", "\\item If $\\mathcal{S}$ is representable the pair $(X, j)$, where $j$ is the", "equivalence $j : \\mathcal{S} \\to \\mathcal{C}/X$, is uniquely determined", "up to isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first assertion follows immediately from", "Lemma \\ref{lemma-setoid-fibres}.", "For the second, suppose that $j' : \\mathcal{S} \\to \\mathcal{C}/X'$ is", "a second such pair. Choose a $1$-morphism", "$t' : \\mathcal{C}/X' \\to \\mathcal{S}$ such that", "$j' \\circ t' \\cong \\text{id}_{\\mathcal{C}/X'}$ and", "$t' \\circ j' \\cong \\text{id}_\\mathcal{S}$. Then", "$j \\circ t' : \\mathcal{C}/X' \\to \\mathcal{C}/X$ is an equivalence.", "Hence it is an isomorphism, see Lemma \\ref{lemma-2-category-fibred-sets}.", "Hence by the Yoneda Lemma \\ref{lemma-yoneda} (via", "Example \\ref{example-fibred-category-from-functor-of-points} for example)", "it is given by an isomorphism $X' \\to X$." ], "refs": [ "categories-lemma-setoid-fibres", "categories-lemma-2-category-fibred-sets", "categories-lemma-yoneda" ], "ref_ids": [ 12312, 12310, 12203 ] } ], "ref_ids": [] }, { "id": 12317, "type": "theorem", "label": "categories-lemma-morphisms-representable-fibred-categories", "categories": [ "categories" ], "title": "categories-lemma-morphisms-representable-fibred-categories", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids", "over $\\mathcal{C}$. Assume that $\\mathcal{X}$, $\\mathcal{Y}$", "are representable by objects $X$, $Y$ of $\\mathcal{C}$.", "Then", "$$", "\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{X}, \\mathcal{Y})", "\\Big/", "2\\text{-isomorphism}", "=", "\\Mor_\\mathcal{C}(X, Y)", "$$", "More precisely, given $\\phi : X \\to Y$ there exists a", "$1$-morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ which induces", "$\\phi$ on isomorphism classes of objects and", "which is unique up to unique $2$-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By", "Example \\ref{example-fibred-category-from-functor-of-points}", "we have $\\mathcal{C}/X = \\mathcal{S}_{h_X}$ and", "$\\mathcal{C}/Y = \\mathcal{S}_{h_Y}$. By", "Lemma \\ref{lemma-2-category-fibred-setoids}", "we have", "$$", "\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{X}, \\mathcal{Y})", "\\Big/", "2\\text{-isomorphism}", "=", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(h_X, h_Y)", "$$", "By the Yoneda", "Lemma \\ref{lemma-yoneda}", "we have $\\Mor_{\\textit{PSh}(\\mathcal{C})}(h_X, h_Y)", "= \\Mor_\\mathcal{C}(X, Y)$." ], "refs": [ "categories-lemma-2-category-fibred-setoids", "categories-lemma-yoneda" ], "ref_ids": [ 12313, 12203 ] } ], "ref_ids": [] }, { "id": 12318, "type": "theorem", "label": "categories-lemma-yoneda-2category", "categories": [ "categories" ], "title": "categories-lemma-yoneda-2category", "contents": [ "Let $\\mathcal{S}\\to \\mathcal{C}$ be fibred in groupoids.", "Let $U \\in \\Ob(\\mathcal{C})$.", "The functor", "$$", "\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{C}/U, \\mathcal{S})", "\\longrightarrow", "\\mathcal{S}_U", "$$", "given by $G \\mapsto G(\\text{id}_U)$ is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Make a choice of pullbacks for $\\mathcal{S}$", "(see Definition \\ref{definition-pullback-functor-fibred-category}).", "We define a functor", "$$", "\\mathcal{S}_U", "\\longrightarrow", "\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{C}/U, \\mathcal{S})", "$$", "as follows. Given", "$x \\in \\Ob(\\mathcal{S}_U)$", "the associated functor is", "\\begin{enumerate}", "\\item on objects: $(f : V \\to U) \\mapsto f^*x$, and", "\\item on morphisms: the arrow $(g : V'/U \\to V/U)$ maps to", "the composition", "$$", "(f \\circ g)^*x \\xrightarrow{(\\alpha_{g, f})_x} g^*f^*x \\rightarrow f^*x", "$$", "where $\\alpha_{g, f}$ is as in Lemma \\ref{lemma-fibred-groupoids}.", "\\end{enumerate}", "We omit the verification that this is an inverse to the functor", "of the lemma." ], "refs": [ "categories-definition-pullback-functor-fibred-category", "categories-lemma-fibred-groupoids" ], "ref_ids": [ 12389, 12294 ] } ], "ref_ids": [] }, { "id": 12319, "type": "theorem", "label": "categories-lemma-identify-fibre-product", "categories": [ "categories" ], "title": "categories-lemma-identify-fibre-product", "contents": [ "In the situation above the fibre category of", "$(\\mathcal{C}/U) \\times_\\mathcal{Y} \\mathcal{X}$ over", "an object $f : V \\to U$ of $\\mathcal{C}/U$", "is the category described as follows:", "\\begin{enumerate}", "\\item objects are pairs $(x, \\phi)$,", "where $x \\in \\Ob(\\mathcal{X}_V)$, and", "$\\phi : f^*y \\to F(x)$ is a morphism in $\\mathcal{Y}_V$,", "\\item the set of morphisms between $(x, \\phi)$ and $(x', \\phi')$", "is the set of morphisms $\\psi : x \\to x'$ in $\\mathcal{X}_V$", "such that $F(\\psi) = \\phi' \\circ \\phi^{-1}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12320, "type": "theorem", "label": "categories-lemma-prepare-representable-map-stack-in-groupoids", "categories": [ "categories" ], "title": "categories-lemma-prepare-representable-map-stack-in-groupoids", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids", "over $\\mathcal{C}$.", "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism.", "Let $G : \\mathcal{C}/U \\to \\mathcal{Y}$ be a $1$-morphism.", "Then", "$$", "(\\mathcal{C}/U) \\times_\\mathcal{Y} \\mathcal{X}", "\\longrightarrow", "\\mathcal{C}/U", "$$", "is a category fibred in groupoids." ], "refs": [], "proofs": [ { "contents": [ "We have already seen in Lemma \\ref{lemma-2-product-fibred-categories}", "that the composition", "$$", "(\\mathcal{C}/U) \\times_\\mathcal{Y} \\mathcal{X}", "\\longrightarrow", "\\mathcal{C}/U", "\\longrightarrow", "\\mathcal{C}", "$$", "is a category fibred in groupoids. Then the lemma follows from", "Lemma \\ref{lemma-cute-groupoids}." ], "refs": [ "categories-lemma-2-product-fibred-categories", "categories-lemma-cute-groupoids" ], "ref_ids": [ 12296, 12301 ] } ], "ref_ids": [] }, { "id": 12321, "type": "theorem", "label": "categories-lemma-spell-out-representable-map-stack-in-groupoids", "categories": [ "categories" ], "title": "categories-lemma-spell-out-representable-map-stack-in-groupoids", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids", "over $\\mathcal{C}$.", "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism.", "If $F$ is representable then every one of the functors", "$$", "F_U : \\mathcal{X}_U \\longrightarrow \\mathcal{Y}_U", "$$", "between fibre categories is faithful." ], "refs": [], "proofs": [ { "contents": [ "Clear from the description of fibre categories in", "Lemma \\ref{lemma-identify-fibre-product} and the characterization", "of representable fibred categories in", "Lemma \\ref{lemma-characterize-representable-fibred-category}." ], "refs": [ "categories-lemma-identify-fibre-product", "categories-lemma-characterize-representable-fibred-category" ], "ref_ids": [ 12319, 12316 ] } ], "ref_ids": [] }, { "id": 12322, "type": "theorem", "label": "categories-lemma-criterion-representable-map-stack-in-groupoids", "categories": [ "categories" ], "title": "categories-lemma-criterion-representable-map-stack-in-groupoids", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids", "over $\\mathcal{C}$.", "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism.", "Make a choice of pullbacks for $\\mathcal{Y}$.", "Assume", "\\begin{enumerate}", "\\item each functor $F_U : \\mathcal{X}_U \\longrightarrow \\mathcal{Y}_U$", "between fibre categories is faithful, and", "\\item for each $U$ and each $y \\in \\mathcal{Y}_U$ the presheaf", "$$", "(f : V \\to U)", "\\longmapsto", "\\{(x, \\phi) \\mid x \\in \\mathcal{X}_V, \\phi : f^*y \\to F(x)\\}/\\cong", "$$", "is a representable presheaf on $\\mathcal{C}/U$.", "\\end{enumerate}", "Then $F$ is representable." ], "refs": [], "proofs": [ { "contents": [ "Clear from the description of fibre categories in", "Lemma \\ref{lemma-identify-fibre-product} and the characterization", "of representable fibred categories in", "Lemma \\ref{lemma-characterize-representable-fibred-category}." ], "refs": [ "categories-lemma-identify-fibre-product", "categories-lemma-characterize-representable-fibred-category" ], "ref_ids": [ 12319, 12316 ] } ], "ref_ids": [] }, { "id": 12323, "type": "theorem", "label": "categories-lemma-representable-diagonal-groupoids", "categories": [ "categories" ], "title": "categories-lemma-representable-diagonal-groupoids", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids.", "Assume $\\mathcal{C}$ has products of pairs of objects and fibre products.", "The following are equivalent:", "\\begin{enumerate}", "\\item The diagonal $\\mathcal{S} \\to \\mathcal{S} \\times \\mathcal{S}$", "is representable.", "\\item For every $U$ in $\\mathcal{C}$, any $G : \\mathcal{C}/U \\to \\mathcal{S}$", "is representable.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose the diagonal is representable, and let $U, G$ be given.", "Consider any $V \\in \\Ob(\\mathcal{C})$ and any", "$G' : \\mathcal{C}/V \\to \\mathcal{S}$.", "Note that $\\mathcal{C}/U \\times \\mathcal{C}/V = \\mathcal{C}/U \\times V$", "is representable. Hence the fibre product", "$$", "\\xymatrix{", "(\\mathcal{C}/U \\times V)", "\\times_{(\\mathcal{S} \\times \\mathcal{S})}", "\\mathcal{S}", "\\ar[r] \\ar[d] &", "\\mathcal{S} \\ar[d] \\\\", "\\mathcal{C}/U \\times V \\ar[r]^{(G, G')} &", "\\mathcal{S} \\times \\mathcal{S}", "}", "$$", "is representable by assumption.", "This means there exists $W \\to U \\times V$ in $\\mathcal{C}$,", "such that", "$$", "\\xymatrix{", "\\mathcal{C}/W \\ar[d] \\ar[r] & \\mathcal{S} \\ar[d] \\\\", "\\mathcal{C}/U \\times \\mathcal{C}/V \\ar[r] & \\mathcal{S} \\times \\mathcal{S}", "}", "$$", "is cartesian. This implies that", "$\\mathcal{C}/W \\cong \\mathcal{C}/U \\times_\\mathcal{S} \\mathcal{C}/V$", "(see Lemma \\ref{lemma-diagonal-1})", "as desired.", "\\medskip\\noindent", "Assume (2) holds. Consider any $V \\in \\Ob(\\mathcal{C})$", "and any $(G, G') : \\mathcal{C}/V \\to \\mathcal{S} \\times \\mathcal{S}$.", "We have to show that", "$\\mathcal{C}/V \\times_{\\mathcal{S} \\times \\mathcal{S}} \\mathcal{S}$", "is representable. What we know is that", "$\\mathcal{C}/V \\times_{G, \\mathcal{S}, G'} \\mathcal{C}/V$", "is representable, say by $a : W \\to V$ in $\\mathcal{C}/V$.", "The equivalence", "$$", "\\mathcal{C}/W \\to \\mathcal{C}/V \\times_{G, \\mathcal{S}, G'} \\mathcal{C}/V", "$$", "followed by the second projection to $\\mathcal{C}/V$ gives a", "second morphism $a' : W \\to V$. Consider", "$W' = W \\times_{(a, a'), V \\times V} V$.", "There exists an equivalence", "$$", "\\mathcal{C}/W' \\cong", "\\mathcal{C}/V \\times_{\\mathcal{S} \\times \\mathcal{S}} \\mathcal{S}", "$$", "namely", "\\begin{eqnarray*}", "\\mathcal{C}/W' & \\cong &", "\\mathcal{C}/W \\times_{(\\mathcal{C}/V \\times \\mathcal{C}/V)} \\mathcal{C}/V \\\\", "& \\cong &", "\\left(\\mathcal{C}/V \\times_{(G, \\mathcal{S}, G')} \\mathcal{C}/V\\right)", "\\times_{(\\mathcal{C}/V \\times \\mathcal{C}/V)} \\mathcal{C}/V \\\\", "& \\cong &", "\\mathcal{C}/V \\times_{(\\mathcal{S} \\times \\mathcal{S})} \\mathcal{S}", "\\end{eqnarray*}", "(for the last isomorphism see Lemma \\ref{lemma-diagonal-2})", "which proves the lemma." ], "refs": [ "categories-lemma-diagonal-1", "categories-lemma-diagonal-2" ], "ref_ids": [ 12276, 12277 ] } ], "ref_ids": [] }, { "id": 12324, "type": "theorem", "label": "categories-lemma-invertible", "categories": [ "categories" ], "title": "categories-lemma-invertible", "contents": [ "Let $\\mathcal{C}$ be a monoidal category. Let $X$ be an object of", "$\\mathcal{C}$. The following are equivalent", "\\begin{enumerate}", "\\item the functor $L : Y \\mapsto X \\otimes Y$ is an equivalence,", "\\item the functor $R : Y \\mapsto Y \\otimes X$ is an equivalence,", "\\item there exists an object $X'$ such that", "$X \\otimes X' \\cong X' \\otimes X \\cong \\mathbf{1}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Choose $X'$ such that $L(X') = \\mathbf{1}$, i.e.,", "$X \\otimes X' \\cong \\mathbf{1}$. Denote $L'$ and $R'$ the functors", "corresponding to $X'$. The equation $X \\otimes X' \\cong \\mathbf{1}$", "implies $L \\circ L' \\cong \\text{id}$. Thus $L'$ must be the quasi-inverse", "to $L$ (which exists by assumption). Hence $L' \\circ L \\cong \\text{id}$.", "Hence $X' \\otimes X \\cong \\mathbf{1}$. Thus (3) holds.", "\\medskip\\noindent", "The proof of (2) $\\Rightarrow$ (3) is dual to what we just said.", "\\medskip\\noindent", "Assume (3). Then it is clear that $L'$ and $L$ are quasi-inverse", "to each other and it is clear that $R'$ and $R$ are quasi-inverse", "to each other. Thus (1) and (2) hold." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12325, "type": "theorem", "label": "categories-lemma-left-dual", "categories": [ "categories" ], "title": "categories-lemma-left-dual", "contents": [ "Let $\\mathcal{C}$ be a monoidal category. If $Y$ is a left dual to $X$,", "then", "$$", "\\Mor(Z' \\otimes X, Z) = \\Mor(Z', Z \\otimes Y)", "\\quad\\text{and}\\quad", "\\Mor(Y \\otimes Z', Z) = \\Mor(Z', X \\otimes Z)", "$$", "functorially in $Z$ and $Z'$." ], "refs": [], "proofs": [ { "contents": [ "Consider the maps", "$$", "\\Mor(Z' \\otimes X, Z) \\to", "\\Mor(Z' \\otimes X \\otimes Y, Z \\otimes Y) \\to", "\\Mor(Z', Z \\otimes Y)", "$$", "where we use $\\eta$ in the second arrow", "and the sequence of maps", "$$", "\\Mor(Z', Z \\otimes Y) \\to", "\\Mor(Z' \\otimes X, Z \\otimes Y \\otimes X) \\to", "\\Mor(Z' \\otimes X, Z)", "$$", "where we use $\\epsilon$ in the second arrow. A straightforward calculation", "using the properties of $\\eta$ and $\\epsilon$", "shows that the compositions of these are mutually inverse.", "Similarly for the other equality." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12326, "type": "theorem", "label": "categories-lemma-tensor-dual", "categories": [ "categories" ], "title": "categories-lemma-tensor-dual", "contents": [ "Let $\\mathcal{C}$ be a monoidal category. If $Y_i$, $i = 1, 2$", "are left duals of $X_i$, $i = 1, 2$, then $Y_2 \\otimes Y_1$ is", "a left dual of $X_1 \\otimes X_2$." ], "refs": [], "proofs": [ { "contents": [ "Follows from uniqueness of adjoints and Remark \\ref{remark-left-dual-adjoint}." ], "refs": [ "categories-remark-left-dual-adjoint" ], "ref_ids": [ 12430 ] } ], "ref_ids": [] }, { "id": 12327, "type": "theorem", "label": "categories-lemma-dual-symmetric", "categories": [ "categories" ], "title": "categories-lemma-dual-symmetric", "contents": [ "Let $(\\mathcal{C}, \\otimes, \\phi, \\psi)$ be a symmetric monoidal category.", "Let $X$ be an object of $\\mathcal{C}$ and let $Y$,", "$\\eta : \\mathbf{1} \\to X \\otimes Y$, and", "$\\epsilon : Y \\otimes X \\to \\mathbf{1}$", "be a left dual of $X$ as in Definition \\ref{definition-dual}.", "Then $\\eta' = \\psi \\circ \\eta : \\mathbf{1} \\to Y \\otimes X$", "and $\\epsilon' = \\epsilon \\circ \\psi : X \\otimes Y \\to \\mathbf{1}$", "makes $X$ into a left dual of $Y$." ], "refs": [ "categories-definition-dual" ], "proofs": [ { "contents": [ "Omitted. Hint: pleasant exercise in the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12407 ] }, { "id": 12431, "type": "theorem", "label": "topologies-lemma-zariski", "categories": [ "topologies" ], "title": "topologies-lemma-zariski", "contents": [ "Let $T$ be a scheme.", "\\begin{enumerate}", "\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$", "is a Zariski covering of $T$.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a Zariski covering and for each", "$i$ we have a Zariski covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then", "$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a Zariski covering.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a Zariski covering", "and $T' \\to T$ is a morphism of schemes then", "$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a Zariski covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12432, "type": "theorem", "label": "topologies-lemma-zariski-affine", "categories": [ "topologies" ], "title": "topologies-lemma-zariski-affine", "contents": [ "Let $T$ be an affine scheme. Let $\\{T_i \\to T\\}_{i \\in I}$ be a", "Zariski covering of $T$. Then there exists a Zariski covering", "$\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement", "of $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is a standard", "open of $T$, see", "Schemes, Definition \\ref{schemes-definition-standard-covering}.", "Moreover, we may choose each $U_j$ to be an open of one of the $T_i$." ], "refs": [ "schemes-definition-standard-covering" ], "proofs": [ { "contents": [ "Follows as $T$ is quasi-compact and standard opens form a basis", "for its topology. This is also proved in", "Schemes, Lemma \\ref{schemes-lemma-standard-open}." ], "refs": [ "schemes-lemma-standard-open" ], "ref_ids": [ 7650 ] } ], "ref_ids": [ 7739 ] }, { "id": 12433, "type": "theorem", "label": "topologies-lemma-zariski-induced", "categories": [ "topologies" ], "title": "topologies-lemma-zariski-induced", "contents": [ "Let $\\Sch_{Zar}$ be a big Zariski site as in", "Definition \\ref{definition-big-zariski-site}.", "Let $T \\in \\Ob(\\Sch_{Zar})$.", "Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary Zariski covering of $T$.", "There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site", "$\\Sch_{Zar}$ which is tautologically equivalent (see", "Sites, Definition \\ref{sites-definition-combinatorial-tautological})", "to $\\{T_i \\to T\\}_{i \\in I}$." ], "refs": [ "topologies-definition-big-zariski-site", "sites-definition-combinatorial-tautological" ], "proofs": [ { "contents": [ "Since each $T_i \\to T$ is an open immersion, we see by", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}", "that each $T_i$ is isomorphic to an object $V_i$ of $\\Sch_{Zar}$.", "The covering $\\{V_i \\to T\\}_{i \\in I}$ is tautologically equivalent", "to $\\{T_i \\to T\\}_{i \\in I}$ (using the identity map on $I$ both ways).", "Moreover, $\\{V_i \\to T\\}_{i \\in I}$ is combinatorially equivalent to a", "covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site $\\Sch_{Zar}$ by", "Sets, Lemma \\ref{sets-lemma-coverings-site}." ], "refs": [ "sets-lemma-what-is-in-it", "sets-lemma-coverings-site" ], "ref_ids": [ 8795, 8800 ] } ], "ref_ids": [ 12523, 8657 ] }, { "id": 12434, "type": "theorem", "label": "topologies-lemma-verify-site-Zariski", "categories": [ "topologies" ], "title": "topologies-lemma-verify-site-Zariski", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski", "site containing $S$.", "Both $S_{Zar}$ and $(\\textit{Aff}/S)_{Zar}$ are sites." ], "refs": [], "proofs": [ { "contents": [ "Let us show that $S_{Zar}$ is a site. It is a category with a", "given set of families of morphisms with fixed target. Thus we", "have to show properties (1), (2) and (3) of", "Sites, Definition \\ref{sites-definition-site}.", "Since $(\\Sch/S)_{Zar}$ is a site, it suffices to prove", "that given any covering $\\{U_i \\to U\\}$ of $(\\Sch/S)_{Zar}$", "with $U \\in \\Ob(S_{Zar})$ we also have $U_i \\in \\Ob(S_{Zar})$.", "This follows from the definitions", "as the composition of open immersions is an open immersion.", "\\medskip\\noindent", "Let us show that $(\\textit{Aff}/S)_{Zar}$ is a site.", "Reasoning as above, it suffices to show that the collection", "of standard Zariski coverings of affines satisfies properties", "(1), (2) and (3) of", "Sites, Definition \\ref{sites-definition-site}.", "Let $R$ be a ring. Let $f_1, \\ldots, f_n \\in R$ generate the unit ideal.", "For each $i \\in \\{1, \\ldots, n\\}$ let $g_{i1}, \\ldots, g_{in_i} \\in R_{f_i}$", "be elements generating the unit ideal of $R_{f_i}$. Write", "$g_{ij} = f_{ij}/f_i^{e_{ij}}$ which is possible. After replacing", "$f_{ij}$ by $f_i f_{ij}$ if necessary, we have that", "$D(f_{ij}) \\subset D(f_i) \\cong \\Spec(R_{f_i})$ is", "equal to $D(g_{ij}) \\subset \\Spec(R_{f_i})$. Hence we see that", "the family of morphisms $\\{D(g_{ij}) \\to \\Spec(R)\\}$", "is a standard Zariski covering. From these considerations", "it follows that (2) holds for standard Zariski coverings.", "We omit the verification of (1) and (3)." ], "refs": [ "sites-definition-site", "sites-definition-site" ], "ref_ids": [ 8652, 8652 ] } ], "ref_ids": [] }, { "id": 12435, "type": "theorem", "label": "topologies-lemma-fibre-products-Zariski", "categories": [ "topologies" ], "title": "topologies-lemma-fibre-products-Zariski", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski", "site containing $S$. The underlying categories of the sites", "$\\Sch_{Zar}$, $(\\Sch/S)_{Zar}$,", "$S_{Zar}$, and $(\\textit{Aff}/S)_{Zar}$ have fibre products.", "In each case the obvious functor into the category $\\Sch$ of", "all schemes commutes with taking fibre products. The categories", "$(\\Sch/S)_{Zar}$, and $S_{Zar}$ both have a final object,", "namely $S/S$." ], "refs": [], "proofs": [ { "contents": [ "For $\\Sch_{Zar}$ it is true by construction, see", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "Suppose we have $U \\to S$, $V \\to U$, $W \\to U$ morphisms", "of schemes with $U, V, W \\in \\Ob(\\Sch_{Zar})$.", "The fibre product $V \\times_U W$ in $\\Sch_{Zar}$", "is a fibre product in $\\Sch$ and", "is the fibre product of $V/S$ with $W/S$ over $U/S$ in", "the category of all schemes over $S$, and hence also a", "fibre product in $(\\Sch/S)_{Zar}$.", "This proves the result for $(\\Sch/S)_{Zar}$.", "If $U \\to S$, $V \\to U$ and $W \\to U$ are open immersions then so is", "$V \\times_U W \\to S$ and hence we get the result for $S_{Zar}$.", "If $U, V, W$ are affine, so is $V \\times_U W$ and hence the", "result for $(\\textit{Aff}/S)_{Zar}$." ], "refs": [ "sets-lemma-what-is-in-it" ], "ref_ids": [ 8795 ] } ], "ref_ids": [] }, { "id": 12436, "type": "theorem", "label": "topologies-lemma-affine-big-site-Zariski", "categories": [ "topologies" ], "title": "topologies-lemma-affine-big-site-Zariski", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski", "site containing $S$.", "The functor $(\\textit{Aff}/S)_{Zar} \\to (\\Sch/S)_{Zar}$", "is a special cocontinuous functor. Hence it induces an equivalence", "of topoi from $\\Sh((\\textit{Aff}/S)_{Zar})$ to", "$\\Sh((\\Sch/S)_{Zar})$." ], "refs": [], "proofs": [ { "contents": [ "The notion of a special cocontinuous functor is introduced in", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}.", "Thus we have to verify assumptions (1) -- (5) of", "Sites, Lemma \\ref{sites-lemma-equivalence}.", "Denote the inclusion functor", "$u : (\\textit{Aff}/S)_{Zar} \\to (\\Sch/S)_{Zar}$.", "Being cocontinuous just means that any Zariski covering of", "$T/S$, $T$ affine, can be refined by a standard Zariski covering of $T$.", "This is the content of", "Lemma \\ref{lemma-zariski-affine}.", "Hence (1) holds. We see $u$ is continuous simply because a standard", "Zariski covering is a Zariski covering. Hence (2) holds.", "Parts (3) and (4) follow immediately from the fact that $u$ is", "fully faithful. And finally condition (5) follows from the", "fact that every scheme has an affine open covering." ], "refs": [ "sites-definition-special-cocontinuous-functor", "sites-lemma-equivalence", "topologies-lemma-zariski-affine" ], "ref_ids": [ 8672, 8578, 12432 ] } ], "ref_ids": [] }, { "id": 12437, "type": "theorem", "label": "topologies-lemma-Zariski-usual", "categories": [ "topologies" ], "title": "topologies-lemma-Zariski-usual", "contents": [ "The category of sheaves on $S_{Zar}$ is equivalent to the", "category of sheaves on the underlying topological space of $S$." ], "refs": [], "proofs": [ { "contents": [ "We will use repeatedly that for any object", "$U/S$ of $S_{Zar}$ the morphism $U \\to S$ is an isomorphism", "onto an open subscheme.", "Let $\\mathcal{F}$ be a sheaf on $S$. Then we define a sheaf", "on $S_{Zar}$ by the rule $\\mathcal{F}'(U/S) = \\mathcal{F}(\\Im(U \\to S))$.", "For the converse, we choose for every open subscheme $U \\subset S$ an object", "$U'/S \\in \\Ob(S_{Zar})$ with $\\Im(U' \\to S) = U$", "(here you have to use Sets, Lemma \\ref{sets-lemma-what-is-in-it}).", "Given a sheaf $\\mathcal{G}$ on $S_{Zar}$ we define a sheaf on $S$ by setting", "$\\mathcal{G}'(U) = \\mathcal{G}(U'/S)$. To see that $\\mathcal{G}'$ is", "a sheaf we use that for any open covering $U = \\bigcup_{i \\in I} U_i$", "the covering $\\{U_i \\to U\\}_{i \\in I}$", "is combinatorially equivalent to a covering $\\{U_j' \\to U'\\}_{j \\in J}$", "in $S_{Zar}$ by Sets, Lemma \\ref{sets-lemma-coverings-site},", "and we use Sites, Lemma \\ref{sites-lemma-tautological-same-sheaf}.", "Details omitted." ], "refs": [ "sets-lemma-what-is-in-it", "sets-lemma-coverings-site", "sites-lemma-tautological-same-sheaf" ], "ref_ids": [ 8795, 8800, 8503 ] } ], "ref_ids": [] }, { "id": 12438, "type": "theorem", "label": "topologies-lemma-put-in-T", "categories": [ "topologies" ], "title": "topologies-lemma-put-in-T", "contents": [ "Let $\\Sch_{Zar}$ be a big Zariski site.", "Let $f : T \\to S$ be a morphism in $\\Sch_{Zar}$.", "The functor $T_{Zar} \\to (\\Sch/S)_{Zar}$", "is cocontinuous and induces a morphism of topoi", "$$", "i_f :", "\\Sh(T_{Zar})", "\\longrightarrow", "\\Sh((\\Sch/S)_{Zar})", "$$", "For a sheaf $\\mathcal{G}$ on $(\\Sch/S)_{Zar}$", "we have the formula $(i_f^{-1}\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.", "The functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes", "with fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "Denote the functor $u : T_{Zar} \\to (\\Sch/S)_{Zar}$.", "In other words, given and open immersion $j : U \\to T$ corresponding", "to an object of $T_{Zar}$ we set $u(U \\to T) = (f \\circ j : U \\to S)$.", "This functor commutes with fibre products, see", "Lemma \\ref{lemma-fibre-products-Zariski}.", "Moreover, $T_{Zar}$ has equalizers (as any two morphisms with the same", "source and target are the same) and $u$ commutes with them.", "It is clearly cocontinuous.", "It is also continuous as $u$ transforms coverings to coverings and", "commutes with fibre products. Hence the lemma follows from", "Sites, Lemmas \\ref{sites-lemma-when-shriek}", "and \\ref{sites-lemma-preserve-equalizers}." ], "refs": [ "topologies-lemma-fibre-products-Zariski", "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers" ], "ref_ids": [ 12435, 8545, 8546 ] } ], "ref_ids": [] }, { "id": 12439, "type": "theorem", "label": "topologies-lemma-at-the-bottom", "categories": [ "topologies" ], "title": "topologies-lemma-at-the-bottom", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski", "site containing $S$.", "The inclusion functor $S_{Zar} \\to (\\Sch/S)_{Zar}$", "satisfies the hypotheses of Sites, Lemma \\ref{sites-lemma-bigger-site}", "and hence induces a morphism of sites", "$$", "\\pi_S : (\\Sch/S)_{Zar} \\longrightarrow S_{Zar}", "$$", "and a morphism of topoi", "$$", "i_S : \\Sh(S_{Zar}) \\longrightarrow \\Sh((\\Sch/S)_{Zar})", "$$", "such that $\\pi_S \\circ i_S = \\text{id}$. Moreover, $i_S = i_{\\text{id}_S}$", "with $i_{\\text{id}_S}$ as in Lemma \\ref{lemma-put-in-T}. In particular the", "functor $i_S^{-1} = \\pi_{S, *}$ is described by the rule", "$i_S^{-1}(\\mathcal{G})(U/S) = \\mathcal{G}(U/S)$." ], "refs": [ "sites-lemma-bigger-site", "topologies-lemma-put-in-T" ], "proofs": [ { "contents": [ "In this case the functor $u : S_{Zar} \\to (\\Sch/S)_{Zar}$,", "in addition to the properties seen in the proof of", "Lemma \\ref{lemma-put-in-T} above, also is fully faithful", "and transforms the final object into the final object.", "The lemma follows." ], "refs": [ "topologies-lemma-put-in-T" ], "ref_ids": [ 12438 ] } ], "ref_ids": [ 8548, 12438 ] }, { "id": 12440, "type": "theorem", "label": "topologies-lemma-morphism-big", "categories": [ "topologies" ], "title": "topologies-lemma-morphism-big", "contents": [ "Let $\\Sch_{Zar}$ be a big Zariski site.", "Let $f : T \\to S$ be a morphism in $\\Sch_{Zar}$.", "The functor", "$$", "u : (\\Sch/T)_{Zar} \\longrightarrow (\\Sch/S)_{Zar},", "\\quad", "V/T \\longmapsto V/S", "$$", "is cocontinuous, and has a continuous right adjoint", "$$", "v : (\\Sch/S)_{Zar} \\longrightarrow (\\Sch/T)_{Zar},", "\\quad", "(U \\to S) \\longmapsto (U \\times_S T \\to T).", "$$", "They induce the same morphism of topoi", "$$", "f_{big} :", "\\Sh((\\Sch/T)_{Zar})", "\\longrightarrow", "\\Sh((\\Sch/S)_{Zar})", "$$", "We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.", "We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.", "Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with", "fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "The functor $u$ is cocontinuous, continuous, and commutes with fibre products", "and equalizers (details omitted; compare with proof of", "Lemma \\ref{lemma-put-in-T}).", "Hence", "Sites, Lemmas \\ref{sites-lemma-when-shriek} and", "\\ref{sites-lemma-preserve-equalizers}", "apply and we deduce the formula", "for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover,", "the functor $v$ is a right adjoint because given $U/T$ and $V/S$", "we have $\\Mor_S(u(U), V) = \\Mor_T(U, V \\times_S T)$", "as desired. Thus we may apply", "Sites, Lemmas \\ref{sites-lemma-have-functor-other-way} and", "\\ref{sites-lemma-have-functor-other-way-morphism}", "to get the formula for $f_{big, *}$." ], "refs": [ "topologies-lemma-put-in-T", "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers", "sites-lemma-have-functor-other-way", "sites-lemma-have-functor-other-way-morphism" ], "ref_ids": [ 12438, 8545, 8546, 8549, 8550 ] } ], "ref_ids": [] }, { "id": 12441, "type": "theorem", "label": "topologies-lemma-morphism-big-small", "categories": [ "topologies" ], "title": "topologies-lemma-morphism-big-small", "contents": [ "Let $\\Sch_{Zar}$ be a big Zariski site.", "Let $f : T \\to S$ be a morphism in $\\Sch_{Zar}$.", "\\begin{enumerate}", "\\item We have $i_f = f_{big} \\circ i_T$ with $i_f$ as in", "Lemma \\ref{lemma-put-in-T} and $i_T$ as in", "Lemma \\ref{lemma-at-the-bottom}.", "\\item The functor $S_{Zar} \\to T_{Zar}$,", "$(U \\to S) \\mapsto (U \\times_S T \\to T)$ is continuous and induces", "a morphism of topoi", "$$", "f_{small} :", "\\Sh(T_{Zar})", "\\longrightarrow", "\\Sh(S_{Zar}).", "$$", "The functors $f_{small}^{-1}$ and $f_{small, *}$ agree with", "the usual notions $f^{-1}$ and $f_*$ is we identify sheaves", "on $T_{Zar}$, resp.\\ $S_{Zar}$ with sheaves on $T$, resp.\\ $S$", "via Lemma \\ref{lemma-Zariski-usual}.", "\\item We have a commutative diagram of morphisms of sites", "$$", "\\xymatrix{", "T_{Zar} \\ar[d]_{f_{small}} &", "(\\Sch/T)_{Zar} \\ar[d]^{f_{big}} \\ar[l]^{\\pi_T} \\\\", "S_{Zar} &", "(\\Sch/S)_{Zar} \\ar[l]_{\\pi_S}", "}", "$$", "so that $f_{small} \\circ \\pi_T = \\pi_S \\circ f_{big}$ as morphisms of topoi.", "\\item We have $f_{small} = \\pi_S \\circ f_{big} \\circ i_T = \\pi_S \\circ i_f$.", "\\end{enumerate}" ], "refs": [ "topologies-lemma-put-in-T", "topologies-lemma-at-the-bottom", "topologies-lemma-Zariski-usual" ], "proofs": [ { "contents": [ "The equality $i_f = f_{big} \\circ i_T$ follows from the", "equality $i_f^{-1} = i_T^{-1} \\circ f_{big}^{-1}$ which is", "clear from the descriptions of these functors above.", "Thus we see (1).", "\\medskip\\noindent", "Statement (2): See Sites, Example \\ref{sites-example-continuous-map}.", "\\medskip\\noindent", "Part (3) follows because $\\pi_S$ and $\\pi_T$ are given by", "the inclusion functors and $f_{small}$ and $f_{big}$ by the", "base change functor $U \\mapsto U \\times_S T$.", "\\medskip\\noindent", "Statement (4) follows from (3) by precomposing with $i_T$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12438, 12439, 12437 ] }, { "id": 12442, "type": "theorem", "label": "topologies-lemma-composition", "categories": [ "topologies" ], "title": "topologies-lemma-composition", "contents": [ "Given schemes $X$, $Y$, $Y$ in $(\\Sch/S)_{Zar}$", "and morphisms $f : X \\to Y$, $g : Y \\to Z$ we have", "$g_{big} \\circ f_{big} = (g \\circ f)_{big}$ and", "$g_{small} \\circ f_{small} = (g \\circ f)_{small}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the simple description of pushforward", "and pullback for the functors on the big sites from", "Lemma \\ref{lemma-morphism-big}. For the functors", "on the small sites this is", "Sheaves, Lemma \\ref{sheaves-lemma-pushforward-composition}", "via the identification of Lemma \\ref{lemma-Zariski-usual}." ], "refs": [ "topologies-lemma-morphism-big", "sheaves-lemma-pushforward-composition", "topologies-lemma-Zariski-usual" ], "ref_ids": [ 12440, 14504, 12437 ] } ], "ref_ids": [] }, { "id": 12443, "type": "theorem", "label": "topologies-lemma-morphism-big-small-cartesian-diagram", "categories": [ "topologies" ], "title": "topologies-lemma-morphism-big-small-cartesian-diagram", "contents": [ "Let $\\Sch_{Zar}$ be a big Zariski site. Consider a cartesian diagram", "$$", "\\xymatrix{", "T' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "in $\\Sch_{Zar}$. Then", "$i_g^{-1} \\circ f_{big, *} = f'_{small, *} \\circ (i_{g'})^{-1}$", "and $g_{big}^{-1} \\circ f_{big, *} = f'_{big, *} \\circ (g'_{big})^{-1}$." ], "refs": [], "proofs": [ { "contents": [ "Since the diagram is cartesian, we have for $U'/S'$", "that $U' \\times_{S'} T' = U' \\times_S T$. Hence both", "$i_g^{-1} \\circ f_{big, *}$ and $f'_{small, *} \\circ (i_{g'})^{-1}$", "send a sheaf $\\mathcal{F}$ on $(\\Sch/T)_{Zar}$ to the sheaf", "$U' \\mapsto \\mathcal{F}(U' \\times_{S'} T')$ on $S'_{Zar}$", "(use Lemmas \\ref{lemma-put-in-T} and \\ref{lemma-morphism-big-small}).", "The second equality can be proved in the same manner or can be", "deduced from the very general", "Sites, Lemma \\ref{sites-lemma-localize-morphism}." ], "refs": [ "topologies-lemma-put-in-T", "topologies-lemma-morphism-big-small", "sites-lemma-localize-morphism" ], "ref_ids": [ 12438, 12441, 8571 ] } ], "ref_ids": [] }, { "id": 12444, "type": "theorem", "label": "topologies-lemma-characterize-sheaf-big", "categories": [ "topologies" ], "title": "topologies-lemma-characterize-sheaf-big", "contents": [ "Let $S$ be a scheme contained in a big Zariski site $\\Sch_{Zar}$.", "A sheaf $\\mathcal{F}$ on the big Zariski site $(\\Sch/S)_{Zar}$", "is given by the following data:", "\\begin{enumerate}", "\\item for every $T/S \\in \\Ob((\\Sch/S)_{Zar})$ a sheaf", "$\\mathcal{F}_T$ on $T$,", "\\item for every $f : T' \\to T$ in", "$(\\Sch/S)_{Zar}$ a map", "$c_f : f^{-1}\\mathcal{F}_T \\to \\mathcal{F}_{T'}$.", "\\end{enumerate}", "These data are subject to the following conditions:", "\\begin{enumerate}", "\\item[(a)] given any $f : T' \\to T$ and $g : T'' \\to T'$ in", "$(\\Sch/S)_{Zar}$ the composition $c_g \\circ g^{-1}c_f$", "is equal to $c_{f \\circ g}$, and", "\\item[(b)] if $f : T' \\to T$ in $(\\Sch/S)_{Zar}$ is an", "open immersion then $c_f$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Given a sheaf $\\mathcal{F}$ on $\\Sh((\\Sch/S)_{Zar})$", "we set $\\mathcal{F}_T = i_p^{-1}\\mathcal{F}$ where $p : T \\to S$", "is the structure morphism. Note that", "$\\mathcal{F}_T(U) = \\mathcal{F}(U'/S)$ for any open $U \\subset T$,", "and $U' \\to T$ an open immersion in $(\\Sch/T)_{Zar}$", "with image $U$, see Lemmas \\ref{lemma-Zariski-usual} and \\ref{lemma-put-in-T}.", "Hence given $f : T' \\to T$ over $S$ and $U, U' \\to T$ we get a canonical", "map $\\mathcal{F}_T(U) = \\mathcal{F}(U'/S) \\to \\mathcal{F}(U'\\times_T T'/S)", "= \\mathcal{F}_{T'}(f^{-1}(U))$ where the middle is the restriction map", "of $\\mathcal{F}$ with respect to the morphism", "$U' \\times_T T' \\to U'$ over $S$. The collection of these maps are", "compatible with restrictions, and hence define an $f$-map $c_f$", "from $\\mathcal{F}_T$ to $\\mathcal{F}_{T'}$, see", "Sheaves, Definition \\ref{sheaves-definition-f-map} and the discussion", "surrounding it. It is clear that $c_{f \\circ g}$ is the composition of", "$c_f$ and $c_g$, since composition of restriction maps of $\\mathcal{F}$", "gives restriction maps.", "\\medskip\\noindent", "Conversely, given a system $(\\mathcal{F}_T, c_f)$ as in the lemma", "we may define a presheaf $\\mathcal{F}$ on $\\Sh((\\Sch/S)_{Zar})$", "by simply setting $\\mathcal{F}(T/S) = \\mathcal{F}_T(T)$. As restriction", "mapping, given $f : T' \\to T$ we set for $s \\in \\mathcal{F}(T)$", "the pullback $f^*(s)$ equal to $c_f(s)$ (where we think of $c_f$ as", "an $f$-map again). The condition on the $c_f$ guarantees that", "pullbacks satisfy the required functoriality property.", "We omit the verification that this is a sheaf.", "It is clear that the constructions so defined are mutually inverse." ], "refs": [ "topologies-lemma-Zariski-usual", "topologies-lemma-put-in-T", "sheaves-definition-f-map" ], "ref_ids": [ 12437, 12438, 14573 ] } ], "ref_ids": [] }, { "id": 12445, "type": "theorem", "label": "topologies-lemma-zariski-etale", "categories": [ "topologies" ], "title": "topologies-lemma-zariski-etale", "contents": [ "Any Zariski covering is an \\'etale covering." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions and the fact that an open immersion", "is an \\'etale morphism, see", "Morphisms, Lemma \\ref{morphisms-lemma-open-immersion-etale}." ], "refs": [ "morphisms-lemma-open-immersion-etale" ], "ref_ids": [ 5366 ] } ], "ref_ids": [] }, { "id": 12446, "type": "theorem", "label": "topologies-lemma-etale", "categories": [ "topologies" ], "title": "topologies-lemma-etale", "contents": [ "Let $T$ be a scheme.", "\\begin{enumerate}", "\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$", "is an \\'etale covering of $T$.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an \\'etale covering and for each", "$i$ we have an \\'etale covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then", "$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is an \\'etale covering.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an \\'etale covering", "and $T' \\to T$ is a morphism of schemes then", "$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is an \\'etale covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12447, "type": "theorem", "label": "topologies-lemma-etale-affine", "categories": [ "topologies" ], "title": "topologies-lemma-etale-affine", "contents": [ "Let $T$ be an affine scheme.", "Let $\\{T_i \\to T\\}_{i \\in I}$ be an \\'etale covering of $T$.", "Then there exists an \\'etale covering", "$\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement", "of $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine", "scheme. Moreover, we may choose each $U_j$ to be open affine", "in one of the $T_i$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12448, "type": "theorem", "label": "topologies-lemma-etale-induced", "categories": [ "topologies" ], "title": "topologies-lemma-etale-induced", "contents": [ "Let $\\Sch_\\etale$ be a big \\'etale site as in", "Definition \\ref{definition-big-etale-site}.", "Let $T \\in \\Ob(\\Sch_\\etale)$.", "Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary \\'etale covering of $T$.", "\\begin{enumerate}", "\\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site", "$\\Sch_\\etale$ which refines $\\{T_i \\to T\\}_{i \\in I}$.", "\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard \\'etale covering, then", "it is tautologically equivalent to a covering in $\\Sch_\\etale$.", "\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then", "it is tautologically equivalent to a covering in $\\Sch_\\etale$.", "\\end{enumerate}" ], "refs": [ "topologies-definition-big-etale-site" ], "proofs": [ { "contents": [ "For each $i$ choose an affine open covering $T_i = \\bigcup_{j \\in J_i} T_{ij}$", "such that each $T_{ij}$ maps into an affine open subscheme of $T$. By", "Lemma \\ref{lemma-etale}", "the refinement $\\{T_{ij} \\to T\\}_{i \\in I, j \\in J_i}$ is an \\'etale covering", "of $T$ as well. Hence we may assume each $T_i$ is affine, and maps into", "an affine open $W_i$ of $T$. Applying", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}", "we see that $W_i$ is isomorphic to an object of $\\Sch_\\etale$.", "But then $T_i$ as a finite type scheme over $W_i$", "is isomorphic to an object $V_i$ of $\\Sch_\\etale$ by a second", "application of", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "The covering $\\{V_i \\to T\\}_{i \\in I}$ refines $\\{T_i \\to T\\}_{i \\in I}$", "(because they are isomorphic).", "Moreover, $\\{V_i \\to T\\}_{i \\in I}$ is combinatorially equivalent to a", "covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site", "$\\Sch_\\etale$ by", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "The covering $\\{U_j \\to T\\}_{j \\in J}$ is a refinement as in (1).", "In the situation of (2), (3) each of the", "schemes $T_i$ is isomorphic to an object of $\\Sch_\\etale$ by", "Sets, Lemma \\ref{sets-lemma-what-is-in-it},", "and another application of", "Sets, Lemma \\ref{sets-lemma-coverings-site}", "gives what we want." ], "refs": [ "topologies-lemma-etale", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-coverings-site" ], "ref_ids": [ 12446, 8795, 8795, 8795, 8795, 8800 ] } ], "ref_ids": [ 12528 ] }, { "id": 12449, "type": "theorem", "label": "topologies-lemma-verify-site-etale", "categories": [ "topologies" ], "title": "topologies-lemma-verify-site-etale", "contents": [ "Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale", "site containing $S$.", "Both $S_\\etale$ and $(\\textit{Aff}/S)_\\etale$ are sites." ], "refs": [], "proofs": [ { "contents": [ "Let us show that $S_\\etale$ is a site. It is a category with a", "given set of families of morphisms with fixed target. Thus we", "have to show properties (1), (2) and (3) of", "Sites, Definition \\ref{sites-definition-site}.", "Since $(\\Sch/S)_\\etale$ is a site, it suffices to prove", "that given any covering $\\{U_i \\to U\\}$ of $(\\Sch/S)_\\etale$", "with $U \\in \\Ob(S_\\etale)$ we also have", "$U_i \\in \\Ob(S_\\etale)$.", "This follows from the definitions as the composition of \\'etale morphisms", "is an \\'etale morphism.", "\\medskip\\noindent", "Let us show that $(\\textit{Aff}/S)_\\etale$ is a site.", "Reasoning as above, it suffices to show that the collection", "of standard \\'etale coverings of affines satisfies properties", "(1), (2) and (3) of", "Sites, Definition \\ref{sites-definition-site}.", "This is clear since for example, given a standard \\'etale", "covering $\\{T_i \\to T\\}_{i\\in I}$ and for each", "$i$ we have a standard \\'etale covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then", "$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a standard \\'etale covering", "because $\\bigcup_{i\\in I} J_i$ is finite and each $T_{ij}$ is affine." ], "refs": [ "sites-definition-site", "sites-definition-site" ], "ref_ids": [ 8652, 8652 ] } ], "ref_ids": [] }, { "id": 12450, "type": "theorem", "label": "topologies-lemma-fibre-products-etale", "categories": [ "topologies" ], "title": "topologies-lemma-fibre-products-etale", "contents": [ "Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale", "site containing $S$. The underlying categories of the sites", "$\\Sch_\\etale$, $(\\Sch/S)_\\etale$,", "$S_\\etale$, and $(\\textit{Aff}/S)_\\etale$ have fibre products.", "In each case the obvious functor into the category $\\Sch$ of", "all schemes commutes with taking fibre products. The categories", "$(\\Sch/S)_\\etale$, and $S_\\etale$ both have a", "final object, namely $S/S$." ], "refs": [], "proofs": [ { "contents": [ "For $\\Sch_\\etale$ it is true by construction, see", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "Suppose we have $U \\to S$, $V \\to U$, $W \\to U$ morphisms", "of schemes with $U, V, W \\in \\Ob(\\Sch_\\etale)$.", "The fibre product $V \\times_U W$ in $\\Sch_\\etale$", "is a fibre product in $\\Sch$ and", "is the fibre product of $V/S$ with $W/S$ over $U/S$ in", "the category of all schemes over $S$, and hence also a", "fibre product in $(\\Sch/S)_\\etale$.", "This proves the result for $(\\Sch/S)_\\etale$.", "If $U \\to S$, $V \\to U$ and $W \\to U$ are \\'etale then so is", "$V \\times_U W \\to S$ and hence we get the result for $S_\\etale$.", "If $U, V, W$ are affine, so is $V \\times_U W$ and hence the", "result for $(\\textit{Aff}/S)_\\etale$." ], "refs": [ "sets-lemma-what-is-in-it" ], "ref_ids": [ 8795 ] } ], "ref_ids": [] }, { "id": 12451, "type": "theorem", "label": "topologies-lemma-affine-big-site-etale", "categories": [ "topologies" ], "title": "topologies-lemma-affine-big-site-etale", "contents": [ "Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale", "site containing $S$.", "The functor", "$(\\textit{Aff}/S)_\\etale \\to (\\Sch/S)_\\etale$", "is special cocontinuous and induces an equivalence of topoi from", "$\\Sh((\\textit{Aff}/S)_\\etale)$ to", "$\\Sh((\\Sch/S)_\\etale)$." ], "refs": [], "proofs": [ { "contents": [ "The notion of a special cocontinuous functor is introduced in", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}.", "Thus we have to verify assumptions (1) -- (5) of", "Sites, Lemma \\ref{sites-lemma-equivalence}.", "Denote the inclusion functor", "$u : (\\textit{Aff}/S)_\\etale \\to (\\Sch/S)_\\etale$.", "Being cocontinuous just means that any \\'etale covering of", "$T/S$, $T$ affine, can be refined by a standard \\'etale covering of $T$.", "This is the content of", "Lemma \\ref{lemma-etale-affine}.", "Hence (1) holds. We see $u$ is continuous simply because a standard", "\\'etale covering is a \\'etale covering. Hence (2) holds.", "Parts (3) and (4) follow immediately from the fact that $u$ is", "fully faithful. And finally condition (5) follows from the", "fact that every scheme has an affine open covering." ], "refs": [ "sites-definition-special-cocontinuous-functor", "sites-lemma-equivalence", "topologies-lemma-etale-affine" ], "ref_ids": [ 8672, 8578, 12447 ] } ], "ref_ids": [] }, { "id": 12452, "type": "theorem", "label": "topologies-lemma-put-in-T-etale", "categories": [ "topologies" ], "title": "topologies-lemma-put-in-T-etale", "contents": [ "Let $\\Sch_\\etale$ be a big \\'etale site.", "Let $f : T \\to S$ be a morphism in $\\Sch_\\etale$.", "The functor $T_\\etale \\to (\\Sch/S)_\\etale$", "is cocontinuous and induces a morphism of topoi", "$$", "i_f :", "\\Sh(T_\\etale)", "\\longrightarrow", "\\Sh((\\Sch/S)_\\etale)", "$$", "For a sheaf $\\mathcal{G}$ on $(\\Sch/S)_\\etale$", "we have the formula $(i_f^{-1}\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.", "The functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes", "with fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "Denote the functor $u : T_\\etale \\to (\\Sch/S)_\\etale$.", "In other words, given an \\'etale morphism $j : U \\to T$ corresponding", "to an object of $T_\\etale$ we set $u(U \\to T) = (f \\circ j : U \\to S)$.", "This functor commutes with fibre products, see", "Lemma \\ref{lemma-fibre-products-etale}.", "Let $a, b : U \\to V$ be two morphisms in $T_\\etale$.", "In this case the equalizer of $a$ and $b$ (in the category of schemes) is", "$$", "V \\times_{\\Delta_{V/T}, V \\times_T V, (a, b)} U \\times_T U", "$$", "which is a fibre product of schemes \\'etale over $T$, hence \\'etale", "over $T$. Thus $T_\\etale$ has equalizers and $u$ commutes with them.", "It is clearly cocontinuous.", "It is also continuous as $u$ transforms coverings to coverings and", "commutes with fibre products. Hence the Lemma follows from", "Sites, Lemmas \\ref{sites-lemma-when-shriek}", "and \\ref{sites-lemma-preserve-equalizers}." ], "refs": [ "topologies-lemma-fibre-products-etale", "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers" ], "ref_ids": [ 12450, 8545, 8546 ] } ], "ref_ids": [] }, { "id": 12453, "type": "theorem", "label": "topologies-lemma-at-the-bottom-etale", "categories": [ "topologies" ], "title": "topologies-lemma-at-the-bottom-etale", "contents": [ "Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale", "site containing $S$.", "The inclusion functor $S_\\etale \\to (\\Sch/S)_\\etale$", "satisfies the hypotheses of Sites, Lemma \\ref{sites-lemma-bigger-site}", "and hence induces a morphism of sites", "$$", "\\pi_S : (\\Sch/S)_\\etale \\longrightarrow S_\\etale", "$$", "and a morphism of topoi", "$$", "i_S : \\Sh(S_\\etale) \\longrightarrow \\Sh((\\Sch/S)_\\etale)", "$$", "such that $\\pi_S \\circ i_S = \\text{id}$. Moreover, $i_S = i_{\\text{id}_S}$", "with $i_{\\text{id}_S}$ as in Lemma \\ref{lemma-put-in-T-etale}.", "In particular the functor $i_S^{-1} = \\pi_{S, *}$ is described by the rule", "$i_S^{-1}(\\mathcal{G})(U/S) = \\mathcal{G}(U/S)$." ], "refs": [ "sites-lemma-bigger-site", "topologies-lemma-put-in-T-etale" ], "proofs": [ { "contents": [ "In this case the functor", "$u : S_\\etale \\to (\\Sch/S)_\\etale$,", "in addition to the properties seen in the proof of", "Lemma \\ref{lemma-put-in-T-etale} above, also is fully faithful", "and transforms the final object into the final object.", "The lemma follows from Sites, Lemma \\ref{sites-lemma-bigger-site}." ], "refs": [ "topologies-lemma-put-in-T-etale", "sites-lemma-bigger-site" ], "ref_ids": [ 12452, 8548 ] } ], "ref_ids": [ 8548, 12452 ] }, { "id": 12454, "type": "theorem", "label": "topologies-lemma-morphism-big-etale", "categories": [ "topologies" ], "title": "topologies-lemma-morphism-big-etale", "contents": [ "Let $\\Sch_\\etale$ be a big \\'etale site.", "Let $f : T \\to S$ be a morphism in $\\Sch_\\etale$.", "The functor", "$$", "u :", "(\\Sch/T)_\\etale", "\\longrightarrow", "(\\Sch/S)_\\etale,", "\\quad", "V/T \\longmapsto V/S", "$$", "is cocontinuous, and has a continuous right adjoint", "$$", "v :", "(\\Sch/S)_\\etale", "\\longrightarrow", "(\\Sch/T)_\\etale,", "\\quad", "(U \\to S) \\longmapsto (U \\times_S T \\to T).", "$$", "They induce the same morphism of topoi", "$$", "f_{big} :", "\\Sh((\\Sch/T)_\\etale)", "\\longrightarrow", "\\Sh((\\Sch/S)_\\etale)", "$$", "We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.", "We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.", "Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with", "fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "The functor $u$ is cocontinuous, continuous and commutes with fibre products", "and equalizers (details omitted; compare with the proof of", "Lemma \\ref{lemma-put-in-T-etale}).", "Hence", "Sites, Lemmas \\ref{sites-lemma-when-shriek} and", "\\ref{sites-lemma-preserve-equalizers}", "apply and we deduce the formula", "for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover,", "the functor $v$ is a right adjoint because given $U/T$ and $V/S$", "we have $\\Mor_S(u(U), V) = \\Mor_T(U, V \\times_S T)$", "as desired. Thus we may apply", "Sites, Lemmas \\ref{sites-lemma-have-functor-other-way} and", "\\ref{sites-lemma-have-functor-other-way-morphism} to get the", "formula for $f_{big, *}$." ], "refs": [ "topologies-lemma-put-in-T-etale", "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers", "sites-lemma-have-functor-other-way", "sites-lemma-have-functor-other-way-morphism" ], "ref_ids": [ 12452, 8545, 8546, 8549, 8550 ] } ], "ref_ids": [] }, { "id": 12455, "type": "theorem", "label": "topologies-lemma-morphism-big-small-etale", "categories": [ "topologies" ], "title": "topologies-lemma-morphism-big-small-etale", "contents": [ "Let $\\Sch_\\etale$ be a big \\'etale site.", "Let $f : T \\to S$ be a morphism in $\\Sch_\\etale$.", "\\begin{enumerate}", "\\item We have $i_f = f_{big} \\circ i_T$ with $i_f$ as in", "Lemma \\ref{lemma-put-in-T-etale} and $i_T$ as in", "Lemma \\ref{lemma-at-the-bottom-etale}.", "\\item The functor $S_\\etale \\to T_\\etale$,", "$(U \\to S) \\mapsto (U \\times_S T \\to T)$ is continuous and induces", "a morphism of sites", "$$", "f_{small} : T_\\etale \\longrightarrow S_\\etale", "$$", "We have $f_{small, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.", "\\item We have a commutative diagram of morphisms of sites", "$$", "\\xymatrix{", "T_\\etale \\ar[d]_{f_{small}} &", "(\\Sch/T)_\\etale \\ar[d]^{f_{big}} \\ar[l]^{\\pi_T}\\\\", "S_\\etale &", "(\\Sch/S)_\\etale \\ar[l]_{\\pi_S}", "}", "$$", "so that $f_{small} \\circ \\pi_T = \\pi_S \\circ f_{big}$ as morphisms of topoi.", "\\item We have $f_{small} = \\pi_S \\circ f_{big} \\circ i_T = \\pi_S \\circ i_f$.", "\\end{enumerate}" ], "refs": [ "topologies-lemma-put-in-T-etale", "topologies-lemma-at-the-bottom-etale" ], "proofs": [ { "contents": [ "The equality $i_f = f_{big} \\circ i_T$ follows from the", "equality $i_f^{-1} = i_T^{-1} \\circ f_{big}^{-1}$ which is", "clear from the descriptions of these functors above.", "Thus we see (1).", "\\medskip\\noindent", "The functor", "$u :", "S_\\etale", "\\to", "T_\\etale$, $u(U \\to S) = (U \\times_S T \\to T)$", "transforms coverings into coverings and commutes with fibre products,", "see Lemma \\ref{lemma-etale} (3) and \\ref{lemma-fibre-products-etale}.", "Moreover, both $S_\\etale$, $T_\\etale$ have final objects,", "namely $S/S$ and $T/T$ and $u(S/S) = T/T$. Hence by", "Sites, Proposition \\ref{sites-proposition-get-morphism}", "the functor $u$ corresponds to a morphism of sites", "$T_\\etale \\to S_\\etale$. This in turn gives rise to the", "morphism of topoi, see", "Sites, Lemma \\ref{sites-lemma-morphism-sites-topoi}. The description", "of the pushforward is clear from these references.", "\\medskip\\noindent", "Part (3) follows because $\\pi_S$ and $\\pi_T$ are given by the", "inclusion functors and $f_{small}$ and $f_{big}$ by the", "base change functors $U \\mapsto U \\times_S T$.", "\\medskip\\noindent", "Statement (4) follows from (3) by precomposing with $i_T$." ], "refs": [ "topologies-lemma-etale", "topologies-lemma-fibre-products-etale", "sites-proposition-get-morphism", "sites-lemma-morphism-sites-topoi" ], "ref_ids": [ 12446, 12450, 8641, 8528 ] } ], "ref_ids": [ 12452, 12453 ] }, { "id": 12456, "type": "theorem", "label": "topologies-lemma-composition-etale", "categories": [ "topologies" ], "title": "topologies-lemma-composition-etale", "contents": [ "Given schemes $X$, $Y$, $Y$ in $\\Sch_\\etale$", "and morphisms $f : X \\to Y$, $g : Y \\to Z$ we have", "$g_{big} \\circ f_{big} = (g \\circ f)_{big}$ and", "$g_{small} \\circ f_{small} = (g \\circ f)_{small}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the simple description of pushforward", "and pullback for the functors on the big sites from", "Lemma \\ref{lemma-morphism-big-etale}. For the functors", "on the small sites this follows from the description of", "the pushforward functors in Lemma \\ref{lemma-morphism-big-small-etale}." ], "refs": [ "topologies-lemma-morphism-big-etale", "topologies-lemma-morphism-big-small-etale" ], "ref_ids": [ 12454, 12455 ] } ], "ref_ids": [] }, { "id": 12457, "type": "theorem", "label": "topologies-lemma-morphism-big-small-cartesian-diagram-etale", "categories": [ "topologies" ], "title": "topologies-lemma-morphism-big-small-cartesian-diagram-etale", "contents": [ "Let $\\Sch_\\etale$ be a big \\'etale site. Consider a cartesian diagram", "$$", "\\xymatrix{", "T' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "in $\\Sch_\\etale$. Then", "$i_g^{-1} \\circ f_{big, *} = f'_{small, *} \\circ (i_{g'})^{-1}$", "and $g_{big}^{-1} \\circ f_{big, *} = f'_{big, *} \\circ (g'_{big})^{-1}$." ], "refs": [], "proofs": [ { "contents": [ "Since the diagram is cartesian, we have for $U'/S'$", "that $U' \\times_{S'} T' = U' \\times_S T$. Hence both", "$i_g^{-1} \\circ f_{big, *}$ and $f'_{small, *} \\circ (i_{g'})^{-1}$", "send a sheaf $\\mathcal{F}$ on $(\\Sch/T)_\\etale$ to the sheaf", "$U' \\mapsto \\mathcal{F}(U' \\times_{S'} T')$ on $S'_\\etale$", "(use Lemmas \\ref{lemma-put-in-T-etale} and \\ref{lemma-morphism-big-etale}).", "The second equality can be proved in the same manner or can be", "deduced from the very general", "Sites, Lemma \\ref{sites-lemma-localize-morphism}." ], "refs": [ "topologies-lemma-put-in-T-etale", "topologies-lemma-morphism-big-etale", "sites-lemma-localize-morphism" ], "ref_ids": [ 12452, 12454, 8571 ] } ], "ref_ids": [] }, { "id": 12458, "type": "theorem", "label": "topologies-lemma-characterize-sheaf-big-etale", "categories": [ "topologies" ], "title": "topologies-lemma-characterize-sheaf-big-etale", "contents": [ "Let $S$ be a scheme contained in a big \\'etale site", "$\\Sch_\\etale$.", "A sheaf $\\mathcal{F}$ on the big \\'etale site", "$(\\Sch/S)_\\etale$ is given by the following data:", "\\begin{enumerate}", "\\item for every $T/S \\in \\Ob((\\Sch/S)_\\etale)$ a sheaf", "$\\mathcal{F}_T$ on $T_\\etale$,", "\\item for every $f : T' \\to T$ in", "$(\\Sch/S)_\\etale$ a map", "$c_f : f_{small}^{-1}\\mathcal{F}_T \\to \\mathcal{F}_{T'}$.", "\\end{enumerate}", "These data are subject to the following conditions:", "\\begin{enumerate}", "\\item[(a)] given any $f : T' \\to T$ and $g : T'' \\to T'$ in", "$(\\Sch/S)_\\etale$ the composition", "$c_g \\circ g_{small}^{-1}c_f$ is equal to $c_{f \\circ g}$, and", "\\item[(b)] if $f : T' \\to T$ in $(\\Sch/S)_\\etale$", "is \\'etale then $c_f$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Given a sheaf $\\mathcal{F}$ on $\\Sh((\\Sch/S)_\\etale)$", "we set $\\mathcal{F}_T = i_p^{-1}\\mathcal{F}$ where $p : T \\to S$", "is the structure morphism. Note that", "$\\mathcal{F}_T(U) = \\mathcal{F}(U/S)$ for any $U \\to T$", "in $T_\\etale$ see Lemma \\ref{lemma-put-in-T-etale}.", "Hence given $f : T' \\to T$ over $S$ and $U \\to T$ we get a canonical", "map $\\mathcal{F}_T(U) = \\mathcal{F}(U/S) \\to \\mathcal{F}(U \\times_T T'/S)", "= \\mathcal{F}_{T'}(U \\times_T T')$ where the middle is the restriction map", "of $\\mathcal{F}$ with respect to the morphism", "$U \\times_T T' \\to U$ over $S$. The collection of these maps are", "compatible with restrictions, and hence define a map", "$c'_f : \\mathcal{F}_T \\to f_{small, *}\\mathcal{F}_{T'}$ where", "$u : T_\\etale \\to T'_\\etale$ is the base change functor", "associated to $f$. By adjunction of $f_{small, *}$ (see", "Sites, Section \\ref{sites-section-continuous-functors}) with", "$f_{small}^{-1}$ this is the same as a map", "$c_f : f_{small}^{-1}\\mathcal{F}_T \\to \\mathcal{F}_{T'}$.", "It is clear that $c'_{f \\circ g}$ is the composition of", "$c'_f$ and $f_{small, *}c'_g$, since composition of restriction maps", "of $\\mathcal{F}$ gives restriction maps, and this gives the desired", "relationship among $c_f$, $c_g$ and $c_{f \\circ g}$.", "\\medskip\\noindent", "Conversely, given a system $(\\mathcal{F}_T, c_f)$ as in the lemma", "we may define a presheaf $\\mathcal{F}$ on", "$\\Sh((\\Sch/S)_\\etale)$", "by simply setting $\\mathcal{F}(T/S) = \\mathcal{F}_T(T)$. As restriction", "mapping, given $f : T' \\to T$ we set for $s \\in \\mathcal{F}(T)$", "the pullback $f^*(s)$ equal to $c_f(s)$ where we think of $c_f$ as", "a map $\\mathcal{F}_T \\to f_{small, *}\\mathcal{F}_{T'}$ again.", "The condition on the $c_f$ guarantees that", "pullbacks satisfy the required functoriality property.", "We omit the verification that this is a sheaf.", "It is clear that the constructions so defined are mutually inverse." ], "refs": [ "topologies-lemma-put-in-T-etale" ], "ref_ids": [ 12452 ] } ], "ref_ids": [] }, { "id": 12459, "type": "theorem", "label": "topologies-lemma-zariski-etale-smooth", "categories": [ "topologies" ], "title": "topologies-lemma-zariski-etale-smooth", "contents": [ "Any \\'etale covering is a smooth covering, and a fortiori,", "any Zariski covering is a smooth covering." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions, the fact that an \\'etale morphism is", "smooth see", "Morphisms, Definition \\ref{morphisms-definition-etale}", "and Lemma \\ref{lemma-zariski-etale}." ], "refs": [ "morphisms-definition-etale", "topologies-lemma-zariski-etale" ], "ref_ids": [ 5567, 12445 ] } ], "ref_ids": [] }, { "id": 12460, "type": "theorem", "label": "topologies-lemma-smooth", "categories": [ "topologies" ], "title": "topologies-lemma-smooth", "contents": [ "Let $T$ be a scheme.", "\\begin{enumerate}", "\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$", "is a smooth covering of $T$.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a smooth covering and for each", "$i$ we have a smooth covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then", "$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a smooth covering.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a smooth covering", "and $T' \\to T$ is a morphism of schemes then", "$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a smooth covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12461, "type": "theorem", "label": "topologies-lemma-smooth-affine", "categories": [ "topologies" ], "title": "topologies-lemma-smooth-affine", "contents": [ "Let $T$ be an affine scheme.", "Let $\\{T_i \\to T\\}_{i \\in I}$ be a smooth covering of $T$.", "Then there exists a smooth covering", "$\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement", "of $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine", "scheme, and such that each morphism $U_j \\to T$ is standard", "smooth, see Morphisms, Definition \\ref{morphisms-definition-smooth}.", "Moreover, we may choose each $U_j$ to be open affine in one of the $T_i$." ], "refs": [ "morphisms-definition-smooth" ], "proofs": [ { "contents": [ "Omitted, but see Algebra, Lemma \\ref{algebra-lemma-smooth-syntomic}." ], "refs": [ "algebra-lemma-smooth-syntomic" ], "ref_ids": [ 1195 ] } ], "ref_ids": [ 5564 ] }, { "id": 12462, "type": "theorem", "label": "topologies-lemma-smooth-induced", "categories": [ "topologies" ], "title": "topologies-lemma-smooth-induced", "contents": [ "Let $\\Sch_{smooth}$ be a big smooth site as in", "Definition \\ref{definition-big-smooth-site}.", "Let $T \\in \\Ob(\\Sch_{smooth})$.", "Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary smooth covering of $T$.", "\\begin{enumerate}", "\\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site", "$\\Sch_{smooth}$ which refines $\\{T_i \\to T\\}_{i \\in I}$.", "\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard smooth covering, then", "it is tautologically equivalent to a covering of $\\Sch_{smooth}$.", "\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then", "it is tautologically equivalent to a covering of $\\Sch_{smooth}$.", "\\end{enumerate}" ], "refs": [ "topologies-definition-big-smooth-site" ], "proofs": [ { "contents": [ "For each $i$ choose an affine open covering $T_i = \\bigcup_{j \\in J_i} T_{ij}$", "such that each $T_{ij}$ maps into an affine open subscheme of $T$. By", "Lemma \\ref{lemma-smooth}", "the refinement $\\{T_{ij} \\to T\\}_{i \\in I, j \\in J_i}$ is a smooth covering", "of $T$ as well. Hence we may assume each $T_i$ is affine, and maps into", "an affine open $W_i$ of $T$. Applying", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}", "we see that $W_i$ is isomorphic to an object of $\\Sch_{smooth}$.", "But then $T_i$ as a finite type scheme over $W_i$", "is isomorphic to an object $V_i$ of $\\Sch_{smooth}$ by a second", "application of", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "The covering $\\{V_i \\to T\\}_{i \\in I}$ refines $\\{T_i \\to T\\}_{i \\in I}$", "(because they are isomorphic).", "Moreover, $\\{V_i \\to T\\}_{i \\in I}$ is combinatorially equivalent to a", "covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site", "$\\Sch_{smooth}$ by", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "The covering $\\{U_j \\to T\\}_{j \\in J}$ is a refinement as in (1).", "In the situation of (2), (3) each of the", "schemes $T_i$ is isomorphic to an object of $\\Sch_{smooth}$ by", "Sets, Lemma \\ref{sets-lemma-what-is-in-it},", "and another application of", "Sets, Lemma \\ref{sets-lemma-coverings-site}", "gives what we want." ], "refs": [ "topologies-lemma-smooth", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-coverings-site" ], "ref_ids": [ 12460, 8795, 8795, 8795, 8795, 8800 ] } ], "ref_ids": [ 12533 ] }, { "id": 12463, "type": "theorem", "label": "topologies-lemma-affine-big-site-smooth", "categories": [ "topologies" ], "title": "topologies-lemma-affine-big-site-smooth", "contents": [ "Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big smooth", "site containing $S$.", "The functor", "$(\\textit{Aff}/S)_{smooth} \\to (\\Sch/S)_{smooth}$", "is special cocontinuous and induces an equivalence of topoi from", "$\\Sh((\\textit{Aff}/S)_{smooth})$ to", "$\\Sh((\\Sch/S)_{smooth})$." ], "refs": [], "proofs": [ { "contents": [ "The notion of a special cocontinuous functor is introduced in", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}.", "Thus we have to verify assumptions (1) -- (5) of", "Sites, Lemma \\ref{sites-lemma-equivalence}.", "Denote the inclusion functor", "$u : (\\textit{Aff}/S)_{smooth} \\to (\\Sch/S)_{smooth}$.", "Being cocontinuous just means that any smooth covering of", "$T/S$, $T$ affine, can be refined by a standard smooth covering of $T$.", "This is the content of", "Lemma \\ref{lemma-smooth-affine}.", "Hence (1) holds. We see $u$ is continuous simply because a standard", "smooth covering is a smooth covering. Hence (2) holds.", "Parts (3) and (4) follow immediately from the fact that $u$ is", "fully faithful. And finally condition (5) follows from the", "fact that every scheme has an affine open covering." ], "refs": [ "sites-definition-special-cocontinuous-functor", "sites-lemma-equivalence", "topologies-lemma-smooth-affine" ], "ref_ids": [ 8672, 8578, 12461 ] } ], "ref_ids": [] }, { "id": 12464, "type": "theorem", "label": "topologies-lemma-morphism-big-smooth", "categories": [ "topologies" ], "title": "topologies-lemma-morphism-big-smooth", "contents": [ "Let $\\Sch_{smooth}$ be a big smooth site.", "Let $f : T \\to S$ be a morphism in $\\Sch_{smooth}$.", "The functor", "$$", "u : (\\Sch/T)_{smooth} \\longrightarrow (\\Sch/S)_{smooth},", "\\quad", "V/T \\longmapsto V/S", "$$", "is cocontinuous, and has a continuous right adjoint", "$$", "v : (\\Sch/S)_{smooth} \\longrightarrow (\\Sch/T)_{smooth},", "\\quad", "(U \\to S) \\longmapsto (U \\times_S T \\to T).", "$$", "They induce the same morphism of topoi", "$$", "f_{big} :", "\\Sh((\\Sch/T)_{smooth})", "\\longrightarrow", "\\Sh((\\Sch/S)_{smooth})", "$$", "We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.", "We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.", "Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with", "fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "The functor $u$ is cocontinuous, continuous, and commutes with fibre products", "and equalizers. Hence", "Sites, Lemmas \\ref{sites-lemma-when-shriek} and", "\\ref{sites-lemma-preserve-equalizers}", "apply and we deduce the formula", "for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover,", "the functor $v$ is a right adjoint because given $U/T$ and $V/S$", "we have $\\Mor_S(u(U), V) = \\Mor_T(U, V \\times_S T)$", "as desired. Thus we may apply", "Sites, Lemmas \\ref{sites-lemma-have-functor-other-way} and", "\\ref{sites-lemma-have-functor-other-way-morphism} to get the", "formula for $f_{big, *}$." ], "refs": [ "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers", "sites-lemma-have-functor-other-way", "sites-lemma-have-functor-other-way-morphism" ], "ref_ids": [ 8545, 8546, 8549, 8550 ] } ], "ref_ids": [] }, { "id": 12465, "type": "theorem", "label": "topologies-lemma-zariski-etale-smooth-syntomic", "categories": [ "topologies" ], "title": "topologies-lemma-zariski-etale-smooth-syntomic", "contents": [ "Any smooth covering is a syntomic covering, and a fortiori,", "any \\'etale or Zariski covering is a syntomic covering." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions and the fact that a smooth", "morphism is syntomic, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-syntomic}", "and Lemma \\ref{lemma-zariski-etale-smooth}." ], "refs": [ "morphisms-lemma-smooth-syntomic", "topologies-lemma-zariski-etale-smooth" ], "ref_ids": [ 5329, 12459 ] } ], "ref_ids": [] }, { "id": 12466, "type": "theorem", "label": "topologies-lemma-syntomic", "categories": [ "topologies" ], "title": "topologies-lemma-syntomic", "contents": [ "Let $T$ be a scheme.", "\\begin{enumerate}", "\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$", "is a syntomic covering of $T$.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a syntomic covering and for each", "$i$ we have a syntomic covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then", "$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a syntomic covering.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a syntomic covering", "and $T' \\to T$ is a morphism of schemes then", "$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a syntomic covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12467, "type": "theorem", "label": "topologies-lemma-syntomic-affine", "categories": [ "topologies" ], "title": "topologies-lemma-syntomic-affine", "contents": [ "Let $T$ be an affine scheme.", "Let $\\{T_i \\to T\\}_{i \\in I}$ be a syntomic covering of $T$.", "Then there exists a syntomic covering", "$\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement", "of $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine", "scheme, and such that each morphism $U_j \\to T$ is standard", "syntomic, see Morphisms, Definition \\ref{morphisms-definition-syntomic}.", "Moreover, we may choose each $U_j$ to be open affine in one of the $T_i$." ], "refs": [ "morphisms-definition-syntomic" ], "proofs": [ { "contents": [ "Omitted, but see Algebra, Lemma \\ref{algebra-lemma-syntomic}." ], "refs": [ "algebra-lemma-syntomic" ], "ref_ids": [ 1185 ] } ], "ref_ids": [ 5560 ] }, { "id": 12468, "type": "theorem", "label": "topologies-lemma-syntomic-induced", "categories": [ "topologies" ], "title": "topologies-lemma-syntomic-induced", "contents": [ "Let $\\Sch_{syntomic}$ be a big syntomic site as in", "Definition \\ref{definition-big-syntomic-site}.", "Let $T \\in \\Ob(\\Sch_{syntomic})$.", "Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary syntomic covering of $T$.", "\\begin{enumerate}", "\\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site", "$\\Sch_{syntomic}$ which refines $\\{T_i \\to T\\}_{i \\in I}$.", "\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard syntomic covering, then", "it is tautologically equivalent to a covering in $\\Sch_{syntomic}$.", "\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then", "it is tautologically equivalent to a covering in $\\Sch_{syntomic}$.", "\\end{enumerate}" ], "refs": [ "topologies-definition-big-syntomic-site" ], "proofs": [ { "contents": [ "For each $i$ choose an affine open covering $T_i = \\bigcup_{j \\in J_i} T_{ij}$", "such that each $T_{ij}$ maps into an affine open subscheme of $T$. By", "Lemma \\ref{lemma-syntomic}", "the refinement $\\{T_{ij} \\to T\\}_{i \\in I, j \\in J_i}$ is a syntomic covering", "of $T$ as well. Hence we may assume each $T_i$ is affine, and maps into", "an affine open $W_i$ of $T$. Applying", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}", "we see that $W_i$ is isomorphic to an object of $\\Sch_{syntomic}$.", "But then $T_i$ as a finite type scheme over $W_i$", "is isomorphic to an object $V_i$ of $\\Sch_{syntomic}$ by a second", "application of", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "The covering $\\{V_i \\to T\\}_{i \\in I}$ refines $\\{T_i \\to T\\}_{i \\in I}$", "(because they are isomorphic).", "Moreover, $\\{V_i \\to T\\}_{i \\in I}$ is combinatorially equivalent to a", "covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site", "$\\Sch_{syntomic}$ by", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "The covering $\\{U_j \\to T\\}_{j \\in J}$ is a covering as in (1).", "In the situation of (2), (3) each of the", "schemes $T_i$ is isomorphic to an object of $\\Sch_{syntomic}$ by", "Sets, Lemma \\ref{sets-lemma-what-is-in-it},", "and another application of", "Sets, Lemma \\ref{sets-lemma-coverings-site}", "gives what we want." ], "refs": [ "topologies-lemma-syntomic", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-coverings-site" ], "ref_ids": [ 12466, 8795, 8795, 8795, 8795, 8800 ] } ], "ref_ids": [ 12537 ] }, { "id": 12469, "type": "theorem", "label": "topologies-lemma-affine-big-site-syntomic", "categories": [ "topologies" ], "title": "topologies-lemma-affine-big-site-syntomic", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{syntomic}$ be a big syntomic", "site containing $S$.", "The functor", "$(\\textit{Aff}/S)_{syntomic} \\to (\\Sch/S)_{syntomic}$", "is special cocontinuous and induces an equivalence of topoi from", "$\\Sh((\\textit{Aff}/S)_{syntomic})$ to", "$\\Sh((\\Sch/S)_{syntomic})$." ], "refs": [], "proofs": [ { "contents": [ "The notion of a special cocontinuous functor is introduced in", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}.", "Thus we have to verify assumptions (1) -- (5) of", "Sites, Lemma \\ref{sites-lemma-equivalence}.", "Denote the inclusion functor", "$u : (\\textit{Aff}/S)_{syntomic} \\to (\\Sch/S)_{syntomic}$.", "Being cocontinuous just means that any syntomic covering of", "$T/S$, $T$ affine, can be refined by a standard syntomic covering of $T$.", "This is the content of", "Lemma \\ref{lemma-syntomic-affine}.", "Hence (1) holds. We see $u$ is continuous simply because a standard", "syntomic covering is a syntomic covering. Hence (2) holds.", "Parts (3) and (4) follow immediately from the fact that $u$ is", "fully faithful. And finally condition (5) follows from the", "fact that every scheme has an affine open covering." ], "refs": [ "sites-definition-special-cocontinuous-functor", "sites-lemma-equivalence", "topologies-lemma-syntomic-affine" ], "ref_ids": [ 8672, 8578, 12467 ] } ], "ref_ids": [] }, { "id": 12470, "type": "theorem", "label": "topologies-lemma-morphism-big-syntomic", "categories": [ "topologies" ], "title": "topologies-lemma-morphism-big-syntomic", "contents": [ "Let $\\Sch_{syntomic}$ be a big syntomic site.", "Let $f : T \\to S$ be a morphism in $\\Sch_{syntomic}$.", "The functor", "$$", "u : (\\Sch/T)_{syntomic} \\longrightarrow (\\Sch/S)_{syntomic},", "\\quad", "V/T \\longmapsto V/S", "$$", "is cocontinuous, and has a continuous right adjoint", "$$", "v : (\\Sch/S)_{syntomic} \\longrightarrow (\\Sch/T)_{syntomic},", "\\quad", "(U \\to S) \\longmapsto (U \\times_S T \\to T).", "$$", "They induce the same morphism of topoi", "$$", "f_{big} :", "\\Sh((\\Sch/T)_{syntomic})", "\\longrightarrow", "\\Sh((\\Sch/S)_{syntomic})", "$$", "We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.", "We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.", "Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with", "fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "The functor $u$ is cocontinuous, continuous, and commutes with fibre products", "and equalizers. Hence", "Sites, Lemmas \\ref{sites-lemma-when-shriek} and", "\\ref{sites-lemma-preserve-equalizers}", "apply and we deduce the formula", "for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover,", "the functor $v$ is a right adjoint because given $U/T$ and $V/S$", "we have $\\Mor_S(u(U), V) = \\Mor_T(U, V \\times_S T)$", "as desired. Thus we may apply", "Sites, Lemmas \\ref{sites-lemma-have-functor-other-way} and", "\\ref{sites-lemma-have-functor-other-way-morphism} to get the", "formula for $f_{big, *}$." ], "refs": [ "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers", "sites-lemma-have-functor-other-way", "sites-lemma-have-functor-other-way-morphism" ], "ref_ids": [ 8545, 8546, 8549, 8550 ] } ], "ref_ids": [] }, { "id": 12471, "type": "theorem", "label": "topologies-lemma-zariski-etale-smooth-syntomic-fppf", "categories": [ "topologies" ], "title": "topologies-lemma-zariski-etale-smooth-syntomic-fppf", "contents": [ "Any syntomic covering is an fppf covering, and a fortiori,", "any smooth, \\'etale, or Zariski covering is an fppf covering." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions, the fact that a syntomic morphism", "is flat and locally of finite presentation, see", "Morphisms, Lemmas", "\\ref{morphisms-lemma-syntomic-locally-finite-presentation} and", "\\ref{morphisms-lemma-syntomic-flat},", "and", "Lemma \\ref{lemma-zariski-etale-smooth-syntomic}." ], "refs": [ "morphisms-lemma-syntomic-locally-finite-presentation", "morphisms-lemma-syntomic-flat", "topologies-lemma-zariski-etale-smooth-syntomic" ], "ref_ids": [ 5293, 5294, 12465 ] } ], "ref_ids": [] }, { "id": 12472, "type": "theorem", "label": "topologies-lemma-fppf", "categories": [ "topologies" ], "title": "topologies-lemma-fppf", "contents": [ "Let $T$ be a scheme.", "\\begin{enumerate}", "\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$", "is an fppf covering of $T$.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an fppf covering and for each", "$i$ we have an fppf covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then", "$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is an fppf covering.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an fppf covering", "and $T' \\to T$ is a morphism of schemes then", "$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is an fppf covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first assertion is clear.", "The second follows as the composition of flat morphisms is flat", "(see Morphisms, Lemma \\ref{morphisms-lemma-composition-flat})", "and the composition of morphisms of finite presentation is", "of finite presentation", "(see Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-presentation}).", "The third follows as the base change of a flat morphism is flat", "(see Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat})", "and the base change of a morphism of finite presentation is", "of finite presentation", "(see Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-presentation}).", "Moreover, the base change of a surjective family of morphisms is surjective", "(proof omitted)." ], "refs": [ "morphisms-lemma-composition-flat", "morphisms-lemma-composition-finite-presentation", "morphisms-lemma-base-change-flat", "morphisms-lemma-base-change-finite-presentation" ], "ref_ids": [ 5263, 5239, 5265, 5240 ] } ], "ref_ids": [] }, { "id": 12473, "type": "theorem", "label": "topologies-lemma-fppf-affine", "categories": [ "topologies" ], "title": "topologies-lemma-fppf-affine", "contents": [ "Let $T$ be an affine scheme.", "Let $\\{T_i \\to T\\}_{i \\in I}$ be an fppf covering of $T$.", "Then there exists an fppf covering", "$\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement", "of $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine", "scheme. Moreover, we may choose each $U_j$ to be open affine", "in one of the $T_i$." ], "refs": [], "proofs": [ { "contents": [ "This follows directly from the definitions using that a", "morphism which is flat and locally of finite presentation is open,", "see Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}." ], "refs": [ "morphisms-lemma-fppf-open" ], "ref_ids": [ 5267 ] } ], "ref_ids": [] }, { "id": 12474, "type": "theorem", "label": "topologies-lemma-fppf-induced", "categories": [ "topologies" ], "title": "topologies-lemma-fppf-induced", "contents": [ "Let $\\Sch_{fppf}$ be a big fppf site as in", "Definition \\ref{definition-big-fppf-site}.", "Let $T \\in \\Ob(\\Sch_{fppf})$.", "Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary fppf covering of $T$.", "\\begin{enumerate}", "\\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site", "$\\Sch_{fppf}$ which refines $\\{T_i \\to T\\}_{i \\in I}$.", "\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard fppf covering, then", "it is tautologically equivalent to a covering of $\\Sch_{fppf}$.", "\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then", "it is tautologically equivalent to a covering of $\\Sch_{fppf}$.", "\\end{enumerate}" ], "refs": [ "topologies-definition-big-fppf-site" ], "proofs": [ { "contents": [ "For each $i$ choose an affine open covering $T_i = \\bigcup_{j \\in J_i} T_{ij}$", "such that each $T_{ij}$ maps into an affine open subscheme of $T$. By", "Lemma \\ref{lemma-fppf}", "the refinement $\\{T_{ij} \\to T\\}_{i \\in I, j \\in J_i}$ is an fppf covering", "of $T$ as well. Hence we may assume each $T_i$ is affine, and maps into", "an affine open $W_i$ of $T$. Applying", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}", "we see that $W_i$ is isomorphic to an object of $\\Sch_{fppf}$.", "But then $T_i$ as a finite type scheme over $W_i$", "is isomorphic to an object $V_i$ of $\\Sch_{fppf}$ by a second", "application of", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "The covering $\\{V_i \\to T\\}_{i \\in I}$ refines $\\{T_i \\to T\\}_{i \\in I}$", "(because they are isomorphic).", "Moreover, $\\{V_i \\to T\\}_{i \\in I}$ is combinatorially equivalent to a", "covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site", "$\\Sch_{fppf}$ by", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "The covering $\\{U_j \\to T\\}_{j \\in J}$ is a refinement as in (1).", "In the situation of (2), (3) each of the", "schemes $T_i$ is isomorphic to an object of $\\Sch_{fppf}$ by", "Sets, Lemma \\ref{sets-lemma-what-is-in-it},", "and another application of", "Sets, Lemma \\ref{sets-lemma-coverings-site}", "gives what we want." ], "refs": [ "topologies-lemma-fppf", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-coverings-site" ], "ref_ids": [ 12472, 8795, 8795, 8795, 8795, 8800 ] } ], "ref_ids": [ 12541 ] }, { "id": 12475, "type": "theorem", "label": "topologies-lemma-verify-site-fppf", "categories": [ "topologies" ], "title": "topologies-lemma-verify-site-fppf", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{fppf}$ be a big fppf", "site containing $S$. Then $(\\textit{Aff}/S)_{fppf}$ is a site." ], "refs": [], "proofs": [ { "contents": [ "Let us show that $(\\textit{Aff}/S)_{fppf}$ is a site.", "Reasoning as in the proof of Lemma \\ref{lemma-verify-site-etale}", "it suffices to show that the collection", "of standard fppf coverings of affines satisfies properties", "(1), (2) and (3) of", "Sites, Definition \\ref{sites-definition-site}.", "This is clear since for example, given a standard fppf", "covering $\\{T_i \\to T\\}_{i\\in I}$ and for each", "$i$ we have a standard fppf covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then", "$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a standard fppf covering", "because $\\bigcup_{i\\in I} J_i$ is finite and each $T_{ij}$ is affine." ], "refs": [ "topologies-lemma-verify-site-etale", "sites-definition-site" ], "ref_ids": [ 12449, 8652 ] } ], "ref_ids": [] }, { "id": 12476, "type": "theorem", "label": "topologies-lemma-fibre-products-fppf", "categories": [ "topologies" ], "title": "topologies-lemma-fibre-products-fppf", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{fppf}$ be a big fppf", "site containing $S$. The underlying categories of the sites", "$\\Sch_{fppf}$, $(\\Sch/S)_{fppf}$,", "and $(\\textit{Aff}/S)_{fppf}$ have fibre products.", "In each case the obvious functor into the category $\\Sch$ of", "all schemes commutes with taking fibre products. The category", "$(\\Sch/S)_{fppf}$ has a final object, namely $S/S$." ], "refs": [], "proofs": [ { "contents": [ "For $\\Sch_{fppf}$ it is true by construction, see", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "Suppose we have $U \\to S$, $V \\to U$, $W \\to U$ morphisms", "of schemes with $U, V, W \\in \\Ob(\\Sch_{fppf})$.", "The fibre product $V \\times_U W$ in $\\Sch_{fppf}$", "is a fibre product in $\\Sch$ and", "is the fibre product of $V/S$ with $W/S$ over $U/S$ in", "the category of all schemes over $S$, and hence also a", "fibre product in $(\\Sch/S)_{fppf}$.", "This proves the result for $(\\Sch/S)_{fppf}$.", "If $U, V, W$ are affine, so is $V \\times_U W$ and hence the", "result for $(\\textit{Aff}/S)_{fppf}$." ], "refs": [ "sets-lemma-what-is-in-it" ], "ref_ids": [ 8795 ] } ], "ref_ids": [] }, { "id": 12477, "type": "theorem", "label": "topologies-lemma-affine-big-site-fppf", "categories": [ "topologies" ], "title": "topologies-lemma-affine-big-site-fppf", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{fppf}$ be a big fppf", "site containing $S$.", "The functor $(\\textit{Aff}/S)_{fppf} \\to (\\Sch/S)_{fppf}$", "is cocontinuous and induces an equivalence of topoi from", "$\\Sh((\\textit{Aff}/S)_{fppf})$ to", "$\\Sh((\\Sch/S)_{fppf})$." ], "refs": [], "proofs": [ { "contents": [ "The notion of a special cocontinuous functor is introduced in", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}.", "Thus we have to verify assumptions (1) -- (5) of", "Sites, Lemma \\ref{sites-lemma-equivalence}.", "Denote the inclusion functor", "$u : (\\textit{Aff}/S)_{fppf} \\to (\\Sch/S)_{fppf}$.", "Being cocontinuous just means that any fppf covering of", "$T/S$, $T$ affine, can be refined by a standard fppf covering of $T$.", "This is the content of", "Lemma \\ref{lemma-fppf-affine}.", "Hence (1) holds. We see $u$ is continuous simply because a standard", "fppf covering is a fppf covering. Hence (2) holds.", "Parts (3) and (4) follow immediately from the fact that $u$ is", "fully faithful. And finally condition (5) follows from the", "fact that every scheme has an affine open covering." ], "refs": [ "sites-definition-special-cocontinuous-functor", "sites-lemma-equivalence", "topologies-lemma-fppf-affine" ], "ref_ids": [ 8672, 8578, 12473 ] } ], "ref_ids": [] }, { "id": 12478, "type": "theorem", "label": "topologies-lemma-morphism-big-fppf", "categories": [ "topologies" ], "title": "topologies-lemma-morphism-big-fppf", "contents": [ "Let $\\Sch_{fppf}$ be a big fppf site.", "Let $f : T \\to S$ be a morphism in $\\Sch_{fppf}$.", "The functor", "$$", "u : (\\Sch/T)_{fppf} \\longrightarrow (\\Sch/S)_{fppf},", "\\quad", "V/T \\longmapsto V/S", "$$", "is cocontinuous, and has a continuous right adjoint", "$$", "v : (\\Sch/S)_{fppf} \\longrightarrow (\\Sch/T)_{fppf},", "\\quad", "(U \\to S) \\longmapsto (U \\times_S T \\to T).", "$$", "They induce the same morphism of topoi", "$$", "f_{big} :", "\\Sh((\\Sch/T)_{fppf})", "\\longrightarrow", "\\Sh((\\Sch/S)_{fppf})", "$$", "We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.", "We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.", "Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with", "fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "The functor $u$ is cocontinuous, continuous, and commutes with fibre products", "and equalizers. Hence", "Sites, Lemmas \\ref{sites-lemma-when-shriek} and", "\\ref{sites-lemma-preserve-equalizers}", "apply and we deduce the formula", "for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover,", "the functor $v$ is a right adjoint because given $U/T$ and $V/S$", "we have $\\Mor_S(u(U), V) = \\Mor_T(U, V \\times_S T)$", "as desired. Thus we may apply", "Sites, Lemmas \\ref{sites-lemma-have-functor-other-way} and", "\\ref{sites-lemma-have-functor-other-way-morphism} to get the", "formula for $f_{big, *}$." ], "refs": [ "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers", "sites-lemma-have-functor-other-way", "sites-lemma-have-functor-other-way-morphism" ], "ref_ids": [ 8545, 8546, 8549, 8550 ] } ], "ref_ids": [] }, { "id": 12479, "type": "theorem", "label": "topologies-lemma-composition-fppf", "categories": [ "topologies" ], "title": "topologies-lemma-composition-fppf", "contents": [ "Given schemes $X$, $Y$, $Y$ in $(\\Sch/S)_{fppf}$", "and morphisms $f : X \\to Y$, $g : Y \\to Z$ we have", "$g_{big} \\circ f_{big} = (g \\circ f)_{big}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the simple description of pushforward", "and pullback for the functors on the big sites from", "Lemma \\ref{lemma-morphism-big-fppf}." ], "refs": [ "topologies-lemma-morphism-big-fppf" ], "ref_ids": [ 12478 ] } ], "ref_ids": [] }, { "id": 12480, "type": "theorem", "label": "topologies-lemma-base-change-standard-ph", "categories": [ "topologies" ], "title": "topologies-lemma-base-change-standard-ph", "contents": [ "Let $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ be a standard ph covering.", "Let $T' \\to T$ be a morphism of affine schemes.", "Then $\\{U_j \\times_T T' \\to T'\\}_{j = 1, \\ldots, m}$ is a", "standard ph covering." ], "refs": [], "proofs": [ { "contents": [ "Let $f : U \\to T$ be proper surjective and let an affine open covering", "$U = \\bigcup_{j = 1, \\ldots, m} U_j$ be given as in", "Definition \\ref{definition-standard-ph-covering}.", "Then $U \\times_T T' \\to T'$ is proper surjective", "(Morphisms, Lemmas \\ref{morphisms-lemma-base-change-surjective} and", "\\ref{morphisms-lemma-base-change-proper}).", "Also, $U \\times_T T' = \\bigcup_{j = 1, \\ldots, m} U_j \\times_T T'$", "is an affine open covering.", "This concludes the proof." ], "refs": [ "topologies-definition-standard-ph-covering", "morphisms-lemma-base-change-surjective", "morphisms-lemma-base-change-proper" ], "ref_ids": [ 12543, 5165, 5409 ] } ], "ref_ids": [] }, { "id": 12481, "type": "theorem", "label": "topologies-lemma-refine-by-standard-ph", "categories": [ "topologies" ], "title": "topologies-lemma-refine-by-standard-ph", "contents": [ "Let $T$ be an affine scheme. Each of the following types of families", "of maps with target $T$ has a refinement by a standard ph covering:", "\\begin{enumerate}", "\\item any Zariski open covering of $T$,", "\\item $\\{W_{ji} \\to T\\}_{j = 1, \\ldots, m, i = 1, \\ldots n_j}$", "where $\\{W_{ji} \\to U_j\\}_{i = 1, \\ldots, n_j}$", "and $\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ are standard ph coverings.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from the fact that any Zariski open covering of $T$", "can be refined by a finite affine open covering.", "\\medskip\\noindent", "Proof of (3). Choose $U \\to T$ proper surjective and", "$U = \\bigcup_{j = 1, \\ldots, m} U_j$ as in", "Definition \\ref{definition-standard-ph-covering}.", "Choose $W_j \\to U_j$ proper surjective and $W_j = \\bigcup W_{ji}$", "as in Definition \\ref{definition-standard-ph-covering}.", "By Chow's lemma (Limits, Lemma \\ref{limits-lemma-chow-finite-type})", "we can find $W'_j \\to W_j$ proper surjective", "and closed immersions $W'_j \\to \\mathbf{P}^{e_j}_{U_j}$.", "Thus, after replacing $W_j$ by $W'_j$ and $W_j = \\bigcup W_{ji}$", "by a suitable affine open covering of $W'_j$, we may assume", "there is a closed immersion $W_j \\subset \\mathbf{P}^{e_j}_{U_j}$", "for all $j = 1, \\ldots, m$.", "\\medskip\\noindent", "Let $\\overline{W}_j \\subset \\mathbf{P}^{e_j}_U$ be the", "scheme theoretic closure of $W_j$. Then $W_j \\subset \\overline{W}_j$", "is an open subscheme; in fact $W_j$ is the inverse image of", "$U_j \\subset U$ under the morphism $\\overline{W}_j \\to U$.", "(To see this use that $W_j \\to \\mathbf{P}^{e_j}_U$ is quasi-compact", "and hence formation of the scheme theoretic image commutes", "with restriction to opens, see", "Morphisms, Section \\ref{morphisms-section-scheme-theoretic-image}.)", "Let $Z_j = U \\setminus U_j$", "with reduced induced closed subscheme structure.", "Then", "$$", "V_j = \\overline{W}_j \\amalg Z_j \\to U", "$$", "is proper surjective and the open subscheme $W_j \\subset V_j$", "is the inverse image of $U_j$. Hence for $v \\in V_j$, $v \\not \\in W_j$", "we can pick an affine open neighbourhood $v \\in V_{j, v} \\subset V_j$", "which maps into $U_{j'}$ for some $1 \\leq j' \\leq m$.", "\\medskip\\noindent", "To finish the proof we consider the proper surjective morphism", "$$", "V = V_1 \\times_U V_2 \\times_U \\ldots \\times_U V_m", "\\longrightarrow U \\longrightarrow T", "$$", "and the covering of $V$ by the affine opens", "$$", "V_{1, v_1} \\times_U \\ldots \\times_U V_{j - 1, v_{j - 1}}", "\\times_U W_{j i} \\times_U", "V_{j + 1, v_{j + 1}} \\times_U \\ldots \\times_U V_{m, v_m}", "$$", "These do indeed form a covering, because each point of $U$ is", "in some $U_j$ and the inverse image of $U_j$ in $V$ is equal to", "$V_1 \\times \\ldots \\times V_{j - 1} \\times W_j \\times V_{j + 1} \\times", "\\ldots \\times V_m$. Observe that the morphism from", "the affine open displayed above to $T$ factors through $W_{ji}$", "thus we obtain a refinement. Finally, we only need a finite number", "of these affine opens as $V$ is quasi-compact (as a scheme proper", "over the affine scheme $T$)." ], "refs": [ "topologies-definition-standard-ph-covering", "topologies-definition-standard-ph-covering", "limits-lemma-chow-finite-type" ], "ref_ids": [ 12543, 12543, 15087 ] } ], "ref_ids": [] }, { "id": 12482, "type": "theorem", "label": "topologies-lemma-zariski-ph", "categories": [ "topologies" ], "title": "topologies-lemma-zariski-ph", "contents": [ "A Zariski covering is a ph covering\\footnote{We will see", "in More on Morphisms, Lemma \\ref{more-morphisms-lemma-fppf-ph} that", "fppf coverings (and hence syntomic, smooth, or \\'etale coverings)", "are ph coverings as well.}." ], "refs": [ "more-morphisms-lemma-fppf-ph" ], "proofs": [ { "contents": [ "This is true because a Zariski covering of an affine scheme", "can be refined by a standard ph covering by", "Lemma \\ref{lemma-refine-by-standard-ph}." ], "refs": [ "topologies-lemma-refine-by-standard-ph" ], "ref_ids": [ 12481 ] } ], "ref_ids": [ 13927 ] }, { "id": 12483, "type": "theorem", "label": "topologies-lemma-surjective-proper-ph", "categories": [ "topologies" ], "title": "topologies-lemma-surjective-proper-ph", "contents": [ "Let $f : Y \\to X$ be a surjective proper morphism of schemes.", "Then $\\{Y \\to X\\}$ is a ph covering." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12484, "type": "theorem", "label": "topologies-lemma-refine-by-ph", "categories": [ "topologies" ], "title": "topologies-lemma-refine-by-ph", "contents": [ "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family", "of morphisms such that $f_i$ is locally of finite type for all $i$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\{T_i \\to T\\}_{i \\in I}$ is a ph covering,", "\\item there is a ph covering which refines $\\{T_i \\to T\\}_{i \\in I}$, and", "\\item $\\{\\coprod_{i \\in I} T_i \\to T\\}$ is a ph covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) follows immediately from", "Definition \\ref{definition-ph-covering}", "and the fact that a refinement of a refinement is a refinement.", "Because of the equivalence of (1) and (2) and since", "$\\{T_i \\to T\\}_{i \\in I}$ refines $\\{\\coprod_{i \\in I} T_i \\to T\\}$", "we see that (1) implies (3). Finally, assume (3) holds.", "Let $U \\subset T$ be an affine open and let", "$\\{U_j \\to U\\}_{j = 1, \\ldots, m}$ be a standard ph covering", "which refines $\\{U \\times_T \\coprod_{i \\in I} T_i \\to U\\}$.", "This means that for each $j$ we have a morphism", "$$", "h_j :", "U_j", "\\longrightarrow", "U \\times_T \\coprod\\nolimits_{i \\in I} T_i =", "\\coprod\\nolimits_{i \\in I} U \\times_T T_i", "$$", "over $U$. Since $U_j$ is quasi-compact, we get", "disjoint union decompositions $U_j = \\coprod_{i \\in I} U_{j, i}$", "by open and closed subschemes almost all of which are empty", "such that $h_j|_{U_{j, i}}$ maps $U_{j, i}$ into $U \\times_T T_i$.", "It follows that", "$$", "\\{U_{j, i} \\to U\\}_{j = 1, \\ldots, m,\\ i \\in I,\\ U_{j, i} \\not = \\emptyset}", "$$", "is a standard ph covering (small detail omitted) refining", "$\\{U \\times_T T_i \\to U\\}_{i \\in I}$. Thus (1) holds." ], "refs": [ "topologies-definition-ph-covering" ], "ref_ids": [ 12544 ] } ], "ref_ids": [] }, { "id": 12485, "type": "theorem", "label": "topologies-lemma-ph", "categories": [ "topologies" ], "title": "topologies-lemma-ph", "contents": [ "Let $T$ be a scheme.", "\\begin{enumerate}", "\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$", "is a ph covering of $T$.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a ph covering and for each", "$i$ we have a ph covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then", "$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a ph covering.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a ph covering", "and $T' \\to T$ is a morphism of schemes then", "$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a ph covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assertion (1) is clear.", "\\medskip\\noindent", "Proof of (3). The base change $T_i \\times_T T' \\to T'$ is", "locally of finite type by", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-type}.", "hence we only need to check the condition on affine opens.", "Let $U' \\subset T'$ be an affine open subscheme.", "Since $U'$ is quasi-compact we can find a finite affine open", "covering $U' = U'_1 \\cup \\ldots \\cup U'$ such that", "$U'_j \\to T$ maps into an affine open $U_j \\subset T$.", "Choose a standard ph covering $\\{U_{jl} \\to U_j\\}_{l = 1, \\ldots, n_j}$", "refining $\\{T_i \\times_T U_j \\to U_j\\}$.", "By Lemma \\ref{lemma-base-change-standard-ph}", "the base change $\\{U_{jl} \\times_{U_j} U'_j \\to U'_j\\}$", "is a standard ph covering. Note that $\\{U'_j \\to U'\\}$", "is a standard ph covering as well.", "By Lemma \\ref{lemma-refine-by-standard-ph} ", "the family $\\{U_{jl} \\times_{U_j} U'_j \\to U'\\}$", "can be refined by a standard ph covering. Since", "$\\{U_{jl} \\times_{U_j} U'_j \\to U'\\}$ refines $\\{T_i \\times_T U' \\to U'\\}$", "we conclude.", "\\medskip\\noindent", "Proof of (2). Composition preserves being locally of finite type,", "see Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-type}.", "Hence we only need to check the condition on affine opens.", "Let $U \\subset T$ be affine open. First we pick a standard ph covering", "$\\{U_k \\to U\\}_{k = 1, \\ldots, m}$ refining", "$\\{T_i \\times_T U \\to U\\}$.", "Say the refinement is given by morphisms $U_k \\to T_{i_k}$ over $T$.", "Then", "$$", "\\{T_{i_kj} \\times_{T_{i_k}} U_k \\to U_k\\}_{j \\in J_{i_k}}", "$$", "is a ph covering by part (3). As $U_k$ is affine,", "we can find a standard ph covering", "$\\{U_{ka} \\to U_k\\}_{a = 1, \\ldots, b_k}$ refining this family.", "Then we apply Lemma \\ref{lemma-refine-by-standard-ph}", "to see that $\\{U_{ka} \\to U\\}$ can be refined by a", "standard ph covering. Since $\\{U_{ka} \\to U\\}$", "refines $\\{T_{ij} \\times_T U \\to U\\}$ this finishes the proof." ], "refs": [ "morphisms-lemma-base-change-finite-type", "topologies-lemma-base-change-standard-ph", "topologies-lemma-refine-by-standard-ph", "morphisms-lemma-composition-finite-type", "topologies-lemma-refine-by-standard-ph" ], "ref_ids": [ 5200, 12480, 12481, 5199, 12481 ] } ], "ref_ids": [] }, { "id": 12486, "type": "theorem", "label": "topologies-lemma-ph-induced", "categories": [ "topologies" ], "title": "topologies-lemma-ph-induced", "contents": [ "Let $\\Sch_{ph}$ be a big ph site as in", "Definition \\ref{definition-big-ph-site}.", "Let $T \\in \\Ob(\\Sch_{ph})$.", "Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary ph covering of $T$.", "\\begin{enumerate}", "\\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site", "$\\Sch_{ph}$ which refines $\\{T_i \\to T\\}_{i \\in I}$.", "\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard ph covering, then", "it is tautologically equivalent to a covering of $\\Sch_{ph}$.", "\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then", "it is tautologically equivalent to a covering of $\\Sch_{ph}$.", "\\end{enumerate}" ], "refs": [ "topologies-definition-big-ph-site" ], "proofs": [ { "contents": [ "For each $i$ choose an affine open covering $T_i = \\bigcup_{j \\in J_i} T_{ij}$", "such that each $T_{ij}$ maps into an affine open subscheme of $T$. By", "Lemmas \\ref{lemma-zariski-ph} and \\ref{lemma-ph}", "the refinement $\\{T_{ij} \\to T\\}_{i \\in I, j \\in J_i}$ is a ph covering", "of $T$ as well. Hence we may assume each $T_i$ is affine, and maps into", "an affine open $W_i$ of $T$. Applying", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}", "we see that $W_i$ is isomorphic to an object of $\\Sch_{ph}$.", "But then $T_i$ as a finite type scheme over $W_i$", "is isomorphic to an object $V_i$ of $\\Sch_{ph}$ by a second", "application of", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "The covering $\\{V_i \\to T\\}_{i \\in I}$ refines $\\{T_i \\to T\\}_{i \\in I}$", "(because they are isomorphic).", "Moreover, $\\{V_i \\to T\\}_{i \\in I}$ is combinatorially equivalent to a", "covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site", "$\\Sch_{ph}$ by", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "The covering $\\{U_j \\to T\\}_{j \\in J}$ is a refinement as in (1).", "In the situation of (2), (3) each of the", "schemes $T_i$ is isomorphic to an object of $\\Sch_{ph}$ by", "Sets, Lemma \\ref{sets-lemma-what-is-in-it},", "and another application of", "Sets, Lemma \\ref{sets-lemma-coverings-site}", "gives what we want." ], "refs": [ "topologies-lemma-zariski-ph", "topologies-lemma-ph", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "sets-lemma-coverings-site" ], "ref_ids": [ 12482, 12485, 8795, 8795, 8795, 8795, 8800 ] } ], "ref_ids": [ 12545 ] }, { "id": 12487, "type": "theorem", "label": "topologies-lemma-verify-site-ph", "categories": [ "topologies" ], "title": "topologies-lemma-verify-site-ph", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{ph}$ be a big ph", "site containing $S$. Then $(\\textit{Aff}/S)_{ph}$ is a site." ], "refs": [], "proofs": [ { "contents": [ "Reasoning as in the proof of Lemma \\ref{lemma-verify-site-etale}", "it suffices to show that the collection of finite ph coverings", "$\\{U_i \\to U\\}$ with $U$, $U_i$ affine satisfies properties", "(1), (2) and (3) of", "Sites, Definition \\ref{sites-definition-site}.", "This is clear since for example, given a finite ph covering", "$\\{T_i \\to T\\}_{i\\in I}$ with $T_i, T$ affine, and for each", "$i$ a finite ph covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$ with $T_{ij}$", "affine , then $\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a ph covering", "(Lemma \\ref{lemma-ph}), $\\bigcup_{i\\in I} J_i$ is finite and", "each $T_{ij}$ is affine." ], "refs": [ "topologies-lemma-verify-site-etale", "sites-definition-site", "topologies-lemma-ph" ], "ref_ids": [ 12449, 8652, 12485 ] } ], "ref_ids": [] }, { "id": 12488, "type": "theorem", "label": "topologies-lemma-fibre-products-ph", "categories": [ "topologies" ], "title": "topologies-lemma-fibre-products-ph", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{ph}$ be a big ph", "site containing $S$. The underlying categories of the sites", "$\\Sch_{ph}$, $(\\Sch/S)_{ph}$, and $(\\textit{Aff}/S)_{ph}$ have fibre products.", "In each case the obvious functor into the category $\\Sch$ of", "all schemes commutes with taking fibre products. The category", "$(\\Sch/S)_{ph}$ has a final object, namely $S/S$." ], "refs": [], "proofs": [ { "contents": [ "For $\\Sch_{ph}$ it is true by construction, see", "Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "Suppose we have $U \\to S$, $V \\to U$, $W \\to U$ morphisms", "of schemes with $U, V, W \\in \\Ob(\\Sch_{ph})$.", "The fibre product $V \\times_U W$ in $\\Sch_{ph}$", "is a fibre product in $\\Sch$ and", "is the fibre product of $V/S$ with $W/S$ over $U/S$ in", "the category of all schemes over $S$, and hence also a", "fibre product in $(\\Sch/S)_{ph}$.", "This proves the result for $(\\Sch/S)_{ph}$.", "If $U, V, W$ are affine, so is $V \\times_U W$ and hence the", "result for $(\\textit{Aff}/S)_{ph}$." ], "refs": [ "sets-lemma-what-is-in-it" ], "ref_ids": [ 8795 ] } ], "ref_ids": [] }, { "id": 12489, "type": "theorem", "label": "topologies-lemma-affine-big-site-ph", "categories": [ "topologies" ], "title": "topologies-lemma-affine-big-site-ph", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{ph}$ be a big ph", "site containing $S$.", "The functor $(\\textit{Aff}/S)_{ph} \\to (\\Sch/S)_{ph}$", "is cocontinuous and induces an equivalence of topoi from", "$\\Sh((\\textit{Aff}/S)_{ph})$ to", "$\\Sh((\\Sch/S)_{ph})$." ], "refs": [], "proofs": [ { "contents": [ "The notion of a special cocontinuous functor is introduced in", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}.", "Thus we have to verify assumptions (1) -- (5) of", "Sites, Lemma \\ref{sites-lemma-equivalence}.", "Denote the inclusion functor", "$u : (\\textit{Aff}/S)_{ph} \\to (\\Sch/S)_{ph}$.", "Being cocontinuous follows because any ph covering of", "$T/S$, $T$ affine, can be refined by a standard ph covering of $T$", "by definition. Hence (1) holds. We see $u$ is continuous simply", "because a finite ph covering of an affine by affines is a ph covering.", "Hence (2) holds.", "Parts (3) and (4) follow immediately from the fact that $u$ is", "fully faithful. And finally condition (5) follows from the", "fact that every scheme has an affine open covering (which is", "a ph covering)." ], "refs": [ "sites-definition-special-cocontinuous-functor", "sites-lemma-equivalence" ], "ref_ids": [ 8672, 8578 ] } ], "ref_ids": [] }, { "id": 12490, "type": "theorem", "label": "topologies-lemma-characterize-sheaf", "categories": [ "topologies" ], "title": "topologies-lemma-characterize-sheaf", "contents": [ "Let $\\mathcal{F}$ be a presheaf on $(\\Sch/S)_{ph}$.", "Then $\\mathcal{F}$ is a sheaf if and only if", "\\begin{enumerate}", "\\item $\\mathcal{F}$ satisfies the sheaf condition for", "Zariski coverings, and", "\\item if $f : V \\to U$ is proper surjective, then", "$\\mathcal{F}(U)$ maps bijectively to the equalizer", "of the two maps $\\mathcal{F}(V) \\to \\mathcal{F}(V \\times_U V)$.", "\\end{enumerate}", "Moreover, in the presence of (1) property (2) is equivalent to property", "\\begin{enumerate}", "\\item[(2')] the sheaf property for $\\{V \\to U\\}$ as in (2) with $U$ affine.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will show that if (1) and (2) hold, then $\\mathcal{F}$ is sheaf.", "Let $\\{T_i \\to T\\}$ be a ph covering, i.e., a covering in $(\\Sch/S)_{ph}$.", "We will verify the sheaf condition for this covering.", "Let $s_i \\in \\mathcal{F}(T_i)$ be sections which restrict to the same", "section over $T_i \\times_T T_{i'}$. We will show that there exists a", "unique section $s \\in \\mathcal{F}(T)$ restricting to $s_i$ over $T_i$.", "Let $T = \\bigcup U_j$ be an affine open covering.", "By property (1) it suffices to produce sections $s_j \\in \\mathcal{F}(U_j)$", "which agree on $U_j \\cap U_{j'}$ in order to produce $s$.", "Consider the ph coverings $\\{T_i \\times_T U_j \\to U_j\\}$.", "Then $s_{ji} = s_i|_{T_i \\times_T U_j}$ are sections agreeing", "over $(T_i \\times_T U_j) \\times_{U_j} (T_{i'} \\times_T U_j)$.", "Choose a proper surjective morphism $V_j \\to U_j$ and a finite affine", "open covering $V_j = \\bigcup V_{jk}$ such that the standard ph covering", "$\\{V_{jk} \\to U_j\\}$ refines $\\{T_i \\times_T U_j \\to U_j\\}$.", "If $s_{jk} \\in \\mathcal{F}(V_{jk})$", "denotes the pullback of $s_{ji}$ to $V_{jk}$ by the", "implied morphisms, then we find that $s_{jk}$ glue to a section", "$s'_j \\in \\mathcal{F}(V_j)$. Using the agreement on overlaps", "once more, we find that $s'_j$ is in the equalizer of the two", "maps $\\mathcal{F}(V_j) \\to \\mathcal{F}(V_j \\times_{U_j} V_j)$.", "Hence by (2) we find that $s'_j$ comes from a unique section", "$s_j \\in \\mathcal{F}(U_j)$. We omit the verification that these", "sections $s_j$ have all the desired properties.", "\\medskip\\noindent", "Proof of the equivalence of (2) and (2') in the presence of (1).", "Suppose $V \\to U$ is a morphism of $(\\Sch/S)_{ph}$ which is", "proper and surjective. Choose an", "affine open covering $U = \\bigcup U_i$ and set $V_i = V \\times_U U_i$.", "Then we see that $\\mathcal{F}(U) \\to \\mathcal{F}(V)$", "is injective because we know $\\mathcal{F}(U_i) \\to \\mathcal{F}(V_i)$", "is injective by (2') and we know $\\mathcal{F}(U) \\to \\prod \\mathcal{F}(U_i)$", "is injective by (1). Finally, suppose that we are given an", "$t \\in \\mathcal{F}(V)$ in the equalizer of the two maps", "$\\mathcal{F}(V) \\to \\mathcal{F}(V \\times_U V)$.", "Then $t|_{V_i}$ is in the equalizer of the two maps", "$\\mathcal{F}(V_i) \\to \\mathcal{F}(V_i \\times_{U_i} V_i)$", "for all $i$. Hence we obtain a unique section $s_i \\in \\mathcal{F}(U_i)$", "mapping to $t|_{V_i}$ for all $i$ by (2').", "We omit the verification that $s_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}$", "for all $i, j$; this uses the uniqueness property just shown.", "By the sheaf property for the covering $U = \\bigcup U_i$ we obtain", "a section $s \\in \\mathcal{F}(U)$. We omit the proof that $s$", "maps to $t$ in $\\mathcal{F}(V)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12491, "type": "theorem", "label": "topologies-lemma-morphism-big-ph", "categories": [ "topologies" ], "title": "topologies-lemma-morphism-big-ph", "contents": [ "Let $\\Sch_{ph}$ be a big ph site.", "Let $f : T \\to S$ be a morphism in $\\Sch_{ph}$.", "The functor", "$$", "u : (\\Sch/T)_{ph} \\longrightarrow (\\Sch/S)_{ph},", "\\quad", "V/T \\longmapsto V/S", "$$", "is cocontinuous, and has a continuous right adjoint", "$$", "v : (\\Sch/S)_{ph} \\longrightarrow (\\Sch/T)_{ph},", "\\quad", "(U \\to S) \\longmapsto (U \\times_S T \\to T).", "$$", "They induce the same morphism of topoi", "$$", "f_{big} :", "\\Sh((\\Sch/T)_{ph})", "\\longrightarrow", "\\Sh((\\Sch/S)_{ph})", "$$", "We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.", "We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.", "Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with", "fibre products and equalizers." ], "refs": [], "proofs": [ { "contents": [ "The functor $u$ is cocontinuous, continuous, and commutes with fibre products", "and equalizers. Hence", "Sites, Lemmas \\ref{sites-lemma-when-shriek} and", "\\ref{sites-lemma-preserve-equalizers}", "apply and we deduce the formula", "for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover,", "the functor $v$ is a right adjoint because given $U/T$ and $V/S$", "we have $\\Mor_S(u(U), V) = \\Mor_T(U, V \\times_S T)$", "as desired. Thus we may apply", "Sites, Lemmas \\ref{sites-lemma-have-functor-other-way} and", "\\ref{sites-lemma-have-functor-other-way-morphism} to get the", "formula for $f_{big, *}$." ], "refs": [ "sites-lemma-when-shriek", "sites-lemma-preserve-equalizers", "sites-lemma-have-functor-other-way", "sites-lemma-have-functor-other-way-morphism" ], "ref_ids": [ 8545, 8546, 8549, 8550 ] } ], "ref_ids": [] }, { "id": 12492, "type": "theorem", "label": "topologies-lemma-composition-ph", "categories": [ "topologies" ], "title": "topologies-lemma-composition-ph", "contents": [ "Given schemes $X$, $Y$, $Y$ in $(\\Sch/S)_{ph}$", "and morphisms $f : X \\to Y$, $g : Y \\to Z$ we have", "$g_{big} \\circ f_{big} = (g \\circ f)_{big}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the simple description of pushforward", "and pullback for the functors on the big sites from", "Lemma \\ref{lemma-morphism-big-ph}." ], "refs": [ "topologies-lemma-morphism-big-ph" ], "ref_ids": [ 12491 ] } ], "ref_ids": [] }, { "id": 12493, "type": "theorem", "label": "topologies-lemma-recognize-fpqc-covering", "categories": [ "topologies" ], "title": "topologies-lemma-recognize-fpqc-covering", "contents": [ "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of", "morphisms of schemes with target $T$. The following are equivalent", "\\begin{enumerate}", "\\item $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering,", "\\item each $f_i$ is flat and for every affine open $U \\subset T$", "there exist quasi-compact opens", "$U_i \\subset T_i$ which are almost all empty,", "such that $U = \\bigcup f_i(U_i)$,", "\\item each $f_i$ is flat and there exists an affine open covering", "$T = \\bigcup_{\\alpha \\in A} U_\\alpha$ and for each $\\alpha \\in A$", "there exist $i_{\\alpha, 1}, \\ldots, i_{\\alpha, n(\\alpha)} \\in I$", "and quasi-compact opens $U_{\\alpha, j} \\subset T_{i_{\\alpha, j}}$ such that", "$U_\\alpha =", "\\bigcup_{j = 1, \\ldots, n(\\alpha)} f_{i_{\\alpha, j}}(U_{\\alpha, j})$.", "\\end{enumerate}", "If $T$ is quasi-separated, these are also equivalent to", "\\begin{enumerate}", "\\item[(4)] each $f_i$ is flat, and for every $t \\in T$ there exist", "$i_1, \\ldots, i_n \\in I$ and quasi-compact opens $U_j \\subset T_{i_j}$", "such that $\\bigcup_{j = 1, \\ldots, n} f_{i_j}(U_j)$ is a", "(not necessarily open) neighbourhood of $t$ in $T$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We omit the proof of the equivalence of (1), (2), and (3).", "From now on assume $T$ is quasi-separated.", "We prove (4) implies (2). Let $U \\subset T$ be an affine open.", "To prove (2) it suffices to show that for every $t \\in U$ there exist", "finitely many quasi-compact opens $U_j \\subset T_{i_j}$ such that", "$f_{i_j}(U_j) \\subset U$ and such that $\\bigcup f_{i_j}(U_j)$", "is a neighbourhood of $t$ in $U$. By assumption there do exist", "finitely many quasi-compact opens $U'_j \\subset T_{i_j}$ such that", "such that $\\bigcup f_{i_j}(U'_j)$ is a neighbourhood of $t$ in $T$.", "Since $T$ is quasi-separated we see that $U_j = U'_j \\cap f_j^{-1}(U)$", "is quasi-compact open as desired. Since it is clear that (2) implies", "(4) the proof is finished." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12494, "type": "theorem", "label": "topologies-lemma-disjoint-union-is-fpqc-covering", "categories": [ "topologies" ], "title": "topologies-lemma-disjoint-union-is-fpqc-covering", "contents": [ "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of", "morphisms of schemes with target $T$. The following are equivalent", "\\begin{enumerate}", "\\item $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering, and", "\\item setting $T' = \\coprod_{i \\in I} T_i$, and $f = \\coprod_{i \\in I} f_i$", "the family $\\{f : T' \\to T\\}$ is an fpqc covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $U \\subset T$ is an affine open. If (1) holds, then we find", "$i_1, \\ldots, i_n \\in I$ and affine opens $U_j \\subset T_{i_j}$ such that", "$U = \\bigcup_{j = 1, \\ldots, n} f_{i_j}(U_j)$. Then", "$U_1 \\amalg \\ldots \\amalg U_n \\subset T'$ is a quasi-compact open surjecting", "onto $U$. Thus $\\{f : T' \\to T\\}$ is an fpqc covering by", "Lemma \\ref{lemma-recognize-fpqc-covering}.", "Conversely, if (2) holds then there exists a quasi-compact open", "$U' \\subset T'$ with $U = f(U')$. Then $U_j = U' \\cap T_j$ is quasi-compact", "open in $T_j$ and empty for almost all $j$. By", "Lemma \\ref{lemma-recognize-fpqc-covering} we see that (1) holds." ], "refs": [ "topologies-lemma-recognize-fpqc-covering", "topologies-lemma-recognize-fpqc-covering" ], "ref_ids": [ 12493, 12493 ] } ], "ref_ids": [] }, { "id": 12495, "type": "theorem", "label": "topologies-lemma-family-flat-dominated-covering", "categories": [ "topologies" ], "title": "topologies-lemma-family-flat-dominated-covering", "contents": [ "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of", "morphisms of schemes with target $T$. Assume that", "\\begin{enumerate}", "\\item each $f_i$ is flat, and", "\\item the family $\\{f_i : T_i \\to T\\}_{i \\in I}$ can be refined by an", "fpqc covering of $T$.", "\\end{enumerate}", "Then $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering of $T$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{g_j : X_j \\to T\\}_{j \\in J}$ be an fpqc covering refining", "$\\{f_i : T_i \\to T\\}$. Suppose that $U \\subset T$ is affine open.", "Choose $j_1, \\ldots, j_m \\in J$ and $V_k \\subset X_{j_k}$ affine", "open such that $U = \\bigcup g_{j_k}(V_k)$. For each $j$ pick $i_j \\in I$", "and a morphism $h_j : X_j \\to T_{i_j}$ such that $g_j = f_{i_j} \\circ h_j$.", "Since $h_{j_k}(V_k)$ is quasi-compact we can find a quasi-compact", "open $h_{j_k}(V_k) \\subset U_k \\subset f_{i_{j_k}}^{-1}(U)$.", "Then $U = \\bigcup f_{i_{j_k}}(U_k)$. We conclude that", "$\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering by", "Lemma \\ref{lemma-recognize-fpqc-covering}." ], "refs": [ "topologies-lemma-recognize-fpqc-covering" ], "ref_ids": [ 12493 ] } ], "ref_ids": [] }, { "id": 12496, "type": "theorem", "label": "topologies-lemma-family-flat-fpqc-local-covering", "categories": [ "topologies" ], "title": "topologies-lemma-family-flat-fpqc-local-covering", "contents": [ "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of", "morphisms of schemes with target $T$. Assume that", "\\begin{enumerate}", "\\item each $f_i$ is flat, and", "\\item there exists an fpqc covering", "$\\{g_j : S_j \\to T\\}_{j \\in J}$ such that each", "$\\{S_j \\times_T T_i \\to S_j\\}_{i \\in I}$ is an fpqc covering.", "\\end{enumerate}", "Then $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering of $T$." ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-recognize-fpqc-covering} without further", "mention. Let $U \\subset T$ be an affine open. By (2) we can find", "quasi-compact opens $V_j \\subset S_j$ for $j \\in J$, almost all empty, such that", "$U = \\bigcup g_j(V_j)$. Then for each $j$ we can choose quasi-compact", "opens $W_{ij} \\subset S_j \\times_T T_i$ for $i \\in I$, almost all empty,", "with $V_j = \\bigcup_i \\text{pr}_1(W_{ij})$. Thus", "$\\{S_j \\times_T T_i \\to T\\}$ is an fpqc covering.", "Since this covering refines $\\{f_i : T_i \\to T\\}$ we conclude by", "Lemma \\ref{lemma-family-flat-dominated-covering}." ], "refs": [ "topologies-lemma-recognize-fpqc-covering", "topologies-lemma-family-flat-dominated-covering" ], "ref_ids": [ 12493, 12495 ] } ], "ref_ids": [] }, { "id": 12497, "type": "theorem", "label": "topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc", "categories": [ "topologies" ], "title": "topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc", "contents": [ "Any fppf covering is an fpqc covering, and a fortiori,", "any syntomic, smooth, \\'etale or Zariski covering is an fpqc covering." ], "refs": [], "proofs": [ { "contents": [ "We will show that an fppf covering is an fpqc covering, and then the", "rest follows from", "Lemma \\ref{lemma-zariski-etale-smooth-syntomic-fppf}.", "Let $\\{f_i : U_i \\to U\\}_{i \\in I}$ be an fppf covering.", "By definition this means that the $f_i$ are flat which checks the first", "condition of Definition \\ref{definition-fpqc-covering}. To check the", "second, let $V \\subset U$ be an affine open subset.", "Write $f_i^{-1}(V) = \\bigcup_{j \\in J_i} V_{ij}$", "for some affine opens $V_{ij} \\subset U_i$. Since each $f_i$ is open", "(Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}), we see that", "$V = \\bigcup_{i\\in I} \\bigcup_{j \\in J_i} f_i(V_{ij})$", "is an open covering of $V$.", "Since $V$ is quasi-compact, this covering has a finite", "refinement. This finishes the proof." ], "refs": [ "topologies-lemma-zariski-etale-smooth-syntomic-fppf", "topologies-definition-fpqc-covering", "morphisms-lemma-fppf-open" ], "ref_ids": [ 12471, 12547, 5267 ] } ], "ref_ids": [] }, { "id": 12498, "type": "theorem", "label": "topologies-lemma-fpqc", "categories": [ "topologies" ], "title": "topologies-lemma-fpqc", "contents": [ "Let $T$ be a scheme.", "\\begin{enumerate}", "\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$", "is an fpqc covering of $T$.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an fpqc covering and for each", "$i$ we have an fpqc covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then", "$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is an fpqc covering.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an fpqc covering", "and $T' \\to T$ is a morphism of schemes then", "$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is an fpqc covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is immediate. Recall that the composition of flat morphisms", "is flat and that the base change of a flat morphism is flat", "(Morphisms, Lemmas \\ref{morphisms-lemma-base-change-flat} and", "\\ref{morphisms-lemma-composition-flat}).", "Thus we can apply Lemma \\ref{lemma-recognize-fpqc-covering}", "in each case to check that our families of morphisms are fpqc coverings.", "\\medskip\\noindent", "Proof of (2). Assume $\\{T_i \\to T\\}_{i\\in I}$ is an fpqc covering and for each", "$i$ we have an fpqc covering $\\{f_{ij} : T_{ij} \\to T_i\\}_{j\\in J_i}$.", "Let $U \\subset T$ be an affine open. We can find", "quasi-compact opens $U_i \\subset T_i$ for $i \\in I$, almost all empty,", "such that $U = \\bigcup f_i(U_i)$. Then for each $i$ we can choose", "quasi-compact opens $W_{ij} \\subset T_{ij}$ for $j \\in J_i$, almost all empty,", "with $U_i = \\bigcup_j f_{ij}(U_{ij})$. Thus", "$\\{T_{ij} \\to T\\}$ is an fpqc covering.", "\\medskip\\noindent", "Proof of (3). Assume $\\{T_i \\to T\\}_{i\\in I}$ is an fpqc covering", "and $T' \\to T$ is a morphism of schemes. Let $U' \\subset T'$ be an affine", "open which maps into the affine open $U \\subset T$. Choose", "quasi-compact opens $U_i \\subset T_i$, almost all empty,", "such that $U = \\bigcup f_i(U_i)$. Then $U' \\times_U U_i$ is", "a quasi-compact open of $T' \\times_T T_i$ and", "$U' = \\bigcup \\text{pr}_1(U' \\times_U U_i)$. Since $T'$", "can be covered by such affine opens $U' \\subset T'$ we see", "that $\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is an fpqc covering by", "Lemma \\ref{lemma-recognize-fpqc-covering}." ], "refs": [ "morphisms-lemma-base-change-flat", "morphisms-lemma-composition-flat", "topologies-lemma-recognize-fpqc-covering", "topologies-lemma-recognize-fpqc-covering" ], "ref_ids": [ 5265, 5263, 12493, 12493 ] } ], "ref_ids": [] }, { "id": 12499, "type": "theorem", "label": "topologies-lemma-fpqc-affine", "categories": [ "topologies" ], "title": "topologies-lemma-fpqc-affine", "contents": [ "Let $T$ be an affine scheme.", "Let $\\{T_i \\to T\\}_{i \\in I}$ be an fpqc covering of $T$.", "Then there exists an fpqc covering", "$\\{U_j \\to T\\}_{j = 1, \\ldots, n}$ which is a refinement", "of $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine", "scheme. Moreover, we may choose each $U_j$ to be open affine", "in one of the $T_i$." ], "refs": [], "proofs": [ { "contents": [ "This follows directly from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12500, "type": "theorem", "label": "topologies-lemma-fpqc-affine-axioms", "categories": [ "topologies" ], "title": "topologies-lemma-fpqc-affine-axioms", "contents": [ "Let $T$ be an affine scheme.", "\\begin{enumerate}", "\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$", "is a standard fpqc covering of $T$.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a standard fpqc covering and for each", "$i$ we have a standard fpqc covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then", "$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a standard fpqc covering.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a standard fpqc covering", "and $T' \\to T$ is a morphism of affine schemes then", "$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a standard fpqc covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows formally from the fact that compositions and base changes", "of flat morphisms are flat", "(Morphisms, Lemmas \\ref{morphisms-lemma-base-change-flat} and", "\\ref{morphisms-lemma-composition-flat})", "and that fibre products of affine schemes are affine", "(Schemes, Lemma \\ref{schemes-lemma-fibre-product-affines})." ], "refs": [ "morphisms-lemma-base-change-flat", "morphisms-lemma-composition-flat", "schemes-lemma-fibre-product-affines" ], "ref_ids": [ 5265, 5263, 7690 ] } ], "ref_ids": [] }, { "id": 12501, "type": "theorem", "label": "topologies-lemma-fpqc-covering-affines-mapping-in", "categories": [ "topologies" ], "title": "topologies-lemma-fpqc-covering-affines-mapping-in", "contents": [ "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of", "morphisms of schemes with target $T$. Assume that", "\\begin{enumerate}", "\\item each $f_i$ is flat, and", "\\item every affine scheme", "$Z$ and morphism $h : Z \\to T$ there exists a standard fpqc covering", "$\\{Z_j \\to Z\\}_{j = 1, \\ldots, n}$ which refines the family", "$\\{T_i \\times_T Z \\to Z\\}_{i \\in I}$.", "\\end{enumerate}", "Then $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering of $T$." ], "refs": [], "proofs": [ { "contents": [ "Let $T = \\bigcup U_\\alpha$ be an affine open covering.", "For each $\\alpha$ the pullback family $\\{T_i \\times_T U_\\alpha \\to U_\\alpha\\}$", "can be refined by a standard fpqc covering, hence is an", "fpqc covering by Lemma", "\\ref{lemma-family-flat-dominated-covering}.", "As $\\{U_\\alpha \\to T\\}$ is an fpqc covering we conclude that", "$\\{T_i \\to T\\}$ is an fpqc covering by", "Lemma \\ref{lemma-family-flat-fpqc-local-covering}." ], "refs": [ "topologies-lemma-family-flat-dominated-covering", "topologies-lemma-family-flat-fpqc-local-covering" ], "ref_ids": [ 12495, 12496 ] } ], "ref_ids": [] }, { "id": 12502, "type": "theorem", "label": "topologies-lemma-sheaf-property-fpqc", "categories": [ "topologies" ], "title": "topologies-lemma-sheaf-property-fpqc", "contents": [ "Let $F$ be a contravariant functor on the category", "of schemes with values in sets. Then $F$ satisfies", "the sheaf property for the fpqc topology if and only", "if it satisfies", "\\begin{enumerate}", "\\item the sheaf property for every Zariski covering, and", "\\item the sheaf property for any standard fpqc covering.", "\\end{enumerate}", "Moreover, in the presence of (1) property (2) is equivalent to", "property", "\\begin{enumerate}", "\\item[(2')] the sheaf property for $\\{V \\to U\\}$", "with $V$, $U$ affine and $V \\to U$ faithfully flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and (2) hold.", "Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be an fpqc covering. Let $s_i \\in F(T_i)$", "be a family of elements such that $s_i$ and $s_j$ map to the same element", "of $F(T_i \\times_T T_j)$. Let $W \\subset T$ be the maximal open subset", "such that there exists a unique $s \\in F(W)$ with", "$s|_{f_i^{-1}(W)} = s_i|_{f_i^{-1}(W)}$ for all $i$.", "Such a maximal open exists because $F$ satisfies the", "sheaf property for Zariski coverings; in fact $W$ is the", "union of all opens with this property. Let $t \\in T$.", "We will show $t \\in W$. To do this we pick an affine open", "$t \\in U \\subset T$ and we will show there is a unique", "$s \\in F(U)$ with", "$s|_{f_i^{-1}(U)} = s_i|_{f_i^{-1}(U)}$ for all $i$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-fpqc-affine} we can find a standard fpqc covering", "$\\{U_j \\to U\\}_{j = 1, \\ldots, n}$ refining $\\{U \\times_T T_i \\to U\\}$,", "say by morphisms $h_j : U_j \\to T_{i_j}$. By (2) we obtain a unique element", "$s \\in F(U)$ such that $s|_{U_j} = F(h_j)(s_{i_j})$. Note that for any", "scheme $V \\to U$ over $U$ there is a unique section $s_V \\in F(V)$", "which restricts to $F(h_j \\circ \\text{pr}_2)(s_{i_j})$ on", "$V \\times_U U_j$ for $j = 1, \\ldots, n$. Namely, this is true if $V$", "is affine by (2) as $\\{V \\times_U U_j \\to V\\}$ is a standard fpqc covering", "and in general this follows from (1) and the affine case by choosing an", "affine open covering of $V$. In particular, $s_V = s|_V$.", "Now, taking $V = U \\times_T T_i$ and using that", "$s_{i_j}|_{T_{i_j} \\times_T T_i} = s_i|_{T_{i_j} \\times_T T_i}$", "we conclude that $s|_{U \\times_T T_i} = s_V = s_i|_{U \\times_T T_i}$", "which is what we had to show.", "\\medskip\\noindent", "Proof of the equivalence of (2) and (2') in the presence of (1).", "Suppose $\\{T_i \\to T\\}$ is a standard fpqc covering, then", "$\\coprod T_i \\to T$ is a faithfully flat morphism of affine schemes.", "In the presence of (1) we have $F(\\coprod T_i) = \\prod F(T_i)$", "and similarly", "$F((\\coprod T_i) \\times_T (\\coprod T_i)) = \\prod F(T_i \\times_T T_{i'})$.", "Thus the sheaf condition for $\\{T_i \\to T\\}$ and $\\{\\coprod T_i \\to T\\}$", "is the same." ], "refs": [ "topologies-lemma-fpqc-affine" ], "ref_ids": [ 12499 ] } ], "ref_ids": [] }, { "id": 12503, "type": "theorem", "label": "topologies-lemma-no-set-of-fpqc-covers-is-initial", "categories": [ "topologies" ], "title": "topologies-lemma-no-set-of-fpqc-covers-is-initial", "contents": [ "Let $R$ be a nonzero ring. There does not exist a set $A$ of", "fpqc-coverings of $\\Spec(R)$ such that every fpqc-covering can", "be refined by an element of $A$." ], "refs": [], "proofs": [ { "contents": [ "Let us first explain this when $R = k$ is a field. For any set $I$ consider", "the purely transcendental field extension", "$k \\subset k_I = k(\\{t_i\\}_{i \\in I})$. Since $k \\to k_I$ is faithfully flat", "we see that $\\{\\Spec(k_I) \\to \\Spec(k)\\}$ is an fpqc covering.", "Let $A$ be a set and for each $\\alpha \\in A$ let", "$\\mathcal{U}_\\alpha = \\{S_{\\alpha, j} \\to \\Spec(k)\\}_{j \\in J_\\alpha}$ be an", "fpqc covering. If $\\mathcal{U}_\\alpha$ refines $\\{\\Spec(k_I) \\to \\Spec(k)\\}$", "then the morphisms $S_{\\alpha, j} \\to \\Spec(k)$ factor through", "$\\Spec(k_I)$. Since $\\mathcal{U}_\\alpha$ is a covering,", "at least some $S_{\\alpha, j}$ is nonempty. Pick a", "point $s \\in S_{\\alpha, j}$. Since we have the factorization", "$S_{\\alpha, j} \\to \\Spec(k_I) \\to \\Spec(k)$", "we obtain a homomorphism of fields $k_I \\to \\kappa(s)$.", "In particular, we see that the cardinality of $\\kappa(s)$", "is at least the cardinality of $I$. Thus if we take $I$ to be a set", "of cardinality bigger than the cardinalities of the residue fields", "of all the schemes $S_{\\alpha, j}$, then such a factorization does", "not exist and the lemma holds for $R = k$.", "\\medskip\\noindent", "General case. Since $R$ is nonzero it has a maximal prime ideal", "$\\mathfrak m$ with residue field $\\kappa$. Let $I$ be a set and", "consider $R_I = S_I^{-1} R[\\{t_i\\}_{i \\in I}]$", "where $S_I \\subset R[\\{t_i\\}_{i \\in I}]$ is the multiplicative", "subset of $f \\in R[\\{t_i\\}_{i \\in I}]$ such that $f$ maps to", "a nonzero element of $R/\\mathfrak p[\\{t_i\\}_{i \\in I}]$ for", "all primes $\\mathfrak p$ of $R$. Then $R_I$ is a faithfully", "flat $R$-algebra and $\\{\\Spec(R_I) \\to \\Spec(R)\\}$ is an", "fpqc covering. We leave it as an exercise to the reader to show that", "$R_I \\otimes_R \\kappa \\cong \\kappa(\\{t_i\\}_{i \\in I}) = \\kappa_I$", "with notation as above (hint: use that $R \\to \\kappa$ is surjective", "and that any $f \\in R[\\{t_i\\}_{i \\in I}]$ one of whose monomials occurs", "with coefficient $1$ is an element of $S_I$). Let $A$ be a set and", "for each $\\alpha \\in A$ let", "$\\mathcal{U}_\\alpha = \\{S_{\\alpha, j} \\to \\Spec(R)\\}_{j \\in J_\\alpha}$ be an", "fpqc covering. If $\\mathcal{U}_\\alpha$ refines $\\{\\Spec(R_I) \\to \\Spec(R)\\}$,", "then by base change we conclude that", "$\\{S_{\\alpha, j} \\times_{\\Spec(R)} \\Spec(\\kappa) \\to \\Spec(\\kappa)\\}$", "refines $\\{\\Spec(\\kappa_I) \\to \\Spec(\\kappa)\\}$.", "Hence by the result of the previous paragraph, there exists an $I$", "such that this is not the case and the lemma is proved." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12504, "type": "theorem", "label": "topologies-lemma-standard-fpqc-standard-V", "categories": [ "topologies" ], "title": "topologies-lemma-standard-fpqc-standard-V", "contents": [ "A standard fpqc covering is a standard V covering." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{X_i \\to X\\}_{i = 1, \\ldots, n}$ be a standard fpqc covering", "(Definition \\ref{definition-standard-fpqc}). Let $g : \\Spec(V) \\to X$", "be a morphism where $V$ is a valuation ring. Let $x \\in X$ be the", "image of the closed point of $\\Spec(V)$. Choose an $i$ and a point", "$x_i \\in X_i$ mapping to $x$. Then $\\Spec(V) \\times_X X_i$", "has a point $x'_i$ mapping to the closed point of $\\Spec(V)$.", "Since $\\Spec(V) \\times_X X_i \\to \\Spec(V)$ is flat", "we can find a specialization $x''_i \\leadsto x'_i$ of", "points of $\\Spec(V) \\times_X X_i$ with $x''_i$ mapping to", "the generic point of $\\Spec(V)$, see", "Morphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat}. By", "Schemes, Lemma \\ref{schemes-lemma-points-specialize}", "we can choose a valuation ring $W$ and a morphism", "$h : \\Spec(W) \\to \\Spec(V) \\times_X X_i$ such that $h$", "maps the generic point of $\\Spec(W)$ to $x''_i$ and the", "closed point of $\\Spec(W)$ to $x'_i$. We obtain a", "commutative diagram", "$$", "\\xymatrix{", "\\Spec(W) \\ar[r] \\ar[d] & X_i \\ar[d] \\\\", "\\Spec(V) \\ar[r] & X", "}", "$$", "where $V \\to W$ is an extension of valuation rings.", "This proves the lemma." ], "refs": [ "topologies-definition-standard-fpqc", "morphisms-lemma-generalizations-lift-flat", "schemes-lemma-points-specialize" ], "ref_ids": [ 12548, 5266, 7704 ] } ], "ref_ids": [] }, { "id": 12505, "type": "theorem", "label": "topologies-lemma-standard-ph-standard-V", "categories": [ "topologies" ], "title": "topologies-lemma-standard-ph-standard-V", "contents": [ "A standard ph covering is a standard V covering." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be an affine scheme. Let $f : U \\to T$ be a proper surjective", "morphism. Let $U = \\bigcup_{j = 1, \\ldots, m} U_j$ be a finite", "affine open covering. We have to show that $\\{U_j \\to T\\}$", "is a standard V covering, see", "Definition \\ref{definition-standard-ph-covering}.", "Let $g : \\Spec(V) \\to T$", "be a morphism where $V$ is a valuation ring with fraction field $K$.", "Since $U \\to T$ is surjective, we may choose a field extension", "$L/K$ and a commutative diagram", "$$", "\\xymatrix{", "\\Spec(L) \\ar[rr] \\ar[d] & & U \\ar[d] \\\\", "\\Spec(K) \\ar[r] & \\Spec(V) \\ar[r]^g & T", "}", "$$", "By Algebra, Lemma \\ref{algebra-lemma-dominate} we can choose a valuation ring", "$W \\subset L$ dominating $V$. By the valuative criterion of", "properness (Morphisms, Lemma \\ref{morphisms-lemma-characterize-proper})", "we can then find the morphism $h$ in the commutative", "diagram", "$$", "\\xymatrix{", "\\Spec(L) \\ar[r] \\ar[d] & \\Spec(W) \\ar[r]_h \\ar[d] & U \\ar[d] \\\\", "\\Spec(K) \\ar[r] & \\Spec(V) \\ar[r]^g & X", "}", "$$", "Since $\\Spec(W)$ has a unique closed point, we see that $\\Im(h)$", "is contained in $U_j$ for some $j$. Thus $h : \\Spec(W) \\to U_j$", "is the desired lift and we conclude $\\{U_j \\to T\\}$ is a standard V covering." ], "refs": [ "topologies-definition-standard-ph-covering", "algebra-lemma-dominate", "morphisms-lemma-characterize-proper" ], "ref_ids": [ 12543, 608, 5416 ] } ], "ref_ids": [] }, { "id": 12506, "type": "theorem", "label": "topologies-lemma-base-change-standard-V", "categories": [ "topologies" ], "title": "topologies-lemma-base-change-standard-V", "contents": [ "Let $\\{T_j \\to T\\}_{j = 1, \\ldots, m}$ be a standard V covering.", "Let $T' \\to T$ be a morphism of affine schemes.", "Then $\\{T_j \\times_T T' \\to T'\\}_{j = 1, \\ldots, m}$ is a", "standard V covering." ], "refs": [], "proofs": [ { "contents": [ "Let $\\Spec(V) \\to T'$ be a morphism where $V$ is a valuation ring.", "By assumption we can find an extension of valuation rings $V \\subset W$,", "an $i$, and a commutative diagram", "$$", "\\xymatrix{", "\\Spec(W) \\ar[r] \\ar[d] & T_i \\ar[d] \\\\", "\\Spec(V) \\ar[r] & T", "}", "$$", "By the universal property of fibre products", "we obtain a morphism $\\Spec(W) \\to T' \\times_T T_i$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12507, "type": "theorem", "label": "topologies-lemma-composition-standard-V", "categories": [ "topologies" ], "title": "topologies-lemma-composition-standard-V", "contents": [ "Let $T$ be an affine scheme. Let $\\{T_j \\to T\\}_{j = 1, \\ldots, m}$", "be a standard V covering. Let $\\{T_{ji} \\to T_j\\}_{i = 1, \\ldots n_j}$", "be a standard V covering. Then $\\{T_{ji} \\to T\\}_{i, j}$", "is a standard V covering." ], "refs": [], "proofs": [ { "contents": [ "This follows formally from the observation that if", "$V \\subset W$ and $W \\subset \\Omega$ are extensions of", "valuation rings, then $V \\subset \\Omega$ is an extension", "of valuation rings." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12508, "type": "theorem", "label": "topologies-lemma-refine-standard-V", "categories": [ "topologies" ], "title": "topologies-lemma-refine-standard-V", "contents": [ "Let $T$ be an affine scheme. Let $\\{T_j \\to T\\}_{j = 1, \\ldots, m}$", "be a family of morphisms with $T_j$ affine for all $j$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\{T_j \\to T\\}_{j = 1, \\ldots, m}$ is a standard V covering,", "\\item there is a standard V covering which refines", "$\\{T_j \\to T\\}_{j = 1, \\ldots, m}$, and", "\\item $\\{\\coprod_{j = 1, \\ldots, m} T_j \\to T\\}$ is a standard", "V covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints: This follows almost immediately from the definition.", "The only slightly interesting point is that", "a morphism from the spectrum of a local ring into", "$\\coprod_{j = 1, \\ldots, m} T_j$ must factor through some $T_j$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12509, "type": "theorem", "label": "topologies-lemma-refine-by-V", "categories": [ "topologies" ], "title": "topologies-lemma-refine-by-V", "contents": [ "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family", "of morphisms. The following are equivalent", "\\begin{enumerate}", "\\item $\\{T_i \\to T\\}_{i \\in I}$ is a V covering,", "\\item there is a V covering which refines $\\{T_i \\to T\\}_{i \\in I}$, and", "\\item $\\{\\coprod_{i \\in I} T_i \\to T\\}$ is a V covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: compare with the proof of", "Lemma \\ref{lemma-refine-by-ph}." ], "refs": [ "topologies-lemma-refine-by-ph" ], "ref_ids": [ 12484 ] } ], "ref_ids": [] }, { "id": 12510, "type": "theorem", "label": "topologies-lemma-V", "categories": [ "topologies" ], "title": "topologies-lemma-V", "contents": [ "Let $T$ be a scheme.", "\\begin{enumerate}", "\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$", "is a V covering of $T$.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a V covering and for each", "$i$ we have a V covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then", "$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a V covering.", "\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a V covering", "and $T' \\to T$ is a morphism of schemes then", "$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a V covering.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assertion (1) is clear.", "\\medskip\\noindent", "Proof of (3).", "Let $U' \\subset T'$ be an affine open subscheme.", "Since $U'$ is quasi-compact we can find a finite affine open", "covering $U' = U'_1 \\cup \\ldots \\cup U'$ such that", "$U'_j \\to T$ maps into an affine open $U_j \\subset T$.", "Choose a standard V covering $\\{U_{jl} \\to U_j\\}_{l = 1, \\ldots, n_j}$", "refining $\\{T_i \\times_T U_j \\to U_j\\}$.", "By Lemma \\ref{lemma-base-change-standard-V}", "the base change $\\{U_{jl} \\times_{U_j} U'_j \\to U'_j\\}$", "is a standard V covering. Note that $\\{U'_j \\to U'\\}$", "is a standard V covering", "(for example by Lemma \\ref{lemma-standard-fpqc-standard-V}).", "By Lemma \\ref{lemma-composition-standard-V} ", "the family $\\{U_{jl} \\times_{U_j} U'_j \\to U'\\}$", "is a standard V covering. Since", "$\\{U_{jl} \\times_{U_j} U'_j \\to U'\\}$ refines $\\{T_i \\times_T U' \\to U'\\}$", "we conclude.", "\\medskip\\noindent", "Proof of (2).", "Let $U \\subset T$ be affine open. First we pick a standard V covering", "$\\{U_k \\to U\\}_{k = 1, \\ldots, m}$ refining", "$\\{T_i \\times_T U \\to U\\}$.", "Say the refinement is given by morphisms $U_k \\to T_{i_k}$ over $T$.", "Then", "$$", "\\{T_{i_kj} \\times_{T_{i_k}} U_k \\to U_k\\}_{j \\in J_{i_k}}", "$$", "is a V covering by part (3). As $U_k$ is affine,", "we can find a standard V covering", "$\\{U_{ka} \\to U_k\\}_{a = 1, \\ldots, b_k}$ refining this family.", "Then we apply Lemma \\ref{lemma-composition-standard-V}", "to see that $\\{U_{ka} \\to U\\}$ is a standard V covering which", "refines $\\{T_{ij} \\times_T U \\to U\\}$. This finishes the proof." ], "refs": [ "topologies-lemma-base-change-standard-V", "topologies-lemma-standard-fpqc-standard-V", "topologies-lemma-composition-standard-V", "topologies-lemma-composition-standard-V" ], "ref_ids": [ 12506, 12504, 12507, 12507 ] } ], "ref_ids": [] }, { "id": 12511, "type": "theorem", "label": "topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc-ph-V", "categories": [ "topologies" ], "title": "topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc-ph-V", "contents": [ "Any fpqc covering is a V covering. A fortiori,", "any fppf, syntomic, smooth, \\'etale or Zariski covering is a V covering.", "Also, a ph covering is a V covering." ], "refs": [], "proofs": [ { "contents": [ "An fpqc covering can affine locally be refined by a", "standard fpqc covering, see Lemmas \\ref{lemma-fpqc-affine}.", "A standard fpqc covering is a standard V covering, see", "Lemma \\ref{lemma-standard-fpqc-standard-V}. Hence the first statement", "follows from our definition of V covers in terms of standard V coverings.", "The conclusion for fppf, syntomic, smooth, \\'etale or Zariski coverings", "follows as these are fpqc coverings, see", "Lemma \\ref{lemma-zariski-etale-smooth-syntomic-fppf-fpqc}.", "\\medskip\\noindent", "The statement on ph coverings follows from", "Lemma \\ref{lemma-standard-ph-standard-V} in the same manner." ], "refs": [ "topologies-lemma-fpqc-affine", "topologies-lemma-standard-fpqc-standard-V", "topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc", "topologies-lemma-standard-ph-standard-V" ], "ref_ids": [ 12499, 12504, 12497, 12505 ] } ], "ref_ids": [] }, { "id": 12512, "type": "theorem", "label": "topologies-lemma-sheaf-property-V", "categories": [ "topologies" ], "title": "topologies-lemma-sheaf-property-V", "contents": [ "Let $F$ be a contravariant functor on the category", "of schemes with values in sets. Then $F$ satisfies", "the sheaf property for the V topology if and only", "if it satisfies", "\\begin{enumerate}", "\\item the sheaf property for every Zariski covering, and", "\\item the sheaf property for any standard V covering.", "\\end{enumerate}", "Moreover, in the presence of (1) property (2) is equivalent to", "property", "\\begin{enumerate}", "\\item[(2')] the sheaf property for a standard V covering", "of the form $\\{V \\to U\\}$, i.e., consisting of a single arrow.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and (2) hold.", "Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a V covering. Let $s_i \\in F(T_i)$", "be a family of elements such that $s_i$ and $s_j$ map to the same element", "of $F(T_i \\times_T T_j)$. Let $W \\subset T$ be the maximal open subset", "such that there exists a unique $s \\in F(W)$ with", "$s|_{f_i^{-1}(W)} = s_i|_{f_i^{-1}(W)}$ for all $i$.", "Such a maximal open exists because $F$ satisfies the", "sheaf property for Zariski coverings; in fact $W$ is the", "union of all opens with this property. Let $t \\in T$.", "We will show $t \\in W$. To do this we pick an affine open", "$t \\in U \\subset T$ and we will show there is a unique", "$s \\in F(U)$ with", "$s|_{f_i^{-1}(U)} = s_i|_{f_i^{-1}(U)}$ for all $i$.", "\\medskip\\noindent", "We can find a standard V covering", "$\\{U_j \\to U\\}_{j = 1, \\ldots, n}$ refining $\\{U \\times_T T_i \\to U\\}$,", "say by morphisms $h_j : U_j \\to T_{i_j}$. By (2) we obtain a unique element", "$s \\in F(U)$ such that $s|_{U_j} = F(h_j)(s_{i_j})$. Note that for any", "scheme $V \\to U$ over $U$ there is a unique section $s_V \\in F(V)$", "which restricts to $F(h_j \\circ \\text{pr}_2)(s_{i_j})$ on", "$V \\times_U U_j$ for $j = 1, \\ldots, n$. Namely, this is true if $V$", "is affine by (2) as $\\{V \\times_U U_j \\to V\\}$ is a standard V covering", "(Lemma \\ref{lemma-base-change-standard-V})", "and in general this follows from (1) and the affine case by choosing an", "affine open covering of $V$. In particular, $s_V = s|_V$.", "Now, taking $V = U \\times_T T_i$ and using that", "$s_{i_j}|_{T_{i_j} \\times_T T_i} = s_i|_{T_{i_j} \\times_T T_i}$", "we conclude that $s|_{U \\times_T T_i} = s_V = s_i|_{U \\times_T T_i}$", "which is what we had to show.", "\\medskip\\noindent", "Proof of the equivalence of (2) and (2') in the presence of (1).", "Suppose $\\{T_i \\to T\\}_{i = 1, \\ldots, n}$ is a standard V covering, then", "$\\coprod_{i = 1, \\ldots, n} T_i \\to T$ is a morphism of affine schemes", "which is clearly also a standard V covering.", "In the presence of (1) we have $F(\\coprod T_i) = \\prod F(T_i)$", "and similarly", "$F((\\coprod T_i) \\times_T (\\coprod T_i)) = \\prod F(T_i \\times_T T_{i'})$.", "Thus the sheaf condition for $\\{T_i \\to T\\}$ and $\\{\\coprod T_i \\to T\\}$", "is the same." ], "refs": [ "topologies-lemma-base-change-standard-V" ], "ref_ids": [ 12506 ] } ], "ref_ids": [] }, { "id": 12513, "type": "theorem", "label": "topologies-lemma-refine-qcqs-V", "categories": [ "topologies" ], "title": "topologies-lemma-refine-qcqs-V", "contents": [ "Let $X \\to Y$ be a quasi-compact morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\{X \\to Y\\}$ is a V covering,", "\\item for any valuation ring $V$ and morphism $g : \\Spec(V) \\to Y$", "there exists an extension of valuation rings $V \\subset W$", "and a commutative diagram", "$$", "\\xymatrix{", "\\Spec(W) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(V) \\ar[r] & Y", "}", "$$", "\\item for any morphism $Z \\to Y$ and specialization $z' \\leadsto z$", "of points in $Z$, there is a specialization $w' \\leadsto w$", "of points in $Z \\times_Y X$ mapping to $z' \\leadsto z$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and let $g : \\Spec(V) \\to Y$ be as in (2).", "Since $V$ is a local ring there is an affine open $U \\subset Y$", "such that $g$ factors through $U$. By Definition \\ref{definition-V-covering}", "we can find a standard V covering $\\{U_j \\to U\\}$ refining", "$\\{X \\times_Y U \\to U\\}$. By Definition \\ref{definition-standard-V-covering}", "we can find a $j$, an extension of valuation rings $V \\subset W$", "and a commutative diagram", "$$", "\\xymatrix{", "\\Spec(W) \\ar[r] \\ar[d] & U_j \\ar[d] \\ar@{..>}[r] & X \\ar[ld] \\\\", "\\Spec(V) \\ar[r] & Y", "}", "$$", "We have the dotted arrow making the diagram commute by the refinement", "property of the covering and we see that (2) holds.", "\\medskip\\noindent", "Assume (2) and let $Z \\to Y$ and $z' \\leadsto z$ be as in (3).", "By Schemes, Lemma \\ref{schemes-lemma-points-specialize}", "we can find a valuation ring $V$ and a morphism $\\Spec(V) \\to Z$", "such that the closed point of $\\Spec(V)$ maps to $z$ and the", "generic point of $\\Spec(V)$ maps to $z'$. By (2) we can find an", "extension of valuation rings $V \\subset W$ and a", "commutative diagram", "$$", "\\xymatrix{", "\\Spec(W) \\ar[rr] \\ar[d] & & X \\ar[d] \\\\", "\\Spec(V) \\ar[r] & Z \\ar[r] & Y", "}", "$$", "The generic and closed points of $\\Spec(W)$ map to points", "$w' \\leadsto w$ in $Z \\times_Y X$ via the", "induced morphism $\\Spec(W) \\to Z \\times_Y X$.", "This shows that (3) holds.", "\\medskip\\noindent", "Assume (3) holds and let $U \\subset Y$ be an affine open.", "Choose a finite affine open covering", "$U \\times_Y X = \\bigcup_{j = 1, \\ldots, m} U_j$.", "This is possible as $X \\to Y$ is quasi-compact.", "We claim that $\\{U_j \\to U\\}$ is a standard V covering.", "The claim implies (1) is true and finishes the proof of the lemma.", "In order to prove the claim, let $V$ be a valuation ring", "and let $g : \\Spec(V) \\to U$ be a morphism.", "By (3) we find a specialization $w' \\leadsto w$ of", "points of", "$$", "T = \\Spec(V) \\times_X Y = \\Spec(V) \\times_U (U \\times_X Y)", "$$", "such that $w'$ maps to the generic point of $\\Spec(V)$", "and $w$ maps to the closed point of $\\Spec(V)$.", "By Schemes, Lemma \\ref{schemes-lemma-points-specialize}", "we can find a valuation ring $W$ and a morphism", "$\\Spec(W) \\to T$ such that the generic point of $\\Spec(W)$", "maps to $w'$ and the closed point of $\\Spec(W)$ maps to $w$.", "The composition $\\Spec(W) \\to T \\to \\Spec(V)$ corresponds", "to an inclusion $V \\subset W$ which presents $W$ as an", "extension of the valuation ring $V$.", "Since $T = \\bigcup \\Spec(V) \\times_U U_j$ is an open", "covering, we see that $\\Spec(W) \\to T$ factors through", "$\\Spec(V) \\times_U U_j$ for some $j$. Thus we obtain", "a commutative diagram", "$$", "\\xymatrix{", "\\Spec(W) \\ar[d] \\ar[r] & U_j \\ar[d] \\\\", "\\Spec(V) \\ar[r] & U", "}", "$$", "and the proof of the claim is complete." ], "refs": [ "topologies-definition-V-covering", "topologies-definition-standard-V-covering", "schemes-lemma-points-specialize", "schemes-lemma-points-specialize" ], "ref_ids": [ 12551, 12550, 7704, 7704 ] } ], "ref_ids": [] }, { "id": 12514, "type": "theorem", "label": "topologies-lemma-V-covering-universally-submersive", "categories": [ "topologies" ], "title": "topologies-lemma-V-covering-universally-submersive", "contents": [ "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a V covering.", "Then", "$$", "\\coprod\\nolimits_{i \\in I} f_i :", "\\coprod\\nolimits_{i \\in I} X_i", "\\longrightarrow", "X", "$$", "is a universally submersive morphism of schemes (Morphisms, Definition", "\\ref{morphisms-definition-submersive})." ], "refs": [ "morphisms-definition-submersive" ], "proofs": [ { "contents": [ "We will use without further mention that the base change of a V covering", "is a V covering (Lemma \\ref{lemma-V}). In particular it suffices", "to show that the morphism is submersive.", "Being submersive is clearly Zariski local on the base.", "Thus we may assume $X$ is affine. Then $\\{X_i \\to X\\}$", "can be refined by a standard V covering $\\{Y_j \\to X\\}$.", "If we can show that $\\coprod Y_j \\to X$ is submersive,", "then since there is a factorization $\\coprod Y_j \\to \\coprod X_i \\to X$", "we conclude that $\\coprod X_i \\to X$ is submersive.", "Set $Y = \\coprod Y_j$ and consider the morphism of affines", "$f : Y \\to X$.", "By Lemma \\ref{lemma-refine-qcqs-V} we know that we can lift any", "specialization $x' \\leadsto x$ in $X$ to some specialization", "$y' \\leadsto y$ in $Y$.", "Thus if $T \\subset X$ is a subset such that $f^{-1}(T)$", "is closed in $Y$, then $T \\subset X$ is closed under specialization.", "Since $f^{-1}(T) \\subset Y$", "with the reduced induced closed subscheme structure", "is an affine scheme, we conclude that $T \\subset X$ is closed by", "Algebra, Lemma \\ref{algebra-lemma-image-stable-specialization-closed}.", "Hence $f$ is submersive." ], "refs": [ "topologies-lemma-V", "topologies-lemma-refine-qcqs-V", "algebra-lemma-image-stable-specialization-closed" ], "ref_ids": [ 12510, 12513, 551 ] } ], "ref_ids": [ 5556 ] }, { "id": 12515, "type": "theorem", "label": "topologies-lemma-contained-in", "categories": [ "topologies" ], "title": "topologies-lemma-contained-in", "contents": [ "Any set of big Zariski sites is contained in a common big Zariski site.", "The same is true, mutatis mutandis, for big fppf and big \\'etale sites." ], "refs": [], "proofs": [ { "contents": [ "This is true because the union of a set of sets is a set, and the", "constructions in Sets, Lemmas \\ref{sets-lemma-construct-category} and", "\\ref{sets-lemma-coverings-site}", "allow one to start with any initially given set of schemes", "and coverings." ], "refs": [ "sets-lemma-construct-category", "sets-lemma-coverings-site" ], "ref_ids": [ 8789, 8800 ] } ], "ref_ids": [] }, { "id": 12516, "type": "theorem", "label": "topologies-lemma-change-alpha", "categories": [ "topologies" ], "title": "topologies-lemma-change-alpha", "contents": [ "Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.", "Suppose given big sites $\\Sch_\\tau$ and $\\Sch'_\\tau$.", "Assume that $\\Sch_\\tau$ is contained in $\\Sch'_\\tau$.", "The inclusion functor $\\Sch_\\tau \\to \\Sch'_\\tau$ satisfies", "the assumptions of Sites, Lemma \\ref{sites-lemma-bigger-site}.", "There are morphisms of topoi", "\\begin{eqnarray*}", "g : \\Sh(\\Sch_\\tau) &", "\\longrightarrow &", "\\Sh(\\Sch'_\\tau) \\\\", "f : \\Sh(\\Sch'_\\tau) &", "\\longrightarrow &", "\\Sh(\\Sch_\\tau)", "\\end{eqnarray*}", "such that $f \\circ g \\cong \\text{id}$. For any object $S$", "of $\\Sch_\\tau$ the inclusion functor", "$(\\Sch/S)_\\tau \\to (\\Sch'/S)_\\tau$ satisfies", "the assumptions of Sites, Lemma \\ref{sites-lemma-bigger-site}", "also. Hence similarly we obtain morphisms", "\\begin{eqnarray*}", "g : \\Sh((\\Sch/S)_\\tau) &", "\\longrightarrow &", "\\Sh((\\Sch'/S)_\\tau) \\\\", "f : \\Sh((\\Sch'/S)_\\tau) &", "\\longrightarrow &", "\\Sh((\\Sch/S)_\\tau)", "\\end{eqnarray*}", "with $f \\circ g \\cong \\text{id}$." ], "refs": [ "sites-lemma-bigger-site", "sites-lemma-bigger-site" ], "proofs": [ { "contents": [ "Assumptions (b), (c), and (e) of", "Sites, Lemma \\ref{sites-lemma-bigger-site}", "are immediate for the functors", "$\\Sch_\\tau \\to \\Sch'_\\tau$ and", "$(\\Sch/S)_\\tau \\to (\\Sch'/S)_\\tau$. Property (a) holds by", "Lemma \\ref{lemma-zariski-induced},", "\\ref{lemma-etale-induced},", "\\ref{lemma-smooth-induced},", "\\ref{lemma-syntomic-induced}, or", "\\ref{lemma-fppf-induced}.", "Property (d) holds because", "fibre products in the categories $\\Sch_\\tau$, $\\Sch'_\\tau$", "exist and are compatible with fibre products in the category of schemes." ], "refs": [ "sites-lemma-bigger-site", "topologies-lemma-zariski-induced", "topologies-lemma-etale-induced", "topologies-lemma-smooth-induced", "topologies-lemma-syntomic-induced", "topologies-lemma-fppf-induced" ], "ref_ids": [ 8548, 12433, 12448, 12462, 12468, 12474 ] } ], "ref_ids": [ 8548, 8548 ] }, { "id": 12517, "type": "theorem", "label": "topologies-lemma-extend", "categories": [ "topologies" ], "title": "topologies-lemma-extend", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{C}$ be a full subcategory", "of the category $\\Sch/S$ of all schemes over $S$. Assume", "\\begin{enumerate}", "\\item if $X \\to S$ is an object of $\\mathcal{C}$ and", "$U \\subset X$ is an affine open, then $U \\to S$ is isomorphic", "to an object of $\\mathcal{C}$,", "\\item if $V$ is an affine scheme lying over an affine open $U \\subset S$", "such that $V \\to U$ is of finite presentation, then $V \\to S$ is isomorphic", "to an object of $\\mathcal{C}$.", "\\end{enumerate}", "Let $F : \\mathcal{C}^{opp} \\to \\textit{Sets}$ be a functor.", "Assume", "\\begin{enumerate}", "\\item[(a)] for any Zariski covering $\\{f_i : X_i \\to X\\}_{i \\in I}$", "with $X, X_i$ objects of $\\mathcal{C}$ we have", "the sheaf condition for $F$ and this family\\footnote{As we", "do not know that $X_i \\times_X X_j$ is in $\\mathcal{C}$", "this has to be interpreted as follows: by property (1)", "there exist Zariski coverings $\\{U_{ijk} \\to X_i \\times_X X_j\\}_{k \\in K_{ij}}$", "with $U_{ijk}$ an object of $\\mathcal{C}$. Then the sheaf condition", "says that $F(X)$ is the equalizer of the two maps", "from $\\prod F(X_i)$ to $\\prod F(U_{ijk})$.},", "\\item[(b)] if $X = \\lim X_i$ is a directed limit of affine schemes", "over $S$ with $X, X_i$ objects of $\\mathcal{C}$, then", "$F(X) = \\colim F(X_i)$.", "\\end{enumerate}", "Then there is a unique way to extend $F$ to a functor", "$F' : (\\Sch/S)^{opp} \\to \\textit{Sets}$ satisfying the analogues of", "(a) and (b), i.e., $F'$ satisfies the sheaf condition for any", "Zariski covering and $F'(X) = \\colim F'(X_i)$ whenever $X = \\lim X_i$", "is a directed limit of affine schemes over $S$." ], "refs": [], "proofs": [ { "contents": [ "The idea will be to first extend $F$ to a sufficiently large collection of", "affine schemes over $S$ and then use the Zariski sheaf property to extend", "to all schemes.", "\\medskip\\noindent", "Suppose that $V$ is an affine scheme over $S$ whose structure morphism", "$V \\to S$ factors through some affine open $U \\subset S$. In this case", "we can write", "$$", "V = \\lim V_i", "$$", "as a cofiltered limit with $V_i \\to U$ of finite presentation", "and $V_i$ affine. See Algebra, Lemma \\ref{algebra-lemma-ring-colimit-fp}.", "By conditions (1) and (2)", "we may replace our $V_i$ by objects of $\\mathcal{C}$.", "Observe that $V_i \\to S$ is locally of finite presentation", "(if $S$ is quasi-separated, then these morphisms are actually", "of finite presentation). Then we set", "$$", "F'(V) = \\colim F(V_i)", "$$", "Actually, we can give a more canonical expression, namely", "$$", "F'(V) = \\colim_{V \\to V'} F(V')", "$$", "where the colimit is over the category of morphisms $V \\to V'$ over $S$", "where $V'$ is an object of $\\mathcal{C}$ whose structure", "morphism $V' \\to S$ is locally of finite presentation.", "The reason this is the same as the first formula is that by", "Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}", "our inverse system $V_i$ is cofinal in this category!", "Finally, note that if $V$ were an object of $\\mathcal{C}$,", "then $F'(V) = F(V)$ by assumption (b).", "\\medskip\\noindent", "The second formula turns $F'$ into a contravariant functor", "on the category formed by affine schemes $V$ over $S$ whose", "structure morphism factors through an affine open of $S$.", "Let $V$ be such an affine scheme over $S$ and", "suppose that $V = \\bigcup_{k = 1, \\ldots, n} V_k$ is a finite open covering", "by affines. Then it makes sense to ask if the sheaf condition", "holds for $F'$ and this open covering.", "This is true and easy to show: write $V = \\lim V_i$ as in", "the previous paragraph. By Limits, Lemma \\ref{limits-lemma-descend-opens}", "for all sufficiently large $i$ we can find affine opens", "$V_{i, k} \\subset V_i$ compatible with transition maps", "pulling back to $V_k$ in $V$. Thus", "$$", "F'(V_k) = \\colim F(V_{i, k})", "\\quad\\text{and}\\quad", "F'(V_k \\cap V_l) = \\colim F(V_{i, k} \\cap V_{i, l})", "$$", "Strictly speaking in these formulas we need to replace $V_{i, k}$ and", "$V_{i, k} \\cap V_{i, l}$ by isomorphic affine objects of $\\mathcal{C}$", "before applying the functor $F$.", "Since $I$ is directed the colimits pass through equalizers.", "Hence the sheaf condition (b) for $F$ and the Zariski coverings", "$\\{V_{i, k} \\to V_i\\}$ implies the sheaf", "condition for $F'$ and this covering.", "\\medskip\\noindent", "Let $X$ be a general scheme over $S$. Let $\\mathcal{B}_X$ denote", "the collection of affine opens of $X$ whose structure morphism", "to $S$ maps into an affine open of $S$. It is clear that", "$\\mathcal{B}_X$ is a basis for the topology of $X$.", "By the result of the previous paragraph and Sheaves, Lemma", "\\ref{sheaves-lemma-cofinal-systems-coverings-standard-case}", "we see that $F'$ is a sheaf on $\\mathcal{B}_X$.", "Hence $F'$ restricted to $\\mathcal{B}_X$", "extends uniquely to a sheaf $F'_X$ on $X$, see", "Sheaves, Lemma \\ref{sheaves-lemma-extend-off-basis}.", "If $X$ is an object of $\\mathcal{C}$ then we have a canonical", "identification $F'_X(X) = F(X)$ by the agreement of $F'$ and $F$", "on the objects for which they are both defined and", "the fact that $F$ satisfies the sheaf condition for", "Zariski coverings.", "\\medskip\\noindent", "Let $f : X \\to Y$ be a morphism of schemes over $S$.", "We get a unique $f$-map from $F'_Y$ to $F'_X$ compatible", "with the maps $F'(V) \\to F'(U)$ for all", "$U \\in \\mathcal{B}_X$ and $V \\in \\mathcal{B}_Y$", "with $f(U) \\subset V$, see", "Sheaves, Lemma \\ref{sheaves-lemma-f-map-basis-above-and-below-structures}.", "We omit the verification that these maps compose", "correctly given morphisms $X \\to Y \\to Z$ of schemes over $S$.", "We also omit the verification that if $f$ is a morphism", "of $\\mathcal{C}$, then the induced map $F'_Y(Y) \\to F'_X(X)$", "is the same as the map $F(Y) \\to F(X)$ via the identifications", "$F'_X(X) = F(X)$ and $F'_Y(Y) = F(Y)$ above.", "In this way we see that the desired extension of", "$F$ is the functor which sends $X/S$ to $F'_X(X)$.", "\\medskip\\noindent", "Property (a) for the functor $X \\mapsto F'_X(X)$ is almost immediate", "from the construction; we omit the details.", "Suppose that $X = \\lim_{i \\in I} X_i$", "is a directed limit of affine schemes over $S$. We have to show that", "$$", "F'_X(X) = \\colim_{i \\in I} F'_{X_i}(X_i)", "$$", "First assume that there is some $i \\in I$ such that", "$X_i \\to S$ factors through an affine open $U \\subset S$.", "Then $F'$ is defined on $X$ and on $X_{i'}$ for $i' \\geq i$", "and we see that $F'_{X_{i'}}(X_{i'}) = F'(X_{i'})$ for", "$i' \\geq i$ and $F'_X(X) = F'(X)$. In this case every arrow", "$X \\to V$ with $V$ locally of finite presentation", "over $S$ factors as $X \\to X_{i'} \\to V$ for some", "$i' \\geq i$, see Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}.", "Thus we have", "\\begin{align*}", "F'_X(X)", "& =", "F'(X) \\\\", "& = \\colim_{X \\to V} F(V) \\\\", "& =", "\\colim_{i' \\geq i} \\colim_{X_{i'} \\to V} F(V) \\\\", "& =", "\\colim_{i' \\geq i} F'(X_{i'}) \\\\", "& =", "\\colim_{i' \\geq i} F'_{X_{i'}}(X_{i'}) \\\\", "& =", "\\colim_{i' \\in I} F'_{X_{i'}}(X_{i'})", "\\end{align*}", "as desired. Finally, in general we pick any $i \\in I$ and we choose", "a finite affine open covering $V_i = V_{i, 1} \\cup \\ldots \\cup V_{i, n}$", "such that $V_{i, k} \\to S$ factors through an affine open of $S$.", "Let $V_k \\subset V$ and $V_{i', k}$ for $i' \\geq i$", "be the inverse images of $V_{i, k}$.", "By the previous case we see that", "$$", "F'_{V_k}(V_k) = \\colim_{i' \\geq i} F'_{V_{i', k}}(V_{i', k})", "$$", "and", "$$", "F'_{V_k \\cap V_l}(V_k \\cap V_l) =", "\\colim_{i' \\geq i}", "F'_{V_{i', k} \\cap V_{i', l}}(V_{i', k} \\cap V_{i', l})", "$$", "By the sheaf property and exactness of filtered colimits", "we find that $F'_X(X) = \\colim_{i \\in I} F'_{X_i}(X_i)$", "also in this case. This finishes the proof of property (b)", "and hence finishes the proof of the lemma." ], "refs": [ "algebra-lemma-ring-colimit-fp", "limits-proposition-characterize-locally-finite-presentation", "limits-lemma-descend-opens", "sheaves-lemma-cofinal-systems-coverings-standard-case", "sheaves-lemma-extend-off-basis", "sheaves-lemma-f-map-basis-above-and-below-structures", "limits-proposition-characterize-locally-finite-presentation" ], "ref_ids": [ 1091, 15127, 15041, 14530, 14532, 14540, 15127 ] } ], "ref_ids": [] }, { "id": 12518, "type": "theorem", "label": "topologies-lemma-limit-fppf-topology", "categories": [ "topologies" ], "title": "topologies-lemma-limit-fppf-topology", "contents": [ "Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.", "Let $T$ be an affine scheme which is written as a limit", "$T = \\lim_{i \\in I} T_i$ of a directed inverse system of affine schemes.", "\\begin{enumerate}", "\\item Let $\\mathcal{V} = \\{V_j \\to T\\}_{j = 1, \\ldots, m}$ be a", "standard $\\tau$-covering of $T$, see Definitions", "\\ref{definition-standard-Zariski},", "\\ref{definition-standard-etale},", "\\ref{definition-standard-smooth},", "\\ref{definition-standard-syntomic}, and", "\\ref{definition-standard-fppf}.", "Then there exists an index $i$ and a standard $\\tau$-covering", "$\\mathcal{V}_i = \\{V_{i, j} \\to T_i\\}_{j = 1, \\ldots, m}$", "whose base change $T \\times_{T_i} \\mathcal{V}_i$ to $T$", "is isomorphic to $\\mathcal{V}$.", "\\item Let $\\mathcal{V}_i$, $\\mathcal{V}'_i$ be a pair of standard", "$\\tau$-coverings of $T_i$. If", "$f : T \\times_{T_i} \\mathcal{V}_i \\to T \\times_{T_i} \\mathcal{V}'_i$ is", "a morphism of coverings of $T$, then there exists an index", "$i' \\geq i$ and a morphism", "$f_{i'} : T_{i'} \\times_{T_i} \\mathcal{V} \\to", "T_{i'} \\times_{T_i} \\mathcal{V}'_i$", "whose base change to $T$ is $f$.", "\\item If", "$f, g : \\mathcal{V} \\to \\mathcal{V}'_i$", "are morphisms of standard $\\tau$-coverings of $T_i$ whose", "base changes $f_T, g_T$ to $T$ are equal then there exists an", "index $i' \\geq i$ such that $f_{T_{i'}} = g_{T_{i'}}$.", "\\end{enumerate}", "In other words, the category of standard $\\tau$-coverings of $T$ is", "the colimit over $I$ of the categories of standard $\\tau$-coverings of $T_i$." ], "refs": [ "topologies-definition-standard-Zariski", "topologies-definition-standard-etale", "topologies-definition-standard-smooth", "topologies-definition-standard-syntomic", "topologies-definition-standard-fppf" ], "proofs": [ { "contents": [ "Let us prove this for $\\tau = fppf$.", "By Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "the category of schemes of finite presentation over $T$ is the", "colimit over $I$ of the categories of finite presentation over $T_i$. By", "Limits, Lemmas \\ref{limits-lemma-descend-affine-finite-presentation}", "and \\ref{limits-lemma-descend-flat-finite-presentation}", "the same is true for category of schemes which are affine, flat and", "of finite presentation over $T$.", "To finish the proof of the lemma it suffices to show that if", "$\\{V_{j, i} \\to T_i\\}_{j = 1, \\ldots, m}$ is a finite family of", "flat finitely presented morphisms with $V_{j, i}$ affine, and the", "base change $\\coprod_j T \\times_{T_i} V_{j, i} \\to T$ is surjective,", "then for some $i' \\geq i$ the morphism", "$\\coprod T_{i'} \\times_{T_i} V_{j, i} \\to T_{i'}$ is surjective.", "Denote $W_{i'} \\subset T_{i'}$, resp.\\ $W \\subset T$ the image.", "Of course $W = T$ by assumption.", "Since the morphisms are flat and of finite presentation we see that", "$W_i$ is a quasi-compact open of $T_i$, see", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}.", "Moreover, $W = T \\times_{T_i} W_i$ (formation of image commutes", "with base change). Hence by", "Limits, Lemma \\ref{limits-lemma-descend-opens}", "we conclude that $W_{i'} = T_{i'}$ for some large enough $i'$", "and we win.", "\\medskip\\noindent", "For $\\tau \\in \\{Zariski, \\etale, smooth, syntomic\\}$ a standard $\\tau$-covering", "is a standard fppf covering. Hence the fully faithfulness of the functor", "holds. The only issue is to show that given a standard fppf covering", "$\\mathcal{V}_i$ for some $i$ such that $\\mathcal{V}_i \\times_{T_i} T$", "is a standard $\\tau$-covering, then $\\mathcal{V}_i \\times_{T_i} T_{i'}$", "is a standard $\\tau$-covering for all $i' \\gg i$. This follows immediately", "from Limits, Lemmas", "\\ref{limits-lemma-descend-open-immersion},", "\\ref{limits-lemma-descend-etale},", "\\ref{limits-lemma-descend-smooth}, and", "\\ref{limits-lemma-descend-syntomic}." ], "refs": [ "limits-lemma-descend-finite-presentation", "limits-lemma-descend-affine-finite-presentation", "limits-lemma-descend-flat-finite-presentation", "morphisms-lemma-fppf-open", "limits-lemma-descend-opens", "limits-lemma-descend-open-immersion", "limits-lemma-descend-etale", "limits-lemma-descend-smooth", "limits-lemma-descend-syntomic" ], "ref_ids": [ 15077, 15057, 15062, 5267, 15041, 15067, 15065, 15064, 15070 ] } ], "ref_ids": [ 12522, 12527, 12532, 12536, 12540 ] }, { "id": 12519, "type": "theorem", "label": "topologies-lemma-extend-sheaf-general", "categories": [ "topologies" ], "title": "topologies-lemma-extend-sheaf-general", "contents": [ "Let $S$, $\\mathcal{C}$, $F$ satisfy conditions (1), (2), (a), and (b) of", "Lemma \\ref{lemma-extend} and denote $F' : (\\Sch/S)^{opp} \\to \\textit{Sets}$", "the unique extension constructed in the lemma. Let", "$\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. Assume", "\\begin{enumerate}", "\\item[(c)] for any standard $\\tau$-covering $\\{V_i \\to V\\}_{i = 1, \\ldots, n}$", "of affines in $\\Sch/S$ such that $V \\to S$ factors through an affine open", "$U \\subset S$ and $V \\to U$ is of finite presentation, the sheaf condition", "hold for $F$ and $\\{V_i \\to V\\}_{i = 1, \\ldots, n}$\\footnote{This makes", "sense as $V$, $V_i$, and $V_i \\times_V V_j$ are isomorphic to objects", "of $\\mathcal{C}$ by (2).}.", "\\end{enumerate}", "Then $F'$ satisfies the sheaf condition for all $\\tau$-coverings." ], "refs": [ "topologies-lemma-extend" ], "proofs": [ { "contents": [ "Let $X$ be a scheme over $S$ and let $\\{X_i \\to X\\}_{i \\in I}$", "be a $\\tau$-covering. Let $s_i \\in F'(X_i)$ be elements such that", "$s_i$ and $s_j$ map to the same element of $F'(X_i \\times_X X_j)$", "for all $i, j \\in I$. We have to show that there is a", "unique element $s \\in F'(X)$ restricting to $s_i \\in F'(X_i)$ for all $i \\in I$.", "\\medskip\\noindent", "Special case: $X$ is an affine such that the structure morphism", "maps into an affine open $U$ of $S$ and the covering", "$\\{X_i \\to X\\}_{i \\in I}$ is a standard $\\tau$-covering.", "In this case we can write", "$$", "X = \\lim V_k", "$$", "as a cofiltered limit with $V_k \\to U$ of finite presentation", "and $V_k$ affine. See Algebra, Lemma \\ref{algebra-lemma-ring-colimit-fp}.", "By Lemma \\ref{lemma-limit-fppf-topology} there exists a $k$ and a standard", "$\\tau$-covering $\\{V_{k, i} \\to V_k\\}_{i \\in I}$ whose base change", "to $X$ is the given covering. For $k' \\geq k$ denote", "$\\{V_{k', i} \\to V_{k'}\\}_{i \\in I}$ the base change to $V_{k'}$", "of our covering. Then we see that", "\\begin{align*}", "F'(X)", "& =", "\\colim_{k' \\geq k} F(V_k) \\\\", "& =", "\\colim_{k' \\geq k}", "\\text{Equalizer}(", "\\xymatrix{", "\\prod F(V_{k', i})", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\prod F(V_{k', i} \\times_{V_{k'}} V_{k', j})", "}", "\\\\", "& =", "\\text{Equalizer}(", "\\xymatrix{", "\\colim_{k' \\geq k}", "\\prod F(V_{k', i})", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\colim_{k' \\geq k}", "\\prod F(V_{k', i} \\times_{V_{k'}} V_{k', j})", "}", "\\\\", "& =", "\\text{Equalizer}(", "\\xymatrix{", "\\prod F'(X_i)", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\prod F'(X_i \\times_X X_j)", "}", "\\end{align*}", "The first equality holds by construction of $F'$. The second holds by", "assumption (c). The third holds because filtered colimits are exact.", "The fourth again holds by construction of $F'$.", "In this way we find that the sheaf property holds for $F'$", "with respect to $\\{X_i \\to X\\}_{i \\in I}$.", "\\medskip\\noindent", "General case. Choose an affine open covering $X = \\bigcup U_k$ such that", "each $U_k$ maps into an affine open of $S$. For every $k$ we may choose", "a standard $\\tau$-covering $\\{V_{k, j} \\to U_k\\}_{j = 1, \\ldots, m_k}$", "which refines $\\{X_i \\times_X U_k \\to U_k\\}_{i \\in I}$. For each", "$j \\in \\{1, \\ldots, m_k\\}$ choose an index $i_{k, j} \\in I$", "and a morphism $g_{k, j} : V_{k, j} \\to X_{i_{k, j}}$ over $X$.", "Let $s_{k, j}$ be the element of $F'(V_{k, j})$ we get by restricting", "$s_{i_{k, j}}$ via $g_{k, j}$. Observe that $s_{k, j}$ and $s_{k', j'}$", "restrict to the same element of $F'(V_{k, j} \\times_X V_{k', j'})$", "for all $k$ and $k'$ and all $j \\in \\{1, \\ldots, m_k\\}$", "and $j' \\in \\{1, \\ldots, m_{k'}\\}$; verification omitted.", "In particular, by the result of the previous paragraph there is a unique", "element $s_k \\in F'(U_k)$ restricting to $s_{k, j}$ for all $j$.", "With this notation we are ready to finish the proof.", "\\medskip\\noindent", "Proof of uniqueness of $s$: this is true because $F'$ satisfies", "the sheaf property for Zariski coverings and $s|_{U_k}$ must be equal", "to $s_k$ because both restrict to $s_{k, j}$ for all $j$.", "This uniqueness then shows that $s_k$ and $s_{k'}$ must restrict", "to the same section of $F'$ over (the non-affine scheme) $U_k \\cap U_{k'}$", "because these sections restrict to the same section over the", "$\\tau$-covering $\\{V_{k, j} \\times_X V_{k', j'} \\to U_k \\cap U_{k'}\\}$.", "Thus by the sheaf property for Zariski coverings, there is a unique", "section $s$ of $F'$ over $X$ whose restriction to $U_k$ is $s_k$.", "We omit the verification (similar to the above) that $s$ restricts", "to $s_i$ over $X_i$." ], "refs": [ "algebra-lemma-ring-colimit-fp", "topologies-lemma-limit-fppf-topology" ], "ref_ids": [ 1091, 12518 ] } ], "ref_ids": [ 12517 ] }, { "id": 12520, "type": "theorem", "label": "topologies-lemma-extend-sheaf", "categories": [ "topologies" ], "title": "topologies-lemma-extend-sheaf", "contents": [ "Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.", "Let $S$ be a scheme contained in a big site $\\Sch_\\tau$.", "Let $F : (\\Sch/S)_\\tau^{opp} \\to \\textit{Sets}$ be a $\\tau$-sheaf", "satisfying property (b) of Lemma \\ref{lemma-extend} with", "$\\mathcal{C} = (\\Sch/S)_\\tau$. Then the extension", "$F'$ of $F$ to the category of all schemes over $S$", "satisfies the sheaf condition for all $\\tau$-coverings." ], "refs": [ "topologies-lemma-extend" ], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-extend-sheaf-general}", "applied with $\\mathcal{C} = (\\Sch/S)_\\tau$.", "Conditions (1), (2), (a), and (b) of Lemma \\ref{lemma-extend}", "hold; we omit the details. Thus we get our unique extension $F'$", "to the category of all schemes over $S$. Finally, observe that", "any standard $\\tau$-covering is tautologically equivalent to a", "covering in $(\\Sch/S)_\\tau$, see Sets, Lemma \\ref{sets-lemma-what-is-in-it}", "as well as Lemmas \\ref{lemma-zariski-induced},", "\\ref{lemma-etale-induced},", "\\ref{lemma-smooth-induced},", "\\ref{lemma-syntomic-induced}, and", "\\ref{lemma-fppf-induced}. By", "Sites, Lemma \\ref{sites-lemma-tautological-same-sheaf}", "the sheaf property passes through tautological equivalence", "of coverings. Hence the fact that $F$ is a $\\tau$-sheaf implies", "that property (c) of Lemma \\ref{lemma-extend-sheaf-general} holds", "and we conclude." ], "refs": [ "topologies-lemma-extend-sheaf-general", "topologies-lemma-extend", "sets-lemma-what-is-in-it", "topologies-lemma-zariski-induced", "topologies-lemma-etale-induced", "topologies-lemma-smooth-induced", "topologies-lemma-syntomic-induced", "topologies-lemma-fppf-induced", "sites-lemma-tautological-same-sheaf", "topologies-lemma-extend-sheaf-general" ], "ref_ids": [ 12519, 12517, 8795, 12433, 12448, 12462, 12468, 12474, 8503, 12519 ] } ], "ref_ids": [ 12517 ] }, { "id": 12554, "type": "theorem", "label": "pic-lemma-hilb-d-sheaf", "categories": [ "pic" ], "title": "pic-lemma-hilb-d-sheaf", "contents": [ "Let $X \\to S$ be a morphism of schemes. The functor $\\Hilbfunctor^d_{X/S}$", "satisfies the sheaf property for the fpqc topology", "(Topologies, Definition \\ref{topologies-definition-sheaf-property-fpqc})." ], "refs": [ "topologies-definition-sheaf-property-fpqc" ], "proofs": [ { "contents": [ "Let $\\{T_i \\to T\\}_{i \\in I}$ be an fpqc covering of schemes over $S$.", "Set $X_i = X_{T_i} = X \\times_S T_i$.", "Note that $\\{X_i \\to X_T\\}_{i \\in I}$ is an fpqc covering of", "$X_T$ (Topologies, Lemma \\ref{topologies-lemma-fpqc})", "and that $X_{T_i \\times_T T_{i'}} = X_i \\times_{X_T} X_{i'}$.", "Suppose that $Z_i \\in \\Hilbfunctor^d_{X/S}(T_i)$ is a collection of", "elements such that $Z_i$ and $Z_{i'}$ map to the same element of", "$\\Hilbfunctor^d_{X/S}(T_i \\times_T T_{i'})$. By effective descent", "for closed immersions (Descent, Lemma \\ref{descent-lemma-closed-immersion})", "there is a closed immersion $Z \\to X_T$ whose base change by", "$X_i \\to X_T$ is equal to $Z_i \\to X_i$. The morphism $Z \\to T$", "then has the property that its base change to $T_i$ is the morphism", "$Z_i \\to T_i$. Hence $Z \\to T$ is finite locally free of degree $d$", "by Descent, Lemma \\ref{descent-lemma-descending-property-finite-locally-free}." ], "refs": [ "topologies-lemma-fpqc", "descent-lemma-closed-immersion", "descent-lemma-descending-property-finite-locally-free" ], "ref_ids": [ 12498, 14749, 14695 ] } ], "ref_ids": [ 12549 ] }, { "id": 12555, "type": "theorem", "label": "pic-lemma-hilb-d-limit-preserving", "categories": [ "pic" ], "title": "pic-lemma-hilb-d-limit-preserving", "contents": [ "Let $X \\to S$ be a morphism of schemes. If $X \\to S$ is", "of finite presentation, then the functor $\\Hilbfunctor^d_{X/S}$", "is limit preserving (Limits, Remark \\ref{limits-remark-limit-preserving})." ], "refs": [ "limits-remark-limit-preserving" ], "proofs": [ { "contents": [ "Let $T = \\lim T_i$ be a limit of affine schemes over $S$. We have to show", "that $\\Hilbfunctor^d_{X/S}(T) = \\colim \\Hilbfunctor^d_{X/S}(T_i)$.", "Observe that if $Z \\to X_T$ is an element of $\\Hilbfunctor^d_{X/S}(T)$,", "then $Z \\to T$ is of finite presentation. Hence by", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "there exists an $i$, a scheme $Z_i$ of finite presentation over $T_i$,", "and a morphism $Z_i \\to X_{T_i}$ over $T_i$ whose base change to $T$", "gives $Z \\to X_T$. We apply Limits, Lemma", "\\ref{limits-lemma-descend-closed-immersion-finite-presentation}", "to see that we may assume $Z_i \\to X_{T_i}$ is a closed immersion", "after increasing $i$.", "We apply Limits, Lemma \\ref{limits-lemma-descend-finite-locally-free}", "to see that $Z_i \\to T_i$ is finite locally free of degree $d$", "after possibly increasing $i$.", "Then $Z_i \\in \\Hilbfunctor^d_{X/S}(T_i)$ as desired." ], "refs": [ "limits-lemma-descend-finite-presentation", "limits-lemma-descend-closed-immersion-finite-presentation", "limits-lemma-descend-finite-locally-free" ], "ref_ids": [ 15077, 15060, 15063 ] } ], "ref_ids": [ 15130 ] }, { "id": 12556, "type": "theorem", "label": "pic-lemma-hilb-d-of-closed", "categories": [ "pic" ], "title": "pic-lemma-hilb-d-of-closed", "contents": [ "Let $S$ be a scheme. Let $i : X \\to Y$ be a closed immersion of schemes.", "If $\\Hilbfunctor^d_{Y/S}$ is representable by a scheme, so is", "$\\Hilbfunctor^d_{X/S}$ and the corresponding morphism of schemes", "$\\underline{\\Hilbfunctor}^d_{X/S} \\to \\underline{\\Hilbfunctor}^d_{Y/S}$", "is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be a scheme over $S$ and let $Z \\in \\Hilbfunctor^d_{Y/S}(T)$.", "Claim: there is a closed subscheme $T_X \\subset T$ such", "that a morphism of schemes $T' \\to T$ factors through $T_X$ if", "and only if $Z_{T'} \\to Y_{T'}$ factors through $X_{T'}$.", "Applying this to a scheme $T_{univ}$ representing $\\Hilbfunctor^d_{Y/S}$ and the", "universal object\\footnote{See", "Categories, Section \\ref{categories-section-opposite}}", "$Z_{univ} \\in \\Hilbfunctor^d_{Y/S}(T_{univ})$", "we get a closed subscheme $T_{univ, X} \\subset T_{univ}$ such that", "$Z_{univ, X} = Z_{univ} \\times_{T_{univ}} T_{univ, X}$", "is a closed subscheme of $X \\times_S T_{univ, X}$ and hence", "defines an element of $\\Hilbfunctor^d_{X/S}(T_{univ, X})$.", "A formal argument then shows that $T_{univ, X}$ is a scheme", "representing $\\Hilbfunctor^d_{X/S}$ with universal object $Z_{univ, X}$.", "\\medskip\\noindent", "Proof of the claim. Consider $Z' = X_T \\times_{Y_T} Z$. Given $T' \\to T$", "we see that $Z_{T'} \\to Y_{T'}$ factors through $X_{T'}$ if and", "only if $Z'_{T'} \\to Z_{T'}$ is an isomorphism. Thus the claim follows", "from the very general", "More on Flatness, Lemma \\ref{flat-lemma-Weil-restriction-closed-subschemes}.", "However, in this special case one can prove the statement directly as", "follows: first reduce to the case $T = \\Spec(A)$ and $Z = \\Spec(B)$.", "After shrinking $T$ further we may assume there is an isomorphism", "$\\varphi : B \\to A^{\\oplus d}$ as $A$-modules. Then $Z' = \\Spec(B/J)$", "for some ideal $J \\subset B$. Let $g_\\beta \\in J$ be a collection of", "generators and write $\\varphi(g_\\beta) = (g_\\beta^1, \\ldots, g_\\beta^d)$.", "Then it is clear that $T_X$ is given by $\\Spec(A/(g_\\beta^j))$." ], "refs": [ "flat-lemma-Weil-restriction-closed-subschemes" ], "ref_ids": [ 6088 ] } ], "ref_ids": [] }, { "id": 12557, "type": "theorem", "label": "pic-lemma-hilb-d-separated", "categories": [ "pic" ], "title": "pic-lemma-hilb-d-separated", "contents": [ "Let $X \\to S$ be a morphism of schemes. If $X \\to S$ is separated and", "$\\Hilbfunctor^d_{X/S}$ is representable,", "then $\\underline{\\Hilbfunctor}^d_{X/S} \\to S$ is separated." ], "refs": [], "proofs": [ { "contents": [ "In this proof all unadorned products are over $S$.", "Let $H = \\underline{\\Hilbfunctor}^d_{X/S}$ and let", "$Z \\in \\Hilbfunctor^d_{X/S}(H)$ be the universal object.", "Consider the two objects $Z_1, Z_2 \\in \\Hilbfunctor^d_{X/S}(H \\times H)$", "we get by pulling back $Z$ by the two projections $H \\times H \\to H$.", "Then $Z_1 = Z \\times H \\subset X_{H \\times H}$ and $Z_2 = H \\times Z", "\\subset X_{H \\times H}$. Since $H$ represents the functor", "$\\Hilbfunctor^d_{X/S}$, the diagonal morphism $\\Delta : H \\to H \\times H$", "has the following universal property: A morphism of schemes", "$T \\to H \\times H$ factors through $\\Delta$ if and only if", "$Z_{1, T} = Z_{2, T}$ as elements of $\\Hilbfunctor^d_{X/S}(T)$.", "Set $Z = Z_1 \\times_{X_{H \\times H}} Z_2$. Then we see that", "$T \\to H \\times H$ factors through $\\Delta$ if and only if", "the morphisms $Z_T \\to Z_{1, T}$ and $Z_T \\to Z_{2, T}$ are", "isomorphisms. It follows from the very general", "More on Flatness, Lemma \\ref{flat-lemma-Weil-restriction-closed-subschemes}", "that $\\Delta$ is a closed immersion. In the proof of", "Lemma \\ref{lemma-hilb-d-of-closed}", "the reader finds an alternative easier proof of the needed result", "in our special case." ], "refs": [ "flat-lemma-Weil-restriction-closed-subschemes", "pic-lemma-hilb-d-of-closed" ], "ref_ids": [ 6088, 12556 ] } ], "ref_ids": [] }, { "id": 12558, "type": "theorem", "label": "pic-lemma-hilb-d-An", "categories": [ "pic" ], "title": "pic-lemma-hilb-d-An", "contents": [ "Let $X \\to S$ be a morphism of affine schemes. Let $d \\geq 0$. Then", "$\\Hilbfunctor^d_{X/S}$ is representable." ], "refs": [], "proofs": [ { "contents": [ "Say $S = \\Spec(R)$. Then we can choose a closed immersion of $X$", "into the spectrum of $R[x_i; i \\in I]$ for some set $I$ (of sufficiently", "large cardinality. Hence by Lemma \\ref{lemma-hilb-d-of-closed}", "we may assume that $X = \\Spec(A)$ where $A = R[x_i; i \\in I]$.", "We will use Schemes, Lemma \\ref{schemes-lemma-glue-functors} to prove the", "lemma in this case.", "\\medskip\\noindent", "Condition (1) of the lemma follows from Lemma \\ref{lemma-hilb-d-sheaf}.", "\\medskip\\noindent", "For every subset $W \\subset A$ of cardinality $d$ we will", "construct a subfunctor $F_W$ of $\\Hilbfunctor^d_{X/S}$.", "(It would be enough to consider the case where $W$ consists of a", "collection of monomials in the $x_i$ but we do not need this.)", "Namely, we will say that $Z \\in \\Hilbfunctor^d_{X/S}(T)$ is in $F_W(T)$", "if and only if the $\\mathcal{O}_T$-linear map", "$$", "\\bigoplus\\nolimits_{f \\in W} \\mathcal{O}_T", "\\longrightarrow", "(Z \\to T)_*\\mathcal{O}_Z,\\quad", "(g_f) \\longmapsto \\sum g_f f|_Z", "$$", "is surjective (equivalently an isomorphism). Here for $f \\in A$", "and $Z \\in \\Hilbfunctor^d_{X/S}(T)$ we denote $f|_Z$ the pullback of $f$", "by the morphism $Z \\to X_T \\to X$.", "\\medskip\\noindent", "Openness, i.e., condition (2)(b) of the lemma. This follows from", "Algebra, Lemma \\ref{algebra-lemma-cokernel-flat}.", "\\medskip\\noindent", "Covering, i.e., condition (2)(c) of the lemma. Since", "$$", "A \\otimes_R \\mathcal{O}_T =", "(X_T \\to T)_*\\mathcal{O}_{X_T} \\to (Z \\to T)_*\\mathcal{O}_Z", "$$", "is surjective and since $(Z \\to T)_*\\mathcal{O}_Z$ is finite locally", "free of rank $d$, for every point $t \\in T$ we can find a finite", "subset $W \\subset A$ of cardinality $d$ whose images form a basis", "of the $d$-dimensional $\\kappa(t)$-vector space", "$((Z \\to T)_*\\mathcal{O}_Z)_t \\otimes_{\\mathcal{O}_{T, t}} \\kappa(t)$.", "By Nakayama's lemma there is an open neighbourhood $V \\subset T$", "of $t$ such that $Z_V \\in F_W(V)$.", "\\medskip\\noindent", "Representable, i.e., condition (2)(a) of the lemma. Let $W \\subset A$", "have cardinality $d$. We claim that $F_W$ is representable by an affine", "scheme over $R$. We will construct this affine scheme here, but we encourage", "the reader to think it trough for themselves. Choose a numbering", "$f_1, \\ldots, f_d$ of the elements of $W$. We will construct a universal", "element $Z_{univ} = \\Spec(B_{univ})$ of $F_W$ over $T_{univ} = \\Spec(R_{univ})$", "which will be the spectrum of", "$$", "B_{univ} = R_{univ}[e_1, \\ldots, e_d]/", "(e_ke_l - \\sum c_{kl}^m e_m)", "$$", "where the $e_l$ will be the images of the $f_l$", "and where the closed immersion $Z_{univ} \\to X_{T_{univ}}$ is given", "by the ring map", "$$", "A \\otimes_R R_{univ} \\longrightarrow B_{univ}", "$$", "mapping $1 \\otimes 1$ to $\\sum b^le_l$ and $x_i$ to $\\sum b_i^le_l$.", "In fact, we claim that $F_W$ is represented by the spectrum of the ring", "$$", "R_{univ} = R[c_{kl}^m, b^l, b_i^l]/\\mathfrak a_{univ}", "$$", "where the ideal $\\mathfrak a_{univ}$ is generated by the", "following elements:", "\\begin{enumerate}", "\\item multiplication on $B_{univ}$ is commutative, i.e.,", "$c_{lk}^m - c_{kl}^m \\in \\mathfrak a_{univ}$,", "\\item multiplication on $B_{univ}$ is associative, i.e.,", "$c_{lk}^m c_{m n}^p - c_{lq}^p c_{kn}^q \\in \\mathfrak a_{univ}$,", "\\item $\\sum b^le_l$ is a multiplicative $1$ in $B_{univ}$,", "in other words, we should have $(\\sum b^le_l)e_k = e_k$ for all $k$,", "which means $\\sum b^lc_{lk}^m - \\delta_{km} \\in \\mathfrak a_{univ}$", "(Kronecker delta).", "\\end{enumerate}", "After dividing out by the ideal $\\mathfrak a'_{univ}$ of the elements", "listed sofar we obtain a well defined ring map", "$$", "\\Psi :", "A \\otimes_R R[c_{kl}^m, b^l, b_i^l]/\\mathfrak a'_{univ}", "\\longrightarrow", "\\left(R[c_{kl}^m, b^l, b_i^l]/\\mathfrak a'_{univ}\\right)", "[e_1, \\ldots, e_d]/(e_ke_l - \\sum c_{kl}^m e_m)", "$$", "sending $1 \\otimes 1$ to $\\sum b^le_l$ and $x_i \\otimes 1$ to $\\sum b_i^le_l$.", "We need to add some more elements to our ideal because we need", "\\begin{enumerate}", "\\item[(5)] $f_l$ to map to $e_l$ in $B_{univ}$. Write", "$\\Psi(f_l) - e_l = \\sum h_l^me_m$ with", "$h_l^m \\in R[c_{kl}^m, b^l, b_i^l]/\\mathfrak a'_{univ}$", "then we need to set $h_l^m$ equal to zero.", "\\end{enumerate}", "Thus setting $\\mathfrak a_{univ} \\subset R[c_{kl}^m, b^l, b_i^l]$", "equal to $\\mathfrak a'_{univ} + $ ideal generated by", "lifts of $h_l^m$ to $R[c_{kl}^m, b^l, b_i^l]$, then", "it is clear that $F_W$ is represented by $\\Spec(R_{univ})$." ], "refs": [ "pic-lemma-hilb-d-of-closed", "schemes-lemma-glue-functors", "pic-lemma-hilb-d-sheaf", "algebra-lemma-cokernel-flat" ], "ref_ids": [ 12556, 7688, 12554, 804 ] } ], "ref_ids": [] }, { "id": 12559, "type": "theorem", "label": "pic-lemma-divisors-on-curves", "categories": [ "pic" ], "title": "pic-lemma-divisors-on-curves", "contents": [ "Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$.", "Let $D \\subset X$ be a closed subscheme. Consider the following conditions", "\\begin{enumerate}", "\\item $D \\to S$ is finite locally free,", "\\item $D$ is a relative effective Cartier divisor on $X/S$,", "\\item $D \\to S$ is locally quasi-finite, flat, and", "locally of finite presentation, and", "\\item $D \\to S$ is locally quasi-finite and flat.", "\\end{enumerate}", "We always have the implications", "$$", "(1) \\Rightarrow (2) \\Leftrightarrow (3) \\Rightarrow (4)", "$$", "If $S$ is locally Noetherian, then the last arrow is an if and only if.", "If $X \\to S$ is proper (and $S$ arbitrary), then the first arrow is", "an if and only if." ], "refs": [], "proofs": [ { "contents": [ "Equivalence of (2) and (3). This follows from", "Divisors, Lemma \\ref{divisors-lemma-fibre-Cartier}", "if we can show the equivalence of (2) and (3) when", "$S$ is the spectrum of a field $k$. Let $x \\in X$ be a closed point.", "As $X$ is smooth of relative dimension $1$ over $k$ and we see that", "$\\mathcal{O}_{X, x}$ is a regular local ring of dimension $1$", "(see Varieties, Lemma \\ref{varieties-lemma-smooth-regular}).", "Thus $\\mathcal{O}_{X, x}$ is a discrete valuation ring", "(Algebra, Lemma \\ref{algebra-lemma-characterize-dvr})", "and hence a PID. It follows that every sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_X$ which is nonvanishing at all", "the generic points of $X$ is invertible", "(Divisors, Lemma \\ref{divisors-lemma-effective-Cartier-in-points}).", "In other words, every closed subscheme of $X$ which does not contain", "a generic point is an effective Cartier divisor.", "It follows that (2) and (3) are equivalent.", "\\medskip\\noindent", "If $S$ is Noetherian, then any locally quasi-finite morphism", "$D \\to S$ is locally of finite presentation (Morphisms, Lemma", "\\ref{morphisms-lemma-noetherian-finite-type-finite-presentation}),", "whence (3) is equivalent to (4).", "\\medskip\\noindent", "If $X \\to S$ is proper (and $S$ is arbitrary), then $D \\to S$ is", "proper as well. Since a proper locally quasi-finite morphism is finite", "(More on Morphisms, Lemma \\ref{more-morphisms-lemma-characterize-finite})", "and a finite, flat, and finitely presented morphism is finite locally free", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}), we see that", "(1) is equivalent to (2)." ], "refs": [ "divisors-lemma-fibre-Cartier", "varieties-lemma-smooth-regular", "algebra-lemma-characterize-dvr", "divisors-lemma-effective-Cartier-in-points", "morphisms-lemma-noetherian-finite-type-finite-presentation", "more-morphisms-lemma-characterize-finite", "morphisms-lemma-finite-flat" ], "ref_ids": [ 7978, 11004, 1023, 7946, 5245, 13903, 5471 ] } ], "ref_ids": [] }, { "id": 12560, "type": "theorem", "label": "pic-lemma-sum-divisors-on-curves", "categories": [ "pic" ], "title": "pic-lemma-sum-divisors-on-curves", "contents": [ "Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$.", "Let $D_1, D_2 \\subset X$ be closed subschemes finite locally free of", "degrees $d_1$, $d_2$ over $S$. Then $D_1 + D_2$ is finite locally free", "of degree $d_1 + d_2$ over $S$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-divisors-on-curves} we see that $D_1$", "and $D_2$ are relative effective Cartier divisors on $X/S$.", "Thus $D = D_1 + D_2$ is a relative effective Cartier divisor", "on $X/S$ by", "Divisors, Lemma \\ref{divisors-lemma-sum-relative-effective-Cartier-divisor}.", "Hence $D \\to S$ is locally quasi-finite, flat, and", "locally of finite presentation by", "Lemma \\ref{lemma-divisors-on-curves}.", "Applying", "Morphisms, Lemma \\ref{morphisms-lemma-image-universally-closed-separated}", "the surjective integral morphism $D_1 \\amalg D_2 \\to D$", "we find that $D \\to S$ is separated. Then", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-is-proper}", "implies that $D \\to S$ is proper.", "This implies that $D \\to S$ is finite", "(More on Morphisms, Lemma \\ref{more-morphisms-lemma-characterize-finite})", "and in turn we see that $D \\to S$ is finite locally free", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}).", "Thus it suffice to show that the degree of $D \\to S$ is $d_1 + d_2$.", "To do this we may base change to a fibre of $X \\to S$, hence we may", "assume that $S = \\Spec(k)$ for some field $k$.", "In this case, there exists a finite set of closed points", "$x_1, \\ldots, x_n \\in X$ such that $D_1$ and $D_2$", "are supported on $\\{x_1, \\ldots, x_n\\}$.", "In fact, there are nonzerodivisors $f_{i, j} \\in \\mathcal{O}_{X, x_i}$", "such that", "$$", "D_1 = \\coprod \\Spec(\\mathcal{O}_{X, x_i}/(f_{i, 1}))", "\\quad\\text{and}\\quad", "D_2 = \\coprod \\Spec(\\mathcal{O}_{X, x_i}/(f_{i, 2}))", "$$", "Then we see that", "$$", "D = \\coprod \\Spec(\\mathcal{O}_{X, x_i}/(f_{i, 1}f_{i, 2}))", "$$", "From this one sees easily that $D$ has degree $d_1 + d_2$", "over $k$ (if need be, use Algebra, Lemma \\ref{algebra-lemma-ord-additive})." ], "refs": [ "pic-lemma-divisors-on-curves", "divisors-lemma-sum-relative-effective-Cartier-divisor", "pic-lemma-divisors-on-curves", "morphisms-lemma-image-universally-closed-separated", "morphisms-lemma-image-proper-is-proper", "more-morphisms-lemma-characterize-finite", "morphisms-lemma-finite-flat", "algebra-lemma-ord-additive" ], "ref_ids": [ 12559, 7973, 12559, 5415, 5413, 13903, 5471, 1043 ] } ], "ref_ids": [] }, { "id": 12561, "type": "theorem", "label": "pic-lemma-difference-divisors-on-curves", "categories": [ "pic" ], "title": "pic-lemma-difference-divisors-on-curves", "contents": [ "Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$.", "Let $D_1, D_2 \\subset X$ be closed subschemes finite locally free of", "degrees $d_1$, $d_2$ over $S$. If $D_1 \\subset D_2$ (as closed subschemes)", "then there is a closed subscheme $D \\subset X$ finite locally free of", "degree $d_2 - d_1$ over $S$ such that $D_2 = D_1 + D$." ], "refs": [], "proofs": [ { "contents": [ "This proof is almost exactly the same as the proof of", "Lemma \\ref{lemma-sum-divisors-on-curves}.", "By Lemma \\ref{lemma-divisors-on-curves} we see that $D_1$", "and $D_2$ are relative effective Cartier divisors on $X/S$.", "By Divisors, Lemma", "\\ref{divisors-lemma-difference-relative-effective-Cartier-divisor}", "there is a relative effective Cartier divisor $D \\subset X$", "such that $D_2 = D_1 + D$. Hence $D \\to S$ is locally quasi-finite, flat, and", "locally of finite presentation by", "Lemma \\ref{lemma-divisors-on-curves}.", "Since $D$ is a closed subscheme of $D_2$, we see that", "$D \\to S$ is finite. It follows that $D \\to S$ is finite locally free", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}).", "Thus it suffice to show that the degree of $D \\to S$ is $d_2 - d_1$.", "This follows from Lemma \\ref{lemma-sum-divisors-on-curves}." ], "refs": [ "pic-lemma-sum-divisors-on-curves", "pic-lemma-divisors-on-curves", "divisors-lemma-difference-relative-effective-Cartier-divisor", "pic-lemma-divisors-on-curves", "morphisms-lemma-finite-flat", "pic-lemma-sum-divisors-on-curves" ], "ref_ids": [ 12560, 12559, 7974, 12559, 5471, 12560 ] } ], "ref_ids": [] }, { "id": 12562, "type": "theorem", "label": "pic-lemma-universal-object", "categories": [ "pic" ], "title": "pic-lemma-universal-object", "contents": [ "Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$", "such that the functors $\\Hilbfunctor^d_{X/S}$ are representable. The morphism", "$\\underline{\\Hilbfunctor}^d_{X/S} \\times_S X \\to", "\\underline{\\Hilbfunctor}^{d + 1}_{X/S}$", "is finite locally free of degree $d + 1$." ], "refs": [], "proofs": [ { "contents": [ "Let $D_{univ} \\subset X \\times_S \\underline{\\Hilbfunctor}^{d + 1}_{X/S}$", "be the universal object. There is a commutative diagram", "$$", "\\xymatrix{", "\\underline{\\Hilbfunctor}^d_{X/S} \\times_S X \\ar[rr] \\ar[rd] & &", "D_{univ} \\ar[ld] \\ar@{^{(}->}[r] &", "\\underline{\\Hilbfunctor}^{d + 1}_{X/S} \\times_S X \\\\", "& \\underline{\\Hilbfunctor}^{d + 1}_{X/S}", "}", "$$", "where the top horizontal arrow maps $(D', x)$ to $(D' + x, x)$.", "We claim this morphism is an isomorphism", "which certainly proves the lemma. Namely, given a scheme $T$ over $S$,", "a $T$-valued point $\\xi$ of $D_{univ}$ is given by a pair $\\xi = (D, x)$", "where $D \\subset X_T$ is a closed subscheme finite locally free", "of degree $d + 1$ over $T$ and $x : T \\to X$ is a morphism whose", "graph $x : T \\to X_T$ factors through $D$. Then by", "Lemma \\ref{lemma-difference-divisors-on-curves}", "we can write $D = D' + x$ for some $D' \\subset X_T$ finite locally", "free of degree $d$ over $T$. Sending $\\xi = (D, x)$ to the pair", "$(D', x)$ is the desired inverse." ], "refs": [ "pic-lemma-difference-divisors-on-curves" ], "ref_ids": [ 12561 ] } ], "ref_ids": [] }, { "id": 12563, "type": "theorem", "label": "pic-lemma-hilb-d-smooth", "categories": [ "pic" ], "title": "pic-lemma-hilb-d-smooth", "contents": [ "Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$", "such that the functors $\\Hilbfunctor^d_{X/S}$ are representable. The", "schemes $\\underline{\\Hilbfunctor}^d_{X/S}$ are smooth over $S$ of", "relative dimension $d$." ], "refs": [], "proofs": [ { "contents": [ "We have $\\underline{\\Hilbfunctor}^0_{X/S} = S$ and", "$\\underline{\\Hilbfunctor}^1_{X/S} = X$ thus the result is true for $d = 0, 1$.", "Assuming the result for $d$, we see that", "$\\underline{\\Hilbfunctor}^d_{X/S} \\times_S X$ is smooth over $S$", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-smooth} and", "\\ref{morphisms-lemma-composition-smooth}). Since", "$\\underline{\\Hilbfunctor}^d_{X/S} \\times_S X \\to", "\\underline{\\Hilbfunctor}^{d + 1}_{X/S}$", "is finite locally free of degree $d + 1$ by", "Lemma \\ref{lemma-universal-object}", "the result follows from", "Descent, Lemma \\ref{descent-lemma-smooth-permanence}.", "We omit the verification that the relative dimension is", "as claimed (you can do this by looking at fibres, or by", "keeping track of the dimensions in the argument above)." ], "refs": [ "morphisms-lemma-base-change-smooth", "morphisms-lemma-composition-smooth", "pic-lemma-universal-object", "descent-lemma-smooth-permanence" ], "ref_ids": [ 5327, 5326, 12562, 14644 ] } ], "ref_ids": [] }, { "id": 12564, "type": "theorem", "label": "pic-lemma-flat-geometrically-connected-fibres", "categories": [ "pic" ], "title": "pic-lemma-flat-geometrically-connected-fibres", "contents": [ "Let $f : X \\to S$ be as in Definition \\ref{definition-picard-functor}.", "If $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism", "for all $T \\in \\Ob((\\Sch/S)_{fppf})$, then", "$$", "0 \\to \\Pic(T) \\to \\Pic(X_T) \\to \\Picardfunctor_{X/S}(T)", "$$", "is an exact sequence for all $T$." ], "refs": [ "pic-definition-picard-functor" ], "proofs": [ { "contents": [ "We may replace $S$ by $T$ and $X$ by $X_T$ and assume that $S = T$", "to simplify the notation. Let $\\mathcal{N}$ be an invertible", "$\\mathcal{O}_S$-module. If $f^*\\mathcal{N} \\cong \\mathcal{O}_X$, then", "we see that $f_*f^*\\mathcal{N} \\cong f_*\\mathcal{O}_X \\cong \\mathcal{O}_S$", "by assumption. Since $\\mathcal{N}$ is locally trivial, we see that", "the canonical map $\\mathcal{N} \\to f_*f^*\\mathcal{N}$ is locally", "an isomorphism (because $\\mathcal{O}_S \\to f_*f^*\\mathcal{O}_S$", "is an isomorphism by assumption). Hence we conclude that", "$\\mathcal{N} \\to f_*f^*\\mathcal{N} \\to \\mathcal{O}_S$ is an isomorphism", "and we see that $\\mathcal{N}$ is trivial. This proves the first arrow", "is injective.", "\\medskip\\noindent", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module which is in", "the kernel of $\\Pic(X) \\to \\Picardfunctor_{X/S}(S)$. Then there exists", "an fppf covering $\\{S_i \\to S\\}$ such that $\\mathcal{L}$ pulls back", "to the trivial invertible sheaf on $X_{S_i}$. Choose a trivializing", "section $s_i$. Then $\\text{pr}_0^*s_i$ and $\\text{pr}_1^*s_j$ are both", "trivialising sections of $\\mathcal{L}$ over $X_{S_i \\times_S S_j}$", "and hence differ by a multiplicative unit", "$$", "f_{ij} \\in", "\\Gamma(X_{S_i \\times_S S_j}, \\mathcal{O}_{X_{S_i \\times_S S_j}}^*) =", "\\Gamma(S_i \\times_S S_j, \\mathcal{O}_{S_i \\times_S S_j}^*)", "$$", "(equality by our assumption on pushforward of structure sheaves).", "Of course these elements satisfy the cocycle condition on", "$S_i \\times_S S_j \\times_S S_k$, hence they define a descent datum", "on invertible sheaves for the fppf covering $\\{S_i \\to S\\}$.", "By Descent, Proposition \\ref{descent-proposition-fpqc-descent-quasi-coherent}", "there is an invertible $\\mathcal{O}_S$-module $\\mathcal{N}$", "with trivializations over $S_i$ whose associated descent datum is", "$\\{f_{ij}\\}$. Then $f^*\\mathcal{N} \\cong \\mathcal{L}$ as the", "functor from descent data to modules is fully faithful (see proposition", "cited above)." ], "refs": [ "descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 14753 ] } ], "ref_ids": [ 12577 ] }, { "id": 12565, "type": "theorem", "label": "pic-lemma-flat-geometrically-connected-fibres-with-section", "categories": [ "pic" ], "title": "pic-lemma-flat-geometrically-connected-fibres-with-section", "contents": [ "Let $f : X \\to S$ be as in Definition \\ref{definition-picard-functor}.", "Assume $f$ has a section $\\sigma$ and that", "$\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism", "for all $T \\in \\Ob((\\Sch/S)_{fppf})$. Then", "$$", "0 \\to \\Pic(T) \\to \\Pic(X_T) \\to \\Picardfunctor_{X/S}(T) \\to 0", "$$", "is a split exact sequence with splitting given by", "$\\sigma_T^* : \\Pic(X_T) \\to \\Pic(T)$." ], "refs": [ "pic-definition-picard-functor" ], "proofs": [ { "contents": [ "Denote $K(T) = \\Ker(\\sigma_T^* : \\Pic(X_T) \\to \\Pic(T))$.", "Since $\\sigma$ is a section of $f$ we see that $\\Pic(X_T)$ is the direct", "sum of $\\Pic(T)$ and $K(T)$.", "Thus by Lemma \\ref{lemma-flat-geometrically-connected-fibres} we see that", "$K(T) \\subset \\Picardfunctor_{X/S}(T)$ for all $T$. Moreover, it is clear", "from the construction that $\\Picardfunctor_{X/S}$ is the sheafification", "of the presheaf $K$. To finish the proof it suffices to show that", "$K$ satisfies the sheaf condition for fppf coverings which we do", "in the next paragraph.", "\\medskip\\noindent", "Let $\\{T_i \\to T\\}$ be an fppf covering. Let $\\mathcal{L}_i$ be", "elements of $K(T_i)$ which map to the same elements of $K(T_i \\times_T T_j)$", "for all $i$ and $j$. Choose an isomorphism", "$\\alpha_i : \\mathcal{O}_{T_i} \\to \\sigma_{T_i}^*\\mathcal{L}_i$", "for all $i$. Choose an isomorphism", "$$", "\\varphi_{ij} :", "\\mathcal{L}_i|_{X_{T_i \\times_T T_j}}", "\\longrightarrow", "\\mathcal{L}_j|_{X_{T_i \\times_T T_j}}", "$$", "If the map", "$$", "\\alpha_j|_{T_i \\times_T T_j} \\circ", "\\sigma_{T_i \\times_T T_j}^*\\varphi_{ij} \\circ", "\\alpha_i|_{T_i \\times_T T_j} :", "\\mathcal{O}_{T_i \\times_T T_j} \\to \\mathcal{O}_{T_i \\times_T T_j}", "$$", "is not equal to multiplication by $1$ but some $u_{ij}$, then we can scale", "$\\varphi_{ij}$ by $u_{ij}^{-1}$ to correct this. Having done this, consider", "the self map", "$$", "\\varphi_{ki}|_{X_{T_i \\times_T T_j \\times_T T_k}} \\circ", "\\varphi_{jk}|_{X_{T_i \\times_T T_j \\times_T T_k}} \\circ", "\\varphi_{ij}|_{X_{T_i \\times_T T_j \\times_T T_k}}", "\\quad\\text{on}\\quad", "\\mathcal{L}_i|_{X_{T_i \\times_T T_j \\times_T T_k}}", "$$", "which is given by multiplication by some regular function $f_{ijk}$", "on the scheme $X_{T_i \\times_T T_j \\times_T T_k}$.", "By our choice of $\\varphi_{ij}$ we see that the pullback of", "this map by $\\sigma$ is equal to multiplication by $1$. By", "our assumption on functions on $X$, we see that $f_{ijk} = 1$.", "Thus we obtain a descent datum for the fppf covering", "$\\{X_{T_i} \\to X\\}$. By", "Descent, Proposition \\ref{descent-proposition-fpqc-descent-quasi-coherent}", "there is an invertible $\\mathcal{O}_{X_T}$-module $\\mathcal{L}$", "and an isomorphism $\\alpha : \\mathcal{O}_T \\to \\sigma_T^*\\mathcal{L}$", "whose pullback to $X_{T_i}$ recovers $(\\mathcal{L}_i, \\alpha_i)$", "(small detail omitted). Thus $\\mathcal{L}$ defines an object", "of $K(T)$ as desired." ], "refs": [ "pic-lemma-flat-geometrically-connected-fibres", "descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 12564, 14753 ] } ], "ref_ids": [ 12577 ] }, { "id": 12566, "type": "theorem", "label": "pic-lemma-criterion", "categories": [ "pic" ], "title": "pic-lemma-criterion", "contents": [ "Let $k$ be a field. Let $G : (\\Sch/k)^{opp} \\to \\textit{Groups}$ be a", "functor. With terminology as in", "Schemes, Definition \\ref{schemes-definition-representable-by-open-immersions},", "assume that", "\\begin{enumerate}", "\\item $G$ satisfies the sheaf property for the Zariski topology,", "\\item there exists a subfunctor $F \\subset G$ such that", "\\begin{enumerate}", "\\item $F$ is representable,", "\\item $F \\subset G$ is representable by open immersion,", "\\item for every field extension $K$ of $k$ and $g \\in G(K)$", "there exists a $g' \\in G(k)$ such that $g'g \\in F(K)$.", "\\end{enumerate}", "\\end{enumerate}", "Then $G$ is representable by a group scheme over $k$." ], "refs": [ "schemes-definition-representable-by-open-immersions" ], "proofs": [ { "contents": [ "This follows from Schemes, Lemma \\ref{schemes-lemma-glue-functors}.", "Namely, take $I = G(k)$ and for $i = g' \\in I$ take $F_i \\subset G$", "the subfunctor which associates to $T$ over $k$ the set of elements", "$g \\in G(T)$ with $g'g \\in F(T)$. Then $F_i \\cong F$ by multiplication", "by $g'$. The map $F_i \\to G$ is isomorphic to the map $F \\to G$", "by multiplication by $g'$, hence is representable by open immersions.", "Finally, the collection $(F_i)_{i \\in I}$ covers $G$ by assumption (2)(c).", "Thus the lemma mentioned above applies and the proof is complete." ], "refs": [ "schemes-lemma-glue-functors" ], "ref_ids": [ 7688 ] } ], "ref_ids": [ 7747 ] }, { "id": 12567, "type": "theorem", "label": "pic-lemma-check-conditions", "categories": [ "pic" ], "title": "pic-lemma-check-conditions", "contents": [ "Let $k$ be a field. Let $X$ be a smooth projective curve over $k$", "which has a $k$-rational point. Then the hypotheses of", "Lemma \\ref{lemma-flat-geometrically-connected-fibres-with-section}", "are satisfied." ], "refs": [], "proofs": [ { "contents": [ "The meaning of the phrase ``has a $k$-rational point'' is exactly that", "the structure morphism $f : X \\to \\Spec(k)$ has a section, which", "verifies the first condition.", "By Varieties, Lemma \\ref{varieties-lemma-regular-functions-proper-variety}", "we see that $k' = H^0(X, \\mathcal{O}_X)$ is a field extension of $k$.", "Since $X$ has a $k$-rational point there is a $k$-algebra homomorphism", "$k' \\to k$ and we conclude $k' = k$.", "Since $k$ is a field, any morphism $T \\to \\Spec(k)$ is flat.", "Hence we see by cohomology and base change", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology})", "that $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism.", "This finishes the proof." ], "refs": [ "varieties-lemma-regular-functions-proper-variety", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 11012, 3298 ] } ], "ref_ids": [] }, { "id": 12568, "type": "theorem", "label": "pic-lemma-define-open", "categories": [ "pic" ], "title": "pic-lemma-define-open", "contents": [ "Let $k$ be a field. Let $X$ be a smooth projective curve over $k$", "with a $k$-rational point $\\sigma$. For a scheme $T$ over $k$,", "consider the subset $F(T) \\subset \\Picardfunctor_{X/k, \\sigma}(T)$ consisting of", "$\\mathcal{L}$ such that $Rf_{T, *}\\mathcal{L}$ is isomorphic to an invertible", "$\\mathcal{O}_T$-module placed in degree $0$. Then", "$F \\subset \\Picardfunctor_{X/k, \\sigma}$ is a subfunctor and the inclusion is", "representable by open immersions." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric}", "applied with $i = 0$ and $r = 1$ and", "Schemes, Definition \\ref{schemes-definition-representable-by-open-immersions}." ], "refs": [ "perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric", "schemes-definition-representable-by-open-immersions" ], "ref_ids": [ 7064, 7747 ] } ], "ref_ids": [] }, { "id": 12569, "type": "theorem", "label": "pic-lemma-open-representable", "categories": [ "pic" ], "title": "pic-lemma-open-representable", "contents": [ "Let $k$ be a field. Let $X$ be a smooth projective curve of genus $g$", "over $k$ with a $k$-rational point $\\sigma$. The open subfunctor $F$ defined", "in Lemma \\ref{lemma-define-open} is representable by an open subscheme of", "$\\underline{\\Hilbfunctor}^g_{X/k}$." ], "refs": [ "pic-lemma-define-open" ], "proofs": [ { "contents": [ "In this proof unadorned products are over $\\Spec(k)$.", "By Proposition \\ref{proposition-hilb-d} the scheme", "$H = \\underline{\\Hilbfunctor}^g_{X/k}$ exists.", "Consider the universal divisor $D_{univ} \\subset H \\times X$", "and the associated invertible sheaf $\\mathcal{O}(D_{univ})$, see", "Remark \\ref{remark-universal-object-hilb-d}.", "We adjust by tensoring with the pullback via", "$\\sigma_H : H \\to H \\times X$ to get", "$$", "\\mathcal{L}_H =", "\\mathcal{O}(D_{univ})", "\\otimes_{\\mathcal{O}_{H \\times X}}", "\\text{pr}_H^*\\sigma_H^*\\mathcal{O}(D_{univ})^{\\otimes -1}", "\\in", "\\Picardfunctor_{X/k, \\sigma}(H)", "$$", "By the Yoneda lemma (Categories, Lemma \\ref{categories-lemma-yoneda})", "the invertible sheaf $\\mathcal{L}_H$ defines a natural transformation", "$$", "h_H \\longrightarrow \\Picardfunctor_{X/k, \\sigma}", "$$", "Because $F$ is an open subfuctor, there exists a maximal open", "$W \\subset H$ such that $\\mathcal{L}_H|_{W \\times X}$ is in", "$F(W)$. Of course, this open is nothing else than the", "open subscheme constructed in", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric}", "with $i = 0$ and $r = 1$ for the morphism $H \\times X \\to H$ and the sheaf", "$\\mathcal{F} = \\mathcal{O}(D_{univ})$. Applying the Yoneda lemma", "again we obtain a commutative diagram", "$$", "\\xymatrix{", "h_W \\ar[d] \\ar[r] & F \\ar[d] \\\\", "h_H \\ar[r] & \\Picardfunctor_{X/k, \\sigma}", "}", "$$", "To finish the proof we will show that the top horizontal arrow is an", "isomorphism.", "\\medskip\\noindent", "Let $\\mathcal{L} \\in F(T) \\subset \\Picardfunctor_{X/k, \\sigma}(T)$.", "Let $\\mathcal{N}$ be the invertible $\\mathcal{O}_T$-module", "such that $Rf_{T, *}\\mathcal{L} \\cong \\mathcal{N}[0]$.", "The adjunction map", "$$", "f_T^*\\mathcal{N} \\longrightarrow \\mathcal{L}", "\\quad\\text{corresponds to a section }s\\text{ of}\\quad", "\\mathcal{L} \\otimes f_T^*\\mathcal{N}^{\\otimes -1}", "$$", "on $X_T$. Claim: The zero scheme of $s$ is a relative effective Cartier", "divisor $D$ on $(T \\times X)/T$ finite locally free of degree $g$ over $T$.", "\\medskip\\noindent", "Let us finish the proof of the lemma admitting the claim.", "Namely, $D$ defines a morphism $m : T \\to H$ such that $D$ is the pullback of", "$D_{univ}$. Then", "$$", "(m \\times \\text{id}_X)^*\\mathcal{O}(D_{univ}) \\cong", "\\mathcal{O}_{T \\times X}(D)", "$$", "Hence $(m \\times \\text{id}_X)^*\\mathcal{L}_H$ and $\\mathcal{O}(D)$", "differ by the pullback of an invertible sheaf on $H$. This in particular", "shows that $m : T \\to H$ factors through the open $W \\subset H$ above.", "Moreover, it follows that these invertible modules define, after adjusting", "by pullback via $\\sigma_T$ as above, the same element of", "$\\Picardfunctor_{X/k, \\sigma}(T)$. Chasing diagrams using Yoneda's lemma", "we see that $m \\in h_W(T)$ maps to $\\mathcal{L} \\in F(T)$. We omit", "the verification that the rule $F(T) \\to h_W(T)$,", "$\\mathcal{L} \\mapsto m$ defines an inverse of the transformation", "of functors above.", "\\medskip\\noindent", "Proof of the claim. Since $D$ is a locally principal closed subscheme", "of $T \\times X$, it suffices to show that the fibres of $D$ over $T$ are", "effective Cartier divisors, see Lemma \\ref{lemma-divisors-on-curves} and", "Divisors, Lemma \\ref{divisors-lemma-fibre-Cartier}. Because taking", "cohomology of $\\mathcal{L}$ commutes with base change", "(Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image-general})", "we reduce to $T = \\Spec(K)$ where $K/k$ is a field extension.", "Then $\\mathcal{L}$ is an invertible sheaf on $X_K$ with", "$H^0(X_K, \\mathcal{L}) = K$ and $H^1(X_K, \\mathcal{L}) = 0$. Thus", "$$", "\\deg(\\mathcal{L}) = \\chi(X_K, \\mathcal{L}) - \\chi(X_K, \\mathcal{O}_{X_K})", "= 1 - (1 - g) = g", "$$", "See Varieties, Definition \\ref{varieties-definition-degree-invertible-sheaf}.", "To finish the proof we have to show a nonzero section of $\\mathcal{L}$", "defines an effective Cartier divisor on $X_K$.", "This is clear." ], "refs": [ "pic-proposition-hilb-d", "pic-remark-universal-object-hilb-d", "categories-lemma-yoneda", "perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric", "pic-lemma-divisors-on-curves", "divisors-lemma-fibre-Cartier", "perfect-lemma-flat-proper-perfect-direct-image-general", "varieties-definition-degree-invertible-sheaf" ], "ref_ids": [ 12575, 12580, 12203, 7064, 12559, 7978, 7054, 11161 ] } ], "ref_ids": [ 12568 ] }, { "id": 12570, "type": "theorem", "label": "pic-lemma-twist-with-general-divisor", "categories": [ "pic" ], "title": "pic-lemma-twist-with-general-divisor", "contents": [ "Let $k$ be a separably closed field. Let $X$ be a smooth projective", "curve of genus $g$ over $k$. Let $K/k$ be a field extension and let", "$\\mathcal{L}$ be an invertible sheaf on $X_K$. Then there exists an", "invertible sheaf $\\mathcal{L}_0$ on $X$ such that", "$\\dim_K H^0(X_K,", "\\mathcal{L} \\otimes_{\\mathcal{O}_{X_K}} \\mathcal{L}_0|_{X_K}) = 1$ and", "$\\dim_K H^1(X_K,", "\\mathcal{L} \\otimes_{\\mathcal{O}_{X_K}} \\mathcal{L}_0|_{X_K}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "This proof is a variant of the proof of", "Varieties, Lemma \\ref{varieties-lemma-general-degree-g-line-bundle}.", "We encourage the reader to read that proof first.", "\\medskip\\noindent", "First we pick an ample invertible sheaf $\\mathcal{L}_0$ and", "we replace $\\mathcal{L}$ by", "$\\mathcal{L} \\otimes_{\\mathcal{O}_{X_K}} \\mathcal{L}_0^{\\otimes n}|_{X_K}$", "for some $n \\gg 0$. The result will be that we may assume that", "$H^0(X_K, \\mathcal{L}) \\not = 0$ and $H^1(X_K, \\mathcal{L}) = 0$.", "Namely, we will get the vanishing by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-vanshing-gives-ample} and the nonvanishing because", "the degree of the tensor product is $\\gg 0$.", "We will finish the proof by descending induction on", "$t = \\dim_K H^0(X_K, \\mathcal{L})$. The base case $t = 1$ is trivial.", "Assume $t > 1$.", "\\medskip\\noindent", "Observe that for a $k$-rational point $x$ of $X$, the inverse image $x_K$", "is a $K$-rational point of $X_K$. Moreover, there are infinitely many", "$k$-rational points by Varieties, Lemma", "\\ref{varieties-lemma-smooth-separable-closed-points-dense}. Therefore", "the points $x_K$ form a Zariski dense collection of points of $X_K$.", "\\medskip\\noindent", "Let $s \\in H^0(X_K, \\mathcal{L})$ be nonzero. From the previous paragraph", "we deduce there exists a $k$-rational", "point $x$ such that $s$ does not vanish in $x_K$. Let $\\mathcal{I}$", "be the ideal sheaf of $i : x_K \\to X_K$ as in", "Varieties, Lemma \\ref{varieties-lemma-regular-point-on-curve}. Look at the", "short exact sequence", "$$", "0 \\to \\mathcal{I} \\otimes_{\\mathcal{O}_{X_K}} \\mathcal{L} \\to", "\\mathcal{L} \\to i_*i^*\\mathcal{L} \\to 0", "$$", "Observe that $H^0(X_K, i_*i^*\\mathcal{L}) = H^0(x_K, i^*\\mathcal{L})$", "has dimension $1$ over $K$. Since $s$ does not vanish at $x$ we conclude that", "$$", "H^0(X_K, \\mathcal{L}) \\longrightarrow H^0(X, i_*i^*\\mathcal{L})", "$$", "is surjective. Hence", "$\\dim_K H^0(X_K, \\mathcal{I} \\otimes_{\\mathcal{O}_{X_K}} \\mathcal{L}) = t - 1$.", "Finally, the long exact sequence of cohomology also shows that", "$H^1(X_K, \\mathcal{I} \\otimes_{\\mathcal{O}_{X_K}} \\mathcal{L}) = 0$", "thereby finishing the proof of the induction step." ], "refs": [ "varieties-lemma-general-degree-g-line-bundle", "coherent-lemma-vanshing-gives-ample", "varieties-lemma-smooth-separable-closed-points-dense", "varieties-lemma-regular-point-on-curve" ], "ref_ids": [ 11119, 3346, 11007, 11118 ] } ], "ref_ids": [] }, { "id": 12571, "type": "theorem", "label": "pic-lemma-picard-pieces", "categories": [ "pic" ], "title": "pic-lemma-picard-pieces", "contents": [ "Let $k$ be a separably closed field. Let $X$ be a smooth projective", "curve of genus $g$ over $k$.", "\\begin{enumerate}", "\\item $\\underline{\\Picardfunctor}_{X/k}$ is a disjoint union of", "$g$-dimensional smooth proper varieties $\\underline{\\Picardfunctor}^d_{X/k}$,", "\\item $k$-points of $\\underline{\\Picardfunctor}^d_{X/k}$", "correspond to invertible $\\mathcal{O}_X$-modules of degree $d$,", "\\item $\\underline{\\Picardfunctor}^0_{X/k}$", "is an open and closed subgroup scheme,", "\\item for $d \\geq 0$ there is a canonical morphism", "$\\gamma_d :", "\\underline{\\Hilbfunctor}^d_{X/k} \\to \\underline{\\Picardfunctor}^d_{X/k}$", "\\item the morphisms $\\gamma_d$", "are surjective for $d \\geq g$ and smooth for $d \\geq 2g - 1$,", "\\item the morphism", "$\\underline{\\Hilbfunctor}^g_{X/k} \\to \\underline{\\Picardfunctor}^g_{X/k}$", "is birational.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Pick a $k$-rational point $\\sigma$ of $X$. Recall that $\\Picardfunctor_{X/k}$", "is isomorphic to the functor $\\Picardfunctor_{X/k, \\sigma}$. By", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-chi-locally-constant-geometric}", "for every $d \\in \\mathbf{Z}$ there is an open subfunctor", "$$", "\\Picardfunctor^d_{X/k, \\sigma} \\subset \\Picardfunctor_{X/k, \\sigma}", "$$", "whose value on a scheme $T$ over $k$ consists of those", "$\\mathcal{L} \\in \\Picardfunctor_{X/k, \\sigma}(T)$ such that", "$\\chi(X_t, \\mathcal{L}_t) = d + 1 - g$ and moreover we have", "$$", "\\Picardfunctor_{X/k, \\sigma} =", "\\coprod\\nolimits_{d \\in \\mathbf{Z}} \\Picardfunctor^d_{X/k, \\sigma}", "$$", "as fppf sheaves. It follows that the scheme $\\underline{\\Picardfunctor}_{X/k}$", "(which exists by Proposition \\ref{proposition-pic-curve})", "has a corresponding decomposition", "$$", "\\underline{\\Picardfunctor}_{X/k, \\sigma} =", "\\coprod\\nolimits_{d \\in \\mathbf{Z}} \\underline{\\Picardfunctor}^d_{X/k, \\sigma}", "$$", "where the points of $\\underline{\\Picardfunctor}^d_{X/k, \\sigma}$ correspond", "to isomorphism classes of invertible modules of degree $d$ on $X$.", "\\medskip\\noindent", "Fix $d \\geq 0$. There is a morphism", "$$", "\\gamma_d :", "\\underline{\\Hilbfunctor}^d_{X/k}", "\\longrightarrow", "\\underline{\\Picardfunctor}^d_{X/k}", "$$", "coming from the invertible sheaf $\\mathcal{O}(D_{univ})$ on", "$\\underline{\\Hilbfunctor}^d_{X/k} \\times_k X$", "(Remark \\ref{remark-universal-object-hilb-d}) by the Yoneda lemma", "(Categories, Lemma \\ref{categories-lemma-yoneda}).", "Our proof of the representability of the Picard functor of $X/k$", "in Proposition \\ref{proposition-pic-curve} and", "Lemma \\ref{lemma-open-representable} shows that $\\gamma_g$", "induces an open immersion on a nonempty open of", "$\\underline{\\Hilbfunctor}^g_{X/k}$. Moreover, the proof shows", "that the translates of this open by $k$-rational points of", "the group scheme $\\underline{\\Picardfunctor}_{X/k}$ define", "an open covering. Since", "$\\underline{\\Hilbfunctor}^g_{X/K}$ is smooth of dimension $g$", "(Proposition \\ref{proposition-hilb-d})", "over $k$, we conclude that the", "group scheme $\\underline{\\Picardfunctor}_{X/k}$ is smooth of dimension $g$", "over $k$.", "\\medskip\\noindent", "By Groupoids, Lemma \\ref{groupoids-lemma-group-scheme-over-field-separated}", "we see that $\\underline{\\Picardfunctor}_{X/k}$ is separated.", "Hence, for every $d \\geq 0$, the image of $\\gamma_d$", "is a proper variety over $k$", "(Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-image-is-proper}).", "\\medskip\\noindent", "Let $d \\geq g$. Then for any field extension $K/k$ and any invertible", "$\\mathcal{O}_{X_K}$-module $\\mathcal{L}$ of degree $d$, we see that", "$\\chi(X_K, \\mathcal{L}) = d + 1 - g > 0$. Hence $\\mathcal{L}$ has a", "nonzero section and we conclude that $\\mathcal{L} = \\mathcal{O}_{X_K}(D)$", "for some divisor $D \\subset X_K$ of degree $d$. It follows that", "$\\gamma_d$ is surjective.", "\\medskip\\noindent", "Combining the facts mentioned above we see that", "$\\underline{\\Picardfunctor}^d_{X/k}$ is proper for $d \\geq g$.", "This finishes the proof of (2) because now we see that", "$\\underline{\\Picardfunctor}^d_{X/k}$ is proper for $d \\geq g$ but", "then all $\\underline{\\Picardfunctor}^d_{X/k}$ are proper by translation.", "\\medskip\\noindent", "It remains to prove that $\\gamma_d$ is smooth for $d \\geq 2g - 1$.", "Consider an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ of degree", "$d$. Then the fibre of the point corresponding to $\\mathcal{L}$ is", "$$", "Z = \\{D \\subset X \\mid \\mathcal{O}_X(D) \\cong \\mathcal{L}\\} \\subset", "\\underline{\\Hilbfunctor}^d_{X/k}", "$$", "with its natural scheme structure. Since any isomorphism", "$\\mathcal{O}_X(D) \\to \\mathcal{L}$ is well defined up", "to multiplying by a nonzero scalar, we see that the canonical", "section $1 \\in \\mathcal{O}_X(D)$ is mapped to a section", "$s \\in \\Gamma(X, \\mathcal{L})$ well defined up to multiplication", "by a nonzero scalar. In this way we obtain a morphism", "$$", "Z \\longrightarrow", "\\text{Proj}(\\text{Sym}(\\Gamma(X, \\mathcal{L})^*))", "$$", "(dual because of our conventions). This morphism is an isomorphism,", "because given an section of $\\mathcal{L}$ we can take the associated", "effective Cartier divisor, in other words we can construct an inverse of", "the displayed morphism; we omit the precise formulation and proof.", "Since $\\dim H^0(X, \\mathcal{L}) = d + 1 - g$ for every", "$\\mathcal{L}$ of degree $d \\geq 2g - 1$ by", "Varieties, Lemma \\ref{varieties-lemma-vanishing-degree-2g-and-1-line-bundle}", "we see that $\\text{Proj}(\\text{Sym}(\\Gamma(X, \\mathcal{L})^*))", "\\cong \\mathbf{P}^{d - g}_k$.", "We conclude that $\\dim(Z) = \\dim(\\mathbf{P}^{d - g}_k) = d - g$.", "We conclude that the fibres of the morphism $\\gamma_d$ all", "have dimension equal to the difference of the dimensions of", "$\\underline{\\Hilbfunctor}^d_{X/k}$ and $\\underline{\\Picardfunctor}^d_{X/k}$.", "It follows that $\\gamma_d$ is flat, see", "Algebra, Lemma \\ref{algebra-lemma-CM-over-regular-flat}.", "As moreover the fibres are smooth, we conclude that $\\gamma_d$", "is smooth by Morphisms, Lemma \\ref{morphisms-lemma-smooth-flat-smooth-fibres}." ], "refs": [ "perfect-lemma-chi-locally-constant-geometric", "pic-proposition-pic-curve", "pic-remark-universal-object-hilb-d", "categories-lemma-yoneda", "pic-proposition-pic-curve", "pic-lemma-open-representable", "pic-proposition-hilb-d", "groupoids-lemma-group-scheme-over-field-separated", "morphisms-lemma-scheme-theoretic-image-is-proper", "varieties-lemma-vanishing-degree-2g-and-1-line-bundle", "algebra-lemma-CM-over-regular-flat", "morphisms-lemma-smooth-flat-smooth-fibres" ], "ref_ids": [ 7063, 12576, 12580, 12203, 12576, 12569, 12575, 9589, 5414, 11120, 1107, 5325 ] } ], "ref_ids": [] }, { "id": 12572, "type": "theorem", "label": "pic-lemma-pic-descends", "categories": [ "pic" ], "title": "pic-lemma-pic-descends", "contents": [ "Let $k$ be a field. Let $X$ be a quasi-compact and quasi-separated scheme", "over $k$ with $H^0(X, \\mathcal{O}_X) = k$. If $X$ has a $k$-rational point,", "then for any Galois extension $k'/k$ we have", "$$", "\\Pic(X) = \\Pic(X_{k'})^{\\text{Gal}(k'/k)}", "$$", "Moreover the action of $\\text{Gal}(k'/k)$ on $\\Pic(X_{k'})$", "is continuous." ], "refs": [], "proofs": [ { "contents": [ "Since $\\text{Gal}(k'/k) = \\text{Aut}(k'/k)$ it acts (from the right)", "on $\\Spec(k')$, hence it acts (from the right) on", "$X_{k'} = X \\times_{\\Spec(k)} \\Spec(k')$, and since $\\Pic(-)$", "is a contravariant functor, it acts (from the left) on $\\Pic(X_{k'})$.", "If $k'/k$ is an infinite Galois extension, then we write", "$k' = \\colim k'_\\lambda$ as a filtered colimit of finite Galois", "extensions, see Fields, Lemma \\ref{fields-lemma-infinite-galois-limit}.", "Then $X_{k'} = \\lim X_{k_\\lambda}$", "(as in Limits, Section \\ref{limits-section-limits})", "and we obtain", "$$", "\\Pic(X_{k'}) = \\colim \\Pic(X_{k_\\lambda})", "$$", "by Limits, Lemma \\ref{limits-lemma-descend-invertible-modules}.", "Moreover, the transition maps in this system of abelian groups", "are injective by Varieties, Lemma \\ref{varieties-lemma-change-fields-pic}.", "It follows that every element of $\\Pic(X_{k'})$ is fixed by", "one of the open subgroups $\\text{Gal}(k'/k_\\lambda)$, which exactly", "means that the action is continuous. Injectivity of the transition maps", "implies that it suffices to prove the statement on fixed points in the", "case that $k'/k$ is finite Galois.", "\\medskip\\noindent", "Assume $k'/k$ is finite Galois with Galois group $G = \\text{Gal}(k'/k)$.", "Let $\\mathcal{L}$ be an element of $\\Pic(X_{k'})$ fixed by $G$.", "We will use Galois descent (Descent, Lemma \\ref{descent-lemma-galois-descent})", "to prove that $\\mathcal{L}$ is the pullback of an invertible sheaf on $X$.", "Recall that $f_\\sigma = \\text{id}_X \\times \\Spec(\\sigma) : X_{k'} \\to X_{k'}$", "and that $\\sigma$ acts on $\\Pic(X_{k'})$ by pulling back by $f_\\sigma$.", "Hence for each $\\sigma \\in G$ we can choose an isomorphism", "$\\varphi_\\sigma : \\mathcal{L} \\to f_\\sigma^*\\mathcal{L}$", "because $\\mathcal{L}$ is a fixed by the $G$-action.", "The trouble is that we don't know if we can choose", "$\\varphi_\\sigma$ such that the cocycle condition", "$\\varphi_{\\sigma\\tau} = f_\\sigma^*\\varphi_\\tau \\circ \\varphi_\\sigma$", "holds. To see that this is possible we use that $X$ has a $k$-rational point", "$x \\in X(k)$. Of course, $x$ similarly determines a $k'$-rational point", "$x' \\in X_{k'}$ which is fixed by $f_\\sigma$ for all $\\sigma$.", "Pick a nonzero element $s$ in the fibre of $\\mathcal{L}$ at $x'$;", "the fibre is the $1$-dimensional $k' = \\kappa(x')$-vector space", "$$", "\\mathcal{L}_{x'} \\otimes_{\\mathcal{O}_{X_{k'}, x'}} \\kappa(x').", "$$", "Then $f_\\sigma^*s$ is a nonzero element of the fibre of", "$f_\\sigma^*\\mathcal{L}$ at $x'$. Since we can multiply $\\varphi_\\sigma$", "by an element of $(k')^*$ we may assume that $\\varphi_\\sigma$ sends", "$s$ to $f_\\sigma^*s$. Then we see that both $\\varphi_{\\sigma\\tau}$ and", "$f_\\sigma^*\\varphi_\\tau \\circ \\varphi_\\sigma$", "send $s$ to $f_{\\sigma\\tau}^*s = f_\\tau^*f_\\sigma^*s$.", "Since $H^0(X_{k'}, \\mathcal{O}_{X_{k'}}) = k'$ these two isomorphisms", "have to be the same (as one is a global unit times the other", "and they agree in $x'$) and the proof is complete." ], "refs": [ "fields-lemma-infinite-galois-limit", "limits-lemma-descend-invertible-modules", "varieties-lemma-change-fields-pic", "descent-lemma-galois-descent" ], "ref_ids": [ 4511, 15079, 11027, 14610 ] } ], "ref_ids": [] }, { "id": 12573, "type": "theorem", "label": "pic-lemma-torsion-descends", "categories": [ "pic" ], "title": "pic-lemma-torsion-descends", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $X$ be a", "quasi-compact and quasi-separated scheme over $k$ with", "$H^0(X, \\mathcal{O}_X) = k$. Let $n$ be an integer prime to $p$.", "Then the map", "$$", "\\Pic(X)[n] \\longrightarrow \\Pic(X_{k'})[n]", "$$", "is bijective for any purely inseparable extension $k'/k$." ], "refs": [], "proofs": [ { "contents": [ "First we observe that the map $\\Pic(X) \\to \\Pic(X_{k'})$", "is injective by Varieties, Lemma \\ref{varieties-lemma-change-fields-pic}.", "Hence we have to show the map in the lemma is surjective.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_{X_{k'}}$-module", "which has order dividing $n$ in $\\Pic(X_{k'})$.", "Choose an isomorphism", "$\\alpha : \\mathcal{L}^{\\otimes n} \\to \\mathcal{O}_{X_{k'}}$", "of invertible modules.", "We will prove that we can descend the pair $(\\mathcal{L}, \\alpha)$", "to $X$.", "\\medskip\\noindent", "Set $A = k' \\otimes_k k'$. Since $k'/k$ is purely inseparable,", "the kernel of the multiplication map $A \\to k'$ is a", "locally nilpotent ideal $I$ of $A$. Observe that", "$$", "X_A = X \\times_{\\Spec(k)} \\Spec(A) = X_{k'} \\times_X X_{k'}", "$$", "comes with two projections $\\text{pr}_i : X_A \\to X_{k'}$, $i = 0, 1$", "which agree over $A/I$. Hence the invertible modules", "$\\mathcal{L}_i = \\text{pr}_i^*\\mathcal{L}$", "agree over the closed subscheme $X_{A/I} = X_{k'}$.", "Since $X_{A/I} \\to X_A$ is a thickening and since", "$\\mathcal{L}_i$ are $n$-torsion, we see that there", "exists an isomorphism $\\varphi : \\mathcal{L}_0 \\to \\mathcal{L}_1$ by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-torsion-pic-thickening}.", "We may pick $\\varphi$ to reduce to the identity modulo $I$. Namely,", "$H^0(X, \\mathcal{O}_X) = k$ implies", "$H^0(X_{k'}, \\mathcal{O}_{X_{k'}}) = k'$ by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}", "and $A \\to k'$ is surjective hence we can adjust $\\varphi$ by multiplying by", "a suitable element of $A$. Consider the map", "$$", "\\lambda : \\mathcal{O}_{X_A}", "\\xrightarrow{\\text{pr}_0^*\\alpha^{-1}}", "\\mathcal{L}_0^{\\otimes n}", "\\xrightarrow{\\varphi^{\\otimes n}}", "\\mathcal{L}_1^{\\otimes n}", "\\xrightarrow{\\text{pr}_0^*\\alpha}", "\\mathcal{O}_{X_A}", "$$", "We can view $\\lambda$ as an element of $A$ because", "$H^0(X_A, \\mathcal{O}_{X_A}) = A$ (same reference as above).", "Since $\\varphi$ reduces to the identity modulo $I$ we", "see that $\\lambda = 1 \\bmod I$. Then there is a unique", "$n$th root of $\\lambda$ in $1 + I$", "(Algebra, Lemma \\ref{algebra-lemma-lift-nth-roots})", "and after multiplying $\\varphi$ by its inverse we get $\\lambda = 1$.", "We claim that $(\\mathcal{L}, \\varphi)$ is a descent datum", "for the fpqc covering $\\{X_{k'} \\to X\\}$", "(Descent, Definition \\ref{descent-definition-descent-datum-quasi-coherent}).", "If true, then $\\mathcal{L}$ is the pullback of an invertible", "$\\mathcal{O}_X$-module $\\mathcal{N}$ by", "Descent, Proposition \\ref{descent-proposition-fpqc-descent-quasi-coherent}.", "Injectivity of the map on Picard groups shows that $\\mathcal{N}$", "is a torsion element of $\\Pic(X)$ of the same order as $\\mathcal{L}$.", "\\medskip\\noindent", "Proof of the claim. To see this we have to verify that", "$$", "\\text{pr}_{12}^*\\varphi \\circ", "\\text{pr}_{01}^*\\varphi =", "\\text{pr}_{02}^*\\varphi", "\\quad\\text{on}\\quad", "X_{k'} \\times_X X_{k'} \\times_X X_{k'} = X_{k' \\otimes_k k' \\otimes_k k'}", "$$", "As before the diagonal morphism", "$\\Delta : X_{k'} \\to X_{k' \\otimes_k k' \\otimes_k k'}$", "is a thickening. The left and right hand sides of the equality signs", "are maps $a, b : p_0^*\\mathcal{L} \\to p_2^*\\mathcal{L}$ compatible with", "$p_0^*\\alpha$ and $p_2^*\\alpha$ where", "$p_i : X_{k' \\otimes_k k' \\otimes_k k'} \\to X_{k'}$", "are the projection morphisms. Finally, $a, b$ pull back to the", "same map under $\\Delta$.", "Affine locally (in local trivializations) this means that", "$a, b$ are given by multiplication by invertible functions", "which reduce to the same function modulo a locally nilpotent", "ideal and which have the same $n$th powers. Then it follows", "from Algebra, Lemma \\ref{algebra-lemma-lift-nth-roots}", "that these functions are the same." ], "refs": [ "varieties-lemma-change-fields-pic", "more-morphisms-lemma-torsion-pic-thickening", "coherent-lemma-flat-base-change-cohomology", "algebra-lemma-lift-nth-roots", "descent-definition-descent-datum-quasi-coherent", "descent-proposition-fpqc-descent-quasi-coherent", "algebra-lemma-lift-nth-roots" ], "ref_ids": [ 11027, 13687, 3298, 463, 14757, 14753, 463 ] } ], "ref_ids": [] }, { "id": 12574, "type": "theorem", "label": "pic-proposition-hilb-d-representable", "categories": [ "pic" ], "title": "pic-proposition-hilb-d-representable", "contents": [ "Let $X \\to S$ be a morphism of schemes. Let $d \\geq 0$. Assume", "for all $(s, x_1, \\ldots, x_d)$ where $s \\in S$ and", "$x_1, \\ldots, x_d \\in X_s$ there exists an affine open $U \\subset X$", "with $x_1, \\ldots, x_d \\in U$. Then $\\Hilbfunctor^d_{X/S}$ is", "representable by a scheme." ], "refs": [], "proofs": [ { "contents": [ "Either using relative glueing (Constructions, Section", "\\ref{constructions-section-relative-glueing}) or using", "the functorial point of view", "(Schemes, Lemma \\ref{schemes-lemma-glue-functors})", "we reduce to the case where $S$ is affine. Details omitted.", "\\medskip\\noindent", "Assume $S$ is affine. For $U \\subset X$ affine open, denote", "$F_U \\subset \\Hilbfunctor^d_{X/S}$ the subfunctor such that", "for a scheme $T/S$ an element $Z \\in \\Hilbfunctor^d_{X/S}(T)$", "is in $F_U(T)$ if and only if $Z \\subset U_T$. We will use", "Schemes, Lemma \\ref{schemes-lemma-glue-functors}", "and the subfunctors $F_U$ to conclude.", "\\medskip\\noindent", "Condition (1) is Lemma \\ref{lemma-hilb-d-sheaf}.", "\\medskip\\noindent", "Condition (2)(a) follows from the fact that $F_U = \\Hilbfunctor^d_{U/S}$", "and that this is representable by Lemma \\ref{lemma-hilb-d-An}.", "Namely, if $Z \\in F_U(T)$, then $Z$ can be viewed as a closed subscheme", "of $U_T$ which is finite locally free of degree $d$ over $T$ and hence", "$Z \\in \\Hilbfunctor^d_{U/S}(T)$. Conversely, if $Z \\in \\Hilbfunctor^d_{U/S}(T)$", "then $Z \\to U_T \\to X_T$ is a closed immersion\\footnote{This is clear", "if $X \\to S$ is separated as in this case Morphisms, Lemma", "\\ref{morphisms-lemma-image-proper-scheme-closed}", "tells us that the immersion $\\varphi : Z \\to X_T$ has closed image", "and hence is a closed immersion by", "Schemes, Lemma \\ref{schemes-lemma-immersion-when-closed}. We suggest the", "reader skip the rest of this footnote as we don't know of any instance", "where the assumptions on $X \\to S$ hold but $X \\to S$ is not separated.", "In the general case, let $x \\in X_T$ be a point in the closure of", "$\\varphi(Z)$. We have to show that $x \\in \\varphi(Z)$. Let $t \\in T$ be the", "image of $x$. By assumption on $X \\to S$ we can choose an affine open", "$W \\subset X_T$ containing $x$ and $\\varphi(Z_t)$. Then $\\varphi^{-1}(W)$", "is an open containing the whole fibre $Z_t$ and since $Z \\to T$ is closed,", "we may after replacing $T$ by an open neighbourhood of $t$ assume that", "$Z = \\varphi^{-1}(W)$. Then $\\varphi(Z) \\subset W$ is closed by the", "separated case (as $W \\to T$ is separated) and we conclude $x \\in \\varphi(Z)$.}", "and we may view $Z$ as an element of $F_U(T)$.", "\\medskip\\noindent", "Let $Z \\in \\Hilbfunctor^d_{X/S}(T)$ for some scheme $T$ over $S$. Let", "$$", "B = (Z \\to T)\\left((Z \\to X_T \\to X)^{-1}(X \\setminus U)\\right)", "$$", "This is a closed subset of $T$ and it is clear that over the open", "$T_{Z, U} = T \\setminus B$ the restriction $Z_{t'}$ maps into $U_{T'}$.", "On the other hand, for any $b \\in B$ the fibre $Z_b$ does not map", "into $U$. Thus we see that given a morphism $T' \\to T$ we", "have $Z_{T'} \\in F_U(T')$ $\\Leftrightarrow$ $T' \\to T$ factors through", "the open $T_{Z, U}$. This proves condition (2)(b).", "\\medskip\\noindent", "Condition (2)(c) follows from our assumption on $X/S$. All we have", "to do is show the following: If $T$ is the spectrum of a field", "and $Z \\subset X_T$ is a closed subscheme, finite flat of degree", "$d$ over $T$, then $Z \\to X_T \\to X$ factors through an affine open", "$U$ of $X$. This is clear because $Z$ will have at most $d$ points", "and these will all map into the fibre of $X$ over the image point", "of $T \\to S$." ], "refs": [ "schemes-lemma-glue-functors", "schemes-lemma-glue-functors", "pic-lemma-hilb-d-sheaf", "pic-lemma-hilb-d-An", "morphisms-lemma-image-proper-scheme-closed", "schemes-lemma-immersion-when-closed" ], "ref_ids": [ 7688, 7688, 12554, 12558, 5411, 7671 ] } ], "ref_ids": [] }, { "id": 12575, "type": "theorem", "label": "pic-proposition-hilb-d", "categories": [ "pic" ], "title": "pic-proposition-hilb-d", "contents": [ "Let $X$ be a geometrically irreducible smooth proper curve over a field $k$.", "\\begin{enumerate}", "\\item The functors $\\Hilbfunctor^d_{X/k}$ are representable by smooth", "proper varieties $\\underline{\\Hilbfunctor}^d_{X/k}$ of dimension", "$d$ over $k$.", "\\item For a field extension $k'/k$ the $k'$-rational points", "of $\\underline{\\Hilbfunctor}^d_{X/k}$ are in $1$-to-$1$ bijection", "with effective Cartier divisors of degree $d$ on $X_{k'}$.", "\\item For $d_1, d_2 \\geq 0$ there is a morphism", "$$", "\\underline{\\Hilbfunctor}^{d_1}_{X/k}", "\\times_k", "\\underline{\\Hilbfunctor}^{d_2}_{X/k}", "\\longrightarrow", "\\underline{\\Hilbfunctor}^{d_1 + d_2}_{X/k}", "$$", "which is finite locally free of degree ${d_1 + d_2 \\choose d_1}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The functors $\\Hilbfunctor^d_{X/k}$ are representable by", "Proposition \\ref{proposition-hilb-d-representable}", "(see also Remark \\ref{remark-when-proposition-applies})", "and the fact that $X$ is projective", "(Varieties, Lemma \\ref{varieties-lemma-dim-1-proper-projective}).", "The schemes $\\underline{\\Hilbfunctor}^d_{X/k}$ are separated", "over $k$ by Lemma \\ref{lemma-hilb-d-separated}.", "The schemes $\\underline{\\Hilbfunctor}^d_{X/k}$ are smooth", "over $k$ by Lemma \\ref{lemma-hilb-d-smooth}.", "Starting with $X = \\underline{\\Hilbfunctor}^1_{X/k}$,", "the morphisms of Lemma \\ref{lemma-universal-object},", "and induction we find a morphism", "$$", "X^d = X \\times_k X \\times_k \\ldots \\times_k X \\longrightarrow", "\\underline{\\Hilbfunctor}^d_{X/k},\\quad", "(x_1, \\ldots, x_d) \\longrightarrow x_1 + \\ldots + x_d", "$$", "which is finite locally free of degree $d!$. Since $X$ is", "proper over $k$, so is $X^d$, hence", "$\\underline{\\Hilbfunctor}^d_{X/k}$ is proper over $k$ by", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-is-proper}.", "Since $X$ is geometrically irreducible over $k$, the product", "$X^d$ is irreducible", "(Varieties, Lemma \\ref{varieties-lemma-bijection-irreducible-components})", "hence the image is irreducible (in fact geometrically irreducible).", "This proves (1). Part (2) follows from the definitions. Part (3) follows", "from the commutative diagram", "$$", "\\xymatrix{", "X^{d_1} \\times_k X^{d_2} \\ar[d] \\ar@{=}[r] & X^{d_1 + d_2} \\ar[d] \\\\", "\\underline{\\Hilbfunctor}^{d_1}_{X/k}", "\\times_k", "\\underline{\\Hilbfunctor}^{d_2}_{X/k}", "\\ar[r] &", "\\underline{\\Hilbfunctor}^{d_1 + d_2}_{X/k}", "}", "$$", "and multiplicativity of degrees of finite locally free morphisms." ], "refs": [ "pic-proposition-hilb-d-representable", "pic-remark-when-proposition-applies", "varieties-lemma-dim-1-proper-projective", "pic-lemma-hilb-d-separated", "pic-lemma-hilb-d-smooth", "pic-lemma-universal-object", "morphisms-lemma-image-proper-is-proper", "varieties-lemma-bijection-irreducible-components" ], "ref_ids": [ 12574, 12579, 11099, 12557, 12563, 12562, 5413, 10934 ] } ], "ref_ids": [] }, { "id": 12576, "type": "theorem", "label": "pic-proposition-pic-curve", "categories": [ "pic" ], "title": "pic-proposition-pic-curve", "contents": [ "Let $k$ be a separably closed field. Let $X$ be a smooth projective", "curve over $k$. The Picard functor $\\Picardfunctor_{X/k}$ is representable." ], "refs": [], "proofs": [ { "contents": [ "Since $k$ is separably closed there exists a $k$-rational point $\\sigma$", "of $X$, see", "Varieties, Lemma \\ref{varieties-lemma-smooth-separable-closed-points-dense}.", "As discussed above, it suffices to show that the functor", "$\\Picardfunctor_{X/k, \\sigma}$ classifying invertible modules trivial along", "$\\sigma$ is representable. To do this we will check conditions (1),", "(2)(a), (2)(b), and (2)(c) of", "Lemma \\ref{lemma-criterion}.", "\\medskip\\noindent", "The functor $\\Picardfunctor_{X/k, \\sigma}$ satisfies the sheaf condition", "for the fppf topology because it is isomorphic to $\\Picardfunctor_{X/k}$.", "It would be more correct to say that we've shown the sheaf condition", "for $\\Picardfunctor_{X/k, \\sigma}$ in the proof of", "Lemma \\ref{lemma-flat-geometrically-connected-fibres-with-section}", "which applies by Lemma \\ref{lemma-check-conditions}.", "This proves condition (1)", "\\medskip\\noindent", "As our subfunctor we use $F$ as defined in Lemma \\ref{lemma-define-open}.", "Condition (2)(b) follows.", "Condition (2)(a) is Lemma \\ref{lemma-open-representable}.", "Condition (2)(c) is Lemma \\ref{lemma-twist-with-general-divisor}." ], "refs": [ "varieties-lemma-smooth-separable-closed-points-dense", "pic-lemma-criterion", "pic-lemma-check-conditions", "pic-lemma-define-open", "pic-lemma-open-representable", "pic-lemma-twist-with-general-divisor" ], "ref_ids": [ 11007, 12566, 12567, 12568, 12569, 12570 ] } ], "ref_ids": [] }, { "id": 12581, "type": "theorem", "label": "constructions-lemma-relative-glueing", "categories": [ "constructions" ], "title": "constructions-lemma-relative-glueing", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{B}$ be a basis for the topology of $S$.", "Suppose given the following data:", "\\begin{enumerate}", "\\item For every $U \\in \\mathcal{B}$ a scheme $f_U : X_U \\to U$ over $U$.", "\\item For $U, V \\in \\mathcal{B}$ with $V \\subset U$ a morphism", "$\\rho^U_V : X_V \\to X_U$ over $U$.", "\\end{enumerate}", "Assume that", "\\begin{enumerate}", "\\item[(a)] each $\\rho^U_V$ induces an isomorphism", "$X_V \\to f_U^{-1}(V)$ of schemes over $V$,", "\\item[(b)] whenever $W, V, U \\in \\mathcal{B}$, with", "$W \\subset V \\subset U$ we have $\\rho^U_W = \\rho^U_V \\circ \\rho ^V_W$.", "\\end{enumerate}", "Then there exists a morphism $f : X \\to S$ of schemes", "and isomorphisms $i_U : f^{-1}(U) \\to X_U$ over $U \\in \\mathcal{B}$", "such that for $V, U \\in \\mathcal{B}$ with $V \\subset U$ the composition", "$$", "\\xymatrix{", "X_V \\ar[r]^{i_V^{-1}} &", "f^{-1}(V) \\ar[rr]^{inclusion} & &", "f^{-1}(U) \\ar[r]^{i_U} &", "X_U", "}", "$$", "is the morphism $\\rho^U_V$. Moreover $X$ is unique up to", "unique isomorphism over $S$." ], "refs": [], "proofs": [ { "contents": [ "To prove this we will use Schemes, Lemma \\ref{schemes-lemma-glue-functors}.", "First we define a contravariant functor $F$ from the category of schemes", "to the category of sets. Namely, for a scheme $T$ we set", "$$", "F(T) =", "\\left\\{", "\\begin{matrix}", "(g, \\{h_U\\}_{U \\in \\mathcal{B}}),", "\\ g : T \\to S, \\ h_U : g^{-1}(U) \\to X_U, \\\\", "f_U \\circ h_U = g|_{g^{-1}(U)},", "\\ h_U|_{g^{-1}(V)} = \\rho^U_V \\circ h_V", "\\ \\forall\\ V, U \\in \\mathcal{B}, V \\subset U", "\\end{matrix}", "\\right\\}.", "$$", "The restriction mapping $F(T) \\to F(T')$ given a morphism", "$T' \\to T$ is just gotten by composition.", "For any $W \\in \\mathcal{B}$ we consider the subfunctor", "$F_W \\subset F$ consisting of those systems $(g, \\{h_U\\})$", "such that $g(T) \\subset W$.", "\\medskip\\noindent", "First we show $F$ satisfies the sheaf property for the Zariski topology.", "Suppose that $T$ is a scheme, $T = \\bigcup V_i$ is an open covering,", "and $\\xi_i \\in F(V_i)$ is an element such that", "$\\xi_i|_{V_i \\cap V_j} = \\xi_j|_{V_i \\cap V_j}$.", "Say $\\xi_i = (g_i, \\{h_{i, U}\\})$. Then we immediately see that", "the morphisms $g_i$ glue to a unique global morphism", "$g : T \\to S$. Moreover, it is clear that", "$g^{-1}(U) = \\bigcup g_i^{-1}(U)$. Hence the morphisms", "$h_{i, U} : g_i^{-1}(U) \\to X_U$ glue to a unique morphism", "$h_U : U \\to X_U$. It is easy to verify that the system", "$(g, \\{f_U\\})$ is an element of $F(T)$. Hence $F$ satisfies the", "sheaf property for the Zariski topology.", "\\medskip\\noindent", "Next we verify that each $F_W$, $W \\in \\mathcal{B}$ is representable.", "Namely, we claim that the transformation of functors", "$$", "F_W \\longrightarrow \\Mor(-, X_W), \\ (g, \\{h_U\\}) \\longmapsto h_W", "$$", "is an isomorphism. To see this suppose that $T$ is a scheme and", "$\\alpha : T \\to X_W$ is a morphism. Set $g = f_W \\circ \\alpha$.", "For any $U \\in \\mathcal{B}$ such that $U \\subset W$ we can", "define $h_U : g^{-1}(U) \\to X_U$ be the composition", "$(\\rho^W_U)^{-1} \\circ \\alpha|_{g^{-1}(U)}$. This works because", "the image $\\alpha(g^{-1}(U))$ is contained in $f_W^{-1}(U)$ and", "condition (a) of the lemma. It is clear that", "$f_U \\circ h_U = g|_{g^{-1}(U)}$ for such a $U$.", "Moreover, if also $V \\in \\mathcal{B}$ and $V \\subset U \\subset W$,", "then $\\rho^U_V \\circ h_V = h_U|_{g^{-1}(V)}$ by property (b)", "of the lemma. We still have to define $h_U$ for an arbitrary", "element $U \\in \\mathcal{B}$. Since $\\mathcal{B}$ is a basis for", "the topology on $S$ we can find an open covering", "$U \\cap W = \\bigcup U_i$ with $U_i \\in \\mathcal{B}$. Since $g$ maps into $W$", "we have", "$g^{-1}(U) = g^{-1}(U \\cap W) = \\bigcup g^{-1}(U_i)$.", "Consider the morphisms", "$h_i = \\rho^U_{U_i} \\circ h_{U_i} : g^{-1}(U_i) \\to X_U$.", "It is a simple matter to use condition (b) of the lemma", "to prove that", "$h_i|_{g^{-1}(U_i) \\cap g^{-1}(U_j)} = h_j|_{g^{-1}(U_i) \\cap g^{-1}(U_j)}$.", "Hence these morphisms glue to give the desired morphism", "$h_U : g^{-1}(U) \\to X_U$. We omit the (easy) verification that", "the system $(g, \\{h_U\\})$ is an element of $F_W(T)$ which", "maps to $\\alpha$ under the displayed arrow above.", "\\medskip\\noindent", "Next, we verify each $F_W \\subset F$ is representable by open immersions.", "This is clear from the definitions.", "\\medskip\\noindent", "Finally we have to verify", "the collection $(F_W)_{W \\in \\mathcal{B}}$ covers $F$.", "This is clear by construction and the fact that $\\mathcal{B}$ is", "a basis for the topology of $S$.", "\\medskip\\noindent", "Let $X$ be a scheme representing the functor $F$.", "Let $(f, \\{i_U\\}) \\in F(X)$ be a ``universal family''.", "Since each $F_W$ is representable by $X_W$ (via the morphism of functors", "displayed above) we see that $i_W : f^{-1}(W) \\to X_W$", "is an isomorphism as desired. The lemma is proved." ], "refs": [ "schemes-lemma-glue-functors" ], "ref_ids": [ 7688 ] } ], "ref_ids": [] }, { "id": 12582, "type": "theorem", "label": "constructions-lemma-relative-glueing-sheaves", "categories": [ "constructions" ], "title": "constructions-lemma-relative-glueing-sheaves", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{B}$ be a basis for the topology of $S$.", "Suppose given the following data:", "\\begin{enumerate}", "\\item For every $U \\in \\mathcal{B}$ a scheme $f_U : X_U \\to U$ over $U$.", "\\item For every $U \\in \\mathcal{B}$ a quasi-coherent sheaf $\\mathcal{F}_U$", "over $X_U$.", "\\item For every pair $U, V \\in \\mathcal{B}$ such that", "$V \\subset U$ a morphism $\\rho^U_V : X_V \\to X_U$.", "\\item For every pair $U, V \\in \\mathcal{B}$ such that", "$V \\subset U$ a morphism", "$\\theta^U_V : (\\rho^U_V)^*\\mathcal{F}_U \\to \\mathcal{F}_V$.", "\\end{enumerate}", "Assume that", "\\begin{enumerate}", "\\item[(a)] each $\\rho^U_V$ induces an isomorphism", "$X_V \\to f_U^{-1}(V)$ of schemes over $V$,", "\\item[(b)] each $\\theta^U_V$ is an isomorphism,", "\\item[(c)] whenever $W, V, U \\in \\mathcal{B}$, with", "$W \\subset V \\subset U$ we have $\\rho^U_W = \\rho^U_V \\circ \\rho ^V_W$,", "\\item[(d)] whenever $W, V, U \\in \\mathcal{B}$, with", "$W \\subset V \\subset U$ we have", "$\\theta^U_W = \\theta^V_W \\circ (\\rho^V_W)^*\\theta^U_V$.", "\\end{enumerate}", "Then there exists a morphism of schemes $f : X \\to S$", "together with a quasi-coherent sheaf $\\mathcal{F}$ on $X$", "and isomorphisms $i_U : f^{-1}(U) \\to X_U$ and", "$\\theta_U : i_U^*\\mathcal{F}_U \\to \\mathcal{F}|_{f^{-1}(U)}$", "over $U \\in \\mathcal{B}$ such that", "for $V, U \\in \\mathcal{B}$ with $V \\subset U$ the composition", "$$", "\\xymatrix{", "X_V \\ar[r]^{i_V^{-1}} &", "f^{-1}(V) \\ar[rr]^{inclusion} & &", "f^{-1}(U) \\ar[r]^{i_U} &", "X_U", "}", "$$", "is the morphism $\\rho^U_V$, and the composition", "\\begin{equation}", "\\label{equation-glue}", "(\\rho^U_V)^*\\mathcal{F}_U", "=", "(i_V^{-1})^*((i_U^*\\mathcal{F}_U)|_{f^{-1}(V)})", "\\xrightarrow{\\theta_U|_{f^{-1}(V)}}", "(i_V^{-1})^*(\\mathcal{F}|_{f^{-1}(V)})", "\\xrightarrow{\\theta_V^{-1}}", "\\mathcal{F}_V", "\\end{equation}", "is equal to $\\theta^U_V$. Moreover $(X, \\mathcal{F})$ is unique", "up to unique isomorphism over $S$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-relative-glueing} we get the scheme $X$ over $S$", "and the isomorphisms $i_U$.", "Set $\\mathcal{F}'_U = i_U^*\\mathcal{F}_U$ for $U \\in \\mathcal{B}$.", "This is a quasi-coherent $\\mathcal{O}_{f^{-1}(U)}$-module.", "The maps", "$$", "\\mathcal{F}'_U|_{f^{-1}(V)} =", "i_U^*\\mathcal{F}_U|_{f^{-1}(V)} =", "i_V^*(\\rho^U_V)^*\\mathcal{F}_U \\xrightarrow{i_V^*\\theta^U_V}", "i_V^*\\mathcal{F}_V = \\mathcal{F}'_V", "$$", "define isomorphisms", "$(\\theta')^U_V : \\mathcal{F}'_U|_{f^{-1}(V)} \\to \\mathcal{F}'_V$", "whenever $V \\subset U$ are elements of $\\mathcal{B}$.", "Condition (d) says exactly that this is compatible in case", "we have a triple of elements $W \\subset V \\subset U$ of $\\mathcal{B}$.", "This allows us to get well defined isomorphisms", "$$", "\\varphi_{12} :", "\\mathcal{F}'_{U_1}|_{f^{-1}(U_1 \\cap U_2)}", "\\longrightarrow", "\\mathcal{F}'_{U_2}|_{f^{-1}(U_1 \\cap U_2)}", "$$", "whenever $U_1, U_2 \\in \\mathcal{B}$ by covering the intersection", "$U_1 \\cap U_2 = \\bigcup V_j$ by elements $V_j$ of $\\mathcal{B}$", "and taking", "$$", "\\varphi_{12}|_{V_j} =", "\\left((\\theta')^{U_2}_{V_j}\\right)^{-1}", "\\circ", "(\\theta')^{U_1}_{V_j}.", "$$", "We omit the verification that these maps do indeed glue to a", "$\\varphi_{12}$ and we omit the verification of the", "cocycle condition of a glueing datum for sheaves", "(as in Sheaves, Section \\ref{sheaves-section-glueing-sheaves}).", "By Sheaves, Lemma \\ref{sheaves-lemma-glue-sheaves}", "we get our $\\mathcal{F}$ on $X$. We omit the verification", "of (\\ref{equation-glue})." ], "refs": [ "constructions-lemma-relative-glueing", "sheaves-lemma-glue-sheaves" ], "ref_ids": [ 12581, 14556 ] } ], "ref_ids": [] }, { "id": 12583, "type": "theorem", "label": "constructions-lemma-spec-inclusion", "categories": [ "constructions" ], "title": "constructions-lemma-spec-inclusion", "contents": [ "In Situation \\ref{situation-relative-spec}.", "Suppose $U \\subset U' \\subset S$ are affine opens.", "Let $A = \\mathcal{A}(U)$ and $A' = \\mathcal{A}(U')$.", "The map of rings $A' \\to A$ induces a morphism", "$\\Spec(A) \\to \\Spec(A')$, and the diagram", "$$", "\\xymatrix{", "\\Spec(A) \\ar[r] \\ar[d] &", "\\Spec(A') \\ar[d] \\\\", "U \\ar[r] &", "U'", "}", "$$", "is cartesian." ], "refs": [], "proofs": [ { "contents": [ "Let $R = \\mathcal{O}_S(U)$ and $R' = \\mathcal{O}_S(U')$.", "Note that the map $R \\otimes_{R'} A' \\to A$ is an isomorphism as", "$\\mathcal{A}$ is quasi-coherent", "(see Schemes, Lemma \\ref{schemes-lemma-widetilde-pullback} for example).", "The result follows from the description of the fibre product of", "affine schemes in", "Schemes, Lemma \\ref{schemes-lemma-fibre-product-affine-schemes}." ], "refs": [ "schemes-lemma-widetilde-pullback", "schemes-lemma-fibre-product-affine-schemes" ], "ref_ids": [ 7662, 7658 ] } ], "ref_ids": [] }, { "id": 12584, "type": "theorem", "label": "constructions-lemma-transitive-spec", "categories": [ "constructions" ], "title": "constructions-lemma-transitive-spec", "contents": [ "In Situation \\ref{situation-relative-spec}.", "Suppose $U \\subset U' \\subset U'' \\subset S$ are affine opens.", "Let $A = \\mathcal{A}(U)$, $A' = \\mathcal{A}(U')$ and $A'' = \\mathcal{A}(U'')$.", "The composition of the morphisms", "$\\Spec(A) \\to \\Spec(A')$, and", "$\\Spec(A') \\to \\Spec(A'')$ of", "Lemma \\ref{lemma-spec-inclusion} gives the", "morphism $\\Spec(A) \\to \\Spec(A'')$", "of Lemma \\ref{lemma-spec-inclusion}." ], "refs": [ "constructions-lemma-spec-inclusion", "constructions-lemma-spec-inclusion" ], "proofs": [ { "contents": [ "This follows as the map $A'' \\to A$ is the composition of $A'' \\to A'$ and", "$A' \\to A$ (because $\\mathcal{A}$ is a sheaf)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12583, 12583 ] }, { "id": 12585, "type": "theorem", "label": "constructions-lemma-glue-relative-spec", "categories": [ "constructions" ], "title": "constructions-lemma-glue-relative-spec", "contents": [ "In Situation \\ref{situation-relative-spec}.", "There exists a morphism of schemes", "$$", "\\pi : \\underline{\\Spec}_S(\\mathcal{A}) \\longrightarrow S", "$$", "with the following properties:", "\\begin{enumerate}", "\\item for every affine open $U \\subset S$ there exists an isomorphism", "$i_U : \\pi^{-1}(U) \\to \\Spec(\\mathcal{A}(U))$, and", "\\item for $U \\subset U' \\subset S$ affine open the composition", "$$", "\\xymatrix{", "\\Spec(\\mathcal{A}(U)) \\ar[r]^{i_U^{-1}} &", "\\pi^{-1}(U) \\ar[rr]^{inclusion} & &", "\\pi^{-1}(U') \\ar[r]^{i_{U'}} &", "\\Spec(\\mathcal{A}(U'))", "}", "$$", "is the open immersion of Lemma \\ref{lemma-spec-inclusion} above.", "\\end{enumerate}" ], "refs": [ "constructions-lemma-spec-inclusion" ], "proofs": [ { "contents": [ "Follows immediately from", "Lemmas \\ref{lemma-relative-glueing},", "\\ref{lemma-spec-inclusion}, and", "\\ref{lemma-transitive-spec}." ], "refs": [ "constructions-lemma-relative-glueing", "constructions-lemma-spec-inclusion", "constructions-lemma-transitive-spec" ], "ref_ids": [ 12581, 12583, 12584 ] } ], "ref_ids": [ 12583 ] }, { "id": 12586, "type": "theorem", "label": "constructions-lemma-spec-base-change", "categories": [ "constructions" ], "title": "constructions-lemma-spec-base-change", "contents": [ "In Situation \\ref{situation-relative-spec}.", "Let $F$ be the functor", "associated to $(S, \\mathcal{A})$ above.", "Let $g : S' \\to S$ be a morphism of schemes.", "Set $\\mathcal{A}' = g^*\\mathcal{A}$. Let $F'$ be the", "functor associated to $(S', \\mathcal{A}')$ above.", "Then there is a canonical isomorphism", "$$", "F' \\cong h_{S'} \\times_{h_S} F", "$$", "of functors." ], "refs": [], "proofs": [ { "contents": [ "A pair $(f' : T \\to S', \\varphi' : (f')^*\\mathcal{A}' \\to \\mathcal{O}_T)$", "is the same as a pair $(f, \\varphi : f^*\\mathcal{A} \\to \\mathcal{O}_T)$", "together with a factorization of $f$ as $f = g \\circ f'$. Namely with", "this notation we have", "$(f')^* \\mathcal{A}' = (f')^*g^*\\mathcal{A} = f^*\\mathcal{A}$.", "Hence the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12587, "type": "theorem", "label": "constructions-lemma-spec-affine", "categories": [ "constructions" ], "title": "constructions-lemma-spec-affine", "contents": [ "In Situation \\ref{situation-relative-spec}.", "Let $F$ be the functor associated to $(S, \\mathcal{A})$ above.", "If $S$ is affine, then $F$ is representable by the", "affine scheme $\\Spec(\\Gamma(S, \\mathcal{A}))$." ], "refs": [], "proofs": [ { "contents": [ "Write $S = \\Spec(R)$ and $A = \\Gamma(S, \\mathcal{A})$.", "Then $A$ is an $R$-algebra and $\\mathcal{A} = \\widetilde A$.", "The ring map $R \\to A$ gives rise to a canonical map", "$$", "f_{univ} : \\Spec(A)", "\\longrightarrow", "S = \\Spec(R).", "$$", "We have", "$f_{univ}^*\\mathcal{A} = \\widetilde{A \\otimes_R A}$", "by Schemes, Lemma \\ref{schemes-lemma-widetilde-pullback}.", "Hence there is a canonical map", "$$", "\\varphi_{univ} :", "f_{univ}^*\\mathcal{A} = \\widetilde{A \\otimes_R A}", "\\longrightarrow", "\\widetilde A = \\mathcal{O}_{\\Spec(A)}", "$$", "coming from the $A$-module map $A \\otimes_R A \\to A$,", "$a \\otimes a' \\mapsto aa'$. We claim that the pair", "$(f_{univ}, \\varphi_{univ})$ represents $F$ in this case.", "In other words we claim that for any scheme $T$ the map", "$$", "\\Mor(T, \\Spec(A)) \\longrightarrow \\{\\text{pairs } (f, \\varphi)\\},\\quad", "a \\longmapsto (f_{univ} \\circ a, a^*\\varphi)", "$$", "is bijective.", "\\medskip\\noindent", "Let us construct the inverse map.", "For any pair $(f : T \\to S, \\varphi)$ we get the induced", "ring map", "$$", "\\xymatrix{", "A = \\Gamma(S, \\mathcal{A}) \\ar[r]^{f^*} &", "\\Gamma(T, f^*\\mathcal{A}) \\ar[r]^{\\varphi} &", "\\Gamma(T, \\mathcal{O}_T)", "}", "$$", "This induces a morphism of schemes $T \\to \\Spec(A)$", "by Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}.", "\\medskip\\noindent", "The verification that this map is inverse to the map", "displayed above is omitted." ], "refs": [ "schemes-lemma-widetilde-pullback", "schemes-lemma-morphism-into-affine" ], "ref_ids": [ 7662, 7655 ] } ], "ref_ids": [] }, { "id": 12588, "type": "theorem", "label": "constructions-lemma-spec", "categories": [ "constructions" ], "title": "constructions-lemma-spec", "contents": [ "In Situation \\ref{situation-relative-spec}.", "The functor $F$ is representable by a scheme." ], "refs": [], "proofs": [ { "contents": [ "We are going to use Schemes, Lemma \\ref{schemes-lemma-glue-functors}.", "\\medskip\\noindent", "First we check that $F$ satisfies the sheaf property for the", "Zariski topology. Namely, suppose that $T$ is a scheme,", "that $T = \\bigcup_{i \\in I} U_i$ is an open covering,", "and that $(f_i, \\varphi_i) \\in F(U_i)$ such that", "$(f_i, \\varphi_i)|_{U_i \\cap U_j} = (f_j, \\varphi_j)|_{U_i \\cap U_j}$.", "This implies that the morphisms $f_i : U_i \\to S$", "glue to a morphism of schemes $f : T \\to S$ such that", "$f|_{I_i} = f_i$, see Schemes, Section \\ref{schemes-section-glueing-schemes}.", "Thus $f_i^*\\mathcal{A} = f^*\\mathcal{A}|_{U_i}$ and by assumption the", "morphisms $\\varphi_i$ agree on $U_i \\cap U_j$. Hence by Sheaves,", "Section \\ref{sheaves-section-glueing-sheaves} these glue to a", "morphism of $\\mathcal{O}_T$-algebras $f^*\\mathcal{A} \\to \\mathcal{O}_T$.", "This proves that $F$ satisfies the sheaf condition with respect to", "the Zariski topology.", "\\medskip\\noindent", "Let $S = \\bigcup_{i \\in I} U_i$ be an affine open covering.", "Let $F_i \\subset F$ be the subfunctor consisting of", "those pairs $(f : T \\to S, \\varphi)$ such that", "$f(T) \\subset U_i$.", "\\medskip\\noindent", "We have to show each $F_i$ is representable.", "This is the case because $F_i$ is identified with", "the functor associated to $U_i$ equipped with", "the quasi-coherent $\\mathcal{O}_{U_i}$-algebra $\\mathcal{A}|_{U_i}$,", "by Lemma \\ref{lemma-spec-base-change}.", "Thus the result follows from Lemma \\ref{lemma-spec-affine}.", "\\medskip\\noindent", "Next we show that $F_i \\subset F$ is representable by open immersions.", "Let $(f : T \\to S, \\varphi) \\in F(T)$. Consider $V_i = f^{-1}(U_i)$.", "It follows from the definition of $F_i$ that given $a : T' \\to T$", "we gave $a^*(f, \\varphi) \\in F_i(T')$ if and only if $a(T') \\subset V_i$.", "This is what we were required to show.", "\\medskip\\noindent", "Finally, we have to show that the collection $(F_i)_{i \\in I}$", "covers $F$. Let $(f : T \\to S, \\varphi) \\in F(T)$.", "Consider $V_i = f^{-1}(U_i)$. Since $S = \\bigcup_{i \\in I} U_i$", "is an open covering of $S$ we see that $T = \\bigcup_{i \\in I} V_i$", "is an open covering of $T$. Moreover $(f, \\varphi)|_{V_i} \\in F_i(V_i)$.", "This finishes the proof of the lemma." ], "refs": [ "schemes-lemma-glue-functors", "constructions-lemma-spec-base-change", "constructions-lemma-spec-affine" ], "ref_ids": [ 7688, 12586, 12587 ] } ], "ref_ids": [] }, { "id": 12589, "type": "theorem", "label": "constructions-lemma-glueing-gives-functor-spec", "categories": [ "constructions" ], "title": "constructions-lemma-glueing-gives-functor-spec", "contents": [ "In Situation \\ref{situation-relative-spec}.", "The scheme $\\pi : \\underline{\\Spec}_S(\\mathcal{A}) \\to S$", "constructed in Lemma \\ref{lemma-glue-relative-spec}", "and the scheme representing the functor $F$ are", "canonically isomorphic as schemes over $S$." ], "refs": [ "constructions-lemma-glue-relative-spec" ], "proofs": [ { "contents": [ "Let $X \\to S$ be the scheme representing the functor $F$.", "Consider the sheaf of $\\mathcal{O}_S$-algebras", "$\\mathcal{R} = \\pi_*\\mathcal{O}_{\\underline{\\Spec}_S(\\mathcal{A})}$.", "By construction of $\\underline{\\Spec}_S(\\mathcal{A})$", "we have isomorphisms $\\mathcal{A}(U) \\to \\mathcal{R}(U)$", "for every affine open $U \\subset S$; this follows from", "Lemma \\ref{lemma-glue-relative-spec} part (1).", "For $U \\subset U' \\subset S$ open these isomorphisms are", "compatible with the restriction mappings; this follows from", "Lemma \\ref{lemma-glue-relative-spec} part (2).", "Hence by Sheaves, Lemma \\ref{sheaves-lemma-restrict-basis-equivalence-modules}", "these isomorphisms result from an isomorphism of $\\mathcal{O}_S$-algebras", "$\\varphi : \\mathcal{A} \\to \\mathcal{R}$. Hence this gives an element", "$(\\underline{\\Spec}_S(\\mathcal{A}), \\varphi)", "\\in F(\\underline{\\Spec}_S(\\mathcal{A}))$.", "Since $X$ represents the functor $F$ we get a corresponding", "morphism of schemes $can : \\underline{\\Spec}_S(\\mathcal{A}) \\to X$", "over $S$.", "\\medskip\\noindent", "Let $U \\subset S$ be any affine open. Let $F_U \\subset F$ be", "the subfunctor of $F$ corresponding to pairs $(f, \\varphi)$ over", "schemes $T$ with $f(T) \\subset U$. Clearly the base change", "$X_U$ represents $F_U$. Moreover, $F_U$ is represented by", "$\\Spec(\\mathcal{A}(U)) = \\pi^{-1}(U)$ according to", "Lemma \\ref{lemma-spec-affine}. In other words $X_U \\cong \\pi^{-1}(U)$.", "We omit the verification that this identification is brought about", "by the base change of the morphism $can$ to $U$." ], "refs": [ "constructions-lemma-glue-relative-spec", "constructions-lemma-glue-relative-spec", "sheaves-lemma-restrict-basis-equivalence-modules", "constructions-lemma-spec-affine" ], "ref_ids": [ 12585, 12585, 14537, 12587 ] } ], "ref_ids": [ 12585 ] }, { "id": 12590, "type": "theorem", "label": "constructions-lemma-spec-properties", "categories": [ "constructions" ], "title": "constructions-lemma-spec-properties", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent", "sheaf of $\\mathcal{O}_S$-algebras. Let", "$\\pi : \\underline{\\Spec}_S(\\mathcal{A}) \\to S$", "be the relative spectrum of $\\mathcal{A}$ over $S$.", "\\begin{enumerate}", "\\item For every affine open $U \\subset S$ the inverse image", "$\\pi^{-1}(U)$ is affine.", "\\item For every morphism $g : S' \\to S$ we have", "$S' \\times_S \\underline{\\Spec}_S(\\mathcal{A}) =", "\\underline{\\Spec}_{S'}(g^*\\mathcal{A})$.", "\\item", "The universal map", "$$", "\\mathcal{A}", "\\longrightarrow", "\\pi_*\\mathcal{O}_{\\underline{\\Spec}_S(\\mathcal{A})}", "$$", "is an isomorphism of $\\mathcal{O}_S$-algebras.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) comes from the description of the relative spectrum", "by glueing, see Lemma \\ref{lemma-glue-relative-spec}.", "Part (2) follows immediately from Lemma \\ref{lemma-spec-base-change}.", "Part (3) follows because it is local on $S$ and it is clear in case $S$", "is affine by Lemma \\ref{lemma-spec-affine} for example." ], "refs": [ "constructions-lemma-glue-relative-spec", "constructions-lemma-spec-base-change", "constructions-lemma-spec-affine" ], "ref_ids": [ 12585, 12586, 12587 ] } ], "ref_ids": [] }, { "id": 12591, "type": "theorem", "label": "constructions-lemma-canonical-morphism", "categories": [ "constructions" ], "title": "constructions-lemma-canonical-morphism", "contents": [ "Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism", "of schemes. By Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "the sheaf $f_*\\mathcal{O}_X$ is a quasi-coherent sheaf of", "$\\mathcal{O}_S$-algebras. There is a canonical morphism", "$$", "can : X \\longrightarrow \\underline{\\Spec}_S(f_*\\mathcal{O}_X)", "$$", "of schemes over $S$.", "For any affine open $U \\subset S$ the restriction $can|_{f^{-1}(U)}$", "is identified with the canonical morphism", "$$", "f^{-1}(U) \\longrightarrow \\Spec(\\Gamma(f^{-1}(U), \\mathcal{O}_X))", "$$", "coming from Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}." ], "refs": [ "schemes-lemma-push-forward-quasi-coherent", "schemes-lemma-morphism-into-affine" ], "proofs": [ { "contents": [ "The morphism comes, via the definition of $\\underline{\\Spec}$", "as the scheme representing the functor $F$, from the canonical map", "$\\varphi : f^*f_*\\mathcal{O}_X \\to \\mathcal{O}_X$ (which by adjointness of", "push and pull corresponds to", "$\\text{id} : f_*\\mathcal{O}_X \\to f_*\\mathcal{O}_X$).", "The statement on the restriction to $f^{-1}(U)$", "follows from the description of the relative spectrum over", "affines, see Lemma \\ref{lemma-spec-affine}." ], "refs": [ "constructions-lemma-spec-affine" ], "ref_ids": [ 12587 ] } ], "ref_ids": [ 7730, 7655 ] }, { "id": 12592, "type": "theorem", "label": "constructions-lemma-category-vector-bundles", "categories": [ "constructions" ], "title": "constructions-lemma-category-vector-bundles", "contents": [ "The category of vector bundles over a scheme $S$ is", "anti-equivalent to the category of quasi-coherent $\\mathcal{O}_S$-modules." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: In one direction one uses the functor", "$\\underline{\\Spec}_S(\\text{Sym}^*_{\\mathcal{O}_S}(-))$", "and in the other the functor", "$(\\pi : V \\to S) \\leadsto (\\pi_*\\mathcal{O}_V)_1$ where the subscript", "indicates we take the degree $1$ part." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12593, "type": "theorem", "label": "constructions-lemma-standard-open", "categories": [ "constructions" ], "title": "constructions-lemma-standard-open", "contents": [ "Let $S$ be a graded ring. Let $f \\in S$ homogeneous of positive degree.", "\\begin{enumerate}", "\\item If $g\\in S$ homogeneous of positive degree", "and $D_{+}(g) \\subset D_{+}(f)$, then", "\\begin{enumerate}", "\\item $f$ is invertible in $S_g$, and", "$f^{\\deg(g)}/g^{\\deg(f)}$ is invertible in $S_{(g)}$,", "\\item $g^e = af$ for some $e \\geq 1$ and $a \\in S$ homogeneous,", "\\item there is a canonical $S$-algebra map $S_f \\to S_g$,", "\\item there is a canonical $S_0$-algebra map $S_{(f)} \\to S_{(g)}$", "compatible with the map $S_f \\to S_g$,", "\\item the map $S_{(f)} \\to S_{(g)}$ induces an isomorphism", "$$", "(S_{(f)})_{g^{\\deg(f)}/f^{\\deg(g)}} \\cong S_{(g)},", "$$", "\\item these maps induce a commutative diagram of", "topological spaces", "$$", "\\xymatrix{", "D_{+}(g) \\ar[d] &", "\\{\\mathbf{Z}\\text{-graded primes of }S_g\\} \\ar[l] \\ar[r] \\ar[d] &", "\\Spec(S_{(g)}) \\ar[d] \\\\", "D_{+}(f) &", "\\{\\mathbf{Z}\\text{-graded primes of }S_f\\} \\ar[l] \\ar[r] &", "\\Spec(S_{(f)})", "}", "$$", "where the horizontal maps are homeomorphisms and the vertical maps", "are open immersions,", "\\item there are compatible canonical $S_f$ and $S_{(f)}$-module", "maps $M_f \\to M_g$ and $M_{(f)} \\to M_{(g)}$ for any graded $S$-module $M$,", "and", "\\item the map $M_{(f)} \\to M_{(g)}$ induces an isomorphism", "$$", "(M_{(f)})_{g^{\\deg(f)}/f^{\\deg(g)}} \\cong M_{(g)}.", "$$", "\\end{enumerate}", "\\item Any open covering of $D_{+}(f)$ can be refined to a finite", "open covering of the form $D_{+}(f) = \\bigcup_{i = 1}^n D_{+}(g_i)$.", "\\item Let $g_1, \\ldots, g_n \\in S$ be homogeneous of positive degree.", "Then $D_{+}(f) \\subset \\bigcup D_{+}(g_i)$", "if and only if", "$g_1^{\\deg(f)}/f^{\\deg(g_1)}, \\ldots, g_n^{\\deg(f)}/f^{\\deg(g_n)}$", "generate the unit ideal in $S_{(f)}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that $D_{+}(g) = \\Spec(S_{(g)})$ with identification", "given by the ring maps $S \\to S_g \\leftarrow S_{(g)}$, see", "Algebra, Lemma \\ref{algebra-lemma-topology-proj}.", "Thus $f^{\\deg(g)}/g^{\\deg(f)}$ is an element of $S_{(g)}$ which is not", "contained in any prime ideal, and hence invertible,", "see Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}.", "We conclude that (a) holds.", "Write the inverse of $f$ in $S_g$ as $a/g^d$.", "We may replace $a$ by its homogeneous part of degree $d\\deg(g) - \\deg(f)$.", "This means $g^d - af$ is annihilated by a power of $g$, whence", "$g^e = af$ for some $a \\in S$ homogeneous of degree $e\\deg(g) - \\deg(f)$.", "This proves (b).", "For (c), the map $S_f \\to S_g$ exists by (a) from the universal property", "of localization, or we can define it by mapping $b/f^n$", "to $a^nb/g^{ne}$. This clearly induces a map of the subrings", "$S_{(f)} \\to S_{(g)}$ of degree zero elements as well.", "We can similarly define $M_f \\to M_g$ and $M_{(f)} \\to M_{(g)}$ by mapping", "$x/f^n$ to $a^nx/g^{ne}$. The statements writing $S_{(g)}$", "resp.\\ $M_{(g)}$ as principal localizations of $S_{(f)}$ resp.\\ $M_{(f)}$", "are clear from the formulas above. The maps in the commutative diagram", "of topological spaces correspond to the ring maps given above. The", "horizontal arrows are homeomorphisms by", "Algebra, Lemma \\ref{algebra-lemma-topology-proj}.", "The vertical arrows are open immersions since the left", "one is the inclusion of an open subset.", "\\medskip\\noindent", "The open $D_{+}(f)$ is quasi-compact because it is homeomorphic", "to $\\Spec(S_{(f)})$, see Algebra, Lemma \\ref{algebra-lemma-quasi-compact}.", "Hence the second statement follows directly", "from the fact that the standard opens form", "a basis for the topology.", "\\medskip\\noindent", "The third statement follows directly from", "Algebra, Lemma \\ref{algebra-lemma-Zariski-topology}." ], "refs": [ "algebra-lemma-topology-proj", "algebra-lemma-Zariski-topology", "algebra-lemma-topology-proj", "algebra-lemma-quasi-compact", "algebra-lemma-Zariski-topology" ], "ref_ids": [ 660, 389, 660, 395, 389 ] } ], "ref_ids": [] }, { "id": 12594, "type": "theorem", "label": "constructions-lemma-proj-sheaves", "categories": [ "constructions" ], "title": "constructions-lemma-proj-sheaves", "contents": [ "Let $S$ be a graded ring. Let $M$ be a graded $S$-module.", "Let $\\widetilde M$ be the sheaf of $\\mathcal{O}_{\\text{Proj}(S)}$-modules", "associated to $M$.", "\\begin{enumerate}", "\\item For every $f \\in S$ homogeneous of positive degree we have", "$$", "\\Gamma(D_{+}(f), \\mathcal{O}_{\\text{Proj}(S)}) = S_{(f)}.", "$$", "\\item For every $f\\in S$ homogeneous of positive degree", "we have $\\Gamma(D_{+}(f), \\widetilde M) = M_{(f)}$", "as an $S_{(f)}$-module.", "\\item Whenever $D_{+}(g) \\subset D_{+}(f)$ the restriction mappings", "on $\\mathcal{O}_{\\text{Proj}(S)}$ and $\\widetilde M$", "are the maps", "$S_{(f)} \\to S_{(g)}$ and $M_{(f)} \\to M_{(g)}$ from Lemma", "\\ref{lemma-standard-open}.", "\\item Let $\\mathfrak p$ be a homogeneous prime of $S$ not containing", "$S_{+}$, and let $x \\in \\text{Proj}(S)$", "be the corresponding point. We have", "$\\mathcal{O}_{\\text{Proj}(S), x} = S_{(\\mathfrak p)}$.", "\\item Let $\\mathfrak p$ be a homogeneous prime of $S$ not containing", "$S_{+}$, and let $x \\in \\text{Proj}(S)$", "be the corresponding point. We have $\\mathcal{F}_x = M_{(\\mathfrak p)}$", "as an $S_{(\\mathfrak p)}$-module.", "\\item", "\\label{item-map}", "There is a canonical ring map", "$", "S_0 \\longrightarrow \\Gamma(\\text{Proj}(S), \\widetilde S)", "$", "and a canonical $S_0$-module map", "$", "M_0 \\longrightarrow \\Gamma(\\text{Proj}(S), \\widetilde M)", "$", "compatible with the descriptions of sections over standard opens", "and stalks above.", "\\end{enumerate}", "Moreover, all these identifications are functorial in the graded", "$S$-module $M$. In particular, the functor $M \\mapsto \\widetilde M$", "is an exact functor from the category of graded $S$-modules", "to the category of $\\mathcal{O}_{\\text{Proj}(S)}$-modules." ], "refs": [ "constructions-lemma-standard-open" ], "proofs": [ { "contents": [ "Assertions (1) - (5) are clear from the discussion above.", "We see (6) since there are canonical maps $M_0 \\to M_{(f)}$,", "$x \\mapsto x/1$ compatible with the restriction maps", "described in (3). The exactness of the functor $M \\mapsto \\widetilde M$", "follows from the fact that the functor $M \\mapsto M_{(\\mathfrak p)}$", "is exact (see Algebra, Lemma \\ref{algebra-lemma-proj-prime})", "and the fact that exactness of short exact sequences", "may be checked on stalks, see", "Modules, Lemma \\ref{modules-lemma-abelian}." ], "refs": [ "algebra-lemma-proj-prime", "modules-lemma-abelian" ], "ref_ids": [ 661, 13221 ] } ], "ref_ids": [ 12593 ] }, { "id": 12595, "type": "theorem", "label": "constructions-lemma-standard-open-proj", "categories": [ "constructions" ], "title": "constructions-lemma-standard-open-proj", "contents": [ "Let $S$ be a graded ring. Let $f \\in S$ be homogeneous of positive degree.", "Suppose that $D(g) \\subset \\Spec(S_{(f)})$ is a standard open.", "Then there exists a $h \\in S$ homogeneous of positive degree such that", "$D(g)$ corresponds to $D_{+}(h) \\subset D_{+}(f)$ via the homeomorphism", "of Algebra, Lemma \\ref{algebra-lemma-topology-proj}. In fact we can", "take $h$ such that $g = h/f^n$ for some $n$." ], "refs": [ "algebra-lemma-topology-proj" ], "proofs": [ { "contents": [ "Write $g = h/f^n$ for some $h$ homogeneous of positive degree", "and some $n \\geq 1$. If $D_{+}(h)$ is not contained in", "$D_{+}(f)$ then we replace $h$ by $hf$ and $n$ by $n + 1$.", "Then $h$ has the required shape and $D_{+}(h) \\subset D_{+}(f)$", "corresponds to $D(g) \\subset \\Spec(S_{(f)})$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 660 ] }, { "id": 12596, "type": "theorem", "label": "constructions-lemma-proj-scheme", "categories": [ "constructions" ], "title": "constructions-lemma-proj-scheme", "contents": [ "Let $S$ be a graded ring.", "The locally ringed space $\\text{Proj}(S)$ is a scheme.", "The standard opens $D_{+}(f)$ are affine opens.", "For any graded $S$-module $M$ the sheaf", "$\\widetilde M$ is a quasi-coherent sheaf of", "$\\mathcal{O}_{\\text{Proj}(S)}$-modules." ], "refs": [], "proofs": [ { "contents": [ "Consider a standard open $D_{+}(f) \\subset \\text{Proj}(S)$.", "By Lemmas \\ref{lemma-standard-open} and \\ref{lemma-proj-sheaves}", "we have $\\Gamma(D_{+}(f), \\mathcal{O}_{\\text{Proj}(S)}) = S_{(f)}$, and", "we have a homeomorphism $\\varphi : D_{+}(f) \\to \\Spec(S_{(f)})$.", "For any standard open $D(g) \\subset \\Spec(S_{(f)})$ we may", "pick a $h \\in S_{+}$ as in Lemma \\ref{lemma-standard-open-proj}.", "Then $\\varphi^{-1}(D(g)) = D_{+}(h)$, and by", "Lemmas \\ref{lemma-proj-sheaves} and \\ref{lemma-standard-open} we see", "$$", "\\Gamma(D_{+}(h), \\mathcal{O}_{\\text{Proj}(S)})", "=", "S_{(h)}", "=", "(S_{(f)})_{h^{\\deg(f)}/f^{\\deg(h)}}", "=", "(S_{(f)})_g", "=", "\\Gamma(D(g), \\mathcal{O}_{\\Spec(S_{(f)})}).", "$$", "Thus the restriction of $\\mathcal{O}_{\\text{Proj}(S)}$ to", "$D_{+}(f)$ corresponds via the homeomorphism $\\varphi$", "exactly to the sheaf $\\mathcal{O}_{\\Spec(S_{(f)})}$", "as defined in Schemes, Section \\ref{schemes-section-affine-schemes}.", "We conclude that $D_{+}(f)$ is an affine scheme isomorphic to", "$\\Spec(S_{(f)})$ via $\\varphi$ and", "hence that $\\text{Proj}(S)$ is a scheme.", "\\medskip\\noindent", "In exactly the same way we show that $\\widetilde M$ is a", "quasi-coherent sheaf of $\\mathcal{O}_{\\text{Proj}(S)}$-modules.", "Namely, the argument above will show that", "$$", "\\widetilde M|_{D_{+}(f)} \\cong \\varphi^*\\left(\\widetilde{M_{(f)}}\\right)", "$$", "which shows that $\\widetilde M$ is quasi-coherent." ], "refs": [ "constructions-lemma-standard-open", "constructions-lemma-proj-sheaves", "constructions-lemma-standard-open-proj", "constructions-lemma-proj-sheaves", "constructions-lemma-standard-open" ], "ref_ids": [ 12593, 12594, 12595, 12594, 12593 ] } ], "ref_ids": [] }, { "id": 12597, "type": "theorem", "label": "constructions-lemma-proj-separated", "categories": [ "constructions" ], "title": "constructions-lemma-proj-separated", "contents": [ "Let $S$ be a graded ring.", "The scheme $\\text{Proj}(S)$ is separated." ], "refs": [], "proofs": [ { "contents": [ "We have to show that the canonical morphism", "$\\text{Proj}(S) \\to \\Spec(\\mathbf{Z})$", "is separated.", "We will use Schemes, Lemma \\ref{schemes-lemma-characterize-separated}.", "Thus it suffices to show given any pair of standard opens", "$D_{+}(f)$ and $D_{+}(g)$ that $D_{+}(f) \\cap D_{+}(g) = D_{+}(fg)$", "is affine (clear) and that the ring map", "$$", "S_{(f)} \\otimes_{\\mathbf{Z}} S_{(g)} \\longrightarrow S_{(fg)}", "$$", "is surjective. Any element $s$ in $S_{(fg)}$ is of", "the form $s = h/(f^ng^m)$ with $h \\in S$ homogeneous of degree", "$n\\deg(f) + m\\deg(g)$. We may multiply $h$ by a suitable", "monomial $f^ig^j$ and assume that $n = n' \\deg(g)$, and", "$m = m' \\deg(f)$. Then we can rewrite $s$ as", "$s = h/f^{(n' + m')\\deg(g)} \\cdot f^{m'\\deg(g)}/g^{m'\\deg(f)}$.", "So $s$ is indeed in the image of the displayed arrow." ], "refs": [ "schemes-lemma-characterize-separated" ], "ref_ids": [ 7710 ] } ], "ref_ids": [] }, { "id": 12598, "type": "theorem", "label": "constructions-lemma-proj-quasi-compact", "categories": [ "constructions" ], "title": "constructions-lemma-proj-quasi-compact", "contents": [ "Let $S$ be a graded ring.", "The scheme $\\text{Proj}(S)$ is quasi-compact if and only", "if there exist finitely many homogeneous elements", "$f_1, \\ldots, f_n \\in S_{+}$ such that", "$S_{+} \\subset \\sqrt{(f_1, \\ldots, f_n)}$. In this case", "$\\text{Proj}(S) = D_+(f_1) \\cup \\ldots \\cup D_+(f_n)$." ], "refs": [], "proofs": [ { "contents": [ "Given such a collection of elements the standard affine opens", "$D_{+}(f_i)$ cover $\\text{Proj}(S)$ by", "Algebra, Lemma \\ref{algebra-lemma-topology-proj}.", "Conversely, if $\\text{Proj}(S)$ is quasi-compact, then we", "may cover it by finitely many standard opens", "$D_{+}(f_i)$, $i = 1, \\ldots, n$ and we see that", "$S_{+} \\subset \\sqrt{(f_1, \\ldots, f_n)}$ by the", "lemma referenced above." ], "refs": [ "algebra-lemma-topology-proj" ], "ref_ids": [ 660 ] } ], "ref_ids": [] }, { "id": 12599, "type": "theorem", "label": "constructions-lemma-structure-morphism-proj", "categories": [ "constructions" ], "title": "constructions-lemma-structure-morphism-proj", "contents": [ "Let $S$ be a graded ring. The scheme $\\text{Proj}(S)$ has a canonical morphism", "towards the affine scheme $\\Spec(S_0)$, agreeing with the map on", "topological spaces coming from", "Algebra, Definition \\ref{algebra-definition-proj}." ], "refs": [ "algebra-definition-proj" ], "proofs": [ { "contents": [ "We saw above that our construction of $\\widetilde S$,", "resp.\\ $\\widetilde M$ gives a sheaf of $S_0$-algebras, resp.\\ $S_0$-modules.", "Hence we get a morphism by", "Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}.", "This morphism, when restricted to $D_{+}(f)$ comes from the", "canonical ring map $S_0 \\to S_{(f)}$. The maps", "$S \\to S_f$, $S_{(f)} \\to S_f$ are $S_0$-algebra maps, see", "Lemma \\ref{lemma-standard-open}.", "Hence if the homogeneous prime $\\mathfrak p \\subset S$", "corresponds to the $\\mathbf{Z}$-graded prime $\\mathfrak p' \\subset S_f$", "and the (usual) prime $\\mathfrak p'' \\subset S_{(f)}$, then", "each of these has the same inverse image in $S_0$." ], "refs": [ "schemes-lemma-morphism-into-affine", "constructions-lemma-standard-open" ], "ref_ids": [ 7655, 12593 ] } ], "ref_ids": [ 1473 ] }, { "id": 12600, "type": "theorem", "label": "constructions-lemma-proj-valuative-criterion", "categories": [ "constructions" ], "title": "constructions-lemma-proj-valuative-criterion", "contents": [ "Let $S$ be a graded ring. If $S$ is finitely generated as", "an algebra over $S_0$, then", "the morphism $\\text{Proj}(S) \\to \\Spec(S_0)$ satisfies", "the existence and uniqueness parts of the valuative criterion,", "see Schemes, Definition \\ref{schemes-definition-valuative-criterion}." ], "refs": [ "schemes-definition-valuative-criterion" ], "proofs": [ { "contents": [ "The uniqueness part follows from the fact that $\\text{Proj}(S)$ is", "separated (Lemma \\ref{lemma-proj-separated} and", "Schemes, Lemma \\ref{schemes-lemma-separated-implies-valuative}).", "Choose $x_i \\in S_{+}$ homogeneous, $i = 1, \\ldots, n$", "which generate $S$ over $S_0$. Let $d_i = \\deg(x_i)$ and", "set $d = \\text{lcm}\\{d_i\\}$. Suppose we are given a diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & \\text{Proj}(S) \\ar[d] \\\\", "\\Spec(A) \\ar[r] & \\Spec(S_0)", "}", "$$", "as in Schemes, Definition \\ref{schemes-definition-valuative-criterion}.", "Denote $v : K^* \\to \\Gamma$ the valuation of $A$, see", "Algebra, Definition \\ref{algebra-definition-value-group}.", "We may choose an $f \\in S_{+}$ homogeneous such that", "$\\Spec(K)$ maps into $D_{+}(f)$. Then we get a commutative", "diagram of ring maps", "$$", "\\xymatrix{", "K & S_{(f)} \\ar[l]^{\\varphi} \\\\", "A \\ar[u] & S_0 \\ar[l] \\ar[u]", "}", "$$", "After renumbering we may assume that $\\varphi(x_i^{\\deg(f)}/f^{d_i})$", "is nonzero for $i = 1, \\ldots, r$ and zero for $i = r + 1, \\ldots, n$.", "Since the open sets $D_{+}(x_i)$ cover $\\text{Proj}(S)$ we see that $r \\geq 1$.", "Let $i_0 \\in \\{1, \\ldots, r\\}$ be an index minimizing", "$\\gamma_i = (d/d_i)v(\\varphi(x_i^{\\deg(f)}/f^{d_i}))$ in $\\Gamma$.", "For convenience set $x_0 = x_{i_0}$ and $d_0 = d_{i_0}$.", "The ring map $\\varphi$ factors though a map $\\varphi' : S_{(fx_0)} \\to K$", "which gives a ring map $S_{(x_0)} \\to S_{(fx_0)} \\to K$.", "The algebra $S_{(x_0)}$ is generated over $S_0$ by the elements", "$x_1^{e_1} \\ldots x_n^{e_n}/x_0^{e_0}$, where $\\sum e_i d_i = e_0 d_0$.", "If $e_i > 0$ for some $i > r$, then", "$\\varphi'(x_1^{e_1} \\ldots x_n^{e_n}/x_0^{e_0}) = 0$.", "If $e_i = 0$ for $i > r$, then we have", "\\begin{align*}", "\\deg(f) v(\\varphi'(x_1^{e_1} \\ldots x_r^{e_r}/x_0^{e_0}))", "& =", "v(\\varphi'(x_1^{e_1 \\deg(f)} \\ldots x_r^{e_r \\deg(f)}/x_0^{e_0 \\deg(f)})) \\\\", "& =", "\\sum e_i v(\\varphi'(x_i^{\\deg(f)}/f^{d_i}))", "- e_0 v(\\varphi'(x_0^{\\deg(f)}/f^{d_0})) \\\\", "& =", "\\sum e_i d_i \\gamma_i - e_0 d_0 \\gamma_0 \\\\", "& \\geq", "\\sum e_i d_i \\gamma_0 - e_0 d_0 \\gamma_0 = 0", "\\end{align*}", "because $\\gamma_0$ is minimal among the $\\gamma_i$.", "This implies that $S_{(x_0)}$ maps into $A$ via $\\varphi'$.", "The corresponding morphism of schemes", "$\\Spec(A) \\to \\Spec(S_{(x_0)}) = D_{+}(x_0)", "\\subset \\text{Proj}(S)$ provides the morphism fitting into", "the first commutative diagram of this proof." ], "refs": [ "constructions-lemma-proj-separated", "schemes-lemma-separated-implies-valuative", "schemes-definition-valuative-criterion", "algebra-definition-value-group" ], "ref_ids": [ 12597, 7719, 7755, 1468 ] } ], "ref_ids": [ 7755 ] }, { "id": 12601, "type": "theorem", "label": "constructions-lemma-widetilde-tensor", "categories": [ "constructions" ], "title": "constructions-lemma-widetilde-tensor", "contents": [ "Let $S$ be a graded ring.", "Let $(X, \\mathcal{O}_X) = (\\text{Proj}(S), \\mathcal{O}_{\\text{Proj}(S)})$", "be the scheme of Lemma \\ref{lemma-proj-scheme}.", "Let $f \\in S_{+}$ be homogeneous. Let $x \\in X$ be a point", "corresponding to the homogeneous prime $\\mathfrak p \\subset S$.", "Let $M$, $N$ be graded $S$-modules.", "There is a canonical map of $\\mathcal{O}_{\\text{Proj}(S)}$-modules", "$$", "\\widetilde M \\otimes_{\\mathcal{O}_X} \\widetilde N", "\\longrightarrow", "\\widetilde{M \\otimes_S N}", "$$", "which induces the canonical map", "$", "M_{(f)} \\otimes_{S_{(f)}} N_{(f)}", "\\to", "(M \\otimes_S N)_{(f)}", "$", "on sections over $D_{+}(f)$ and the canonical map", "$", "M_{(\\mathfrak p)} \\otimes_{S_{(\\mathfrak p)}} N_{(\\mathfrak p)}", "\\to", "(M \\otimes_S N)_{(\\mathfrak p)}", "$", "on stalks at $x$. Moreover, the following diagram", "$$", "\\xymatrix{", "M_0 \\otimes_{S_0} N_0 \\ar[r] \\ar[d] &", "(M \\otimes_S N)_0 \\ar[d] \\\\", "\\Gamma(X, \\widetilde M \\otimes_{\\mathcal{O}_X} \\widetilde N) \\ar[r] &", "\\Gamma(X, \\widetilde{M \\otimes_R N})", "}", "$$", "is commutative where the vertical maps are given by", "(\\ref{equation-map-global-sections})." ], "refs": [ "constructions-lemma-proj-scheme" ], "proofs": [ { "contents": [ "To construct a morphism as displayed is the same as constructing", "a $\\mathcal{O}_X$-bilinear map", "$$", "\\widetilde M \\times \\widetilde N", "\\longrightarrow", "\\widetilde{M \\otimes_R N}", "$$", "see Modules, Section \\ref{modules-section-tensor-product}.", "It suffices to define this on sections over the opens $D_{+}(f)$", "compatible with restriction mappings. On $D_{+}(f)$ we use the", "$S_{(f)}$-bilinear map", "$M_{(f)} \\times N_{(f)} \\to (M \\otimes_S N)_{(f)}$,", "$(x/f^n, y/f^m) \\mapsto (x \\otimes y)/f^{n + m}$. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12596 ] }, { "id": 12602, "type": "theorem", "label": "constructions-lemma-when-invertible", "categories": [ "constructions" ], "title": "constructions-lemma-when-invertible", "contents": [ "Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$.", "Let $f \\in S$ be homogeneous of degree $d > 0$.", "The sheaves $\\mathcal{O}_X(nd)|_{D_{+}(f)}$ are invertible,", "and in fact trivial for all $n \\in \\mathbf{Z}$", "(see Modules, Definition \\ref{modules-definition-invertible}).", "The maps (\\ref{equation-multiply}) restricted to $D_{+}(f)$", "$$", "\\mathcal{O}_X(nd)|_{D_{+}(f)} \\otimes_{\\mathcal{O}_{D_{+}(f)}}", "\\mathcal{O}_X(m)|_{D_{+}(f)}", "\\longrightarrow", "\\mathcal{O}_X(nd + m)|_{D_{+}(f)},", "$$", "the maps (\\ref{equation-multiply-on-sheaf}) restricted to $D_+(f)$", "$$", "\\mathcal{O}_X(nd)|_{D_{+}(f)} \\otimes_{\\mathcal{O}_{D_{+}(f)}}", "\\mathcal{F}(m)|_{D_{+}(f)}", "\\longrightarrow", "\\mathcal{F}(nd + m)|_{D_{+}(f)},", "$$", "and the maps (\\ref{equation-multiply-more-generally})", "restricted to $D_{+}(f)$", "$$", "\\widetilde M(nd)|_{D_{+}(f)}", "=", "\\widetilde M|_{D_{+}(f)}", "\\otimes_{\\mathcal{O}_{D_{+}(f)}}", "\\mathcal{O}_X(nd)|_{D_{+}(f)}", "\\longrightarrow", "\\widetilde{M(nd)}|_{D_{+}(f)}", "$$", "are isomorphisms for all $n, m \\in \\mathbf{Z}$." ], "refs": [ "modules-definition-invertible" ], "proofs": [ { "contents": [ "The (not graded) $S$-module maps $S \\to S(nd)$, and $M \\to M(nd)$, given by", "$x \\mapsto f^n x$ become isomorphisms after inverting $f$. The first shows that", "$S_{(f)} \\cong S(nd)_{(f)}$ which gives an isomorphism", "$\\mathcal{O}_{D_{+}(f)} \\cong \\mathcal{O}_X(nd)|_{D_{+}(f)}$.", "The second shows that the map", "$S(nd)_{(f)} \\otimes_{S_{(f)}} M_{(f)} \\to M(nd)_{(f)}$", "is an isomorphism. The case of the map (\\ref{equation-multiply-on-sheaf})", "is a consequence of the case of the map (\\ref{equation-multiply})." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13349 ] }, { "id": 12603, "type": "theorem", "label": "constructions-lemma-apply-modules", "categories": [ "constructions" ], "title": "constructions-lemma-apply-modules", "contents": [ "Let $S$ be a graded ring. Let $M$ be a graded $S$-module.", "Set $X = \\text{Proj}(S)$. Assume $X$ is covered by the standard", "opens $D_+(f)$ with $f \\in S_1$, e.g., if $S$ is generated by $S_1$", "over $S_0$. Then the sheaves $\\mathcal{O}_X(n)$", "are invertible and the maps", "(\\ref{equation-multiply}), (\\ref{equation-multiply-on-sheaf}), and", "(\\ref{equation-multiply-more-generally}) are isomorphisms.", "In particular, these maps induce isomorphisms", "$$", "\\mathcal{O}_X(1)^{\\otimes n} \\cong", "\\mathcal{O}_X(n)", "\\quad", "\\text{and}", "\\quad", "\\widetilde{M} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(n) =", "\\widetilde{M}(n) \\cong \\widetilde{M(n)}", "$$", "Thus (\\ref{equation-map-global-sections-degree-n}) becomes a map", "\\begin{equation}", "\\label{equation-map-global-sections-degree-n-simplified}", "M_n \\longrightarrow \\Gamma(X, \\widetilde{M}(n))", "\\end{equation}", "and (\\ref{equation-global-sections-more-generally}) becomes a map", "\\begin{equation}", "\\label{equation-global-sections-more-generally-simplified}", "M \\longrightarrow", "\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\Gamma(X, \\widetilde{M}(n)).", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Under the assumptions of the lemma $X$ is covered by the", "open subsets $D_{+}(f)$ with $f \\in S_1$ and the", "lemma is a consequence of Lemma \\ref{lemma-when-invertible} above." ], "refs": [ "constructions-lemma-when-invertible" ], "ref_ids": [ 12602 ] } ], "ref_ids": [] }, { "id": 12604, "type": "theorem", "label": "constructions-lemma-where-invertible", "categories": [ "constructions" ], "title": "constructions-lemma-where-invertible", "contents": [ "Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$. Fix $d \\geq 1$ an", "integer. The following open subsets of $X$ are equal:", "\\begin{enumerate}", "\\item The largest open subset $W = W_d \\subset X$ such that", "each $\\mathcal{O}_X(dn)|_W$ is invertible and all the", "multiplication maps", "$\\mathcal{O}_X(nd)|_W \\otimes_{\\mathcal{O}_W} \\mathcal{O}_X(md)|_W", "\\to \\mathcal{O}_X(nd + md)|_W$", "(see \\ref{equation-multiply}) are isomorphisms.", "\\item The union of the open subsets $D_{+}(fg)$ with", "$f, g \\in S$ homogeneous and $\\deg(f) = \\deg(g) + d$.", "\\end{enumerate}", "Moreover, all the maps", "$\\widetilde M(nd)|_W = \\widetilde M|_W \\otimes_{\\mathcal{O}_W}", "\\mathcal{O}_X(nd)|_W \\to \\widetilde{M(nd)}|_W$", "(see \\ref{equation-multiply-more-generally}) are isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "If $x \\in D_{+}(fg)$ with $\\deg(f) = \\deg(g) + d$ then", "on $D_{+}(fg)$ the sheaves $\\mathcal{O}_X(dn)$", "are generated by the element $(f/g)^n = f^{2n}/(fg)^n$. This implies $x$", "is in the open subset $W$ defined in (1) by arguing as in the", "proof of Lemma \\ref{lemma-when-invertible}.", "\\medskip\\noindent", "Conversely, suppose that $\\mathcal{O}_X(d)$ is free of rank 1", "in an open neighbourhood $V$ of $x \\in X$ and all the", "multiplication maps", "$\\mathcal{O}_X(nd)|_V \\otimes_{\\mathcal{O}_V} \\mathcal{O}_X(md)|_V", "\\to \\mathcal{O}_X(nd + md)|_V$ are isomorphisms.", "We may choose $h \\in S_{+}$ homogeneous such that $D_{+}(h) \\subset V$.", "By the definition of the twists of the structure sheaf we conclude there", "exists an element $s$ of $(S_h)_d$ such that $s^n$ is a basis of $(S_h)_{nd}$", "as a module over $S_{(h)}$ for all $n \\in \\mathbf{Z}$.", "We may write", "$s = f/h^m$ for some $m \\geq 1$ and $f \\in S_{d + m \\deg(h)}$.", "Set $g = h^m$ so $s = f/g$. Note that $x \\in D(g)$ by construction.", "Note that $g^d \\in (S_h)_{-d\\deg(g)}$.", "By assumption we can write this as a multiple of", "$s^{\\deg(g)} = f^{\\deg(g)}/g^{\\deg(g)}$, say", "$g^d = a/g^e \\cdot f^{\\deg(g)}/g^{\\deg(g)}$.", "Then we conclude that $g^{d + e + \\deg(g)} = a f^{\\deg(g)}$", "and hence also $x \\in D_{+}(f)$. So $x$ is an element of the set defined", "in (2).", "\\medskip\\noindent", "The existence of the generating section $s = f/g$ over", "the affine open $D_{+}(fg)$ whose", "powers freely generate the sheaves of modules", "$\\mathcal{O}_X(nd)$ easily implies that the multiplication maps", "$\\widetilde M(nd)|_W = \\widetilde M|_W \\otimes_{\\mathcal{O}_W}", "\\mathcal{O}_X(nd)|_W \\to \\widetilde{M(nd)}|_W$", "(see \\ref{equation-multiply-more-generally})", "are isomorphisms. Compare with the proof of Lemma \\ref{lemma-when-invertible}." ], "refs": [ "constructions-lemma-when-invertible", "constructions-lemma-when-invertible" ], "ref_ids": [ 12602, 12602 ] } ], "ref_ids": [] }, { "id": 12605, "type": "theorem", "label": "constructions-lemma-principal-open", "categories": [ "constructions" ], "title": "constructions-lemma-principal-open", "contents": [ "Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$. Fix $d \\geq 1$ an", "integer. Let $W = W_d \\subset X$ be the open subscheme defined in", "Lemma \\ref{lemma-where-invertible}. Let $n \\geq 1$ and $f \\in S_{nd}$.", "Denote $s \\in \\Gamma(W, \\mathcal{O}_W(nd))$ the section which is", "the image of $f$ via (\\ref{equation-global-sections}) restricted to $W$. Then", "$$", "W_s = D_{+}(f) \\cap W.", "$$" ], "refs": [ "constructions-lemma-where-invertible" ], "proofs": [ { "contents": [ "Let $D_{+}(ab) \\subset W$ be a standard affine open with", "$a, b \\in S$ homogeneous and $\\deg(a) = \\deg(b) + d$.", "Note that $D_{+}(ab) \\cap D_{+}(f) = D_{+}(abf)$.", "On the other hand the restriction of $s$ to $D_{+}(ab)$", "corresponds to the element $f/1 = b^nf/a^n (a/b)^n \\in (S_{ab})_{nd}$.", "We have seen in the proof of Lemma \\ref{lemma-where-invertible} that", "$(a/b)^n$ is a generator for $\\mathcal{O}_W(nd)$ over $D_{+}(ab)$.", "We conclude that $W_s \\cap D_{+}(ab)$ is the principal open", "associated to $b^nf/a^n \\in \\mathcal{O}_X(D_{+}(ab))$.", "Thus the result of the lemma is clear." ], "refs": [ "constructions-lemma-where-invertible" ], "ref_ids": [ 12604 ] } ], "ref_ids": [ 12604 ] }, { "id": 12606, "type": "theorem", "label": "constructions-lemma-ample-on-proj", "categories": [ "constructions" ], "title": "constructions-lemma-ample-on-proj", "contents": [ "Let $S$ be a graded ring.", "Let $X = \\text{Proj}(S)$.", "Let $Y \\subset X$ be a quasi-compact open subscheme.", "Denote $\\mathcal{O}_Y(n)$ the restriction of", "$\\mathcal{O}_X(n)$ to $Y$.", "There exists an integer $d \\geq 1$ such that", "\\begin{enumerate}", "\\item the subscheme $Y$ is contained in the open $W_d$ defined", "in Lemma \\ref{lemma-where-invertible},", "\\item the sheaf $\\mathcal{O}_Y(dn)$ is invertible for all $n \\in \\mathbf{Z}$,", "\\item all the maps", "$\\mathcal{O}_Y(nd) \\otimes_{\\mathcal{O}_Y} \\mathcal{O}_Y(m)", "\\longrightarrow", "\\mathcal{O}_Y(nd + m)$", "of Equation (\\ref{equation-multiply}) are isomorphisms,", "\\item all the maps", "$\\widetilde M(nd)|_Y = \\widetilde M|_Y \\otimes_{\\mathcal{O}_Y}", "\\mathcal{O}_X(nd)|_Y \\to \\widetilde{M(nd)}|_Y$", "(see \\ref{equation-multiply-more-generally}) are isomorphisms,", "\\item given $f \\in S_{nd}$ denote $s \\in \\Gamma(Y, \\mathcal{O}_Y(nd))$", "the image of $f$ via (\\ref{equation-global-sections})", "restricted to $Y$, then $D_{+}(f) \\cap Y = Y_s$,", "\\item a basis for the topology on $Y$ is given", "by the collection of opens $Y_s$, where $s \\in \\Gamma(Y, \\mathcal{O}_Y(nd))$,", "$n \\geq 1$, and", "\\item a basis for the topology of $Y$ is given", "by those opens $Y_s \\subset Y$, for", "$s \\in \\Gamma(Y, \\mathcal{O}_Y(nd))$, $n \\geq 1$ which are affine.", "\\end{enumerate}" ], "refs": [ "constructions-lemma-where-invertible" ], "proofs": [ { "contents": [ "Since $Y$ is quasi-compact there exist finitely many homogeneous", "$f_i \\in S_{+}$, $i = 1, \\ldots, n$ such that the standard opens", "$D_{+}(f_i)$ give an open covering of $Y$. Let $d_i = \\deg(f_i)$ and set", "$d = d_1 \\ldots d_n$. Note that $D_{+}(f_i) = D_{+}(f_i^{d/d_i})$", "and hence we see immediately that $Y \\subset W_d$, by characterization", "(2) in Lemma \\ref{lemma-where-invertible} or", "by (1) using Lemma \\ref{lemma-when-invertible}.", "Note that (1) implies (2), (3) and (4) by Lemma \\ref{lemma-where-invertible}.", "(Note that (3) is a special case of (4).)", "Assertion (5) follows from Lemma \\ref{lemma-principal-open}.", "Assertions (6) and (7) follow because the open subsets $D_{+}(f)$", "form a basis for the topology of $X$ and are affine." ], "refs": [ "constructions-lemma-where-invertible", "constructions-lemma-when-invertible", "constructions-lemma-where-invertible", "constructions-lemma-principal-open" ], "ref_ids": [ 12604, 12602, 12604, 12605 ] } ], "ref_ids": [ 12604 ] }, { "id": 12607, "type": "theorem", "label": "constructions-lemma-comparison-proj-quasi-coherent", "categories": [ "constructions" ], "title": "constructions-lemma-comparison-proj-quasi-coherent", "contents": [ "Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Set $M = \\bigoplus_{n \\in \\mathbf{Z}} \\Gamma(X, \\mathcal{F}(n))$ as", "a graded $S$-module, using", "(\\ref{equation-global-sections-module}) and (\\ref{equation-global-sections}).", "Then there is a canonical $\\mathcal{O}_X$-module map", "$$", "\\widetilde{M} \\longrightarrow \\mathcal{F}", "$$", "functorial in $\\mathcal{F}$ such that the induced map", "$M_0 \\to \\Gamma(X, \\mathcal{F})$ is the identity." ], "refs": [], "proofs": [ { "contents": [ "Let $f \\in S$ be homogeneous of degree $d > 0$. Recall that", "$\\widetilde{M}|_{D_{+}(f)}$ corresponds to the", "$S_{(f)}$-module $M_{(f)}$ by Lemma \\ref{lemma-proj-sheaves}.", "Thus we can define a canonical map", "$$", "M_{(f)} \\longrightarrow \\Gamma(D_+(f), \\mathcal{F}),\\quad", "m/f^n \\longmapsto m|_{D_+(f)} \\otimes f|_{D_+(f)}^{-n}", "$$", "which makes sense because $f|_{D_+(f)}$ is a trivializing", "section of the invertible sheaf $\\mathcal{O}_X(d)|_{D_+(f)}$, see", "Lemma \\ref{lemma-when-invertible} and its proof.", "Since $\\widetilde{M}$ is quasi-coherent, this leads to a canonical", "map", "$$", "\\widetilde{M}|_{D_+(f)} \\longrightarrow \\mathcal{F}|_{D_+(f)}", "$$", "via Schemes, Lemma \\ref{schemes-lemma-compare-constructions}.", "We obtain a global map if we prove that the displayed maps glue on overlaps.", "Proof of this is omitted. We also omit the proof of the final statement." ], "refs": [ "constructions-lemma-proj-sheaves", "constructions-lemma-when-invertible", "schemes-lemma-compare-constructions" ], "ref_ids": [ 12594, 12602, 7660 ] } ], "ref_ids": [] }, { "id": 12608, "type": "theorem", "label": "constructions-lemma-morphism-proj", "categories": [ "constructions" ], "title": "constructions-lemma-morphism-proj", "contents": [ "Let $A$, $B$ be two graded rings.", "Set $X = \\text{Proj}(A)$ and $Y = \\text{Proj}(B)$.", "Let $\\psi : A \\to B$ be a graded ring map.", "Set", "$$", "U(\\psi)", "=", "\\bigcup\\nolimits_{f \\in A_{+}\\ \\text{homogeneous}} D_{+}(\\psi(f))", "\\subset Y.", "$$", "Then there is a canonical morphism of schemes", "$$", "r_\\psi :", "U(\\psi)", "\\longrightarrow", "X", "$$", "and a map of $\\mathbf{Z}$-graded $\\mathcal{O}_{U(\\psi)}$-algebras", "$$", "\\theta = \\theta_\\psi :", "r_\\psi^*\\left(", "\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_X(d)", "\\right)", "\\longrightarrow", "\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_{U(\\psi)}(d).", "$$", "The triple $(U(\\psi), r_\\psi, \\theta)$ is", "characterized by the following properties:", "\\begin{enumerate}", "\\item For every $d \\geq 0$ the diagram", "$$", "\\xymatrix{", "A_d \\ar[d] \\ar[rr]_{\\psi} & &", "B_d \\ar[d] \\\\", "\\Gamma(X, \\mathcal{O}_X(d)) \\ar[r]^-\\theta &", "\\Gamma(U(\\psi), \\mathcal{O}_Y(d)) &", "\\Gamma(Y, \\mathcal{O}_Y(d)) \\ar[l]", "}", "$$", "is commutative.", "\\item For any $f \\in A_{+}$ homogeneous", "we have $r_\\psi^{-1}(D_{+}(f)) = D_{+}(\\psi(f))$ and", "the restriction of $r_\\psi$ to $D_{+}(\\psi(f))$", "corresponds to the ring map", "$A_{(f)} \\to B_{(\\psi(f))}$ induced by $\\psi$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Clearly condition (2) uniquely determines the morphism of schemes", "and the open subset $U(\\psi)$. Pick $f \\in A_d$ with $d \\geq 1$.", "Note that", "$\\mathcal{O}_X(n)|_{D_{+}(f)}$ corresponds to the", "$A_{(f)}$-module $(A_f)_n$ and that", "$\\mathcal{O}_Y(n)|_{D_{+}(\\psi(f))}$ corresponds to the", "$B_{(\\psi(f))}$-module $(B_{\\psi(f)})_n$. In other words $\\theta$", "when restricted to $D_{+}(\\psi(f))$ corresponds to a map of", "$\\mathbf{Z}$-graded $B_{(\\psi(f))}$-algebras", "$$", "A_f \\otimes_{A_{(f)}} B_{(\\psi(f))}", "\\longrightarrow", "B_{\\psi(f)}", "$$", "Condition (1) determines the images of all elements of $A$.", "Since $f$ is an invertible element which is mapped to $\\psi(f)$", "we see that $1/f^m$ is mapped to $1/\\psi(f)^m$. It easily follows", "from this that $\\theta$ is uniquely determined, namely it is", "given by the rule", "$$", "a/f^m \\otimes b/\\psi(f)^e \\longmapsto \\psi(a)b/\\psi(f)^{m + e}.", "$$", "To show existence we remark that the proof of uniqueness above gave", "a well defined prescription for the morphism $r$ and the map $\\theta$", "when restricted to every standard open of the form", "$D_{+}(\\psi(f)) \\subset U(\\psi)$ into $D_{+}(f)$.", "Call these $r_f$ and $\\theta_f$.", "Hence we only need to verify that if $D_{+}(f) \\subset D_{+}(g)$", "for some $f, g \\in A_{+}$ homogeneous, then the restriction of", "$r_g$ to $D_{+}(\\psi(f))$ matches $r_f$. This is clear from the", "formulas given for $r$ and $\\theta$ above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12609, "type": "theorem", "label": "constructions-lemma-morphism-proj-transitive", "categories": [ "constructions" ], "title": "constructions-lemma-morphism-proj-transitive", "contents": [ "Let $A$, $B$, and $C$ be graded rings.", "Set $X = \\text{Proj}(A)$, $Y = \\text{Proj}(B)$ and $Z = \\text{Proj}(C)$.", "Let $\\varphi : A \\to B$, $\\psi : B \\to C$ be graded ring maps.", "Then we have", "$$", "U(\\psi \\circ \\varphi) = r_\\varphi^{-1}(U(\\psi))", "\\quad", "\\text{and}", "\\quad", "r_{\\psi \\circ \\varphi}", "=", "r_\\varphi \\circ r_\\psi|_{U(\\psi \\circ \\varphi)}.", "$$", "In addition we have", "$$", "\\theta_\\psi \\circ r_\\psi^*\\theta_\\varphi", "=", "\\theta_{\\psi \\circ \\varphi}", "$$", "with obvious notation." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12610, "type": "theorem", "label": "constructions-lemma-surjective-graded-rings-map-proj", "categories": [ "constructions" ], "title": "constructions-lemma-surjective-graded-rings-map-proj", "contents": [ "With hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above.", "Assume $A_d \\to B_d$ is surjective for all $d \\gg 0$. Then", "\\begin{enumerate}", "\\item $U(\\psi) = Y$,", "\\item $r_\\psi : Y \\to X$ is a closed immersion, and", "\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$", "are surjective but not isomorphisms in general (even if $A \\to B$ is", "surjective).", "\\end{enumerate}" ], "refs": [ "constructions-lemma-morphism-proj" ], "proofs": [ { "contents": [ "Part (1) follows from the definition of $U(\\psi)$ and the fact that", "$D_{+}(f) = D_{+}(f^n)$ for any $n > 0$. For $f \\in A_{+}$ homogeneous", "we see that $A_{(f)} \\to B_{(\\psi(f))}$ is surjective because", "any element of $B_{(\\psi(f))}$ can be represented by a fraction", "$b/\\psi(f)^n$ with $n$ arbitrarily large (which forces the degree of", "$b \\in B$ to be large). This proves (2).", "The same argument shows the map", "$$", "A_f \\to B_{\\psi(f)}", "$$", "is surjective which proves the surjectivity of $\\theta$.", "For an example where this map is not an isomorphism", "consider the graded ring $A = k[x, y]$ where $k$ is a field", "and $\\deg(x) = 1$, $\\deg(y) = 2$. Set $I = (x)$, so that", "$B = k[y]$. Note that $\\mathcal{O}_Y(1) = 0$ in this case.", "But it is easy to see that $r_\\psi^*\\mathcal{O}_X(1)$", "is not zero. (There are less silly examples.)" ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12608 ] }, { "id": 12611, "type": "theorem", "label": "constructions-lemma-eventual-iso-graded-rings-map-proj", "categories": [ "constructions" ], "title": "constructions-lemma-eventual-iso-graded-rings-map-proj", "contents": [ "With hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above.", "Assume $A_d \\to B_d$ is an isomorphism for all $d \\gg 0$. Then", "\\begin{enumerate}", "\\item $U(\\psi) = Y$,", "\\item $r_\\psi : Y \\to X$ is an isomorphism, and", "\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$", "are isomorphisms.", "\\end{enumerate}" ], "refs": [ "constructions-lemma-morphism-proj" ], "proofs": [ { "contents": [ "We have (1) by Lemma \\ref{lemma-surjective-graded-rings-map-proj}.", "Let $f \\in A_{+}$ be homogeneous. The assumption on $\\psi$ implies that", "$A_f \\to B_f$ is an isomorphism (details omitted). Thus it is clear that", "$r_\\psi$ and $\\theta$ restrict to isomorphisms over $D_{+}(f)$.", "The lemma follows." ], "refs": [ "constructions-lemma-surjective-graded-rings-map-proj" ], "ref_ids": [ 12610 ] } ], "ref_ids": [ 12608 ] }, { "id": 12612, "type": "theorem", "label": "constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj", "categories": [ "constructions" ], "title": "constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj", "contents": [ "With hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above.", "Assume $A_d \\to B_d$ is surjective for $d \\gg 0$ and that $A$ is generated", "by $A_1$ over $A_0$. Then", "\\begin{enumerate}", "\\item $U(\\psi) = Y$,", "\\item $r_\\psi : Y \\to X$ is a closed immersion, and", "\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$", "are isomorphisms.", "\\end{enumerate}" ], "refs": [ "constructions-lemma-morphism-proj" ], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-eventual-iso-graded-rings-map-proj} and", "\\ref{lemma-morphism-proj-transitive}", "we may replace $B$ by the image of $A \\to B$", "without changing $X$ or the sheaves $\\mathcal{O}_X(n)$.", "Thus we may assume that $A \\to B$ is surjective. By", "Lemma \\ref{lemma-surjective-graded-rings-map-proj} we get (1) and (2)", "and surjectivity in (3).", "By Lemma \\ref{lemma-apply-modules} we see that both", "$\\mathcal{O}_X(n)$ and $\\mathcal{O}_Y(n)$", "are invertible. Hence $\\theta$ is an isomorphism." ], "refs": [ "constructions-lemma-eventual-iso-graded-rings-map-proj", "constructions-lemma-morphism-proj-transitive", "constructions-lemma-surjective-graded-rings-map-proj", "constructions-lemma-apply-modules" ], "ref_ids": [ 12611, 12609, 12610, 12603 ] } ], "ref_ids": [ 12608 ] }, { "id": 12613, "type": "theorem", "label": "constructions-lemma-base-change-map-proj", "categories": [ "constructions" ], "title": "constructions-lemma-base-change-map-proj", "contents": [ "With hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above.", "Assume there exists a ring map $R \\to A_0$ and a ring map", "$R \\to R'$ such that $B = R' \\otimes_R A$. Then", "\\begin{enumerate}", "\\item $U(\\psi) = Y$,", "\\item the diagram", "$$", "\\xymatrix{", "Y = \\text{Proj}(B) \\ar[r]_{r_\\psi} \\ar[d] &", "\\text{Proj}(A) = X \\ar[d] \\\\", "\\Spec(R') \\ar[r] &", "\\Spec(R)", "}", "$$", "is a fibre product square, and", "\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$", "are isomorphisms.", "\\end{enumerate}" ], "refs": [ "constructions-lemma-morphism-proj" ], "proofs": [ { "contents": [ "This follows immediately by looking at what happens over the standard", "opens $D_{+}(f)$ for $f \\in A_{+}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12608 ] }, { "id": 12614, "type": "theorem", "label": "constructions-lemma-localization-map-proj", "categories": [ "constructions" ], "title": "constructions-lemma-localization-map-proj", "contents": [ "With hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above.", "Assume there exists a $g \\in A_0$ such that $\\psi$ induces an", "isomorphism $A_g \\to B$. Then", "$U(\\psi) = Y$, $r_\\psi : Y \\to X$ is an open immersion", "which induces an isomorphism of $Y$ with the inverse image", "of $D(g) \\subset \\Spec(A_0)$. Moreover the map $\\theta$", "is an isomorphism." ], "refs": [ "constructions-lemma-morphism-proj" ], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-base-change-map-proj} above." ], "refs": [ "constructions-lemma-base-change-map-proj" ], "ref_ids": [ 12613 ] } ], "ref_ids": [ 12608 ] }, { "id": 12615, "type": "theorem", "label": "constructions-lemma-d-uple", "categories": [ "constructions" ], "title": "constructions-lemma-d-uple", "contents": [ "Let $S$ be a graded ring. Let $d \\geq 1$. Set $S' = S^{(d)}$ with notation", "as in Algebra, Section \\ref{algebra-section-graded}. Set", "$X = \\text{Proj}(S)$ and $X' = \\text{Proj}(S')$. There is a canonical", "isomorphism $i : X \\to X'$ of schemes such that", "\\begin{enumerate}", "\\item for any graded $S$-module $M$ setting $M' = M^{(d)}$,", "we have a canonical isomorphism $\\widetilde{M} \\to i^*\\widetilde{M'}$,", "\\item we have canonical isomorphisms", "$\\mathcal{O}_{X}(nd) \\to i^*\\mathcal{O}_{X'}(n)$", "\\end{enumerate}", "and these isomorphisms are compatible with the multiplication maps", "of Lemma \\ref{lemma-widetilde-tensor} and hence with the maps", "(\\ref{equation-multiply}),", "(\\ref{equation-multiply-on-sheaf}),", "(\\ref{equation-global-sections}),", "(\\ref{equation-global-sections-module}),", "(\\ref{equation-multiply-more-generally}), and", "(\\ref{equation-global-sections-more-generally}) (see proof for precise", "statements." ], "refs": [ "constructions-lemma-widetilde-tensor" ], "proofs": [ { "contents": [ "The injective ring map $S' \\to S$ (which is not a homomorphism of graded rings", "due to our conventions), induces a map $j : \\Spec(S) \\to \\Spec(S')$.", "Given a graded prime ideal $\\mathfrak p \\subset S$ we see that", "$\\mathfrak p' = j(\\mathfrak p) = S' \\cap \\mathfrak p$", "is a graded prime ideal of $S'$.", "Moreover, if $f \\in S_+$ is homogeneous and $f \\not \\in \\mathfrak p$, then", "$f^d \\in S'_+$ and $f^d \\not \\in \\mathfrak p'$. Conversely, if", "$\\mathfrak p' \\subset S'$ is a graded prime ideal not containing some", "homogeneous element $f \\in S'_+$, then", "$\\mathfrak p = \\{g \\in S \\mid g^d \\in \\mathfrak p'\\}$ is a", "graded prime ideal of $S$ not containing $f$ whose image under $j$", "is $\\mathfrak p'$. To see that $\\mathfrak p$ is an ideal, note", "that if $g, h \\in \\mathfrak p$, then", "$(g + h)^{2d} \\in \\mathfrak p'$ by the binomial formula", "and hence $g + h \\in \\mathfrak p'$ as $\\mathfrak p'$ is a prime.", "In this way we see that $j$ induces a homeomorphism $i : X \\to X'$.", "Moreover, given $f \\in S_+$ homogeneous, then we have", "$S_{(f)} \\cong S'_{(f^d)}$. Since these isomorphisms are compatible", "with the restrictions mappings of", "Lemma \\ref{lemma-standard-open}, we see that there exists an", "isomorphism $i^\\sharp : i^{-1}\\mathcal{O}_{X'} \\to \\mathcal{O}_X$ of", "structure sheaves on $X$ and $X'$, hence $i$ is an isomorphism", "of schemes.", "\\medskip\\noindent", "Let $M$ be a graded $S$-module. Given $f \\in S_+$ homogeneous, we have", "$M_{(f)} \\cong M'_{(f^d)}$, hence in exactly the same manner as above", "we obtain the isomorphism in (1). The isomorphisms in (2) are a special", "case of (1) for $M = S(nd)$ which gives $M' = S'(n)$. Let $M$ and $N$", "be graded $S$-modules. Then we have", "$$", "M' \\otimes_{S'} N' =", "(M \\otimes_S N)^{(d)} =", "(M \\otimes_S N)'", "$$", "as can be verified directly from the definitions. Having said this", "the compatibility with the multiplication maps of", "Lemma \\ref{lemma-widetilde-tensor} is the commutativity of the diagram", "$$", "\\xymatrix{", "\\widetilde M \\otimes_{\\mathcal{O}_X} \\widetilde N", "\\ar[d]_{(1) \\otimes (1)} \\ar[r] &", "\\widetilde{M \\otimes_S N} \\ar[d]^{(1)} \\\\", "i^*(\\widetilde{M'} \\otimes_{\\mathcal{O}_{X'}} \\widetilde{N'}) \\ar[r] &", "i^*(\\widetilde{M' \\otimes_{S'} N'})", "}", "$$", "This can be seen by looking at the construction of the maps", "over the open $D_+(f) = D_+(f^d)$ where the top horizontal", "arrow is given by the map", "$M_{(f)} \\times N_{(f)} \\to (M \\otimes_S N)_{(f)}$", "and the lower horizontal arrow by the map", "$M'_{(f^d)} \\times N'_{(f^d)} \\to (M' \\otimes_{S'} N')_{(f^d)}$.", "Since these maps agree via the identifications", "$M_{(f)} = M'_{(f^d)}$, etc, we get the desired compatibility.", "We omit the proof of the other compatibilities." ], "refs": [ "constructions-lemma-standard-open", "constructions-lemma-widetilde-tensor" ], "ref_ids": [ 12593, 12601 ] } ], "ref_ids": [ 12601 ] }, { "id": 12616, "type": "theorem", "label": "constructions-lemma-converse-construction", "categories": [ "constructions" ], "title": "constructions-lemma-converse-construction", "contents": [ "Let $S$ be a graded ring, and $X = \\text{Proj}(S)$.", "Let $d \\geq 1$ and $U_d \\subset X$ as above.", "Let $Y$ be a scheme.", "Let $\\mathcal{L}$ be an invertible sheaf on $Y$.", "Let $\\psi : S^{(d)} \\to \\Gamma_*(Y, \\mathcal{L})$ be", "a graded ring homomorphism such that $\\mathcal{L}$ is", "generated by the sections in the image of", "$\\psi|_{S_d} : S_d \\to \\Gamma(Y, \\mathcal{L})$.", "Then there exists a morphism", "$\\varphi : Y \\to X$ such that $\\varphi(Y) \\subset U_d$ and", "an isomorphism $\\alpha : \\varphi^*\\mathcal{O}_{U_d}(d) \\to \\mathcal{L}$", "such that $\\psi_\\varphi^d$ agrees with $\\psi$ via $\\alpha$:", "$$", "\\xymatrix{", "\\Gamma_*(Y, \\mathcal{L}) &", "\\Gamma_*(Y, \\varphi^*\\mathcal{O}_{U_d}(d)) \\ar[l]^-\\alpha &", "\\Gamma_*(U_d, \\mathcal{O}_{U_d}(d)) \\ar[l]^-{\\varphi^*} \\\\", "S^{(d)} \\ar[u]^\\psi & &", "S^{(d)} \\ar[u]^{\\psi^d} \\ar[ul]^{\\psi^d_\\varphi} \\ar[ll]_{\\text{id}}", "}", "$$", "commutes. Moreover, the pair $(\\varphi, \\alpha)$ is unique." ], "refs": [], "proofs": [ { "contents": [ "Pick $f \\in S_d$. Denote $s = \\psi(f) \\in \\Gamma(Y, \\mathcal{L})$.", "On the open set $Y_s$ where $s$ does not vanish multiplication", "by $s$ induces an isomorphism $\\mathcal{O}_{Y_s} \\to \\mathcal{L}|_{Y_s}$,", "see Modules, Lemma \\ref{modules-lemma-s-open}. We will denote", "the inverse of this map $x \\mapsto x/s$, and similarly for", "powers of $\\mathcal{L}$. Using this we", "define a ring map $\\psi_{(f)} : S_{(f)} \\to \\Gamma(Y_s, \\mathcal{O})$", "by mapping the fraction $a/f^n$ to $\\psi(a)/s^n$.", "By Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}", "this corresponds to a morphism", "$\\varphi_f : Y_s \\to \\Spec(S_{(f)}) = D_{+}(f)$.", "We also introduce the isomorphism", "$\\alpha_f : \\varphi_f^*\\mathcal{O}_{D_{+}(f)}(d) \\to \\mathcal{L}|_{Y_s}$", "which maps the pullback of the trivializing section", "$f$ over $D_{+}(f)$ to the trivializing section $s$ over $Y_s$.", "With this choice the commutativity of the diagram in the lemma", "holds with $Y$ replace by $Y_s$, $\\varphi$ replaced by $\\varphi_f$,", "and $\\alpha$ replaced by $\\alpha_f$; verification omitted.", "\\medskip\\noindent", "Suppose that $f' \\in S_d$ is a second element, and denote", "$s' = \\psi(f') \\in \\Gamma(Y, \\mathcal{L})$. Then", "$Y_s \\cap Y_{s'} = Y_{ss'}$ and similarly", "$D_{+}(f) \\cap D_{+}(f') = D_{+}(ff')$.", "In Lemma \\ref{lemma-ample-on-proj} we saw that", "$D_{+}(f') \\cap D_{+}(f)$ is the same as the set", "of points of $D_{+}(f)$ where the section of", "$\\mathcal{O}_X(d)$ defined by $f'$ does not vanish.", "Hence", "$\\varphi_f^{-1}(D_{+}(f') \\cap D_{+}(f)) = Y_s \\cap Y_{s'}", "= \\varphi_{f'}^{-1}(D_{+}(f') \\cap D_{+}(f))$.", "On $D_{+}(f) \\cap D_{+}(f')$ the fraction $f/f'$ is an", "invertible section of the structure sheaf with inverse", "$f'/f$. Note that $\\psi_{(f')}(f/f') = \\psi(f)/s' = s/s'$", "and $\\psi_{(f)}(f'/f) = \\psi(f')/s = s'/s$. We claim there", "is a unique ring map", "$S_{(ff')} \\to \\Gamma(Y_{ss'}, \\mathcal{O})$ making the", "following diagram commute", "$$", "\\xymatrix{", "\\Gamma(Y_s, \\mathcal{O}) \\ar[r] &", "\\Gamma(Y_{ss'}, \\mathcal{O}) &", "\\Gamma(Y_{s, '} \\mathcal{O}) \\ar[l]\\\\", "S_{(f)} \\ar[r] \\ar[u]^{\\psi_{(f)}} &", "S_{(ff')} \\ar[u] &", "S_{(f')} \\ar[l] \\ar[u]^{\\psi_{(f')}}", "}", "$$", "It exists because we may use the rule", "$x/(ff')^n \\mapsto \\psi(x)/(ss')^n$, which ``works'' by the formulas", "above. Uniqueness follows as $\\text{Proj}(S)$ is separated, see", "Lemma \\ref{lemma-proj-separated} and its proof. This shows that the", "morphisms $\\varphi_f$ and $\\varphi_{f'}$ agree over $Y_s \\cap Y_{s'}$.", "The restrictions of $\\alpha_f$ and $\\alpha_{f'}$ agree over", "$Y_s \\cap Y_{s'}$ because the regular functions $s/s'$ and", "$\\psi_{(f')}(f)$ agree. This proves that the morphisms $\\psi_f$", "glue to a global morphism from $Y$ into $U_d \\subset X$, and", "that the maps $\\alpha_f$ glue to an isomorphism satisfying the", "conditions of the lemma.", "\\medskip\\noindent", "We still have to show the pair $(\\varphi, \\alpha)$ is unique.", "Suppose $(\\varphi', \\alpha')$ is a second such pair.", "Let $f \\in S_d$. By the commutativity of the diagrams in the lemma we have", "that the inverse images of $D_{+}(f)$ under both $\\varphi$ and $\\varphi'$", "are equal to $Y_{\\psi(f)}$. Since the opens $D_{+}(f)$ are a basis", "for the topology on $X$, and since $X$ is a sober topological", "space (see Schemes, Lemma \\ref{schemes-lemma-scheme-sober})", "this means the maps $\\varphi$ and $\\varphi'$ are the same", "on underlying topological spaces. Let us use $s = \\psi(f)$ to", "trivialize the invertible sheaf $\\mathcal{L}$ over $Y_{\\psi(f)}$.", "By the commutativity of the diagrams we have that", "$\\alpha^{\\otimes n}(\\psi^d_{\\varphi}(x)) =", "\\psi(x) = (\\alpha')^{\\otimes n}(\\psi^d_{\\varphi'}(x))$", "for all $x \\in S_{nd}$. By construction of $\\psi^d_{\\varphi}$", "and $\\psi^d_{\\varphi'}$ we have", "$\\psi^d_{\\varphi}(x) = \\varphi^\\sharp(x/f^n) \\psi^d_{\\varphi}(f^n)$", "over $Y_{\\psi(f)}$,", "and similarly for $\\psi^d_{\\varphi'}$. by the commutativity of", "the diagrams of the lemma we deduce that", "$\\varphi^\\sharp(x/f^n) = (\\varphi')^\\sharp(x/f^n)$.", "This proves that $\\varphi$ and $\\varphi'$ induce the same morphism", "from $Y_{\\psi(f)}$ into the affine scheme $D_{+}(f) = \\Spec(S_{(f)})$.", "Hence $\\varphi$ and $\\varphi'$ are the same as morphisms.", "Finally, it remains to show that the commutativity of the", "diagram of the lemma singles out, given $\\varphi$, a unique", "$\\alpha$. We omit the verification." ], "refs": [ "modules-lemma-s-open", "schemes-lemma-morphism-into-affine", "constructions-lemma-ample-on-proj", "constructions-lemma-proj-separated", "schemes-lemma-scheme-sober" ], "ref_ids": [ 13305, 7655, 12606, 12597, 7672 ] } ], "ref_ids": [] }, { "id": 12617, "type": "theorem", "label": "constructions-lemma-proj-functor-strict", "categories": [ "constructions" ], "title": "constructions-lemma-proj-functor-strict", "contents": [ "Let $S$ be a graded ring.", "Let $X = \\text{Proj}(S)$.", "The open subscheme $U_d \\subset X$ (\\ref{equation-Ud}) represents the", "functor $F_d$ and the triple $(d, \\mathcal{O}_{U_d}(d), \\psi^d)$", "defined above is the universal family (see", "Schemes, Section \\ref{schemes-section-representable})." ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of Lemma \\ref{lemma-converse-construction}" ], "refs": [ "constructions-lemma-converse-construction" ], "ref_ids": [ 12616 ] } ], "ref_ids": [] }, { "id": 12618, "type": "theorem", "label": "constructions-lemma-apply", "categories": [ "constructions" ], "title": "constructions-lemma-apply", "contents": [ "Let $S$ be a graded ring generated as an $S_0$-algebra by", "the elements of $S_1$. In this case the scheme $X = \\text{Proj}(S)$", "represents the functor which associates to a scheme", "$Y$ the set of pairs $(\\mathcal{L}, \\psi)$, where", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is an invertible $\\mathcal{O}_Y$-module, and", "\\item $\\psi : S \\to \\Gamma_*(Y, \\mathcal{L})$ is a graded", "ring homomorphism such that $\\mathcal{L}$ is generated by", "the global sections $\\psi(f)$, with $f \\in S_1$", "\\end{enumerate}", "up to strict equivalence as above." ], "refs": [], "proofs": [ { "contents": [ "Under the assumptions of the lemma we have $X = U_1$ and the", "lemma is a reformulation of Lemma \\ref{lemma-proj-functor-strict} above." ], "refs": [ "constructions-lemma-proj-functor-strict" ], "ref_ids": [ 12617 ] } ], "ref_ids": [] }, { "id": 12619, "type": "theorem", "label": "constructions-lemma-equivalent", "categories": [ "constructions" ], "title": "constructions-lemma-equivalent", "contents": [ "Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$. Let $T$ be a scheme.", "Let $(d, \\mathcal{L}, \\psi)$ and $(d', \\mathcal{L}', \\psi')$", "be two triples over $T$. The following are equivalent:", "\\begin{enumerate}", "\\item Let $n = \\text{lcm}(d, d')$. Write $n = ad = a'd'$. There exists", "an isomorphism", "$\\beta : \\mathcal{L}^{\\otimes a} \\to (\\mathcal{L}')^{\\otimes a'}$", "with the property that", "$\\beta \\circ \\psi|_{S^{(n)}}$ and $\\psi'|_{S^{(n)}}$ agree", "as graded ring maps $S^{(n)} \\to \\Gamma_*(Y, (\\mathcal{L}')^{\\otimes n})$.", "\\item The triples $(d, \\mathcal{L}, \\psi)$ and $(d', \\mathcal{L}', \\psi')$", "are equivalent.", "\\item For some positive integer $n = ad = a'd'$ there exists", "an isomorphism", "$\\beta : \\mathcal{L}^{\\otimes a} \\to (\\mathcal{L}')^{\\otimes a'}$", "with the property that", "$\\beta \\circ \\psi|_{S^{(n)}}$ and $\\psi'|_{S^{(n)}}$ agree", "as graded ring maps $S^{(n)} \\to \\Gamma_*(Y, (\\mathcal{L}')^{\\otimes n})$.", "\\item The morphisms $\\varphi : T \\to X$ and $\\varphi' : T \\to X$", "associated to $(d, \\mathcal{L}, \\psi)$ and $(d', \\mathcal{L}', \\psi')$", "are equal.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Clearly (1) implies (2) and (2) implies (3) by restricting to", "more divisible degrees and powers of invertible sheaves.", "Also (3) implies (4) by the uniqueness statement", "in Lemma \\ref{lemma-converse-construction}.", "Thus we have to prove that (4) implies (1). Assume (4),", "in other words $\\varphi = \\varphi'$.", "Note that this implies that we may write", "$\\mathcal{L} = \\varphi^*\\mathcal{O}_X(d)$ and", "$\\mathcal{L}' = \\varphi^*\\mathcal{O}_X(d')$.", "Moreover, via these identifications we have that the graded ring", "maps $\\psi$ and $\\psi'$ correspond to the restriction of the canonical", "graded ring map", "$$", "S \\longrightarrow \\bigoplus\\nolimits_{n \\geq 0} \\Gamma(X, \\mathcal{O}_X(n))", "$$", "to $S^{(d)}$ and $S^{(d')}$ composed with pullback by $\\varphi$", "(by Lemma \\ref{lemma-converse-construction} again). Hence taking", "$\\beta$ to be the isomorphism", "$$", "(\\varphi^*\\mathcal{O}_X(d))^{\\otimes a} =", "\\varphi^*\\mathcal{O}_X(n) =", "(\\varphi^*\\mathcal{O}_X(d'))^{\\otimes a'}", "$$", "works." ], "refs": [ "constructions-lemma-converse-construction", "constructions-lemma-converse-construction" ], "ref_ids": [ 12616, 12616 ] } ], "ref_ids": [] }, { "id": 12620, "type": "theorem", "label": "constructions-lemma-proj-functor", "categories": [ "constructions" ], "title": "constructions-lemma-proj-functor", "contents": [ "Let $S$ be a graded ring.", "Let $X = \\text{Proj}(S)$.", "The functor $F$ defined above is representable by the scheme $X$." ], "refs": [], "proofs": [ { "contents": [ "We have seen above that the functor $F_d$ corresponds to the", "open subscheme $U_d \\subset X$. Moreover the transformation", "of functors $F_d \\to F_{d'}$ (if $d | d'$) defined above", "corresponds to the inclusion morphism $U_d \\to U_{d'}$", "(see discussion above). Hence to show that $F$ is represented", "by $X$ it suffices to show that $T \\to X$ for a quasi-compact", "scheme $T$ ends up in some $U_d$, and that for a general scheme", "$T$ we have", "$$", "\\Mor(T, X)", "=", "\\lim_{V \\subset T\\text{ quasi-compact open}} \\Mor(V, X).", "$$", "These verifications are omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12621, "type": "theorem", "label": "constructions-lemma-projective-space", "categories": [ "constructions" ], "title": "constructions-lemma-projective-space", "contents": [ "Let $S = \\mathbf{Z}[T_0, \\ldots, T_n]$ with $\\deg(T_i) = 1$.", "The scheme", "$$", "\\mathbf{P}^n_{\\mathbf{Z}} = \\text{Proj}(S)", "$$", "represents the functor which associates to a scheme $Y$ the pairs", "$(\\mathcal{L}, (s_0, \\ldots, s_n))$ where", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is an invertible $\\mathcal{O}_Y$-module, and", "\\item $s_0, \\ldots, s_n$ are global sections of $\\mathcal{L}$", "which generate $\\mathcal{L}$", "\\end{enumerate}", "up to the following equivalence:", "$(\\mathcal{L}, (s_0, \\ldots, s_n)) \\sim", "(\\mathcal{N}, (t_0, \\ldots, t_n))$ $\\Leftrightarrow$ there exists", "an isomorphism $\\beta : \\mathcal{L} \\to \\mathcal{N}$", "with $\\beta(s_i) = t_i$ for $i = 0, \\ldots, n$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-apply} above.", "Namely, for any graded ring $A$ we have", "\\begin{eqnarray*}", "\\Mor_{graded rings}(\\mathbf{Z}[T_0, \\ldots, T_n], A)", "& = &", "A_1 \\times \\ldots \\times A_1 \\\\", "\\psi & \\mapsto & (\\psi(T_0), \\ldots, \\psi(T_n))", "\\end{eqnarray*}", "and the degree $1$ part of $\\Gamma_*(Y, \\mathcal{L})$ is", "just $\\Gamma(Y, \\mathcal{L})$." ], "refs": [ "constructions-lemma-apply" ], "ref_ids": [ 12618 ] } ], "ref_ids": [] }, { "id": 12622, "type": "theorem", "label": "constructions-lemma-standard-covering-projective-space", "categories": [ "constructions" ], "title": "constructions-lemma-standard-covering-projective-space", "contents": [ "Projective $n$-space over $\\mathbf{Z}$ is covered by", "$n + 1$ standard opens", "$$", "\\mathbf{P}^n_{\\mathbf{Z}} =", "\\bigcup\\nolimits_{i = 0, \\ldots, n} D_{+}(T_i)", "$$", "where each $D_{+}(T_i)$ is isomorphic to $\\mathbf{A}^n_{\\mathbf{Z}}$", "affine $n$-space over $\\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$\\mathbf{Z}[T_0, \\ldots, T_n]_{+} = (T_0, \\ldots, T_n)$ and", "since", "$$", "\\Spec", "\\left(", "\\mathbf{Z}", "\\left[\\frac{T_0}{T_i}, \\ldots, \\frac{T_n}{T_i}", "\\right]", "\\right)", "\\cong", "\\mathbf{A}^n_{\\mathbf{Z}}", "$$", "in an obvious way." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12623, "type": "theorem", "label": "constructions-lemma-projective-space-separated", "categories": [ "constructions" ], "title": "constructions-lemma-projective-space-separated", "contents": [ "Let $S$ be a scheme.", "The structure morphism $\\mathbf{P}^n_S \\to S$ is", "\\begin{enumerate}", "\\item separated,", "\\item quasi-compact,", "\\item satisfies the existence and uniqueness parts of the valuative criterion,", "and", "\\item universally closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "All these properties are stable under base change (this is clear for the", "last two and for the other two see", "Schemes, Lemmas", "\\ref{schemes-lemma-separated-permanence} and", "\\ref{schemes-lemma-quasi-compact-preserved-base-change}).", "Hence it suffices to prove them for the morphism", "$\\mathbf{P}^n_{\\mathbf{Z}} \\to \\Spec(\\mathbf{Z})$.", "Separatedness is Lemma \\ref{lemma-proj-separated}. Quasi-compactness follows", "from Lemma \\ref{lemma-standard-covering-projective-space}.", "Existence and uniqueness of the valuative criterion follow from", "Lemma \\ref{lemma-proj-valuative-criterion}.", "Universally closed follows from the above and", "Schemes, Proposition \\ref{schemes-proposition-characterize-universally-closed}." ], "refs": [ "schemes-lemma-separated-permanence", "schemes-lemma-quasi-compact-preserved-base-change", "constructions-lemma-proj-separated", "constructions-lemma-standard-covering-projective-space", "constructions-lemma-proj-valuative-criterion", "schemes-proposition-characterize-universally-closed" ], "ref_ids": [ 7714, 7698, 12597, 12622, 12600, 7733 ] } ], "ref_ids": [] }, { "id": 12624, "type": "theorem", "label": "constructions-lemma-segre-embedding", "categories": [ "constructions" ], "title": "constructions-lemma-segre-embedding", "contents": [ "Let $S$ be a scheme. There exists a closed immersion", "$$", "\\mathbf{P}^n_S \\times_S \\mathbf{P}^m_S", "\\longrightarrow", "\\mathbf{P}^{nm + n + m}_S", "$$", "called the {\\it Segre embedding}." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove this when $S = \\Spec(\\mathbf{Z})$.", "Hence we will drop the index $S$ and work in the absolute setting.", "Write $\\mathbf{P}^n = \\text{Proj}(\\mathbf{Z}[X_0, \\ldots, X_n])$,", "$\\mathbf{P}^m = \\text{Proj}(\\mathbf{Z}[Y_0, \\ldots, Y_m])$,", "and", "$\\mathbf{P}^{nm + n + m} =", "\\text{Proj}(\\mathbf{Z}[Z_0, \\ldots, Z_{nm + n + m}])$.", "In order to map into $\\mathbf{P}^{nm + n + m}$ we have to", "write down an invertible sheaf $\\mathcal{L}$ on the left hand", "side and $(n + 1)(m + 1)$ sections $s_i$ which generate it.", "See Lemma \\ref{lemma-projective-space}.", "The invertible sheaf we take is", "$$", "\\mathcal{L} =", "\\text{pr}_1^*\\mathcal{O}_{\\mathbf{P}^n}(1)", "\\otimes", "\\text{pr}_2^*\\mathcal{O}_{\\mathbf{P}^m}(1)", "$$", "The sections we take are", "$$", "s_0 = X_0Y_0, \\ s_1 = X_1Y_0, \\ldots, \\ s_n = X_nY_0,", "\\ s_{n + 1} = X_0Y_1, \\ldots, \\ s_{nm + n + m} = X_nY_m.", "$$", "These generate $\\mathcal{L}$ since the sections $X_i$ generate", "$\\mathcal{O}_{\\mathbf{P}^n}(1)$ and the sections $Y_j$ generate", "$\\mathcal{O}_{\\mathbf{P}^m}(1)$. The induced morphism", "$\\varphi$ has the property that", "$$", "\\varphi^{-1}(D_{+}(Z_{i + (n + 1)j})) = D_{+}(X_i) \\times D_{+}(Y_j).", "$$", "Hence it is an affine morphism. The corresponding ring map in case", "$(i, j) = (0, 0)$ is the map", "$$", "\\mathbf{Z}[Z_1/Z_0, \\ldots, Z_{nm + n + m}/Z_0]", "\\longrightarrow", "\\mathbf{Z}[X_1/X_0, \\ldots, X_n/X_0, Y_1/Y_0, \\ldots, Y_n/Y_0]", "$$", "which maps $Z_i/Z_0$ to the element $X_i/X_0$ for $i \\leq n$ and", "the element $Z_{(n + 1)j}/Z_0$ to the element $Y_j/Y_0$. Hence it", "is surjective. A similar argument works for the other affine", "open subsets. Hence the morphism $\\varphi$ is a closed immersion", "(see Schemes, Lemma \\ref{schemes-lemma-closed-local-target} and", "Example \\ref{schemes-example-closed-immersion-affines}.)" ], "refs": [ "constructions-lemma-projective-space", "schemes-lemma-closed-local-target" ], "ref_ids": [ 12621, 7646 ] } ], "ref_ids": [] }, { "id": 12625, "type": "theorem", "label": "constructions-lemma-closed-in-projective-space", "categories": [ "constructions" ], "title": "constructions-lemma-closed-in-projective-space", "contents": [ "Let $R$ be a ring. Let $Z \\subset \\mathbf{P}^n_R$ be a closed subscheme.", "Let", "$$", "I_d = \\Ker\\left(", "R[T_0, \\ldots, T_n]_d", "\\longrightarrow", "\\Gamma(Z, \\mathcal{O}_{\\mathbf{P}^n_R}(d)|_Z)\\right)", "$$", "Then $I = \\bigoplus I_d \\subset R[T_0, \\ldots, T_n]$ is", "a graded ideal and $Z = \\text{Proj}(R[T_0, \\ldots, T_n]/I)$." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $I$ is a graded ideal.", "Set $Z' = \\text{Proj}(R[T_0, \\ldots, T_n]/I)$.", "By Lemma \\ref{lemma-surjective-graded-rings-generated-degree-1-map-proj}", "we see that $Z'$ is a closed subscheme of $\\mathbf{P}^n_R$.", "To see the equality $Z = Z'$", "it suffices to check on an standard affine open", "$D_{+}(T_i)$. By renumbering the homogeneous coordinates we", "may assume $i = 0$. Say $Z \\cap D_{+}(T_0)$, resp.\\ $Z' \\cap D_{+}(T_0)$", "is cut out by the ideal $J$, resp.\\ $J'$ of $R[T_1/T_0, \\ldots, T_n/T_0]$.", "Then $J'$ is the ideal generated by the elements $F/T_0^{\\deg(F)}$ where", "$F \\in I$ is homogeneous.", "Suppose the degree of $F \\in I$ is $d$. Since $F$ vanishes as a section", "of $\\mathcal{O}_{\\mathbf{P}^n_R}(d)$ restricted to $Z$ we see that", "$F/T_0^d$ is an element of $J$. Thus $J' \\subset J$.", "\\medskip\\noindent", "Conversely, suppose that $f \\in J$. If $f$ has total degree", "$d$ in $T_1/T_0, \\ldots, T_n/T_0$, then we can write", "$f = F/T_0^d$ for some $F \\in R[T_0, \\ldots, T_n]_d$.", "Pick $i \\in \\{1, \\ldots, n\\}$. Then $Z \\cap D_{+}(T_i)$ is", "cut out by some ideal $J_i \\subset R[T_0/T_i, \\ldots, T_n/T_i]$.", "Moreover,", "$$", "J \\cdot", "R\\left[", "\\frac{T_1}{T_0}, \\ldots, \\frac{T_n}{T_0},", "\\frac{T_0}{T_i}, \\ldots, \\frac{T_n}{T_i}", "\\right]", "=", "J_i \\cdot", "R\\left[", "\\frac{T_1}{T_0}, \\ldots, \\frac{T_n}{T_0},", "\\frac{T_0}{T_i}, \\ldots, \\frac{T_n}{T_i}", "\\right]", "$$", "The left hand side is the localization of $J$ with respect to the", "element $T_i/T_0$ and the right hand side is the localization of $J_i$", "with respect to the element $T_0/T_i$. It follows that", "$T_0^{d_i}F/T_i^{d + d_i}$ is an element of $J_i$ for some $d_i$", "sufficiently large. This proves that $T_0^{\\max(d_i)}F$ is an", "element of $I$, because its restriction to each standard affine", "open $D_{+}(T_i)$ vanishes on the closed subscheme", "$Z \\cap D_{+}(T_i)$. Hence $f \\in J'$ and we conclude $J \\subset J'$", "as desired." ], "refs": [ "constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj" ], "ref_ids": [ 12612 ] } ], "ref_ids": [] }, { "id": 12626, "type": "theorem", "label": "constructions-lemma-quasi-coherent-projective-space", "categories": [ "constructions" ], "title": "constructions-lemma-quasi-coherent-projective-space", "contents": [ "Let $R$ be a ring.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $\\mathbf{P}^n_R$.", "For $d \\geq 0$ set", "$$", "M_d", "=", "\\Gamma(\\mathbf{P}^n_R,", "\\mathcal{F} \\otimes_{\\mathcal{O}_{\\mathbf{P}^n_R}}", "\\mathcal{O}_{\\mathbf{P}^n_R}(d))", "=", "\\Gamma(\\mathbf{P}^n_R, \\mathcal{F}(d))", "$$", "Then $M = \\bigoplus_{d \\geq 0} M_d$ is a graded $R[T_0, \\ldots, R_n]$-module", "and there is a canonical isomorphism $\\mathcal{F} = \\widetilde{M}$." ], "refs": [], "proofs": [ { "contents": [ "The multiplication maps", "$$", "R[T_0, \\ldots, R_n]_e \\times M_d \\longrightarrow M_{d + e}", "$$", "come from the natural isomorphisms", "$$", "\\mathcal{O}_{\\mathbf{P}^n_R}(e)", "\\otimes_{\\mathcal{O}_{\\mathbf{P}^n_R}}", "\\mathcal{F}(d)", "\\longrightarrow", "\\mathcal{F}(e + d)", "$$", "see Equation (\\ref{equation-global-sections-module}). Let us construct the", "map $c : \\widetilde{M} \\to \\mathcal{F}$. On each of the standard affines", "$U_i = D_{+}(T_i)$ we see that $\\Gamma(U_i, \\widetilde{M}) = (M[1/T_i])_0$", "where the subscript ${}_0$ means degree $0$ part. An element of this", "can be written as $m/T_i^d$ with $m \\in M_d$. Since $T_i$ is a generator", "of $\\mathcal{O}(1)$ over $U_i$ we can always write", "$m|_{U_i} = m_i \\otimes T_i^d$ where $m_i \\in \\Gamma(U_i, \\mathcal{F})$", "is a unique section. Thus a natural guess is $c(m/T_i^d) = m_i$.", "A small argument, which is omitted here, shows that this gives a", "well defined map $c : \\widetilde{M} \\to \\mathcal{F}$ if we can", "show that", "$$", "(T_i/T_j)^d m_i|_{U_i \\cap U_j} = m_j|_{U_i \\cap U_j}", "$$", "in $M[1/T_iT_j]$.", "But this is clear since on the overlap the generators $T_i$ and", "$T_j$ of $\\mathcal{O}(1)$ differ by the invertible function $T_i/T_j$.", "\\medskip\\noindent", "Injectivity of $c$. We may check for injectivity over the affine opens", "$U_i$. Let $i \\in \\{0, \\ldots, n\\}$", "and let $s$ be an element $s = m/T_i^d \\in \\Gamma(U_i, \\widetilde{M})$", "such that $c(m/T_i^d) = 0$. By the description of $c$ above this means that", "$m_i = 0$, hence $m|_{U_i} = 0$. Hence $T_i^em = 0$ in $M$ for some", "$e$. Hence $s = m/T_i^d = T_i^e/T_i^{e + d} = 0$ as desired.", "\\medskip\\noindent", "Surjectivity of $c$. We may check for surjectivity over the affine opens", "$U_i$. By renumbering it suffices to check it over $U_0$.", "Let $s \\in \\mathcal{F}(U_0)$.", "Let us write $\\mathcal{F}|_{U_i} = \\widetilde{N_i}$ for some", "$R[T_0/T_i, \\ldots, T_0/T_i]$-module $N_i$, which is possible because", "$\\mathcal{F}$ is quasi-coherent. So $s$ corresponds to an element", "$x \\in N_0$. Then we have that", "$$", "(N_i)_{T_j/T_i} \\cong (N_j)_{T_i/T_j}", "$$", "(where the subscripts mean ``principal localization at'')", "as modules over the ring", "$$", "R\\left[", "\\frac{T_0}{T_i}, \\ldots, \\frac{T_n}{T_i},", "\\frac{T_0}{T_j}, \\ldots, \\frac{T_n}{T_j}", "\\right].", "$$", "This means that for some large integer $d$ there exist elements", "$s_i \\in N_i$, $i = 1, \\ldots, n$ such that", "$$", "s = (T_i/T_0)^d s_i", "$$", "on $U_0 \\cap U_i$. Next, we look at the difference", "$$", "t_{ij} = s_i - (T_j/T_i)^d s_j", "$$", "on $U_i \\cap U_j$, $0 < i < j$. By our choice of $s_i$ we know that", "$t_{ij}|_{U_0 \\cap U_i \\cap U_j} = 0$. Hence there exists a large integer", "$e$ such that $(T_0/T_i)^et_{ij} = 0$. Set $s_i' = (T_0/T_i)^es_i$,", "and $s_0' = s$. Then we will have", "$$", "s_a' = (T_b/T_a)^{e + d} s_b'", "$$", "on $U_a \\cap U_b$ for all $a, b$. This is exactly the condition that the", "elements $s'_a$ glue to a global section", "$m \\in \\Gamma(\\mathbf{P}^n_R, \\mathcal{F}(e + d))$.", "And moreover $c(m/T_0^{e + d}) = s$ by construction. Hence $c$ is", "surjective and we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12627, "type": "theorem", "label": "constructions-lemma-globally-generated-omega-twist-1", "categories": [ "constructions" ], "title": "constructions-lemma-globally-generated-omega-twist-1", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible sheaf", "and let $s_0, \\ldots, s_n$ be global sections of $\\mathcal{L}$", "which generate it. Let $\\mathcal{F}$ be the kernel of the induced", "map $\\mathcal{O}_X^{\\oplus n + 1} \\to \\mathcal{L}$.", "Then $\\mathcal{F} \\otimes \\mathcal{L}$ is globally generated." ], "refs": [], "proofs": [ { "contents": [ "In fact the result is true if $X$ is any locally ringed space.", "The sheaf $\\mathcal{F}$ is a finite locally free $\\mathcal{O}_X$-module", "of rank $n$. The elements", "$$", "s_{ij} = (0, \\ldots, 0, s_j, 0, \\ldots, 0, -s_i, 0, \\ldots, 0)", "\\in \\Gamma(X, \\mathcal{L}^{\\oplus n + 1})", "$$", "with $s_j$ in the $i$th spot and $-s_i$ in the $j$th spot map to zero", "in $\\mathcal{L}^{\\otimes 2}$. Hence", "$s_{ij} \\in \\Gamma(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L})$.", "A local computation shows that these sections generate", "$\\mathcal{F} \\otimes \\mathcal{L}$.", "\\medskip\\noindent", "Alternative proof. Consider the morphism", "$\\varphi : X \\to \\mathbf{P}^n_\\mathbf{Z}$ associated to", "the pair $(\\mathcal{L}, (s_0, \\ldots, s_n))$. Since the pullback", "of $\\mathcal{O}(1)$ is $\\mathcal{L}$ and since the pullback", "of $T_i$ is $s_i$, it suffices to prove the lemma in the", "case of $\\mathbf{P}^n_\\mathbf{Z}$. In this case the sheaf", "$\\mathcal{F}$ corresponds to the graded $S = \\mathbf{Z}[T_0, \\ldots, T_n]$", "module $M$ which fits into the short exact sequence", "$$", "0 \\to M \\to S^{\\oplus n + 1} \\to S(1) \\to 0", "$$", "where the second map is given by $T_0, \\ldots, T_n$. In this", "case the statement above translates into the statement that", "the elements", "$$", "T_{ij} = (0, \\ldots, 0, T_j, 0, \\ldots, 0, -T_i, 0, \\ldots, 0)", "\\in M(1)_0", "$$", "generate the graded module $M(1)$ over $S$. We omit the details." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12628, "type": "theorem", "label": "constructions-lemma-invertible-map-into-proj", "categories": [ "constructions" ], "title": "constructions-lemma-invertible-map-into-proj", "contents": [ "Let $A$ be a graded ring.", "Set $X = \\text{Proj}(A)$.", "Let $T$ be a scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_T$-module.", "Let $\\psi : A \\to \\Gamma_*(T, \\mathcal{L})$ be a homomorphism", "of graded rings. Set", "$$", "U(\\psi) = \\bigcup\\nolimits_{f \\in A_{+}\\text{ homogeneous}} T_{\\psi(f)}", "$$", "The morphism $\\psi$ induces a canonical morphism of schemes", "$$", "r_{\\mathcal{L}, \\psi} :", "U(\\psi) \\longrightarrow X", "$$", "together with a map of $\\mathbf{Z}$-graded $\\mathcal{O}_T$-algebras", "$$", "\\theta :", "r_{\\mathcal{L}, \\psi}^*\\left(", "\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_X(d)", "\\right)", "\\longrightarrow", "\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{L}^{\\otimes d}|_{U(\\psi)}.", "$$", "The triple $(U(\\psi), r_{\\mathcal{L}, \\psi}, \\theta)$ is", "characterized by the following properties:", "\\begin{enumerate}", "\\item For $f \\in A_{+}$ homogeneous we have", "$r_{\\mathcal{L}, \\psi}^{-1}(D_{+}(f)) = T_{\\psi(f)}$.", "\\item For every $d \\geq 0$ the diagram", "$$", "\\xymatrix{", "A_d \\ar[d]_{(\\ref{equation-global-sections})} \\ar[r]_{\\psi} &", "\\Gamma(T, \\mathcal{L}^{\\otimes d}) \\ar[d]^{restrict} \\\\", "\\Gamma(X, \\mathcal{O}_X(d)) \\ar[r]^{\\theta} &", "\\Gamma(U(\\psi), \\mathcal{L}^{\\otimes d})", "}", "$$", "is commutative.", "\\end{enumerate}", "Moreover, for any $d \\geq 1$ and any open subscheme $V \\subset T$", "such that the sections in $\\psi(A_d)$ generate $\\mathcal{L}^{\\otimes d}|_V$", "the morphism $r_{\\mathcal{L}, \\psi}|_V$ agrees with the morphism", "$\\varphi : V \\to \\text{Proj}(A)$ and the map $\\theta|_V$ agrees with the map", "$\\alpha : \\varphi^*\\mathcal{O}_X(d) \\to \\mathcal{L}^{\\otimes d}|_V$", "where $(\\varphi, \\alpha)$ is the pair", "of Lemma \\ref{lemma-converse-construction}", "associated to", "$\\psi|_{A^{(d)}} : A^{(d)} \\to \\Gamma_*(V, \\mathcal{L}^{\\otimes d})$." ], "refs": [ "constructions-lemma-converse-construction" ], "proofs": [ { "contents": [ "Suppose that we have two triples $(U, r : U \\to X, \\theta)$", "and $(U', r' : U' \\to X, \\theta')$ satisfying (1) and (2).", "Property (1) implies that $U = U' = U(\\psi)$ and that", "$r = r'$ as maps of underlying topological", "spaces, since the opens $D_{+}(f)$ form a basis for the topology", "on $X$, and since $X$ is a sober topological space (see", "Algebra, Section \\ref{algebra-section-proj}", "and", "Schemes, Lemma \\ref{schemes-lemma-scheme-sober}).", "Let $f \\in A_{+}$ be homogeneous. Note that", "$\\Gamma(D_{+}(f), \\bigoplus_{n \\in \\mathbf{Z}} \\mathcal{O}_X(n)) = A_f$", "as a $\\mathbf{Z}$-graded algebra. Consider the two", "$\\mathbf{Z}$-graded ring maps", "$$", "\\theta, \\theta' :", "A_f", "\\longrightarrow", "\\Gamma(T_{\\psi(f)}, \\bigoplus \\mathcal{L}^{\\otimes n}).", "$$", "We know that multiplication by $f$ (resp.\\ $\\psi(f)$)", "is an isomorphism on the left (resp.\\ right) hand side.", "We also know that $\\theta(x/1) = \\theta'(x/1) = \\psi(x)|_{T_{\\psi(f)}}$", "by (2) for all $x \\in A$. Hence we deduce easily that $\\theta = \\theta'$", "as desired. Considering the degree $0$ parts we deduce that", "$r^\\sharp = (r')^\\sharp$, i.e., that $r = r'$ as morphisms of schemes.", "This proves the uniqueness.", "\\medskip\\noindent", "Now we come to existence. By the uniqueness just proved, it is enough to", "construct the pair $(r, \\theta)$ locally on $T$. Hence we may assume", "that $T = \\Spec(R)$ is affine, that $\\mathcal{L} = \\mathcal{O}_T$", "and that for some $f \\in A_{+}$ homogeneous we have", "$\\psi(f)$ generates $\\mathcal{O}_T = \\mathcal{O}_T^{\\otimes \\deg(f)}$.", "In other words, $\\psi(f) = u \\in R^*$ is a unit. In this case the map", "$\\psi$ is a graded ring map", "$$", "A \\longrightarrow R[x] = \\Gamma_*(T, \\mathcal{O}_T)", "$$", "which maps $f$ to $ux^{\\deg(f)}$. Clearly this extends (uniquely) to", "a $\\mathbf{Z}$-graded ring map $\\theta : A_f \\to R[x, x^{-1}]$ by", "mapping $1/f$ to $u^{-1}x^{-\\deg(f)}$. This map in degree zero gives", "the ring map $A_{(f)} \\to R$ which gives the morphism", "$r : T = \\Spec(R) \\to \\Spec(A_{(f)}) = D_{+}(f) \\subset X$.", "Hence we have constructed $(r, \\theta)$ in this special case.", "\\medskip\\noindent", "Let us show the last statement of the lemma.", "According to Lemma \\ref{lemma-converse-construction}", "the morphism constructed there is the unique one such that", "the displayed diagram in its statement commutes.", "The commutativity of the diagram in the lemma implies the", "commutativity when restricted to $V$ and $A^{(d)}$.", "Whence the result." ], "refs": [ "schemes-lemma-scheme-sober", "constructions-lemma-converse-construction" ], "ref_ids": [ 7672, 12616 ] } ], "ref_ids": [ 12616 ] }, { "id": 12629, "type": "theorem", "label": "constructions-lemma-proj-inclusion", "categories": [ "constructions" ], "title": "constructions-lemma-proj-inclusion", "contents": [ "In Situation \\ref{situation-relative-proj}.", "Suppose $U \\subset U' \\subset S$ are affine opens.", "Let $A = \\mathcal{A}(U)$ and $A' = \\mathcal{A}(U')$.", "The map of graded rings $A' \\to A$ induces a morphism", "$r : \\text{Proj}(A) \\to \\text{Proj}(A')$, and the diagram", "$$", "\\xymatrix{", "\\text{Proj}(A) \\ar[r] \\ar[d] &", "\\text{Proj}(A') \\ar[d] \\\\", "U \\ar[r] &", "U'", "}", "$$", "is cartesian. Moreover there are canonical isomorphisms", "$\\theta : r^*\\mathcal{O}_{\\text{Proj}(A')}(n) \\to", "\\mathcal{O}_{\\text{Proj}(A)}(n)$ compatible with multiplication maps." ], "refs": [], "proofs": [ { "contents": [ "Let $R = \\mathcal{O}_S(U)$ and $R' = \\mathcal{O}_S(U')$.", "Note that the map $R \\otimes_{R'} A' \\to A$ is an isomorphism as", "$\\mathcal{A}$ is quasi-coherent", "(see Schemes, Lemma \\ref{schemes-lemma-widetilde-pullback} for example).", "Hence the lemma follows from", "Lemma \\ref{lemma-base-change-map-proj}." ], "refs": [ "schemes-lemma-widetilde-pullback", "constructions-lemma-base-change-map-proj" ], "ref_ids": [ 7662, 12613 ] } ], "ref_ids": [] }, { "id": 12630, "type": "theorem", "label": "constructions-lemma-transitive-proj", "categories": [ "constructions" ], "title": "constructions-lemma-transitive-proj", "contents": [ "In Situation \\ref{situation-relative-proj}.", "Suppose $U \\subset U' \\subset U'' \\subset S$ are affine opens.", "Let $A = \\mathcal{A}(U)$, $A' = \\mathcal{A}(U')$ and $A'' = \\mathcal{A}(U'')$.", "The composition of the morphisms", "$r : \\text{Proj}(A) \\to \\text{Proj}(A')$, and", "$r' : \\text{Proj}(A') \\to \\text{Proj}(A'')$ of", "Lemma \\ref{lemma-proj-inclusion} gives the", "morphism $r'' : \\text{Proj}(A) \\to \\text{Proj}(A'')$", "of Lemma \\ref{lemma-proj-inclusion}. A similar statement", "holds for the isomorphisms $\\theta$." ], "refs": [ "constructions-lemma-proj-inclusion", "constructions-lemma-proj-inclusion" ], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-morphism-proj-transitive} since", "the map $A'' \\to A$ is the composition of $A'' \\to A'$ and", "$A' \\to A$." ], "refs": [ "constructions-lemma-morphism-proj-transitive" ], "ref_ids": [ 12609 ] } ], "ref_ids": [ 12629, 12629 ] }, { "id": 12631, "type": "theorem", "label": "constructions-lemma-glue-relative-proj", "categories": [ "constructions" ], "title": "constructions-lemma-glue-relative-proj", "contents": [ "In Situation \\ref{situation-relative-proj}.", "There exists a morphism of schemes", "$$", "\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\longrightarrow S", "$$", "with the following properties:", "\\begin{enumerate}", "\\item for every affine open $U \\subset S$ there exists an isomorphism", "$i_U : \\pi^{-1}(U) \\to \\text{Proj}(A)$ with $A = \\mathcal{A}(U)$, and", "\\item for $U \\subset U' \\subset S$ affine open the composition", "$$", "\\xymatrix{", "\\text{Proj}(A) \\ar[r]^{i_U^{-1}} &", "\\pi^{-1}(U) \\ar[rr]^{inclusion} & &", "\\pi^{-1}(U') \\ar[r]^{i_{U'}} &", "\\text{Proj}(A')", "}", "$$", "with $A = \\mathcal{A}(U)$, $A' = \\mathcal{A}(U')$", "is the open immersion of Lemma \\ref{lemma-proj-inclusion} above.", "\\end{enumerate}" ], "refs": [ "constructions-lemma-proj-inclusion" ], "proofs": [ { "contents": [ "Follows immediately from", "Lemmas \\ref{lemma-relative-glueing},", "\\ref{lemma-proj-inclusion}, and", "\\ref{lemma-transitive-proj}." ], "refs": [ "constructions-lemma-relative-glueing", "constructions-lemma-proj-inclusion", "constructions-lemma-transitive-proj" ], "ref_ids": [ 12581, 12629, 12630 ] } ], "ref_ids": [ 12629 ] }, { "id": 12632, "type": "theorem", "label": "constructions-lemma-glue-relative-proj-twists", "categories": [ "constructions" ], "title": "constructions-lemma-glue-relative-proj-twists", "contents": [ "In Situation \\ref{situation-relative-proj}.", "The morphism $\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$", "of Lemma \\ref{lemma-glue-relative-proj} comes with the following", "additional structure.", "There exists a quasi-coherent $\\mathbf{Z}$-graded sheaf", "of $\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}$-algebras", "$\\bigoplus\\nolimits_{n \\in \\mathbf{Z}}", "\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n)$,", "and a morphism of graded $\\mathcal{O}_S$-algebras", "$$", "\\psi :", "\\mathcal{A}", "\\longrightarrow", "\\bigoplus\\nolimits_{n \\geq 0}", "\\pi_*\\left(\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n)\\right)", "$$", "uniquely determined by the following property:", "For every affine open $U \\subset S$ with $A = \\mathcal{A}(U)$", "there is an isomorphism", "$$", "\\theta_U :", "i_U^*\\left(", "\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_{\\text{Proj}(A)}(n)", "\\right)", "\\longrightarrow", "\\left(", "\\bigoplus\\nolimits_{n \\in \\mathbf{Z}}", "\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n)", "\\right)|_{\\pi^{-1}(U)}", "$$", "of $\\mathbf{Z}$-graded $\\mathcal{O}_{\\pi^{-1}(U)}$-algebras", "such that", "$$", "\\xymatrix{", "A_n", "\\ar[rr]_\\psi", "\\ar[dr]_-{(\\ref{equation-global-sections})}", "& &", "\\Gamma(\\pi^{-1}(U),", "\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n)) \\\\", "&", "\\Gamma(\\text{Proj}(A),", "\\mathcal{O}_{\\text{Proj}(A)}(n))", "\\ar[ru]_-{\\theta_U}", "&", "}", "$$", "is commutative." ], "refs": [ "constructions-lemma-glue-relative-proj" ], "proofs": [ { "contents": [ "We are going to use Lemma \\ref{lemma-relative-glueing-sheaves}", "to glue the sheaves of $\\mathbf{Z}$-graded algebras", "$\\bigoplus_{n \\in \\mathbf{Z}} \\mathcal{O}_{\\text{Proj}(A)}(n)$", "for $A = \\mathcal{A}(U)$, $U \\subset S$ affine open", "over the scheme $\\underline{\\text{Proj}}_S(\\mathcal{A})$.", "We have constructed the data necessary for this in", "Lemma \\ref{lemma-proj-inclusion} and we have checked condition (d) of", "Lemma \\ref{lemma-relative-glueing-sheaves} in", "Lemma \\ref{lemma-transitive-proj}. Hence we get the", "sheaf of $\\mathbf{Z}$-graded", "$\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}$-algebras", "$\\bigoplus_{n \\in \\mathbf{Z}}", "\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n)$", "together with the isomorphisms $\\theta_U$ for all", "$U \\subset S$ affine open and all $n \\in \\mathbf{Z}$.", "For every affine open $U \\subset S$ with $A = \\mathcal{A}(U)$ we have a map", "$A \\to \\Gamma(\\text{Proj}(A),", "\\bigoplus_{n \\geq 0} \\mathcal{O}_{\\text{Proj}(A)}(n))$.", "Hence the map $\\psi$ exists by functoriality", "of relative glueing, see Remark \\ref{remark-relative-glueing-functorial}.", "The diagram of the lemma commutes by construction.", "This characterizes the sheaf of $\\mathbf{Z}$-graded", "$\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}$-algebras", "$\\bigoplus \\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n)$", "because the proof of Lemma \\ref{lemma-morphism-proj} shows that", "having these diagrams commute uniquely determines the maps $\\theta_U$.", "Some details omitted." ], "refs": [ "constructions-lemma-relative-glueing-sheaves", "constructions-lemma-proj-inclusion", "constructions-lemma-relative-glueing-sheaves", "constructions-lemma-transitive-proj", "constructions-remark-relative-glueing-functorial", "constructions-lemma-morphism-proj" ], "ref_ids": [ 12582, 12629, 12582, 12630, 12668, 12608 ] } ], "ref_ids": [ 12631 ] }, { "id": 12633, "type": "theorem", "label": "constructions-lemma-proj-base-change", "categories": [ "constructions" ], "title": "constructions-lemma-proj-base-change", "contents": [ "In Situation \\ref{situation-relative-proj}. Let $d \\geq 1$.", "Let $F_d$ be the functor", "associated to $(S, \\mathcal{A})$ above.", "Let $g : S' \\to S$ be a morphism of schemes.", "Set $\\mathcal{A}' = g^*\\mathcal{A}$. Let $F_d'$ be the", "functor associated to $(S', \\mathcal{A}')$ above.", "Then there is a canonical isomorphism", "$$", "F'_d \\cong h_{S'} \\times_{h_S} F_d", "$$", "of functors." ], "refs": [], "proofs": [ { "contents": [ "A quadruple", "$(d, f' : T \\to S', \\mathcal{L}',", "\\psi' : (f')^*(\\mathcal{A}')^{(d)} \\to", "\\bigoplus_{n \\geq 0} (\\mathcal{L}')^{\\otimes n})$", "is the same as a quadruple", "$(d, f, \\mathcal{L},", "\\psi : f^*\\mathcal{A}^{(d)} \\to", "\\bigoplus_{n \\geq 0} \\mathcal{L}^{\\otimes n})$", "together with a factorization of $f$ as $f = g \\circ f'$. Namely,", "the correspondence is $f = g \\circ f'$, $\\mathcal{L} = \\mathcal{L}'$", "and $\\psi = \\psi'$ via the identifications", "$(f')^*(\\mathcal{A}')^{(d)} = (f')^*g^*(\\mathcal{A}^{(d)}) =", "f^*\\mathcal{A}^{(d)}$. Hence the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12634, "type": "theorem", "label": "constructions-lemma-relative-proj-affine", "categories": [ "constructions" ], "title": "constructions-lemma-relative-proj-affine", "contents": [ "In Situation \\ref{situation-relative-proj}. Let $F_d$ be the functor", "associated to $(d, S, \\mathcal{A})$ above.", "If $S$ is affine, then $F_d$ is representable by the open subscheme", "$U_d$ (\\ref{equation-Ud})", "of the scheme $\\text{Proj}(\\Gamma(S, \\mathcal{A}))$." ], "refs": [], "proofs": [ { "contents": [ "Write $S = \\Spec(R)$ and $A = \\Gamma(S, \\mathcal{A})$.", "Then $A$ is a graded $R$-algebra and $\\mathcal{A} = \\widetilde A$.", "To prove the lemma we have to identify the functor $F_d$", "with the functor $F_d^{triples}$ of triples defined in Section", "\\ref{section-morphisms-proj}.", "\\medskip\\noindent", "Let $(d, f : T \\to S, \\mathcal{L}, \\psi)$ be a quadruple.", "We may think of $\\psi$ as a $\\mathcal{O}_S$-module map", "$\\mathcal{A}^{(d)} \\to \\bigoplus_{n \\geq 0} f_*\\mathcal{L}^{\\otimes n}$.", "Since $\\mathcal{A}^{(d)}$ is quasi-coherent this is the same", "thing as an $R$-linear homomorphism of graded rings", "$A^{(d)} \\to \\Gamma(S, \\bigoplus_{n \\geq 0} f_*\\mathcal{L}^{\\otimes n})$.", "Clearly, $\\Gamma(S, \\bigoplus_{n \\geq 0} f_*\\mathcal{L}^{\\otimes n}) =", "\\Gamma_*(T, \\mathcal{L})$. Thus we may associate to", "the quadruple the triple $(d, \\mathcal{L}, \\psi)$.", "\\medskip\\noindent", "Conversely, let $(d, \\mathcal{L}, \\psi)$ be a triple.", "The composition $R \\to A_0 \\to \\Gamma(T, \\mathcal{O}_T)$", "determines a morphism $f : T \\to S = \\Spec(R)$, see", "Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}.", "With this choice of $f$ the map", "$A^{(d)} \\to \\Gamma(S, \\bigoplus_{n \\geq 0} f_*\\mathcal{L}^{\\otimes n})$", "is $R$-linear, and hence corresponds to a $\\psi$ which we", "can use for a quadruple $(d, f : T \\to S, \\mathcal{L}, \\psi)$.", "We omit the verification that this establishes an isomorphism", "of functors $F_d = F_d^{triples}$." ], "refs": [ "schemes-lemma-morphism-into-affine" ], "ref_ids": [ 7655 ] } ], "ref_ids": [] }, { "id": 12635, "type": "theorem", "label": "constructions-lemma-relative-proj-d", "categories": [ "constructions" ], "title": "constructions-lemma-relative-proj-d", "contents": [ "In Situation \\ref{situation-relative-proj}.", "The functor $F_d$ is representable by a scheme." ], "refs": [], "proofs": [ { "contents": [ "We are going to use Schemes, Lemma \\ref{schemes-lemma-glue-functors}.", "\\medskip\\noindent", "First we check that $F_d$ satisfies the sheaf property for the", "Zariski topology. Namely, suppose that $T$ is a scheme,", "that $T = \\bigcup_{i \\in I} U_i$ is an open covering,", "and that $(d, f_i, \\mathcal{L}_i, \\psi_i) \\in F_d(U_i)$ such that", "$(d, f_i, \\mathcal{L}_i, \\psi_i)|_{U_i \\cap U_j}$ and", "$(d, f_j, \\mathcal{L}_j, \\psi_j)|_{U_i \\cap U_j}$ are strictly", "equivalent. This implies that the morphisms $f_i : U_i \\to S$", "glue to a morphism of schemes $f : T \\to S$ such that", "$f|_{I_i} = f_i$, see Schemes, Section \\ref{schemes-section-glueing-schemes}.", "Thus $f_i^*\\mathcal{A}^{(d)} = f^*\\mathcal{A}^{(d)}|_{U_i}$.", "It also implies there exist isomorphisms", "$\\beta_{ij} : \\mathcal{L}_i|_{U_i \\cap U_j} \\to \\mathcal{L}_j|_{U_i \\cap U_j}$", "such that $\\beta_{ij} \\circ \\psi_i = \\psi_j$ on $U_i \\cap U_j$.", "Note that the isomorphisms $\\beta_{ij}$ are uniquely determined", "by this requirement because the maps $f_i^*\\mathcal{A}_d \\to \\mathcal{L}_i$", "are surjective. In particular we see that", "$\\beta_{jk} \\circ \\beta_{ij} = \\beta_{ik}$ on $U_i \\cap U_j \\cap U_k$.", "Hence by Sheaves,", "Section \\ref{sheaves-section-glueing-sheaves} the invertible sheaves", "$\\mathcal{L}_i$ glue to an invertible $\\mathcal{O}_T$-module", "$\\mathcal{L}$ and the morphisms $\\psi_i$ glue to", "morphism of $\\mathcal{O}_T$-algebras", "$\\psi : f^*\\mathcal{A}^{(d)} \\to \\bigoplus_{n \\geq 0} \\mathcal{L}^{\\otimes n}$.", "This proves that $F_d$ satisfies the sheaf condition with respect to", "the Zariski topology.", "\\medskip\\noindent", "Let $S = \\bigcup_{i \\in I} U_i$ be an affine open covering.", "Let $F_{d, i} \\subset F_d$ be the subfunctor consisting of", "those pairs $(f : T \\to S, \\varphi)$ such that", "$f(T) \\subset U_i$.", "\\medskip\\noindent", "We have to show each $F_{d, i}$ is representable.", "This is the case because $F_{d, i}$ is identified with", "the functor associated to $U_i$ equipped with", "the quasi-coherent graded $\\mathcal{O}_{U_i}$-algebra", "$\\mathcal{A}|_{U_i}$ by Lemma \\ref{lemma-proj-base-change}.", "Thus the result follows from Lemma \\ref{lemma-relative-proj-affine}.", "\\medskip\\noindent", "Next we show that $F_{d, i} \\subset F_d$ is representable by open immersions.", "Let $(f : T \\to S, \\varphi) \\in F_d(T)$. Consider $V_i = f^{-1}(U_i)$.", "It follows from the definition of $F_{d, i}$ that given $a : T' \\to T$", "we gave $a^*(f, \\varphi) \\in F_{d, i}(T')$ if and only if $a(T') \\subset V_i$.", "This is what we were required to show.", "\\medskip\\noindent", "Finally, we have to show that the collection $(F_{d, i})_{i \\in I}$", "covers $F_d$. Let $(f : T \\to S, \\varphi) \\in F_d(T)$.", "Consider $V_i = f^{-1}(U_i)$. Since $S = \\bigcup_{i \\in I} U_i$", "is an open covering of $S$ we see that $T = \\bigcup_{i \\in I} V_i$", "is an open covering of $T$. Moreover $(f, \\varphi)|_{V_i} \\in F_{d, i}(V_i)$.", "This finishes the proof of the lemma." ], "refs": [ "schemes-lemma-glue-functors", "constructions-lemma-proj-base-change", "constructions-lemma-relative-proj-affine" ], "ref_ids": [ 7688, 12633, 12634 ] } ], "ref_ids": [] }, { "id": 12636, "type": "theorem", "label": "constructions-lemma-equivalent-relative", "categories": [ "constructions" ], "title": "constructions-lemma-equivalent-relative", "contents": [ "In Situation \\ref{situation-relative-proj}.", "Let $T$ be a scheme.", "Let $(d, f, \\mathcal{L}, \\psi)$, $(d', f', \\mathcal{L}', \\psi')$", "be two quadruples over $T$. The following are equivalent:", "\\begin{enumerate}", "\\item Let $m = \\text{lcm}(d, d')$. Write $m = ad = a'd'$.", "We have $f = f'$ and there exists", "an isomorphism", "$\\beta : \\mathcal{L}^{\\otimes a} \\to (\\mathcal{L}')^{\\otimes a'}$", "with the property that $\\beta \\circ \\psi|_{f^*\\mathcal{A}^{(m)}}$", "and $\\psi'|_{f^*\\mathcal{A}^{(m)}}$ agree", "as graded ring maps", "$f^*\\mathcal{A}^{(m)} \\to \\bigoplus_{n \\geq 0} (\\mathcal{L}')^{\\otimes mn}$.", "\\item The quadruples $(d, f, \\mathcal{L}, \\psi)$ and", "$(d', f', \\mathcal{L}', \\psi')$ are equivalent.", "\\item We have $f = f'$ and", "for some positive integer $m = ad = a'd'$ there exists an isomorphism", "$\\beta : \\mathcal{L}^{\\otimes a} \\to (\\mathcal{L}')^{\\otimes a'}$", "with the property that $\\beta \\circ \\psi|_{f^*\\mathcal{A}^{(m)}}$", "and $\\psi'|_{f^*\\mathcal{A}^{(m)}}$ agree", "as graded ring maps", "$f^*\\mathcal{A}^{(m)} \\to \\bigoplus_{n \\geq 0} (\\mathcal{L}')^{\\otimes mn}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Clearly (1) implies (2) and (2) implies (3) by restricting to", "more divisible degrees and powers of invertible sheaves.", "Assume (3) for some integer $m = ad = a'd'$. Let", "$m_0 = \\text{lcm}(d, d')$ and write it as $m_0 = a_0d = a'_0d'$.", "We are given an isomorphism", "$\\beta : \\mathcal{L}^{\\otimes a} \\to (\\mathcal{L}')^{\\otimes a'}$", "with the property described in (3). We want to find an isomorphism", "$\\beta_0 : \\mathcal{L}^{\\otimes a_0} \\to (\\mathcal{L}')^{\\otimes a'_0}$", "having that property as well.", "Since by assumption the maps $\\psi : f^*\\mathcal{A}_d \\to \\mathcal{L}$", "and $\\psi' : (f')^*\\mathcal{A}_{d'} \\to \\mathcal{L}'$ are surjective the", "same is true for the maps", "$\\psi : f^*\\mathcal{A}_{m_0} \\to \\mathcal{L}^{\\otimes a_0}$", "and $\\psi' : (f')^*\\mathcal{A}_{m_0} \\to (\\mathcal{L}')^{\\otimes a_0}$.", "Hence if $\\beta_0$ exists it is uniquely determined by the", "condition that $\\beta_0 \\circ \\psi = \\psi'$. This means that", "we may work locally on $T$. Hence we may assume that", "$f = f' : T \\to S$ maps into an affine open, in other words", "we may assume that $S$ is affine. In this case the result follows", "from the corresponding result for triples (see Lemma \\ref{lemma-equivalent})", "and the fact that triples and quadruples correspond in the", "affine base case (see proof of Lemma \\ref{lemma-relative-proj-affine})." ], "refs": [ "constructions-lemma-equivalent", "constructions-lemma-relative-proj-affine" ], "ref_ids": [ 12619, 12634 ] } ], "ref_ids": [] }, { "id": 12637, "type": "theorem", "label": "constructions-lemma-relative-proj", "categories": [ "constructions" ], "title": "constructions-lemma-relative-proj", "contents": [ "In Situation \\ref{situation-relative-proj}.", "The functor $F$ above is representable by a scheme." ], "refs": [], "proofs": [ { "contents": [ "Let $U_d \\to S$ be the scheme representing the functor $F_d$", "defined above. Let $\\mathcal{L}_d$,", "$\\psi^d : \\pi_d^*\\mathcal{A}^{(d)} \\to", "\\bigoplus_{n \\geq 0} \\mathcal{L}_d^{\\otimes n}$ be the universal object.", "If $d | d'$, then we may consider the quadruple", "$(d', \\pi_d, \\mathcal{L}_d^{\\otimes d'/d}, \\psi^d|_{\\mathcal{A}^{(d')}})$", "which determines a canonical morphism $U_d \\to U_{d'}$ over $S$.", "By construction this morphism corresponds to the transformation", "of functors $F_d \\to F_{d'}$ defined above.", "\\medskip\\noindent", "For every affine open $\\Spec(R) = V \\subset S$", "setting $A = \\Gamma(V, \\mathcal{A})$ we have a canonical", "identification of the base change $U_{d, V}$ with the corresponding open", "subscheme of $\\text{Proj}(A)$, see Lemma \\ref{lemma-relative-proj-affine}.", "Moreover, the morphisms $U_{d, V} \\to U_{d', V}$ constructed above", "correspond to the inclusions of opens in $\\text{Proj}(A)$.", "Thus we conclude that $U_d \\to U_{d'}$ is an open immersion.", "\\medskip\\noindent", "This allows us to construct $X$", "by glueing the schemes $U_d$ along the open immersions $U_d \\to U_{d'}$.", "Technically, it is convenient to choose a sequence", "$d_1 | d_2 | d_3 | \\ldots$ such that every positive integer", "divides one of the $d_i$ and to simply take", "$X = \\bigcup U_{d_i}$ using the open immersions above.", "It is then a simple matter to prove that $X$ represents the", "functor $F$." ], "refs": [ "constructions-lemma-relative-proj-affine" ], "ref_ids": [ 12634 ] } ], "ref_ids": [] }, { "id": 12638, "type": "theorem", "label": "constructions-lemma-glueing-gives-functor-proj", "categories": [ "constructions" ], "title": "constructions-lemma-glueing-gives-functor-proj", "contents": [ "In Situation \\ref{situation-relative-proj}.", "The scheme $\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$", "constructed in Lemma \\ref{lemma-glue-relative-proj}", "and the scheme representing the functor $F$", "are canonically isomorphic as schemes over $S$." ], "refs": [ "constructions-lemma-glue-relative-proj" ], "proofs": [ { "contents": [ "Let $X$ be the scheme representing the functor $F$.", "Note that $X$ is a scheme over $S$ since the functor $F$", "comes equipped with a natural transformation $F \\to h_S$.", "Write $Y = \\underline{\\text{Proj}}_S(\\mathcal{A})$.", "We have to show that $X \\cong Y$ as $S$-schemes.", "We give two arguments.", "\\medskip\\noindent", "The first argument uses the construction of $X$ as the union", "of the schemes $U_d$ representing $F_d$ in the", "proof of Lemma \\ref{lemma-relative-proj}.", "Over each affine open of $S$ we can identify $X$ with the homogeneous spectrum", "of the sections of $\\mathcal{A}$ over that open, since this was", "true for the opens $U_d$. Moreover, these identifications", "are compatible with further restrictions to smaller affine opens.", "On the other hand, $Y$ was constructed by glueing these", "homogeneous spectra.", "Hence we can glue these isomorphisms to an isomorphism", "between $X$ and $\\underline{\\text{Proj}}_S(\\mathcal{A})$ as", "desired. Details omitted.", "\\medskip\\noindent", "Here is the second argument.", "Lemma \\ref{lemma-glue-relative-proj-twists}", "shows that there exists a morphism of graded algebras", "$$", "\\psi : \\pi^*\\mathcal{A}", "\\longrightarrow", "\\bigoplus\\nolimits_{n \\geq 0} \\mathcal{O}_Y(n)", "$$", "over $Y$ which on sections over affine opens of $S$ agrees with", "(\\ref{equation-global-sections}). Hence for every $y \\in Y$", "there exists an open neighbourhood $V \\subset Y$ of $y$", "and an integer $d \\geq 1$ such that for $d | n$ the sheaf", "$\\mathcal{O}_Y(n)|_V$ is invertible and the multiplication maps", "$\\mathcal{O}_Y(n)|_V \\otimes_{\\mathcal{O}_V} \\mathcal{O}_Y(m)|_V", "\\to \\mathcal{O}_Y(n + m)|_V$ are isomorphisms. Thus", "$\\psi$ restricted to the sheaf $\\pi^*\\mathcal{A}^{(d)}|_V$", "gives an element of $F_d(V)$. Since the opens $V$ cover $Y$", "we see ``$\\psi$'' gives rise to an element of $F(Y)$.", "Hence a canonical morphism $Y \\to X$ over $S$.", "Because this construction is completely canonical to see", "that it is an isomorphism we may work locally on $S$.", "Hence we reduce to the case $S$ affine where the result is clear." ], "refs": [ "constructions-lemma-relative-proj", "constructions-lemma-glue-relative-proj-twists" ], "ref_ids": [ 12637, 12632 ] } ], "ref_ids": [ 12631 ] }, { "id": 12639, "type": "theorem", "label": "constructions-lemma-tie-up-psi", "categories": [ "constructions" ], "title": "constructions-lemma-tie-up-psi", "contents": [ "In Situation \\ref{situation-relative-proj}.", "Let $(f : T \\to S, d, \\mathcal{L}, \\psi)$", "be a quadruple. Let", "$r_{d, \\mathcal{L}, \\psi} : T \\to \\underline{\\text{Proj}}_S(\\mathcal{A})$", "be the associated $S$-morphism.", "There exists an isomorphism", "of $\\mathbf{Z}$-graded $\\mathcal{O}_T$-algebras", "$$", "\\theta :", "r_{d, \\mathcal{L}, \\psi}^*\\left(", "\\bigoplus\\nolimits_{n \\in \\mathbf{Z}}", "\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(nd)", "\\right)", "\\longrightarrow", "\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{L}^{\\otimes n}", "$$", "such that the following diagram commutes", "$$", "\\xymatrix{", "\\mathcal{A}^{(d)} \\ar[rr]_-{\\psi}", " \\ar[rd]_-{\\psi_{univ}} & &", "f_*\\left(", "\\bigoplus\\nolimits_{n \\in \\mathbf{Z}}", "\\mathcal{L}^{\\otimes n}", "\\right) \\\\", " &", "\\pi_*\\left(", "\\bigoplus\\nolimits_{n \\geq 0}", "\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(nd)", "\\right) \\ar[ru]_\\theta", "}", "$$", "The commutativity of this diagram uniquely determines $\\theta$." ], "refs": [], "proofs": [ { "contents": [ "Note that the quadruple $(f : T \\to S, d, \\mathcal{L}, \\psi)$", "defines an element of $F_d(T)$. Let", "$U_d \\subset \\underline{\\text{Proj}}_S(\\mathcal{A})$", "be the locus", "where the sheaf $\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(d)$", "is invertible and generated by the image of", "$\\psi_{univ} : \\pi^*\\mathcal{A}_d \\to", "\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(d)$.", "Recall that $U_d$ represents the functor $F_d$, see the proof", "of Lemma \\ref{lemma-relative-proj}. Hence the result will follow", "if we can show the quadruple", "$(U_d \\to S, d, \\mathcal{O}_{U_d}(d), \\psi_{univ}|_{\\mathcal{A}^{(d)}})$", "is the universal family, i.e., the representing object in $F_d(U_d)$.", "We may do this after restricting to an affine open of $S$ because", "(a) the formation of the functors $F_d$ commutes with base change", "(see Lemma \\ref{lemma-proj-base-change}), and (b) the pair", "$(\\bigoplus_{n \\in \\mathbf{Z}}", "\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n),", "\\psi_{univ})$", "is constructed by glueing over affine opens in $S$", "(see Lemma \\ref{lemma-glue-relative-proj-twists}).", "Hence we may assume that $S$ is affine. In this case the functor", "of quadruples $F_d$ and the functor of triples $F_d$ agree", "(see proof of Lemma \\ref{lemma-relative-proj-affine}) and moreover", "Lemma \\ref{lemma-proj-functor-strict}", "shows that $(d, \\mathcal{O}_{U_d}(d), \\psi^d)$", "is the universal triple over $U_d$.", "Going backwards through the identifications in the proof of", "Lemma \\ref{lemma-relative-proj-affine} shows that", "$(U_d \\to S, d, \\mathcal{O}_{U_d}(d), \\psi_{univ}|_{\\mathcal{A}^{(d)}})$", "is the universal quadruple as desired." ], "refs": [ "constructions-lemma-relative-proj", "constructions-lemma-proj-base-change", "constructions-lemma-glue-relative-proj-twists", "constructions-lemma-relative-proj-affine", "constructions-lemma-proj-functor-strict", "constructions-lemma-relative-proj-affine" ], "ref_ids": [ 12637, 12633, 12632, 12634, 12617, 12634 ] } ], "ref_ids": [] }, { "id": 12640, "type": "theorem", "label": "constructions-lemma-relative-proj-separated", "categories": [ "constructions" ], "title": "constructions-lemma-relative-proj-separated", "contents": [ "Let $S$ be a scheme and $\\mathcal{A}$ be a quasi-coherent sheaf", "of graded $\\mathcal{O}_S$-algebras. The morphism", "$\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$", "is separated." ], "refs": [], "proofs": [ { "contents": [ "To prove a morphism is separated we may work locally on the base,", "see Schemes, Section \\ref{schemes-section-separation-axioms}.", "By construction $\\underline{\\text{Proj}}_S(\\mathcal{A})$ is", "over any affine $U \\subset S$ isomorphic to", "$\\text{Proj}(A)$ with $A = \\mathcal{A}(U)$. By", "Lemma \\ref{lemma-proj-separated} we see that $\\text{Proj}(A)$ is separated.", "Hence $\\text{Proj}(A) \\to U$ is separated (see", "Schemes, Lemma \\ref{schemes-lemma-compose-after-separated}) as desired." ], "refs": [ "constructions-lemma-proj-separated", "schemes-lemma-compose-after-separated" ], "ref_ids": [ 12597, 7715 ] } ], "ref_ids": [] }, { "id": 12641, "type": "theorem", "label": "constructions-lemma-relative-proj-base-change", "categories": [ "constructions" ], "title": "constructions-lemma-relative-proj-base-change", "contents": [ "Let $S$ be a scheme and $\\mathcal{A}$ be a quasi-coherent sheaf", "of graded $\\mathcal{O}_S$-algebras. Let $g : S' \\to S$ be any morphism", "of schemes. Then there is a canonical isomorphism", "$$", "r :", "\\underline{\\text{Proj}}_{S'}(g^*\\mathcal{A})", "\\longrightarrow", "S' \\times_S \\underline{\\text{Proj}}_S(\\mathcal{A})", "$$", "as well as a corresponding isomorphism", "$$", "\\theta :", "r^*\\text{pr}_2^*\\left(\\bigoplus\\nolimits_{d \\in \\mathbf{Z}}", "\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(d)\\right)", "\\longrightarrow", "\\bigoplus\\nolimits_{d \\in \\mathbf{Z}}", "\\mathcal{O}_{\\underline{\\text{Proj}}_{S'}(g^*\\mathcal{A})}(d)", "$$", "of $\\mathbf{Z}$-graded", "$\\mathcal{O}_{\\underline{\\text{Proj}}_{S'}(g^*\\mathcal{A})}$-algebras." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-proj-base-change} and the construction", "of $\\underline{\\text{Proj}}_S(\\mathcal{A})$ in", "Lemma \\ref{lemma-relative-proj} as the union", "of the schemes $U_d$ representing the functors $F_d$.", "In terms of the construction of relative Proj via glueing", "this isomorphism is given by the isomorphisms constructed", "in Lemma \\ref{lemma-base-change-map-proj} which provides us with", "the isomorphism $\\theta$. Some details omitted." ], "refs": [ "constructions-lemma-proj-base-change", "constructions-lemma-relative-proj", "constructions-lemma-base-change-map-proj" ], "ref_ids": [ 12633, 12637, 12613 ] } ], "ref_ids": [] }, { "id": 12642, "type": "theorem", "label": "constructions-lemma-apply-relative", "categories": [ "constructions" ], "title": "constructions-lemma-apply-relative", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{A}$ be a quasi-coherent sheaf of graded $\\mathcal{O}_S$-modules", "generated as an $\\mathcal{A}_0$-algebra by $\\mathcal{A}_1$.", "In this case the scheme $X = \\underline{\\text{Proj}}_S(\\mathcal{A})$", "represents the functor $F_1$ which associates to a scheme", "$f : T \\to S$ over $S$ the set of pairs $(\\mathcal{L}, \\psi)$, where", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is an invertible $\\mathcal{O}_T$-module, and", "\\item $\\psi : f^*\\mathcal{A} \\to \\bigoplus_{n \\geq 0} \\mathcal{L}^{\\otimes n}$", "is a graded $\\mathcal{O}_T$-algebra homomorphism such that", "$f^*\\mathcal{A}_1 \\to \\mathcal{L}$ is surjective", "\\end{enumerate}", "up to strict equivalence as above. Moreover, in this case all the", "quasi-coherent sheaves", "$\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}(n)$", "are invertible", "$\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}$-modules", "and the multiplication maps induce isomorphisms", "$", "\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}(n)", "\\otimes_{\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}}", "\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}(m) =", "\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}(n + m)$." ], "refs": [], "proofs": [ { "contents": [ "Under the assumptions of the lemma the sheaves", "$\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}(n)$", "are invertible and the multiplication maps isomorphisms", "by Lemma \\ref{lemma-relative-proj} and", "Lemma \\ref{lemma-apply}", "over affine opens of $S$. Thus $X$ actually represents the", "functor $F_1$, see proof of Lemma \\ref{lemma-relative-proj}." ], "refs": [ "constructions-lemma-relative-proj", "constructions-lemma-apply", "constructions-lemma-relative-proj" ], "ref_ids": [ 12637, 12618, 12637 ] } ], "ref_ids": [] }, { "id": 12643, "type": "theorem", "label": "constructions-lemma-relative-proj-modules", "categories": [ "constructions" ], "title": "constructions-lemma-relative-proj-modules", "contents": [ "In Situation \\ref{situation-relative-proj}.", "For any quasi-coherent sheaf of graded $\\mathcal{A}$-modules", "$\\mathcal{M}$ on $S$, there exists a canonical associated sheaf", "of $\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}$-modules", "$\\widetilde{\\mathcal{M}}$ with the following properties:", "\\begin{enumerate}", "\\item Given a scheme $T$ and a quadruple", "$(T \\to S, d, \\mathcal{L}, \\psi)$ over $T$", "corresponding to a morphism", "$h : T \\to \\underline{\\text{Proj}}_S(\\mathcal{A})$ there is", "a canonical isomorphism", "$\\widetilde{\\mathcal{M}}_T = h^*\\widetilde{\\mathcal{M}}$", "where $\\widetilde{\\mathcal{M}}_T$ is defined by (\\ref{equation-widetilde-M}).", "\\item The isomorphisms of (1) are compatible with pullbacks.", "\\item There is a canonical map", "$$", "\\pi^*\\mathcal{M}_0 \\longrightarrow \\widetilde{\\mathcal{M}}.", "$$", "\\item The construction $\\mathcal{M} \\mapsto \\widetilde{\\mathcal{M}}$", "is functorial in $\\mathcal{M}$.", "\\item The construction $\\mathcal{M} \\mapsto \\widetilde{\\mathcal{M}}$", "is exact.", "\\item There are canonical maps", "$$", "\\widetilde{\\mathcal{M}}", "\\otimes_{\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}}", "\\widetilde{\\mathcal{N}}", "\\longrightarrow", "\\widetilde{\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}}", "$$", "as in", "Lemma \\ref{lemma-widetilde-tensor}.", "\\item There exist canonical maps", "$$", "\\pi^*\\mathcal{M}", "\\longrightarrow", "\\bigoplus\\nolimits_{n \\in \\mathbf{Z}}", "\\widetilde{\\mathcal{M}(n)}", "$$", "generalizing (\\ref{equation-global-sections-more-generally}).", "\\item The formation of $\\widetilde{\\mathcal{M}}$ commutes with base change.", "\\end{enumerate}" ], "refs": [ "constructions-lemma-widetilde-tensor" ], "proofs": [ { "contents": [ "Omitted. We should split this lemma into parts and prove the parts separately." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12601 ] }, { "id": 12644, "type": "theorem", "label": "constructions-lemma-morphism-relative-proj", "categories": [ "constructions" ], "title": "constructions-lemma-morphism-relative-proj", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{A}$, $\\mathcal{B}$ be two graded", "quasi-coherent $\\mathcal{O}_S$-algebras. Set", "$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ and", "$q : Y = \\underline{\\text{Proj}}_S(\\mathcal{B}) \\to S$. Let", "$\\psi : \\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of", "graded $\\mathcal{O}_S$-algebras. There is a canonical open", "$U(\\psi) \\subset Y$ and a canonical morphism of schemes", "$$", "r_\\psi :", "U(\\psi)", "\\longrightarrow", "X", "$$", "over $S$ and a map of $\\mathbf{Z}$-graded $\\mathcal{O}_{U(\\psi)}$-algebras", "$$", "\\theta = \\theta_\\psi :", "r_\\psi^*\\left(", "\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_X(d)", "\\right)", "\\longrightarrow", "\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_{U(\\psi)}(d).", "$$", "The triple $(U(\\psi), r_\\psi, \\theta)$ is characterized by the property", "that for any affine open $W \\subset S$ the triple", "$$", "(U(\\psi) \\cap p^{-1}W,\\quad", "r_\\psi|_{U(\\psi) \\cap p^{-1}W} : U(\\psi) \\cap p^{-1}W \\to q^{-1}W,\\quad", "\\theta|_{U(\\psi) \\cap p^{-1}W})", "$$", "is equal to the triple associated to", "$\\psi : \\mathcal{A}(W) \\to \\mathcal{B}(W)$ in", "Lemma \\ref{lemma-morphism-proj} via the identifications", "$p^{-1}W = \\text{Proj}(\\mathcal{A}(W))$ and", "$q^{-1}W = \\text{Proj}(\\mathcal{B}(W))$ of", "Section \\ref{section-relative-proj-via-glueing}." ], "refs": [ "constructions-lemma-morphism-proj" ], "proofs": [ { "contents": [ "This lemma proves itself by glueing the local triples." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12608 ] }, { "id": 12645, "type": "theorem", "label": "constructions-lemma-morphism-relative-proj-transitive", "categories": [ "constructions" ], "title": "constructions-lemma-morphism-relative-proj-transitive", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{A}$, $\\mathcal{B}$, and $\\mathcal{C}$ be", "quasi-coherent graded $\\mathcal{O}_S$-algebras.", "Set $X = \\underline{\\text{Proj}}_S(\\mathcal{A})$,", "$Y = \\underline{\\text{Proj}}_S(\\mathcal{B})$ and", "$Z = \\underline{\\text{Proj}}_S(\\mathcal{C})$.", "Let $\\varphi : \\mathcal{A} \\to \\mathcal{B}$,", "$\\psi : \\mathcal{B} \\to \\mathcal{C}$ be graded $\\mathcal{O}_S$-algebra maps.", "Then we have", "$$", "U(\\psi \\circ \\varphi) = r_\\varphi^{-1}(U(\\psi))", "\\quad", "\\text{and}", "\\quad", "r_{\\psi \\circ \\varphi}", "=", "r_\\varphi \\circ r_\\psi|_{U(\\psi \\circ \\varphi)}.", "$$", "In addition we have", "$$", "\\theta_\\psi \\circ r_\\psi^*\\theta_\\varphi", "=", "\\theta_{\\psi \\circ \\varphi}", "$$", "with obvious notation." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12646, "type": "theorem", "label": "constructions-lemma-surjective-graded-rings-map-relative-proj", "categories": [ "constructions" ], "title": "constructions-lemma-surjective-graded-rings-map-relative-proj", "contents": [ "With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj}", "above. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is surjective for", "$d \\gg 0$. Then", "\\begin{enumerate}", "\\item $U(\\psi) = Y$,", "\\item $r_\\psi : Y \\to X$ is a closed immersion, and", "\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$", "are surjective but not isomorphisms in general (even if", "$\\mathcal{A} \\to \\mathcal{B}$ is surjective).", "\\end{enumerate}" ], "refs": [ "constructions-lemma-morphism-relative-proj" ], "proofs": [ { "contents": [ "Follows on combining", "Lemma \\ref{lemma-morphism-relative-proj}", "with", "Lemma \\ref{lemma-surjective-graded-rings-map-proj}." ], "refs": [ "constructions-lemma-morphism-relative-proj", "constructions-lemma-surjective-graded-rings-map-proj" ], "ref_ids": [ 12644, 12610 ] } ], "ref_ids": [ 12644 ] }, { "id": 12647, "type": "theorem", "label": "constructions-lemma-eventual-iso-graded-rings-map-relative-proj", "categories": [ "constructions" ], "title": "constructions-lemma-eventual-iso-graded-rings-map-relative-proj", "contents": [ "With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj}", "above. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is an isomorphism for all", "$d \\gg 0$. Then", "\\begin{enumerate}", "\\item $U(\\psi) = Y$,", "\\item $r_\\psi : Y \\to X$ is an isomorphism, and", "\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$", "are isomorphisms.", "\\end{enumerate}" ], "refs": [ "constructions-lemma-morphism-relative-proj" ], "proofs": [ { "contents": [ "Follows on combining", "Lemma \\ref{lemma-morphism-relative-proj}", "with", "Lemma \\ref{lemma-eventual-iso-graded-rings-map-proj}." ], "refs": [ "constructions-lemma-morphism-relative-proj", "constructions-lemma-eventual-iso-graded-rings-map-proj" ], "ref_ids": [ 12644, 12611 ] } ], "ref_ids": [ 12644 ] }, { "id": 12648, "type": "theorem", "label": "constructions-lemma-surjective-generated-degree-1-map-relative-proj", "categories": [ "constructions" ], "title": "constructions-lemma-surjective-generated-degree-1-map-relative-proj", "contents": [ "With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj}", "above. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is surjective for $d \\gg 0$", "and that $\\mathcal{A}$ is generated by $\\mathcal{A}_1$ over $\\mathcal{A}_0$.", "Then", "\\begin{enumerate}", "\\item $U(\\psi) = Y$,", "\\item $r_\\psi : Y \\to X$ is a closed immersion, and", "\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$", "are isomorphisms.", "\\end{enumerate}" ], "refs": [ "constructions-lemma-morphism-relative-proj" ], "proofs": [ { "contents": [ "Follows on combining", "Lemma \\ref{lemma-morphism-relative-proj}", "with", "Lemma \\ref{lemma-surjective-graded-rings-generated-degree-1-map-proj}." ], "refs": [ "constructions-lemma-morphism-relative-proj", "constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj" ], "ref_ids": [ 12644, 12612 ] } ], "ref_ids": [ 12644 ] }, { "id": 12649, "type": "theorem", "label": "constructions-lemma-invertible-map-into-relative-proj", "categories": [ "constructions" ], "title": "constructions-lemma-invertible-map-into-relative-proj", "contents": [ "With assumptions and notation as above. The morphism", "$\\psi$ induces a canonical morphism of schemes over $S$", "$$", "r_{\\mathcal{L}, \\psi} :", "U(\\psi) \\longrightarrow \\underline{\\text{Proj}}_S(\\mathcal{A})", "$$", "together with a map of graded $\\mathcal{O}_{U(\\psi)}$-algebras", "$$", "\\theta :", "r_{\\mathcal{L}, \\psi}^*\\left(", "\\bigoplus\\nolimits_{d \\geq 0}", "\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(d)", "\\right)", "\\longrightarrow", "\\bigoplus\\nolimits_{d \\geq 0} \\mathcal{L}^{\\otimes d}|_{U(\\psi)}", "$$", "characterized by the following properties:", "\\begin{enumerate}", "\\item For every open $V \\subset S$ and every $d \\geq 0$ the diagram", "$$", "\\xymatrix{", "\\mathcal{A}_d(V) \\ar[d]_{\\psi} \\ar[r]_{\\psi} &", "\\Gamma(f^{-1}(V), \\mathcal{L}^{\\otimes d}) \\ar[d]^{restrict} \\\\", "\\Gamma(\\pi^{-1}(V),", "\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(d)) \\ar[r]^{\\theta} &", "\\Gamma(f^{-1}(V) \\cap U(\\psi), \\mathcal{L}^{\\otimes d})", "}", "$$", "is commutative.", "\\item For any $d \\geq 1$ and any open subscheme $W \\subset X$", "such that $\\psi|_W : f^*\\mathcal{A}_d|_W \\to \\mathcal{L}^{\\otimes d}|_W$", "is surjective the restriction of the morphism $r_{\\mathcal{L}, \\psi}$", "agrees with the morphism $W \\to \\underline{\\text{Proj}}_S(\\mathcal{A})$", "which exists by the construction of the relative homogeneous spectrum,", "see Definition \\ref{definition-relative-proj}.", "\\item For any affine open $V \\subset S$, the restriction", "$$", "(U(\\psi) \\cap f^{-1}(V), r_{\\mathcal{L}, \\psi}|_{U(\\psi) \\cap f^{-1}(V)},", "\\theta|_{U(\\psi) \\cap f^{-1}(V)})", "$$", "agrees via $i_V$ (see Lemma \\ref{lemma-glue-relative-proj}) with the triple", "$(U(\\psi'), r_{\\mathcal{L}, \\psi'}, \\theta')$", "of Lemma \\ref{lemma-invertible-map-into-proj} associated to the map", "$\\psi' : A = \\mathcal{A}(V) \\to \\Gamma_*(f^{-1}(V), \\mathcal{L}|_{f^{-1}(V)})$", "induced by $\\psi$.", "\\end{enumerate}" ], "refs": [ "constructions-definition-relative-proj", "constructions-lemma-glue-relative-proj", "constructions-lemma-invertible-map-into-proj" ], "proofs": [ { "contents": [ "Use characterization (3) to construct the morphism $r_{\\mathcal{L}, \\psi}$", "and $\\theta$ locally over $S$. Use the uniqueness of", "Lemma \\ref{lemma-invertible-map-into-proj}", "to show that the construction glues. Details omitted." ], "refs": [ "constructions-lemma-invertible-map-into-proj" ], "ref_ids": [ 12628 ] } ], "ref_ids": [ 12665, 12631, 12628 ] }, { "id": 12650, "type": "theorem", "label": "constructions-lemma-twisting-and-proj", "categories": [ "constructions" ], "title": "constructions-lemma-twisting-and-proj", "contents": [ "With notation $S$, $\\mathcal{A}$, $\\mathcal{L}$ and $\\mathcal{B}$ as", "above. There is a canonical isomorphism", "$$", "\\xymatrix{", "P = \\underline{\\text{Proj}}_S(\\mathcal{A})", "\\ar[rr]_g \\ar[rd]_\\pi & &", "\\underline{\\text{Proj}}_S(\\mathcal{B}) = P'", "\\ar[ld]^{\\pi'} \\\\", "& S &", "}", "$$", "with the following properties", "\\begin{enumerate}", "\\item There are isomorphisms", "$\\theta_n : g^*\\mathcal{O}_{P'}(n)", "\\to", "\\mathcal{O}_P(n) \\otimes \\pi^*\\mathcal{L}^{\\otimes n}$", "which fit together to give an isomorphism of $\\mathbf{Z}$-graded", "algebras", "$$", "\\theta :", "g^*\\left(", "\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_{P'}(n)", "\\right)", "\\longrightarrow", "\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_P(n)", "\\otimes \\pi^*\\mathcal{L}^{\\otimes n}", "$$", "\\item For every open $V \\subset S$ the diagrams", "$$", "\\xymatrix{", "\\mathcal{A}_n(V) \\otimes \\mathcal{L}^{\\otimes n}(V)", "\\ar[r]_{multiply} \\ar[d]^{\\psi \\otimes \\pi^*}", "&", "\\mathcal{B}_n(V) \\ar[dd]^\\psi \\\\", "\\Gamma(\\pi^{-1}V, \\mathcal{O}_P(n)) \\otimes", "\\Gamma(\\pi^{-1}V, \\pi^*\\mathcal{L}^{\\otimes n})", "\\ar[d]^{multiply} \\\\", "\\Gamma(\\pi^{-1}V, \\mathcal{O}_P(n) \\otimes \\pi^*\\mathcal{L}^{\\otimes n})", "&", "\\Gamma(\\pi'^{-1}V, \\mathcal{O}_{P'}(n)) \\ar[l]_-{\\theta_n}", "}", "$$", "are commutative.", "\\item Add more here as necessary.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is the identity map when $\\mathcal{L} \\cong \\mathcal{O}_S$.", "In general choose an open covering of $S$ such that $\\mathcal{L}$", "is trivialized over the pieces and glue the corresponding maps.", "Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12651, "type": "theorem", "label": "constructions-lemma-projective-bundle-separated", "categories": [ "constructions" ], "title": "constructions-lemma-projective-bundle-separated", "contents": [ "Let $S$ be a scheme.", "The structure morphism $\\mathbf{P}(\\mathcal{E}) \\to S$ of a", "projective bundle over $S$ is separated." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-relative-proj-separated}." ], "refs": [ "constructions-lemma-relative-proj-separated" ], "ref_ids": [ 12640 ] } ], "ref_ids": [] }, { "id": 12652, "type": "theorem", "label": "constructions-lemma-projective-space-bundle", "categories": [ "constructions" ], "title": "constructions-lemma-projective-space-bundle", "contents": [ "Let $S$ be a scheme. Let $n \\geq 0$. Then", "$\\mathbf{P}^n_S$ is a projective bundle over $S$." ], "refs": [], "proofs": [ { "contents": [ "Note that", "$$", "\\mathbf{P}^n_{\\mathbf{Z}} =", "\\text{Proj}(\\mathbf{Z}[T_0, \\ldots, T_n]) =", "\\underline{\\text{Proj}}_{\\Spec(\\mathbf{Z})}", "\\left(\\widetilde{\\mathbf{Z}[T_0, \\ldots, T_n]}\\right)", "$$", "where the grading on the ring $\\mathbf{Z}[T_0, \\ldots, T_n]$ is given by", "$\\deg(T_i) = 1$ and the elements of $\\mathbf{Z}$ are in degree $0$.", "Recall that $\\mathbf{P}^n_S$ is defined as", "$\\mathbf{P}^n_{\\mathbf{Z}} \\times_{\\Spec(\\mathbf{Z})} S$.", "Moreover, forming the relative homogeneous spectrum commutes with base change,", "see Lemma \\ref{lemma-relative-proj-base-change}.", "For any scheme $g : S \\to \\Spec(\\mathbf{Z})$ we have", "$g^*\\mathcal{O}_{\\Spec(\\mathbf{Z})}[T_0, \\ldots, T_n]", "= \\mathcal{O}_S[T_0, \\ldots, T_n]$.", "Combining the above we see that", "$$", "\\mathbf{P}^n_S = \\underline{\\text{Proj}}_S(\\mathcal{O}_S[T_0, \\ldots, T_n]).", "$$", "Finally, note that", "$\\mathcal{O}_S[T_0, \\ldots, T_n] = \\text{Sym}(\\mathcal{O}_S^{\\oplus n + 1})$.", "Hence we see that $\\mathbf{P}^n_S$ is a projective bundle over $S$." ], "refs": [ "constructions-lemma-relative-proj-base-change" ], "ref_ids": [ 12641 ] } ], "ref_ids": [] }, { "id": 12653, "type": "theorem", "label": "constructions-lemma-gkn-representable", "categories": [ "constructions" ], "title": "constructions-lemma-gkn-representable", "contents": [ "Let $0 < k < n$.", "The functor $G(k, n)$ of (\\ref{equation-gkn}) is representable by a scheme." ], "refs": [], "proofs": [ { "contents": [ "Set $F = G(k, n)$. To prove the lemma we will use the criterion of", "Schemes, Lemma \\ref{schemes-lemma-glue-functors}.", "The reason $F$ satisfies the sheaf property for the", "Zariski topology is that we can glue sheaves, see Sheaves,", "Section \\ref{sheaves-section-glueing-sheaves} (some details omitted).", "\\medskip\\noindent", "The family of subfunctors $F_i$.", "Let $I$ be the set of subsets of $\\{1, \\ldots, n\\}$ of cardinality $n - k$.", "Given a scheme $S$ and $j \\in \\{1, \\ldots, n\\}$ we denote $e_j$", "the global section", "$$", "e_j = (0, \\ldots, 0, 1, 0, \\ldots, 0)\\quad(1\\text{ in }j\\text{th spot})", "$$", "of $\\mathcal{O}_S^{\\oplus n}$. Of course these sections freely generate", "$\\mathcal{O}_S^{\\oplus n}$. Similarly, for $j \\in \\{1, \\ldots, n - k\\}$", "we denote $f_j$ the global section of $\\mathcal{O}_S^{\\oplus n - k}$", "which is zero in all summands except the $j$th where we put a $1$.", "For $i \\in I$ we let", "$$", "s_i : \\mathcal{O}_S^{\\oplus n - k} \\longrightarrow \\mathcal{O}_S^{\\oplus n}", "$$", "which is the direct sum of the coprojections", "$\\mathcal{O}_S \\to \\mathcal{O}_S^{\\oplus n}$ corresponding to elements of $i$.", "More precisely, if $i = \\{i_1, \\ldots, i_{n - k}\\}$ with", "$i_1 < i_2 < \\ldots < i_{n - k}$", "then $s_i$ maps $f_j$ to $e_{i_j}$ for $j \\in \\{1, \\ldots, n - k\\}$.", "With this notation we can set", "$$", "F_i(S) = \\{q : \\mathcal{O}_S^{\\oplus n} \\to \\mathcal{Q} \\in F(S) \\mid", "q \\circ s_i \\text{ is surjective}\\}", "\\subset F(S)", "$$", "Given a morphism $f : T \\to S$ of schemes the pullback $f^*s_i$", "is the corresponding map over $T$. Since $f^*$ is right exact", "(Modules, Lemma \\ref{modules-lemma-exactness-pushforward-pullback})", "we conclude that $F_i$ is a subfunctor of $F$.", "\\medskip\\noindent", "Representability of $F_i$. To prove this we may assume (after renumbering)", "that $i = \\{1, \\ldots, n - k\\}$. This means $s_i$ is the inclusion of", "the first $n - k$ summands. Observe that if $q \\circ s_i$ is surjective,", "then $q \\circ s_i$ is an isomorphism as a surjective map between finite", "locally free modules of the same rank", "(Modules, Lemma \\ref{modules-lemma-map-finite-locally-free}).", "Thus if $q : \\mathcal{O}_S^{\\oplus n} \\to \\mathcal{Q}$ is an element of", "$F_i(S)$, then we can use $q \\circ s_i$ to identify $\\mathcal{Q}$ with", "$\\mathcal{O}_S^{\\oplus n - k}$. After doing so we obtain", "$$", "q : \\mathcal{O}_S^{\\oplus n} \\longrightarrow \\mathcal{O}_S^{\\oplus n - k}", "$$", "mapping $e_j$ to $f_j$ (notation as above) for $j = 1, \\ldots, n - k$.", "To determine $q$ completely we have to fix the images", "$q(e_{n - k + 1}), \\ldots, q(e_n)$ in", "$\\Gamma(S, \\mathcal{O}_S^{\\oplus n - k})$.", "It follows that $F_i$ is isomorphic to the functor", "$$", "S \\longmapsto", "\\prod\\nolimits_{j = n - k + 1, \\ldots, n}", "\\Gamma(S, \\mathcal{O}_S^{\\oplus n - k})", "$$", "This functor is isomorphic to the $k(n - k)$-fold self product of the functor", "$S \\mapsto \\Gamma(S, \\mathcal{O}_S)$. By", "Schemes, Example \\ref{schemes-example-global-sections}", "the latter is representable by $\\mathbf{A}^1_\\mathbf{Z}$. It follows $F_i$", "is representable by $\\mathbf{A}^{k(n - k)}_\\mathbf{Z}$ since fibred product", "over $\\Spec(\\mathbf{Z})$ is the product in the category of schemes.", "\\medskip\\noindent", "The inclusion $F_i \\subset F$ is representable by open immersions.", "Let $S$ be a scheme and let", "$q : \\mathcal{O}_S^{\\oplus n} \\to \\mathcal{Q}$ be an element of", "$F(S)$. By", "Modules, Lemma \\ref{modules-lemma-finite-type-surjective-on-stalk}.", "the set $U_i = \\{s \\in S \\mid (q \\circ s_i)_s\\text{ surjective}\\}$", "is open in $S$. Since $\\mathcal{O}_{S, s}$ is a local ring", "and $\\mathcal{Q}_s$ a finite $\\mathcal{O}_{S, s}$-module", "by Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK}) we have", "$$", "s \\in U_i \\Leftrightarrow", "\\left(", "\\text{the map }", "\\kappa(s)^{\\oplus n - k} \\to \\mathcal{Q}_s/\\mathfrak m_s\\mathcal{Q}_s", "\\text{ induced by }", "(q \\circ s_i)_s", "\\text{ is surjective}", "\\right)", "$$", "Let $f : T \\to S$ be a morphism of schemes and let $t \\in T$ be a point", "mapping to $s \\in S$. We have", "$(f^*\\mathcal{Q})_t =", "\\mathcal{Q}_s \\otimes_{\\mathcal{O}_{S, s}} \\mathcal{O}_{T, t}$", "(Sheaves, Lemma \\ref{sheaves-lemma-stalk-pullback-modules})", "and so on. Thus the map", "$$", "\\kappa(t)^{\\oplus n - k} \\to (f^*\\mathcal{Q})_t/\\mathfrak m_t(f^*\\mathcal{Q})_t", "$$", "induced by $(f^*q \\circ f^*s_i)_t$ is the base change of the map", "$\\kappa(s)^{\\oplus n - k} \\to \\mathcal{Q}_s/\\mathfrak m_s\\mathcal{Q}_s$", "above by the field extension $\\kappa(s) \\subset \\kappa(t)$. It follows", "that $s \\in U_i$ if and only if $t$ is in the corresponding open", "for $f^*q$. In particular $T \\to S$ factors through $U_i$ if", "and only if $f^*q \\in F_i(T)$ as desired.", "\\medskip\\noindent", "The collection $F_i$, $i \\in I$ covers $F$. Let", "$q : \\mathcal{O}_S^{\\oplus n} \\to \\mathcal{Q}$ be an element of", "$F(S)$. We have to show that for every point $s$ of $S$ there exists", "an $i \\in I$ such that $s_i$ is surjective in a neighbourhood of $s$.", "Thus we have to show that one of the compositions", "$$", "\\kappa(s)^{\\oplus n - k} \\xrightarrow{s_i}", "\\kappa(s)^{\\oplus n} \\rightarrow", "\\mathcal{Q}_s/\\mathfrak m_s\\mathcal{Q}_s", "$$", "is surjective (see previous paragraph). As", "$\\mathcal{Q}_s/\\mathfrak m_s\\mathcal{Q}_s$ is a vector space of", "dimension $n - k$ this follows from the theory of vector spaces." ], "refs": [ "schemes-lemma-glue-functors", "modules-lemma-exactness-pushforward-pullback", "modules-lemma-map-finite-locally-free", "modules-lemma-finite-type-surjective-on-stalk", "algebra-lemma-NAK", "sheaves-lemma-stalk-pullback-modules" ], "ref_ids": [ 7688, 13223, 13265, 13238, 401, 14523 ] } ], "ref_ids": [] }, { "id": 12654, "type": "theorem", "label": "constructions-lemma-projective-space-grassmannian", "categories": [ "constructions" ], "title": "constructions-lemma-projective-space-grassmannian", "contents": [ "Let $n \\geq 1$. There is a canonical isomorphism", "$\\mathbf{G}(n, n + 1) = \\mathbf{P}^n_\\mathbf{Z}$." ], "refs": [], "proofs": [ { "contents": [ "According to Lemma \\ref{lemma-projective-space} the scheme", "$\\mathbf{P}^n_\\mathbf{Z}$ represents the functor", "which assigns to a scheme $S$ the set of isomorphisms classes", "of pairs $(\\mathcal{L}, (s_0, \\ldots, s_n))$ consisting of", "an invertible module $\\mathcal{L}$ and an $(n + 1)$-tuple", "of global sections generating $\\mathcal{L}$.", "Given such a pair we obtain a quotient", "$$", "\\mathcal{O}_S^{\\oplus n + 1} \\longrightarrow \\mathcal{L},\\quad", "(h_0, \\ldots, h_n) \\longmapsto \\sum h_i s_i.", "$$", "Conversely, given an element", "$q : \\mathcal{O}_S^{\\oplus n + 1} \\to \\mathcal{Q}$ of $G(n, n + 1)(S)$", "we obtain such a pair, namely $(\\mathcal{Q}, (q(e_1), \\ldots, q(e_{n + 1})))$.", "Here $e_i$, $i = 1, \\ldots, n + 1$ are the standard generating sections", "of the free module $\\mathcal{O}_S^{\\oplus n + 1}$.", "We omit the verification that these constructions define mutually", "inverse transformations of functors." ], "refs": [ "constructions-lemma-projective-space" ], "ref_ids": [ 12621 ] } ], "ref_ids": [] }, { "id": 12673, "type": "theorem", "label": "algebraization-theorem-final-bootstrap", "categories": [ "algebraization" ], "title": "algebraization-theorem-final-bootstrap", "contents": [ "In Situation \\ref{situation-bootstrap} the inverse system", "$\\{H^i_T(M/I^nM)\\}_{n \\geq 0}$ satisfies the", "Mittag-Leffler condition for $i \\leq s$, the map", "$$", "H^i_T(M) \\longrightarrow \\lim H^i_T(M/I^nM)", "$$", "is an isomorphism for $i \\leq s$, and $H^i_T(M)$", "is annihilated by a power of $I$ for $i \\leq s$." ], "refs": [], "proofs": [ { "contents": [ "To prove the final assertion of the theorem we apply Local Cohomology,", "Proposition \\ref{local-cohomology-proposition-annihilator} with", "$T \\subset V(I) \\subset \\Spec(A)$. Namely, suppose", "that $\\mathfrak p \\not \\in V(I)$, $\\mathfrak q \\in T$", "with $\\mathfrak p \\subset \\mathfrak q$.", "Then either there exists a prime", "$\\mathfrak p \\subset \\mathfrak r \\subset \\mathfrak q$", "with $\\mathfrak r \\in V(I) \\setminus T$ and we get", "$$", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s", "\\quad\\text{or}\\quad", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > d + s", "$$", "by (4) in Situation \\ref{situation-bootstrap} or there does", "not exist an $\\mathfrak r$ and we get", "$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) > s$ by", "Lemma \\ref{lemma-helper-bootstrap}.", "In all three cases we see that", "$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > s$.", "Thus Local Cohomology, Proposition", "\\ref{local-cohomology-proposition-annihilator} (2)", "holds and we find that a power of $I$ annihilates", "$H^i_T(M)$ for $i \\leq s$.", "\\medskip\\noindent", "We already know the other two assertions of the theorem hold", "for $i < s$ by Lemma \\ref{lemma-final-bootstrap} and for the", "module $I^{m_0}M$ for $i = s$ and $m_0$ large enough.", "To finish of the proof we will show that in fact these", "assertions for $i = s$ holds for $M$.", "\\medskip\\noindent", "Let $M' = H^0_I(M)$ and $M'' = M/M'$ so that we have a short exact", "sequence", "$$", "0 \\to M' \\to M \\to M'' \\to 0", "$$", "and $M''$ has $H^0_I(M') = 0$ by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-divide-by-torsion}.", "By Artin-Rees (Algebra, Lemma \\ref{algebra-lemma-Artin-Rees})", "we get short exact sequences", "$$", "0 \\to M' \\to M/I^n M \\to M''/I^n M'' \\to 0", "$$", "for $n$ large enough. Consider the long exact sequences", "$$", "H^s_T(M') \\to", "H^s_T(M/I^nM) \\to", "H^s_T(M''/I^nM'') \\to", "H^{s + 1}_T(M')", "$$", "Now it is a simple matter to see that if we have Mittag-Leffler", "for the inverse system $\\{H^s_T(M''/I^nM'')\\}_{n \\geq 0}$", "then we have Mittag-Leffler for the inverse system", "$\\{H^s_T(M/I^nM)\\}_{n \\geq 0}$.", "(Note that the ML condition for an inverse system of groups $G_n$", "only depends on the values of the inverse system for sufficiently large $n$.)", "Moreover the sequence", "$$", "H^s_T(M') \\to", "\\lim H^s_T(M/I^nM) \\to", "\\lim H^s_T(M''/I^nM'') \\to", "H^{s + 1}_T(M')", "$$", "is exact because we have ML in the required spots, see", "Homology, Lemma \\ref{homology-lemma-apply-Mittag-Leffler}.", "Hence, if $H^s_T(M'') \\to \\lim H^s_T(M''/I^nM'')$", "is an isomorphism, then", "$H^s_T(M) \\to \\lim H^s_T(M/I^nM)$", "is an isomorphism too by the five lemma", "(Homology, Lemma \\ref{homology-lemma-five-lemma}).", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume that $H^0_I(M) = 0$. Choose generators", "$f_1, \\ldots, f_r$ of $I^{m_0}$ where $m_0$ is the", "integer found for $M$ in Lemma \\ref{lemma-final-bootstrap}.", "Then we consider the exact sequence", "$$", "0 \\to M \\xrightarrow{f_1, \\ldots, f_r}", "(I^{m_0}M)^{\\oplus r} \\to Q \\to 0", "$$", "defining $Q$. Some observations: the first map is injective", "exactly because $H^0_I(M) = 0$. The cokernel $Q$ of this injection", "is a finite $A$-module such that for every $1 \\leq j \\leq r$", "we have $Q_{f_j} \\cong (M_{f_j})^{\\oplus r - 1}$.", "In particular, for a prime $\\mathfrak p \\subset A$", "with $\\mathfrak p \\not \\in V(I)$ we have", "$Q_\\mathfrak p \\cong (M_\\mathfrak p)^{\\oplus r - 1}$.", "Similarly, given $\\mathfrak q \\in T$ and", "$\\mathfrak p' \\subset A' = (A_\\mathfrak q)^\\wedge$", "not contained in $V(IA')$, we have", "$Q'_{\\mathfrak p'} \\cong (M'_{\\mathfrak p'})^{\\oplus r - 1}$", "where $Q' = (Q_\\mathfrak q)^\\wedge$ and $M' = (M_\\mathfrak q)^\\wedge$.", "Thus the conditions in Situation \\ref{situation-bootstrap}", "hold for $A, I, T, Q$. (Observe that $Q$ may have", "nonvanishing $H^0_I(Q)$ but this won't matter.)", "\\medskip\\noindent", "For any $n \\geq 0$ we set $F^nM = M \\cap I^n(I^{m_0}M)^{\\oplus r}$", "so that we get short exact sequences", "$$", "0 \\to F^nM \\to I^n(I^{m_0}M)^{\\oplus r} \\to I^nQ \\to 0", "$$", "By Artin-Rees (Algebra, Lemma \\ref{algebra-lemma-Artin-Rees})", "there exists a $c \\geq 0$ such that", "$I^n M \\subset F^nM \\subset I^{n - c}M$ for all $n \\geq c$.", "Let $m_0$ be the integer and let $m'(m)$", "be the function defined for $m \\geq m_0$", "found in Lemma \\ref{lemma-bootstrap}", "applied to $M$. Note that the integer $m_0$", "is the same as our integer $m_0$ chosen above (you don't need to", "check this: you can just take the maximum of the two integers if", "you like). Finally, by Lemma \\ref{lemma-bootstrap}", "applied to $Q$ for every integer $m$ there exists an integer", "$m''(m) \\geq m$ such that $H^s_T(I^kQ) \\to H^s_T(I^mQ)$", "is zero for all $k \\geq m''(m)$.", "\\medskip\\noindent", "Fix $m \\geq m_0$. Choose $k \\geq m'(m''(m + c))$.", "Choose $\\xi \\in H^{s + 1}_T(I^kM)$", "which maps to zero in $H^{s + 1}_T(M)$.", "We want to show that $\\xi$ maps to zero in $H^{s + 1}_T(I^mM)$.", "Namely, this will show that $\\{H^s_T(M/I^nM)\\}_{n \\geq 0}$", "is Mittag-Leffler exactly as in the proof of Lemma \\ref{lemma-final-bootstrap}.", "Picture to help vizualize the argument:", "$$", "\\xymatrix{", "&", "H^{s + 1}_T(I^kM) \\ar[r] \\ar[d] &", "H^{s + 1}_T(I^k(I^{m_0}M)^{\\oplus r}) \\ar[d] &", "\\\\", "H^s_T(I^{m''(m + c)}Q) \\ar[r]_-\\delta \\ar[d] &", "H^{s + 1}_T(F^{m''(m + c)}M) \\ar[r] \\ar[d] &", "H^{s + 1}_T(I^{m''(m + c)}(I^{m_0}M)^{\\oplus r}) \\\\", "H^s_T(I^{m + c}Q) \\ar[r] &", "H^{s + 1}_T(F^{m + c}M) \\ar[d] &", "\\\\", "&", "H^{s + 1}_T(I^mM)", "}", "$$", "The image of $\\xi$ in $H^{s + 1}_T(I^k(I^{m_0}M)^{\\oplus r})$", "maps to zero in $H^{s + 1}_T((I^{m_0}M)^{\\oplus r})$", "and hence maps to zero in", "$H^{s + 1}_T(I^{m''(m + c)}(I^{m_0}M)^{\\oplus r})$", "by choice of $m'(-)$.", "Thus the image $\\xi' \\in H^{s + 1}_T(F^{m''(m + c)}M)$", "maps to zero in $H^{s + 1}_T(I^{m''(m + c)}(I^{m_0}M)^{\\oplus r})$", "and hence $\\xi' = \\delta(\\eta)$ for some", "$\\eta \\in H^s_T(I^{m''(m + c)}Q)$.", "By our choice of $m''(-)$ we find that $\\eta$ maps to", "zero in $H^s_T(I^{m + c}Q)$.", "This in turn means that $\\xi'$ maps to zero in", "$H^{s + 1}_T(F^{m + c}M)$.", "Since $F^{m + c}M \\subset I^mM$ we conclude.", "\\medskip\\noindent", "Finally, we prove the statement on limits. Consider the short", "exact sequences", "$$", "0 \\to M/F^nM \\to (I^{m_0}M)^{\\oplus r}/I^n (I^{m_0}M)^{\\oplus r}", "\\to Q/I^nQ \\to 0", "$$", "We have $\\lim H^s_T(M/I^nM) = \\lim H^s_T(M/F^nM)$", "as these inverse systems are pro-isomorphic. We obtain a commutative diagram", "$$", "\\xymatrix{", "H^{s - 1}_T(Q) \\ar[r] \\ar[d] &", "\\lim H^{s - 1}_T(Q/I^nQ) \\ar[d] \\\\", "H^s_T(M) \\ar[r] \\ar[d] &", "\\lim H^s_T(M/I^nM) \\ar[d] \\\\", "H^s_T((I^{m_0}M)^{\\oplus r}) \\ar[r] \\ar[d] &", "\\lim H^s_T((I^{m_0}M)^{\\oplus r}/I^n(I^{m_0}M)^{\\oplus r}) \\ar[d] \\\\", "H^s_T(Q) \\ar[r] &", "\\lim H^s_T(Q/I^nQ)", "}", "$$", "The right column is exact because we have ML in the required spots, see", "Homology, Lemma \\ref{homology-lemma-apply-Mittag-Leffler}.", "The lowest horizontal arrow is injective (!) by", "part (5) of Lemma \\ref{lemma-final-bootstrap}.", "The horizontal arrow above it is bijective by", "part (4) of Lemma \\ref{lemma-final-bootstrap}.", "The arrows in cohomological degrees $\\leq s - 1$ are isomorphisms.", "Thus we conclude $H^s_T(M) \\to \\lim H^s_T(M/I^nM)$", "is an isomorphism by the five lemma", "(Homology, Lemma \\ref{homology-lemma-five-lemma}).", "This finishes the proof of the theorem." ], "refs": [ "local-cohomology-proposition-annihilator", "algebraization-lemma-helper-bootstrap", "local-cohomology-proposition-annihilator", "algebraization-lemma-final-bootstrap", "dualizing-lemma-divide-by-torsion", "algebra-lemma-Artin-Rees", "homology-lemma-apply-Mittag-Leffler", "homology-lemma-five-lemma", "algebraization-lemma-final-bootstrap", "algebra-lemma-Artin-Rees", "algebraization-lemma-bootstrap", "algebraization-lemma-bootstrap", "algebraization-lemma-final-bootstrap", "homology-lemma-apply-Mittag-Leffler", "algebraization-lemma-final-bootstrap", "algebraization-lemma-final-bootstrap", "homology-lemma-five-lemma" ], "ref_ids": [ 9786, 12722, 9786, 12727, 2831, 625, 12125, 12030, 12727, 625, 12726, 12726, 12727, 12125, 12727, 12727, 12030 ] } ], "ref_ids": [] }, { "id": 12674, "type": "theorem", "label": "algebraization-theorem-algebraization-formal-sections", "categories": [ "algebraization" ], "title": "algebraization-theorem-algebraization-formal-sections", "contents": [ "\\begin{reference}", "The method of proof follows roughly the method of", "proof of \\cite[Theorem 1]{Faltings-algebraisation}", "and \\cite[Satz 2]{Faltings-uber}.", "The result is almost the same as", "\\cite[Theorem 1.1]{MRaynaud-paper} (affine complement case) and", "\\cite[Theorem 3.9]{MRaynaud-book} (complement is union of few affines).", "\\end{reference}", "Let $(A, \\mathfrak m)$ be a Noetherian local ring which has a", "dualizing complex and is complete with respect to an ideal $I$.", "Set $X = \\Spec(A)$, $Y = V(I)$, and $U = X \\setminus \\{\\mathfrak m\\}$.", "Let $\\mathcal{F}$ be a coherent sheaf on $U$.", "Assume", "\\begin{enumerate}", "\\item $\\text{cd}(A, I) \\leq d$, i.e.,", "$H^i(X \\setminus Y, \\mathcal{G}) = 0$ for $i \\geq d$ and", "quasi-coherent $\\mathcal{G}$ on $X$,", "\\item for any $x \\in X \\setminus Y$ whose closure $\\overline{\\{x\\}}$", "in $X$ meets $U \\cap Y$ we have", "$$", "\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) \\geq s", "\\quad\\text{or}\\quad", "\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x)", "+ \\dim(\\overline{\\{x\\}}) > d + s", "$$", "\\end{enumerate}", "Then there exists an open $V_0 \\subset U$ containing $U \\cap Y$", "such that for any open $V \\subset V_0$ containing $U \\cap Y$", "the map", "$$", "H^i(V, \\mathcal{F}) \\to \\lim H^i(U, \\mathcal{F}/I^n\\mathcal{F})", "$$", "is an isomorphism for $i < s$. If in addition", "$", "\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) +", "\\dim(\\overline{\\{x\\}}) > s", "$", "for all $x \\in U \\cap Y$, then these cohomology groups are finite $A$-modules." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite $A$-module $M$ such that $\\mathcal{F}$ is the", "restriction to $U$ of the", "coherent $\\mathcal{O}_X$-module associated to $M$, see Local Cohomology,", "Lemma \\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}.", "Then the assumptions of", "Lemma \\ref{lemma-algebraize-local-cohomology}", "are satisfied.", "Pick $J_0$ as in that lemma and set $V_0 = X \\setminus V(J_0)$.", "Then opens $V \\subset V_0$ containing $U \\cap Y$", "correspond $1$-to-$1$ with ideals $J \\subset J_0$ with", "$V(J) \\cap V(I) = \\{\\mathfrak m\\}$.", "Moreover, for such a choice we have a distinguished triangle", "$$", "R\\Gamma_J(M) \\to M \\to R\\Gamma(V, \\mathcal{F}) \\to", "R\\Gamma_J(M)[1]", "$$", "We similarly have a distinguished triangle", "$$", "R\\Gamma_\\mathfrak m(M)^\\wedge \\to", "M \\to", "R\\Gamma(U, \\mathcal{F})^\\wedge \\to", "R\\Gamma_\\mathfrak m(M)^\\wedge[1]", "$$", "involving derived $I$-adic completions.", "The cohomology groups of $R\\Gamma(U, \\mathcal{F})^\\wedge$ are", "equal to the limits in the statement of the theorem by", "Lemma \\ref{lemma-compare-with-derived-completion}.", "The canonical map between these triangles", "and some easy arguments show that our", "theorem follows from the main Lemma \\ref{lemma-algebraize-local-cohomology}", "(note that we have $i < s$ here whereas we have", "$i \\leq s$ in the lemma; this is because of the shift).", "The finiteness of the cohomology groups", "(under the additional assumption) follows from", "Lemma \\ref{lemma-kill-colimit}." ], "refs": [ "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "algebraization-lemma-algebraize-local-cohomology", "algebraization-lemma-compare-with-derived-completion", "algebraization-lemma-algebraize-local-cohomology", "algebraization-lemma-kill-colimit" ], "ref_ids": [ 9729, 12721, 12729, 12721, 12719 ] } ], "ref_ids": [] }, { "id": 12675, "type": "theorem", "label": "algebraization-lemma-ML-general", "categories": [ "algebraization" ], "title": "algebraization-lemma-ML-general", "contents": [ "Let $I$ be an ideal of a ring $A$. Let $X$ be a scheme over $\\Spec(A)$. Let", "$$", "\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1", "$$", "be an inverse system of $\\mathcal{O}_X$-modules", "such that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$.", "Assume", "$$", "\\bigoplus\\nolimits_{n \\geq 0} H^1(X, I^n\\mathcal{F}_{n + 1})", "$$", "satisfies the ascending chain condition as a graded", "$\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-module.", "Then the inverse system $M_n = \\Gamma(X, \\mathcal{F}_n)$ satisfies the", "Mittag-Leffler condition." ], "refs": [], "proofs": [ { "contents": [ "Set $H^1_n = H^1(X, I^n\\mathcal{F}_{n + 1})$ and let", "$\\delta_n : M_n \\to H^1_n$ be the boundary map on cohomology. Then", "$\\bigoplus \\Im(\\delta_n) \\subset \\bigoplus H^1_n$ is a graded submodule.", "Namely, if $s \\in M_n$ and $f \\in I^m$, then we have a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "I^n\\mathcal{F}_{n + 1} \\ar[d]_f \\ar[r] &", "\\mathcal{F}_{n + 1} \\ar[d]_f \\ar[r] &", "\\mathcal{F}_n \\ar[d]_f \\ar[r] & 0 \\\\", "0 \\ar[r] &", "I^{n + m}\\mathcal{F}_{n + m + 1} \\ar[r] &", "\\mathcal{F}_{n + m + 1} \\ar[r] &", "\\mathcal{F}_{n + m} \\ar[r] & 0", "}", "$$", "The middle vertical map is given by lifting a local section of", "$\\mathcal{F}_{n + 1}$ to a section of $\\mathcal{F}_{n + m + 1}$", "and then multiplying by $f$; similarly for the other vertical arrows.", "We conclude that $\\delta_{n + m}(fs) = f \\delta_n(s)$.", "By assumption we can find $s_j \\in M_{n_j}$, $j = 1, \\ldots, N$", "such that $\\delta_{n_j}(s_j)$", "generate $\\bigoplus \\Im(\\delta_n)$ as a graded module. Let $n > c = \\max(n_j)$.", "Let $s \\in M_n$. Then we can find $f_j \\in I^{n - n_j}$ such that", "$\\delta_n(s) = \\sum f_j \\delta_{n_j}(s_j)$. We conclude that", "$\\delta(s - \\sum f_j s_j) = 0$, i.e., we can find $s' \\in M_{n + 1}$", "mapping to $s - \\sum f_js_j$ in $M_n$. It follows that", "$$", "\\Im(M_{n + 1} \\to M_{n - c}) = \\Im(M_n \\to M_{n - c})", "$$", "This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12676, "type": "theorem", "label": "algebraization-lemma-ML-general-better", "categories": [ "algebraization" ], "title": "algebraization-lemma-ML-general-better", "contents": [ "Let $I$ be an ideal of a ring $A$. Let $X$ be a scheme over $\\Spec(A)$. Let", "$$", "\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1", "$$", "be an inverse system of $\\mathcal{O}_X$-modules", "such that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$.", "Given $n$ define", "$$", "H^1_n =", "\\bigcap\\nolimits_{m \\geq n}", "\\Im\\left(", "H^1(X, I^n\\mathcal{F}_{m + 1}) \\to H^1(X, I^n\\mathcal{F}_{n + 1})", "\\right)", "$$", "If $\\bigoplus H^1_n$ satisfies the ascending chain condition as a graded", "$\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-module, then the inverse system", "$M_n = \\Gamma(X, \\mathcal{F}_n)$ satisfies the Mittag-Leffler condition." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of Lemma \\ref{lemma-ML-general}.", "In fact, the result will follow from the arguments given there", "as soon as we show that", "$\\bigoplus H^1_n$ is a graded $\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-submodule", "of $\\bigoplus H^1(X, I^n\\mathcal{F}_{n + 1})$", "and that the boundary maps $\\delta_n$ have image contained in $H^1_n$.", "\\medskip\\noindent", "Suppose that $\\xi \\in H^1_n$ and $f \\in I^k$.", "Choose $m \\gg n + k$. Choose", "$\\xi' \\in H^1(X, I^n\\mathcal{F}_{m + 1})$ lifting", "$\\xi$. We consider the diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "I^n\\mathcal{F}_{m + 1} \\ar[d]_f \\ar[r] &", "\\mathcal{F}_{m + 1} \\ar[d]_f \\ar[r] &", "\\mathcal{F}_n \\ar[d]_f \\ar[r] & 0 \\\\", "0 \\ar[r] &", "I^{n + k}\\mathcal{F}_{m + 1} \\ar[r] &", "\\mathcal{F}_{m + 1} \\ar[r] &", "\\mathcal{F}_{n + k} \\ar[r] & 0", "}", "$$", "constructed as in the proof of Lemma \\ref{lemma-ML-general}.", "We get an induced map on cohomology and we see that", "$f \\xi' \\in H^1(X, I^{n + k}\\mathcal{F}_{m + 1})$", "maps to $f \\xi$. Since this is true for all $m \\gg n + k$", "we see that $f\\xi$ is in $H^1_{n + k}$ as desired.", "\\medskip\\noindent", "To see the boundary maps $\\delta_n$ have image contained in $H^1_n$", "we consider the diagrams", "$$", "\\xymatrix{", "0 \\ar[r] &", "I^n\\mathcal{F}_{m + 1} \\ar[d] \\ar[r] &", "\\mathcal{F}_{m + 1} \\ar[d] \\ar[r] &", "\\mathcal{F}_n \\ar[d] \\ar[r] & 0 \\\\", "0 \\ar[r] &", "I^n\\mathcal{F}_{n + 1} \\ar[r] &", "\\mathcal{F}_{n + 1} \\ar[r] &", "\\mathcal{F}_n \\ar[r] & 0", "}", "$$", "for $m \\geq n$. Looking at the induced maps on cohomology we conclude." ], "refs": [ "algebraization-lemma-ML-general", "algebraization-lemma-ML-general" ], "ref_ids": [ 12675, 12675 ] } ], "ref_ids": [] }, { "id": 12677, "type": "theorem", "label": "algebraization-lemma-topology-I-adic-general", "categories": [ "algebraization" ], "title": "algebraization-lemma-topology-I-adic-general", "contents": [ "Let $I$ be a finitely generated ideal of a ring $A$.", "Let $X$ be a scheme over $\\Spec(A)$. Let", "$$", "\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1", "$$", "be an inverse system of $\\mathcal{O}_X$-modules such that", "$\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$. Assume", "$$", "\\bigoplus\\nolimits_{n \\geq 0} H^0(X, I^n\\mathcal{F}_{n + 1})", "$$", "satisfies the ascending chain condition as a graded", "$\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-module.", "Then the limit topology on $M = \\lim \\Gamma(X, \\mathcal{F}_n)$", "is the $I$-adic topology." ], "refs": [], "proofs": [ { "contents": [ "Set $F^n = \\Ker(M \\to H^0(X, \\mathcal{F}_n))$ for $n \\geq 1$ and $F^0 = M$.", "Observe that $I F^n \\subset F^{n + 1}$. In particular $I^n M \\subset F^n$", "and we are trying to show that given $n$", "there exists an $m \\geq n$ such that $F^m \\subset I^nM$.", "We have an injective map of graded modules", "$$", "\\bigoplus\\nolimits_{n \\geq 0} F^n/F^{n + 1}", "\\longrightarrow", "\\bigoplus\\nolimits_{n \\geq 0} H^0(X, I^n\\mathcal{F}_{n + 1})", "$$", "By assumption the left hand side is generated by finitely many", "homogeneous elements. Hence we can find $r$ and", "$c_1, \\ldots, c_r \\geq 0$ and $a_i \\in F^{c_i}$ whose", "images in $\\bigoplus F^n/F^{n + 1}$ generate.", "Set $c = \\max(c_i)$.", "\\medskip\\noindent", "For $n \\geq c$ we claim that $I F^n = F^{n + 1}$.", "Namely, suppose $a \\in F^{n + 1}$. The image of", "$a$ in $F^{n + 1}/F^{n + 2}$ is a linear combination", "of our $a_i$. Therefore $a - \\sum f_i a_i \\in F^{n + 2}$", "for some $f_i \\in I^{n + 1 - c_i}$. Since", "$I^{n + 1 - c_i} = I \\cdot I^{n - c_i}$ as $n \\geq c_i$ we can write", "$f_i = \\sum g_{i, j} h_{i, j}$ with $g_{i, j} \\in I$", "and $h_{i, j}a_i \\in F^n$. Thus we see that", "$F^{n + 1} = F^{n + 2} + IF^n$.", "A simple induction argument gives $F^{n + 1} = F^{n + e} + IF^n$", "for all $e > 0$. It follows that $IF^n$ is dense in $F^{n + 1}$.", "Choose generators $k_1, \\ldots, k_r$ of $I$ and consider", "the continuous map", "$$", "u : (F^n)^{\\oplus r} \\longrightarrow F^{n + 1},\\quad", "(x_1, \\ldots, x_r) \\mapsto \\sum k_i x_i", "$$", "(in the limit topology).", "By the above the image of $(F^m)^{\\oplus r}$ under $u$ is dense in", "$F^{m + 1}$ for all $m \\geq n$. By the open mapping lemma", "(More on Algebra, Lemma \\ref{more-algebra-lemma-open-mapping}) we find", "that $u$ is open. Hence $u$ is surjective. Hence $IF^n = F^{n + 1}$", "for $n \\geq c$. This concludes the proof." ], "refs": [ "more-algebra-lemma-open-mapping" ], "ref_ids": [ 10013 ] } ], "ref_ids": [] }, { "id": 12678, "type": "theorem", "label": "algebraization-lemma-properties-system", "categories": [ "algebraization" ], "title": "algebraization-lemma-properties-system", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be a quasi-coherent sheaf of ideals. Let", "$$", "\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1", "$$", "be an inverse system of quasi-coherent $\\mathcal{O}_X$-modules", "such that", "$\\mathcal{F}_n = \\mathcal{F}_{n + 1}/\\mathcal{I}^n\\mathcal{F}_{n + 1}$.", "Set $\\mathcal{F} = \\lim \\mathcal{F}_n$. Then", "\\begin{enumerate}", "\\item $\\mathcal{F} = R\\lim \\mathcal{F}_n$,", "\\item for any affine open $U \\subset X$ we have", "$H^p(U, \\mathcal{F}) = 0$ for $p > 0$, and", "\\item for each $p$ there is a short exact sequence", "$0 \\to R^1\\lim H^{p - 1}(X, \\mathcal{F}_n) \\to", "H^p(X, \\mathcal{F}) \\to \\lim H^p(X, \\mathcal{F}_n) \\to 0$.", "\\end{enumerate}", "If moreover $\\mathcal{I}$ is of finite type, then", "\\begin{enumerate}", "\\item[(4)]", "$\\mathcal{F}_n = \\mathcal{F}/\\mathcal{I}^n\\mathcal{F}$, and", "\\item[(5)]", "$\\mathcal{I}^n \\mathcal{F} = \\lim_{m \\geq n} \\mathcal{I}^n\\mathcal{F}_m$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1), (2), and (3) are general facts about inverse systems of", "quasi-coherent modules with surjective transition maps, see", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-Rlim-quasi-coherent}", "and Cohomology, Lemma \\ref{cohomology-lemma-RGamma-commutes-with-Rlim}.", "Next, assume $\\mathcal{I}$ is of finite type.", "Let $U \\subset X$ be affine open. Say $U = \\Spec(A)$ and $\\mathcal{I}|_U$", "corresponds to $I \\subset A$. Observe that $I$ is a finitely generated ideal.", "By the equivalence of categories between quasi-coherent $\\mathcal{O}_U$-modules", "and $A$-modules (Schemes, Lemma \\ref{schemes-lemma-equivalence-quasi-coherent})", "we find that $M_n = \\mathcal{F}_n(U)$ is an inverse system", "of $A$-modules with $M_n = M_{n + 1}/I^nM_{n + 1}$. Thus", "$$", "M = \\mathcal{F}(U) = \\lim \\mathcal{F}_n(U) = \\lim M_n", "$$", "is an $I$-adically complete module with $M/I^nM = M_n$ by", "Algebra, Lemma \\ref{algebra-lemma-limit-complete}. This proves (4).", "Part (5) translates into the statement that", "$\\lim_{m \\geq n} I^nM/I^mM = I^nM$.", "Since $I^mM = I^{m - n} \\cdot I^nM$ this is just the statement that", "$I^mM$ is $I$-adically complete. This follows from", "Algebra, Lemma \\ref{algebra-lemma-hathat-finitely-generated}", "and the fact that $M$ is complete." ], "refs": [ "perfect-lemma-Rlim-quasi-coherent", "cohomology-lemma-RGamma-commutes-with-Rlim", "schemes-lemma-equivalence-quasi-coherent", "algebra-lemma-limit-complete", "algebra-lemma-hathat-finitely-generated" ], "ref_ids": [ 6938, 2160, 7664, 880, 859 ] } ], "ref_ids": [] }, { "id": 12679, "type": "theorem", "label": "algebraization-lemma-equivalent-f-good", "categories": [ "algebraization" ], "title": "algebraization-lemma-equivalent-f-good", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $f \\in \\Gamma(X, \\mathcal{O}_X)$. Let", "$$", "\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1", "$$", "be inverse system of $\\mathcal{O}_X$-modules.", "The following are equivalent", "\\begin{enumerate}", "\\item for all $n \\geq 1$ the map", "$f : \\mathcal{F}_{n + 1} \\to \\mathcal{F}_{n + 1}$ factors", "through $\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n$ to give a", "short exact sequence", "$0 \\to \\mathcal{F}_n \\to \\mathcal{F}_{n + 1} \\to \\mathcal{F}_1 \\to 0$,", "\\item for all $n \\geq 1$ the map", "$f^n : \\mathcal{F}_{n + 1} \\to \\mathcal{F}_{n + 1}$", "factors through $\\mathcal{F}_{n + 1} \\to \\mathcal{F}_1$", "to give a short exact sequence", "$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_{n + 1} \\to \\mathcal{F}_n \\to 0$", "\\item there exists an $\\mathcal{O}_X$-module $\\mathcal{G}$", "which is $f$-divisible such that $\\mathcal{F}_n = \\mathcal{G}[f^n]$.", "\\end{enumerate}", "If $X$ is a scheme and $\\mathcal{F}_n$ is quasi-coherent, then these", "are also equivalent to", "\\begin{enumerate}", "\\item[(4)] there exists an $\\mathcal{O}_X$-module $\\mathcal{F}$", "which is $f$-torsion free such that", "$\\mathcal{F}_n = \\mathcal{F}/f^n\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We omit the proof of the equivalence of (1) and (2).", "The condition that $\\mathcal{G}$ is $f$-divisible means that", "$f : \\mathcal{G} \\to \\mathcal{G}$ is surjective.", "Thus given $\\mathcal{F}_n$ as in (1) we set", "$\\mathcal{G} = \\colim \\mathcal{F}_n$ where the maps", "$\\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to \\ldots$", "are as in (1). This produces an $f$-divisible $\\mathcal{O}_X$-module", "with $\\mathcal{F}_n = \\mathcal{G}[f^n]$ as can be seen by", "checking on stalks.", "The condition that $\\mathcal{F}$ is $f$-torsion free means that", "$f : \\mathcal{F} \\to \\mathcal{F}$ is injective.", "If $X$ is a scheme and $\\mathcal{F}_n$ is quasi-coherent,", "then we set $\\mathcal{F} = \\lim \\mathcal{F}_n$. Namely, for an", "affine open $U \\subset X$ the transition maps", "$\\mathcal{F}_{n + 1}(U) \\to \\mathcal{F}_n(U)$ are surjective", "by vanishing of higher cohomology. This produces an $f$-torsion free", "$\\mathcal{O}_X$-module with", "$\\mathcal{F}_n = \\mathcal{F}/f^n\\mathcal{F}$", "(Lemma \\ref{lemma-properties-system})." ], "refs": [ "algebraization-lemma-properties-system" ], "ref_ids": [ 12678 ] } ], "ref_ids": [] }, { "id": 12680, "type": "theorem", "label": "algebraization-lemma-topology-I-adic-f", "categories": [ "algebraization" ], "title": "algebraization-lemma-topology-I-adic-f", "contents": [ "Suppose $X$, $f$, $(\\mathcal{F}_n)$ is as in", "Lemma \\ref{lemma-equivalent-f-good}. Then the limit topology on", "$H^p = \\lim H^p(X, \\mathcal{F}_n)$ is the $f$-adic topology." ], "refs": [ "algebraization-lemma-equivalent-f-good" ], "proofs": [ { "contents": [ "Namely, it is clear that $f^t H^p$ maps to zero in $H^p(X, \\mathcal{F}_t)$.", "On the other hand, let $c \\geq 1$. If $\\xi = (\\xi_n) \\in H^p$ is small in the", "limit topology, then $\\xi_c = 0$, and hence $\\xi_n$", "maps to zero in $H^p(X, \\mathcal{F}_c)$ for $n \\geq c$.", "Consider the inverse system of short exact sequences", "$$", "0 \\to \\mathcal{F}_{n - c} \\xrightarrow{f^c} \\mathcal{F}_n \\to", "\\mathcal{F}_c \\to 0", "$$", "and the corresponding inverse system of long exact cohomology sequences", "$$", "H^{p - 1}(X, \\mathcal{F}_c) \\to", "H^p(X, \\mathcal{F}_{n - c}) \\to", "H^p(X, \\mathcal{F}_n) \\to", "H^p(X, \\mathcal{F}_c)", "$$", "Since the term $H^{p - 1}(X, \\mathcal{F}_c)$ is independent of", "$n$ we can choose a compatible sequence of elements", "$\\xi'_n \\in H^1(X, \\mathcal{F}_{n - c})$", "lifting $\\xi_n$. Setting $\\xi' = (\\xi'_n)$ we see that", "$\\xi = f^{c + 1} \\xi'$. This even shows that", "$f^c H^p = \\Ker(H^p \\to H^p(X, \\mathcal{F}_c))$ on the nose." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12679 ] }, { "id": 12681, "type": "theorem", "label": "algebraization-lemma-limit-finite", "categories": [ "algebraization" ], "title": "algebraization-lemma-limit-finite", "contents": [ "Let $A$ be a Noetherian ring complete with respect to a principal ideal $(f)$.", "Let $X$ be a scheme over $\\Spec(A)$. Let", "$$", "\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1", "$$", "be an inverse system of $\\mathcal{O}_X$-modules. Assume", "\\begin{enumerate}", "\\item $\\Gamma(X, \\mathcal{F}_1)$ is a finite $A$-module,", "\\item the equivalent conditions of Lemma \\ref{lemma-equivalent-f-good} hold.", "\\end{enumerate}", "Then", "$$", "M = \\lim \\Gamma(X, \\mathcal{F}_n)", "$$", "is a finite $A$-module, $f$ is a nonzerodivisor on $M$, and", "$M/fM$ is the image of $M$ in $\\Gamma(X, \\mathcal{F}_1)$." ], "refs": [ "algebraization-lemma-equivalent-f-good" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-topology-I-adic-f} and its proof we have", "$M/fM \\subset H^0(X, \\mathcal{F}_1)$. From (1) and the Noetherian", "property of $A$ we get that $M/fM$ is a finite $A$-module.", "Observe that $\\bigcap f^nM = 0$ as $f^nM$ maps to zero in", "$H^0(X, \\mathcal{F}_n)$. By", "Algebra, Lemma \\ref{algebra-lemma-finite-over-complete-ring}", "we conclude that $M$ is finite over $A$." ], "refs": [ "algebraization-lemma-topology-I-adic-f", "algebra-lemma-finite-over-complete-ring" ], "ref_ids": [ 12680, 868 ] } ], "ref_ids": [ 12679 ] }, { "id": 12682, "type": "theorem", "label": "algebraization-lemma-ML", "categories": [ "algebraization" ], "title": "algebraization-lemma-ML", "contents": [ "Let $A$ be a ring. Let $f \\in A$. Let $X$ be a scheme over $\\Spec(A)$. Let", "$$", "\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1", "$$", "be an inverse system of $\\mathcal{O}_X$-modules. Assume", "\\begin{enumerate}", "\\item either $H^1(X, \\mathcal{F}_1)$ is an $A$-module of finite length", "or $A$ is Noetherian and $H^1(X, \\mathcal{F}_1)$ is a finite $A$-module,", "\\item the equivalent conditions of Lemma \\ref{lemma-equivalent-f-good} hold.", "\\end{enumerate}", "Then the inverse system $M_n = \\Gamma(X, \\mathcal{F}_n)$ satisfies the", "Mittag-Leffler condition." ], "refs": [ "algebraization-lemma-equivalent-f-good" ], "proofs": [ { "contents": [ "Set $I = (f)$. We will use the criterion of Lemma \\ref{lemma-ML-general}.", "Observe that $f^n : \\mathcal{F}_0 \\to I^n\\mathcal{F}_{n + 1}$", "is an isomorphism for all $n \\geq 0$.", "Thus it suffices to show that", "$$", "\\bigoplus\\nolimits_{n \\geq 1} H^1(X, \\mathcal{F}_1) \\cdot f^{n + 1}", "$$", "is a graded $S = \\bigoplus_{n \\geq 0} A/(f) \\cdot f^n$-module satisfying the", "ascending chain condition. If $A$ is not Noetherian, then", "$H^1(X, \\mathcal{F}_1)$ has finite length and the result holds.", "If $A$ is Noetherian, then $S$ is a Noetherian ring and the result", "holds as the module is finite over $S$ by the assumed finiteness", "of $H^1(X, \\mathcal{F}_1)$. Some details omitted." ], "refs": [ "algebraization-lemma-ML-general" ], "ref_ids": [ 12675 ] } ], "ref_ids": [ 12679 ] }, { "id": 12683, "type": "theorem", "label": "algebraization-lemma-ML-better", "categories": [ "algebraization" ], "title": "algebraization-lemma-ML-better", "contents": [ "Let $A$ be a ring. Let $f \\in A$. Let $X$ be a scheme over $\\Spec(A)$. Let", "$$", "\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1", "$$", "be an inverse system of $\\mathcal{O}_X$-modules. Assume", "\\begin{enumerate}", "\\item either there is an $m \\geq 1$ such that the image of", "$H^1(X, \\mathcal{F}_m) \\to H^1(X, \\mathcal{F}_1)$", "is an $A$-module of finite length or $A$ is Noetherian", "and the intersection of the images of", "$H^1(X, \\mathcal{F}_m) \\to H^1(X, \\mathcal{F}_1)$", "is a finite $A$-module,", "\\item the equivalent conditions of Lemma \\ref{lemma-equivalent-f-good} hold.", "\\end{enumerate}", "Then the inverse system $M_n = \\Gamma(X, \\mathcal{F}_n)$ satisfies the", "Mittag-Leffler condition." ], "refs": [ "algebraization-lemma-equivalent-f-good" ], "proofs": [ { "contents": [ "Set $I = (f)$. We will use the criterion of Lemma \\ref{lemma-ML-general-better}", "involving the modules $H^1_n$. For $m \\geq n$ we have", "$I^n\\mathcal{F}_{m + 1} = \\mathcal{F}_{m + 1 - n}$. Thus we see that", "$$", "H^1_n = \\bigcap\\nolimits_{m \\geq 1} \\Im\\left(", "H^1(X, \\mathcal{F}_m) \\to H^1(X, \\mathcal{F}_1)", "\\right)", "$$", "is independent of $n$ and", "$\\bigoplus H^1_n = \\bigoplus H^1_1 \\cdot f^{n + 1}$.", "Thus we conclude exactly as in the proof of Lemma \\ref{lemma-ML}." ], "refs": [ "algebraization-lemma-ML-general-better", "algebraization-lemma-ML" ], "ref_ids": [ 12676, 12682 ] } ], "ref_ids": [ 12679 ] }, { "id": 12684, "type": "theorem", "label": "algebraization-lemma-formal-functions-principal", "categories": [ "algebraization" ], "title": "algebraization-lemma-formal-functions-principal", "contents": [ "\\begin{reference}", "\\cite[Lemma 1.6]{Bhatt-local}", "\\end{reference}", "Let $A$ be a ring and $f \\in A$. Let $X$ be a scheme over $A$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume that $\\mathcal{F}[f^n] = \\Ker(f^n : \\mathcal{F} \\to \\mathcal{F})$", "stabilizes. Then", "$$", "R\\Gamma(X, \\lim \\mathcal{F}/f^n\\mathcal{F}) =", "R\\Gamma(X, \\mathcal{F})^\\wedge", "$$", "where the right hand side indicates the derived completion", "with respect to the ideal $(f) \\subset A$. Let $H^p$ be the", "$p$th cohomology group of this complex. Then there are short", "exact sequences", "$$", "0 \\to R^1\\lim H^{p - 1}(X, \\mathcal{F}/f^n\\mathcal{F})", "\\to H^p \\to \\lim H^p(X, \\mathcal{F}/f^n\\mathcal{F}) \\to 0", "$$", "and", "$$", "0 \\to H^0(H^p(X, \\mathcal{F})^\\wedge) \\to H^p \\to", "T_f(H^{p + 1}(X, \\mathcal{F})) \\to 0", "$$", "where $T_f(-)$ denote the $f$-adic Tate module as in", "More on Algebra, Example", "\\ref{more-algebra-example-spectral-sequence-principal}." ], "refs": [], "proofs": [ { "contents": [ "We start with the canonical identifications", "\\begin{align*}", "R\\Gamma(X, \\mathcal{F})^\\wedge", "& =", "R\\lim R\\Gamma(X, \\mathcal{F}) \\otimes_A^\\mathbf{L} (A \\xrightarrow{f^n} A) \\\\", "& =", "R\\lim R\\Gamma(X, \\mathcal{F} \\xrightarrow{f^n} \\mathcal{F}) \\\\", "& =", "R\\Gamma(X, R\\lim (\\mathcal{F} \\xrightarrow{f^n} \\mathcal{F}))", "\\end{align*}", "The first equality holds by", "More on Algebra, Lemma \\ref{more-algebra-lemma-derived-completion-koszul}.", "The second by the projection formula, see ", "Cohomology, Lemma \\ref{cohomology-lemma-projection-formula-perfect}.", "The third by Cohomology, Lemma", "\\ref{cohomology-lemma-Rf-commutes-with-Rlim}.", "Note that by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-Rlim-quasi-coherent}", "we have", "$\\lim \\mathcal{F}/f^n\\mathcal{F} = R\\lim \\mathcal{F}/f^n \\mathcal{F}$.", "Thus to finish the proof of the first statement of the lemma it suffices to", "show that the pro-objects $(f^n : \\mathcal{F} \\to \\mathcal{F})$", "and $(\\mathcal{F}/f^n \\mathcal{F})$ are isomorphic. There is clearly", "a map from the first inverse system to the second. Suppose that", "$\\mathcal{F}[f^c] = \\mathcal{F}[f^{c + 1}] = \\mathcal{F}[f^{c + 2}] = \\ldots$.", "Then we can define an arrow of inverse systems in $D(\\mathcal{O}_X)$", "in the other direction by the diagrams", "$$", "\\xymatrix{", "\\mathcal{F}/\\mathcal{F}[f^c] \\ar[r]_-{f^{n + c}} \\ar[d]_{f^c} &", "\\mathcal{F} \\ar[d]^1 \\\\", "\\mathcal{F} \\ar[r]^{f^n} & \\mathcal{F}", "}", "$$", "Since the top horizontal arrow is injective the complex", "in the top row is quasi-isomorphic to $\\mathcal{F}/f^{n + c}\\mathcal{F}$.", "Some details omitted.", "\\medskip\\noindent", "Since $R\\Gamma(X, -)$ commutes with derived limits", "(Injectives, Lemma \\ref{injectives-lemma-RF-commutes-with-Rlim})", "we see that", "$$", "R\\Gamma(X, \\lim \\mathcal{F}/f^n\\mathcal{F}) =", "R\\Gamma(X, R\\lim \\mathcal{F}/f^n\\mathcal{F}) =", "R\\lim R\\Gamma(X, \\mathcal{F}/f^n\\mathcal{F})", "$$", "(for first equality see first paragraph of proof).", "By More on Algebra, Remark \\ref{more-algebra-remark-compare-derived-limit}", "we obtain exact sequences", "$$", "0 \\to", "R^1\\lim H^{p - 1}(X, \\mathcal{F}/f^n\\mathcal{F}) \\to", "H^p(X, \\lim \\mathcal{F}/I^n\\mathcal{F}) \\to", "\\lim H^p(X, \\mathcal{F}/I^n\\mathcal{F}) \\to 0", "$$", "of $A$-modules. The second set of short exact sequences follow immediately", "from the discussion in More on Algebra, Example", "\\ref{more-algebra-example-spectral-sequence-principal}." ], "refs": [ "more-algebra-lemma-derived-completion-koszul", "cohomology-lemma-projection-formula-perfect", "cohomology-lemma-Rf-commutes-with-Rlim", "perfect-lemma-Rlim-quasi-coherent", "injectives-lemma-RF-commutes-with-Rlim", "more-algebra-remark-compare-derived-limit" ], "ref_ids": [ 10378, 2244, 2161, 6938, 7796, 10658 ] } ], "ref_ids": [] }, { "id": 12685, "type": "theorem", "label": "algebraization-lemma-cd-one", "categories": [ "algebraization" ], "title": "algebraization-lemma-cd-one", "contents": [ "Let $I = (f_1, \\ldots, f_r)$ be an ideal of a Noetherian ring $A$.", "If $\\text{cd}(A, I) = 1$, then there exist $c \\geq 1$ and maps", "$\\varphi_j : I^c \\to A$ such that $\\sum f_j \\varphi_j : I^c \\to I$", "is the inclusion map." ], "refs": [], "proofs": [ { "contents": [ "Since $\\text{cd}(A, I) = 1$ the complement $U = \\Spec(A) \\setminus V(I)$", "is affine (Local Cohomology, Lemma \\ref{local-cohomology-lemma-cd-is-one}).", "Say $U = \\Spec(B)$. Then $IB = B$", "and we can write $1 = \\sum_{j = 1, \\ldots, r} f_j b_j$", "for some $b_j \\in B$. By", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-homs-over-open}", "we can represent $b_j$ by maps $\\varphi_j : I^c \\to A$", "for some $c \\geq 0$. Then $\\sum f_j \\varphi_j : I^c \\to I \\subset A$", "is the canonical embedding, after possibly replacing $c$ by a larger", "integer, by the same lemma." ], "refs": [ "local-cohomology-lemma-cd-is-one", "coherent-lemma-homs-over-open" ], "ref_ids": [ 9709, 3322 ] } ], "ref_ids": [] }, { "id": 12686, "type": "theorem", "label": "algebraization-lemma-cd-one-extend", "categories": [ "algebraization" ], "title": "algebraization-lemma-cd-one-extend", "contents": [ "Let $I = (f_1, \\ldots, f_r)$ be an ideal of a Noetherian ring $A$", "with $\\text{cd}(A, I) = 1$. Let $c \\geq 1$ and $\\varphi_j : I^c \\to A$,", "$j = 1, \\ldots, r$ be as in Lemma \\ref{lemma-cd-one}.", "Then there is a unique graded $A$-algebra map", "$$", "\\Phi : \\bigoplus\\nolimits_{n \\geq 0} I^{nc} \\to A[T_1, \\ldots, T_r]", "$$", "with $\\Phi(g) = \\sum \\varphi_j(g) T_j$ for $g \\in I^c$.", "Moreover, the composition of $\\Phi$ with the map", "$A[T_1, \\ldots, T_r] \\to \\bigoplus_{n \\geq 0} I^n$,", "$T_j \\mapsto f_j$ is the inclusion map", "$\\bigoplus_{n \\geq 0} I^{nc} \\to \\bigoplus_{n \\geq 0} I^n$." ], "refs": [ "algebraization-lemma-cd-one" ], "proofs": [ { "contents": [ "For each $j$ and $m \\geq c$ the restriction of $\\varphi_j$ to", "$I^m$ is a map $\\varphi_j : I^m \\to I^{m - c}$.", "Given $j_1, \\ldots, j_n \\in \\{1, \\ldots, r\\}$ we claim that the", "composition", "$$", "\\varphi_{j_1} \\ldots \\varphi_{j_n} :", "I^{nc} \\to I^{(n - 1)c} \\to \\ldots \\to I^c \\to A", "$$", "is independent of the order of the indices $j_1, \\ldots, j_n$.", "Namely, if $g = g_1 \\ldots g_n$ with $g_i \\in I^c$, then", "we see that", "$$", "(\\varphi_{j_1} \\ldots \\varphi_{j_n})(g) =", "\\varphi_{j_1}(g_1) \\ldots \\varphi_{j_n}(g_n)", "$$", "is independent of the ordering as multiplication in $A$ is commutative.", "Thus we can define $\\Phi$ by sending $g \\in I^{nc}$ to", "$$", "\\Phi(g) = \\sum\\nolimits_{e_1 + \\ldots + e_r = n}", "(\\varphi_1^{e_1} \\circ \\ldots \\circ \\varphi_r^{e_r})(g)", "T_1^{e_1} \\ldots T_r^{e_r}", "$$", "It is straightforward to prove that this is a graded $A$-algebra", "homomorphism with the desired property. Uniqueness is immediate", "as is the final property. This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12685 ] }, { "id": 12687, "type": "theorem", "label": "algebraization-lemma-cd-one-extend-to-module", "categories": [ "algebraization" ], "title": "algebraization-lemma-cd-one-extend-to-module", "contents": [ "Let $I = (f_1, \\ldots, f_r)$ be an ideal of a Noetherian ring $A$", "with $\\text{cd}(A, I) = 1$. Let $c \\geq 1$ and $\\varphi_j : I^c \\to A$,", "$j = 1, \\ldots, r$ be as in Lemma \\ref{lemma-cd-one}.", "Let $A \\to B$ be a ring map with $B$ Noetherian and let $N$ be", "a finite $B$-module. Then, after possibly increasing $c$", "and adjusting $\\varphi_j$ accordingly, there is a unique", "unique graded $B$-module map", "$$", "\\Phi_N : \\bigoplus\\nolimits_{n \\geq 0} I^{nc}N \\to N[T_1, \\ldots, T_r]", "$$", "with $\\Phi_N(g x) = \\Phi(g) x$ for $g \\in I^{nc}$ and $x \\in N$", "where $\\Phi$ is as in Lemma \\ref{lemma-cd-one-extend}.", "The composition of $\\Phi_N$ with the map", "$N[T_1, \\ldots, T_r] \\to \\bigoplus_{n \\geq 0} I^nN$,", "$T_j \\mapsto f_j$ is the inclusion map", "$\\bigoplus_{n \\geq 0} I^{nc}N \\to \\bigoplus_{n \\geq 0} I^nN$." ], "refs": [ "algebraization-lemma-cd-one", "algebraization-lemma-cd-one-extend" ], "proofs": [ { "contents": [ "The uniqueness is clear from the formula and the uniqueness of $\\Phi$ in", "Lemma \\ref{lemma-cd-one-extend}. Consider the Noetherian $A$-algebra", "$B' = B \\oplus N$ where $N$ is an ideal of square zero. To show", "the existence of $\\Phi_N$ it is enough", "(via Lemma \\ref{lemma-cd-one}) to show that $\\varphi_j$ extends to", "a map $\\varphi'_j : I^cB' \\to B'$ after possibly increasing $c$", "to some $c'$ (and replacing $\\varphi_j$ by the composition of the inclusion", "$I^{c'} \\to I^c$ with $\\varphi_j$). Recall that $\\varphi_j$ corresponds to a", "section", "$$", "h_j \\in \\Gamma(\\Spec(A) \\setminus V(I), \\mathcal{O}_{\\Spec(A)})", "$$", "see Cohomology of Schemes, Lemma \\ref{coherent-lemma-homs-over-open}.", "(This is in fact how we chose our $\\varphi_j$ in the proof of", "Lemma \\ref{lemma-cd-one}.) Let us use the same lemma to represent the pullback", "$$", "h'_j \\in \\Gamma(\\Spec(B') \\setminus V(IB'), \\mathcal{O}_{\\Spec(B')})", "$$", "of $h_j$ by a $B'$-linear map", "$\\varphi'_j : I^{c'}B' \\to B'$ for some $c' \\geq c$.", "The agreement with $\\varphi_j$ will hold for $c'$", "sufficiently large by a further application of the lemma:", "namely we can test agreement on a finite list of generators of $I^{c'}$.", "Small detail omitted." ], "refs": [ "algebraization-lemma-cd-one-extend", "algebraization-lemma-cd-one", "coherent-lemma-homs-over-open", "algebraization-lemma-cd-one" ], "ref_ids": [ 12686, 12685, 3322, 12685 ] } ], "ref_ids": [ 12685, 12686 ] }, { "id": 12688, "type": "theorem", "label": "algebraization-lemma-cd-is-one-for-system", "categories": [ "algebraization" ], "title": "algebraization-lemma-cd-is-one-for-system", "contents": [ "Let $I = (f_1, \\ldots, f_r)$ be an ideal of a Noetherian ring $A$ with", "$\\text{cd}(A, I) = 1$. Let $c \\geq 1$ and $\\varphi_j : I^c \\to A$,", "$j = 1, \\ldots, r$ be as in Lemma \\ref{lemma-cd-one}.", "Let $X$ be a Noetherian scheme over $\\Spec(A)$. Let", "$$", "\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1", "$$", "be an inverse system of coherent $\\mathcal{O}_X$-modules", "such that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$.", "Set $\\mathcal{F} = \\lim \\mathcal{F}_n$.", "Then, after possibly increasing $c$ and adjusting $\\varphi_j$ accordingly,", "there exists a unique graded $\\mathcal{O}_X$-module map", "$$", "\\Phi_\\mathcal{F} :", "\\bigoplus\\nolimits_{n \\geq 0} I^{nc}\\mathcal{F}", "\\longrightarrow", "\\mathcal{F}[T_1, \\ldots, T_r]", "$$", "with $\\Phi_\\mathcal{F}(g s) = \\Phi(g) s$ for $g \\in I^{nc}$ and", "$s$ a local section of $\\mathcal{F}$ where $\\Phi$ is as in", "Lemma \\ref{lemma-cd-one-extend}. The composition of $\\Phi_\\mathcal{F}$", "with the map", "$\\mathcal{F}[T_1, \\ldots, T_r] \\to \\bigoplus_{n \\geq 0} I^n\\mathcal{F}$,", "$T_j \\mapsto f_j$", "is the canonical inclusion", "$\\bigoplus_{n \\geq 0} I^{nc}\\mathcal{F} \\to", "\\bigoplus_{n \\geq 0} I^n\\mathcal{F}$." ], "refs": [ "algebraization-lemma-cd-one", "algebraization-lemma-cd-one-extend" ], "proofs": [ { "contents": [ "The uniqueness is immediate from the $\\mathcal{O}_X$-linearity", "and the requirement that $\\Phi_\\mathcal{F}(g s) = \\Phi(g) s$ for", "$g \\in I^{nc}$ and $s$ a local section of $\\mathcal{F}$.", "Thus we may assume $X = \\Spec(B)$ is affine.", "Observe that $(\\mathcal{F}_n)$ is an object of the category", "$\\textit{Coh}(X, I\\mathcal{O}_X)$ introduced", "in Cohomology of Schemes, Section \\ref{coherent-section-coherent-formal}.", "Let $B' = B^\\wedge$ be the $I$-adic completion of $B$.", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-affine}", "the object $(\\mathcal{F}_n)$ corresponds to a finite $B'$-module $N$", "in the sense that $\\mathcal{F}_n$ is the coherent", "module associated to the finite $B$-module $N/I^n N$.", "Applying Lemma \\ref{lemma-cd-one-extend-to-module}", "to $I \\subset A \\to B'$ and $N$", "we see that, after possibly increasing $c$ and adjusting", "$\\varphi_j$ accordingly, we get unique maps", "$$", "\\Phi_N : \\bigoplus\\nolimits_{n \\geq 0} I^{nc}N \\to N[T_1, \\ldots, T_r]", "$$", "with the corresponding properties. Note that in degree $n$ we obtain", "an inverse system of maps $N/I^mN \\to \\bigoplus_{e_1 + \\ldots + e_r = n}", "N/I^{m - nc}N \\cdot T_1^{e_1} \\ldots T_r^{e_r}$ for $m \\geq nc$.", "Translating back into coherent", "sheaves we see that $\\Phi_N$ corresponds to a system of maps", "$$", "\\Phi^n_m :", "I^{nc}\\mathcal{F}_m", "\\longrightarrow", "\\bigoplus\\nolimits_{e_1 + \\ldots + e_r = n}", "\\mathcal{F}_{m - nc} \\cdot T_1^{e_1} \\ldots T_r^{e_r}", "$$", "for varying $m \\geq nc$ and $n \\geq 1$. Taking the inverse limit of", "these maps over $m$ we obtain $\\Phi_\\mathcal{F} = \\bigoplus_n \\lim_m \\Phi^n_m$.", "Note that $\\lim_m I^t\\mathcal{F}_m = I^t \\mathcal{F}$ as can be seen by", "evaluating on affines for example, but in fact we don't need this because", "it is clear there is a map $I^t\\mathcal{F} \\to \\lim_m I^t\\mathcal{F}_m$." ], "refs": [ "coherent-lemma-inverse-systems-affine", "algebraization-lemma-cd-one-extend-to-module" ], "ref_ids": [ 3370, 12687 ] } ], "ref_ids": [ 12685, 12686 ] }, { "id": 12689, "type": "theorem", "label": "algebraization-lemma-topology-I-adic", "categories": [ "algebraization" ], "title": "algebraization-lemma-topology-I-adic", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$. Let $X$ be a Noetherian scheme", "over $\\Spec(A)$. Let", "$$", "\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1", "$$", "be an inverse system of coherent $\\mathcal{O}_X$-modules", "such that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$.", "If $\\text{cd}(A, I) = 1$, then for all $p \\in \\mathbf{Z}$ the limit topology on", "$\\lim H^p(X, \\mathcal{F}_n)$ is $I$-adic." ], "refs": [], "proofs": [ { "contents": [ "First it is clear that $I^t \\lim H^p(X, \\mathcal{F}_n)$", "maps to zero in $H^p(X, \\mathcal{F}_t)$. Thus the $I$-adic topology", "is finer than the limit topology. For the converse we set", "$\\mathcal{F} = \\lim \\mathcal{F}_n$, we pick generators $f_1, \\ldots, f_r$", "of $I$, we pick $c \\geq 1$, and we choose", "$\\Phi_\\mathcal{F}$ as in Lemma \\ref{lemma-cd-is-one-for-system}.", "We will use the results of Lemma \\ref{lemma-properties-system}", "without further mention. In particular we have a short exact", "sequence", "$$", "0 \\to R^1\\lim H^{p - 1}(X, \\mathcal{F}_n) \\to H^p(X, \\mathcal{F})", "\\to \\lim H^p(X, \\mathcal{F}_n) \\to 0", "$$", "Thus we can lift any element $\\xi$ of $\\lim H^p(X, \\mathcal{F}_n)$", "to an element $\\xi' \\in H^p(X, \\mathcal{F})$. Suppose $\\xi$ maps to zero", "in $H^p(X, \\mathcal{F}_{nc})$ for some $n$, in other", "words, suppose $\\xi$ is ``small'' in the limit topology. We have a", "short exact sequence", "$$", "0 \\to I^{nc}\\mathcal{F} \\to \\mathcal{F} \\to \\mathcal{F}_{nc} \\to 0", "$$", "and hence the assumption means we can lift $\\xi'$ to an element", "$\\xi'' \\in H^p(X, I^{nc}\\mathcal{F})$. Applying $\\Phi_\\mathcal{F}$", "we get", "$$", "\\Phi_\\mathcal{F}(\\xi'') = \\sum\\nolimits_{e_1 + \\ldots + e_r = n}", "\\xi'_{e_1, \\ldots, e_r} \\cdot T_1^{e_1} \\ldots T_r^{e_r}", "$$", "for some $\\xi'_{e_1, \\ldots, e_r} \\in H^p(X, \\mathcal{F})$.", "Letting $\\xi_{e_1, \\ldots, e_r} \\in \\lim H^p(X, \\mathcal{F}_n)$", "be the images and using the final assertion of", "Lemma \\ref{lemma-cd-is-one-for-system}", "we conclude that", "$$", "\\xi = \\sum f_1^{e_1} \\ldots f_r^{e_r} \\xi_{e_1, \\ldots, e_r}", "$$", "is in $I^n \\lim H^p(X, \\mathcal{F}_n)$ as desired." ], "refs": [ "algebraization-lemma-cd-is-one-for-system", "algebraization-lemma-properties-system", "algebraization-lemma-cd-is-one-for-system" ], "ref_ids": [ 12688, 12678, 12688 ] } ], "ref_ids": [] }, { "id": 12690, "type": "theorem", "label": "algebraization-lemma-descending-chain", "categories": [ "algebraization" ], "title": "algebraization-lemma-descending-chain", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "\\begin{enumerate}", "\\item Let $M$ be a finite $A$-module. Then the $A$-module", "$H^i_\\mathfrak m(M)$ satisfies the descending chain condition", "for any $i$.", "\\item Let $U = \\Spec(A) \\setminus \\{\\mathfrak m\\}$ be the", "punctured spectrum of $A$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module.", "Then the $A$-module $H^i(U, \\mathcal{F})$", "satisfies the descending chain condition for $i > 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $A^\\wedge$ be the completion of $A$. Since", "$H^i_\\mathfrak m(M)$ is $\\mathfrak m$-power torsion, we see that", "$H^i_\\mathfrak m(M) = H^i_\\mathfrak m(M) \\otimes_A A^\\wedge$. Moreover,", "we have $H^i_\\mathfrak m(M) \\otimes_A A^\\wedge =", "H^i_{\\mathfrak mA^\\wedge}(M \\otimes_A A^\\wedge)$ by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-torsion-change-rings}.", "Thus", "$$", "H^i_\\mathfrak m(M) = H^i_{\\mathfrak mA^\\wedge}(M \\otimes_A A^\\wedge)", "$$", "and $A$-submodules of the left hand side are the same thing as", "$A^\\wedge$-submodules of the right hand side. Thus we reduce", "to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $A$ is complete. Then $A$ has a normalized dualizing complex", "$\\omega_A^\\bullet$ (Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-ubiquity-dualizing}).", "By the local duality theorem (Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-special-case-local-duality}) we find an isomorphism", "$$", "\\Hom_A(H^i_\\mathfrak m(M), E) =", "\\text{Ext}^{-i}_A(M, \\omega_A^\\bullet)^\\wedge", "$$", "where $E$ is an injective hull of the residue field of $A$. The module", "$\\text{Ext}^{-i}_A(M, \\omega_A^\\bullet)$", "on the right hand side is a finite $A$-module by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-dualizing}.", "Since $A$ is complete, the completion isn't necessary.", "Thus $H^i_\\mathfrak m(M)$ has the descending chain condition", "by Matlis duality, see ", "Dualizing Complexes, Proposition \\ref{dualizing-proposition-matlis}", "and its addendum Remark \\ref{dualizing-remark-matlis}.", "\\medskip\\noindent", "Part (2) follows from (1) via Local Cohomology,", "Lemma \\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}." ], "refs": [ "dualizing-lemma-torsion-change-rings", "dualizing-lemma-ubiquity-dualizing", "dualizing-lemma-special-case-local-duality", "dualizing-lemma-dualizing", "dualizing-proposition-matlis", "dualizing-remark-matlis", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local" ], "ref_ids": [ 2817, 2890, 2873, 2848, 2924, 2934, 9729 ] } ], "ref_ids": [] }, { "id": 12691, "type": "theorem", "label": "algebraization-lemma-ML-local", "categories": [ "algebraization" ], "title": "algebraization-lemma-ML-local", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "\\begin{enumerate}", "\\item Let $(M_n)$ be an inverse system of finite $A$-modules. Then the", "inverse system $H^i_\\mathfrak m(M_n)$ satisfies the Mittag-Leffler", "condition for any $i$.", "\\item Let $U = \\Spec(A) \\setminus \\{\\mathfrak m\\}$ be the", "punctured spectrum of $A$.", "Let $\\mathcal{F}_n$ be an inverse system of", "coherent $\\mathcal{O}_U$-modules.", "Then the inverse system $H^i(U, \\mathcal{F}_n)$", "satisfies the Mittag-Leffler condition for $i > 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-descending-chain}." ], "refs": [ "algebraization-lemma-descending-chain" ], "ref_ids": [ 12690 ] } ], "ref_ids": [] }, { "id": 12692, "type": "theorem", "label": "algebraization-lemma-terrific", "categories": [ "algebraization" ], "title": "algebraization-lemma-terrific", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "Let $(M_n)$ be an inverse system of finite $A$-modules.", "Let $M \\to \\lim M_n$ be a map where $M$ is a finite $A$-module", "such that for some $i$ the map", "$H^i_\\mathfrak m(M) \\to \\lim H^i_\\mathfrak m(M_n)$", "is an isomorphism.", "Then the inverse system $H^i_\\mathfrak m(M_n)$", "is essentially constant with value $H^i_\\mathfrak m(M)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-ML-local} the inverse system $H^i_\\mathfrak m(M_n)$", "satisfies the Mittag-Leffler condition. Let $E_n \\subset H^i_\\mathfrak m(M_n)$", "be the image of $H^i_\\mathfrak m(M_{n'})$ for $n' \\gg n$.", "Then $(E_n)$ is an inverse system with surjective transition maps", "and $H^i_\\mathfrak m(M) = \\lim E_n$. Since $H^i_\\mathfrak m(M)$", "has the descending chain condition by", "Lemma \\ref{lemma-descending-chain}", "we find there can only be a finite number of nontrivial", "kernels of the surjections $H^i_\\mathfrak m(M) \\to E_n$.", "Thus $E_n \\to E_{n - 1}$ is an isomorphism for all $n \\gg 0$", "as desired." ], "refs": [ "algebraization-lemma-ML-local", "algebraization-lemma-descending-chain" ], "ref_ids": [ 12691, 12690 ] } ], "ref_ids": [] }, { "id": 12693, "type": "theorem", "label": "algebraization-lemma-local-cohomology-derived-completion", "categories": [ "algebraization" ], "title": "algebraization-lemma-local-cohomology-derived-completion", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module.", "Then", "$$", "H^i(R\\Gamma_\\mathfrak m(M)^\\wedge) = \\lim H^i_\\mathfrak m(M/I^nM)", "$$", "for all $i$ where $R\\Gamma_\\mathfrak m(M)^\\wedge$ denotes", "the derived $I$-adic completion." ], "refs": [], "proofs": [ { "contents": [ "Apply Dualizing Complexes, Lemma \\ref{dualizing-lemma-completion-local}", "and Lemma \\ref{lemma-ML-local} to see the vanishing of the $R^1\\lim$ terms." ], "refs": [ "dualizing-lemma-completion-local", "algebraization-lemma-ML-local" ], "ref_ids": [ 2834, 12691 ] } ], "ref_ids": [] }, { "id": 12694, "type": "theorem", "label": "algebraization-lemma-map-twice-localize", "categories": [ "algebraization" ], "title": "algebraization-lemma-map-twice-localize", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $f$ be a global", "section of $\\mathcal{O}$.", "\\begin{enumerate}", "\\item For $L, N \\in D(\\mathcal{O}_f)$ we have", "$R\\SheafHom_\\mathcal{O}(L, N) = R\\SheafHom_{\\mathcal{O}_f}(L, N)$.", "In particular the two $\\mathcal{O}_f$-structures on", "$R\\SheafHom_\\mathcal{O}(L, N)$ agree.", "\\item For $K \\in D(\\mathcal{O})$ and", "$L \\in D(\\mathcal{O}_f)$ we have", "$$", "R\\SheafHom_\\mathcal{O}(L, K) =", "R\\SheafHom_{\\mathcal{O}_f}(L, R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K))", "$$", "In particular", "$R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f,", "R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K)) =", "R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K)$.", "\\item If $g$ is a second global", "section of $\\mathcal{O}$, then", "$$", "R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, R\\SheafHom_\\mathcal{O}(\\mathcal{O}_g, K))", "= R\\SheafHom_\\mathcal{O}(\\mathcal{O}_{gf}, K).", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $\\mathcal{J}^\\bullet$ be a K-injective complex of", "$\\mathcal{O}_f$-modules representing $N$. By Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-K-injective-flat} it follows that", "$\\mathcal{J}^\\bullet$ is a K-injective complex of", "$\\mathcal{O}$-modules as well. Let $\\mathcal{F}^\\bullet$ be a complex of", "$\\mathcal{O}_f$-modules representing $L$. Then", "$$", "R\\SheafHom_\\mathcal{O}(L, N) =", "R\\SheafHom_\\mathcal{O}(\\mathcal{F}^\\bullet, \\mathcal{J}^\\bullet) =", "R\\SheafHom_{\\mathcal{O}_f}(\\mathcal{F}^\\bullet, \\mathcal{J}^\\bullet)", "$$", "by", "Modules on Sites, Lemma \\ref{sites-modules-lemma-epimorphism-modules}", "because $\\mathcal{J}^\\bullet$ is a K-injective complex of $\\mathcal{O}$", "and of $\\mathcal{O}_f$-modules.", "\\medskip\\noindent", "Proof of (2). Let $\\mathcal{I}^\\bullet$ be a K-injective complex of", "$\\mathcal{O}$-modules representing $K$.", "Then $R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K)$ is represented by", "$\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, \\mathcal{I}^\\bullet)$ which is", "a K-injective complex of $\\mathcal{O}_f$-modules and of", "$\\mathcal{O}$-modules by", "Cohomology on Sites, Lemmas \\ref{sites-cohomology-lemma-hom-K-injective} and", "\\ref{sites-cohomology-lemma-K-injective-flat}.", "Let $\\mathcal{F}^\\bullet$ be a complex of $\\mathcal{O}_f$-modules", "representing $L$. Then", "$$", "R\\SheafHom_\\mathcal{O}(L, K) =", "R\\SheafHom_\\mathcal{O}(\\mathcal{F}^\\bullet, \\mathcal{I}^\\bullet) =", "R\\SheafHom_{\\mathcal{O}_f}(\\mathcal{F}^\\bullet,", "\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, \\mathcal{I}^\\bullet))", "$$", "by Modules on Sites, Lemma \\ref{sites-modules-lemma-adjoint-hom-restrict}", "and because $\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, \\mathcal{I}^\\bullet)$ is a", "K-injective complex of $\\mathcal{O}_f$-modules.", "\\medskip\\noindent", "Proof of (3). This follows from the fact that", "$R\\SheafHom_\\mathcal{O}(\\mathcal{O}_g, \\mathcal{I}^\\bullet)$", "is K-injective as a complex of $\\mathcal{O}$-modules and the fact that", "$\\SheafHom_\\mathcal{O}(\\mathcal{O}_f,", "\\SheafHom_\\mathcal{O}(\\mathcal{O}_g, \\mathcal{H})) = ", "\\SheafHom_\\mathcal{O}(\\mathcal{O}_{gf}, \\mathcal{H})$", "for all sheaves of $\\mathcal{O}$-modules $\\mathcal{H}$." ], "refs": [ "sites-cohomology-lemma-K-injective-flat", "sites-modules-lemma-epimorphism-modules", "sites-cohomology-lemma-hom-K-injective", "sites-cohomology-lemma-K-injective-flat", "sites-modules-lemma-adjoint-hom-restrict" ], "ref_ids": [ 4261, 14150, 4262, 4261, 14196 ] } ], "ref_ids": [] }, { "id": 12695, "type": "theorem", "label": "algebraization-lemma-hom-from-Af", "categories": [ "algebraization" ], "title": "algebraization-lemma-hom-from-Af", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $f$ be a global", "section of $\\mathcal{O}$. Let $K \\in D(\\mathcal{O})$.", "The following are equivalent", "\\begin{enumerate}", "\\item $R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K) = 0$,", "\\item $R\\SheafHom_\\mathcal{O}(L, K) = 0$ for all $L$ in $D(\\mathcal{O}_f)$,", "\\item $T(K, f) = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (2) implies (1). The implication (1) $\\Rightarrow$ (2)", "follows from Lemma \\ref{lemma-map-twice-localize}.", "A free resolution of the $\\mathcal{O}$-module $\\mathcal{O}_f$ is given by", "$$", "0 \\to \\bigoplus\\nolimits_{n \\in \\mathbf{N}} \\mathcal{O} \\to", "\\bigoplus\\nolimits_{n \\in \\mathbf{N}} \\mathcal{O}", "\\to \\mathcal{O}_f \\to 0", "$$", "where the first map sends a local section $(x_0, x_1, \\ldots)$ to", "$(fx_0 - x_1, fx_1 - x_2, \\ldots)$ and the second map sends", "$(x_0, x_1, \\ldots)$ to $x_0 + x_1/f + x_2/f^2 + \\ldots$.", "Applying $\\SheafHom_\\mathcal{O}(-, \\mathcal{I}^\\bullet)$", "where $\\mathcal{I}^\\bullet$ is a K-injective complex of $\\mathcal{O}$-modules", "representing $K$ we get a short exact sequence of complexes", "$$", "0 \\to \\SheafHom_\\mathcal{O}(\\mathcal{O}_f, \\mathcal{I}^\\bullet) \\to", "\\prod \\mathcal{I}^\\bullet \\to \\prod \\mathcal{I}^\\bullet \\to 0", "$$", "because $\\mathcal{I}^n$ is an injective $\\mathcal{O}$-module.", "The products are products in $D(\\mathcal{O})$, see", "Injectives, Lemma \\ref{injectives-lemma-derived-products}.", "This means that the object $T(K, f)$ is a representative of", "$R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K)$ in $D(\\mathcal{O})$.", "Thus the equivalence of (1) and (3)." ], "refs": [ "algebraization-lemma-map-twice-localize", "injectives-lemma-derived-products" ], "ref_ids": [ 12694, 7795 ] } ], "ref_ids": [] }, { "id": 12696, "type": "theorem", "label": "algebraization-lemma-ideal-of-elements-complete-wrt", "categories": [ "algebraization" ], "title": "algebraization-lemma-ideal-of-elements-complete-wrt", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K \\in D(\\mathcal{O})$.", "The rule which associates to $U$ the set $\\mathcal{I}(U)$", "of sections $f \\in \\mathcal{O}(U)$ such that $T(K|_U, f) = 0$", "is a sheaf of ideals in $\\mathcal{O}$." ], "refs": [], "proofs": [ { "contents": [ "We will use the results of Lemma \\ref{lemma-hom-from-Af} without further", "mention. If $f \\in \\mathcal{I}(U)$, and $g \\in \\mathcal{O}(U)$, then", "$\\mathcal{O}_{U, gf}$ is an $\\mathcal{O}_{U, f}$-module", "hence $R\\SheafHom_\\mathcal{O}(\\mathcal{O}_{U, gf}, K|_U) = 0$, hence", "$gf \\in \\mathcal{I}(U)$. Suppose $f, g \\in \\mathcal{O}(U)$.", "Then there is a short exact sequence", "$$", "0 \\to \\mathcal{O}_{U, f + g} \\to", "\\mathcal{O}_{U, f(f + g)} \\oplus \\mathcal{O}_{U, g(f + g)} \\to", "\\mathcal{O}_{U, gf(f + g)} \\to 0", "$$", "because $f, g$ generate the unit ideal in $\\mathcal{O}(U)_{f + g}$.", "This follows from", "Algebra, Lemma \\ref{algebra-lemma-standard-covering}", "and the easy fact that the last arrow is surjective.", "Because $R\\SheafHom_\\mathcal{O}( - , K|_U)$ is an exact functor", "of triangulated categories the vanishing of", "$R\\SheafHom_{\\mathcal{O}_U}(\\mathcal{O}_{U, f(f + g)}, K|_U)$,", "$R\\SheafHom_{\\mathcal{O}_U}(\\mathcal{O}_{U, g(f + g)}, K|_U)$, and", "$R\\SheafHom_{\\mathcal{O}_U}(\\mathcal{O}_{U, gf(f + g)}, K|_U)$,", "implies the vanishing of ", "$R\\SheafHom_{\\mathcal{O}_U}(\\mathcal{O}_{U, f + g}, K|_U)$.", "We omit the verification of the sheaf condition." ], "refs": [ "algebraization-lemma-hom-from-Af", "algebra-lemma-standard-covering" ], "ref_ids": [ 12695, 414 ] } ], "ref_ids": [] }, { "id": 12697, "type": "theorem", "label": "algebraization-lemma-derived-complete-internal-hom", "categories": [ "algebraization" ], "title": "algebraization-lemma-derived-complete-internal-hom", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{I} \\subset \\mathcal{O}$ be a sheaf of ideals.", "If $K \\in D(\\mathcal{O})$ and $L \\in D_{comp}(\\mathcal{O})$, then", "$R\\SheafHom_\\mathcal{O}(K, L) \\in D_{comp}(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be an object of $\\mathcal{C}$ and let $f \\in \\mathcal{I}(U)$.", "Recall that", "$$", "\\Hom_{D(\\mathcal{O}_U)}(\\mathcal{O}_{U, f}, R\\SheafHom_\\mathcal{O}(K, L)|_U)", "=", "\\Hom_{D(\\mathcal{O}_U)}(", "K|_U \\otimes_{\\mathcal{O}_U}^\\mathbf{L} \\mathcal{O}_{U, f}, L|_U)", "$$", "by Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-internal-hom}.", "The right hand side is zero by Lemma \\ref{lemma-hom-from-Af}", "and the relationship between internal hom and actual hom, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-section-RHom-over-U}.", "The same vanishing holds for all $U'/U$. Thus the object", "$R\\SheafHom_{\\mathcal{O}_U}(\\mathcal{O}_{U, f},", "R\\SheafHom_\\mathcal{O}(K, L)|_U)$ of $D(\\mathcal{O}_U)$ has vanishing", "$0$th cohomology sheaf (by locus citatus). Similarly for the other", "cohomology sheaves, i.e., $R\\SheafHom_{\\mathcal{O}_U}(\\mathcal{O}_{U, f},", "R\\SheafHom_\\mathcal{O}(K, L)|_U)$ is zero in $D(\\mathcal{O}_U)$.", "By Lemma \\ref{lemma-hom-from-Af} we conclude." ], "refs": [ "sites-cohomology-lemma-internal-hom", "algebraization-lemma-hom-from-Af", "algebraization-lemma-hom-from-Af" ], "ref_ids": [ 4328, 12695, 12695 ] } ], "ref_ids": [] }, { "id": 12698, "type": "theorem", "label": "algebraization-lemma-restriction-derived-complete", "categories": [ "algebraization" ], "title": "algebraization-lemma-restriction-derived-complete", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}'$", "be a homomorphism of sheaves of rings. Let $\\mathcal{I} \\subset \\mathcal{O}$", "be a sheaf of ideals. The inverse image of $D_{comp}(\\mathcal{O}, \\mathcal{I})$", "under the restriction functor $D(\\mathcal{O}') \\to D(\\mathcal{O})$ is", "$D_{comp}(\\mathcal{O}', \\mathcal{I}\\mathcal{O}')$." ], "refs": [], "proofs": [ { "contents": [ "Using Lemma \\ref{lemma-ideal-of-elements-complete-wrt}", "we see that $K' \\in D(\\mathcal{O}')$ is in", "$D_{comp}(\\mathcal{O}', \\mathcal{I}\\mathcal{O}')$", "if and only if $T(K'|_U, f)$ is zero for every local section", "$f \\in \\mathcal{I}(U)$. Observe that the cohomology sheaves of", "$T(K'|_U, f)$ are computed in the category of abelian sheaves,", "so it doesn't matter whether we think of $f$ as a section of", "$\\mathcal{O}$ or take the image of $f$ as a section of $\\mathcal{O}'$.", "The lemma follows immediately from this and the", "definition of derived complete objects." ], "refs": [ "algebraization-lemma-ideal-of-elements-complete-wrt" ], "ref_ids": [ 12696 ] } ], "ref_ids": [] }, { "id": 12699, "type": "theorem", "label": "algebraization-lemma-pushforward-derived-complete", "categories": [ "algebraization" ], "title": "algebraization-lemma-pushforward-derived-complete", "contents": [ "Let $f : (\\Sh(\\mathcal{D}), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$", "be a morphism of ringed topoi. Let $\\mathcal{I} \\subset \\mathcal{O}$", "and $\\mathcal{I}' \\subset \\mathcal{O}'$ be sheaves of ideals such", "that $f^\\sharp$ sends $f^{-1}\\mathcal{I}$ into $\\mathcal{I}'$.", "Then $Rf_*$ sends $D_{comp}(\\mathcal{O}', \\mathcal{I}')$", "into $D_{comp}(\\mathcal{O}, \\mathcal{I})$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $f$ is given by a morphism of ringed sites corresponding", "to a continuous functor $\\mathcal{C} \\to \\mathcal{D}$", "(Modules on Sites, Lemma", "\\ref{sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites}", ").", "Let $U$ be an object of $\\mathcal{C}$ and let $g$ be a section of", "$\\mathcal{I}$ over $U$. We have to show that", "$\\Hom_{D(\\mathcal{O}_U)}(\\mathcal{O}_{U, g}, Rf_*K|_U) = 0$", "whenever $K$ is derived complete with respect to $\\mathcal{I}'$.", "Namely, by Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-section-RHom-over-U}", "this, applied to all objects over $U$ and all shifts of $K$,", "will imply that $R\\SheafHom_{\\mathcal{O}_U}(\\mathcal{O}_{U, g}, Rf_*K|_U)$", "is zero, which implies that $T(Rf_*K|_U, g)$ is zero", "(Lemma \\ref{lemma-hom-from-Af}) which is what we have to show", "(Definition \\ref{definition-derived-complete}).", "Let $V$ in $\\mathcal{D}$ be the image of $U$. Then", "$$", "\\Hom_{D(\\mathcal{O}_U)}(\\mathcal{O}_{U, g}, Rf_*K|_U) =", "\\Hom_{D(\\mathcal{O}'_V)}(\\mathcal{O}'_{V, g'}, K|_V) = 0", "$$", "where $g' = f^\\sharp(g) \\in \\mathcal{I}'(V)$. The second equality", "because $K$ is derived complete and the first equality because", "the derived pullback of $\\mathcal{O}_{U, g}$ is $\\mathcal{O}'_{V, g'}$", "and", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-adjoint}." ], "refs": [ "sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites", "algebraization-lemma-hom-from-Af", "algebraization-definition-derived-complete", "sites-cohomology-lemma-adjoint" ], "ref_ids": [ 14145, 12695, 12803, 4249 ] } ], "ref_ids": [] }, { "id": 12700, "type": "theorem", "label": "algebraization-lemma-derived-completion", "categories": [ "algebraization" ], "title": "algebraization-lemma-derived-completion", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed on a site. Let $f_1, \\ldots, f_r$", "be global sections of $\\mathcal{O}$. Let $\\mathcal{I} \\subset \\mathcal{O}$ be", "the ideal sheaf generated by $f_1, \\ldots, f_r$.", "Then the inclusion functor $D_{comp}(\\mathcal{O}) \\to D(\\mathcal{O})$", "has a left adjoint, i.e., given any object $K$ of $D(\\mathcal{O})$", "there exists a map $K \\to K^\\wedge$ with $K^\\wedge$ in $D_{comp}(\\mathcal{O})$", "such that the map", "$$", "\\Hom_{D(\\mathcal{O})}(K^\\wedge, E) \\longrightarrow \\Hom_{D(\\mathcal{O})}(K, E)", "$$", "is bijective whenever $E$ is in $D_{comp}(\\mathcal{O})$. In fact", "we have", "$$", "K^\\wedge =", "R\\SheafHom_\\mathcal{O}", "(\\mathcal{O} \\to \\prod\\nolimits_{i_0} \\mathcal{O}_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} \\mathcal{O}_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to \\mathcal{O}_{f_1\\ldots f_r}, K)", "$$", "functorially in $K$." ], "refs": [], "proofs": [ { "contents": [ "Define $K^\\wedge$ by the last displayed formula of the lemma.", "There is a map of complexes", "$$", "(\\mathcal{O} \\to \\prod\\nolimits_{i_0} \\mathcal{O}_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} \\mathcal{O}_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to \\mathcal{O}_{f_1\\ldots f_r}) \\longrightarrow \\mathcal{O}", "$$", "which induces a map $K \\to K^\\wedge$. It suffices to prove that", "$K^\\wedge$ is derived complete and that $K \\to K^\\wedge$ is an", "isomorphism if $K$ is derived complete.", "\\medskip\\noindent", "Let $f$ be a global section of $\\mathcal{O}$.", "By Lemma \\ref{lemma-map-twice-localize} the object", "$R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K^\\wedge)$", "is equal to", "$$", "R\\SheafHom_\\mathcal{O}(", "(\\mathcal{O}_f \\to \\prod\\nolimits_{i_0} \\mathcal{O}_{ff_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} \\mathcal{O}_{ff_{i_0}f_{i_1}} \\to", "\\ldots \\to \\mathcal{O}_{ff_1\\ldots f_r}), K)", "$$", "If $f = f_i$ for some $i$, then $f_1, \\ldots, f_r$ generate the", "unit ideal in $\\mathcal{O}_f$, hence the extended alternating", "{\\v C}ech complex", "$$", "\\mathcal{O}_f \\to \\prod\\nolimits_{i_0} \\mathcal{O}_{ff_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} \\mathcal{O}_{ff_{i_0}f_{i_1}} \\to", "\\ldots \\to \\mathcal{O}_{ff_1\\ldots f_r}", "$$", "is zero (even homotopic to zero). In this way we see that $K^\\wedge$", "is derived complete.", "\\medskip\\noindent", "If $K$ is derived complete, then $R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K)$", "is zero for all $f = f_{i_0} \\ldots f_{i_p}$, $p \\geq 0$. Thus", "$K \\to K^\\wedge$ is an isomorphism in $D(\\mathcal{O})$." ], "refs": [ "algebraization-lemma-map-twice-localize" ], "ref_ids": [ 12694 ] } ], "ref_ids": [] }, { "id": 12701, "type": "theorem", "label": "algebraization-lemma-derived-completion-koszul", "categories": [ "algebraization" ], "title": "algebraization-lemma-derived-completion-koszul", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed on a site. Let $f_1, \\ldots, f_r$", "be global sections of $\\mathcal{O}$. Let $\\mathcal{I} \\subset \\mathcal{O}$ be", "the ideal sheaf generated by $f_1, \\ldots, f_r$. Let $K \\in D(\\mathcal{O})$.", "The derived completion $K^\\wedge$ of Lemma \\ref{lemma-derived-completion}", "is given by the formula", "$$", "K^\\wedge = R\\lim K \\otimes^\\mathbf{L}_\\mathcal{O} K_n", "$$", "where $K_n = K(\\mathcal{O}, f_1^n, \\ldots, f_r^n)$", "is the Koszul complex on $f_1^n, \\ldots, f_r^n$ over $\\mathcal{O}$." ], "refs": [ "algebraization-lemma-derived-completion" ], "proofs": [ { "contents": [ "In More on Algebra, Lemma", "\\ref{more-algebra-lemma-extended-alternating-Cech-is-colimit-koszul}", "we have seen that the extended alternating {\\v C}ech complex", "$$", "\\mathcal{O} \\to \\prod\\nolimits_{i_0} \\mathcal{O}_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} \\mathcal{O}_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to \\mathcal{O}_{f_1\\ldots f_r}", "$$", "is a colimit of the Koszul complexes", "$K^n = K(\\mathcal{O}, f_1^n, \\ldots, f_r^n)$ sitting in", "degrees $0, \\ldots, r$. Note that $K^n$ is a finite chain complex", "of finite free $\\mathcal{O}$-modules with dual", "$\\SheafHom_\\mathcal{O}(K^n, \\mathcal{O}) = K_n$ where $K_n$ is the Koszul", "cochain complex sitting in degrees $-r, \\ldots, 0$ (as usual). By", "Lemma \\ref{lemma-derived-completion}", "the functor $E \\mapsto E^\\wedge$ is gotten by taking", "$R\\SheafHom$ from the extended alternating {\\v C}ech complex into $E$:", "$$", "E^\\wedge = R\\SheafHom(\\colim K^n, E)", "$$", "This is equal to $R\\lim (E \\otimes_\\mathcal{O}^\\mathbf{L} K_n)$", "by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-colim-and-lim-of-duals}." ], "refs": [ "more-algebra-lemma-extended-alternating-Cech-is-colimit-koszul", "algebraization-lemma-derived-completion", "sites-cohomology-lemma-colim-and-lim-of-duals" ], "ref_ids": [ 9972, 12700, 4393 ] } ], "ref_ids": [ 12700 ] }, { "id": 12702, "type": "theorem", "label": "algebraization-lemma-all-rings", "categories": [ "algebraization" ], "title": "algebraization-lemma-all-rings", "contents": [ "There exist a way to construct", "\\begin{enumerate}", "\\item for every pair $(A, I)$ consisting of a ring $A$ and a finitely", "generated ideal $I \\subset A$ a complex $K(A, I)$ of $A$-modules,", "\\item a map $K(A, I) \\to A$ of complexes of $A$-modules,", "\\item for every ring map $A \\to B$ and finitely generated ideal $I \\subset A$", "a map of complexes $K(A, I) \\to K(B, IB)$,", "\\end{enumerate}", "such that", "\\begin{enumerate}", "\\item[(a)] for $A \\to B$ and $I \\subset A$ finitely generated the diagram", "$$", "\\xymatrix{", "K(A, I) \\ar[r] \\ar[d] & A \\ar[d] \\\\", "K(B, IB) \\ar[r] & B", "}", "$$", "commutes,", "\\item[(b)] for $A \\to B \\to C$ and $I \\subset A$ finitely generated", "the composition of the maps", "$K(A, I) \\to K(B, IB) \\to K(C, IC)$ is the map $K(A, I) \\to K(C, IC)$.", "\\item[(c)] for $A \\to B$ and a finitely generated ideal $I \\subset A$", "the induced map $K(A, I) \\otimes_A^\\mathbf{L} B \\to K(B, IB)$", "is an isomorphism in $D(B)$, and", "\\item[(d)] if $I = (f_1, \\ldots, f_r) \\subset A$ then there is a commutative", "diagram", "$$", "\\xymatrix{", "(A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to A_{f_1\\ldots f_r}) \\ar[r] \\ar[d] & K(A, I) \\ar[d] \\\\", "A \\ar[r]^1 & A", "}", "$$", "in $D(A)$ whose horizontal arrows are isomorphisms.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $S$ be the set of rings $A_0$ of the form", "$A_0 = \\mathbf{Z}[x_1, \\ldots, x_n]/J$.", "Every finite type $\\mathbf{Z}$-algebra is isomorphic to", "an element of $S$. Let $\\mathcal{A}_0$ be the category whose objects", "are pairs $(A_0, I_0)$ where $A_0 \\in S$ and $I_0 \\subset A_0$", "is an ideal and whose morphisms $(A_0, I_0) \\to (B_0, J_0)$ are", "ring maps $\\varphi : A_0 \\to B_0$ such that $J_0 = \\varphi(I_0)B_0$.", "\\medskip\\noindent", "Suppose we can construct $K(A_0, I_0) \\to A_0$ functorially for", "objects of $\\mathcal{A}_0$ having properties (a), (b), (c), and (d).", "Then we take", "$$", "K(A, I) = \\colim_{\\varphi : (A_0, I_0) \\to (A, I)} K(A_0, I_0)", "$$", "where the colimit is over ring maps $\\varphi : A_0 \\to A$ such", "that $\\varphi(I_0)A = I$ with $(A_0, I_0)$ in $\\mathcal{A}_0$.", "A morphism between $(A_0, I_0) \\to (A, I)$ and $(A_0', I_0') \\to (A, I)$", "are given by maps $(A_0, I_0) \\to (A_0', I_0')$ in $\\mathcal{A}_0$", "commuting with maps to $A$.", "The category of these $(A_0, I_0) \\to (A, I)$ is filtered", "(details omitted). Moreover, $\\colim_{\\varphi : (A_0, I_0) \\to (A, I)} A_0 = A$", "so that $K(A, I)$ is a complex of $A$-modules.", "Finally, given $\\varphi : A \\to B$ and $I \\subset A$", "for every $(A_0, I_0) \\to (A, I)$ in the colimit, the composition", "$(A_0, I_0) \\to (B, IB)$ lives in the colimit for $(B, IB)$.", "In this way we get a map on colimits. Properties (a), (b), (c), and (d)", "follow readily from this and the corresponding", "properties of the complexes $K(A_0, I_0)$.", "\\medskip\\noindent", "Endow $\\mathcal{C}_0 = \\mathcal{A}_0^{opp}$ with the chaotic topology.", "We equip $\\mathcal{C}_0$ with the sheaf of rings", "$\\mathcal{O} : (A, I) \\mapsto A$. The ideals $I$ fit together to give a", "sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}$.", "Choose an injective resolution $\\mathcal{O} \\to \\mathcal{J}^\\bullet$.", "Consider the object", "$$", "\\mathcal{F}^\\bullet = \\bigcup\\nolimits_n \\mathcal{J}^\\bullet[\\mathcal{I}^n]", "$$", "Let $U = (A, I) \\in \\Ob(\\mathcal{C}_0)$.", "Since the topology in $\\mathcal{C}_0$ is chaotic, the value", "$\\mathcal{J}^\\bullet(U)$ is a resolution of $A$ by injective", "$A$-modules. Hence the value $\\mathcal{F}^\\bullet(U)$ is an", "object of $D(A)$ representing the image of $R\\Gamma_I(A)$ in $D(A)$, see", "Dualizing Complexes, Section \\ref{dualizing-section-local-cohomology}.", "Choose a complex of $\\mathcal{O}$-modules $\\mathcal{K}^\\bullet$", "and a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{O} \\ar[r] & \\mathcal{J}^\\bullet \\\\", "\\mathcal{K}^\\bullet \\ar[r] \\ar[u] & \\mathcal{F}^\\bullet \\ar[u]", "}", "$$", "where the horizontal arrows are quasi-isomorphisms. This is possible", "by the construction of the derived category $D(\\mathcal{O})$.", "Set $K(A, I) = \\mathcal{K}^\\bullet(U)$ where $U = (A, I)$.", "Properties (a) and (b) are clear and properties (c) and (d)", "follow from Dualizing Complexes, Lemmas", "\\ref{dualizing-lemma-compute-local-cohomology-noetherian} and", "\\ref{dualizing-lemma-local-cohomology-change-rings}." ], "refs": [ "dualizing-lemma-compute-local-cohomology-noetherian", "dualizing-lemma-local-cohomology-change-rings" ], "ref_ids": [ 2824, 2825 ] } ], "ref_ids": [] }, { "id": 12703, "type": "theorem", "label": "algebraization-lemma-global-extended-cech-complex", "categories": [ "algebraization" ], "title": "algebraization-lemma-global-extended-cech-complex", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of ideals.", "There exists a map $K \\to \\mathcal{O}$ in $D(\\mathcal{O})$", "such that for every $U \\in \\Ob(\\mathcal{C})$ such that", "$\\mathcal{I}|_U$ is generated by $f_1, \\ldots, f_r \\in \\mathcal{I}(U)$", "there is an isomorphism", "$$", "(\\mathcal{O}_U \\to \\prod\\nolimits_{i_0} \\mathcal{O}_{U, f_{i_0}} \\to", "\\prod\\nolimits_{i_0 < i_1} \\mathcal{O}_{U, f_{i_0}f_{i_1}} \\to", "\\ldots \\to \\mathcal{O}_{U, f_1\\ldots f_r}) \\longrightarrow K|_U", "$$", "compatible with maps to $\\mathcal{O}_U$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{C}' \\subset \\mathcal{C}$ be the full subcategory", "of objects $U$ such that $\\mathcal{I}|_U$ is generated by", "finitely many sections. Then $\\mathcal{C}' \\to \\mathcal{C}$", "is a special cocontinuous functor", "(Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}).", "Hence it suffices to work with $\\mathcal{C}'$, see", "Sites, Lemma \\ref{sites-lemma-equivalence}.", "In other words we may assume that for every", "object $U$ of $\\mathcal{C}$ there exists a finitely generated", "ideal $I \\subset \\mathcal{I}(U)$ such that", "$\\mathcal{I}|_U = \\Im(I \\otimes \\mathcal{O}_U \\to \\mathcal{O}_U)$.", "We will say that $I$ generates $\\mathcal{I}|_U$.", "Warning: We do not know that $\\mathcal{I}(U)$ is a finitely generated", "ideal in $\\mathcal{O}(U)$.", "\\medskip\\noindent", "Let $U$ be an object and $I \\subset \\mathcal{O}(U)$ a finitely", "generated ideal which generates $\\mathcal{I}|_U$.", "On the category $\\mathcal{C}/U$ consider the complex of presheaves", "$$", "K_{U, I}^\\bullet : U'/U \\longmapsto K(\\mathcal{O}(U'), I\\mathcal{O}(U'))", "$$", "with $K(-, -)$ as in Lemma \\ref{lemma-all-rings}.", "We claim that the sheafification of this is independent of", "the choice of $I$. Indeed, if $I' \\subset \\mathcal{O}(U)$", "is a finitely generated ideal which also generates $\\mathcal{I}|_U$, then", "there exists a covering $\\{U_j \\to U\\}$ such that", "$I\\mathcal{O}(U_j) = I'\\mathcal{O}(U_j)$. (Hint: this works because", "both $I$ and $I'$ are finitely generated and generate $\\mathcal{I}|_U$.)", "Hence $K_{U, I}^\\bullet$ and $K_{U, I'}^\\bullet$ are the {\\it same}", "for any object lying over one of the $U_j$. The statement", "on sheafifications follows. Denote $K_U^\\bullet$ the common value.", "\\medskip\\noindent", "The independence of choice of $I$ also shows that", "$K_U^\\bullet|_{\\mathcal{C}/U'} = K_{U'}^\\bullet$", "whenever we are given a morphism", "$U' \\to U$ and hence a localization morphism", "$\\mathcal{C}/U' \\to \\mathcal{C}/U$. Thus the complexes", "$K_U^\\bullet$ glue to give a single well defined complex $K^\\bullet$", "of $\\mathcal{O}$-modules. The existence of the map $K^\\bullet \\to \\mathcal{O}$", "and the quasi-isomorphism of the lemma follow immediately from", "the corresponding properties of the complexes $K(-, -)$ in", "Lemma \\ref{lemma-all-rings}." ], "refs": [ "sites-definition-special-cocontinuous-functor", "sites-lemma-equivalence", "algebraization-lemma-all-rings", "algebraization-lemma-all-rings" ], "ref_ids": [ 8672, 8578, 12702, 12702 ] } ], "ref_ids": [] }, { "id": 12704, "type": "theorem", "label": "algebraization-lemma-map-identifies-koszul-and-cech-complexes", "categories": [ "algebraization" ], "title": "algebraization-lemma-map-identifies-koszul-and-cech-complexes", "contents": [ "Let $\\mathcal{C}$ be a site.", "Assume $\\varphi : \\mathcal{O} \\to \\mathcal{O}'$ is a flat homomorphism", "of sheaves of rings. Let $f_1, \\ldots, f_r$ be global sections", "of $\\mathcal{O}$ such that", "$\\mathcal{O}/(f_1, \\ldots, f_r) \\cong \\mathcal{O}'/(f_1, \\ldots, f_r)$.", "Then the map of extended alternating {\\v C}ech complexes", "$$", "\\xymatrix{", "\\mathcal{O} \\to", "\\prod_{i_0} \\mathcal{O}_{f_{i_0}} \\to", "\\prod_{i_0 < i_1} \\mathcal{O}_{f_{i_0}f_{i_1}} \\to \\ldots \\to", "\\mathcal{O}_{f_1\\ldots f_r} \\ar[d] \\\\", "\\mathcal{O}' \\to", "\\prod_{i_0} \\mathcal{O}'_{f_{i_0}} \\to", "\\prod_{i_0 < i_1} \\mathcal{O}'_{f_{i_0}f_{i_1}} \\to \\ldots \\to", "\\mathcal{O}'_{f_1\\ldots f_r}", "}", "$$", "is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Observe that the second complex is the tensor product of the first", "complex with $\\mathcal{O}'$. We can write the first extended", "alternating {\\v C}ech complex as a colimit of the Koszul complexes", "$K_n = K(\\mathcal{O}, f_1^n, \\ldots, f_r^n)$, see", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-extended-alternating-Cech-is-colimit-koszul}.", "Hence it suffices to prove $K_n \\to K_n \\otimes_\\mathcal{O} \\mathcal{O}'$", "is a quasi-isomorphism. Since $\\mathcal{O} \\to \\mathcal{O}'$ is flat", "it suffices to show that $H^i \\to H^i \\otimes_\\mathcal{O} \\mathcal{O}'$", "is an isomorphism where $H^i$ is the $i$th cohomology sheaf", "$H^i = H^i(K_n)$. These sheaves are annihilated by $f_1^n, \\ldots, f_r^n$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-homotopy-koszul}.", "Thus it suffices to show that", "$\\mathcal{O}/(f_1^n, \\ldots, f_r^n) \\to \\mathcal{O}'/(f_1^n, \\ldots, f_r^n)$", "is an isomorphism. Equivalently, we will show that", "$\\mathcal{O}/(f_1, \\ldots, f_r)^n \\to \\mathcal{O}'/(f_1, \\ldots, f_r)^n$", "is an isomorphism for all $n$. This holds for $n = 1$ by assumption.", "It follows for all $n$ by induction using", "Modules on Sites, Lemma \\ref{sites-modules-lemma-flat-over-thickening}", "applied to the ring map", "$\\mathcal{O}/(f_1, \\ldots, f_r)^{n + 1} \\to \\mathcal{O}/(f_1, \\ldots, f_r)^n$", "and the module $\\mathcal{O}'/(f_1, \\ldots, f_r)^{n + 1}$." ], "refs": [ "more-algebra-lemma-extended-alternating-Cech-is-colimit-koszul", "more-algebra-lemma-homotopy-koszul", "sites-modules-lemma-flat-over-thickening" ], "ref_ids": [ 9972, 9960, 14210 ] } ], "ref_ids": [] }, { "id": 12705, "type": "theorem", "label": "algebraization-lemma-restriction-derived-complete-equivalence", "categories": [ "algebraization" ], "title": "algebraization-lemma-restriction-derived-complete-equivalence", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}'$ be a", "homomorphism of sheaves of rings. Let $\\mathcal{I} \\subset \\mathcal{O}$", "be a finite type sheaf of ideals.", "If $\\mathcal{O} \\to \\mathcal{O}'$ is flat and", "$\\mathcal{O}/\\mathcal{I} \\cong \\mathcal{O}'/\\mathcal{I}\\mathcal{O}'$,", "then the restriction functor $D(\\mathcal{O}') \\to D(\\mathcal{O})$", "induces an equivalence", "$D_{comp}(\\mathcal{O}', \\mathcal{I}\\mathcal{O}') \\to", "D_{comp}(\\mathcal{O}, \\mathcal{I})$." ], "refs": [], "proofs": [ { "contents": [ "Lemma \\ref{lemma-pushforward-derived-complete} implies", "restriction $r : D(\\mathcal{O}') \\to D(\\mathcal{O})$", "sends $D_{comp}(\\mathcal{O}', \\mathcal{I}\\mathcal{O}')$", "into $D_{comp}(\\mathcal{O}, \\mathcal{I})$. We will construct a", "quasi-inverse $E \\mapsto E'$.", "\\medskip\\noindent", "Let $K \\to \\mathcal{O}$ be the morphism of $D(\\mathcal{O})$", "constructed in Lemma \\ref{lemma-global-extended-cech-complex}. ", "Set $K' = K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}'$ in $D(\\mathcal{O}')$.", "Then $K' \\to \\mathcal{O}'$ is a map in $D(\\mathcal{O}')$ which", "satisfies the conclusions of Lemma \\ref{lemma-global-extended-cech-complex}", "with respect to $\\mathcal{I}' = \\mathcal{I}\\mathcal{O}'$.", "The map $K \\to r(K')$ is a quasi-isomorphism by", "Lemma \\ref{lemma-map-identifies-koszul-and-cech-complexes}.", "Now, for $E \\in D_{comp}(\\mathcal{O}, \\mathcal{I})$ we set", "$$", "E' = R\\SheafHom_\\mathcal{O}(r(K'), E)", "$$", "viewed as an object in $D(\\mathcal{O}')$ using the $\\mathcal{O}'$-module", "structure on $K'$. Since $E$ is derived complete", "we have $E = R\\SheafHom_\\mathcal{O}(K, E)$, see", "proof of Proposition \\ref{proposition-derived-completion}.", "On the other hand, since $K \\to r(K')$ is an isomorphism in", "we see that there is an isomorphism", "$E \\to r(E')$ in $D(\\mathcal{O})$. To finish the proof we", "have to show that, if $E = r(M')$ for an object $M'$ of", "$D_{comp}(\\mathcal{O}', \\mathcal{I}')$, then", "$E' \\cong M'$. To get a map we use", "$$", "M' = R\\SheafHom_{\\mathcal{O}'}(\\mathcal{O}', M') \\to", "R\\SheafHom_\\mathcal{O}(r(\\mathcal{O}'), r(M')) \\to", "R\\SheafHom_\\mathcal{O}(r(K'), r(M')) = E'", "$$", "where the second arrow uses the map $K' \\to \\mathcal{O}'$.", "To see that this is an isomorphism, one shows that $r$ applied", "to this arrow is the same as the isomorphism $E \\to r(E')$ above.", "Details omitted." ], "refs": [ "algebraization-lemma-pushforward-derived-complete", "algebraization-lemma-global-extended-cech-complex", "algebraization-lemma-global-extended-cech-complex", "algebraization-lemma-map-identifies-koszul-and-cech-complexes", "algebraization-proposition-derived-completion" ], "ref_ids": [ 12699, 12703, 12703, 12704, 12790 ] } ], "ref_ids": [] }, { "id": 12706, "type": "theorem", "label": "algebraization-lemma-pushforward-derived-complete-adjoint", "categories": [ "algebraization" ], "title": "algebraization-lemma-pushforward-derived-complete-adjoint", "contents": [ "Let $f : (\\Sh(\\mathcal{D}), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$", "be a morphism of ringed topoi. Let $\\mathcal{I} \\subset \\mathcal{O}$", "and $\\mathcal{I}' \\subset \\mathcal{O}'$ ", "be finite type sheaves of ideals such that $f^\\sharp$ sends", "$f^{-1}\\mathcal{I}$ into $\\mathcal{I}'$.", "Then $Rf_*$ sends $D_{comp}(\\mathcal{O}', \\mathcal{I}')$", "into $D_{comp}(\\mathcal{O}, \\mathcal{I})$ and has a left adjoint", "$Lf_{comp}^*$ which is $Lf^*$ followed by derived completion." ], "refs": [], "proofs": [ { "contents": [ "The first statement we have seen in", "Lemma \\ref{lemma-pushforward-derived-complete}.", "Note that the second statement makes sense as we have a derived", "completion functor $D(\\mathcal{O}') \\to D_{comp}(\\mathcal{O}', \\mathcal{I}')$", "by Proposition \\ref{proposition-derived-completion}.", "OK, so now let $K \\in D_{comp}(\\mathcal{O}, \\mathcal{I})$", "and $M \\in D_{comp}(\\mathcal{O}', \\mathcal{I}')$. Then we have", "$$", "\\Hom(K, Rf_*M) = \\Hom(Lf^*K, M) = \\Hom(Lf_{comp}^*K, M)", "$$", "by the universal property of derived completion." ], "refs": [ "algebraization-lemma-pushforward-derived-complete", "algebraization-proposition-derived-completion" ], "ref_ids": [ 12699, 12790 ] } ], "ref_ids": [] }, { "id": 12707, "type": "theorem", "label": "algebraization-lemma-pushforward-commutes-with-derived-completion", "categories": [ "algebraization" ], "title": "algebraization-lemma-pushforward-commutes-with-derived-completion", "contents": [ "\\begin{reference}", "Generalization of \\cite[Lemma 6.5.9 (2)]{BS}. Compare with", "\\cite[Theorem 6.5]{HL-P} in the setting of quasi-coherent modules", "and morphisms of (derived) algebraic stacks.", "\\end{reference}", "Let $f : (\\Sh(\\mathcal{D}), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$", "be a morphism of ringed topoi. Let $\\mathcal{I} \\subset \\mathcal{O}$", "be a finite type sheaf of ideals. Let $\\mathcal{I}' \\subset \\mathcal{O}'$", "be the ideal generated by $f^\\sharp(f^{-1}\\mathcal{I})$.", "Then $Rf_*$ commutes with derived completion, i.e.,", "$Rf_*(K^\\wedge) = (Rf_*K)^\\wedge$." ], "refs": [], "proofs": [ { "contents": [ "By Proposition \\ref{proposition-derived-completion} the derived completion", "functors exist. By Lemma \\ref{lemma-pushforward-derived-complete} the object", "$Rf_*(K^\\wedge)$ is derived complete, and hence we obtain a canonical map", "$(Rf_*K)^\\wedge \\to Rf_*(K^\\wedge)$ by the universal property of derived", "completion. We may check this map is an isomorphism locally on $\\mathcal{C}$.", "Thus, since derived completion commutes with localization", "(Remark \\ref{remark-localization-and-completion}) we may assume", "that $\\mathcal{I}$ is generated by global sections $f_1, \\ldots, f_r$.", "Then $\\mathcal{I}'$ is generated by $g_i = f^\\sharp(f_i)$. By", "Lemma \\ref{lemma-derived-completion-koszul}", "we have to prove that", "$$", "R\\lim \\left(", "Rf_*K \\otimes^\\mathbf{L}_\\mathcal{O} K(\\mathcal{O}, f_1^n, \\ldots, f_r^n)", "\\right)", "=", "Rf_*\\left(", "R\\lim", "K \\otimes^\\mathbf{L}_{\\mathcal{O}'} K(\\mathcal{O}', g_1^n, \\ldots, g_r^n)", "\\right)", "$$", "Because $Rf_*$ commutes with $R\\lim$", "(Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-Rf-commutes-with-Rlim})", "it suffices to prove that", "$$", "Rf_*K \\otimes^\\mathbf{L}_\\mathcal{O} K(\\mathcal{O}, f_1^n, \\ldots, f_r^n) =", "Rf_*\\left(", "K \\otimes^\\mathbf{L}_{\\mathcal{O}'} K(\\mathcal{O}', g_1^n, \\ldots, g_r^n)", "\\right)", "$$", "This follows from the projection formula (Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-projection-formula}) and the fact that", "$Lf^*K(\\mathcal{O}, f_1^n, \\ldots, f_r^n) =", "K(\\mathcal{O}', g_1^n, \\ldots, g_r^n)$." ], "refs": [ "algebraization-proposition-derived-completion", "algebraization-lemma-pushforward-derived-complete", "algebraization-remark-localization-and-completion", "algebraization-lemma-derived-completion-koszul", "sites-cohomology-lemma-Rf-commutes-with-Rlim", "sites-cohomology-lemma-projection-formula" ], "ref_ids": [ 12790, 12699, 12808, 12701, 4267, 4396 ] } ], "ref_ids": [] }, { "id": 12708, "type": "theorem", "label": "algebraization-lemma-formal-functions-general", "categories": [ "algebraization" ], "title": "algebraization-lemma-formal-functions-general", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "Let $\\mathcal{C}$ be a site and let $\\mathcal{O}$ be a sheaf", "of $A$-algebras. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.", "Then we have", "$$", "R\\Gamma(\\mathcal{C}, \\mathcal{F})^\\wedge =", "R\\Gamma(\\mathcal{C}, \\mathcal{F}^\\wedge)", "$$", "in $D(A)$ where $\\mathcal{F}^\\wedge$ is the derived", "completion of $\\mathcal{F}$ with respect to $I\\mathcal{O}$ and on the", "left hand wide we have the derived completion with respect to $I$.", "This produces two spectral sequences", "$$", "E_2^{i, j} = H^i(H^j(\\mathcal{C}, \\mathcal{F})^\\wedge)", "\\quad\\text{and}\\quad", "E_2^{p, q} = H^p(\\mathcal{C}, H^q(\\mathcal{F}^\\wedge))", "$$", "both converging to", "$H^*(R\\Gamma(\\mathcal{C}, \\mathcal{F})^\\wedge) =", "H^*(\\mathcal{C}, \\mathcal{F}^\\wedge)$" ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-pushforward-commutes-with-derived-completion}", "to the morphism of ringed topoi $(\\mathcal{C}, \\mathcal{O}) \\to (pt, A)$", "and take cohomology to get the first statement. The second spectral sequence", "is the second spectral sequence of", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}.", "The first spectral sequence is the spectral sequence of", "More on Algebra, Example", "\\ref{more-algebra-example-derived-completion-spectral-sequence}", "applied to $R\\Gamma(\\mathcal{C}, \\mathcal{F})^\\wedge$." ], "refs": [ "algebraization-lemma-pushforward-commutes-with-derived-completion", "derived-lemma-two-ss-complex-functor" ], "ref_ids": [ 12707, 1871 ] } ], "ref_ids": [] }, { "id": 12709, "type": "theorem", "label": "algebraization-lemma-sections-derived-completion-pseudo-coherent", "categories": [ "algebraization" ], "title": "algebraization-lemma-sections-derived-completion-pseudo-coherent", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be a quasi-coherent sheaf of ideals. Let $K$ be a", "pseudo-coherent object of $D(\\mathcal{O}_X)$ with derived completion", "$K^\\wedge$. Then", "$$", "H^p(U, K^\\wedge) = \\lim H^p(U, K)/I^nH^p(U, K) =", "H^p(U, K)^\\wedge", "$$", "for any affine open $U \\subset X$", "where $I = \\mathcal{I}(U)$ and where on the right we have the derived", "completion with respect to $I$." ], "refs": [], "proofs": [ { "contents": [ "Write $U = \\Spec(A)$. The ring $A$ is Noetherian", "and hence $I \\subset A$ is finitely generated. Then we have", "$$", "R\\Gamma(U, K^\\wedge) = R\\Gamma(U, K)^\\wedge", "$$", "by Remark \\ref{remark-local-calculation-derived-completion}.", "Now $R\\Gamma(U, K)$ is a pseudo-coherent complex of $A$-modules", "(Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-pseudo-coherent-affine}).", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-derived-completion-pseudo-coherent}", "we conclude that the $p$th cohomology module of $R\\Gamma(U, K^\\wedge)$", "is equal to the $I$-adic completion of $H^p(U, K)$.", "This proves the first equality. The second (less important) equality", "follows immediately from a second application of the lemma just used." ], "refs": [ "algebraization-remark-local-calculation-derived-completion", "perfect-lemma-pseudo-coherent-affine", "more-algebra-lemma-derived-completion-pseudo-coherent" ], "ref_ids": [ 12810, 6975, 10393 ] } ], "ref_ids": [] }, { "id": 12710, "type": "theorem", "label": "algebraization-lemma-derived-completion-pseudo-coherent", "categories": [ "algebraization" ], "title": "algebraization-lemma-derived-completion-pseudo-coherent", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be a quasi-coherent sheaf of ideals.", "Let $K$ be an object of $D(\\mathcal{O}_X)$. Then", "\\begin{enumerate}", "\\item the derived completion $K^\\wedge$ is equal to", "$R\\lim (K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_X/\\mathcal{I}^n)$.", "\\end{enumerate}", "Let $K$ is a pseudo-coherent object of $D(\\mathcal{O}_X)$. Then", "\\begin{enumerate}", "\\item[(2)] the cohomology sheaf $H^q(K^\\wedge)$ is equal to", "$\\lim H^q(K)/\\mathcal{I}^nH^q(K)$.", "\\end{enumerate}", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module\\footnote{For example", "$H^q(K)$ for $K$ pseudo-coherent on our locally Noetherian $X$.}. Then", "\\begin{enumerate}", "\\item[(3)] the derived completion $\\mathcal{F}^\\wedge$ is equal to", "$\\lim \\mathcal{F}/\\mathcal{I}^n\\mathcal{F}$,", "\\item[(4)]", "$\\lim \\mathcal{F}/I^n \\mathcal{F} = R\\lim \\mathcal{F}/I^n \\mathcal{F}$,", "\\item[(5)] $H^p(U, \\mathcal{F}^\\wedge) = 0$ for $p \\not = 0$ for all", "affine opens $U \\subset X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). There is a canonical map", "$$", "K \\longrightarrow", "R\\lim (K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_X/\\mathcal{I}^n),", "$$", "see Remark \\ref{remark-compare-with-completion}.", "Derived completion commutes with passing to open subschemes", "(Remark \\ref{remark-localization-and-completion}).", "Formation of $R\\lim$ commutes with passsing to open subschemes.", "It follows that to check our map is an isomorphism, we may work locally.", "Thus we may assume $X = U = \\Spec(A)$.", "Say $I = (f_1, \\ldots, f_r)$. Let", "$K_n = K(A, f_1^n, \\ldots, f_r^n)$ be the Koszul complex.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-sequence-Koszul-complexes}", "we have seen that the pro-systems $\\{K_n\\}$ and", "$\\{A/I^n\\}$ of $D(A)$ are isomorphic.", "Using the equivalence $D(A) = D_{\\QCoh}(\\mathcal{O}_X)$", "of Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded}", "we see that the pro-systems $\\{K(\\mathcal{O}_X, f_1^n, \\ldots, f_r^n)\\}$", "and $\\{\\mathcal{O}_X/\\mathcal{I}^n\\}$ are isomorphic in", "$D(\\mathcal{O}_X)$. This proves the second equality in", "$$", "K^\\wedge = R\\lim \\left(", "K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K(\\mathcal{O}_X, f_1^n, \\ldots, f_r^n)", "\\right) =", "R\\lim (K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_X/\\mathcal{I}^n)", "$$", "The first equality is", "Lemma \\ref{lemma-derived-completion-koszul}.", "\\medskip\\noindent", "Assume $K$ is pseudo-coherent. For $U \\subset X$ affine open", "we have $H^q(U, K^\\wedge) = \\lim H^q(U, K)/\\mathcal{I}^n(U)H^q(U, K)$", "by Lemma \\ref{lemma-sections-derived-completion-pseudo-coherent}.", "As this is true for every $U$ we see that", "$H^q(K^\\wedge) = \\lim H^q(K)/\\mathcal{I}^nH^q(K)$ as sheaves.", "This proves (2).", "\\medskip\\noindent", "Part (3) is a special case of (2).", "Parts (4) and (5) follow from", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-Rlim-quasi-coherent}." ], "refs": [ "algebraization-remark-compare-with-completion", "algebraization-remark-localization-and-completion", "more-algebra-lemma-sequence-Koszul-complexes", "perfect-lemma-affine-compare-bounded", "algebraization-lemma-derived-completion-koszul", "algebraization-lemma-sections-derived-completion-pseudo-coherent", "perfect-lemma-Rlim-quasi-coherent" ], "ref_ids": [ 12807, 12808, 10391, 6941, 12701, 12709, 6938 ] } ], "ref_ids": [] }, { "id": 12711, "type": "theorem", "label": "algebraization-lemma-formal-functions", "categories": [ "algebraization" ], "title": "algebraization-lemma-formal-functions", "contents": [ "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. Let $X$ be a", "Noetherian scheme over $A$. Let $\\mathcal{F}$ be a coherent", "$\\mathcal{O}_X$-module. Assume that $H^p(X, \\mathcal{F})$ is", "a finite $A$-module for all $p$. Then there are short exact sequences", "$$", "0 \\to R^1\\lim H^{p - 1}(X, \\mathcal{F}/I^n\\mathcal{F}) \\to", "H^p(X, \\mathcal{F})^\\wedge \\to \\lim H^p(X, \\mathcal{F}/I^n\\mathcal{F}) \\to 0", "$$", "of $A$-modules where $H^p(X, \\mathcal{F})^\\wedge$ is the usual $I$-adic", "completion. If $f$ is proper, then the $R^1\\lim$ term is zero." ], "refs": [], "proofs": [ { "contents": [ "Consider the two spectral sequences of", "Lemma \\ref{lemma-formal-functions-general}.", "The first degenerates by More on Algebra, Lemma", "\\ref{more-algebra-lemma-derived-completion-pseudo-coherent}.", "We obtain $H^p(X, \\mathcal{F})^\\wedge$ in degree $p$.", "This is where we use the assumption that $H^p(X, \\mathcal{F})$ is", "a finite $A$-module. The second degenerates because", "$$", "\\mathcal{F}^\\wedge = \\lim \\mathcal{F}/I^n\\mathcal{F} =", "R\\lim \\mathcal{F}/I^n\\mathcal{F}", "$$", "is a sheaf by Lemma \\ref{lemma-derived-completion-pseudo-coherent}.", "We obtain $H^p(X, \\lim \\mathcal{F}/I^n\\mathcal{F})$ in degree $p$.", "Since $R\\Gamma(X, -)$ commutes with derived limits", "(Injectives, Lemma \\ref{injectives-lemma-RF-commutes-with-Rlim})", "we also get", "$$", "R\\Gamma(X, \\lim \\mathcal{F}/I^n\\mathcal{F}) =", "R\\Gamma(X, R\\lim \\mathcal{F}/I^n\\mathcal{F}) =", "R\\lim R\\Gamma(X, \\mathcal{F}/I^n\\mathcal{F})", "$$", "By More on Algebra, Remark", "\\ref{more-algebra-remark-how-unique}", "we obtain exact sequences", "$$", "0 \\to", "R^1\\lim H^{p - 1}(X, \\mathcal{F}/I^n\\mathcal{F}) \\to", "H^p(X, \\lim \\mathcal{F}/I^n\\mathcal{F}) \\to", "\\lim H^p(X, \\mathcal{F}/I^n\\mathcal{F}) \\to 0", "$$", "of $A$-modules. Combining the above we get the first statement of the lemma.", "The vanishing of the $R^1\\lim$ term follows from", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-ML-cohomology-powers-ideal}." ], "refs": [ "algebraization-lemma-formal-functions-general", "more-algebra-lemma-derived-completion-pseudo-coherent", "algebraization-lemma-derived-completion-pseudo-coherent", "injectives-lemma-RF-commutes-with-Rlim", "more-algebra-remark-how-unique", "coherent-lemma-ML-cohomology-powers-ideal" ], "ref_ids": [ 12708, 10393, 12710, 7796, 10660, 3360 ] } ], "ref_ids": [] }, { "id": 12712, "type": "theorem", "label": "algebraization-lemma-kill-completion-general", "categories": [ "algebraization" ], "title": "algebraization-lemma-kill-completion-general", "contents": [ "Let $I, J$ be ideals of a Noetherian ring $A$.", "Let $M$ be a finite $A$-module. Let $\\mathfrak p \\subset A$ be a prime.", "Let $s$ and $d$ be integers. Assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex,", "\\item $\\mathfrak p \\not \\in V(J) \\cap V(I)$,", "\\item $\\text{cd}(A, I) \\leq d$, and", "\\item for all primes $\\mathfrak p' \\subset \\mathfrak p$", "we have", "$\\text{depth}_{A_{\\mathfrak p'}}(M_{\\mathfrak p'}) +", "\\dim((A/\\mathfrak p')_\\mathfrak q) > d + s$", "for all $\\mathfrak q \\in V(\\mathfrak p') \\cap V(J) \\cap V(I)$.", "\\end{enumerate}", "Then there exists an $f \\in A$, $f \\not \\in \\mathfrak p$ which annihilates", "$H^i(R\\Gamma_J(M)^\\wedge)$ for $i \\leq s$ where ${}^\\wedge$", "indicates $I$-adic completion." ], "refs": [], "proofs": [ { "contents": [ "We will use that $R\\Gamma_J = R\\Gamma_{V(J)}$ and similarly for", "$I + J$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-local-cohomology-noetherian}.", "Observe that", "$R\\Gamma_J(M)^\\wedge = R\\Gamma_I(R\\Gamma_J(M))^\\wedge =", "R\\Gamma_{I + J}(M)^\\wedge$, see", "Dualizing Complexes, Lemmas", "\\ref{dualizing-lemma-complete-and-local} and", "\\ref{dualizing-lemma-local-cohomology-ss}.", "Thus we may replace $J$ by $I + J$ and assume $I \\subset J$", "and $\\mathfrak p \\not \\in V(J)$.", "Recall that", "$$", "R\\Gamma_J(M)^\\wedge = R\\Hom_A(R\\Gamma_I(A), R\\Gamma_J(M))", "$$", "by the description of derived completion in", "More on Algebra, Lemma \\ref{more-algebra-lemma-derived-completion}", "combined with the description of local cohomology in", "Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-compute-local-cohomology-noetherian}.", "Assumption (3) means that $R\\Gamma_I(A)$ has nonzero cohomology", "only in degrees $\\leq d$. Using the canonical truncations of", "$R\\Gamma_I(A)$ we find it suffices to show that", "$$", "\\text{Ext}^i(N, R\\Gamma_J(M))", "$$", "is annihilated by an $f \\in A$, $f \\not \\in \\mathfrak p$ for", "$i \\leq s + d$ and any $A$-module $N$.", "In turn using the canonical truncations for $R\\Gamma_J(M)$", "we see that it suffices to show", "$H^i_J(M)$ is annihilated by an $f \\in A$, $f \\not \\in \\mathfrak p$", "for $i \\leq s + d$.", "This follows from Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-kill-local-cohomology-at-prime}." ], "refs": [ "dualizing-lemma-local-cohomology-noetherian", "dualizing-lemma-complete-and-local", "dualizing-lemma-local-cohomology-ss", "more-algebra-lemma-derived-completion", "dualizing-lemma-compute-local-cohomology-noetherian", "local-cohomology-lemma-kill-local-cohomology-at-prime" ], "ref_ids": [ 2823, 2832, 2820, 10372, 2824, 9738 ] } ], "ref_ids": [] }, { "id": 12713, "type": "theorem", "label": "algebraization-lemma-kill-colimit-weak-general", "categories": [ "algebraization" ], "title": "algebraization-lemma-kill-colimit-weak-general", "contents": [ "Let $I, J$ be ideals of a Noetherian ring. Let $M$ be a finite $A$-module.", "Let $s$ and $d$ be integers. With $T$ as in", "(\\ref{equation-associated-subset}) assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex,", "\\item if $\\mathfrak p \\in V(I)$, then no condition,", "\\item if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\in T$, then", "$\\dim((A/\\mathfrak p)_\\mathfrak q) \\leq d$ for some", "$\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$,", "\\item if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\not \\in T$, then", "$$", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s", "\\quad\\text{or}\\quad", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > d + s", "$$", "for all $\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$.", "\\end{enumerate}", "Then there exists an ideal $J_0 \\subset J$ with", "$V(J_0) \\cap V(I) = V(J) \\cap V(I)$ such that for any $J' \\subset J_0$ with", "$V(J') \\cap V(I) = V(J) \\cap V(I)$ the map", "$$", "R\\Gamma_{J'}(M) \\longrightarrow R\\Gamma_{J_0}(M)", "$$", "induces an isomorphism in cohomology in degrees $\\leq s$", "and moreover these modules are annihilated by a power of $J_0I$." ], "refs": [], "proofs": [ { "contents": [ "Let us consider the set", "$$", "B = \\{\\mathfrak p \\not \\in V(I),\\ \\mathfrak p \\in T,\\text{ and }", "\\text{depth}(M_\\mathfrak p) \\leq s\\}", "$$", "Choose $J_0 \\subset J$ such that $V(J_0)$ is the closure of $B \\cup V(J)$.", "\\medskip\\noindent", "Claim I: $V(J_0) \\cap V(I) = V(J) \\cap V(I)$.", "\\medskip\\noindent", "Proof of Claim I. The inclusion $\\supset$ holds by construction.", "Let $\\mathfrak p$ be a minimal prime of $V(J_0)$.", "If $\\mathfrak p \\in B \\cup V(J)$, then either $\\mathfrak p \\in T$", "or $\\mathfrak p \\in V(J)$ and in both cases", "$V(\\mathfrak p) \\cap V(I) \\subset V(J) \\cap V(I)$ as desired.", "If $\\mathfrak p \\not \\in B \\cup V(J)$, then", "$V(\\mathfrak p) \\cap B$ is dense, hence infinite, and we conclude that", "$\\text{depth}(M_\\mathfrak p) < s$ by", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-depth-function}.", "In fact, let", "$V(\\mathfrak p) \\cap B = \\{\\mathfrak p_\\lambda\\}_{\\lambda \\in \\Lambda}$.", "Pick $\\mathfrak q_\\lambda \\in V(\\mathfrak p_\\lambda) \\cap V(J) \\cap V(I)$", "as in (3).", "Let $\\delta : \\Spec(A) \\to \\mathbf{Z}$ be the dimension function", "associated to a dualizing complex $\\omega_A^\\bullet$ for $A$.", "Since $\\Lambda$ is infinite and $\\delta$ is bounded,", "there exists an infinite subset $\\Lambda' \\subset \\Lambda$ on which", "$\\delta(\\mathfrak q_\\lambda)$ is constant. For", "$\\lambda \\in \\Lambda'$ we have", "$$", "\\text{depth}(M_{\\mathfrak p_\\lambda}) +", "\\delta(\\mathfrak p_\\lambda) - \\delta(\\mathfrak q_\\lambda) =", "\\text{depth}(M_{\\mathfrak p_\\lambda}) +", "\\dim((A/\\mathfrak p_\\lambda)_{\\mathfrak q_\\lambda})", "\\leq d + s", "$$", "by (3) and the definition of $B$. By the semi-continuity of", "the function $\\text{depth} + \\delta$ proved in", "Duality for Schemes, Lemma \\ref{duality-lemma-sitting-in-degrees}", "we conclude that", "$$", "\\text{depth}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_{\\mathfrak q_\\lambda}) =", "\\text{depth}(M_\\mathfrak p) + \\delta(\\mathfrak p) - \\delta(\\mathfrak q_\\lambda)", "\\leq d + s", "$$", "Since also $\\mathfrak p \\not \\in V(I)$ we read off from (4) that", "$\\mathfrak p \\in T$, i.e.,", "$V(\\mathfrak p) \\cap V(I) \\subset V(J) \\cap V(I)$. This finishes the", "proof of Claim I.", "\\medskip\\noindent", "Claim II: $H^i_{J_0}(M) \\to H^i_J(M)$ is an isomorphism for $i \\leq s$", "and $J' \\subset J_0$ with $V(J') \\cap V(I) = V(J) \\cap V(I)$.", "\\medskip\\noindent", "Proof of claim II. Choose $\\mathfrak p \\in V(J')$ not in $V(J_0)$.", "It suffices to show that $H^i_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) = 0$", "for $i \\leq s$, see", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-isomorphism}.", "Observe that $\\mathfrak p \\in T$. Hence since $\\mathfrak p$ is not in $B$", "we see that $\\text{depth}(M_\\mathfrak p) > s$ and the groups vanish by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth}.", "\\medskip\\noindent", "Claim III. The final statement of the lemma is true.", "\\medskip\\noindent", "By Claim II for $i \\leq s$ we have", "$$", "H^i_T(M) = H^i_{J_0}(M) = H^i_{J'}(M)", "$$", "for all ideals $J' \\subset J_0$ with $V(J') \\cap V(I) = V(J) \\cap V(I)$.", "See Local Cohomology, Lemma \\ref{local-cohomology-lemma-adjoint-ext}.", "Let us check the hypotheses of Local Cohomology,", "Proposition \\ref{local-cohomology-proposition-annihilator}", "for the subsets $T \\subset T \\cup V(I)$, the module $M$, and the integer $s$.", "We have to show that given $\\mathfrak p \\subset \\mathfrak q$", "with $\\mathfrak p \\not \\in T \\cup V(I)$ and $\\mathfrak q \\in T$", "we have", "$$", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > s", "$$", "If $\\text{depth}(M_\\mathfrak p) \\geq s$, then this is true because", "the dimension of $(A/\\mathfrak p)_\\mathfrak q$ is at least $1$.", "Thus we may assume $\\text{depth}(M_\\mathfrak p) < s$.", "If $\\mathfrak q \\in V(I)$, then $\\mathfrak q \\in V(J) \\cap V(I)$", "and the inequality holds by (4). If $\\mathfrak q \\not \\in V(I)$,", "then we can use (3) to pick", "$\\mathfrak q' \\in V(\\mathfrak q) \\cap V(J) \\cap V(I)$ with", "$\\dim((A/\\mathfrak q)_{\\mathfrak q'}) \\leq d$.", "Then assumption (4) gives", "$$", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_{\\mathfrak q'}) > s + d", "$$", "Since $A$ is catenary this implies the inequality we want.", "Applying Local Cohomology,", "Proposition \\ref{local-cohomology-proposition-annihilator} we", "find $J'' \\subset A$ with $V(J'') \\subset T \\cup V(I)$", "such that $J''$ annihilates $H^i_T(M)$ for $i \\leq s$.", "Then we can write $V(J'') \\cup V(J_0) \\cup V(I) = V(J'I)$", "for some $J' \\subset J_0$ with $V(J') \\cap V(I) = V(J) \\cap V(I)$.", "Replacing $J_0$ by $J'$ the proof is complete." ], "refs": [ "local-cohomology-lemma-depth-function", "duality-lemma-sitting-in-degrees", "local-cohomology-lemma-isomorphism", "dualizing-lemma-depth", "local-cohomology-lemma-adjoint-ext", "local-cohomology-proposition-annihilator", "local-cohomology-proposition-annihilator" ], "ref_ids": [ 9735, 13502, 9700, 2826, 9715, 9786, 9786 ] } ], "ref_ids": [] }, { "id": 12714, "type": "theorem", "label": "algebraization-lemma-kill-colimit-general", "categories": [ "algebraization" ], "title": "algebraization-lemma-kill-colimit-general", "contents": [ "In Lemma \\ref{lemma-kill-colimit-weak-general} if instead of the empty", "condition (2) we assume", "\\begin{enumerate}", "\\item[(2')] if $\\mathfrak p \\in V(I)$, $\\mathfrak p \\not \\in V(J) \\cap V(I)$,", "then", "$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > s$", "for all $\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$,", "\\end{enumerate}", "then the conditions also imply that $H^i_{J_0}(M)$ is a finite", "$A$-module for $i \\leq s$." ], "refs": [ "algebraization-lemma-kill-colimit-weak-general" ], "proofs": [ { "contents": [ "Recall that $H^i_{J_0}(M) = H^i_T(M)$, see proof of", "Lemma \\ref{lemma-kill-colimit-weak-general}. Thus it suffices to", "check that for $\\mathfrak p \\not \\in T$ and $\\mathfrak q \\in T$", "with $\\mathfrak p \\subset \\mathfrak q$ we have", "$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > s$, see Local Cohomology,", "Proposition \\ref{local-cohomology-proposition-finiteness}.", "Condition (2') tells us this is true for $\\mathfrak p \\in V(I)$.", "Since we know $H^i_T(M)$ is annihilated by a power of $IJ_0$", "we know the condition holds if $\\mathfrak p \\not \\in V(IJ_0)$", "by Local Cohomology, Proposition \\ref{local-cohomology-proposition-annihilator}.", "This covers all cases and the proof is complete." ], "refs": [ "algebraization-lemma-kill-colimit-weak-general", "local-cohomology-proposition-finiteness", "local-cohomology-proposition-annihilator" ], "ref_ids": [ 12713, 9787, 9786 ] } ], "ref_ids": [ 12713 ] }, { "id": 12715, "type": "theorem", "label": "algebraization-lemma-kill-colimit-support-general", "categories": [ "algebraization" ], "title": "algebraization-lemma-kill-colimit-support-general", "contents": [ "If in Lemma \\ref{lemma-kill-colimit-weak-general} we additionally assume", "\\begin{enumerate}", "\\item[(6)] if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\in T$, then", "$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) > s$,", "\\end{enumerate}", "then $H^i_{J_0}(M) = H^i_J(M) = H^i_{J + I}(M)$ for $i \\leq s$ and these", "modules are annihilated by a power of $I$." ], "refs": [ "algebraization-lemma-kill-colimit-weak-general" ], "proofs": [ { "contents": [ "Choose $\\mathfrak p \\in V(J)$ or $\\mathfrak p \\in V(J_0)$ but", "$\\mathfrak p \\not \\in V(J + I) = V(J_0 + I)$.", "It suffices to show that $H^i_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) = 0$", "for $i \\leq s$, see", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-isomorphism}.", "These groups vanish by condition (6) and", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth}.", "The final statement follows from", "Local Cohomology, Proposition \\ref{local-cohomology-proposition-annihilator}." ], "refs": [ "local-cohomology-lemma-isomorphism", "dualizing-lemma-depth", "local-cohomology-proposition-annihilator" ], "ref_ids": [ 9700, 2826, 9786 ] } ], "ref_ids": [ 12713 ] }, { "id": 12716, "type": "theorem", "label": "algebraization-lemma-algebraize-local-cohomology-general", "categories": [ "algebraization" ], "title": "algebraization-lemma-algebraize-local-cohomology-general", "contents": [ "Let $I, J$ be ideals of a Noetherian ring $A$.", "Let $M$ be a finite $A$-module.", "Let $s$ and $d$ be integers. With $T$ as in", "(\\ref{equation-associated-subset}) assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete and has a dualizing complex,", "\\item if $\\mathfrak p \\in V(I)$ no condition,", "\\item $\\text{cd}(A, I) \\leq d$,", "\\item if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\not \\in T$ then", "$$", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s", "\\quad\\text{or}\\quad", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > d + s", "$$", "for all $\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$,", "\\item if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\not \\in T$,", "$V(\\mathfrak p) \\cap V(J) \\cap V(I) \\not = \\emptyset$, and", "$\\text{depth}(M_\\mathfrak p) < s$, then one", "of the following holds\\footnote{Our method", "forces this additional condition. We will return to this", "(insert future reference).}:", "\\begin{enumerate}", "\\item $\\dim(\\text{Supp}(M_\\mathfrak p)) < s + 2$\\footnote{For example", "if $M$ satisfies Serre's condition $(S_s)$", "on the complement of $V(I) \\cup T$.}, or", "\\item $\\delta(\\mathfrak p) > d + \\delta_{max} - 1$", "where $\\delta$ is a dimension function and $\\delta_{max}$", "is the maximum of $\\delta$ on $V(J) \\cap V(I)$, or", "\\item $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > d + s + \\delta_{max} - \\delta_{min} - 2$", "for all $\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$.", "\\end{enumerate}", "\\end{enumerate}", "Then there exists an ideal $J_0 \\subset J$ with", "$V(J_0) \\cap V(I) = V(J) \\cap V(I)$", "such that for any $J' \\subset J_0$ with", "$V(J') \\cap V(I) = V(J) \\cap V(I)$ the map", "$$", "R\\Gamma_{J'}(M) \\longrightarrow R\\Gamma_J(M)^\\wedge", "$$", "induces an isomorphism on cohomology in degrees $\\leq s$.", "Here ${}^\\wedge$ denotes derived $I$-adic completion." ], "refs": [], "proofs": [ { "contents": [ "For an ideal $\\mathfrak a \\subset A$ we have", "$R\\Gamma_\\mathfrak a = R\\Gamma_{V(\\mathfrak a)}$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-local-cohomology-noetherian}.", "Next, we observe that", "$$", "R\\Gamma_J(M)^\\wedge =", "R\\Gamma_I(R\\Gamma_J(M))^\\wedge =", "R\\Gamma_{I + J}(M)^\\wedge =", "R\\Gamma_{I + J'}(M)^\\wedge =", "R\\Gamma_I(R\\Gamma_{J'}(M))^\\wedge =", "R\\Gamma_{J'}(M)^\\wedge", "$$", "by Dualizing Complexes, Lemmas \\ref{dualizing-lemma-local-cohomology-ss} and", "\\ref{dualizing-lemma-complete-and-local}.", "This explains how we define the arrow in the statement of the lemma.", "\\medskip\\noindent", "We claim that the hypotheses of Lemma \\ref{lemma-kill-colimit-weak-general}", "are implied by our current hypotheses on $M$.", "The only thing to verify is hypothesis (3).", "Thus let $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\in T$.", "Then $V(\\mathfrak p) \\cap V(I)$ is nonempty as $I$ is", "contained in the Jacobson radical of $A$", "(Algebra, Lemma \\ref{algebra-lemma-radical-completion}).", "Since $\\mathfrak p \\in T$ we have", "$V(\\mathfrak p) \\cap V(I) = V(\\mathfrak p) \\cap V(J) \\cap V(I)$.", "Let $\\mathfrak q \\in V(\\mathfrak p) \\cap V(I)$ be the", "generic point of an irreducible component.", "We have $\\text{cd}(A_\\mathfrak q, I_\\mathfrak q) \\leq d$", "by Local Cohomology, Lemma \\ref{local-cohomology-lemma-cd-local}.", "We have $V(\\mathfrak pA_\\mathfrak q) \\cap V(I_\\mathfrak q) =", "\\{\\mathfrak qA_\\mathfrak q\\}$ by our choice of $\\mathfrak q$", "and we conclude $\\dim((A/\\mathfrak p)_\\mathfrak q) \\leq d$", "by Local Cohomology, Lemma \\ref{local-cohomology-lemma-cd-bound-dim-local}.", "\\medskip\\noindent", "Observe that the lemma holds for $s < 0$. This is not a trivial case because", "it is not a priori clear that $H^i(R\\Gamma_J(M)^\\wedge)$", "is zero for $i < 0$. However, this vanishing was established in", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-completion-local}.", "We will prove the lemma by induction for $s \\geq 0$.", "\\medskip\\noindent", "The lemma for $s = 0$ follows immediately from", "the conclusion of Lemma \\ref{lemma-kill-colimit-weak-general}", "and Dualizing Complexes, Lemma \\ref{dualizing-lemma-completion-local-H0}.", "\\medskip\\noindent", "Assume $s > 0$ and the lemma has been shown for smaller values of $s$.", "Let $M' \\subset M$ be the maximal submodule whose support is contained", "in $V(I) \\cup T$. Then $M'$ is a finite $A$-module whose support", "is contained in $V(J') \\cup V(I)$ for some ideal $J' \\subset J$", "with $V(J') \\cap V(I) = V(J) \\cap V(I)$.", "We claim that", "$$", "R\\Gamma_{J'}(M') \\to R\\Gamma_J(M')^\\wedge", "$$", "is an isomorphism for any choice of $J'$.", "Namely, we can choose a short exact sequence", "$0 \\to M_1 \\oplus M_2 \\to M' \\to N \\to 0$ with", "$M_1$ annihilated by a power of $J'$, with $M_2$ annihilated", "by a power of $I$, and with $N$ annihilated by a power of $I + J'$.", "Thus it suffices to show that the claim holds for $M_1$, $M_2$, and $N$.", "In the case of $M_1$ we see that $R\\Gamma_{J'}(M_1) = M_1$ and", "since $M_1$ is a finite $A$-module and $I$-adically complete", "we have $M_1^\\wedge = M_1$. This proves the claim for $M_1$", "by the initial remarks of the proof. In the case of $M_2$ we see that", "$H^i_J(M_2) = H^i_{I + J}(M) = H^i_{I + J'}(M) = H^i_{J'}(M_2)$", "are annihilated by a power of $I$ and hence derived complete.", "Thus the claim in this case also. For $N$ we can use either of", "the arguments just given. Considering the short exact sequence", "$0 \\to M' \\to M \\to M/M' \\to 0$", "we see that it suffices to prove the lemma for $M/M'$.", "Thus we may assume $\\text{Ass}(M) \\cap (V(I) \\cup T) = \\emptyset$.", "\\medskip\\noindent", "Let $\\mathfrak p \\in \\text{Ass}(M)$ be such that", "$V(\\mathfrak p) \\cap V(J) \\cap V(I) = \\emptyset$.", "Since $I$ is contained in the Jacobson radical of $A$ this implies", "that $V(\\mathfrak p) \\cap V(J') = \\emptyset$ for any", "$J' \\subset J$ with $V(J') \\cap V(I) = V(J) \\cap V(I)$.", "Thus setting $N = H^0_\\mathfrak p(M)$ we see that", "$R\\Gamma_J(N) = R\\Gamma_{J'}(N) = 0$ for all", "$J' \\subset J$ with $V(J') \\cap V(I) = V(J) \\cap V(I)$.", "In particular $R\\Gamma_J(N)^\\wedge = 0$.", "Thus we may replace $M$ by $M/N$ as this changes the", "structure of $M$ only in primes which do not play", "a role in conditions (4) or (5). Repeating we may assume that", "$V(\\mathfrak p) \\cap V(J) \\cap V(I) \\not = \\emptyset$", "for all $\\mathfrak p \\in \\text{Ass}(M)$.", "\\medskip\\noindent", "Assume $\\text{Ass}(M) \\cap (V(I) \\cup T) = \\emptyset$ and that", "$V(\\mathfrak p) \\cap V(J) \\cap V(I) \\not = \\emptyset$", "for all $\\mathfrak p \\in \\text{Ass}(M)$.", "Let $\\mathfrak p \\in \\text{Ass}(M)$. We want to show that we may apply", "Lemma \\ref{lemma-kill-completion-general}.", "It is in the verification of this that we will use the supplemental", "condition (5). Choose $\\mathfrak p' \\subset \\mathfrak p$", "and $\\mathfrak q' \\subset V(\\mathfrak p) \\cap V(J) \\cap V(I)$.", "\\begin{enumerate}", "\\item If $M_{\\mathfrak p'} = 0$, then", "$\\text{depth}(M_{\\mathfrak p'}) = \\infty$ and", "$\\text{depth}(M_{\\mathfrak p'}) +", "\\dim((A/\\mathfrak p')_{\\mathfrak q'}) > d + s$.", "\\item If $\\text{depth}(M_{\\mathfrak p'}) < s$, then", "$\\text{depth}(M_{\\mathfrak p'}) +", "\\dim((A/\\mathfrak p')_{\\mathfrak q'}) > d + s$ by (4).", "\\end{enumerate}", "In the remaining cases we have $M_{\\mathfrak p'} \\not = 0$ and", "$\\text{depth}(M_{\\mathfrak p'}) \\geq s$. In particular, we see that", "$\\mathfrak p'$ is in the support of $M$ and we can choose", "$\\mathfrak p'' \\subset \\mathfrak p'$ with $\\mathfrak p'' \\in \\text{Ass}(M)$.", "\\begin{enumerate}", "\\item[(a)] Observe that", "$\\dim((A/\\mathfrak p'')_{\\mathfrak p'}) \\geq \\text{depth}(M_{\\mathfrak p'})$", "by Algebra, Lemma \\ref{algebra-lemma-depth-dim-associated-primes}.", "If equality holds, then we have", "$$", "\\text{depth}(M_{\\mathfrak p'}) + \\dim((A/\\mathfrak p')_{\\mathfrak q'}) =", "\\text{depth}(M_{\\mathfrak p''}) + \\dim((A/\\mathfrak p'')_{\\mathfrak q'})", "> s + d", "$$", "by (4) applied to $\\mathfrak p''$ and we are done. This means we are", "only in trouble if", "$\\dim((A/\\mathfrak p'')_{\\mathfrak p'}) > \\text{depth}(M_{\\mathfrak p'})$.", "This implies that $\\dim(M_\\mathfrak p) \\geq s + 2$.", "Thus if (5)(a) holds, then this does not occur.", "\\item[(b)] If (5)(b) holds, then we get", "$$", "\\text{depth}(M_{\\mathfrak p'}) + \\dim((A/\\mathfrak p')_{\\mathfrak q'})", "\\geq s + \\delta(\\mathfrak p') - \\delta(\\mathfrak q')", "\\geq s + 1 + \\delta(\\mathfrak p) - \\delta_{max}", "> s + d", "$$", "as desired.", "\\item[(c)] If (5)(c) holds, then we get", "\\begin{align*}", "\\text{depth}(M_{\\mathfrak p'}) + \\dim((A/\\mathfrak p')_{\\mathfrak q'})", "& \\geq", "s + \\delta(\\mathfrak p') - \\delta(\\mathfrak q') \\\\", "& \\geq", "s + 1 + \\delta(\\mathfrak p) - \\delta(\\mathfrak q') \\\\", "& =", "s + 1 + \\delta(\\mathfrak p) - \\delta(\\mathfrak q) +", "\\delta(\\mathfrak q) - \\delta(\\mathfrak q') \\\\", "& >", "s + 1 + (s + d + \\delta_{max} - \\delta_{min} - 2) +", "\\delta(\\mathfrak q) - \\delta(\\mathfrak q') \\\\", "& \\geq ", "2s + d - 1 \\geq s + d", "\\end{align*}", "as desired. Observe that this argument works because", "we know that a prime $\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$", "exists.", "\\end{enumerate}", "Now we are ready to do the induction step.", "\\medskip\\noindent", "Choose an ideal $J_0$ as in Lemma \\ref{lemma-kill-colimit-weak-general}", "and an integer $t > 0$ such that $(J_0I)^t$ annihilates $H^s_J(M)$.", "The assumptions of Lemma \\ref{lemma-kill-completion-general}", "are satisfied for every $\\mathfrak p \\in \\text{Ass}(M)$", "(see previous paragraph).", "Thus the annihilator $\\mathfrak a \\subset A$ of", "$H^s(R\\Gamma_J(M)^\\wedge)$", "is not contained in $\\mathfrak p$ for $\\mathfrak p \\in \\text{Ass}(M)$.", "Thus we can find an $f \\in \\mathfrak a(J_0I)^t$", "not in any associated prime of $M$ which is an annihilator", "of both $H^s(R\\Gamma_J(M)^\\wedge)$ and $H^s_J(M)$.", "Then $f$ is a nonzerodivisor on $M$ and we can consider the", "short exact sequence", "$$", "0 \\to M \\xrightarrow{f} M \\to M/fM \\to 0", "$$", "Our choice of $f$ shows that we obtain", "$$", "\\xymatrix{", "H^{s - 1}_{J'}(M) \\ar[d] \\ar[r] &", "H^{s - 1}_{J'}(M/fM) \\ar[d] \\ar[r] &", "H^s_{J'}(M) \\ar[d] \\ar[r] & 0 \\\\", "H^{s - 1}(R\\Gamma_J(M)^\\wedge) \\ar[r] &", "H^{s - 1}(R\\Gamma_J(M/fM)^\\wedge) \\ar[r] &", "H^s(R\\Gamma_J(M)^\\wedge) \\ar[r] & 0", "}", "$$", "for any $J' \\subset J_0$ with $V(J') \\cap V(I) = V(J) \\cap V(I)$.", "Thus if we choose $J'$ such that it works for", "$M$ and $M/fM$ and $s - 1$ (possible by induction hypothesis --", "see next paragraph), then we conclude that the lemma is true.", "\\medskip\\noindent", "To finish the proof we have to show that the module", "$M/fM$ satisfies the hypotheses (4) and (5) for $s - 1$.", "Thus we let $\\mathfrak p$ be a prime in the support", "of $M/fM$ with $\\text{depth}((M/fM)_\\mathfrak p) < s - 1$", "and with $V(\\mathfrak p) \\cap V(J) \\cap V(I)$ nonempty.", "Then $\\dim(M_\\mathfrak p) = \\dim((M/fM)_\\mathfrak p) + 1$", "and $\\text{depth}(M_\\mathfrak p) = \\text{depth}((M/fM)_\\mathfrak p) + 1$.", "In particular, we know (4) and (5) hold for $\\mathfrak p$ and $M$", "with the original value $s$.", "The desired inequalities then follow by inspection." ], "refs": [ "dualizing-lemma-local-cohomology-noetherian", "dualizing-lemma-local-cohomology-ss", "dualizing-lemma-complete-and-local", "algebraization-lemma-kill-colimit-weak-general", "algebra-lemma-radical-completion", "local-cohomology-lemma-cd-local", "local-cohomology-lemma-cd-bound-dim-local", "dualizing-lemma-completion-local", "algebraization-lemma-kill-colimit-weak-general", "dualizing-lemma-completion-local-H0", "algebraization-lemma-kill-completion-general", "algebra-lemma-depth-dim-associated-primes", "algebraization-lemma-kill-colimit-weak-general", "algebraization-lemma-kill-completion-general" ], "ref_ids": [ 2823, 2820, 2832, 12713, 862, 9707, 9711, 2834, 12713, 2835, 12712, 776, 12713, 12712 ] } ], "ref_ids": [] }, { "id": 12717, "type": "theorem", "label": "algebraization-lemma-kill-completion", "categories": [ "algebraization" ], "title": "algebraization-lemma-kill-completion", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module and", "let $\\mathfrak p \\subset A$ be a prime. Let $s$ and $d$ be integers. Assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex,", "\\item $\\text{cd}(A, I) \\leq d$, and", "\\item", "$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > d + s$.", "\\end{enumerate}", "Then there exists an $f \\in A \\setminus \\mathfrak p$ which annihilates", "$H^i(R\\Gamma_\\mathfrak m(M)^\\wedge)$ for $i \\leq s$ where ${}^\\wedge$", "indicates $I$-adic completion." ], "refs": [], "proofs": [ { "contents": [ "According to Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-sitting-in-degrees}", "the function", "$$", "\\mathfrak p' \\longmapsto", "\\text{depth}_{A_{\\mathfrak p'}}(M_{\\mathfrak p'}) + \\dim(A/\\mathfrak p')", "$$", "is lower semi-continuous on $\\Spec(A)$. Thus the value", "of this function on $\\mathfrak p' \\subset \\mathfrak p$", "is $> s + d$. Thus our lemma is a special case of", "Lemma \\ref{lemma-kill-completion-general}", "provided that $\\mathfrak p \\not = \\mathfrak m$.", "If $\\mathfrak p = \\mathfrak m$,", "then we have $H^i_\\mathfrak m(M) = 0$ for $i \\leq s + d$ by", "the relationship between depth and local cohomology", "(Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth}).", "Thus the argument given in the proof of", "Lemma \\ref{lemma-kill-completion-general}", "shows that $H^i(R\\Gamma_\\mathfrak m(M)^\\wedge) = 0$", "for $i \\leq s$ in this (degenerate) case." ], "refs": [ "local-cohomology-lemma-sitting-in-degrees", "algebraization-lemma-kill-completion-general", "dualizing-lemma-depth", "algebraization-lemma-kill-completion-general" ], "ref_ids": [ 9737, 12712, 2826, 12712 ] } ], "ref_ids": [] }, { "id": 12718, "type": "theorem", "label": "algebraization-lemma-kill-colimit-weak", "categories": [ "algebraization" ], "title": "algebraization-lemma-kill-colimit-weak", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module.", "Let $s$ and $d$ be integers. Assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex,", "\\item if $\\mathfrak p \\in V(I)$, then no condition,", "\\item if $\\mathfrak p \\not \\in V(I)$ and", "$V(\\mathfrak p) \\cap V(I) = \\{\\mathfrak m\\}$, then", "$\\dim(A/\\mathfrak p) \\leq d$,", "\\item if $\\mathfrak p \\not \\in V(I)$ and", "$V(\\mathfrak p) \\cap V(I) \\not = \\{\\mathfrak m\\}$, then", "$$", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s", "\\quad\\text{or}\\quad", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > d + s", "$$", "\\end{enumerate}", "Then there exists an ideal $J_0 \\subset A$ with", "$V(J_0) \\cap V(I) = \\{\\mathfrak m\\}$ such that for any $J \\subset J_0$ with", "$V(J) \\cap V(I) = \\{\\mathfrak m\\}$ the map", "$$", "R\\Gamma_J(M) \\longrightarrow R\\Gamma_{J_0}(M)", "$$", "induces an isomorphism in cohomology in degrees $\\leq s$", "and moreover these modules are annihilated by a power of $J_0I$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-kill-colimit-weak-general}." ], "refs": [ "algebraization-lemma-kill-colimit-weak-general" ], "ref_ids": [ 12713 ] } ], "ref_ids": [] }, { "id": 12719, "type": "theorem", "label": "algebraization-lemma-kill-colimit", "categories": [ "algebraization" ], "title": "algebraization-lemma-kill-colimit", "contents": [ "In Lemma \\ref{lemma-kill-colimit-weak} if instead of the empty", "condition (2) we assume", "\\begin{enumerate}", "\\item[(2')] if $\\mathfrak p \\in V(I)$ and $\\mathfrak p \\not = \\mathfrak m$,", "then $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > s$,", "\\end{enumerate}", "then the conditions also imply that $H^i_{J_0}(M)$ is a finite", "$A$-module for $i \\leq s$." ], "refs": [ "algebraization-lemma-kill-colimit-weak" ], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-kill-colimit-general}." ], "refs": [ "algebraization-lemma-kill-colimit-general" ], "ref_ids": [ 12714 ] } ], "ref_ids": [ 12718 ] }, { "id": 12720, "type": "theorem", "label": "algebraization-lemma-kill-colimit-support", "categories": [ "algebraization" ], "title": "algebraization-lemma-kill-colimit-support", "contents": [ "If in Lemma \\ref{lemma-kill-colimit-weak} we additionally assume", "\\begin{enumerate}", "\\item[(6)] if $\\mathfrak p \\not \\in V(I)$ and", "$V(\\mathfrak p) \\cap V(I) = \\{\\mathfrak m\\}$, then", "$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) > s$,", "\\end{enumerate}", "then $H^i_{J_0}(M) = H^i_J(M) = H^i_\\mathfrak m(M)$ for $i \\leq s$", "and these modules are annihilated by a power of $I$." ], "refs": [ "algebraization-lemma-kill-colimit-weak" ], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-kill-colimit-support-general}." ], "refs": [ "algebraization-lemma-kill-colimit-support-general" ], "ref_ids": [ 12715 ] } ], "ref_ids": [ 12718 ] }, { "id": 12721, "type": "theorem", "label": "algebraization-lemma-algebraize-local-cohomology", "categories": [ "algebraization" ], "title": "algebraization-lemma-algebraize-local-cohomology", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring.", "Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module.", "Let $s$ and $d$ be integers. Assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete and has a dualizing complex,", "\\item if $\\mathfrak p \\in V(I)$, no condition,", "\\item $\\text{cd}(A, I) \\leq d$,", "\\item if $\\mathfrak p \\not \\in V(I)$ and", "$V(\\mathfrak p) \\cap V(I) \\not = \\{\\mathfrak m\\}$ then", "$$", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s", "\\quad\\text{or}\\quad", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > d + s", "$$", "\\end{enumerate}", "Then there exists an ideal $J_0 \\subset A$ with", "$V(J_0) \\cap V(I) = \\{\\mathfrak m\\}$ such that for any $J \\subset J_0$ with", "$V(J) \\cap V(I) = \\{\\mathfrak m\\}$ the map", "$$", "R\\Gamma_J(M) \\longrightarrow", "R\\Gamma_J(M)^\\wedge = R\\Gamma_\\mathfrak m(M)^\\wedge", "$$", "induces an isomorphism in cohomology in degrees $\\leq s$.", "Here ${}^\\wedge$ denotes derived $I$-adic completion." ], "refs": [], "proofs": [ { "contents": [ "This lemma is a special case of", "Lemma \\ref{lemma-algebraize-local-cohomology-general}", "since condition (5)(c) is implied by condition (4)", "as $\\delta_{max} = \\delta_{min} = \\delta(\\mathfrak m)$.", "We will give the proof of this important special case", "as it is somewhat easier (fewer things to check).", "\\medskip\\noindent", "There is no difference between $R\\Gamma_\\mathfrak a$ and", "$R\\Gamma_{V(\\mathfrak a)}$ in our current situation, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-local-cohomology-noetherian}.", "Next, we observe that", "$$", "R\\Gamma_\\mathfrak m(M)^\\wedge =", "R\\Gamma_I(R\\Gamma_J(M))^\\wedge =", "R\\Gamma_J(M)^\\wedge", "$$", "by Dualizing Complexes, Lemmas \\ref{dualizing-lemma-local-cohomology-ss} and", "\\ref{dualizing-lemma-complete-and-local}", "which explains the equality sign in the statement of the lemma.", "\\medskip\\noindent", "Observe that the lemma holds for $s < 0$. This is not a trivial case because", "it is not a priori clear that $H^s(R\\Gamma_\\mathfrak m(M)^\\wedge)$", "is zero for negative $s$. However, this vanishing was established", "in Lemma \\ref{lemma-local-cohomology-derived-completion}.", "We will prove the lemma by induction for $s \\geq 0$.", "\\medskip\\noindent", "The assumptions of Lemma \\ref{lemma-kill-colimit-weak}", "are satisfied by Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-cd-bound-dim-local}.", "The lemma for $s = 0$ follows from Lemma \\ref{lemma-kill-colimit-weak} and", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-completion-local-H0}.", "\\medskip\\noindent", "Assume $s > 0$ and the lemma holds for smaller values of $s$.", "Let $M' \\subset M$ be the submodule of elements whose", "support is condained in $V(I) \\cup V(J)$ for some", "ideal $J$ with $V(J) \\cap V(I) = \\{\\mathfrak m\\}$.", "Then $M'$ is a finite $A$-module.", "We claim that", "$$", "R\\Gamma_J(M') \\to R\\Gamma_\\mathfrak m(M')^\\wedge", "$$", "is an isomorphism for any choice of $J$.", "Namely, for any such module there is a short exact sequence", "$0 \\to M_1 \\oplus M_2 \\to M' \\to N \\to 0$ with", "$M_1$ annihilated by a power of $J$, with $M_2$ annihilated", "by a power of $I$ and with $N$ annihilated by a power of $\\mathfrak m$.", "In the case of $M_1$ we see that $R\\Gamma_J(M_1) = M_1$ and", "since $M_1$ is a finite $A$-module and $I$-adically complete", "we have $M_1^\\wedge = M_1$. Thus the claim holds for $M_1$.", "In the case of $M_2$ we see that $H^i_J(M_2)$ is annihilated", "by a power of $I$ and hence derived complete. Thus the claim", "for $M_2$. By the same arguments the claim holds for $N$", "and we conclude that the claim holds. Considering the", "short exact sequence $0 \\to M' \\to M \\to M/M' \\to 0$", "we see that it suffices to prove the lemma for $M/M'$.", "This we may assume $\\mathfrak p \\in \\text{Ass}(M)$", "implies $V(\\mathfrak p) \\cap V(I) \\not = \\{\\mathfrak m\\}$, i.e.,", "$\\mathfrak p$ is a prime as in (4).", "\\medskip\\noindent", "Choose an ideal $J_0$ as in Lemma \\ref{lemma-kill-colimit-weak}", "and an integer $t > 0$ such that $(J_0I)^t$ annihilates $H^s_J(M)$.", "Here $J$ denotes an arbitrary ideal $J \\subset J_0$ with", "$V(J) \\cap V(I) = \\{\\mathfrak m\\}$.", "The assumptions of Lemma \\ref{lemma-kill-completion}", "are satisfied for every $\\mathfrak p \\in \\text{Ass}(M)$", "(see previous paragraph). Thus the annihilator $\\mathfrak a \\subset A$ of", "$H^s(R\\Gamma_\\mathfrak m(M)^\\wedge)$", "is not contained in $\\mathfrak p$ for $\\mathfrak p \\in \\text{Ass}(M)$.", "Thus we can find an $f \\in \\mathfrak a(J_0I)^t$", "not in any associated prime of $M$ which is an annihilator", "of both $H^s(R\\Gamma_\\mathfrak m(M)^\\wedge)$ and $H^s_J(M)$.", "Then $f$ is a nonzerodivisor on $M$ and we can consider the", "short exact sequence", "$$", "0 \\to M \\xrightarrow{f} M \\to M/fM \\to 0", "$$", "Our choice of $f$ shows that we obtain", "$$", "\\xymatrix{", "H^{s - 1}_J(M) \\ar[d] \\ar[r] &", "H^{s - 1}_J(M/fM) \\ar[d] \\ar[r] &", "H^s_J(M) \\ar[d] \\ar[r] & 0 \\\\", "H^{s - 1}(R\\Gamma_\\mathfrak m(M)^\\wedge) \\ar[r] &", "H^{s - 1}(R\\Gamma_\\mathfrak m(M/fM)^\\wedge) \\ar[r] &", "H^s(R\\Gamma_\\mathfrak m(M)^\\wedge) \\ar[r] & 0", "}", "$$", "for any $J \\subset J_0$ with $V(J) \\cap V(I) = \\{\\mathfrak m\\}$.", "Thus if we choose $J$ such that it works for", "$M$ and $M/fM$ and $s - 1$ (possible by induction hypothesis),", "then we conclude that the lemma is true." ], "refs": [ "algebraization-lemma-algebraize-local-cohomology-general", "dualizing-lemma-local-cohomology-noetherian", "dualizing-lemma-local-cohomology-ss", "dualizing-lemma-complete-and-local", "algebraization-lemma-local-cohomology-derived-completion", "algebraization-lemma-kill-colimit-weak", "local-cohomology-lemma-cd-bound-dim-local", "algebraization-lemma-kill-colimit-weak", "dualizing-lemma-completion-local-H0", "algebraization-lemma-kill-colimit-weak", "algebraization-lemma-kill-completion" ], "ref_ids": [ 12716, 2823, 2820, 2832, 12693, 12718, 9711, 12718, 2835, 12718, 12717 ] } ], "ref_ids": [] }, { "id": 12722, "type": "theorem", "label": "algebraization-lemma-helper-bootstrap", "categories": [ "algebraization" ], "title": "algebraization-lemma-helper-bootstrap", "contents": [ "In Situation \\ref{situation-bootstrap} let $\\mathfrak p \\subset \\mathfrak q$", "be primes of $A$ with $\\mathfrak p \\not \\in V(I)$ and", "$\\mathfrak q \\in T$. If there does not exist an", "$\\mathfrak r \\in V(I) \\setminus T$ with", "$\\mathfrak p \\subset \\mathfrak r \\subset \\mathfrak q$", "then $\\text{depth}(M_\\mathfrak p) > s$." ], "refs": [], "proofs": [ { "contents": [ "Choose $\\mathfrak q' \\in T$ with", "$\\mathfrak p \\subset \\mathfrak q' \\subset \\mathfrak q$", "such that there is no prime in $T$ strictly", "in between $\\mathfrak p$ and $\\mathfrak q'$. To prove the lemma", "we may and do replace $\\mathfrak q$ by $\\mathfrak q'$.", "Next, let $\\mathfrak p' \\subset A_\\mathfrak q$ be the prime corresponding to", "$\\mathfrak p$. After doing this we obtain that", "$V(\\mathfrak p') \\cap V(IA_\\mathfrak q) = \\{\\mathfrak q A_\\mathfrak q\\}$", "because of the nonexistence of a prime $\\mathfrak r$ as in the lemma.", "Let $A', I', \\mathfrak m', M'$ be the $I$-adic completions of", "$A_\\mathfrak q, I_\\mathfrak q, \\mathfrak qA_\\mathfrak q, M_\\mathfrak q$.", "Since $A_\\mathfrak q \\to A'$ is faithfully flat", "(Algebra, Lemma \\ref{algebra-lemma-completion-faithfully-flat})", "we can choose $\\mathfrak p'' \\subset A'$ lying over $\\mathfrak p'$", "with $\\dim(A'_{\\mathfrak p''}/\\mathfrak p' A'_{\\mathfrak p''}) = 0$.", "Then we see that", "$$", "\\text{depth}(M'_{\\mathfrak p''}) =", "\\text{depth}((M_\\mathfrak q \\otimes_{A_\\mathfrak q} A')_{\\mathfrak p''}) =", "\\text{depth}(M_\\mathfrak p \\otimes_{A_\\mathfrak p} A'_{\\mathfrak p''}) =", "\\text{depth}(M_\\mathfrak p)", "$$", "by flatness of $A \\to A'$ and our choice of $\\mathfrak p''$, see", "Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck-module}.", "Since $\\mathfrak p''$ lies over $\\mathfrak p'$ we have", "$V(\\mathfrak p'') \\cap V(I') = \\{\\mathfrak m'\\}$. Thus", "condition (6) in Situation \\ref{situation-bootstrap} implies", "$\\text{depth}(M'_{\\mathfrak p''}) > s$ which finishes the proof." ], "refs": [ "algebra-lemma-completion-faithfully-flat", "algebra-lemma-apply-grothendieck-module" ], "ref_ids": [ 871, 1360 ] } ], "ref_ids": [] }, { "id": 12723, "type": "theorem", "label": "algebraization-lemma-bootstrap-inherited", "categories": [ "algebraization" ], "title": "algebraization-lemma-bootstrap-inherited", "contents": [ "In Situation \\ref{situation-bootstrap} we have", "\\begin{enumerate}", "\\item[(E)] if $T' \\subset T$ is a smaller specialization stable subset, then", "$A, I, T', M$ satisfies the assumptions of Situation \\ref{situation-bootstrap},", "\\item[(F)] if $S \\subset A$ is a multiplicative subset, then", "$S^{-1}A, S^{-1}I, T', S^{-1}M$", "satisfies the assumptions of Situation \\ref{situation-bootstrap}", "where $T' \\subset V(S^{-1}I)$ is the inverse image of $T$,", "\\item[(G)] the quadruple $A', I', T', M'$", "satisfies the assumptions of Situation \\ref{situation-bootstrap}", "where $A', I', M'$ are the usual $I$-adic completions of $A, I, M$", "and $T' \\subset V(I')$ is the inverse image of $T$.", "\\end{enumerate}", "Let $I \\subset \\mathfrak a \\subset A$ be an ideal such that", "$V(\\mathfrak a) \\subset T$. Then", "\\begin{enumerate}", "\\item[(A)] if $I$ is contained in the Jacobson radical of $A$,", "then all hypotheses of", "Lemmas \\ref{lemma-kill-colimit-weak-general} and", "\\ref{lemma-kill-colimit-support-general} are satisfied", "for $A, I, \\mathfrak a, M$,", "\\item[(B)] if $A$ is complete with respect to $I$, then", "all hypotheses except for possibly (5) of", "Lemma \\ref{lemma-algebraize-local-cohomology-general}", "are satisfied for $A, I, \\mathfrak a, M$,", "\\item[(C)] if $A$ is local with maximal ideal $\\mathfrak m = \\mathfrak a$,", "then all hypotheses of", "Lemmas \\ref{lemma-kill-colimit-weak} and \\ref{lemma-kill-colimit-support}", "hold for $A, \\mathfrak m, I, M$,", "\\item[(D)] if $A$ is local with maximal ideal $\\mathfrak m = \\mathfrak a$", "and $I$-adically complete, then all hypotheses of", "Lemma \\ref{lemma-algebraize-local-cohomology}", "hold for $A, \\mathfrak m, I, M$,", "\\end{enumerate}" ], "refs": [ "algebraization-lemma-kill-colimit-weak-general", "algebraization-lemma-kill-colimit-support-general", "algebraization-lemma-algebraize-local-cohomology-general", "algebraization-lemma-kill-colimit-weak", "algebraization-lemma-kill-colimit-support", "algebraization-lemma-algebraize-local-cohomology" ], "proofs": [ { "contents": [ "Proof of (E). We have to prove assumptions (1), (3), (4), (6)", "of Situation \\ref{situation-bootstrap} hold for", "$A, I, T, M$. Shrinking $T$ to $T'$", "weakens assumption (6) and strengthens assumption (4). However, if we have", "$\\mathfrak p \\subset \\mathfrak r \\subset \\mathfrak q$ with", "$\\mathfrak p \\not \\in V(I)$, $\\mathfrak r \\in V(I) \\setminus T'$,", "$\\mathfrak q \\in T'$ as in assumption (4) for $A, I, T', M$, then", "either we can pick $\\mathfrak r \\in V(I) \\setminus T$ and", "condition (4) for $A, I, T, M$ kicks in or we cannot", "find such an $\\mathfrak r$ in which case we get", "$\\text{depth}(M_\\mathfrak p) > s$ by Lemma \\ref{lemma-helper-bootstrap}.", "This proves (4) holds for $A, I, T', M$ as desired.", "\\medskip\\noindent", "Proof of (F). This is straightforward and we omit the details.", "\\medskip\\noindent", "Proof of (G). We have to prove assumptions (1), (3), (4), (6)", "of Situation \\ref{situation-bootstrap} hold for the $I$-adic", "completions $A', I', T', M'$. Please keep in mind that", "$\\Spec(A') \\to \\Spec(A)$ induces an isomorphism $V(I') \\to V(I)$.", "\\medskip\\noindent", "Assumption (1): The ring $A'$ has a dualizing complex, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-ubiquity-dualizing}.", "\\medskip\\noindent", "Assumption (3): Since $I' = IA'$ this follows from Local Cohomology,", "Lemma \\ref{local-cohomology-lemma-cd-change-rings}.", "\\medskip\\noindent", "Assumption (4): If we have primes", "$\\mathfrak p' \\subset \\mathfrak r' \\subset \\mathfrak q'$ in $A'$", "with $\\mathfrak p' \\not \\in V(I')$,", "$\\mathfrak r' \\in V(I') \\setminus T'$,", "$\\mathfrak q' \\in T'$ then their images", "$\\mathfrak p \\subset \\mathfrak r \\subset \\mathfrak q$ in", "the spectrum of $A$", "satisfy", "$\\mathfrak p \\not \\in V(I)$, $\\mathfrak r \\in V(I) \\setminus T$,", "$\\mathfrak q \\in T$.", "Then we have", "$$", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s", "\\quad\\text{or}\\quad", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > d + s", "$$", "by assumption (4) for $A, I, T, M$. We have", "$\\text{depth}(M'_{\\mathfrak p'}) \\geq \\text{depth}(M_\\mathfrak p)$ and", "$\\text{depth}(M'_{\\mathfrak p'}) +", "\\dim((A'/\\mathfrak p')_{\\mathfrak q'}) =", "\\text{depth}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q)$", "by Local Cohomology, Lemma \\ref{local-cohomology-lemma-change-completion}.", "Thus assumption (4) holds for $A', I', T', M'$.", "\\medskip\\noindent", "Assumption (6): Let $\\mathfrak q' \\in T'$ lying over the", "prime $\\mathfrak q \\in T$. Then $A'_{\\mathfrak q'}$", "and $A_\\mathfrak q$ have isomorphic $I$-adic completions", "and similarly for $M_\\mathfrak q$ and $M'_{\\mathfrak q'}$.", "Thus assumption (6) for $A', I', T', M'$ is equivalent", "to assumption (6) for $A, I, T, M$.", "\\medskip\\noindent", "Proof of (A). We have to check conditions (1), (2), (3), (4), and (6)", "of Lemmas \\ref{lemma-kill-colimit-weak-general} and", "\\ref{lemma-kill-colimit-support-general} for", "$(A, I, \\mathfrak a, M)$. Warning: the set $T$ in the statement of", "these lemmas is not the same as the set $T$ above.", "\\medskip\\noindent", "Condition (1): This holds because we have assumed $A$ has a dualizing complex in", "Situation \\ref{situation-bootstrap}.", "\\medskip\\noindent", "Condition (2): This is empty.", "\\medskip\\noindent", "Condition (3): Let $\\mathfrak p \\subset A$ with", "$V(\\mathfrak p) \\cap V(I) \\subset V(\\mathfrak a)$.", "Since $I$ is contained in the Jacobson radical of $A$ we see", "that $V(\\mathfrak p) \\cap V(I) \\not = \\emptyset$.", "Let $\\mathfrak q \\in V(\\mathfrak p) \\cap V(I)$ be a generic point.", "Since $\\text{cd}(A_\\mathfrak q, I_\\mathfrak q) \\leq d$", "(Local Cohomology, Lemma \\ref{local-cohomology-lemma-cd-local}) and since", "$V(\\mathfrak p A_\\mathfrak q) \\cap V(I_\\mathfrak q) =", "\\{\\mathfrak q A_\\mathfrak q\\}$ we get", "$\\dim((A/\\mathfrak p)_\\mathfrak q) \\leq d$ by Local Cohomology,", "Lemma \\ref{local-cohomology-lemma-cd-bound-dim-local} which proves (3).", "\\medskip\\noindent", "Condition (4): Suppose $\\mathfrak p \\not \\in V(I)$ and", "$\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$.", "It suffices to show", "$$", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s", "\\quad\\text{or}\\quad", "\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > d + s", "$$", "If there exists a prime $\\mathfrak p \\subset \\mathfrak r \\subset \\mathfrak q$", "with $\\mathfrak r \\in V(I) \\setminus T$, then this follows", "immediately from assumption (4) in Situation \\ref{situation-bootstrap}.", "If not, then $\\text{depth}(M_\\mathfrak p) > s$ by", "Lemma \\ref{lemma-helper-bootstrap}.", "\\medskip\\noindent", "Condition (6): Let $\\mathfrak p \\not \\in V(I)$ with", "$V(\\mathfrak p) \\cap V(I) \\subset V(\\mathfrak a)$.", "Since $I$ is contained in the Jacobson radical of $A$ we see", "that $V(\\mathfrak p) \\cap V(I) \\not = \\emptyset$.", "Choose $\\mathfrak q \\in V(\\mathfrak p) \\cap V(I) \\subset V(\\mathfrak a)$.", "It is clear there does not exist a prime", "$\\mathfrak p \\subset \\mathfrak r \\subset \\mathfrak q$", "with $\\mathfrak r \\in V(I) \\setminus T$.", "By Lemma \\ref{lemma-helper-bootstrap} we have", "$\\text{depth}(M_\\mathfrak p) > s$ which proves (6).", "\\medskip\\noindent", "Proof of (B). We have to check conditions (1), (2), (3), (4) of", "Lemma \\ref{lemma-algebraize-local-cohomology-general}. Warning:", "the set $T$ in the statement of", "this lemma is not the same as the set $T$ above.", "\\medskip\\noindent", "Condition (1): This holds because $A$ is complete and has a dualizing complex.", "\\medskip\\noindent", "Condition (2): This is empty.", "\\medskip\\noindent", "Condition (3): This is the same as assumption (3) in", "Situation \\ref{situation-bootstrap}.", "\\medskip\\noindent", "Condition (4): This is the same as assumption (4) in", "Lemma \\ref{lemma-kill-colimit-weak-general} which we proved in (A).", "\\medskip\\noindent", "Proof of (C). This is true because the assumptions in", "Lemmas \\ref{lemma-kill-colimit-weak} and \\ref{lemma-kill-colimit-support}", "are the same as the assumptions in", "Lemmas \\ref{lemma-kill-colimit-weak-general} and", "\\ref{lemma-kill-colimit-support-general} in the local case", "and we proved these hold in (A).", "\\medskip\\noindent", "Proof of (D). This is true because the assumptions in", "Lemma \\ref{lemma-algebraize-local-cohomology}", "are the same as the assumptions (1), (2), (3), (4) in", "Lemma \\ref{lemma-algebraize-local-cohomology-general}", "and we proved these hold in (B)." ], "refs": [ "algebraization-lemma-helper-bootstrap", "dualizing-lemma-ubiquity-dualizing", "local-cohomology-lemma-cd-change-rings", "local-cohomology-lemma-change-completion", "algebraization-lemma-kill-colimit-weak-general", "algebraization-lemma-kill-colimit-support-general", "local-cohomology-lemma-cd-local", "local-cohomology-lemma-cd-bound-dim-local", "algebraization-lemma-helper-bootstrap", "algebraization-lemma-helper-bootstrap", "algebraization-lemma-algebraize-local-cohomology-general", "algebraization-lemma-kill-colimit-weak-general", "algebraization-lemma-kill-colimit-weak", "algebraization-lemma-kill-colimit-support", "algebraization-lemma-kill-colimit-weak-general", "algebraization-lemma-kill-colimit-support-general", "algebraization-lemma-algebraize-local-cohomology", "algebraization-lemma-algebraize-local-cohomology-general" ], "ref_ids": [ 12722, 2890, 9706, 9740, 12713, 12715, 9707, 9711, 12722, 12722, 12716, 12713, 12718, 12720, 12713, 12715, 12721, 12716 ] } ], "ref_ids": [ 12713, 12715, 12716, 12718, 12720, 12721 ] }, { "id": 12724, "type": "theorem", "label": "algebraization-lemma-algebraize-local-cohomology-bis", "categories": [ "algebraization" ], "title": "algebraization-lemma-algebraize-local-cohomology-bis", "contents": [ "In Situation \\ref{situation-bootstrap} assume $A$ is local with", "maximal ideal $\\mathfrak m$ and $T = \\{\\mathfrak m\\}$. Then", "$H^i_\\mathfrak m(M) \\to \\lim H^i_\\mathfrak m(M/I^nM)$", "is an isomorphism for $i \\leq s$ and these modules are", "annihilated by a power of $I$." ], "refs": [], "proofs": [ { "contents": [ "Let $A', I', \\mathfrak m', M'$ be the usual $I$-adic completions", "of $A, I, \\mathfrak m, M$. Recall that we have", "$H^i_\\mathfrak m(M) \\otimes_A A' = H^i_{\\mathfrak m'}(M')$", "by flatness of $A \\to A'$ and Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-torsion-change-rings}.", "Since $H^i_\\mathfrak m(M)$ is $\\mathfrak m$-power torsion we have", "$H^i_\\mathfrak m(M) = H^i_\\mathfrak m(M) \\otimes_A A'$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-neighbourhood-equivalence}.", "We conclude that $H^i_\\mathfrak m(M) = H^i_{\\mathfrak m'}(M')$.", "The exact same arguments will show that", "$H^i_\\mathfrak m(M/I^nM) = H^i_{\\mathfrak m'}(M'/(I')^nM')$", "for all $n$ and $i$.", "\\medskip\\noindent", "Lemmas \\ref{lemma-algebraize-local-cohomology},", "\\ref{lemma-kill-colimit-weak}, and", "\\ref{lemma-kill-colimit-support}", "apply to $A', \\mathfrak m', I', M'$ by", "Lemma \\ref{lemma-bootstrap-inherited} parts (C) and (D).", "Thus we get an isomorphism", "$$", "H^i_{\\mathfrak m'}(M') \\longrightarrow H^i(R\\Gamma_{\\mathfrak m'}(M')^\\wedge)", "$$", "for $i \\leq s$ where ${}^\\wedge$ is derived $I'$-adic completion and these", "modules are annihilated by a power of $I'$.", "By Lemma \\ref{lemma-local-cohomology-derived-completion}", "we obtain isomorphisms", "$$", "H^i_{\\mathfrak m'}(M') \\longrightarrow", "\\lim H^i_{\\mathfrak m'}(M'/(I')^nM'))", "$$", "for $i \\leq s$. Combined with the already established comparison", "with local cohomology over $A$ we conclude the lemma is true." ], "refs": [ "dualizing-lemma-torsion-change-rings", "more-algebra-lemma-neighbourhood-equivalence", "algebraization-lemma-algebraize-local-cohomology", "algebraization-lemma-kill-colimit-weak", "algebraization-lemma-kill-colimit-support", "algebraization-lemma-bootstrap-inherited", "algebraization-lemma-local-cohomology-derived-completion" ], "ref_ids": [ 2817, 10341, 12721, 12718, 12720, 12723, 12693 ] } ], "ref_ids": [] }, { "id": 12725, "type": "theorem", "label": "algebraization-lemma-bootstrap-bis-bis", "categories": [ "algebraization" ], "title": "algebraization-lemma-bootstrap-bis-bis", "contents": [ "Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.", "Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers.", "If we assume", "\\begin{enumerate}", "\\item[(a)] $A$ has a dualizing complex,", "\\item[(b)] $\\text{cd}(A, I) \\leq d$,", "\\item[(c)] if $\\mathfrak p \\not \\in V(I)$ and", "$\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$ then", "$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) > s$ or", "$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q) > d + s$.", "\\end{enumerate}", "Then $A, I, V(\\mathfrak a), M, s, d$ are as in", "Situation \\ref{situation-bootstrap}." ], "refs": [], "proofs": [ { "contents": [ "We have to show that assumptions (1), (3), (4), and (6) of", "Situation \\ref{situation-bootstrap} hold.", "It is clear that (a) $\\Rightarrow$ (1),", "(b) $\\Rightarrow$ (3), and (c) $\\Rightarrow$ (4).", "To finish the proof in the next paragraph we show (6) holds.", "\\medskip\\noindent", "Let $\\mathfrak q \\in V(\\mathfrak a)$.", "Denote $A', I', \\mathfrak m', M'$", "the $I$-adic completions of", "$A_\\mathfrak q, I_\\mathfrak q, \\mathfrak qA_\\mathfrak q, M_\\mathfrak q$.", "Let $\\mathfrak p' \\subset A'$ be a nonmaximal prime with", "$V(\\mathfrak p') \\cap V(I') = \\{\\mathfrak m'\\}$.", "Observe that this implies $\\dim(A'/\\mathfrak p') \\leq d$", "by Local Cohomology, Lemma \\ref{local-cohomology-lemma-cd-bound-dim-local}.", "Denote $\\mathfrak p \\subset A$ the image of $\\mathfrak p'$.", "We have", "$\\text{depth}(M'_{\\mathfrak p'}) \\geq \\text{depth}(M_\\mathfrak p)$ and", "$\\text{depth}(M'_{\\mathfrak p'}) +", "\\dim(A'/\\mathfrak p') =", "\\text{depth}(M_\\mathfrak p) +", "\\dim((A/\\mathfrak p)_\\mathfrak q)$", "by Local Cohomology, Lemma \\ref{local-cohomology-lemma-change-completion}.", "By assumption (c) either we have", "$\\text{depth}(M'_{\\mathfrak p'}) \\geq \\text{depth}(M_\\mathfrak p) > s$", "and we're done or we have", "$\\text{depth}(M'_{\\mathfrak p'}) +", "\\dim(A'/\\mathfrak p') > s + d$ which implies", "$\\text{depth}(M'_{\\mathfrak p'}) > s$ because of the already shown", "inequality $\\dim(A'/\\mathfrak p') \\leq d$. In both cases we", "obtain what we want." ], "refs": [ "local-cohomology-lemma-cd-bound-dim-local", "local-cohomology-lemma-change-completion" ], "ref_ids": [ 9711, 9740 ] } ], "ref_ids": [] }, { "id": 12726, "type": "theorem", "label": "algebraization-lemma-bootstrap", "categories": [ "algebraization" ], "title": "algebraization-lemma-bootstrap", "contents": [ "In Situation \\ref{situation-bootstrap} the inverse systems", "$\\{H^i_T(I^nM)\\}_{n \\geq 0}$ are pro-zero for $i \\leq s$.", "Moreover, there exists an integer $m_0$ such that for all", "$m \\geq m_0$ there exists an integer $m'(m) \\geq m$ such that for", "$k \\geq m'(m)$ the image of", "$H^{s + 1}_T(I^kM) \\to H^{s + 1}_T(I^mM)$", "maps injectively to $H^{s + 1}_T(I^{m_0}M)$." ], "refs": [], "proofs": [ { "contents": [ "Fix $m$. Let $\\mathfrak q \\in T$.", "By Lemmas \\ref{lemma-bootstrap-inherited} and", "\\ref{lemma-algebraize-local-cohomology-bis}", "we see that", "$$", "H^i_\\mathfrak q(M_\\mathfrak q)", "\\longrightarrow", "\\lim H^i_\\mathfrak q(M_\\mathfrak q/I^nM_\\mathfrak q)", "$$", "is an isomorphism for $i \\leq s$. The inverse systems", "$\\{H^i_\\mathfrak q(I^nM_\\mathfrak q)\\}_{n \\geq 0}$ and", "$\\{H^i_\\mathfrak q(M/I^nM)\\}_{n \\geq 0}$", "satisfy the Mittag-Leffler condition for all $i$, see", "Lemma \\ref{lemma-ML-local}. Thus looking at the inverse system of", "long exact sequences", "$$", "0 \\to H^0_\\mathfrak q(I^nM_\\mathfrak q) \\to", "H^0_\\mathfrak q(M_\\mathfrak q) \\to", "H^0_\\mathfrak q(M_\\mathfrak q/I^nM_\\mathfrak q) \\to", "H^1_\\mathfrak q(I^nM_\\mathfrak q) \\to", "H^1_\\mathfrak q(M_\\mathfrak q) \\to \\ldots", "$$", "we conclude (some details omitted) that there exists an integer", "$m'(m, \\mathfrak q) \\geq m$ such that for all $k \\geq m'(m, \\mathfrak q)$", "the map", "$H^i_\\mathfrak q(I^kM_\\mathfrak q) \\to H^i_\\mathfrak q(I^mM_\\mathfrak q)$", "is zero for $i \\leq s$ and the image of", "$H^{s + 1}_\\mathfrak q(I^kM_\\mathfrak q) \\to", "H^{s + 1}_\\mathfrak q(I^mM_\\mathfrak q)$", "is independent of $k \\geq m'(m, \\mathfrak q)$ and", "maps injectively into $H^{s + 1}_\\mathfrak q(M_\\mathfrak q)$.", "\\medskip\\noindent", "Suppose we can show that $m'(m, \\mathfrak q)$ can be chosen", "independently of $\\mathfrak q \\in T$.", "Then the lemma follows immediately from", "Local Cohomology, Lemmas \\ref{local-cohomology-lemma-zero} and", "\\ref{local-cohomology-lemma-essential-image}.", "\\medskip\\noindent", "Let $\\omega_A^\\bullet$ be a dualizing complex. Let", "$\\delta : \\Spec(A) \\to \\mathbf{Z}$ be the corresponding", "dimension function. Recall that $\\delta$ attains only a", "finite number of values, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-universally-catenary}.", "Claim: for each $d \\in \\mathbf{Z}$ the integer", "$m'(m, \\mathfrak q)$ can be chosen independently", "of $\\mathfrak q \\in T$ with $\\delta(\\mathfrak q) = d$.", "Clearly the claim implies the lemma by what we said above.", "\\medskip\\noindent", "Pick $\\mathfrak q \\in T$ with $\\delta(\\mathfrak q) = d$.", "Consider the ext modules", "$$", "E(n, j) = \\text{Ext}^j_A(I^nM, \\omega_A^\\bullet)", "$$", "A key feature we will use is that these are finite $A$-modules.", "Recall that $(\\omega_A^\\bullet)_\\mathfrak q[-d]$ is a normalized", "dualizing complex for $A_\\mathfrak q$ by definition of the", "dimension function associated to a dualizing complex, see", "Dualizing Complexes, Section \\ref{dualizing-section-dimension-function}.", "The local duality theorem (Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-special-case-local-duality}) tells us that", "the $\\mathfrak qA_\\mathfrak q$-adic completion of", "$E(n, -d - i)_\\mathfrak q$ is Matlis dual to", "$H^i_\\mathfrak q(I^nM_\\mathfrak q)$. Thus the choice of", "$m'(m, \\mathfrak q)$ for $i \\leq s$ in the first paragraph tells us that", "for $k \\geq m'(m, \\mathfrak q)$ and $j \\geq -d - s$ the map", "$$", "E(m, j)_\\mathfrak q \\to E(k, j)_\\mathfrak q", "$$", "is zero. Since these modules are finite and nonzero only", "for a finite number of possible $j$ (small detail omitted),", "we can find an open neighbourhood $W \\subset \\Spec(A)$ of $\\mathfrak q$", "such that", "$$", "E(m, j)_{\\mathfrak q'} \\to E(m'(m, \\mathfrak q), j)_{\\mathfrak q'}", "$$", "is zero for $j \\geq -d - s$ for all $\\mathfrak q' \\in W$.", "Then of course the maps $E(m, j)_{\\mathfrak q'} \\to E(k, j)_{\\mathfrak q'}$", "for $k \\geq m'(m, \\mathfrak q)$ are zero as well.", "\\medskip\\noindent", "For $i = s + 1$ corresponding to $j = - d - s - 1$ we obtain", "from local duality and the results of the first paragraph that", "$$", "K_{k, \\mathfrak q} =", "\\Ker(E(m, -d - s - 1)_\\mathfrak q \\to E(k, -d - s - 1)_\\mathfrak q)", "$$", "is independent of $k \\geq m'(m, \\mathfrak q)$ and that", "$$", "E(0, -d - s - 1)_\\mathfrak q \\to", "E(m, -d - s - 1)_\\mathfrak q/K_{m'(m, \\mathfrak q), \\mathfrak q}", "$$", "is surjective. For $k \\geq m'(m, \\mathfrak q)$ set", "$$", "K_k = \\Ker(E(m, -d - s - 1) \\to E(k, -d - s - 1))", "$$", "Since $K_k$ is an increasing sequence of submodules of the finite", "module $E(m, -d - s - 1)$ we see that, at the cost of increasing", "$m'(m, \\mathfrak q)$ a little bit, we may assume", "$K_{m'(m, \\mathfrak q)} = K_k$ for $k \\geq m'(m, \\mathfrak q)$.", "After shrinking $W$ further if necessary, we may also assume that", "$$", "E(0, -d - s - 1)_{\\mathfrak q'} \\to", "E(m, -d - s - 1)_{\\mathfrak q'}/K_{m'(m, \\mathfrak q), \\mathfrak q'}", "$$", "is surjective for all $\\mathfrak q' \\in W$ (as before use that", "these modules are finite", "and that the map is surjective after localization at $\\mathfrak q$).", "\\medskip\\noindent", "Any subset, in particular", "$T_d = \\{\\mathfrak q \\in T \\text{ with }\\delta(\\mathfrak q) = d\\}$,", "of the Noetherian topological space $\\Spec(A)$", "with the endowed topology is Noetherian and hence quasi-compact.", "Above we have seen that for every $\\mathfrak q \\in T_d$", "there is an open neighbourhood $W$ where", "$m'(m, \\mathfrak q)$ works for all $\\mathfrak q' \\in T_d \\cap W$.", "We conclude that we can find an integer $m'(m, d)$ such that for all", "$\\mathfrak q \\in T_d$ we have", "$$", "E(m, j)_\\mathfrak q \\to E(m'(m, d), j)_\\mathfrak q", "$$", "is zero for $j \\geq -d - s$ and with", "$K_{m'(m, d)} = \\Ker(E(m, -d - s - 1) \\to E(m'(m, d), -d - s - 1))$", "we have", "$$", "K_{m'(m, d), \\mathfrak q} =", "\\Ker(E(m, -d - s - 1)_{\\mathfrak q} \\to E(k, -d - s - 1)_{\\mathfrak q})", "$$", "for all $k \\geq m'(m, d)$ and the map", "$$", "E(0, -d - s - 1)_\\mathfrak q \\to", "E(m, -d - s - 1)_\\mathfrak q/K_{m'(m, d), \\mathfrak q}", "$$", "is surjective. Using the local duality theorem again (in the opposite", "direction) we conclude that the claim is correct. This finishes the proof." ], "refs": [ "algebraization-lemma-bootstrap-inherited", "algebraization-lemma-algebraize-local-cohomology-bis", "algebraization-lemma-ML-local", "local-cohomology-lemma-zero", "local-cohomology-lemma-essential-image", "dualizing-lemma-universally-catenary", "dualizing-lemma-special-case-local-duality" ], "ref_ids": [ 12723, 12724, 12691, 9722, 9723, 2870, 2873 ] } ], "ref_ids": [] }, { "id": 12727, "type": "theorem", "label": "algebraization-lemma-final-bootstrap", "categories": [ "algebraization" ], "title": "algebraization-lemma-final-bootstrap", "contents": [ "In Situation \\ref{situation-bootstrap} there exists an integer $m_0 \\geq 0$", "such that", "\\begin{enumerate}", "\\item $\\{H^i_T(M/I^nM)\\}_{n \\geq 0}$", "satisfies the Mittag-Leffler condition for $i < s$.", "\\item $\\{H^i_T(I^{m_0}M/I^nM)\\}_{n \\geq m_0}$", "satisfies the Mittag-Leffler condition for $i \\leq s$,", "\\item $H^i_T(M) \\to \\lim H^i_T(M/I^nM)$", "is an isomorphism for $i < s$,", "\\item $H^s_T(I^{m_0}M) \\to \\lim H^s_T(I^{m_0}M/I^nM)$", "is an isomorphism for $i \\leq s$,", "\\item $H^s_T(M) \\to \\lim H^s_T(M/I^nM)$ is", "injective with cokernel killed by $I^{m_0}$, and", "\\item $R^1\\lim H^s_T(M/I^nM)$ is killed by $I^{m_0}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the long exact sequences", "$$", "0 \\to H^0_T(I^nM) \\to H^0_T(M) \\to", "H^0_T(M/I^nM) \\to H^1_T(I^nM) \\to", "H^1_T(M) \\to \\ldots", "$$", "Parts (1) and (3) follows easily from this and Lemma \\ref{lemma-bootstrap}.", "\\medskip\\noindent", "Let $m_0$ and $m'(-)$ be as in Lemma \\ref{lemma-bootstrap}.", "For $m \\geq m_0$ consider the long exact sequence", "$$", "H^s_T(I^mM) \\to H^s_T(I^{m_0}M) \\to", "H^s_T(I^{m_0}M/I^mM) \\to H^{s + 1}_T(I^mM) \\to", "H^1_T(I^{m_0}M)", "$$", "Then for $k \\geq m'(m)$ the image of", "$H^{s + 1}_T(I^kM) \\to H^{s + 1}_T(I^mM)$", "maps injectively to $H^{s + 1}_T(I^{m_0}M)$.", "Hence the image of", "$H^s_T(I^{m_0}M/I^kM) \\to H^s_T(I^{m_0}M/I^mM)$", "maps to zero in $H^{s + 1}_T(I^mM)$ for all $k \\geq m'(m)$.", "We conclude that (2) and (4) hold.", "\\medskip\\noindent", "Consider the short exact sequences", "$0 \\to I^{m_0}M \\to M \\to M/I^{m_0} M \\to 0$ and", "$0 \\to I^{m_0}M/I^nM \\to M/I^nM \\to M/I^{m_0} M \\to 0$.", "We obtain a diagram", "$$", "\\xymatrix{", "H^{s - 1}_T(M/I^{m_0}M) \\ar[r] &", "\\lim H^s_T(I^{m_0}M/I^nM) \\ar[r] &", "\\lim H^s_T(M/I^nM) \\ar[r] &", "H^s_T(M/I^{m_0}M) \\\\", "H^{s - 1}_T(M/I^{m_0}M) \\ar[r] \\ar@{=}[u] &", "H^s_T(I^{m_0}M) \\ar[r] \\ar[u]_{\\cong} &", "H^s_T(M) \\ar[r] \\ar[u] &", "H^s_T(M/I^{m_0}M) \\ar@{=}[u]", "}", "$$", "whose lower row is exact. The top row is also exact", "(at the middle two spots) by", "Homology, Lemma \\ref{homology-lemma-apply-Mittag-Leffler}.", "Part (5) follows.", "\\medskip\\noindent", "Write $B_n = H^s_T(M/I^nM)$. Let $A_n \\subset B_n$", "be the image of $H^s_T(I^{m_0}M/I^nM) \\to H^s_T(M/I^nM)$.", "Then $(A_n)$ satisfies the Mittag-Leffler condition by (2) and", "Homology, Lemma \\ref{homology-lemma-Mittag-Leffler}.", "Also $C_n = B_n/A_n$ is killed by $I^{m_0}$. Thus", "$R^1\\lim B_n \\cong R^1\\lim C_n$ is killed by $I^{m_0}$ and we get (6)." ], "refs": [ "algebraization-lemma-bootstrap", "algebraization-lemma-bootstrap", "homology-lemma-apply-Mittag-Leffler", "homology-lemma-Mittag-Leffler" ], "ref_ids": [ 12726, 12726, 12125, 12124 ] } ], "ref_ids": [] }, { "id": 12728, "type": "theorem", "label": "algebraization-lemma-combine-two", "categories": [ "algebraization" ], "title": "algebraization-lemma-combine-two", "contents": [ "Let $I \\subset \\mathfrak a \\subset A$ be ideals of a Noetherian ring $A$", "and let $M$ be a finite $A$-module. Let $s$ and $d$ be integers.", "Suppose that", "\\begin{enumerate}", "\\item $A, I, V(\\mathfrak a), M$ satisfy the assumptions of", "Situation \\ref{situation-bootstrap} for $s$ and $d$, and", "\\item $A, I, \\mathfrak a, M$ satisfy the conditions of", "Lemma \\ref{lemma-algebraize-local-cohomology-general}", "for $s + 1$ and $d$ with $J = \\mathfrak a$.", "\\end{enumerate}", "Then there exists an ideal", "$J_0 \\subset \\mathfrak a$ with $V(J_0) \\cap V(I) = V(\\mathfrak a)$", "such that for any $J \\subset J_0$ with $V(J) \\cap V(I) = V(\\mathfrak a)$", "the map", "$$", "H^{s + 1}_J(M) \\longrightarrow \\lim H^{s + 1}_\\mathfrak a(M/I^nM)", "$$", "is an isomorphism." ], "refs": [ "algebraization-lemma-algebraize-local-cohomology-general" ], "proofs": [ { "contents": [ "Namely, we have the existence of $J_0$", "and the isomorphism", "$H^{s + 1}_J(M) = H^{s + 1}(R\\Gamma_\\mathfrak a(M)^\\wedge)$", "by Lemma \\ref{lemma-algebraize-local-cohomology-general},", "we have a short exact sequence", "$$", "0 \\to R^1\\lim H^s_\\mathfrak a(M/I^nM) \\to", "H^{s + 1}(R\\Gamma_\\mathfrak a(M)^\\wedge) \\to", "\\lim H^{s + 1}_\\mathfrak a(M/I^nM) \\to 0", "$$", "by Dualizing Complexes, Lemma \\ref{dualizing-lemma-completion-local},", "and the module $R^1\\lim H^s_\\mathfrak a(M/I^nM)$ is zero because", "$\\{H^s_\\mathfrak a(M/I^nM)\\}_{n \\geq 0}$ has Mittag-Leffler", "by Theorem \\ref{theorem-final-bootstrap}." ], "refs": [ "algebraization-lemma-algebraize-local-cohomology-general", "dualizing-lemma-completion-local", "algebraization-theorem-final-bootstrap" ], "ref_ids": [ 12716, 2834, 12673 ] } ], "ref_ids": [ 12716 ] }, { "id": 12729, "type": "theorem", "label": "algebraization-lemma-compare-with-derived-completion", "categories": [ "algebraization" ], "title": "algebraization-lemma-compare-with-derived-completion", "contents": [ "Let $U$ be the punctured spectrum of a Noetherian local ring $A$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module.", "Let $I \\subset A$ be an ideal. Then", "$$", "H^i(R\\Gamma(U, \\mathcal{F})^\\wedge) =", "\\lim H^i(U, \\mathcal{F}/I^n\\mathcal{F})", "$$", "for all $i$ where $R\\Gamma(U, \\mathcal{F})^\\wedge$ denotes", "the derived $I$-adic completion." ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-formal-functions-general} and", "\\ref{lemma-derived-completion-pseudo-coherent} we have", "$$", "R\\Gamma(U, \\mathcal{F})^\\wedge =", "R\\Gamma(U, \\mathcal{F}^\\wedge) =", "R\\Gamma(U, R\\lim \\mathcal{F}/I^n\\mathcal{F})", "$$", "Thus we obtain short exact sequences", "$$", "0 \\to R^1\\lim H^{i - 1}(U, \\mathcal{F}/I^n\\mathcal{F}) \\to", "H^i(R\\Gamma(U, \\mathcal{F})^\\wedge) \\to", "\\lim H^i(U, \\mathcal{F}/I^n\\mathcal{F}) \\to 0", "$$", "by Cohomology, Lemma \\ref{cohomology-lemma-RGamma-commutes-with-Rlim}.", "The $R^1\\lim$ terms vanish because the inverse systems of groups", "$H^i(U, \\mathcal{F}/I^n\\mathcal{F})$ satisfy the Mittag-Leffler condition", "by Lemma \\ref{lemma-ML-local}." ], "refs": [ "algebraization-lemma-formal-functions-general", "algebraization-lemma-derived-completion-pseudo-coherent", "cohomology-lemma-RGamma-commutes-with-Rlim", "algebraization-lemma-ML-local" ], "ref_ids": [ 12708, 12710, 2160, 12691 ] } ], "ref_ids": [] }, { "id": 12730, "type": "theorem", "label": "algebraization-lemma-application-theorem", "categories": [ "algebraization" ], "title": "algebraization-lemma-application-theorem", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring which has a", "dualizing complex and is complete with respect to an ideal $I$.", "Set $X = \\Spec(A)$, $Y = V(I)$, and $U = X \\setminus \\{\\mathfrak m\\}$.", "Let $\\mathcal{F}$ be a coherent sheaf on $U$.", "Assume for any associated point $x \\in U$ of $\\mathcal{F}$", "we have $\\dim(\\overline{\\{x\\}}) > \\text{cd}(A, I) + 1$", "where $\\overline{\\{x\\}}$ is the closure in $X$.", "Then the map", "$$", "\\colim H^0(V, \\mathcal{F})", "\\longrightarrow", "\\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})", "$$", "is an isomorphism of finite $A$-modules", "where the colimit is over opens $V \\subset U$", "containing $U \\cap Y$." ], "refs": [], "proofs": [ { "contents": [ "Apply Theorem \\ref{theorem-algebraization-formal-sections} with $s = 1$", "(we get finiteness too)." ], "refs": [ "algebraization-theorem-algebraization-formal-sections" ], "ref_ids": [ 12674 ] } ], "ref_ids": [] }, { "id": 12731, "type": "theorem", "label": "algebraization-lemma-application-H0-pre", "categories": [ "algebraization" ], "title": "algebraization-lemma-application-H0-pre", "contents": [ "Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.", "Let $\\mathcal{F}$ be a coherent module on", "$U = \\Spec(A) \\setminus V(\\mathfrak a)$.", "Assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete and has a dualizing complex,", "\\item if $x \\in \\text{Ass}(\\mathcal{F})$, $x \\not \\in V(I)$,", "$\\overline{\\{x\\}} \\cap V(I) \\not \\subset V(\\mathfrak a)$", "and $z \\in \\overline{\\{x\\}} \\cap V(\\mathfrak a)$, then", "$\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + 1$,", "\\item one of the following holds:", "\\begin{enumerate}", "\\item the restriction of $\\mathcal{F}$ to $U \\setminus V(I)$ is $(S_1)$", "\\item the dimension of $V(\\mathfrak a)$ is at most $2$\\footnote{In", "the sense that the difference of the maximal and minimal values", "on $V(\\mathfrak a)$ of a dimension function on $\\Spec(A)$ is at most $2$.}.", "\\end{enumerate}", "\\end{enumerate}", "Then we obtain an isomorphism", "$$", "\\colim H^0(V, \\mathcal{F})", "\\longrightarrow", "\\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})", "$$", "where the colimit is over opens $V \\subset U$ containing $U \\cap V(I)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite $A$-module $M$ such that $\\mathcal{F}$ is the restriction", "to $U$ of the coherent module associated to $M$, see Local Cohomology,", "Lemma \\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}.", "Set $d = \\text{cd}(A, I)$.", "Let $\\mathfrak p$ be a prime of $A$ not contained in $V(I)$", "and let $\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$.", "Then either $\\mathfrak p$ is not an associated prime of $M$", "and hence $\\text{depth}(M_\\mathfrak p) \\geq 1$", "or we have $\\dim((A/\\mathfrak p)_\\mathfrak q) > d + 1$ by (2).", "Thus the hypotheses of", "Lemma \\ref{lemma-algebraize-local-cohomology-general}", "are satisfied for $s = 1$ and $d$; here we use condition (3).", "Thus we find there exists an ideal", "$J_0 \\subset \\mathfrak a$ with $V(J_0) \\cap V(I) = V(\\mathfrak a)$", "such that for any $J \\subset J_0$ with $V(J) \\cap V(I) = V(\\mathfrak a)$", "the maps", "$$", "H^i_J(M) \\longrightarrow H^i(R\\Gamma_\\mathfrak a(M)^\\wedge)", "$$", "are isomorphisms for $i = 0, 1$. Consider the morphisms of", "exact triangles", "$$", "\\xymatrix{", "R\\Gamma_J(M) \\ar[d] \\ar[r] &", "M \\ar[r] \\ar[d] &", "R\\Gamma(V, \\mathcal{F}) \\ar[d] \\ar[r] &", "R\\Gamma_J(M)[1] \\ar[d] \\\\", "R\\Gamma_J(M)^\\wedge \\ar[r] &", "M \\ar[r] &", "R\\Gamma(V, \\mathcal{F})^\\wedge \\ar[r] &", "R\\Gamma_J(M)^\\wedge[1] \\\\", "R\\Gamma_\\mathfrak a(M)^\\wedge \\ar[r] \\ar[u] &", "M \\ar[r] \\ar[u] &", "R\\Gamma(U, \\mathcal{F})^\\wedge \\ar[r] \\ar[u] &", "R\\Gamma_\\mathfrak a(M)^\\wedge[1] \\ar[u]", "}", "$$", "where $V = \\Spec(A) \\setminus V(J)$. Recall that", "$R\\Gamma_\\mathfrak a(M)^\\wedge \\to R\\Gamma_J(M)^\\wedge$", "is an isomorphism (because $\\mathfrak a$, $\\mathfrak a + I$, and $J + I$", "cut out the same closed subscheme, for example", "see proof of Lemma \\ref{lemma-algebraize-local-cohomology-general}).", "Hence", "$R\\Gamma(U, \\mathcal{F})^\\wedge = R\\Gamma(V, \\mathcal{F})^\\wedge$.", "This produces a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "H^0_J(M) \\ar[r] \\ar[d] &", "M \\ar[r] \\ar[d] \\ar[r] &", "\\Gamma(V, \\mathcal{F}) \\ar[d] \\ar[r] &", "H^1_J(M) \\ar[d] \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "H^0(R\\Gamma_J(M)^\\wedge) \\ar[r] &", "M \\ar[r] &", "H^0(R\\Gamma(V, \\mathcal{F})^\\wedge) \\ar[r] &", "H^1(R\\Gamma_J(M)^\\wedge) \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "H^0(R\\Gamma_\\mathfrak a(M)^\\wedge) \\ar[r] \\ar[u] &", "M \\ar[r] \\ar[u] &", "H^0(R\\Gamma(U, \\mathcal{F})^\\wedge) \\ar[r] \\ar[u] &", "H^1(R\\Gamma_\\mathfrak a(M)^\\wedge) \\ar[r] \\ar[u] &", "0", "}", "$$", "with exact rows and isomorphisms for the lower vertical arrows. Hence", "we obtain an isomorphism", "$\\Gamma(V, \\mathcal{F}) \\to H^0(R\\Gamma(U, \\mathcal{F})^\\wedge)$.", "By Lemmas \\ref{lemma-formal-functions-general}", "and \\ref{lemma-derived-completion-pseudo-coherent} we have", "$$", "R\\Gamma(U, \\mathcal{F})^\\wedge =", "R\\Gamma(U, \\mathcal{F}^\\wedge) =", "R\\Gamma(U, R\\lim \\mathcal{F}/I^n\\mathcal{F})", "$$", "and we find $H^0(R\\Gamma(U, \\mathcal{F})^\\wedge) =", "\\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})$ by", "Cohomology, Lemma \\ref{cohomology-lemma-RGamma-commutes-with-Rlim}." ], "refs": [ "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "algebraization-lemma-algebraize-local-cohomology-general", "algebraization-lemma-algebraize-local-cohomology-general", "algebraization-lemma-formal-functions-general", "algebraization-lemma-derived-completion-pseudo-coherent", "cohomology-lemma-RGamma-commutes-with-Rlim" ], "ref_ids": [ 9729, 12716, 12716, 12708, 12710, 2160 ] } ], "ref_ids": [] }, { "id": 12732, "type": "theorem", "label": "algebraization-lemma-application-H0", "categories": [ "algebraization" ], "title": "algebraization-lemma-application-H0", "contents": [ "Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.", "Let $\\mathcal{F}$ be a coherent module on", "$U = \\Spec(A) \\setminus V(\\mathfrak a)$.", "Assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete and has a dualizing complex,", "\\item if $x \\in \\text{Ass}(\\mathcal{F})$, $x \\not \\in V(I)$,", "$z \\in V(\\mathfrak a) \\cap \\overline{\\{x\\}}$, then", "$\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + 1$,", "\\item for $x \\in U$ with $\\overline{\\{x\\}} \\cap V(I) \\subset V(\\mathfrak a)$", "we have $\\text{depth}(\\mathcal{F}_x) \\geq 2$,", "\\end{enumerate}", "Then we obtain an isomorphism", "$$", "H^0(U, \\mathcal{F})", "\\longrightarrow", "\\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $\\hat s \\in \\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})$.", "By Proposition \\ref{proposition-application-H0}", "we find that $\\hat s$ is the image of an element $s \\in \\mathcal{F}(V)$", "for some $V \\subset U$ open containing $U \\cap V(I)$.", "However, condition (3) shows that $\\text{depth}(\\mathcal{F}_x) \\geq 2$", "for all $x \\in U \\setminus V$ and hence we find that", "$\\mathcal{F}(V) = \\mathcal{F}(U)$ by", "Divisors, Lemma \\ref{divisors-lemma-depth-2-hartog}", "and the proof is complete." ], "refs": [ "algebraization-proposition-application-H0", "divisors-lemma-depth-2-hartog" ], "ref_ids": [ 12791, 7881 ] } ], "ref_ids": [] }, { "id": 12733, "type": "theorem", "label": "algebraization-lemma-alternative-colim-H0", "categories": [ "algebraization" ], "title": "algebraization-lemma-alternative-colim-H0", "contents": [ "Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$", "be an element of an ideal of $A$. Let $M$ be a finite $A$-module.", "Assume", "\\begin{enumerate}", "\\item $A$ is $f$-adically complete,", "\\item $f$ is a nonzerodivisor on $M$,", "\\item $H^1_\\mathfrak a(M/fM)$ is a finite $A$-module.", "\\end{enumerate}", "Then with $U = \\Spec(A) \\setminus V(\\mathfrak a)$ the map", "$$", "\\colim_V \\Gamma(V, \\widetilde{M})", "\\longrightarrow", "\\lim \\Gamma(U, \\widetilde{M/f^nM})", "$$", "is an isomorphism where the colimit is over opens $V \\subset U$", "containing $U \\cap V(f)$." ], "refs": [], "proofs": [ { "contents": [ "Set $\\mathcal{F} = \\widetilde{M}|_U$.", "The finiteness of $H^1_\\mathfrak a(M/fM)$ implies that", "$H^0(U, \\mathcal{F}/f\\mathcal{F})$ is finite, see", "Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}.", "By Lemma \\ref{lemma-limit-finite} (which applies as $f$ is a", "nonzerodivisor on $\\mathcal{F}$)", "we see that $N = \\lim H^0(U, \\mathcal{F}/f^n\\mathcal{F})$", "is a finite $A$-module, is $f$-torsion free, and", "$N/fN \\subset H^0(U, \\mathcal{F}/f\\mathcal{F})$.", "On the other hand, we have $M \\to N$ and the map", "$$", "M/fM \\longrightarrow H^0(U, \\mathcal{F}/f\\mathcal{F})", "$$", "is an isomorphism upon localization at any prime $\\mathfrak q$ in", "$U_0 = V(f) \\setminus \\{\\mathfrak m\\}$ (details omitted). Thus", "$M_\\mathfrak q \\to N_\\mathfrak q$ induces an isomorphism", "$$", "M_\\mathfrak q/fM_\\mathfrak q =", "(M/fM)_\\mathfrak q \\to (N/fN)_\\mathfrak q =", "N_\\mathfrak q/fN_\\mathfrak q", "$$", "Since $f$ is a nonzerodivisor on both $N$ and $M$ we conclude", "that $M_\\mathfrak q \\to N_\\mathfrak q$ is an isomorphism (use", "Nakayama to see surjectivity). We conclude that $M$ and $N$", "determine isomorphic coherent modules over an open $V$", "as in the statement of the lemma. This finishes the proof." ], "refs": [ "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "algebraization-lemma-limit-finite" ], "ref_ids": [ 9729, 12681 ] } ], "ref_ids": [] }, { "id": 12734, "type": "theorem", "label": "algebraization-lemma-alternative-H0", "categories": [ "algebraization" ], "title": "algebraization-lemma-alternative-H0", "contents": [ "Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$", "be an element of an ideal of $A$. Let $M$ be a finite $A$-module.", "Assume", "\\begin{enumerate}", "\\item $A$ is $f$-adically complete,", "\\item $H^1_\\mathfrak a(M)$ and $H^2_\\mathfrak a(M)$ are", "annihilated by a power of $f$.", "\\end{enumerate}", "Then with $U = \\Spec(A) \\setminus V(\\mathfrak a)$ the map", "$$", "\\Gamma(U, \\widetilde{M})", "\\longrightarrow", "\\lim \\Gamma(U, \\widetilde{M/f^nM})", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "We may apply", "Lemma \\ref{lemma-formal-functions-principal}", "to $U$ and $\\mathcal{F} = \\widetilde{M}|_U$", "because $\\mathcal{F}$ is a Noetherian object in", "the category of coherent $\\mathcal{O}_U$-modules.", "Since $H^1(U, \\mathcal{F}) = H^2_\\mathfrak a(M)$", "(Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local})", "is annihilated by a power of $f$, we see that", "its $f$-adic Tate module is zero.", "Hence the lemma shows $\\lim H^0(U, \\mathcal{F}/f^n \\mathcal{F})$", "is the $0$th cohomology group of the", "derived $f$-adic completion of $H^0(U, \\mathcal{F})$.", "Consider the exact sequence", "$$", "0 \\to H^0_\\mathfrak a(M) \\to M \\to", "\\Gamma(U, \\mathcal{F}) \\to H^1_\\mathfrak a(M) \\to 0", "$$", "of Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}.", "Since $H^1_\\mathfrak a(M)$ is annihilated by a power of $f$", "it is derived complete with respect to $(f)$.", "Since $M$ and $H^0_\\mathfrak a(M)$ are finite $A$-modules", "they are complete", "(Algebra, Lemma \\ref{algebra-lemma-completion-tensor})", "hence derived complete", "(More on Algebra,", "Proposition \\ref{more-algebra-proposition-derived-complete-modules}).", "By More on Algebra, Lemma \\ref{more-algebra-lemma-serre-subcategory}", "we conclude that $\\Gamma(U, \\mathcal{F})$ is derived complete", "as desired." ], "refs": [ "algebraization-lemma-formal-functions-principal", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "algebra-lemma-completion-tensor", "more-algebra-proposition-derived-complete-modules", "more-algebra-lemma-serre-subcategory" ], "ref_ids": [ 12684, 9729, 9729, 869, 10589, 10368 ] } ], "ref_ids": [] }, { "id": 12735, "type": "theorem", "label": "algebraization-lemma-alternative-higher", "categories": [ "algebraization" ], "title": "algebraization-lemma-alternative-higher", "contents": [ "Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$", "be an element of an ideal of $A$. Let $M$ be a finite $A$-module.", "Let $s \\geq 0$. Assume", "\\begin{enumerate}", "\\item $A$ is $f$-adically complete,", "\\item $H^i_\\mathfrak a(M)$ is annihilated by a power of $f$", "for $i \\leq s + 1$.", "\\end{enumerate}", "Then with $U = \\Spec(A) \\setminus V(\\mathfrak a)$ the map", "$$", "H^i(U, \\widetilde{M})", "\\longrightarrow", "\\lim H^i(U, \\widetilde{M/f^nM})", "$$", "is an isomorphism for $i < s$." ], "refs": [], "proofs": [ { "contents": [ "The proof is the same as the proof of Lemma \\ref{lemma-alternative-H0}.", "We may apply Lemma \\ref{lemma-formal-functions-principal}", "to $U$ and $\\mathcal{F} = \\widetilde{M}|_U$", "because $\\mathcal{F}$ is a Noetherian object in", "the category of coherent $\\mathcal{O}_U$-modules.", "Since $H^i(U, \\mathcal{F}) = H^{i + 1}_\\mathfrak a(M)$", "(Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local})", "is annihilated by a power of $f$ for $i \\leq s$, we see that", "its $f$-adic Tate module is zero.", "Hence the lemma shows $\\lim H^{i - 1}(U, \\mathcal{F}/f^n \\mathcal{F})$", "is the $0$th cohomology group of the", "derived $f$-adic completion of $H^{i - 1}(U, \\mathcal{F})$.", "However, if $s \\geq i > 1$, then this equal to the $f$-power torsion", "module $H^i_\\mathfrak a(M)$ and hence equal to its own", "(derived) completion. For $i = 0$, we refer to", "Lemma \\ref{lemma-alternative-H0}." ], "refs": [ "algebraization-lemma-alternative-H0", "algebraization-lemma-formal-functions-principal", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "algebraization-lemma-alternative-H0" ], "ref_ids": [ 12734, 12684, 9729, 12734 ] } ], "ref_ids": [] }, { "id": 12736, "type": "theorem", "label": "algebraization-lemma-punctured-still-connected", "categories": [ "algebraization" ], "title": "algebraization-lemma-punctured-still-connected", "contents": [ "\\begin{reference}", "\\cite[Theorem 1.6]{Varbaro}", "\\end{reference}", "Let $(A, \\mathfrak m)$ be a Noetherian complete local ring.", "Let $I$ be a proper ideal of $A$.", "Set $X = \\Spec(A)$ and $Y = V(I)$.", "Denote", "\\begin{enumerate}", "\\item $d$ the minimal dimension of an irreducible component of $X$, and", "\\item $c$ the minimal dimension of a closed subset $Z \\subset X$", "such that $X \\setminus Z$ is disconnected.", "\\end{enumerate}", "Then for $Z \\subset Y$ closed we have $Y \\setminus Z$ is connected if", "$\\dim(Z) < \\min(c, d - 1) - \\text{cd}(A, I)$. In particular, the punctured", "spectrum of $A/I$ is connected if $\\text{cd}(A, I) < \\min(c, d - 1)$." ], "refs": [], "proofs": [ { "contents": [ "Let us first prove the final assertion. As a first case, if the punctured", "spectrum of $A/I$ is empty, then", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-cd-bound-dim-local}", "shows every irreducible component of $X$ has dimension", "$\\leq \\text{cd}(A, I)$ and we get $\\min(c, d - 1) - \\text{cd}(A, I) < 0$", "which implies the lemma holds in this case. Thus we may assume", "$U \\cap Y$ is nonempty where $U = X \\setminus \\{\\mathfrak m\\}$", "is the punctured spectrum of $A$. We may replace $A$ by its reduction.", "Observe that $A$ has a dualizing complex", "(Dualizing Complexes, Lemma \\ref{dualizing-lemma-ubiquity-dualizing})", "and that $A$ is complete with respect to $I$", "(Algebra, Lemma \\ref{algebra-lemma-complete-by-sub}).", "If we assume $d - 1 > \\text{cd}(A, I)$, then we may apply", "Lemma \\ref{lemma-application-theorem} to see that", "$$", "\\colim H^0(V, \\mathcal{O}_V)", "\\longrightarrow", "\\lim H^0(U, \\mathcal{O}_U/I^n\\mathcal{O}_U)", "$$", "is an isomorphism where the colimit is over opens $V \\subset U$", "containing $U \\cap Y$. If $U \\cap Y$ is disconnected, then", "its $n$th infinitesimal neighbourhood in $U$ is disconnected", "for all $n$ and we find the", "right hand side has a nontrivial idempotent (here we use", "that $U \\cap Y$ is nonempty).", "Thus we can find a $V$ which is disconnected.", "Set $Z = X \\setminus V$. By", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-cd-bound-dim-local}", "we see that every irreducible component of $Z$ has dimension", "$\\leq \\text{cd}(A, I)$. Hence $c \\leq \\text{cd}(A, I)$ and this", "indeed proves the final statement.", "\\medskip\\noindent", "We can deduce the statement of the lemma from what we just proved", "as follows. Suppose that $Z \\subset Y$ closed and $Y \\setminus Z$ is", "disconnected and $\\dim(Z) = e$. Recall that a connected space is nonempty", "by convention. Hence we conclude either (a) $Y = Z$ or (b)", "$Y \\setminus Z = W_1 \\amalg W_2$ with $W_i$ nonempty, open, and closed", "in $Y \\setminus Z$. In case (b) we may pick points $w_i \\in W_i$", "which are closed in $U$, see", "Morphisms, Lemma \\ref{morphisms-lemma-ubiquity-Jacobson-schemes}.", "Then we can find $f_1, \\ldots, f_e \\in \\mathfrak m$", "such that $V(f_1, \\ldots, f_e) \\cap Z = \\{\\mathfrak m\\}$", "and in case (b) we may assume $w_i \\in V(f_1, \\ldots, f_e)$.", "Namely, we can inductively using prime avoidance", "choose $f_i$ such that $\\dim V(f_1, \\ldots, f_i) \\cap Z = e - i$", "and such that in case (b) we have $w_1, w_2 \\in V(f_i)$.", "It follows that the punctured spectrum of $A/I + (f_1, \\ldots, f_e)$", "is disconnected (small detail omitted). Since", "$\\text{cd}(A, I + (f_1, \\ldots, f_e)) \\leq \\text{cd}(A, I) + e$ by", "Local Cohomology, Lemmas \\ref{local-cohomology-lemma-cd-sum} and", "\\ref{local-cohomology-lemma-bound-cd} we conclude that", "$$", "\\text{cd}(A, I) + e \\geq \\min(c, d - 1)", "$$", "by the first part of the proof. This implies", "$e \\geq \\min(c, d - 1) - \\text{cd}(A, I)$ which is what we had to show." ], "refs": [ "local-cohomology-lemma-cd-bound-dim-local", "dualizing-lemma-ubiquity-dualizing", "algebra-lemma-complete-by-sub", "algebraization-lemma-application-theorem", "local-cohomology-lemma-cd-bound-dim-local", "morphisms-lemma-ubiquity-Jacobson-schemes", "local-cohomology-lemma-cd-sum", "local-cohomology-lemma-bound-cd" ], "ref_ids": [ 9711, 2890, 864, 12730, 9711, 5213, 9705, 9704 ] } ], "ref_ids": [] }, { "id": 12737, "type": "theorem", "label": "algebraization-lemma-connected", "categories": [ "algebraization" ], "title": "algebraization-lemma-connected", "contents": [ "Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.", "Assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete and has a dualizing complex,", "\\item if $\\mathfrak p \\subset A$ is a minimal prime not contained", "in $V(I)$ and $\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$, then", "$\\dim((A/\\mathfrak p)_\\mathfrak q) > \\text{cd}(A, I) + 1$,", "\\item any nonempty open $V \\subset \\Spec(A)$ which contains", "$V(I) \\setminus V(\\mathfrak a)$ is connected\\footnote{For example", "if $A$ is a domain.}.", "\\end{enumerate}", "Then $V(I) \\setminus V(\\mathfrak a)$ is either empty or connected." ], "refs": [], "proofs": [ { "contents": [ "We may replace $A$ by its reduction. Then we have the inequality", "in (2) for all associated primes of $A$. By", "Proposition \\ref{proposition-application-H0} we see that", "$$", "\\colim H^0(V, \\mathcal{O}_V) = \\lim H^0(T_n, \\mathcal{O}_{T_n})", "$$", "where the colimit is over the opens $V$ as in (3) and $T_n$ is the", "$n$th infinitesimal neighbourhood of $T = V(I) \\setminus V(\\mathfrak a)$", "in $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Thus $T$ is either empty", "or connected, since if not, then the right hand side would have a", "nontrivial idempotent and we've assumed the left hand side does not.", "Some details omitted." ], "refs": [ "algebraization-proposition-application-H0" ], "ref_ids": [ 12791 ] } ], "ref_ids": [] }, { "id": 12738, "type": "theorem", "label": "algebraization-lemma-completion-fully-faithful", "categories": [ "algebraization" ], "title": "algebraization-lemma-completion-fully-faithful", "contents": [ "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme.", "Let $Y_n \\subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$.", "Consider the following conditions", "\\begin{enumerate}", "\\item $X$ is quasi-affine and", "$\\Gamma(X, \\mathcal{O}_X) \\to \\lim \\Gamma(Y_n, \\mathcal{O}_{Y_n})$", "is an isomorphism,", "\\item $X$ has an ample invertible module $\\mathcal{L}$ and", "$\\Gamma(X, \\mathcal{L}^{\\otimes m}) \\to", "\\lim \\Gamma(Y_n, \\mathcal{L}^{\\otimes m}|_{Y_n})$", "is an isomorphism for all $m \\gg 0$,", "\\item for every finite locally free $\\mathcal{O}_X$-module", "$\\mathcal{E}$ the map", "$\\Gamma(X, \\mathcal{E}) \\to \\lim \\Gamma(Y_n, \\mathcal{E}|_{Y_n})$", "is an isomorphism, and", "\\item the completion functor", "$\\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(X, \\mathcal{I})$", "is fully faithful on the full subcategory of finite locally free", "objects.", "\\end{enumerate}", "Then (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3) $\\Rightarrow$ (4)", "and (4) $\\Rightarrow$ (3)." ], "refs": [], "proofs": [ { "contents": [ "Proof of (3) $\\Rightarrow$ (4). If $\\mathcal{F}$ and $\\mathcal{G}$", "are finite locally free on $X$, then considering", "$\\mathcal{H} = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{F})$", "and using Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-completion-internal-hom}", "we see that (3) implies (4).", "\\medskip\\noindent", "Proof of (2) $\\rightarrow$ (3). Namely, let $\\mathcal{L}$ be ample", "on $X$ and suppose that $\\mathcal{E}$ is a", "finite locally free $\\mathcal{O}_X$-module.", "We claim we can find a universally exact sequence", "$$", "0 \\to \\mathcal{E} \\to", "(\\mathcal{L}^{\\otimes p})^{\\oplus r} \\to", "(\\mathcal{L}^{\\otimes q})^{\\oplus s}", "$$", "for some $r, s \\geq 0$ and $0 \\ll p \\ll q$. If this holds, then", "using the exact sequence", "$$", "0 \\to \\lim \\Gamma(\\mathcal{E}|_{Y_n}) \\to", "\\lim \\Gamma((\\mathcal{L}^{\\otimes p})^{\\oplus r}|_{Y_n}) \\to", "\\lim \\Gamma((\\mathcal{L}^{\\otimes q})^{\\oplus s}|_{Y_n})", "$$", "and the isomorphisms in (2) we get the isomorphism in (3).", "To prove the claim, consider the dual locally free module", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}, \\mathcal{O}_X)$", "and apply", "Properties, Proposition \\ref{properties-proposition-characterize-ample}", "to find a surjection", "$$", "(\\mathcal{L}^{\\otimes -p})^{\\oplus r}", "\\longrightarrow", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}, \\mathcal{O}_X)", "$$", "Taking duals we obtain the first map in the exact sequence", "(it is universally injective because being a surjection is universal).", "Repeat with the cokernel to get the second. Some details omitted.", "\\medskip\\noindent", "Proof of (1) $\\Rightarrow$ (2). This is true because if $X$ is quasi-affine", "then $\\mathcal{O}_X$ is an ample invertible module, see", "Properties, Lemma \\ref{properties-lemma-quasi-affine-O-ample}.", "\\medskip\\noindent", "We omit the proof of (4) $\\Rightarrow$ (3)." ], "refs": [ "coherent-lemma-completion-internal-hom", "properties-proposition-characterize-ample", "properties-lemma-quasi-affine-O-ample" ], "ref_ids": [ 3374, 3067, 3053 ] } ], "ref_ids": [] }, { "id": 12739, "type": "theorem", "label": "algebraization-lemma-completion-fully-faithful-general", "categories": [ "algebraization" ], "title": "algebraization-lemma-completion-fully-faithful-general", "contents": [ "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme", "with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.", "Let $Y_n \\subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$.", "Let $\\mathcal{V}$ be the set of open subschemes $V \\subset X$ containing $Y$", "ordered by reverse inclusion.", "\\begin{enumerate}", "\\item $X$ is quasi-affine and", "$$", "\\colim_\\mathcal{V} \\Gamma(V, \\mathcal{O}_V)", "\\longrightarrow", "\\lim \\Gamma(Y_n, \\mathcal{O}_{Y_n})", "$$", "is an isomorphism,", "\\item $X$ has an ample invertible module $\\mathcal{L}$ and", "$$", "\\colim_\\mathcal{V} \\Gamma(V, \\mathcal{L}^{\\otimes m})", "\\longrightarrow", "\\lim \\Gamma(Y_n, \\mathcal{L}^{\\otimes m}|_{Y_n})", "$$", "is an isomorphism for all $m \\gg 0$,", "\\item for every $V \\in \\mathcal{V}$ and every finite locally free", "$\\mathcal{O}_V$-module $\\mathcal{E}$ the map", "$$", "\\colim_{V' \\geq V} \\Gamma(V', \\mathcal{E}|_{V'})", "\\longrightarrow", "\\lim \\Gamma(Y_n, \\mathcal{E}|_{Y_n})", "$$", "is an isomorphism, and", "\\item the completion functor", "$$", "\\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V)", "\\longrightarrow", "\\textit{Coh}(X, \\mathcal{I}),", "\\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge", "$$", "is fully faithful on the full subcategory of", "finite locally free objects (see explanation above).", "\\end{enumerate}", "Then (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3) $\\Rightarrow$ (4)", "and (4) $\\Rightarrow$ (3)." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\mathcal{V}$ is a directed set, so the colimits are", "as in Categories, Section \\ref{categories-section-directed-colimits}.", "The rest of the argument is almost exactly the same as the argument", "in the proof of Lemma \\ref{lemma-completion-fully-faithful}; we urge", "the reader to skip it.", "\\medskip\\noindent", "Proof of (3) $\\Rightarrow$ (4). If $\\mathcal{F}$ and $\\mathcal{G}$", "are finite locally free on $V \\in \\mathcal{V}$, then considering", "$\\mathcal{H} = \\SheafHom_{\\mathcal{O}_V}(\\mathcal{G}, \\mathcal{F})$", "and using Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-completion-internal-hom}", "we see that (3) implies (4).", "\\medskip\\noindent", "Proof of (2) $\\Rightarrow$ (3). Let $\\mathcal{L}$ be ample", "on $X$ and suppose that $\\mathcal{E}$ is a", "finite locally free $\\mathcal{O}_V$-module", "for some $V \\in \\mathcal{V}$.", "We claim we can find a universally exact sequence", "$$", "0 \\to \\mathcal{E} \\to", "(\\mathcal{L}^{\\otimes p})^{\\oplus r}|_{V} \\to", "(\\mathcal{L}^{\\otimes q})^{\\oplus s}|_{V}", "$$", "for some $r, s \\geq 0$ and $0 \\ll p \\ll q$. If this is true, then", "the isomorphism in (2) will imply the isomorphism in (3).", "To prove the claim, recall that $\\mathcal{L}|_V$ is ample, see", "Properties, Lemma \\ref{properties-lemma-ample-on-locally-closed}.", "Consider the dual locally free module", "$\\SheafHom_{\\mathcal{O}_V}(\\mathcal{E}, \\mathcal{O}_V)$", "and apply", "Properties, Proposition \\ref{properties-proposition-characterize-ample}", "to find a surjection", "$$", "(\\mathcal{L}^{\\otimes -p})^{\\oplus r}|_V \\longrightarrow", "\\SheafHom_{\\mathcal{O}_V}(\\mathcal{E}, \\mathcal{O}_V)", "$$", "(it is universally injective because being a surjection is universal).", "Taking duals we obtain the first map in the exact sequence.", "Repeat with the cokernel to get the second. Some details omitted.", "\\medskip\\noindent", "Proof of (1) $\\Rightarrow$ (2). This is true because if $X$ is quasi-affine", "then $\\mathcal{O}_X$ is an ample invertible module, see", "Properties, Lemma \\ref{properties-lemma-quasi-affine-O-ample}.", "\\medskip\\noindent", "We omit the proof of (4) $\\Rightarrow$ (3)." ], "refs": [ "algebraization-lemma-completion-fully-faithful", "coherent-lemma-completion-internal-hom", "properties-lemma-ample-on-locally-closed", "properties-proposition-characterize-ample", "properties-lemma-quasi-affine-O-ample" ], "ref_ids": [ 12738, 3374, 3051, 3067, 3053 ] } ], "ref_ids": [] }, { "id": 12740, "type": "theorem", "label": "algebraization-lemma-recognize-formal-coherent-modules", "categories": [ "algebraization" ], "title": "algebraization-lemma-recognize-formal-coherent-modules", "contents": [ "Let $X$ be a Noetherian scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be a quasi-coherent sheaf of ideals. The functor", "$$", "\\textit{Coh}(X, \\mathcal{I}) \\longrightarrow \\text{Pro-}\\QCoh(\\mathcal{O}_X)", "$$", "is fully faithful, see Categories, Remark \\ref{categories-remark-pro-category}." ], "refs": [ "categories-remark-pro-category" ], "proofs": [ { "contents": [ "Let $(\\mathcal{F}_n)$ and $(\\mathcal{G}_n)$ be objects of", "$\\textit{Coh}(X, \\mathcal{I})$. A morphism of pro-objects", "$\\alpha$ from $(\\mathcal{F}_n)$ to $(\\mathcal{G}_n)$ is given", "by a system of maps", "$\\alpha_n : \\mathcal{F}_{n'(n)} \\to \\mathcal{G}_n$", "where $\\mathbf{N} \\to \\mathbf{N}$, $n \\mapsto n'(n)$", "is an increasing function. Since", "$\\mathcal{F}_n = \\mathcal{F}_{n'(n)}/\\mathcal{I}^n\\mathcal{F}_{n'(n)}$", "and since $\\mathcal{G}_n$ is annihilated by $\\mathcal{I}^n$", "we see that $\\alpha_n$ induces a map $\\mathcal{F}_n \\to \\mathcal{G}_n$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12420 ] }, { "id": 12741, "type": "theorem", "label": "algebraization-lemma-fully-faithful", "categories": [ "algebraization" ], "title": "algebraization-lemma-fully-faithful", "contents": [ "Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.", "Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete and has a dualizing complex,", "\\item for any associated prime $\\mathfrak p \\subset A$,", "$I \\not \\subset \\mathfrak p$ and", "$\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$ we have", "$\\dim((A/\\mathfrak p)_\\mathfrak q) > \\text{cd}(A, I) + 1$.", "\\item for $\\mathfrak p \\subset A$, $I \\not \\subset \\mathfrak p$ with", "with $V(\\mathfrak p) \\cap V(I) \\subset V(\\mathfrak a)$", "we have $\\text{depth}(A_\\mathfrak p) \\geq 2$.", "\\end{enumerate}", "Then the completion functor", "$$", "\\textit{Coh}(\\mathcal{O}_U)", "\\longrightarrow", "\\textit{Coh}(U, I\\mathcal{O}_U),", "\\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge", "$$", "is fully faithful on the full subcategory of", "finite locally free objects." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-completion-fully-faithful}", "it suffices to show that", "$$", "\\Gamma(U, \\mathcal{O}_U) =", "\\lim \\Gamma(U, \\mathcal{O}_U/I^n\\mathcal{O}_U)", "$$", "This follows immediately from", "Lemma \\ref{lemma-application-H0}." ], "refs": [ "algebraization-lemma-completion-fully-faithful", "algebraization-lemma-application-H0" ], "ref_ids": [ 12738, 12732 ] } ], "ref_ids": [] }, { "id": 12742, "type": "theorem", "label": "algebraization-lemma-fully-faithful-simple-one", "categories": [ "algebraization" ], "title": "algebraization-lemma-fully-faithful-simple-one", "contents": [ "Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a$ be an element of", "an ideal of $A$. Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex and is complete with respect to $f$,", "\\item $A_f$ is $(S_2)$ and for every minimal prime $\\mathfrak p \\subset A$,", "$f \\not \\in \\mathfrak p$ and", "$\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$ we have", "$\\dim((A/\\mathfrak p)_\\mathfrak q) \\geq 3$.", "\\end{enumerate}", "Then the completion functor", "$$", "\\textit{Coh}(\\mathcal{O}_U)", "\\longrightarrow", "\\textit{Coh}(U, I\\mathcal{O}_U),", "\\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge", "$$", "is fully faithful on the full subcategory of finite locally free objects." ], "refs": [], "proofs": [ { "contents": [ "We will show that Lemma \\ref{lemma-fully-faithful} applies.", "Assumption (1) of Lemma \\ref{lemma-fully-faithful} holds.", "Observe that $\\text{cd}(A, (f)) \\leq 1$, see", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-bound-cd}.", "Since $A_f$ is $(S_2)$ we see that every associated prime", "$\\mathfrak p \\subset A$, $f \\not \\in \\mathfrak p$ is a minimal prime.", "Thus we get assumption (2) of Lemma \\ref{lemma-fully-faithful}.", "If $\\mathfrak p \\subset A$, $f \\not \\in \\mathfrak p$ satisfies", "$V(\\mathfrak p) \\cap V(I) \\subset V(\\mathfrak a)$ and if", "$\\mathfrak q \\in V(\\mathfrak p) \\cap V(f)$ is a generic point,", "then $\\dim((A/\\mathfrak p)_\\mathfrak q) = 1$.", "Then we obtain $\\dim(A_\\mathfrak p) \\geq 2$ by looking at the minimal primes", "$\\mathfrak p_0 \\subset \\mathfrak p$ and using that", "$\\dim((A/\\mathfrak p_0)_\\mathfrak q) \\geq 3$ by assumption. Thus", "$\\text{depth}(A_\\mathfrak p) \\geq 2$ by the $(S_2)$ assumption.", "This verifies assumption (3) of Lemma \\ref{lemma-fully-faithful}", "and the proof is complete." ], "refs": [ "algebraization-lemma-fully-faithful", "algebraization-lemma-fully-faithful", "local-cohomology-lemma-bound-cd", "algebraization-lemma-fully-faithful", "algebraization-lemma-fully-faithful" ], "ref_ids": [ 12741, 12741, 9704, 12741, 12741 ] } ], "ref_ids": [] }, { "id": 12743, "type": "theorem", "label": "algebraization-lemma-fully-faithful-alternative", "categories": [ "algebraization" ], "title": "algebraization-lemma-fully-faithful-alternative", "contents": [ "Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$", "be an element of an ideal of $A$. Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$.", "Assume", "\\begin{enumerate}", "\\item $A$ is $f$-adically complete,", "\\item $H^1_\\mathfrak a(A)$ and $H^2_\\mathfrak a(A)$ are", "annihilated by a power of $f$.", "\\end{enumerate}", "Then the completion functor", "$$", "\\textit{Coh}(\\mathcal{O}_U)", "\\longrightarrow", "\\textit{Coh}(U, I\\mathcal{O}_U),", "\\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge", "$$", "is fully faithful on the full subcategory of", "finite locally free objects." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-completion-fully-faithful}", "it suffices to show that", "$$", "\\Gamma(U, \\mathcal{O}_U) =", "\\lim \\Gamma(U, \\mathcal{O}_U/I^n\\mathcal{O}_U)", "$$", "This follows immediately from", "Lemma \\ref{lemma-alternative-H0}." ], "refs": [ "algebraization-lemma-completion-fully-faithful", "algebraization-lemma-alternative-H0" ], "ref_ids": [ 12738, 12734 ] } ], "ref_ids": [] }, { "id": 12744, "type": "theorem", "label": "algebraization-lemma-fully-faithful-simple-two", "categories": [ "algebraization" ], "title": "algebraization-lemma-fully-faithful-simple-two", "contents": [ "Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a$ be an element of", "an ideal of $A$. Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex and is complete with respect to $f$,", "\\item for every prime $\\mathfrak p \\subset A$, $f \\not \\in \\mathfrak p$", "and $\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$ we have", "$\\text{depth}(A_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) > 2$.", "\\end{enumerate}", "Then the completion functor", "$$", "\\textit{Coh}(\\mathcal{O}_U)", "\\longrightarrow", "\\textit{Coh}(U, I\\mathcal{O}_U),", "\\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge", "$$", "is fully faithful on the full subcategory of finite locally free objects." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-fully-faithful-alternative} and", "Local Cohomology, Proposition \\ref{local-cohomology-proposition-annihilator}." ], "refs": [ "algebraization-lemma-fully-faithful-alternative", "local-cohomology-proposition-annihilator" ], "ref_ids": [ 12743, 9786 ] } ], "ref_ids": [] }, { "id": 12745, "type": "theorem", "label": "algebraization-lemma-fully-faithful-general", "categories": [ "algebraization" ], "title": "algebraization-lemma-fully-faithful-general", "contents": [ "Let $I \\subset \\mathfrak a \\subset A$ be ideals of a Noetherian ring $A$.", "Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Let $\\mathcal{V}$ be", "the set of open subschemes of $U$ containing $U \\cap V(I)$", "ordered by reverse inclusion. Assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete and has a dualizing complex,", "\\item for any associated prime", "$\\mathfrak p \\subset A$ with", "$I \\not \\subset \\mathfrak p$ and", "$V(\\mathfrak p) \\cap V(I) \\not \\subset V(\\mathfrak a)$", "and $\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$ we have", "$\\dim((A/\\mathfrak p)_\\mathfrak q) > \\text{cd}(A, I) + 1$.", "\\end{enumerate}", "Then the completion functor", "$$", "\\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V)", "\\longrightarrow", "\\textit{Coh}(U, I\\mathcal{O}_U),", "\\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge", "$$", "is fully faithful on the full subcategory of", "finite locally free objects." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-completion-fully-faithful-general}", "it suffices to show that", "$$", "\\colim_\\mathcal{V} \\Gamma(V, \\mathcal{O}_V) =", "\\lim \\Gamma(U, \\mathcal{O}_U/I^n\\mathcal{O}_U)", "$$", "This follows immediately from Proposition \\ref{proposition-application-H0}." ], "refs": [ "algebraization-lemma-completion-fully-faithful-general", "algebraization-proposition-application-H0" ], "ref_ids": [ 12739, 12791 ] } ], "ref_ids": [] }, { "id": 12746, "type": "theorem", "label": "algebraization-lemma-fully-faithful-general-alternative", "categories": [ "algebraization" ], "title": "algebraization-lemma-fully-faithful-general-alternative", "contents": [ "Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$", "be an element of an ideal of $A$. Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$.", "Let $\\mathcal{V}$ be the set of open subschemes of $U$ containing $U \\cap V(f)$", "ordered by reverse inclusion. Assume", "\\begin{enumerate}", "\\item $A$ is $f$-adically complete,", "\\item $f$ is a nonzerodivisor,", "\\item $H^1_\\mathfrak a(A/fA)$ is a finite $A$-module.", "\\end{enumerate}", "Then the completion functor", "$$", "\\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V)", "\\longrightarrow", "\\textit{Coh}(U, f\\mathcal{O}_U),", "\\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge", "$$", "is fully faithful on the full subcategory of finite locally free objects." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-completion-fully-faithful-general}", "it suffices to show that", "$$", "\\colim_\\mathcal{V} \\Gamma(V, \\mathcal{O}_V) =", "\\lim \\Gamma(U, \\mathcal{O}_U/I^n\\mathcal{O}_U)", "$$", "This follows immediately from Lemma \\ref{lemma-alternative-colim-H0}." ], "refs": [ "algebraization-lemma-completion-fully-faithful-general", "algebraization-lemma-alternative-colim-H0" ], "ref_ids": [ 12739, 12733 ] } ], "ref_ids": [] }, { "id": 12747, "type": "theorem", "label": "algebraization-lemma-fully-faithful-very-general", "categories": [ "algebraization" ], "title": "algebraization-lemma-fully-faithful-very-general", "contents": [ "Let $I \\subset \\mathfrak a \\subset A$ be ideals of a Noetherian ring $A$.", "Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Let $\\mathcal{V}$ be the set", "of open subschemes of $U$ containing $U \\cap V(I)$ ordered by reverse", "inclusion. Let $\\mathcal{F}$ and", "$\\mathcal{G}$ be coherent $\\mathcal{O}_V$-modules for some", "$V \\in \\mathcal{V}$. The map", "$$", "\\colim_{V' \\geq V} \\Hom_V(\\mathcal{G}|_{V'}, \\mathcal{F}|_{V'})", "\\longrightarrow", "\\Hom_{\\textit{Coh}(U, I\\mathcal{O}_U)}(\\mathcal{G}^\\wedge, \\mathcal{F}^\\wedge)", "$$", "is bijective if the following assumptions hold:", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete and has a dualizing complex,", "\\item if $x \\in \\text{Ass}(\\mathcal{F})$, $x \\not \\in V(I)$,", "$\\overline{\\{x\\}} \\cap V(I) \\not \\subset V(\\mathfrak a)$", "and $z \\in \\overline{\\{x\\}} \\cap V(\\mathfrak a)$, then", "$\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + 1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may choose coherent $\\mathcal{O}_U$-modules", "$\\mathcal{F}'$ and $\\mathcal{G}'$ whose restriction to $V$", "is $\\mathcal{F}$ and $\\mathcal{G}$, see", "Properties, Lemma \\ref{properties-lemma-lift-finite-presentation}.", "We may modify our choice of $\\mathcal{F}'$ to ensure that", "$\\text{Ass}(\\mathcal{F}') \\subset V$, see for example", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-get-depth-1-along-Z}.", "Thus we may and do replace $V$ by $U$ and $\\mathcal{F}$ and $\\mathcal{G}$", "by $\\mathcal{F}'$ and $\\mathcal{G}'$.", "Set $\\mathcal{H} = \\SheafHom_{\\mathcal{O}_U}(\\mathcal{G}, \\mathcal{F})$.", "This is a coherent $\\mathcal{O}_U$-module. We have", "$$", "\\Hom_V(\\mathcal{G}|_V, \\mathcal{F}|_V) =", "H^0(V, \\mathcal{H})", "\\quad\\text{and}\\quad", "\\lim H^0(U, \\mathcal{H}/\\mathcal{I}^n\\mathcal{H}) =", "\\Mor_{\\textit{Coh}(U, I\\mathcal{O}_U)}", "(\\mathcal{G}^\\wedge, \\mathcal{F}^\\wedge)", "$$", "See Cohomology of Schemes, Lemma \\ref{coherent-lemma-completion-internal-hom}.", "Thus if we can show that the assumptions of", "Proposition \\ref{proposition-application-H0}", "hold for $\\mathcal{H}$, then the proof is complete.", "This holds because", "$\\text{Ass}(\\mathcal{H}) \\subset \\text{Ass}(\\mathcal{F})$.", "See Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-hom-into-depth}." ], "refs": [ "properties-lemma-lift-finite-presentation", "local-cohomology-lemma-get-depth-1-along-Z", "coherent-lemma-completion-internal-hom", "algebraization-proposition-application-H0", "coherent-lemma-hom-into-depth" ], "ref_ids": [ 3022, 9747, 3374, 12791, 3324 ] } ], "ref_ids": [] }, { "id": 12748, "type": "theorem", "label": "algebraization-lemma-system-of-modules", "categories": [ "algebraization" ], "title": "algebraization-lemma-system-of-modules", "contents": [ "In Situation \\ref{situation-algebraize}.", "Consider an inverse system $(M_n)$ of $A$-modules such", "that", "\\begin{enumerate}", "\\item $M_n$ is a finite $A$-module,", "\\item $M_n$ is annihilated by $I^n$,", "\\item the kernel and cokernel of $M_{n + 1}/I^nM_{n + 1} \\to M_n$", "are $\\mathfrak a$-power torsion.", "\\end{enumerate}", "Then $(\\widetilde{M}_n|_U)$ is in $\\textit{Coh}(U, I\\mathcal{O}_U)$.", "Conversely, every object of $\\textit{Coh}(U, I\\mathcal{O}_U)$", "arises in this manner." ], "refs": [], "proofs": [ { "contents": [ "We omit the verification that $(\\widetilde{M}_n|_U)$ is in", "$\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $(\\mathcal{F}_n)$", "be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$.", "By Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}", "we see that $\\mathcal{F}_n = \\widetilde{M_n}$ for some finite", "$A/I^n$-module $M_n$. After dividing $M_n$ by $H^0_\\mathfrak a(M_n)$", "we may assume $M_n \\subset H^0(U, \\mathcal{F}_n)$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-divide-by-torsion}", "and the already referenced lemma.", "After replacing inductively $M_{n + 1}$ by the inverse image", "of $M_n$ under the map $M_{n + 1} \\to H^0(U, \\mathcal{F}_{n + 1})", "\\to H^0(U, \\mathcal{F}_n)$, we may assume $M_{n + 1}$ maps into", "$M_n$. This gives a inverse system $(M_n)$ satisfying (1) and (2)", "such that $\\mathcal{F}_n = \\widetilde{M_n}$. To see that (3)", "holds, use that $M_{n + 1}/I^nM_{n + 1} \\to M_n$ is a map", "of finite $A$-modules which induces an isomorphism after", "applying $\\widetilde{\\ }$ and restriction to $U$", "(here we use the first referenced lemma one more time)." ], "refs": [ "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "dualizing-lemma-divide-by-torsion" ], "ref_ids": [ 9729, 2831 ] } ], "ref_ids": [] }, { "id": 12749, "type": "theorem", "label": "algebraization-lemma-essential-image-completion", "categories": [ "algebraization" ], "title": "algebraization-lemma-essential-image-completion", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$", "be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Consider the", "following conditions:", "\\begin{enumerate}", "\\item $(\\mathcal{F}_n)$ is in the essential image", "of the functor (\\ref{equation-completion}),", "\\item $(\\mathcal{F}_n)$ is the completion of a", "coherent $\\mathcal{O}_U$-module,", "\\item $(\\mathcal{F}_n)$ is the completion of a coherent", "$\\mathcal{O}_V$-module for $U \\cap Y \\subset V \\subset U$ open,", "\\item $(\\mathcal{F}_n)$ is the completion of", "the restriction to $U$ of a coherent $\\mathcal{O}_X$-module,", "\\item $(\\mathcal{F}_n)$ is the restriction to $U$ of", "the completion of a coherent $\\mathcal{O}_X$-module,", "\\item there exists an object $(\\mathcal{G}_n)$ of", "$\\textit{Coh}(X, I\\mathcal{O}_X)$ whose restriction", "to $U$ is $(\\mathcal{F}_n)$.", "\\end{enumerate}", "Then conditions (1), (2), (3), (4), and (5) are equivalent and imply (6).", "If $A$ is $I$-adically complete then condition (6) implies the others." ], "refs": [], "proofs": [ { "contents": [ "Parts (1) and (2) are equivalent, because the completion of a coherent", "$\\mathcal{O}_U$-module $\\mathcal{F}$ is by definition the image of", "$\\mathcal{F}$ under the functor (\\ref{equation-completion}).", "If $V \\subset U$ is an open subscheme containing $U \\cap Y$, then we have", "$$", "\\textit{Coh}(V, I\\mathcal{O}_V) =", "\\textit{Coh}(U, I\\mathcal{O}_U)", "$$", "since the category of coherent $\\mathcal{O}_V$-modules supported on", "$V \\cap Y$ is the same as the category of coherent $\\mathcal{O}_U$-modules", "supported on $U \\cap Y$. Thus the completion of a coherent", "$\\mathcal{O}_V$-module is an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$.", "Having said this the equivalence of (2), (3), (4), and (5)", "holds because the functors", "$\\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(\\mathcal{O}_U) \\to", "\\textit{Coh}(\\mathcal{O}_V)$ are essentially surjective.", "See Properties, Lemma \\ref{properties-lemma-lift-finite-presentation}.", "\\medskip\\noindent", "It is always the case that (5) implies (6). Assume $A$ is $I$-adically complete.", "Then any object of $\\textit{Coh}(X, I\\mathcal{O}_X)$ corresponds to a finite", "$A$-module by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-inverse-systems-affine}.", "Thus we see that (6) implies (5) in this case." ], "refs": [ "properties-lemma-lift-finite-presentation", "coherent-lemma-inverse-systems-affine" ], "ref_ids": [ 3022, 3370 ] } ], "ref_ids": [] }, { "id": 12750, "type": "theorem", "label": "algebraization-lemma-algebraizable", "categories": [ "algebraization" ], "title": "algebraization-lemma-algebraizable", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an", "object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $A', I', \\mathfrak a'$", "be the $I$-adic completions of $A, I, \\mathfrak a$. Set $X' = \\Spec(A')$", "and $U' = X' \\setminus V(\\mathfrak a')$. The following are equivalent", "\\begin{enumerate}", "\\item $(\\mathcal{F}_n)$ extends to $X$, and", "\\item the pullback of $(\\mathcal{F}_n)$ to $U'$ is the completion", "of a coherent $\\mathcal{O}_{U'}$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that $A \\to A'$ is a flat ring map which induces an isomorphism", "$A/I \\to A'/I'$. See", "Algebra, Lemmas \\ref{algebra-lemma-completion-flat} and", "\\ref{algebra-lemma-completion-complete}.", "Thus $X' \\to X$ is a", "flat morphism inducing an isomorphism $Y' \\to Y$. Thus $U' \\to U$", "is a flat morphism which induces an isomorphism $U' \\cap Y' \\to U \\cap Y$.", "This implies that in the commutative diagram", "$$", "\\xymatrix{", "\\textit{Coh}(X', I\\mathcal{O}_{X'}) \\ar[r] &", "\\textit{Coh}(U', I\\mathcal{O}_{U'}) \\\\", "\\textit{Coh}(X, I\\mathcal{O}_X) \\ar[u] \\ar[r] &", "\\textit{Coh}(U, I\\mathcal{O}_U) \\ar[u]", "}", "$$", "the vertical functors are equivalences. See", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-inverse-systems-pullback-equivalence}.", "The lemma follows formally from this and the results of", "Lemma \\ref{lemma-essential-image-completion}." ], "refs": [ "algebra-lemma-completion-flat", "algebra-lemma-completion-complete", "coherent-lemma-inverse-systems-pullback-equivalence", "algebraization-lemma-essential-image-completion" ], "ref_ids": [ 870, 872, 3379, 12749 ] } ], "ref_ids": [] }, { "id": 12751, "type": "theorem", "label": "algebraization-lemma-canonically-algebraizable", "categories": [ "algebraization" ], "title": "algebraization-lemma-canonically-algebraizable", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an", "object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. If $(\\mathcal{F}_n)$", "canonically extends to $X$, then", "\\begin{enumerate}", "\\item $(\\widetilde{H^0(U, \\mathcal{F}_n)})$ is pro-isomorphic to an", "object $(\\mathcal{G}_n)$ of $\\textit{Coh}(X, I \\mathcal{O}_X)$", "unique up to unique isomorphism,", "\\item the restriction of $(\\mathcal{G}_n)$ to $U$ is isomorphic", "to $(\\mathcal{F}_n)$, i.e., $(\\mathcal{F}_n)$ extends to $X$,", "\\item the inverse system $\\{H^0(U, \\mathcal{F}_n)\\}$", "satisfies the Mittag-Leffler condition, and", "\\item the module $M$ in (\\ref{equation-guess}) is finite over the", "$I$-adic completion of $A$ and the limit topology on", "$M$ is the $I$-adic topology.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The existence of $(\\mathcal{G}_n)$ in (1) follows from", "Definition \\ref{definition-canonically-algebraizable}.", "The uniqueness of $(\\mathcal{G}_n)$ in (1) follows from", "Lemma \\ref{lemma-recognize-formal-coherent-modules}.", "Write $\\mathcal{G}_n = \\widetilde{M_n}$.", "Then $\\{M_n\\}$ is an inverse system of finite $A$-modules", "with $M_n = M_{n + 1}/I^n M_{n + 1}$.", "By Definition \\ref{definition-canonically-algebraizable}", "the inverse system $\\{H^0(U, \\mathcal{F}_n)\\}$", "is pro-isomorphic to $\\{M_n\\}$.", "Hence we see that the inverse system $\\{H^0(U, \\mathcal{F}_n)\\}$", "satisfies the Mittag-Leffler condition and that", "$M = \\lim M_n$ (as topological modules).", "Thus the properties of $M$ in (4) follow from", "Algebra, Lemmas \\ref{algebra-lemma-limit-complete},", "\\ref{algebra-lemma-finite-over-complete-ring}, and", "\\ref{algebra-lemma-hathat-finitely-generated}.", "Since $U$ is quasi-affine the canonical maps", "$$", "\\widetilde{H^0(U, \\mathcal{F}_n)}|_U \\to \\mathcal{F}_n", "$$", "are isomorphisms (Properties, Lemma", "\\ref{properties-lemma-quasi-coherent-quasi-affine}).", "We conclude that $(\\mathcal{G}_n|_U)$ and $(\\mathcal{F}_n)$ are", "pro-isomorphic and hence isomorphic by", "Lemma \\ref{lemma-recognize-formal-coherent-modules}." ], "refs": [ "algebraization-definition-canonically-algebraizable", "algebraization-lemma-recognize-formal-coherent-modules", "algebraization-definition-canonically-algebraizable", "algebra-lemma-limit-complete", "algebra-lemma-finite-over-complete-ring", "algebra-lemma-hathat-finitely-generated", "properties-lemma-quasi-coherent-quasi-affine", "algebraization-lemma-recognize-formal-coherent-modules" ], "ref_ids": [ 12805, 12740, 12805, 880, 868, 859, 3007, 12740 ] } ], "ref_ids": [] }, { "id": 12752, "type": "theorem", "label": "algebraization-lemma-canonically-extend-base-change", "categories": [ "algebraization" ], "title": "algebraization-lemma-canonically-extend-base-change", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an", "object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $A \\to A'$ be a flat ring", "map. Set $X' = \\Spec(A')$, let $U' \\subset X'$ be the inverse image of $U$,", "and denote $g : U' \\to U$ the induced morphism. Set", "$(\\mathcal{F}'_n) = (g^*\\mathcal{F}_n)$, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-pullback}.", "If $(\\mathcal{F}_n)$ canonically extends to $X$, then", "$(\\mathcal{F}'_n)$ canonically extends to $X'$.", "Moreover, the extension found in Lemma \\ref{lemma-canonically-algebraizable}", "for $(\\mathcal{F}_n)$ pulls back to the extension for", "$(\\mathcal{F}'_n)$." ], "refs": [ "coherent-lemma-inverse-systems-pullback", "algebraization-lemma-canonically-algebraizable" ], "proofs": [ { "contents": [ "Let $f : X' \\to X$ be the induced morphism.", "We have $H^0(U', \\mathcal{F}'_n) = H^0(U, \\mathcal{F}_n) \\otimes_A A'$ by", "flat base change, see Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology}.", "Thus if $(\\mathcal{G}_n)$ in $\\textit{Coh}(X, I\\mathcal{O}_X)$", "is pro-isomorphic to $(\\widetilde{H^0(U, \\mathcal{F}_n)})$, then", "$(f^*\\mathcal{G}_n)$ is pro-isomorphic to", "$$", "(f^*\\widetilde{H^0(U, \\mathcal{F}_n)}) =", "(\\widetilde{H^0(U, \\mathcal{F}_n) \\otimes_A A'}) =", "(\\widetilde{H^0(U', \\mathcal{F}'_n)})", "$$", "This finishes the proof." ], "refs": [ "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 3298 ] } ], "ref_ids": [ 3378, 12751 ] }, { "id": 12753, "type": "theorem", "label": "algebraization-lemma-when-done", "categories": [ "algebraization" ], "title": "algebraization-lemma-when-done", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object", "of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $M$ be as in (\\ref{equation-guess}).", "Assume", "\\begin{enumerate}", "\\item[(a)] the inverse system $H^0(U, \\mathcal{F}_n)$ has Mittag-Leffler,", "\\item[(b)] the limit topology on $M$ agrees with the $I$-adic topology, and", "\\item[(c)] the image of $M \\to H^0(U, \\mathcal{F}_n)$ is a finite $A$-module", "for all $n$.", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ extends canonically to $X$.", "In particular, if $A$ is $I$-adically complete, then", "$(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module." ], "refs": [], "proofs": [ { "contents": [ "Since $H^0(U, \\mathcal{F}_n)$ has the Mittag-Leffler condition", "and since the limit topology on $M$ is the $I$-adic topology", "we see that $\\{M/I^nM\\}$ and $\\{H^0(U, \\mathcal{F}_n)\\}$", "are pro-isomorphic inverse systems of $A$-modules.", "Thus if we set", "$$", "\\mathcal{G}_n = \\widetilde{M/I^n M}", "$$", "then we see that to verify the condition in", "Definition \\ref{definition-canonically-algebraizable}", "it suffices to show that $M$ is a finite module over the", "$I$-adic completion of $A$. This follows from", "the fact that $M/I^n M$ is finite by condition (c)", "and the above and", "Algebra, Lemma \\ref{algebra-lemma-finite-over-complete-ring}." ], "refs": [ "algebraization-definition-canonically-algebraizable", "algebra-lemma-finite-over-complete-ring" ], "ref_ids": [ 12805, 868 ] } ], "ref_ids": [] }, { "id": 12754, "type": "theorem", "label": "algebraization-lemma-algebraization-principal-variant", "categories": [ "algebraization" ], "title": "algebraization-lemma-algebraization-principal-variant", "contents": [ "In Situation \\ref{situation-algebraize} let", "$(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$.", "Assume", "\\begin{enumerate}", "\\item $I = (f)$ is a principal ideal for a nonzerodivisor $f \\in \\mathfrak a$,", "\\item $\\mathcal{F}_n$ is a finite locally free", "$\\mathcal{O}_U/f^n\\mathcal{O}_U$-module,", "\\item $H^1_\\mathfrak a(A/fA)$ and $H^2_\\mathfrak a(A/fA)$", "are finite $A$-modules.", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$", "is complete, then $(\\mathcal{F}_n)$ is the completion of a coherent", "$\\mathcal{O}_U$-module." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by verifying hypotheses (a), (b), and (c) of", "Lemma \\ref{lemma-when-done}.", "\\medskip\\noindent", "Since $\\mathcal{F}_n$ is locally free over $\\mathcal{O}_U/f^n\\mathcal{O}_U$", "we see that we have short exact sequences", "$0 \\to \\mathcal{F}_n \\to \\mathcal{F}_{n + 1} \\to \\mathcal{F}_1 \\to 0$", "for all $n$. Thus condition (b) holds by Lemma \\ref{lemma-topology-I-adic-f}.", "\\medskip\\noindent", "As $f$ is a nonzerodivisor we obtain short exact sequences", "$$", "0 \\to A/f^nA \\xrightarrow{f} A/f^{n + 1}A \\to A/fA \\to 0", "$$", "and we have corresponding short exact sequences", "$0 \\to \\mathcal{F}_n \\to \\mathcal{F}_{n + 1} \\to \\mathcal{F}_1 \\to 0$.", "We will use", "Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}", "without further mention. Our assumptions imply that", "$H^0(U, \\mathcal{O}_U/f\\mathcal{O}_U)$ and", "$H^1(U, \\mathcal{O}_U/f\\mathcal{O}_U)$", "are finite $A$-modules. Hence the same thing is true for $\\mathcal{F}_1$, see", "Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-for-finite-locally-free}.", "Using induction and the short exact sequences we find that", "$H^0(U, \\mathcal{F}_n)$ are finite $A$-modules for all $n$.", "In this way we see hypothesis (c) is satisfied.", "\\medskip\\noindent", "Finally, as $H^1(U, \\mathcal{F}_1)$ is a finite $A$-module", "we can apply Lemma \\ref{lemma-ML} to see hypothesis (a) holds." ], "refs": [ "algebraization-lemma-when-done", "algebraization-lemma-topology-I-adic-f", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "local-cohomology-lemma-finiteness-for-finite-locally-free", "algebraization-lemma-ML" ], "ref_ids": [ 12753, 12680, 9729, 9744, 12682 ] } ], "ref_ids": [] }, { "id": 12755, "type": "theorem", "label": "algebraization-lemma-map-kernel-cokernel-on-closed", "categories": [ "algebraization" ], "title": "algebraization-lemma-map-kernel-cokernel-on-closed", "contents": [ "In Situation \\ref{situation-algebraize}. Let", "$(\\mathcal{F}_n) \\to (\\mathcal{F}'_n)$ be a morphism of", "$\\textit{Coh}(U, I\\mathcal{O}_U)$", "whose kernel and cokernel are annihilated by a power of $I$. Then", "\\begin{enumerate}", "\\item $(\\mathcal{F}_n)$ extends to $X$ if and only if", "$(\\mathcal{F}'_n)$ extends to $X$, and", "\\item $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module", "if and only if $(\\mathcal{F}'_n)$ is.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) follows immediately from", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-existence-easy}.", "To see part (1), we first use Lemma \\ref{lemma-algebraizable}", "to reduce to the case where $A$ is $I$-adically complete.", "However, in that case (1) reduces to (2) by", "Lemma \\ref{lemma-essential-image-completion}." ], "refs": [ "coherent-lemma-existence-easy", "algebraization-lemma-algebraizable", "algebraization-lemma-essential-image-completion" ], "ref_ids": [ 3375, 12750, 12749 ] } ], "ref_ids": [] }, { "id": 12756, "type": "theorem", "label": "algebraization-lemma-when-ML", "categories": [ "algebraization" ], "title": "algebraization-lemma-when-ML", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object", "of $\\textit{Coh}(U, I\\mathcal{O}_U)$. If the inverse system", "$H^0(U, \\mathcal{F}_n)$ has Mittag-Leffler, then the canonical maps", "$$", "\\widetilde{M/I^nM}|_U \\to \\mathcal{F}_n", "$$", "are surjective for all $n$ where $M$ is as in (\\ref{equation-guess})." ], "refs": [], "proofs": [ { "contents": [ "Surjectivity may be checked on the stalk at some point $y \\in Y \\setminus Z$.", "If $y$ corresponds to the prime $\\mathfrak q \\subset A$, then we can", "choose $f \\in \\mathfrak a$, $f \\not \\in \\mathfrak q$. Then it suffices", "to show", "$$", "M_f \\longrightarrow H^0(U, \\mathcal{F}_n)_f = H^0(D(f), \\mathcal{F}_n)", "$$", "is surjective as $D(f)$ is affine (equality holds by Properties,", "Lemma \\ref{properties-lemma-invert-f-sections}). Since we have the", "Mittag-Leffler property, we find that", "$$", "\\Im(M \\to H^0(U, \\mathcal{F}_n)) =", "\\Im(H^0(U, \\mathcal{F}_m) \\to H^0(U, \\mathcal{F}_n))", "$$", "for some $m \\geq n$. Using the long exact sequence of cohomology we see", "that", "$$", "\\Coker(H^0(U, \\mathcal{F}_m) \\to H^0(U, \\mathcal{F}_n))", "\\subset", "H^1(U, \\Ker(\\mathcal{F}_m \\to \\mathcal{F}_n))", "$$", "Since $U = X \\setminus V(\\mathfrak a)$ this $H^1$ is $\\mathfrak a$-power", "torsion. Hence after inverting $f$ the cokernel becomes zero." ], "refs": [ "properties-lemma-invert-f-sections" ], "ref_ids": [ 3004 ] } ], "ref_ids": [] }, { "id": 12757, "type": "theorem", "label": "algebraization-lemma-when-topology", "categories": [ "algebraization" ], "title": "algebraization-lemma-when-topology", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object", "of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $M$ be as in (\\ref{equation-guess}).", "Set", "$$", "\\mathcal{G}_n = \\widetilde{M/I^nM}.", "$$", "If the limit topology on $M$ agrees with the $I$-adic topology, then", "$\\mathcal{G}_n|_U$ is a coherent", "$\\mathcal{O}_U$-module and the map of inverse systems", "$$", "(\\mathcal{G}_n|_U) \\longrightarrow (\\mathcal{F}_n)", "$$", "is injective in the abelian category $\\textit{Coh}(U, I\\mathcal{O}_U)$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\mathcal{G}_n$ is a quasi-coherent $\\mathcal{O}_X$-module", "annihilated by $I^n$ and that", "$\\mathcal{G}_{n + 1}/I^n\\mathcal{G}_{n + 1} = \\mathcal{G}_n$.", "Consider", "$$", "M_n = \\Im(M \\longrightarrow H^0(U, \\mathcal{F}_n))", "$$", "The assumption says that the inverse systems $(M_n)$ and", "$(M/I^nM)$ are isomorphic as pro-objects of $\\text{Mod}_A$.", "Pick $f \\in \\mathfrak a$ so $D(f) \\subset U$ is an affine open. Then we have", "$$", "(M_n)_f \\subset H^0(U, \\mathcal{F}_n)_f = H^0(D(f), \\mathcal{F}_n)", "$$", "Equality holds by Properties, Lemma \\ref{properties-lemma-invert-f-sections}.", "Thus $\\widetilde{M_n}|_U \\to \\mathcal{F}_n$ is injective.", "It follows that $\\widetilde{M_n}|_U$ is a coherent module", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient}).", "Since $M \\to M/I^nM$ is surjective and factors as", "$M_{n'} \\to M/I^nM$ for some $n' \\geq n$ we find that $\\mathcal{G}_n|_U$", "is coherent as the quotient of a coherent module.", "Combined with the initical remarks of the proof we conclude", "that $(\\mathcal{G}_n|_U)$ indeed forms an object", "of $\\textit{Coh}(U, I\\mathcal{O}_U)$.", "Finally, to show the injectivity of the map", "it suffices to show that", "$$", "\\lim (M/I^nM)_f = \\lim H^0(D(f), \\mathcal{G}_n) \\to", "\\lim H^0(D(f), \\mathcal{F}_n)", "$$", "is injective, see Cohomology of Schemes, Lemmas", "\\ref{coherent-lemma-inverse-systems-abelian} and", "\\ref{coherent-lemma-inverse-systems-affine}.", "The injectivity of $\\lim (M_n)_f \\to \\lim H^0(D(f), \\mathcal{F}_n)$", "is clear (see above) and by our remark on pro-systems we have", "$\\lim (M_n)_f = \\lim (M/I^nM)_f$. This finishes the proof." ], "refs": [ "properties-lemma-invert-f-sections", "coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient", "coherent-lemma-inverse-systems-abelian", "coherent-lemma-inverse-systems-affine" ], "ref_ids": [ 3004, 3310, 3371, 3370 ] } ], "ref_ids": [] }, { "id": 12758, "type": "theorem", "label": "algebraization-lemma-discussion", "categories": [ "algebraization" ], "title": "algebraization-lemma-discussion", "contents": [ "Let $Y$ be a Noetherian scheme and let $Z \\subset Y$ be a closed subset.", "\\begin{enumerate}", "\\item For $y \\in Y$ we have $\\delta_Z(y) = 0 \\Leftrightarrow y \\in Z$.", "\\item The subsets $\\{y \\in Y \\mid \\delta_Z(y) \\leq k\\}$ are", "stable under specialization.", "\\item For $y \\in Y$ and $z \\in \\overline{\\{y\\}} \\cap Z$ we have", "$\\dim(\\mathcal{O}_{\\overline{\\{y\\}}, z}) \\geq \\delta_Z(y)$.", "\\item If $\\delta$ is a dimension function on $Y$, then", "$\\delta(y) \\leq \\delta_Z(y) + \\delta_{max}$ where $\\delta_{max}$", "is the maximum value of $\\delta$ on $Z$.", "\\item If $Y = \\Spec(A)$ is the spectrum of a catenary Noetherian local ring", "with maximal ideal $\\mathfrak m$ and $Z = \\{\\mathfrak m\\}$, then", "$\\delta_Z(y) = \\dim(\\overline{\\{y\\}})$.", "\\item Given a pattern of specializations", "$$", "\\xymatrix{", "& y'_0 \\ar@{~>}[ld] \\ar@{~>}[rd] &", "& y'_1 \\ar@{~>}[ld] & \\ldots", "& y'_{k - 1} \\ar@{~>}[rd] &", "\\\\", "y_0 & &", "y_1 & &", "\\ldots & &", "y_k = y", "}", "$$", "between points of $Y$ with $y_0 \\in Z$ and $y_i' \\leadsto y_i$", "an immediate specialization, then $\\delta_Z(y_k) \\leq k$.", "\\item If $Y' \\subset Y$ is an open subscheme, then", "$\\delta^{Y'}_{Y' \\cap Z}(y') \\geq \\delta^Y_Z(y')$ for $y' \\in Y'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is essentially true by definition. Namely, if $y \\in Z$,", "then we can take $k = 0$ and $V_0 = W_0 = \\overline{\\{y\\}}$.", "\\medskip\\noindent", "Proof of (2). Let $y \\leadsto y'$ be a nontrivial specialization and let", "$V_0 \\subset W_0 \\supset V_1 \\subset W_1 \\supset \\ldots \\subset W_k$", "is a system for $y$. Here there are two cases.", "Case I: $V_k = W_k$, i.e., $c_k = 0$. In this case", "we can set $V'_k = W'_k = \\overline{\\{y'\\}}$.", "An easy computation shows that", "$\\delta(V_0, W_0, \\ldots, V'_k, W'_k) \\leq", "\\delta(V_0, W_0, \\ldots, V_k, W_k)$", "because only $b_{k - 1}$ is changed into a bigger integer.", "Case II: $V_k \\not = W_k$, i.e., $c_k > 0$. Observe that", "in this case $\\max_{i = 0, 1, \\ldots, k}", "(c_i + c_{i + 1} + \\ldots + c_k - b_i - b_{i + 1} - \\ldots - b_{k - 1}) > 0$.", "Hence if we set $V'_{k + 1} = W_{k + 1} = \\overline{\\{y'\\}}$,", "then although $k$ is replaced by $k + 1$, the maximum now looks like", "$$", "\\max_{i = 0, 1, \\ldots, k + 1}", "(c_i + c_{i + 1} + \\ldots + c_k + c_{k + 1}", "- b_i - b_{i + 1} - \\ldots - b_{k - 1} - b_k)", "$$", "with $c_{k + 1} = 0$ and $b_k = \\text{codim}(V_{k + 1}, W_k) > 0$.", "This is strictly smaller than", "$\\max_{i = 0, 1, \\ldots, k}", "(c_i + c_{i + 1} + \\ldots + c_k - b_i - b_{i + 1} - \\ldots - b_{k - 1})$", "and hence", "$\\delta(V_0, W_0, \\ldots, V'_{k + 1}, W'_{k + 1}) \\leq", "\\delta(V_0, W_0, \\ldots, V_k, W_k)$ as desired.", "\\medskip\\noindent", "Proof of (3). Given $y \\in Y$ and $z \\in \\overline{\\{y\\}} \\cap Z$", "we get the system", "$$", "V_0 = \\overline{\\{z\\}} \\subset W_0 = \\overline{\\{y\\}}", "$$", "and $c_0 = \\text{codim}(V_0, W_0) = \\dim(\\mathcal{O}_{\\overline{\\{y\\}}, z})$", "by Properties, Lemma \\ref{properties-lemma-codimension-local-ring}.", "Thus we see that $\\delta(V_0, W_0) = 0 + c_0 = c_0$ which proves", "what we want.", "\\medskip\\noindent", "Proof of (4). Let $\\delta$ be a dimension function on $Y$.", "Let $V_0 \\subset W_0 \\supset V_1 \\subset W_1 \\supset \\ldots \\subset W_k$", "be a system for $y$. Let $y'_i \\in W_i$ and $y_i \\in V_i$ be the", "generic points, so $y_0 \\in Z$ and $y_k = y$. Then we see that", "$$", "\\delta(y_i) - \\delta(y_{i - 1}) =", "\\delta(y'_{i - 1}) - \\delta(y_{i - 1}) - \\delta(y'_{i - 1}) + \\delta(y_i) =", "c_{i - 1} - b_{i - 1}", "$$", "Finally, we have $\\delta(y'_k) - \\delta(y_{k - 1}) = c_k$.", "Thus we see that", "$$", "\\delta(y) - \\delta(y_0) =", "c_0 + \\ldots + c_k - b_0 - \\ldots - b_{k - 1}", "$$", "We conclude", "$\\delta(V_0, W_0, \\ldots, W_k) \\geq k + \\delta(y) - \\delta(y_0)$", "which proves what we want.", "\\medskip\\noindent", "Proof of (5). The function $\\delta(y) = \\dim(\\overline{\\{y\\}})$", "is a dimension function. Hence $\\delta(y) \\leq \\delta_Z(y)$ by", "part (4). By part (3) we have $\\delta_Z(y) \\leq \\delta(y)$", "and we are done.", "\\medskip\\noindent", "Proof of (6). Given such a sequence of points, we may assume", "all the specializations $y'_i \\leadsto y_{i + 1}$ are nontrivial", "(otherwise we can shorten the chain of specializations).", "Then we set $V_i = \\overline{\\{y_i\\}}$ and $W_i = \\overline{\\{y'_i\\}}$", "and we compute $\\delta(V_0, W_1, V_1, \\ldots, W_{k - 1}) = k$ because all", "the codimensions $c_i$ of $V_i \\subset W_i$ are $1$ and all $b_i > 0$.", "This implies $\\delta_Z(y'_{k - 1}) \\leq k$ as $y'_{k - 1}$ is the generic", "point of $W_k$. Then $\\delta_Z(y) \\leq k$ by part (2) as $y$ is a", "specialization of $y_{k - 1}$.", "\\medskip\\noindent", "Proof of (7). This is clear as their are fewer systems to consider", "in the computation of $\\delta^{Y'}_{Y' \\cap Z}$." ], "refs": [ "properties-lemma-codimension-local-ring" ], "ref_ids": [ 2979 ] } ], "ref_ids": [] }, { "id": 12759, "type": "theorem", "label": "algebraization-lemma-change-distance-function", "categories": [ "algebraization" ], "title": "algebraization-lemma-change-distance-function", "contents": [ "Let $Y$ be a universally catenary Noetherian scheme. Let $Z \\subset Y$", "be a closed subscheme. Let $f : Y' \\to Y$ be a finite type", "morphism all of whose fibres have dimension $\\leq e$. Set $Z' = f^{-1}(Z)$.", "Then", "$$", "\\delta_Z(y) \\leq \\delta_{Z'}(y') + e - \\text{trdeg}_{\\kappa(y)}(\\kappa(y'))", "$$", "for $y' \\in Y'$ with image $y \\in Y$." ], "refs": [], "proofs": [ { "contents": [ "If $\\delta_{Z'}(y') = \\infty$, then there is nothing to prove.", "If $\\delta_{Z'}(y') < \\infty$, then we choose a system", "of integral closed subschemes", "$$", "V'_0 \\subset W'_0 \\supset V'_1 \\subset W'_1 \\supset \\ldots \\subset W'_k", "$$", "of $Y'$ with $V'_0 \\subset Z'$ and $y'$ the generic point of $W'_k$", "such that $\\delta_{Z'}(y') = \\delta(V'_0, W'_0, \\ldots, W'_k)$.", "Denote", "$$", "V_0 \\subset W_0 \\supset V_1 \\subset W_1 \\supset \\ldots \\subset W_k", "$$", "the scheme theoretic images of the above schemes in $Y$. Observe", "that $y$ is the generic point of $W_k$ and that $V_0 \\subset Z$.", "For each $i$ we look at the diagram", "$$", "\\xymatrix{", "V'_i \\ar[r] \\ar[d] & W'_i \\ar[d] & V'_{i + 1} \\ar[l] \\ar[d] \\\\", "V_i \\ar[r] & W_i & V_{i + 1} \\ar[l]", "}", "$$", "Denote $n_i$ the relative dimension of $V'_i/V_i$ and", "$m_i$ the relative dimension of $W'_i/W_i$; more precisely", "these are the transcendence degrees of the corresponding extensions of", "the function fields. Set", "$c_i = \\text{codim}(V_i, W_i)$,", "$c'_i = \\text{codim}(V'_i, W'_i)$,", "$b_i = \\text{codim}(V_{i + 1}, W_i)$, and", "$b'_i = \\text{codim}(V'_{i + 1}, W'_i)$.", "By the dimension formula we have", "$$", "c_i = c'_i + n_i - m_i", "\\quad\\text{and}\\quad", "b_i = b'_i + n_{i + 1} - m_i", "$$", "See Morphisms, Lemma \\ref{morphisms-lemma-dimension-formula}.", "Hence $c_i - b_i = c'_i - b'_i + n_i - n_{i + 1}$. Thus we see that", "\\begin{align*}", "& c_i + c_{i + 1} + \\ldots + c_k - b_i - b_{i + 1} - \\ldots - b_{k - 1} \\\\", "& =", "c'_i + c'_{i + 1} + \\ldots + c'_k - b'_i - b'_{i + 1} - \\ldots - b'_{k - 1}", "+ n_i - n_k + c_k - c'_k \\\\", "& =", "c'_i + c'_{i + 1} + \\ldots + c'_k - b'_i - b'_{i + 1} - \\ldots - b'_{k - 1}", "+ n_i - m_k", "\\end{align*}", "Thus we see that", "\\begin{align*}", "\\max_{i = 0, \\ldots, k}", "& (c_i + c_{i + 1} + \\ldots + c_k - b_i - b_{i + 1} - \\ldots - b_{k - 1}) \\\\", "& =", "\\max_{i = 0, \\ldots, k}", "(c'_i + c'_{i + 1} + \\ldots + c'_k - b'_i - b'_{i + 1} - \\ldots - b'_{k - 1}", "+ n_i - m_k) \\\\", "& =", "\\max_{i = 0, \\ldots, k}", "(c'_i + c'_{i + 1} + \\ldots + c'_k - b'_i - b'_{i + 1} - \\ldots - b'_{k - 1}", "+ n_i) - m_k \\\\", "& \\leq", "\\max_{i = 0, \\ldots, k}", "(c'_i + c'_{i + 1} + \\ldots + c'_k - b'_i - b'_{i + 1} - \\ldots - b'_{k - 1})", "+ e - m_k", "\\end{align*}", "Since $m_k = \\text{trdeg}_{\\kappa(y)}(\\kappa(y'))$ we conclude that", "$$", "\\delta(V_0, W_0, \\ldots, W_k) \\leq", "\\delta(V'_0, W'_0, \\ldots, W'_k) + e - \\text{trdeg}_{\\kappa(y)}(\\kappa(y'))", "$$", "as desired." ], "refs": [ "morphisms-lemma-dimension-formula" ], "ref_ids": [ 5493 ] } ], "ref_ids": [] }, { "id": 12760, "type": "theorem", "label": "algebraization-lemma-elementary", "categories": [ "algebraization" ], "title": "algebraization-lemma-elementary", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object", "of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $a, b$ be integers.", "\\begin{enumerate}", "\\item If $(\\mathcal{F}_n)$ is annihilated by a power of $I$, then", "$(\\mathcal{F}_n)$ satisfies the $(a, b)$-inequalities for any $a, b$.", "\\item If $(\\mathcal{F}_n)$ satisfies the $(a + 1, b)$-inequalities, then", "$(\\mathcal{F}_n)$ satisfies the strict $(a, b)$-inequalities.", "\\end{enumerate}", "If $\\text{cd}(A, I) \\leq d$ and $A$ has a dualizing complex, then", "\\begin{enumerate}", "\\item[(3)] $(\\mathcal{F}_n)$ satisfies the $(s, s + d)$-inequalities", "if and only if for all $y \\in U \\cap Y$ the tuple", "$\\mathcal{O}_{X, y}^\\wedge, I\\mathcal{O}_{X, y}^\\wedge,", "\\{\\mathfrak m_y^\\wedge\\}, \\mathcal{F}_y^\\wedge, s - \\delta^Y_Z(y), d$", "is as in Situation \\ref{situation-bootstrap}.", "\\item[(4)]", "If $(\\mathcal{F}_n)$ satisfies the strict $(s, s + d)$-inequalities, then", "$(\\mathcal{F}_n)$ satisfies the $(s, s + d)$-inequalities.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Immediate except for part (4) which is a consequence of", "Lemma \\ref{lemma-bootstrap-bis-bis} and the translation in (3)." ], "refs": [ "algebraization-lemma-bootstrap-bis-bis" ], "ref_ids": [ 12725 ] } ], "ref_ids": [] }, { "id": 12761, "type": "theorem", "label": "algebraization-lemma-explain-2-3-cd-1", "categories": [ "algebraization" ], "title": "algebraization-lemma-explain-2-3-cd-1", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object", "of $\\textit{Coh}(U, I\\mathcal{O}_U)$. If $\\text{cd}(A, I) = 1$, then", "$\\mathcal{F}$ satisfies the $(2, 3)$-inequalities if and only if", "$$", "\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > 3", "$$", "for all $y \\in U \\cap Y$ and $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$", "with $\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$." ], "refs": [], "proofs": [ { "contents": [ "Observe that for a prime $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$,", "$\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$", "we have $V(\\mathfrak p) \\cap V(I\\mathcal{O}_{X, y}^\\wedge) =", "\\{\\mathfrak m_y^\\wedge\\}", "\\Leftrightarrow \\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) = 1$", "as $\\text{cd}(A, I) = 1$.", "See Local Cohomology, Lemmas", "\\ref{local-cohomology-lemma-cd-change-rings} and", "\\ref{local-cohomology-lemma-cd-bound-dim-local}.", "OK, consider the three numbers", "$\\alpha = \\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) \\geq 0$,", "$\\beta = \\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) \\geq 1$, and", "$\\gamma = \\delta^Y_Z(y) \\geq 1$.", "Then we see Definition \\ref{definition-s-d-inequalities} requires", "\\begin{enumerate}", "\\item if $\\beta > 1$, then", "$\\alpha + \\gamma \\geq 2$ or $\\alpha + \\beta + \\gamma > 3$, and", "\\item if $\\beta = 1$, then $\\alpha + \\gamma > 2$.", "\\end{enumerate}", "It is trivial to see that this is equivalent to", "$\\alpha + \\beta + \\gamma > 3$." ], "refs": [ "local-cohomology-lemma-cd-change-rings", "local-cohomology-lemma-cd-bound-dim-local", "algebraization-definition-s-d-inequalities" ], "ref_ids": [ 9706, 9711, 12806 ] } ], "ref_ids": [] }, { "id": 12762, "type": "theorem", "label": "algebraization-lemma-sanity", "categories": [ "algebraization" ], "title": "algebraization-lemma-sanity", "contents": [ "In Situation \\ref{situation-algebraize} let $\\mathcal{F}$ be a", "coherent $\\mathcal{O}_U$-module and $d \\geq 1$. Assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete, has a dualizing complex, and", "$\\text{cd}(A, I) \\leq d$,", "\\item the completion $\\mathcal{F}^\\wedge$ of $\\mathcal{F}$", "satisfies the strict $(1, 1 + d)$-inequalities.", "\\end{enumerate}", "Let $x \\in X$ be a point. Let $W = \\overline{\\{x\\}}$.", "If $W \\cap Y$ has an irreducible component contained in $Z$", "and one which is not, then $\\text{depth}(\\mathcal{F}_x) \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "Let $W \\cap Y = W_1 \\cup \\ldots \\cup W_n$ be the decomposition into", "irreducible components. By assumption, after renumbering, we can find", "$0 < m < n$ such that $W_1, \\ldots, W_m \\subset Z$ and", "$W_{m + 1}, \\ldots, W_n \\not \\subset Z$. We conclude that", "$$", "W \\cap Y \\setminus", "\\left((W_1 \\cup \\ldots \\cup W_m) \\cap (W_{m + 1} \\cup \\ldots \\cup W_n)\\right)", "$$", "is disconnected. By Lemma \\ref{lemma-connected} we can find", "$1 \\leq i \\leq m < j \\leq n$ and", "$z \\in W_i \\cap W_j$ such that $\\dim(\\mathcal{O}_{W, z}) \\leq d + 1$.", "Choose an immediate specialization $y \\leadsto z$ with", "$y \\in W_j$, $y \\not \\in Z$; existence of $y$ follows from", "Properties, Lemma \\ref{properties-lemma-complement-closed-point-Jacobson}.", "Observe that $\\delta^Y_Z(y) = 1$ and $\\dim(\\mathcal{O}_{W, y}) \\leq d$.", "Let $\\mathfrak p \\subset \\mathcal{O}_{X, y}$ be the prime corresponding to $x$.", "Let $\\mathfrak p' \\subset \\mathcal{O}_{X, y}^\\wedge$ be a minimal prime", "over $\\mathfrak p\\mathcal{O}_{X, y}^\\wedge$. Then we have", "$$", "\\text{depth}(\\mathcal{F}_x) =", "\\text{depth}((\\mathcal{F}^\\wedge_y)_{\\mathfrak p'})", "\\quad\\text{and}\\quad", "\\dim(\\mathcal{O}_{W, y}) = \\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p')", "$$", "See Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck-module} and", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-change-completion}.", "Now we read off the conclusion from the inequalities given to us." ], "refs": [ "algebraization-lemma-connected", "properties-lemma-complement-closed-point-Jacobson", "algebra-lemma-apply-grothendieck-module", "local-cohomology-lemma-change-completion" ], "ref_ids": [ 12737, 2965, 1360, 9740 ] } ], "ref_ids": [] }, { "id": 12763, "type": "theorem", "label": "algebraization-lemma-recover", "categories": [ "algebraization" ], "title": "algebraization-lemma-recover", "contents": [ "In Situation \\ref{situation-algebraize} let $\\mathcal{F}$ be a", "coherent $\\mathcal{O}_U$-module and $d \\geq 1$. Assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete, has a dualizing complex, and", "$\\text{cd}(A, I) \\leq d$,", "\\item the completion $\\mathcal{F}^\\wedge$ of $\\mathcal{F}$", "satisfies the strict $(1, 1+ d)$-inequalities, and", "\\item for $x \\in U$ with $\\overline{\\{x\\}} \\cap Y \\subset Z$", "we have $\\text{depth}(\\mathcal{F}_x) \\geq 2$.", "\\end{enumerate}", "Then $H^0(U, \\mathcal{F}) \\to \\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by showing that Lemma \\ref{lemma-application-H0} applies.", "Thus we let $x \\in \\text{Ass}(\\mathcal{F})$ with $x \\not \\in Y$.", "Set $W = \\overline{\\{x\\}}$.", "By condition (3) we see that $W \\cap Y \\not \\subset Z$.", "By Lemma \\ref{lemma-sanity} we see that no irreducible", "component of $W \\cap Y$ is contained in $Z$.", "Thus if $z \\in W \\cap Z$, then there is an immediate", "specialization $y \\leadsto z$, $y \\in W \\cap Y$, $y \\not \\in Z$.", "For existence of $y$ use", "Properties, Lemma \\ref{properties-lemma-complement-closed-point-Jacobson}.", "Then $\\delta^Y_Z(y) = 1$ and the assumption", "implies that $\\dim(\\mathcal{O}_{W, y}) > d$.", "Hence $\\dim(\\mathcal{O}_{W, z}) > 1 + d$ and we win." ], "refs": [ "algebraization-lemma-application-H0", "algebraization-lemma-sanity", "properties-lemma-complement-closed-point-Jacobson" ], "ref_ids": [ 12732, 12762, 2965 ] } ], "ref_ids": [] }, { "id": 12764, "type": "theorem", "label": "algebraization-lemma-fully-faithful-inequalities", "categories": [ "algebraization" ], "title": "algebraization-lemma-fully-faithful-inequalities", "contents": [ "In Situation \\ref{situation-algebraize} let $\\mathcal{F}$ be a", "coherent $\\mathcal{O}_U$-module and $d \\geq 1$. Assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete, has a dualizing complex, and", "$\\text{cd}(A, I) \\leq d$,", "\\item the completion $\\mathcal{F}^\\wedge$ of $\\mathcal{F}$", "satisfies the strict $(1, 1 + d)$-inequalities, and", "\\item for $x \\in U$ with $\\overline{\\{x\\}} \\cap Y \\subset Z$", "we have $\\text{depth}(\\mathcal{F}_x) \\geq 2$.", "\\end{enumerate}", "Then the map", "$$", "\\Hom_U(\\mathcal{G}, \\mathcal{F})", "\\longrightarrow", "\\Hom_{\\textit{Coh}(U, I\\mathcal{O}_U)}(\\mathcal{G}^\\wedge, \\mathcal{F}^\\wedge)", "$$", "is bijective for every coherent $\\mathcal{O}_U$-module $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Set $\\mathcal{H} = \\SheafHom_{\\mathcal{O}_U}(\\mathcal{G}, \\mathcal{F})$.", "Using Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-hom-into-depth} or", "More on Algebra, Lemma \\ref{more-algebra-lemma-hom-into-depth}", "we see that the completion of $\\mathcal{H}$", "satisfies the strict $(1, 1 + d)$-inequalities and that for", "$x \\in U$ with $\\overline{\\{x\\}} \\cap Y \\subset Z$", "we have $\\text{depth}(\\mathcal{H}_x) \\geq 2$. Details omitted.", "Thus by Lemma \\ref{lemma-recover} we have", "$$", "\\Hom_U(\\mathcal{G}, \\mathcal{F}) =", "H^0(U, \\mathcal{H}) =", "\\lim H^0(U, \\mathcal{H}/\\mathcal{I}^n\\mathcal{H}) =", "\\Mor_{\\textit{Coh}(U, I\\mathcal{O}_U)}", "(\\mathcal{G}^\\wedge, \\mathcal{F}^\\wedge)", "$$", "See Cohomology of Schemes, Lemma \\ref{coherent-lemma-completion-internal-hom}", "for the final equality." ], "refs": [ "coherent-lemma-hom-into-depth", "more-algebra-lemma-hom-into-depth", "algebraization-lemma-recover", "coherent-lemma-completion-internal-hom" ], "ref_ids": [ 3324, 9930, 12763, 3374 ] } ], "ref_ids": [] }, { "id": 12765, "type": "theorem", "label": "algebraization-lemma-construct-unique", "categories": [ "algebraization" ], "title": "algebraization-lemma-construct-unique", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an", "object of $\\textit{Coh}(U, I\\mathcal{O}_U)$ and $d \\geq 1$. Assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete, has a dualizing complex, and", "$\\text{cd}(A, I) \\leq d$,", "\\item $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module,", "\\item $(\\mathcal{F}_n)$ satisfies the strict $(1, 1 + d)$-inequalities.", "\\end{enumerate}", "Then there exists a unique coherent $\\mathcal{O}_U$-module $\\mathcal{F}$", "whose completion is $(\\mathcal{F}_n)$ such that for", "$x \\in U$ with $\\overline{\\{x\\}} \\cap Y \\subset Z$", "we have $\\text{depth}(\\mathcal{F}_x) \\geq 2$." ], "refs": [], "proofs": [ { "contents": [ "Choose a coherent $\\mathcal{O}_U$-module $\\mathcal{F}$ whose", "completion is $(\\mathcal{F}_n)$. Let", "$T = \\{x \\in U \\mid \\overline{\\{x\\}} \\cap Y \\subset Z\\}$.", "We will construct $\\mathcal{F}$ by applying Local Cohomology,", "Lemma \\ref{local-cohomology-lemma-make-S2-along-T-simple}", "with $\\mathcal{F}$ and $T$.", "Then uniqueness will follow from the mapping property", "of Lemma \\ref{lemma-fully-faithful-inequalities}.", "\\medskip\\noindent", "Since $T$ is stable under specialization in $U$ the only", "thing to check is the following. If $x' \\leadsto x$ is an", "immediate specialization of points of $U$ with $x \\in T$", "and $x' \\not \\in T$, then $\\text{depth}(\\mathcal{F}_{x'}) \\geq 1$.", "Set $W = \\overline{\\{x\\}}$ and $W' = \\overline{\\{x'\\}}$.", "Since $x' \\not \\in T$ we see that $W' \\cap Y$ is not contained in $Z$.", "If $W' \\cap Y$ contains an irreducible component contained in $Z$,", "then we are done by Lemma \\ref{lemma-sanity}.", "If not, we choose an irreducible component $W_1$ of $W \\cap Y$ and", "an irreducible component $W'_1$ of $W' \\cap Y$ with $W_1 \\subset W'_1$.", "Let $z \\in W_1$ be the generic point. Let $y \\leadsto z$, $y \\in W'_1$", "be an immediate specialization with $y \\not \\in Z$; existence of $y$", "follows from $W'_1 \\not \\subset Z$ (see above) and", "Properties, Lemma \\ref{properties-lemma-complement-closed-point-Jacobson}.", "Then we have the following $z \\in Z$, $x \\leadsto z$,", "$x' \\leadsto y \\leadsto z$, $y \\in Y \\setminus Z$, and $\\delta^Y_Z(y) = 1$.", "By Local Cohomology, Lemma \\ref{local-cohomology-lemma-cd-bound-dim-local}", "and the fact that $z$", "is a generic point of $W \\cap Y$ we have", "$\\dim(\\mathcal{O}_{W, z}) \\leq d$.", "Since $x' \\leadsto x$ is an immediate specialization we have", "$\\dim(\\mathcal{O}_{W', z}) \\leq d + 1$.", "Since $y \\not = z$ we conclude", "$\\dim(\\mathcal{O}_{W', y}) \\leq d$.", "If $\\text{depth}(\\mathcal{F}_{x'}) = 0$ then we would get", "a contradiction with assumption (3); details about passage", "from $\\mathcal{O}_{X, y}$ to its completion omitted.", "This finishes the proof." ], "refs": [ "local-cohomology-lemma-make-S2-along-T-simple", "algebraization-lemma-fully-faithful-inequalities", "algebraization-lemma-sanity", "properties-lemma-complement-closed-point-Jacobson", "local-cohomology-lemma-cd-bound-dim-local" ], "ref_ids": [ 9750, 12764, 12762, 2965, 9711 ] } ], "ref_ids": [] }, { "id": 12766, "type": "theorem", "label": "algebraization-lemma-algebraization-principal", "categories": [ "algebraization" ], "title": "algebraization-lemma-algebraization-principal", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object", "of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume", "\\begin{enumerate}", "\\item $A$ is local and $\\mathfrak a = \\mathfrak m$ is the maximal ideal,", "\\item $A$ has a dualizing complex,", "\\item $I = (f)$ is a principal ideal for a nonzerodivisor $f \\in \\mathfrak m$,", "\\item $\\mathcal{F}_n$ is a finite locally free", "$\\mathcal{O}_U/f^n\\mathcal{O}_U$-module,", "\\item if $\\mathfrak p \\in V(f) \\setminus \\{\\mathfrak m\\}$, then", "$\\text{depth}((A/f)_\\mathfrak p) + \\dim(A/\\mathfrak p) > 1$, and", "\\item if $\\mathfrak p \\not \\in V(f)$ and", "$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then", "$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$", "is complete, then $(\\mathcal{F}_n)$ is the completion of a coherent", "$\\mathcal{O}_U$-module." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by verifying hypotheses (a), (b), and (c) of", "Lemma \\ref{lemma-when-done}.", "\\medskip\\noindent", "Since $\\mathcal{F}_n$ is locally free over $\\mathcal{O}_U/f^n\\mathcal{O}_U$", "we see that we have short exact sequences", "$0 \\to \\mathcal{F}_n \\to \\mathcal{F}_{n + 1} \\to \\mathcal{F}_1 \\to 0$", "for all $n$. Thus condition (b) holds by Lemma \\ref{lemma-topology-I-adic-f}.", "\\medskip\\noindent", "By induction on $n$ and the short exact sequences", "$0 \\to A/f^n \\to A/f^{n + 1} \\to A/f \\to 0$ we see that", "the associated primes of $A/f^nA$ agree with the associated", "primes of $A/fA$. Since the associated points of $\\mathcal{F}_n$", "correspond to the associated primes of $A/f^nA$ not equal to $\\mathfrak m$", "by assumption (3), we conclude that", "$M_n = H^0(U, \\mathcal{F}_n)$ is a finite $A$-module by (5) and", "Local Cohomology, Proposition \\ref{local-cohomology-proposition-kollar}.", "Thus hypothesis (c) holds.", "\\medskip\\noindent", "To finish the proof it suffices to show that there exists an $n > 1$", "such that the image of", "$$", "H^1(U, \\mathcal{F}_n) \\longrightarrow H^1(U, \\mathcal{F}_1)", "$$", "has finite length as an $A$-module. Namely, this will imply hypothesis (a)", "by Lemma \\ref{lemma-ML-better}. The image is independent", "of $n$ for $n$ large enough by Lemma \\ref{lemma-ML-local}.", "Let $\\omega_A^\\bullet$ be a normalized dualizing complex for $A$.", "By the local duality theorem and Matlis duality", "(Dualizing Complexes, Lemma \\ref{dualizing-lemma-special-case-local-duality}", "and Proposition \\ref{dualizing-proposition-matlis})", "our claim is equivalent to: the image of", "$$", "\\text{Ext}^{-2}_A(M_1, \\omega_A^\\bullet) \\to", "\\text{Ext}^{-2}_A(M_n, \\omega_A^\\bullet)", "$$", "has finite length for $n \\gg 1$. The modules in question are", "finite $A$-modules supported at $V(f)$. Thus it suffices to show that this", "map is zero after localization at a prime $\\mathfrak q$", "containing $f$ and different from $\\mathfrak m$.", "Let $\\omega_{A_\\mathfrak q}^\\bullet$ be a normalized", "dualizing complex on $A_\\mathfrak q$ and recall that", "$\\omega_{A_\\mathfrak q}^\\bullet =", "(\\omega_A^\\bullet)_\\mathfrak q[\\dim(A/\\mathfrak q)]$ by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-dimension-function}.", "Using the local structure of $\\mathcal{F}_n$ given in (4)", "we find that it suffices to show the vanishing of", "$$", "\\text{Ext}^{-2 + \\dim(A/\\mathfrak q)}_{A_\\mathfrak q}(", "A_\\mathfrak q/f, \\omega_{A_\\mathfrak q}^\\bullet)", "\\to", "\\text{Ext}^{-2 + \\dim(A/\\mathfrak q)}_{A_\\mathfrak q}(", "A_\\mathfrak q/f^n, \\omega_{A_\\mathfrak q}^\\bullet)", "$$", "for $n$ large enough. If $\\dim(A/\\mathfrak q) > 3$, then this is immediate from", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-sitting-in-degrees}.", "For the other cases we will use the long exact sequence", "$$", "\\ldots", "\\xrightarrow{f^n}", "H^{-1}(\\omega_{A_\\mathfrak q}^\\bullet)", "\\to", "\\text{Ext}^{-1}_{A_\\mathfrak q}(", "A_\\mathfrak q/f^n, \\omega_{A_\\mathfrak q}^\\bullet) \\to", "H^0(\\omega_{A_\\mathfrak q}^\\bullet)", "\\xrightarrow{f^n}", "H^0(\\omega_{A_\\mathfrak q}^\\bullet)", "\\to", "\\text{Ext}^0_{A_\\mathfrak q}(", "A_\\mathfrak q/f^n, \\omega_{A_\\mathfrak q}^\\bullet) \\to 0", "$$", "If $\\dim(A/\\mathfrak q) = 2$, then", "$H^0(\\omega_{A_\\mathfrak q}^\\bullet) = 0$", "because $\\text{depth}(A_\\mathfrak q) \\geq 1$ as", "$f$ is a nonzerodivisor.", "Thus the long exact sequence shows the condition is that", "$$", "f^{n - 1} :", "H^{-1}(\\omega_{A_\\mathfrak q}^\\bullet)/f \\to", "H^{-1}(\\omega_{A_\\mathfrak q}^\\bullet)/f^n", "$$", "is zero. Now $H^{-1}(\\omega^\\bullet_\\mathfrak q)$ is a finite", "module supported in the primes $\\mathfrak p \\subset A_\\mathfrak q$ such that", "$\\text{depth}(A_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) \\leq 1$.", "Since $\\dim((A/\\mathfrak p)_\\mathfrak q) = \\dim(A/\\mathfrak p) - 2$", "condition (6) tells us these primes are contained in $V(f)$.", "Thus the desired vanishing for $n$ large enough.", "Finally, if $\\dim(A/\\mathfrak q) = 1$, then condition (5) combined", "with the fact that $f$ is a nonzerodivisor", "insures that $A_\\mathfrak q$ has depth at least $2$. Hence", "$H^0(\\omega_{A_\\mathfrak q}^\\bullet) =", "H^{-1}(\\omega_{A_\\mathfrak q}^\\bullet) = 0$", "and the long exact sequence shows the claim is", "equivalent to the vanishing of", "$$", "f^{n - 1} :", "H^{-2}(\\omega_{A_\\mathfrak q}^\\bullet)/f \\to", "H^{-2}(\\omega_{A_\\mathfrak q}^\\bullet)/f^n", "$$", "Now $H^{-2}(\\omega^\\bullet_\\mathfrak q)$ is a finite", "module supported in the primes $\\mathfrak p \\subset A_\\mathfrak q$", "such that $\\text{depth}(A_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q)", "\\leq 2$. By condition (6) all of these primes are contained in $V(f)$.", "Thus the desired vanishing for $n$ large enough." ], "refs": [ "algebraization-lemma-when-done", "algebraization-lemma-topology-I-adic-f", "local-cohomology-proposition-kollar", "algebraization-lemma-ML-better", "algebraization-lemma-ML-local", "dualizing-lemma-special-case-local-duality", "dualizing-proposition-matlis", "dualizing-lemma-dimension-function", "local-cohomology-lemma-sitting-in-degrees" ], "ref_ids": [ 12753, 12680, 9785, 12683, 12691, 2873, 2924, 2869, 9737 ] } ], "ref_ids": [] }, { "id": 12767, "type": "theorem", "label": "algebraization-lemma-helper-algebraize", "categories": [ "algebraization" ], "title": "algebraization-lemma-helper-algebraize", "contents": [ "In Situation \\ref{situation-algebraize} let $(M_n)$ be an inverse system of", "$A$-modules as in Lemma \\ref{lemma-system-of-modules} and let", "$(\\mathcal{F}_n)$ be the corresponding object of", "$\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $d \\geq \\text{cd}(A, I)$", "and $s \\geq 0$ be integers.", "With notation as above assume", "\\begin{enumerate}", "\\item $A$ is local with maximal ideal $\\mathfrak m = \\mathfrak a$,", "\\item $A$ has a dualizing complex, and", "\\item $(\\mathcal{F}_n)$ satisfies the $(s, s + d)$-inequalities", "(Definition \\ref{definition-s-d-inequalities}).", "\\end{enumerate}", "Let $E$ be an injective hull of the residue field of $A$. Then for $i \\leq s$", "there exists a finite $A$-module $N$ annihilated by a power", "of $I$ and for $n \\gg 0$ compatible maps", "$$", "H^i_\\mathfrak m(M_n) \\to \\Hom_A(N, E)", "$$", "whose cokernels are finite length $A$-modules and whose kernels $K_n$", "form an inverse system such that $\\Im(K_{n''} \\to K_{n'})$ has finite", "length for $n'' \\gg n' \\gg 0$." ], "refs": [ "algebraization-lemma-system-of-modules", "algebraization-definition-s-d-inequalities" ], "proofs": [ { "contents": [ "Let $\\omega_A^\\bullet$ be a normalized dualizing complex. Then", "$\\delta^Y_Z = \\delta$ is the dimension function associated with", "this dualizing complex.", "Observe that $\\Ext^{-i}_A(M_n, \\omega_A^\\bullet)$ is a finite $A$-module", "annihilated by $I^n$. Fix $0 \\leq i \\leq s$.", "Below we will find $n_1 > n_0 > 0$ such that if we set", "$$", "N = \\Im(\\Ext^{-i}_A(M_{n_0}, \\omega_A^\\bullet) \\to", "\\Ext^{-i}_A(M_{n_1}, \\omega_A^\\bullet))", "$$", "then the kernels of the maps", "$$", "N \\to \\Ext^{-i}_A(M_n, \\omega_A^\\bullet),\\quad n \\geq n_1", "$$", "are finite length $A$-modules and the cokernels $Q_n$ form a", "system such that $\\Im(Q_{n'} \\to Q_{n''})$ has finite length", "for $n'' \\gg n' \\gg n_1$. This is equivalent to the statement that", "the system $\\{\\Ext^{-i}_A(M_n, \\omega_A^\\bullet)\\}_{n \\geq 1}$", "is essentially constant in the quotient of the category of finite", "$A$-modules modulo the Serre subcategory of finite length $A$-modules.", "By the local duality theorem", "(Dualizing Complexes, Lemma \\ref{dualizing-lemma-special-case-local-duality})", "and Matlis duality", "(Dualizing Complexes, Proposition \\ref{dualizing-proposition-matlis})", "we conclude that there are maps", "$$", "H^i_\\mathfrak m(M_n) \\to \\Hom_A(N, E),\\quad n \\geq n_1", "$$", "as in the statement of the lemma.", "\\medskip\\noindent", "Pick $f \\in \\mathfrak m$. Let $B = A_f^\\wedge$ be the $I$-adic completion", "of the localization $A_f$. Recall that", "$\\omega_{A_f}^\\bullet = \\omega_A^\\bullet \\otimes_A A_f$", "and $\\omega_B^\\bullet = \\omega_A^\\bullet \\otimes_A B$ are dualizing", "complexes (Dualizing Complexes, Lemma \\ref{dualizing-lemma-dualizing-localize}", "and \\ref{dualizing-lemma-completion-henselization-dualizing}).", "Let $M$ be the finite $B$-module $\\lim M_{n, f}$ (compare with", "discussion in Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-inverse-systems-affine}). Then", "$$", "\\Ext^{-i}_A(M_n, \\omega_A^\\bullet)_f =", "\\Ext^{-i}_{A_f}(M_{n, f}, \\omega_{A_f}^\\bullet) =", "\\Ext^{-i}_B(M/I^n M, \\omega_B^\\bullet)", "$$", "Since $\\mathfrak m$ can be generated by finitely many $f \\in \\mathfrak m$", "it suffices to show that for each $f$ the system", "$$", "\\{\\Ext^{-i}_B(M/I^n M, \\omega_B^\\bullet)\\}_{n \\geq 1}", "$$", "is essentially constant. Some details omitted.", "\\medskip\\noindent", "Let $\\mathfrak q \\subset IB$ be a prime ideal. Then $\\mathfrak q$ corresponds", "to a point $y \\in U \\cap Y$. Observe that", "$\\delta(\\mathfrak q) = \\dim(\\overline{\\{y\\}})$", "is also the value of the dimension function associated to $\\omega_B^\\bullet$", "(we omit the details; use that $\\omega_B^\\bullet$ is gotten from", "$\\omega_A^\\bullet$ by tensoring up with $B$). Assumption", "(3) guarantees via Lemma \\ref{lemma-elementary}", "that Lemma \\ref{lemma-algebraize-local-cohomology-bis}", "applies to", "$B_\\mathfrak q, IB_\\mathfrak q, \\mathfrak qB_\\mathfrak q, M_\\mathfrak q$", "with $s$ replaced by $s - \\delta(y)$. We obtain that", "$$", "H^{i - \\delta(\\mathfrak q)}_{\\mathfrak qB_\\mathfrak q}(M_\\mathfrak q) =", "\\lim H^{i - \\delta(\\mathfrak q)}_{\\mathfrak qB_\\mathfrak q}(", "(M/I^nM)_\\mathfrak q)", "$$", "and this module is annihilated by a power of $I$.", "By Lemma \\ref{lemma-terrific} we find that the inverse systems", "$H^{i - \\delta(\\mathfrak q)}_{\\mathfrak qB_\\mathfrak q}((M/I^nM)_\\mathfrak q)$", "are essentially constant with value", "$H^{i - \\delta(\\mathfrak q)}_{\\mathfrak qB_\\mathfrak q}(M_\\mathfrak q)$.", "Since $(\\omega_B^\\bullet)_\\mathfrak q[-\\delta(\\mathfrak q)]$ is a normalized", "dualizing complex on $B_\\mathfrak q$ the local duality theorem", "shows that the system", "$$", "\\Ext^{-i}_B(M/I^n M, \\omega_B^\\bullet)_\\mathfrak q", "$$", "is essentially constant with value", "$\\Ext^{-i}_B(M, \\omega_B^\\bullet)_\\mathfrak q$.", "\\medskip\\noindent", "To finish the proof we globalize as in the proof of", "Lemma \\ref{lemma-bootstrap}; the argument here is easier", "because we know the value of our system already. Namely, consider the maps", "$$", "\\alpha_n :", "\\Ext^{-i}_B(M/I^n M, \\omega_B^\\bullet)", "\\longrightarrow", "\\Ext^{-i}_B(M, \\omega_B^\\bullet)", "$$", "for varying $n$. By the above, for every $\\mathfrak q$ we can find an", "$n$ such that $\\alpha_n$ is surjective after localization at $\\mathfrak q$.", "Since $B$ is Noetherian and $\\Ext^{-i}_B(M, \\omega_B^\\bullet)$", "a finite module, we can find an $n$ such that $\\alpha_n$ is surjective.", "For any $n$ such that $\\alpha_n$ is surjective, given a prime", "$\\mathfrak q \\in V(IB)$ we can find an $n' > n$ such that", "$\\Ker(\\alpha_n)$ maps to zero in $\\Ext^{-i}(M/I^{n'}M, \\omega_B^\\bullet)$", "at least after localizing at $\\mathfrak q$.", "Since $\\Ker(\\alpha_n)$ is a finite $A$-module and since supports of", "sections are quasi-compact, we can find an $n'$ such that", "$\\Ker(\\alpha_n)$ maps to zero in $\\Ext^{-i}(M/I^{n'}M, \\omega_B^\\bullet)$.", "In this way we see that $\\Ext^{-i}(M/I^n M, \\omega_B^\\bullet)$", "is essentially constant with value $\\Ext^{-i}(M, \\omega_B^\\bullet)$.", "This finishes the proof." ], "refs": [ "dualizing-lemma-special-case-local-duality", "dualizing-proposition-matlis", "dualizing-lemma-dualizing-localize", "dualizing-lemma-completion-henselization-dualizing", "coherent-lemma-inverse-systems-affine", "algebraization-lemma-elementary", "algebraization-lemma-algebraize-local-cohomology-bis", "algebraization-lemma-terrific", "algebraization-lemma-bootstrap" ], "ref_ids": [ 2873, 2924, 2851, 2889, 3370, 12760, 12724, 12692, 12726 ] } ], "ref_ids": [ 12748, 12806 ] }, { "id": 12768, "type": "theorem", "label": "algebraization-lemma-algebraization-principal-bis", "categories": [ "algebraization" ], "title": "algebraization-lemma-algebraization-principal-bis", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object", "of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume", "\\begin{enumerate}", "\\item $A$ is local and $\\mathfrak a = \\mathfrak m$ is the maximal ideal,", "\\item $A$ has a dualizing complex,", "\\item $I = (f)$ is a principal ideal,", "\\item $(\\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities.", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ extends to $X$. In particular, if $A$ is", "$I$-adically complete, then $(\\mathcal{F}_n)$ is the completion", "of a coherent $\\mathcal{O}_U$-module." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\textit{Coh}(U, I\\mathcal{O}_U)$ is an abelian category, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-abelian}.", "Over affine opens of $U$ the object $(\\mathcal{F}_n)$", "corresponds to a finite module over a Noetherian ring", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-affine}).", "Thus the kernels of the maps $f^N : (\\mathcal{F}_n) \\to (\\mathcal{F}_n)$", "stabilize for $N$ large enough. By", "Lemmas \\ref{lemma-map-kernel-cokernel-on-closed} and", "\\ref{lemma-essential-image-completion}", "in order to prove the lemma", "we may replace $(\\mathcal{F}_n)$ by the image of such a map.", "Thus we may assume $f$ is injective on $(\\mathcal{F}_n)$.", "After this replacement the equivalent conditions of", "Lemma \\ref{lemma-equivalent-f-good} hold for the inverse system", "$(\\mathcal{F}_n)$ on $U$. We will use this without further mention", "in the rest of the proof.", "\\medskip\\noindent", "We will check hypotheses (a), (b), and (c) of", "Lemma \\ref{lemma-when-done}.", "Hypothesis (b) holds by Lemma \\ref{lemma-topology-I-adic-f}.", "\\medskip\\noindent", "Pick a inverse system of modules $\\{M_n\\}$ as in", "Lemma \\ref{lemma-system-of-modules}.", "We may assume $H^0_\\mathfrak m(M_n) = 0$ by replacing $M_n$ by", "$M_n/H^0_\\mathfrak m(M_n)$ if necessary. Then we obtain short exact", "sequences", "$$", "0 \\to M_n \\to H^0(U, \\mathcal{F}_n) \\to H^1_\\mathfrak m(M_n) \\to 0", "$$", "for all $n$. Let $E$ be an injective hull of the residue field of $A$.", "By Lemma \\ref{lemma-helper-algebraize} and our current assumption (4)", "we can choose, an integer $m \\geq 0$, finite $A$-modules", "$N_1$ and $N_2$ annihilated by $f^c$ for some $c \\geq 0$ and", "compatible systems of maps", "$$", "H^i_\\mathfrak m(M_n) \\to \\Hom_A(N_i, E), \\quad i = 1, 2", "$$", "for $n \\geq m$", "with the properties stated in the lemma.", "\\medskip\\noindent", "We know that $M = \\lim H^0(U, \\mathcal{F}_n)$ is an $A$-module whose", "limit topology is the $f$-adic topology. Thus, given $n$, the module", "$M/f^nM$ is a subquotient of $H^0(U, \\mathcal{F}_N)$ for some $N \\gg n$.", "Looking at the information obtained above we see that", "$f^cM/f^nM$ is a finite $A$-module. Since $f$ is a nonzerodivisor", "on $M$ we conclude that $M/f^{n - c}M$ is a finite $A$-module.", "In this way we see that hypothesis (c) of Lemma \\ref{lemma-when-done} holds.", "\\medskip\\noindent", "Next, we study the module", "$$", "Ob = \\lim H^1(U, \\mathcal{F}_n) = \\lim H^2_\\mathfrak m(M_n)", "$$", "For $n \\geq m$ let $K_n$ be the kernel of the map", "$H^2_\\mathfrak m(M_n) \\to \\Hom_A(N_2, E)$.", "Set $K = \\lim K_n$. We obtain an exact sequence", "$$", "0 \\to K \\to Ob \\to \\Hom_A(N_2, E)", "$$", "By the above the limit topology on $Ob = \\lim H^2_\\mathfrak m(M_n)$", "is the $f$-adic topology. Since $N_2$ is annihilated by $f^c$", "we conclude the same is true for the limit topology on $K = \\lim K_n$.", "Thus $K/fK$ is a subquotient of $K_n$ for $n \\gg 1$.", "However, since $\\{K_n\\}$ is pro-isomorphic to a inverse system of", "finite length $A$-modules (by the conclusion of", "Lemma \\ref{lemma-helper-algebraize})", "we conclude that $K/fK$ is a subquotient of a finite length", "$A$-module. It follows that $K$ is a finite $A$-module, see", "Algebra, Lemma \\ref{algebra-lemma-finite-over-complete-ring}.", "(In fact, we even see that $\\dim(\\text{Supp}(K)) = 1$ but", "we will not need this.)", "\\medskip\\noindent", "Given $n \\geq 1$ consider the boundary map", "$$", "\\delta_n :", "H^0(U, \\mathcal{F}_n)", "\\longrightarrow", "\\lim_N H^1(U, f^n\\mathcal{F}_N) \\xrightarrow{f^{-n}} Ob", "$$", "(the second map is an isomorphism)", "coming from the short exact sequences", "$$", "0 \\to f^n\\mathcal{F}_N \\to \\mathcal{F}_N \\to \\mathcal{F}_n \\to 0", "$$", "For each $n$ set", "$$", "P_n = \\Im(H^0(U, \\mathcal{F}_{n + m}) \\to H^0(U, \\mathcal{F}_n))", "$$", "where $m$ is as above. Observe that $\\{P_n\\}$ is an inverse", "system and that the map $f : \\mathcal{F}_n \\to \\mathcal{F}_{n + 1}$", "on global sections maps $P_n$ into $P_{n + 1}$.", "If $p \\in P_n$, then $\\delta_n(p) \\in K \\subset Ob$", "because $\\delta_n(p)$ maps to zero in", "$H^1(U, f^n\\mathcal{F}_{n + m}) = H^2_\\mathfrak m(M_m)$", "and the composition of $\\delta_n$ and $Ob \\to \\Hom_A(N_2, E)$", "factors through $H^2_\\mathfrak m(M_m)$ by our choice of $m$.", "Hence", "$$", "\\bigoplus\\nolimits_{n \\geq 0} \\Im(P_n \\to Ob)", "$$", "is a finite graded $A[T]$-module where $T$ acts via multiplication by $f$.", "Namely, it is a graded submodule of $K[T]$ and $K$ is finite over $A$.", "Arguing as in the proof of", "Lemma \\ref{lemma-ML-general}\\footnote{Choose homogeneous generators", "of the form $\\delta_{n_j}(p_j)$ for the displayed module.", "Then if $k = \\max(n_j)$ we find that for $n \\geq k$", "and any $p \\in P_n$ we can find $a_j \\in A$ such that", "$p - \\sum a_j f^{n - n_j} p_j$ is in the kernel of $\\delta_n$", "and hence in the image of $P_{n'}$ for all $n' \\geq n$.", "Thus $\\Im(P_n \\to P_{n - k}) = \\Im(P_{n'} \\to P_{n - k})$", "for all $n' \\geq n$.}", "we find that the inverse system $\\{P_n\\}$ satisfies ML.", "Since $\\{P_n\\}$ is pro-isomorphic to $\\{H^0(U, \\mathcal{F}_n)\\}$", "we conclude that $\\{H^0(U, \\mathcal{F}_n)\\}$ has ML.", "Thus hypothesis (a) of Lemma \\ref{lemma-when-done}", "holds and the proof is complete." ], "refs": [ "coherent-lemma-inverse-systems-abelian", "coherent-lemma-inverse-systems-affine", "algebraization-lemma-map-kernel-cokernel-on-closed", "algebraization-lemma-essential-image-completion", "algebraization-lemma-equivalent-f-good", "algebraization-lemma-when-done", "algebraization-lemma-topology-I-adic-f", "algebraization-lemma-system-of-modules", "algebraization-lemma-helper-algebraize", "algebraization-lemma-when-done", "algebraization-lemma-helper-algebraize", "algebra-lemma-finite-over-complete-ring", "algebraization-lemma-ML-general", "algebraization-lemma-when-done" ], "ref_ids": [ 3371, 3370, 12755, 12749, 12679, 12753, 12680, 12748, 12767, 12753, 12767, 868, 12675, 12753 ] } ], "ref_ids": [] }, { "id": 12769, "type": "theorem", "label": "algebraization-lemma-unwinding-conditions", "categories": [ "algebraization" ], "title": "algebraization-lemma-unwinding-conditions", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object", "of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume", "\\begin{enumerate}", "\\item $A$ is local with maximal ideal $\\mathfrak a = \\mathfrak m$,", "\\item $\\text{cd}(A, I) = 1$.", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities if and only", "if for all $y \\in U \\cap Y$ with $\\dim(\\{y\\}) = 1$ and every prime", "$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$,", "$\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$ we have", "$$", "\\text{depth}((\\mathcal{F}_y^\\wedge)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) > 2", "$$" ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-explain-2-3-cd-1}", "without further mention. In particular, we see the condition is necessary.", "Conversely, suppose the condition is true.", "Note that $\\delta^Y_Z(y) = \\dim(\\overline{\\{y\\}})$ by", "Lemma \\ref{lemma-discussion}. Let us write $\\delta$ for this function.", "Let $y \\in U \\cap Y$. If $\\delta(y) > 2$, then the inequality", "of Lemma \\ref{lemma-explain-2-3-cd-1} holds.", "Finally, suppose $\\delta(y) = 2$. We have to show that", "$$", "\\text{depth}((\\mathcal{F}_y^\\wedge)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) > 1", "$$", "Choose a specialization $y \\leadsto y'$ with $\\delta(y') = 1$. Then", "there is a ring map $\\mathcal{O}_{X, y'}^\\wedge \\to \\mathcal{O}_{X, y}^\\wedge$", "which identifies the target with the completion of the localization", "of $\\mathcal{O}_{X, y'}^\\wedge$ at a prime $\\mathfrak q$", "with $\\dim(\\mathcal{O}_{X, y'}^\\wedge/\\mathfrak q) = 1$.", "Moreover, we then obtain", "$$", "\\mathcal{F}_y^\\wedge =", "\\mathcal{F}_{y'}^\\wedge", "\\otimes_{\\mathcal{O}_{X, y'}^\\wedge}", "\\mathcal{O}_{X, y}^\\wedge", "$$", "Let $\\mathfrak p' \\subset \\mathcal{O}_{X, y'}^\\wedge$ be the image", "of $\\mathfrak p$.", "By Local Cohomology, Lemma \\ref{local-cohomology-lemma-change-completion}", "we have", "\\begin{align*}", "\\text{depth}((\\mathcal{F}_y^\\wedge)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p)", "& =", "\\text{depth}((\\mathcal{F}_{y'}^\\wedge)_{\\mathfrak p'}) +", "\\dim((\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p)_{\\mathfrak p'}) \\\\", "& =", "\\text{depth}((\\mathcal{F}_{y'}^\\wedge)_{\\mathfrak p'}) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p') - 1", "\\end{align*}", "the last equality because the specialization is immediate.", "Thus the lemma is prove by the assumed inequality for $y', \\mathfrak p'$." ], "refs": [ "algebraization-lemma-explain-2-3-cd-1", "algebraization-lemma-discussion", "algebraization-lemma-explain-2-3-cd-1", "local-cohomology-lemma-change-completion" ], "ref_ids": [ 12761, 12758, 12761, 9740 ] } ], "ref_ids": [] }, { "id": 12770, "type": "theorem", "label": "algebraization-lemma-unwinding-conditions-bis", "categories": [ "algebraization" ], "title": "algebraization-lemma-unwinding-conditions-bis", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object", "of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume", "\\begin{enumerate}", "\\item $A$ is local with maximal ideal $\\mathfrak a = \\mathfrak m$,", "\\item $A$ has a dualizing complex,", "\\item $\\text{cd}(A, I) = 1$,", "\\item for $y \\in U \\cap Y$ the module $\\mathcal{F}_y^\\wedge$", "is finite locally free outside $V(I\\mathcal{O}_{X, y}^\\wedge)$,", "for example if $\\mathcal{F}_n$ is a finite locally free", "$\\mathcal{O}_U/I^n\\mathcal{O}_U$-module, and", "\\item one of the following is true", "\\begin{enumerate}", "\\item $A_f$ is $(S_2)$ and every irreducible component of $X$", "not contained in $Y$ has dimension $\\geq 4$, or", "\\item if $\\mathfrak p \\not \\in V(f)$ and", "$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then", "$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.", "\\end{enumerate}", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities." ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion of Lemma \\ref{lemma-unwinding-conditions}.", "Let $y \\in U \\cap Y$ with $\\dim(\\overline{\\{y\\}} = 1$ and let", "$\\mathfrak p$ be a prime $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with", "$\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$.", "Condition (4) shows that", "$\\text{depth}((\\mathcal{F}_y^\\wedge)_\\mathfrak p) =", "\\text{depth}((\\mathcal{O}_{X, y}^\\wedge)_\\mathfrak p)$.", "Thus we have to prove", "$$", "\\text{depth}((\\mathcal{O}_{X, y}^\\wedge)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) > 2", "$$", "Let $\\mathfrak p_0 \\subset A$ be the image of $\\mathfrak p$.", "Let $\\mathfrak q \\subset A$ be the prime corresponding to $y$.", "By Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-change-completion}", "we have", "\\begin{align*}", "\\text{depth}((\\mathcal{O}_{X, y}^\\wedge)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p)", "& =", "\\text{depth}(A_{\\mathfrak p_0}) + \\dim((A/\\mathfrak p_0)_\\mathfrak q) \\\\", "& =", "\\text{depth}(A_{\\mathfrak p_0}) + \\dim(A/\\mathfrak p_0) - 1", "\\end{align*}", "If (5)(a) holds, then we get that this is", "$$", "\\geq \\min(2, \\dim(A_{\\mathfrak p_0})) + \\dim(A/\\mathfrak p_0) - 1", "$$", "Note that in any case $\\dim(A/\\mathfrak p_0) \\geq 2$. Hence if", "we get $2$ for the minimum, then we are done. If not we get", "$$", "\\dim(A_{\\mathfrak p_0}) + \\dim(A/\\mathfrak p_0) - 1 \\geq 4 - 1", "$$", "because every component of $\\Spec(A)$ passing through $\\mathfrak p_0$", "has dimension $\\geq 4$. If (5)(b) holds, then we win immediately." ], "refs": [ "algebraization-lemma-unwinding-conditions", "local-cohomology-lemma-change-completion" ], "ref_ids": [ 12769, 9740 ] } ], "ref_ids": [] }, { "id": 12771, "type": "theorem", "label": "algebraization-lemma-divide-torsion-formal-coherent-module", "categories": [ "algebraization" ], "title": "algebraization-lemma-divide-torsion-formal-coherent-module", "contents": [ "In the situation above assume $X$ locally has a dualizing complex.", "Let $T \\subset Y$ be a subset stable under specialization.", "Assume for $y \\in T$ and for a nonmaximal prime", "$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with", "$V(\\mathfrak p) \\cap V(\\mathcal{I}^\\wedge_y) = \\{\\mathfrak m_y^\\wedge\\}$", "we have", "$$", "\\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p}", "((\\mathcal{F}^\\wedge_y)_\\mathfrak p) > 0", "$$", "Then there exists a canonical map", "$(\\mathcal{F}_n) \\to (\\mathcal{F}_n')$", "of inverse systems of coherent $\\mathcal{O}_X$-modules", "with the following properties", "\\begin{enumerate}", "\\item for $y \\in T$ we have $\\text{depth}(\\mathcal{F}'_{n, y}) \\geq 1$,", "\\item $(\\mathcal{F}'_n)$ is isomorphic as a pro-system to an object", "$(\\mathcal{G}_n)$ of $\\textit{Coh}(X, \\mathcal{I})$,", "\\item the induced morphism", "$(\\mathcal{F}_n) \\to (\\mathcal{G}_n)$ of", "$\\textit{Coh}(X, \\mathcal{I})$ is surjective with kernel", "annihilated by a power of $\\mathcal{I}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For every $n$ we let $\\mathcal{F}_n \\to \\mathcal{F}'_n$ be the surjection", "constructed in", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-get-depth-1-along-Z}.", "Since this is the quotient of $\\mathcal{F}_n$ by the subsheaf", "of sections supported on $T$ we see that we get canonical maps", "$\\mathcal{F}'_{n + 1} \\to \\mathcal{F}'_n$ such that we obtain", "a map $(\\mathcal{F}_n) \\to (\\mathcal{F}_n')$", "of inverse systems of coherent $\\mathcal{O}_X$-modules.", "Property (1) holds by construction.", "\\medskip\\noindent", "To prove properties (2) and (3) we may assume that $X = \\Spec(A_0)$ is", "affine and $A_0$ has a dualizing complex. Let $I_0 \\subset A_0$ be the", "ideal corresponding to $Y$. Let $A, I$ be the $I$-adic completions of", "$A_0, I_0$. For later use we observe that $A$ has a dualizing complex", "(Dualizing Complexes, Lemma \\ref{dualizing-lemma-ubiquity-dualizing}).", "Let $M$ be the finite $A$-module corresponding to $(\\mathcal{F}_n)$, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-affine}.", "Then $\\mathcal{F}_n$ corresponds to $M_n = M/I^nM$. Recall that", "$\\mathcal{F}'_n$ corresponds to the quotient $M'_n = M_n / H^0_T(M_n)$,", "see Local Cohomology, Lemma \\ref{local-cohomology-lemma-get-depth-1-along-Z}", "and its proof.", "\\medskip\\noindent", "Set $s = 0$ and $d = \\text{cd}(A, I)$.", "We claim that $A, I, T, M, s, d$ satisfy assumptions (1), (3), (4), (6)", "of Situation \\ref{situation-bootstrap}.", "Namely, (1) and (3) are immediate from the above, (4) is", "the empty condition as $s = 0$, and (6) is the assumption", "we made in the statement of the lemma.", "\\medskip\\noindent", "By Theorem \\ref{theorem-final-bootstrap} we see that $\\{H^0_T(M_n)\\}$", "is Mittag-Leffler, that $\\lim H^0_T(M_n) = H^0_T(M)$, and that", "$H^0_T(M)$ is killed by a power of $I$. Thus the", "limit of the short exact sequences $0 \\to H^0_T(M_n) \\to M_n \\to M'_n \\to 0$", "is the short exact sequence", "$$", "0 \\to H^0_T(M) \\to M \\to \\lim M'_n \\to 0", "$$", "Setting $M' = \\lim M'_n$ we find that $\\mathcal{G}_n$ corresponds to", "the finite $A_0$-module $M'/I^nM'$. To finish the prove we have to show", "that the canonical map $\\{M'/I^nM'\\} \\to \\{M'_n\\}$ is a pro-isomorphism.", "This is equivalent to saying that", "$\\{H^0_T(M) + I^nM\\} \\to \\{\\ker(M \\to M'_n)\\}$ is a", "pro-isomorphism. Which in turn says that", "$\\{H^0_T(M)/H^0_T(M) \\cap I^nM\\} \\to \\{H^0_T(M_n)\\}$", "is a pro-isomorphism. This is true because $\\{H^0_T(M_n)\\}$", "is Mittag-Leffler, $\\lim H^0_T(M_n) = H^0_T(M)$, and", "$H^0_T(M)$ is killed by a power of $I$ (so that Artin-Rees", "tells us that $H^0_T(M) \\cap I^nM = 0$ for $n$ large enough)." ], "refs": [ "local-cohomology-lemma-get-depth-1-along-Z", "dualizing-lemma-ubiquity-dualizing", "coherent-lemma-inverse-systems-affine", "local-cohomology-lemma-get-depth-1-along-Z", "algebraization-theorem-final-bootstrap" ], "ref_ids": [ 9747, 2890, 3370, 9747, 12673 ] } ], "ref_ids": [] }, { "id": 12772, "type": "theorem", "label": "algebraization-lemma-improvement-formal-coherent-module-better", "categories": [ "algebraization" ], "title": "algebraization-lemma-improvement-formal-coherent-module-better", "contents": [ "In the situation above assume $X$ locally has a dualizing complex.", "Let $T' \\subset T \\subset Y$ be subsets stable under specialization.", "Let $d \\geq 0$ be an integer. Assume", "\\begin{enumerate}", "\\item[(a)] affine locally we have $X = \\Spec(A_0)$ and $Y = V(I_0)$", "and $\\text{cd}(A_0, I_0) \\leq d$,", "\\item[(b)] for $y \\in T$ and a nonmaximal prime", "$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with", "$V(\\mathfrak p) \\cap V(\\mathcal{I}_y^\\wedge) = \\{\\mathfrak m_y^\\wedge\\}$", "we have", "$$", "\\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p}", "((\\mathcal{F}^\\wedge_y)_\\mathfrak p) > 0", "$$", "\\item[(c)] for $y \\in T'$ and for a prime", "$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with", "$\\mathfrak p \\not \\in V(\\mathcal{I}_y^\\wedge)$", "and $V(\\mathfrak p) \\cap V(\\mathcal{I}_y^\\wedge) \\not =", "\\{\\mathfrak m_y^\\wedge\\}$ we have", "$$", "\\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p}", "((\\mathcal{F}^\\wedge_y)_\\mathfrak p) \\geq 1", "\\quad\\text{or}\\quad", "\\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p}", "((\\mathcal{F}^\\wedge_y)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) > 1 + d", "$$", "\\item[(d)] for $y \\in T'$ and a nonmaximal prime", "$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with", "$V(\\mathfrak p) \\cap V(\\mathcal{I}_y^\\wedge) = \\{\\mathfrak m_y^\\wedge\\}$", "we have", "$$", "\\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p}", "((\\mathcal{F}^\\wedge_y)_\\mathfrak p) > 1", "$$", "\\item[(e)] if $y \\leadsto y'$ is an immediate specialization and", "$y' \\in T'$, then $y \\in T$.", "\\end{enumerate}", "Then there exists a canonical map $(\\mathcal{F}_n) \\to (\\mathcal{F}_n'')$", "of inverse systems of coherent $\\mathcal{O}_X$-modules", "with the following properties", "\\begin{enumerate}", "\\item for $y \\in T$ we have $\\text{depth}(\\mathcal{F}''_{n, y}) \\geq 1$,", "\\item for $y' \\in T'$ we have $\\text{depth}(\\mathcal{F}''_{n, y'}) \\geq 2$,", "\\item $(\\mathcal{F}''_n)$ is isomorphic as a pro-system to an object", "$(\\mathcal{H}_n)$ of $\\textit{Coh}(X, \\mathcal{I})$,", "\\item the induced morphism $(\\mathcal{F}_n) \\to (\\mathcal{H}_n)$ of", "$\\textit{Coh}(X, \\mathcal{I})$ has kernel and cokernel", "annihilated by a power of $\\mathcal{I}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As in Lemma \\ref{lemma-divide-torsion-formal-coherent-module} and its proof", "for every $n$ we let $\\mathcal{F}_n \\to \\mathcal{F}'_n$ be the surjection", "constructed in", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-get-depth-1-along-Z}.", "Next, we let $\\mathcal{F}'_n \\to \\mathcal{F}''_n$ be the injection", "constructed in", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-make-S2-along-T}", "and its proof. The constructions show that we get canonical maps", "$\\mathcal{F}''_{n + 1} \\to \\mathcal{F}''_n$ such that we obtain", "maps", "$$", "(\\mathcal{F}_n) \\longrightarrow (\\mathcal{F}_n') \\longrightarrow", "(\\mathcal{F}''_n)", "$$", "of inverse systems of coherent $\\mathcal{O}_X$-modules.", "Properties (1) and (2) hold by construction.", "\\medskip\\noindent", "To prove properties (3) and (4) we may assume that $X = \\Spec(A_0)$ is", "affine and $A_0$ has a dualizing complex. Let $I_0 \\subset A_0$ be the", "ideal corresponding to $Y$. Let $A, I$ be the $I$-adic completions of", "$A_0, I_0$. For later use we observe that $A$ has a dualizing complex", "(Dualizing Complexes, Lemma \\ref{dualizing-lemma-ubiquity-dualizing}).", "Let $M$ be the finite $A$-module corresponding to $(\\mathcal{F}_n)$, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-affine}.", "Then $\\mathcal{F}_n$ corresponds to $M_n = M/I^nM$. Recall that", "$\\mathcal{F}'_n$ corresponds to the quotient $M'_n = M_n / H^0_T(M_n)$.", "Also, recall that $M' = \\lim M'_n$ is the quotient of $M$ by", "$H^0_T(M)$ and that $\\{M'_n\\}$ and $\\{M'/I^nM'\\}$ are isomorphic", "as pro-systems. Finally, we see that $\\mathcal{F}''_n$ corresponds", "to an extension", "$$", "0 \\to M'_n \\to M''_n \\to H^1_{T'}(M'_n) \\to 0", "$$", "see proof of", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-make-S2-along-T}.", "\\medskip\\noindent", "Set $s = 1$. We claim that $A, I, T', M', s, d$ satisfy assumptions", "(1), (3), (4), (6) of Situation \\ref{situation-bootstrap}. Namely, (1) and (3)", "are immediate, (4) is implied by (c), and (6) follows from (d).", "We omit the details of the verification (c) $\\Rightarrow$ (4).", "\\medskip\\noindent", "By Theorem \\ref{theorem-final-bootstrap} we see that $\\{H^1_{T'}(M'/I^nM')\\}$", "is Mittag-Leffler, that $H^1_{T'}(M') = \\lim H^1_{T'}(M'/I^nM')$, and that", "$H^1_{T'}(M')$ is killed by a power of $I$. We deduce", "$\\{H^1_{T'}(M'_n)\\}$ is Mittag-Leffler and $H^1_{T'}(M') = \\lim H^1_{T'}(M'_n)$.", "Thus the limit of the short exact sequences displayed above", "is the short exact sequence", "$$", "0 \\to M' \\to \\lim M''_n \\to H^1_{T'}(M') \\to 0", "$$", "Set $M'' = \\lim M''_n$. It follows from", "Local Cohomology, Proposition \\ref{local-cohomology-proposition-finiteness}", "that $H^1_{T'}(M')$", "and hence $M''$ are finite $A$-modules.", "Thus we find that $\\mathcal{H}_n$ corresponds to", "the finite $A_0$-module $M''/I^nM''$. To finish the prove we have to show", "that the canonical map $\\{M''/I^nM''\\} \\to \\{M''_n\\}$ is a pro-isomorphism.", "Since we already know that $\\{M'/I^nM'\\}$ is pro-isomorphic to", "$\\{M'_n\\}$ the reader verifies (omitted) this is equivalent to asking", "$\\{H^1_{T'}(M')/I^nH^1_{T'}(M')\\} \\to \\{H^1_{T'}(M'_n)\\}$", "to be a pro-isomorphism. This is true because $\\{H^1_{T'}(M'_n)\\}$", "is Mittag-Leffler, $H^1_{T'}(M') = \\lim H^1_{T'}(M'_n)$, and", "$H^1_{T'}(M')$ is killed by a power of $I$." ], "refs": [ "algebraization-lemma-divide-torsion-formal-coherent-module", "local-cohomology-lemma-get-depth-1-along-Z", "local-cohomology-lemma-make-S2-along-T", "dualizing-lemma-ubiquity-dualizing", "coherent-lemma-inverse-systems-affine", "local-cohomology-lemma-make-S2-along-T", "algebraization-theorem-final-bootstrap", "local-cohomology-proposition-finiteness" ], "ref_ids": [ 12771, 9747, 9751, 2890, 3370, 9751, 12673, 9787 ] } ], "ref_ids": [] }, { "id": 12773, "type": "theorem", "label": "algebraization-lemma-improvement-application", "categories": [ "algebraization" ], "title": "algebraization-lemma-improvement-application", "contents": [ "In Situation \\ref{situation-algebraize} assume that $A$ has", "a dualizing complex. Let $d \\geq \\text{cd}(A, I)$. Let $(\\mathcal{F}_n)$", "be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume", "$(\\mathcal{F}_n)$ satisfies the $(2, 2 + d)$-inequalities, see", "Definition \\ref{definition-s-d-inequalities}.", "Then there exists a canonical map $(\\mathcal{F}_n) \\to (\\mathcal{F}_n'')$", "of inverse systems of coherent $\\mathcal{O}_U$-modules", "with the following properties", "\\begin{enumerate}", "\\item if $\\text{depth}(\\mathcal{F}''_{n, y}) + \\delta^Y_Z(y) \\geq 3$", "for all $y \\in U \\cap Y$,", "\\item $(\\mathcal{F}''_n)$ is isomorphic as a pro-system to an object", "$(\\mathcal{H}_n)$ of $\\textit{Coh}(U, I\\mathcal{O}_U)$,", "\\item the induced morphism $(\\mathcal{F}_n) \\to (\\mathcal{H}_n)$ of", "$\\textit{Coh}(U, I\\mathcal{O}_U)$ has kernel and cokernel", "annihilated by a power of $I$,", "\\item the modules $H^0(U, \\mathcal{F}''_n)$ and $H^1(U, \\mathcal{F}''_n)$", "are finite $A$-modules for all $n$.", "\\end{enumerate}" ], "refs": [ "algebraization-definition-s-d-inequalities" ], "proofs": [ { "contents": [ "The existence and properties (2), (3), (4) follow immediately from", "Lemma \\ref{lemma-improvement-formal-coherent-module-better} applied", "to $U$, $U \\cap Y$, $T = \\{y \\in U \\cap Y : \\delta^Y_Z(y) \\leq 2\\}$,", "$T' = \\{y \\in U \\cap Y : \\delta^Y_Z(y) \\leq 1\\}$, and $(\\mathcal{F}_n)$.", "The finiteness of the modules $H^0(U, \\mathcal{F}''_n)$ and", "$H^1(U, \\mathcal{F}''_n)$ follows from", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-finiteness-Rjstar}", "and the elementary properties of the function $\\delta^Y_Z(-)$", "proved in Lemma \\ref{lemma-discussion}." ], "refs": [ "algebraization-lemma-improvement-formal-coherent-module-better", "local-cohomology-lemma-finiteness-Rjstar", "algebraization-lemma-discussion" ], "ref_ids": [ 12772, 9743, 12758 ] } ], "ref_ids": [ 12806 ] }, { "id": 12774, "type": "theorem", "label": "algebraization-lemma-cd-1-canonical", "categories": [ "algebraization" ], "title": "algebraization-lemma-cd-1-canonical", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$", "be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex and $\\text{cd}(A, I) = 1$,", "\\item $(\\mathcal{F}_n)$ is pro-isomorphic to an inverse system", "$(\\mathcal{F}_n'')$ of coherent $\\mathcal{O}_U$-modules such that", "$\\text{depth}(\\mathcal{F}''_{n, y}) + \\delta^Y_Z(y) \\geq 3$", "for all $y \\in U \\cap Y$.", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ extends canonically to $X$, see", "Definition \\ref{definition-canonically-algebraizable}." ], "refs": [ "algebraization-definition-canonically-algebraizable" ], "proofs": [ { "contents": [ "We will check hypotheses (a), (b), and (c) of Lemma \\ref{lemma-when-done}.", "Before we start, let us point out that the modules", "$H^0(U, \\mathcal{F}''_n)$ and $H^1(U, \\mathcal{F}''_n)$", "are finite $A$-modules for all $n$ by", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-finiteness-Rjstar}.", "\\medskip\\noindent", "Observe that for each $p \\geq 0$", "the limit topology on $\\lim H^p(U, \\mathcal{F}_n)$", "is the $I$-adic topology by Lemma \\ref{lemma-topology-I-adic}.", "In particular, hypothesis (b) holds.", "\\medskip\\noindent", "We know that $M = \\lim H^0(U, \\mathcal{F}_n)$ is an $A$-module whose", "limit topology is the $I$-adic topology. Thus, given $n$, the module", "$M/I^nM$ is a subquotient of $H^0(U, \\mathcal{F}_N)$ for some $N \\gg n$.", "Since the inverse system $\\{H^0(U, \\mathcal{F}_N)\\}$ is pro-isomorphic to an", "inverse system of finite $A$-modules, namely $\\{H^0(U, \\mathcal{F}''_N)\\}$,", "we conclude that $M/I^nM$ is finite. It follows that $M$ is finite, see", "Algebra, Lemma \\ref{algebra-lemma-finite-over-complete-ring}.", "In particular hypothesis (c) holds.", "\\medskip\\noindent", "For each $n \\geq 0$ let us write $Ob_n = \\lim_N H^1(U, I^n\\mathcal{F}_N)$.", "A special case is $Ob = Ob_0 = \\lim_N H^1(U, \\mathcal{F}_N)$.", "Arguing exactly as in the previous paragraph we find that $Ob$", "is a finite $A$-module. (In fact, we also know that $Ob/I Ob$ is annihilated", "by a power of $\\mathfrak a$, but it seems somewhat difficult to use this.)", "\\medskip\\noindent", "We set $\\mathcal{F} = \\lim \\mathcal{F}_n$, we pick generators", "$f_1, \\ldots, f_r$ of $I$, we pick $c \\geq 1$, and we choose", "$\\Phi_\\mathcal{F}$ as in Lemma \\ref{lemma-cd-is-one-for-system}.", "We will use the results of Lemma \\ref{lemma-properties-system}", "without further mention. In particular, for each $n \\geq 1$ there are maps", "$$", "\\delta_n :", "H^0(U, \\mathcal{F}_n)", "\\longrightarrow", "H^1(U, I^n\\mathcal{F})", "\\longrightarrow", "Ob_n", "$$", "The first comes from the short exact sequence", "$0 \\to I^n\\mathcal{F} \\to \\mathcal{F} \\to \\mathcal{F}_n \\to 0$", "and the second from $I^n\\mathcal{F} = \\lim I^n\\mathcal{F}_N$.", "We will later use that if $\\delta_n(s) = 0$ for $s \\in H^0(U, \\mathcal{F}_n)$", "then we can for each $n' \\geq n$ find $s' \\in H^0(U, \\mathcal{F}_{n'})$", "mapping to $s$.", "Observe that there are commutative diagrams", "$$", "\\xymatrix{", "H^0(U, \\mathcal{F}_{nc}) \\ar[r] \\ar[dd] &", "H^1(U, I^{nc}\\mathcal{F}) \\ar[dd] \\ar[rd]^{\\Phi_\\mathcal{F}} \\\\", "& &", "\\bigoplus_{e_1 + \\ldots + e_r = n}", "H^1(U, \\mathcal{F}) \\cdot T_1^{e_1} \\ldots T_r^{e_r} \\ar[ld] \\\\", "H^0(U, \\mathcal{F}_n) \\ar[r] &", "H^1(U, I^n\\mathcal{F})", "}", "$$", "We conclude that the obstruction map", "$H^0(U, \\mathcal{F}_n) \\to Ob_n$", "sends the image of", "$H^0(U, \\mathcal{F}_{nc}) \\to H^0(U, \\mathcal{F}_n)$", "into the submodule", "$$", "Ob'_n =", "\\Im\\left(", "\\bigoplus\\nolimits_{e_1 + \\ldots + e_r = n}", "Ob \\cdot T_1^{e_1} \\ldots T_r^{e_r} \\to Ob_n", "\\right)", "$$", "where on the summand $Ob \\cdot T_1^{e_1} \\ldots T_r^{e_r}$", "we use the map on cohomology coming from the reductions modulo", "powers of $I$ of the multiplication map", "$f_1^{e_1} \\ldots f_r^{e_r} : \\mathcal{F} \\to I^n\\mathcal{F}$.", "By construction", "$$", "\\bigoplus\\nolimits_{n \\geq 0} Ob'_n", "$$", "is a finite graded module over the Rees algebra $\\bigoplus_{n \\geq 0} I^n$.", "For each $n$ we set", "$$", "M_n = \\{s \\in H^0(U, \\mathcal{F}_n) \\mid \\delta_n(s) \\in Ob'_n\\}", "$$", "Observe that $\\{M_n\\}$ is an inverse system and that", "$f_j : \\mathcal{F}_n \\to \\mathcal{F}_{n + 1}$ on global", "sections maps $M_n$ into $M_{n + 1}$.", "By exactly the same argument as in the proof of", "Lemma \\ref{lemma-ML-general}", "we find that $\\{M_n\\}$ is ML. Namely, because the Rees algebra", "is Noetherian we can choose a finite number of homogeneous generators", "of the form $\\delta_{n_j}(z_j)$ with $z_j \\in M_{n_j}$ for the graded submodule", "$\\bigoplus_{n \\geq 0} \\Im(M_n \\to Ob'_n)$.", "Then if $k = \\max(n_j)$ we find that for $n \\geq k$", "and any $z \\in M_n$ we can find $a_j \\in I^{n - n_j}$ such that", "$z - \\sum a_j z_j$ is in the kernel of $\\delta_n$", "and hence in the image of $M_{n'}$ for all $n' \\geq n$", "(because the vanishing of $\\delta_n$ means that we can", "lift $z - \\sum a_j z_j$ to an element $z' \\in H^0(U, \\mathcal{F}_{n'c})$", "for all $n' \\ge n$ and then the image of $z'$ in $H^0(U, \\mathcal{F}_{n'})$", "is in $M_{n'}$ by what we proved above).", "Thus $\\Im(M_n \\to M_{n - k}) = \\Im(M_{n'} \\to M_{n - k})$", "for all $n' \\geq n$.", "\\medskip\\noindent", "Choose $n$. By the Mittag-Leffler property of $\\{M_n\\}$ we just established", "we can find an $n' \\geq n$ such that the image of $M_{n'} \\to M_n$", "is the same as the image of $M' \\to M_n$. By the above we see that", "the image of $M' \\to M_n$ contains the image of", "$H^0(U, \\mathcal{F}_{n'c}) \\to H^0(U, \\mathcal{F}_n)$.", "Thus we see that $\\{M_n\\}$ and $\\{H^0(U, \\mathcal{F}_n)\\}$", "are pro-isomorphic. Therefore $\\{H^0(U, \\mathcal{F}_n)\\}$", "has ML and we finally conclude that hypothesis (a) holds.", "This concludes the proof." ], "refs": [ "algebraization-lemma-when-done", "local-cohomology-lemma-finiteness-Rjstar", "algebraization-lemma-topology-I-adic", "algebra-lemma-finite-over-complete-ring", "algebraization-lemma-cd-is-one-for-system", "algebraization-lemma-properties-system", "algebraization-lemma-ML-general" ], "ref_ids": [ 12753, 9743, 12689, 868, 12688, 12678, 12675 ] } ], "ref_ids": [ 12805 ] }, { "id": 12775, "type": "theorem", "label": "algebraization-lemma-blowup", "categories": [ "algebraization" ], "title": "algebraization-lemma-blowup", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$", "be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex,", "\\item all fibres of the blowing up $b : X' \\to X$ of $I$", "have dimension $\\leq d - 1$,", "\\item one of the following is true", "\\begin{enumerate}", "\\item $(\\mathcal{F}_n)$ satisfies the $(d + 1, d + 2)$-inequalities", "(Definition \\ref{definition-s-d-inequalities}), or", "\\item for $y \\in U \\cap Y$ and a prime", "$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with", "$\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$", "we have", "$$", "\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > d + 2", "$$", "\\end{enumerate}", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ extends to $X$." ], "refs": [ "algebraization-definition-s-d-inequalities" ], "proofs": [ { "contents": [ "Let $Y' \\subset X'$ be the exceptional divisor.", "Let $Z' \\subset Y'$ be the inverse image of $Z \\subset Y$.", "Then $U' = X' \\setminus Z'$ is the inverse image of $U$.", "With $\\delta^{Y'}_{Z'}$ as in (\\ref{equation-delta-Z}) we set", "$$", "T' = \\{y' \\in Y' \\mid \\delta^{Y'}_{Z'}(y') = 1\\text{ or }2\\}", "\\subset", "T = \\{y' \\in Y' \\mid \\delta^{Y'}_{Z'}(y') = 1\\}", "$$", "These are specialization stable subsets of", "$U' \\cap Y' = Y' \\setminus Z'$. Consider the", "object $(b|_{U'}^*\\mathcal{F}_n)$ of $\\textit{Coh}(U', I\\mathcal{O}_{U'})$,", "see Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-pullback}.", "For $y' \\in U' \\cap Y'$ let us denote", "$$", "\\mathcal{F}_{y'}^\\wedge = \\lim (b|_{U'}^*\\mathcal{F}_n)_{y'}", "$$", "the ``stalk'' of this pullback at $y'$. We claim that conditions", "(a), (b), (c), (d), and (e) of", "Lemma \\ref{lemma-improvement-formal-coherent-module-better}", "hold for the object $(b|_{U'}^*\\mathcal{F}_n)$ on $U'$ with $d$", "replaced by $1$ and the subsets $T' \\subset T \\subset U' \\cap Y'$.", "Condition (a) holds because $Y'$ is an effective Cartier divisor", "and hence locally cut out by $1$ equation. Condition (e) holds", "by Lemma \\ref{lemma-discussion} parts (1) and (2).", "To prove (b), (c), and (d) we need some preparation.", "\\medskip\\noindent", "Let $y' \\in U' \\cap Y'$ and let", "$\\mathfrak p' \\subset \\mathcal{O}_{X', y'}^\\wedge$", "be a prime ideal not contained in $V(I\\mathcal{O}_{X', y'}^\\wedge)$.", "Denote $y = b(y') \\in U \\cap Y$. Choose $f \\in I$ such that", "$y'$ is contained in the spectrum of the affine blowup algebra", "$A[\\frac{I}{f}]$, see Divisors, Lemma \\ref{divisors-lemma-blowing-up-affine}.", "For any $A$-algebra $B$ denote $B' = B[\\frac{IB}{f}]$ the corresponding affine", "blowup algebra. Denote $I$-adic completion by ${\\ }^\\wedge$.", "By our choice of $f$ we get a ring map", "$(\\mathcal{O}_{X, y}^\\wedge)' \\to \\mathcal{O}_{X', y'}^\\wedge$.", "If we let $\\mathfrak q' \\subset (\\mathcal{O}_{X, y}^\\wedge)'$", "be the inverse image of $\\mathfrak m_{y'}^\\wedge$, then", "we see that", "$((\\mathcal{O}_{X, y}^\\wedge)'_{\\mathfrak q'})^\\wedge =", "\\mathcal{O}_{X', y'}^\\wedge$.", "Let $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ be the corresponding", "prime. At this point we have a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{O}_{X, y}^\\wedge \\ar[d] \\ar[r] &", "(\\mathcal{O}_{X, y}^\\wedge)' \\ar[d]_\\alpha \\ar[r] &", "(\\mathcal{O}_{X, y}^\\wedge)'_{\\mathfrak q'} \\ar[d] \\ar[r]_\\beta &", "\\mathcal{O}_{X', y'}^\\wedge \\ar[d] \\\\", "\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p \\ar[r] &", "(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p)' \\ar[r] &", "(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p)'_{\\mathfrak q'} \\ar[r]^\\gamma &", "((\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p)'_{\\mathfrak q'})^\\wedge \\ar[d] \\\\", " & & &", "\\mathcal{O}_{X', y'}^\\wedge/\\mathfrak p'", "}", "$$", "whose vertical arrows are surjective. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-completion-dimension}", "and the dimension formula", "(Algebra, Lemma \\ref{algebra-lemma-dimension-formula})", "we have", "$$", "\\dim(((\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p)'_{\\mathfrak q'})^\\wedge) =", "\\dim((\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p)'_{\\mathfrak q'}) =", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p)", "- \\text{trdeg}(\\kappa(y')/\\kappa(y))", "$$", "Tracing through the definitions of pullbacks, stalks, localizations,", "and completions we find", "$$", "(\\mathcal{F}_y^\\wedge)_{\\mathfrak p}", "\\otimes_{(\\mathcal{O}_{X, y}^\\wedge)_\\mathfrak p}", "(\\mathcal{O}_{X', y'}^\\wedge)_{\\mathfrak p'}", "=", "(\\mathcal{F}_{y'}^\\wedge)_{\\mathfrak p'}", "$$", "Details omitted. The ring maps $\\beta$ and $\\gamma$ in the diagram", "are flat with Gorenstein (hence Cohen-Macaulay) fibres, as these are", "completions of rings having a dualizing complex. See", "Dualizing Complexes, Lemmas", "\\ref{dualizing-lemma-formal-fibres-gorenstein} and", "\\ref{dualizing-lemma-dualizing-gorenstein-formal-fibres}", "and the discussion in More on Algebra, Section", "\\ref{more-algebra-section-properties-formal-fibres}.", "Observe that $(\\mathcal{O}_{X, y}^\\wedge)_\\mathfrak p =", "(\\mathcal{O}_{X, y}^\\wedge)'_{\\tilde{\\mathfrak p}}$", "where $\\tilde{\\mathfrak p}$ is the kernel of $\\alpha$", "in the diagram. On the other hand,", "$(\\mathcal{O}_{X, y}^\\wedge)'_{\\tilde{\\mathfrak p}}", "\\to (\\mathcal{O}_{X', y'}^\\wedge)_{\\mathfrak p'}$", "is flat with CM fibres by the above. Whence", "$(\\mathcal{O}_{X, y}^\\wedge)_\\mathfrak p \\to", "(\\mathcal{O}_{X', y'}^\\wedge)_{\\mathfrak p'}$ is flat with CM fibres.", "Using Algebra, Lemma \\ref{algebra-lemma-apply-grothendieck-module}", "we see that", "$$", "\\text{depth}((\\mathcal{F}_{y'}^\\wedge)_{\\mathfrak p'}) =", "\\text{depth}((\\mathcal{F}_y^\\wedge)_{\\mathfrak p}) +", "\\dim(F_\\mathfrak r)", "$$", "where $F$ is the generic formal fibre of", "$(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p)'_{\\mathfrak q'}$", "and $\\mathfrak r$ is the prime corresponding to $\\mathfrak p'$.", "Since $(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p)'_{\\mathfrak q'}$", "is a universally catenary local domain, its $I$-adic completion", "is equidimensional and (universally) catenary by Ratliff's theorem", "(More on Algebra, Proposition \\ref{more-algebra-proposition-ratliff}).", "It then follows that", "$$", "\\dim(((\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p)'_{\\mathfrak q'})^\\wedge) =", "\\dim(F_\\mathfrak r) + \\dim(\\mathcal{O}_{X', y'}^\\wedge/\\mathfrak p')", "$$", "Combined with Lemma \\ref{lemma-change-distance-function}", "we get", "\\begin{equation}", "\\label{equation-one}", "\\begin{aligned}", "&", "\\text{depth}((\\mathcal{F}_{y'}^\\wedge)_{\\mathfrak p'}) +", "\\delta^{Y'}_{Z'}(y') \\\\", "& =", "\\text{depth}((\\mathcal{F}_y^\\wedge)_{\\mathfrak p}) +", "\\dim(F_\\mathfrak r) + \\delta^{Y'}_{Z'}(y') \\\\", "& \\geq", "\\text{depth}((\\mathcal{F}_y^\\wedge)_{\\mathfrak p}) + \\delta^Y_Z(y) +", "\\dim(F_\\mathfrak r) + \\text{trdeg}(\\kappa(y')/\\kappa(y)) - (d - 1) \\\\", "& =", "\\text{depth}((\\mathcal{F}_y^\\wedge)_{\\mathfrak p}) + \\delta^Y_Z(y) - (d - 1)", "+ \\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) -", "\\dim(\\mathcal{O}_{X', y'}^\\wedge/\\mathfrak p')", "\\end{aligned}", "\\end{equation}", "Please keep in mind that", "$\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) \\geq", "\\dim(\\mathcal{O}_{X', y'}^\\wedge/\\mathfrak p')$. Rewriting this we get", "\\begin{equation}", "\\label{equation-two}", "\\begin{aligned}", "&", "\\text{depth}((\\mathcal{F}_{y'}^\\wedge)_{\\mathfrak p'}) +", "\\dim(\\mathcal{O}_{X', y'}^\\wedge/\\mathfrak p') +", "\\delta^{Y'}_{Z'}(y') \\\\", "& \\geq", "\\text{depth}((\\mathcal{F}_y^\\wedge)_{\\mathfrak p}) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) +", "\\delta^Y_Z(y) - (d - 1)", "\\end{aligned}", "\\end{equation}", "This inequality will allow us to check the remaning conditions.", "\\medskip\\noindent", "Conditions (b) and (d) of", "Lemma \\ref{lemma-improvement-formal-coherent-module-better}. Assume", "$V(\\mathfrak p') \\cap V(I\\mathcal{O}_{X', y'}^\\wedge) =", "\\{\\mathfrak m_{y'}^\\wedge\\}$.", "This implies that $\\dim(\\mathcal{O}_{X', y'}^\\wedge/\\mathfrak p') = 1$", "because $Z'$ is an effective Cartier divisor.", "The combination of (b) and (d) is equivalent with", "$$", "\\text{depth}((\\mathcal{F}_{y'}^\\wedge)_{\\mathfrak p'}) + \\delta^{Y'}_{Z'}(y')", "> 2", "$$", "If $(\\mathcal{F}_n)$ satisfies the inequalities in (3)(b)", "then we immediately conclude this is true by applying (\\ref{equation-two}).", "If $(\\mathcal{F}_n)$ satisfies (3)(a), i.e., the", "$(d + 1, d + 2)$-inequalities, then we see that in any case", "$$", "\\text{depth}((\\mathcal{F}_y^\\wedge)_{\\mathfrak p}) + \\delta^Y_Z(y)", "\\geq d + 1", "\\quad\\text{or}\\quad", "\\text{depth}((\\mathcal{F}_y^\\wedge)_{\\mathfrak p}) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) +", "\\delta^Y_Z(y) > d + 2", "$$", "Looking at (\\ref{equation-one}) and (\\ref{equation-two}) above this gives", "what we want except possibly if", "$\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) = 1$.", "However, if $\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) = 1$, then we have", "$V(\\mathfrak p) \\cap V(I\\mathcal{O}_{X, y}^\\wedge) = \\{\\mathfrak m_y^\\wedge\\}$", "and we see that actually", "$$", "\\text{depth}((\\mathcal{F}_y^\\wedge)_{\\mathfrak p}) + \\delta^Y_Z(y) > d + 1", "$$", "as $(\\mathcal{F}_n)$ satisfies the $(d + 1, d + 2)$-inequalities and we", "conclude again.", "\\medskip\\noindent", "Condition (c) of", "Lemma \\ref{lemma-improvement-formal-coherent-module-better}. Assume", "$V(\\mathfrak p') \\cap V(I\\mathcal{O}_{X', y'}^\\wedge) \\not =", "\\{\\mathfrak m_{y'}^\\wedge\\}$. Then condition (c) is equivalent to", "$$", "\\text{depth}((\\mathcal{F}_{y'}^\\wedge)_{\\mathfrak p'}) + \\delta^{Y'}_{Z'}(y')", "\\geq 2", "\\quad\\text{or}\\quad", "\\text{depth}((\\mathcal{F}_{y'}^\\wedge)_{\\mathfrak p'}) +", "\\dim(\\mathcal{O}_{X', y'}^\\wedge/\\mathfrak p') +", "\\delta^{Y'}_{Z'}(y') > 3", "$$", "If $(\\mathcal{F}_n)$ satisfies the inequalities in (3)(b)", "then we see the second of the two displayed inequalities holds true", "by applying (\\ref{equation-two}). If $(\\mathcal{F}_n)$ satisfies (3)(a), i.e.,", "the $(d + 1, d + 2)$-inequalities, then this follows immediately from", "(\\ref{equation-one}) and (\\ref{equation-two}).", "This finishes the proof of our claim.", "\\medskip\\noindent", "Choose $(b|_{U'}^*\\mathcal{F}_n) \\to (\\mathcal{F}_n'')$", "and $(\\mathcal{H}_n)$ in $\\textit{Coh}(U', I\\mathcal{O}_{U'})$", "as in Lemma \\ref{lemma-improvement-formal-coherent-module-better}.", "For any affine open $W \\subset X'$ observe that", "$\\delta^{W \\cap Y'}_{W \\cap Z'}(y') \\geq \\delta^{Y'}_{Z'}(y')$ by", "Lemma \\ref{lemma-discussion} part (7). Hence we see that", "$(\\mathcal{H}_n|_W)$ satisfies the assumptions of", "Lemma \\ref{lemma-cd-1-canonical}.", "Thus $(\\mathcal{H}_n|_W)$ extends canonically to $W$.", "Let $(\\mathcal{G}_{W, n})$ in $\\textit{Coh}(W, I\\mathcal{O}_W)$", "be the canonical extension as in", "Lemma \\ref{lemma-canonically-algebraizable}.", "By Lemma \\ref{lemma-canonically-extend-base-change}", "we see that for $W' \\subset W$ there is a unique isomorphism", "$$", "(\\mathcal{G}_{W, n}|_{W'}) \\longrightarrow", "(\\mathcal{G}_{W', n})", "$$", "compatible with the given isomorphisms", "$(\\mathcal{G}_{W, n}|_{W \\cap U}) \\cong (\\mathcal{H}_n|_{W \\cap U})$.", "We conclude that there exists an object", "$(\\mathcal{G}_n)$ of $\\textit{Coh}(X', I\\mathcal{O}_{X'})$", "whose restriction to $U$ is isomorphic to $(\\mathcal{H}_n)$.", "\\medskip\\noindent", "If $A$ is $I$-adically complete we can finish the proof as follows.", "By Grothedieck's existence theorem", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-existence-projective})", "we see that $(\\mathcal{G}_n)$ is the completion of a coherent", "$\\mathcal{O}_{X'}$-module. Then by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-existence-easy}", "we see that $(b|_{U'}^*\\mathcal{F}_n)$", "is the completion of a coherent $\\mathcal{O}_{U'}$-module", "$\\mathcal{F}'$.", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-push-pull}", "we see that there is a map", "$$", "(\\mathcal{F}_n) \\longrightarrow ((b|_{U'})_*\\mathcal{F}')^\\wedge", "$$", "whose kernel and cokernel is annihilated by a power of $I$.", "Then finally, we win by applying", "Lemma \\ref{lemma-map-kernel-cokernel-on-closed}.", "\\medskip\\noindent", "If $A$ is not complete, then, before starting the proof, we may replace $A$", "by its completion, see Lemma \\ref{lemma-algebraizable}.", "After completion the assumptions still hold: this is immediate", "for condition (3), follows from", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-ubiquity-dualizing}", "for condition (1), and from", "Divisors, Lemma \\ref{divisors-lemma-flat-base-change-blowing-up}", "for condition (2).", "Thus the complete case implies the general case." ], "refs": [ "coherent-lemma-inverse-systems-pullback", "algebraization-lemma-improvement-formal-coherent-module-better", "algebraization-lemma-discussion", "divisors-lemma-blowing-up-affine", "more-algebra-lemma-completion-dimension", "algebra-lemma-dimension-formula", "dualizing-lemma-formal-fibres-gorenstein", "dualizing-lemma-dualizing-gorenstein-formal-fibres", "algebra-lemma-apply-grothendieck-module", "more-algebra-proposition-ratliff", "algebraization-lemma-change-distance-function", "algebraization-lemma-improvement-formal-coherent-module-better", "algebraization-lemma-improvement-formal-coherent-module-better", "algebraization-lemma-improvement-formal-coherent-module-better", "algebraization-lemma-discussion", "algebraization-lemma-cd-1-canonical", "algebraization-lemma-canonically-algebraizable", "algebraization-lemma-canonically-extend-base-change", "coherent-lemma-existence-projective", "coherent-lemma-existence-easy", "coherent-lemma-inverse-systems-push-pull", "algebraization-lemma-map-kernel-cokernel-on-closed", "algebraization-lemma-algebraizable", "dualizing-lemma-ubiquity-dualizing", "divisors-lemma-flat-base-change-blowing-up" ], "ref_ids": [ 3378, 12772, 12758, 8052, 10042, 990, 2891, 2892, 1360, 10591, 12759, 12772, 12772, 12772, 12758, 12774, 12751, 12752, 3383, 3375, 3385, 12755, 12750, 2890, 8053 ] } ], "ref_ids": [ 12806 ] }, { "id": 12776, "type": "theorem", "label": "algebraization-lemma-interesting-case-final", "categories": [ "algebraization" ], "title": "algebraization-lemma-interesting-case-final", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$", "be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume", "\\begin{enumerate}", "\\item $A$ is a local ring which has a dualizing complex,", "\\item all irreducible components of $X$ have the same dimension,", "\\item the scheme $X \\setminus Y$ is Cohen-Macaulay,", "\\item $I$ is generated by $d$ elements,", "\\item $\\dim(X) - \\dim(Z) > d + 2$, and", "\\item for $y \\in U \\cap Y$ the module $\\mathcal{F}_y^\\wedge$", "is finite locally free outside $V(I\\mathcal{O}_{X, y}^\\wedge)$,", "for example if $\\mathcal{F}_n$ is a finite locally free", "$\\mathcal{O}_U/I^n\\mathcal{O}_U$-module.", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ extends to $X$. In particular if $A$ is $I$-adically", "complete, then $(\\mathcal{F}_n)$ is the completion of a coherent", "$\\mathcal{O}_U$-module." ], "refs": [], "proofs": [ { "contents": [ "We will show that the hypotheses (1), (2), (3)(b) of", "Proposition \\ref{proposition-d-generators} are satisfied.", "This is clear for (1) and (2).", "\\medskip\\noindent", "Let $y \\in U \\cap Y$ and let $\\mathfrak p$ be a prime", "$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with", "$\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$.", "The last condition shows that", "$\\text{depth}((\\mathcal{F}_y^\\wedge)_\\mathfrak p) =", "\\text{depth}((\\mathcal{O}_{X, y}^\\wedge)_\\mathfrak p)$.", "Since $X \\setminus Y$ is Cohen-Macaulay we see that", "$(\\mathcal{O}_{X, y}^\\wedge)_\\mathfrak p$ is Cohen-Macaulay.", "Thus we see that", "\\begin{align*}", "& \\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) \\\\", "& =", "\\dim((\\mathcal{O}_{X, y}^\\wedge)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) \\\\", "& =", "\\dim(\\mathcal{O}_{X, y}^\\wedge) + \\delta^Y_Z(y)", "\\end{align*}", "The final equality because $\\mathcal{O}_{X, y}$ is equidimensional", "by the second condition.", "Let $\\delta(y) = \\dim(\\overline{\\{y\\}})$. This is a dimension function", "as $A$ is a catenary local ring.", "By Lemma \\ref{lemma-discussion}", "we have $\\delta^Y_Z(y) \\geq \\delta(y) - \\dim(Z)$. Since $X$ is", "equidimensional we get", "$$", "\\dim(\\mathcal{O}_{X, y}^\\wedge) + \\delta^Y_Z(y)", "\\geq \\dim(\\mathcal{O}_{X, y}^\\wedge) + \\delta(y) - \\dim(Z)", "= \\dim(X) - \\dim(Z)", "$$", "Thus we get the desired inequality and we win." ], "refs": [ "algebraization-proposition-d-generators", "algebraization-lemma-discussion" ], "ref_ids": [ 12794, 12758 ] } ], "ref_ids": [] }, { "id": 12777, "type": "theorem", "label": "algebraization-lemma-equivalence-better", "categories": [ "algebraization" ], "title": "algebraization-lemma-equivalence-better", "contents": [ "In Situation \\ref{situation-algebraize} assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex and is $I$-adically complete,", "\\item $I = (f)$ generated by a single element,", "\\item $A$ is local with maximal ideal $\\mathfrak a = \\mathfrak m$,", "\\item one of the following is true", "\\begin{enumerate}", "\\item $A_f$ is $(S_2)$ and for $\\mathfrak p \\subset A$,", "$f \\not \\in \\mathfrak p$ minimal we have $\\dim(A/\\mathfrak p) \\geq 4$, or", "\\item if $\\mathfrak p \\not \\in V(f)$ and", "$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then", "$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.", "\\end{enumerate}", "\\end{enumerate}", "Then with $U_0 = U \\cap V(f)$ the completion functor", "$$", "\\colim_{U_0 \\subset U' \\subset U\\text{ open}}", "\\textit{Coh}(\\mathcal{O}_{U'})", "\\longrightarrow", "\\textit{Coh}(U, f\\mathcal{O}_U)", "$$", "is an equivalence on the full subcategories of finite locally free objects." ], "refs": [], "proofs": [ { "contents": [ "It follows from Lemma \\ref{lemma-fully-faithful-general}", "that the functor is fully faithful (details omitted).", "Let us prove essential surjectivity. Let $(\\mathcal{F}_n)$ be a finite locally", "free object of $\\textit{Coh}(U, f\\mathcal{O}_U)$. By either", "Lemma \\ref{lemma-algebraization-principal-bis} or", "Proposition \\ref{proposition-cd-1}", "there exists a coherent $\\mathcal{O}_U$-module $\\mathcal{F}$", "such that $(\\mathcal{F}_n)$ is the completion of $\\mathcal{F}$.", "Namely, for the application of either result the only thing to", "check is that $(\\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities.", "This is done in Lemma \\ref{lemma-unwinding-conditions-bis}. If $y \\in U_0$,", "then the $f$-adic completion of the stalk $\\mathcal{F}_y$ is isomorphic to", "a finite free module over the $f$-adic completion of $\\mathcal{O}_{U, y}$.", "Hence $\\mathcal{F}$ is finite locally free in an open neighbourhood", "$U'$ of $U_0$. This finishes the proof." ], "refs": [ "algebraization-lemma-fully-faithful-general", "algebraization-lemma-algebraization-principal-bis", "algebraization-proposition-cd-1", "algebraization-lemma-unwinding-conditions-bis" ], "ref_ids": [ 12745, 12768, 12793, 12770 ] } ], "ref_ids": [] }, { "id": 12778, "type": "theorem", "label": "algebraization-lemma-equivalence", "categories": [ "algebraization" ], "title": "algebraization-lemma-equivalence", "contents": [ "In Situation \\ref{situation-algebraize} assume", "\\begin{enumerate}", "\\item $I = (f)$ is principal,", "\\item $A$ is $f$-adically complete,", "\\item $f$ is a nonzerodivisor,", "\\item $H^1_\\mathfrak a(A/fA)$ and $H^2_\\mathfrak a(A/fA)$", "are finite $A$-modules.", "\\end{enumerate}", "Then with $U_0 = U \\cap V(f)$ the completion functor", "$$", "\\colim_{U_0 \\subset U' \\subset U\\text{ open}}", "\\textit{Coh}(\\mathcal{O}_{U'})", "\\longrightarrow", "\\textit{Coh}(U, f\\mathcal{O}_U)", "$$", "is an equivalence on the full subcategories of finite locally free objects." ], "refs": [], "proofs": [ { "contents": [ "The functor is fully faithful by", "Lemma \\ref{lemma-fully-faithful-general-alternative}.", "Essential surjectivity follows from", "Lemma \\ref{lemma-algebraization-principal-variant}." ], "refs": [ "algebraization-lemma-fully-faithful-general-alternative", "algebraization-lemma-algebraization-principal-variant" ], "ref_ids": [ 12746, 12754 ] } ], "ref_ids": [] }, { "id": 12779, "type": "theorem", "label": "algebraization-lemma-prepare-chi-triple", "categories": [ "algebraization" ], "title": "algebraization-lemma-prepare-chi-triple", "contents": [ "For any coherent triple $(\\mathcal{F}, \\mathcal{F}_0, \\alpha)$", "there exists a coherent $\\mathcal{O}_X$-module $\\mathcal{F}'$", "such that $f : \\mathcal{F}' \\to \\mathcal{F}'$ is injective,", "an isomorphism $\\alpha' : \\mathcal{F}'|_U \\to \\mathcal{F}$, and a map", "$\\alpha'_0 : \\mathcal{F}'/f\\mathcal{F}' \\to \\mathcal{F}_0$", "such that $\\alpha \\circ (\\alpha' \\bmod f) = \\alpha'_0|_{U_0}$." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite $A$-module $M$ such that $\\mathcal{F}$ is the restriction", "to $U$ of the coherent $\\mathcal{O}_X$-module associated to $M$, see", "Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}.", "Since $\\mathcal{F}$ is $f$-torsion free, we may replace $M$ by", "its quotient by $f$-power torsion.", "On the other hand, let $M_0 = \\Gamma(X_0, \\mathcal{F}_0)$", "so that $\\mathcal{F}_0$ is the coherent $\\mathcal{O}_{X_0}$-module", "associated to the finite $A/fA$-module $M_0$.", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-homs-over-open}", "there exists an $n$ such that the", "isomorphism $\\alpha_0$ corresponds to an $A/fA$-module homomorphism", "$\\mathfrak m^n M/fM \\to M_0$ (whose kernel and cokernel", "are annihilated by a power of $\\mathfrak m$, but we don't need this).", "Thus if we take $M' = \\mathfrak m^n M$ and we let", "$\\mathcal{F}'$ be the coherent $\\mathcal{O}_X$-module", "associated to $M'$, then the lemma is clear." ], "refs": [ "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "coherent-lemma-homs-over-open" ], "ref_ids": [ 9729, 3322 ] } ], "ref_ids": [] }, { "id": 12780, "type": "theorem", "label": "algebraization-lemma-well-defined-chi-triple", "categories": [ "algebraization" ], "title": "algebraization-lemma-well-defined-chi-triple", "contents": [ "The quantity $\\chi(\\mathcal{F}, \\mathcal{F}_0, \\alpha)$ in", "(\\ref{equation-chi-triple}) does not depend on the choice of", "$\\mathcal{F}', \\alpha', \\alpha'_0$ as in Lemma \\ref{lemma-prepare-chi-triple}." ], "refs": [ "algebraization-lemma-prepare-chi-triple" ], "proofs": [ { "contents": [ "Let $\\mathcal{F}', \\alpha', \\alpha'_0$ and", "$\\mathcal{F}'', \\alpha'', \\alpha''_0$ be two such choices.", "For $n > 0$ set $\\mathcal{F}'_n = \\mathfrak m^n \\mathcal{F}'$.", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-homs-over-open}", "for some $n$", "there exists an $\\mathcal{O}_X$-module map $\\mathcal{F}'_n \\to \\mathcal{F}''$", "agreeing with the identification", "$\\mathcal{F}''|_U = \\mathcal{F}'|_U$ determined by $\\alpha'$ and $\\alpha''$.", "Then the diagram", "$$", "\\xymatrix{", "\\mathcal{F}'_n/f\\mathcal{F}'_n \\ar[r] \\ar[d] &", "\\mathcal{F}'/f\\mathcal{F}' \\ar[d]^{\\alpha_0'} \\\\", "\\mathcal{F}''/f\\mathcal{F}'' \\ar[r]^{\\alpha_0''} &", "\\mathcal{F}_0", "}", "$$", "is commutative after restricting to $U_0$. Hence by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-homs-over-open}", "it is commutative after restricting to", "$\\mathfrak m^l(\\mathcal{F}'_n/f\\mathcal{F}'_n)$ for some $l > 0$. Since", "$\\mathcal{F}'_{n + l}/f\\mathcal{F}'_{n + l} \\to \\mathcal{F}'_n/f\\mathcal{F}'_n$", "factors through $\\mathfrak m^l(\\mathcal{F}'_n/f\\mathcal{F}'_n)$", "we see that after replacing $n$ by $n + l$ the diagram", "is commutative. In other words, we have found a third choice", "$\\mathcal{F}''', \\alpha''', \\alpha'''_0$", "such that there are maps $\\mathcal{F}''' \\to \\mathcal{F}''$", "and $\\mathcal{F}''' \\to \\mathcal{F}'$ over $X$", "compatible with the maps over $U$ and $X_0$. This reduces us to", "the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume we have a map $\\mathcal{F}'' \\to \\mathcal{F}'$ over $X$ compatible with", "$\\alpha', \\alpha''$ over $U$ and with $\\alpha'_0, \\alpha''_0$ over $X_0$.", "Observe that $\\mathcal{F}'' \\to \\mathcal{F}'$ is injective as it is an", "isomorphism over $U$ and since $f : \\mathcal{F}'' \\to \\mathcal{F}''$", "is injective. Clearly $\\mathcal{F}'/\\mathcal{F}''$ is supported on", "$\\{\\mathfrak m\\}$ hence has finite length. We have the maps", "of coherent $\\mathcal{O}_{X_0}$-modules", "$$", "\\mathcal{F}''/f\\mathcal{F}'' \\to", "\\mathcal{F}'/f\\mathcal{F}' \\xrightarrow{\\alpha'_0}", "\\mathcal{F}_0", "$$", "whose composition is $\\alpha''_0$ and which are isomorphisms over $U_0$.", "Elementary homological algebra gives a $6$-term exact sequence", "$$", "\\begin{matrix}", "0 \\to", "\\Ker(\\mathcal{F}''/f\\mathcal{F}'' \\to \\mathcal{F}'/f\\mathcal{F}') \\to", "\\Ker(\\alpha''_0) \\to", "\\Ker(\\alpha'_0) \\to \\\\", "\\Coker(\\mathcal{F}''/f\\mathcal{F}'' \\to \\mathcal{F}'/f\\mathcal{F}') \\to", "\\Coker(\\alpha''_0) \\to", "\\Coker(\\alpha'_0) \\to 0", "\\end{matrix}", "$$", "By additivity of lengths (Algebra, Lemma \\ref{algebra-lemma-length-additive})", "we find that it suffices to show that", "$$", "\\text{length}_A(", "\\Coker(\\mathcal{F}''/f\\mathcal{F}'' \\to \\mathcal{F}'/f\\mathcal{F}')) -", "\\text{length}_A(", "\\Ker(\\mathcal{F}''/f\\mathcal{F}'' \\to \\mathcal{F}'/f\\mathcal{F}')) = 0", "$$", "This follows from applying the snake lemma to", "the diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{F}'' \\ar[r]_f \\ar[d] &", "\\mathcal{F}'' \\ar[r] \\ar[d] &", "\\mathcal{F}''/f\\mathcal{F}'' \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{F}' \\ar[r]^f &", "\\mathcal{F}' \\ar[r] &", "\\mathcal{F}'/f\\mathcal{F}' \\ar[r] &", "0", "}", "$$", "and the fact that $\\mathcal{F}'/\\mathcal{F}''$ has finite length." ], "refs": [ "coherent-lemma-homs-over-open", "coherent-lemma-homs-over-open", "algebra-lemma-length-additive" ], "ref_ids": [ 3322, 3322, 631 ] } ], "ref_ids": [ 12779 ] }, { "id": 12781, "type": "theorem", "label": "algebraization-lemma-ses-chi-triple", "categories": [ "algebraization" ], "title": "algebraization-lemma-ses-chi-triple", "contents": [ "We have", "$\\chi(\\mathcal{G}, \\mathcal{G}_0, \\beta) =", "\\chi(\\mathcal{F}, \\mathcal{F}_0, \\alpha) +", "\\chi(\\mathcal{H}, \\mathcal{H}_0, \\gamma)$ if", "$$", "0 \\to", "(\\mathcal{F}, \\mathcal{F}_0, \\alpha) \\to", "(\\mathcal{G}, \\mathcal{G}_0, \\beta) \\to", "(\\mathcal{H}, \\mathcal{H}_0, \\gamma)", "\\to 0", "$$", "is a short exact sequence of coherent triples." ], "refs": [], "proofs": [ { "contents": [ "Choose $\\mathcal{G}', \\beta', \\beta'_0$ as in", "Lemma \\ref{lemma-prepare-chi-triple}", "for the triple $(\\mathcal{G}, \\mathcal{G}_0, \\beta)$.", "Denote $j : U \\to X$ the inclusion morphism.", "Let $\\mathcal{F}' \\subset \\mathcal{G}'$", "be the kernel of the composition", "$$", "\\mathcal{G}' \\xrightarrow{\\beta'} j_*\\mathcal{G} \\to j_*\\mathcal{H}", "$$", "Observe that $\\mathcal{H}' = \\mathcal{G}'/\\mathcal{F}'$", "is a coherent subsheaf of $j_*\\mathcal{H}$ and hence", "$f : \\mathcal{H}' \\to \\mathcal{H}'$ is injective.", "Hence by the snake lemma we obtain a short exact sequence", "$$", "0 \\to \\mathcal{F}'/f\\mathcal{F}' \\to", "\\mathcal{G}'/f\\mathcal{G}' \\to", "\\mathcal{H}'/f\\mathcal{H}' \\to 0", "$$", "We have isomorphisms", "$\\alpha' : \\mathcal{F}'|_U \\to \\mathcal{F}$,", "$\\beta' : \\mathcal{G}'|_U \\to \\mathcal{G}$, and", "$\\gamma' : \\mathcal{H}'|_U \\to \\mathcal{H}$ by construction.", "To finish the proof we'll need to construct maps", "$\\alpha'_0 : \\mathcal{F}'/f\\mathcal{F}' \\to \\mathcal{F}_0$ and", "$\\gamma'_0 : \\mathcal{H}'/f\\mathcal{H}' \\to \\mathcal{H}_0$ as in", "Lemma \\ref{lemma-prepare-chi-triple} and fitting into", "a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{F}'/f\\mathcal{F}' \\ar[r] \\ar@{..>}[d]^{\\alpha'_0} &", "\\mathcal{G}'/f\\mathcal{G}' \\ar[r] \\ar[d]^{\\beta'_0} &", "\\mathcal{H}'/f\\mathcal{H}' \\ar[r] \\ar@{..>}[d]^{\\gamma'_0} &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{F}_0 \\ar[r] &", "\\mathcal{G}_0 \\ar[r] &", "\\mathcal{H}_0 \\ar[r] &", "0", "}", "$$", "However, this may not be possible with our initial choice of $\\mathcal{G}'$.", "From the displayed diagram we see the obstruction is", "exactly the composition", "$$", "\\delta :", "\\mathcal{F}'/f\\mathcal{F}' \\to", "\\mathcal{G}'/f\\mathcal{G}' \\xrightarrow{\\beta'_0}", "\\mathcal{G}_0 \\to", "\\mathcal{H}_0", "$$", "Note that the restriction of $\\delta$ to $U_0$ is zero by our choice of", "$\\mathcal{F}'$ and $\\mathcal{H}'$. Hence by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-homs-over-open}", "there exists an $k > 0$ such that", "$\\delta$ vanishes on $\\mathfrak m^k \\cdot (\\mathcal{F}'/f\\mathcal{F}')$.", "For $n > k$ set $\\mathcal{G}'_n = \\mathfrak m^n \\mathcal{G}'$,", "$\\mathcal{F}'_n = \\mathcal{G}'_n \\cap \\mathcal{F}'$, and", "$\\mathcal{H}'_n = \\mathcal{G}'_n/\\mathcal{F}'_n$.", "Observe that $\\beta'_0$ can be composed with", "$\\mathcal{G}'_n/f\\mathcal{G}'_n \\to \\mathcal{G}'/f\\mathcal{G}'$", "to give a map", "$\\beta'_{n, 0} : \\mathcal{G}'_n/f\\mathcal{G}'_n \\to \\mathcal{G}_0$", "as in Lemma \\ref{lemma-prepare-chi-triple}.", "By Artin-Rees (Algebra, Lemma \\ref{algebra-lemma-Artin-Rees})", "we may choose $n$ such that", "$\\mathcal{F}'_n \\subset \\mathfrak m^k \\mathcal{F}'$.", "As above the maps", "$f : \\mathcal{F}'_n \\to \\mathcal{F}'_n$,", "$f : \\mathcal{G}'_n \\to \\mathcal{G}'_n$, and", "$f : \\mathcal{H}'_n \\to \\mathcal{H}'_n$ are injective", "and as above using the snake lemma we obtain a short exact", "sequence", "$$", "0 \\to \\mathcal{F}'_n/f\\mathcal{F}'_n \\to", "\\mathcal{G}'_n/f\\mathcal{G}'_n \\to", "\\mathcal{H}'_n/f\\mathcal{H}'_n \\to 0", "$$", "As above we have isomorphisms", "$\\alpha'_n : \\mathcal{F}'_n|_U \\to \\mathcal{F}$,", "$\\beta'_n : \\mathcal{G}'_n|_U \\to \\mathcal{G}$, and", "$\\gamma'_n : \\mathcal{H}'_n|_U \\to \\mathcal{H}$.", "We consider the obstruction", "$$", "\\delta_n :", "\\mathcal{F}'_n/f\\mathcal{F}'_n \\to", "\\mathcal{G}'_n/f\\mathcal{G}'_n", "\\xrightarrow{\\beta'_{n, 0}}", "\\mathcal{G}_0 \\to", "\\mathcal{H}_0", "$$", "as before. However, the commutative diagram", "$$", "\\xymatrix{", "\\mathcal{F}'_n/f\\mathcal{F}'_n \\ar[r] \\ar[d] &", "\\mathcal{G}'_n/f\\mathcal{G}'_n \\ar[r]_{\\beta'_{n, 0}} \\ar[d] &", "\\mathcal{G}_0 \\ar[r] \\ar[d] &", "\\mathcal{H}_0 \\ar[d] \\\\", "\\mathcal{F}'/f\\mathcal{F}' \\ar[r] &", "\\mathcal{G}'/f\\mathcal{G}' \\ar[r]^{\\beta'_0} &", "\\mathcal{G}_0 \\ar[r] &", "\\mathcal{H}_0", "}", "$$", "our choice of $n$ and our observation about $\\delta$", "show that $\\delta_n = 0$.", "This produces the desired maps", "$\\alpha'_{n, 0} : \\mathcal{F}'_n/f\\mathcal{F}'_n \\to \\mathcal{F}_0$, and", "$\\gamma'_{n, 0} : \\mathcal{H}'_n/f\\mathcal{H}'_n \\to \\mathcal{H}_0$.", "OK, so we may use", "$\\mathcal{F}'_n, \\alpha'_n, \\alpha'_{n, 0}$,", "$\\mathcal{G}'_n, \\beta'_n, \\beta'_{n, 0}$, and", "$\\mathcal{H}'_n, \\gamma'_n, \\gamma'_{n, 0}$", "to compute", "$\\chi(\\mathcal{F}, \\mathcal{F}_0, \\alpha)$,", "$\\chi(\\mathcal{G}, \\mathcal{G}_0, \\beta)$, and", "$\\chi(\\mathcal{H}, \\mathcal{H}_0, \\gamma)$.", "Now finally the lemma follows from", "an application of the snake lemma to", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{F}'_n/f\\mathcal{F}'_n \\ar[r] \\ar[d] &", "\\mathcal{G}'_n/f\\mathcal{G}'_n \\ar[r] \\ar[d] &", "\\mathcal{H}'_n/f\\mathcal{H}'_n \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{F}_0 \\ar[r] &", "\\mathcal{G}_0 \\ar[r] &", "\\mathcal{H}_0 \\ar[r] &", "0", "}", "$$", "and additivity of lengths (Algebra, Lemma \\ref{algebra-lemma-length-additive})." ], "refs": [ "algebraization-lemma-prepare-chi-triple", "algebraization-lemma-prepare-chi-triple", "coherent-lemma-homs-over-open", "algebraization-lemma-prepare-chi-triple", "algebra-lemma-Artin-Rees", "algebra-lemma-length-additive" ], "ref_ids": [ 12779, 12779, 3322, 12779, 625, 631 ] } ], "ref_ids": [] }, { "id": 12782, "type": "theorem", "label": "algebraization-lemma-nonnegative-chi-triple", "categories": [ "algebraization" ], "title": "algebraization-lemma-nonnegative-chi-triple", "contents": [ "Assume $\\text{depth}(A) \\geq 3$ or equivalently", "$\\text{depth}(A/fA) \\geq 2$. Let $(\\mathcal{L}, \\mathcal{L}_0, \\lambda)$", "be an invertible coherent triple. Then", "$$", "\\chi(\\mathcal{L}, \\mathcal{L}_0, \\lambda) =", "\\text{length}_A \\Coker(\\Gamma(U, \\mathcal{L}) \\to \\Gamma(U_0, \\mathcal{L}_0))", "$$", "and in particular this is $\\geq 0$. Moreover, ", "$\\chi(\\mathcal{L}, \\mathcal{L}_0, \\lambda) = 0$ if and only if", "$\\mathcal{L} \\cong \\mathcal{O}_U$." ], "refs": [], "proofs": [ { "contents": [ "The equivalence of the depth conditions follows from", "Algebra, Lemma \\ref{algebra-lemma-depth-drops-by-one}.", "By the depth condition we see that", "$\\Gamma(U, \\mathcal{O}_U) = A$ and", "$\\Gamma(U_0, \\mathcal{O}_{U_0}) = A/fA$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth} and", "Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}.", "Using Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-for-finite-locally-free}", "we find that $M = \\Gamma(U, \\mathcal{L})$ is a finite $A$-module.", "This in turn implies $\\text{depth}(M) \\geq 2$ for example by", "part (4) of Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}", "or by Divisors, Lemma \\ref{divisors-lemma-depth-pushforward}.", "Also, we have $\\mathcal{L}_0 \\cong \\mathcal{O}_{X_0}$", "as $X_0$ is a local scheme. Hence we also see that", "$M_0 = \\Gamma(X_0, \\mathcal{L}_0) = \\Gamma(U_0, \\mathcal{L}_0|_{U_0})$", "and that this module is isomorphic to $A/fA$.", "\\medskip\\noindent", "By the above $\\mathcal{F}' = \\widetilde{M}$ is a coherent", "$\\mathcal{O}_X$-module whose restriction to $U$ is isomorphic to $\\mathcal{L}$.", "The isomorphism $\\lambda : \\mathcal{L}/f\\mathcal{L} \\to \\mathcal{L}_0|_{U_0}$", "determines a map $M/fM \\to M_0$ on global sections", "which is an isomorphism over $U_0$.", "Since $\\text{depth}(M) \\geq 2$ we see", "that $H^0_\\mathfrak m(M/fM) = 0$ and it follows that", "$M/fM \\to M_0$ is injective. Thus by definition", "$$", "\\chi(\\mathcal{L}, \\mathcal{L}_0, \\lambda) =", "\\text{length}_A \\Coker(M/fM \\to M_0)", "$$", "which gives the first statement of the lemma.", "\\medskip\\noindent", "Finally, if this length is $0$, then $M \\to M_0$ is surjective.", "Hence we can find $s \\in M = \\Gamma(U, \\mathcal{L})$", "mapping to a trivializing section of $\\mathcal{L}_0$.", "Consider the finite $A$-modules $K$, $Q$ defined by the exact", "sequence", "$$", "0 \\to K \\to A \\xrightarrow{s} M \\to Q \\to 0", "$$", "The supports of $K$ and $Q$ do not meet $U_0$ because $s$", "is nonzero at points of $U_0$. Using", "Algebra, Lemma \\ref{algebra-lemma-depth-in-ses}", "we see that $\\text{depth}(K) \\geq 2$ (observe that", "$As \\subset M$ has $\\text{depth} \\geq 1$ as a submodule of $M$).", "Thus the support of $K$ if nonempty has dimension $\\geq 2$ by", "Algebra, Lemma \\ref{algebra-lemma-bound-depth}.", "This contradicts $\\text{Supp}(M) \\cap V(f) \\subset \\{\\mathfrak m\\}$", "unless $K = 0$. When $K = 0$ we find that", "$\\text{depth}(Q) \\geq 2$ and we conclude", "$Q = 0$ as before. Hence $A \\cong M$ and", "$\\mathcal{L}$ is trivial." ], "refs": [ "algebra-lemma-depth-drops-by-one", "dualizing-lemma-depth", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "local-cohomology-lemma-finiteness-for-finite-locally-free", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "divisors-lemma-depth-pushforward", "algebra-lemma-depth-in-ses", "algebra-lemma-bound-depth" ], "ref_ids": [ 774, 2826, 9729, 9744, 9729, 7888, 773, 770 ] } ], "ref_ids": [] }, { "id": 12783, "type": "theorem", "label": "algebraization-lemma-injective-torsion-in-pic", "categories": [ "algebraization" ], "title": "algebraization-lemma-injective-torsion-in-pic", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $f \\in \\mathfrak m$", "be a nonzerodivisor and assume that $\\text{depth}(A/fA) \\geq 2$, or equivalently", "$\\text{depth}(A) \\geq 3$. Let $U$, resp.\\ $U_0$ be the punctured", "spectrum of $A$, resp.\\ $A/fA$. The map", "$$", "\\Pic(U) \\to \\Pic(U_0)", "$$", "is injective on torsion." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_U$-module.", "Observe that $\\mathcal{L}$ maps to $0$ in $\\text{Pic}(U_0)$", "if and only if we can extend $\\mathcal{L}$ to an invertible", "coherent triple $(\\mathcal{L}, \\mathcal{L}_0, \\lambda)$", "as in Section \\ref{section-coherent-triples}.", "By Proposition \\ref{proposition-hilbert-triple}", "the function", "$$", "n \\longmapsto \\chi((\\mathcal{L}, \\mathcal{L}_0, \\lambda)^{\\otimes n})", "$$", "is a polynomial. By Lemma \\ref{lemma-nonnegative-chi-triple}", "the value of this polynomial is zero if and only if", "$\\mathcal{L}^{\\otimes n}$ is trivial.", "Thus if $\\mathcal{L}$ is torsion, then this", "polynomial has infinitely many zeros, hence is", "identically zero, hence $\\mathcal{L}$ is trivial." ], "refs": [ "algebraization-proposition-hilbert-triple", "algebraization-lemma-nonnegative-chi-triple" ], "ref_ids": [ 12797, 12782 ] } ], "ref_ids": [] }, { "id": 12784, "type": "theorem", "label": "algebraization-lemma-surjective-Pic-first", "categories": [ "algebraization" ], "title": "algebraization-lemma-surjective-Pic-first", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring and $f \\in \\mathfrak m$.", "Assume", "\\begin{enumerate}", "\\item $A$ is $f$-adically complete,", "\\item $f$ is a nonzerodivisor,", "\\item $H^1_\\mathfrak m(A/fA)$ and $H^2_\\mathfrak m(A/fA)$", "are finite $A$-modules, and", "\\item $H^3_\\mathfrak m(A/fA) = 0$\\footnote{Observe that (3) and (4) hold", "if $\\text{depth}(A/fA) \\geq 4$, or equivalently $\\text{depth}(A) \\geq 5$.}.", "\\end{enumerate}", "Let $U$, resp.\\ $U_0$ be the punctured spectrum of $A$, resp.\\ $A/fA$.", "Then", "$$", "\\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\Pic(U')", "\\longrightarrow", "\\Pic(U_0)", "$$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Let $U_0 \\subset U_n \\subset U$ be the $n$th infinitesimal neighbourhood", "of $U_0$. Observe that the ideal sheaf of $U_n$ in $U_{n + 1}$ is", "isomorphic to $\\mathcal{O}_{U_0}$ as $U_0 \\subset U$ is the principal", "closed subscheme cut out by the nonzerodivisor $f$. Hence we have", "an exact sequence of abelian groups", "$$", "\\Pic(U_{n + 1}) \\to \\Pic(U_n) \\to", "H^2(U_0, \\mathcal{O}_{U_0}) = H^3_\\mathfrak m(A/fA) = 0", "$$", "see More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-picard-group-first-order-thickening}.", "Thus every invertible $\\mathcal{O}_{U_0}$-module is the restriction", "of an invertible coherent formal module, i.e., an invertible object of", "$\\textit{Coh}(U, f\\mathcal{O}_U)$. We conclude by applying", "Lemma \\ref{lemma-equivalence}." ], "refs": [ "more-morphisms-lemma-picard-group-first-order-thickening", "algebraization-lemma-equivalence" ], "ref_ids": [ 13686, 12778 ] } ], "ref_ids": [] }, { "id": 12785, "type": "theorem", "label": "algebraization-lemma-surjective-Pic-first-better", "categories": [ "algebraization" ], "title": "algebraization-lemma-surjective-Pic-first-better", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring and $f \\in \\mathfrak m$.", "Assume", "\\begin{enumerate}", "\\item the conditions of Lemma \\ref{lemma-surjective-Pic-first} hold, and", "\\item for every maximal ideal $\\mathfrak p \\subset A_f$", "the punctured spectrum of $(A_f)_\\mathfrak p$ has trivial Picard group.", "\\end{enumerate}", "Let $U$, resp.\\ $U_0$ be the punctured spectrum of $A$, resp.\\ $A/fA$.", "Then", "$$", "\\Pic(U) \\longrightarrow \\Pic(U_0)", "$$", "is surjective." ], "refs": [ "algebraization-lemma-surjective-Pic-first" ], "proofs": [ { "contents": [ "Let $\\mathcal{L}_0 \\in \\Pic(U_0)$. By", "Lemma \\ref{lemma-surjective-Pic-first}", "there exists an open $U_0 \\subset U' \\subset U$", "and $\\mathcal{L}' \\in \\Pic(U')$ whose restriction", "to $U_0$ is $\\mathcal{L}_0$.", "Since $U' \\supset U_0$ we see that $U \\setminus U'$", "consists of points corresponding to prime ideals", "$\\mathfrak p_1, \\ldots, \\mathfrak p_n$ as in (2).", "By assumption we can find invertible modules", "$\\mathcal{L}'_i$ on $\\Spec(A_{\\mathfrak p_i})$ agreeing with", "$\\mathcal{L}'$ over the punctured spectrum", "$U' \\times_U \\Spec(A_{\\mathfrak p_i})$ since", "trivial invertible modules always extend.", "By Limits, Lemma \\ref{limits-lemma-glueing-near-closed-point-modules}", "applied $n$ times we see that $\\mathcal{L}'$ extends to an", "invertible module on $U$." ], "refs": [ "algebraization-lemma-surjective-Pic-first", "limits-lemma-glueing-near-closed-point-modules" ], "ref_ids": [ 12784, 15113 ] } ], "ref_ids": [ 12784 ] }, { "id": 12786, "type": "theorem", "label": "algebraization-lemma-local-pic-to-completion", "categories": [ "algebraization" ], "title": "algebraization-lemma-local-pic-to-completion", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring of depth $\\geq 2$.", "Let $A^\\wedge$ be its completion. Let $U$, resp.\\ $U^\\wedge$", "be the punctured spectrum of $A$, resp.\\ $A^\\wedge$. Then", "$\\Pic(U) \\to \\Pic(U^\\wedge)$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_U$-module", "with pullback $\\mathcal{L}^\\wedge$ on $U^\\wedge$.", "We have $H^0(U, \\mathcal{O}_U) = A$ by our assumption on depth and", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-depth} and", "Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}.", "Thus $\\mathcal{L}$ is trivial if and only if", "$M = H^0(U, \\mathcal{L})$ is isomorphic to $A$ as an $A$-module.", "(Details omitted.) Since $A \\to A^\\wedge$ is flat", "we have $M \\otimes_A A^\\wedge = \\Gamma(U^\\wedge, \\mathcal{L}^\\wedge)$", "by flat base change, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}.", "Finally, it is easy to see that $M \\cong A$ if and only if", "$M \\otimes_A A^\\wedge \\cong A^\\wedge$." ], "refs": [ "dualizing-lemma-depth", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 2826, 9729, 3298 ] } ], "ref_ids": [] }, { "id": 12787, "type": "theorem", "label": "algebraization-lemma-trivial-local-pic-regular", "categories": [ "algebraization" ], "title": "algebraization-lemma-trivial-local-pic-regular", "contents": [ "Let $(A, \\mathfrak m)$ be a regular local ring. Then the Picard", "group of the punctured spectrum of $A$ is trivial." ], "refs": [], "proofs": [ { "contents": [ "Combine Divisors, Lemma \\ref{divisors-lemma-extend-invertible-module}", "with More on Algebra, Lemma \\ref{more-algebra-lemma-regular-local-UFD}." ], "refs": [ "divisors-lemma-extend-invertible-module", "more-algebra-lemma-regular-local-UFD" ], "ref_ids": [ 8032, 10544 ] } ], "ref_ids": [] }, { "id": 12788, "type": "theorem", "label": "algebraization-lemma-lefschetz-addendum", "categories": [ "algebraization" ], "title": "algebraization-lemma-lefschetz-addendum", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$.", "Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{L})$. Let $Y = Z(s)$ be the", "zero scheme of $s$ with $n$th infinitesimal neighbourhood $Y_n = Z(s^n)$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Assume that for all $x \\in X \\setminus Y$ we have", "$$", "\\text{depth}(\\mathcal{F}_x) + \\dim(\\overline{\\{x\\}}) > 1", "$$", "Then $\\Gamma(V, \\mathcal{F}) \\to \\lim \\Gamma(Y_n, \\mathcal{F}|_{Y_n})$", "is an isomorphism for any open subscheme $V \\subset X$ containing $Y$." ], "refs": [], "proofs": [ { "contents": [ "By Proposition \\ref{proposition-lefschetz} this is true for $V = X$.", "Thus it suffices to show that the map", "$\\Gamma(V, \\mathcal{F}) \\to \\lim \\Gamma(Y_n, \\mathcal{F}|_{Y_n})$", "is injective. If $\\sigma \\in \\Gamma(V, \\mathcal{F})$", "maps to zero, then its support is disjoint from $Y$ (details omitted; hint:", "use Krull's intersection theorem). Then the closure $T \\subset X$", "of $\\text{Supp}(\\sigma)$ is disjoint from $Y$.", "Whence $T$ is proper over $k$ (being closed in $X$)", "and affine (being closed in the affine scheme $X \\setminus Y$, see", "Morphisms, Lemma \\ref{morphisms-lemma-proper-ample-delete-affine})", "and hence finite over $k$", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-proper}).", "Thus $T$ is a finite set of closed points of $X$.", "Thus $\\text{depth}(\\mathcal{F}_x) \\geq 2$ is at least $1$", "for $x \\in T$ by our assumption. We conclude that", "$\\Gamma(V, \\mathcal{F}) \\to \\Gamma(V \\setminus T, \\mathcal{F})$", "is injective and $\\sigma = 0$ as desired." ], "refs": [ "algebraization-proposition-lefschetz", "morphisms-lemma-proper-ample-delete-affine", "morphisms-lemma-finite-proper" ], "ref_ids": [ 12800, 5435, 5445 ] } ], "ref_ids": [] }, { "id": 12789, "type": "theorem", "label": "algebraization-lemma-Gm-equivariant-extend-canonically", "categories": [ "algebraization" ], "title": "algebraization-lemma-Gm-equivariant-extend-canonically", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an", "object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume", "\\begin{enumerate}", "\\item $A$ is a graded ring, $\\mathfrak a = A_+$, and", "$I$ is a homogeneous ideal,", "\\item $(\\mathcal{F}_n) = (\\widetilde{M_n}|_U)$ where $(M_n)$", "is an inverse system of graded $A$-modules, and", "\\item $(\\mathcal{F}_n)$ extends canonically to $X$.", "\\end{enumerate}", "Then there is a finite graded $A$-module $N$ such that", "\\begin{enumerate}", "\\item[(a)] the inverse systems $(N/I^nN)$ and $(M_n)$ are pro-isomorphic", "in the category of graded $A$-modules modulo $A_+$-power torsion", "modules, and", "\\item[(b)] $(\\mathcal{F}_n)$ is the completion of of the coherent", "module associated to $N$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $(\\mathcal{G}_n)$ be the canonical extension as in", "Lemma \\ref{lemma-canonically-algebraizable}.", "The grading on $A$ and $M_n$ determines an action", "$$", "a : \\mathbf{G}_m \\times X \\longrightarrow X", "$$", "of the group scheme $\\mathbf{G}_m$ on $X$ such that", "$(\\widetilde{M_n})$ becomes an inverse system of", "$\\mathbf{G}_m$-equivariant quasi-coherent $\\mathcal{O}_X$-modules, see", "Groupoids, Example \\ref{groupoids-example-Gm-on-affine}.", "Since $\\mathfrak a$ and $I$ are homogeneous ideals", "the closed subschemes $Z$, $Y$ and the open subscheme $U$", "are $\\mathbf{G}_m$-invariant closed and open subschemes.", "The restriction $(\\mathcal{F}_n)$ of $(\\widetilde{M_n})$", "is an inverse system of $\\mathbf{G}_m$-equivariant", "coherent $\\mathcal{O}_U$-modules. In other words, $(\\mathcal{F}_n)$", "is a $\\mathbf{G}_m$-equivariant coherent formal module,", "in the sense that there is an isomorphism", "$$", "\\alpha : (a^*\\mathcal{F}_n) \\longrightarrow (p^*\\mathcal{F}_n)", "$$", "over $\\mathbf{G}_m \\times U$ satisfying a suitable cocycle condition.", "Since $a$ and $p$ are flat morphisms of affine schemes,", "by Lemma \\ref{lemma-canonically-extend-base-change}", "we conclude that there exists a unique isomorphism", "$$", "\\beta : (a^*\\mathcal{G}_n) \\longrightarrow (p^*\\mathcal{G}_n)", "$$", "over $\\mathbf{G}_m \\times X$ restricting to $\\alpha$ on", "$\\mathbf{G}_m \\times U$. The uniqueness guarantees that", "$\\beta$ satisfies the corresponding cocycle condition.", "In this way each $\\mathcal{G}_n$ becomes", "a $\\mathbf{G}_m$-equivariant coherent $\\mathcal{O}_X$-module", "in a manner compatible with transition maps.", "\\medskip\\noindent", "By Groupoids, Lemma \\ref{groupoids-lemma-Gm-equivariant-module}", "we see that $\\mathcal{G}_n$ with its $\\mathbf{G}_m$-equivariant", "structure corresponds to a graded $A$-module $N_n$. The transition maps", "$N_{n + 1} \\to N_n$ are graded module maps. Note that $N_n$ is a finite", "$A$-module and $N_n = N_{n + 1}/I^n N_{n + 1}$ because", "$(\\mathcal{G}_n)$ is an object of $\\textit{Coh}(X, I\\mathcal{O}_X)$.", "Let $N$ be the finite graded $A$-module foud in", "Algebra, Lemma \\ref{algebra-lemma-finiteness-graded}.", "Then $N_n = N/I^nN$, whence $(\\mathcal{G}_n)$", "is the completion of the coherent module", "associated to $N$, and a fortiori we see that (b) is true.", "\\medskip\\noindent", "To see (a) we have to unwind the situation described above a bit more.", "First, observe that the kernel and cokernel of $M_n \\to H^0(U, \\mathcal{F}_n)$", "is $A_+$-power torsion (Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}).", "Observe that $H^0(U, \\mathcal{F}_n)$ comes with a natural grading", "such that these maps and the transition maps of the system are", "graded $A$-module map; for example we can use that", "$(U \\to X)_*\\mathcal{F}_n$ is a $\\mathbf{G}_m$-equivariant module", "on $X$ and use ", "Groupoids, Lemma \\ref{groupoids-lemma-Gm-equivariant-module}.", "Next, recall that $(N_n)$ and $(H^0(U, \\mathcal{F}_n))$", "are pro-isomorphic by Definition \\ref{definition-canonically-algebraizable}", "and Lemma \\ref{lemma-canonically-algebraizable}.", "We omit the verification that the maps defining this", "pro-isomorphism are graded module maps.", "Thus $(N_n)$ and $(M_n)$ are pro-isomorphic in the category of", "graded $A$-modules modulo $A_+$-power torsion modules." ], "refs": [ "algebraization-lemma-canonically-algebraizable", "algebraization-lemma-canonically-extend-base-change", "groupoids-lemma-Gm-equivariant-module", "algebra-lemma-finiteness-graded", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "groupoids-lemma-Gm-equivariant-module", "algebraization-definition-canonically-algebraizable", "algebraization-lemma-canonically-algebraizable" ], "ref_ids": [ 12751, 12752, 9619, 881, 9729, 9619, 12805, 12751 ] } ], "ref_ids": [] }, { "id": 12790, "type": "theorem", "label": "algebraization-proposition-derived-completion", "categories": [ "algebraization" ], "title": "algebraization-proposition-derived-completion", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of", "ideals. There exists a left adjoint to the inclusion", "functor $D_{comp}(\\mathcal{O}) \\to D(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "Let $K \\to \\mathcal{O}$ in $D(\\mathcal{O})$ be as constructed in", "Lemma \\ref{lemma-global-extended-cech-complex}. Let $E \\in D(\\mathcal{O})$.", "Then $E^\\wedge = R\\SheafHom(K, E)$ together with the map $E \\to E^\\wedge$", "will do the job. Namely, locally on the site $\\mathcal{C}$ we", "recover the adjoint of Lemma \\ref{lemma-derived-completion}.", "This shows that $E^\\wedge$ is always derived complete and that", "$E \\to E^\\wedge$ is an isomorphism if $E$ is derived complete." ], "refs": [ "algebraization-lemma-global-extended-cech-complex", "algebraization-lemma-derived-completion" ], "ref_ids": [ 12703, 12700 ] } ], "ref_ids": [] }, { "id": 12791, "type": "theorem", "label": "algebraization-proposition-application-H0", "categories": [ "algebraization" ], "title": "algebraization-proposition-application-H0", "contents": [ "Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.", "Let $\\mathcal{F}$ be a coherent module on", "$U = \\Spec(A) \\setminus V(\\mathfrak a)$.", "Assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete and has a dualizing complex,", "\\item if $x \\in \\text{Ass}(\\mathcal{F})$, $x \\not \\in V(I)$,", "$\\overline{\\{x\\}} \\cap V(I) \\not \\subset V(\\mathfrak a)$", "and $z \\in \\overline{\\{x\\}} \\cap V(\\mathfrak a)$, then", "$\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + 1$.", "\\end{enumerate}", "Then we obtain an isomorphism", "$$", "\\colim H^0(V, \\mathcal{F})", "\\longrightarrow", "\\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})", "$$", "where the colimit is over opens $V \\subset U$ containing $U \\cap V(I)$." ], "refs": [], "proofs": [ { "contents": [ "Let $T \\subset U$ be the set of points $x$ with", "$\\overline{\\{x\\}} \\cap V(I) \\subset V(\\mathfrak a)$.", "Let $\\mathcal{F} \\to \\mathcal{F}'$ be the surjection", "of coherent modules on $U$ constructed in", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-get-depth-1-along-Z}.", "Since $\\mathcal{F} \\to \\mathcal{F}'$ is an isomorphism", "over an open $V \\subset U$ containing $U \\cap V(I)$", "it suffices to prove the lemma with $\\mathcal{F}$ replaced", "by $\\mathcal{F}'$. Hence we may and do assume", "for $x \\in U$ with $\\overline{\\{x\\}} \\cap V(I) \\subset V(\\mathfrak a)$", "we have $\\text{depth}(\\mathcal{F}_x) \\geq 1$.", "\\medskip\\noindent", "Let $\\mathcal{V}$ be the set of open subschemes $V \\subset U$", "containing $U \\cap V(I)$ ordered by reverse inclusion.", "This is a directed set. We first claim that", "$$", "\\mathcal{F}(V)", "\\longrightarrow", "\\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})", "$$", "is injective for any $V \\in \\mathcal{F}$ (and in particular the map", "of the lemma is injective). Namely, an associated point $x$ of $\\mathcal{F}$", "must have $\\overline{\\{x\\}} \\cap U \\cap Y \\not = \\emptyset$", "by the previous paragraph. If $y \\in \\overline{\\{x\\}} \\cap U \\cap Y$ then", "$\\mathcal{F}_x$ is a localization of $\\mathcal{F}_y$", "and $\\mathcal{F}_y \\subset \\lim \\mathcal{F}_y/I^n \\mathcal{F}_y$", "by Krull's intersection theorem", "(Algebra, Lemma \\ref{algebra-lemma-intersect-powers-ideal-module-zero}).", "This proves the claim as a section $s \\in \\mathcal{F}(V)$", "in the kernel would have to have empty support, hence would have to be zero.", "\\medskip\\noindent", "Choose a finite $A$-module $M$ such that $\\mathcal{F}$ is the restriction", "of $\\widetilde{M}$ to $U$, see Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}.", "We may and do assume that $H^0_\\mathfrak a(M) = 0$.", "Let $\\text{Ass}(M) \\setminus V(I) = \\{\\mathfrak p_1, \\ldots, \\mathfrak p_n\\}$.", "We will prove the lemma by induction on $n$. After reordering we", "may assume that $\\mathfrak p_n$ is a minimal element of the set", "$\\{\\mathfrak p_1, \\ldots, \\mathfrak p_n\\}$ with respect to inclusion, i.e,", "$\\mathfrak p_n$ is a generic point of the support of $M$.", "Set", "$$", "M' = H^0_{\\mathfrak p_1 \\ldots \\mathfrak p_{n - 1} I}(M)", "$$", "and $M'' = M/M'$. Let $\\mathcal{F}'$ and $\\mathcal{F}''$ be the", "coherent $\\mathcal{O}_U$-modules corresponding to $M'$ and $M''$.", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-divide-by-torsion}", "implies that $M''$ has only one associated prime, namely $\\mathfrak p_n$.", "On the other hand, since", "$\\mathfrak p_n \\not \\in V(\\mathfrak p_1 \\ldots \\mathfrak p_{n - 1} I)$", "we see that $\\mathfrak p_n$ is not an associated prime of $M'$.", "Hence the induction hypothesis applies to $M'$; note", "that since $\\mathcal{F}' \\subset \\mathcal{F}$", "the condition $\\text{depth}(\\mathcal{F}'_x) \\geq 1$ at points $x$ with", "$\\overline{\\{x\\}} \\cap V(I) \\subset V(\\mathfrak a)$ holds, see", "Algebra, Lemma \\ref{algebra-lemma-depth-in-ses}.", "\\medskip\\noindent", "Let $\\hat s$ be an element of $\\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})$.", "Let $\\hat s''$ be the image in $\\lim H^0(U, \\mathcal{F}''/I^n\\mathcal{F}'')$.", "Since $\\mathcal{F}''$ has only one associated point, namely the point", "corresponding to $\\mathfrak p_n$, we see that", "Lemma \\ref{lemma-application-H0-pre} applies and we find an open", "$U \\cap V(I) \\subset V \\subset U$", "and a section $s'' \\in \\mathcal{F}''(V)$ mapping to $\\hat s''$.", "Let $J \\subset A$ be an ideal such that $V(J) = \\Spec(A) \\setminus V$.", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-homs-over-open}", "after replacing $J$ by a power, we may assume", "there is an $A$-linear map $\\varphi : J \\to M''$", "corresponding to $s''$. Since $M \\to M''$ is surjective, for", "each $g \\in J$ we can choose $m_g \\in M$ mapping to", "$\\varphi(g) \\in M''$. Then $\\hat s'_g = g \\hat s - m_g$", "is in $\\lim H^0(U, \\mathcal{F}'/I^n\\mathcal{F}')$.", "By induction hypothesis there is a $V' \\geq V$", "section $s'_g \\in \\mathcal{F}'(V')$", "mapping to $\\hat s'_g$. All in all we conclude that", "$g \\hat s$ is in the image of", "$\\mathcal{F}(V') \\to \\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})$", "for some $V' \\subset V$ possibly depending on $g$.", "However, since $J$ is finitely generated we can find a single", "$V' \\in \\mathcal{V}$ which works for each of the generators", "and it follows that $V'$ works for all $g$.", "\\medskip\\noindent", "Combining the previous paragraph with the injectivity", "shown in the second paragraph we find there exists", "a $V' \\geq V$ and an $A$-module map $\\psi : J \\to \\mathcal{F}(V')$", "such that $\\psi(g)$ maps to $g\\hat s$. This determines a", "map $\\widetilde{J} \\to (V' \\to \\Spec(A))_*\\mathcal{F}|_{V'}$", "whose restriction to $V'$ provides an element", "$s \\in \\mathcal{F}(V')$ mapping to $\\hat s$.", "This finishes the proof." ], "refs": [ "local-cohomology-lemma-get-depth-1-along-Z", "algebra-lemma-intersect-powers-ideal-module-zero", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "dualizing-lemma-divide-by-torsion", "algebra-lemma-depth-in-ses", "algebraization-lemma-application-H0-pre", "coherent-lemma-homs-over-open" ], "ref_ids": [ 9747, 627, 9729, 2831, 773, 12731, 3322 ] } ], "ref_ids": [] }, { "id": 12792, "type": "theorem", "label": "algebraization-proposition-application-higher", "categories": [ "algebraization" ], "title": "algebraization-proposition-application-higher", "contents": [ "Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.", "Let $\\mathcal{F}$ be a coherent module on", "$U = \\Spec(A) \\setminus V(\\mathfrak a)$.", "Let $s \\geq 0$.", "Assume", "\\begin{enumerate}", "\\item $A$ is $I$-adically complete and has a dualizing complex,", "\\item if $x \\in U \\setminus V(I)$ then", "$\\text{depth}(\\mathcal{F}_x) > s$ or", "$$", "\\text{depth}(\\mathcal{F}_x) +", "\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + s + 1", "$$", "for all $z \\in V(\\mathfrak a) \\cap \\overline{\\{x\\}}$,", "\\item one of the following conditions holds:", "\\begin{enumerate}", "\\item the restriction of $\\mathcal{F}$ to $U \\setminus V(I)$", "is $(S_{s + 1})$, or", "\\item the dimension of $V(\\mathfrak a)$ is at most $2$\\footnote{In", "the sense that the difference of the maximal and minimal values", "on $V(\\mathfrak a)$ of a dimension function on $\\Spec(A)$ is at most $2$.}.", "\\end{enumerate}", "\\end{enumerate}", "Then the maps", "$$", "H^i(U, \\mathcal{F})", "\\longrightarrow", "\\lim H^i(U, \\mathcal{F}/I^n\\mathcal{F})", "$$", "are isomorphisms for $i < s$. Moreover we have an isomorphism", "$$", "\\colim H^s(V, \\mathcal{F})", "\\longrightarrow", "\\lim H^s(U, \\mathcal{F}/I^n\\mathcal{F})", "$$", "where the colimit is over opens $V \\subset U$ containing $U \\cap V(I)$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $s > 0$ as the case $s = 0$ was done in", "Proposition \\ref{proposition-application-H0}.", "\\medskip\\noindent", "Choose a finite $A$-module $M$ such that $\\mathcal{F}$ is the restriction", "to $U$ of the coherent module associated to $M$, see Local Cohomology,", "Lemma \\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}.", "Set $d = \\text{cd}(A, I)$.", "Let $\\mathfrak p$ be a prime of $A$ not contained in $V(I)$", "and let $\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$.", "Then either $\\text{depth}(M_\\mathfrak p) \\geq s + 1 > s$", "or we have $\\dim((A/\\mathfrak p)_\\mathfrak q) > d + s + 1$ by (2).", "By Lemma \\ref{lemma-bootstrap-bis-bis} we conclude that the", "assumptions of Situation \\ref{situation-bootstrap}", "are satisfied for $A, I, V(\\mathfrak a), M, s, d$.", "On the other hand, the hypotheses of", "Lemma \\ref{lemma-algebraize-local-cohomology-general}", "are satisfied for $s + 1$ and $d$; this is where condition (3) is used.", "\\medskip\\noindent", "Applying Lemma \\ref{lemma-algebraize-local-cohomology-general}", "we find there exists an ideal", "$J_0 \\subset \\mathfrak a$ with $V(J_0) \\cap V(I) = V(\\mathfrak a)$", "such that for any $J \\subset J_0$ with $V(J) \\cap V(I) = V(\\mathfrak a)$", "the maps", "$$", "H^i_J(M) \\longrightarrow H^i(R\\Gamma_\\mathfrak a(M)^\\wedge)", "$$", "is an isomorphism for $i \\leq s + 1$.", "\\medskip\\noindent", "For $i \\leq s$ the map $H^i_\\mathfrak a(M) \\to H^i_J(M)$", "is an isomorphism by Lemmas \\ref{lemma-bootstrap-inherited} and", "\\ref{lemma-kill-colimit-support-general}.", "Using the comparison of cohomology and local cohomology", "(Local Cohomology, Lemma \\ref{local-cohomology-lemma-local-cohomology})", "we deduce", "$H^i(U, \\mathcal{F}) \\to H^i(V,\\mathcal{F})$", "is an isomorphism for $V = \\Spec(A) \\setminus V(J)$ and", "$i < s$.", "\\medskip\\noindent", "By Theorem \\ref{theorem-final-bootstrap} we have", "$H^i_\\mathfrak a(M) = \\lim H^i_\\mathfrak a(M/I^nM)$", "for $i \\leq s$. By Lemma \\ref{lemma-combine-two} we have", "$H^{s + 1}_\\mathfrak a(M) = \\lim H^{s + 1}_\\mathfrak a(M/I^nM)$.", "\\medskip\\noindent", "The isomorphism $H^0(U, \\mathcal{F}) = H^0(V, \\mathcal{F}) =", "\\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})$ follows from the above and", "Proposition \\ref{proposition-application-H0}.", "For $0 < i < s$ we get the desired isomorphisms", "$H^i(U, \\mathcal{F}) = H^i(V, \\mathcal{F}) =", "\\lim H^i(U, \\mathcal{F}/I^n\\mathcal{F})$ in", "the same manner using the relation between local cohomology", "and cohomology; it is easier than the case $i = 0$", "because for $i > 0$ we have", "$$", "H^i(U, \\mathcal{F}) = H^{i + 1}_\\mathfrak a(M),", "\\quad", "H^i(V, \\mathcal{F}) = H^{i + 1}_J(M),", "\\quad", "H^i(R\\Gamma(U, \\mathcal{F})^\\wedge) = ", "H^{i + 1}(R\\Gamma_\\mathfrak a(M)^\\wedge)", "$$", "Similarly for the final statement." ], "refs": [ "algebraization-proposition-application-H0", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "algebraization-lemma-bootstrap-bis-bis", "algebraization-lemma-algebraize-local-cohomology-general", "algebraization-lemma-algebraize-local-cohomology-general", "algebraization-lemma-bootstrap-inherited", "algebraization-lemma-kill-colimit-support-general", "local-cohomology-lemma-local-cohomology", "algebraization-theorem-final-bootstrap", "algebraization-lemma-combine-two", "algebraization-proposition-application-H0" ], "ref_ids": [ 12791, 9729, 12725, 12716, 12716, 12723, 12715, 9696, 12673, 12728, 12791 ] } ], "ref_ids": [] }, { "id": 12793, "type": "theorem", "label": "algebraization-proposition-cd-1", "categories": [ "algebraization" ], "title": "algebraization-proposition-cd-1", "contents": [ "\\begin{reference}", "The local case of this result is \\cite[IV Corollaire 2.9]{MRaynaud-book}.", "\\end{reference}", "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$", "be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex and $\\text{cd}(A, I) = 1$,", "\\item $(\\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities, see", "Definition \\ref{definition-s-d-inequalities}.", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ extends to $X$. In particular, if $A$ is", "$I$-adically complete, then $(\\mathcal{F}_n)$ is the completion", "of a coherent $\\mathcal{O}_U$-module." ], "refs": [ "algebraization-definition-s-d-inequalities" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-map-kernel-cokernel-on-closed}", "we may replace $(\\mathcal{F}_n)$ by the object $(\\mathcal{H}_n)$", "of $\\textit{Coh}(U, I\\mathcal{O}_U)$ found in", "Lemma \\ref{lemma-improvement-application}.", "Thus we may assume that $(\\mathcal{F}_n)$ is pro-isomorphic", "to a inverse system $(\\mathcal{F}_n'')$ with the properties", "mentioned in Lemma \\ref{lemma-improvement-application}.", "In Lemma \\ref{lemma-cd-1-canonical} we proved that", "$(\\mathcal{F}_n)$ canonically extends to $X$.", "The final statement follows from Lemma \\ref{lemma-canonically-algebraizable}." ], "refs": [ "algebraization-lemma-map-kernel-cokernel-on-closed", "algebraization-lemma-improvement-application", "algebraization-lemma-improvement-application", "algebraization-lemma-cd-1-canonical", "algebraization-lemma-canonically-algebraizable" ], "ref_ids": [ 12755, 12773, 12773, 12774, 12751 ] } ], "ref_ids": [ 12806 ] }, { "id": 12794, "type": "theorem", "label": "algebraization-proposition-d-generators", "categories": [ "algebraization" ], "title": "algebraization-proposition-d-generators", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$", "be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex,", "\\item $V(I) = V(f_1, \\ldots, f_d)$ for some $d \\geq 1$ and", "$f_1, \\ldots, f_d \\in A$,", "\\item one of the following is true", "\\begin{enumerate}", "\\item $(\\mathcal{F}_n)$ satisfies the $(d + 1, d + 2)$-inequalities", "(Definition \\ref{definition-s-d-inequalities}), or", "\\item for $y \\in U \\cap Y$ and a prime", "$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with", "$\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$", "we have", "$$", "\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > d + 2", "$$", "\\end{enumerate}", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ extends to $X$. In particular, if $A$ is", "$I$-adically complete, then $(\\mathcal{F}_n)$ is the completion", "of a coherent $\\mathcal{O}_U$-module." ], "refs": [ "algebraization-definition-s-d-inequalities" ], "proofs": [ { "contents": [ "We may assume $I = (f_1, \\ldots, f_d)$, see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-inverse-systems-ideals-equivalence}.", "Then we see that", "all fibres of the blowup of $X$ in $I$ have dimension at most $d - 1$.", "Thus we get the extension from Lemma \\ref{lemma-blowup}.", "The final statement follows from Lemma \\ref{lemma-essential-image-completion}." ], "refs": [ "coherent-lemma-inverse-systems-ideals-equivalence", "algebraization-lemma-blowup", "algebraization-lemma-essential-image-completion" ], "ref_ids": [ 3380, 12775, 12749 ] } ], "ref_ids": [ 12806 ] }, { "id": 12795, "type": "theorem", "label": "algebraization-proposition-algebraization-regular-sequence", "categories": [ "algebraization" ], "title": "algebraization-proposition-algebraization-regular-sequence", "contents": [ "In Situation \\ref{situation-algebraize} let", "$(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$.", "Assume", "\\begin{enumerate}", "\\item there exist $f_1, \\ldots, f_d \\in I$ such that", "for $y \\in U \\cap Y$ the ideal $I\\mathcal{O}_{X, y}$", "is generated by $f_1, \\ldots, f_d$ and", "$f_1, \\ldots, f_d$ form a $\\mathcal{F}_y^\\wedge$-regular sequence,", "\\item $H^0(U, \\mathcal{F}_1)$ and $H^1(U, \\mathcal{F}_1)$", "are finite $A$-modules.", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$", "is complete, then $(\\mathcal{F}_n)$ is the completion of a coherent", "$\\mathcal{O}_U$-module." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by verifying hypotheses (a), (b), and (c) of", "Lemma \\ref{lemma-when-done}.", "For every $n$ we have a short exact sequence", "$$", "0 \\to I^n\\mathcal{F}_{n + 1} \\to \\mathcal{F}_{n + 1} \\to \\mathcal{F}_n \\to 0", "$$", "Since $f_1, \\ldots, f_d$ forms a regular sequence (and hence", "quasi-regular, see Algebra, Lemma \\ref{algebra-lemma-regular-quasi-regular})", "on each of the ``stalks'' $\\mathcal{F}_y^\\wedge$ and since we have", "$I\\mathcal{F}_n = (f_1, \\ldots, f_d)\\mathcal{F}_n$ for all $n$,", "we find that", "$$", "I^n\\mathcal{F}_{n + 1} =", "\\bigoplus\\nolimits_{e_1 + \\ldots + e_d = n} \\mathcal{F}_1 \\cdot", "f_1^{e_1} \\ldots f_d^{e_d}", "$$", "by checking on stalks. Using the assumption of finiteness of", "$H^0(U, \\mathcal{F}_1)$ and induction, we first conclude that", "$M_n = H^0(U, \\mathcal{F}_n)$ is a finite $A$-module for all $n$.", "In this way we see that condition (c) of Lemma \\ref{lemma-when-done} holds.", "We also see that", "$$", "\\bigoplus\\nolimits_{n \\geq 0} H^1(U, I^n\\mathcal{F}_{n + 1})", "$$", "is a finite graded $R = \\bigoplus I^n/I^{n +1}$-module.", "By Lemma \\ref{lemma-ML-general} we conclude that condition (a) of", "Lemma \\ref{lemma-when-done} is satisfied. Finally, condition (b) of", "Lemma \\ref{lemma-when-done} is satisfied because", "$\\bigoplus H^0(U, I^n\\mathcal{F}_{n + 1})$ is a finite graded $R$-module", "and we can apply Lemma \\ref{lemma-topology-I-adic-general}." ], "refs": [ "algebraization-lemma-when-done", "algebra-lemma-regular-quasi-regular", "algebraization-lemma-when-done", "algebraization-lemma-ML-general", "algebraization-lemma-when-done", "algebraization-lemma-when-done", "algebraization-lemma-topology-I-adic-general" ], "ref_ids": [ 12753, 746, 12753, 12675, 12753, 12753, 12677 ] } ], "ref_ids": [] }, { "id": 12796, "type": "theorem", "label": "algebraization-proposition-algebraization-flat", "categories": [ "algebraization" ], "title": "algebraization-proposition-algebraization-flat", "contents": [ "In Situation \\ref{situation-algebraize} let", "$(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$.", "Assume there is Noetherian local ring $(R, \\mathfrak m)$ and a ring", "map $R \\to A$ such that", "\\begin{enumerate}", "\\item $I = \\mathfrak m A$,", "\\item for $y \\in U \\cap Y$ the stalk $\\mathcal{F}_y^\\wedge$ is $R$-flat,", "\\item $H^0(U, \\mathcal{F}_1)$ and $H^1(U, \\mathcal{F}_1)$ are finite", "$A$-modules.", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$", "is complete, then $(\\mathcal{F}_n)$ is the completion of a coherent", "$\\mathcal{O}_U$-module." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Proposition \\ref{proposition-algebraization-regular-sequence}.", "Namely, if $\\kappa = R/\\mathfrak m$ then for $n \\geq 0$", "there is an isomorphism", "$$", "I^n \\mathcal{F}_{n + 1} \\cong", "\\mathcal{F}_1 \\otimes_\\kappa \\mathfrak m^n/\\mathfrak m^{n + 1}", "$$", "and the right hand side is a finite direct sum of copies", "of $\\mathcal{F}_1$. This can be checked by looking at stalks.", "Everything else is exactly the same." ], "refs": [ "algebraization-proposition-algebraization-regular-sequence" ], "ref_ids": [ 12795 ] } ], "ref_ids": [] }, { "id": 12797, "type": "theorem", "label": "algebraization-proposition-hilbert-triple", "categories": [ "algebraization" ], "title": "algebraization-proposition-hilbert-triple", "contents": [ "Let $(\\mathcal{F}, \\mathcal{F}_0, \\alpha)$ be a coherent triple.", "Let $(\\mathcal{L}, \\mathcal{L}_0, \\lambda)$ be an invertible coherent", "triple. Then the function", "$$", "\\mathbf{Z} \\longrightarrow \\mathbf{Z},\\quad", "n \\longmapsto", "\\chi((\\mathcal{F}, \\mathcal{F}_0, \\alpha) \\otimes", "(\\mathcal{L}, \\mathcal{L}_0, \\lambda)^{\\otimes n})", "$$", "is a polynomial of degree $\\leq \\dim(\\text{Supp}(\\mathcal{F}))$." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by induction on the dimension of the support of", "$\\mathcal{F}$.", "\\medskip\\noindent", "The base case is when $\\mathcal{F} = 0$. Then either", "$\\mathcal{F}_0$ is zero or its support is $\\{\\mathfrak m\\}$.", "In this case we have", "$$", "(\\mathcal{F}, \\mathcal{F}_0, \\alpha) \\otimes", "(\\mathcal{L}, \\mathcal{L}_0, \\lambda)^{\\otimes n} =", "(0, \\mathcal{F}_0 \\otimes \\mathcal{L}_0^{\\otimes n}, 0) \\cong", "(0, \\mathcal{F}_0, 0)", "$$", "Thus the function of the lemma is constant with value equal", "to the length of $\\mathcal{F}_0$.", "\\medskip\\noindent", "Induction step. Assume the support of $\\mathcal{F}$ is nonempty.", "Let $\\mathcal{G}_0 \\subset \\mathcal{F}_0$ denote the submodule", "of sections supported on $\\{\\mathfrak m\\}$. Then we get a short", "exact sequence", "$$", "0 \\to (0, \\mathcal{G}_0, 0) \\to", "(\\mathcal{F}, \\mathcal{F}_0, \\alpha) \\to", "(\\mathcal{F}, \\mathcal{F}_0/\\mathcal{G}_0, \\alpha) \\to 0", "$$", "This sequence remains exact if we tensor by the invertible", "coherent triple $(\\mathcal{L}, \\mathcal{L}_0, \\lambda)$, see", "discussion above. Thus by additivity of $\\chi$", "(Lemma \\ref{lemma-ses-chi-triple})", "and the base case explained above, it suffices to prove", "the induction step for", "$(\\mathcal{F}, \\mathcal{F}_0/\\mathcal{G}_0, \\alpha)$.", "In this way we see that we may assume $\\mathfrak m$ is not", "an associated point of $\\mathcal{F}_0$.", "\\medskip\\noindent", "Let $T = \\text{Ass}(\\mathcal{F}) \\cup \\text{Ass}(\\mathcal{F}/f\\mathcal{F})$.", "Since $U$ is quasi-affine, we can find $s \\in \\Gamma(U, \\mathcal{L})$", "which does not vanish at any $u \\in T$, see", "Properties, Lemma", "\\ref{properties-lemma-quasi-affine-invertible-nonvanishing-section}.", "After multiplying $s$ by a suitable element of $\\mathfrak m$", "we may assume $\\lambda(s \\bmod f) = s_0|_{U_0}$ for some", "$s_0 \\in \\Gamma(X_0, \\mathcal{L}_0)$; details omitted.", "We obtain a morphism", "$$", "(s, s_0) :", "(\\mathcal{O}_U, \\mathcal{O}_{X_0}, 1)", "\\longrightarrow", "(\\mathcal{L}, \\mathcal{L}_0, \\lambda)", "$$", "in the category of coherent triples. Let", "$\\mathcal{G} = \\Coker(s : \\mathcal{F} \\to \\mathcal{F} \\otimes \\mathcal{L})$", "and", "$\\mathcal{G}_0 = \\Coker(s_0 : \\mathcal{F}_0 \\to", "\\mathcal{F}_0 \\otimes \\mathcal{L}_0)$. Observe that $s_0 : \\mathcal{F}_0 \\to", "\\mathcal{F}_0 \\otimes \\mathcal{L}_0$ is injective as it is injective", "on $U_0$ by our choice of $s$ and as $\\mathfrak m$ isn't an", "associated point of $\\mathcal{F}_0$. It follows that", "there exists an", "isomorphism $\\beta : \\mathcal{G}/f\\mathcal{G} \\to \\mathcal{G}_0|_{U_0}$", "such that we obtain a short exact sequence", "$$", "0 \\to", "(\\mathcal{F}, \\mathcal{F}_0, \\alpha) \\to", "(\\mathcal{F}, \\mathcal{F}_0, \\alpha) \\otimes", "(\\mathcal{L}, \\mathcal{L}_0, \\lambda) \\to", "(\\mathcal{G}, \\mathcal{G}_0, \\beta) \\to 0", "$$", "By induction on the dimension of the support we know the proposition", "holds for the coherent triple $(\\mathcal{G}, \\mathcal{G}_0, \\beta)$.", "Using the additivity of Lemma \\ref{lemma-ses-chi-triple}", "we see that", "$$", "n \\longmapsto ", "\\chi((\\mathcal{F}, \\mathcal{F}_0, \\alpha) \\otimes", "(\\mathcal{L}, \\mathcal{L}_0, \\lambda)^{\\otimes n + 1})", "-", "\\chi((\\mathcal{F}, \\mathcal{F}_0, \\alpha) \\otimes", "(\\mathcal{L}, \\mathcal{L}_0, \\lambda)^{\\otimes n})", "$$", "is a polynomial. We conclude by a variant of", "Algebra, Lemma \\ref{algebra-lemma-numerical-polynomial}", "for functions defined for all integers (details omitted)." ], "refs": [ "algebraization-lemma-ses-chi-triple", "algebraization-lemma-ses-chi-triple", "algebra-lemma-numerical-polynomial" ], "ref_ids": [ 12781, 12781, 670 ] } ], "ref_ids": [] }, { "id": 12798, "type": "theorem", "label": "algebraization-proposition-injective-pic", "categories": [ "algebraization" ], "title": "algebraization-proposition-injective-pic", "contents": [ "\\begin{reference}", "\\cite[Theorem 1.9]{Kollar-map-pic}", "\\end{reference}", "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $f \\in \\mathfrak m$.", "Assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex,", "\\item $f$ is a nonzerodivisor,", "\\item $\\text{depth}(A/fA) \\geq 2$, or equivalently $\\text{depth}(A) \\geq 3$,", "\\item if $f \\in \\mathfrak p \\subset A$ is a prime ideal with", "$\\dim(A/\\mathfrak p) = 2$, then $\\text{depth}(A_\\mathfrak p) \\geq 2$.", "\\end{enumerate}", "Let $U$, resp.\\ $U_0$ be the punctured spectrum of $A$, resp.\\ $A/fA$. The map", "$$", "\\Pic(U) \\to \\Pic(U_0)", "$$", "is injective. Finally, if (1), (2), (3), $A$ is $(S_2)$, and", "$\\dim(A) \\geq 4$, then (4) holds." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_U$-module.", "Observe that $\\mathcal{L}$ maps to $0$ in $\\text{Pic}(U_0)$", "if and only if we can extend $\\mathcal{L}$ to an invertible", "coherent triple $(\\mathcal{L}, \\mathcal{L}_0, \\lambda)$", "as in Section \\ref{section-coherent-triples}.", "By Proposition \\ref{proposition-hilbert-triple}", "the function", "$$", "n \\longmapsto \\chi((\\mathcal{L}, \\mathcal{L}_0, \\lambda)^{\\otimes n})", "$$", "is a polynomial $P$. By Lemma \\ref{lemma-nonnegative-chi-triple}", "we have", "$P(n) \\geq 0$ for all $n \\in \\mathbf{Z}$ with equality if and only if", "$\\mathcal{L}^{\\otimes n}$ is trivial. In particular $P(0) = 0$", "and $P$ is either identically zero and we win or $P$ has even degree $\\geq 2$.", "\\medskip\\noindent", "Set $M = \\Gamma(U, \\mathcal{L})$ and", "$M_0 = \\Gamma(X_0, \\mathcal{L}_0) = \\Gamma(U_0, \\mathcal{L}_0)$.", "Then $M$ is a finite $A$-module of depth $\\geq 2$", "and $M_0 \\cong A/fA$, see proof of Lemma \\ref{lemma-nonnegative-chi-triple}.", "Note that $H^2_\\mathfrak m(M)$ is finite $A$-module by", "Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-local-finiteness-for-finite-locally-free}", "and the fact that $H^i_\\mathfrak m(A) = 0$ for $i = 0, 1, 2$", "since $\\text{depth}(A) \\geq 3$.", "Consider the short exact sequence", "$$", "0 \\to M/fM \\to M_0 \\to Q \\to 0", "$$", "Lemma \\ref{lemma-nonnegative-chi-triple} tells us $Q$ has finite length", "equal to $\\chi(\\mathcal{L}, \\mathcal{L}_0, \\lambda)$.", "We obtain $Q = H^1_\\mathfrak m(M/fM)$ and", "$H^i_\\mathfrak m(M/fM) = H^i_\\mathfrak m(M_0) \\cong H^i_\\mathfrak m(A/fA)$", "for $i > 1$ from the long exact sequence of local cohomology", "associated to the displayed short exact sequence. Consider the long", "exact sequence of local cohomology associated to the sequence", "$0 \\to M \\to M \\to M/fM \\to 0$. It starts with", "$$", "0 \\to Q \\to H^2_\\mathfrak m(M) \\to H^2_\\mathfrak m(M) \\to", "H^2_\\mathfrak m(A/fA)", "$$", "Using additivity of lengths we see that", "$\\chi(\\mathcal{L}, \\mathcal{L}_0, \\lambda)$", "is equal to the length of the image of", "$H^2_\\mathfrak m(M) \\to H^2_\\mathfrak m(A/fA)$.", "\\medskip\\noindent", "Let prove the lemma in a special case to elucidate the rest of the proof.", "Namely, assume for a moment that $H^2_\\mathfrak m(A/fA)$ is", "a finite length module. Then", "we would have $P(1) \\leq \\text{length}_A H^2_\\mathfrak m(A/fA)$.", "The exact same argument applied to $\\mathcal{L}^{\\otimes n}$ shows that", "$P(n) \\leq \\text{length}_A H^2_\\mathfrak m(A/fA)$ for all $n$.", "Thus $P$ cannot have positive degree and we win.", "In the rest of the proof we will modify this argument to give", "a linear upper bound for $P(n)$ which suffices.", "\\medskip\\noindent", "Let us study the map", "$H^2_\\mathfrak m(M) \\to H^2_\\mathfrak m(M_0) \\cong H^2_\\mathfrak m(A/fA)$.", "Choose a normalized dualizing complex $\\omega_A^\\bullet$ for $A$.", "By local duality", "(Dualizing Complexes, Lemma \\ref{dualizing-lemma-special-case-local-duality})", "this map is Matlis dual to the map", "$$", "\\text{Ext}^{-2}_A(M, \\omega_A^\\bullet)", "\\longleftarrow", "\\text{Ext}^{-2}_A(M_0, \\omega_A^\\bullet)", "$$", "whose image therefore has the same (finite) length.", "The support (if nonempty) of the finite $A$-module", "$\\text{Ext}^{-2}_A(M_0, \\omega_A^\\bullet)$ consists of", "$\\mathfrak m$ and a finite number of primes", "$\\mathfrak p_1, \\ldots, \\mathfrak p_r$ containing $f$ with", "$\\dim(A/\\mathfrak p_i) = 1$. Namely, by", "Local Cohomology, Lemma \\ref{local-cohomology-lemma-sitting-in-degrees}", "the support is contained in the set of primes $\\mathfrak p \\subset A$ with", "$\\text{depth}_{A_\\mathfrak p}(M_{0, \\mathfrak p}) + \\dim(A/\\mathfrak p) \\leq 2$.", "Thus it suffices to show there is no prime $\\mathfrak p$ containing $f$ with", "$\\dim(A/\\mathfrak p) = 2$ and", "$\\text{depth}_{A_\\mathfrak p}(M_{0, \\mathfrak p}) = 0$.", "However, because $M_{0, \\mathfrak p} \\cong (A/fA)_\\mathfrak p$", "this would give $\\text{depth}(A_\\mathfrak p) = 1$ which contradicts", "assumption (4).", "Choose a section $t \\in \\Gamma(U, \\mathcal{L}^{\\otimes -1})$", "which does not vanish in the points $\\mathfrak p_1, \\ldots, \\mathfrak p_r$, see", "Properties, Lemma", "\\ref{properties-lemma-quasi-affine-invertible-nonvanishing-section}.", "Multiplication by $t$ determines a map $t : M \\to A$ which defines an", "isomorphism $M_{\\mathfrak p_i} \\to A_{\\mathfrak p_i}$ for", "$i = 1, \\ldots, r$.", "Via $M_0 \\cong A/fA$ we may and do", "view $t \\bmod f$ as an element $t_0 \\in A/fA$.", "We conclude that there is a commutative diagram", "$$", "\\xymatrix{", "\\text{Ext}^{-2}_A(M, \\omega_A^\\bullet) &", "\\text{Ext}^{-2}_A(M_0, \\omega_A^\\bullet) \\ar[l] \\\\", "\\text{Ext}^{-2}_A(A, \\omega_A^\\bullet) \\ar[u]^t &", "\\text{Ext}^{-2}_A(A/fA, \\omega_A^\\bullet) \\ar[l] \\ar[u]_{t_0}", "}", "$$", "It follows that the length of the image of the top horizontal", "map is at most the length of $\\text{Ext}^{-2}_A(A/fA, \\omega_A^\\bullet)$", "plus the length of the cokernel of $t_0$.", "\\medskip\\noindent", "However, if we replace $\\mathcal{L}$ by $\\mathcal{L}^n$ for $n > 1$,", "then we can use", "$$", "t^n :", "M_n = \\Gamma(U, \\mathcal{L}^{\\otimes n})", "\\longrightarrow", "\\Gamma(U_0, \\mathcal{L}_0^{\\otimes n}) = M_{n, 0}", "$$", "instead of $t$. This replaces $t_0 \\in A/fA$ by its $n$th power.", "Thus the length of the image of the map", "$\\text{Ext}^{-2}_A(M_n, \\omega_A^\\bullet) \\leftarrow", "\\text{Ext}^{-2}_A(M_{n, 0}, \\omega_A^\\bullet)$", "is at most the length of $\\text{Ext}^{-2}_A(A/fA, \\omega_A^\\bullet)$", "plus the length of the cokernel of", "$$", "t_0^n : ", "\\text{Ext}^{-2}_A(A/fA, \\omega_A^\\bullet)", "\\longrightarrow", "\\text{Ext}^{-2}_A(A/fA, \\omega_A^\\bullet)", "$$", "Since $\\text{Ext}^{-2}_A(A/fA, \\omega_A^\\bullet)$ is a finite $A$-module", "with support of dimension $1$ as indicated above this length", "grows linearly in $n$ by", "Algebra, Lemma \\ref{algebra-lemma-support-dimension-d}.", "\\medskip\\noindent", "To finish the proof we prove the final assertion. Assume", "$f \\in \\mathfrak m \\subset A$ satisfies", "(1), (2), (3), $A$ is $(S_2)$, and $\\dim(A) \\geq 4$.", "Condition (1) implies $A$ is catenary, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-universally-catenary}.", "Then $\\Spec(A)$ is equidimensional by Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-catenary-S2-equidimensional}.", "Thus $\\dim(A_\\mathfrak p) + \\dim(A/\\mathfrak p) \\geq 4$", "for every prime $\\mathfrak p$ of $A$. Then", "$\\text{depth}(A_\\mathfrak p) \\geq \\min(2, \\dim(A_\\mathfrak p))", "\\geq \\min(2, 4 - \\dim(A/\\mathfrak p))$ and hence (4) holds." ], "refs": [ "algebraization-proposition-hilbert-triple", "algebraization-lemma-nonnegative-chi-triple", "algebraization-lemma-nonnegative-chi-triple", "local-cohomology-lemma-local-finiteness-for-finite-locally-free", "algebraization-lemma-nonnegative-chi-triple", "dualizing-lemma-special-case-local-duality", "local-cohomology-lemma-sitting-in-degrees", "algebra-lemma-support-dimension-d", "dualizing-lemma-universally-catenary", "local-cohomology-lemma-catenary-S2-equidimensional" ], "ref_ids": [ 12797, 12782, 12782, 9727, 12782, 2873, 9737, 696, 2870, 9702 ] } ], "ref_ids": [] }, { "id": 12799, "type": "theorem", "label": "algebraization-proposition-trivial-local-pic-complete-intersection", "categories": [ "algebraization" ], "title": "algebraization-proposition-trivial-local-pic-complete-intersection", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring. If $A$ is a", "complete intersection of dimension $\\geq 4$, then the Picard", "group of the punctured spectrum of $A$ is trivial." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-local-pic-to-completion} we may assume that $A$ is", "a complete local ring. By assumption we can write", "$A = B/(f_1, \\ldots, f_r)$ where $B$ is a complete regular local", "ring and $f_1, \\ldots, f_r$ is a regular sequence.", "We will finish the proof by induction on $r$.", "The base case is $r = 0$ which follows from", "Lemma \\ref{lemma-trivial-local-pic-regular}.", "\\medskip\\noindent", "Assume that $A = B/(f_1, \\ldots, f_r)$ and that the proposition", "holds for $r - 1$. Set $A' = B/(f_1, \\ldots, f_{r - 1})$ and apply", "Lemma \\ref{lemma-surjective-Pic-first-better} to $f_r \\in A'$.", "This is permissible:", "\\begin{enumerate}", "\\item condition (1) of Lemma \\ref{lemma-surjective-Pic-first} holds", "because our local rings are complete,", "\\item condition (2) of Lemma \\ref{lemma-surjective-Pic-first} holds", "holds as $f_1, \\ldots, f_r$ is a regular sequence,", "\\item condition (3) and (4) of Lemma \\ref{lemma-surjective-Pic-first} hold", "as $A = A'/f_r A'$ is Cohen-Macaulay of dimension $\\dim(A) \\geq 4$,", "\\item condition (2) of Lemma \\ref{lemma-surjective-Pic-first-better}", "holds by induction hypothesis as", "$\\dim((A'_{f_r})_\\mathfrak p) \\geq 4$ for a maximal", "prime $\\mathfrak p$ of $A'_{f_r}$ and as", "$(A'_{f_r})_\\mathfrak p = B_\\mathfrak q/(f_1, \\ldots, f_{r - 1})$", "for some prime ideal $\\mathfrak q \\subset B$ and $B_\\mathfrak q$ is regular.", "\\end{enumerate}", "This finishes the proof." ], "refs": [ "algebraization-lemma-local-pic-to-completion", "algebraization-lemma-trivial-local-pic-regular", "algebraization-lemma-surjective-Pic-first-better", "algebraization-lemma-surjective-Pic-first", "algebraization-lemma-surjective-Pic-first", "algebraization-lemma-surjective-Pic-first", "algebraization-lemma-surjective-Pic-first-better" ], "ref_ids": [ 12786, 12787, 12785, 12784, 12784, 12784, 12785 ] } ], "ref_ids": [] }, { "id": 12800, "type": "theorem", "label": "algebraization-proposition-lefschetz", "categories": [ "algebraization" ], "title": "algebraization-proposition-lefschetz", "contents": [ "In the situation above assume for all points $p \\in P \\setminus Q$ we have", "$$", "\\text{depth}(\\mathcal{F}_p) + \\dim(\\overline{\\{p\\}}) > s", "$$", "Then the map", "$$", "H^i(P, \\mathcal{F}) \\longrightarrow \\lim H^i(Q_n, \\mathcal{F}_n)", "$$", "is an isomorphism for $0 \\leq i < s$." ], "refs": [], "proofs": [ { "contents": [ "We will use More on Morphisms, Lemma \\ref{more-morphisms-lemma-apply-proj-spec}", "and we will use the notation used and results found", "More on Morphisms, Section \\ref{more-morphisms-section-proj-spec}", "without further mention; this proof will not make sense without", "at least understanding the statement of the lemma.", "Observe that in our case", "$A = \\bigoplus_{m \\geq 0} \\Gamma(P, \\mathcal{L}^{\\otimes m})$", "is a finite type $k$-algebra", "all of whose graded parts are finite dimensional $k$-vector spaces, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-proper-ample}.", "\\medskip\\noindent", "We may and do think of $s$ as an element $f \\in A_1 \\subset A$, i.e.,", "a homogeneous element of degree $1$ of $A$. Denote", "$Y = V(f) \\subset X$ the closed subscheme defined by $f$.", "Then $U \\cap Y = (\\pi|_U)^{-1}(Q)$ scheme theoretically.", "Recall the notation", "$\\mathcal{F}_U = \\pi^*\\mathcal{F}|_U = (\\pi|_U)^*\\mathcal{F}$.", "This is a coherent $\\mathcal{O}_U$-module.", "Choose a finite $A$-module $M$ such that", "$\\mathcal{F}_U = \\widetilde{M}|_U$", "(for existence see Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}).", "We claim that $H^i_Z(M)$ is annihilated by", "a power of $f$ for $i \\leq s + 1$.", "\\medskip\\noindent", "To prove the claim we will apply", "Local Cohomology, Proposition \\ref{local-cohomology-proposition-annihilator}.", "Translating into geometry we see that it suffices to prove", "for $u \\in U$, $u \\not \\in Y$ and $z \\in \\overline{\\{u\\}} \\cap Z$", "that", "$$", "\\text{depth}(\\mathcal{F}_{U, u}) +", "\\dim(\\mathcal{O}_{\\overline{\\{u\\}}, z}) > s + 1", "$$", "This requires only a small amount of thought.", "\\medskip\\noindent", "Observe that $Z = \\Spec(A_0)$ is a finite set of closed points of $X$ because", "$A_0$ is a finite dimensional $k$-algebra.", "(The reader who would like $Z$ to be a singleton can replace the", "finite $k$-algebra $A_0$ by $k$; it won't affect anything else in the proof.)", "\\medskip\\noindent", "The morphism $\\pi : L \\to P$ and its restriction $\\pi|_U : U \\to P$", "are smooth of relative dimension $1$.", "Let $u \\in U$, $u \\not \\in Y$ and $z \\in \\overline{\\{u\\}} \\cap Z$.", "Let $p = \\pi(u) \\in P \\setminus Q$ be its image.", "Then either $u$ is a generic", "point of the fibre of $\\pi$ over $p$ or a closed point of the fibre.", "If $u$ is a generic point of the fibre, then", "$\\text{depth}(\\mathcal{F}_{U, u}) = \\text{depth}(\\mathcal{F}_p)$", "and", "$\\dim(\\overline{\\{u\\}}) = \\dim(\\overline{\\{p\\}}) + 1$.", "If $u$ is a closed point of the fibre, then", "$\\text{depth}(\\mathcal{F}_{U, u}) = \\text{depth}(\\mathcal{F}_p) + 1$", "and", "$\\dim(\\overline{\\{u\\}}) = \\dim(\\overline{\\{p\\}})$.", "In both cases we have", "$\\dim(\\overline{\\{u\\}}) = \\dim(\\mathcal{O}_{\\overline{\\{u\\}}, z})$", "because every point of $Z$ is closed. Thus the desired", "inequality follows from the assumption in the statement of", "the lemma.", "\\medskip\\noindent", "Let $A'$ be the $f$-adic completion of $A$. So $A \\to A'$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}.", "Denote $U' \\subset X' = \\Spec(A')$ the inverse image of", "$U$ and similarly for $Y'$ and $Z'$. Let $\\mathcal{F}'$", "on $U'$ be the pullback of $\\mathcal{F}_U$ and let", "$M' = M \\otimes_A A'$.", "By flat base change for local cohomology", "(Local Cohomology, Lemma \\ref{local-cohomology-lemma-torsion-change-rings})", "we have", "$$", "H^i_{Z'}(M') = H^i_Z(M) \\otimes_A A'", "$$", "and we find that for $i \\leq s + 1$ these are annihilated by a power of $f$.", "Consider the diagram", "$$", "\\xymatrix{", "& H^i(U, \\mathcal{F}_U) \\ar[ld] \\ar[d] \\ar[r] &", "\\lim H^i(U, \\mathcal{F}_U/f^n\\mathcal{F}_U) \\ar@{=}[d] \\\\", "H^i(U, \\mathcal{F}_U) \\otimes_A A' \\ar@{=}[r] &", "H^i(U', \\mathcal{F}') \\ar[r] &", "\\lim H^i(U', \\mathcal{F}'/f^n\\mathcal{F}')", "}", "$$", "The lower horizontal arrow is an isomorphism for $i < s$ by", "Lemma \\ref{lemma-alternative-higher} and the torsion", "property we just proved. The horizontal equal sign is flat base change", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology})", "and the vertical equal sign is because $U \\cap Y$ and $U' \\cap Y'$", "as well as their $n$th infinitesimal neighbourhoods", "are mapped isomorphically onto each other (as we are", "completing with respect to $f$).", "\\medskip\\noindent", "Applying More on Morphisms, Equation", "(\\ref{more-morphisms-equation-cohomology-torsor})", "we have compatible direct sum decompositions", "$$", "\\lim H^i(U, \\mathcal{F}_U/f^n\\mathcal{F}_U) =", "\\lim", "\\left(", "\\bigoplus\\nolimits_{m \\in \\mathbf{Z}}", "H^i(Q_n, \\mathcal{F}_n \\otimes \\mathcal{L}^{\\otimes m})", "\\right)", "$$", "and", "$$", "H^i(U, \\mathcal{F}_U) =", "\\bigoplus\\nolimits_{m \\in \\mathbf{Z}}", "H^i(P, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes m})", "$$", "Thus we conclude by Algebra, Lemma \\ref{algebra-lemma-daniel-litt}." ], "refs": [ "more-morphisms-lemma-apply-proj-spec", "coherent-lemma-coherent-proper-ample", "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "local-cohomology-proposition-annihilator", "algebra-lemma-completion-flat", "local-cohomology-lemma-torsion-change-rings", "algebraization-lemma-alternative-higher", "coherent-lemma-flat-base-change-cohomology", "algebra-lemma-daniel-litt" ], "ref_ids": [ 13934, 3343, 9729, 9786, 870, 9718, 12735, 3298, 882 ] } ], "ref_ids": [] }, { "id": 12801, "type": "theorem", "label": "algebraization-proposition-lefschetz-existence", "categories": [ "algebraization" ], "title": "algebraization-proposition-lefschetz-existence", "contents": [ "In the situation above let $(\\mathcal{F}_n)$ be an object of", "$\\textit{Coh}(P, \\mathcal{I})$. Assume for all $q \\in Q$ and for", "all primes $\\mathfrak p \\in \\mathcal{O}_{P, q}^\\wedge$,", "$\\mathfrak p \\not \\in V(\\mathcal{I}_q^\\wedge)$ we have", "$$", "\\text{depth}((\\mathcal{F}_q^\\wedge)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{P, q}^\\wedge/\\mathfrak p) +", "\\dim(\\overline{\\{q\\}}) > 2", "$$", "Then $(\\mathcal{F}_n)$ is the completion of a coherent", "$\\mathcal{O}_P$-module." ], "refs": [], "proofs": [ { "contents": [ "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-existence-easy}", "to prove the lemma, we may replace $(\\mathcal{F}_n)$ by an object", "differing from it by $\\mathcal{I}$-torsion (see below for more precision).", "Let $T' = \\{q \\in Q \\mid \\dim(\\overline{\\{q\\}}) = 0\\}$", "and $T = \\{q \\in Q \\mid \\dim(\\overline{\\{q\\}}) \\leq 1\\}$.", "The assumption in the proposition is exactly that", "$Q \\subset P$, $(\\mathcal{F}_n)$, and $T' \\subset T \\subset Q$", "satisfy the conditions of", "Lemma \\ref{lemma-improvement-formal-coherent-module-better}", "with $d = 1$; besides trivial manipulations of inequalities, use that", "$V(\\mathfrak p) \\cap V(\\mathcal{I}^\\wedge_y) = \\{\\mathfrak m^\\wedge_y\\}", "\\Leftrightarrow \\dim(\\mathcal{O}_{P, q}^\\wedge/\\mathfrak p) = 1$", "as $\\mathcal{I}_y^\\wedge$ is generated by $1$ element.", "Combining these two remarks, we may replace $(\\mathcal{F}_n)$ by the", "object $(\\mathcal{H}_n)$ of $\\textit{Coh}(P, \\mathcal{I})$ found in", "Lemma \\ref{lemma-improvement-formal-coherent-module-better}.", "Thus we may and do assume $(\\mathcal{F}_n)$ is pro-isomorphic to", "an inverse system $(\\mathcal{F}_n'')$ of coherent $\\mathcal{O}_P$-modules", "such that $\\text{depth}(\\mathcal{F}''_{n, q}) + \\dim(\\overline{\\{q\\}}) \\geq 2$", "for all $q \\in Q$.", "\\medskip\\noindent", "We will use More on Morphisms, Lemma \\ref{more-morphisms-lemma-apply-proj-spec}", "and we will use the notation used and results found", "More on Morphisms, Section \\ref{more-morphisms-section-proj-spec}", "without further mention; this proof will not make sense without", "at least understanding the statement of the lemma.", "Observe that in our case", "$A = \\bigoplus_{m \\geq 0} \\Gamma(P, \\mathcal{L}^{\\otimes m})$", "is a finite type $k$-algebra", "all of whose graded parts are finite dimensional $k$-vector spaces, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-proper-ample}.", "\\medskip\\noindent", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-pullback}", "the pull back by $\\pi|_U : U \\to P$ is an object", "$(\\pi|_U^*\\mathcal{F}_n)$ of $\\textit{Coh}(U, f\\mathcal{O}_U)$", "which is pro-isomorphic to the inverse system", "$(\\pi|_U^*\\mathcal{F}_n'')$ of coherent $\\mathcal{O}_U$-modules.", "We claim", "$$", "\\text{depth}(\\pi|_U^*\\mathcal{F}''_{n, y}) + \\delta_Z^Y(y) \\geq 3", "$$", "for all $y \\in U \\cap Y$. Since all the points of $Z$ are closed, we", "see that $\\delta_Z^Y(y) \\geq \\dim(\\overline{\\{y\\}})$ for all", "$y \\in U \\cap Y$, see Lemma \\ref{lemma-discussion}.", "Let $q \\in Q$ be the image of $y$. Since the morphism $\\pi : U \\to P$ is", "smooth of relative dimension $1$ we see that either $y$ is a closed point", "of a fibre of $\\pi$ or a generic point.", "Thus we see that", "$$", "\\text{depth}(\\pi^*\\mathcal{F}''_{n, y}) + \\delta_Z^Y(y)", "\\geq", "\\text{depth}(\\pi^*\\mathcal{F}''_{n, y}) + \\dim(\\overline{\\{y\\}}) =", "\\text{depth}(\\mathcal{F}''_{n, q}) + \\dim(\\overline{\\{q\\}}) + 1", "$$", "because either the depth goes up by $1$ or the dimension.", "This proves the claim.", "\\medskip\\noindent", "By Lemma \\ref{lemma-cd-1-canonical} we conclude that", "$(\\pi|_U^*\\mathcal{F}_n)$ canonically extends to $X$.", "Observe that", "$$", "M_n = \\Gamma(U, \\pi|_U^*\\mathcal{F}_n) =", "\\bigoplus\\nolimits_{m \\in \\mathbf{Z}}", "\\Gamma(P, \\mathcal{F}_n \\otimes_{\\mathcal{O}_P} \\mathcal{L}^{\\otimes m})", "$$", "is canonically a graded $A$-module, see", "More on Morphisms, Equation (\\ref{more-morphisms-equation-cohomology-torsor}).", "By Properties, Lemma \\ref{properties-lemma-quasi-coherent-quasi-affine}", "we have $\\pi|_U^*\\mathcal{F}_n = \\widetilde{M_n}|_U$.", "Thus we may apply Lemma \\ref{lemma-Gm-equivariant-extend-canonically}", "to find a finite graded $A$-module $N$ such that", "$(M_n)$ and $(N/I^nN)$ are pro-isomorphic in the category", "of graded $A$-modules modulo $A_+$-torsion modules.", "Let $\\mathcal{F}$ be the coherent $\\mathcal{O}_P$-module", "associated to $N$, see", "Cohomology of Schemes, Proposition", "\\ref{coherent-proposition-coherent-modules-on-proj-general}.", "The same proposition tells us that $(\\mathcal{F}/\\mathcal{I}^n\\mathcal{F})$", "is pro-isomorphic to $(\\mathcal{F}_n)$.", "Since both are objects of $\\textit{Coh}(P, \\mathcal{I})$", "we win by Lemma \\ref{lemma-recognize-formal-coherent-modules}." ], "refs": [ "coherent-lemma-existence-easy", "algebraization-lemma-improvement-formal-coherent-module-better", "algebraization-lemma-improvement-formal-coherent-module-better", "more-morphisms-lemma-apply-proj-spec", "coherent-lemma-coherent-proper-ample", "coherent-lemma-inverse-systems-pullback", "algebraization-lemma-discussion", "algebraization-lemma-cd-1-canonical", "properties-lemma-quasi-coherent-quasi-affine", "algebraization-lemma-Gm-equivariant-extend-canonically", "coherent-proposition-coherent-modules-on-proj-general", "algebraization-lemma-recognize-formal-coherent-modules" ], "ref_ids": [ 3375, 12772, 12772, 13934, 3343, 3378, 12758, 12774, 3007, 12789, 3400, 12740 ] } ], "ref_ids": [] }, { "id": 12802, "type": "theorem", "label": "algebraization-proposition-lefschetz-equivalence", "categories": [ "algebraization" ], "title": "algebraization-proposition-lefschetz-equivalence", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$.", "Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module", "and let $s \\in \\Gamma(X, \\mathcal{L})$. Let $Y = Z(s)$", "be the zero scheme of $s$ and denote $\\mathcal{I} \\subset \\mathcal{O}_X$", "the corresponding sheaf of ideals.", "Let $\\mathcal{V}$ be the set of open subschemes of $X$ containing $Y$", "ordered by reverse inclusion.", "Assume that for all $x \\in X \\setminus Y$ we have", "$$", "\\text{depth}(\\mathcal{O}_{X, x}) + \\dim(\\overline{\\{x\\}}) > 2", "$$", "Then the completion functor", "$$", "\\colim_\\mathcal{V}", "\\textit{Coh}(\\mathcal{O}_V)", "\\longrightarrow", "\\textit{Coh}(X, \\mathcal{I})", "$$", "is an equivalence on the full subcategories of finite locally free objects." ], "refs": [], "proofs": [ { "contents": [ "To prove fully faithfulness it suffices to prove that", "$$", "\\colim_\\mathcal{V} \\Gamma(V, \\mathcal{L}^{\\otimes m})", "\\longrightarrow", "\\lim \\Gamma(Y_n, \\mathcal{L}^{\\otimes m}|_{Y_n})", "$$", "is an isomorphism for all $m$, see", "Lemma \\ref{lemma-completion-fully-faithful-general}.", "This follows from Lemma \\ref{lemma-lefschetz-addendum}.", "\\medskip\\noindent", "Essential surjectivity. Let $(\\mathcal{F}_n)$ be a finite locally", "free object of $\\textit{Coh}(X, \\mathcal{I})$. Then for $y \\in Y$ we have", "$\\mathcal{F}_y^\\wedge = \\lim \\mathcal{F}_{n, y}$ is", "is a finite free $\\mathcal{O}_{X, y}^\\wedge$-module.", "Let $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$", "be a prime with $\\mathfrak p \\not \\in V(\\mathcal{I}_y^\\wedge)$.", "Then $\\mathfrak p$ lies over a prime $\\mathfrak p_0 \\subset \\mathcal{O}_{X, y}$", "which corresponds to a specialization $x \\leadsto y$ with", "$x \\not \\in Y$. By Local Cohomology, Lemma", "\\ref{local-cohomology-lemma-change-completion}", "and some dimension theory", "(see Varieties, Section \\ref{varieties-section-algebraic-schemes})", "we have", "$$", "\\text{depth}((\\mathcal{O}_{X, y}^\\wedge)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) =", "\\text{depth}(\\mathcal{O}_{X, x}) +", "\\dim(\\overline{\\{x\\}}) - \\dim(\\overline{\\{y\\}})", "$$", "Thus our assumptions imply the assumptions of", "Proposition \\ref{proposition-lefschetz-existence}", "are satisfied and we find that $(\\mathcal{F}_n)$", "is the completion of a coherent $\\mathcal{O}_X$-module $\\mathcal{F}$.", "It then follows that $\\mathcal{F}_y$ is finite free for all $y \\in Y$", "and hence $\\mathcal{F}$ is finite locally free in an open", "neighbourhood $V$ of $Y$. This finishes the proof." ], "refs": [ "algebraization-lemma-completion-fully-faithful-general", "algebraization-lemma-lefschetz-addendum", "local-cohomology-lemma-change-completion", "algebraization-proposition-lefschetz-existence" ], "ref_ids": [ 12739, 12788, 9740, 12801 ] } ], "ref_ids": [] }, { "id": 12821, "type": "theorem", "label": "spaces-over-fields-lemma-quasi-finite-in-codim-1", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-quasi-finite-in-codim-1", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is locally of finite type and $Y$ is locally Noetherian.", "Let $y \\in |Y|$ be a point of codimension $\\leq 1$ on $Y$.", "Let $X^0 \\subset |X|$ be the set of points of codimension $0$ on $X$.", "Assume in addition one of the following conditions is satisfied", "\\begin{enumerate}", "\\item for every $x \\in X^0$ the transcendence degree of $x/f(x)$ is $0$,", "\\item for every $x \\in X^0$ with $f(x) \\leadsto y$", "the transcendence degree of $x/f(x)$ is $0$,", "\\item $f$ is quasi-finite at every $x \\in X^0$,", "\\item $f$ is quasi-finite at a dense set of points of $|X|$,", "\\item add more here.", "\\end{enumerate}", "Then $f$ is quasi-finite at every point of $X$ lying over $y$." ], "refs": [], "proofs": [ { "contents": [ "We want to reduce the proof to the case of schemes. To do this we", "choose a commutative diagram", "$$", "\\xymatrix{", "U \\ar[r] \\ar[d]_g & X \\ar[d]^f \\\\", "V \\ar[r] & Y", "}", "$$", "where $U$, $V$ are schemes and where the horizontal arrows are \\'etale", "and surjective. Pick $v \\in V$ mapping to $y$. Observe that", "$V$ is locally Noetherian and that $\\dim(\\mathcal{O}_{V, v}) \\leq 1$", "(see Properties of Spaces, Definitions", "\\ref{spaces-properties-definition-dimension-local-ring} and", "Remark \\ref{spaces-properties-remark-list-properties-local-etale-topology}).", "The fibre $U_v$ of $U \\to V$ over $v$ surjects onto", "$f^{-1}(\\{y\\}) \\subset |X|$. The inverse image of $X^0$ in $U$", "is exactly the set of", "generic points of irreducible components of $U$ (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-codimension-0-points}).", "If $\\eta \\in U$ is such a point with image $x \\in X^0$, then", "the transcendence degree of $x / f(x)$ is the transcendence", "degree of $\\kappa(\\eta)$ over $\\kappa(g(\\eta))$", "(Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-dimension-fibre}).", "Observe that $U \\to V$ is quasi-finite at $u \\in U$ if and only if", "$f$ is quasi-finite at the image of $u$ in $X$.", "\\medskip\\noindent", "Case (1). Here case (1) of", "Varieties, Lemma \\ref{varieties-lemma-quasi-finite-in-codim-1} applies", "and we conclude that $U \\to V$ is quasi-finite at all points of $U_v$.", "Hence $f$ is quasi-finite at every point lying over $y$.", "\\medskip\\noindent", "Case (2). Let $u \\in U$ be a generic point of an irreducible component", "whose image in $V$ specializes to $v$. Then the image $x \\in X^0$ of", "$u$ has the property that $f(x) \\leadsto y$. Hence we see that", "case (2) of", "Varieties, Lemma \\ref{varieties-lemma-quasi-finite-in-codim-1} applies", "and we conclude as before.", "\\medskip\\noindent", "Case (3) follows from case (3) of", "Varieties, Lemma \\ref{varieties-lemma-quasi-finite-in-codim-1}.", "\\medskip\\noindent", "In case (4), since $|U| \\to |X|$ is open, we see that", "the set of points where $U \\to V$ is quasi-finite is dense as well.", "Hence case (4) of", "Varieties, Lemma \\ref{varieties-lemma-quasi-finite-in-codim-1} applies." ], "refs": [ "spaces-properties-definition-dimension-local-ring", "spaces-properties-remark-list-properties-local-etale-topology", "spaces-properties-lemma-codimension-0-points", "spaces-morphisms-definition-dimension-fibre", "varieties-lemma-quasi-finite-in-codim-1", "varieties-lemma-quasi-finite-in-codim-1", "varieties-lemma-quasi-finite-in-codim-1", "varieties-lemma-quasi-finite-in-codim-1" ], "ref_ids": [ 11931, 11950, 11842, 5009, 10977, 10977, 10977, 10977 ] } ], "ref_ids": [] }, { "id": 12822, "type": "theorem", "label": "spaces-over-fields-lemma-finite-in-codim-1", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-finite-in-codim-1", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume $f$ is proper and $Y$ is locally Noetherian.", "Let $y \\in Y$ be a point of codimension $\\leq 1$ in $Y$.", "Let $X^0 \\subset |X|$ be the set of points of codimension $0$ on $X$.", "Assume in addition one of the", "following conditions is satisfied", "\\begin{enumerate}", "\\item for every $x \\in X^0$ the transcendence degree of $x/f(x)$ is $0$,", "\\item for every $x \\in X^0$ with $f(x) \\leadsto y$ the transcendence degree", "of $x/f(x)$ is $0$,", "\\item $f$ is quasi-finite at every $x \\in X^0$,", "\\item $f$ is quasi-finite at a dense set of points of $|X|$,", "\\item add more here.", "\\end{enumerate}", "Then there exists an open subspace $Y' \\subset Y$ containing $y$ such that", "$Y' \\times_Y X \\to Y'$ is finite." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-quasi-finite-in-codim-1} the morphism $f$ is", "quasi-finite at every point lying over $y$. Let $\\overline{y} : \\Spec(k) \\to Y$", "be a geometric point lying over $y$. Then $|X_{\\overline{y}}|$ is a", "discrete space (Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-conditions-on-fibre-and-qf}).", "Since $X_{\\overline{y}}$ is quasi-compact as $f$ is proper we conclude", "that $|X_{\\overline{y}}|$ is finite.", "Thus we can apply Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-finite-fibre-finite-in-neighbourhood}", "to conclude." ], "refs": [ "spaces-over-fields-lemma-quasi-finite-in-codim-1", "decent-spaces-lemma-conditions-on-fibre-and-qf", "spaces-cohomology-lemma-proper-finite-fibre-finite-in-neighbourhood" ], "ref_ids": [ 12821, 9529, 11344 ] } ], "ref_ids": [] }, { "id": 12823, "type": "theorem", "label": "spaces-over-fields-lemma-modification-normal-iso-over-codimension-1", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-modification-normal-iso-over-codimension-1", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $f : Y \\to X$ be a birational proper morphism of algebraic spaces", "with $Y$ reduced.", "Let $U \\subset X$ be the maximal open over which $f$ is an isomorphism.", "Then $U$ contains", "\\begin{enumerate}", "\\item every point of codimension $0$ in $X$,", "\\item every $x \\in |X|$ of codimension $1$ on $X$ such that the local ring of", "$X$ at $x$ is normal (Properties of Spaces, Remark", "\\ref{spaces-properties-remark-list-properties-local-ring-local-etale-topology}),", "and", "\\item every $x \\in |X|$ such that the fibre of $|Y| \\to |X|$ over $x$ is", "finite and such that the local ring of $X$ at $x$ is normal.", "\\end{enumerate}" ], "refs": [ "spaces-properties-remark-list-properties-local-ring-local-etale-topology" ], "proofs": [ { "contents": [ "Part (1) follows from Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-birational-isomorphism-over-dense-open}", "(and the fact that the Noetherian algebraic spaces $X$ and $Y$", "are quasi-separated and hence decent).", "Part (2) follows from part (3) and Lemma \\ref{lemma-finite-in-codim-1}", "(and the fact that finite morphisms have finite fibres).", "Let $x \\in |X|$ be as in (3). By", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-finite-fibre-finite-in-neighbourhood}", "(which applies by Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-conditions-on-fibre-and-qf})", "we may assume $f$ is finite. Choose an affine scheme $X'$ and", "an \\'etale morphism $X' \\to X$ and a point $x' \\in X$ mapping to $x$.", "It suffices to show there exists an open neighbourhood $U'$ of $x' \\in X'$", "such that $Y \\times_X X' \\to X'$ is an isomorphism over $U'$", "(namely, then $U$ contains the image of $U'$ in $X$, see Spaces, Lemma", "\\ref{spaces-lemma-descent-representable-transformations-property}).", "Then $Y \\times_X X' \\to X$ is a finite birational", "(Decent Spaces, Lemma \\ref{decent-spaces-lemma-birational-etale-localization})", "morphism. Since a finite morphism is affine we reduce to", "the case of a finite birational morphism of Noetherian affine schemes", "$Y \\to X$ and $x \\in X$ such that $\\mathcal{O}_{X, x}$ is a", "normal domain. This is treated in Varieties, Lemma", "\\ref{varieties-lemma-modification-normal-iso-over-codimension-1}." ], "refs": [ "decent-spaces-lemma-birational-isomorphism-over-dense-open", "spaces-over-fields-lemma-finite-in-codim-1", "spaces-cohomology-lemma-proper-finite-fibre-finite-in-neighbourhood", "decent-spaces-lemma-conditions-on-fibre-and-qf", "spaces-lemma-descent-representable-transformations-property", "decent-spaces-lemma-birational-etale-localization", "varieties-lemma-modification-normal-iso-over-codimension-1" ], "ref_ids": [ 9542, 12822, 11344, 9529, 8134, 9543, 10979 ] } ], "ref_ids": [ 11951 ] }, { "id": 12824, "type": "theorem", "label": "spaces-over-fields-lemma-integral-algebraic-space-rational-functions", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-integral-algebraic-space-rational-functions", "contents": [ "Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$.", "Let $\\eta \\in |X|$ be the generic point of $X$.", "There are canonical identifications", "$$", "R(X) = \\mathcal{O}_{X, \\eta}^h = \\kappa(\\eta)", "$$", "where $R(X)$ is the ring of rational functions defined in", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-ring-of-rational-functions},", "$\\kappa(\\eta)$ is the residue field defined in", "Decent Spaces, Definition \\ref{decent-spaces-definition-residue-field},", "and $\\mathcal{O}_{X, \\eta}^h$ is the henselian local ring defined in", "Decent Spaces, Definition", "\\ref{decent-spaces-definition-elemenary-etale-neighbourhood}.", "In particular, these rings are fields." ], "refs": [ "spaces-morphisms-definition-ring-of-rational-functions", "decent-spaces-definition-residue-field", "decent-spaces-definition-elemenary-etale-neighbourhood" ], "proofs": [ { "contents": [ "Since $X$ is a scheme in an open neighbourhood of $\\eta$ (see discussion", "above), this follows immediately from the corresponding result for", "schemes, see Morphisms, Lemma", "\\ref{morphisms-lemma-integral-scheme-rational-functions}.", "We also use: the henselianization of a field is itself", "and that our definitions of these objects", "for algebraic spaces are compatible with those for schemes.", "Details omitted." ], "refs": [ "morphisms-lemma-integral-scheme-rational-functions" ], "ref_ids": [ 5478 ] } ], "ref_ids": [ 5021, 9563, 9564 ] }, { "id": 12825, "type": "theorem", "label": "spaces-over-fields-lemma-integral-sections", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-integral-sections", "contents": [ "Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$.", "Then $\\Gamma(X, \\mathcal{O}_X)$ is a domain." ], "refs": [], "proofs": [ { "contents": [ "Set $R = \\Gamma(X, \\mathcal{O}_X)$. If $f, g \\in R$ are nonzero and", "$fg = 0$ then $X = V(f) \\cup V(g)$ where $V(f)$ denotes the closed subspace", "of $X$ cut out by $f$. Since $X$ is irreducible, we see that either", "$V(f) = X$ or $V(g) = X$. Then either $f = 0$ or $g = 0$ by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-reduced-space}." ], "refs": [ "spaces-properties-lemma-reduced-space" ], "ref_ids": [ 11883 ] } ], "ref_ids": [] }, { "id": 12826, "type": "theorem", "label": "spaces-over-fields-lemma-normal-integral-cover-by-affines", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-normal-integral-cover-by-affines", "contents": [ "Let $S$ be a scheme. Let $X$ be a normal integral algebraic space over $S$.", "For every $x \\in |X|$ there exists a normal integral affine scheme $U$", "and an \\'etale morphism $U \\to X$ such that $x$ is in the image." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine scheme $U$ and an \\'etale morphism $U \\to X$ such that", "$x$ is in the image. Let $u_i$, $i \\in I$ be the generic points of irreducible", "components of $U$. Then each $u_i$ maps to the generic point of $X$", "(Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-generic-points}). By ", "our definition of a decent space", "(Decent Spaces, Definition \\ref{decent-spaces-definition-very-reasonable}),", "we see that $I$ is finite. Hence $U = \\Spec(A)$ where $A$ is a normal ring", "with finitely many minimal primes.", "Thus $A = \\prod_{i \\in I} A_i$ is a product of normal domains by", "Algebra, Lemma \\ref{algebra-lemma-characterize-reduced-ring-normal}.", "Then $U = \\coprod U_i$ with $U_i = \\Spec(A_i)$ and $x$ is in the image of", "$U_i \\to X$ for some $i$. This proves the lemma." ], "refs": [ "decent-spaces-lemma-decent-generic-points", "decent-spaces-definition-very-reasonable", "algebra-lemma-characterize-reduced-ring-normal" ], "ref_ids": [ 9531, 9562, 515 ] } ], "ref_ids": [] }, { "id": 12827, "type": "theorem", "label": "spaces-over-fields-lemma-normal-integral-sections", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-normal-integral-sections", "contents": [ "Let $S$ be a scheme. Let $X$ be a normal integral algebraic space over $S$.", "Then $\\Gamma(X, \\mathcal{O}_X)$ is a normal domain." ], "refs": [], "proofs": [ { "contents": [ "Set $R = \\Gamma(X, \\mathcal{O}_X)$. Then $R$ is a domain by", "Lemma \\ref{lemma-integral-sections}.", "Let $f = a/b$ be an element of the fraction field of $R$", "which is integral over $R$.", "For any $U \\to X$ \\'etale with $U$ a scheme there is at most one", "$f_U \\in \\Gamma(U, \\mathcal{O}_U)$ with $b|_U f_U = a|_U$.", "Namely, $U$ is reduced and the generic points of $U$ map to", "the generic point of $X$ which implies that $b|_U$ is a", "nonzerodivisor.", "For every $x \\in |X|$ we choose $U \\to X$ as in", "Lemma \\ref{lemma-normal-integral-cover-by-affines}.", "Then there is a unique $f_U \\in \\Gamma(U, \\mathcal{O}_U)$", "with $b|_U f_U = a|_U$ because", "$\\Gamma(U, \\mathcal{O}_U)$ is a normal domain by", "Properties, Lemma \\ref{properties-lemma-normal-integral-sections}.", "By the uniqueness mentioned above these $f_U$", "glue and define a global section $f$ of the structure", "sheaf, i.e., of $R$." ], "refs": [ "spaces-over-fields-lemma-integral-sections", "spaces-over-fields-lemma-normal-integral-cover-by-affines", "properties-lemma-normal-integral-sections" ], "ref_ids": [ 12825, 12826, 2972 ] } ], "ref_ids": [] }, { "id": 12828, "type": "theorem", "label": "spaces-over-fields-lemma-decent-irreducible-closed", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-decent-irreducible-closed", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "There are canonical bijections between the following sets:", "\\begin{enumerate}", "\\item the set of points of $X$, i.e., $|X|$,", "\\item the set of irreducible closed subsets of $|X|$,", "\\item the set of integral closed subspaces of $X$.", "\\end{enumerate}", "The bijection from (1) to (2) sends $x$ to $\\overline{\\{x\\}}$.", "The bijection from (3) to (2) sends $Z$ to $|Z|$." ], "refs": [], "proofs": [ { "contents": [ "Our map defines a bijection between (1) and (2) as $|X|$ is", "sober by ", "Decent Spaces, Proposition \\ref{decent-spaces-proposition-reasonable-sober}.", "Given $T \\subset |X|$ closed and irreducible, there is a", "unique reduced closed subspace $Z \\subset X$ such that", "$|Z| = T$, namely, $Z$ is the reduced induced subspace structure", "on $T$, see Properties of Spaces, Definition", "\\ref{spaces-properties-definition-reduced-induced-space}.", "This is an integral algebraic space because it is decent,", "reduced, and irreducible." ], "refs": [ "decent-spaces-proposition-reasonable-sober", "spaces-properties-definition-reduced-induced-space" ], "ref_ids": [ 9559, 11932 ] } ], "ref_ids": [] }, { "id": 12829, "type": "theorem", "label": "spaces-over-fields-lemma-finite-degree", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-finite-degree", "contents": [ "Let $S$ be a scheme. Let $X$, $Y$ be integral algebraic spaces over $S$", "Let $x \\in |X|$ and $y \\in |Y|$ be the generic points. Let $f : X \\to Y$", "be locally of finite type. Assume $f$ is dominant", "(Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-dominant}).", "The following are equivalent:", "\\begin{enumerate}", "\\item the transcendence degree of $x/y$ is $0$,", "\\item the extension $\\kappa(x) \\supset \\kappa(y)$ (see proof) is finite,", "\\item there exist nonempty affine opens $U \\subset X$ and $V \\subset Y$", "such that $f(U) \\subset V$ and $f|_U : U \\to V$ is finite,", "\\item $f$ is quasi-finite at $x$, and", "\\item $x$ is the only point of $|X|$ mapping to $y$.", "\\end{enumerate}", "If $f$ is separated or if $f$ is quasi-compact, then these are", "also equivalent to", "\\begin{enumerate}", "\\item[(6)] there exists a nonempty affine open $V \\subset Y$ such", "that $f^{-1}(V) \\to V$ is finite.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-definition-dominant" ], "proofs": [ { "contents": [ "By elementary topology, we see that $f(x) = y$ as $f$ is dominant.", "Let $Y' \\subset Y$ be the schematic locus of $Y$ and let", "$X' \\subset f^{-1}(Y')$ be the schematic locus of $f^{-1}(Y')$.", "By the discussion above, using", "Decent Spaces, Proposition \\ref{decent-spaces-proposition-reasonable-sober} and", "Theorem \\ref{decent-spaces-theorem-decent-open-dense-scheme},", "we see that $x \\in |X'|$ and $y \\in |Y'|$.", "Then $f|_{X'} : X' \\to Y'$ is a morphism of integral schemes", "which is locally of finite type. Thus we see that (1), (2), (3)", "are equivalent by Morphisms, Lemma \\ref{morphisms-lemma-finite-degree}.", "\\medskip\\noindent", "Condition (4) implies condition (1) by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-compare-tr-deg}", "applied to $X \\to Y \\to Y$.", "On the other hand, condition (3) implies condition (4) as", "a finite morphism is quasi-finite and as $x \\in U$ because $x$", "is the generic point. Thus (1) -- (4) are equivalent.", "\\medskip\\noindent", "Assume the equivalent conditions (1) -- (4). Suppose that", "$x' \\mapsto y$. Then $x \\leadsto x'$ is a specialization in the", "fibre of $|X| \\to |Y|$ over $y$. If $x' \\not = x$, then $f$ is not", "quasi-finite at $x$ by Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-conditions-on-point-in-fibre-and-qf}.", "Hence $x = x'$ and (5) holds. Conversely, if (5) holds, then", "(5) holds for the morphism of schemes $X' \\to Y'$ (see above)", "and we can use", "Morphisms, Lemma \\ref{morphisms-lemma-finite-degree}", "to see that (1) holds.", "\\medskip\\noindent", "Observe that (6) implies the equivalent conditions (1) -- (5)", "without any further assumptions on $f$. To finish the proof", "we have to show the equivalent conditions (1) -- (5) imply (6).", "This follows from Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-finite-over-dense-open}." ], "refs": [ "decent-spaces-proposition-reasonable-sober", "decent-spaces-theorem-decent-open-dense-scheme", "morphisms-lemma-finite-degree", "spaces-morphisms-lemma-compare-tr-deg", "decent-spaces-lemma-conditions-on-point-in-fibre-and-qf", "morphisms-lemma-finite-degree", "decent-spaces-lemma-finite-over-dense-open" ], "ref_ids": [ 9559, 9454, 5491, 4867, 9528, 5491, 9538 ] } ], "ref_ids": [ 4996 ] }, { "id": 12830, "type": "theorem", "label": "spaces-over-fields-lemma-degree-composition", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-degree-composition", "contents": [ "Let $S$ be a scheme.", "Let $X$, $Y$, $Z$ be integral algebraic spaces over $S$.", "Let $f : X \\to Y$ and $g : Y \\to Z$ be dominant morphisms locally", "of finite type. Assume any of the equivalent conditions", "(1) -- (5) of Lemma \\ref{lemma-finite-degree} hold for $f$ and $g$. Then", "$$", "\\deg(X/Z) = \\deg(X/Y) \\deg(Y/Z).", "$$" ], "refs": [ "spaces-over-fields-lemma-finite-degree" ], "proofs": [ { "contents": [ "This comes from the multiplicativity of degrees in towers", "of finite extensions of fields, see", "Fields, Lemma \\ref{fields-lemma-multiplicativity-degrees}." ], "refs": [ "fields-lemma-multiplicativity-degrees" ], "ref_ids": [ 4450 ] } ], "ref_ids": [ 12829 ] }, { "id": 12831, "type": "theorem", "label": "spaces-over-fields-lemma-components-locally-finite", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-components-locally-finite", "contents": [ "Let $S$ be a scheme and let $X$ be a locally Noetherian", "algebraic space over $S$. If $T \\subset |X|$ is a closed subset,", "then the collection of irreducible components of $T$ is locally finite." ], "refs": [], "proofs": [ { "contents": [ "The topological space $|X|$ is locally Noetherian", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-Noetherian-topology}).", "A Noetherian topological space has a finite number of", "irreducible components and a subspace of a Noetherian space is Noetherian", "(Topology, Lemma \\ref{topology-lemma-Noetherian}).", "Thus the lemma follows from the definition of locally finite", "(Topology, Definition \\ref{topology-definition-locally-finite})." ], "refs": [ "spaces-properties-lemma-Noetherian-topology", "topology-lemma-Noetherian", "topology-definition-locally-finite" ], "ref_ids": [ 11891, 8220, 8376 ] } ], "ref_ids": [] }, { "id": 12832, "type": "theorem", "label": "spaces-over-fields-lemma-order-vanishing", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-order-vanishing", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space", "over $S$. Let $Z \\subset X$ be a prime divisor and let $\\xi \\in |Z|$ be", "the generic point. Then the henselian local ring $\\mathcal{O}_{X, \\xi}^h$", "is a reduced $1$-dimensional Noetherian local ring and", "there is a canonical injective map", "$$", "R(X) \\longrightarrow Q(\\mathcal{O}_{X, \\xi}^h)", "$$", "from the function field $R(X)$ of $X$ into the total ring of fractions." ], "refs": [], "proofs": [ { "contents": [ "We will use the results of Decent Spaces, Section", "\\ref{decent-spaces-section-residue-fields-henselian-local-rings}.", "Let $(U, u) \\to (X, \\xi)$ be an elementary \\'etale neighbourhood.", "Observe that $U$ is locally Noetherian and reduced.", "Thus $\\mathcal{O}_{U, u}$ is a $1$-dimensional", "(by our definition of prime divisors)", "reduced Noetherian ring.", "After replacing $U$ by an affine open neighbourhood of $u$", "we may assume $U$ is Noetherian and affine.", "After replacing $U$ by a smaller open, we may assume every irreducible", "component of $U$ passes through $u$.", "Since $U \\to X$ is open and $X$ irreducible, $U \\to X$ is dominant.", "Hence we obtain a ring map $R(X) \\to R(U)$ by composing rational maps, see", "Morphisms of Spaces, Section \\ref{spaces-morphisms-section-rational-maps}.", "Since $R(X)$ is a field, this map is injective.", "By our choice of $U$ we see that $R(U)$ is the total quotient", "ring $Q(\\mathcal{O}_{U, u})$, see", "Morphisms, Lemma \\ref{morphisms-lemma-integral-scheme-rational-functions}", "and", "Algebra, Lemma \\ref{algebra-lemma-total-ring-fractions-no-embedded-points}.", "\\medskip\\noindent", "At this point we have proved all the statements in the lemma with", "$\\mathcal{O}_{U, u}$ in stead of $\\mathcal{O}_{X, \\xi}^h$.", "However, $\\mathcal{O}_{X, \\xi}^h$ is the henselization of", "$\\mathcal{O}_{U, u}$. Thus $\\mathcal{O}_{X, \\xi}^h$ is a $1$-dimensional", "reduced Noetherian ring, see", "More on Algebra, Lemmas", "\\ref{more-algebra-lemma-henselization-reduced},", "\\ref{more-algebra-lemma-henselization-dimension}, and", "\\ref{more-algebra-lemma-henselization-noetherian}.", "Since $\\mathcal{O}_{U, u} \\to \\mathcal{O}_{X, \\xi}^h$ is", "faithfully flat by More on Algebra, Lemma", "\\ref{more-algebra-lemma-dumb-properties-henselization}", "it sends nonzerodivisors to nonzerodivisors.", "Therefore we obtain a canonical map", "$Q(\\mathcal{O}_{U, u}) \\to Q(\\mathcal{O}_{X, \\xi}^h)$", "and we obtain our map.", "We omit the verification that the map is independent of", "the choice of $(U, u) \\to (X, x)$; a slightly better", "approach would be to first observe that", "$\\colim Q(\\mathcal{O}_{U, u}) = Q(\\mathcal{O}_{X, \\xi}^h)$." ], "refs": [ "morphisms-lemma-integral-scheme-rational-functions", "algebra-lemma-total-ring-fractions-no-embedded-points", "more-algebra-lemma-henselization-reduced", "more-algebra-lemma-henselization-dimension", "more-algebra-lemma-henselization-noetherian", "more-algebra-lemma-dumb-properties-henselization" ], "ref_ids": [ 5478, 421, 10058, 10061, 10057, 10055 ] } ], "ref_ids": [] }, { "id": 12833, "type": "theorem", "label": "spaces-over-fields-lemma-order-vanishing-agrees", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-order-vanishing-agrees", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic", "space over $S$. Let $f \\in R(X)^*$. If the prime divisor", "$Z \\subset X$ meets the schematic locus of $X$, then the order", "of vanishing $\\text{ord}_Z(f)$ of Definition \\ref{definition-order-vanishing}", "agrees with the order of vanishing of", "Divisors, Definition \\ref{divisors-definition-order-vanishing}." ], "refs": [ "spaces-over-fields-definition-order-vanishing", "divisors-definition-order-vanishing" ], "proofs": [ { "contents": [ "After shrinking $X$ we may assume $X$ is an integral Noetherian scheme.", "If $\\xi \\in Z$ denotes the generic point, then we find that", "$\\mathcal{O}_{X, \\xi}^h$ is the henselization of $\\mathcal{O}_{X, \\xi}$", "(Decent Spaces, Lemma \\ref{decent-spaces-lemma-describe-henselian-local-ring}).", "To prove the lemma it suffices and is necessary to show that", "$$", "\\text{length}_{\\mathcal{O}_{X, \\xi}}", "(\\mathcal{O}_{X, \\xi}/a \\mathcal{O}_{X, \\xi}) =", "\\text{length}_{\\mathcal{O}_{X, \\xi}^h}", "(\\mathcal{O}_{X, \\xi}^h/a \\mathcal{O}_{X, \\xi}^h)", "$$", "This follows immediately from", "Algebra, Lemma \\ref{algebra-lemma-pullback-module}", "(and the fact that $\\mathcal{O}_{X, \\xi} \\to \\mathcal{O}_{X, \\xi}^h$", "is a flat local ring homomorphism of local Noetherian rings)." ], "refs": [ "decent-spaces-lemma-describe-henselian-local-ring", "algebra-lemma-pullback-module" ], "ref_ids": [ 9490, 640 ] } ], "ref_ids": [ 12887, 8107 ] }, { "id": 12834, "type": "theorem", "label": "spaces-over-fields-lemma-divisor-locally-finite", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-divisor-locally-finite", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space", "over $S$. Let $f \\in R(X)^*$. Then the collections", "$$", "\\{Z \\subset X \\mid Z\\text{ a prime divisor with generic point }\\xi", "\\text{ and }f\\text{ not in }\\mathcal{O}_{X, \\xi}\\}", "$$", "and", "$$", "\\{Z \\subset X \\mid Z \\text{ a prime divisor and }\\text{ord}_Z(f) \\not = 0\\}", "$$", "are locally finite in $X$." ], "refs": [], "proofs": [ { "contents": [ "There exists a nonempty open subspace $U \\subset X$", "such that $f$ corresponds to a section of $\\Gamma(U, \\mathcal{O}_X^*)$.", "Hence the prime divisors which can occur in the sets of the lemma all", "correspond to irreducible components of $|X| \\setminus |U|$.", "Hence Lemma \\ref{lemma-components-locally-finite} gives the desired result." ], "refs": [ "spaces-over-fields-lemma-components-locally-finite" ], "ref_ids": [ 12831 ] } ], "ref_ids": [] }, { "id": 12835, "type": "theorem", "label": "spaces-over-fields-lemma-div-additive", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-div-additive", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a locally Noetherian integral algebraic space over $S$.", "Let $f, g \\in R(X)^*$. Then", "$$", "\\text{div}_X(fg) = \\text{div}_X(f) + \\text{div}_X(g)", "$$", "as Weil divisors on $X$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the additivity of the $\\text{ord}$ functions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12836, "type": "theorem", "label": "spaces-over-fields-lemma-divisor-meromorphic-locally-finite", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-divisor-meromorphic-locally-finite", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space", "over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\mathcal{K}_X(\\mathcal{L})$ be a", "regular (i.e., nonzero) meromorphic section of $\\mathcal{L}$. Then the sets", "$$", "\\{Z \\subset X \\mid Z \\text{ a prime divisor with generic point }\\xi", "\\text{ and }s\\text{ not in }\\mathcal{L}_\\xi\\}", "$$", "and", "$$", "\\{Z \\subset X \\mid Z \\text{ is a prime divisor and }", "\\text{ord}_{Z, \\mathcal{L}}(s) \\not = 0\\}", "$$", "are locally finite in $X$." ], "refs": [], "proofs": [ { "contents": [ "There exists a nonempty open subspace $U \\subset X$ such that $s$", "corresponds to a section of $\\Gamma(U, \\mathcal{L})$ which generates", "$\\mathcal{L}$ over $U$. Hence the prime divisors which can occur", "in the sets of the lemma all correspond to irreducible components of", "$|X| \\setminus |U|$. Hence Lemma \\ref{lemma-components-locally-finite}.", "gives the desired result." ], "refs": [ "spaces-over-fields-lemma-components-locally-finite" ], "ref_ids": [ 12831 ] } ], "ref_ids": [] }, { "id": 12837, "type": "theorem", "label": "spaces-over-fields-lemma-divisor-meromorphic-well-defined", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-divisor-meromorphic-well-defined", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space", "over $S$ Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s, s' \\in \\mathcal{K}_X(\\mathcal{L})$ be nonzero", "meromorphic sections of $\\mathcal{L}$. Then $f = s/s'$", "is an element of $R(X)^*$ and we have", "$$", "\\sum \\text{ord}_{Z, \\mathcal{L}}(s)[Z]", "=", "\\sum \\text{ord}_{Z, \\mathcal{L}}(s')[Z]", "+", "\\text{div}(f)", "$$", "as Weil divisors." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions.", "Note that Lemma \\ref{lemma-divisor-meromorphic-locally-finite}", "guarantees that the sums are indeed Weil divisors." ], "refs": [ "spaces-over-fields-lemma-divisor-meromorphic-locally-finite" ], "ref_ids": [ 12836 ] } ], "ref_ids": [] }, { "id": 12838, "type": "theorem", "label": "spaces-over-fields-lemma-c1-additive", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-c1-additive", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space", "over $S$. Let $\\mathcal{L}$, $\\mathcal{N}$ be invertible", "$\\mathcal{O}_X$-modules. Let $s$, resp.\\ $t$ be a nonzero meromorphic section", "of $\\mathcal{L}$, resp.\\ $\\mathcal{N}$. Then $st$ is a nonzero", "meromorphic section of $\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}$ and", "$$", "\\text{div}_{\\mathcal{L} \\otimes \\mathcal{N}}(st)", "=", "\\text{div}_\\mathcal{L}(s) + \\text{div}_\\mathcal{N}(t)", "$$", "in $\\text{Div}(X)$. In particular, the Weil divisor class of", "$\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}$ is the sum", "of the Weil divisor classes of $\\mathcal{L}$ and $\\mathcal{N}$." ], "refs": [], "proofs": [ { "contents": [ "Let $s$, resp.\\ $t$ be a nonzero meromorphic section", "of $\\mathcal{L}$, resp.\\ $\\mathcal{N}$. Then $st$ is a nonzero", "meromorphic section of $\\mathcal{L} \\otimes \\mathcal{N}$.", "Let $Z \\subset X$ be a prime divisor. Let $\\xi \\in |Z|$ be its generic", "point. Choose generators $s_\\xi \\in \\mathcal{L}_\\xi$, and", "$t_\\xi \\in \\mathcal{N}_\\xi$ with notation as described earlier", "in this section. Then $s_\\xi \\otimes t_\\xi$ is a generator", "for $(\\mathcal{L} \\otimes \\mathcal{N})_\\xi$.", "So $st/(s_\\xi t_\\xi) = (s/s_\\xi)(t/t_\\xi)$ in", "$Q(\\mathcal{O}_{X, \\xi}^h)$. Applying the additivity of", "Algebra, Lemma \\ref{algebra-lemma-ord-additive}", "we conclude that", "$$", "\\text{div}_{\\mathcal{L} \\otimes \\mathcal{N}, Z}(st)", "=", "\\text{div}_{\\mathcal{L}, Z}(s) + \\text{div}_{\\mathcal{N}, Z}(t)", "$$", "Some details omitted." ], "refs": [ "algebra-lemma-ord-additive" ], "ref_ids": [ 1043 ] } ], "ref_ids": [] }, { "id": 12839, "type": "theorem", "label": "spaces-over-fields-lemma-normal-c1-injective", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-normal-c1-injective", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space", "over $S$. If $X$ is normal, then the map (\\ref{equation-c1})", "$\\Pic(X) \\to \\text{Cl}(X)$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module whose", "associated Weil divisor class is trivial. Let $s$ be a regular", "meromorphic section of $\\mathcal{L}$. The assumption means that", "$\\text{div}_\\mathcal{L}(s) = \\text{div}(f)$ for some", "$f \\in R(X)^*$. Then we see that $t = f^{-1}s$ is a regular", "meromorphic section of $\\mathcal{L}$ with", "$\\text{div}_\\mathcal{L}(t) = 0$, see", "Lemma \\ref{lemma-divisor-meromorphic-well-defined}.", "We claim that $t$ defines a trivialization of $\\mathcal{L}$.", "The claim finishes the proof of the lemma.", "Our proof of the claim is a bit awkward as we", "don't yet have a lot of theory at our dispposal; we suggest", "the reader skip the proof.", "\\medskip\\noindent", "We may check our claim \\'etale locally. Let $U \\in X_\\etale$ be affine", "such that $\\mathcal{L}|_U$ is trivial. Say $s_U \\in \\Gamma(U, \\mathcal{L}|_U)$", "is a trivialization. By", "Properties, Lemma \\ref{properties-lemma-normal-locally-finite-nr-irreducibles}", "we may also assume $U$ is integral. Write $U = \\Spec(A)$ as the spectrum of a", "normal Noetherian domain $A$ with fraction field $K$.", "We may write $t|_U = f s_U$ for some element $f$ of $K$, see", "Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-meromorphic-quasi-coherent}", "for example. Let $\\mathfrak p \\subset A$ be a height one prime", "corresponding to a codimension $1$ point $u \\in U$ which maps", "to a codimension $1$ point $\\xi \\in |X|$. Choose a trivialization", "$s_\\xi$ of $\\mathcal{L}_\\xi$ as in the beginning of this section.", "Choose a geometric point $\\overline{u}$ of $U$ lying over $u$.", "Then", "$$", "(\\mathcal{O}_{X, \\xi}^h)^{sh} =", "\\mathcal{O}_{X, \\overline{u}} =", "\\mathcal{O}_{U, u}^{sh} = (A_\\mathfrak p)^{sh}", "$$", "see Decent Spaces, Lemmas", "\\ref{decent-spaces-lemma-henselian-local-ring-strict}", "and", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring}.", "The normality of $X$ shows that all of these are", "discrete valuation rings. The trivializations $s_U$ and $s_\\xi$", "differ by a unit as sections of $\\mathcal{L}$ pulled back to", "$\\Spec(\\mathcal{O}_{X, \\overline{u}})$.", "Write $t = f_\\xi s_\\xi$ with $f_\\xi \\in Q(\\mathcal{O}_{X, \\xi}^h)$.", "We conclude that $f_\\xi$ and $f$ differ by a unit", "in $Q(\\mathcal{O}_{X, \\overline{u}})$.", "If $Z \\subset X$ denotes the prime divisor corresponding to $\\xi$", "(Lemma \\ref{lemma-decent-irreducible-closed}), then", "$0 = \\text{ord}_{Z, \\mathcal{L}}(t) =", "\\text{ord}_{\\mathcal{O}_{X, \\xi}^h}(f_\\xi)$", "and since $\\mathcal{O}_{X, \\xi}^h$ is a discrete valuation ring", "we see that $f_\\xi$ is a unit.", "Thus $f$ is a unit in $\\mathcal{O}_{X, \\overline{u}}$", "and hence in particular $f \\in A_\\mathfrak p$.", "This implies $f \\in A$ by Algebra, Lemma", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}.", "We conclude that $t \\in \\Gamma(X, \\mathcal{L})$.", "Repeating the argument with $t^{-1}$ viewed as a meromorphic", "section of $\\mathcal{L}^{\\otimes -1}$ finishes the proof." ], "refs": [ "spaces-over-fields-lemma-divisor-meromorphic-well-defined", "properties-lemma-normal-locally-finite-nr-irreducibles", "spaces-divisors-lemma-meromorphic-quasi-coherent", "decent-spaces-lemma-henselian-local-ring-strict", "spaces-properties-lemma-describe-etale-local-ring", "spaces-over-fields-lemma-decent-irreducible-closed", "algebra-lemma-normal-domain-intersection-localizations-height-1" ], "ref_ids": [ 12837, 2969, 12956, 9491, 11884, 12828, 1313 ] } ], "ref_ids": [] }, { "id": 12840, "type": "theorem", "label": "spaces-over-fields-lemma-modification-iso-over-open", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-modification-iso-over-open", "contents": [ "Let $f : X' \\to X$ be a modification as in", "Definition \\ref{definition-modification}.", "There exists a nonempty open $U \\subset X$ such that $f^{-1}(U) \\to U$", "is an isomorphism." ], "refs": [ "spaces-over-fields-definition-modification" ], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-finite-degree} there exists a nonempty $U \\subset X$ such", "that $f^{-1}(U) \\to U$ is finite. By generic flatness", "(Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-generic-flatness-reduced})", "we may assume $f^{-1}(U) \\to U$ is flat and of finite presentation.", "So $f^{-1}(U) \\to U$ is finite locally free", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-flat}).", "Since $f$ is birational, the degree of $X'$ over $X$ is $1$.", "Hence $f^{-1}(U) \\to U$ is finite locally free of degree $1$,", "in other words it is an isomorphism." ], "refs": [ "spaces-over-fields-lemma-finite-degree", "spaces-morphisms-proposition-generic-flatness-reduced", "spaces-morphisms-lemma-finite-flat" ], "ref_ids": [ 12829, 4981, 4954 ] } ], "ref_ids": [ 12892 ] }, { "id": 12841, "type": "theorem", "label": "spaces-over-fields-lemma-alteration-generically-finite", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-alteration-generically-finite", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper dominant morphism of", "integral algebraic spaces over $S$. Then $f$ is an alteration", "if and only if any of the equivalent conditions (1) -- (6) of", "Lemma \\ref{lemma-finite-degree} hold." ], "refs": [ "spaces-over-fields-lemma-finite-degree" ], "proofs": [ { "contents": [ "Immediate consequence of the lemma referenced in the statement." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12829 ] }, { "id": 12842, "type": "theorem", "label": "spaces-over-fields-lemma-alteration-contained-in", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-alteration-contained-in", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper surjective morphism of", "algebraic spaces over $S$. Assume $Y$ is integral. Then", "there exists an integral closed subspace $X' \\subset X$ such that", "$f' = f|_{X'} : X' \\to Y$ is an alteration." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset Y$ be a nonempty open affine", "(Decent Spaces, Theorem \\ref{decent-spaces-theorem-decent-open-dense-scheme}).", "Let $\\eta \\in V$ be the generic point. Then", "$X_\\eta$ is a nonempty proper algebraic space over $\\eta$.", "Choose a closed point $x \\in |X_\\eta|$", "(exists because $|X_\\eta|$ is a quasi-compact, sober", "topological space, see Decent Spaces, Proposition", "\\ref{decent-spaces-proposition-reasonable-sober}", "and Topology, Lemma \\ref{topology-lemma-quasi-compact-closed-point}.)", "Let $X'$ be the reduced induced closed subspace structure on", "$\\overline{\\{x\\}} \\subset |X|$ (Properties of Spaces, Definition", "\\ref{spaces-properties-definition-reduced-induced-space}.", "Then $f' : X' \\to Y$ is surjective as the image contains $\\eta$.", "Also $f'$ is proper as a composition of a closed immersion", "and a proper morphism. Finally, the fibre $X'_\\eta$ has a", "single point; to see this use", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-topology-fibre}", "for both $X \\to Y$ and $X' \\to Y$ and the point $\\eta$.", "Since $Y$ is decent and $X' \\to Y$ is separated we see that $X'$ is decent", "(Decent Spaces, Lemmas", "\\ref{decent-spaces-lemma-properties-trivial-implications} and", "\\ref{decent-spaces-lemma-property-over-property}).", "Thus $f'$ is an alteration by", "Lemma \\ref{lemma-alteration-generically-finite}." ], "refs": [ "decent-spaces-theorem-decent-open-dense-scheme", "decent-spaces-proposition-reasonable-sober", "topology-lemma-quasi-compact-closed-point", "spaces-properties-definition-reduced-induced-space", "decent-spaces-lemma-topology-fibre", "decent-spaces-lemma-properties-trivial-implications", "decent-spaces-lemma-property-over-property", "spaces-over-fields-lemma-alteration-generically-finite" ], "ref_ids": [ 9454, 9559, 8234, 11932, 9525, 9513, 9516, 12841 ] } ], "ref_ids": [] }, { "id": 12843, "type": "theorem", "label": "spaces-over-fields-lemma-locally-finite-type-dim-zero", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-locally-finite-type-dim-zero", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "In each of the following cases $X$ is a scheme:", "\\begin{enumerate}", "\\item $X$ is quasi-compact and quasi-separated and $\\dim(X) = 0$,", "\\item $X$ is locally of finite type over a field $k$ and $\\dim(X) = 0$,", "\\item $X$ is Noetherian and $\\dim(X) = 0$, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Cases (2) and (3) follow immediately from case (1) but we will give a separate", "proofs of (2) and (3) as these proofs use significantly less theory.", "\\medskip\\noindent", "Proof of (3). Let $U$ be an affine scheme and let $U \\to X$ be an", "\\'etale morphism. Set $R = U \\times_X U$. The two projection", "morphisms $s, t : R \\to U$ are \\'etale morphisms of schemes. By", "Properties of Spaces, Definition \\ref{spaces-properties-definition-dimension}", "we see that $\\dim(U) = 0$ and $\\dim(R) = 0$.", "Since $R$ is a locally Noetherian scheme of dimension $0$,", "we see that $R$ is a disjoint union of spectra of", "Artinian local rings", "(Properties, Lemma \\ref{properties-lemma-locally-Noetherian-dimension-0}).", "Since we assumed that $X$ is Noetherian (so quasi-separated) we", "conclude that $R$ is quasi-compact. Hence $R$ is an affine scheme", "(use Schemes, Lemma \\ref{schemes-lemma-disjoint-union-affines}).", "The \\'etale morphisms $s, t : R \\to U$ induce finite residue field", "extensions. Hence $s$ and $t$ are finite by", "Algebra, Lemma", "\\ref{algebra-lemma-essentially-of-finite-type-into-artinian-local}", "(small detail omitted). ", "Thus", "Groupoids, Proposition \\ref{groupoids-proposition-finite-flat-equivalence}", "shows that $X = U/R$ is an affine scheme.", "\\medskip\\noindent", "Proof of (2) -- almost identical to the proof of (4).", "Let $U$ be an affine scheme and let $U \\to X$ be an \\'etale morphism.", "Set $R = U \\times_X U$. The two projection morphisms", "$s, t : R \\to U$ are \\'etale morphisms of schemes. By", "Properties of Spaces, Definition \\ref{spaces-properties-definition-dimension}", "we see that $\\dim(U) = 0$ and similarly $\\dim(R) = 0$.", "On the other hand, the morphism $U \\to \\Spec(k)$ is locally of finite", "type as the composition of the \\'etale morphism $U \\to X$ and", "$X \\to \\Spec(k)$, see", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-composition-finite-type} and", "\\ref{spaces-morphisms-lemma-etale-locally-finite-type}.", "Similarly, $R \\to \\Spec(k)$ is locally of finite type.", "Hence by", "Varieties, Lemma \\ref{varieties-lemma-algebraic-scheme-dim-0}", "we see that $U$ and $R$ are disjoint unions of spectra of", "local Artinian $k$-algebras finite over $k$. The same thing", "is therefore true of $U \\times_{\\Spec(k)} U$. As", "$$", "R = U \\times_X U \\longrightarrow U \\times_{\\Spec(k)} U", "$$", "is a monomorphism, we see that $R$ is a finite(!) union of spectra of", "finite $k$-algebras. It follows that $R$ is affine, see", "Schemes, Lemma \\ref{schemes-lemma-disjoint-union-affines}.", "Applying", "Varieties, Lemma \\ref{varieties-lemma-algebraic-scheme-dim-0}", "once more we see that $R$ is finite over $k$. Hence $s, t$", "are finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-permanence}.", "Thus", "Groupoids, Proposition \\ref{groupoids-proposition-finite-flat-equivalence}", "shows that the open subspace $U/R$ of $X$ is an affine scheme. Since the", "schematic locus of $X$ is an open subspace (see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme}),", "and since $U \\to X$ was an arbitrary \\'etale morphism from an affine scheme", "we conclude that $X$ is a scheme.", "\\medskip\\noindent", "Proof of (1). By Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-vanishing-above-dimension}", "we have vanishing of higher cohomology groups for all", "quasi-coherent sheaves $\\mathcal{F}$ on $X$. Hence $X$", "is affine (in particular a scheme) by", "Cohomology of Spaces, Proposition", "\\ref{spaces-cohomology-proposition-vanishing-affine}." ], "refs": [ "spaces-properties-definition-dimension", "properties-lemma-locally-Noetherian-dimension-0", "schemes-lemma-disjoint-union-affines", "algebra-lemma-essentially-of-finite-type-into-artinian-local", "groupoids-proposition-finite-flat-equivalence", "spaces-properties-definition-dimension", "spaces-morphisms-lemma-composition-finite-type", "spaces-morphisms-lemma-etale-locally-finite-type", "varieties-lemma-algebraic-scheme-dim-0", "schemes-lemma-disjoint-union-affines", "varieties-lemma-algebraic-scheme-dim-0", "morphisms-lemma-finite-permanence", "groupoids-proposition-finite-flat-equivalence", "spaces-properties-lemma-subscheme", "spaces-cohomology-lemma-vanishing-above-dimension", "spaces-cohomology-proposition-vanishing-affine" ], "ref_ids": [ 11930, 2981, 7659, 649, 9669, 11930, 4814, 4912, 10988, 7659, 10988, 5448, 9669, 11848, 11294, 11346 ] } ], "ref_ids": [] }, { "id": 12844, "type": "theorem", "label": "spaces-over-fields-lemma-generic-point-in-schematic-locus", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-generic-point-in-schematic-locus", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$.", "Let $x \\in |X|$. The following are equivalent", "\\begin{enumerate}", "\\item $x$ is a point of codimension $0$ on $X$,", "\\item the local ring of $X$ at $x$ has dimension $0$, and", "\\item $x$ is a generic point of an irreducible component of $|X|$.", "\\end{enumerate}", "If true, then there exists an open subspace of $X$", "containing $x$ which is a scheme." ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1), (2), and (3) follows from", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-generic-points}", "and the fact that a quasi-separated algebraic space is decent", "(Decent Spaces, Section \\ref{decent-spaces-section-reasonable-decent}).", "However in the next paragraph we will give a more elementary proof of the", "equivalence.", "\\medskip\\noindent", "Note that (1) and (2) are equivalent by definition", "(Properties of Spaces, Definition", "\\ref{spaces-properties-definition-dimension-local-ring}).", "To prove the equivalence of (1) and (3) we may assume $X$ is quasi-compact.", "Choose", "$$", "\\emptyset = U_{n + 1} \\subset", "U_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X", "$$", "and $f_i : V_i \\to U_i$ as in Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated}.", "Say $x \\in U_i$, $x \\not \\in U_{i + 1}$. Then $x = f_i(y)$ for", "a unique $y \\in V_i$. If (1) holds, then $y$ is a generic point of", "an irreducible component of $V_i$ (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-codimension-0-points}).", "Since $f_i^{-1}(U_{i + 1})$ is a quasi-compact open of $V_i$", "not containing $y$, there is an open neighbourhood $W \\subset V_i$", "of $y$ disjoint from $f_i^{-1}(V_i)$", "(see", "Properties, Lemma \\ref{properties-lemma-generic-point-in-constructible}", "or more simply Algebra, Lemma", "\\ref{algebra-lemma-standard-open-containing-maximal-point}).", "Then $f_i|_W : W \\to X$ is an isomorphism onto its image and hence", "$x = f_i(y)$ is a generic point of $|X|$. Conversely, assume (3) holds.", "Then $f_i$ maps $\\overline{\\{y\\}}$ onto the irreducible component", "$\\overline{\\{x\\}}$ of $|U_i|$. Since $|f_i|$ is bijective over", "$\\overline{\\{x\\}}$, it follows that $\\overline{\\{y\\}}$", "is an irreducible component of $U_i$. Thus $x$ is a point of", "codimension $0$.", "\\medskip\\noindent", "The final statement of the lemma is", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}." ], "refs": [ "decent-spaces-lemma-decent-generic-points", "spaces-properties-definition-dimension-local-ring", "decent-spaces-lemma-filter-quasi-compact-quasi-separated", "spaces-properties-lemma-codimension-0-points", "properties-lemma-generic-point-in-constructible", "algebra-lemma-standard-open-containing-maximal-point", "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme" ], "ref_ids": [ 9531, 11931, 9480, 11842, 2939, 425, 11917 ] } ], "ref_ids": [] }, { "id": 12845, "type": "theorem", "label": "spaces-over-fields-lemma-codim-1-point-in-schematic-locus", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-codim-1-point-in-schematic-locus", "contents": [ "\\begin{slogan}", "Separated algebraic spaces are schemes in codimension 1.", "\\end{slogan}", "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$. If $X$ is separated, locally Noetherian, and", "the dimension of the local ring of $X$ at $x$ is $\\leq 1$", "(Properties of Spaces, Definition", "\\ref{spaces-properties-definition-dimension-local-ring}),", "then there exists an open subspace of $X$ containing $x$ which is a scheme." ], "refs": [ "spaces-properties-definition-dimension-local-ring" ], "proofs": [ { "contents": [ "(Please see the remark below for a different approach avoiding the material on", "finite groupoids.) We can replace $X$ by an quasi-compact neighbourhood of", "$x$, hence we may assume $X$ is quasi-compact, separated, and Noetherian.", "There exists a scheme $U$ and a finite surjective morphism $U \\to X$,", "see Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-there-is-a-scheme-finite-over}.", "Let $R = U \\times_X U$. Then $j : R \\to U \\times_S U$ is an equivalence", "relation and we obtain a groupoid scheme $(U, R, s, t, c)$ over $S$", "with $s, t$ finite and $U$ Noetherian and separated.", "Let $\\{u_1, \\ldots, u_n\\} \\subset U$ be the set of points mapping to $x$. ", "Then $\\dim(\\mathcal{O}_{U, u_i}) \\leq 1$ by", "Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-dimension-local-ring-quasi-finite}.", "\\medskip\\noindent", "By More on Groupoids, Lemma", "\\ref{more-groupoids-lemma-find-affine-codimension-1}", "there exists an $R$-invariant affine open $W \\subset U$ containing", "the orbit $\\{u_1, \\ldots, u_n\\}$. Since $U \\to X$ is finite surjective", "the continuous map $|U| \\to |X|$ is closed surjective, hence", "submersive by Topology, Lemma", "\\ref{topology-lemma-closed-morphism-quotient-topology}.", "Thus $f(W)$ is open and there is an open subspace $X' \\subset X$", "with $f : W \\to X'$ a surjective finite morphism.", "Then $X'$ is an affine scheme by", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian}", "and the proof is finished." ], "refs": [ "spaces-limits-proposition-there-is-a-scheme-finite-over", "decent-spaces-lemma-dimension-local-ring-quasi-finite", "more-groupoids-lemma-find-affine-codimension-1", "topology-lemma-closed-morphism-quotient-topology", "spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian" ], "ref_ids": [ 4659, 9497, 2502, 8204, 11326 ] } ], "ref_ids": [ 11931 ] }, { "id": 12846, "type": "theorem", "label": "spaces-over-fields-lemma-scheme-after-purely-inseparable-base-change", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-scheme-after-purely-inseparable-base-change", "contents": [ "Let $k$ be a field. Let $X$ be an algebraic space over $k$.", "If there exists a purely inseparable field extension $k \\subset k'$", "such that $X_{k'}$ is a scheme, then $X$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "The morphism $X_{k'} \\to X$ is integral, surjective, and", "universally injective. Hence this lemma follows from", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-integral-universally-bijective-scheme}." ], "refs": [ "spaces-limits-lemma-integral-universally-bijective-scheme" ], "ref_ids": [ 4628 ] } ], "ref_ids": [] }, { "id": 12847, "type": "theorem", "label": "spaces-over-fields-lemma-when-scheme-after-base-change", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-when-scheme-after-base-change", "contents": [ "Let $k$ be a field with algebraic closure $\\overline{k}$.", "Let $X$ be a quasi-separated algebraic space over $k$.", "\\begin{enumerate}", "\\item If there exists a field extension $k \\subset K$ such that", "$X_K$ is a scheme, then $X_{\\overline{k}}$ is a scheme.", "\\item If $X$ is quasi-compact and there exists a field extension", "$k \\subset K$ such that $X_K$ is a scheme, then $X_{k'}$", "is a scheme for some finite separable extension $k'$ of $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since every algebraic space is the union of its quasi-compact open", "subspaces, we see that the first part of the lemma follows from", "the second part (some details omitted). Thus we assume $X$ is quasi-compact", "and we assume given an extension $k \\subset K$ with $K_K$ representable.", "Write $K = \\bigcup A$ as the colimit of finitely generated $k$-subalgebras", "$A$. By Limits of Spaces, Lemma \\ref{spaces-limits-lemma-limit-is-scheme}", "we see that $X_A$ is a scheme for some $A$. Choose a maximal ideal", "$\\mathfrak m \\subset A$. By the Hilbert Nullstellensatz", "(Algebra, Theorem \\ref{algebra-theorem-nullstellensatz})", "the residue field $k' = A/\\mathfrak m$ is a finite extension of $k$.", "Thus we see that $X_{k'}$ is a scheme. If $k' \\supset k$ is not", "separable, let $k' \\supset k'' \\supset k$ be the subextension", "found in Fields, Lemma \\ref{fields-lemma-separable-first}.", "Since $k'/k''$ is purely inseparable, by", "Lemma \\ref{lemma-scheme-after-purely-inseparable-base-change}", "the algebraic space $X_{k''}$ is a scheme. Since $k''|k$ is separable", "the proof is complete." ], "refs": [ "spaces-limits-lemma-limit-is-scheme", "algebra-theorem-nullstellensatz", "fields-lemma-separable-first", "spaces-over-fields-lemma-scheme-after-purely-inseparable-base-change" ], "ref_ids": [ 4579, 316, 4482, 12846 ] } ], "ref_ids": [] }, { "id": 12848, "type": "theorem", "label": "spaces-over-fields-lemma-base-change-by-Galois", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-base-change-by-Galois", "contents": [ "Let $k \\subset k'$ be a finite Galois extension with Galois group $G$.", "Let $X$ be an algebraic space over $k$. Then $G$ acts freely on the", "algebraic space $X_{k'}$ and $X = X_{k'}/G$ in the sense of", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quotient}." ], "refs": [ "spaces-properties-lemma-quotient" ], "proofs": [ { "contents": [ "Omitted. Hints: First show that $\\Spec(k) = \\Spec(k')/G$.", "Then use compatibility of taking quotients with base change." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 11916 ] }, { "id": 12849, "type": "theorem", "label": "spaces-over-fields-lemma-when-quotient-scheme-at-point", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-when-quotient-scheme-at-point", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ and", "let $G$ be a finite group acting freely on $X$. Set $Y = X/G$ as", "in Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quotient}.", "For $y \\in |Y|$ the following are equivalent", "\\begin{enumerate}", "\\item $y$ is in the schematic locus of $Y$, and", "\\item there exists an affine open $U \\subset X$", "containing the preimage of $y$.", "\\end{enumerate}" ], "refs": [ "spaces-properties-lemma-quotient" ], "proofs": [ { "contents": [ "It follows from the construction of $Y = X/G$ in", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quotient}", "that the morphism $X \\to Y$ is surjective and \\'etale.", "Of course we have $X \\times_Y X = X \\times G$ hence the morphism", "$X \\to Y$ is even finite \\'etale. It is also surjective.", "Thus the lemma follows from", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-when-quotient-scheme-at-point}." ], "refs": [ "spaces-properties-lemma-quotient", "decent-spaces-lemma-when-quotient-scheme-at-point" ], "ref_ids": [ 11916, 9484 ] } ], "ref_ids": [ 11916 ] }, { "id": 12850, "type": "theorem", "label": "spaces-over-fields-lemma-scheme-after-purely-transcendental-base-change", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-scheme-after-purely-transcendental-base-change", "contents": [ "Let $k$ be a field. Let $X$ be a quasi-separated", "algebraic space over $k$. If there exists a purely transcendental", "field extension $k \\subset K$ such that $X_K$ is a scheme, then", "$X$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "Since every algebraic space is the union of its quasi-compact open", "subspaces, we may assume $X$ is quasi-compact (some details omitted).", "Recall (Fields, Definition \\ref{fields-definition-transcendence})", "that the assumption on the extension $K/k$ signifies that", "$K$ is the fraction field of a polynomial ring (in possibly infinitely", "many variables) over $k$. Thus $K = \\bigcup A$ is the union of subalgebras", "each of which is a localization of a finite polynomial algebra over $k$.", "By Limits of Spaces, Lemma \\ref{spaces-limits-lemma-limit-is-scheme}", "we see that $X_A$ is a scheme for some $A$. Write", "$$", "A = k[x_1, \\ldots, x_n][1/f]", "$$", "for some nonzero $f \\in k[x_1, \\ldots, x_n]$.", "\\medskip\\noindent", "If $k$ is infinite then we can finish the proof as follows: choose", "$a_1, \\ldots, a_n \\in k$ with $f(a_1, \\ldots, a_n) \\not = 0$.", "Then $(a_1, \\ldots, a_n)$ define an $k$-algebra map $A \\to k$", "mapping $x_i$ to $a_i$ and $1/f$ to $1/f(a_1, \\ldots, a_n)$.", "Thus the base change $X_A \\times_{\\Spec(A)} \\Spec(k) \\cong X$ is a", "scheme as desired.", "\\medskip\\noindent", "In this paragraph we finish the proof in case $k$ is finite. In this", "case we write $X = \\lim X_i$ with $X_i$ of finite presentation over $k$", "and with affine transition morphisms", "(Limits of Spaces, Lemma \\ref{spaces-limits-lemma-relative-approximation}).", "Using Limits of Spaces, Lemma \\ref{spaces-limits-lemma-limit-is-scheme}", "we see that $X_{i, A}$ is a scheme for some $i$. Thus we may assume", "$X \\to \\Spec(k)$ is of finite presentation. Let $x \\in |X|$ be a closed", "point. We may represent $x$ by a closed immersion", "$\\Spec(\\kappa) \\to X$", "(Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-space-closed-point}).", "Then $\\Spec(\\kappa) \\to \\Spec(k)$ is of finite type, hence $\\kappa$", "is a finite extension of $k$ (by the Hilbert Nullstellensatz, see", "Algebra, Theorem \\ref{algebra-theorem-nullstellensatz};", "some details omitted). Say $[\\kappa : k] = d$. Choose an integer", "$n \\gg 0$ prime to $d$ and let $k \\subset k'$ be the extension", "of degree $n$. Then $k'/k$ is Galois with $G = \\text{Aut}(k'/k)$", "cyclic of order $n$. If $n$ is large enough there will be $k$-algebra", "homomorphism $A \\to k'$ by the same reason as above.", "Then $X_{k'}$ is a scheme and $X = X_{k'}/G$", "(Lemma \\ref{lemma-base-change-by-Galois}).", "On the other hand, since $n$ and $d$ are relatively prime we see that", "$$", "\\Spec(\\kappa) \\times_{X} X_{k'} =", "\\Spec(\\kappa) \\times_{\\Spec(k)} \\Spec(k') =", "\\Spec(\\kappa \\otimes_k k')", "$$", "is the spectrum of a field. In other words, the fibre of $X_{k'} \\to X$", "over $x$ consists of a single point. Thus by", "Lemma \\ref{lemma-when-quotient-scheme-at-point}", "we see that $x$ is in the schematic locus of $X$ as desired." ], "refs": [ "fields-definition-transcendence", "spaces-limits-lemma-limit-is-scheme", "spaces-limits-lemma-relative-approximation", "spaces-limits-lemma-limit-is-scheme", "decent-spaces-lemma-decent-space-closed-point", "algebra-theorem-nullstellensatz", "spaces-over-fields-lemma-base-change-by-Galois", "spaces-over-fields-lemma-when-quotient-scheme-at-point" ], "ref_ids": [ 4550, 4579, 4609, 4579, 9510, 316, 12848, 12849 ] } ], "ref_ids": [] }, { "id": 12851, "type": "theorem", "label": "spaces-over-fields-lemma-scheme-over-algebraic-closure-enough-affines", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-scheme-over-algebraic-closure-enough-affines", "contents": [ "Let $k$ be a field with algebraic closure $\\overline{k}$. Let $X$", "be an algebraic space over $k$ such that", "\\begin{enumerate}", "\\item $X$ is decent and locally of finite type over $k$,", "\\item $X_{\\overline{k}}$ is a scheme, and", "\\item any finite set of $\\overline{k}$-rational points of $X_{\\overline{k}}$", "are contained in an affine.", "\\end{enumerate}", "Then $X$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "If $k \\subset K$ is an extension, then the base change $X_K$ is", "decent (Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-representable-named-properties})", "and locally of finite type", "over $K$ (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-base-change-finite-type}).", "By Lemma \\ref{lemma-scheme-after-purely-inseparable-base-change}", "it suffices to prove that $X$ becomes a scheme after base change to", "the perfection of $k$, hence we may assume $k$ is a perfect field", "(this step isn't strictly necessary, but makes the other arguments", "easier to think about).", "By covering $X$ by quasi-compact opens we see that it suffices to prove", "the lemma in case $X$ is quasi-compact (small detail omitted).", "In this case $|X|$ is a sober topological space", "(Decent Spaces, Proposition", "\\ref{decent-spaces-proposition-reasonable-sober}).", "Hence it suffices to show that every closed point in $|X|$", "is contained in the schematic locus of $X$", "(use Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme} and", "Topology, Lemma \\ref{topology-lemma-quasi-compact-closed-point}).", "\\medskip\\noindent", "Let $x \\in |X|$ be a closed point. By Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-decent-space-closed-point}", "we can find a closed immersion $\\Spec(l) \\to X$ representing $x$.", "Then $\\Spec(l) \\to \\Spec(k)$ is of finite type (Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-composition-finite-type}) and we", "conclude that $l$ is a finite extension of $k$", "by the Hilbert Nullstellensatz (Algebra, Theorem", "\\ref{algebra-theorem-nullstellensatz}). It is separable because", "$k$ is perfect. Thus the scheme", "$$", "\\Spec(l) \\times_X X_{\\overline{k}} =", "\\Spec(l) \\times_{\\Spec(k)} \\Spec(\\overline{k}) =", "\\Spec(l \\otimes_k \\overline{k})", "$$", "is the disjoint union of a finite number of $\\overline{k}$-rational points.", "By assumption (3) we can find an affine open $W \\subset X_{\\overline{k}}$", "containing these points.", "\\medskip\\noindent", "By Lemma \\ref{lemma-when-scheme-after-base-change} we see that $X_{k'}$", "is a scheme for some finite extension $k'/k$. After enlarging", "$k'$ we may assume that there exists an affine open $U' \\subset X_{k'}$", "whose base change to $\\overline{k}$ recovers $W$", "(use that $X_{\\overline{k}}$ is the limit of the schemes $X_{k''}$", "for $k' \\subset k'' \\subset \\overline{k}$ finite and use", "Limits, Lemmas \\ref{limits-lemma-descend-opens} and", "\\ref{limits-lemma-limit-affine}). We may assume", "that $k'/k$ is a Galois extension (take the normal closure", "Fields, Lemma \\ref{fields-lemma-normal-closure} and use", "that $k$ is perfect). Set $G = \\text{Gal}(k'/k)$.", "By construction the $G$-invariant closed subscheme", "$\\Spec(l) \\times_X X_{k'}$ is contained in $U'$.", "Thus $x$ is in the schematic locus by", "Lemmas \\ref{lemma-base-change-by-Galois} and", "\\ref{lemma-when-quotient-scheme-at-point}." ], "refs": [ "decent-spaces-lemma-representable-named-properties", "spaces-morphisms-lemma-base-change-finite-type", "spaces-over-fields-lemma-scheme-after-purely-inseparable-base-change", "decent-spaces-proposition-reasonable-sober", "spaces-properties-lemma-subscheme", "topology-lemma-quasi-compact-closed-point", "decent-spaces-lemma-decent-space-closed-point", "spaces-morphisms-lemma-composition-finite-type", "algebra-theorem-nullstellensatz", "spaces-over-fields-lemma-when-scheme-after-base-change", "limits-lemma-descend-opens", "limits-lemma-limit-affine", "fields-lemma-normal-closure", "spaces-over-fields-lemma-base-change-by-Galois", "spaces-over-fields-lemma-when-quotient-scheme-at-point" ], "ref_ids": [ 9470, 4815, 12846, 9559, 11848, 8234, 9510, 4814, 316, 12847, 15041, 15043, 4494, 12848, 12849 ] } ], "ref_ids": [] }, { "id": 12852, "type": "theorem", "label": "spaces-over-fields-lemma-locally-quasi-finite-over-field", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-locally-quasi-finite-over-field", "contents": [ "Let $k$ be a field. Let $X$ be an algebraic space over $k$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is locally quasi-finite over $k$,", "\\item $X$ is locally of finite type over $k$ and has dimension $0$,", "\\item $X$ is a scheme and is locally quasi-finite over $k$,", "\\item $X$ is a scheme and is locally of finite type over $k$ and has", "dimension $0$, and", "\\item $X$ is a disjoint union of spectra of Artinian local $k$-algebras", "$A$ over $k$ with $\\dim_k(A) < \\infty$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Because we are over a field relative dimension of $X/k$ is the same as", "the dimension of $X$. Hence by", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0}", "we see that (1) and (2) are equivalent. Hence it follows from", "Lemma \\ref{lemma-locally-finite-type-dim-zero}", "(and trivial implications) that (1) -- (4) are equivalent.", "Finally,", "Varieties, Lemma \\ref{varieties-lemma-algebraic-scheme-dim-0}", "shows that (1) -- (4) are equivalent with (5)." ], "refs": [ "spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0", "spaces-over-fields-lemma-locally-finite-type-dim-zero", "varieties-lemma-algebraic-scheme-dim-0" ], "ref_ids": [ 4875, 12843, 10988 ] } ], "ref_ids": [] }, { "id": 12853, "type": "theorem", "label": "spaces-over-fields-lemma-mono-towards-locally-quasi-finite-over-field", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-mono-towards-locally-quasi-finite-over-field", "contents": [ "Let $k$ be a field. Let $f : X \\to Y$ be a monomorphism of algebraic spaces", "over $k$. If $Y$ is locally quasi-finite over $k$ so is $X$." ], "refs": [], "proofs": [ { "contents": [ "Assume $Y$ is locally quasi-finite over $k$. By", "Lemma \\ref{lemma-locally-quasi-finite-over-field}", "we see that $Y = \\coprod \\Spec(A_i)$ where each $A_i$ is an", "Artinian local ring finite over $k$. By", "Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-monomorphism-toward-disjoint-union-dim-0-rings}", "we see that $X$ is a scheme. Consider $X_i = f^{-1}(\\Spec(A_i))$.", "Then $X_i$ has either one or zero points. If $X_i$ has zero points there", "is nothing to prove. If $X_i$ has one point, then", "$X_i = \\Spec(B_i)$ with $B_i$ a zero dimensional local ring", "and $A_i \\to B_i$ is an epimorphism of rings. In particular", "$A_i/\\mathfrak m_{A_i} = B_i/\\mathfrak m_{A_i}B_i$ and we see that", "$A_i \\to B_i$ is surjective by Nakayama's lemma,", "Algebra, Lemma \\ref{algebra-lemma-NAK}", "(because $\\mathfrak m_{A_i}$ is a nilpotent ideal!).", "Thus $B_i$ is a finite local $k$-algebra, and we conclude by", "Lemma \\ref{lemma-locally-quasi-finite-over-field}", "that $X \\to \\Spec(k)$ is locally quasi-finite." ], "refs": [ "spaces-over-fields-lemma-locally-quasi-finite-over-field", "decent-spaces-lemma-monomorphism-toward-disjoint-union-dim-0-rings", "algebra-lemma-NAK", "spaces-over-fields-lemma-locally-quasi-finite-over-field" ], "ref_ids": [ 12852, 9530, 401, 12852 ] } ], "ref_ids": [] }, { "id": 12854, "type": "theorem", "label": "spaces-over-fields-lemma-geometrically-reduced-at-point", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-geometrically-reduced-at-point", "contents": [ "Let $k$ be a field. Let $X$ be an algebraic space over $k$.", "Let $x \\in |X|$. The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically reduced at $x$,", "\\item for some \\'etale neighbourhood $(U, u) \\to (X, x)$", "where $U$ is a scheme, $U$ is geometrically reduced at $u$,", "\\item for any \\'etale neighbourhood $(U, u) \\to (X, x)$", "where $U$ is a scheme, $U$ is geometrically reduced at $u$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that the local ring $\\mathcal{O}_{X, \\overline{x}}$", "is the strict henselization of $\\mathcal{O}_{U, u}$, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring}.", "By Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced-at-point}", "we find that $U$ is geometrically reduced at $u$ if and only", "if $\\mathcal{O}_{U, u}$ is geometrically reduced over $k$.", "Thus we have to show: if $A$ is a local $k$-algebra, then", "$A$ is geometrically reduced over $k$ if and only if", "$A^{sh}$ is geometrically reduced over $k$.", "We check this using the definition of geometrically reduced", "algebras (Algebra, Definition \\ref{algebra-definition-geometrically-reduced}).", "Let $K/k$ be a field extension.", "Since $A \\to A^{sh}$ is faithfully flat", "(More on Algebra, Lemma \\ref{more-algebra-lemma-dumb-properties-henselization})", "we see that", "$A \\otimes_k K \\to A^{sh} \\otimes_k K$ is faithfully flat", "(Algebra, Lemma \\ref{algebra-lemma-flat-base-change}).", "Hence if $A^{sh} \\otimes_k K$ is reduced, so is", "$A \\otimes_k K$ by Algebra, Lemma \\ref{algebra-lemma-descent-reduced}.", "Conversely, recall that $A^{sh}$ is a colimit of", "\\'etale $A$-algebra, see", "Algebra, Lemma \\ref{algebra-lemma-strict-henselization}.", "Thus $A^{sh} \\otimes_k K$ is a filtered", "colimit of \\'etale $A \\otimes_k K$-algebras.", "We conclude by Algebra, Lemma \\ref{algebra-lemma-reduced-goes-up}." ], "refs": [ "spaces-properties-lemma-describe-etale-local-ring", "varieties-lemma-geometrically-reduced-at-point", "algebra-definition-geometrically-reduced", "more-algebra-lemma-dumb-properties-henselization", "algebra-lemma-flat-base-change", "algebra-lemma-descent-reduced", "algebra-lemma-strict-henselization", "algebra-lemma-reduced-goes-up" ], "ref_ids": [ 11884, 10906, 1461, 10055, 527, 1371, 1295, 1366 ] } ], "ref_ids": [] }, { "id": 12855, "type": "theorem", "label": "spaces-over-fields-lemma-geometrically-reduced", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-geometrically-reduced", "contents": [ "Let $k$ be a field. Let $X$ be an algebraic space over $k$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically reduced,", "\\item for some surjective \\'etale morphism $U \\to X$ where $U$", "is a scheme, $U$ is geometrically reduced,", "\\item for any \\'etale morphism $U \\to X$", "where $U$ is a scheme, $U$ is geometrically reduced.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions and", "Lemma \\ref{lemma-geometrically-reduced-at-point}." ], "refs": [ "spaces-over-fields-lemma-geometrically-reduced-at-point" ], "ref_ids": [ 12854 ] } ], "ref_ids": [] }, { "id": 12856, "type": "theorem", "label": "spaces-over-fields-lemma-perfect-reduced", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-perfect-reduced", "contents": [ "Let $X$ be an algebraic space over a perfect field $k$ (for example", "$k$ has characteristic zero).", "\\begin{enumerate}", "\\item For $x \\in |X|$, if $\\mathcal{O}_{X, \\overline{x}}$ is", "reduced, then $X$ is geometrically reduced at $x$.", "\\item If $X$ is reduced, then $X$ is geometrically reduced over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first statement follows from", "Algebra, Lemma \\ref{algebra-lemma-separable-extension-preserves-reducedness}", "and the definition of a perfect field", "(Algebra, Definition \\ref{algebra-definition-perfect}).", "The second statement follows from the first." ], "refs": [ "algebra-lemma-separable-extension-preserves-reducedness", "algebra-definition-perfect" ], "ref_ids": [ 565, 1462 ] } ], "ref_ids": [] }, { "id": 12857, "type": "theorem", "label": "spaces-over-fields-lemma-geometrically-reduced-positive-characteristic", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-geometrically-reduced-positive-characteristic", "contents": [ "Let $k$ be a field of characteristic $p > 0$. Let $X$ be an algebraic space", "over $k$. The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically reduced over $k$,", "\\item $X_{k'}$ is reduced for every field extension $k'/k$,", "\\item $X_{k'}$ is reduced for every", "finite purely inseparable field extension $k'/k$,", "\\item $X_{k^{1/p}}$ is reduced,", "\\item $X_{k^{perf}}$ is reduced, and", "\\item $X_{\\bar k}$ is reduced.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a surjective \\'etale morphism $U \\to X$ where $U$ is a scheme.", "Via Lemma \\ref{lemma-geometrically-reduced} the lemma follows from", "the result for $U$ over $k$.", "See Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced}." ], "refs": [ "spaces-over-fields-lemma-geometrically-reduced", "varieties-lemma-geometrically-reduced" ], "ref_ids": [ 12855, 10908 ] } ], "ref_ids": [] }, { "id": 12858, "type": "theorem", "label": "spaces-over-fields-lemma-geometrically-reduced-upstairs", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-geometrically-reduced-upstairs", "contents": [ "Let $k$ be a field. Let $X$ be an algebraic space over $k$.", "Let $k'/k$ be a field extension. Let $x \\in |X|$ be a point and let", "$x' \\in |X_{k'}|$ be a point lying over $x$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically reduced at $x$,", "\\item $X_{k'}$ is geometrically reduced at $x'$.", "\\end{enumerate}", "In particular, $X$ is geometrically reduced over $k$ if and only if", "$X_{k'}$ is geometrically reduced over $k'$." ], "refs": [], "proofs": [ { "contents": [ "Choose an \\'etale morphism $U \\to X$ where $U$ is a scheme", "and a point $u \\in U$ mapping to $x \\in |X|$.", "By Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}", "we may choose a point $u' \\in U_{k'} = U \\times_X X_{k'}$ mapping", "to both $u$ and $x'$. By Lemma \\ref{lemma-geometrically-reduced-at-point}", "the lemma follows from the lemma for $U, u, u'$ which is", "Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced-upstairs}." ], "refs": [ "spaces-properties-lemma-points-cartesian", "spaces-over-fields-lemma-geometrically-reduced-at-point", "varieties-lemma-geometrically-reduced-upstairs" ], "ref_ids": [ 11819, 12854, 10910 ] } ], "ref_ids": [] }, { "id": 12859, "type": "theorem", "label": "spaces-over-fields-lemma-geometrically-reduced-etale-local", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-geometrically-reduced-etale-local", "contents": [ "Let $k$ be a field. Let $f : X \\to Y$ be a morphism of", "algebraic spaces over $k$. Let $x \\in |X|$ be a point", "with image $y \\in |Y|$.", "\\begin{enumerate}", "\\item if $f$ is \\'etale at $x$, then", "$X$ is geometrically reduced at $x$ $\\Leftrightarrow$", "$Y$ is geometrically reduced at $y$,", "\\item if $f$ is surjective \\'etale, then", "$X$ is geometrically reduced $\\Leftrightarrow$", "$Y$ is geometrically reduced.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is clear because", "$\\mathcal{O}_{X, \\overline{x}} = \\mathcal{O}_{Y, \\overline{y}}$", "if $f$ is \\'etale at $x$.", "Part (2) follows immediately from part (1)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12860, "type": "theorem", "label": "spaces-over-fields-lemma-geometrically-connected-check-after-extension", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-geometrically-connected-check-after-extension", "contents": [ "Let $X$ be an algebraic space over the field $k$.", "Let $k \\subset k'$ be a field extension.", "Then $X$ is geometrically connected over $k$ if and only if", "$X_{k'}$ is geometrically connected over $k'$." ], "refs": [], "proofs": [ { "contents": [ "If $X$ is geometrically connected over $k$, then it is clear that", "$X_{k'}$ is geometrically connected over $k'$. For the converse, note", "that for any field extension $k \\subset k''$ there exists a common", "field extension $k' \\subset k'''$ and $k'' \\subset k'''$. As the", "morphism $X_{k'''} \\to X_{k''}$ is surjective (as a base change of", "a surjective morphism between spectra of fields) we see that the", "connectedness of $X_{k'''}$ implies the connectedness of $X_{k''}$.", "Thus if $X_{k'}$ is geometrically connected over $k'$ then", "$X$ is geometrically connected over $k$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12861, "type": "theorem", "label": "spaces-over-fields-lemma-bijection-connected-components", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-bijection-connected-components", "contents": [ "Let $k$ be a field. Let $X$, $Y$ be algebraic spaces over $k$.", "Assume $X$ is geometrically connected over $k$.", "Then the projection morphism", "$$", "p : X \\times_k Y \\longrightarrow Y", "$$", "induces a bijection between connected components." ], "refs": [], "proofs": [ { "contents": [ "Let $y \\in |Y|$ be represented by a morphism $\\Spec(K) \\to Y$ be a morphism", "where $K$ is a field. The fibre of $|X \\times_k Y| \\to |Y|$ over $y$", "is the image of $|Y_K| \\to |X \\times_k Y|$ by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}.", "Thus these fibres are connected by our assumption that $Y$ is", "geometrically connected. By", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-space-over-field-universally-open}", "the map $|p|$ is open.", "Thus we may apply Topology,", "Lemma \\ref{topology-lemma-connected-fibres-connected-components}", "to conclude." ], "refs": [ "spaces-properties-lemma-points-cartesian", "spaces-morphisms-lemma-space-over-field-universally-open", "topology-lemma-connected-fibres-connected-components" ], "ref_ids": [ 11819, 4732, 8208 ] } ], "ref_ids": [] }, { "id": 12862, "type": "theorem", "label": "spaces-over-fields-lemma-separably-closed-field-connected-components", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-separably-closed-field-connected-components", "contents": [ "Let $k \\subset k'$ be an extension of fields. Let $X$ be an algebraic space", "over $k$. Assume $k$ separably algebraically closed. Then the morphism", "$X_{k'} \\to X$ induces a bijection of connected components. In particular,", "$X$ is geometrically connected over $k$ if and only if $X$ is connected." ], "refs": [], "proofs": [ { "contents": [ "Since $k$ is separably algebraically closed we see that", "$k'$ is geometrically connected over $k$, see", "Algebra,", "Lemma \\ref{algebra-lemma-separably-closed-connected-implies-geometric}.", "Hence $Z = \\Spec(k')$ is geometrically connected over $k$ by", "Varieties, Lemma \\ref{varieties-lemma-affine-geometrically-connected}.", "Since $X_{k'} = Z \\times_k X$ the result is a special case of", "Lemma \\ref{lemma-bijection-connected-components}." ], "refs": [ "algebra-lemma-separably-closed-connected-implies-geometric", "varieties-lemma-affine-geometrically-connected", "spaces-over-fields-lemma-bijection-connected-components" ], "ref_ids": [ 602, 10917, 12861 ] } ], "ref_ids": [] }, { "id": 12863, "type": "theorem", "label": "spaces-over-fields-lemma-characterize-geometrically-connected", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-characterize-geometrically-connected", "contents": [ "Let $k$ be a field. Let $X$ be an algebraic space over $k$.", "Let $\\overline{k}$ be a separable algebraic closure of $k$.", "Then $X$ is geometrically connected if and only if the base change", "$X_{\\overline{k}}$ is connected." ], "refs": [], "proofs": [ { "contents": [ "Assume $X_{\\overline{k}}$ is connected. Let $k \\subset k'$ be a field", "extension. There exists a field extension $\\overline{k} \\subset \\overline{k}'$", "such that $k'$ embeds into $\\overline{k}'$ as an extension of $k$.", "By Lemma \\ref{lemma-separably-closed-field-connected-components}", "we see that $X_{\\overline{k}'}$ is connected.", "Since $X_{\\overline{k}'} \\to X_{k'}$ is surjective we conclude", "that $X_{k'}$ is connected as desired." ], "refs": [ "spaces-over-fields-lemma-separably-closed-field-connected-components" ], "ref_ids": [ 12862 ] } ], "ref_ids": [] }, { "id": 12864, "type": "theorem", "label": "spaces-over-fields-lemma-Galois-action-quasi-compact-open", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-Galois-action-quasi-compact-open", "contents": [ "Let $k$ be a field. Let $X$ be an algebraic space over $k$.", "Let $\\overline{k}$ be a (possibly infinite) Galois extension of $k$.", "Let $V \\subset X_{\\overline{k}}$ be a quasi-compact open.", "Then", "\\begin{enumerate}", "\\item there exists a finite subextension $k \\subset k' \\subset \\overline{k}$", "and a quasi-compact open $V' \\subset X_{k'}$ such that", "$V = (V')_{\\overline{k}}$,", "\\item there exists an open subgroup $H \\subset \\text{Gal}(\\overline{k}/k)$", "such that $\\sigma(V) = V$ for all $\\sigma \\in H$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "Choose a quasi-compact open $W \\subset U_{\\overline{k}}$ whose", "image in $X_{\\overline{k}}$ is $V$. This is possible because", "$|U_{\\overline{k}}| \\to |X_{\\overline{k}}|$ is continuous and because", "$|U_{\\overline{k}}|$ has a basis of quasi-compact opens. We can apply", "Varieties, Lemma", "\\ref{varieties-lemma-Galois-action-quasi-compact-open}", "to $W \\subset U_{\\overline{k}}$ to obtain the lemma." ], "refs": [ "varieties-lemma-Galois-action-quasi-compact-open" ], "ref_ids": [ 10921 ] } ], "ref_ids": [] }, { "id": 12865, "type": "theorem", "label": "spaces-over-fields-lemma-closed-fixed-by-Galois", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-closed-fixed-by-Galois", "contents": [ "Let $k$ be a field. Let $k \\subset \\overline{k}$ be a (possibly infinite)", "Galois extension. Let $X$ be an algebraic space over $k$. Let", "$\\overline{T} \\subset |X_{\\overline{k}}|$ have the following properties", "\\begin{enumerate}", "\\item $\\overline{T}$ is a closed subset of $|X_{\\overline{k}}|$,", "\\item for every $\\sigma \\in \\text{Gal}(\\overline{k}/k)$", "we have $\\sigma(\\overline{T}) = \\overline{T}$.", "\\end{enumerate}", "Then there exists a closed subset $T \\subset |X|$ whose inverse image", "in $|X_{k'}|$ is $\\overline{T}$." ], "refs": [], "proofs": [ { "contents": [ "Let $T \\subset |X|$ be the image of $\\overline{T}$.", "Since $|X_{\\overline{k}}| \\to |X|$ is surjective, the statement means", "that $T$ is closed and that its inverse image is $\\overline{T}$.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "By the case of schemes", "(see Varieties, Lemma \\ref{varieties-lemma-closed-fixed-by-Galois})", "there exists a closed subset $T' \\subset |U|$ whose inverse image", "in $|U_{\\overline{k}}|$ is the inverse image of $\\overline{T}$.", "Since $|U_{\\overline{k}}| \\to |X_{\\overline{k}}|$ is surjective,", "we see that $T'$ is the inverse image of $T$ via $|U| \\to |X|$.", "By our construction of the topology on $|X|$ this means that $T$ is", "closed. In the same manner one sees that $\\overline{T}$ is the inverse", "image of $T$." ], "refs": [ "varieties-lemma-closed-fixed-by-Galois" ], "ref_ids": [ 10922 ] } ], "ref_ids": [] }, { "id": 12866, "type": "theorem", "label": "spaces-over-fields-lemma-characterize-geometrically-disconnected", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-characterize-geometrically-disconnected", "contents": [ "Let $k$ be a field. Let $X$ be an algebraic space over $k$.", "The following are equivalent", "\\begin{enumerate}", "\\item $X$ is geometrically connected,", "\\item for every finite separable field extension $k \\subset k'$", "the algebraic space $X_{k'}$ is connected.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This proof is identical to the proof of", "Varieties, Lemma \\ref{varieties-lemma-characterize-geometrically-disconnected}", "except that", "we replace", "Varieties, Lemma \\ref{varieties-lemma-characterize-geometrically-connected}", "by Lemma \\ref{lemma-characterize-geometrically-connected},", "we replace", "Varieties, Lemma \\ref{varieties-lemma-Galois-action-quasi-compact-open}", "by Lemma \\ref{lemma-Galois-action-quasi-compact-open}, and", "we replace", "Varieties, Lemma \\ref{varieties-lemma-closed-fixed-by-Galois}", "by Lemma \\ref{lemma-closed-fixed-by-Galois}.", "We urge the reader to read that proof in stead of this one.", "\\medskip\\noindent", "It follows immediately from the definition that (1) implies (2).", "Assume that $X$ is not geometrically connected.", "Let $k \\subset \\overline{k}$ be a separable algebraic", "closure of $k$. By", "Lemma \\ref{lemma-characterize-geometrically-connected}", "it follows that $X_{\\overline{k}}$ is disconnected.", "Say $X_{\\overline{k}} = \\overline{U} \\amalg \\overline{V}$", "with $\\overline{U}$ and $\\overline{V}$ open, closed, and nonempty", "algebraic subspaces of $X_{\\overline{k}}$.", "\\medskip\\noindent", "Suppose that $W \\subset X$ is any quasi-compact open subspace.", "Then $W_{\\overline{k}} \\cap \\overline{U}$ and", "$W_{\\overline{k}} \\cap \\overline{V}$ are open and closed subspaces of", "$W_{\\overline{k}}$. In particular $W_{\\overline{k}} \\cap \\overline{U}$ and", "$W_{\\overline{k}} \\cap \\overline{V}$ are quasi-compact, and by", "Lemma \\ref{lemma-Galois-action-quasi-compact-open}", "both $W_{\\overline{k}} \\cap \\overline{U}$ and", "$W_{\\overline{k}} \\cap \\overline{V}$", "are defined over a finite subextension and invariant under an", "open subgroup of $\\text{Gal}(\\overline{k}/k)$.", "We will use this without further mention in the following.", "\\medskip\\noindent", "Pick $W_0 \\subset X$ quasi-compact open subspace such that both", "$W_{0, \\overline{k}} \\cap \\overline{U}$ and", "$W_{0, \\overline{k}} \\cap \\overline{V}$ are nonempty.", "Choose a finite subextension $k \\subset k' \\subset \\overline{k}$", "and a decomposition $W_{0, k'} = U_0' \\amalg V_0'$ into open and closed", "subsets such that", "$W_{0, \\overline{k}} \\cap \\overline{U} = (U'_0)_{\\overline{k}}$ and", "$W_{0, \\overline{k}} \\cap \\overline{V} = (V'_0)_{\\overline{k}}$.", "Let $H = \\text{Gal}(\\overline{k}/k') \\subset \\text{Gal}(\\overline{k}/k)$.", "In particular", "$\\sigma(W_{0, \\overline{k}} \\cap \\overline{U}) =", "W_{0, \\overline{k}} \\cap \\overline{U}$ and similarly for", "$\\overline{V}$.", "\\medskip\\noindent", "Having chosen $W_0$, $k'$ as above, for every quasi-compact open subspace", "$W \\subset X$ we set", "$$", "U_W =", "\\bigcap\\nolimits_{\\sigma \\in H} \\sigma(W_{\\overline{k}} \\cap \\overline{U}),", "\\quad", "V_W =", "\\bigcup\\nolimits_{\\sigma \\in H} \\sigma(W_{\\overline{k}} \\cap \\overline{V}).", "$$", "Now, since $W_{\\overline{k}} \\cap \\overline{U}$ and", "$W_{\\overline{k}} \\cap \\overline{V}$ are fixed by an open subgroup of", "$\\text{Gal}(\\overline{k}/k)$ we see that the union and intersection", "above are finite. Hence $U_W$ and $V_W$ are both open and closed subspaces.", "Also, by construction $W_{\\bar k} = U_W \\amalg V_W$.", "\\medskip\\noindent", "We claim that if $W \\subset W' \\subset X$ are quasi-compact", "open subspaces, then $W_{\\overline{k}} \\cap U_{W'} = U_W$ and", "$W_{\\overline{k}} \\cap V_{W'} = V_W$. Verification omitted.", "Hence we see that upon defining $U = \\bigcup_{W \\subset X} U_W$", "and $V = \\bigcup_{W \\subset X} V_W$ we obtain", "$X_{\\overline{k}} = U \\amalg V$ is a disjoint union of open", "and closed subsets.", "It is clear that $V$ is nonempty as it is constructed by taking", "unions (locally). On the other hand, $U$ is nonempty since it contains", "$W_0 \\cap \\overline{U}$ by construction. Finally, $U, V \\subset X_{\\bar k}$", "are closed and $H$-invariant by construction. Hence by", "Lemma \\ref{lemma-closed-fixed-by-Galois}", "we have $U = (U')_{\\bar k}$, and $V = (V')_{\\bar k}$ for some", "closed $U', V' \\subset X_{k'}$. Clearly $X_{k'} = U' \\amalg V'$", "and we see that $X_{k'}$ is disconnected as desired." ], "refs": [ "varieties-lemma-characterize-geometrically-disconnected", "varieties-lemma-characterize-geometrically-connected", "spaces-over-fields-lemma-characterize-geometrically-connected", "varieties-lemma-Galois-action-quasi-compact-open", "spaces-over-fields-lemma-Galois-action-quasi-compact-open", "varieties-lemma-closed-fixed-by-Galois", "spaces-over-fields-lemma-closed-fixed-by-Galois", "spaces-over-fields-lemma-characterize-geometrically-connected", "spaces-over-fields-lemma-Galois-action-quasi-compact-open", "spaces-over-fields-lemma-closed-fixed-by-Galois" ], "ref_ids": [ 10923, 10919, 12863, 10921, 12864, 10922, 12865, 12863, 12864, 12865 ] } ], "ref_ids": [] }, { "id": 12867, "type": "theorem", "label": "spaces-over-fields-lemma-geometrically-integral", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-geometrically-integral", "contents": [ "Let $k$ be a field. Let $X$ be a decent algebraic space over $k$.", "Then $X$ is geometrically integral over $k$ if and only if", "$X$ is both geometrically reduced and geometrically irreducible", "over $k$." ], "refs": [], "proofs": [ { "contents": [ "This is an immediate consequence of the definitions because our", "notion of integral (in the presence of decency) is equivalent to", "reduced and irreducible." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12868, "type": "theorem", "label": "spaces-over-fields-lemma-proper-geometrically-reduced-global-sections", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-proper-geometrically-reduced-global-sections", "contents": [ "Let $k$ be a field. Let $X$ be a proper algebraic space over $k$.", "\\begin{enumerate}", "\\item $A = H^0(X, \\mathcal{O}_X)$ is a finite dimensional $k$-algebra,", "\\item $A = \\prod_{i = 1, \\ldots, n} A_i$ is a product of Artinian", "local $k$-algebras, one factor for each connected component of $|X|$,", "\\item if $X$ is reduced, then $A = \\prod_{i = 1, \\ldots, n} k_i$", "is a product of fields, each a finite extension of $k$,", "\\item if $X$ is geometrically reduced, then $k_i$ is finite separable", "over $k$,", "\\item if $X$ is geometrically connected, then $A$ is geometrically", "irreducible over $k$,", "\\item if $X$ is geometrically irreducible, then $A$ is geometrically", "irreducible over $k$,", "\\item if $X$ is geometrically reduced and connected, then $A = k$, and", "\\item if $X$ is geometrically integral, then $A = k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-over-affine-cohomology-finite}", "we see that $A = H^0(X, \\mathcal{O}_X)$ is a finite dimensional", "$k$-algebra. This proves (1).", "\\medskip\\noindent", "Then $A$ is a product of local rings by", "Algebra, Lemma \\ref{algebra-lemma-finite-dimensional-algebra} and", "Algebra, Proposition \\ref{algebra-proposition-dimension-zero-ring}.", "If $X = Y \\amalg Z$ with $Y$ and $Z$ open subspaces of $X$, then we obtain", "an idempotent $e \\in A$ by taking the section of $\\mathcal{O}_X$", "which is $1$ on $Y$ and $0$ on $Z$. Conversely, if $e \\in A$", "is an idempotent, then we get a corresponding decomposition of $|X|$.", "Finally, as $|X|$ is a Noetherian topological space", "(by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-finite-presentation-noetherian} and", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-Noetherian-topology})", "its connected components are open. Hence the connected components", "of $|X|$ correspond $1$-to-$1$ with primitive idempotents of $A$.", "This proves (2).", "\\medskip\\noindent", "If $X$ is reduced, then $A$ is reduced", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-reduced-space}).", "Hence the local rings $A_i = k_i$ are reduced and therefore fields", "(for example by Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring}).", "This proves (3).", "\\medskip\\noindent", "If $X$ is geometrically reduced, then same thing is true for", "$A \\otimes_k \\overline{k} =", "H^0(X_{\\overline{k}}, \\mathcal{O}_{X_{\\overline{k}}})$", "(see Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-flat-base-change-cohomology} for equality).", "This implies that $k_i \\otimes_k \\overline{k}$ is a product", "of fields and hence $k_i/k$ is separable for example by", "Algebra,", "Lemmas \\ref{algebra-lemma-characterize-separable-field-extensions} and", "\\ref{algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension}.", "This proves (4).", "\\medskip\\noindent", "If $X$ is geometrically connected, then $A \\otimes_k \\overline{k} =", "H^0(X_{\\overline{k}}, \\mathcal{O}_{X_{\\overline{k}}})$", "is a zero dimensional local ring by part (2) and hence its", "spectrum has one point, in particular it is irreducible.", "Thus $A$ is geometrically irreducible. This proves (5).", "Of course (5) implies (6).", "\\medskip\\noindent", "If $X$ is geometrically reduced and connected, then", "$A = k_1$ is a field and the extension $k_1/k$ is finite separable and", "geometrically irreducible. However, then $k_1 \\otimes_k \\overline{k}$", "is a product of $[k_1 : k]$ copies of $\\overline{k}$ and we conclude", "that $k_1 = k$. This proves (7). Of course (7) implies (8)." ], "refs": [ "spaces-cohomology-lemma-proper-over-affine-cohomology-finite", "algebra-lemma-finite-dimensional-algebra", "algebra-proposition-dimension-zero-ring", "spaces-morphisms-lemma-finite-presentation-noetherian", "spaces-properties-lemma-Noetherian-topology", "spaces-properties-lemma-reduced-space", "algebra-lemma-minimal-prime-reduced-ring", "spaces-cohomology-lemma-flat-base-change-cohomology", "algebra-lemma-characterize-separable-field-extensions", "algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension" ], "ref_ids": [ 11332, 642, 1410, 4843, 11891, 11883, 418, 11296, 569, 571 ] } ], "ref_ids": [] }, { "id": 12869, "type": "theorem", "label": "spaces-over-fields-lemma-characterize-trivial-pic-integral", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-characterize-trivial-pic-integral", "contents": [ "Let $k$ be a field. Let $X$ be a proper integral algebraic space over $k$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "If $H^0(X, \\mathcal{L})$ and $H^0(X, \\mathcal{L}^{\\otimes - 1})$", "are both nonzero, then $\\mathcal{L} \\cong \\mathcal{O}_X$." ], "refs": [], "proofs": [ { "contents": [ "Let $s \\in H^0(X, \\mathcal{L})$ and $t \\in H^0(X, \\mathcal{L}^{\\otimes - 1})$", "be nonzero sections. Let $x \\in |X|$ be a point in the support of $s$.", "Choose an affine \\'etale neighbourhood $(U, u) \\to (X, x)$ such that", "$\\mathcal{L}|_U \\cong \\mathcal{O}_U$. Then $s|_U$ corresponds to a nonzero", "regular function on the reduced (because $X$ is reduced) scheme $U$ and hence", "is nonvanishing in a generic point of an irreducible component of $U$. By", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-generic-points}", "we conclude that the generic point $\\eta$ of $|X|$ is in the support of $s$.", "The same is true for $t$. Then of course $st$ must be nonzero because", "the local ring of $X$ at $\\eta$ is a field (by aforementioned lemma", "the local ring has dimension zero, as $X$ is reduced the local ring is", "reduced, and Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring}).", "However, we have seen that $K = H^0(X, \\mathcal{O}_X)$", "is a field in Lemma \\ref{lemma-proper-geometrically-reduced-global-sections}.", "Thus $st$ is everywhere nonzero and we see that", "$s : \\mathcal{O}_X \\to \\mathcal{L}$ is an isomorphism." ], "refs": [ "decent-spaces-lemma-decent-generic-points", "algebra-lemma-minimal-prime-reduced-ring", "spaces-over-fields-lemma-proper-geometrically-reduced-global-sections" ], "ref_ids": [ 9531, 418, 12868 ] } ], "ref_ids": [] }, { "id": 12870, "type": "theorem", "label": "spaces-over-fields-lemma-integral-dimension", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-integral-dimension", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be an integral morphism", "of algebraic spaces. Then $\\dim(X) \\leq \\dim(Y)$.", "If $f$ is surjective then $\\dim(X) = \\dim(Y)$." ], "refs": [], "proofs": [ { "contents": [ "Choose $V \\to Y$ surjective \\'etale with $V$ a scheme.", "Then $U = X \\times_Y V$ is a scheme and $U \\to V$ is integral", "(and surjective if $f$ is surjective).", "By Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-dimension-decent-invariant-under-etale}", "we have $\\dim(X) = \\dim(U)$ and $\\dim(Y) = \\dim(V)$.", "Thus the result follows from the case of schemes", "which is Morphisms, Lemma \\ref{morphisms-lemma-integral-dimension}." ], "refs": [ "spaces-properties-lemma-dimension-decent-invariant-under-etale", "morphisms-lemma-integral-dimension" ], "ref_ids": [ 11887, 5443 ] } ], "ref_ids": [] }, { "id": 12871, "type": "theorem", "label": "spaces-over-fields-lemma-alteration-dimension", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-alteration-dimension", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Assume that", "\\begin{enumerate}", "\\item $Y$ is locally Noetherian,", "\\item $X$ and $Y$ are integral algebraic spaces,", "\\item $f$ is dominant, and", "\\item $f$ is locally of finite type.", "\\end{enumerate}", "If $x \\in |X|$ and $y \\in |Y|$ are the generic points, then", "$$", "\\dim(X) \\leq \\dim(Y) + \\text{transcendence degree of }x/y.", "$$", "If $f$ is proper, then equality holds." ], "refs": [], "proofs": [ { "contents": [ "Recall that $|X|$ and $|Y|$ are irreducible sober topological spaces, see", "discussion following Definition \\ref{definition-integral-algebraic-space}.", "Thus the fact that $f$ is dominant means that $|f|$ maps $x$ to $y$.", "Moreover, $x \\in |X|$ is the unique point at which the", "local ring of $X$ has dimension $0$, see", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-generic-points}.", "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-dimension-formula-general}", "we see that the dimension of the local ring of $X$ at", "any point $x' \\in |X|$ is at most the dimension of the local", "ring of $Y$ at $y' = f(x')$ plus the transcendence degree of $x/y$.", "Since the dimension of $X$, resp.\\ dimension of $Y$ is the", "supremum of the dimensions of the local rings at $x'$, resp.\\ $y'$", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-dimension})", "we conclude the inequality holds.", "\\medskip\\noindent", "Assume $f$ is proper.", "Let $V \\subset Y$ be a nonempty quasi-compact open subspace.", "If we can prove the equality for the morphism $f^{-1}(V) \\to V$,", "then we get the equality for $X \\to Y$. Thus we may assume that", "$X$ and $Y$ are quasi-compact.", "Observe that $X$ is quasi-separated as", "a locally Noetherian decent algebraic space, see", "Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-locally-Noetherian-decent-quasi-separated}.", "Thus we may choose $Y' \\to Y$ finite surjective where $Y'$", "is a scheme, see Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-there-is-a-scheme-finite-over}.", "After replacing $Y'$ by a suitable closed subscheme, we", "may assume $Y'$ is integral, see for example the more general", "Lemma \\ref{lemma-alteration-contained-in}.", "By the same lemma, we may choose a closed subspace", "$X' \\subset X \\times_Y Y'$ such that $X'$ is integral", "and $X' \\to X$ is finite surjective.", "Now $X'$ is also locally Noetherian", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian})", "and we can use Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-there-is-a-scheme-finite-over}", "once more to choose a finite surjective morphism $X'' \\to X'$", "with $X''$ a scheme. As before we may assume that $X''$", "is integral. Picture", "$$", "\\xymatrix{", "X'' \\ar[d] \\ar[r] & X \\ar[d]^f \\\\", "Y' \\ar[r] & Y", "}", "$$", "By Lemma \\ref{lemma-integral-dimension} we have $\\dim(X'') = \\dim(X)$", "and $\\dim(Y') = \\dim(Y)$. Since $X$ and $Y$ have open neighbourhoods", "of $x$, resp.\\ $y$ which are schemes, we readily see that the generic", "points $x'' \\in X''$, resp.\\ $y' \\in Y'$ are the unique points mapping", "to $x$, resp.\\ $y$ and that the residue field extensions", "$\\kappa(x'')/\\kappa(x)$ and $\\kappa(y')/\\kappa(y)$ are finite.", "This implies that the transcendence degree of $x''/y'$ is the", "same as the transcendence degree of $x/y$. Thus the equality follows", "from the case of schemes whicn is", "Morphisms, Lemma \\ref{morphisms-lemma-alteration-dimension}." ], "refs": [ "spaces-over-fields-definition-integral-algebraic-space", "decent-spaces-lemma-decent-generic-points", "spaces-morphisms-lemma-dimension-formula-general", "spaces-properties-lemma-dimension", "decent-spaces-lemma-locally-Noetherian-decent-quasi-separated", "spaces-limits-proposition-there-is-a-scheme-finite-over", "spaces-over-fields-lemma-alteration-contained-in", "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "spaces-limits-proposition-there-is-a-scheme-finite-over", "spaces-over-fields-lemma-integral-dimension", "morphisms-lemma-alteration-dimension" ], "ref_ids": [ 12883, 9531, 4878, 11841, 9506, 4659, 12842, 4817, 4659, 12870, 5496 ] } ], "ref_ids": [] }, { "id": 12872, "type": "theorem", "label": "spaces-over-fields-lemma-smooth-regular", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-smooth-regular", "contents": [ "Let $k$ be a field.", "Let $X$ be an algebraic space smooth over $k$.", "Then $X$ is a regular algebraic space." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "The morphism $U \\to \\Spec(k)$ is smooth as a composition of", "an \\'etale (hence smooth) morphism and a smooth morphism (see", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-etale-smooth}", "and \\ref{spaces-morphisms-lemma-composition-smooth}).", "Hence $U$ is regular by", "Varieties, Lemma \\ref{varieties-lemma-smooth-regular}.", "By", "Properties of Spaces, Definition", "\\ref{spaces-properties-definition-type-property}", "this means that $X$ is regular." ], "refs": [ "spaces-morphisms-lemma-etale-smooth", "spaces-morphisms-lemma-composition-smooth", "varieties-lemma-smooth-regular", "spaces-properties-definition-type-property" ], "ref_ids": [ 4909, 4886, 11004, 11926 ] } ], "ref_ids": [] }, { "id": 12873, "type": "theorem", "label": "spaces-over-fields-lemma-smooth-separable-closed-points-dense", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-smooth-separable-closed-points-dense", "contents": [ "Let $k$ be a field. Let $X$ be an algebraic space smooth over $\\Spec(k)$.", "The set of $x \\in |X|$ which are image of morphisms $\\Spec(k') \\to X$", "with $k' \\supset k$ finite separable is dense in $|X|$." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "The morphism $U \\to \\Spec(k)$ is smooth as a composition of", "an \\'etale (hence smooth) morphism and a smooth morphism (see", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-etale-smooth}", "and \\ref{spaces-morphisms-lemma-composition-smooth}).", "Hence we can apply Varieties, Lemma", "\\ref{varieties-lemma-smooth-separable-closed-points-dense} to see that", "the closed points of $U$ whose residue fields are finite separable over", "$k$ are dense. This implies the lemma by our definition of the", "topology on $|X|$." ], "refs": [ "spaces-morphisms-lemma-etale-smooth", "spaces-morphisms-lemma-composition-smooth", "varieties-lemma-smooth-separable-closed-points-dense" ], "ref_ids": [ 4909, 4886, 11007 ] } ], "ref_ids": [] }, { "id": 12874, "type": "theorem", "label": "spaces-over-fields-lemma-euler-characteristic-additive", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-euler-characteristic-additive", "contents": [ "Let $k$ be a field. Let $X$ be a proper algebraic space over $k$.", "Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "be a short exact sequence of coherent modules on $X$. Then", "$$", "\\chi(X, \\mathcal{F}_2) = \\chi(X, \\mathcal{F}_1) + \\chi(X, \\mathcal{F}_3)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Consider the long exact sequence of cohomology", "$$", "0 \\to H^0(X, \\mathcal{F}_1) \\to H^0(X, \\mathcal{F}_2) \\to", "H^0(X, \\mathcal{F}_3) \\to H^1(X, \\mathcal{F}_1) \\to \\ldots", "$$", "associated to the short exact sequence of the lemma. The rank-nullity theorem", "in linear algebra shows that", "$$", "0 = \\dim H^0(X, \\mathcal{F}_1) - \\dim H^0(X, \\mathcal{F}_2)", "+ \\dim H^0(X, \\mathcal{F}_3) - \\dim H^1(X, \\mathcal{F}_1) + \\ldots", "$$", "This immediately implies the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12875, "type": "theorem", "label": "spaces-over-fields-lemma-euler-characteristic-morphism", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-euler-characteristic-morphism", "contents": [ "Let $k$ be a field. Let $f : Y \\to X$ be a morphism of", "algebraic spaces proper over $k$. Let $\\mathcal{G}$ be a", "coherent $\\mathcal{O}_Y$-module. Then", "$$", "\\chi(Y, \\mathcal{G}) = \\sum (-1)^i \\chi(X, R^if_*\\mathcal{G})", "$$" ], "refs": [], "proofs": [ { "contents": [ "The formula makes sense: the sheaves $R^if_*\\mathcal{G}$ are coherent", "and only a finite number of them are nonzero, see", "Cohomology of Spaces, Lemmas", "\\ref{spaces-cohomology-lemma-proper-pushforward-coherent} and", "\\ref{spaces-cohomology-lemma-vanishing-higher-direct-images}.", "By Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-Leray}", "there is a spectral sequence with", "$$", "E_2^{p, q} = H^p(X, R^qf_*\\mathcal{G})", "$$", "converging to $H^{p + q}(Y, \\mathcal{G})$. By finiteness of cohomology", "on $X$ we see that only a finite number of $E_2^{p, q}$ are nonzero", "and each $E_2^{p, q}$ is a finite dimensional vector space. It follows", "that the same is true for $E_r^{p, q}$ for $r \\geq 2$ and that", "$$", "\\sum (-1)^{p + q} \\dim_k E_r^{p, q}", "$$", "is independent of $r$. Since for $r$ large enough we have", "$E_r^{p, q} = E_\\infty^{p, q}$ and since convergence means there", "is a filtration on $H^n(Y, \\mathcal{G})$ whose graded pieces are", "$E_\\infty^{p, q}$ with $p + 1 = n$ (this is the meaning of convergence", "of the spectral sequence), we conclude." ], "refs": [ "spaces-cohomology-lemma-proper-pushforward-coherent", "spaces-cohomology-lemma-vanishing-higher-direct-images", "sites-cohomology-lemma-Leray" ], "ref_ids": [ 11331, 11287, 4220 ] } ], "ref_ids": [] }, { "id": 12876, "type": "theorem", "label": "spaces-over-fields-lemma-numerical-polynomial-from-euler", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-numerical-polynomial-from-euler", "contents": [ "Let $k$ be a field. Let $X$ be a proper algebraic space over $k$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let", "$\\mathcal{L}_1, \\ldots, \\mathcal{L}_r$ be invertible $\\mathcal{O}_X$-modules.", "The map", "$$", "(n_1, \\ldots, n_r) \\longmapsto", "\\chi(X, \\mathcal{F} \\otimes", "\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r})", "$$", "is a numerical polynomial in $n_1, \\ldots, n_r$ of total degree at", "most the dimension of the scheme theoretic support of $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset X$ be the scheme theoretic support of $\\mathcal{F}$.", "Then $\\mathcal{F} = i_*\\mathcal{G}$ for some coherent", "$\\mathcal{O}_Z$-module $\\mathcal{G}$", "(Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-coherent-support-closed})", "and we have", "$$", "\\chi(X, \\mathcal{F} \\otimes", "\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}) =", "\\chi(Z, \\mathcal{G} \\otimes", "i^*\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "i^*\\mathcal{L}_r^{\\otimes n_r})", "$$", "by the projection formula", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-projection-formula})", "and Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-relative-affine-cohomology}.", "Since $|Z| = \\text{Supp}(\\mathcal{F})$ we see that it suffices", "to show", "$$", "P_\\mathcal{F}(n_1, \\ldots, n_r) :", "(n_1, \\ldots, n_r)", "\\longmapsto", "\\chi(X, \\mathcal{F} \\otimes", "\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r})", "$$", "is a numerical polynomial in $n_1, \\ldots, n_r$ of total degree at", "most $\\dim(X)$. Let us say property $\\mathcal{P}$ holds for the", "coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ if the above is true.", "\\medskip\\noindent", "We will prove this statement by devissage, more precisely we will", "check conditions (1), (2), and (3) of", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-property-higher-rank-cohomological-variant}", "are satisfied.", "\\medskip\\noindent", "Verification of condition (1). Let", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0", "$$", "be a short exact sequence of coherent sheaves on $X$.", "By Lemma \\ref{lemma-euler-characteristic-additive} we have", "$$", "P_{\\mathcal{F}_2}(n_1, \\ldots, n_r) =", "P_{\\mathcal{F}_1}(n_1, \\ldots, n_r) +", "P_{\\mathcal{F}_3}(n_1, \\ldots, n_r)", "$$", "Then it is clear that if 2-out-of-3 of the sheaves $\\mathcal{F}_i$", "have property $\\mathcal{P}$, then so does the third.", "\\medskip\\noindent", "Condition (2) follows because", "$P_{\\mathcal{F}^{\\oplus m}}(n_1, \\ldots, n_r) =", "mP_\\mathcal{F}(n_1, \\ldots, n_r)$.", "\\medskip\\noindent", "Proof of (3). Let $i : Z \\to X$ be a reduced closed subspace with", "$|Z|$ irreducible. We have to find a coherent module $\\mathcal{G}$", "on $X$ whose support is $Z$ such that $\\mathcal{P}$ holds for $\\mathcal{G}$.", "We will give two constructions: one using Chow's lemma and one", "using a finite cover by a scheme.", "\\medskip\\noindent", "Proof existence $\\mathcal{G}$ using a finite cover by a scheme.", "Choose $\\pi : Z' \\to Z$ finite surjective where $Z'$ is a scheme, see", "Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-there-is-a-scheme-finite-over}.", "Set $\\mathcal{G} = i_*\\pi_*\\mathcal{O}_{Z'} = (i \\circ \\pi)_*\\mathcal{O}_{Z'}$.", "Note that $Z'$ is proper over $k$ and that the support of $\\mathcal{G}$ is $Y$", "(details omitted). We have", "$$", "R(\\pi \\circ i)_*(\\mathcal{O}_{Z'}) = \\mathcal{G}", "\\quad\\text{and}\\quad", "R(\\pi \\circ i)_*(\\pi^*i^*(\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r})", ") = \\mathcal{G} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}", "$$", "The first equality holds because $i \\circ \\pi$ is affine", "(Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-affine-vanishing-higher-direct-images})", "and the second equality follows from the first and the projection formula", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-projection-formula}).", "Using Leray", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-apply-Leray})", "we obtain", "$$", "P_\\mathcal{G}(n_1, \\ldots, n_r) =", "\\chi(Z', \\pi^*i^*(\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}))", "$$", "By the case of schemes", "(Varieties, Lemma \\ref{varieties-lemma-numerical-polynomial-from-euler})", "this is a numerical polynomial in", "$n_1, \\ldots, n_r$ of degree at most $\\dim(Z')$.", "We conclude because $\\dim(Z') \\leq \\dim(Z) \\leq \\dim(X)$.", "The first inequality follows from", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-dimension-quasi-finite}.", "\\medskip\\noindent", "Proof existence $\\mathcal{G}$ using Chow's lemma. We apply", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-weak-chow}", "to the morphism $Z \\to \\Spec(k)$. Thus we get a", "surjective proper morphism $f : Y \\to Z$ over $\\Spec(k)$", "where $Y$ is a closed subscheme of $\\mathbf{P}^m_k$ for some $m$.", "After replacing $Y$ by a closed subscheme we may assume that $Y$", "is integral and $f : Y \\to Z$ is an alteration, see", "Lemma \\ref{lemma-alteration-contained-in}.", "Denote $\\mathcal{O}_Y(n)$ the pullback of $\\mathcal{O}_{\\mathbf{P}^m_k}(n)$.", "Pick $n > 0$ such that $R^pf_*\\mathcal{O}_Y(n) = 0$", "for $p > 0$, see Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-kill-by-twisting}.", "We claim that $\\mathcal{G} = i_*f_*\\mathcal{O}_Y(n)$ satisfies $\\mathcal{P}$.", "Namely, by the case of schemes", "(Varieties, Lemma \\ref{varieties-lemma-numerical-polynomial-from-euler})", "we know that", "$$", "(n_1, \\ldots, n_r)", "\\longmapsto", "\\chi(Y, \\mathcal{O}_Y(n) \\otimes", "f^*i^*(\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}))", "$$", "is a numerical polynomial in $n_1, \\ldots, n_r$ of total degree at", "most $\\dim(Y)$. On the other hand, by the projection formula", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-projection-formula})", "\\begin{align*}", "i_*Rf_*\\left(", "\\mathcal{O}_Y(n) \\otimes", "f^*i^*(\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r})\\right)", "& =", "i_*Rf_*\\mathcal{O}_Y(n) \\otimes", "\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r} \\\\", "& =", "\\mathcal{G} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}", "\\end{align*}", "the last equality by our choice of $n$. By", "Leray (Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-apply-Leray})", "we get", "$$", "\\chi(Y, \\mathcal{O}_Y(n) \\otimes", "f^*i^*(\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r})) =", "P_\\mathcal{G}(n_1, \\ldots, n_r)", "$$", "and we conclude because $\\dim(Y) \\leq \\dim(Z) \\leq \\dim(X)$.", "The first inequality holds by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-alteration-dimension-general}", "and the fact that $Y \\to Z$ is an alteration (and hence the", "induced extension of residue fields in generic points is finite)." ], "refs": [ "spaces-cohomology-lemma-coherent-support-closed", "sites-cohomology-lemma-projection-formula", "spaces-cohomology-lemma-relative-affine-cohomology", "spaces-cohomology-lemma-property-higher-rank-cohomological-variant", "spaces-over-fields-lemma-euler-characteristic-additive", "spaces-limits-proposition-there-is-a-scheme-finite-over", "spaces-cohomology-lemma-affine-vanishing-higher-direct-images", "sites-cohomology-lemma-projection-formula", "sites-cohomology-lemma-apply-Leray", "varieties-lemma-numerical-polynomial-from-euler", "decent-spaces-lemma-dimension-quasi-finite", "spaces-cohomology-lemma-weak-chow", "spaces-over-fields-lemma-alteration-contained-in", "spaces-cohomology-lemma-kill-by-twisting", "varieties-lemma-numerical-polynomial-from-euler", "sites-cohomology-lemma-projection-formula", "sites-cohomology-lemma-apply-Leray", "spaces-morphisms-lemma-alteration-dimension-general" ], "ref_ids": [ 11302, 4396, 11289, 11314, 12874, 4659, 11288, 4396, 4221, 11121, 9498, 11327, 12842, 11330, 11121, 4396, 4221, 4879 ] } ], "ref_ids": [] }, { "id": 12877, "type": "theorem", "label": "spaces-over-fields-lemma-numerical-polynomial-leading-term", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-numerical-polynomial-leading-term", "contents": [ "Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let", "$\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let", "$\\mathcal{L}_1, \\ldots, \\mathcal{L}_r$ be invertible $\\mathcal{O}_X$-modules.", "Let $d = \\dim(\\text{Supp}(\\mathcal{F}))$.", "Let $Z_i \\subset X$ be the irreducible components", "of $\\text{Supp}(\\mathcal{F})$ of dimension $d$. Let $\\overline{x}_i$", "be a geometric generic point of $Z_i$ and set", "$m_i = \\text{length}_{\\mathcal{O}_{X, \\overline{x}_i}}", "(\\mathcal{F}_{\\overline{x}_i})$.", "Then", "$$", "\\chi(X, \\mathcal{F} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}) -", "\\sum\\nolimits_i", "m_i\\ \\chi(Z_i, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}|_{Z_i})", "$$", "is a numerical polynomial in $n_1, \\ldots, n_r$ of total degree $< d$." ], "refs": [], "proofs": [ { "contents": [ "We first prove a slightly weaker statement. Namely, say", "$\\dim(X) = N$ and let $X_i \\subset X$ be the irreducible", "components of dimension $N$. Let $\\overline{x}_i$ be a geometric", "generic point of $X_i$. The \\'etale local ring", "$\\mathcal{O}_{X, \\overline{x}_i}$ is Noetherian of", "dimension $0$, hence for every coherent $\\mathcal{O}_X$-module", "$\\mathcal{F}$ the length", "$$", "m_i(\\mathcal{F}) = \\text{length}_{\\mathcal{O}_{X, \\overline{x}_i}}", "(\\mathcal{F}_{\\overline{x}_i})", "$$", "is an integer $\\geq 0$. We claim that", "$$", "E(\\mathcal{F}) =", "\\chi(X, \\mathcal{F} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}) -", "\\sum\\nolimits_i", "m_i(\\mathcal{F})\\ \\chi(Z_i, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}|_{Z_i})", "$$", "is a numerical polynomial in $n_1, \\ldots, n_r$ of total degree $< N$.", "We will prove this using Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-property-higher-rank-cohomological-variant}.", "For any short exact sequence $0 \\to \\mathcal{F}' \\to \\mathcal{F} \\to", "\\mathcal{F}'' \\to 0$ we have", "$E(\\mathcal{F}) = E(\\mathcal{F}') + E(\\mathcal{F}'')$.", "This follows from additivity of Euler characteristics", "(Lemma \\ref{lemma-euler-characteristic-additive})", "and additivity of lengths", "(Algebra, Lemma \\ref{algebra-lemma-length-additive}).", "This immediately implies properties (1) and (2) of Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-property-higher-rank-cohomological-variant}.", "Finally, property (3) holds because for $\\mathcal{G} = \\mathcal{O}_Z$", "for any $Z \\subset X$ irreducible reduced closed subspace.", "Namely, if $Z = Z_{i_0}$ for some $i_0$, then", "$m_i(\\mathcal{G}) = \\delta_{i_0i}$ and we conclude $E(\\mathcal{G}) = 0$.", "If $Z \\not = Z_i$ for any $i$, then $m_i(\\mathcal{G}) = 0$ for all $i$,", "$\\dim(Z) < N$ and we get the result from", "Lemma \\ref{lemma-numerical-polynomial-from-euler}.", "\\medskip\\noindent", "Proof of the statement as in the lemma.", "Let $Z \\subset X$ be the scheme theoretic support of $\\mathcal{F}$.", "Then $\\mathcal{F} = i_*\\mathcal{G}$ for some coherent", "$\\mathcal{O}_Z$-module $\\mathcal{G}$", "(Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-coherent-support-closed})", "and we have", "$$", "\\chi(X, \\mathcal{F} \\otimes", "\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}) =", "\\chi(Z, \\mathcal{G} \\otimes", "i^*\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "i^*\\mathcal{L}_r^{\\otimes n_r})", "$$", "by the projection formula", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-projection-formula})", "and Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-relative-affine-cohomology}.", "Since $|Z| = \\text{Supp}(\\mathcal{F})$ we see that $Z_i \\subset Z$", "for all $i$ and we see that these are the irreducible components", "of $Z$ of dimension $d$. We may and do think of $\\overline{x}_i$", "as a geometric point of $Z$. The map", "$i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$", "determines a surjection", "$$", "\\mathcal{O}_{X, \\overline{x}_i} \\to \\mathcal{O}_{Z, \\overline{x}_i}", "$$", "Via this map we have an isomorphism of modules", "$\\mathcal{G}_{\\overline{x}_i} = \\mathcal{F}_{\\overline{x}_i}$", "as $\\mathcal{F} = i_*\\mathcal{G}$. This implies that", "$$", "m_i = \\text{length}_{\\mathcal{O}_{X, \\overline{x}_i}}", "(\\mathcal{F}_{\\overline{x}_i}) =", "\\text{length}_{\\mathcal{O}_{Z, \\overline{x}_i}}", "(\\mathcal{G}_{\\overline{x}_i})", "$$", "Thus we see that the expression in the lemma is equal to", "$$", "\\chi(Z, \\mathcal{G} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}) -", "\\sum\\nolimits_i", "m_i\\ \\chi(Z_i, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_r^{\\otimes n_r}|_{Z_i})", "$$", "and the result follows from the discussion in the first paragraph", "(applied with $Z$ in stead of $X$)." ], "refs": [ "spaces-cohomology-lemma-property-higher-rank-cohomological-variant", "spaces-over-fields-lemma-euler-characteristic-additive", "algebra-lemma-length-additive", "spaces-cohomology-lemma-property-higher-rank-cohomological-variant", "spaces-over-fields-lemma-numerical-polynomial-from-euler", "spaces-cohomology-lemma-coherent-support-closed", "sites-cohomology-lemma-projection-formula", "spaces-cohomology-lemma-relative-affine-cohomology" ], "ref_ids": [ 11314, 12874, 631, 11314, 12876, 11302, 4396, 11289 ] } ], "ref_ids": [] }, { "id": 12878, "type": "theorem", "label": "spaces-over-fields-lemma-intersection-number-integer", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-intersection-number-integer", "contents": [ "In the situation of Definition \\ref{definition-intersection-number}", "the intersection number", "$(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$", "is an integer." ], "refs": [ "spaces-over-fields-definition-intersection-number" ], "proofs": [ { "contents": [ "Any numerical polynomial of degree $e$ in $n_1, \\ldots, n_d$ can be", "written uniquely as a $\\mathbf{Z}$-linear combination of the functions", "${n_1 \\choose k_1}{n_2 \\choose k_2} \\ldots {n_d \\choose k_d}$ with", "$k_1 + \\ldots + k_d \\leq e$. Apply this with $e = d$.", "Left as an exercise." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12899 ] }, { "id": 12879, "type": "theorem", "label": "spaces-over-fields-lemma-intersection-number-additive", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-intersection-number-additive", "contents": [ "In the situation of Definition \\ref{definition-intersection-number}", "the intersection number", "$(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$", "is additive: if $\\mathcal{L}_i = \\mathcal{L}_i' \\otimes \\mathcal{L}_i''$,", "then we have", "$$", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_i \\cdots \\mathcal{L}_d \\cdot Z) =", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_i' \\cdots \\mathcal{L}_d \\cdot Z) +", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_i'' \\cdots \\mathcal{L}_d \\cdot Z)", "$$" ], "refs": [ "spaces-over-fields-definition-intersection-number" ], "proofs": [ { "contents": [ "This is true because by Lemma \\ref{lemma-numerical-polynomial-from-euler}", "the function", "$$", "(n_1, \\ldots, n_{i - 1}, n_i', n_i'', n_{i + 1}, \\ldots, n_d)", "\\mapsto", "\\chi(Z, \\mathcal{L}_1^{\\otimes n_1} \\otimes", "\\ldots \\otimes (\\mathcal{L}_i')^{\\otimes n_i'} \\otimes", "(\\mathcal{L}_i'')^{\\otimes n_i''} \\otimes \\ldots \\otimes", "\\mathcal{L}_d^{\\otimes n_d}|_Z)", "$$", "is a numerical polynomial of total degree at most $d$ in $d + 1$ variables." ], "refs": [ "spaces-over-fields-lemma-numerical-polynomial-from-euler" ], "ref_ids": [ 12876 ] } ], "ref_ids": [ 12899 ] }, { "id": 12880, "type": "theorem", "label": "spaces-over-fields-lemma-intersection-number-in-terms-of-components", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-intersection-number-in-terms-of-components", "contents": [ "In the situation of Definition \\ref{definition-intersection-number}", "let $Z_i \\subset Z$ be the irreducible components of dimension $d$. Let", "$m_i = \\text{length}_{\\mathcal{O}_{X, \\overline{x}_i}}", "(\\mathcal{O}_{Z, \\overline{x}_i})$", "where $\\overline{x}_i$ is a geometric generic point of $Z_i$. Then", "$$", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) =", "\\sum m_i(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z_i)", "$$" ], "refs": [ "spaces-over-fields-definition-intersection-number" ], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-numerical-polynomial-leading-term}", "and the definitions." ], "refs": [ "spaces-over-fields-lemma-numerical-polynomial-leading-term" ], "ref_ids": [ 12877 ] } ], "ref_ids": [ 12899 ] }, { "id": 12881, "type": "theorem", "label": "spaces-over-fields-lemma-intersection-number-and-pullback", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-intersection-number-and-pullback", "contents": [ "Let $k$ be a field. Let $f : Y \\to X$ be a morphism of", "algebraic spaces proper over $k$.", "Let $Z \\subset Y$ be an integral closed subspace of dimension $d$ and let", "$\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible $\\mathcal{O}_X$-modules.", "Then", "$$", "(f^*\\mathcal{L}_1 \\cdots f^*\\mathcal{L}_d \\cdot Z) =", "\\deg(f|_Z : Z \\to f(Z)) (\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot f(Z))", "$$", "where $\\deg(Z \\to f(Z))$ is as in Definition \\ref{definition-degree}", "or $0$ if $\\dim(f(Z)) < d$." ], "refs": [ "spaces-over-fields-definition-degree" ], "proofs": [ { "contents": [ "In the statement $f(Z) \\subset X$ is the scheme theoretic image of $f$", "and it is also the reduced induced algebraic space structure on the", "closed subset $f(|Z|) \\subset X$, see Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-scheme-theoretic-image-reduced}.", "Then $Z$ and $f(Z)$ are reduced, proper (hence decent) algebraic spaces", "over $k$, whence integral", "(Definition \\ref{definition-integral-algebraic-space}).", "The left hand side is computed using the coefficient of $n_1 \\ldots n_d$", "in the function", "$$", "\\chi(Y, \\mathcal{O}_Z \\otimes f^*\\mathcal{L}_1^{\\otimes n_1} \\otimes", "\\ldots \\otimes f^*\\mathcal{L}_d^{\\otimes n_d}) =", "\\sum (-1)^i", "\\chi(X, R^if_*\\mathcal{O}_Z \\otimes", "\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_d^{\\otimes n_d})", "$$", "The equality follows from Lemma \\ref{lemma-euler-characteristic-morphism}", "and the projection formula", "(Cohomology, Lemma \\ref{cohomology-lemma-projection-formula}).", "If $f(Z)$ has dimension $< d$, then the right hand side", "is a polynomial of total degree $< d$ by", "Lemma \\ref{lemma-numerical-polynomial-from-euler}", "and the result is true. Assume $\\dim(f(Z)) = d$. Then", "by dimension theory (Lemma \\ref{lemma-alteration-dimension})", "we find that the equivalent conditions (1) -- (5) of", "Lemma \\ref{lemma-finite-degree} hold. Thus", "$\\deg(Z \\to f(Z))$ is well defined.", "By the already used Lemma \\ref{lemma-finite-degree}", "we find $f : Z \\to f(Z)$ is finite over a nonempty open", "$V$ of $f(Z)$; after possibly shrinking $V$ we may assume", "$V$ is a scheme. Let $\\xi \\in V$ be the generic point.", "Thus $\\deg(f : Z \\to f(Z))$ the length of the stalk of", "$f_*\\mathcal{O}_Z$ at $\\xi$ over $\\mathcal{O}_{X, \\xi}$", "and the stalk of $R^if_*\\mathcal{O}_X$ at $\\xi$ is zero for $i > 0$", "(for example by Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-finite-higher-direct-image-zero}).", "Thus the terms $\\chi(X, R^if_*\\mathcal{O}_Z \\otimes", "\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_d^{\\otimes n_d})$ with $i > 0$ have total", "degree $< d$ and", "$$", "\\chi(X, f_*\\mathcal{O}_Z \\otimes", "\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_d^{\\otimes n_d})", "=", "\\deg(f : Z \\to f(Z)) \\chi(f(Z),", "\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_d^{\\otimes n_d}|_{f(Z)})", "$$", "modulo a polynomial of total degree $< d$ by", "Lemma \\ref{lemma-numerical-polynomial-leading-term}.", "The desired result follows." ], "refs": [ "spaces-morphisms-lemma-scheme-theoretic-image-reduced", "spaces-over-fields-definition-integral-algebraic-space", "spaces-over-fields-lemma-euler-characteristic-morphism", "cohomology-lemma-projection-formula", "spaces-over-fields-lemma-numerical-polynomial-from-euler", "spaces-over-fields-lemma-alteration-dimension", "spaces-over-fields-lemma-finite-degree", "spaces-over-fields-lemma-finite-degree", "spaces-cohomology-lemma-finite-higher-direct-image-zero", "spaces-over-fields-lemma-numerical-polynomial-leading-term" ], "ref_ids": [ 4781, 12883, 12875, 2243, 12876, 12871, 12829, 12829, 11273, 12877 ] } ], "ref_ids": [ 12885 ] }, { "id": 12882, "type": "theorem", "label": "spaces-over-fields-lemma-numerical-intersection-effective-Cartier-divisor", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-lemma-numerical-intersection-effective-Cartier-divisor", "contents": [ "Let $k$ be a field. Let $X$ be a proper algebraic space over $k$.", "Let $Z \\subset X$ be a closed subspace of dimension $d$.", "Let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$", "be invertible $\\mathcal{O}_X$-modules. Assume there exists an", "effective Cartier divisor $D \\subset Z$ such that", "$\\mathcal{L}_1|_Z \\cong \\mathcal{O}_Z(D)$. Then", "$$", "(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) =", "(\\mathcal{L}_2 \\cdots \\mathcal{L}_d \\cdot D)", "$$" ], "refs": [], "proofs": [ { "contents": [ "We may replace $X$ by $Z$ and $\\mathcal{L}_i$ by $\\mathcal{L}_i|_Z$.", "Thus we may assume $X = Z$ and $\\mathcal{L}_1 = \\mathcal{O}_X(D)$.", "Then $\\mathcal{L}_1^{-1}$ is the ideal sheaf of $D$ and we can", "consider the short exact sequence", "$$", "0 \\to \\mathcal{L}_1^{\\otimes -1} \\to \\mathcal{O}_X \\to \\mathcal{O}_D \\to 0", "$$", "Set", "$P(n_1, \\ldots, n_d) =", "\\chi(X, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_d^{\\otimes n_d})$", "and", "$Q(n_1, \\ldots, n_d) =", "\\chi(D, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes", "\\mathcal{L}_d^{\\otimes n_d}|_D)$.", "We conclude from additivity", "(Lemma \\ref{lemma-euler-characteristic-additive})", "that", "$$", "P(n_1, \\ldots, n_d) - P(n_1 - 1, n_2, \\ldots, n_d) =", "Q(n_1, \\ldots, n_d)", "$$", "Because the total degree of $P$ is at most $d$, we see that", "the coefficient of $n_1 \\ldots n_d$ in $P$ is equal to the coefficient", "of $n_2 \\ldots n_d$ in $Q$." ], "refs": [ "spaces-over-fields-lemma-euler-characteristic-additive" ], "ref_ids": [ 12874 ] } ], "ref_ids": [] }, { "id": 12902, "type": "theorem", "label": "spaces-divisors-lemma-associated", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-associated", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $x \\in |X|$. The following are equivalent", "\\begin{enumerate}", "\\item for some \\'etale morphism $f : U \\to X$ with $U$ a scheme", "and $u \\in U$ mapping to $x$, the point $u$ is weakly associated", "to $f^*\\mathcal{F}$,", "\\item for every \\'etale morphism $f : U \\to X$ with $U$ a scheme", "and $u \\in U$ mapping to $x$, the point $u$ is weakly associated", "to $f^*\\mathcal{F}$,", "\\item the maximal ideal of $\\mathcal{O}_{X, \\overline{x}}$", "is a weakly associated prime of the stalk $\\mathcal{F}_{\\overline{x}}$.", "\\end{enumerate}", "If $X$ is locally Noetherian, then these are also equivalent to", "\\begin{enumerate}", "\\item[(4)] for some \\'etale morphism $f : U \\to X$ with $U$ a scheme", "and $u \\in U$ mapping to $x$, the point $u$ is associated", "to $f^*\\mathcal{F}$,", "\\item[(5)] for every \\'etale morphism $f : U \\to X$ with $U$ a scheme", "and $u \\in U$ mapping to $x$, the point $u$ is associated", "to $f^*\\mathcal{F}$,", "\\item[(6)] the maximal ideal of $\\mathcal{O}_{X, \\overline{x}}$", "is an associated prime of the stalk $\\mathcal{F}_{\\overline{x}}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ with a point $u$ and an \\'etale morphism", "$f : U \\to X$ mapping $u$ to $x$. Lift $\\overline{x}$ to a geometric", "point of $U$ over $u$. Recall that", "$\\mathcal{O}_{X, \\overline{x}} = \\mathcal{O}_{U, u}^{sh}$", "where the strict henselization is with respect to our chosen", "lift of $\\overline{x}$, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring}.", "Finally, we have", "$$", "\\mathcal{F}_{\\overline{x}} =", "(f^*\\mathcal{F})_u \\otimes_{\\mathcal{O}_{U, u}}", "\\mathcal{O}_{X, \\overline{x}} =", "(f^*\\mathcal{F})_u \\otimes_{\\mathcal{O}_{U, u}}", "\\mathcal{O}_{U, u}^{sh}", "$$", "by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-stalk-quasi-coherent}.", "Hence the equivalence of (1), (2), and (3) follows from", "More on Flatness, Lemma \\ref{flat-lemma-weakly-associated-henselization}.", "If $X$ is locally Noetherian, then", "any $U$ as above is locally Noetherian,", "hence we see that (1), resp.\\ (2) are equivalent to (4), resp.\\ (5) by", "Divisors, Lemma \\ref{divisors-lemma-ass-weakly-ass}.", "On the other hand, in the locally Noetherian case the", "local ring $\\mathcal{O}_{X, \\overline{x}}$ is Noetherian too", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-Noetherian-local-ring-Noetherian}).", "Hence the equivalence of (3) and (6) by the same lemma", "(or by Algebra, Lemma \\ref{algebra-lemma-ass-weakly-ass})." ], "refs": [ "spaces-properties-lemma-describe-etale-local-ring", "spaces-properties-lemma-stalk-quasi-coherent", "flat-lemma-weakly-associated-henselization", "divisors-lemma-ass-weakly-ass", "spaces-properties-lemma-Noetherian-local-ring-Noetherian", "algebra-lemma-ass-weakly-ass" ], "ref_ids": [ 11884, 11909, 5986, 7878, 11893, 727 ] } ], "ref_ids": [] }, { "id": 12903, "type": "theorem", "label": "spaces-divisors-lemma-weakly-ass-support", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-weakly-ass-support", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then $\\text{WeakAss}(\\mathcal{F}) \\subset \\text{Supp}(\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions. The support of an abelian sheaf", "on $X$ is defined in Properties of Spaces, Definition", "\\ref{spaces-properties-definition-support}." ], "refs": [ "spaces-properties-definition-support" ], "ref_ids": [ 11941 ] } ], "ref_ids": [] }, { "id": 12904, "type": "theorem", "label": "spaces-divisors-lemma-ses-weakly-ass", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-ses-weakly-ass", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "be a short exact sequence of quasi-coherent sheaves on $X$.", "Then", "$\\text{WeakAss}(\\mathcal{F}_2) \\subset", "\\text{WeakAss}(\\mathcal{F}_1) \\cup \\text{WeakAss}(\\mathcal{F}_3)$", "and", "$\\text{WeakAss}(\\mathcal{F}_1) \\subset \\text{WeakAss}(\\mathcal{F}_2)$." ], "refs": [], "proofs": [ { "contents": [ "For every geometric point $\\overline{x} \\in X$", "the sequence of stalks", "$0 \\to \\mathcal{F}_{1, \\overline{x}} \\to", "\\mathcal{F}_{2, \\overline{x}} \\to", "\\mathcal{F}_{3, \\overline{x}} \\to 0$", "is a short exact sequence of $\\mathcal{O}_{X, \\overline{x}}$-modules.", "Hence the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-weakly-ass}." ], "refs": [ "algebra-lemma-weakly-ass" ], "ref_ids": [ 722 ] } ], "ref_ids": [] }, { "id": 12905, "type": "theorem", "label": "spaces-divisors-lemma-weakly-ass-zero", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-weakly-ass-zero", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then", "$$", "\\mathcal{F} = (0) \\Leftrightarrow \\text{WeakAss}(\\mathcal{F}) = \\emptyset", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $f : U \\to X$.", "Then $\\mathcal{F}$ is zero if and only if $f^*\\mathcal{F}$ is zero.", "Hence the lemma follows from the definition and the lemma in the", "case of schemes, see", "Divisors, Lemma \\ref{divisors-lemma-weakly-ass-zero}." ], "refs": [ "divisors-lemma-weakly-ass-zero" ], "ref_ids": [ 7875 ] } ], "ref_ids": [] }, { "id": 12906, "type": "theorem", "label": "spaces-divisors-lemma-minimal-support-in-weakly-ass", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-minimal-support-in-weakly-ass", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $x \\in |X|$. If", "\\begin{enumerate}", "\\item $x \\in \\text{Supp}(\\mathcal{F})$", "\\item $x$ is a codimension $0$ point of $X$", "(Properties of Spaces, Definition", "\\ref{spaces-properties-definition-dimension-local-ring}).", "\\end{enumerate}", "Then $x \\in \\text{WeakAss}(\\mathcal{F})$. If $\\mathcal{F}$", "is a finite type $\\mathcal{O}_X$-module with scheme theoretic support $Z$", "(Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-scheme-theoretic-support})", "and $x$ is a codimension $0$ point of $Z$, then", "$x \\in \\text{WeakAss}(\\mathcal{F})$." ], "refs": [ "spaces-properties-definition-dimension-local-ring", "spaces-morphisms-definition-scheme-theoretic-support" ], "proofs": [ { "contents": [ "Since $x \\in \\text{Supp}(\\mathcal{F})$ the stalk", "$\\mathcal{F}_{\\overline{x}}$ is not zero. Hence", "$\\text{WeakAss}(\\mathcal{F}_{\\overline{x}})$", "is nonempty by", "Algebra, Lemma \\ref{algebra-lemma-weakly-ass-zero}.", "On the other hand, the spectrum of $\\mathcal{O}_{X, \\overline{x}}$", "is a singleton. Hence $x$ is a weakly associated point of", "$\\mathcal{F}$ by definition. The final statement follows", "as $\\mathcal{O}_{X, \\overline{x}} \\to \\mathcal{O}_{Z, \\overline{z}}$", "is a surjection, the spectrum of $\\mathcal{O}_{Z, \\overline{z}}$", "is a singleton, and $\\mathcal{F}_{\\overline{x}}$ is a nonzero", "module over $\\mathcal{O}_{Z, \\overline{z}}$." ], "refs": [ "algebra-lemma-weakly-ass-zero" ], "ref_ids": [ 723 ] } ], "ref_ids": [ 11931, 4993 ] }, { "id": 12907, "type": "theorem", "label": "spaces-divisors-lemma-minimal-support-in-weakly-ass-decent", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-minimal-support-in-weakly-ass-decent", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $x \\in |X|$. If", "\\begin{enumerate}", "\\item $X$ is decent (for example quasi-separated or locally separated),", "\\item $x \\in \\text{Supp}(\\mathcal{F})$", "\\item $x$ is not a specialization of another point in", "$\\text{Supp}(\\mathcal{F})$.", "\\end{enumerate}", "Then $x \\in \\text{WeakAss}(\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "(A quasi-separated algebraic space is decent, see", "Decent Spaces, Section \\ref{decent-spaces-section-reasonable-decent}.", "A locally separated algebraic space is decent, see", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-locally-separated-decent}.)", "Choose a scheme $U$, a point $u \\in U$, and an \\'etale morphism", "$f : U \\to X$ mapping $u$ to $x$. By", "Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-decent-no-specializations-map-to-same-point}", "if $u' \\leadsto u$ is a nontrivial specialization, then", "$f(u') \\not = x$. Hence we see that $u \\in \\text{Supp}(f^*\\mathcal{F})$", "is not a specialization of another point of", "$\\text{Supp}(f^*\\mathcal{F})$.", "Hence $u \\in \\text{WeakAss}(f^*\\mathcal{F})$ by", "Divisors, Lemma \\ref{lemma-minimal-support-in-weakly-ass}." ], "refs": [ "decent-spaces-lemma-locally-separated-decent", "decent-spaces-lemma-decent-no-specializations-map-to-same-point", "spaces-divisors-lemma-minimal-support-in-weakly-ass" ], "ref_ids": [ 9512, 9493, 12906 ] } ], "ref_ids": [] }, { "id": 12908, "type": "theorem", "label": "spaces-divisors-lemma-finite-ass", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-finite-ass", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Then $\\text{Ass}(\\mathcal{F}) \\cap W$ is finite for", "every quasi-compact open $W \\subset |X|$." ], "refs": [], "proofs": [ { "contents": [ "Choose a quasi-compact scheme $U$ and an \\'etale morphism $U \\to X$", "such that $W$ is the image of $|U| \\to |X|$. Then $U$ is a", "Noetherian scheme and we may apply", "Divisors, Lemma \\ref{divisors-lemma-finite-ass} to conclude." ], "refs": [ "divisors-lemma-finite-ass" ], "ref_ids": [ 7859 ] } ], "ref_ids": [] }, { "id": 12909, "type": "theorem", "label": "spaces-divisors-lemma-restriction-injective-open-contains-weakly-ass", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-restriction-injective-open-contains-weakly-ass", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "If $U \\to X$ is an \\'etale morphism such that", "$\\text{WeakAss}(\\mathcal{F}) \\subset \\Im(|U| \\to |X|)$, then", "$\\Gamma(X, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{F})$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Let $s \\in \\Gamma(X, \\mathcal{F})$ be a section which restricts to zero on $U$.", "Let $\\mathcal{F}' \\subset \\mathcal{F}$ be the image of the map", "$\\mathcal{O}_X \\to \\mathcal{F}$ defined by $s$. Then $\\mathcal{F}'|_U = 0$.", "This implies that", "$\\text{WeakAss}(\\mathcal{F}') \\cap \\Im(|U| \\to |X|) = \\emptyset$", "(by the definition of weakly associated points).", "On the other hand,", "$\\text{WeakAss}(\\mathcal{F}') \\subset \\text{WeakAss}(\\mathcal{F})$", "by Lemma \\ref{lemma-ses-weakly-ass}. We conclude", "$\\text{WeakAss}(\\mathcal{F}') = \\emptyset$.", "Hence $\\mathcal{F}' = 0$ by Lemma \\ref{lemma-weakly-ass-zero}." ], "refs": [ "spaces-divisors-lemma-ses-weakly-ass", "spaces-divisors-lemma-weakly-ass-zero" ], "ref_ids": [ 12904, 12905 ] } ], "ref_ids": [] }, { "id": 12910, "type": "theorem", "label": "spaces-divisors-lemma-weakass-pushforward", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-weakass-pushforward", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and quasi-separated", "morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. Let $y \\in |Y|$ be a point which is not in the", "image of $|f|$. Then $y$ is not weakly associated to $f_*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-pushforward}", "the $\\mathcal{O}_Y$-module $f_*\\mathcal{F}$ is quasi-coherent hence", "the lemma makes sense.", "Choose an affine scheme $V$, a point $v \\in V$, and an \\'etale morphism", "$V \\to Y$ mapping $v$ to $y$. We may replace", "$f : X \\to Y$, $\\mathcal{F}$, $y$ by", "$X \\times_Y V \\to V$, $\\mathcal{F}|_{X \\times_Y V}$, $v$.", "Thus we may assume $Y$ is an affine scheme.", "In this case $X$ is quasi-compact, hence we can choose", "an affine scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "Denote $g : U \\to Y$ the composition.", "Then $f_*\\mathcal{F} \\subset g_*(\\mathcal{F}|_U)$.", "By Lemma \\ref{lemma-ses-weakly-ass}", "we reduce to the case of schemes which is", "Divisors, Lemma \\ref{divisors-lemma-weakass-pushforward}." ], "refs": [ "spaces-morphisms-lemma-pushforward", "spaces-divisors-lemma-ses-weakly-ass", "divisors-lemma-weakass-pushforward" ], "ref_ids": [ 4760, 12904, 7879 ] } ], "ref_ids": [] }, { "id": 12911, "type": "theorem", "label": "spaces-divisors-lemma-check-injective-on-weakass", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-check-injective-on-weakass", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of", "quasi-coherent $\\mathcal{O}_X$-modules. Assume that for every", "$x \\in |X|$ at least one of the following happens", "\\begin{enumerate}", "\\item $\\mathcal{F}_{\\overline{x}} \\to \\mathcal{G}_{\\overline{x}}$", "is injective, or", "\\item $x \\not \\in \\text{WeakAss}(\\mathcal{F})$.", "\\end{enumerate}", "Then $\\varphi$ is injective." ], "refs": [], "proofs": [ { "contents": [ "The assumptions imply that $\\text{WeakAss}(\\Ker(\\varphi)) = \\emptyset$", "and hence $\\Ker(\\varphi) = 0$ by Lemma \\ref{lemma-weakly-ass-zero}." ], "refs": [ "spaces-divisors-lemma-weakly-ass-zero" ], "ref_ids": [ 12905 ] } ], "ref_ids": [] }, { "id": 12912, "type": "theorem", "label": "spaces-divisors-lemma-weakass-reduced", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-weakass-reduced", "contents": [ "Let $S$ be a scheme. Let $X$ be a reduced algebraic space over $S$.", "Then the weakly associated point of $X$ are exactly the codimension $0$", "points of $X$." ], "refs": [], "proofs": [ { "contents": [ "Working \\'etale locally this follows from", "Divisors, Lemma \\ref{divisors-lemma-weakass-reduced}", "and", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-codimension-0-points}." ], "refs": [ "divisors-lemma-weakass-reduced", "spaces-properties-lemma-codimension-0-points" ], "ref_ids": [ 7882, 11842 ] } ], "ref_ids": [] }, { "id": 12913, "type": "theorem", "label": "spaces-divisors-lemma-weakly-ass-reverse-functorial", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-weakly-ass-reverse-functorial", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be an affine morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then we have", "$$", "\\text{WeakAss}_S(f_*\\mathcal{F}) \\subset f(\\text{WeakAss}_X(\\mathcal{F}))", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Set $U = X \\times_Y V$. Then $U \\to V$ is an affine morphism", "of schemes. By our definition of weakly associated points", "the problem is reduced to the morphism of schemes $U \\to V$. This case is", "treated in Divisors, Lemma \\ref{divisors-lemma-weakly-ass-reverse-functorial}." ], "refs": [ "divisors-lemma-weakly-ass-reverse-functorial" ], "ref_ids": [ 7883 ] } ], "ref_ids": [] }, { "id": 12914, "type": "theorem", "label": "spaces-divisors-lemma-ass-functorial-equal", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-ass-functorial-equal", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be an affine morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "If $X$ is locally Noetherian, then we have", "$$", "\\text{WeakAss}_Y(f_*\\mathcal{F}) =", "f(\\text{WeakAss}_X(\\mathcal{F}))", "$$" ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Set $U = X \\times_Y V$. Then $U \\to V$ is an affine morphism", "of schemes and $U$ is locally Noetherian.", "By our definition of weakly associated points", "the problem is reduced to the morphism of schemes $U \\to V$. This case is", "treated in Divisors, Lemma \\ref{divisors-lemma-ass-functorial-equal}." ], "refs": [ "divisors-lemma-ass-functorial-equal" ], "ref_ids": [ 7884 ] } ], "ref_ids": [] }, { "id": 12915, "type": "theorem", "label": "spaces-divisors-lemma-weakly-associated-finite", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-weakly-associated-finite", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a finite morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Then $\\text{WeakAss}(f_*\\mathcal{F}) = f(\\text{WeakAss}(\\mathcal{F}))$." ], "refs": [], "proofs": [ { "contents": [ "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Set $U = X \\times_Y V$. Then $U \\to V$ is a finite morphism", "of schemes. By our definition of weakly associated points", "the problem is reduced to the morphism of schemes $U \\to V$. This case is", "treated in Divisors, Lemma \\ref{divisors-lemma-weakly-associated-finite}." ], "refs": [ "divisors-lemma-weakly-associated-finite" ], "ref_ids": [ 7885 ] } ], "ref_ids": [] }, { "id": 12916, "type": "theorem", "label": "spaces-divisors-lemma-weakly-ass-pullback", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-weakly-ass-pullback", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.", "Let $x \\in |X|$ and $y = f(x) \\in |Y|$. If", "\\begin{enumerate}", "\\item $y \\in \\text{WeakAss}_S(\\mathcal{G})$,", "\\item $f$ is flat at $x$, and", "\\item the dimension of the local ring of the fibre of $f$ at $x$", "is zero (Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-dimension-fibre}),", "\\end{enumerate}", "then $x \\in \\text{WeakAss}(f^*\\mathcal{G})$." ], "refs": [ "spaces-morphisms-definition-dimension-fibre" ], "proofs": [ { "contents": [ "Choose a scheme $V$, a point $v \\in V$, and an \\'etale morphism $V \\to Y$", "mapping $v$ to $y$. Choose a scheme $U$, a point $u \\in U$, and an", "\\'etale morphism $U \\to V \\times_Y X$ mapping $v$ to a point lying over", "$v$ and $x$. This is possible because there is a $t \\in |V \\times_Y X|$", "mapping to $(v, y)$ by Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-points-cartesian}.", "By definition we see that the dimension of $\\mathcal{O}_{U_v, u}$ is zero.", "Hence $u$ is a generic point of the fiber $U_v$.", "By our definition of weakly associated points", "the problem is reduced to the morphism of schemes $U \\to V$.", "This case is treated in", "Divisors, Lemma \\ref{divisors-lemma-weakly-ass-pullback}." ], "refs": [ "spaces-properties-lemma-points-cartesian", "divisors-lemma-weakly-ass-pullback" ], "ref_ids": [ 11819, 7886 ] } ], "ref_ids": [ 5009 ] }, { "id": 12917, "type": "theorem", "label": "spaces-divisors-lemma-weakly-ass-change-fields", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-weakly-ass-change-fields", "contents": [ "Let $K/k$ be a field extension. Let $X$ be an algebraic space over $k$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $y \\in X_K$ with image $x \\in X$. If $y$ is a weakly", "associated point of the pullback $\\mathcal{F}_K$, then $x$", "is a weakly associated point of $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "This is the translation of", "Divisors, Lemma \\ref{divisors-lemma-weakly-ass-change-fields}", "into the language of algebraic spaces. We omit the details of the", "translation." ], "refs": [ "divisors-lemma-weakly-ass-change-fields" ], "ref_ids": [ 7887 ] } ], "ref_ids": [] }, { "id": 12918, "type": "theorem", "label": "spaces-divisors-lemma-finite-flat-weak-assassin-up-down", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-finite-flat-weak-assassin-up-down", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a finite flat morphism of algebraic spaces.", "Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.", "Let $x \\in |X|$ be a point with image $y \\in |Y|$. Then", "$$", "x \\in \\text{WeakAss}(g^*\\mathcal{G})", "\\Leftrightarrow", "y \\in \\text{WeakAss}(\\mathcal{G})", "$$" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from the case of schemes", "(More on Flatness, Lemma \\ref{flat-lemma-finite-flat-weak-assassin-up-down})", "by \\'etale localization." ], "refs": [ "flat-lemma-finite-flat-weak-assassin-up-down" ], "ref_ids": [ 5984 ] } ], "ref_ids": [] }, { "id": 12919, "type": "theorem", "label": "spaces-divisors-lemma-etale-weak-assassin-up-down", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-etale-weak-assassin-up-down", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be an \\'etale morphism of algebraic spaces.", "Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.", "Let $x \\in |X|$ be a point with image $y \\in |Y|$. Then", "$$", "x \\in \\text{WeakAss}(f^*\\mathcal{G})", "\\Leftrightarrow", "y \\in \\text{WeakAss}(\\mathcal{G})", "$$" ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definition of weakly associated points", "and in fact the corresponding lemma for the case of schemes", "(More on Flatness, Lemma \\ref{flat-lemma-etale-weak-assassin-up-down})", "is the basis for our definition." ], "refs": [ "flat-lemma-etale-weak-assassin-up-down" ], "ref_ids": [ 5985 ] } ], "ref_ids": [] }, { "id": 12920, "type": "theorem", "label": "spaces-divisors-lemma-locally-noetherian-fibre", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-locally-noetherian-fibre", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $y \\in |Y|$. The following are equivalent", "\\begin{enumerate}", "\\item for some scheme $V$, point $v \\in V$, and \\'etale morphism $V \\to Y$", "mapping $v$ to $y$, the algebraic space $X_v$ is locally Noetherian,", "\\item for every scheme $V$, point $v \\in V$, and \\'etale morphism $V \\to Y$", "mapping $v$ to $y$, the algebraic space $X_v$ is locally Noetherian, and", "\\item there exists a field $k$ and a morphism $\\Spec(k) \\to Y$ representing", "$y$ such that $X_k$ is locally Noetherian.", "\\end{enumerate}", "If there exists a field $k_0$ and a monomorphism $\\Spec(k_0) \\to Y$", "representing $y$, then these are also equivalent to", "\\begin{enumerate}", "\\item[(4)] the algebraic space $X_{k_0}$ is locally Noetherian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $X_v = v \\times_Y X = \\Spec(\\kappa(v)) \\times_Y X$.", "Hence the implications (2) $\\Rightarrow$ (1) $\\Rightarrow$ (3) are clear.", "Assume that $\\Spec(k) \\to Y$ is a morphism from the spectrum of a field", "such that $X_k$ is locally Noetherian. Let $V \\to Y$ be an \\'etale morphism", "from a scheme $V$ and let $v \\in V$ a point mapping to $y$.", "Then the scheme $v \\times_Y \\Spec(k)$ is nonempty. Choose a", "point $w \\in v \\times_Y \\Spec(k)$. Consider the morphisms", "$$", "X_v \\longleftarrow X_w \\longrightarrow X_k", "$$", "Since $V \\to Y$ is \\'etale and since $w$ may be viewed as a point of", "$V \\times_Y \\Spec(k)$, we see that $\\kappa(w) \\supset k$", "is a finite separable extension of fields", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}).", "Thus $X_w \\to X_k$ is a finite \\'etale morphism as a base change of", "$w \\to \\Spec(k)$. Hence $X_w$ is locally Noetherian", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}).", "The morphism $X_w \\to X_v$ is a surjective, affine, flat morphism", "as a base change of the surjective, affine, flat morphism $w \\to v$.", "Then the fact that $X_w$ is locally Noetherian implies that", "$X_v$ is locally Noetherian. This can be seen by picking a", "surjective \\'etale morphism $U \\to X$ and then using that", "$U_w \\to U_v$ is surjective, affine, and flat. Working", "affine locally on the scheme $U_v$ we conclude", "that $U_w$ is locally Noetherian by", "Algebra, Lemma \\ref{algebra-lemma-descent-Noetherian}.", "\\medskip\\noindent", "Finally, it suffices to prove that (3) implies (4)", "in case we have a monomorphism $\\Spec(k_0) \\to Y$ in the class of $y$.", "Then $\\Spec(k) \\to Y$ factors as $\\Spec(k) \\to \\Spec(k_0) \\to Y$.", "The argument given above then shows that $X_k$ being", "locally Noetherian impies that $X_{k_0}$ is locally Noetherian." ], "refs": [ "morphisms-lemma-etale-over-field", "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "algebra-lemma-descent-Noetherian" ], "ref_ids": [ 5364, 4817, 1370 ] } ], "ref_ids": [] }, { "id": 12921, "type": "theorem", "label": "spaces-divisors-lemma-locally-finite-type-locally-Noetherian-fibres", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-locally-finite-type-locally-Noetherian-fibres", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. If $f$ is locally of finite type, then ", "the fibres of $f$ are locally Noetherian." ], "refs": [], "proofs": [ { "contents": [ "This follows from Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}", "and the fact that the spectrum of a field is Noetherian." ], "refs": [ "spaces-morphisms-lemma-locally-finite-type-locally-noetherian" ], "ref_ids": [ 4817 ] } ], "ref_ids": [] }, { "id": 12922, "type": "theorem", "label": "spaces-divisors-lemma-relative-assassin", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relative-assassin", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $x \\in |X|$ and $y = f(x) \\in |Y|$.", "Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. Consider commutative diagrams", "$$", "\\xymatrix{", "X \\ar[d] & X \\times_Y V \\ar[d] \\ar[l] & X_v \\ar[d] \\ar[l] \\\\", "Y & V \\ar[l] & v \\ar[l]", "}", "\\quad", "\\xymatrix{", "X \\ar[d] & U \\ar[d] \\ar[l] & U_v \\ar[d] \\ar[l] \\\\", "Y & V \\ar[l] & v \\ar[l]", "}", "\\quad", "\\xymatrix{", "x \\ar@{|->}[d] &", "x' \\ar@{|->}[d] \\ar@{|->}[l] &", "u \\ar@{|->}[ld] \\ar@{|->}[l] \\\\", "y &", "v \\ar@{|->}[l]", "}", "$$", "where $V$ and $U$ are schemes, $V \\to Y$ and $U \\to X \\times_Y V$", "are \\'etale, $v \\in V$, $x' \\in |X_v|$, $u \\in U$ are points", "related as in the last diagram.", "Denote $\\mathcal{F}|_{X_v}$ and $\\mathcal{F}|_{U_v}$", "the pullbacks of $\\mathcal{F}$.", "The following are equivalent", "\\begin{enumerate}", "\\item for some $V, v, x'$ as above $x'$ is a weakly associated", "point of $\\mathcal{F}|_{X_v}$,", "\\item for every $V \\to Y, v, x'$ as above $x'$ is a weakly associated", "point of $\\mathcal{F}|_{X_v}$,", "\\item for some $U, V, u, v$ as above $u$ is a weakly associated", "point of $\\mathcal{F}|_{U_v}$,", "\\item for every $U, V, u, v$ as above $u$ is a weakly associated", "point of $\\mathcal{F}|_{U_v}$,", "\\item for some field $k$ and morphism $\\Spec(k) \\to Y$ representing $y$", "and some $t \\in |X_k|$ mapping to $x$, the point $t$ is a weakly", "associated point of $\\mathcal{F}|_{X_k}$.", "\\end{enumerate}", "If there exists a field $k_0$ and a monomorphism $\\Spec(k_0) \\to Y$", "representing $y$, then these are also equivalent to", "\\begin{enumerate}", "\\item[(6)] $x_0$ is a weakly associated point of $\\mathcal{F}|_{X_{k_0}}$", "where $x_0 \\in |X_{k_0}|$ is the unique point mapping to $x$.", "\\end{enumerate}", "If the fibre of $f$ over $y$ is locally Noetherian, then in", "conditions (1), (2), (3), (4), and (6) we may replace", "``weakly associated'' with ``associated''." ], "refs": [], "proofs": [ { "contents": [ "Observe that given $V, v, x'$ as in the lemma we can find", "$U \\to X \\times_Y V$ and $u \\in U$ mapping to $x'$", "and then the morphism $U_v \\to X_v$ is \\'etale.", "Thus it is clear that (1) and (3) are equivalent", "as well as (2) and (4). Each of these implies (5).", "We will show that (5) implies (2).", "Suppose given $V, v, x'$ as well as $\\Spec(k) \\to X$ and $t \\in |X_k|$", "such that the point $t$ is a weakly", "associated point of $\\mathcal{F}|_{X_k}$.", "We can choose a point $w \\in v \\times_Y \\Spec(k)$.", "Then we obtain the morphisms", "$$", "X_v \\longleftarrow X_w \\longrightarrow X_k", "$$", "Since $V \\to Y$ is \\'etale and since $w$ may be viewed as a point of", "$V \\times_Y \\Spec(k)$, we see that $\\kappa(w) \\supset k$", "is a finite separable extension of fields", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}).", "Thus $X_w \\to X_k$ is a finite \\'etale morphism as a base change of", "$w \\to \\Spec(k)$. Thus any point $x''$ of $X_w$ lying over $t$", "is a weakly associated point of $\\mathcal{F}|_{X_w}$ by", "Lemma \\ref{lemma-etale-weak-assassin-up-down}.", "We may pick $x''$ mapping to $x'$", "(Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}).", "Then Lemma \\ref{lemma-weakly-ass-change-fields}", "implies that $x'$ is a weakly associated", "point of $\\mathcal{F}|_{X_v}$.", "\\medskip\\noindent", "To finish the proof it suffices to show that the equivalent", "conditions (1) -- (5) imply (6) if we are given", "$\\Spec(k_0) \\to Y$ as in (6). In this case the morphism", "$\\Spec(k) \\to Y$ of (5) factors uniquely as $\\Spec(k) \\to \\Spec(k_0) \\to Y$.", "Then $x_0$ is the image of $t$ under the morphism $X_k \\to X_{k_0}$.", "Hence the same lemma as above shows that (6) is true." ], "refs": [ "morphisms-lemma-etale-over-field", "spaces-divisors-lemma-etale-weak-assassin-up-down", "spaces-properties-lemma-points-cartesian", "spaces-divisors-lemma-weakly-ass-change-fields" ], "ref_ids": [ 5364, 12919, 11819, 12917 ] } ], "ref_ids": [] }, { "id": 12923, "type": "theorem", "label": "spaces-divisors-lemma-bourbaki", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-bourbaki", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is flat over $Y$,", "\\item $X$ and $Y$ are locally Noetherian, and", "\\item the fibres of $f$ are locally Noetherian.", "\\end{enumerate}", "Then", "$$", "\\text{Ass}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}) =", "\\{x \\in \\text{Ass}_{X/Y}(\\mathcal{F})\\text{ such that }", "f(x) \\in \\text{Ass}_Y(\\mathcal{G}) \\}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Via \\'etale localization, this is an immediate consequence of the result", "for schemes, see", "Divisors, Lemma \\ref{divisors-lemma-bourbaki}.", "The result for schemes is more general only because", "we haven't defined associated points for", "non-Noetherian algebraic spaces (hence we need to assume $X$", "and the fibres of $X \\to Y$ are locally Noetherian to even", "be able to formulate this result)." ], "refs": [ "divisors-lemma-bourbaki" ], "ref_ids": [ 7865 ] } ], "ref_ids": [] }, { "id": 12924, "type": "theorem", "label": "spaces-divisors-lemma-base-change-relative-assassin", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-base-change-relative-assassin", "contents": [ "Let $S$ be a scheme. Let", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "be a cartesian diagram of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module", "and set $\\mathcal{F}' = (g')^*\\mathcal{F}$.", "If $f$ is locally of finite type, then", "\\begin{enumerate}", "\\item $x' \\in \\text{Ass}_{X'/Y'}(\\mathcal{F}')", "\\Rightarrow g'(x') \\in \\text{Ass}_{X/Y}(\\mathcal{F})$", "\\item if $x \\in \\text{Ass}_{X/Y}(\\mathcal{F})$, then given", "$y' \\in |Y'|$ with $f(x) = g(y')$, there exists an", "$x' \\in \\text{Ass}_{X'/Y'}(\\mathcal{F}')$", "with $g'(x') = x$ and $f'(x') = y'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from the case of schemes by \\'etale localization.", "We write out the details completely. Choose a scheme", "$V$ and a surjective \\'etale morphism $V \\to Y$.", "Choose a scheme $U$ and a surjective", "\\'etale morphism $U \\to V \\times_Y X$. Choose a scheme $V'$", "and a surjective \\'etale morphism $V' \\to V \\times_Y Y'$.", "Then $U' = V' \\times_V U$ is a scheme and the morphism", "$U' \\to X'$ is surjective and \\'etale.", "\\medskip\\noindent", "Proof of (1). Choose $u' \\in U'$ mapping to $x'$.", "Denote $v' \\in V'$ the image of $u'$.", "Then $x' \\in \\text{Ass}_{X'/Y'}(\\mathcal{F}')$ is", "equivalent to $u' \\in \\text{Ass}(\\mathcal{F}|_{U'_{v'}})$", "by definition (writing $\\text{Ass}$ instead of $\\text{WeakAss}$", "makes sense as $U'_{v'}$ is locally Noetherian).", "Applying Divisors, Lemma \\ref{divisors-lemma-base-change-relative-assassin}", "we see that the image $u \\in U$ of $u'$ is in", "$\\text{Ass}(\\mathcal{F}|_{U_v})$ where $v \\in V$ is the image of $u$.", "This in turn means $g'(x') \\in \\text{Ass}_{X/Y}(\\mathcal{F})$.", "\\medskip\\noindent", "Proof of (2). Choose $u \\in U$ mapping to $x$.", "Denote $v \\in V$ the image of $u$.", "Then $x \\in \\text{Ass}_{X/Y}(\\mathcal{F})$ is", "equivalent to $u \\in \\text{Ass}(\\mathcal{F}|_{U_v})$", "by definition. Choose a point $v' \\in V'$ mapping", "to $y' \\in |Y'|$ and to $v \\in V$ (possible by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}).", "Let $t \\in \\Spec(\\kappa(v') \\otimes_{\\kappa(v)} \\kappa(u))$", "be a generic point of an irreducible component.", "Let $u' \\in U'$ be the image of $t$.", "Applying Divisors, Lemma \\ref{divisors-lemma-base-change-relative-assassin}", "we see that $u' \\in \\text{Ass}(\\mathcal{F}'|_{U'_{v'}})$.", "This in turn means $x' \\in \\text{Ass}_{X'/Y'}(\\mathcal{F}')$", "where $x' \\in |X'|$ is the image of $u'$." ], "refs": [ "divisors-lemma-base-change-relative-assassin", "spaces-properties-lemma-points-cartesian", "divisors-lemma-base-change-relative-assassin" ], "ref_ids": [ 7890, 11819, 7890 ] } ], "ref_ids": [] }, { "id": 12925, "type": "theorem", "label": "spaces-divisors-lemma-base-change-relative-assassin-quasi-finite", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-base-change-relative-assassin-quasi-finite", "contents": [ "With notation and assumptions as in", "Lemma \\ref{lemma-base-change-relative-assassin}.", "Assume $g$ is locally quasi-finite, or more generally that", "for every $y' \\in |Y'|$ the transcendence degree of $y'/g(y')$ is $0$.", "Then $\\text{Ass}_{X'/Y'}(\\mathcal{F}')$ is the inverse image of", "$\\text{Ass}_{X/Y}(\\mathcal{F})$." ], "refs": [ "spaces-divisors-lemma-base-change-relative-assassin" ], "proofs": [ { "contents": [ "The transcendence degree of a point over its image is defined in", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-dimension-fibre}.", "Let $x' \\in |X'|$ with image $x \\in |X|$.", "Choose a scheme $V$ and a surjective \\'etale morphism $V \\to Y$.", "Choose a scheme $U$ and a surjective", "\\'etale morphism $U \\to V \\times_Y X$. Choose a scheme $V'$", "and a surjective \\'etale morphism $V' \\to V \\times_Y Y'$.", "Then $U' = V' \\times_V U$ is a scheme and the morphism", "$U' \\to X'$ is surjective and \\'etale.", "Choose $u \\in U$ mapping to $x$.", "Denote $v \\in V$ the image of $u$.", "Then $x \\in \\text{Ass}_{X/Y}(\\mathcal{F})$ is", "equivalent to $u \\in \\text{Ass}(\\mathcal{F}|_{U_v})$", "by definition. Choose a point $u' \\in U'$ mapping", "to $x' \\in |X'|$ and to $u \\in U$ (possible by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-points-cartesian}).", "Let $v' \\in V'$ be the image of $u'$.", "Then $x' \\in \\text{Ass}_{X'/Y'}(\\mathcal{F}')$ is", "equivalent to $u' \\in \\text{Ass}(\\mathcal{F}'|_{U'_{v'}})$", "by definition.", "Now the lemma follows from the discussion in", "Divisors, Remark \\ref{divisors-remark-base-change-relative-assassin}", "applied to $u' \\in \\Spec(\\kappa(v') \\otimes_{\\kappa(v)} \\kappa(u))$." ], "refs": [ "spaces-morphisms-definition-dimension-fibre", "spaces-properties-lemma-points-cartesian", "divisors-remark-base-change-relative-assassin" ], "ref_ids": [ 5009, 11819, 8115 ] } ], "ref_ids": [ 12924 ] }, { "id": 12926, "type": "theorem", "label": "spaces-divisors-lemma-relative-weak-assassin-finite", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relative-weak-assassin-finite", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $i : Z \\to X$ be a finite morphism.", "Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Z$-module.", "Then $\\text{WeakAss}_{X/Y}(i_*\\mathcal{G}) =", "i(\\text{WeakAss}_{Z/Y}(\\mathcal{G}))$." ], "refs": [], "proofs": [ { "contents": [ "Follows from the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-relative-weak-assassin-finite})", "by \\'etale localization. Details omitted." ], "refs": [ "divisors-lemma-relative-weak-assassin-finite" ], "ref_ids": [ 7892 ] } ], "ref_ids": [] }, { "id": 12927, "type": "theorem", "label": "spaces-divisors-lemma-relative-assassin-constructible", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relative-assassin-constructible", "contents": [ "Let $Y$ be a scheme. Let $X$ be an algebraic space of finite presentation", "over $Y$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module", "of finite presentation. Let $U \\subset X$ be an open subspace", "such that $U \\to Y$ is quasi-compact. Then the set", "$$", "E = \\{y \\in Y \\mid \\text{Ass}_{X_y}(\\mathcal{F}_y) \\subset |U_y|\\}", "$$", "is locally constructible in $Y$." ], "refs": [], "proofs": [ { "contents": [ "Note that since $Y$ is a scheme, it makes sense to take the fibres", "$X_y = \\Spec(\\kappa(y)) \\times_Y X$. (Also, by our definitions, the", "set $\\text{Ass}_{X_y}(\\mathcal{F}_y)$ is exactly the fibre of", "$\\text{Ass}_{X/Y}(\\mathcal{F}) \\to Y$ over $y$, but we won't need this.)", "The question is local on $Y$, indeed, we have to show that", "$E$ is constructible if $Y$ is affine.", "In this case $X$ is quasi-compact. Choose an affine scheme $W$", "and a surjective \\'etale morphism $\\varphi : W \\to X$.", "Then $\\text{Ass}_{X_y}(\\mathcal{F}_y)$ is the image of", "$\\text{Ass}_{W_y}(\\varphi^*\\mathcal{F}_y)$ for all $y \\in Y$.", "Hence the lemma follows from the case of schemes for", "the open $\\varphi^{-1}(U) \\subset W$ and the morphism $W \\to Y$.", "The case of schemes is", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-relative-assassin-constructible}." ], "refs": [ "more-morphisms-lemma-relative-assassin-constructible" ], "ref_ids": [ 13812 ] } ], "ref_ids": [] }, { "id": 12928, "type": "theorem", "label": "spaces-divisors-lemma-base-change-fitting-ideal", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-base-change-fitting-ideal", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_Y$-module.", "Then", "$f^{-1}\\text{Fit}_i(\\mathcal{F}) \\cdot \\mathcal{O}_X =", "\\text{Fit}_i(f^*\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Reduces to", "Divisors, Lemma \\ref{divisors-lemma-base-change-fitting-ideal}", "by \\'etale localization." ], "refs": [ "divisors-lemma-base-change-fitting-ideal" ], "ref_ids": [ 7893 ] } ], "ref_ids": [] }, { "id": 12929, "type": "theorem", "label": "spaces-divisors-lemma-fitting-ideal-of-finitely-presented", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-fitting-ideal-of-finitely-presented", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module.", "Then $\\text{Fit}_r(\\mathcal{F})$ is a quasi-coherent ideal of finite type." ], "refs": [], "proofs": [ { "contents": [ "Reduces to", "Divisors, Lemma \\ref{divisors-lemma-fitting-ideal-of-finitely-presented}", "by \\'etale localization." ], "refs": [ "divisors-lemma-fitting-ideal-of-finitely-presented" ], "ref_ids": [ 7894 ] } ], "ref_ids": [] }, { "id": 12930, "type": "theorem", "label": "spaces-divisors-lemma-on-subscheme-cut-out-by-Fit-0", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-on-subscheme-cut-out-by-Fit-0", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module.", "Let $Z_0 \\subset X$ be the closed subspace cut out by", "$\\text{Fit}_0(\\mathcal{F})$.", "Let $Z \\subset X$ be the scheme theoretic support of $\\mathcal{F}$.", "Then", "\\begin{enumerate}", "\\item $Z \\subset Z_0 \\subset X$ as closed subspaces,", "\\item $|Z| = |Z_0| = \\text{Supp}(\\mathcal{F})$ as closed subsets of $|X|$,", "\\item there exists a finite type, quasi-coherent $\\mathcal{O}_{Z_0}$-module", "$\\mathcal{G}_0$ with", "$$", "(Z_0 \\to X)_*\\mathcal{G}_0 = \\mathcal{F}.", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that formation of $Z$ commutes with \\'etale localization, see", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-scheme-theoretic-support}", "(which uses Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-scheme-theoretic-support}", "to define $Z$). Hence (1) and (2) follow from the case of schemes, see", "Divisors, Lemma \\ref{divisors-lemma-on-subscheme-cut-out-by-Fit-0}.", "To get $\\mathcal{G}_0$ as in part (3) we can use", "that we have $\\mathcal{G}$ on $Z$ as in Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-scheme-theoretic-support}", "and set $\\mathcal{G}_0 = (Z \\to Z_0)_*\\mathcal{G}$." ], "refs": [ "spaces-morphisms-definition-scheme-theoretic-support", "spaces-morphisms-lemma-scheme-theoretic-support", "divisors-lemma-on-subscheme-cut-out-by-Fit-0", "spaces-morphisms-lemma-scheme-theoretic-support" ], "ref_ids": [ 4993, 4778, 7895, 4778 ] } ], "ref_ids": [] }, { "id": 12931, "type": "theorem", "label": "spaces-divisors-lemma-fitting-ideal-generate-locally", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-fitting-ideal-generate-locally", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a finite type, quasi-coherent", "$\\mathcal{O}_X$-module. Let $x \\in |X|$. Then $\\mathcal{F}$ can be", "generated by $r$ elements in an \\'etale neighbourhood of $x$ if and only", "if $\\text{Fit}_r(\\mathcal{F})_{\\overline{x}} = \\mathcal{O}_{X, \\overline{x}}$." ], "refs": [], "proofs": [ { "contents": [ "Reduces to", "Divisors, Lemma \\ref{divisors-lemma-fitting-ideal-generate-locally}", "by \\'etale localization (as well as the description of the local", "ring in Properties of Spaces, Section", "\\ref{spaces-properties-section-stalks-structure-sheaf}", "and the fact that the strict henselization of a local ring", "is faithfully flat to see that the equality over the strict", "henselization is equivalent to the equality over the local ring)." ], "refs": [ "divisors-lemma-fitting-ideal-generate-locally" ], "ref_ids": [ 7896 ] } ], "ref_ids": [] }, { "id": 12932, "type": "theorem", "label": "spaces-divisors-lemma-fitting-ideal-finite-locally-free", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-fitting-ideal-finite-locally-free", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a finite type, quasi-coherent", "$\\mathcal{O}_X$-module. Let $r \\geq 0$. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is finite locally free of rank $r$", "\\item $\\text{Fit}_{r - 1}(\\mathcal{F}) = 0$ and", "$\\text{Fit}_r(\\mathcal{F}) = \\mathcal{O}_X$, and", "\\item $\\text{Fit}_k(\\mathcal{F}) = 0$ for $k < r$ and", "$\\text{Fit}_k(\\mathcal{F}) = \\mathcal{O}_X$ for $k \\geq r$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Reduces to", "Divisors, Lemma \\ref{divisors-lemma-fitting-ideal-finite-locally-free}", "by \\'etale localization." ], "refs": [ "divisors-lemma-fitting-ideal-finite-locally-free" ], "ref_ids": [ 7897 ] } ], "ref_ids": [] }, { "id": 12933, "type": "theorem", "label": "spaces-divisors-lemma-locally-free-rank-r-pullback", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-locally-free-rank-r-pullback", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a finite type, quasi-coherent", "$\\mathcal{O}_X$-module. The closed subspaces", "$$", "X = Z_{-1} \\supset Z_0 \\supset Z_1 \\supset Z_2 \\ldots", "$$", "defined by the Fitting ideals of $\\mathcal{F}$ have the following", "properties", "\\begin{enumerate}", "\\item The intersection $\\bigcap Z_r$ is empty.", "\\item The functor $(\\Sch/X)^{opp} \\to \\textit{Sets}$ defined by the rule", "$$", "T \\longmapsto", "\\left\\{", "\\begin{matrix}", "\\{*\\} & \\text{if }\\mathcal{F}_T\\text{ is locally generated by }", "\\leq r\\text{ sections} \\\\", "\\emptyset & \\text{otherwise}", "\\end{matrix}", "\\right.", "$$", "is representable by the open subspace $X \\setminus Z_r$.", "\\item The functor $F_r : (\\Sch/X)^{opp} \\to \\textit{Sets}$ defined by the rule", "$$", "T \\longmapsto", "\\left\\{", "\\begin{matrix}", "\\{*\\} & \\text{if }\\mathcal{F}_T\\text{ locally free rank }r\\\\", "\\emptyset & \\text{otherwise}", "\\end{matrix}", "\\right.", "$$", "is representable by the locally closed subspace $Z_{r - 1} \\setminus Z_r$", "of $X$.", "\\end{enumerate}", "If $\\mathcal{F}$ is of finite presentation, then", "$Z_r \\to X$, $X \\setminus Z_r \\to X$, and $Z_{r - 1} \\setminus Z_r \\to X$", "are of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Reduces to", "Divisors, Lemma \\ref{divisors-lemma-locally-free-rank-r-pullback}", "by \\'etale localization." ], "refs": [ "divisors-lemma-locally-free-rank-r-pullback" ], "ref_ids": [ 7898 ] } ], "ref_ids": [] }, { "id": 12934, "type": "theorem", "label": "spaces-divisors-lemma-finite-presentation-module", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-finite-presentation-module", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module", "of finite presentation. Let $X = Z_{-1} \\subset Z_0 \\subset Z_1 \\subset \\ldots$", "be as in Lemma \\ref{lemma-locally-free-rank-r-pullback}.", "Set $X_r = Z_{r - 1} \\setminus Z_r$.", "Then $X' = \\coprod_{r \\geq 0} X_r$ represents the functor", "$$", "F_{flat} : \\Sch/X \\longrightarrow \\textit{Sets},\\quad\\quad", "T \\longmapsto", "\\left\\{", "\\begin{matrix}", "\\{*\\} & \\text{if }\\mathcal{F}_T\\text{ flat over }T\\\\", "\\emptyset & \\text{otherwise}", "\\end{matrix}", "\\right.", "$$", "Moreover, $\\mathcal{F}|_{X_r}$ is locally free of rank $r$ and the", "morphisms $X_r \\to X$ and $X' \\to X$ are of finite presentation." ], "refs": [ "spaces-divisors-lemma-locally-free-rank-r-pullback" ], "proofs": [ { "contents": [ "Reduces to", "Divisors, Lemma \\ref{divisors-lemma-finite-presentation-module}", "by \\'etale localization." ], "refs": [ "divisors-lemma-finite-presentation-module" ], "ref_ids": [ 7899 ] } ], "ref_ids": [ 12933 ] }, { "id": 12935, "type": "theorem", "label": "spaces-divisors-lemma-characterize-effective-Cartier-divisor", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-characterize-effective-Cartier-divisor", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $D \\subset X$ be a closed subspace.", "The following are equivalent:", "\\begin{enumerate}", "\\item The subspace $D$ is an effective Cartier divisor on $X$.", "\\item For some scheme $U$ and surjective \\'etale morphism $U \\to X$", "the inverse image $D \\times_X U$ is an effective Cartier divisor on $U$.", "\\item For every scheme $U$ and every \\'etale morphism $U \\to X$", "the inverse image $D \\times_X U$ is an effective Cartier divisor on $U$.", "\\item For every $x \\in |D|$ there exists an \\'etale morphism", "$(U, u) \\to (X, x)$ of pointed algebraic spaces such that $U = \\Spec(A)$", "and $D \\times_X U = \\Spec(A/(f))$ with $f \\in A$ not a zerodivisor.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) -- (3) follows from", "Definition \\ref{definition-effective-Cartier-divisor}", "and the references preceding it.", "Assume (1) and let $x \\in |D|$. Choose a scheme $W$ and a", "surjective \\'etale morphism", "$W \\to X$. Choose $w \\in D \\times_X W$ mapping to $x$.", "By (3) $D \\times_X W$ is an effective Cartier", "divisor on $W$. Hence we can find affine \\'etale neighbourhood $U$", "by choosing an affine open neighbourhood of $w$ in $W$ as in", "Divisors, Lemma \\ref{divisors-lemma-characterize-effective-Cartier-divisor}.", "\\medskip\\noindent", "Assume (4). Then we see that $\\mathcal{I}_D|_U$ is invertible by", "Divisors, Lemma \\ref{divisors-lemma-characterize-effective-Cartier-divisor}.", "Since we can find an \\'etale covering of $X$ by the collection of", "all such $U$ and $X \\setminus D$, we conclude that", "$\\mathcal{I}_D$ is an invertible $\\mathcal{O}_X$-module." ], "refs": [ "spaces-divisors-definition-effective-Cartier-divisor", "divisors-lemma-characterize-effective-Cartier-divisor", "divisors-lemma-characterize-effective-Cartier-divisor" ], "ref_ids": [ 13016, 7927, 7927 ] } ], "ref_ids": [] }, { "id": 12936, "type": "theorem", "label": "spaces-divisors-lemma-complement-locally-principal-closed-subscheme", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-complement-locally-principal-closed-subscheme", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $Z \\subset X$ be a locally principal closed", "subspace. Let $U = X \\setminus Z$. Then $U \\to X$ is an affine morphism." ], "refs": [], "proofs": [ { "contents": [ "The question is \\'etale local on $X$, see", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-affine-local}", "and", "Lemma \\ref{lemma-characterize-effective-Cartier-divisor}.", "Thus this follows from the case of schemes which is", "Divisors, Lemma", "\\ref{divisors-lemma-complement-locally-principal-closed-subscheme}." ], "refs": [ "spaces-morphisms-lemma-affine-local", "spaces-divisors-lemma-characterize-effective-Cartier-divisor", "divisors-lemma-complement-locally-principal-closed-subscheme" ], "ref_ids": [ 4798, 12935, 7928 ] } ], "ref_ids": [] }, { "id": 12937, "type": "theorem", "label": "spaces-divisors-lemma-complement-effective-Cartier-divisor", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-complement-effective-Cartier-divisor", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $D \\subset X$ be an effective Cartier divisor.", "Let $U = X \\setminus D$. Then $U \\to X$ is an affine morphism and $U$", "is scheme theoretically dense in $X$." ], "refs": [], "proofs": [ { "contents": [ "Affineness is Lemma \\ref{lemma-complement-locally-principal-closed-subscheme}.", "The density question is \\'etale local on $X$ by", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-scheme-theoretically-dense}.", "Thus this follows from the case of schemes which is", "Divisors, Lemma", "\\ref{divisors-lemma-complement-effective-Cartier-divisor}." ], "refs": [ "spaces-divisors-lemma-complement-locally-principal-closed-subscheme", "spaces-morphisms-definition-scheme-theoretically-dense", "divisors-lemma-complement-effective-Cartier-divisor" ], "ref_ids": [ 12936, 4995, 7929 ] } ], "ref_ids": [] }, { "id": 12938, "type": "theorem", "label": "spaces-divisors-lemma-effective-Cartier-makes-dimension-drop", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-effective-Cartier-makes-dimension-drop", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $D \\subset X$ be an effective Cartier divisor.", "Let $x \\in |D|$.", "If $\\dim_x(X) < \\infty$, then $\\dim_x(D) < \\dim_x(X)$." ], "refs": [], "proofs": [ { "contents": [ "Both the definition of an effective Cartier divisor and of the", "dimension of an algebraic space at a point", "(Properties of Spaces, Definition", "\\ref{spaces-properties-definition-dimension-at-point})", "are \\'etale local. Hence this lemma follows from the case of schemes", "which is", "Divisors, Lemma \\ref{divisors-lemma-effective-Cartier-makes-dimension-drop}." ], "refs": [ "spaces-properties-definition-dimension-at-point", "divisors-lemma-effective-Cartier-makes-dimension-drop" ], "ref_ids": [ 11929, 7930 ] } ], "ref_ids": [] }, { "id": 12939, "type": "theorem", "label": "spaces-divisors-lemma-sum-effective-Cartier-divisors", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-sum-effective-Cartier-divisors", "contents": [ "The sum of two effective Cartier divisors is an effective", "Cartier divisor." ], "refs": [], "proofs": [ { "contents": [ "Omitted. \\'Etale locally this reduces to the following simple", "algebra fact: if $f_1, f_2 \\in A$ are nonzerodivisors of a ring $A$, then", "$f_1f_2 \\in A$ is a nonzerodivisor." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12940, "type": "theorem", "label": "spaces-divisors-lemma-sum-closed-subschemes-effective-Cartier", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-sum-closed-subschemes-effective-Cartier", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $Z, Y$ be two closed subspaces of $X$", "with ideal sheaves $\\mathcal{I}$ and $\\mathcal{J}$. If $\\mathcal{I}\\mathcal{J}$", "defines an effective Cartier divisor $D \\subset X$, then $Z$ and $Y$", "are effective Cartier divisors and $D = Z + Y$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-characterize-effective-Cartier-divisor}", "this reduces to the case of schemes which is", "Divisors, Lemma \\ref{divisors-lemma-sum-closed-subschemes-effective-Cartier}." ], "refs": [ "spaces-divisors-lemma-characterize-effective-Cartier-divisor", "divisors-lemma-sum-closed-subschemes-effective-Cartier" ], "ref_ids": [ 12935, 7933 ] } ], "ref_ids": [] }, { "id": 12941, "type": "theorem", "label": "spaces-divisors-lemma-pullback-locally-principal", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-pullback-locally-principal", "contents": [ "Let $S$ be a scheme.", "Let $f : X' \\to X$ be a morphism of algebraic spaces over $S$.", "Let $Z \\subset X$ be a locally principal closed subspace.", "Then the inverse image $f^{-1}(Z)$ is a locally principal closed", "subspace of $X'$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12942, "type": "theorem", "label": "spaces-divisors-lemma-pullback-effective-Cartier-defined", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-pullback-effective-Cartier-defined", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $D \\subset Y$ be an effective Cartier divisor.", "The pullback of $D$ by $f$ is defined in each of the following cases:", "\\begin{enumerate}", "\\item $f(x) \\not \\in |D|$ for any weakly associated point $x$ of $X$,", "\\item $f$ is flat, and", "\\item add more here as needed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Working \\'etale locally this lemma reduces to the case of schemes, see", "Divisors, Lemma \\ref{divisors-lemma-pullback-effective-Cartier-defined}." ], "refs": [ "divisors-lemma-pullback-effective-Cartier-defined" ], "ref_ids": [ 7936 ] } ], "ref_ids": [] }, { "id": 12943, "type": "theorem", "label": "spaces-divisors-lemma-pullback-effective-Cartier-divisors-additive", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-pullback-effective-Cartier-divisors-additive", "contents": [ "Let $S$ be a scheme.", "Let $f : X' \\to X$ be a morphism of algebraic spaces over $S$.", "Let $D_1$, $D_2$ be effective Cartier divisors on $X$.", "If the pullbacks of $D_1$ and $D_2$ are defined then the", "pullback of $D = D_1 + D_2$ is defined and", "$f^*D = f^*D_1 + f^*D_2$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12944, "type": "theorem", "label": "spaces-divisors-lemma-conormal-effective-Cartier-divisor", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-conormal-effective-Cartier-divisor", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $D \\subset X$ be an effective Cartier divisor.", "Then for the conormal sheaf we have $\\mathcal{C}_{D/X} = \\mathcal{I}_D|D =", "\\mathcal{O}_X(D)^{\\otimes -1}|_D$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12945, "type": "theorem", "label": "spaces-divisors-lemma-invertible-sheaf-sum-effective-Cartier-divisors", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-invertible-sheaf-sum-effective-Cartier-divisors", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $D_1$, $D_2$ be effective Cartier divisors on $X$.", "Let $D = D_1 + D_2$.", "Then there is a unique isomorphism", "$$", "\\mathcal{O}_X(D_1) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(D_2)", "\\longrightarrow", "\\mathcal{O}_X(D)", "$$", "which maps $1_{D_1} \\otimes 1_{D_2}$ to $1_D$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12946, "type": "theorem", "label": "spaces-divisors-lemma-regular-section-structure-sheaf", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-regular-section-structure-sheaf", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $f \\in \\Gamma(X, \\mathcal{O}_X)$.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is a regular section, and", "\\item for any $x \\in X$ the image $f \\in \\mathcal{O}_{X, \\overline{x}}$", "is not a zerodivisor.", "\\item for any affine $U = \\Spec(A)$ \\'etale over $X$", "the restriction $f|_U$ is a nonzerodivisor of $A$, and", "\\item there exists a scheme $U$ and a surjective \\'etale morphism", "$U \\to X$ such that $f|_U$ is a regular section of $\\mathcal{O}_U$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12947, "type": "theorem", "label": "spaces-divisors-lemma-zero-scheme", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-zero-scheme", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{L})$.", "\\begin{enumerate}", "\\item Consider closed immersions $i : Z \\to X$ such that", "$i^*s \\in \\Gamma(Z, i^*\\mathcal{L}))$ is zero", "ordered by inclusion. The zero scheme $Z(s)$ is the", "maximal element of this ordered set.", "\\item For any morphism of algebraic spaces $f : Y \\to X$ over $S$", "we have $f^*s = 0$ in $\\Gamma(Y, f^*\\mathcal{L})$ if and only if", "$f$ factors through $Z(s)$.", "\\item The zero scheme $Z(s)$ is a locally principal closed subspace of $X$.", "\\item The zero scheme $Z(s)$ is an effective Cartier divisor on $X$", "if and only if $s$ is a regular section of $\\mathcal{L}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12948, "type": "theorem", "label": "spaces-divisors-lemma-characterize-OD", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-characterize-OD", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item If $D \\subset X$ is an effective Cartier divisor, then", "the canonical section $1_D$ of $\\mathcal{O}_X(D)$ is regular.", "\\item Conversely, if $s$ is a regular section of the invertible", "sheaf $\\mathcal{L}$, then there exists a unique effective", "Cartier divisor $D = Z(s) \\subset X$ and a unique isomorphism", "$\\mathcal{O}_X(D) \\to \\mathcal{L}$ which maps $1_D$ to $s$.", "\\end{enumerate}", "The constructions", "$D \\mapsto (\\mathcal{O}_X(D), 1_D)$ and $(\\mathcal{L}, s) \\mapsto Z(s)$", "give mutually inverse maps", "$$", "\\left\\{", "\\begin{matrix}", "\\text{effective Cartier divisors on }X", "\\end{matrix}", "\\right\\}", "\\leftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{pairs }(\\mathcal{L}, s)\\text{ consisting of an invertible}\\\\", "\\mathcal{O}_X\\text{-module and a regular global section}", "\\end{matrix}", "\\right\\}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12949, "type": "theorem", "label": "spaces-divisors-lemma-effective-Cartier-divisor-Sk", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-effective-Cartier-divisor-Sk", "contents": [ "Let $S$ be a scheme and let $X$ be a locally Noetherian algebraic space", "over $S$. Let $D \\subset X$ be an effective Cartier divisor. If $X$ is", "$(S_k)$, then $D$ is $(S_{k - 1})$." ], "refs": [], "proofs": [ { "contents": [ "By our definition of the property $(S_k)$ for algebraic spaces", "(Properties of Spaces, Section", "\\ref{spaces-properties-section-types-properties})", "and", "Lemma \\ref{lemma-characterize-effective-Cartier-divisor}", "this follows from the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-effective-Cartier-divisor-Sk})." ], "refs": [ "spaces-divisors-lemma-characterize-effective-Cartier-divisor", "divisors-lemma-effective-Cartier-divisor-Sk" ], "ref_ids": [ 12935, 7949 ] } ], "ref_ids": [] }, { "id": 12950, "type": "theorem", "label": "spaces-divisors-lemma-normal-effective-Cartier-divisor-S1", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-normal-effective-Cartier-divisor-S1", "contents": [ "Let $S$ be a scheme and let $X$ be a locally Noetherian normal", "algebraic space over $S$. Let $D \\subset X$ be an", "effective Cartier divisor. Then $D$ is $(S_1)$." ], "refs": [], "proofs": [ { "contents": [ "By our definition of normality for algebraic spaces", "(Properties of Spaces, Section", "\\ref{spaces-properties-section-types-properties})", "and", "Lemma \\ref{lemma-characterize-effective-Cartier-divisor}", "this follows from the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-normal-effective-Cartier-divisor-S1})." ], "refs": [ "spaces-divisors-lemma-characterize-effective-Cartier-divisor", "divisors-lemma-normal-effective-Cartier-divisor-S1" ], "ref_ids": [ 12935, 7950 ] } ], "ref_ids": [] }, { "id": 12951, "type": "theorem", "label": "spaces-divisors-lemma-complement-open-affine-effective-cartier-divisor", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-complement-open-affine-effective-cartier-divisor", "contents": [ "Let $S$ be a scheme. Let $X$ be a regular Noetherian separated algebraic space", "over $S$. Let $U \\subset X$ be a dense affine open. Then there exists an", "effective Cartier divisor $D \\subset X$ with $U = X \\setminus D$." ], "refs": [], "proofs": [ { "contents": [ "We claim that the reduced induced algebraic space structure $D$", "on $X \\setminus U$ (Properties of Spaces, Definition", "\\ref{spaces-properties-definition-reduced-induced-space})", "is the desired effective Cartier divisor. The construction", "of $D$ commutes with \\'etale localization, see proof of", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-reduced-closed-subspace}.", "Let $X' \\to X$ be a surjective \\'etale morphism with $X'$ affine.", "Since $X$ is separated, we see that $U' = X' \\times_X U$ is", "affine. Since $|X'| \\to |X|$ is open, we see that $U'$", "is dense in $X'$. Since $D' = X' \\times_X D$ is the reduced induced", "scheme structure on $X' \\setminus U'$, we conclude that", "$D'$ is an effective Cartier divisor by", "Divisors, Lemma", "\\ref{divisors-lemma-complement-open-affine-effective-cartier-divisor}", "and its proof. This is what we had to show." ], "refs": [ "spaces-properties-definition-reduced-induced-space", "spaces-properties-lemma-reduced-closed-subspace", "divisors-lemma-complement-open-affine-effective-cartier-divisor" ], "ref_ids": [ 11932, 11846, 7961 ] } ], "ref_ids": [] }, { "id": 12952, "type": "theorem", "label": "spaces-divisors-lemma-Noetherian-regular-separated-pic-effective-Cartier", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-Noetherian-regular-separated-pic-effective-Cartier", "contents": [ "Let $S$ be a scheme. Let $X$ be a regular Noetherian separated algebraic space", "over $S$. Then every invertible $\\mathcal{O}_X$-module is isomorphic to", "$$", "\\mathcal{O}_X(D - D') =", "\\mathcal{O}_X(D) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(D')^{\\otimes -1}", "$$", "for some effective Cartier divisors $D, D'$ in $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Choose a dense affine open $U \\subset X$ such that", "$\\mathcal{L}|_U$ is trivial. This is possible because", "$X$ has a dense open subspace which is a scheme, see", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}.", "Denote $s : \\mathcal{O}_U \\to \\mathcal{L}|_U$ the trivialization.", "The complement of $U$ is an effective Cartier divisor", "$D$. We claim that for some $n > 0$ the map $s$ extends uniquely to a map", "$$", "s : \\mathcal{O}_X(-nD) \\longrightarrow \\mathcal{L}", "$$", "The claim implies the lemma because it shows that", "$\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(nD)$", "has a regular global section hence is isomorphic to", "$\\mathcal{O}_X(D')$ for some effective Cartier divisor $D'$", "by Lemma \\ref{lemma-characterize-OD}.", "To prove the claim we may work \\'etale locally. Thus we may assume", "$X$ is an affine Noetherian scheme. Since", "$\\mathcal{O}_X(-nD) = \\mathcal{I}^n$ where $\\mathcal{I} = \\mathcal{O}_X(-D)$", "is the ideal sheaf of $D$ in $X$, this case follows from", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-homs-over-open}." ], "refs": [ "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme", "spaces-divisors-lemma-characterize-OD", "coherent-lemma-homs-over-open" ], "ref_ids": [ 11917, 12948, 3322 ] } ], "ref_ids": [] }, { "id": 12953, "type": "theorem", "label": "spaces-divisors-lemma-smooth-over-valuation-ring-effective-Cartier", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-smooth-over-valuation-ring-effective-Cartier", "contents": [ "Let $R$ be a valuation ring with fraction field $K$.", "Let $X$ be an algebraic space over $R$ such that $X \\to \\Spec(R)$", "is smooth. For every effective Cartier divisor $D \\subset X_K$", "there exists an effective Cartier divisor $D' \\subset X$", "with $D'_K = D$." ], "refs": [], "proofs": [ { "contents": [ "Let $D' \\subset X$ be the scheme theoretic image of $D \\to X_K \\to X$.", "Since this morphism is quasi-compact, formation of $D'$", "commutes with flat base change, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image}.", "In particular we find that $D'_K = D$. Hence,", "we may assume $X$ is affine. Say $X = \\Spec(A)$.", "Then $X_K = \\Spec(A \\otimes_R K)$ and $D$ corresponds to", "an ideal $I \\subset A \\otimes_R K$. We have to show that", "$J = I \\cap A$ cuts out an effective Cartier divisor in $X$.", "First, observe that $A/J$ is flat over $R$ (as a torsion", "free $R$-module, see More on Algebra, Lemma", "\\ref{more-algebra-lemma-valuation-ring-torsion-free-flat}),", "hence $J$ is finitely generated by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-flat-finite-type-valuation-ring-finite-presentation}", "and", "Algebra, Lemma \\ref{algebra-lemma-extension}.", "Thus it suffices to show that $J_\\mathfrak q \\subset A_\\mathfrak q$", "is generated by a single element for each prime $\\mathfrak q \\subset A$.", "Let $\\mathfrak p = R \\cap \\mathfrak q$. Then", "$R_\\mathfrak p$ is a valuation ring", "(Algebra, Lemma \\ref{algebra-lemma-make-valuation-rings}).", "Observe further that $A_\\mathfrak q/\\mathfrak p A_\\mathfrak q$", "is a regular ring by Algebra, Lemma", "\\ref{algebra-lemma-characterize-smooth-over-field}.", "Thus we may apply More on Algebra, Lemma", "\\ref{more-algebra-lemma-picard-group-generic-fibre-regular}", "to see that $I(A_\\mathfrak q \\otimes_R K)$ is generated by", "a single element $f \\in A_\\mathfrak p \\otimes_R K$.", "After clearing denominators we may assume $f \\in A_\\mathfrak q$.", "Let $\\mathfrak c \\subset R_\\mathfrak p$ be the content ideal of $f$", "(see More on Algebra, Definition \\ref{more-algebra-definition-content-ideal}", "and More on Flatness, Lemma", "\\ref{flat-lemma-flat-finite-type-local-valuation-ring-has-content}).", "Since $R_\\mathfrak p$ is a valuation ring and", "since $\\mathfrak c$ is finitely generated", "(More on Algebra, Lemma \\ref{more-algebra-lemma-content-finitely-generated})", "we see $\\mathfrak c = (\\pi)$ for some $\\pi \\in R_\\mathfrak p$", "(Algebra, Lemma \\ref{algebra-lemma-characterize-valuation-ring}).", "After relacing $f$ by $\\pi^{-1}f$ we see that $f \\in A_\\mathfrak q$", "and $f \\not \\in \\mathfrak pA_\\mathfrak q$.", "Claim: $I_\\mathfrak q = (f)$ which finishes the proof.", "To see the claim, observe that $f \\in I_\\mathfrak q$.", "Hence we have a surjection $A_\\mathfrak q/(f) \\to A_\\mathfrak q/I_\\mathfrak q$", "which is an isomorphism after tensoring over $R$ with $K$.", "Thus we are done if", "$A_\\mathfrak q/(f)$ is $R_\\mathfrak p$-flat.", "This follows from", "Algebra, Lemma \\ref{algebra-lemma-grothendieck-general}", "and our choice of $f$." ], "refs": [ "spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image", "more-algebra-lemma-valuation-ring-torsion-free-flat", "more-algebra-lemma-flat-finite-type-valuation-ring-finite-presentation", "algebra-lemma-extension", "algebra-lemma-make-valuation-rings", "algebra-lemma-characterize-smooth-over-field", "more-algebra-lemma-picard-group-generic-fibre-regular", "more-algebra-definition-content-ideal", "flat-lemma-flat-finite-type-local-valuation-ring-has-content", "more-algebra-lemma-content-finitely-generated", "algebra-lemma-characterize-valuation-ring", "algebra-lemma-grothendieck-general" ], "ref_ids": [ 4861, 9920, 9947, 330, 614, 1223, 10545, 10603, 6075, 9940, 620, 1111 ] } ], "ref_ids": [] }, { "id": 12954, "type": "theorem", "label": "spaces-divisors-lemma-relative-Cartier", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relative-Cartier", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "algebraic spaces over $S$. Let $D \\subset X$ be a closed subspace.", "Assume", "\\begin{enumerate}", "\\item $D$ is an effective Cartier divisor, and", "\\item $D \\to Y$ is a flat morphism.", "\\end{enumerate}", "Then for every morphism of schemes $g : Y' \\to Y$ the pullback", "$(g')^{-1}D$ is an effective Cartier divisor on $X' = Y' \\times_Y X$", "where $g' : X' \\to X$ is the projection." ], "refs": [], "proofs": [ { "contents": [ "Using Lemma \\ref{lemma-characterize-effective-Cartier-divisor}", "the property of being an effective Cartier divisor is \\'etale local.", "Thus this lemmma immediately reduces to the case of schemes", "which is Divisors, Lemma \\ref{divisors-lemma-relative-Cartier}." ], "refs": [ "spaces-divisors-lemma-characterize-effective-Cartier-divisor", "divisors-lemma-relative-Cartier" ], "ref_ids": [ 12935, 7972 ] } ], "ref_ids": [] }, { "id": 12955, "type": "theorem", "label": "spaces-divisors-lemma-describe-S", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-describe-S", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "For $U$ affine and \\'etale over $X$ the set", "$\\mathcal{S}_X(U)$ is the set of nonzerodivisors in", "$\\mathcal{O}_X(U)$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-regular-section-structure-sheaf}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12956, "type": "theorem", "label": "spaces-divisors-lemma-meromorphic-quasi-coherent", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-meromorphic-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume", "\\begin{enumerate}", "\\item[(a)] every weakly associated point of $X$", "is a point of codimension $0$, and", "\\item[(b)] $X$ satisfies the equivalent conditions", "of Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-prepare-normalization}.", "\\end{enumerate}", "Then", "\\begin{enumerate}", "\\item $\\mathcal{K}_X$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras,", "\\item for $U \\in X_\\etale$ affine $\\mathcal{K}_X(U)$", "is the total ring of fractions of $\\mathcal{O}_X(U)$,", "\\item for a geometric point $\\overline{x}$ the set", "$\\mathcal{S}_{\\overline{x}}$", "the set of nonzerodivisors of $\\mathcal{O}_{X, \\overline{x}}$, and", "\\item for a geometric point $\\overline{x}$ the ring", "$\\mathcal{K}_{X, \\overline{x}}$ is the total ring of fractions of", "$\\mathcal{O}_{X, \\overline{x}}$.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-lemma-prepare-normalization" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-regular-section-structure-sheaf}", "we see that $U \\in X_\\etale$ affine", "$\\mathcal{S}_X(U) \\subset \\mathcal{O}_X(U)$", "is the set of nonzerodivisors in $\\mathcal{O}_X(U)$.", "Thus the presheaf $\\mathcal{S}^{-1}\\mathcal{O}_X$ is equal to", "$$", "U \\longmapsto Q(\\mathcal{O}_X(U))", "$$", "on $X_{affine, \\etale}$, with notation as in", "Algebra, Example \\ref{algebra-example-localize-at-prime}.", "Observe that the codimension $0$ points of $X$ correspond", "to the generic points of $U$, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-codimension-0-points}.", "Hence if $U = \\Spec(A)$, then $A$ is a ring with finitely many minimal primes", "such that any weakly associated prime of $A$ is minimal.", "The same is true for any \\'etale extension of $A$ (because the", "spectrum of such is an affine scheme \\'etale over $X$ hence can", "play the role of $A$ in the previous sentence).", "In order to show that our presheaf is a sheaf and quasi-coherent", "it suffices to show that", "$$", "Q(A) \\otimes_A B \\longrightarrow Q(B)", "$$", "is an isomorphism when $A \\to B$ is an \\'etale ring map, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-characterize-quasi-coherent-small-etale}.", "(To define the displayed arrow, observe that since $A \\to B$", "is flat it maps nonzerodivisors to nonzerodivisors.)", "By Algebra, Lemmas", "\\ref{algebra-lemma-total-ring-fractions-no-embedded-points} and", "\\ref{algebra-lemma-weakly-ass-zero-divisors}.", "we have", "$$", "Q(A) = \\prod\\nolimits_{\\mathfrak p \\subset A\\text{ minimal}} A_\\mathfrak p", "\\quad\\text{and}\\quad", "Q(B) = \\prod\\nolimits_{\\mathfrak q \\subset B\\text{ minimal}} B_\\mathfrak q", "$$", "Since $A \\to B$ is \\'etale, the minimal primes of $B$ are exactly the", "primes of $B$ lying over the minimal primes", "of $A$ (for example by More on Algebra, Lemma", "\\ref{more-algebra-lemma-dimension-etale-extension}).", "By Algebra, Lemmas \\ref{algebra-lemma-local-dimension-zero-henselian},", "\\ref{algebra-lemma-characterize-henselian} (13), and", "\\ref{algebra-lemma-mop-up} we see that $A_\\mathfrak p \\otimes_A B$", "is a finite product of local rings finite \\'etale over $A_\\mathfrak p$.", "This cleary implies that $A_\\mathfrak p \\otimes_A B =", "\\prod_{\\mathfrak q\\text{ lies over }\\mathfrak p} B_\\mathfrak q$", "as desired.", "\\medskip\\noindent", "At this point we know that (1) and (2) hold.", "Proof of (3). Let $s \\in \\mathcal{O}_{X, \\overline{x}}$", "be a nonzerodivisor. Then we can find an \\'etale neighbourhood", "$(U, \\overline{u}) \\to (X, \\overline{x})$", "and $f \\in \\mathcal{O}_X(U)$ mapping to $s$.", "Let $u \\in U$ be the point determined by $\\overline{u}$.", "Since $\\mathcal{O}_{U, u} \\to \\mathcal{O}_{X, \\overline{x}}$", "is faithfully flat (as a strict henselization), we see that", "$f$ maps to a nonzerodivisor in $\\mathcal{O}_{U, u}$.", "By Divisors, Lemma \\ref{divisors-lemma-meromorphic-weakass-finite}", "after shrinking $U$ we find that $f$ is a nonzerodivisor", "and hence a section of $\\mathcal{S}_X(U)$.", "Part (4) follows from (3) by computing stalks." ], "refs": [ "spaces-properties-lemma-codimension-0-points", "spaces-properties-lemma-characterize-quasi-coherent-small-etale", "algebra-lemma-total-ring-fractions-no-embedded-points", "algebra-lemma-weakly-ass-zero-divisors", "more-algebra-lemma-dimension-etale-extension", "algebra-lemma-local-dimension-zero-henselian", "algebra-lemma-characterize-henselian", "algebra-lemma-mop-up", "divisors-lemma-meromorphic-weakass-finite" ], "ref_ids": [ 11842, 11908, 421, 725, 10052, 1283, 1276, 1278, 8010 ] } ], "ref_ids": [ 4966 ] }, { "id": 12957, "type": "theorem", "label": "spaces-divisors-lemma-meromorphic-quasi-coherent-agree", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-meromorphic-quasi-coherent-agree", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume", "\\begin{enumerate}", "\\item[(a)] every weakly associated point of $X$", "is a point of codimension $0$, and", "\\item[(b)] $X$ satisfies the equivalent conditions", "of Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-prepare-normalization}.", "\\item[(c)] $X$ is representable by a scheme $X_0$", "(awkward but temporary notation).", "\\end{enumerate}", "Then the sheaf of meromorphic functions $\\mathcal{K}_X$", "is the quasi-coherent sheaf of $\\mathcal{O}_X$-algebras", "associated to the quasi-coherent sheaf of meromorphic", "functions $\\mathcal{K}_{X_0}$." ], "refs": [ "spaces-morphisms-lemma-prepare-normalization" ], "proofs": [ { "contents": [ "For the equivalence between $\\QCoh(\\mathcal{O}_X)$ and", "$\\QCoh(\\mathcal{O}_{X_0})$, please see", "Properties of Spaces, Section \\ref{spaces-properties-section-quasi-coherent}.", "The lemma is true because $\\mathcal{K}_X$ and $\\mathcal{K}_{X_0}$", "are quasi-coherent and have the same value on corresponding affine", "opens of $X$ and $X_0$ by Lemma \\ref{lemma-meromorphic-quasi-coherent} and", "Divisors, Lemma \\ref{divisors-lemma-meromorphic-weakass-finite}." ], "refs": [ "spaces-divisors-lemma-meromorphic-quasi-coherent", "divisors-lemma-meromorphic-weakass-finite" ], "ref_ids": [ 12956, 8010 ] } ], "ref_ids": [ 4966 ] }, { "id": 12958, "type": "theorem", "label": "spaces-divisors-lemma-pullback-meromorphic-sections-defined", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-pullback-meromorphic-sections-defined", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Pullbacks of meromorphic sections are defined", "in each of the following cases", "\\begin{enumerate}", "\\item weakly associated points of $X$ are mapped", "to points of codimension $0$ on $Y$,", "\\item $f$ is flat,", "\\item add more here as needed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Working \\'etale locally, this translates into the case of schemes, see", "Divisors, Lemma \\ref{divisors-lemma-pullback-meromorphic-sections-defined}.", "To do the translation use Lemma \\ref{lemma-regular-section-structure-sheaf}", "(description of regular sections),", "Definition \\ref{definition-weakly-associated} (definition", "of weakly associated points), and", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-codimension-0-points}", "(description of codimension $0$ points)." ], "refs": [ "divisors-lemma-pullback-meromorphic-sections-defined", "spaces-divisors-definition-weakly-associated", "spaces-properties-lemma-codimension-0-points" ], "ref_ids": [ 8009, 13013, 11842 ] } ], "ref_ids": [] }, { "id": 12959, "type": "theorem", "label": "spaces-divisors-lemma-compute-meromorphic", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-compute-meromorphic", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume", "\\begin{enumerate}", "\\item[(a)] every weakly associated point of $X$", "is a point of codimension $0$, and", "\\item[(b)] $X$ satisfies the equivalent conditions", "of Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-prepare-normalization},", "\\item[(c)] every codimension $0$ point of $X$ can be represented", "by a monomorphism $\\Spec(k) \\to X$.", "\\end{enumerate}", "Let $X^0 \\subset |X|$ be the set of codimension $0$ points of $X$.", "Then we have", "$$", "\\mathcal{K}_X =", "\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta} =", "\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta}", "$$", "where $j_\\eta : \\Spec(\\mathcal{O}_{X, \\eta}) \\to X$ is the canonical map", "of Schemes, Section \\ref{schemes-section-points}; this makes sense because", "$X^0$ is contained in the schematic locus of $X$. Similarly,", "for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "we obtain the formula", "$$", "\\mathcal{K}_X(\\mathcal{F}) =", "\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta =", "\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta", "$$", "for the sheaf of meromorphic sections of $\\mathcal{F}$.", "Finally, the ring of rational functions of $X$ is the ring of meromorphic", "functions on $X$, in a formula: $R(X) = \\Gamma(X, \\mathcal{K}_X)$." ], "refs": [ "spaces-morphisms-lemma-prepare-normalization" ], "proofs": [ { "contents": [ "By Decent Spaces, Lemma \\ref{decent-spaces-lemma-get-reasonable} and", "Section \\ref{decent-spaces-section-reasonable-decent}", "we see that $X$ is decent\\footnote{Conversely, if $X$ is decent,", "then condition (c) holds automatically.}.", "Thus $X^0 \\subset |X|$ is the set of generic points of irreducible", "components", "(Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-generic-points})", "and $X^0$ is locally finite in $|X|$ by (b).", "It follows that $X^0$ is contained in every dense open subset of $|X|$.", "In particular, $X^0$ is contained in the schematic locus", "(Decent Spaces, Theorem \\ref{decent-spaces-theorem-decent-open-dense-scheme}).", "Thus the local rings $\\mathcal{O}_{X, \\eta}$ and the morphisms $j_\\eta$", "are defined.", "\\medskip\\noindent", "Observe that a locally finite direct sum of sheaves of modules", "is equal to the product. This and the fact that $X^0$ is locally", "finite in $|X|$ explains the equalities between", "direct sums and products in the statement.", "Then since $\\mathcal{K}_X(\\mathcal{F}) =", "\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{K}_X$", "we see that the second equality follows from the first.", "\\medskip\\noindent", "Let", "$j : Y = \\coprod\\nolimits_{\\eta \\in X^0} \\Spec(\\mathcal{O}_{X, \\eta}) \\to X$", "be the product of the morphisms $j_\\eta$.", "We have to show that $\\mathcal{K}_X = j_*\\mathcal{O}_Y$.", "Observe that $\\mathcal{K}_Y = \\mathcal{O}_Y$ as $Y$ is a disjoint", "union of spectra of local rings of dimension $0$: in a local", "ring of dimension zero any nonzerodivisor is a unit.", "Next, note that pullbacks of meromorphic", "functions are defined for $j$ by", "Lemma \\ref{lemma-pullback-meromorphic-sections-defined}.", "This gives a map", "$$", "\\mathcal{K}_X \\longrightarrow j_*\\mathcal{O}_Y.", "$$", "Let $U \\in X_\\etale$ be affine. By Lemma \\ref{lemma-meromorphic-quasi-coherent}", "the left hand side evaluates to total ring of fractions of $\\mathcal{O}_X(U)$.", "On the other hand, the right hand side is equal to the product of the", "local rings of $U$ at the codimension $0$ points, i.e., the generic points", "of $U$. These two rings are equal (as we already saw in the proof", "of Lemma \\ref{lemma-meromorphic-quasi-coherent}) by", "Algebra, Lemmas", "\\ref{algebra-lemma-total-ring-fractions-no-embedded-points} and", "\\ref{algebra-lemma-weakly-ass-zero-divisors}.", "Thus our map is an isomorphism.", "\\medskip\\noindent", "Finally, we have to show that $R(X) = \\Gamma(X, \\mathcal{K}_X)$.", "This follows from the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-meromorphic-weakass-finite})", "applied to the schematic locus $X' \\subset X$.", "Namely, the ring of rational functions of $X$ is by definition", "the same as the ring of rational functions on $X'$", "as it is a dense open subspace of $X$ (see above).", "Certainly, $R(X')$ agrees with the ring of rational functions", "when $X'$ is viewed as a scheme.", "On the other hand, by our description of $\\mathcal{K}_X$ above,", "and the fact, seen above, that $X^0 \\subset |X'|$", "is contained in any dense open, we see that", "$\\Gamma(X, \\mathcal{K}_X) = \\Gamma(X', \\mathcal{K}_{X'})$.", "Finally, use the compatibility recorded in", "Lemma \\ref{lemma-meromorphic-quasi-coherent-agree}." ], "refs": [ "decent-spaces-lemma-get-reasonable", "decent-spaces-lemma-decent-generic-points", "decent-spaces-theorem-decent-open-dense-scheme", "spaces-divisors-lemma-pullback-meromorphic-sections-defined", "spaces-divisors-lemma-meromorphic-quasi-coherent", "spaces-divisors-lemma-meromorphic-quasi-coherent", "algebra-lemma-total-ring-fractions-no-embedded-points", "algebra-lemma-weakly-ass-zero-divisors", "divisors-lemma-meromorphic-weakass-finite", "spaces-divisors-lemma-meromorphic-quasi-coherent-agree" ], "ref_ids": [ 9533, 9531, 9454, 12958, 12956, 12956, 421, 725, 8010, 12957 ] } ], "ref_ids": [ 4966 ] }, { "id": 12960, "type": "theorem", "label": "spaces-divisors-lemma-meromorphic-sections-pullback", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-meromorphic-sections-pullback", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume that pullbacks of meromorphic functions are defined", "for $f$ (see", "Definition \\ref{definition-pullback-meromorphic-sections}).", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_Y$-modules.", "There is a canonical pullback map", "$f^* : \\Gamma(Y, \\mathcal{K}_Y(\\mathcal{F})) \\to", "\\Gamma(X, \\mathcal{K}_X(f^*\\mathcal{F}))$", "for meromorphic sections of $\\mathcal{F}$.", "\\item Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "A regular meromorphic section $s$ of $\\mathcal{L}$ pulls back", "to a regular meromorphic section $f^*s$ of $f^*\\mathcal{L}$.", "\\end{enumerate}" ], "refs": [ "spaces-divisors-definition-pullback-meromorphic-sections" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13025 ] }, { "id": 12961, "type": "theorem", "label": "spaces-divisors-lemma-regular-meromorphic-section-exists", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-regular-meromorphic-section-exists", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$", "satisfying (a), (b), and (c) of", "Lemma \\ref{lemma-compute-meromorphic}.", "Then every invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ has", "a regular meromorphic section." ], "refs": [ "spaces-divisors-lemma-compute-meromorphic" ], "proofs": [ { "contents": [ "With notation as in Lemma \\ref{lemma-compute-meromorphic}", "the stalk $\\mathcal{L}_\\eta$ of $\\mathcal{L}$ at is defined for all", "$\\eta \\in X^0$ and it is a rank $1$ free $\\mathcal{O}_{X, \\eta}$-module.", "Pick a generator $s_\\eta \\in \\mathcal{L}_\\eta$ for all $\\eta \\in X^0$.", "It follows immediately from the description of", "$\\mathcal{K}_X$ and $\\mathcal{K}_X(\\mathcal{L})$", "in Lemma \\ref{lemma-compute-meromorphic}", "that $s = \\prod s_\\eta$ is a regular meromorphic section of $\\mathcal{L}$." ], "refs": [ "spaces-divisors-lemma-compute-meromorphic", "spaces-divisors-lemma-compute-meromorphic" ], "ref_ids": [ 12959, 12959 ] } ], "ref_ids": [ 12959 ] }, { "id": 12962, "type": "theorem", "label": "spaces-divisors-lemma-relative-proj", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relative-proj", "contents": [ "In Situation \\ref{situation-relative-proj}. The functor $F$ above is an", "algebraic space. For any morphism $g : Z \\to X$ where $Z$ is a scheme", "there is a canonical isomorphism", "$\\underline{\\text{Proj}}_Z(g^*\\mathcal{A}) = Z \\times_X F$", "compatible with further base change." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove the second assertion, see", "Spaces, Lemma \\ref{spaces-lemma-representable-over-space}.", "Let $g : Z \\to X$ be a morphism where $Z$ is a scheme.", "Let $F'$ be the functor of quadruples associated", "to the graded quasi-coherent $\\mathcal{O}_Z$-algebra $g^*\\mathcal{A}$.", "Then there is a canonical isomorphism $F' = Z \\times_X F$, sending", "a quadruple $(d, f : T \\to Z, \\mathcal{L}, \\psi)$ for $F'$", "to $(d, g \\circ f, \\mathcal{L}, \\psi)$ (details omitted, see proof of", "Constructions, Lemma \\ref{constructions-lemma-proj-base-change}).", "By Constructions, Lemmas", "\\ref{constructions-lemma-equivalent-relative},", "\\ref{constructions-lemma-relative-proj}, and", "\\ref{constructions-lemma-glueing-gives-functor-proj} and", "Definition \\ref{constructions-definition-relative-proj}", "we see that $F'$ is representable by", "$\\underline{\\text{Proj}}_Z(g^*\\mathcal{A})$." ], "refs": [ "spaces-lemma-representable-over-space", "constructions-lemma-proj-base-change", "constructions-lemma-equivalent-relative", "constructions-lemma-relative-proj", "constructions-lemma-glueing-gives-functor-proj", "constructions-definition-relative-proj" ], "ref_ids": [ 8156, 12633, 12636, 12637, 12638, 12665 ] } ], "ref_ids": [] }, { "id": 12963, "type": "theorem", "label": "spaces-divisors-lemma-twists-of-structure-sheaf", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-twists-of-structure-sheaf", "contents": [ "In Situation \\ref{situation-relative-proj}. The relative Proj comes", "equipped with a quasi-coherent sheaf of $\\mathbf{Z}$-graded algebras", "$\\bigoplus_{n \\in \\mathbf{Z}}", "\\mathcal{O}_{\\underline{\\text{Proj}}_X(\\mathcal{A})}(n)$", "and a canonical homomorphism of graded algebras", "$$", "\\psi :", "\\pi^*\\mathcal{A}", "\\longrightarrow", "\\bigoplus\\nolimits_{n \\geq 0}", "\\mathcal{O}_{\\underline{\\text{Proj}}_X(\\mathcal{A})}(n)", "$$", "whose base change to any scheme over $X$ agrees with", "Constructions, Lemma \\ref{constructions-lemma-glue-relative-proj-twists}." ], "refs": [ "constructions-lemma-glue-relative-proj-twists" ], "proofs": [ { "contents": [ "As in the discussion following Definition \\ref{definition-relative-proj}", "choose a scheme $U$ and a surjective \\'etale morphism", "$U \\to X$, set $R = U \\times_X U$ with projections $s, t : R \\to U$,", "$\\mathcal{A}_U = \\mathcal{A}|_U$, $\\mathcal{A}_R = \\mathcal{A}|_R$,", "and $\\pi : P = \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$,", "$\\pi_U : P_U = \\underline{\\text{Proj}}_U(\\mathcal{A}_U)$ and", "$\\pi_R : P_R = \\underline{\\text{Proj}}_U(\\mathcal{A}_R)$.", "By the", "Constructions, Lemma \\ref{constructions-lemma-glue-relative-proj-twists}", "we have a quasi-coherent sheaf of $\\mathbf{Z}$-graded", "$\\mathcal{O}_{P_U}$-algebras", "$\\bigoplus_{n \\in \\mathbf{Z}} \\mathcal{O}_{P_U}(n)$", "and a canonical map", "$\\psi_U : \\pi_U^*\\mathcal{A}_U \\to \\bigoplus_{n \\geq 0} \\mathcal{O}_{P_U}(n)$", "and similarly for $P_R$. By", "Constructions, Lemma \\ref{constructions-lemma-relative-proj-base-change}", "the pullback of $\\mathcal{O}_{P_U}(n)$ and $\\psi_U$ by either projection", "$P_R \\to P_U$ is equal to $\\mathcal{O}_{P_R}(n)$ and $\\psi_R$.", "By Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-quasi-coherent}", "we obtain $\\mathcal{O}_{P}(n)$ and $\\psi$.", "We omit the verification of compatibility with pullback to", "arbitrary schemes over $X$." ], "refs": [ "spaces-divisors-definition-relative-proj", "constructions-lemma-glue-relative-proj-twists", "constructions-lemma-relative-proj-base-change", "spaces-properties-proposition-quasi-coherent" ], "ref_ids": [ 13027, 12632, 12641, 11920 ] } ], "ref_ids": [ 12632 ] }, { "id": 12964, "type": "theorem", "label": "spaces-divisors-lemma-relative-proj-base-change", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relative-proj-base-change", "contents": [ "Let $S$ be a scheme. Let $g : X' \\to X$ be a morphism of algebraic spaces", "over $S$ and let $\\mathcal{A}$ be a quasi-coherent sheaf", "of graded $\\mathcal{O}_X$-algebras. Then there is a canonical isomorphism", "$$", "r :", "\\underline{\\text{Proj}}_{X'}(g^*\\mathcal{A})", "\\longrightarrow", "X' \\times_X \\underline{\\text{Proj}}_X(\\mathcal{A})", "$$", "as well as a corresponding isomorphism", "$$", "\\theta :", "r^*\\text{pr}_2^*\\left(\\bigoplus\\nolimits_{d \\in \\mathbf{Z}}", "\\mathcal{O}_{\\underline{\\text{Proj}}_X(\\mathcal{A})}(d)\\right)", "\\longrightarrow", "\\bigoplus\\nolimits_{d \\in \\mathbf{Z}}", "\\mathcal{O}_{\\underline{\\text{Proj}}_{X'}(g^*\\mathcal{A})}(d)", "$$", "of $\\mathbf{Z}$-graded", "$\\mathcal{O}_{\\underline{\\text{Proj}}_{X'}(g^*\\mathcal{A})}$-algebras." ], "refs": [], "proofs": [ { "contents": [ "Let $F$ be the functor (\\ref{equation-proj}) and let $F'$ be the", "corresponding functor defined using $g^*\\mathcal{A}$ on $X'$.", "We claim there is a canonical isomorphism $r : F' \\to X' \\times_X F$", "of functors (and of course $r$ is the isomorphism of the lemma).", "It suffices to construct the bijection", "$r : F'(T) \\to X'(T) \\times_{X(T)} F(T)$ for quasi-compact schemes $T$", "over $S$. First, if $\\xi = (d', f', \\mathcal{L}', \\psi')$ is a", "quadruple over $T$ for $F'$, then we can set", "$r(\\xi) = (f', (d', g \\circ f', \\mathcal{L}', \\psi'))$. This makes sense", "as $(g \\circ f')^*\\mathcal{A}^{(d)} = (f')^*(g^*\\mathcal{A})^{(d)}$.", "The inverse map sends the pair $(f', (d, f, \\mathcal{L}, \\psi))$", "to the quadruple $(d, f', \\mathcal{L}, \\psi)$. We omit the proof", "of the final assertion (hint: reduce to the case of schemes by \\'etale", "localization and apply Constructions, Lemma", "\\ref{constructions-lemma-relative-proj-base-change})." ], "refs": [ "constructions-lemma-relative-proj-base-change" ], "ref_ids": [ 12641 ] } ], "ref_ids": [] }, { "id": 12965, "type": "theorem", "label": "spaces-divisors-lemma-relative-proj-separated", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relative-proj-separated", "contents": [ "In Situation \\ref{situation-relative-proj} the morphism", "$\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$", "is separated." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-separated-local}", "and the construction of the relative Proj this follows from the", "case of schemes which is", "Constructions, Lemma \\ref{constructions-lemma-relative-proj-separated}." ], "refs": [ "spaces-morphisms-lemma-separated-local", "constructions-lemma-relative-proj-separated" ], "ref_ids": [ 4722, 12640 ] } ], "ref_ids": [] }, { "id": 12966, "type": "theorem", "label": "spaces-divisors-lemma-relative-proj-quasi-compact", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relative-proj-quasi-compact", "contents": [ "In Situation \\ref{situation-relative-proj}. If one of the following holds", "\\begin{enumerate}", "\\item $\\mathcal{A}$ is of finite type as a sheaf of", "$\\mathcal{A}_0$-algebras,", "\\item $\\mathcal{A}$ is generated by $\\mathcal{A}_1$ as an", "$\\mathcal{A}_0$-algebra and $\\mathcal{A}_1$ is a finite type", "$\\mathcal{A}_0$-module,", "\\item there exists a finite type quasi-coherent $\\mathcal{A}_0$-submodule", "$\\mathcal{F} \\subset \\mathcal{A}_{+}$ such that", "$\\mathcal{A}_{+}/\\mathcal{F}\\mathcal{A}$ is a locally nilpotent", "sheaf of ideals of $\\mathcal{A}/\\mathcal{F}\\mathcal{A}$,", "\\end{enumerate}", "then $\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ is quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-quasi-compact-local}", "and the construction of the relative Proj this follows from the", "case of schemes which is", "Divisors, Lemma \\ref{divisors-lemma-relative-proj-quasi-compact}." ], "refs": [ "spaces-morphisms-lemma-quasi-compact-local", "divisors-lemma-relative-proj-quasi-compact" ], "ref_ids": [ 4742, 8040 ] } ], "ref_ids": [] }, { "id": 12967, "type": "theorem", "label": "spaces-divisors-lemma-relative-proj-finite-type", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relative-proj-finite-type", "contents": [ "In Situation \\ref{situation-relative-proj}.", "If $\\mathcal{A}$ is of finite type as a sheaf of", "$\\mathcal{O}_X$-algebras, then", "$\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ is of finite type." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-type-local}", "and the construction of the relative Proj this follows from the", "case of schemes which is", "Divisors, Lemma \\ref{divisors-lemma-relative-proj-finite-type}." ], "refs": [ "spaces-morphisms-lemma-finite-type-local", "divisors-lemma-relative-proj-finite-type" ], "ref_ids": [ 4816, 8041 ] } ], "ref_ids": [] }, { "id": 12968, "type": "theorem", "label": "spaces-divisors-lemma-relative-proj-universally-closed", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relative-proj-universally-closed", "contents": [ "In Situation \\ref{situation-relative-proj}. If", "$\\mathcal{O}_X \\to \\mathcal{A}_0$", "is an integral algebra map\\footnote{In other words, the integral", "closure of $\\mathcal{O}_X$ in $\\mathcal{A}_0$, see", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-integral-closure}, equals", "$\\mathcal{A}_0$.} and $\\mathcal{A}$ is of finite type as an", "$\\mathcal{A}_0$-algebra, then", "$\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ is universally closed." ], "refs": [ "spaces-morphisms-definition-integral-closure" ], "proofs": [ { "contents": [ "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-universally-closed-local}", "and the construction of the relative Proj this follows from the", "case of schemes which is", "Divisors, Lemma \\ref{divisors-lemma-relative-proj-universally-closed}." ], "refs": [ "spaces-morphisms-lemma-universally-closed-local", "divisors-lemma-relative-proj-universally-closed" ], "ref_ids": [ 4748, 8042 ] } ], "ref_ids": [ 5025 ] }, { "id": 12969, "type": "theorem", "label": "spaces-divisors-lemma-relative-proj-proper", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relative-proj-proper", "contents": [ "In Situation \\ref{situation-relative-proj}.", "The following conditions are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{A}_0$ is a finite type $\\mathcal{O}_X$-module", "and $\\mathcal{A}$ is of finite type as an $\\mathcal{A}_0$-algebra,", "\\item $\\mathcal{A}_0$ is a finite type $\\mathcal{O}_X$-module ", "and $\\mathcal{A}$ is of finite type as an $\\mathcal{O}_X$-algebra.", "\\end{enumerate}", "If these conditions hold, then", "$\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$", "is proper." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-proper-local}", "and the construction of the relative Proj this follows from the", "case of schemes which is", "Divisors, Lemma \\ref{divisors-lemma-relative-proj-universally-closed}." ], "refs": [ "spaces-morphisms-lemma-proper-local", "divisors-lemma-relative-proj-universally-closed" ], "ref_ids": [ 4916, 8042 ] } ], "ref_ids": [] }, { "id": 12970, "type": "theorem", "label": "spaces-divisors-lemma-relative-proj-generated-in-degree-1", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relative-proj-generated-in-degree-1", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{A}$ be a quasi-coherent sheaf of graded $\\mathcal{O}_X$-modules", "generated as an $\\mathcal{A}_0$-algebra by $\\mathcal{A}_1$.", "With $P = \\underline{\\text{Proj}}_X(\\mathcal{A})$ we have", "\\begin{enumerate}", "\\item $P$ represents the functor $F_1$ which associates to", "$T$ over $S$ the set of isomorphism classes of", "triples $(f, \\mathcal{L}, \\psi)$, where $f : T \\to X$ is a morphism", "over $S$, $\\mathcal{L}$ is an invertible $\\mathcal{O}_T$-module, and", "$\\psi : f^*\\mathcal{A} \\to \\bigoplus_{n \\geq 0} \\mathcal{L}^{\\otimes n}$", "is a map of graded $\\mathcal{O}_T$-algebras inducing a surjection", "$f^*\\mathcal{A}_1 \\to \\mathcal{L}$,", "\\item the canonical map $\\pi^*\\mathcal{A}_1 \\to \\mathcal{O}_P(1)$ is", "surjective, and", "\\item each $\\mathcal{O}_P(n)$ is invertible", "and the multiplication maps induce isomorphisms", "$\\mathcal{O}_P(n) \\otimes_{\\mathcal{O}_P} \\mathcal{O}_P(m) =", "\\mathcal{O}_P(n + m)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted.", "See Constructions, Lemma \\ref{constructions-lemma-apply-relative}", "for the case of schemes." ], "refs": [ "constructions-lemma-apply-relative" ], "ref_ids": [ 12642 ] } ], "ref_ids": [] }, { "id": 12971, "type": "theorem", "label": "spaces-divisors-lemma-morphism-relative-proj", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-morphism-relative-proj", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\psi : \\mathcal{A} \\to \\mathcal{B}$ be a map of", "quasi-coherent graded $\\mathcal{O}_X$-algebras. Set", "$P = \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ and", "$Q = \\underline{\\text{Proj}}_X(\\mathcal{B}) \\to X$.", "There is a canonical open subspace", "$U(\\psi) \\subset Q$ and a canonical morphism of", "algebraic spaces", "$$", "r_\\psi :", "U(\\psi)", "\\longrightarrow", "P", "$$", "over $X$ and a map of $\\mathbf{Z}$-graded $\\mathcal{O}_{U(\\psi)}$-algebras", "$$", "\\theta = \\theta_\\psi :", "r_\\psi^*\\left(", "\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_P(d)", "\\right)", "\\longrightarrow", "\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_{U(\\psi)}(d).", "$$", "The triple $(U(\\psi), r_\\psi, \\theta)$ is characterized by the property", "that for any scheme $W$ \\'etale over $X$ the triple", "$$", "(U(\\psi) \\times_X W,\\quad", "r_\\psi|_{U(\\psi) \\times_X W} : U(\\psi) \\times_X W \\to P \\times_X W,\\quad", "\\theta|_{U(\\psi) \\times_X W})", "$$", "is equal to the triple associated to $\\psi : \\mathcal{A}|_W \\to \\mathcal{B}|_W$", "of Constructions, Lemma \\ref{constructions-lemma-morphism-relative-proj}." ], "refs": [ "constructions-lemma-morphism-relative-proj" ], "proofs": [ { "contents": [ "This lemma follows from \\'etale localization and the case of schemes, see", "discussion following", "Definition \\ref{definition-relative-proj}. Details omitted." ], "refs": [ "spaces-divisors-definition-relative-proj" ], "ref_ids": [ 13027 ] } ], "ref_ids": [ 12644 ] }, { "id": 12972, "type": "theorem", "label": "spaces-divisors-lemma-morphism-relative-proj-transitive", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-morphism-relative-proj-transitive", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{A}$, $\\mathcal{B}$, and $\\mathcal{C}$ be", "quasi-coherent graded $\\mathcal{O}_X$-algebras.", "Set $P = \\underline{\\text{Proj}}_X(\\mathcal{A})$,", "$Q = \\underline{\\text{Proj}}_X(\\mathcal{B})$ and", "$R = \\underline{\\text{Proj}}_X(\\mathcal{C})$.", "Let $\\varphi : \\mathcal{A} \\to \\mathcal{B}$,", "$\\psi : \\mathcal{B} \\to \\mathcal{C}$ be graded $\\mathcal{O}_X$-algebra maps.", "Then we have", "$$", "U(\\psi \\circ \\varphi) = r_\\varphi^{-1}(U(\\psi))", "\\quad", "\\text{and}", "\\quad", "r_{\\psi \\circ \\varphi}", "=", "r_\\varphi \\circ r_\\psi|_{U(\\psi \\circ \\varphi)}.", "$$", "In addition we have", "$$", "\\theta_\\psi \\circ r_\\psi^*\\theta_\\varphi", "=", "\\theta_{\\psi \\circ \\varphi}", "$$", "with obvious notation." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12973, "type": "theorem", "label": "spaces-divisors-lemma-surjective-graded-rings-map-relative-proj", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-surjective-graded-rings-map-relative-proj", "contents": [ "With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj}", "above. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is surjective for", "$d \\gg 0$. Then", "\\begin{enumerate}", "\\item $U(\\psi) = Q$,", "\\item $r_\\psi : Q \\to R$ is a closed immersion, and", "\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_P(n) \\to \\mathcal{O}_Q(n)$", "are surjective but not isomorphisms in general (even if", "$\\mathcal{A} \\to \\mathcal{B}$ is surjective).", "\\end{enumerate}" ], "refs": [ "spaces-divisors-lemma-morphism-relative-proj" ], "proofs": [ { "contents": [ "Follows from the case of schemes", "(Constructions, Lemma", "\\ref{constructions-lemma-surjective-graded-rings-map-relative-proj})", "by \\'etale localization." ], "refs": [ "constructions-lemma-surjective-graded-rings-map-relative-proj" ], "ref_ids": [ 12646 ] } ], "ref_ids": [ 12971 ] }, { "id": 12974, "type": "theorem", "label": "spaces-divisors-lemma-eventual-iso-graded-rings-map-relative-proj", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-eventual-iso-graded-rings-map-relative-proj", "contents": [ "With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj}", "above. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is an isomorphism for all", "$d \\gg 0$. Then", "\\begin{enumerate}", "\\item $U(\\psi) = Q$,", "\\item $r_\\psi : Q \\to P$ is an isomorphism, and", "\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_P(n) \\to \\mathcal{O}_Q(n)$", "are isomorphisms.", "\\end{enumerate}" ], "refs": [ "spaces-divisors-lemma-morphism-relative-proj" ], "proofs": [ { "contents": [ "Follows from the case of schemes", "(Constructions, Lemma", "\\ref{constructions-lemma-eventual-iso-graded-rings-map-relative-proj})", "by \\'etale localization." ], "refs": [ "constructions-lemma-eventual-iso-graded-rings-map-relative-proj" ], "ref_ids": [ 12647 ] } ], "ref_ids": [ 12971 ] }, { "id": 12975, "type": "theorem", "label": "spaces-divisors-lemma-surjective-generated-degree-1-map-relative-proj", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-surjective-generated-degree-1-map-relative-proj", "contents": [ "With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj}", "above. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is surjective for $d \\gg 0$", "and that $\\mathcal{A}$ is generated by $\\mathcal{A}_1$ over $\\mathcal{A}_0$.", "Then", "\\begin{enumerate}", "\\item $U(\\psi) = Q$,", "\\item $r_\\psi : Q \\to P$ is a closed immersion, and", "\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_P(n) \\to \\mathcal{O}_Q(n)$", "are isomorphisms.", "\\end{enumerate}" ], "refs": [ "spaces-divisors-lemma-morphism-relative-proj" ], "proofs": [ { "contents": [ "Follows from the case of schemes", "(Constructions, Lemma", "\\ref{constructions-lemma-surjective-generated-degree-1-map-relative-proj})", "by \\'etale localization." ], "refs": [ "constructions-lemma-surjective-generated-degree-1-map-relative-proj" ], "ref_ids": [ 12648 ] } ], "ref_ids": [ 12971 ] }, { "id": 12976, "type": "theorem", "label": "spaces-divisors-lemma-invertible-map-into-relative-proj", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-invertible-map-into-relative-proj", "contents": [ "With assumptions and notation as above. The morphism", "$\\psi$ induces a canonical morphism of algebraic spaces over $Y$", "$$", "r_{\\mathcal{L}, \\psi} :", "U(\\psi) \\longrightarrow \\underline{\\text{Proj}}_Y(\\mathcal{A})", "$$", "together with a map of graded $\\mathcal{O}_{U(\\psi)}$-algebras", "$$", "\\theta :", "r_{\\mathcal{L}, \\psi}^*\\left(", "\\bigoplus\\nolimits_{d \\geq 0}", "\\mathcal{O}_{\\underline{\\text{Proj}}_Y(\\mathcal{A})}(d)", "\\right)", "\\longrightarrow", "\\bigoplus\\nolimits_{d \\geq 0} \\mathcal{L}^{\\otimes d}|_{U(\\psi)}", "$$", "characterized by the following properties:", "\\begin{enumerate}", "\\item For $V \\to Y$ \\'etale and $d \\geq 0$ the diagram", "$$", "\\xymatrix{", "\\mathcal{A}_d(V) \\ar[d]_{\\psi} \\ar[r]_{\\psi} &", "\\Gamma(V \\times_Y X, \\mathcal{L}^{\\otimes d}) \\ar[d]^{restrict} \\\\", "\\Gamma(V \\times_Y \\underline{\\text{Proj}}_Y(\\mathcal{A}),", "\\mathcal{O}_{\\underline{\\text{Proj}}_Y(\\mathcal{A})}(d)) \\ar[r]^-\\theta &", "\\Gamma(V \\times_Y U(\\psi), \\mathcal{L}^{\\otimes d})", "}", "$$", "is commutative.", "\\item For any $d \\geq 1$ and any morphism $W \\to X$ where $W$ is a scheme", "such that $\\psi|_W : f^*\\mathcal{A}_d|_W \\to \\mathcal{L}^{\\otimes d}|_W$", "is surjective we have (a) $W \\to X$ factors through $U(\\psi)$ and", "(b) composition of $W \\to U(\\psi)$ with $r_{\\mathcal{L}, \\psi}$", "agrees with the morphism $W \\to \\underline{\\text{Proj}}_Y(\\mathcal{A})$", "which exists by the construction of $\\underline{\\text{Proj}}_Y(\\mathcal{A})$,", "see Definition \\ref{definition-relative-proj}.", "\\item Consider a commutative diagram", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "where $X'$ and $Y'$ are schemes, set $\\mathcal{A}' = g^*\\mathcal{A}$", "and $\\mathcal{L}' = (g')^*\\mathcal{L}$ and denote", "$\\psi' : (f')^*\\mathcal{A} \\to \\bigoplus_{d \\geq 0} (\\mathcal{L}')^{\\otimes d}$", "the pullback of $\\psi$. Let $U(\\psi')$, $r_{\\psi', \\mathcal{L}'}$,", "and $\\theta'$ be the open, morphism, and homomorphism constructed", "in Constructions, Lemma \\ref{lemma-invertible-map-into-relative-proj}.", "Then $U(\\psi') = (g')^{-1}(U(\\psi))$", "and $r_{\\psi', \\mathcal{L}'}$ agrees with the base change", "of $r_{\\psi, \\mathcal{L}}$ via the isomorphism", "$\\underline{\\text{Proj}}_{Y'}(\\mathcal{A}') =", "Y' \\times_Y \\underline{\\text{Proj}}_Y(\\mathcal{A})$", "of Lemma \\ref{lemma-relative-proj-base-change}.", "Moreover, $\\theta'$ is the pullback of $\\theta$.", "\\end{enumerate}" ], "refs": [ "spaces-divisors-definition-relative-proj", "spaces-divisors-lemma-invertible-map-into-relative-proj", "spaces-divisors-lemma-relative-proj-base-change" ], "proofs": [ { "contents": [ "Omitted. Hints:", "First we observe that for a quasi-compact scheme", "$W$ over $X$ the following are equivalent", "\\begin{enumerate}", "\\item $W \\to X$ factors through $U(\\psi)$, and", "\\item there exists a $d$ such that", "$\\psi|_W : f^*\\mathcal{A}_d|_W \\to \\mathcal{L}^{\\otimes d}|_W$", "is surjective.", "\\end{enumerate}", "This gives a description of $U(\\psi)$ as a subfunctor of $X$", "on our base category $(\\Sch/S)_{fppf}$. For such a $W$ and $d$", "we consider the quadruple", "$(d, W \\to Y, \\mathcal{L}|_W, \\psi^{(d)}|_W)$.", "By definition of $\\underline{\\text{Proj}}_Y(\\mathcal{A})$", "we obtain a morphism $W \\to \\underline{\\text{Proj}}_Y(\\mathcal{A})$.", "By our notion of equivalence of quadruples one sees that", "this morphism is independent of the choice of $d$.", "This clearly defines a transformation of functors", "$r_{\\psi, \\mathcal{L}} : U(\\psi) \\to \\underline{\\text{Proj}}_Y(\\mathcal{A})$,", "i.e., a morphism of algebraic spaces.", "By construction this morphism satisfies (2).", "Since the morphism constructed in", "Constructions, Lemma \\ref{constructions-lemma-invertible-map-into-relative-proj}", "satisfies the same property, we see that (3) is true.", "\\medskip\\noindent", "To construct $\\theta$ and check the compatibility (1) of the", "lemma, work \\'etale locally on $Y$ and $X$, arguing as", "in the discussion following", "Definition \\ref{definition-relative-proj}." ], "refs": [ "constructions-lemma-invertible-map-into-relative-proj", "spaces-divisors-definition-relative-proj" ], "ref_ids": [ 12649, 13027 ] } ], "ref_ids": [ 13027, 12976, 12964 ] }, { "id": 12977, "type": "theorem", "label": "spaces-divisors-lemma-relatively-ample-sanity-check", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relatively-ample-sanity-check", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Assume $Y$ is a scheme. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is ample on $X/Y$ in the sense of", "Definition \\ref{definition-relatively-ample}, and", "\\item $X$ is a scheme and $\\mathcal{L}$ is ample on $X/Y$", "in the sense of", "Morphisms, Definition \\ref{morphisms-definition-relatively-ample}.", "\\end{enumerate}" ], "refs": [ "spaces-divisors-definition-relatively-ample", "morphisms-definition-relatively-ample" ], "proofs": [ { "contents": [ "This follows from the definitions and", "Morphisms, Lemma \\ref{morphisms-lemma-ample-base-change}", "(which says that being relatively ample for schemes", "is preserved under base change)." ], "refs": [ "morphisms-lemma-ample-base-change" ], "ref_ids": [ 5385 ] } ], "ref_ids": [ 13028, 5568 ] }, { "id": 12978, "type": "theorem", "label": "spaces-divisors-lemma-ample-base-change", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-ample-base-change", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $Y' \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $f' : X' \\to Y'$ be the base change of $f$ and denote", "$\\mathcal{L}'$ the pullback of $\\mathcal{L}$ to $X'$.", "If $\\mathcal{L}$ is $f$-ample, then $\\mathcal{L}'$ is $f'$-ample." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the definition!", "(Hint: transitivity of base change.)" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12979, "type": "theorem", "label": "spaces-divisors-lemma-relatively-ample-properties", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relatively-ample-properties", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If there exists an $f$-ample invertible sheaf, then", "$f$ is representable, quasi-compact, and separated." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions and", "Morphisms, Lemma \\ref{morphisms-lemma-relatively-ample-separated}.", "(If in doubt, take a look at the principle of", "Algebraic Spaces, Lemma", "\\ref{spaces-lemma-representable-transformations-property-implication}.)" ], "refs": [ "morphisms-lemma-relatively-ample-separated", "spaces-lemma-representable-transformations-property-implication" ], "ref_ids": [ 5379, 8136 ] } ], "ref_ids": [] }, { "id": 12980, "type": "theorem", "label": "spaces-divisors-lemma-descend-relatively-ample", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-descend-relatively-ample", "contents": [ "Let $V \\to U$ be a surjective \\'etale morphism of affine schemes.", "Let $X$ be an algebraic space over $U$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $Y = V \\times_U X$ and let $\\mathcal{N}$", "be the pullback of $\\mathcal{L}$ to $Y$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is ample on $X/U$, and", "\\item $\\mathcal{N}$ is ample on $Y/V$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) follows from", "Lemma \\ref{lemma-ample-base-change}.", "Assume (2). This implies that $Y \\to V$ is", "quasi-compact and separated (Lemma \\ref{lemma-relatively-ample-properties})", "and $Y$ is a scheme. Then we conclude that $X \\to U$ is", "quasi-compact and separated", "(Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-quasi-compact-local} and", "\\ref{spaces-morphisms-lemma-separated-local}).", "Set $\\mathcal{A} = \\bigoplus_{d \\geq 0} f_*\\mathcal{L}^{\\otimes d}$.", "Thus is a quasi-coherent sheaf of graded $\\mathcal{O}_U$-algebras", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-pushforward}).", "By adjunction we have a map", "$\\psi : f^*\\mathcal{A} \\to \\bigoplus_{d \\geq 0} \\mathcal{L}^{\\otimes d}$.", "Applying Lemma \\ref{lemma-invertible-map-into-relative-proj}", "we obtain an open subspace $U(\\psi) \\subset X$ and a morphism", "$$", "r_{\\mathcal{L}, \\psi} : U(\\psi) \\to \\underline{\\text{Proj}}_U(\\mathcal{A})", "$$", "Since $h : V \\to U$ is \\'etale we have", "$\\mathcal{A}|_V = (Y \\to V)_*(\\bigoplus_{d \\geq 0} \\mathcal{N}^{\\otimes d})$,", "see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-pushforward-etale-base-change-modules}.", "It follows that the pullback $\\psi'$ of $\\psi$ to", "$Y$ is the adjunction map for the situation $(Y \\to V, \\mathcal{N})$ as in", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-relatively-ample} part (5).", "Since $\\mathcal{N}$ is ample on $Y/V$ we conclude from the lemma just", "cited that $U(\\psi') = Y$ and that $r_{\\mathcal{N}, \\psi'}$", "is an open immersion.", "Since Lemma \\ref{lemma-invertible-map-into-relative-proj}", "tells us that the formation of $r_{\\mathcal{L}, \\psi}$", "commutes with base change, we conclude that", "$U(\\psi) = X$ and that we have a commutative diagram", "$$", "\\xymatrix{", "Y \\ar[r]_-{r'} \\ar[d] &", "\\underline{\\text{Proj}}_V(\\mathcal{A}|_V) \\ar[d] \\ar[r] &", "V \\ar[d] \\\\", "X \\ar[r]^-r &", "\\underline{\\text{Proj}}_U(\\mathcal{A}) \\ar[r] &", "U", "}", "$$", "whose squares are fibre products. We conclude that $r$ is an", "open immersion by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-closed-immersion-local}.", "Thus $X$ is a scheme. Then we can apply", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-relatively-ample} part (5)", "to conclude that $\\mathcal{L}$ is ample on $X/U$." ], "refs": [ "spaces-divisors-lemma-ample-base-change", "spaces-divisors-lemma-relatively-ample-properties", "spaces-morphisms-lemma-quasi-compact-local", "spaces-morphisms-lemma-separated-local", "spaces-morphisms-lemma-pushforward", "spaces-divisors-lemma-invertible-map-into-relative-proj", "spaces-properties-lemma-pushforward-etale-base-change-modules", "morphisms-lemma-characterize-relatively-ample", "spaces-divisors-lemma-invertible-map-into-relative-proj", "spaces-morphisms-lemma-closed-immersion-local", "morphisms-lemma-characterize-relatively-ample" ], "ref_ids": [ 12978, 12979, 4742, 4722, 4760, 12976, 11898, 5380, 12976, 4761, 5380 ] } ], "ref_ids": [] }, { "id": 12981, "type": "theorem", "label": "spaces-divisors-lemma-relatively-ample-local", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-relatively-ample-local", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is ample on $X/Y$,", "\\item for every scheme $Z$ and every morphism $Z \\to Y$", "the algebraic space $X_Z = Z \\times_Y X$ is a scheme", "and the pullback $\\mathcal{L}_Z$ is ample on $X_Z/Z$,", "\\item for every affine scheme $Z$ and every morphism $Z \\to Y$", "the algebraic space $X_Z = Z \\times_Y X$ is a scheme", "and the pullback $\\mathcal{L}_Z$ is ample on $X_Z/Z$,", "\\item there exists a scheme $V$ and a surjective \\'etale morphism", "$V \\to Y$ such that the algebraic space $X_V = V \\times_Y X$ is a scheme", "and the pullback $\\mathcal{L}_V$ is ample on $X_V/V$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1) and (2) are equivalent by definition.", "The implication (2) $\\Rightarrow$ (3) is immediate.", "If (3) holds and $Z \\to Y$ is as in (2), then we see", "that $X_Z \\to Z$ is affine locally on $Z$ representable.", "Hence $X_Z$ is a scheme for example by", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-subscheme}.", "Then it follows that $\\mathcal{L}_Z$ is ample on $X_Z/Z$ because", "it holds locally on $Z$ and we can use", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-relatively-ample}.", "Thus (1), (2), and (3) are equivalent. Clearly these conditions", "imply (4).", "\\medskip\\noindent", "Assume (4). Let $Z \\to Y$ be a morphism with $Z$ affine.", "Then $U = V \\times_Y Z \\to Z$ is a surjective \\'etale morphism", "such that the pullback of $\\mathcal{L}_Z$ by $X_U \\to X_Z$", "is relatively ample on $X_U/U$.", "Of course we may replace $U$ by an affine open.", "It follows that $\\mathcal{L}_Z$ is ample on $X_Z/Z$ by", "Lemma \\ref{lemma-descend-relatively-ample}.", "Thus (4) $\\Rightarrow$ (3) and the proof is complete." ], "refs": [ "spaces-properties-lemma-subscheme", "morphisms-lemma-characterize-relatively-ample", "spaces-divisors-lemma-descend-relatively-ample" ], "ref_ids": [ 11848, 5380, 12980 ] } ], "ref_ids": [] }, { "id": 12982, "type": "theorem", "label": "spaces-divisors-lemma-vanshing-gives-ample", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-vanshing-gives-ample", "contents": [ "Let $R$ be a Noetherian ring. Let $X$ be an algebraic space over $R$", "such that the structure morphism $f : X \\to \\Spec(R)$ is proper.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is ample on $X/R$", "(Definition \\ref{definition-relatively-ample}),", "\\item for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "there exists an $n_0 \\geq 0$ such that", "$H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}) = 0$", "for all $n \\geq n_0$ and $p > 0$.", "\\end{enumerate}" ], "refs": [ "spaces-divisors-definition-relatively-ample" ], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) follows from", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-proper-ample}", "because assumption (1) implies that $X$ is a scheme.", "The implication (2) $\\Rightarrow$ (1) is", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-Noetherian-h1-zero-invertible}." ], "refs": [ "coherent-lemma-coherent-proper-ample", "spaces-cohomology-lemma-Noetherian-h1-zero-invertible" ], "ref_ids": [ 3343, 11323 ] } ], "ref_ids": [ 13028 ] }, { "id": 12983, "type": "theorem", "label": "spaces-divisors-lemma-ample-on-fibre", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-ample-on-fibre", "contents": [ "Let $Y$ be a Noetherian scheme. Let $X$ be an algebraic space over $Y$", "such that the structure morphism $f : X \\to Y$ is proper.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Let $y \\in Y$ be a point such that $X_y$ is a scheme and", "$\\mathcal{L}_y$ is ample on $X_y$.", "Then there exists a $d_0$ such that for all $d \\geq d_0$ we have", "$$", "R^pf_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})_y = 0", "\\text{ for }p > 0", "$$", "and the map", "$$", "f_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})_y", "\\longrightarrow", "H^0(X_y, \\mathcal{F}_y \\otimes_{\\mathcal{O}_{X_y}} \\mathcal{L}_y^{\\otimes d})", "$$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Note that $\\mathcal{O}_{Y, y}$ is a Noetherian local ring.", "Consider the canonical morphism", "$c : \\Spec(\\mathcal{O}_{Y, y}) \\to Y$, see", "Schemes, Equation (\\ref{schemes-equation-canonical-morphism}).", "This is a flat morphism as it identifies local rings.", "Denote momentarily $f' : X' \\to \\Spec(\\mathcal{O}_{Y, y})$", "the base change of $f$ to this local ring. We see that", "$c^*R^pf_*\\mathcal{F} = R^pf'_*\\mathcal{F}'$ by", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-flat-base-change-cohomology}.", "Moreover, the fibres $X_y$ and $X'_y$ are identified.", "Hence we may assume that $Y = \\Spec(A)$ is the spectrum of", "a Noetherian local ring $(A, \\mathfrak m, \\kappa)$ and $y \\in Y$", "corresponds to $\\mathfrak m$. In this case", "$R^pf_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})_y =", "H^p(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})$", "for all $p \\geq 0$. Denote $f_y : X_y \\to \\Spec(\\kappa)$ the projection.", "\\medskip\\noindent", "Let $B = \\text{Gr}_\\mathfrak m(A) =", "\\bigoplus_{n \\geq 0} \\mathfrak m^n/\\mathfrak m^{n + 1}$.", "Consider the sheaf $\\mathcal{B} = f_y^*\\widetilde{B}$", "of quasi-coherent graded $\\mathcal{O}_{X_y}$-algebras.", "We will use notation as in Cohomology of Spaces, Section", "\\ref{spaces-cohomology-section-theorem-formal-functions}", "with $I$ replaced by $\\mathfrak m$.", "Since $X_y$ is the closed subspace of $X$ cut out by", "$\\mathfrak m\\mathcal{O}_X$ we may think of", "$\\mathfrak m^n\\mathcal{F}/\\mathfrak m^{n + 1}\\mathcal{F}$", "as a coherent $\\mathcal{O}_{X_y}$-module, see", "Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-i-star-equivalence}.", "Then", "$\\bigoplus_{n \\geq 0} \\mathfrak m^n\\mathcal{F}/\\mathfrak m^{n + 1}\\mathcal{F}$", "is a quasi-coherent graded $\\mathcal{B}$-module of finite type", "because it is generated in degree zero over $\\mathcal{B}$", "abd because the degree zero part is", "$\\mathcal{F}_y = \\mathcal{F}/\\mathfrak m \\mathcal{F}$", "which is a coherent $\\mathcal{O}_{X_y}$-module.", "Hence by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-graded-finiteness} part (2)", "there exists a $d_0$ such that", "$$", "H^p(X_y, \\mathfrak m^n \\mathcal{F}/ \\mathfrak m^{n + 1}\\mathcal{F}", "\\otimes_{\\mathcal{O}_{X_y}} \\mathcal{L}_y^{\\otimes d}) = 0", "$$", "for all $p > 0$, $d \\geq d_0$, and $n \\geq 0$. By", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-relative-affine-cohomology}", "this is the same as the statement that", "$", "H^p(X, \\mathfrak m^n \\mathcal{F}/ \\mathfrak m^{n + 1}\\mathcal{F}", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) = 0", "$", "for all $p > 0$, $d \\geq d_0$, and $n \\geq 0$.", "\\medskip\\noindent", "Consider the short exact sequences", "$$", "0 \\to \\mathfrak m^n\\mathcal{F}/\\mathfrak m^{n + 1} \\mathcal{F}", "\\to \\mathcal{F}/\\mathfrak m^{n + 1} \\mathcal{F}", "\\to \\mathcal{F}/\\mathfrak m^n \\mathcal{F} \\to 0", "$$", "of coherent $\\mathcal{O}_X$-modules. Tensoring with $\\mathcal{L}^{\\otimes d}$", "is an exact functor and we obtain short exact sequences", "$$", "0 \\to", "\\mathfrak m^n\\mathcal{F}/\\mathfrak m^{n + 1} \\mathcal{F}", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}", "\\to \\mathcal{F}/\\mathfrak m^{n + 1} \\mathcal{F}", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}", "\\to \\mathcal{F}/\\mathfrak m^n \\mathcal{F}", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d} \\to 0", "$$", "Using the long exact cohomology sequence and the vanishing above", "we conclude (using induction) that", "\\begin{enumerate}", "\\item $H^p(X, \\mathcal{F}/\\mathfrak m^n \\mathcal{F}", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) = 0$", "for all $p > 0$, $d \\geq d_0$, and $n \\geq 0$, and", "\\item $H^0(X, \\mathcal{F}/\\mathfrak m^n \\mathcal{F}", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) \\to", "H^0(X_y, \\mathcal{F}_y \\otimes_{\\mathcal{O}_{X_y}} \\mathcal{L}_y^{\\otimes d})$", "is surjective for all $d \\geq d_0$ and $n \\geq 1$.", "\\end{enumerate}", "By the theorem on formal functions", "(Cohomology of Spaces, Theorem", "\\ref{spaces-cohomology-theorem-formal-functions})", "we find that the $\\mathfrak m$-adic completion of", "$H^p(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})$", "is zero for all $d \\geq d_0$ and $p > 0$.", "Since $H^p(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})$", "is a finite $A$-module by", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-over-affine-cohomology-finite}", "it follows from Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "that $H^p(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})$", "is zero for all $d \\geq d_0$ and $p > 0$.", "For $p = 0$ we deduce from", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-ML-cohomology-powers-ideal} part (3)", "that $H^0(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) \\to", "H^0(X_y, \\mathcal{F}_y \\otimes_{\\mathcal{O}_{X_y}} \\mathcal{L}_y^{\\otimes d})$", "is surjective, which gives the final statement of the lemma." ], "refs": [ "spaces-cohomology-lemma-flat-base-change-cohomology", "spaces-cohomology-lemma-i-star-equivalence", "coherent-lemma-graded-finiteness", "spaces-cohomology-lemma-relative-affine-cohomology", "spaces-cohomology-theorem-formal-functions", "spaces-cohomology-lemma-proper-over-affine-cohomology-finite", "algebra-lemma-NAK", "spaces-cohomology-lemma-ML-cohomology-powers-ideal" ], "ref_ids": [ 11296, 11303, 3356, 11289, 11270, 11332, 401, 11338 ] } ], "ref_ids": [] }, { "id": 12984, "type": "theorem", "label": "spaces-divisors-lemma-ample-in-neighbourhood", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-ample-in-neighbourhood", "contents": [ "(For a more general version see", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-ample-in-neighbourhood}).", "Let $Y$ be a Noetherian scheme. Let $X$ be an algebraic space over $Y$", "such that the structure morphism $f : X \\to Y$ is proper.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $y \\in Y$ be a point such that $X_y$ is a scheme and", "$\\mathcal{L}_y$ is ample on $X_y$.", "Then there is an open neighbourhood $V \\subset Y$", "of $y$ such that $\\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$", "(as in Definition \\ref{definition-relatively-ample})." ], "refs": [ "spaces-descent-lemma-ample-in-neighbourhood", "spaces-divisors-definition-relatively-ample" ], "proofs": [ { "contents": [ "Pick $d_0$ as in Lemma \\ref{lemma-ample-on-fibre} for", "$\\mathcal{F} = \\mathcal{O}_X$. Pick $d \\geq d_0$", "so that we can find $r \\geq 0$ and sections", "$s_{y, 0}, \\ldots, s_{y, r} \\in H^0(X_y, \\mathcal{L}_y^{\\otimes d})$", "which define a closed immersion", "$$", "\\varphi_y =", "\\varphi_{\\mathcal{L}_y^{\\otimes d}, (s_{y, 0}, \\ldots, s_{y, r})} :", "X_y \\to \\mathbf{P}^r_{\\kappa(y)}.", "$$", "This is possible by Morphisms, Lemma", "\\ref{morphisms-lemma-finite-type-over-affine-ample-very-ample}", "but we also use", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}", "to see that $\\varphi_y$ is a closed immersion and", "Constructions, Section \\ref{constructions-section-projective-space}", "for the description of morphisms into projective", "space in terms of invertible sheaves and sections.", "By our choice of $d_0$, after replacing $Y$ by an open neighbourhood", "of $y$, we can choose", "$s_0, \\ldots, s_r \\in H^0(X, \\mathcal{L}^{\\otimes d})$", "mapping to $s_{y, 0}, \\ldots, s_{y, r}$.", "Let $X_{s_i} \\subset X$ be the open subspace where $s_i$", "is a generator of $\\mathcal{L}^{\\otimes d}$. Since", "the $s_{y, i}$ generate $\\mathcal{L}_y^{\\otimes d}$ we see that", "$|X_y| \\subset U = \\bigcup |X_{s_i}|$. Since $X \\to Y$ is closed,", "we see that there is an open neighbourhood $y \\in V \\subset Y$", "such that $|f|^{-1}(V) \\subset U$. After replacing $Y$ by $V$ we may", "assume that the $s_i$ generate $\\mathcal{L}^{\\otimes d}$. Thus we", "obtain a morphism", "$$", "\\varphi = \\varphi_{\\mathcal{L}^{\\otimes d}, (s_0, \\ldots, s_r)} :", "X \\longrightarrow \\mathbf{P}^r_Y", "$$", "with $\\mathcal{L}^{\\otimes d} \\cong \\varphi^*\\mathcal{O}_{\\mathbf{P}^r_Y}(1)$", "whose base change to $y$ gives $\\varphi_y$ (strictly speaking we need", "to write out a proof that the construction of morphisms into projective", "space given in", "Constructions, Section \\ref{constructions-section-projective-space}", "also works to describe morphisms of algebraic spaces into projective", "space; we omit the details).", "\\medskip\\noindent", "We will finish the proof by a sleight of hand; the ``correct'' proof", "proceeds by directly showing that $\\varphi$ is a closed", "immersion after base changing to an open neighbourhood of $y$.", "Namely, by", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-finite-fibre-finite-in-neighbourhood}", "we see that $\\varphi$ is a finite over an open neighbourhood", "of the fibre $\\mathbf{P}^r_{\\kappa(y)}$ of $\\mathbf{P}^r_Y \\to Y$", "above $y$. Using that $\\mathbf{P}^r_Y \\to Y$ is closed, after", "shrinking $Y$ we may assume that $\\varphi$ is finite.", "In particular $X$ is a scheme.", "Then $\\mathcal{L}^{\\otimes d} \\cong \\varphi^*\\mathcal{O}_{\\mathbf{P}^r_Y}(1)$", "is ample by the very general", "Morphisms, Lemma \\ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}." ], "refs": [ "spaces-divisors-lemma-ample-on-fibre", "morphisms-lemma-finite-type-over-affine-ample-very-ample", "morphisms-lemma-image-proper-scheme-closed", "spaces-cohomology-lemma-proper-finite-fibre-finite-in-neighbourhood", "morphisms-lemma-pullback-ample-tensor-relatively-ample" ], "ref_ids": [ 12983, 5394, 5411, 11344, 5383 ] } ], "ref_ids": [ 9414, 13028 ] }, { "id": 12985, "type": "theorem", "label": "spaces-divisors-lemma-closed-subscheme-proj", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-closed-subscheme-proj", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_X$-algebra. Let", "$\\pi : P = \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ be the relative", "Proj of $\\mathcal{A}$. Let $i : Z \\to P$ be a closed subspace. Denote", "$\\mathcal{I} \\subset \\mathcal{A}$ the kernel of the canonical map", "$$", "\\mathcal{A}", "\\longrightarrow", "\\bigoplus\\nolimits_{d \\geq 0} \\pi_*\\left((i_*\\mathcal{O}_Z)(d)\\right)", "$$", "If $\\pi$ is quasi-compact, then there is an isomorphism", "$Z = \\underline{\\text{Proj}}_X(\\mathcal{A}/\\mathcal{I})$." ], "refs": [], "proofs": [ { "contents": [ "The morphism $\\pi$ is separated by", "Lemma \\ref{lemma-relative-proj-separated}.", "As $\\pi$ is quasi-compact, $\\pi_*$ transforms quasi-coherent modules", "into quasi-coherent modules, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-pushforward}.", "Hence $\\mathcal{I}$ is a quasi-coherent $\\mathcal{O}_X$-module.", "In particular, $\\mathcal{B} = \\mathcal{A}/\\mathcal{I}$ is a", "quasi-coherent graded $\\mathcal{O}_X$-algebra. The functoriality", "morphism $Z' = \\underline{\\text{Proj}}_X(\\mathcal{B}) \\to", "\\underline{\\text{Proj}}_X(\\mathcal{A})$ is everywhere defined and", "a closed immersion, see Lemma", "\\ref{lemma-surjective-graded-rings-map-relative-proj}.", "Hence it suffices to prove $Z = Z'$ as closed subspaces of $P$.", "\\medskip\\noindent", "Having said this, the question is \\'etale local on the base and we", "reduce to the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-closed-subscheme-proj})", "by \\'etale localization." ], "refs": [ "spaces-divisors-lemma-relative-proj-separated", "spaces-morphisms-lemma-pushforward", "spaces-divisors-lemma-surjective-graded-rings-map-relative-proj", "divisors-lemma-closed-subscheme-proj" ], "ref_ids": [ 12965, 4760, 12973, 8047 ] } ], "ref_ids": [] }, { "id": 12986, "type": "theorem", "label": "spaces-divisors-lemma-closed-subscheme-proj-finite", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-closed-subscheme-proj-finite", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$.", "Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_X$-algebra. Let", "$\\pi : P = \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ be the relative", "Proj of $\\mathcal{A}$. Let $i : Z \\to P$ be a closed subscheme.", "If $\\pi$ is quasi-compact and $i$ of finite presentation, then there exists", "a $d > 0$ and a quasi-coherent finite type $\\mathcal{O}_X$-submodule", "$\\mathcal{F} \\subset \\mathcal{A}_d$ such that", "$Z = \\underline{\\text{Proj}}_X(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$." ], "refs": [], "proofs": [ { "contents": [ "The reader can redo the arguments used in the case of schemes. However, we", "will show the lemma follows from the case of schemes by a trick.", "Let $\\mathcal{I} \\subset \\mathcal{A}$ be the quasi-coherent graded", "ideal cutting out $Z$ of Lemma \\ref{lemma-closed-subscheme-proj}.", "Choose an affine scheme $U$ and a surjective \\'etale morphism", "$U \\to X$, see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}.", "By the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-closed-subscheme-proj-finite})", "there exists a $d > 0$ and a quasi-coherent finite type", "$\\mathcal{O}_U$-submodule", "$\\mathcal{F}' \\subset \\mathcal{I}_d|_U \\subset \\mathcal{A}_d|_U$", "such that $Z \\times_X U$ is equal to", "$\\underline{\\text{Proj}}_U(\\mathcal{A}|_U/\\mathcal{F}'\\mathcal{A}|_U)$.", "By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-directed-colimit-finite-type}", "we can find a finite type quasi-coherent submodule", "$\\mathcal{F} \\subset \\mathcal{I}_d$ such that", "$\\mathcal{F}' \\subset \\mathcal{F}|_U$. Let", "$Z' = \\underline{\\text{Proj}}_X(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$.", "Then $Z' \\to P$ is a closed immersion", "(Lemma \\ref{lemma-surjective-generated-degree-1-map-relative-proj})", "and $Z \\subset Z'$ as $\\mathcal{F}\\mathcal{A} \\subset \\mathcal{I}$.", "On the other hand, $Z' \\times_X U \\subset Z \\times_X U$ by our", "choice of $\\mathcal{F}$. Thus $Z = Z'$ as desired." ], "refs": [ "spaces-divisors-lemma-closed-subscheme-proj", "spaces-properties-lemma-quasi-compact-affine-cover", "divisors-lemma-closed-subscheme-proj-finite", "spaces-limits-lemma-directed-colimit-finite-type", "spaces-divisors-lemma-surjective-generated-degree-1-map-relative-proj" ], "ref_ids": [ 12985, 11832, 8049, 4602, 12975 ] } ], "ref_ids": [] }, { "id": 12987, "type": "theorem", "label": "spaces-divisors-lemma-closed-subscheme-proj-finite-type", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-closed-subscheme-proj-finite-type", "contents": [ "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated", "algebraic space over $S$.", "Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_X$-algebra.", "Let $\\pi : P = \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ be the relative", "Proj of $\\mathcal{A}$. Let $i : Z \\to X$ be a closed subspace.", "Let $U \\subset X$ be an open. Assume that", "\\begin{enumerate}", "\\item $\\pi$ is quasi-compact,", "\\item $i$ of finite presentation,", "\\item $|U| \\cap |\\pi|(|i|(|Z|)) = \\emptyset$,", "\\item $U$ is quasi-compact,", "\\item $\\mathcal{A}_n$ is a finite type $\\mathcal{O}_X$-module for all $n$.", "\\end{enumerate}", "Then there exists a $d > 0$ and a quasi-coherent finite type", "$\\mathcal{O}_X$-submodule $\\mathcal{F} \\subset \\mathcal{A}_d$ with (a)", "$Z = \\underline{\\text{Proj}}_X(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$", "and (b) the support of $\\mathcal{A}_d/\\mathcal{F}$ is disjoint from $U$." ], "refs": [], "proofs": [ { "contents": [ "We use the same trick as in the proof of", "Lemma \\ref{lemma-closed-subscheme-proj-finite}", "to reduce to the case of schemes.", "Let $\\mathcal{I} \\subset \\mathcal{A}$ be the quasi-coherent graded", "ideal cutting out $Z$ of Lemma \\ref{lemma-closed-subscheme-proj}.", "Choose an affine scheme $W$ and a surjective \\'etale morphism", "$W \\to X$, see Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-quasi-compact-affine-cover}.", "By the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-closed-subscheme-proj-finite-type})", "there exists a $d > 0$ and a quasi-coherent finite type", "$\\mathcal{O}_W$-submodule", "$\\mathcal{F}' \\subset \\mathcal{I}_d|_W \\subset \\mathcal{A}_d|_W$", "such that (a) $Z \\times_X W$ is equal to", "$\\underline{\\text{Proj}}_W(\\mathcal{A}|_W/\\mathcal{F}'\\mathcal{A}|_W)$", "and (b) the support of $\\mathcal{A}_d|_W/\\mathcal{F}'$ is disjoint from", "$U \\times_X W$. By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-directed-colimit-finite-type}", "we can find a finite type quasi-coherent submodule", "$\\mathcal{F} \\subset \\mathcal{I}_d$ such that", "$\\mathcal{F}' \\subset \\mathcal{F}|_W$. Let", "$Z' = \\underline{\\text{Proj}}_X(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$.", "Then $Z' \\to P$ is a closed immersion", "(Lemma \\ref{lemma-surjective-generated-degree-1-map-relative-proj})", "and $Z \\subset Z'$ as $\\mathcal{F}\\mathcal{A} \\subset \\mathcal{I}$.", "On the other hand, $Z' \\times_X W \\subset Z \\times_X W$ by our", "choice of $\\mathcal{F}$. Thus $Z = Z'$.", "Finally, we see that $\\mathcal{A}_d/\\mathcal{F}$ is supported on", "$X \\setminus U$ as $\\mathcal{A}_d|_W/\\mathcal{F}|_W$ is a quotient", "of $\\mathcal{A}_d|_W/\\mathcal{F}'$ which is supported on", "$W \\setminus U \\times_X W$. Thus the lemma follows." ], "refs": [ "spaces-divisors-lemma-closed-subscheme-proj-finite", "spaces-divisors-lemma-closed-subscheme-proj", "spaces-properties-lemma-quasi-compact-affine-cover", "divisors-lemma-closed-subscheme-proj-finite-type", "spaces-limits-lemma-directed-colimit-finite-type", "spaces-divisors-lemma-surjective-generated-degree-1-map-relative-proj" ], "ref_ids": [ 12986, 12985, 11832, 8050, 4602, 12975 ] } ], "ref_ids": [] }, { "id": 12988, "type": "theorem", "label": "spaces-divisors-lemma-conormal-sheaf-section-projective-bundle", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-conormal-sheaf-section-projective-bundle", "contents": [ "Let $S$ be a scheme and let $X$ be an algebraic space over $S$.", "Let $\\mathcal{E}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "There is a bijection", "$$", "\\left\\{", "\\begin{matrix}", "\\text{sections }\\sigma\\text{ of the } \\\\", "\\text{morphism } \\mathbf{P}(\\mathcal{E}) \\to X", "\\end{matrix}", "\\right\\}", "\\leftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{surjections }\\mathcal{E} \\to \\mathcal{L}\\text{ where} \\\\", "\\mathcal{L}\\text{ is an invertible }\\mathcal{O}_X\\text{-module}", "\\end{matrix}", "\\right\\}", "$$", "In this case $\\sigma$ is a closed immersion and there is a canonical", "isomorphism", "$$", "\\Ker(\\mathcal{E} \\to \\mathcal{L})", "\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes -1}", "\\longrightarrow", "\\mathcal{C}_{\\sigma(X)/\\mathbf{P}(\\mathcal{E})}", "$$", "Both the bijection and isomorphism are compatible with base change." ], "refs": [], "proofs": [ { "contents": [ "Because the constructions are compatible with base change, it suffices to", "check the statement \\'etale locally on $X$. Thus we may assume $X$ is", "a scheme and the result is", "Divisors, Lemma \\ref{divisors-lemma-conormal-sheaf-section-projective-bundle}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 12989, "type": "theorem", "label": "spaces-divisors-lemma-blowing-up-affine", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-blowing-up-affine", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a", "quasi-coherent sheaf of ideals. Let $U = \\Spec(A)$ be an affine scheme", "\\'etale over $X$ and let $I \\subset A$ be the ideal corresponding to", "$\\mathcal{I}|_U$. If $X' \\to X$ is the blowup of $X$ in $\\mathcal{I}$,", "then there is a canonical isomorphism", "$$", "U \\times_X X' = \\text{Proj}(\\bigoplus\\nolimits_{d \\geq 0} I^d)", "$$", "of schemes over $U$, where the right hand side is", "the homogeneous spectrum of the Rees algebra of $I$ in $A$.", "Moreover, $U \\times_X X'$ has an affine open covering by", "spectra of the affine blowup algebras $A[\\frac{I}{a}]$." ], "refs": [], "proofs": [ { "contents": [ "Note that the restriction $\\mathcal{I}|_U$ is equal to the pullback", "of $\\mathcal{I}$ via the morphism $U \\to X$, see", "Properties of Spaces, Section \\ref{spaces-properties-section-modules}.", "Thus the lemma follows on combining Lemma \\ref{lemma-relative-proj} with", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-affine}." ], "refs": [ "spaces-divisors-lemma-relative-proj", "divisors-lemma-blowing-up-affine" ], "ref_ids": [ 12962, 8052 ] } ], "ref_ids": [] }, { "id": 12990, "type": "theorem", "label": "spaces-divisors-lemma-flat-base-change-blowing-up", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-flat-base-change-blowing-up", "contents": [ "Let $S$ be a scheme.", "Let $X_1 \\to X_2$ be a flat morphism of algebraic spaces over $S$.", "Let $Z_2 \\subset X_2$ be a closed subspace.", "Let $Z_1$ be the inverse image of $Z_2$ in $X_1$.", "Let $X'_i$ be the blowup of $Z_i$ in $X_i$. Then there exists a cartesian", "diagram", "$$", "\\xymatrix{", "X_1' \\ar[r] \\ar[d] & X_2' \\ar[d] \\\\", "X_1 \\ar[r] & X_2", "}", "$$", "of algebraic spaces over $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I}_2$ be the ideal sheaf of $Z_2$ in $X_2$.", "Denote $g : X_1 \\to X_2$ the given morphism. Then the ideal sheaf", "$\\mathcal{I}_1$ of $Z_1$ is the image of", "$g^*\\mathcal{I}_2 \\to \\mathcal{O}_{X_1}$", "(see Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-inverse-image-closed-subspace}", "and discussion following the definition).", "By Lemma \\ref{lemma-relative-proj-base-change}", "we see that $X_1 \\times_{X_2} X_2'$ is the relative Proj of", "$\\bigoplus_{n \\geq 0} g^*\\mathcal{I}_2^n$. Because $g$ is flat the map", "$g^*\\mathcal{I}_2^n \\to \\mathcal{O}_{X_1}$ is injective with image", "$\\mathcal{I}_1^n$. Thus we see that $X_1 \\times_{X_2} X_2' = X_1'$." ], "refs": [ "spaces-morphisms-definition-inverse-image-closed-subspace", "spaces-divisors-lemma-relative-proj-base-change" ], "ref_ids": [ 4991, 12964 ] } ], "ref_ids": [] }, { "id": 12991, "type": "theorem", "label": "spaces-divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $Z \\subset X$ be a closed subspace.", "The blowing up $b : X' \\to X$ of $Z$ in $X$", "has the following properties:", "\\begin{enumerate}", "\\item $b|_{b^{-1}(X \\setminus Z)} : b^{-1}(X \\setminus Z) \\to X \\setminus Z$", "is an isomorphism,", "\\item the exceptional divisor $E = b^{-1}(Z)$ is an effective Cartier divisor", "on $X'$,", "\\item there is a canonical isomorphism", "$\\mathcal{O}_{X'}(-1) = \\mathcal{O}_{X'}(E)$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $U \\to X$ be a surjective \\'etale morphism.", "As blowing up commutes with flat base change", "(Lemma \\ref{lemma-flat-base-change-blowing-up})", "we can prove each of these statements after base change to $U$.", "This reduces us to the case of schemes.", "In this case the result is", "Divisors, Lemma", "\\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}." ], "refs": [ "spaces-divisors-lemma-flat-base-change-blowing-up", "divisors-lemma-blowing-up-gives-effective-Cartier-divisor" ], "ref_ids": [ 12990, 8054 ] } ], "ref_ids": [] }, { "id": 12992, "type": "theorem", "label": "spaces-divisors-lemma-universal-property-blowing-up", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-universal-property-blowing-up", "contents": [ "\\begin{slogan}", "Blow up a closed subset to make it Cartier.", "\\end{slogan}", "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $Z \\subset X$ be a closed subspace.", "Let $\\mathcal{C}$ be the full subcategory of $(\\textit{Spaces}/X)$ consisting", "of $Y \\to X$ such that the inverse image of $Z$ is an effective", "Cartier divisor on $Y$. Then the blowing up $b : X' \\to X$ of $Z$ in $X$", "is a final object of $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "We see that $b : X' \\to X$ is an object of $\\mathcal{C}$ according to", "Lemma \\ref{lemma-blowing-up-gives-effective-Cartier-divisor}.", "Let $f : Y \\to X$ be an object of $\\mathcal{C}$. We have to show there exists", "a unique morphism $Y \\to X'$ over $X$. Let $D = f^{-1}(Z)$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the ideal sheaf of $Z$", "and let $\\mathcal{I}_D$ be the ideal sheaf of $D$. Then", "$f^*\\mathcal{I} \\to \\mathcal{I}_D$ is a surjection", "to an invertible $\\mathcal{O}_Y$-module. This extends to a map", "$\\psi : \\bigoplus f^*\\mathcal{I}^d \\to \\bigoplus \\mathcal{I}_D^d$", "of graded $\\mathcal{O}_Y$-algebras. (We observe that", "$\\mathcal{I}_D^d = \\mathcal{I}_D^{\\otimes d}$ as $D$ is an", "effective Cartier divisor.) By", "Lemma \\ref{lemma-relative-proj-generated-in-degree-1}.", "the triple $(f : Y \\to X, \\mathcal{I}_D, \\psi)$ defines a", "morphism $Y \\to X'$ over $X$. The restriction", "$$", "Y \\setminus D \\longrightarrow X' \\setminus b^{-1}(Z) = X \\setminus Z", "$$", "is unique. The open $Y \\setminus D$ is scheme theoretically dense in $Y$", "according to Lemma \\ref{lemma-complement-effective-Cartier-divisor}. ", "Thus the morphism $Y \\to X'$ is unique by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-equality-of-morphisms}", "(also $b$ is separated by Lemma", "\\ref{lemma-relative-proj-separated})." ], "refs": [ "spaces-divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "spaces-divisors-lemma-relative-proj-generated-in-degree-1", "spaces-divisors-lemma-complement-effective-Cartier-divisor", "spaces-morphisms-lemma-equality-of-morphisms", "spaces-divisors-lemma-relative-proj-separated" ], "ref_ids": [ 12991, 12970, 12937, 4791, 12965 ] } ], "ref_ids": [] }, { "id": 12993, "type": "theorem", "label": "spaces-divisors-lemma-blow-up-effective-Cartier-divisor", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-blow-up-effective-Cartier-divisor", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $Z \\subset X$ be an effective Cartier divisor.", "The blowup of $X$ in $Z$ is the identity morphism of $X$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the universal property of blowups", "(Lemma \\ref{lemma-universal-property-blowing-up})." ], "refs": [ "spaces-divisors-lemma-universal-property-blowing-up" ], "ref_ids": [ 12992 ] } ], "ref_ids": [] }, { "id": 12994, "type": "theorem", "label": "spaces-divisors-lemma-blow-up-reduced-space", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-blow-up-reduced-space", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a", "quasi-coherent sheaf of ideals. If $X$ is reduced, then the", "blowup $X'$ of $X$ in $\\mathcal{I}$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $U \\to X$ be a surjective \\'etale morphism.", "As blowing up commutes with flat base change", "(Lemma \\ref{lemma-flat-base-change-blowing-up})", "we can prove each of these statements after base change to $U$.", "This reduces us to the case of schemes.", "In this case the result is", "Divisors, Lemma \\ref{divisors-lemma-blow-up-reduced-scheme}." ], "refs": [ "spaces-divisors-lemma-flat-base-change-blowing-up", "divisors-lemma-blow-up-reduced-scheme" ], "ref_ids": [ 12990, 8058 ] } ], "ref_ids": [] }, { "id": 12995, "type": "theorem", "label": "spaces-divisors-lemma-blowup-finite-nr-irreducibles", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-blowup-finite-nr-irreducibles", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let", "$b : X' \\to X$ be the blowup of $X$ in a closed subspace. If", "$X$ satisfies the equivalent conditions of", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-prepare-normalization}", "then so does $X'$." ], "refs": [ "spaces-morphisms-lemma-prepare-normalization" ], "proofs": [ { "contents": [ "Follows immediately from the lemma cited in the statement,", "the \\'etale local description of blowing ups in", "Lemma \\ref{lemma-blowing-up-affine}, and", "Divisors, Lemma \\ref{divisors-lemma-blow-up-and-irreducible-components}." ], "refs": [ "spaces-divisors-lemma-blowing-up-affine", "divisors-lemma-blow-up-and-irreducible-components" ], "ref_ids": [ 12989, 8060 ] } ], "ref_ids": [ 4966 ] }, { "id": 12996, "type": "theorem", "label": "spaces-divisors-lemma-blow-up-pullback-effective-Cartier", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-blow-up-pullback-effective-Cartier", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $b : X' \\to X$ be a blowup of $X$ in a closed subspace.", "For any effective Cartier divisor $D$ on $X$ the pullback", "$b^{-1}D$ is defined (see Definition", "\\ref{definition-pullback-effective-Cartier-divisor})." ], "refs": [ "spaces-divisors-definition-pullback-effective-Cartier-divisor" ], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-blowing-up-affine} and", "\\ref{lemma-characterize-effective-Cartier-divisor}", "this reduces to the following algebra fact:", "Let $A$ be a ring, $I \\subset A$ an ideal, $a \\in I$, and $x \\in A$", "a nonzerodivisor. Then the image of $x$ in $A[\\frac{I}{a}]$ is a", "nonzerodivisor. Namely, suppose that $x (y/a^n) = 0$ in $A[\\frac{I}{a}]$.", "Then $a^mxy = 0$ in $A$ for some $m$. Hence $a^my = 0$ as $x$ is a", "nonzerodivisor. Whence $y/a^n$ is zero in $A[\\frac{I}{a}]$ as desired." ], "refs": [ "spaces-divisors-lemma-blowing-up-affine", "spaces-divisors-lemma-characterize-effective-Cartier-divisor" ], "ref_ids": [ 12989, 12935 ] } ], "ref_ids": [ 13018 ] }, { "id": 12997, "type": "theorem", "label": "spaces-divisors-lemma-blowing-up-two-ideals", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-blowing-up-two-ideals", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ and $\\mathcal{J}$ be", "quasi-coherent sheaves of ideals. Let $b : X' \\to X$ be the blowing up", "of $X$ in $\\mathcal{I}$. Let $b' : X'' \\to X'$ be the blowing up of", "$X'$ in $b^{-1}\\mathcal{J} \\mathcal{O}_{X'}$. Then $X'' \\to X$", "is canonically isomorphic to the blowing up of $X$ in $\\mathcal{I}\\mathcal{J}$." ], "refs": [], "proofs": [ { "contents": [ "Let $E \\subset X'$ be the exceptional divisor of $b$ which is an effective", "Cartier divisor by", "Lemma \\ref{lemma-blowing-up-gives-effective-Cartier-divisor}.", "Then $(b')^{-1}E$ is an effective Cartier divisor on $X''$ by", "Lemma \\ref{lemma-blow-up-pullback-effective-Cartier}.", "Let $E' \\subset X''$ be the exceptional divisor of $b'$ (also an effective", "Cartier divisor). Consider the effective Cartier divisor", "$E'' = E' + (b')^{-1}E$. By construction the ideal of $E''$ is", "$(b \\circ b')^{-1}\\mathcal{I} (b \\circ b')^{-1}\\mathcal{J} \\mathcal{O}_{X''}$.", "Hence according to Lemma \\ref{lemma-universal-property-blowing-up}", "there is a canonical morphism from $X''$ to the blowup $c : Y \\to X$", "of $X$ in $\\mathcal{I}\\mathcal{J}$. Conversely, as $\\mathcal{I}\\mathcal{J}$", "pulls back to an invertible ideal we see that", "$c^{-1}\\mathcal{I}\\mathcal{O}_Y$ defines", "an effective Cartier divisor, see", "Lemma \\ref{lemma-sum-closed-subschemes-effective-Cartier}.", "Thus a morphism $c' : Y \\to X'$ over $X$ by", "Lemma \\ref{lemma-universal-property-blowing-up}.", "Then $(c')^{-1}b^{-1}\\mathcal{J}\\mathcal{O}_Y = c^{-1}\\mathcal{J}\\mathcal{O}_Y$", "which also defines an effective Cartier divisor. Thus a morphism", "$c'' : Y \\to X''$ over $X'$. We omit the verification that this", "morphism is inverse to the morphism $X'' \\to Y$ constructed earlier." ], "refs": [ "spaces-divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "spaces-divisors-lemma-blow-up-pullback-effective-Cartier", "spaces-divisors-lemma-universal-property-blowing-up", "spaces-divisors-lemma-sum-closed-subschemes-effective-Cartier", "spaces-divisors-lemma-universal-property-blowing-up" ], "ref_ids": [ 12991, 12996, 12992, 12940, 12992 ] } ], "ref_ids": [] }, { "id": 12998, "type": "theorem", "label": "spaces-divisors-lemma-blowing-up-projective", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-blowing-up-projective", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent", "sheaf of ideals. Let $b : X' \\to X$ be the blowing up of $X$", "in the ideal sheaf $\\mathcal{I}$. If $\\mathcal{I}$ is of finite type, then", "$b : X' \\to X$ is a proper morphism." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be a scheme and let $U \\to X$ be a surjective \\'etale morphism.", "As blowing up commutes with flat base change", "(Lemma \\ref{lemma-flat-base-change-blowing-up})", "we can prove each of these statements after base change to $U$", "(see Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-proper-local}).", "This reduces us to the case of schemes.", "In this case the morphism $b$ is projective by", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-projective}", "hence proper by", "Morphisms, Lemma \\ref{morphisms-lemma-locally-projective-proper}." ], "refs": [ "spaces-divisors-lemma-flat-base-change-blowing-up", "spaces-morphisms-lemma-proper-local", "divisors-lemma-blowing-up-projective", "morphisms-lemma-locally-projective-proper" ], "ref_ids": [ 12990, 4916, 8063, 5422 ] } ], "ref_ids": [] }, { "id": 12999, "type": "theorem", "label": "spaces-divisors-lemma-composition-finite-type-blowups", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-composition-finite-type-blowups", "contents": [ "Let $S$ be a scheme and let $X$ be an algebraic space over $S$.", "Assume $X$ is quasi-compact and quasi-separated.", "Let $Z \\subset X$ be a closed subspace of finite presentation.", "Let $b : X' \\to X$ be the blowing up with center $Z$.", "Let $Z' \\subset X'$ be a closed subspace of finite presentation.", "Let $X'' \\to X'$ be the blowing up with center $Z'$.", "There exists a closed subspace $Y \\subset X$ of finite presentation,", "such that", "\\begin{enumerate}", "\\item $|Y| = |Z| \\cup |b|(|Z'|)$, and", "\\item the composition $X'' \\to X$ is isomorphic to the blowing up", "of $X$ in $Y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The condition that $Z \\to X$ is of finite presentation means that", "$Z$ is cut out by a finite type quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_X$, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-closed-immersion-finite-presentation}.", "Write $\\mathcal{A} = \\bigoplus_{n \\geq 0} \\mathcal{I}^n$ so that", "$X' = \\underline{\\text{Proj}}(\\mathcal{A})$.", "Note that $X \\setminus Z$ is a quasi-compact open subspace of $X$ by", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-quasi-coherent-finite-type-ideals}.", "Since $b^{-1}(X \\setminus Z) \\to X \\setminus Z$ is an isomorphism", "(Lemma \\ref{lemma-blowing-up-gives-effective-Cartier-divisor}) the same", "result shows that", "$b^{-1}(X \\setminus Z) \\setminus Z'$ is quasi-compact open subspace in $X'$.", "Hence $U = X \\setminus (Z \\cup b(Z'))$ is quasi-compact open subspace in $X$.", "By Lemma \\ref{lemma-closed-subscheme-proj-finite-type}", "there exist a $d > 0$ and a finite type", "$\\mathcal{O}_X$-submodule $\\mathcal{F} \\subset \\mathcal{I}^d$ such", "that $Z' = \\underline{\\text{Proj}}(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$", "and such that the support of $\\mathcal{I}^d/\\mathcal{F}$ is contained", "in $X \\setminus U$.", "\\medskip\\noindent", "Since $\\mathcal{F} \\subset \\mathcal{I}^d$ is an $\\mathcal{O}_X$-submodule", "we may think of $\\mathcal{F} \\subset \\mathcal{I}^d \\subset \\mathcal{O}_X$", "as a finite type quasi-coherent sheaf of ideals on $X$. Let's denote this", "$\\mathcal{J} \\subset \\mathcal{O}_X$ to prevent confusion. Since", "$\\mathcal{I}^d / \\mathcal{J}$ and $\\mathcal{O}/\\mathcal{I}^d$ are", "supported on $|X| \\setminus |U|$ we see that $|V(\\mathcal{J})|$ is contained", "in $|X| \\setminus |U|$. Conversely, as $\\mathcal{J} \\subset \\mathcal{I}^d$", "we see that $|Z| \\subset |V(\\mathcal{J})|$. Over", "$X \\setminus Z \\cong X' \\setminus b^{-1}(Z)$ the sheaf of ideals", "$\\mathcal{J}$ cuts out $Z'$ (see displayed formula below). Hence", "$|V(\\mathcal{J})|$ equals $|Z| \\cup |b|(|Z'|)$. It follows that also", "$|V(\\mathcal{I}\\mathcal{J})| = |Z| \\cup |b|(|Z'|)$. Moreover,", "$\\mathcal{I}\\mathcal{J}$ is an ideal of finite type as a product of two such.", "We claim that $X'' \\to X$ is isomorphic to the blowing up of $X$ in", "$\\mathcal{I}\\mathcal{J}$ which finishes the proof of the lemma by setting", "$Y = V(\\mathcal{I}\\mathcal{J})$.", "\\medskip\\noindent", "First, recall that the blowup of $X$ in $\\mathcal{I}\\mathcal{J}$", "is the same as the blowup of $X'$ in $b^{-1}\\mathcal{J} \\mathcal{O}_{X'}$,", "see Lemma \\ref{lemma-blowing-up-two-ideals}.", "Hence it suffices to show that the blowup of $X'$ in", "$b^{-1}\\mathcal{J} \\mathcal{O}_{X'}$ agrees with the blowup of $X'$", "in $Z'$. We will show that", "$$", "b^{-1}\\mathcal{J} \\mathcal{O}_{X'} = \\mathcal{I}_E^d \\mathcal{I}_{Z'}", "$$", "as ideal sheaves on $X''$. This will prove what we want as", "$\\mathcal{I}_E^d$ cuts out the effective Cartier divisor $dE$", "and we can use Lemmas \\ref{lemma-blow-up-effective-Cartier-divisor} and", "\\ref{lemma-blowing-up-two-ideals}.", "\\medskip\\noindent", "To see the displayed equality of the ideals we may work locally.", "With notation $A$, $I$, $a \\in I$ as in Lemma \\ref{lemma-blowing-up-affine}", "we see that $\\mathcal{F}$ corresponds to an $R$-submodule $M \\subset I^d$", "mapping isomorphically to an ideal $J \\subset R$. The condition", "$Z' = \\underline{\\text{Proj}}(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$", "means that $Z' \\cap \\Spec(A[\\frac{I}{a}])$ is cut out by the ideal", "generated by the elements $m/a^d$, $m \\in M$. Say the element $m \\in M$", "corresponds to the function $f \\in J$. Then in the affine blowup algebra", "$A' = A[\\frac{I}{a}]$ we see that $f = (a^dm)/a^d = a^d (m/a^d)$.", "Thus the equality holds." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-finite-presentation", "spaces-limits-lemma-quasi-coherent-finite-type-ideals", "spaces-divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "spaces-divisors-lemma-closed-subscheme-proj-finite-type", "spaces-divisors-lemma-blowing-up-two-ideals", "spaces-divisors-lemma-blow-up-effective-Cartier-divisor", "spaces-divisors-lemma-blowing-up-two-ideals", "spaces-divisors-lemma-blowing-up-affine" ], "ref_ids": [ 4849, 4621, 12991, 12987, 12997, 12993, 12997, 12989 ] } ], "ref_ids": [] }, { "id": 13000, "type": "theorem", "label": "spaces-divisors-lemma-strict-transform-local", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-strict-transform-local", "contents": [ "In the situation of Definition \\ref{definition-strict-transform}.", "Let", "$$", "\\xymatrix{", "U \\ar[r] \\ar[d] & X \\ar[d] \\\\", "V \\ar[r] & B", "}", "$$", "be a commutative diagram of morphisms with $U$ and $V$ schemes and", "\\'etale horizontal arrows. Let $V' \\to V$ be the blowup of $V$", "in $Z \\times_B V$. Then", "\\begin{enumerate}", "\\item $V' = V \\times_B B'$ and the maps", "$V' \\to B'$ and $U \\times_V V' \\to X \\times_B B'$ are \\'etale,", "\\item the strict transform $U'$ of $U$ relative to $V' \\to V$", "is equal to $X' \\times_X U$ where $X'$ is the strict transform of $X$", "relative to $B' \\to B$, and", "\\item for a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the", "restriction of the strict transform $\\mathcal{F}'$ to", "$U \\times_V V'$ is the strict transform of $\\mathcal{F}|_U$ relative", "to $V' \\to V$.", "\\end{enumerate}" ], "refs": [ "spaces-divisors-definition-strict-transform" ], "proofs": [ { "contents": [ "Part (1) follows from the fact that blowup commutes with flat base", "change (Lemma \\ref{lemma-flat-base-change-blowing-up}), the fact that", "\\'etale morphisms are flat, and that the base change of an \\'etale", "morphism is \\'etale. Part (3) then follows from the fact that taking", "the sheaf of sections supported on a closed commutes with pullback", "by \\'etale morphisms, see Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-sections-supported-on-closed-subset}.", "Part (2) follows from (3) applied to $\\mathcal{F} = \\mathcal{O}_X$." ], "refs": [ "spaces-divisors-lemma-flat-base-change-blowing-up", "spaces-limits-lemma-sections-supported-on-closed-subset" ], "ref_ids": [ 12990, 4624 ] } ], "ref_ids": [ 13030 ] }, { "id": 13001, "type": "theorem", "label": "spaces-divisors-lemma-strict-transform", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-strict-transform", "contents": [ "In the situation of Definition \\ref{definition-strict-transform}.", "\\begin{enumerate}", "\\item The strict transform $X'$ of $X$ is the blowup of $X$ in the closed", "subspace $f^{-1}Z$ of $X$.", "\\item For a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the", "strict transform $\\mathcal{F}'$ is canonically isomorphic to", "the pushforward along $X' \\to X \\times_B B'$ of the strict transform of", "$\\mathcal{F}$ relative to the blowing up $X' \\to X$.", "\\end{enumerate}" ], "refs": [ "spaces-divisors-definition-strict-transform" ], "proofs": [ { "contents": [ "Let $X'' \\to X$ be the blowup of $X$ in $f^{-1}Z$. By the universal", "property of blowing up (Lemma \\ref{lemma-universal-property-blowing-up})", "there exists a commutative diagram", "$$", "\\xymatrix{", "X'' \\ar[r] \\ar[d] & X \\ar[d] \\\\", "B' \\ar[r] & B", "}", "$$", "whence a morphism $i : X'' \\to X \\times_B B'$. The first assertion", "of the lemma is that $i$ is a closed immersion with image $X'$.", "The second assertion of the lemma is that $\\mathcal{F}' = i_*\\mathcal{F}''$", "where $\\mathcal{F}''$ is the strict transform of $\\mathcal{F}$ with", "respect to the blowing up $X'' \\to X$. We can check these assertions", "\\'etale locally on $X$, hence we reduce to the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-strict-transform}).", "Some details omitted." ], "refs": [ "spaces-divisors-lemma-universal-property-blowing-up", "divisors-lemma-strict-transform" ], "ref_ids": [ 12992, 8065 ] } ], "ref_ids": [ 13030 ] }, { "id": 13002, "type": "theorem", "label": "spaces-divisors-lemma-strict-transform-flat", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-strict-transform-flat", "contents": [ "In the situation of Definition \\ref{definition-strict-transform}.", "\\begin{enumerate}", "\\item If $X$ is flat over $B$ at all points lying over $Z$, then", "the strict transform of $X$ is equal to the base change $X \\times_B B'$.", "\\item Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "If $\\mathcal{F}$ is flat over $B$ at all points lying over $Z$, then", "the strict transform $\\mathcal{F}'$ of $\\mathcal{F}$ is equal to the", "pullback $\\text{pr}_X^*\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [ "spaces-divisors-definition-strict-transform" ], "proofs": [ { "contents": [ "Omitted. Hint: Follows from the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-strict-transform-flat})", "by \\'etale localization", "(Lemma \\ref{lemma-strict-transform-local})." ], "refs": [ "divisors-lemma-strict-transform-flat", "spaces-divisors-lemma-strict-transform-local" ], "ref_ids": [ 8066, 13000 ] } ], "ref_ids": [ 13030 ] }, { "id": 13003, "type": "theorem", "label": "spaces-divisors-lemma-strict-transform-affine", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-strict-transform-affine", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $Z \\subset B$ be a closed subspace.", "Let $b : B' \\to B$ be the blowing up of $Z$ in $B$. Let", "$g : X \\to Y$ be an affine morphism of spaces over $B$.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $g' : X \\times_B B' \\to Y \\times_B B'$ be the base change", "of $g$. Let $\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$", "relative to $b$. Then $g'_*\\mathcal{F}'$ is the strict transform", "of $g_*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Follows from the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-strict-transform-affine})", "by \\'etale localization (Lemma \\ref{lemma-strict-transform-local})." ], "refs": [ "divisors-lemma-strict-transform-affine", "spaces-divisors-lemma-strict-transform-local" ], "ref_ids": [ 8067, 13000 ] } ], "ref_ids": [] }, { "id": 13004, "type": "theorem", "label": "spaces-divisors-lemma-strict-transform-different-centers", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-strict-transform-different-centers", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $Z \\subset B$ be a closed subspace.", "Let $D \\subset B$ be an effective Cartier divisor.", "Let $Z' \\subset B$ be the closed subspace cut out by the product", "of the ideal sheaves of $Z$ and $D$.", "Let $B' \\to B$ be the blowup of $B$ in $Z$.", "\\begin{enumerate}", "\\item The blowup of $B$ in $Z'$ is isomorphic to $B' \\to B$.", "\\item Let $f : X \\to B$ be a morphism of algebraic spaces and let $\\mathcal{F}$", "be a quasi-coherent $\\mathcal{O}_X$-module. If the subsheaf of $\\mathcal{F}$ of", "sections supported on $|f^{-1}D|$ is zero, then the", "strict transform of $\\mathcal{F}$ relative to the blowing up", "in $Z$ agrees with the strict transform of $\\mathcal{F}$ relative", "to the blowing up of $B$ in $Z'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Follows from the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-strict-transform-different-centers})", "by \\'etale localization (Lemma \\ref{lemma-strict-transform-local})." ], "refs": [ "divisors-lemma-strict-transform-different-centers", "spaces-divisors-lemma-strict-transform-local" ], "ref_ids": [ 8068, 13000 ] } ], "ref_ids": [] }, { "id": 13005, "type": "theorem", "label": "spaces-divisors-lemma-strict-transform-composition-blowups", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-strict-transform-composition-blowups", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $Z \\subset B$ be a closed subspace.", "Let $b : B' \\to B$ be the blowing up with center $Z$.", "Let $Z' \\subset B'$ be a closed subspace.", "Let $B'' \\to B'$ be the blowing up with center $Z'$.", "Let $Y \\subset B$ be a closed subscheme such that", "$|Y| = |Z| \\cup |b|(|Z'|)$ and the composition $B'' \\to B$", "is isomorphic to the blowing up of $B$ in $Y$.", "In this situation, given any scheme $X$ over $B$ and", "$\\mathcal{F} \\in \\QCoh(\\mathcal{O}_X)$ we have", "\\begin{enumerate}", "\\item the strict transform of $\\mathcal{F}$ with respect to the blowing", "up of $B$ in $Y$ is equal to the strict transform with respect to the", "blowup $B'' \\to B'$ in $Z'$ of the strict transform of $\\mathcal{F}$", "with respect to the blowup $B' \\to B$ of $B$ in $Z$, and", "\\item the strict transform of $X$ with respect to the blowing", "up of $B$ in $Y$ is equal to the strict transform with respect to the", "blowup $B'' \\to B'$ in $Z'$ of the strict transform of $X$", "with respect to the blowup $B' \\to B$ of $B$ in $Z$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Follows from the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-strict-transform-composition-blowups})", "by \\'etale localization (Lemma \\ref{lemma-strict-transform-local})." ], "refs": [ "divisors-lemma-strict-transform-composition-blowups", "spaces-divisors-lemma-strict-transform-local" ], "ref_ids": [ 8069, 13000 ] } ], "ref_ids": [] }, { "id": 13006, "type": "theorem", "label": "spaces-divisors-lemma-strict-transform-universally-injective", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-strict-transform-universally-injective", "contents": [ "In the situation of Definition \\ref{definition-strict-transform}.", "Suppose that", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0", "$$", "is an exact sequence of quasi-coherent sheaves on $X$ which remains", "exact after any base change $T \\to B$. Then the strict transforms of", "$\\mathcal{F}_i'$ relative to any blowup $B' \\to B$", "form a short exact sequence", "$0 \\to \\mathcal{F}'_1 \\to \\mathcal{F}'_2 \\to \\mathcal{F}'_3 \\to 0$ too." ], "refs": [ "spaces-divisors-definition-strict-transform" ], "proofs": [ { "contents": [ "Omitted. Hint: Follows from the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-strict-transform-universally-injective})", "by \\'etale localization (Lemma \\ref{lemma-strict-transform-local})." ], "refs": [ "divisors-lemma-strict-transform-universally-injective", "spaces-divisors-lemma-strict-transform-local" ], "ref_ids": [ 8070, 13000 ] } ], "ref_ids": [ 13030 ] }, { "id": 13007, "type": "theorem", "label": "spaces-divisors-lemma-strict-transform-blowup-fitting-ideal", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-strict-transform-blowup-fitting-ideal", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_B$-module.", "Let $Z_k \\subset S$ be the closed subscheme cut out by", "$\\text{Fit}_k(\\mathcal{F})$, see Section \\ref{section-fitting-ideals}.", "Let $B' \\to B$ be the blowup of $B$ in $Z_k$ and let", "$\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$.", "Then $\\mathcal{F}'$ can locally be generated by $\\leq k$", "sections." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Follows from the case of schemes", "(Divisors, Lemma \\ref{divisors-lemma-strict-transform-blowup-fitting-ideal})", "by \\'etale localization (Lemma \\ref{lemma-strict-transform-local})." ], "refs": [ "divisors-lemma-strict-transform-blowup-fitting-ideal", "spaces-divisors-lemma-strict-transform-local" ], "ref_ids": [ 8076, 13000 ] } ], "ref_ids": [] }, { "id": 13008, "type": "theorem", "label": "spaces-divisors-lemma-strict-transform-blowup-fitting-ideal-locally-free", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-strict-transform-blowup-fitting-ideal-locally-free", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_B$-module.", "Let $Z_k \\subset S$ be the closed subscheme cut out by", "$\\text{Fit}_k(\\mathcal{F})$, see Section \\ref{section-fitting-ideals}.", "Assume that $\\mathcal{F}$ is locally free of rank $k$ on $B \\setminus Z_k$.", "Let $B' \\to B$ be the blowup of $B$ in $Z_k$ and let", "$\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$.", "Then $\\mathcal{F}'$ is locally free of rank $k$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Follows from the case of schemes", "(Divisors, Lemma", "\\ref{divisors-lemma-strict-transform-blowup-fitting-ideal-locally-free})", "by \\'etale localization (Lemma \\ref{lemma-strict-transform-local})." ], "refs": [ "divisors-lemma-strict-transform-blowup-fitting-ideal-locally-free", "spaces-divisors-lemma-strict-transform-local" ], "ref_ids": [ 8077, 13000 ] } ], "ref_ids": [] }, { "id": 13009, "type": "theorem", "label": "spaces-divisors-lemma-composition-admissible-blowups", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-composition-admissible-blowups", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $U \\subset X$ be a quasi-compact open subspace.", "Let $b : X' \\to X$ be a $U$-admissible blowup.", "Let $X'' \\to X'$ be a $U$-admissible blowup.", "Then the composition $X'' \\to X$ is a $U$-admissible blowup." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the more precise", "Lemma \\ref{lemma-composition-finite-type-blowups}." ], "refs": [ "spaces-divisors-lemma-composition-finite-type-blowups" ], "ref_ids": [ 12999 ] } ], "ref_ids": [] }, { "id": 13010, "type": "theorem", "label": "spaces-divisors-lemma-extend-admissible-blowups", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-extend-admissible-blowups", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space.", "Let $U, V \\subset X$ be quasi-compact open subspaces.", "Let $b : V' \\to V$ be a $U \\cap V$-admissible blowup.", "Then there exists a $U$-admissible blowup $X' \\to X$", "whose restriction to $V$ is $V'$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\subset \\mathcal{O}_V$ be the finite type", "quasi-coherent sheaf of ideals such that $V(\\mathcal{I})$ is", "disjoint from $U \\cap V$ and such that $V'$ is isomorphic to the", "blowup of $V$ in $\\mathcal{I}$. Let", "$\\mathcal{I}' \\subset \\mathcal{O}_{U \\cup V}$ be the quasi-coherent", "sheaf of ideals whose restriction to $U$ is $\\mathcal{O}_U$ and", "whose restriction to $V$ is $\\mathcal{I}$.", "By Limits of Spaces, Lemma \\ref{spaces-limits-lemma-extend}", "there exists a finite type quasi-coherent sheaf of ideals", "$\\mathcal{J} \\subset \\mathcal{O}_X$ whose restriction to $U \\cup V$ is", "$\\mathcal{I}'$. The lemma follows." ], "refs": [ "spaces-limits-lemma-extend" ], "ref_ids": [ 4608 ] } ], "ref_ids": [] }, { "id": 13011, "type": "theorem", "label": "spaces-divisors-lemma-dominate-admissible-blowups", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-dominate-admissible-blowups", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $U \\subset X$ be a quasi-compact open subspace.", "Let $b_i : X_i \\to X$, $i = 1, \\ldots, n$ be $U$-admissible blowups.", "There exists a $U$-admissible blowup $b : X' \\to X$ such that", "(a) $b$ factors as $X' \\to X_i \\to X$ for $i = 1, \\ldots, n$ and", "(b) each of the morphisms $X' \\to X_i$ is a $U$-admissible blowup." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I}_i \\subset \\mathcal{O}_X$ be the finite type", "quasi-coherent sheaf of ideals such that $V(\\mathcal{I}_i)$ is", "disjoint from $U$ and such that $X_i$ is isomorphic to the", "blowup of $X$ in $\\mathcal{I}_i$. Set", "$\\mathcal{I} = \\mathcal{I}_1 \\cdot \\ldots \\cdot \\mathcal{I}_n$", "and let $X'$ be the blowup of $X$ in $\\mathcal{I}$. Then", "$X' \\to X$ factors through $b_i$ by Lemma \\ref{lemma-blowing-up-two-ideals}." ], "refs": [ "spaces-divisors-lemma-blowing-up-two-ideals" ], "ref_ids": [ 12997 ] } ], "ref_ids": [] }, { "id": 13012, "type": "theorem", "label": "spaces-divisors-lemma-separate-disjoint-opens-by-blowing-up", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-lemma-separate-disjoint-opens-by-blowing-up", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $U, V$ be quasi-compact disjoint open subspaces of $X$.", "Then there exist a $U \\cup V$-admissible blowup $b : X' \\to X$", "such that $X'$ is a disjoint union of open subspaces", "$X' = X'_1 \\amalg X'_2$ with $b^{-1}(U) \\subset X'_1$ and", "$b^{-1}(V) \\subset X'_2$." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite type quasi-coherent sheaf of ideals $\\mathcal{I}$,", "resp.\\ $\\mathcal{J}$ such that $X \\setminus U = V(\\mathcal{I})$,", "resp.\\ $X \\setminus V = V(\\mathcal{J})$, see", "Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-quasi-coherent-finite-type-ideals}.", "Then $|V(\\mathcal{I}\\mathcal{J})| = |X|$. Hence", "$\\mathcal{I}\\mathcal{J}$ is a locally nilpotent sheaf of ideals.", "Since $\\mathcal{I}$ and $\\mathcal{J}$ are of finite type and $X$", "is quasi-compact there exists an $n > 0$ such that", "$\\mathcal{I}^n \\mathcal{J}^n = 0$. We may and do replace $\\mathcal{I}$", "by $\\mathcal{I}^n$ and $\\mathcal{J}$ by $\\mathcal{J}^n$. Whence", "$\\mathcal{I} \\mathcal{J} = 0$. Let $b : X' \\to X$ be the blowing", "up in $\\mathcal{I} + \\mathcal{J}$. This is $U \\cup V$-admissible", "as $|V(\\mathcal{I} + \\mathcal{J})| = |X| \\setminus |U| \\cup |V|$.", "We will show that $X'$ is a disjoint union of open subspaces", "$X' = X'_1 \\amalg X'_2$ as in the statement of the lemma.", "\\medskip\\noindent", "Since $|V(\\mathcal{I} + \\mathcal{J})|$ is the complement of", "$|U \\cup V|$ we conclude that $V \\cup U$ is scheme theoretically", "dense in $X'$, see", "Lemmas \\ref{lemma-blowing-up-gives-effective-Cartier-divisor} and", "\\ref{lemma-complement-effective-Cartier-divisor}.", "Thus if such a decomposition $X' = X'_1 \\amalg X'_2$", "into open and closed subspaces exists, then $X'_1$ is the", "scheme theoretic closure of $U$ in $X'$ and similarly $X'_2$ is", "the scheme theoretic closure of $V$ in $X'$. Since $U \\to X'$", "and $V \\to X'$ are quasi-compact taking scheme theoretic", "closures commutes with \\'etale localization (Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image}).", "Hence to verify the existence of $X'_1$ and $X'_2$ we may work \\'etale", "locally on $X$. This reduces us to the case of schemes which is", "treated in the proof of Divisors, Lemma", "\\ref{divisors-lemma-separate-disjoint-opens-by-blowing-up}." ], "refs": [ "spaces-limits-lemma-quasi-coherent-finite-type-ideals", "spaces-divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "spaces-divisors-lemma-complement-effective-Cartier-divisor", "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image", "divisors-lemma-separate-disjoint-opens-by-blowing-up" ], "ref_ids": [ 4621, 12991, 12937, 4780, 8074 ] } ], "ref_ids": [] }, { "id": 13032, "type": "theorem", "label": "dga-lemma-total-complex-tensor-product", "categories": [ "dga" ], "title": "dga-lemma-total-complex-tensor-product", "contents": [ "Let $R$ be a ring.", "Let $(A, \\text{d})$, $(B, \\text{d})$ be differential graded algebras over $R$.", "Denote $A^\\bullet$, $B^\\bullet$ the underlying cochain complexes.", "As cochain complexes of $R$-modules we have", "$$", "(A \\otimes_R B)^\\bullet = \\text{Tot}(A^\\bullet \\otimes_R B^\\bullet).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Recall that the differential of the total complex is given by", "$\\text{d}_1^{p, q} + (-1)^p \\text{d}_2^{p, q}$ on $A^p \\otimes_R B^q$.", "And this is exactly the same as the rule for the differential", "on $A \\otimes_R B$ in", "Definition \\ref{definition-tensor-product}." ], "refs": [ "dga-definition-tensor-product" ], "ref_ids": [ 13137 ] } ], "ref_ids": [] }, { "id": 13033, "type": "theorem", "label": "dga-lemma-dgm-abelian", "categories": [ "dga" ], "title": "dga-lemma-dgm-abelian", "contents": [ "Let $(A, d)$ be a differential graded algebra. The category", "$\\text{Mod}_{(A, \\text{d})}$ is abelian and has arbitrary limits and colimits." ], "refs": [], "proofs": [ { "contents": [ "Kernels and cokernels commute with taking underlying $A$-modules.", "Similarly for direct sums and colimits. In other words, these operations", "in $\\text{Mod}_{(A, \\text{d})}$ commute with the forgetful functor to the", "category of $A$-modules. This is not the case for products and limits.", "Namely, if $N_i$, $i \\in I$ is a family of", "differential graded $A$-modules, then the product $\\prod N_i$ in", "$\\text{Mod}_{(A, \\text{d})}$ is given by setting $(\\prod N_i)^n = \\prod N_i^n$", "and $\\prod N_i = \\bigoplus_n (\\prod N_i)^n$. Thus we see that the product", "does commute with the forgetful functor to the category of graded $A$-modules.", "A category with products and equalizers has limits, see", "Categories, Lemma \\ref{categories-lemma-limits-products-equalizers}." ], "refs": [ "categories-lemma-limits-products-equalizers" ], "ref_ids": [ 12213 ] } ], "ref_ids": [] }, { "id": 13034, "type": "theorem", "label": "dga-lemma-compose-homotopy", "categories": [ "dga" ], "title": "dga-lemma-compose-homotopy", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $f, g : L \\to M$ be homomorphisms of differential graded $A$-modules.", "Suppose given further homomorphisms $a : K \\to L$, and $c : M \\to N$.", "If $h : L \\to M$ is an $A$-module map which defines a homotopy between", "$f$ and $g$, then $c \\circ h \\circ a$ defines a homotopy between", "$c \\circ f \\circ a$ and $c \\circ g \\circ a$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Homology, Lemma \\ref{homology-lemma-compose-homotopy-cochain}." ], "refs": [ "homology-lemma-compose-homotopy-cochain" ], "ref_ids": [ 12058 ] } ], "ref_ids": [] }, { "id": 13035, "type": "theorem", "label": "dga-lemma-homotopy-direct-sums", "categories": [ "dga" ], "title": "dga-lemma-homotopy-direct-sums", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "The homotopy category $K(\\text{Mod}_{(A, \\text{d})})$", "has direct sums and products." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Just use the direct sums and products as in", "Lemma \\ref{lemma-dgm-abelian}. This works because we saw that", "these functors commute with the forgetful functor to the category", "of graded $A$-modules and because $\\prod$ is an exact functor", "on the category of families of abelian groups." ], "refs": [ "dga-lemma-dgm-abelian" ], "ref_ids": [ 13033 ] } ], "ref_ids": [] }, { "id": 13036, "type": "theorem", "label": "dga-lemma-functorial-cone", "categories": [ "dga" ], "title": "dga-lemma-functorial-cone", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Suppose that", "$$", "\\xymatrix{", "K_1 \\ar[r]_{f_1} \\ar[d]_a & L_1 \\ar[d]^b \\\\", "K_2 \\ar[r]^{f_2} & L_2", "}", "$$", "is a diagram of homomorphisms of differential graded $A$-modules which is", "commutative up to homotopy.", "Then there exists a morphism $c : C(f_1) \\to C(f_2)$ which gives rise to", "a morphism of triangles", "$$", "(a, b, c) : (K_1, L_1, C(f_1), f_1, i_1, p_1) \\to", "(K_1, L_1, C(f_1), f_2, i_2, p_2)", "$$", "in $K(\\text{Mod}_{(A, \\text{d})})$." ], "refs": [], "proofs": [ { "contents": [ "Let $h : K_1 \\to L_2$ be a homotopy between $f_2 \\circ a$ and $b \\circ f_1$.", "Define $c$ by the matrix", "$$", "c =", "\\left(", "\\begin{matrix}", "b & h \\\\", "0 & a", "\\end{matrix}", "\\right) :", "L_1 \\oplus K_1 \\to L_2 \\oplus K_2", "$$", "A matrix computation show that $c$ is a morphism of differential", "graded modules. It is trivial that $c \\circ i_1 = i_2 \\circ b$, and it is", "trivial also to check that $p_2 \\circ c = a \\circ p_1$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13037, "type": "theorem", "label": "dga-lemma-admissible-ses", "categories": [ "dga" ], "title": "dga-lemma-admissible-ses", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $0 \\to K \\to L \\to M \\to 0$ be an admissible short exact sequence", "of differential graded $A$-modules. Let $s : M \\to L$ and $\\pi : L \\to K$", "be splittings such that $\\Ker(\\pi) = \\Im(s)$.", "Then we obtain a morphism", "$$", "\\delta = \\pi \\circ \\text{d}_L \\circ s : M \\to K[1]", "$$", "of $\\text{Mod}_{(A, \\text{d})}$ which induces the boundary maps", "in the long exact sequence of cohomology (\\ref{equation-les})." ], "refs": [], "proofs": [ { "contents": [ "The map $\\pi \\circ \\text{d}_L \\circ s$ is compatible with the $A$-module", "structure and the gradings by construction. It is compatible with", "differentials by Homology, Lemmas", "\\ref{homology-lemma-ses-termwise-split-cochain}.", "Let $R$ be the ring that $A$ is a differential graded algebra over.", "The equality of maps is a statement about $R$-modules. Hence this", "follows from Homology, Lemmas", "\\ref{homology-lemma-ses-termwise-split-cochain} and", "\\ref{homology-lemma-ses-termwise-split-long-cochain}." ], "refs": [ "homology-lemma-ses-termwise-split-cochain", "homology-lemma-ses-termwise-split-cochain", "homology-lemma-ses-termwise-split-long-cochain" ], "ref_ids": [ 12067, 12067, 12068 ] } ], "ref_ids": [] }, { "id": 13038, "type": "theorem", "label": "dga-lemma-make-commute-map", "categories": [ "dga" ], "title": "dga-lemma-make-commute-map", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. Let", "$$", "\\xymatrix{", "K \\ar[r]_f \\ar[d]_a & L \\ar[d]^b \\\\", "M \\ar[r]^g & N", "}", "$$", "be a diagram of homomorphisms of differential graded $A$-modules", "commuting up to homotopy.", "\\begin{enumerate}", "\\item If $f$ is an admissible monomorphism, then $b$ is homotopic to a", "homomorphism which makes the diagram commute.", "\\item If $g$ is an admissible epimorphism, then $a$ is homotopic to a", "morphism which makes the diagram commute.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $h : K \\to N$ be a homotopy between $bf$ and $ga$, i.e.,", "$bf - ga = \\text{d}h + h\\text{d}$. Suppose that $\\pi : L \\to K$", "is a graded $A$-module map left inverse to $f$. Take", "$b' = b - \\text{d}h\\pi - h\\pi \\text{d}$.", "Suppose $s : N \\to M$ is a graded $A$-module map right inverse to $g$.", "Take $a' = a + \\text{d}sh + sh\\text{d}$.", "Computations omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13039, "type": "theorem", "label": "dga-lemma-make-injective", "categories": [ "dga" ], "title": "dga-lemma-make-injective", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $\\alpha : K \\to L$ be a homomorphism of differential graded", "$A$-modules. There exists a factorization", "$$", "\\xymatrix{", "K \\ar[r]^{\\tilde \\alpha} \\ar@/_1pc/[rr]_\\alpha &", "\\tilde L \\ar[r]^\\pi & L", "}", "$$", "in $\\text{Mod}_{(A, \\text{d})}$ such that", "\\begin{enumerate}", "\\item $\\tilde \\alpha$ is an admissible monomorphism (see", "Definition \\ref{definition-admissible-ses}),", "\\item there is a morphism $s : L \\to \\tilde L$", "such that $\\pi \\circ s = \\text{id}_L$ and such that", "$s \\circ \\pi$ is homotopic to $\\text{id}_{\\tilde L}$.", "\\end{enumerate}" ], "refs": [ "dga-definition-admissible-ses" ], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Derived Categories, Lemma \\ref{derived-lemma-make-injective}.", "Namely, we set $\\tilde L = L \\oplus C(1_K)$ and we use elementary", "properties of the cone construction." ], "refs": [ "derived-lemma-make-injective" ], "ref_ids": [ 1797 ] } ], "ref_ids": [ 13143 ] }, { "id": 13040, "type": "theorem", "label": "dga-lemma-sequence-maps-split", "categories": [ "dga" ], "title": "dga-lemma-sequence-maps-split", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $L_1 \\to L_2 \\to \\ldots \\to L_n$", "be a sequence of composable homomorphisms of", "differential graded $A$-modules.", "There exists a commutative diagram", "$$", "\\xymatrix{", "L_1 \\ar[r] &", "L_2 \\ar[r] &", "\\ldots \\ar[r] &", "L_n \\\\", "M_1 \\ar[r] \\ar[u] &", "M_2 \\ar[r] \\ar[u] &", "\\ldots \\ar[r] &", "M_n \\ar[u]", "}", "$$", "in $\\text{Mod}_{(A, \\text{d})}$ such that each $M_i \\to M_{i + 1}$", "is an admissible monomorphism and each $M_i \\to L_i$", "is a homotopy equivalence." ], "refs": [], "proofs": [ { "contents": [ "The case $n = 1$ is without content.", "Lemma \\ref{lemma-make-injective} is the case $n = 2$.", "Suppose we have constructed the diagram", "except for $M_n$. Apply Lemma \\ref{lemma-make-injective} to", "the composition $M_{n - 1} \\to L_{n - 1} \\to L_n$.", "The result is a factorization $M_{n - 1} \\to M_n \\to L_n$", "as desired." ], "refs": [ "dga-lemma-make-injective", "dga-lemma-make-injective" ], "ref_ids": [ 13039, 13039 ] } ], "ref_ids": [] }, { "id": 13041, "type": "theorem", "label": "dga-lemma-nilpotent", "categories": [ "dga" ], "title": "dga-lemma-nilpotent", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $0 \\to K_i \\to L_i \\to M_i \\to 0$, $i = 1, 2, 3$", "be admissible short exact sequence of differential graded $A$-modules.", "Let $b : L_1 \\to L_2$ and $b' : L_2 \\to L_3$", "be homomorphisms of differential graded modules such that", "$$", "\\vcenter{", "\\xymatrix{", "K_1 \\ar[d]_0 \\ar[r] &", "L_1 \\ar[r] \\ar[d]_b &", "M_1 \\ar[d]_0 \\\\", "K_2 \\ar[r] & L_2 \\ar[r] & M_2", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "K_2 \\ar[d]^0 \\ar[r] &", "L_2 \\ar[r] \\ar[d]^{b'} &", "M_2 \\ar[d]^0 \\\\", "K_3 \\ar[r] & L_3 \\ar[r] & M_3", "}", "}", "$$", "commute up to homotopy. Then $b' \\circ b$ is homotopic to $0$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-make-commute-map} we can replace $b$ and $b'$ by", "homotopic maps such that the right square of the left diagram commutes", "and the left square of the right diagram commutes. In other words, we have", "$\\Im(b) \\subset \\Im(K_2 \\to L_2)$ and", "$\\Ker((b')^n) \\supset \\Im(K_2 \\to L_2)$.", "Then $b \\circ b' = 0$ as a map of modules." ], "refs": [ "dga-lemma-make-commute-map" ], "ref_ids": [ 13038 ] } ], "ref_ids": [] }, { "id": 13042, "type": "theorem", "label": "dga-lemma-triangle-independent-splittings", "categories": [ "dga" ], "title": "dga-lemma-triangle-independent-splittings", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. Let", "$0 \\to K \\to L \\to M \\to 0$ be an admissible short exact sequence", "of differential graded $A$-modules. The triangle", "\\begin{equation}", "\\label{equation-triangle-associated-to-admissible-ses}", "K \\to L \\to M \\xrightarrow{\\delta} K[1]", "\\end{equation}", "with $\\delta$ as in Lemma \\ref{lemma-admissible-ses} is, up to canonical", "isomorphism in $K(\\text{Mod}_{(A, \\text{d})})$, independent of the choices", "made in Lemma \\ref{lemma-admissible-ses}." ], "refs": [ "dga-lemma-admissible-ses", "dga-lemma-admissible-ses" ], "proofs": [ { "contents": [ "Namely, let $(s', \\pi')$ be a second choice of splittings as in", "Lemma \\ref{lemma-admissible-ses}. Then we claim that $\\delta$ and $\\delta'$", "are homotopic. Namely, write $s' = s + \\alpha \\circ h$ and", "$\\pi' = \\pi + g \\circ \\beta$ for some unique homomorphisms", "of $A$-modules $h : M \\to K$ and $g : M \\to K$ of degree $-1$.", "Then $g = -h$ and $g$ is a homotopy between $\\delta$ and $\\delta'$.", "The computations are done in the proof of", "Homology, Lemma \\ref{homology-lemma-ses-termwise-split-homotopy-cochain}." ], "refs": [ "dga-lemma-admissible-ses", "homology-lemma-ses-termwise-split-homotopy-cochain" ], "ref_ids": [ 13037, 12069 ] } ], "ref_ids": [ 13037, 13037 ] }, { "id": 13043, "type": "theorem", "label": "dga-lemma-rotate-cone", "categories": [ "dga" ], "title": "dga-lemma-rotate-cone", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $f : K \\to L$ be a homomorphism of differential graded modules.", "The triangle $(L, C(f), K[1], i, p, f[1])$ is", "the triangle associated to the admissible short exact sequence", "$$", "0 \\to L \\to C(f) \\to K[1] \\to 0", "$$", "coming from the definition of the cone of $f$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13044, "type": "theorem", "label": "dga-lemma-rotate-triangle", "categories": [ "dga" ], "title": "dga-lemma-rotate-triangle", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $\\alpha : K \\to L$ and $\\beta : L \\to M$", "define an admissible short exact sequence", "$$", "0 \\to K \\to L \\to M \\to 0", "$$", "of differential graded $A$-modules.", "Let $(K, L, M, \\alpha, \\beta, \\delta)$", "be the associated triangle. Then the triangles", "$$", "(M[-1], K, L, \\delta[-1], \\alpha, \\beta)", "\\quad\\text{and}\\quad", "(M[-1], K, C(\\delta[-1]), \\delta[-1], i, p)", "$$", "are isomorphic." ], "refs": [], "proofs": [ { "contents": [ "Using a choice of splittings we write $L = K \\oplus M$ and we identify", "$\\alpha$ and $\\beta$ with the natural inclusion and projection maps.", "By construction of $\\delta$ we have", "$$", "d_B =", "\\left(", "\\begin{matrix}", "d_K & \\delta \\\\", "0 & d_M", "\\end{matrix}", "\\right)", "$$", "On the other hand the cone of $\\delta[-1] : M[-1] \\to K$", "is given as $C(\\delta[-1]) = K \\oplus M$ with differential identical", "with the matrix above! Whence the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13045, "type": "theorem", "label": "dga-lemma-third-isomorphism", "categories": [ "dga" ], "title": "dga-lemma-third-isomorphism", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $f_1 : K_1 \\to L_1$ and $f_2 : K_2 \\to L_2$ be homomorphisms of", "differential graded $A$-modules. Let", "$$", "(a, b, c) :", "(K_1, L_1, C(f_1), f_1, i_1, p_1)", "\\longrightarrow", "(K_1, L_1, C(f_1), f_2, i_2, p_2)", "$$", "be any morphism of triangles of $K(\\text{Mod}_{(A, \\text{d})})$.", "If $a$ and $b$ are homotopy equivalences then so is $c$." ], "refs": [], "proofs": [ { "contents": [ "Let $a^{-1} : K_2 \\to K_1$ be a homomorphism of differential graded $A$-modules", "which is inverse to $a$ in $K(\\text{Mod}_{(A, \\text{d})})$.", "Let $b^{-1} : L_2 \\to L_1$ be a homomorphism of differential graded $A$-modules", "which is inverse to $b$ in $K(\\text{Mod}_{(A, \\text{d})})$.", "Let $c' : C(f_2) \\to C(f_1)$ be the morphism from", "Lemma \\ref{lemma-functorial-cone} applied to", "$f_1 \\circ a^{-1} = b^{-1} \\circ f_2$.", "If we can show that $c \\circ c'$ and $c' \\circ c$ are isomorphisms in", "$K(\\text{Mod}_{(A, \\text{d})})$", "then we win. Hence it suffices to prove the following: Given", "a morphism of triangles", "$(1, 1, c) : (K, L, C(f), f, i, p)$", "in $K(\\text{Mod}_{(A, \\text{d})})$ the morphism $c$ is an isomorphism", "in $K(\\text{Mod}_{(A, \\text{d})})$.", "By assumption the two squares in the diagram", "$$", "\\xymatrix{", "L \\ar[r] \\ar[d]_1 &", "C(f) \\ar[r] \\ar[d]_c &", "K[1] \\ar[d]_1 \\\\", "L \\ar[r] &", "C(f) \\ar[r] &", "K[1]", "}", "$$", "commute up to homotopy. By construction of $C(f)$ the rows", "form admissible short exact sequences. Thus we see that", "$(c - 1)^2 = 0$ in $K(\\text{Mod}_{(A, \\text{d})})$ by", "Lemma \\ref{lemma-nilpotent}.", "Hence $c$ is an isomorphism in $K(\\text{Mod}_{(A, \\text{d})})$", "with inverse $2 - c$." ], "refs": [ "dga-lemma-functorial-cone", "dga-lemma-nilpotent" ], "ref_ids": [ 13036, 13041 ] } ], "ref_ids": [] }, { "id": 13046, "type": "theorem", "label": "dga-lemma-the-same-up-to-isomorphisms", "categories": [ "dga" ], "title": "dga-lemma-the-same-up-to-isomorphisms", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "\\begin{enumerate}", "\\item Given an admissible short exact sequence", "$0 \\to K \\xrightarrow{\\alpha} L \\to M \\to 0$", "of differential graded $A$-modules there exists a homotopy equivalence", "$C(\\alpha) \\to M$ such that the diagram", "$$", "\\xymatrix{", "K \\ar[r] \\ar[d] & L \\ar[d] \\ar[r] &", "C(\\alpha) \\ar[r]_{-p} \\ar[d] & K[1] \\ar[d] \\\\", "K \\ar[r]^\\alpha & L \\ar[r]^\\beta &", "M \\ar[r]^\\delta & K[1]", "}", "$$", "defines an isomorphism of triangles in $K(\\text{Mod}_{(A, \\text{d})})$.", "\\item Given a morphism of complexes $f : K \\to L$", "there exists an isomorphism of triangles", "$$", "\\xymatrix{", "K \\ar[r] \\ar[d] & \\tilde L \\ar[d] \\ar[r] &", "M \\ar[r]_{\\delta} \\ar[d] & K[1] \\ar[d] \\\\", "K \\ar[r] & L \\ar[r] &", "C(f) \\ar[r]^{-p} & K[1]", "}", "$$", "where the upper triangle is the triangle associated to a", "admissible short exact sequence $K \\to \\tilde L \\to M$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). We have $C(\\alpha) = L \\oplus K$ and we simply define", "$C(\\alpha) \\to M$ via the projection onto $L$ followed by $\\beta$.", "This defines a morphism of differential graded modules because the", "compositions $K^{n + 1} \\to L^{n + 1} \\to M^{n + 1}$ are zero.", "Choose splittings $s : M \\to L$ and $\\pi : L \\to K$ with", "$\\Ker(\\pi) = \\Im(s)$ and set", "$\\delta = \\pi \\circ \\text{d}_L \\circ s$ as usual.", "To get a homotopy inverse we take", "$M \\to C(\\alpha)$ given by $(s , -\\delta)$. This is compatible with", "differentials because $\\delta^n$ can be characterized as the", "unique map $M^n \\to K^{n + 1}$ such that", "$\\text{d} \\circ s^n - s^{n + 1} \\circ \\text{d} = \\alpha \\circ \\delta^n$,", "see proof of", "Homology, Lemma \\ref{homology-lemma-ses-termwise-split-cochain}.", "The composition $M \\to C(f) \\to M$ is the identity.", "The composition $C(f) \\to M \\to C(f)$ is equal to the morphism", "$$", "\\left(", "\\begin{matrix}", "s \\circ \\beta & 0 \\\\", "-\\delta \\circ \\beta & 0", "\\end{matrix}", "\\right)", "$$", "To see that this is homotopic to the identity map", "use the homotopy $h : C(\\alpha) \\to C(\\alpha)$", "given by the matrix", "$$", "\\left(", "\\begin{matrix}", "0 & 0 \\\\", "\\pi & 0", "\\end{matrix}", "\\right) :", "C(\\alpha) = L \\oplus K", "\\to", "L \\oplus K = C(\\alpha)", "$$", "It is trivial to verify that", "$$", "\\left(", "\\begin{matrix}", "1 & 0 \\\\", "0 & 1", "\\end{matrix}", "\\right)", "-", "\\left(", "\\begin{matrix}", "s \\\\", "-\\delta", "\\end{matrix}", "\\right)", "\\left(", "\\begin{matrix}", "\\beta & 0", "\\end{matrix}", "\\right)", "=", "\\left(", "\\begin{matrix}", "\\text{d} & \\alpha \\\\", "0 & -\\text{d}", "\\end{matrix}", "\\right)", "\\left(", "\\begin{matrix}", "0 & 0 \\\\", "\\pi & 0", "\\end{matrix}", "\\right)", "+", "\\left(", "\\begin{matrix}", "0 & 0 \\\\", "\\pi & 0", "\\end{matrix}", "\\right)", "\\left(", "\\begin{matrix}", "\\text{d} & \\alpha \\\\", "0 & -\\text{d}", "\\end{matrix}", "\\right)", "$$", "To finish the proof of (1) we have to show that the morphisms", "$-p : C(\\alpha) \\to K[1]$ (see", "Definition \\ref{definition-cone})", "and $C(\\alpha) \\to M \\to K[1]$ agree up", "to homotopy. This is clear from the above. Namely, we can use the homotopy", "inverse $(s, -\\delta) : M \\to C(\\alpha)$", "and check instead that the two maps", "$M \\to K[1]$ agree. And note that", "$p \\circ (s, -\\delta) = -\\delta$ as desired.", "\\medskip\\noindent", "Proof of (2). We let $\\tilde f : K \\to \\tilde L$,", "$s : L \\to \\tilde L$", "and $\\pi : L \\to L$ be as in", "Lemma \\ref{lemma-make-injective}. By", "Lemmas \\ref{lemma-functorial-cone} and \\ref{lemma-third-isomorphism}", "the triangles $(K, L, C(f), i, p)$ and", "$(K, \\tilde L, C(\\tilde f), \\tilde i, \\tilde p)$", "are isomorphic. Note that we can compose isomorphisms of", "triangles. Thus we may replace $L$ by", "$\\tilde L$ and $f$ by $\\tilde f$. In other words", "we may assume that $f$ is an admissible monomorphism.", "In this case the result follows from part (1)." ], "refs": [ "homology-lemma-ses-termwise-split-cochain", "dga-definition-cone", "dga-lemma-make-injective", "dga-lemma-functorial-cone", "dga-lemma-third-isomorphism" ], "ref_ids": [ 12067, 13142, 13039, 13036, 13045 ] } ], "ref_ids": [] }, { "id": 13047, "type": "theorem", "label": "dga-lemma-homotopy-category-pre-triangulated", "categories": [ "dga" ], "title": "dga-lemma-homotopy-category-pre-triangulated", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "The homotopy category $K(\\text{Mod}_{(A, \\text{d})})$", "with its natural translation functors and distinguished triangles", "is a pre-triangulated category." ], "refs": [], "proofs": [ { "contents": [ "Proof of TR1. By definition every triangle isomorphic to a distinguished", "one is distinguished. Also, any triangle $(K, K, 0, 1, 0, 0)$", "is distinguished since $0 \\to K \\to K \\to 0 \\to 0$ is", "an admissible short exact sequence. Finally, given any homomorphism", "$f : K \\to L$ of differential graded $A$-modules the triangle", "$(K, L, C(f), f, i, -p)$ is distinguished by", "Lemma \\ref{lemma-the-same-up-to-isomorphisms}.", "\\medskip\\noindent", "Proof of TR2. Let $(X, Y, Z, f, g, h)$ be a triangle.", "Assume $(Y, Z, X[1], g, h, -f[1])$ is distinguished.", "Then there exists an admissible short exact sequence", "$0 \\to K \\to L \\to M \\to 0$ such that the associated", "triangle $(K, L, M, \\alpha, \\beta, \\delta)$", "is isomorphic to $(Y, Z, X[1], g, h, -f[1])$. Rotating back we see", "that $(X, Y, Z, f, g, h)$ is isomorphic to", "$(M[-1], K, L, -\\delta[-1], \\alpha, \\beta)$.", "It follows from Lemma \\ref{lemma-rotate-triangle} that the triangle", "$(M[-1], K, L, \\delta[-1], \\alpha, \\beta)$", "is isomorphic to", "$(M[-1], K, C(\\delta[-1]), \\delta[-1], i, p)$.", "Precomposing the previous isomorphism of triangles with $-1$ on $Y$", "it follows that $(X, Y, Z, f, g, h)$ is isomorphic to", "$(M[-1], K, C(\\delta[-1]), \\delta[-1], i, -p)$.", "Hence it is distinguished by", "Lemma \\ref{lemma-the-same-up-to-isomorphisms}.", "On the other hand, suppose that $(X, Y, Z, f, g, h)$ is distinguished.", "By Lemma \\ref{lemma-the-same-up-to-isomorphisms} this means that it is", "isomorphic to a triangle of the form", "$(K, L, C(f), f, i, -p)$ for some morphism $f$ of", "$\\text{Mod}_{(A, \\text{d})}$. Then the rotated triangle", "$(Y, Z, X[1], g, h, -f[1])$ is", "isomorphic to $(L, C(f), K[1], i, -p, -f[1])$ which is", "isomorphic to the triangle", "$(L, C(f), K[1], i, p, f[1])$.", "By Lemma \\ref{lemma-rotate-cone} this triangle is distinguished.", "Hence $(Y, Z, X[1], g, h, -f[1])$ is distinguished as desired.", "\\medskip\\noindent", "Proof of TR3. Let $(X, Y, Z, f, g, h)$ and $(X', Y', Z', f', g', h')$", "be distinguished triangles of $K(\\mathcal{A})$ and let $a : X \\to X'$", "and $b : Y \\to Y'$ be morphisms such that $f' \\circ a = b \\circ f$. By", "Lemma \\ref{lemma-the-same-up-to-isomorphisms} we may assume that", "$(X, Y, Z, f, g, h) = (X, Y, C(f), f, i, -p)$ and", "$(X', Y', Z', f', g', h') = (X', Y', C(f'), f', i', -p')$.", "At this point we simply apply Lemma \\ref{lemma-functorial-cone}", "to the commutative diagram given by $f, f', a, b$." ], "refs": [ "dga-lemma-the-same-up-to-isomorphisms", "dga-lemma-rotate-triangle", "dga-lemma-the-same-up-to-isomorphisms", "dga-lemma-the-same-up-to-isomorphisms", "dga-lemma-rotate-cone", "dga-lemma-the-same-up-to-isomorphisms", "dga-lemma-functorial-cone" ], "ref_ids": [ 13046, 13044, 13046, 13046, 13043, 13046, 13036 ] } ], "ref_ids": [] }, { "id": 13048, "type": "theorem", "label": "dga-lemma-two-split-injections", "categories": [ "dga" ], "title": "dga-lemma-two-split-injections", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. Suppose that", "$\\alpha : K \\to L$ and $\\beta : L \\to M$ are admissible monomorphisms", "of differential graded $A$-modules. Then there exist distinguished triangles", "$(K, L, Q_1, \\alpha, p_1, d_1)$, $(K, M, Q_2, \\beta \\circ \\alpha, p_2, d_2)$", "and $(L, M, Q_3, \\beta, p_3, d_3)$ for which TR4 holds." ], "refs": [], "proofs": [ { "contents": [ "Say $\\pi_1 : L \\to K$ and $\\pi_3 : M \\to L$ are homomorphisms", "of graded $A$-modules which are left inverse to $\\alpha$ and $\\beta$.", "Then also $K \\to M$ is an admissible monomorphism with left", "inverse $\\pi_2 = \\pi_1 \\circ \\pi_3$. Let us write $Q_1$, $Q_2$", "and $Q_3$ for the cokernels of $K \\to L$, $K \\to M$, and $L \\to M$.", "Then we obtain identifications (as graded $A$-modules)", "$Q_1 = \\Ker(\\pi_1)$, $Q_3 = \\Ker(\\pi_3)$ and", "$Q_2 = \\Ker(\\pi_2)$. Then $L = K \\oplus Q_1$ and", "$M = L \\oplus Q_3$ as graded $A$-modules. This implies", "$M = K \\oplus Q_1 \\oplus Q_3$. Note that $\\pi_2 = \\pi_1 \\circ \\pi_3$", "is zero on both $Q_1$ and $Q_3$. Hence $Q_2 = Q_1 \\oplus Q_3$.", "Consider the commutative diagram", "$$", "\\begin{matrix}", "0 & \\to & K & \\to & L & \\to & Q_1 & \\to & 0 \\\\", " & & \\downarrow & & \\downarrow & & \\downarrow & \\\\", "0 & \\to & K & \\to & M & \\to & Q_2 & \\to & 0 \\\\", " & & \\downarrow & & \\downarrow & & \\downarrow & \\\\", "0 & \\to & L & \\to & M & \\to & Q_3 & \\to & 0", "\\end{matrix}", "$$", "The rows of this diagram are admissible short exact sequences, and", "hence determine distinguished triangles by definition. Moreover", "downward arrows in the diagram above are compatible with the chosen", "splittings and hence define morphisms of triangles", "$$", "(K \\to L \\to Q_1 \\to K[1])", "\\longrightarrow", "(K \\to M \\to Q_2 \\to K[1])", "$$", "and", "$$", "(K \\to M \\to Q_2 \\to K[1])", "\\longrightarrow", "(L \\to M \\to Q_3 \\to L[1]).", "$$", "Note that the splittings $Q_3 \\to M$ of the bottom sequence in the", "diagram provides a splitting for the split sequence", "$0 \\to Q_1 \\to Q_2 \\to Q_3 \\to 0$ upon composing with $M \\to Q_2$.", "It follows easily from this that the morphism $\\delta : Q_3 \\to Q_1[1]$", "in the corresponding distinguished triangle", "$$", "(Q_1 \\to Q_2 \\to Q_3 \\to Q_1[1])", "$$", "is equal to the composition $Q_3 \\to L[1] \\to Q_1[1]$.", "Hence we get a structure as in the conclusion of axiom TR4." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13049, "type": "theorem", "label": "dga-lemma-left-right", "categories": [ "dga" ], "title": "dga-lemma-left-right", "contents": [ "Let $(A, \\text{d})$ be a differential graded $R$-algebra.", "The functor $M \\mapsto M^{opp}$ from the category of", "left differential graded $A$-modules to the category of right", "differential graded $A^{opp}$-modules is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13050, "type": "theorem", "label": "dga-lemma-left-module-structure", "categories": [ "dga" ], "title": "dga-lemma-left-module-structure", "contents": [ "In the situation above, let $A$ be a differential graded $R$-algebra.", "To give a left $A$-module structure on $M$ is the same thing as", "giving a homomorphism $A \\to E$ of differential graded $R$-algebras." ], "refs": [], "proofs": [ { "contents": [ "Proof omitted. Observe that no signs intervene in this correspondence." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13051, "type": "theorem", "label": "dga-lemma-characterize-hom", "categories": [ "dga" ], "title": "dga-lemma-characterize-hom", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ be a differential graded $R$-algebra.", "Let $M'$ be a right differential graded $A$-module and let", "$M$ be a left differential graded $A$-module.", "Let $N^\\bullet$ be a complex of $R$-modules. Then we have", "$$", "\\Hom_{\\text{Mod}_{(A, d)}}(M', \\Hom(M, N^\\bullet)) =", "\\Hom_{\\text{Comp}(R)}(M' \\otimes_A M, N^\\bullet)", "$$", "where $M \\otimes_A M$ is viewed as a complex of $R$-modules", "as in Section \\ref{section-tensor-product}." ], "refs": [], "proofs": [ { "contents": [ "Let us show that both sides correspond to graded $A$-bilinear maps", "$$", "M' \\times M \\longrightarrow N^\\bullet", "$$", "compatible with differentials. We have seen this is true for the right", "hand side in Section \\ref{section-tensor-product}. Given an element", "$g$ of the left hand side, the equality of", "More on Algebra, Lemma \\ref{more-algebra-lemma-compose}", "determines a map of complexes of $R$-modules", "$g' : \\text{Tot}(M' \\otimes_R M) \\to N^\\bullet$.", "In other words, we obtain a graded $R$-bilinear", "map $g'' : M' \\times M \\to N^\\bullet$ compatible with differentials.", "The $A$-linearity of $g$ translates immediately", "into $A$-bilinarity of $g''$." ], "refs": [ "more-algebra-lemma-compose" ], "ref_ids": [ 10198 ] } ], "ref_ids": [] }, { "id": 13052, "type": "theorem", "label": "dga-lemma-right-module-structure", "categories": [ "dga" ], "title": "dga-lemma-right-module-structure", "contents": [ "In the situation above, let $A$ be a differential graded $R$-algebra.", "To give a right $A$-module structure on $M$ is the same thing as", "giving a homomorphism $\\tau : A \\to E^{opp}$", "of differential graded $R$-algebras." ], "refs": [], "proofs": [ { "contents": [ "See discussion above and note that the construction of $\\tau$", "from the multiplication map $M^n \\times A^m \\to M^{n + m}$", "uses signs." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13053, "type": "theorem", "label": "dga-lemma-characterize-hom-other-side", "categories": [ "dga" ], "title": "dga-lemma-characterize-hom-other-side", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ be a differential graded $R$-algebra.", "Let $M$ be a right differential graded $A$-module and let", "$M'$ be a left differential graded $A$-module.", "Let $N^\\bullet$ be a complex of $R$-modules. Then we have", "$$", "\\Hom_{\\text{left diff graded }A\\text{-modules}}(M', \\Hom(M, N^\\bullet)) =", "\\Hom_{\\text{Comp}(R)}(M \\otimes_A M', N^\\bullet)", "$$", "where $M \\otimes_A M'$ is viewed as a complex of $R$-modules", "as in Section \\ref{section-tensor-product}." ], "refs": [], "proofs": [ { "contents": [ "Let us show that both sides correspond to graded $A$-bilinear maps", "$$", "M \\times M' \\longrightarrow N^\\bullet", "$$", "compatible with differentials. We have seen this is true for the right", "hand side in Section \\ref{section-tensor-product}. Given an element", "$g$ of the left hand side, the equality of", "More on Algebra, Lemma \\ref{more-algebra-lemma-compose}", "determines a map of complexes", "$g' : \\text{Tot}(M' \\otimes_R M) \\to N^\\bullet$.", "We precompose with the commutativity constraint to get", "$$", "\\text{Tot}(M \\otimes_R M') \\xrightarrow{\\psi}", "\\text{Tot}(M' \\otimes_R M) \\xrightarrow{g'}", "N^\\bullet", "$$", "which corresponds to a graded $R$-bilinear", "map $g'' : M \\times M' \\to N^\\bullet$ compatible with differentials.", "The $A$-linearity of $g$ translates immediately into $A$-bilinarity of $g''$.", "Namely, say $x \\in M^e$ and $x' \\in (M')^{e'}$ and $a \\in A^n$. Then", "on the one hand we have", "\\begin{align*}", "g''(x, ax')", "& =", "(-1)^{e(n + e')} g'(ax' \\otimes x) \\\\", "& =", "(-1)^{e(n + e')} g(ax')(x) \\\\", "& =", "(-1)^{e(n + e')} (a \\cdot g(x'))(x) \\\\", "& =", "(-1)^{e(n + e') + n(n + e + e') + n} g(x')(xa)", "\\end{align*}", "and on the other hand we have", "$$", "g''(xa, x') = (-1)^{(e + n)e'} g'(x' \\otimes xa) =", "(-1)^{(e + n)e'} g(x')(xa) ", "$$", "which is the same thing by a trivial mod $2$ calculation of the exponents." ], "refs": [ "more-algebra-lemma-compose" ], "ref_ids": [ 10198 ] } ], "ref_ids": [] }, { "id": 13054, "type": "theorem", "label": "dga-lemma-target-graded-projective", "categories": [ "dga" ], "title": "dga-lemma-target-graded-projective", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $M \\to P$ be a surjective homomorphism of differential graded", "$A$-modules. If $P$ is projective as a graded $A$-module, then", "$M \\to P$ is an admissible epimorphism." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13055, "type": "theorem", "label": "dga-lemma-hom-from-shift-free", "categories": [ "dga" ], "title": "dga-lemma-hom-from-shift-free", "contents": [ "Let $(A, d)$ be a differential graded algebra. Then we have", "$$", "\\Hom_{\\text{Mod}_{(A, \\text{d})}}(A[k], M) =", "\\Ker(\\text{d} : M^{-k} \\to M^{-k + 1})", "$$", "and", "$$", "\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(A[k], M) = H^{-k}(M)", "$$", "for any differential graded $A$-module $M$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13056, "type": "theorem", "label": "dga-lemma-source-graded-injective", "categories": [ "dga" ], "title": "dga-lemma-source-graded-injective", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $I \\to M$ be an injective homomorphism of differential graded", "$A$-modules. If $I$ is graded injective, then", "$I \\to M$ is an admissible monomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13057, "type": "theorem", "label": "dga-lemma-map-into-dual", "categories": [ "dga" ], "title": "dga-lemma-map-into-dual", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. If", "$M$ is a left differential graded $A$-module and $N$ is a", "right differential graded $A$-module, then", "\\begin{align*}", "\\Hom_{\\text{Mod}_{(A, \\text{d})}}(N, M^\\vee)", "& =", "\\Hom_{\\text{Comp}(\\mathbf{Z})}(N \\otimes_A M, \\mathbf{Q}/\\mathbf{Z}) \\\\", "& =", "\\text{DifferentialGradedBilinear}_A(N \\times M, \\mathbf{Q}/\\mathbf{Z})", "\\end{align*}" ], "refs": [], "proofs": [ { "contents": [ "The first equality is Lemma \\ref{lemma-characterize-hom}", "and the second equality was shown in Section \\ref{section-tensor-product}." ], "refs": [ "dga-lemma-characterize-hom" ], "ref_ids": [ 13051 ] } ], "ref_ids": [] }, { "id": 13058, "type": "theorem", "label": "dga-lemma-hom-into-shift-dual-free", "categories": [ "dga" ], "title": "dga-lemma-hom-into-shift-dual-free", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. Then we have", "$$", "\\Hom_{\\text{Mod}_{(A, \\text{d})}}(M, A^\\vee[k]) =", "\\Ker(\\text{d} : (M^\\vee)^k \\to (M^\\vee)^{k + 1})", "$$", "and", "$$", "\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(M, A^\\vee[k]) = H^k(M^\\vee)", "$$", "as functors in the differential graded $A$-module $M$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13059, "type": "theorem", "label": "dga-lemma-property-P-sequence", "categories": [ "dga" ], "title": "dga-lemma-property-P-sequence", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $P$ be a differential graded $A$-module. If $F_\\bullet$", "is a filtration as in property (P), then we obtain an", "admissible short exact sequence", "$$", "0 \\to", "\\bigoplus\\nolimits F_iP \\to", "\\bigoplus\\nolimits F_iP \\to P \\to 0", "$$", "of differential graded $A$-modules." ], "refs": [], "proofs": [ { "contents": [ "The second map is the direct sum of the inclusion maps.", "The first map on the summand $F_iP$ of the source is the sum", "of the identity $F_iP \\to F_iP$ and the negative of the inclusion", "map $F_iP \\to F_{i + 1}P$. Choose homomorphisms $s_i : F_{i + 1}P \\to F_iP$", "of graded $A$-modules which are left inverse to the inclusion maps.", "Composing gives maps $s_{j, i} : F_jP \\to F_iP$ for all $j > i$.", "Then a left inverse of the first arrow maps $x \\in F_jP$ to", "$(s_{j, 0}(x), s_{j, 1}(x), \\ldots, s_{j, j - 1}(x), 0, \\ldots)$", "in $\\bigoplus F_iP$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13060, "type": "theorem", "label": "dga-lemma-property-P-K-projective", "categories": [ "dga" ], "title": "dga-lemma-property-P-K-projective", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $P$ be a differential graded $A$-module with property (P).", "Then", "$$", "\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(P, N) = 0", "$$", "for all acyclic differential graded $A$-modules $N$." ], "refs": [], "proofs": [ { "contents": [ "We will use that $K(\\text{Mod}_{(A, \\text{d})})$ is a triangulated", "category (Proposition \\ref{proposition-homotopy-category-triangulated}).", "Let $F_\\bullet$ be a filtration on $P$ as in property (P).", "The short exact sequence of Lemma \\ref{lemma-property-P-sequence}", "produces a distinguished triangle. Hence by", "Derived Categories, Lemma \\ref{derived-lemma-representable-homological}", "it suffices to show that", "$$", "\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(F_iP, N) = 0", "$$", "for all acyclic differential graded $A$-modules $N$ and all $i$.", "Each of the differential graded modules $F_iP$ has a finite filtration", "by admissible monomorphisms, whose graded pieces are direct sums", "of shifts $A[k]$. Thus it suffices to prove that", "$$", "\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(A[k], N) = 0", "$$", "for all acyclic differential graded $A$-modules $N$ and all $k$.", "This follows from Lemma \\ref{lemma-hom-from-shift-free}." ], "refs": [ "dga-proposition-homotopy-category-triangulated", "dga-lemma-property-P-sequence", "derived-lemma-representable-homological", "dga-lemma-hom-from-shift-free" ], "ref_ids": [ 13130, 13059, 1758, 13055 ] } ], "ref_ids": [] }, { "id": 13061, "type": "theorem", "label": "dga-lemma-good-quotient", "categories": [ "dga" ], "title": "dga-lemma-good-quotient", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $M$ be a differential graded $A$-module. There exists a homomorphism", "$P \\to M$ of differential graded $A$-modules with the following", "properties", "\\begin{enumerate}", "\\item $P \\to M$ is surjective,", "\\item $\\Ker(\\text{d}_P) \\to \\Ker(\\text{d}_M)$ is surjective, and", "\\item $P$ sits in an admissible short exact sequence", "$0 \\to P' \\to P \\to P'' \\to 0$ where $P'$, $P''$ are direct sums", "of shifts of $A$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $P_k$ be the free $A$-module with generators $x, y$ in degrees", "$k$ and $k + 1$. Define the structure of a differential graded", "$A$-module on $P_k$ by setting $\\text{d}(x) = y$ and $\\text{d}(y) = 0$.", "For every element $m \\in M^k$ there is a homomorphism", "$P_k \\to M$ sending $x$ to $m$ and $y$ to $\\text{d}(m)$.", "Thus we see that there is a surjection from a direct sum", "of copies of $P_k$ to $M$. This clearly produces $P \\to M$", "having properties (1) and (3). To obtain property (2) note", "that if $m \\in \\Ker(\\text{d}_M)$ has degree $k$, then there is a map", "$A[k] \\to M$ mapping $1$ to $m$. Hence we can achieve (2) by adding", "a direct sum of copies of shifts of $A$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13062, "type": "theorem", "label": "dga-lemma-resolve", "categories": [ "dga" ], "title": "dga-lemma-resolve", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $M$ be a differential graded $A$-module. There exists a homomorphism", "$P \\to M$ of differential graded $A$-modules such that", "\\begin{enumerate}", "\\item $P \\to M$ is a quasi-isomorphism, and", "\\item $P$ has property (P).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Set $M = M_0$. We inductively choose short exact sequences", "$$", "0 \\to M_{i + 1} \\to P_i \\to M_i \\to 0", "$$", "where the maps $P_i \\to M_i$ are chosen as in Lemma \\ref{lemma-good-quotient}.", "This gives a ``resolution''", "$$", "\\ldots \\to P_2 \\xrightarrow{f_2} P_1 \\xrightarrow{f_1} P_0 \\to M \\to 0", "$$", "Then we set", "$$", "P = \\bigoplus\\nolimits_{i \\geq 0} P_i", "$$", "as an $A$-module with grading given by", "$P^n = \\bigoplus_{a + b = n} P_{-a}^b$ and", "differential (as in the construction of the total complex associated", "to a double complex) by", "$$", "\\text{d}_P(x) = f_{-a}(x) + (-1)^a \\text{d}_{P_{-a}}(x)", "$$", "for $x \\in P_{-a}^b$. With these conventions $P$ is indeed a differential", "graded $A$-module. Recalling that each $P_i$ has a two step filtration", "$0 \\to P_i' \\to P_i \\to P_i'' \\to 0$ we set", "$$", "F_{2i}P = \\bigoplus\\nolimits_{i \\geq j \\geq 0} P_j", "\\subset", "\\bigoplus\\nolimits_{i \\geq 0} P_i = P", "$$", "and we add $P'_{i + 1}$ to $F_{2i}P$ to get $F_{2i + 1}$.", "These are differential graded submodules and the successive quotients", "are direct sums of shifts of $A$. By", "Lemma \\ref{lemma-target-graded-projective} we see that", "the inclusions $F_iP \\to F_{i + 1}P$ are admissible monomorphisms.", "Finally, we have to show that the map $P \\to M$ (given by the", "augmentation $P_0 \\to M$) is a quasi-isomorphism. This follows from", "Homology, Lemma \\ref{homology-lemma-good-resolution-gives-qis}." ], "refs": [ "dga-lemma-good-quotient", "dga-lemma-target-graded-projective", "homology-lemma-good-resolution-gives-qis" ], "ref_ids": [ 13061, 13054, 12109 ] } ], "ref_ids": [] }, { "id": 13063, "type": "theorem", "label": "dga-lemma-property-I-sequence", "categories": [ "dga" ], "title": "dga-lemma-property-I-sequence", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $I$ be a differential graded $A$-module. If $F_\\bullet$", "is a filtration as in property (I), then we obtain an", "admissible short exact sequence", "$$", "0 \\to I \\to", "\\prod\\nolimits I/F_iI \\to", "\\prod\\nolimits I/F_iI \\to 0", "$$", "of differential graded $A$-modules." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is dual to Lemma \\ref{lemma-property-P-sequence}." ], "refs": [ "dga-lemma-property-P-sequence" ], "ref_ids": [ 13059 ] } ], "ref_ids": [] }, { "id": 13064, "type": "theorem", "label": "dga-lemma-property-I-K-injective", "categories": [ "dga" ], "title": "dga-lemma-property-I-K-injective", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $I$ be a differential graded $A$-module with property (I).", "Then", "$$", "\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(N, I) = 0", "$$", "for all acyclic differential graded $A$-modules $N$." ], "refs": [], "proofs": [ { "contents": [ "We will use that $K(\\text{Mod}_{(A, \\text{d})})$ is a triangulated", "category (Proposition \\ref{proposition-homotopy-category-triangulated}).", "Let $F_\\bullet$ be a filtration on $I$ as in property (I).", "The short exact sequence of Lemma \\ref{lemma-property-I-sequence}", "produces a distinguished triangle. Hence by", "Derived Categories, Lemma \\ref{derived-lemma-representable-homological}", "it suffices to show that", "$$", "\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(N, I/F_iI) = 0", "$$", "for all acyclic differential graded $A$-modules $N$ and all $i$.", "Each of the differential graded modules $I/F_iI$ has a finite filtration", "by admissible monomorphisms, whose graded pieces are", "products of $A^\\vee[k]$. Thus it suffices to prove that", "$$", "\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(N, A^\\vee[k]) = 0", "$$", "for all acyclic differential graded $A$-modules $N$ and all $k$.", "This follows from Lemma \\ref{lemma-hom-into-shift-dual-free}", "and the fact that $(-)^\\vee$ is an exact functor." ], "refs": [ "dga-proposition-homotopy-category-triangulated", "dga-lemma-property-I-sequence", "derived-lemma-representable-homological", "dga-lemma-hom-into-shift-dual-free" ], "ref_ids": [ 13130, 13063, 1758, 13058 ] } ], "ref_ids": [] }, { "id": 13065, "type": "theorem", "label": "dga-lemma-good-sub", "categories": [ "dga" ], "title": "dga-lemma-good-sub", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $M$ be a differential graded $A$-module. There exists a homomorphism", "$M \\to I$ of differential graded $A$-modules with the following", "properties", "\\begin{enumerate}", "\\item $M \\to I$ is injective,", "\\item $\\Coker(\\text{d}_M) \\to \\Coker(\\text{d}_I)$ is injective,", "and", "\\item $I$ sits in an admissible short exact sequence", "$0 \\to I' \\to I \\to I'' \\to 0$ where $I'$, $I''$ are products", "of shifts of $A^\\vee$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use the functors $N \\mapsto N^\\vee$ (from left to right", "differential graded modules and from right to left differential", "graded modules) constructed in", "Section \\ref{section-modules-noncommutative-differential-graded}", "and all of their properties.", "For every $k \\in \\mathbf{Z}$ let $Q_k$ be the free left $A$-module with", "generators $x, y$ in degrees $k$ and $k + 1$. Define the structure of a", "left differential graded $A$-module on $Q_k$ by setting $\\text{d}(x) = y$", "and $\\text{d}(y) = 0$. Arguing exactly as in the proof of", "Lemma \\ref{lemma-good-quotient} we find a surjection", "$$", "\\bigoplus\\nolimits_{i \\in I} Q_{k_i} \\longrightarrow M^\\vee", "$$", "of left differential graded $A$-modules. Then we can consider the injection", "$$", "M \\to (M^\\vee)^\\vee \\to (\\bigoplus\\nolimits_{i \\in I} Q_{k_i})^\\vee =", "\\prod\\nolimits_{i \\in I} I_{k_i}", "$$", "where $I_k = Q_{-k}^\\vee$ is the ``dual'' right differential graded $A$-module.", "Further, the short exact sequence $0 \\to A[-k - 1] \\to Q_k \\to A[-k] \\to 0$", "produces a short exact sequence", "$0 \\to A^\\vee[k] \\to I_k \\to A^\\vee[k + 1] \\to 0$.", "\\medskip\\noindent", "The result of the previous paragraph produces $M \\to I$", "having properties (1) and (3). To obtain property (2), suppose", "$\\overline{m} \\in \\Coker(\\text{d}_M)$ is a nonzero element of", "degree $k$. Pick a map $\\lambda : M^k \\to \\mathbf{Q}/\\mathbf{Z}$", "which vanishes on $\\Im(M^{k - 1} \\to M^k)$ but not on $m$. By", "Lemma \\ref{lemma-hom-into-shift-dual-free}", "this corresponds to a homomorphism $M \\to A^\\vee[k]$ of", "differential graded $A$-modules which does not vanish on $m$.", "Hence we can achieve (2) by adding", "a product of copies of shifts of $A^\\vee$." ], "refs": [ "dga-lemma-good-quotient", "dga-lemma-hom-into-shift-dual-free" ], "ref_ids": [ 13061, 13058 ] } ], "ref_ids": [] }, { "id": 13066, "type": "theorem", "label": "dga-lemma-right-resolution", "categories": [ "dga" ], "title": "dga-lemma-right-resolution", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $M$ be a differential graded $A$-module. There exists a homomorphism", "$M \\to I$ of differential graded $A$-modules such that", "\\begin{enumerate}", "\\item $M \\to I$ is a quasi-isomorphism, and", "\\item $I$ has property (I).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Set $M = M_0$. We inductively choose short exact sequences", "$$", "0 \\to M_i \\to I_i \\to M_{i + 1} \\to 0", "$$", "where the maps $M_i \\to I_i$ are chosen as in Lemma \\ref{lemma-good-sub}.", "This gives a ``resolution''", "$$", "0 \\to M \\to I_0 \\xrightarrow{f_0} I_1 \\xrightarrow{f_1} I_1 \\to \\ldots", "$$", "Denote $I$ the differential graded $A$-module with graded parts", "$$", "I^n = \\prod\\nolimits_{i \\geq 0} I^{n - i}_i", "$$", "and differential defined by", "$$", "\\text{d}_I(x) = f_i(x) + (-1)^i \\text{d}_{I_i}(x)", "$$", "for $x \\in I_i^{n - i}$. With these conventions $I$ is indeed a differential", "graded $A$-module. Recalling that each $I_i$ has a two step filtration", "$0 \\to I_i' \\to I_i \\to I_i'' \\to 0$ we set", "$$", "F_{2i}I^n = \\prod\\nolimits_{j \\geq i} I^{n - j}_j", "\\subset", "\\prod\\nolimits_{i \\geq 0} I^{n - i}_i = I^n", "$$", "and we add a factor $I'_{i + 1}$ to $F_{2i}I$ to get $F_{2i + 1}I$.", "These are differential graded submodules and the successive quotients", "are products of shifts of $A^\\vee$. By", "Lemma \\ref{lemma-source-graded-injective} we see that", "the inclusions $F_{i + 1}I \\to F_iI$ are admissible monomorphisms.", "Finally, we have to show that the map $M \\to I$ (given by the", "augmentation $M \\to I_0$) is a quasi-isomorphism. This follows from", "Homology, Lemma \\ref{homology-lemma-good-right-resolution-gives-qis}." ], "refs": [ "dga-lemma-good-sub", "dga-lemma-source-graded-injective", "homology-lemma-good-right-resolution-gives-qis" ], "ref_ids": [ 13065, 13056, 12110 ] } ], "ref_ids": [] }, { "id": 13067, "type": "theorem", "label": "dga-lemma-acyclic", "categories": [ "dga" ], "title": "dga-lemma-acyclic", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "The full subcategory $\\text{Ac}$ of $K(\\text{Mod}_{(A, \\text{d})})$", "consisting of acyclic modules is a strictly full saturated triangulated", "subcategory of $K(\\text{Mod}_{(A, \\text{d})})$.", "The corresponding saturated multiplicative system", "(see Derived Categories, Lemma \\ref{derived-lemma-operations})", "of $K(\\text{Mod}_{(A, \\text{d})})$ is the class $\\text{Qis}$", "of quasi-isomorphisms. In particular, the kernel of the localization", "functor", "$$", "Q : K(\\text{Mod}_{(A, \\text{d})}) \\to", "\\text{Qis}^{-1}K(\\text{Mod}_{(A, \\text{d})})", "$$", "is $\\text{Ac}$. Moreover, the functor $H^0$ factors through $Q$." ], "refs": [ "derived-lemma-operations" ], "proofs": [ { "contents": [ "We know that $H^0$ is a homological functor by the long exact", "sequence of homology (\\ref{equation-les}).", "The kernel of $H^0$ is the subcategory of acyclic objects and", "the arrows with induce isomorphisms on all $H^i$ are the", "quasi-isomorphisms. Thus this lemma is a special case of", "Derived Categories, Lemma \\ref{derived-lemma-acyclic-general}.", "\\medskip\\noindent", "Set theoretical remark. The construction of the localization in", "Derived Categories, Proposition", "\\ref{derived-proposition-construct-localization}", "assumes the given triangulated category is ``small'', i.e., that the", "underlying collection of objects forms a set. Let $V_\\alpha$ be a", "partial universe (as in Sets, Section \\ref{sets-section-sets-hierarchy})", "containing $(A, \\text{d})$ and where the cofinality of $\\alpha$", "is bigger than $\\aleph_0$", "(see Sets, Proposition \\ref{sets-proposition-exist-ordinals-large-cofinality}).", "Then we can consider the category $\\text{Mod}_{(A, \\text{d}), \\alpha}$", "of differential graded $A$-modules contained in $V_\\alpha$.", "A straightforward check shows that all the constructions used in", "the proof of Proposition \\ref{proposition-homotopy-category-triangulated}", "work inside of $\\text{Mod}_{(A, \\text{d}), \\alpha}$", "(because at worst we take finite direct sums of differential graded modules).", "Thus we obtain a triangulated category", "$\\text{Qis}_\\alpha^{-1}K(\\text{Mod}_{(A, \\text{d}), \\alpha})$.", "We will see below that if $\\beta > \\alpha$, then the transition functors", "$$", "\\text{Qis}_\\alpha^{-1}K(\\text{Mod}_{(A, \\text{d}), \\alpha})", "\\longrightarrow", "\\text{Qis}_\\beta^{-1}K(\\text{Mod}_{(A, \\text{d}), \\beta})", "$$", "are fully faithful as the morphism sets in the quotient categories", "are computed by maps in the homotopy categories from P-resolutions", "(the construction of a P-resolution in the proof of Lemma \\ref{lemma-resolve}", "takes countable direct sums as well as direct sums indexed over subsets", "of the given module). The reader should therefore think of the category", "of the lemma as the union of these subcategories." ], "refs": [ "derived-lemma-acyclic-general", "derived-proposition-construct-localization", "sets-proposition-exist-ordinals-large-cofinality", "dga-proposition-homotopy-category-triangulated", "dga-lemma-resolve" ], "ref_ids": [ 1791, 1959, 8802, 13130, 13062 ] } ], "ref_ids": [ 1790 ] }, { "id": 13068, "type": "theorem", "label": "dga-lemma-hom-derived", "categories": [ "dga" ], "title": "dga-lemma-hom-derived", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $M$ and $N$ be differential graded $A$-modules.", "\\begin{enumerate}", "\\item Let $P \\to M$ be a P-resolution as in", "Lemma \\ref{lemma-resolve}. Then", "$$", "\\Hom_{D(A, \\text{d})}(M, N) =", "\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(P, N)", "$$", "\\item Let $N \\to I$ be an I-resolution as in", "Lemma \\ref{lemma-right-resolution}. Then", "$$", "\\Hom_{D(A, \\text{d})}(M, N) =", "\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(M, I)", "$$", "\\end{enumerate}" ], "refs": [ "dga-lemma-resolve", "dga-lemma-right-resolution" ], "proofs": [ { "contents": [ "Let $P \\to M$ be as in (1). Since $P \\to M$ is a quasi-isomorphism we see that", "$$", "\\Hom_{D(A, \\text{d})}(P, N) = \\Hom_{D(A, \\text{d})}(M, N)", "$$", "by definition of the derived category. A morphism", "$f : P \\to N$ in $D(A, \\text{d})$ is equal to", "$s^{-1}f'$ where $f' : P \\to N'$ is a morphism and", "$s : N \\to N'$ is a quasi-isomorphism. Choose a distinguished triangle", "$$", "N \\to N' \\to Q \\to N[1]", "$$", "As $s$ is a quasi-isomorphism, we see that $Q$ is acyclic. Thus", "$\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(P, Q[k]) = 0$ for all $k$ by", "Lemma \\ref{lemma-property-P-K-projective}. Since", "$\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(P, -)$", "is cohomological, we conclude that we can lift $f' : P \\to N'$", "uniquely to a morphism $f : P \\to N$. This finishes the proof.", "\\medskip\\noindent", "The proof of (2) is dual to that of (1) using", "Lemma \\ref{lemma-property-I-K-injective} in stead of", "Lemma \\ref{lemma-property-P-K-projective}." ], "refs": [ "dga-lemma-property-P-K-projective", "dga-lemma-property-I-K-injective", "dga-lemma-property-P-K-projective" ], "ref_ids": [ 13060, 13064, 13060 ] } ], "ref_ids": [ 13062, 13066 ] }, { "id": 13069, "type": "theorem", "label": "dga-lemma-derived-products", "categories": [ "dga" ], "title": "dga-lemma-derived-products", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. Then", "\\begin{enumerate}", "\\item $D(A, \\text{d})$ has both direct sums and products,", "\\item direct sums are obtained by taking direct sums of differential graded", "modules,", "\\item products are obtained by taking products of differential", "graded modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use that $\\text{Mod}_{(A, \\text{d})}$ is an abelian category", "with arbitrary direct sums and products, and that these give rise", "to direct sums and products in $K(\\text{Mod}_{(A, \\text{d})})$.", "See Lemmas \\ref{lemma-dgm-abelian} and \\ref{lemma-homotopy-direct-sums}.", "\\medskip\\noindent", "Let $M_j$ be a family of differential graded $A$-modules.", "Consider the graded direct sum $M = \\bigoplus M_j$ which is a", "differential graded $A$-module with the obvious.", "For a differential graded $A$-module $N$ choose a quasi-isomorphism", "$N \\to I$ where $I$ is a differential graded $A$-module with property (I).", "See Lemma \\ref{lemma-right-resolution}.", "Using Lemma \\ref{lemma-hom-derived} we have", "\\begin{align*}", "\\Hom_{D(A, \\text{d})}(M, N)", "& =", "\\Hom_{K(A, \\text{d})}(M, I) \\\\", "& =", "\\prod \\Hom_{K(A, \\text{d})}(M_j, I) \\\\", "& =", "\\prod \\Hom_{D(A, \\text{d})}(M_j, N)", "\\end{align*}", "whence the existence of direct sums in $D(A, \\text{d})$ as given in", "part (2) of the lemma.", "\\medskip\\noindent", "Let $M_j$ be a family of differential graded $A$-modules.", "Consider the product $M = \\prod M_j$ of differential graded $A$-modules.", "For a differential graded $A$-module $N$ choose a quasi-isomorphism", "$P \\to N$ where $P$ is a differential graded $A$-module with property (P).", "See Lemma \\ref{lemma-resolve}.", "Using Lemma \\ref{lemma-hom-derived} we have", "\\begin{align*}", "\\Hom_{D(A, \\text{d})}(N, M)", "& =", "\\Hom_{K(A, \\text{d})}(P, M) \\\\", "& =", "\\prod \\Hom_{K(A, \\text{d})}(P, M_j) \\\\", "& =", "\\prod \\Hom_{D(A, \\text{d})}(N, M_j)", "\\end{align*}", "whence the existence of direct sums in $D(A, \\text{d})$ as given in", "part (3) of the lemma." ], "refs": [ "dga-lemma-dgm-abelian", "dga-lemma-homotopy-direct-sums", "dga-lemma-right-resolution", "dga-lemma-hom-derived", "dga-lemma-resolve", "dga-lemma-hom-derived" ], "ref_ids": [ 13033, 13035, 13066, 13068, 13062, 13068 ] } ], "ref_ids": [] }, { "id": 13070, "type": "theorem", "label": "dga-lemma-derived-canonical-delta-functor", "categories": [ "dga" ], "title": "dga-lemma-derived-canonical-delta-functor", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. The functor", "$\\text{Mod}_{(A, \\text{d})} \\to D(A, \\text{d})$", "defined has the natural structure of a $\\delta$-functor, with", "$$", "\\delta_{K \\to L \\to M} = - p \\circ q^{-1}", "$$", "with $p$ and $q$ as explained above." ], "refs": [], "proofs": [ { "contents": [ "We have already seen that this choice leads to a distinguished", "triangle whenever given a short exact sequence of complexes.", "We have to show functoriality of this construction, see", "Derived Categories, Definition \\ref{derived-definition-delta-functor}.", "This follows from Lemma \\ref{lemma-functorial-cone} with a bit of", "work. Compare with", "Derived Categories, Lemma \\ref{derived-lemma-derived-canonical-delta-functor}." ], "refs": [ "derived-definition-delta-functor", "dga-lemma-functorial-cone", "derived-lemma-derived-canonical-delta-functor" ], "ref_ids": [ 1972, 13036, 1814 ] } ], "ref_ids": [] }, { "id": 13071, "type": "theorem", "label": "dga-lemma-homotopy-colimit", "categories": [ "dga" ], "title": "dga-lemma-homotopy-colimit", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. Let", "$M_n$ be a system of differential graded modules. Then the derived", "colimit $\\text{hocolim} M_n$ in $D(A, \\text{d})$ is represented", "by the differential graded module $\\colim M_n$." ], "refs": [], "proofs": [ { "contents": [ "Set $M = \\colim M_n$. We have an exact sequence of differential graded modules", "$$", "0 \\to \\bigoplus M_n \\to \\bigoplus M_n \\to M \\to 0", "$$", "by Derived Categories, Lemma \\ref{derived-lemma-compute-colimit}", "(applied the underlying complexes of abelian groups).", "The direct sums are direct sums in $D(\\mathcal{A})$ by", "Lemma \\ref{lemma-derived-products}.", "Thus the result follows from the definition", "of derived colimits in Derived Categories,", "Definition \\ref{derived-definition-derived-colimit}", "and the fact that a short exact sequence of complexes", "gives a distinguished triangle", "(Lemma \\ref{lemma-derived-canonical-delta-functor})." ], "refs": [ "derived-lemma-compute-colimit", "dga-lemma-derived-products", "derived-definition-derived-colimit", "dga-lemma-derived-canonical-delta-functor" ], "ref_ids": [ 1921, 13069, 2001, 13070 ] } ], "ref_ids": [] }, { "id": 13072, "type": "theorem", "label": "dga-lemma-functorial", "categories": [ "dga" ], "title": "dga-lemma-functorial", "contents": [ "Let $R$ be a ring. A functor $F : \\mathcal{A} \\to \\mathcal{B}$", "of differential graded categories over $R$ induces functors", "$\\text{Comp}(\\mathcal{A}) \\to \\text{Comp}(\\mathcal{B})$", "and $K(\\mathcal{A}) \\to K(\\mathcal{B})$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13073, "type": "theorem", "label": "dga-lemma-additive-functor-induces-dga-functor", "categories": [ "dga" ], "title": "dga-lemma-additive-functor-induces-dga-functor", "contents": [ "Let $F : \\mathcal{B} \\to \\mathcal{B}'$ be an additive functor between", "additive categories. Then $F$ induces a functor of differential", "graded categories", "$$", "F : \\text{Comp}^{dg}(\\mathcal{B}) \\to \\text{Comp}^{dg}(\\mathcal{B}')", "$$", "of Example \\ref{example-category-complexes}", "inducing the usual functors on the category of complexes and the", "homotopy categories." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13074, "type": "theorem", "label": "dga-lemma-homomorphism-induces-dga-functor", "categories": [ "dga" ], "title": "dga-lemma-homomorphism-induces-dga-functor", "contents": [ "Let $\\varphi : (A, \\text{d}) \\to (E, \\text{d})$ be a homomorphism of", "differential graded algebras. Then $\\varphi$ induces a functor of differential", "graded categories", "$$", "F :", "\\text{Mod}^{dg}_{(E, \\text{d})}", "\\longrightarrow", "\\text{Mod}^{dg}_{(A, \\text{d})}", "$$", "of Example \\ref{example-dgm-dg-cat} inducing obvious restriction functors", "on the categories of differential graded modules and homotopy categories." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13075, "type": "theorem", "label": "dga-lemma-construction", "categories": [ "dga" ], "title": "dga-lemma-construction", "contents": [ "Let $R$ be a ring. Let $\\mathcal{A}$ be a differential graded category", "over $R$. Let $x$ be an object of $\\mathcal{A}$. Let", "$$", "(E, \\text{d}) = \\Hom_\\mathcal{A}(x, x)", "$$", "be the differential graded $R$-algebra of endomorphisms of $x$.", "We obtain a functor", "$$", "\\mathcal{A} \\longrightarrow \\text{Mod}^{dg}_{(E, \\text{d})},\\quad", "y \\longmapsto \\Hom_\\mathcal{A}(x, y)", "$$", "of differential graded categories by letting $E$ act on", "$\\Hom_\\mathcal{A}(x, y)$ via composition in $\\mathcal{A}$.", "This functor induces functors", "$$", "\\text{Comp}(\\mathcal{A}) \\to \\text{Mod}_{(A, \\text{d})}", "\\quad\\text{and}\\quad", "K(\\mathcal{A}) \\to K(\\text{Mod}_{(A, \\text{d})})", "$$", "by an application of Lemma \\ref{lemma-functorial}." ], "refs": [ "dga-lemma-functorial" ], "proofs": [ { "contents": [ "This lemma proves itself." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13072 ] }, { "id": 13076, "type": "theorem", "label": "dga-lemma-get-triangle", "categories": [ "dga" ], "title": "dga-lemma-get-triangle", "contents": [ "Let $\\mathcal{A}$ be a differential graded category satisfying", "axioms (A) and (B). Given an admissible short exact sequence", "$x \\to y \\to z$ we obtain (see proof) a triangle", "$$", "x \\to y \\to z \\to x[1]", "$$", "in $\\text{Comp}(\\mathcal{A})$ with the property that any two compositions", "in $z[-1] \\to x \\to y \\to z \\to x[1]$ are zero in $K(\\mathcal{A})$." ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram", "$$", "\\xymatrix{", "x \\ar[rr]_1 \\ar[rd]_a & & x \\\\", "& y \\ar[ru]_\\pi \\ar[rd]^b & \\\\", "z \\ar[rr]^1 \\ar[ru]^s & & z", "}", "$$", "giving the isomorphism of graded objects $y \\cong x \\oplus z$ as in the", "definition of an admissible short exact sequence. Here are some equations", "that hold in this situation", "\\begin{enumerate}", "\\item $1 = \\pi a$ and hence $\\text{d}(\\pi) a = 0$,", "\\item $1 = b s$ and hence $b \\text{d}(s) = 0$,", "\\item $1 = a \\pi + s b$ and hence $a \\text{d}(\\pi) + \\text{d}(s) b = 0$,", "\\item $\\pi s = 0$ and hence $\\text{d}(\\pi)s + \\pi \\text{d}(s) = 0$,", "\\item $\\text{d}(s) = a \\pi \\text{d}(s)$ because", "$\\text{d}(s) = (a \\pi + s b)\\text{d}(s)$ and $b\\text{d}(s) = 0$,", "\\item $\\text{d}(\\pi) = \\text{d}(\\pi) s b$ because", "$\\text{d}(\\pi) = \\text{d}(\\pi)(a \\pi + s b)$ and $\\text{d}(\\pi)a = 0$,", "\\item $\\text{d}(\\pi \\text{d}(s)) = 0$ because if we postcompose it", "with the monomorphism $a$ we get", "$\\text{d}(a\\pi \\text{d}(s)) = \\text{d}(\\text{d}(s)) = 0$, and", "\\item $\\text{d}(\\text{d}(\\pi)s) = 0$ as by (4) it is the negative", "of $\\text{d}(\\pi\\text{d}(s))$ which is $0$ by (7).", "\\end{enumerate}", "We've used repeatedly that $\\text{d}(a) = 0$, $\\text{d}(b) = 0$,", "and that $\\text{d}(1) = 0$. By (7) we see that", "$$", "\\delta = \\pi \\text{d}(s) = - \\text{d}(\\pi) s : z \\to x[1]", "$$", "is a morphism in $\\text{Comp}(\\mathcal{A})$. By (5) we see that", "the composition $a \\delta = a \\pi \\text{d}(s) = \\text{d}(s)$", "is homotopic to zero. By (6) we see that the composition", "$\\delta b = - \\text{d}(\\pi)sb = \\text{d}(-\\pi)$ is homotopic to zero." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13077, "type": "theorem", "label": "dga-lemma-cone", "categories": [ "dga" ], "title": "dga-lemma-cone", "contents": [ "\\begin{slogan}", "The homotopy category is a triangulated category.", "This lemma proves a part of the axioms of a triangulated category.", "\\end{slogan}", "In Situation \\ref{situation-ABC} suppose that", "$$", "\\xymatrix{", "x_1 \\ar[r]_{f_1} \\ar[d]_a & y_1 \\ar[d]^b \\\\", "x_2 \\ar[r]^{f_2} & y_2", "}", "$$", "is a diagram of $\\text{Comp}(\\mathcal{A})$ commutative up to homotopy.", "Then there exists a morphism $c : c(f_1) \\to c(f_2)$ which gives rise to", "a morphism of triangles", "$$", "(a, b, c) : (x_1, y_1, c(f_1)) \\to (x_1, y_1, c(f_1))", "$$", "in $K(\\mathcal{A})$." ], "refs": [], "proofs": [ { "contents": [ "The assumption means there exists a morphism $h : x_1 \\to y_2$ of degree", "$-1$ such that $\\text{d}(h) = b f_1 - f_2 a$. Choose isomorphisms", "$c(f_i) = y_i \\oplus x_i[1]$ of graded objects compatible with the", "morphisms $y_i \\to c(f_i) \\to x_i[1]$. Let's denote", "$a_i : y_i \\to c(f_i)$, $b_i : c(f_i) \\to x_i[1]$, $s_i : x_i[1] \\to c(f_i)$,", "and $\\pi_i : c(f_i) \\to y_i$ the given morphisms. Recall that", "$x_i[1] \\to y_i[1]$ is given by $\\pi_i \\text{d}(s_i)$. By axiom (C)", "this means that", "$$", "f_i = \\pi_i \\text{d}(s_i) = - \\text{d}(\\pi_i) s_i", "$$", "(we identify $\\Hom(x_i, y_i)$ with $\\Hom(x_i[1], y_i[1])$", "using the shift functor $[1]$).", "Set $c = a_2 b \\pi_1 + s_2 a b_1 + a_2hb$. Then, using the", "equalities found in the proof of Lemma \\ref{lemma-get-triangle}", "we obtain", "\\begin{align*}", "\\text{d}(c)", "& =", "a_2 b \\text{d}(\\pi_1) + \\text{d}(s_2) a b_1 + a_2 \\text{d}(h) b_1 \\\\", "& =", "- a_2 b f_1 b_1 + a_2 f_2 a b_1 + a_2 (b f_1 - f_2 a) b_1 \\\\", "& = 0", "\\end{align*}", "(where we have used in particular that", "$\\text{d}(\\pi_1) = \\text{d}(\\pi_1) s_1 b_1 = f_1 b_1$ and", "$\\text{d}(s_2) = a_2 \\pi_2 \\text{d}(s_2) = a_2 f_2$).", "Thus $c$ is a degree $0$ morphism $c : c(f_1) \\to c(f_2)$ of $\\mathcal{A}$", "compatible with the given morphisms $y_i \\to c(f_i) \\to x_i[1]$." ], "refs": [ "dga-lemma-get-triangle" ], "ref_ids": [ 13076 ] } ], "ref_ids": [] }, { "id": 13078, "type": "theorem", "label": "dga-lemma-id-cone-null", "categories": [ "dga" ], "title": "dga-lemma-id-cone-null", "contents": [ "In Situation \\ref{situation-ABC}", "given any object $x$ of $\\mathcal{A}$, and the cone $C(1_x)$ of the", "identity morphism $1_x : x \\to x$, the identity morphism on", "$C(1_x)$ is homotopic to zero." ], "refs": [], "proofs": [ { "contents": [ "Consider the admissible short exact sequence given by axiom (C).", "$$", "\\xymatrix{", "x \\ar@<0.5ex>[r]^a &", "C(1_x) \\ar@<0.5ex>[l]^{\\pi} \\ar@<0.5ex>[r]^b &", "x[1]\\ar@<0.5ex>[l]^s", "}", "$$", "Then by Lemma \\ref{lemma-get-triangle}, identifying hom-sets under", "shifting, we have $1_x=\\pi d(s)=-d(\\pi)s$ where $s$ is regarded as", "a morphism in $\\Hom_{\\mathcal{A}}^{-1}(x,C(1_x))$. Therefore", "$a=a\\pi d(s)=d(s)$ using formula (5) of Lemma \\ref{lemma-get-triangle},", "and $b=-d(\\pi)sb=-d(\\pi)$ by formula (6) of Lemma \\ref{lemma-get-triangle}.", "Hence", "$$", "1_{C(1_x)} = a\\pi + sb = d(s)\\pi - sd(\\pi) = d(s\\pi)", "$$", "since $s$ is of degree $-1$." ], "refs": [ "dga-lemma-get-triangle", "dga-lemma-get-triangle", "dga-lemma-get-triangle" ], "ref_ids": [ 13076, 13076, 13076 ] } ], "ref_ids": [] }, { "id": 13079, "type": "theorem", "label": "dga-lemma-homo-change", "categories": [ "dga" ], "title": "dga-lemma-homo-change", "contents": [ "In Situation \\ref{situation-ABC} given a diagram", "$$", "\\xymatrix{x\\ar[r]^f\\ar[d]_a & y\\ar[d]^b\\\\", "z\\ar[r]^g & w}", "$$", "in $\\text{Comp}(\\mathcal{A})$ commuting up to homotopy. Then", "\\begin{enumerate}", "\\item If $f$ is an admissible monomorphism, then $b$ is homotopic", "to a morphism $b'$ which makes the diagram commute.", "\\item If $g$ is an admissible epimorphism, then $a$ is homotopic", "to a morphism $a'$ which makes the diagram commute.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove (1), observe that the hypothesis implies that there is some", "$h\\in\\Hom_{\\mathcal{A}}(x,w)$ of degree $-1$ such that $bf-ga=d(h)$.", "Since $f$ is an admissible monomorphism, there is a morphism", "$\\pi : y \\to x$ in the category $\\mathcal{A}$ of degree $0$.", "Let $b' = b - d(h\\pi)$. Then", "\\begin{align*}", "b'f = bf - d(h\\pi)f", "= &", "bf - d(h\\pi f) \\quad (\\text{since }d(f) = 0) \\\\", "= &", "bf-d(h) \\\\", "= &", "ga", "\\end{align*}", "as desired. The proof for (2) is omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13080, "type": "theorem", "label": "dga-lemma-factor", "categories": [ "dga" ], "title": "dga-lemma-factor", "contents": [ "In Situation \\ref{situation-ABC} let $\\alpha : x \\to y$", "be a morphism in $\\text{Comp}(\\mathcal{A})$. Then there exists", "a factorization in $\\text{Comp}(\\mathcal{A})$:", "$$", "\\xymatrix{", "x \\ar[r]^{\\tilde{\\alpha}} &", "\\tilde{y} \\ar@<0.5ex>[r]^{\\pi} &", "y\\ar@<0.5ex>[l]^s", "}", "$$", "such that", "\\begin{enumerate}", "\\item $\\tilde{\\alpha}$ is an admissible monomorphism, and", "$\\pi\\tilde{\\alpha}=\\alpha$.", "\\item There exists a morphism", "$s:y\\to\\tilde{y}$ in $\\text{Comp}(\\mathcal{A})$", "such that $\\pi s=1_y$ and $s\\pi$ is homotopic to $1_{\\tilde{y}}$. ", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By axiom (A), we may let $\\tilde{y}$ be the differential graded direct", "sum of $y$ and $C(1_x)$, i.e., there exists a diagram", "$$", "\\xymatrix@C=3pc{", "y \\ar@<0.5ex>[r]^s &", "y\\oplus C(1_x) \\ar@<0.5ex>[l]^{\\pi} \\ar@<0.5ex>[r]^{p} &", "C(1_x)\\ar@<0.5ex>[l]^t", "}", "$$", "where all morphisms are of degree zero, and in", "$\\text{Comp}(\\mathcal{A})$. Let $\\tilde{y} = y \\oplus C(1_x)$.", "Then $1_{\\tilde{y}} = s\\pi + tp$. Consider now the diagram", "$$", "\\xymatrix{", "x \\ar[r]^{\\tilde{\\alpha}} &", "\\tilde{y} \\ar@<0.5ex>[r]^{\\pi} &", "y\\ar@<0.5ex>[l]^s", "}", "$$", "where $\\tilde{\\alpha}$ is induced by the morphism $x\\xrightarrow{\\alpha}y$", "and the natural morphism $x\\to C(1_x)$ fitting in the admissible", "short exact sequence", "$$", "\\xymatrix{", "x \\ar@<0.5ex>[r] &", "C(1_x) \\ar@<0.5ex>[l] \\ar@<0.5ex>[r] &", "x[1]\\ar@<0.5ex>[l]", "}", "$$", "So the morphism $C(1_x)\\to x$ of degree 0 in this diagram,", "together with the zero morphism $y\\to x$, induces a degree-0", "morphism $\\beta : \\tilde{y} \\to x$. Then $\\tilde{\\alpha}$ is an", "admissible monomorphism since it fits into the admissible short", "exact sequence", "$$", "\\xymatrix{", "x\\ar[r]^{\\tilde{\\alpha}} &", "\\tilde{y} \\ar[r] &", "x[1]", "}", "$$", "Furthermore, $\\pi\\tilde{\\alpha} = \\alpha$ by the construction of", "$\\tilde{\\alpha}$, and $\\pi s = 1_y$ by the first diagram. It", "remains to show that $s\\pi$ is homotopic to $1_{\\tilde{y}}$.", "Write $1_x$ as $d(h)$ for some degree $-1$ map. Then, our", "last statement follows from", "\\begin{align*}", "1_{\\tilde{y}} - s\\pi", "= &", "tp \\\\", "= &", "t(dh)p\\quad\\text{(by Lemma \\ref{lemma-id-cone-null})} \\\\", "= &", "d(thp)", "\\end{align*}", "since $dt = dp = 0$, and $t$ is of degree zero." ], "refs": [ "dga-lemma-id-cone-null" ], "ref_ids": [ 13078 ] } ], "ref_ids": [] }, { "id": 13081, "type": "theorem", "label": "dga-lemma-analogue-sequence-maps-split", "categories": [ "dga" ], "title": "dga-lemma-analogue-sequence-maps-split", "contents": [ "In Situation \\ref{situation-ABC}", "let $x_1 \\to x_2 \\to \\ldots \\to x_n$", "be a sequence of composable morphisms in $\\text{Comp}(\\mathcal{A})$.", "Then there exists a commutative diagram in $\\text{Comp}(\\mathcal{A})$:", "$$", "\\xymatrix{x_1\\ar[r] & x_2\\ar[r] & \\ldots\\ar[r] & x_n\\\\", "y_1\\ar[r]\\ar[u] & y_2\\ar[r]\\ar[u] & \\ldots\\ar[r] & y_n\\ar[u]}", "$$", "such that each $y_i\\to y_{i+1}$ is an admissible monomorphism", "and each $y_i\\to x_i$ is a homotopy equivalence." ], "refs": [], "proofs": [ { "contents": [ "The case for $n=1$ is trivial: one simply takes $y_1 = x_1$ and the", "identity morphism on $x_1$ is in particular a homotopy equivalence.", "The case $n = 2$ is given by Lemma \\ref{lemma-factor}. Suppose we have", "constructed the diagram up to $x_{n - 1}$. We apply", "Lemma \\ref{lemma-factor} to the composition", "$y_{n - 1} \\to x_{n-1} \\to x_n$ to obtain $y_n$. Then", "$y_{n - 1} \\to y_n$ will be an admissible monomorphism, and", "$y_n \\to x_n$ a homotopy equivalence." ], "refs": [ "dga-lemma-factor", "dga-lemma-factor" ], "ref_ids": [ 13080, 13080 ] } ], "ref_ids": [] }, { "id": 13082, "type": "theorem", "label": "dga-lemma-triseq", "categories": [ "dga" ], "title": "dga-lemma-triseq", "contents": [ "In Situation \\ref{situation-ABC} let $x_i \\to y_i \\to z_i$", "be morphisms in $\\mathcal{A}$ ($i=1,2,3$) such that", "$x_2 \\to y_2\\to z_2$ is an admissible short exact sequence.", "Let $b : y_1 \\to y_2$ and $b' : y_2\\to y_3$ be morphisms", "in $\\text{Comp}(\\mathcal{A})$ such that", "$$", "\\vcenter{", "\\xymatrix{", "x_1 \\ar[d]_0 \\ar[r] &", "y_1 \\ar[r] \\ar[d]_b &", "z_1 \\ar[d]_0 \\\\", "x_2 \\ar[r] & y_2 \\ar[r] & z_2", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "x_2 \\ar[d]^0 \\ar[r] &", "y_2 \\ar[r] \\ar[d]^{b'} &", "z_2 \\ar[d]^0 \\\\", "x_3 \\ar[r] & y_3 \\ar[r] & z_3", "}", "}", "$$", "commute up to homotopy. Then $b'\\circ b$ is homotopic to $0$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-homo-change}, we can replace $b$ and $b'$", "by homotopic maps $\\tilde{b}$ and $\\tilde{b}'$, such that the right", "square of the left diagram commutes and the left square of the right", "diagram commutes. Say $b = \\tilde{b} + d(h)$ and $b'=\\tilde{b}'+d(h')$", "for degree $-1$ morphisms $h$ and $h'$ in $\\mathcal{A}$. Hence", "$$", "b'b = \\tilde{b}'\\tilde{b} + d(\\tilde{b}'h + h'\\tilde{b} + h'd(h))", "$$", "since $d(\\tilde{b})=d(\\tilde{b}')=0$, i.e. $b'b$ is homotopic to", "$\\tilde{b}'\\tilde{b}$. We now want to show that $\\tilde{b}'\\tilde{b}=0$.", "Because $x_2\\xrightarrow{f} y_2\\xrightarrow{g} z_2$ is an admissible", "short exact sequence, there exist degree $0$ morphisms", "$\\pi : y_2 \\to x_2$ and $s : z_2 \\to y_2$ such that", "$\\text{id}_{y_2} = f\\pi + sg$. Therefore", "$$", "\\tilde{b}'\\tilde{b} = \\tilde{b}'(f\\pi + sg)\\tilde{b} = 0", "$$", "since $g\\tilde{b} = 0$ and $\\tilde{b}'f = 0$ as consequences", "of the two commuting squares." ], "refs": [ "dga-lemma-homo-change" ], "ref_ids": [ 13079 ] } ], "ref_ids": [] }, { "id": 13083, "type": "theorem", "label": "dga-lemma-analogue-triangle-independent-splittings", "categories": [ "dga" ], "title": "dga-lemma-analogue-triangle-independent-splittings", "contents": [ "In Situation \\ref{situation-ABC}", "let $0 \\to x \\to y \\to z \\to 0$ be an admissible short", "exact sequence in $\\text{Comp}(\\mathcal{A})$. The triangle", "$$", "\\xymatrix{x\\ar[r] & y\\ar[r] & z\\ar[r]^{\\delta} & x[1]}", "$$", "with $\\delta : z \\to x[1]$ as defined in Lemma \\ref{lemma-get-triangle}", "is up to canonical isomorphism in $K(\\mathcal{A})$, independent of the", "choices made in Lemma \\ref{lemma-get-triangle}." ], "refs": [ "dga-lemma-get-triangle", "dga-lemma-get-triangle" ], "proofs": [ { "contents": [ "Suppose $\\delta$ is defined by the splitting", "$$", "\\xymatrix{", "x \\ar@<0.5ex>[r]^{a} &", "y \\ar@<0.5ex>[r]^b\\ar@<0.5ex>[l]^{\\pi} &", "z \\ar@<0.5ex>[l]^s", "}", "$$", "and $\\delta'$ is defined by the splitting with $\\pi',s'$", "in place of $\\pi,s$. Then", "$$", "s'-s = (a\\pi + sb)(s'-s) = a\\pi s'", "$$", "since $bs' = bs = 1_z$ and $\\pi s = 0$. Similarly,", "$$", "\\pi' - \\pi = (\\pi' - \\pi)(a\\pi + sb) = \\pi'sb", "$$", "Since $\\delta = \\pi d(s)$ and $\\delta' = \\pi'd(s')$", "as constructed in Lemma \\ref{lemma-get-triangle}, we may compute", "$$", "\\delta' = \\pi'd(s') = (\\pi + \\pi'sb)d(s + a\\pi s') = \\delta + d(\\pi s')", "$$", "using $\\pi a = 1_x$, $ba = 0$, and $\\pi'sbd(s') = \\pi'sba\\pi d(s') = 0$", "by formula (5) in Lemma \\ref{lemma-get-triangle}." ], "refs": [ "dga-lemma-get-triangle", "dga-lemma-get-triangle" ], "ref_ids": [ 13076, 13076 ] } ], "ref_ids": [ 13076, 13076 ] }, { "id": 13084, "type": "theorem", "label": "dga-lemma-restate-axiom-c", "categories": [ "dga" ], "title": "dga-lemma-restate-axiom-c", "contents": [ "In Situation \\ref{situation-ABC}", "let $f: x \\to y$ be a morphism in $\\text{Comp}(\\mathcal{A})$.", "The triangle $(y, c(f), x[1], i, p, f[1])$ is the triangle associated", "to the admissible short exact sequence ", "$$", "\\xymatrix{y\\ar[r] & c(f) \\ar[r] & x[1]}", "$$", "where the cone $c(f)$ is defined as in Lemma \\ref{lemma-get-triangle}." ], "refs": [ "dga-lemma-get-triangle" ], "proofs": [ { "contents": [ "This follows from axiom (C)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13076 ] }, { "id": 13085, "type": "theorem", "label": "dga-lemma-cone-rotate-isom", "categories": [ "dga" ], "title": "dga-lemma-cone-rotate-isom", "contents": [ "In Situation \\ref{situation-ABC} let $\\alpha : x \\to y$ and $\\beta : y \\to z$", "define an admissible short exact sequence", "$$", "\\xymatrix{", "x \\ar[r] &", "y\\ar[r] &", "z", "}", "$$", "in $\\text{Comp}(\\mathcal{A})$. Let $(x, y, z, \\alpha, \\beta, \\delta)$", "be the associated triangle in $K(\\mathcal{A})$. Then, the triangles", "$$", "(z[-1], x, y, \\delta[-1], \\alpha, \\beta)", "\\quad\\text{and}\\quad", "(z[-1], x, c(\\delta[-1]), \\delta[-1], i, p)", "$$", "are isomorphic." ], "refs": [], "proofs": [ { "contents": [ "We have a diagram of the form", "$$", "\\xymatrix{", "z[-1]\\ar[r]^{\\delta[-1]}\\ar[d]^1 &", "x\\ar@<0.5ex>[r]^{\\alpha}\\ar[d]^1 &", "y\\ar@<0.5ex>[r]^{\\beta}\\ar@{.>}[d]\\ar@<0.5ex>[l]^{\\tilde{\\alpha}} &", "z\\ar[d]^1\\ar@<0.5ex>[l]^{\\tilde\\beta} \\\\", "z[-1] \\ar[r]^{\\delta[-1]} &", "x\\ar@<0.5ex>[r]^i &", "c(\\delta[-1]) \\ar@<0.5ex>[r]^p\\ar@<0.5ex>[l]^{\\tilde i} &", "z\\ar@<0.5ex>[l]^{\\tilde p}", "}", "$$", "with splittings to $\\alpha, \\beta, i$, and $p$ given by", "$\\tilde{\\alpha}, \\tilde{\\beta}, \\tilde{i},$ and $\\tilde{p}$ respectively.", "Define a morphism $y \\to c(\\delta[-1])$ by", "$i\\tilde{\\alpha} + \\tilde{p}\\beta$ and a morphism", "$c(\\delta[-1]) \\to y$ by $\\alpha \\tilde{i} + \\tilde{\\beta} p$.", "Let us first check that these define morphisms in $\\text{Comp}(\\mathcal{A})$.", "We remark that by identities from Lemma \\ref{lemma-get-triangle},", "we have the relation", "$\\delta[-1] = \\tilde{\\alpha}d(\\tilde{\\beta}) = -d(\\tilde{\\alpha})\\tilde{\\beta}$", "and the relation $\\delta[-1] = \\tilde{i}d(\\tilde{p})$. Then", "\\begin{align*}", "d(\\tilde{\\alpha})", "& =", "d(\\tilde{\\alpha})\\tilde{\\beta}\\beta \\\\", "& =", "-\\delta[-1]\\beta", "\\end{align*}", "where we have used equation (6) of", "Lemma \\ref{lemma-get-triangle} for the first equality and", "the preceeding remark for the second. Similarly, we obtain", "$d(\\tilde{p}) = i\\delta[-1]$. Hence", "\\begin{align*}", "d(i\\tilde{\\alpha} + \\tilde{p}\\beta)", "& =", "d(i)\\tilde{\\alpha} + id(\\tilde{\\alpha}) +", "d(\\tilde{p})\\beta + \\tilde{p}d(\\beta) \\\\", "& =", "id(\\tilde{\\alpha}) + d(\\tilde{p})\\beta \\\\", "& =", "-i\\delta[-1]\\beta + i\\delta[-1]\\beta \\\\", "& =", "0", "\\end{align*}", "so $i\\tilde{\\alpha} + \\tilde{p}\\beta$ is indeed a morphism of", "$\\text{Comp}(\\mathcal{A})$. By a similar calculation,", "$\\alpha \\tilde{i} + \\tilde{\\beta} p$ is also a morphism of", "$\\text{Comp}(\\mathcal{A})$. It is immediate that these morphisms", "fit in the commutative diagram. We compute:", "\\begin{align*}", "(i\\tilde{\\alpha} + \\tilde{p}\\beta)(\\alpha \\tilde{i} + \\tilde{\\beta} p)", "& =", "i\\tilde{\\alpha}\\alpha\\tilde{i} + i\\tilde{\\alpha}\\tilde{\\beta}p", "+ \\tilde{p}\\beta\\alpha\\tilde{i} + \\tilde{p}\\beta\\tilde{\\beta}p \\\\", "& =", "i\\tilde{i} + \\tilde{p}p \\\\", "& =", "1_{c(\\delta[-1])}", "\\end{align*}", "where we have freely used the identities of", "Lemma \\ref{lemma-get-triangle}. Similarly, we compute", "$(\\alpha \\tilde{i} + \\tilde{\\beta} p)(i\\tilde{\\alpha} + \\tilde{p}\\beta) = 1_y$,", "so we conclude $y \\cong c(\\delta[-1])$. Hence, the two triangles in question", "are isomorphic." ], "refs": [ "dga-lemma-get-triangle", "dga-lemma-get-triangle", "dga-lemma-get-triangle" ], "ref_ids": [ 13076, 13076, 13076 ] } ], "ref_ids": [] }, { "id": 13086, "type": "theorem", "label": "dga-lemma-analogue-third-isomorphism", "categories": [ "dga" ], "title": "dga-lemma-analogue-third-isomorphism", "contents": [ "In Situation \\ref{situation-ABC} let $f_1 : x_1 \\to y_1$ and", "$f_2 : x_2 \\to y_2$ be morphisms in $\\text{Comp}(\\mathcal{A})$. Let ", "$$", "(a,b,c): (x_1,y_1,c(f_1), f_1, i_1, p_1) \\to (x_2,y_2, c(f_2), f_2, i_1, p_1)", "$$", "be any morphism of triangles in $K(\\mathcal{A})$.", "If $a$ and $b$ are homotopy equivalences, then so is $c$." ], "refs": [], "proofs": [ { "contents": [ "Since $a$ and $b$ are homotopy equivalences, they are invertible in", "$K(\\mathcal{A})$ so let $a^{-1}$ and $b^{-1}$ denote their inverses", "in $K(\\mathcal{A})$, giving us a commutative diagram ", "$$", "\\xymatrix{", "x_2\\ar[d]^{a^{-1}}\\ar[r]^{f_2} &", "y_2\\ar[d]^{b^{-1}}\\ar[r]^{i_2} &", "c(f_2)\\ar[d]^{c'} \\\\", "x_1\\ar[r]^{f_1} &", "y_1 \\ar[r]^{i_1} &", "c(f_1)", "}", "$$", "where the map $c'$ is defined via Lemma \\ref{lemma-cone} applied to the left", "commutative box of the above diagram. Since the diagram commutes", "in $K(\\mathcal{A})$, it suffices by Lemma \\ref{lemma-triseq} to", "prove the following: given a morphism of triangle", "$(1,1,c): (x,y,c(f),f,i,p)\\to (x,y,c(f),f,i,p)$", "in $K(\\mathcal{A})$, the map $c$ is an isomorphism in", "$K(\\mathcal{A})$. We have the commutative diagrams in $K(\\mathcal{A})$:", "$$", "\\vcenter{", "\\xymatrix{", "y\\ar[d]^{1}\\ar[r] &", "c(f)\\ar[d]^{c}\\ar[r] &", "x[1]\\ar[d]^{1} \\\\", "y\\ar[r] &", "c(f) \\ar[r] &", "x[1]", "}", "}", "\\quad\\Rightarrow\\quad", "\\vcenter{", "\\xymatrix{", "y\\ar[d]^{0}\\ar[r] &", "c(f)\\ar[d]^{c-1}\\ar[r] &", "x[1]\\ar[d]^{0} \\\\", "y\\ar[r] &", "c(f) \\ar[r] &", "x[1]", "}", "}", "$$", "Since the rows are admissible short exact sequences, we obtain", "the identity $(c-1)^2 = 0$ by Lemma \\ref{lemma-triseq}, from", "which we conclude that $2-c$ is inverse to $c$ in $K(\\mathcal{A})$", "so that $c$ is an isomorphism." ], "refs": [ "dga-lemma-cone", "dga-lemma-triseq", "dga-lemma-triseq" ], "ref_ids": [ 13077, 13082, 13082 ] } ], "ref_ids": [] }, { "id": 13087, "type": "theorem", "label": "dga-lemma-cone-homotopy", "categories": [ "dga" ], "title": "dga-lemma-cone-homotopy", "contents": [ "In Situation \\ref{situation-ABC}.", "\\begin{enumerate}", "\\item Given an admissible short exact sequence", "$x\\xrightarrow{\\alpha} y\\xrightarrow{\\beta} z$.", "Then there exists a homotopy equivalence", "$e:C(\\alpha)\\to z$ such that the diagram", "\\begin{equation}", "\\label{equation-cone-isom-triangle}", "\\vcenter{", "\\xymatrix{", "x\\ar[r]^{\\alpha}\\ar[d] &", "y\\ar[r]^{b}\\ar[d] &", "C(\\alpha)\\ar[r]^{-c}\\ar@{.>}[d]^{e} &", "x[1]\\ar[d] \\\\", "x\\ar[r]^{\\alpha} &", "y\\ar[r]^{\\beta} &", "z\\ar[r]^{\\delta} & x[1]", "}", "}", "\\end{equation}", "defines an isomorphism of triangles in $K(\\mathcal{A})$. Here", "$y\\xrightarrow{b}C(\\alpha)\\xrightarrow{c}x[1]$", "is the admissible short exact sequence given as in axiom (C).", "\\item Given a morphism", "$\\alpha : x \\to y$ in $\\text{Comp}(\\mathcal{A})$, let", "$x \\xrightarrow{\\tilde{\\alpha}} \\tilde{y} \\to y$ be the", "factorization given as in Lemma \\ref{lemma-factor}, where the admissible", "monomorphism $x \\xrightarrow{\\tilde{\\alpha}} y$ extends to the", "admissible short exact sequence", "$$", "\\xymatrix{", "x \\ar[r]^{\\tilde{\\alpha}} &", "\\tilde{y} \\ar[r] & z", "}", "$$", "Then there exists an isomorphism of triangles", "$$", "\\xymatrix{", "x \\ar[r]^{\\tilde{\\alpha}} \\ar[d] &", "\\tilde{y} \\ar[r] \\ar[d] &", "z \\ar[r]^{\\delta} \\ar@{.>}[d]^{e} &", "x[1] \\ar[d] \\\\", "x \\ar[r]^{\\alpha} &", "y \\ar[r] &", "C(\\alpha) \\ar[r]^{-c} &", "x[1]", "}", "$$", "where the upper triangle is the triangle", "associated to the sequence", "$x \\xrightarrow{\\tilde{\\alpha}} \\tilde{y} \\to z$.", "\\end{enumerate}" ], "refs": [ "dga-lemma-factor" ], "proofs": [ { "contents": [ "For (1), we consider the more complete diagram, \\emph{without} the", "sign change on $c$:", "$$", "\\xymatrix{", "x\\ar@<0.5ex>[r]^{\\alpha} \\ar[d] &", "y\\ar@<0.5ex>[l]^{\\pi} \\ar@<0.5ex>[r]^{b}\\ar[d] &", "C(\\alpha)\\ar@<0.5ex>[l]^{p} \\ar@<0.5ex>[r]^{c}\\ar@{.>}@<0.5ex>[d]^{e} &", "x[1]\\ar@<0.5ex>[l]^{\\sigma} \\ar[d]\\ar@<0.5ex>[r]^{\\alpha} &", "y[1]\\ar@<0.5ex>[l]^{\\pi} \\\\", "x\\ar@<0.5ex>[r]^{\\alpha} &", "y\\ar@<0.5ex>[r]^{\\beta} \\ar@<0.5ex>[l]^{\\pi} &", "z\\ar[r]^{\\delta}\\ar@<0.5ex>[l]^{s} \\ar@{.>}@<0.5ex>[u]^{f} &", "x[1]", "}", "$$", "where the admissible short exact sequence", "$x\\xrightarrow{\\alpha} y\\xrightarrow{\\beta} z$", "is given the splitting $\\pi$, $s$, and the admissible short exact sequence", "$y\\xrightarrow{b}C(\\alpha)\\xrightarrow{c}x[1]$ is given the splitting", "$p$, $\\sigma$. Note that (identifying hom-sets under shifting)", "$$", "\\alpha = pd(\\sigma) = -d(p)\\sigma,\\quad", "\\delta = \\pi d(s) = -d(\\pi)s", "$$", "by the construction in Lemma \\ref{lemma-get-triangle}.", "\\medskip\\noindent", "We define $e=\\beta p$ and $f=bs-\\sigma\\delta$. We first check that they are", "morphisms in $\\text{Comp}(\\mathcal{A})$. To show that $d(e)=\\beta d(p)$", "vanishes, it suffices to show that $\\beta d(p)b$ and $\\beta d(p)\\sigma$", "both vanish, whereas", "$$", "\\beta d(p)b = \\beta d(pb) = \\beta d(1_y) = 0,\\quad", "\\beta d(p)\\sigma = -\\beta\\alpha = 0", "$$", "Similarly, to check that $d(f)=bd(s)-d(\\sigma)\\delta$ vanishes,", "it suffices to check the post-compositions by $p$ and $c$ both vanish,", "whereas", "\\begin{align*}", "pbd(s) - pd(\\sigma)\\delta", "= &", "d(s)-\\alpha\\delta = d(s)-\\alpha\\pi d(s) = 0 \\\\", "cbd(s)-cd(\\sigma)\\delta", "= &", "-cd(\\sigma)\\delta = -d(c\\sigma)\\delta = 0", "\\end{align*}", "The commutativity of left two squares of the", "diagram \\ref{equation-cone-isom-triangle} follows directly from definition.", "Before we prove the commutativity of the right square (up to homotopy),", "we first check that $e$ is a homotopy equivalence. Clearly,", "$$", "ef=\\beta p (bs-\\sigma\\delta)=\\beta s=1_z", "$$", "To check that $fe$ is homotopic to $1_{C(\\alpha)}$, we first observe", "$$", "b\\alpha = bpd(\\alpha) = d(\\sigma),\\quad", "\\alpha c = -d(p)\\sigma c = -d(p),\\quad", "d(\\pi)p = d(\\pi)s\\beta p = -\\delta\\beta p", "$$", "Using these identities, we compute", "\\begin{align*}", "1_{C(\\alpha)} = &", "bp + \\sigma c", "\\quad (\\text{from }y \\xrightarrow{b} C(\\alpha) \\xrightarrow{c} x[1]) \\\\", "= &", "b(\\alpha\\pi + s\\beta)p + \\sigma(\\pi\\alpha)c", "\\quad (\\text{from }x \\xrightarrow{\\alpha} y \\xrightarrow{\\beta} z) \\\\", "= &", "d(\\sigma)\\pi p + bs\\beta p - \\sigma\\pi d(p)", "\\quad (\\text{by the first two identities above}) \\\\", "= &", "d(\\sigma)\\pi p + bs\\beta p - \\sigma\\delta\\beta p", "+ \\sigma\\delta\\beta p - \\sigma\\pi d(p) \\\\", "= &", "(bs - \\sigma\\delta)\\beta p + d(\\sigma)\\pi p", "- \\sigma d(\\pi)p - \\sigma\\pi d(p)\\quad", "(\\text{by the third identity above}) \\\\", "= &", "fe + d(\\sigma \\pi p)", "\\end{align*}", "since $\\sigma \\in \\Hom^{-1}(x, C(\\alpha))$", "(cf. proof of Lemma \\ref{lemma-id-cone-null}).", "Hence $e$ and $f$ are homotopy inverses.", "Finally, to check that the right square of", "diagram \\ref{equation-cone-isom-triangle} commutes up to homotopy,", "it suffices to check that $-cf=\\delta$. This follows from", "$$", "-cf = -c(bs-\\sigma\\delta) = c\\sigma\\delta = \\delta", "$$", "since $cb=0$.", "\\medskip\\noindent", "For (2), consider the factorization", "$x\\xrightarrow{\\tilde{\\alpha}}\\tilde{y}\\to y$", "given as in Lemma \\ref{lemma-factor}, so the second morphism", "is a homotopy equivalence. By Lemmas \\ref{lemma-cone} and", "\\ref{lemma-analogue-third-isomorphism}, there", "exists an isomorphism of triangles between", "$$", "x \\xrightarrow{\\alpha} y \\to C(\\alpha) \\to x[1]", "\\quad\\text{and}\\quad", "x \\xrightarrow{\\tilde{\\alpha}} \\tilde{y} \\to C(\\tilde{\\alpha}) \\to x[1]", "$$", "Since we can compose isomorphisms of triangles, by replacing", "$\\alpha$ by $\\tilde{\\alpha}$, $y$ by $\\tilde{y}$, and $C(\\alpha)$ by", "$C(\\tilde{\\alpha})$, we may assume $\\alpha$ is an admissible monomorphism.", "In this case, the result follows from (1)." ], "refs": [ "dga-lemma-get-triangle", "dga-lemma-id-cone-null", "dga-lemma-factor", "dga-lemma-cone", "dga-lemma-analogue-third-isomorphism" ], "ref_ids": [ 13076, 13078, 13080, 13077, 13086 ] } ], "ref_ids": [ 13080 ] }, { "id": 13088, "type": "theorem", "label": "dga-lemma-analogue-homotopy-category-pre-triangulated", "categories": [ "dga" ], "title": "dga-lemma-analogue-homotopy-category-pre-triangulated", "contents": [ "In Situation \\ref{situation-ABC} the homotopy category $K(\\mathcal{A})$", "with its natural translation functors and distinguished triangles", "is a pre-triangulated category." ], "refs": [], "proofs": [ { "contents": [ "We will verify each of TR1, TR2, and TR3. ", "\\medskip\\noindent", "Proof of TR1. By definition every triangle isomorphic to a distinguished", "one is distinguished. Since", "$$", "\\xymatrix{x\\ar[r]^{1_x} & x\\ar[r] & 0}", "$$", "is an admissible short exact sequence, $(x, x, 0, 1_x, 0, 0)$", "is a distinguished triangle. Moreover, given a morphism", "$\\alpha : x \\to y$ in $\\text{Comp}(\\mathcal{A})$, the triangle", "given by $(x, y, c(\\alpha), \\alpha, i, -p)$ is distinguished by", "Lemma \\ref{lemma-cone-homotopy}.", "\\medskip\\noindent", "Proof of TR2. Let $(x,y,z,\\alpha,\\beta,\\gamma)$ be a triangle and", "suppose $(y,z,x[1],\\beta,\\gamma,-\\alpha[1])$ is distinguished.", "Then there exists an admissible short exact sequence", "$0 \\to x' \\to y' \\to z' \\to 0$ such that the associated triangle", "$(x',y',z',\\alpha',\\beta',\\gamma')$ is isomorphic to", "$(y,z,x[1],\\beta,\\gamma,-\\alpha[1])$. After rotating, we conclude", "that $(x,y,z,\\alpha,\\beta,\\gamma)$ is isomorphic to", "$(z'[-1],x',y', \\gamma'[-1], \\alpha',\\beta')$. By", "Lemma \\ref{lemma-cone-rotate-isom},", "we deduce that $(z'[-1],x',y', \\gamma'[-1], \\alpha',\\beta')$ is", "isomorphic to $(z'[-1],x',c(\\gamma'[-1]), \\gamma'[-1], i, p)$.", "Composing the two isomorphisms with sign changes as indicated in", "the following diagram:", "$$", "\\xymatrix@C=3pc{", "x\\ar[r]^{\\alpha}\\ar[d] &", "y\\ar[r]^{\\beta}\\ar[d] &", "z\\ar[r]^{\\gamma}\\ar[d] &", "x[1]\\ar[d] \\\\", "z'[-1]\\ar[r]^{-\\gamma'[-1]}\\ar[d]_{-1_{z'[-1]}} &", "x \\ar[r]^{\\alpha'}\\ar@{=}[d] &", "y' \\ar[r]^{\\beta'} \\ar[d] &", "z'\\ar[d]^{-1_{z'}} \\\\", "z'[-1]\\ar[r]^{\\gamma'[-1]} &", "x \\ar[r]^{\\alpha'} &", "c(\\gamma'[-1]) \\ar[r]^{-p} &", "z'", "}", "$$", "We conclude that $(x,y,z,\\alpha,\\beta,\\gamma)$ is distinguished by", "Lemma \\ref{lemma-cone-homotopy} (2). Conversely, suppose that", "$(x,y,z,\\alpha,\\beta,\\gamma)$ is distinguished, so that by", "Lemma \\ref{lemma-cone-homotopy} (1), it is isomorphic to a", "triangle of the form $(x',y', c(\\alpha'), \\alpha', i, -p)$", "for some morphism $\\alpha': x' \\to y'$ in $\\text{Comp}(\\mathcal{A})$.", "The rotated triangle $(y,z,x[1],\\beta,\\gamma, -\\alpha[1])$ is", "isomorphic to the triangle $(y',c(\\alpha'), x'[1], i, -p, -\\alpha[1])$", "which is isomorphic to $(y',c(\\alpha'), x'[1], i, p, \\alpha[1])$.", "By Lemma \\ref{lemma-restate-axiom-c}, this triangle is distinguished,", "from which it follows that $(y,z,x[1], \\beta,\\gamma, -\\alpha[1])$", "is distinguished.", "\\medskip\\noindent", "Proof of TR3: Suppose $(x,y,z, \\alpha,\\beta,\\gamma)$ and", "$(x',y',z',\\alpha',\\beta',\\gamma')$ are distinguished triangles", "of $\\text{Comp}(\\mathcal{A})$ and let $f: x \\to x'$ and", "$g: y \\to y'$ be morphisms such that", "$\\alpha' \\circ f = g \\circ \\alpha$. By", "Lemma \\ref{lemma-cone-homotopy}, we may assume that", "$(x,y,z,\\alpha,\\beta,\\gamma)= (x,y,c(\\alpha),\\alpha, i, -p)$", "and $(x',y',z', \\alpha',\\beta',\\gamma')= (x',y',c(\\alpha'), \\alpha',i',-p')$.", "Now apply Lemma \\ref{lemma-cone}", "and we are done." ], "refs": [ "dga-lemma-cone-homotopy", "dga-lemma-cone-rotate-isom", "dga-lemma-cone-homotopy", "dga-lemma-cone-homotopy", "dga-lemma-restate-axiom-c", "dga-lemma-cone-homotopy", "dga-lemma-cone" ], "ref_ids": [ 13087, 13085, 13087, 13087, 13084, 13087, 13077 ] } ], "ref_ids": [] }, { "id": 13089, "type": "theorem", "label": "dga-lemma-dgc-analogue-tr4", "categories": [ "dga" ], "title": "dga-lemma-dgc-analogue-tr4", "contents": [ "In Situation \\ref{situation-ABC} given admissible monomorphisms", "$x \\xrightarrow{\\alpha} y$, $y \\xrightarrow{\\beta} z$ in $\\mathcal{A}$,", "there exist distinguished triangles", "$(x,y,q_1,\\alpha,p_1,\\delta_1)$, $(x,z,q_2,\\beta\\alpha,p_2,\\delta_2)$", "and $(y,z,q_3,\\beta,p_3,\\delta_3)$ for which TR4 holds." ], "refs": [], "proofs": [ { "contents": [ "Given admissible monomorphisms $x\\xrightarrow{\\alpha} y$ and", "$y\\xrightarrow{\\beta}z$, we can find distinguished triangles,", "via their extensions to admissible short exact sequences,", "$$", "\\xymatrix{", "x\\ar@<0.5ex>[r]^{\\alpha} &", "y\\ar@<0.5ex>[l]^{\\pi_1} \\ar@<0.5ex>[r]^{p_1} &", "q_1 \\ar[r]^{\\delta_1} \\ar@<0.5ex>[l]^{s_1} &", "x[1]", "}", "$$", "$$", "\\xymatrix{", "x\\ar@<0.5ex>[r]^{\\beta\\alpha} &", "z\\ar@<0.5ex>[l]^{\\pi_1\\pi_3} \\ar@<0.5ex>[r]^{p_2} &", "q_2 \\ar[r]^{\\delta_2} \\ar@<0.5ex>[l]^{s_2} &", "x[1]", "}", "$$", "$$", "\\xymatrix{", "y\\ar@<0.5ex>[r]^{\\beta} &", "z\\ar@<0.5ex>[l]^{\\pi_3} \\ar@<0.5ex>[r]^{p_3} &", "q_3 \\ar[r]^{\\delta_3} \\ar@<0.5ex>[l]^{s_3} &", "x[1]", "}", "$$", "In these diagrams, the maps $\\delta_i$ are defined as", "$\\delta_i = \\pi_i d(s_i)$ analagous to the maps defined in", "Lemma \\ref{lemma-get-triangle}.", "They fit in the following solid commutative diagram", "$$", "\\xymatrix@C=5pc@R=3pc{", "x\\ar@<0.5ex>[r]^{\\alpha} \\ar@<0.5ex>[dr]^{\\beta\\alpha} &", "y\\ar@<0.5ex>[d]^{\\beta} \\ar@<0.5ex>[l]^{\\pi_1} \\ar@<0.5ex>[r]^{p_1} &", "q_1 \\ar[r]^{\\delta_1} \\ar@<0.5ex>[l]^{s_1} \\ar@{.>}[dd]^{p_2\\beta s_1} &", "x[1] \\\\", " &", "z \\ar@<0.5ex>[u]^{\\pi_3}\\ar@<0.5ex>[d]^{p_3}", "\\ar@<0.5ex>[dr]^{p_2} \\ar@<0.5ex>[ul]^{\\pi_1\\pi_3} & & \\\\", " &", "q_3\\ar@<0.5ex>[u]^{s_3} \\ar[d]^{\\delta_3} &", "q_2 \\ar@{.>}[l]^{p_3s_2} \\ar@<0.5ex>[ul]^{s_2} \\ar[dr]^{\\delta_2} \\\\", " &", "y[1] & & x[1]}", "$$", "where we have defined the dashed arrows as indicated.", "Clearly, their composition $p_3s_2p_2\\beta s_1 = 0$", "since $s_2p_2 = 0$. We claim that they both are morphisms of", "$\\text{Comp}(\\mathcal{A})$. We can check this using equations in", "Lemma \\ref{lemma-get-triangle}:", "$$", "d(p_2\\beta s_1) = p_2\\beta d(s_1) = p_2\\beta\\alpha\\pi_1 d(s_1) = 0", "$$", "since $p_2\\beta\\alpha = 0$, and", "$$", "d(p_3s_2) = p_3d(s_2) = p_3\\beta\\alpha\\pi_1\\pi_3 d(s_2) = 0", "$$", "since $p_3\\beta = 0$. To check that $q_1\\to q_2\\to q_3$", "is an admissible short exact sequence, it remains to show", "that in the underlying graded category, $q_2 = q_1\\oplus q_3$", "with the above two morphisms as coprojection and projection.", "To do this, observe that in the underlying graded category", "$\\mathcal{C}$, there hold", "$$", "y = x\\oplus q_1,\\quad", "z = y\\oplus q_3 = x\\oplus q_1\\oplus q_3", "$$", "where $\\pi_1\\pi_3$ gives the projection morphism onto the first", "factor: $x\\oplus q_1\\oplus q_3\\to z$. By axiom (A) on", "$\\mathcal{A}$, $\\mathcal{C}$ is an additive category, hence", "we may apply", "Homology, Lemma \\ref{homology-lemma-additive-cat-biproduct-kernel}", "and conclude that", "$$", "\\Ker(\\pi_1\\pi_3) = q_1\\oplus q_3", "$$", "in $\\mathcal{C}$. Another application of", "Homology, Lemma \\ref{homology-lemma-additive-cat-biproduct-kernel}", "to $z = x\\oplus q_2$ gives $\\Ker(\\pi_1\\pi_3) = q_2$.", "Hence $q_2\\cong q_1\\oplus q_3$ in $\\mathcal{C}$.", "It is clear that the dashed morphisms defined above give", "coprojection and projection.", "\\medskip\\noindent", "Finally, we have to check that the morphism", "$\\delta : q_3 \\to q_1[1]$ induced by the admissible", "short exact sequence $q_1\\to q_2\\to q_3$ agrees with", "$p_1\\delta_3$. By the construction in", "Lemma \\ref{lemma-get-triangle}, the morphism $\\delta$ is given by", "\\begin{align*}", "p_1\\pi_3s_2d(p_2s_3)", "= &", "p_1\\pi_3s_2p_2d(s_3) \\\\", "= &", "p_1\\pi_3(1-\\beta\\alpha\\pi_1\\pi_3)d(s_3) \\\\", "= &", "p_1\\pi_3d(s_3)\\quad (\\text{since }\\pi_3\\beta = 0) \\\\", "= &", "p_1\\delta_3", "\\end{align*}", "as desired. The proof is complete." ], "refs": [ "dga-lemma-get-triangle", "dga-lemma-get-triangle", "homology-lemma-additive-cat-biproduct-kernel", "homology-lemma-additive-cat-biproduct-kernel", "dga-lemma-get-triangle" ], "ref_ids": [ 13076, 13076, 12011, 12011, 13076 ] } ], "ref_ids": [] }, { "id": 13090, "type": "theorem", "label": "dga-lemma-functor-between-ABC", "categories": [ "dga" ], "title": "dga-lemma-functor-between-ABC", "contents": [ "Let $R$ be a ring. Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor", "between differential graded categories over $R$ satisfying axioms", "(A), (B), and (C) such that $F(x[1]) = F(x)[1]$.", "Then $F$ induces an exact functor", "$K(\\mathcal{A}) \\to K(\\mathcal{B})$ of triangulated categories." ], "refs": [], "proofs": [ { "contents": [ "Namely, if $x \\to y \\to z$ is an admissible short exact sequence", "in $\\text{Comp}(\\mathcal{A})$, then $F(x) \\to F(y) \\to F(z)$", "is an admissible short exact sequence in $\\text{Comp}(\\mathcal{B})$.", "Moreover, the ``boundary'' morphism $\\delta = \\pi\\text{d}(s) : z \\to x[1]$", "constructed in Lemma \\ref{lemma-get-triangle} produces the morphism", "$F(\\delta) : F(z) \\to F(x[1]) = F(x)[1]$ which is equal to the boundary", "map $F(\\pi) \\text{d}(F(s))$ for the admissible short exact sequence", "$F(x) \\to F(y) \\to F(z)$." ], "refs": [ "dga-lemma-get-triangle" ], "ref_ids": [ 13076 ] } ], "ref_ids": [] }, { "id": 13091, "type": "theorem", "label": "dga-lemma-what-makes-a-bimodule-dg", "categories": [ "dga" ], "title": "dga-lemma-what-makes-a-bimodule-dg", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be", "differential graded algebras over $R$. Let $M$ be a right differential", "graded $B$-module. There is a $1$-to-$1$ correspondence", "between $(A, B)$-bimodule structures on $M$ compatible with the given", "differential graded $B$-module structure and homomorphisms", "$$", "A", "\\longrightarrow", "\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(M, M)", "$$", "of differential graded $R$-algebras." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mu : A \\times M \\to M$ define a left differential graded $A$-module", "structure on the underlying complex of $R$-modules $M^\\bullet$ of $M$.", "By Lemma \\ref{lemma-left-module-structure} the structure $\\mu$ corresponds", "to a map $\\gamma : A \\to \\Hom^\\bullet(M^\\bullet, M^\\bullet)$", "of differential graded $R$-algebras. The assertion of the lemma is simply", "that $\\mu$ commutes with the $B$-action, if and only if $\\gamma$ ends", "up inside", "$$", "\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(M, M) \\subset", "\\Hom^\\bullet(M^\\bullet, M^\\bullet)", "$$", "We omit the detailed calculation." ], "refs": [ "dga-lemma-left-module-structure" ], "ref_ids": [ 13050 ] } ], "ref_ids": [] }, { "id": 13092, "type": "theorem", "label": "dga-lemma-bimodule-over-tensor", "categories": [ "dga" ], "title": "dga-lemma-bimodule-over-tensor", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$", "be differential graded algebras over $R$. The construction above", "defines an equivalence of categories", "$$", "\\begin{matrix}", "\\text{differential graded}\\\\", "(A, B)\\text{-bimodules}", "\\end{matrix}", "\\longleftrightarrow", "\\begin{matrix}", "\\text{right differential graded }\\\\", "A^{opp} \\otimes_R B\\text{-modules}", "\\end{matrix}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Immediate from discussion the above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13093, "type": "theorem", "label": "dga-lemma-bimodule-resolve", "categories": [ "dga" ], "title": "dga-lemma-bimodule-resolve", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be", "differential graded $R$-algebras. Let $M$ be a differential graded", "$(A, B)$-bimodule. There exists a homomorphism $P \\to M$", "of differential graded $(A, B)$-bimodules which is a quasi-isomorphism", "such that $P$ has property (P) as defined above." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemmas \\ref{lemma-bimodule-over-tensor} and", "\\ref{lemma-resolve}." ], "refs": [ "dga-lemma-bimodule-over-tensor", "dga-lemma-resolve" ], "ref_ids": [ 13092, 13062 ] } ], "ref_ids": [] }, { "id": 13094, "type": "theorem", "label": "dga-lemma-bimodule-property-P-sequence", "categories": [ "dga" ], "title": "dga-lemma-bimodule-property-P-sequence", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be", "differential graded $R$-algebras. Let $P$ be a", "differential graded $(A, B)$-bimodule having property (P)", "with corresponding filtration $F_\\bullet$, then we obtain a", "short exact sequence", "$$", "0 \\to", "\\bigoplus\\nolimits F_iP \\to", "\\bigoplus\\nolimits F_iP \\to P \\to 0", "$$", "of differential graded $(A, B)$-bimodules which is split as a sequence", "of graded $(A, B)$-bimodules." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemmas \\ref{lemma-bimodule-over-tensor} and", "\\ref{lemma-property-P-sequence}." ], "refs": [ "dga-lemma-bimodule-over-tensor", "dga-lemma-property-P-sequence" ], "ref_ids": [ 13092, 13059 ] } ], "ref_ids": [] }, { "id": 13095, "type": "theorem", "label": "dga-lemma-tensor", "categories": [ "dga" ], "title": "dga-lemma-tensor", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$", "be differential graded algebras over $R$. Let $N$ be a", "differential graded $(A, B)$-bimodule. Then", "$M \\mapsto M \\otimes_A N$ defines a functor", "$$", "- \\otimes_A N :", "\\text{Mod}^{dg}_{(A, \\text{d})}", "\\longrightarrow", "\\text{Mod}^{dg}_{(B, \\text{d})}", "$$", "of differential graded categories. This functor induces functors", "$$", "\\text{Mod}_{(A, \\text{d})} \\to \\text{Mod}_{(B, \\text{d})}", "\\quad\\text{and}\\quad", "K(\\text{Mod}_{(A, \\text{d})}) \\to K(\\text{Mod}_{(B, \\text{d})})", "$$", "by an application of Lemma \\ref{lemma-functorial}." ], "refs": [ "dga-lemma-functorial" ], "proofs": [ { "contents": [ "Above we have seen how the construction defines a functor of underlying", "graded categories. Thus it suffices to show that the construction is", "compatible with differentials. Let $M$ and $M'$ be differential", "graded $A$-modules and let $f : M \\to M'$ be an $A$-module homomorphism", "which is homogeneous of degree $n$. Then we have", "$$", "\\text{d}(f) = \\text{d}_{M'} \\circ f - (-1)^n f \\circ \\text{d}_M", "$$", "On the other hand, we have", "$$", "\\text{d}(f \\otimes \\text{id}_N) =", "\\text{d}_{M' \\otimes_A N} \\circ", "(f \\otimes \\text{id}_N)", "- (-1)^n", "(f \\otimes \\text{id}_N) \\circ \\text{d}_{M \\otimes_A N}", "$$", "Applying this to an element $x \\otimes y$ with $x \\in M$ and", "$y \\in N$ homogeneous we get", "\\begin{align*}", "\\text{d}(f \\otimes \\text{id}_N)(x \\otimes y)", "= &", "\\text{d}_{M'}(f(x)) \\otimes y + (-1)^{n + \\deg(x)}f(x) \\otimes \\text{d}_N(y) \\\\", "& - (-1)^n f(\\text{d}_M(x)) \\otimes y", "- (-1)^{n + \\deg(x)}f(x) \\otimes \\text{d}_N(y) \\\\", "= &", "\\text{d}(f) (x \\otimes y)", "\\end{align*}", "Thus we see that $\\text{d}(f) \\otimes \\text{id}_N =", "\\text{d}(f \\otimes \\text{id}_N)$ and the proof is complete." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13072 ] }, { "id": 13096, "type": "theorem", "label": "dga-lemma-hom", "categories": [ "dga" ], "title": "dga-lemma-hom", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$", "be differential graded algebras over $R$. Let $N$ be a", "differential graded $(A, B)$-bimodule. The construction above", "defines a functor", "$$", "\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, -) :", "\\text{Mod}^{dg}_{(B, \\text{d})}", "\\longrightarrow", "\\text{Mod}^{dg}_{(A, \\text{d})}", "$$", "of differential graded categories. This functor induces functors", "$$", "\\text{Mod}_{(B, \\text{d})} \\to \\text{Mod}_{(A, \\text{d})}", "\\quad\\text{and}\\quad", "K(\\text{Mod}_{(B, \\text{d})}) \\to K(\\text{Mod}_{(A, \\text{d})})", "$$", "by an application of Lemma \\ref{lemma-functorial}." ], "refs": [ "dga-lemma-functorial" ], "proofs": [ { "contents": [ "Above we have seen how the construction defines a functor of underlying", "graded categories. Thus it suffices to show that the construction is", "compatible with differentials. Let $N_1$ and $N_2$ be differential", "graded $B$-modules. Write", "$$", "H_{12} = \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N_1, N_2),\\quad", "H_1 = \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, N_1),\\quad", "H_2 = \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, N_2)", "$$", "Consider the composition", "$$", "c : H_{12} \\otimes_R H_1 \\longrightarrow H_2", "$$", "in the differential graded category $\\text{Mod}^{dg}_{(B, \\text{d})}$.", "Let $f : N_1 \\to N_2$ be a $B$-module homomorphism which is homogeneous", "of degree $n$, in other words, $f \\in H_{12}^n$.", "The functor in the lemma sends $f$ to $c_f : H_1 \\to H_2$, $g \\mapsto c(f, g)$.", "Simlarly for $\\text{d}(f)$. On the other hand, the differential on", "$$", "\\Hom_{\\text{Mod}^{dg}_{(A, \\text{d})}}(H_1, H_2)", "$$", "sends $c_f$ to $\\text{d}_{H_2} \\circ c_f - (-1)^n c_f \\circ \\text{d}_{H_1}$.", "As $c$ is a morphism of complexes of $R$-modules we have", "$\\text{d} c(f, g) = c(\\text{d}f, g) + (-1)^n c(f, \\text{d}g)$.", "Hence we see that", "\\begin{align*}", "(\\text{d}c_f)(g)", "& =", "\\text{d}c(f,g) - (-1)^n c(f, \\text{d}g) \\\\", "& =", "c(\\text{d}f, g) + (-1)^n c(f, \\text{d}g) - (-1)^n c(f, \\text{d}g) \\\\", "& =", "c(\\text{d}f, g) = c_{\\text{d}f}(g)", "\\end{align*}", "and the proof is complete." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13072 ] }, { "id": 13097, "type": "theorem", "label": "dga-lemma-tensor-hom-adjunction", "categories": [ "dga" ], "title": "dga-lemma-tensor-hom-adjunction", "contents": [ "Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras.", "Let $M$ be a right $A$-module, $N$ an $(A, B)$-bimodule, and", "$N'$ a right $B$-module. Then we have a canonical isomorphism", "$$", "\\Hom_B(M \\otimes_A N, N') = \\Hom_A(M, \\Hom_B(N, N'))", "$$", "of $R$-modules.", "If $A$, $B$, $M$, $N$, $N'$ are compatibly graded, then we have a", "canonical isomorphism", "$$", "\\Hom_{\\text{Mod}_B^{gr}}(M \\otimes_A N, N') =", "\\Hom_{\\text{Mod}_A^{gr}}(M, \\Hom_{\\text{Mod}_B^{gr}}(N, N'))", "$$", "of graded $R$-modules", "If $A$, $B$, $M$, $N$, $N'$ are compatibly differential graded, then", "we have a canonical isomorphism", "$$", "\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(M \\otimes_A N, N') =", "\\Hom_{\\text{Mod}^{dg}_{(A, \\text{d})}}(M,", "\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, N'))", "$$", "of complexes of $R$-modules." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: in the ungraded case interpret both sides as $A$-bilinear maps", "$\\psi : M \\times N \\to N'$ which are $B$-linear on the right.", "In the (differential) graded case, use the isomorphism of", "More on Algebra, Lemma \\ref{more-algebra-lemma-compose}", "and check it is compatible with the module structures.", "Alternatively, use the isomorphism of Lemma \\ref{lemma-characterize-hom}", "and show that it is compatible with the $B$-module structures." ], "refs": [ "more-algebra-lemma-compose", "dga-lemma-characterize-hom" ], "ref_ids": [ 10198, 13051 ] } ], "ref_ids": [] }, { "id": 13098, "type": "theorem", "label": "dga-lemma-restriction-homotopy", "categories": [ "dga" ], "title": "dga-lemma-restriction-homotopy", "contents": [ "The functor (\\ref{equation-restriction}) defines an exact functor", "$K(\\text{Mod}_{(B, \\text{d})}) \\to K(\\text{Mod}_{(A, \\text{d})})$", "of triangulated categories." ], "refs": [], "proofs": [ { "contents": [ "Via Lemma \\ref{lemma-hom} and", "Remark \\ref{remark-shift-hom-no-sign}", "this follows from the general principle of", "Lemma \\ref{lemma-functor-between-ABC}." ], "refs": [ "dga-lemma-hom", "dga-remark-shift-hom-no-sign", "dga-lemma-functor-between-ABC" ], "ref_ids": [ 13096, 13165, 13090 ] } ], "ref_ids": [] }, { "id": 13099, "type": "theorem", "label": "dga-lemma-derived-restriction", "categories": [ "dga" ], "title": "dga-lemma-derived-restriction", "contents": [ "In the situation above, the right derived functor of $F$ exists.", "We denote it $R\\Hom(N, -) : D(B, \\text{d}) \\to D(A, \\text{d})$." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Derived Categories, Lemma \\ref{derived-lemma-find-existence-computes}", "to prove this. As our collection $\\mathcal{I}$", "of objects we will use the objects with property (I).", "Property (1) was shown in Lemma \\ref{lemma-right-resolution}.", "Property (2) holds because if $s : I \\to I'$ is a quasi-isomorphism", "of modules with property (I), then $s$ is a homotopy equivalence", "by Lemma \\ref{lemma-hom-derived}." ], "refs": [ "derived-lemma-find-existence-computes", "dga-lemma-right-resolution", "dga-lemma-hom-derived" ], "ref_ids": [ 1832, 13066, 13068 ] } ], "ref_ids": [] }, { "id": 13100, "type": "theorem", "label": "dga-lemma-functoriality-derived-restriction", "categories": [ "dga" ], "title": "dga-lemma-functoriality-derived-restriction", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be", "differential graded $R$-algebras. Let $f : N \\to N'$ be a", "homomorphism of differential graded $(A, B)$-bimodules.", "Then $f$ induces a morphism of functors", "$$", "- \\circ f : R\\Hom(N', -) \\longrightarrow R\\Hom(N, -)", "$$", "If $f$ is a quasi-isomorphism, then $f \\circ -$ is an isomorphism of", "functors." ], "refs": [], "proofs": [ { "contents": [ "Write $\\mathcal{B} = \\text{Mod}^{dg}_{(B, \\text{d})}$ the", "differential graded category of differential graded $B$-modules, see", "Example \\ref{example-dgm-dg-cat}.", "Let $I$ be a differential graded $B$-module with property (I).", "Then $f \\circ - : \\Hom_\\mathcal{B}(N', I) \\to \\Hom_\\mathcal{B}(N, I)$", "is a map of differential graded $A$-modules. Moreover, this is functorial", "with respect to $I$. Since the functors", "$ R\\Hom(N', -)$ and $R\\Hom(N, -)$ are", "computed by applying $\\Hom_\\mathcal{B}$ into objects with property (I)", "(Lemma \\ref{lemma-derived-restriction}) we obtain a transformation of functors", "as indicated.", "\\medskip\\noindent", "Assume that $f$ is a quasi-isomorphism. Let $F_\\bullet$ be the", "given filtration on $I$. Since $I = \\lim I/F_pI$ we see that", "$\\Hom_\\mathcal{B}(N', I) = \\lim \\Hom_\\mathcal{B}(N', I/F_pI)$ and", "$\\Hom_\\mathcal{B}(N, I) = \\lim \\Hom_\\mathcal{B}(N, I/F_pI)$.", "Since the transition maps in the system $I/F_pI$ are split", "as graded modules, we see that the transition maps in the", "systems $\\Hom_\\mathcal{B}(N', I/F_pI)$ and $\\Hom_\\mathcal{B}(N, I/F_pI)$", "are surjective. Hence $\\Hom_\\mathcal{B}(N', I)$, resp. $\\Hom_\\mathcal{B}(N, I)$", "viewed as a complex of abelian groups computes $R\\lim$ of the system", "of complexes", "$\\Hom_\\mathcal{B}(N', I/F_pI)$, resp. $\\Hom_\\mathcal{B}(N, I/F_pI)$.", "See More on Algebra, Lemma \\ref{more-algebra-lemma-compute-Rlim}.", "Thus it suffices to prove each", "$$", "\\Hom_\\mathcal{B}(N', I/F_pI) \\to \\Hom_\\mathcal{B}(N, I/F_pI)", "$$", "is a quasi-isomorphism. Since the surjections $I/F_{p + 1}I \\to I/F_pI$", "are split as maps of graded $B$-modules we see that", "$$", "0 \\to \\Hom_\\mathcal{B}(N', F_pI/F_{p + 1}I) \\to", "\\Hom_\\mathcal{B}(N', I/F_{p + 1}I) \\to", "\\Hom_\\mathcal{B}(N', I/F_pI) \\to 0", "$$", "is a short exact sequence of differential graded $A$-modules.", "There is a similar sequence for $N$ and $f$ induces a map", "of short exact sequences. Hence by induction on $p$ (starting with $p = 0$", "when $I/F_0I = 0$) we conclude that it suffices to show that", "the map", "$\\Hom_\\mathcal{B}(N', F_pI/F_{p + 1}I) \\to \\Hom_\\mathcal{B}(N, F_pI/F_{p + 1}I)$", "is a quasi-isomorphism. Since $F_pI/F_{p + 1}I$ is a product of shifts of", "$A^\\vee$ it suffice to prove", "$\\Hom_\\mathcal{B}(N', B^\\vee[k]) \\to \\Hom_\\mathcal{B}(N, B^\\vee[k])$", "is a quasi-isomorphism. By Lemma \\ref{lemma-hom-into-shift-dual-free}", "it suffices to show $(N')^\\vee \\to N^\\vee$ is a quasi-isomorphism.", "This is true because $f$ is a quasi-isomorphism and $(\\ )^\\vee$", "is an exact functor." ], "refs": [ "dga-lemma-derived-restriction", "more-algebra-lemma-compute-Rlim", "dga-lemma-hom-into-shift-dual-free" ], "ref_ids": [ 13099, 10313, 13058 ] } ], "ref_ids": [] }, { "id": 13101, "type": "theorem", "label": "dga-lemma-derived-restriction-exts", "categories": [ "dga" ], "title": "dga-lemma-derived-restriction-exts", "contents": [ "Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded algebras", "over a ring $R$. Let $N$ be a differential graded $(A, B)$-bimodule.", "Then for every $n \\in \\mathbf{Z}$ there are isomorphisms", "$$", "H^n(R\\Hom(N, M)) = \\Ext^n_{D(B, \\text{d})}(N, M)", "$$", "of $R$-modules functorial in $M$. It is also functorial in $N$", "with respect to the operation described in", "Lemma \\ref{lemma-functoriality-derived-restriction}." ], "refs": [ "dga-lemma-functoriality-derived-restriction" ], "proofs": [ { "contents": [ "In the proof of Lemma \\ref{lemma-derived-restriction}", "we have seen", "$$", "R\\Hom(N, M) = ", "\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, I)", "$$", "as a differential graded $A$-module", "where $M \\to I$ is a quasi-isomorphism of $M$ into a differential", "graded $B$-module with property (I). Hence this complex has the", "correct cohomology modules by Lemma \\ref{lemma-hom-derived}.", "We omit a discussion of the functorial nature of these", "identifications." ], "refs": [ "dga-lemma-derived-restriction", "dga-lemma-hom-derived" ], "ref_ids": [ 13099, 13068 ] } ], "ref_ids": [ 13100 ] }, { "id": 13102, "type": "theorem", "label": "dga-lemma-compute-derived-restriction", "categories": [ "dga" ], "title": "dga-lemma-compute-derived-restriction", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be", "differential graded $R$-algebras. Let $N$ be a differential", "graded $(A, B)$-bimodule. If", "$\\Hom_{D(B, \\text{d})}(N, N') = \\Hom_{K(\\text{Mod}_{(B, \\text{d})})}(N, N')$", "for all $N' \\in K(B, \\text{d})$, for example if $N$", "has property (P) as a differential graded $B$-module, then", "$$", "R\\Hom(N, M) = \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, M)", "$$", "functorially in $M$ in $D(B, \\text{d})$." ], "refs": [], "proofs": [ { "contents": [ "By construction (Lemma \\ref{lemma-derived-restriction})", "to find $R\\Hom(N, M)$ we choose a quasi-isomorphism", "$M \\to I$ where $I$ is a differential graded $B$-module", "with property (I) and we set", "$R\\Hom(N, M) = \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, I)$.", "By assumption the map", "$$", "\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, M) \\longrightarrow", "\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, I)", "$$", "induced by $M \\to I$ is a quasi-isomorphism, see discussion in", "Example \\ref{example-dgm-dg-cat}. This proves the lemma.", "If $N$ has property (P) as a $B$-module, then we see that the", "assumption is satisfied by Lemma \\ref{lemma-hom-derived}." ], "refs": [ "dga-lemma-derived-restriction", "dga-lemma-hom-derived" ], "ref_ids": [ 13099, 13068 ] } ], "ref_ids": [] }, { "id": 13103, "type": "theorem", "label": "dga-lemma-existence-of-derived", "categories": [ "dga" ], "title": "dga-lemma-existence-of-derived", "contents": [ "In the situation above. If the right derived functor $R\\Hom(K^\\bullet, -)$", "of $\\Hom(K^\\bullet, -) : K(\\mathcal{A}) \\to D(\\textit{Ab})$", "is everywhere defined on $D(\\mathcal{A})$, then we obtain a canonical exact", "functor", "$$", "R\\Hom(K^\\bullet, -) : D(\\mathcal{A}) \\longrightarrow D(E, \\text{d})", "$$", "of triangulated categories which reduces to the usual one on taking", "associated complexes of abelian groups." ], "refs": [], "proofs": [ { "contents": [ "Note that we have an associated functor", "$K(\\mathcal{A}) \\to K(\\text{Mod}_{(E, \\text{d})})$ by", "Lemma \\ref{lemma-construction}.", "We claim this functor is an exact functor of triangulated categories.", "Namely, let $f : A^\\bullet \\to B^\\bullet$ be a map of complexes", "of $\\mathcal{A}$. Then a computation shows that", "$$", "\\Hom_{\\text{Comp}^{dg}(\\mathcal{A})}(K^\\bullet, C(f)^\\bullet)", "=", "C\\left(", "\\Hom_{\\text{Comp}^{dg}(\\mathcal{A})}(K^\\bullet, A^\\bullet) \\to", "\\Hom_{\\text{Comp}^{dg}(\\mathcal{A})}(K^\\bullet, B^\\bullet)", "\\right)", "$$", "where the right hand side is the cone in $\\text{Mod}_{(E, \\text{d})}$", "defined earlier in this chapter.", "This shows that our functor is compatible with cones, hence with", "distinguished triangles. Let $X^\\bullet$ be an object of $K(\\mathcal{A})$.", "Consider the category of quasi-isomorphisms $s : X^\\bullet \\to Y^\\bullet$.", "We are given that the functor", "$(s : X^\\bullet \\to Y^\\bullet) \\mapsto \\Hom_\\mathcal{A}(K^\\bullet, Y^\\bullet)$", "is essentially constant when viewed in $D(\\textit{Ab})$.", "But since the forgetful functor $D(E, \\text{d}) \\to D(\\textit{Ab})$", "is compatible with taking cohomology, the same thing is true in", "$D(E, \\text{d})$. This proves the lemma." ], "refs": [ "dga-lemma-construction" ], "ref_ids": [ 13075 ] } ], "ref_ids": [] }, { "id": 13104, "type": "theorem", "label": "dga-lemma-bc-homotopy", "categories": [ "dga" ], "title": "dga-lemma-bc-homotopy", "contents": [ "The functor (\\ref{equation-bc}) defines an exact functor", "of triangulated categories", "$K(\\text{Mod}_{(A, \\text{d})}) \\to K(\\text{Mod}_{(B, \\text{d})})$." ], "refs": [], "proofs": [ { "contents": [ "Via Lemma \\ref{lemma-tensor} and", "Remark \\ref{remark-shift-tensor-no-sign}", "this follows from the general principle of", "Lemma \\ref{lemma-functor-between-ABC}." ], "refs": [ "dga-lemma-tensor", "dga-remark-shift-tensor-no-sign", "dga-lemma-functor-between-ABC" ], "ref_ids": [ 13095, 13164, 13090 ] } ], "ref_ids": [] }, { "id": 13105, "type": "theorem", "label": "dga-lemma-derived-bc", "categories": [ "dga" ], "title": "dga-lemma-derived-bc", "contents": [ "In the situation above, the left derived functor of $F$ exists.", "We denote it", "$- \\otimes_A^\\mathbf{L} N : D(A, \\text{d}) \\to D(B, \\text{d})$." ], "refs": [], "proofs": [ { "contents": [ "We will use", "Derived Categories, Lemma \\ref{derived-lemma-find-existence-computes}", "to prove this. As our collection $\\mathcal{P}$", "of objects we will use the objects with property (P).", "Property (1) was shown in Lemma \\ref{lemma-resolve}.", "Property (2) holds because if $s : P \\to P'$ is a quasi-isomorphism", "of modules with property (P), then $s$ is a homotopy equivalence", "by Lemma \\ref{lemma-hom-derived}." ], "refs": [ "derived-lemma-find-existence-computes", "dga-lemma-resolve", "dga-lemma-hom-derived" ], "ref_ids": [ 1832, 13062, 13068 ] } ], "ref_ids": [] }, { "id": 13106, "type": "theorem", "label": "dga-lemma-functoriality-bc", "categories": [ "dga" ], "title": "dga-lemma-functoriality-bc", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be", "differential graded $R$-algebras. Let $f : N \\to N'$ be a", "homomorphism of differential graded $(A, B)$-bimodules.", "Then $f$ induces a morphism of functors", "$$", "1\\otimes f :", "- \\otimes_A^\\mathbf{L} N", "\\longrightarrow", "- \\otimes_A^\\mathbf{L} N'", "$$", "If $f$ is a quasi-isomorphism, then $1 \\otimes f$ is an isomorphism of", "functors." ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be a differential graded $A$-module with property (P).", "Then $1 \\otimes f : M \\otimes_A N \\to M \\otimes_A N'$ is a", "map of differential graded $B$-modules. Moreover, this is functorial", "with respect to $M$. Since the functors", "$- \\otimes_A^\\mathbf{L} N$ and $- \\otimes_A^\\mathbf{L} N'$ are", "computed by tensoring on objects with property (P)", "(Lemma \\ref{lemma-derived-bc}) we obtain a transformation of functors", "as indicated.", "\\medskip\\noindent", "Assume that $f$ is a quasi-isomorphism. Let $F_\\bullet$ be the", "given filtration on $M$. Observe that", "$M \\otimes_A N = \\colim F_i(M) \\otimes_A N$ and", "$M \\otimes_A N' = \\colim F_i(M) \\otimes_A N'$.", "Hence it suffices to show that", "$F_n(M) \\otimes_A N \\to F_n(M) \\otimes_A N'$", "is a quasi-isomorphism (filtered colimits are exact, see", "Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}).", "Since the inclusions $F_n(M) \\to F_{n + 1}(M)$", "are split as maps of graded $A$-modules we see that", "$$", "0 \\to F_n(M) \\otimes_A N \\to F_{n + 1}(M) \\otimes_A N \\to", "F_{n + 1}(M)/F_n(M) \\otimes_A N \\to 0", "$$", "is a short exact sequence of differential graded $B$-modules.", "There is a similar sequence for $N'$ and $f$ induces a map", "of short exact sequences. Hence by induction on $n$ (starting with $n = -1$", "when $F_{-1}(M) = 0$) we conclude that it suffices to show that", "the map $F_{n + 1}(M)/F_n(M) \\otimes_A N \\to F_{n + 1}(M)/F_n(M) \\otimes_A N'$", "is a quasi-isomorphism. This is true because $F_{n + 1}(M)/F_n(M)$", "is a direct sum of shifts of $A$ and the result is true for $A[k]$", "as $f : N \\to N'$ is a quasi-isomorphism." ], "refs": [ "dga-lemma-derived-bc", "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 13105, 343 ] } ], "ref_ids": [] }, { "id": 13107, "type": "theorem", "label": "dga-lemma-tensor-hom-adjoint", "categories": [ "dga" ], "title": "dga-lemma-tensor-hom-adjoint", "contents": [ "Let $R$ be a ring.", "Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras.", "Let $N$ be a differential graded $(A, B)$-bimodule.", "Then the functor", "$$", "- \\otimes_A^\\mathbf{L} N : D(A, \\text{d}) \\longrightarrow D(B, \\text{d})", "$$", "of Lemma \\ref{lemma-derived-bc} is a left adjoint to the functor", "$$", "R\\Hom(N, -) : D(B, \\text{d}) \\longrightarrow D(A, \\text{d})", "$$", "of Lemma \\ref{lemma-derived-restriction}." ], "refs": [ "dga-lemma-derived-bc", "dga-lemma-derived-restriction" ], "proofs": [ { "contents": [ "This follows from Derived Categories, Lemma", "\\ref{derived-lemma-pre-derived-adjoint-functors-general}", "and the fact that $- \\otimes_A N$ and", "$\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, -)$ are adjoint by", "Lemma \\ref{lemma-tensor-hom-adjunction}." ], "refs": [ "derived-lemma-pre-derived-adjoint-functors-general", "dga-lemma-tensor-hom-adjunction" ], "ref_ids": [ 1905, 13097 ] } ], "ref_ids": [ 13105, 13099 ] }, { "id": 13108, "type": "theorem", "label": "dga-lemma-tensor-with-compact-fully-faithful", "categories": [ "dga" ], "title": "dga-lemma-tensor-with-compact-fully-faithful", "contents": [ "With notation and assumptions as in Lemma \\ref{lemma-tensor-hom-adjoint}.", "Assume", "\\begin{enumerate}", "\\item $N$ defines a compact object of $D(B, \\text{d})$, and", "\\item the map $H^k(A) \\to \\Hom_{D(B, \\text{d})}(N, N[k])$ is an", "isomorphism for all $k \\in \\mathbf{Z}$.", "\\end{enumerate}", "Then the functor $-\\otimes_A^\\mathbf{L} N$ is fully faithful." ], "refs": [ "dga-lemma-tensor-hom-adjoint" ], "proofs": [ { "contents": [ "Our functor has a left adjoint given by", "$R\\Hom(N, -)$ by Lemma \\ref{lemma-tensor-hom-adjoint}.", "By Categories, Lemma \\ref{categories-lemma-adjoint-fully-faithful}", "it suffices to show that for a differential graded $A$-module $M$", "the map", "$$", "M \\longrightarrow R\\Hom(N, M \\otimes_A^\\mathbf{L} N)", "$$", "is an isomorphism in $D(A, \\text{d})$. For this it suffices to show that", "$$", "H^n(M) \\longrightarrow", "\\text{Ext}^n_{D(B, \\text{d})}(N, M \\otimes_A^\\mathbf{L} N)", "$$", "is an isomorphism, see Lemma \\ref{lemma-derived-restriction-exts}.", "Since $N$ is a compact object the right hand side commutes", "with direct sums. Thus by Remark \\ref{remark-P-resolution}", "it suffices to prove this map is an isomorphism for $M = A[k]$.", "Since $(A[k] \\otimes_A^\\mathbf{L} N) = N[k]$ by", "Remark \\ref{remark-shift-tensor-no-sign},", "assumption (2) on $N$ is that the result holds for these." ], "refs": [ "dga-lemma-tensor-hom-adjoint", "categories-lemma-adjoint-fully-faithful", "dga-lemma-derived-restriction-exts", "dga-remark-P-resolution", "dga-remark-shift-tensor-no-sign" ], "ref_ids": [ 13107, 12248, 13101, 13162, 13164 ] } ], "ref_ids": [ 13107 ] }, { "id": 13109, "type": "theorem", "label": "dga-lemma-base-change-K-flat", "categories": [ "dga" ], "title": "dga-lemma-base-change-K-flat", "contents": [ "Let $R \\to R'$ be a ring map. Let $(A, \\text{d})$ be a differential", "graded $R$-algebra. Let $(A', \\text{d})$ be the base change, i.e.,", "$A' = A \\otimes_R R'$. If $A$ is K-flat as a complex of $R$-modules,", "then", "\\begin{enumerate}", "\\item $- \\otimes_A^\\mathbf{L} A' : D(A, \\text{d}) \\to D(A', \\text{d})$", "is equal to the right derived functor of", "$$", "K(A, \\text{d}) \\longrightarrow K(A', \\text{d}),\\quad", "M \\longmapsto M \\otimes_R R'", "$$", "\\item the diagram", "$$", "\\xymatrix{", "D(A, \\text{d}) \\ar[r]_{- \\otimes_A^\\mathbf{L} A'} \\ar[d]_{restriction} &", "D(A', \\text{d}) \\ar[d]^{restriction} \\\\", "D(R) \\ar[r]^{- \\otimes_R^\\mathbf{L} R'} & D(R')", "}", "$$", "commutes, and", "\\item if $M$ is K-flat as a complex of $R$-modules, then the", "differential graded $A'$-module $M \\otimes_R R'$ represents", "$M \\otimes_A^\\mathbf{L} A'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For any differential graded $A$-module $M$ there is a canonical map", "$$", "c_M : M \\otimes_R R' \\longrightarrow M \\otimes_A A'", "$$", "Let $P$ be a differential graded $A$-module with property (P).", "We claim that $c_P$ is an isomorphism and that $P$ is K-flat", "as a complex of $R$-modules. This will prove all the results", "stated in the lemma by formal arguments using the definition", "of derived tensor product in Lemma \\ref{lemma-derived-bc} and", "More on Algebra, Section \\ref{more-algebra-section-derived-tensor-product}.", "\\medskip\\noindent", "Let $F_\\bullet$ be the filtration on $P$ showing that $P$ has property (P).", "Note that $c_A$ is an isomorphism and $A$ is K-flat as a complex", "of $R$-modules by assumption. Hence the same is true for", "direct sums of shifts of $A$ (you can use", "More on Algebra, Lemma \\ref{more-algebra-lemma-colimit-K-flat}", "to deal with direct sums if you like).", "Hence this holds for the complexes $F_{p + 1}P/F_pP$.", "Since the short exact sequences", "$$", "0 \\to F_pP \\to F_{p + 1}P \\to F_{p + 1}P/F_pP \\to 0", "$$", "are split exact as sequences of graded modules, we can argue", "by induction that $c_{F_pP}$ is an isomorphism for all $p$", "and that $F_pP$ is K-flat as a complex of $R$-modules (use", "More on Algebra, Lemma \\ref{more-algebra-lemma-K-flat-two-out-of-three}).", "Finally, using that $P = \\colim F_pP$ we conclude that", "$c_P$ is an isomorphism and that $P$ is K-flat", "as a complex of $R$-modules (use", "More on Algebra, Lemma \\ref{more-algebra-lemma-colimit-K-flat})." ], "refs": [ "dga-lemma-derived-bc", "more-algebra-lemma-colimit-K-flat", "more-algebra-lemma-K-flat-two-out-of-three", "more-algebra-lemma-colimit-K-flat" ], "ref_ids": [ 13105, 10129, 10126, 10129 ] } ], "ref_ids": [] }, { "id": 13110, "type": "theorem", "label": "dga-lemma-RHom-is-tensor", "categories": [ "dga" ], "title": "dga-lemma-RHom-is-tensor", "contents": [ "Let $R$ be a ring.", "Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras.", "Let $T$ be a differential graded $(A, B)$-bimodule.", "Assume", "\\begin{enumerate}", "\\item $T$ defines a compact object of $D(B, \\text{d})$, and", "\\item $S = \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(T, B)$", "represents $R\\Hom(T, B)$ in $D(A, \\text{d})$.", "\\end{enumerate}", "Then $S$ has a structure of a differential graded $(B, A)$-bimodule", "and there is an isomorphism", "$$", "N \\otimes_B^\\mathbf{L} S \\longrightarrow R\\Hom(T, N)", "$$", "functorial in $N$ in $D(B, \\text{d})$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\mathcal{B} = \\text{Mod}^{dg}_{(B, \\text{d})}$.", "The right $A$-module structure on $S$ comes from the map", "$A \\to \\Hom_\\mathcal{B}(T, T)$ and the composition", "$\\Hom_\\mathcal{B}(T, B) \\otimes \\Hom_\\mathcal{B}(T, T)", "\\to \\Hom_\\mathcal{B}(T, B)$ defined in Example \\ref{example-dgm-dg-cat}.", "Using this multiplication a second time there is a map", "$$", "c_N :", "N \\otimes_B S = \\Hom_\\mathcal{B}(B, N) \\otimes_B \\Hom_\\mathcal{B}(T, B)", "\\longrightarrow", "\\Hom_\\mathcal{B}(T, N)", "$$", "functorial in $N$. Given $N$ we can choose quasi-isomorphisms", "$P \\to N \\to I$ where $P$, resp.\\ $I$ is a differential graded $B$-module", "with property (P), resp.\\ (I). Then using $c_N$ we obtain a map", "$P \\otimes_B S \\to \\Hom_\\mathcal{B}(T, I)$ between the objects", "representing $S \\otimes_B^\\mathbf{L} N$ and $R\\Hom(T, N)$.", "Clearly this defines a transformation of functors $c$ as in the lemma.", "\\medskip\\noindent", "To prove that $c$ is an isomorphism of functors, we may", "assume $N$ is a differential graded $B$-module which", "has property (P). Since $T$ defines a compact object in", "$D(B, \\text{d})$ and since both sides of the arrow define", "exact functors of triangulated categories, we reduce using", "Lemma \\ref{lemma-property-P-sequence}", "to the case where $N$ has a finite filtration whose", "graded pieces are direct sums of $B[k]$. Using again that", "both sides of the arrow are exact functors of triangulated", "categories and compactness of $T$ we reduce to", "the case $N = B[k]$. Assumption (2) is exactly the", "assumption that $c$ is an isomorphism in this case." ], "refs": [ "dga-lemma-property-P-sequence" ], "ref_ids": [ 13059 ] } ], "ref_ids": [] }, { "id": 13111, "type": "theorem", "label": "dga-lemma-compose-tensor-functors-general", "categories": [ "dga" ], "title": "dga-lemma-compose-tensor-functors-general", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$, $(B, \\text{d})$, and", "$(C, \\text{d})$ be differential graded $R$-algebras.", "Let $N$ be a differential graded $(A, B)$-bimodule.", "Let $N'$ be a differential graded $(B, C)$-module.", "Assume (\\ref{equation-plain-versus-derived}) is an isomorphism.", "Then the composition", "$$", "\\xymatrix{", "D(A, \\text{d}) \\ar[rr]^{- \\otimes_A^\\mathbf{L} N} & &", "D(B, \\text{d}) \\ar[rr]^{- \\otimes_B^\\mathbf{L} N'} & &", "D(C, \\text{d})", "}", "$$", "is isomorphic to $- \\otimes_A^\\mathbf{L} N''$ with", "$N'' = N \\otimes_B N'$ viewed as $(A, C)$-bimodule." ], "refs": [], "proofs": [ { "contents": [ "Let us define a transformation of functors", "$$", "(- \\otimes_A^\\mathbf{L} N) \\otimes_B^\\mathbf{L} N'", "\\longrightarrow", "- \\otimes_A^\\mathbf{L} N''", "$$", "To do this, let", "$M$ be a differential graded $A$-module with property (P).", "According to the construction of the functor $- \\otimes_A^\\mathbf{L} N''$", "of the proof of Lemma \\ref{lemma-derived-bc} the plain tensor", "product $M \\otimes_A N''$ represents $M \\otimes_A^\\mathbf{L} N''$", "in $D(C, \\text{d})$. Then we write", "$$", "M \\otimes_A N'' =", "M \\otimes_A (N \\otimes_B N') =", "(M \\otimes_A N) \\otimes_B N'", "$$", "The module $M \\otimes_A N$ represents $M \\otimes_A^\\mathbf{L} N$", "in $D(B, \\text{d})$. Choose a quasi-isomorphism $Q \\to M \\otimes_A N$", "where $Q$ is a differential graded $B$-module with property (P). Then", "$Q \\otimes_B N'$ represents", "$(M \\otimes_A^\\mathbf{L} N) \\otimes_B^\\mathbf{L} N'$ in", "$D(C, \\text{d})$.", "Thus we can define our map via", "$$", "(M \\otimes_A^\\mathbf{L} N) \\otimes_B^\\mathbf{L} N' =", "Q \\otimes_B N' \\to", "M \\otimes_A N \\otimes_B N' =", "M \\otimes_A^\\mathbf{L} N''", "$$", "The construction of this map is functorial in $M$ and compatible", "with distinguished triangles and direct sums; we omit the details.", "Consider the property $T$ of objects $M$ of $D(A, \\text{d})$ ", "expressing that this map is an isomorphism. Then", "\\begin{enumerate}", "\\item if $T$ holds for $M_i$ then $T$ holds for $\\bigoplus M_i$,", "\\item if $T$ holds for $2$-out-of-$3$ in a distinguished", "triangle, then it holds for the third, and", "\\item $T$ holds for $A[k]$ because here we obtain a shift", "of the map (\\ref{equation-plain-versus-derived}) which we", "have assumed is an isomorphism.", "\\end{enumerate}", "Thus by Remark \\ref{remark-P-resolution} property $T$", "always holds and the proof is complete." ], "refs": [ "dga-lemma-derived-bc", "dga-remark-P-resolution" ], "ref_ids": [ 13105, 13162 ] } ], "ref_ids": [] }, { "id": 13112, "type": "theorem", "label": "dga-lemma-compose-tensor-functors-general-algebra", "categories": [ "dga" ], "title": "dga-lemma-compose-tensor-functors-general-algebra", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$, $(B, \\text{d})$, and", "$(C, \\text{d})$ be differential graded $R$-algebras. Assume", "that (\\ref{equation-plain-versus-derived-algebras}) is an isomorphism.", "Let $N$ be a differential graded $(A, B)$-bimodule.", "Let $N'$ be a differential graded $(B, C)$-bimodule.", "Then the composition", "$$", "\\xymatrix{", "D(A, \\text{d}) \\ar[rr]^{- \\otimes_A^\\mathbf{L} N} & &", "D(B, \\text{d}) \\ar[rr]^{- \\otimes_B^\\mathbf{L} N'} & &", "D(C, \\text{d})", "}", "$$", "is isomorphic to $- \\otimes_A^\\mathbf{L} N''$ for a differential graded", "$(A, C)$-bimodule $N''$ described in the proof." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-functoriality-bc} we may replace $N$ and $N'$ by", "quasi-isomorphic bimodules. Thus we may assume $N$, resp.\\ $N'$", "has property (P) as differential graded", "$(A, B)$-bimodule, resp.\\ $(B, C)$-bimodule, see", "Lemma \\ref{lemma-bimodule-resolve}. We claim the lemma holds", "with the $(A, C)$-bimodule $N'' = N \\otimes_B N'$.", "To prove this, it suffices to show that", "$$", "N_B \\otimes_B^\\mathbf{L} N' \\longrightarrow (N \\otimes_B N')_C", "$$", "is an isomorphism in $D(C, \\text{d})$, see", "Lemma \\ref{lemma-compose-tensor-functors-general}.", "\\medskip\\noindent", "Let $F_\\bullet$ be the filtration on $N$ as in property (P) for bimodules.", "By Lemma \\ref{lemma-bimodule-property-P-sequence}", "there is a short exact sequence", "$$", "0 \\to", "\\bigoplus\\nolimits F_iN \\to", "\\bigoplus\\nolimits F_iN \\to N \\to 0", "$$", "of differential graded $(A, B)$-bimodules which is split as a sequence", "of graded $(A, B)$-bimodules. A fortiori this is an admissible short exact", "sequence of differential graded $B$-modules and this produces a distinguished", "triangle", "$$", "\\bigoplus\\nolimits F_iN_B \\to", "\\bigoplus\\nolimits F_iN_B \\to N_B \\to", "\\bigoplus\\nolimits F_iN_B[1]", "$$", "in $D(B, \\text{d})$. Using that $- \\otimes_B^\\mathbf{L} N'$", "is an exact functor of triangulated categories and commutes", "with direct sums and using that $- \\otimes_B N'$ transforms", "admissible exact sequences into admissible exact sequences", "and commutes with direct sums we reduce to proving", "that", "$$", "(F_pN)_B \\otimes_B^\\mathbf{L} N' \\longrightarrow (F_pN)_B \\otimes_B N'", "$$", "is a quasi-isomorphism for all $p$. Repeating the argument", "with the short exact sequences of $(A, B)$-bimodules", "$$", "0 \\to F_pN \\to F_{p + 1}N \\to F_{p + 1}N/F_pN \\to 0", "$$", "which are split as graded $(A, B)$-bimodules", "we reduce to showing the same statement for $F_{p + 1}N/F_pN$.", "Since these modules are direct sums of shifts of $(A \\otimes_R B)_B$", "we reduce to showing that", "$$", "(A \\otimes_R B)_B \\otimes_B^\\mathbf{L} N'", "\\longrightarrow", "(A \\otimes_R B)_B \\otimes_B N'", "$$", "is a quasi-isomorphism.", "\\medskip\\noindent", "Choose a filtration $F_\\bullet$ on $N'$ as in property (P) for bimodules.", "Choose a quasi-isomorphism $P \\to (A \\otimes_R B)_B$", "of differential graded $B$-modules where $P$ has property (P).", "We have to show that", "$P \\otimes_B N' \\to (A \\otimes_R B)_B \\otimes_B N'$ is", "a quasi-isomorphism because $P \\otimes_B N'$ represents", "$(A \\otimes_R B)_B \\otimes_B^\\mathbf{L} N'$ in $D(C, \\text{d})$", "by the construction in Lemma \\ref{lemma-derived-bc}.", "As $N' = \\colim F_pN'$ we find", "that it suffices to show that", "$P \\otimes_B F_pN' \\to (A \\otimes_R B)_B \\otimes_B F_pN'$", "is a quasi-isomorphism. Using the short exact sequences", "$0 \\to F_pN' \\to F_{p + 1}N' \\to F_{p + 1}N'/F_pN' \\to 0$", "which are split as graded $(B, C)$-bimodules we reduce to showing", "$P \\otimes_B F_{p + 1}N'/F_pN' \\to", "(A \\otimes_R B)_B \\otimes_B F_{p + 1}N'/F_pN'$", "is a quasi-isomorphism for all $p$.", "Then finally using that $F_{p + 1}N'/F_pN'$", "is a direct sum of shifts of ${}_B(B \\otimes_R C)_C$", "we conclude that it suffices to show that", "$$", "P \\otimes_B {}_B(B \\otimes_R C)_C \\to", "(A \\otimes_R B)_B \\otimes_B {}_B(B \\otimes_R C)_C", "$$", "is a quasi-isomorphism. Since $P \\to (A \\otimes_R B)_B$", "is a resolution by a module satisfying property (P)", "this map of differential graded $C$-modules", "represents the morphism (\\ref{equation-plain-versus-derived-algebras})", "in $D(C, \\text{d})$ and the proof is complete." ], "refs": [ "dga-lemma-functoriality-bc", "dga-lemma-bimodule-resolve", "dga-lemma-compose-tensor-functors-general", "dga-lemma-bimodule-property-P-sequence", "dga-lemma-derived-bc" ], "ref_ids": [ 13106, 13093, 13111, 13094, 13105 ] } ], "ref_ids": [] }, { "id": 13113, "type": "theorem", "label": "dga-lemma-compose-tensor-functors", "categories": [ "dga" ], "title": "dga-lemma-compose-tensor-functors", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$, $(B, \\text{d})$, and", "$(C, \\text{d})$ be differential graded $R$-algebras.", "If $C$ is K-flat as a complex of $R$-modules, then", "(\\ref{equation-plain-versus-derived-algebras})", "is an isomorphism and the conclusion of", "Lemma \\ref{lemma-compose-tensor-functors-general-algebra} is valid." ], "refs": [ "dga-lemma-compose-tensor-functors-general-algebra" ], "proofs": [ { "contents": [ "Choose a quasi-isomorphism $P \\to (A \\otimes_R B)_B$ of differential", "graded $B$-modules, where $P$ has property (P). Then we have to show", "that", "$$", "P \\otimes_B (B \\otimes_R C) \\longrightarrow", "(A \\otimes_R B) \\otimes_B (B \\otimes_R C)", "$$", "is a quasi-isomorphism. Equivalently we are looking at", "$$", "P \\otimes_R C \\longrightarrow", "A \\otimes_R B \\otimes_R C", "$$", "This is a quasi-isomorphism if $C$ is K-flat as a complex of $R$-modules by", "More on Algebra, Lemma \\ref{more-algebra-lemma-K-flat-quasi-isomorphism}." ], "refs": [ "more-algebra-lemma-K-flat-quasi-isomorphism" ], "ref_ids": [ 10123 ] } ], "ref_ids": [ 13112 ] }, { "id": 13114, "type": "theorem", "label": "dga-lemma-tensor-with-complex", "categories": [ "dga" ], "title": "dga-lemma-tensor-with-complex", "contents": [ "In the situation above there is a functor", "$$", "- \\otimes_E K^\\bullet :", "\\text{Mod}^{dg}_{(E, \\text{d})}", "\\longrightarrow", "\\text{Comp}^{dg}(\\mathcal{O})", "$$", "of differential graded categories. This functor sends $E$ to $K^\\bullet$", "and commutes with direct sums." ], "refs": [], "proofs": [ { "contents": [ "Let $M$ be a differential graded $E$-module. For every object $U$ of", "$\\mathcal{C}$ the complex $K^\\bullet(U)$ is a left differential", "graded $E$-module as well as a right $\\mathcal{O}(U)$-module.", "The actions commute, so we have a bimodule.", "Thus, by the constructions in", "Sections \\ref{section-tensor-product} and \\ref{section-bimodules}", "we can form the tensor product", "$$", "M \\otimes_E K^\\bullet(U)", "$$", "which is a differential graded $\\mathcal{O}(U)$-module, i.e., a complex", "of $\\mathcal{O}(U)$-modules. This construction is functorial with respect", "to $U$, hence we can sheafify to get a complex of $\\mathcal{O}$-modules", "which we denote", "$$", "M \\otimes_E K^\\bullet", "$$", "Moreover, for each $U$ the construction determines a functor", "$\\text{Mod}^{dg}_{(E, \\text{d})} \\to \\text{Comp}^{dg}(\\mathcal{O}(U))$", "of differential graded categories by Lemma \\ref{lemma-tensor}.", "It is therefore clear that we obtain a functor as stated in the lemma." ], "refs": [ "dga-lemma-tensor" ], "ref_ids": [ 13095 ] } ], "ref_ids": [] }, { "id": 13115, "type": "theorem", "label": "dga-lemma-tensor-with-complex-homotopy", "categories": [ "dga" ], "title": "dga-lemma-tensor-with-complex-homotopy", "contents": [ "The functor of Lemma \\ref{lemma-tensor-with-complex} defines an exact functor", "of triangulated categories", "$K(\\text{Mod}_{(E \\text{d})}) \\to K(\\mathcal{O})$." ], "refs": [ "dga-lemma-tensor-with-complex" ], "proofs": [ { "contents": [ "The functor induces a functor between homotopy categories by", "Lemma \\ref{lemma-functorial}.", "We have to show that $- \\otimes_E K^\\bullet$ transforms distinguished", "triangles into distinguished triangles.", "Suppose that $0 \\to K \\to L \\to M \\to 0$ is an admissible short", "exact sequence of differential graded $E$-modules. Let $s : M \\to L$ be", "a graded $E$-module homomorphism which is left inverse to $L \\to M$.", "Then $s$ defines a map $M \\otimes_E K^\\bullet \\to L \\otimes_E K^\\bullet$", "of graded $\\mathcal{O}$-modules (i.e., respecting $\\mathcal{O}$-module", "structure and grading, but not differentials)", "which is left inverse to $L \\otimes_E K^\\bullet \\to M \\otimes_E K^\\bullet$.", "Thus we see that", "$$", "0 \\to K \\otimes_E K^\\bullet \\to L \\otimes_E K^\\bullet \\to", "M \\otimes_E K^\\bullet \\to 0", "$$", "is a termwise split short exact sequences of complexes, i.e., a", "defines a distinguished triangle in $K(\\mathcal{O})$." ], "refs": [ "dga-lemma-functorial" ], "ref_ids": [ 13072 ] } ], "ref_ids": [ 13114 ] }, { "id": 13116, "type": "theorem", "label": "dga-lemma-tensor-with-complex-derived", "categories": [ "dga" ], "title": "dga-lemma-tensor-with-complex-derived", "contents": [ "The functor $K(\\text{Mod}_{(E, \\text{d})}) \\to K(\\mathcal{O})$", "of Lemma \\ref{lemma-tensor-with-complex-homotopy} has a left derived", "version defined on all of $D(E, \\text{d})$. We denote it", "$- \\otimes_E^\\mathbf{L} K^\\bullet : D(E, \\text{d}) \\to D(\\mathcal{O})$." ], "refs": [ "dga-lemma-tensor-with-complex-homotopy" ], "proofs": [ { "contents": [ "We will use", "Derived Categories, Lemma \\ref{derived-lemma-find-existence-computes}", "to prove this. As our collection $\\mathcal{P}$", "of objects we will use the objects with property (P).", "Property (1) was shown in Lemma \\ref{lemma-resolve}.", "Property (2) holds because if $s : P \\to P'$ is a quasi-isomorphism", "of modules with property (P), then $s$ is a homotopy equivalence", "by Lemma \\ref{lemma-hom-derived}." ], "refs": [ "derived-lemma-find-existence-computes", "dga-lemma-resolve", "dga-lemma-hom-derived" ], "ref_ids": [ 1832, 13062, 13068 ] } ], "ref_ids": [ 13115 ] }, { "id": 13117, "type": "theorem", "label": "dga-lemma-upgrade-tensor-with-complex-derived", "categories": [ "dga" ], "title": "dga-lemma-upgrade-tensor-with-complex-derived", "contents": [ "Let $R$ be a ring. Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}$", "be a sheaf of commutative $R$-algebras. Let $K^\\bullet$", "be a complex of $\\mathcal{O}$-modules.", "The functor", "of Lemma \\ref{lemma-tensor-with-complex-derived} has the following", "property: For every $M$, $N$ in $D(E, \\text{d})$ there is a", "canonical map", "$$", "R\\Hom(M, N)", "\\longrightarrow", "R\\Hom_\\mathcal{O}(M \\otimes_E^\\mathbf{L} K^\\bullet,", "N \\otimes_E^\\mathbf{L} K^\\bullet)", "$$", "in $D(R)$ which on cohomology modules gives the maps", "$$", "\\Ext^n_{D(E, \\text{d})}(M, N) \\to", "\\Ext^n_{D(\\mathcal{O})}", "(M \\otimes_E^\\mathbf{L} K^\\bullet, N \\otimes_E^\\mathbf{L} K^\\bullet)", "$$", "induced by the functor $- \\otimes_E^\\mathbf{L} K^\\bullet$." ], "refs": [ "dga-lemma-tensor-with-complex-derived" ], "proofs": [ { "contents": [ "The right hand side of the arrow is the global derived hom introduced", "in Cohomology on Sites, Section \\ref{sites-cohomology-section-global-RHom}", "which has the correct cohomology modules.", "For the left hand side we think of $M$ as a $(R, A)$-bimodule and", "we have the derived $\\Hom$ introduced in Section \\ref{section-restriction}", "which also has the correct cohomology modules.", "To prove the lemma we may assume $M$ and $N$ are differential graded", "$E$-modules with property (P); this does not change the left hand", "side of the arrow by", "Lemma \\ref{lemma-functoriality-derived-restriction}.", "By Lemma \\ref{lemma-compute-derived-restriction}", "this means that the left hand side of the arrow becomes", "$\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(M, N)$.", "In Lemmas \\ref{lemma-tensor-with-complex},", "\\ref{lemma-tensor-with-complex-homotopy}, and", "\\ref{lemma-tensor-with-complex-derived}", "we have constructed a functor", "$$", "- \\otimes_E K^\\bullet :", "\\text{Mod}^{dg}_{(E, \\text{d})}", "\\longrightarrow", "\\text{Comp}^{dg}(\\mathcal{O})", "$$", "of differential graded categories", "and we have shown that $- \\otimes_E^\\mathbf{L} K^\\bullet$ is computed", "by evaluating this functor", "on differential graded $E$-modules with property (P).", "Hence we obtain a map of complexes of $R$-modules", "$$", "\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(M, N)", "\\longrightarrow", "\\Hom_{\\text{Comp}^{dg}(\\mathcal{O})}", "(M \\otimes_E K^\\bullet, N \\otimes_E K^\\bullet)", "$$", "For any complexes of $\\mathcal{O}$-modules", "$\\mathcal{F}^\\bullet$, $\\mathcal{G}^\\bullet$ there", "is a canonical map", "$$", "\\Hom_{\\text{Comp}^{dg}(\\mathcal{O})}", "(\\mathcal{F}^\\bullet, \\mathcal{G}^\\bullet) =", "\\Gamma(\\mathcal{C},", "\\SheafHom^\\bullet(\\mathcal{F}^\\bullet, \\mathcal{G}^\\bullet))", "\\longrightarrow", "R\\Hom_\\mathcal{O}(\\mathcal{F}^\\bullet, \\mathcal{G}^\\bullet).", "$$", "Combining these maps", "we obtain the desired map of the lemma." ], "refs": [ "dga-lemma-functoriality-derived-restriction", "dga-lemma-compute-derived-restriction", "dga-lemma-tensor-with-complex", "dga-lemma-tensor-with-complex-homotopy", "dga-lemma-tensor-with-complex-derived" ], "ref_ids": [ 13100, 13102, 13114, 13115, 13116 ] } ], "ref_ids": [ 13116 ] }, { "id": 13118, "type": "theorem", "label": "dga-lemma-tensor-with-complex-hom-adjoint", "categories": [ "dga" ], "title": "dga-lemma-tensor-with-complex-hom-adjoint", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $K^\\bullet$ be a complex of $\\mathcal{O}$-modules.", "Then the functor", "$$", "- \\otimes_E^\\mathbf{L} K^\\bullet :", "D(E, \\text{d})", "\\longrightarrow", "D(\\mathcal{O})", "$$", "of Lemma \\ref{lemma-tensor-with-complex-derived} is a left adjoint", "of the functor", "$$", "R\\Hom(K^\\bullet, -) : D(\\mathcal{O}) \\longrightarrow D(E, \\text{d})", "$$", "of Lemma \\ref{lemma-existence-of-derived}." ], "refs": [ "dga-lemma-tensor-with-complex-derived", "dga-lemma-existence-of-derived" ], "proofs": [ { "contents": [ "The statement means that we have", "$$", "\\Hom_{D(E, \\text{d})}(M, R\\Hom(K^\\bullet, L^\\bullet)) =", "\\Hom_{D(\\mathcal{O})}(M \\otimes^\\mathbf{L}_E K^\\bullet, L^\\bullet)", "$$", "bifunctorially in $M$ and $L^\\bullet$. To see this we may replace $M$", "by a differential graded $E$-module $P$ with property (P).", "We also may replace $L^\\bullet$ by a K-injective complex of", "$\\mathcal{O}$-modules $I^\\bullet$. The computation", "of the derived functors given in the lemmas referenced in the statement", "combined with Lemma \\ref{lemma-hom-derived} translates the above into", "$$", "\\Hom_{K(\\text{Mod}_{(E, \\text{d})})}", "(P, \\Hom_\\mathcal{B}(K^\\bullet, I^\\bullet)) =", "\\Hom_{K(\\mathcal{O})}(P \\otimes_E K^\\bullet, I^\\bullet)", "$$", "where $\\mathcal{B} = \\text{Comp}^{dg}(\\mathcal{O})$. ", "There is an evaluation map from right to left functorial", "in $P$ and $I^\\bullet$ (details omitted).", "Choose a filtration $F_\\bullet$ on $P$ as in the definition of property (P).", "By Lemma \\ref{lemma-property-P-sequence} and the fact that", "both sides of the equation are homological functors in $P$", "on $K(\\text{Mod}_{(E, \\text{d})})$", "we reduce to the case where $P$ is replaced by", "the differential graded $E$-module $\\bigoplus F_iP$.", "Since both sides turn direct sums in the variable $P$", "into direct products we reduce to the case where $P$ is one of the", "differential graded $E$-modules $F_iP$.", "Since each $F_iP$ has a finite filtration (given by admissible", "monomorphisms) whose graded pieces are graded projective $E$-modules", "we reduce to the case where $P$ is a graded projective $E$-module.", "In this case we clearly have", "$$", "\\Hom_{\\text{Mod}^{dg}_{(E, \\text{d})}}", "(P, \\Hom_\\mathcal{B}(K^\\bullet, I^\\bullet)) =", "\\Hom_{\\text{Comp}^{dg}(\\mathcal{O})}(P \\otimes_E K^\\bullet, I^\\bullet)", "$$", "as graded $\\mathbf{Z}$-modules (because this statement reduces to the case", "$P = E[k]$ where it is obvious). As the isomorphism is compatible with", "differentials we conclude." ], "refs": [ "dga-lemma-hom-derived", "dga-lemma-property-P-sequence" ], "ref_ids": [ 13068, 13059 ] } ], "ref_ids": [ 13116, 13103 ] }, { "id": 13119, "type": "theorem", "label": "dga-lemma-fully-faithful-in-compact-case", "categories": [ "dga" ], "title": "dga-lemma-fully-faithful-in-compact-case", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $K^\\bullet$ be a complex of $\\mathcal{O}$-modules.", "Assume", "\\begin{enumerate}", "\\item $K^\\bullet$ represents a compact object of $D(\\mathcal{O})$, and", "\\item $E = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O})}(K^\\bullet, K^\\bullet)$", "computes the ext groups of $K^\\bullet$ in $D(\\mathcal{O})$.", "\\end{enumerate}", "Then the functor", "$$", "- \\otimes_E^\\mathbf{L} K^\\bullet :", "D(E, \\text{d})", "\\longrightarrow", "D(\\mathcal{O})", "$$", "of Lemma \\ref{lemma-tensor-with-complex-derived} is fully faithful." ], "refs": [ "dga-lemma-tensor-with-complex-derived" ], "proofs": [ { "contents": [ "Because our functor has a left adjoint given by", "$R\\Hom(K^\\bullet, -)$ by Lemma \\ref{lemma-tensor-with-complex-hom-adjoint}", "it suffices to show for a differential graded $E$-module $M$ that the map", "$$", "H^0(M) \\longrightarrow", "\\Hom_{D(\\mathcal{O})}(K^\\bullet, M \\otimes_E^\\mathbf{L} K^\\bullet)", "$$", "is an isomorphism. We may assume that $M = P$ is a differential graded", "$E$-module which has property (P). Since $K^\\bullet$ defines a", "compact object, we reduce using", "Lemma \\ref{lemma-property-P-sequence}", "to the case where $P$ has a finite filtration whose graded pieces", "are direct sums of $E[k]$. Again using compactness we reduce", "to the case $P = E[k]$. The assumption on $K^\\bullet$ is that", "the result holds for these." ], "refs": [ "dga-lemma-tensor-with-complex-hom-adjoint", "dga-lemma-property-P-sequence" ], "ref_ids": [ 13118, 13059 ] } ], "ref_ids": [ 13116 ] }, { "id": 13120, "type": "theorem", "label": "dga-lemma-factor-through-nicer", "categories": [ "dga" ], "title": "dga-lemma-factor-through-nicer", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. Let $E$ be a compact", "object of $D(A, \\text{d})$. Let $P$ be a differential graded $A$-module", "which has a finite filtration", "$$", "0 = F_{-1}P \\subset F_0P \\subset F_1P \\subset \\ldots \\subset F_nP = P", "$$", "by differential graded submodules such that", "$$", "F_{i + 1}P/F_iP \\cong \\bigoplus\\nolimits_{j \\in J_i} A[k_{i, j}]", "$$", "as differential graded $A$-modules for some sets $J_i$ and integers $k_{i, j}$.", "Let $E \\to P$ be a morphism of $D(A, \\text{d})$.", "Then there exists a differential graded submodule $P' \\subset P$ such that", "$F_{i + 1}P \\cap P'/(F_iP \\cap P')$ is equal to", "$\\bigoplus_{j \\in J'_i} A[k_{i, j}]$ for some finite subsets", "$J'_i \\subset J_i$ and such that $E \\to P$ factors through $P'$." ], "refs": [], "proofs": [ { "contents": [ "We will prove by induction on $-1 \\leq m \\leq n$ that there exists", "a differential graded submodule $P' \\subset P$ such that", "\\begin{enumerate}", "\\item $F_mP \\subset P'$,", "\\item for $i \\geq m$ the quotient $F_{i + 1}P \\cap P'/(F_iP \\cap P')$ is", "isomorphic to $\\bigoplus_{j \\in J'_i} A[k_{i, j}]$ for some finite subsets", "$J'_i \\subset J_i$, and", "\\item $E \\to P$ factors through $P'$.", "\\end{enumerate}", "The base case is $m = n$ where we can take $P' = P$.", "\\medskip\\noindent", "Induction step. Assume $P'$ works for $m$.", "For $i \\geq m$ and $j \\in J'_i$ let $x_{i, j} \\in F_{i + 1}P \\cap P'$", "be a homogeneous element of degree $k_{i, j}$ whose image in", "$F_{i + 1}P \\cap P'/(F_iP \\cap P')$ is the generator in", "the summand corresponding to $j \\in J_i$. The", "$x_{i, j}$ generate $P'/F_mP$ as an $A$-module. Write", "$$", "\\text{d}(x_{i, j}) = \\sum x_{i', j'} a_{i, j}^{i', j'} + y_{i, j}", "$$", "with $y_{i, j} \\in F_mP$ and $a_{i, j}^{i', j'} \\in A$.", "There exists a finite subset", "$J'_{m - 1} \\subset J_{m - 1}$ such that each $y_{i, j}$ maps to", "an element of the submodule $\\bigoplus_{j \\in J'_{m - 1}} A[k_{m - 1, j}]$", "of $F_mP/F_{m - 1}P$. Let $P'' \\subset F_mP$ be the inverse", "image of $\\bigoplus_{j \\in J'_{m - 1}} A[k_{m - 1, j}]$ under", "the map $F_mP \\to F_mP/F_{m - 1}P$. Then we see that the $A$-submodule", "$$", "P'' + \\sum x_{i, j}A", "$$", "is a differential graded submodule of the type we are looking for. Moreover", "$$", "P'/(P'' + \\sum x_{i, j}A) =", "\\bigoplus\\nolimits_{j \\in J_{m - 1} \\setminus J'_{m - 1}} A[k_{m - 1, j}]", "$$", "Since $E$ is compact, the composition of the given map $E \\to P'$", "with the quotient map, factors through a finite direct subsum of", "the module displayed above. Hence after enlarging $J'_{m - 1}$", "we may assume $E \\to P'$ factors through", "$P'' + \\sum x_{i, j}A$ as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13121, "type": "theorem", "label": "dga-lemma-compact-implies-bounded", "categories": [ "dga" ], "title": "dga-lemma-compact-implies-bounded", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "For every compact object $E$ of $D(A, \\text{d})$ there", "exist integers $a \\leq b$ such that $\\Hom_{D(A, \\text{d})}(E, M) = 0$", "if $H^i(M) = 0$ for $i \\in [a, b]$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the collection of objects of $D(A, \\text{d})$ for which", "such a pair of integers exists is a saturated, strictly full triangulated", "subcategory of $D(A, \\text{d})$.", "Thus by Proposition \\ref{proposition-compact} it suffices to prove", "this when $E$ is represented by a differential graded module $P$ which", "has a finite filtration $F_\\bullet$ by differential graded submodules", "such that $F_iP/F_{i - 1}P$ are finite direct sums of shifts of $A$.", "Using the compatibility with triangles, we see that it suffices", "to prove it for $P = A$. In this case $\\Hom_{D(A, \\text{d})}(A, M) = H^0(M)$", "and the result holds with $a = b = 0$." ], "refs": [ "dga-proposition-compact" ], "ref_ids": [ 13132 ] } ], "ref_ids": [] }, { "id": 13122, "type": "theorem", "label": "dga-lemma-compact", "categories": [ "dga" ], "title": "dga-lemma-compact", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. Assume that $A^n = 0$", "for $|n| \\gg 0$. Let $E$ be an object of $D(A, \\text{d})$.", "The following are equivalent", "\\begin{enumerate}", "\\item $E$ is a compact object, and", "\\item $E$ can be represented by a differential graded $A$-module $P$", "which is finite projective as a graded $A$-module and satisfies", "$\\Hom_{K(A, \\text{d})}(P, M) = \\Hom_{D(A, \\text{d})}(P, M)$", "for every differential graded $A$-module $M$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{D} \\subset K(A, \\text{d})$ be the triangulated subcategory", "discussed in Remark \\ref{remark-source-graded-projective}.", "Let $P$ be an object of $\\mathcal{D}$ which is finite projective", "as a graded $A$-module. Then $P$ represents a compact object of", "$D(A, \\text{d})$ by Remark \\ref{remark-graded-projective-is-compact}.", "\\medskip\\noindent", "To prove the converse, let $E$ be a compact object of $D(A, \\text{d})$.", "Fix $a \\leq b$ as in Lemma \\ref{lemma-compact-implies-bounded}.", "After decreasing $a$ and increasing $b$ if necessary, we may also", "assume that $H^i(E) = 0$ for $i \\not \\in [a, b]$ (this follows", "from Proposition \\ref{proposition-compact} and our assumption on $A$).", "Moreover, fix an integer $c > 0$ such that $A^n = 0$ if $|n| \\geq c$.", "\\medskip\\noindent", "By Proposition \\ref{proposition-compact} we see that $E$ is a direct", "summand, in $D(A, \\text{d})$, of a differential graded $A$-module $P$", "which has a finite filtration $F_\\bullet$ by differential", "graded submodules such that $F_iP/F_{i - 1}P$ are finite direct sums", "of shifts of $A$. In particular, $P$ has property (P) and we have", "$\\Hom_{D(A, \\text{d})}(P, M) = \\Hom_{K(A, \\text{d})}(P, M)$ for any", "differential graded module $M$ by Lemma \\ref{lemma-hom-derived}.", "In other words, $P$ is an object of the triangulated", "subcategory $\\mathcal{D} \\subset K(A, \\text{d})$ discussed in", "Remark \\ref{remark-source-graded-projective}.", "Note that $P$ is finite free as a graded $A$-module.", "\\medskip\\noindent", "Choose $n > 0$ such that $b + 4c - n < a$.", "Represent the projector onto $E$ by an endomorphism $\\varphi : P \\to P$ of", "differential graded $A$-modules. Consider the distinguished triangle", "$$", "P \\xrightarrow{1 - \\varphi} P \\to C \\to P[1]", "$$", "in $K(A, \\text{d})$ where $C$ is the cone of the first arrow. Then", "$C$ is an object of $\\mathcal{D}$,", "we have $C \\cong E \\oplus E[1]$ in $D(A, \\text{d})$, and", "$C$ is a finite graded free $A$-module.", "Next, consider a distinguished triangle", "$$", "C[1] \\to C \\to C' \\to C[2]", "$$", "in $K(A, \\text{d})$ where $C'$ is the cone on a morphism $C[1] \\to C$", "representing the composition", "$$", "C[1] \\cong E[1] \\oplus E[2] \\to E[1] \\to E \\oplus E[1] \\cong C", "$$", "in $D(A, \\text{d})$. Then we see that $C'$ represents $E \\oplus E[2]$.", "Continuing in this manner we see that we can find a differential", "graded $A$-module $P$ which is an object of $\\mathcal{D}$,", "is a finite free as a graded $A$-module, and represents $E \\oplus E[n]$.", "\\medskip\\noindent", "Choose a basis $x_i$, $i \\in I$ of homogeneous elements for $P$ as an", "$A$-module. Let $d_i = \\deg(x_i)$.", "Let $P_1$ be the $A$-submodule of $P$ generated by $x_i$ and", "$\\text{d}(x_i)$ for $d_i \\leq a - c - 1$.", "Let $P_2$ be the $A$-submodule of $P$ generated by $x_i$ and", "$\\text{d}(x_i)$ for $d_i \\geq b - n + c$.", "We observe", "\\begin{enumerate}", "\\item $P_1$ and $P_2$ are differential graded submodules of $P$,", "\\item $P_1^t = 0$ for $t \\geq a$,", "\\item $P_1^t = P^t$ for $t \\leq a - 2c$,", "\\item $P_2^t = 0$ for $t \\leq b - n$,", "\\item $P_2^t = P^t$ for $t \\geq b - n + 2c$.", "\\end{enumerate}", "As $b - n + 2c \\geq a - 2c$ by our choice of $n$", "we obtain a short exact sequence of differential graded $A$-modules", "$$", "0 \\to P_1 \\cap P_2 \\to P_1 \\oplus P_2 \\xrightarrow{\\pi} P \\to 0", "$$", "Since $P$ is projective as a graded $A$-module this is an admissible", "short exact sequence (Lemma \\ref{lemma-target-graded-projective}).", "Hence we obtain a boundary map", "$\\delta : P \\to (P_1 \\cap P_2)[1]$ in $K(A, \\text{d})$, see", "Lemma \\ref{lemma-admissible-ses}.", "Since $P = E \\oplus E[n]$ and since $P_1 \\cap P_2$ lives in", "degrees $(b - n, a)$ we find that", "$\\Hom_{D(A, \\text{d})}(E \\oplus E[n], (P_1 \\cap P_2)[1])$ is", "zero. Therefore $\\delta = 0$ as a morphism in $K(A, \\text{d})$", "as $P$ is an object of $\\mathcal{D}$.", "By Derived Categories, Lemma \\ref{derived-lemma-split}", "we can find a map $s : P \\to P_1 \\oplus P_2$ such that", "$\\pi \\circ s = \\text{id}_P + \\text{d}h + h\\text{d}$ for some $h : P \\to P$", "of degree $-1$. Since $P_1 \\oplus P_2 \\to P$ is surjective and since $P$", "is projective as a graded $A$-module we can choose a homogeneous", "lift $\\tilde h : P \\to P_1 \\oplus P_2$ of $h$. Then we change", "$s$ into $s + \\text{d} \\tilde h + \\tilde h \\text{d}$ to get", "$\\pi \\circ s = \\text{id}_P$. This means we obtain a direct", "sum decomposition $P = s^{-1}(P_1) \\oplus s^{-1}(P_2)$.", "Since $s^{-1}(P_2)$ is equal to $P$ in degrees $\\geq b - n + 2c$", "we see that $s^{-1}(P_2) \\to P \\to E$ is a quasi-isomorphism,", "i.e., an isomorphism in $D(A, \\text{d})$. This finishes the proof." ], "refs": [ "dga-remark-source-graded-projective", "dga-remark-graded-projective-is-compact", "dga-lemma-compact-implies-bounded", "dga-proposition-compact", "dga-proposition-compact", "dga-lemma-hom-derived", "dga-remark-source-graded-projective", "dga-lemma-target-graded-projective", "dga-lemma-admissible-ses", "derived-lemma-split" ], "ref_ids": [ 13166, 13167, 13121, 13132, 13132, 13068, 13166, 13054, 13037, 1766 ] } ], "ref_ids": [] }, { "id": 13123, "type": "theorem", "label": "dga-lemma-qis-equivalence", "categories": [ "dga" ], "title": "dga-lemma-qis-equivalence", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d}) \\to (B, \\text{d})$ be a", "homomorphism of differential graded algebras over $R$, which induces", "an isomorphism on cohomology algebras. Then", "$$", "- \\otimes_A^\\mathbf{L} B : D(A, \\text{d}) \\to D(B, \\text{d})", "$$", "gives an $R$-linear equivalence of triangulated categories with", "quasi-inverse the restriction functor $N \\mapsto N_A$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-tensor-with-compact-fully-faithful}", "the functor $M \\longmapsto M \\otimes_A^\\mathbf{L} B$ is", "fully faithful. By Lemma \\ref{lemma-tensor-hom-adjoint}", "the functor $N \\longmapsto R\\Hom(B, N) = N_A$ is a right adjoint, see", "Example \\ref{example-map-hom-tensor}.", "It is clear that the kernel of $R\\Hom(B, -)$ is zero.", "Hence the result follows from", "Derived Categories, Lemma", "\\ref{derived-lemma-fully-faithful-adjoint-kernel-zero}." ], "refs": [ "dga-lemma-tensor-with-compact-fully-faithful", "dga-lemma-tensor-hom-adjoint", "derived-lemma-fully-faithful-adjoint-kernel-zero" ], "ref_ids": [ 13108, 13107, 1793 ] } ], "ref_ids": [] }, { "id": 13124, "type": "theorem", "label": "dga-lemma-tilting-equivalence", "categories": [ "dga" ], "title": "dga-lemma-tilting-equivalence", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be ", "differential graded algebras over $R$. Let $N$ be a", "differential graded $(A, B)$-bimodule. Assume that", "\\begin{enumerate}", "\\item $N$ defines a compact object of $D(B, \\text{d})$,", "\\item if $N' \\in D(B, \\text{d})$ and", "$\\Hom_{D(B, \\text{d})}(N, N'[n]) = 0$ for $n \\in \\mathbf{Z}$,", "then $N' = 0$, and", "\\item the map $H^k(A) \\to \\Hom_{D(B, \\text{d})}(N, N[k])$ is an", "isomorphism for all $k \\in \\mathbf{Z}$.", "\\end{enumerate}", "Then", "$$", "- \\otimes_A^\\mathbf{L} N : D(A, \\text{d}) \\to D(B, \\text{d})", "$$", "gives an $R$-linear equivalence of triangulated categories." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-tensor-with-compact-fully-faithful}", "the functor $M \\longmapsto M \\otimes_A^\\mathbf{L} N$ is", "fully faithful. By Lemma \\ref{lemma-tensor-hom-adjoint}", "the functor $N' \\longmapsto R\\Hom(N, N')$ is a right adjoint.", "By assumption (3) the kernel of $R\\Hom(N, -)$ is zero.", "Hence the result follows from", "Derived Categories, Lemma", "\\ref{derived-lemma-fully-faithful-adjoint-kernel-zero}." ], "refs": [ "dga-lemma-tensor-with-compact-fully-faithful", "dga-lemma-tensor-hom-adjoint", "derived-lemma-fully-faithful-adjoint-kernel-zero" ], "ref_ids": [ 13108, 13107, 1793 ] } ], "ref_ids": [] }, { "id": 13125, "type": "theorem", "label": "dga-lemma-rickard", "categories": [ "dga" ], "title": "dga-lemma-rickard", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be", "differential graded $R$-algebras. Assume that $A = H^0(A)$.", "The following are equivalent", "\\begin{enumerate}", "\\item $D(A, \\text{d})$ and $D(B, \\text{d})$ are equivalent as $R$-linear", "triangulated categories, and", "\\item there exists an object $P$ of $D(B, \\text{d})$ such that", "\\begin{enumerate}", "\\item $P$ is a compact object of $D(B, \\text{d})$,", "\\item if $N \\in D(B, \\text{d})$ with $\\Hom_{D(B, \\text{d})}(P, N[i]) = 0$", "for $i \\in \\mathbf{Z}$, then $N = 0$,", "\\item $\\Hom_{D(B, \\text{d})}(P, P[i]) = 0$ for $i \\not = 0$ and", "equal to $A$ for $i = 0$.", "\\end{enumerate}", "\\end{enumerate}", "The equivalence $D(A, \\text{d}) \\to D(B, \\text{d})$", "constructed in (2) sends $A$ to $P$." ], "refs": [], "proofs": [ { "contents": [ "Let $F : D(A, \\text{d}) \\to D(B, \\text{d})$ be an equivalence.", "Then $F$ maps compact objects to compact objects. Hence $P = F(A)$ is", "compact, i.e., (2)(a) holds. Conditions (2)(b) and (2)(c) are immediate", "from the fact that $F$ is an equivalence.", "\\medskip\\noindent", "Let $P$ be an object as in (2). Represent $P$ by a", "differential graded module with property (P). Set", "$$", "(E, \\text{d}) = \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(P, P)", "$$", "Then $H^0(E) = A$ and $H^k(E) = 0$ for $k \\not = 0$ by", "Lemma \\ref{lemma-hom-derived} and assumption (2)(c).", "Viewing $P$ as a $(E, B)$-bimodule and using", "Lemma \\ref{lemma-tilting-equivalence} and assumption (2)(b)", "we obtain an equivalence", "$$", "D(E, \\text{d}) \\to D(B, \\text{d})", "$$", "sending $E$ to $P$.", "Let $E' \\subset E$ be the differential graded $R$-subalgebra", "with", "$$", "(E')^i = \\left\\{", "\\begin{matrix}", "E^i & \\text{if }i < 0 \\\\", "\\Ker(E^0 \\to E^1) & \\text{if }i = 0 \\\\", "0 & \\text{if }i > 0", "\\end{matrix}", "\\right.", "$$", "Then there are quasi-isomorphisms of differential graded", "algebras $(A, \\text{d}) \\leftarrow (E', \\text{d}) \\rightarrow (E, \\text{d})$.", "Thus we obtain equivalences", "$$", "D(A, \\text{d}) \\leftarrow D(E', \\text{d}) \\rightarrow D(E, \\text{d})", "\\rightarrow D(B, \\text{d})", "$$", "by Lemma \\ref{lemma-qis-equivalence}." ], "refs": [ "dga-lemma-hom-derived", "dga-lemma-tilting-equivalence", "dga-lemma-qis-equivalence" ], "ref_ids": [ 13068, 13124, 13123 ] } ], "ref_ids": [] }, { "id": 13126, "type": "theorem", "label": "dga-lemma-rickard-rings", "categories": [ "dga" ], "title": "dga-lemma-rickard-rings", "contents": [ "Let $R$ be a ring.", "Let $A$ and $B$ be $R$-algebras. The following are equivalent", "\\begin{enumerate}", "\\item there is an $R$-linear equivalence $D(A) \\to D(B)$", "of triangulated categories,", "\\item there exists an object $P$ of $D(B)$ such that", "\\begin{enumerate}", "\\item $P$ can be represented by a finite complex", "of finite projective $B$-modules,", "\\item if $K \\in D(B)$ with $\\Ext^i_B(P, K) = 0$ for", "$i \\in \\mathbf{Z}$, then $K = 0$, and", "\\item $\\Ext^i_B(P, P) = 0$ for $i \\not = 0$ and", "equal to $A$ for $i= 0$.", "\\end{enumerate}", "\\end{enumerate}", "Moreover, if $B$ is flat as an $R$-module, then this is also", "equivalent to", "\\begin{enumerate}", "\\item[(3)] there exists an $(A, B)$-bimodule $N$ such that", "$- \\otimes_A^\\mathbf{L} N : D(A) \\to D(B)$ is an equivalence.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is a special case of", "Lemma \\ref{lemma-rickard} combined with the result of", "Lemma \\ref{lemma-compact} characterizing compact objects of $D(B)$", "(small detail omitted).", "The equivalence with (3) if $B$ is $R$-flat follows from", "Proposition \\ref{proposition-rickard}." ], "refs": [ "dga-lemma-rickard", "dga-lemma-compact", "dga-proposition-rickard" ], "ref_ids": [ 13125, 13122, 13133 ] } ], "ref_ids": [] }, { "id": 13127, "type": "theorem", "label": "dga-lemma-K-flat-resolution", "categories": [ "dga" ], "title": "dga-lemma-K-flat-resolution", "contents": [ "Let $R$ be a ring. Let $(B, \\text{d})$ be a differential graded $R$-algebra.", "There exists a quasi-isomorphism $(A, \\text{d}) \\to (B, \\text{d})$ of", "differential graded $R$-algebras with the following properties", "\\begin{enumerate}", "\\item $A$ is K-flat as a complex of $R$-modules,", "\\item $A$ is a free graded $R$-algebra.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First we claim we can find $(A_0, \\text{d}) \\to (B, \\text{d})$", "having (1) and (2) inducing a surjection on cohomology.", "Namely, take a graded set $S$ and for each $s \\in S$", "a homogeneous element $b_s \\in \\Ker(d : B \\to B)$ of degree $\\deg(s)$", "such that the classes $\\overline{b}_s$ in $H^*(B)$", "generate $H^*(B)$ as an $R$-module.", "Then we can set $A_0 = R\\langle S \\rangle$ with zero differential", "and $A_0 \\to B$ given by mapping $s$ to $b_s$.", "\\medskip\\noindent", "Given $A_0 \\to B$ inducing a surjection on cohomology we construct", "a sequence", "$$", "A_0 \\to A_1 \\to A_2 \\to \\ldots B", "$$", "by induction. Given $A_n \\to B$ we set $S_n$ be a graded set", "and for each $s \\in S_n$ we let $a_s \\in \\Ker(A_n \\to A_n)$", "be a homogeneous element of degree $\\deg(s) + 1$", "mapping to a class $\\overline{a}_s$ in $H^*(A_n)$", "which maps to zero in $H^*(B)$. We choose $S_n$ large enough", "so that the elements $\\overline{a}_s$ generate $\\Ker(H^*(A_n) \\to H^*(B))$", "as an $R$-module. Then we set", "$$", "A_{n + 1} = A_n\\langle S_n \\rangle", "$$", "with differential given by $\\text{d}(s) = a_s$ see discussion above.", "Then each $(A_n, \\text{d})$ satisfies (1) and (2), we omit the details.", "The map $H^*(A_n) \\to H^*(B)$ is surjective as this was true for $n = 0$.", "\\medskip\\noindent", "It is clear that $A = \\colim A_n$ is a free graded $R$-algebra.", "It is K-flat by More on Algebra, Lemma \\ref{more-algebra-lemma-colimit-K-flat}.", "The map $H^*(A) \\to H^*(B)$ is an isomorphism as it is surjective", "and injective: every element of $H^*(A)$ comes from an element of", "$H^*(A_n)$ for some $n$ and if it dies in $H^*(B)$, then it dies", "in $H^*(A_{n + 1})$ hence in $H^*(A)$." ], "refs": [ "more-algebra-lemma-colimit-K-flat" ], "ref_ids": [ 10129 ] } ], "ref_ids": [] }, { "id": 13128, "type": "theorem", "label": "dga-lemma-compose-tensor-functors-tor", "categories": [ "dga" ], "title": "dga-lemma-compose-tensor-functors-tor", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$, $(B, \\text{d})$, and", "$(C, \\text{d})$ be differential graded $R$-algebras. Assume", "$A \\otimes_R C$ represents $A \\otimes^\\mathbf{L}_R C$ in $D(R)$.", "Let $N$ be a differential graded $(A, B)$-bimodule.", "Let $N'$ be a differential graded $(B, C)$-bimodule.", "Then the composition", "$$", "\\xymatrix{", "D(A, \\text{d}) \\ar[rr]^{- \\otimes_A^\\mathbf{L} N} & &", "D(B, \\text{d}) \\ar[rr]^{- \\otimes_B^\\mathbf{L} N'} & &", "D(C, \\text{d})", "}", "$$", "is isomorphic to $- \\otimes_A^\\mathbf{L} N''$ for some differential graded", "$(A, C)$-bimodule $N''$." ], "refs": [], "proofs": [ { "contents": [ "Using Lemma \\ref{lemma-K-flat-resolution}", "we choose a quasi-isomorphism $(B', \\text{d}) \\to (B, \\text{d})$", "with $B'$ K-flat as a complex of $R$-modules.", "By Lemma \\ref{lemma-qis-equivalence}", "the functor $-\\otimes^\\mathbf{L}_{B'} B : D(B', \\text{d}) \\to D(B, \\text{d})$", "is an equivalence with quasi-inverse given by restriction.", "Note that restriction is canonically isomorphic to the functor", "$- \\otimes^\\mathbf{L}_B B : D(B, \\text{d}) \\to D(B', \\text{d})$", "where $B$ is viewed as a $(B, B')$-bimodule.", "Thus it suffices to prove the lemma for the compositions", "$$", "D(A) \\to D(B) \\to D(B'),\\quad", "D(B') \\to D(B) \\to D(C),\\quad", "D(A) \\to D(B') \\to D(C).", "$$", "The first one is Lemma \\ref{lemma-compose-tensor-functors}", "because $B'$ is K-flat as a complex of $R$-modules.", "The second one is true because", "$B \\otimes_B^\\mathbf{L} N' = N' = B \\otimes_B N'$", "and hence Lemma \\ref{lemma-compose-tensor-functors-general} applies.", "Thus we reduce to the case where $B$ is K-flat as a complex", "of $R$-modules.", "\\medskip\\noindent", "Assume $B$ is K-flat as a complex of $R$-modules. It suffices to", "show that (\\ref{equation-plain-versus-derived-algebras}) is an", "isomorphism, see", "Lemma \\ref{lemma-compose-tensor-functors-general-algebra}.", "Choose a quasi-isomorphism $L \\to A$ where $L$ is a differential", "graded $R$-module which has property (P). Then it is clear that", "$P = L \\otimes_R B$ has property (P) as a differential graded $B$-module.", "Hence we have to show that $P \\to A \\otimes_R B$", "induces a quasi-isomorphism", "$$", "P \\otimes_B (B \\otimes_R C)", "\\longrightarrow", "(A \\otimes_R B) \\otimes_B (B \\otimes_R C)", "$$", "We can rewrite this as", "$$", "P \\otimes_R B \\otimes_R C \\longrightarrow A \\otimes_R B \\otimes_R C", "$$", "Since $B$ is K-flat as a complex of $R$-modules, it", "follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-K-flat-quasi-isomorphism}", "that it is enough", "to show that", "$$", "P \\otimes_R C \\to A \\otimes_R C", "$$", "is a quasi-isomorphism, which is exactly our assumption." ], "refs": [ "dga-lemma-K-flat-resolution", "dga-lemma-qis-equivalence", "dga-lemma-compose-tensor-functors", "dga-lemma-compose-tensor-functors-general", "dga-lemma-compose-tensor-functors-general-algebra", "more-algebra-lemma-K-flat-quasi-isomorphism" ], "ref_ids": [ 13127, 13123, 13113, 13111, 13112, 10123 ] } ], "ref_ids": [] }, { "id": 13129, "type": "theorem", "label": "dga-lemma-countable", "categories": [ "dga" ], "title": "dga-lemma-countable", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra with", "$H^i(A)$ countable for each $i$. Let $M$ be an object of $D(A, \\text{d})$.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $M = \\text{hocolim} E_n$ with $E_n$ compact in $D(A, \\text{d})$, and", "\\item $H^i(M)$ is countable for each $i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) holds. Then we have $H^i(M) = \\colim H^i(E_n)$ by", "Derived Categories, Lemma \\ref{derived-lemma-cohomology-of-hocolim}.", "Thus it suffices to prove that $H^i(E_n)$ is countable for each $n$.", "By Proposition \\ref{proposition-compact} we see that $E_n$", "is isomorphic in $D(A, \\text{d})$ to a direct summand of a", "differential graded module $P$ which has a finite filtration", "$F_\\bullet$ by differential graded submodules", "such that $F_jP/F_{j - 1}P$ are finite direct sums of shifts of $A$.", "By assumption the groups $H^i(F_jP/F_{j - 1}P)$ are countable.", "Arguing by induction on the length of the filtration and using", "the long exact cohomology sequence we conclude that (2) is true.", "The interesting implication is the other one.", "\\medskip\\noindent", "We claim there is a countable differential graded", "subalgebra $A' \\subset A$ such that the inclusion map", "$A' \\to A$ defines an isomorphism on cohomology.", "To construct $A'$ we choose countable differential graded", "subalgebras", "$$", "A_1 \\subset A_2 \\subset A_3 \\subset \\ldots", "$$", "such that (a) $H^i(A_1) \\to H^i(A)$ is surjective, and (b)", "for $n > 1$ the kernel of the map $H^i(A_{n - 1}) \\to H^i(A_n)$", "is the same as the kernel of the map $H^i(A_{n - 1}) \\to H^i(A)$.", "To construct $A_1$ take any countable collection of cochains", "$S \\subset A$ generating the cohomology of $A$ (as a ring or as", "a graded abelian group) and let", "$A_1$ be the differential graded subalgebra of $A$ generated by $S$.", "To construct $A_n$ given $A_{n - 1}$ for each cochain $a \\in A_{n - 1}^i$", "which maps to zero in $H^i(A)$ choose $s_a \\in A^{i - 1}$", "with $\\text{d}(s_a) = a$ and let $A_n$ be the differential graded", "subalgebra of $A$ generated by $A_{n - 1}$ and the elements $s_a$.", "Finally, take $A' = \\bigcup A_n$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-qis-equivalence}", "the restriction map $D(A, \\text{d}) \\to D(A', \\text{d})$,", "$M \\mapsto M_{A'}$ is an equivalence. Since the cohomology", "groups of $M$ and $M_{A'}$ are the same, we see that it", "suffices to prove the implication (2) $\\Rightarrow$ (1)", "for $(A', \\text{d})$.", "\\medskip\\noindent", "Assume $A$ is countable. By the exact same type of argument as", "given above we see that for $M$ in $D(A, \\text{d})$", "the following are equivalent: $H^i(M)$ is countable for each $i$", "and $M$ can be represented by a countable differential graded module.", "Hence in order to prove the implication (2) $\\Rightarrow$ (1)", "we reduce to the situation described in the next paragraph.", "\\medskip\\noindent", "Assume $A$ is countable and that $M$ is a countable differential graded", "module over $A$. We claim there exists a homomorphism", "$P \\to M$ of differential graded $A$-modules such that", "\\begin{enumerate}", "\\item $P \\to M$ is a quasi-isomorphism,", "\\item $P$ has property (P), and", "\\item $P$ is countable.", "\\end{enumerate}", "Looking at the proof of the construction of P-resolutions in", "Lemma \\ref{lemma-resolve} we see that it suffices to show that", "we can prove Lemma \\ref{lemma-good-quotient}", "in the setting of countable differential graded modules.", "This is immediate from the proof.", "\\medskip\\noindent", "Assume that $A$ is countable and that $M$ is a countable", "differential graded module with property (P). Choose a filtration", "$$", "0 = F_{-1}P \\subset F_0P \\subset F_1P \\subset \\ldots \\subset P", "$$", "by differential graded submodules such that we have", "\\begin{enumerate}", "\\item $P = \\bigcup F_pP$,", "\\item $F_iP \\to F_{i + 1}P$ is an admissible monomorphism,", "\\item isomorphisms of differential graded modules", "$F_iP/F_{i - 1}P \\to \\bigoplus_{j \\in J_i} A[k_j]$", "for some sets $J_i$ and integers $k_j$.", "\\end{enumerate}", "Of course $J_i$ is countable for each $i$. For each $i$ and", "$j \\in J_i$ choose $x_{i, j} \\in F_iP$ of degree $k_j$ whose", "image in $F_iP/F_{i - 1}P$ generates the summand corresponding", "to $j$.", "\\medskip\\noindent", "Claim: Given $n$ and finite subsets $S_i \\subset J_i$, $i = 1, \\ldots, n$", "there exist finite subsets $S_i \\subset T_i \\subset J_i$, $i = 1, \\ldots, n$", "such that $P' = \\bigoplus_{i \\leq n} \\bigoplus_{j \\in T_i} Ax_{i, j}$", "is a differential graded submodule of $P$. This was shown in the", "proof of Lemma \\ref{lemma-factor-through-nicer} but it is also", "easily shown directly: the elements $x_{i, j}$ freely generate", "$P$ as a right $A$-module. The structure of $P$ shows that", "$$", "\\text{d}(x_{i, j}) = \\sum\\nolimits_{i' < i} x_{i', j'}a_{i', j'}", "$$", "where of course the sum is finite.", "Thus given $S_0, \\ldots, S_n$ we can first choose", "$S_0 \\subset S'_0, \\ldots, S_{n - 1} \\subset S'_{n - 1}$ with", "$\\text{d}(x_{n, j}) \\in \\bigoplus_{i' < n, j' \\in S'_{i'}} x_{i', j'}A$", "for all $j \\in S_n$. Then by induction on $n$ we can choose", "$S'_0 \\subset T_0, \\ldots, S'_{n - 1} \\subset T_{n - 1}$", "to make sure that $\\bigoplus_{i' < n, j' \\in T_{i'}} x_{i', j'}A$", "is a differential graded $A$-submodule. Setting $T_n = S_n$ we find that", "$P' = \\bigoplus_{i \\leq n, j \\in T_i} x_{i, j}A$ is as desired.", "\\medskip\\noindent", "From the claim it is clear that $P = \\bigcup P'_n$", "is a countable rising union of $P'_n$ as above.", "By construction each $P'_n$ is a differential graded module with", "property (P) such that the filtration is finite and the succesive", "quotients are finite direct sums of shifts of $A$. Hence $P'_n$", "defines a compact object of $D(A, \\text{d})$, see for example", "Proposition \\ref{proposition-compact}. Since", "$P = \\text{hocolim} P'_n$ in $D(A, \\text{d})$", "by Lemma \\ref{lemma-homotopy-colimit}", "the proof of the implication (2) $\\Rightarrow$ (1) is complete." ], "refs": [ "derived-lemma-cohomology-of-hocolim", "dga-proposition-compact", "dga-lemma-qis-equivalence", "dga-lemma-resolve", "dga-lemma-good-quotient", "dga-lemma-factor-through-nicer", "dga-proposition-compact", "dga-lemma-homotopy-colimit" ], "ref_ids": [ 1923, 13132, 13123, 13062, 13061, 13120, 13132, 13071 ] } ], "ref_ids": [] }, { "id": 13130, "type": "theorem", "label": "dga-proposition-homotopy-category-triangulated", "categories": [ "dga" ], "title": "dga-proposition-homotopy-category-triangulated", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. The homotopy category", "$K(\\text{Mod}_{(A, \\text{d})})$ of differential graded $A$-modules with its", "natural translation functors and distinguished triangles is a triangulated", "category." ], "refs": [], "proofs": [ { "contents": [ "We know that $K(\\text{Mod}_{(A, \\text{d})})$ is a pre-triangulated category.", "Hence it suffices to prove TR4 and to prove it we can use", "Derived Categories, Lemma \\ref{derived-lemma-easier-axiom-four}.", "Let $K \\to L$ and $L \\to M$ be composable morphisms of", "$K(\\text{Mod}_{(A, \\text{d})})$. By", "Lemma \\ref{lemma-sequence-maps-split} we may assume that", "$K \\to L$ and $L \\to M$ are admissible monomorphisms.", "In this case the result follows from", "Lemma \\ref{lemma-two-split-injections}." ], "refs": [ "derived-lemma-easier-axiom-four", "dga-lemma-sequence-maps-split", "dga-lemma-two-split-injections" ], "ref_ids": [ 1770, 13040, 13048 ] } ], "ref_ids": [] }, { "id": 13131, "type": "theorem", "label": "dga-proposition-ABC-homotopy-category-triangulated", "categories": [ "dga" ], "title": "dga-proposition-ABC-homotopy-category-triangulated", "contents": [ "In Situation \\ref{situation-ABC} the homotopy category $K(\\mathcal{A})$", "with its natural translation functors and distinguished triangles is a", "triangulated category." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-analogue-homotopy-category-pre-triangulated} we know that", "$K(\\mathcal{A})$ is pre-triangulated. Combining", "Lemmas \\ref{lemma-analogue-sequence-maps-split} and", "\\ref{lemma-dgc-analogue-tr4} with", "Derived Categories, Lemma \\ref{derived-lemma-easier-axiom-four},", "we conclude that $K(\\mathcal{A})$ is a triangulated category." ], "refs": [ "dga-lemma-analogue-homotopy-category-pre-triangulated", "dga-lemma-analogue-sequence-maps-split", "dga-lemma-dgc-analogue-tr4", "derived-lemma-easier-axiom-four" ], "ref_ids": [ 13088, 13081, 13089, 1770 ] } ], "ref_ids": [] }, { "id": 13132, "type": "theorem", "label": "dga-proposition-compact", "categories": [ "dga" ], "title": "dga-proposition-compact", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. Let $E$ be an", "object of $D(A, \\text{d})$. Then the following are equivalent", "\\begin{enumerate}", "\\item $E$ is a compact object,", "\\item $E$ is a direct summand of an object of $D(A, \\text{d})$", "which is represented by a differential graded module $P$ which", "has a finite filtration $F_\\bullet$ by differential graded submodules", "such that $F_iP/F_{i - 1}P$ are finite direct sums of shifts of $A$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $E$ is compact. By Lemma \\ref{lemma-resolve} we may assume that $E$", "is represented by a differential graded $A$-module $P$ with property (P).", "Consider the distinguished triangle", "$$", "\\bigoplus F_iP \\to \\bigoplus F_iP \\to P", "\\xrightarrow{\\delta} \\bigoplus F_iP[1]", "$$", "coming from the admissible short exact sequence of", "Lemma \\ref{lemma-property-P-sequence}. Since $E$ is compact we have", "$\\delta = \\sum_{i = 1, \\ldots, n} \\delta_i$ for some", "$\\delta_i : P \\to F_iP[1]$. Since the composition of $\\delta$", "with the map $\\bigoplus F_iP[1] \\to \\bigoplus F_iP[1]$ is zero", "(Derived Categories, Lemma \\ref{derived-lemma-composition-zero})", "it follows that $\\delta = 0$ (follows as $\\bigoplus F_iP \\to \\bigoplus F_iP$", "maps the summand $F_iP$ via the difference of $\\text{id}$ and the inclusion", "map into $F_{i - 1}P$).", "Thus we see that the identity on $E$ factors through", "$\\bigoplus F_iP$ in $D(A, \\text{d})$ (by", "Derived Categories, Lemma \\ref{derived-lemma-split}).", "Next, we use that $P$ is compact again to see that the map", "$E \\to \\bigoplus F_iP$ factors through $\\bigoplus_{i = 1, \\ldots, n} F_iP$", "for some $n$. In other words, the identity on $E$ factors through", "$\\bigoplus_{i = 1, \\ldots, n} F_iP$. By", "Lemma \\ref{lemma-factor-through-nicer}", "we see that the identity of $E$ factors as $E \\to P \\to E$", "where $P$ is as in part (2) of the statement of the lemma.", "In other words, we have proven that (1) implies (2).", "\\medskip\\noindent", "Assume (2). By", "Derived Categories, Lemma \\ref{derived-lemma-compact-objects-subcategory}", "it suffices to show that $P$ gives a compact object. Observe that", "$P$ has property (P), hence we have", "$$", "\\Hom_{D(A, \\text{d})}(P, M) = \\Hom_{K(A, \\text{d})}(P, M)", "$$", "for any differential graded module $M$ by Lemma \\ref{lemma-hom-derived}.", "As direct sums in $D(A, \\text{d})$ are given by direct sums of", "graded modules (Lemma \\ref{lemma-derived-products}) we reduce", "to showing that $\\Hom_{K(A, \\text{d})}(P, M)$ commutes with direct", "sums. Using that $K(A, \\text{d})$ is a triangulated category,", "that $\\Hom$ is a cohomological functor in the first", "variable, and the filtration on $P$, we reduce to the case that", "$P$ is a finite direct sum of shifts of $A$. Thus we reduce to", "the case $P = A[k]$ which is clear." ], "refs": [ "dga-lemma-resolve", "dga-lemma-property-P-sequence", "derived-lemma-composition-zero", "derived-lemma-split", "dga-lemma-factor-through-nicer", "derived-lemma-compact-objects-subcategory", "dga-lemma-hom-derived", "dga-lemma-derived-products" ], "ref_ids": [ 13062, 13059, 1757, 1766, 13120, 1940, 13068, 13069 ] } ], "ref_ids": [] }, { "id": 13133, "type": "theorem", "label": "dga-proposition-rickard", "categories": [ "dga" ], "title": "dga-proposition-rickard", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be", "differential graded $R$-algebras. Let $F : D(A, \\text{d}) \\to D(B, \\text{d})$", "be an $R$-linear equivalence of triangulated categories. Assume that", "\\begin{enumerate}", "\\item $A = H^0(A)$, and", "\\item $B$ is K-flat as a complex of $R$-modules.", "\\end{enumerate}", "Then there exists an $(A, B)$-bimodule $N$ as in", "Lemma \\ref{lemma-tilting-equivalence}." ], "refs": [ "dga-lemma-tilting-equivalence" ], "proofs": [ { "contents": [ "As in Remark \\ref{remark-lift-equivalence-to-dga} above, we set $N = F(A)$", "in $D(B, \\text{d})$. We may assume that $N$ is a differential graded", "$B$-module with property (P). Set", "$$", "(E, \\text{d}) = \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, N)", "$$", "Then $H^0(E) = A$ and $H^k(E) = 0$ for $k \\not = 0$ by", "Lemma \\ref{lemma-hom-derived}.", "Moreover, by the discussion in Remark \\ref{remark-lift-equivalence-to-dga}", "and by Lemma \\ref{lemma-tilting-equivalence}", "we see that $N$ as a $(E, B)$-bimodule induces an", "equivalence $- \\otimes_E^\\mathbf{L} N : D(E, \\text{d}) \\to D(B, \\text{d})$.", "Let $E' \\subset E$ be the differential graded $R$-subalgebra", "with", "$$", "(E')^i = \\left\\{", "\\begin{matrix}", "E^i & \\text{if }i < 0 \\\\", "\\Ker(E^0 \\to E^1) & \\text{if }i = 0 \\\\", "0 & \\text{if }i > 0", "\\end{matrix}", "\\right.", "$$", "Then there are quasi-isomorphisms of differential graded", "algebras $(A, \\text{d}) \\leftarrow (E', \\text{d}) \\rightarrow (E, \\text{d})$.", "Thus we obtain equivalences", "$$", "D(A, \\text{d}) \\leftarrow D(E', \\text{d}) \\rightarrow D(E, \\text{d})", "\\rightarrow D(B, \\text{d})", "$$", "by Lemma \\ref{lemma-qis-equivalence}.", "Note that the quasi-inverse $D(A, \\text{d}) \\to D(E', \\text{d})$", "of the left vertical arrow is given", "by $M \\mapsto M \\otimes_A^\\mathbf{L} A$ where $A$ is viewed as a", "$(A, E')$-bimodule, see Example \\ref{example-map-hom-tensor}.", "On the other hand the functor $D(E', \\text{d}) \\to D(B, \\text{d})$ is given by", "$M \\mapsto M \\otimes_{E'}^\\mathbf{L} N$ where $N$ is as above.", "We conclude by Lemma \\ref{lemma-compose-tensor-functors}." ], "refs": [ "dga-remark-lift-equivalence-to-dga", "dga-lemma-hom-derived", "dga-remark-lift-equivalence-to-dga", "dga-lemma-tilting-equivalence", "dga-lemma-qis-equivalence", "dga-lemma-compose-tensor-functors" ], "ref_ids": [ 13169, 13068, 13169, 13124, 13123, 13113 ] } ], "ref_ids": [ 13124 ] }, { "id": 13173, "type": "theorem", "label": "spaces-more-groupoids-lemma-first-order-structure-c", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-first-order-structure-c", "contents": [ "The map $I/I^2 \\to J/J^2$ induced by $c$ is the composition", "$$", "I/I^2 \\xrightarrow{(1, 1)} I/I^2 \\oplus I/I^2 \\to J/J^2", "$$", "where the second arrow comes from the equality", "$J = (I \\otimes B + B \\otimes I)C$.", "The map $i : B \\to B$ induces the map $-1 : I/I^2 \\to I/I^2$." ], "refs": [], "proofs": [ { "contents": [ "To describe a local homomorphism from $C$ to another", "henselian local ring it is enough to say what happens", "to elements of the form $b_1 \\otimes b_2$ by", "Algebra, Lemma \\ref{algebra-lemma-henselian-functorial}", "for example. Keeping this in mind we have the two canonical maps", "$$", "e_2 : C \\to B,\\ b_1 \\otimes b_2 \\mapsto b_1s(e(b_2)),\\quad", "e_1 : C \\to B,\\ b_1 \\otimes b_2 \\mapsto t(e(b_1))b_2", "$$", "corresponding to the embeddings", "$R \\to R \\times_{s, U, t} R$ given by", "$r \\mapsto (r, e(s(r)))$ and $r \\mapsto (e(t(r)), r)$.", "These maps define maps $J/J^2 \\to I/I^2$ which jointly", "give an inverse to the map $I/I^2 \\oplus I/I^2 \\to J/J^2$", "of the lemma. Thus to prove statement we only have to show", "that $e_1 \\circ c : B \\to B$ and $e_2 \\circ c : B \\to B$", "are the identity maps. This follows from the fact that both", "compositions $R \\to R \\times_{s, U, t} R \\to R$ are identities.", "\\medskip\\noindent", "The statement on $i$ follows from the statement on $c$ and the", "fact that $c \\circ (1, i) = e \\circ t$. Some details omitted." ], "refs": [ "algebra-lemma-henselian-functorial" ], "ref_ids": [ 1297 ] } ], "ref_ids": [] }, { "id": 13174, "type": "theorem", "label": "spaces-more-groupoids-lemma-idenity-on-conormal", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-idenity-on-conormal", "contents": [ "In the situation discussed in this section, let $\\delta \\in \\Gamma_0$", "and $f = t \\circ \\delta : U \\to U$. If $s, t$ are flat, then the", "canonical map $\\mathcal{C}_{U_0/U} \\to \\mathcal{C}_{U_0/U}$ induced by $f$", "(More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-conormal-functorial})", "is the identity map." ], "refs": [ "spaces-more-morphisms-lemma-conormal-functorial" ], "proofs": [ { "contents": [ "To see this we extend the bottom of the diagram (\\ref{equation-pull})", "as follows", "$$", "\\xymatrix{", "Y \\ar[r] \\ar[d] &", "R \\times_{s, U, t} R", "\\ar@<1ex>[r]^-c \\ar@<-1ex>[r]_-{\\text{pr}_0} \\ar[d]_{\\text{pr}_1} &", "R \\ar[r]^t \\ar[d]^s &", "U \\\\", "U \\ar[r]_\\delta &", "R \\ar@<1ex>[r]^s \\ar@<-1ex>[r]_t &", "U", "}", "$$", "where the left square is cartesian and this is our definition", "of $Y$; we will not need to know more about $Y$.", "There is a similar diagram with similar properties obtained by", "base change to $U_0$ everywhere.", "We are trying to show that $\\text{id}_U = s \\circ \\delta$", "and $f = t \\circ \\delta$ induce the same maps on conormal sheaves.", "Since $s$ is flat and surjective, it suffices to prove the same", "thing for the two compositions $a, b : Y \\to R$ along the top row.", "Observe that $a_0 = b_0$ and that one of $a$ and $b$ is an isomorphism", "as we know that $s \\circ \\delta$ is an isomorphism. Therefore", "the two morphisms $a, b : Y \\to R$ are morphisms between", "algebraic spaces flat over $U$ (via the morphism $t : R \\to U$", "and the morphism $t \\circ a = t \\circ b : Y \\to U$).", "This implies what we want. Namely, by the compatibility with compositions in", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-conormal-functorial-more}", "we conclude that both maps", "$a_0^*\\mathcal{C}_{R_0/R} \\to \\mathcal{C}_{Y_0/Y}$", "fit into a commutative diagram", "$$", "\\xymatrix{", "a_0^*\\mathcal{C}_{R_0/R} \\ar[rr] & & \\mathcal{C}_{Y_0/Y} \\\\", "a_0^*t_0^*\\mathcal{C}_{U_0/U} \\ar[u] \\ar@{=}[rr] & &", "(t_0 \\circ a_0)^*\\mathcal{C}_{U_0/U} \\ar[u]", "}", "$$", "whose vertical arrows are isomorphisms by", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-deform}.", "Thus the lemma holds." ], "refs": [ "spaces-more-morphisms-lemma-conormal-functorial-more", "spaces-more-morphisms-lemma-deform" ], "ref_ids": [ 25, 101 ] } ], "ref_ids": [ 24 ] }, { "id": 13175, "type": "theorem", "label": "spaces-more-groupoids-lemma-composition-is-addition", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-composition-is-addition", "contents": [ "The bijection (\\ref{equation-isomorphism}) is an isomorphism", "of groups." ], "refs": [], "proofs": [ { "contents": [ "Let $\\delta_1, \\delta_2 \\in \\Gamma_0$ correspond to $\\theta_1, \\theta_2$", "as above and the composition $\\delta = \\delta_1 \\circ \\delta_2$", "in $\\Gamma_0$ correspond to $\\theta$. We have to show that", "$\\theta = \\theta_1 + \\theta_2$. Recall", "(More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-action-by-derivations})", "that $\\theta_1, \\theta_2, \\theta$ correspond to derivations", "$D_1, D_2, D : e_0^{-1}\\mathcal{O}_{R_0} \\to \\mathcal{C}_{U_0/U}$", "given by $D_1 = \\theta_1 \\circ \\text{d}_{R_0/U_0}$ and so on.", "It suffices to check that $D = D_1 + D_2$.", "\\medskip\\noindent", "We may check equality on stalks.", "Let $\\overline{u}$ be a geometric point of $U$ and let us use the local", "rings $A, B, C$ introduced in Section \\ref{section-local}.", "The morphisms $\\delta_i$ correspond to ring", "maps $\\delta_i : B \\to A$. Let $K \\subset A$ be the ideal of", "square zero such that $A/K = \\mathcal{O}_{U_0, \\overline{u}}$.", "In other words, $K$ is the stalk of $\\mathcal{C}_{U_0/U}$ at $\\overline{u}$.", "The fact that $\\delta_i \\in \\Gamma_0$", "means exactly that $\\delta_i(I) \\subset K$.", "The derivation $D_i$ is just the map $\\delta_i - e : B \\to A$.", "Since $B = s(A) \\oplus I$ we see that $D_i$ is determined by", "its restriction to $I$ and that this is just given by", "$\\delta_i|_I$. Moreover $D_i$ and hence $\\delta_i$ annihilates $I^2$", "because $I = \\Ker(I)$.", "\\medskip\\noindent", "To finish the proof we observe that $\\delta$ corresponds to the", "composition", "$$", "B \\to C =", "(B \\otimes_{s, A, t} B)^h_{\\mathfrak m_B \\otimes B + B \\otimes \\mathfrak m_B}", "\\to A", "$$", "where the first arrow is $c$ and the second arrow is determined", "by the rule", "$b_1 \\otimes b_2 \\mapsto \\delta_2(t(\\delta_1(b_1))) \\delta_2(b_2)$", "as follows from (\\ref{equation-composition}).", "By Lemma \\ref{lemma-first-order-structure-c}", "we see that an element $\\zeta$ of $I$ maps to", "$\\zeta \\otimes 1 + 1 \\otimes \\zeta$ plus higher order terms.", "Hence we conclude that", "$$", "D(\\zeta) = (\\delta_2 \\circ t)\\left(D_1(\\zeta)\\right) + D_2(\\zeta)", "$$", "However, by Lemma \\ref{lemma-idenity-on-conormal}", "the action of $\\delta_2 \\circ t$ on", "$K = \\mathcal{C}_{U_0/U, \\overline{u}}$ is the identity and", "we win." ], "refs": [ "spaces-more-morphisms-lemma-action-by-derivations", "spaces-more-groupoids-lemma-first-order-structure-c", "spaces-more-groupoids-lemma-idenity-on-conormal" ], "ref_ids": [ 97, 13173, 13174 ] } ], "ref_ids": [] }, { "id": 13176, "type": "theorem", "label": "spaces-more-groupoids-lemma-property-invariant", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-property-invariant", "contents": [ "Let $B \\to S$ be as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "Let", "$\\tau \\in \\{fppf, \\linebreak[0] \\etale, \\linebreak[0]", "smooth, \\linebreak[0] syntomic\\}$.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces", "which is $\\tau$-local on the target", "(Descent on Spaces,", "Definition \\ref{spaces-descent-definition-property-morphisms-local}).", "Assume $\\{s : R \\to U\\}$ and $\\{t : R \\to U\\}$ are coverings for the", "$\\tau$-topology. Let $W \\subset U$ be the maximal open subspace such that", "$s^{-1}(W) \\to W$ has property $\\mathcal{P}$.", "Then $W$ is $R$-invariant", "(Groupoids in Spaces,", "Definition \\ref{spaces-groupoids-definition-invariant-open})." ], "refs": [ "spaces-descent-definition-property-morphisms-local", "spaces-groupoids-definition-invariant-open" ], "proofs": [ { "contents": [ "The existence and properties of the open $W \\subset U$ are described in", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-largest-open-of-the-base}.", "In", "Diagram (\\ref{equation-diagram})", "let $W_1 \\subset R$ be the maximal open subscheme over which the morphism", "$\\text{pr}_1 : R \\times_{s, U, t} R \\to R$ has property $\\mathcal{P}$.", "It follows from the aforementioned", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-largest-open-of-the-base}", "and the assumption that $\\{s : R \\to U\\}$ and $\\{t : R \\to U\\}$ are coverings", "for the $\\tau$-topology that $t^{-1}(W) = W_1 = s^{-1}(W)$ as desired." ], "refs": [ "spaces-descent-lemma-largest-open-of-the-base", "spaces-descent-lemma-largest-open-of-the-base" ], "ref_ids": [ 9379, 9379 ] } ], "ref_ids": [ 9440, 9351 ] }, { "id": 13177, "type": "theorem", "label": "spaces-more-groupoids-lemma-property-G-invariant", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-property-G-invariant", "contents": [ "Let $B \\to S$ be as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "Let $G \\to U$ be its stabilizer group algebraic space.", "Let", "$\\tau \\in \\{fppf, \\linebreak[0] \\etale, \\linebreak[0]", "smooth, \\linebreak[0] syntomic\\}$.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces", "which is $\\tau$-local on the target.", "Assume $\\{s : R \\to U\\}$ and $\\{t : R \\to U\\}$ are coverings for the", "$\\tau$-topology. Let $W \\subset U$ be the maximal open subspace such that", "$G_W \\to W$ has property $\\mathcal{P}$.", "Then $W$ is $R$-invariant (see", "Groupoids in Spaces,", "Definition \\ref{spaces-groupoids-definition-invariant-open})." ], "refs": [ "spaces-groupoids-definition-invariant-open" ], "proofs": [ { "contents": [ "The existence and properties of the open $W \\subset U$ are described in", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-largest-open-of-the-base}.", "The morphism", "$$", "G \\times_{U, t} R \\longrightarrow R \\times_{s, U} G, \\quad", "(g, r) \\longmapsto (r, r^{-1} \\circ g \\circ r)", "$$", "is an isomorphism of algebraic spaces over $R$ (where $\\circ$ denotes", "composition in the groupoid). Hence $s^{-1}(W) = t^{-1}(W)$ by the", "properties of $W$ proved in the aforementioned", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-largest-open-of-the-base}." ], "refs": [ "spaces-descent-lemma-largest-open-of-the-base", "spaces-descent-lemma-largest-open-of-the-base" ], "ref_ids": [ 9379, 9379 ] } ], "ref_ids": [ 9351 ] }, { "id": 13178, "type": "theorem", "label": "spaces-more-groupoids-lemma-two-fibres", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-two-fibres", "contents": [ "Let $B \\to S$ be as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "Let $K$ be a field and let $r, r' : \\Spec(K) \\to R$", "be morphisms such that $t \\circ r = t \\circ r' : \\Spec(K) \\to U$.", "Set $u = s \\circ r$, $u' = s \\circ r'$ and denote", "$F_u = \\Spec(K) \\times_{u, U, s} R$ and", "$F_{u'} = \\Spec(K) \\times_{u', U, s} R$ the fibre products.", "Then $F_u \\cong F_{u'}$ as algebraic spaces over $K$." ], "refs": [], "proofs": [ { "contents": [ "We use the properties and the existence of", "Diagram (\\ref{equation-diagram}).", "There exists a morphism $\\xi : \\Spec(K) \\to R \\times_{s, U, t} R$", "with $\\text{pr}_0 \\circ \\xi = r$ and $c \\circ \\xi = r'$.", "Let $\\tilde r = \\text{pr}_1 \\circ \\xi : \\Spec(K) \\to R$.", "Then looking at the bottom two squares of", "Diagram (\\ref{equation-diagram})", "we see that both $F_u$ and $F_{u'}$ are identified with the algebraic space", "$\\Spec(K) \\times_{\\tilde r, R, \\text{pr}_1} (R \\times_{s, U, t} R)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13179, "type": "theorem", "label": "spaces-more-groupoids-lemma-restrict-preserves-type", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-restrict-preserves-type", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "Let $g : U' \\to U$ be a morphism of algebraic spaces over $B$.", "Let $(U', R', s', t', c')$ be the restriction of", "$(U, R, s, t, c)$ via $g$.", "\\begin{enumerate}", "\\item If $s, t$ are locally of finite type and $g$ is locally of finite", "type, then $s', t'$ are locally of finite type.", "\\item If $s, t$ are locally of finite presentation and $g$ is locally of finite", "presentation, then $s', t'$ are locally of finite presentation.", "\\item If $s, t$ are flat and $g$ is flat, then $s', t'$ are flat.", "\\item Add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The property of being locally of finite type is stable under composition", "and arbitrary base change, see", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-composition-finite-type} and", "\\ref{spaces-morphisms-lemma-base-change-finite-type}.", "Hence (1) is clear from Diagram (\\ref{equation-restriction}).", "For the other cases, see", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-composition-finite-presentation},", "\\ref{spaces-morphisms-lemma-base-change-finite-presentation},", "\\ref{spaces-morphisms-lemma-composition-flat}, and", "\\ref{spaces-morphisms-lemma-base-change-flat}." ], "refs": [ "spaces-morphisms-lemma-composition-finite-type", "spaces-morphisms-lemma-base-change-finite-type", "spaces-morphisms-lemma-composition-finite-presentation", "spaces-morphisms-lemma-base-change-finite-presentation", "spaces-morphisms-lemma-composition-flat", "spaces-morphisms-lemma-base-change-flat" ], "ref_ids": [ 4814, 4815, 4839, 4840, 4852, 4853 ] } ], "ref_ids": [] }, { "id": 13180, "type": "theorem", "label": "spaces-more-groupoids-lemma-groupoid-on-field-open-multiplication", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-groupoid-on-field-open-multiplication", "contents": [ "In", "Situation \\ref{situation-groupoid-on-field}", "the composition morphism $c : R \\times_{s, U, t} R \\to R$ is flat and", "universally open.", "In", "Situation \\ref{situation-group-over-field}", "the group law $m : G \\times_k G \\to G$ is flat and", "universally open." ], "refs": [], "proofs": [ { "contents": [ "The composition is isomorphic to the projection map", "$\\text{pr}_1 : R \\times_{t, U, t} R \\to R$ by", "Diagram (\\ref{equation-pull}).", "The projection is flat as a base change of the flat morphism $t$", "and open by", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-space-over-field-universally-open}.", "The second assertion follows immediately from the first because", "$m$ matches $c$ in (\\ref{equation-groupoid-from-group})." ], "refs": [ "spaces-morphisms-lemma-space-over-field-universally-open" ], "ref_ids": [ 4732 ] } ], "ref_ids": [] }, { "id": 13181, "type": "theorem", "label": "spaces-more-groupoids-lemma-group-scheme-over-field-separated", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-group-scheme-over-field-separated", "contents": [ "In Situation \\ref{situation-groupoid-on-field}", "assume $R$ is a decent space. Then $R$ is a separated algebraic space.", "In Situation \\ref{situation-group-over-field} assume that", "$G$ is a decent algebraic space. Then $G$ is separated algebraic space." ], "refs": [], "proofs": [ { "contents": [ "We first prove the second assertion. By Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-group-scheme-separated}", "we have to show that $e : S \\to G$ is a closed immersion.", "This follows from Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-finite-residue-field-extension-finite}.", "\\medskip\\noindent", "Next, we prove the first assertion. To do this we may replace $B$ by $S$.", "By the paragraph above the stabilizer group scheme $G \\to U$ is separated. By", "Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-diagonal}", "the morphism $j = (t, s) : R \\to U \\times_S U$ is separated.", "As $U$ is the spectrum of a field the scheme", "$U \\times_S U$ is affine (by the construction of fibre products in", "Schemes, Section \\ref{schemes-section-fibre-products}).", "Hence $R$ is separated, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-separated-over-separated}." ], "refs": [ "spaces-groupoids-lemma-group-scheme-separated", "decent-spaces-lemma-finite-residue-field-extension-finite", "spaces-groupoids-lemma-diagonal", "spaces-morphisms-lemma-separated-over-separated" ], "ref_ids": [ 9289, 9509, 9338, 4719 ] } ], "ref_ids": [] }, { "id": 13182, "type": "theorem", "label": "spaces-more-groupoids-lemma-restrict-groupoid-on-field", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-restrict-groupoid-on-field", "contents": [ "In", "Situation \\ref{situation-groupoid-on-field}.", "Let $k \\subset k'$ be a field extension, $U' = \\Spec(k')$", "and let $(U', R', s', t', c')$ be the restriction of", "$(U, R, s, t, c)$ via $U' \\to U$. In the defining diagram", "$$", "\\xymatrix{", "R' \\ar[d] \\ar[r] \\ar@/_3pc/[dd]_{t'} \\ar@/^1pc/[rr]^{s'} \\ar@{..>}[rd] &", "R \\times_{s, U} U' \\ar[r] \\ar[d] &", "U' \\ar[d] \\\\", "U' \\times_{U, t} R \\ar[d] \\ar[r] &", "R \\ar[r]^s \\ar[d]_t &", "U \\\\", "U' \\ar[r] &", "U", "}", "$$", "all the morphisms are surjective, flat, and universally open.", "The dotted arrow $R' \\to R$ is in addition affine." ], "refs": [], "proofs": [ { "contents": [ "The morphism $U' \\to U$ equals $\\Spec(k') \\to \\Spec(k)$,", "hence is affine, surjective and flat. The morphisms $s, t : R \\to U$", "and the morphism $U' \\to U$ are universally open by", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open}.", "Since $R$ is not empty and $U$ is the spectrum of a field the morphisms", "$s, t : R \\to U$ are surjective and flat. Then you conclude by using", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-base-change-surjective},", "\\ref{spaces-morphisms-lemma-composition-surjective},", "\\ref{spaces-morphisms-lemma-composition-open},", "\\ref{spaces-morphisms-lemma-base-change-affine},", "\\ref{spaces-morphisms-lemma-composition-affine},", "\\ref{spaces-morphisms-lemma-base-change-flat}, and", "\\ref{spaces-morphisms-lemma-composition-flat}." ], "refs": [ "morphisms-lemma-scheme-over-field-universally-open", "spaces-morphisms-lemma-base-change-surjective", "spaces-morphisms-lemma-composition-surjective", "spaces-morphisms-lemma-composition-open", "spaces-morphisms-lemma-base-change-affine", "spaces-morphisms-lemma-composition-affine", "spaces-morphisms-lemma-base-change-flat", "spaces-morphisms-lemma-composition-flat" ], "ref_ids": [ 5254, 4727, 4726, 4730, 4800, 4799, 4853, 4852 ] } ], "ref_ids": [] }, { "id": 13183, "type": "theorem", "label": "spaces-more-groupoids-lemma-groupoid-on-field-explain-points", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-groupoid-on-field-explain-points", "contents": [ "In", "Situation \\ref{situation-groupoid-on-field}.", "For any point $r \\in |R|$ there exist", "\\begin{enumerate}", "\\item a field extension $k \\subset k'$ with $k'$ algebraically closed,", "\\item a point $r' : \\Spec(k') \\to R'$ where", "$(U', R', s', t', c')$ is the restriction of $(U, R, s, t, c)$", "via $\\Spec(k') \\to \\Spec(k)$", "\\end{enumerate}", "such that", "\\begin{enumerate}", "\\item the point $r'$ maps to $r$ under the morphism $R' \\to R$, and", "\\item the maps", "$s' \\circ r', t' \\circ r' : \\Spec(k') \\to \\Spec(k')$", "are automorphisms.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let's represent $r$ by a morphism $r : \\Spec(K) \\to R$ for some", "field $K$. To prove the lemma we have to find an algebraically closed", "field $k'$ and a commutative diagram", "$$", "\\xymatrix{", "k' & k' \\ar[l]^1 & \\\\", "k' \\ar[u]^\\tau & K \\ar[lu]^\\sigma & k \\ar[l]^-s \\ar[lu]_i \\\\", "& k \\ar[lu]^i \\ar[u]_t", "}", "$$", "where $s, t : k \\to K$ are the field maps coming from", "$s \\circ r$ and $t \\circ r$. In the proof of", "More on Groupoids,", "Lemma \\ref{more-groupoids-lemma-groupoid-on-field-explain-points}", "it is shown how to construct such a diagram." ], "refs": [ "more-groupoids-lemma-groupoid-on-field-explain-points" ], "ref_ids": [ 2472 ] } ], "ref_ids": [] }, { "id": 13184, "type": "theorem", "label": "spaces-more-groupoids-lemma-groupoid-on-field-move-point", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-groupoid-on-field-move-point", "contents": [ "In", "Situation \\ref{situation-groupoid-on-field}.", "If $r : \\Spec(k) \\to R$ is a morphism such that", "$s \\circ r, t \\circ r$ are automorphisms of $\\Spec(k)$, then the map", "$$", "R \\longrightarrow R, \\quad", "x \\longmapsto c(r, x)", "$$", "is an automorphism $R \\to R$ which maps $e$ to $r$." ], "refs": [], "proofs": [ { "contents": [ "Proof is identical to the proof of", "More on Groupoids,", "Lemma \\ref{more-groupoids-lemma-groupoid-on-field-move-point}." ], "refs": [ "more-groupoids-lemma-groupoid-on-field-move-point" ], "ref_ids": [ 2473 ] } ], "ref_ids": [] }, { "id": 13185, "type": "theorem", "label": "spaces-more-groupoids-lemma-groupoid-on-field-geometrically-irreducible", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-groupoid-on-field-geometrically-irreducible", "contents": [ "In", "Situation \\ref{situation-groupoid-on-field}", "the algebraic space $R$ is geometrically unibranch. In", "Situation \\ref{situation-group-over-field}", "the algebraic space $G$ is geometrically unibranch." ], "refs": [], "proofs": [ { "contents": [ "Let $r \\in |R|$. We have to show that $R$ is geometrically unibranch", "at $r$. Combining", "Lemma \\ref{lemma-restrict-groupoid-on-field}", "with", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-descend-unibranch}", "we see that it suffices to prove this in case $k$ is algebraically closed", "and $r$ comes from a morphism $r : \\Spec(k) \\to R$ such that", "$s \\circ r$ and $t \\circ r$", "are automorphisms of $\\Spec(k)$. By", "Lemma \\ref{lemma-groupoid-on-field-move-point}", "we reduce to the case that $r = e$ is the identity of $R$ and $k$ is", "algebraically closed.", "\\medskip\\noindent", "Assume $r = e$ and $k$ is algebraically closed. Let", "$A = \\mathcal{O}_{R, e}$ be the \\'etale local ring of", "$R$ at $e$ and let", "$C = \\mathcal{O}_{R \\times_{s, U, t} R, (e, e)}$", "be the \\'etale local ring of $R \\times_{s, U, t} R$ at $(e, e)$.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-minimal-primes-tensor-strictly-henselian}", "the minimal prime ideals $\\mathfrak q$ of $C$ correspond $1$-to-$1$", "to pairs of minimal primes $\\mathfrak p, \\mathfrak p' \\subset A$.", "On the other hand, the composition law induces a flat ring map", "$$", "\\xymatrix{", "A \\ar[r]_{c^\\sharp} & C & \\mathfrak q \\\\", "& A \\otimes_{s^\\sharp, k, t^\\sharp} A \\ar[u] &", "\\mathfrak p \\otimes A + A \\otimes \\mathfrak p' \\ar@{|}[u]", "}", "$$", "Note that $(c^\\sharp)^{-1}(\\mathfrak q)$ contains both $\\mathfrak p$ and", "$\\mathfrak p'$ as the diagrams", "$$", "\\xymatrix{", "A \\ar[r]_{c^\\sharp} & C \\\\", "A \\otimes_{s^\\sharp, k} k \\ar[u] &", "A \\otimes_{s^\\sharp, k, t^\\sharp} A \\ar[l]_{1 \\otimes e^\\sharp} \\ar[u]", "}", "\\quad\\quad", "\\xymatrix{", "A \\ar[r]_{c^\\sharp} & C \\\\", "k \\otimes_{k, t^\\sharp} A \\ar[u] &", "A \\otimes_{s^\\sharp, k, t^\\sharp} A \\ar[l]_{e^\\sharp \\otimes 1} \\ar[u]", "}", "$$", "commute by (\\ref{equation-diagram}).", "Since $c^\\sharp$ is flat (as $c$ is a flat morphism by", "Lemma \\ref{lemma-groupoid-on-field-open-multiplication}),", "we see that $(c^\\sharp)^{-1}(\\mathfrak q)$ is a minimal prime", "of $A$. Hence $\\mathfrak p = (c^\\sharp)^{-1}(\\mathfrak q) = \\mathfrak p'$." ], "refs": [ "spaces-more-groupoids-lemma-restrict-groupoid-on-field", "spaces-descent-lemma-descend-unibranch", "spaces-more-groupoids-lemma-groupoid-on-field-move-point", "more-algebra-lemma-minimal-primes-tensor-strictly-henselian", "spaces-more-groupoids-lemma-groupoid-on-field-open-multiplication" ], "ref_ids": [ 13182, 9373, 13184, 10471, 13180 ] } ], "ref_ids": [] }, { "id": 13186, "type": "theorem", "label": "spaces-more-groupoids-lemma-groupoid-on-field-locally-finite-type-dimension", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-groupoid-on-field-locally-finite-type-dimension", "contents": [ "In", "Situation \\ref{situation-groupoid-on-field}", "assume $s, t$ are locally of finite type.", "For all $r \\in |R|$", "\\begin{enumerate}", "\\item $\\dim(R) = \\dim_r(R)$,", "\\item the transcendence degree of $r$ over $\\Spec(k)$", "via $s$ equals the transcendence degree of $r$ over $\\Spec(k)$", "via $t$, and", "\\item if the transcendence degree mentioned in (2) is $0$, then", "$\\dim(R) = \\dim(\\mathcal{O}_{R, \\overline{r}})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $r \\in |R|$. Denote $\\text{trdeg}(r/_{\\!\\! s}k)$ the transcendence", "degree of $r$ over $\\Spec(k)$ via $s$. Choose an \\'etale morphism", "$\\varphi : V \\to R$ where $V$ is a scheme and $v \\in V$ mapping to $r$.", "Using the definitions mentioned above the lemma we see that", "$$", "\\dim_r(R) = \\dim_v(V) =", "\\dim(\\mathcal{O}_{V, v}) + \\text{trdeg}_{s(k)}(\\kappa(v)) =", "\\dim(\\mathcal{O}_{R, \\overline{r}}) + \\text{trdeg}(r/_{\\!\\! s}k)", "$$", "and similarly for $t$ (the second equality by", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}).", "Hence we see that $\\text{trdeg}(r/_{\\!\\! s}k) = \\text{trdeg}(r/_{\\!\\! t}k)$,", "i.e., (2) holds.", "\\medskip\\noindent", "Let $k \\subset k'$ be a field extension. Note that the restriction $R'$", "of $R$ to $\\Spec(k')$ (see", "Lemma \\ref{lemma-restrict-groupoid-on-field})", "is obtained from $R$ by two base changes by morphisms of fields. Thus", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}", "shows the dimension of $R$ at a point is unchanged by this operation.", "Hence in order to prove (1) we may assume, by", "Lemma \\ref{lemma-groupoid-on-field-explain-points},", "that $r$ is represented by a morphism $r : \\Spec(k) \\to R$ such", "that both $s \\circ r$ and $t \\circ r$ are automorphisms of $\\Spec(k)$.", "In this case there exists an automorphism $R \\to R$ which maps $r$ to $e$", "(Lemma \\ref{lemma-groupoid-on-field-move-point}).", "Hence we see that $\\dim_r(R) = \\dim_e(R)$ for any $r$. By definition this", "means that $\\dim_r(R) = \\dim(R)$.", "\\medskip\\noindent", "Part (3) is a formal consequence of the results obtained in the discussion", "above." ], "refs": [ "morphisms-lemma-dimension-fibre-at-a-point", "spaces-more-groupoids-lemma-restrict-groupoid-on-field", "spaces-morphisms-lemma-dimension-fibre-after-base-change", "spaces-more-groupoids-lemma-groupoid-on-field-explain-points", "spaces-more-groupoids-lemma-groupoid-on-field-move-point" ], "ref_ids": [ 5277, 13182, 4872, 13183, 13184 ] } ], "ref_ids": [] }, { "id": 13187, "type": "theorem", "label": "spaces-more-groupoids-lemma-group-over-field-locally-finite-type-dimension", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-group-over-field-locally-finite-type-dimension", "contents": [ "In", "Situation \\ref{situation-group-over-field}", "assume $G$ locally of finite type.", "For all $g \\in |G|$", "\\begin{enumerate}", "\\item $\\dim(G) = \\dim_g(G)$,", "\\item if the transcendence degree of $g$ over $k$ is $0$, then", "$\\dim(G) = \\dim(\\mathcal{O}_{G, \\overline{g}})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Immediate from", "Lemma \\ref{lemma-groupoid-on-field-locally-finite-type-dimension}", "via (\\ref{equation-groupoid-from-group})." ], "refs": [ "spaces-more-groupoids-lemma-groupoid-on-field-locally-finite-type-dimension" ], "ref_ids": [ 13186 ] } ], "ref_ids": [] }, { "id": 13188, "type": "theorem", "label": "spaces-more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer", "contents": [ "In", "Situation \\ref{situation-groupoid-on-field}", "assume $s, t$ are locally of finite type.", "Let", "$G = \\Spec(k)", "\\times_{\\Delta, \\Spec(k) \\times_B \\Spec(k), t \\times s} R$", "be the stabilizer group algebraic space.", "Then we have $\\dim(R) = \\dim(G)$." ], "refs": [], "proofs": [ { "contents": [ "Since $G$ and $R$ are equidimensional (see", "Lemmas \\ref{lemma-groupoid-on-field-locally-finite-type-dimension} and", "\\ref{lemma-group-over-field-locally-finite-type-dimension})", "it suffices to prove that $\\dim_e(R) = \\dim_e(G)$. Let $V$ be an affine scheme,", "$v \\in V$, and let $\\varphi : V \\to R$ be an \\'etale morphism of schemes", "such that $\\varphi(v) = e$. Note that $V$ is a Noetherian scheme as", "$s \\circ \\varphi$ is locally of finite type as a composition of morphisms", "locally of finite type and as $V$ is quasi-compact (use", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-composition-finite-type},", "\\ref{spaces-morphisms-lemma-etale-locally-finite-presentation}, and", "\\ref{spaces-morphisms-lemma-finite-presentation-finite-type}", "and", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}).", "Hence $V$ is locally connected (see", "Properties, Lemma \\ref{properties-lemma-Noetherian-topology}", "and", "Topology, Lemma \\ref{topology-lemma-locally-Noetherian-locally-connected}).", "Thus we may replace $V$ by the connected component containing $v$ (it", "is still affine as it is an open and closed subscheme of $V$).", "Set $T = V_{red}$ equal to the reduction of $V$. Consider the two", "morphisms $a, b : T \\to \\Spec(k)$ given by", "$a = s \\circ \\varphi|_T$ and $b = t \\circ \\varphi|_T$. Note that", "$a, b$ induce the same field map $k \\to \\kappa(v)$ because $\\varphi(v) = e$!", "Let $k_a \\subset \\Gamma(T, \\mathcal{O}_T)$ be the integral closure of", "$a^\\sharp(k) \\subset \\Gamma(T, \\mathcal{O}_T)$. Similarly, let", "$k_b \\subset \\Gamma(T, \\mathcal{O}_T)$ be the integral closure of", "$b^\\sharp(k) \\subset \\Gamma(T, \\mathcal{O}_T)$. By", "Varieties, Proposition \\ref{varieties-proposition-unique-base-field}", "we see that $k_a = k_b$. Thus we obtain the following commutative diagram", "$$", "\\xymatrix{", "k \\ar[rd]^a \\ar[rrrd] \\\\", "& k_a = k_b \\ar[r] & \\Gamma(T, \\mathcal{O}_T) \\ar[r] & \\kappa(v) \\\\", "k \\ar[ru]_b \\ar[rrru]", "}", "$$", "As discussed above the long arrows are equal.", "Since $k_a = k_b \\to \\kappa(v)$ is injective we conclude that", "the two morphisms $a$ and $b$ agree. Hence $T \\to R$ factors through $G$.", "It follows that $R_{red} = G_{red}$ in an open neighbourhood of $e$", "which certainly implies that $\\dim_e(R) = \\dim_e(G)$." ], "refs": [ "spaces-more-groupoids-lemma-groupoid-on-field-locally-finite-type-dimension", "spaces-more-groupoids-lemma-group-over-field-locally-finite-type-dimension", "spaces-morphisms-lemma-composition-finite-type", "spaces-morphisms-lemma-etale-locally-finite-presentation", "spaces-morphisms-lemma-finite-presentation-finite-type", "morphisms-lemma-finite-type-noetherian", "properties-lemma-Noetherian-topology", "topology-lemma-locally-Noetherian-locally-connected", "varieties-proposition-unique-base-field" ], "ref_ids": [ 13186, 13187, 4814, 4911, 4842, 5202, 2954, 8223, 11137 ] } ], "ref_ids": [] }, { "id": 13189, "type": "theorem", "label": "spaces-more-groupoids-lemma-group-space-scheme-over-kbar", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-group-space-scheme-over-kbar", "contents": [ "Let $k$ be a field with algebraic closure $\\overline{k}$.", "Let $G$ be a group algebraic space over $k$", "which is separated\\footnote{It is enough to assume $G$ is decent,", "e.g., locally separated or quasi-separated by", "Lemma \\ref{lemma-group-scheme-over-field-separated}.}.", "Then $G_{\\overline{k}}$ is a scheme." ], "refs": [ "spaces-more-groupoids-lemma-group-scheme-over-field-separated" ], "proofs": [ { "contents": [ "By Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-when-scheme-after-base-change}", "it suffices to show that $G_K$ is a scheme for some field", "extension $K/k$. Denote $G_K' \\subset G_K$ the schematic", "locus of $G_K$ as in Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-subscheme}.", "By Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}", "we see that $G_K' \\subset G_K$ is dense open, in particular not empty.", "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to G$. By", "Varieties, Lemma \\ref{varieties-lemma-make-Jacobson}", "if $K$ is an algebraically closed field of large enough", "transcendence degree, then $U_K$ is a Jacobson scheme and", "every closed point of $U_K$ is $K$-rational.", "Hence $G_K'$ has a $K$-rational point and it suffices", "to show that every $K$-rational point of $G_K$ is in $G_K'$.", "If $g \\in G_K(K)$ is a $K$-rational point and $g' \\in G_K'(K)$", "a $K$-rational point in the schematic locus, then we see that", "$g$ is in the image of $G_K'$ under the automorphism", "$$", "G_K \\longrightarrow G_K,\\quad", "h \\longmapsto g(g')^{-1}h", "$$", "of $G_K$. Since automorphisms of $G_K$ as an algebraic space preserve", "$G_K'$, we conclude that $g \\in G_K'$ as desired." ], "refs": [ "spaces-over-fields-lemma-when-scheme-after-base-change", "spaces-properties-lemma-subscheme", "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme", "varieties-lemma-make-Jacobson" ], "ref_ids": [ 12847, 11848, 11917, 10965 ] } ], "ref_ids": [ 13181 ] }, { "id": 13190, "type": "theorem", "label": "spaces-more-groupoids-lemma-group-space-scheme-locally-finite-type-over-k", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-group-space-scheme-locally-finite-type-over-k", "contents": [ "Let $k$ be a field. Let $G$ be a group algebraic space over $k$.", "If $G$ is separated and locally of finite type over $k$,", "then $G$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-group-space-scheme-over-kbar},", "Groupoids, Lemma \\ref{groupoids-lemma-points-in-affine}, and", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-scheme-over-algebraic-closure-enough-affines}." ], "refs": [ "spaces-more-groupoids-lemma-group-space-scheme-over-kbar", "groupoids-lemma-points-in-affine", "spaces-over-fields-lemma-scheme-over-algebraic-closure-enough-affines" ], "ref_ids": [ 13189, 9602, 12851 ] } ], "ref_ids": [] }, { "id": 13191, "type": "theorem", "label": "spaces-more-groupoids-lemma-factor-through-over-open", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-factor-through-over-open", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $f : X \\to Y$ and $g : X \\to Z$ be morphisms of algebraic", "spaces over $B$. Assume", "\\begin{enumerate}", "\\item $Y \\to B$ is separated,", "\\item $g$ is surjective, flat, and locally of finite presentation,", "\\item there is a scheme theoretically dense open $V \\subset Z$", "such that $f|_{g^{-1}(V)} : g^{-1}(V) \\to Y$ factors through $V$.", "\\end{enumerate}", "Then $f$ factors through $g$." ], "refs": [], "proofs": [ { "contents": [ "Set $R = X \\times_Z X$. By (2) we see that $Z = X/R$ as sheaves.", "Also (2) implies that the inverse image of $V$ in $R$ is scheme", "theoretically dense in $R$ (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-flat-morphism-scheme-theoretically-dense-open}).", "The we see that the two compositions", "$R \\to X \\to Y$ are equal by Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-equality-of-morphisms}.", "The lemma follows." ], "refs": [ "spaces-morphisms-lemma-flat-morphism-scheme-theoretically-dense-open", "spaces-morphisms-lemma-equality-of-morphisms" ], "ref_ids": [ 4860, 4791 ] } ], "ref_ids": [] }, { "id": 13192, "type": "theorem", "label": "spaces-more-groupoids-lemma-quotient-power-P1", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-quotient-power-P1", "contents": [ "\\begin{slogan}", "A morphism from a nonempty product of projective lines over a field to", "a separated finite type algebraic space over a field factors as a", "finite morphism after a projection to a product of projective lines.", "\\end{slogan}", "Let $k$ be a field. Let $n \\geq 1$ and let $(\\mathbf{P}^1_k)^n$", "be the $n$-fold self product over $\\Spec(k)$. Let", "$f : (\\mathbf{P}^1_k)^n \\to Z$ be a morphism of algebraic spaces over $k$.", "If $Z$ is separated of finite type over $k$, then $f$ factors as", "$$", "(\\mathbf{P}^1_k)^n \\xrightarrow{projection}", "(\\mathbf{P}^1_k)^m \\xrightarrow{finite} Z.", "$$" ], "refs": [], "proofs": [ { "contents": [ "We may assume $k$ is algebraically closed (details omitted); we only", "do this so we may argue using rational points, but the reader can work", "around this if she/he so desires. In the proof products are over $k$.", "The automorphism group algebraic space of $(\\mathbf{P}^1_k)^n$ contains", "$G = (\\text{GL}_{2, k})^n$. If $C \\subset (\\mathbf{P}^1_k)^n$ is a", "closed subvariety (in particular irreducible over $k$) which is mapped", "to a point, then we can apply", "More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-flat-proper-family-cannot-collapse-fibre}", "to the morphism", "$$", "G \\times C \\to G \\times Z,\\quad (g, c) \\mapsto (g, f(g \\cdot c))", "$$", "over $G$. Hence $g(C)$ is mapped to a point for $g \\in G(k)$", "lying in a Zariski open $U \\subset G$. Suppose", "$x = (x_1, \\ldots, x_n)$, $y = (y_1, \\ldots, y_n)$", "are $k$-valued points of $(\\mathbf{P}^1_k)^n$. Let", "$I \\subset \\{1, \\ldots, n\\}$ be the set of indices $i$", "such that $x_i = y_i$. Then", "$$", "\\{g(x) \\mid g(y) = y,\\ g \\in U(k)\\}", "$$", "is Zariski dense in the fibre of the projection", "$\\pi_I : (\\mathbf{P}^1_k)^n \\to \\prod_{i \\in I} \\mathbf{P}^1_k$", "(exercise). Hence if $x, y \\in C(k)$ are distinct, we conclude", "that $f$ maps the whole fibre of $\\pi_I$ containing $x, y$ to a", "single point. Moreover, the $U(k)$-orbit of $C$ meets a Zariski", "open set of fibres of $\\pi_I$. By Lemma \\ref{lemma-factor-through-over-open}", "the morphism $f$ factors through $\\pi_I$.", "After repeating this process finitely many times we reach", "the stage where all fibres of $f$ over $k$ points are finite.", "In this case $f$ is finite by", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood}", "and the fact that $k$ points are dense in $Z$", "(Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-smooth-separable-closed-points-dense})." ], "refs": [ "spaces-more-morphisms-lemma-flat-proper-family-cannot-collapse-fibre", "spaces-more-groupoids-lemma-factor-through-over-open", "spaces-more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood", "spaces-over-fields-lemma-smooth-separable-closed-points-dense" ], "ref_ids": [ 175, 13191, 174, 12873 ] } ], "ref_ids": [] }, { "id": 13193, "type": "theorem", "label": "spaces-more-groupoids-lemma-no-nonconstant-morphism-from-P1-to-group", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-no-nonconstant-morphism-from-P1-to-group", "contents": [ "\\begin{slogan}", "No complete rational curves on groups.", "\\end{slogan}", "Let $k$ be a field. Let $G$ be a separated group algebraic space locally", "of finite type over $k$. There does not exist a nonconstant", "morphism $f : \\mathbf{P}^1_k \\to G$ over $\\Spec(k)$." ], "refs": [], "proofs": [ { "contents": [ "Assume $f$ is nonconstant. Consider the morphisms", "$$", "\\mathbf{P}^1_k \\times_{\\Spec(k)} \\ldots \\times_{\\Spec(k)} \\mathbf{P}^1_k", "\\longrightarrow G,", "\\quad (t_1, \\ldots, t_n) \\longmapsto f(g_1) \\ldots f(g_n)", "$$", "where on the right hand side we use multiplication in the group.", "By Lemma \\ref{lemma-quotient-power-P1} and the assumption that $f$", "is nonconstant this morphism is finite onto its image.", "Hence $\\dim(G) \\geq n$ for all $n$, which is impossible by", "Lemma \\ref{lemma-group-over-field-locally-finite-type-dimension}", "and the fact that $G$ is locally of finite type over $k$." ], "refs": [ "spaces-more-groupoids-lemma-quotient-power-P1", "spaces-more-groupoids-lemma-group-over-field-locally-finite-type-dimension" ], "ref_ids": [ 13192, 13187 ] } ], "ref_ids": [] }, { "id": 13194, "type": "theorem", "label": "spaces-more-groupoids-lemma-finite-sheaf", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-finite-sheaf", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Then we have", "\\begin{enumerate}", "\\item The presheaf $(X/Y)_{fin}$ satisfies the sheaf condition for", "the fppf topology.", "\\item If $T$ is an algebraic space over $S$, then there is a", "canonical bijection", "$$", "\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, (X/Y)_{fin})", "=", "\\{(a, Z)\\text{ satisfying \\ref{equation-finite-conditions}}\\}", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be an algebraic space over $S$.", "Let $\\{T_i \\to T\\}$ be an fppf covering (by algebraic spaces).", "Let $s_i = (a_i, Z_i)$ be pairs over $T_i$", "satisfying \\ref{equation-finite-conditions}", "such that we have $s_i|_{T_i \\times_T T_j} = s_j|_{T_i \\times_T T_j}$.", "First, this implies in particular that $a_i$ and $a_j$ define the same", "morphism $T_i \\times_T T_j \\to Y$. By", "Descent on Spaces,", "Lemma \\ref{spaces-descent-lemma-fpqc-universal-effective-epimorphisms}", "we deduce that there exists a unique morphism $a : T \\to Y$", "such that $a_i$ equals the composition $T_i \\to T \\to Y$.", "Second, this implies that $Z_i \\subset T_i \\times_Y X$ are open subspaces", "whose inverse images in $(T_i \\times_T T_j) \\times_Y X$ are equal.", "Since $\\{T_i \\times_Y X \\to T \\times_Y X\\}$ is an fppf covering", "we deduce that there exists a unique open subspace $Z \\subset T \\times_Y X$", "which restricts back to $Z_i$ over $T_i$, see", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-open-fpqc-covering}.", "We claim that the projection $Z \\to T$ is finite.", "This follows as being finite is local for the fpqc topology, see", "Descent on Spaces, Lemma \\ref{spaces-descent-lemma-descending-property-finite}.", "\\medskip\\noindent", "Note that the result of the preceding paragraph in particular implies (1).", "\\medskip\\noindent", "Let $T$ be an algebraic space over $S$. In order to prove (2) we will", "construct mutually inverse maps between the displayed sets. In the", "following when we say ``pair'' we mean a pair satisfying", "conditions \\ref{equation-finite-conditions}.", "\\medskip\\noindent", "Let $v : T \\to (X/Y)_{fin}$ be a natural transformation.", "Choose a scheme $U$ and a surjective \\'etale morphism $p : U \\to T$.", "Then $v(p) \\in (X/Y)_{fin}(U)$ corresponds to a pair $(a_U, Z_U)$", "over $U$. Let $R = U \\times_T U$ with projections $t, s : R \\to U$.", "As $v$ is a transformation of functors we see that the pullbacks of", "$(a_U, Z_U)$ by $s$ and $t$ agree. Hence, since $\\{U \\to T\\}$ is an", "fppf covering, we may apply the result of the first paragraph that", "deduce that there exists a unique pair $(a, Z)$ over $T$.", "\\medskip\\noindent", "Conversely, let $(a, Z)$ be a pair over $T$.", "Let $U \\to T$, $R = U \\times_T U$, and $t, s : R \\to U$ be as", "above. Then the restriction $(a, Z)|_U$ gives rise to a", "transformation of functors $v : h_U \\to (X/Y)_{fin}$ by the", "Yoneda lemma", "(Categories, Lemma \\ref{categories-lemma-yoneda}).", "As the two pullbacks $s^*(a, Z)|_U$ and $t^*(a, Z)|_U$", "are equal, we see that $v$ coequalizes the two maps", "$h_t, h_s : h_R \\to h_U$. Since $T = U/R$ is the fppf quotient sheaf by", "Spaces, Lemma \\ref{spaces-lemma-space-presentation}", "and since $(X/Y)_{fin}$ is an fppf sheaf by (1) we conclude", "that $v$ factors through a map $T \\to (X/Y)_{fin}$.", "\\medskip\\noindent", "We omit the verification that the two constructions above are mutually", "inverse." ], "refs": [ "spaces-descent-lemma-fpqc-universal-effective-epimorphisms", "spaces-descent-lemma-open-fpqc-covering", "spaces-descent-lemma-descending-property-finite", "categories-lemma-yoneda", "spaces-lemma-space-presentation" ], "ref_ids": [ 9367, 9366, 9403, 12203, 8149 ] } ], "ref_ids": [] }, { "id": 13195, "type": "theorem", "label": "spaces-more-groupoids-lemma-finite-open", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-finite-open", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "X' \\ar[rr]_j \\ar[rd] & & X \\ar[ld] \\\\", "& Y", "}", "$$", "of algebraic spaces over $S$. If $j$ is an open immersion, then", "there is a canonical injective map of sheaves", "$j : (X'/Y)_{fin} \\to (X/Y)_{fin}$." ], "refs": [], "proofs": [ { "contents": [ "If $(a, Z)$ is a pair over $T$ for $X'/Y$, then", "$(a, j(Z))$ is a pair over $T$ for $X/Y$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13196, "type": "theorem", "label": "spaces-more-groupoids-lemma-finite-lives-on-locally-quasi-finite-part", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-finite-lives-on-locally-quasi-finite-part", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is", "locally of finite type.", "Let $X' \\subset X$ be the maximal open subspace over which $f$ is", "locally quasi-finite, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part}.", "Then $(X/Y)_{fin} = (X'/Y)_{fin}$." ], "refs": [ "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part" ], "proofs": [ { "contents": [ "Lemma \\ref{lemma-finite-open}", "gives us an injective map $(X'/Y)_{fin} \\to (X/Y)_{fin}$.", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part}", "assures us that formation of $X'$ commutes with base change.", "Hence everything comes down to proving that if", "$Z \\subset X$ is an open subspace such that $f|_Z : Z \\to Y$ is finite,", "then $Z \\subset X'$. This is true because a finite morphism", "is locally quasi-finite, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-quasi-finite}." ], "refs": [ "spaces-more-groupoids-lemma-finite-open", "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part", "spaces-morphisms-lemma-finite-quasi-finite" ], "ref_ids": [ 13195, 4876, 4945 ] } ], "ref_ids": [ 4876 ] }, { "id": 13197, "type": "theorem", "label": "spaces-more-groupoids-lemma-finite-separated", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-finite-separated", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $T$ be an algebraic space over $S$, and let $(a, Z)$ be", "a pair as in \\ref{equation-finite-conditions}.", "If $f$ is separated, then $Z$ is closed in $T \\times_Y X$." ], "refs": [], "proofs": [ { "contents": [ "A finite morphism of algebraic spaces is universally closed by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-proper}.", "Since $f$ is separated so is the morphism $T \\times_Y X \\to T$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-separated}.", "Thus the closedness of $Z$ follows from", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-universally-closed-permanence}." ], "refs": [ "spaces-morphisms-lemma-finite-proper", "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-universally-closed-permanence" ], "ref_ids": [ 4946, 4714, 4920 ] } ], "ref_ids": [] }, { "id": 13198, "type": "theorem", "label": "spaces-more-groupoids-lemma-finite-diagonal", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-finite-diagonal", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The diagonal of $(X/Y)_{fin} \\to Y$", "$$", "(X/Y)_{fin} \\longrightarrow (X/Y)_{fin} \\times_Y (X/Y)_{fin}", "$$", "is representable (by schemes) and an open immersion and the ``absolute''", "diagonal", "$$", "(X/Y)_{fin} \\longrightarrow (X/Y)_{fin} \\times (X/Y)_{fin}", "$$", "is representable (by schemes)." ], "refs": [], "proofs": [ { "contents": [ "The second statement follows from the first as the absolute diagonal", "is the composition of the relative diagonal and a base change", "of the diagonal of $Y$ (which is representable by schemes), see", "Spaces, Section \\ref{spaces-section-representable}.", "To prove the first assertion we have to show the following:", "Given a scheme $T$ and two pairs $(a, Z_1)$ and $(a, Z_2)$ over $T$", "with identical first component", "satisfying \\ref{equation-finite-conditions}", "there is an open subscheme $V \\subset T$ with the following", "property: For any morphism of schemes $h : T' \\to T$ we have", "$$", "h(T') \\subset V \\Leftrightarrow", "\\Big(T' \\times_T Z_1 = T' \\times_T Z_2", "\\text{ as subspaces of }T' \\times_Y X\\Big)", "$$", "Let us construct $V$. Note that $Z_1 \\cap Z_2$ is open in $Z_1$", "and in $Z_2$. Since $\\text{pr}_0|_{Z_i} : Z_i \\to T$ is finite,", "hence proper (see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-proper})", "we see that", "$$", "E =", "\\text{pr}_0|_{Z_1}\\left(Z_1 \\setminus Z_1 \\cap Z_2)\\right)", "\\cup", "\\text{pr}_0|_{Z_2}\\left(Z_2 \\setminus Z_1 \\cap Z_2)\\right)", "$$", "is closed in $T$. Now it is clear that $V = T \\setminus E$ works." ], "refs": [ "spaces-morphisms-lemma-finite-proper" ], "ref_ids": [ 4946 ] } ], "ref_ids": [] }, { "id": 13199, "type": "theorem", "label": "spaces-more-groupoids-lemma-finite-criterion-etale", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-finite-criterion-etale", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Suppose that $U$ is a scheme, $U \\to Y$ is an \\'etale morphism and", "$Z \\subset U \\times_Y X$ is an open subspace finite over $U$.", "Then the induced morphism $U \\to (X/Y)_{fin}$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "This is formal from the description of the diagonal in", "Lemma \\ref{lemma-finite-diagonal}", "but we write it out since it is an important step in the development", "of the theory. We have to check that for any scheme $T$ over $S$ and a morphism", "$T \\to (X/Y)_{fin}$ the projection map", "$$", "T \\times_{(X/Y)_{fin}} U \\longrightarrow T", "$$", "is \\'etale. Note that", "$$", "T \\times_{(X/Y)_{fin}} U", "=", "(X/Y)_{fin} \\times_{((X/Y)_{fin} \\times_Y (X/Y)_{fin})} (T \\times_Y U)", "$$", "Applying the result of", "Lemma \\ref{lemma-finite-diagonal}", "we see that $T \\times_{(X/Y)_{fin}} U$ is represented by an open subscheme of", "$T \\times_Y U$. As the projection $T \\times_Y U \\to T$ is \\'etale by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-base-change-etale}", "we conclude." ], "refs": [ "spaces-more-groupoids-lemma-finite-diagonal", "spaces-more-groupoids-lemma-finite-diagonal", "spaces-morphisms-lemma-base-change-etale" ], "ref_ids": [ 13198, 13198, 4907 ] } ], "ref_ids": [] }, { "id": 13200, "type": "theorem", "label": "spaces-more-groupoids-lemma-finite-pullback", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-finite-pullback", "contents": [ "Let $S$ be a scheme.", "Let", "$$", "\\xymatrix{", "X' \\ar[d] \\ar[r] & X \\ar[d] \\\\", "Y' \\ar[r] & Y", "}", "$$", "be a fibre product square of algebraic spaces over $S$. Then", "$$", "\\xymatrix{", "(X'/Y')_{fin} \\ar[d] \\ar[r] & (X/Y)_{fin} \\ar[d] \\\\", "Y' \\ar[r] & Y", "}", "$$", "is a fibre product square of sheaves on $(\\Sch/S)_{fppf}$." ], "refs": [], "proofs": [ { "contents": [ "It follows immediately from the definitions that", "the sheaf $(X'/Y')_{fin}$ is equal to the sheaf", "$Y' \\times_Y (X/Y)_{fin}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13201, "type": "theorem", "label": "spaces-more-groupoids-lemma-finite-surjective-etale-cover", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-finite-surjective-etale-cover", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "If $f$ is separated and locally quasi-finite, then there exists a", "scheme $U$ \\'etale over $Y$ and a surjective \\'etale morphism", "$U \\to (X/Y)_{fin}$ over $Y$." ], "refs": [], "proofs": [ { "contents": [ "Note that the assertion makes sense by the result of", "Lemma \\ref{lemma-finite-diagonal}", "on the diagonal of $(X/Y)_{fin}$, see", "Spaces, Lemma \\ref{spaces-lemma-representable-diagonal}.", "Let $V$ be a scheme and let $V \\to Y$ be a surjective \\'etale morphism. By", "Lemma \\ref{lemma-finite-pullback}", "the morphism $(V \\times_Y X/V)_{fin} \\to (X/Y)_{fin}$ is", "a base change of the map $V \\to Y$ and hence is surjective and \\'etale, see", "Spaces,", "Lemma \\ref{spaces-lemma-base-change-representable-transformations-property}.", "Hence it suffices to prove the lemma for $(V \\times_Y X/V)_{fin}$.", "(Here we implicitly use that the composition of representable, surjective, and", "\\'etale transformations of functors is again representable, surjective, and", "\\'etale, see", "Spaces, Lemmas \\ref{spaces-lemma-composition-representable-transformations} and", "\\ref{spaces-lemma-composition-representable-transformations-property}, and", "Morphisms, Lemmas \\ref{morphisms-lemma-composition-surjective} and", "\\ref{morphisms-lemma-composition-etale}.)", "Note that the properties of being separated and locally quasi-finite", "are preserved under base change, see", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-base-change-separated} and", "\\ref{spaces-morphisms-lemma-base-change-quasi-finite}.", "Hence $V \\times_Y X \\to V$ is separated and locally quasi-finite as well,", "and by", "Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}", "we see that $V \\times_Y X$ is a scheme as well.", "Thus we may assume that $f : X \\to Y$ is a separated and locally quasi-finite", "morphism of schemes.", "\\medskip\\noindent", "Pick a point $y \\in Y$. Pick $x_1, \\ldots, x_n \\in X$ points", "lying over $y$. Pick an \\'etale neighbourhood $a : (U, u) \\to (Y, y)$ and a", "decomposition", "$$", "U \\times_S X =", "W \\amalg", "\\ \\coprod\\nolimits_{i = 1, \\ldots, n}", "\\ \\coprod\\nolimits_{j = 1, \\ldots, m_j}", "V_{i, j}", "$$", "as in", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant}.", "Pick any subset", "$$", "I \\subset \\{(i, j) \\mid 1 \\leq i \\leq n, \\ 1 \\leq j \\leq m_i\\}.", "$$", "Given these choices we obtain a pair $(a, Z)$ with", "$Z = \\bigcup_{(i, j) \\in I} V_{i, j}$", "which satisfies conditions \\ref{equation-finite-conditions}. In other words", "we obtain a morphism $U \\to (X/Y)_{fin}$. The construction of this morphism", "depends on all the things we picked above, so we should really write", "$$", "U(y, n, x_1, \\ldots, x_n, a, I) \\longrightarrow (X/Y)_{fin}", "$$", "This morphism is \\'etale by Lemma \\ref{lemma-finite-criterion-etale}.", "\\medskip\\noindent", "Claim: The disjoint union of all of these is surjective onto $(X/Y)_{fin}$.", "It is clear that if the claim holds, then the lemma is true.", "\\medskip\\noindent", "To show surjectivity we have to show the following (see", "Spaces, Remark \\ref{spaces-remark-warning}): Given a scheme $T$ over", "$S$, a point $t \\in T$, and a map $T \\to (X/Y)_{fin}$ we can find a datum", "$(y, n, x_1, \\ldots, x_n, a, I)$ as above such that", "$t$ is in the image of the projection map", "$$", "U(y, n, x_1, \\ldots, x_n, a, I) \\times_{(X/Y)_{fin}} T \\longrightarrow T.", "$$", "To prove this we may clearly replace $T$ by", "$\\Spec(\\overline{\\kappa(t)})$ and", "$T \\to (X/Y)_{fin}$ by the composition", "$\\Spec(\\overline{\\kappa(t)}) \\to T \\to (X/Y)_{fin}$.", "In other words, we may assume that $T$ is", "the spectrum of an algebraically closed field.", "\\medskip\\noindent", "Let $T = \\Spec(k)$ be the spectrum of an algebraically closed", "field $k$. The morphism $T \\to (X/Y)_{fin}$ is given by a pair", "$(T \\to Y, Z)$ satisfying conditions \\ref{equation-finite-conditions}.", "Here is a picture:", "$$", "\\xymatrix{", "& Z \\ar[d] \\ar[r] & X \\ar[d] \\\\", "\\Spec(k) \\ar@{=}[r] & T \\ar[r] & Y", "}", "$$", "Let $y \\in Y$ be the image point of $T \\to Y$.", "Since $Z$ is finite over $k$ it has finitely many points.", "Thus there exist finitely many points $x_1, \\ldots, x_n \\in X$", "such that the image of $Z$ in $X$ is contained in $\\{x_1, \\ldots, x_n\\}$.", "Choose $a : (U, u) \\to (Y, y)$ adapted to $y$ and $x_1, \\ldots, x_n$ as", "above, which gives the diagram", "$$", "\\xymatrix{", "W \\amalg", "\\ \\coprod\\nolimits_{i = 1, \\ldots, n}", "\\ \\coprod\\nolimits_{j = 1, \\ldots, m_j}", "V_{i, j} \\ar[d] \\ar[r] &", "X \\ar[d] \\\\", "U \\ar[r] & Y.", "}", "$$", "Since $k$ is algebraically closed and", "$\\kappa(y) \\subset \\kappa(u)$ is finite separable", "we may factor the morphism", "$T = \\Spec(k) \\to Y$ through the morphism", "$u = \\Spec(\\kappa(u)) \\to \\Spec(\\kappa(y)) = y \\subset Y$.", "With this choice we obtain the commutative diagram:", "$$", "\\xymatrix{", "Z \\ar[d] \\ar[r] &", "W \\amalg", "\\ \\coprod\\nolimits_{i = 1, \\ldots, n}", "\\ \\coprod\\nolimits_{j = 1, \\ldots, m_j}", "V_{i, j} \\ar[d] \\ar[r] &", "X \\ar[d] \\\\", "\\Spec(k) \\ar[r] &", "U \\ar[r] & Y", "}", "$$", "We know that the image of the left upper arrow ends up in", "$\\coprod V_{i, j}$. Recall also that $Z$ is an open subscheme", "of $\\Spec(k) \\times_Y X$ by definition of $(X/Y)_{fin}$", "and that the right hand square is a fibre product square.", "Thus we see that", "$$", "Z \\subset", "\\coprod\\nolimits_{i = 1, \\ldots, n}\\ \\coprod\\nolimits_{j = 1, \\ldots, m_j}", "\\Spec(k) \\times_U V_{i, j}", "$$", "is an open subscheme. By construction (see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant})", "each $V_{i, j}$ has a unique point $v_{i, j}$ lying over $u$", "with purely inseparable residue field extension", "$\\kappa(u) \\subset \\kappa(v_{i, j})$. Hence each", "scheme $\\Spec(k) \\times_U V_{i, j}$ has exactly one", "point. Thus we see that", "$$", "Z = \\coprod\\nolimits_{(i, j) \\in I} \\Spec(k) \\times_U V_{i, j}", "$$", "for a unique subset", "$I \\subset \\{(i, j) \\mid 1 \\leq i \\leq n, \\ 1 \\leq j \\leq m_i\\}$.", "Unwinding the definitions this shows that", "$$", "U(y, n, x_1, \\ldots, x_n, a, I) \\times_{(X/Y)_{fin}} T", "$$", "with $I$ as found above is nonempty as desired." ], "refs": [ "spaces-more-groupoids-lemma-finite-diagonal", "spaces-lemma-representable-diagonal", "spaces-more-groupoids-lemma-finite-pullback", "spaces-lemma-base-change-representable-transformations-property", "spaces-lemma-composition-representable-transformations", "spaces-lemma-composition-representable-transformations-property", "morphisms-lemma-composition-surjective", "morphisms-lemma-composition-etale", "spaces-morphisms-lemma-base-change-separated", "spaces-morphisms-lemma-base-change-quasi-finite", "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant", "spaces-more-groupoids-lemma-finite-criterion-etale", "spaces-remark-warning", "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant" ], "ref_ids": [ 13198, 8138, 13200, 8133, 8126, 8132, 5163, 5360, 4714, 4832, 4983, 13896, 13199, 8187, 13896 ] } ], "ref_ids": [] }, { "id": 13202, "type": "theorem", "label": "spaces-more-groupoids-lemma-finite-separated-flat-locally-finite-presentation", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-finite-separated-flat-locally-finite-presentation", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which", "is separated, flat, and locally of finite presentation.", "In this case", "\\begin{enumerate}", "\\item $(X/Y)_{fin} \\to Y$ is separated, representable, and \\'etale, and", "\\item if $Y$ is a scheme, then $(X/Y)_{fin}$ is (representable by) a scheme.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is in particular separated and locally of finite type (see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-finite-presentation-finite-type})", "we see that $(X/Y)_{fin}$ is an algebraic space by", "Proposition \\ref{proposition-finite-algebraic-space}.", "To prove that $(X/Y)_{fin} \\to Y$ is separated we have to show", "the following: Given a scheme $T$ and two pairs $(a, Z_1)$ and $(a, Z_2)$", "over $T$", "with identical first component satisfying \\ref{equation-finite-conditions}", "there is a closed subscheme $V \\subset T$ with the following", "property: For any morphism of schemes $h : T' \\to T$ we have", "$$", "h \\text{ factors through } V \\Leftrightarrow", "\\Big(T' \\times_T Z_1 = T' \\times_T Z_2", "\\text{ as subspaces of }T' \\times_Y X\\Big)", "$$", "In the proof of", "Lemma \\ref{lemma-finite-diagonal}", "we have seen that $V = T' \\setminus E$ is an open subscheme of $T'$", "with closed complement", "$$", "E =", "\\text{pr}_0|_{Z_1}\\left(Z_1 \\setminus Z_1 \\cap Z_2)\\right)", "\\cup", "\\text{pr}_0|_{Z_2}\\left(Z_2 \\setminus Z_1 \\cap Z_2)\\right).", "$$", "Thus everything comes down to showing that $E$ is also open. By", "Lemma \\ref{lemma-finite-separated}", "we see that $Z_1$ and $Z_2$ are closed in $T' \\times_Y X$. Hence", "$Z_1 \\setminus Z_1 \\cap Z_2$ is open in $Z_1$. As $f$ is flat and", "locally of finite presentation, so is $\\text{pr}_0|_{Z_1}$.", "This is true as $Z_1$ is an open subspace of the base change", "$T' \\times_Y X$, and", "Morphisms of Spaces,", "Lemmas \\ref{spaces-morphisms-lemma-base-change-finite-presentation} and", "Lemmas \\ref{spaces-morphisms-lemma-base-change-flat}.", "Hence $\\text{pr}_0|_{Z_1}$ is open, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-fppf-open}.", "Thus $\\text{pr}_0|_{Z_1}\\left(Z_1 \\setminus Z_1 \\cap Z_2)\\right)$ is", "open and it follows that $E$ is open as desired.", "\\medskip\\noindent", "We have already seen that $(X/Y)_{fin} \\to Y$ is \\'etale, see", "Proposition \\ref{proposition-finite-algebraic-space}.", "Hence now we know it is locally quasi-finite (see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-etale-locally-quasi-finite})", "and separated, hence representable by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable}.", "The final assertion is clear (if you like you can use", "Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme})." ], "refs": [ "spaces-morphisms-lemma-finite-presentation-finite-type", "spaces-more-groupoids-proposition-finite-algebraic-space", "spaces-more-groupoids-lemma-finite-diagonal", "spaces-more-groupoids-lemma-finite-separated", "spaces-morphisms-lemma-base-change-finite-presentation", "spaces-morphisms-lemma-base-change-flat", "spaces-morphisms-lemma-fppf-open", "spaces-more-groupoids-proposition-finite-algebraic-space", "spaces-morphisms-lemma-etale-locally-quasi-finite", "spaces-morphisms-lemma-locally-quasi-finite-separated-representable", "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme" ], "ref_ids": [ 4842, 13215, 13198, 13197, 4840, 4853, 4855, 13215, 4908, 4972, 4983 ] } ], "ref_ids": [] }, { "id": 13203, "type": "theorem", "label": "spaces-more-groupoids-lemma-finite-plus-section", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-finite-plus-section", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\sigma : Y \\to X$ be a section of $f$. Consider the", "transformation of functors", "$$", "t : (X/Y, \\sigma)_{fin} \\longrightarrow (X/Y)_{fin}.", "$$", "defined above. Then", "\\begin{enumerate}", "\\item $t$ is representable by open immersions,", "\\item if $f$ is separated, then $t$ is representable by open", "and closed immersions,", "\\item if $(X/Y)_{fin}$ is an algebraic space, then", "$(X/Y, \\sigma)_{fin}$ is an algebraic space and", "an open subspace of $(X/Y)_{fin}$, and", "\\item if $(X/Y)_{fin}$ is a scheme, then $(X/Y, \\sigma)_{fin}$ is an", "open subscheme of it.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Given a pair $(a, Z)$ over $T$ as in", "(\\ref{equation-finite-conditions}) the inverse image of", "$Z$ by $(1_T, \\sigma \\circ a) : T \\to T \\times_Y X$ is the open", "subscheme of $T$ we are looking for." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13204, "type": "theorem", "label": "spaces-more-groupoids-lemma-finite-part-groupoid", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-finite-part-groupoid", "contents": [ "Let $S$ be a scheme.", "Let $B$ be an algebraic space over $S$.", "Let $(U, R, s, t, c, e, i)$ be a groupoid in algebraic spaces over $B$.", "Assume the morphisms $s, t$ are separated and locally of finite type.", "There exists a canonical morphism", "$$", "(U', Z_{univ}, s', t', c', e', i')", "\\longrightarrow", "(U, R, s, t, c, e, i)", "$$", "of groupoids in algebraic spaces over $B$ where", "\\begin{enumerate}", "\\item $g : U' \\to U$ is identified with $(R_s/U, e)_{fin} \\to U$, and", "\\item $Z_{univ} \\subset R \\times_{s, U, g} U'$ is the universal", "open (and closed) subspace finite over $U'$ which contains the base", "change of the unit $e$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13205, "type": "theorem", "label": "spaces-more-groupoids-lemma-strong-splitting", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-strong-splitting", "contents": [ "In Situation \\ref{situation-strong-splitting}", "there exists an algebraic space $U'$, an \\'etale morphism", "$U' \\to U$, and a point $u' : \\Spec(\\kappa(u)) \\to U'$", "lying over $u : \\Spec(\\kappa(u)) \\to U$", "such that the restriction $R' = R|_{U'}$ of $R$ to $U'$", "is strongly split over $u'$." ], "refs": [], "proofs": [ { "contents": [ "Let $f : (U', Z_{univ}, s', t', c') \\to (U, R, s, t, c)$ be as constructed in", "Lemma \\ref{lemma-finite-part-groupoid}.", "Recall that $R' = R \\times_{(U \\times_S U)} (U' \\times_S U')$.", "Thus we get a morphism $(f, t', s') : Z_{univ} \\to R'$ of groupoids", "in algebraic spaces", "$$", "(U', Z_{univ}, s', t', c') \\to (U', R', s', t', c')", "$$", "(by abuse of notation we indicate the morphisms in the two groupoids", "by the same symbols). Now, as $Z_{univ} \\subset R \\times_{s, U, g} U'$ is open", "and $R' \\to R \\times_{s, U, g} U'$ is \\'etale (as a base change", "of $U' \\to U$) we see that $Z_{univ} \\to R'$ is an open immersion.", "By construction the morphisms $s', t' : Z_{univ} \\to U'$ are finite.", "It remains to find the point $u'$ of $U'$.", "\\medskip\\noindent", "We think of $u$ as a morphism $\\Spec(\\kappa(u)) \\to U$ as in", "the statement of the lemma. Set $F_u = R \\times_{s, U} \\Spec(\\kappa(u))$.", "The set $\\{r \\in R : s(r) = u, t(r) = u\\}$ is finite by assumption", "and $F_u \\to \\Spec(\\kappa(u))$ is quasi-finite at each", "of its elements by assumption. Hence we can find a decomposition into", "open and closed subschemes", "$$", "F_u = Z_u \\amalg Rest", "$$", "for some scheme $Z_u$ finite over $\\kappa(u)$ whose support is", "$\\{r \\in R : s(r) = u, t(r) = u\\}$. Note that $e(u) \\in Z_u$.", "Hence by the construction of $U'$ in", "Section \\ref{section-finite-part-groupoid}", "$(u, Z_u)$ defines a $\\Spec(\\kappa(u))$-valued", "point $u'$ of $U'$.", "\\medskip\\noindent", "We still have to show that the set", "$\\{r' \\in |R'| : s'(r') = u', t'(r') = u'\\}$", "is contained in $|Z_{univ}|$.", "Pick any point $r'$ in this set and represent it by a morphism", "$z' : \\Spec(k) \\to R'$. Denote $z : \\Spec(k) \\to R$", "the composition of $z'$ with the map $R' \\to R$.", "Clearly, $z$ defines an element of the set", "$\\{r \\in R : s(r) = u, t(r) = u\\}$.", "Also, the compositions $s \\circ z, t \\circ z : \\Spec(k) \\to U$", "factor through $u$, so we may think of $s \\circ z, t \\circ z$", "as a morphism $\\Spec(k) \\to \\Spec(\\kappa(u))$. Then", "$z' = (z, u' \\circ t \\circ z, u'\\circ s \\circ u)$ as morphisms into", "$R' = R \\times_{(U \\times_S U)} (U' \\times_S U')$.", "Consider the triple", "$$", "(s \\circ z, Z_u \\times_{\\Spec(\\kappa(u)), s \\circ z} \\Spec(k), z)", "$$", "where $Z_u$ is as above. This defines a $\\Spec(k)$-valued point", "of $Z_{univ}$ whose image via $s', t'$ in $U'$ is $u'$ and", "whose image via", "$Z_{univ} \\to R'$ is the point $r'$ by the relationship between", "$z$ and $z'$ mentioned above.", "This finishes the proof." ], "refs": [ "spaces-more-groupoids-lemma-finite-part-groupoid" ], "ref_ids": [ 13204 ] } ], "ref_ids": [] }, { "id": 13206, "type": "theorem", "label": "spaces-more-groupoids-lemma-splitting", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-splitting", "contents": [ "In Situation \\ref{situation-splitting}", "there exists an algebraic space $U'$, an \\'etale morphism", "$U' \\to U$, and a point $u' : \\Spec(\\kappa(u)) \\to U'$", "lying over $u : \\Spec(\\kappa(u)) \\to U$", "such that the restriction $R' = R|_{U'}$ of $R$ to $U'$", "is split over $u'$." ], "refs": [], "proofs": [ { "contents": [ "Let $f : (U', Z_{univ}, s', t', c') \\to (U, R, s, t, c)$ be as constructed in", "Lemma \\ref{lemma-finite-part-groupoid}.", "Recall that $R' = R \\times_{(U \\times_S U)} (U' \\times_S U')$.", "Thus we get a morphism $(f, t', s') : Z_{univ} \\to R'$ of groupoids", "in algebraic spaces", "$$", "(U', Z_{univ}, s', t', c') \\to (U', R', s', t', c')", "$$", "(by abuse of notation we indicate the morphisms in the two groupoids", "by the same symbols). Now, as $Z_{univ} \\subset R \\times_{s, U, g} U'$ is open", "and $R' \\to R \\times_{s, U, g} U'$ is \\'etale (as a base change", "of $U' \\to U$) we see that $Z_{univ} \\to R'$ is an open immersion.", "By construction the morphisms $s', t' : Z_{univ} \\to U'$ are finite.", "It remains to find the point $u'$ of $U'$.", "\\medskip\\noindent", "We think of $u$ as a morphism $\\Spec(\\kappa(u)) \\to U$ as in", "the statement of the lemma. Set $F_u = R \\times_{s, U} \\Spec(\\kappa(u))$.", "Let $G_u \\subset F_u$ be the scheme theoretic fibre of $G \\to U$ over $u$.", "By assumption $G_u$ is finite and $F_u \\to \\Spec(\\kappa(u))$", "is quasi-finite at each point of $G_u$ by assumption.", "Hence we can find a decomposition into open and closed subschemes", "$$", "F_u = Z_u \\amalg Rest", "$$", "for some scheme $Z_u$ finite over $\\kappa(u)$ whose support is $G_u$.", "Note that $e(u) \\in Z_u$. Hence by the construction of $U'$ in", "Section \\ref{section-finite-part-groupoid}", "$(u, Z_u)$ defines a $\\Spec(\\kappa(u))$-valued", "point $u'$ of $U'$.", "\\medskip\\noindent", "We still have to show that the set $\\{g' \\in |G'| : g'\\text{ maps to }u'\\}$", "is contained in $|Z_{univ}|$. Pick any point $g'$ in this set and", "represent it by a morphism $z' : \\Spec(k) \\to G'$. Denote", "$z : \\Spec(k) \\to G$ the composition of $z'$ with the map $G' \\to G$.", "Clearly, $z$ defines a point of $G_u$. In fact, let us write", "$\\tilde u : \\Spec(k) \\to u \\to U$ for the corresponding map to $u$ or $U$.", "Consider the triple", "$$", "(\\tilde u, Z_u \\times_{u, \\tilde u} \\Spec(k), z)", "$$", "where $Z_u$ is as above. This defines a $\\Spec(k)$-valued point", "of $Z_{univ}$ whose image via $s', t'$ in $U'$ is $u'$ and", "whose image via $Z_{univ} \\to R'$ is the point $z'$", "(because the image in $R$ is $z$).", "This finishes the proof." ], "refs": [ "spaces-more-groupoids-lemma-finite-part-groupoid" ], "ref_ids": [ 13204 ] } ], "ref_ids": [] }, { "id": 13207, "type": "theorem", "label": "spaces-more-groupoids-lemma-quasi-splitting", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-quasi-splitting", "contents": [ "In Situation \\ref{situation-quasi-splitting}", "there exists an algebraic space $U'$, an \\'etale morphism", "$U' \\to U$, and a point $u' : \\Spec(\\kappa(u)) \\to U'$", "lying over $u : \\Spec(\\kappa(u)) \\to U$", "such that the restriction $R' = R|_{U'}$ of $R$ to $U'$", "is quasi-split over $u'$." ], "refs": [], "proofs": [ { "contents": [ "Let $f : (U', Z_{univ}, s', t', c') \\to (U, R, s, t, c)$ be as constructed in", "Lemma \\ref{lemma-finite-part-groupoid}.", "Recall that $R' = R \\times_{(U \\times_S U)} (U' \\times_S U')$.", "Thus we get a morphism $(f, t', s') : Z_{univ} \\to R'$ of groupoids", "in algebraic spaces", "$$", "(U', Z_{univ}, s', t', c') \\to (U', R', s', t', c')", "$$", "(by abuse of notation we indicate the morphisms in the two groupoids", "by the same symbols). Now, as $Z_{univ} \\subset R \\times_{s, U, g} U'$ is open", "and $R' \\to R \\times_{s, U, g} U'$ is \\'etale (as a base change", "of $U' \\to U$) we see that $Z_{univ} \\to R'$ is an open immersion.", "By construction the morphisms $s', t' : Z_{univ} \\to U'$ are finite.", "It remains to find the point $u'$ of $U'$.", "\\medskip\\noindent", "We think of $u$ as a morphism $\\Spec(\\kappa(u)) \\to U$ as in", "the statement of the lemma. Set $F_u = R \\times_{s, U} \\Spec(\\kappa(u))$.", "The morphism $F_u \\to \\Spec(\\kappa(u))$ is quasi-finite at $e(u)$", "by assumption. Hence we can find a decomposition into open and closed", "subschemes", "$$", "F_u = Z_u \\amalg Rest", "$$", "for some scheme $Z_u$ finite over $\\kappa(u)$ whose support is $e(u)$.", "Hence by the construction of $U'$ in", "Section \\ref{section-finite-part-groupoid}", "$(u, Z_u)$ defines a $\\Spec(\\kappa(u))$-valued", "point $u'$ of $U'$. To finish the proof we have to show that", "$e'(u') \\in Z_{univ}$ which is clear." ], "refs": [ "spaces-more-groupoids-lemma-finite-part-groupoid" ], "ref_ids": [ 13204 ] } ], "ref_ids": [] }, { "id": 13208, "type": "theorem", "label": "spaces-more-groupoids-lemma-strong-splitting-scheme", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-strong-splitting-scheme", "contents": [ "In Situation \\ref{situation-strong-splitting} assume in addition that", "$s, t$ are flat and locally of finite presentation.", "Then there exists a scheme $U'$, a separated \\'etale morphism", "$U' \\to U$, and a point $u' \\in U'$", "lying over $u$ with $\\kappa(u) = \\kappa(u')$", "such that the restriction $R' = R|_{U'}$ of $R$ to $U'$", "is strongly split over $u'$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the construction of $U'$ in the proof of", "Lemma \\ref{lemma-strong-splitting}", "because in this case $U' = (R_s/U, e)_{fin}$ is a scheme separated over", "$U$ by", "Lemmas \\ref{lemma-finite-separated-flat-locally-finite-presentation} and", "\\ref{lemma-finite-plus-section}." ], "refs": [ "spaces-more-groupoids-lemma-strong-splitting", "spaces-more-groupoids-lemma-finite-separated-flat-locally-finite-presentation" ], "ref_ids": [ 13205, 13202 ] } ], "ref_ids": [] }, { "id": 13209, "type": "theorem", "label": "spaces-more-groupoids-lemma-splitting-scheme", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-splitting-scheme", "contents": [ "In Situation \\ref{situation-splitting} assume in addition that", "$s, t$ are flat and locally of finite presentation.", "Then there exists a scheme $U'$, a separated \\'etale morphism", "$U' \\to U$, and a point $u' \\in U'$", "lying over $u$ with $\\kappa(u) = \\kappa(u')$", "such that the restriction $R' = R|_{U'}$ of $R$ to $U'$", "is split over $u'$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the construction of $U'$ in the proof of", "Lemma \\ref{lemma-splitting}", "because in this case $U' = (R_s/U, e)_{fin}$ is a scheme separated over", "$U$ by", "Lemmas \\ref{lemma-finite-separated-flat-locally-finite-presentation} and", "\\ref{lemma-finite-plus-section}." ], "refs": [ "spaces-more-groupoids-lemma-splitting", "spaces-more-groupoids-lemma-finite-separated-flat-locally-finite-presentation" ], "ref_ids": [ 13206, 13202 ] } ], "ref_ids": [] }, { "id": 13210, "type": "theorem", "label": "spaces-more-groupoids-lemma-quasi-splitting-scheme", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-quasi-splitting-scheme", "contents": [ "In Situation \\ref{situation-quasi-splitting} assume in addition that", "$s, t$ are flat and locally of finite presentation.", "Then there exists a scheme $U'$, a separated \\'etale morphism", "$U' \\to U$, and a point $u' \\in U'$ lying over $u$ with", "$\\kappa(u) = \\kappa(u')$ such that the restriction $R' = R|_{U'}$ of", "$R$ to $U'$ is quasi-split over $u'$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the construction of $U'$ in the proof of", "Lemma \\ref{lemma-quasi-splitting}", "because in this case $U' = (R_s/U, e)_{fin}$ is a scheme separated", "over $U$ by", "Lemmas \\ref{lemma-finite-separated-flat-locally-finite-presentation} and", "\\ref{lemma-finite-plus-section}." ], "refs": [ "spaces-more-groupoids-lemma-quasi-splitting", "spaces-more-groupoids-lemma-finite-separated-flat-locally-finite-presentation" ], "ref_ids": [ 13207, 13202 ] } ], "ref_ids": [] }, { "id": 13211, "type": "theorem", "label": "spaces-more-groupoids-lemma-strong-splitting-affine-scheme", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-strong-splitting-affine-scheme", "contents": [ "In Situation \\ref{situation-strong-splitting} assume in addition that", "$s, t$ are flat and locally of finite presentation and that $U$ is affine.", "Then there exists an affine scheme $U'$, an \\'etale morphism", "$U' \\to U$, and a point $u' \\in U'$ lying over $u$ with", "$\\kappa(u) = \\kappa(u')$ such that the restriction $R' = R|_{U'}$ of", "$R$ to $U'$ is strongly split over $u'$." ], "refs": [], "proofs": [ { "contents": [ "Let $U' \\to U$ and $u' \\in U'$ be the separated \\'etale morphism of schemes", "we found in Lemma \\ref{lemma-strong-splitting-scheme}.", "Let $P \\subset R'$ be the strong splitting of $R'$ over $u'$. By", "More on Groupoids, Lemma \\ref{more-groupoids-lemma-restrict-preserves-type}", "the morphisms $s', t' : R' \\to U'$ are flat and locally of finite presentation.", "They are finite by assumption. Hence $s', t'$ are finite locally", "free, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}.", "In particular $t(s^{-1}(u'))$ is a finite set of points", "$\\{u'_1, u'_2, \\ldots, u'_n\\}$ of $U'$. Choose a quasi-compact open", "$W \\subset U'$ containing each $u'_i$. As $U$ is affine the morphism", "$W \\to U$ is quasi-compact (see", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-affine}).", "The morphism $W \\to U$ is also locally quasi-finite (see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-locally-quasi-finite})", "and separated. Hence by", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-quasi-finite-separated-quasi-affine}", "(a version of Zariski's Main Theorem)", "we conclude that $W$ is quasi-affine. By", "Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-affine}", "we see that $\\{u'_1, \\ldots, u'_n\\}$ are contained in an affine", "open of $U'$. Thus we may apply", "Groupoids, Lemma \\ref{groupoids-lemma-find-invariant-affine}", "to conclude that there exists an affine $P$-invariant open", "$U'' \\subset U'$ which contains $u'$.", "\\medskip\\noindent", "To finish the proof denote $R'' = R|_{U''}$ the restriction of $R$", "to $U''$. This is the same as the restriction of $R'$ to $U''$.", "As $P \\subset R'$ is an open and closed subscheme, so is", "$P|_{U''} \\subset R''$. By construction the open subscheme $U'' \\subset U'$", "is $P$-invariant which means that", "$P|_{U''} = (s'|_P)^{-1}(U'') = (t'|_P)^{-1}(U'')$", "(see discussion in", "Groupoids, Section \\ref{groupoids-section-invariant})", "so the restrictions of $s''$ and $t''$ to $P|_{U''}$ are still finite.", "The sub groupoid scheme $P|_{U''}$ is still a strong splitting of", "$R''$ over $u''$; above we verified (a), (b) and (c) holds as", "$\\{r' \\in R': t'(r') = u', s'(r') = u'\\} =", "\\{r'' \\in R'': t''(r'') = u', s''(r'') = u'\\}$ trivially.", "The lemma is proved." ], "refs": [ "spaces-more-groupoids-lemma-strong-splitting-scheme", "more-groupoids-lemma-restrict-preserves-type", "morphisms-lemma-finite-flat", "schemes-lemma-quasi-compact-affine", "morphisms-lemma-etale-locally-quasi-finite", "more-morphisms-lemma-quasi-finite-separated-quasi-affine", "properties-lemma-ample-finite-set-in-affine", "groupoids-lemma-find-invariant-affine" ], "ref_ids": [ 13208, 2464, 5471, 7697, 5363, 13900, 3062, 9664 ] } ], "ref_ids": [] }, { "id": 13212, "type": "theorem", "label": "spaces-more-groupoids-lemma-splitting-affine-scheme", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-splitting-affine-scheme", "contents": [ "In Situation \\ref{situation-splitting} assume in addition that", "$s, t$ are flat and locally of finite presentation and that $U$ is affine.", "Then there exists an affine scheme $U'$, an \\'etale morphism", "$U' \\to U$, and a point $u' \\in U'$ lying over $u$ with", "$\\kappa(u) = \\kappa(u')$ such that the restriction $R' = R|_{U'}$ of", "$R$ to $U'$ is split over $u'$." ], "refs": [], "proofs": [ { "contents": [ "The proof of this lemma is literally the same as the proof of", "Lemma \\ref{lemma-strong-splitting-affine-scheme}", "except that ``strong splitting'' needs to be replaced by ``splitting''", "(2 times) and that the reference to", "Lemma \\ref{lemma-strong-splitting-scheme}", "needs to be replaced by a reference to", "Lemma \\ref{lemma-splitting-scheme}." ], "refs": [ "spaces-more-groupoids-lemma-strong-splitting-affine-scheme", "spaces-more-groupoids-lemma-strong-splitting-scheme", "spaces-more-groupoids-lemma-splitting-scheme" ], "ref_ids": [ 13211, 13208, 13209 ] } ], "ref_ids": [] }, { "id": 13213, "type": "theorem", "label": "spaces-more-groupoids-lemma-quasi-splitting-affine-scheme", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-lemma-quasi-splitting-affine-scheme", "contents": [ "In Situation \\ref{situation-quasi-splitting} assume in addition that", "$s, t$ are flat and locally of finite presentation and that $U$ is affine.", "Then there exists an affine scheme $U'$, an \\'etale morphism", "$U' \\to U$, and a point $u' \\in U'$ lying over $u$ with", "$\\kappa(u) = \\kappa(u')$ such that the restriction $R' = R|_{U'}$ of", "$R$ to $U'$ is quasi-split over $u'$." ], "refs": [], "proofs": [ { "contents": [ "The proof of this lemma is literally the same as the proof of", "Lemma \\ref{lemma-strong-splitting-affine-scheme}", "except that ``strong splitting'' needs to be replaced by ``quasi-splitting''", "(2 times) and that the reference to", "Lemma \\ref{lemma-strong-splitting-scheme}", "needs to be replaced by a reference to", "Lemma \\ref{lemma-quasi-splitting-scheme}." ], "refs": [ "spaces-more-groupoids-lemma-strong-splitting-affine-scheme", "spaces-more-groupoids-lemma-strong-splitting-scheme", "spaces-more-groupoids-lemma-quasi-splitting-scheme" ], "ref_ids": [ 13211, 13208, 13210 ] } ], "ref_ids": [] }, { "id": 13214, "type": "theorem", "label": "spaces-more-groupoids-proposition-group-space-scheme-over-field", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-proposition-group-space-scheme-over-field", "contents": [ "Let $k$ be a field. Let $G$ be a group algebraic space over $k$.", "If $G$ is separated, then $G$ is a scheme." ], "refs": [], "proofs": [ { "contents": [ "This lemma generalizes", "Lemma \\ref{lemma-group-space-scheme-locally-finite-type-over-k}", "(which covers all cases one cares about in practice).", "The proof is very similar to the proof of", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-scheme-over-algebraic-closure-enough-affines}", "used in the proof of", "Lemma \\ref{lemma-group-space-scheme-locally-finite-type-over-k}", "and we encourage the reader to read that proof first.", "\\medskip\\noindent", "By Lemma \\ref{lemma-group-space-scheme-over-kbar} the base", "change $G_{\\overline{k}}$ is a scheme.", "Let $K/k$ be a purely transcendental extension of very large", "transcendence degree. By Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-scheme-after-purely-transcendental-base-change}", "it suffices to show that $G_K$ is a scheme.", "Let $K^{perf}$ be the perfect closure of $K$. By", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-scheme-after-purely-inseparable-base-change}", "it suffices to show that $G_{K^{perf}}$ is a scheme.", "Let $K \\subset K^{perf} \\subset \\overline{K}$ be the algebraic closure", "of $K$. We may choose an embedding $\\overline{k} \\to \\overline{K}$ over", "$k$, so that $G_{\\overline{K}}$ is the base change of the scheme", "$G_{\\overline{k}}$ by $\\overline{k} \\to \\overline{K}$. By", "Varieties, Lemma \\ref{varieties-lemma-make-Jacobson}", "we see that $G_{\\overline{K}}$ is a Jacobson scheme all of whose", "closed points have residue field $\\overline{K}$.", "\\medskip\\noindent", "Since $G_{\\overline{K}} \\to G_{K^{perf}}$ is surjective, it suffices to", "show that the image $g \\in |G_{K^{perf}}|$ of an arbitrary closed point of", "$G_{\\overline{K}}$ is in the schematic locus of $G_K$.", "In particular, we may represent $g$ by a morphism", "$g : \\Spec(L) \\to G_{K^{perf}}$ where $L/K^{perf}$ is separable algebraic", "(for example we can take $L = \\overline{K}$). Thus the scheme", "\\begin{align*}", "T & = \\Spec(L) \\times_{G_{K^{perf}}} G_{\\overline{K}} \\\\", "& =", "\\Spec(L) \\times_{\\Spec(K^{perf})} \\Spec(\\overline{K}) \\\\", "& =", "\\Spec(L \\otimes_{K^{perf}} \\overline{K})", "\\end{align*}", "is the spectrum of a $\\overline{K}$-algebra which is a filtered colimit", "of algebras which are finite products of copies of $\\overline{K}$.", "Thus by Groupoids, Lemma \\ref{groupoids-lemma-compact-set-in-affine}", "we can find an affine open $W \\subset G_{\\overline{K}}$ containing", "the image of $g_{\\overline{K}} : T \\to G_{\\overline{K}}$.", "\\medskip\\noindent", "Choose a quasi-compact open $V \\subset G_{K^{perf}}$ containing the", "image of $W$. By Spaces over Fields,", "Lemma \\ref{spaces-over-fields-lemma-when-scheme-after-base-change}", "we see that $V_{K'}$ is a scheme for some finite extension $K'/K^{perf}$.", "After enlarging $K'$ we may assume that there exists an affine open", "$U' \\subset V_{K'} \\subset G_{K'}$ whose base change to $\\overline{K}$", "recovers $W$ (use that $V_{\\overline{K}}$ is the limit of the schemes", "$V_{K''}$ for $K' \\subset K'' \\subset \\overline{K}$ finite and use", "Limits, Lemmas \\ref{limits-lemma-descend-opens} and", "\\ref{limits-lemma-limit-affine}). We may assume", "that $K'/K^{perf}$ is a Galois extension (take the normal closure", "Fields, Lemma \\ref{fields-lemma-normal-closure} and use", "that $K^{perf}$ is perfect). Set $H = \\text{Gal}(K'/K^{perf})$.", "By construction the $H$-invariant closed subscheme", "$\\Spec(L) \\times_{G_{K^{perf}}} G_{K'}$ is contained in $U'$.", "By", "Spaces over Fields, Lemmas", "\\ref{spaces-over-fields-lemma-base-change-by-Galois} and", "\\ref{spaces-over-fields-lemma-when-quotient-scheme-at-point} we conclude." ], "refs": [ "spaces-more-groupoids-lemma-group-space-scheme-locally-finite-type-over-k", "spaces-over-fields-lemma-scheme-over-algebraic-closure-enough-affines", "spaces-more-groupoids-lemma-group-space-scheme-locally-finite-type-over-k", "spaces-more-groupoids-lemma-group-space-scheme-over-kbar", "spaces-over-fields-lemma-scheme-after-purely-transcendental-base-change", "spaces-over-fields-lemma-scheme-after-purely-inseparable-base-change", "varieties-lemma-make-Jacobson", "groupoids-lemma-compact-set-in-affine", "spaces-over-fields-lemma-when-scheme-after-base-change", "limits-lemma-descend-opens", "limits-lemma-limit-affine", "fields-lemma-normal-closure", "spaces-over-fields-lemma-base-change-by-Galois", "spaces-over-fields-lemma-when-quotient-scheme-at-point" ], "ref_ids": [ 13190, 12851, 13190, 13189, 12850, 12846, 10965, 9597, 12847, 15041, 15043, 4494, 12848, 12849 ] } ], "ref_ids": [] }, { "id": 13215, "type": "theorem", "label": "spaces-more-groupoids-proposition-finite-algebraic-space", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-proposition-finite-algebraic-space", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which", "is separated and locally of finite type. Then $(X/Y)_{fin}$", "is an algebraic space. Moreover, the morphism", "$(X/Y)_{fin} \\to Y$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-finite-lives-on-locally-quasi-finite-part}", "we may replace $X$ by the open subscheme which is locally quasi-finite", "over $Y$. Hence we may assume that $f$ is separated and locally quasi-finite.", "We will check the three conditions of", "Spaces, Definition \\ref{spaces-definition-algebraic-space}.", "Condition (1) follows from", "Lemma \\ref{lemma-finite-sheaf}.", "Condition (2) follows from", "Lemma \\ref{lemma-finite-diagonal}.", "Finally, condition (3) follows from", "Lemma \\ref{lemma-finite-surjective-etale-cover}.", "Thus $(X/Y)_{fin}$ is an algebraic space.", "Moreover, that lemma shows that there exists a commutative", "diagram", "$$", "\\xymatrix{", "U \\ar[rr] \\ar[rd] & & (X/Y)_{fin} \\ar[ld] \\\\", "& Y", "}", "$$", "with horizontal arrow surjective and \\'etale and south-east arrow", "\\'etale. By", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-etale-local}", "this implies that the south-west arrow is \\'etale as well." ], "refs": [ "spaces-more-groupoids-lemma-finite-lives-on-locally-quasi-finite-part", "spaces-definition-algebraic-space", "spaces-more-groupoids-lemma-finite-sheaf", "spaces-more-groupoids-lemma-finite-diagonal", "spaces-more-groupoids-lemma-finite-surjective-etale-cover", "spaces-properties-lemma-etale-local" ], "ref_ids": [ 13196, 8174, 13194, 13198, 13201, 11856 ] } ], "ref_ids": [] }, { "id": 13221, "type": "theorem", "label": "modules-lemma-abelian", "categories": [ "modules" ], "title": "modules-lemma-abelian", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. The category", "$\\textit{Mod}(\\mathcal{O}_X)$ is an abelian category. Moreover", "a complex", "$$", "\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}", "$$", "is exact at $\\mathcal{G}$ if and only if for all $x \\in X$ the", "complex", "$$", "\\mathcal{F}_x \\to \\mathcal{G}_x \\to \\mathcal{H}_x", "$$", "is exact at $\\mathcal{G}_x$." ], "refs": [], "proofs": [ { "contents": [ "By Homology, Definition \\ref{homology-definition-abelian-category}", "we have to show that image and coimage agree. By Sheaves,", "Lemma \\ref{sheaves-lemma-points-exactness} it is enough to show", "that image and coimage have the same stalk at every $x \\in X$.", "By the constructions of kernels and cokernels above these stalks", "are the coimage and image in the categories of $\\mathcal{O}_{X, x}$-modules.", "Thus we get the result from the fact that the category of modules", "over a ring is abelian." ], "refs": [ "homology-definition-abelian-category", "sheaves-lemma-points-exactness" ], "ref_ids": [ 12137, 14490 ] } ], "ref_ids": [] }, { "id": 13222, "type": "theorem", "label": "modules-lemma-limits-colimits", "categories": [ "modules" ], "title": "modules-lemma-limits-colimits", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "\\begin{enumerate}", "\\item All limits exist in $\\textit{Mod}(\\mathcal{O}_X)$.", "Limits are the same as the corresponding limits of presheaves of", "$\\mathcal{O}_X$-modules (i.e., commute with taking", "sections over opens).", "\\item All colimits exist in $\\textit{Mod}(\\mathcal{O}_X)$.", "Colimits are the sheafification of the corresponding colimit in", "the category of presheaves. Taking colimits commutes with taking", "stalks.", "\\item Filtered colimits are exact.", "\\item Finite direct sums are the same as the corresponding", "finite direct sums of presheaves of $\\mathcal{O}_X$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As $\\textit{Mod}(\\mathcal{O}_X)$ is abelian (Lemma \\ref{lemma-abelian})", "it has all finite limits and colimits", "(Homology, Lemma \\ref{homology-lemma-colimit-abelian-category}).", "Thus the existence of limits and colimits and their description follows from", "the existence of products and coproducts and their description", "(see discussion above) and", "Categories, Lemmas \\ref{categories-lemma-limits-products-equalizers} and", "\\ref{categories-lemma-colimits-coproducts-coequalizers}.", "Since sheafification commutes with taking stalks we see that", "colimits commute with taking stalks. Part (3) signifies that given", "a system $0 \\to \\mathcal{F}_i \\to \\mathcal{G}_i \\to \\mathcal{H}_i \\to 0$", "of exact sequences of $\\mathcal{O}_X$-modules over a directed set $I$", "the sequence $0 \\to \\colim \\mathcal{F}_i \\to \\colim \\mathcal{G}_i \\to", "\\colim \\mathcal{H}_i \\to 0$ is exact as well. Since we can check", "exactness on stalks (Lemma \\ref{lemma-abelian}) this follows from the case", "of modules which is", "Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}.", "We omit the proof of (4)." ], "refs": [ "modules-lemma-abelian", "homology-lemma-colimit-abelian-category", "categories-lemma-limits-products-equalizers", "categories-lemma-colimits-coproducts-coequalizers", "modules-lemma-abelian", "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 13221, 12018, 12213, 12214, 13221, 343 ] } ], "ref_ids": [] }, { "id": 13223, "type": "theorem", "label": "modules-lemma-exactness-pushforward-pullback", "categories": [ "modules" ], "title": "modules-lemma-exactness-pushforward-pullback", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces.", "\\begin{enumerate}", "\\item The functor", "$f_* : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_Y)$", "is left exact. In fact it commutes with all limits.", "\\item The functor", "$f^* : \\textit{Mod}(\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X)$", "is right exact. In fact it commutes with all colimits.", "\\item Pullback $f^{-1} : \\textit{Ab}(Y) \\to \\textit{Ab}(X)$", "on abelian sheaves is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1) and (2) hold because $(f^*, f_*)$ is an adjoint pair", "of functors, see", "Sheaves, Lemma \\ref{sheaves-lemma-adjoint-pullback-pushforward-modules}", "and", "Categories, Section \\ref{categories-section-adjoint}.", "Part (3) holds because exactness can be checked on stalks", "(Lemma \\ref{lemma-abelian})", "and the description of stalks of the pullback, see", "Sheaves, Lemma \\ref{sheaves-lemma-pullback-abelian-stalk}." ], "refs": [ "sheaves-lemma-adjoint-pullback-pushforward-modules", "modules-lemma-abelian", "sheaves-lemma-pullback-abelian-stalk" ], "ref_ids": [ 14521, 13221, 14511 ] } ], "ref_ids": [] }, { "id": 13224, "type": "theorem", "label": "modules-lemma-j-shriek-exact", "categories": [ "modules" ], "title": "modules-lemma-j-shriek-exact", "contents": [ "Let $j : U \\to X$ be an open immersion of topological spaces.", "The functor $j_! : \\textit{Ab}(U) \\to \\textit{Ab}(X)$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "Follows from the description of stalks", "given in Sheaves, Lemma \\ref{sheaves-lemma-j-shriek-abelian}." ], "refs": [ "sheaves-lemma-j-shriek-abelian" ], "ref_ids": [ 14544 ] } ], "ref_ids": [] }, { "id": 13225, "type": "theorem", "label": "modules-lemma-section-direct-sum-quasi-compact", "categories": [ "modules" ], "title": "modules-lemma-section-direct-sum-quasi-compact", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $I$ be a set. For $i \\in I$, let $\\mathcal{F}_i$", "be a sheaf of $\\mathcal{O}_X$-modules.", "For $U \\subset X$ quasi-compact open the map", "$$", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{F}_i(U)", "\\longrightarrow", "\\left(\\bigoplus\\nolimits_{i \\in I} \\mathcal{F}_i\\right)(U)", "$$", "is bijective." ], "refs": [], "proofs": [ { "contents": [ "If $s$ is an element of the right hand side, then", "there exists an open covering $U = \\bigcup_{j \\in J} U_j$", "such that $s|_{U_j}$ is a finite sum", "$\\sum_{i \\in I_j} s_{ji}$ with $s_{ji} \\in \\mathcal{F}_i(U_j)$.", "Because $U$ is quasi-compact we may assume that the", "covering is finite, i.e., that $J$ is finite.", "Then $I' = \\bigcup_{j \\in J} I_j$ is a finite subset of", "$I$. Clearly, $s$ is a section of the subsheaf", "$\\bigoplus_{i \\in I'} \\mathcal{F}_i$. The result follows", "from the fact that for a finite direct sum sheafification", "is not needed, see Lemma \\ref{lemma-limits-colimits} above." ], "refs": [ "modules-lemma-limits-colimits" ], "ref_ids": [ 13222 ] } ], "ref_ids": [] }, { "id": 13226, "type": "theorem", "label": "modules-lemma-globally-generated", "categories": [ "modules" ], "title": "modules-lemma-globally-generated", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "Let $I$ be a set. Let", "$s_i \\in \\Gamma(X, \\mathcal{F})$, $i \\in I$", "be global sections. The sections $s_i$ generate", "$\\mathcal{F}$ if and only if for all $x\\in X$ the", "elements $s_{i, x} \\in \\mathcal{F}_x$ generate", "the $\\mathcal{O}_{X, x}$-module $\\mathcal{F}_x$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13227, "type": "theorem", "label": "modules-lemma-tensor-product-globally-generated", "categories": [ "modules" ], "title": "modules-lemma-tensor-product-globally-generated", "contents": [ "\\begin{slogan}", "The tensor product of globally generated sheaves of modules is", "globally generated.", "\\end{slogan}", "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of $\\mathcal{O}_X$-modules.", "If $\\mathcal{F}$ and $\\mathcal{G}$ are generated by global sections", "then so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13228, "type": "theorem", "label": "modules-lemma-generated-by-local-sections", "categories": [ "modules" ], "title": "modules-lemma-generated-by-local-sections", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "Let $I$ be a set. Let $s_i$, $i \\in I$ be a collection", "of local sections of $\\mathcal{F}$, i.e., $s_i \\in \\mathcal{F}(U_i)$", "for some opens $U_i \\subset X$. There exists a unique smallest", "subsheaf of $\\mathcal{O}_X$-modules $\\mathcal{G}$ such", "that each $s_i$ corresponds to a local section of", "$\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the subpresheaf of $\\mathcal{O}_X$-modules", "defined by the rule", "$$", "U", "\\longmapsto", "\\{", "\\text{sums } \\sum\\nolimits_{i \\in J} f_i (s_i|_U)", "\\text{ where } J \\text{ is finite, }", "U \\subset U_i \\text{ for } i\\in J, \\text{ and }", "f_i \\in \\mathcal{O}_X(U)", "\\}", "$$", "Let $\\mathcal{G}$ be the sheafification of this subpresheaf.", "This is a subsheaf of $\\mathcal{F}$ by", "Sheaves, Lemma \\ref{sheaves-lemma-characterize-epi-mono}.", "Since all the finite sums clearly have to be in $\\mathcal{G}$", "this is the smallest subsheaf as desired." ], "refs": [ "sheaves-lemma-characterize-epi-mono" ], "ref_ids": [ 14491 ] } ], "ref_ids": [] }, { "id": 13229, "type": "theorem", "label": "modules-lemma-generated-by-local-sections-stalk", "categories": [ "modules" ], "title": "modules-lemma-generated-by-local-sections-stalk", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "Given a set $I$, and", "local sections $s_i$, $i \\in I$ of $\\mathcal{F}$.", "Let $\\mathcal{G}$ be the subsheaf generated by the", "$s_i$ and let $x\\in X$.", "Then $\\mathcal{G}_x$ is the $\\mathcal{O}_{X, x}$-submodule of", "$\\mathcal{F}_x$ generated by the elements $s_{i, x}$", "for those $i$ such that $s_i$ is defined at $x$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the construction of $\\mathcal{G}$", "in the proof of Lemma \\ref{lemma-generated-by-local-sections}." ], "refs": [ "modules-lemma-generated-by-local-sections" ], "ref_ids": [ 13228 ] } ], "ref_ids": [] }, { "id": 13230, "type": "theorem", "label": "modules-lemma-support-section-closed", "categories": [ "modules" ], "title": "modules-lemma-support-section-closed", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "Let $U \\subset X$ open.", "\\begin{enumerate}", "\\item The support of $s \\in \\mathcal{F}(U)$ is closed in $U$.", "\\item The support of $fs$ is contained in the intersections", "of the supports of $f \\in \\mathcal{O}_X(U)$ and $s \\in \\mathcal{F}(U)$.", "\\item The support of $s + s'$ is contained in the union of", "the supports of $s, s' \\in \\mathcal{F}(U)$.", "\\item The support of $\\mathcal{F}$ is the union of the supports", "of all local sections of $\\mathcal{F}$.", "\\item If $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of", "$\\mathcal{O}_X$-modules, then the support of $\\varphi(s)$ is", "contained in the support of $s \\in \\mathcal{F}(U)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is true because if $s_x = 0$, then $s$ is zero", "in an open neighbourhood of $x$ by definition of stalks.", "Similarly for $f$. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13231, "type": "theorem", "label": "modules-lemma-support-sheaf-rings-closed", "categories": [ "modules" ], "title": "modules-lemma-support-sheaf-rings-closed", "contents": [ "Let $X$ be a topological space.", "The support of a sheaf of rings is closed." ], "refs": [], "proofs": [ { "contents": [ "This is true because (according to our conventions)", "a ring is $0$ if and only if", "$1 = 0$, and hence the support of a sheaf of rings", "is the support of the unit section." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13232, "type": "theorem", "label": "modules-lemma-i-star-exact", "categories": [ "modules" ], "title": "modules-lemma-i-star-exact", "contents": [ "Let $X$ be a topological space. Let $Z \\subset X$ be a closed subset.", "Denote $i : Z \\to X$ the inclusion map. The functor", "$$", "i_* : \\textit{Ab}(Z) \\longrightarrow \\textit{Ab}(X)", "$$", "is exact, fully faithful, with essential image exactly those", "abelian sheaves whose support is contained in $Z$. The functor $i^{-1}$", "is a left inverse to $i_*$." ], "refs": [], "proofs": [ { "contents": [ "Exactness follows from the description of", "stalks in Sheaves, Lemma \\ref{sheaves-lemma-stalks-closed-pushforward}", "and Lemma \\ref{lemma-abelian}. The rest was shown in", "Sheaves, Lemma \\ref{sheaves-lemma-equivalence-categories-closed-abelian}." ], "refs": [ "sheaves-lemma-stalks-closed-pushforward", "modules-lemma-abelian", "sheaves-lemma-equivalence-categories-closed-abelian" ], "ref_ids": [ 14551, 13221, 14553 ] } ], "ref_ids": [] }, { "id": 13233, "type": "theorem", "label": "modules-lemma-i-star-right-adjoint", "categories": [ "modules" ], "title": "modules-lemma-i-star-right-adjoint", "contents": [ "Let $i : Z \\to X$ be the inclusion of a closed subset into the", "topological space $X$. The functor $\\textit{Ab}(X) \\to \\textit{Ab}(Z)$,", "$\\mathcal{F} \\mapsto \\mathcal{H}_Z(\\mathcal{F})$ of", "Remark \\ref{remark-sections-support-in-closed}", "is a right adjoint to $i_* : \\textit{Ab}(Z) \\to \\textit{Ab}(Z)$.", "In particular $i_*$ commutes with arbitrary colimits." ], "refs": [ "modules-remark-sections-support-in-closed" ], "proofs": [ { "contents": [ "We have to show that for any abelian sheaf $\\mathcal{F}$ on $X$ and any", "abelian sheaf $\\mathcal{G}$ on $Z$ we have", "$$", "\\Hom_{\\textit{Ab}(X)}(i_*\\mathcal{G}, \\mathcal{F}) =", "\\Hom_{\\textit{Ab}(Z)}(\\mathcal{G}, \\mathcal{H}_Z(\\mathcal{F}))", "$$", "This is clear because after all any section of $i_*\\mathcal{G}$", "has support in $Z$. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13363 ] }, { "id": 13234, "type": "theorem", "label": "modules-lemma-canonical-exact-sequence", "categories": [ "modules" ], "title": "modules-lemma-canonical-exact-sequence", "contents": [ "Let $X$ be a topological space.", "Let $U \\subset X$ be an open subset with complement $Z \\subset X$.", "Denote $j : U \\to X$ the open immersion and", "$i : Z \\to X$ the closed immersion.", "For any sheaf of abelian groups $\\mathcal{F}$ on $X$", "the adjunction mappings $j_{!}j^*\\mathcal{F} \\to \\mathcal{F}$ and", "$\\mathcal{F} \\to i_*i^*\\mathcal{F}$ give a short exact", "sequence", "$$", "0 \\to j_{!}j^*\\mathcal{F} \\to \\mathcal{F} \\to i_*i^*\\mathcal{F} \\to 0", "$$", "of sheaves of abelian groups. For any morphism", "$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ of abelian sheaves on $X$", "we obtain a morphism of short exact sequences", "$$", "\\xymatrix{", "0 \\ar[r] &", "j_{!}j^*\\mathcal{F} \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[r] \\ar[d] &", "i_*i^*\\mathcal{F} \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "j_{!}j^*\\mathcal{G} \\ar[r] &", "\\mathcal{G} \\ar[r] &", "i_*i^*\\mathcal{G} \\ar[r] &", "0", "}", "$$" ], "refs": [], "proofs": [ { "contents": [ "The functoriality of the short exact sequence is", "immediate from the naturality of the adjunction mappings.", "We may check exactness on stalks (Lemma \\ref{lemma-abelian}).", "For a description of the stalks in question see", "Sheaves, Lemmas \\ref{sheaves-lemma-j-shriek-abelian}", "and \\ref{sheaves-lemma-stalks-closed-pushforward}." ], "refs": [ "modules-lemma-abelian", "sheaves-lemma-j-shriek-abelian", "sheaves-lemma-stalks-closed-pushforward" ], "ref_ids": [ 13221, 14544, 14551 ] } ], "ref_ids": [] }, { "id": 13235, "type": "theorem", "label": "modules-lemma-pullback-locally-generated", "categories": [ "modules" ], "title": "modules-lemma-pullback-locally-generated", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces.", "The pullback $f^*\\mathcal{G}$ is locally generated by sections", "if $\\mathcal{G}$ is locally generated by sections." ], "refs": [], "proofs": [ { "contents": [ "Given an open subspace $V$ of $Y$ we may", "consider the commutative diagram of ringed spaces", "$$", "\\xymatrix{", "(f^{-1}V, \\mathcal{O}_{f^{-1}V}) \\ar[r]_{j'} \\ar[d]_{f'} &", "(X, \\mathcal{O}_X) \\ar[d]^f \\\\", "(V, \\mathcal{O}_V) \\ar[r]^j &", "(Y, \\mathcal{O}_Y)", "}", "$$", "We know that $f^*\\mathcal{G}|_{f^{-1}V} \\cong (f')^*(\\mathcal{G}|_V)$,", "see Sheaves, Lemma \\ref{sheaves-lemma-push-pull-composition-modules}.", "Thus we may assume that $\\mathcal{G}$ is globally generated.", "\\medskip\\noindent", "We have seen that $f^*$ commutes with all colimits,", "and is right exact, see Lemma \\ref{lemma-exactness-pushforward-pullback}.", "Thus if we have a surjection", "$$", "\\bigoplus\\nolimits_{i \\in I}", "\\mathcal{O}_Y", "\\to", "\\mathcal{G}", "\\to", "0", "$$", "then upon applying $f^*$ we obtain the surjection", "$$", "\\bigoplus\\nolimits_{i \\in I}", "\\mathcal{O}_X", "\\to", "f^*\\mathcal{G}", "\\to", "0.", "$$", "This implies the lemma." ], "refs": [ "sheaves-lemma-push-pull-composition-modules", "modules-lemma-exactness-pushforward-pullback" ], "ref_ids": [ 14522, 13223 ] } ], "ref_ids": [] }, { "id": 13236, "type": "theorem", "label": "modules-lemma-pullback-finite-type", "categories": [ "modules" ], "title": "modules-lemma-pullback-finite-type", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces.", "The pullback $f^*\\mathcal{G}$ of a finite type", "$\\mathcal{O}_Y$-module is a finite type $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "Arguing as in the proof of Lemma \\ref{lemma-pullback-locally-generated}", "we may assume $\\mathcal{G}$ is globally generated by finitely", "many sections.", "We have seen that $f^*$ commutes with all colimits,", "and is right exact, see Lemma \\ref{lemma-exactness-pushforward-pullback}.", "Thus if we have a surjection", "$$", "\\bigoplus\\nolimits_{i = 1, \\ldots, n}", "\\mathcal{O}_Y", "\\to", "\\mathcal{G}", "\\to", "0", "$$", "then upon applying $f^*$ we obtain the surjection", "$$", "\\bigoplus\\nolimits_{i = 1, \\ldots, n}", "\\mathcal{O}_X", "\\to", "f^*\\mathcal{G}", "\\to", "0.", "$$", "This implies the lemma." ], "refs": [ "modules-lemma-pullback-locally-generated", "modules-lemma-exactness-pushforward-pullback" ], "ref_ids": [ 13235, 13223 ] } ], "ref_ids": [] }, { "id": 13237, "type": "theorem", "label": "modules-lemma-extension-finite-type", "categories": [ "modules" ], "title": "modules-lemma-extension-finite-type", "contents": [ "Let $X$ be a ringed space.", "The image of a morphism of $\\mathcal{O}_X$-modules of finite", "type is of finite type.", "Let", "$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "be a short exact sequence of $\\mathcal{O}_X$-modules.", "If $\\mathcal{F}_1$ and $\\mathcal{F}_3$ are of finite type,", "so is $\\mathcal{F}_2$." ], "refs": [], "proofs": [ { "contents": [ "The statement on images is trivial.", "The statement on short exact sequences comes from the", "fact that sections of $\\mathcal{F}_3$ locally lift to sections", "of $\\mathcal{F}_2$ and the corresponding result in", "the category of modules over a ring (applied to the stalks", "for example)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13238, "type": "theorem", "label": "modules-lemma-finite-type-surjective-on-stalk", "categories": [ "modules" ], "title": "modules-lemma-finite-type-surjective-on-stalk", "contents": [ "Let $X$ be a ringed space.", "Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a homomorphism", "of $\\mathcal{O}_X$-modules.", "Let $x \\in X$. Assume $\\mathcal{F}$ of finite type and", "the map on stalks", "$\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ surjective.", "Then there exists an open neighbourhood", "$x \\in U \\subset X$ such that $\\varphi|_U$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "Choose an open neighbourhood $U \\subset X$ of $x$ such that $\\mathcal{F}$ is", "generated by $s_1, \\ldots, s_n \\in \\mathcal{F}(U)$ over $U$.", "By assumption of surjectivity of $\\varphi_x$,", "after shrinking $U$ we may assume that $s_i = \\varphi(t_i)$", "for some $t_i \\in \\mathcal{G}(U)$.", "Then $U$ works." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13239, "type": "theorem", "label": "modules-lemma-finite-type-stalk-zero", "categories": [ "modules" ], "title": "modules-lemma-finite-type-stalk-zero", "contents": [ "Let $X$ be a ringed space.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "Let $x \\in X$.", "Assume $\\mathcal{F}$ of finite type and $\\mathcal{F}_x = 0$.", "Then there exists an open neighbourhood", "$x \\in U \\subset X$ such that $\\mathcal{F}|_U$ is zero." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-finite-type-surjective-on-stalk}", "applied to the morphism $0 \\to \\mathcal{F}$." ], "refs": [ "modules-lemma-finite-type-surjective-on-stalk" ], "ref_ids": [ 13238 ] } ], "ref_ids": [] }, { "id": 13240, "type": "theorem", "label": "modules-lemma-support-finite-type-closed", "categories": [ "modules" ], "title": "modules-lemma-support-finite-type-closed", "contents": [ "\\begin{slogan}", "Over any ringed space, sheaves of modules of finite type have closed support.", "\\end{slogan}", "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "If $\\mathcal{F}$ is of finite type then support of $\\mathcal{F}$ is closed." ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of Lemma \\ref{lemma-finite-type-stalk-zero}." ], "refs": [ "modules-lemma-finite-type-stalk-zero" ], "ref_ids": [ 13239 ] } ], "ref_ids": [] }, { "id": 13241, "type": "theorem", "label": "modules-lemma-finite-type-quasi-compact-colimit", "categories": [ "modules" ], "title": "modules-lemma-finite-type-quasi-compact-colimit", "contents": [ "Let $X$ be a ringed space. Let $I$ be a preordered set and", "let $(\\mathcal{F}_i, f_{ii'})$ be a system over $I$ consisting of sheaves", "of $\\mathcal{O}_X$-modules (see", "Categories, Section \\ref{categories-section-posets-limits}).", "Let $\\mathcal{F} = \\colim \\mathcal{F}_i$ be the colimit. Assume", "(a) $I$ is directed,", "(b) $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module, and", "(c) $X$ is quasi-compact. Then there exists an $i$ such that", "$\\mathcal{F}_i \\to \\mathcal{F}$ is surjective.", "If the transition maps $f_{ii'}$ are injective", "then we conclude that $\\mathcal{F} = \\mathcal{F}_i$ for some $i \\in I$." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$. There exists an open neighbourhood", "$U \\subset X$ of $x$ and finitely many sections", "$s_j \\in \\mathcal{F}(U)$, $j = 1, \\ldots, m$ such that", "$s_1, \\ldots, s_m$ generate $\\mathcal{F}$ as $\\mathcal{O}_U$-module.", "After possibly shrinking $U$ to a smaller open neighbourhood of $x$", "we may assume that each $s_j$ comes from a section of $\\mathcal{F}_i$", "for some $i \\in I$.", "Hence, since $X$ is quasi-compact we can find a finite open", "covering $X = \\bigcup_{j = 1, \\ldots, m} U_j$, and for each $j$", "an index $i_j$ and finitely many sections $s_{jl} \\in \\mathcal{F}_{i_j}(U_j)$", "whose images generate the restriction of $\\mathcal{F}$ to", "$U_j$. Clearly, the lemma holds for any index $i \\in I$ which", "is $\\geq$ all $i_j$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13242, "type": "theorem", "label": "modules-lemma-set-isomorphism-classes-finite-type-modules", "categories": [ "modules" ], "title": "modules-lemma-set-isomorphism-classes-finite-type-modules", "contents": [ "Let $X$ be a ringed space.", "There exists a set of $\\mathcal{O}_X$-modules", "$\\{\\mathcal{F}_i\\}_{i \\in I}$ of finite type", "such that each finite type $\\mathcal{O}_X$-module", "on $X$ is isomorphic to exactly one of the $\\mathcal{F}_i$." ], "refs": [], "proofs": [ { "contents": [ "For each open covering $\\mathcal{U} : X = \\bigcup U_j$ consider the", "sheaves of $\\mathcal{O}_X$-modules $\\mathcal{F}$ such that each", "restriction $\\mathcal{F}|_{U_j}$ is a quotient of", "$\\mathcal{O}_{U_j}^{\\oplus r}$ for some $r_j \\geq 0$.", "These are parametrized by subsheaves", "$\\mathcal{K}_i \\subset \\mathcal{O}_{U_j}^{\\oplus r_j}$ and glueing", "data", "$$", "\\varphi_{jj'} :", "\\mathcal{O}_{U_j \\cap U_{j'}}^{\\oplus r_j}/", "(\\mathcal{K}_j|_{U_j \\cap U_{j'}})", "\\longrightarrow", "\\mathcal{O}_{U_j \\cap U_{j'}}^{\\oplus r_{j'}}/", "(\\mathcal{K}_{j'}|_{U_j \\cap U_{j'}})", "$$", "see Sheaves, Section \\ref{sheaves-section-glueing-sheaves}.", "Note that the collection of all glueing data forms a set.", "The collection of all coverings $\\mathcal{U} : X = \\bigcup_{j \\in J} U_i$", "where $J \\to \\mathcal{P}(X)$, $j \\mapsto U_j$ is injective forms a set as", "well. Hence the collection of all sheaves of $\\mathcal{O}_X$-modules", "gotten from glueing quotients as above forms a set $\\mathcal{I}$.", "By definition every finite type $\\mathcal{O}_X$-module", "is isomorphic to an element of $\\mathcal{I}$. Choosing an", "element out of each isomorphism class inside $\\mathcal{I}$", "gives the desired set of sheaves (uses axiom of choice)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13243, "type": "theorem", "label": "modules-lemma-direct-sum-quasi-coherent", "categories": [ "modules" ], "title": "modules-lemma-direct-sum-quasi-coherent", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "The direct sum of two quasi-coherent $\\mathcal{O}_X$-modules is", "a quasi-coherent $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13244, "type": "theorem", "label": "modules-lemma-pullback-quasi-coherent", "categories": [ "modules" ], "title": "modules-lemma-pullback-quasi-coherent", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces.", "The pullback $f^*\\mathcal{G}$ of a quasi-coherent", "$\\mathcal{O}_Y$-module is quasi-coherent." ], "refs": [], "proofs": [ { "contents": [ "Arguing as in the proof of Lemma \\ref{lemma-pullback-locally-generated}", "we may assume $\\mathcal{G}$ has a global presentation by", "direct sums of copies of $\\mathcal{O}_Y$.", "We have seen that $f^*$ commutes with all colimits,", "and is right exact, see Lemma \\ref{lemma-exactness-pushforward-pullback}.", "Thus if we have an exact sequence", "$$", "\\bigoplus\\nolimits_{j \\in J}", "\\mathcal{O}_Y", "\\longrightarrow", "\\bigoplus\\nolimits_{i \\in I}", "\\mathcal{O}_Y", "\\longrightarrow", "\\mathcal{G}", "\\longrightarrow", "0", "$$", "then upon applying $f^*$ we obtain the exact sequence", "$$", "\\bigoplus\\nolimits_{j \\in J}", "\\mathcal{O}_X", "\\longrightarrow", "\\bigoplus\\nolimits_{i \\in I}", "\\mathcal{O}_X", "\\longrightarrow", "f^*\\mathcal{G}", "\\longrightarrow", "0.", "$$", "This implies the lemma." ], "refs": [ "modules-lemma-pullback-locally-generated", "modules-lemma-exactness-pushforward-pullback" ], "ref_ids": [ 13235, 13223 ] } ], "ref_ids": [] }, { "id": 13245, "type": "theorem", "label": "modules-lemma-construct-quasi-coherent-sheaves", "categories": [ "modules" ], "title": "modules-lemma-construct-quasi-coherent-sheaves", "contents": [ "Let $(X, \\mathcal{O}_X)$ be ringed space.", "Let $\\alpha : R \\to \\Gamma(X, \\mathcal{O}_X)$ be a ring homomorphism from", "a ring $R$ into the ring of global sections on $X$.", "Let $M$ be an $R$-module.", "The following three constructions give canonically isomorphic", "sheaves of $\\mathcal{O}_X$-modules:", "\\begin{enumerate}", "\\item Let $\\pi : (X, \\mathcal{O}_X) \\longrightarrow (\\{*\\}, R)$", "be the morphism of ringed spaces with $\\pi : X \\to \\{*\\}$", "the unique map and with $\\pi$-map $\\pi^\\sharp$ the given map", "$\\alpha : R \\to \\Gamma(X, \\mathcal{O}_X)$. Set $\\mathcal{F}_1 = \\pi^*M$.", "\\item Choose a presentation", "$\\bigoplus_{j \\in J} R \\to \\bigoplus_{i \\in I} R \\to M \\to 0$.", "Set", "$$", "\\mathcal{F}_2 = \\Coker\\left(", "\\bigoplus\\nolimits_{j \\in J} \\mathcal{O}_X", "\\to", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{O}_X", "\\right).", "$$", "Here the map on the component $\\mathcal{O}_X$ corresponding to $j \\in J$", "given by the section $\\sum_i \\alpha(r_{ij})$ where the $r_{ij}$", "are the matrix coefficients of the map in the presentation of $M$.", "\\item Set $\\mathcal{F}_3$ equal to the sheaf associated to the presheaf", "$U \\mapsto \\mathcal{O}_X(U) \\otimes_R M$, where the map", "$R \\to \\mathcal{O}_X(U)$ is the composition of $\\alpha$ and", "the restriction map $\\mathcal{O}_X(X) \\to \\mathcal{O}_X(U)$.", "\\end{enumerate}", "This construction has the following properties:", "\\begin{enumerate}", "\\item The resulting sheaf of $\\mathcal{O}_X$-modules", "$\\mathcal{F}_M = \\mathcal{F}_1 = \\mathcal{F}_2 = \\mathcal{F}_3$", "is quasi-coherent.", "\\item The construction gives a functor from", "the category of $R$-modules to the category of quasi-coherent", "sheaves on $X$ which commutes with arbitrary colimits.", "\\item For any $x \\in X$ we have", "$\\mathcal{F}_{M, x} = \\mathcal{O}_{X, x} \\otimes_R M$", "functorial in $M$.", "\\item Given any $\\mathcal{O}_X$-module", "$\\mathcal{G}$ we have", "$$", "\\Mor_{\\mathcal{O}_X}(\\mathcal{F}_M, \\mathcal{G})", "=", "\\Hom_R(M, \\Gamma(X, \\mathcal{G}))", "$$", "where the $R$-module structure on $\\Gamma(X, \\mathcal{G})$", "comes from the $\\Gamma(X, \\mathcal{O}_X)$-module structure via", "$\\alpha$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The isomorphism between $\\mathcal{F}_1$ and $\\mathcal{F}_3$", "comes from the fact that $\\pi^*$ is defined as the sheafification", "of the presheaf in (3), see Sheaves, Section", "\\ref{sheaves-section-ringed-spaces-functoriality-modules}.", "The isomorphism between the constructions in (2) and (1) comes", "from the fact that the functor $\\pi^*$ is right exact, so", "$\\pi^*(\\bigoplus_{j \\in J} R) \\to \\pi^*(\\bigoplus_{i \\in I} R) \\to", "\\pi^*M \\to 0$ is exact, $\\pi^*$ commutes with arbitrary", "direct sums, see Lemma \\ref{lemma-exactness-pushforward-pullback},", "and finally the fact that $\\pi^*(R) = \\mathcal{O}_X$.", "\\medskip\\noindent", "Assertion (1) is clear from construction (2).", "Assertion (2) is clear since $\\pi^*$ has these properties.", "Assertion (3) follows from the description of stalks of", "pullback sheaves, see", "Sheaves, Lemma \\ref{sheaves-lemma-stalk-pullback-modules}.", "Assertion (4) follows from adjointness of $\\pi_*$ and", "$\\pi^*$." ], "refs": [ "modules-lemma-exactness-pushforward-pullback", "sheaves-lemma-stalk-pullback-modules" ], "ref_ids": [ 13223, 14523 ] } ], "ref_ids": [] }, { "id": 13246, "type": "theorem", "label": "modules-lemma-restrict-quasi-coherent", "categories": [ "modules" ], "title": "modules-lemma-restrict-quasi-coherent", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Set $R = \\Gamma(X, \\mathcal{O}_X)$.", "Let $M$ be an $R$-module.", "Let $\\mathcal{F}_M$ be the quasi-coherent sheaf of", "$\\mathcal{O}_X$-modules associated to $M$.", "If $g : (Y, \\mathcal{O}_Y) \\to (X, \\mathcal{O}_X)$", "is a morphism of ringed spaces, then", "$g^*\\mathcal{F}_M$ is the sheaf associated", "to the $\\Gamma(Y, \\mathcal{O}_Y)$-module", "$\\Gamma(Y, \\mathcal{O}_Y) \\otimes_R M$." ], "refs": [], "proofs": [ { "contents": [ "The assertion follows from the first description", "of $\\mathcal{F}_M$ in Lemma \\ref{lemma-construct-quasi-coherent-sheaves}", "as $\\pi^*M$, and the following commutative diagram", "of ringed spaces", "$$", "\\xymatrix{", "(Y, \\mathcal{O}_Y) \\ar[r]_-\\pi \\ar[d]_g &", "(\\{*\\}, \\Gamma(Y, \\mathcal{O}_Y)) \\ar[d]^{\\text{induced by }g^\\sharp} \\\\", "(X, \\mathcal{O}_X) \\ar[r]^-\\pi &", "(\\{*\\}, \\Gamma(X, \\mathcal{O}_X))", "}", "$$", "(Also use Sheaves, Lemma \\ref{sheaves-lemma-push-pull-composition-modules}.)" ], "refs": [ "modules-lemma-construct-quasi-coherent-sheaves", "sheaves-lemma-push-pull-composition-modules" ], "ref_ids": [ 13245, 14522 ] } ], "ref_ids": [] }, { "id": 13247, "type": "theorem", "label": "modules-lemma-quasi-coherent-module", "categories": [ "modules" ], "title": "modules-lemma-quasi-coherent-module", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $x \\in X$ be a point.", "Assume that $x$ has a fundamental system of quasi-compact neighbourhoods.", "Consider any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$.", "Then there exists an open neighbourhood $U$ of $x$", "such that $\\mathcal{F}|_U$ is isomorphic to the", "sheaf of modules $\\mathcal{F}_M$ on $(U, \\mathcal{O}_U)$", "associated to some $\\Gamma(U, \\mathcal{O}_U)$-module $M$." ], "refs": [], "proofs": [ { "contents": [ "First we may replace $X$ by an open neighbourhood of $x$", "and assume that $\\mathcal{F}$ is isomorphic to the", "cokernel of a map", "$$", "\\Psi :", "\\bigoplus\\nolimits_{j \\in J}", "\\mathcal{O}_X", "\\longrightarrow", "\\bigoplus\\nolimits_{i \\in I}", "\\mathcal{O}_X.", "$$", "The problem is that this map may not be given by", "a ``matrix'', because the module of global sections of a direct sum", "is in general different from the direct sum of the modules", "of global sections.", "\\medskip\\noindent", "Let $x \\in E \\subset X$ be a quasi-compact", "neighbourhood of $x$ (note: $E$ may not be open).", "Let $x \\in U \\subset E$ be an open neighbourhood of", "$x$ contained in $E$.", "Next, we proceed as in the proof of", "Lemma \\ref{lemma-section-direct-sum-quasi-compact}.", "For each $j \\in J$ denote", "$s_j \\in \\Gamma(X, \\bigoplus\\nolimits_{i \\in I} \\mathcal{O}_X)$", "the image of the section $1$ in the summand $\\mathcal{O}_X$", "corresponding to $j$. There exists a finite collection of opens", "$U_{jk}$, $k \\in K_j$ such that $E \\subset \\bigcup_{k \\in K_j} U_{jk}$", "and such that each restriction $s_j|_{U_{jk}}$", "is a finite sum $\\sum_{i \\in I_{jk}} f_{jki}$", "with $I_{jk} \\subset I$, and $f_{jki}$ in the summand", "$\\mathcal{O}_X$ corresponding to $i \\in I$. Set", "$I_j = \\bigcup_{k \\in K_j} I_{jk}$. This is a finite set.", "Since $U \\subset E \\subset \\bigcup_{k \\in K_j} U_{jk}$", "the section $s_j|_U$ is a section of the finite direct sum", "$\\bigoplus_{i \\in I_j} \\mathcal{O}_X$.", "By Lemma \\ref{lemma-limits-colimits}", "we see that actually $s_j|_U$ is a sum", "$\\sum_{i \\in I_j} f_{ij}$ and", "$f_{ij} \\in \\mathcal{O}_X(U) = \\Gamma(U, \\mathcal{O}_U)$.", "\\medskip\\noindent", "At this point we can define a module $M$ as the cokernel of the map", "$$", "\\bigoplus\\nolimits_{j \\in J}", "\\Gamma(U, \\mathcal{O}_U)", "\\longrightarrow", "\\bigoplus\\nolimits_{i \\in I}", "\\Gamma(U, \\mathcal{O}_U)", "$$", "with matrix given by the $(f_{ij})$. By construction (2) of", "Lemma \\ref{lemma-construct-quasi-coherent-sheaves} we see that", "$\\mathcal{F}_M$ has the same presentation as $\\mathcal{F}|_U$", "and therefore $\\mathcal{F}_M \\cong \\mathcal{F}|_U$." ], "refs": [ "modules-lemma-limits-colimits", "modules-lemma-construct-quasi-coherent-sheaves" ], "ref_ids": [ 13222, 13245 ] } ], "ref_ids": [] }, { "id": 13248, "type": "theorem", "label": "modules-lemma-finite-presentation-quasi-coherent", "categories": [ "modules" ], "title": "modules-lemma-finite-presentation-quasi-coherent", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Any $\\mathcal{O}_X$-module of finite presentation", "is quasi-coherent." ], "refs": [], "proofs": [ { "contents": [ "Immediate from definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13249, "type": "theorem", "label": "modules-lemma-kernel-surjection-finite-free-onto-finite-presentation", "categories": [ "modules" ], "title": "modules-lemma-kernel-surjection-finite-free-onto-finite-presentation", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a $\\mathcal{O}_X$-module of finite presentation.", "\\begin{enumerate}", "\\item If $\\psi : \\mathcal{O}_X^{\\oplus r} \\to \\mathcal{F}$ is a surjection,", "then $\\Ker(\\psi)$ is of finite type.", "\\item If $\\theta : \\mathcal{G} \\to \\mathcal{F}$ is surjective with", "$\\mathcal{G}$ of finite type, then $\\Ker(\\theta)$ is of finite type.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1).", "Let $x \\in X$. Choose an open neighbourhood $U \\subset X$ of $x$", "such that there exists a presentation", "$$", "\\mathcal{O}_U^{\\oplus m}", "\\xrightarrow{\\chi}", "\\mathcal{O}_U^{\\oplus n}", "\\xrightarrow{\\varphi}", "\\mathcal{F}|_U", "\\to", "0.", "$$", "Let $e_k$ be the section generating the $k$th factor of", "$\\mathcal{O}_X^{\\oplus r}$. For every $k = 1, \\ldots, r$", "we can, after shrinking $U$ to a small neighbourhood of $x$,", "lift $\\psi(e_k)$ to a section $\\tilde e_k$ of", "$\\mathcal{O}_U^{\\oplus n}$ over $U$. This gives a morphism", "of sheaves $\\alpha : \\mathcal{O}_U^{\\oplus r} \\to \\mathcal{O}_U^{\\oplus n}$", "such that $\\varphi \\circ \\alpha = \\psi$.", "Similarly, after shrinking $U$, we can find a morphism", "$\\beta : \\mathcal{O}_U^{\\oplus n} \\to \\mathcal{O}_U^{\\oplus r}$", "such that $\\psi \\circ \\beta = \\varphi$. Then the map", "$$", "\\mathcal{O}_U^{\\oplus m} \\oplus", "\\mathcal{O}_U^{\\oplus r}", "\\xrightarrow{\\beta \\circ \\chi, 1 - \\beta \\circ \\alpha}", "\\mathcal{O}_U^{\\oplus r}", "$$", "is a surjection onto the kernel of $\\psi$.", "\\medskip\\noindent", "To prove (2) we may locally choose a surjection", "$\\eta : \\mathcal{O}_X^{\\oplus r} \\to \\mathcal{G}$.", "By part (1) we see $\\Ker(\\theta \\circ \\eta)$ is of finite type.", "Since $\\Ker(\\theta) = \\eta(\\Ker(\\theta \\circ \\eta))$ we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13250, "type": "theorem", "label": "modules-lemma-pullback-finite-presentation", "categories": [ "modules" ], "title": "modules-lemma-pullback-finite-presentation", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces.", "The pullback $f^*\\mathcal{G}$ of a module of finite presentation", "is of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Exactly the same as the proof of Lemma \\ref{lemma-pullback-quasi-coherent}", "but with finite index sets." ], "refs": [ "modules-lemma-pullback-quasi-coherent" ], "ref_ids": [ 13244 ] } ], "ref_ids": [] }, { "id": 13251, "type": "theorem", "label": "modules-lemma-quasi-coherent-limit-finite-presentation", "categories": [ "modules" ], "title": "modules-lemma-quasi-coherent-limit-finite-presentation", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Set $R = \\Gamma(X, \\mathcal{O}_X)$.", "Let $M$ be an $R$-module.", "The $\\mathcal{O}_X$-module $\\mathcal{F}_M$ associated to $M$", "is a directed colimit of finitely presented $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from", "Lemma \\ref{lemma-construct-quasi-coherent-sheaves}", "and the fact that any module is a directed colimit", "of finitely presented modules, see", "Algebra, Lemma \\ref{algebra-lemma-module-colimit-fp}." ], "refs": [ "modules-lemma-construct-quasi-coherent-sheaves", "algebra-lemma-module-colimit-fp" ], "ref_ids": [ 13245, 355 ] } ], "ref_ids": [] }, { "id": 13252, "type": "theorem", "label": "modules-lemma-finite-presentation-quasi-compact-colimit", "categories": [ "modules" ], "title": "modules-lemma-finite-presentation-quasi-compact-colimit", "contents": [ "Let $X$ be a ringed space.", "Let $I$ be a preordered set and", "let $(\\mathcal{F}_i, \\varphi_{ii'})$ be a system over $I$", "consisting of sheaves of $\\mathcal{O}_X$-modules", "(see Categories, Section \\ref{categories-section-posets-limits}).", "Assume", "\\begin{enumerate}", "\\item $I$ is directed,", "\\item $\\mathcal{G}$ is an $\\mathcal{O}_X$-module of finite presentation, and", "\\item $X$ has a cofinal system of open coverings", "$\\mathcal{U} : X = \\bigcup_{j\\in J} U_j$ with", "$J$ finite and $U_j \\cap U_{j'}$ quasi-compact", "for all $j, j' \\in J$.", "\\end{enumerate}", "Then we have", "$$", "\\colim_i \\Hom_X(\\mathcal{G}, \\mathcal{F}_i)", "=", "\\Hom_X(\\mathcal{G}, \\colim_i \\mathcal{F}_i).", "$$" ], "refs": [], "proofs": [ { "contents": [ "An element of the left hand side is given by the equivalence classe", "of a pair $(i, \\alpha_i)$ where $i \\in I$ and", "$\\alpha_i : \\mathcal{G} \\to \\mathcal{F}_i$ is a morphism", "of $\\mathcal{O}_X$-modules, see", "Categories, Section \\ref{categories-section-directed-colimits}.", "Postcomposing with the coprojection", "$p_i : \\mathcal{F}_i \\to \\colim_{i' \\in I} \\mathcal{F}_{i'}$", "we get $\\alpha = p_i \\circ \\alpha_i$ in the right hand side.", "We obtain a map", "$$", "\\colim_i \\Hom_X(\\mathcal{G}, \\mathcal{F}_i) \\to", "\\Hom_X(\\mathcal{G}, \\colim_i \\mathcal{F}_i)", "$$", "\\medskip\\noindent", "Let us show this map is injective. Let $\\alpha_i$ be as above", "such that $\\alpha = p_i \\circ \\alpha_i$ is zero.", "By the assumption that $\\mathcal{G}$ is of finite presentation,", "for every $x \\in X$ we can choose an open neighbourhood", "$U_x \\subset X$ of $x$ and a finite set", "$s_{x, 1}, \\ldots, s_{x, n_x} \\in \\mathcal{G}(U_x)$", "generating $\\mathcal{G}|_{U_x}$. These sections map to", "zero in the stalk", "$(\\colim_{i'} \\mathcal{F}_i)_x = \\colim_{i'} \\mathcal{F}_{i', x}$. Hence", "for each $x$ we can pick $i(x) \\geq i$ such that after replacing", "$U_x$ by a smaller open we have that $s_{x, 1}, \\ldots, s_{x, n_x}$", "map to zero in $\\mathcal{F}_{i(x)}(U_x)$.", "Then $X = \\bigcup U_x$. By condition (3) we can refine this", "open covering by a finite open covering $X = \\bigcup_{j \\in J} U_j$.", "For $j \\in J$ pick $x_j \\in X$ with $U_j \\subset U_{x_j}$.", "Set $i' = \\max(i(x_j); j \\in J)$. Then $\\mathcal{G}|_{U_j}$ is generated", "by the sections $s_{x_j, k}$ which are mapped to zero in", "$\\mathcal{F}_{i(x)}$ and hence in $\\mathcal{F}_{i'}$.", "Hence the composition", "$\\mathcal{G} \\to \\mathcal{F}_i \\to \\mathcal{F}_{i'}$", "is zero as desired.", "\\medskip\\noindent", "Proof of surjectivity. Let $\\alpha$ be an element of the right hand side.", "For every point $x \\in X$ we may choose an open neighbourhood", "$U \\subset X$ and finitely many sections", "$s_j \\in \\mathcal{G}(U)$ which generate $\\mathcal{G}$ over $U$", "and finitely many relations $\\sum f_{kj} s_j = 0$, $k = 1, \\ldots, n$", "with $f_{kj} \\in \\mathcal{O}_X(U)$ which generate the kernel", "of $\\bigoplus_{j = 1, \\ldots, m} \\mathcal{O}_U \\to \\mathcal{G}$.", "After possibly shrinking $U$ to a smaller open neighbourhood of $x$", "we may assume there exists an index $i \\in I$ such that", "the sections $\\alpha(s_j)$ all come from sections", "$s_j' \\in \\mathcal{F}_i(U)$.", "After possibly shrinking $U$ to a smaller open neighbourhood of $x$", "and increasing $i$ we may assume the relations", "$\\sum f_{kj} s'_j = 0$ hold in $\\mathcal{F}_i(U)$.", "Hence we see that $\\alpha|_U$ lifts to a morphism", "$\\mathcal{G}|_U \\to \\mathcal{F}_i|_U$ for some index $i \\in I$.", "\\medskip\\noindent", "By condition (3) and the preceding arguments, we may choose", "a finite open covering $X = \\bigcup_{j = 1, \\ldots, m} U_j$", "such that (a) $\\mathcal{G}|_{U_j}$ is generated by finitely", "many sections $s_{jk} \\in \\mathcal{G}(U_j)$, (b) the restriction", "$\\alpha|_{U_j}$ comes from a morphism", "$\\alpha_j : \\mathcal{G} \\to \\mathcal{F}_{i_j}$", "for some $i_j \\in I$, and (c) the intersections", "$U_j \\cap U_{j'}$ are all quasi-compact.", "For every pair $(j, j') \\in \\{1, \\ldots, m\\}^2$", "and any $k$ we can find", "we can find an index $i \\geq \\max(i_j, i_{j'})$ such", "that", "$$", "\\varphi_{i_ji}(\\alpha_j(s_{jk}|_{U_j \\cap U_{j'}})) =", "\\varphi_{i_{j'}i}(\\alpha_{j'}(s_{jk}|_{U_j \\cap U_{j'}}))", "$$", "see Sheaves, Lemma \\ref{sheaves-lemma-directed-colimits-sections} (2).", "Since there are finitely many of these pairs $(j, j')$ and", "finitely many $s_{jk}$ we see that we can find a single $i$", "which works for all of them. For this index $i$ all of the maps", "$\\varphi_{i_ji} \\circ \\alpha_j$ agree on the overlaps $U_j \\cap U_{j'}$", "as the sections $s_{jk}$ generate $\\mathcal{G}$ over this overlap.", "Hence we get a morphism $\\mathcal{G} \\to \\mathcal{F}_i$ as desired." ], "refs": [ "sheaves-lemma-directed-colimits-sections" ], "ref_ids": [ 14526 ] } ], "ref_ids": [] }, { "id": 13253, "type": "theorem", "label": "modules-lemma-finite-presentation-stalk-free", "categories": [ "modules" ], "title": "modules-lemma-finite-presentation-stalk-free", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be", "a finitely presented $\\mathcal{O}_X$-module. Let $x \\in X$ such that", "$\\mathcal{F}_x \\cong \\mathcal{O}_{X, x}^{\\oplus r}$. Then there exists", "an open neighbourhood $U$ of $x$ such that", "$\\mathcal{F}|_U \\cong \\mathcal{O}_U^{\\oplus r}$." ], "refs": [], "proofs": [ { "contents": [ "Choose $s_1, \\ldots, s_r \\in \\mathcal{F}_x$ mapping to a basis of", "$\\mathcal{O}_{X, x}^{\\oplus r}$ by the isomorphism. Choose an open", "neighbourhood $U$ of $x$ such that $s_i$ lifts to $s_i \\in \\mathcal{F}(U)$.", "After shrinking $U$ we see that the induced map", "$\\psi : \\mathcal{O}_U^{\\oplus r} \\to \\mathcal{F}|_U$ is surjective", "(Lemma \\ref{lemma-finite-type-surjective-on-stalk}).", "By Lemma \\ref{lemma-kernel-surjection-finite-free-onto-finite-presentation}", "we see that $\\Ker(\\psi)$ is of finite type.", "Then $\\Ker(\\psi)_x = 0$ implies that $\\Ker(\\psi)$ becomes zero", "after shrinking $U$ once more (Lemma \\ref{lemma-finite-type-stalk-zero})." ], "refs": [ "modules-lemma-finite-type-surjective-on-stalk", "modules-lemma-kernel-surjection-finite-free-onto-finite-presentation", "modules-lemma-finite-type-stalk-zero" ], "ref_ids": [ 13238, 13249, 13239 ] } ], "ref_ids": [] }, { "id": 13254, "type": "theorem", "label": "modules-lemma-coherent-finite-presentation", "categories": [ "modules" ], "title": "modules-lemma-coherent-finite-presentation", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Any coherent $\\mathcal{O}_X$-module is of finite presentation", "and hence quasi-coherent." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be a coherent sheaf on $X$.", "Pick a point $x \\in X$.", "By (1) of the definition of coherent, we may find an open neighbourhood $U$", "and sections $s_i$, $i = 1, \\ldots, n$ of $\\mathcal{F}$ over $U$", "such that $\\Psi : \\bigoplus_{i = 1, \\ldots, n} \\mathcal{O}_U \\to \\mathcal{F}$", "is surjective. By (2) of the definition of coherent, we may find", "an open neighbourhood $V$, $x \\in V \\subset U$ and sections", "$t_1, \\ldots, t_m$ of $\\bigoplus_{i = 1, \\ldots, n} \\mathcal{O}_V$", "which generate the kernel of $\\Psi|_V$. Then over $V$ we get the", "presentation", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, m}", "\\mathcal{O}_V", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n}", "\\mathcal{O}_V", "\\to", "\\mathcal{F}|_V", "\\to", "0", "$$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13255, "type": "theorem", "label": "modules-lemma-coherent-abelian", "categories": [ "modules" ], "title": "modules-lemma-coherent-abelian", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "\\begin{enumerate}", "\\item Any finite type subsheaf of a coherent sheaf is coherent.", "\\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "be a morphism from a finite type sheaf $\\mathcal{F}$", "to a coherent sheaf $\\mathcal{G}$. Then $\\Ker(\\varphi)$ is finite type.", "\\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism", "of coherent $\\mathcal{O}_X$-modules. Then", "$\\Ker(\\varphi)$ and", "$\\Coker(\\varphi)$ are coherent.", "\\item Given a short exact sequence of $\\mathcal{O}_X$-modules", "$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "if two out of three are coherent so is the third.", "\\item The category $\\textit{Coh}(\\mathcal{O}_X)$ is a weak Serre subcategory", "of $\\textit{Mod}(\\mathcal{O}_X)$. In particular, the category of", "coherent modules is abelian and the inclusion functor", "$\\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X)$", "is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Condition (2) of Definition \\ref{definition-coherent}", "holds for any subsheaf of a coherent sheaf. Thus we get (1).", "\\medskip\\noindent", "Assume the hypotheses of (2).", "Let us show that $\\Ker(\\varphi)$ is of finite type. Pick $x \\in X$.", "Choose an open neighbourhood $U$ of $x$ in $X$ such", "that $\\mathcal{F}|_U$ is generated by $s_1, \\ldots, s_n$.", "By Definition \\ref{definition-coherent} the kernel $\\mathcal{K}$", "of the induced map", "$\\bigoplus_{i = 1}^n \\mathcal{O}_U \\to \\mathcal{G}$,", "$e_i \\mapsto \\varphi(s_i)$ is of finite type.", "Hence $\\Ker(\\varphi)$ which is the image of the", "composition", "$\\mathcal{K} \\to \\bigoplus_{i = 1}^n \\mathcal{O}_U \\to \\mathcal{F}$", "is of finite type.", "\\medskip\\noindent", "Assume the hypotheses of (3).", "By (2) the kernel of $\\varphi$ is of finite type and", "hence by (1) it is coherent.", "\\medskip\\noindent", "With the same hypotheses let us show that $\\Coker(\\varphi)$ is coherent.", "Since $\\mathcal{G}$ is of finite type so is $\\Coker(\\varphi)$.", "Let $U \\subset X$ be open and let $\\overline{s}_i \\in \\Coker(\\varphi)(U)$,", "$i = 1, \\ldots, n$ be sections. We have to show that the kernel of the", "associated morphism", "$\\overline{\\Psi} : \\bigoplus_{i = 1}^n \\mathcal{O}_U \\to \\Coker(\\varphi)$", "has finite type. There exists an open covering", "of $U$ such that on each open all the sections $\\overline{s}_i$", "lift to sections $s_i$ of $\\mathcal{G}$. Hence we may assume", "this is the case over $U$. We may in addition assume there are", "sections $t_j$, $j = 1, \\ldots, m$ of $\\Im(\\varphi)$ over $U$", "which generate $\\Im(\\varphi)$ over $U$.", "Let $\\Phi : \\bigoplus_{j = 1}^m \\mathcal{O}_U \\to \\Im(\\varphi)$", "be defined using $t_j$ and", "$\\Psi :", "\\bigoplus_{j = 1}^m \\mathcal{O}_U \\oplus", "\\bigoplus_{i = 1}^n \\mathcal{O}_U \\to \\mathcal{G}$", "using $t_j$ and $s_i$.", "Consider the following commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\bigoplus_{j = 1}^m \\mathcal{O}_U \\ar[d]_\\Phi \\ar[r] &", "\\bigoplus_{j = 1}^m \\mathcal{O}_U \\oplus", "\\bigoplus_{i = 1}^n \\mathcal{O}_U \\ar[d]_\\Psi \\ar[r] &", "\\bigoplus_{i = 1}^n \\mathcal{O}_U \\ar[d]_{\\overline{\\Psi}} \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "\\Im(\\varphi) \\ar[r] &", "\\mathcal{G} \\ar[r] &", "\\Coker(\\varphi) \\ar[r] &", "0", "}", "$$", "By the snake lemma we get an exact sequence", "$\\Ker(\\Psi) \\to \\Ker(\\overline{\\Psi}) \\to 0$. Since $\\Ker(\\Psi)$", "is a finite type module, we see that $\\Ker(\\overline{\\Psi})$ has finite type.", "\\medskip\\noindent", "Proof of part (4).", "Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "be a short exact sequence of $\\mathcal{O}_X$-modules. By part", "(3) it suffices", "to prove that if $\\mathcal{F}_1$ and $\\mathcal{F}_3$ are coherent", "so is $\\mathcal{F}_2$. By Lemma \\ref{lemma-extension-finite-type} we", "see that $\\mathcal{F}_2$ has finite type. Let", "$s_1, \\ldots, s_n$ be finitely many local", "sections of $\\mathcal{F}_2$ defined over a common open $U$ of $X$.", "We have to show that the module of relations $\\mathcal{K}$", "between them is of finite type.", "Consider the following commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "0 \\ar[r] \\ar[d] &", "\\bigoplus_{i = 1}^{n} \\mathcal{O}_U \\ar[r] \\ar[d] &", "\\bigoplus_{i = 1}^{n} \\mathcal{O}_U \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{F}_1 \\ar[r] &", "\\mathcal{F}_2 \\ar[r] &", "\\mathcal{F}_3 \\ar[r] &", "0", "}", "$$", "with obvious notation. By the snake lemma", "we get a short exact sequence", "$0 \\to \\mathcal{K} \\to \\mathcal{K}_3 \\to \\mathcal{F}_1$", "where $\\mathcal{K}_3$ is the module of relations among", "the images of the sections $s_i$ in $\\mathcal{F}_3$.", "Since $\\mathcal{F}_1$ is coherent we see that", "$\\mathcal{K}$ is the kernel of a map from a finite type module", "to a coherent module and hence finite type by (2).", "\\medskip\\noindent", "Proof of (5). This follows because (3) and (4) show that", "Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}", "applies." ], "refs": [ "modules-definition-coherent", "modules-definition-coherent", "modules-lemma-extension-finite-type", "homology-lemma-characterize-weak-serre-subcategory" ], "ref_ids": [ 13340, 13340, 13237, 12046 ] } ], "ref_ids": [] }, { "id": 13256, "type": "theorem", "label": "modules-lemma-coherent-structure-sheaf", "categories": [ "modules" ], "title": "modules-lemma-coherent-structure-sheaf", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "Assume $\\mathcal{O}_X$ is a coherent $\\mathcal{O}_X$-module.", "Then $\\mathcal{F}$ is coherent if and only if it is", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13257, "type": "theorem", "label": "modules-lemma-finite-type-to-coherent-injective-on-stalk", "categories": [ "modules" ], "title": "modules-lemma-finite-type-to-coherent-injective-on-stalk", "contents": [ "Let $X$ be a ringed space.", "Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a homomorphism", "of $\\mathcal{O}_X$-modules.", "Let $x \\in X$. Assume $\\mathcal{G}$ of finite type,", "$\\mathcal{F}$ coherent and the map on stalks", "$\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ injective.", "Then there exists an open neighbourhood", "$x \\in U \\subset X$ such that $\\varphi|_U$ is injective." ], "refs": [], "proofs": [ { "contents": [ "Denote $\\mathcal{K} \\subset \\mathcal{G}$ the kernel of $\\varphi$.", "By Lemma \\ref{lemma-coherent-abelian} we see that $\\mathcal{K}$ is", "a finite type $\\mathcal{O}_X$-module. Our assumption is that", "$\\mathcal{K}_x = 0$. By Lemma \\ref{lemma-finite-type-stalk-zero}", "there exists an open neighbourhood $U$ of $x$ such that $\\mathcal{K}|_U = 0$.", "Then $U$ works." ], "refs": [ "modules-lemma-coherent-abelian", "modules-lemma-finite-type-stalk-zero" ], "ref_ids": [ 13255, 13239 ] } ], "ref_ids": [] }, { "id": 13258, "type": "theorem", "label": "modules-lemma-i-star-quasi-coherent", "categories": [ "modules" ], "title": "modules-lemma-i-star-quasi-coherent", "contents": [ "Let $i : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$", "be a closed immersion of ringed spaces.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_Z$-module.", "Then $i_*\\mathcal{F}$ is locally on $X$ the cokernel of", "a map of quasi-coherent $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "This is true because $i_*\\mathcal{O}_Z$ is quasi-coherent", "by definition. And locally on $Z$ the sheaf $\\mathcal{F}$", "is a cokernel of a map between direct sums of copies", "of $\\mathcal{O}_Z$. Moreover, any direct sum of copies of the", "{\\it the same} quasi-coherent sheaf is quasi-coherent.", "And finally, $i_*$ commutes with arbitrary colimits,", "see Lemma \\ref{lemma-i-star-right-adjoint}. Some details omitted." ], "refs": [ "modules-lemma-i-star-right-adjoint" ], "ref_ids": [ 13233 ] } ], "ref_ids": [] }, { "id": 13259, "type": "theorem", "label": "modules-lemma-i-star-reflects-finite-type", "categories": [ "modules" ], "title": "modules-lemma-i-star-reflects-finite-type", "contents": [ "Let $i : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$ be a morphism of ringed", "spaces. Assume $i$ is a homeomorphism onto a closed subset of $X$ and", "that $\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_Z$-module.", "Then $i_*\\mathcal{F}$ is of finite type if and only if", "$\\mathcal{F}$ is of finite type." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $\\mathcal{F}$ is of finite type.", "Pick $x \\in X$. If $x \\not \\in Z$, then $i_*\\mathcal{F}$", "is zero in a neighbourhood of $x$ and hence finitely generated in", "a neighbourhood of $x$. If $x = i(z)$, then choose an open neighbourhood", "$z \\in V \\subset Z$ and sections $s_1, \\ldots, s_n \\in \\mathcal{F}(V)$", "which generate $\\mathcal{F}$ over $V$. Write $V = Z \\cap U$ for some open", "$U \\subset X$. Note that $U$ is a neighbourhood of $x$. Clearly the", "sections $s_i$ give sections $s_i$ of $i_*\\mathcal{F}$ over $U$.", "The resulting map", "$$", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} \\mathcal{O}_U", "\\longrightarrow", "i_*\\mathcal{F}|_U", "$$", "is surjective by inspection of what it does on stalks", "(here we use that $\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$", "is surjective). Hence $i_*\\mathcal{F}$", "is of finite type.", "\\medskip\\noindent", "Conversely, suppose that $i_*\\mathcal{F}$ is of finite type.", "Choose $z \\in Z$. Set $x = i(z)$. By assumption there exists", "an open neighbourhood $U \\subset X$ of $x$, and sections", "$s_1, \\ldots, s_n \\in (i_*\\mathcal{F})(U)$ which generate $i_*\\mathcal{F}$", "over $U$. Set $V = Z \\cap U$. By definition of $i_*$ the sections", "$s_i$ correspond to sections $s_i$ of $\\mathcal{F}$ over $V$.", "The resulting map", "$$", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} \\mathcal{O}_V", "\\longrightarrow", "\\mathcal{F}|_V", "$$", "is surjective by inspection of what it does on stalks. Hence", "$\\mathcal{F}$ is of finite type." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13260, "type": "theorem", "label": "modules-lemma-i-star-equivalence", "categories": [ "modules" ], "title": "modules-lemma-i-star-equivalence", "contents": [ "Let $i : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$ be a morphism", "of ringed spaces. Assume $i$ is a homeomorphism onto a closed subset of $X$", "and $i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective.", "Denote $\\mathcal{I} \\subset \\mathcal{O}_X$ the kernel of $i^\\sharp$.", "The functor", "$$", "i_* :", "\\textit{Mod}(\\mathcal{O}_Z)", "\\longrightarrow", "\\textit{Mod}(\\mathcal{O}_X)", "$$", "is exact, fully faithful, with essential image those", "$\\mathcal{O}_X$-modules $\\mathcal{G}$ such that $\\mathcal{I}\\mathcal{G} = 0$." ], "refs": [], "proofs": [ { "contents": [ "We claim that for a $\\mathcal{O}_Z$-module $\\mathcal{F}$ the canonical map", "$$", "i^*i_*\\mathcal{F} \\longrightarrow \\mathcal{F}", "$$", "is an isomorphism. We check this on stalks. Say $z \\in Z$ and $x = i(z)$.", "We have", "$$", "(i^*i_*\\mathcal{F})_z =", "(i_*\\mathcal{F})_x \\otimes_{\\mathcal{O}_{X, x}} \\mathcal{O}_{Z, z} =", "\\mathcal{F}_z \\otimes_{\\mathcal{O}_{X, x}} \\mathcal{O}_{Z, z} =", "\\mathcal{F}_z", "$$", "by Sheaves, Lemma \\ref{sheaves-lemma-stalk-pullback-modules},", "the fact that $\\mathcal{O}_{Z, z}$ is a quotient of $\\mathcal{O}_{X, x}$, and", "Sheaves, Lemma \\ref{sheaves-lemma-stalks-closed-pushforward}.", "It follows that $i_*$ is fully faithful.", "\\medskip\\noindent", "Let $\\mathcal{G}$ be a $\\mathcal{O}_X$-module with", "$\\mathcal{I}\\mathcal{G} = 0$. We will prove the canonical map", "$$", "\\mathcal{G} \\longrightarrow i_*i^*\\mathcal{G}", "$$", "is an isomorphism. This proves that $\\mathcal{G} = i_*\\mathcal{F}$", "with $\\mathcal{F} = i^*\\mathcal{G}$ which finishes the proof.", "We check the displayed map induces an isomorphism on stalks.", "If $x \\in X$, $x \\not \\in i(Z)$, then $\\mathcal{G}_x = 0$", "because $\\mathcal{I}_x = \\mathcal{O}_{X, x}$ in this", "case. As above $(i_*i^*\\mathcal{G})_x = 0$ by", "Sheaves, Lemma \\ref{sheaves-lemma-stalks-closed-pushforward}.", "On the other hand, if $x \\in Z$, then we obtain the map", "$$", "\\mathcal{G}_x", "\\longrightarrow", "\\mathcal{G}_x \\otimes_{\\mathcal{O}_{X, x}} \\mathcal{O}_{Z, x}", "$$", "by Sheaves, Lemmas \\ref{sheaves-lemma-stalk-pullback-modules} and", "\\ref{sheaves-lemma-stalks-closed-pushforward}. This map is an isomorphism", "because $\\mathcal{O}_{Z, x} = \\mathcal{O}_{X, x}/\\mathcal{I}_x$", "and because $\\mathcal{G}_x$ is annihilated by $\\mathcal{I}_x$ by assumption." ], "refs": [ "sheaves-lemma-stalk-pullback-modules", "sheaves-lemma-stalks-closed-pushforward", "sheaves-lemma-stalks-closed-pushforward", "sheaves-lemma-stalk-pullback-modules", "sheaves-lemma-stalks-closed-pushforward" ], "ref_ids": [ 14523, 14551, 14551, 14523, 14551 ] } ], "ref_ids": [] }, { "id": 13261, "type": "theorem", "label": "modules-lemma-adjoint-section-with-support", "categories": [ "modules" ], "title": "modules-lemma-adjoint-section-with-support", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the", "inclusion of a closed subset. The functor", "$\\mathcal{H}_Z : \\textit{Mod}(\\mathcal{O}_X) \\to", "\\textit{Mod}(\\mathcal{O}_X|_Z)$ of", "Remark \\ref{remark-sections-support-in-closed-modules}", "is right adjoint to", "$i_* : \\textit{Mod}(\\mathcal{O}_X|_Z) \\to \\textit{Mod}(\\mathcal{O}_X)$." ], "refs": [ "modules-remark-sections-support-in-closed-modules" ], "proofs": [ { "contents": [ "We have to show that for any $\\mathcal{O}_X$-module $\\mathcal{F}$", "and any $\\mathcal{O}_X|_Z$-module $\\mathcal{G}$ we have", "$$", "\\Hom_{\\mathcal{O}_X|_Z}(\\mathcal{G}, \\mathcal{H}_Z(\\mathcal{F})) =", "\\Hom_{\\mathcal{O}_X}(i_*\\mathcal{G}, \\mathcal{F})", "$$", "This is clear", "because after all any section of $i_*\\mathcal{G}$ has support in $Z$.", "Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13367 ] }, { "id": 13262, "type": "theorem", "label": "modules-lemma-locally-free-quasi-coherent", "categories": [ "modules" ], "title": "modules-lemma-locally-free-quasi-coherent", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "If $\\mathcal{F}$ is locally free then it is quasi-coherent." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13263, "type": "theorem", "label": "modules-lemma-pullback-locally-free", "categories": [ "modules" ], "title": "modules-lemma-pullback-locally-free", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces. If $\\mathcal{G}$ is", "a locally free $\\mathcal{O}_Y$-module, then", "$f^*\\mathcal{G}$ is a locally free $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13264, "type": "theorem", "label": "modules-lemma-rank", "categories": [ "modules" ], "title": "modules-lemma-rank", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Suppose that the support of $\\mathcal{O}_X$ is $X$,", "i.e., all stalk of $\\mathcal{O}_X$ are nonzero rings.", "Let $\\mathcal{F}$ be a locally free sheaf of $\\mathcal{O}_X$-modules.", "There exists a locally constant function", "$$", "\\text{rank}_\\mathcal{F} :", "X \\longrightarrow \\{0, 1, 2, \\ldots\\}\\cup\\{\\infty\\}", "$$", "such that for any point $x \\in X$ the cardinality of any", "set $I$ such that $\\mathcal{F}$ is isomorphic to", "$\\bigoplus_{i\\in I} \\mathcal{O}_X$ in a neighbourhood", "of $x$ is $\\text{rank}_\\mathcal{F}(x)$." ], "refs": [], "proofs": [ { "contents": [ "Under the assumption of the lemma the cardinality of $I$ can be read off", "from the rank of the free module $\\mathcal{F}_x$ over the nonzero ring", "$\\mathcal{O}_{X, x}$, and it is constant in a neighbourhood of $x$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13265, "type": "theorem", "label": "modules-lemma-map-finite-locally-free", "categories": [ "modules" ], "title": "modules-lemma-map-finite-locally-free", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $r \\geq 0$.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of finite", "locally free $\\mathcal{O}_X$-modules of rank $r$.", "Then $\\varphi$ is an isomorphism if and only if $\\varphi$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\varphi$ is surjective. Pick $x \\in X$.", "There exists an open neighbourhood $U$ of $x$", "such that both $\\mathcal{F}|_U$ and $\\mathcal{G}|_U$ are", "isomorphic to $\\mathcal{O}_U^{\\oplus r}$.", "Pick lifts of the free generators of $\\mathcal{G}|_U$", "to obtain a map $\\psi : \\mathcal{G}|_U \\to \\mathcal{F}|_U$", "such that $\\varphi|_U \\circ \\psi = \\text{id}$.", "Hence we conclude that the map", "$\\Gamma(U, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{G})$", "induced by $\\varphi$ is surjective. Since both", "$\\Gamma(U, \\mathcal{F})$ and $\\Gamma(U, \\mathcal{G})$", "are isomorphic to $\\Gamma(U, \\mathcal{O}_U)^{\\oplus r}$", "as an $\\Gamma(U, \\mathcal{O}_U)$-module we may", "apply Algebra, Lemma \\ref{algebra-lemma-fun} to see that", "$\\Gamma(U, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{G})$", "is injective. This finishes the proof." ], "refs": [ "algebra-lemma-fun" ], "ref_ids": [ 388 ] } ], "ref_ids": [] }, { "id": 13266, "type": "theorem", "label": "modules-lemma-direct-summand-of-locally-free-is-locally-free", "categories": [ "modules" ], "title": "modules-lemma-direct-summand-of-locally-free-is-locally-free", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. If all stalks $\\mathcal{O}_{X, x}$", "are local rings, then any direct summand of a finite locally free", "$\\mathcal{O}_X$-module is finite locally free." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{F}$ is a direct summand of the finite locally free", "$\\mathcal{O}_X$-module $\\mathcal{H}$.", "Let $x \\in X$ be a point. Then $\\mathcal{H}_x$ is a finite free", "$\\mathcal{O}_{X, x}$-module.", "Because $\\mathcal{O}_{X, x}$ is local, we see that", "$\\mathcal{F}_x \\cong \\mathcal{O}_{X, x}^{\\oplus r}$ for some $r$, see", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}.", "By Lemma \\ref{lemma-finite-presentation-stalk-free}", "we see that $\\mathcal{F}$ is free of rank $r$ in an open neighbourhood of $x$.", "(Note that $\\mathcal{F}$ is of finite presentation as a summand of", "$\\mathcal{H}$.)" ], "refs": [ "algebra-lemma-finite-projective", "modules-lemma-finite-presentation-stalk-free" ], "ref_ids": [ 795, 13253 ] } ], "ref_ids": [] }, { "id": 13267, "type": "theorem", "label": "modules-lemma-stalk-tensor-product", "categories": [ "modules" ], "title": "modules-lemma-stalk-tensor-product", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules.", "Let $x \\in X$. There is a canonical isomorphism", "of $\\mathcal{O}_{X, x}$-modules", "$$", "(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G})_x", "=", "\\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\mathcal{G}_x", "$$", "functorial in $\\mathcal{F}$ and $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13268, "type": "theorem", "label": "modules-lemma-tensor-product-sheafification", "categories": [ "modules" ], "title": "modules-lemma-tensor-product-sheafification", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}'$, $\\mathcal{G}'$ be presheaves of $\\mathcal{O}_X$-modules", "with sheafifications $\\mathcal{F}$, $\\mathcal{G}$. Then", "$\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G} =", "(\\mathcal{F}' \\otimes_{p, \\mathcal{O}_X} \\mathcal{G}')^\\#$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13269, "type": "theorem", "label": "modules-lemma-tensor-product-exact", "categories": [ "modules" ], "title": "modules-lemma-tensor-product-exact", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{G}$ be an $\\mathcal{O}_X$-module.", "If", "$\\mathcal{F}_1", "\\to \\mathcal{F}_2", "\\to \\mathcal{F}_3", "\\to 0$", "is an exact sequence of $\\mathcal{O}_X$-modules then", "the induced sequence", "$$", "\\mathcal{F}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to", "\\mathcal{F}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to", "\\mathcal{F}_3 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to", "0", "$$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that exactness may be checked at stalks", "(Lemma \\ref{lemma-abelian}), the description of stalks", "(Lemma \\ref{lemma-stalk-tensor-product}) and the corresponding", "result for tensor products of modules", "(Algebra, Lemma \\ref{algebra-lemma-tensor-product-exact})." ], "refs": [ "modules-lemma-abelian", "modules-lemma-stalk-tensor-product", "algebra-lemma-tensor-product-exact" ], "ref_ids": [ 13221, 13267, 364 ] } ], "ref_ids": [] }, { "id": 13270, "type": "theorem", "label": "modules-lemma-tensor-product-pullback", "categories": [ "modules" ], "title": "modules-lemma-tensor-product-pullback", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be", "a morphism of ringed spaces. Let $\\mathcal{F}$, $\\mathcal{G}$", "be $\\mathcal{O}_Y$-modules. Then", "$f^*(\\mathcal{F} \\otimes_{\\mathcal{O}_Y} \\mathcal{G})", "= f^*\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}$", "functorially in $\\mathcal{F}$, $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13271, "type": "theorem", "label": "modules-lemma-tensor-product-permanence", "categories": [ "modules" ], "title": "modules-lemma-tensor-product-permanence", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$, $\\mathcal{G}$ are locally generated", "by sections, so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$.", "\\item If $\\mathcal{F}$, $\\mathcal{G}$ are of finite type,", "so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$.", "\\item If $\\mathcal{F}$, $\\mathcal{G}$ are quasi-coherent,", "so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$.", "\\item If $\\mathcal{F}$, $\\mathcal{G}$ are of finite presentation,", "so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$.", "\\item If $\\mathcal{F}$ is of finite presentation and $\\mathcal{G}$ is coherent,", "then $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$ is coherent.", "\\item If $\\mathcal{F}$, $\\mathcal{G}$ are coherent,", "so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$.", "\\item If $\\mathcal{F}$, $\\mathcal{G}$ are locally free,", "so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We first prove that the tensor product of locally free", "$\\mathcal{O}_X$-modules is locally free. This follows if we show", "that", "$(\\bigoplus_{i \\in I} \\mathcal{O}_X) \\otimes_{\\mathcal{O}_X}", "(\\bigoplus_{j \\in J} \\mathcal{O}_X) \\cong", "\\bigoplus_{(i, j) \\in I \\times J} \\mathcal{O}_X$.", "The sheaf $\\bigoplus_{i \\in I} \\mathcal{O}_X$ is the sheaf associated", "to the presheaf $U \\mapsto \\bigoplus_{i \\in I} \\mathcal{O}_X(U)$.", "Hence the tensor product is the sheaf associated", "to the presheaf", "$$", "U \\longmapsto", "(\\bigoplus\\nolimits_{i \\in I} \\mathcal{O}_X(U))", "\\otimes_{\\mathcal{O}_X(U)}", "(\\bigoplus\\nolimits_{j \\in J} \\mathcal{O}_X(U)).", "$$", "We deduce what we want since for any ring $R$ we have", "$(\\bigoplus_{i \\in I} R) \\otimes_R (\\bigoplus_{j \\in J} R) =", "\\bigoplus_{(i, j) \\in I \\times J} R$.", "\\medskip\\noindent", "If $\\mathcal{F}_2 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to 0$", "is exact, then by Lemma \\ref{lemma-tensor-product-exact}", "the complex", "$\\mathcal{F}_2 \\otimes \\mathcal{G} \\to", "\\mathcal{F}_1 \\otimes \\mathcal{G} \\to", "\\mathcal{F} \\otimes \\mathcal{G} \\to 0$", "is exact. Using this we can prove (5). Namely, in this case there", "exists locally such an exact sequence with $\\mathcal{F}_i$, $i = 1, 2$", "finite free. Hence the two terms $\\mathcal{F}_2 \\otimes \\mathcal{G}$", "are isomorphic to finite direct sums of $\\mathcal{G}$.", "Since finite direct sums are coherent sheaves, these are coherent", "and so is the cokernel of the map, see Lemma \\ref{lemma-coherent-abelian}.", "\\medskip\\noindent", "And if also", "$\\mathcal{G}_2 \\to \\mathcal{G}_1 \\to \\mathcal{G} \\to 0$", "is exact, then we see that", "$$", "\\mathcal{F}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{G}_1", "\\oplus", "\\mathcal{F}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{G}_2", "\\to", "\\mathcal{F}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{G}_1", "\\to", "\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}", "\\to 0", "$$", "is exact. Using this we can for example prove (3).", "Namely, the assumption means that we can locally find presentations", "as above with $\\mathcal{F}_i$ and $\\mathcal{G}_i$", "free $\\mathcal{O}_X$-modules. Hence the displayed presentation", "is a presentation of the tensor product by free sheaves as well.", "\\medskip\\noindent", "The proof of the other statements is omitted." ], "refs": [ "modules-lemma-tensor-product-exact", "modules-lemma-coherent-abelian" ], "ref_ids": [ 13269, 13255 ] } ], "ref_ids": [] }, { "id": 13272, "type": "theorem", "label": "modules-lemma-tensor-commute-colimits", "categories": [ "modules" ], "title": "modules-lemma-tensor-commute-colimits", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "For any $\\mathcal{O}_X$-module $\\mathcal{F}$ the functor", "$$", "\\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\textit{Mod}(\\mathcal{O}_X)", ", \\quad", "\\mathcal{G} \\longmapsto \\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}", "$$", "commutes with arbitrary colimits." ], "refs": [], "proofs": [ { "contents": [ "Let $I$ be a preordered set and let $\\{\\mathcal{G}_i\\}$ be", "a system over $I$. Set $\\mathcal{G} = \\colim_i \\mathcal{G}_i$.", "Recall that $\\mathcal{G}$ is the sheaf associated to the presheaf", "$\\mathcal{G}' : U \\mapsto \\colim_i \\mathcal{G}_i(U)$, see", "Sheaves, Section \\ref{sheaves-section-limits-sheaves}.", "By", "Lemma \\ref{lemma-tensor-product-sheafification}", "the tensor product $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$", "is the sheafification of the presheaf", "$$", "U \\longmapsto", "\\mathcal{F}(U) \\otimes_{\\mathcal{O}_X(U)} \\colim_i \\mathcal{G}_i(U) =", "\\colim_i \\mathcal{F}(U) \\otimes_{\\mathcal{O}_X(U)} \\mathcal{G}_i(U)", "$$", "where the equality sign is", "Algebra, Lemma \\ref{algebra-lemma-tensor-products-commute-with-limits}.", "Hence the lemma follows from the description of colimits in", "$\\textit{Mod}(\\mathcal{O}_X)$." ], "refs": [ "modules-lemma-tensor-product-sheafification", "algebra-lemma-tensor-products-commute-with-limits" ], "ref_ids": [ 13268, 363 ] } ], "ref_ids": [] }, { "id": 13273, "type": "theorem", "label": "modules-lemma-flat-stalks-flat", "categories": [ "modules" ], "title": "modules-lemma-flat-stalks-flat", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "An $\\mathcal{O}_X$-module $\\mathcal{F}$ is flat", "if and only if the stalk $\\mathcal{F}_x$ is a flat", "$\\mathcal{O}_{X, x}$-module for all $x \\in X$." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{F}_x$ is a flat $\\mathcal{O}_{X, x}$-module for all", "$x \\in X$. In this case, if $\\mathcal{G} \\to \\mathcal{H} \\to \\mathcal{K}$", "is exact, then also", "$\\mathcal{G} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "\\mathcal{H} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "\\mathcal{K} \\otimes_{\\mathcal{O}_X} \\mathcal{F}$", "is exact because we can check exactness at stalks and because", "tensor product commutes with taking stalks, see", "Lemma \\ref{lemma-stalk-tensor-product}.", "Conversely, suppose that $\\mathcal{F}$ is flat, and let $x \\in X$.", "Consider the skyscraper sheaves $i_{x, *} M$ where $M$ is a", "$\\mathcal{O}_{X, x}$-module. Note that", "$$", "M \\otimes_{\\mathcal{O}_{X, x}} \\mathcal{F}_x =", "\\left(i_{x, *} M \\otimes_{\\mathcal{O}_X} \\mathcal{F}\\right)_x", "$$", "again by", "Lemma \\ref{lemma-stalk-tensor-product}.", "Since $i_{x, *}$ is exact, we see that the fact that $\\mathcal{F}$", "is flat implies that $M \\mapsto M \\otimes_{\\mathcal{O}_{X, x}} \\mathcal{F}_x$", "is exact. Hence $\\mathcal{F}_x$ is a flat $\\mathcal{O}_{X, x}$-module." ], "refs": [ "modules-lemma-stalk-tensor-product", "modules-lemma-stalk-tensor-product" ], "ref_ids": [ 13267, 13267 ] } ], "ref_ids": [] }, { "id": 13274, "type": "theorem", "label": "modules-lemma-colimits-flat", "categories": [ "modules" ], "title": "modules-lemma-colimits-flat", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "A filtered colimit of flat $\\mathcal{O}_X$-modules is flat.", "A direct sum of flat $\\mathcal{O}_X$-modules is flat." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-tensor-commute-colimits},", "Lemma \\ref{lemma-stalk-tensor-product},", "Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact},", "and the fact that we can check exactness at stalks." ], "refs": [ "modules-lemma-tensor-commute-colimits", "modules-lemma-stalk-tensor-product", "algebra-lemma-directed-colimit-exact" ], "ref_ids": [ 13272, 13267, 343 ] } ], "ref_ids": [] }, { "id": 13275, "type": "theorem", "label": "modules-lemma-j-shriek-flat", "categories": [ "modules" ], "title": "modules-lemma-j-shriek-flat", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $U \\subset X$ be open. The sheaf $j_{U!}\\mathcal{O}_U$", "is a flat sheaf of $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "The stalks of $j_{U!}\\mathcal{O}_U$ are either zero or equal", "to $\\mathcal{O}_{X, x}$. Apply", "Lemma \\ref{lemma-flat-stalks-flat}." ], "refs": [ "modules-lemma-flat-stalks-flat" ], "ref_ids": [ 13273 ] } ], "ref_ids": [] }, { "id": 13276, "type": "theorem", "label": "modules-lemma-module-quotient-flat", "categories": [ "modules" ], "title": "modules-lemma-module-quotient-flat", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "\\begin{enumerate}", "\\item Any sheaf of $\\mathcal{O}_X$-modules is a quotient of", "a direct sum $\\bigoplus j_{U_i!}\\mathcal{O}_{U_i}$.", "\\item Any $\\mathcal{O}_X$-module is a quotient of", "a flat $\\mathcal{O}_X$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "For every open $U \\subset X$ and every", "$s \\in \\mathcal{F}(U)$ we get a morphism", "$j_{U!}\\mathcal{O}_U \\to \\mathcal{F}$, namely the adjoint to", "the morphism $\\mathcal{O}_U \\to \\mathcal{F}|_U$, $1 \\mapsto s$.", "Clearly the map", "$$", "\\bigoplus\\nolimits_{(U, s)} j_{U!}\\mathcal{O}_U", "\\longrightarrow", "\\mathcal{F}", "$$", "is surjective, and the source is flat by combining Lemmas", "\\ref{lemma-colimits-flat} and \\ref{lemma-j-shriek-flat}." ], "refs": [ "modules-lemma-colimits-flat", "modules-lemma-j-shriek-flat" ], "ref_ids": [ 13274, 13275 ] } ], "ref_ids": [] }, { "id": 13277, "type": "theorem", "label": "modules-lemma-flat-tor-zero", "categories": [ "modules" ], "title": "modules-lemma-flat-tor-zero", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let", "$$", "0 \\to \\mathcal{F}'' \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0", "$$", "be a short exact sequence of $\\mathcal{O}_X$-modules.", "Assume $\\mathcal{F}$ is flat. Then for any $\\mathcal{O}_X$-module", "$\\mathcal{G}$ the sequence", "$$", "0 \\to", "\\mathcal{F}'' \\otimes_\\mathcal{O} \\mathcal{G} \\to", "\\mathcal{F}' \\otimes_\\mathcal{O} \\mathcal{G} \\to", "\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G} \\to 0", "$$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "Using that $\\mathcal{F}_x$ is a flat $\\mathcal{O}_{X, x}$-module", "for every $x \\in X$ and that exactness can be checked on stalks, this", "follows from", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}." ], "refs": [ "algebra-lemma-flat-tor-zero" ], "ref_ids": [ 532 ] } ], "ref_ids": [] }, { "id": 13278, "type": "theorem", "label": "modules-lemma-flat-ses", "categories": [ "modules" ], "title": "modules-lemma-flat-ses", "contents": [ "\\begin{slogan}", "Kernels of epimorphisms and extensions of flat sheaves of modules over", "a ringed space are again flat.", "\\end{slogan}", "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let", "$$", "0 \\to", "\\mathcal{F}_2 \\to", "\\mathcal{F}_1 \\to", "\\mathcal{F}_0 \\to 0", "$$", "be a short exact sequence of $\\mathcal{O}_X$-modules.", "\\begin{enumerate}", "\\item If $\\mathcal{F}_2$ and $\\mathcal{F}_0$ are flat so is", "$\\mathcal{F}_1$.", "\\item If $\\mathcal{F}_1$ and $\\mathcal{F}_0$ are flat so is", "$\\mathcal{F}_2$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since exactness and flatness may be checked at the level of stalks", "this follows from", "Algebra, Lemma \\ref{algebra-lemma-flat-ses}." ], "refs": [ "algebra-lemma-flat-ses" ], "ref_ids": [ 533 ] } ], "ref_ids": [] }, { "id": 13279, "type": "theorem", "label": "modules-lemma-flat-resolution-of-flat", "categories": [ "modules" ], "title": "modules-lemma-flat-resolution-of-flat", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let", "$$", "\\ldots \\to", "\\mathcal{F}_2 \\to", "\\mathcal{F}_1 \\to", "\\mathcal{F}_0 \\to", "\\mathcal{Q} \\to 0", "$$", "be an exact complex of $\\mathcal{O}_X$-modules.", "If $\\mathcal{Q}$ and all $\\mathcal{F}_i$ are flat $\\mathcal{O}_X$-modules,", "then for any $\\mathcal{O}_X$-module $\\mathcal{G}$ the complex", "$$", "\\ldots \\to", "\\mathcal{F}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to", "\\mathcal{F}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to", "\\mathcal{F}_0 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to", "\\mathcal{Q} \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to 0", "$$", "is exact also." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-flat-tor-zero} by splitting the complex", "into short exact sequences and using Lemma \\ref{lemma-flat-ses} to", "prove inductively that $\\Im(\\mathcal{F}_{i + 1} \\to \\mathcal{F}_i)$", "is flat." ], "refs": [ "modules-lemma-flat-tor-zero", "modules-lemma-flat-ses" ], "ref_ids": [ 13277, 13278 ] } ], "ref_ids": [] }, { "id": 13280, "type": "theorem", "label": "modules-lemma-flat-eq", "categories": [ "modules" ], "title": "modules-lemma-flat-eq", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a flat", "$\\mathcal{O}_X$-module. Let $U \\subset X$ be open and let", "$$", "\\mathcal{O}_U \\xrightarrow{(f_1, \\ldots, f_n)}", "\\mathcal{O}_U^{\\oplus n} \\xrightarrow{(s_1, \\ldots, s_n)}", "\\mathcal{F}|_U", "$$", "be a complex of $\\mathcal{O}_U$-modules. For every $x \\in U$ there", "exists an open neighbourhood $V \\subset U$ of $x$ and a factorization", "$$", "\\mathcal{O}_V^{\\oplus n}", "\\xrightarrow{A}", "\\mathcal{O}_V^{\\oplus m} \\xrightarrow{(t_1, \\ldots, t_m)}", "\\mathcal{F}|_V", "$$", "of $(s_1, \\ldots, s_n)|_V$ such that $A \\circ (f_1, \\ldots, f_n)|_V = 0$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\subset \\mathcal{O}_U$ be the sheaf of ideals", "generated by $f_1, \\ldots, f_n$. Then $\\sum f_i \\otimes s_i$ is", "a section of $\\mathcal{I} \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U$", "which maps to zero in $\\mathcal{F}|_U$. As $\\mathcal{F}|_U$ is flat", "the map", "$\\mathcal{I} \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U \\to \\mathcal{F}|_U$", "is injective. Since $\\mathcal{I} \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U$", "is the sheaf associated to the presheaf tensor product, we see", "there exists an open neighbourhood $V \\subset U$ of $x$ such", "that $\\sum f_i|_V \\otimes s_i|_V$ is zero in", "$\\mathcal{I}(V) \\otimes_{\\mathcal{O}(V)} \\mathcal{F}(V)$.", "Unwinding the definitions using Algebra, Lemma \\ref{algebra-lemma-relations}", "we find $t_1, \\ldots, t_m \\in \\mathcal{F}(V)$ and $a_{ij} \\in \\mathcal{O}(V)$", "such that $\\sum a_{ij}f_i|_V = 0$ and $s_i|_V = \\sum a_{ij}t_j$." ], "refs": [ "algebra-lemma-relations" ], "ref_ids": [ 956 ] } ], "ref_ids": [] }, { "id": 13281, "type": "theorem", "label": "modules-lemma-left-dual-module", "categories": [ "modules" ], "title": "modules-lemma-left-dual-module", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be an", "$\\mathcal{O}_X$-module. Let $\\mathcal{G}, \\eta, \\epsilon$", "be a left dual of $\\mathcal{F}$ in the monoidal category of", "$\\mathcal{O}_X$-modules, see", "Categories, Definition \\ref{categories-definition-dual}. Then", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is locally a direct summand of a finite free", "$\\mathcal{O}_X$-module,", "\\item the map", "$e : \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X) \\to \\mathcal{G}$", "sending a local section $\\lambda$ to $(\\lambda \\otimes 1)(\\eta)$", "is an isomorphism,", "\\item we have $\\epsilon(f, g) = e^{-1}(g)(f)$ for local sections", "$f$ and $g$ of $\\mathcal{F}$ and $\\mathcal{G}$.", "\\end{enumerate}" ], "refs": [ "categories-definition-dual" ], "proofs": [ { "contents": [ "The assumptions mean that", "$$", "\\mathcal{F} \\xrightarrow{\\eta \\otimes 1}", "\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}", "\\otimes_{\\mathcal{O}_X} \\mathcal{F}", "\\xrightarrow{1 \\otimes \\epsilon} \\mathcal{F}", "\\quad\\text{and}\\quad", "\\mathcal{G} \\xrightarrow{1 \\otimes \\eta}", "\\mathcal{G} \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "\\otimes_{\\mathcal{O}_X} \\mathcal{G}", "\\xrightarrow{\\epsilon \\otimes 1} \\mathcal{G}", "$$", "are the identity map. Let $x \\in X$. We can find an open neighbourhood", "$U$ of $x$, a finite number of sections $f_1, \\ldots, f_n$", "and $g_1, \\ldots, g_n$ of", "$\\mathcal{F}$ and $\\mathcal{G}$ over $U$ such that", "$\\eta(1) = \\sum f_i g_i$. Denote", "$$", "\\mathcal{O}_U^{\\oplus n} \\to \\mathcal{F}|_U", "$$", "the map sending the $i$th basis vector to $f_i$. Then we", "can factor the map $\\eta|_U$ over a map", "$\\tilde \\eta : \\mathcal{O}_U \\to", "\\mathcal{O}_U^{\\oplus n} \\otimes_{\\mathcal{O}_U} \\mathcal{G}|_U$.", "We obtain a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{F}|_U", "\\ar[rr]_-{\\eta \\otimes 1} \\ar[rrd]_-{\\tilde \\eta \\otimes 1} & &", "\\mathcal{F}|_U \\otimes \\mathcal{G}|_U \\otimes \\mathcal{F}|_U", "\\ar[r]_-{1 \\otimes \\epsilon} &", "\\mathcal{F}|_U \\\\", "& &", "\\mathcal{O}_U^{\\oplus n} \\otimes \\mathcal{G}|_U \\otimes \\mathcal{F}|_U", "\\ar[u] \\ar[r]^-{1 \\otimes \\epsilon} &", "\\mathcal{O}_U^{\\oplus n} \\ar[u]", "}", "$$", "This shows that the identity on $\\mathcal{F}$ locally on $X$", "factors through a finite free module. This proves (1). Part (2) follows from", "Categories, Lemma \\ref{categories-lemma-left-dual} and its proof.", "Part (3) follows from the first equality of the proof.", "You can also deduce (2) and (3) from the uniqueness of left duals", "(Categories, Remark \\ref{categories-remark-left-dual-adjoint})", "and the construction of the left dual in", "Example \\ref{example-dual}." ], "refs": [ "categories-lemma-left-dual", "categories-remark-left-dual-adjoint" ], "ref_ids": [ 12325, 12430 ] } ], "ref_ids": [ 12407 ] }, { "id": 13282, "type": "theorem", "label": "modules-lemma-flat-locally-finite-presentation", "categories": [ "modules" ], "title": "modules-lemma-flat-locally-finite-presentation", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a", "flat $\\mathcal{O}_X$-module of finite presentation. Then $\\mathcal{F}$ is", "locally a direct summand of a finite free $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "After replacing $X$ by the members of an open covering, we may", "assume there exists a presentation", "$$", "\\mathcal{O}_X^{\\oplus r} \\to", "\\mathcal{O}_X^{\\oplus n} \\to \\mathcal{F} \\to 0", "$$", "Let $x \\in X$. By Lemma \\ref{lemma-flat-eq}", "we can, after shrinking $X$ to an open", "neighbourhood of $x$, assume there exists a factorization", "$$", "\\mathcal{O}_X^{\\oplus n} \\to", "\\mathcal{O}_X^{\\oplus n_1} \\to \\mathcal{F}", "$$", "such that the composition", "$\\mathcal{O}_X^{\\oplus r} \\to \\mathcal{O}_X^{\\oplus n} \\to", "\\mathcal{O}_X^{\\oplus n_1}$", "annihilates the first summand of $\\mathcal{O}_X^{\\oplus r}$.", "Repeating this argument $r - 1$ more times we obtain a factorization", "$$", "\\mathcal{O}_X^{\\oplus n} \\to", "\\mathcal{O}_X^{\\oplus n_r} \\to \\mathcal{F}", "$$", "such that the composition", "$\\mathcal{O}_X^{\\oplus r} \\to \\mathcal{O}_X^{\\oplus n}", "\\to \\mathcal{O}_X^{\\oplus n_r}$ is zero.", "This means that the surjection $\\mathcal{O}_X^{\\oplus n_r} \\to \\mathcal{F}$", "has a section and we win." ], "refs": [ "modules-lemma-flat-eq" ], "ref_ids": [ 13280 ] } ], "ref_ids": [] }, { "id": 13283, "type": "theorem", "label": "modules-lemma-surjection", "categories": [ "modules" ], "title": "modules-lemma-surjection", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the", "topology on $X$. Let $\\mathcal{F}$ be a sheaf of sets on $X$.", "There exists a set $I$ and for each $i \\in I$ an element", "$U_i \\in \\mathcal{B}$ and a finite set $S_i$ such that there exists", "a surjection $\\coprod_{i \\in I} j_{U_i!}\\underline{S_i} \\to \\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Let $S$ be a singleton set. We will prove the result with $S_i = S$.", "For every $x \\in X$ and element $s \\in \\mathcal{F}_x$ we can choose", "a $U(x, s) \\in \\mathcal{B}$ and $s(x, s) \\in \\mathcal{F}(U(x, s))$", "which maps to $s$ in $\\mathcal{F}_x$. By", "Sheaves, Lemma \\ref{sheaves-lemma-j-shriek}", "the section $s(x, s)$", "corresponds to a map of sheaves $j_{U(x, s)!}\\underline{S} \\to \\mathcal{F}$.", "Then", "$$", "\\coprod\\nolimits_{(x, s)} j_{U(x, s)!}\\underline{S} \\to \\mathcal{F}", "$$", "is surjective on stalks and hence surjective." ], "refs": [ "sheaves-lemma-j-shriek" ], "ref_ids": [ 14543 ] } ], "ref_ids": [] }, { "id": 13284, "type": "theorem", "label": "modules-lemma-filtered-colimit-constructibles", "categories": [ "modules" ], "title": "modules-lemma-filtered-colimit-constructibles", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the", "topology of $X$ and assume that each $U \\in \\mathcal{B}$ is quasi-compact.", "Then every sheaf of sets on $X$ is a filtered colimit of sheaves of the form", "\\begin{equation}", "\\label{equation-towards-constructible-sets}", "\\text{Coequalizer}\\left(", "\\xymatrix{", "\\coprod\\nolimits_{b = 1, \\ldots, m} j_{V_b!}\\underline{S_b}", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\coprod\\nolimits_{a = 1, \\ldots, n} j_{U_a!}\\underline{S_a}", "}", "\\right)", "\\end{equation}", "with $U_a$ and $V_b$ in $\\mathcal{B}$ and $S_a$ and $S_b$ finite sets." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-surjection} every sheaf of sets $\\mathcal{F}$", "is the target of a surjection whose source $\\mathcal{F}_0$ is a coproduct", "of sheaves the form $j_{U!}\\underline{S}$ with $U \\in \\mathcal{B}$", "and $S$ finite. Applying this to", "$\\mathcal{F}_0 \\times_\\mathcal{F} \\mathcal{F}_0$", "we find that $\\mathcal{F}$ is a coequalizer of a pair of maps", "$$", "\\xymatrix{", "\\coprod\\nolimits_{b \\in B} j_{V_b!}\\underline{S_b}", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\coprod\\nolimits_{a \\in A} j_{U_a!}\\underline{S_a}", "}", "$$", "for some index sets $A$, $B$ and $V_b$ and $U_a$ in $\\mathcal{B}$ and", "$S_a$ and $S_b$ finite. For every finite subset $B' \\subset B$", "there is a finite subset $A' \\subset A$ such that the coproduct", "over $b \\in B'$ maps into the coproduct over $a \\in A'$ via both maps.", "Namely, we can view the right hand side as a filtered colimit with", "injective transition maps. Hence taking sections over the quasi-compact", "opens $V_b$, $b \\in B'$ commutes with this coproduct,", "see Sheaves, Lemma \\ref{sheaves-lemma-directed-colimits-sections}.", "Thus our sheaf is the colimit of the cokernels of these maps", "between finite coproducts." ], "refs": [ "modules-lemma-surjection", "sheaves-lemma-directed-colimits-sections" ], "ref_ids": [ 13283, 14526 ] } ], "ref_ids": [] }, { "id": 13285, "type": "theorem", "label": "modules-lemma-constructible-comes-from-finite", "categories": [ "modules" ], "title": "modules-lemma-constructible-comes-from-finite", "contents": [ "Let $X$ be a spectral topological space. Let $\\mathcal{B}$ be", "the set of quasi-compact open subsets of $X$.", "Let $\\mathcal{F}$ be a sheaf of sets as in", "Equation (\\ref{equation-towards-constructible-sets}).", "Then there exists a continuous spectral map $f : X \\to Y$", "to a finite sober topological space $Y$ and a sheaf", "of sets $\\mathcal{G}$ on $Y$ with finite stalks", "such that $f^{-1}\\mathcal{G} \\cong \\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "We can write $X = \\lim X_i$ as a directed limit", "of finite sober spaces, see Topology, Lemma", "\\ref{topology-lemma-spectral-inverse-limit-finite-sober-spaces}.", "Of course the transition maps $X_{i'} \\to X_i$ are spectral and hence", "by Topology, Lemma \\ref{topology-lemma-directed-inverse-limit-spectral-spaces}", "the maps $p_i : X \\to X_i$ are spectral.", "For some $i$ we can find opens $U_{a, i}$ and $V_{b, i}$", "of $X_i$ whose inverse images are $U_a$ and $V_b$, see", "Topology, Lemma \\ref{topology-lemma-descend-opens}.", "The two maps", "$$", "\\beta, \\gamma :", "\\coprod\\nolimits_{b \\in B} j_{V_b!}\\underline{S_b}", "\\longrightarrow", "\\coprod\\nolimits_{a \\in A} j_{U_a!}\\underline{S_a}", "$$", "whose coequalizer is $\\mathcal{F}$ correspond by adjunction to two families", "$$", "\\beta_b, \\gamma_b :", "S_b", "\\longrightarrow", "\\Gamma(V_b, \\coprod\\nolimits_{a \\in A} j_{U_a!}\\underline{S_a}), \\quad", "b \\in B", "$$", "of maps of sets. Observe that", "$p_i^{-1}(j_{U_{a, i}!}\\underline{S_a}) = j_{U_a!}\\underline{S_a}$", "and $(X_{i'} \\to X_i)^{-1}(j_{U_{a, i}!}\\underline{S_a}) =", "j_{U_{a, i'}!}\\underline{S_a}$. It follows from", "Sheaves, Lemma \\ref{sheaves-lemma-compute-pullback-to-limit}", "(and using that $S_b$ and $B$ are finite sets) that", "after increasing $i$ we find maps", "$$", "\\beta_{b, i}, \\gamma_{b, i} :", "S_b", "\\longrightarrow", "\\Gamma(V_{b, i}, \\coprod\\nolimits_{a \\in A} j_{U_{a, i}!}\\underline{S_a})", ", \\quad b \\in B", "$$", "which give rise to the maps $\\beta_b$ and $\\gamma_b$ after pulling", "back by $p_i$. These maps correspond in turn to maps of sheaves", "$$", "\\beta_i, \\gamma_i :", "\\coprod\\nolimits_{b \\in B} j_{V_{b, i}!}\\underline{S_b}", "\\longrightarrow", "\\coprod\\nolimits_{a \\in A} j_{U_{a, i}!}\\underline{S_a}", "$$", "on $X_i$. Then we can take $Y = X_i$ and", "$$", "\\mathcal{G} =", "\\text{Coequalizer}\\left(", "\\xymatrix{", "\\coprod\\nolimits_{b = 1, \\ldots, m} j_{V_{b, i}!}\\underline{S_b}", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\coprod\\nolimits_{a = 1, \\ldots, n} j_{U_{a, i}!}\\underline{S_a}", "}", "\\right)", "$$", "We omit some details." ], "refs": [ "topology-lemma-spectral-inverse-limit-finite-sober-spaces", "topology-lemma-directed-inverse-limit-spectral-spaces", "topology-lemma-descend-opens", "sheaves-lemma-compute-pullback-to-limit" ], "ref_ids": [ 8315, 8321, 8322, 14527 ] } ], "ref_ids": [] }, { "id": 13286, "type": "theorem", "label": "modules-lemma-constructible-in-constant", "categories": [ "modules" ], "title": "modules-lemma-constructible-in-constant", "contents": [ "Let $X$ be a spectral topological space. Let $\\mathcal{B}$ be", "the set of quasi-compact open subsets of $X$.", "Let $\\mathcal{F}$ be a sheaf of sets as in", "Equation (\\ref{equation-towards-constructible-sets}).", "Then there exist finitely many constructible closed subsets", "$Z_1, \\ldots, Z_n \\subset X$ and finite sets $S_i$", "such that $\\mathcal{F}$ is isomorphic to a subsheaf of", "$\\prod (Z_i \\to X)_*\\underline{S_i}$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-constructible-comes-from-finite}", "we reduce to the case of a finite sober topological space", "and a sheaf with finite stalks.", "In this case $\\mathcal{F} \\subset \\prod_{x \\in X} i_{x, *}\\mathcal{F}_x$", "where $i_x : \\{x\\} \\to X$ is the embedding. We omit the proof", "that $i_{x, *}\\mathcal{F}_x$ is a constant sheaf on $\\overline{\\{x\\}}$." ], "refs": [ "modules-lemma-constructible-comes-from-finite" ], "ref_ids": [ 13285 ] } ], "ref_ids": [] }, { "id": 13287, "type": "theorem", "label": "modules-lemma-pullback-flat", "categories": [ "modules" ], "title": "modules-lemma-pullback-flat", "contents": [ "Let $f : X \\to Y$ be a flat morphism of ringed spaces.", "Then the pullback functor", "$f^* : \\textit{Mod}(\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X)$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "The functor $f^*$ is the composition of the exact functor", "$f^{-1} : \\textit{Mod}(\\mathcal{O}_Y) \\to \\textit{Mod}(f^{-1}\\mathcal{O}_Y)$", "and the change of rings functor", "$$", "\\textit{Mod}(f^{-1}\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X), \\quad", "\\mathcal{F} \\longmapsto", "\\mathcal{F} \\otimes_{f^{-1}\\mathcal{O}_Y} \\mathcal{O}_X.", "$$", "Thus the result follows from the discussion following", "Definition \\ref{definition-flat-morphism}." ], "refs": [ "modules-definition-flat-morphism" ], "ref_ids": [ 13345 ] } ], "ref_ids": [] }, { "id": 13288, "type": "theorem", "label": "modules-lemma-local-tensor-algebra", "categories": [ "modules" ], "title": "modules-lemma-local-tensor-algebra", "contents": [ "In the situation described above.", "The sheaf $\\wedge^n\\mathcal{F}$ is the sheafification of the", "presheaf", "$$", "U \\longmapsto \\wedge^n_{\\mathcal{O}_X(U)}(\\mathcal{F}(U)).", "$$", "See Algebra, Section \\ref{algebra-section-tensor-algebra}.", "Similarly, the sheaf $\\text{Sym}^n\\mathcal{F}$ is the sheafification", "of the presheaf", "$$", "U \\longmapsto \\text{Sym}^n_{\\mathcal{O}_X(U)}(\\mathcal{F}(U)).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted. It may be more efficient to define $\\text{Sym}(\\mathcal{F})$", "and $\\wedge(\\mathcal{F})$ in this way instead of the method", "given above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13289, "type": "theorem", "label": "modules-lemma-stalk-tensor-algebra", "categories": [ "modules" ], "title": "modules-lemma-stalk-tensor-algebra", "contents": [ "In the situation described above. Let $x \\in X$.", "There are canonical isomorphisms of $\\mathcal{O}_{X, x}$-modules", "$\\text{T}(\\mathcal{F})_x = \\text{T}(\\mathcal{F}_x)$,", "$\\text{Sym}(\\mathcal{F})_x = \\text{Sym}(\\mathcal{F}_x)$, and", "$\\wedge(\\mathcal{F})_x = \\wedge(\\mathcal{F}_x)$." ], "refs": [], "proofs": [ { "contents": [ "Clear from Lemma \\ref{lemma-local-tensor-algebra} above, and", "Algebra, Lemma \\ref{algebra-lemma-colimit-tensor-algebra}." ], "refs": [ "modules-lemma-local-tensor-algebra", "algebra-lemma-colimit-tensor-algebra" ], "ref_ids": [ 13288, 371 ] } ], "ref_ids": [] }, { "id": 13290, "type": "theorem", "label": "modules-lemma-pullback-tensor-algebra", "categories": [ "modules" ], "title": "modules-lemma-pullback-tensor-algebra", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of", "ringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_Y$-modules.", "Then $f^*\\text{T}(\\mathcal{F}) = \\text{T}(f^*\\mathcal{F})$,", "and similarly for the exterior and symmetric algebras associated", "to $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13291, "type": "theorem", "label": "modules-lemma-presentation-sym-exterior", "categories": [ "modules" ], "title": "modules-lemma-presentation-sym-exterior", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}_2 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to 0$", "be an exact sequence of sheaves of $\\mathcal{O}_X$-modules.", "For each $n \\geq 1$ there is an exact sequence", "$$", "\\mathcal{F}_2 \\otimes_{\\mathcal{O}_X} \\text{Sym}^{n - 1}(\\mathcal{F}_1)", "\\to", "\\text{Sym}^n(\\mathcal{F}_1)", "\\to", "\\text{Sym}^n(\\mathcal{F})", "\\to", "0", "$$", "and similarly an exact sequence", "$$", "\\mathcal{F}_2 \\otimes_{\\mathcal{O}_X} \\wedge^{n - 1}(\\mathcal{F}_1)", "\\to", "\\wedge^n(\\mathcal{F}_1)", "\\to", "\\wedge^n(\\mathcal{F})", "\\to", "0", "$$" ], "refs": [], "proofs": [ { "contents": [ "See Algebra, Lemma \\ref{algebra-lemma-presentation-sym-exterior}." ], "refs": [ "algebra-lemma-presentation-sym-exterior" ], "ref_ids": [ 369 ] } ], "ref_ids": [] }, { "id": 13292, "type": "theorem", "label": "modules-lemma-tensor-algebra-permanence", "categories": [ "modules" ], "title": "modules-lemma-tensor-algebra-permanence", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is locally generated by sections,", "then so is each $\\text{T}^n(\\mathcal{F})$,", "$\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$.", "\\item If $\\mathcal{F}$ is of finite type,", "then so is each $\\text{T}^n(\\mathcal{F})$,", "$\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$.", "\\item If $\\mathcal{F}$ is of finite presentation,", "then so is each $\\text{T}^n(\\mathcal{F})$,", "$\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$.", "\\item If $\\mathcal{F}$ is coherent,", "then for $n > 0$ each $\\text{T}^n(\\mathcal{F})$,", "$\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$", "is coherent.", "\\item If $\\mathcal{F}$ is quasi-coherent,", "then so is each $\\text{T}^n(\\mathcal{F})$,", "$\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$.", "\\item If $\\mathcal{F}$ is locally free,", "then so is each $\\text{T}^n(\\mathcal{F})$,", "$\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "These statements for $\\text{T}^n(\\mathcal{F})$ follow", "from Lemma \\ref{lemma-tensor-product-permanence}.", "\\medskip\\noindent", "Statements (1) and (2) follow from the fact that", "$\\wedge^n(\\mathcal{F})$ and $\\text{Sym}^n(\\mathcal{F})$", "are quotients of $\\text{T}^n(\\mathcal{F})$.", "\\medskip\\noindent", "Statement (6) follows from", "Algebra, Lemma \\ref{algebra-lemma-free-tensor-algebra}.", "\\medskip\\noindent", "For (3) and (5) we will use", "Lemma \\ref{lemma-presentation-sym-exterior} above.", "By locally choosing a presentation", "$\\mathcal{F}_2 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to 0$", "with $\\mathcal{F}_i$ free, or finite free and applying the", "lemma we see that $\\text{Sym}^n(\\mathcal{F})$, $\\wedge^n(\\mathcal{F})$", "has a similar presentation; here we use (6) and", "Lemma \\ref{lemma-tensor-product-permanence}.", "\\medskip\\noindent", "To prove (4) we will use", "Algebra, Lemma \\ref{algebra-lemma-present-sym-wedge}.", "We may localize on $X$ and assume that", "$\\mathcal{F}$ is generated by a finite set", "$(s_i)_{i \\in I}$ of global sections.", "The lemma mentioned above", "combined with Lemma \\ref{lemma-local-tensor-algebra} above", "implies that for $n \\geq 2$", "there exists an exact sequence", "$$", "\\bigoplus\\nolimits_{j \\in J}", "\\text{T}^{n - 2}(\\mathcal{F})", "\\to", "\\text{T}^n(\\mathcal{F})", "\\to", "\\text{Sym}^n(\\mathcal{F})", "\\to", "0", "$$", "where the index set $J$ is finite. Now we know that", "$\\text{T}^{n - 2}(\\mathcal{F})$ is finitely generated", "and hence the image of the first arrow is a coherent", "subsheaf of $\\text{T}^n(\\mathcal{F})$, see Lemma \\ref{lemma-coherent-abelian}.", "By that same lemma we conclude that $\\text{Sym}^n(\\mathcal{F})$ is", "coherent." ], "refs": [ "modules-lemma-tensor-product-permanence", "algebra-lemma-free-tensor-algebra", "modules-lemma-presentation-sym-exterior", "modules-lemma-tensor-product-permanence", "algebra-lemma-present-sym-wedge", "modules-lemma-local-tensor-algebra", "modules-lemma-coherent-abelian" ], "ref_ids": [ 13271, 368, 13291, 13271, 370, 13288, 13255 ] } ], "ref_ids": [] }, { "id": 13293, "type": "theorem", "label": "modules-lemma-whole-tensor-algebra-permanence", "categories": [ "modules" ], "title": "modules-lemma-whole-tensor-algebra-permanence", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is quasi-coherent,", "then so is each $\\text{T}(\\mathcal{F})$,", "$\\wedge(\\mathcal{F})$, and $\\text{Sym}(\\mathcal{F})$.", "\\item If $\\mathcal{F}$ is locally free,", "then so is each $\\text{T}(\\mathcal{F})$,", "$\\wedge(\\mathcal{F})$, and $\\text{Sym}(\\mathcal{F})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is not true that an infinite direct sum $\\bigoplus \\mathcal{G}_i$ of", "locally free modules is locally free, or that an", "infinite direct sum of quasi-coherent modules", "is quasi-coherent. The problem is that given a", "point $x \\in X$ the open neighbourhoods $U_i$ of $x$ on which $\\mathcal{G}_i$", "becomes free (resp.\\ has a suitable presentation) may have an intersection", "which is not an open neighbourhood of $x$. However, in the", "proof of Lemma \\ref{lemma-tensor-algebra-permanence} we saw that", "once a suitable open neighbourhood for $\\mathcal{F}$ has been chosen,", "then this open neighbourhood works for each of the sheaves", "$\\text{T}^n(\\mathcal{F})$, $\\wedge^n(\\mathcal{F})$ and", "$\\text{Sym}^n(\\mathcal{F})$.", "The lemma follows." ], "refs": [ "modules-lemma-tensor-algebra-permanence" ], "ref_ids": [ 13292 ] } ], "ref_ids": [] }, { "id": 13294, "type": "theorem", "label": "modules-lemma-internal-hom", "categories": [ "modules" ], "title": "modules-lemma-internal-hom", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$, $\\mathcal{G}$, $\\mathcal{H}$ be $\\mathcal{O}_X$-modules.", "There is a canonical isomorphism", "$$", "\\SheafHom_{\\mathcal{O}_X}", "(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}, \\mathcal{H})", "\\longrightarrow", "\\SheafHom_{\\mathcal{O}_X}", "(\\mathcal{F}, \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{H}))", "$$", "which is functorial in all three entries (sheaf Hom in", "all three spots). In particular, to give a", "morphism $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to \\mathcal{H}$", "is the same as giving a morphism", "$\\mathcal{F} \\to \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{H})$." ], "refs": [], "proofs": [ { "contents": [ "This is the analogue of", "Algebra, Lemma \\ref{algebra-lemma-hom-from-tensor-product}.", "The proof is the same, and is omitted." ], "refs": [ "algebra-lemma-hom-from-tensor-product" ], "ref_ids": [ 362 ] } ], "ref_ids": [] }, { "id": 13295, "type": "theorem", "label": "modules-lemma-internal-hom-exact", "categories": [ "modules" ], "title": "modules-lemma-internal-hom-exact", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules.", "\\begin{enumerate}", "\\item If $\\mathcal{F}_2 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to 0$", "is an exact sequence of $\\mathcal{O}_X$-modules, then", "$$", "0 \\to", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) \\to", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}_1, \\mathcal{G}) \\to", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}_2, \\mathcal{G})", "$$", "is exact.", "\\item If $0 \\to \\mathcal{G} \\to \\mathcal{G}_1 \\to \\mathcal{G}_2$", "is an exact sequence of $\\mathcal{O}_X$-modules, then", "$$", "0 \\to", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) \\to", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}_1) \\to", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}_2)", "$$", "is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13296, "type": "theorem", "label": "modules-lemma-stalk-internal-hom", "categories": [ "modules" ], "title": "modules-lemma-stalk-internal-hom", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules.", "If $\\mathcal{F}$ is finitely presented then the canonical map", "$$", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})_x", "\\to", "\\Hom_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x, \\mathcal{G}_x)", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By localizing on $X$ we may assume that $\\mathcal{F}$ has a presentation", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, m}", "\\mathcal{O}_X", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n}", "\\mathcal{O}_X", "\\to", "\\mathcal{F}", "\\to", "0.", "$$", "By Lemma \\ref{lemma-internal-hom-exact} this gives an exact sequence", "$", "0 \\to", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) \\to", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} \\mathcal{G}", "\\longrightarrow", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} \\mathcal{G}.", "$", "Taking stalks we get an exact sequence", "$", "0 \\to", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})_x \\to", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} \\mathcal{G}_x", "\\longrightarrow", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} \\mathcal{G}_x", "$", "and the result follows since $\\mathcal{F}_x$ sits in", "an exact sequence", "$", "\\bigoplus\\nolimits_{j = 1, \\ldots, m}", "\\mathcal{O}_{X, x}", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n}", "\\mathcal{O}_{X, x}", "\\to", "\\mathcal{F}_x", "\\to", "0", "$", "which induces the exact sequence", "$", "0 \\to", "\\Hom_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x, \\mathcal{G}_x) \\to", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} \\mathcal{G}_x", "\\longrightarrow", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} \\mathcal{G}_x", "$", "which is the same as the one above." ], "refs": [ "modules-lemma-internal-hom-exact" ], "ref_ids": [ 13295 ] } ], "ref_ids": [] }, { "id": 13297, "type": "theorem", "label": "modules-lemma-pullback-internal-hom", "categories": [ "modules" ], "title": "modules-lemma-pullback-internal-hom", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism", "of ringed spaces. Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_Y$-modules.", "If $\\mathcal{F}$ is finitely presented and $f$ is flat,", "then the canonical map", "$$", "f^*\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{F}, \\mathcal{G})", "\\longrightarrow", "\\SheafHom_{\\mathcal{O}_X}(f^*\\mathcal{F}, f^*\\mathcal{G})", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Note that $f^*\\mathcal{F}$ is also finitely presented", "(Lemma \\ref{lemma-pullback-finite-presentation}).", "Let $x \\in X$ map to $y \\in Y$. Looking at the stalks", "at $x$ we get an isomorphism by", "Lemma \\ref{lemma-stalk-internal-hom} and", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-pseudo-coherence-and-base-change-ext}", "to see that in this case $\\Hom$ commutes with base change by", "$\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$.", "Second proof: use the exact same argument as given", "in the proof of Lemma \\ref{lemma-stalk-internal-hom}." ], "refs": [ "modules-lemma-pullback-finite-presentation", "modules-lemma-stalk-internal-hom", "more-algebra-lemma-pseudo-coherence-and-base-change-ext", "modules-lemma-stalk-internal-hom" ], "ref_ids": [ 13250, 13296, 10165, 13296 ] } ], "ref_ids": [] }, { "id": 13298, "type": "theorem", "label": "modules-lemma-internal-hom-locally-kernel-direct-sum", "categories": [ "modules" ], "title": "modules-lemma-internal-hom-locally-kernel-direct-sum", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules.", "If $\\mathcal{F}$ is finitely presented then the sheaf", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is", "locally a kernel of a map between finite direct sums", "of copies of $\\mathcal{G}$.", "In particular, if $\\mathcal{G}$ is coherent then", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$", "is coherent too." ], "refs": [], "proofs": [ { "contents": [ "The first assertion", "we saw in the proof of Lemma \\ref{lemma-stalk-internal-hom}.", "And the result for coherent sheaves then follows from", "Lemma \\ref{lemma-coherent-abelian}." ], "refs": [ "modules-lemma-stalk-internal-hom", "modules-lemma-coherent-abelian" ], "ref_ids": [ 13296, 13255 ] } ], "ref_ids": [] }, { "id": 13299, "type": "theorem", "label": "modules-lemma-adjoint-tensor-restrict", "categories": [ "modules" ], "title": "modules-lemma-adjoint-tensor-restrict", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings. Then we have", "$$", "\\Hom_{\\mathcal{O}_1}(\\mathcal{F}_{\\mathcal{O}_1}, \\mathcal{G}) =", "\\Hom_{\\mathcal{O}_2}(\\mathcal{F},", "\\SheafHom_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{G}))", "$$", "bifunctorially in $\\mathcal{F} \\in \\textit{Mod}(\\mathcal{O}_2)$", "and $\\mathcal{G} \\in \\textit{Mod}(\\mathcal{O}_1)$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. This is the analogue of", "Algebra, Lemma \\ref{algebra-lemma-adjoint-hom-restrict}", "and is proved in exactly the same way." ], "refs": [ "algebra-lemma-adjoint-hom-restrict" ], "ref_ids": [ 375 ] } ], "ref_ids": [] }, { "id": 13300, "type": "theorem", "label": "modules-lemma-invertible", "categories": [ "modules" ], "title": "modules-lemma-invertible", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{L}$", "be an $\\mathcal{O}_X$-module. Equivalent are", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is invertible, and", "\\item there exists an $\\mathcal{O}_X$-module $\\mathcal{N}$", "such that", "$\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N} \\cong \\mathcal{O}_X$.", "\\end{enumerate}", "In this case $\\mathcal{L}$ is locally a direct summand of a finite free", "$\\mathcal{O}_X$-module and the module $\\mathcal{N}$ in (2) is isomorphic to", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{L}, \\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Then the functor $- \\otimes_{\\mathcal{O}_X} \\mathcal{L}$", "is essentially surjective, hence there exists an $\\mathcal{O}_X$-module", "$\\mathcal{N}$ as in (2). If (2) holds, then the functor", "$- \\otimes_{\\mathcal{O}_X} \\mathcal{N}$ is a quasi-inverse", "to the functor $- \\otimes_{\\mathcal{O}_X} \\mathcal{L}$ and", "we see that (1) holds.", "\\medskip\\noindent", "Assume (1) and (2) hold. Denote", "$\\psi : \\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N} \\to \\mathcal{O}_X$", "the given isomorphism. Let $x \\in X$. Choose an open neighbourhood", "$U$ an integer $n \\geq 1$ and sections $s_i \\in \\mathcal{L}(U)$,", "$t_i \\in \\mathcal{N}(U)$ such that $\\psi(\\sum s_i \\otimes t_i) = 1$.", "Consider the isomorphisms", "$$", "\\mathcal{L}|_U \\to", "\\mathcal{L}|_U \\otimes_{\\mathcal{O}_U}", "\\mathcal{L}|_U \\otimes_{\\mathcal{O}_U} \\mathcal{N}|_U \\to \\mathcal{L}|_U", "$$", "where the first arrow sends $s$ to $\\sum s_i \\otimes s \\otimes t_i$", "and the second arrow sends $s \\otimes s' \\otimes t$ to $\\psi(s' \\otimes t)s$.", "We conclude that $s \\mapsto \\sum \\psi(s \\otimes t_i)s_i$ is", "an automorphism of $\\mathcal{L}|_U$. This automorphism factors as", "$$", "\\mathcal{L}|_U \\to \\mathcal{O}_U^{\\oplus n} \\to \\mathcal{L}|_U", "$$", "where the first arrow is given by", "$s \\mapsto (\\psi(s \\otimes t_1), \\ldots, \\psi(s \\otimes t_n))$", "and the second arrow by $(a_1, \\ldots, a_n) \\mapsto \\sum a_i s_i$.", "In this way we conclude that $\\mathcal{L}|_U$ is a direct summand", "of a finite free $\\mathcal{O}_U$-module.", "\\medskip\\noindent", "Assume (1) and (2) hold. Consider the evaluation map", "$$", "\\mathcal{L} \\otimes_{\\mathcal{O}_X}", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{L}, \\mathcal{O}_X)", "\\longrightarrow \\mathcal{O}_X", "$$", "To finish the proof of the lemma", "we will show this is an isomorphism by checking it induces", "isomorphisms on stalks. Let $x \\in X$.", "Since we know (by the previous paragraph)", "that $\\mathcal{L}$ is a finitely presented", "$\\mathcal{O}_X$-module", "we can use Lemma \\ref{lemma-stalk-internal-hom}", "to see that it suffices to show that", "$$", "\\mathcal{L}_x \\otimes_{\\mathcal{O}_{X, x}}", "\\Hom_{\\mathcal{O}_{X, x}}(\\mathcal{L}_x, \\mathcal{O}_{X, x})", "\\longrightarrow \\mathcal{O}_{X, x}", "$$", "is an isomorphism. Since", "$\\mathcal{L}_x \\otimes_{\\mathcal{O}_{X, x}} \\mathcal{N}_x =", "(\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N})_x =", "\\mathcal{O}_{X, x}$ (Lemma \\ref{lemma-stalk-tensor-product})", "the desired result follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-invertible}." ], "refs": [ "modules-lemma-stalk-internal-hom", "modules-lemma-stalk-tensor-product", "more-algebra-lemma-invertible" ], "ref_ids": [ 13296, 13267, 10533 ] } ], "ref_ids": [] }, { "id": 13301, "type": "theorem", "label": "modules-lemma-pullback-invertible", "categories": [ "modules" ], "title": "modules-lemma-pullback-invertible", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a", "morphism of ringed spaces. The pullback $f^*\\mathcal{L}$ of an", "invertible $\\mathcal{O}_Y$-module is invertible." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-invertible}", "there exists an $\\mathcal{O}_Y$-module $\\mathcal{N}$ such that", "$\\mathcal{L} \\otimes_{\\mathcal{O}_Y} \\mathcal{N} \\cong \\mathcal{O}_Y$.", "Pulling back we get", "$f^*\\mathcal{L} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{N} \\cong \\mathcal{O}_X$", "by Lemma \\ref{lemma-tensor-product-pullback}.", "Thus $f^*\\mathcal{L}$ is invertible by Lemma \\ref{lemma-invertible}." ], "refs": [ "modules-lemma-invertible", "modules-lemma-tensor-product-pullback", "modules-lemma-invertible" ], "ref_ids": [ 13300, 13270, 13300 ] } ], "ref_ids": [] }, { "id": 13302, "type": "theorem", "label": "modules-lemma-invertible-is-locally-free-rank-1", "categories": [ "modules" ], "title": "modules-lemma-invertible-is-locally-free-rank-1", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Any locally free", "$\\mathcal{O}_X$-module of rank $1$ is invertible.", "If all stalks $\\mathcal{O}_{X, x}$ are local rings, then", "the converse holds as well (but in general this is not the case)." ], "refs": [], "proofs": [ { "contents": [ "The parenthetical statement follows by considering a one point", "space $X$ with sheaf of rings $\\mathcal{O}_X$ given by a ring $R$.", "Then invertible $\\mathcal{O}_X$-modules correspond to invertible", "$R$-modules, hence as soon as $\\Pic(R)$ is not the trivial group,", "then we get an example.", "\\medskip\\noindent", "Assume $\\mathcal{L}$ is locally free of rank $1$ and consider the", "evaluation map", "$$", "\\mathcal{L} \\otimes_{\\mathcal{O}_X}", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{L}, \\mathcal{O}_X)", "\\longrightarrow \\mathcal{O}_X", "$$", "Looking over an open covering trivialization $\\mathcal{L}$, we see", "that this map is an isomorphism. Hence $\\mathcal{L}$ is invertible", "by Lemma \\ref{lemma-invertible}.", "\\medskip\\noindent", "Assume all stalks $\\mathcal{O}_{X, x}$ are local rings and $\\mathcal{L}$", "invertible. In the proof of Lemma \\ref{lemma-invertible}", "we have seen that $\\mathcal{L}_x$ is an invertible", "$\\mathcal{O}_{X, x}$-module for all $x \\in X$. Since", "$\\mathcal{O}_{X, x}$ is local, we see that", "$\\mathcal{L}_x \\cong \\mathcal{O}_{X, x}$", "(More on Algebra, Section \\ref{more-algebra-section-picard}).", "Since $\\mathcal{L}$ is of finite presentation by", "Lemma \\ref{lemma-invertible} we conclude that $\\mathcal{L}$", "is locally free of rank $1$ by", "Lemma \\ref{lemma-finite-presentation-stalk-free}." ], "refs": [ "modules-lemma-invertible", "modules-lemma-invertible", "modules-lemma-invertible", "modules-lemma-finite-presentation-stalk-free" ], "ref_ids": [ 13300, 13300, 13300, 13253 ] } ], "ref_ids": [] }, { "id": 13303, "type": "theorem", "label": "modules-lemma-constructions-invertible", "categories": [ "modules" ], "title": "modules-lemma-constructions-invertible", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "\\begin{enumerate}", "\\item If $\\mathcal{L}$, $\\mathcal{N}$ are invertible", "$\\mathcal{O}_X$-modules, then so is", "$\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}$.", "\\item If $\\mathcal{L}$ is an invertible $\\mathcal{O}_X$-module, then so is", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{L}, \\mathcal{O}_X)$ and the evaluation map", "$\\mathcal{L} \\otimes_{\\mathcal{O}_X}", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{L}, \\mathcal{O}_X) \\to \\mathcal{O}_X$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is clear from the definition and part (2) follows from", "Lemma \\ref{lemma-invertible} and its proof." ], "refs": [ "modules-lemma-invertible" ], "ref_ids": [ 13300 ] } ], "ref_ids": [] }, { "id": 13304, "type": "theorem", "label": "modules-lemma-pic-set", "categories": [ "modules" ], "title": "modules-lemma-pic-set", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "There exists a set of invertible modules $\\{\\mathcal{L}_i\\}_{i \\in I}$", "such that each invertible module on $X$ is isomorphic to exactly", "one of the $\\mathcal{L}_i$." ], "refs": [], "proofs": [ { "contents": [ "Recall that any invertible $\\mathcal{O}_X$-module is locally", "a direct summand of a finite free $\\mathcal{O}_X$-module, see", "Lemma \\ref{lemma-invertible}.", "For each open covering $\\mathcal{U} : X = \\bigcup_{j \\in J} U_j$", "and map $r : J \\to \\mathbf{N}$ consider the sheaves of", "$\\mathcal{O}_X$-modules $\\mathcal{F}$ such that", "$\\mathcal{F}_j = \\mathcal{F}|_{U_j}$ is a direct summand of", "$\\mathcal{O}_{U_j}^{\\oplus r(j)}$.", "The collection of isomorphism classes of $\\mathcal{F}_j$ is a set, because", "$\\Hom_{\\mathcal{O}_U}(\\mathcal{O}_U^{\\oplus r}, \\mathcal{O}_U^{\\oplus r})$", "is a set. The sheaf $\\mathcal{F}$ is gotten by glueing $\\mathcal{F}_j$,", "see Sheaves, Section", "\\ref{sheaves-section-glueing-sheaves}. Note that the collection of", "all glueing data forms a set. The collection of all coverings", "$\\mathcal{U} : X = \\bigcup_{j \\in J} U_i$ where $J \\to \\mathcal{P}(X)$,", "$j \\mapsto U_j$ is injective forms a set as well. For each covering", "there is a set of maps $r : J \\to \\mathbf{N}$. Hence the collection", "of all $\\mathcal{F}$ forms a set." ], "refs": [ "modules-lemma-invertible" ], "ref_ids": [ 13300 ] } ], "ref_ids": [] }, { "id": 13305, "type": "theorem", "label": "modules-lemma-s-open", "categories": [ "modules" ], "title": "modules-lemma-s-open", "contents": [ "\\begin{slogan}", "A (local) trivialisation of a linebundle", "is the same as a (local) nonvanishing section.", "\\end{slogan}", "Let $X$ be a ringed space. Assume that each stalk $\\mathcal{O}_{X, x}$", "is a local ring with maximal ideal $\\mathfrak m_x$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "For any section $s \\in \\Gamma(X, \\mathcal{L})$ the set", "$$", "X_s = \\{x \\in X \\mid \\text{image }s \\not\\in \\mathfrak m_x\\mathcal{L}_x\\}", "$$", "is open in $X$. The map $s : \\mathcal{O}_{X_s} \\to \\mathcal{L}|_{X_s}$", "is an isomorphism, and there exists a section $s'$", "of $\\mathcal{L}^{\\otimes -1}$ over $X_s$ such that $s' (s|_{X_s}) = 1$." ], "refs": [], "proofs": [ { "contents": [ "Suppose $x \\in X_s$.", "We have an isomorphism", "$$", "\\mathcal{L}_x \\otimes_{\\mathcal{O}_{X, x}} (\\mathcal{L}^{\\otimes -1})_x", "\\longrightarrow", "\\mathcal{O}_{X, x}", "$$", "by Lemma \\ref{lemma-constructions-invertible}.", "Both $\\mathcal{L}_x$ and $(\\mathcal{L}^{\\otimes -1})_x$", "are free $\\mathcal{O}_{X, x}$-modules of rank $1$. We conclude", "from Algebra, Nakayama's Lemma \\ref{algebra-lemma-NAK} that", "$s_x$ is a basis for $\\mathcal{L}_x$. Hence there exists", "a basis element $t_x \\in (\\mathcal{L}^{\\otimes -1})_x$", "such that $s_x \\otimes t_x$ maps to $1$.", "Choose an open neighbourhood $U$ of", "$x$ such that $t_x$ comes from a section $t$", "of $\\mathcal{L}^{\\otimes -1}$ over $U$ and such that", "$s \\otimes t$ maps to $1 \\in \\mathcal{O}_X(U)$.", "Clearly, for every $x' \\in U$ we see that $s$ generates", "the module $\\mathcal{L}_{x'}$. Hence $U \\subset X_s$.", "This proves that $X_s$ is open. Moreover, the section", "$t$ constructed over $U$ above is unique, and hence", "these glue to give the section $s'$ of the lemma." ], "refs": [ "modules-lemma-constructions-invertible", "algebra-lemma-NAK" ], "ref_ids": [ 13303, 401 ] } ], "ref_ids": [] }, { "id": 13306, "type": "theorem", "label": "modules-lemma-det-ses", "categories": [ "modules" ], "title": "modules-lemma-det-ses", "contents": [ "Let $X$ be a ringed space. Let", "$0 \\to \\mathcal{E}' \\to \\mathcal{E} \\to \\mathcal{E}'' \\to 0$", "be a short exact sequence of finite locally free $\\mathcal{O}_X$-modules,", "Then there is a canonical isomorphism", "$$", "\\det(\\mathcal{E}') \\otimes_{\\mathcal{O}_X}\\det(\\mathcal{E}'')", "\\longrightarrow", "\\det(\\mathcal{E})", "$$", "of $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "We can decompose $X$ into disjoint open and closed subsets such that", "both $\\mathcal{E}'$ and $\\mathcal{E}''$ have constant rank on them.", "Thus we reduce to the case where $\\mathcal{E}'$ and $\\mathcal{E}''$", "have constant rank, say $r'$ and $r''$. In this situation we", "define", "$$", "\\wedge^{r'}(\\mathcal{E}') \\otimes_{\\mathcal{O}_X} \\wedge^{r''}(\\mathcal{E}'')", "\\longrightarrow", "\\wedge^{r' + r''}(\\mathcal{E})", "$$", "as follows. Given local sections $s'_1, \\ldots, s'_{r'}$ of $\\mathcal{E}'$", "and local sections $s''_1, \\ldots, s''_{r''}$ of $\\mathcal{E}''$", "we map", "$$", "s'_1 \\wedge \\ldots \\wedge s'_{r'} \\otimes", "s''_1 \\wedge \\ldots \\wedge s''_{r''}", "\\quad\\text{to}\\quad", "s'_1 \\wedge \\ldots \\wedge s'_{r'} \\wedge", "\\tilde s''_1 \\wedge \\ldots \\wedge \\tilde s''_{r''}", "$$", "where $\\tilde s''_i$ is a local lift of the section", "$s''_i$ to a section of $\\mathcal{E}$. We omit the details." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13307, "type": "theorem", "label": "modules-lemma-determinant-as-socle", "categories": [ "modules" ], "title": "modules-lemma-determinant-as-socle", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$", "be a flat and finitely presented $\\mathcal{O}_X$-module.", "Denote", "$$", "\\det(\\mathcal{F}) \\subset", "\\wedge^*_{\\mathcal{O}_X}(\\mathcal{F})", "$$", "the annihilator of $\\mathcal{F} \\subset \\wedge^*_{\\mathcal{O}_X}(\\mathcal{F})$.", "Then $\\det(\\mathcal{F})$ is an invertible $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "To prove this we may work locally on $X$. Hence we may assume $\\mathcal{F}$", "is a direct summand of a finite free module, see", "Lemma \\ref{lemma-flat-locally-finite-presentation}. Say", "$\\mathcal{F} \\oplus \\mathcal{G} = \\mathcal{O}_X^{\\oplus n}$.", "Set $R = \\mathcal{O}_X(X)$. Then we see", "$\\mathcal{F}(X) \\oplus \\mathcal{G}(X) = R^{\\oplus n}$", "and correspondingly", "$\\mathcal{F}(U) \\oplus \\mathcal{G}(U) = \\mathcal{O}_X(U)^{\\oplus n}$", "for all opens $U \\subset X$.", "We conclude that $\\mathcal{F} = \\mathcal{F}_M$ as in", "Lemma \\ref{lemma-construct-quasi-coherent-sheaves}", "with $M = \\mathcal{F}(X)$ a finite projective $R$-module.", "In other words, we have $\\mathcal{F}(U) = M \\otimes_R \\mathcal{O}_X(U)$.", "This implies that", "$\\det(M) \\otimes_R \\mathcal{O}_X(U) = \\det(\\mathcal{F}(U))$", "for all open $U \\subset X$ with $\\det$ as in", "More on Algebra, Section \\ref{more-algebra-section-determinants}. By", "More on Algebra, Remark \\ref{more-algebra-remark-determinant-as-socle}", "we see that", "$$", "\\det(M) \\otimes_R \\mathcal{O}_X(U) =", "\\det(\\mathcal{F}(U)) \\subset", "\\wedge^*_{\\mathcal{O}_X(U)}(\\mathcal{F}(U))", "$$", "is the annihilator of $\\mathcal{F}(U)$. We conclude that", "$\\det(\\mathcal{F})$ as defined in the statement of the lemma", "is equal to $\\mathcal{F}_{\\det(M)}$. Some details omitted; one has", "to be careful as annihilators cannot be defined as the sheafification", "of taking annihilators on sections over opens. Thus $\\det(\\mathcal{F})$", "is the pullback of an invertible module and we conclude." ], "refs": [ "modules-lemma-flat-locally-finite-presentation", "modules-lemma-construct-quasi-coherent-sheaves", "more-algebra-remark-determinant-as-socle" ], "ref_ids": [ 13282, 13245, 10680 ] } ], "ref_ids": [] }, { "id": 13308, "type": "theorem", "label": "modules-lemma-simple-invert", "categories": [ "modules" ], "title": "modules-lemma-simple-invert", "contents": [ "Let $X$ be a topological space and let $\\mathcal{O}_X$ be a", "presheaf of rings. Let $\\mathcal{S} \\subset \\mathcal{O}_X$", "be a pre-sheaf of sets contained in $\\mathcal{O}_X$.", "Suppose that for every open $U \\subset X$ the set", "$\\mathcal{S}(U) \\subset \\mathcal{O}_X(U)$ is a multiplicative subset.", "\\begin{enumerate}", "\\item There is a map of presheaves of rings", "$\\mathcal{O}_X \\to \\mathcal{S}^{-1}\\mathcal{O}_X$", "such that every local section of $\\mathcal{S}$ maps to an invertible", "section of $\\mathcal{O}_X$.", "\\item For any homomorphism of presheaves of rings", "$\\mathcal{O}_X \\to \\mathcal{A}$ such that each local section", "of $\\mathcal{S}$ maps to an invertible section of $\\mathcal{A}$", "there exists a unique factorization", "$\\mathcal{S}^{-1}\\mathcal{O}_X \\to \\mathcal{A}$.", "\\item For any $x \\in X$ we have", "$$", "(\\mathcal{S}^{-1}\\mathcal{O}_X)_x = \\mathcal{S}_x^{-1} \\mathcal{O}_{X, x}.", "$$", "\\item The sheafification $(\\mathcal{S}^{-1}\\mathcal{O}_X)^\\#$ is a sheaf", "of rings with a map of sheaves of rings", "$(\\mathcal{O}_X)^\\# \\to (\\mathcal{S}^{-1}\\mathcal{O}_X)^\\#$", "which is universal for maps of $(\\mathcal{O}_X)^\\#$ into sheaves", "of rings such that each local section of $\\mathcal{S}$ maps", "to an invertible section.", "\\item For any $x \\in X$ we have", "$$", "(\\mathcal{S}^{-1}\\mathcal{O}_X)^\\#_x = \\mathcal{S}_x^{-1} \\mathcal{O}_{X, x}.", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13309, "type": "theorem", "label": "modules-lemma-simple-invert-module", "categories": [ "modules" ], "title": "modules-lemma-simple-invert-module", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{O}_X$ be a presheaf of rings.", "Let $\\mathcal{S} \\subset \\mathcal{O}_X$ be a pre-sheaf of sets contained", "in $\\mathcal{O}_X$. Suppose that for every open $U \\subset X$ the set", "$\\mathcal{S}(U) \\subset \\mathcal{O}_X(U)$ is a multiplicative subset.", "For any presheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ we", "have", "$$", "\\mathcal{S}^{-1}\\mathcal{F}", "=", "\\mathcal{S}^{-1}\\mathcal{O}_X \\otimes_{p, \\mathcal{O}_X} \\mathcal{F}", "$$", "(see Sheaves, Section \\ref{sheaves-section-presheaves-modules} for notation)", "and if $\\mathcal{F}$ and $\\mathcal{O}_X$ are sheaves then", "$$", "(\\mathcal{S}^{-1}\\mathcal{F})^\\#", "=", "(\\mathcal{S}^{-1}\\mathcal{O}_X)^\\# \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "$$", "(see Sheaves, Section \\ref{sheaves-section-sheafification-presheaves-modules}", "for notation)." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13310, "type": "theorem", "label": "modules-lemma-universal-module", "categories": [ "modules" ], "title": "modules-lemma-universal-module", "contents": [ "Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings. The functor", "$$", "\\textit{Mod}(\\mathcal{O}_2) \\longrightarrow \\textit{Ab}, \\quad", "\\mathcal{F} \\longmapsto \\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})", "$$", "is representable." ], "refs": [], "proofs": [ { "contents": [ "This is proved in exactly the same way as the analogous statement in algebra.", "During this proof, for any sheaf of sets $\\mathcal{F}$ on $X$,", "let us denote $\\mathcal{O}_2[\\mathcal{F}]$ the sheafification of the", "presheaf $U \\mapsto \\mathcal{O}_2(U)[\\mathcal{F}(U)]$ where this denotes", "the free $\\mathcal{O}_2(U)$-module on the set $\\mathcal{F}(U)$.", "For $s \\in \\mathcal{F}(U)$ we denote $[s]$ the corresponding section", "of $\\mathcal{O}_2[\\mathcal{F}]$ over $U$. If $\\mathcal{F}$ is a sheaf of", "$\\mathcal{O}_2$-modules, then there is a canonical map", "$$", "c : \\mathcal{O}_2[\\mathcal{F}] \\longrightarrow \\mathcal{F}", "$$", "which on the presheaf level is given by the rule", "$\\sum f_s[s] \\mapsto \\sum f_s s$. We will employ the short hand", "$[s] \\mapsto s$ to", "describe this map and similarly for other maps below. Consider", "the map of $\\mathcal{O}_2$-modules", "\\begin{equation}", "\\label{equation-define-module-differentials}", "\\begin{matrix}", "\\mathcal{O}_2[\\mathcal{O}_2 \\times \\mathcal{O}_2] \\oplus", "\\mathcal{O}_2[\\mathcal{O}_2 \\times \\mathcal{O}_2] \\oplus", "\\mathcal{O}_2[\\mathcal{O}_1] &", "\\longrightarrow &", "\\mathcal{O}_2[\\mathcal{O}_2] \\\\", "[(a, b)] \\oplus [(f, g)] \\oplus [h] & \\longmapsto & [a + b] - [a] - [b] + \\\\", "& & [fg] - g[f] - f[g] + \\\\", "& & [\\varphi(h)]", "\\end{matrix}", "\\end{equation}", "with short hand notation as above. Set $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$", "equal to the cokernel of this map. Then it is clear that there exists", "a map of sheaves of sets", "$$", "\\text{d} : \\mathcal{O}_2 \\longrightarrow \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}", "$$", "mapping a local section $f$ to the image of $[f]$ in", "$\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$. By construction $\\text{d}$", "is a $\\mathcal{O}_1$-derivation. Next, let $\\mathcal{F}$", "be a sheaf of $\\mathcal{O}_2$-modules and let", "$D : \\mathcal{O}_2 \\to \\mathcal{F}$ be a $\\mathcal{O}_1$-derivation.", "Then we can consider the $\\mathcal{O}_2$-linear map", "$\\mathcal{O}_2[\\mathcal{O}_2] \\to \\mathcal{F}$ which sends $[g]$ to $D(g)$.", "It follows from the definition of a derivation that this map annihilates", "sections in the image of the map (\\ref{equation-define-module-differentials})", "and hence defines a map", "$$", "\\alpha_D : \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\longrightarrow \\mathcal{F}", "$$", "Since it is clear that $D = \\alpha_D \\circ \\text{d}$ the lemma is proved." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13311, "type": "theorem", "label": "modules-lemma-differentials-sheafify", "categories": [ "modules" ], "title": "modules-lemma-differentials-sheafify", "contents": [ "Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings on $X$. Then", "$\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$ is the sheaf associated to the", "presheaf $U \\mapsto \\Omega_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the map (\\ref{equation-define-module-differentials}). There is", "a similar map of presheaves whose value on the open $U$ is", "$$", "\\mathcal{O}_2(U)[\\mathcal{O}_2(U) \\times \\mathcal{O}_2(U)] \\oplus", "\\mathcal{O}_2(U)[\\mathcal{O}_2(U) \\times \\mathcal{O}_2(U)] \\oplus", "\\mathcal{O}_2(U)[\\mathcal{O}_1(U)]", "\\longrightarrow", "\\mathcal{O}_2(U)[\\mathcal{O}_2(U)]", "$$", "The cokernel of this map has value $\\Omega_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}$", "over $U$ by the construction of the module of differentials in ", "Algebra, Definition \\ref{algebra-definition-differentials}.", "On the other hand, the sheaves in (\\ref{equation-define-module-differentials})", "are the sheafifications of the presheaves above. Thus the result follows", "as sheafification is exact." ], "refs": [ "algebra-definition-differentials" ], "ref_ids": [ 1526 ] } ], "ref_ids": [] }, { "id": 13312, "type": "theorem", "label": "modules-lemma-localize-differentials", "categories": [ "modules" ], "title": "modules-lemma-localize-differentials", "contents": [ "Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings. For $U \\subset X$ open", "there is a canonical isomorphism", "$$", "\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}|_U =", "\\Omega_{(\\mathcal{O}_2|_U)/(\\mathcal{O}_1|_U)}", "$$", "compatible with universal derivations." ], "refs": [], "proofs": [ { "contents": [ "Holds because $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$", "is the cokernel of the map (\\ref{equation-define-module-differentials})." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13313, "type": "theorem", "label": "modules-lemma-pullback-differentials", "categories": [ "modules" ], "title": "modules-lemma-pullback-differentials", "contents": [ "Let $f : Y \\to X$ be a continuous map of topological spaces.", "Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings on $X$.", "Then there is a canonical identification", "$f^{-1}\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} =", "\\Omega_{f^{-1}\\mathcal{O}_2/f^{-1}\\mathcal{O}_1}$", "compatible with universal derivations." ], "refs": [], "proofs": [ { "contents": [ "This holds because the sheaf $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$", "is the cokernel of the map (\\ref{equation-define-module-differentials})", "and a similar statement holds for", "$\\Omega_{f^{-1}\\mathcal{O}_2/f^{-1}\\mathcal{O}_1}$,", "because the functor $f^{-1}$ is exact, and because", "$f^{-1}(\\mathcal{O}_2[\\mathcal{O}_2]) =", "f^{-1}\\mathcal{O}_2[f^{-1}\\mathcal{O}_2]$,", "$f^{-1}(\\mathcal{O}_2[\\mathcal{O}_2 \\times \\mathcal{O}_2]) =", "f^{-1}\\mathcal{O}_2[f^{-1}\\mathcal{O}_2 \\times f^{-1}\\mathcal{O}_2]$, and", "$f^{-1}(\\mathcal{O}_2[\\mathcal{O}_1]) =", "f^{-1}\\mathcal{O}_2[f^{-1}\\mathcal{O}_1]$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13314, "type": "theorem", "label": "modules-lemma-stalk-module-differentials", "categories": [ "modules" ], "title": "modules-lemma-stalk-module-differentials", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings on $X$. Let $x \\in X$. Then we have", "$\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1, x} =", "\\Omega_{\\mathcal{O}_{2, x}/\\mathcal{O}_{1, x}}$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-pullback-differentials}", "for the inclusion map $\\{x\\} \\to X$. An alternative proof is to use", "Lemma \\ref{lemma-differentials-sheafify},", "Sheaves, Lemma \\ref{sheaves-lemma-stalk-sheafification}, and", "Algebra, Lemma \\ref{algebra-lemma-colimit-differentials}" ], "refs": [ "modules-lemma-pullback-differentials", "modules-lemma-differentials-sheafify", "sheaves-lemma-stalk-sheafification", "algebra-lemma-colimit-differentials" ], "ref_ids": [ 13313, 13311, 14494, 1130 ] } ], "ref_ids": [] }, { "id": 13315, "type": "theorem", "label": "modules-lemma-functoriality-differentials", "categories": [ "modules" ], "title": "modules-lemma-functoriality-differentials", "contents": [ "Let $X$ be a topological space. Let", "$$", "\\xymatrix{", "\\mathcal{O}_2 \\ar[r]_\\varphi & \\mathcal{O}_2' \\\\", "\\mathcal{O}_1 \\ar[r] \\ar[u] & \\mathcal{O}'_1 \\ar[u]", "}", "$$", "be a commutative diagram of sheaves of rings on $X$. The map", "$\\mathcal{O}_2 \\to \\mathcal{O}'_2$ composed with the map", "$\\text{d} : \\mathcal{O}'_2 \\to \\Omega_{\\mathcal{O}'_2/\\mathcal{O}'_1}$", "is a $\\mathcal{O}_1$-derivation. Hence we obtain a canonical map of", "$\\mathcal{O}_2$-modules", "$\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\to", "\\Omega_{\\mathcal{O}'_2/\\mathcal{O}'_1}$.", "It is uniquely characterized by the property that", "$\\text{d}(f) \\mapsto \\text{d}(\\varphi(f))$", "for any local section $f$ of $\\mathcal{O}_2$.", "In this way $\\Omega_{-/-}$ becomes a functor on the category", "of arrows of sheaves of rings." ], "refs": [], "proofs": [ { "contents": [ "This lemma proves itself." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13316, "type": "theorem", "label": "modules-lemma-differential-seq", "categories": [ "modules" ], "title": "modules-lemma-differential-seq", "contents": [ "In Lemma \\ref{lemma-functoriality-differentials} suppose that", "$\\mathcal{O}_2 \\to \\mathcal{O}'_2$ is surjective with kernel", "$\\mathcal{I} \\subset \\mathcal{O}_2$ and assume that", "$\\mathcal{O}_1 = \\mathcal{O}'_1$. Then there is a canonical exact", "sequence of $\\mathcal{O}'_2$-modules", "$$", "\\mathcal{I}/\\mathcal{I}^2", "\\longrightarrow", "\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\otimes_{\\mathcal{O}_2} \\mathcal{O}'_2", "\\longrightarrow", "\\Omega_{\\mathcal{O}'_2/\\mathcal{O}_1}", "\\longrightarrow", "0", "$$", "The leftmost map is characterized by the rule that a local section", "$f$ of $\\mathcal{I}$ maps to $\\text{d}f \\otimes 1$." ], "refs": [ "modules-lemma-functoriality-differentials" ], "proofs": [ { "contents": [ "For a local section $f$ of $\\mathcal{I}$ denote $\\overline{f}$ the image of", "$f$ in $\\mathcal{I}/\\mathcal{I}^2$. To show that the map", "$\\overline{f} \\mapsto \\text{d}f \\otimes 1$ is well defined we just have to", "check that $\\text{d} f_1f_2 \\otimes 1 = 0$ if $f_1, f_2$ are local sections", "of $\\mathcal{I}$. And this is clear from the Leibniz rule", "$\\text{d} f_1f_2 \\otimes 1 =", "(f_1 \\text{d}f_2 + f_2 \\text{d} f_1 )\\otimes 1 =", "\\text{d}f_2 \\otimes f_1 + \\text{d}f_1 \\otimes f_2 = 0$.", "A similar computation show this map is", "$\\mathcal{O}'_2 = \\mathcal{O}_2/\\mathcal{I}$-linear. The map on the right", "is the one from Lemma \\ref{lemma-functoriality-differentials}. To see", "that the sequence is exact, we can check on stalks", "(Lemma \\ref{lemma-abelian}). By", "Lemma \\ref{lemma-stalk-module-differentials}", "this follows from", "Algebra, Lemma \\ref{algebra-lemma-differential-seq}." ], "refs": [ "modules-lemma-functoriality-differentials", "modules-lemma-abelian", "modules-lemma-stalk-module-differentials", "algebra-lemma-differential-seq" ], "ref_ids": [ 13315, 13221, 13314, 1135 ] } ], "ref_ids": [ 13315 ] }, { "id": 13317, "type": "theorem", "label": "modules-lemma-double-structure-gives-derivation", "categories": [ "modules" ], "title": "modules-lemma-double-structure-gives-derivation", "contents": [ "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$", "be a morphism of ringed spaces. Consider a short exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{A} \\to \\mathcal{O}_X \\to 0", "$$", "Here $\\mathcal{A}$ is a sheaf of $f^{-1}\\mathcal{O}_S$-algebras,", "$\\pi : \\mathcal{A} \\to \\mathcal{O}_X$ is a surjection", "of sheaves of $f^{-1}\\mathcal{O}_S$-algebras, and", "$\\mathcal{I} = \\Ker(\\pi)$ is its kernel. Assume $\\mathcal{I}$ an ideal", "sheaf with square zero in $\\mathcal{A}$. So $\\mathcal{I}$", "has a natural structure of an $\\mathcal{O}_X$-module.", "A section $s : \\mathcal{O}_X \\to \\mathcal{A}$ of $\\pi$", "is a $f^{-1}\\mathcal{O}_S$-algebra map such that $\\pi \\circ s = \\text{id}$.", "Given any section $s : \\mathcal{O}_X \\to \\mathcal{A}$", "of $\\pi$ and any $S$-derivation $D : \\mathcal{O}_X \\to \\mathcal{I}$", "the map", "$$", "s + D : \\mathcal{O}_X \\to \\mathcal{A}", "$$", "is a section of $\\pi$ and every section $s'$ is of the form $s + D$", "for a unique $S$-derivation $D$." ], "refs": [], "proofs": [ { "contents": [ "Recall that the $\\mathcal{O}_X$-module structure on $\\mathcal{I}$", "is given by $h \\tau = \\tilde h \\tau$ (multiplication in $\\mathcal{A}$)", "where $h$ is a local section of $\\mathcal{O}_X$, and", "$\\tilde h$ is a local lift of $h$ to a local", "section of $\\mathcal{A}$, and $\\tau$ is a local section of $\\mathcal{I}$.", "In particular, given $s$, we may use $\\tilde h = s(h)$.", "To verify that $s + D$ is a homomorphism of sheaves of rings we", "compute", "\\begin{eqnarray*}", "(s + D)(ab) & = & s(ab) + D(ab) \\\\", "& = & s(a)s(b) + aD(b) + D(a)b \\\\", "& = & s(a) s(b) + s(a)D(b) + D(a)s(b) \\\\", "& = & (s(a) + D(a))(s(b) + D(b))", "\\end{eqnarray*}", "by the Leibniz rule. In the same manner one shows", "$s + D$ is a $f^{-1}\\mathcal{O}_S$-algebra", "map because $D$ is an $S$-derivation. Conversely, given $s'$ we set", "$D = s' - s$. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13318, "type": "theorem", "label": "modules-lemma-functoriality-differentials-ringed-spaces", "categories": [ "modules" ], "title": "modules-lemma-functoriality-differentials-ringed-spaces", "contents": [ "Let", "$$", "\\xymatrix{", "X' \\ar[d]_{h'} \\ar[r]_f & X \\ar[d]^h \\\\", "S' \\ar[r]^g & S", "}", "$$", "be a commutative diagram of ringed spaces.", "\\begin{enumerate}", "\\item The canonical map $\\mathcal{O}_X \\to f_*\\mathcal{O}_{X'}$ composed with", "$f_*\\text{d}_{X'/S'} : f_*\\mathcal{O}_{X'} \\to f_*\\Omega_{X'/S'}$ is a", "$S$-derivation and we obtain a canonical map of $\\mathcal{O}_X$-modules", "$\\Omega_{X/S} \\to f_*\\Omega_{X'/S'}$.", "\\item The commutative diagram", "$$", "\\xymatrix{", "f^{-1}\\mathcal{O}_X \\ar[r] & \\mathcal{O}_{X'} \\\\", "f^{-1}h^{-1}\\mathcal{O}_S \\ar[u] \\ar[r] & (h')^{-1}\\mathcal{O}_{S'} \\ar[u]", "}", "$$", "induces by Lemmas \\ref{lemma-pullback-differentials} and", "\\ref{lemma-functoriality-differentials}", "a canonical map $f^{-1}\\Omega_{X/S} \\to \\Omega_{X'/S'}$.", "\\end{enumerate}", "These two maps correspond (via adjointness of $f_*$ and $f^*$ and", "via $f^*\\Omega_{X/S} =", "f^{-1}\\Omega_{X/S} \\otimes_{f^{-1}\\mathcal{O}_X} \\mathcal{O}_{X'}$ and", "Sheaves, Lemma \\ref{sheaves-lemma-adjointness-tensor-restrict})", "to the same $\\mathcal{O}_{X'}$-module homomorphism", "$$", "c_f : f^*\\Omega_{X/S} \\longrightarrow \\Omega_{X'/S'}", "$$", "which is uniquely characterized by the property that", "$f^*\\text{d}_{X/S}(a)$ maps to $\\text{d}_{X'/S'}(f^*a)$", "for any local section $a$ of $\\mathcal{O}_X$." ], "refs": [ "modules-lemma-pullback-differentials", "modules-lemma-functoriality-differentials", "sheaves-lemma-adjointness-tensor-restrict" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13313, 13315, 14501 ] }, { "id": 13319, "type": "theorem", "label": "modules-lemma-check-functoriality-differentials", "categories": [ "modules" ], "title": "modules-lemma-check-functoriality-differentials", "contents": [ "Let", "$$", "\\xymatrix{", "X'' \\ar[d] \\ar[r]_g & X' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\", "S'' \\ar[r] & S' \\ar[r] & S", "}", "$$", "be a commutative diagram of ringed spaces. With notation as in", "Lemma \\ref{lemma-functoriality-differentials-ringed-spaces} we have", "$$", "c_{f \\circ g} = c_g \\circ g^* c_f", "$$", "as maps $(f \\circ g)^*\\Omega_{X/S} \\to \\Omega_{X''/S''}$." ], "refs": [ "modules-lemma-functoriality-differentials-ringed-spaces" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13318 ] }, { "id": 13320, "type": "theorem", "label": "modules-lemma-composition-differential-operators", "categories": [ "modules" ], "title": "modules-lemma-composition-differential-operators", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings on $X$.", "Let $\\mathcal{E}, \\mathcal{F}, \\mathcal{G}$ be sheaves of", "$\\mathcal{O}_2$-modules.", "If $D : \\mathcal{E} \\to \\mathcal{F}$ and $D' : \\mathcal{F} \\to \\mathcal{G}$", "are differential operators of order $k$ and $k'$, then $D' \\circ D$ is a", "differential operator of order $k + k'$." ], "refs": [], "proofs": [ { "contents": [ "Let $g$ be a local section of $\\mathcal{O}_2$.", "Then the map which sends a local section $x$ of $\\mathcal{E}$ to", "$$", "D'(D(gx)) - gD'(D(x)) = D'(D(gx)) - D'(gD(x)) + D'(gD(x)) - gD'(D(x))", "$$", "is a sum of two compositions of differential operators of lower order.", "Hence the lemma follows by induction on $k + k'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13321, "type": "theorem", "label": "modules-lemma-module-principal-parts", "categories": [ "modules" ], "title": "modules-lemma-module-principal-parts", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings on $X$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_2$-modules.", "Let $k \\geq 0$. There exists a sheaf of $\\mathcal{O}_2$-modules", "$\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F})$", "and a canonical isomorphism", "$$", "\\text{Diff}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}, \\mathcal{G}) =", "\\Hom_{\\mathcal{O}_2}(", "\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}), \\mathcal{G})", "$$", "functorial in the $\\mathcal{O}_2$-module $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "The existence follows from general category theoretic arguments", "(insert future reference here), but we will also give a direct", "construction as this construction will be useful in the future proofs.", "We will freely use the notation introduced in the proof of", "Lemma \\ref{lemma-universal-module}.", "Given any differential operator $D : \\mathcal{F} \\to \\mathcal{G}$", "we obtain an $\\mathcal{O}_2$-linear map", "$L_D : \\mathcal{O}_2[\\mathcal{F}] \\to \\mathcal{G}$", "sending $[m]$ to $D(m)$. If $D$ has order $0$", "then $L_D$ annihilates the local sections", "$$", "[m + m'] - [m] - [m'],\\quad", "g_0[m] - [g_0m]", "$$", "where $g_0$ is a local section of $\\mathcal{O}_2$ and $m, m'$", "are local sections of $\\mathcal{F}$. If $D$ has order $1$, then $L_D$", "annihilates the local sections", "$$", "[m + m' - [m] - [m'],\\quad", "f[m] - [fm], \\quad", "g_0g_1[m] - g_0[g_1m] - g_1[g_0m] + [g_1g_0m]", "$$", "where $f$ is a local section of $\\mathcal{O}_1$,", "$g_0, g_1$ are local sections of $\\mathcal{O}_2$, and", "$m, m'$ are local sections of $\\mathcal{F}$.", "If $D$ has order $k$, then $L_D$ annihilates the local sections", "$[m + m'] - [m] - [m']$, $f[m] - [fm]$, and the local sections", "$$", "g_0g_1\\ldots g_k[m] - \\sum g_0 \\ldots \\hat g_i \\ldots g_k[g_im] + \\ldots", "+(-1)^{k + 1}[g_0\\ldots g_km]", "$$", "Conversely, if $L : \\mathcal{O}_2[\\mathcal{F}] \\to \\mathcal{G}$ is an", "$\\mathcal{O}_2$-linear map annihilating all the local sections", "listed in the previous sentence, then $m \\mapsto L([m])$ is a", "differential operator of order $k$. Thus we see that", "$\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F})$", "is the quotient of $\\mathcal{O}_2[\\mathcal{F}]$", "by the $\\mathcal{O}_2$-submodule generated by these local sections." ], "refs": [ "modules-lemma-universal-module" ], "ref_ids": [ 13310 ] } ], "ref_ids": [] }, { "id": 13322, "type": "theorem", "label": "modules-lemma-differential-operators-sheafify", "categories": [ "modules" ], "title": "modules-lemma-differential-operators-sheafify", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of presheaves of rings on $X$. Let $\\mathcal{F}$ be a", "presheaf of $\\mathcal{O}_2$-modules. Then", "$\\mathcal{P}^k_{\\mathcal{O}_2^\\#/\\mathcal{O}_1^\\#}(\\mathcal{F}^\\#)$", "is the sheaf associated to the presheaf", "$U \\mapsto P^k_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}(\\mathcal{F}(U))$." ], "refs": [], "proofs": [ { "contents": [ "This can be proved in exactly the same way as is done for the sheaf", "of differentials in Lemma \\ref{lemma-differentials-sheafify}.", "Perhaps a more pleasing approach is to use the universal property", "of Lemma \\ref{lemma-module-principal-parts} directly to see the equality.", "We omit the details." ], "refs": [ "modules-lemma-differentials-sheafify", "modules-lemma-module-principal-parts" ], "ref_ids": [ 13311, 13321 ] } ], "ref_ids": [] }, { "id": 13323, "type": "theorem", "label": "modules-lemma-sequence-of-principal-parts", "categories": [ "modules" ], "title": "modules-lemma-sequence-of-principal-parts", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings on $X$. Let $\\mathcal{F}$ be a", "sheaf of $\\mathcal{O}_2$-modules. There is a", "canonical short exact sequence", "$$", "0 \\to", "\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\otimes_{\\mathcal{O}_2} \\mathcal{F} \\to", "\\mathcal{P}^1_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}) \\to", "\\mathcal{F} \\to 0", "$$", "functorial in $\\mathcal{F}$ called the {\\it sequence of principal parts}." ], "refs": [], "proofs": [ { "contents": [ "Follows from the commutative algebra version", "(Algebra, Lemma \\ref{algebra-lemma-sequence-of-principal-parts})", "and Lemmas \\ref{lemma-differentials-sheafify} and", "\\ref{lemma-differential-operators-sheafify}." ], "refs": [ "algebra-lemma-sequence-of-principal-parts", "modules-lemma-differentials-sheafify", "modules-lemma-differential-operators-sheafify" ], "ref_ids": [ 1146, 13311, 13322 ] } ], "ref_ids": [] }, { "id": 13324, "type": "theorem", "label": "modules-lemma-pullback-de-rham-complex", "categories": [ "modules" ], "title": "modules-lemma-pullback-de-rham-complex", "contents": [ "Let $f : Y \\to X$ be a continuous map of topological spaces.", "Let $\\mathcal{A} \\to \\mathcal{B}$", "be a homomorphism of sheaves of rings on $X$.", "Then there is a canonical identification", "$f^{-1}\\Omega^\\bullet_{\\mathcal{B}/\\mathcal{A}} =", "\\Omega^\\bullet_{f^{-1}\\mathcal{B}/f^{-1}\\mathcal{A}}$", "of de Rham complexes." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: compare with Lemma \\ref{lemma-pullback-differentials}." ], "refs": [ "modules-lemma-pullback-differentials" ], "ref_ids": [ 13313 ] } ], "ref_ids": [] }, { "id": 13325, "type": "theorem", "label": "modules-lemma-differentials-de-rham-complex-order-1", "categories": [ "modules" ], "title": "modules-lemma-differentials-de-rham-complex-order-1", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{A} \\to \\mathcal{B}$", "be a homomorphism of sheaves of rings on $X$. The differentials", "$\\text{d} : \\Omega^i_{\\mathcal{B}/\\mathcal{A}} \\to", "\\Omega^{i + 1}_{\\mathcal{B}/\\mathcal{A}}$", "are differential operators of order $1$." ], "refs": [], "proofs": [ { "contents": [ "Via our construction of the de Rham complex above as the sheafification", "of the rule $U \\mapsto \\Omega^\\bullet_{\\mathcal{B}(U)/\\mathcal{A}(U)}$", "this follows from", "Algebra, Lemma \\ref{algebra-lemma-differentials-de-rham-complex-order-1}." ], "refs": [ "algebra-lemma-differentials-de-rham-complex-order-1" ], "ref_ids": [ 1147 ] } ], "ref_ids": [] }, { "id": 13326, "type": "theorem", "label": "modules-lemma-differentials-relative-de-rham-complex-order-1", "categories": [ "modules" ], "title": "modules-lemma-differentials-relative-de-rham-complex-order-1", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces. The differentials", "$\\text{d} : \\Omega^i_{X/Y} \\to \\Omega^{i + 1}_{X/Y}$", "are differential operators of order $1$ on $X/Y$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-differentials-de-rham-complex-order-1}", "and the definition." ], "refs": [ "modules-lemma-differentials-de-rham-complex-order-1" ], "ref_ids": [ 13325 ] } ], "ref_ids": [] }, { "id": 13327, "type": "theorem", "label": "modules-lemma-NL-up-to-qis", "categories": [ "modules" ], "title": "modules-lemma-NL-up-to-qis", "contents": [ "In the situation above there is a canonical isomorphism", "$\\NL(\\alpha) = \\NL_{\\mathcal{B}/\\mathcal{A}}$ in $D(\\mathcal{B})$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\NL_{\\mathcal{B}/\\mathcal{A}} = \\NL(\\text{id}_\\mathcal{B})$.", "Thus it suffices to show that given two maps", "$\\alpha_i : \\mathcal{E}_i \\to \\mathcal{B}$ as above, there is a", "canonical quasi-isomorphism $\\NL(\\alpha_1) = \\NL(\\alpha_2)$ in $D(\\mathcal{B})$.", "To see this set $\\mathcal{E} = \\mathcal{E}_1 \\amalg \\mathcal{E}_2$ and", "$\\alpha = \\alpha_1 \\amalg \\alpha_2 : \\mathcal{E} \\to \\mathcal{B}$.", "Set", "$\\mathcal{J}_i = \\Ker(\\mathcal{A}[\\mathcal{E}_i] \\to \\mathcal{B})$", "and", "$\\mathcal{J} = \\Ker(\\mathcal{A}[\\mathcal{E}] \\to \\mathcal{B})$.", "We obtain maps $\\mathcal{A}[\\mathcal{E}_i] \\to \\mathcal{A}[\\mathcal{E}]$", "which send $\\mathcal{J}_i$ into $\\mathcal{J}$.", "Thus we obtain canonical maps of complexes", "$$", "\\NL(\\alpha_i) \\longrightarrow \\NL(\\alpha)", "$$", "and it suffices to show these maps are quasi-isomorphism. To see this", "it suffices to check on stalks (Lemma \\ref{lemma-abelian}). If $x \\in X$", "then the stalk of $\\NL(\\alpha)$ is the complex $\\NL(\\alpha_x)$ of", "Algebra, Section \\ref{algebra-section-netherlander}", "associated to the presentation $\\mathcal{A}_x[\\mathcal{E}_x] \\to \\mathcal{B}_x$", "coming from the map $\\alpha_x : \\mathcal{E}_x \\to \\mathcal{B}_x$.", "(Some details omitted; use Lemma \\ref{lemma-stalk-module-differentials}", "to see compatibility of forming differentials and taking stalks.)", "We conclude the result holds by", "Algebra, Lemma \\ref{algebra-lemma-NL-homotopy}." ], "refs": [ "modules-lemma-abelian", "modules-lemma-stalk-module-differentials", "algebra-lemma-NL-homotopy" ], "ref_ids": [ 13221, 13314, 1151 ] } ], "ref_ids": [] }, { "id": 13328, "type": "theorem", "label": "modules-lemma-pullback-NL", "categories": [ "modules" ], "title": "modules-lemma-pullback-NL", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings", "on $Y$. Then $f^{-1}\\NL_{\\mathcal{B}/\\mathcal{A}} =", "\\NL_{f^{-1}\\mathcal{B}/f^{-1}\\mathcal{A}}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use Lemma \\ref{lemma-pullback-differentials}." ], "refs": [ "modules-lemma-pullback-differentials" ], "ref_ids": [ 13313 ] } ], "ref_ids": [] }, { "id": 13329, "type": "theorem", "label": "modules-lemma-stalk-NL", "categories": [ "modules" ], "title": "modules-lemma-stalk-NL", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{A} \\to \\mathcal{B}$", "be a homomorphism of sheaves of rings on $X$. Let $x \\in X$.", "Then we have $\\NL_{\\mathcal{B}/\\mathcal{A}, x} =", "\\NL_{\\mathcal{B}_x/\\mathcal{A}_x}$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-pullback-NL}", "for the inclusion map $\\{x\\} \\to X$." ], "refs": [ "modules-lemma-pullback-NL" ], "ref_ids": [ 13328 ] } ], "ref_ids": [] }, { "id": 13330, "type": "theorem", "label": "modules-lemma-exact-sequence-NL", "categories": [ "modules" ], "title": "modules-lemma-exact-sequence-NL", "contents": [ "Let $X$ be a topological space. Let", "$\\mathcal{A} \\to \\mathcal{B} \\to \\mathcal{C}$", "be maps of sheaves of rings. Let $C$ be the cone", "(Derived Categories, Definition \\ref{derived-definition-cone})", "of the map of complexes", "$\\NL_{\\mathcal{C}/\\mathcal{A}} \\to \\NL_{\\mathcal{C}/\\mathcal{B}}$.", "There is a canonical map", "$$", "c :", "\\NL_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B} \\mathcal{C}", "\\longrightarrow", "C[-1]", "$$", "of complexes of $\\mathcal{C}$-modules", "which produces a canonical six term exact sequence", "$$", "\\xymatrix{", "H^0(\\NL_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B} \\mathcal{C}) \\ar[r] &", "H^0(\\NL_{\\mathcal{C}/\\mathcal{A}}) \\ar[r] &", "H^0(\\NL_{\\mathcal{C}/\\mathcal{B}}) \\ar[r] &", "0 \\\\", "H^{-1}(\\NL_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B} \\mathcal{C}) \\ar[r] &", "H^{-1}(\\NL_{\\mathcal{C}/\\mathcal{A}}) \\ar[r] &", "H^{-1}(\\NL_{\\mathcal{C}/\\mathcal{B}}) \\ar[llu]", "}", "$$", "of cohomology sheaves." ], "refs": [ "derived-definition-cone" ], "proofs": [ { "contents": [ "To give the map $c$ we have to give a map", "$c_1 : \\NL_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B} \\mathcal{C}", "\\to \\NL_{\\mathcal{C}/\\mathcal{A}}$ and an explicity homotopy", "between the composition", "$$", "\\NL_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B} \\mathcal{C} \\to", "\\NL_{\\mathcal{C}/\\mathcal{A}} \\to \\NL_{\\mathcal{C}/\\mathcal{B}}", "$$", "and the zero map, see", "Derived Categories, Lemma \\ref{derived-lemma-map-from-cone}.", "For $c_1$ we use the functoriality described", "above for the obvious diagram. For the homotopy we use the map", "$$", "\\NL_{\\mathcal{B}/\\mathcal{A}}^0 \\otimes_\\mathcal{B} \\mathcal{C}", "\\longrightarrow", "\\NL_{\\mathcal{C}/\\mathcal{B}}^{-1},\\quad", "\\text{d}[b] \\otimes 1 \\longmapsto [\\varphi(b)] - b[1]", "$$", "where $\\varphi : \\mathcal{B} \\to \\mathcal{C}$ is the given map.", "Please compare with", "Algebra, Remark \\ref{algebra-remark-composition-homotopy-equivalent-to-zero}.", "To see the consequence for cohomology sheaves, it suffices to show", "that $H^0(c)$ is an isomorphism and $H^{-1}(c)$ surjective.", "To see this we can look at stalks, see Lemma \\ref{lemma-stalk-NL},", "and then we can use the corresponding result in commutative algebra,", "see Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL}.", "Some details omitted." ], "refs": [ "derived-lemma-map-from-cone", "algebra-remark-composition-homotopy-equivalent-to-zero", "modules-lemma-stalk-NL", "algebra-lemma-exact-sequence-NL" ], "ref_ids": [ 1795, 1579, 13329, 1153 ] } ], "ref_ids": [ 1978 ] }, { "id": 13331, "type": "theorem", "label": "modules-lemma-exact-sequence-NL-ringed-topoi", "categories": [ "modules" ], "title": "modules-lemma-exact-sequence-NL-ringed-topoi", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces.", "Let $C$ be the cone of the map $\\NL_{X/Z} \\to \\NL_{X/Y}$ of complexes", "of $\\mathcal{O}_X$-modules. There is a canonical map", "$$", "f^*\\NL_{Y/Z} \\to C[-1]", "$$", "which produces a canonical six term exact sequence", "$$", "\\xymatrix{", "H^0(f^*\\NL_{Y/Z}) \\ar[r] &", "H^0(\\NL_{X/Z}) \\ar[r] &", "H^0(\\NL_{X/Y}) \\ar[r] &", "0 \\\\", "H^{-1}(f^*\\NL_{Y/Z}) \\ar[r] &", "H^{-1}(\\NL_{X/Z}) \\ar[r] &", "H^{-1}(\\NL_{X/Y}) \\ar[llu]", "}", "$$", "of cohomology sheaves." ], "refs": [], "proofs": [ { "contents": [ "Consider the maps of sheaves rings", "$$", "(g \\circ f)^{-1}\\mathcal{O}_Z \\to f^{-1}\\mathcal{O}_Y \\to \\mathcal{O}_X", "$$", "and apply Lemma \\ref{lemma-exact-sequence-NL}." ], "refs": [ "modules-lemma-exact-sequence-NL" ], "ref_ids": [ 13330 ] } ], "ref_ids": [] }, { "id": 13369, "type": "theorem", "label": "defos-lemma-huge-diagram", "categories": [ "defos" ], "title": "defos-lemma-huge-diagram", "contents": [ "Given a commutative diagram", "$$", "\\xymatrix{", "& 0 \\ar[r] & N_2 \\ar[r] & B'_2 \\ar[r] & B_2 \\ar[r] & 0 \\\\", "& 0 \\ar[r]|\\hole & I_2 \\ar[u]_{c_2} \\ar[r] &", "A'_2 \\ar[u] \\ar[r]|\\hole & A_2 \\ar[u] \\ar[r] & 0 \\\\", "0 \\ar[r] & N_1 \\ar[ruu] \\ar[r] & B'_1 \\ar[r] & B_1 \\ar[ruu] \\ar[r] & 0 \\\\", "0 \\ar[r] & I_1 \\ar[ruu]|\\hole \\ar[u]^{c_1} \\ar[r] &", "A'_1 \\ar[ruu]|\\hole \\ar[u] \\ar[r] & A_1 \\ar[ruu]|\\hole \\ar[u] \\ar[r] & 0", "}", "$$", "with front and back solutions to (\\ref{equation-to-solve}) we have", "\\begin{enumerate}", "\\item There exist a canonical element in", "$\\Ext^1_{B_1}(\\NL_{B_1/A_1}, N_2)$", "whose vanishing is a necessary and sufficient condition for the existence", "of a ring map $B'_1 \\to B'_2$ fitting into the diagram.", "\\item If there exists a map $B'_1 \\to B'_2$ fitting into the diagram", "the set of all such maps is a principal homogeneous space under", "$\\Hom_{B_1}(\\Omega_{B_1/A_1}, N_2)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $E = B_1$ viewed as a set.", "Consider the surjection $A_1[E] \\to B_1$ with kernel $J$ used", "to define the naive cotangent complex by the formula", "$$", "\\NL_{B_1/A_1} = (J/J^2 \\to \\Omega_{A_1[E]/A_1} \\otimes_{A_1[E]} B_1)", "$$", "in", "Algebra, Section \\ref{algebra-section-netherlander}.", "Since $\\Omega_{A_1[E]/A_1} \\otimes B_1$ is a free", "$B_1$-module we have", "$$", "\\Ext^1_{B_1}(\\NL_{B_1/A_1}, N_2) =", "\\frac{\\Hom_{B_1}(J/J^2, N_2)}", "{\\Hom_{B_1}(\\Omega_{A_1[E]/A_1} \\otimes B_1, N_2)}", "$$", "We will construct an obstruction in the module on the right.", "Let $J' = \\Ker(A'_1[E] \\to B_1)$. Note that there is a surjection", "$J' \\to J$ whose kernel is $I_1A_1[E]$.", "For every $e \\in E$ denote $x_e \\in A_1[E]$ the corresponding variable.", "Choose a lift $y_e \\in B'_1$ of the image of $x_e$ in $B_1$ and", "a lift $z_e \\in B'_2$ of the image of $x_e$ in $B_2$.", "These choices determine $A'_1$-algebra maps", "$$", "A'_1[E] \\to B'_1 \\quad\\text{and}\\quad A'_1[E] \\to B'_2", "$$", "The first of these gives a map $J' \\to N_1$, $f' \\mapsto f'(y_e)$", "and the second gives a map $J' \\to N_2$, $f' \\mapsto f'(z_e)$.", "A calculation shows that these maps annihilate $(J')^2$.", "Because the left square of the diagram (involving $c_1$ and $c_2$)", "commutes we see that these maps agree on $I_1A_1[E]$ as maps into $N_2$.", "Observe that $B'_1$ is the pushout of $J' \\to A'_1[B_1]$ and $J' \\to N_1$. ", "Thus, if the maps $J' \\to N_1 \\to N_2$ and $J' \\to N_2$ agree, then we", "obtain a map $B'_1 \\to B'_2$ fitting into the diagram.", "Thus we let the obstruction be the class of the map", "$$", "J/J^2 \\to N_2,\\quad f \\mapsto f'(z_e) - \\nu(f'(y_e))", "$$", "where $\\nu : N_1 \\to N_2$ is the given map and where $f' \\in J'$", "is a lift of $f$. This is well defined by our remarks above.", "Note that we have the freedom", "to modify our choices of $z_e$ into $z_e + \\delta_{2, e}$", "and $y_e$ into $y_e + \\delta_{1, e}$ for some $\\delta_{i, e} \\in N_i$.", "This will modify the map above into", "$$", "f \\mapsto f'(z_e + \\delta_{2, e}) - \\nu(f'(y_e + \\delta_{1, e})) =", "f'(z_e) - \\nu(f'(z_e)) +", "\\sum (\\delta_{2, e} - \\nu(\\delta_{1, e}))\\frac{\\partial f}{\\partial x_e}", "$$", "This means exactly that we are modifying the map $J/J^2 \\to N_2$", "by the composition $J/J^2 \\to \\Omega_{A_1[E]/A_1} \\otimes B_1 \\to N_2$", "where the second map sends $\\text{d}x_e$ to", "$\\delta_{2, e} - \\nu(\\delta_{1, e})$. Thus our obstruction is well defined", "and is zero if and only if a lift exists.", "\\medskip\\noindent", "Part (2) comes from the observation that given two maps", "$\\varphi, \\psi : B'_1 \\to B'_2$ fitting into the diagram, then", "$\\varphi - \\psi$ factors through a map $D : B_1 \\to N_2$ which", "is an $A_1$-derivation:", "\\begin{align*}", "D(fg) & = \\varphi(f'g') - \\psi(f'g') \\\\", "& =", "\\varphi(f')\\varphi(g') - \\psi(f')\\psi(g') \\\\", "& =", "(\\varphi(f') - \\psi(f'))\\varphi(g') + \\psi(f')(\\varphi(g') - \\psi(g')) \\\\", "& =", "gD(f) + fD(g)", "\\end{align*}", "Thus $D$ corresponds to a unique $B_1$-linear map", "$\\Omega_{B_1/A_1} \\to N_2$. Conversely, given such a linear map", "we get a derivation $D$ and given a ring map $\\psi : B'_1 \\to B'_2$", "fitting into the diagram", "the map $\\psi + D$ is another ring map fitting into the diagram." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13370, "type": "theorem", "label": "defos-lemma-existence-lci", "categories": [ "defos" ], "title": "defos-lemma-existence-lci", "contents": [ "If $A \\to B$ is a local complete intersection ring map, then", "there exists a solution to (\\ref{equation-to-solve})." ], "refs": [], "proofs": [ { "contents": [ "Write $B = A[x_1, \\ldots, x_n]/J$. Let $J' \\subset A'[x_1, \\ldots, x_n]$", "be the inverse image of $J$. Denote $I[x_1, \\ldots, x_n]$ the", "kernel of $A'[x_1, \\ldots, x_n] \\to A[x_1, \\ldots, x_n]$.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-conormal-sequence-H1-regular-ideal} we have", "$I[x_1, \\ldots, x_n] \\cap (J')^2 = J'I[x_1, \\ldots, x_n] =", "JI[x_1, \\ldots, x_n]$. Hence we obtain a short exact sequence", "$$", "0 \\to I \\otimes_A B \\to J'/(J')^2 \\to J/J^2 \\to 0", "$$", "Since $J/J^2$ is projective (More on Algebra, Lemma", "\\ref{more-algebra-lemma-quasi-regular-ideal-finite-projective})", "we can choose a splitting of this sequence", "$$", "J'/(J')^2 = I \\otimes_A B \\oplus J/J^2 ", "$$", "Let $(J')^2 \\subset J'' \\subset J'$ be the elements which map to the", "second summand in the decomposition above. Then", "$$", "0 \\to I \\otimes_A B \\to A'[x_1, \\ldots, x_n]/J'' \\to B \\to 0", "$$", "is a solution to (\\ref{equation-to-solve}) with $N = I \\otimes_A B$.", "The general case is obtained by doing a pushout along the given", "map $I \\otimes_A B \\to N$." ], "refs": [ "more-algebra-lemma-conormal-sequence-H1-regular-ideal", "more-algebra-lemma-quasi-regular-ideal-finite-projective" ], "ref_ids": [ 9998, 9996 ] } ], "ref_ids": [] }, { "id": 13371, "type": "theorem", "label": "defos-lemma-choices", "categories": [ "defos" ], "title": "defos-lemma-choices", "contents": [ "If there exists a solution to (\\ref{equation-to-solve}), then the set of", "isomorphism classes of solutions is principal homogeneous under", "$\\Ext^1_B(\\NL_{B/A}, N)$." ], "refs": [], "proofs": [ { "contents": [ "We observe right away that given two solutions $B'_1$ and $B'_2$", "to (\\ref{equation-to-solve}) we obtain by Lemma \\ref{lemma-huge-diagram} an", "obstruction element $o(B'_1, B'_2) \\in \\Ext^1_B(\\NL_{B/A}, N)$", "to the existence of a map $B'_1 \\to B'_2$. Clearly, this element", "is the obstruction to the existence of an isomorphism, hence separates", "the isomorphism classes. To finish the proof it therefore suffices to", "show that given a solution $B'$ and an element", "$\\xi \\in \\Ext^1_B(\\NL_{B/A}, N)$", "we can find a second solution $B'_\\xi$ such that", "$o(B', B'_\\xi) = \\xi$.", "\\medskip\\noindent", "Let $E = B$ viewed as a set. Consider the surjection $A[E] \\to B$ with kernel", "$J$ used to define the naive cotangent complex by the formula", "$$", "\\NL_{B/A} = (J/J^2 \\to \\Omega_{A[E]/A} \\otimes_{A[E]} B)", "$$", "in Algebra, Section \\ref{algebra-section-netherlander}.", "Since $\\Omega_{A[E]/A} \\otimes B$ is a free $B$-module we have", "$$", "\\Ext^1_B(\\NL_{B/A}, N) =", "\\frac{\\Hom_B(J/J^2, N)}", "{\\Hom_B(\\Omega_{A[E]/A} \\otimes B, N)}", "$$", "Thus we may represent $\\xi$ as the class of a morphism $\\delta : J/J^2 \\to N$.", "\\medskip\\noindent", "For every $e \\in E$ denote $x_e \\in A[E]$ the corresponding variable.", "Choose a lift $y_e \\in B'$ of the image of $x_e$ in $B$.", "These choices determine an $A'$-algebra map $\\varphi : A'[E] \\to B'$.", "Let $J' = \\Ker(A'[E] \\to B)$. Observe that $\\varphi$ induces a map", "$\\varphi|_{J'} : J' \\to N$ and that $B'$ is the pushout, as in the following", "diagram", "$$", "\\xymatrix{", "0 \\ar[r] & N \\ar[r] & B' \\ar[r] & B \\ar[r] & 0 \\\\", "0 \\ar[r] & J' \\ar[u]^{\\varphi|_{J'}} \\ar[r] & A'[E] \\ar[u] \\ar[r] &", "B \\ar[u]_{=} \\ar[r] & 0", "}", "$$", "Let $\\psi : J' \\to N$ be the sum of the map $\\varphi|_{J'}$ and the", "composition", "$$", "J' \\to J'/(J')^2 \\to J/J^2 \\xrightarrow{\\delta} N.", "$$", "Then the pushout along $\\psi$ is an other ring extension $B'_\\xi$", "fitting into a diagram as above. A calculation shows that", "$o(B', B'_\\xi) = \\xi$ as desired." ], "refs": [ "defos-lemma-huge-diagram" ], "ref_ids": [ 13369 ] } ], "ref_ids": [] }, { "id": 13372, "type": "theorem", "label": "defos-lemma-extensions-of-rings", "categories": [ "defos" ], "title": "defos-lemma-extensions-of-rings", "contents": [ "Let $A$ be a ring and let $I$ be an $A$-module.", "\\begin{enumerate}", "\\item The set of extensions of rings $0 \\to I \\to A' \\to A \\to 0$", "where $I$ is an ideal of square zero is canonically bijective to", "$\\Ext^1_A(\\NL_{A/\\mathbf{Z}}, I)$.", "\\item Given a ring map $A \\to B$, a $B$-module $N$, an $A$-module", "map $c : I \\to N$, and given extensions of rings with square zero kernels:", "\\begin{enumerate}", "\\item[(a)] $0 \\to I \\to A' \\to A \\to 0$ corresponding to", "$\\alpha \\in \\Ext^1_A(\\NL_{A/\\mathbf{Z}}, I)$, and", "\\item[(b)] $0 \\to N \\to B' \\to B \\to 0$ corresponding to", "$\\beta \\in \\Ext^1_B(\\NL_{B/\\mathbf{Z}}, N)$", "\\end{enumerate}", "then there is a map $A' \\to B'$ fitting into a diagram", "(\\ref{equation-to-solve}) if and only if $\\beta$ and $\\alpha$", "map to the same element of", "$\\Ext^1_A(\\NL_{A/\\mathbf{Z}}, N)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove this we apply the previous results where we work over", "$0 \\to 0 \\to \\mathbf{Z} \\to \\mathbf{Z} \\to 0$, in order words,", "we work over the extension of $\\mathbf{Z}$ by $0$.", "Part (1) follows from Lemma \\ref{lemma-choices}", "and the fact that there exists a solution, namely $I \\oplus A$.", "Part (2) follows from Lemma \\ref{lemma-huge-diagram}", "and a compatibility between the constructions in the proofs", "of Lemmas \\ref{lemma-choices} and \\ref{lemma-huge-diagram}", "whose statement and proof we omit." ], "refs": [ "defos-lemma-choices", "defos-lemma-huge-diagram", "defos-lemma-choices", "defos-lemma-huge-diagram" ], "ref_ids": [ 13371, 13369, 13371, 13369 ] } ], "ref_ids": [] }, { "id": 13373, "type": "theorem", "label": "defos-lemma-strict-morphism-thickenings", "categories": [ "defos" ], "title": "defos-lemma-strict-morphism-thickenings", "contents": [ "In Situation \\ref{situation-morphism-thickenings} the morphism $(f, f')$", "is a strict morphism of thickenings if and only if", "(\\ref{equation-morphism-thickenings}) is cartesian in the category", "of ringed spaces." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13374, "type": "theorem", "label": "defos-lemma-inf-map", "categories": [ "defos" ], "title": "defos-lemma-inf-map", "contents": [ "Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$", "be a first order thickening of ringed spaces. Assume given", "extensions", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0", "\\quad\\text{and}\\quad", "0 \\to \\mathcal{L} \\to \\mathcal{G}' \\to \\mathcal{G} \\to 0", "$$", "as in (\\ref{equation-extension})", "and maps $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and", "$\\psi : \\mathcal{K} \\to \\mathcal{L}$.", "\\begin{enumerate}", "\\item If there exists an $\\mathcal{O}_{X'}$-module", "map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$", "and $\\psi$, then the diagram", "$$", "\\xymatrix{", "\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "\\ar[r]_-{c_{\\mathcal{F}'}} \\ar[d]_{1 \\otimes \\varphi} &", "\\mathcal{K} \\ar[d]^\\psi \\\\", "\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{G}", "\\ar[r]^-{c_{\\mathcal{G}'}} &", "\\mathcal{L}", "}", "$$", "is commutative.", "\\item The set of $\\mathcal{O}_{X'}$-module", "maps $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$", "and $\\psi$ is, if nonempty, a principal homogeneous space under", "$\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{L})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is immediate from the description of the maps.", "For (2), if $\\varphi'$ and $\\varphi''$ are two maps", "$\\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$", "and $\\psi$, then $\\varphi' - \\varphi''$ factors as", "$$", "\\mathcal{F}' \\to \\mathcal{F} \\to \\mathcal{L} \\to \\mathcal{G}'", "$$", "The map in the middle comes from a unique element of", "$\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{L})$ by", "Modules, Lemma \\ref{modules-lemma-i-star-equivalence}.", "Conversely, given an element $\\alpha$ of this group we can add the", "composition (as displayed above with $\\alpha$ in the middle)", "to $\\varphi'$. Some details omitted." ], "refs": [ "modules-lemma-i-star-equivalence" ], "ref_ids": [ 13260 ] } ], "ref_ids": [] }, { "id": 13375, "type": "theorem", "label": "defos-lemma-inf-obs-map", "categories": [ "defos" ], "title": "defos-lemma-inf-obs-map", "contents": [ "Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$", "be a first order thickening of ringed spaces. Assume given", "extensions", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0", "\\quad\\text{and}\\quad", "0 \\to \\mathcal{L} \\to \\mathcal{G}' \\to \\mathcal{G} \\to 0", "$$", "as in (\\ref{equation-extension})", "and maps $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and", "$\\psi : \\mathcal{K} \\to \\mathcal{L}$. Assume the diagram", "$$", "\\xymatrix{", "\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "\\ar[r]_-{c_{\\mathcal{F}'}} \\ar[d]_{1 \\otimes \\varphi} &", "\\mathcal{K} \\ar[d]^\\psi \\\\", "\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{G}", "\\ar[r]^-{c_{\\mathcal{G}'}} &", "\\mathcal{L}", "}", "$$", "is commutative. Then there exists an element", "$$", "o(\\varphi, \\psi) \\in", "\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{L})", "$$", "whose vanishing is a necessary and sufficient condition for the existence", "of a map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with", "$\\varphi$ and $\\psi$." ], "refs": [], "proofs": [ { "contents": [ "We can construct explicitly an extension", "$$", "0 \\to \\mathcal{L} \\to \\mathcal{H} \\to \\mathcal{F} \\to 0", "$$", "by taking $\\mathcal{H}$ to be the cohomology of the complex", "$$", "\\mathcal{K}", "\\xrightarrow{1, - \\psi}", "\\mathcal{F}' \\oplus \\mathcal{G}' \\xrightarrow{\\varphi, 1}", "\\mathcal{G}", "$$", "in the middle (with obvious notation). A calculation with local sections", "using the assumption that the diagram of the lemma commutes", "shows that $\\mathcal{H}$ is annihilated by $\\mathcal{I}$. Hence", "$\\mathcal{H}$ defines a class in", "$$", "\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{L})", "\\subset", "\\Ext^1_{\\mathcal{O}_{X'}}(\\mathcal{F}, \\mathcal{L})", "$$", "Finally, the class of $\\mathcal{H}$ is the difference of the pushout", "of the extension $\\mathcal{F}'$ via $\\psi$ and the pullback", "of the extension $\\mathcal{G}'$ via $\\varphi$ (calculations omitted).", "Thus the vanishing of the class of $\\mathcal{H}$ is equivalent to the", "existence of a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{K} \\ar[r] \\ar[d]_{\\psi} &", "\\mathcal{F}' \\ar[r] \\ar[d]_{\\varphi'} &", "\\mathcal{F} \\ar[r] \\ar[d]_\\varphi & 0\\\\", "0 \\ar[r] &", "\\mathcal{L} \\ar[r] &", "\\mathcal{G}' \\ar[r] &", "\\mathcal{G} \\ar[r] & 0", "}", "$$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13376, "type": "theorem", "label": "defos-lemma-inf-ext", "categories": [ "defos" ], "title": "defos-lemma-inf-ext", "contents": [ "Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$ be a first order", "thickening of ringed spaces.", "Assume given $\\mathcal{O}_X$-modules $\\mathcal{F}$, $\\mathcal{K}$", "and an $\\mathcal{O}_X$-linear map", "$c : \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{K}$.", "If there exists a sequence (\\ref{equation-extension}) with", "$c_{\\mathcal{F}'} = c$ then the set of isomorphism classes of these", "extensions is principal homogeneous under", "$\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K})$." ], "refs": [], "proofs": [ { "contents": [ "Assume given extensions", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{F}'_1 \\to \\mathcal{F} \\to 0", "\\quad\\text{and}\\quad", "0 \\to \\mathcal{K} \\to \\mathcal{F}'_2 \\to \\mathcal{F} \\to 0", "$$", "with $c_{\\mathcal{F}'_1} = c_{\\mathcal{F}'_2} = c$. Then the difference", "(in the extension group, see", "Homology, Section \\ref{homology-section-extensions})", "is an extension", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{E} \\to \\mathcal{F} \\to 0", "$$", "where $\\mathcal{E}$ is annihilated by $\\mathcal{I}$ (local computation", "omitted). Hence the sequence is an extension of $\\mathcal{O}_X$-modules,", "see Modules, Lemma \\ref{modules-lemma-i-star-equivalence}.", "Conversely, given such an extension $\\mathcal{E}$ we can add the extension", "$\\mathcal{E}$ to the $\\mathcal{O}_{X'}$-extension $\\mathcal{F}'$ without", "affecting the map $c_{\\mathcal{F}'}$. Some details omitted." ], "refs": [ "modules-lemma-i-star-equivalence" ], "ref_ids": [ 13260 ] } ], "ref_ids": [] }, { "id": 13377, "type": "theorem", "label": "defos-lemma-inf-obs-ext", "categories": [ "defos" ], "title": "defos-lemma-inf-obs-ext", "contents": [ "Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$", "be a first order thickening of ringed spaces. Assume given", "$\\mathcal{O}_X$-modules $\\mathcal{F}$, $\\mathcal{K}$", "and an $\\mathcal{O}_X$-linear map", "$c : \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{K}$.", "Then there exists an element", "$$", "o(\\mathcal{F}, \\mathcal{K}, c) \\in", "\\Ext^2_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K})", "$$", "whose vanishing is a necessary and sufficient condition for the existence", "of a sequence (\\ref{equation-extension}) with $c_{\\mathcal{F}'} = c$." ], "refs": [], "proofs": [ { "contents": [ "We first show that if $\\mathcal{K}$ is an injective $\\mathcal{O}_X$-module,", "then there does exist a sequence (\\ref{equation-extension}) with", "$c_{\\mathcal{F}'} = c$. To do this, choose a flat", "$\\mathcal{O}_{X'}$-module $\\mathcal{H}'$ and a surjection", "$\\mathcal{H}' \\to \\mathcal{F}$", "(Modules, Lemma \\ref{modules-lemma-module-quotient-flat}).", "Let $\\mathcal{J} \\subset \\mathcal{H}'$ be the kernel. Since $\\mathcal{H}'$", "is flat we have", "$$", "\\mathcal{I} \\otimes_{\\mathcal{O}_{X'}} \\mathcal{H}' =", "\\mathcal{I}\\mathcal{H}'", "\\subset \\mathcal{J} \\subset \\mathcal{H}'", "$$", "Observe that the map", "$$", "\\mathcal{I}\\mathcal{H}' =", "\\mathcal{I} \\otimes_{\\mathcal{O}_{X'}} \\mathcal{H}'", "\\longrightarrow", "\\mathcal{I} \\otimes_{\\mathcal{O}_{X'}} \\mathcal{F} =", "\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "$$", "annihilates $\\mathcal{I}\\mathcal{J}$. Namely, if $f$ is a local section", "of $\\mathcal{I}$ and $s$ is a local section of $\\mathcal{H}$, then", "$fs$ is mapped to $f \\otimes \\overline{s}$ where $\\overline{s}$ is", "the image of $s$ in $\\mathcal{F}$. Thus we obtain", "$$", "\\xymatrix{", "\\mathcal{I}\\mathcal{H}'/\\mathcal{I}\\mathcal{J}", "\\ar@{^{(}->}[r] \\ar[d] &", "\\mathcal{J}/\\mathcal{I}\\mathcal{J} \\ar@{..>}[d]_\\gamma \\\\", "\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\ar[r]^-c &", "\\mathcal{K}", "}", "$$", "a diagram of $\\mathcal{O}_X$-modules. If $\\mathcal{K}$ is injective", "as an $\\mathcal{O}_X$-module, then we obtain the dotted arrow.", "Denote $\\gamma' : \\mathcal{J} \\to \\mathcal{K}$ the composition", "of $\\gamma$ with $\\mathcal{J} \\to \\mathcal{J}/\\mathcal{I}\\mathcal{J}$.", "A local calculation shows the pushout", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{J} \\ar[r] \\ar[d]_{\\gamma'} &", "\\mathcal{H}' \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[r] \\ar@{=}[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{K} \\ar[r] &", "\\mathcal{F}' \\ar[r] &", "\\mathcal{F} \\ar[r] &", "0", "}", "$$", "is a solution to the problem posed by the lemma.", "\\medskip\\noindent", "General case. Choose an embedding $\\mathcal{K} \\subset \\mathcal{K}'$", "with $\\mathcal{K}'$ an injective $\\mathcal{O}_X$-module. Let $\\mathcal{Q}$", "be the quotient, so that we have an exact sequence", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{K}' \\to \\mathcal{Q} \\to 0", "$$", "Denote", "$c' : \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{K}'$", "be the composition. By the paragraph above there exists a sequence", "$$", "0 \\to \\mathcal{K}' \\to \\mathcal{E}' \\to \\mathcal{F} \\to 0", "$$", "as in (\\ref{equation-extension}) with $c_{\\mathcal{E}'} = c'$.", "Note that $c'$ composed with the map $\\mathcal{K}' \\to \\mathcal{Q}$", "is zero, hence the pushout of $\\mathcal{E}'$ by", "$\\mathcal{K}' \\to \\mathcal{Q}$ is an extension", "$$", "0 \\to \\mathcal{Q} \\to \\mathcal{D}' \\to \\mathcal{F} \\to 0", "$$", "as in (\\ref{equation-extension}) with $c_{\\mathcal{D}'} = 0$.", "This means exactly that $\\mathcal{D}'$ is annihilated by", "$\\mathcal{I}$, in other words, the $\\mathcal{D}'$ is an extension", "of $\\mathcal{O}_X$-modules, i.e., defines an element", "$$", "o(\\mathcal{F}, \\mathcal{K}, c) \\in", "\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{Q}) =", "\\Ext^2_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K})", "$$", "(the equality holds by the long exact cohomology sequence associated", "to the exact sequence above and the vanishing of higher ext groups", "into the injective module $\\mathcal{K}'$). If", "$o(\\mathcal{F}, \\mathcal{K}, c) = 0$, then we can choose a splitting", "$s : \\mathcal{F} \\to \\mathcal{D}'$ and we can set", "$$", "\\mathcal{F}' = \\Ker(\\mathcal{E}' \\to \\mathcal{D}'/s(\\mathcal{F}))", "$$", "so that we obtain the following diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{K} \\ar[r] \\ar[d] &", "\\mathcal{F}' \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[r] \\ar@{=}[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{K}' \\ar[r] &", "\\mathcal{E}' \\ar[r] &", "\\mathcal{F} \\ar[r] & 0", "}", "$$", "with exact rows which shows that $c_{\\mathcal{F}'} = c$. Conversely, if", "$\\mathcal{F}'$ exists, then the pushout of $\\mathcal{F}'$ by the map", "$\\mathcal{K} \\to \\mathcal{K}'$ is isomorphic to $\\mathcal{E}'$ by", "Lemma \\ref{lemma-inf-ext} and the vanishing of higher ext groups", "into the injective module $\\mathcal{K}'$. This gives a diagram", "as above, which implies that $\\mathcal{D}'$ is split as an extension, i.e.,", "the class $o(\\mathcal{F}, \\mathcal{K}, c)$ is zero." ], "refs": [ "modules-lemma-module-quotient-flat", "defos-lemma-inf-ext" ], "ref_ids": [ 13276, 13376 ] } ], "ref_ids": [] }, { "id": 13378, "type": "theorem", "label": "defos-lemma-inf-map-special", "categories": [ "defos" ], "title": "defos-lemma-inf-map-special", "contents": [ "Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$", "be a first order thickening of ringed spaces.", "Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}_{X'}$-modules.", "Set $\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}_X$-linear map.", "The set of lifts of $\\varphi$ to an $\\mathcal{O}_{X'}$-linear map", "$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ is, if nonempty, a principal", "homogeneous space under", "$\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{I}\\mathcal{G}')$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-inf-map} but we also", "give a direct proof. We have short exact sequences of modules", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_{X'} \\to \\mathcal{O}_X \\to 0", "\\quad\\text{and}\\quad", "0 \\to \\mathcal{I}\\mathcal{G}' \\to \\mathcal{G}' \\to \\mathcal{G} \\to 0", "$$", "and similarly for $\\mathcal{F}'$.", "Since $\\mathcal{I}$ has square zero the $\\mathcal{O}_{X'}$-module", "structure on $\\mathcal{I}$ and $\\mathcal{I}\\mathcal{G}'$ comes from", "a unique $\\mathcal{O}_X$-module structure. It follows that", "$$", "\\Hom_{\\mathcal{O}_{X'}}(\\mathcal{F}', \\mathcal{I}\\mathcal{G}') =", "\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{I}\\mathcal{G}')", "\\quad\\text{and}\\quad", "\\Hom_{\\mathcal{O}_{X'}}(\\mathcal{F}', \\mathcal{G}) =", "\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})", "$$", "The lemma now follows from the exact sequence", "$$", "0 \\to \\Hom_{\\mathcal{O}_{X'}}(\\mathcal{F}', \\mathcal{I}\\mathcal{G}') \\to", "\\Hom_{\\mathcal{O}_{X'}}(\\mathcal{F}', \\mathcal{G}') \\to", "\\Hom_{\\mathcal{O}_{X'}}(\\mathcal{F}', \\mathcal{G})", "$$", "see Homology, Lemma \\ref{homology-lemma-check-exactness}." ], "refs": [ "defos-lemma-inf-map", "homology-lemma-check-exactness" ], "ref_ids": [ 13374, 12019 ] } ], "ref_ids": [] }, { "id": 13379, "type": "theorem", "label": "defos-lemma-deform-module", "categories": [ "defos" ], "title": "defos-lemma-deform-module", "contents": [ "Let $(f, f')$ be a morphism of first order thickenings of ringed spaces", "as in Situation \\ref{situation-morphism-thickenings}.", "Let $\\mathcal{F}'$ be an $\\mathcal{O}_{X'}$-module", "and set $\\mathcal{F} = i^*\\mathcal{F}'$.", "Assume that $\\mathcal{F}$ is flat over $S$", "and that $(f, f')$ is a strict morphism of thickenings", "(Definition \\ref{definition-strict-morphism-thickenings}).", "Then the following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}'$ is flat over $S'$, and", "\\item the canonical map", "$f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "\\mathcal{I}\\mathcal{F}'$", "is an isomorphism.", "\\end{enumerate}", "Moreover, in this case the maps", "$$", "f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "\\mathcal{I}\\mathcal{F}'", "$$", "are isomorphisms." ], "refs": [ "defos-definition-strict-morphism-thickenings" ], "proofs": [ { "contents": [ "The map $f^*\\mathcal{J} \\to \\mathcal{I}$ is surjective", "as $(f, f')$ is a strict morphism of thickenings.", "Hence the final statement is a consequence of (2).", "\\medskip\\noindent", "Proof of the equivalence of (1) and (2). We may check these conditions", "at stalks. Let $x \\in X \\subset X'$", "be a point with image $s = f(x) \\in S \\subset S'$.", "Set $A' = \\mathcal{O}_{S', s}$, $B' = \\mathcal{O}_{X', x}$,", "$A = \\mathcal{O}_{S, s}$, and $B = \\mathcal{O}_{X, x}$.", "Then $A = A'/J$ and $B = B'/I$ for some square zero ideals.", "Since $(f, f')$ is a strict morphism of thickenings we have $I = JB'$.", "Let $M' = \\mathcal{F}'_x$ and $M = \\mathcal{F}_x$.", "Then $M'$ is a $B'$-module and $M$ is a $B$-module.", "Since $\\mathcal{F} = i^*\\mathcal{F}'$ we see that the kernel of the", "surjection $M' \\to M$ is $IM' = JM'$. Thus we have a short exact", "sequence", "$$", "0 \\to JM' \\to M' \\to M \\to 0", "$$", "Using", "Sheaves, Lemma \\ref{sheaves-lemma-stalk-pullback-modules}", "and", "Modules, Lemma \\ref{modules-lemma-stalk-tensor-product}", "to identify stalks of pullbacks and tensor products we see", "that the stalk at $x$ of the canonical map of the lemma is the map", "$$", "(J \\otimes_A B) \\otimes_B M = J \\otimes_A M = J \\otimes_{A'} M'", "\\longrightarrow JM'", "$$", "The assumption that $\\mathcal{F}$ is flat over $S$ signifies that", "$M$ is a flat $A$-module.", "\\medskip\\noindent", "Assume (1). Flatness implies $\\text{Tor}_1^{A'}(M', A) = 0$ by", "Algebra, Lemma \\ref{algebra-lemma-characterize-flat}.", "This means $J \\otimes_{A'} M' \\to M'$ is injective by", "Algebra, Remark \\ref{algebra-remark-Tor-ring-mod-ideal}.", "Hence $J \\otimes_A M \\to JM'$ is an isomorphism.", "\\medskip\\noindent", "Assume (2). Then $J \\otimes_{A'} M' \\to M'$ is injective. Hence", "$\\text{Tor}_1^{A'}(M', A) = 0$ by", "Algebra, Remark \\ref{algebra-remark-Tor-ring-mod-ideal}.", "Hence $M'$ is flat over $A'$ by", "Algebra, Lemma \\ref{algebra-lemma-what-does-it-mean}." ], "refs": [ "sheaves-lemma-stalk-pullback-modules", "modules-lemma-stalk-tensor-product", "algebra-lemma-characterize-flat", "algebra-remark-Tor-ring-mod-ideal", "algebra-remark-Tor-ring-mod-ideal", "algebra-lemma-what-does-it-mean" ], "ref_ids": [ 14523, 13267, 786, 1570, 1570, 890 ] } ], "ref_ids": [ 13423 ] }, { "id": 13380, "type": "theorem", "label": "defos-lemma-inf-map-rel", "categories": [ "defos" ], "title": "defos-lemma-inf-map-rel", "contents": [ "Let $(f, f')$ be a morphism of first order thickenings as in", "Situation \\ref{situation-morphism-thickenings}.", "Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}_{X'}$-modules and set", "$\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}_X$-linear map.", "Assume that $\\mathcal{G}'$ is flat over $S'$ and that", "$(f, f')$ is a strict morphism of thickenings.", "The set of lifts of $\\varphi$ to an $\\mathcal{O}_{X'}$-linear map", "$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ is, if nonempty, a principal", "homogeneous space under", "$$", "\\Hom_{\\mathcal{O}_X}(\\mathcal{F},", "\\mathcal{G} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{J})", "$$" ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-inf-map-special} and \\ref{lemma-deform-module}." ], "refs": [ "defos-lemma-inf-map-special", "defos-lemma-deform-module" ], "ref_ids": [ 13378, 13379 ] } ], "ref_ids": [] }, { "id": 13381, "type": "theorem", "label": "defos-lemma-inf-obs-map-special", "categories": [ "defos" ], "title": "defos-lemma-inf-obs-map-special", "contents": [ "Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$", "be a first order thickening of ringed spaces.", "Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}_{X'}$-modules and set", "$\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}_X$-linear map.", "There exists an element", "$$", "o(\\varphi) \\in", "\\Ext^1_{\\mathcal{O}_X}(Li^*\\mathcal{F}',", "\\mathcal{I}\\mathcal{G}')", "$$", "whose vanishing is a necessary and sufficient condition for the", "existence of a lift of $\\varphi$ to an $\\mathcal{O}_{X'}$-linear map", "$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$." ], "refs": [], "proofs": [ { "contents": [ "It is clear from the proof of Lemma \\ref{lemma-inf-map-special} that the", "vanishing of the boundary of $\\varphi$ via the map", "$$", "\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) =", "\\Hom_{\\mathcal{O}_{X'}}(\\mathcal{F}', \\mathcal{G}) \\longrightarrow", "\\Ext^1_{\\mathcal{O}_{X'}}(\\mathcal{F}', \\mathcal{I}\\mathcal{G}')", "$$", "is a necessary and sufficient condition for the existence of a lift. We", "conclude as", "$$", "\\Ext^1_{\\mathcal{O}_{X'}}(\\mathcal{F}', \\mathcal{I}\\mathcal{G}') =", "\\Ext^1_{\\mathcal{O}_X}(Li^*\\mathcal{F}', \\mathcal{I}\\mathcal{G}')", "$$", "the adjointness of $i_* = Ri_*$ and $Li^*$ on the derived category", "(Cohomology, Lemma \\ref{cohomology-lemma-adjoint})." ], "refs": [ "defos-lemma-inf-map-special", "cohomology-lemma-adjoint" ], "ref_ids": [ 13378, 2121 ] } ], "ref_ids": [] }, { "id": 13382, "type": "theorem", "label": "defos-lemma-inf-obs-map-rel", "categories": [ "defos" ], "title": "defos-lemma-inf-obs-map-rel", "contents": [ "Let $(f, f')$ be a morphism of first", "order thickenings as in Situation \\ref{situation-morphism-thickenings}.", "Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}_{X'}$-modules and set", "$\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}_X$-linear map.", "Assume that $\\mathcal{F}'$ and $\\mathcal{G}'$ are flat over $S'$ and", "that $(f, f')$ is a strict morphism of thickenings. There exists an element", "$$", "o(\\varphi) \\in \\Ext^1_{\\mathcal{O}_X}(\\mathcal{F},", "\\mathcal{G} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{J})", "$$", "whose vanishing is a necessary and sufficient condition for the", "existence of a lift of $\\varphi$ to an $\\mathcal{O}_{X'}$-linear map", "$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "This follows from Lemma \\ref{lemma-inf-obs-map-special}", "as we claim that under the assumptions of the lemma we have", "$$", "\\Ext^1_{\\mathcal{O}_X}(Li^*\\mathcal{F}',", "\\mathcal{I}\\mathcal{G}') =", "\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F},", "\\mathcal{G} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{J})", "$$", "Namely, we have", "$\\mathcal{I}\\mathcal{G}' =", "\\mathcal{G} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{J}$", "by Lemma \\ref{lemma-deform-module}.", "On the other hand, observe that", "$$", "H^{-1}(Li^*\\mathcal{F}') =", "\\text{Tor}_1^{\\mathcal{O}_{X'}}(\\mathcal{F}', \\mathcal{O}_X)", "$$", "(local computation omitted). Using the short exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_{X'} \\to \\mathcal{O}_X \\to 0", "$$", "we see that this $\\text{Tor}_1$ is computed by the kernel of the map", "$\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{I}\\mathcal{F}'$", "which is zero by the final assertion of Lemma \\ref{lemma-deform-module}.", "Thus $\\tau_{\\geq -1}Li^*\\mathcal{F}' = \\mathcal{F}$.", "On the other hand, we have", "$$", "\\Ext^1_{\\mathcal{O}_X}(Li^*\\mathcal{F}',", "\\mathcal{I}\\mathcal{G}') =", "\\Ext^1_{\\mathcal{O}_X}(\\tau_{\\geq -1}Li^*\\mathcal{F}',", "\\mathcal{I}\\mathcal{G}')", "$$", "by the dual of", "Derived Categories, Lemma \\ref{derived-lemma-negative-vanishing}." ], "refs": [ "defos-lemma-inf-obs-map-special", "defos-lemma-deform-module", "defos-lemma-deform-module", "derived-lemma-negative-vanishing" ], "ref_ids": [ 13381, 13379, 13379, 1839 ] } ], "ref_ids": [] }, { "id": 13383, "type": "theorem", "label": "defos-lemma-inf-ext-rel", "categories": [ "defos" ], "title": "defos-lemma-inf-ext-rel", "contents": [ "Let $(f, f')$ be a morphism of first order thickenings as in", "Situation \\ref{situation-morphism-thickenings}.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "Assume $(f, f')$ is a strict morphism of thickenings and", "$\\mathcal{F}$ flat over $S$. If there exists a pair", "$(\\mathcal{F}', \\alpha)$ consisting of an", "$\\mathcal{O}_{X'}$-module $\\mathcal{F}'$ flat over $S'$ and an isomorphism", "$\\alpha : i^*\\mathcal{F}' \\to \\mathcal{F}$, then the set of", "isomorphism classes of such pairs is principal homogeneous", "under", "$\\Ext^1_{\\mathcal{O}_X}(", "\\mathcal{F}, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "If we assume there exists one such module, then the canonical map", "$$", "f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "$$", "is an isomorphism by Lemma \\ref{lemma-deform-module}. Apply", "Lemma \\ref{lemma-inf-ext} with $\\mathcal{K} = ", "\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F}$", "and $c = 1$. By Lemma \\ref{lemma-deform-module} the corresponding extensions", "$\\mathcal{F}'$ are all flat over $S'$." ], "refs": [ "defos-lemma-deform-module", "defos-lemma-inf-ext", "defos-lemma-deform-module" ], "ref_ids": [ 13379, 13376, 13379 ] } ], "ref_ids": [] }, { "id": 13384, "type": "theorem", "label": "defos-lemma-inf-obs-ext-rel", "categories": [ "defos" ], "title": "defos-lemma-inf-obs-ext-rel", "contents": [ "Let $(f, f')$ be a morphism of first order thickenings as in", "Situation \\ref{situation-morphism-thickenings}.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Assume", "$(f, f')$ is a strict morphism of thickenings", "and $\\mathcal{F}$ flat over $S$. There exists an", "$\\mathcal{O}_{X'}$-module $\\mathcal{F}'$ flat over $S'$ with", "$i^*\\mathcal{F}' \\cong \\mathcal{F}$, if and only if", "\\begin{enumerate}", "\\item the canonical map $", "f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F}$", "is an isomorphism, and", "\\item the class", "$o(\\mathcal{F}, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F}, 1)", "\\in \\Ext^2_{\\mathcal{O}_X}(", "\\mathcal{F}, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$", "of Lemma \\ref{lemma-inf-obs-ext} is zero.", "\\end{enumerate}" ], "refs": [ "defos-lemma-inf-obs-ext" ], "proofs": [ { "contents": [ "This follows immediately from the characterization of", "$\\mathcal{O}_{X'}$-modules flat over $S'$ of ", "Lemma \\ref{lemma-deform-module} and", "Lemma \\ref{lemma-inf-obs-ext}." ], "refs": [ "defos-lemma-deform-module", "defos-lemma-inf-obs-ext" ], "ref_ids": [ 13379, 13377 ] } ], "ref_ids": [ 13377 ] }, { "id": 13385, "type": "theorem", "label": "defos-lemma-flat", "categories": [ "defos" ], "title": "defos-lemma-flat", "contents": [ "In the situation above.", "\\begin{enumerate}", "\\item There exists an $\\mathcal{O}_{X'}$-module $\\mathcal{F}'$ flat over", "$S'$ with $i^*\\mathcal{F}' \\cong \\mathcal{F}$, if and only if", "the class", "$o(\\mathcal{F}, f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F}, 1)", "\\in \\Ext^2_{\\mathcal{O}_X}(", "\\mathcal{F}, f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$", "of Lemma \\ref{lemma-inf-obs-ext} is zero.", "\\item If such a module exists, then the set of isomorphism classes", "of lifts is principal homogeneous under", "$\\Ext^1_{\\mathcal{O}_X}(", "\\mathcal{F}, f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$.", "\\item Given a lift $\\mathcal{F}'$, the set of automorphisms of", "$\\mathcal{F}'$ which pull back to $\\text{id}_\\mathcal{F}$ is canonically", "isomorphic to $\\Ext^0_{\\mathcal{O}_X}(", "\\mathcal{F}, f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$.", "\\end{enumerate}" ], "refs": [ "defos-lemma-inf-obs-ext" ], "proofs": [ { "contents": [ "Part (1) follows from Lemma \\ref{lemma-inf-obs-ext-rel}", "as we have seen above that $\\mathcal{I} = f^*\\mathcal{J}$.", "Part (2) follows from Lemma \\ref{lemma-inf-ext-rel}.", "Part (3) follows from Lemma \\ref{lemma-inf-map-rel}." ], "refs": [ "defos-lemma-inf-obs-ext-rel", "defos-lemma-inf-ext-rel", "defos-lemma-inf-map-rel" ], "ref_ids": [ 13384, 13383, 13380 ] } ], "ref_ids": [ 13377 ] }, { "id": 13386, "type": "theorem", "label": "defos-lemma-verify-iv", "categories": [ "defos" ], "title": "defos-lemma-verify-iv", "contents": [ "In Situation \\ref{situation-ses-flat-thickenings} the modules", "$\\pi^*\\mathcal{F}$ and $h^*\\mathcal{F}'_2$ are $\\mathcal{O}'_1$-modules", "flat over $S'_1$ restricting to $\\mathcal{F}$ on $X$.", "Their difference (Lemma \\ref{lemma-flat}) is an element", "$\\theta$ of $\\Ext^1_{\\mathcal{O}_X}(", "\\mathcal{F}, f^*\\mathcal{J}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{F})$", "whose boundary in", "$\\Ext^2_{\\mathcal{O}_X}(", "\\mathcal{F}, f^*\\mathcal{J}_3 \\otimes_{\\mathcal{O}_X} \\mathcal{F})$", "equals the obstruction (Lemma \\ref{lemma-flat})", "to lifting $\\mathcal{F}$ to an $\\mathcal{O}'_3$-module flat over $S'_3$." ], "refs": [ "defos-lemma-flat", "defos-lemma-flat" ], "proofs": [ { "contents": [ "Note that both $\\pi^*\\mathcal{F}$ and $h^*\\mathcal{F}'_2$", "restrict to $\\mathcal{F}$ on $X$ and that the kernels of", "$\\pi^*\\mathcal{F} \\to \\mathcal{F}$ and $h^*\\mathcal{F}'_2 \\to \\mathcal{F}$", "are given by $f^*\\mathcal{J}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{F}$.", "Hence flatness by Lemma \\ref{lemma-deform-module}.", "Taking the boundary makes sense as the sequence of modules", "$$", "0 \\to f^*\\mathcal{J}_3 \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "f^*\\mathcal{J}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "f^*\\mathcal{J}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to 0", "$$", "is short exact due to the assumptions in", "Situation \\ref{situation-ses-flat-thickenings}", "and the fact that $\\mathcal{F}$ is flat over $S$.", "The statement on the obstruction class is a direct translation", "of the result of", "Remark \\ref{remark-complex-thickenings-and-ses-modules}", "to this particular situation." ], "refs": [ "defos-lemma-deform-module", "defos-remark-complex-thickenings-and-ses-modules" ], "ref_ids": [ 13379, 13430 ] } ], "ref_ids": [ 13385, 13385 ] }, { "id": 13387, "type": "theorem", "label": "defos-lemma-huge-diagram-ringed-spaces", "categories": [ "defos" ], "title": "defos-lemma-huge-diagram-ringed-spaces", "contents": [ "Assume given a commutative diagram of morphisms of ringed spaces", "\\begin{equation}", "\\label{equation-huge-1}", "\\vcenter{", "\\xymatrix{", "& (X_2, \\mathcal{O}_{X_2}) \\ar[r]_{i_2} \\ar[d]_{f_2} \\ar[ddl]_g &", "(X'_2, \\mathcal{O}_{X'_2}) \\ar[d]^{f'_2} \\\\", "& (S_2, \\mathcal{O}_{S_2}) \\ar[r]^{t_2} \\ar[ddl]|\\hole &", "(S'_2, \\mathcal{O}_{S'_2}) \\ar[ddl] \\\\", "(X_1, \\mathcal{O}_{X_1}) \\ar[r]_{i_1} \\ar[d]_{f_1} &", "(X'_1, \\mathcal{O}_{X'_1}) \\ar[d]^{f'_1} \\\\", "(S_1, \\mathcal{O}_{S_1}) \\ar[r]^{t_1} &", "(S'_1, \\mathcal{O}_{S'_1})", "}", "}", "\\end{equation}", "whose horizontal arrows are first order thickenings. Set", "$\\mathcal{G}_j = \\Ker(i_j^\\sharp)$ and assume given", "a $g$-map $\\nu : \\mathcal{G}_1 \\to \\mathcal{G}_2$ of modules", "giving rise to the commutative diagram", "\\begin{equation}", "\\label{equation-huge-2}", "\\vcenter{", "\\xymatrix{", "& 0 \\ar[r] & \\mathcal{G}_2 \\ar[r] &", "\\mathcal{O}_{X'_2} \\ar[r] &", "\\mathcal{O}_{X_2} \\ar[r] & 0 \\\\", "& 0 \\ar[r]|\\hole &", "\\mathcal{J}_2 \\ar[u]_{c_2} \\ar[r] &", "\\mathcal{O}_{S'_2} \\ar[u] \\ar[r]|\\hole &", "\\mathcal{O}_{S_2} \\ar[u] \\ar[r] & 0 \\\\", "0 \\ar[r] & \\mathcal{G}_1 \\ar[ruu] \\ar[r] &", "\\mathcal{O}_{X'_1} \\ar[r] &", "\\mathcal{O}_{X_1} \\ar[ruu] \\ar[r] & 0 \\\\", "0 \\ar[r] & \\mathcal{J}_1 \\ar[ruu]|\\hole \\ar[u]^{c_1} \\ar[r] &", "\\mathcal{O}_{S'_1} \\ar[ruu]|\\hole \\ar[u] \\ar[r] &", "\\mathcal{O}_{S_1} \\ar[ruu]|\\hole \\ar[u] \\ar[r] & 0", "}", "}", "\\end{equation}", "with front and back solutions to (\\ref{equation-to-solve-ringed-spaces}).", "\\begin{enumerate}", "\\item There exist a canonical element in", "$\\Ext^1_{\\mathcal{O}_{X_2}}(Lg^*\\NL_{X_1/S_1}, \\mathcal{G}_2)$", "whose vanishing is a necessary and sufficient condition for the existence", "of a morphism of ringed spaces $X'_2 \\to X'_1$ fitting into", "(\\ref{equation-huge-1}) compatibly with $\\nu$.", "\\item If there exists a morphism $X'_2 \\to X'_1$ fitting into", "(\\ref{equation-huge-1}) compatibly with $\\nu$ the set of all such morphisms", "is a principal homogeneous space under", "$$", "\\Hom_{\\mathcal{O}_{X_1}}(\\Omega_{X_1/S_1}, g_*\\mathcal{G}_2) =", "\\Hom_{\\mathcal{O}_{X_2}}(g^*\\Omega_{X_1/S_1}, \\mathcal{G}_2) =", "\\Ext^0_{\\mathcal{O}_{X_2}}(Lg^*\\NL_{X_1/S_1}, \\mathcal{G}_2).", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The naive cotangent complex $\\NL_{X_1/S_1}$ is defined in Modules, Definition", "\\ref{modules-definition-cotangent-complex-morphism-ringed-topoi}.", "The equalities in the last statement of the lemma follow from", "the fact that $g^*$ is adjoint to $g_*$, the fact that", "$H^0(\\NL_{X_1/S_1}) = \\Omega_{X_1/S_1}$ (by construction of the", "naive cotangent complex) and the fact that $Lg^*$ is the left", "derived functor of $g^*$. Thus we will work with the groups", "$\\Ext^k_{\\mathcal{O}_{X_2}}(Lg^*\\NL_{X_1/S_1}, \\mathcal{G}_2)$,", "$k = 0, 1$ in the rest of the proof. We first argue that we can reduce", "to the case where the underlying topological spaces of all ringed", "spaces in the lemma is the same.", "\\medskip\\noindent", "To do this, observe that $g^{-1}\\NL_{X_1/S_1}$ is equal to the naive", "cotangent complex of the homomorphism of sheaves of rings", "$g^{-1}f_1^{-1}\\mathcal{O}_{S_1} \\to g^{-1}\\mathcal{O}_{X_1}$, see", "Modules, Lemma \\ref{modules-lemma-pullback-NL}.", "Moreover, the degree $0$ term of $\\NL_{X_1/S_1}$ is a flat", "$\\mathcal{O}_{X_1}$-module, hence the canonical map", "$$", "Lg^*\\NL_{X_1/S_1}", "\\longrightarrow", "g^{-1}\\NL_{X_1/S_1} \\otimes_{g^{-1}\\mathcal{O}_{X_1}} \\mathcal{O}_{X_2}", "$$", "induces an isomorphism on cohomology sheaves in degrees $0$ and $-1$.", "Thus we may replace the Ext groups of the lemma with", "$$", "\\Ext^k_{g^{-1}\\mathcal{O}_{X_1}}(g^{-1}\\NL_{X_1/S_1}, \\mathcal{G}_2) =", "\\Ext^k_{g^{-1}\\mathcal{O}_{X_1}}(", "\\NL_{g^{-1}\\mathcal{O}_{X_1}/g^{-1}f_1^{-1}\\mathcal{O}_{S_1}}, \\mathcal{G}_2)", "$$", "The set of morphism of ringed spaces $X'_2 \\to X'_1$ fitting into", "(\\ref{equation-huge-1}) compatibly with $\\nu$", "is in one-to-one bijection with", "the set of homomorphisms of $g^{-1}f_1^{-1}\\mathcal{O}_{S'_1}$-algebras", "$g^{-1}\\mathcal{O}_{X'_1} \\to \\mathcal{O}_{X'_2}$ which are compatible with", "$f^\\sharp$ and $\\nu$. In this way we see that we may assume we have a", "diagram (\\ref{equation-huge-2}) of sheaves on $X$ and we are looking to", "find a homomorphism of sheaves of rings", "$\\mathcal{O}_{X'_1} \\to \\mathcal{O}_{X'_2}$ fitting into it.", "\\medskip\\noindent", "In the rest of the proof of the lemma we assume", "all underlying topological spaces are the", "same, i.e., we have a diagram (\\ref{equation-huge-2}) of sheaves on", "a space $X$ and we are looking for homomorphisms of sheaves of rings", "$\\mathcal{O}_{X'_1} \\to \\mathcal{O}_{X'_2}$ fitting into it.", "As ext groups we will use", "$\\Ext^k_{\\mathcal{O}_{X_1}}(", "\\NL_{\\mathcal{O}_{X_1}/\\mathcal{O}_{S_1}}, \\mathcal{G}_2)$, $k = 0, 1$.", "\\medskip\\noindent", "Step 1. Construction of the obstruction class. Consider the sheaf", "of sets", "$$", "\\mathcal{E} = \\mathcal{O}_{X'_1} \\times_{\\mathcal{O}_{X_2}} \\mathcal{O}_{X'_2}", "$$", "This comes with a surjective map $\\alpha : \\mathcal{E} \\to \\mathcal{O}_{X_1}$", "and hence we can use $\\NL(\\alpha)$ instead of", "$\\NL_{\\mathcal{O}_{X_1}/\\mathcal{O}_{S_1}}$, see", "Modules, Lemma \\ref{modules-lemma-NL-up-to-qis}.", "Set", "$$", "\\mathcal{I}' =", "\\Ker(\\mathcal{O}_{S'_1}[\\mathcal{E}] \\to \\mathcal{O}_{X_1})", "\\quad\\text{and}\\quad", "\\mathcal{I} =", "\\Ker(\\mathcal{O}_{S_1}[\\mathcal{E}] \\to \\mathcal{O}_{X_1})", "$$", "There is a surjection $\\mathcal{I}' \\to \\mathcal{I}$ whose kernel", "is $\\mathcal{J}_1\\mathcal{O}_{S'_1}[\\mathcal{E}]$.", "We obtain two homomorphisms of $\\mathcal{O}_{S'_2}$-algebras", "$$", "a : \\mathcal{O}_{S'_1}[\\mathcal{E}] \\to \\mathcal{O}_{X'_1}", "\\quad\\text{and}\\quad", "b : \\mathcal{O}_{S'_1}[\\mathcal{E}] \\to \\mathcal{O}_{X'_2}", "$$", "which induce maps $a|_{\\mathcal{I}'} : \\mathcal{I}' \\to \\mathcal{G}_1$ and", "$b|_{\\mathcal{I}'} : \\mathcal{I}' \\to \\mathcal{G}_2$. Both $a$ and $b$", "annihilate $(\\mathcal{I}')^2$. Moreover $a$ and $b$ agree on", "$\\mathcal{J}_1\\mathcal{O}_{S'_1}[\\mathcal{E}]$ as maps into $\\mathcal{G}_2$", "because the left hand square of (\\ref{equation-huge-2}) is commutative.", "Thus the difference", "$b|_{\\mathcal{I}'} - \\nu \\circ a|_{\\mathcal{I}'}$", "induces a well defined $\\mathcal{O}_{X_1}$-linear map", "$$", "\\xi : \\mathcal{I}/\\mathcal{I}^2 \\longrightarrow \\mathcal{G}_2", "$$", "which sends the class of a local section $f$ of $\\mathcal{I}$ to", "$a(f') - \\nu(b(f'))$ where $f'$ is a lift of $f$ to a local", "section of $\\mathcal{I}'$. We let", "$[\\xi] \\in \\Ext^1_{\\mathcal{O}_{X_1}}(\\NL(\\alpha), \\mathcal{G}_2)$", "be the image (see below).", "\\medskip\\noindent", "Step 2. Vanishing of $[\\xi]$ is necessary. Let us write", "$\\Omega = \\Omega_{\\mathcal{O}_{S_1}[\\mathcal{E}]/\\mathcal{O}_{S_1}}", "\\otimes_{\\mathcal{O}_{S_1}[\\mathcal{E}]} \\mathcal{O}_{X_1}$.", "Observe that $\\NL(\\alpha) = (\\mathcal{I}/\\mathcal{I}^2 \\to \\Omega)$", "fits into a distinguished triangle", "$$", "\\Omega[0] \\to", "\\NL(\\alpha) \\to", "\\mathcal{I}/\\mathcal{I}^2[1] \\to", "\\Omega[1]", "$$", "Thus we see that $[\\xi]$ is zero if and only if $\\xi$", "is a composition $\\mathcal{I}/\\mathcal{I}^2 \\to \\Omega \\to \\mathcal{G}_2$", "for some map $\\Omega \\to \\mathcal{G}_2$. Suppose there exists a", "homomorphisms of sheaves of rings", "$\\varphi : \\mathcal{O}_{X'_1} \\to \\mathcal{O}_{X'_2}$ fitting into", "(\\ref{equation-huge-2}). In this case consider the map", "$\\mathcal{O}_{S'_1}[\\mathcal{E}] \\to \\mathcal{G}_2$,", "$f' \\mapsto b(f') - \\varphi(a(f'))$. A calculation", "shows this annihilates $\\mathcal{J}_1\\mathcal{O}_{S'_1}[\\mathcal{E}]$", "and induces a derivation $\\mathcal{O}_{S_1}[\\mathcal{E}] \\to \\mathcal{G}_2$.", "The resulting linear map $\\Omega \\to \\mathcal{G}_2$ witnesses the", "fact that $[\\xi] = 0$ in this case.", "\\medskip\\noindent", "Step 3. Vanishing of $[\\xi]$ is sufficient. Let", "$\\theta : \\Omega \\to \\mathcal{G}_2$ be a $\\mathcal{O}_{X_1}$-linear map", "such that $\\xi$ is equal to", "$\\theta \\circ (\\mathcal{I}/\\mathcal{I}^2 \\to \\Omega)$.", "Then a calculation shows that", "$$", "b + \\theta \\circ d : \\mathcal{O}_{S'_1}[\\mathcal{E}] \\to \\mathcal{O}_{X'_2}", "$$", "annihilates $\\mathcal{I}'$ and hence defines a map", "$\\mathcal{O}_{X'_1} \\to \\mathcal{O}_{X'_2}$ fitting into", "(\\ref{equation-huge-2}).", "\\medskip\\noindent", "Proof of (2) in the special case above. Omitted. Hint:", "This is exactly the same as the proof of (2) of Lemma \\ref{lemma-huge-diagram}." ], "refs": [ "modules-definition-cotangent-complex-morphism-ringed-topoi", "modules-lemma-pullback-NL", "modules-lemma-NL-up-to-qis", "defos-lemma-huge-diagram" ], "ref_ids": [ 13362, 13328, 13327, 13369 ] } ], "ref_ids": [] }, { "id": 13388, "type": "theorem", "label": "defos-lemma-NL-represent-ext-class", "categories": [ "defos" ], "title": "defos-lemma-NL-represent-ext-class", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{A} \\to \\mathcal{B}$ be a", "homomorphism of sheaves of rings. Let $\\mathcal{G}$ be a $\\mathcal{B}$-module.", "Let", "$\\xi \\in \\Ext^1_\\mathcal{B}(\\NL_{\\mathcal{B}/\\mathcal{A}}, \\mathcal{G})$. ", "There exists a map of sheaves of sets $\\alpha : \\mathcal{E} \\to \\mathcal{B}$", "such that $\\xi \\in \\Ext^1_\\mathcal{B}(\\NL(\\alpha), \\mathcal{G})$", "is the class of a map $\\mathcal{I}/\\mathcal{I}^2 \\to \\mathcal{G}$", "(see proof for notation)." ], "refs": [], "proofs": [ { "contents": [ "Recall that given $\\alpha : \\mathcal{E} \\to \\mathcal{B}$", "such that $\\mathcal{A}[\\mathcal{E}] \\to \\mathcal{B}$ is surjective", "with kernel $\\mathcal{I}$ the complex", "$\\NL(\\alpha) = (\\mathcal{I}/\\mathcal{I}^2 \\to ", "\\Omega_{\\mathcal{A}[\\mathcal{E}]/\\mathcal{A}}", "\\otimes_{\\mathcal{A}[\\mathcal{E}]} \\mathcal{B})$ is canonically", "isomorphic to $\\NL_{\\mathcal{B}/\\mathcal{A}}$, see", "Modules, Lemma \\ref{modules-lemma-NL-up-to-qis}.", "Observe moreover, that", "$\\Omega = \\Omega_{\\mathcal{A}[\\mathcal{E}]/\\mathcal{A}}", "\\otimes_{\\mathcal{A}[\\mathcal{E}]} \\mathcal{B}$ is the sheaf", "associated to the presheaf", "$U \\mapsto \\bigoplus_{e \\in \\mathcal{E}(U)} \\mathcal{B}(U)$.", "In other words, $\\Omega$ is the free $\\mathcal{B}$-module on the", "sheaf of sets $\\mathcal{E}$ and in particular there is a canonical", "map $\\mathcal{E} \\to \\Omega$.", "\\medskip\\noindent", "Having said this, pick some $\\mathcal{E}$ (for example", "$\\mathcal{E} = \\mathcal{B}$ as in the definition of the naive cotangent", "complex). The obstruction to writing $\\xi$ as the class of a map", "$\\mathcal{I}/\\mathcal{I}^2 \\to \\mathcal{G}$ is an element in", "$\\Ext^1_\\mathcal{B}(\\Omega, \\mathcal{G})$. Say this is represented", "by the extension $0 \\to \\mathcal{G} \\to \\mathcal{H} \\to \\Omega \\to 0$", "of $\\mathcal{B}$-modules. Consider the sheaf of sets", "$\\mathcal{E}' = \\mathcal{E} \\times_\\Omega \\mathcal{H}$", "which comes with an induced map $\\alpha' : \\mathcal{E}' \\to \\mathcal{B}$.", "Let $\\mathcal{I}' = \\Ker(\\mathcal{A}[\\mathcal{E}'] \\to \\mathcal{B})$", "and $\\Omega' = \\Omega_{\\mathcal{A}[\\mathcal{E}']/\\mathcal{A}}", "\\otimes_{\\mathcal{A}[\\mathcal{E}']} \\mathcal{B}$.", "The pullback of $\\xi$ under the quasi-isomorphism", "$\\NL(\\alpha') \\to \\NL(\\alpha)$ maps to zero in", "$\\Ext^1_\\mathcal{B}(\\Omega', \\mathcal{G})$", "because the pullback of the extension $\\mathcal{H}$", "by the map $\\Omega' \\to \\Omega$ is split as $\\Omega'$ is the free", "$\\mathcal{B}$-module on the sheaf of sets $\\mathcal{E}'$ and since", "by construction there is a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{E}' \\ar[r] \\ar[d] & \\mathcal{E} \\ar[d] \\\\", "\\mathcal{H} \\ar[r] & \\Omega", "}", "$$", "This finishes the proof." ], "refs": [ "modules-lemma-NL-up-to-qis" ], "ref_ids": [ 13327 ] } ], "ref_ids": [] }, { "id": 13389, "type": "theorem", "label": "defos-lemma-choices-ringed-spaces", "categories": [ "defos" ], "title": "defos-lemma-choices-ringed-spaces", "contents": [ "If there exists a solution to (\\ref{equation-to-solve-ringed-spaces}),", "then the set of isomorphism classes of solutions is principal homogeneous", "under $\\Ext^1_{\\mathcal{O}_X}(\\NL_{X/S}, \\mathcal{G})$." ], "refs": [], "proofs": [ { "contents": [ "We observe right away that given two solutions $X'_1$ and $X'_2$", "to (\\ref{equation-to-solve-ringed-spaces}) we obtain by", "Lemma \\ref{lemma-huge-diagram-ringed-spaces} an obstruction element", "$o(X'_1, X'_2) \\in \\Ext^1_{\\mathcal{O}_X}(\\NL_{X/S}, \\mathcal{G})$", "to the existence of a map $X'_1 \\to X'_2$. Clearly, this element", "is the obstruction to the existence of an isomorphism, hence separates", "the isomorphism classes. To finish the proof it therefore suffices to", "show that given a solution $X'$ and an element", "$\\xi \\in \\Ext^1_{\\mathcal{O}_X}(\\NL_{X/S}, \\mathcal{G})$", "we can find a second solution $X'_\\xi$ such that", "$o(X', X'_\\xi) = \\xi$.", "\\medskip\\noindent", "Pick $\\alpha : \\mathcal{E} \\to \\mathcal{O}_X$ as in", "Lemma \\ref{lemma-NL-represent-ext-class}", "for the class $\\xi$. Consider the surjection", "$f^{-1}\\mathcal{O}_S[\\mathcal{E}] \\to \\mathcal{O}_X$", "with kernel $\\mathcal{I}$ and corresponding naive cotangent complex", "$\\NL(\\alpha) = (\\mathcal{I}/\\mathcal{I}^2 \\to", "\\Omega_{f^{-1}\\mathcal{O}_S[\\mathcal{E}]/f^{-1}\\mathcal{O}_S}", "\\otimes_{f^{-1}\\mathcal{O}_S[\\mathcal{E}]} \\mathcal{O}_X)$.", "By the lemma $\\xi$ is the class of a morphism", "$\\delta : \\mathcal{I}/\\mathcal{I}^2 \\to \\mathcal{G}$.", "After replacing $\\mathcal{E}$ by", "$\\mathcal{E} \\times_{\\mathcal{O}_X} \\mathcal{O}_{X'}$ we may also assume", "that $\\alpha$ factors through a map", "$\\alpha' : \\mathcal{E} \\to \\mathcal{O}_{X'}$.", "\\medskip\\noindent", "These choices determine an $f^{-1}\\mathcal{O}_{S'}$-algebra map", "$\\varphi : \\mathcal{O}_{S'}[\\mathcal{E}] \\to \\mathcal{O}_{X'}$.", "Let $\\mathcal{I}' = \\Ker(\\varphi)$.", "Observe that $\\varphi$ induces a map", "$\\varphi|_{\\mathcal{I}'} : \\mathcal{I}' \\to \\mathcal{G}$", "and that $\\mathcal{O}_{X'}$ is the pushout, as in the following", "diagram", "$$", "\\xymatrix{", "0 \\ar[r] & \\mathcal{G} \\ar[r] & \\mathcal{O}_{X'} \\ar[r] &", "\\mathcal{O}_X \\ar[r] & 0 \\\\", "0 \\ar[r] & \\mathcal{I}' \\ar[u]^{\\varphi|_{\\mathcal{I}'}} \\ar[r] &", "f^{-1}\\mathcal{O}_{S'}[\\mathcal{E}] \\ar[u] \\ar[r] &", "\\mathcal{O}_X \\ar[u]_{=} \\ar[r] & 0", "}", "$$", "Let $\\psi : \\mathcal{I}' \\to \\mathcal{G}$ be the sum of the map", "$\\varphi|_{\\mathcal{I}'}$ and the composition", "$$", "\\mathcal{I}' \\to \\mathcal{I}'/(\\mathcal{I}')^2 \\to", "\\mathcal{I}/\\mathcal{I}^2 \\xrightarrow{\\delta} \\mathcal{G}.", "$$", "Then the pushout along $\\psi$ is an other ring extension", "$\\mathcal{O}_{X'_\\xi}$ fitting into a diagram as above.", "A calculation (omitted) shows that $o(X', X'_\\xi) = \\xi$ as desired." ], "refs": [ "defos-lemma-huge-diagram-ringed-spaces", "defos-lemma-NL-represent-ext-class" ], "ref_ids": [ 13387, 13388 ] } ], "ref_ids": [] }, { "id": 13390, "type": "theorem", "label": "defos-lemma-extensions-of-ringed-spaces", "categories": [ "defos" ], "title": "defos-lemma-extensions-of-ringed-spaces", "contents": [ "Let $(S, \\mathcal{O}_S)$ be a ringed space and let $\\mathcal{J}$", "be an $\\mathcal{O}_S$-module.", "\\begin{enumerate}", "\\item The set of extensions of sheaves of rings", "$0 \\to \\mathcal{J} \\to \\mathcal{O}_{S'} \\to \\mathcal{O}_S \\to 0$", "where $\\mathcal{J}$ is an ideal of square zero is canonically bijective to", "$\\Ext^1_{\\mathcal{O}_S}(\\NL_{S/\\mathbf{Z}}, \\mathcal{J})$.", "\\item Given a morphism of ringed spaces", "$f : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$, an $\\mathcal{O}_X$-module", "$\\mathcal{G}$, an $f$-map $c : \\mathcal{J} \\to \\mathcal{G}$, and", "given extensions of sheaves of rings with square zero kernels:", "\\begin{enumerate}", "\\item[(a)] $0 \\to \\mathcal{J} \\to \\mathcal{O}_{S'} \\to \\mathcal{O}_S \\to 0$", "corresponding to", "$\\alpha \\in \\Ext^1_{\\mathcal{O}_S}(\\NL_{S/\\mathbf{Z}}, \\mathcal{J})$,", "\\item[(b)] $0 \\to \\mathcal{G} \\to \\mathcal{O}_{X'} \\to \\mathcal{O}_X \\to 0$", "corresponding to", "$\\beta \\in \\Ext^1_{\\mathcal{O}_X}(\\NL_{X/\\mathbf{Z}}, \\mathcal{G})$", "\\end{enumerate}", "then there is a morphism $X' \\to S'$ fitting into a diagram", "(\\ref{equation-to-solve-ringed-spaces}) if and only if $\\beta$ and $\\alpha$", "map to the same element of", "$\\Ext^1_{\\mathcal{O}_X}(Lf^*\\NL_{S/\\mathbf{Z}}, \\mathcal{G})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove this we apply the previous results where we work over", "the base ringed space $(*, \\mathbf{Z})$ with trivial thickening.", "Part (1) follows from Lemma \\ref{lemma-choices-ringed-spaces}", "and the fact that there exists a solution, namely", "$\\mathcal{J} \\oplus \\mathcal{O}_S$.", "Part (2) follows from Lemma \\ref{lemma-huge-diagram-ringed-spaces}", "and a compatibility between the constructions in the proofs", "of Lemmas \\ref{lemma-choices-ringed-spaces} and", "\\ref{lemma-huge-diagram-ringed-spaces}", "whose statement and proof we omit." ], "refs": [ "defos-lemma-choices-ringed-spaces", "defos-lemma-huge-diagram-ringed-spaces", "defos-lemma-choices-ringed-spaces", "defos-lemma-huge-diagram-ringed-spaces" ], "ref_ids": [ 13389, 13387, 13389, 13387 ] } ], "ref_ids": [] }, { "id": 13391, "type": "theorem", "label": "defos-lemma-deform", "categories": [ "defos" ], "title": "defos-lemma-deform", "contents": [ "Let $S \\subset S'$ be a first order thickening of schemes.", "Let $f : X \\to S$ be a flat morphism of schemes.", "If there exists a flat morphism $f' : X' \\to S'$ of schemes", "and an isomorphism $a : X \\to X' \\times_{S'} S$ over $S$, then", "\\begin{enumerate}", "\\item the set of isomorphism classes of pairs $(f' : X' \\to S', a)$ is", "principal homogeneous under", "$\\Ext^1_{\\mathcal{O}_X}(\\NL_{X/S}, f^*\\mathcal{C}_{S/S'})$, and", "\\item the set of automorphisms of $\\varphi : X' \\to X'$", "over $S'$ which reduce to the identity on $X' \\times_{S'} S$", "is $\\Ext^0_{\\mathcal{O}_X}(\\NL_{X/S}, f^*\\mathcal{C}_{S/S'})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First we observe that thickenings of schemes as defined in", "More on Morphisms, Section \\ref{more-morphisms-section-thickenings}", "are the same things as morphisms of schemes which", "are thickenings in the sense of", "Section \\ref{section-thickenings-spaces}.", "We may think of $X$ as a closed subscheme of $X'$", "so that $(f, f') : (X \\subset X') \\to (S \\subset S')$", "is a morphism of first order thickenings. Then we see", "from More on Morphisms, Lemma \\ref{more-morphisms-lemma-deform}", "(or from the more general Lemma \\ref{lemma-deform-module})", "that the ideal sheaf of $X$ in $X'$ is equal to $f^*\\mathcal{C}_{S/S'}$.", "Hence we have a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] & f^*\\mathcal{C}_{S/S'} \\ar[r] &", "\\mathcal{O}_{X'} \\ar[r] &", "\\mathcal{O}_X \\ar[r] & 0 \\\\", "0 \\ar[r] & \\mathcal{C}_{S/S'} \\ar[u] \\ar[r] &", "\\mathcal{O}_{S'} \\ar[u] \\ar[r] &", "\\mathcal{O}_S \\ar[u] \\ar[r] & 0", "}", "$$", "where the vertical arrows are $f$-maps; please compare with", "(\\ref{equation-to-solve-ringed-spaces}).", "Thus part (1) follows from", "Lemma \\ref{lemma-choices-ringed-spaces}", "and part (2) from part (2) of", "Lemma \\ref{lemma-huge-diagram-ringed-spaces}.", "(Note that $\\NL_{X/S}$ as defined for a morphism of schemes in", "More on Morphisms, Section \\ref{more-morphisms-section-netherlander}", "agrees with $\\NL_{X/S}$ as used in", "Section \\ref{section-deformations-ringed-spaces}.)" ], "refs": [ "more-morphisms-lemma-deform", "defos-lemma-deform-module", "defos-lemma-choices-ringed-spaces", "defos-lemma-huge-diagram-ringed-spaces" ], "ref_ids": [ 13723, 13379, 13389, 13387 ] } ], "ref_ids": [] }, { "id": 13392, "type": "theorem", "label": "defos-lemma-inf-map-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-inf-map-ringed-topoi", "contents": [ "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$", "be a first order thickening of ringed topoi. Assume given", "extensions", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0", "\\quad\\text{and}\\quad", "0 \\to \\mathcal{L} \\to \\mathcal{G}' \\to \\mathcal{G} \\to 0", "$$", "as in (\\ref{equation-extension-ringed-topoi})", "and maps $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and", "$\\psi : \\mathcal{K} \\to \\mathcal{L}$.", "\\begin{enumerate}", "\\item If there exists an $\\mathcal{O}'$-module", "map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$", "and $\\psi$, then the diagram", "$$", "\\xymatrix{", "\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F}", "\\ar[r]_-{c_{\\mathcal{F}'}} \\ar[d]_{1 \\otimes \\varphi} &", "\\mathcal{K} \\ar[d]^\\psi \\\\", "\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{G}", "\\ar[r]^-{c_{\\mathcal{G}'}} &", "\\mathcal{L}", "}", "$$", "is commutative.", "\\item The set of $\\mathcal{O}'$-module", "maps $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$", "and $\\psi$ is, if nonempty, a principal homogeneous space under", "$\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{L})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is immediate from the description of the maps.", "For (2), if $\\varphi'$ and $\\varphi''$ are two maps", "$\\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$", "and $\\psi$, then $\\varphi' - \\varphi''$ factors as", "$$", "\\mathcal{F}' \\to \\mathcal{F} \\to \\mathcal{L} \\to \\mathcal{G}'", "$$", "The map in the middle comes from a unique element of", "$\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{L})$ by", "Modules on Sites, Lemma \\ref{sites-modules-lemma-i-star-equivalence}.", "Conversely, given an element $\\alpha$ of this group we can add the", "composition (as displayed above with $\\alpha$ in the middle)", "to $\\varphi'$. Some details omitted." ], "refs": [ "sites-modules-lemma-i-star-equivalence" ], "ref_ids": [ 14188 ] } ], "ref_ids": [] }, { "id": 13393, "type": "theorem", "label": "defos-lemma-inf-obs-map-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-inf-obs-map-ringed-topoi", "contents": [ "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$", "be a first order thickening of ringed topoi. Assume given extensions", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0", "\\quad\\text{and}\\quad", "0 \\to \\mathcal{L} \\to \\mathcal{G}' \\to \\mathcal{G} \\to 0", "$$", "as in (\\ref{equation-extension-ringed-topoi})", "and maps $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and", "$\\psi : \\mathcal{K} \\to \\mathcal{L}$. Assume the diagram", "$$", "\\xymatrix{", "\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F}", "\\ar[r]_-{c_{\\mathcal{F}'}} \\ar[d]_{1 \\otimes \\varphi} &", "\\mathcal{K} \\ar[d]^\\psi \\\\", "\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{G}", "\\ar[r]^-{c_{\\mathcal{G}'}} &", "\\mathcal{L}", "}", "$$", "is commutative. Then there exists an element", "$$", "o(\\varphi, \\psi) \\in", "\\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{L})", "$$", "whose vanishing is a necessary and sufficient condition for the existence", "of a map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with", "$\\varphi$ and $\\psi$." ], "refs": [], "proofs": [ { "contents": [ "We can construct explicitly an extension", "$$", "0 \\to \\mathcal{L} \\to \\mathcal{H} \\to \\mathcal{F} \\to 0", "$$", "by taking $\\mathcal{H}$ to be the cohomology of the complex", "$$", "\\mathcal{K}", "\\xrightarrow{1, - \\psi}", "\\mathcal{F}' \\oplus \\mathcal{G}' \\xrightarrow{\\varphi, 1}", "\\mathcal{G}", "$$", "in the middle (with obvious notation). A calculation with local sections", "using the assumption that the diagram of the lemma commutes", "shows that $\\mathcal{H}$ is annihilated by $\\mathcal{I}$. Hence", "$\\mathcal{H}$ defines a class in", "$$", "\\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{L})", "\\subset", "\\Ext^1_{\\mathcal{O}'}(\\mathcal{F}, \\mathcal{L})", "$$", "Finally, the class of $\\mathcal{H}$ is the difference of the pushout", "of the extension $\\mathcal{F}'$ via $\\psi$ and the pullback", "of the extension $\\mathcal{G}'$ via $\\varphi$ (calculations omitted).", "Thus the vanishing of the class of $\\mathcal{H}$ is equivalent to the", "existence of a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{K} \\ar[r] \\ar[d]_{\\psi} &", "\\mathcal{F}' \\ar[r] \\ar[d]_{\\varphi'} &", "\\mathcal{F} \\ar[r] \\ar[d]_\\varphi & 0\\\\", "0 \\ar[r] &", "\\mathcal{L} \\ar[r] &", "\\mathcal{G}' \\ar[r] &", "\\mathcal{G} \\ar[r] & 0", "}", "$$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13394, "type": "theorem", "label": "defos-lemma-inf-ext-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-inf-ext-ringed-topoi", "contents": [ "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$", "be a first order thickening of ringed topoi. Assume given", "$\\mathcal{O}$-modules $\\mathcal{F}$, $\\mathcal{K}$", "and an $\\mathcal{O}$-linear map", "$c : \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{K}$.", "If there exists a sequence (\\ref{equation-extension-ringed-topoi}) with", "$c_{\\mathcal{F}'} = c$ then the set of isomorphism classes of these", "extensions is principal homogeneous under", "$\\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K})$." ], "refs": [], "proofs": [ { "contents": [ "Assume given extensions", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{F}'_1 \\to \\mathcal{F} \\to 0", "\\quad\\text{and}\\quad", "0 \\to \\mathcal{K} \\to \\mathcal{F}'_2 \\to \\mathcal{F} \\to 0", "$$", "with $c_{\\mathcal{F}'_1} = c_{\\mathcal{F}'_2} = c$. Then the difference", "(in the extension group, see", "Homology, Section \\ref{homology-section-extensions})", "is an extension", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{E} \\to \\mathcal{F} \\to 0", "$$", "where $\\mathcal{E}$ is annihilated by $\\mathcal{I}$ (local computation", "omitted). Hence the sequence is an extension of $\\mathcal{O}$-modules,", "see Modules on Sites, Lemma \\ref{sites-modules-lemma-i-star-equivalence}.", "Conversely, given such an extension $\\mathcal{E}$ we can add the extension", "$\\mathcal{E}$ to the $\\mathcal{O}'$-extension $\\mathcal{F}'$ without", "affecting the map $c_{\\mathcal{F}'}$. Some details omitted." ], "refs": [ "sites-modules-lemma-i-star-equivalence" ], "ref_ids": [ 14188 ] } ], "ref_ids": [] }, { "id": 13395, "type": "theorem", "label": "defos-lemma-inf-obs-ext-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-inf-obs-ext-ringed-topoi", "contents": [ "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$", "be a first order thickening of ringed topoi. Assume given", "$\\mathcal{O}$-modules $\\mathcal{F}$, $\\mathcal{K}$", "and an $\\mathcal{O}$-linear map", "$c : \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{K}$.", "Then there exists an element", "$$", "o(\\mathcal{F}, \\mathcal{K}, c) \\in", "\\Ext^2_\\mathcal{O}(\\mathcal{F}, \\mathcal{K})", "$$", "whose vanishing is a necessary and sufficient condition for the existence", "of a sequence (\\ref{equation-extension-ringed-topoi})", "with $c_{\\mathcal{F}'} = c$." ], "refs": [], "proofs": [ { "contents": [ "We first show that if $\\mathcal{K}$ is an injective $\\mathcal{O}$-module,", "then there does exist a sequence (\\ref{equation-extension-ringed-topoi}) with", "$c_{\\mathcal{F}'} = c$. To do this, choose a flat", "$\\mathcal{O}'$-module $\\mathcal{H}'$ and a surjection", "$\\mathcal{H}' \\to \\mathcal{F}$", "(Modules on Sites, Lemma \\ref{sites-modules-lemma-module-quotient-flat}).", "Let $\\mathcal{J} \\subset \\mathcal{H}'$ be the kernel. Since $\\mathcal{H}'$", "is flat we have", "$$", "\\mathcal{I} \\otimes_{\\mathcal{O}'} \\mathcal{H}' =", "\\mathcal{I}\\mathcal{H}'", "\\subset \\mathcal{J} \\subset \\mathcal{H}'", "$$", "Observe that the map", "$$", "\\mathcal{I}\\mathcal{H}' =", "\\mathcal{I} \\otimes_{\\mathcal{O}'} \\mathcal{H}'", "\\longrightarrow", "\\mathcal{I} \\otimes_{\\mathcal{O}'} \\mathcal{F} =", "\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F}", "$$", "annihilates $\\mathcal{I}\\mathcal{J}$. Namely, if $f$ is a local section", "of $\\mathcal{I}$ and $s$ is a local section of $\\mathcal{H}$, then", "$fs$ is mapped to $f \\otimes \\overline{s}$ where $\\overline{s}$ is", "the image of $s$ in $\\mathcal{F}$. Thus we obtain", "$$", "\\xymatrix{", "\\mathcal{I}\\mathcal{H}'/\\mathcal{I}\\mathcal{J}", "\\ar@{^{(}->}[r] \\ar[d] &", "\\mathcal{J}/\\mathcal{I}\\mathcal{J} \\ar@{..>}[d]_\\gamma \\\\", "\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\ar[r]^-c &", "\\mathcal{K}", "}", "$$", "a diagram of $\\mathcal{O}$-modules. If $\\mathcal{K}$ is injective", "as an $\\mathcal{O}$-module, then we obtain the dotted arrow.", "Denote $\\gamma' : \\mathcal{J} \\to \\mathcal{K}$ the composition", "of $\\gamma$ with $\\mathcal{J} \\to \\mathcal{J}/\\mathcal{I}\\mathcal{J}$.", "A local calculation shows the pushout", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{J} \\ar[r] \\ar[d]_{\\gamma'} &", "\\mathcal{H}' \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[r] \\ar@{=}[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{K} \\ar[r] &", "\\mathcal{F}' \\ar[r] &", "\\mathcal{F} \\ar[r] &", "0", "}", "$$", "is a solution to the problem posed by the lemma.", "\\medskip\\noindent", "General case. Choose an embedding $\\mathcal{K} \\subset \\mathcal{K}'$", "with $\\mathcal{K}'$ an injective $\\mathcal{O}$-module. Let $\\mathcal{Q}$", "be the quotient, so that we have an exact sequence", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{K}' \\to \\mathcal{Q} \\to 0", "$$", "Denote", "$c' : \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{K}'$", "be the composition. By the paragraph above there exists a sequence", "$$", "0 \\to \\mathcal{K}' \\to \\mathcal{E}' \\to \\mathcal{F} \\to 0", "$$", "as in (\\ref{equation-extension-ringed-topoi}) with $c_{\\mathcal{E}'} = c'$.", "Note that $c'$ composed with the map $\\mathcal{K}' \\to \\mathcal{Q}$", "is zero, hence the pushout of $\\mathcal{E}'$ by", "$\\mathcal{K}' \\to \\mathcal{Q}$ is an extension", "$$", "0 \\to \\mathcal{Q} \\to \\mathcal{D}' \\to \\mathcal{F} \\to 0", "$$", "as in (\\ref{equation-extension-ringed-topoi}) with $c_{\\mathcal{D}'} = 0$.", "This means exactly that $\\mathcal{D}'$ is annihilated by", "$\\mathcal{I}$, in other words, the $\\mathcal{D}'$ is an extension", "of $\\mathcal{O}$-modules, i.e., defines an element", "$$", "o(\\mathcal{F}, \\mathcal{K}, c) \\in", "\\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{Q}) =", "\\Ext^2_\\mathcal{O}(\\mathcal{F}, \\mathcal{K})", "$$", "(the equality holds by the long exact cohomology sequence associated", "to the exact sequence above and the vanishing of higher ext groups", "into the injective module $\\mathcal{K}'$). If", "$o(\\mathcal{F}, \\mathcal{K}, c) = 0$, then we can choose a splitting", "$s : \\mathcal{F} \\to \\mathcal{D}'$ and we can set", "$$", "\\mathcal{F}' = \\Ker(\\mathcal{E}' \\to \\mathcal{D}'/s(\\mathcal{F}))", "$$", "so that we obtain the following diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{K} \\ar[r] \\ar[d] &", "\\mathcal{F}' \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[r] \\ar@{=}[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{K}' \\ar[r] &", "\\mathcal{E}' \\ar[r] &", "\\mathcal{F} \\ar[r] & 0", "}", "$$", "with exact rows which shows that $c_{\\mathcal{F}'} = c$. Conversely, if", "$\\mathcal{F}'$ exists, then the pushout of $\\mathcal{F}'$ by the map", "$\\mathcal{K} \\to \\mathcal{K}'$ is isomorphic to $\\mathcal{E}'$ by", "Lemma \\ref{lemma-inf-ext-ringed-topoi} and the vanishing of higher ext groups", "into the injective module $\\mathcal{K}'$. This gives a diagram", "as above, which implies that $\\mathcal{D}'$ is split as an extension, i.e.,", "the class $o(\\mathcal{F}, \\mathcal{K}, c)$ is zero." ], "refs": [ "sites-modules-lemma-module-quotient-flat", "defos-lemma-inf-ext-ringed-topoi" ], "ref_ids": [ 14203, 13394 ] } ], "ref_ids": [] }, { "id": 13396, "type": "theorem", "label": "defos-lemma-inf-map-special-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-inf-map-special-ringed-topoi", "contents": [ "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$", "be a first order thickening of ringed topoi.", "Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}'$-modules.", "Set $\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}$-linear map.", "The set of lifts of $\\varphi$ to an $\\mathcal{O}'$-linear map", "$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ is, if nonempty, a principal", "homogeneous space under", "$\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{I}\\mathcal{G}')$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-inf-map-ringed-topoi} but we also", "give a direct proof. We have short exact sequences of modules", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}' \\to \\mathcal{O} \\to 0", "\\quad\\text{and}\\quad", "0 \\to \\mathcal{I}\\mathcal{G}' \\to \\mathcal{G}' \\to \\mathcal{G} \\to 0", "$$", "and similarly for $\\mathcal{F}'$.", "Since $\\mathcal{I}$ has square zero the $\\mathcal{O}'$-module", "structure on $\\mathcal{I}$ and $\\mathcal{I}\\mathcal{G}'$ comes from", "a unique $\\mathcal{O}$-module structure. It follows that", "$$", "\\Hom_{\\mathcal{O}'}(\\mathcal{F}', \\mathcal{I}\\mathcal{G}') =", "\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{I}\\mathcal{G}')", "\\quad\\text{and}\\quad", "\\Hom_{\\mathcal{O}'}(\\mathcal{F}', \\mathcal{G}) =", "\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})", "$$", "The lemma now follows from the exact sequence", "$$", "0 \\to \\Hom_{\\mathcal{O}'}(\\mathcal{F}', \\mathcal{I}\\mathcal{G}') \\to", "\\Hom_{\\mathcal{O}'}(\\mathcal{F}', \\mathcal{G}') \\to", "\\Hom_{\\mathcal{O}'}(\\mathcal{F}', \\mathcal{G})", "$$", "see Homology, Lemma \\ref{homology-lemma-check-exactness}." ], "refs": [ "defos-lemma-inf-map-ringed-topoi", "homology-lemma-check-exactness" ], "ref_ids": [ 13392, 12019 ] } ], "ref_ids": [] }, { "id": 13397, "type": "theorem", "label": "defos-lemma-deform-module-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-deform-module-ringed-topoi", "contents": [ "Let $(f, f')$ be a morphism of first order thickenings of ringed topoi", "as in Situation \\ref{situation-morphism-thickenings-ringed-topoi}.", "Let $\\mathcal{F}'$ be an $\\mathcal{O}'$-module", "and set $\\mathcal{F} = i^*\\mathcal{F}'$.", "Assume that $\\mathcal{F}$ is flat over $\\mathcal{O}_\\mathcal{B}$", "and that $(f, f')$ is a strict morphism of thickenings", "(Definition \\ref{definition-strict-morphism-thickenings-ringed-topoi}).", "Then the following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}'$ is flat over $\\mathcal{O}_{\\mathcal{B}'}$, and", "\\item the canonical map", "$f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F} \\to", "\\mathcal{I}\\mathcal{F}'$", "is an isomorphism.", "\\end{enumerate}", "Moreover, in this case the maps", "$$", "f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F} \\to", "\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to", "\\mathcal{I}\\mathcal{F}'", "$$", "are isomorphisms." ], "refs": [ "defos-definition-strict-morphism-thickenings-ringed-topoi" ], "proofs": [ { "contents": [ "The map $f^*\\mathcal{J} \\to \\mathcal{I}$ is surjective", "as $(f, f')$ is a strict morphism of thickenings.", "Hence the final statement is a consequence of (2).", "\\medskip\\noindent", "Proof of the equivalence of (1) and (2). By definition flatness over", "$\\mathcal{O}_\\mathcal{B}$ means flatness over $f^{-1}\\mathcal{O}_\\mathcal{B}$.", "Similarly for flatness over $f^{-1}\\mathcal{O}_{\\mathcal{B}'}$.", "Note that the strictness of $(f, f')$ and the assumption that", "$\\mathcal{F} = i^*\\mathcal{F}'$ imply that", "$$", "\\mathcal{F} = \\mathcal{F}'/(f^{-1}\\mathcal{J})\\mathcal{F}'", "$$", "as sheaves on $\\mathcal{C}$. Moreover, observe that", "$f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F} =", "f^{-1}\\mathcal{J} \\otimes_{f^{-1}\\mathcal{O}_\\mathcal{B}} \\mathcal{F}$.", "Hence the equivalence of (1) and (2) follows from", "Modules on Sites, Lemma \\ref{sites-modules-lemma-flat-over-thickening}." ], "refs": [ "sites-modules-lemma-flat-over-thickening" ], "ref_ids": [ 14210 ] } ], "ref_ids": [ 13424 ] }, { "id": 13398, "type": "theorem", "label": "defos-lemma-deform-fp-module-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-deform-fp-module-ringed-topoi", "contents": [ "Let $(f, f')$ be a morphism of first order thickenings of ringed topoi", "as in Situation \\ref{situation-morphism-thickenings-ringed-topoi}.", "Let $\\mathcal{F}'$ be an $\\mathcal{O}'$-module", "and set $\\mathcal{F} = i^*\\mathcal{F}'$.", "Assume that $\\mathcal{F}'$ is flat over $\\mathcal{O}_{\\mathcal{B}'}$", "and that $(f, f')$ is a strict morphism of thickenings.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}'$ is an $\\mathcal{O}'$-module of finite presentation, and", "\\item $\\mathcal{F}$ is an $\\mathcal{O}$-module of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) follows from", "Modules on Sites, Lemma \\ref{sites-modules-lemma-local-pullback}.", "For the converse, assume $\\mathcal{F}$ of finite presentation.", "We may and do assume that $\\mathcal{C} = \\mathcal{C}'$.", "By Lemma \\ref{lemma-deform-module-ringed-topoi} we have a short exact sequence", "$$", "0 \\to \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "\\mathcal{F}' \\to \\mathcal{F} \\to 0", "$$", "Let $U$ be an object of $\\mathcal{C}$ such that $\\mathcal{F}|_U$ has a", "presentation", "$$", "\\mathcal{O}_U^{\\oplus m} \\to \\mathcal{O}_U^{\\oplus n} \\to \\mathcal{F}|_U \\to 0", "$$", "After replacing $U$ by the members of a covering we may assume the", "map $\\mathcal{O}_U^{\\oplus n} \\to \\mathcal{F}|_U$ lifts to a map", "$(\\mathcal{O}'_U)^{\\oplus n} \\to \\mathcal{F}'|_U$. The induced map", "$\\mathcal{I}^{\\oplus n} \\to \\mathcal{I} \\otimes \\mathcal{F}$ is", "surjective by right exactness of $\\otimes$. Thus after replacing $U$", "by the members of a covering we can find a lift", "$(\\mathcal{O}'|_U)^{\\oplus m} \\to (\\mathcal{O}'|_U)^{\\oplus n}$", "of the given map $\\mathcal{O}_U^{\\oplus m} \\to \\mathcal{O}_U^{\\oplus n}$", "such that", "$$", "(\\mathcal{O}'_U)^{\\oplus m} \\to (\\mathcal{O}'_U)^{\\oplus n} \\to", "\\mathcal{F}'|_U \\to 0", "$$", "is a complex. Using right exactness of $\\otimes$ once more it is seen", "that this complex is exact." ], "refs": [ "sites-modules-lemma-local-pullback", "defos-lemma-deform-module-ringed-topoi" ], "ref_ids": [ 14186, 13397 ] } ], "ref_ids": [] }, { "id": 13399, "type": "theorem", "label": "defos-lemma-inf-map-rel-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-inf-map-rel-ringed-topoi", "contents": [ "Let $(f, f')$ be a morphism of first order thickenings as in", "Situation \\ref{situation-morphism-thickenings-ringed-topoi}.", "Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}'$-modules and set", "$\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}$-linear map.", "Assume that $\\mathcal{G}'$ is flat over $\\mathcal{O}_{\\mathcal{B}'}$ and that", "$(f, f')$ is a strict morphism of thickenings.", "The set of lifts of $\\varphi$ to an $\\mathcal{O}'$-linear map", "$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ is, if nonempty, a principal", "homogeneous space under", "$$", "\\Hom_\\mathcal{O}(\\mathcal{F},", "\\mathcal{G} \\otimes_\\mathcal{O} f^*\\mathcal{J})", "$$" ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-inf-map-special-ringed-topoi} and", "\\ref{lemma-deform-module-ringed-topoi}." ], "refs": [ "defos-lemma-inf-map-special-ringed-topoi", "defos-lemma-deform-module-ringed-topoi" ], "ref_ids": [ 13396, 13397 ] } ], "ref_ids": [] }, { "id": 13400, "type": "theorem", "label": "defos-lemma-inf-obs-map-special-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-inf-obs-map-special-ringed-topoi", "contents": [ "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$", "be a first order thickening of ringed topoi.", "Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}'$-modules and set", "$\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}$-linear map.", "There exists an element", "$$", "o(\\varphi) \\in", "\\Ext^1_\\mathcal{O}(Li^*\\mathcal{F}', \\mathcal{I}\\mathcal{G}')", "$$", "whose vanishing is a necessary and sufficient condition for the", "existence of a lift of $\\varphi$ to an $\\mathcal{O}'$-linear map", "$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$." ], "refs": [], "proofs": [ { "contents": [ "It is clear from the proof of Lemma \\ref{lemma-inf-map-special-ringed-topoi}", "that the vanishing of the boundary of $\\varphi$ via the map", "$$", "\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G}) =", "\\Hom_{\\mathcal{O}'}(\\mathcal{F}', \\mathcal{G}) \\longrightarrow", "\\Ext^1_{\\mathcal{O}'}(\\mathcal{F}', \\mathcal{I}\\mathcal{G}')", "$$", "is a necessary and sufficient condition for the existence of a lift. We", "conclude as", "$$", "\\Ext^1_{\\mathcal{O}'}(\\mathcal{F}', \\mathcal{I}\\mathcal{G}') =", "\\Ext^1_\\mathcal{O}(Li^*\\mathcal{F}', \\mathcal{I}\\mathcal{G}')", "$$", "the adjointness of $i_* = Ri_*$ and $Li^*$ on the derived category", "(Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-adjoint})." ], "refs": [ "defos-lemma-inf-map-special-ringed-topoi", "sites-cohomology-lemma-adjoint" ], "ref_ids": [ 13396, 4249 ] } ], "ref_ids": [] }, { "id": 13401, "type": "theorem", "label": "defos-lemma-inf-obs-map-rel-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-inf-obs-map-rel-ringed-topoi", "contents": [ "Let $(f, f')$ be a morphism of first order thickenings as in", "Situation \\ref{situation-morphism-thickenings-ringed-topoi}.", "Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}'$-modules and set", "$\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}$-linear map.", "Assume that $\\mathcal{F}'$ and $\\mathcal{G}'$ are flat over", "$\\mathcal{O}_{\\mathcal{B}'}$ and", "that $(f, f')$ is a strict morphism of thickenings. There exists an element", "$$", "o(\\varphi) \\in", "\\Ext^1_\\mathcal{O}(\\mathcal{F},", "\\mathcal{G} \\otimes_\\mathcal{O} f^*\\mathcal{J})", "$$", "whose vanishing is a necessary and sufficient condition for the", "existence of a lift of $\\varphi$ to an $\\mathcal{O}'$-linear map", "$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "This follows from Lemma \\ref{lemma-inf-obs-map-special-ringed-topoi}", "as we claim that under the assumptions of the lemma we have", "$$", "\\Ext^1_\\mathcal{O}(Li^*\\mathcal{F}', \\mathcal{I}\\mathcal{G}') =", "\\Ext^1_\\mathcal{O}(\\mathcal{F},", "\\mathcal{G} \\otimes_\\mathcal{O} f^*\\mathcal{J})", "$$", "Namely, we have", "$\\mathcal{I}\\mathcal{G}' =", "\\mathcal{G} \\otimes_\\mathcal{O} f^*\\mathcal{J}$", "by Lemma \\ref{lemma-deform-module-ringed-topoi}.", "On the other hand, observe that", "$$", "H^{-1}(Li^*\\mathcal{F}') =", "\\text{Tor}_1^{\\mathcal{O}'}(\\mathcal{F}', \\mathcal{O})", "$$", "(local computation omitted). Using the short exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}' \\to \\mathcal{O} \\to 0", "$$", "we see that this $\\text{Tor}_1$ is computed by the kernel of the map", "$\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{I}\\mathcal{F}'$", "which is zero by the final assertion of", "Lemma \\ref{lemma-deform-module-ringed-topoi}.", "Thus $\\tau_{\\geq -1}Li^*\\mathcal{F}' = \\mathcal{F}$.", "On the other hand, we have", "$$", "\\Ext^1_\\mathcal{O}(Li^*\\mathcal{F}',", "\\mathcal{I}\\mathcal{G}') =", "\\Ext^1_\\mathcal{O}(\\tau_{\\geq -1}Li^*\\mathcal{F}',", "\\mathcal{I}\\mathcal{G}')", "$$", "by the dual of", "Derived Categories, Lemma \\ref{derived-lemma-negative-vanishing}." ], "refs": [ "defos-lemma-inf-obs-map-special-ringed-topoi", "defos-lemma-deform-module-ringed-topoi", "defos-lemma-deform-module-ringed-topoi", "derived-lemma-negative-vanishing" ], "ref_ids": [ 13400, 13397, 13397, 1839 ] } ], "ref_ids": [] }, { "id": 13402, "type": "theorem", "label": "defos-lemma-inf-ext-rel-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-inf-ext-rel-ringed-topoi", "contents": [ "Let $(f, f')$ be a morphism of first order thickenings as in", "Situation \\ref{situation-morphism-thickenings-ringed-topoi}.", "Let $\\mathcal{F}$ be an $\\mathcal{O}$-module.", "Assume $(f, f')$ is a strict morphism of thickenings and", "$\\mathcal{F}$ flat over $\\mathcal{O}_\\mathcal{B}$. If there exists a pair", "$(\\mathcal{F}', \\alpha)$ consisting of an", "$\\mathcal{O}'$-module $\\mathcal{F}'$ flat over $\\mathcal{O}_{\\mathcal{B}'}$", "and an isomorphism", "$\\alpha : i^*\\mathcal{F}' \\to \\mathcal{F}$, then the set of", "isomorphism classes of such pairs is principal homogeneous", "under", "$\\Ext^1_\\mathcal{O}(", "\\mathcal{F}, \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "If we assume there exists one such module, then the canonical map", "$$", "f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F} \\to", "\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F}", "$$", "is an isomorphism by Lemma \\ref{lemma-deform-module-ringed-topoi}. Apply", "Lemma \\ref{lemma-inf-ext-ringed-topoi} with $\\mathcal{K} = ", "\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F}$", "and $c = 1$. By Lemma \\ref{lemma-deform-module-ringed-topoi}", "the corresponding extensions", "$\\mathcal{F}'$ are all flat over $\\mathcal{O}_{\\mathcal{B}'}$." ], "refs": [ "defos-lemma-deform-module-ringed-topoi", "defos-lemma-inf-ext-ringed-topoi", "defos-lemma-deform-module-ringed-topoi" ], "ref_ids": [ 13397, 13394, 13397 ] } ], "ref_ids": [] }, { "id": 13403, "type": "theorem", "label": "defos-lemma-inf-obs-ext-rel-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-inf-obs-ext-rel-ringed-topoi", "contents": [ "Let $(f, f')$ be a morphism of first order thickenings as in", "Situation \\ref{situation-morphism-thickenings-ringed-topoi}.", "Let $\\mathcal{F}$ be an $\\mathcal{O}$-module. Assume", "$(f, f')$ is a strict morphism of thickenings", "and $\\mathcal{F}$ flat over $\\mathcal{O}_\\mathcal{B}$. There exists an", "$\\mathcal{O}'$-module $\\mathcal{F}'$ flat over $\\mathcal{O}_{\\mathcal{B}'}$", "with $i^*\\mathcal{F}' \\cong \\mathcal{F}$, if and only if", "\\begin{enumerate}", "\\item the canonical map", "$f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F} \\to", "\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F}$", "is an isomorphism, and", "\\item the class", "$o(\\mathcal{F}, \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F}, 1)", "\\in \\Ext^2_\\mathcal{O}(", "\\mathcal{F}, \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F})$", "of Lemma \\ref{lemma-inf-obs-ext-ringed-topoi} is zero.", "\\end{enumerate}" ], "refs": [ "defos-lemma-inf-obs-ext-ringed-topoi" ], "proofs": [ { "contents": [ "This follows immediately from the characterization of", "$\\mathcal{O}'$-modules flat over $\\mathcal{O}_{\\mathcal{B}'}$ of ", "Lemma \\ref{lemma-deform-module-ringed-topoi} and", "Lemma \\ref{lemma-inf-obs-ext-ringed-topoi}." ], "refs": [ "defos-lemma-deform-module-ringed-topoi", "defos-lemma-inf-obs-ext-ringed-topoi" ], "ref_ids": [ 13397, 13395 ] } ], "ref_ids": [ 13395 ] }, { "id": 13404, "type": "theorem", "label": "defos-lemma-flat-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-flat-ringed-topoi", "contents": [ "In the situation above.", "\\begin{enumerate}", "\\item There exists an $\\mathcal{O}'$-module $\\mathcal{F}'$ flat over", "$\\mathcal{O}_{\\mathcal{B}'}$ with $i^*\\mathcal{F}' \\cong \\mathcal{F}$,", "if and only if", "the class $o(\\mathcal{F}, f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F}, 1)", "\\in \\Ext^2_\\mathcal{O}(", "\\mathcal{F}, f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F})$", "of Lemma \\ref{lemma-inf-obs-ext-ringed-topoi} is zero.", "\\item If such a module exists, then the set of isomorphism classes", "of lifts is principal homogeneous under", "$\\Ext^1_\\mathcal{O}(", "\\mathcal{F}, f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F})$.", "\\item Given a lift $\\mathcal{F}'$, the set of automorphisms of", "$\\mathcal{F}'$ which pull back to $\\text{id}_\\mathcal{F}$ is canonically", "isomorphic to $\\Ext^0_\\mathcal{O}(", "\\mathcal{F}, f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F})$.", "\\end{enumerate}" ], "refs": [ "defos-lemma-inf-obs-ext-ringed-topoi" ], "proofs": [ { "contents": [ "Part (1) follows from Lemma \\ref{lemma-inf-obs-ext-rel-ringed-topoi}", "as we have seen above that $\\mathcal{I} = f^*\\mathcal{J}$.", "Part (2) follows from Lemma \\ref{lemma-inf-ext-rel-ringed-topoi}.", "Part (3) follows from Lemma \\ref{lemma-inf-map-rel-ringed-topoi}." ], "refs": [ "defos-lemma-inf-obs-ext-rel-ringed-topoi", "defos-lemma-inf-ext-rel-ringed-topoi", "defos-lemma-inf-map-rel-ringed-topoi" ], "ref_ids": [ 13403, 13402, 13399 ] } ], "ref_ids": [ 13395 ] }, { "id": 13405, "type": "theorem", "label": "defos-lemma-functorial-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-functorial-ringed-topoi", "contents": [ "In Situation \\ref{situation-morphism-flat-thickenings-ringed-topoi}", "the obstruction class", "$o(\\mathcal{F}, f^*\\mathcal{J}_2 \\otimes_\\mathcal{O} \\mathcal{F}, 1)$", "maps to the obstruction class", "$o(\\mathcal{F}, f^*\\mathcal{J}_1 \\otimes_\\mathcal{O} \\mathcal{F}, 1)$", "under the canonical map", "$$", "\\Ext^2_\\mathcal{O}(", "\\mathcal{F}, f^*\\mathcal{J}_2 \\otimes_\\mathcal{O} \\mathcal{F})", "\\to \\Ext^2_\\mathcal{O}(", "\\mathcal{F}, f^*\\mathcal{J}_1 \\otimes_\\mathcal{O} \\mathcal{F})", "$$" ], "refs": [], "proofs": [ { "contents": [ "Follows from Remark \\ref{remark-obstruction-extension-functorial-ringed-topoi}." ], "refs": [ "defos-remark-obstruction-extension-functorial-ringed-topoi" ], "ref_ids": [ 13435 ] } ], "ref_ids": [] }, { "id": 13406, "type": "theorem", "label": "defos-lemma-verify-iv-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-verify-iv-ringed-topoi", "contents": [ "In Situation \\ref{situation-ses-flat-thickenings-ringed-topoi} the modules", "$\\pi^*\\mathcal{F}$ and $h^*\\mathcal{F}'_2$ are $\\mathcal{O}'_1$-modules", "flat over $\\mathcal{O}_{\\mathcal{B}'_1}$ restricting to $\\mathcal{F}$ on", "$(\\Sh(\\mathcal{C}), \\mathcal{O})$.", "Their difference (Lemma \\ref{lemma-flat-ringed-topoi}) is an element", "$\\theta$ of", "$\\Ext^1_\\mathcal{O}(\\mathcal{F},", "f^*\\mathcal{J}_1 \\otimes_\\mathcal{O} \\mathcal{F})$", "whose boundary in", "$\\Ext^2_\\mathcal{O}(\\mathcal{F},", "f^*\\mathcal{J}_3 \\otimes_\\mathcal{O} \\mathcal{F})$", "equals the obstruction (Lemma \\ref{lemma-flat-ringed-topoi})", "to lifting $\\mathcal{F}$ to an $\\mathcal{O}'_3$-module flat over", "$\\mathcal{O}_{\\mathcal{B}'_3}$." ], "refs": [ "defos-lemma-flat-ringed-topoi", "defos-lemma-flat-ringed-topoi" ], "proofs": [ { "contents": [ "Note that both $\\pi^*\\mathcal{F}$ and $h^*\\mathcal{F}'_2$", "restrict to $\\mathcal{F}$ on $(\\Sh(\\mathcal{C}), \\mathcal{O})$", "and that the kernels of", "$\\pi^*\\mathcal{F} \\to \\mathcal{F}$ and $h^*\\mathcal{F}'_2 \\to \\mathcal{F}$", "are given by $f^*\\mathcal{J}_1 \\otimes_\\mathcal{O} \\mathcal{F}$.", "Hence flatness by Lemma \\ref{lemma-deform-module-ringed-topoi}.", "Taking the boundary makes sense as the sequence of modules", "$$", "0 \\to f^*\\mathcal{J}_3 \\otimes_\\mathcal{O} \\mathcal{F} \\to", "f^*\\mathcal{J}_2 \\otimes_\\mathcal{O} \\mathcal{F} \\to", "f^*\\mathcal{J}_1 \\otimes_\\mathcal{O} \\mathcal{F} \\to 0", "$$", "is short exact due to the assumptions in", "Situation \\ref{situation-ses-flat-thickenings-ringed-topoi}", "and the fact that $\\mathcal{F}$ is flat over $\\mathcal{O}_\\mathcal{B}$.", "The statement on the obstruction class is a direct translation", "of the result of", "Remark \\ref{remark-complex-thickenings-and-ses-modules-ringed-topoi}", "to this particular situation." ], "refs": [ "defos-lemma-deform-module-ringed-topoi", "defos-remark-complex-thickenings-and-ses-modules-ringed-topoi" ], "ref_ids": [ 13397, 13437 ] } ], "ref_ids": [ 13404, 13404 ] }, { "id": 13407, "type": "theorem", "label": "defos-lemma-huge-diagram-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-huge-diagram-ringed-topoi", "contents": [ "Assume given a commutative diagram of morphisms ringed topoi", "\\begin{equation}", "\\label{equation-huge-1-ringed-topoi}", "\\vcenter{", "\\xymatrix{", "& (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\ar[r]_{i_2} \\ar[d]_{f_2} \\ar[ddl]_g &", "(\\Sh(\\mathcal{C}'_2), \\mathcal{O}'_2) \\ar[d]^{f'_2} \\\\", "&", "(\\Sh(\\mathcal{B}_2), \\mathcal{O}_{\\mathcal{B}_2}) \\ar[r]^{t_2} \\ar[ddl]|\\hole &", "(\\Sh(\\mathcal{B}'_2), \\mathcal{O}_{\\mathcal{B}'_2}) \\ar[ddl] \\\\", "(\\Sh(\\mathcal{C}_1), \\mathcal{O}_1) \\ar[r]_{i_1} \\ar[d]_{f_1} &", "(\\Sh(\\mathcal{C}'_1), \\mathcal{O}'_1) \\ar[d]^{f'_1} \\\\", "(\\Sh(\\mathcal{B}_1), \\mathcal{O}_{\\mathcal{B}_1}) \\ar[r]^{t_1} &", "(\\Sh(\\mathcal{B}'_1), \\mathcal{O}_{\\mathcal{B}'_1})", "}", "}", "\\end{equation}", "whose horizontal arrows are first order thickenings. Set", "$\\mathcal{G}_j = \\Ker(i_j^\\sharp)$ and assume given a", "map of $g^{-1}\\mathcal{O}_1$-modules", "$\\nu : g^{-1}\\mathcal{G}_1 \\to \\mathcal{G}_2$", "giving rise to the commutative diagram", "\\begin{equation}", "\\label{equation-huge-2-ringed-topoi}", "\\vcenter{", "\\xymatrix{", "& 0 \\ar[r] & \\mathcal{G}_2 \\ar[r] &", "\\mathcal{O}'_2 \\ar[r] &", "\\mathcal{O}_2 \\ar[r] & 0 \\\\", "& 0 \\ar[r]|\\hole &", "f_2^{-1}\\mathcal{J}_2 \\ar[u]_{c_2} \\ar[r] &", "f_2^{-1}\\mathcal{O}_{\\mathcal{B}'_2} \\ar[u] \\ar[r]|\\hole &", "f_2^{-1}\\mathcal{O}_{\\mathcal{B}_2} \\ar[u] \\ar[r] & 0 \\\\", "0 \\ar[r] &", "\\mathcal{G}_1 \\ar[ruu] \\ar[r] &", "\\mathcal{O}'_1 \\ar[r] &", "\\mathcal{O}_1 \\ar[ruu] \\ar[r] & 0 \\\\", "0 \\ar[r] &", "f_1^{-1}\\mathcal{J}_1 \\ar[ruu]|\\hole \\ar[u]^{c_1} \\ar[r] &", "f_1^{-1}\\mathcal{O}_{\\mathcal{B}'_1} \\ar[ruu]|\\hole \\ar[u] \\ar[r] &", "f_1^{-1}\\mathcal{O}_{\\mathcal{B}_1} \\ar[ruu]|\\hole \\ar[u] \\ar[r] & 0", "}", "}", "\\end{equation}", "with front and back solutions to (\\ref{equation-to-solve-ringed-topoi}).", "(The north-north-west arrows are maps on $\\mathcal{C}_2$ after applying", "$g^{-1}$ to the source.)", "\\begin{enumerate}", "\\item There exist a canonical element in", "$\\Ext^1_{\\mathcal{O}_2}(", "Lg^*\\NL_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, \\mathcal{G}_2)$", "whose vanishing is a necessary and sufficient condition for the existence", "of a morphism of ringed topoi", "$(\\Sh(\\mathcal{C}'_2), \\mathcal{O}'_2) \\to", "(\\Sh(\\mathcal{C}'_1), \\mathcal{O}'_1)$ fitting into", "(\\ref{equation-huge-1-ringed-topoi}) compatibly with $\\nu$.", "\\item If there exists a morphism", "$(\\Sh(\\mathcal{C}'_2), \\mathcal{O}'_2) \\to", "(\\Sh(\\mathcal{C}'_1), \\mathcal{O}'_1)$", "fitting into", "(\\ref{equation-huge-1-ringed-topoi}) compatibly with $\\nu$ the set", "of all such morphisms is a principal homogeneous space under", "$$", "\\Hom_{\\mathcal{O}_1}(", "\\Omega_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, g_*\\mathcal{G}_2) =", "\\Hom_{\\mathcal{O}_2}(", "g^*\\Omega_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, \\mathcal{G}_2) =", "\\Ext^0_{\\mathcal{O}_2}(", "Lg^*\\NL_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, \\mathcal{G}_2).", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The proof of this lemma is identical to the proof of", "Lemma \\ref{lemma-huge-diagram-ringed-spaces}.", "We urge the reader to read that proof instead of this one.", "We will identify the underlying topoi for every", "thickening in sight (we have already used this convention", "in the statement). The equalities in the last statement of the", "lemma are immediate from the definitions. Thus we will work with the groups", "$\\Ext^k_{\\mathcal{O}_2}(", "Lg^*\\NL_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, \\mathcal{G}_2)$,", "$k = 0, 1$ in the rest of the proof. We first argue that we can reduce", "to the case where the underlying topos of all ringed topoi in the lemma", "is the same.", "\\medskip\\noindent", "To do this, observe that", "$g^{-1}\\NL_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}$ is equal to the naive", "cotangent complex of the homomorphism of sheaves of rings", "$g^{-1}f_1^{-1}\\mathcal{O}_{\\mathcal{B}_1} \\to g^{-1}\\mathcal{O}_1$, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-pullback-differentials}.", "Moreover, the degree $0$ term of", "$\\NL_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}$ is a flat", "$\\mathcal{O}_1$-module, hence the canonical map", "$$", "Lg^*\\NL_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}", "\\longrightarrow", "g^{-1}\\NL_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}", "\\otimes_{g^{-1}\\mathcal{O}_1} \\mathcal{O}_2", "$$", "induces an isomorphism on cohomology sheaves in degrees $0$ and $-1$.", "Thus we may replace the Ext groups of the lemma with", "$$", "\\Ext^k_{g^{-1}\\mathcal{O}_1}(", "g^{-1}\\NL_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, \\mathcal{G}_2) =", "\\Ext^k_{g^{-1}\\mathcal{O}_1}(", "\\NL_{g^{-1}\\mathcal{O}_1/g^{-1}f_1^{-1}\\mathcal{O}_{\\mathcal{B}_1}},", "\\mathcal{G}_2)", "$$", "The set of morphism of ringed topoi", "$(\\Sh(\\mathcal{C}'_2), \\mathcal{O}'_2) \\to", "(\\Sh(\\mathcal{C}'_1), \\mathcal{O}'_1)$ fitting into", "(\\ref{equation-huge-1-ringed-topoi}) compatibly with $\\nu$ is in", "one-to-one bijection with the set of homomorphisms of", "$g^{-1}f_1^{-1}\\mathcal{O}_{\\mathcal{B}'_1}$-algebras", "$g^{-1}\\mathcal{O}'_1 \\to \\mathcal{O}'_2$ which are compatible with", "$f^\\sharp$ and $\\nu$. In this way we see that we may assume we have a", "diagram (\\ref{equation-huge-2-ringed-topoi}) of sheaves on a site", "$\\mathcal{C}$ (with $f_1 = f_2 = \\text{id}$ on underlying topoi)", "and we are looking to find a homomorphism of sheaves of rings", "$\\mathcal{O}'_1 \\to \\mathcal{O}'_2$ fitting into it.", "\\medskip\\noindent", "In the rest of the proof of the lemma we assume", "all underlying topological spaces are the", "same, i.e., we have a diagram (\\ref{equation-huge-2-ringed-topoi})", "of sheaves on a site $\\mathcal{C}$ (with $f_1 = f_2 = \\text{id}$", "on underlying topoi) and we are looking for", "homomorphisms of sheaves of rings", "$\\mathcal{O}'_1 \\to \\mathcal{O}'_2$ fitting into it.", "As ext groups we will use", "$\\Ext^k_{\\mathcal{O}_1}(", "\\NL_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, \\mathcal{G}_2)$, $k = 0, 1$.", "\\medskip\\noindent", "Step 1. Construction of the obstruction class. Consider the sheaf", "of sets", "$$", "\\mathcal{E} = \\mathcal{O}'_1 \\times_{\\mathcal{O}_2} \\mathcal{O}'_2", "$$", "This comes with a surjective map $\\alpha : \\mathcal{E} \\to \\mathcal{O}_1$", "and hence we can use $\\NL(\\alpha)$ instead of", "$\\NL_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}$, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-NL-up-to-qis}.", "Set", "$$", "\\mathcal{I}' =", "\\Ker(\\mathcal{O}_{\\mathcal{B}'_1}[\\mathcal{E}] \\to \\mathcal{O}_1)", "\\quad\\text{and}\\quad", "\\mathcal{I} =", "\\Ker(\\mathcal{O}_{\\mathcal{B}_1}[\\mathcal{E}] \\to \\mathcal{O}_1)", "$$", "There is a surjection $\\mathcal{I}' \\to \\mathcal{I}$ whose kernel", "is $\\mathcal{J}_1\\mathcal{O}_{\\mathcal{B}'_1}[\\mathcal{E}]$.", "We obtain two homomorphisms of $\\mathcal{O}_{\\mathcal{B}'_2}$-algebras", "$$", "a : \\mathcal{O}_{\\mathcal{B}'_1}[\\mathcal{E}] \\to \\mathcal{O}'_1", "\\quad\\text{and}\\quad", "b : \\mathcal{O}_{\\mathcal{B}'_1}[\\mathcal{E}] \\to \\mathcal{O}'_2", "$$", "which induce maps $a|_{\\mathcal{I}'} : \\mathcal{I}' \\to \\mathcal{G}_1$ and", "$b|_{\\mathcal{I}'} : \\mathcal{I}' \\to \\mathcal{G}_2$. Both $a$ and $b$", "annihilate $(\\mathcal{I}')^2$. Moreover $a$ and $b$ agree on", "$\\mathcal{J}_1\\mathcal{O}_{\\mathcal{B}'_1}[\\mathcal{E}]$", "as maps into $\\mathcal{G}_2$", "because the left hand square of (\\ref{equation-huge-2-ringed-topoi})", "is commutative. Thus the difference", "$b|_{\\mathcal{I}'} - \\nu \\circ a|_{\\mathcal{I}'}$", "induces a well defined $\\mathcal{O}_1$-linear map", "$$", "\\xi : \\mathcal{I}/\\mathcal{I}^2 \\longrightarrow \\mathcal{G}_2", "$$", "which sends the class of a local section $f$ of $\\mathcal{I}$ to", "$a(f') - \\nu(b(f'))$ where $f'$ is a lift of $f$ to a local", "section of $\\mathcal{I}'$. We let", "$[\\xi] \\in \\Ext^1_{\\mathcal{O}_1}(\\NL(\\alpha), \\mathcal{G}_2)$", "be the image (see below).", "\\medskip\\noindent", "Step 2. Vanishing of $[\\xi]$ is necessary. Let us write $\\Omega =", "\\Omega_{\\mathcal{O}_{\\mathcal{B}_1}[\\mathcal{E}]/\\mathcal{O}_{\\mathcal{B}_1}}", "\\otimes_{\\mathcal{O}_{\\mathcal{B}_1}[\\mathcal{E}]} \\mathcal{O}_1$.", "Observe that $\\NL(\\alpha) = (\\mathcal{I}/\\mathcal{I}^2 \\to \\Omega)$", "fits into a distinguished triangle", "$$", "\\Omega[0] \\to", "\\NL(\\alpha) \\to", "\\mathcal{I}/\\mathcal{I}^2[1] \\to", "\\Omega[1]", "$$", "Thus we see that $[\\xi]$ is zero if and only if $\\xi$", "is a composition $\\mathcal{I}/\\mathcal{I}^2 \\to \\Omega \\to \\mathcal{G}_2$", "for some map $\\Omega \\to \\mathcal{G}_2$. Suppose there exists a", "homomorphisms of sheaves of rings", "$\\varphi : \\mathcal{O}'_1 \\to \\mathcal{O}'_2$ fitting into", "(\\ref{equation-huge-2-ringed-topoi}). In this case consider the map", "$\\mathcal{O}'_1[\\mathcal{E}] \\to \\mathcal{G}_2$,", "$f' \\mapsto b(f') - \\varphi(a(f'))$. A calculation", "shows this annihilates $\\mathcal{J}_1\\mathcal{O}_{\\mathcal{B}'_1}[\\mathcal{E}]$", "and induces a derivation", "$\\mathcal{O}_{\\mathcal{B}_1}[\\mathcal{E}] \\to \\mathcal{G}_2$.", "The resulting linear map $\\Omega \\to \\mathcal{G}_2$ witnesses the", "fact that $[\\xi] = 0$ in this case.", "\\medskip\\noindent", "Step 3. Vanishing of $[\\xi]$ is sufficient. Let", "$\\theta : \\Omega \\to \\mathcal{G}_2$ be a $\\mathcal{O}_1$-linear map", "such that $\\xi$ is equal to", "$\\theta \\circ (\\mathcal{I}/\\mathcal{I}^2 \\to \\Omega)$.", "Then a calculation shows that", "$$", "b + \\theta \\circ d :", "\\mathcal{O}_{\\mathcal{B}'_1}[\\mathcal{E}]", "\\longrightarrow", "\\mathcal{O}'_2", "$$", "annihilates $\\mathcal{I}'$ and hence defines a map", "$\\mathcal{O}'_1 \\to \\mathcal{O}'_2$ fitting into", "(\\ref{equation-huge-2-ringed-topoi}).", "\\medskip\\noindent", "Proof of (2) in the special case above. Omitted. Hint:", "This is exactly the same as the proof of (2) of", "Lemma \\ref{lemma-huge-diagram}." ], "refs": [ "defos-lemma-huge-diagram-ringed-spaces", "sites-modules-lemma-pullback-differentials", "sites-modules-lemma-NL-up-to-qis", "defos-lemma-huge-diagram" ], "ref_ids": [ 13387, 14230, 14240, 13369 ] } ], "ref_ids": [] }, { "id": 13408, "type": "theorem", "label": "defos-lemma-NL-represent-ext-class-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-NL-represent-ext-class-ringed-topoi", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a", "homomorphism of sheaves of rings on $\\mathcal{C}$.", "Let $\\mathcal{G}$ be a $\\mathcal{B}$-module.", "Let", "$\\xi \\in \\Ext^1_\\mathcal{B}(\\NL_{\\mathcal{B}/\\mathcal{A}}, \\mathcal{G})$. ", "There exists a map of sheaves of sets $\\alpha : \\mathcal{E} \\to \\mathcal{B}$", "such that $\\xi \\in \\Ext^1_\\mathcal{B}(\\NL(\\alpha), \\mathcal{G})$", "is the class of a map $\\mathcal{I}/\\mathcal{I}^2 \\to \\mathcal{G}$", "(see proof for notation)." ], "refs": [], "proofs": [ { "contents": [ "Recall that given $\\alpha : \\mathcal{E} \\to \\mathcal{B}$", "such that $\\mathcal{A}[\\mathcal{E}] \\to \\mathcal{B}$ is surjective", "with kernel $\\mathcal{I}$ the complex", "$\\NL(\\alpha) = (\\mathcal{I}/\\mathcal{I}^2 \\to ", "\\Omega_{\\mathcal{A}[\\mathcal{E}]/\\mathcal{A}}", "\\otimes_{\\mathcal{A}[\\mathcal{E}]} \\mathcal{B})$ is canonically", "isomorphic to $\\NL_{\\mathcal{B}/\\mathcal{A}}$, see", "Modules on Sites, Lemma \\ref{sites-modules-lemma-NL-up-to-qis}.", "Observe moreover, that", "$\\Omega = \\Omega_{\\mathcal{A}[\\mathcal{E}]/\\mathcal{A}}", "\\otimes_{\\mathcal{A}[\\mathcal{E}]} \\mathcal{B}$ is the sheaf", "associated to the presheaf", "$U \\mapsto \\bigoplus_{e \\in \\mathcal{E}(U)} \\mathcal{B}(U)$.", "In other words, $\\Omega$ is the free $\\mathcal{B}$-module on the", "sheaf of sets $\\mathcal{E}$ and in particular there is a canonical", "map $\\mathcal{E} \\to \\Omega$.", "\\medskip\\noindent", "Having said this, pick some $\\mathcal{E}$ (for example", "$\\mathcal{E} = \\mathcal{B}$ as in the definition of the naive cotangent", "complex). The obstruction to writing $\\xi$ as the class of a map", "$\\mathcal{I}/\\mathcal{I}^2 \\to \\mathcal{G}$ is an element in", "$\\Ext^1_\\mathcal{B}(\\Omega, \\mathcal{G})$. Say this is represented", "by the extension $0 \\to \\mathcal{G} \\to \\mathcal{H} \\to \\Omega \\to 0$", "of $\\mathcal{B}$-modules. Consider the sheaf of sets", "$\\mathcal{E}' = \\mathcal{E} \\times_\\Omega \\mathcal{H}$", "which comes with an induced map $\\alpha' : \\mathcal{E}' \\to \\mathcal{B}$.", "Let $\\mathcal{I}' = \\Ker(\\mathcal{A}[\\mathcal{E}'] \\to \\mathcal{B})$", "and $\\Omega' = \\Omega_{\\mathcal{A}[\\mathcal{E}']/\\mathcal{A}}", "\\otimes_{\\mathcal{A}[\\mathcal{E}']} \\mathcal{B}$.", "The pullback of $\\xi$ under the quasi-isomorphism", "$\\NL(\\alpha') \\to \\NL(\\alpha)$ maps to zero in", "$\\Ext^1_\\mathcal{B}(\\Omega', \\mathcal{G})$", "because the pullback of the extension $\\mathcal{H}$", "by the map $\\Omega' \\to \\Omega$ is split as $\\Omega'$ is the free", "$\\mathcal{B}$-module on the sheaf of sets $\\mathcal{E}'$ and since", "by construction there is a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{E}' \\ar[r] \\ar[d] & \\mathcal{E} \\ar[d] \\\\", "\\mathcal{H} \\ar[r] & \\Omega", "}", "$$", "This finishes the proof." ], "refs": [ "sites-modules-lemma-NL-up-to-qis" ], "ref_ids": [ 14240 ] } ], "ref_ids": [] }, { "id": 13409, "type": "theorem", "label": "defos-lemma-choices-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-choices-ringed-topoi", "contents": [ "If there exists a solution to (\\ref{equation-to-solve-ringed-topoi}),", "then the set of isomorphism classes of solutions is principal homogeneous", "under $\\Ext^1_\\mathcal{O}(", "\\NL_{\\mathcal{O}/\\mathcal{O}_\\mathcal{B}}, \\mathcal{G})$." ], "refs": [], "proofs": [ { "contents": [ "We observe right away that given two solutions $\\mathcal{O}'_1$ and", "$\\mathcal{O}'_2$ to (\\ref{equation-to-solve-ringed-topoi}) we obtain by", "Lemma \\ref{lemma-huge-diagram-ringed-topoi} an obstruction element", "$o(\\mathcal{O}'_1, \\mathcal{O}'_2) \\in \\Ext^1_\\mathcal{O}(", "\\NL_{\\mathcal{O}/\\mathcal{O}_\\mathcal{B}}, \\mathcal{G})$", "to the existence of a map $\\mathcal{O}'_1 \\to \\mathcal{O}'_2$.", "Clearly, this element", "is the obstruction to the existence of an isomorphism, hence separates", "the isomorphism classes. To finish the proof it therefore suffices to", "show that given a solution $\\mathcal{O}'$ and an element", "$\\xi \\in \\Ext^1_\\mathcal{O}(", "\\NL_{\\mathcal{O}/\\mathcal{O}_\\mathcal{B}}, \\mathcal{G})$", "we can find a second solution $\\mathcal{O}'_\\xi$ such that", "$o(\\mathcal{O}', \\mathcal{O}'_\\xi) = \\xi$.", "\\medskip\\noindent", "Pick $\\alpha : \\mathcal{E} \\to \\mathcal{O}$ as in", "Lemma \\ref{lemma-NL-represent-ext-class-ringed-topoi}", "for the class $\\xi$. Consider the surjection", "$f^{-1}\\mathcal{O}_\\mathcal{B}[\\mathcal{E}] \\to \\mathcal{O}$", "with kernel $\\mathcal{I}$ and corresponding naive cotangent complex", "$\\NL(\\alpha) = (\\mathcal{I}/\\mathcal{I}^2 \\to", "\\Omega_{f^{-1}\\mathcal{O}_\\mathcal{B}[\\mathcal{E}]/", "f^{-1}\\mathcal{O}_\\mathcal{B}}", "\\otimes_{f^{-1}\\mathcal{O}_\\mathcal{B}[\\mathcal{E}]} \\mathcal{O})$.", "By the lemma $\\xi$ is the class of a morphism", "$\\delta : \\mathcal{I}/\\mathcal{I}^2 \\to \\mathcal{G}$.", "After replacing $\\mathcal{E}$ by", "$\\mathcal{E} \\times_\\mathcal{O} \\mathcal{O}'$ we may also assume", "that $\\alpha$ factors through a map", "$\\alpha' : \\mathcal{E} \\to \\mathcal{O}'$.", "\\medskip\\noindent", "These choices determine an $f^{-1}\\mathcal{O}_{\\mathcal{B}'}$-algebra map", "$\\varphi : \\mathcal{O}_{\\mathcal{B}'}[\\mathcal{E}] \\to \\mathcal{O}'$.", "Let $\\mathcal{I}' = \\Ker(\\varphi)$.", "Observe that $\\varphi$ induces a map", "$\\varphi|_{\\mathcal{I}'} : \\mathcal{I}' \\to \\mathcal{G}$", "and that $\\mathcal{O}'$ is the pushout, as in the following", "diagram", "$$", "\\xymatrix{", "0 \\ar[r] & \\mathcal{G} \\ar[r] & \\mathcal{O}' \\ar[r] &", "\\mathcal{O} \\ar[r] & 0 \\\\", "0 \\ar[r] & \\mathcal{I}' \\ar[u]^{\\varphi|_{\\mathcal{I}'}} \\ar[r] &", "f^{-1}\\mathcal{O}_{\\mathcal{B}'}[\\mathcal{E}] \\ar[u] \\ar[r] &", "\\mathcal{O} \\ar[u]_{=} \\ar[r] & 0", "}", "$$", "Let $\\psi : \\mathcal{I}' \\to \\mathcal{G}$ be the sum of the map", "$\\varphi|_{\\mathcal{I}'}$ and the composition", "$$", "\\mathcal{I}' \\to \\mathcal{I}'/(\\mathcal{I}')^2 \\to", "\\mathcal{I}/\\mathcal{I}^2 \\xrightarrow{\\delta} \\mathcal{G}.", "$$", "Then the pushout along $\\psi$ is an other ring extension", "$\\mathcal{O}'_\\xi$ fitting into a diagram as above.", "A calculation (omitted) shows that $o(\\mathcal{O}', \\mathcal{O}'_\\xi) = \\xi$", "as desired." ], "refs": [ "defos-lemma-huge-diagram-ringed-topoi", "defos-lemma-NL-represent-ext-class-ringed-topoi" ], "ref_ids": [ 13407, 13408 ] } ], "ref_ids": [] }, { "id": 13410, "type": "theorem", "label": "defos-lemma-extensions-of-ringed-topoi", "categories": [ "defos" ], "title": "defos-lemma-extensions-of-ringed-topoi", "contents": [ "Let $(\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$ be a ringed topos", "and let $\\mathcal{J}$ be an $\\mathcal{O}_\\mathcal{B}$-module.", "\\begin{enumerate}", "\\item The set of extensions of sheaves of rings", "$0 \\to \\mathcal{J} \\to \\mathcal{O}_{\\mathcal{B}'} \\to", "\\mathcal{O}_\\mathcal{B} \\to 0$", "where $\\mathcal{J}$ is an ideal of square zero is canonically bijective to", "$\\Ext^1_{\\mathcal{O}_\\mathcal{B}}(", "\\NL_{\\mathcal{O}_\\mathcal{B}/\\mathbf{Z}}, \\mathcal{J})$.", "\\item Given a morphism of ringed topoi", "$f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to", "(\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$, an $\\mathcal{O}$-module", "$\\mathcal{G}$, an $f^{-1}\\mathcal{O}_\\mathcal{B}$-module map", "$c : f^{-1}\\mathcal{J} \\to \\mathcal{G}$, and", "given extensions of sheaves of rings with square zero kernels:", "\\begin{enumerate}", "\\item[(a)] $0 \\to \\mathcal{J} \\to \\mathcal{O}_{\\mathcal{B}'} \\to", "\\mathcal{O}_\\mathcal{B} \\to 0$ corresponding to", "$\\alpha \\in \\Ext^1_{\\mathcal{O}_\\mathcal{B}}(", "\\NL_{\\mathcal{O}_\\mathcal{B}/\\mathbf{Z}}, \\mathcal{J})$,", "\\item[(b)] $0 \\to \\mathcal{G} \\to \\mathcal{O}' \\to \\mathcal{O} \\to 0$", "corresponding to", "$\\beta \\in \\Ext^1_\\mathcal{O}(\\NL_{\\mathcal{O}/\\mathbf{Z}}, \\mathcal{G})$", "\\end{enumerate}", "then there is a morphism $(\\Sh(\\mathcal{C}), \\mathcal{O}') \\to", "(\\Sh(\\mathcal{B}, \\mathcal{O}_{\\mathcal{B}'})$ fitting into a diagram", "(\\ref{equation-to-solve-ringed-topoi}) if and only if $\\beta$ and $\\alpha$", "map to the same element of", "$\\Ext^1_\\mathcal{O}(", "Lf^*\\NL_{\\mathcal{O}_\\mathcal{B}/\\mathbf{Z}}, \\mathcal{G})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To prove this we apply the previous results where we work over", "the base ringed topos $(\\Sh(*), \\mathbf{Z})$ with trivial thickening.", "Part (1) follows from Lemma \\ref{lemma-choices-ringed-topoi}", "and the fact that there exists a solution, namely", "$\\mathcal{J} \\oplus \\mathcal{O}_\\mathcal{B}$.", "Part (2) follows from Lemma \\ref{lemma-huge-diagram-ringed-topoi}", "and a compatibility between the constructions in the proofs", "of Lemmas \\ref{lemma-choices-ringed-topoi} and", "\\ref{lemma-huge-diagram-ringed-topoi}", "whose statement and proof we omit." ], "refs": [ "defos-lemma-choices-ringed-topoi", "defos-lemma-huge-diagram-ringed-topoi", "defos-lemma-choices-ringed-topoi", "defos-lemma-huge-diagram-ringed-topoi" ], "ref_ids": [ 13409, 13407, 13409, 13407 ] } ], "ref_ids": [] }, { "id": 13411, "type": "theorem", "label": "defos-lemma-match-thickenings", "categories": [ "defos" ], "title": "defos-lemma-match-thickenings", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to Z'$ be a morphism of algebraic spaces", "over $S$. The following are equivalent", "\\begin{enumerate}", "\\item $i$ is a thickening of algebraic spaces as defined", "in More on Morphisms of Spaces, Section", "\\ref{spaces-more-morphisms-section-thickenings}, and", "\\item the associated morphism", "$i_{small} : (\\Sh(Z_\\etale), \\mathcal{O}_Z) \\to", "(\\Sh(Z'_\\etale), \\mathcal{O}_{Z'})$", "of ringed topoi (Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-morphism-ringed-topoi})", "is a thickening in the sense of", "Section \\ref{section-thickenings-ringed-topoi}.", "\\end{enumerate}" ], "refs": [ "spaces-properties-lemma-morphism-ringed-topoi" ], "proofs": [ { "contents": [ "We stress that this is not a triviality.", "\\medskip\\noindent", "Assume (1). By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-thickening-equivalence}", "the morphism $i$ induces an equivalence of small \\'etale", "sites and in particular of topoi. Of course $i^\\sharp$", "is surjective with locally nilpotent kernel by definition", "of thickenings.", "\\medskip\\noindent", "Assume (2). (This direction is less important and more of", "a curiosity.) For any \\'etale morphism $Y' \\to Z'$ we see", "that $Y = Z \\times_{Z'} Y'$ has the same \\'etale topos", "as $Y'$. In particular, $Y'$ is quasi-compact if and only if", "$Y$ is quasi-compact because being quasi-compact", "is a topos theoretic notion (Sites, Lemma \\ref{sites-lemma-quasi-compact}).", "Having said this we see that $Y'$ is quasi-compact and quasi-separated", "if and only if $Y$ is quasi-compact and quasi-separated", "(because you can characterize $Y'$ being quasi-separated by saying", "that for all $Y'_1, Y'_2$ quasi-compact algebraic spaces \\'etale over $Y'$", "we have that $Y'_1 \\times_{Y'} Y'_2$ is quasi-compact).", "Take $Y'$ affine. Then the algebraic space $Y$ is", "quasi-compact and quasi-separated. For any", "quasi-coherent $\\mathcal{O}_Y$-module $\\mathcal{F}$ we have", "$H^q(Y, \\mathcal{F}) = H^q(Y', (Y \\to Y')_*\\mathcal{F})$", "because the \\'etale topoi are the same.", "Then $H^q(Y', (Y \\to Y')_*\\mathcal{F}) = 0$", "because the pushforward is quasi-coherent", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-pushforward})", "and $Y$ is affine. It follows that $Y'$ is affine by", "Cohomology of Spaces, Proposition", "\\ref{spaces-cohomology-proposition-vanishing-affine}", "(there surely is a proof of this direction of the lemma", "avoiding this reference).", "Hence $i$ is an affine morphism. In the affine case it", "follows easily from the conditions in", "Section \\ref{section-thickenings-ringed-topoi}", "that $i$ is a thickening of algebraic spaces." ], "refs": [ "spaces-more-morphisms-lemma-thickening-equivalence", "sites-lemma-quasi-compact", "spaces-morphisms-lemma-pushforward", "spaces-cohomology-proposition-vanishing-affine" ], "ref_ids": [ 50, 8530, 4760, 11346 ] } ], "ref_ids": [ 11882 ] }, { "id": 13412, "type": "theorem", "label": "defos-lemma-deform-spaces", "categories": [ "defos" ], "title": "defos-lemma-deform-spaces", "contents": [ "Let $S$ be a scheme.", "Let $Y \\subset Y'$ be a first order thickening of algebraic spaces", "over $S$.", "Let $f : X \\to Y$ be a flat morphism of algebraic spaces over $S$.", "If there exists a flat morphism $f' : X' \\to Y'$ of algebraic spaces over $S$", "and an isomorphsm $a : X \\to X' \\times_{Y'} Y$ over $Y$, then", "\\begin{enumerate}", "\\item the set of isomorphism classes of pairs $(f' : X' \\to Y', a)$ is", "principal homogeneous under", "$\\Ext^1_{\\mathcal{O}_X}(\\NL_{X/Y}, f^*\\mathcal{C}_{Y/Y'})$, and", "\\item the set of automorphisms of $\\varphi : X' \\to X'$", "over $Y'$ which reduce to the identity on $X' \\times_{Y'} Y$", "is $\\Ext^0_{\\mathcal{O}_X}(\\NL_{X/Y}, f^*\\mathcal{C}_{Y/Y'})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will apply the material on deformations of ringed topoi", "to the small \\'etale topoi of the algebraic spaces in the lemma.", "We may think of $X$ as a closed subspace of $X'$", "so that $(f, f') : (X \\subset X') \\to (Y \\subset Y')$", "is a morphism of first order thickenings.", "By Lemma \\ref{lemma-match-thickenings}", "this translates into a morphism of thickenings of ringed topoi.", "Then we see from More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-deform}", "(or from the more general Lemma \\ref{lemma-deform-module-ringed-topoi})", "that the ideal sheaf of $X$ in $X'$ is equal to $f^*\\mathcal{C}_{Y'/Y}$", "and this is in fact equivalent to flatness of $X'$ over $Y'$.", "Hence we have a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] & f^*\\mathcal{C}_{Y/Y'} \\ar[r] &", "\\mathcal{O}_{X'} \\ar[r] &", "\\mathcal{O}_X \\ar[r] & 0 \\\\", "0 \\ar[r] &", "f_{small}^{-1}\\mathcal{C}_{Y/Y'} \\ar[u] \\ar[r] &", "f_{small}^{-1}\\mathcal{O}_{Y'} \\ar[u] \\ar[r] &", "f_{small}^{-1}\\mathcal{O}_Y \\ar[u] \\ar[r] & 0", "}", "$$", "Please compare with (\\ref{equation-to-solve-ringed-topoi}).", "Observe that automorphisms $\\varphi$ as in (2)", "give automorphisms $\\varphi^\\sharp : \\mathcal{O}_{X'} \\to \\mathcal{O}_{X'}$", "fitting in the diagram above. Conversely, an automorphism", "$\\alpha : \\mathcal{O}_{X'} \\to \\mathcal{O}_{X'}$", "fitting into the diagram of sheaves above is equal to $\\varphi^\\sharp$", "for some automorphism $\\varphi$ as in (2)", "by More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-first-order-thickening-maps}.", "Finally, by More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-first-order-thickening}", "if we find another sheaf of rings $\\mathcal{A}$ on $X_\\etale$", "fitting into the diagram", "$$", "\\xymatrix{", "0 \\ar[r] & f^*\\mathcal{C}_{Y/Y'} \\ar[r] &", "\\mathcal{A} \\ar[r] &", "\\mathcal{O}_X \\ar[r] & 0 \\\\", "0 \\ar[r] &", "f_{small}^{-1}\\mathcal{C}_{Y/Y'} \\ar[u] \\ar[r] &", "f_{small}^{-1}\\mathcal{O}_{Y'} \\ar[u] \\ar[r] &", "f_{small}^{-1}\\mathcal{O}_Y \\ar[u] \\ar[r] & 0", "}", "$$", "then there exists a first order thickening $X \\subset X''$", "with $\\mathcal{O}_{X''} = \\mathcal{A}$ and applying ", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-first-order-thickening-maps}", "once more, we obtain a morphism", "$(f, f'') : (X \\subset X'') \\to (Y \\subset Y')$ with all the", "desired properties.", "Thus part (1) follows from", "Lemma \\ref{lemma-choices-ringed-topoi}", "and part (2) from part (2) of", "Lemma \\ref{lemma-huge-diagram-ringed-topoi}.", "(Note that $\\NL_{X/Y}$ as defined for a morphism of algebraic spaces in", "More on Morphisms of Spaces, Section", "\\ref{spaces-more-morphisms-section-netherlander}", "agrees with $\\NL_{X/Y}$ as used in", "Section \\ref{section-deformations-ringed-topoi}.)" ], "refs": [ "defos-lemma-match-thickenings", "spaces-more-morphisms-lemma-deform", "defos-lemma-deform-module-ringed-topoi", "spaces-more-morphisms-lemma-first-order-thickening-maps", "spaces-more-morphisms-lemma-first-order-thickening", "spaces-more-morphisms-lemma-first-order-thickening-maps", "defos-lemma-choices-ringed-topoi", "defos-lemma-huge-diagram-ringed-topoi" ], "ref_ids": [ 13411, 101, 13397, 46, 51, 46, 13409, 13407 ] } ], "ref_ids": [] }, { "id": 13413, "type": "theorem", "label": "defos-lemma-thickening-space-quasi-coherent", "categories": [ "defos" ], "title": "defos-lemma-thickening-space-quasi-coherent", "contents": [ "In the situation above assume that $X$ is quasi-compact and quasi-separated", "and that $DQ_X(\\mathcal{F}) \\to DQ_X(\\mathcal{G})$", "(Derived Categories of Spaces, Section", "\\ref{spaces-perfect-section-better-coherator})", "is an isomorphism. Then the functor $F$ is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\NL_{X/B}$ is an object of $D_\\QCoh(\\mathcal{O}_X)$, see", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-netherlander-quasi-coherent}.", "Hence our assumption implies the maps", "$$", "\\Ext^i_X(\\NL_{X/B}, \\mathcal{F}) \\longrightarrow", "\\Ext^i_X(\\NL_{X/B}, \\mathcal{G})", "$$", "are isomorphisms for all $i$. This implies our functor is fully", "faithful by Lemma \\ref{lemma-huge-diagram-ringed-topoi}.", "On the other hand, the functor is essentially surjective by", "Lemma \\ref{lemma-choices-ringed-topoi} because", "we have the solutions $\\mathcal{O}_X \\oplus \\mathcal{F}$", "and $\\mathcal{O}_X \\oplus \\mathcal{G}$ in both categories." ], "refs": [ "spaces-more-morphisms-lemma-netherlander-quasi-coherent", "defos-lemma-huge-diagram-ringed-topoi", "defos-lemma-choices-ringed-topoi" ], "ref_ids": [ 124, 13407, 13409 ] } ], "ref_ids": [] }, { "id": 13414, "type": "theorem", "label": "defos-lemma-thickening-over-thickening-space-quasi-coherent", "categories": [ "defos" ], "title": "defos-lemma-thickening-over-thickening-space-quasi-coherent", "contents": [ "In the situation above assume that $X$ is quasi-compact and quasi-separated", "and that $DQ_X(\\mathcal{F}) \\to DQ_X(\\mathcal{G})$", "(Derived Categories of Spaces, Section", "\\ref{spaces-perfect-section-better-coherator})", "is an isomorphism. Then the functor $FT$ is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "A solution of (\\ref{equation-to-solve-ringed-topoi}) for $\\mathcal{F}$", "in particular gives an extension of $f^{-1}\\mathcal{O}_{B'}$-algebras", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{O}' \\to \\mathcal{O}_X \\to 0", "$$", "where $\\mathcal{F}$ is an ideal of square zero. Similarly for $\\mathcal{G}$.", "Moreover, given such an extension, we obtain a map", "$c_{\\mathcal{O}'} : f^{-1}\\mathcal{J} \\to \\mathcal{F}$.", "Thus we are looking at the full subcategory of such extensions", "of $f^{-1}\\mathcal{O}_{B'}$-algebras with $c = c_{\\mathcal{O}'}$.", "Clearly, if $\\mathcal{O}'' = F(\\mathcal{O}')$ where", "$F$ is the equivalence of Lemma \\ref{lemma-thickening-space-quasi-coherent}", "(applied to $X \\to B'$ this time),", "then $c_{\\mathcal{O}''}$ is the composition of", "$c_{\\mathcal{O}'}$ and the map $\\mathcal{F} \\to \\mathcal{G}$.", "This proves the lemma." ], "refs": [ "defos-lemma-thickening-space-quasi-coherent" ], "ref_ids": [ 13413 ] } ], "ref_ids": [] }, { "id": 13415, "type": "theorem", "label": "defos-lemma-canonical-class-algebra", "categories": [ "defos" ], "title": "defos-lemma-canonical-class-algebra", "contents": [ "Let $R' \\to R$ be a surjection of rings whose kernel is an ideal", "$I$ of square zero. For every $K \\in D^-(R)$ there is a canonical", "map", "$$", "\\omega(K) : K \\longrightarrow K \\otimes_R^\\mathbf{L} I[2]", "$$", "in $D(R)$ with the following properties", "\\begin{enumerate}", "\\item $\\omega(K) = 0$ if and only if there exists", "$K' \\in D(R')$ with $K' \\otimes_{R'}^\\mathbf{L} R = K$,", "\\item given $K \\to L$ in $D^-(R)$ the diagram", "$$", "\\xymatrix{", "K \\ar[d] \\ar[rr]_-{\\omega(K)} & &", "K \\otimes^\\mathbf{L}_R I[2] \\ar[d] \\\\", "L \\ar[rr]^-{\\omega(L)} & &", "L \\otimes^\\mathbf{L}_R I[2]", "}", "$$", "commutes, and", "\\item formation of $\\omega(K)$ is compatible with ring maps $R' \\to S'$", "(see proof for a precise statement).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a bounded above complex $K^\\bullet$ of free $R$-modules representing", "$K$. Then we can choose free $R'$-modules $(K')^n$ lifting $K^n$.", "We can choose $R'$-module maps $(d')^n_K : (K')^n \\to (K')^{n + 1}$", "lifting the differentials $d^n_K : K^n \\to K^{n + 1}$ of $K^\\bullet$.", "Although the compositions", "$$", "(d')^{n + 1}_K \\circ (d')^n_K : (K')^n \\to (K')^{n + 2}", "$$", "may not be zero, they do factor as", "$$", "(K')^n \\to K^n \\xrightarrow{\\omega^n_K}", "K^{n + 2} \\otimes_R I = I(K')^{n + 2} \\to (K')^{n + 2}", "$$", "because $d^{n + 1} \\circ d^n = 0$.", "A calculation shows that $\\omega^n_K$ defines a map of complexes.", "This map of complexes defines $\\omega(K)$.", "\\medskip\\noindent", "Let us prove this construction is compatible with a map of complexes", "$\\alpha^\\bullet : K^\\bullet \\to L^\\bullet$ of bounded above free $R$-modules", "and given choices of lifts $(K')^n, (L')^n, (d')^n_K, (d')^n_L$.", "Namely, choose $(\\alpha')^n : (K')^n \\to (L')^n$ lifting the", "components $\\alpha^n : K^n \\to L^n$. As before we get a", "factorization", "$$", "(K')^n \\to K^n \\xrightarrow{h^n}", "L^{n + 1} \\otimes_R I = I(L')^{n + 1} \\to (L')^{n + 2}", "$$", "of $(d')^n_L \\circ (\\alpha')^n - (\\alpha')^{n + 1} \\circ (d')_K^n$.", "Then it is an pleasant calculation to show that", "$$", "\\omega^n_L \\circ \\alpha^n =", "(d_L^{n + 1} \\otimes \\text{id}_I) \\circ h^n + h^{n + 1} \\circ d_K^n +", "(\\alpha^{n + 2} \\otimes \\text{id}_I) \\circ \\omega^n_K", "$$", "This proves the commutativity of the diagram in (2) of the lemma", "in this particular case. Using this for two different choices", "of bounded above free complexes representing $K$, we find that", "$\\omega(K)$ is well defined! And of course (2) holds in general as well.", "\\medskip\\noindent", "If $K$ lifts to $K'$ in $D^-(R')$, then we can represent", "$K'$ by a bounded above complex of free $R'$-modules", "and we see immediately that $\\omega(K) = 0$.", "Conversely, going back to our choices $K^\\bullet$,", "$(K')^n$, $(d')^n_K$, if $\\omega(K) = 0$, then we can find", "$g^n : K^n \\to K^{n + 1} \\otimes_R I$ with", "$$", "\\omega^n = (d_K^{n + 1} \\otimes \\text{id}_I) \\circ g^n +", "g^{n + 1} \\circ d_K^n", "$$", "This means that with differentials", "$(d')^n_K + g^n : (K')^n \\to (K')^{n + 1}$", "we obtain a complex of free $R'$-modules lifting $K^\\bullet$.", "This proves (1).", "\\medskip\\noindent", "Finally, part (3) means the following: Let $R' \\to S'$ be a map of", "rings. Set $S = S' \\otimes_{R'} R$ and denote $J = IS' \\subset S'$", "the square zero kernel of $S' \\to S$. Then given $K \\in D^-(R)$", "the statement is that we get a commutative diagram", "$$", "\\xymatrix{", "K \\otimes_R^\\mathbf{L} S \\ar[d] \\ar[rr]_-{\\omega(K) \\otimes \\text{id}} & &", "(K \\otimes^\\mathbf{L}_R I[2]) \\otimes_R^\\mathbf{L} S \\ar[d] \\\\", "K \\otimes_R^\\mathbf{L} S \\ar[rr]^-{\\omega(K \\otimes_R^\\mathbf{L} S)} & &", "(K \\otimes_R^\\mathbf{L} S) \\otimes^\\mathbf{L}_S J[2]", "}", "$$", "Here the right vertical arrow comes from", "$$", "(K \\otimes^\\mathbf{L}_R I[2]) \\otimes_R^\\mathbf{L} S =", "(K \\otimes_R^\\mathbf{L} S) \\otimes_S^\\mathbf{L}", "(I \\otimes_R^\\mathbf{L} S)[2] \\longrightarrow", "(K \\otimes_R^\\mathbf{L} S) \\otimes_S^\\mathbf{L} J[2]", "$$", "Choose $K^\\bullet$, $(K')^n$, and $(d')^n_K$ as above.", "Then we can use $K^\\bullet \\otimes_R S$, $(K')^n \\otimes_{R'} S'$, and", "$(d')^n_K \\otimes \\text{id}_{S'}$ for the construction of", "$\\omega(K \\otimes_R^\\mathbf{L} S)$.", "With these choices commutativity", "is immediately verified on the level of maps of complexes." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13416, "type": "theorem", "label": "defos-lemma-lift-complex", "categories": [ "defos" ], "title": "defos-lemma-lift-complex", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$", "be a surjection of sheaves of rings. Assume given the following data", "\\begin{enumerate}", "\\item flat $\\mathcal{O}$-modules $\\mathcal{G}^n$,", "\\item maps of $\\mathcal{O}$-modules $\\mathcal{G}^n \\to \\mathcal{G}^{n + 1}$,", "\\item a complex $\\mathcal{K}_0^\\bullet$ of $\\mathcal{O}_0$-modules,", "\\item maps of $\\mathcal{O}$-modules $\\mathcal{G}^n \\to \\mathcal{K}_0^n$", "\\end{enumerate}", "such that", "\\begin{enumerate}", "\\item[(a)] $H^n(\\mathcal{K}_0^\\bullet) = 0$ for $n \\gg 0$,", "\\item[(b)] $\\mathcal{G}^n = 0$ for $n \\gg 0$,", "\\item[(c)] with", "$\\mathcal{G}^n_0 = \\mathcal{G}^n \\otimes_\\mathcal{O} \\mathcal{O}_0$", "the induced maps determine a complex $\\mathcal{G}_0^\\bullet$ and a map", "of complexes $\\mathcal{G}_0^\\bullet \\to \\mathcal{K}_0^\\bullet$.", "\\end{enumerate}", "Then there exist", "\\begin{enumerate}", "\\item[(\\romannumeral1)]", "flat $\\mathcal{O}$-modules $\\mathcal{F}^n$,", "\\item[(\\romannumeral2)]", "maps of $\\mathcal{O}$-modules $\\mathcal{F}^n \\to \\mathcal{F}^{n + 1}$,", "\\item[(\\romannumeral3)]", "maps of $\\mathcal{O}$-modules $\\mathcal{F}^n \\to \\mathcal{K}_0^n$,", "\\item[(\\romannumeral4)]", "maps of $\\mathcal{O}$-modules $\\mathcal{G}^n \\to \\mathcal{F}^n$,", "\\end{enumerate}", "such that $\\mathcal{F}^n = 0$ for $n \\gg 0$, such that the diagrams", "$$", "\\xymatrix{", "\\mathcal{G}^n \\ar[r] \\ar[d] & \\mathcal{G}^{n + 1} \\ar[d] \\\\", "\\mathcal{F}^n \\ar[r] & \\mathcal{F}^{n + 1}", "}", "$$", "commute for all $n$, such that the composition", "$\\mathcal{G}^n \\to \\mathcal{F}^n \\to \\mathcal{K}_0^n$", "is the given map $\\mathcal{G}^n \\to \\mathcal{K}_0^n$, and such that with", "$\\mathcal{F}^n_0 = \\mathcal{F}^n \\otimes_\\mathcal{O} \\mathcal{O}_0$", "we obtain a complex $\\mathcal{F}_0^\\bullet$ and map of complexes", "$\\mathcal{F}_0^\\bullet \\to \\mathcal{K}_0^\\bullet$ which is a", "quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "We will prove by descending induction on $e$ that we can find $\\mathcal{F}^n$,", "$\\mathcal{G}^n \\to \\mathcal{F}^n$, and", "$\\mathcal{F}^n \\to \\mathcal{F}^{n + 1}$ for $n \\geq e$", "fitting into a commutative diagram", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "\\mathcal{G}^{e - 1} \\ar[r] \\ar@/_2pc/[dd] &", "\\mathcal{G}^e \\ar[d] \\ar[r] \\ar@/_2pc/[dd] &", "\\mathcal{G}^{e + 1} \\ar[d] \\ar[r] \\ar@/_2pc/[dd]|\\hole &", "\\ldots \\\\", "& &", "\\mathcal{F}^e \\ar[d] \\ar[r] &", "\\mathcal{F}^{e + 1} \\ar[d] \\ar[r] & \\ldots \\\\", "\\ldots \\ar[r] &", "\\mathcal{K}_0^{e - 1} \\ar[r] &", "\\mathcal{K}_0^e \\ar[r] &", "\\mathcal{K}_0^{e + 1} \\ar[r] & \\ldots", "}", "$$", "such that $\\mathcal{F}_0^\\bullet$ is a complex,", "the induced map $\\mathcal{F}_0^\\bullet \\to \\mathcal{K}_0^\\bullet$", "induces an isomorphism on $H^n$ for $n > e$ and a surjection", "for $n = e$. For $e \\gg 0$ this is true because we can take", "$\\mathcal{F}^n = 0$ for $n \\geq e$ in that case by assumptions", "(a) and (b).", "\\medskip\\noindent", "Induction step. We have to construct $\\mathcal{F}^{e - 1}$", "and the maps $\\mathcal{G}^{e - 1} \\to \\mathcal{F}^{e - 1}$,", "$\\mathcal{F}^{e - 1} \\to \\mathcal{F}^e$, and", "$\\mathcal{F}^{e - 1} \\to \\mathcal{K}_0^{e - 1}$.", "We will choose $\\mathcal{F}^{e - 1} = A \\oplus B \\oplus C$", "as a direct sum of three pieces.", "\\medskip\\noindent", "For the first we take $A = \\mathcal{G}^{e - 1}$ and we choose our map", "$\\mathcal{G}^{e - 1} \\to \\mathcal{F}^{e - 1}$ to be the inclusion of", "the first summand. The maps $A \\to \\mathcal{K}^{e - 1}_0$", "and $A \\to \\mathcal{F}^e$ will be the obvious ones.", "\\medskip\\noindent", "To choose $B$ we consider the surjection (by induction hypothesis)", "$$", "\\gamma :", "\\Ker(\\mathcal{F}^e_0 \\to \\mathcal{F}^{e + 1}_0)", "\\longrightarrow", "\\Ker(\\mathcal{K}^e_0 \\to \\mathcal{K}^{e + 1}_0)/", "\\Im(\\mathcal{K}^{e - 1}_0 \\to \\mathcal{K}^e_0)", "$$", "We can choose a set $I$, for each $i \\in I$", "an object $U_i$ of $\\mathcal{C}$, and sections", "$s_i \\in \\mathcal{F}^e(U_i)$, $t_i \\in \\mathcal{K}^{e - 1}_0(U_i)$", "such that", "\\begin{enumerate}", "\\item $s_i$ maps to a section of $\\Ker(\\gamma) \\subset", "\\Ker(\\mathcal{F}^e_0 \\to \\mathcal{F}^{e + 1}_0)$,", "\\item $s_i$ and $t_i$ map to the same section of", "$\\mathcal{K}^e_0$,", "\\item the sections $s_i$ generate $\\Ker(\\gamma)$ as an $\\mathcal{O}_0$-module.", "\\end{enumerate}", "We omit giving the full justification for this;", "one uses that $\\mathcal{F}^e \\to \\mathcal{F}^e_0$", "is a surjective maps of sheaves of sets. Then we set", "to put", "$$", "B = \\bigoplus\\nolimits_{i \\in I} j_{U_i!}\\mathcal{O}_{U_i}", "$$", "and define the maps $B \\to \\mathcal{F}^e$ and $B \\to \\mathcal{K}_0^{e - 1}$", "by using $s_i$ and $t_i$ to determine where to send the summand", "$j_{U_i!}\\mathcal{O}_{U_i}$.", "\\medskip\\noindent", "With $\\mathcal{F}^{e - 1} = A \\oplus B$ and maps as above,", "this produces a diagram as above for $e - 1$ such that", "$\\mathcal{F}_0^\\bullet \\to \\mathcal{K}_0^\\bullet$", "induces an isomorphism on $H^n$ for $n \\geq e$.", "To get the map to be surjective on $H^{e - 1}$ we choose", "the summand $C$ as follows.", "Choose a set $J$, for each $j \\in J$ an object $U_j$ of $\\mathcal{C}$", "and a section $t_j$ of $\\Ker(\\mathcal{K}^{e - 1}_0 \\to \\mathcal{K}^e_0)$", "over $U_j$ such that these sections generate this kernel over", "$\\mathcal{O}_0$. Then we put", "$$", "C = \\bigoplus\\nolimits_{j \\in J} j_{U_j!}\\mathcal{O}_{U_j}", "$$", "and the zero map $C \\to \\mathcal{F}^e$ and the map", "$C \\to \\mathcal{K}_0^{e - 1}$ by using $s_j$ to determine where to the summand", "$j_{U_j!}\\mathcal{O}_{U_j}$. This finishes the induction step", "by taking $\\mathcal{F}^{e - 1} = A \\oplus B \\oplus C$ and", "maps as indicated." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13417, "type": "theorem", "label": "defos-lemma-canonical-class", "categories": [ "defos" ], "title": "defos-lemma-canonical-class", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$", "be a surjection of sheaves of rings whose kernel is an ideal sheaf", "$\\mathcal{I}$ of square zero. For every object", "$K_0$ in $D^-(\\mathcal{O}_0)$ there is a canonical map", "$$", "\\omega(K_0) :", "K_0 \\longrightarrow", "K_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I}[2]", "$$", "in $D(\\mathcal{O}_0)$ such that for any map", "$K_0 \\to L_0$ in $D^-(\\mathcal{O}_0)$ the diagram", "$$", "\\xymatrix{", "K_0 \\ar[d] \\ar[rr]_-{\\omega(K_0)} & &", "(K_0 \\otimes^\\mathbf{L}_{\\mathcal{O}_0} \\mathcal{I})[2] \\ar[d] \\\\", "L_0 \\ar[rr]^-{\\omega(L_0)} & &", "(L_0 \\otimes^\\mathbf{L}_{\\mathcal{O}_0} \\mathcal{I})[2]", "}", "$$", "commutes." ], "refs": [], "proofs": [ { "contents": [ "Represent $K_0$ by any complex", "$\\mathcal{K}_0^\\bullet$ of $\\mathcal{O}_0$-modules.", "Apply Lemma \\ref{lemma-lift-complex}", "with $\\mathcal{G}^n = 0$ for all $n$.", "Denote $d : \\mathcal{F}^n \\to \\mathcal{F}^{n + 1}$", "the maps produced by the lemma. Then we see that", "$d \\circ d : \\mathcal{F}^n \\to \\mathcal{F}^{n + 2}$", "is zero modulo $\\mathcal{I}$. Since $\\mathcal{F}^n$ is flat,", "we see that", "$\\mathcal{I}\\mathcal{F}^n =", "\\mathcal{F}^n \\otimes_{\\mathcal{O}} \\mathcal{I} =", "\\mathcal{F}^n_0 \\otimes_{\\mathcal{O}_0} \\mathcal{I}$.", "Hence we obtain a canonical map of complexes", "$$", "d \\circ d : \\mathcal{F}_0^\\bullet", "\\longrightarrow", "(\\mathcal{F}_0^\\bullet \\otimes_{\\mathcal{O}_0} \\mathcal{I})[2]", "$$", "Since $\\mathcal{F}_0^\\bullet$ is a bounded above complex", "of flat $\\mathcal{O}_0$-modules, it is K-flat and may be used", "to compute derived tensor product. Moreover, the map of", "complexes $\\mathcal{F}_0^\\bullet \\to \\mathcal{K}_0^\\bullet$", "is a quasi-isomorphism by construction. Therefore the source and", "target of the map just constructed represent $K_0$ and", "$K_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I}[2]$", "and we obtain our map $\\omega(K_0)$.", "\\medskip\\noindent", "Let us show that this procedure is compatible with maps of complexes.", "Namely, let $\\mathcal{L}_0^\\bullet$ represent another object of", "$D^-(\\mathcal{O}_0)$ and suppose that", "$$", "\\mathcal{K}_0^\\bullet \\longrightarrow \\mathcal{L}_0^\\bullet", "$$", "is a map of complexes. Apply Lemma \\ref{lemma-lift-complex}", "for the complex $\\mathcal{L}_0^\\bullet$, the flat modules", "$\\mathcal{F}^n$, the maps $\\mathcal{F}^n \\to \\mathcal{F}^{n + 1}$,", "and the compositions", "$\\mathcal{F}^n \\to \\mathcal{K}_0^n \\to \\mathcal{L}_0^n$", "(we apologize for the reversal of letters used).", "We obtain flat modules $\\mathcal{G}^n$, maps", "$\\mathcal{F}^n \\to \\mathcal{G}^n$, maps", "$\\mathcal{G}^n \\to \\mathcal{G}^{n + 1}$, and maps", "$\\mathcal{G}^n \\to \\mathcal{L}_0^n$ with all properties", "as in the lemma. Then it is clear that", "$$", "\\xymatrix{", "\\mathcal{F}_0^\\bullet \\ar[d] \\ar[r] &", "(\\mathcal{F}_0^\\bullet \\otimes_{\\mathcal{O}_0} \\mathcal{I})[2] \\ar[d] \\\\", "\\mathcal{G}_0^\\bullet \\ar[r] &", "(\\mathcal{G}_0^\\bullet \\otimes_{\\mathcal{O}_0} \\mathcal{I})[2]", "}", "$$", "is a commutative diagram of complexes.", "\\medskip\\noindent", "To see that $\\omega(K_0)$ is well defined, suppose that we have two complexes", "$\\mathcal{K}_0^\\bullet$ and $(\\mathcal{K}'_0)^\\bullet$", "of $\\mathcal{O}_0$-modules representing $K_0$ and two systems", "$(\\mathcal{F}^n, d : \\mathcal{F}^n \\to \\mathcal{F}^{n + 1},", "\\mathcal{F}^n \\to \\mathcal{K}_0^n)$", "and", "$((\\mathcal{F}')^n, d : (\\mathcal{F}')^n \\to (\\mathcal{F}')^{n + 1},", "(\\mathcal{F}')^n \\to \\mathcal{K}_0^n)$", "as above. Then we can choose a complex $(\\mathcal{K}''_0)^\\bullet$", "and quasi-isomorphisms", "$\\mathcal{K}_0^\\bullet \\to (\\mathcal{K}''_0)^\\bullet$", "and", "$(\\mathcal{K}'_0)^\\bullet \\to (\\mathcal{K}''_0)^\\bullet$", "realizing the fact that both complexes represent $K_0$ in the", "derived category. Next, we apply the result of the previous paragraph", "to", "$$", "(\\mathcal{K}_0)^\\bullet \\oplus (\\mathcal{K}'_0)^\\bullet", "\\longrightarrow", "(\\mathcal{K}''_0)^\\bullet", "$$", "This produces a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{F}_0^\\bullet \\oplus (\\mathcal{F}'_0)^\\bullet", "\\ar[d] \\ar[r] &", "(\\mathcal{F}_0^\\bullet \\otimes_{\\mathcal{O}_0} \\mathcal{I})[2] \\oplus", "((\\mathcal{F}'_0)^\\bullet \\otimes_{\\mathcal{O}_0} \\mathcal{I})[2] \\ar[d] \\\\", "\\mathcal{G}_0^\\bullet \\ar[r] &", "(\\mathcal{G}_0^\\bullet \\otimes_{\\mathcal{O}_0} \\mathcal{I})[2]", "}", "$$", "Since the vertical arrows give quasi-isomorphisms on the summands", "we conclude the desired commutativity in $D(\\mathcal{O}_0)$.", "\\medskip\\noindent", "Having established well-definedness, the statement on compatibility", "with maps is a consequence of the result in the second", "paragraph." ], "refs": [ "defos-lemma-lift-complex", "defos-lemma-lift-complex" ], "ref_ids": [ 13416, 13416 ] } ], "ref_ids": [] }, { "id": 13418, "type": "theorem", "label": "defos-lemma-induced-map", "categories": [ "defos" ], "title": "defos-lemma-induced-map", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\alpha : K \\to L$ be a map of $D^-(\\mathcal{O})$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.", "Let $n \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item If $H^i(\\alpha)$ is an isomorphism for $i \\geq n$,", "then $H^i(\\alpha \\otimes_\\mathcal{O}^\\mathbf{L} \\text{id}_\\mathcal{F})$", "is an isomorphism for $i \\geq n$.", "\\item If $H^i(\\alpha)$ is an isomorphism for $i > n$ ", "and surjective for $i = n$,", "then $H^i(\\alpha \\otimes_\\mathcal{O}^\\mathbf{L} \\text{id}_\\mathcal{F})$", "is an isomorphism for $i > n$ and surjective for $i = n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a distinguished triangle", "$$", "K \\to L \\to C \\to K[1]", "$$", "In case (2) we see that $H^i(C) = 0$ for $i \\geq n$.", "Hence $H^i(C \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{F}) = 0$", "for $i \\geq n$ by (the dual of)", "Derived Categories, Lemma \\ref{derived-lemma-negative-vanishing}.", "This in turn shows that", "$H^i(\\alpha \\otimes_\\mathcal{O}^\\mathbf{L} \\text{id}_\\mathcal{F})$", "is an isomorphism for $i > n$ and surjective for $i = n$.", "In case (1) we moreover see that $H^{n - 1}(L) \\to H^{n - 1}(C)$", "is surjective. Considering the diagram", "$$", "\\xymatrix{", "H^{n - 1}(L) \\otimes_\\mathcal{O} \\mathcal{F} \\ar[r] \\ar[d] &", "H^{n - 1}(C) \\otimes_\\mathcal{O} \\mathcal{F} \\ar@{=}[d] \\\\", "H^{n - 1}(L \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{F}) \\ar[r] &", "H^{n - 1}(C \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{F})", "}", "$$", "we conclude the lower horizontal arrow is surjective. Combined with what", "was said before this implies that", "$H^n(\\alpha \\otimes_\\mathcal{O}^\\mathbf{L} \\text{id}_\\mathcal{F})$", "is an isomorphism." ], "refs": [ "derived-lemma-negative-vanishing" ], "ref_ids": [ 1839 ] } ], "ref_ids": [] }, { "id": 13419, "type": "theorem", "label": "defos-lemma-canonical-class-obstruction", "categories": [ "defos" ], "title": "defos-lemma-canonical-class-obstruction", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$", "be a surjection of sheaves of rings whose kernel is an ideal sheaf", "$\\mathcal{I}$ of square zero. For every object", "$K_0$ in $D^-(\\mathcal{O}_0)$ the following are equivalent", "\\begin{enumerate}", "\\item the class", "$\\omega(K_0) \\in", "\\Ext^2_{\\mathcal{O}_0}(K_0, K_0 \\otimes_{\\mathcal{O}_0} \\mathcal{I})$", "constructed in Lemma \\ref{lemma-canonical-class} is zero,", "\\item there exists $K \\in D^-(\\mathcal{O})$ with", "$K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0 = K_0$", "in $D(\\mathcal{O}_0)$.", "\\end{enumerate}" ], "refs": [ "defos-lemma-canonical-class" ], "proofs": [ { "contents": [ "Let $K$ be as in (2). Then we can represent $K$ by a bounded above", "complex $\\mathcal{F}^\\bullet$ of flat $\\mathcal{O}$-modules.", "Then $\\mathcal{F}_0^\\bullet =", "\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}} \\mathcal{O}_0$", "represents $K_0$ in $D(\\mathcal{O}_0)$.", "Since $d_{\\mathcal{F}^\\bullet} \\circ d_{\\mathcal{F}^\\bullet} = 0$", "as $\\mathcal{F}^\\bullet$ is a complex, we see from the very construction", "of $\\omega(K_0)$ that it is zero.", "\\medskip\\noindent", "Assume (1). Let $\\mathcal{F}^n$, $d : \\mathcal{F}^n \\to \\mathcal{F}^{n + 1}$", "be as in the construction of $\\omega(K_0)$. The nullity of", "$\\omega(K_0)$ implies that the map", "$$", "\\omega = d \\circ d : \\mathcal{F}_0^\\bullet", "\\longrightarrow", "(\\mathcal{F}_0^\\bullet \\otimes_{\\mathcal{O}_0} \\mathcal{I})[2]", "$$", "is zero in $D(\\mathcal{O}_0)$. By definition of the derived category", "as the localization of the homotopy category of complexes", "of $\\mathcal{O}_0$-modules, there exists a quasi-isomorphism", "$\\alpha : \\mathcal{G}_0^\\bullet \\to \\mathcal{F}_0^\\bullet$", "such that there exist $\\mathcal{O}_0$-modules maps", "$h^n : \\mathcal{G}_0^n \\to", "\\mathcal{F}_0^{n + 1} \\otimes_\\mathcal{O} \\mathcal{I}$", "with", "$$", "\\omega \\circ \\alpha =", "d_{\\mathcal{F}_0^\\bullet \\otimes \\mathcal{I}} \\circ h +", "h \\circ d_{\\mathcal{G}_0^\\bullet}", "$$", "We set", "$$", "\\mathcal{H}^n = \\mathcal{F}^n \\times_{\\mathcal{F}^n_0} \\mathcal{G}_0^n", "$$", "and we define", "$$", "d' : \\mathcal{H}^n \\longrightarrow \\mathcal{H}^{n + 1},\\quad", "(f^n, g_0^n) \\longmapsto (d(f^n) - h^n(g_0^n), d(g_0^n))", "$$", "with obvious notation using that", "$\\mathcal{F}_0^{n + 1} \\otimes_{\\mathcal{O}_0} \\mathcal{I} =", "\\mathcal{F}^{n + 1} \\otimes_\\mathcal{O} \\mathcal{I} =", "\\mathcal{I}\\mathcal{F}^{n + 1} \\subset \\mathcal{F}^{n + 1}$.", "Then one checks $d' \\circ d' = 0$ by our choice of $h^n$", "and definition of $\\omega$.", "Hence $\\mathcal{H}^\\bullet$ defines an object in $D(\\mathcal{O})$.", "On the other hand, there is a short exact sequence of complexes", "of $\\mathcal{O}$-modules", "$$", "0 \\to \\mathcal{F}_0^\\bullet \\otimes_{\\mathcal{O}_0} \\mathcal{I} \\to", "\\mathcal{H}^\\bullet \\to \\mathcal{G}_0^\\bullet \\to 0", "$$", "We still have to show that", "$\\mathcal{H}^\\bullet \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0$", "is isomorphic to $K_0$.", "Choose a quasi-isomorphism", "$\\mathcal{E}^\\bullet \\to \\mathcal{H}^\\bullet$", "where $\\mathcal{E}^\\bullet$ is a bounded above complex of flat", "$\\mathcal{O}$-modules. We obtain a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{E}^\\bullet \\otimes_\\mathcal{O} \\mathcal{I} \\ar[d]^\\beta \\ar[r] &", "\\mathcal{E}^\\bullet \\ar[d]^\\gamma \\ar[r] &", "\\mathcal{E}_0^\\bullet \\ar[d]^\\delta \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{F}_0^\\bullet \\otimes_{\\mathcal{O}_0} \\mathcal{I} \\ar[r] &", "\\mathcal{H}^\\bullet \\ar[r] &", "\\mathcal{G}_0^\\bullet \\ar[r] &", "0", "}", "$$", "We claim that $\\delta$ is a quasi-isomorphism. Since $H^i(\\delta)$", "is an isomorphism for $i \\gg 0$, we can use descending induction", "on $n$ such that $H^i(\\delta)$ is an isomorphism for $i \\geq n$.", "Observe that", "$\\mathcal{E}^\\bullet \\otimes_\\mathcal{O} \\mathcal{I}$", "represents", "$\\mathcal{E}_0^\\bullet \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I}$,", "that", "$\\mathcal{F}_0^\\bullet \\otimes_{\\mathcal{O}_0} \\mathcal{I}$", "represents", "$\\mathcal{G}_0^\\bullet \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I}$,", "and that", "$\\beta = \\delta \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\text{id}_\\mathcal{I}$", "as maps in $D(\\mathcal{O}_0)$. This is true because", "$\\beta =", "(\\alpha \\otimes \\text{id}_\\mathcal{I})", "\\circ", "(\\delta \\otimes \\text{id}_\\mathcal{I})$.", "Suppose that $H^i(\\delta)$ is an isomorphism in degrees $\\geq n$.", "Then the same is true for $\\beta$ by what we just said", "and Lemma \\ref{lemma-induced-map}.", "Then we can look at the diagram", "$$", "\\xymatrix{", "H^{n - 1}(\\mathcal{E}^\\bullet \\otimes_\\mathcal{O} \\mathcal{I})", "\\ar[r] \\ar[d]^{H^{n - 1}(\\beta)} &", "H^{n - 1}(\\mathcal{E}^\\bullet) \\ar[r] \\ar[d] &", "H^{n - 1}(\\mathcal{E}_0^\\bullet) \\ar[r] \\ar[d]^{H^{n - 1}(\\delta)} &", "H^n(\\mathcal{E}^\\bullet \\otimes_\\mathcal{O} \\mathcal{I})", "\\ar[r] \\ar[d]^{H^n(\\beta)} &", "H^n(\\mathcal{E}^\\bullet) \\ar[d] \\\\", "H^{n - 1}(\\mathcal{F}_0^\\bullet \\otimes_\\mathcal{O} \\mathcal{I}) \\ar[r] &", "H^{n - 1}(\\mathcal{H}^\\bullet) \\ar[r] &", "H^{n - 1}(\\mathcal{G}_0^\\bullet) \\ar[r] &", "H^n(\\mathcal{F}_0^\\bullet \\otimes_\\mathcal{O} \\mathcal{I}) \\ar[r] &", "H^n(\\mathcal{H}^\\bullet)", "}", "$$", "Using Homology, Lemma \\ref{homology-lemma-four-lemma}", "we see that $H^{n - 1}(\\delta)$ is surjective.", "This in turn implies that $H^{n - 1}(\\beta)$ is surjective", "by Lemma \\ref{lemma-induced-map}.", "Using Homology, Lemma \\ref{homology-lemma-four-lemma}", "again we see that $H^{n - 1}(\\delta)$ is an isomorphism.", "The claim holds by induction, so $\\delta$ is a quasi-isomorphism", "which is what we wanted to show." ], "refs": [ "defos-lemma-induced-map", "homology-lemma-four-lemma", "defos-lemma-induced-map", "homology-lemma-four-lemma" ], "ref_ids": [ 13418, 12029, 13418, 12029 ] } ], "ref_ids": [ 13417 ] }, { "id": 13420, "type": "theorem", "label": "defos-lemma-lift-map-complexes", "categories": [ "defos" ], "title": "defos-lemma-lift-map-complexes", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$", "be a surjection of sheaves of rings. Assume given the following data", "\\begin{enumerate}", "\\item a complex of $\\mathcal{O}$-modules $\\mathcal{F}^\\bullet$,", "\\item a complex $\\mathcal{K}_0^\\bullet$ of $\\mathcal{O}_0$-modules,", "\\item a quasi-isomorphism $\\mathcal{K}_0^\\bullet \\to", "\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{O}_0$,", "\\end{enumerate}", "Then there exist a quasi-isomorphism", "$\\mathcal{G}^\\bullet \\to \\mathcal{F}^\\bullet$ such that the map", "of complexes", "$\\mathcal{G}^\\bullet \\otimes_\\mathcal{O} \\mathcal{O}_0 \\to", "\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{O}_0$ factors", "through $\\mathcal{K}_0^\\bullet$ in the homotopy category", "of complexes of $\\mathcal{O}_0$-modules." ], "refs": [], "proofs": [ { "contents": [ "Set $\\mathcal{F}_0^\\bullet =", "\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{O}_0$.", "By Derived Categories, Lemma \\ref{derived-lemma-make-surjective}", "there exists a factorization", "$$", "\\mathcal{K}_0^\\bullet \\to \\mathcal{L}_0^\\bullet \\to \\mathcal{F}_0^\\bullet", "$$", "of the given map such that the first arrow has an inverse up", "to homotopy and the second arrow is termwise split surjective.", "Hence we may assume that $\\mathcal{K}_0^\\bullet \\to \\mathcal{F}_0^\\bullet$", "is termwise surjective.", "In that case we take", "$$", "\\mathcal{G}^n = \\mathcal{F}^n \\times_{\\mathcal{F}^n_0} \\mathcal{K}_0^n", "$$", "and everything is clear." ], "refs": [ "derived-lemma-make-surjective" ], "ref_ids": [ 1798 ] } ], "ref_ids": [] }, { "id": 13421, "type": "theorem", "label": "defos-lemma-inf-obs-map-defo-complex", "categories": [ "defos" ], "title": "defos-lemma-inf-obs-map-defo-complex", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$", "be a surjection of sheaves of rings whose kernel is an ideal sheaf", "$\\mathcal{I}$ of square zero. Let $K, L \\in D^-(\\mathcal{O})$.", "Set $K_0 = K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0$", "and $L_0 = L \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0$", "in $D^-(\\mathcal{O}_0)$. Given $\\alpha_0 : K_0 \\to L_0$ in $D(\\mathcal{O}_0)$", "there is a canonical element", "$$", "o(\\alpha_0) \\in \\Ext^1_{\\mathcal{O}_0}(K_0,", "L_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I})", "$$", "whose vanishing is necessary and sufficient for the", "existence of a map $\\alpha : K \\to L$ in $D(\\mathcal{O})$", "with $\\alpha_0 = \\alpha \\otimes_\\mathcal{O}^\\mathbf{L} \\text{id}$." ], "refs": [], "proofs": [ { "contents": [ "Finding $\\alpha : K \\to L$ lifing $\\alpha_0$ is the same as finding", "$\\alpha : K \\to L$ such that the composition $K \\xrightarrow{\\alpha} L \\to L_0$", "is equal to the composition $K \\to K_0 \\xrightarrow{\\alpha_0} L_0$.", "The short exact sequence", "$0 \\to \\mathcal{I} \\to \\mathcal{O} \\to \\mathcal{O}_0 \\to 0$", "gives rise to a canonical distinguished triangle", "$$", "L \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{I} \\to", "L \\to", "L_0 \\to", "(L \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{I})[1]", "$$", "in $D(\\mathcal{O})$.", "By Derived Categories, Lemma \\ref{derived-lemma-representable-homological}", "the composition", "$$", "K \\to K_0 \\xrightarrow{\\alpha_0} L_0 \\to", "(L \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{I})[1]", "$$", "is zero if and only if we can find $\\alpha : K \\to L$", "lifting $\\alpha_0$. The composition is an element in", "$$", "\\Hom_{D(\\mathcal{O})}(K, (L \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{I})[1]) =", "\\Hom_{D(\\mathcal{O}_0)}(K_0,", "(L \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{I})[1]) =", "\\Ext^1_{\\mathcal{O}_0}(K_0,", "L_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I})", "$$", "by adjunction." ], "refs": [ "derived-lemma-representable-homological" ], "ref_ids": [ 1758 ] } ], "ref_ids": [] }, { "id": 13422, "type": "theorem", "label": "defos-lemma-first-order-defos-complex", "categories": [ "defos" ], "title": "defos-lemma-first-order-defos-complex", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$", "be a surjection of sheaves of rings whose kernel is an ideal sheaf", "$\\mathcal{I}$ of square zero. Let $K_0 \\in D^-(\\mathcal{O})$.", "A lift of $K_0$ is a pair $(K, \\alpha_0)$ consisting of an object", "$K$ in $D^-(\\mathcal{O})$ and an isomorphism", "$\\alpha_0 : K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0 \\to K_0$", "in $D(\\mathcal{O}_0)$.", "\\begin{enumerate}", "\\item Given a lift $(K, \\alpha)$ the group of automorphism of the pair", "is canonically the cokernel of a map", "$$", "\\Ext^{-1}_{\\mathcal{O}_0}(K_0, K_0)", "\\longrightarrow", "\\Hom_{\\mathcal{O}_0}(K_0, K_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I})", "$$", "\\item If there is a lift, then the set of isomorphism classes of lifts", "is principal homogenenous under", "$\\Ext^1_{\\mathcal{O}_0}(K_0,", "K_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "An automorphism of $(K, \\alpha)$ is a map $\\varphi : K \\to K$", "in $D(\\mathcal{O})$ with", "$\\varphi \\otimes_\\mathcal{O} \\text{id}_{\\mathcal{O}_0} = \\text{id}$.", "This is the same thing as saying that", "$$", "K \\xrightarrow{\\varphi - \\text{id}} K \\to", "K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0", "$$", "is zero. We conclude the group of automorphisms is", "the cokernel of a map", "$$", "\\Hom_\\mathcal{O}(K, K_0[-1])", "\\longrightarrow", "\\Hom_\\mathcal{O}(K, K_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I})", "$$", "by the distinguished triangle", "$$", "K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{I} \\to", "K \\to", "K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0 \\to", "(K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{I})[1]", "$$", "in $D(\\mathcal{O})$ and ", "Derived Categories, Lemma \\ref{derived-lemma-representable-homological}.", "To translate into the groups in the lemma use adjunction", "of the restriction functor $D(\\mathcal{O}_0) \\to D(\\mathcal{O})$ and", "$- \\otimes_\\mathcal{O} \\mathcal{O}_0 : D(\\mathcal{O}) \\to D(\\mathcal{O}_0)$.", "This proves (1).", "\\medskip\\noindent", "Proof of (2).", "Assume that $K_0 = K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0$", "in $D(\\mathcal{O})$. By Lemma \\ref{lemma-inf-obs-map-defo-complex}", "the map sending a lift $(K', \\alpha_0)$ to the obstruction $o(\\alpha_0)$", "to lifting $\\alpha_0$ defines a canonical injective map", "from the set of isomomorphism classes of pairs", "to $\\Ext^1_{\\mathcal{O}_0}(K_0,", "K_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I})$.", "To finish the proof we show that it is surjective.", "Pick $\\xi : K_0 \\to (K_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I})[1]$", "in the $\\Ext^1$ of the lemma.", "Choose a bounded above complex $\\mathcal{F}^\\bullet$", "of flat $\\mathcal{O}$-modules representing $K$.", "The map $\\xi$ can be represented as $t \\circ s^{-1}$", "where $s : \\mathcal{K}_0^\\bullet \\to \\mathcal{F}_0^\\bullet$", "is a quasi-isomorphism and", "$t : \\mathcal{K}_0^\\bullet \\to", "\\mathcal{F}_0^\\bullet \\otimes_{\\mathcal{O}_0} \\mathcal{I}[1]$", "is a map of complexes.", "By Lemma \\ref{lemma-lift-map-complexes}", "we can assume there exists a quasi-isomorphism", "$\\mathcal{G}^\\bullet \\to \\mathcal{F}^\\bullet$", "of complexes of $\\mathcal{O}$-modules", "such that $\\mathcal{G}_0^\\bullet \\to \\mathcal{F}_0^\\bullet$", "factors through $s$ up to homotopy.", "We may and do replace $\\mathcal{G}^\\bullet$ by a bounded", "above complex of flat $\\mathcal{O}$-modules (by picking a qis", "from such to $\\mathcal{G}^\\bullet$ and replacing).", "Then we see that $\\xi$ is represented by", "a map of complexes", "$t : \\mathcal{G}_0^\\bullet \\to", "\\mathcal{F}_0^\\bullet \\otimes_{\\mathcal{O}_0} \\mathcal{I}[1]$", "and the quasi-isomorphism $\\mathcal{G}_0^\\bullet \\to \\mathcal{F}_0^\\bullet$.", "Set", "$$", "\\mathcal{H}^n = \\mathcal{F}^n \\times_{\\mathcal{F}_0^n} \\mathcal{G}_0^n", "$$", "with differentials", "$$", "\\mathcal{H}^n \\to \\mathcal{H}^{n + 1},\\quad", "(f^n, g_0^n) \\mapsto (d(f^n) + t(g_0^n), d(g_0^n))", "$$", "This makes sense as", "$\\mathcal{F}_0^{n + 1} \\otimes_{\\mathcal{O}_0} \\mathcal{I} =", "\\mathcal{F}^{n + 1} \\otimes_\\mathcal{O} \\mathcal{I} =", "\\mathcal{I}\\mathcal{F}^{n + 1} \\subset \\mathcal{F}^{n + 1}$.", "We omit the computation that shows that $\\mathcal{H}^\\bullet$", "is a complex of $\\mathcal{O}$-modules. By construction there is", "a short exact sequence", "$$", "0 \\to \\mathcal{F}_0^\\bullet \\otimes_{\\mathcal{O}_0} \\mathcal{I} \\to", "\\mathcal{H}^\\bullet \\to \\mathcal{G}_0^\\bullet \\to 0", "$$", "of complexes of $\\mathcal{O}$-modules.", "Exactly as in the proof of Lemma \\ref{lemma-canonical-class-obstruction}", "one shows that this sequence induces an isomorphism", "$\\alpha_0 :", "\\mathcal{H}^\\bullet \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0 \\to", "\\mathcal{G}_0^\\bullet$ in $D(\\mathcal{O}_0)$.", "In other words, we have produced a pair $(\\mathcal{H}^\\bullet, \\alpha_0)$.", "We omit the verification that $o(\\alpha_0) = \\xi$; hint: $o(\\alpha_0)$", "can be computed explitly in this case as we have maps", "$\\mathcal{H}^n \\to \\mathcal{F}^n$ (not compatible with differentials)", "lifting the components of $\\alpha_0$. This finishes the proof." ], "refs": [ "derived-lemma-representable-homological", "defos-lemma-inf-obs-map-defo-complex", "defos-lemma-lift-map-complexes", "defos-lemma-canonical-class-obstruction" ], "ref_ids": [ 1758, 13421, 13420, 13419 ] } ], "ref_ids": [] }, { "id": 13438, "type": "theorem", "label": "groupoids-quotients-lemma-invariant", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-invariant", "contents": [ "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.", "Let $j = (t, s) : R \\to U \\times_B U$ be a pre-relation of algebraic", "spaces over $B$. A morphism of algebraic spaces $\\phi : U \\to X$ is", "$R$-invariant if and only if it factors as", "$U \\to U/R \\to X$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definition of the quotient sheaf in", "Groupoids in Spaces, Section \\ref{spaces-groupoids-section-quotient-sheaves}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13439, "type": "theorem", "label": "groupoids-quotients-lemma-base-change-on-invariant", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-base-change-on-invariant", "contents": [ "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.", "Let $j = (t, s) : R \\to U \\times_B U$ be a pre-relation of algebraic", "spaces over $B$. Let $U \\to X$ be an $R$-invariant morphism of algebraic", "spaces over $B$. Let $X' \\to X$ be any morphism of algebraic spaces.", "\\begin{enumerate}", "\\item Setting $U' = X' \\times_X U$, $R' = X' \\times_X R$ we obtain", "a pre-relation $j' : R' \\to U' \\times_B U'$.", "\\item If $j$ is a relation, then $j'$ is a relation.", "\\item If $j$ is a pre-equivalence relation, then $j'$ is a", "pre-equivalence relation.", "\\item If $j$ is an equivalence relation, then $j'$ is an equivalence", "relation.", "\\item If $j$ comes from a groupoid in algebraic spaces", "$(U, R, s, t, c)$ over $B$, then", "\\begin{enumerate}", "\\item $(U, R, s, t, c)$ is a groupoid in algebraic spaces over $X$, and", "\\item $j'$ comes from the base change $(U', R', s', t', c')$", "of this groupoid to $X'$, see", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-base-change-groupoid}.", "\\end{enumerate}", "\\item If $j$ comes from the action of a group algebraic space $G/B$ on $U$", "as in Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-groupoid-from-action}", "then $j'$ comes from the induced action of $G$ on $U'$.", "\\end{enumerate}" ], "refs": [ "spaces-groupoids-lemma-base-change-groupoid", "spaces-groupoids-lemma-groupoid-from-action" ], "proofs": [ { "contents": [ "Omitted. Hint: Functorial point of view combined with the picture:", "$$", "\\xymatrix{", "R' = X' \\times_X R \\ar[dd] \\ar[rr] \\ar[rd] & &", "X' \\times_X U = U' \\ar'[d][dd] \\ar[rd] \\\\", "& R \\ar[dd] \\ar[rr] & & U \\ar[dd] \\\\", "U' = X' \\times_X U \\ar'[r][rr] \\ar[rd] & & X' \\ar[rd] \\\\", "& U \\ar[rr] & & X", "}", "$$" ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 9301, 9308 ] }, { "id": 13440, "type": "theorem", "label": "groupoids-quotients-lemma-base-change-quotient-sheaf", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-base-change-quotient-sheaf", "contents": [ "In the situation of Lemma \\ref{lemma-base-change-on-invariant}", "there is an isomorphism of sheaves", "$$", "U'/R' = X' \\times_X U/R", "$$", "For the construction of quotient sheaves, see", "Groupoids in Spaces, Section \\ref{spaces-groupoids-section-quotient-sheaves}." ], "refs": [ "groupoids-quotients-lemma-base-change-on-invariant" ], "proofs": [ { "contents": [ "Since $U \\to X$ is $R$-invariant, it is clear that the map", "$U \\to X$ factors through the quotient sheaf $U/R$.", "Recall that by definition", "$$", "\\xymatrix{", "R \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "U \\ar[r] &", "U/R", "}", "$$", "is a coequalizer diagram in the category $\\Sh$ of sheaves of sets on", "$(\\Sch/S)_{fppf}$. In fact, this is a coequalizer diagram in the", "comma category $\\Sh/X$. Since the base change functor", "$X' \\times_X - : \\Sh/X \\to \\Sh/X'$ is exact (true in any topos),", "we conclude." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13439 ] }, { "id": 13441, "type": "theorem", "label": "groupoids-quotients-lemma-base-change-quotient-stack", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-base-change-quotient-stack", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "Let $U \\to X$ be an $R$-invariant morphism of algebraic spaces over", "$B$. Let $g : X' \\to X$ be a morphism of algebraic spaces over $B$", "and let $(U', R', s', t', c')$ be the base change as in", "Lemma \\ref{lemma-base-change-on-invariant}. Then", "$$", "\\xymatrix{", "[U'/R'] \\ar[r] \\ar[d] & [U/R] \\ar[d] \\\\", "\\mathcal{S}_{X'} \\ar[r] & \\mathcal{S}_X", "}", "$$", "is a $2$-fibre product of stacks in groupoids over $(\\Sch/S)_{fppf}$.", "For the construction of quotient stacks and the morphisms in this", "diagram, see", "Groupoids in Spaces, Section \\ref{spaces-groupoids-section-stacks}." ], "refs": [ "groupoids-quotients-lemma-base-change-on-invariant" ], "proofs": [ { "contents": [ "We will prove this by using the explicit", "description of the quotient stacks given in", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-objects}.", "However, we strongly urge the reader to find their own proof.", "First, we may view $(U, R, s, t, c)$ as a groupoid in", "algebraic spaces over $X$, hence we obtain a map", "$f : [U/R] \\to \\mathcal{S}_X$, see", "Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-quotient-stack-arrows}.", "Similarly, we have $f' : [U'/R'] \\to X'$.", "\\medskip\\noindent", "An object of the $2$-fibre product", "$\\mathcal{S}_{X'} \\times_{\\mathcal{S}_X} [U/R]$ over a scheme $T$ over $S$", "is the same as a morphism $x' : T \\to X'$ and an object $y$ of $[U/R]$ over $T$", "such that such that the composition $g \\circ x'$ is equal to $f(y)$.", "This makes sense because objects of $\\mathcal{S}_X$ over $T$", "are morphisms $T \\to X$. By Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-objects}", "we may assume $y$ is given by a $[U/R]$-descent datum $(u_i, r_{ij})$", "relative to an fppf covering $\\{T_i \\to T\\}$.", "The agreement of $g \\circ x' = f(y)$ means that the diagrams", "$$", "\\vcenter{", "\\xymatrix{", "T_i \\ar[rr]_{u_i} \\ar[d] & & U \\ar[d] \\\\", "T \\ar[r]^{x'} & X' \\ar[r]^g & X", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "T_i \\times_T T_j \\ar[rr]_{r_{ij}} \\ar[d] & & R \\ar[d] \\\\", "T \\ar[r]^{x'} & X' \\ar[r]^g & X", "}", "}", "$$", "are commutative.", "\\medskip\\noindent", "On the other hand, an object $y'$ of $[U'/R']$ over a scheme $T$ over $S$", "by Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-objects}", "is given by a $[U'/R']$-descent datum $(u'_i, r'_{ij})$", "relative to an fppf covering $\\{T_i \\to T\\}$.", "Setting $f'(y') = x' : T \\to X'$ we see that", "the diagrams", "$$", "\\vcenter{", "\\xymatrix{", "T_i \\ar[r]_{u'_i} \\ar[d] & U' \\ar[d] \\\\", "T \\ar[r]^{x'} & X'", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "T_i \\times_T T_j \\ar[r]_{r'_{ij}} \\ar[d] & U' \\ar[d] \\\\", "T \\ar[r]^{x'} & X'", "}", "}", "$$", "are commutative.", "\\medskip\\noindent", "With this notation in place, we define a functor", "$$", "[U'/R'] \\longrightarrow \\mathcal{S}_{X'} \\times_{\\mathcal{S}_X} [U/R]", "$$", "by sending $y' = (u'_i, r'_{ij})$ as above to the object", "$(x', (u_i, r_{ij}))$ where $x' = f'(y')$, where", "$u_i$ is the composition $T_i \\to U' \\to U$, and where", "$r_{ij}$ is the composition $T_i \\times_T T_j \\to R' \\to R$.", "Conversely, given an object $(x', (u_i, r_{ij})$", "of the right hand side we can send this to the object", "$((x', u_i), (x', r_{ij}))$ of the left hand side.", "We omit the discussion of what to do with morphisms (works", "in exactly the same manner)." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-objects", "spaces-groupoids-lemma-quotient-stack-arrows", "spaces-groupoids-lemma-quotient-stack-objects", "spaces-groupoids-lemma-quotient-stack-objects" ], "ref_ids": [ 9328, 9319, 9328, 9328 ] } ], "ref_ids": [ 13439 ] }, { "id": 13442, "type": "theorem", "label": "groupoids-quotients-lemma-categorical-unique", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-categorical-unique", "contents": [ "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-relation in algebraic spaces over $B$.", "If a categorical quotient in the category of algebraic spaces", "over $B$ exists, then it is unique up to unique isomorphism.", "Similarly for categorical quotients in full subcategories of", "$\\textit{Spaces}/B$." ], "refs": [], "proofs": [ { "contents": [ "See Categories, Section \\ref{categories-section-coequalizers}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13443, "type": "theorem", "label": "groupoids-quotients-lemma-categorical-reduced", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-categorical-reduced", "contents": [ "In the situation of", "Definition \\ref{definition-categorical}.", "If $\\phi : U \\to X$ is a categorical quotient and $U$ is reduced,", "then $X$ is reduced. The same holds for categorical quotients in", "a category of spaces $\\mathcal{C}$ listed in", "Example \\ref{example-categories}." ], "refs": [ "groupoids-quotients-definition-categorical" ], "proofs": [ { "contents": [ "Let $X_{red}$ be the reduction of the algebraic space $X$.", "Since $U$ is reduced the morphism $\\phi : U \\to X$ factors through", "$i : X_{red} \\to X$ (insert future reference here). Denote this morphism", "by $\\phi_{red} : U \\to X_{red}$. Since $\\phi \\circ s = \\phi \\circ t$ we", "see that also $\\phi_{red} \\circ s = \\phi_{red} \\circ t$ (as", "$i : X_{red} \\to X$ is a monomorphism). Hence by the universal property", "of $\\phi$ there exists a morphism $\\chi : X \\to X_{red}$ such that", "$\\phi_{red} = \\phi \\circ \\chi$. By uniqueness we see that", "$i \\circ \\chi = \\text{id}_X$ and $\\chi \\circ i = \\text{id}_{X_{red}}$.", "Hence $i$ is an isomorphism and $X$ is reduced.", "\\medskip\\noindent", "To show that this argument works in a category $\\mathcal{C}$ one", "just needs to show that the reduction of an object of $\\mathcal{C}$", "is an object of $\\mathcal{C}$. We omit the verification that this", "holds for each of the standard examples." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13460 ] }, { "id": 13444, "type": "theorem", "label": "groupoids-quotients-lemma-pre-equivalence-equivalence-relation-points", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-pre-equivalence-equivalence-relation-points", "contents": [ "Let $B \\to S$ as in Section \\ref{section-conventions-notation}.", "Let $j : R \\to U \\times_B U$ be a pre-equivalence relation", "of algebraic spaces over $B$. Then", "$$", "O_u =", "\\{u' \\in |U| \\text{ such that } \\exists r \\in |R|, \\ s(r) = u, \\ t(r) = u'\\}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "By the aforementioned", "Groupoids in Spaces,", "Lemma \\ref{spaces-groupoids-lemma-pre-equivalence-equivalence-relation-points}", "we see that the orbits $O_u$ as defined in the lemma give a disjoint", "union decomposition of $|U|$. Thus we see they are equal to the", "orbits as defined in Definition \\ref{definition-orbit}." ], "refs": [ "spaces-groupoids-lemma-pre-equivalence-equivalence-relation-points", "groupoids-quotients-definition-orbit" ], "ref_ids": [ 9287, 13462 ] } ], "ref_ids": [] }, { "id": 13445, "type": "theorem", "label": "groupoids-quotients-lemma-invariant-map-constant-on-orbit", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-invariant-map-constant-on-orbit", "contents": [ "In the situation of Definition \\ref{definition-orbit}.", "Let $\\phi : U \\to X$ be an $R$-invariant morphism of algebraic spaces over", "$B$. Then $|\\phi| : |U| \\to |X|$ is constant on the orbits." ], "refs": [ "groupoids-quotients-definition-orbit" ], "proofs": [ { "contents": [ "To see this we just have to show that $\\phi(u) = \\phi(u')$", "for all $u, u' \\in |U|$ such that", "there exists an $r \\in |R|$ such that $s(r) = u$ and $t(r) = u'$.", "And this is clear since $\\phi$ equalizes $s$ and $t$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13462 ] }, { "id": 13446, "type": "theorem", "label": "groupoids-quotients-lemma-weak-orbit-pre-equivalence", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-weak-orbit-pre-equivalence", "contents": [ "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.", "Let $\\Spec(k) \\to B$ be a geometric point of $B$.", "Let $j : R \\to U \\times_B U$ be a pre-equivalence relation over $B$.", "In this case the weak orbit of $\\overline{u} \\in U(k)$ is simply", "$$", "\\{", "\\overline{u}' \\in U(k)", "\\text{ such that }", "\\exists \\overline{r} \\in R(k),", "\\ s(\\overline{r}) = \\overline{u},", "\\ t(\\overline{r}) = \\overline{u}'", "\\}", "$$", "and the orbit of $\\overline{u} \\in U(k)$ is", "$$", "\\{", "\\overline{u}' \\in U(k) :", "\\exists\\text{ field extension }k \\subset K, \\ \\exists\\ r \\in R(K),", "\\ s(r) = \\overline{u}, \\ t(r) = \\overline{u}'\\}", "$$" ], "refs": [], "proofs": [ { "contents": [ "This is true because by definition of a pre-equivalence relation the image", "$j(R(k)) \\subset U(k) \\times U(k)$ is an equivalence relation." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13447, "type": "theorem", "label": "groupoids-quotients-lemma-make-pre-equivalence", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-make-pre-equivalence", "contents": [ "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-relation over $B$.", "Then $j_\\infty : R_\\infty \\to U \\times_B U$ is a", "pre-equivalence relation over $B$. Moreover", "\\begin{enumerate}", "\\item $\\phi : U \\to X$ is $R$-invariant if and only if it is", "$R_\\infty$-invariant,", "\\item the canonical map of quotient sheaves $U/R \\to U/R_\\infty$ (see", "Groupoids in Spaces, Section \\ref{spaces-groupoids-section-quotient-sheaves})", "is an isomorphism,", "\\item weak $R$-orbits agree with weak $R_\\infty$-orbits,", "\\item $R$-orbits agree with $R_\\infty$-orbits,", "\\item if $s, t$ are locally of finite type, then $s_\\infty$, $t_\\infty$", "are locally of finite type,", "\\item add more here as needed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint for (5): Any property of $s, t$ which is stable under composition", "and stable under base change, and Zariski local on the source", "will be inherited by $s_\\infty, t_\\infty$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13448, "type": "theorem", "label": "groupoids-quotients-lemma-geometric-orbits", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-geometric-orbits", "contents": [ "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-relation over $B$.", "Let $\\Spec(k) \\to B$ be a geometric point of $B$.", "\\begin{enumerate}", "\\item If $s, t : R \\to U$ are locally of finite type", "then weak $R$-equivalence on $U(k)$ agrees with $R$-equivalence, and", "weak $R$-orbits agree with $R$-orbits on $U(k)$.", "\\item If $k$ has sufficiently large cardinality then weak $R$-equivalence", "on $U(k)$ agrees with $R$-equivalence, and weak $R$-orbits agree", "with $R$-orbits on $U(k)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We first prove (1). Assume $s, t$ locally of finite type. By", "Lemma \\ref{lemma-make-pre-equivalence}", "we may assume that $R$ is a pre-equivalence relation.", "Let $k$ be an algebraically closed field over $B$.", "Suppose $\\overline{u}, \\overline{u}' \\in U(k)$ are $R$-equivalent.", "Then for some extension field $k \\subset \\Omega$ there exists", "a point $\\overline{r} \\in R(\\Omega)$ mapping to", "$(\\overline{u}, \\overline{u}') \\in (U \\times_B U)(\\Omega)$, see", "Lemma \\ref{lemma-weak-orbit-pre-equivalence}.", "Hence", "$$", "Z = R \\times_{j, U \\times_B U, (\\overline{u}, \\overline{u}')} \\Spec(k)", "$$", "is nonempty. As $s$ is locally of finite type we see that", "also $j$ is locally of finite type, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-permanence-finite-type}.", "This implies $Z$ is a nonempty algebraic space locally of finite type", "over the algebraically closed field $k$ (use", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-base-change-finite-type}).", "Thus $Z$ has a $k$-valued point, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points}.", "Hence we conclude there exists a $\\overline{r} \\in R(k)$ with", "$j(\\overline{r}) = (\\overline{u}, \\overline{u}')$, and we conclude that", "$\\overline{u}, \\overline{u}'$ are $R$-equivalent as desired.", "\\medskip\\noindent", "The proof of part (2) is the same, except that it uses", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-large-enough}", "instead of", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points}.", "This shows that the assertion holds as soon as $|k| > \\lambda(R)$ with", "$\\lambda(R)$ as introduced just above", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points}." ], "refs": [ "groupoids-quotients-lemma-make-pre-equivalence", "groupoids-quotients-lemma-weak-orbit-pre-equivalence", "spaces-morphisms-lemma-permanence-finite-type", "spaces-morphisms-lemma-base-change-finite-type", "spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points", "spaces-morphisms-lemma-large-enough", "spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points", "spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points" ], "ref_ids": [ 13447, 13446, 4818, 4815, 4820, 4821, 4820, 4820 ] } ], "ref_ids": [] }, { "id": 13449, "type": "theorem", "label": "groupoids-quotients-lemma-set-theoretic-invariant", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-set-theoretic-invariant", "contents": [ "In the situation of Definition \\ref{definition-set-theoretically-invariant}.", "A morphism $\\phi : U \\to X$ is set-theoretically $R$-invariant if and", "only if for any algebraically closed field $k$ over $B$ the map", "$U(k) \\to X(k)$ is constant on orbits." ], "refs": [ "groupoids-quotients-definition-set-theoretically-invariant" ], "proofs": [ { "contents": [ "This is true because the condition is supposed to hold for all algebraically", "closed fields over $B$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13464 ] }, { "id": 13450, "type": "theorem", "label": "groupoids-quotients-lemma-invariant-set-theoretically-invariant", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-invariant-set-theoretically-invariant", "contents": [ "In the situation of Definition \\ref{definition-set-theoretically-invariant}.", "An invariant morphism is set-theoretically invariant." ], "refs": [ "groupoids-quotients-definition-set-theoretically-invariant" ], "proofs": [ { "contents": [ "This is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13464 ] }, { "id": 13451, "type": "theorem", "label": "groupoids-quotients-lemma-set-theoretically-invariant-invariant-when-reduced", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-set-theoretically-invariant-invariant-when-reduced", "contents": [ "In the situation of Definition \\ref{definition-set-theoretically-invariant}.", "Let $\\phi : U \\to X$ be a morphism of algebraic spaces over $B$.", "Assume", "\\begin{enumerate}", "\\item $\\phi$ is set-theoretically $R$-invariant,", "\\item $R$ is reduced, and", "\\item $X$ is locally separated over $B$.", "\\end{enumerate}", "Then $\\phi$ is $R$-invariant." ], "refs": [ "groupoids-quotients-definition-set-theoretically-invariant" ], "proofs": [ { "contents": [ "Consider the equalizer", "$$", "Z = R \\times_{(\\phi, \\phi) \\circ j, X \\times_B X, \\Delta_{X/B}} X", "$$", "algebraic space. Then $Z \\to R$ is an immersion by assumption (3).", "By assumption (1) $|Z| \\to |R|$ is surjective. This implies that", "$Z \\to R$ is a bijective closed immersion (use", "Schemes, Lemma \\ref{schemes-lemma-immersion-when-closed})", "and by assumption (2) we conclude that $Z = R$." ], "refs": [ "schemes-lemma-immersion-when-closed" ], "ref_ids": [ 7671 ] } ], "ref_ids": [ 13464 ] }, { "id": 13452, "type": "theorem", "label": "groupoids-quotients-lemma-set-theoretic-pre-equivalence-geometric", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-set-theoretic-pre-equivalence-geometric", "contents": [ "In the situation of Definition \\ref{definition-set-theoretic-equivalence}.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $j$ is a set-theoretic pre-equivalence relation.", "\\item The subset $j(|R|) \\subset |U \\times_B U|$ contains the image of", "$|j'|$ for any of the morphisms $j'$ as in Equation (\\ref{equation-list}).", "\\item For every algebraically closed field $k$ over $B$ of sufficiently large", "cardinality the subset $j(R(k)) \\subset U(k) \\times U(k)$ is an equivalence", "relation.", "\\end{enumerate}", "If $s, t$ are locally of finite type these are also equivalent to", "\\begin{enumerate}", "\\item[(4)] For every algebraically closed field $k$ over $B$", "the subset $j(R(k)) \\subset U(k) \\times U(k)$ is an equivalence relation.", "\\end{enumerate}" ], "refs": [ "groupoids-quotients-definition-set-theoretic-equivalence" ], "proofs": [ { "contents": [ "Assume (2). Let $k$ be an algebraically closed field over $B$.", "We are going to show that $\\sim_R$ is an equivalence relation.", "Suppose that $\\overline{u}_i : \\Spec(k) \\to U$, $i = 1, 2$", "are $k$-valued points of $U$. Suppose that $(\\overline{u}_1, \\overline{u}_2)$", "is the image of a $K$-valued point $r \\in R(K)$. Consider the", "solid commutative diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar@{..>}[r] \\ar@{..>}[d]", "&", "\\Spec(k) \\ar[d]_{(\\overline{u}_2, \\overline{u}_1)} &", "\\Spec(K) \\ar[d] \\ar[l] \\\\", "R \\ar[r]^-j &", "U \\times_B U &", "R \\ar[l]_-{j_{flip}}", "}", "$$", "We also denote $r \\in |R|$ the image of $r$.", "By assumption the image of $|j_{flip}|$ is contained in the image of", "$|j|$, in other words there exists a $r' \\in |R|$ such that", "$|j|(r') = |j_{flip}|(r)$. But note that $(\\overline{u}_2, \\overline{u}_1)$", "is in the equivalence class that defines $|j|(r')$ (by the commutativity", "of the solid part of the diagram). This means there exists a field", "extension $k \\subset K'$ and a morphism $r' : \\Spec(K) \\to R$", "(abusively denoted $r'$ as well) with", "$j \\circ r' = (\\overline{u}_2, \\overline{u}_1) \\circ i$", "where $i : \\Spec(K') \\to \\Spec(K)$ is the obvious map.", "In other words the dotted part of the diagram commutes.", "This proves that $\\sim_R$ is a symmetric relation on $U(k)$.", "In the similar way, using that the image of $|j_{diag}|$ is contained", "in the image of $|j|$ we see that $\\sim_R$ is reflexive (details omitted).", "\\medskip\\noindent", "To show that $\\sim_R$ is transitive assume given", "$\\overline{u}_i : \\Spec(k) \\to U$, $i = 1, 2, 3$", "and field extensions $k \\subset K_i$ and points", "$r_i : \\Spec(K_i) \\to R$, $i = 1, 2$ such that", "$j(r_1) = (\\overline{u}_1, \\overline{u}_2)$ and", "$j(r_1) = (\\overline{u}_2, \\overline{u}_3)$. Then we may choose a", "commutative diagram of fields", "$$", "\\xymatrix{", "K & K_2 \\ar[l] \\\\", "K_1 \\ar[u] & k \\ar[l] \\ar[u]", "}", "$$", "and we may think of $r_1, r_2 \\in R(K)$. We consider the", "commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar@{..>}[r] \\ar@{..>}[d]", "&", "\\Spec(k) \\ar[d]_{(\\overline{u}_1, \\overline{u}_3)} &", "\\Spec(K) \\ar[d]^{(r_1, r_2)} \\ar[l]", "\\\\", "R \\ar[r]^-j &", "U \\times_B U &", "R \\times_{s, U, t} R \\ar[l]_-{j_{comp}}", "}", "$$", "By exactly the same reasoning as in the first part of the proof, but", "this time using that $|j_{comp}|((r_1, r_2))$ is in the image of $|j|$,", "we conclude that a field $K'$ and dotted arrows exist making the", "diagram commute. This proves that $\\sim_R$ is transitive and concludes", "the proof that (2) implies (1).", "\\medskip\\noindent", "Assume (1) and let $k$ be an algebraically closed field over $B$ whose", "cardinality is larger than $\\lambda(R)$, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-large-enough}.", "Suppose that $\\overline{u} \\sim_R \\overline{u}'$ with", "$\\overline{u}, \\overline{u}' \\in U(k)$. By assumption there exists", "a point in $|R|$ mapping to $(\\overline{u}, \\overline{u}') \\in |U \\times_B U|$.", "Hence by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-large-enough}", "we conclude there exists an $\\overline{r} \\in R(k)$ with", "$j(\\overline{r}) = (\\overline{u}, \\overline{u}')$. In this way we see", "that (1) implies (3).", "\\medskip\\noindent", "Assume (3). Let us show that", "$\\Im(|j_{comp}|) \\subset \\Im(|j|)$. Pick any point", "$c \\in |R \\times_{s, U, t} R|$. We may represent this by a morphism", "$\\overline{c} : \\Spec(k) \\to R \\times_{s, U, t} R$, with $k$ over $B$", "having sufficiently large cardinality. By assumption we see that", "$j_{comp}(\\overline{c}) \\in U(k) \\times U(k) = (U \\times_B U)(k)$", "is also the image $j(\\overline{r})$ for some $\\overline{r} \\in R(k)$.", "Hence $j_{comp}(c) = j(r)$ in $|U \\times_B U|$ as desired (with", "$r \\in |R|$ the equivalence class of $\\overline{r}$). The same argument", "shows also that $\\Im(|j_{diag}|) \\subset \\Im(|j|)$ and", "$\\Im(|j_{flip}|) \\subset \\Im(|j|)$ (details omitted).", "In this way we see that (3) implies (2). At this point we have", "shown that (1), (2) and (3) are all equivalent.", "\\medskip\\noindent", "It is clear that (4) implies (3) (without any assumptions on $s$, $t$).", "To finish the proof of the lemma we show that (1) implies (4) if $s, t$", "are locally of finite type. Namely, let $k$ be an algebraically closed", "field over $B$. Suppose that $\\overline{u} \\sim_R \\overline{u}'$ with", "$\\overline{u}, \\overline{u}' \\in U(k)$. By assumption the algebraic space", "$Z = R \\times_{j, U \\times_B U, (\\overline{u}, \\overline{u}')} \\Spec(k)$", "is nonempty. On the other hand, since $j = (t, s)$ is locally of finite type", "the morphism $Z \\to \\Spec(k)$ is locally of finite type as well (use", "Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-permanence-finite-type}", "and \\ref{spaces-morphisms-lemma-base-change-finite-type}).", "Hence $Z$ has a $k$ point by", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points}", "and we conclude that $(\\overline{u}, \\overline{u}') \\in j(R(k))$", "as desired. This finishes the proof of the lemma." ], "refs": [ "spaces-morphisms-lemma-large-enough", "spaces-morphisms-lemma-large-enough", "spaces-morphisms-lemma-permanence-finite-type", "spaces-morphisms-lemma-base-change-finite-type", "spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points" ], "ref_ids": [ 4821, 4821, 4818, 4815, 4820 ] } ], "ref_ids": [ 13465 ] }, { "id": 13453, "type": "theorem", "label": "groupoids-quotients-lemma-set-theoretic-equivalence-geometric", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-set-theoretic-equivalence-geometric", "contents": [ "In the situation of Definition \\ref{definition-set-theoretic-equivalence}.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $j$ is a set-theoretic equivalence relation.", "\\item The morphism $j$ is universally injective and", "$j(|R|) \\subset |U \\times_B U|$ contains the image of", "$|j'|$ for any of the morphisms $j'$ as in Equation (\\ref{equation-list}).", "\\item For every algebraically closed field $k$ over $B$ of sufficiently large", "cardinality the map $j : R(k) \\to U(k) \\times U(k)$ is injective and", "its image is an equivalence relation.", "\\end{enumerate}", "If $j$ is decent, or locally separated, or quasi-separated", "these are also equivalent to", "\\begin{enumerate}", "\\item[(4)] For every algebraically closed field $k$ over $B$", "the map $j : R(k) \\to U(k) \\times U(k)$ is injective and its image", "is an equivalence relation.", "\\end{enumerate}" ], "refs": [ "groupoids-quotients-definition-set-theoretic-equivalence" ], "proofs": [ { "contents": [ "The implications (1) $\\Rightarrow$ (2) and (2) $\\Rightarrow$ (3) follow from", "Lemma \\ref{lemma-set-theoretic-pre-equivalence-geometric}", "and the definitions. The same lemma shows that (3) implies", "$j$ is a set-theoretic pre-equivalence relation. But of course condition", "(3) also implies that $j$ is universally injective, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-universally-injective},", "so that $j$ is indeed a set-theoretic equivalence relation.", "At this point we know that (1), (2), (3) are all equivalent.", "\\medskip\\noindent", "Condition (4) implies (3) without any further hypotheses on $j$. Assume $j$", "is decent, or locally separated, or quasi-separated and the equivalent", "conditions (1), (2), (3) hold. By", "More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-when-universally-injective-radicial}", "we see that $j$ is radicial.", "Let $k$ be any algebraically closed field over $B$. Let", "$\\overline{u}, \\overline{u}' \\in U(k)$ with", "$\\overline{u} \\sim_R \\overline{u}'$. We see that", "$R \\times_{U \\times_B U, (\\overline{u}, \\overline{u}')} \\Spec(k)$", "is nonempty. Hence, as $j$ is radicial, its reduction is the spectrum of a", "field purely inseparable over $k$. As $k = \\overline{k}$ we see that", "it is the spectrum of $k$. Whence a point $\\overline{r} \\in R(k)$", "with $t(\\overline{r}) = \\overline{u}$ and $s(\\overline{r}) = \\overline{u}'$", "as desired." ], "refs": [ "groupoids-quotients-lemma-set-theoretic-pre-equivalence-geometric", "spaces-morphisms-lemma-universally-injective", "spaces-more-morphisms-lemma-when-universally-injective-radicial" ], "ref_ids": [ 13452, 4793, 18 ] } ], "ref_ids": [ 13465 ] }, { "id": 13454, "type": "theorem", "label": "groupoids-quotients-lemma-set-theoretic-equivalence", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-set-theoretic-equivalence", "contents": [ "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-relation over $B$.", "\\begin{enumerate}", "\\item If $j$ is a pre-equivalence relation, then $j$ is a", "set-theoretic pre-equivalence relation. This holds in particular", "when $j$ comes from a groupoid in algebraic spaces, or from an", "action of a group algebraic space on $U$.", "\\item If $j$ is an equivalence relation, then $j$ is a", "set-theoretic equivalence relation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13455, "type": "theorem", "label": "groupoids-quotients-lemma-separates-orbits", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-separates-orbits", "contents": [ "Let $B \\to S$ be as in Section \\ref{section-conventions-notation}.", "Let $j : R \\to U \\times_B U$ be a pre-relation.", "Let $\\phi : U \\to X$ be a morphism of algebraic spaces over $B$.", "Consider the diagram", "$$", "\\xymatrix{", "(U \\times_X U) \\times_{(U \\times_B U)} R \\ar[d]^q \\ar[r]_-p & R \\ar[d]^j \\\\", "U \\times_X U \\ar[r]^c & U \\times_B U", "}", "$$", "Then we have:", "\\begin{enumerate}", "\\item The morphism $\\phi$ is set-theoretically invariant if and only", "if $p$ is surjective.", "\\item If $j$ is a set-theoretic pre-equivalence relation then", "$\\phi$ separates orbits if and only if $p$ and $q$ are surjective.", "\\item If $p$ and $q$ are surjective, then $j$ is a set-theoretic", "pre-equivalence relation (and $\\phi$ separates orbits).", "\\item If $\\phi$ is $R$-invariant and $j$ is a set-theoretic pre-equivalence", "relation, then $\\phi$ separates orbits if and only if the induced morphism", "$R \\to U \\times_X U$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $\\phi$ is set-theoretically invariant. This means that for any", "algebraically closed field $k$ over $B$ and any $\\overline{r} \\in R(k)$", "we have $\\phi(s(\\overline{r})) = \\phi(t(\\overline{r}))$. Hence", "$((\\phi(t(\\overline{r})), \\phi(s(\\overline{r}))), \\overline{r})$", "defines a point in the fibre product mapping to $\\overline{r}$ via", "$p$. This shows that $p$ is surjective. Conversely, assume $p$ is", "surjective. Pick $\\overline{r} \\in R(k)$. As $p$ is surjective, we", "can find a field extension $k \\subset K$ and a $K$-valued point", "$\\tilde r$ of the fibre product with $p(\\tilde r) = \\overline{r}$.", "Then $q(\\tilde r) \\in U \\times_X U$ maps to", "$(t(\\overline{r}), s(\\overline{r}))$ in $U \\times_B U$ and we conclude", "that $\\phi(s(\\overline{r})) = \\phi(t(\\overline{r}))$. This proves", "that $\\phi$ is set-theoretically invariant.", "\\medskip\\noindent", "The proofs of (2), (3), and (4) are omitted. Hint: Assume $k$ is an", "algebraically closed field over $B$ of large cardinality. Consider the", "associated diagram of sets", "$$", "\\xymatrix{", "(U(k) \\times_{X(k)} U(k)) \\times_{U(k) \\times U(k)} R(k) \\ar[d]^q \\ar[r]_-p &", "R(k) \\ar[d]^j \\\\", "U(k) \\times_{X(k)} U(k) \\ar[r]^c & U(k) \\times U(k)", "}", "$$", "By the lemmas above the equivalences posed in (2), (3), and (4) become", "set-theoretic questions related to the diagram we just displayed, using", "that surjectivity translates into surjectivity on $k$-valued points by", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-large-enough}." ], "refs": [ "spaces-morphisms-lemma-large-enough" ], "ref_ids": [ 4821 ] } ], "ref_ids": [] }, { "id": 13456, "type": "theorem", "label": "groupoids-quotients-lemma-orbit-space", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-orbit-space", "contents": [ "Let $B \\to S$ as in Section \\ref{section-conventions-notation}.", "Let $j : R \\to U \\times_B U$ be a set-theoretic pre-equivalence", "relation. A morphism $\\phi : U \\to X$ is an orbit space for $R$ if and only if", "\\begin{enumerate}", "\\item $\\phi \\circ s = \\phi \\circ t$, i.e., $\\phi$ is invariant,", "\\item the induced morphism $(t, s) : R \\to U \\times_X U$ is surjective, and", "\\item the morphism $\\phi : U \\to X$ is surjective.", "\\end{enumerate}", "This characterization applies for example if $j$ is a pre-equivalence relation,", "or comes from a groupoid in algebraic spaces over $B$, or comes from the action", "of a group algebraic space over $B$ on $U$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-separates-orbits} part (4)." ], "refs": [ "groupoids-quotients-lemma-separates-orbits" ], "ref_ids": [ 13455 ] } ], "ref_ids": [] }, { "id": 13457, "type": "theorem", "label": "groupoids-quotients-lemma-orbit-space-locally-finite-type-over-base", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-lemma-orbit-space-locally-finite-type-over-base", "contents": [ "Let $B \\to S$ as in Section \\ref{section-conventions-notation}.", "Let $j = (t, s) : R \\to U \\times_B U$ be a pre-relation.", "Assume $R, U$ are locally of finite type over $B$.", "Let $\\phi : U \\to X$ be an $R$-invariant morphism of algebraic spaces over $B$.", "Then $\\phi$ is an orbit space for $R$ if and only if the natural map", "$$", "U(k)/\\big(\\text{equivalence relation generated by }j(R(k))\\big)", "\\longrightarrow", "X(k)", "$$", "is bijective for all algebraically closed fields $k$ over $B$." ], "refs": [], "proofs": [ { "contents": [ "Note that since $U$, $R$ are locally of finite type over $B$ all of the", "morphisms $s, t, j, \\phi$ are locally of finite type, see", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-permanence-finite-type}.", "We will also use without further mention", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points}.", "Assume $\\phi$ is an orbit space. Let $k$ be any algebraically closed", "field over $B$. Let $\\overline{x} \\in X(k)$. Consider", "$U \\times_{\\phi, X, \\overline{x}} \\Spec(k)$.", "This is a nonempty algebraic space", "which is locally of finite type over $k$. Hence it has a $k$-valued point.", "This shows the displayed map of the lemma is surjective.", "Suppose that $\\overline{u}, \\overline{u}' \\in U(k)$ map to the same", "element of $X(k)$. By", "Definition \\ref{definition-set-theoretically-invariant}", "this means that $\\overline{u}, \\overline{u}'$ are in the same", "$R$-orbit. By Lemma \\ref{lemma-geometric-orbits} this means that", "they are equivalent under the equivalence relation generated by", "$j(R(k))$. Thus the displayed morphism is injective.", "\\medskip\\noindent", "Conversely, assume the displayed map is bijective for all algebraically", "closed fields $k$ over $B$. This condition clearly implies that $\\phi$", "is surjective. We have already assumed that $\\phi$ is $R$-invariant.", "Finally, the injectivity of all the displayed maps implies that", "$\\phi$ separates orbits. Hence $\\phi$ is an orbit space." ], "refs": [ "spaces-morphisms-lemma-permanence-finite-type", "spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points", "groupoids-quotients-definition-set-theoretically-invariant", "groupoids-quotients-lemma-geometric-orbits" ], "ref_ids": [ 4818, 4820, 13464, 13448 ] } ], "ref_ids": [] }, { "id": 13472, "type": "theorem", "label": "spaces-resolve-theorem-resolve", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-theorem-resolve", "contents": [ "Let $S$ be a scheme. Let $Y$ be a two dimensional integral", "Noetherian algebraic space over $S$. The following are equivalent", "\\begin{enumerate}", "\\item there exists an alteration $X \\to Y$ with $X$ regular,", "\\item there exists a resolution of singularities of $Y$,", "\\item $Y$ has a resolution of singularities by normalized blowups,", "\\item the normalization $Y^\\nu \\to Y$ is finite and $Y^\\nu$ has", "finitely many singular points $y_1, \\ldots, y_m \\in |Y|$ such that the", "completions of the henselian local rings $\\mathcal{O}_{Y^\\nu, y_i}^h$", "are normal.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implications (3) $\\Rightarrow$ (2) $\\Rightarrow$ (1) are immediate.", "\\medskip\\noindent", "Let $X \\to Y$ be an alteration with $X$ regular. Then $Y^\\nu \\to Y$", "is finite by Lemma \\ref{lemma-regular-alteration-implies}.", "Consider the factorization $f : X \\to Y^\\nu$ from ", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-normalization-normal}.", "The morphism $f$ is finite over an open $V \\subset Y^\\nu$ containing", "every point of codimension $\\leq 1$ in $Y^\\nu$", "by Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-finite-in-codim-1}.", "Then $f$ is flat over $V$ by", "Algebra, Lemma \\ref{algebra-lemma-CM-over-regular-flat}", "and the fact that a normal local ring", "of dimension $\\leq 2$ is Cohen-Macaulay by Serre's criterion", "(Algebra, Lemma \\ref{algebra-lemma-criterion-normal}).", "Then $V$ is regular by Algebra, Lemma \\ref{algebra-lemma-descent-regular}.", "As $Y^\\nu$ is Noetherian we conclude that", "$Y^\\nu \\setminus V = \\{y_1, \\ldots, y_m\\}$ is finite.", "For each $i$ let $\\mathcal{O}_{Y^\\nu, y_i}^h$ be the henselian", "local ring. Then $X \\times_Y \\Spec(\\mathcal{O}_{Y^\\nu, y_i}^h)$", "is a regular alteration of $\\Spec(\\mathcal{O}_{Y^\\nu, y_i}^h)$", "(some details omitted).", "By Lemma \\ref{lemma-regular-alteration-implies-local}", "the completion of $\\mathcal{O}_{Y^\\nu, y_i}^h$ is normal.", "In this way we see that (1) $\\Rightarrow$ (4).", "\\medskip\\noindent", "Assume (4). We have to prove (3). We may immediately replace", "$Y$ by its normalization. Let $y_1, \\ldots, y_m \\in |Y|$ be the", "singular points. Choose a collection of elementary \\'etale neighbourhoods", "$(V_i, v_i) \\to (Y, y_i)$ as in Section \\ref{section-strategy}.", "For each $i$ the henselian local ring $\\mathcal{O}_{Y^\\nu, y_i}^h$", "is the henselization of $\\mathcal{O}_{V_i, v_i}$.", "Hence these rings have isomorphic completions.", "Thus by the result for schemes", "(Resolution of Surfaces, Theorem \\ref{resolve-theorem-resolve})", "we see that there exist finite sequences of normalized blowups", "$$", "X_{i, n_i} \\to X_{i, n_i - 1} \\to \\ldots \\to V_i", "$$", "blowing up only in points lying over $v_i$ such that $X_{i, n_i}$", "is regular. By Lemma \\ref{lemma-equivalence-sequence-normalized-blowups}", "there is a sequence of normalized blowing ups", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to Y", "$$", "and of course $X_n$ is regular too (look at the local rings).", "This completes the proof." ], "refs": [ "spaces-resolve-lemma-regular-alteration-implies", "spaces-morphisms-lemma-normalization-normal", "spaces-over-fields-lemma-finite-in-codim-1", "algebra-lemma-CM-over-regular-flat", "algebra-lemma-criterion-normal", "algebra-lemma-descent-regular", "spaces-resolve-lemma-regular-alteration-implies-local", "resolve-theorem-resolve", "spaces-resolve-lemma-equivalence-sequence-normalized-blowups" ], "ref_ids": [ 13489, 4969, 12822, 1107, 1311, 1373, 13490, 11635, 13484 ] } ], "ref_ids": [] }, { "id": 13473, "type": "theorem", "label": "spaces-resolve-lemma-modification", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-modification", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a $2$-dimensional Noetherian", "local domain such that $U = \\Spec(A) \\setminus \\{\\mathfrak m\\}$", "is a normal scheme. Then any modification $f : X \\to \\Spec(A)$", "is a morphism as in (\\ref{equation-modification})." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be a modification. We have to show that", "$f^{-1}(U) \\to U$ is an isomorphism. Since every closed point $u$ of $U$", "has codimension $1$, this follows from", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-modification-normal-iso-over-codimension-1}." ], "refs": [ "spaces-over-fields-lemma-modification-normal-iso-over-codimension-1" ], "ref_ids": [ 12823 ] } ], "ref_ids": [] }, { "id": 13474, "type": "theorem", "label": "spaces-resolve-lemma-closed-immersion-on-fibre", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-closed-immersion-on-fibre", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Let $g : X \\to Y$ be a morphism in the category (\\ref{equation-modification}).", "If the induced morphism $X_\\kappa \\to Y_\\kappa$ of special fibres is", "a closed immersion, then $g$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-where-closed-immersion}." ], "refs": [ "spaces-more-morphisms-lemma-where-closed-immersion" ], "ref_ids": [ 249 ] } ], "ref_ids": [] }, { "id": 13475, "type": "theorem", "label": "spaces-resolve-lemma-dimension-special-fibre", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-dimension-special-fibre", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local domain", "of dimension $\\geq 1$.", "Let $f : X \\to \\Spec(A)$ be a morphism of algebraic spaces.", "Assume at least one of the following conditions is satisfied", "\\begin{enumerate}", "\\item $f$ is a modification (Spaces over Fields, Definition", "\\ref{spaces-over-fields-definition-modification}),", "\\item $f$ is an alteration (Spaces over Fields, Definition", "\\ref{spaces-over-fields-definition-alteration}),", "\\item $f$ is locally of finite type, quasi-separated, $X$ is integral,", "and there is exactly one point of $|X|$ mapping to the generic point", "of $\\Spec(A)$,", "\\item $f$ is locally of finite type, $X$ is decent, and the points", "of $|X|$ mapping to the generic point of $\\Spec(A)$ are", "the generic points of irreducible components of $|X|$,", "\\item add more here.", "\\end{enumerate}", "Then $\\dim(X_\\kappa) \\leq \\dim(A) - 1$." ], "refs": [ "spaces-over-fields-definition-modification", "spaces-over-fields-definition-alteration" ], "proofs": [ { "contents": [ "Cases (1), (2), and (3) are special cases of (4). Choose an affine scheme", "$U = \\Spec(B)$ and an \\'etale morphism $U \\to X$. The ring map $A \\to B$", "is of finite type. We have to show that", "$\\dim(U_\\kappa) \\leq \\dim(A) - 1$. Since $X$ is decent, the generic", "points of irreducible components of $U$ are the points lying over", "generic points of irreducible components of $|X|$, see", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-generic-points}.", "Hence the fibre of $\\Spec(B) \\to \\Spec(A)$ over $(0)$", "is the (finite) set of minimal primes $\\mathfrak q_1, \\ldots, \\mathfrak q_r$", "of $B$. Thus $A_f \\to B_f$ is finite for some nonzero $f \\in A$", "(Algebra, Lemma \\ref{algebra-lemma-generically-finite}).", "We conclude $\\kappa(\\mathfrak q_i)$ is a finite extension of the", "fraction field of $A$.", "Let $\\mathfrak q \\subset B$ be a prime lying over $\\mathfrak m$. Then", "$$", "\\dim(B_\\mathfrak q) = \\max \\dim((B/\\mathfrak q_i)_{\\mathfrak q})", "\\leq \\dim(A)", "$$", "the inequality by the dimension formula for $A \\subset B/\\mathfrak q_i$, see", "Algebra, Lemma \\ref{algebra-lemma-dimension-formula}.", "However, the dimension of $B_\\mathfrak q/\\mathfrak m B_\\mathfrak q$", "(which is the local ring of $U_\\kappa$ at the corresponding point)", "is at least one less because the minimal primes $\\mathfrak q_i$", "are not in $V(\\mathfrak m)$. We conclude by", "Properties, Lemma \\ref{properties-lemma-dimension}." ], "refs": [ "decent-spaces-lemma-decent-generic-points", "algebra-lemma-generically-finite", "algebra-lemma-dimension-formula", "properties-lemma-dimension" ], "ref_ids": [ 9531, 1056, 990, 2978 ] } ], "ref_ids": [ 12892, 12893 ] }, { "id": 13476, "type": "theorem", "label": "spaces-resolve-lemma-modification-of-dim-2-is-projective-over-complete", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-modification-of-dim-2-is-projective-over-complete", "contents": [ "If $(A, \\mathfrak m, \\kappa)$ is a complete Noetherian local domain", "of dimension $2$, then every modification of $\\Spec(A)$ is projective over $A$." ], "refs": [], "proofs": [ { "contents": [ "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-projective-over-complete}", "it suffices to show that the special fibre of any modification", "$X$ of $\\Spec(A)$ has dimension $\\leq 1$.", "This follows from Lemma \\ref{lemma-dimension-special-fibre}." ], "refs": [ "spaces-more-morphisms-lemma-projective-over-complete", "spaces-resolve-lemma-dimension-special-fibre" ], "ref_ids": [ 213, 13475 ] } ], "ref_ids": [] }, { "id": 13477, "type": "theorem", "label": "spaces-resolve-lemma-equivalence", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-equivalence", "contents": [ "The functor $F$ (\\ref{equation-equivalence}) is an equivalence." ], "refs": [], "proofs": [ { "contents": [ "For $n = 1$ this is Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-excision-modifications}.", "For $n > 1$ the lemma can be proved in exactly the same way or it", "can be deduced from it. For example, suppose that", "$g_i : Y_i \\to U_i$ are objects of $\\mathcal{C}_{U_i, u_i}$.", "Then by the case $n = 1$ we can find $f'_i : Y'_i \\to X$", "which are isomorphisms over $X \\setminus \\{x_i\\}$ and whose", "base change to $U_i$ is $f_i$. Then we can set", "$$", "f : Y = Y'_1 \\times_X \\ldots \\times_X Y'_n \\to X", "$$", "This is an object of $\\mathcal{C}_{X, \\{x_1, \\ldots, x_n\\}}$", "whose base change by $U_i \\to X$ recovers $g_i$. Thus the functor", "is essentially surjective. We omit the proof of", "fully faithfulness." ], "refs": [ "spaces-limits-lemma-excision-modifications" ], "ref_ids": [ 4638 ] } ], "ref_ids": [] }, { "id": 13478, "type": "theorem", "label": "spaces-resolve-lemma-equivalence-properties", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-equivalence-properties", "contents": [ "Let $X, x_i, U_i \\to X, u_i$ be as in (\\ref{equation-equivalence}).", "If $f : Y \\to X$ corresponds to $g_i : Y_i \\to U_i$ under $F$,", "then $f$ is quasi-compact, quasi-separated, separated, locally of finite", "presentation, of finite presentation, locally of finite type, of finite type,", "proper, integral, finite, if and only if $g_i$ is so", "for $i = 1, \\ldots, n$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-excision-modifications-properties}." ], "refs": [ "spaces-limits-lemma-excision-modifications-properties" ], "ref_ids": [ 4639 ] } ], "ref_ids": [] }, { "id": 13479, "type": "theorem", "label": "spaces-resolve-lemma-equivalence-fibre", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-equivalence-fibre", "contents": [ "Let $X, x_i, U_i \\to X, u_i$ be as in (\\ref{equation-equivalence}).", "If $f : Y \\to X$ corresponds to $g_i : Y_i \\to U_i$ under $F$,", "then $Y_{x_i} \\cong (Y_i)_{u_i}$ as algebraic spaces." ], "refs": [], "proofs": [ { "contents": [ "This is clear because $u_i \\to x_i$ is an isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13480, "type": "theorem", "label": "spaces-resolve-lemma-equivalence-sequence-blowups", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-equivalence-sequence-blowups", "contents": [ "Let $X, x_i, U_i \\to X, u_i$ be as in (\\ref{equation-equivalence})", "and assume $f : Y \\to X$ corresponds to $g_i : Y_i \\to U_i$ under $F$.", "Then there exists a factorization", "$$", "Y = Z_m \\to Z_{m - 1} \\to \\ldots \\to Z_1 \\to Z_0 = X", "$$", "of $f$ where $Z_{j + 1} \\to Z_j$ is the blowing up of $Z_j$ at a closed", "point $z_j$ lying over $\\{x_1, \\ldots, x_n\\}$ if and only if for each", "$i$ there exists a factorization", "$$", "Y_i = Z_{i, m_i} \\to Z_{i, m_i - 1} \\to \\ldots \\to Z_{i, 1} \\to Z_{i, 0} = U_i", "$$", "of $g_i$ where $Z_{i, j + 1} \\to Z_{i, j}$ is the blowing up of $Z_{i, j}$", "at a closed point $z_{i, j}$ lying over $u_i$." ], "refs": [], "proofs": [ { "contents": [ "A blowing up is a representable morphism. Hence in either case", "we inductively see that $Z_j \\to X$ or $Z_{i, j} \\to U_i$ is", "representable. Whence each $Z_j$ or $Z_{i, j}$ is a decent", "algebraic space by Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-representable-named-properties}.", "This shows that the assertions make sense (since blowing up", "is only defined for decent spaces).", "To prove the equivalence, let's start with a sequence of blowups", "$Z_m \\to Z_{m - 1} \\to \\ldots \\to Z_1 \\to Z_0 = X$.", "The first morphism $Z_1 \\to X$ is given", "by blowing up one of the $x_i$, say $x_1$. Applying $F$", "to $Z_1 \\to X$ we find a blowup $Z_{1, 1} \\to U_1$ at $u_1$", "is the blowing up at $u_1$ and otherwise $Z_{i, 0} = U_i$ for $i > 1$.", "In the next step, we either blow up one of the $x_i$, $i \\geq 2$", "on $Z_1$ or we pick a closed point $z_1$ of the fibre of $Z_1 \\to X$", "over $x_1$. In the first case it is clear what to do and in", "the second case we use that $(Z_1)_{x_1} \\cong (Z_{1, 1})_{u_1}$", "(Lemma \\ref{lemma-equivalence-fibre})", "to get a closed point $z_{1, 1} \\in Z_{1, 1}$ corresponding to $z_1$.", "Then we set $Z_{1, 2} \\to Z_{1, 1}$ equal to the blowing up", "in $z_{1, 1}$. Continuing in this manner we construct the factorizations", "of each $g_i$.", "\\medskip\\noindent", "Conversely, given sequences of blowups", "$Z_{i, m_i} \\to Z_{i, m_i - 1} \\to \\ldots \\to Z_{i, 1} \\to Z_{i, 0} = U_i$", "we construct the sequence of blowing ups of $X$ in exactly the same manner." ], "refs": [ "decent-spaces-lemma-representable-named-properties", "spaces-resolve-lemma-equivalence-fibre" ], "ref_ids": [ 9470, 13479 ] } ], "ref_ids": [] }, { "id": 13481, "type": "theorem", "label": "spaces-resolve-lemma-make-ideal-principal", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-make-ideal-principal", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $T \\subset |X|$ be a finite set of closed points $x$ such that", "(1) $X$ is regular at $x$ and (2) the local ring of $X$ at $x$ has", "dimension $2$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent", "sheaf of ideals such that $\\mathcal{O}_X/\\mathcal{I}$ is supported on $T$.", "Then there exists a sequence", "$$", "X_m \\to X_{m - 1} \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "where $X_{j + 1} \\to X_j$ is the blowing up of $X_j$ at a closed", "point $x_j$ lying above a point of $T$ such that", "$\\mathcal{I}\\mathcal{O}_{X_n}$ is an invertible ideal sheaf." ], "refs": [], "proofs": [ { "contents": [ "Say $T = \\{x_1, \\ldots, x_r\\}$. Pick elementary \\'etale neighbourhoods", "$(U_i, u_i) \\to (X, x_i)$ as in Section \\ref{section-strategy}.", "For each $i$ the restriction", "$\\mathcal{I}_i = \\mathcal{I}|_{U_i} \\subset \\mathcal{O}_{U_i}$", "is a quasi-coherent sheaf of ideals supported at $u_i$.", "The local ring of $U_i$ at $u_i$ is regular and has dimension $2$.", "Thus we may apply", "Resolution of Surfaces, Lemma \\ref{resolve-lemma-make-ideal-principal}", "to find a sequence", "$$", "X_{i, m_i} \\to X_{i, m_i - 1} \\to \\ldots \\to X_1 \\to X_{i, 0} = U_i", "$$", "of blowing ups in closed points lying over $u_i$ such that", "$\\mathcal{I}_i \\mathcal{O}_{X_{i, m_i}}$ is invertible.", "By Lemma \\ref{lemma-equivalence-sequence-blowups}", "we find a sequence of blowing ups", "$$", "X_m \\to X_{m - 1} \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "as in the statement of the lemma whose base change to our $U_i$", "produces the given sequences. It follows that", "$\\mathcal{I}\\mathcal{O}_{X_n}$ is an invertible ideal sheaf.", "Namely, we know this is true over $X \\setminus \\{x_1, \\ldots, x_n\\}$", "and in an \\'etale neighbourhood of the fibre of each $x_i$", "it is true by construction." ], "refs": [ "resolve-lemma-make-ideal-principal", "spaces-resolve-lemma-equivalence-sequence-blowups" ], "ref_ids": [ 11645, 13480 ] } ], "ref_ids": [] }, { "id": 13482, "type": "theorem", "label": "spaces-resolve-lemma-dominate-by-blowing-up-in-points", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-dominate-by-blowing-up-in-points", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.", "Let $T \\subset |X|$ be a finite set of closed points $x$ such that", "(1) $X$ is regular at $x$ and (2) the local ring of $X$ at $x$ has", "dimension $2$. Let $f : Y \\to X$ be a proper morphism of", "algebraic spaces which is an isomorphism over $U = X \\setminus T$.", "Then there exists a sequence", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "where $X_{i + 1} \\to X_i$ is the blowing up of $X_i$ at a closed", "point $x_i$ lying above a point of $T$ and a factorization $X_n \\to Y \\to X$", "of the composition." ], "refs": [], "proofs": [ { "contents": [ "By More on Morphisms of Spaces,", "Lemma \\ref{spaces-more-morphisms-lemma-dominate-modification-by-blowup} ", "there exists a $U$-admissible blowup $X' \\to X$ which dominates", "$Y \\to X$. Hence we may assume there exists an ideal sheaf", "$\\mathcal{I} \\subset \\mathcal{O}_X$ such that", "$\\mathcal{O}_X/\\mathcal{I}$ is supported on $T$ and such that", "$Y$ is the blowing up of $X$ in $\\mathcal{I}$.", "By Lemma \\ref{lemma-make-ideal-principal} ", "there exists a sequence", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "where $X_{i + 1} \\to X_i$ is the blowing up of $X_i$ at a closed", "point $x_i$ lying above a point of $T$ such that", "$\\mathcal{I}\\mathcal{O}_{X_n}$ is an invertible ideal sheaf.", "By the universal property of blowing up", "(Divisors on Spaces, Lemma", "\\ref{spaces-divisors-lemma-universal-property-blowing-up})", "we find the desired factorization." ], "refs": [ "spaces-more-morphisms-lemma-dominate-modification-by-blowup", "spaces-resolve-lemma-make-ideal-principal", "spaces-divisors-lemma-universal-property-blowing-up" ], "ref_ids": [ 193, 13481, 12992 ] } ], "ref_ids": [] }, { "id": 13483, "type": "theorem", "label": "spaces-resolve-lemma-Nagata-normalized-blowup", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-Nagata-normalized-blowup", "contents": [ "In Definition \\ref{definition-normalized-blowup} if $X$ is Nagata,", "then the normalized blowing up of $X$ at $x$ is a", "normal Nagata algebraic space proper over $X$." ], "refs": [ "spaces-resolve-definition-normalized-blowup" ], "proofs": [ { "contents": [ "The blowup morphism $X' \\to X$ is proper", "(as $X$ is locally Noetherian we may apply", "Divisors on Spaces, Lemma \\ref{spaces-divisors-lemma-blowing-up-projective}).", "Thus $X'$ is Nagata", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-finite-type-nagata}).", "Therefore the normalization $X'' \\to X'$ is finite", "(Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-nagata-normalization})", "and we conclude that $X'' \\to X$ is proper as well", "(Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-finite-proper} and", "\\ref{spaces-morphisms-lemma-composition-proper}).", "It follows that the normalized blowing up", "is a normal (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-normalization-normal})", "Nagata algebraic space." ], "refs": [ "spaces-divisors-lemma-blowing-up-projective", "spaces-morphisms-lemma-finite-type-nagata", "spaces-morphisms-lemma-nagata-normalization", "spaces-morphisms-lemma-finite-proper", "spaces-morphisms-lemma-composition-proper", "spaces-morphisms-lemma-normalization-normal" ], "ref_ids": [ 12998, 4828, 4970, 4946, 4918, 4969 ] } ], "ref_ids": [ 13492 ] }, { "id": 13484, "type": "theorem", "label": "spaces-resolve-lemma-equivalence-sequence-normalized-blowups", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-equivalence-sequence-normalized-blowups", "contents": [ "Let $X, x_i, U_i \\to X, u_i$ be as in (\\ref{equation-equivalence})", "and assume $f : Y \\to X$ corresponds to $g_i : Y_i \\to U_i$ under $F$.", "Assume $X$ satisfies the equivalent conditions of", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-prepare-normalization}.", "Then there exists a factorization", "$$", "Y = Z_m \\to Z_{m - 1} \\to \\ldots \\to Z_1 \\to Z_0 = X", "$$", "of $f$ where $Z_{j + 1} \\to Z_j$ is the normalized blowing up of $Z_j$", "at a closed point $z_j$ lying over $\\{x_1, \\ldots, x_n\\}$ if and only if", "for each $i$ there exists a factorization", "$$", "Y_i = Z_{i, m_i} \\to Z_{i, m_i - 1} \\to \\ldots \\to Z_{i, 1} \\to Z_{i, 0} = U_i", "$$", "of $g_i$ where $Z_{i, j + 1} \\to Z_{i, j}$ is the normalized blowing up of", "$Z_{i, j}$ at a closed point $z_{i, j}$ lying over $u_i$." ], "refs": [ "spaces-morphisms-lemma-prepare-normalization" ], "proofs": [ { "contents": [ "This follows by the exact same argument as used to prove", "Lemma \\ref{lemma-equivalence-sequence-blowups}." ], "refs": [ "spaces-resolve-lemma-equivalence-sequence-blowups" ], "ref_ids": [ 13480 ] } ], "ref_ids": [ 4966 ] }, { "id": 13485, "type": "theorem", "label": "spaces-resolve-lemma-dominate-by-normalized-blowing-up", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-dominate-by-normalized-blowing-up", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian Nagata algebraic space over $S$", "with $\\dim(X) = 2$. Let $f : Y \\to X$ be a proper birational morphism.", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "X_n \\ar[r] \\ar[d] &", "X_{n - 1} \\ar[r] &", "\\ldots \\ar[r] &", "X_1 \\ar[r] &", "X_0 \\ar[d] \\\\", "Y \\ar[rrrr] & & & & X", "}", "$$", "where $X_0 \\to X$ is the normalization and", "where $X_{i + 1} \\to X_i$ is the normalized blowing up of $X_i$ at a closed", "point." ], "refs": [], "proofs": [ { "contents": [ "Although one can prove this lemma directly for algebraic spaces,", "we will continue the approach used above to reduce it to the case", "of schemes.", "\\medskip\\noindent", "We will use that Noetherian algebraic spaces are quasi-separated", "and hence points have well defined residue fields (for example by", "Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-decent-space-elementary-etale-neighbourhood}).", "We will use the results of Morphisms of Spaces, Sections", "\\ref{spaces-morphisms-section-nagata},", "\\ref{spaces-morphisms-section-dimension-formula}, and", "\\ref{spaces-morphisms-section-normalization} without further mention.", "We may replace $Y$ by its normalization. Let $X_0 \\to X$ be the normalization.", "The morphism $Y \\to X$ factors through $X_0$.", "Thus we may assume that both $X$ and $Y$ are normal.", "\\medskip\\noindent", "Assume $X$ and $Y$ are normal. The morphism $f : Y \\to X$ is an isomorphism", "over an open which contains every point of codimension $0$ and $1$ in $Y$ and", "every point of $Y$ over which the fibre is finite, see", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-modification-normal-iso-over-codimension-1}.", "Hence we see that there is a finite set of closed points $T \\subset |X|$", "such that $f$ is an isomorphism over $X \\setminus T$.", "By More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-dominate-modification-by-blowup}", "there exists an $X \\setminus T$-admissible blowup $Y' \\to X$", "which dominates $Y$. After replacing $Y$ by the normalization of", "$Y'$ we see that we may assume that $Y \\to X$ is representable.", "\\medskip\\noindent", "Say $T = \\{x_1, \\ldots, x_r\\}$. Pick elementary \\'etale neighbourhoods", "$(U_i, u_i) \\to (X, x_i)$ as in Section \\ref{section-strategy}.", "For each $i$ the morphism $Y_i = Y \\times_X U_i \\to U_i$", "is a proper birational morphism which is an isomorphism over", "$U_i \\setminus \\{u_i\\}$. Thus we may apply", "Resolution of Surfaces, Lemma", "\\ref{resolve-lemma-dominate-by-normalized-blowing-up}", "to find a sequence", "$$", "X_{i, m_i} \\to X_{i, m_i - 1} \\to \\ldots \\to X_1 \\to X_{i, 0} = U_i", "$$", "of normalized blowing ups in closed points lying over $u_i$ such that", "$X_{i, m_i}$ dominates $Y_i$.", "By Lemma \\ref{lemma-equivalence-sequence-normalized-blowups}", "we find a sequence of normalized blowing ups", "$$", "X_m \\to X_{m - 1} \\to \\ldots \\to X_1 \\to X_0 = X", "$$", "as in the statement of the lemma whose base change to our $U_i$", "produces the given sequences. It follows that $X_m$ dominates", "$Y$ by the equivalence of categories of", "Lemma \\ref{lemma-equivalence}." ], "refs": [ "decent-spaces-lemma-decent-space-elementary-etale-neighbourhood", "spaces-over-fields-lemma-modification-normal-iso-over-codimension-1", "spaces-more-morphisms-lemma-dominate-modification-by-blowup", "resolve-lemma-dominate-by-normalized-blowing-up", "spaces-resolve-lemma-equivalence-sequence-normalized-blowups", "spaces-resolve-lemma-equivalence" ], "ref_ids": [ 9488, 12823, 193, 11649, 13484, 13477 ] } ], "ref_ids": [] }, { "id": 13486, "type": "theorem", "label": "spaces-resolve-lemma-iso-completions", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-iso-completions", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a local ring with finitely generated", "maximal ideal $\\mathfrak m$. Let $X$ be a decent algebraic", "space over $A$. Let $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$ where", "$A^\\wedge$ is the $\\mathfrak m$-adic completion of $A$.", "For a point $q \\in |Y|$ with image $p \\in |X|$ lying", "over the closed point of $\\Spec(A)$ the map", "$\\mathcal{O}_{X, p}^h \\to \\mathcal{O}_{Y, q}^h$", "of henselian local rings induces an isomorphism on completions." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the case of schemes by", "choosing an elementary \\'etale neighbourhood $(U, u) \\to (X, p)$", "as in Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-decent-space-elementary-etale-neighbourhood},", "setting $V = U \\times_X Y = U \\times_{\\Spec(A)} \\Spec(A^\\wedge)$", "and $v = (u, q)$.", "The case of schemes is", "Resolution of Surfaces, Lemma \\ref{resolve-lemma-iso-completions}." ], "refs": [ "decent-spaces-lemma-decent-space-elementary-etale-neighbourhood", "resolve-lemma-iso-completions" ], "ref_ids": [ 9488, 11679 ] } ], "ref_ids": [] }, { "id": 13487, "type": "theorem", "label": "spaces-resolve-lemma-port-regularity-to-completion", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-port-regularity-to-completion", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Let $X \\to \\Spec(A)$ be a morphism which is locally of finite type", "with $X$ a decent algebraic space. Set", "$Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$. Let $y \\in |Y|$", "with image $x \\in |X|$. Then", "\\begin{enumerate}", "\\item if $\\mathcal{O}_{Y, y}^h$ is regular, then", "$\\mathcal{O}_{X, x}^h$ is regular,", "\\item if $y$ is in the closed fibre, then $\\mathcal{O}_{Y, y}^h$ is regular", "$\\Leftrightarrow \\mathcal{O}_{X, x}^h$ is regular, and", "\\item If $X$ is proper over $A$, then $X$ is regular", "if and only if $Y$ is regular.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By \\'etale localization the first two statements follow", "immediately from the counter part to this lemma for schemes, see", "Resolution of Surfaces, Lemma \\ref{resolve-lemma-port-regularity-to-completion}.", "For part (3), since $Y \\to X$ is surjective (as $A \\to A^\\wedge$", "is faithfully flat) we see that $Y$ regular implies $X$ regular", "by part (1). Conversely, if $X$ is regular, then the henselian", "local rings of $Y$ are regular for all points of the special fibre.", "Let $y \\in |Y|$ be a general point.", "Since $|Y| \\to |\\Spec(A^\\wedge)|$ is closed in the proper", "case, we can find a specialization $y \\leadsto y_0$ with", "$y_0$ in the closed fibre. Choose an elementary \\'etale", "neighbourhood $(V, v_0) \\to (Y, y_0)$ as in", "Decent Spaces, Lemma", "\\ref{decent-spaces-lemma-decent-space-elementary-etale-neighbourhood}.", "Since $Y$ is decent we can lift $y \\leadsto y_0$ to a specialization", "$v \\leadsto v_0$ in $V$", "(Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-specialization}).", "Then we conclude that", "$\\mathcal{O}_{V, v}$ is a localization of $\\mathcal{O}_{V, v_0}$", "hence regular and the proof is complete." ], "refs": [ "resolve-lemma-port-regularity-to-completion", "decent-spaces-lemma-decent-space-elementary-etale-neighbourhood", "decent-spaces-lemma-decent-specialization" ], "ref_ids": [ 11680, 9488, 9494 ] } ], "ref_ids": [] }, { "id": 13488, "type": "theorem", "label": "spaces-resolve-lemma-formally-unramified", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-formally-unramified", "contents": [ "Let $(A, \\mathfrak m)$ be a local Noetherian ring. Let", "$X$ be an algebraic space over $A$. Assume", "\\begin{enumerate}", "\\item $A$ is analytically unramified", "(Algebra, Definition \\ref{algebra-definition-analytically-unramified}),", "\\item $X$ is locally of finite type over $A$,", "\\item $X \\to \\Spec(A)$ is \\'etale at every point of codimension $0$ in $X$.", "\\end{enumerate}", "Then the normalization of $X$ is finite over $X$." ], "refs": [ "algebra-definition-analytically-unramified" ], "proofs": [ { "contents": [ "Choose a scheme $U$ and a surjective \\'etale morphism $U \\to X$.", "Then $U \\to \\Spec(A)$ satisfies the assumptions and hence the", "conclusions of", "Resolution of Surfaces, Lemma \\ref{resolve-lemma-formally-unramified}." ], "refs": [ "resolve-lemma-formally-unramified" ], "ref_ids": [ 11683 ] } ], "ref_ids": [ 1553 ] }, { "id": 13489, "type": "theorem", "label": "spaces-resolve-lemma-regular-alteration-implies", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-regular-alteration-implies", "contents": [ "Let $S$ be a scheme. Let $Y$ be a Noetherian integral algebraic space", "over $S$. Assume there exists an alteration", "$f : X \\to Y$ with $X$ regular. Then the normalization $Y^\\nu \\to Y$", "is finite and $Y$ has a dense open which is regular." ], "refs": [], "proofs": [ { "contents": [ "By \\'etale localization, it suffices to prove this when", "$Y = \\Spec(A)$ where $A$ is a Noetherian domain.", "Let $B$ be the integral closure of $A$ in its fraction field.", "Set $C = \\Gamma(X, \\mathcal{O}_X)$. By", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-pushforward-coherent}", "we see that $C$ is a finite $A$-module.", "As $X$ is normal", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-regular-normal})", "we see that $C$ is normal domain", "(Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-normal-integral-sections}).", "Thus $B \\subset C$ and we conclude that $B$ is finite over $A$", "as $A$ is Noetherian.", "\\medskip\\noindent", "There exists a nonempty open $V \\subset Y$ such that $f^{-1}V \\to V$", "is finite, see Spaces over Fields, Definition", "\\ref{spaces-over-fields-definition-alteration}.", "After shrinking $V$ we may assume that $f^{-1}V \\to V$ is flat", "(Morphisms of Spaces, Proposition", "\\ref{spaces-morphisms-proposition-generic-flatness-reduced}).", "Thus $f^{-1}V \\to V$ is faithfully flat. Then $V$ is regular by", "Algebra, Lemma \\ref{algebra-lemma-descent-regular}." ], "refs": [ "spaces-cohomology-lemma-proper-pushforward-coherent", "spaces-properties-lemma-regular-normal", "spaces-over-fields-lemma-normal-integral-sections", "spaces-over-fields-definition-alteration", "spaces-morphisms-proposition-generic-flatness-reduced", "algebra-lemma-descent-regular" ], "ref_ids": [ 11331, 11896, 12827, 12893, 4981, 1373 ] } ], "ref_ids": [] }, { "id": 13490, "type": "theorem", "label": "spaces-resolve-lemma-regular-alteration-implies-local", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-lemma-regular-alteration-implies-local", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a local Noetherian domain.", "Assume there exists an alteration $f : X \\to \\Spec(A)$", "with $X$ regular. Then", "\\begin{enumerate}", "\\item there exists a nonzero $f \\in A$ such that $A_f$ is regular,", "\\item the integral closure $B$ of $A$ in its fraction field is finite over $A$,", "\\item the $\\mathfrak m$-adic completion of $B$ is a normal ring, i.e., the", "completions of $B$ at its maximal ideals are normal domains, and", "\\item the generic formal fibre of $A$ is regular.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (1) and (2) follow from Lemma \\ref{lemma-regular-alteration-implies}.", "We have to redo part of the proof of that lemma in order to set up notation", "for the proof of (3). Set $C = \\Gamma(X, \\mathcal{O}_X)$. By", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-pushforward-coherent}", "we see that $C$ is a finite $A$-module.", "As $X$ is normal", "(Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-regular-normal})", "we see that $C$ is normal domain", "(Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-normal-integral-sections}).", "Thus $B \\subset C$ and we conclude that $B$ is finite over $A$", "as $A$ is Noetherian. By", "Resolution of Surfaces, Lemma \\ref{resolve-lemma-algebra-helper}", "in order to prove (3) it suffices to show", "that the $\\mathfrak m$-adic completion $C^\\wedge$ is normal.", "\\medskip\\noindent", "By Algebra, Lemma \\ref{algebra-lemma-completion-finite-extension}", "the completion $C^\\wedge$ is the product of the completions of", "$C$ at the prime ideals of $C$ lying over $\\mathfrak m$.", "There are finitely many of these and these are the maximal", "ideals $\\mathfrak m_1, \\ldots, \\mathfrak m_r$ of $C$.", "(The corresponding result for $B$ explains the final statement of the lemma.)", "Thus replacing $A$ by $C_{\\mathfrak m_i}$ and $X$ by", "$X_i = X \\times_{\\Spec(C)} \\Spec(C_{\\mathfrak m_i})$", "we reduce to the case discussed in the next paragraph.", "(Note that $\\Gamma(X_i, \\mathcal{O}) = C_{\\mathfrak m_i}$ by", "Cohomology of Spaces,", "Lemma \\ref{spaces-cohomology-lemma-flat-base-change-cohomology}.)", "\\medskip\\noindent", "Here $A$ is a Noetherian local normal domain and $f : X \\to \\Spec(A)$", "is a regular alteration with $\\Gamma(X, \\mathcal{O}_X) = A$.", "We have to show that the completion $A^\\wedge$", "of $A$ is a normal domain. By", "Lemma \\ref{lemma-port-regularity-to-completion}", "$Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$ is regular.", "Since $\\Gamma(Y, \\mathcal{O}_Y) = A^\\wedge$", "by Cohomology of Spaces,", "Lemma \\ref{spaces-cohomology-lemma-flat-base-change-cohomology}.", "We conclude that $A^\\wedge$ is normal as before.", "Namely, $Y$ is normal by Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-regular-normal}.", "It is connected because $\\Gamma(Y, \\mathcal{O}_Y) = A^\\wedge$ is local.", "Hence $Y$ is normal and integral (as connected and normal", "implies integral for separated algebraic spaces). Thus", "$\\Gamma(Y, \\mathcal{O}_Y) = A^\\wedge$ is a normal domain by", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-normal-integral-sections}.", "This proves (3).", "\\medskip\\noindent", "Proof of (4). Let $\\eta \\in \\Spec(A)$ denote the generic point", "and denote by a subscript $\\eta$ the base change to $\\eta$.", "Since $f$ is an alteration, the scheme $X_\\eta$ is finite and", "faithfully flat over $\\eta$. Since $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$", "is regular by Lemma \\ref{lemma-port-regularity-to-completion}", "we see that $Y_\\eta$ is regular (as a limit of opens in $Y$).", "Then $Y_\\eta \\to \\Spec(A^\\wedge \\otimes_A \\kappa(\\eta))$ is finite", "faithfully flat onto the generic formal fibre. We conclude by", "Algebra, Lemma \\ref{algebra-lemma-descent-regular}." ], "refs": [ "spaces-resolve-lemma-regular-alteration-implies", "spaces-cohomology-lemma-proper-pushforward-coherent", "spaces-properties-lemma-regular-normal", "spaces-over-fields-lemma-normal-integral-sections", "resolve-lemma-algebra-helper", "algebra-lemma-completion-finite-extension", "spaces-cohomology-lemma-flat-base-change-cohomology", "spaces-resolve-lemma-port-regularity-to-completion", "spaces-cohomology-lemma-flat-base-change-cohomology", "spaces-properties-lemma-regular-normal", "spaces-over-fields-lemma-normal-integral-sections", "spaces-resolve-lemma-port-regularity-to-completion", "algebra-lemma-descent-regular" ], "ref_ids": [ 13489, 11331, 11896, 12827, 11689, 876, 11296, 13487, 11296, 11896, 12827, 13487, 1373 ] } ], "ref_ids": [] }, { "id": 13495, "type": "theorem", "label": "duality-theorem-lichtenbaum", "categories": [ "duality" ], "title": "duality-theorem-lichtenbaum", "contents": [ "Let $X$ be a nonempty separated scheme of finite type over a field $k$.", "Let $d = \\dim(X)$. The following are equivalent", "\\begin{enumerate}", "\\item $H^d(X, \\mathcal{F}) = 0$ for all coherent $\\mathcal{O}_X$-modules", "$\\mathcal{F}$ on $X$,", "\\item $H^d(X, \\mathcal{F}) = 0$ for all quasi-coherent $\\mathcal{O}_X$-modules", "$\\mathcal{F}$ on $X$, and", "\\item no irreducible component $X' \\subset X$ of dimension $d$", "is proper over $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume there exists an irreducible component $X' \\subset X$ (which we view", "as an integral closed subscheme) which is proper and has dimension $d$.", "Let $\\omega_{X'}$ be a dualizing module of $X'$ over $k$, see", "Lemma \\ref{lemma-duality-proper-over-field}. Then", "$H^d(X', \\omega_{X'})$ is nonzero as it is dual to $H^0(X', \\mathcal{O}_{X'})$", "by the lemma. Hence we see that $H^d(X, \\omega_{X'}) = H^d(X', \\omega_{X'})$", "is nonzero and we conclude that (1) does not hold.", "In this way we see that (1) implies (3).", "\\medskip\\noindent", "Let us prove that (3) implies (1).", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module such that", "$H^d(X, \\mathcal{F})$ is nonzero. Choose a filtration", "$$", "0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset", "\\ldots \\subset \\mathcal{F}_m = \\mathcal{F}", "$$", "as in Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-filter}.", "We obtain exact sequences", "$$", "H^d(X, \\mathcal{F}_i) \\to H^d(X, \\mathcal{F}_{i + 1}) \\to", "H^d(X, \\mathcal{F}_{i + 1}/\\mathcal{F}_i)", "$$", "Thus for some $i \\in \\{1, \\ldots, m\\}$ we find that", "$H^d(X, \\mathcal{F}_{i + 1}/\\mathcal{F}_i)$ is nonzero.", "By our choice of the filtration this means that there exists", "an integral closed subscheme $Z \\subset X$", "and a sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$", "such that $H^d(Z, \\mathcal{I})$ is nonzero.", "By Lemma \\ref{lemma-lichtenbaum}", "we conclude $\\dim(Z) = d$ and $Z$ is proper over $k$", "contradicting (3). Hence (3) implies (1).", "\\medskip\\noindent", "Finally, let us show that (1) and (2) are equivalent for any Noetherian scheme", "$X$. Namely, (2) trivially implies (1). On the other hand, assume (1) and", "let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then we can write", "$\\mathcal{F} = \\colim \\mathcal{F}_i$ as the filtered colimit of its coherent", "submodules, see", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-colimit-finite-type}.", "Then we have $H^d(X, \\mathcal{F}) = \\colim H^d(X, \\mathcal{F}_i) = 0$", "by Cohomology, Lemma \\ref{cohomology-lemma-quasi-separated-cohomology-colimit}.", "Thus (2) is true." ], "refs": [ "duality-lemma-duality-proper-over-field", "coherent-lemma-coherent-filter", "duality-lemma-lichtenbaum", "properties-lemma-quasi-coherent-colimit-finite-type", "cohomology-lemma-quasi-separated-cohomology-colimit" ], "ref_ids": [ 13606, 3329, 13638, 3020, 2082 ] } ], "ref_ids": [] }, { "id": 13496, "type": "theorem", "label": "duality-lemma-equivalent-definitions", "categories": [ "duality" ], "title": "duality-lemma-equivalent-definitions", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $K$ be an object of", "$D(\\mathcal{O}_X)$. The following are equivalent", "\\begin{enumerate}", "\\item For every affine open $U = \\Spec(A) \\subset X$ there exists", "a dualizing complex $\\omega_A^\\bullet$ for $A$ such that", "$K|_U$ is isomorphic to the image of $\\omega_A^\\bullet$ by", "the functor $\\widetilde{} : D(A) \\to D(\\mathcal{O}_U)$.", "\\item There is an affine open covering $X = \\bigcup U_i$, $U_i = \\Spec(A_i)$", "such that for each $i$ there exists a dualizing complex $\\omega_i^\\bullet$ for", "$A_i$ such that $K|_{U_i}$ is isomorphic to the image of $\\omega_i^\\bullet$ by", "the functor $\\widetilde{} : D(A_i) \\to D(\\mathcal{O}_{U_i})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (2) and let $U = \\Spec(A)$ be an affine open of $X$.", "Since condition (2) implies that $K$ is in $D_\\QCoh(\\mathcal{O}_X)$", "we find an object $\\omega_A^\\bullet$ in $D(A)$ whose associated", "complex of quasi-coherent sheaves is isomorphic to $K|_U$, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded}.", "We will show that $\\omega_A^\\bullet$ is a dualizing complex for $A$", "which will finish the proof.", "\\medskip\\noindent", "Since $X = \\bigcup U_i$ is an open covering, we can find a standard", "open covering $U = D(f_1) \\cup \\ldots \\cup D(f_m)$ such that", "each $D(f_j)$ is a standard open in one of the affine opens $U_i$, see", "Schemes, Lemma \\ref{schemes-lemma-standard-open-two-affines}.", "Say $D(f_j) = D(g_j)$ for $g_j \\in A_{i_j}$.", "Then $A_{f_j} \\cong (A_{i_j})_{g_j}$ and we have", "$$", "(\\omega_A^\\bullet)_{f_j} \\cong (\\omega_i^\\bullet)_{g_j}", "$$", "in the derived category by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded}.", "By Dualizing Complexes, Lemma \\ref{dualizing-lemma-dualizing-localize}", "we find that", "the complex $(\\omega_A^\\bullet)_{f_j}$ is a dualizing complex over", "$A_{f_j}$ for $j = 1, \\ldots, m$. This implies that $\\omega_A^\\bullet$", "is dualizing by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-dualizing-glue}." ], "refs": [ "perfect-lemma-affine-compare-bounded", "schemes-lemma-standard-open-two-affines", "perfect-lemma-affine-compare-bounded", "dualizing-lemma-dualizing-localize", "dualizing-lemma-dualizing-glue" ], "ref_ids": [ 6941, 7675, 6941, 2851, 2852 ] } ], "ref_ids": [] }, { "id": 13497, "type": "theorem", "label": "duality-lemma-affine-duality", "categories": [ "duality" ], "title": "duality-lemma-affine-duality", "contents": [ "Let $A$ be a Noetherian ring and let $X = \\Spec(A)$. Let $K, L$ be objects", "of $D(A)$. If $K \\in D_{\\textit{Coh}}(A)$ and $L$ has finite injective", "dimension, then", "$$", "R\\SheafHom_{\\mathcal{O}_X}(\\widetilde{K}, \\widetilde{L})", "=", "\\widetilde{R\\Hom_A(K, L)}", "$$", "in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "We may assume that $L$ is given by a finite complex $I^\\bullet$", "of injective $A$-modules. By induction on the length of $I^\\bullet$", "and compatibility of the constructions with distinguished triangles,", "we reduce to the case that $L = I[0]$ where $I$ is an injective $A$-module.", "In this case, Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-internal-hom}, tells us that", "the $n$th cohomology sheaf of", "$R\\SheafHom_{\\mathcal{O}_X}(\\widetilde{K}, \\widetilde{L})$", "is the sheaf associated to the presheaf", "$$", "D(f) \\longmapsto \\Ext^n_{A_f}(K \\otimes_A A_f, I \\otimes_A A_f)", "$$", "Since $A$ is Noetherian, the $A_f$-module $I \\otimes_A A_f$ is injective", "(Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-localization-injective-modules}). Hence we see that", "\\begin{align*}", "\\Ext^n_{A_f}(K \\otimes_A A_f, I \\otimes_A A_f)", "& =", "\\Hom_{A_f}(H^{-n}(K \\otimes_A A_f), I \\otimes_A A_f) \\\\", "& =", "\\Hom_{A_f}(H^{-n}(K) \\otimes_A A_f, I \\otimes_A A_f) \\\\", "& =", "\\Hom_A(H^{-n}(K), I) \\otimes_A A_f", "\\end{align*}", "The last equality because $H^{-n}(K)$ is a finite $A$-module, see", "Algebra, Lemma \\ref{algebra-lemma-hom-from-finitely-presented}.", "This proves that the canonical map", "$$", "\\widetilde{R\\Hom_A(K, L)}", "\\longrightarrow", "R\\SheafHom_{\\mathcal{O}_X}(\\widetilde{K}, \\widetilde{L})", "$$", "is a quasi-isomorphism in this case and the proof is done." ], "refs": [ "perfect-lemma-quasi-coherence-internal-hom", "dualizing-lemma-localization-injective-modules", "algebra-lemma-hom-from-finitely-presented" ], "ref_ids": [ 6981, 2790, 353 ] } ], "ref_ids": [] }, { "id": 13498, "type": "theorem", "label": "duality-lemma-internal-hom-evaluate-isom", "categories": [ "duality" ], "title": "duality-lemma-internal-hom-evaluate-isom", "contents": [ "Let $X$ be a Noetherian scheme. Let $K, L, M \\in D_\\QCoh(\\mathcal{O}_X)$. ", "Then the map", "$$", "R\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K", "\\longrightarrow", "R\\SheafHom(R\\SheafHom(K, L), M)", "$$", "of Cohomology, Lemma \\ref{cohomology-lemma-internal-hom-evaluate}", "is an isomorphism in the following two cases", "\\begin{enumerate}", "\\item $K \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$,", "$L \\in D^+_{\\textit{Coh}}(\\mathcal{O}_X)$, and $M$ affine locally has", "finite injective dimension (see proof), or", "\\item $K$ and $L$ are in $D_{\\textit{Coh}}(\\mathcal{O}_X)$,", "the object $R\\SheafHom(L, M)$ has finite tor dimension, and", "$L$ and $M$ affine locally have finite injective dimension", "(in particular $L$ and $M$ are bounded).", "\\end{enumerate}" ], "refs": [ "cohomology-lemma-internal-hom-evaluate" ], "proofs": [ { "contents": [ "Proof of (1). We say $M$ has affine locally finite injective dimension", "if $X$ has an open covering by affines $U = \\Spec(A)$ such that the object", "of $D(A)$ corresponding to $M|_U$ (Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded})", "has finite injective dimension\\footnote{This condition is independent of the", "choice of the affine open cover of the Noetherian scheme $X$.", "Details omitted.}. To prove the lemma we may", "replace $X$ by $U$, i.e., we may assume $X = \\Spec(A)$", "for some Noetherian ring $A$. Observe that", "$R\\SheafHom(K, L)$ is in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-coherent-internal-hom}.", "Moreover, the formation of the left and right hand side", "of the arrow commutes with the functor $D(A) \\to D_\\QCoh(\\mathcal{O}_X)$ by", "Lemma \\ref{lemma-affine-duality} and", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-internal-hom}", "(to be sure this uses the assumptions on $K$, $L$, $M$ and what we just", "proved about $R\\SheafHom(K, L)$).", "Then finally the arrow is an isomorphism by", "More on Algebra, Lemmas", "\\ref{more-algebra-lemma-internal-hom-evaluate-isomorphism} part (2).", "\\medskip\\noindent", "Proof of (2). We argue as above. A small change is that here we get", "$R\\SheafHom(K, L)$ in $D_{\\textit{Coh}}(\\mathcal{O}_X)$ because", "affine locally (which is allowable by Lemma \\ref{lemma-affine-duality})", "we may appeal to Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-finite-ext-into-bounded-injective}.", "Then we finally conclude by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-internal-hom-evaluate-isomorphism-technical}." ], "refs": [ "perfect-lemma-affine-compare-bounded", "perfect-lemma-coherent-internal-hom", "duality-lemma-affine-duality", "perfect-lemma-quasi-coherence-internal-hom", "more-algebra-lemma-internal-hom-evaluate-isomorphism", "duality-lemma-affine-duality", "dualizing-lemma-finite-ext-into-bounded-injective", "more-algebra-lemma-internal-hom-evaluate-isomorphism-technical" ], "ref_ids": [ 6941, 6986, 13497, 6981, 10413, 13497, 2847, 10414 ] } ], "ref_ids": [ 2190 ] }, { "id": 13499, "type": "theorem", "label": "duality-lemma-dualizing-schemes", "categories": [ "duality" ], "title": "duality-lemma-dualizing-schemes", "contents": [ "Let $K$ be a dualizing complex on a locally Noetherian scheme $X$.", "Then $K$ is an object of $D_{\\textit{Coh}}(\\mathcal{O}_X)$", "and $D = R\\SheafHom_{\\mathcal{O}_X}(-, K)$ induces an anti-equivalence", "$$", "D :", "D_{\\textit{Coh}}(\\mathcal{O}_X)", "\\longrightarrow", "D_{\\textit{Coh}}(\\mathcal{O}_X)", "$$", "which comes equipped with a canonical isomorphism", "$\\text{id} \\to D \\circ D$. If $X$ is quasi-compact, then", "$D$ exchanges $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and", "$D^-_{\\textit{Coh}}(\\mathcal{O}_X)$ and induces an equivalence", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be an affine open. Say $U = \\Spec(A)$ and", "let $\\omega_A^\\bullet$ be a dualizing complex for $A$", "corresponding to $K|_U$", "as in Lemma \\ref{lemma-equivalent-definitions}.", "By Lemma \\ref{lemma-affine-duality} the diagram", "$$", "\\xymatrix{", "D_{\\textit{Coh}}(A) \\ar[r] \\ar[d]_{R\\Hom_A(-, \\omega_A^\\bullet)} &", "D_{\\textit{Coh}}(\\mathcal{O}_U) \\ar[d]^{R\\SheafHom_{\\mathcal{O}_X}(-, K|_U)} \\\\", "D_{\\textit{Coh}}(A) \\ar[r] &", "D(\\mathcal{O}_U)", "}", "$$", "commutes. We conclude that $D$ sends $D_{\\textit{Coh}}(\\mathcal{O}_X)$ into", "$D_{\\textit{Coh}}(\\mathcal{O}_X)$. Moreover, the canonical map", "$$", "L", "\\longrightarrow", "R\\SheafHom_{\\mathcal{O}_X}(K, K) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L", "\\longrightarrow", "R\\SheafHom_{\\mathcal{O}_X}(R\\SheafHom_{\\mathcal{O}_X}(L, K), K)", "$$", "(using Cohomology, Lemma \\ref{cohomology-lemma-internal-hom-evaluate}", "for the second arrow)", "is an isomorphism for all $L$ because this is true on affines by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-dualizing}\\footnote{An", "alternative is to first show that", "$R\\SheafHom_{\\mathcal{O}_X}(K, K) = \\mathcal{O}_X$ by", "working affine locally and then use", "Lemma \\ref{lemma-internal-hom-evaluate-isom} part (2)", "to see the map is an isomorphism.}", "and we have already seen on affines that we recover what", "happens in algebra.", "The statement on boundedness properties of the functor $D$", "in the quasi-compact case also follows from the corresponding", "statements of Dualizing Complexes, Lemma \\ref{dualizing-lemma-dualizing}." ], "refs": [ "duality-lemma-equivalent-definitions", "duality-lemma-affine-duality", "cohomology-lemma-internal-hom-evaluate", "dualizing-lemma-dualizing", "duality-lemma-internal-hom-evaluate-isom", "dualizing-lemma-dualizing" ], "ref_ids": [ 13496, 13497, 2190, 2848, 13498, 2848 ] } ], "ref_ids": [] }, { "id": 13500, "type": "theorem", "label": "duality-lemma-dualizing-unique-schemes", "categories": [ "duality" ], "title": "duality-lemma-dualizing-unique-schemes", "contents": [ "Let $X$ be a locally Noetherian scheme. If $K$ and $K'$ are dualizing", "complexes on $X$, then $K'$ is isomorphic to", "$K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$", "for some invertible object $L$ of $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Set", "$$", "L = R\\SheafHom_{\\mathcal{O}_X}(K, K')", "$$", "This is an invertible object of $D(\\mathcal{O}_X)$, because affine locally", "this is true, see Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-dualizing-unique} and its proof.", "The evaluation map $L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K \\to K'$", "is an isomorphism for the same reason." ], "refs": [ "dualizing-lemma-dualizing-unique" ], "ref_ids": [ 2850 ] } ], "ref_ids": [] }, { "id": 13501, "type": "theorem", "label": "duality-lemma-dimension-function-scheme", "categories": [ "duality" ], "title": "duality-lemma-dimension-function-scheme", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\omega_X^\\bullet$", "be a dualizing complex on $X$. Then $X$ is universally catenary", "and the function", "$X \\to \\mathbf{Z}$ defined by", "$$", "x \\longmapsto \\delta(x)\\text{ such that }", "\\omega_{X, x}^\\bullet[-\\delta(x)]", "\\text{ is a normalized dualizing complex over }", "\\mathcal{O}_{X, x}", "$$", "is a dimension function." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the affine case", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-dimension-function}", "and the definitions." ], "refs": [ "dualizing-lemma-dimension-function" ], "ref_ids": [ 2869 ] } ], "ref_ids": [] }, { "id": 13502, "type": "theorem", "label": "duality-lemma-sitting-in-degrees", "categories": [ "duality" ], "title": "duality-lemma-sitting-in-degrees", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\omega_X^\\bullet$", "be a dualizing complex on $X$ with associated dimension function $\\delta$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Set", "$\\mathcal{E}^i = \\SheafExt^{-i}_{\\mathcal{O}_X}(\\mathcal{F}, \\omega_X^\\bullet)$.", "Then $\\mathcal{E}^i$ is a coherent $\\mathcal{O}_X$-module and", "for $x \\in X$ we have", "\\begin{enumerate}", "\\item $\\mathcal{E}^i_x$ is nonzero only for", "$\\delta(x) \\leq i \\leq \\delta(x) + \\dim(\\text{Supp}(\\mathcal{F}_x))$,", "\\item $\\dim(\\text{Supp}(\\mathcal{E}^{i + \\delta(x)}_x)) \\leq i$,", "\\item $\\text{depth}(\\mathcal{F}_x)$ is the smallest integer", "$i \\geq 0$ such that $\\mathcal{E}^{i + \\delta(x)} \\not = 0$, and", "\\item we have", "$x \\in \\text{Supp}(\\bigoplus_{j \\leq i} \\mathcal{E}^j)", "\\Leftrightarrow", "\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) + \\delta(x) \\leq i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Lemma \\ref{lemma-dualizing-schemes} tells us that $\\mathcal{E}^i$", "is coherent. Choosing an affine neighbourhood of $x$ and using", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-internal-hom}", "and", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-base-change-RHom} part (3)", "we have", "$$", "\\mathcal{E}^i_x =", "\\SheafExt^{-i}_{\\mathcal{O}_X}(\\mathcal{F}, \\omega_X^\\bullet)_x =", "\\Ext^{-i}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x,", "\\omega_{X, x}^\\bullet) =", "\\Ext^{\\delta(x) - i}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x,", "\\omega_{X, x}^\\bullet[-\\delta(x)])", "$$", "By construction of $\\delta$ in Lemma \\ref{lemma-dimension-function-scheme}", "this reduces parts (1), (2), and (3) to", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-sitting-in-degrees}.", "Part (4) is a formal consequence of (3) and (1)." ], "refs": [ "duality-lemma-dualizing-schemes", "perfect-lemma-quasi-coherence-internal-hom", "more-algebra-lemma-base-change-RHom", "duality-lemma-dimension-function-scheme", "dualizing-lemma-sitting-in-degrees" ], "ref_ids": [ 13499, 6981, 10418, 13501, 2861 ] } ], "ref_ids": [] }, { "id": 13503, "type": "theorem", "label": "duality-lemma-twisted-inverse-image", "categories": [ "duality" ], "title": "duality-lemma-twisted-inverse-image", "contents": [ "\\begin{reference}", "This is almost the same as \\cite[Example 4.2]{Neeman-Grothendieck}.", "\\end{reference}", "Let $f : X \\to Y$ be a morphism between quasi-separated and quasi-compact", "schemes. The functor $Rf_* : D_\\QCoh(X) \\to D_\\QCoh(Y)$ has a", "right adjoint." ], "refs": [], "proofs": [ { "contents": [ "We will prove a right adjoint exists by verifying the hypotheses of", "Derived Categories, Proposition \\ref{derived-proposition-brown}.", "First off, the category $D_\\QCoh(\\mathcal{O}_X)$ has direct sums, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-direct-sums}.", "The category $D_\\QCoh(\\mathcal{O}_X)$ is compactly generated by", "Derived Categories of Schemes, Theorem", "\\ref{perfect-theorem-bondal-van-den-Bergh}.", "Since $X$ and $Y$ are quasi-compact and quasi-separated, so is $f$, see", "Schemes, Lemmas \\ref{schemes-lemma-compose-after-separated} and", "\\ref{schemes-lemma-quasi-compact-permanence}.", "Hence the functor $Rf_*$ commutes with direct sums, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-pushforward-direct-sums}.", "This finishes the proof." ], "refs": [ "derived-proposition-brown", "perfect-lemma-quasi-coherence-direct-sums", "perfect-theorem-bondal-van-den-Bergh", "schemes-lemma-compose-after-separated", "schemes-lemma-quasi-compact-permanence", "perfect-lemma-quasi-coherence-pushforward-direct-sums" ], "ref_ids": [ 1966, 6937, 6935, 7715, 7716, 6950 ] } ], "ref_ids": [] }, { "id": 13504, "type": "theorem", "label": "duality-lemma-twisted-inverse-image-bounded-below", "categories": [ "duality" ], "title": "duality-lemma-twisted-inverse-image-bounded-below", "contents": [ "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated", "schemes. Let $a : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$", "be the right adjoint to $Rf_*$ of Lemma \\ref{lemma-twisted-inverse-image}.", "Then $a$ maps $D^+_\\QCoh(\\mathcal{O}_Y)$ into $D^+_\\QCoh(\\mathcal{O}_X)$.", "In fact, there exists an integer $N$ such that", "$H^i(K) = 0$ for $i \\leq c$ implies $H^i(a(K)) = 0$ for $i \\leq c - N$." ], "refs": [ "duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-direct-image}", "the functor $Rf_*$ has finite cohomological dimension. In other words,", "there exist an integer $N$ such that", "$H^i(Rf_*L) = 0$ for $i \\geq N + c$ if $H^i(L) = 0$ for $i \\geq c$.", "Say $K \\in D^+_\\QCoh(\\mathcal{O}_Y)$ has $H^i(K) = 0$ for $i \\leq c$.", "Then", "$$", "\\Hom_{D(\\mathcal{O}_X)}(\\tau_{\\leq c - N}a(K), a(K)) =", "\\Hom_{D(\\mathcal{O}_Y)}(Rf_*\\tau_{\\leq c - N}a(K), K) = 0", "$$", "by what we said above. Clearly, this implies that", "$H^i(a(K)) = 0$ for $i \\leq c - N$." ], "refs": [ "perfect-lemma-quasi-coherence-direct-image" ], "ref_ids": [ 6946 ] } ], "ref_ids": [ 13503 ] }, { "id": 13505, "type": "theorem", "label": "duality-lemma-iso-on-RSheafHom", "categories": [ "duality" ], "title": "duality-lemma-iso-on-RSheafHom", "contents": [ "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated schemes.", "Let $a$ be the right adjoint to", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$.", "Let $L \\in D_\\QCoh(\\mathcal{O}_X)$ and $K \\in D_\\QCoh(\\mathcal{O}_Y)$.", "Then the map (\\ref{equation-sheafy-trace})", "$$", "Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K))", "\\longrightarrow", "R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K)", "$$", "becomes an isomorphism after applying the functor", "$DQ_Y : D(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_Y)$", "discussed in Derived Categories of Schemes, Section", "\\ref{perfect-section-better-coherator}." ], "refs": [], "proofs": [ { "contents": [ "The statement makes sense as $DQ_Y$ exists by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-better-coherator}.", "Since $DQ_Y$ is the right adjoint to the inclusion", "functor $D_\\QCoh(\\mathcal{O}_Y) \\to D(\\mathcal{O}_Y)$", "to prove the lemma we have to show that for any $M \\in D_\\QCoh(\\mathcal{O}_Y)$", "the map (\\ref{equation-sheafy-trace}) induces an bijection", "$$", "\\Hom_Y(M, Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K)))", "\\longrightarrow", "\\Hom_Y(M, R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K))", "$$", "To see this we use the following string of equalities", "\\begin{align*}", "\\Hom_Y(M, Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K)))", "& =", "\\Hom_X(Lf^*M, R\\SheafHom_{\\mathcal{O}_X}(L, a(K))) \\\\", "& =", "\\Hom_X(Lf^*M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L, a(K)) \\\\", "& =", "\\Hom_Y(Rf_*(Lf^*M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L), K) \\\\", "& =", "\\Hom_Y(M \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L, K) \\\\", "& =", "\\Hom_Y(M, R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K))", "\\end{align*}", "The first equality holds by Cohomology, Lemma \\ref{cohomology-lemma-adjoint}.", "The second equality by Cohomology, Lemma \\ref{cohomology-lemma-internal-hom}.", "The third equality by construction of $a$.", "The fourth equality by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-cohomology-base-change} (this is the important step).", "The fifth by Cohomology, Lemma \\ref{cohomology-lemma-internal-hom}." ], "refs": [ "perfect-lemma-better-coherator", "cohomology-lemma-adjoint", "cohomology-lemma-internal-hom", "perfect-lemma-cohomology-base-change", "cohomology-lemma-internal-hom" ], "ref_ids": [ 7022, 2121, 2183, 7025, 2183 ] } ], "ref_ids": [] }, { "id": 13506, "type": "theorem", "label": "duality-lemma-iso-global-hom", "categories": [ "duality" ], "title": "duality-lemma-iso-global-hom", "contents": [ "Let $f : X \\to Y$ be a morphism of quasi-separated and quasi-compact", "schemes.", "For all $L \\in D_\\QCoh(\\mathcal{O}_X)$ and $K \\in D_\\QCoh(\\mathcal{O}_Y)$", "(\\ref{equation-sheafy-trace}) induces an isomorphism", "$R\\Hom_X(L, a(K)) \\to R\\Hom_Y(Rf_*L, K)$ of global derived homs." ], "refs": [], "proofs": [ { "contents": [ "By the construction in", "Cohomology, Section \\ref{cohomology-section-global-RHom}", "we have", "$$", "R\\Hom_X(L, a(K)) =", "R\\Gamma(X, R\\SheafHom_{\\mathcal{O}_X}(L, a(K))) =", "R\\Gamma(Y, Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K)))", "$$", "and", "$$", "R\\Hom_Y(Rf_*L, K) = R\\Gamma(Y, R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K))", "$$", "Thus the lemma is a consequence of Lemma \\ref{lemma-iso-on-RSheafHom}.", "Namely, a map $E \\to E'$ in $D(\\mathcal{O}_Y)$ which induces", "an isomorphism $DQ_Y(E) \\to DQ_Y(E')$ induces a quasi-isomorphism", "$R\\Gamma(Y, E) \\to R\\Gamma(Y, E')$. Indeed we have", "$H^i(Y, E) = \\Ext^i_Y(\\mathcal{O}_Y, E) = \\Hom(\\mathcal{O}_Y[-i], E) =", "\\Hom(\\mathcal{O}_Y[-i], DQ_Y(E))$ because $\\mathcal{O}_Y[-i]$", "is in $D_\\QCoh(\\mathcal{O}_Y)$ and $DQ_Y$ is the right adjoint", "to the inclusion functor $D_\\QCoh(\\mathcal{O}_Y) \\to D(\\mathcal{O}_Y)$." ], "refs": [ "duality-lemma-iso-on-RSheafHom" ], "ref_ids": [ 13505 ] } ], "ref_ids": [] }, { "id": 13507, "type": "theorem", "label": "duality-lemma-flat-precompose-pus", "categories": [ "duality" ], "title": "duality-lemma-flat-precompose-pus", "contents": [ "In diagram (\\ref{equation-base-change}) assume that $g$ is flat or", "more generally that $f$ and $g$ are Tor independent. Then", "$a \\circ Rg_* \\leftarrow Rg'_* \\circ a'$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "In this case the base change map", "$Lg^* \\circ Rf_* K \\longrightarrow Rf'_* \\circ L(g')^*K$", "is an isomorphism for every $K$ in $D_\\QCoh(\\mathcal{O}_X)$ by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-compare-base-change}.", "Thus the corresponding transformation between adjoint functors", "is an isomorphism as well." ], "refs": [ "perfect-lemma-compare-base-change" ], "ref_ids": [ 7028 ] } ], "ref_ids": [] }, { "id": 13508, "type": "theorem", "label": "duality-lemma-when-sheafy", "categories": [ "duality" ], "title": "duality-lemma-when-sheafy", "contents": [ "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated", "schemes. Let $a$ be the right adjoint to", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$.", "Let $V \\subset Y$ be quasi-compact open with inverse image $U \\subset X$.", "\\begin{enumerate}", "\\item For every $Q \\in D_\\QCoh^+(\\mathcal{O}_Y)$", "supported on $Y \\setminus V$ the image $a(Q)$ is supported on", "$X \\setminus U$ if and only if (\\ref{equation-sheafy})", "is an isomorphism on all $K$ in $D_\\QCoh^+(\\mathcal{O}_Y)$.", "\\item For every $Q \\in D_\\QCoh(\\mathcal{O}_Y)$", "supported on $Y \\setminus V$ the image $a(Q)$ is supported on", "$X \\setminus U$ if and only if (\\ref{equation-sheafy})", "is an isomorphism on all $K$ in $D_\\QCoh(\\mathcal{O}_Y)$.", "\\item If $a$ commutes with direct sums, then the equivalent conditions of", "(1) imply the equivalent conditions of (2).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Let $K \\in D_\\QCoh^+(\\mathcal{O}_Y)$.", "Choose a distinguished triangle", "$$", "K \\to Rj_*K|_V \\to Q \\to K[1]", "$$", "Observe that $Q$ is in $D_\\QCoh^+(\\mathcal{O}_Y)$", "(Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-direct-image})", "and is supported on $Y \\setminus V$", "(Derived Categories of Schemes, Definition", "\\ref{perfect-definition-supported-on}).", "Applying $a$ we obtain a distinguished triangle", "$$", "a(K) \\to a(Rj_*K|_V) \\to a(Q) \\to a(K)[1]", "$$", "on $X$. If $a(Q)$ is supported on $X \\setminus U$, then", "restricting to $U$ the map $a(K)|_U \\to a(Rj_*K|_V)|_U$ is an", "isomorphism, i.e., (\\ref{equation-sheafy}) is an isomorphism on $K$.", "The converse is immediate.", "\\medskip\\noindent", "The proof of (2) is exactly the same as the proof of (1).", "\\medskip\\noindent", "Proof of (3). Assume the equivalent conditions of (1) hold.", "Set $T = Y \\setminus V$.", "We will use the notation $D_{\\QCoh, T}(\\mathcal{O}_Y)$ and", "$D_{\\QCoh, f^{-1}(T)}(\\mathcal{O}_X)$ to denote complexes", "whose cohomology sheaves are supported on $T$ and $f^{-1}(T)$.", "Since $a$ commutes with direct sums, the strictly full, saturated, triangulated", "subcategory $\\mathcal{D}$ with objects", "$$", "\\{Q \\in D_{\\QCoh, T}(\\mathcal{O}_Y) \\mid", "a(Q) \\in D_{\\QCoh, f^{-1}(T)}(\\mathcal{O}_X)\\}", "$$", "is preserved by direct sums and hence derived colimits.", "On the other hand, the category $D_{\\QCoh, T}(\\mathcal{O}_Y)$", "is generated by a perfect object $E$", "(see Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-generator-with-support}).", "By assumption we see that $E \\in \\mathcal{D}$.", "By Derived Categories, Lemma \\ref{derived-lemma-write-as-colimit}", "every object $Q$ of $D_{\\QCoh, T}(\\mathcal{O}_Y)$ is a derived", "colimit of a system $Q_1 \\to Q_2 \\to Q_3 \\to \\ldots$", "such that the cones of the transition maps are direct sums", "of shifts of $E$. Arguing by induction we see that", "$Q_n \\in \\mathcal{D}$ for all $n$ and finally that $Q$ is", "in $\\mathcal{D}$. Thus the equivalent conditions of (2) hold." ], "refs": [ "perfect-lemma-quasi-coherence-direct-image", "perfect-definition-supported-on", "perfect-lemma-generator-with-support", "derived-lemma-write-as-colimit" ], "ref_ids": [ 6946, 7115, 7011, 1941 ] } ], "ref_ids": [] }, { "id": 13509, "type": "theorem", "label": "duality-lemma-proper-noetherian", "categories": [ "duality" ], "title": "duality-lemma-proper-noetherian", "contents": [ "Let $Y$ be a quasi-compact and quasi-separated scheme.", "Let $f : X \\to Y$ be a proper morphism. If\\footnote{This proof works for those", "morphisms of quasi-compact and quasi-separated schemes such that", "$Rf_*P$ is pseudo-coherent for all $P$ perfect on $X$. It follows", "easily from a theorem of Kiehl \\cite{Kiehl} that this holds if", "$f$ is proper and pseudo-coherent. This is the correct generality", "for this lemma and some of the other results in this chapter.}", "\\begin{enumerate}", "\\item $f$ is flat and of finite presentation, or", "\\item $Y$ is Noetherian", "\\end{enumerate}", "then the equivalent conditions of Lemma \\ref{lemma-when-sheafy} part (1)", "hold for all quasi-compact opens $V$ of $Y$." ], "refs": [ "duality-lemma-when-sheafy" ], "proofs": [ { "contents": [ "Let $Q \\in D^+_\\QCoh(\\mathcal{O}_Y)$ be supported on $Y \\setminus V$.", "To get a contradiction, assume that $a(Q)$ is not supported on", "$X \\setminus U$. Then we can find a perfect complex $P_U$ on $U$", "and a nonzero map $P_U \\to a(Q)|_U$ (follows from", "Derived Categories of Schemes, Theorem", "\\ref{perfect-theorem-bondal-van-den-Bergh}). Then using", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-lift-map-from-perfect-complex-with-support}", "we may assume there is a perfect complex $P$ on $X$ and a map", "$P \\to a(Q)$ whose restriction to $U$ is nonzero.", "By definition of $a$ this map", "is adjoint to a map $Rf_*P \\to Q$.", "\\medskip\\noindent", "The complex $Rf_*P$ is pseudo-coherent. In case (1) this follows", "from Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-pseudo-coherent-direct-image-general}.", "In case (2) this follows from", "Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-direct-image-coherent} and", "\\ref{perfect-lemma-identify-pseudo-coherent-noetherian}.", "Thus we may apply", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-map-from-pseudo-coherent-to-complex-with-support}", "and get a map $I \\to \\mathcal{O}_Y$ of perfect complexes", "whose restriction to $V$ is an isomorphism such that the composition", "$I \\otimes^\\mathbf{L}_{\\mathcal{O}_Y} Rf_*P \\to Rf_*P \\to Q$ is zero.", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-cohomology-base-change}", "we have $I \\otimes^\\mathbf{L}_{\\mathcal{O}_Y} Rf_*P =", "Rf_*(Lf^*I \\otimes^\\mathbf{L}_{\\mathcal{O}_X} P)$.", "We conclude that the composition", "$$", "Lf^*I \\otimes^\\mathbf{L}_{\\mathcal{O}_X} P \\to P \\to a(Q)", "$$", "is zero. However, the restriction to $U$ is the map", "$P|_U \\to a(Q)|_U$ which we assumed to be nonzero.", "This contradiction finishes the proof." ], "refs": [ "perfect-theorem-bondal-van-den-Bergh", "perfect-lemma-lift-map-from-perfect-complex-with-support", "perfect-lemma-flat-proper-pseudo-coherent-direct-image-general", "perfect-lemma-direct-image-coherent", "perfect-lemma-identify-pseudo-coherent-noetherian", "perfect-lemma-map-from-pseudo-coherent-to-complex-with-support", "perfect-lemma-cohomology-base-change" ], "ref_ids": [ 6935, 7005, 7055, 6984, 6976, 7016, 7025 ] } ], "ref_ids": [ 13508 ] }, { "id": 13510, "type": "theorem", "label": "duality-lemma-compose-base-change-maps", "categories": [ "duality" ], "title": "duality-lemma-compose-base-change-maps", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^l \\ar[d]_{g'} & Y \\ar[d]^g \\\\", "Z' \\ar[r]^m & Z", "}", "$$", "of quasi-compact and quasi-separated schemes where", "both diagrams are cartesian and where $f$ and $l$", "as well as $g$ and $m$ are Tor independent.", "Then the maps (\\ref{equation-base-change-map})", "for the two squares compose to give the base", "change map for the outer rectangle (see proof for a precise statement)." ], "refs": [], "proofs": [ { "contents": [ "It follows from the assumptions that $g \\circ f$ and $m$ are Tor", "independent (details omitted), hence the statement makes sense.", "In this proof we write $k^*$ in place of $Lk^*$ and $f_*$ instead", "of $Rf_*$. Let $a$, $b$, and $c$ be the right adjoints of", "Lemma \\ref{lemma-twisted-inverse-image}", "for $f$, $g$, and $g \\circ f$ and similarly for the primed versions.", "The arrow corresponding to the top square is the composition", "$$", "\\gamma_{top} :", "k^* \\circ a \\to k^* \\circ a \\circ l_* \\circ l^*", "\\xleftarrow{\\xi_{top}} k^* \\circ k_* \\circ a' \\circ l^* \\to a' \\circ l^*", "$$", "where $\\xi_{top} : k_* \\circ a' \\to a \\circ l_*$", "is an isomorphism (hence can be inverted)", "and is the arrow ``dual'' to the base change map", "$l^* \\circ f_* \\to f'_* \\circ k^*$. The outer arrows come", "from the canonical maps $1 \\to l_* \\circ l^*$ and $k^* \\circ k_* \\to 1$.", "Similarly for the second square we have", "$$", "\\gamma_{bot} :", "l^* \\circ b \\to l^* \\circ b \\circ m_* \\circ m^*", "\\xleftarrow{\\xi_{bot}} l^* \\circ l_* \\circ b' \\circ m^* \\to b' \\circ m^*", "$$", "For the outer rectangle we get", "$$", "\\gamma_{rect} :", "k^* \\circ c \\to k^* \\circ c \\circ m_* \\circ m^*", "\\xleftarrow{\\xi_{rect}} k^* \\circ k_* \\circ c' \\circ m^* \\to c' \\circ m^*", "$$", "We have $(g \\circ f)_* = g_* \\circ f_*$ and hence", "$c = a \\circ b$ and similarly $c' = a' \\circ b'$.", "The statement of the lemma is that $\\gamma_{rect}$", "is equal to the composition", "$$", "k^* \\circ c = k^* \\circ a \\circ b \\xrightarrow{\\gamma_{top}}", "a' \\circ l^* \\circ b \\xrightarrow{\\gamma_{bot}}", "a' \\circ b' \\circ m^* = c' \\circ m^*", "$$", "To see this we contemplate the following diagram:", "$$", "\\xymatrix{", "& & k^* \\circ a \\circ b \\ar[d] \\ar[lldd] \\\\", "& & k^* \\circ a \\circ l_* \\circ l^* \\circ b \\ar[ld] \\\\", "k^* \\circ a \\circ b \\circ m_* \\circ m^* \\ar[r] &", "k^* \\circ a \\circ l_* \\circ l^* \\circ b \\circ m_* \\circ m^* &", "k^* \\circ k_* \\circ a' \\circ l^* \\circ b \\ar[u]_{\\xi_{top}} \\ar[d] \\ar[ld] \\\\", "& k^*\\circ k_* \\circ a' \\circ l^* \\circ b \\circ m_* \\circ m^*", "\\ar[u]_{\\xi_{top}} \\ar[rd] &", "a' \\circ l^* \\circ b \\ar[d] \\\\", "k^* \\circ k_* \\circ a' \\circ b' \\circ m^* \\ar[uu]_{\\xi_{rect}} \\ar[ddrr] &", "k^*\\circ k_* \\circ a' \\circ l^* \\circ l_* \\circ b' \\circ m^*", "\\ar[u]_{\\xi_{bot}} \\ar[l] \\ar[dr] &", "a' \\circ l^* \\circ b \\circ m_* \\circ m^* \\\\", "& & a' \\circ l^* \\circ l_* \\circ b' \\circ m^* \\ar[u]_{\\xi_{bot}} \\ar[d] \\\\", "& & a' \\circ b' \\circ m^*", "}", "$$", "Going down the right hand side we have the composition and going", "down the left hand side we have $\\gamma_{rect}$.", "All the quadrilaterals on the right hand side of this diagram commute", "by Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats}", "or more simply the discussion preceding", "Categories, Definition \\ref{categories-definition-horizontal-composition}.", "Hence we see that it suffices to show the diagram", "$$", "\\xymatrix{", "a \\circ l_* \\circ l^* \\circ b \\circ m_* &", "a \\circ b \\circ m_* \\ar[l] \\\\", "k_* \\circ a' \\circ l^* \\circ b \\circ m_* \\ar[u]_{\\xi_{top}} & \\\\", "k_* \\circ a' \\circ l^* \\circ l_* \\circ b' \\ar[u]_{\\xi_{bot}} \\ar[r] &", "k_* \\circ a' \\circ b' \\ar[uu]_{\\xi_{rect}}", "}", "$$", "becomes commutative if we invert the arrows $\\xi_{top}$, $\\xi_{bot}$,", "and $\\xi_{rect}$ (note that this is different from asking the", "diagram to be commutative). However, the diagram", "$$", "\\xymatrix{", "& a \\circ l_* \\circ l^* \\circ b \\circ m_* \\\\", "a \\circ l_* \\circ l^* \\circ l_* \\circ b'", "\\ar[ru]^{\\xi_{bot}} & &", "k_* \\circ a' \\circ l^* \\circ b \\circ m_* \\ar[ul]_{\\xi_{top}} \\\\", "& k_* \\circ a' \\circ l^* \\circ l_* \\circ b'", "\\ar[ul]^{\\xi_{top}} \\ar[ur]_{\\xi_{bot}}", "}", "$$", "commutes by Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats}.", "Since the diagrams", "$$", "\\vcenter{", "\\xymatrix{", "a \\circ l_* \\circ l^* \\circ b \\circ m_* & a \\circ b \\circ m \\ar[l] \\\\", "a \\circ l_* \\circ l^* \\circ l_* \\circ b' \\ar[u] &", "a \\circ l_* \\circ b' \\ar[l] \\ar[u]", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "a \\circ l_* \\circ l^* \\circ l_* \\circ b' \\ar[r] & a \\circ l_* \\circ b' \\\\", "k_* \\circ a' \\circ l^* \\circ l_* \\circ b' \\ar[u] \\ar[r] &", "k_* \\circ a' \\circ b' \\ar[u]", "}", "}", "$$", "commute (see references cited) and since the composition of", "$l_* \\to l_* \\circ l^* \\circ l_* \\to l_*$ is the identity,", "we find that it suffices to prove that", "$$", "k \\circ a' \\circ b' \\xrightarrow{\\xi_{bot}} a \\circ l_* \\circ b", "\\xrightarrow{\\xi_{top}} a \\circ b \\circ m_*", "$$", "is equal to $\\xi_{rect}$ (via the identifications $a \\circ b = c$", "and $a' \\circ b' = c'$). This is the statement dual to", "Cohomology, Remark \\ref{cohomology-remark-compose-base-change}", "and the proof is complete." ], "refs": [ "duality-lemma-twisted-inverse-image", "categories-lemma-properties-2-cat-cats", "categories-definition-horizontal-composition", "categories-lemma-properties-2-cat-cats", "cohomology-remark-compose-base-change" ], "ref_ids": [ 13503, 12269, 12377, 12269, 2270 ] } ], "ref_ids": [] }, { "id": 13511, "type": "theorem", "label": "duality-lemma-compose-base-change-maps-horizontal", "categories": [ "duality" ], "title": "duality-lemma-compose-base-change-maps-horizontal", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X'' \\ar[r]_{g'} \\ar[d]_{f''} & X' \\ar[r]_g \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y'' \\ar[r]^{h'} & Y' \\ar[r]^h & Y", "}", "$$", "of quasi-compact and quasi-separated schemes where", "both diagrams are cartesian and where $f$ and $h$", "as well as $f'$ and $h'$ are Tor independent.", "Then the maps (\\ref{equation-base-change-map})", "for the two squares compose to give the base", "change map for the outer rectangle (see proof for a precise statement)." ], "refs": [], "proofs": [ { "contents": [ "It follows from the assumptions that $f$ and $h \\circ h'$ are Tor", "independent (details omitted), hence the statement makes sense.", "In this proof we write $g^*$ in place of $Lg^*$ and $f_*$ instead", "of $Rf_*$. Let $a$, $a'$, and $a''$ be the right adjoints of", "Lemma \\ref{lemma-twisted-inverse-image}", "for $f$, $f'$, and $f''$. The arrow corresponding to the right", "square is the composition", "$$", "\\gamma_{right} :", "g^* \\circ a \\to g^* \\circ a \\circ h_* \\circ h^*", "\\xleftarrow{\\xi_{right}} g^* \\circ g_* \\circ a' \\circ h^* \\to a' \\circ h^*", "$$", "where $\\xi_{right} : g_* \\circ a' \\to a \\circ h_*$", "is an isomorphism (hence can be inverted)", "and is the arrow ``dual'' to the base change map", "$h^* \\circ f_* \\to f'_* \\circ g^*$. The outer arrows come", "from the canonical maps $1 \\to h_* \\circ h^*$ and $g^* \\circ g_* \\to 1$.", "Similarly for the left square we have", "$$", "\\gamma_{left} :", "(g')^* \\circ a' \\to (g')^* \\circ a' \\circ (h')_* \\circ (h')^*", "\\xleftarrow{\\xi_{left}}", "(g')^* \\circ (g')_* \\circ a'' \\circ (h')^* \\to a'' \\circ (h')^*", "$$", "For the outer rectangle we get", "$$", "\\gamma_{rect} :", "k^* \\circ a \\to", "k^* \\circ a \\circ m_* \\circ m^* \\xleftarrow{\\xi_{rect}}", "k^* \\circ k_* \\circ a'' \\circ m^* \\to", "a'' \\circ m^*", "$$", "where $k = g \\circ g'$ and $m = h \\circ h'$.", "We have $k^* = (g')^* \\circ g^*$ and $m^* = (h')^* \\circ h^*$.", "The statement of the lemma is that $\\gamma_{rect}$", "is equal to the composition", "$$", "k^* \\circ a =", "(g')^* \\circ g^* \\circ a \\xrightarrow{\\gamma_{right}}", "(g')^* \\circ a' \\circ h^* \\xrightarrow{\\gamma_{left}}", "a'' \\circ (h')^* \\circ h^* = a'' \\circ m^*", "$$", "To see this we contemplate the following diagram", "$$", "\\xymatrix{", "& (g')^* \\circ g^* \\circ a \\ar[d] \\ar[ddl] \\\\", "& (g')^* \\circ g^* \\circ a \\circ h_* \\circ h^* \\ar[ld] \\\\", "(g')^* \\circ g^* \\circ a \\circ h_* \\circ (h')_* \\circ (h')^* \\circ h^* &", "(g')^* \\circ g^* \\circ g_* \\circ a' \\circ h^*", "\\ar[u]_{\\xi_{right}} \\ar[d] \\ar[ld] \\\\", "(g')^* \\circ g^* \\circ g_* \\circ a' \\circ (h')_* \\circ (h')^* \\circ h^*", "\\ar[u]_{\\xi_{right}} \\ar[dr] &", "(g')^* \\circ a' \\circ h^* \\ar[d] \\\\", "(g')^* \\circ g^* \\circ g_* \\circ (g')_* \\circ a'' \\circ (h')^* \\circ h^*", "\\ar[u]_{\\xi_{left}} \\ar[ddr] \\ar[dr] &", "(g')^* \\circ a' \\circ (h')_* \\circ (h')^* \\circ h^* \\\\", "& (g')^*\\circ (g')_* \\circ a'' \\circ (h')^* \\circ h^*", "\\ar[u]_{\\xi_{left}} \\ar[d] \\\\", "& a'' \\circ (h')^* \\circ h^*", "}", "$$", "Going down the right hand side we have the composition and going", "down the left hand side we have $\\gamma_{rect}$.", "All the quadrilaterals on the right hand side of this diagram commute", "by Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats}", "or more simply the discussion preceding", "Categories, Definition \\ref{categories-definition-horizontal-composition}.", "Hence we see that it suffices to show that", "$$", "g_* \\circ (g')_* \\circ a'' \\xrightarrow{\\xi_{left}}", "g_* \\circ a' \\circ (h')_* \\xrightarrow{\\xi_{right}}", "a \\circ h_* \\circ (h')_*", "$$", "is equal to $\\xi_{rect}$. This is the statement dual to", "Cohomology, Remark \\ref{cohomology-remark-compose-base-change-horizontal}", "and the proof is complete." ], "refs": [ "duality-lemma-twisted-inverse-image", "categories-lemma-properties-2-cat-cats", "categories-definition-horizontal-composition", "cohomology-remark-compose-base-change-horizontal" ], "ref_ids": [ 13503, 12269, 12377, 2271 ] } ], "ref_ids": [] }, { "id": 13512, "type": "theorem", "label": "duality-lemma-more-base-change", "categories": [ "duality" ], "title": "duality-lemma-more-base-change", "contents": [ "In diagram (\\ref{equation-base-change}) assume", "\\begin{enumerate}", "\\item $g : Y' \\to Y$ is a morphism of affine schemes,", "\\item $f : X \\to Y$ is proper, and", "\\item $f$ and $g$ are Tor independent.", "\\end{enumerate}", "Then the base change map (\\ref{equation-base-change-map}) induces an", "isomorphism", "$$", "L(g')^*a(K) \\longrightarrow a'(Lg^*K)", "$$", "in the following cases", "\\begin{enumerate}", "\\item for all $K \\in D_\\QCoh(\\mathcal{O}_X)$ if $f$", "is flat of finite presentation,", "\\item for all $K \\in D_\\QCoh(\\mathcal{O}_X)$ if $f$", "is perfect and $Y$ Noetherian,", "\\item for $K \\in D_\\QCoh^+(\\mathcal{O}_X)$ if $g$ has finite Tor dimension", "and $Y$ Noetherian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $Y = \\Spec(A)$ and $Y' = \\Spec(A')$. As a base change of an affine", "morphism, the morphism $g'$ is affine. Let $M$ be a perfect generator", "for $D_\\QCoh(\\mathcal{O}_X)$, see Derived Categories of Schemes, Theorem", "\\ref{perfect-theorem-bondal-van-den-Bergh}. Then $L(g')^*M$ is a", "generator for $D_\\QCoh(\\mathcal{O}_{X'})$, see", "Derived Categories of Schemes, Remark \\ref{perfect-remark-pullback-generator}.", "Hence it suffices to show that (\\ref{equation-base-change-map})", "induces an isomorphism", "\\begin{equation}", "\\label{equation-iso}", "R\\Hom_{X'}(L(g')^*M, L(g')^*a(K))", "\\longrightarrow", "R\\Hom_{X'}(L(g')^*M, a'(Lg^*K))", "\\end{equation}", "of global hom complexes, see", "Cohomology, Section \\ref{cohomology-section-global-RHom},", "as this will imply the cone of $L(g')^*a(K) \\to a'(Lg^*K)$", "is zero.", "The structure of the proof is as follows: we will first show that", "these Hom complexes are isomorphic and in the last part of the proof", "we will show that the isomorphism is induced by (\\ref{equation-iso}).", "\\medskip\\noindent", "The left hand side. Because $M$ is perfect, the canonical map", "$$", "R\\Hom_X(M, a(K)) \\otimes^\\mathbf{L}_A A'", "\\longrightarrow", "R\\Hom_{X'}(L(g')^*M, L(g')^*a(K))", "$$", "is an isomorphism by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-morphism-and-hom-out-of-perfect}.", "We can combine this with the isomorphism", "$R\\Hom_Y(Rf_*M, K) = R\\Hom_X(M, a(K))$", "of Lemma \\ref{lemma-iso-global-hom}", "to get that the left hand side equals", "$R\\Hom_Y(Rf_*M, K) \\otimes^\\mathbf{L}_A A'$.", "\\medskip\\noindent", "The right hand side. Here we first use the isomorphism", "$$", "R\\Hom_{X'}(L(g')^*M, a'(Lg^*K)) = R\\Hom_{Y'}(Rf'_*L(g')^*M, Lg^*K)", "$$", "of Lemma \\ref{lemma-iso-global-hom}. Then we use the base change", "map $Lg^*Rf_*M \\to Rf'_*L(g')^*M$ is an isomorphism by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-compare-base-change}.", "Hence we may rewrite this as $R\\Hom_{Y'}(Lg^*Rf_*M, Lg^*K)$.", "Since $Y$, $Y'$ are affine and $K$, $Rf_*M$ are in $D_\\QCoh(\\mathcal{O}_Y)$", "(Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-direct-image})", "we have a canonical map", "$$", "\\beta :", "R\\Hom_Y(Rf_*M, K) \\otimes^\\mathbf{L}_A A'", "\\longrightarrow", "R\\Hom_{Y'}(Lg^*Rf_*M, Lg^*K)", "$$", "in $D(A')$. This is the arrow", "More on Algebra, Equation (\\ref{more-algebra-equation-base-change-RHom})", "where we have used Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-affine-compare-bounded} and", "\\ref{perfect-lemma-quasi-coherence-internal-hom}", "to translate back and forth into algebra.", "\\begin{enumerate}", "\\item If $f$ is flat and of finite presentation, the complex $Rf_*M$", "is perfect on $Y$ by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image-general}", "and $\\beta$ is an isomorphism by", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom} part (1).", "\\item If $f$ is perfect and $Y$ Noetherian, the complex $Rf_*M$", "is perfect on $Y$ by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-perfect-proper-perfect-direct-image}", "and $\\beta$ is an isomorphism as before.", "\\item If $g$ has finite tor dimension and $Y$ is Noetherian,", "the complex $Rf_*M$ is pseudo-coherent on $Y$", "(Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-direct-image-coherent} and", "\\ref{perfect-lemma-identify-pseudo-coherent-noetherian})", "and $\\beta$ is an isomorphism by", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-RHom} part (4).", "\\end{enumerate}", "We conclude that we obtain the same answer as in the previous paragraph.", "\\medskip\\noindent", "In the rest of the proof we show that the identifications of", "the left and right hand side of (\\ref{equation-iso})", "given in the second and third paragraph are in fact given by", "(\\ref{equation-iso}). To make our formulas manageable", "we will use $(-, -)_X = R\\Hom_X(-, -)$, use $- \\otimes A'$", "in stead of $- \\otimes_A^\\mathbf{L} A'$, and we will abbreviate", "$g^* = Lg^*$ and $f_* = Rf_*$. Consider the following", "commutative diagram", "$$", "\\xymatrix{", "((g')^*M, (g')^*a(K))_{X'} \\ar[d] &", "(M, a(K))_X \\otimes A' \\ar[l]^-\\alpha \\ar[d] &", "(f_*M, K)_Y \\otimes A' \\ar@{=}[l] \\ar[d] \\\\", "((g')^*M, (g')^*a(g_*g^*K))_{X'} &", "(M, a(g_*g^*K))_X \\otimes A' \\ar[l]^-\\alpha &", "(f_*M, g_*g^*K)_Y \\otimes A' \\ar@{=}[l] \\ar@/_4pc/[dd]_{\\mu'} \\\\", "((g')^*M, (g')^*g'_*a'(g^*K))_{X'} \\ar[u] \\ar[d] &", "(M, g'_*a'(g^*K))_X \\otimes A' \\ar[u] \\ar[l]^-\\alpha \\ar[ld]^\\mu &", "(f_*M, K) \\otimes A' \\ar[d]^\\beta \\\\", "((g')^*M, a'(g^*K))_{X'} &", "(f'_*(g')^*M, g^*K)_{Y'} \\ar@{=}[l] \\ar[r] &", "(g^*f_*M, g^*K)_{Y'}", "}", "$$", "The arrows labeled $\\alpha$ are the maps from", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-morphism-and-hom-out-of-perfect}", "for the diagram with corners $X', X, Y', Y$.", "The upper part of the diagram is commutative as the horizontal arrows are", "functorial in the entries.", "The middle vertical arrows come from the invertible transformation", "$g'_* \\circ a' \\to a \\circ g_*$ of Lemma \\ref{lemma-flat-precompose-pus}", "and therefore the middle square is commutative.", "Going down the left hand side is (\\ref{equation-iso}).", "The upper horizontal arrows provide the identifications used in the", "second paragraph of the proof.", "The lower horizontal arrows including $\\beta$ provide the identifications", "used in the third paragraph of the proof. Given $E \\in D(A)$,", "$E' \\in D(A')$, and $c : E \\to E'$ in $D(A)$ we will denote", "$\\mu_c : E \\otimes A' \\to E'$ the map induced by $c$", "and the adjointness of restriction and base change;", "if $c$ is clear we write $\\mu = \\mu_c$, i.e., we", "drop $c$ from the notation. The map $\\mu$ in the diagram is of this", "form with $c$ given by the identification", "$(M, g'_*a(g^*K))_X = ((g')^*M, a'(g^*K))_{X'}$", "; the triangle involving $\\mu$ is commutative by", "Derived Categories of Schemes, Remark \\ref{perfect-remark-multiplication-map}.", "\\medskip\\noindent", "Observe that", "$$", "\\xymatrix{", "(M, a(g_*g^*K))_X &", "(f_*M, g_* g^*K)_Y \\ar@{=}[l] &", "(g^*f_*M, g^*K)_{Y'} \\ar@{=}[l] \\\\", "(M, g'_* a'(g^*K))_X \\ar[u] &", "((g')^*M, a'(g^*K))_{X'} \\ar@{=}[l] &", "(f'_*(g')^*M, g^*K)_{Y'} \\ar@{=}[l] \\ar[u]", "}", "$$", "is commutative by the very definition of the transformation", "$g'_* \\circ a' \\to a \\circ g_*$. Letting $\\mu'$ be as above", "corresponding to the identification", "$(f_*M, g_*g^*K)_X = (g^*f_*M, g^*K)_{Y'}$, then the", "hexagon commutes as well. Thus it suffices to show that", "$\\beta$ is equal to the composition of", "$(f_*M, K)_Y \\otimes A' \\to (f_*M, g_*g^*K)_X \\otimes A'$", "and $\\mu'$. To do this, it suffices to prove the two induced maps", "$(f_*M, K)_Y \\to (g^*f_*M, g^*K)_{Y'}$ are the same.", "In other words, it suffices to show the diagram", "$$", "\\xymatrix{", "R\\Hom_A(E, K) \\ar[rr]_{\\text{induced by }\\beta} \\ar[rd] & &", "R\\Hom_{A'}(E \\otimes_A^\\mathbf{L} A', K \\otimes_A^\\mathbf{L} A') \\\\", "& R\\Hom_A(E, K \\otimes_A^\\mathbf{L} A') \\ar[ru]", "}", "$$", "commutes for all $E, K \\in D(A)$. Since this is how $\\beta$ is constructed in", "More on Algebra, Section \\ref{more-algebra-section-base-change-RHom}", "the proof is complete." ], "refs": [ "perfect-theorem-bondal-van-den-Bergh", "perfect-remark-pullback-generator", "perfect-lemma-affine-morphism-and-hom-out-of-perfect", "duality-lemma-iso-global-hom", "duality-lemma-iso-global-hom", "perfect-lemma-compare-base-change", "perfect-lemma-quasi-coherence-direct-image", "perfect-lemma-affine-compare-bounded", "perfect-lemma-quasi-coherence-internal-hom", "perfect-lemma-flat-proper-perfect-direct-image-general", "more-algebra-lemma-base-change-RHom", "more-morphisms-lemma-perfect-proper-perfect-direct-image", "perfect-lemma-direct-image-coherent", "perfect-lemma-identify-pseudo-coherent-noetherian", "more-algebra-lemma-base-change-RHom", "perfect-lemma-affine-morphism-and-hom-out-of-perfect", "duality-lemma-flat-precompose-pus", "perfect-remark-multiplication-map" ], "ref_ids": [ 6935, 7128, 7029, 13506, 13506, 7028, 6946, 6941, 6981, 7054, 10418, 13997, 6984, 6976, 10418, 7029, 13507, 7134 ] } ], "ref_ids": [] }, { "id": 13513, "type": "theorem", "label": "duality-lemma-trace-map-and-base-change", "categories": [ "duality" ], "title": "duality-lemma-trace-map-and-base-change", "contents": [ "Suppose we have a diagram (\\ref{equation-base-change}) where $f$ and $g$", "are tor independent. Then the maps", "$1 \\star \\text{Tr}_f : Lg^* \\circ Rf_* \\circ a \\to Lg^*$ and", "$\\text{Tr}_{f'} \\star 1 : Rf'_* \\circ a' \\circ Lg^* \\to Lg^*$", "agree via the base change maps", "$\\beta : Lg^* \\circ Rf_* \\to Rf'_* \\circ L(g')^*$", "(Cohomology, Remark \\ref{cohomology-remark-base-change})", "and $\\alpha : L(g')^* \\circ a \\to a' \\circ Lg^*$", "(\\ref{equation-base-change-map}).", "More precisely, the diagram", "$$", "\\xymatrix{", "Lg^* \\circ Rf_* \\circ a", "\\ar[d]_{\\beta \\star 1} \\ar[r]_-{1 \\star \\text{Tr}_f} &", "Lg^* \\\\", "Rf'_* \\circ L(g')^* \\circ a \\ar[r]^{1 \\star \\alpha} &", "Rf'_* \\circ a' \\circ Lg^* \\ar[u]_{\\text{Tr}_{f'} \\star 1}", "}", "$$", "of transformations of functors commutes." ], "refs": [ "cohomology-remark-base-change" ], "proofs": [ { "contents": [ "In this proof we write $f_*$ for $Rf_*$ and $g^*$ for $Lg^*$ and we", "drop $\\star$ products with identities as one can figure out which ones", "to add as long as the source and target of the transformation is known.", "Recall that $\\beta : g^* \\circ f_* \\to f'_* \\circ (g')^*$ is an isomorphism", "and that $\\alpha$ is defined using", "the isomorphism $\\beta^\\vee : g'_* \\circ a' \\to a \\circ g_*$", "which is the adjoint of $\\beta$, see Lemma \\ref{lemma-flat-precompose-pus}", "and its proof. First we note that the top horizontal arrow", "of the diagram in the lemma is equal to the composition", "$$", "g^* \\circ f_* \\circ a \\to", "g^* \\circ f_* \\circ a \\circ g_* \\circ g^* \\to", "g^* \\circ g_* \\circ g^* \\to g^*", "$$", "where the first arrow is the unit for $(g^*, g_*)$, the second arrow", "is $\\text{Tr}_f$, and the third arrow is the counit for $(g^*, g_*)$.", "This is a simple consequence of the fact that the composition", "$g^* \\to g^* \\circ g_* \\circ g^* \\to g^*$ of unit and counit is the identity.", "Consider the diagram", "$$", "\\xymatrix{", "& g^* \\circ f_* \\circ a \\ar[ld]_\\beta \\ar[d] \\ar[r]_{\\text{Tr}_f} & g^* \\\\", "f'_* \\circ (g')^* \\circ a \\ar[dr] &", "g^* \\circ f_* \\circ a \\circ g_* \\circ g^* \\ar[d]_\\beta \\ar[ru] &", "g^* \\circ f_* \\circ g'_* \\circ a' \\circ g^* \\ar[l]_{\\beta^\\vee} \\ar[d]_\\beta &", "f'_* \\circ a' \\circ g^* \\ar[lu]_{\\text{Tr}_{f'}} \\\\", "& f'_* \\circ (g')^* \\circ a \\circ g_* \\circ g^* &", "f'_* \\circ (g')^* \\circ g'_* \\circ a' \\circ g^* \\ar[ru] \\ar[l]_{\\beta^\\vee}", "}", "$$", "In this diagram the two squares commute ", "Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats}", "or more simply the discussion preceding", "Categories, Definition \\ref{categories-definition-horizontal-composition}.", "The triangle commutes by the discussion above. By", "Categories, Lemma", "\\ref{categories-lemma-transformation-between-functors-and-adjoints}", "the square", "$$", "\\xymatrix{", "g^* \\circ f_* \\circ g'_* \\circ a' \\ar[d]_{\\beta^\\vee} \\ar[r]_-\\beta &", "f'_* \\circ (g')^* \\circ g'_* \\circ a' \\ar[d] \\\\", "g^* \\circ f_* \\circ a \\circ g_* \\ar[r] &", "\\text{id}", "}", "$$", "commutes which implies the pentagon in the big diagram commutes.", "Since $\\beta$ and $\\beta^\\vee$ are isomorphisms, and since going on", "the outside of the big diagram equals", "$\\text{Tr}_f \\circ \\alpha \\circ \\beta$ by definition this proves the lemma." ], "refs": [ "duality-lemma-flat-precompose-pus", "categories-lemma-properties-2-cat-cats", "categories-definition-horizontal-composition", "categories-lemma-transformation-between-functors-and-adjoints" ], "ref_ids": [ 13507, 12269, 12377, 12251 ] } ], "ref_ids": [ 2269 ] }, { "id": 13514, "type": "theorem", "label": "duality-lemma-unit-and-base-change", "categories": [ "duality" ], "title": "duality-lemma-unit-and-base-change", "contents": [ "Suppose we have a diagram (\\ref{equation-base-change}) where $f$ and $g$", "are tor independent. Then the maps", "$1 \\star \\eta_f : L(g')^* \\to L(g')^* \\circ a \\circ Rf_*$ and", "$\\eta_{f'} \\star 1 : L(g')^* \\to a' \\circ Rf'_* \\circ L(g')^*$", "agree via the base change maps", "$\\beta : Lg^* \\circ Rf_* \\to Rf'_* \\circ L(g')^*$", "(Cohomology, Remark \\ref{cohomology-remark-base-change})", "and $\\alpha : L(g')^* \\circ a \\to a' \\circ Lg^*$", "(\\ref{equation-base-change-map}).", "More precisely, the diagram", "$$", "\\xymatrix{", "L(g')^* \\ar[r]_-{1 \\star \\eta_f} \\ar[d]_{\\eta_{f'} \\star 1} &", "L(g')^* \\circ a \\circ Rf_* \\ar[d]^\\alpha \\\\", "a' \\circ Rf'_* \\circ L(g')^* &", "a' \\circ Lg^* \\circ Rf_* \\ar[l]_-\\beta", "}", "$$", "of transformations of functors commutes." ], "refs": [ "cohomology-remark-base-change" ], "proofs": [ { "contents": [ "This proof is dual to the proof of Lemma \\ref{lemma-trace-map-and-base-change}.", "In this proof we write $f_*$ for $Rf_*$ and $g^*$ for $Lg^*$ and we", "drop $\\star$ products with identities as one can figure out which ones", "to add as long as the source and target of the transformation is known.", "Recall that $\\beta : g^* \\circ f_* \\to f'_* \\circ (g')^*$ is an isomorphism", "and that $\\alpha$ is defined using", "the isomorphism $\\beta^\\vee : g'_* \\circ a' \\to a \\circ g_*$", "which is the adjoint of $\\beta$, see Lemma \\ref{lemma-flat-precompose-pus}", "and its proof. First we note that the left vertical arrow", "of the diagram in the lemma is equal to the composition", "$$", "(g')^* \\to (g')^* \\circ g'_* \\circ (g')^* \\to", "(g')^* \\circ g'_* \\circ a' \\circ f'_* \\circ (g')^* \\to", "a' \\circ f'_* \\circ (g')^*", "$$", "where the first arrow is the unit for $((g')^*, g'_*)$, the second arrow", "is $\\eta_{f'}$, and the third arrow is the counit for $((g')^*, g'_*)$.", "This is a simple consequence of the fact that the composition", "$(g')^* \\to (g')^* \\circ (g')_* \\circ (g')^* \\to (g')^*$", "of unit and counit is the identity. Consider the diagram", "$$", "\\xymatrix{", "& (g')^* \\circ a \\circ f_* \\ar[r] &", "(g')^* \\circ a \\circ g_* \\circ g^* \\circ f_*", "\\ar[ld]_\\beta \\\\", "(g')^* \\ar[ru]^{\\eta_f} \\ar[dd]_{\\eta_{f'}} \\ar[rd] &", "(g')^* \\circ a \\circ g_* \\circ f'_* \\circ (g')^* &", "(g')^* \\circ g'_* \\circ a' \\circ g^* \\circ f_*", "\\ar[u]_{\\beta^\\vee} \\ar[ld]_\\beta \\ar[d] \\\\", "& (g')^* \\circ g'_* \\circ a' \\circ f'_* \\circ (g')^*", "\\ar[ld] \\ar[u]_{\\beta^\\vee} &", "a' \\circ g^* \\circ f_* \\ar[lld]^\\beta \\\\", "a' \\circ f'_* \\circ (g')^*", "}", "$$", "In this diagram the two squares commute ", "Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats}", "or more simply the discussion preceding", "Categories, Definition \\ref{categories-definition-horizontal-composition}.", "The triangle commutes by the discussion above. By the dual of", "Categories, Lemma", "\\ref{categories-lemma-transformation-between-functors-and-adjoints}", "the square", "$$", "\\xymatrix{", "\\text{id} \\ar[r] \\ar[d] &", "g'_* \\circ a' \\circ g^* \\circ f_* \\ar[d]^\\beta \\\\", "g'_* \\circ a' \\circ g^* \\circ f_* \\ar[r]^{\\beta^\\vee} &", "a \\circ g_* \\circ f'_* \\circ (g')^*", "}", "$$", "commutes which implies the pentagon in the big diagram commutes.", "Since $\\beta$ and $\\beta^\\vee$ are isomorphisms, and since going on", "the outside of the big diagram equals", "$\\beta \\circ \\alpha \\circ \\eta_f$ by definition this proves the lemma." ], "refs": [ "duality-lemma-trace-map-and-base-change", "duality-lemma-flat-precompose-pus", "categories-lemma-properties-2-cat-cats", "categories-definition-horizontal-composition", "categories-lemma-transformation-between-functors-and-adjoints" ], "ref_ids": [ 13513, 13507, 12269, 12377, 12251 ] } ], "ref_ids": [ 2269 ] }, { "id": 13515, "type": "theorem", "label": "duality-lemma-compare-with-pullback-perfect", "categories": [ "duality" ], "title": "duality-lemma-compare-with-pullback-perfect", "contents": [ "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated", "schemes. The map", "$Lf^*K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} a(L) \\to", "a(K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} L)$", "defined above for $K, L \\in D_\\QCoh(\\mathcal{O}_Y)$", "is an isomorphism if $K$ is perfect. In particular,", "(\\ref{equation-compare-with-pullback}) is an isomorphism if $K$ is perfect." ], "refs": [], "proofs": [ { "contents": [ "Let $K^\\vee$ be the ``dual'' to $K$, see", "Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "For $M \\in D_\\QCoh(\\mathcal{O}_X)$ we have", "\\begin{align*}", "\\Hom_{D(\\mathcal{O}_Y)}(Rf_*M, K \\otimes^\\mathbf{L}_{\\mathcal{O}_Y} L)", "& =", "\\Hom_{D(\\mathcal{O}_Y)}(", "Rf_*M \\otimes^\\mathbf{L}_{\\mathcal{O}_Y} K^\\vee, L) \\\\", "& =", "\\Hom_{D(\\mathcal{O}_X)}(", "M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} Lf^*K^\\vee, a(L)) \\\\", "& =", "\\Hom_{D(\\mathcal{O}_X)}(M,", "Lf^*K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} a(L))", "\\end{align*}", "Second equality by the definition of $a$ and the projection formula", "(Cohomology, Lemma \\ref{cohomology-lemma-projection-formula-perfect})", "or the more general Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-cohomology-base-change}.", "Hence the result by the Yoneda lemma." ], "refs": [ "cohomology-lemma-dual-perfect-complex", "cohomology-lemma-projection-formula-perfect", "perfect-lemma-cohomology-base-change" ], "ref_ids": [ 2233, 2244, 7025 ] } ], "ref_ids": [] }, { "id": 13516, "type": "theorem", "label": "duality-lemma-restriction-compare-with-pullback", "categories": [ "duality" ], "title": "duality-lemma-restriction-compare-with-pullback", "contents": [ "Suppose we have a diagram (\\ref{equation-base-change}) where $f$ and $g$", "are tor independent. Let $K \\in D_\\QCoh(\\mathcal{O}_Y)$. The diagram", "$$", "\\xymatrix{", "L(g')^*(Lf^*K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} a(\\mathcal{O}_Y))", "\\ar[r] \\ar[d] & L(g')^*a(K) \\ar[d] \\\\", "L(f')^*Lg^*K \\otimes_{\\mathcal{O}_{X'}}^\\mathbf{L} a'(\\mathcal{O}_{Y'})", "\\ar[r] & a'(Lg^*K)", "}", "$$", "commutes where the horizontal arrows are the maps", "(\\ref{equation-compare-with-pullback}) for $K$ and $Lg^*K$", "and the vertical maps are constructed using", "Cohomology, Remark \\ref{cohomology-remark-base-change} and", "(\\ref{equation-base-change-map})." ], "refs": [ "cohomology-remark-base-change" ], "proofs": [ { "contents": [ "In this proof we will write $f_*$ for $Rf_*$ and $f^*$ for $Lf^*$, etc,", "and we will write $\\otimes$ for $\\otimes^\\mathbf{L}_{\\mathcal{O}_X}$, etc.", "Let us write (\\ref{equation-compare-with-pullback}) as the composition", "\\begin{align*}", "f^*K \\otimes a(\\mathcal{O}_Y)", "& \\to", "a(f_*(f^*K \\otimes a(\\mathcal{O}_Y))) \\\\", "& \\leftarrow", "a(K \\otimes f_*a(\\mathcal{O}_K)) \\\\", "& \\to", "a(K \\otimes \\mathcal{O}_Y) \\\\", "& \\to", "a(K)", "\\end{align*}", "Here the first arrow is the unit $\\eta_f$, the second arrow is $a$", "applied to Cohomology, Equation", "(\\ref{cohomology-equation-projection-formula-map}) which is an", "isomorphism by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-cohomology-base-change}, the third arrow is", "$a$ applied to $\\text{id}_K \\otimes \\text{Tr}_f$, and the fourth", "arrow is $a$ applied to the isomorphism $K \\otimes \\mathcal{O}_Y = K$.", "The proof of the lemma consists in showing that each of these", "maps gives rise to a commutative square as in the statement of the lemma.", "For $\\eta_f$ and $\\text{Tr}_f$ this is", "Lemmas \\ref{lemma-unit-and-base-change} and", "\\ref{lemma-trace-map-and-base-change}.", "For the arrow using Cohomology, Equation", "(\\ref{cohomology-equation-projection-formula-map})", "this is Cohomology, Remark \\ref{cohomology-remark-compatible-with-diagram}.", "For the multiplication map it is clear. This finishes the proof." ], "refs": [ "perfect-lemma-cohomology-base-change", "duality-lemma-unit-and-base-change", "duality-lemma-trace-map-and-base-change", "cohomology-remark-compatible-with-diagram" ], "ref_ids": [ 7025, 13514, 13513, 2285 ] } ], "ref_ids": [ 2269 ] }, { "id": 13517, "type": "theorem", "label": "duality-lemma-compare-on-open", "categories": [ "duality" ], "title": "duality-lemma-compare-on-open", "contents": [ "Let $f : X \\to Y$ be a proper morphism of Noetherian schemes. Let $V \\subset Y$", "be an open such that $f^{-1}(V) \\to V$ is an isomorphism. Then for", "$K \\in D_\\QCoh^+(\\mathcal{O}_Y)$ the map (\\ref{equation-compare-with-pullback})", "restricts to an isomorphism over $f^{-1}(V)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-proper-noetherian} the map (\\ref{equation-sheafy}) is an", "isomorphism for objects of $D_\\QCoh^+(\\mathcal{O}_Y)$. Hence", "Lemma \\ref{lemma-restriction-compare-with-pullback} tells us the", "restriction of (\\ref{equation-compare-with-pullback}) for $K$", "to $f^{-1}(V)$ is the map (\\ref{equation-compare-with-pullback})", "for $K|_V$ and $f^{-1}(V) \\to V$. Thus it suffices to show that", "the map is an isomorphism when $f$ is the identity morphism. This is clear." ], "refs": [ "duality-lemma-proper-noetherian", "duality-lemma-restriction-compare-with-pullback" ], "ref_ids": [ 13509, 13516 ] } ], "ref_ids": [] }, { "id": 13518, "type": "theorem", "label": "duality-lemma-transitivity-compare-with-pullback", "categories": [ "duality" ], "title": "duality-lemma-transitivity-compare-with-pullback", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be composable morphisms of quasi-compact", "and quasi-separated schemes and set $h = g \\circ f$. Let $a, b, c$ be the", "adjoints of Lemma \\ref{lemma-twisted-inverse-image} for $f, g, h$.", "For any $K \\in D_\\QCoh(\\mathcal{O}_Z)$ the diagram", "$$", "\\xymatrix{", "Lf^*(Lg^*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "b(\\mathcal{O}_Z)) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} a(\\mathcal{O}_Y)", "\\ar@{=}[d] \\ar[r] &", "a(Lg^*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} b(\\mathcal{O}_Z)) \\ar[r] &", "a(b(K)) \\ar@{=}[d] \\\\", "Lh^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*b(\\mathcal{O}_Z)", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L} a(\\mathcal{O}_Y) \\ar[r] &", "Lh^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} c(\\mathcal{O}_Z) \\ar[r] &", "c(K)", "}", "$$", "is commutative where the arrows are (\\ref{equation-compare-with-pullback})", "and we have used $Lh^* = Lf^* \\circ Lg^*$ and $c = a \\circ b$." ], "refs": [ "duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "In this proof we will write $f_*$ for $Rf_*$ and $f^*$ for $Lf^*$, etc,", "and we will write $\\otimes$ for $\\otimes^\\mathbf{L}_{\\mathcal{O}_X}$, etc.", "The composition of the top arrows is adjoint to a map", "$$", "g_*f_*(f^*(g^*K \\otimes b(\\mathcal{O}_Z)) \\otimes a(\\mathcal{O}_Y)) \\to K", "$$", "The left hand side is equal to", "$K \\otimes g_*f_*(f^*b(\\mathcal{O}_Z) \\otimes a(\\mathcal{O}_Y))$ by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-cohomology-base-change}", "and inspection of the definitions shows the map comes from the map", "$$", "g_*f_*(f^*b(\\mathcal{O}_Z) \\otimes a(\\mathcal{O}_Y))", "\\xleftarrow{g_*\\epsilon}", "g_*(b(\\mathcal{O}_Z) \\otimes f_*a(\\mathcal{O}_Y)) \\xrightarrow{g_*\\alpha}", "g_*(b(\\mathcal{O}_Z)) \\xrightarrow{\\beta} \\mathcal{O}_Z", "$$", "tensored with $\\text{id}_K$. Here $\\epsilon$ is the isomorphism from", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-cohomology-base-change} and", "$\\beta$ comes from the counit map", "$g_*b \\to \\text{id}$. Similarly, the composition of the lower", "horizontal arrows is adjoint to $\\text{id}_K$ tensored with the composition ", "$$", "g_*f_*(f^*b(\\mathcal{O}_Z) \\otimes a(\\mathcal{O}_Y)) \\xrightarrow{g_*f_*\\delta}", "g_*f_*(ab(\\mathcal{O}_Z)) \\xrightarrow{g_*\\gamma}", "g_*(b(\\mathcal{O}_Z)) \\xrightarrow{\\beta}", "\\mathcal{O}_Z", "$$", "where $\\gamma$ comes from the counit map $f_*a \\to \\text{id}$", "and $\\delta$ is the map whose adjoint is the composition", "$$", "f_*(f^*b(\\mathcal{O}_Z) \\otimes a(\\mathcal{O}_Y))", "\\xleftarrow{\\epsilon}", "b(\\mathcal{O}_Z) \\otimes f_*a(\\mathcal{O}_Y) \\xrightarrow{\\alpha}", "b(\\mathcal{O}_Z)", "$$", "By general properties of adjoint functors, adjoint maps, and counits", "(see Categories, Section \\ref{categories-section-adjoint})", "we have $\\gamma \\circ f_*\\delta = \\alpha \\circ \\epsilon^{-1}$ as desired." ], "refs": [ "perfect-lemma-cohomology-base-change", "perfect-lemma-cohomology-base-change" ], "ref_ids": [ 7025, 7025 ] } ], "ref_ids": [ 13503 ] }, { "id": 13519, "type": "theorem", "label": "duality-lemma-compute-sheaf-with-exact-support", "categories": [ "duality" ], "title": "duality-lemma-compute-sheaf-with-exact-support", "contents": [ "With notation as above. The functor $\\SheafHom(\\mathcal{O}_Z, -)$ is a", "right adjoint to the functor", "$i_* : \\textit{Mod}(\\mathcal{O}_Z) \\to \\textit{Mod}(\\mathcal{O}_X)$.", "For $V \\subset Z$ open we have", "$$", "\\Gamma(V, \\SheafHom(\\mathcal{O}_Z, \\mathcal{F})) =", "\\{s \\in \\Gamma(U, \\mathcal{F}) \\mid \\mathcal{I}s = 0\\}", "$$", "where $U \\subset X$ is an open whose intersection with $Z$ is $V$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Z$-modules. Then", "$$", "\\Hom_{\\mathcal{O}_X}(i_*\\mathcal{G}, \\mathcal{F}) =", "\\Hom_{i_*\\mathcal{O}_Z}(i_*\\mathcal{G},", "\\SheafHom_{\\mathcal{O}_X}(i_*\\mathcal{O}_Z, \\mathcal{F})) =", "\\Hom_{\\mathcal{O}_Z}(\\mathcal{G}, \\SheafHom(\\mathcal{O}_Z, \\mathcal{F}))", "$$", "The first equality by", "Modules, Lemma \\ref{modules-lemma-adjoint-tensor-restrict}", "and the second by the fully faithfulness of $i_*$, see", "Modules, Lemma \\ref{modules-lemma-i-star-equivalence}.", "The description of sections is left to the reader." ], "refs": [ "modules-lemma-adjoint-tensor-restrict", "modules-lemma-i-star-equivalence" ], "ref_ids": [ 13299, 13260 ] } ], "ref_ids": [] }, { "id": 13520, "type": "theorem", "label": "duality-lemma-sheaf-with-exact-support-adjoint", "categories": [ "duality" ], "title": "duality-lemma-sheaf-with-exact-support-adjoint", "contents": [ "With notation as above. The functor $R\\SheafHom(\\mathcal{O}_Z, -)$", "is the right adjoint of the functor", "$Ri_* : D(\\mathcal{O}_Z) \\to D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "This is a consequence of the fact that $i_*$ and", "$\\SheafHom(\\mathcal{O}_Z, -)$ are adjoint functors by", "Lemma \\ref{lemma-compute-sheaf-with-exact-support}. See", "Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors}." ], "refs": [ "duality-lemma-compute-sheaf-with-exact-support", "derived-lemma-derived-adjoint-functors" ], "ref_ids": [ 13519, 1907 ] } ], "ref_ids": [] }, { "id": 13521, "type": "theorem", "label": "duality-lemma-sheaf-with-exact-support-ext", "categories": [ "duality" ], "title": "duality-lemma-sheaf-with-exact-support-ext", "contents": [ "With notation as above. We have", "$$", "Ri_*R\\SheafHom(\\mathcal{O}_Z, K) =", "R\\SheafHom_{\\mathcal{O}_X}(i_*\\mathcal{O}_Z, K)", "$$", "in $D(\\mathcal{O}_X)$ for all $K$ in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the construction of the functor", "$R\\SheafHom(\\mathcal{O}_Z, -)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13522, "type": "theorem", "label": "duality-lemma-sheaf-with-exact-support-internal-home", "categories": [ "duality" ], "title": "duality-lemma-sheaf-with-exact-support-internal-home", "contents": [ "With notation as above. For $M \\in D(\\mathcal{O}_Z)$ we have", "$$", "R\\SheafHom_{\\mathcal{O}_X}(Ri_*M, K) =", "Ri_*R\\SheafHom_{\\mathcal{O}_Z}(M, R\\SheafHom(\\mathcal{O}_Z, K))", "$$", "in $D(\\mathcal{O}_Z)$ for all $K$ in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the construction of the functor", "$R\\SheafHom(\\mathcal{O}_Z, -)$ and the fact that if", "$\\mathcal{K}^\\bullet$ is a K-injective complex of", "$\\mathcal{O}_X$-modules, then $\\SheafHom(\\mathcal{O}_Z, \\mathcal{K}^\\bullet)$", "is a K-injective complex of $\\mathcal{O}_Z$-modules, see", "Derived Categories, Lemma \\ref{derived-lemma-adjoint-preserve-K-injectives}." ], "refs": [ "derived-lemma-adjoint-preserve-K-injectives" ], "ref_ids": [ 1915 ] } ], "ref_ids": [] }, { "id": 13523, "type": "theorem", "label": "duality-lemma-sheaf-with-exact-support-quasi-coherent", "categories": [ "duality" ], "title": "duality-lemma-sheaf-with-exact-support-quasi-coherent", "contents": [ "Let $i : Z \\to X$ be a pseudo-coherent closed immersion of schemes", "(any closed immersion if $X$ is locally Noetherian).", "Then", "\\begin{enumerate}", "\\item $R\\SheafHom(\\mathcal{O}_Z, -)$ maps $D^+_\\QCoh(\\mathcal{O}_X)$", "into $D^+_\\QCoh(\\mathcal{O}_Z)$, and", "\\item if $X = \\Spec(A)$ and $Z = \\Spec(B)$, then the diagram", "$$", "\\xymatrix{", "D^+(B) \\ar[r] & D_\\QCoh^+(\\mathcal{O}_Z) \\\\", "D^+(A) \\ar[r] \\ar[u]^{R\\Hom(B, -)} &", "D_\\QCoh^+(\\mathcal{O}_X) \\ar[u]_{R\\SheafHom(\\mathcal{O}_Z, -)}", "}", "$$", "is commutative.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To explain the parenthetical remark, if $X$ is locally Noetherian, then", "$i$ is pseudo-coherent by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-Noetherian-pseudo-coherent}.", "\\medskip\\noindent", "Let $K$ be an object of $D^+_\\QCoh(\\mathcal{O}_X)$. To prove (1), by", "Morphisms, Lemma \\ref{morphisms-lemma-i-star-equivalence}", "it suffices to show that $i_*$ applied to", "$H^n(R\\SheafHom(\\mathcal{O}_Z, K))$ produces a", "quasi-coherent module on $X$. By", "Lemma \\ref{lemma-sheaf-with-exact-support-ext}", "this means we have to show that", "$R\\SheafHom_{\\mathcal{O}_X}(i_*\\mathcal{O}_Z, K)$", "is in $D_\\QCoh(\\mathcal{O}_X)$. Since $i$ is pseudo-coherent", "the sheaf $\\mathcal{O}_Z$ is a pseudo-coherent $\\mathcal{O}_X$-module.", "Hence the result follows from", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-internal-hom}.", "\\medskip\\noindent", "Assume $X = \\Spec(A)$ and $Z = \\Spec(B)$ as in (2).", "Let $I^\\bullet$ be a bounded below complex of injective $A$-modules", "representing an object $K$ of $D^+(A)$.", "Then we know that $R\\Hom(B, K) = \\Hom_A(B, I^\\bullet)$ viewed", "as a complex of $B$-modules. Choose a quasi-isomorphism", "$$", "\\widetilde{I^\\bullet} \\longrightarrow \\mathcal{I}^\\bullet", "$$", "where $\\mathcal{I}^\\bullet$ is a bounded below complex of injective", "$\\mathcal{O}_X$-modules. It follows from the description of", "the functor $\\SheafHom(\\mathcal{O}_Z, -)$ in", "Lemma \\ref{lemma-compute-sheaf-with-exact-support}", "that there is a map", "$$", "\\Hom_A(B, I^\\bullet)", "\\longrightarrow", "\\Gamma(Z, \\SheafHom(\\mathcal{O}_Z, \\mathcal{I}^\\bullet))", "$$", "Observe that $\\SheafHom(\\mathcal{O}_Z, \\mathcal{I}^\\bullet)$", "represents $R\\SheafHom(\\mathcal{O}_Z, \\widetilde{K})$.", "Applying the universal property of the $\\widetilde{\\ }$ functor we", "obtain a map", "$$", "\\widetilde{\\Hom_A(B, I^\\bullet)}", "\\longrightarrow", "R\\SheafHom(\\mathcal{O}_Z, \\widetilde{K})", "$$", "in $D(\\mathcal{O}_Z)$. We may check that this map is an isomorphism in", "$D(\\mathcal{O}_Z)$ after applying $i_*$. However, once we apply", "$i_*$ we obtain the isomorphism of Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-quasi-coherence-internal-hom}", "via the identification of", "Lemma \\ref{lemma-sheaf-with-exact-support-ext}." ], "refs": [ "more-morphisms-lemma-Noetherian-pseudo-coherent", "morphisms-lemma-i-star-equivalence", "duality-lemma-sheaf-with-exact-support-ext", "perfect-lemma-quasi-coherence-internal-hom", "duality-lemma-compute-sheaf-with-exact-support", "perfect-lemma-quasi-coherence-internal-hom", "duality-lemma-sheaf-with-exact-support-ext" ], "ref_ids": [ 13982, 5136, 13521, 6981, 13519, 6981, 13521 ] } ], "ref_ids": [] }, { "id": 13524, "type": "theorem", "label": "duality-lemma-sheaf-with-exact-support-coherent", "categories": [ "duality" ], "title": "duality-lemma-sheaf-with-exact-support-coherent", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes.", "Assume $X$ is a locally Noetherian.", "Then $R\\SheafHom(\\mathcal{O}_Z, -)$ maps $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$", "into $D^+_{\\textit{Coh}}(\\mathcal{O}_Z)$." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$, hence we may assume that $X$ is affine.", "Say $X = \\Spec(A)$ and $Z = \\Spec(B)$ with $A$ Noetherian and", "$A \\to B$ surjective. In this case, we can apply", "Lemma \\ref{lemma-sheaf-with-exact-support-quasi-coherent}", "to translate the question into algebra.", "The corresponding algebra result is a consequence of", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-exact-support-coherent}." ], "refs": [ "duality-lemma-sheaf-with-exact-support-quasi-coherent", "dualizing-lemma-exact-support-coherent" ], "ref_ids": [ 13523, 2839 ] } ], "ref_ids": [] }, { "id": 13525, "type": "theorem", "label": "duality-lemma-twisted-inverse-image-closed", "categories": [ "duality" ], "title": "duality-lemma-twisted-inverse-image-closed", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $i : Z \\to X$ be a pseudo-coherent closed immersion", "(if $X$ is Noetherian, then any closed immersion is pseudo-coherent).", "Let $a : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Z)$ be the", "right adjoint to $Ri_*$. Then there is a functorial isomorphism", "$$", "a(K) = R\\SheafHom(\\mathcal{O}_Z, K)", "$$", "for $K \\in D_\\QCoh^+(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "(The parenthetical statement follows from More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-Noetherian-pseudo-coherent}.)", "By Lemma \\ref{lemma-sheaf-with-exact-support-adjoint}", "the functor $R\\SheafHom(\\mathcal{O}_Z, -)$ is a right adjoint", "to $Ri_* : D(\\mathcal{O}_Z) \\to D(\\mathcal{O}_X)$. Moreover,", "by Lemma \\ref{lemma-sheaf-with-exact-support-quasi-coherent}", "and Lemma \\ref{lemma-twisted-inverse-image-bounded-below}", "both $R\\SheafHom(\\mathcal{O}_Z, -)$ and $a$ map", "$D_\\QCoh^+(\\mathcal{O}_X)$ into $D_\\QCoh^+(\\mathcal{O}_Z)$.", "Hence we obtain the isomorphism by uniqueness of adjoint", "functors." ], "refs": [ "more-morphisms-lemma-Noetherian-pseudo-coherent", "duality-lemma-sheaf-with-exact-support-adjoint", "duality-lemma-sheaf-with-exact-support-quasi-coherent", "duality-lemma-twisted-inverse-image-bounded-below" ], "ref_ids": [ 13982, 13520, 13523, 13504 ] } ], "ref_ids": [] }, { "id": 13526, "type": "theorem", "label": "duality-lemma-check-base-change-is-iso", "categories": [ "duality" ], "title": "duality-lemma-check-base-change-is-iso", "contents": [ "In the situation above, the map (\\ref{equation-base-change-exact-support})", "is an isomorphism if and only if the base change map", "$$", "Lf^*R\\SheafHom_{\\mathcal{O}_X}(\\mathcal{O}_Z, K)", "\\longrightarrow", "R\\SheafHom_{\\mathcal{O}_{X'}}(\\mathcal{O}_{Z'}, Lf^*K)", "$$", "of Cohomology, Remark \\ref{cohomology-remark-prepare-fancy-base-change}", "is an isomorphism." ], "refs": [ "cohomology-remark-prepare-fancy-base-change" ], "proofs": [ { "contents": [ "The statement makes sense because $\\mathcal{O}_{Z'} = Lf^*\\mathcal{O}_Z$", "by the assumed tor independence.", "Since $i'_*$ is exact and faithful we see that it suffices to show", "the map (\\ref{equation-base-change-exact-support})", "is an isomorphism after applying $Ri'_*$. Since", "$Ri'_* \\circ Lg^* = Lf^* \\circ Ri_*$ by the assumed tor indepence and", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-compare-base-change-closed-immersion}", "we obtain a map", "$$", "Lf^*Ri_*R\\SheafHom(\\mathcal{O}_Z, K)", "\\longrightarrow", "Ri'_*R\\SheafHom(\\mathcal{O}_{Z'}, Lf^*K)", "$$", "whose source and target are as in the statement of the lemma by", "Lemma \\ref{lemma-sheaf-with-exact-support-ext}. We omit the", "verification that this is the same map as the one constructed", "in Cohomology, Remark \\ref{cohomology-remark-prepare-fancy-base-change}." ], "refs": [ "perfect-lemma-compare-base-change-closed-immersion", "duality-lemma-sheaf-with-exact-support-ext", "cohomology-remark-prepare-fancy-base-change" ], "ref_ids": [ 7031, 13521, 2282 ] } ], "ref_ids": [ 2282 ] }, { "id": 13527, "type": "theorem", "label": "duality-lemma-flat-bc-sheaf-with-exact-support", "categories": [ "duality" ], "title": "duality-lemma-flat-bc-sheaf-with-exact-support", "contents": [ "In the situation above, assume $f$ is flat and $i$ pseudo-coherent.", "Then (\\ref{equation-base-change-exact-support}) is an isomorphism", "for $K$ in $D^+_\\QCoh(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "First proof. To prove this map is an isomorphism, we may work locally.", "Hence we may assume $X$, $X'$, $Z$, $Z'$ are affine, say corresponding", "to the rings $A$, $A'$, $B$, $B'$. Then $B$ and $A'$ are tor independent", "over $A$. By Lemma \\ref{lemma-check-base-change-is-iso} it suffices", "to check that", "$$", "R\\Hom_A(B, K) \\otimes_A^\\mathbf{L} A' =", "R\\Hom_{A'}(B', K \\otimes_A^\\mathbf{L} A')", "$$", "in $D(A')$ for all $K \\in D^+(A)$. Here we use", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-internal-hom}", "and the fact that $B$, resp.\\ $B'$ is pseudo-coherent as an", "$A$-module, resp.\\ $A'$-module", "to compare derived hom on the level of rings and schemes.", "The displayed equality follows from", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism}", "part (3). See also the discussion in Dualizing Complexes, Section", "\\ref{dualizing-section-base-change-trivial-duality}.", "\\medskip\\noindent", "Second proof\\footnote{This proof shows", "it suffices to assume $K$ is in $D^+(\\mathcal{O}_X)$.}.", "Let $z' \\in Z'$ with image $z \\in Z$.", "First show that (\\ref{equation-base-change-exact-support})", "on stalks at $z'$ induces the map", "$$", "R\\Hom(\\mathcal{O}_{Z, z}, K_z)", "\\otimes_{\\mathcal{O}_{Z, x}}^\\mathbf{L} \\mathcal{O}_{Z', z'}", "\\longrightarrow", "R\\Hom(\\mathcal{O}_{Z', z'},", "K_z \\otimes_{\\mathcal{O}_{X, z}}^\\mathbf{L} \\mathcal{O}_{X', z'})", "$$", "from Dualizing Complexes, Equation (\\ref{dualizing-equation-base-change}).", "Namely, the constructions of these maps are identical.", "Then apply Dualizing Complexes, Lemma \\ref{dualizing-lemma-flat-bc-surjection}." ], "refs": [ "duality-lemma-check-base-change-is-iso", "perfect-lemma-quasi-coherence-internal-hom", "more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism", "dualizing-lemma-flat-bc-surjection" ], "ref_ids": [ 13526, 6981, 10415, 2845 ] } ], "ref_ids": [] }, { "id": 13528, "type": "theorem", "label": "duality-lemma-sheaf-with-exact-support-tensor", "categories": [ "duality" ], "title": "duality-lemma-sheaf-with-exact-support-tensor", "contents": [ "Let $i : Z \\to X$ be a pseudo-coherent closed immersion of schemes.", "Let $M \\in D_\\QCoh(\\mathcal{O}_X)$ locally have tor-amplitude in $[a, \\infty)$.", "Let $K \\in D_\\QCoh^+(\\mathcal{O}_X)$. Then there is a canonical isomorphism", "$$", "R\\SheafHom(\\mathcal{O}_Z, K) \\otimes_{\\mathcal{O}_Z}^\\mathbf{L} Li^*M =", "R\\SheafHom(\\mathcal{O}_Z, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)", "$$", "in $D(\\mathcal{O}_Z)$." ], "refs": [], "proofs": [ { "contents": [ "A map from LHS to RHS is the same thing as a map", "$$", "Ri_*R\\SheafHom(\\mathcal{O}_Z, K) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M", "\\longrightarrow", "K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M", "$$", "by Lemmas \\ref{lemma-sheaf-with-exact-support-adjoint} and", "\\ref{lemma-sheaf-with-exact-support-ext}. For this map we take the", "counit $Ri_*R\\SheafHom(\\mathcal{O}_Z, K) \\to K$ tensored with $\\text{id}_M$.", "To see this map is an isomorphism under the hypotheses given,", "translate into algebra using", "Lemma \\ref{lemma-sheaf-with-exact-support-quasi-coherent}", "and then for example use More on Algebra, Lemma", "\\ref{more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism} part (3).", "Instead of using Lemma \\ref{lemma-sheaf-with-exact-support-quasi-coherent}", "you can look at stalks as in the second proof of", "Lemma \\ref{lemma-flat-bc-sheaf-with-exact-support}." ], "refs": [ "duality-lemma-sheaf-with-exact-support-adjoint", "duality-lemma-sheaf-with-exact-support-ext", "duality-lemma-sheaf-with-exact-support-quasi-coherent", "more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism", "duality-lemma-sheaf-with-exact-support-quasi-coherent", "duality-lemma-flat-bc-sheaf-with-exact-support" ], "ref_ids": [ 13520, 13521, 13523, 10415, 13523, 13527 ] } ], "ref_ids": [] }, { "id": 13529, "type": "theorem", "label": "duality-lemma-compute-sheafhom-affine", "categories": [ "duality" ], "title": "duality-lemma-compute-sheafhom-affine", "contents": [ "With notation as above. The functor $\\SheafHom(f_*\\mathcal{O}_Y, -)$ is a", "right adjoint to the restriction functor", "$\\textit{Mod}(f_*\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X)$.", "For an affine open $U \\subset X$ we have", "$$", "\\Gamma(U, \\SheafHom(f_*\\mathcal{O}_Y, \\mathcal{F})) =", "\\Hom_A(B, \\mathcal{F}(U))", "$$", "where $A = \\mathcal{O}_X(U)$ and $B = \\mathcal{O}_Y(f^{-1}(U))$." ], "refs": [], "proofs": [ { "contents": [ "Adjointness follows from", "Modules, Lemma \\ref{modules-lemma-adjoint-tensor-restrict}.", "As $f$ is affine we see that $f_*\\mathcal{O}_Y$ is", "the quasi-coherent sheaf corresponding to $B$ viewed", "as an $A$-module. Hence the description of sections over $U$ follows from", "Schemes, Lemma \\ref{schemes-lemma-compare-constructions}." ], "refs": [ "modules-lemma-adjoint-tensor-restrict", "schemes-lemma-compare-constructions" ], "ref_ids": [ 13299, 7660 ] } ], "ref_ids": [] }, { "id": 13530, "type": "theorem", "label": "duality-lemma-sheafhom-affine-adjoint", "categories": [ "duality" ], "title": "duality-lemma-sheafhom-affine-adjoint", "contents": [ "With notation as above. The functor $R\\SheafHom(f_*\\mathcal{O}_Y, -)$", "is the right adjoint of the functor $D(f_*\\mathcal{O}_Y) \\to D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-compute-sheafhom-affine}", "and", "Derived Categories, Lemma \\ref{derived-lemma-derived-adjoint-functors}." ], "refs": [ "duality-lemma-compute-sheafhom-affine", "derived-lemma-derived-adjoint-functors" ], "ref_ids": [ 13529, 1907 ] } ], "ref_ids": [] }, { "id": 13531, "type": "theorem", "label": "duality-lemma-sheafhom-affine-ext", "categories": [ "duality" ], "title": "duality-lemma-sheafhom-affine-ext", "contents": [ "With notation as above. The composition", "$$", "D(\\mathcal{O}_X) \\xrightarrow{R\\SheafHom(f_*\\mathcal{O}_Y, -)}", "D(f_*\\mathcal{O}_Y) \\to D(\\mathcal{O}_X)", "$$", "is the functor $K \\mapsto R\\SheafHom_{\\mathcal{O}_X}(f_*\\mathcal{O}_Y, K)$." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the construction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13532, "type": "theorem", "label": "duality-lemma-finite-twisted", "categories": [ "duality" ], "title": "duality-lemma-finite-twisted", "contents": [ "Let $f : Y \\to X$ be a finite pseudo-coherent morphism of schemes", "(a finite morphism of Noetherian schemes is pseudo-coherent).", "The functor $R\\SheafHom(f_*\\mathcal{O}_Y, -)$ maps", "$D_\\QCoh^+(\\mathcal{O}_X)$ into $D_\\QCoh^+(f_*\\mathcal{O}_Y)$.", "If $X$ is quasi-compact and quasi-separated, then the diagram", "$$", "\\xymatrix{", "D_\\QCoh^+(\\mathcal{O}_X) \\ar[rr]_a \\ar[rd]_{R\\SheafHom(f_*\\mathcal{O}_Y, -)}", "& & D_\\QCoh^+(\\mathcal{O}_Y) \\ar[ld]^\\Phi \\\\", "& D_\\QCoh^+(f_*\\mathcal{O}_Y)", "}", "$$", "is commutative, where $a$ is the right adjoint of", "Lemma \\ref{lemma-twisted-inverse-image} for $f$ and $\\Phi$ is the equivalence", "of Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-morphism-equivalence}." ], "refs": [ "duality-lemma-twisted-inverse-image", "perfect-lemma-affine-morphism-equivalence" ], "proofs": [ { "contents": [ "(The parenthetical remark follows from More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-Noetherian-pseudo-coherent}.)", "Since $f$ is pseudo-coherent, the $\\mathcal{O}_X$-module $f_*\\mathcal{O}_Y$", "is pseudo-coherent, see More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-finite-pseudo-coherent}.", "Thus $R\\SheafHom(f_*\\mathcal{O}_Y, -)$ maps", "$D_\\QCoh^+(\\mathcal{O}_X)$ into", "$D_\\QCoh^+(f_*\\mathcal{O}_Y)$, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-internal-hom}.", "Then $\\Phi \\circ a$ and $R\\SheafHom(f_*\\mathcal{O}_Y, -)$", "agree on $D_\\QCoh^+(\\mathcal{O}_X)$ because these functors are", "both right adjoint to the restriction functor", "$D_\\QCoh^+(f_*\\mathcal{O}_Y) \\to D_\\QCoh^+(\\mathcal{O}_X)$. To see this", "use Lemmas \\ref{lemma-twisted-inverse-image-bounded-below} and", "\\ref{lemma-sheafhom-affine-adjoint}." ], "refs": [ "more-morphisms-lemma-Noetherian-pseudo-coherent", "more-morphisms-lemma-finite-pseudo-coherent", "perfect-lemma-quasi-coherence-internal-hom", "duality-lemma-twisted-inverse-image-bounded-below", "duality-lemma-sheafhom-affine-adjoint" ], "ref_ids": [ 13982, 13981, 6981, 13504, 13530 ] } ], "ref_ids": [ 13503, 6954 ] }, { "id": 13533, "type": "theorem", "label": "duality-lemma-proper-flat", "categories": [ "duality" ], "title": "duality-lemma-proper-flat", "contents": [ "Let $Y$ be a quasi-compact and quasi-separated scheme.", "Let $f : X \\to Y$ be a morphism of schemes which is proper, flat, and", "of finite presentation.", "Let $a$ be the right adjoint for", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of", "Lemma \\ref{lemma-twisted-inverse-image}. Then $a$ commutes with direct sums." ], "refs": [ "duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "Let $P$ be a perfect object of $D(\\mathcal{O}_X)$. By", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image-general}", "the complex $Rf_*P$ is perfect on $Y$.", "Let $K_i$ be a family of objects of $D_\\QCoh(\\mathcal{O}_Y)$.", "Then", "\\begin{align*}", "\\Hom_{D(\\mathcal{O}_X)}(P, a(\\bigoplus K_i))", "& =", "\\Hom_{D(\\mathcal{O}_Y)}(Rf_*P, \\bigoplus K_i) \\\\", "& =", "\\bigoplus \\Hom_{D(\\mathcal{O}_Y)}(Rf_*P, K_i) \\\\", "& =", "\\bigoplus \\Hom_{D(\\mathcal{O}_X)}(P, a(K_i))", "\\end{align*}", "because a perfect object is compact (Derived Categories of Schemes,", "Proposition \\ref{perfect-proposition-compact-is-perfect}).", "Since $D_\\QCoh(\\mathcal{O}_X)$ has a perfect generator", "(Derived Categories of Schemes, Theorem", "\\ref{perfect-theorem-bondal-van-den-Bergh})", "we conclude that the map $\\bigoplus a(K_i) \\to a(\\bigoplus K_i)$", "is an isomorphism, i.e., $a$ commutes with direct sums." ], "refs": [ "perfect-lemma-flat-proper-perfect-direct-image-general", "perfect-proposition-compact-is-perfect", "perfect-theorem-bondal-van-den-Bergh" ], "ref_ids": [ 7054, 7111, 6935 ] } ], "ref_ids": [ 13503 ] }, { "id": 13534, "type": "theorem", "label": "duality-lemma-proper-flat-relative", "categories": [ "duality" ], "title": "duality-lemma-proper-flat-relative", "contents": [ "Let $Y$ be a quasi-compact and quasi-separated scheme.", "Let $f : X \\to Y$ be a morphism of schemes which is proper, flat, and", "of finite presentation.", "Let $a$ be the right adjoint for", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of", "Lemma \\ref{lemma-twisted-inverse-image}. Then", "\\begin{enumerate}", "\\item for every closed $T \\subset Y$ if $Q \\in D_\\QCoh(Y)$ is supported on $T$,", "then $a(Q)$ is supported on $f^{-1}(T)$,", "\\item for every open $V \\subset Y$ and any $K \\in D_\\QCoh(\\mathcal{O}_Y)$", "the map (\\ref{equation-sheafy}) is an isomorphism, and", "\\end{enumerate}" ], "refs": [ "duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-when-sheafy},", "\\ref{lemma-proper-noetherian}, and", "\\ref{lemma-proper-flat}." ], "refs": [ "duality-lemma-when-sheafy", "duality-lemma-proper-noetherian", "duality-lemma-proper-flat" ], "ref_ids": [ 13508, 13509, 13533 ] } ], "ref_ids": [ 13503 ] }, { "id": 13535, "type": "theorem", "label": "duality-lemma-compare-with-pullback-flat-proper", "categories": [ "duality" ], "title": "duality-lemma-compare-with-pullback-flat-proper", "contents": [ "Let $Y$ be a quasi-compact and quasi-separated scheme.", "Let $f : X \\to Y$ be a morphism of schemes which is proper, flat, and", "of finite presentation.", "The map (\\ref{equation-compare-with-pullback}) is an isomorphism", "for every object $K$ of $D_\\QCoh(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-proper-flat} we know that $a$ commutes", "with direct sums. Hence the collection of objects of", "$D_\\QCoh(\\mathcal{O}_Y)$ for which (\\ref{equation-compare-with-pullback})", "is an isomorphism is a strictly full, saturated, triangulated", "subcategory of $D_\\QCoh(\\mathcal{O}_Y)$ which is moreover", "preserved under taking direct sums. Since $D_\\QCoh(\\mathcal{O}_Y)$", "is a module category (Derived Categories of Schemes, Theorem", "\\ref{perfect-theorem-DQCoh-is-Ddga}) generated by a single", "perfect object (Derived Categories of Schemes, Theorem", "\\ref{perfect-theorem-bondal-van-den-Bergh})", "we can argue as in", "More on Algebra, Remark \\ref{more-algebra-remark-P-resolution}", "to see that it suffices to prove (\\ref{equation-compare-with-pullback})", "is an isomorphism for a single perfect object.", "However, the result holds for perfect objects, see", "Lemma \\ref{lemma-compare-with-pullback-perfect}." ], "refs": [ "duality-lemma-proper-flat", "perfect-theorem-DQCoh-is-Ddga", "perfect-theorem-bondal-van-den-Bergh", "more-algebra-remark-P-resolution", "duality-lemma-compare-with-pullback-perfect" ], "ref_ids": [ 13533, 6936, 6935, 10653, 13515 ] } ], "ref_ids": [] }, { "id": 13536, "type": "theorem", "label": "duality-lemma-proper-flat-base-change", "categories": [ "duality" ], "title": "duality-lemma-proper-flat-base-change", "contents": [ "Let $g : Y' \\to Y$ be a morphism of quasi-compact and quasi-separated schemes.", "Let $f : X \\to Y$ be a proper, flat morphism of finite presentation.", "Then the base change map (\\ref{equation-base-change-map}) is an isomorphism", "for all $K \\in D_\\QCoh(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-proper-flat-relative} formation of the", "functors $a$ and $a'$ commutes with restriction to opens of $Y$ and $Y'$.", "Hence we may assume $Y' \\to Y$ is a morphism of affine schemes, see", "Remark \\ref{remark-check-over-affines}. In this", "case the statement follows from Lemma \\ref{lemma-more-base-change}." ], "refs": [ "duality-lemma-proper-flat-relative", "duality-remark-check-over-affines", "duality-lemma-more-base-change" ], "ref_ids": [ 13534, 13647, 13512 ] } ], "ref_ids": [] }, { "id": 13537, "type": "theorem", "label": "duality-lemma-properties-relative-dualizing", "categories": [ "duality" ], "title": "duality-lemma-properties-relative-dualizing", "contents": [ "Let $Y$ be a quasi-compact and quasi-separated scheme.", "Let $f : X \\to Y$ be a morphism of schemes which is", "proper, flat, and of finite presentation with", "relative dualizing complex $\\omega_{X/Y}^\\bullet$", "(Remark \\ref{remark-relative-dualizing-complex}).", "Then", "\\begin{enumerate}", "\\item $\\omega_{X/Y}^\\bullet$ is a $Y$-perfect object of $D(\\mathcal{O}_X)$,", "\\item $Rf_*\\omega_{X/Y}^\\bullet$ has vanishing cohomology sheaves", "in positive degrees,", "\\item $\\mathcal{O}_X \\to", "R\\SheafHom_{\\mathcal{O}_X}(\\omega_{X/Y}^\\bullet, \\omega_{X/Y}^\\bullet)$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [ "duality-remark-relative-dualizing-complex" ], "proofs": [ { "contents": [ "In view of the fact that formation of $\\omega_{X/Y}^\\bullet$ commutes", "with base change (see Remark \\ref{remark-relative-dualizing-complex}),", "we may and do assume that $Y$ is affine. For a perfect object $E$ of", "$D(\\mathcal{O}_X)$ we have", "\\begin{align*}", "Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_{X/Y}^\\bullet)", "& =", "Rf_*R\\SheafHom_{\\mathcal{O}_X}(E^\\vee, \\omega_{X/Y}^\\bullet) \\\\", "& =", "R\\SheafHom_{\\mathcal{O}_Y}(Rf_*E^\\vee, \\mathcal{O}_Y) \\\\", "& =", "(Rf_*E^\\vee)^\\vee", "\\end{align*}", "For the first equality, see", "Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "For the second equality, see Lemma \\ref{lemma-iso-on-RSheafHom},", "Remark \\ref{remark-iso-on-RSheafHom}, and ", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image-general}.", "The third equality is the definition of the dual. In particular", "these references also show that the outcome is a perfect object", "of $D(\\mathcal{O}_Y)$. We conclude that $\\omega_{X/Y}^\\bullet$", "is $Y$-perfect by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-characterize-relatively-perfect}.", "This proves (1).", "\\medskip\\noindent", "Let $M$ be an object of $D_\\QCoh(\\mathcal{O}_Y)$. Then", "\\begin{align*}", "\\Hom_Y(M, Rf_*\\omega_{X/Y}^\\bullet) & =", "\\Hom_X(Lf^*M, \\omega_{X/Y}^\\bullet) \\\\", "& =", "\\Hom_Y(Rf_*Lf^*M, \\mathcal{O}_Y) \\\\", "& =", "\\Hom_Y(M \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*\\mathcal{O}_Y, \\mathcal{O}_Y)", "\\end{align*}", "The first equality holds by Cohomology, Lemma", "\\ref{cohomology-lemma-adjoint}.", "The second equality by construction of $a$.", "The third equality by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-cohomology-base-change}.", "Recall $Rf_*\\mathcal{O}_X$ is perfect of tor amplitude in $[0, N]$", "for some $N$, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image-general}.", "Thus we can represent $Rf_*\\mathcal{O}_X$ by a complex of", "finite projective modules sitting in degrees $[0, N]$", "(using More on Algebra, Lemma \\ref{more-algebra-lemma-perfect}", "and the fact that $Y$ is affine).", "Hence if $M = \\mathcal{O}_Y[-i]$ for some $i > 0$, then the last", "group is zero. Since $Y$ is affine we conclude that", "$H^i(Rf_*\\omega_{X/Y}^\\bullet) = 0$ for $i > 0$.", "This proves (2).", "\\medskip\\noindent", "Let $E$ be a perfect object of $D_\\QCoh(\\mathcal{O}_X)$. Then", "we have", "\\begin{align*}", "\\Hom_X(E, R\\SheafHom_{\\mathcal{O}_X}(\\omega_{X/Y}^\\bullet, \\omega_{X/Y}^\\bullet)", "& =", "\\Hom_X(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_{X/Y}^\\bullet,", "\\omega_{X/Y}^\\bullet) \\\\", "& =", "\\Hom_Y(Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_{X/Y}^\\bullet),", "\\mathcal{O}_Y) \\\\", "& =", "\\Hom_Y(Rf_*(R\\SheafHom_{\\mathcal{O}_X}(E^\\vee, \\omega_{X/Y}^\\bullet)),", "\\mathcal{O}_Y) \\\\", "& =", "\\Hom_Y(R\\SheafHom_{\\mathcal{O}_Y}(Rf_*E^\\vee, \\mathcal{O}_Y),", "\\mathcal{O}_Y) \\\\", "& =", "R\\Gamma(Y, Rf_*E^\\vee) \\\\", "& =", "\\Hom_X(E, \\mathcal{O}_X)", "\\end{align*}", "The first equality holds by Cohomology, Lemma", "\\ref{cohomology-lemma-internal-hom}.", "The second equality is the definition of $\\omega_{X/Y}^\\bullet$.", "The third equality comes from the construction of the dual perfect", "complex $E^\\vee$, see Cohomology, Lemma", "\\ref{cohomology-lemma-dual-perfect-complex}.", "The fourth equality follows from the equality", "$Rf_*R\\SheafHom_{\\mathcal{O}_X}(E^\\vee, \\omega_{X/Y}^\\bullet) =", "R\\SheafHom_{\\mathcal{O}_Y}(Rf_*E^\\vee, \\mathcal{O}_Y)$", "shown in the first paragraph of the proof.", "The fifth equality holds by double duality for perfect complexes", "(Cohomology, Lemma", "\\ref{cohomology-lemma-dual-perfect-complex})", "and the fact that $Rf_*E$ is perfect by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image-general}.", "The last equality is Leray for $f$.", "This string of equalities essentially shows (3)", "holds by the Yoneda lemma. Namely, the object", "$R\\SheafHom(\\omega_{X/Y}^\\bullet, \\omega_{X/Y}^\\bullet)$", "is in $D_\\QCoh(\\mathcal{O}_X)$ by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-internal-hom}.", "Taking $E = \\mathcal{O}_X$ in the above we get a map", "$\\alpha : \\mathcal{O}_X \\to", "R\\SheafHom_{\\mathcal{O}_X}(\\omega_{X/Y}^\\bullet, \\omega_{X/Y}^\\bullet)$", "corresponding to", "$\\text{id}_{\\mathcal{O}_X} \\in \\Hom_X(\\mathcal{O}_X, \\mathcal{O}_X)$.", "Since all the isomorphisms above are functorial in $E$ we", "see that the cone on $\\alpha$ is an object $C$ of $D_\\QCoh(\\mathcal{O}_X)$", "such that $\\Hom(E, C) = 0$ for all perfect $E$.", "Since the perfect objects generate", "(Derived Categories of Schemes, Theorem", "\\ref{perfect-theorem-bondal-van-den-Bergh})", "we conclude that $\\alpha$ is an isomorphism." ], "refs": [ "duality-remark-relative-dualizing-complex", "cohomology-lemma-dual-perfect-complex", "duality-lemma-iso-on-RSheafHom", "duality-remark-iso-on-RSheafHom", "perfect-lemma-flat-proper-perfect-direct-image-general", "more-morphisms-lemma-characterize-relatively-perfect", "cohomology-lemma-adjoint", "perfect-lemma-cohomology-base-change", "perfect-lemma-flat-proper-perfect-direct-image-general", "more-algebra-lemma-perfect", "cohomology-lemma-internal-hom", "cohomology-lemma-dual-perfect-complex", "cohomology-lemma-dual-perfect-complex", "perfect-lemma-flat-proper-perfect-direct-image-general", "perfect-lemma-quasi-coherence-internal-hom", "perfect-theorem-bondal-van-den-Bergh" ], "ref_ids": [ 13649, 2233, 13505, 13645, 7054, 14062, 2121, 7025, 7054, 10212, 2183, 2233, 2233, 7054, 6981, 6935 ] } ], "ref_ids": [ 13649 ] }, { "id": 13538, "type": "theorem", "label": "duality-lemma-van-den-bergh", "categories": [ "duality" ], "title": "duality-lemma-van-den-bergh", "contents": [ "Let $Y$ be a quasi-compact and quasi-separated scheme.", "Let $f : X \\to Y$ be a proper, flat morphism of finite presentation", "with relative dualizing complex $\\omega_{X/Y}^\\bullet$", "(Remark \\ref{remark-relative-dualizing-complex}).", "There is a canonical isomorphism", "\\begin{equation}", "\\label{equation-pre-rigid}", "\\mathcal{O}_X =", "c(L\\text{pr}_1^*\\omega_{X/Y}^\\bullet) =", "c(L\\text{pr}_2^*\\omega_{X/Y}^\\bullet)", "\\end{equation}", "and a canonical isomorphism", "\\begin{equation}", "\\label{equation-rigid}", "\\omega_{X/Y}^\\bullet =", "c\\left(L\\text{pr}_1^*\\omega_{X/Y}^\\bullet", "\\otimes_{\\mathcal{O}_{X \\times_Y X}}^\\mathbf{L}", "L\\text{pr}_2^*\\omega_{X/Y}^\\bullet\\right)", "\\end{equation}", "where $c$ is the right adjoint of", "Lemma \\ref{lemma-twisted-inverse-image}", "for the diagonal $\\Delta : X \\to X \\times_Y X$." ], "refs": [ "duality-remark-relative-dualizing-complex", "duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "Let $a$ be the right adjoint to $Rf_*$ as in", "Lemma \\ref{lemma-twisted-inverse-image}.", "Consider the cartesian square", "$$", "\\xymatrix{", "X \\times_Y X \\ar[r]_q \\ar[d]_p & X \\ar[d]_f \\\\", "X \\ar[r]^f & Y", "}", "$$", "Let $b$ be the right adjoint for $p$", "as in Lemma \\ref{lemma-twisted-inverse-image}. Then", "\\begin{align*}", "\\omega_{X/Y}^\\bullet", "& =", "c(b(\\omega_{X/Y}^\\bullet)) \\\\", "& =", "c(Lp^*\\omega_{X/Y}^\\bullet", "\\otimes_{\\mathcal{O}_{X \\times_Y X}}^\\mathbf{L} b(\\mathcal{O}_X)) \\\\", "& =", "c(Lp^*\\omega_{X/Y}^\\bullet", "\\otimes_{\\mathcal{O}_{X \\times_Y X}}^\\mathbf{L}", "Lq^*a(\\mathcal{O}_Y)) \\\\", "& =", "c(Lp^*\\omega_{X/Y}^\\bullet", "\\otimes_{\\mathcal{O}_{X \\times_Y X}}^\\mathbf{L}", "Lq^*\\omega_{X/Y}^\\bullet)", "\\end{align*}", "as in (\\ref{equation-rigid}). Explanation as follows:", "\\begin{enumerate}", "\\item The first equality holds as $\\text{id} = c \\circ b$ because", "$\\text{id}_X = p \\circ \\Delta$.", "\\item The second equality holds by", "Lemma \\ref{lemma-compare-with-pullback-flat-proper}.", "\\item The third holds by Lemma \\ref{lemma-proper-flat-base-change}", "and the fact that $\\mathcal{O}_X = Lf^*\\mathcal{O}_Y$.", "\\item The fourth holds because $\\omega_{X/Y}^\\bullet = a(\\mathcal{O}_Y)$.", "\\end{enumerate}", "Equation (\\ref{equation-pre-rigid}) is proved in exactly the same way." ], "refs": [ "duality-lemma-twisted-inverse-image", "duality-lemma-twisted-inverse-image", "duality-lemma-compare-with-pullback-flat-proper", "duality-lemma-proper-flat-base-change" ], "ref_ids": [ 13503, 13503, 13535, 13536 ] } ], "ref_ids": [ 13649, 13503 ] }, { "id": 13539, "type": "theorem", "label": "duality-lemma-proper-flat-noetherian", "categories": [ "duality" ], "title": "duality-lemma-proper-flat-noetherian", "contents": [ "Let $f : X \\to Y$ be a perfect proper morphism of Noetherian schemes.", "Let $a$ be the right adjoint for", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of", "Lemma \\ref{lemma-twisted-inverse-image}. Then $a$ commutes with direct sums." ], "refs": [ "duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "Let $P$ be a perfect object of $D(\\mathcal{O}_X)$. By", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-perfect-proper-perfect-direct-image}", "the complex $Rf_*P$ is perfect on $Y$.", "Let $K_i$ be a family of objects of $D_\\QCoh(\\mathcal{O}_Y)$.", "Then", "\\begin{align*}", "\\Hom_{D(\\mathcal{O}_X)}(P, a(\\bigoplus K_i))", "& =", "\\Hom_{D(\\mathcal{O}_Y)}(Rf_*P, \\bigoplus K_i) \\\\", "& =", "\\bigoplus \\Hom_{D(\\mathcal{O}_Y)}(Rf_*P, K_i) \\\\", "& =", "\\bigoplus \\Hom_{D(\\mathcal{O}_X)}(P, a(K_i))", "\\end{align*}", "because a perfect object is compact (Derived Categories of Schemes,", "Proposition \\ref{perfect-proposition-compact-is-perfect}).", "Since $D_\\QCoh(\\mathcal{O}_X)$ has a perfect generator", "(Derived Categories of Schemes, Theorem", "\\ref{perfect-theorem-bondal-van-den-Bergh})", "we conclude that the map $\\bigoplus a(K_i) \\to a(\\bigoplus K_i)$", "is an isomorphism, i.e., $a$ commutes with direct sums." ], "refs": [ "more-morphisms-lemma-perfect-proper-perfect-direct-image", "perfect-proposition-compact-is-perfect", "perfect-theorem-bondal-van-den-Bergh" ], "ref_ids": [ 13997, 7111, 6935 ] } ], "ref_ids": [ 13503 ] }, { "id": 13540, "type": "theorem", "label": "duality-lemma-proper-flat-noetherian-relative", "categories": [ "duality" ], "title": "duality-lemma-proper-flat-noetherian-relative", "contents": [ "Let $f : X \\to Y$ be a perfect proper morphism of Noetherian schemes.", "Let $a$ be the right adjoint for", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of", "Lemma \\ref{lemma-twisted-inverse-image}. Then", "\\begin{enumerate}", "\\item for every closed $T \\subset Y$ if $Q \\in D_\\QCoh(Y)$ is supported on $T$,", "then $a(Q)$ is supported on $f^{-1}(T)$,", "\\item for every open $V \\subset Y$ and any $K \\in D_\\QCoh(\\mathcal{O}_Y)$", "the map (\\ref{equation-sheafy}) is an isomorphism, and", "\\end{enumerate}" ], "refs": [ "duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "This follows from Lemmas \\ref{lemma-when-sheafy},", "\\ref{lemma-proper-noetherian}, and", "\\ref{lemma-proper-flat-noetherian}." ], "refs": [ "duality-lemma-when-sheafy", "duality-lemma-proper-noetherian", "duality-lemma-proper-flat-noetherian" ], "ref_ids": [ 13508, 13509, 13539 ] } ], "ref_ids": [ 13503 ] }, { "id": 13541, "type": "theorem", "label": "duality-lemma-compare-with-pullback-flat-proper-noetherian", "categories": [ "duality" ], "title": "duality-lemma-compare-with-pullback-flat-proper-noetherian", "contents": [ "Let $f : X \\to Y$ be a perfect proper morphism of Noetherian", "schemes. The map (\\ref{equation-compare-with-pullback}) is an isomorphism", "for every object $K$ of $D_\\QCoh(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-proper-flat-noetherian} we know that $a$ commutes", "with direct sums. Hence the collection of objects of", "$D_\\QCoh(\\mathcal{O}_Y)$ for which (\\ref{equation-compare-with-pullback})", "is an isomorphism is a strictly full, saturated, triangulated", "subcategory of $D_\\QCoh(\\mathcal{O}_Y)$ which is moreover", "preserved under taking direct sums. Since $D_\\QCoh(\\mathcal{O}_Y)$", "is a module category (Derived Categories of Schemes, Theorem", "\\ref{perfect-theorem-DQCoh-is-Ddga}) generated by a single", "perfect object (Derived Categories of Schemes, Theorem", "\\ref{perfect-theorem-bondal-van-den-Bergh})", "we can argue as in", "More on Algebra, Remark \\ref{more-algebra-remark-P-resolution}", "to see that it suffices to prove (\\ref{equation-compare-with-pullback})", "is an isomorphism for a single perfect object.", "However, the result holds for perfect objects, see", "Lemma \\ref{lemma-compare-with-pullback-perfect}." ], "refs": [ "duality-lemma-proper-flat-noetherian", "perfect-theorem-DQCoh-is-Ddga", "perfect-theorem-bondal-van-den-Bergh", "more-algebra-remark-P-resolution", "duality-lemma-compare-with-pullback-perfect" ], "ref_ids": [ 13539, 6936, 6935, 10653, 13515 ] } ], "ref_ids": [] }, { "id": 13542, "type": "theorem", "label": "duality-lemma-proper-perfect-base-change", "categories": [ "duality" ], "title": "duality-lemma-proper-perfect-base-change", "contents": [ "Let $f : X \\to Y$ be a perfect proper morphism of Noetherian schemes.", "Let $g : Y' \\to Y$ be a morphism with $Y'$ Noetherian. If $X$ and", "$Y'$ are tor independent over $Y$, then the base", "change map (\\ref{equation-base-change-map}) is an isomorphism", "for all $K \\in D_\\QCoh(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-proper-flat-noetherian-relative} formation of the", "functors $a$ and $a'$ commutes with restriction to opens of $Y$ and $Y'$.", "Hence we may assume $Y' \\to Y$ is a morphism of affine schemes, see", "Remark \\ref{remark-check-over-affines}. In this", "case the statement follows from Lemma \\ref{lemma-more-base-change}." ], "refs": [ "duality-lemma-proper-flat-noetherian-relative", "duality-remark-check-over-affines", "duality-lemma-more-base-change" ], "ref_ids": [ 13540, 13647, 13512 ] } ], "ref_ids": [] }, { "id": 13543, "type": "theorem", "label": "duality-lemma-compute-for-effective-Cartier", "categories": [ "duality" ], "title": "duality-lemma-compute-for-effective-Cartier", "contents": [ "As above, let $X$ be a scheme and let $D \\subset X$ be an", "effective Cartier divisor. There is a canonical isomorphism", "$R\\SheafHom(\\mathcal{O}_D, \\mathcal{O}_X) = \\mathcal{N}[-1]$", "in $D(\\mathcal{O}_D)$." ], "refs": [], "proofs": [ { "contents": [ "Equivalently, we are saying that $R\\SheafHom(\\mathcal{O}_D, \\mathcal{O}_X)$", "has a unique nonzero cohomology sheaf in degree $1$ and that this", "sheaf is isomorphic to $\\mathcal{N}$. Since $i_*$ is exact and fully", "faithful, it suffices to prove that", "$i_*R\\SheafHom(\\mathcal{O}_D, \\mathcal{O}_X)$ is isomorphic", "to $i_*\\mathcal{N}[-1]$. We have", "$i_*R\\SheafHom(\\mathcal{O}_D, \\mathcal{O}_X) =", "R\\SheafHom_{\\mathcal{O}_X}(i_*\\mathcal{O}_D, \\mathcal{O}_X)$", "by Lemma \\ref{lemma-sheaf-with-exact-support-ext}. We have a resolution", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_X \\to i_*\\mathcal{O}_D \\to 0", "$$", "where $\\mathcal{I}$ is the ideal sheaf of $D$", "which we can use to compute. Since", "$R\\SheafHom_{\\mathcal{O}_X}(\\mathcal{O}_X, \\mathcal{O}_X) = \\mathcal{O}_X$ and", "$R\\SheafHom_{\\mathcal{O}_X}(\\mathcal{I}, \\mathcal{O}_X) = \\mathcal{O}_X(D)$ by", "a local computation, we see that", "$$", "R\\SheafHom_{\\mathcal{O}_X}(i_*\\mathcal{O}_D, \\mathcal{O}_X) =", "(\\mathcal{O}_X \\to \\mathcal{O}_X(D))", "$$", "where on the right hand side we have $\\mathcal{O}_X$ in degree $0$", "and $\\mathcal{O}_X(D)$ in degree $1$. The result follows from the", "short exact sequence", "$$", "0 \\to \\mathcal{O}_X \\to \\mathcal{O}_X(D) \\to i_*\\mathcal{N} \\to 0", "$$", "coming from the fact that $D$ is the zero scheme of the canonical section", "of $\\mathcal{O}_X(D)$ and from the fact that", "$\\mathcal{N} = i^*\\mathcal{O}_X(D)$." ], "refs": [ "duality-lemma-sheaf-with-exact-support-ext" ], "ref_ids": [ 13521 ] } ], "ref_ids": [] }, { "id": 13544, "type": "theorem", "label": "duality-lemma-sheaf-with-exact-support-effective-Cartier", "categories": [ "duality" ], "title": "duality-lemma-sheaf-with-exact-support-effective-Cartier", "contents": [ "As above, let $X$ be a scheme and let $D \\subset X$ be an", "effective Cartier divisor. Then (\\ref{equation-map-effective-Cartier})", "combined with Lemma \\ref{lemma-compute-for-effective-Cartier}", "defines an isomorphism", "$$", "Li^*K \\otimes_{\\mathcal{O}_D}^\\mathbf{L} \\mathcal{N}[-1]", "\\longrightarrow", "R\\SheafHom(\\mathcal{O}_D, K)", "$$", "functorial in $K$ in $D(\\mathcal{O}_X)$." ], "refs": [ "duality-lemma-compute-for-effective-Cartier" ], "proofs": [ { "contents": [ "Since $i_*$ is exact and fully faithful on modules, to prove the map is an", "isomorphism, it suffices to show that it is an isomorphism after applying", "$i_*$. We will use the short exact sequences", "$0 \\to \\mathcal{I} \\to \\mathcal{O}_X \\to i_*\\mathcal{O}_D \\to 0$", "and", "$0 \\to \\mathcal{O}_X \\to \\mathcal{O}_X(D) \\to i_*\\mathcal{N} \\to 0$", "used in the proof of Lemma \\ref{lemma-compute-for-effective-Cartier}", "without further mention. By", "Cohomology, Lemma \\ref{cohomology-lemma-projection-formula-closed-immersion}", "which was used to define the map (\\ref{equation-map-effective-Cartier})", "the left hand side becomes", "$$", "K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} i_*\\mathcal{N}[-1] =", "K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} (\\mathcal{O}_X \\to \\mathcal{O}_X(D))", "$$", "The right hand side becomes", "\\begin{align*}", "R\\SheafHom_{\\mathcal{O}_X}(i_*\\mathcal{O}_D, K) & =", "R\\SheafHom_{\\mathcal{O}_X}((\\mathcal{I} \\to \\mathcal{O}_X), K) \\\\", "& =", "R\\SheafHom_{\\mathcal{O}_X}((\\mathcal{I} \\to \\mathcal{O}_X), \\mathcal{O}_X)", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L} K", "\\end{align*}", "the final equality by", "Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "Since the map comes from the isomorphism", "$$", "R\\SheafHom_{\\mathcal{O}_X}((\\mathcal{I} \\to \\mathcal{O}_X), \\mathcal{O}_X)", "= (\\mathcal{O}_X \\to \\mathcal{O}_X(D))", "$$", "the lemma is clear." ], "refs": [ "duality-lemma-compute-for-effective-Cartier", "cohomology-lemma-projection-formula-closed-immersion", "cohomology-lemma-dual-perfect-complex" ], "ref_ids": [ 13543, 2245, 2233 ] } ], "ref_ids": [ 13543 ] }, { "id": 13545, "type": "theorem", "label": "duality-lemma-upper-shriek-P1", "categories": [ "duality" ], "title": "duality-lemma-upper-shriek-P1", "contents": [ "Let $Y$ be a Noetherian scheme. Let $\\mathcal{E}$ be a finite locally", "free $\\mathcal{O}_Y$-module of rank $n + 1$ with determinant", "$\\mathcal{L} = \\wedge^{n + 1}(\\mathcal{E})$.", "Let $f : X = \\mathbf{P}(\\mathcal{E}) \\to Y$ be the projection.", "Let $a$ be the right adjoint for", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of", "Lemma \\ref{lemma-twisted-inverse-image}.", "Then there is an isomorphism", "$$", "c : f^*\\mathcal{L}(-n - 1)[n] \\longrightarrow a(\\mathcal{O}_Y)", "$$", "In particular, if $\\mathcal{E} = \\mathcal{O}_Y^{\\oplus n + 1}$, then", "$X = \\mathbf{P}^n_Y$ and we obtain", "$a(\\mathcal{O}_Y) = \\mathcal{O}_X(-n - 1)[n]$." ], "refs": [ "duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "In (the proof of) Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cohomology-projective-bundle}", "we constructed a canonical isomorphism", "$$", "R^nf_*(f^*\\mathcal{L}(-n - 1)) \\longrightarrow \\mathcal{O}_Y", "$$", "Moreover, $Rf_*(f^*\\mathcal{L}(-n - 1))[n] = R^nf_*(f^*\\mathcal{L}(-n - 1))$,", "i.e., the other higher direct images are zero. Thus we find an isomorphism", "$$", "Rf_*(f^*\\mathcal{L}(-n - 1)[n]) \\longrightarrow \\mathcal{O}_Y", "$$", "This isomorphism determines $c$ as in the statement of the lemma", "because $a$ is the right adjoint of $Rf_*$.", "By Lemma \\ref{lemma-proper-noetherian} construction of the $a$", "is local on the base. In particular, to check that", "$c$ is an isomorphism, we may work locally on $Y$.", "In other words, we may assume $Y$ is affine and", "$\\mathcal{E} = \\mathcal{O}_Y^{\\oplus n + 1}$.", "In this case the sheaves $\\mathcal{O}_X, \\mathcal{O}_X(-1), \\ldots,", "\\mathcal{O}_X(-n)$ generate $D_\\QCoh(X)$, see", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-generator-P1}.", "Hence it suffices to show that", "$c : \\mathcal{O}_X(-n - 1)[n] \\to a(\\mathcal{O}_Y)$", "is transformed into an isomorphism under the functors", "$$", "F_{i, p}(-) = \\Hom_{D(\\mathcal{O}_X)}(\\mathcal{O}_X(i), (-)[p])", "$$", "for $i \\in \\{-n, \\ldots, 0\\}$ and $p \\in \\mathbf{Z}$.", "For $F_{0, p}$ this holds by construction of the arrow $c$!", "For $i \\in \\{-n, \\ldots, -1\\}$ we have", "$$", "\\Hom_{D(\\mathcal{O}_X)}(\\mathcal{O}_X(i), \\mathcal{O}_X(-n - 1)[n + p]) =", "H^p(X, \\mathcal{O}_X(-n - 1 - i)) = 0", "$$", "by the computation of cohomology of projective space", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cohomology-projective-space-over-ring})", "and we have", "$$", "\\Hom_{D(\\mathcal{O}_X)}(\\mathcal{O}_X(i), a(\\mathcal{O}_Y)[p]) =", "\\Hom_{D(\\mathcal{O}_Y)}(Rf_*\\mathcal{O}_X(i), \\mathcal{O}_Y[p]) = 0", "$$", "because $Rf_*\\mathcal{O}_X(i) = 0$ by the same lemma.", "Hence the source and the target of $F_{i, p}(c)$ vanish", "and $F_{i, p}(c)$ is necessarily an isomorphism.", "This finishes the proof." ], "refs": [ "coherent-lemma-cohomology-projective-bundle", "duality-lemma-proper-noetherian", "perfect-lemma-generator-P1", "coherent-lemma-cohomology-projective-space-over-ring" ], "ref_ids": [ 3307, 13509, 7014, 3304 ] } ], "ref_ids": [ 13503 ] }, { "id": 13546, "type": "theorem", "label": "duality-lemma-ext", "categories": [ "duality" ], "title": "duality-lemma-ext", "contents": [ "Let $Y$ be a ringed space. Let $\\mathcal{I} \\subset \\mathcal{O}_Y$", "be a sheaf of ideals. Set $\\mathcal{O}_X = \\mathcal{O}_Y/\\mathcal{I}$ and", "$\\mathcal{N} =", "\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{I}/\\mathcal{I}^2, \\mathcal{O}_X)$.", "There is a canonical isomorphism", "$c : \\mathcal{N} \\to", "\\SheafExt^1_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)", "$." ], "refs": [], "proofs": [ { "contents": [ "Consider the canonical short exact sequence", "\\begin{equation}", "\\label{equation-second-order-thickening}", "0 \\to \\mathcal{I}/\\mathcal{I}^2 \\to \\mathcal{O}_Y/\\mathcal{I}^2 \\to", "\\mathcal{O}_X \\to 0", "\\end{equation}", "Let $U \\subset X$ be open and let $s \\in \\mathcal{N}(U)$. Then we can", "pushout (\\ref{equation-second-order-thickening}) via $s$ to", "get an extension $E_s$ of $\\mathcal{O}_X|_U$ by $\\mathcal{O}_X|_U$.", "This in turn defines a section $c(s)$ of", "$\\SheafExt^1_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)$", "over $U$.", "See Cohomology, Lemma \\ref{cohomology-lemma-section-RHom-over-U}", "and Derived Categories, Lemma \\ref{derived-lemma-ext-1}.", "Conversely, given an extension", "$$", "0 \\to \\mathcal{O}_X|_U \\to \\mathcal{E} \\to \\mathcal{O}_X|_U \\to 0", "$$", "of $\\mathcal{O}_U$-modules, we can find an open covering", "$U = \\bigcup U_i$ and sections $e_i \\in \\mathcal{E}(U_i)$", "mapping to $1 \\in \\mathcal{O}_X(U_i)$. Then $e_i$ defines a map", "$\\mathcal{O}_Y|_{U_i} \\to \\mathcal{E}|_{U_i}$ whose kernel", "contains $\\mathcal{I}^2$. In this way we see that", "$\\mathcal{E}|_{U_i}$ comes from a pushout as above.", "This shows that $c$ is surjective. We omit the proof", "of injectivity." ], "refs": [ "derived-lemma-ext-1" ], "ref_ids": [ 1895 ] } ], "ref_ids": [] }, { "id": 13547, "type": "theorem", "label": "duality-lemma-regular-ideal-ext", "categories": [ "duality" ], "title": "duality-lemma-regular-ideal-ext", "contents": [ "Let $Y$ be a ringed space. Let $\\mathcal{I} \\subset \\mathcal{O}_Y$", "be a sheaf of ideals. Set $\\mathcal{O}_X = \\mathcal{O}_Y/\\mathcal{I}$.", "If $\\mathcal{I}$ is Koszul-regular", "(Divisors, Definition \\ref{divisors-definition-regular-ideal-sheaf})", "then composition on $R\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)$", "defines isomorphisms", "$$", "\\wedge^i(\\SheafExt^1_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X))", "\\longrightarrow", "\\SheafExt^i_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)", "$$", "for all $i$." ], "refs": [ "divisors-definition-regular-ideal-sheaf" ], "proofs": [ { "contents": [ "By composition we mean the map", "$$", "R\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "R\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)", "\\longrightarrow", "R\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)", "$$", "of Cohomology, Lemma \\ref{cohomology-lemma-internal-hom-composition}.", "This induces multiplication maps", "$$", "\\SheafExt^a_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)", "\\otimes_{\\mathcal{O}_Y}", "\\SheafExt^b_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)", "\\longrightarrow", "\\SheafExt^{a + b}_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)", "$$", "Please compare with", "More on Algebra, Equation (\\ref{more-algebra-equation-simple-tor-product}).", "The statement of the lemma means that the induced map", "$$", "\\SheafExt^1_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)", "\\otimes \\ldots \\otimes", "\\SheafExt^1_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)", "\\longrightarrow", "\\SheafExt^i_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)", "$$", "factors through the wedge product and then induces an isomorphism.", "To see this is true we may work locally on $Y$. Hence we may assume", "that we have global sections $f_1, \\ldots, f_r$ of $\\mathcal{O}_Y$", "which generate $\\mathcal{I}$ and which form a Koszul regular sequence.", "Denote", "$$", "\\mathcal{A} = \\mathcal{O}_Y\\langle \\xi_1, \\ldots, \\xi_r\\rangle", "$$", "the sheaf of strictly commutative differential graded $\\mathcal{O}_Y$-algebras", "which is a (divided power) polynomial algebra on", "$\\xi_1, \\ldots, \\xi_r$ in degree $-1$ over $\\mathcal{O}_Y$", "with differential $\\text{d}$ given by the rule $\\text{d}\\xi_i = f_i$.", "Let us denote $\\mathcal{A}^\\bullet$ the underlying", "complex of $\\mathcal{O}_Y$-modules which is the Koszul complex", "mentioned above. Thus the canonical map", "$\\mathcal{A}^\\bullet \\to \\mathcal{O}_X$", "is a quasi-isomorphism. We obtain quasi-isomorphisms", "$$", "R\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X) \\to", "\\SheafHom^\\bullet(\\mathcal{A}^\\bullet, \\mathcal{A}^\\bullet) \\to", "\\SheafHom^\\bullet(\\mathcal{A}^\\bullet, \\mathcal{O}_X)", "$$", "by Cohomology, Lemma \\ref{cohomology-lemma-Rhom-strictly-perfect}.", "The differentials of the latter complex are zero, and hence", "$$", "\\SheafExt^i_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)", "\\cong \\SheafHom_{\\mathcal{O}_Y}(\\mathcal{A}^{-i}, \\mathcal{O}_X)", "$$", "For $j \\in \\{1, \\ldots, r\\}$ let $\\delta_j : \\mathcal{A} \\to \\mathcal{A}$", "be the derivation of degree $1$ with $\\delta_j(\\xi_i) = \\delta_{ij}$", "(Kronecker delta). A computation shows that", "$\\delta_j \\circ \\text{d} = - \\text{d} \\circ \\delta_j$ which shows that", "we get a morphism of complexes.", "$$", "\\delta_j : \\mathcal{A}^\\bullet \\to \\mathcal{A}^\\bullet[1].", "$$", "Whence $\\delta_j$ defines a section of the corresponding", "$\\SheafExt$-sheaf.", "Another computation shows that $\\delta_1, \\ldots, \\delta_r$", "map to a basis for $\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{A}^{-1}, \\mathcal{O}_X)$", "over $\\mathcal{O}_X$.", "Since it is clear that $\\delta_j \\circ \\delta_j = 0$", "and $\\delta_j \\circ \\delta_{j'} = - \\delta_{j'} \\circ \\delta_j$", "as endomorphisms of $\\mathcal{A}$ and hence in the", "$\\SheafExt$-sheaves", "we obtain the statement that our map above factors through", "the exterior power. To see we get the desired isomorphism", "the reader checks that the elements", "$$", "\\delta_{j_1} \\circ \\ldots \\circ \\delta_{j_i}", "$$", "for $j_1 < \\ldots < j_i$ map to a basis of the sheaf", "$\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{A}^{-i}, \\mathcal{O}_X)$", "over $\\mathcal{O}_X$." ], "refs": [ "cohomology-lemma-internal-hom-composition", "cohomology-lemma-Rhom-strictly-perfect" ], "ref_ids": [ 2186, 2203 ] } ], "ref_ids": [ 8098 ] }, { "id": 13548, "type": "theorem", "label": "duality-lemma-regular-immersion-ext", "categories": [ "duality" ], "title": "duality-lemma-regular-immersion-ext", "contents": [ "Let $Y$ be a ringed space. Let $\\mathcal{I} \\subset \\mathcal{O}_Y$", "be a sheaf of ideals. Set $\\mathcal{O}_X = \\mathcal{O}_Y/\\mathcal{I}$ and", "$\\mathcal{N} =", "\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{I}/\\mathcal{I}^2, \\mathcal{O}_X)$.", "If $\\mathcal{I}$ is Koszul-regular", "(Divisors, Definition \\ref{divisors-definition-regular-ideal-sheaf}) then", "$$", "R\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_Y) =", "\\wedge^r \\mathcal{N}[r]", "$$", "where $r : Y \\to \\{1, 2, 3, \\ldots \\}$ sends $y$ to", "the minimal number of generators of $\\mathcal{I}$ needed in a neighbourhood", "of $y$." ], "refs": [ "divisors-definition-regular-ideal-sheaf" ], "proofs": [ { "contents": [ "We can use Lemmas \\ref{lemma-ext} and \\ref{lemma-regular-ideal-ext}", "to see that we have isomorphisms", "$\\wedge^i\\mathcal{N} \\to", "\\SheafExt^i_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)$", "for $i \\geq 0$. Thus it suffices to show that the map", "$\\mathcal{O}_Y \\to \\mathcal{O}_X$ induces an isomorphism", "$$", "\\SheafExt^r_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_Y)", "\\longrightarrow", "\\SheafExt^r_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)", "$$", "and that", "$\\SheafExt^i_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_Y)$", "is zero for $i \\not = r$. These statements are local on $Y$. Thus", "we may assume that we have global sections $f_1, \\ldots, f_r$ of", "$\\mathcal{O}_Y$ which generate $\\mathcal{I}$ and which form a", "Koszul regular sequence. Let $\\mathcal{A}^\\bullet$", "be the Koszul complex on $f_1, \\ldots, f_r$ as introduced in the proof of", "Lemma \\ref{lemma-regular-ideal-ext}. Then", "$$", "R\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_Y) =", "\\SheafHom^\\bullet(\\mathcal{A}^\\bullet, \\mathcal{O}_Y)", "$$", "by Cohomology, Lemma \\ref{cohomology-lemma-Rhom-strictly-perfect}.", "Denote $1 \\in H^0(\\SheafHom^\\bullet(\\mathcal{A}^\\bullet, \\mathcal{O}_Y))$", "the identity map of $\\mathcal{A}^0 = \\mathcal{O}_Y \\to \\mathcal{O}_Y$.", "With $\\delta_j$ as in the proof of Lemma \\ref{lemma-regular-ideal-ext}", "we get an isomorphism of graded $\\mathcal{O}_Y$-modules", "$$", "\\mathcal{O}_Y\\langle \\delta_1, \\ldots, \\delta_r\\rangle", "\\longrightarrow", "\\SheafHom^\\bullet(\\mathcal{A}^\\bullet, \\mathcal{O}_Y)", "$$", "by mapping $\\delta_{j_1} \\ldots \\delta_{j_i}$ to", "$1 \\circ \\delta_{j_1} \\circ \\ldots \\circ \\delta_{j_i}$ in degree $i$.", "Via this isomorphism the differential on the right hand side", "induces a differential $\\text{d}$ on the left hand side.", "By our sign rules we have $\\text{d}(1) = - \\sum f_j \\delta_j$.", "Since $\\delta_j : \\mathcal{A}^\\bullet \\to \\mathcal{A}^\\bullet[1]$", "is a morphism of complexes, it follows that", "$$", "\\text{d}(\\delta_{j_1} \\ldots \\delta_{j_i}) =", "(- \\sum f_j \\delta_j )\\delta_{j_1} \\ldots \\delta_{j_i}", "$$", "Observe that we have $\\text{d} = \\sum f_j \\delta_j$ on the differential", "graded algebra $\\mathcal{A}$. Therefore the map defined by the rule", "$$", "1 \\circ \\delta_{j_1} \\ldots \\delta_{j_i} \\longmapsto", "(\\delta_{j_1} \\circ \\ldots \\circ \\delta_{j_i})(\\xi_1 \\ldots \\xi_r)", "$$", "will define an isomorphism of complexes", "$$", "\\SheafHom^\\bullet(\\mathcal{A}^\\bullet, \\mathcal{O}_Y)", "\\longrightarrow \\mathcal{A}^\\bullet[-r]", "$$", "if $r$ is odd and commuting with differentials up to sign if $r$ is even.", "In any case these complexes have isomorphic cohomology, which shows the", "desired vanishing. The isomorphism on cohomology in degree $r$", "under the map", "$$", "\\SheafHom^\\bullet(\\mathcal{A}^\\bullet, \\mathcal{O}_Y)", "\\longrightarrow", "\\SheafHom^\\bullet(\\mathcal{A}^\\bullet, \\mathcal{O}_X)", "$$", "also follows in a straightforward manner from this.", "(We observe that our choice of conventions regarding", "Koszul complexes does intervene in the definition", "of the isomorphism", "$R\\SheafHom_{\\mathcal{O}_X}(\\mathcal{O}_X, \\mathcal{O}_Y) =", "\\wedge^r \\mathcal{N}[r]$.)" ], "refs": [ "duality-lemma-ext", "duality-lemma-regular-ideal-ext", "duality-lemma-regular-ideal-ext", "cohomology-lemma-Rhom-strictly-perfect", "duality-lemma-regular-ideal-ext" ], "ref_ids": [ 13546, 13547, 13547, 2203, 13547 ] } ], "ref_ids": [ 8098 ] }, { "id": 13549, "type": "theorem", "label": "duality-lemma-regular-immersion", "categories": [ "duality" ], "title": "duality-lemma-regular-immersion", "contents": [ "Let $Y$ be a quasi-compact and quasi-separated scheme.", "Let $i : X \\to Y$ be a Koszul-regular closed immersion.", "Let $a$ be the right adjoint of", "$Ri_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of", "Lemma \\ref{lemma-twisted-inverse-image}. Then there is an isomorphism", "$$", "\\wedge^r\\mathcal{N}[-r] \\longrightarrow a(\\mathcal{O}_Y)", "$$", "where", "$\\mathcal{N} = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{C}_{X/Y}, \\mathcal{O}_X)$", "is the normal sheaf of $i$", "(Morphisms, Section \\ref{morphisms-section-conormal-sheaf})", "and $r$ is its rank viewed as a locally constant", "function on $X$." ], "refs": [ "duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "Recall, from Lemmas \\ref{lemma-twisted-inverse-image-closed}", "and \\ref{lemma-sheaf-with-exact-support-ext},", "that $a(\\mathcal{O}_Y)$ is an object of $D_\\QCoh(\\mathcal{O}_X)$ whose", "pushforward to $Y$ is", "$R\\SheafHom_{\\mathcal{O}_Y}(i_*\\mathcal{O}_X, \\mathcal{O}_Y)$.", "Thus the result follows from Lemma \\ref{lemma-regular-immersion-ext}." ], "refs": [ "duality-lemma-twisted-inverse-image-closed", "duality-lemma-sheaf-with-exact-support-ext", "duality-lemma-regular-immersion-ext" ], "ref_ids": [ 13525, 13521, 13548 ] } ], "ref_ids": [ 13503 ] }, { "id": 13550, "type": "theorem", "label": "duality-lemma-smooth-proper", "categories": [ "duality" ], "title": "duality-lemma-smooth-proper", "contents": [ "Let $S$ be a Noetherian scheme.", "Let $f : X \\to S$ be a smooth proper morphism of relative dimension $d$.", "Let $a$ be the right adjoint of", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_S)$ as in", "Lemma \\ref{lemma-twisted-inverse-image}. Then there is an isomorphism", "$$", "\\wedge^d \\Omega_{X/S}[d] \\longrightarrow a(\\mathcal{O}_S)", "$$", "in $D(\\mathcal{O}_X)$." ], "refs": [ "duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "Set $\\omega_{X/S}^\\bullet = a(\\mathcal{O}_S)$ as in", "Remark \\ref{remark-relative-dualizing-complex}.", "Let $c$ be the right adjoint of Lemma \\ref{lemma-twisted-inverse-image} for", "$\\Delta : X \\to X \\times_S X$. Because $\\Delta$", "is the diagonal of a smooth morphism it is a", "Koszul-regular immersion, see Divisors, Lemma", "\\ref{divisors-lemma-immersion-smooth-into-smooth-regular-immersion}.", "In particular, $\\Delta$ is a perfect proper morphism", "(More on Morphisms, Lemma \\ref{more-morphisms-lemma-regular-immersion-perfect})", "and we obtain", "\\begin{align*}", "\\mathcal{O}_X", "& =", "c(L\\text{pr}_1^*\\omega_{X/S}^\\bullet) \\\\", "& =", "L\\Delta^*(L\\text{pr}_1^*\\omega_{X/S}^\\bullet)", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "c(\\mathcal{O}_{X \\times_S X}) \\\\", "& =", "\\omega_{X/S}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "c(\\mathcal{O}_{X \\times_S X}) \\\\", "& =", "\\omega_{X/S}^\\bullet", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "\\wedge^d(\\mathcal{N}_\\Delta)[-d]", "\\end{align*}", "The first equality is (\\ref{equation-pre-rigid}) because", "$\\omega_{X/S}^\\bullet = a(\\mathcal{O}_S)$. The second equality by", "Lemma \\ref{lemma-compare-with-pullback-flat-proper-noetherian}.", "The third equality because $p \\circ \\Delta = \\text{id}_X$.", "The fourth equality by Lemma \\ref{lemma-regular-immersion}.", "Observe that $\\wedge^d(\\mathcal{N}_\\Delta)$ is an invertible", "$\\mathcal{O}_X$-module. Hence $\\wedge^d(\\mathcal{N}_\\Delta)[-d]$", "is an invertible object of $D(\\mathcal{O}_X)$ and we conclude that", "$a(\\mathcal{O}_S) = \\omega_{X/S}^\\bullet = \\wedge^d(\\mathcal{C}_\\Delta)[d]$.", "Since the conormal sheaf $\\mathcal{C}_\\Delta$ of $\\Delta$ is", "$\\Omega_{X/S}$ by", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-diagonal}", "the proof is complete." ], "refs": [ "duality-remark-relative-dualizing-complex", "duality-lemma-twisted-inverse-image", "divisors-lemma-immersion-smooth-into-smooth-regular-immersion", "more-morphisms-lemma-regular-immersion-perfect", "duality-lemma-compare-with-pullback-flat-proper-noetherian", "duality-lemma-regular-immersion", "morphisms-lemma-differentials-diagonal" ], "ref_ids": [ 13649, 13503, 8007, 13992, 13541, 13549, 5311 ] } ], "ref_ids": [ 13503 ] }, { "id": 13551, "type": "theorem", "label": "duality-lemma-shriek-well-defined", "categories": [ "duality" ], "title": "duality-lemma-shriek-well-defined", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism", "of $\\textit{FTS}_S$. The functor $f^!$ is, up to canonical isomorphism,", "independent of the choice of the compactification." ], "refs": [], "proofs": [ { "contents": [ "The category of compactifications of $X$ over $Y$ is defined", "in More on Flatness, Section \\ref{flat-section-compactify}.", "By More on Flatness, Theorem \\ref{flat-theorem-nagata} and", "Lemma \\ref{flat-lemma-compactifyable} it is nonempty.", "To every choice of a compactification", "$$", "j : X \\to \\overline{X},\\quad \\overline{f} : \\overline{X} \\to Y", "$$", "the construction above associates the functor $j^* \\circ \\overline{a} :", "D_\\QCoh^+(\\mathcal{O}_Y) \\to D_\\QCoh^+(\\mathcal{O}_X)$", "where $\\overline{a}$ is the right adjoint of $R\\overline{f}_*$", "constructed in Lemma \\ref{lemma-twisted-inverse-image}.", "\\medskip\\noindent", "Suppose given a morphism $g : \\overline{X}_1 \\to \\overline{X}_2$", "between compactifications $j_i : X \\to \\overline{X}_i$ over $Y$", "such that $g^{-1}(j_2(X)) = j_1(X)$\\footnote{This may fail", "with our definition of compactification. See", "More on Flatness, Section \\ref{flat-section-compactify}.}.", "Let $\\overline{c}$ be the right adjoint of", "Lemma \\ref{lemma-twisted-inverse-image} for $g$.", "Then $\\overline{c} \\circ \\overline{a}_2 = \\overline{a}_1$", "because these functors are adjoint to", "$R\\overline{f}_{2, *} \\circ Rg_* = R(\\overline{f}_2 \\circ g)_*$.", "By (\\ref{equation-sheafy}) we have a canonical transformation", "$$", "j_1^* \\circ \\overline{c} \\longrightarrow j_2^*", "$$", "of functors", "$D^+_\\QCoh(\\mathcal{O}_{\\overline{X}_2}) \\to D^+_\\QCoh(\\mathcal{O}_X)$", "which is an isomorphism by Lemma \\ref{lemma-proper-noetherian}.", "The composition", "$$", "j_1^* \\circ \\overline{a}_1 \\longrightarrow", "j_1^* \\circ \\overline{c} \\circ \\overline{a}_2 \\longrightarrow", "j_2^* \\circ \\overline{a}_2", "$$", "is an isomorphism of functors which we will denote by $\\alpha_g$.", "\\medskip\\noindent", "Consider two compactifications $j_i : X \\to \\overline{X}_i$, $i = 1, 2$", "of $X$ over $Y$. By More on Flatness, Lemma", "\\ref{flat-lemma-compactifications-cofiltered} part (b)", "we can find a compactification $j : X \\to \\overline{X}$", "with dense image and morphisms", "$g_i : \\overline{X} \\to \\overline{X}_i$ of compactififcatins.", "By More on Flatness, Lemma", "\\ref{flat-lemma-compactifications-cofiltered} part (c)", "we have $g_i^{-1}(j_i(X)) = j(X)$. Hence we get isomorpisms", "$$", "\\alpha_{g_i} :", "j^* \\circ \\overline{a}", "\\longrightarrow", "j_i^* \\circ \\overline{a}_i", "$$", "by the previous paragraph. We obtain an isomorphism", "$$", "\\alpha_{g_2} \\circ \\alpha_{g_1}^{-1} :", "j_1^* \\circ \\overline{a}_1 \\to j_2^* \\circ \\overline{a}_2", "$$", "To finish the proof we have to show that these isomorphisms are well defined.", "We claim it suffices to show the composition of isomorphisms constructed", "in the previous paragraph is another (for a precise statement see the next", "paragraph). We suggest the reader check this is true on a napkin, but we", "will also completely spell it out in the rest of this paragraph.", "Namely, consider a second choice of a compactification", "$j' : X \\to \\overline{X}'$ with dense image", "and morphisms of compactifications $g'_i : \\overline{X}' \\to \\overline{X}_i$.", "By More on Flatness, Lemma \\ref{flat-lemma-compactifications-cofiltered}", "we can find a compactification $j'' : X \\to \\overline{X}''$", "with dense image and morphisms of compactifications", "$h : \\overline{X}'' \\to \\overline{X}$ and", "$h' : \\overline{X}'' \\to \\overline{X}'$. We may even assume", "$g_1 \\circ h = g'_1 \\circ h'$ and $g_2 \\circ h = g'_2 \\circ h'$.", "The result of the next paragraph gives", "$$", "\\alpha_{g_i} \\circ \\alpha_h = \\alpha_{g_i \\circ h} =", "\\alpha_{g'_i \\circ h'} = \\alpha_{g'_i} \\circ \\alpha_{h'}", "$$", "for $i = 1, 2$. Since these are all isomorphisms of functors", "we conclude that $\\alpha_{g_2} \\circ \\alpha_{g_1}^{-1} =", "\\alpha_{g'_2} \\circ \\alpha_{g'_1}^{-1}$ as desired.", "\\medskip\\noindent", "Suppose given compactifications $j_i : X \\to \\overline{X}_i$", "for $i = 1, 2, 3$. Suppose given morphisms", "$g : \\overline{X}_1 \\to \\overline{X}_2$ and", "$h : \\overline{X}_2 \\to \\overline{X}_3$ of compactifications", "such that $g^{-1}(j_2(X)) = j_1(X)$ and $h^{-1}(j_2(X)) = j_3(X)$.", "Let $\\overline{a}_i$ be as above. The claim above means that", "$$", "\\alpha_g \\circ \\alpha_h = \\alpha_{g \\circ h} :", "j_1^* \\circ \\overline{a}_1 \\to j_3^* \\circ \\overline{a}_3", "$$", "Let $\\overline{c}$, resp.\\ $\\overline{d}$ be the right adjoint of", "Lemma \\ref{lemma-twisted-inverse-image} for $g$, resp.\\ $h$.", "Then $\\overline{c} \\circ \\overline{a}_2 = \\overline{a}_1$ and", "$\\overline{d} \\circ \\overline{a}_3 = \\overline{a}_2$", "and there are canonical transformations", "$$", "j_1^* \\circ \\overline{c} \\longrightarrow j_2^*", "\\quad\\text{and}\\quad", "j_2^* \\circ \\overline{d} \\longrightarrow j_3^*", "$$", "of functors", "$D^+_\\QCoh(\\mathcal{O}_{\\overline{X}_2}) \\to D^+_\\QCoh(\\mathcal{O}_X)$", "and", "$D^+_\\QCoh(\\mathcal{O}_{\\overline{X}_3}) \\to D^+_\\QCoh(\\mathcal{O}_X)$", "for the same reasons as above. Denote $\\overline{e}$ the", "right adjoint of Lemma \\ref{lemma-twisted-inverse-image}", "for $h \\circ g$. There is a canonical transformation", "$$", "j_1^* \\circ \\overline{e} \\longrightarrow j_3^*", "$$", "of functors", "$D^+_\\QCoh(\\mathcal{O}_{\\overline{X}_3}) \\to D^+_\\QCoh(\\mathcal{O}_X)$", "given by (\\ref{equation-sheafy}). Spelling things out we have to", "show that the composition", "$$", "\\alpha_h \\circ \\alpha_g :", "j_1^* \\circ \\overline{a}_1 \\to", "j_1^* \\circ \\overline{c} \\circ \\overline{a}_2 \\to", "j_2^* \\circ \\overline{a}_2 \\to", "j_2^* \\circ \\overline{d} \\circ \\overline{a}_3 \\to", "j_3^* \\circ \\overline{a}_3", "$$", "is the same as the composition", "$$", "\\alpha_{h \\circ g} :", "j_1^* \\circ \\overline{a}_1 \\to", "j_1^* \\circ \\overline{e} \\circ \\overline{a}_3 \\to", "j_3^* \\circ \\overline{a}_3", "$$", "We split this into two parts. The first is to show that the diagram", "$$", "\\xymatrix{", "\\overline{a}_1 \\ar[r] \\ar[d] & \\overline{c} \\circ \\overline{a}_2 \\ar[d] \\\\", "\\overline{e} \\circ \\overline{a}_3 \\ar[r] &", "\\overline{c} \\circ \\overline{d} \\circ \\overline{a}_3", "}", "$$", "commutes where the lower horizontal arrow comes from the identification", "$\\overline{e} = \\overline{c} \\circ \\overline{d}$. This is true", "because the corresponding diagram of total direct image functors", "$$", "\\xymatrix{", "R\\overline{f}_{1, *} \\ar[r] \\ar[d] & Rg_* \\circ R\\overline{f}_{2, *} \\ar[d] \\\\", "R(h \\circ g)_* \\circ R\\overline{f}_{3, *} \\ar[r] &", "Rg_* \\circ Rh_* \\circ R\\overline{f}_{3, *}", "}", "$$", "is commutative (insert future reference here). The second part", "is to show that the composition", "$$", "j_1^* \\circ \\overline{c} \\circ \\overline{d} \\to", "j_2^* \\circ \\overline{d} \\to j_3^*", "$$", "is equal to the map", "$$", "j_1^* \\circ \\overline{e} \\to j_3^*", "$$", "via the identification $\\overline{e} = \\overline{c} \\circ \\overline{d}$.", "This was proven in Lemma \\ref{lemma-compose-base-change-maps}", "(note that in the current case the morphisms $f', g'$ of that", "lemma are equal to $\\text{id}_X$)." ], "refs": [ "flat-theorem-nagata", "flat-lemma-compactifyable", "duality-lemma-twisted-inverse-image", "duality-lemma-twisted-inverse-image", "duality-lemma-proper-noetherian", "flat-lemma-compactifications-cofiltered", "flat-lemma-compactifications-cofiltered", "flat-lemma-compactifications-cofiltered", "duality-lemma-twisted-inverse-image", "duality-lemma-twisted-inverse-image", "duality-lemma-compose-base-change-maps" ], "ref_ids": [ 5976, 6129, 13503, 13503, 13509, 6128, 6128, 6128, 13503, 13503, 13510 ] } ], "ref_ids": [] }, { "id": 13552, "type": "theorem", "label": "duality-lemma-upper-shriek-composition", "categories": [ "duality" ], "title": "duality-lemma-upper-shriek-composition", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ and $g : Y \\to Z$", "be composable morphisms of $\\textit{FTS}_S$. Then there is a canonical", "isomorphism $(g \\circ f)^! \\to f^! \\circ g^!$." ], "refs": [], "proofs": [ { "contents": [ "Choose a compactification $i : Y \\to \\overline{Y}$ of $Y$ over $Z$.", "Choose a compactification $X \\to \\overline{X}$ of $X$ over", "$\\overline{Y}$. This uses More on Flatness, Theorem \\ref{flat-theorem-nagata}", "and Lemma \\ref{flat-lemma-compactifyable} twice.", "Let $\\overline{a}$ be the", "right adjoint of Lemma \\ref{lemma-twisted-inverse-image} for", "$\\overline{X} \\to \\overline{Y}$ and let $\\overline{b}$", "be the", "right adjoint of Lemma \\ref{lemma-twisted-inverse-image} for", "$\\overline{Y} \\to Z$.", "Then $\\overline{a} \\circ \\overline{b}$ is the", "right adjoint of Lemma \\ref{lemma-twisted-inverse-image} for", "the composition $\\overline{X} \\to Z$.", "Hence $g^! = i^* \\circ \\overline{b}$ and", "$(g \\circ f)^! = (X \\to \\overline{X})^* \\circ \\overline{a} \\circ \\overline{b}$.", "Let $U$ be the inverse image of $Y$ in $\\overline{X}$", "so that we get the commutative diagram", "$$", "\\xymatrix{", "X \\ar[r]_j \\ar[d] & U \\ar[dl] \\ar[r]_{j'} & \\overline{X} \\ar[dl] \\\\", "Y \\ar[r]_i \\ar[d] & \\overline{Y} \\ar[dl] \\\\", "Z", "}", "$$", "Let $\\overline{a}'$ be the", "right adjoint of Lemma \\ref{lemma-twisted-inverse-image} for", "$U \\to Y$.", "Then $f^! = j^* \\circ \\overline{a}'$. We obtain", "$$", "\\gamma : (j')^* \\circ \\overline{a} \\to \\overline{a}' \\circ i^*", "$$", "by (\\ref{equation-sheafy}) and we can use it to define", "$$", "(g \\circ f)^! =", "(j' \\circ j)^* \\circ \\overline{a} \\circ \\overline{b} =", "j^* \\circ (j')^* \\circ \\overline{a} \\circ \\overline{b}", "\\to", "j^* \\circ \\overline{a}' \\circ i^* \\circ \\overline{b} =", "f^! \\circ g^!", "$$", "which is an isomorphism on objects of $D_\\QCoh^+(\\mathcal{O}_Z)$ by", "Lemma \\ref{lemma-proper-noetherian}. To finish the proof we show that", "this isomorphism is independent of choices made.", "\\medskip\\noindent", "Suppose we have two diagrams", "$$", "\\vcenter{", "\\xymatrix{", "X \\ar[r]_{j_1} \\ar[d] & U_1 \\ar[dl] \\ar[r]_{j'_1} & \\overline{X}_1 \\ar[dl] \\\\", "Y \\ar[r]_{i_1} \\ar[d] & \\overline{Y}_1 \\ar[dl] \\\\", "Z", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "X \\ar[r]_{j_2} \\ar[d] & U_2 \\ar[dl] \\ar[r]_{j'_2} & \\overline{X}_2 \\ar[dl] \\\\", "Y \\ar[r]_{i_2} \\ar[d] & \\overline{Y}_2 \\ar[dl] \\\\", "Z", "}", "}", "$$", "We can first choose a compactification $i : Y \\to \\overline{Y}$", "with dense image of $Y$ over $Z$ which dominates both", "$\\overline{Y}_1$ and $\\overline{Y}_2$,", "see More on Flatness, Lemma \\ref{flat-lemma-compactifications-cofiltered}.", "By More on Flatness, Lemma \\ref{flat-lemma-right-multiplicative-system} and", "Categories, Lemmas \\ref{categories-lemma-morphisms-right-localization} and", "\\ref{categories-lemma-equality-morphisms-right-localization}", "we can choose a compactification $X \\to \\overline{X}$ with dense image of", "$X$ over $\\overline{Y}$ with morphisms $\\overline{X} \\to \\overline{X}_1$", "and $\\overline{X} \\to \\overline{X}_2$ and such that the composition", "$\\overline{X} \\to \\overline{Y} \\to \\overline{Y}_1$ is equal to", "the composition $\\overline{X} \\to \\overline{X}_1 \\to \\overline{Y}_1$", "and such that the composition", "$\\overline{X} \\to \\overline{Y} \\to \\overline{Y}_2$ is equal to", "the composition $\\overline{X} \\to \\overline{X}_2 \\to \\overline{Y}_2$.", "Thus we see that it suffices to compare the maps", "determined by our diagrams when we have a commutative diagram", "as follows", "$$", "\\xymatrix{", "X \\ar[rr]_{j_1} \\ar@{=}[d] & &", "U_1 \\ar[d] \\ar[ddll] \\ar[rr]_{j'_1} & &", "\\overline{X}_1 \\ar[d] \\ar[ddll] \\\\", "X \\ar'[r][rr]^-{j_2} \\ar[d] & &", "U_2 \\ar'[dl][ddll] \\ar'[r][rr]^-{j'_2} & &", "\\overline{X}_2 \\ar[ddll] \\\\", "Y \\ar[rr]^{i_1} \\ar@{=}[d] & & \\overline{Y}_1 \\ar[d] \\\\", "Y \\ar[rr]^{i_2} \\ar[d] & & \\overline{Y}_2 \\ar[dll] \\\\", "Z", "}", "$$", "and moreover the compactifications $X \\to \\overline{X}_1$ and", "$Y \\to \\overline{Y}_2$ have dense image.", "We use $\\overline{a}_i$, $\\overline{a}'_i$, $\\overline{c}$, and", "$\\overline{c}'$ for the", "right adjoint of Lemma \\ref{lemma-twisted-inverse-image} for", "$\\overline{X}_i \\to \\overline{Y}_i$, $U_i \\to Y$,", "$\\overline{X}_1 \\to \\overline{X}_2$, and $U_1 \\to U_2$.", "Each of the squares", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d] \\ar@{}[dr]|A & U_1 \\ar[d] \\\\", "X \\ar[r] & U_2", "}", "\\quad", "\\xymatrix{", "U_2 \\ar[r] \\ar[d] \\ar@{}[dr]|B & \\overline{X}_2 \\ar[d] \\\\", "Y \\ar[r] & \\overline{Y}_2", "}", "\\quad", "\\xymatrix{", "U_1 \\ar[r] \\ar[d] \\ar@{}[dr]|C & \\overline{X}_1 \\ar[d] \\\\", "Y \\ar[r] & \\overline{Y}_1", "}", "\\quad", "\\xymatrix{", "Y \\ar[r] \\ar[d] \\ar@{}[dr]|D & \\overline{Y}_1 \\ar[d] \\\\", "Y \\ar[r] & \\overline{Y}_2", "}", "\\quad", "\\xymatrix{", "X \\ar[r] \\ar[d] \\ar@{}[dr]|E & \\overline{X}_1 \\ar[d] \\\\", "X \\ar[r] & \\overline{X}_2", "}", "$$", "is cartesian (see", "More on Flatness, Lemma \\ref{flat-lemma-compactifications-cofiltered} part (c)", "for A, D, E and recall that $U_i$ is the inverse image of $Y$", "by $\\overline{X}_i \\to \\overline{Y}_i$ for B, C) and hence", "gives rise to a base change map (\\ref{equation-sheafy}) as follows", "$$", "\\begin{matrix}", "\\gamma_A : j_1^* \\circ \\overline{c}' \\to j_2^* &", "\\gamma_B : (j_2')^* \\circ \\overline{a}_2 \\to \\overline{a}'_2 \\circ i_2^* &", "\\gamma_C : (j_1')^* \\circ \\overline{a}_1 \\to \\overline{a}'_1 \\circ i_1^* \\\\", "\\gamma_D : i_1^* \\circ \\overline{d} \\to i_2^* &", "\\gamma_E : (j'_1 \\circ j_1)^* \\circ \\overline{c} \\to (j'_2 \\circ j_2)^*", "\\end{matrix}", "$$", "Denote $f_1^! = j_1^* \\circ \\overline{a}'_1$,", "$f_2^! = j_2^* \\circ \\overline{a}'_2$,", "$g_1^! = i_1^* \\circ \\overline{b}_1$,", "$g_2^! = i_2^* \\circ \\overline{b}_2$,", "$(g \\circ f)_1^! =", "(j_1' \\circ j_1)^* \\circ \\overline{a}_1 \\circ \\overline{b}_1$, and", "$(g \\circ f)^!_2 =", "(j_2' \\circ j_2)^* \\circ \\overline{a}_2 \\circ \\overline{b}_2$.", "The construction given in the first paragraph of the proof", "and in Lemma \\ref{lemma-shriek-well-defined} uses", "\\begin{enumerate}", "\\item $\\gamma_C$ for the map $(g \\circ f)^!_1 \\to f_1^! \\circ g_1^!$,", "\\item $\\gamma_B$ for the map $(g \\circ f)^!_2 \\to f_2^! \\circ g_2^!$,", "\\item $\\gamma_A$ for the map $f_1^! \\to f_2^!$,", "\\item $\\gamma_D$ for the map $g_1^! \\to g_2^!$, and", "\\item $\\gamma_E$ for the map $(g \\circ f)^!_1 \\to (g \\circ f)^!_2$.", "\\end{enumerate}", "We have to show that the diagram", "$$", "\\xymatrix{", "(g \\circ f)^!_1 \\ar[r]_{\\gamma_E} \\ar[d]_{\\gamma_C} &", "(g \\circ f)^!_2 \\ar[d]_{\\gamma_B} \\\\", "f_1^! \\circ g_1^! \\ar[r]^{\\gamma_A \\circ \\gamma_D} & f_2^! \\circ g_2^!", "}", "$$", "is commutative. We will use", "Lemmas \\ref{lemma-compose-base-change-maps} and", "\\ref{lemma-compose-base-change-maps-horizontal}", "and with (abuse of) notation as in", "Remark \\ref{remark-going-around} (in particular", "dropping $\\star$ products with identity transformations", "from the notation).", "We can write $\\gamma_E = \\gamma_A \\circ \\gamma_F$ where", "$$", "\\xymatrix{", "U_1 \\ar[r] \\ar[d] \\ar@{}[rd]|F & \\overline{X}_1 \\ar[d] \\\\", "U_2 \\ar[r] & \\overline{X}_2", "}", "$$", "Thus we see that", "$$", "\\gamma_B \\circ \\gamma_E = \\gamma_B \\circ \\gamma_A \\circ \\gamma_F", "= \\gamma_A \\circ \\gamma_B \\circ \\gamma_F", "$$", "the last equality because the two squares $A$ and $B$ only", "intersect in one point (similar to the last argument in", "Remark \\ref{remark-going-around}). Thus it suffices to prove that", "$\\gamma_D \\circ \\gamma_C = \\gamma_B \\circ \\gamma_F$.", "Since both of these are equal to the map (\\ref{equation-sheafy})", "for the square", "$$", "\\xymatrix{", "U_1 \\ar[r] \\ar[d] & \\overline{X}_1 \\ar[d] \\\\", "Y \\ar[r] & \\overline{Y}_2", "}", "$$", "we conclude." ], "refs": [ "flat-theorem-nagata", "flat-lemma-compactifyable", "duality-lemma-twisted-inverse-image", "duality-lemma-twisted-inverse-image", "duality-lemma-twisted-inverse-image", "duality-lemma-twisted-inverse-image", "duality-lemma-proper-noetherian", "flat-lemma-compactifications-cofiltered", "flat-lemma-right-multiplicative-system", "categories-lemma-morphisms-right-localization", "categories-lemma-equality-morphisms-right-localization", "duality-lemma-twisted-inverse-image", "flat-lemma-compactifications-cofiltered", "duality-lemma-shriek-well-defined", "duality-lemma-compose-base-change-maps", "duality-lemma-compose-base-change-maps-horizontal", "duality-remark-going-around", "duality-remark-going-around" ], "ref_ids": [ 5976, 6129, 13503, 13503, 13503, 13503, 13509, 6128, 6130, 12262, 12263, 13503, 6128, 13551, 13510, 13511, 13646, 13646 ] } ], "ref_ids": [] }, { "id": 13553, "type": "theorem", "label": "duality-lemma-pseudo-functor", "categories": [ "duality" ], "title": "duality-lemma-pseudo-functor", "contents": [ "In Situation \\ref{situation-shriek} the constructions of", "Lemmas \\ref{lemma-shriek-well-defined} and \\ref{lemma-upper-shriek-composition}", "define a pseudo functor from the category $\\textit{FTS}_S$", "into the $2$-category of categories (see Categories, Definition", "\\ref{categories-definition-functor-into-2-category})." ], "refs": [ "duality-lemma-shriek-well-defined", "duality-lemma-upper-shriek-composition", "categories-definition-functor-into-2-category" ], "proofs": [ { "contents": [ "To show this we have to prove given morphisms", "$f : X \\to Y$, $g : Y \\to Z$, $h : Z \\to T$", "that", "$$", "\\xymatrix{", "(h \\circ g \\circ f)^! \\ar[r]_{\\gamma_{A + B}} \\ar[d]_{\\gamma_{B + C}} &", "f^! \\circ (h \\circ g)^! \\ar[d]^{\\gamma_C} \\\\", "(g \\circ f)^! \\circ h^! \\ar[r]^{\\gamma_A} & f^! \\circ g^! \\circ h^!", "}", "$$", "is commutative (for the meaning of the $\\gamma$'s, see below).", "To do this we choose a compactification $\\overline{Z}$", "of $Z$ over $T$, then a compactification $\\overline{Y}$ of $Y$ over", "$\\overline{Z}$, and then a compactification $\\overline{X}$ of", "$X$ over $\\overline{Y}$. This uses", "More on Flatness, Theorem \\ref{flat-theorem-nagata} and", "Lemma \\ref{flat-lemma-compactifyable}.", "Let $W \\subset \\overline{Y}$ be the inverse image of $Z$ under", "$\\overline{Y} \\to \\overline{Z}$ and let $U \\subset V \\subset \\overline{X}$", "be the inverse images of $Y \\subset W$ under $\\overline{X} \\to \\overline{Y}$.", "This produces the following diagram", "$$", "\\xymatrix{", "X \\ar[d]_f \\ar[r] & U \\ar[r] \\ar[d] \\ar@{}[dr]|A &", "V \\ar[d] \\ar[r] \\ar@{}[rd]|B & \\overline{X} \\ar[d] \\\\", "Y \\ar[d]_g \\ar[r] & Y \\ar[r] \\ar[d] & W \\ar[r] \\ar[d] \\ar@{}[rd]|C &", "\\overline{Y} \\ar[d] \\\\", "Z \\ar[d]_h \\ar[r] & Z \\ar[d] \\ar[r] & Z \\ar[d] \\ar[r] & \\overline{Z} \\ar[d] \\\\", "T \\ar[r] & T \\ar[r] & T \\ar[r] & T", "}", "$$", "Without introducing tons of notation but arguing exactly", "as in the proof of Lemma \\ref{lemma-upper-shriek-composition}", "we see that the maps in the first displayed diagram use the", "maps (\\ref{equation-sheafy}) for the rectangles", "$A + B$, $B + C$, $A$, and $C$ as indicated. Since by", "Lemmas \\ref{lemma-compose-base-change-maps} and", "\\ref{lemma-compose-base-change-maps-horizontal}", "we have $\\gamma_{A + B} = \\gamma_A \\circ \\gamma_B$ and", "$\\gamma_{B + C} = \\gamma_C \\circ \\gamma_B$ we conclude", "that the desired equality holds provided", "$\\gamma_A \\circ \\gamma_C = \\gamma_C \\circ \\gamma_A$.", "This is true because the two squares $A$ and $C$ only", "intersect in one point (similar to the last argument in", "Remark \\ref{remark-going-around})." ], "refs": [ "flat-theorem-nagata", "flat-lemma-compactifyable", "duality-lemma-upper-shriek-composition", "duality-lemma-compose-base-change-maps", "duality-lemma-compose-base-change-maps-horizontal", "duality-remark-going-around" ], "ref_ids": [ 5976, 6129, 13552, 13510, 13511, 13646 ] } ], "ref_ids": [ 13551, 13552, 12381 ] }, { "id": 13554, "type": "theorem", "label": "duality-lemma-map-pullback-to-shriek-well-defined", "categories": [ "duality" ], "title": "duality-lemma-map-pullback-to-shriek-well-defined", "contents": [ "In Situation \\ref{situation-shriek} let", "$f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. There are canonical maps", "$$", "\\mu_{f, K} :", "Lf^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!\\mathcal{O}_Y", "\\longrightarrow", "f^!K", "$$", "functorial in $K$ in $D^+_\\QCoh(\\mathcal{O}_Y)$.", "If $g : Y \\to Z$ is another morphism of $\\textit{FTS}_S$, then", "the diagram", "$$", "\\xymatrix{", "Lf^*(Lg^*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} g^!\\mathcal{O}_Z)", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!\\mathcal{O}_Y", "\\ar@{=}[d] \\ar[r]_-{\\mu_f} &", "f^!(Lg^*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} g^!\\mathcal{O}_Z)", "\\ar[r]_-{f^!\\mu_g} &", "f^!g^!K \\ar@{=}[d] \\\\", "Lf^*Lg^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^* g^!\\mathcal{O}_Z", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!\\mathcal{O}_Y \\ar[r]^-{\\mu_f} &", "Lf^*Lg^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!g^!\\mathcal{O}_Z", "\\ar[r]^-{\\mu_{g \\circ f}} & f^!g^!K", "}", "$$", "commutes for all $K \\in D^+_\\QCoh(\\mathcal{O}_Z)$." ], "refs": [], "proofs": [ { "contents": [ "If $f$ is proper, then $f^! = a$ and we can use", "(\\ref{equation-compare-with-pullback}) and if $g$ is also proper,", "then Lemma \\ref{lemma-transitivity-compare-with-pullback} proves", "the commutativity of the diagram (in greater generality).", "\\medskip\\noindent", "Let us define the map $\\mu_{f, K}$. Choose a compactification", "$j : X \\to \\overline{X}$ of $X$ over $Y$. Since $f^!$ is defined", "as $j^* \\circ \\overline{a}$ we obtain $\\mu_{f, K}$ as the restriction", "of the map (\\ref{equation-compare-with-pullback})", "$$", "L\\overline{f}^*K \\otimes_{\\mathcal{O}_{\\overline{X}}}^\\mathbf{L}", "\\overline{a}(\\mathcal{O}_Y)", "\\longrightarrow", "\\overline{a}(K)", "$$", "to $X$. To see this is independent of the choice of the compactification", "we argue as in the proof of Lemma \\ref{lemma-shriek-well-defined}.", "We urge the reader to read the proof of that lemma first.", "\\medskip\\noindent", "Assume given a morphism $g : \\overline{X}_1 \\to \\overline{X}_2$", "between compactifications $j_i : X \\to \\overline{X}_i$ over $Y$", "such that $g^{-1}(j_2(X)) = j_1(X)$. Denote $\\overline{c}$ the", "right adjoint for pushforward of Lemma \\ref{lemma-twisted-inverse-image}", "for the morphism $g$. The maps", "$$", "L\\overline{f}_1^*K \\otimes_{\\mathcal{O}_{\\overline{X}}}^\\mathbf{L}", "\\overline{a}_1(\\mathcal{O}_Y)", "\\longrightarrow", "\\overline{a}_1(K)", "\\quad\\text{and}\\quad", "L\\overline{f}_2^*K \\otimes_{\\mathcal{O}_{\\overline{X}}}^\\mathbf{L}", "\\overline{a}_2(\\mathcal{O}_Y)", "\\longrightarrow", "\\overline{a}_2(K)", "$$", "fit into the commutative diagram", "$$", "\\xymatrix{", "Lg^*(L\\overline{f}_2^*K \\otimes^\\mathbf{L}", "\\overline{a}_2(\\mathcal{O}_Y))", "\\otimes^\\mathbf{L} \\overline{c}(\\mathcal{O}_{\\overline{X}_2})", "\\ar@{=}[d] \\ar[r]_-\\sigma &", "\\overline{c}(L\\overline{f}_2^*K \\otimes^\\mathbf{L}", "\\overline{a}_2(\\mathcal{O}_Y)) \\ar[r] &", "\\overline{c}(\\overline{a}_2(K)) \\ar@{=}[d] \\\\", "L\\overline{f}_1^*K \\otimes^\\mathbf{L} Lg^*\\overline{a}_2(\\mathcal{O}_Y)", "\\otimes^\\mathbf{L} \\overline{c}(\\mathcal{O}_{\\overline{X}_2})", "\\ar[r]^-{1 \\otimes \\tau} &", "L\\overline{f}_1^*K \\otimes^\\mathbf{L} \\overline{a}_1(\\mathcal{O}_Y) \\ar[r] &", "\\overline{a}_1(K)", "}", "$$", "by Lemma \\ref{lemma-transitivity-compare-with-pullback}. By", "Lemma \\ref{lemma-compare-on-open} the maps $\\sigma$ and $\\tau$", "restrict to an isomorphism over $X$. In fact, we can say more.", "Recall that in the proof of Lemma \\ref{lemma-shriek-well-defined} we used", "the map (\\ref{equation-sheafy}) $\\gamma : j_1^* \\circ \\overline{c} \\to j_2^*$", "to construct our isomorphism", "$\\alpha_g : j_1^* \\circ \\overline{a}_1 \\to j_2^* \\circ \\overline{a}_2$.", "Pulling back to map $\\sigma$ by $j_1$ we obtain the identity", "map on $j_2^*\\left(L\\overline{f}_2^*K \\otimes^\\mathbf{L}", "\\overline{a}_2(\\mathcal{O}_Y)\\right)$ if we identify", "$j_1^*\\overline{c}(\\mathcal{O}_{\\overline{X}_2})$", "with $\\mathcal{O}_X$ via $j_1^* \\circ \\overline{c} \\to j_2^*$, see", "Lemma \\ref{lemma-restriction-compare-with-pullback}.", "Similarly, the map $\\tau : Lg^*\\overline{a}_2(\\mathcal{O}_Y)", "\\otimes^\\mathbf{L} \\overline{c}(\\mathcal{O}_{\\overline{X}_2}) \\to", "\\overline{a}_1(\\mathcal{O}_Y) = \\overline{c}(\\overline{a}_2(\\mathcal{O}_Y))$", "pulls back to the identity map on $j_2^*\\overline{a}_2(\\mathcal{O}_Y)$.", "We conclude that pulling back by $j_1$ and applying $\\gamma$ wherever", "we can we obtain a commutative diagram", "$$", "\\xymatrix{", "j_2^*\\left(L\\overline{f}_2^*K \\otimes^\\mathbf{L}", "\\overline{a}_2(\\mathcal{O}_Y)\\right) \\ar[r] \\ar[d] &", "j_2^*\\overline{a}_2(K) \\\\", "j_1^*L\\overline{f}_1^*K \\otimes^\\mathbf{L} j_2^*\\overline{a}_2(\\mathcal{O}_Y) &", "j_1^*(L\\overline{f}_1^*K \\otimes^\\mathbf{L} \\overline{a}_1(\\mathcal{O}_Y))", "\\ar[r] \\ar[l]_{1 \\otimes \\alpha_g} &", "j_1^* \\overline{a}_1(K) \\ar[lu]_{\\alpha_g}", "}", "$$", "The commutativity of this diagram exactly tells us that the map", "$\\mu_{f, K}$ constructed using the compactification $\\overline{X}_1$", "is the same as the map $\\mu_{f, K}$ constructed using the compactification", "$\\overline{X}_2$ via the identification $\\alpha_g$ used in the proof", "of Lemma \\ref{lemma-shriek-well-defined}. Some categorical arguments", "exactly as in the proof of Lemma \\ref{lemma-shriek-well-defined}", "now show that $\\mu_{f, K}$ is well defined (small detail omitted).", "\\medskip\\noindent", "Having said this, the commutativity of the diagram in the statement", "of our lemma follows from the construction of the isomorphism", "$(g \\circ f)^! \\to f^! \\circ g^!$ (first part of the proof of", "Lemma \\ref{lemma-upper-shriek-composition} using", "$\\overline{X} \\to \\overline{Y} \\to Z$) and the result", "of Lemma \\ref{lemma-transitivity-compare-with-pullback}", "for $\\overline{X} \\to \\overline{Y} \\to Z$." ], "refs": [ "duality-lemma-transitivity-compare-with-pullback", "duality-lemma-shriek-well-defined", "duality-lemma-twisted-inverse-image", "duality-lemma-transitivity-compare-with-pullback", "duality-lemma-compare-on-open", "duality-lemma-shriek-well-defined", "duality-lemma-restriction-compare-with-pullback", "duality-lemma-shriek-well-defined", "duality-lemma-shriek-well-defined", "duality-lemma-upper-shriek-composition", "duality-lemma-transitivity-compare-with-pullback" ], "ref_ids": [ 13518, 13551, 13503, 13518, 13517, 13551, 13516, 13551, 13551, 13552, 13518 ] } ], "ref_ids": [] }, { "id": 13555, "type": "theorem", "label": "duality-lemma-shriek-open-immersion", "categories": [ "duality" ], "title": "duality-lemma-shriek-open-immersion", "contents": [ "In Situation \\ref{situation-shriek} let $Y$ be an object", "of $\\textit{FTS}_S$ and let $j : X \\to Y$ be an open immersion.", "Then there is a canonical isomorphism $j^! = j^*$ of functors." ], "refs": [], "proofs": [ { "contents": [ "In this case we may choose $\\overline{X} = Y$ as our compactification.", "Then the", "right adjoint of Lemma \\ref{lemma-twisted-inverse-image} for", "$\\text{id} : Y \\to Y$ is the", "identity functor and hence $j^! = j^*$ by definition." ], "refs": [ "duality-lemma-twisted-inverse-image" ], "ref_ids": [ 13503 ] } ], "ref_ids": [] }, { "id": 13556, "type": "theorem", "label": "duality-lemma-restrict-before-or-after", "categories": [ "duality" ], "title": "duality-lemma-restrict-before-or-after", "contents": [ "In Situation \\ref{situation-shriek} let", "$$", "\\xymatrix{", "U \\ar[r]_j \\ar[d]_g & X \\ar[d]^f \\\\", "V \\ar[r]^{j'} & Y", "}", "$$", "be a commutative diagram of $\\textit{FTS}_S$ where $j$ and $j'$ are", "open immersions. Then $j^* \\circ f^! = g^! \\circ (j')^*$ as functors", "$D^+_\\QCoh(\\mathcal{O}_Y) \\to D^+(\\mathcal{O}_U)$." ], "refs": [], "proofs": [ { "contents": [ "Let $h = f \\circ j = j' \\circ g$. By", "Lemma \\ref{lemma-upper-shriek-composition} we have", "$h^! = j^! \\circ f^! = g^! \\circ (j')^!$. By", "Lemma \\ref{lemma-shriek-open-immersion}", "we have $j^! = j^*$ and $(j')^! = (j')^*$." ], "refs": [ "duality-lemma-upper-shriek-composition", "duality-lemma-shriek-open-immersion" ], "ref_ids": [ 13552, 13555 ] } ], "ref_ids": [] }, { "id": 13557, "type": "theorem", "label": "duality-lemma-shriek-affine-line", "categories": [ "duality" ], "title": "duality-lemma-shriek-affine-line", "contents": [ "In Situation \\ref{situation-shriek} let $Y$ be an object of $\\textit{FTS}_S$", "and let $f : X = \\mathbf{A}^1_Y \\to Y$ be", "the projection. Then there is a (noncanonical) isomorphism", "$f^!(-) \\cong Lf^*(-) [1]$ of functors." ], "refs": [], "proofs": [ { "contents": [ "Since $X = \\mathbf{A}^1_Y \\subset \\mathbf{P}^1_Y$", "and since $\\mathcal{O}_{\\mathbf{P}^1_Y}(-2)|_X \\cong \\mathcal{O}_X$", "this follows from Lemmas \\ref{lemma-upper-shriek-P1} and", "\\ref{lemma-compare-with-pullback-flat-proper-noetherian}." ], "refs": [ "duality-lemma-upper-shriek-P1", "duality-lemma-compare-with-pullback-flat-proper-noetherian" ], "ref_ids": [ 13545, 13541 ] } ], "ref_ids": [] }, { "id": 13558, "type": "theorem", "label": "duality-lemma-shriek-closed-immersion", "categories": [ "duality" ], "title": "duality-lemma-shriek-closed-immersion", "contents": [ "In Situation \\ref{situation-shriek} let $Y$ be an object of", "$\\textit{FTS}_S$ and let $i : X \\to Y$ be a closed immersion.", "Then there is a canonical isomorphism", "$i^!(-) = R\\SheafHom(\\mathcal{O}_X, -)$ of functors." ], "refs": [], "proofs": [ { "contents": [ "This is a restatement of Lemma \\ref{lemma-twisted-inverse-image-closed}." ], "refs": [ "duality-lemma-twisted-inverse-image-closed" ], "ref_ids": [ 13525 ] } ], "ref_ids": [] }, { "id": 13559, "type": "theorem", "label": "duality-lemma-shriek-coherent", "categories": [ "duality" ], "title": "duality-lemma-shriek-coherent", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. Then $f^!$ maps", "$D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$ into $D_{\\textit{Coh}}^+(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$ hence we may assume that $X$ and $Y$ are", "affine schemes. In this case we can factor $f : X \\to Y$ as", "$$", "X \\xrightarrow{i} \\mathbf{A}^n_Y \\to \\mathbf{A}^{n - 1}_Y \\to \\ldots \\to", "\\mathbf{A}^1_Y \\to Y", "$$", "where $i$ is a closed immersion. The lemma follows from", "By Lemmas \\ref{lemma-shriek-affine-line} and", "\\ref{lemma-sheaf-with-exact-support-coherent} and", "Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-dualizing-polynomial-ring}", "and induction." ], "refs": [ "duality-lemma-shriek-affine-line", "duality-lemma-sheaf-with-exact-support-coherent", "dualizing-lemma-dualizing-polynomial-ring" ], "ref_ids": [ 13557, 13524, 2855 ] } ], "ref_ids": [] }, { "id": 13560, "type": "theorem", "label": "duality-lemma-shriek-dualizing", "categories": [ "duality" ], "title": "duality-lemma-shriek-dualizing", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. If $K$ is a dualizing complex", "for $Y$, then $f^!K$ is a dualizing complex for $X$." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$ hence we may assume that $X$ and $Y$ are", "affine schemes. In this case we can factor $f : X \\to Y$ as", "$$", "X \\xrightarrow{i} \\mathbf{A}^n_Y \\to \\mathbf{A}^{n - 1}_Y \\to \\ldots \\to", "\\mathbf{A}^1_Y \\to Y", "$$", "where $i$ is a closed immersion. By Lemma \\ref{lemma-shriek-affine-line} and", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-dualizing-polynomial-ring}", "and induction we see that", "the $p^!K$ is a dualizing complex on $\\mathbf{A}^n_Y$ where", "$p : \\mathbf{A}^n_Y \\to Y$ is the projection. Similarly, by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-dualizing-quotient}", "and Lemmas", "\\ref{lemma-sheaf-with-exact-support-quasi-coherent} and", "\\ref{lemma-shriek-closed-immersion} we see that $i^!$", "transforms dualizing complexes into dualizing complexes." ], "refs": [ "duality-lemma-shriek-affine-line", "dualizing-lemma-dualizing-polynomial-ring", "dualizing-lemma-dualizing-quotient", "duality-lemma-sheaf-with-exact-support-quasi-coherent", "duality-lemma-shriek-closed-immersion" ], "ref_ids": [ 13557, 2855, 2854, 13523, 13558 ] } ], "ref_ids": [] }, { "id": 13561, "type": "theorem", "label": "duality-lemma-shriek-via-duality", "categories": [ "duality" ], "title": "duality-lemma-shriek-via-duality", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. Let $K$ be a dualizing complex", "on $Y$. Set $D_Y(M) = R\\SheafHom_{\\mathcal{O}_Y}(M, K)$ for", "$M \\in D_{\\textit{Coh}}(\\mathcal{O}_Y)$ and", "$D_X(E) = R\\SheafHom_{\\mathcal{O}_X}(E, f^!K)$ for", "$E \\in D_{\\textit{Coh}}(\\mathcal{O}_X)$. Then there is a canonical", "isomorphism", "$$", "f^!M \\longrightarrow D_X(Lf^*D_Y(M))", "$$", "for $M \\in D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "Choose compactification $j : X \\subset \\overline{X}$ of $X$ over $Y$", "(More on Flatness, Theorem \\ref{flat-theorem-nagata} and", "Lemma \\ref{flat-lemma-compactifyable}). Let $a$ be the", "right adjoint of Lemma \\ref{lemma-twisted-inverse-image} for", "$\\overline{X} \\to Y$. Set", "$D_{\\overline{X}}(E) = R\\SheafHom_{\\mathcal{O}_{\\overline{X}}}(E, a(K))$", "for $E \\in D_{\\textit{Coh}}(\\mathcal{O}_{\\overline{X}})$.", "Since formation of $R\\SheafHom$ commutes with restriction to opens", "and since $f^! = j^* \\circ a$ we see that it suffices to prove that", "there is a canonical isomorphism", "$$", "a(M) \\longrightarrow D_{\\overline{X}}(L\\overline{f}^*D_Y(M))", "$$", "for $M \\in D_{\\textit{Coh}}(\\mathcal{O}_Y)$. For", "$F \\in D_\\QCoh(\\mathcal{O}_X)$ we have", "\\begin{align*}", "\\Hom_{\\overline{X}}(", "F, D_{\\overline{X}}(L\\overline{f}^*D_Y(M)))", "& =", "\\Hom_{\\overline{X}}(", "F \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L\\overline{f}^*D_Y(M), a(K)) \\\\", "& =", "\\Hom_Y(", "R\\overline{f}_*(F \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L\\overline{f}^*D_Y(M)),", "K) \\\\", "& =", "\\Hom_Y(", "R\\overline{f}_*(F) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} D_Y(M),", "K) \\\\", "& =", "\\Hom_Y(", "R\\overline{f}_*(F), D_Y(D_Y(M))) \\\\", "& =", "\\Hom_Y(R\\overline{f}_*(F), M) \\\\", "& = \\Hom_{\\overline{X}}(F, a(M))", "\\end{align*}", "The first equality by Cohomology, Lemma \\ref{cohomology-lemma-internal-hom}.", "The second by definition of $a$.", "The third by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-cohomology-base-change}.", "The fourth equality by Cohomology, Lemma \\ref{cohomology-lemma-internal-hom}", "and the definition of $D_Y$.", "The fifth equality by Lemma \\ref{lemma-dualizing-schemes}.", "The final equality by definition of $a$.", "Hence we see that $a(M) = D_{\\overline{X}}(L\\overline{f}^*D_Y(M))$", "by Yoneda's lemma." ], "refs": [ "flat-theorem-nagata", "flat-lemma-compactifyable", "duality-lemma-twisted-inverse-image", "cohomology-lemma-internal-hom", "perfect-lemma-cohomology-base-change", "cohomology-lemma-internal-hom", "duality-lemma-dualizing-schemes" ], "ref_ids": [ 5976, 6129, 13503, 2183, 7025, 2183, 13499 ] } ], "ref_ids": [] }, { "id": 13562, "type": "theorem", "label": "duality-lemma-perfect-comparison-shriek", "categories": [ "duality" ], "title": "duality-lemma-perfect-comparison-shriek", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. Assume $f$ is perfect (e.g., flat). Then", "\\begin{enumerate}", "\\item[(a)] $f^!$ maps $D_{\\textit{Coh}}^b(\\mathcal{O}_Y)$ into", "$D_{\\textit{Coh}}^b(\\mathcal{O}_X)$,", "\\item[(b)] the map", "$\\mu_{f, K} :", "Lf^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!\\mathcal{O}_Y", "\\to", "f^!K$", "of Lemma \\ref{lemma-map-pullback-to-shriek-well-defined}", "is an isomorphism for all $K \\in D_\\QCoh^+(\\mathcal{O}_Y)$.", "\\end{enumerate}" ], "refs": [ "duality-lemma-map-pullback-to-shriek-well-defined" ], "proofs": [ { "contents": [ "(A flat morphism of finite presentation is perfect, see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-flat-finite-presentation-perfect}.)", "We begin with a series of preliminary remarks.", "\\begin{enumerate}", "\\item We already know that $f^!$ sends $D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$", "into $D_{\\textit{Coh}}^+(\\mathcal{O}_X)$, see", "Lemma \\ref{lemma-shriek-coherent}.", "\\item If $f$ is an open immersion, then (a) and (b) are true because", "we can take $\\overline{X} = Y$ in the construction of $f^!$ and $\\mu_f$.", "See also Lemma \\ref{lemma-shriek-open-immersion}.", "\\item If $f$ is a perfect proper morphism, then (b) is true by", "Lemma \\ref{lemma-compare-with-pullback-flat-proper-noetherian}.", "\\item If there exists an open covering $X = \\bigcup U_i$ and (a) is", "true for $U_i \\to Y$, then (a) is true for $X \\to Y$. Same for (b).", "This holds because the construction of $f^!$ and $\\mu_f$ commutes", "with passing to open subschemes.", "\\item If $g : Y \\to Z$ is a second perfect morphism in $\\textit{FTS}_S$", "and (b) holds for $f$ and $g$, then", "$f^!g^!\\mathcal{O}_Z =", "Lf^*g^!\\mathcal{O}_Z \\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!\\mathcal{O}_Y$", "and (b) holds for $g \\circ f$ by the commutative diagram", "of Lemma \\ref{lemma-map-pullback-to-shriek-well-defined}.", "\\item If (a) and (b) hold for both $f$ and $g$, then", "(a) and (b) hold for $g \\circ f$. Namely, then $f^!g^!\\mathcal{O}_Z$", "is bounded above (by the previous point) and $L(g \\circ f)^*$ has finite", "cohomological dimension and (a) follows from (b) which we saw above.", "\\end{enumerate}", "From these points we see it suffices to prove the result in case $X$ is affine.", "Choose an immersion $X \\to \\mathbf{A}^n_Y$", "(Morphisms, Lemma \\ref{morphisms-lemma-quasi-affine-finite-type-over-S})", "which we factor as $X \\to U \\to \\mathbf{A}^n_Y \\to Y$ where $X \\to U$", "is a closed immersion and $U \\subset \\mathbf{A}^n_Y$ is open.", "Note that $X \\to U$ is a perfect closed immersion by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-perfect-permanence}.", "Thus it suffices to prove the lemma for a perfect closed immersion", "and for the projection $\\mathbf{A}^n_Y \\to Y$.", "\\medskip\\noindent", "Let $f : X \\to Y$ be a perfect closed immersion. We already know (b) holds.", "Let $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$.", "Then $f^!K = R\\SheafHom(\\mathcal{O}_X, K)$", "(Lemma \\ref{lemma-shriek-closed-immersion})", "and $f_*f^!K = R\\SheafHom_{\\mathcal{O}_Y}(f_*\\mathcal{O}_X, K)$.", "Since $f$ is perfect, the complex $f_*\\mathcal{O}_X$ is perfect", "and hence $R\\SheafHom_{\\mathcal{O}_Y}(f_*\\mathcal{O}_X, K)$ is bounded above.", "This proves that (a) holds. Some details omitted.", "\\medskip\\noindent", "Let $f : \\mathbf{A}^n_Y \\to Y$ be the projection. Then (a) holds", "by repeated application of Lemma \\ref{lemma-shriek-affine-line}.", "Finally, (b) is true because it holds for $\\mathbf{P}^n_Y \\to Y$", "(flat and proper) and because $\\mathbf{A}^n_Y \\subset \\mathbf{P}^n_Y$", "is an open." ], "refs": [ "more-morphisms-lemma-flat-finite-presentation-perfect", "duality-lemma-shriek-coherent", "duality-lemma-shriek-open-immersion", "duality-lemma-compare-with-pullback-flat-proper-noetherian", "duality-lemma-map-pullback-to-shriek-well-defined", "morphisms-lemma-quasi-affine-finite-type-over-S", "more-morphisms-lemma-perfect-permanence", "duality-lemma-shriek-closed-immersion", "duality-lemma-shriek-affine-line" ], "ref_ids": [ 13990, 13559, 13555, 13541, 13554, 5392, 13993, 13558, 13557 ] } ], "ref_ids": [ 13554 ] }, { "id": 13563, "type": "theorem", "label": "duality-lemma-flat-shriek-relatively-perfect", "categories": [ "duality" ], "title": "duality-lemma-flat-shriek-relatively-perfect", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. If $f$ is flat, then", "$f^!\\mathcal{O}_Y$ is a $Y$-perfect object of $D(\\mathcal{O}_X)$ and", "$\\mathcal{O}_X \\to", "R\\SheafHom_{\\mathcal{O}_X}(f^!\\mathcal{O}_Y, f^!\\mathcal{O}_Y)$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Both assertions are local on $X$. Thus we may assume $X$ and $Y$ are", "affine. Then Remark \\ref{remark-local-calculation-shriek}", "turns the lemma into an algebra lemma, namely", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-relative-dualizing-algebraic}.", "(Use Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-locally-rel-perfect} to match the languages.)" ], "refs": [ "duality-remark-local-calculation-shriek", "dualizing-lemma-relative-dualizing-algebraic", "perfect-lemma-affine-locally-rel-perfect" ], "ref_ids": [ 13652, 2906, 7077 ] } ], "ref_ids": [] }, { "id": 13564, "type": "theorem", "label": "duality-lemma-lci-shriek", "categories": [ "duality" ], "title": "duality-lemma-lci-shriek", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. Assume $f : X \\to Y$ is a local complete", "intersection morphism. Then", "\\begin{enumerate}", "\\item $f^!\\mathcal{O}_Y$ is an invertible object of $D(\\mathcal{O}_X)$, and", "\\item $f^!$ maps perfect complexes to perfect complexes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Recall that a local complete intersection morphism is perfect, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-lci-properties}.", "By Lemma \\ref{lemma-perfect-comparison-shriek} it suffices to show", "that $f^!\\mathcal{O}_Y$ is an invertible object in $D(\\mathcal{O}_X)$.", "This question is local on $X$ and $Y$. Hence we may assume that $X \\to Y$", "factors as $X \\to \\mathbf{A}^n_Y \\to Y$ where the first arrow is a", "Koszul regular immersion. See More on Morphisms, Section", "\\ref{more-morphisms-section-lci}.", "The result holds for $\\mathbf{A}^n_Y \\to Y$", "by Lemma \\ref{lemma-shriek-affine-line}. Thus it suffices to prove", "the lemma when $f$ is a Koszul regular immersion.", "Working locally once again we reduce to the case", "$X = \\Spec(A)$ and $Y = \\Spec(B)$, where $A = B/(f_1, \\ldots, f_r)$", "for some regular sequence $f_1, \\ldots, f_r \\in B$", "(use that for Noetherian local rings the notion of Koszul", "regular and regular are the same, see", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-noetherian-finite-all-equivalent}).", "Thus $X \\to Y$ is a composition", "$$", "X = X_r \\to X_{r - 1} \\to \\ldots \\to X_1 \\to X_0 = Y", "$$", "where each arrow is the inclusion of an effective Cartier divisor.", "In this way we reduce to the case of an inclusion of an effective", "Cartier divisor $i : D \\to X$. In this case", "$i^!\\mathcal{O}_X = \\mathcal{N}[1]$ by", "Lemma \\ref{lemma-compute-for-effective-Cartier} and the proof is complete." ], "refs": [ "more-morphisms-lemma-lci-properties", "duality-lemma-perfect-comparison-shriek", "duality-lemma-shriek-affine-line", "more-algebra-lemma-noetherian-finite-all-equivalent", "duality-lemma-compute-for-effective-Cartier" ], "ref_ids": [ 14002, 13562, 13557, 9978, 13543 ] } ], "ref_ids": [] }, { "id": 13565, "type": "theorem", "label": "duality-lemma-base-change-shriek-flat", "categories": [ "duality" ], "title": "duality-lemma-base-change-shriek-flat", "contents": [ "In Situation \\ref{situation-shriek} let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "be a cartesian diagram of $\\textit{FTS}_S$ with $g$ flat.", "Then there is an isomorphism", "$L(g')^* \\circ f^! \\to (f')^! \\circ Lg^*$ on", "$D_\\QCoh^+(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "Namely, because $g$ is flat, for every choice of", "compactification $j : X \\to \\overline{X}$ of $X$ over $Y$", "the scheme $\\overline{X}$ is Tor independent of $Y'$.", "Denote $j' : X' \\to \\overline{X}'$ the", "base change of $j$ and $\\overline{g}' : \\overline{X}' \\to \\overline{X}$", "the projection. We define the base change map as the composition", "$$", "L(g')^* \\circ f^! = L(g')^* \\circ j^* \\circ a =", "(j')^* \\circ L(\\overline{g}')^* \\circ a \\longrightarrow", "(j')^* \\circ a' \\circ Lg^* = (f')^! \\circ Lg^*", "$$", "where the middle arrow is the base change map", "(\\ref{equation-base-change-map})", "and $a$ and $a'$ are the right adjoints to pushforward", "of Lemma \\ref{lemma-twisted-inverse-image}", "for $\\overline{X} \\to Y$ and $\\overline{X}' \\to Y'$.", "This construction is independent of the choice of", "compactification (we will formulate a precise lemma", "and prove it, if we ever need this result).", "\\medskip\\noindent", "To finish the proof it suffices to show that the base change", "map $L(g')^* \\circ a \\to a' \\circ Lg^*$ is an isomorphism", "on $D_\\QCoh^+(\\mathcal{O}_Y)$.", "By Lemma \\ref{lemma-proper-noetherian} formation of $a$ and $a'$", "commutes with restriction to affine opens of $Y$ and $Y'$.", "Thus by Remark \\ref{remark-check-over-affines}", "we may assume that $Y$ and $Y'$ are affine.", "Thus the result by Lemma \\ref{lemma-more-base-change}." ], "refs": [ "duality-lemma-twisted-inverse-image", "duality-lemma-proper-noetherian", "duality-remark-check-over-affines", "duality-lemma-more-base-change" ], "ref_ids": [ 13503, 13509, 13647, 13512 ] } ], "ref_ids": [] }, { "id": 13566, "type": "theorem", "label": "duality-lemma-shriek-etale", "categories": [ "duality" ], "title": "duality-lemma-shriek-etale", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be an \\'etale", "morphism of $\\textit{FTS}_S$. Then $f^! \\cong f^*$ as functors on", "$D^+_\\QCoh(\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "We are going to use that an \\'etale morphism is flat, syntomic,", "and a local complete intersection morphism", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-syntomic} and", "\\ref{morphisms-lemma-etale-flat} and", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-flat-lci}).", "By Lemma \\ref{lemma-perfect-comparison-shriek} it suffices", "to show $f^!\\mathcal{O}_Y = \\mathcal{O}_X$.", "By Lemma \\ref{lemma-lci-shriek} we know that $f^!\\mathcal{O}_Y$", "is an invertible module. Consider the commutative diagram", "$$", "\\xymatrix{", "X \\times_Y X \\ar[r]_{p_2} \\ar[d]_{p_1} & X \\ar[d]^f \\\\", "X \\ar[r]^f & Y", "}", "$$", "and the diagonal $\\Delta : X \\to X \\times_Y X$. Since $\\Delta$", "is an open immersion (by Morphisms, Lemmas", "\\ref{morphisms-lemma-diagonal-unramified-morphism} and", "\\ref{morphisms-lemma-etale-smooth-unramified}), by", "Lemma \\ref{lemma-shriek-open-immersion} we have $\\Delta^! = \\Delta^*$.", "By Lemma \\ref{lemma-upper-shriek-composition} we have", "$\\Delta^! \\circ p_1^! \\circ f^! = f^!$.", "By Lemma \\ref{lemma-base-change-shriek-flat} applied to", "the diagram we have $p_1^!\\mathcal{O}_X = p_2^*f^!\\mathcal{O}_Y$.", "Hence we conclude", "$$", "f^!\\mathcal{O}_X = \\Delta^!p_1^!f^!\\mathcal{O}_Y =", "\\Delta^*(p_1^*f^!\\mathcal{O}_Y \\otimes p_1^!\\mathcal{O}_X) =", "\\Delta^*(p_2^*f^!\\mathcal{O}_Y \\otimes p_1^*f^!\\mathcal{O}_Y) =", "(f^!\\mathcal{O}_Y)^{\\otimes 2}", "$$", "where in the second step we have used", "Lemma \\ref{lemma-perfect-comparison-shriek} once more.", "Thus $f^!\\mathcal{O}_Y = \\mathcal{O}_X$ as desired." ], "refs": [ "morphisms-lemma-etale-syntomic", "morphisms-lemma-etale-flat", "more-morphisms-lemma-flat-lci", "duality-lemma-perfect-comparison-shriek", "duality-lemma-lci-shriek", "morphisms-lemma-diagonal-unramified-morphism", "morphisms-lemma-etale-smooth-unramified", "duality-lemma-shriek-open-immersion", "duality-lemma-upper-shriek-composition", "duality-lemma-base-change-shriek-flat", "duality-lemma-perfect-comparison-shriek" ], "ref_ids": [ 5367, 5369, 14006, 13562, 13564, 5354, 5362, 13555, 13552, 13565, 13562 ] } ], "ref_ids": [] }, { "id": 13567, "type": "theorem", "label": "duality-lemma-base-change-locally", "categories": [ "duality" ], "title": "duality-lemma-base-change-locally", "contents": [ "In Situation \\ref{situation-shriek} let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "be a cartesian diagram of $\\textit{FTS}_S$.", "Let $E \\in D^+_\\QCoh(\\mathcal{O}_Y)$ be an object", "such that $Lg^*E$ is in $D^+(\\mathcal{O}_Y)$.", "If $f$ is flat, then $L(g')^*f^!E$ and $(f')^!Lg^*E$", "restrict to isomorphic objects of $D(\\mathcal{O}_{U'})$", "for $U' \\subset X'$ affine open mapping into affine opens of $Y$, $Y'$, and $X$." ], "refs": [], "proofs": [ { "contents": [ "By our assumptions we immediately reduce to the case where", "$X$, $Y$, $Y'$, and $X'$ are affine.", "Say $Y = \\Spec(R)$, $Y' = \\Spec(R')$, $X = \\Spec(A)$, and $X' = \\Spec(A')$.", "Then $A' = A \\otimes_R R'$. Let", "$E$ correspond to $K \\in D^+(R)$.", "Denoting $\\varphi : R \\to A$ and $\\varphi' : R' \\to A'$", "the given maps we see from", "Remark \\ref{remark-local-calculation-shriek}", "that $L(g')^*f^!E$ and $(f')^!Lg^*E$ correspond to", "$\\varphi^!(K) \\otimes_A^\\mathbf{L} A'$ and", "$(\\varphi')^!(K \\otimes_R^\\mathbf{L} R')$", "where $\\varphi^!$ and $(\\varphi')^!$ are the functors from", "Dualizing Complexes, Section", "\\ref{dualizing-section-relative-dualizing-complex-algebraic}.", "The result follows from", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-bc-flat}." ], "refs": [ "duality-remark-local-calculation-shriek", "dualizing-lemma-bc-flat" ], "ref_ids": [ 13652, 2899 ] } ], "ref_ids": [] }, { "id": 13568, "type": "theorem", "label": "duality-lemma-relative-dualizing-fibres", "categories": [ "duality" ], "title": "duality-lemma-relative-dualizing-fibres", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. Assume $f$ is flat. Set", "$\\omega_{X/Y}^\\bullet = f^!\\mathcal{O}_Y$ in $D^b_{\\textit{Coh}}(X)$.", "Let $y \\in Y$ and $h : X_y \\to X$ the projection.", "Then $Lh^*\\omega_{X/Y}^\\bullet$ is a dualizing complex", "on $X_y$." ], "refs": [], "proofs": [ { "contents": [ "The complex $\\omega_{X/Y}^\\bullet$ is in $D^b_{\\textit{Coh}}$", "by Lemma \\ref{lemma-perfect-comparison-shriek}.", "Being a dualizing complex is a local property.", "Hence by Lemma \\ref{lemma-base-change-locally}", "it suffices to show that $(X_y \\to y)^!\\mathcal{O}_y$", "is a dualizing complex on $X_y$.", "This follows from Lemma \\ref{lemma-shriek-dualizing}." ], "refs": [ "duality-lemma-perfect-comparison-shriek", "duality-lemma-base-change-locally", "duality-lemma-shriek-dualizing" ], "ref_ids": [ 13562, 13567, 13560 ] } ], "ref_ids": [] }, { "id": 13569, "type": "theorem", "label": "duality-lemma-good-dualizing-unique", "categories": [ "duality" ], "title": "duality-lemma-good-dualizing-unique", "contents": [ "In Situation \\ref{situation-dualizing} let $X$ be a scheme of finite type", "over $S$ and let $\\mathcal{U}$ be a finite open covering of $X$", "by schemes separated over $S$. If there exists a dualizing complex", "normalized relative to $\\omega_S^\\bullet$ and $\\mathcal{U}$, then it is unique", "up to unique isomorphism." ], "refs": [], "proofs": [ { "contents": [ "If $(K, \\alpha_i)$ and $(K', \\alpha_i')$ are two, then we consider", "$L = R\\SheafHom_{\\mathcal{O}_X}(K, K')$.", "By Lemma \\ref{lemma-dualizing-unique-schemes}", "and its proof, this is an invertible object of $D(\\mathcal{O}_X)$.", "Using $\\alpha_i$ and $\\alpha'_i$ we obtain an isomorphism", "$$", "\\alpha_i^t \\otimes \\alpha'_i :", "L|_{U_i} \\longrightarrow", "R\\SheafHom_{\\mathcal{O}_X}(\\omega_i^\\bullet, \\omega_i^\\bullet) =", "\\mathcal{O}_{U_i}[0]", "$$", "This already implies that $L = H^0(L)[0]$ in $D(\\mathcal{O}_X)$.", "Moreover, $H^0(L)$ is an invertible sheaf with given trivializations", "on the opens $U_i$ of $X$. Finally, the condition that", "$\\alpha_j|_{U_i \\cap U_j} \\circ \\alpha_i^{-1}|_{U_i \\cap U_j}$", "and", "$\\alpha'_j|_{U_i \\cap U_j} \\circ (\\alpha'_i)^{-1}|_{U_i \\cap U_j}$", "both give $\\varphi_{ij}$ implies that the transition maps", "are $1$ and we get an isomorphism $H^0(L) = \\mathcal{O}_X$." ], "refs": [ "duality-lemma-dualizing-unique-schemes" ], "ref_ids": [ 13500 ] } ], "ref_ids": [] }, { "id": 13570, "type": "theorem", "label": "duality-lemma-good-dualizing-independence-covering", "categories": [ "duality" ], "title": "duality-lemma-good-dualizing-independence-covering", "contents": [ "In Situation \\ref{situation-dualizing} let $X$ be a scheme of finite type", "over $S$ and let $\\mathcal{U}$, $\\mathcal{V}$ be two finite open coverings", "of $X$ by schemes separated over $S$.", "If there exists a dualizing complex normalized", "relative to $\\omega_S^\\bullet$ and $\\mathcal{U}$, then", "there exists a dualizing complex normalized relative to", "$\\omega_S^\\bullet$ and $\\mathcal{V}$ and these complexes are", "canonically isomorphic." ], "refs": [], "proofs": [ { "contents": [ "It suffices to prove this when $\\mathcal{U}$ is given by the opens", "$U_1, \\ldots, U_n$ and $\\mathcal{V}$ by the opens $U_1, \\ldots, U_{n + m}$.", "In fact, we may and do even assume $m = 1$.", "To go from a dualizing complex $(K, \\alpha_i)$ normalized", "relative to $\\omega_S^\\bullet$ and $\\mathcal{V}$ to a", "dualizing complex normalized relative to $\\omega_S^\\bullet$ and $\\mathcal{U}$", "is achieved by forgetting about $\\alpha_i$ for $i = n + 1$. Conversely, let", "$(K, \\alpha_i)$ be a dualizing complex normalized relative to", "$\\omega_S^\\bullet$ and $\\mathcal{U}$.", "To finish the proof we need to construct a map", "$\\alpha_{n + 1} : K|_{U_{n + 1}} \\to \\omega_{n + 1}^\\bullet$ satisfying", "the desired conditions.", "To do this we observe that $U_{n + 1} = \\bigcup U_i \\cap U_{n + 1}$", "is an open covering.", "It is clear that $(K|_{U_{n + 1}}, \\alpha_i|_{U_i \\cap U_{n + 1}})$", "is a dualizing complex normalized relative to $\\omega_S^\\bullet$", "and the covering $U_{n + 1} = \\bigcup U_i \\cap U_{n + 1}$.", "On the other hand, by condition (\\ref{item-cocycle-glueing}) the pair", "$(\\omega_{n + 1}^\\bullet|_{U_{n + 1}}, \\varphi_{n + 1i})$", "is another dualizing complex normalized relative to $\\omega_S^\\bullet$", "and the covering", "$U_{n + 1} = \\bigcup U_i \\cap U_{n + 1}$.", "By Lemma \\ref{lemma-good-dualizing-unique} we obtain a unique isomorphism", "$$", "\\alpha_{n + 1} : K|_{U_{n + 1}} \\longrightarrow \\omega_{n + 1}^\\bullet", "$$", "compatible with the given local isomorphisms.", "It is a pleasant exercise to show that this means it satisfies", "the required property." ], "refs": [ "duality-lemma-good-dualizing-unique" ], "ref_ids": [ 13569 ] } ], "ref_ids": [] }, { "id": 13571, "type": "theorem", "label": "duality-lemma-existence-good-dualizing", "categories": [ "duality" ], "title": "duality-lemma-existence-good-dualizing", "contents": [ "In Situation \\ref{situation-dualizing} let $X$ be a scheme of finite type", "over $S$ and let $\\mathcal{U}$ be a finite open covering", "of $X$ by schemes separated over $S$. Then there exists", "a dualizing complex normalized relative to $\\omega_S^\\bullet$ and", "$\\mathcal{U}$." ], "refs": [], "proofs": [ { "contents": [ "Say $\\mathcal{U} : X = \\bigcup_{i = 1, \\ldots, n} U_i$.", "We prove the lemma by induction on $n$. The base case $n = 1$ is immediate.", "Assume $n > 1$. Set $X' = U_1 \\cup \\ldots \\cup U_{n - 1}$", "and let $(K', \\{\\alpha'_i\\}_{i = 1, \\ldots, n - 1})$", "be a dualizing complex normalized relative to $\\omega_S^\\bullet$", "and $\\mathcal{U}' : X' = \\bigcup_{i = 1, \\ldots, n - 1} U_i$.", "It is clear that $(K'|_{X' \\cap U_n}, \\alpha'_i|_{U_i \\cap U_n})$", "is a dualizing complex normalized relative to $\\omega_S^\\bullet$", "and the covering", "$X' \\cap U_n = \\bigcup_{i = 1, \\ldots, n - 1} U_i \\cap U_n$.", "On the other hand, by condition (\\ref{item-cocycle-glueing}) the pair", "$(\\omega_n^\\bullet|_{X' \\cap U_n}, \\varphi_{ni})$", "is another dualizing complex normalized relative to $\\omega_S^\\bullet$", "and the covering", "$X' \\cap U_n = \\bigcup_{i = 1, \\ldots, n - 1} U_i \\cap U_n$.", "By Lemma \\ref{lemma-good-dualizing-unique} we obtain a unique isomorphism", "$$", "\\epsilon : K'|_{X' \\cap U_n} \\longrightarrow \\omega_i^\\bullet|_{X' \\cap U_n}", "$$", "compatible with the given local isomorphisms.", "By Cohomology, Lemma \\ref{cohomology-lemma-glue}", "we obtain $K \\in D(\\mathcal{O}_X)$ together with", "isomorphisms $\\beta : K|_{X'} \\to K'$ and", "$\\gamma : K|_{U_n} \\to \\omega_n^\\bullet$ such that", "$\\epsilon = \\gamma|_{X'\\cap U_n} \\circ \\beta|_{X' \\cap U_n}^{-1}$.", "Then we define", "$$", "\\alpha_i = \\alpha'_i \\circ \\beta|_{U_i}, i = 1, \\ldots, n - 1,", "\\text{ and }", "\\alpha_n = \\gamma", "$$", "We still need to verify that $\\varphi_{ij}$ is given by", "$\\alpha_j|_{U_i \\cap U_j} \\circ \\alpha_i^{-1}|_{U_i \\cap U_j}$.", "For $i, j \\leq n - 1$ this follows from the corresponding", "condition for $\\alpha_i'$. For $i = j = n$ it is clear as well.", "If $i < j = n$, then we get", "$$", "\\alpha_n|_{U_i \\cap U_n} \\circ \\alpha_i^{-1}|_{U_i \\cap U_n} =", "\\gamma|_{U_i \\cap U_n} \\circ \\beta^{-1}|_{U_i \\cap U_n}", "\\circ (\\alpha'_i)^{-1}|_{U_i \\cap U_n} =", "\\epsilon|_{U_i \\cap U_n} \\circ (\\alpha'_i)^{-1}|_{U_i \\cap U_n}", "$$", "This is equal to $\\alpha_{in}$ exactly because $\\epsilon$", "is the unique map compatible with the maps", "$\\alpha_i'$ and $\\alpha_{ni}$." ], "refs": [ "duality-lemma-good-dualizing-unique", "cohomology-lemma-glue" ], "ref_ids": [ 13569, 2191 ] } ], "ref_ids": [] }, { "id": 13572, "type": "theorem", "label": "duality-lemma-good-over-both", "categories": [ "duality" ], "title": "duality-lemma-good-over-both", "contents": [ "Let $(S, \\omega_S^\\bullet)$ be as in Situation \\ref{situation-dualizing}.", "Let $f : X \\to Y$ be a morphism of finite type schemes over $S$.", "Let $\\omega_X^\\bullet$ and $\\omega_Y^\\bullet$ be dualizing complexes", "normalized relative to $\\omega_S^\\bullet$. Then $\\omega_X^\\bullet$", "is a dualizing complex normalized relative to $\\omega_Y^\\bullet$." ], "refs": [], "proofs": [ { "contents": [ "This is just a matter of bookkeeping.", "Choose a finite affine open covering $\\mathcal{V} : Y = \\bigcup V_j$.", "For each $j$ choose a finite affine open covering $f^{-1}(V_j) = U_{ji}$.", "Set $\\mathcal{U} : X = \\bigcup U_{ji}$. The schemes $V_j$ and $U_{ji}$ are", "separated over $S$, hence we have the upper shriek functors for", "$q_j : V_j \\to S$, $p_{ji} : U_{ji} \\to S$ and", "$f_{ji} : U_{ji} \\to V_j$ and $f_{ji}' : U_{ji} \\to Y$.", "Let $(L, \\beta_j)$ be a dualizing complex normalized relative to", "$\\omega_S^\\bullet$ and $\\mathcal{V}$.", "Let $(K, \\gamma_{ji})$ be a dualizing complex normalized relative to", "$\\omega_S^\\bullet$ and $\\mathcal{U}$.", "(In other words, $L = \\omega_Y^\\bullet$ and $K = \\omega_X^\\bullet$.)", "We can define", "$$", "\\alpha_{ji} :", "K|_{U_{ji}} \\xrightarrow{\\gamma_{ji}}", "p_{ji}^!\\omega_S^\\bullet = f_{ji}^!q_j^!\\omega_S^\\bullet", "\\xrightarrow{f_{ji}^!\\beta_j^{-1}} f_{ji}^!(L|_{V_j}) =", "(f_{ji}')^!(L)", "$$", "To finish the proof we have to show that", "$\\alpha_{ji}|_{U_{ji} \\cap U_{j'i'}}", "\\circ \\alpha_{j'i'}^{-1}|_{U_{ji} \\cap U_{j'i'}}$", "is the canonical isomorphism", "$(f_{ji}')^!(L)|_{U_{ji} \\cap U_{j'i'}} \\to", "(f_{j'i'}')^!(L)|_{U_{ji} \\cap U_{j'i'}}$. This is formal and we", "omit the details." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13573, "type": "theorem", "label": "duality-lemma-open-immersion-good-dualizing-complex", "categories": [ "duality" ], "title": "duality-lemma-open-immersion-good-dualizing-complex", "contents": [ "Let $(S, \\omega_S^\\bullet)$ be as in Situation \\ref{situation-dualizing}.", "Let $j : X \\to Y$ be an open immersion of schemes of finite type over $S$.", "Let $\\omega_X^\\bullet$ and $\\omega_Y^\\bullet$ be dualizing complexes", "normalized relative to $\\omega_S^\\bullet$. Then there is a canonical", "isomorphism $\\omega_X^\\bullet = \\omega_Y^\\bullet|_X$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the construction of normalized dualizing complexes", "given just above", "Definition \\ref{definition-good-dualizing}." ], "refs": [ "duality-definition-good-dualizing" ], "ref_ids": [ 13641 ] } ], "ref_ids": [] }, { "id": 13574, "type": "theorem", "label": "duality-lemma-proper-map-good-dualizing-complex", "categories": [ "duality" ], "title": "duality-lemma-proper-map-good-dualizing-complex", "contents": [ "Let $(S, \\omega_S^\\bullet)$ be as in Situation \\ref{situation-dualizing}.", "Let $f : X \\to Y$ be a proper morphism of schemes of finite type over $S$.", "Let $\\omega_X^\\bullet$ and $\\omega_Y^\\bullet$ be dualizing complexes", "normalized relative to $\\omega_S^\\bullet$. Let $a$ be the", "right adjoint of Lemma \\ref{lemma-twisted-inverse-image} for", "$f$. Then there is a canonical isomorphism", "$a(\\omega_Y^\\bullet) = \\omega_X^\\bullet$." ], "refs": [ "duality-lemma-twisted-inverse-image" ], "proofs": [ { "contents": [ "Let $p : X \\to S$ and $q : Y \\to S$ be the structure morphisms.", "If $X$ and $Y$ are separated over $S$, then this follows", "from the fact that $\\omega_X^\\bullet = p^!\\omega_S^\\bullet$,", "$\\omega_Y^\\bullet = q^!\\omega_S^\\bullet$, $f^! = a$, and", "$f^! \\circ q^! = p^!$ (Lemma \\ref{lemma-upper-shriek-composition}).", "In the general case we first use Lemma \\ref{lemma-good-over-both}", "to reduce to the case $Y = S$. In this case $X$ and $Y$", "are separated over $S$ and we've just seen the result." ], "refs": [ "duality-lemma-upper-shriek-composition", "duality-lemma-good-over-both" ], "ref_ids": [ 13552, 13572 ] } ], "ref_ids": [ 13503 ] }, { "id": 13575, "type": "theorem", "label": "duality-lemma-duality-bootstrap", "categories": [ "duality" ], "title": "duality-lemma-duality-bootstrap", "contents": [ "Let $(S, \\omega_S^\\bullet)$ be as in Situation \\ref{situation-dualizing}.", "With $f^!_{new}$ and $\\omega_X^\\bullet$ defined for all (morphisms of)", "schemes of finite type over $S$ as above:", "\\begin{enumerate}", "\\item the functors $f^!_{new}$ and the arrows", "$(g \\circ f)^!_{new} \\to f^!_{new} \\circ g^!_{new}$", "turn $D_{\\textit{Coh}}^+$ into a pseudo functor from the category of", "schemes of finite type over $S$ into the $2$-category of categories,", "\\item $\\omega_X^\\bullet = (X \\to S)^!_{new} \\omega_S^\\bullet$,", "\\item the functor $D_X$", "defines an involution of $D_{\\textit{Coh}}(\\mathcal{O}_X)$", "switching $D_{\\textit{Coh}}^+(\\mathcal{O}_X)$ and", "$D_{\\textit{Coh}}^-(\\mathcal{O}_X)$ and fixing", "$D_{\\textit{Coh}}^b(\\mathcal{O}_X)$,", "\\item $\\omega_X^\\bullet = f^!_{new}\\omega_Y^\\bullet$ for", "$f : X \\to Y$ a morphism of finite type schemes over $S$,", "\\item $f^!_{new}M = D_X(Lf^*D_Y(M))$ for", "$M \\in D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$, and", "\\item if in addition $f$ is proper, then $f^!_{new}$ is isomorphic", "to the restriction of the right adjoint of", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ to", "$D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$ and there is a canonical isomorphism", "$$", "Rf_*R\\SheafHom_{\\mathcal{O}_X}(K, f^!_{new}M)", "\\to", "R\\SheafHom_{\\mathcal{O}_Y}(Rf_*K, M)", "$$", "for $K \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$ and", "$M \\in D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$, and", "$$", "Rf_*R\\SheafHom_{\\mathcal{O}_X}(K, \\omega_X^\\bullet) =", "R\\SheafHom_{\\mathcal{O}_Y}(Rf_*K, \\omega_Y^\\bullet)", "$$", "for $K \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$ and", "\\end{enumerate}", "If $X$ is separated over $S$, then", "$\\omega_X^\\bullet$ is canonically isomorphic to", "$(X \\to S)^!\\omega_S^\\bullet$ and", "if $f$ is a morphism between schemes separated", "over $S$, then there is a canonical isomorphism\\footnote{We haven't", "checked that these are compatible with the isomorphisms", "$(g \\circ f)^! \\to f^! \\circ g^!$ and", "$(g \\circ f)^!_{new} \\to f^!_{new} \\circ g^!_{new}$. We will do this", "here if we need this later.}", "$f_{new}^!K = f^!K$ for $K$ in $D_{\\textit{Coh}}^+$." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$, $g : Y \\to Z$, $h : Z \\to T$ be morphisms of schemes", "of finite type over $S$. We have to show that", "$$", "\\xymatrix{", "(h \\circ g \\circ f)^!_{new} \\ar[r] \\ar[d] &", "f^!_{new} \\circ (h \\circ g)^!_{new} \\ar[d] \\\\", "(g \\circ f)^!_{new} \\circ h^!_{new} \\ar[r] &", "f^!_{new} \\circ g^!_{new} \\circ h^!_{new}", "}", "$$", "is commutative. Let $\\eta_Y : \\text{id} \\to D_Y^2$", "and $\\eta_Z : \\text{id} \\to D_Z^2$ be the canonical isomorphisms", "of Lemma \\ref{lemma-dualizing-schemes}. Then, using", "Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats},", "a computation (omitted) shows that both arrows", "$(h \\circ g \\circ f)^!_{new} \\to f^!_{new} \\circ g^!_{new} \\circ h^!_{new}$", "are given by", "$$", "1 \\star \\eta_Y \\star 1 \\star \\eta_Z \\star 1 :", "D_X \\circ Lf^* \\circ Lg^* \\circ Lh^* \\circ D_T", "\\longrightarrow", "D_X \\circ Lf^* \\circ D_Y^2 \\circ Lg^* \\circ D_Z^2 \\circ Lh^* \\circ D_T", "$$", "This proves (1). Part (2) is immediate from the definition of", "$(X \\to S)^!_{new}$ and the fact that $D_S(\\omega_S^\\bullet) = \\mathcal{O}_S$.", "Part (3) is Lemma \\ref{lemma-dualizing-schemes}.", "Part (4) follows by the same argument as part (2).", "Part (5) is the definition of $f^!_{new}$.", "\\medskip\\noindent", "Proof of (6). Let $a$ be the", "right adjoint of Lemma \\ref{lemma-twisted-inverse-image} for the", "proper morphism $f : X \\to Y$ of schemes of finite type over $S$.", "The issue is that we do not know $X$ or $Y$ is", "separated over $S$ (and in general this won't be true)", "hence we cannot immediately apply", "Lemma \\ref{lemma-shriek-via-duality} to $f$ over $S$.", "To get around this we use the canonical identification", "$\\omega_X^\\bullet = a(\\omega_Y^\\bullet)$ of", "Lemma \\ref{lemma-proper-map-good-dualizing-complex}.", "Hence $f^!_{new}$ is the restriction of $a$ to", "$D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$ by Lemma \\ref{lemma-shriek-via-duality}", "applied to $f : X \\to Y$ over the base scheme $Y$!", "The displayed equalities hold by", "Example \\ref{example-iso-on-RSheafHom-noetherian}.", "\\medskip\\noindent", "The final assertions follow from the construction of normalized", "dualizing complexes and the already used Lemma \\ref{lemma-shriek-via-duality}." ], "refs": [ "duality-lemma-dualizing-schemes", "categories-lemma-properties-2-cat-cats", "duality-lemma-dualizing-schemes", "duality-lemma-twisted-inverse-image", "duality-lemma-shriek-via-duality", "duality-lemma-proper-map-good-dualizing-complex", "duality-lemma-shriek-via-duality", "duality-lemma-shriek-via-duality" ], "ref_ids": [ 13499, 12269, 13499, 13503, 13561, 13574, 13561, 13561 ] } ], "ref_ids": [] }, { "id": 13576, "type": "theorem", "label": "duality-lemma-good-dualizing-normalized", "categories": [ "duality" ], "title": "duality-lemma-good-dualizing-normalized", "contents": [ "Let $S$ be a Noetherian scheme and let $\\omega_S^\\bullet$ be a", "dualizing complex. Let $X$ be a scheme of finite type over $S$ and let", "$\\omega_X^\\bullet$ be the dualizing complex normalized relative", "to $\\omega_S^\\bullet$. If $x \\in X$ is a closed point lying over", "a closed point $s$ of $S$, then $\\omega_{X, x}^\\bullet$", "is a normalized dualizing complex over $\\mathcal{O}_{X, x}$", "provided that $\\omega_{S, s}^\\bullet$ is a normalized dualizing", "complex over $\\mathcal{O}_{S, s}$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $X$ by an affine neighbourhood of $x$, hence we may", "and do assume that $f : X \\to S$ is separated.", "Then $\\omega_X^\\bullet = f^!\\omega_S^\\bullet$. We have to show that", "$R\\Hom_{\\mathcal{O}_{X, x}}(\\kappa(x), \\omega_{X, x}^\\bullet)$", "is sitting in degree $0$. Let $i_x : x \\to X$ denote the inclusion", "morphism which is a closed immersion as $x$ is a closed point.", "Hence $R\\Hom_{\\mathcal{O}_{X, x}}(\\kappa(x), \\omega_{X, x}^\\bullet)$", "represents $i_x^!\\omega_X^\\bullet$ by", "Lemma \\ref{lemma-shriek-closed-immersion}.", "Consider the commutative diagram", "$$", "\\xymatrix{", "x \\ar[r]_{i_x} \\ar[d]_\\pi & X \\ar[d]^f \\\\", "s \\ar[r]^{i_s} & S", "}", "$$", "By Morphisms, Lemma", "\\ref{morphisms-lemma-closed-point-fibre-locally-finite-type}", "the extension $\\kappa(s) \\subset \\kappa(x)$ is finite and hence", "$\\pi$ is a finite morphism. We conclude that", "$$", "i_x^!\\omega_X^\\bullet = i_x^! f^! \\omega_S^\\bullet =", "\\pi^! i_s^! \\omega_S^\\bullet", "$$", "Thus if $\\omega_{S, s}^\\bullet$ is a normalized dualizing complex", "over $\\mathcal{O}_{S, s}$, then $i_s^!\\omega_S^\\bullet = \\kappa(s)[0]$", "by the same reasoning as above. We have", "$$", "R\\pi_*(\\pi^!(\\kappa(s)[0])) =", "R\\SheafHom_{\\mathcal{O}_s}(R\\pi_*(\\kappa(x)[0]), \\kappa(s)[0]) =", "\\widetilde{\\Hom_{\\kappa(s)}(\\kappa(x), \\kappa(s))}", "$$", "The first equality by Example \\ref{example-iso-on-RSheafHom-noetherian}", "applied with $L = \\kappa(x)[0]$. The second equality holds because", "$\\pi_*$ is exact.", "Thus $\\pi^!(\\kappa(s)[0])$ is supported in degree $0$ and we win." ], "refs": [ "duality-lemma-shriek-closed-immersion", "morphisms-lemma-closed-point-fibre-locally-finite-type" ], "ref_ids": [ 13558, 5223 ] } ], "ref_ids": [] }, { "id": 13577, "type": "theorem", "label": "duality-lemma-good-dualizing-dimension-function", "categories": [ "duality" ], "title": "duality-lemma-good-dualizing-dimension-function", "contents": [ "Let $S$ be a Noetherian scheme and let $\\omega_S^\\bullet$ be a", "dualizing complex. Let $f : X \\to S$ be of finite type", "and let $\\omega_X^\\bullet$ be the dualizing complex", "normalized relative to $\\omega_S^\\bullet$. For all $x \\in X$ we have", "$$", "\\delta_X(x) - \\delta_S(f(x)) = \\text{trdeg}_{\\kappa(f(x))}(\\kappa(x))", "$$", "where $\\delta_S$, resp.\\ $\\delta_X$", "is the dimension function of", "$\\omega_S^\\bullet$, resp.\\ $\\omega_X^\\bullet$, see", "Lemma \\ref{lemma-dimension-function-scheme}." ], "refs": [ "duality-lemma-dimension-function-scheme" ], "proofs": [ { "contents": [ "We may replace $X$ by an affine neighbourhood of $x$. Hence we may", "and do assume there is a compactification $X \\subset \\overline{X}$", "over $S$. Then we may replace $X$ by $\\overline{X}$ and assume", "that $X$ is proper over $S$. We may also assume $X$ is connected", "by replacing $X$ by the connected component of $X$ containing $x$.", "Next, recall that both $\\delta_X$ and the function", "$x \\mapsto \\delta_S(f(x)) + \\text{trdeg}_{\\kappa(f(x))}(\\kappa(x))$", "are dimension functions on $X$, see", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-function-propagates}", "(and the fact that $S$ is universally catenary by", "Lemma \\ref{lemma-dimension-function-scheme}).", "By Topology, Lemma \\ref{topology-lemma-dimension-function-unique}", "we see that the difference is locally constant, hence constant as $X$ is", "connected. Thus it suffices to prove equality in any point of $X$.", "By Properties, Lemma \\ref{properties-lemma-locally-Noetherian-closed-point}", "the scheme $X$ has a closed point $x$. Since $X \\to S$ is proper", "the image $s$ of $x$ is closed in $S$. Thus we may apply", "Lemma \\ref{lemma-good-dualizing-normalized} to conclude." ], "refs": [ "morphisms-lemma-dimension-function-propagates", "duality-lemma-dimension-function-scheme", "topology-lemma-dimension-function-unique", "properties-lemma-locally-Noetherian-closed-point", "duality-lemma-good-dualizing-normalized" ], "ref_ids": [ 5495, 13501, 8292, 2958, 13576 ] } ], "ref_ids": [ 13501 ] }, { "id": 13578, "type": "theorem", "label": "duality-lemma-shriek", "categories": [ "duality" ], "title": "duality-lemma-shriek", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. Let $x \\in X$ with image $y \\in Y$. Then", "$$", "H^i(f^!\\mathcal{O}_Y)_x \\not = 0", "\\Rightarrow - \\dim_x(X_y) \\leq i.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since the statement is local on $X$ we may assume $X$", "and $Y$ are affine schemes. Write", "$X = \\Spec(A)$ and $Y = \\Spec(R)$.", "Then $f^!\\mathcal{O}_Y$ corresponds to the relative dualizing", "complex $\\omega_{A/R}^\\bullet$ of", "Dualizing Complexes, Section", "\\ref{dualizing-section-relative-dualizing-complexes-Noetherian}", "by Remark \\ref{remark-local-calculation-shriek}.", "Thus the lemma follows from Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-relative-dualizing-trivial-vanishing}." ], "refs": [ "duality-remark-local-calculation-shriek", "dualizing-lemma-relative-dualizing-trivial-vanishing" ], "ref_ids": [ 13652, 2911 ] } ], "ref_ids": [] }, { "id": 13579, "type": "theorem", "label": "duality-lemma-flat-shriek", "categories": [ "duality" ], "title": "duality-lemma-flat-shriek", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. Let $x \\in X$ with image $y \\in Y$.", "If $f$ is flat, then", "$$", "H^i(f^!\\mathcal{O}_Y)_x \\not = 0", "\\Rightarrow - \\dim_x(X_y) \\leq i \\leq 0.", "$$", "In fact, if all fibres of $f$ have dimension $\\leq d$, then", "$f^!\\mathcal{O}_Y$ has tor-amplitude in $[-d, 0]$ as an object", "of $D(X, f^{-1}\\mathcal{O}_Y)$." ], "refs": [], "proofs": [ { "contents": [ "Arguing exactly as in the proof of Lemma \\ref{lemma-shriek}", "this follows from Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-relative-dualizing-flat-vanishing}." ], "refs": [ "duality-lemma-shriek", "dualizing-lemma-relative-dualizing-flat-vanishing" ], "ref_ids": [ 13578, 2912 ] } ], "ref_ids": [] }, { "id": 13580, "type": "theorem", "label": "duality-lemma-shriek-over-CM", "categories": [ "duality" ], "title": "duality-lemma-shriek-over-CM", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. Let $x \\in X$ with image $y \\in Y$. Assume", "\\begin{enumerate}", "\\item $\\mathcal{O}_{Y, y}$ is Cohen-Macaulay, and", "\\item $\\text{trdeg}_{\\kappa(f(\\xi))}(\\kappa(\\xi)) \\leq r$", "for any generic point $\\xi$ of an irreducible component", "of $X$ containing $x$.", "\\end{enumerate}", "Then", "$$", "H^i(f^!\\mathcal{O}_Y)_x \\not = 0", "\\Rightarrow - r \\leq i", "$$", "and the stalk $H^{-r}(f^!\\mathcal{O}_Y)_x$ is $(S_2)$ as an", "$\\mathcal{O}_{X, x}$-module." ], "refs": [], "proofs": [ { "contents": [ "After replacing $X$ by an open neighbourhood of $x$, we may", "assume every irreducible component of $X$ passes through $x$.", "Then arguing exactly as in the proof of Lemma \\ref{lemma-shriek}", "this follows from Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-relative-dualizing-CM-vanishing}." ], "refs": [ "duality-lemma-shriek", "dualizing-lemma-relative-dualizing-CM-vanishing" ], "ref_ids": [ 13578, 2913 ] } ], "ref_ids": [] }, { "id": 13581, "type": "theorem", "label": "duality-lemma-flat-quasi-finite-shriek", "categories": [ "duality" ], "title": "duality-lemma-flat-quasi-finite-shriek", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. If $f$ is flat and quasi-finite, then", "$$", "f^!\\mathcal{O}_Y = \\omega_{X/Y}[0]", "$$", "for some coherent $\\mathcal{O}_X$-module $\\omega_{X/Y}$ flat over $Y$." ], "refs": [], "proofs": [ { "contents": [ "Consequence of Lemma \\ref{lemma-flat-shriek} and the fact that the", "cohomology sheaves of $f^!\\mathcal{O}_Y$ are coherent by", "Lemma \\ref{lemma-shriek-coherent}." ], "refs": [ "duality-lemma-flat-shriek", "duality-lemma-shriek-coherent" ], "ref_ids": [ 13579, 13559 ] } ], "ref_ids": [] }, { "id": 13582, "type": "theorem", "label": "duality-lemma-CM-shriek", "categories": [ "duality" ], "title": "duality-lemma-CM-shriek", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. If $f$ is Cohen-Macaulay (More on Morphisms, Definition", "\\ref{more-morphisms-definition-CM}), then", "$$", "f^!\\mathcal{O}_Y = \\omega_{X/Y}[d]", "$$", "for some coherent $\\mathcal{O}_X$-module $\\omega_{X/Y}$ flat over $Y$", "where $d$ is the locally constant", "function on $X$ which gives the relative dimension of $X$ over $Y$." ], "refs": [ "more-morphisms-definition-CM" ], "proofs": [ { "contents": [ "The relative dimension $d$ is well defined and locally constant by", "Morphisms, Lemma", "\\ref{morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension}.", "The cohomology sheaves of $f^!\\mathcal{O}_Y$ are coherent by", "Lemma \\ref{lemma-shriek-coherent}.", "We will get flatness of $\\omega_{X/Y}$ from Lemma \\ref{lemma-flat-shriek}", "if we can show the other cohomology sheaves of $f^!\\mathcal{O}_Y$", "are zero.", "\\medskip\\noindent", "The question is local on $X$, hence we may assume $X$ and $Y$ are affine", "and the morphism has relative dimension $d$. If $d = 0$, then the", "result follows directly from Lemma \\ref{lemma-flat-quasi-finite-shriek}.", "If $d > 0$, then we may assume there is a factorization", "$$", "X \\xrightarrow{g} \\mathbf{A}^d_Y \\xrightarrow{p} Y", "$$", "with $g$ quasi-finite and flat, see More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-flat-finite-presentation-characterize-CM}.", "Then $f^! = g^! \\circ p^!$. By Lemma \\ref{lemma-shriek-affine-line}", "we see that $p^!\\mathcal{O}_Y \\cong \\mathcal{O}_{\\mathbf{A}^d_Y}[-d]$.", "We conclude by the case $d = 0$." ], "refs": [ "morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension", "duality-lemma-shriek-coherent", "duality-lemma-flat-shriek", "duality-lemma-flat-quasi-finite-shriek", "more-morphisms-lemma-flat-finite-presentation-characterize-CM", "duality-lemma-shriek-affine-line" ], "ref_ids": [ 5286, 13559, 13579, 13581, 13790, 13557 ] } ], "ref_ids": [ 14115 ] }, { "id": 13583, "type": "theorem", "label": "duality-lemma-dualizing-module", "categories": [ "duality" ], "title": "duality-lemma-dualizing-module", "contents": [ "Let $X$ be a connected Noetherian scheme and let $\\omega_X$ be a dualizing", "module on $X$. The support of $\\omega_X$ is the union of the irreducible", "components of maximal dimension with respect to any dimension function", "and $\\omega_X$ is a coherent $\\mathcal{O}_X$-module having property $(S_2)$." ], "refs": [], "proofs": [ { "contents": [ "By our conventions discussed above there exists a dualizing complex", "$\\omega_X^\\bullet$ such that $\\omega_X$ is the leftmost nonvanishing", "cohomology sheaf. Since $X$ is connected, any two dimension functions", "differ by a constant", "(Topology, Lemma \\ref{topology-lemma-dimension-function-unique}).", "Hence we may use the", "dimension function associated to $\\omega_X^\\bullet$", "(Lemma \\ref{lemma-dimension-function-scheme}).", "With these remarks in place, the lemma now", "follows from Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-depth-dualizing-module}", "and the definitions (in particular", "Cohomology of Schemes, Definition \\ref{coherent-definition-depth})." ], "refs": [ "topology-lemma-dimension-function-unique", "duality-lemma-dimension-function-scheme", "dualizing-lemma-depth-dualizing-module", "coherent-definition-depth" ], "ref_ids": [ 8292, 13501, 2871, 3403 ] } ], "ref_ids": [] }, { "id": 13584, "type": "theorem", "label": "duality-lemma-vanishing-good-dualizing", "categories": [ "duality" ], "title": "duality-lemma-vanishing-good-dualizing", "contents": [ "Let $X/A$ with $\\omega_X^\\bullet$ and $\\omega_X$ be as in", "Example \\ref{example-proper-over-local}. Then", "\\begin{enumerate}", "\\item $H^i(\\omega_X^\\bullet) \\not = 0 \\Rightarrow", "i \\in \\{-\\dim(X), \\ldots, 0\\}$,", "\\item the dimension of the support of $H^i(\\omega_X^\\bullet)$ is at most $-i$,", "\\item $\\text{Supp}(\\omega_X)$ is the union of", "the components of dimension $\\dim(X)$, and", "\\item $\\omega_X$ has property $(S_2)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\delta_X$ and $\\delta_S$ be the dimension functions associated to", "$\\omega_X^\\bullet$ and $\\omega_S^\\bullet$ as in", "Lemma \\ref{lemma-good-dualizing-dimension-function}.", "As $X$ is proper over $A$, every closed subscheme of $X$ contains", "a closed point $x$ which maps to the closed point $s \\in S$", "and $\\delta_X(x) = \\delta_S(s) = 0$. Hence", "$\\delta_X(\\xi) = \\dim(\\overline{\\{\\xi\\}})$ for any point", "$\\xi \\in X$. Hence we can check each of", "the statements of the lemma by looking at what happens over", "$\\Spec(\\mathcal{O}_{X, x})$ in which case the result follows", "from Dualizing Complexes, Lemmas \\ref{dualizing-lemma-sitting-in-degrees} and", "\\ref{dualizing-lemma-depth-dualizing-module}.", "Some details omitted.", "The last two statements can also be deduced from", "Lemma \\ref{lemma-dualizing-module}." ], "refs": [ "duality-lemma-good-dualizing-dimension-function", "dualizing-lemma-sitting-in-degrees", "dualizing-lemma-depth-dualizing-module", "duality-lemma-dualizing-module" ], "ref_ids": [ 13577, 2861, 2871, 13583 ] } ], "ref_ids": [] }, { "id": 13585, "type": "theorem", "label": "duality-lemma-dualizing-module-proper-over-A", "categories": [ "duality" ], "title": "duality-lemma-dualizing-module-proper-over-A", "contents": [ "Let $X/A$ with dualizing module $\\omega_X$ be as in", "Example \\ref{example-proper-over-local}.", "Let $d = \\dim(X_s)$ be the dimension", "of the closed fibre. If $\\dim(X) = d + \\dim(A)$, then", "the dualizing module $\\omega_X$ represents the functor", "$$", "\\mathcal{F} \\longmapsto \\Hom_A(H^d(X, \\mathcal{F}), \\omega_A)", "$$", "on the category of coherent $\\mathcal{O}_X$-modules." ], "refs": [], "proofs": [ { "contents": [ "We have", "\\begin{align*}", "\\Hom_X(\\mathcal{F}, \\omega_X)", "& =", "\\Ext^{-\\dim(X)}_X(\\mathcal{F}, \\omega_X^\\bullet) \\\\", "& =", "\\Hom_X(\\mathcal{F}[\\dim(X)], \\omega_X^\\bullet) \\\\", "& =", "\\Hom_X(\\mathcal{F}[\\dim(X)], f^!(\\omega_A^\\bullet)) \\\\", "& =", "\\Hom_S(Rf_*\\mathcal{F}[\\dim(X)], \\omega_A^\\bullet) \\\\", "& =", "\\Hom_A(H^d(X, \\mathcal{F}), \\omega_A)", "\\end{align*}", "The first equality because $H^i(\\omega_X^\\bullet) = 0$ for", "$i < -\\dim(X)$, see Lemma \\ref{lemma-vanishing-good-dualizing} and", "Derived Categories, Lemma \\ref{derived-lemma-negative-exts}.", "The second equality is follows from the definition of Ext groups.", "The third equality is our choice of $\\omega_X^\\bullet$.", "The fourth equality holds because $f^!$ is the", "right adjoint of Lemma \\ref{lemma-twisted-inverse-image} for", "$f$, see Section \\ref{section-duality}.", "The final equality holds because $R^if_*\\mathcal{F}$ is zero", "for $i > d$ (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-higher-direct-images-zero-above-dimension-fibre})", "and $H^j(\\omega_A^\\bullet)$ is zero for $j < -\\dim(A)$." ], "refs": [ "duality-lemma-vanishing-good-dualizing", "derived-lemma-negative-exts", "duality-lemma-twisted-inverse-image", "coherent-lemma-higher-direct-images-zero-above-dimension-fibre" ], "ref_ids": [ 13584, 1893, 13503, 3364 ] } ], "ref_ids": [] }, { "id": 13586, "type": "theorem", "label": "duality-lemma-dualizing-module-CM-scheme", "categories": [ "duality" ], "title": "duality-lemma-dualizing-module-CM-scheme", "contents": [ "Let $X$ be a locally Noetherian scheme with dualizing complex", "$\\omega_X^\\bullet$.", "\\begin{enumerate}", "\\item $X$ is Cohen-Macaulay $\\Leftrightarrow$ $\\omega_X^\\bullet$", "locally has a unique nonzero cohomology sheaf,", "\\item $\\mathcal{O}_{X, x}$ is Cohen-Macaulay $\\Leftrightarrow$", "$\\omega_{X, x}^\\bullet$ has a unique nonzero cohomology,", "\\item $U = \\{x \\in X \\mid \\mathcal{O}_{X, x}\\text{ is Cohen-Macaulay}\\}$", "is open and Cohen-Macaulay.", "\\end{enumerate}", "If $X$ is connected and Cohen-Macaulay, then there is an integer $n$", "and a coherent Cohen-Macaulay $\\mathcal{O}_X$-module $\\omega_X$", "such that $\\omega_X^\\bullet = \\omega_X[-n]$." ], "refs": [], "proofs": [ { "contents": [ "By definition and Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-dualizing-localize} for every $x \\in X$", "the complex $\\omega_{X, x}^\\bullet$ is a dualizing complex over", "$\\mathcal{O}_{X, x}$. By", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-apply-CM}", "we see that (2) holds.", "\\medskip\\noindent", "To see (3) assume that $\\mathcal{O}_{X, x}$ is Cohen-Macaulay.", "Let $n_x$ be the unique integer such that", "$H^{n_{x}}(\\omega_{X, x}^\\bullet)$ is nonzero.", "For an affine neighbourhood $V \\subset X$", "of $x$ we have $\\omega_X^\\bullet|_V$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_V)$", "hence there are finitely many nonzero coherent modules", "$H^i(\\omega_X^\\bullet)|_V$. Thus after shrinking $V$ we may assume", "only $H^{n_x}$ is nonzero, see", "Modules, Lemma \\ref{modules-lemma-finite-type-stalk-zero}.", "In this way we see that $\\mathcal{O}_{X, v}$ is Cohen-Macaulay", "for every $v \\in V$. This proves that $U$ is open as well", "as a Cohen-Macaulay scheme.", "\\medskip\\noindent", "Proof of (1). The implication $\\Leftarrow$ follows from (2).", "The implication $\\Rightarrow$ follows from the discussion", "in the previous paragraph, where we showed that if $\\mathcal{O}_{X, x}$", "is Cohen-Macaulay, then in a neighbourhood of $x$ the complex", "$\\omega_X^\\bullet$ has only one nonzero cohomology sheaf.", "\\medskip\\noindent", "Assume $X$ is connected and Cohen-Macaulay. The above shows that", "the map $x \\mapsto n_x$ is locally constant.", "Since $X$ is connected it is constant, say equal to $n$.", "Setting $\\omega_X = H^n(\\omega_X^\\bullet)$ we see that the lemma", "holds because $\\omega_X$ is Cohen-Macaulay by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-apply-CM}", "(and Cohomology of Schemes, Definition", "\\ref{coherent-definition-Cohen-Macaulay})." ], "refs": [ "dualizing-lemma-dualizing-localize", "dualizing-lemma-apply-CM", "modules-lemma-finite-type-stalk-zero", "dualizing-lemma-apply-CM", "coherent-definition-Cohen-Macaulay" ], "ref_ids": [ 2851, 2875, 13239, 2875, 3404 ] } ], "ref_ids": [] }, { "id": 13587, "type": "theorem", "label": "duality-lemma-has-dualizing-module-CM-scheme", "categories": [ "duality" ], "title": "duality-lemma-has-dualizing-module-CM-scheme", "contents": [ "Let $X$ be a locally Noetherian scheme. If there exists a coherent sheaf", "$\\omega_X$ such that $\\omega_X[0]$ is a dualizing complex on $X$, then", "$X$ is a Cohen-Macaulay scheme." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-has-dualizing-module-CM}", "and our definitions." ], "refs": [ "dualizing-lemma-has-dualizing-module-CM" ], "ref_ids": [ 2876 ] } ], "ref_ids": [] }, { "id": 13588, "type": "theorem", "label": "duality-lemma-affine-flat-Noetherian-CM", "categories": [ "duality" ], "title": "duality-lemma-affine-flat-Noetherian-CM", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism", "of $\\textit{FTS}_S$. Let $x \\in X$. If $f$ is flat, then", "the following are equivalent", "\\begin{enumerate}", "\\item $f$ is Cohen-Macaulay at $x$,", "\\item $f^!\\mathcal{O}_Y$ has a unique nonzero cohomology sheaf", "in a neighbourhood of $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "One direction of the lemma follows from Lemma \\ref{lemma-CM-shriek}.", "To prove the converse, we may assume $f^!\\mathcal{O}_Y$ has a unique", "nonzero cohomology sheaf. Let $y = f(x)$. Let $\\xi_1, \\ldots, \\xi_n \\in X_y$", "be the generic points of the fibre $X_y$ specializing to $x$.", "Let $d_1, \\ldots, d_n$ be the dimensions of the corresponding", "irreducible components of $X_y$. The morphism $f : X \\to Y$ is Cohen-Macaulay", "at $\\eta_i$ by More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-flat-finite-presentation-CM-open}.", "Hence by Lemma \\ref{lemma-CM-shriek} we see that", "$d_1 = \\ldots = d_n$. If $d$ denotes the common value, then $d = \\dim_x(X_y)$.", "After shrinking $X$ we may assume all fibres have dimension at most $d$", "(Morphisms, Lemma \\ref{morphisms-lemma-openness-bounded-dimension-fibres}).", "Then the only nonzero cohomology sheaf $\\omega = H^{-d}(f^!\\mathcal{O}_Y)$", "is flat over $Y$ by Lemma \\ref{lemma-flat-shriek}.", "Hence, if $h : X_y \\to X$ denotes the canonical morphism, then", "$Lh^*(f^!\\mathcal{O}_Y) = Lh^*(\\omega[d]) = (h^*\\omega)[d]$", "by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-tor-independence-and-tor-amplitude}.", "Thus $h^*\\omega[d]$ is the dualizing complex of $X_y$ by", "Lemma \\ref{lemma-relative-dualizing-fibres}.", "Hence $X_y$ is Cohen-Macaulay by", "Lemma \\ref{lemma-dualizing-module-CM-scheme}.", "This proves $f$ is Cohen-Macaulay at $x$ as desired." ], "refs": [ "duality-lemma-CM-shriek", "more-morphisms-lemma-flat-finite-presentation-CM-open", "duality-lemma-CM-shriek", "morphisms-lemma-openness-bounded-dimension-fibres", "duality-lemma-flat-shriek", "perfect-lemma-tor-independence-and-tor-amplitude", "duality-lemma-relative-dualizing-fibres", "duality-lemma-dualizing-module-CM-scheme" ], "ref_ids": [ 13582, 13789, 13582, 5280, 13579, 7030, 13568, 13586 ] } ], "ref_ids": [] }, { "id": 13589, "type": "theorem", "label": "duality-lemma-gorenstein-CM", "categories": [ "duality" ], "title": "duality-lemma-gorenstein-CM", "contents": [ "A Gorenstein scheme is Cohen-Macaulay." ], "refs": [], "proofs": [ { "contents": [ "Looking affine locally this follows from the corresponding", "result in algebra, namely", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-gorenstein-CM}." ], "refs": [ "dualizing-lemma-gorenstein-CM" ], "ref_ids": [ 2879 ] } ], "ref_ids": [] }, { "id": 13590, "type": "theorem", "label": "duality-lemma-regular-gorenstein", "categories": [ "duality" ], "title": "duality-lemma-regular-gorenstein", "contents": [ "A regular scheme is Gorenstein." ], "refs": [], "proofs": [ { "contents": [ "Looking affine locally this follows from the corresponding", "result in algebra, namely", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-regular-gorenstein}." ], "refs": [ "dualizing-lemma-regular-gorenstein" ], "ref_ids": [ 2880 ] } ], "ref_ids": [] }, { "id": 13591, "type": "theorem", "label": "duality-lemma-gorenstein", "categories": [ "duality" ], "title": "duality-lemma-gorenstein", "contents": [ "Let $X$ be a locally Noetherian scheme.", "\\begin{enumerate}", "\\item If $X$ has a dualizing complex $\\omega_X^\\bullet$, then", "\\begin{enumerate}", "\\item $X$ is Gorenstein $\\Leftrightarrow$ $\\omega_X^\\bullet$ is an invertible", "object of $D(\\mathcal{O}_X)$,", "\\item $\\mathcal{O}_{X, x}$ is Gorenstein $\\Leftrightarrow$", "$\\omega_{X, x}^\\bullet$ is an invertible object of $D(\\mathcal{O}_{X, x})$,", "\\item $U = \\{x \\in X \\mid \\mathcal{O}_{X, x}\\text{ is Gorenstein}\\}$", "is an open Gorenstein subscheme.", "\\end{enumerate}", "\\item If $X$ is Gorenstein, then $X$ has a dualizing complex if and", "only if $\\mathcal{O}_X[0]$ is a dualizing complex.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Looking affine locally this follows from the corresponding", "result in algebra, namely", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-gorenstein}." ], "refs": [ "dualizing-lemma-gorenstein" ], "ref_ids": [ 2881 ] } ], "ref_ids": [] }, { "id": 13592, "type": "theorem", "label": "duality-lemma-gorenstein-lci", "categories": [ "duality" ], "title": "duality-lemma-gorenstein-lci", "contents": [ "If $f : Y \\to X$ is a local complete intersection morphism", "with $X$ a Gorenstein scheme, then $Y$ is Gorenstein." ], "refs": [], "proofs": [ { "contents": [ "By More on Morphisms, Lemma \\ref{more-morphisms-lemma-affine-lci}", "it suffices to prove the corresponding statement about ring maps.", "This is Dualizing Complexes, Lemma \\ref{dualizing-lemma-gorenstein-lci}." ], "refs": [ "more-morphisms-lemma-affine-lci", "dualizing-lemma-gorenstein-lci" ], "ref_ids": [ 14003, 2884 ] } ], "ref_ids": [] }, { "id": 13593, "type": "theorem", "label": "duality-lemma-gorenstein-local-syntomic", "categories": [ "duality" ], "title": "duality-lemma-gorenstein-local-syntomic", "contents": [ "The property $\\mathcal{P}(S) =$``$S$ is Gorenstein''", "is local in the syntomic topology." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{S_i \\to S\\}$ be a syntomic covering. The scheme $S$ is locally", "Noetherian if and only if each $S_i$ is Noetherian, see", "Descent, Lemma \\ref{descent-lemma-Noetherian-local-fppf}.", "Thus we may now assume $S$ and $S_i$ are locally Noetherian.", "If $S$ is Gorenstein, then", "each $S_i$ is Gorenstein by Lemma \\ref{lemma-gorenstein-lci}.", "Conversely, if each $S_i$ is Gorenstein, then for each point", "$s \\in S$ we can pick $i$ and $t \\in S_i$ mapping to $s$.", "Then $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{S_i, t}$", "is a flat local ring homomorphism with $\\mathcal{O}_{S_i, t}$", "Gorenstein. Hence $\\mathcal{O}_{S, s}$ is Gorenstein by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-flat-under-gorenstein}." ], "refs": [ "descent-lemma-Noetherian-local-fppf", "duality-lemma-gorenstein-lci", "dualizing-lemma-flat-under-gorenstein" ], "ref_ids": [ 14648, 13592, 2885 ] } ], "ref_ids": [] }, { "id": 13594, "type": "theorem", "label": "duality-lemma-gorenstein-base-change", "categories": [ "duality" ], "title": "duality-lemma-gorenstein-base-change", "contents": [ "Let $X$ be a locally Noetherian scheme over the field $k$.", "Let $k \\subset k'$ be a finitely generated field extension.", "Let $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying", "over $x$. Then we have", "$$", "\\mathcal{O}_{X, x}\\text{ is Gorenstein}", "\\Leftrightarrow", "\\mathcal{O}_{X_{k'}, x'}\\text{ is Gorenstein}", "$$", "If $X$ is locally of finite type over $k$, the same holds for any", "field extension $k \\subset k'$." ], "refs": [], "proofs": [ { "contents": [ "In both cases the ring map $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X_{k'}, x'}$", "is a faithfully flat local homomorphism of Noetherian local rings.", "Thus if $\\mathcal{O}_{X_{k'}, x'}$ is Gorenstein, then so is", "$\\mathcal{O}_{X, x}$ by", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-flat-under-gorenstein}.", "To go up, we use", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-flat-under-gorenstein} as well.", "Thus we have to show that", "$$", "\\mathcal{O}_{X_{k'}, x'}/\\mathfrak m_x \\mathcal{O}_{X_{k'}, x'} =", "\\kappa(x) \\otimes_k k'", "$$", "is Gorenstein. Note that in the first case $k \\to k'$ is finitely", "generated and in the second case $k \\to \\kappa(x)$ is finitely", "generated. Hence this follows as property (A) holds for", "Gorenstein, see Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-formal-fibres-gorenstein}." ], "refs": [ "dualizing-lemma-flat-under-gorenstein", "dualizing-lemma-flat-under-gorenstein", "dualizing-lemma-formal-fibres-gorenstein" ], "ref_ids": [ 2885, 2885, 2891 ] } ], "ref_ids": [] }, { "id": 13595, "type": "theorem", "label": "duality-lemma-gorenstein-morphism", "categories": [ "duality" ], "title": "duality-lemma-gorenstein-morphism", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume all fibres of $f$ are locally Noetherian.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is Gorenstein, and", "\\item $f$ is flat and its fibres are Gorenstein schemes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows directly from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13596, "type": "theorem", "label": "duality-lemma-gorenstein-CM-morphism", "categories": [ "duality" ], "title": "duality-lemma-gorenstein-CM-morphism", "contents": [ "A Gorenstein morphism is Cohen-Macaulay." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-gorenstein-CM} and the definitions." ], "refs": [ "duality-lemma-gorenstein-CM" ], "ref_ids": [ 13589 ] } ], "ref_ids": [] }, { "id": 13597, "type": "theorem", "label": "duality-lemma-lci-gorenstein", "categories": [ "duality" ], "title": "duality-lemma-lci-gorenstein", "contents": [ "A syntomic morphism is Gorenstein. Equivalently a flat", "local complete intersection morphism is Gorenstein." ], "refs": [], "proofs": [ { "contents": [ "Recall that a syntomic morphism is flat and its fibres", "are local complete intersections over fields, see", "Morphisms, Lemma \\ref{morphisms-lemma-syntomic-flat-fibres}.", "Since a local complete intersection over a field is a Gorenstein scheme", "by Lemma \\ref{lemma-gorenstein-lci} we conclude.", "The properties ``syntomic'' and ``flat and local", "complete intersection morphism'' are equivalent by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-flat-lci}." ], "refs": [ "morphisms-lemma-syntomic-flat-fibres", "duality-lemma-gorenstein-lci", "more-morphisms-lemma-flat-lci" ], "ref_ids": [ 5298, 13592, 14006 ] } ], "ref_ids": [] }, { "id": 13598, "type": "theorem", "label": "duality-lemma-composition-gorenstein", "categories": [ "duality" ], "title": "duality-lemma-composition-gorenstein", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms. Assume that the", "fibres $X_y$, $Y_z$ and $X_z$ of $f$, $g$, and $g \\circ f$ are", "locally Noetherian.", "\\begin{enumerate}", "\\item If $f$ is Gorenstein at $x$ and $g$ is Gorenstein", "at $f(x)$, then $g \\circ f$ is Gorenstein at $x$.", "\\item If $f$ and $g$ are Gorenstein, then $g \\circ f$ is Gorenstein.", "\\item If $g \\circ f$ is Gorenstein at $x$ and $f$ is flat at $x$,", "then $f$ is Gorenstein at $x$ and $g$ is Gorenstein at $f(x)$.", "\\item If $f \\circ g$ is Gorenstein and $f$ is flat, then", "$f$ is Gorenstein and $g$ is Gorenstein at every point in", "the image of $f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "After translating into algebra this follows from", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-flat-under-gorenstein}." ], "refs": [ "dualizing-lemma-flat-under-gorenstein" ], "ref_ids": [ 2885 ] } ], "ref_ids": [] }, { "id": 13599, "type": "theorem", "label": "duality-lemma-flat-morphism-from-gorenstein-scheme", "categories": [ "duality" ], "title": "duality-lemma-flat-morphism-from-gorenstein-scheme", "contents": [ "\\begin{slogan}", "Gorensteinnes of the total space of a flat fibration implies", "same for base and fibres", "\\end{slogan}", "Let $f : X \\to Y$ be a flat morphism of locally Noetherian schemes.", "If $X$ is Gorenstein, then $f$ is Gorenstein and $\\mathcal{O}_{Y, f(x)}$", "is Gorenstein for all $x \\in X$." ], "refs": [], "proofs": [ { "contents": [ "After translating into algebra this follows from", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-flat-under-gorenstein}." ], "refs": [ "dualizing-lemma-flat-under-gorenstein" ], "ref_ids": [ 2885 ] } ], "ref_ids": [] }, { "id": 13600, "type": "theorem", "label": "duality-lemma-base-change-gorenstein", "categories": [ "duality" ], "title": "duality-lemma-base-change-gorenstein", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume that all the fibres $X_y$ are locally Noetherian schemes.", "Let $Y' \\to Y$ be locally of finite type. Let $f' : X' \\to Y'$", "be the base change of $f$.", "Let $x' \\in X'$ be a point with image $x \\in X$.", "\\begin{enumerate}", "\\item If $f$ is Gorenstein at $x$, then", "$f' : X' \\to Y'$ is Gorenstein at $x'$.", "\\item If $f$ is flat and $x$ and $f'$ is Gorenstein at $x'$, then $f$", "is Gorenstein at $x$.", "\\item If $Y' \\to Y$ is flat at $f'(x')$ and $f'$ is Gorenstein at", "$x'$, then $f$ is Gorenstein at $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that the assumption on $Y' \\to Y$ implies that for $y' \\in Y'$", "mapping to $y \\in Y$ the field extension $\\kappa(y) \\subset \\kappa(y')$", "is finitely generated. Hence also all the fibres", "$X'_{y'} = (X_y)_{\\kappa(y')}$ are locally Noetherian, see", "Varieties, Lemma \\ref{varieties-lemma-locally-Noetherian-base-change}.", "Thus the lemma makes sense. Set $y' = f'(x')$ and $y = f(x)$.", "Hence we get the following commutative diagram of local rings", "$$", "\\xymatrix{", "\\mathcal{O}_{X', x'} & \\mathcal{O}_{X, x} \\ar[l] \\\\", "\\mathcal{O}_{Y', y'} \\ar[u] & \\mathcal{O}_{Y, y} \\ar[l] \\ar[u]", "}", "$$", "where the upper left corner is a localization of the tensor product", "of the upper right and lower left corners over the lower right corner.", "\\medskip\\noindent", "Assume $f$ is Gorenstein at $x$.", "The flatness of $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$", "implies the flatness of $\\mathcal{O}_{Y', y'} \\to \\mathcal{O}_{X', x'}$, see", "Algebra, Lemma \\ref{algebra-lemma-base-change-flat-up-down}.", "The fact that $\\mathcal{O}_{X, x}/\\mathfrak m_y\\mathcal{O}_{X, x}$", "is Gorenstein implies that", "$\\mathcal{O}_{X', x'}/\\mathfrak m_{y'}\\mathcal{O}_{X', x'}$", "is Gorenstein, see", "Lemma \\ref{lemma-gorenstein-base-change}. Hence we see that $f'$", "is Gorenstein at $x'$.", "\\medskip\\noindent", "Assume $f$ is flat at $x$ and $f'$ is Gorenstein at $x'$.", "The fact that $\\mathcal{O}_{X', x'}/\\mathfrak m_{y'}\\mathcal{O}_{X', x'}$", "is Gorenstein implies that", "$\\mathcal{O}_{X, x}/\\mathfrak m_y\\mathcal{O}_{X, x}$", "is Gorenstein, see", "Lemma \\ref{lemma-gorenstein-base-change}. Hence we see that $f$", "is Gorenstein at $x$.", "\\medskip\\noindent", "Assume $Y' \\to Y$ is flat at $y'$ and $f'$ is Gorenstein at", "$x'$. The flatness of $\\mathcal{O}_{Y', y'} \\to \\mathcal{O}_{X', x'}$", "and $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{Y', y'}$ implies the flatness", "of $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$, see", "Algebra, Lemma \\ref{algebra-lemma-base-change-flat-up-down}.", "The fact that $\\mathcal{O}_{X', x'}/\\mathfrak m_{y'}\\mathcal{O}_{X', x'}$", "is Gorenstein implies that", "$\\mathcal{O}_{X, x}/\\mathfrak m_y\\mathcal{O}_{X, x}$", "is Gorenstein, see", "Lemma \\ref{lemma-gorenstein-base-change}. Hence we see that $f$", "is Gorenstein at $x$." ], "refs": [ "varieties-lemma-locally-Noetherian-base-change", "algebra-lemma-base-change-flat-up-down", "duality-lemma-gorenstein-base-change", "duality-lemma-gorenstein-base-change", "algebra-lemma-base-change-flat-up-down", "duality-lemma-gorenstein-base-change" ], "ref_ids": [ 10957, 898, 13594, 13594, 898, 13594 ] } ], "ref_ids": [] }, { "id": 13601, "type": "theorem", "label": "duality-lemma-flat-lft-base-change-gorenstein", "categories": [ "duality" ], "title": "duality-lemma-flat-lft-base-change-gorenstein", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes which is flat and", "locally of finite type. Then formation of the set", "$\\{x \\in X \\mid f\\text{ is Gorenstein at }x\\}$", "commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "The assumption implies any fibre of $f$ is locally of finite type", "over a field and hence locally Noetherian and the same is true for", "any base change. Thus the statement makes sense. Looking at", "fibres we reduce to the following problem: let $X$ be a scheme", "locally of finite type over a field $k$,", "let $K/k$ be a field extension, and", "let $x_K \\in X_K$ be a point with image $x \\in X$.", "Problem: show that $\\mathcal{O}_{X_K, x_K}$ is Gorenstein if and only if", "$\\mathcal{O}_{X, x}$ is Gorenstein.", "\\medskip\\noindent", "The problem can be solved using a bit of algebra as follows.", "Choose an affine open $\\Spec(A) \\subset X$ containing $x$.", "Say $x$ corresponds to $\\mathfrak p \\subset A$.", "With $A_K = A \\otimes_k K$ we see that $\\Spec(A_K) \\subset X_K$", "contains $x_K$. Say $x_K$ corresponds to $\\mathfrak p_K \\subset A_K$.", "Let $\\omega_A^\\bullet$ be a dualizing complex for $A$.", "By Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-base-change-dualizing-over-field}", "$\\omega_A^\\bullet \\otimes_A A_K$ is a dualizing complex for $A_K$.", "Now we are done because", "$A_\\mathfrak p \\to (A_K)_{\\mathfrak p_K}$ is a flat local", "homomorphism of Noetherian rings and hence", "$(\\omega_A^\\bullet)_\\mathfrak p$ is an invertible object", "of $D(A_\\mathfrak p)$ if and only if", "$(\\omega_A^\\bullet)_\\mathfrak p \\otimes_{A_\\mathfrak p} (A_K)_{\\mathfrak p_K}$", "is an invertible object of $D((A_K)_{\\mathfrak p_K})$.", "Some details omitted; hint: look at cohomology modules." ], "refs": [ "dualizing-lemma-base-change-dualizing-over-field" ], "ref_ids": [ 2907 ] } ], "ref_ids": [] }, { "id": 13602, "type": "theorem", "label": "duality-lemma-affine-flat-Noetherian-gorenstein", "categories": [ "duality" ], "title": "duality-lemma-affine-flat-Noetherian-gorenstein", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism", "of $\\textit{FTS}_S$. Let $x \\in X$. If $f$ is flat, then", "the following are equivalent", "\\begin{enumerate}", "\\item $f$ is Gorenstein at $x$,", "\\item $f^!\\mathcal{O}_X$ is isomorphic to an invertible object", "in a neighbourhood of $x$.", "\\end{enumerate}", "In particular, the set of points where $f$ is Gorenstein is", "open in $X$." ], "refs": [], "proofs": [ { "contents": [ "Set $\\omega^\\bullet = f^!\\mathcal{O}_Y$. By", "Lemma \\ref{lemma-relative-dualizing-fibres}", "this is a bounded complex with coherent cohomology", "sheaves whose derived restriction $Lh^*\\omega^\\bullet$", "to the fibre $X_y$ is a dualizing complex on $X_y$.", "Denote $i : x \\to X_y$ the inclusion of a point.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $f$ is Gorenstein at $x$,", "\\item $\\mathcal{O}_{X_y, x}$ is Gorenstein,", "\\item $Lh^*\\omega^\\bullet$ is invertible in a neighbourhood of $x$,", "\\item $Li^* Lh^* \\omega^\\bullet$ has exactly one nonzero", "cohomology of dimension $1$ over $\\kappa(x)$,", "\\item $L(h \\circ i)^* \\omega^\\bullet$ has exactly one nonzero", "cohomology of dimension $1$ over $\\kappa(x)$,", "\\item $\\omega^\\bullet$ is invertible in a neighbourhood of $x$.", "\\end{enumerate}", "The equivalence of (1) and (2) is by definition (as $f$ is flat).", "The equivalence of (2) and (3) follows from", "Lemma \\ref{lemma-gorenstein}.", "The equivalence of (3) and (4) follows from", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-bounded-pseudo-coherent-to-perfect}.", "The equivalence of (4) and (5) holds because", "$Li^* Lh^* = L(h \\circ i)^*$.", "The equivalence of (5) and (6) holds by", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-bounded-pseudo-coherent-to-perfect}.", "Thus the lemma is clear." ], "refs": [ "duality-lemma-relative-dualizing-fibres", "duality-lemma-gorenstein", "more-algebra-lemma-lift-bounded-pseudo-coherent-to-perfect", "more-algebra-lemma-lift-bounded-pseudo-coherent-to-perfect" ], "ref_ids": [ 13568, 13591, 10241, 10241 ] } ], "ref_ids": [] }, { "id": 13603, "type": "theorem", "label": "duality-lemma-flat-finite-presentation-characterize-gorenstein", "categories": [ "duality" ], "title": "duality-lemma-flat-finite-presentation-characterize-gorenstein", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat and locally", "of finite presentation. Let $x \\in X$ with image $s \\in S$.", "Set $d = \\dim_x(X_s)$. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is Gorenstein at $x$,", "\\item there exists an open neighbourhood $U \\subset X$ of $x$", "and a locally quasi-finite morphism $U \\to \\mathbf{A}^d_S$ over $S$", "which is Gorenstein at $x$,", "\\item there exists an open neighbourhood $U \\subset X$ of $x$", "and a locally quasi-finite Gorenstein morphism $U \\to \\mathbf{A}^d_S$ over $S$,", "\\item for any $S$-morphism $g : U \\to \\mathbf{A}^d_S$", "of an open neighbourhood $U \\subset X$ of $x$ we have:", "$g$ is quasi-finite at $x$ $\\Rightarrow$ $g$ is Gorenstein at $x$.", "\\end{enumerate}", "In particular, the set of points where $f$ is Gorenstein is open in $X$." ], "refs": [], "proofs": [ { "contents": [ "Choose affine open $U = \\Spec(A) \\subset X$ with $x \\in U$ and", "$V = \\Spec(R) \\subset S$ with $f(U) \\subset V$. Then $R \\to A$", "is a flat ring map of finite presentation. Let $\\mathfrak p \\subset A$", "be the prime ideal corresponding to $x$. After replacing $A$ by a", "principal localization we may assume there exists a quasi-finite map", "$R[x_1, \\ldots, x_d] \\to A$, see", "Algebra, Lemma \\ref{algebra-lemma-quasi-finite-over-polynomial-algebra}.", "Thus there exists at least one pair $(U, g)$ consisting of an", "open neighbourhood $U \\subset X$ of $x$ and a locally\\footnote{If $S$", "is quasi-separated, then $g$ will be quasi-finite.} quasi-finite morphism", "$g : U \\to \\mathbf{A}^d_S$.", "\\medskip\\noindent", "Having said this, the lemma translates into the following algebra", "problem (translation omitted). Given $R \\to A$ flat and of finite", "presentation, a prime $\\mathfrak p \\subset A$ and", "$\\varphi : R[x_1, \\ldots, x_d] \\to A$ quasi-finite at $\\mathfrak p$", "the following are equivalent", "\\begin{enumerate}", "\\item[(a)] $\\Spec(\\varphi)$ is Gorenstein at $\\mathfrak p$, and", "\\item[(b)] $\\Spec(A) \\to \\Spec(R)$ is Gorenstein at $\\mathfrak p$.", "\\item[(c)] $\\Spec(A) \\to \\Spec(R)$ is Gorenstein in an open neighbourhood", "of $\\mathfrak p$.", "\\end{enumerate}", "In each case $R[x_1, \\ldots, x_n] \\to A$ is flat at $\\mathfrak p$", "hence by openness of flatness", "(Algebra, Theorem \\ref{algebra-theorem-openness-flatness}),", "we may assume $R[x_1, \\ldots, x_n] \\to A$", "is flat (replace $A$ by a suitable principal localization).", "By Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "there exists $R_0 \\subset R$ and $R_0[x_1, \\ldots, x_n] \\to A_0$", "such that $R_0$ is of finite type over $\\mathbf{Z}$ and", "$R_0 \\to A_0$ is of finite type and $R_0[x_1, \\ldots, x_n] \\to A_0$ is flat.", "Note that the set of points where a flat finite type morphism", "is Gorenstein commutes with base change by", "Lemma \\ref{lemma-base-change-gorenstein}.", "In this way we reduce to the case where $R$ is Noetherian.", "\\medskip\\noindent", "Thus we may assume $X$ and $S$ affine and that", "we have a factorization of $f$ of the form", "$$", "X \\xrightarrow{g} \\mathbf{A}^n_S \\xrightarrow{p} S", "$$", "with $g$ flat and quasi-finite and $S$ Noetherian. Then $X$ and", "$\\mathbf{A}^n_S$ are separated over $S$ and we have", "$$", "f^!\\mathcal{O}_S = g^!p^!\\mathcal{O}_S = g^!\\mathcal{O}_{\\mathbf{A}^n_S}[n]", "$$", "by know properties of upper shriek functors", "(Lemmas \\ref{lemma-upper-shriek-composition} and", "\\ref{lemma-shriek-affine-line}).", "Hence the equivalence of (a), (b), and (c) by", "Lemma \\ref{lemma-affine-flat-Noetherian-gorenstein}." ], "refs": [ "algebra-lemma-quasi-finite-over-polynomial-algebra", "algebra-theorem-openness-flatness", "algebra-lemma-flat-finite-presentation-limit-flat", "duality-lemma-base-change-gorenstein", "duality-lemma-upper-shriek-composition", "duality-lemma-shriek-affine-line", "duality-lemma-affine-flat-Noetherian-gorenstein" ], "ref_ids": [ 1071, 326, 1389, 13600, 13552, 13557, 13602 ] } ], "ref_ids": [] }, { "id": 13604, "type": "theorem", "label": "duality-lemma-gorenstein-local-source-and-target", "categories": [ "duality" ], "title": "duality-lemma-gorenstein-local-source-and-target", "contents": [ "The property", "$\\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian and $f$ is", "Gorenstein'' is local in the fppf topology on the target and", "local in the syntomic topology on the source." ], "refs": [], "proofs": [ { "contents": [ "We have", "$\\mathcal{P}(f) =", "\\mathcal{P}_1(f) \\wedge \\mathcal{P}_2(f)$", "where", "$\\mathcal{P}_1(f)=$``$f$ is flat'', and", "$\\mathcal{P}_2(f)=$``the fibres of $f$ are locally Noetherian", "and Gorenstein''.", "We know that $\\mathcal{P}_1$ is", "local in the fppf topology on the source and the target, see", "Descent, Lemmas \\ref{descent-lemma-descending-property-flat} and", "\\ref{descent-lemma-flat-fpqc-local-source}. Thus we have to deal", "with $\\mathcal{P}_2$.", "\\medskip\\noindent", "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\{\\varphi_i : Y_i \\to Y\\}_{i \\in I}$ be an fppf covering of $Y$.", "Denote $f_i : X_i \\to Y_i$ the base change of $f$ by $\\varphi_i$.", "Let $i \\in I$ and let $y_i \\in Y_i$ be a point.", "Set $y = \\varphi_i(y_i)$. Note that", "$$", "X_{i, y_i} = \\Spec(\\kappa(y_i)) \\times_{\\Spec(\\kappa(y))} X_y.", "$$", "and that $\\kappa(y) \\subset \\kappa(y_i)$ is a finitely generated field", "extension. Hence if $X_y$ is locally Noetherian, then", "$X_{i, y_i}$ is locally Noetherian, see", "Varieties, Lemma \\ref{varieties-lemma-locally-Noetherian-base-change}.", "And if in addition $X_y$ is Gorenstein,", "then $X_{i, y_i}$ is Gorenstein, see", "Lemma \\ref{lemma-gorenstein-base-change}.", "Thus $\\mathcal{P}_2$ is fppf local on the target.", "\\medskip\\noindent", "Let $\\{X_i \\to X\\}$ be a syntomic covering of $X$.", "Let $y \\in Y$. In this case $\\{X_{i, y} \\to X_y\\}$ is a", "syntomic covering of the fibre. Hence the locality of $\\mathcal{P}_2$", "for the syntomic topology on the source follows from", "Lemma \\ref{lemma-gorenstein-local-syntomic}." ], "refs": [ "descent-lemma-descending-property-flat", "descent-lemma-flat-fpqc-local-source", "varieties-lemma-locally-Noetherian-base-change", "duality-lemma-gorenstein-base-change", "duality-lemma-gorenstein-local-syntomic" ], "ref_ids": [ 14680, 14708, 10957, 13594, 13593 ] } ], "ref_ids": [] }, { "id": 13605, "type": "theorem", "label": "duality-lemma-descent-ascent", "categories": [ "duality" ], "title": "duality-lemma-descent-ascent", "contents": [ "Let $f : X \\to Y$ be a morphism of locally Noetherian schemes. Assume", "\\begin{enumerate}", "\\item $f$ is syntomic and surjective, or", "\\item $f$ is a surjective flat local complete intersection morphism, or", "\\item $f$ is a surjective Gorenstein morphism of finite type.", "\\end{enumerate}", "Then $K \\in D_\\QCoh(\\mathcal{O}_Y)$ is a dualizing complex on $Y$ if and only", "if $Lf^*K$ is a dualizing complex on $X$." ], "refs": [], "proofs": [ { "contents": [ "Taking affine opens and using", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-affine-compare-bounded}", "this translates into", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-descent-ascent}." ], "refs": [ "perfect-lemma-affine-compare-bounded", "dualizing-lemma-descent-ascent" ], "ref_ids": [ 6941, 2915 ] } ], "ref_ids": [] }, { "id": 13606, "type": "theorem", "label": "duality-lemma-duality-proper-over-field", "categories": [ "duality" ], "title": "duality-lemma-duality-proper-over-field", "contents": [ "Let $X$ be a proper scheme over a field $k$. There exists a dualizing complex", "$\\omega_X^\\bullet$ with the following properties", "\\begin{enumerate}", "\\item $H^i(\\omega_X^\\bullet)$ is nonzero only for $i \\in [-\\dim(X), 0]$,", "\\item $\\omega_X = H^{-\\dim(X)}(\\omega_X^\\bullet)$ is a coherent", "$(S_2)$-module whose support is the irreducible components of dimension $d$,", "\\item the dimension of the support of $H^i(\\omega_X^\\bullet)$ is at most $-i$,", "\\item for $x \\in X$ closed the module", "$H^i(\\omega_{X, x}^\\bullet) \\oplus \\ldots \\oplus H^0(\\omega_{X, x}^\\bullet)$", "is nonzero if and only if $\\text{depth}(\\mathcal{O}_{X, x}) \\leq -i$,", "\\item for $K \\in D_\\QCoh(\\mathcal{O}_X)$ there are functorial", "isomorphisms\\footnote{This property", "characterizes $\\omega_X^\\bullet$ in $D_\\QCoh(\\mathcal{O}_X)$", "up to unique isomorphism by the Yoneda lemma. Since $\\omega_X^\\bullet$", "is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ in fact it suffices to consider", "$K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.}", "$$", "\\Ext^i_X(K, \\omega_X^\\bullet) = \\Hom_k(H^{-i}(X, K), k)", "$$", "compatible with shifts and distinguished triangles,", "\\item there are functorial isomorphisms", "$\\Hom(\\mathcal{F}, \\omega_X) = \\Hom_k(H^{\\dim(X)}(X, \\mathcal{F}), k)$", "for $\\mathcal{F}$ quasi-coherent on $X$, and", "\\item if $X \\to \\Spec(k)$ is smooth of relative dimension $d$,", "then $\\omega_X^\\bullet \\cong \\wedge^d\\Omega_{X/k}[d]$ and", "$\\omega_X \\cong \\wedge^d\\Omega_{X/k}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $f : X \\to \\Spec(k)$ the structure morphism. Let $a$ be the", "right adjoint of pushforward of this morphism, see", "Lemma \\ref{lemma-twisted-inverse-image}. Consider the relative dualizing", "complex", "$$", "\\omega_X^\\bullet = a(\\mathcal{O}_{\\Spec(k)})", "$$", "Compare with Remark \\ref{remark-relative-dualizing-complex}.", "Since $f$ is proper we have", "$f^!(\\mathcal{O}_{\\Spec(k)}) = a(\\mathcal{O}_{\\Spec(k)})$ by", "definition, see Section \\ref{section-upper-shriek}.", "Applying Lemma \\ref{lemma-shriek-dualizing} we find that", "$\\omega_X^\\bullet$ is a dualizing complex. Moreover, we see that", "$\\omega_X^\\bullet$ and $\\omega_X$ are as in", "Example \\ref{example-proper-over-local} and as in", "Example \\ref{example-equidimensional-over-field}.", "\\medskip\\noindent", "Parts (1), (2), and (3) follow from", "Lemma \\ref{lemma-vanishing-good-dualizing}.", "\\medskip\\noindent", "For a closed point $x \\in X$ we see that $\\omega_{X, x}^\\bullet$ is a", "normalized dualizing complex over $\\mathcal{O}_{X, x}$, see", "Lemma \\ref{lemma-good-dualizing-normalized}.", "Part (4) then follows from Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-depth-in-terms-dualizing-complex}.", "\\medskip\\noindent", "Part (5) holds by construction as $a$ is the right adjoint to", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D(\\mathcal{O}_{\\Spec(k)}) = D(k)$", "which we can identify with $K \\mapsto R\\Gamma(X, K)$. We also use", "that the derived category $D(k)$ of $k$-modules is the same as the", "category of graded $k$-vector spaces.", "\\medskip\\noindent", "Part (6) follows from Lemma \\ref{lemma-dualizing-module-proper-over-A}", "for coherent $\\mathcal{F}$ and in general by unwinding", "(5) for $K = \\mathcal{F}[0]$ and $i = -\\dim(X)$.", "\\medskip\\noindent", "Part (7) follows from Lemma \\ref{lemma-smooth-proper}." ], "refs": [ "duality-lemma-twisted-inverse-image", "duality-remark-relative-dualizing-complex", "duality-lemma-shriek-dualizing", "duality-lemma-vanishing-good-dualizing", "duality-lemma-good-dualizing-normalized", "dualizing-lemma-depth-in-terms-dualizing-complex", "duality-lemma-dualizing-module-proper-over-A", "duality-lemma-smooth-proper" ], "ref_ids": [ 13503, 13649, 13560, 13584, 13576, 2874, 13585, 13550 ] } ], "ref_ids": [] }, { "id": 13607, "type": "theorem", "label": "duality-lemma-duality-proper-over-field-perfect", "categories": [ "duality" ], "title": "duality-lemma-duality-proper-over-field-perfect", "contents": [ "Let $k$, $X$, and $\\omega_X^\\bullet$", "be as in Lemma \\ref{lemma-duality-proper-over-field}.", "Let $t : H^0(X, \\omega_X^\\bullet) \\to k$ be as in", "Remark \\ref{remark-duality-proper-over-field}.", "Let $E \\in D(\\mathcal{O}_X)$ be perfect. Then the pairings", "$$", "H^i(X, \\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E^\\vee)", "\\times", "H^{-i}(X, E) \\longrightarrow k, \\quad", "(\\xi, \\eta) \\longmapsto", "t((1_{\\omega_X^\\bullet} \\otimes \\epsilon)(\\xi \\cup \\eta))", "$$", "are perfect for all $i$. Here $\\cup$ denotes the cupproduct", "of Cohomology, Section \\ref{cohomology-section-cup-product} and", "$\\epsilon : E^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E \\to \\mathcal{O}_X$", "is as in Cohomology, Example \\ref{cohomology-example-dual-derived}." ], "refs": [ "duality-lemma-duality-proper-over-field", "duality-remark-duality-proper-over-field" ], "proofs": [ { "contents": [ "By replacing $E$ with $E[-i]$ this reduces to the case $i = 0$.", "By Cohomology, Lemma \\ref{cohomology-lemma-ext-composition-is-cup}", "we see that the pairing is the same as the one discussed", "in Remark \\ref{remark-duality-proper-over-field}", "whence the result by the discussion in that remark." ], "refs": [ "cohomology-lemma-ext-composition-is-cup", "duality-remark-duality-proper-over-field" ], "ref_ids": [ 2237, 13658 ] } ], "ref_ids": [ 13606, 13658 ] }, { "id": 13608, "type": "theorem", "label": "duality-lemma-duality-proper-over-field-CM", "categories": [ "duality" ], "title": "duality-lemma-duality-proper-over-field-CM", "contents": [ "Let $X$ be a proper scheme over a field $k$ which is Cohen-Macaulay", "and equidimensional of dimension $d$. The module $\\omega_X$", "of Lemma \\ref{lemma-duality-proper-over-field} has the following properties", "\\begin{enumerate}", "\\item $\\omega_X$ is a dualizing module on $X$", "(Section \\ref{section-dualizing-module}),", "\\item $\\omega_X$ is a coherent Cohen-Macaulay module whose support is $X$,", "\\item there are functorial isomorphisms", "$\\Ext^i_X(K, \\omega_X[d]) = \\Hom_k(H^{-i}(X, K), k)$", "compatible with shifts and distinguished triangles for $K \\in D_\\QCoh(X)$,", "\\item there are functorial isomorphisms", "$\\Ext^{d - i}(\\mathcal{F}, \\omega_X) = \\Hom_k(H^i(X, \\mathcal{F}), k)$", "for $\\mathcal{F}$ quasi-coherent on $X$.", "\\end{enumerate}" ], "refs": [ "duality-lemma-duality-proper-over-field" ], "proofs": [ { "contents": [ "It is clear from Lemma \\ref{lemma-duality-proper-over-field}", "that $\\omega_X$ is a dualizing module (as it is the left most", "nonvanishing cohomology sheaf of a dualizing complex).", "We have $\\omega_X^\\bullet = \\omega_X[d]$ and $\\omega_X$ is Cohen-Macaulay", "as $X$ is Cohen-Macualay, see Lemma \\ref{lemma-dualizing-module-CM-scheme}.", "The other statements follow from this combined with the", "corresponding statements of Lemma \\ref{lemma-duality-proper-over-field}." ], "refs": [ "duality-lemma-duality-proper-over-field", "duality-lemma-dualizing-module-CM-scheme", "duality-lemma-duality-proper-over-field" ], "ref_ids": [ 13606, 13586, 13606 ] } ], "ref_ids": [ 13606 ] }, { "id": 13609, "type": "theorem", "label": "duality-lemma-sanity-check-duality", "categories": [ "duality" ], "title": "duality-lemma-sanity-check-duality", "contents": [ "Let $X$ be a proper scheme over a field $k$. Let $\\omega_X^\\bullet$ and", "$\\omega_X$ be as in Lemma \\ref{lemma-duality-proper-over-field}.", "\\begin{enumerate}", "\\item If $X \\to \\Spec(k)$ factors as $X \\to \\Spec(k') \\to \\Spec(k)$", "for some field $k'$, then $\\omega_X^\\bullet$ and $\\omega_X$", "are as in Lemma \\ref{lemma-duality-proper-over-field} for the morphism", "$X \\to \\Spec(k')$.", "\\item If $K/k$ is a field extension, then the pullback of", "$\\omega_X^\\bullet$ and $\\omega_X$ to the base change $X_K$", "are as in Lemma \\ref{lemma-duality-proper-over-field} for the morphism", "$X_K \\to \\Spec(K)$.", "\\end{enumerate}" ], "refs": [ "duality-lemma-duality-proper-over-field", "duality-lemma-duality-proper-over-field", "duality-lemma-duality-proper-over-field" ], "proofs": [ { "contents": [ "Denote $f : X \\to \\Spec(k)$ the structure morphism and denote", "$f' : X \\to \\Spec(k')$ the given factorization.", "In the proof of Lemma \\ref{lemma-duality-proper-over-field}", "we took $\\omega_X^\\bullet = a(\\mathcal{O}_{\\Spec(k)})$", "where $a$ be is the right adjoint of", "Lemma \\ref{lemma-twisted-inverse-image} for $f$.", "Thus we have to show", "$a(\\mathcal{O}_{\\Spec(k)}) \\cong a'(\\mathcal{O}_{\\Spec(k)})$", "where $a'$ be is the right adjoint of", "Lemma \\ref{lemma-twisted-inverse-image} for $f'$.", "Since $k' \\subset H^0(X, \\mathcal{O}_X)$ we see that $k'/k$ is a finite", "extension (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-over-affine-cohomology-finite}).", "By uniqueness of adjoints we have $a = a' \\circ b$ where", "$b$ is the right adjoint of Lemma", "\\ref{lemma-twisted-inverse-image} for $g : \\Spec(k') \\to \\Spec(k)$.", "Another way to say this: we have $f^! = (f')^! \\circ g^!$.", "Thus it suffices to show that $\\Hom_k(k', k) \\cong k'$ as", "$k'$-modules, see Example \\ref{example-affine-twisted-inverse-image}.", "This holds because these are $k'$-vector spaces of", "the same dimension (namely dimension $1$).", "\\medskip\\noindent", "Proof of (2). This holds because we have base change for $a$ by", "Lemma \\ref{lemma-more-base-change}. See discussion in", "Remark \\ref{remark-relative-dualizing-complex}." ], "refs": [ "duality-lemma-duality-proper-over-field", "duality-lemma-twisted-inverse-image", "duality-lemma-twisted-inverse-image", "coherent-lemma-proper-over-affine-cohomology-finite", "duality-lemma-twisted-inverse-image", "duality-lemma-more-base-change", "duality-remark-relative-dualizing-complex" ], "ref_ids": [ 13606, 13503, 13503, 3355, 13503, 13512, 13649 ] } ], "ref_ids": [ 13606, 13606, 13606 ] }, { "id": 13610, "type": "theorem", "label": "duality-lemma-relative-dualizing-complex-algebra", "categories": [ "duality" ], "title": "duality-lemma-relative-dualizing-complex-algebra", "contents": [ "Let $X \\to S$ be a morphism of schemes which is flat and", "locally of finite presentation. Let $(K, \\xi)$ be a relative", "dualizing complex. Then for any commutative diagram", "$$", "\\xymatrix{", "\\Spec(A) \\ar[d] \\ar[r] & X \\ar[d] \\\\", "\\Spec(R) \\ar[r] & S", "}", "$$", "whose horizontal arrows are open immersions, the", "restriction of $K$ to $\\Spec(A)$ corresponds via", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded}", "to a relative dualizing complex for $R \\to A$", "in the sense of Dualizing Complexes, Definition", "\\ref{dualizing-definition-relative-dualizing-complex}." ], "refs": [ "perfect-lemma-affine-compare-bounded", "dualizing-definition-relative-dualizing-complex" ], "proofs": [ { "contents": [ "Since formation of $R\\SheafHom$ commutes with restrictions to", "opens we may as well assume $X = \\Spec(A)$ and $S = \\Spec(R)$.", "Observe that relatively perfect objects of $D(\\mathcal{O}_X)$", "are pseudo-coherent and hence are in $D_\\QCoh(\\mathcal{O}_X)$", "(Derived Categories of Schemes, Lemma \\ref{perfect-lemma-pseudo-coherent}).", "Thus the statement makes sense.", "Observe that taking $\\Delta_*$, $L\\text{pr}_1^*$, and", "$R\\SheafHom$ is compatible with what happens on the algebraic", "side by", "Derived Categories of Schemes,", "Lemmas \\ref{perfect-lemma-quasi-coherence-pushforward},", "\\ref{perfect-lemma-quasi-coherence-pullback},", "\\ref{perfect-lemma-quasi-coherence-internal-hom}.", "For the last one we observe that $L\\text{pr}_1^*K$", "is $S$-perfect (hence bounded below) and that $\\Delta_*\\mathcal{O}_X$", "is a pseudo-coherent object of $D(\\mathcal{O}_W)$;", "translated into algebra this means that $A$ is pseudo-coherent", "as an $A \\otimes_R A$-module which follows from", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-more-relative-pseudo-coherent-is-moot}", "applied to $R \\to A \\otimes_R A \\to A$.", "Thus we recover exactly the conditions in", "Dualizing Complexes, Definition", "\\ref{dualizing-definition-relative-dualizing-complex}." ], "refs": [ "perfect-lemma-pseudo-coherent", "perfect-lemma-quasi-coherence-pushforward", "perfect-lemma-quasi-coherence-pullback", "perfect-lemma-quasi-coherence-internal-hom", "more-algebra-lemma-more-relative-pseudo-coherent-is-moot", "dualizing-definition-relative-dualizing-complex" ], "ref_ids": [ 6974, 6943, 6944, 6981, 10287, 2933 ] } ], "ref_ids": [ 6941, 2933 ] }, { "id": 13611, "type": "theorem", "label": "duality-lemma-relative-dualizing-RHom", "categories": [ "duality" ], "title": "duality-lemma-relative-dualizing-RHom", "contents": [ "Let $X \\to S$ be a morphism of schemes which is flat and", "locally of finite presentation. Let $(K, \\xi)$ be a relative", "dualizing complex. Then", "$\\mathcal{O}_X \\to R\\SheafHom_{\\mathcal{O}_X}(K, K)$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Looking affine locally this reduces using", "Lemma \\ref{lemma-relative-dualizing-complex-algebra}", "to the algebraic case which is", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-relative-dualizing-RHom}." ], "refs": [ "duality-lemma-relative-dualizing-complex-algebra", "dualizing-lemma-relative-dualizing-RHom" ], "ref_ids": [ 13610, 2921 ] } ], "ref_ids": [] }, { "id": 13612, "type": "theorem", "label": "duality-lemma-uniqueness-relative-dualizing", "categories": [ "duality" ], "title": "duality-lemma-uniqueness-relative-dualizing", "contents": [ "Let $X \\to S$ be a morphism of schemes which is flat and", "locally of finite presentation. If $(K, \\xi)$ and $(L, \\eta)$", "are two relative dualizing complexes on $X/S$, then there is a unique", "isomorphism $K \\to L$ sending $\\xi$ to $\\eta$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be an affine open mapping into an", "affine open of $S$. Then there is an isomorphism", "$K|_U \\to L|_U$ by Lemma \\ref{lemma-relative-dualizing-complex-algebra} and", "Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-uniqueness-relative-dualizing}.", "The reader can reuse the argument of that lemma", "in the schemes case to obtain a proof in this case.", "We will instead use a glueing argument.", "\\medskip\\noindent", "Suppose we have an isomorphism $\\alpha : K \\to L$.", "Then $\\alpha(\\xi) = u \\eta$ for some invertible section", "$u \\in H^0(W, \\Delta_*\\mathcal{O}_X) = H^0(X, \\mathcal{O}_X)$.", "(Because both $\\eta$ and $\\alpha(\\xi)$ are generators", "of an invertible $\\Delta_*\\mathcal{O}_X$-module by assumption.)", "Hence after replacing $\\alpha$ by $u^{-1}\\alpha$", "we see that $\\alpha(\\xi) = \\eta$.", "Since the automorphism group of", "$K$ is $H^0(X, \\mathcal{O}_X^*)$ by Lemma \\ref{lemma-relative-dualizing-RHom}", "there is at most one such $\\alpha$.", "\\medskip\\noindent", "Let $\\mathcal{B}$ be the collection of affine opens of", "$X$ which map into an affine open of $S$. For each $U \\in \\mathcal{B}$", "we have a unique isomorphism $\\alpha_U : K|_U \\to L|_U$", "mapping $\\xi$ to $\\eta$ by the discussion in the previous", "two paragraphs.", "Observe that $\\text{Ext}^i(K|_U, K|_U) = 0$ for $i < 0$", "and any open $U$ of $X$ by Lemma \\ref{lemma-relative-dualizing-RHom}.", "By Cohomology, Lemma \\ref{cohomology-lemma-vanishing-and-glueing}", "applied to $\\text{id} : X \\to X$ we get a unique morphism", "$\\alpha : K \\to L$ agreeing", "with $\\alpha_U$ for all $U \\in \\mathcal{B}$.", "Then $\\alpha$ sends $\\xi$ to $\\eta$ as this is true locally." ], "refs": [ "duality-lemma-relative-dualizing-complex-algebra", "dualizing-lemma-uniqueness-relative-dualizing", "duality-lemma-relative-dualizing-RHom", "duality-lemma-relative-dualizing-RHom", "cohomology-lemma-vanishing-and-glueing" ], "ref_ids": [ 13610, 2918, 13611, 13611, 2192 ] } ], "ref_ids": [] }, { "id": 13613, "type": "theorem", "label": "duality-lemma-existence-relative-dualizing", "categories": [ "duality" ], "title": "duality-lemma-existence-relative-dualizing", "contents": [ "Let $X \\to S$ be a morphism of schemes which is", "flat and locally of finite presentation.", "There exists a relative dualizing complex $(K, \\xi)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{B}$ be the collection of affine opens of", "$X$ which map into an affine open of $S$. For each $U$", "we have a relative dualizing complex $(K_U, \\xi_U)$ for", "$U$ over $S$. Namely, choose an affine open", "$V \\subset S$ such that $U \\to X \\to S$ factors through $V$.", "Write $U = \\Spec(A)$ and $V = \\Spec(R)$. By", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-base-change-relative-dualizing}", "there exists a relative dualizing complex $K_A \\in D(A)$", "for $R \\to A$. Arguing backwards through the proof of", "Lemma \\ref{lemma-relative-dualizing-complex-algebra}", "this determines an $V$-perfect object $K_U \\in D(\\mathcal{O}_U)$", "and a map", "$$", "\\xi : \\Delta_*\\mathcal{O}_U \\to L\\text{pr}_1^*K_U", "$$", "in $D(\\mathcal{O}_{U \\times_V U})$. Since being $V$-perfect is the", "same as being $S$-perfect and since $U \\times_V U = U \\times_S U$", "we find that $(K_U, \\xi_U)$ is as desired.", "\\medskip\\noindent", "If $U' \\subset U \\subset X$ with $U', U \\in \\mathcal{B}$, then", "we have a unique isomorphism $\\rho_{U'}^U : K_U|_{U'} \\to K_{U'}$", "in $D(\\mathcal{O}_{U'})$ sending $\\xi_U|_{U' \\times_S U'}$", "to $\\xi_{U'}$ by Lemma \\ref{lemma-uniqueness-relative-dualizing}", "(note that trivially the restriction of a relative dualizing", "complex to an open is a relative dualizing complex).", "The uniqueness guarantees that", "$\\rho^U_{U''} = \\rho^V_{U''} \\circ \\rho ^U_{U'}|_{U''}$", "for $U'' \\subset U' \\subset U$ in $\\mathcal{B}$.", "Observe that $\\text{Ext}^i(K_U, K_U) = 0$ for $i < 0$", "for $U \\in \\mathcal{B}$ by Lemma \\ref{lemma-relative-dualizing-RHom}", "applied to $U/S$ and $K_U$.", "Thus the BBD glueing lemma", "(Cohomology, Theorem \\ref{cohomology-theorem-glueing-bbd-general})", "tells us there is a unique solution, namely, an object", "$K \\in D(\\mathcal{O}_X)$ and isomorphisms $\\rho_U : K|_U \\to K_U$", "such that we have", "$\\rho^U_{U'} \\circ \\rho_U|_{U'} = \\rho_{U'}$ for all $U' \\subset U$,", "$U, U' \\in \\mathcal{B}$.", "\\medskip\\noindent", "To finish the proof we have to construct the map", "$$", "\\xi : \\Delta_*\\mathcal{O}_X \\longrightarrow L\\text{pr}_1^*K|_W", "$$", "in $D(\\mathcal{O}_W)$ inducing an isomorphism from $\\Delta_*\\mathcal{O}_X$ to", "$R\\SheafHom_{\\mathcal{O}_W}(\\Delta_*\\mathcal{O}_X, L\\text{pr}_1^*K|_W)$.", "Since we may change $W$, we choose", "$W = \\bigcup_{U \\in \\mathcal{B}} U \\times_S U$.", "We can use $\\rho_U$ to get isomorphisms", "$$", "R\\SheafHom_{\\mathcal{O}_W}(", "\\Delta_*\\mathcal{O}_X, L\\text{pr}_1^*K|_W)|_{U \\times_S U}", "\\xrightarrow{\\rho_U}", "R\\SheafHom_{\\mathcal{O}_{U \\times_S U}}(", "\\Delta_*\\mathcal{O}_U, L\\text{pr}_1^*K_U)", "$$", "As $W$ is covered by the opens $U \\times_S U$", "we conclude that the cohomology sheaves of", "$R\\SheafHom_{\\mathcal{O}_W}(\\Delta_*\\mathcal{O}_X, L\\text{pr}_1^*K|_W)$", "are zero except in degree $0$. Moreover, we obtain isomorphisms", "$$", "H^0\\left(U \\times_S U, R\\SheafHom_{\\mathcal{O}_W}(\\Delta_*\\mathcal{O}_X,", "L\\text{pr}_1^*K|_W)\\right)", "\\xrightarrow{\\rho_U}", "H^0\\left((R\\SheafHom_{\\mathcal{O}_{U \\times_S U}}(", "\\Delta_*\\mathcal{O}_U, L\\text{pr}_1^*K_U)\\right)", "$$", "Let $\\tau_U$ in the LHS be an element mapping to $\\xi_U$ under this map.", "The compatibilities between", "$\\rho^U_{U'}$, $\\xi_U$, $\\xi_{U'}$, $\\rho_U$, and $\\rho_{U'}$", "for $U' \\subset U \\subset X$ open $U', U \\in \\mathcal{B}$", "imply that $\\tau_U|_{U' \\times_S U'} = \\tau_{U'}$.", "Thus we get a global section $\\tau$ of the $0$th cohomology sheaf", "$H^0(R\\SheafHom_{\\mathcal{O}_W}(\\Delta_*\\mathcal{O}_X, L\\text{pr}_1^*K|_W))$.", "Since the other cohomology sheaves of", "$R\\SheafHom_{\\mathcal{O}_W}(\\Delta_*\\mathcal{O}_X, L\\text{pr}_1^*K|_W)$", "are zero, this global section $\\tau$", "determines a morphism $\\xi$ as desired. Since the restriction", "of $\\xi$ to $U \\times_S U$ gives $\\xi_U$, we see that it", "satisfies the final condition of", "Definition \\ref{definition-relative-dualizing-complex}." ], "refs": [ "dualizing-lemma-base-change-relative-dualizing", "duality-lemma-relative-dualizing-complex-algebra", "duality-lemma-uniqueness-relative-dualizing", "duality-lemma-relative-dualizing-RHom", "cohomology-theorem-glueing-bbd-general", "duality-definition-relative-dualizing-complex" ], "ref_ids": [ 2920, 13610, 13612, 13611, 2032, 13644 ] } ], "ref_ids": [] }, { "id": 13614, "type": "theorem", "label": "duality-lemma-base-change-relative-dualizing", "categories": [ "duality" ], "title": "duality-lemma-base-change-relative-dualizing", "contents": [ "Consider a cartesian square", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "of schemes. Assume $X \\to S$ is flat and locally of finite presentation.", "Let $(K, \\xi)$ be a relative dualizing complex for $f$.", "Set $K' = L(g')^*K$. Let $\\xi'$ be the derived base change of $\\xi$", "(see proof). Then $(K', \\xi')$ is a relative dualizing complex for $f'$." ], "refs": [], "proofs": [ { "contents": [ "Consider the cartesian square", "$$", "\\xymatrix{", "X' \\ar[d]_{\\Delta_{X'/S'}} \\ar[r] & X \\ar[d]^{\\Delta_{X/S}} \\\\", "X' \\times_{S'} X' \\ar[r]^{g' \\times g'} & X \\times_S X", "}", "$$", "Choose $W \\subset X \\times_S X$ open such that $\\Delta_{X/S}$", "factors through a closed immersion $\\Delta : X \\to W$.", "Choose $W' \\subset X' \\times_{S'} X'$ open such that $\\Delta_{X'/S'}$", "factors through a closed immersion $\\Delta' : X \\to W'$", "and such that $(g' \\times g')(W') \\subset W$. Let us still denote", "$g' \\times g' : W' \\to W$ the induced morphism. We have", "$$", "L(g' \\times g')^*\\Delta_*\\mathcal{O}_X =", "\\Delta'_*\\mathcal{O}_{X'}", "\\quad\\text{and}\\quad", "L(g' \\times g')^*L\\text{pr}_1^*K|_W =", "L\\text{pr}_1^*K'|_{W'}", "$$", "The first equality holds because $X$ and $X' \\times_{S'} X'$", "are tor independent over $X \\times_S X$ (see for example", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-case-of-tor-independence}).", "The second holds by transitivity of derived pullback", "(Cohomology, Lemma \\ref{cohomology-lemma-derived-pullback-composition}).", "Thus $\\xi' = L(g' \\times g')^*\\xi$ can be viewed as a map", "$$", "\\xi' : \\Delta'_*\\mathcal{O}_{X'} \\longrightarrow L\\text{pr}_1^*K'|_{W'}", "$$", "Having said this the proof of the lemma is straightforward.", "First, $K'$ is $S'$-perfect by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-base-change-relatively-perfect}.", "To check that $\\xi'$ induces an isomorphism", "of $\\Delta'_*\\mathcal{O}_{X'}$ to", "$R\\SheafHom_{\\mathcal{O}_{W'}}(", "\\Delta'_*\\mathcal{O}_{X'}, L\\text{pr}_1^*K'|_{W'})$", "we may work affine locally. By", "Lemma \\ref{lemma-relative-dualizing-complex-algebra}", "we reduce to the corresponding statement in algebra", "which is proven in Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-base-change-relative-dualizing}." ], "refs": [ "more-morphisms-lemma-case-of-tor-independence", "cohomology-lemma-derived-pullback-composition", "perfect-lemma-base-change-relatively-perfect", "duality-lemma-relative-dualizing-complex-algebra", "dualizing-lemma-base-change-relative-dualizing" ], "ref_ids": [ 14057, 2117, 7080, 13610, 2920 ] } ], "ref_ids": [] }, { "id": 13615, "type": "theorem", "label": "duality-lemma-flat-proper-relative-dualizing", "categories": [ "duality" ], "title": "duality-lemma-flat-proper-relative-dualizing", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $f : X \\to S$ be a proper, flat morphism of finite presentation.", "The relative dualizing complex $\\omega_{X/S}^\\bullet$ of", "Remark \\ref{remark-relative-dualizing-complex}", "together with (\\ref{equation-pre-rigid}) is a relative", "dualizing complex in the sense of", "Definition \\ref{definition-relative-dualizing-complex}." ], "refs": [ "duality-remark-relative-dualizing-complex", "duality-definition-relative-dualizing-complex" ], "proofs": [ { "contents": [ "In Lemma \\ref{lemma-properties-relative-dualizing} we proved that", "$\\omega_{X/S}^\\bullet$ is $S$-perfect.", "Let $c$ be the right adjoint of", "Lemma \\ref{lemma-twisted-inverse-image}", "for the diagonal $\\Delta : X \\to X \\times_S X$.", "Then we can apply $\\Delta_*$ to (\\ref{equation-pre-rigid})", "to get an isomorphism", "$$", "\\Delta_*\\mathcal{O}_X \\to", "\\Delta_*(c(L\\text{pr}_1^*\\omega_{X/S}^\\bullet)) =", "R\\SheafHom_{\\mathcal{O}_{X \\times_S X}}(", "\\Delta_*\\mathcal{O}_X, L\\text{pr}_1^*\\omega_{X/S}^\\bullet)", "$$", "The equality holds by", "Lemmas \\ref{lemma-twisted-inverse-image-closed} and", "\\ref{lemma-sheaf-with-exact-support-ext}.", "This finishes the proof." ], "refs": [ "duality-lemma-properties-relative-dualizing", "duality-lemma-twisted-inverse-image", "duality-lemma-twisted-inverse-image-closed", "duality-lemma-sheaf-with-exact-support-ext" ], "ref_ids": [ 13537, 13503, 13525, 13521 ] } ], "ref_ids": [ 13649, 13644 ] }, { "id": 13616, "type": "theorem", "label": "duality-lemma-compactifyable-relative-dualizing", "categories": [ "duality" ], "title": "duality-lemma-compactifyable-relative-dualizing", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$", "be a morphism of $\\textit{FTS}_S$. If $f$ is flat, then", "$f^!\\mathcal{O}_Y$ is (the first component of)", "a relative dualizing complex for $X$ over $Y$ in the sense of", "Definition \\ref{definition-relative-dualizing-complex}." ], "refs": [ "duality-definition-relative-dualizing-complex" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-flat-shriek-relatively-perfect}", "we have that $f^!\\mathcal{O}_Y$", "is $Y$-perfect. As $f$ is separated the diagonal", "$\\Delta : X \\to X \\times_Y X$ is a closed immersion and", "$\\Delta_*\\Delta^!(-) =", "R\\SheafHom_{\\mathcal{O}_{X \\times_Y X}}(\\mathcal{O}_X, -)$, see", "Lemmas", "\\ref{lemma-twisted-inverse-image-closed} and", "\\ref{lemma-sheaf-with-exact-support-ext}.", "Hence to finish the proof it suffices to show", "$\\Delta^!(L\\text{pr}_1^*f^!(\\mathcal{O}_Y)) \\cong \\mathcal{O}_X$", "where $\\text{pr}_1 : X \\times_Y X \\to X$ is the first projection.", "We have", "$$", "\\mathcal{O}_X = \\Delta^! \\text{pr}_1^!\\mathcal{O}_X =", "\\Delta^! \\text{pr}_1^! L\\text{pr}_2^*\\mathcal{O}_Y =", "\\Delta^!(L\\text{pr}_1^* f^!\\mathcal{O}_Y)", "$$", "where $\\text{pr}_2 : X \\times_Y X \\to X$ is the second projection", "and where we have used the base change isomorphism", "$\\text{pr}_1^! \\circ L\\text{pr}_2^* = L\\text{pr}_1^* \\circ f^!$ of", "Lemma \\ref{lemma-base-change-shriek-flat}." ], "refs": [ "duality-lemma-flat-shriek-relatively-perfect", "duality-lemma-twisted-inverse-image-closed", "duality-lemma-sheaf-with-exact-support-ext", "duality-lemma-base-change-shriek-flat" ], "ref_ids": [ 13563, 13525, 13521, 13565 ] } ], "ref_ids": [ 13644 ] }, { "id": 13617, "type": "theorem", "label": "duality-lemma-relative-dualizing-composition", "categories": [ "duality" ], "title": "duality-lemma-relative-dualizing-composition", "contents": [ "Let $f : Y \\to X$ and $X \\to S$ be morphisms of schemes", "which are flat and of finite presentation.", "Let $(K, \\xi)$ and $(M, \\eta)$", "be a relative dualizing complex for $X \\to S$ and $Y \\to X$.", "Set $E = M \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Lf^*K$.", "Then $(E, \\zeta)$ is a relative dualizing complex for $Y \\to S$ for", "a suitable $\\zeta$." ], "refs": [], "proofs": [ { "contents": [ "Using Lemma \\ref{lemma-relative-dualizing-complex-algebra}", "and the algebraic version of this lemma (Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-relative-dualizing-composition})", "we see that $E$", "is affine locally the first component of a relative dualizing complex.", "In particular we see that $E$", "is $S$-perfect since this may be checked affine locally, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-locally-rel-perfect}.", "\\medskip\\noindent", "Let us first prove the existence of $\\zeta$ in case the", "morphisms $X \\to S$ and $Y \\to X$ are separated so that", "$\\Delta_{X/S}$, $\\Delta_{Y/X}$, and $\\Delta_{Y/S}$", "are closed immersions. Consider the following diagram", "$$", "\\xymatrix{", "& & Y \\ar@{=}[r] & Y \\ar[d]^f \\\\", "Y \\ar[r]_{\\Delta_{Y/X}} &", "Y \\times_X Y \\ar[d]_m \\ar[r]_\\delta \\ar[ru]_q &", "Y \\times_S Y \\ar[d]^{f \\times f} \\ar[ru]_p & X\\\\", "& X \\ar[r]^{\\Delta_{X/S}} & X \\times_S X \\ar[ru]_r", "}", "$$", "where $p$, $q$, $r$ are the first projections.", "By Lemma \\ref{lemma-sheaf-with-exact-support-internal-home}", "we have", "$$", "R\\SheafHom_{\\mathcal{O}_{Y \\times_S Y}}(", "\\Delta_{Y/S, *}\\mathcal{O}_Y, Lp^*E) =", "R\\delta_*\\left(R\\SheafHom_{\\mathcal{O}_{Y \\times_X Y}}(", "\\Delta_{Y/X, *}\\mathcal{O}_Y,", "R\\SheafHom(\\mathcal{O}_{Y \\times_X Y}, Lp^*E))\\right)", "$$", "By Lemma \\ref{lemma-sheaf-with-exact-support-tensor} we have", "$$", "R\\SheafHom(\\mathcal{O}_{Y \\times_X Y}, Lp^*E) =", "R\\SheafHom(\\mathcal{O}_{Y \\times_X Y}, L(f \\times f)^*Lr^*K)", "\\otimes_{\\mathcal{O}_{Y \\times_S Y}}^\\mathbf{L} Lq^*M", "$$", "By Lemma \\ref{lemma-flat-bc-sheaf-with-exact-support} we have", "$$", "R\\SheafHom(\\mathcal{O}_{Y \\times_X Y}, L(f \\times f)^*Lr^*K) =", "Lm^*R\\SheafHom(\\mathcal{O}_X, Lr^*K)", "$$", "The last expression is isomorphic (via $\\xi$) to", "$Lm^*\\mathcal{O}_X = \\mathcal{O}_{Y \\times_X Y}$.", "Hence the expression preceding is isomorphic to", "$Lq^*M$. Hence", "$$", "R\\SheafHom_{\\mathcal{O}_{Y \\times_S Y}}(", "\\Delta_{Y/S, *}\\mathcal{O}_Y, Lp^*E) =", "R\\delta_*\\left(R\\SheafHom_{\\mathcal{O}_{Y \\times_X Y}}(", "\\Delta_{Y/X, *}\\mathcal{O}_Y, Lq^*M)\\right)", "$$", "The material inside the parentheses is isomorphic to", "$\\Delta_{Y/X, *}*\\mathcal{O}_X$ via $\\eta$.", "This finishes the proof in the separated case.", "\\medskip\\noindent", "In the general case we choose an open $W \\subset X \\times_S X$", "such that $\\Delta_{X/S}$ factors through a closed immersion", "$\\Delta : X \\to W$ and we choose an open $V \\subset Y \\times_X Y$", "such that $\\Delta_{Y/X}$ factors through a closed immersion", "$\\Delta' : Y \\to V$. Finally, choose an open", "$W' \\subset Y \\times_S Y$ whose intersection with $Y \\times_X Y$", "gives $V$ and which maps into $W$. Then we consider the diagram", "$$", "\\xymatrix{", "& & Y \\ar@{=}[r] & Y \\ar[d]^f \\\\", "Y \\ar[r]_{\\Delta'} &", "V \\ar[d]_m \\ar[r]_\\delta \\ar[ru]_q &", "W' \\ar[d]^{f \\times f} \\ar[ru]_p & X\\\\", "& X \\ar[r]^\\Delta & W \\ar[ru]_r", "}", "$$", "and we use exactly the same argument as before." ], "refs": [ "duality-lemma-relative-dualizing-complex-algebra", "dualizing-lemma-relative-dualizing-composition", "perfect-lemma-affine-locally-rel-perfect", "duality-lemma-sheaf-with-exact-support-internal-home", "duality-lemma-sheaf-with-exact-support-tensor", "duality-lemma-flat-bc-sheaf-with-exact-support" ], "ref_ids": [ 13610, 2922, 7077, 13522, 13528, 13527 ] } ], "ref_ids": [] }, { "id": 13618, "type": "theorem", "label": "duality-lemma-determinant", "categories": [ "duality" ], "title": "duality-lemma-determinant", "contents": [ "Let $X$ be a locally ringed space. Let", "$$", "\\mathcal{E}_1 \\xrightarrow{\\alpha} \\mathcal{E}_0 \\to \\mathcal{F} \\to 0", "$$", "be a short exact sequence of $\\mathcal{O}_X$-modules.", "Assume $\\mathcal{E}_1$ and $\\mathcal{E}_0$ are locally", "free of ranks $r_1, r_0$. Then there is a canonical map", "$$", "\\wedge^{r_0 - r_1}\\mathcal{F}", "\\longrightarrow", "\\wedge^{r_1}(\\mathcal{E}_1^\\vee) \\otimes \\wedge^{r_0}\\mathcal{E}_0", "$$", "which is an isomorphism on the stalk at $x \\in X$", "if and only if $\\mathcal{F}$ is locally free of rank $r_0 - r_1$", "in an open neighbourhood of $x$." ], "refs": [], "proofs": [ { "contents": [ "If $r_1 > r_0$ then $\\wedge^{r_0 - r_1}\\mathcal{F} = 0$ by convention", "and the unique map cannot be an isomorphism. Thus we may assume", "$r = r_0 - r_1 \\geq 0$. Define the map by the formula", "$$", "s_1 \\wedge \\ldots \\wedge s_r \\mapsto", "t_1^\\vee \\wedge \\ldots \\wedge t_{r_1}^\\vee \\otimes", "\\alpha(t_1) \\wedge \\ldots \\wedge \\alpha(t_{r_1}) \\wedge", "\\tilde s_1 \\wedge \\ldots \\wedge \\tilde s_r", "$$", "where $t_1, \\ldots, t_{r_1}$ is a local basis for $\\mathcal{E}_1$,", "correspondingly", "$t_1^\\vee, \\ldots, t_{r_1}^\\vee$ is the dual basis for $\\mathcal{E}_1^\\vee$,", "and $s'_i$ is a local lift of $s_i$ to a section of $\\mathcal{E}_0$.", "We omit the proof that this is well defined.", "\\medskip\\noindent", "If $\\mathcal{F}$ is locally free of rank $r$, then it is straightforward", "to verify that the map is an isomorphism. Conversely, assume the map", "is an isomorphism on stalks at $x$. Then $\\wedge^r\\mathcal{F}_x$", "is invertible. This implies that $\\mathcal{F}_x$ is generated by", "at most $r$ elements. This can only happen if $\\alpha$ has rank", "$r$ modulo $\\mathfrak m_x$, i.e., $\\alpha$ has maximal rank modulo", "$\\mathfrak m_x$. This implies that $\\alpha$ has maximal rank", "in a neighbourhood of $x$ and hence $\\mathcal{F}$ is locally free", "of rank $r$ in a neighbourhood as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13619, "type": "theorem", "label": "duality-lemma-fundamental-class-lci", "categories": [ "duality" ], "title": "duality-lemma-fundamental-class-lci", "contents": [ "Let $Y$ be a Noetherian scheme. Let $f : X \\to Y$ be a", "local complete intersection morphism which factors", "as an immersion $X \\to P$ followed by a proper smooth morphism $P \\to Y$.", "Let $r$ be the locally constant function on", "$X$ such that $\\omega_{Y/X} = H^{-r}(f^!\\mathcal{O}_Y)$", "is the unique nonzero cohomology sheaf of $f^!\\mathcal{O}_Y$, see", "Lemma \\ref{lemma-lci-shriek}.", "Then there is a map", "$$", "\\wedge^r\\Omega_{X/Y} \\longrightarrow \\omega_{Y/X}", "$$", "which is an isomorphism on the stalk at a point $x$ if and only", "if $f$ is smooth at $x$." ], "refs": [ "duality-lemma-lci-shriek" ], "proofs": [ { "contents": [ "The assumption implies that $X$ is compactifyable over $Y$ hence $f^!$", "is defined, see Section \\ref{section-upper-shriek}.", "Let $j : W \\to P$ be an open subscheme such that", "$X \\to P$ factors through a closed immersion $i : X \\to W$.", "Moreover, we have $f^! = i^! \\circ j^! \\circ g^!$ where", "$g : P \\to Y$ is the given morphism.", "We have $g^!\\mathcal{O}_Y = \\wedge^d\\Omega_{P/Y}[d]$ by", "Lemma \\ref{lemma-smooth-proper} where $d$ is the locally", "constant function giving the relative dimension of $P$ over $Y$.", "We have $j^! = j^*$. We have $i^!\\mathcal{O}_W = \\wedge^c\\mathcal{N}[-c]$", "where $c$ is the codimension of $X$ in $W$ (a locally constant", "function on $X$) and where $\\mathcal{N}$ is the normal sheaf of", "the Koszul-regular immersion $i$, see Lemma \\ref{lemma-regular-immersion}.", "Combining the above we find", "$$", "f^!\\mathcal{O}_Y =", "\\left(\\wedge^c\\mathcal{N} \\otimes_{\\mathcal{O}_X}", "\\wedge^d\\Omega_{P/Y}|_X\\right)[d - c]", "$$", "where we have also used Lemma \\ref{lemma-perfect-comparison-shriek}.", "Thus $r = d|_X - c$ as locally constant functions on $X$.", "The conormal sheaf of $X \\to P$ is the module", "$\\mathcal{I}/\\mathcal{I}^2$ where $\\mathcal{I} \\subset \\mathcal{O}_W$", "is the ideal sheaf of $i$, see", "Morphisms, Section \\ref{morphisms-section-conormal-sheaf}.", "Consider the canonical exact sequence", "$$", "\\mathcal{I}/\\mathcal{I}^2 \\to", "\\Omega_{P/Y}|_X \\to \\Omega_{X/Y} \\to 0", "$$", "of Morphisms, Lemma \\ref{morphisms-lemma-differentials-relative-immersion}.", "We obtain our map by an application of Lemma \\ref{lemma-determinant}.", "\\medskip\\noindent", "If $f$ is smooth at $x$, then the map is an isomorphism by an application of", "Lemma \\ref{lemma-determinant}", "and the fact that $\\Omega_{X/Y}$ is locally free at $x$", "of rank $r$. Conversely, assume that our map is an isomorphism on stalks", "at $x$. Then the lemma shows that $\\Omega_{X/Y}$ is free of rank $r$", "after replacing $X$ by an open neighbourhood of $x$.", "On the other hand, we may also assume that $X = \\Spec(A)$ and", "$Y = \\Spec(R)$ where $A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$", "and where $f_1, \\ldots, f_m$ is a Koszul regular sequence", "(this follows from the definition of local complete intersection morphisms).", "Clearly this implies $r = n - m$. We conclude that the rank of the matrix", "of partials $\\partial f_j/\\partial x_i$ in the residue field at $x$ is $m$.", "Thus after reordering the variables we may assume", "the determinant of $(\\partial f_j/\\partial x_i)_{1 \\leq i, j \\leq m}$", "is invertible in an open neighbourhood of $x$. It follows", "that $R \\to A$ is smooth at this point, see for example", "Algebra, Example \\ref{algebra-example-make-standard-smooth}." ], "refs": [ "duality-lemma-smooth-proper", "duality-lemma-regular-immersion", "duality-lemma-perfect-comparison-shriek", "morphisms-lemma-differentials-relative-immersion", "duality-lemma-determinant", "duality-lemma-determinant" ], "ref_ids": [ 13550, 13549, 13562, 5319, 13618, 13618 ] } ], "ref_ids": [ 13564 ] }, { "id": 13620, "type": "theorem", "label": "duality-lemma-fundamental-class-almost-lci", "categories": [ "duality" ], "title": "duality-lemma-fundamental-class-almost-lci", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $r \\geq 0$. Assume", "\\begin{enumerate}", "\\item $Y$ is Cohen-Macaulay (Properties, Definition", "\\ref{properties-definition-Cohen-Macaulay}),", "\\item $f$ factors as $X \\to P \\to Y$ where the first morphism is", "an immersion and the second is smooth and proper,", "\\item if $x \\in X$ and $\\dim(\\mathcal{O}_{X, x}) \\leq 1$,", "then $f$ is Koszul at $x$ (More on Morphisms, Definition", "\\ref{more-morphisms-definition-lci}), and", "\\item if $\\xi$ is a generic point of an irreducible component of $X$, then", "we have", "$\\text{trdeg}_{\\kappa(f(\\xi))} \\kappa(\\xi) = r$.", "\\end{enumerate}", "Then with $\\omega_{Y/X} = H^{-r}(f^!\\mathcal{O}_Y)$ there is a map", "$$", "\\wedge^r\\Omega_{X/Y} \\longrightarrow \\omega_{Y/X}", "$$", "which is an isomorphism on the locus where $f$ is smooth." ], "refs": [ "properties-definition-Cohen-Macaulay", "more-morphisms-definition-lci" ], "proofs": [ { "contents": [ "Let $U \\subset X$ be the open subscheme over which $f$ is a", "local complete intersection morphism. Since $f$ has relative", "dimension $r$ at all generic points by assumption (4) we", "see that the locally constant function of", "Lemma \\ref{lemma-fundamental-class-lci}", "is constant with value $r$ and we obtain a map", "$$", "\\wedge^r\\Omega_{X/Y}|_U = \\wedge^r \\Omega_{U/Y}", "\\longrightarrow", "\\omega_{U/Y} = \\omega_{X/Y}|_U", "$$", "which is an isomorphism in the smooth points of $f$ (this locus", "is contained in $U$ because a smooth morphism is a local complete", "intersection morphism). By Lemma \\ref{lemma-shriek-over-CM}", "and the assumption that $Y$ is Cohen-Macaulay", "the module $\\omega_{X/Y}$ is $(S_2)$.", "Since $U$ contains all the points of codimension $1$ by condition (3)", "and using Divisors, Lemma \\ref{divisors-lemma-depth-2-hartog}", "we see that $j_*\\omega_{U/Y} = \\omega_{X/Y}$.", "Hence the map over $U$ extends to $X$ and the proof", "is complete." ], "refs": [ "duality-lemma-fundamental-class-lci", "duality-lemma-shriek-over-CM", "divisors-lemma-depth-2-hartog" ], "ref_ids": [ 13619, 13580, 7881 ] } ], "ref_ids": [ 3074, 14121 ] }, { "id": 13621, "type": "theorem", "label": "duality-lemma-lift-map", "categories": [ "duality" ], "title": "duality-lemma-lift-map", "contents": [ "Let $j : U \\to X$ be an open immersion of Noetherian schemes.", "Let $(K_n)$ be a Deligne system and denote", "$K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_U)$ the value", "of the constant system $(K_n|_U)$. Let $L$ be an object of", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.", "Then $\\colim \\Hom_X(K_n, L) = \\Hom_U(K, L|_U)$." ], "refs": [], "proofs": [ { "contents": [ "Let $L \\to M \\to N \\to L[1]$ be a distinguished triangle in", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$. Then we obtain", "a commutative diagram", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "\\colim \\Hom_X(K_n, L) \\ar[r] \\ar[d] &", "\\colim \\Hom_X(K_n, M) \\ar[r] \\ar[d] &", "\\colim \\Hom_X(K_n, N) \\ar[r] \\ar[d] &", "\\ldots \\\\", "\\ldots \\ar[r] &", "\\Hom_U(K, L|_U) \\ar[r] &", "\\Hom_U(K, M|_U) \\ar[r] &", "\\Hom_U(K, N|_U) \\ar[r] &", "\\ldots", "}", "$$", "whose rows are exact by Derived Categories, Lemma", "\\ref{derived-lemma-representable-homological} and", "Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}.", "Hence if the statement of the lemma holds for", "$N[-1]$, $L$, $N$, and $L[1]$ then it holds for $M$ by the 5-lemma.", "Thus, using the distinguished triangles", "for the canonical truncations of $L$ (see Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle})", "we reduce to the case that $L$ has only one nonzero cohomology sheaf.", "\\medskip\\noindent", "Choose a bounded complex $\\mathcal{F}^\\bullet$ of coherent", "$\\mathcal{O}_X$-modules and a quasi-coherent ideal", "$\\mathcal{I} \\subset \\mathcal{O}_X$ cutting out $X \\setminus U$", "such that $K_n$ is represented by $\\mathcal{I}^n\\mathcal{F}^\\bullet$.", "Using ``stupid'' truncations we obtain compatible termwise split short", "exact sequences of complexes", "$$", "0 \\to \\sigma_{\\geq a + 1} \\mathcal{I}^n\\mathcal{F}^\\bullet \\to", "\\mathcal{I}^n\\mathcal{F}^\\bullet \\to", "\\sigma_{\\leq a} \\mathcal{I}^n\\mathcal{F}^\\bullet \\to 0", "$$", "which in turn correspond to compatible systems of distinguished", "triangles in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.", "Arguing as above we reduce to the case where $\\mathcal{F}^\\bullet$", "has only one nonzero term. This reduces us to the case discussed in", "the next paragraph.", "\\medskip\\noindent", "Given a coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ and a coherent", "$\\mathcal{O}_X$-module $\\mathcal{G}$ we", "have to show that the canonical map", "$$", "\\colim \\Ext^i_X(\\mathcal{I}^n\\mathcal{F}, \\mathcal{G})", "\\longrightarrow", "\\Ext^i_U(\\mathcal{F}|_U, \\mathcal{G}|_U)", "$$", "is an isomorphism for all $i \\geq 0$. For $i = 0$ this is", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-homs-over-open}.", "Assume $i > 0$.", "\\medskip\\noindent", "Injectivity. Let $\\xi \\in \\Ext^i_X(\\mathcal{I}^n\\mathcal{F}, \\mathcal{G})$", "be an element whose restriction to $U$ is zero. We have to show there exists", "an $m \\geq n$ such that the restriction of $\\xi$ to", "$\\mathcal{I}^m\\mathcal{F} = \\mathcal{I}^{m - n}\\mathcal{I}^n\\mathcal{F}$", "is zero. After replacing $\\mathcal{F}$ by $\\mathcal{I}^n\\mathcal{F}$", "we may assume $n = 0$, i.e., we have", "$\\xi \\in \\Ext^i_X(\\mathcal{F}, \\mathcal{G})$", "whose restriction to $U$ is zero. By", "Derived Categories of Schemes, Proposition \\ref{perfect-proposition-DCoh}", "we have", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X) = D^b(\\textit{Coh}(\\mathcal{O}_X))$.", "Hence we can compute the $\\Ext$ group in the abelian category of", "coherent $\\mathcal{O}_X$-modules. This implies there exists an", "surjection $\\alpha : \\mathcal{F}'' \\to \\mathcal{F}$ such that", "$\\xi \\circ \\alpha = 0$ (this is where we use that $i > 0$).", "Set $\\mathcal{F}' = \\Ker(\\alpha)$ so that we have a short exact", "sequence", "$$", "0 \\to \\mathcal{F}' \\to \\mathcal{F}'' \\to \\mathcal{F} \\to 0", "$$", "It follows that $\\xi$ is the image of an element", "$\\xi' \\in \\Ext^{i - 1}_X(\\mathcal{F}', \\mathcal{G})$", "whose restriction to $U$ is in the image of", "$\\Ext^{i - 1}_U(\\mathcal{F}''|_U, \\mathcal{G}|_U) \\to", "\\Ext^{i - 1}_U(\\mathcal{F}'|_U, \\mathcal{G}|_U)$.", "By Artin-Rees the inverse systems $(\\mathcal{I}^n\\mathcal{F}')$ and", "$(\\mathcal{I}^n \\mathcal{F}'' \\cap \\mathcal{F}')$ are pro-isomorphic, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-Artin-Rees}.", "Since we have the compatible system of short exact sequences", "$$", "0 \\to", "\\mathcal{F}' \\cap \\mathcal{I}^n\\mathcal{F}'' \\to", "\\mathcal{I}^n\\mathcal{F}'' \\to", "\\mathcal{I}^n\\mathcal{F} \\to 0", "$$", "we obtain a commutativew diagram", "$$", "\\xymatrix{", "\\colim \\Ext^{i - 1}_X(\\mathcal{I}^n\\mathcal{F}'', \\mathcal{G})", "\\ar[r] \\ar[d] &", "\\colim \\Ext^{i - 1}_X(\\mathcal{F}' \\cap \\mathcal{I}^n\\mathcal{F}'', \\mathcal{G})", "\\ar[r] \\ar[d] &", "\\colim \\Ext^i_X(\\mathcal{I}^n\\mathcal{F}, \\mathcal{G})", "\\ar[d] \\\\", "\\Ext^{i - 1}_U(\\mathcal{F}''|_U, \\mathcal{G}|_U) \\ar[r] &", "\\Ext^{i - 1}_U(\\mathcal{F}'|_U, \\mathcal{G}|_U) \\ar[r] &", "\\Ext^{i - 1}_U(\\mathcal{F}|_U, \\mathcal{G}|_U)", "}", "$$", "with exact rows. By induction on $i$ and the comment on inverse systems above", "we find that the left two vertical arrows are isomorphisms.", "Now $\\xi$ gives an element in the top right group which is the image", "of $\\xi'$ in the middle top group, which in turn maps to an element of", "the bottom middle group coming from some element in the left bottom group.", "We conclude that $\\xi$ maps to zero in", "$\\Ext^i_X(\\mathcal{I}^n\\mathcal{F}, \\mathcal{G})$", "for some $n$ as desired.", "\\medskip\\noindent", "Surjectivity. Let $\\xi \\in \\Ext^i_U(\\mathcal{F}|_U, \\mathcal{G}|_U)$.", "Arguing as above using that $i > 0$ we can find an surjection", "$\\mathcal{H} \\to \\mathcal{F}|_U$ of coherent $\\mathcal{O}_U$-modules", "such that $\\xi$ maps to zero in $\\Ext^i_U(\\mathcal{H}, \\mathcal{G}|_U)$.", "Then we can find a map $\\varphi : \\mathcal{F}'' \\to \\mathcal{F}$", "of coherent $\\mathcal{O}_X$-modules whose restriction to $U$ is", "$\\mathcal{H} \\to \\mathcal{F}|_U$, see", "Properties, Lemma \\ref{properties-lemma-extend-finite-presentation}.", "Observe that the lemma doesn't guarantee $\\varphi$ is surjective", "but this won't matter (it is possible to pick a surjective $\\varphi$", "with a little bit of additional work).", "Denote $\\mathcal{F}' = \\Ker(\\varphi)$. The short exact sequence", "$$", "0 \\to \\mathcal{F}'|_U \\to \\mathcal{F}''|_U \\to \\mathcal{F}|_U \\to 0", "$$", "shows that $\\xi$ is the image of $\\xi'$ in", "$\\Ext^{i - 1}_U(\\mathcal{F}'|_U, \\mathcal{G}|_U)$.", "By induction on $i$ we can find an $n$ such that", "$\\xi'$ is the image of some $\\xi'_n$ in", "$\\Ext^{i - 1}_X(\\mathcal{I}^n\\mathcal{F}', \\mathcal{G})$.", "By Artin-Rees we can find an $m \\geq n$ such that", "$\\mathcal{F}' \\cap \\mathcal{I}^m\\mathcal{F}'' \\subset", "\\mathcal{I}^n\\mathcal{F}'$. Using the short exact sequence", "$$", "0 \\to \\mathcal{F}' \\cap \\mathcal{I}^m\\mathcal{F}'' \\to", "\\mathcal{I}^m\\mathcal{F}'' \\to \\mathcal{I}^m\\Im(\\varphi) \\to 0", "$$", "the image of $\\xi'_n$ in", "$\\Ext^{i - 1}_X(\\mathcal{F}' \\cap \\mathcal{I}^m\\mathcal{F}'', \\mathcal{G})$", "maps by the boundary map to an element $\\xi_m$ of", "$\\Ext^i_X(\\mathcal{I}^m\\Im(\\varphi), \\mathcal{G})$ which maps", "to $\\xi$. Since $\\Im(\\varphi)$ and $\\mathcal{F}$ agree over $U$", "we see that $\\mathcal{F}/\\mathcal{I}^m\\Im(\\varphi)$ is supported", "on $X \\setminus U$. Hence there exists an $l \\geq m$", "such that $\\mathcal{I}^l\\mathcal{F} \\subset \\mathcal{I}^m\\Im(\\varphi)$, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-power-ideal-kills-sheaf}.", "Taking the image of $\\xi_m$ in", "$\\Ext^i_X(\\mathcal{I}^l\\mathcal{F}, \\mathcal{G})$ we win." ], "refs": [ "derived-lemma-representable-homological", "algebra-lemma-directed-colimit-exact", "derived-remark-truncation-distinguished-triangle", "coherent-lemma-homs-over-open", "perfect-proposition-DCoh", "coherent-lemma-Artin-Rees", "properties-lemma-extend-finite-presentation", "coherent-lemma-power-ideal-kills-sheaf" ], "ref_ids": [ 1758, 343, 2016, 3322, 7110, 3321, 3021, 3320 ] } ], "ref_ids": [] }, { "id": 13622, "type": "theorem", "label": "duality-lemma-lift-map-plus", "categories": [ "duality" ], "title": "duality-lemma-lift-map-plus", "contents": [ "The result of Lemma \\ref{lemma-lift-map} holds even for", "$L \\in D^+_{\\textit{Coh}}(\\mathcal{O}_X)$." ], "refs": [ "duality-lemma-lift-map" ], "proofs": [ { "contents": [ "Namely, if $(K_n)$ is a Deligne system then there exists a", "$b \\in \\mathbf{Z}$ such that $H^i(K_n) = 0$ for $i > b$.", "Then $\\Hom(K_n, L) = \\Hom(K_n, \\tau_{\\leq b}L)$ and", "$\\Hom(K, L) = \\Hom(K, \\tau_{\\leq b}L)$. Hence using", "the result of the lemma for $\\tau_{\\leq b}L$ we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 13621 ] }, { "id": 13623, "type": "theorem", "label": "duality-lemma-extension-by-zero", "categories": [ "duality" ], "title": "duality-lemma-extension-by-zero", "contents": [ "Let $j : U \\to X$ be an open immersion of Noetherian schemes.", "\\begin{enumerate}", "\\item Let $(K_n)$ and $(L_n)$ be Deligne systems.", "Let $K$ and $L$ be the values of the constant systems", "$(K_n|_U)$ and $(L_n|_U)$. Given a morphism $\\alpha : K \\to L$", "of $D(\\mathcal{O}_U)$", "there is a unique morphism of pro-systems $(K_n) \\to (L_n)$", "of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ whose restriction to $U$ is $\\alpha$.", "\\item Given $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_U)$ there exists a", "Deligne system $(K_n)$ such that $(K_n|_U)$ is constant", "with value $K$.", "\\item The pro-object $(K_n)$ of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$", "of (2) is unique up to unique isomorphism (as a pro-object).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is an immediate consequence of Lemma \\ref{lemma-lift-map}", "and the fact that morphisms between pro-systems are", "the same as morphisms between the functors they corepresent, see", "Categories, Remark \\ref{categories-remark-pro-category-copresheaves}.", "\\medskip\\noindent", "Let $K$ be as in (2). We can choose $K' \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$", "whose restriction to $U$ is isomorphic to $K$, see", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-lift-coherent}.", "By Derived Categories of Schemes, Proposition \\ref{perfect-proposition-DCoh}", "we can represent $K'$ by a bounded complex $\\mathcal{F}^\\bullet$", "of coherent $\\mathcal{O}_X$-modules. Choose a quasi-coherent sheaf", "of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$ whose vanishing", "locus is $X \\setminus U$ (for example choose $\\mathcal{I}$ to correspond", "to the reduced induced subscheme structure on $X \\setminus U$).", "Then we can set $K_n$ equal to the object represented by the complex", "$\\mathcal{I}^n\\mathcal{F}^\\bullet$ as in the introduction", "to this section.", "\\medskip\\noindent", "Part (3) is immediate from parts (1) and (2)." ], "refs": [ "duality-lemma-lift-map", "categories-remark-pro-category-copresheaves", "perfect-lemma-lift-coherent", "perfect-proposition-DCoh" ], "ref_ids": [ 13621, 12421, 6997, 7110 ] } ], "ref_ids": [] }, { "id": 13624, "type": "theorem", "label": "duality-lemma-extension-by-zero-triangle", "categories": [ "duality" ], "title": "duality-lemma-extension-by-zero-triangle", "contents": [ "Let $j : U \\to X$ be an open immersion of Noetherian schemes. Let", "$$", "K \\to L \\to M \\to K[1]", "$$", "be a distinguished triangle of $D^b_{\\textit{Coh}}(\\mathcal{O}_U)$.", "Then there exists an inverse system of distinguished triangles", "$$", "K_n \\to L_n \\to M_n \\to K_n[1]", "$$", "in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ such that $(K_n)$, $(L_n)$, $(M_n)$", "are Deligne systems and such that the restriction of these", "distinguished triangles to $U$ is isomorphic to the distinguished triangle", "we started out with." ], "refs": [], "proofs": [ { "contents": [ "Let $(K_n)$ be as in Lemma \\ref{lemma-extension-by-zero} part (2).", "Choose an object $L'$ of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$", "whose restriction to $U$ is $L$ (we can do this as the lemma shows).", "By Lemma \\ref{lemma-lift-map} we can find an $n$ and a morphism", "$K_n \\to L'$ on $X$ whose restriction to $U$ is the given arrow", "$K \\to L$. We conclude there is a morphism $K' \\to L'$ of", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ whose restriction to $U$", "is the given arrow $K \\to L$.", "\\medskip\\noindent", "By Derived Categories of Schemes, Proposition \\ref{perfect-proposition-DCoh}", "we can find a morphism $\\alpha^\\bullet : \\mathcal{F}^\\bullet \\to", "\\mathcal{G}^\\bullet$ of bounded complexes", "of coherent $\\mathcal{O}_X$-modules representing $K' \\to L'$.", "Choose a quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$", "whose vanishing locus is $X \\setminus U$.", "Then we let", "$K_n = \\mathcal{I}^n\\mathcal{F}^\\bullet$", "and", "$L_n = \\mathcal{I}^n\\mathcal{G}^\\bullet$.", "Observe that $\\alpha^\\bullet$ induces a morphism of", "complexes $\\alpha_n^\\bullet : \\mathcal{I}^n\\mathcal{F}^\\bullet \\to", "\\mathcal{I}^n\\mathcal{G}^\\bullet$. From the construction of cones in", "Derived Categories, Section \\ref{derived-section-cones}", "it is clear that", "$$", "C(\\alpha_n)^\\bullet = \\mathcal{I}^nC(\\alpha^\\bullet)", "$$", "and hence we can set $M_n = C(\\alpha_n)^\\bullet$. Namely, we", "have a compatible system of distinguished triangles", "(see discussion in Derived Categories, Section", "\\ref{derived-section-canonical-delta-functor})", "$$", "K_n \\to L_n \\to M_n \\to K_n[1]", "$$", "whose restriction to $U$ is isomorphic to the distinguished ", "triangle we started out with by axiom TR3 and Derived Categories,", "Lemma \\ref{derived-lemma-third-isomorphism-triangle}." ], "refs": [ "duality-lemma-extension-by-zero", "duality-lemma-lift-map", "perfect-proposition-DCoh", "derived-lemma-third-isomorphism-triangle" ], "ref_ids": [ 13623, 13621, 7110, 1759 ] } ], "ref_ids": [] }, { "id": 13625, "type": "theorem", "label": "duality-lemma-deligne-system-2-out-of-3", "categories": [ "duality" ], "title": "duality-lemma-deligne-system-2-out-of-3", "contents": [ "Let $j : U \\to X$ be an open immersion of Noetherian schemes.", "Let", "$$", "K_n \\to L_n \\to M_n \\to K_n[1]", "$$", "be an inverse system of distinguished triangles in", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$. If $(K_n)$ and $(M_n)$", "are pro-isomorphic to Deligne systems, then so is $(L_n)$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the systems $(K_n|_U)$ and $(M_n|_U)$ are essentially constant", "as they are pro-isomorphic to constant systems.", "Denote $K$ and $M$ their values. By Derived Categories, Lemma", "\\ref{derived-lemma-essentially-constant-2-out-of-3}", "we see that the inverse system $L_n|_U$ is essentially constant as well.", "Denote $L$ its value.", "Let $N \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$. Consider the commutative", "diagram", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "\\colim \\Hom_X(M_n, N) \\ar[r] \\ar[d] &", "\\colim \\Hom_X(L_n, N) \\ar[r] \\ar[d] &", "\\colim \\Hom_X(K_n, N) \\ar[r] \\ar[d] &", "\\ldots \\\\", "\\ldots \\ar[r] &", "\\Hom_U(M, N|_U) \\ar[r] &", "\\Hom_U(L, N|_U) \\ar[r] &", "\\Hom_U(K, N|_U) \\ar[r] &", "\\ldots", "}", "$$", "By Lemma \\ref{lemma-lift-map} and the fact that isomorphic ind-systems", "have the same colimit, we see that the vertical arrows two to the right", "and two to the left of the middle one are isomorphisms. By the 5-lemma", "we conclude that the", "middle vertical arrow is an isomorphism. Now, if $(L'_n)$ is a Deligne system", "whose restriction to $U$ has constant value $L$ (which", "exists by Lemma \\ref{lemma-extension-by-zero}), then", "we have $\\colim \\Hom_X(L'_n, N) = \\Hom_U(L, N|_U)$ as well.", "Hence the pro-systems $(L_n)$ and $(L'_n)$ are", "pro-isomorphic by", "Categories, Remark \\ref{categories-remark-pro-category-copresheaves}." ], "refs": [ "derived-lemma-essentially-constant-2-out-of-3", "duality-lemma-lift-map", "duality-lemma-extension-by-zero", "categories-remark-pro-category-copresheaves" ], "ref_ids": [ 1954, 13621, 13623, 12421 ] } ], "ref_ids": [] }, { "id": 13626, "type": "theorem", "label": "duality-lemma-consequence-Artin-Rees-bis", "categories": [ "duality" ], "title": "duality-lemma-consequence-Artin-Rees-bis", "contents": [ "Let $X$ be a Noetherian scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$", "be a quasi-coherent sheaf of ideals. Let $\\mathcal{F}^\\bullet$ be a", "complex of coherent $\\mathcal{O}_X$-modules. Let $p \\in \\mathbf{Z}$.", "Set $\\mathcal{H} = H^p(\\mathcal{F}^\\bullet)$ and", "$\\mathcal{H}_n = H^p(\\mathcal{I}^n\\mathcal{F}^\\bullet)$.", "Then there are canonical $\\mathcal{O}_X$-module maps", "$$", "\\ldots \\to \\mathcal{H}_3 \\to \\mathcal{H}_2 \\to \\mathcal{H}_1 \\to \\mathcal{H}", "$$", "There exists a $c > 0$ such that for $n \\geq c$ the image of", "$\\mathcal{H}_n \\to \\mathcal{H}$ is contained in", "$\\mathcal{I}^{n - c}\\mathcal{H}$ and there is a canonical", "$\\mathcal{O}_X$-module map", "$\\mathcal{I}^n\\mathcal{H} \\to \\mathcal{H}_{n - c}$ such that the compositions", "$$", "\\mathcal{I}^n \\mathcal{H} \\to \\mathcal{H}_{n - c} \\to", "\\mathcal{I}^{n - 2c}\\mathcal{H}", "\\quad\\text{and}\\quad", "\\mathcal{H}_n \\to \\mathcal{I}^{n - c}\\mathcal{H} \\to \\mathcal{H}_{n - 2c}", "$$", "are the canonical ones. In particular, the inverse systems", "$(\\mathcal{H}_n)$ and $(\\mathcal{I}^n\\mathcal{H})$", "are isomorphic as pro-objects of $\\textit{Mod}(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "If $X$ is affine, translated into algebra this is More on Algebra, Lemma", "\\ref{more-algebra-lemma-consequence-Artin-Rees-bis}.", "In the general case, argue exactly as in the proof of that lemma", "replacing the reference to Artin-Rees in algebra with a reference to", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-Artin-Rees}.", "Details omitted." ], "refs": [ "more-algebra-lemma-consequence-Artin-Rees-bis", "coherent-lemma-Artin-Rees" ], "ref_ids": [ 10427, 3321 ] } ], "ref_ids": [] }, { "id": 13627, "type": "theorem", "label": "duality-lemma-characterize-extension-by-zero-algebra", "categories": [ "duality" ], "title": "duality-lemma-characterize-extension-by-zero-algebra", "contents": [ "Let $j : U \\to X$ be an open immersion of Noetherian schemes.", "Let $a \\leq b$ be integers. Let $(K_n)$ be an inverse system of", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$", "such that $H^i(K_n) = 0$ for $i \\not \\in [a, b]$.", "The following are equivalent", "\\begin{enumerate}", "\\item $(K_n)$ is pro-isomorphic to a Deligne system,", "\\item for every $p \\in \\mathbf{Z}$ there exists a coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ such that the pro-systems", "$(H^p(K_n))$ and $(\\mathcal{I}^n\\mathcal{F})$ are pro-isomorphic.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). To prove (2) holds we may assume $(K_n)$ is a Deligne system.", "By definition we may choose a bounded complex $\\mathcal{F}^\\bullet$", "of coherent $\\mathcal{O}_X$-modules and a quasi-coherent", "sheaf of ideals cutting out $X \\setminus U$ such that", "$K_n$ is represented by $\\mathcal{I}^n\\mathcal{F}^\\bullet$.", "Thus the result follows from Lemma \\ref{lemma-consequence-Artin-Rees-bis}.", "\\medskip\\noindent", "Assume (2). We will prove that $(K_n)$ is as in (1) by induction on", "$b - a$. If $a = b$ then (1) holds essentially by assumption.", "If $a < b$ then we consider the compatible system of", "distinguished triangles", "$$", "\\tau_{\\leq a}K_n \\to K_n \\to \\tau_{\\geq a + 1}K_n \\to (\\tau_{\\leq a}K_n)[1]", "$$", "See Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle}.", "By induction on $b - a$ we know that $\\tau_{\\leq a}K_n$ and", "$\\tau_{\\geq a + 1}K_n$ are pro-isomorphic to Deligne systems.", "We conclude by Lemma \\ref{lemma-deligne-system-2-out-of-3}." ], "refs": [ "duality-lemma-consequence-Artin-Rees-bis", "derived-remark-truncation-distinguished-triangle", "duality-lemma-deligne-system-2-out-of-3" ], "ref_ids": [ 13626, 2016, 13625 ] } ], "ref_ids": [] }, { "id": 13628, "type": "theorem", "label": "duality-lemma-characterize-extension-by-zero", "categories": [ "duality" ], "title": "duality-lemma-characterize-extension-by-zero", "contents": [ "Let $j : U \\to X$ be an open immersion of Noetherian schemes. Let", "$(K_n)$ be an inverse system in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.", "Let $X = W_1 \\cup \\ldots \\cup W_r$ be an open covering.", "The following are equivalent", "\\begin{enumerate}", "\\item $(K_n)$ is pro-isomorphic to a Deligne system,", "\\item for each $i$ the restriction $(K_n|_{W_i})$", "is pro-isomorphic to a Deligne system with respect to", "the open immersion $U \\cap W_i \\to W_i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By induction on $r$. If $r = 1$ then the result is clear.", "Assume $r > 1$. Set $V = W_1 \\cup \\ldots \\cup W_{r - 1}$.", "By induction we see that $(K_n|_V)$ is a Deligne system.", "This reduces us to the discussion in the next paragraph.", "\\medskip\\noindent", "Assume $X = V \\cup W$ is an open covering and", "$(K_n|_W)$ and $(K_n|_V)$ are pro-isomorphic to Deligne systems.", "We have to show that $(K_n)$ is pro-isomorphic to a Deligne system.", "Observe that $(K_n|_{V \\cap W})$ is pro-isomorphic to a Deligne system", "(it follows immediately from the construction of Deligne systems", "that restrictions to opens preserves them). In particular the pro-systems", "$(K_n|_{U \\cap V})$,", "$(K_n|_{U \\cap W})$, and", "$(K_n|_{U \\cap V \\cap W})$", "are essentially constant. It follows from the distinguished triangles", "in Cohomology, Lemma \\ref{cohomology-lemma-exact-sequence-j-star} and", "Derived Categories, Lemma \\ref{derived-lemma-essentially-constant-2-out-of-3}", "that $(K_n|_U)$ is essentially constant.", "Denote $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_U)$ the value of this system.", "Let $L$ be an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.", "Consider the diagram", "$$", "\\xymatrix{", "\\colim \\Ext^{-1}(K_n|_V, L|_V) \\oplus", "\\colim \\Ext^{-1}(K_n|_W, L|_W) \\ar[r] \\ar[d] &", "\\Ext^{-1}(K|_{U \\cap V}, L|_{U \\cap V}) \\oplus", "\\Ext^{-1}(K|_{U \\cap W}, L|_{U \\cap W}) \\ar[d] \\\\", "\\colim \\Ext^{-1}(K_n|_{V \\cap W}, L|_{V \\cap W}) \\ar[r] \\ar[d] &", "\\Ext^{-1}(K|_{U \\cap V \\cap W}, L|_{U \\cap V \\cap W}) \\ar[d] \\\\", "\\colim \\Hom(K_n, L) \\ar[d] \\ar[r] &", "\\Hom(K|_U, L|_U) \\ar[d] \\\\", "\\colim \\Hom(K_n|_V, L|_V) \\oplus \\colim \\Hom(K_n|_W, L|_W) \\ar[r] \\ar[d] &", "\\Hom(K|_{U \\cap V}, L|_{U \\cap V}) \\oplus", "\\Hom(K|_{U \\cap W}, L|_{U \\cap W}) \\ar[d] \\\\", "\\colim \\Hom(K_n|_{V \\cap W}, L|_{V \\cap W}) \\ar[r] &", "\\Hom(K|_{U \\cap V \\cap W}, L|_{U \\cap V \\cap W})", "}", "$$", "The vertical sequences are exact by", "Cohomology, Lemma \\ref{cohomology-lemma-mayer-vietoris-hom}", "and the fact that filtered colimits are exact.", "All horizontal arrows except for the middle one are isomorphisms", "by Lemma \\ref{lemma-lift-map} and the fact that pro-isomorphic systems", "have the same colimits. Hence the middle one is an isomorphism too by", "the 5-lemma. It follows that $(K_n)$ is pro-isomorphic to", "a Deligne system for $K$. Namey, if $(K'_n)$ is a Deligne system", "whose restriction to $U$ has constant value $K$ (which", "exists by Lemma \\ref{lemma-extension-by-zero}), then", "we have $\\colim \\Hom_X(K'_n, L) = \\Hom_U(K, L|_U)$ as well.", "Hence the pro-systems $(K_n)$ and $(K'_n)$ are", "pro-isomorphic by", "Categories, Remark \\ref{categories-remark-pro-category-copresheaves}." ], "refs": [ "cohomology-lemma-exact-sequence-j-star", "derived-lemma-essentially-constant-2-out-of-3", "cohomology-lemma-mayer-vietoris-hom", "duality-lemma-lift-map", "duality-lemma-extension-by-zero", "categories-remark-pro-category-copresheaves" ], "ref_ids": [ 2144, 1954, 2145, 13621, 13623, 12421 ] } ], "ref_ids": [] }, { "id": 13629, "type": "theorem", "label": "duality-lemma-tensoring-Deligne-system", "categories": [ "duality" ], "title": "duality-lemma-tensoring-Deligne-system", "contents": [ "Let $j : U \\to X$ be an open immersion of Noetherian schemes. Let", "$\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of", "ideals with $V(\\mathcal{I}) = X \\setminus U$.", "Let $K$ be in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.", "Then", "$$", "K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{I}^n", "$$", "is pro-isomorphic to a Deligne system with constant value $K|_U$ over $U$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-characterize-extension-by-zero} the question is", "local on $X$. Thus we may assume $X$ is the spectrum of a Noetherian", "ring. In this case the statement follows from the algebra version which is", "More on Algebra, Lemma \\ref{more-algebra-lemma-tensoring-Deligne-system}." ], "refs": [ "duality-lemma-characterize-extension-by-zero", "more-algebra-lemma-tensoring-Deligne-system" ], "ref_ids": [ 13628, 10432 ] } ], "ref_ids": [] }, { "id": 13630, "type": "theorem", "label": "duality-lemma-well-defined-pre", "categories": [ "duality" ], "title": "duality-lemma-well-defined-pre", "contents": [ "Let $f : X \\to Y$ be a proper morphism of Noetherian schemes.", "Let $V \\subset Y$ be an open subscheme and set $U = f^{-1}(V)$.", "Picture", "$$", "\\xymatrix{", "U \\ar[r]_j \\ar[d]_g & X \\ar[d]^f \\\\", "V \\ar[r]^{j'} & Y", "}", "$$", "Then we have a canonical isomorphism $Rj'_! \\circ Rg_* \\to Rf_* \\circ Rj_!$", "of functors $D^b_{\\textit{Coh}}(\\mathcal{O}_U) \\to", "\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ where", "$Rj_!$ and $Rj'_!$ are as in Remark \\ref{remark-extension-by-zero}." ], "refs": [ "duality-remark-extension-by-zero" ], "proofs": [ { "contents": [ "[First proof]", "Let $K$ be an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_U)$. Let $(K_n)$", "be a Deligne system for $U \\to X$ whose restriction to $U$ is constant", "with value $K$. Of course this means that $(K_n)$ represents $Rj_!K$", "in $\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_X)$. Observe that", "both $Rj'_!Rg_*K$ and $Rf_*Rj_!K$ restrict to the constant pro-object with", "value $Rg_*K$ on $V$. This is immediate for the first one and for the second", "one it follows from the fact that $(Rf_*K_n)|_V = Rg_*(K_n|_U) = Rg_*K$.", "By the uniqueness of Deligne systems in Lemma \\ref{lemma-extension-by-zero}", "it suffices to show that $(Rf_*K_n)$ is pro-isomorphic to a", "Deligne system. The lemma referenced will also show that the", "isomorphism we obtain is functorial.", "\\medskip\\noindent", "Proof that $(Rf_*K_n)$ is pro-isomorphic to a Deligne system.", "First, we observe that the question is independent of the choice", "of the Deligne system $(K_n)$ corresponding to $K$ (by the aforementioned", "uniqueness). By Lemmas \\ref{lemma-extension-by-zero-triangle} and", "\\ref{lemma-deligne-system-2-out-of-3}", "if we have a distinguished triangle", "$$", "K \\to L \\to M \\to K[1]", "$$", "in $D^b_{\\textit{Coh}}(\\mathcal{O}_U)$ and the result holds for", "$K$ and $M$, then the result holds for $L$. Using the distinguished", "triangles of canonical truncations (Derived Categories, Remark", "\\ref{derived-remark-truncation-distinguished-triangle})", "we reduce to the problem studied in the next paragraph.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Let $\\mathcal{J} \\subset \\mathcal{O}_Y$ be a quasi-coherent", "sheaf of ideals cutting out $Y \\setminus V$. Denote", "$\\mathcal{J}^n\\mathcal{F}$ the image of", "$f^*\\mathcal{J}^n \\otimes \\mathcal{F} \\to \\mathcal{F}$.", "We have to show that $(Rf_*(\\mathcal{J}^n\\mathcal{F}))$", "is a Deligne system. By Lemma \\ref{lemma-characterize-extension-by-zero}", "the question is local on $Y$. Thus we may assume $Y = \\Spec(A)$ is affine", "and $\\mathcal{J}$ corresponds to an ideal $I \\subset A$. By", "Lemma \\ref{lemma-characterize-extension-by-zero-algebra}", "it suffices to show that the inverse system of cohomology modules", "$(H^p(X, I^n\\mathcal{F}))$ is pro-isomorphic to the inverse system", "$(I^n M)$ for some finite $A$-module $M$.", "This is shown in Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cohomology-powers-ideal-application}." ], "refs": [ "duality-lemma-extension-by-zero", "duality-lemma-extension-by-zero-triangle", "duality-lemma-deligne-system-2-out-of-3", "derived-remark-truncation-distinguished-triangle", "duality-lemma-characterize-extension-by-zero", "duality-lemma-characterize-extension-by-zero-algebra", "coherent-lemma-cohomology-powers-ideal-application" ], "ref_ids": [ 13623, 13624, 13625, 2016, 13628, 13627, 3359 ] } ], "ref_ids": [ 13662 ] }, { "id": 13631, "type": "theorem", "label": "duality-lemma-well-defined", "categories": [ "duality" ], "title": "duality-lemma-well-defined", "contents": [ "Let $j : U \\to X$ be an open immersion of Noetherian schemes.", "Let $j' : U \\to X'$ be a compactification of $U$ over $X$ (see proof)", "and denote $f : X' \\to X$ the structure morphism.", "Then we have a canonical isomorphism $Rj_! \\to Rf_* \\circ R(j')_!$", "of functors $D^b_{\\textit{Coh}}(\\mathcal{O}_U) \\to", "\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ where", "$Rj_!$ and $Rj'_!$ are as in Remark \\ref{remark-extension-by-zero}." ], "refs": [ "duality-remark-extension-by-zero" ], "proofs": [ { "contents": [ "The fact that $X'$ is a compactification of $U$ over $X$ means", "precisely that $f : X' \\to X$ is proper, that $j'$ is an open immersion,", "and $j = f \\circ j'$. See", "More on Flatness, Section \\ref{flat-section-compactify}.", "If $j'(U) = f^{-1}(j(U))$, then the lemma follows immediately from", "Lemma \\ref{lemma-well-defined-pre}.", "If $j'(U) \\not = f^{-1}(j(U))$, then denote $X'' \\subset X'$ the", "scheme theoretic closure of $j' : U \\to X'$ and denote", "$j'' : U \\to X''$ the corresponding open immersion.", "Picture", "$$", "\\xymatrix{", "& & X'' \\ar[d]^{f'} \\\\", "& & X' \\ar[d]^f \\\\", "U \\ar[rr]^j \\ar[rru]^{j'} \\ar[rruu]^{j''} & & X", "}", "$$", "By", "More on Flatness, Lemma \\ref{flat-lemma-compactifications-cofiltered} part (c)", "and the discussion above we have isomorphisms", "$Rf'_* \\circ Rj''_! = Rj'_!$ and $R(f \\circ f')_* \\circ Rj''_! = Rj_!$.", "Since $R(f \\circ f')_* = Rf_* \\circ Rf'_*$ we conclude." ], "refs": [ "duality-lemma-well-defined-pre", "flat-lemma-compactifications-cofiltered" ], "ref_ids": [ 13630, 6128 ] } ], "ref_ids": [ 13662 ] }, { "id": 13632, "type": "theorem", "label": "duality-lemma-lower-shriek-well-defined", "categories": [ "duality" ], "title": "duality-lemma-lower-shriek-well-defined", "contents": [ "The functor $Rf_!$ is, up to isomorphism, independent", "of the choice of the compactification." ], "refs": [], "proofs": [ { "contents": [ "Consider the category of compactifications of $X$ over $Y$, which is", "cofiltered according to More on Flatness, Theorem \\ref{flat-theorem-nagata} and", "Lemmas \\ref{flat-lemma-compactifications-cofiltered} and", "\\ref{flat-lemma-compactifyable}.", "To every choice of a compactification", "$$", "j : X \\to \\overline{X},\\quad \\overline{f} : \\overline{X} \\to Y", "$$", "the construction above associates the functor $R\\overline{f}_* \\circ Rj_!$.", "Suppose given a morphism $g : \\overline{X}_1 \\to \\overline{X}_2$", "between compactifications $j_i : X \\to \\overline{X}_i$ over $Y$.", "Then we get an isomorphism", "$$", "R\\overline{f}_{2, *} \\circ Rj_{2, !} =", "R\\overline{f}_{2, *} \\circ Rg_* \\circ j_{1, !} =", "R\\overline{f}_{1, *} \\circ Rj_{1, !}", "$$", "using Lemma \\ref{lemma-well-defined} in the first equality. In this way", "we see our functor is independent of the choice of compactification", "up to isomorphism." ], "refs": [ "flat-theorem-nagata", "flat-lemma-compactifications-cofiltered", "flat-lemma-compactifyable", "duality-lemma-well-defined" ], "ref_ids": [ 5976, 6128, 6129, 13631 ] } ], "ref_ids": [] }, { "id": 13633, "type": "theorem", "label": "duality-lemma-compactly-supported-triangle", "categories": [ "duality" ], "title": "duality-lemma-compactly-supported-triangle", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. Let", "$$", "K \\to L \\to M \\to K[1]", "$$", "be a distinguished triangle of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.", "Then there exists an inverse system of distinguished triangles", "$$", "K_n \\to L_n \\to M_n \\to K_n[1]", "$$", "in $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ such that the pro-systems", "$(K_n)$, $(L_n)$, and $(M_n)$ give $Rf_!K$, $Rf_!L$, and $Rf_!M$." ], "refs": [], "proofs": [ { "contents": [ "Choose a compactification $j : X \\to \\overline{X}$ over $Y$ and", "denote $\\overline{f} : \\overline{X} \\to Y$ the structure morphism.", "Choose an inverse system of distinguished triangles", "$$", "\\overline{K}_n \\to \\overline{L}_n \\to \\overline{M}_n \\to \\overline{K}_n[1]", "$$", "in $D^b_{\\textit{Coh}}(\\mathcal{O}_{\\overline{X}})$ as in", "Lemma \\ref{lemma-extension-by-zero-triangle} corresponding to the", "open immersion $j$ and the given distinguished triangle. Take", "$K_n = R\\overline{f}_*\\overline{K}_n$", "and similarly for $L_n$ and $M_n$. This works by the very definition", "of $Rf_!$." ], "refs": [ "duality-lemma-extension-by-zero-triangle" ], "ref_ids": [ 13624 ] } ], "ref_ids": [] }, { "id": 13634, "type": "theorem", "label": "duality-lemma-composition-lower-shriek", "categories": [ "duality" ], "title": "duality-lemma-composition-lower-shriek", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ and $g : Y \\to Z$", "be composable morphisms of $\\textit{FTS}_S$. With notation as in", "Remark \\ref{remark-composition-lower-shriek} we have", "$Rg_! \\circ Rf_! = R(g \\circ f)_!$." ], "refs": [ "duality-remark-composition-lower-shriek" ], "proofs": [ { "contents": [ "By the discussion in", "Categories, Remark \\ref{categories-remark-pro-category-copresheaves}", "it suffices to show that we obtain the same answer if we compute", "$\\Hom$ into $L$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_Z)$. To do this", "we compute, using the notation in", "Remark \\ref{remark-composition-lower-shriek}, as follows", "\\begin{align*}", "\\Hom_Z(Rg_!Rf_!K, L)", "& =", "\\colim_n \\Hom_Z(Rg_!M_n, L) \\\\", "& =", "\\colim_n \\Hom_Y(M_n, g^!L) \\\\", "& =", "\\Hom_Y(Rf_!K, g^!L) \\\\", "& =", "\\Hom_X(K, f^!g^!L) \\\\", "& =", "\\Hom_X(K, (g \\circ f)^!L) \\\\", "& =", "\\Hom_Z(R(g \\circ f)_!K, L)", "\\end{align*}", "The first equality is the definition of $Rg_!Rf_!K$. The second", "equality is Proposition \\ref{proposition-duality-compactly-supported}", "for $g$.", "The third equality is the fact that $Rf_!K$ is given by $(M_n)$.", "The fourth equality is", "Proposition \\ref{proposition-duality-compactly-supported} for $f$.", "The fifth equality is Lemma \\ref{lemma-upper-shriek-composition}.", "The sixth is ", "Proposition \\ref{proposition-duality-compactly-supported} for $g \\circ f$." ], "refs": [ "categories-remark-pro-category-copresheaves", "duality-remark-composition-lower-shriek", "duality-proposition-duality-compactly-supported", "duality-proposition-duality-compactly-supported", "duality-lemma-upper-shriek-composition", "duality-proposition-duality-compactly-supported" ], "ref_ids": [ 12421, 13666, 13639, 13639, 13552, 13639 ] } ], "ref_ids": [ 13666 ] }, { "id": 13635, "type": "theorem", "label": "duality-lemma-duality-compact-support", "categories": [ "duality" ], "title": "duality-lemma-duality-compact-support", "contents": [ "Let $p : U \\to \\Spec(k)$ be separated of finite type where $k$ is a field.", "Let $\\omega_{U/k}^\\bullet = p^!\\mathcal{O}_{\\Spec(k)}$.", "There are canonical isomorphisms", "$$", "\\Hom_k(H^i(U, K), k) =", "H^{-i}_c(U, R\\SheafHom_{\\mathcal{O}_U}(K, \\omega_{U/k}^\\bullet))", "$$", "of topological $k$-vector spaces", "functorial for $K$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_U)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a compactification $j : U \\to X$ over $k$. Let", "$\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent ideal", "sheaf with $V(\\mathcal{I}) = X \\setminus U$.", "By Derived Categories of Schemes, Proposition \\ref{perfect-proposition-DCoh}", "we may choose $M \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$", "with $K = M|_U$. We have", "$$", "H^i(U, K) =", "\\Ext^i_U(\\mathcal{O}_U, M|_U) =", "\\colim \\Ext^i_X(\\mathcal{I}^n, M) =", "\\colim H^i(X, R\\SheafHom_{\\mathcal{O}_X}(\\mathcal{I}^n, M))", "$$", "by Lemma \\ref{lemma-lift-map}.", "Since $\\mathcal{I}^n$ is a coherent $\\mathcal{O}_X$-module,", "we have $\\mathcal{I}^n$ in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$,", "hence $R\\SheafHom_{\\mathcal{O}_X}(\\mathcal{I}^n, M)$ is in", "$D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-coherent-internal-hom}.", "\\medskip\\noindent", "Let $\\omega_{X/k}^\\bullet = q^!\\mathcal{O}_{\\Spec(k)}$ where", "$q : X \\to \\Spec(k)$ is the structure morphism, see", "Section \\ref{section-duality-proper-over-field}. We find that", "\\begin{align*}", "\\Hom_k(", "&", "H^i(X, R\\SheafHom_{\\mathcal{O}_X}(\\mathcal{I}^n, M)), k) \\\\", "& =", "\\Ext^{-i}_X(R\\SheafHom_{\\mathcal{O}_X}(\\mathcal{I}^n, M),", "\\omega_{X/k}^\\bullet) \\\\", "& =", "H^{-i}(X, R\\SheafHom_{\\mathcal{O}_X}(R\\SheafHom_{\\mathcal{O}_X}(", "\\mathcal{I}^n, M), \\omega_{X/k}^\\bullet))", "\\end{align*}", "by Lemma \\ref{lemma-duality-proper-over-field}. By", "Lemma \\ref{lemma-internal-hom-evaluate-isom} part (1) the canonical map", "$$", "R\\SheafHom_{\\mathcal{O}_X}(M, \\omega_{X/k}^\\bullet)", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{I}^n", "\\longrightarrow", "R\\SheafHom_{\\mathcal{O}_X}(R\\SheafHom_{\\mathcal{O}_X}(", "\\mathcal{I}^n, M), \\omega_{X/k}^\\bullet)", "$$", "is an isomorphism. Observe that", "$\\omega^\\bullet_{U/k} = \\omega^\\bullet_{X/k}|_U$", "because $p^!$ is constructed as $q^!$ composed with restriction to $U$.", "Hence $R\\SheafHom_{\\mathcal{O}_X}(M, \\omega_{X/k}^\\bullet)$ is", "an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ which restricts", "to $R\\SheafHom_{\\mathcal{O}_U}(K, \\omega_{U/k}^\\bullet)$ on $U$.", "Hence by Lemma \\ref{lemma-tensoring-Deligne-system} we conclude that", "$$", "\\lim H^{-i}(X, R\\SheafHom_{\\mathcal{O}_X}(M, \\omega_{X/k}^\\bullet)", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{I}^n)", "$$", "is an avatar for the right hand side of the equality of the lemma.", "Combining all the isomorphisms obtained in this manner we get", "the isomorphism of the lemma." ], "refs": [ "perfect-proposition-DCoh", "duality-lemma-lift-map", "perfect-lemma-coherent-internal-hom", "duality-lemma-duality-proper-over-field", "duality-lemma-internal-hom-evaluate-isom", "duality-lemma-tensoring-Deligne-system" ], "ref_ids": [ 7110, 13621, 6986, 13606, 13498, 13629 ] } ], "ref_ids": [] }, { "id": 13636, "type": "theorem", "label": "duality-lemma-duality-compact-support-restrict-open", "categories": [ "duality" ], "title": "duality-lemma-duality-compact-support-restrict-open", "contents": [ "With notation as in Lemma \\ref{lemma-duality-compact-support}", "suppose $U' \\subset U$ is an open subscheme. Then the diagram", "$$", "\\xymatrix{", "\\Hom_k(H^i(U, K), k) \\ar[rr] & &", "H^{-i}_c(U, R\\SheafHom_{\\mathcal{O}_U}(K, \\omega_{U/k}^\\bullet)) \\\\", "\\Hom_k(H^i(U', K|_{U'}), k) \\ar[rr] \\ar[u] & &", "H^{-i}_c(U', R\\SheafHom_{\\mathcal{O}_{U'}}(K, \\omega_{U'/k}^\\bullet)) \\ar[u]", "}", "$$", "is commutative. Here the horizontal arrows are the isomorphisms of", "Lemma \\ref{lemma-duality-compact-support}, the vertical arrow on the", "left is the contragredient to the restriction map", "$H^i(U, K) \\to H^i(U', K|_{U'})$, and the right vertical arrow", "is Remark \\ref{remark-covariance-open-lower-shriek} (see discussion", "before the lemma)." ], "refs": [ "duality-lemma-duality-compact-support", "duality-lemma-duality-compact-support", "duality-remark-covariance-open-lower-shriek" ], "proofs": [ { "contents": [ "We strongly urge the reader to skip this proof. Choose $X$ and $M$ as", "in the proof of Lemma \\ref{lemma-duality-compact-support}. We are going to drop", "the subscript $\\mathcal{O}_X$ from $R\\SheafHom$ and $\\otimes^\\mathbf{L}$.", "We write", "$$", "H^i(U, K) = \\colim H^i(X, R\\SheafHom(\\mathcal{I}^n, M))", "$$", "and", "$$", "H^i(U', K|_{U'}) = \\colim H^i(X, R\\SheafHom((\\mathcal{I}')^n, M))", "$$", "as in the proof of Lemma \\ref{lemma-duality-compact-support} where we choose", "$\\mathcal{I}' \\subset \\mathcal{I}$ as in the discussion in", "Remark \\ref{remark-covariance-open-j-lower-shriek} so that the map", "$H^i(U, K) \\to H^i(U', K|_{U'})$ is induced by the maps", "$(\\mathcal{I}')^n \\to \\mathcal{I}^n$.", "We similarly write", "$$", "H^i_c(U, R\\SheafHom(K, \\omega_{U/k}^\\bullet)) =", "\\lim H^i(X, R\\SheafHom(M, \\omega_{X/k}^\\bullet)", "\\otimes^\\mathbf{L} \\mathcal{I}^n)", "$$", "and", "$$", "H^i_c(U', R\\SheafHom(K|_{U'}, \\omega_{U'/k}^\\bullet)) =", "\\lim H^i(X, R\\SheafHom(M, \\omega_{X/k}^\\bullet)", "\\otimes^\\mathbf{L} (\\mathcal{I}')^n)", "$$", "so that the arrow $H^i_c(U', R\\SheafHom(K|_{U'}, \\omega_{U'/k}^\\bullet))", "\\to H^i_c(U, R\\SheafHom(K, \\omega_{U/k}^\\bullet))$ is similarly", "deduced from the maps $(\\mathcal{I}')^n \\to \\mathcal{I}^n$.", "The diagrams", "$$", "\\xymatrix{", "R\\SheafHom(M, \\omega_{X/k}^\\bullet)", "\\otimes^\\mathbf{L} \\mathcal{I}^n", "\\ar[rr] & &", "R\\SheafHom(R\\SheafHom(\\mathcal{I}^n, M), \\omega_{X/k}^\\bullet) \\\\", "R\\SheafHom(M, \\omega_{X/k}^\\bullet) \\otimes^\\mathbf{L} (\\mathcal{I}')^n", "\\ar[rr] \\ar[u] & &", "R\\SheafHom(R\\SheafHom((\\mathcal{I}')^n, M), \\omega_{X/k}^\\bullet) \\ar[u]", "}", "$$", "commute because the construction of the horizontal arrows in", "Cohomology, Lemma \\ref{cohomology-lemma-internal-hom-evaluate}", "is functorial in all three entries. Hence we finally come down", "to the assertion that the diagrams", "$$", "\\xymatrix{", "\\Hom_k(H^i(X, R\\SheafHom(\\mathcal{I}^n, M)), k) \\ar[r] &", "H^{-i}(X, R\\SheafHom(R\\SheafHom(", "\\mathcal{I}^n, M), \\omega_{X/k}^\\bullet)) \\\\", "\\Hom_k(H^i(X, R\\SheafHom((\\mathcal{I}')^n, M)), k) \\ar[r] \\ar[u] &", "H^{-i}(X, R\\SheafHom(R\\SheafHom(", "(\\mathcal{I}')^n, M), \\omega_{X/k}^\\bullet)) \\ar[u]", "}", "$$", "commute. This is true because the duality isomorphism", "$$", "\\Hom_k(H^i(X, L), k) = \\Ext^{-i}_X(L, \\omega_{X/k}^\\bullet) =", "H^{-i}(X, R\\SheafHom(L, \\omega_{X/k}^\\bullet))", "$$", "is functorial for $L$ in $D_\\QCoh(\\mathcal{O}_X)$." ], "refs": [ "duality-lemma-duality-compact-support", "duality-lemma-duality-compact-support", "duality-remark-covariance-open-j-lower-shriek", "cohomology-lemma-internal-hom-evaluate" ], "ref_ids": [ 13635, 13635, 13664, 2190 ] } ], "ref_ids": [ 13635, 13635, 13667 ] }, { "id": 13637, "type": "theorem", "label": "duality-lemma-h0-compactly-supported", "categories": [ "duality" ], "title": "duality-lemma-h0-compactly-supported", "contents": [ "Let $X$ be a proper scheme over a field $k$. Let", "$K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ with $H^i(K) = 0$", "for $i < 0$. Set $\\mathcal{F} = H^0(K)$.", "Let $Z \\subset X$ be closed with complement $U = X \\setminus U$.", "Then", "$$", "H^0_c(U, K|_U) \\subset H^0(X, \\mathcal{F})", "$$", "is given by those global sections of $\\mathcal{F}$ which", "vanish in an open neighbourhood of $Z$." ], "refs": [], "proofs": [ { "contents": [ "Consider the map", "$H^0_c(U, K|_U) \\to H^0_x(X, K) = H^0(X, K) = H^0(X, \\mathcal{F})$ of", "Remark \\ref{remark-covariance-open-lower-shriek}. To study this", "we represent $K$ by a bounded complex $\\mathcal{F}^\\bullet$ with", "$\\mathcal{F}^i = 0$ for $i < 0$. Then we have by definition", "$$", "H^0_c(U, K|_U) = \\lim H^0(X, \\mathcal{I}^n\\mathcal{F}^\\bullet)", "= \\lim \\Ker(", "H^0(X, \\mathcal{I}^n\\mathcal{F}^0) \\to", "H^0(X, \\mathcal{I}^n\\mathcal{F}^1))", "$$", "By Artin-Rees (Cohomology of Schemes, Lemma \\ref{coherent-lemma-Artin-Rees})", "this is the same as $\\lim H^0(X, \\mathcal{I}^n\\mathcal{F})$.", "Thus the arrow $H^0_c(U, K|_U) \\to H^0(X, \\mathcal{F})$", "is injective and the image consists of those global sections of $\\mathcal{F}$", "which are contained in the subsheaf $\\mathcal{I}^n\\mathcal{F}$", "for any $n$. The characterization of these as the sections which", "vanish in a neighbourhood of $Z$ comes from Krull's intersection", "theorem (Algebra, Lemma \\ref{algebra-lemma-intersect-powers-ideal-module-zero})", "by looking at stalks of $\\mathcal{F}$. See discussion in", "Algebra, Remark \\ref{algebra-remark-intersection-powers-ideal}", "for the case of functions." ], "refs": [ "duality-remark-covariance-open-lower-shriek", "coherent-lemma-Artin-Rees", "algebra-lemma-intersect-powers-ideal-module-zero", "algebra-remark-intersection-powers-ideal" ], "ref_ids": [ 13667, 3321, 627, 1560 ] } ], "ref_ids": [] }, { "id": 13638, "type": "theorem", "label": "duality-lemma-lichtenbaum", "categories": [ "duality" ], "title": "duality-lemma-lichtenbaum", "contents": [ "Let $U$ be a variety. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module.", "If $H^d(U, \\mathcal{F})$ is nonzero, then $\\dim(U) \\geq d$ and if", "equality holds, then $U$ is proper." ], "refs": [], "proofs": [ { "contents": [ "By the Grothendieck's vanishing result in", "Cohomology, Proposition \\ref{cohomology-proposition-vanishing-Noetherian}", "we conclude that $\\dim(U) \\geq d$.", "Assume $\\dim(U) = d$. Choose a compactification $U \\to X$", "such that $U$ is dense in $X$. (This is possible by", "More on Flatness, Theorem \\ref{flat-theorem-nagata} and", "Lemma \\ref{flat-lemma-compactifyable}.) After replacing $X$ by its", "reduction we find that $X$ is a proper variety of dimension $d$", "and we see that $U$ is proper if and only if $U = X$.", "Set $Z = X \\setminus U$. We will show that $H^d(U, \\mathcal{F})$", "is zero if $Z$ is nonempty.", "\\medskip\\noindent", "Choose a coherent $\\mathcal{O}_X$-module", "$\\mathcal{G}$ whose restriction to $U$ is $\\mathcal{F}$, see", "Properties, Lemma \\ref{properties-lemma-lift-finite-presentation}.", "Let $\\omega_X^\\bullet$ denote the dualizing complex of $X$ as in", "Section \\ref{section-duality-proper-over-field}.", "Set $\\omega_U^\\bullet = \\omega_X^\\bullet|_U$.", "Then $H^d(U, \\mathcal{F})$ is dual to", "$$", "H^{-d}_c(U, R\\SheafHom_{\\mathcal{O}_U}(\\mathcal{F}, \\omega_U^\\bullet))", "$$", "by Lemma \\ref{lemma-duality-compact-support}. By", "Lemma \\ref{lemma-duality-proper-over-field} we see that", "the cohomology sheaves of $\\omega_X^\\bullet$ vanish in degrees $< -d$", "and $H^{-d}(\\omega_X^\\bullet) = \\omega_X$ is a coherent", "$\\mathcal{O}_X$-module which is $(S_2)$ and whose support is $X$.", "In particular, $\\omega_X$ is torsion free, see", "Divisors, Lemma \\ref{divisors-lemma-torsion-free-finite-noetherian-domain}.", "Thus we see that the cohomology sheaf", "$$", "H^{-d}(R\\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\omega_X^\\bullet)) =", "\\SheafHom(\\mathcal{G}, \\omega_X)", "$$", "is torsion free, see", "Divisors, Lemma \\ref{divisors-lemma-hom-into-torsion-free}.", "Consequently this sheaf has no nonzero sections vanishing", "on any nonempty open of $X$ (those would be torsion sections).", "Thus it follows from Lemma \\ref{lemma-h0-compactly-supported} that", "$H^{-d}_c(U, R\\SheafHom_{\\mathcal{O}_U}(\\mathcal{F}, \\omega_U^\\bullet))$", "is zero, and hence $H^d(U, \\mathcal{F})$ is zero as desired." ], "refs": [ "cohomology-proposition-vanishing-Noetherian", "flat-theorem-nagata", "flat-lemma-compactifyable", "properties-lemma-lift-finite-presentation", "duality-lemma-duality-compact-support", "duality-lemma-duality-proper-over-field", "divisors-lemma-torsion-free-finite-noetherian-domain", "divisors-lemma-hom-into-torsion-free", "duality-lemma-h0-compactly-supported" ], "ref_ids": [ 2246, 5976, 6129, 3022, 13635, 13606, 7911, 7913, 13637 ] } ], "ref_ids": [] }, { "id": 13639, "type": "theorem", "label": "duality-proposition-duality-compactly-supported", "categories": [ "duality" ], "title": "duality-proposition-duality-compactly-supported", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. Then the functors $Rf_!$ and $f^!$ are adjoint in", "the following sense: for all $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$", "and $L \\in D^+_{\\textit{Coh}}(\\mathcal{O}_Y)$ we have", "$$", "\\Hom_X(K, f^!L) =", "\\Hom_{\\text{Pro-}D^+_{\\textit{Coh}}(\\mathcal{O}_Y)}(Rf_!K, L)", "$$", "bifunctorially in $K$ and $L$." ], "refs": [], "proofs": [ { "contents": [ "Choose a compactification $j : X \\to \\overline{X}$ over $Y$ and", "denote $\\overline{f} : \\overline{X} \\to Y$ the structure morphism.", "Then we have", "\\begin{align*}", "\\Hom_X(K, f^!L)", "& =", "\\Hom_X(K, j^*\\overline{f}{}^!L) \\\\", "& =", "\\Hom_{\\text{Pro-}D^+_{\\textit{Coh}}(\\mathcal{O}_{\\overline{X}})}", "(Rj_!K, \\overline{f}{}^!L) \\\\", "& =", "\\Hom_{\\text{Pro-}D^+_{\\textit{Coh}}(\\mathcal{O}_Y)}(Rf_*Rj_!K, L) \\\\", "& =", "\\Hom_{\\text{Pro-}D^+_{\\textit{Coh}}(\\mathcal{O}_Y)}(Rf_!K, L)", "\\end{align*}", "The first equality follows immediately from the construction of $f^!$ in", "Section \\ref{section-upper-shriek}.", "By Lemma \\ref{lemma-shriek-coherent} we have $\\overline{f}{}^!L$", "in $D^+_{\\textit{Coh}}(\\mathcal{O}_{\\overline{X}})$ hence the second", "equality follows from Lemma \\ref{lemma-lift-map-plus}.", "Since $\\overline{f}$ is proper the functor $\\overline{f}{}^!$", "is the right adjoint of pushforward by construction. This is", "why we have the third equality.", "The fourth equality holds because $Rf_! = Rf_* Rj_!$." ], "refs": [ "duality-lemma-shriek-coherent", "duality-lemma-lift-map-plus" ], "ref_ids": [ 13559, 13622 ] } ], "ref_ids": [] }, { "id": 13670, "type": "theorem", "label": "more-morphisms-theorem-openness-flatness", "categories": [ "more-morphisms" ], "title": "more-morphisms-theorem-openness-flatness", "contents": [ "\\begin{reference}", "\\cite[IV Theorem 11.3.1]{EGA}", "\\end{reference}", "Let $S$ be a scheme.", "Let $f : X \\to S$ be a morphism which is locally of finite presentation.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module which is", "locally of finite presentation. Then", "$$", "U = \\{x \\in X \\mid \\mathcal{F}\\text{ is flat over }S\\text{ at }x\\}", "$$", "is open in $X$." ], "refs": [], "proofs": [ { "contents": [ "We may test for openness locally on $X$ hence we may assume", "that $f$ is a morphism of affine schemes. In this case the", "theorem is exactly", "Algebra, Theorem \\ref{algebra-theorem-openness-flatness}." ], "refs": [ "algebra-theorem-openness-flatness" ], "ref_ids": [ 326 ] } ], "ref_ids": [] }, { "id": 13671, "type": "theorem", "label": "more-morphisms-theorem-criterion-flatness-fibre-Noetherian", "categories": [ "more-morphisms" ], "title": "more-morphisms-theorem-criterion-flatness-fibre-Noetherian", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of schemes over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $x \\in X$. Set $y = f(x)$ and $s \\in S$ the image of $x$ in $S$.", "Assume $S$, $X$, $Y$ locally Noetherian,", "$\\mathcal{F}$ coherent, and $\\mathcal{F}_x \\not = 0$.", "Then the following are equivalent:", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is flat over $S$ at $x$, and", "$\\mathcal{F}_s$ is flat over $Y_s$ at $x$, and", "\\item $Y$ is flat over $S$ at $y$ and $\\mathcal{F}$ is", "flat over $Y$ at $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the ring maps", "$$", "\\mathcal{O}_{S, s} \\longrightarrow", "\\mathcal{O}_{Y, y} \\longrightarrow \\mathcal{O}_{X, x}", "$$", "and the module $\\mathcal{F}_x$. The stalk of $\\mathcal{F}_s$ at $x$", "is the module $\\mathcal{F}_x/\\mathfrak m_s \\mathcal{F}_x$ and", "the local ring of $Y_s$ at $y$ is", "$\\mathcal{O}_{Y, y}/\\mathfrak m_s \\mathcal{O}_{Y, y}$.", "Thus the implication (1) $\\Rightarrow$ (2) is", "Algebra, Lemma \\ref{algebra-lemma-criterion-flatness-fibre-Noetherian}.", "If (2) holds, then the first ring map is faithfully flat", "and $\\mathcal{F}_x$ is flat over $\\mathcal{O}_{Y, y}$ so by", "Algebra, Lemma \\ref{algebra-lemma-composition-flat}", "we see that $\\mathcal{F}_x$ is flat over $\\mathcal{O}_{S, s}$.", "Moreover, $\\mathcal{F}_x/\\mathfrak m_s \\mathcal{F}_x$ is the", "base change of the flat module $\\mathcal{F}_x$ by", "$\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{Y, y}/\\mathfrak m_s \\mathcal{O}_{Y, y}$,", "hence flat by", "Algebra, Lemma \\ref{algebra-lemma-flat-base-change}." ], "refs": [ "algebra-lemma-criterion-flatness-fibre-Noetherian", "algebra-lemma-composition-flat", "algebra-lemma-flat-base-change" ], "ref_ids": [ 897, 524, 527 ] } ], "ref_ids": [] }, { "id": 13672, "type": "theorem", "label": "more-morphisms-theorem-criterion-flatness-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-theorem-criterion-flatness-fibre", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of schemes over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume", "\\begin{enumerate}", "\\item $X$ is locally of finite presentation over $S$,", "\\item $\\mathcal{F}$ an $\\mathcal{O}_X$-module of finite presentation, and", "\\item $Y$ is locally of finite type over $S$.", "\\end{enumerate}", "Let $x \\in X$. Set $y = f(x)$ and let $s \\in S$ be the image of $x$ in $S$.", "If $\\mathcal{F}_x \\not = 0$, then the following are equivalent:", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is flat over $S$ at $x$, and", "$\\mathcal{F}_s$ is flat over $Y_s$ at $x$, and", "\\item $Y$ is flat over $S$ at $y$ and $\\mathcal{F}$ is", "flat over $Y$ at $x$.", "\\end{enumerate}", "Moreover, the set of points $x$ where (1) and (2) hold is open in", "$\\text{Supp}(\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "Consider the ring maps", "$$", "\\mathcal{O}_{S, s} \\longrightarrow", "\\mathcal{O}_{Y, y} \\longrightarrow \\mathcal{O}_{X, x}", "$$", "and the module $\\mathcal{F}_x$. The stalk of $\\mathcal{F}_s$ at $x$", "is the module $\\mathcal{F}_x/\\mathfrak m_s \\mathcal{F}_x$ and", "the local ring of $Y_s$ at $y$ is", "$\\mathcal{O}_{Y, y}/\\mathfrak m_s \\mathcal{O}_{Y, y}$.", "Thus the implication (1) $\\Rightarrow$ (2) is", "Algebra, Lemma \\ref{algebra-lemma-criterion-flatness-fibre-fp-over-ft}.", "If (2) holds, then the first ring map is faithfully flat", "and $\\mathcal{F}_x$ is flat over $\\mathcal{O}_{Y, y}$ so by", "Algebra, Lemma \\ref{algebra-lemma-composition-flat}", "we see that $\\mathcal{F}_x$ is flat over $\\mathcal{O}_{S, s}$.", "Moreover, $\\mathcal{F}_x/\\mathfrak m_s \\mathcal{F}_x$ is the", "base change of the flat module $\\mathcal{F}_x$ by", "$\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{Y, y}/\\mathfrak m_s \\mathcal{O}_{Y, y}$,", "hence flat by", "Algebra, Lemma \\ref{algebra-lemma-flat-base-change}.", "\\medskip\\noindent", "By", "Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-permanence}", "the morphism $f$ is locally of finite presentation.", "Consider the set", "\\begin{equation}", "\\label{equation-open}", "U = \\{x \\in X \\mid \\mathcal{F} \\text{ flat at }x", "\\text{ over both }Y\\text{ and }S\\}.", "\\end{equation}", "This set is open in $X$ by", "Theorem \\ref{theorem-openness-flatness}.", "Note that if $x \\in U$, then $\\mathcal{F}_s$ is flat at", "$x$ over $Y_s$ as a base change of a flat module under the", "morphism $Y_s \\to Y$, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-module-flat}.", "Hence at every point of $U \\cap \\text{Supp}(\\mathcal{F})$", "condition (1) is satisfied. On the other hand, it is", "clear that if $x \\in \\text{Supp}(\\mathcal{F})$ satisfies", "(1) and (2), then $x \\in U$. Thus the open set we are", "looking for is $U \\cap \\text{Supp}(\\mathcal{F})$." ], "refs": [ "algebra-lemma-criterion-flatness-fibre-fp-over-ft", "algebra-lemma-composition-flat", "algebra-lemma-flat-base-change", "morphisms-lemma-finite-presentation-permanence", "more-morphisms-theorem-openness-flatness", "morphisms-lemma-base-change-module-flat" ], "ref_ids": [ 1115, 524, 527, 5247, 13670, 5264 ] } ], "ref_ids": [] }, { "id": 13673, "type": "theorem", "label": "more-morphisms-theorem-of-the-cube", "categories": [ "more-morphisms" ], "title": "more-morphisms-theorem-of-the-cube", "contents": [ "Let $S$ be a scheme. Let $X$, $Y$, and $Z$ be schemes over $S$.", "Let $x : S \\to X$ and $y : S \\to Y$ be sections of the structure morphisms.", "Let $\\mathcal{L}$ be an invertible module on $X \\times_S Y \\times_S Z$. If", "\\begin{enumerate}", "\\item $X \\to S$ and $Y \\to S$ are flat, proper morphisms", "of finite presentation with geometrically integral fibres,", "\\item the pullbacks of $\\mathcal{L}$ by", "$x \\times \\text{id}_Y \\times \\text{id}_Z$ and", "$\\text{id}_X \\times y \\times \\text{id}_Z$", "are trivial over $Y \\times_S Z$ and $X \\times_S Z$,", "\\item there is a point $z \\in Z$ such that $\\mathcal{L}$", "restricted to $X \\times_S Y \\times_S z$ is trivial, and", "\\item $Z$ is connected,", "\\end{enumerate}", "then $\\mathcal{L}$ is trivial." ], "refs": [], "proofs": [ { "contents": [ "Observe that the morphism $X \\times_S Y \\to S$ is a flat, proper morphism", "of finite presentation whose geometrically integral fibres", "(see Varieties, Lemmas \\ref{varieties-lemma-geometrically-integral},", "\\ref{varieties-lemma-bijection-irreducible-components},", "and \\ref{varieties-lemma-geometrically-reduced-any-base-change} for the", "statement about the fibres). By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-proper-flat-geom-red-connected}", "we see that the pushforward of the structure sheaf by $X \\to S$, $Y \\to S$, or", "$X \\times_S Y \\to S$ is the structure sheaf of $S$ and the same remains true", "after any base change. Thus we may apply", "Lemma \\ref{lemma-diagonal-picard-flat-proper} to the morphism", "$$", "p : X \\times_S Y \\times_S Z \\longrightarrow Z", "$$", "and the invertible module $\\mathcal{L}$ to get a ``universal'' locally closed", "subscheme $Z' \\subset Z$ such that $\\mathcal{L}|_{X \\times_S Y \\times_S Z'}$", "is the pullback of an invertible module $\\mathcal{N}$ on $Z'$.", "The existence of $z$ shows that $Z'$ is nonempty. By", "Lemma \\ref{lemma-get-a-closed}", "we see that $Z' \\subset Z$ is a closed subscheme.", "Let $z' \\in Z'$ be a point.", "Observe that we may write $p$ as the product morphism", "$$", "(X \\times_S Z) \\times_Z (Y \\times_S Z) \\longrightarrow Z", "$$", "Hence we may apply Lemma \\ref{lemma-pic-of-product}", "to the morphism $p$, the point $z'$, and the sections", "$\\sigma : Z \\to X \\times_S Z$ and $\\tau : Z \\to Y \\times_S Z$", "given by $x$ and $y$. We conclude that $Z'$ is open.", "Hence $Z' = Z$ and $\\mathcal{L} = p^*\\mathcal{N}$", "for some invertible module $\\mathcal{N}$ on $Z$.", "Pulling back via", "$x \\times y \\times \\text{id}_Z : Z \\to X \\times_S Y \\times_S Z$", "we obtain on the one hand $\\mathcal{N}$ and on the other hand", "we obtain the trivial invertible module by assumption (2).", "Thus $\\mathcal{N} = \\mathcal{O}_Z$ and the proof is complete." ], "refs": [ "varieties-lemma-geometrically-integral", "varieties-lemma-bijection-irreducible-components", "varieties-lemma-geometrically-reduced-any-base-change", "perfect-lemma-proper-flat-geom-red-connected", "more-morphisms-lemma-diagonal-picard-flat-proper", "more-morphisms-lemma-get-a-closed", "more-morphisms-lemma-pic-of-product" ], "ref_ids": [ 10947, 10934, 10911, 7067, 13848, 13851, 13854 ] } ], "ref_ids": [] }, { "id": 13674, "type": "theorem", "label": "more-morphisms-theorem-stein-factorization-Noetherian", "categories": [ "more-morphisms" ], "title": "more-morphisms-theorem-stein-factorization-Noetherian", "contents": [ "Let $S$ be a locally Noetherian scheme.", "Let $f : X \\to S$ be a proper morphism.", "There exists a factorization", "$$", "\\xymatrix{", "X \\ar[rr]_{f'} \\ar[rd]_f & & S' \\ar[dl]^\\pi \\\\", "& S &", "}", "$$", "with the following properties:", "\\begin{enumerate}", "\\item the morphism $f'$ is proper with geometrically connected fibres,", "\\item the morphism $\\pi : S' \\to S$ is finite,", "\\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{S'}$,", "\\item we have $S' = \\underline{\\Spec}_S(f_*\\mathcal{O}_X)$, and", "\\item $S'$ is the normalization of $S$ in $X$, see", "Morphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-normalization-X-in-Y" ], "proofs": [ { "contents": [ "Let $f = \\pi \\circ f'$ be the factorization of", "Lemma \\ref{lemma-stein-universally-closed}. Note that besides the", "conclusions of Lemma \\ref{lemma-stein-universally-closed} we", "also have that $f'$ is separated", "(Schemes, Lemma \\ref{schemes-lemma-compose-after-separated})", "and finite type", "(Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type}).", "Hence $f'$ is proper. By", "Cohomology of Schemes, Proposition", "\\ref{coherent-proposition-proper-pushforward-coherent}", "we see that $f_*\\mathcal{O}_X$ is a coherent $\\mathcal{O}_S$-module.", "Hence we see that $\\pi$ is finite, i.e., (2) holds.", "\\medskip\\noindent", "This proves all but the most interesting assertion, namely that", "all the fibres of $f'$ are geometrically connected.", "It is clear from the discussion above that we may replace $S$ by $S'$,", "and we may therefore assume that $S$ is Noetherian, affine,", "$f : X \\to S$ is proper, and $f_*\\mathcal{O}_X = \\mathcal{O}_S$.", "Let $s \\in S$ be a point of $S$. We have to show that $X_s$ is", "geometrically connected. By Lemma", "\\ref{lemma-characterize-geometrically-connected-fibres}", "we see that it suffices to show $X_u$ is connected", "for every \\'etale neighbourhood $(U, u) \\to (S, s)$.", "We may assume $U$ is affine. Thus $U$ is Noetherian", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}),", "the base change $f_U : X_U \\to U$ is proper", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-proper}),", "and that also $(f_U)_*\\mathcal{O}_{X_U} = \\mathcal{O}_U$", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}).", "Hence after replacing", "$(f : X \\to S, s)$ by the base change $(f_U : X_U \\to U, u)$", "it suffices to prove that the fibre $X_s$ is connected when", "$f_*\\mathcal{O}_X = \\mathcal{O}_S$. We can deduce this", "from Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-proper-idempotent-on-fibre}", "(by looking at idempotents in the structure sheaf of $X_s$)", "but we will also give a direct argument below.", "\\medskip\\noindent", "Namely, we apply the theorem on formal functions,", "more precisely Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-formal-functions-stalk}.", "It tells us that", "$$", "\\mathcal{O}^\\wedge_{S, s} = (f_*\\mathcal{O}_X)_s^\\wedge =", "\\lim_n H^0(X_n, \\mathcal{O}_{X_n})", "$$", "where $X_n$ is the $n$th infinitesimal neighbourhood of $X_s$.", "Since the underlying topological space of $X_n$ is equal to that", "of $X_s$ we see that if $X_s = T_1 \\amalg T_2$ is a disjoint union", "of nonempty open and closed subschemes, then similarly", "$X_n = T_{1, n} \\amalg T_{2, n}$ for all $n$. And this in turn means", "$H^0(X_n, \\mathcal{O}_{X_n})$ contains a nontrivial idempotent $e_{1, n}$,", "namely the function which is identically $1$ on $T_{1, n}$ and", "identically $0$ on $T_{2, n}$. It is clear that $e_{1, n + 1}$", "restricts to $e_{1, n}$ on $X_n$. Hence $e_1 = \\lim e_{1, n}$", "is a nontrivial idempotent of the limit. This contradicts the fact", "that $\\mathcal{O}^\\wedge_{S, s}$ is a local ring. Thus the", "assumption was wrong, i.e., $X_s$ is connected, and we win." ], "refs": [ "more-morphisms-lemma-stein-universally-closed", "more-morphisms-lemma-stein-universally-closed", "schemes-lemma-compose-after-separated", "morphisms-lemma-permanence-finite-type", "coherent-proposition-proper-pushforward-coherent", "more-morphisms-lemma-characterize-geometrically-connected-fibres", "morphisms-lemma-finite-type-noetherian", "morphisms-lemma-base-change-proper", "coherent-lemma-flat-base-change-cohomology", "perfect-lemma-proper-idempotent-on-fibre", "coherent-lemma-formal-functions-stalk" ], "ref_ids": [ 13942, 13942, 7715, 5204, 3401, 13944, 5202, 5409, 3298, 7068, 3362 ] } ], "ref_ids": [ 5591 ] }, { "id": 13675, "type": "theorem", "label": "more-morphisms-theorem-stein-factorization-general", "categories": [ "more-morphisms" ], "title": "more-morphisms-theorem-stein-factorization-general", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to S$ be a proper morphism.", "There exists a factorization", "$$", "\\xymatrix{", "X \\ar[rr]_{f'} \\ar[rd]_f & & S' \\ar[dl]^\\pi \\\\", "& S &", "}", "$$", "with the following properties:", "\\begin{enumerate}", "\\item the morphism $f'$ is proper with geometrically connected fibres,", "\\item the morphism $\\pi : S' \\to S$ is integral,", "\\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{S'}$,", "\\item we have $S' = \\underline{\\Spec}_S(f_*\\mathcal{O}_X)$, and", "\\item $S'$ is the normalization of $S$ in $X$, see", "Morphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-normalization-X-in-Y" ], "proofs": [ { "contents": [ "We may apply Lemma \\ref{lemma-stein-universally-closed} to get the", "morphism $f' : X \\to S'$.", "Note that besides the", "conclusions of Lemma \\ref{lemma-stein-universally-closed} we", "also have that $f'$ is separated", "(Schemes, Lemma \\ref{schemes-lemma-compose-after-separated})", "and finite type", "(Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type}).", "Hence $f'$ is proper. At this point we have proved all of the", "statements except for the statement", "that $f'$ has geometrically connected fibres.", "\\medskip\\noindent", "We may assume that $S = \\Spec(R)$ is affine.", "Set $R' = \\Gamma(X, \\mathcal{O}_X)$. Then $S' = \\Spec(R')$.", "Thus we may replace $S$ by $S'$ and assume that", "$S = \\Spec(R)$ is affine $R = \\Gamma(X, \\mathcal{O}_X)$.", "Next, let $s \\in S$ be a point. Let $U \\to S$ be an \\'etale morphism", "of affine schemes and let $u \\in U$ be a point mapping to $s$.", "Let $X_U \\to U$ be the base change of $X$. By", "Lemma \\ref{lemma-characterize-geometrically-connected-fibres}", "it suffices to show that the fibre of $X_U \\to U$ over $u$ is", "connected. By", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}", "we see that", "$\\Gamma(X_U, \\mathcal{O}_{X_U}) = \\Gamma(U, \\mathcal{O}_U)$.", "Hence we have to show: Given", "$S = \\Spec(R)$ affine, $X \\to S$ proper with $\\Gamma(X, \\mathcal{O}_X) = R$", "and $s \\in S$ is a point, the fibre $X_s$ is connected.", "\\medskip\\noindent", "To do this it suffices to show that the only idempotents", "$e \\in H^0(X_s, \\mathcal{O}_{X_s})$ are $0$ and $1$ (we already", "know that $X_s$ is nonempty by Lemma \\ref{lemma-stein-universally-closed}).", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-proper-idempotent-on-fibre}", "after replacing $R$ by a principal localization", "we may assume $e$ is the image of an element of $R$.", "Since $R \\to H^0(X_s, \\mathcal{O}_{X_s})$ factors through", "$\\kappa(s)$ we conclude." ], "refs": [ "more-morphisms-lemma-stein-universally-closed", "more-morphisms-lemma-stein-universally-closed", "schemes-lemma-compose-after-separated", "morphisms-lemma-permanence-finite-type", "more-morphisms-lemma-characterize-geometrically-connected-fibres", "coherent-lemma-flat-base-change-cohomology", "more-morphisms-lemma-stein-universally-closed", "perfect-lemma-proper-idempotent-on-fibre" ], "ref_ids": [ 13942, 13942, 7715, 5204, 13944, 3298, 13942, 7068 ] } ], "ref_ids": [ 5591 ] }, { "id": 13676, "type": "theorem", "label": "more-morphisms-theorem-normalized-base-change-with-reduced-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-theorem-normalized-base-change-with-reduced-fibre", "contents": [ "Let $A$ be a Dedekind ring with fraction field $K$.", "Let $X$ be a scheme flat and of finite type over $A$.", "Assume $A$ is a Nagata ring.", "There exists a finite extension $K \\subset L$ such that", "the normalized base change $Y$ is smooth over $\\Spec(B)$", "at all generic points of all fibres." ], "refs": [], "proofs": [ { "contents": [ "During the proof we will repeatedly use that formation of the set of points", "where a (flat, finitely presented) morphism like $X \\to \\Spec(A)$ is", "smooth commutes with base change, see", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-smooth}.", "\\medskip\\noindent", "We first choose a finite extension $K \\subset L$ such that", "$(X_L)_{red}$ is geometrically reduced over $L$, see", "Varieties, Lemma \\ref{varieties-lemma-finite-extension-geometrically-reduced}.", "Since $Y \\to (X_B)_{red}$ is birational we see applying", "Varieties, Lemma \\ref{varieties-lemma-generic-points-geometrically-reduced}", "that $Y_L$ is geometrically reduced over $L$ as well.", "Hence $Y_L \\to \\Spec(L)$ is smooth on a dense open $V \\subset Y_L$ by", "Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced-dense-smooth-open}.", "Thus the smooth locus $U \\subset Y$ of the morphism $Y \\to \\Spec(B)$", "is open (by Morphisms, Definition \\ref{morphisms-definition-smooth})", "and is dense in the generic fibre. Replacing $A$ by $B$ and $X$ by $Y$", "we reduce to the case treated in the next paragraph.", "\\medskip\\noindent", "Assume $X$ is normal and the smooth locus $U \\subset X$ of $X \\to \\Spec(A)$", "is dense in the generic fibre. This implies that $U$ is dense in all but", "finitely many fibres, see Lemma \\ref{lemma-nowhere-dense-generic-fibre}.", "Let $x_1, \\ldots, x_r \\in X \\setminus U$ be the finitely many generic", "points of irreducible components of $X \\setminus U$ which are moreover", "generic points of irreducible components of fibres of $X \\to \\Spec(A)$.", "Set $\\mathcal{O}_i = \\mathcal{O}_{X, x_i}$. Let $A_i$ be the localization of", "$A$ at the maximal ideal corresponding to the image of $x_i$ in $\\Spec(A)$.", "By", "More on Algebra, Proposition", "\\ref{more-algebra-proposition-epp-essentially-finite-type}", "there exist finite extensions", "$K \\subset K_i$ which are solutions for the extension of discrete valuation", "rings $A_i \\to \\mathcal{O}_i$. Let $K \\subset L$ be a finite extension", "dominating all of the extensions $K \\subset K_i$. Then $K \\subset L$", "is still a solution for $A_i \\to \\mathcal{O}_i$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-formally-smooth-goes-up}.", "\\medskip\\noindent", "Consider the diagram (\\ref{equation-normalized-base-change})", "with the extension $L/K$ we just produced. Note that $U_B \\subset X_B$", "is smooth over $B$, hence normal (for example use", "Algebra, Lemma \\ref{algebra-lemma-normal-goes-up}).", "Thus $Y \\to X_B$ is an isomorphism over $U_B$.", "Let $y \\in Y$ be a generic point of an irreducible", "component of a fibre of $Y \\to \\Spec(B)$ lying over the maximal", "ideal $\\mathfrak m \\subset B$. Assume that $y \\not \\in U_B$.", "Then $y$ maps to one of the points $x_i$. It follows that", "$\\mathcal{O}_{Y, y}$ is a local ring of the integral closure", "of $\\mathcal{O}_i$ in $R(X) \\otimes_K L$ (details omitted).", "Hence because $K \\subset L$ is a solution for", "$A_i \\to \\mathcal{O}_i$ we see that", "$B_\\mathfrak m \\to \\mathcal{O}_{Y, y}$ is formally smooth", "in the $\\mathfrak m_y$-adic topology", "(this is the definition of being a \"solution\").", "In other words, $\\mathfrak m\\mathcal{O}_{Y, y} = \\mathfrak m_y$", "and the residue field extension is separable, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-extension-dvrs-formally-smooth}.", "Hence the local ring", "of the fibre at $y$ is $\\kappa(y)$.", "This implies the fibre is smooth over $\\kappa(\\mathfrak m)$", "at $y$ for example by Algebra, Lemma \\ref{algebra-lemma-separable-smooth}.", "This finishes the proof." ], "refs": [ "morphisms-lemma-set-points-where-fibres-smooth", "varieties-lemma-finite-extension-geometrically-reduced", "varieties-lemma-generic-points-geometrically-reduced", "varieties-lemma-geometrically-reduced-dense-smooth-open", "morphisms-definition-smooth", "more-morphisms-lemma-nowhere-dense-generic-fibre", "more-algebra-proposition-epp-essentially-finite-type", "more-algebra-lemma-formally-smooth-goes-up", "algebra-lemma-normal-goes-up", "more-algebra-lemma-extension-dvrs-formally-smooth", "algebra-lemma-separable-smooth" ], "ref_ids": [ 5336, 10914, 10912, 11008, 5564, 13802, 10592, 10508, 1368, 10495, 1225 ] } ], "ref_ids": [] }, { "id": 13677, "type": "theorem", "label": "more-morphisms-lemma-first-order-thickening", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-first-order-thickening", "contents": [ "Let $X$ be a scheme over a base $S$. Consider a short exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{A} \\to \\mathcal{O}_X \\to 0", "$$", "of sheaves on $X$ where $\\mathcal{A}$ is a sheaf of", "$f^{-1}\\mathcal{O}_S$-algebras,", "$\\mathcal{A} \\to \\mathcal{O}_X$ is a surjection", "of sheaves of $f^{-1}\\mathcal{O}_S$-algebras, and $\\mathcal{I}$ is its kernel.", "If", "\\begin{enumerate}", "\\item $\\mathcal{I}$ is an ideal of square zero in $\\mathcal{A}$, and", "\\item $\\mathcal{I}$ is quasi-coherent as an $\\mathcal{O}_X$-module", "\\end{enumerate}", "then $X' = (X, \\mathcal{A})$ is a scheme and $X \\to X'$ is a first", "order thickening over $S$. Moreover, any first order thickening over", "$S$ is of this form." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $X'$ is a locally ringed space. Let $U = \\Spec(B)$", "be an affine open of $X$. Set $A = \\Gamma(U, \\mathcal{A})$. Note that", "since $H^1(U, \\mathcal{I}) = 0$ (see Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero})", "the map $A \\to B$ is surjective. By assumption the kernel", "$I = \\mathcal{I}(U)$ is an ideal of square zero in the ring $A$.", "By", "Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}", "there is a canonical morphism of locally ringed spaces", "$$", "(U, \\mathcal{A}|_U) \\longrightarrow \\Spec(A)", "$$", "coming from the map $B \\to \\Gamma(U, \\mathcal{A})$. Since this morphism", "fits into the commutative diagram", "$$", "\\xymatrix{", "(U, \\mathcal{O}_X|_U) \\ar[d] \\ar[r] & \\Spec(B) \\ar[d] \\\\", "(U, \\mathcal{A}|_U) \\ar[r] & \\Spec(A)", "}", "$$", "we see that it is a homeomorphism on underlying topological spaces.", "Thus to see that it is an isomorphism, it suffices to check it induces", "an isomorphism on the local rings.", "For $u \\in U$ corresponding to the prime $\\mathfrak p \\subset A$", "we obtain a commutative diagram of short exact sequences", "$$", "\\xymatrix{", "0 \\ar[r] &", "I_{\\mathfrak p} \\ar[r] \\ar[d] &", "A_{\\mathfrak p} \\ar[r] \\ar[d] &", "B_{\\mathfrak p} \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "\\mathcal{I}_u \\ar[r] &", "\\mathcal{A}_u \\ar[r] &", "\\mathcal{O}_{X, u} \\ar[r] & 0.", "}", "$$", "The left and right vertical arrows are isomorphisms because", "$\\mathcal{I}$ and $\\mathcal{O}_X$ are quasi-coherent sheaves.", "Hence also the middle map is an isomorphism. Hence every point", "of $X' = (X, \\mathcal{A})$ has an affine neighbourhood and $X'$ is a", "scheme as desired." ], "refs": [ "coherent-lemma-quasi-coherent-affine-cohomology-zero", "schemes-lemma-morphism-into-affine" ], "ref_ids": [ 3282, 7655 ] } ], "ref_ids": [] }, { "id": 13678, "type": "theorem", "label": "more-morphisms-lemma-thickening-affine-scheme", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-thickening-affine-scheme", "contents": [ "\\begin{slogan}", "Affineness is insensitive to thickenings", "\\end{slogan}", "Any thickening of an affine scheme is affine." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Limits, Proposition \\ref{limits-proposition-affine}." ], "refs": [ "limits-proposition-affine" ], "ref_ids": [ 15129 ] } ], "ref_ids": [] }, { "id": 13679, "type": "theorem", "label": "more-morphisms-lemma-base-change-thickening", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-thickening", "contents": [ "Let $S \\subset S'$ be a thickening of schemes. Let $X' \\to S'$ be a morphism", "and set $X = S \\times_{S'} X'$. Then $(X \\subset X') \\to (S \\subset S')$", "is a morphism of thickenings. If $S \\subset S'$ is a first", "(resp.\\ finite order) thickening, then $X \\subset X'$ is a first", "(resp.\\ finite order) thickening." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13680, "type": "theorem", "label": "more-morphisms-lemma-composition-thickening", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-composition-thickening", "contents": [ "\\begin{slogan}", "Compositions of thickenings are thickenings", "\\end{slogan}", "If $S \\subset S'$ and $S' \\subset S''$ are thickenings, then so is", "$S \\subset S''$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13681, "type": "theorem", "label": "more-morphisms-lemma-descending-property-thickening", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-descending-property-thickening", "contents": [ "The property of being a thickening is fpqc local.", "Similarly for first order thickenings." ], "refs": [], "proofs": [ { "contents": [ "The statement means the following: Let $X \\to X'$ be a morphism", "of schemes and let $\\{g_i : X'_i \\to X'\\}$", "be an fpqc covering such that the base change $X_i \\to X'_i$", "is a thickening for all $i$. Then $X \\to X'$ is a thickening.", "Since the morphisms $g_i$ are jointly surjective we conclude", "that $X \\to X'$ is surjective. By", "Descent, Lemma \\ref{descent-lemma-descending-property-closed-immersion}", "we conclude that $X \\to X'$ is a closed immersion.", "Thus $X \\to X'$ is a thickening. We omit the proof in the", "case of first order thickenings." ], "refs": [ "descent-lemma-descending-property-closed-immersion" ], "ref_ids": [ 14684 ] } ], "ref_ids": [] }, { "id": 13682, "type": "theorem", "label": "more-morphisms-lemma-thicken-property-morphisms", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-thicken-property-morphisms", "contents": [ "Let $(f, f') : (X \\subset X') \\to (S \\subset S')$ be a morphism", "of thickenings. Then", "\\begin{enumerate}", "\\item $f$ is an affine morphism if and only if $f'$ is an affine morphism,", "\\item $f$ is a surjective morphism if and only if $f'$ is a surjective morphism,", "\\item $f$ is quasi-compact if and only if $f'$ quasi-compact,", "\\item $f$ is universally closed if and only if $f'$ is universally closed,", "\\item $f$ is integral if and only if $f'$ is integral,", "\\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,", "\\item $f$ is universally injective if and only if $f'$ is universally injective,", "\\item $f$ is universally open if and only if $f'$ is universally open,", "\\item $f$ is quasi-affine if and only if $f'$ is quasi-affine, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that $S \\to S'$ and $X \\to X'$ are universal homeomorphisms", "(see for example", "Morphisms, Lemma \\ref{morphisms-lemma-reduction-universal-homeomorphism}).", "This immediately implies parts (2), (3), (4), (7), and (8).", "Part (1) follows from Lemma \\ref{lemma-thickening-affine-scheme}", "which tells us that there is a 1-to-1 correspondence between", "affine opens of $S$ and $S'$ and between affine opens of $X$ and $X'$.", "Part (9) follows from", "Limits, Lemma \\ref{limits-lemma-thickening-quasi-affine}", "and the remark just made about affine opens of $S$ and $S'$.", "Part (5) follows from (1) and (4) by", "Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed}.", "Finally, note that", "$$", "S \\times_X S = S \\times_{X'} S \\to S \\times_{X'} S' \\to S' \\times_{X'} S'", "$$", "is a thickening (the two arrows are thickenings by", "Lemma \\ref{lemma-base-change-thickening}).", "Hence applying (3) and (4) to the morphism", "$(S \\subset S') \\to (S \\times_X S \\to S' \\times_{X'} S')$", "we obtain (6)." ], "refs": [ "morphisms-lemma-reduction-universal-homeomorphism", "more-morphisms-lemma-thickening-affine-scheme", "limits-lemma-thickening-quasi-affine", "morphisms-lemma-integral-universally-closed", "more-morphisms-lemma-base-change-thickening" ], "ref_ids": [ 5455, 13678, 15085, 5441, 13679 ] } ], "ref_ids": [] }, { "id": 13683, "type": "theorem", "label": "more-morphisms-lemma-thicken-property-relatively-ample", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-thicken-property-relatively-ample", "contents": [ "Let $(f, f') : (X \\subset X') \\to (S \\subset S')$ be a morphism", "of thickenings. Let $\\mathcal{L}'$ be an invertible sheaf on $X'$", "and denote $\\mathcal{L}$ the restriction to $X$.", "Then $\\mathcal{L}'$ is $f'$-ample if and only if", "$\\mathcal{L}$ is $f$-ample." ], "refs": [], "proofs": [ { "contents": [ "Recall that being relatively ample is a condition for each", "affine open in the base, see", "Morphisms, Definition \\ref{morphisms-definition-relatively-ample}.", "By Lemma \\ref{lemma-thickening-affine-scheme}", "there is a 1-to-1 correspondence between", "affine opens of $S$ and $S'$.", "Thus we may assume $S$ and $S'$ are affine", "and we reduce to proving that", "$\\mathcal{L}'$ is ample if and only if", "$\\mathcal{L}$ is ample.", "This is Limits, Lemma \\ref{limits-lemma-ample-on-reduction}." ], "refs": [ "morphisms-definition-relatively-ample", "more-morphisms-lemma-thickening-affine-scheme", "limits-lemma-ample-on-reduction" ], "ref_ids": [ 5568, 13678, 15084 ] } ], "ref_ids": [] }, { "id": 13684, "type": "theorem", "label": "more-morphisms-lemma-thicken-property-morphisms-cartesian", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-thicken-property-morphisms-cartesian", "contents": [ "Let $(f, f') : (X \\subset X') \\to (S \\subset S')$ be a morphism", "of thickenings such that $X = S \\times_{S'} X'$. If $S \\subset S'$", "is a finite order thickening, then", "\\begin{enumerate}", "\\item $f$ is a closed immersion if and only if $f'$ is a closed immersion,", "\\item $f$ is locally of finite type if and only if $f'$ is", "locally of finite type,", "\\item $f$ is locally quasi-finite if and only if $f'$ is locally", "quasi-finite,", "\\item $f$ is locally of finite type of relative dimension $d$ if and", "only if $f'$ is locally of finite type of relative dimension $d$,", "\\item $\\Omega_{X/S} = 0$ if and only if $\\Omega_{X'/S'} = 0$,", "\\item $f$ is unramified if and only if $f'$ is unramified,", "\\item $f$ is proper if and only if $f'$ is proper,", "\\item $f$ is finite if and only if $f'$ is finite,", "\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,", "\\item $f$ is an immersion if and only if $f'$ is an immersion, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The properties $\\mathcal{P}$ listed in the lemma are all stable", "under base change, hence if $f'$ has property $\\mathcal{P}$, then so", "does $f$. See", "Schemes, Lemmas \\ref{schemes-lemma-base-change-immersion} and", "\\ref{schemes-lemma-base-change-monomorphism}", "and", "Morphisms, Lemmas", "\\ref{morphisms-lemma-base-change-finite-type},", "\\ref{morphisms-lemma-base-change-quasi-finite},", "\\ref{morphisms-lemma-base-change-relative-dimension-d},", "\\ref{morphisms-lemma-base-change-differentials},", "\\ref{morphisms-lemma-base-change-unramified},", "\\ref{morphisms-lemma-base-change-proper}, and", "\\ref{morphisms-lemma-base-change-finite}.", "\\medskip\\noindent", "The interesting direction in each case is therefore to assume", "that $f$ has the property and deduce that $f'$ has it too.", "By induction on the order of the thickening we may", "assume that $S \\subset S'$ is a first order thickening, see", "discussion immediately following", "Definition \\ref{definition-thickening}.", "\\medskip\\noindent", "Most of the proofs will use a reduction to the affine case. Let", "$U' \\subset S'$ be an affine open and let $V' \\subset X'$ be an affine open", "lying over $U'$. Let $U' = \\Spec(A')$ and denote $I \\subset A'$ be the ideal", "defining the closed subscheme $U' \\cap S$. Say $V' = \\Spec(B')$.", "Then $V' \\cap X = \\Spec(B'/IB')$. Setting $A = A'/I$ and", "$B = B'/IB'$ we get a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "IB' \\ar[r] &", "B' \\ar[r] &", "B \\ar[r] & 0 \\\\", "0 \\ar[r] &", "IA' \\ar[r] \\ar[u] &", "A' \\ar[r] \\ar[u] &", "A \\ar[r] \\ar[u] & 0", "}", "$$", "with exact rows and $I^2 = 0$.", "\\medskip\\noindent", "The translation of (1) into algebra: If $A \\to B$ is surjective,", "then $A' \\to B'$ is surjective. This follows from", "Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK}).", "\\medskip\\noindent", "The translation of (2) into algebra: If $A \\to B$ is a finite type ring", "map, then $A' \\to B'$ is a finite type ring map. This follows from", "Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "applied to a map $A'[x_1, \\ldots, x_n] \\to B'$ such that", "$A[x_1, \\ldots, x_n] \\to B$ is surjective.", "\\medskip\\noindent", "Proof of (3). Follows from (2) and that quasi-finiteness of a morphism", "which is locally of finite type can be checked on fibres, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}.", "\\medskip\\noindent", "Proof of (4). Follows from (2) and that the additional property of ``being of", "relative dimension $d$'' can be checked on fibres (by definition, see", "Morphisms, Definition \\ref{morphisms-definition-relative-dimension-d}.", "\\medskip\\noindent", "The translation of (5) into algebra: If $\\Omega_{B/A} = 0$, then", "$\\Omega_{B'/A'} = 0$. By", "Algebra, Lemma \\ref{algebra-lemma-differentials-base-change}", "we have $0 = \\Omega_{B/A} = \\Omega_{B'/A'}/I\\Omega_{B'/A'}$.", "Hence $\\Omega_{B'/A'} = 0$ by", "Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK}).", "\\medskip\\noindent", "The translation of (6) into algebra: If $A \\to B$ is unramified", "map, then $A' \\to B'$ is unramified. Since $A \\to B$ is of finite", "type we see that $A' \\to B'$ is of finite type by (2) above.", "Since $A \\to B$ is unramified we have $\\Omega_{B/A} = 0$. By", "part (5) we have $\\Omega_{B'/A'} = 0$. Thus $A' \\to B'$ is unramified.", "\\medskip\\noindent", "Proof of (7). Follows by combining (2) with", "results of Lemma \\ref{lemma-thicken-property-morphisms}", "and the fact that proper equals quasi-compact $+$", "separated $+$ locally of finite type $+$ universally closed.", "\\medskip\\noindent", "Proof of (8). Follows by combining (2) with", "results of Lemma \\ref{lemma-thicken-property-morphisms}", "and using the fact that finite equals integral $+$ locally", "of finite type (Morphisms, Lemma \\ref{morphisms-lemma-finite-integral}).", "\\medskip\\noindent", "Proof of (9). As $f$ is a monomorphism we have $X = X \\times_S X$.", "We may apply the results proved so far to the morphism of thickenings", "$(X \\subset X') \\to (X \\times_S X \\subset X' \\times_{S'} X')$.", "We conclude $X' \\to X' \\times_{S'} X'$ is a closed immersion by (1).", "In fact, it is a first order thickening as the ideal defining the", "closed immersion", "$X' \\to X' \\times_{S'} X'$ is contained in the pullback of the ideal", "$\\mathcal{I} \\subset \\mathcal{O}_{S'}$ cutting out $S$ in $S'$.", "Indeed, $X = X \\times_S X = (X' \\times_{S'} X') \\times_{S'} S$", "is contained in $X'$. Hence by", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-diagonal}", "it suffices to show that", "$\\Omega_{X'/S'} = 0$ which follows from (5)", "and the corresponding statement for $X/S$.", "\\medskip\\noindent", "Proof of (10). If $f : X \\to S$ is an immersion, then it factors as", "$X \\to U \\to S$ where $U \\to S$ is an open immersion and $X \\to U$ is a", "closed immersion. Let $U' \\subset S'$ be the open subscheme whose", "underlying topological space is the same as $U$. Then $X' \\to S'$", "factors through $U'$ and we conclude that $X' \\to U'$ is a closed", "immersion by part (1). This finishes the proof." ], "refs": [ "schemes-lemma-base-change-immersion", "schemes-lemma-base-change-monomorphism", "morphisms-lemma-base-change-finite-type", "morphisms-lemma-base-change-quasi-finite", "morphisms-lemma-base-change-relative-dimension-d", "morphisms-lemma-base-change-differentials", "morphisms-lemma-base-change-unramified", "morphisms-lemma-base-change-proper", "morphisms-lemma-base-change-finite", "more-morphisms-definition-thickening", "algebra-lemma-NAK", "algebra-lemma-NAK", "morphisms-lemma-quasi-finite-at-point-characterize", "morphisms-definition-relative-dimension-d", "algebra-lemma-differentials-base-change", "algebra-lemma-NAK", "more-morphisms-lemma-thicken-property-morphisms", "more-morphisms-lemma-thicken-property-morphisms", "morphisms-lemma-finite-integral", "morphisms-lemma-differentials-diagonal" ], "ref_ids": [ 7695, 7724, 5200, 5233, 5284, 5314, 5346, 5409, 5440, 14106, 401, 401, 5226, 5559, 1138, 401, 13682, 13682, 5438, 5311 ] } ], "ref_ids": [] }, { "id": 13685, "type": "theorem", "label": "more-morphisms-lemma-properties-that-extend-over-thickenings", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-properties-that-extend-over-thickenings", "contents": [ "Let $(f, f') : (X \\subset X') \\to (Y \\to Y')$ be a morphism", "of thickenings. Assume $f$ and $f'$ are locally of finite type", "and $X = Y \\times_{Y'} X'$. Then", "\\begin{enumerate}", "\\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,", "\\item $f$ is finite if and only if $f'$ is finite,", "\\item $f$ is a closed immersion if and only if $f'$ is a closed immersion,", "\\item $\\Omega_{X/Y} = 0$ if and only if $\\Omega_{X'/Y'} = 0$,", "\\item $f$ is unramified if and only if $f'$ is unramified,", "\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,", "\\item $f$ is an immersion if and only if $f'$ is an immersion,", "\\item $f$ is proper if and only if $f'$ is proper, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The properties $\\mathcal{P}$ listed in the lemma are all stable", "under base change, hence if $f'$ has property $\\mathcal{P}$, then so", "does $f$. See", "Schemes, Lemmas \\ref{schemes-lemma-base-change-immersion} and", "\\ref{schemes-lemma-base-change-monomorphism}", "and", "Morphisms, Lemmas", "\\ref{morphisms-lemma-base-change-quasi-finite},", "\\ref{morphisms-lemma-base-change-relative-dimension-d},", "\\ref{morphisms-lemma-base-change-differentials},", "\\ref{morphisms-lemma-base-change-unramified},", "\\ref{morphisms-lemma-base-change-proper}, and", "\\ref{morphisms-lemma-base-change-finite}.", "Hence in each case we need only to prove that if $f$ has", "the desired property, so does $f'$.", "\\medskip\\noindent", "A morphism is locally quasi-finite if and only if it is locally", "of finite type and the scheme theoretic fibres are discrete spaces, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-fibres}.", "Since the underlying topological space is unchanged by ", "passing to a thickening, we see that $f'$ is locally quasi-finite if", "(and only if) $f$ is. This proves (1).", "\\medskip\\noindent", "Case (2) follows from case (5) of Lemma \\ref{lemma-thicken-property-morphisms}", "and the fact that the finite morphisms are precisely", "the integral morphisms that are locally of finite type", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-integral}).", "\\medskip\\noindent", "Case (3). This follows immediately from", "Morphisms, Lemma \\ref{morphisms-lemma-check-closed-infinitesimally}.", "\\medskip\\noindent", "Case (4) follows from the following algebra statement: Let $A$ be a ring and", "let $I \\subset A$ be a locally nilpotent ideal. Let $B$ be a finite type", "$A$-algebra. If $\\Omega_{(B/IB)/(A/I)} = 0$, then $\\Omega_{B/A} = 0$.", "Namely, the assumption means that $I\\Omega_{B/A} = 0$, see", "Algebra, Lemma \\ref{algebra-lemma-differentials-base-change}.", "On the other hand $\\Omega_{B/A}$ is a finite $B$-module, see", "Algebra, Lemma \\ref{algebra-lemma-differentials-finitely-generated}.", "Hence the vanishing of $\\Omega_{B/A}$ follows from Nakayama's", "lemma (Algebra, Lemma \\ref{algebra-lemma-NAK}) and the fact", "that $IB$ is contained in the Jacobson radical of $B$.", "\\medskip\\noindent", "Case (5) follows immediately from (4) and", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-omega-zero}.", "\\medskip\\noindent", "Proof of (6). As $f$ is a monomorphism we have $X = X \\times_Y X$.", "We may apply the results proved so far to the morphism of thickenings", "$(X \\subset X') \\to (X \\times_Y X \\subset X' \\times_{Y'} X')$.", "We conclude $\\Delta_{X'/Y'} : X' \\to X' \\times_{Y'} X'$", "is a closed immersion by (3). In fact $\\Delta_{X'/Y'}$ is a bijection on", "underlying sets, hence $\\Delta_{X'/Y'}$ is a thickening. On the other hand", "$\\Delta_{X'/Y'}$ is locally of finite presentation by", "Morphisms, Lemma \\ref{morphisms-lemma-diagonal-morphism-finite-type}.", "In other words, $\\Delta_{X'/Y'}(X')$ is cut out by", "a quasi-coherent sheaf of ideals", "$\\mathcal{J} \\subset \\mathcal{O}_{X' \\times_{Y'} X'}$ of finite type.", "Since $\\Omega_{X'/Y'} = 0$ by (5) we see that", "the conormal sheaf of $X' \\to X' \\times_{Y'} X'$ is zero by", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-diagonal}.", "In other words, $\\mathcal{J}/\\mathcal{J}^2 = 0$.", "This implies $\\Delta_{X'/Y'}$ is an isomorphism, for example", "by Algebra, Lemma \\ref{algebra-lemma-ideal-is-squared-union-connected}.", "\\medskip\\noindent", "Proof of (7). If $f : X \\to Y$ is an immersion, then it factors as", "$X \\to V \\to Y$ where $V \\to Y$ is an open immersion and $X \\to V$ is a", "closed immersion. Let $V' \\subset Y'$ be the open subscheme whose", "underlying topological space is the same as $V$. Then $X' \\to V'$", "factors through $V'$ and we conclude that $X' \\to V'$ is a closed", "immersion by part (3).", "\\medskip\\noindent", "Case (8) follows from Lemma \\ref{lemma-thicken-property-morphisms}", "and the definition of proper morphisms as being the quasi-compact,", "universally closed, and separated morphisms that are locally of finite type." ], "refs": [ "schemes-lemma-base-change-immersion", "schemes-lemma-base-change-monomorphism", "morphisms-lemma-base-change-quasi-finite", "morphisms-lemma-base-change-relative-dimension-d", "morphisms-lemma-base-change-differentials", "morphisms-lemma-base-change-unramified", "morphisms-lemma-base-change-proper", "morphisms-lemma-base-change-finite", "morphisms-lemma-locally-quasi-finite-fibres", "more-morphisms-lemma-thicken-property-morphisms", "morphisms-lemma-finite-integral", "morphisms-lemma-check-closed-infinitesimally", "algebra-lemma-differentials-base-change", "algebra-lemma-differentials-finitely-generated", "algebra-lemma-NAK", "morphisms-lemma-unramified-omega-zero", "morphisms-lemma-diagonal-morphism-finite-type", "morphisms-lemma-differentials-diagonal", "algebra-lemma-ideal-is-squared-union-connected", "more-morphisms-lemma-thicken-property-morphisms" ], "ref_ids": [ 7695, 7724, 5233, 5284, 5314, 5346, 5409, 5440, 5228, 13682, 5438, 5456, 1138, 1142, 401, 5343, 5248, 5311, 407, 13682 ] } ], "ref_ids": [] }, { "id": 13686, "type": "theorem", "label": "more-morphisms-lemma-picard-group-first-order-thickening", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-picard-group-first-order-thickening", "contents": [ "Let $X \\subset X'$ be a first order thickening", "with ideal sheaf $\\mathcal{I}$. Then there is a canonical", "exact sequence", "$$", "\\xymatrix{", "0 \\ar[r] &", "H^0(X, \\mathcal{I}) \\ar[r] &", "H^0(X', \\mathcal{O}_{X'}^*) \\ar[r] &", "H^0(X, \\mathcal{O}^*_X) \\ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\\\", "& H^1(X, \\mathcal{I}) \\ar[r] &", "\\Pic(X') \\ar[r] &", "\\Pic(X) \\ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\\\", "& H^2(X, \\mathcal{I}) \\ar[r] & \\ldots \\ar[r] & \\ldots", "}", "$$", "of abelian groups." ], "refs": [], "proofs": [ { "contents": [ "This is the long exact cohomology sequence associated to the", "short exact sequence of sheaves of abelian groups", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_{X'}^* \\to \\mathcal{O}_X^* \\to 0", "$$", "where the first map sends a local section $f$ of $\\mathcal{I}$", "to the invertible section $1 + f$ of $\\mathcal{O}_{X'}$.", "We also use the identification of the Picard group of a", "ringed space with the first cohomology group of the sheaf", "of invertible functions, see", "Cohomology, Lemma \\ref{cohomology-lemma-h1-invertible}." ], "refs": [ "cohomology-lemma-h1-invertible" ], "ref_ids": [ 2036 ] } ], "ref_ids": [] }, { "id": 13687, "type": "theorem", "label": "more-morphisms-lemma-torsion-pic-thickening", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-torsion-pic-thickening", "contents": [ "Let $X \\subset X'$ be a thickening. Let $n$ be an integer", "invertible in $\\mathcal{O}_X$. Then the map", "$\\Pic(X')[n] \\to \\Pic(X)[n]$ is bijective." ], "refs": [], "proofs": [ { "contents": [ "[Proof for a finite order thickening]", "By the general principle explained following", "Definition \\ref{definition-thickening}", "this reduces to the case of a first order thickening.", "Then may use Lemma \\ref{lemma-picard-group-first-order-thickening}", "to see that it suffices to show that", "$H^1(X, \\mathcal{I})[n]$, $H^1(X, \\mathcal{I})/n$, and", "$H^2(X, \\mathcal{I})[n]$ are zero.", "This follows as multiplication by $n$ on $\\mathcal{I}$", "is an isomorphism as it is an $\\mathcal{O}_X$-module." ], "refs": [ "more-morphisms-definition-thickening", "more-morphisms-lemma-picard-group-first-order-thickening" ], "ref_ids": [ 14106, 13686 ] } ], "ref_ids": [] }, { "id": 13688, "type": "theorem", "label": "more-morphisms-lemma-first-order-infinitesimal-neighbourhood", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-first-order-infinitesimal-neighbourhood", "contents": [ "Let $i : Z \\to X$ be an immersion of schemes. The first order infinitesimal", "neighbourhood $Z'$ of $Z$ in $X$ has the following universal property:", "Given any commutative diagram", "$$", "\\xymatrix{", "Z \\ar[d]_i & T \\ar[l]^a \\ar[d] \\\\", "X & T' \\ar[l]_b", "}", "$$", "where $T \\subset T'$ is a first order thickening over $X$, there exists", "a unique morphism $(a', a) : (T \\subset T') \\to (Z \\subset Z')$ of", "thickenings over $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ be the open used in the construction of $Z'$, i.e., an", "open such that $Z$ is identified with a closed subscheme of $U$ cut out by", "the quasi-coherent sheaf of ideals $\\mathcal{I}$.", "Since $|T| = |T'|$ we see that $b(T') \\subset U$. Hence we can", "think of $b$ as a morphism into $U$. Let $\\mathcal{J} \\subset \\mathcal{O}_{T'}$", "be the ideal cutting out $T$. Since $b(T) \\subset Z$ by the diagram above", "we see that $b^\\sharp(b^{-1}\\mathcal{I}) \\subset \\mathcal{J}$. As", "$T'$ is a first order thickening of $T$ we see that $\\mathcal{J}^2 = 0$", "hence $b^\\sharp(b^{-1}(\\mathcal{I}^2)) = 0$. By", "Schemes, Lemma \\ref{schemes-lemma-characterize-closed-subspace}", "this implies that $b$ factors through $Z'$. Denote $a' : T' \\to Z'$", "this factorization and everything is clear." ], "refs": [ "schemes-lemma-characterize-closed-subspace" ], "ref_ids": [ 7648 ] } ], "ref_ids": [] }, { "id": 13689, "type": "theorem", "label": "more-morphisms-lemma-infinitesimal-neighbourhood-conormal", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-infinitesimal-neighbourhood-conormal", "contents": [ "Let $i : Z \\to X$ be an immersion of schemes. Let $Z \\subset Z'$ be", "the first order infinitesimal neighbourhood of $Z$ in $X$.", "Then the diagram", "$$", "\\xymatrix{", "Z \\ar[r] \\ar[d] & Z' \\ar[d] \\\\", "Z \\ar[r] & X", "}", "$$", "induces a map of conormal sheaves $\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Z'}$ by", "Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial}.", "This map is an isomorphism." ], "refs": [ "morphisms-lemma-conormal-functorial" ], "proofs": [ { "contents": [ "This is clear from the construction of $Z'$ above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 5304 ] }, { "id": 13690, "type": "theorem", "label": "more-morphisms-lemma-formally-unramified-not-affine", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-formally-unramified-not-affine", "contents": [ "If $f : X \\to S$ is a formally unramified morphism, then given", "any solid commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\", "S & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "where $T \\subset T'$ is a first order thickening of schemes over $S$", "there exists at most one dotted arrow making the diagram commute.", "In other words, in", "Definition \\ref{definition-formally-unramified}", "the condition that $T$ be affine may be dropped." ], "refs": [ "more-morphisms-definition-formally-unramified" ], "proofs": [ { "contents": [ "This is true because a morphism is determined by its restrictions", "to affine opens." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 14108 ] }, { "id": 13691, "type": "theorem", "label": "more-morphisms-lemma-composition-formally-unramified", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-composition-formally-unramified", "contents": [ "A composition of formally unramified morphisms is formally unramified." ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13692, "type": "theorem", "label": "more-morphisms-lemma-base-change-formally-unramified", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-formally-unramified", "contents": [ "A base change of a formally unramified morphism is formally unramified." ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13693, "type": "theorem", "label": "more-morphisms-lemma-formally-unramified-on-opens", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-formally-unramified-on-opens", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $U \\subset X$ and $V \\subset S$ be open such that", "$f(U) \\subset V$. If $f$ is formally unramified, so is $f|_U : U \\to V$." ], "refs": [], "proofs": [ { "contents": [ "Consider a solid diagram", "$$", "\\xymatrix{", "U \\ar[d]_{f|_U} & T \\ar[d]^i \\ar[l]^a \\\\", "V & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "as in Definition \\ref{definition-formally-unramified}. If $f$ is formally", "ramified, then there exists at most one", "$S$-morphism $a' : T' \\to X$ such that $a'|_T = a$.", "Hence clearly there exists at most one such morphism into $U$." ], "refs": [ "more-morphisms-definition-formally-unramified" ], "ref_ids": [ 14108 ] } ], "ref_ids": [] }, { "id": 13694, "type": "theorem", "label": "more-morphisms-lemma-affine-formally-unramified", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-affine-formally-unramified", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume $X$ and $S$ are affine.", "Then $f$ is formally unramified if and only if", "$\\mathcal{O}_S(S) \\to \\mathcal{O}_X(X)$ is a formally unramified", "ring map." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions", "(Definition \\ref{definition-formally-unramified} and", "Algebra, Definition \\ref{algebra-definition-formally-unramified})", "by the equivalence of categories of rings and affine schemes,", "see", "Schemes, Lemma \\ref{schemes-lemma-category-affine-schemes}." ], "refs": [ "more-morphisms-definition-formally-unramified", "algebra-definition-formally-unramified", "schemes-lemma-category-affine-schemes" ], "ref_ids": [ 14108, 1541, 7656 ] } ], "ref_ids": [] }, { "id": 13695, "type": "theorem", "label": "more-morphisms-lemma-formally-unramified-differentials", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-formally-unramified-differentials", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Then $f$ is formally unramified if and only if $\\Omega_{X/S} = 0$." ], "refs": [], "proofs": [ { "contents": [ "We recall some of the arguments of the proof of", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-affine}.", "Let $W \\subset X \\times_S X$ be an open such that", "$\\Delta : X \\to X \\times_S X$ induces a closed immersion into $W$.", "Let $\\mathcal{J} \\subset \\mathcal{O}_W$ be the ideal sheaf of this", "closed immersion. Let $X' \\subset W$ be the closed subscheme", "defined by the quasi-coherent sheaf of ideals $\\mathcal{J}^2$.", "Consider the two morphisms $p_1, p_2 : X' \\to X$ induced by", "the two projections $X \\times_S X \\to X$.", "Note that $p_1$ and $p_2$ agree when composed with $\\Delta : X \\to X'$", "and that $X \\to X'$ is a closed immersion defined by a an ideal", "whose square is zero. Moreover there is a short exact sequence", "$$", "0 \\to \\mathcal{J}/\\mathcal{J}^2 \\to \\mathcal{O}_{X'} \\to \\mathcal{O}_X \\to 0", "$$", "and $\\Omega_{X/S} = \\mathcal{J}/\\mathcal{J}^2$. Moreover,", "$\\mathcal{J}/\\mathcal{J}^2$ is generated by the local", "sections $p_1^\\sharp(f) - p_2^\\sharp(f)$ for $f$ a local section of", "$\\mathcal{O}_X$.", "\\medskip\\noindent", "Suppose that $f : X \\to S$ is formally unramified.", "By assumption this means that $p_1 = p_2$ when restricted to any", "affine open $T' \\subset X'$. Hence $p_1 = p_2$. By what was said above", "we conclude that $\\Omega_{X/S} = \\mathcal{J}/\\mathcal{J}^2 = 0$.", "\\medskip\\noindent", "Conversely, suppose that $\\Omega_{X/S} = 0$. Then $X' = X$. Take any pair", "of morphisms $f'_1, f'_2 : T' \\to X$ fitting as dotted arrows in", "the diagram of", "Definition \\ref{definition-formally-unramified}.", "This gives a morphism $(f'_1, f'_2) : T' \\to X \\times_S X$.", "Since $f'_1|_T = f'_2|_T$ and $|T| =|T'|$ we see that the image of $T'$", "under $(f'_1, f'_2)$ is contained in the open $W$ chosen above. Since", "$(f'_1, f'_2)(T) \\subset \\Delta(X)$ and since $T$ is defined by an ideal", "of square zero in $T'$ we see that $(f'_1, f'_2)$ factors through $X'$.", "As $X' = X$ we conclude $f_1' = f'_2$ as desired." ], "refs": [ "morphisms-lemma-differentials-affine", "more-morphisms-definition-formally-unramified" ], "ref_ids": [ 5310, 14108 ] } ], "ref_ids": [] }, { "id": 13696, "type": "theorem", "label": "more-morphisms-lemma-unramified-formally-unramified", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-unramified-formally-unramified", "contents": [ "\\begin{slogan}", "Unramified morphisms are the same as formally unramified morphism that", "are locally of finite type.", "\\end{slogan}", "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is unramified (resp.\\ G-unramified), and", "\\item the morphism $f$ is locally of finite type (resp.\\ locally of finite", "presentation) and formally unramified.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Use Lemma \\ref{lemma-formally-unramified-differentials} and", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-omega-zero}." ], "refs": [ "more-morphisms-lemma-formally-unramified-differentials", "morphisms-lemma-unramified-omega-zero" ], "ref_ids": [ 13695, 5343 ] } ], "ref_ids": [] }, { "id": 13697, "type": "theorem", "label": "more-morphisms-lemma-universal-thickening", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-universal-thickening", "contents": [ "Let $h : Z \\to X$ be a formally unramified morphism of schemes.", "There exists a universal first order thickening $Z \\subset Z'$ of", "$Z$ over $X$." ], "refs": [], "proofs": [ { "contents": [ "During this proof we will say $Z \\subset Z'$ is a universal first order", "thickening of $Z$ over $X$ if it satisfies the condition of the lemma.", "We will construct the universal first order thickening $Z \\subset Z'$ over $X$", "by glueing, starting with the affine case which is", "Algebra, Lemma \\ref{algebra-lemma-universal-thickening}.", "We begin with some general remarks.", "\\medskip\\noindent", "If a universal first order thickening of $Z$ over $X$ exists, then it is unique", "up to unique isomorphism. Moreover, suppose that $V \\subset Z$ and", "$U \\subset X$ are open subschemes such that $h(V) \\subset U$. Let", "$Z \\subset Z'$ be a universal first order thickening of $Z$ over $X$.", "Let $V' \\subset Z'$ be the open subscheme such that $V = Z \\cap V'$.", "Then we claim that $V \\subset V'$ is the universal first order thickening of", "$V$ over $U$. Namely, suppose given any diagram", "$$", "\\xymatrix{", "V \\ar[d]_h & T \\ar[l]^a \\ar[d] \\\\", "U & T' \\ar[l]_b", "}", "$$", "where $T \\subset T'$ is a first order thickening over $U$. By the universal", "property of $Z'$ we obtain $(a, a') : (T \\subset T') \\to (Z \\subset Z')$.", "But since we have equality $|T| = |T'|$ of underlying topological spaces", "we see that $a'(T') \\subset V'$. Hence we may think of $(a, a')$", "as a morphism of thickenings $(a, a') : (T \\subset T') \\to (V \\subset V')$", "over $U$. Uniqueness is clear also. In a completely similar manner one proves", "that if $h(Z) \\subset U$ and $Z \\subset Z'$ is a universal first order", "thickening over $U$, then $Z \\subset Z'$ is a universal first order thickening", "over $X$.", "\\medskip\\noindent", "Before we glue affine pieces let us show that the lemma holds if", "$Z$ and $X$ are affine. Say $X = \\Spec(R)$ and $Z = \\Spec(S)$. By", "Algebra, Lemma \\ref{algebra-lemma-universal-thickening}", "there exists a first order thickening $Z \\subset Z'$ over $X$", "which has the universal property of the lemma for diagrams", "$$", "\\xymatrix{", "Z \\ar[d]_h & T \\ar[l]^a \\ar[d] \\\\", "X & T' \\ar[l]_b", "}", "$$", "where $T, T'$ are affine. Given a general diagram we can choose an affine", "open covering $T' = \\bigcup T'_i$ and we obtain morphisms", "$a'_i : T'_i \\to Z'$ over $X$ such that $a'_i|_{T_i} = a|_{T_i}$.", "By uniqueness we see that $a'_i$ and $a'_j$ agree on any affine open", "of $T'_i \\cap T'_j$. Hence the morphisms $a'_i$ glue to a global morphism", "$a' : T' \\to Z'$ over $X$ as desired. Thus the lemma holds if $X$ and $Z$", "are affine.", "\\medskip\\noindent", "Choose an affine open covering $Z = \\bigcup Z_i$ such that each $Z_i$", "maps into an affine open $U_i$ of $X$. By", "Lemma \\ref{lemma-formally-unramified-on-opens}", "the morphisms $Z_i \\to U_i$ are formally unramified.", "Hence by the affine case we obtain universal first order thickenings", "$Z_i \\subset Z_i'$ over $U_i$. By the general remarks above", "$Z_i \\subset Z_i'$ is also a universal first order thickening of", "$Z_i$ over $X$. Let $Z'_{i, j} \\subset Z'_i$ be the open subscheme", "such that $Z_i \\cap Z_j = Z'_{i, j} \\cap Z_i$. By the general remarks", "we see that both $Z'_{i, j}$ and $Z'_{j, i}$ are universal first", "order thickenings of $Z_i \\cap Z_j$ over $X$. Thus, by", "the first of our general remarks, we see that there is a canonical isomorphism", "$\\varphi_{ij} : Z'_{i, j} \\to Z'_{j, i}$ inducing the identity on", "$Z_i \\cap Z_j$. We claim that these morphisms satisfy the cocycle condition of", "Schemes, Section \\ref{schemes-section-glueing-schemes}.", "(Verification omitted. Hint: Use that $Z'_{i, j} \\cap Z'_{i, k}$ is the", "universal first order thickening of $Z_i \\cap Z_j \\cap Z_k$ which determines", "it up to unique isomorphism by what was said above.)", "Hence we can use the results of", "Schemes, Section \\ref{schemes-section-glueing-schemes}", "to get a first order thickening $Z \\subset Z'$ over $X$ which the property", "that the open subscheme $Z'_i \\subset Z'$ with $Z_i = Z'_i \\cap Z$", "is a universal first order thickening of $Z_i$ over $X$.", "\\medskip\\noindent", "It turns out that this implies formally that $Z'$ is a universal first order", "thickening of $Z$ over $X$. Namely, we have the universal property for any", "diagram", "$$", "\\xymatrix{", "Z \\ar[d]_h & T \\ar[l]^a \\ar[d] \\\\", "X & T' \\ar[l]_b", "}", "$$", "where $a(T)$ is contained in some $Z_i$. Given a general diagram we can", "choose an open covering $T' = \\bigcup T'_i$ such that $a(T_i) \\subset Z_i$.", "We obtain morphisms $a'_i : T'_i \\to Z'$ over $X$ such that", "$a'_i|_{T_i} = a|_{T_i}$. We see that $a'_i$ and $a'_j$ necessarily agree", "on $T'_i \\cap T'_j$ since both $a'_i|_{T'_i \\cap T'_j}$ and", "$a'_j|_{T'_i \\cap T'_j}$ are solutions of the problem of mapping into the", "universal first order thickening $Z'_i \\cap Z'_j$ of $Z_i \\cap Z_j$ over $X$.", "Hence the morphisms $a'_i$ glue to a global morphism", "$a' : T' \\to Z'$ over $X$ as desired. This finishes the proof." ], "refs": [ "algebra-lemma-universal-thickening", "algebra-lemma-universal-thickening", "more-morphisms-lemma-formally-unramified-on-opens" ], "ref_ids": [ 1258, 1258, 13693 ] } ], "ref_ids": [] }, { "id": 13698, "type": "theorem", "label": "more-morphisms-lemma-immersion-universal-thickening", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-immersion-universal-thickening", "contents": [ "Let $i : Z \\to X$ be an immersion of schemes. Then", "\\begin{enumerate}", "\\item $i$ is formally unramified,", "\\item the universal first order thickening of $Z$ over $X$ is the first order", "infinitesimal neighbourhood of $Z$ in $X$ of", "Definition \\ref{definition-first-order-infinitesimal-neighbourhood}, and", "\\item the conormal sheaf of $i$ in the sense of", "Morphisms, Definition \\ref{morphisms-definition-conormal-sheaf}", "agrees with the conormal sheaf of $i$ in the sense of", "Definition \\ref{definition-universal-thickening}.", "\\end{enumerate}" ], "refs": [ "more-morphisms-definition-first-order-infinitesimal-neighbourhood", "morphisms-definition-conormal-sheaf", "more-morphisms-definition-universal-thickening" ], "proofs": [ { "contents": [ "By", "Morphisms, Lemmas \\ref{morphisms-lemma-open-immersion-unramified} and", "\\ref{morphisms-lemma-closed-immersion-unramified}", "an immersion is unramified, hence formally unramified by", "Lemma \\ref{lemma-unramified-formally-unramified}.", "The other assertions follow by combining", "Lemmas \\ref{lemma-first-order-infinitesimal-neighbourhood} and", "\\ref{lemma-infinitesimal-neighbourhood-conormal}", "and the definitions." ], "refs": [ "morphisms-lemma-open-immersion-unramified", "morphisms-lemma-closed-immersion-unramified", "more-morphisms-lemma-unramified-formally-unramified", "more-morphisms-lemma-first-order-infinitesimal-neighbourhood", "more-morphisms-lemma-infinitesimal-neighbourhood-conormal" ], "ref_ids": [ 5348, 5349, 13696, 13688, 13689 ] } ], "ref_ids": [ 14107, 5562, 14109 ] }, { "id": 13699, "type": "theorem", "label": "more-morphisms-lemma-universal-thickening-unramified", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-universal-thickening-unramified", "contents": [ "Let $Z \\to X$ be a formally unramified morphism of schemes.", "Then the universal first order thickening $Z'$ is formally", "unramified over $X$." ], "refs": [], "proofs": [ { "contents": [ "There are two proofs. The first is to show that $\\Omega_{Z'/X} = 0$", "by working affine locally and applying", "Algebra, Lemma \\ref{algebra-lemma-differentials-universal-thickening}.", "Then", "Lemma \\ref{lemma-formally-unramified-differentials}", "implies what we want.", "The second is a direct argument as follows.", "\\medskip\\noindent", "Let $T \\subset T'$ be a first order thickening. Let", "$$", "\\xymatrix{", "Z' \\ar[d] & T \\ar[l]^c \\ar[d] \\\\", "X & T' \\ar[l] \\ar[lu]^{a, b}", "}", "$$", "be a commutative diagram. Consider two morphisms $a, b : T' \\to Z'$", "fitting into the diagram. Set $T_0 = c^{-1}(Z) \\subset T$ and", "$T'_a = a^{-1}(Z)$ (scheme theoretically).", "Since $Z'$ is a first order thickening of $Z$, we see that $T'$", "is a first order thickening of $T'_a$. Moreover, since $c = a|_T$ we see that", "$T_0 = T \\cap T'_a$ (scheme theoretically). As $T'$ is a first order", "thickening of $T$ it follows that $T'_a$", "is a first order thickening of $T_0$. Now $a|_{T'_a}$ and $b|_{T'_a}$", "are morphisms of $T'_a$ into $Z'$ over $X$ which agree on $T_0$ as", "morphisms into $Z$. Hence by the universal property of $Z'$ we conclude that", "$a|_{T'_a} = b|_{T'_a}$. Thus $a$ and $b$ are morphism from", "the first order thickening $T'$ of $T'_a$ whose restrictions to", "$T'_a$ agree as morphisms into $Z$. Thus using the universal property of", "$Z'$ once more we conclude that $a = b$. In other words, the defining", "property of a formally unramified morphism holds for $Z' \\to X$ as desired." ], "refs": [ "algebra-lemma-differentials-universal-thickening", "more-morphisms-lemma-formally-unramified-differentials" ], "ref_ids": [ 1261, 13695 ] } ], "ref_ids": [] }, { "id": 13700, "type": "theorem", "label": "more-morphisms-lemma-universal-thickening-functorial", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-universal-thickening-functorial", "contents": [ "Consider a commutative diagram of schemes", "$$", "\\xymatrix{", "Z \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\", "W \\ar[r]^{h'} & Y", "}", "$$", "with $h$ and $h'$ formally unramified. Let $Z \\subset Z'$ be the universal", "first order thickening of $Z$ over $X$. Let $W \\subset W'$ be the universal", "first order thickening of $W$ over $Y$. There exists a canonical morphism", "$(f, f') : (Z, Z') \\to (W, W')$ of thickenings over $Y$ which fits into", "the following commutative diagram", "$$", "\\xymatrix{", "& & & Z' \\ar[ld] \\ar[d]^{f'} \\\\", "Z \\ar[rr] \\ar[d]_f \\ar[rrru] & & X \\ar[d] & W' \\ar[ld] \\\\", "W \\ar[rrru]|!{[rr];[rruu]}\\hole \\ar[rr] & & Y", "}", "$$", "In particular the morphism $(f, f')$ of thickenings induces a morphism", "of conormal sheaves $f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$." ], "refs": [], "proofs": [ { "contents": [ "The first assertion is clear from the universal property of $W'$.", "The induced map on conormal sheaves is the map of", "Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial}", "applied to $(Z \\subset Z') \\to (W \\subset W')$." ], "refs": [ "morphisms-lemma-conormal-functorial" ], "ref_ids": [ 5304 ] } ], "ref_ids": [] }, { "id": 13701, "type": "theorem", "label": "more-morphisms-lemma-universal-thickening-fibre-product", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-universal-thickening-fibre-product", "contents": [ "Let", "$$", "\\xymatrix{", "Z \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\", "W \\ar[r]^{h'} & Y", "}", "$$", "be a fibre product diagram in the category of schemes with", "$h'$ formally unramified. Then $h$ is formally unramified and if", "$W \\subset W'$ is the universal first order thickening of $W$ over $Y$,", "then $Z = X \\times_Y W \\subset X \\times_Y W'$ is the universal", "first order thickening of $Z$ over $X$. In particular the canonical map", "$f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$ of", "Lemma \\ref{lemma-universal-thickening-functorial}", "is surjective." ], "refs": [ "more-morphisms-lemma-universal-thickening-functorial" ], "proofs": [ { "contents": [ "The morphism $h$ is formally unramified by", "Lemma \\ref{lemma-base-change-formally-unramified}.", "It is clear that $X \\times_Y W'$ is a first order thickening.", "It is straightforward to check that it has the universal property", "because $W'$ has the universal property (by mapping properties of", "fibre products). See", "Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial-flat}", "for why this implies that the map of conormal sheaves is surjective." ], "refs": [ "more-morphisms-lemma-base-change-formally-unramified", "morphisms-lemma-conormal-functorial-flat" ], "ref_ids": [ 13692, 5305 ] } ], "ref_ids": [ 13700 ] }, { "id": 13702, "type": "theorem", "label": "more-morphisms-lemma-universal-thickening-fibre-product-flat", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-universal-thickening-fibre-product-flat", "contents": [ "Let", "$$", "\\xymatrix{", "Z \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\", "W \\ar[r]^{h'} & Y", "}", "$$", "be a fibre product diagram in the category of schemes with", "$h'$ formally unramified and $g$ flat. In this case the corresponding", "map $Z' \\to W'$ of universal first order thickenings is flat, and", "$f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Flatness is preserved under base change, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat}.", "Hence the first statement follows from the description of", "$W'$ in Lemma \\ref{lemma-universal-thickening-fibre-product}.", "It is clear that $X \\times_Y W'$ is a first order thickening.", "It is straightforward to check that it has the universal property", "because $W'$ has the universal property (by mapping properties of", "fibre products). See", "Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial-flat}", "for why this implies that the map of conormal sheaves is an isomorphism." ], "refs": [ "morphisms-lemma-base-change-flat", "more-morphisms-lemma-universal-thickening-fibre-product", "morphisms-lemma-conormal-functorial-flat" ], "ref_ids": [ 5265, 13701, 5305 ] } ], "ref_ids": [] }, { "id": 13703, "type": "theorem", "label": "more-morphisms-lemma-universal-thickening-localize", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-universal-thickening-localize", "contents": [ "Taking the universal first order thickenings commutes with taking opens.", "More precisely, let $h : Z \\to X$ be a formally unramified morphism of schemes.", "Let $V \\subset Z$, $U \\subset X$ be opens such that $h(V) \\subset U$.", "Let $Z'$ be the universal first order thickening of $Z$ over $X$.", "Then $h|_V : V \\to U$ is formally unramified and the universal first", "order thickening of $V$ over $U$ is the open subscheme $V' \\subset Z'$", "such that $V = Z \\cap V'$. In particular,", "$\\mathcal{C}_{Z/X}|_V = \\mathcal{C}_{V/U}$." ], "refs": [], "proofs": [ { "contents": [ "The first statement is", "Lemma \\ref{lemma-formally-unramified-on-opens}.", "The compatibility of universal thickenings can be deduced from the proof of", "Lemma \\ref{lemma-universal-thickening},", "or from", "Algebra, Lemma \\ref{algebra-lemma-universal-thickening-localize}", "or deduced from", "Lemma \\ref{lemma-universal-thickening-fibre-product-flat}." ], "refs": [ "more-morphisms-lemma-formally-unramified-on-opens", "more-morphisms-lemma-universal-thickening", "algebra-lemma-universal-thickening-localize", "more-morphisms-lemma-universal-thickening-fibre-product-flat" ], "ref_ids": [ 13693, 13697, 1260, 13702 ] } ], "ref_ids": [] }, { "id": 13704, "type": "theorem", "label": "more-morphisms-lemma-differentials-universally-unramified", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-differentials-universally-unramified", "contents": [ "Let $h : Z \\to X$ be a formally unramified morphism of schemes over $S$.", "Let $Z \\subset Z'$ be the universal first order thickening of $Z$", "over $X$ with structure morphism $h' : Z' \\to X$. The canonical map", "$$", "c_{h'} : (h')^*\\Omega_{X/S} \\longrightarrow \\Omega_{Z'/S}", "$$", "induces an isomorphism", "$h^*\\Omega_{X/S} \\to \\Omega_{Z'/S} \\otimes \\mathcal{O}_Z$." ], "refs": [], "proofs": [ { "contents": [ "The map $c_{h'}$ is the map defined in", "Morphisms, Lemma \\ref{morphisms-lemma-functoriality-differentials}.", "If $i : Z \\to Z'$ is the given closed immersion, then", "$i^*c_{h'}$ is a map", "$h^*\\Omega_{X/S} \\to \\Omega_{Z'/S} \\otimes \\mathcal{O}_Z$.", "Checking that it is an isomorphism reduces to the affine case", "by localization, see", "Lemma \\ref{lemma-universal-thickening-localize}", "and", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-restrict-open}.", "In this case the result is", "Algebra, Lemma \\ref{algebra-lemma-differentials-universal-thickening}." ], "refs": [ "morphisms-lemma-functoriality-differentials", "more-morphisms-lemma-universal-thickening-localize", "morphisms-lemma-differentials-restrict-open", "algebra-lemma-differentials-universal-thickening" ], "ref_ids": [ 5312, 13703, 5308, 1261 ] } ], "ref_ids": [] }, { "id": 13705, "type": "theorem", "label": "more-morphisms-lemma-universally-unramified-differentials-sequence", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-universally-unramified-differentials-sequence", "contents": [ "Let $h : Z \\to X$ be a formally unramified morphism of schemes over $S$.", "There is a canonical exact sequence", "$$", "\\mathcal{C}_{Z/X} \\to h^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0.", "$$", "The first arrow is induced by $\\text{d}_{Z'/S}$ where", "$Z'$ is the universal first order neighbourhood of $Z$ over $X$." ], "refs": [], "proofs": [ { "contents": [ "We know that there is a canonical exact sequence", "$$", "\\mathcal{C}_{Z/Z'} \\to", "\\Omega_{Z'/S} \\otimes \\mathcal{O}_Z \\to", "\\Omega_{Z/S} \\to 0.", "$$", "see", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-relative-immersion}.", "Hence the result follows on applying", "Lemma \\ref{lemma-differentials-universally-unramified}." ], "refs": [ "morphisms-lemma-differentials-relative-immersion", "more-morphisms-lemma-differentials-universally-unramified" ], "ref_ids": [ 5319, 13704 ] } ], "ref_ids": [] }, { "id": 13706, "type": "theorem", "label": "more-morphisms-lemma-two-unramified-morphisms", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-two-unramified-morphisms", "contents": [ "Let", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[rd]_j & X \\ar[d] \\\\", "& Y", "}", "$$", "be a commutative diagram of schemes where $i$ and $j$ are formally", "unramified. Then there is a canonical exact sequence", "$$", "\\mathcal{C}_{Z/Y} \\to", "\\mathcal{C}_{Z/X} \\to", "i^*\\Omega_{X/Y} \\to 0", "$$", "where the first arrow comes from", "Lemma \\ref{lemma-universal-thickening-functorial}", "and the second from", "Lemma \\ref{lemma-universally-unramified-differentials-sequence}." ], "refs": [ "more-morphisms-lemma-universal-thickening-functorial", "more-morphisms-lemma-universally-unramified-differentials-sequence" ], "proofs": [ { "contents": [ "Denote $Z \\to Z'$ the universal first order thickening of $Z$ over $X$.", "Denote $Z \\to Z''$ the universal first order thickening of $Z$ over $Y$.", "By", "Lemma \\ref{lemma-universally-unramified-differentials-sequence}", "here is a canonical morphism $Z' \\to Z''$ so that we have a commutative", "diagram", "$$", "\\xymatrix{", "Z \\ar[r]_{i'} \\ar[rd]_{j'} & Z' \\ar[r] \\ar[d] & X \\ar[d] \\\\", "& Z'' \\ar[r] & Y", "}", "$$", "Apply", "Morphisms, Lemma \\ref{morphisms-lemma-two-immersions}", "to the left triangle to get an exact sequence", "$$", "\\mathcal{C}_{Z/Z''} \\to", "\\mathcal{C}_{Z/Z'} \\to", "(i')^*\\Omega_{Z'/Z''} \\to 0", "$$", "As $Z''$ is formally unramified over $Y$ (see", "Lemma \\ref{lemma-universal-thickening-unramified})", "we have", "$\\Omega_{Z'/Z''} = \\Omega_{Z/Y}$ (by combining", "Lemma \\ref{lemma-formally-unramified-differentials}", "and", "Morphisms, Lemma \\ref{morphisms-lemma-triangle-differentials}).", "Then we have $(i')^*\\Omega_{Z'/Y} = i^*\\Omega_{X/Y}$ by", "Lemma \\ref{lemma-differentials-universally-unramified}." ], "refs": [ "more-morphisms-lemma-universally-unramified-differentials-sequence", "morphisms-lemma-two-immersions", "more-morphisms-lemma-universal-thickening-unramified", "more-morphisms-lemma-formally-unramified-differentials", "morphisms-lemma-triangle-differentials", "more-morphisms-lemma-differentials-universally-unramified" ], "ref_ids": [ 13705, 5321, 13699, 13695, 5313, 13704 ] } ], "ref_ids": [ 13700, 13705 ] }, { "id": 13707, "type": "theorem", "label": "more-morphisms-lemma-transitivity-conormal", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-transitivity-conormal", "contents": [ "Let $Z \\to Y \\to X$ be formally unramified morphisms of schemes.", "\\begin{enumerate}", "\\item If $Z \\subset Z'$ is the universal first order thickening of $Z$", "over $X$ and $Y \\subset Y'$ is the universal first order thickening of $Y$", "over $X$, then there is a morphism $Z' \\to Y'$ and $Y \\times_{Y'} Z'$ is", "the universal first order thickening of $Z$ over $Y$.", "\\item There is a canonical exact sequence", "$$", "i^*\\mathcal{C}_{Y/X} \\to", "\\mathcal{C}_{Z/X} \\to", "\\mathcal{C}_{Z/Y} \\to 0", "$$", "where the maps come from", "Lemma \\ref{lemma-universal-thickening-functorial}", "and $i : Z \\to Y$ is the first morphism.", "\\end{enumerate}" ], "refs": [ "more-morphisms-lemma-universal-thickening-functorial" ], "proofs": [ { "contents": [ "The map $h : Z' \\to Y'$ in (1) comes from", "Lemma \\ref{lemma-universal-thickening-functorial}.", "The assertion that $Y \\times_{Y'} Z'$ is the universal first order", "thickening of $Z$ over $Y$ is clear from the universal properties", "of $Z'$ and $Y'$. By", "Morphisms, Lemma \\ref{morphisms-lemma-transitivity-conormal}", "we have an exact sequence", "$$", "(i')^*\\mathcal{C}_{Y \\times_{Y'} Z'/Z'} \\to", "\\mathcal{C}_{Z/Z'} \\to", "\\mathcal{C}_{Z/Y \\times_{Y'} Z'} \\to 0", "$$", "where $i' : Z \\to Y \\times_{Y'} Z'$ is the given morphism. By", "Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial-flat}", "there exists a surjection", "$h^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{Y \\times_{Y'} Z'/Z'}$.", "Combined with the equalities", "$\\mathcal{C}_{Y/Y'} = \\mathcal{C}_{Y/X}$,", "$\\mathcal{C}_{Z/Z'} = \\mathcal{C}_{Z/X}$, and", "$\\mathcal{C}_{Z/Y \\times_{Y'} Z'} = \\mathcal{C}_{Z/Y}$", "this proves the lemma." ], "refs": [ "more-morphisms-lemma-universal-thickening-functorial", "morphisms-lemma-transitivity-conormal", "morphisms-lemma-conormal-functorial-flat" ], "ref_ids": [ 13700, 5306, 5305 ] } ], "ref_ids": [ 13700 ] }, { "id": 13708, "type": "theorem", "label": "more-morphisms-lemma-formally-etale-not-affine", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-formally-etale-not-affine", "contents": [ "If $f : X \\to S$ is a formally \\'etale morphism, then given", "any solid commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\", "S & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "where $T \\subset T'$ is a first order thickening of schemes over $S$", "there exists exactly one dotted arrow making the diagram commute.", "In other words, in", "Definition \\ref{definition-formally-etale}", "the condition that $T$ be affine may be dropped." ], "refs": [ "more-morphisms-definition-formally-etale" ], "proofs": [ { "contents": [ "Let $T' = \\bigcup T'_i$ be an affine open covering, and let", "$T_i = T \\cap T'_i$. Then we get morphisms $a'_i : T'_i \\to X$ fitting", "into the diagram. By uniqueness we see that $a'_i$ and $a'_j$ agree on", "any affine open subscheme of $T'_i \\cap T'_j$. Hence $a'_i$ and", "$a'_j$ agree on $T'_i \\cap T'_j$. Thus we see that the morphisms $a'_i$", "glue to a global morphism $a' : T' \\to X$. The uniqueness of", "$a'$ we have seen in", "Lemma \\ref{lemma-formally-unramified-not-affine}." ], "refs": [ "more-morphisms-lemma-formally-unramified-not-affine" ], "ref_ids": [ 13690 ] } ], "ref_ids": [ 14110 ] }, { "id": 13709, "type": "theorem", "label": "more-morphisms-lemma-composition-formally-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-composition-formally-etale", "contents": [ "A composition of formally \\'etale morphisms is formally \\'etale." ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13710, "type": "theorem", "label": "more-morphisms-lemma-base-change-formally-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-formally-etale", "contents": [ "A base change of a formally \\'etale morphism is formally \\'etale." ], "refs": [], "proofs": [ { "contents": [ "This is formal." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13711, "type": "theorem", "label": "more-morphisms-lemma-formally-etale-on-opens", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-formally-etale-on-opens", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $U \\subset X$ and $V \\subset S$ be open subschemes such that", "$f(U) \\subset V$. If $f$ is formally \\'etale, so is $f|_U : U \\to V$." ], "refs": [], "proofs": [ { "contents": [ "Consider a solid diagram", "$$", "\\xymatrix{", "U \\ar[d]_{f|_U} & T \\ar[d]^i \\ar[l]^a \\\\", "V & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "as in Definition \\ref{definition-formally-etale}. If $f$ is formally", "ramified, then there exists exactly one $S$-morphism $a' : T' \\to X$", "such that $a'|_T = a$. Since $|T'| = |T|$ we conclude that $a'(T') \\subset U$", "which gives our unique morphism from $T'$ into $U$." ], "refs": [ "more-morphisms-definition-formally-etale" ], "ref_ids": [ 14110 ] } ], "ref_ids": [] }, { "id": 13712, "type": "theorem", "label": "more-morphisms-lemma-characterize-formally-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-characterize-formally-etale", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is formally \\'etale,", "\\item $f$ is formally unramified and the universal first order thickening", "of $X$ over $S$ is equal to $X$,", "\\item $f$ is formally unramified and $\\mathcal{C}_{X/S} = 0$, and", "\\item $\\Omega_{X/S} = 0$ and $\\mathcal{C}_{X/S} = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Actually, the last assertion only make sense because $\\Omega_{X/S} = 0$", "implies that $\\mathcal{C}_{X/S}$ is defined via", "Lemma \\ref{lemma-formally-unramified-differentials}", "and", "Definition \\ref{definition-universal-thickening}.", "This also makes it clear that (3) and (4) are equivalent.", "\\medskip\\noindent", "Either of the assumptions (1), (2), and (3) imply that $f$ is formally", "unramified. Hence we may assume $f$ is formally unramified. The equivalence", "of (1), (2), and (3) follow from the universal property of the universal", "first order thickening $X'$ of $X$ over $S$ and the fact that", "$X = X' \\Leftrightarrow \\mathcal{C}_{X/S} = 0$ since", "after all by definition $\\mathcal{C}_{X/S} = \\mathcal{C}_{X/X'}$", "is the ideal sheaf of $X$ in $X'$." ], "refs": [ "more-morphisms-lemma-formally-unramified-differentials", "more-morphisms-definition-universal-thickening" ], "ref_ids": [ 13695, 14109 ] } ], "ref_ids": [] }, { "id": 13713, "type": "theorem", "label": "more-morphisms-lemma-unramified-flat-formally-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-unramified-flat-formally-etale", "contents": [ "An unramified flat morphism is formally \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Say $X \\to S$ is unramified and flat. Then $\\Delta : X \\to X \\times_S X$", "is an open immersion, see", "Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism}.", "We have to show that $\\mathcal{C}_{X/S}$ is zero.", "Consider the two projections $p, q : X \\times_S X \\to X$.", "As $f$ is formally unramified (see", "Lemma \\ref{lemma-unramified-formally-unramified}),", "$q$ is formally unramified (see", "Lemma \\ref{lemma-base-change-formally-unramified}).", "As $f$ is flat, $p$ is flat, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat}.", "Hence $p^*\\mathcal{C}_{X/S} = \\mathcal{C}_q$ by", "Lemma \\ref{lemma-universal-thickening-fibre-product-flat}", "where $\\mathcal{C}_q$ denotes the conormal sheaf of the formally", "unramified morphism $q : X \\times_S X \\to X$.", "But $\\Delta(X) \\subset X \\times_S X$ is an open subscheme", "which maps isomorphically to $X$ via $q$. Hence by", "Lemma \\ref{lemma-universal-thickening-localize}", "we see that $\\mathcal{C}_q|_{\\Delta(X)} = \\mathcal{C}_{X/X} = 0$.", "In other words, the pullback of $\\mathcal{C}_{X/S}$ to $X$ via", "the identity morphism is zero, i.e., $\\mathcal{C}_{X/S} = 0$." ], "refs": [ "morphisms-lemma-diagonal-unramified-morphism", "more-morphisms-lemma-unramified-formally-unramified", "more-morphisms-lemma-base-change-formally-unramified", "morphisms-lemma-base-change-flat", "more-morphisms-lemma-universal-thickening-fibre-product-flat", "more-morphisms-lemma-universal-thickening-localize" ], "ref_ids": [ 5354, 13696, 13692, 5265, 13702, 13703 ] } ], "ref_ids": [] }, { "id": 13714, "type": "theorem", "label": "more-morphisms-lemma-affine-formally-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-affine-formally-etale", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume $X$ and $S$ are affine.", "Then $f$ is formally \\'etale if and only if", "$\\mathcal{O}_S(S) \\to \\mathcal{O}_X(X)$ is a formally \\'etale", "ring map." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions", "(Definition \\ref{definition-formally-etale} and", "Algebra, Definition \\ref{algebra-definition-formally-etale})", "by the equivalence of categories of rings and affine schemes,", "see", "Schemes, Lemma \\ref{schemes-lemma-category-affine-schemes}." ], "refs": [ "more-morphisms-definition-formally-etale", "algebra-definition-formally-etale", "schemes-lemma-category-affine-schemes" ], "ref_ids": [ 14110, 1543, 7656 ] } ], "ref_ids": [] }, { "id": 13715, "type": "theorem", "label": "more-morphisms-lemma-etale-formally-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-etale-formally-etale", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is \\'etale, and", "\\item the morphism $f$ is locally of finite presentation and", "formally \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f$ is \\'etale.", "An \\'etale morphism is locally of finite presentation, flat and unramified, see", "Morphisms, Section \\ref{morphisms-section-etale}.", "Hence $f$ is locally of finite presentation and formally \\'etale, see", "Lemma \\ref{lemma-unramified-flat-formally-etale}.", "\\medskip\\noindent", "Conversely, suppose that $f$ is locally of finite presentation and", "formally \\'etale. Being \\'etale is local in the Zariski topology on", "$X$ and $S$, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-characterize}.", "By", "Lemma \\ref{lemma-formally-etale-on-opens}", "we can cover $X$ by affine opens $U$ which map into affine opens", "$V$ such that $U \\to V$ is formally \\'etale (and of finite presentation, see", "Morphisms,", "Lemma \\ref{morphisms-lemma-locally-finite-presentation-characterize}).", "By", "Lemma \\ref{lemma-affine-formally-etale}", "we see that the ring maps $\\mathcal{O}(V) \\to \\mathcal{O}(U)$ are", "formally \\'etale (and of finite presentation).", "We win by", "Algebra, Lemma \\ref{algebra-lemma-formally-etale-etale}.", "(We will give another proof of this implication when we discuss", "formally smooth morphisms.)" ], "refs": [ "more-morphisms-lemma-unramified-flat-formally-etale", "morphisms-lemma-etale-characterize", "more-morphisms-lemma-formally-etale-on-opens", "morphisms-lemma-locally-finite-presentation-characterize", "more-morphisms-lemma-affine-formally-etale", "algebra-lemma-formally-etale-etale" ], "ref_ids": [ 13713, 5359, 13711, 5238, 13714, 1262 ] } ], "ref_ids": [] }, { "id": 13716, "type": "theorem", "label": "more-morphisms-lemma-difference-derivation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-difference-derivation", "contents": [ "Let $S$ be a scheme.", "Let $X \\subset X'$ and $Y \\subset Y'$ be two first order thickenings", "over $S$. Let $(a, a'), (b, b') : (X \\subset X') \\to (Y \\subset Y')$", "be two morphisms of thickenings over $S$. Assume that", "\\begin{enumerate}", "\\item $a = b$, and", "\\item the two maps $a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$", "(Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial})", "are equal.", "\\end{enumerate}", "Then the map $(a')^\\sharp - (b')^\\sharp$ factors as", "$$", "\\mathcal{O}_{Y'} \\to \\mathcal{O}_Y \\xrightarrow{D}", "a_*\\mathcal{C}_{X/X'} \\to a_*\\mathcal{O}_{X'}", "$$", "where $D$ is an $\\mathcal{O}_S$-derivation." ], "refs": [ "morphisms-lemma-conormal-functorial" ], "proofs": [ { "contents": [ "Instead of working on $Y$ we work on $X$. The advantage is that the pullback", "functor $a^{-1}$ is exact. Using (1) and (2) we obtain a commutative diagram", "with exact rows", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{C}_{X/X'} \\ar[r] &", "\\mathcal{O}_{X'} \\ar[r] &", "\\mathcal{O}_X \\ar[r] & 0 \\\\", "0 \\ar[r] &", "a^{-1}\\mathcal{C}_{Y/Y'} \\ar[r] \\ar[u] &", "a^{-1}\\mathcal{O}_{Y'}", "\\ar[r] \\ar@<1ex>[u]^{(a')^\\sharp} \\ar@<-1ex>[u]_{(b')^\\sharp} &", "a^{-1}\\mathcal{O}_Y \\ar[r] \\ar[u] & 0", "}", "$$", "Now it is a general fact that in such a situation the difference of the", "$\\mathcal{O}_S$-algebra maps $(a')^\\sharp$ and $(b')^\\sharp$ is an", "$\\mathcal{O}_S$-derivation from $a^{-1}\\mathcal{O}_Y$ to $\\mathcal{C}_{X/X'}$.", "By adjointness of the functors $a^{-1}$ and $a_*$ this is the same", "thing as an $\\mathcal{O}_S$-derivation from", "$\\mathcal{O}_Y$ into $a_*\\mathcal{C}_{X/X'}$. Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 5304 ] }, { "id": 13717, "type": "theorem", "label": "more-morphisms-lemma-action-by-derivations", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-action-by-derivations", "contents": [ "Let $S$ be a scheme.", "Let $(a, a') : (X \\subset X') \\to (Y \\subset Y')$", "be a morphism of first order thickenings over $S$.", "Let", "$$", "\\theta : a^*\\Omega_{Y/S} \\to \\mathcal{C}_{X/X'}", "$$", "be an $\\mathcal{O}_X$-linear map. Then there exists a unique morphism of pairs", "$(b, b') : (X \\subset X') \\to (Y \\subset Y')$ such that", "(1) and (2) of", "Lemma \\ref{lemma-difference-derivation}", "hold and the derivation $D$ and $\\theta$ are related by", "Equation (\\ref{equation-D})." ], "refs": [ "more-morphisms-lemma-difference-derivation" ], "proofs": [ { "contents": [ "We simply set $b = a$ and we define $(b')^\\sharp$ to be the map", "$$", "(a')^\\sharp + D : a^{-1}\\mathcal{O}_{Y'} \\to \\mathcal{O}_{X'}", "$$", "where $D$ is as in Equation (\\ref{equation-D}). We omit the verification", "that $(b')^\\sharp$ is a map of sheaves of $\\mathcal{O}_S$-algebras and", "that (1) and (2) of", "Lemma \\ref{lemma-difference-derivation}", "hold. Equation (\\ref{equation-D}) holds by construction." ], "refs": [ "more-morphisms-lemma-difference-derivation" ], "ref_ids": [ 13716 ] } ], "ref_ids": [ 13716 ] }, { "id": 13718, "type": "theorem", "label": "more-morphisms-lemma-sheaf", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-sheaf", "contents": [ "Let $S$ be a scheme.", "Let $X \\subset X'$ and $Y \\subset Y'$ be first order thickenings", "over $S$. Assume given a morphism $a : X \\to Y$ and a map", "$A : a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$ of", "$\\mathcal{O}_X$-modules. For an open subscheme $U' \\subset X'$", "consider morphisms $a' : U' \\to Y'$ such that", "\\begin{enumerate}", "\\item $a'$ is a morphism over $S$,", "\\item $a'|_U = a|_U$, and", "\\item the induced map", "$a^*\\mathcal{C}_{Y/Y'}|_U \\to \\mathcal{C}_{X/X'}|_U$", "is the restriction of $A$ to $U$.", "\\end{enumerate}", "Here $U = X \\cap U'$. Then the rule", "\\begin{equation}", "\\label{equation-sheaf}", "U' \\mapsto", "\\{a' : U' \\to Y'\\text{ such that (1), (2), (3) hold.}\\}", "\\end{equation}", "defines a sheaf of sets on $X'$." ], "refs": [], "proofs": [ { "contents": [ "Denote $\\mathcal{F}$ the rule of the lemma.", "The restriction mapping $\\mathcal{F}(U') \\to \\mathcal{F}(V')$ for", "$V' \\subset U' \\subset X'$", "of $\\mathcal{F}$ is really the restriction map $a' \\mapsto a'|_{V'}$.", "With this definition in place it is clear that $\\mathcal{F}$ is a", "sheaf since morphisms are defined locally." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13719, "type": "theorem", "label": "more-morphisms-lemma-action-sheaf", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-action-sheaf", "contents": [ "Same notation and assumptions as in Lemma \\ref{lemma-sheaf}.", "There is an action of the sheaf", "$$", "\\SheafHom_{\\mathcal{O}_X}(a^*\\Omega_{Y/S}, \\mathcal{C}_{X/X'})", "$$", "on the sheaf (\\ref{equation-sheaf}). Moreover, the action", "is simply transitive for any open $U' \\subset X'$ over which the sheaf", "(\\ref{equation-sheaf}) has a section." ], "refs": [ "more-morphisms-lemma-sheaf" ], "proofs": [ { "contents": [ "This is a combination of", "Lemmas \\ref{lemma-difference-derivation},", "\\ref{lemma-action-by-derivations},", "and \\ref{lemma-sheaf}." ], "refs": [ "more-morphisms-lemma-difference-derivation", "more-morphisms-lemma-action-by-derivations", "more-morphisms-lemma-sheaf" ], "ref_ids": [ 13716, 13717, 13718 ] } ], "ref_ids": [ 13718 ] }, { "id": 13720, "type": "theorem", "label": "more-morphisms-lemma-omega-deformation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-omega-deformation", "contents": [ "Let $S$ be a scheme. Let $X \\subset X'$ be a first order thickening over", "$S$. Let $Y$ be a scheme over $S$. Let", "$a', b' : X' \\to Y$ be two morphisms over $S$ with", "$a = a'|_X = b'|_X$. This gives rise to a commutative diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d]_a & X' \\ar[d]^{(b', a')} \\\\", "Y \\ar[r]^-{\\Delta_{Y/S}} & Y \\times_S Y", "}", "$$", "Since the horizontal arrows are immersions with conormal sheaves", "$\\mathcal{C}_{X/X'}$ and $\\Omega_{Y/S}$, by", "Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial},", "we obtain a map $\\theta : a^*\\Omega_{Y/S} \\to \\mathcal{C}_{X/X'}$.", "Then this $\\theta$ and the derivation $D$ of", "Lemma \\ref{lemma-difference-derivation}", "are related by Equation (\\ref{equation-D})." ], "refs": [ "morphisms-lemma-conormal-functorial", "more-morphisms-lemma-difference-derivation" ], "proofs": [ { "contents": [ "Omitted. Hint: The equality may be checked on affine opens where it", "comes from the following computation. If $f$ is a local section of", "$\\mathcal{O}_Y$, then $1 \\otimes f - f \\otimes 1$ is a local section", "of $\\mathcal{C}_{Y/(Y \\times_S Y)}$ corresponding to $\\text{d}_{Y/S}(f)$.", "It is mapped to the local section $(a')^\\sharp(f) - (b')^\\sharp(f) = D(f)$", "of $\\mathcal{C}_{X/X'}$. In other words, $\\theta(\\text{d}_{Y/S}(f)) = D(f)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 5304, 13716 ] }, { "id": 13721, "type": "theorem", "label": "more-morphisms-lemma-sheaf-differentials-etale-localization", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-sheaf-differentials-etale-localization", "contents": [ "Let", "$$", "\\xymatrix{", "X_1 \\ar[d] & X_2 \\ar[l]^f \\ar[d] \\\\", "S_1 & S_2 \\ar[l]", "}", "$$", "be a commutative diagram of schemes with $X_2 \\to X_1$ and $S_2 \\to S_1$", "\\'etale. Then the map $c_f : f^*\\Omega_{X_1/S_1} \\to \\Omega_{X_2/S_2}$ of", "Morphisms, Lemma \\ref{morphisms-lemma-functoriality-differentials}", "is an isomorphism." ], "refs": [ "morphisms-lemma-functoriality-differentials" ], "proofs": [ { "contents": [ "We recall that an \\'etale morphism $U \\to V$ is a smooth morphism", "with $\\Omega_{U/V} = 0$. Using this we see that", "Morphisms, Lemma \\ref{morphisms-lemma-triangle-differentials}", "implies $\\Omega_{X_2/S_2} = \\Omega_{X_2/S_1}$ and", "Morphisms, Lemma \\ref{morphisms-lemma-triangle-differentials-smooth}", "implies that the map $f^*\\Omega_{X_1/S_1} \\to \\Omega_{X_2/S_1}$", "(for the morphism $f$ seen as a morphism over $S_1$)", "is an isomorphism. Hence the lemma follows." ], "refs": [ "morphisms-lemma-triangle-differentials", "morphisms-lemma-triangle-differentials-smooth" ], "ref_ids": [ 5313, 5337 ] } ], "ref_ids": [ 5312 ] }, { "id": 13722, "type": "theorem", "label": "more-morphisms-lemma-action-by-derivations-etale-localization", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-action-by-derivations-etale-localization", "contents": [ "Consider a commutative diagram of first order thickenings", "$$", "\\vcenter{", "\\xymatrix{", "(T_2 \\subset T_2') \\ar[d]_{(h, h')} \\ar[rr]_{(a_2, a_2')} & &", "(X_2 \\subset X_2') \\ar[d]^{(f, f')} \\\\", "(T_1 \\subset T_1') \\ar[rr]^{(a_1, a_1')} & &", "(X_1 \\subset X_1')", "}", "}", "\\quad", "\\begin{matrix}", "\\text{and a commutative} \\\\", "\\text{diagram of schemes}", "\\end{matrix}", "\\quad", "\\vcenter{", "\\xymatrix{", "X_2' \\ar[r] \\ar[d] & S_2 \\ar[d] \\\\", "X_1' \\ar[r] & S_1", "}", "}", "$$", "with $X_2 \\to X_1$ and $S_2 \\to S_1$ \\'etale.", "For any $\\mathcal{O}_{T_1}$-linear map", "$\\theta_1 : a_1^*\\Omega_{X_1/S_1} \\to \\mathcal{C}_{T_1/T'_1}$ let", "$\\theta_2$ be the composition", "$$", "\\xymatrix{", "a_2^*\\Omega_{X_2/S_2} \\ar@{=}[r] &", "h^*a_1^*\\Omega_{X_1/S_1} \\ar[r]^-{h^*\\theta_1} &", "h^*\\mathcal{C}_{T_1/T'_1} \\ar[r] &", "\\mathcal{C}_{T_2/T'_2}", "}", "$$", "(equality sign is explained in the proof). Then the diagram", "$$", "\\xymatrix{", "T_2' \\ar[rr]_{\\theta_2 \\cdot a_2'} \\ar[d] & & X'_2 \\ar[d] \\\\", "T_1' \\ar[rr]^{\\theta_1 \\cdot a_1'} & & X'_1", "}", "$$", "commutes where the actions $\\theta_2 \\cdot a_2'$ and $\\theta_1 \\cdot a_1'$", "are as in Remark \\ref{remark-action-by-derivations}." ], "refs": [ "more-morphisms-remark-action-by-derivations" ], "proofs": [ { "contents": [ "The equality sign comes from the identification", "$f^*\\Omega_{X_1/S_1} = \\Omega_{X_2/S_2}$ of", "Lemma \\ref{lemma-sheaf-differentials-etale-localization}.", "Namely, using this we have", "$a_2^*\\Omega_{X_2/S_2} = a_2^*f^*\\Omega_{X_1/S_1} =", "h^*a_1^*\\Omega_{X_1/S_1}$ because $f \\circ a_2 = a_1 \\circ h$.", "Having said this, the commutativity of the diagram may be checked", "on affine opens. Hence we may assume the schemes in the initial", "big diagram are affine. Thus we obtain commutative diagrams", "$$", "\\vcenter{", "\\xymatrix{", "(B'_2, I_2) & & (A'_2, J_2) \\ar[ll]^{a_2'} \\\\", "(B'_1, I_1) \\ar[u]^{h'} & & (A'_1, J_1) \\ar[ll]_{a_1'} \\ar[u]_{f'}", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "A'_2 & & R_2 \\ar[ll] \\\\", "A'_1 \\ar[u] & & R_1 \\ar[ll] \\ar[u]", "}", "}", "$$", "The notation signifies that $I_1, I_2, J_1, J_2$ are ideals of square", "zero and maps of pairs are ring maps sending ideals into ideals.", "Set $A_1 = A'_1/J_1$, $A_2 = A'_2/J_2$, $B_1 = B'_1/I_1$, and", "$B_2 = B'_2/I_2$. We are given that", "$$", "A_2 \\otimes_{A_1} \\Omega_{A_1/R_1} \\longrightarrow \\Omega_{A_2/R_2}", "$$", "is an isomorphism. Then", "$\\theta_1 : B_1 \\otimes_{A_1} \\Omega_{A_1/R_1} \\to I_1$", "is $B_1$-linear. This gives an $R_1$-derivation", "$D_1 = \\theta_1 \\circ \\text{d}_{A_1/R_1} : A_1 \\to I_1$.", "In a similar way we see that", "$\\theta_2 : B_2 \\otimes_{A_2} \\Omega_{A_2/R_2} \\to I_2$", "gives rise to a $R_2$-derivation", "$D_2 = \\theta_2 \\circ \\text{d}_{A_2/R_2} : A_2 \\to I_2$.", "The construction of $\\theta_2$ implies the following compatibility between", "$\\theta_1$ and $\\theta_2$: for every $x \\in A_1$ we have", "$$", "h'(D_1(x)) = D_2(f'(x))", "$$", "as elements of $I_2$. We may view $D_1$ as a map $A'_1 \\to B'_1$", "using $A'_1 \\to A_1 \\xrightarrow{D_1} I_1 \\to B_1$ similarly", "we may view $D_2$ as a map $A'_2 \\to B'_2$. Then the displayed", "equality holds for $x \\in A'_1$.", "By the construction of the action in", "Lemma \\ref{lemma-action-by-derivations} and", "Remark \\ref{remark-action-by-derivations}", "we know that $\\theta_1 \\cdot a_1'$ corresponds to the ring map", "$a_1' + D_1 : A'_1 \\to B'_1$ and $\\theta_2 \\cdot a_2'$ corresponds", "to the ring map $a_2' + D_2 : A'_2 \\to B'_2$. By the displayed equality", "we obtain that", "$h' \\circ (a_1' + D_1) = (a_2' + D_2) \\circ f'$", "as desired." ], "refs": [ "more-morphisms-lemma-sheaf-differentials-etale-localization", "more-morphisms-lemma-action-by-derivations", "more-morphisms-remark-action-by-derivations" ], "ref_ids": [ 13721, 13717, 14127 ] } ], "ref_ids": [ 14127 ] }, { "id": 13723, "type": "theorem", "label": "more-morphisms-lemma-deform", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-deform", "contents": [ "Let $(f, f') : (X \\subset X') \\to (S \\subset S')$ be a morphism", "of first order thickenings. Assume that $f$ is flat.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $f'$ is flat and $X = S \\times_{S'} X'$, and", "\\item the canonical map $f^*\\mathcal{C}_{S/S'} \\to \\mathcal{C}_{X/X'}$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As the problem is local on $X'$ we may assume that $X, X', S, S'$", "are affine schemes. Say $S' = \\Spec(A')$, $X' = \\Spec(B')$,", "$S = \\Spec(A)$, $X = \\Spec(B)$ with $A = A'/I$ and $B = B'/J$", "for some square zero ideals. Then we obtain the following commutative", "diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "J \\ar[r] &", "B' \\ar[r] &", "B \\ar[r] & 0 \\\\", "0 \\ar[r] &", "I \\ar[r] \\ar[u] &", "A' \\ar[r] \\ar[u] &", "A \\ar[r] \\ar[u] & 0", "}", "$$", "with exact rows. The canonical map of the lemma is the map", "$$", "I \\otimes_A B = I \\otimes_{A'} B' \\longrightarrow J.", "$$", "The assumption that $f$ is flat signifies that $A \\to B$ is flat.", "\\medskip\\noindent", "Assume (1). Then $A' \\to B'$ is flat and $J = IB'$. Flatness implies", "$\\text{Tor}_1^{A'}(B', A) = 0$ (see", "Algebra, Lemma \\ref{algebra-lemma-characterize-flat}).", "This means $I \\otimes_{A'} B' \\to B'$ is injective (see", "Algebra, Remark \\ref{algebra-remark-Tor-ring-mod-ideal}).", "Hence we see that $I \\otimes_A B \\to J$ is an isomorphism.", "\\medskip\\noindent", "Assume (2). Then it follows that $J = IB'$, so that $X = S \\times_{S'} X'$.", "Moreover, we get $\\text{Tor}_1^{A'}(B', A'/I) = 0$ by reversing the", "implications in the previous paragraph. Hence $B'$ is flat over $A'$ by", "Algebra, Lemma \\ref{algebra-lemma-what-does-it-mean}." ], "refs": [ "algebra-lemma-characterize-flat", "algebra-remark-Tor-ring-mod-ideal", "algebra-lemma-what-does-it-mean" ], "ref_ids": [ 786, 1570, 890 ] } ], "ref_ids": [] }, { "id": 13724, "type": "theorem", "label": "more-morphisms-lemma-flatness-morphism-thickenings", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flatness-morphism-thickenings", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\", "& (S \\subset S')", "}", "$$", "of thickenings. Assume", "\\begin{enumerate}", "\\item $X'$ is flat over $S'$,", "\\item $f$ is flat,", "\\item $S \\subset S'$ is a finite order thickening, and", "\\item $X = S \\times_{S'} X'$ and $Y = S \\times_{S'} Y'$.", "\\end{enumerate}", "Then $f'$ is flat and $Y'$ is flat over $S'$ at all points in", "the image of $f'$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Algebra, Lemma \\ref{algebra-lemma-criterion-flatness-fibre-nilpotent}." ], "refs": [ "algebra-lemma-criterion-flatness-fibre-nilpotent" ], "ref_ids": [ 907 ] } ], "ref_ids": [] }, { "id": 13725, "type": "theorem", "label": "more-morphisms-lemma-deform-property", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-deform-property", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\", "& (S \\subset S')", "}", "$$", "of thickenings. Assume $S \\subset S'$ is a finite order thickening,", "$X'$ flat over $S'$, $X = S \\times_{S'} X'$, and", "$Y = S \\times_{S'} Y'$. Then", "\\begin{enumerate}", "\\item $f$ is flat if and only if $f'$ is flat,", "\\label{item-flat}", "\\item $f$ is an isomorphism if and only if $f'$ is an isomorphism,", "\\label{item-isomorphism}", "\\item $f$ is an open immersion if and only if $f'$ is an open immersion,", "\\label{item-open-immersion}", "\\item $f$ is quasi-compact if and only if $f'$ is quasi-compact,", "\\label{item-quasi-compact}", "\\item $f$ is universally closed if and only if $f'$ is universally closed,", "\\label{item-universally-closed}", "\\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,", "\\label{item-separated}", "\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,", "\\label{item-monomorphism}", "\\item $f$ is surjective if and only if $f'$ is surjective,", "\\label{item-surjective}", "\\item $f$ is universally injective if and only if $f'$ is universally injective,", "\\label{item-universally-injective}", "\\item $f$ is affine if and only if $f'$ is affine,", "\\label{item-affine}", "\\item", "\\label{item-finite-type}", "$f$ is locally of finite type if and only if $f'$ is locally of finite type,", "\\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,", "\\label{item-quasi-finite}", "\\item", "\\label{item-finite-presentation}", "$f$ is locally of finite presentation if and only if $f'$ is locally of", "finite presentation,", "\\item", "\\label{item-relative-dimension-d}", "$f$ is locally of finite type of relative dimension $d$ if and only if", "$f'$ is locally of finite type of relative dimension $d$,", "\\item $f$ is universally open if and only if $f'$ is universally open,", "\\label{item-universally-open}", "\\item $f$ is syntomic if and only if $f'$ is syntomic,", "\\label{item-syntomic}", "\\item $f$ is smooth if and only if $f'$ is smooth,", "\\label{item-smooth}", "\\item $f$ is unramified if and only if $f'$ is unramified,", "\\label{item-unramified}", "\\item $f$ is \\'etale if and only if $f'$ is \\'etale,", "\\label{item-etale}", "\\item $f$ is proper if and only if $f'$ is proper,", "\\label{item-proper}", "\\item $f$ is integral if and only if $f'$ is integral,", "\\label{item-integral}", "\\item $f$ is finite if and only if $f'$ is finite,", "\\label{item-finite}", "\\item", "\\label{item-finite-locally-free}", "$f$ is finite locally free (of rank $d$) if and only if $f'$", "is finite locally free (of rank $d$), and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The assumptions on $X$ and $Y$ mean that $f$ is the base change of", "$f'$ by $X \\to X'$.", "The properties $\\mathcal{P}$ listed in (1) -- (23) above are all stable", "under base change, hence if $f'$ has property $\\mathcal{P}$, then so", "does $f$. See", "Schemes, Lemmas \\ref{schemes-lemma-base-change-immersion},", "\\ref{schemes-lemma-quasi-compact-preserved-base-change},", "\\ref{schemes-lemma-separated-permanence}, and", "\\ref{schemes-lemma-base-change-monomorphism}", "and", "Morphisms, Lemmas", "\\ref{morphisms-lemma-base-change-surjective},", "\\ref{morphisms-lemma-base-change-universally-injective},", "\\ref{morphisms-lemma-base-change-affine},", "\\ref{morphisms-lemma-base-change-finite-type},", "\\ref{morphisms-lemma-base-change-quasi-finite},", "\\ref{morphisms-lemma-base-change-finite-presentation},", "\\ref{morphisms-lemma-base-change-relative-dimension-d},", "\\ref{morphisms-lemma-base-change-syntomic},", "\\ref{morphisms-lemma-base-change-smooth},", "\\ref{morphisms-lemma-base-change-unramified},", "\\ref{morphisms-lemma-base-change-etale},", "\\ref{morphisms-lemma-base-change-proper},", "\\ref{morphisms-lemma-base-change-finite}, and", "\\ref{morphisms-lemma-base-change-finite-locally-free}.", "\\medskip\\noindent", "The interesting direction in each case is therefore to assume", "that $f$ has the property and deduce that $f'$ has it too.", "By induction on the order of the thickening we may", "assume that $S \\subset S'$ is a first order thickening, see", "discussion immediately following", "Definition \\ref{definition-thickening}.", "We make a couple of general remarks which we will use without further", "mention in the arguments below.", "(I) Let $W' \\subset S'$ be an affine open and let $U' \\subset X'$", "and $V' \\subset Y'$ be affine opens lying over $W'$ with $f'(U') \\subset V'$.", "Let $W' = \\Spec(R')$ and denote $I \\subset R'$ be the ideal", "defining the closed subscheme $W' \\cap S$. Say $U' = \\Spec(B')$", "and $V' = \\Spec(A')$. Then we get a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "IB' \\ar[r] &", "B' \\ar[r] &", "B \\ar[r] & 0 \\\\", "0 \\ar[r] &", "IA' \\ar[r] \\ar[u] &", "A' \\ar[r] \\ar[u] &", "A \\ar[r] \\ar[u] & 0", "}", "$$", "with exact rows. Moreover $IB' \\cong I \\otimes_R B$, see proof of", "Lemma \\ref{lemma-deform}.", "(II) The morphisms $X \\to X'$ and $Y \\to Y'$ are universal homeomorphisms.", "Hence the topology of the maps $f$ and $f'$ (after any base change)", "is identical. (III) If $f$ is flat, then $f'$ is flat and", "$Y' \\to S'$ is flat at every point in the image of $f'$, see", "Lemma \\ref{lemma-flatness-morphism-thickenings}.", "\\medskip\\noindent", "Ad (\\ref{item-flat}). This is general remark (III).", "\\medskip\\noindent", "Ad (\\ref{item-isomorphism}). Assume $f$ is an isomorphism.", "By (III) we see that $Y' \\to S'$ is flat. Choose an", "affine open $V' \\subset Y'$ and set $U' = (f')^{-1}(V')$. Then", "$V = Y \\cap V'$ is affine which implies that", "$V \\cong f^{-1}(V) = U = Y \\times_{Y'} U'$ is affine. By", "Lemma \\ref{lemma-thickening-affine-scheme}", "we see that $U'$ is affine. Thus we have a diagram as in the", "general remark (I) and moreover $IA \\cong I \\otimes_R A$ because", "$R' \\to A'$ is flat. Then", "$IB' \\cong I \\otimes_R B \\cong I \\otimes_R A \\cong IA'$", "and $A \\cong B$. By the exactness of the rows", "in the diagram above we see that $A' \\cong B'$,", "i.e., $U' \\cong V'$. Thus $f'$ is an isomorphism.", "\\medskip\\noindent", "Ad (\\ref{item-open-immersion}). Assume $f$ is an open immersion.", "Then $f$ is an isomorphism of $X$ with an open subscheme $V \\subset Y$.", "Let $V' \\subset Y'$ be the open subscheme whose underlying topological", "space is $V$. Then $f'$ is a map from $X'$ to $V'$ which is an isomorphism by", "(\\ref{item-isomorphism}). Hence $f'$ is an open immersion.", "\\medskip\\noindent", "Ad (\\ref{item-quasi-compact}). Immediate from remark (II). See also", "Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-universally-closed}). Immediate from remark (II). See also", "Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-separated}). Note that", "$X \\times_Y X = Y \\times_{Y'} (X' \\times_{Y'} X')$ so that", "$X' \\times_{Y'} X'$ is a thickening of $X \\times_Y X$.", "Hence the topology of the maps $\\Delta_{X/Y}$ and $\\Delta_{X'/Y'}$", "matches and we win. See also", "Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-monomorphism}). Assume $f$ is a monomorphism.", "Consider the diagonal morphism $\\Delta_{X'/Y'} : X' \\to X' \\times_{Y'} X'$.", "The base change of $\\Delta_{X'/Y'}$ by $S \\to S'$ is $\\Delta_{X/Y}$", "which is an isomorphism by assumption. By (\\ref{item-isomorphism})", "we conclude that $\\Delta_{X'/Y'}$ is an isomorphism.", "\\medskip\\noindent", "Ad (\\ref{item-surjective}). This is clear. See also", "Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-universally-injective}). Immediate from remark (II). See also", "Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-affine}). Assume $f$ is affine. Choose an", "affine open $V' \\subset Y'$ and set $U' = (f')^{-1}(V')$.", "Then $V = Y \\cap V'$ is affine which implies that", "$U = Y \\times_{Y'} U'$ is affine. By", "Lemma \\ref{lemma-thickening-affine-scheme}", "we see that $U'$ is affine. Hence $f'$ is affine. See also", "Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-finite-type}). Via remark (I) comes down to proving $A' \\to B'$", "is of finite type if $A \\to B$ is of finite type. Suppose that", "$x_1, \\ldots, x_n \\in B'$ are elements whose images in $B$ generate $B$", "as an $A$-algebra. Then $A'[x_1, \\ldots, x_n] \\to B$ is surjective", "as both $A'[x_1, \\ldots, x_n] \\to B$ is surjective and", "$I \\otimes_R A[x_1, \\ldots, x_n] \\to I \\otimes_R B$ is surjective. See also", "Lemma \\ref{lemma-thicken-property-morphisms-cartesian}", "for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-quasi-finite}). Follows from (\\ref{item-finite-type}) and that", "quasi-finiteness of a morphism of finite type can be checked on fibres, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}.", "See also Lemma \\ref{lemma-thicken-property-morphisms-cartesian}", "for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-finite-presentation}). Via remark (I) comes down to proving", "$A' \\to B'$ is of finite presentation if $A \\to B$ is of finite presentation.", "We may assume that $B' = A'[x_1, \\ldots, x_n]/K'$ for some ideal $K'$ by", "(\\ref{item-finite-type}). We get a short exact sequence", "$$", "0 \\to K' \\to A'[x_1, \\ldots, x_n] \\to B' \\to 0", "$$", "As $B'$ is flat over $R'$ we see that $K' \\otimes_{R'} R$ is the kernel of", "the surjection $A[x_1, \\ldots, x_n] \\to B$. By assumption on $A \\to B$ there", "exist finitely many $f'_1, \\ldots, f'_m \\in K'$ whose images in", "$A[x_1, \\ldots, x_n]$ generate this kernel. Since $I$ is nilpotent we see", "that $f'_1, \\ldots, f'_m$ generate $K'$ by Nakayama's lemma, see", "Algebra, Lemma \\ref{algebra-lemma-NAK}.", "\\medskip\\noindent", "Ad (\\ref{item-relative-dimension-d}). Follows from (\\ref{item-finite-type})", "and general remark (II). See also", "Lemma \\ref{lemma-thicken-property-morphisms-cartesian}", "for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-universally-open}). Immediate from general remark (II). See also", "Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-syntomic}). Assume $f$ is syntomic. By", "(\\ref{item-finite-presentation}) $f'$ is locally of finite presentation,", "by general remark (III) $f'$ is flat and the fibres of $f'$ are the fibres", "of $f$. Hence $f'$ is syntomic by", "Morphisms, Lemma \\ref{morphisms-lemma-syntomic-flat-fibres}.", "\\medskip\\noindent", "Ad (\\ref{item-smooth}). Assume $f$ is smooth. By", "(\\ref{item-finite-presentation}) $f'$ is locally of finite presentation,", "by general remark (III) $f'$ is flat, and the fibres of $f'$ are the", "fibres of $f$. Hence $f'$ is smooth by", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-flat-smooth-fibres}.", "\\medskip\\noindent", "Ad (\\ref{item-unramified}). Assume $f$ unramified. By", "(\\ref{item-finite-type}) $f'$ is locally of finite type", "and the fibres of $f'$ are the fibres of $f$.", "Hence $f'$ is unramified by", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-etale-fibres}. See also", "Lemma \\ref{lemma-thicken-property-morphisms-cartesian}", "for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-etale}). Assume $f$ \\'etale. By", "(\\ref{item-finite-presentation}) $f'$ is locally of finite presentation,", "by general remark (III) $f'$ is flat, and the fibres of $f'$ are the fibres", "of $f$. Hence $f'$ is \\'etale by", "Morphisms, Lemma \\ref{morphisms-lemma-etale-flat-etale-fibres}.", "\\medskip\\noindent", "Ad (\\ref{item-proper}). This follows from a combination of", "(\\ref{item-separated}), (\\ref{item-finite-type}), (\\ref{item-quasi-compact}),", "and (\\ref{item-universally-closed}). See also", "Lemma \\ref{lemma-thicken-property-morphisms-cartesian}", "for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-integral}). Combine (\\ref{item-universally-closed}) and", "(\\ref{item-affine}) with", "Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed}. See also", "Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-finite}). Combine (\\ref{item-integral}),", "and (\\ref{item-finite-type}) with", "Morphisms, Lemma \\ref{morphisms-lemma-finite-integral}. See also", "Lemma \\ref{lemma-thicken-property-morphisms-cartesian}", "for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-finite-locally-free}). Assume $f$ finite locally free. By", "(\\ref{item-finite}) we see that $f'$ is finite, by general remark (III)", "$f'$ is flat, and by (\\ref{item-finite-presentation}) $f'$ is locally of finite", "presentation. Hence $f'$ is finite locally free by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}." ], "refs": [ "schemes-lemma-base-change-immersion", "schemes-lemma-quasi-compact-preserved-base-change", "schemes-lemma-separated-permanence", "schemes-lemma-base-change-monomorphism", "morphisms-lemma-base-change-surjective", "morphisms-lemma-base-change-universally-injective", "morphisms-lemma-base-change-affine", "morphisms-lemma-base-change-finite-type", "morphisms-lemma-base-change-quasi-finite", "morphisms-lemma-base-change-finite-presentation", "morphisms-lemma-base-change-relative-dimension-d", "morphisms-lemma-base-change-syntomic", "morphisms-lemma-base-change-smooth", "morphisms-lemma-base-change-unramified", "morphisms-lemma-base-change-etale", "morphisms-lemma-base-change-proper", "morphisms-lemma-base-change-finite", "morphisms-lemma-base-change-finite-locally-free", "more-morphisms-definition-thickening", "more-morphisms-lemma-deform", "more-morphisms-lemma-flatness-morphism-thickenings", "more-morphisms-lemma-thickening-affine-scheme", "more-morphisms-lemma-thicken-property-morphisms", "more-morphisms-lemma-thicken-property-morphisms", "more-morphisms-lemma-thicken-property-morphisms", "more-morphisms-lemma-thicken-property-morphisms", "more-morphisms-lemma-thicken-property-morphisms", "more-morphisms-lemma-thickening-affine-scheme", "more-morphisms-lemma-thicken-property-morphisms", "more-morphisms-lemma-thicken-property-morphisms-cartesian", "morphisms-lemma-quasi-finite-at-point-characterize", "more-morphisms-lemma-thicken-property-morphisms-cartesian", "algebra-lemma-NAK", "more-morphisms-lemma-thicken-property-morphisms-cartesian", "more-morphisms-lemma-thicken-property-morphisms", "morphisms-lemma-syntomic-flat-fibres", "morphisms-lemma-smooth-flat-smooth-fibres", "morphisms-lemma-unramified-etale-fibres", "more-morphisms-lemma-thicken-property-morphisms-cartesian", "morphisms-lemma-etale-flat-etale-fibres", "more-morphisms-lemma-thicken-property-morphisms-cartesian", "morphisms-lemma-integral-universally-closed", "more-morphisms-lemma-thicken-property-morphisms", "morphisms-lemma-finite-integral", "more-morphisms-lemma-thicken-property-morphisms-cartesian", "morphisms-lemma-finite-flat" ], "ref_ids": [ 7695, 7698, 7714, 7724, 5165, 5169, 5176, 5200, 5233, 5240, 5284, 5291, 5327, 5346, 5361, 5409, 5440, 5473, 14106, 13723, 13724, 13678, 13682, 13682, 13682, 13682, 13682, 13678, 13682, 13684, 5226, 13684, 401, 13684, 13682, 5298, 5325, 5353, 13684, 5365, 13684, 5441, 13682, 5438, 13684, 5471 ] } ], "ref_ids": [] }, { "id": 13726, "type": "theorem", "label": "more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\", "& (S \\subset S')", "}", "$$", "of thickenings. Assume", "\\begin{enumerate}", "\\item $Y' \\to S'$ is locally of finite type,", "\\item $X' \\to S'$ is flat and locally of finite presentation,", "\\item $f$ is flat, and", "\\item $X = S \\times_{S'} X'$ and $Y = S \\times_{S'} Y'$.", "\\end{enumerate}", "Then $f'$ is flat and for all $y' \\in Y'$ in the image of $f'$", "the local ring $\\mathcal{O}_{Y', y'}$ is", "flat and essentially of finite presentation over $\\mathcal{O}_{S', s'}$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of", "Algebra, Lemma \\ref{algebra-lemma-criterion-flatness-fibre-locally-nilpotent}." ], "refs": [ "algebra-lemma-criterion-flatness-fibre-locally-nilpotent" ], "ref_ids": [ 1116 ] } ], "ref_ids": [] }, { "id": 13727, "type": "theorem", "label": "more-morphisms-lemma-deform-property-fp-over-ft", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-deform-property-fp-over-ft", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\", "& (S \\subset S')", "}", "$$", "of thickenings. Assume $Y' \\to S'$ locally of finite type,", "$X' \\to S'$ flat and locally of finite presentation,", "$X = S \\times_{S'} X'$, and $Y = S \\times_{S'} Y'$. Then", "\\begin{enumerate}", "\\item $f$ is flat if and only if $f'$ is flat,", "\\label{item-flat-fp-over-ft}", "\\item $f$ is an isomorphism if and only if $f'$ is an isomorphism,", "\\label{item-isomorphism-fp-over-ft}", "\\item $f$ is an open immersion if and only if $f'$ is an open immersion,", "\\label{item-open-immersion-fp-over-ft}", "\\item $f$ is quasi-compact if and only if $f'$ is quasi-compact,", "\\label{item-quasi-compact-fp-over-ft}", "\\item $f$ is universally closed if and only if $f'$ is universally closed,", "\\label{item-universally-closed-fp-over-ft}", "\\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,", "\\label{item-separated-fp-over-ft}", "\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,", "\\label{item-monomorphism-fp-over-ft}", "\\item $f$ is surjective if and only if $f'$ is surjective,", "\\label{item-surjective-fp-over-ft}", "\\item $f$ is universally injective if and only if $f'$ is universally injective,", "\\label{item-universally-injective-fp-over-ft}", "\\item $f$ is affine if and only if $f'$ is affine,", "\\label{item-affine-fp-over-ft}", "\\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,", "\\label{item-quasi-finite-fp-over-ft}", "\\item", "\\label{item-relative-dimension-d-fp-over-ft}", "$f$ is locally of finite type of relative dimension $d$ if and only if", "$f'$ is locally of finite type of relative dimension $d$,", "\\item $f$ is universally open if and only if $f'$ is universally open,", "\\label{item-universally-open-fp-over-ft}", "\\item $f$ is syntomic if and only if $f'$ is syntomic,", "\\label{item-syntomic-fp-over-ft}", "\\item $f$ is smooth if and only if $f'$ is smooth,", "\\label{item-smooth-fp-over-ft}", "\\item $f$ is unramified if and only if $f'$ is unramified,", "\\label{item-unramified-fp-over-ft}", "\\item $f$ is \\'etale if and only if $f'$ is \\'etale,", "\\label{item-etale-fp-over-ft}", "\\item $f$ is proper if and only if $f'$ is proper,", "\\label{item-proper-fp-over-ft}", "\\item $f$ is finite if and only if $f'$ is finite,", "\\label{item-finite-fp-over-ft}", "\\item", "\\label{item-finite-locally-free-fp-over-ft}", "$f$ is finite locally free (of rank $d$) if and only if $f'$", "is finite locally free (of rank $d$), and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The assumptions on $X$ and $Y$ mean that $f$ is the base change of", "$f'$ by $X \\to X'$.", "The properties $\\mathcal{P}$ listed in (1) -- (20) above are all stable", "under base change, hence if $f'$ has property $\\mathcal{P}$, then so", "does $f$. See", "Schemes, Lemmas \\ref{schemes-lemma-base-change-immersion},", "\\ref{schemes-lemma-quasi-compact-preserved-base-change},", "\\ref{schemes-lemma-separated-permanence}, and", "\\ref{schemes-lemma-base-change-monomorphism}", "and", "Morphisms, Lemmas", "\\ref{morphisms-lemma-base-change-surjective},", "\\ref{morphisms-lemma-base-change-universally-injective},", "\\ref{morphisms-lemma-base-change-affine},", "\\ref{morphisms-lemma-base-change-quasi-finite},", "\\ref{morphisms-lemma-base-change-relative-dimension-d},", "\\ref{morphisms-lemma-base-change-syntomic},", "\\ref{morphisms-lemma-base-change-smooth},", "\\ref{morphisms-lemma-base-change-unramified},", "\\ref{morphisms-lemma-base-change-etale},", "\\ref{morphisms-lemma-base-change-proper},", "\\ref{morphisms-lemma-base-change-finite}, and", "\\ref{morphisms-lemma-base-change-finite-locally-free}.", "\\medskip\\noindent", "The interesting direction in each case is therefore to assume", "that $f$ has the property and deduce that $f'$ has it too.", "We make a couple of general remarks which we will use without further", "mention in the arguments below.", "(I) Let $W' \\subset S'$ be an affine open and let $U' \\subset X'$", "and $V' \\subset Y'$ be affine opens lying over $W'$ with $f'(U') \\subset V'$.", "Let $W' = \\Spec(R')$ and denote $I \\subset R'$ be the ideal", "defining the closed subscheme $W' \\cap S$. Say $U' = \\Spec(B')$", "and $V' = \\Spec(A')$. Then we get a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "IB' \\ar[r] &", "B' \\ar[r] &", "B \\ar[r] & 0 \\\\", "0 \\ar[r] &", "IA' \\ar[r] \\ar[u] &", "A' \\ar[r] \\ar[u] &", "A \\ar[r] \\ar[u] & 0", "}", "$$", "with exact rows.", "(II) The morphisms $X \\to X'$ and $Y \\to Y'$ are universal homeomorphisms.", "Hence the topology of the maps $f$ and $f'$ (after any base change) is", "identical.", "(III) If $f$ is flat, then $f'$ is flat and $Y' \\to S'$ is flat at every", "point in the image of $f'$, see", "Lemma \\ref{lemma-flatness-morphism-thickenings}.", "\\medskip\\noindent", "Ad (\\ref{item-flat-fp-over-ft}). This is general remark (III).", "\\medskip\\noindent", "Ad (\\ref{item-isomorphism-fp-over-ft}). Assume $f$ is an isomorphism.", "Choose an affine open $V' \\subset Y'$ and set $U' = (f')^{-1}(V')$.", "Then $V = Y \\cap V'$ is affine which implies that", "$V \\cong f^{-1}(V) = U = Y \\times_{Y'} U'$ is affine. By", "Lemma \\ref{lemma-thickening-affine-scheme}", "we see that $U'$ is affine. Thus we have a diagram as in the", "general remark (I). By Algebra, Lemma", "\\ref{algebra-lemma-isomorphism-modulo-locally-nilpotent}", "we see that $A' \\to B'$ is an isomorphism, i.e., $U' \\cong V'$.", "Thus $f'$ is an isomorphism.", "\\medskip\\noindent", "Ad (\\ref{item-open-immersion-fp-over-ft}). Assume $f$ is an open immersion.", "Then $f$ is an isomorphism of $X$ with an open subscheme $V \\subset Y$.", "Let $V' \\subset Y'$ be the open subscheme whose underlying topological", "space is $V$. Then $f'$ is a map from $X'$ to $V'$ which is an isomorphism by", "(\\ref{item-isomorphism-fp-over-ft}). Hence $f'$ is an open immersion.", "\\medskip\\noindent", "Ad (\\ref{item-quasi-compact-fp-over-ft}). Immediate from remark (II). See also", "Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-universally-closed-fp-over-ft}). Immediate from remark (II). See", "also Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-separated-fp-over-ft}). Note that", "$X \\times_Y X = Y \\times_{Y'} (X' \\times_{Y'} X')$ so that", "$X' \\times_{Y'} X'$ is a thickening of $X \\times_Y X$.", "Hence the topology of the maps $\\Delta_{X/Y}$ and $\\Delta_{X'/Y'}$", "matches and we win. See also", "Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-monomorphism-fp-over-ft}). Assume $f$ is a monomorphism.", "Consider the diagonal morphism $\\Delta_{X'/Y'} : X' \\to X' \\times_{Y'} X'$.", "Observe that $X' \\times_{Y'} X' \\to S'$ is locally of finite type.", "The base change of $\\Delta_{X'/Y'}$ by $S \\to S'$ is $\\Delta_{X/Y}$", "which is an isomorphism by assumption. By (\\ref{item-isomorphism-fp-over-ft})", "we conclude that $\\Delta_{X'/Y'}$ is an isomorphism.", "\\medskip\\noindent", "Ad (\\ref{item-surjective-fp-over-ft}). This is clear. See also", "Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-universally-injective-fp-over-ft}).", "Immediate from remark (II). See also", "Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-affine-fp-over-ft}). Assume $f$ is affine. Choose an", "affine open $V' \\subset Y'$ and set $U' = (f')^{-1}(V')$.", "Then $V = Y \\cap V'$ is affine which implies that", "$U = Y \\times_{Y'} U'$ is affine. By", "Lemma \\ref{lemma-thickening-affine-scheme}", "we see that $U'$ is affine. Hence $f'$ is affine. See also", "Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-quasi-finite-fp-over-ft}). Follows from the fact that $f'$", "is locally of finite type", "(by Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type}) and that", "quasi-finiteness of a morphism of finite type can be checked on fibres, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}.", "\\medskip\\noindent", "Ad (\\ref{item-relative-dimension-d-fp-over-ft}).", "Follows from general remark (II) and the fact that $f'$", "is locally of finite type", "(Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type}).", "\\medskip\\noindent", "Ad (\\ref{item-universally-open-fp-over-ft}).", "Immediate from general remark (II). See also", "Lemma \\ref{lemma-thicken-property-morphisms} for a more general statement.", "\\medskip\\noindent", "Ad (\\ref{item-syntomic-fp-over-ft}). Assume $f$ is syntomic. By", "Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-permanence}", "$f'$ is locally of finite presentation.", "By general remark (III) $f'$ is flat. The fibres of $f'$ are the fibres", "of $f$. Hence $f'$ is syntomic by", "Morphisms, Lemma \\ref{morphisms-lemma-syntomic-flat-fibres}.", "\\medskip\\noindent", "Ad (\\ref{item-smooth-fp-over-ft}). Assume $f$ is smooth. By", "Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-permanence}", "$f'$ is locally of finite presentation.", "By general remark (III) $f'$ is flat. The fibres of $f'$ are the", "fibres of $f$. Hence $f'$ is smooth by", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-flat-smooth-fibres}.", "\\medskip\\noindent", "Ad (\\ref{item-unramified-fp-over-ft}). Assume $f$ unramified. By", "Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type}", "$f'$ is locally of finite type. The fibres of $f'$ are the fibres of $f$.", "Hence $f'$ is unramified by", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-etale-fibres}.", "\\medskip\\noindent", "Ad (\\ref{item-etale-fp-over-ft}). Assume $f$ \\'etale. By", "Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-permanence}", "$f'$ is locally of finite presentation.", "By general remark (III) $f'$ is flat.", "The fibres of $f'$ are the fibres of $f$. Hence $f'$ is \\'etale by", "Morphisms, Lemma \\ref{morphisms-lemma-etale-flat-etale-fibres}.", "\\medskip\\noindent", "Ad (\\ref{item-proper-fp-over-ft}). This follows from a combination of", "(\\ref{item-separated-fp-over-ft}), the fact that $f$ is locally", "of finite type (Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type}),", "(\\ref{item-quasi-compact-fp-over-ft}),", "and (\\ref{item-universally-closed-fp-over-ft}).", "\\medskip\\noindent", "Ad (\\ref{item-finite-fp-over-ft}).", "Combine (\\ref{item-universally-closed-fp-over-ft}),", "(\\ref{item-affine-fp-over-ft}),", "Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed},", "the fact that $f$ is locally of finite type", "(Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type}), and", "Morphisms, Lemma \\ref{morphisms-lemma-finite-integral}.", "\\medskip\\noindent", "Ad (\\ref{item-finite-locally-free-fp-over-ft}).", "Assume $f$ finite locally free. By", "(\\ref{item-finite-fp-over-ft}) we see that $f'$ is finite.", "By general remark (III) $f'$ is flat.", "By Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-permanence}", "$f'$ is locally of finite presentation. Hence $f'$ is finite locally free by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}." ], "refs": [ "schemes-lemma-base-change-immersion", "schemes-lemma-quasi-compact-preserved-base-change", "schemes-lemma-separated-permanence", "schemes-lemma-base-change-monomorphism", "morphisms-lemma-base-change-surjective", "morphisms-lemma-base-change-universally-injective", "morphisms-lemma-base-change-affine", "morphisms-lemma-base-change-quasi-finite", "morphisms-lemma-base-change-relative-dimension-d", "morphisms-lemma-base-change-syntomic", "morphisms-lemma-base-change-smooth", "morphisms-lemma-base-change-unramified", "morphisms-lemma-base-change-etale", "morphisms-lemma-base-change-proper", "morphisms-lemma-base-change-finite", "morphisms-lemma-base-change-finite-locally-free", "more-morphisms-lemma-flatness-morphism-thickenings", "more-morphisms-lemma-thickening-affine-scheme", "algebra-lemma-isomorphism-modulo-locally-nilpotent", "more-morphisms-lemma-thicken-property-morphisms", "more-morphisms-lemma-thicken-property-morphisms", "more-morphisms-lemma-thicken-property-morphisms", "more-morphisms-lemma-thicken-property-morphisms", "more-morphisms-lemma-thicken-property-morphisms", "more-morphisms-lemma-thickening-affine-scheme", "more-morphisms-lemma-thicken-property-morphisms", "morphisms-lemma-permanence-finite-type", "morphisms-lemma-quasi-finite-at-point-characterize", "morphisms-lemma-permanence-finite-type", "more-morphisms-lemma-thicken-property-morphisms", "morphisms-lemma-finite-presentation-permanence", "morphisms-lemma-syntomic-flat-fibres", "morphisms-lemma-finite-presentation-permanence", "morphisms-lemma-smooth-flat-smooth-fibres", "morphisms-lemma-permanence-finite-type", "morphisms-lemma-unramified-etale-fibres", "morphisms-lemma-finite-presentation-permanence", "morphisms-lemma-etale-flat-etale-fibres", "morphisms-lemma-permanence-finite-type", "morphisms-lemma-integral-universally-closed", "morphisms-lemma-permanence-finite-type", "morphisms-lemma-finite-integral", "morphisms-lemma-finite-presentation-permanence", "morphisms-lemma-finite-flat" ], "ref_ids": [ 7695, 7698, 7714, 7724, 5165, 5169, 5176, 5233, 5284, 5291, 5327, 5346, 5361, 5409, 5440, 5473, 13724, 13678, 1089, 13682, 13682, 13682, 13682, 13682, 13678, 13682, 5204, 5226, 5204, 13682, 5247, 5298, 5247, 5325, 5204, 5353, 5247, 5365, 5204, 5441, 5204, 5438, 5247, 5471 ] } ], "ref_ids": [] }, { "id": 13728, "type": "theorem", "label": "more-morphisms-lemma-deform-projective", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-deform-projective", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is proper, flat, and", "of finite presentation. Let $\\mathcal{L}$ be $f$-ample. Assume", "$S$ is quasi-compact. There exists a $d_0 \\geq 0$ such that", "for every cartesian diagram", "$$", "\\vcenter{", "\\xymatrix{", "X \\ar[r]_{i'} \\ar[d]_f & X' \\ar[d]^{f'} \\\\", "S \\ar[r]^i & S'", "}", "}", "\\quad\\text{and}\\quad", "\\begin{matrix}", "\\text{invertible }\\mathcal{O}_{X'}\\text{-module}\\\\", "\\mathcal{L}'\\text{ with }\\mathcal{L} \\cong (i')^*\\mathcal{L}'", "\\end{matrix}", "$$", "where $S \\subset S'$ is a thickening and $f'$ is", "proper, flat, of finite presentation we have", "\\begin{enumerate}", "\\item $R^p(f')_*(\\mathcal{L}')^{\\otimes d} = 0$", "for all $p > 0$ and $d \\geq d_0$,", "\\item $\\mathcal{A}'_d = (f')_*(\\mathcal{L}')^{\\otimes d}$", "is finite locally free for $d \\geq d_0$,", "\\item $\\mathcal{A}' =", "\\mathcal{O}_{S'} \\oplus \\bigoplus_{d \\geq d_0} \\mathcal{A}'_d$", "is a quasi-coherent $\\mathcal{O}_{S'}$-algebra of finite presentation,", "\\item there is a canonical isomorphism", "$r' : X' \\to \\underline{\\text{Proj}}_{S'}(\\mathcal{A}')$, and", "\\item there is a canonical isomorphism", "$\\theta' : (r')^*\\mathcal{O}_{\\underline{\\text{Proj}}_{S'}(\\mathcal{A}')}(1)", "\\to \\mathcal{L}'$.", "\\end{enumerate}", "The construction of $\\mathcal{A}'$, $r'$, $\\theta'$", "is functorial in the data $(X', S', i, i', f', \\mathcal{L}')$." ], "refs": [], "proofs": [ { "contents": [ "We first describe the maps $r'$ and $\\theta'$.", "Observe that $\\mathcal{L}'$ is $f'$-ample, see", "Lemma \\ref{lemma-thicken-property-relatively-ample}.", "There is a canonical map of quasi-coherent graded", "$\\mathcal{O}_{S'}$-algebras", "$\\mathcal{A}' \\to \\bigoplus_{d \\geq 0} (f')_*(\\mathcal{L}')^{\\otimes d}$", "which is an isomorphism in degrees $\\geq d_0$.", "Hence this induces an isomorphism on relative Proj", "compatible with the Serre twists of the structure sheaf, see", "Constructions, Lemma", "\\ref{constructions-lemma-eventual-iso-graded-rings-map-relative-proj}.", "Hence we get the morphism $r'$ by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-relatively-ample}", "(which in turn appeals to the construction given in", "Constructions, Lemma", "\\ref{constructions-lemma-invertible-map-into-relative-proj})", "and it is an isomorphism by", "Morphisms, Lemma \\ref{morphisms-lemma-proper-ample-is-proj}.", "We get the map $\\theta'$ from Constructions, Lemma", "\\ref{constructions-lemma-invertible-map-into-relative-proj}.", "By Properties, Lemma \\ref{properties-lemma-ample-gcd-is-one}", "we find that $\\theta'$ is an isomorphism", "(this also uses that the morphism $r'$ over affine", "opens of $S'$ is the same as the morphism from", "Properties, Lemma \\ref{properties-lemma-map-into-proj}", "as is explained in the proof", "of Morphisms, Lemma \\ref{morphisms-lemma-proper-ample-is-proj}).", "\\medskip\\noindent", "Assuming the vanishing and local freeness stated in parts", "(1) and (2), the functoriality of the construction can be seen as follows.", "Suppose that $h : T \\to S'$ is a morphism of schemes, denote", "$f_T : X'_T \\to T$ the base change of $f'$ and", "$\\mathcal{L}_T$ the pullback of $\\mathcal{L}$ to $X'_T$.", "By cohomology and base change", "(as formulated in Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-compare-base-change} for example)", "we have the corresponding vanishing over $T$ and moreover", "$h^*\\mathcal{A}'_d = f_{T, *}\\mathcal{L}_T^{\\otimes d}$", "(and thus the local freeness of pushforwards as well", "as the finite generation of the corresponding graded", "$\\mathcal{O}_T$-algebra $\\mathcal{A}_T$).", "Hence the morphism", "$r_T : X_T \\to", "\\underline{\\text{Proj}}_T(\\bigoplus f_{T, *}\\mathcal{L}_T^{\\otimes d})$", "is simply the base change of $r'$ to $T$ and the pullback of", "$\\theta'$ is the map $\\theta_T$.", "\\medskip\\noindent", "Having said all of the above, we see that it suffices to prove", "(1), (2), and (3). Pick $d_0$ such that", "$R^pf_*\\mathcal{L}^{\\otimes d} = 0$ for all $d \\geq d_0$ and $p > 0$, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-proper-ample}.", "We claim that $d_0$ works.", "\\medskip\\noindent", "By cohomology and base change", "(Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-flat-proper-perfect-direct-image-general})", "we see that $E'_d = Rf'_*(\\mathcal{L}')^{\\otimes d}$", "is a perfect object of $D(\\mathcal{O}_{S'})$", "and its formation commutes with arbitrary base change.", "In particular, $E_d = Li^*E'_d = Rf_*\\mathcal{L}^{\\otimes d}$.", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-vanishing-implies-locally-free}", "we see that for $d \\geq d_0$ the complex $E_d$ is isomorphic to", "the finite locally free $\\mathcal{O}_S$-module", "$f_*\\mathcal{L}^{\\otimes d}$ placed in", "cohomological degree $0$. Then by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-open-where-cohomology-in-degree-i-rank-r}", "we conclude that $E'_d$ is isomorphic to a finite locally free", "module placed in cohomological degree $0$.", "Of course this means that $E'_d = \\mathcal{A}'_d[0]$,", "that $R^pf'_*(\\mathcal{L}')^{\\otimes d} = 0$ for $p > 0$,", "and that $\\mathcal{A}'_d$ is finite locally free.", "This proves (1) and (2).", "\\medskip\\noindent", "The last thing we have to show is finite presentation of", "$\\mathcal{A}'$ as a sheaf of $\\mathcal{O}_{S'}$-algebras", "(this notion was introduced in Properties, Section", "\\ref{properties-section-extending-quasi-coherent-sheaves}).", "Let $U' = \\Spec(R') \\subset S'$ be an affine open.", "Then $A' = \\mathcal{A}'(U')$ is a graded $R'$-algebra", "whose graded parts are finite projective $R'$-modules.", "We have to show that $A'$ is a finitely presented $R'$-algebra.", "We will prove this by reduction to the Noetherian case.", "Namely, we can find a finite type $\\mathbf{Z}$-subalgebra", "$R'_0 \\subset R'$ and a pair\\footnote{With the same properties", "as those enjoyed by $X' \\to S'$ and $\\mathcal{L}'$, i.e.,", "$X'_0 \\to \\Spec(R'_0)$ is flat and proper and $\\mathcal{L}'_0$", "is ample.} $(X'_0, \\mathcal{L}'_0)$ over $R'_0$", "whose base change is $(X'_{U'}, \\mathcal{L}'|_{X'_{U'}})$, see", "Limits, Lemmas", "\\ref{limits-lemma-descend-modules-finite-presentation},", "\\ref{limits-lemma-descend-invertible-modules},", "\\ref{limits-lemma-eventually-proper},", "\\ref{limits-lemma-descend-flat-finite-presentation}, and", "\\ref{limits-lemma-limit-ample}.", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-proper-ample}", "implies", "$A'_0 = \\bigoplus_{d \\geq 0} H^0(X'_0, (\\mathcal{L}'_0)^{\\otimes d})$", "is a finitely generated graded $R'_0$-algebra and implies", "there exists a $d'_0$ such that", "$H^p(X'_0, (\\mathcal{L}'_0)^{\\otimes d}) = 0$, $p > 0$ for $d \\geq d'_0$.", "By the arguments given above applied to $X'_0 \\to \\Spec(R'_0)$ and", "$\\mathcal{L}'_0$ we see that $(A'_0)_d$ is a finite projective $R'_0$-module", "and that", "$$", "A'_d = \\mathcal{A}'_d(U') =", "H^0(X'_{U'}, (\\mathcal{L}')^{\\otimes d}|_{X'_{U'}}) =", "H^0(X'_0, (\\mathcal{L}'_0)^{\\otimes d}) \\otimes_{R'_0} R' =", "(A'_0)_d \\otimes_{R'_0} R'", "$$", "for $d \\geq d'_0$. Now a small twist in the argument is that we", "don't know that we can choose $d'_0$ equal to $d_0$\\footnote{Actually,", "one can reduce to this case by doing more limit arguments.}. To", "get around this we use the following sequence of arguments to finish", "the proof:", "\\begin{enumerate}", "\\item[(a)] The algebra", "$B = R'_0 \\oplus \\bigoplus_{d \\geq \\max(d_0, d'_0)} (A'_0)_d$", "is an $R'_0$-algebra of finite type: apply", "the Artin-Tate lemma to $B \\subset A'_0$, see", "Algebra, Lemma \\ref{algebra-lemma-Artin-Tate}.", "\\item[(b)] As $R'_0$ is Noetherian we see that", "$B$ is an $R'_0$-algebra of finite presentation.", "\\item[(c)] By right exactness of tensor product we see that", "$B \\otimes_{R'_0} R'$ is an $R'$-algebra of finite presentation.", "\\item[(d)] By the displayed equalities this exactly says that", "$C = R' \\oplus \\bigoplus_{d \\geq \\max(d_0, d'_0)} A'_d$", "is an $R'$-algebra of finite presentation.", "\\item[(e)] The quotient $A'/C$ is the direct sum of the finite", "projective $R'$-modules $A'_d$, $d_0 \\leq d \\leq \\max(d_0, d'_0)$,", "hence finitely presented as $R'$-module.", "\\item[(f)] The quotient $A'/C$ is finitely presented", "as a $C$-module by Algebra, Lemma", "\\ref{algebra-lemma-finitely-presented-over-subring}.", "\\item[(g)] Thus $A'$ is finitely presented as a $C$-module by", "Algebra, Lemma \\ref{algebra-lemma-extension}.", "\\item[(h)] By Algebra, Lemma \\ref{algebra-lemma-finite-finite-type}", "this implies $A'$ is finitely presented as a $C$-algebra.", "\\item[(i)] Finally, by", "Algebra, Lemma \\ref{algebra-lemma-compose-finite-type}", "applied to $R' \\to C \\to A'$", "this implies $A'$ is finitely presented as an $R'$-algebra.", "\\end{enumerate}", "This finishes the proof." ], "refs": [ "more-morphisms-lemma-thicken-property-relatively-ample", "constructions-lemma-eventual-iso-graded-rings-map-relative-proj", "morphisms-lemma-characterize-relatively-ample", "constructions-lemma-invertible-map-into-relative-proj", "morphisms-lemma-proper-ample-is-proj", "constructions-lemma-invertible-map-into-relative-proj", "properties-lemma-ample-gcd-is-one", "properties-lemma-map-into-proj", "morphisms-lemma-proper-ample-is-proj", "perfect-lemma-compare-base-change", "coherent-lemma-coherent-proper-ample", "perfect-lemma-flat-proper-perfect-direct-image-general", "perfect-lemma-vanishing-implies-locally-free", "perfect-lemma-open-where-cohomology-in-degree-i-rank-r", "limits-lemma-descend-modules-finite-presentation", "limits-lemma-descend-invertible-modules", "limits-lemma-eventually-proper", "limits-lemma-descend-flat-finite-presentation", "limits-lemma-limit-ample", "coherent-lemma-coherent-proper-ample", "algebra-lemma-Artin-Tate", "algebra-lemma-finitely-presented-over-subring", "algebra-lemma-extension", "algebra-lemma-finite-finite-type", "algebra-lemma-compose-finite-type" ], "ref_ids": [ 13683, 12647, 5380, 12649, 5434, 12649, 3056, 3047, 5434, 7028, 3343, 7054, 7065, 7060, 15078, 15079, 15089, 15062, 15045, 3343, 629, 335, 330, 338, 333 ] } ], "ref_ids": [] }, { "id": 13729, "type": "theorem", "label": "more-morphisms-lemma-composition-formally-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-composition-formally-smooth", "contents": [ "A composition of formally smooth morphisms is formally smooth." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13730, "type": "theorem", "label": "more-morphisms-lemma-base-change-formally-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-formally-smooth", "contents": [ "A base change of a formally smooth morphism is formally smooth." ], "refs": [], "proofs": [ { "contents": [ "Omitted, but see Algebra, Lemma \\ref{algebra-lemma-base-change-fs}", "for the algebraic version." ], "refs": [ "algebra-lemma-base-change-fs" ], "ref_ids": [ 1204 ] } ], "ref_ids": [] }, { "id": 13731, "type": "theorem", "label": "more-morphisms-lemma-formally-etale-unramified-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-formally-etale-unramified-smooth", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Then $f$ is formally \\'etale if and only if", "$f$ is formally smooth and formally unramified." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13732, "type": "theorem", "label": "more-morphisms-lemma-formally-smooth-on-opens", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-formally-smooth-on-opens", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $U \\subset X$ and $V \\subset S$ be open subschemes such that", "$f(U) \\subset V$. If $f$ is formally smooth, so is $f|_U : U \\to V$." ], "refs": [], "proofs": [ { "contents": [ "Consider a solid diagram", "$$", "\\xymatrix{", "U \\ar[d]_{f|_U} & T \\ar[d]^i \\ar[l]^a \\\\", "V & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "as in Definition \\ref{definition-formally-smooth}. If $f$ is formally", "smooth, then there exists an $S$-morphism $a' : T' \\to X$ such that", "$a'|_T = a$. Since the underlying sets of $T$ and $T'$ are the same", "we see that $a'$ is a morphism into $U$ (see Schemes, Section", "\\ref{schemes-section-open-immersion}). And it clearly is a $V$-morphism", "as well. Hence the dotted arrow above as desired." ], "refs": [ "more-morphisms-definition-formally-smooth" ], "ref_ids": [ 14111 ] } ], "ref_ids": [] }, { "id": 13733, "type": "theorem", "label": "more-morphisms-lemma-affine-formally-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-affine-formally-smooth", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume $X$ and $S$ are affine.", "Then $f$ is formally smooth if and only if", "$\\mathcal{O}_S(S) \\to \\mathcal{O}_X(X)$ is a formally smooth", "ring map." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions", "(Definition \\ref{definition-formally-smooth} and", "Algebra, Definition \\ref{algebra-definition-formally-smooth})", "by the equivalence of categories of rings and affine schemes,", "see", "Schemes, Lemma \\ref{schemes-lemma-category-affine-schemes}." ], "refs": [ "more-morphisms-definition-formally-smooth", "algebra-definition-formally-smooth", "schemes-lemma-category-affine-schemes" ], "ref_ids": [ 14111, 1537, 7656 ] } ], "ref_ids": [] }, { "id": 13734, "type": "theorem", "label": "more-morphisms-lemma-smooth-formally-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-smooth-formally-smooth", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is smooth, and", "\\item the morphism $f$ is locally of finite presentation and", "formally smooth.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $f : X \\to S$ is locally of finite presentation and formally smooth.", "Consider a pair of affine opens $\\Spec(A) = U \\subset X$ and", "$\\Spec(R) = V \\subset S$", "such that $f(U) \\subset V$. By Lemma \\ref{lemma-formally-smooth-on-opens}", "we see that $U \\to V$ is formally smooth. By Lemma", "\\ref{lemma-affine-formally-smooth} we see that $R \\to A$ is formally", "smooth. By", "Morphisms, Lemma \\ref{morphisms-lemma-locally-finite-presentation-characterize}", "we see that $R \\to A$ is of finite presentation.", "By Algebra, Proposition \\ref{algebra-proposition-smooth-formally-smooth}", "we see that $R \\to A$ is smooth.", "Hence by the definition of a smooth morphism we see that $X \\to S$ is smooth.", "\\medskip\\noindent", "Conversely, assume that $f : X \\to S$ is smooth. Consider a solid commutative", "diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & T \\ar[d]^i \\ar[l]^a \\\\", "S & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "as in Definition \\ref{definition-formally-smooth}.", "We will show the dotted arrow exists thereby", "proving that $f$ is formally smooth.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be the sheaf of sets on $T'$ of Lemma \\ref{lemma-sheaf}", "in the special case discussed in Remark \\ref{remark-special-case}.", "Let", "$$", "\\mathcal{H} =", "\\SheafHom_{\\mathcal{O}_T}(a^*\\Omega_{X/S}, \\mathcal{C}_{T/T'})", "$$", "be the sheaf of $\\mathcal{O}_T$-modules with action", "$\\mathcal{H} \\times \\mathcal{F} \\to \\mathcal{F}$ as in", "Lemma \\ref{lemma-action-sheaf}. Our goal is simply", "to show that $\\mathcal{F}(T) \\not = \\emptyset$. In other words we", "are trying to show that $\\mathcal{F}$ is a trivial $\\mathcal{H}$-torsor", "on $T$ (see Cohomology, Section \\ref{cohomology-section-h1-torsors}).", "There are two steps: (I) To show that $\\mathcal{F}$ is a torsor", "we have to show that $\\mathcal{F}_t \\not = \\emptyset$ for all $t \\in T$ (see", "Cohomology, Definition \\ref{cohomology-definition-torsor}).", "(II) To show that $\\mathcal{F}$ is the trivial torsor it suffices", "to show that $H^1(T, \\mathcal{H}) = 0$ (see", "Cohomology, Lemma \\ref{cohomology-lemma-torsors-h1} --", "we may use either cohomology", "of $\\mathcal{H}$ as an abelian sheaf or as an $\\mathcal{O}_T$-module,", "see Cohomology, Lemma \\ref{cohomology-lemma-modules-abelian}).", "\\medskip\\noindent", "First we prove (I). To see this, for every $t \\in T$ we can", "choose an affine open $U \\subset T$ neighbourhood of $t$", "such that $a(U)$ is contained", "in an affine open $\\Spec(A) = W \\subset X$", "which maps to an affine open $\\Spec(R) = V \\subset S$.", "By Morphisms, Lemma \\ref{morphisms-lemma-smooth-characterize}", "the ring map $R \\to A$ is smooth.", "Hence by Algebra, Proposition \\ref{algebra-proposition-smooth-formally-smooth}", "the ring map $R \\to A$ is formally smooth.", "Lemma \\ref{lemma-affine-formally-smooth}", "in turn implies that $W \\to V$ is formally smooth.", "Hence we can lift $a|_U : U \\to W$ to a $V$-morphism", "$a' : U' \\to W \\subset X$ showing that $\\mathcal{F}(U) \\not = \\emptyset$.", "\\medskip\\noindent", "Finally we prove (II).", "By Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-differentials}", "we see that $\\Omega_{X/S}$ is of finite presentation", "(it is even finite locally free by", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-omega-finite-locally-free}).", "Hence $a^*\\Omega_{X/S}$ is of finite presentation (see", "Modules, Lemma \\ref{modules-lemma-pullback-finite-presentation}).", "Hence the sheaf", "$\\mathcal{H} =", "\\SheafHom_{\\mathcal{O}_T}(a^*\\Omega_{X/S}, \\mathcal{C}_{T/T'})$", "is quasi-coherent by the discussion in", "Schemes, Section \\ref{schemes-section-quasi-coherent}.", "Thus by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}", "we have $H^1(T, \\mathcal{H}) = 0$ as desired." ], "refs": [ "more-morphisms-lemma-formally-smooth-on-opens", "more-morphisms-lemma-affine-formally-smooth", "morphisms-lemma-locally-finite-presentation-characterize", "algebra-proposition-smooth-formally-smooth", "more-morphisms-definition-formally-smooth", "more-morphisms-lemma-sheaf", "more-morphisms-remark-special-case", "more-morphisms-lemma-action-sheaf", "cohomology-definition-torsor", "cohomology-lemma-torsors-h1", "cohomology-lemma-modules-abelian", "morphisms-lemma-smooth-characterize", "algebra-proposition-smooth-formally-smooth", "more-morphisms-lemma-affine-formally-smooth", "morphisms-lemma-finite-presentation-differentials", "morphisms-lemma-smooth-omega-finite-locally-free", "modules-lemma-pullback-finite-presentation", "coherent-lemma-quasi-coherent-affine-cohomology-zero" ], "ref_ids": [ 13732, 13733, 5238, 1426, 14111, 13718, 14128, 13719, 2248, 2034, 2069, 5324, 1426, 13733, 5317, 5334, 13250, 3282 ] } ], "ref_ids": [] }, { "id": 13735, "type": "theorem", "label": "more-morphisms-lemma-formally-smooth-sheaf-differentials", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-formally-smooth-sheaf-differentials", "contents": [ "Let $f : X \\to Y$ be a formally smooth morphism of schemes.", "Then $\\Omega_{X/Y}$ is locally projective on $X$." ], "refs": [], "proofs": [ { "contents": [ "Choose $U \\subset X$ and $V \\subset Y$ affine open such that", "$f(U) \\subset V$. By", "Lemma \\ref{lemma-formally-smooth-on-opens}", "$f|_U : U \\to V$ is formally smooth. Hence", "$\\Gamma(V, \\mathcal{O}_V) \\to \\Gamma(U, \\mathcal{O}_U)$ is", "a formally smooth ring map, see", "Lemma \\ref{lemma-affine-formally-smooth}.", "Hence by", "Algebra, Lemma \\ref{algebra-lemma-characterize-formally-smooth-again}", "the $\\Gamma(U, \\mathcal{O}_U)$-module", "$\\Omega_{\\Gamma(U, \\mathcal{O}_U)/\\Gamma(V, \\mathcal{O}_V)}$", "is projective. Hence $\\Omega_{U/V}$ is locally projective, see", "Properties, Section \\ref{properties-section-locally-projective}." ], "refs": [ "more-morphisms-lemma-formally-smooth-on-opens", "more-morphisms-lemma-affine-formally-smooth", "algebra-lemma-characterize-formally-smooth-again" ], "ref_ids": [ 13732, 13733, 1208 ] } ], "ref_ids": [] }, { "id": 13736, "type": "theorem", "label": "more-morphisms-lemma-h1-is-zero", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-h1-is-zero", "contents": [ "Let $T$ be an affine scheme. Let $\\mathcal{F}$, $\\mathcal{G}$ be quasi-coherent", "$\\mathcal{O}_T$-modules. Consider", "$\\mathcal{H} = \\SheafHom_{\\mathcal{O}_T}(\\mathcal{F}, \\mathcal{G})$.", "If $\\mathcal{F}$ is locally projective, then $H^1(T, \\mathcal{H}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "By the definition of a locally projective sheaf on a scheme (see", "Properties, Definition \\ref{properties-definition-locally-projective})", "we see that $\\mathcal{F}$ is a direct summand of a free", "$\\mathcal{O}_T$-module. Hence we may assume that", "$\\mathcal{F} = \\bigoplus_{i \\in I} \\mathcal{O}_T$ is a free module.", "In this case $\\mathcal{H} = \\prod_{i \\in I} \\mathcal{G}$ is", "a product of quasi-coherent modules. By", "Cohomology, Lemma \\ref{cohomology-lemma-cohomology-products}", "we conclude that $H^1 = 0$ because the cohomology of a quasi-coherent sheaf", "on an affine scheme is zero, see Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}." ], "refs": [ "properties-definition-locally-projective", "cohomology-lemma-cohomology-products", "coherent-lemma-quasi-coherent-affine-cohomology-zero" ], "ref_ids": [ 3084, 2062, 3282 ] } ], "ref_ids": [] }, { "id": 13737, "type": "theorem", "label": "more-morphisms-lemma-formally-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-formally-smooth", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is formally smooth,", "\\item for every $x \\in X$ there exist opens $x \\in U \\subset X$ and", "$f(x) \\in V \\subset Y$ with $f(U) \\subset V$ such that", "$f|_U : U \\to V$ is formally smooth,", "\\item for every pair of affine opens $U \\subset X$ and $V \\subset Y$", "with $f(U) \\subset V$ the ring map $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$", "is formally smooth, and", "\\item there exists an affine open covering $Y = \\bigcup V_j$ and", "for each $j$ an affine open covering $f^{-1}(V_j) = \\bigcup U_{ji}$", "such that $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$ is a formally smooth", "ring map for all $j$ and $i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implications (1) $\\Rightarrow$ (2),", "(1) $\\Rightarrow$ (3), and (2) $\\Rightarrow$ (4) follow from", "Lemma \\ref{lemma-formally-smooth-on-opens}.", "The implication (3) $\\Rightarrow$ (4) is immediate.", "\\medskip\\noindent", "Assume (4). The proof that $f$ is formally smooth is the same", "as the second part of the proof of Lemma \\ref{lemma-smooth-formally-smooth}.", "Consider a solid commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & T \\ar[d]^i \\ar[l]^a \\\\", "Y & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "as in Definition \\ref{definition-formally-smooth}.", "We will show the dotted arrow exists thereby", "proving that $f$ is formally smooth.", "Let $\\mathcal{F}$ be the sheaf of sets on $T'$ of", "Lemma \\ref{lemma-sheaf} as in the special case discussed in", "Remark \\ref{remark-special-case}.", "Let", "$$", "\\mathcal{H} =", "\\SheafHom_{\\mathcal{O}_T}(a^*\\Omega_{X/Y}, \\mathcal{C}_{T/T'})", "$$", "be the sheaf of $\\mathcal{O}_T$-modules on $T$", "with action $\\mathcal{H} \\times \\mathcal{F} \\to \\mathcal{F}$ as in", "Lemma \\ref{lemma-action-sheaf}.", "The action $\\mathcal{H} \\times \\mathcal{F} \\to \\mathcal{F}$", "turns $\\mathcal{F}$ into a pseudo $\\mathcal{H}$-torsor, see", "Cohomology, Definition \\ref{cohomology-definition-torsor}.", "Our goal is to show that $\\mathcal{F}$ is a trivial $\\mathcal{H}$-torsor.", "There are two steps: (I) To show that $\\mathcal{F}$ is a torsor", "we have to show that $\\mathcal{F}$ locally has a", "section. (II) To show that $\\mathcal{F}$ is the trivial torsor", "it suffices to show that $H^1(T, \\mathcal{H}) = 0$, see", "Cohomology, Lemma \\ref{cohomology-lemma-torsors-h1}.", "\\medskip\\noindent", "First we prove (I). To see this, for every $t \\in T$ we can", "choose an affine open $W \\subset T$ neighbourhood of $t$", "such that $a(W)$ is contained in $U_{ji}$ for some $i, j$.", "Let $W' \\subset T'$ be the corresponding open subscheme.", "By assumption (4) we can lift $a|_W : W \\to U_{ji}$", "to a $V_j$-morphism $a' : W' \\to U_{ji}$ showing that", "$\\mathcal{F}(W')$ is nonempty.", "\\medskip\\noindent", "Finally we prove (II). By", "Lemma \\ref{lemma-formally-smooth-sheaf-differentials}", "we see that $\\Omega_{U_{ji}/V_j}$ locally projective.", "Hence $\\Omega_{X/Y}$ is locally projective, see", "Properties, Lemma \\ref{properties-lemma-locally-projective}.", "Hence $a^*\\Omega_{X/Y}$ is locally projective, see", "Properties, Lemma \\ref{properties-lemma-locally-projective-pullback}.", "Hence", "$$", "H^1(T, \\mathcal{H}) =", "H^1(T, \\SheafHom_{\\mathcal{O}_T}(a^*\\Omega_{X/Y}, \\mathcal{C}_{T/T'}) = 0", "$$", "by", "Lemma \\ref{lemma-h1-is-zero}", "as desired." ], "refs": [ "more-morphisms-lemma-formally-smooth-on-opens", "more-morphisms-lemma-smooth-formally-smooth", "more-morphisms-definition-formally-smooth", "more-morphisms-lemma-sheaf", "more-morphisms-remark-special-case", "more-morphisms-lemma-action-sheaf", "cohomology-definition-torsor", "cohomology-lemma-torsors-h1", "more-morphisms-lemma-formally-smooth-sheaf-differentials", "properties-lemma-locally-projective", "properties-lemma-locally-projective-pullback", "more-morphisms-lemma-h1-is-zero" ], "ref_ids": [ 13732, 13734, 14111, 13718, 14128, 13719, 2248, 2034, 13735, 3016, 3017, 13736 ] } ], "ref_ids": [] }, { "id": 13738, "type": "theorem", "label": "more-morphisms-lemma-triangle-differentials-formally-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-triangle-differentials-formally-smooth", "contents": [ "Let $f : X \\to Y$, $g : Y \\to S$ be morphisms of schemes.", "Assume $f$ is formally smooth. Then", "$$", "0 \\to f^*\\Omega_{Y/S} \\to \\Omega_{X/S} \\to \\Omega_{X/Y} \\to 0", "$$", "(see", "Morphisms, Lemma \\ref{morphisms-lemma-triangle-differentials})", "is short exact." ], "refs": [ "morphisms-lemma-triangle-differentials" ], "proofs": [ { "contents": [ "The algebraic version of this lemma is the following:", "Given ring maps $A \\to B \\to C$ with $B \\to C$ formally smooth, then", "the sequence", "$$", "0 \\to C \\otimes_B \\Omega_{B/A} \\to \\Omega_{C/A} \\to \\Omega_{C/B} \\to 0", "$$", "of", "Algebra, Lemma \\ref{algebra-lemma-exact-sequence-differentials}", "is exact. This is", "Algebra, Lemma \\ref{algebra-lemma-ses-formally-smooth}." ], "refs": [ "algebra-lemma-exact-sequence-differentials", "algebra-lemma-ses-formally-smooth" ], "ref_ids": [ 1133, 1209 ] } ], "ref_ids": [ 5313 ] }, { "id": 13739, "type": "theorem", "label": "more-morphisms-lemma-differentials-formally-unramified-formally-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-differentials-formally-unramified-formally-smooth", "contents": [ "Let $h : Z \\to X$ be a formally unramified morphism of schemes over $S$.", "Assume that $Z$ is formally smooth over $S$. Then the", "canonical exact sequence", "$$", "0 \\to \\mathcal{C}_{Z/X} \\to h^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0", "$$", "of", "Lemma \\ref{lemma-universally-unramified-differentials-sequence}", "is short exact." ], "refs": [ "more-morphisms-lemma-universally-unramified-differentials-sequence" ], "proofs": [ { "contents": [ "Let $Z \\to Z'$ be the universal first order thickening of $Z$ over $X$.", "From the proof of", "Lemma \\ref{lemma-universally-unramified-differentials-sequence}", "we see that our sequence is identified with the sequence", "$$", "\\mathcal{C}_{Z/Z'} \\to \\Omega_{Z'/S} \\otimes \\mathcal{O}_Z \\to", "\\Omega_{Z/S} \\to 0.", "$$", "Since $Z \\to S$ is formally smooth we can locally on $Z'$ find", "a left inverse $Z' \\to Z$ over $S$ to the inclusion map $Z \\to Z'$.", "Thus the sequence is locally split, see", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-relative-immersion-section}." ], "refs": [ "more-morphisms-lemma-universally-unramified-differentials-sequence" ], "ref_ids": [ 13705 ] } ], "ref_ids": [ 13705 ] }, { "id": 13740, "type": "theorem", "label": "more-morphisms-lemma-two-unramified-morphisms-formally-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-two-unramified-morphisms-formally-smooth", "contents": [ "Let", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[rd]_j & X \\ar[d]^f \\\\", "& Y", "}", "$$", "be a commutative diagram of schemes where $i$ and $j$ are formally", "unramified and $f$ is formally smooth. Then the canonical exact sequence", "$$", "0 \\to", "\\mathcal{C}_{Z/Y} \\to", "\\mathcal{C}_{Z/X} \\to", "i^*\\Omega_{X/Y} \\to 0", "$$", "of", "Lemma \\ref{lemma-two-unramified-morphisms}", "is exact and locally split." ], "refs": [ "more-morphisms-lemma-two-unramified-morphisms" ], "proofs": [ { "contents": [ "Denote $Z \\to Z'$ the universal first order thickening of $Z$ over $X$.", "Denote $Z \\to Z''$ the universal first order thickening of $Z$ over $Y$.", "By", "Lemma \\ref{lemma-universally-unramified-differentials-sequence}", "here is a canonical morphism $Z' \\to Z''$ so that we have a commutative", "diagram", "$$", "\\xymatrix{", "Z \\ar[r]_{i'} \\ar[rd]_{j'} & Z' \\ar[r]_a \\ar[d]^k & X \\ar[d]^f \\\\", "& Z'' \\ar[r]^b & Y", "}", "$$", "In the proof of", "Lemma \\ref{lemma-two-unramified-morphisms}", "we identified the sequence above with the sequence", "$$", "\\mathcal{C}_{Z/Z''} \\to", "\\mathcal{C}_{Z/Z'} \\to", "(i')^*\\Omega_{Z'/Z''} \\to 0", "$$", "Let $U'' \\subset Z''$ be an affine open. Denote $U \\subset Z$ and", "$U' \\subset Z'$ the corresponding affine open subschemes.", "As $f$ is formally smooth there exists a morphism $h : U'' \\to X$", "which agrees with $i$ on $U$ and such that $f \\circ h$ equals $b|_{U''}$.", "Since $Z'$ is the universal first order thickening we obtain a unique", "morphism $g : U'' \\to Z'$ such that $g = a \\circ h$. The universal", "property of $Z''$ implies that $k \\circ g$ is the inclusion map", "$U'' \\to Z''$. Hence $g$ is a left inverse to $k$. Picture", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & Z' \\ar[d]^k \\\\", "U'' \\ar[r] \\ar[ru]^g & Z''", "}", "$$", "Thus $g$ induces a map $\\mathcal{C}_{Z/Z'}|_U \\to \\mathcal{C}_{Z/Z''}|_U$", "which is a left inverse to the map", "$\\mathcal{C}_{Z/Z''} \\to \\mathcal{C}_{Z/Z'}$ over $U$." ], "refs": [ "more-morphisms-lemma-universally-unramified-differentials-sequence", "more-morphisms-lemma-two-unramified-morphisms" ], "ref_ids": [ 13705, 13706 ] } ], "ref_ids": [ 13706 ] }, { "id": 13741, "type": "theorem", "label": "more-morphisms-lemma-lifting-along-artinian-at-point", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-lifting-along-artinian-at-point", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$.", "Assume that $S$ is locally Noetherian and $f$ locally of finite type.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is smooth at $x$,", "\\item for every solid commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & \\Spec(B) \\ar[d]^i \\ar[l]^-\\alpha \\\\", "S & \\Spec(B') \\ar[l]_-{\\beta} \\ar@{-->}[lu]", "}", "$$", "where $B' \\to B$ is a surjection of local rings with", "$\\Ker(B' \\to B)$ of square zero, and $\\alpha$ mapping the", "closed point of $\\Spec(B)$ to $x$ there exists", "a dotted arrow making the diagram commute,", "\\item same as in (2) but with $B' \\to B$ ranging over small", "extensions (see Algebra, Definition \\ref{algebra-definition-small-extension}),", "and", "\\item same as in (2) but with $B' \\to B$ ranging over small", "extensions such that $\\alpha$ induces an isomorphism", "$\\kappa(x) \\to \\kappa(\\mathfrak m)$ where $\\mathfrak m \\subset B$", "is the maximal ideal.", "\\end{enumerate}" ], "refs": [ "algebra-definition-small-extension" ], "proofs": [ { "contents": [ "Choose an affine neighbourhood $V \\subset S$ of $f(x)$ and choose an", "affine neighbourhood $U \\subset X$ of $x$ such that $f(U) \\subset V$.", "For any ``test'' diagram as in (2) the morphism $\\alpha$ will map", "$\\Spec(B)$ into $U$ and the morphism $\\beta$ will map $\\Spec(B')$", "into $V$ (see Schemes, Section \\ref{schemes-section-points}).", "Hence the lemma reduces to the morphism $f|_U : U \\to V$ of affines.", "(Indeed, $V$ is Noetherian and $f|_U$ is of finite type, see", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian} and", "Morphisms, Lemma \\ref{morphisms-lemma-locally-finite-type-characterize}.)", "In this affine case the lemma is identical to", "Algebra, Lemma \\ref{algebra-lemma-smooth-test-artinian}." ], "refs": [ "properties-lemma-locally-Noetherian", "morphisms-lemma-locally-finite-type-characterize", "algebra-lemma-smooth-test-artinian" ], "ref_ids": [ 2951, 5198, 1229 ] } ], "ref_ids": [ 1538 ] }, { "id": 13742, "type": "theorem", "label": "more-morphisms-lemma-lifting-along-artinian", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-lifting-along-artinian", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume that $S$ is locally Noetherian and $f$ locally of finite type.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is smooth,", "\\item for every solid commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & \\Spec(B) \\ar[d]^i \\ar[l]^-\\alpha \\\\", "S & \\Spec(B') \\ar[l]_-{\\beta} \\ar@{-->}[lu]", "}", "$$", "where $B' \\to B$ is a small extension of Artinian local rings", "and $\\beta$ of finite type (!) there exists a dotted arrow making", "the diagram commute.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $f$ is smooth, then the infinitesimal lifting criterion", "(Lemma \\ref{lemma-smooth-formally-smooth}) says", "$f$ is formally smooth and (2) holds.", "\\medskip\\noindent", "Assume (2). The set of points $x \\in X$ where $f$ is not smooth", "forms a closed subset $T$ of $X$. By the discussion in Morphisms,", "Section \\ref{morphisms-section-points-finite-type}, if $T \\not = \\emptyset$", "there exists a point $x \\in T \\subset X$ such that the morphism", "$$", "\\Spec(\\kappa(x)) \\to X \\to S", "$$", "is of finite type (namely, pick any point $x$ of $T$ which is closed", "in an affine open of $X$). By", "Morphisms, Lemma \\ref{morphisms-lemma-artinian-finite-type} given any", "local Artinian ring $B'$ with residue field $\\kappa(x)$ then any", "morphism $\\beta : \\Spec(B') \\to S$ is of finite type. Thus", "we see that all the diagrams used in", "Lemma \\ref{lemma-lifting-along-artinian-at-point} (4) correspond", "to diagrams as in the current lemma (2). Whence $X \\to S$ is smooth", "a $x$ a contradiction." ], "refs": [ "more-morphisms-lemma-smooth-formally-smooth", "morphisms-lemma-artinian-finite-type", "more-morphisms-lemma-lifting-along-artinian-at-point" ], "ref_ids": [ 13734, 5206, 13741 ] } ], "ref_ids": [] }, { "id": 13743, "type": "theorem", "label": "more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds", "contents": [ "Let $f : X \\to S$ be a finite type morphism of locally Noetherian schemes.", "Let $Z \\subset S$ be a closed subscheme with $n$th infinitesimal", "neighbourhood $Z_n \\subset S$. Set $X_n = Z_n \\times_S X$.", "\\begin{enumerate}", "\\item If $X_n \\to Z_n$ is smooth for all $n$, then $f$", "is smooth at every point of $f^{-1}(Z)$.", "\\item If $X_n \\to Z_n$ is \\'etale for all $n$, then $f$", "is \\'etale at every point of $f^{-1}(Z)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $X_n \\to Z_n$ is smooth for all $n$.", "Let $x \\in X$ be a point lying over a point of $Z$.", "Given a small extension $B' \\to B$ and morphisms $\\alpha$, $\\beta$ as in", "Lemma \\ref{lemma-lifting-along-artinian-at-point} part (3)", "the maximal ideal of $B'$ is nilpotent (as $B'$ is Artinian)", "and hence the morphism $\\beta$ factors through $Z_n$ and $\\alpha$ factors", "through $X_n$ for a suitable $n$. Thus the lifting property for", "$X_n \\to Z_n$ kicks in to get the desired dotted arrow in the diagram.", "This proves (1). Part (2) follows from (1) and the fact that a morphism", "is \\'etale if and only if it is smooth of relative dimension $0$." ], "refs": [ "more-morphisms-lemma-lifting-along-artinian-at-point" ], "ref_ids": [ 13741 ] } ], "ref_ids": [] }, { "id": 13744, "type": "theorem", "label": "more-morphisms-lemma-check-flatness-on-infinitesimal-nbhds", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-check-flatness-on-infinitesimal-nbhds", "contents": [ "Let $f : X \\to S$ be a morphism of locally Noetherian schemes.", "Let $Z \\subset S$ be a closed subscheme with $n$th infinitesimal", "neighbourhood $Z_n \\subset S$. Set $X_n = Z_n \\times_S X$.", "If $X_n \\to Z_n$ is flat for all $n$, then $f$", "is flat at every point of $f^{-1}(Z)$." ], "refs": [], "proofs": [ { "contents": [ "This is a translation of Algebra, Lemma \\ref{algebra-lemma-flat-module-powers}", "into the language of schemes." ], "refs": [ "algebra-lemma-flat-module-powers" ], "ref_ids": [ 893 ] } ], "ref_ids": [] }, { "id": 13745, "type": "theorem", "label": "more-morphisms-lemma-NL-affine", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-NL-affine", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let", "$\\Spec(A) = U \\subset X$ and $\\Spec(R) = V \\subset S$", "be affine opens with $f(U) \\subset V$.", "There is a canonical map", "$$", "\\widetilde{\\NL_{A/R}} \\longrightarrow \\NL_{X/Y}|_U", "$$", "of complexes which is an isomorphism in $D(\\mathcal{O}_U)$." ], "refs": [], "proofs": [ { "contents": [ "From the construction of $\\NL_{X/Y}$ in", "Modules, Section \\ref{modules-section-netherlander}", "we see there is a canonical map of complexes", "$\\NL_{\\mathcal{O}_X(U)/f^{-1}\\mathcal{O}_Y(U)} \\to \\NL_{X/Y}(U)$", "of $A = \\mathcal{O}_X(U)$-modules, which is compatible", "with further restrictions. Using the canonical map", "$R \\to f^{-1}\\mathcal{O}_Y(U)$ we obtain a canonical map", "$\\NL_{A/R} \\to \\NL_{\\mathcal{O}_X(U)/f^{-1}\\mathcal{O}_Y(U)}$", "of complexes of $A$-modules.", "Using the universal property of the $\\widetilde{\\ }$", "functor (see Schemes, Lemma \\ref{schemes-lemma-compare-constructions})", "we obtain a map as in the statement of the lemma.", "We may check this map is an isomorphism on cohomology sheaves", "by checking it induces isomorphisms on stalks.", "This follows from", "Algebra, Lemma \\ref{algebra-lemma-NL-localize-bottom} and", "\\ref{algebra-lemma-localize-NL}", "and", "Modules, Lemma \\ref{modules-lemma-stalk-NL}", "(and the description of the stalks of", "$\\mathcal{O}_X$ and $f^{-1}\\mathcal{O}_Y$", "at a point $\\mathfrak p \\in \\Spec(A)$ as $A_\\mathfrak p$ and", "$R_\\mathfrak q$ where $\\mathfrak q = R \\cap \\mathfrak p$; references", "used are Schemes, Lemma \\ref{schemes-lemma-spec-sheaves}", "and", "Sheaves, Lemma \\ref{sheaves-lemma-stalk-pullback})." ], "refs": [ "schemes-lemma-compare-constructions", "algebra-lemma-NL-localize-bottom", "algebra-lemma-localize-NL", "modules-lemma-stalk-NL", "schemes-lemma-spec-sheaves", "sheaves-lemma-stalk-pullback" ], "ref_ids": [ 7660, 1159, 1161, 13329, 7651, 14507 ] } ], "ref_ids": [] }, { "id": 13746, "type": "theorem", "label": "more-morphisms-lemma-netherlander-quasi-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-netherlander-quasi-coherent", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. The cohomology sheaves", "of the complex $\\NL_{X/Y}$ are quasi-coherent, zero outside", "degrees $-1$, $0$ and equal to $\\Omega_{X/Y}$ in degree $0$." ], "refs": [], "proofs": [ { "contents": [ "By construction of the naive cotangent complex in", "Modules, Section \\ref{modules-section-netherlander}", "we have that $\\NL_{X/Y}$ is a complex sitting in degrees $-1$, $0$", "and that its cohomology in degree $0$ is $\\Omega_{X/Y}$.", "The sheaf of differentials is quasi-coherent (by", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-diagonal}).", "To finish the proof it suffices to show that $H^{-1}(\\NL_{X/Y})$", "is quasi-coherent. This follows by checking over affines", "using Lemma \\ref{lemma-NL-affine}." ], "refs": [ "morphisms-lemma-differentials-diagonal", "more-morphisms-lemma-NL-affine" ], "ref_ids": [ 5311, 13745 ] } ], "ref_ids": [] }, { "id": 13747, "type": "theorem", "label": "more-morphisms-lemma-netherlander-fp", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-netherlander-fp", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. If $f$ is locally of finite", "presentation, then $\\NL_{X/Y}$ is locally on $X$ quasi-isomorphic to", "a complex", "$$", "\\ldots \\to 0 \\to \\mathcal{F}^{-1} \\to \\mathcal{F}^0 \\to 0 \\to \\ldots", "$$", "of quasi-coherent $\\mathcal{O}_X$-modules", "with $\\mathcal{F}^0$ of finite presentation", "and $\\mathcal{F}^{-1}$ of finite type." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-NL-affine} it suffices to show that $\\NL_{A/R}$", "has this shape if $R \\to A$ is a finitely presented ring map.", "Write $A = R[x_1, \\ldots, x_n]/I$ with $I$ finitely generated.", "Then $I/I^2$ is a finite", "$A$-module and $\\NL_{A/R}$ is quasi-isomorphic to", "$$", "\\ldots \\to 0 \\to I/I^2 \\to", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} A\\text{d}x_i \\to 0 \\to \\ldots", "$$", "by Algebra, Section \\ref{algebra-section-netherlander}", "and in particular", "Algebra, Lemma \\ref{algebra-lemma-NL-homotopy}." ], "refs": [ "more-morphisms-lemma-NL-affine", "algebra-lemma-NL-homotopy" ], "ref_ids": [ 13745, 1151 ] } ], "ref_ids": [] }, { "id": 13748, "type": "theorem", "label": "more-morphisms-lemma-NL-formally-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-NL-formally-smooth", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is formally smooth,", "\\item $H^{-1}(\\NL_{X/Y}) = 0$ and $H^0(\\NL_{X/Y}) = \\Omega_{X/Y}$", "is locally projective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from Algebra, Proposition", "\\ref{algebra-proposition-characterize-formally-smooth}", "and Lemma \\ref{lemma-formally-smooth}." ], "refs": [ "algebra-proposition-characterize-formally-smooth", "more-morphisms-lemma-formally-smooth" ], "ref_ids": [ 1425, 13737 ] } ], "ref_ids": [] }, { "id": 13749, "type": "theorem", "label": "more-morphisms-lemma-NL-formally-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-NL-formally-etale", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is formally \\'etale,", "\\item $H^{-1}(\\NL_{X/Y}) = H^0(\\NL_{X/Y}) = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "A formally \\'etale morphism is formally smooth and hence", "we have $H^{-1}(\\NL_{X/Y}) = 0$ by Lemma \\ref{lemma-NL-formally-smooth}.", "On the other hand, we have $\\Omega_{X/Y} = 0$ by", "Lemma \\ref{lemma-characterize-formally-etale}.", "Conversely, if (2) holds, then $f$ is formally smooth by", "Lemma \\ref{lemma-NL-formally-smooth}", "and formally unramified by", "Lemma \\ref{lemma-formally-unramified-differentials}", "and hence formally \\'etale by", "Lemmas \\ref{lemma-formally-etale-unramified-smooth}." ], "refs": [ "more-morphisms-lemma-NL-formally-smooth", "more-morphisms-lemma-characterize-formally-etale", "more-morphisms-lemma-NL-formally-smooth", "more-morphisms-lemma-formally-unramified-differentials", "more-morphisms-lemma-formally-etale-unramified-smooth" ], "ref_ids": [ 13748, 13712, 13748, 13695, 13731 ] } ], "ref_ids": [] }, { "id": 13750, "type": "theorem", "label": "more-morphisms-lemma-NL-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-NL-smooth", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is smooth, and", "\\item $f$ is locally of finite presentation,", "$H^{-1}(\\NL_{X/Y}) = 0$, and $H^0(\\NL_{X/Y}) = \\Omega_{X/Y}$", "is finite locally free.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from the definition of a smooth ring homomorphism", "(Algebra, Definition \\ref{algebra-definition-smooth}),", "Lemma \\ref{lemma-NL-affine}, and", "the definition of a smooth morphism of schemes", "(Morphisms, Definition \\ref{morphisms-definition-smooth}).", "We also use that finite locally free is the same as", "finite projective for modules over rings", "(Algebra, Lemma \\ref{algebra-lemma-finite-projective})." ], "refs": [ "algebra-definition-smooth", "more-morphisms-lemma-NL-affine", "morphisms-definition-smooth", "algebra-lemma-finite-projective" ], "ref_ids": [ 1534, 13745, 5564, 795 ] } ], "ref_ids": [] }, { "id": 13751, "type": "theorem", "label": "more-morphisms-lemma-NL-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-NL-etale", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is \\'etale, and", "\\item $f$ is locally of finite presentation and", "$H^{-1}(\\NL_{X/Y}) = H^0(\\NL_{X/Y}) = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows from the definition of an \\'etale ring homomorphism", "(Algebra, Definition \\ref{algebra-definition-etale}),", "Lemma \\ref{lemma-NL-affine}, and", "the definition of an \\'etale morphism of schemes", "(Morphisms, Definition \\ref{morphisms-definition-etale})." ], "refs": [ "algebra-definition-etale", "more-morphisms-lemma-NL-affine", "morphisms-definition-etale" ], "ref_ids": [ 1539, 13745, 5567 ] } ], "ref_ids": [] }, { "id": 13752, "type": "theorem", "label": "more-morphisms-lemma-NL-immersion", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-NL-immersion", "contents": [ "Let $i : Z \\to X$ be an immersion of schemes. Then $\\NL_{Z/X}$", "is isomorphic to $\\mathcal{C}_{Z/X}[1]$ in $D(\\mathcal{O}_Z)$", "where $\\mathcal{C}_{Z/X}$ is the conormal sheaf of $Z$ in $X$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Algebra, Lemma \\ref{algebra-lemma-NL-surjection},", "Morphisms, Lemma \\ref{morphisms-lemma-affine-conormal}, and", "Lemma \\ref{lemma-NL-affine}." ], "refs": [ "algebra-lemma-NL-surjection", "morphisms-lemma-affine-conormal", "more-morphisms-lemma-NL-affine" ], "ref_ids": [ 1154, 5303, 13745 ] } ], "ref_ids": [] }, { "id": 13753, "type": "theorem", "label": "more-morphisms-lemma-exact-sequence-NL", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-exact-sequence-NL", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes.", "There is a canonical six term exact sequence", "$$", "H^{-1}(f^*\\NL_{Y/Z}) \\to", "H^{-1}(\\NL_{X/Z}) \\to", "H^{-1}(\\NL_{X/Y}) \\to", "f^*\\Omega_{Y/Z} \\to \\Omega_{X/Z} \\to \\Omega_{X/Y} \\to 0", "$$", "of cohomology sheaves." ], "refs": [], "proofs": [ { "contents": [ "Special case of", "Modules, Lemma \\ref{modules-lemma-exact-sequence-NL-ringed-topoi}." ], "refs": [ "modules-lemma-exact-sequence-NL-ringed-topoi" ], "ref_ids": [ 13331 ] } ], "ref_ids": [] }, { "id": 13754, "type": "theorem", "label": "more-morphisms-lemma-get-triangle-NL", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-get-triangle-NL", "contents": [ "Let $f : X \\to Y$ and $Y \\to Z$ be morphisms of schemes. Assume", "$X \\to Y$ is a complete intersection morphism. Then there is", "a canonical distinguished triangle", "$$", "f^*\\NL_{Y/Z} \\to \\NL_{X/Z} \\to \\NL_{X/Y} \\to f^*\\NL_{Y/Z}[1]", "$$", "in $D(\\mathcal{O}_X)$ which recovers the $6$-term exact sequence of", "Lemma \\ref{lemma-exact-sequence-NL}." ], "refs": [ "more-morphisms-lemma-exact-sequence-NL" ], "proofs": [ { "contents": [ "It suffices to show the canonical map", "$$", "f^*\\NL_{Y/Z} \\to \\text{Cone}(\\NL_{X/Y} \\to \\NL_{X/Z})[-1]", "$$", "of Modules, Lemma \\ref{modules-lemma-exact-sequence-NL-ringed-topoi}", "is an isomorphism in $D(\\mathcal{O}_X)$. In order to show this, it", "suffices to show that the $6$-term sequence has", "a zero on the left, i.e., that $H^{-1}(f^*\\NL_{Y/Z}) \\to H^{-1}(\\NL_{X/Z})$", "is injective. Affine locally this follows from the corresponding", "algebra result in More on Algebra, Lemma", "\\ref{more-algebra-lemma-transitive-lci-at-end}.", "To translate into algebra use Lemma \\ref{lemma-NL-affine}." ], "refs": [ "modules-lemma-exact-sequence-NL-ringed-topoi", "more-algebra-lemma-transitive-lci-at-end", "more-morphisms-lemma-NL-affine" ], "ref_ids": [ 13331, 10003, 13745 ] } ], "ref_ids": [ 13753 ] }, { "id": 13755, "type": "theorem", "label": "more-morphisms-lemma-smooth-etale-permanence", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-smooth-etale-permanence", "contents": [ "Let $X \\to Y \\to Z$ be morphisms of schemes. Assume $X \\to Z$ smooth", "and $Y \\to Z$ \\'etale. Then $X \\to Y$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "The morphism $X \\to Y$ is locally of finite presentation by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-permanence}.", "By Lemma \\ref{lemma-NL-smooth} we have $H^{-1}(\\NL_{X/Z}) = 0$", "and the module $\\Omega_{X/Z}$ is finite locally free.", "By Lemma \\ref{lemma-NL-etale} we have", "$H^{-1}(\\NL_{Y/Z}) = H^0(\\NL_{Y/Z}) = 0$.", "By Lemma \\ref{lemma-exact-sequence-NL} we get", "$H^{-1}(\\NL_{X/Y}) = 0$ and $\\Omega_{X/Y} \\cong \\Omega_{X/Z}$", "is finite locally free.", "By Lemma \\ref{lemma-NL-smooth} the morphism $X \\to Y$ is smooth." ], "refs": [ "morphisms-lemma-finite-presentation-permanence", "more-morphisms-lemma-NL-smooth", "more-morphisms-lemma-NL-etale", "more-morphisms-lemma-exact-sequence-NL", "more-morphisms-lemma-NL-smooth" ], "ref_ids": [ 5247, 13750, 13751, 13753, 13750 ] } ], "ref_ids": [] }, { "id": 13756, "type": "theorem", "label": "more-morphisms-lemma-get-NL", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-get-NL", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes which factors", "as $f = g \\circ i$ with $i$ an immersion and $g : P \\to Y$", "formally smooth (for example smooth). Then there is a canonical isomorphism", "$$", "\\NL_{X/Y} \\cong \\left(\\mathcal{C}_{X/P} \\to i^*\\Omega_{P/Y}\\right)", "$$", "in $D(\\mathcal{O}_X)$ where the conormal sheaf $\\mathcal{C}_{X/P}$", "is placed in degree $-1$." ], "refs": [], "proofs": [ { "contents": [ "(For the parenthetical statement see Lemma \\ref{lemma-smooth-formally-smooth}.)", "By Lemmas \\ref{lemma-NL-immersion} and \\ref{lemma-NL-formally-smooth} we have", "$\\NL_{X/P} = \\mathcal{C}_{X/P}[1]$ and $\\NL_{P/Y} = \\Omega_{P/Y}$ with", "$\\Omega_{P/Y}$ locally projective. This implies that", "$i^*\\NL_{P/Y} \\to i^*\\Omega_{P/Y}$ is a quasi-isomorphism too", "(small detail omitted; the reason is that $i^*\\NL_{P/Y}$ is the", "same thing as $\\tau_{\\geq -1}Li^*\\NL_{P/Y}$, see More on Algebra, Lemma", "\\ref{more-algebra-lemma-tensor-NL}).", "Thus the canonical map", "$$", "i^*\\NL_{P/Y} \\to \\text{Cone}(\\NL_{X/Y} \\to \\NL_{X/P})[-1]", "$$", "of Modules, Lemma \\ref{modules-lemma-exact-sequence-NL-ringed-topoi}", "is an isomorphism in $D(\\mathcal{O}_X)$ because the cohomology", "group $H^{-1}(i^*\\NL_{P/Y})$ is zero by what we said above.", "In other words, we have a distinguished triangle", "$$", "i^*\\NL_{P/Y} \\to \\NL_{X/Y} \\to \\NL_{X/P} \\to i^*\\NL_{P/Y}[1]", "$$", "Clearly, this means that $\\NL_{X/Y}$ is the cone on the map", "$\\NL_{X/P}[-1] \\to i^*\\NL_{P/Y}$ which is equivalent to the", "statement of the lemma by our computation of the cohomology", "sheaves of these objects in the derived category given above." ], "refs": [ "more-morphisms-lemma-smooth-formally-smooth", "more-morphisms-lemma-NL-immersion", "more-morphisms-lemma-NL-formally-smooth", "more-algebra-lemma-tensor-NL", "modules-lemma-exact-sequence-NL-ringed-topoi" ], "ref_ids": [ 13734, 13752, 13748, 10308, 13331 ] } ], "ref_ids": [] }, { "id": 13757, "type": "theorem", "label": "more-morphisms-lemma-base-change-NL", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-NL", "contents": [ "Consider a cartesian diagram of schemes", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d] & X \\ar[d] \\\\", "Y' \\ar[r] & Y", "}", "$$", "The canonical map $(g')^*\\NL_{X/Y} \\to \\NL_{X'/Y'}$ induces", "an isomorphism on $H^0$ and a surjection on $H^{-1}$." ], "refs": [], "proofs": [ { "contents": [ "Translated into algebra this is", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-NL}.", "To do the translation use Lemma \\ref{lemma-NL-affine}." ], "refs": [ "more-algebra-lemma-base-change-NL", "more-morphisms-lemma-NL-affine" ], "ref_ids": [ 10309, 13745 ] } ], "ref_ids": [] }, { "id": 13758, "type": "theorem", "label": "more-morphisms-lemma-flat-base-change-NL", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-base-change-NL", "contents": [ "Consider a cartesian diagram of schemes", "$$", "\\xymatrix{", "X' \\ar[d] \\ar[r]_{g'} & X \\ar[d] \\\\", "Y' \\ar[r] & Y", "}", "$$", "If $Y' \\to Y$ is flat, then the canonical map", "$(g')^*\\NL_{X/Y} \\to \\NL_{X'/Y'}$ is a quasi-isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-NL-affine} this follows from", "Algebra, Lemma \\ref{algebra-lemma-change-base-NL}." ], "refs": [ "more-morphisms-lemma-NL-affine", "algebra-lemma-change-base-NL" ], "ref_ids": [ 13745, 1156 ] } ], "ref_ids": [] }, { "id": 13759, "type": "theorem", "label": "more-morphisms-lemma-base-change-NL-flat", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-NL-flat", "contents": [ "Consider a cartesian diagram of schemes", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d] & X \\ar[d] \\\\", "Y' \\ar[r] & Y", "}", "$$", "If $X \\to Y$ is flat, then the canonical map", "$(g')^*\\NL_{X/Y} \\to \\NL_{X'/Y'}$ is a quasi-isomorphism.", "If in addition $\\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$", "then $L(g')^*\\NL_{X/Y} \\to \\NL_{X'/Y'}$ is a quasi-isomorphism too." ], "refs": [], "proofs": [ { "contents": [ "Translated into algebra this is", "More on Algebra, Lemma \\ref{more-algebra-lemma-base-change-NL-flat}.", "To do the translation use Lemma \\ref{lemma-NL-affine}", "and Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-affine-compare-bounded} and", "\\ref{perfect-lemma-tor-dimension-affine}." ], "refs": [ "more-algebra-lemma-base-change-NL-flat", "more-morphisms-lemma-NL-affine", "perfect-lemma-affine-compare-bounded", "perfect-lemma-tor-dimension-affine" ], "ref_ids": [ 10310, 13745, 6941, 6977 ] } ], "ref_ids": [] }, { "id": 13760, "type": "theorem", "label": "more-morphisms-lemma-basic-example-pushout", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-basic-example-pushout", "contents": [ "Let $A' \\to A$ be a surjection of rings and let $B \\to A$ be a ring map.", "Let $B' = B \\times_A A'$ be the fibre product of rings. Set", "$S = \\Spec(A)$, $S' = \\Spec(A')$, $T = \\Spec(B)$, and $T' = \\Spec(B')$.", "Then", "$$", "\\vcenter{", "\\xymatrix{", "S \\ar[r]_i \\ar[d]_f & S' \\ar[d]^{f'} \\\\", "T \\ar[r]^{i'} & T'", "}", "}", "\\quad\\text{corresponding to}\\quad", "\\vcenter{", "\\xymatrix{", "A & A' \\ar[l] \\\\", "B \\ar[u] & B' \\ar[l] \\ar[u]", "}", "}", "$$", "is a pushout of schemes." ], "refs": [], "proofs": [ { "contents": [ "By More on Algebra, Lemma \\ref{more-algebra-lemma-points-of-fibre-product}", "we have $T' = T \\amalg_S S'$ as topological spaces, i.e., the diagram", "is a pushout in the category of topological spaces. Next, consider", "the map", "$$", "((i')^\\sharp, (f')^\\sharp) :", "\\mathcal{O}_{T'}", "\\longrightarrow", "i'_*\\mathcal{O}_T \\times_{g_*\\mathcal{O}_S} f'_*\\mathcal{O}_{S'}", "$$", "where $g = i' \\circ f = f' \\circ i$. We claim this map is an isomorphism of", "sheaves of rings. Namely, we can view both sides as quasi-coherent", "$\\mathcal{O}_{T'}$-modules (use", "Schemes, Lemmas \\ref{schemes-lemma-push-forward-quasi-coherent}", "for the right hand side) and the map is $\\mathcal{O}_{T'}$-linear.", "Thus it suffices to show the map is an isomorphism on the level", "of global sections", "(Schemes, Lemma \\ref{schemes-lemma-equivalence-quasi-coherent}).", "On global sections we recover the identification", "$B' \\to B \\times_A A'$ from statement of the lemma (this is", "how we chose $B'$).", "\\medskip\\noindent", "Let $X$ be a scheme. Suppose we are given morphisms of schemes", "$m' : S' \\to X$ and $n : T \\to X$ such that $m' \\circ i = n \\circ f$", "(call this $m$). We get a unique map of topological spaces", "$n' : T' \\to X$ compatible with $m'$ and $n$ as", "$T' = T \\amalg_S S'$ (see above).", "By the description of $\\mathcal{O}_{T'}$ in the previous", "paragraph we obtain a unique homomorphism of sheaves of rings", "$$", "(n')^\\sharp :", "\\mathcal{O}_X", "\\longrightarrow", "(n')_*\\mathcal{O}_{T'} =", "m'_*\\mathcal{O}_T \\times_{m_*\\mathcal{O}_T} n_*\\mathcal{O}_S", "$$", "given by $(m')^\\sharp$ and $n^\\sharp$.", "Thus $(n', (n')^\\sharp)$ is the unque morphism of", "ringed spaces $T' \\to X$ compatible with $m'$ and $n$.", "To finish the proof it suffices to show that $n'$", "is a morphism of schemes, i.e., a morphism of locally ringed spaces.", "\\medskip\\noindent", "Let $t' \\in T'$ with image $x \\in X$.", "We have to show that $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{T', t'}$", "is local. If $t' \\not \\in T$, then $t'$ is the image of a unique", "point $s' \\in S'$ and $\\mathcal{O}_{T', t'} = \\mathcal{O}_{S', s'}$.", "Namely, $S' \\setminus S \\to T' \\setminus T$ is an isomorphism of", "schemes as $B' \\to A'$ induces an isomorphism", "$\\Ker(B' \\to B) = \\Ker(A' \\to A)$.", "If $t'$ is the image of $t \\in T$, then", "we know that the composition", "$\\mathcal{O}_{X, x} \\to \\mathcal{O}_{T', t'} \\to \\mathcal{O}_{T, t}$", "is local and we conclude also." ], "refs": [ "more-algebra-lemma-points-of-fibre-product", "schemes-lemma-push-forward-quasi-coherent", "schemes-lemma-equivalence-quasi-coherent" ], "ref_ids": [ 9818, 7730, 7664 ] } ], "ref_ids": [] }, { "id": 13761, "type": "theorem", "label": "more-morphisms-lemma-pushout-fpqc-local", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pushout-fpqc-local", "contents": [ "Let $\\mathcal{I} \\to (\\Sch/S)_{fppf}$, $i \\mapsto X_i$ be a diagram of schemes.", "Let $(W, X_i \\to W)$ be a cocone for the diagram in the category of schemes", "(Categories, Remark \\ref{categories-remark-cones-and-cocones}).", "If there exists a fpqc covering $\\{W_a \\to W\\}_{a \\in A}$ of schemes such that", "\\begin{enumerate}", "\\item for all $a \\in A$ we have", "$W_a = \\colim X_i \\times_W W_a$", "in the category of schemes, and", "\\item for all $a, b \\in A$ we have", "$W_a \\times_W W_b = \\colim X_i \\times_W W_a \\times_W W_b$", "in the category of schemes,", "\\end{enumerate}", "then $W = \\colim X_i$ in the category of schemes." ], "refs": [ "categories-remark-cones-and-cocones" ], "proofs": [ { "contents": [ "Namely, for a scheme $T$ a morphism $W \\to T$ is the same thing as", "collection of morphism $W_a \\to T$, $a \\in A$ which agree on the", "overlaps $W_a \\times_W W_b$, see", "Descent, Lemma \\ref{descent-lemma-fpqc-universal-effective-epimorphisms}." ], "refs": [ "descent-lemma-fpqc-universal-effective-epimorphisms" ], "ref_ids": [ 14638 ] } ], "ref_ids": [ 12416 ] }, { "id": 13762, "type": "theorem", "label": "more-morphisms-lemma-pushout-along-thickening", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pushout-along-thickening", "contents": [ "Let $X \\to X'$ be a thickening of schemes and let $X \\to Y$ be an affine", "morphism of schemes. Then there exists a pushout", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d]_f", "&", "X' \\ar[d]^{f'}", "\\\\", "Y \\ar[r]", "&", "Y'", "}", "$$", "in the category of schemes. Moreover, $Y \\subset Y'$ is a", "thickening, $X = Y \\times_{Y'} X'$, and", "$$", "\\mathcal{O}_{Y'} = \\mathcal{O}_Y \\times_{f_*\\mathcal{O}_X} f'_*\\mathcal{O}_{X'}", "$$", "as sheaves on $|Y| = |Y'|$." ], "refs": [], "proofs": [ { "contents": [ "We first construct $Y'$ as a ringed space. Namely, as topological", "space we take $Y' = Y$. Denote $f' : X' \\to Y'$ the map of topological", "spaces which equals $f$. As structure sheaf $\\mathcal{O}_{Y'}$ we take", "the right hand side of the equation of the lemma. To see that", "$Y'$ is a scheme, we have to show that any point has an affine", "neighbourhood. Since the formation of the fibre product of sheaves", "commutes with restricting to opens, we may assume $Y$ is affine.", "Then $X$ is affine (as $f$ is affine) and $X'$ is affine as well", "(see Lemma \\ref{lemma-thickening-affine-scheme}).", "Say $Y \\leftarrow X \\rightarrow X'$ corresponds", "to $B \\rightarrow A \\leftarrow A'$. Set $B' = B \\times_A A'$; this", "is the global sections of $\\mathcal{O}_{Y'}$. As $A' \\to A$ is surjective", "with locally nilpotent kernel we see that $B' \\to B$ is surjective", "with locally nilpotent kernel. Hence $\\Spec(B') = \\Spec(B)$ (as", "topological spaces). We claim that $Y' = \\Spec(B')$. To see this", "we will show for $g' \\in B'$ with image $g \\in B$ that", "$\\mathcal{O}_{Y'}(D(g)) = B'_{g'}$. Namely, by", "More on Algebra, Lemma \\ref{more-algebra-lemma-diagram-localize} we see that", "$$", "(B')_{g'} = B_g \\times_{A_h} A'_{h'}", "$$", "where $h \\in A$, $h' \\in A'$ are the images of $g'$. Since", "$B_g$, resp.\\ $A_h$, resp.\\ $A'_{h'}$ is equal to $\\mathcal{O}_Y(D(g))$,", "resp.\\ $f_*\\mathcal{O}_X(D(g))$, resp.\\ $f'_*\\mathcal{O}_{X'}(D(g))$ the", "claim follows.", "\\medskip\\noindent", "It remains to show that $Y'$ is the pushout.", "The discussion above shows the scheme $Y'$", "has an affine open covering $Y' = \\bigcup W'_i$", "such that the corresponding opens", "$U'_i \\subset X'$, $W_i \\subset Y$, and", "$U_i \\subset X$ are affine open.", "Moreover, if $A'_i$, $B_i$, $A_i$ are the rings corresponding to", "$U'_i$, $W_i$, $U_i$, then", "$W'_i$ corresponds to $B_i \\times_{A_i} A'_i$.", "Thus we can apply Lemmas \\ref{lemma-basic-example-pushout} and", "\\ref{lemma-pushout-fpqc-local} to conclude our construction is a pushout", "in the category of schemes." ], "refs": [ "more-morphisms-lemma-thickening-affine-scheme", "more-algebra-lemma-diagram-localize", "more-morphisms-lemma-pushout-fpqc-local" ], "ref_ids": [ 13678, 9816, 13761 ] } ], "ref_ids": [] }, { "id": 13763, "type": "theorem", "label": "more-morphisms-lemma-equivalence-categories-schemes-over-pushout", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-equivalence-categories-schemes-over-pushout", "contents": [ "Let $X \\to X'$ be a thickening of schemes and let $X \\to Y$ be an", "affine morphism of schemes. Let $Y' = Y \\amalg_X X'$ be the pushout", "(see Lemma \\ref{lemma-pushout-along-thickening}). Base change gives", "a functor", "$$", "F : (\\Sch/Y') \\longrightarrow (\\Sch/Y) \\times_{(\\Sch/Y')} (\\Sch/X')", "$$", "given by $V' \\longmapsto (V' \\times_{Y'} Y, V' \\times_{Y'} X', 1)$", "which has a left adjoint", "$$", "G : (\\Sch/Y) \\times_{(\\Sch/Y')} (\\Sch/X') \\longrightarrow (\\Sch/Y')", "$$", "which sends the triple $(V, U', \\varphi)$ to the pushout", "$V \\amalg_{(V \\times_Y X)} U'$. Finally, $F \\circ G$ is isomorphic to the", "identity functor." ], "refs": [ "more-morphisms-lemma-pushout-along-thickening" ], "proofs": [ { "contents": [ "Let $(V, U', \\varphi)$ be an object of the fibre product category.", "Set $U = U' \\times_{X'} X$. Note that $U \\to U'$ is a thickening.", "Since $\\varphi : V \\times_Y X \\to U' \\times_{X'} X = U$ is an isomorphism", "we have a morphism $U \\to V$ over $X \\to Y$ which identifies $U$ with", "the fibre product $X \\times_Y V$. In particular $U \\to V$ is affine, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-affine}.", "Hence we can apply Lemma \\ref{lemma-pushout-along-thickening}", "to get a pushout $V' = V \\amalg_U U'$. Denote $V' \\to Y'$ the morphism", "we obtain in virtue of the fact that $V'$ is a pushout and because", "we are given morphisms $V \\to Y$ and $U' \\to X'$ agreeing on $U$", "as morphisms into $Y'$. Setting $G(V, U', \\varphi) = V'$", "gives the functor $G$.", "\\medskip\\noindent", "Let us prove that $G$ is a left adjoint to $F$. Let $Z$ be a scheme", "over $Y'$. We have to show that", "$$", "\\Mor(V', Z) = \\Mor((V, U', \\varphi), F(Z))", "$$", "where the morphism sets are taking in their respective categories.", "Let $g' : V' \\to Z$ be a morphism. Denote $\\tilde g$, resp.\\ $\\tilde f'$", "the composition of $g'$ with the morphism $V \\to V'$, resp.\\ $U' \\to V'$.", "Base change $\\tilde g$, resp.\\ $\\tilde f'$ by $Y \\to Y'$, resp.\\ $X' \\to Y'$", "to get a morphism $g : V \\to Z \\times_{Y'} Y$,", "resp.\\ $f' : U' \\to Z \\times_{Y'} X'$. Then $(g, f')$ is an element", "of the right hand side of the equation above (details omitted).", "Conversely, suppose that $(g, f') : (V, U', \\varphi) \\to F(Z)$ is an", "element of the right hand side.", "We may consider the composition $\\tilde g : V \\to Z$,", "resp.\\ $\\tilde f' : U' \\to Z$ of $g$, resp.\\ $f$ by", "$Z \\times_{Y'} X' \\to Z$, resp.\\ $Z \\times_{Y'} Y \\to Z$.", "Then $\\tilde g$ and $\\tilde f'$ agree as morphism from $U$ to $Z$.", "By the universal property of pushout, we obtain a morphism", "$g' : V' \\to Z$, i.e., an element of the left hand side.", "We omit the verification that these constructions are mutually inverse.", "\\medskip\\noindent", "To prove that $F \\circ G$ is isomorphic to the identity we have to", "show that the adjunction mapping", "$(V, U', \\varphi) \\to F(G(V, U', \\varphi))$ is an isomorphism.", "To do this we may work affine locally. Say $X = \\Spec(A)$, $X' = \\Spec(A')$,", "and $Y = \\Spec(B)$. Then $A' \\to A$ and $B \\to A$ are ring maps as in", "More on Algebra, Lemma \\ref{more-algebra-lemma-module-over-fibre-product}", "and $Y' = \\Spec(B')$ with $B' = B \\times_A A'$. Next, suppose that", "$V = \\Spec(D)$, $U' = \\Spec(C')$ and $\\varphi$ is given by an", "$A$-algebra isomorphism $D \\otimes_B A \\to C' \\otimes_{A'} A = C'/IC'$.", "Set $D' = D \\times_{C'/IC'} C'$. In this case the statement we have to", "prove is that $D' \\otimes_{B'} B \\cong D$ and $D' \\otimes_{B'} A' \\cong C'$.", "This is a special case of More on Algebra, Lemma", "\\ref{more-algebra-lemma-module-over-fibre-product}." ], "refs": [ "morphisms-lemma-base-change-affine", "more-morphisms-lemma-pushout-along-thickening", "more-algebra-lemma-module-over-fibre-product", "more-algebra-lemma-module-over-fibre-product" ], "ref_ids": [ 5176, 13762, 9820, 9820 ] } ], "ref_ids": [ 13762 ] }, { "id": 13764, "type": "theorem", "label": "more-morphisms-lemma-scheme-over-pushout-flat-modules", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-scheme-over-pushout-flat-modules", "contents": [ "Let $X \\to X'$ be a thickening of schemes and let $X \\to Y$ be an", "affine morphism of schemes. Let $Y' = Y \\amalg_X X'$ be the pushout", "(see Lemma \\ref{lemma-pushout-along-thickening}). Let $V' \\to Y'$", "be a morphism of schemes. Set", "$V = Y \\times_{Y'} V'$, $U' = X' \\times_{Y'} V'$, and $U = X \\times_{Y'} V'$.", "There is an equivalence of categories between", "\\begin{enumerate}", "\\item quasi-coherent $\\mathcal{O}_{V'}$-modules flat over $Y'$, and", "\\item the category of triples $(\\mathcal{G}, \\mathcal{F}', \\varphi)$ where", "\\begin{enumerate}", "\\item $\\mathcal{G}$ is a quasi-coherent $\\mathcal{O}_V$-module flat over $Y$,", "\\item $\\mathcal{F}'$ is a quasi-coherent $\\mathcal{O}_{U'}$-module flat", "over $X'$, and", "\\item $\\varphi : (U \\to V)^*\\mathcal{G} \\to (U \\to U')^*\\mathcal{F}'$", "is an isomorphism of $\\mathcal{O}_U$-modules.", "\\end{enumerate}", "\\end{enumerate}", "The equivalence maps $\\mathcal{G}'$ to", "$((V \\to V')^*\\mathcal{G}', (U' \\to V')^*\\mathcal{G}', can)$.", "Suppose $\\mathcal{G}'$ corresponds to the triple", "$(\\mathcal{G}, \\mathcal{F}', \\varphi)$. Then", "\\begin{enumerate}", "\\item[(a)] $\\mathcal{G}'$ is a finite type $\\mathcal{O}_{V'}$-module if and", "only if $\\mathcal{G}$ and $\\mathcal{F}'$ are finite type", "$\\mathcal{O}_Y$ and $\\mathcal{O}_{U'}$-modules.", "\\item[(b)] if $V' \\to Y'$ is locally of finite presentation, then", "$\\mathcal{G}'$ is an $\\mathcal{O}_{V'}$-module of finite", "presentation if and only if $\\mathcal{G}$ and $\\mathcal{F}'$ are", "$\\mathcal{O}_Y$ and $\\mathcal{O}_{U'}$-modules of finite presentation.", "\\end{enumerate}" ], "refs": [ "more-morphisms-lemma-pushout-along-thickening" ], "proofs": [ { "contents": [ "A quasi-inverse functor assigns to the triple", "$(\\mathcal{G}, \\mathcal{F}', \\varphi)$ the fibre product", "$$", "(V \\to V')_*\\mathcal{G}", "\\times_{(U \\to V')_*\\mathcal{F}}", "(U' \\to V')_*\\mathcal{F}'", "$$", "where $\\mathcal{F} = (U \\to U')^*\\mathcal{F}'$. This works, because on", "affines we recover the equivalence of More on Algebra, Lemma", "\\ref{more-algebra-lemma-relative-flat-module-over-fibre-product}.", "Some details omitted.", "\\medskip\\noindent", "Parts (a) and (b) follow from", "More on Algebra, Lemmas", "\\ref{more-algebra-lemma-relative-finite-module-over-fibre-product} and", "\\ref{more-algebra-lemma-relative-finitely-presented-module-over-fibre-product}." ], "refs": [ "more-algebra-lemma-relative-flat-module-over-fibre-product", "more-algebra-lemma-relative-finite-module-over-fibre-product", "more-algebra-lemma-relative-finitely-presented-module-over-fibre-product" ], "ref_ids": [ 9829, 9828, 9830 ] } ], "ref_ids": [ 13762 ] }, { "id": 13765, "type": "theorem", "label": "more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat", "contents": [ "In the situation of", "Lemma \\ref{lemma-equivalence-categories-schemes-over-pushout}.", "If $V' = G(V, U', \\varphi)$ for some triple $(V, U', \\varphi)$, then", "\\begin{enumerate}", "\\item $V' \\to Y'$ is locally of finite type if and only if $V \\to Y$ and", "$U' \\to X'$ are locally of finite type,", "\\item $V' \\to Y'$ is flat if and only if $V \\to Y$ and $U' \\to X'$ are flat,", "\\item $V' \\to Y'$ is flat and locally of finite presentation if and only if", "$V \\to Y$ and $U' \\to X'$ are flat and locally of finite presentation,", "\\item $V' \\to Y'$ is smooth if and only if $V \\to Y$ and $U' \\to X'$ are smooth,", "\\item $V' \\to Y'$ is \\'etale if and only if $V \\to Y$ and $U' \\to X'$", "are \\'etale, and", "\\item add more here as needed.", "\\end{enumerate}", "If $W'$ is flat over $Y'$, then the adjunction mapping", "$G(F(W')) \\to W'$ is an isomorphism. Hence $F$ and $G$ define mutually", "quasi-inverse functors between the category of schemes flat over $Y'$", "and the category of triples $(V, U', \\varphi)$ with $V \\to Y$", "and $U' \\to X'$ flat." ], "refs": [ "more-morphisms-lemma-equivalence-categories-schemes-over-pushout" ], "proofs": [ { "contents": [ "Looking over affine pieces the assertions of this lemma", "are equivalent to the corresponding assertions of", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-properties-algebras-over-fibre-product}." ], "refs": [ "more-algebra-lemma-properties-algebras-over-fibre-product" ], "ref_ids": [ 9831 ] } ], "ref_ids": [ 13763 ] }, { "id": 13766, "type": "theorem", "label": "more-morphisms-lemma-flat-locus-base-change", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-locus-base-change", "contents": [ "Let $S$ be a scheme.", "Let", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "S' \\ar[r]^g & S", "}", "$$", "be a cartesian diagram of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $x' \\in X'$ with images", "$x = g'(x')$ and $s' = g'(x')$.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is flat over $S$ at $x$, then", "$(g')^*\\mathcal{F}$ is flat over $S'$ at $x'$.", "\\item If $g$ is flat at $s'$ and $(g')^*\\mathcal{F}$ is flat over $S'$ at", "$x'$, then $\\mathcal{F}$ is flat over $S$ at $x$.", "\\end{enumerate}", "In particular, if $g$ is flat, $f$ is locally of finite presentation,", "and $\\mathcal{F}$ is locally of finite presentation,", "then formation of the open subset of", "Theorem \\ref{theorem-openness-flatness}", "commutes with base change." ], "refs": [ "more-morphisms-theorem-openness-flatness" ], "proofs": [ { "contents": [ "Consider the commutative diagram of local rings", "$$", "\\xymatrix{", "\\mathcal{O}_{X', x'} & \\mathcal{O}_{X, x} \\ar[l] \\\\", "\\mathcal{O}_{S', s'} \\ar[u] & \\mathcal{O}_{S, s} \\ar[l] \\ar[u]", "}", "$$", "Note that $\\mathcal{O}_{X', x'}$", "is a localization of", "$\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{S, s}} \\mathcal{O}_{S', s'}$,", "and that $((g')^*\\mathcal{F})_{x'}$ is equal to", "$\\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\mathcal{O}_{X', x'}$.", "Hence the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-base-change-flat-up-down}." ], "refs": [ "algebra-lemma-base-change-flat-up-down" ], "ref_ids": [ 898 ] } ], "ref_ids": [ 13670 ] }, { "id": 13767, "type": "theorem", "label": "more-morphisms-lemma-morphism-between-flat-Noetherian", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-morphism-between-flat-Noetherian", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of schemes over $S$.", "Assume", "\\begin{enumerate}", "\\item $S$, $X$, $Y$ are locally Noetherian,", "\\item $X$ is flat over $S$,", "\\item for every $s \\in S$ the morphism", "$f_s : X_s \\to Y_s$ is flat.", "\\end{enumerate}", "Then $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $S$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Theorem \\ref{theorem-criterion-flatness-fibre-Noetherian}." ], "refs": [ "more-morphisms-theorem-criterion-flatness-fibre-Noetherian" ], "ref_ids": [ 13671 ] } ], "ref_ids": [] }, { "id": 13768, "type": "theorem", "label": "more-morphisms-lemma-morphism-between-flat", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-morphism-between-flat", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of schemes over $S$.", "Assume", "\\begin{enumerate}", "\\item $X$ is locally of finite presentation over $S$,", "\\item $X$ is flat over $S$,", "\\item for every $s \\in S$ the morphism", "$f_s : X_s \\to Y_s$ is flat, and", "\\item $Y$ is locally of finite type over $S$.", "\\end{enumerate}", "Then $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $S$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Theorem \\ref{theorem-criterion-flatness-fibre}." ], "refs": [ "more-morphisms-theorem-criterion-flatness-fibre" ], "ref_ids": [ 13672 ] } ], "ref_ids": [] }, { "id": 13769, "type": "theorem", "label": "more-morphisms-lemma-base-change-criterion-flatness-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-criterion-flatness-fibre", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume", "\\begin{enumerate}", "\\item $X$ is locally of finite presentation over $S$,", "\\item $\\mathcal{F}$ an $\\mathcal{O}_X$-module of finite presentation,", "\\item $\\mathcal{F}$ is flat over $S$, and", "\\item $Y$ is locally of finite type over $S$.", "\\end{enumerate}", "Then the set", "$$", "U = \\{x \\in X \\mid \\mathcal{F} \\text{ flat at }x \\text{ over }Y\\}.", "$$", "is open in $X$ and its formation commutes with arbitrary base change:", "If $S' \\to S$ is a morphism of schemes, and $U'$ is the set of points", "of $X' = X \\times_S S'$ where $\\mathcal{F}' = \\mathcal{F} \\times_S S'$", "is flat over $Y' = Y \\times_S S'$, then $U' = U \\times_S S'$." ], "refs": [], "proofs": [ { "contents": [ "By", "Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-permanence}", "the morphism $f$ is locally of finite presentation.", "Hence $U$ is open by", "Theorem \\ref{theorem-openness-flatness}.", "Because we have assumed that $\\mathcal{F}$ is flat over $S$ we see that", "Theorem \\ref{theorem-criterion-flatness-fibre}", "implies", "$$", "U = \\{x \\in X \\mid \\mathcal{F}_s \\text{ flat at }x \\text{ over }Y_s\\}.", "$$", "where $s$ always denotes the image of $x$ in $S$. (This description also", "works trivially when $\\mathcal{F}_x = 0$.) Moreover, the assumptions", "of the lemma remain in force for the morphism $f' : X' \\to Y'$", "and the sheaf $\\mathcal{F}'$. Hence $U'$ has a similar description.", "In other words, it suffices to prove that given", "$s' \\in S'$ mapping to $s \\in S$ we have", "$$", "\\{x' \\in X'_{s'} \\mid", "\\mathcal{F}'_{s'} \\text{ flat at }x' \\text{ over }Y'_{s'}\\}", "$$", "is the inverse image of the corresponding locus in $X_s$.", "This is true by", "Lemma \\ref{lemma-flat-locus-base-change}", "because in the cartesian diagram", "$$", "\\xymatrix{", "X'_{s'} \\ar[d] \\ar[r] & X_s \\ar[d] \\\\", "Y'_{s'} \\ar[r] & Y_s", "}", "$$", "the horizontal morphisms are flat as they are base changes by the flat", "morphism $\\Spec(\\kappa(s')) \\to \\Spec(\\kappa(s))$." ], "refs": [ "morphisms-lemma-finite-presentation-permanence", "more-morphisms-theorem-openness-flatness", "more-morphisms-theorem-criterion-flatness-fibre", "more-morphisms-lemma-flat-locus-base-change" ], "ref_ids": [ 5247, 13670, 13672, 13766 ] } ], "ref_ids": [] }, { "id": 13770, "type": "theorem", "label": "more-morphisms-lemma-base-change-flatness-fibres", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-flatness-fibres", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$.", "Assume", "\\begin{enumerate}", "\\item $X$ is locally of finite presentation over $S$,", "\\item $X$ is flat over $S$, and", "\\item $Y$ is locally of finite type over $S$.", "\\end{enumerate}", "Then the set", "$$", "U = \\{x \\in X \\mid X\\text{ flat at }x \\text{ over }Y\\}.", "$$", "is open in $X$ and its formation commutes with arbitrary base change." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-base-change-criterion-flatness-fibre}." ], "refs": [ "more-morphisms-lemma-base-change-criterion-flatness-fibre" ], "ref_ids": [ 13769 ] } ], "ref_ids": [] }, { "id": 13771, "type": "theorem", "label": "more-morphisms-lemma-flat-and-free-at-point-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-and-free-at-point-fibre", "contents": [ "Let $f : X \\to S$ be a morphism of schemes of finite presentation.", "Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module.", "Let $x \\in X$ with image $s \\in S$.", "If $\\mathcal{F}$ is flat at $x$ over $S$ and $(\\mathcal{F}_s)_x$ is a flat", "$\\mathcal{O}_{X_s, x}$-module, then $\\mathcal{F}$", "is finite free in a neighbourhood of $x$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{F}_x \\otimes \\kappa(x)$ is zero, then $\\mathcal{F}_x = 0$", "by Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK}) and hence", "$\\mathcal{F}$ is zero in a neighbourhood of $x$", "(Modules, Lemma \\ref{modules-lemma-finite-type-stalk-zero})", "and the lemma holds. Thus we may assume $\\mathcal{F}_x \\otimes \\kappa(x)$", "is not zero and we see that", "Theorem \\ref{theorem-criterion-flatness-fibre}", "applies with $f = \\text{id} : X \\to X$. We conclude that $\\mathcal{F}_x$", "is flat over $\\mathcal{O}_{X, x}$. Hence $\\mathcal{F}_x$ is free, see", "Algebra, Lemma \\ref{algebra-lemma-finite-flat-local} for example.", "Choose an open neighbourhood $x \\in U \\subset X$ and sections", "$s_1, \\ldots, s_r \\in \\mathcal{F}(U)$ which map to a basis in", "$\\mathcal{F}_x$. The corresponding map", "$\\psi : \\mathcal{O}_U^{\\oplus r} \\to \\mathcal{F}|_U$ is surjective after", "shrinking $U$ (Modules, Lemma \\ref{modules-lemma-finite-type-stalk-zero}).", "Then $\\Ker(\\psi)$ is of finite type (see Modules, Lemma", "\\ref{modules-lemma-kernel-surjection-finite-free-onto-finite-presentation})", "and $\\Ker(\\psi)_x = 0$. Whence after shrinking $U$ once more", "$\\psi$ is an isomorphism." ], "refs": [ "algebra-lemma-NAK", "modules-lemma-finite-type-stalk-zero", "more-morphisms-theorem-criterion-flatness-fibre", "algebra-lemma-finite-flat-local", "modules-lemma-finite-type-stalk-zero", "modules-lemma-kernel-surjection-finite-free-onto-finite-presentation" ], "ref_ids": [ 401, 13239, 13672, 797, 13239, 13249 ] } ], "ref_ids": [] }, { "id": 13772, "type": "theorem", "label": "more-morphisms-lemma-finite-free-open", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-finite-free-open", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is", "locally of finite presentation.", "Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module", "flat over $S$. Then the set", "$$", "\\{x \\in X : \\mathcal{F}\\text{ free in a neighbourhood of }x\\}", "$$", "is open in $X$ and its formation commutes with arbitrary base change", "$S' \\to S$." ], "refs": [], "proofs": [ { "contents": [ "Openness holds trivially. Let $x \\in X$ mapping to $s \\in S$.", "By Lemma \\ref{lemma-flat-and-free-at-point-fibre}", "we see that $x$ is in our set if and only if", "$\\mathcal{F}|_{X_s}$ is flat at $x$ over $X_s$.", "Clearly this is also equivalent to $\\mathcal{F}$ being", "flat at $x$ over $X$ (because this statement is", "implied by freeness of $\\mathcal{F}_x$ and implies", "flatness of $\\mathcal{F}|_{X_s}$ at $x$ over $X_s$).", "Thus the base change statement follows from", "Lemma \\ref{lemma-base-change-criterion-flatness-fibre}", "applied to $\\text{id} : X \\to X$ over $S$." ], "refs": [ "more-morphisms-lemma-flat-and-free-at-point-fibre", "more-morphisms-lemma-base-change-criterion-flatness-fibre" ], "ref_ids": [ 13771, 13769 ] } ], "ref_ids": [] }, { "id": 13773, "type": "theorem", "label": "more-morphisms-lemma-integral-closure-smooth-pullback", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-integral-closure-smooth-pullback", "contents": [ "Let $f : Y \\to X$ be a smooth morphism of schemes. Let $\\mathcal{A}$ be a", "quasi-coherent sheaf of $\\mathcal{O}_X$-algebras. The integral closure", "of $\\mathcal{O}_Y$ in $f^*\\mathcal{A}$ is equal to $f^*\\mathcal{A}'$", "where $\\mathcal{A}' \\subset \\mathcal{A}$ is the integral closure of", "$\\mathcal{O}_X$ in $\\mathcal{A}$." ], "refs": [], "proofs": [ { "contents": [ "This is a translation of", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-commutes-smooth}", "into the language of schemes. Details omitted." ], "refs": [ "algebra-lemma-integral-closure-commutes-smooth" ], "ref_ids": [ 1252 ] } ], "ref_ids": [] }, { "id": 13774, "type": "theorem", "label": "more-morphisms-lemma-normalization-smooth-localization", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-normalization-smooth-localization", "contents": [ "Let", "$$", "\\xymatrix{", "Y_2 \\ar[r] \\ar[d]_{f_2} & Y_1 \\ar[d]^{f_1} \\\\", "X_2 \\ar[r]^\\varphi & X_1", "}", "$$", "be a fibre square in the category of schemes. Assume $f_1$ is quasi-compact", "and quasi-separated, and $\\varphi$ is smooth.", "Let $Y_i \\to X_i' \\to X_i$ be the normalization of $X_i$ in $Y_i$.", "Then $X_2' \\cong X_2 \\times_{X_1} X_1'$." ], "refs": [], "proofs": [ { "contents": [ "The base change of the factorization $Y_1 \\to X_1' \\to X_1$ to $X_2$", "is a factorization $Y_2 \\to X_2 \\times_{X_1} X_1' \\to X_2$ and", "$X_2 \\times_{X_1} X_1' \\to X_2$ is integral", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite}).", "Hence we get a morphism", "$h : X_2' \\to X_2 \\times_{X_1} X_1'$ by the universal property of", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-normalization}.", "Observe that $X_2'$ is the relative spectrum of the integral closure", "of $\\mathcal{O}_{X_2}$ in $f_{2, *}\\mathcal{O}_{Y_2}$.", "If $\\mathcal{A}' \\subset f_{1, *}\\mathcal{O}_{Y_1}$ denotes the integral", "closure of $\\mathcal{O}_{X_2}$, then $X_2 \\times_{X_1} X_1'$ is the", "relative spectrum of $\\varphi^*\\mathcal{A}'$, see", "Constructions, Lemma \\ref{constructions-lemma-spec-properties}.", "By", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}", "we know that $f_{2, *}\\mathcal{O}_{Y_2} = \\varphi^*f_{1, *}\\mathcal{O}_{Y_1}$.", "Hence the result follows from", "Lemma \\ref{lemma-integral-closure-smooth-pullback}." ], "refs": [ "morphisms-lemma-base-change-finite", "morphisms-lemma-characterize-normalization", "constructions-lemma-spec-properties", "coherent-lemma-flat-base-change-cohomology", "more-morphisms-lemma-integral-closure-smooth-pullback" ], "ref_ids": [ 5440, 5499, 12590, 3298, 13773 ] } ], "ref_ids": [] }, { "id": 13775, "type": "theorem", "label": "more-morphisms-lemma-normalization-and-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-normalization-and-smooth", "contents": [ "Let $X \\to Y$ be a smooth morphism of schemes. Assume every quasi-compact", "open of $Y$ has finitely many irreducible components. Then the same", "is true for $X$ and there is a canonical isomorphism", "$X^\\nu = X \\times_Y Y^\\nu$ where $X^\\nu$, $Y^\\nu$ are the normalizations", "of $X$, $Y$." ], "refs": [], "proofs": [ { "contents": [ "By Descent, Lemma \\ref{descent-lemma-locally-finite-nr-irred-local-fppf}", "every quasi-compact open of $X$ has finitely many irreducible components.", "Note that $X_{red} = X \\times_Y Y_{red}$ as a scheme smooth over a reduced", "scheme is reduced, see", "Descent, Lemma \\ref{descent-lemma-reduced-local-smooth}.", "Hence we may assume that $X$ and $Y$ are reduced (as the normalization", "of a scheme is equal to the normalization of its reduction by", "definition). Next, note that $X' = X \\times_Y Y^\\nu$ is a normal scheme", "by Descent, Lemma \\ref{descent-lemma-normal-local-smooth}.", "The morphism $X' \\to Y^\\nu$ is smooth (hence flat) thus the generic", "points of irreducible components of $X'$ lie over generic points of", "irreducible components of $Y^\\nu$. Since $Y^\\nu \\to Y$ is birational", "we conclude that $X' \\to X$ is birational too (because $X' \\to Y^\\nu$", "induces an isomorphism on fibres over generic points of $Y$).", "We conclude that there exists a factorization", "$X^\\nu \\to X' \\to X$, see", "Morphisms, Lemma \\ref{morphisms-lemma-normalization-normal}", "which is an isomorphism as $X'$ is normal and integral over $X$." ], "refs": [ "descent-lemma-locally-finite-nr-irred-local-fppf", "descent-lemma-reduced-local-smooth", "descent-lemma-normal-local-smooth", "morphisms-lemma-normalization-normal" ], "ref_ids": [ 14650, 14653, 14654, 5515 ] } ], "ref_ids": [] }, { "id": 13776, "type": "theorem", "label": "more-morphisms-lemma-normalization-and-henselization", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-normalization-and-henselization", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $\\nu : X^\\nu \\to X$", "be the normalization morphism. Then for any point $x \\in X$", "the base change", "$$", "X^\\nu \\times_X \\Spec(\\mathcal{O}_{X, x}^h) \\to \\Spec(\\mathcal{O}_{X, x}^h),", "\\quad\\text{resp.}\\quad", "X^\\nu \\times_X \\Spec(\\mathcal{O}_{X, x}^{sh}) \\to \\Spec(\\mathcal{O}_{X, x}^{sh})", "$$", "is the normalization of $\\Spec(\\mathcal{O}_{X, x}^h)$,", "resp.\\ $\\Spec(\\mathcal{O}_{X, x}^{sh})$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\eta_1, \\ldots, \\eta_r$ be the generic points of the irreducible", "components of $X$ passing through $x$. The base change of the normalization to", "$\\Spec(\\mathcal{O}_{X, x})$ is the spectrum of the", "integral closure of $\\mathcal{O}_{X, x}$ in $\\prod \\kappa(\\eta_i)$.", "This follows from our construction of the normalization of $X$ in", "Morphisms, Definition \\ref{morphisms-definition-normalization}", "and Morphisms, Lemma \\ref{morphisms-lemma-integral-closure};", "you can also use the description of the normalization", "in Morphisms, Lemma \\ref{morphisms-lemma-description-normalization}.", "Thus we reduce to the following algebra problem.", "Let $A$ be a Noetherian local ring; recall that this implies", "the henselization $A^h$ and strict henselization $A^{sh}$", "are Noetherian too (More on Algebra, Lemma", "\\ref{more-algebra-lemma-henselization-noetherian}).", "Let $\\mathfrak p_1, \\ldots, \\mathfrak p_r$", "be its minimal primes. Let $A'$ be the integral closure", "of $A$ in $\\prod \\kappa(\\mathfrak p_i)$.", "Problem: show that", "$A' \\otimes_A A^h$, resp.\\ $A' \\otimes_A A^{sh}$ is constructed", "from the Noetherian local ring $A^h$, resp.\\ $A^{sh}$ in the same", "manner.", "\\medskip\\noindent", "Since $A^h$, resp.\\ $A^{sh}$ are colimits of \\'etale $A$-algebras,", "we see that the minimal primes of $A$ and $A^{sh}$ are exactly", "the primes of $A^h$, resp.\\ $A^{sh}$ lying over the minimal primes", "of $A$ (by going down, see", "Algebra, Lemmas \\ref{algebra-lemma-flat-going-down} and", "\\ref{algebra-lemma-minimal-prime-image-minimal-prime}).", "Thus More on Algebra, Lemma \\ref{more-algebra-lemma-fibres-henselization}", "tells us that", "$A^h \\otimes_A \\prod \\kappa(\\mathfrak p_i)$,", "resp.", "$A^{sh} \\otimes_A \\prod \\kappa(\\mathfrak p_i)$", "is the product of the residue fields at the minimal primes", "of $A^h$, resp.\\ $A^{sh}$. We know that taking the", "integral closure in an overring commutes with \\'etale", "base change, see", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-commutes-etale}.", "Writing $A^h$ and $A^{sh}$ as a limit of \\'etale $A$-algebras", "we see that the same thing is true for the base change to", "$A^h$ and $A^{sh}$ (you can also use the more general", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-commutes-colim-smooth})." ], "refs": [ "morphisms-definition-normalization", "morphisms-lemma-integral-closure", "morphisms-lemma-description-normalization", "more-algebra-lemma-henselization-noetherian", "algebra-lemma-flat-going-down", "algebra-lemma-minimal-prime-image-minimal-prime", "more-algebra-lemma-fibres-henselization", "algebra-lemma-integral-closure-commutes-etale", "algebra-lemma-integral-closure-commutes-colim-smooth" ], "ref_ids": [ 5592, 5498, 5513, 10057, 539, 447, 10067, 1251, 1253 ] } ], "ref_ids": [] }, { "id": 13777, "type": "theorem", "label": "more-morphisms-lemma-normal", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-normal", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume all fibres of $f$ are locally Noetherian.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is normal, and", "\\item $f$ is flat and its fibres are geometrically normal schemes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows directly from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13778, "type": "theorem", "label": "more-morphisms-lemma-smooth-normal", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-smooth-normal", "contents": [ "A smooth morphism is normal." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a smooth morphism.", "As $f$ is locally of finite presentation, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-locally-finite-presentation}", "the fibres $X_y$ are locally of finite type over a field, hence", "locally Noetherian. Moreover, $f$ is flat, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-flat}.", "Finally, the fibres $X_y$ are smooth over a field (by", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-smooth})", "and hence geometrically normal by", "Varieties, Lemma \\ref{varieties-lemma-smooth-geometrically-normal}.", "Thus $f$ is normal by", "Lemma \\ref{lemma-normal}." ], "refs": [ "morphisms-lemma-smooth-locally-finite-presentation", "morphisms-lemma-smooth-flat", "morphisms-lemma-base-change-smooth", "varieties-lemma-smooth-geometrically-normal", "more-morphisms-lemma-normal" ], "ref_ids": [ 5330, 5331, 5327, 11005, 13777 ] } ], "ref_ids": [] }, { "id": 13779, "type": "theorem", "label": "more-morphisms-lemma-locally-Noetherian-fibres-fppf-local-source-and-target", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-locally-Noetherian-fibres-fppf-local-source-and-target", "contents": [ "The property $\\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian''", "is local in the fppf topology on the source and the target." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\{\\varphi_i : Y_i \\to Y\\}_{i \\in I}$ be an fppf covering of $Y$.", "Denote $f_i : X_i \\to Y_i$ the base change of $f$ by $\\varphi_i$.", "Let $i \\in I$ and let $y_i \\in Y_i$ be a point.", "Set $y = \\varphi_i(y_i)$. Note that", "$$", "X_{i, y_i} = \\Spec(\\kappa(y_i)) \\times_{\\Spec(\\kappa(y))} X_y.", "$$", "Moreover, as $\\varphi_i$ is of finite presentation the field extension", "$\\kappa(y) \\subset \\kappa(y_i)$ is finitely generated.", "Hence in this situation we have that $X_y$ is locally Noetherian if and", "only if $X_{i, y_i}$ is locally Noetherian, see", "Varieties, Lemma \\ref{varieties-lemma-locally-Noetherian-base-change}.", "This fact implies locality on the target.", "\\medskip\\noindent", "Let $\\{X_i \\to X\\}$ be an fppf covering of $X$.", "Let $y \\in Y$. In this case $\\{X_{i, y} \\to X_y\\}$ is an", "fppf covering of the fibre. Hence the locality on the source", "follows from Descent, Lemma \\ref{descent-lemma-Noetherian-local-fppf}." ], "refs": [ "varieties-lemma-locally-Noetherian-base-change", "descent-lemma-Noetherian-local-fppf" ], "ref_ids": [ 10957, 14648 ] } ], "ref_ids": [] }, { "id": 13780, "type": "theorem", "label": "more-morphisms-lemma-normal-fppf-local-source-and-target", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-normal-fppf-local-source-and-target", "contents": [ "The property", "$\\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian and $f$ is normal''", "is local in the fppf topology on the target and", "local in the smooth topology on the source." ], "refs": [], "proofs": [ { "contents": [ "We have", "$\\mathcal{P}(f) =", "\\mathcal{P}_1(f) \\wedge \\mathcal{P}_2(f) \\wedge \\mathcal{P}_3(f)$", "where", "$\\mathcal{P}_1(f)=$``the fibres of $f$ are locally Noetherian'',", "$\\mathcal{P}_2(f)=$``$f$ is flat'', and", "$\\mathcal{P}_3(f)=$``the fibres of $f$ are geometrically normal''.", "We have already seen that $\\mathcal{P}_1$ and $\\mathcal{P}_2$", "are local in the fppf topology on the source and the target, see", "Lemma \\ref{lemma-locally-Noetherian-fibres-fppf-local-source-and-target},", "and Descent, Lemmas \\ref{descent-lemma-descending-property-flat} and", "\\ref{descent-lemma-flat-fpqc-local-source}. Thus we have to deal", "with $\\mathcal{P}_3$.", "\\medskip\\noindent", "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\{\\varphi_i : Y_i \\to Y\\}_{i \\in I}$ be an fpqc covering of $Y$.", "Denote $f_i : X_i \\to Y_i$ the base change of $f$ by $\\varphi_i$.", "Let $i \\in I$ and let $y_i \\in Y_i$ be a point.", "Set $y = \\varphi_i(y_i)$. Note that", "$$", "X_{i, y_i} = \\Spec(\\kappa(y_i)) \\times_{\\Spec(\\kappa(y))} X_y.", "$$", "Hence in this situation we have that $X_y$ is geometrically normal if and", "only if $X_{i, y_i}$ is geometrically normal, see", "Varieties, Lemma \\ref{varieties-lemma-geometrically-normal-upstairs}.", "This fact implies $\\mathcal{P}_3$ is fpqc local on the target.", "\\medskip\\noindent", "Let $\\{X_i \\to X\\}$ be a smooth covering of $X$.", "Let $y \\in Y$. In this case $\\{X_{i, y} \\to X_y\\}$ is a", "smooth covering of the fibre. Hence the locality of $\\mathcal{P}_3$", "for the smooth topology on the source follows from", "Descent, Lemma \\ref{descent-lemma-normal-local-smooth}.", "Combining the above the lemma follows." ], "refs": [ "more-morphisms-lemma-locally-Noetherian-fibres-fppf-local-source-and-target", "descent-lemma-descending-property-flat", "descent-lemma-flat-fpqc-local-source", "varieties-lemma-geometrically-normal-upstairs", "descent-lemma-normal-local-smooth" ], "ref_ids": [ 13779, 14680, 14708, 10953, 14654 ] } ], "ref_ids": [] }, { "id": 13781, "type": "theorem", "label": "more-morphisms-lemma-regular", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-regular", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume all fibres of $f$ are locally Noetherian.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is regular,", "\\item $f$ is flat and its fibres are geometrically regular schemes,", "\\item for every pair of affine opens $U \\subset X$, $V \\subset Y$", "with $f(U) \\subset V$ the ring map $\\mathcal{O}(V) \\to \\mathcal{O}(U)$", "is regular,", "\\item there exists an open covering $Y = \\bigcup_{j \\in J} V_j$", "and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that each of the morphisms $U_i \\to V_j$ is regular, and", "\\item there exists an affine open covering $Y = \\bigcup_{j \\in J} V_j$", "and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that the ring maps $\\mathcal{O}(V_j) \\to \\mathcal{O}(U_i)$ are regular.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is immediate from the definitions.", "Let $x \\in X$ with $y = f(x)$. By definition $f$ is flat at $x$", "if and only if $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ is a", "flat ring map, and $X_y$ is geometrically regular at $x$ over", "$\\kappa(y)$ if and only if", "$\\mathcal{O}_{X_y, x} = \\mathcal{O}_{X, x}/\\mathfrak m_y\\mathcal{O}_{X, x}$", "is a geometrically regular algebra over $\\kappa(y)$. Hence", "Whether or not $f$ is regular at $x$", "depends only on the local homomorphism of local rings", "$\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$.", "Thus the equivalence of (1) and (4) is clear.", "\\medskip\\noindent", "Recall (More on Algebra, Definition \\ref{more-algebra-definition-regular})", "that a ring map $A \\to B$ is regular if and only if it is flat", "and the fibre rings $B \\otimes_A \\kappa(\\mathfrak p)$ are Noetherian", "and geometrically regular for all primes $\\mathfrak p \\subset A$.", "By Varieties, Lemma \\ref{varieties-lemma-geometrically-regular}", "this is equivalent to $\\Spec(B \\otimes_A \\kappa(\\mathfrak p))$", "being a geometrically regular scheme over $\\kappa(\\mathfrak p)$.", "Thus we see that (2) implies (3). It is clear that (3) implies", "(5). Finally, assume (5). This implies that $f$ is flat", "(see Morphisms, Lemma \\ref{morphisms-lemma-flat-characterize}).", "Moreover, if $y \\in Y$, then $y \\in V_j$ for some $j$ and we see", "that $X_y = \\bigcup_{i \\in I_j} U_{i, y}$ with each $U_{i, y}$", "geometrically regular over $\\kappa(y)$ by", "Varieties, Lemma \\ref{varieties-lemma-geometrically-regular}.", "Another application of", "Varieties, Lemma \\ref{varieties-lemma-geometrically-regular}", "shows that $X_y$ is geometrically regular. Hence (2) holds", "and the proof of the lemma is finished." ], "refs": [ "more-algebra-definition-regular", "varieties-lemma-geometrically-regular", "morphisms-lemma-flat-characterize", "varieties-lemma-geometrically-regular", "varieties-lemma-geometrically-regular" ], "ref_ids": [ 10613, 10959, 5260, 10959, 10959 ] } ], "ref_ids": [] }, { "id": 13782, "type": "theorem", "label": "more-morphisms-lemma-smooth-regular", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-smooth-regular", "contents": [ "A smooth morphism is regular." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a smooth morphism.", "As $f$ is locally of finite presentation, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-locally-finite-presentation}", "the fibres $X_y$ are locally of finite type over a field, hence", "locally Noetherian. Moreover, $f$ is flat, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-flat}.", "Finally, the fibres $X_y$ are smooth over a field (by", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-smooth})", "and hence geometrically regular by", "Varieties, Lemma \\ref{varieties-lemma-smooth-geometrically-normal}.", "Thus $f$ is regular by", "Lemma \\ref{lemma-regular}." ], "refs": [ "morphisms-lemma-smooth-locally-finite-presentation", "morphisms-lemma-smooth-flat", "morphisms-lemma-base-change-smooth", "varieties-lemma-smooth-geometrically-normal", "more-morphisms-lemma-regular" ], "ref_ids": [ 5330, 5331, 5327, 11005, 13781 ] } ], "ref_ids": [] }, { "id": 13783, "type": "theorem", "label": "more-morphisms-lemma-regular-fppf-local-source-and-target", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-regular-fppf-local-source-and-target", "contents": [ "The property $\\mathcal{P}(f)=$``the fibres of $f$ are", "locally Noetherian and $f$ is regular''", "is local in the fppf topology on the target and", "local in the smooth topology on the source." ], "refs": [], "proofs": [ { "contents": [ "We have", "$\\mathcal{P}(f) =", "\\mathcal{P}_1(f) \\wedge \\mathcal{P}_2(f) \\wedge \\mathcal{P}_3(f)$", "where", "$\\mathcal{P}_1(f)=$``the fibres of $f$ are locally Noetherian'',", "$\\mathcal{P}_2(f)=$``$f$ is flat'', and", "$\\mathcal{P}_3(f)=$``the fibres of $f$ are geometrically regular''.", "We have already seen that $\\mathcal{P}_1$ and $\\mathcal{P}_2$", "are local in the fppf topology on the source and the target, see", "Lemma \\ref{lemma-locally-Noetherian-fibres-fppf-local-source-and-target},", "and Descent, Lemmas \\ref{descent-lemma-descending-property-flat} and", "\\ref{descent-lemma-flat-fpqc-local-source}. Thus we have to deal", "with $\\mathcal{P}_3$.", "\\medskip\\noindent", "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\{\\varphi_i : Y_i \\to Y\\}_{i \\in I}$ be an fpqc covering of $Y$.", "Denote $f_i : X_i \\to Y_i$ the base change of $f$ by $\\varphi_i$.", "Let $i \\in I$ and let $y_i \\in Y_i$ be a point.", "Set $y = \\varphi_i(y_i)$. Note that", "$$", "X_{i, y_i} = \\Spec(\\kappa(y_i)) \\times_{\\Spec(\\kappa(y))} X_y.", "$$", "Hence in this situation we have that $X_y$ is geometrically regular if and", "only if $X_{i, y_i}$ is geometrically regular, see", "Varieties, Lemma \\ref{varieties-lemma-geometrically-regular-upstairs}.", "This fact implies $\\mathcal{P}_3$ is fpqc local on the target.", "\\medskip\\noindent", "Let $\\{X_i \\to X\\}$ be a smooth covering of $X$.", "Let $y \\in Y$. In this case $\\{X_{i, y} \\to X_y\\}$ is a", "smooth covering of the fibre. Hence the locality of $\\mathcal{P}_3$", "for the smooth topology on the source follows from", "Descent, Lemma \\ref{descent-lemma-regular-local-smooth}.", "Combining the above the lemma follows." ], "refs": [ "more-morphisms-lemma-locally-Noetherian-fibres-fppf-local-source-and-target", "descent-lemma-descending-property-flat", "descent-lemma-flat-fpqc-local-source", "varieties-lemma-geometrically-regular-upstairs", "descent-lemma-regular-local-smooth" ], "ref_ids": [ 13779, 14680, 14708, 10960, 14656 ] } ], "ref_ids": [] }, { "id": 13784, "type": "theorem", "label": "more-morphisms-lemma-CM", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-CM", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume all fibres of $f$ are locally Noetherian.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is Cohen-Macaulay, and", "\\item $f$ is flat and its fibres are Cohen-Macaulay schemes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows directly from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13785, "type": "theorem", "label": "more-morphisms-lemma-CM-dimension", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-CM-dimension", "contents": [ "Let $f : X \\to Y$ be a morphism of locally Noetherian schemes", "which is locally of finite type and Cohen-Macaulay.", "For every point $x$ in $X$ with image $y$ in $Y$,", "$$", "\\dim_x(X) = \\dim_y(Y) + \\dim_x(X_y),", "$$", "where $X_y$ denotes the fiber over $y$." ], "refs": [], "proofs": [ { "contents": [ "After replacing $X$ by an open neighborhood of $x$,", "there is a natural number $d$ such that all fibers", "of $X \\to Y$ have dimension $d$ at every point, see", "Morphisms, Lemma", "\\ref{morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension}.", "Then $f$ is flat, locally of finite type", "and of relative dimension $d$. Hence the result follows from", "Morphisms, Lemma \\ref{morphisms-lemma-rel-dimension-dimension}." ], "refs": [ "morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension", "morphisms-lemma-rel-dimension-dimension" ], "ref_ids": [ 5286, 5288 ] } ], "ref_ids": [] }, { "id": 13786, "type": "theorem", "label": "more-morphisms-lemma-composition-CM", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-composition-CM", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes. Assume that the", "fibres of $f$, $g$, and $g \\circ f$ are locally Noetherian.", "Let $x \\in X$ with images $y \\in Y$ and $z \\in Z$.", "\\begin{enumerate}", "\\item If $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay", "at $f(x)$, then $g \\circ f$ is Cohen-Macaulay at $x$.", "\\item If $f$ and $g$ are Cohen-Macaulay, then $g \\circ f$ is Cohen-Macaulay.", "\\item If $g \\circ f$ is Cohen-Macaulay at $x$ and $f$ is flat at $x$,", "then $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$.", "\\item If $g \\circ f$ is Cohen-Macaulay and $f$ is flat, then", "$f$ is Cohen-Macaulay and $g$ is Cohen-Macaulay at every point in", "the image of $f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the map of Noetherian local rings", "$$", "\\mathcal{O}_{Y_z, y} \\to \\mathcal{O}_{X_z, x}", "$$", "and observe that its fibre is", "$$", "\\mathcal{O}_{X_z, x}/\\mathfrak m_{Y_z, y}\\mathcal{O}_{X_z, x} =", "\\mathcal{O}_{X_y, x}", "$$", "Thus the lemma this follows from", "Algebra, Lemma \\ref{algebra-lemma-CM-goes-up}." ], "refs": [ "algebra-lemma-CM-goes-up" ], "ref_ids": [ 1362 ] } ], "ref_ids": [] }, { "id": 13787, "type": "theorem", "label": "more-morphisms-lemma-flat-morphism-from-CM-scheme", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-morphism-from-CM-scheme", "contents": [ "Let $f : X \\to Y$ be a flat morphism of locally Noetherian schemes.", "If $X$ is Cohen-Macaulay, then $f$ is Cohen-Macaulay and", "$\\mathcal{O}_{Y, f(x)}$ is Cohen-Macaulay for all $x \\in X$." ], "refs": [], "proofs": [ { "contents": [ "After translating into algebra this follows from", "Algebra, Lemma \\ref{algebra-lemma-CM-goes-up}." ], "refs": [ "algebra-lemma-CM-goes-up" ], "ref_ids": [ 1362 ] } ], "ref_ids": [] }, { "id": 13788, "type": "theorem", "label": "more-morphisms-lemma-base-change-CM", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-CM", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume that all the fibres $X_y$ are locally Noetherian schemes.", "Let $Y' \\to Y$ be locally of finite type. Let $f' : X' \\to Y'$", "be the base change of $f$.", "Let $x' \\in X'$ be a point with image $x \\in X$.", "\\begin{enumerate}", "\\item If $f$ is Cohen-Macaulay at $x$, then", "$f' : X' \\to Y'$ is Cohen-Macaulay at $x'$.", "\\item If $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$, then $f$", "is Cohen-Macaulay at $x$.", "\\item If $Y' \\to Y$ is flat at $f'(x')$ and $f'$ is Cohen-Macaulay at", "$x'$, then $f$ is Cohen-Macaulay at $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that the assumption on $Y' \\to Y$ implies that for $y' \\in Y'$", "mapping to $y \\in Y$ the field extension $\\kappa(y) \\subset \\kappa(y')$", "is finitely generated. Hence also all the fibres", "$X'_{y'} = (X_y)_{\\kappa(y')}$ are locally Noetherian, see", "Varieties, Lemma \\ref{varieties-lemma-locally-Noetherian-base-change}.", "Thus the lemma makes sense. Set $y' = f'(x')$ and $y = f(x)$.", "Hence we get the following commutative diagram of local rings", "$$", "\\xymatrix{", "\\mathcal{O}_{X', x'} & \\mathcal{O}_{X, x} \\ar[l] \\\\", "\\mathcal{O}_{Y', y'} \\ar[u] & \\mathcal{O}_{Y, y} \\ar[l] \\ar[u]", "}", "$$", "where the upper left corner is a localization of the tensor product", "of the upper right and lower left corners over the lower right corner.", "\\medskip\\noindent", "Assume $f$ is Cohen-Macaulay at $x$.", "The flatness of $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$", "implies the flatness of $\\mathcal{O}_{Y', y'} \\to \\mathcal{O}_{X', x'}$, see", "Algebra, Lemma \\ref{algebra-lemma-base-change-flat-up-down}.", "The fact that $\\mathcal{O}_{X, x}/\\mathfrak m_y\\mathcal{O}_{X, x}$", "is Cohen-Macaulay implies that", "$\\mathcal{O}_{X', x'}/\\mathfrak m_{y'}\\mathcal{O}_{X', x'}$", "is Cohen-Macaulay, see", "Varieties, Lemma \\ref{varieties-lemma-CM-base-change}. Hence we see that $f'$", "is Cohen-Macaulay at $x'$.", "\\medskip\\noindent", "Assume $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$.", "The fact that $\\mathcal{O}_{X', x'}/\\mathfrak m_{y'}\\mathcal{O}_{X', x'}$", "is Cohen-Macaulay implies that", "$\\mathcal{O}_{X, x}/\\mathfrak m_y\\mathcal{O}_{X, x}$", "is Cohen-Macaulay, see", "Varieties, Lemma \\ref{varieties-lemma-CM-base-change}.", "Hence we see that $f$ is Cohen-Macaulay at $x$.", "\\medskip\\noindent", "Assume $Y' \\to Y$ is flat at $y'$ and $f'$ is Cohen-Macaulay at", "$x'$. The flatness of $\\mathcal{O}_{Y', y'} \\to \\mathcal{O}_{X', x'}$", "and $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{Y', y'}$ implies the flatness", "of $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$, see", "Algebra, Lemma \\ref{algebra-lemma-base-change-flat-up-down}.", "The fact that $\\mathcal{O}_{X', x'}/\\mathfrak m_{y'}\\mathcal{O}_{X', x'}$", "is Cohen-Macaulay implies that", "$\\mathcal{O}_{X, x}/\\mathfrak m_y\\mathcal{O}_{X, x}$", "is Cohen-Macaulay, see", "Varieties, Lemma \\ref{varieties-lemma-CM-base-change}. Hence we see that $f$", "is Cohen-Macaulay at $x$." ], "refs": [ "varieties-lemma-locally-Noetherian-base-change", "algebra-lemma-base-change-flat-up-down", "varieties-lemma-CM-base-change", "varieties-lemma-CM-base-change", "algebra-lemma-base-change-flat-up-down", "varieties-lemma-CM-base-change" ], "ref_ids": [ 10957, 898, 10963, 10963, 898, 10963 ] } ], "ref_ids": [] }, { "id": 13789, "type": "theorem", "label": "more-morphisms-lemma-flat-finite-presentation-CM-open", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-finite-presentation-CM-open", "contents": [ "\\begin{reference}", "\\cite[IV Corollary 12.1.7(iii)]{EGA}", "\\end{reference}", "Let $f : X \\to S$ be a morphism of schemes which is flat and locally", "of finite presentation. Let", "$$", "W = \\{x \\in X \\mid f\\text{ is Cohen-Macaulay at }x\\}", "$$", "Then", "\\begin{enumerate}", "\\item $W = \\{x \\in X \\mid \\mathcal{O}_{X_{f(x)}, x}\\text{ is Cohen-Macaulay}\\}$,", "\\item $W$ is open in $X$,", "\\item $W$ dense in every fibre of $X \\to S$,", "\\item the formation of $W$ commutes with arbitrary base change of $f$:", "For any morphism $g : S' \\to S$, consider", "the base change $f' : X' \\to S'$ of $f$ and the", "projection $g' : X' \\to X$. Then the corresponding", "set $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "As $f$ is flat with locally Noetherian fibres the equality in (1) holds", "by definition. Parts (2) and (3) follow from", "Algebra, Lemma \\ref{algebra-lemma-generic-CM-flat-finite-presentation}.", "Part (4) follows either from", "Algebra, Lemma \\ref{algebra-lemma-CM-locus-commutes-base-change}", "or", "Varieties, Lemma \\ref{varieties-lemma-CM-base-change}." ], "refs": [ "algebra-lemma-generic-CM-flat-finite-presentation", "algebra-lemma-CM-locus-commutes-base-change", "varieties-lemma-CM-base-change" ], "ref_ids": [ 1125, 1127, 10963 ] } ], "ref_ids": [] }, { "id": 13790, "type": "theorem", "label": "more-morphisms-lemma-flat-finite-presentation-characterize-CM", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-finite-presentation-characterize-CM", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat and locally", "of finite presentation. Let $x \\in X$ with image $s \\in S$.", "Set $d = \\dim_x(X_s)$.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is Cohen-Macaulay at $x$,", "\\item there exists an open neighbourhood $U \\subset X$ of $x$", "and a locally quasi-finite morphism $U \\to \\mathbf{A}^d_S$ over $S$", "which is flat at $x$,", "\\item there exists an open neighbourhood $U \\subset X$ of $x$", "and a locally quasi-finite flat morphism $U \\to \\mathbf{A}^d_S$ over $S$,", "\\item for any $S$-morphism $g : U \\to \\mathbf{A}^d_S$", "of an open neighbourhood $U \\subset X$ of $x$ we have:", "$g$ is quasi-finite at $x$ $\\Rightarrow$ $g$ is flat at $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Openness of flatness shows (2) and (3)", "are equivalent, see Theorem \\ref{theorem-openness-flatness}.", "\\medskip\\noindent", "Choose affine open $U = \\Spec(A) \\subset X$ with $x \\in U$ and", "$V = \\Spec(R) \\subset S$ with $f(U) \\subset V$. Then $R \\to A$", "is a flat ring map of finite presentation. Let $\\mathfrak p \\subset A$", "be the prime ideal corresponding to $x$. After replacing $A$ by a", "principal localization we may assume there exists a quasi-finite map", "$R[x_1, \\ldots, x_d] \\to A$, see", "Algebra, Lemma \\ref{algebra-lemma-quasi-finite-over-polynomial-algebra}.", "Thus there exists at least one pair $(U, g)$ consisting of an", "open neighbourhood $U \\subset X$ of $x$ and a locally\\footnote{If $S$", "is quasi-separated, then $g$ will be quasi-finite.} quasi-finite morphism", "$g : U \\to \\mathbf{A}^d_S$.", "\\medskip\\noindent", "Claim: Given $R \\to A$ flat and of finite presentation, a prime", "$\\mathfrak p \\subset A$ and $\\varphi : R[x_1, \\ldots, x_d] \\to A$", "quasi-finite at $\\mathfrak p$ we have: $\\Spec(\\varphi)$", "is flat at $\\mathfrak p$ if and only if $\\Spec(A) \\to \\Spec(R)$", "is Cohen-Macaulay at $\\mathfrak p$. Namely, by", "Theorem \\ref{theorem-criterion-flatness-fibre}", "flatness may be checked on fibres. The same is true", "for being Cohen-Macaulay (as $A$ is already assumed flat over $R$).", "Thus the claim follows from", "Algebra, Lemma \\ref{algebra-lemma-where-CM}.", "\\medskip\\noindent", "The claim shows that (1) is equivalent to (4) and combined with the", "fact that we have constructed a suitable $(U, g)$ in the second", "paragraph, the claim also shows that (1) is equivalent to (2)." ], "refs": [ "more-morphisms-theorem-openness-flatness", "algebra-lemma-quasi-finite-over-polynomial-algebra", "more-morphisms-theorem-criterion-flatness-fibre", "algebra-lemma-where-CM" ], "ref_ids": [ 13670, 1071, 13672, 1120 ] } ], "ref_ids": [] }, { "id": 13791, "type": "theorem", "label": "more-morphisms-lemma-flat-finite-presentation-CM-pieces", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-finite-presentation-CM-pieces", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat and locally", "of finite presentation. For $d \\geq 0$ there exist opens $U_d \\subset X$", "with the following properties", "\\begin{enumerate}", "\\item $W = \\bigcup_{d \\geq 0} U_d$ is dense in every fibre of $f$, and", "\\item $U_d \\to S$ is of relative dimension $d$ (see", "Morphisms, Definition \\ref{morphisms-definition-relative-dimension-d}).", "\\end{enumerate}" ], "refs": [ "morphisms-definition-relative-dimension-d" ], "proofs": [ { "contents": [ "This follows by combining", "Lemma \\ref{lemma-flat-finite-presentation-CM-open}", "with", "Morphisms, Lemma", "\\ref{morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension}." ], "refs": [ "more-morphisms-lemma-flat-finite-presentation-CM-open", "morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension" ], "ref_ids": [ 13789, 5286 ] } ], "ref_ids": [ 5559 ] }, { "id": 13792, "type": "theorem", "label": "more-morphisms-lemma-flat-finite-presentation-specialization-dimension", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-finite-presentation-specialization-dimension", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat and locally", "of finite presentation.", "Suppose $x' \\leadsto x$ is a specialization of points of $X$", "with image $s' \\leadsto s$ in $S$. If $x$ is a generic point of an", "irreducible component of $X_s$ then $\\dim_{x'}(X_{s'}) = \\dim_x(X_s)$." ], "refs": [], "proofs": [ { "contents": [ "The point $x$ is contained in $U_d$ for some $d$, where $U_d$ as in", "Lemma \\ref{lemma-flat-finite-presentation-CM-pieces}." ], "refs": [ "more-morphisms-lemma-flat-finite-presentation-CM-pieces" ], "ref_ids": [ 13791 ] } ], "ref_ids": [] }, { "id": 13793, "type": "theorem", "label": "more-morphisms-lemma-CM-local-source-and-target", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-CM-local-source-and-target", "contents": [ "The property", "$\\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian and $f$ is", "Cohen-Macaulay'' is local in the fppf topology on the target and", "local in the syntomic topology on the source." ], "refs": [], "proofs": [ { "contents": [ "We have", "$\\mathcal{P}(f) =", "\\mathcal{P}_1(f) \\wedge \\mathcal{P}_2(f)$", "where", "$\\mathcal{P}_1(f)=$``$f$ is flat'', and", "$\\mathcal{P}_2(f)=$``the fibres of $f$ are locally Noetherian", "and Cohen-Macaulay''.", "We know that $\\mathcal{P}_1$ is", "local in the fppf topology on the source and the target, see", "Descent, Lemmas \\ref{descent-lemma-descending-property-flat} and", "\\ref{descent-lemma-flat-fpqc-local-source}. Thus we have to deal", "with $\\mathcal{P}_2$.", "\\medskip\\noindent", "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\{\\varphi_i : Y_i \\to Y\\}_{i \\in I}$ be an fppf covering of $Y$.", "Denote $f_i : X_i \\to Y_i$ the base change of $f$ by $\\varphi_i$.", "Let $i \\in I$ and let $y_i \\in Y_i$ be a point.", "Set $y = \\varphi_i(y_i)$. Note that", "$$", "X_{i, y_i} = \\Spec(\\kappa(y_i)) \\times_{\\Spec(\\kappa(y))} X_y.", "$$", "and that $\\kappa(y) \\subset \\kappa(y_i)$ is a finitely generated field", "extension. Hence if $X_y$ is locally Noetherian, then", "$X_{i, y_i}$ is locally Noetherian, see", "Varieties, Lemma \\ref{varieties-lemma-locally-Noetherian-base-change}.", "And if in addition $X_y$ is Cohen-Macaulay,", "then $X_{i, y_i}$ is Cohen-Macaulay, see", "Varieties, Lemma \\ref{varieties-lemma-CM-base-change}.", "Thus $\\mathcal{P}_2$ is fppf local on the target.", "\\medskip\\noindent", "Let $\\{X_i \\to X\\}$ be a syntomic covering of $X$.", "Let $y \\in Y$. In this case $\\{X_{i, y} \\to X_y\\}$ is a", "syntomic covering of the fibre. Hence the locality of $\\mathcal{P}_2$", "for the syntomic topology on the source follows from", "Descent, Lemma \\ref{descent-lemma-CM-local-syntomic}.", "Combining the above the lemma follows." ], "refs": [ "descent-lemma-descending-property-flat", "descent-lemma-flat-fpqc-local-source", "varieties-lemma-locally-Noetherian-base-change", "varieties-lemma-CM-base-change", "descent-lemma-CM-local-syntomic" ], "ref_ids": [ 14680, 14708, 10957, 10963, 14652 ] } ], "ref_ids": [] }, { "id": 13794, "type": "theorem", "label": "more-morphisms-lemma-slice-given-element", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-slice-given-element", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point with image $s \\in S$.", "Let $h \\in \\mathfrak m_x \\subset \\mathcal{O}_{X, x}$.", "Assume", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $f$ is flat at $x$, and", "\\item the image $\\overline{h}$ of $h$ in", "$\\mathcal{O}_{X_s, x} = \\mathcal{O}_{X, x}/\\mathfrak m_s\\mathcal{O}_{X, x}$", "is a nonzerodivisor.", "\\end{enumerate}", "Then there exists an affine open neighbourhood $U \\subset X$ of $x$", "such that $h$ comes from $h \\in \\Gamma(U, \\mathcal{O}_U)$ and such", "that $D = V(h)$ is an effective Cartier divisor in $U$ with $x \\in D$ and", "$D \\to S$ flat and locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "We are going to prove this by reducing to the Noetherian case.", "By openness of flatness (see", "Theorem \\ref{theorem-openness-flatness})", "we may assume, after replacing $X$ by an", "open neighbourhood of $x$, that $X \\to S$ is flat.", "We may also assume that $X$ and $S$ are affine.", "After possible shrinking $X$ a bit we may assume that there exists", "an $h \\in \\Gamma(X, \\mathcal{O}_X)$ which maps to our given $h$.", "\\medskip\\noindent", "We may write $S = \\Spec(A)$ and we may write $A = \\colim_i A_i$", "as a directed colimit of finite type $\\mathbf{Z}$ algebras.", "Then by", "Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "or", "Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation},", "\\ref{limits-lemma-descend-affine-finite-presentation}, and", "\\ref{limits-lemma-descend-finite-presentation}", "we can find a cartesian diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d]_f & X_0 \\ar[d]^{f_0} \\\\", "S \\ar[r] & S_0", "}", "$$", "with $f_0$ flat and of finite presentation, $X_0$ affine, and", "$S_0$ affine and Noetherian. Let $x_0 \\in X_0$, resp.\\ $s_0 \\in S_0$", "be the image of $x$, resp.\\ $s$. We may also assume there exists an element", "$h_0 \\in \\Gamma(X_0, \\mathcal{O}_{X_0})$ which restricts to $h$ on $X$.", "(If you used the algebra reference above then this is clear; if you used", "the references to the chapter on limits then this follows from", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "by thinking of $h$ as a morphism $X \\to \\mathbf{A}^1_S$.)", "Note that $\\mathcal{O}_{X_s, x}$ is a localization of", "$\\mathcal{O}_{(X_0)_{s_0}, x_0} \\otimes_{\\kappa(s_0)} \\kappa(s)$, so that", "$\\mathcal{O}_{(X_0)_{s_0}, x_0} \\to \\mathcal{O}_{X_s, x}$ is a flat", "local ring map, in particular faithfully flat. Hence the image", "$\\overline{h}_0 \\in \\mathcal{O}_{(X_0)_{s_0}, x_0}$", "is contained in $\\mathfrak m_{(X_0)_{s_0}, x_0}$ and is a nonzerodivisor.", "We claim that after replacing $X_0$ by a principal open neighbourhood of", "$x_0$ the element $h_0$ is a nonzerodivisor in", "$B_0 = \\Gamma(X_0, \\mathcal{O}_{X_0})$ such that $B_0/h_0B_0$ is flat", "over $A_0 = \\Gamma(S_0, \\mathcal{O}_{S_0})$.", "If so then", "$$", "0 \\to B_0 \\xrightarrow{h_0} B_0 \\to B_0/h_0B_0 \\to 0", "$$", "is a short exact sequence of flat $A_0$-modules. Hence this remains exact", "on tensoring with $A$ (by", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero})", "and the lemma follows.", "\\medskip\\noindent", "It remains to prove the claim above. The corresponding algebra statement", "is the following (we drop the subscript ${}_0$ here):", "Let $A \\to B$ be a flat, finite type ring map of Noetherian rings.", "Let $\\mathfrak q \\subset B$ be a prime lying over $\\mathfrak p \\subset A$.", "Assume $h \\in \\mathfrak q$ maps to a nonzerodivisor in", "$B_{\\mathfrak q}/\\mathfrak p B_{\\mathfrak q}$.", "Goal: show that after possible replacing $B$ by $B_g$ for some", "$g \\in B$, $g \\not \\in \\mathfrak q$ the element $h$ becomes a nonzerodivisor", "and $B/hB$ becomes flat over $A$. By", "Algebra, Lemma \\ref{algebra-lemma-grothendieck}", "we see that $h$ is a nonzerodivisor in $B_{\\mathfrak q}$ and that", "$B_{\\mathfrak q}/hB_{\\mathfrak q}$ is flat over $A$.", "By openness of flatness, see", "Algebra, Theorem \\ref{algebra-theorem-openness-flatness}", "or", "Theorem \\ref{theorem-openness-flatness}", "we see that $B/hB$ is flat over $A$ after replacing $B$ by $B_g$ for some", "$g \\in B$, $g \\not \\in \\mathfrak q$. Finally, let", "$I = \\{b \\in B \\mid hb = 0\\}$ be the annihilator of $h$. Then", "$IB_{\\mathfrak q} = 0$ as $h$ is a nonzerodivisor in $B_{\\mathfrak q}$.", "Also $I$ is finitely generated as $B$ is Noetherian.", "Hence there exists a $g \\in B$, $g \\not \\in \\mathfrak q$ such that $IB_g = 0$.", "After replacing $B$ by $B_g$ we see that $h$ is a nonzerodivisor." ], "refs": [ "more-morphisms-theorem-openness-flatness", "algebra-lemma-flat-finite-presentation-limit-flat", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-affine-finite-presentation", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-finite-presentation", "algebra-lemma-flat-tor-zero", "algebra-lemma-grothendieck", "algebra-theorem-openness-flatness", "more-morphisms-theorem-openness-flatness" ], "ref_ids": [ 13670, 1389, 15077, 15057, 15077, 15077, 532, 884, 326, 13670 ] } ], "ref_ids": [] }, { "id": 13795, "type": "theorem", "label": "more-morphisms-lemma-slice-given-elements", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-slice-given-elements", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point with image $s \\in S$.", "Let $h_1, \\ldots, h_r \\in \\mathcal{O}_{X, x}$.", "Assume", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $f$ is flat at $x$, and", "\\item the images of $h_1, \\ldots, h_r$ in", "$\\mathcal{O}_{X_s, x} = \\mathcal{O}_{X, x}/\\mathfrak m_s\\mathcal{O}_{X, x}$", "form a regular sequence.", "\\end{enumerate}", "Then there exists an affine open neighbourhood $U \\subset X$ of $x$", "such that $h_1, \\ldots, h_r$ come from", "$h_1, \\ldots, h_r \\in \\Gamma(U, \\mathcal{O}_U)$ and such", "that $Z = V(h_1, \\ldots, h_r) \\to U$ is a regular immersion with", "$x \\in Z$ and $Z \\to S$ flat and locally of finite presentation.", "Moreover, the base change $Z_{S'} \\to U_{S'}$ is a regular immersion", "for any scheme $S'$ over $S$." ], "refs": [], "proofs": [ { "contents": [ "(Our conventions on regular sequences imply that $h_i \\in \\mathfrak m_x$", "for each $i$.) The case $r = 1$ follows from", "Lemma \\ref{lemma-slice-given-element}", "combined with", "Divisors, Lemma \\ref{divisors-lemma-relative-Cartier}", "to see that $V(h_1)$ remains an effective Cartier divisor after base change.", "The case $r > 1$ follows from a straightforward induction on $r$ (applying", "the result for $r = 1$ exactly $r$ times; details omitted).", "\\medskip\\noindent", "Another way to prove the lemma is using the material from", "Divisors, Section \\ref{divisors-section-relative-regular-immersion}.", "Namely, first by openness of flatness (see", "Theorem \\ref{theorem-openness-flatness})", "we may assume, after replacing $X$ by an", "open neighbourhood of $x$, that $X \\to S$ is flat.", "We may also assume that $X$ and $S$ are affine.", "After possible shrinking $X$ a bit we may assume that we have", "$h_1, \\ldots, h_r \\in \\Gamma(X, \\mathcal{O}_X)$. Set", "$Z = V(h_1, \\ldots, h_r)$. Note that $X_s$ is a Noetherian scheme", "(because it is an algebraic $\\kappa(s)$-scheme, see", "Varieties, Section \\ref{varieties-section-algebraic-schemes})", "and that the topology on $X_s$ is induced from the topology on $X$", "(see", "Schemes, Lemma \\ref{schemes-lemma-fibre-topological}).", "Hence after shrinking $X$ a bit more", "we may assume that $Z_s \\subset X_s$ is a regular immersion", "cut out by the $r$ elements $h_i|_{X_s}$, see", "Divisors, Lemma \\ref{divisors-lemma-Noetherian-scheme-regular-ideal}", "and its proof. It is also clear that $r = \\dim_x(X_s) - \\dim_x(Z_s)$ because", "\\begin{align*}", "\\dim_x(X_s) & = \\dim(\\mathcal{O}_{X_s, x}) +", "\\text{trdeg}_{\\kappa(s)}(\\kappa(x)), \\\\", "\\dim_x(Z_s) & = \\dim(\\mathcal{O}_{Z_s, x}) +", "\\text{trdeg}_{\\kappa(s)}(\\kappa(x)), \\\\", "\\dim(\\mathcal{O}_{X_s, x}) & = \\dim(\\mathcal{O}_{Z_s, x}) + r", "\\end{align*}", "the first two equalities by", "Algebra, Lemma \\ref{algebra-lemma-dimension-at-a-point-finite-type-field}", "and the second by $r$ times applying", "Algebra, Lemma \\ref{algebra-lemma-one-equation}.", "Hence", "Divisors, Lemma \\ref{divisors-lemma-fibre-quasi-regular} part (3)", "applies to show that (after Zariski shrinking $X$) the morphism", "$Z \\to X$ is a regular immersion to which", "Divisors, Lemma", "\\ref{divisors-lemma-relative-regular-immersion-flat-in-neighbourhood}", "applies (which gives the flatness and the statement on base change)." ], "refs": [ "more-morphisms-lemma-slice-given-element", "divisors-lemma-relative-Cartier", "more-morphisms-theorem-openness-flatness", "schemes-lemma-fibre-topological", "divisors-lemma-Noetherian-scheme-regular-ideal", "algebra-lemma-dimension-at-a-point-finite-type-field", "divisors-lemma-fibre-quasi-regular", "divisors-lemma-relative-regular-immersion-flat-in-neighbourhood" ], "ref_ids": [ 13794, 7972, 13670, 7696, 7988, 1007, 8003, 8001 ] } ], "ref_ids": [] }, { "id": 13796, "type": "theorem", "label": "more-morphisms-lemma-slice-once", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-slice-once", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point with image $s \\in S$.", "Assume", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $f$ is flat at $x$, and", "\\item $\\mathcal{O}_{X_s, x}$ has $\\text{depth} \\geq 1$.", "\\end{enumerate}", "Then there exists an affine open neighbourhood $U \\subset X$ of $x$", "and an effective Cartier divisor $D \\subset U$ containing $x$ such that", "$D \\to S$ is flat and of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Pick any $h \\in \\mathfrak m_x \\subset \\mathcal{O}_{X, x}$ which", "maps to a nonzerodivisor in $\\mathcal{O}_{X_s, x}$ and apply", "Lemma \\ref{lemma-slice-given-element}." ], "refs": [ "more-morphisms-lemma-slice-given-element" ], "ref_ids": [ 13794 ] } ], "ref_ids": [] }, { "id": 13797, "type": "theorem", "label": "more-morphisms-lemma-slice-CM", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-slice-CM", "contents": [ "\\begin{reference}", "\\cite[IV Proposition 17.16.1]{EGA}", "\\end{reference}", "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point with image $s \\in S$.", "Assume", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $f$ is Cohen-Macaulay at $x$, and", "\\item $x$ is a closed point of $X_s$.", "\\end{enumerate}", "Then there exists a regular immersion $Z \\to X$ containing $x$ such that", "\\begin{enumerate}", "\\item[(a)] $Z \\to S$ is flat and locally of finite presentation,", "\\item[(b)] $Z \\to S$ is locally quasi-finite, and", "\\item[(c)] $Z_s = \\{x\\}$ set theoretically.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may and do replace $S$ by an affine open neighbourhood of $s$.", "We will prove the lemma for affine $S$ by induction on $d = \\dim_x(X_s)$.", "\\medskip\\noindent", "The case $d = 0$. In this case we show that we may take $Z$ to be", "an open neighbourhood of $x$. (Note that an open immersion is", "a regular immersion.) Namely, if $d = 0$, then $X \\to S$", "is quasi-finite at $x$, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}.", "Hence there exists an affine open neighbourhood $U \\subset X$ such", "that $U \\to S$ is quasi-finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-points-open}.", "Thus after replacing $X$ by $U$ we see that the fibre $X_s$ is a finite", "discrete set. Hence after replacing $X$ by a further affine open neighbourhood", "of $X$ we see that $f^{-1}(\\{s\\}) = \\{x\\}$ (because the topology", "on $X_s$ is induced from the topology on $X$, see", "Schemes, Lemma \\ref{schemes-lemma-fibre-topological}).", "This proves the lemma in this case.", "\\medskip\\noindent", "Next, assume $d > 0$. Note that because $x$ is a closed point of its", "fibre the extension $\\kappa(s) \\subset \\kappa(x)$ is finite (by the", "Hilbert Nullstellensatz, see", "Morphisms, Lemma \\ref{morphisms-lemma-closed-point-fibre-locally-finite-type}).", "Thus we see", "$$", "\\text{depth}(\\mathcal{O}_{X_s, x}) = \\dim(\\mathcal{O}_{X_s, x}) = d > 0", "$$", "the first equality as $\\mathcal{O}_{X_s, x}$ is Cohen-Macaulay and", "the second by", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}.", "Thus we may apply", "Lemma \\ref{lemma-slice-once}", "to find a diagram", "$$", "\\xymatrix{", "D \\ar[r] \\ar[rrd] & U \\ar[r] \\ar[rd] & X \\ar[d] \\\\", "& & S", "}", "$$", "with $x \\in D$. Note that", "$\\mathcal{O}_{D_s, x} = \\mathcal{O}_{X_s, x}/(\\overline{h})$ for some", "nonzerodivisor $\\overline{h}$, see", "Divisors, Lemma \\ref{divisors-lemma-relative-Cartier}.", "Hence $\\mathcal{O}_{D_s, x}$ is Cohen-Macaulay of dimension", "one less than the dimension of $\\mathcal{O}_{X_s, x}$, see", "Algebra, Lemma \\ref{algebra-lemma-reformulate-CM}", "for example. Thus the morphism $D \\to S$ is flat,", "locally of finite presentation, and Cohen-Macaulay at $x$ with", "$\\dim_x(D_s) = \\dim_x(X_s) - 1 = d - 1$. By induction hypothesis", "we can find a regular immersion $Z \\to D$ having properties (a), (b), (c).", "As $Z \\to D \\to U$ are both regular immersions, we see that also", "$Z \\to U$ is a regular immersion by", "Divisors, Lemma \\ref{divisors-lemma-composition-regular-immersion}.", "This finishes the proof." ], "refs": [ "morphisms-lemma-locally-quasi-finite-rel-dimension-0", "morphisms-lemma-quasi-finite-points-open", "schemes-lemma-fibre-topological", "morphisms-lemma-closed-point-fibre-locally-finite-type", "morphisms-lemma-dimension-fibre-at-a-point", "more-morphisms-lemma-slice-once", "divisors-lemma-relative-Cartier", "algebra-lemma-reformulate-CM", "divisors-lemma-composition-regular-immersion" ], "ref_ids": [ 5287, 5521, 7696, 5223, 5277, 13796, 7972, 923, 7994 ] } ], "ref_ids": [] }, { "id": 13798, "type": "theorem", "label": "more-morphisms-lemma-qf-fp-flat-neighbourhood-dominates-fppf", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-qf-fp-flat-neighbourhood-dominates-fppf", "contents": [ "Let $f : X \\to S$ be a flat morphism of schemes which is", "locally of finite presentation. Let $s \\in S$ be a point in the image of $f$.", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "S' \\ar[rr] \\ar[rd]_g & & X \\ar[ld]^f \\\\", "& S", "}", "$$", "where $g : S' \\to S$ is flat, locally of finite presentation,", "locally quasi-finite, and $s \\in g(S')$." ], "refs": [], "proofs": [ { "contents": [ "The fibre $X_s$ is not empty by assumption. Hence there exists a closed", "point $x \\in X_s$ where $f$ is Cohen-Macaulay, see", "Lemma \\ref{lemma-flat-finite-presentation-CM-open}.", "Apply", "Lemma \\ref{lemma-slice-CM}", "and set $S' = S$." ], "refs": [ "more-morphisms-lemma-flat-finite-presentation-CM-open", "more-morphisms-lemma-slice-CM" ], "ref_ids": [ 13789, 13797 ] } ], "ref_ids": [] }, { "id": 13799, "type": "theorem", "label": "more-morphisms-lemma-qf-fp-flat-dominates-fppf", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-qf-fp-flat-dominates-fppf", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{U} = \\{S_i \\to S\\}_{i \\in I}$ be an fppf", "covering of $S$, see", "Topologies, Definition \\ref{topologies-definition-fppf-covering}.", "Then there exists an fppf covering $\\mathcal{V} = \\{T_j \\to S\\}_{j \\in J}$", "which refines (see", "Sites, Definition \\ref{sites-definition-morphism-coverings})", "$\\mathcal{U}$ such that each $T_j \\to S$ is locally quasi-finite." ], "refs": [ "topologies-definition-fppf-covering", "sites-definition-morphism-coverings" ], "proofs": [ { "contents": [ "For every $s \\in S$ there exists an $i \\in I$ such that $s$ is in", "the image of $S_i \\to S$. By", "Lemma \\ref{lemma-qf-fp-flat-neighbourhood-dominates-fppf}", "we can find a morphism $g_s : T_s \\to S$ such that $s \\in g_s(T_s)$", "which is flat, locally of finite presentation and locally quasi-finite", "and such that $g_s$ factors through $S_i \\to S$. Hence", "$\\{T_s \\to S\\}$ is the desired covering of $S$ that refines $\\mathcal{U}$." ], "refs": [ "more-morphisms-lemma-qf-fp-flat-neighbourhood-dominates-fppf" ], "ref_ids": [ 13798 ] } ], "ref_ids": [ 12539, 8656 ] }, { "id": 13800, "type": "theorem", "label": "more-morphisms-lemma-empty-generic-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-empty-generic-fibre", "contents": [ "Let $f : X \\to Y$ be a finite type morphism of schemes. Assume", "$Y$ irreducible with generic point $\\eta$. If $X_\\eta = \\emptyset$", "then there exists a nonempty open $V \\subset Y$ such that", "$X_V = V \\times_Y X = \\emptyset$." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from the more general", "Morphisms,", "Lemma \\ref{morphisms-lemma-quasi-compact-generic-point-not-in-image}." ], "refs": [ "morphisms-lemma-quasi-compact-generic-point-not-in-image" ], "ref_ids": [ 5160 ] } ], "ref_ids": [] }, { "id": 13801, "type": "theorem", "label": "more-morphisms-lemma-nonempty-generic-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-nonempty-generic-fibre", "contents": [ "Let $f : X \\to Y$ be a finite type morphism of schemes. Assume", "$Y$ irreducible with generic point $\\eta$. If $X_\\eta \\not = \\emptyset$", "then there exists a nonempty open $V \\subset Y$ such that", "$X_V = V \\times_Y X \\to V$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "This follows, upon taking affine opens, from", "Algebra, Lemma \\ref{algebra-lemma-characterize-image-finite-type}.", "(Of course it also follows from generic flatness.)" ], "refs": [ "algebra-lemma-characterize-image-finite-type" ], "ref_ids": [ 442 ] } ], "ref_ids": [] }, { "id": 13802, "type": "theorem", "label": "more-morphisms-lemma-nowhere-dense-generic-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-nowhere-dense-generic-fibre", "contents": [ "Let $f : X \\to Y$ be a finite type morphism of schemes. Assume", "$Y$ irreducible with generic point $\\eta$.", "If $Z \\subset X$ is a closed subset with $Z_\\eta$ nowhere dense", "in $X_\\eta$, then there exists a nonempty open $V \\subset Y$ such", "that $Z_y$ is nowhere dense in $X_y$ for all $y \\in V$." ], "refs": [], "proofs": [ { "contents": [ "Let $Y' \\subset Y$ be the reduction of $Y$.", "Set $X' = Y' \\times_Y X$ and $Z' = Y' \\times_Y Z$.", "As $Y' \\to Y$ is a universal homeomorphism by", "Morphisms, Lemma \\ref{morphisms-lemma-reduction-universal-homeomorphism}", "we see that it suffices to prove the lemma for $Z' \\subset X' \\to Y'$.", "Thus we may assume that $Y$ is integral, see", "Properties, Lemma \\ref{properties-lemma-characterize-integral}.", "By", "Morphisms, Proposition \\ref{morphisms-proposition-generic-flatness}", "there exists a nonempty affine open $V \\subset Y$ such that", "$X_V \\to V$ and $Z_V \\to V$ are flat and of finite presentation.", "We claim that $V$ works.", "Pick $y \\in V$. If $Z_y$ has a nonempty interior, then $Z_y$ contains", "a generic point $\\xi$ of an irreducible component of $X_y$.", "Note that $\\eta \\leadsto f(\\xi)$. Since $Z_V \\to V$ is flat we can", "choose a specialization $\\xi' \\leadsto \\xi$, $\\xi' \\in Z$", "with $f(\\xi') = \\eta$, see", "Morphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat}.", "By", "Lemma \\ref{lemma-flat-finite-presentation-specialization-dimension}", "we see that", "$$", "\\dim_{\\xi'}(Z_\\eta) = \\dim_{\\xi}(Z_y) = \\dim_{\\xi}(X_y) = \\dim_{\\xi'}(X_\\eta).", "$$", "Hence some irreducible component of $Z_\\eta$ passing through $\\xi'$ has", "dimension $\\dim_{\\xi'}(X_\\eta)$ which contradicts the assumption that", "$Z_\\eta$ is nowhere dense in $X_\\eta$ and we win." ], "refs": [ "morphisms-lemma-reduction-universal-homeomorphism", "properties-lemma-characterize-integral", "morphisms-proposition-generic-flatness", "morphisms-lemma-generalizations-lift-flat", "more-morphisms-lemma-flat-finite-presentation-specialization-dimension" ], "ref_ids": [ 5455, 2947, 5533, 5266, 13792 ] } ], "ref_ids": [] }, { "id": 13803, "type": "theorem", "label": "more-morphisms-lemma-scheme-theoretically-dense-generic-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-scheme-theoretically-dense-generic-fibre", "contents": [ "Let $f : X \\to Y$ be a finite type morphism of schemes.", "Assume $Y$ irreducible with generic point $\\eta$.", "Let $U \\subset X$ be an open subscheme such that $U_\\eta$ is", "scheme theoretically dense in $X_\\eta$.", "Then there exists a nonempty open $V \\subset Y$ such", "that $U_y$ is scheme theoretically dense in $X_y$ for all $y \\in V$." ], "refs": [], "proofs": [ { "contents": [ "Let $Y' \\subset Y$ be the reduction of $Y$.", "Let $X' = Y' \\times_Y X$ and $U' = Y' \\times_Y U$.", "As $Y' \\to Y$ induces a bijection on points, and as", "$U' \\to U$ and $X' \\to X$ induce isomorphisms of scheme theoretic fibres,", "we may replace $Y$ by $Y'$ and $X$ by $X'$.", "Thus we may assume that $Y$ is integral, see", "Properties, Lemma \\ref{properties-lemma-characterize-integral}.", "We may also replace $Y$ by a nonempty affine open. In other words we", "may assume that $Y = \\Spec(A)$ where $A$ is a domain with fraction", "field $K$.", "\\medskip\\noindent", "As $f$ is of finite type we see that $X$ is quasi-compact.", "Write $X = X_1 \\cup \\ldots \\cup X_n$ for some affine opens $X_i$. By", "Morphisms, Definition \\ref{morphisms-definition-scheme-theoretically-dense}", "we see that $U_i = X_i \\cap U$ is an open subscheme of $X_i$ such that", "$U_{i, \\eta}$ is scheme theoretically dense in $X_{i, \\eta}$.", "Thus it suffices to prove the result for the pairs $(X_i, U_i)$,", "in other words we may assume that $X$ is affine.", "\\medskip\\noindent", "Write $X = \\Spec(B)$. Note that $B_K$ is Noetherian as it is a", "finite type $K$-algebra. Hence $U_\\eta$ is quasi-compact. Thus we can", "find finitely many $g_1, \\ldots, g_m \\in B$ such that $D(g_j) \\subset U$", "and such that $U_\\eta = D(g_1)_\\eta \\cup \\ldots \\cup D(g_m)_\\eta$.", "The fact that $U_\\eta$ is scheme theoretically dense in", "$X_\\eta$ means that $B_K \\to \\bigoplus_j (B_K)_{g_j}$", "is injective, see", "Morphisms, Example \\ref{morphisms-example-scheme-theoretic-closure}.", "By", "Algebra, Lemma \\ref{algebra-lemma-when-injective-covering}", "this is equivalent to the injectivity of", "$B_K \\to \\bigoplus\\nolimits_{j = 1, \\ldots, m} B_K$,", "$b \\mapsto (g_1b, \\ldots, g_mb)$. Let $M$ be the cokernel of this", "map over $A$, i.e., such that we have an exact sequence", "$$", "0 \\to I \\to B \\xrightarrow{(g_1, \\ldots, g_m)}", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} B \\to M \\to 0", "$$", "After replacing $A$ by $A_h$ for some nonzero $h$ we may assume that $B$", "is a flat, finitely presented $A$-algebra, and that $M$", "is flat over $A$, see", "Algebra, Lemma \\ref{algebra-lemma-generic-flatness}.", "The flatness of $B$ over $A$ implies that $B$ is torsion free as an", "$A$-module, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-torsion-free}.", "Hence $B \\subset B_K$. By assumption $I_K = 0$ which implies that $I = 0$", "(as $I \\subset B \\subset B_K$ is a subset of $I_K$). Hence now", "we have a short exact sequence", "$$", "0 \\to B \\xrightarrow{(g_1, \\ldots, g_m)}", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} B \\to M \\to 0", "$$", "with $M$ flat over $A$. Hence for every homomorphism $A \\to \\kappa$ where", "$\\kappa$ is a field, we obtain a short exact sequence", "$$", "0 \\to B \\otimes_A \\kappa \\xrightarrow{(g_1 \\otimes 1, \\ldots, g_m \\otimes 1)}", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} B \\otimes_A \\kappa \\to", "M \\otimes_A \\kappa \\to 0", "$$", "see", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}.", "Reversing the arguments above", "this means that $\\bigcup D(g_j \\otimes 1)$ is scheme", "theoretically dense in $\\Spec(B \\otimes_A \\kappa)$.", "As $\\bigcup D(g_j \\otimes 1) = \\bigcup D(g_j)_\\kappa \\subset U_\\kappa$", "we obtain that $U_\\kappa$ is scheme theoretically dense in $X_\\kappa$", "which is what we wanted to prove." ], "refs": [ "properties-lemma-characterize-integral", "morphisms-definition-scheme-theoretically-dense", "algebra-lemma-when-injective-covering", "algebra-lemma-generic-flatness", "more-algebra-lemma-flat-torsion-free", "algebra-lemma-flat-tor-zero" ], "ref_ids": [ 2947, 5540, 416, 1015, 9919, 532 ] } ], "ref_ids": [] }, { "id": 13804, "type": "theorem", "label": "more-morphisms-lemma-cover-generic-fibre-neighbourhood", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-cover-generic-fibre-neighbourhood", "contents": [ "Let $f : X \\to Y$ be a finite type morphism of schemes. Assume", "$Y$ irreducible with generic point $\\eta$. Let", "$X_\\eta = Z_{1, \\eta} \\cup \\ldots \\cup Z_{n, \\eta}$ be a covering of", "the generic fibre by closed subsets of $X_\\eta$.", "Let $Z_i$ be the closure of $Z_{i, \\eta}$ in $X$ (see discussion above).", "Then there exists a nonempty open $V \\subset Y$ such", "that $X_y = Z_{1, y} \\cup \\ldots \\cup Z_{n, y}$ for all $y \\in V$." ], "refs": [], "proofs": [ { "contents": [ "If $Y$ is Noetherian then $U = X \\setminus (Z_1 \\cup \\ldots \\cup Z_n)$", "is of finite type over $Y$ and we can directly apply", "Lemma \\ref{lemma-empty-generic-fibre}", "to get that $U_V = \\emptyset$ for a nonempty open $V \\subset Y$.", "In general we argue as follows. As the question is topological", "we may replace $Y$ by its reduction. Thus $Y$ is integral, see", "Properties, Lemma \\ref{properties-lemma-characterize-integral}.", "After shrinking $Y$ we may assume that $X \\to Y$ is flat, see", "Morphisms, Proposition \\ref{morphisms-proposition-generic-flatness}.", "In this case every point $x$ in $X_y$ is a specialization of a point", "$x' \\in X_\\eta$ by", "Morphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat}.", "As the $Z_i$ are closed in $X$ and cover the generic fibre this", "implies that $X_y = \\bigcup Z_{i, y}$ for $y \\in Y$ as desired." ], "refs": [ "more-morphisms-lemma-empty-generic-fibre", "properties-lemma-characterize-integral", "morphisms-proposition-generic-flatness", "morphisms-lemma-generalizations-lift-flat" ], "ref_ids": [ 13800, 2947, 5533, 5266 ] } ], "ref_ids": [] }, { "id": 13805, "type": "theorem", "label": "more-morphisms-lemma-reduction-generic-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-reduction-generic-fibre", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $\\eta \\in Y$ be a generic", "point of an irreducible component of $Y$. Then", "$(X_\\eta)_{red} = (X_{red})_\\eta$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine neighbourhood $\\Spec(A) \\subset Y$ of $\\eta$.", "Choose an affine open $\\Spec(B) \\subset X$ mapping into $\\Spec(A)$", "via the morphism $f$. Let $\\mathfrak p \\subset A$ be the minimal prime", "corresponding to $\\eta$. Let $B_{red}$ be the quotient of $B$ by", "the nilradical $\\sqrt{(0)}$. The algebraic content of the lemma is that", "$C = B_{red} \\otimes_A \\kappa(\\mathfrak p)$ is reduced.", "Denote $I \\subset A$ the nilradical so that $A_{red} = A/I$.", "Denote $\\mathfrak p_{red} = \\mathfrak p/I$", "which is a minimal prime of $A_{red}$ with", "$\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p_{red})$.", "Since $A \\to B_{red}$ and $A \\to \\kappa(\\mathfrak p)$", "both factor through $A \\to A_{red}$ we have", "$C = B_{red} \\otimes_{A_{red}} \\kappa(\\mathfrak p_{red})$.", "Now $\\kappa(\\mathfrak p_{red}) = (A_{red})_{\\mathfrak p_{red}}$", "is a localization by ", "Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring}.", "Hence $C$ is a localization of $B_{red}$", "(Algebra, Lemma \\ref{algebra-lemma-tensor-localization})", "and hence reduced." ], "refs": [ "algebra-lemma-minimal-prime-reduced-ring", "algebra-lemma-tensor-localization" ], "ref_ids": [ 418, 366 ] } ], "ref_ids": [] }, { "id": 13806, "type": "theorem", "label": "more-morphisms-lemma-make-generic-fibre-geometrically-reduced", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-make-generic-fibre-geometrically-reduced", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume that $Y$ is irreducible and $f$ is of finite type.", "There exists a diagram", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X_V \\ar[r] \\ar[d] & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & V \\ar[r] & Y", "}", "$$", "where", "\\begin{enumerate}", "\\item $V$ is a nonempty open of $Y$,", "\\item $X_V = V \\times_Y X$,", "\\item $g : Y' \\to V$ is a finite universal homeomorphism,", "\\item $X' = (Y' \\times_Y X)_{red} = (Y' \\times_V X_V)_{red}$,", "\\item $g'$ is a finite universal homeomorphism,", "\\item $Y'$ is an integral affine scheme,", "\\item $f'$ is flat and of finite presentation, and", "\\item the generic fibre of $f'$ is geometrically reduced.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $V = \\Spec(A)$ be a nonempty affine open of $Y$.", "By assumption the Jacobson radical of $A$ is a prime ideal $\\mathfrak p$.", "Let $K = \\kappa(\\mathfrak p)$.", "Let $p$ be the characteristic of $K$ if positive and $1$", "if the characteristic is zero. By", "Varieties, Lemma \\ref{varieties-lemma-finite-extension-geometrically-reduced}", "there exists a finite purely inseparable field extension", "$K \\subset K'$ such that $X_{K'}$ is geometrically reduced over $K'$.", "Choose elements $x_1, \\ldots, x_n \\in K'$ which generate $K'$ over", "$K$ and such that some $p$-power of $x_i$ is in $A/\\mathfrak p$.", "Let $A' \\subset K'$ be the finite $A$-subalgebra of $K'$ generated by", "$x_1, \\ldots, x_n$. Note that $A'$ is a domain with fraction field $K'$. By", "Algebra, Lemma \\ref{algebra-lemma-p-ring-map}", "we see that $A \\to A'$ is a universal homeomorphism.", "Set $Y' = \\Spec(A')$. Set $X' = (Y' \\times_Y X)_{red}$.", "The generic fibre of $X' \\to Y'$ is $(X_K)_{red}$ by", "Lemma \\ref{lemma-reduction-generic-fibre}", "which is geometrically reduced by construction.", "Note that $X' \\to X_V$ is a finite universal homeomorphism as the", "composition of the reduction morphism $X' \\to Y' \\times_Y X$ (see", "Morphisms, Lemma \\ref{morphisms-lemma-reduction-universal-homeomorphism})", "and the base change of $g$.", "At this point all of the properties of the lemma hold except for", "possibly (7). This can be achieved by shrinking $Y'$ and hence $V$, see", "Morphisms, Proposition \\ref{morphisms-proposition-generic-flatness}." ], "refs": [ "varieties-lemma-finite-extension-geometrically-reduced", "algebra-lemma-p-ring-map", "more-morphisms-lemma-reduction-generic-fibre", "morphisms-lemma-reduction-universal-homeomorphism", "morphisms-proposition-generic-flatness" ], "ref_ids": [ 10914, 582, 13805, 5455, 5533 ] } ], "ref_ids": [] }, { "id": 13807, "type": "theorem", "label": "more-morphisms-lemma-make-components-generic-fibre-geometrically-irreducible", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-make-components-generic-fibre-geometrically-irreducible", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume that $Y$ is irreducible and $f$ is of finite type.", "There exists a diagram", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X_V \\ar[r] \\ar[d] & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & V \\ar[r] & Y", "}", "$$", "where", "\\begin{enumerate}", "\\item $V$ is a nonempty open of $Y$,", "\\item $X_V = V \\times_Y X$,", "\\item $g : Y' \\to V$ is surjective finite \\'etale,", "\\item $X' = Y' \\times_Y X = Y' \\times_V X_V$,", "\\item $g'$ is surjective finite \\'etale,", "\\item $Y'$ is an irreducible affine scheme, and", "\\item all irreducible components of the generic fibre of $f'$", "are geometrically irreducible.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $V = \\Spec(A)$ be a nonempty affine open of $Y$.", "By assumption the Jacobson radical of $A$ is a prime ideal $\\mathfrak p$.", "Let $K = \\kappa(\\mathfrak p)$. By", "Varieties, Lemma", "\\ref{varieties-lemma-finite-extension-geometrically-irreducible-components}", "there exists a finite separable field extension", "$K \\subset K'$ such that all irreducible components of $X_{K'}$ are", "geometrically irreducible over $K'$.", "Choose an element $\\alpha \\in K'$ which generates $K'$ over", "$K$, see", "Fields, Lemma \\ref{fields-lemma-primitive-element}.", "Let $P(T) \\in K[T]$ be the minimal polynomial for $\\alpha$ over $K$.", "After replacing $\\alpha$ by $f \\alpha$ for some", "$f \\in A$, $f \\not \\in \\mathfrak p$", "we may assume that there exists a monic polynomial", "$T^d + a_1T^{d - 1} + \\ldots + a_d \\in A[T]$ which maps to", "$P(T) \\in K[T]$ under the map $A[T] \\to K[T]$.", "Set $A' = A[T]/(P)$. Then $A \\to A'$ is a finite free ring map", "such that there exists a unique prime $\\mathfrak q$ lying over", "$\\mathfrak p$, such that", "$K = \\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q) = K'$", "is finite separable, and such that $\\mathfrak pA'_{\\mathfrak q}$", "is the maximal ideal of $A'_{\\mathfrak q}$.", "Hence $g : Y' = \\Spec(A') \\to V = \\Spec(A)$", "is \\'etale at $\\mathfrak q$, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-etale}.", "This means that there exists an open $W \\subset \\Spec(A')$ such", "that $g|_W : W \\to \\Spec(A)$ is \\'etale.", "Since $g$ is finite and since $\\mathfrak q$ is the only point lying over", "$\\mathfrak p$ we see that $Z = g(Y' \\setminus W)$ is a closed subset of $V$", "not containing $\\mathfrak p$. Hence after replacing $V$ by a principal", "affine open of $V$ which does not meet $Z$ we obtain that $g$ is finite", "\\'etale." ], "refs": [ "varieties-lemma-finite-extension-geometrically-irreducible-components", "fields-lemma-primitive-element", "algebra-lemma-characterize-etale" ], "ref_ids": [ 10945, 4498, 1235 ] } ], "ref_ids": [] }, { "id": 13808, "type": "theorem", "label": "more-morphisms-lemma-relative-assassin-in-neighbourhood", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-relative-assassin-in-neighbourhood", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\xi \\in \\text{Ass}_{X/S}(\\mathcal{F})$ and set", "$Z = \\overline{\\{\\xi\\}} \\subset X$.", "If $f$ is locally of finite type and $\\mathcal{F}$ is a", "finite type $\\mathcal{O}_X$-module, then there exists a nonempty", "open $V \\subset Z$ such that for every $s \\in f(V)$ the generic", "points of $V_s$ are elements of $\\text{Ass}_{X/S}(\\mathcal{F})$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $S$ by an affine open neighbourhood of $f(\\xi)$", "and $X$ by an affine open neighbourhood of $\\xi$. Hence we may assume", "$S = \\Spec(A)$, $X = \\Spec(B)$ and that $f$ is given by", "the finite type ring map $A \\to B$, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-finite-type-characterize}.", "Moreover, we may write $\\mathcal{F} = \\widetilde{M}$ for some finite", "$B$-module $M$, see", "Properties, Lemma \\ref{properties-lemma-finite-type-module}.", "Let $\\mathfrak q \\subset B$ be the prime corresponding to $\\xi$ and", "let $\\mathfrak p \\subset A$ be the corresponding prime of $A$.", "By assumption $\\mathfrak q \\in \\text{Ass}_B(M \\otimes_A \\kappa(\\mathfrak p))$,", "see", "Algebra, Remark \\ref{algebra-remark-bourbaki}", "and", "Divisors, Lemma \\ref{divisors-lemma-associated-affine-open}.", "With this notation $Z = V(\\mathfrak q) \\subset \\Spec(B)$.", "In particular $f(Z) \\subset V(\\mathfrak p)$. Hence clearly it suffices to", "prove the lemma after replacing $A$, $B$, and $M$ by", "$A/\\mathfrak pA$, $B/\\mathfrak pB$, and $M/\\mathfrak pM$.", "In other words we may assume that $A$ is a domain with fraction field", "$K$ and $\\mathfrak q \\subset B$ is an associated prime of $M \\otimes_A K$.", "\\medskip\\noindent", "At this point we can use generic flatness. Namely, by", "Algebra, Lemma \\ref{algebra-lemma-generic-flatness}", "there exists a nonzero $g \\in A$ such that $M_g$ is flat", "as an $A_g$-module. After replacing $A$ by $A_g$ we may assume that", "$M$ is flat as an $A$-module.", "\\medskip\\noindent", "In this case, by", "Algebra, Lemma \\ref{algebra-lemma-post-bourbaki}", "we see that $\\mathfrak q$ is also an associated prime of $M$.", "Hence we obtain an injective $B$-module map $B/\\mathfrak q \\to M$.", "Let $Q$ be the cokernel so that we obtain a short exact sequence", "$$", "0 \\to B/\\mathfrak q \\to M \\to Q \\to 0", "$$", "of finite $B$-modules. After applying generic flatness", "Algebra, Lemma \\ref{algebra-lemma-generic-flatness}", "once more, this time to the $B$-module $Q$, we may assume that $Q$", "is a flat $A$-module. In particular we may assume the short exact", "sequence above is universally injective, see", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}.", "In this situation", "$(B/\\mathfrak q) \\otimes_A \\kappa(\\mathfrak p')", "\\subset M \\otimes_A \\kappa(\\mathfrak p')$", "for any prime $\\mathfrak p'$ of $A$. The lemma follows as a minimal", "prime $\\mathfrak q'$ of the support of", "$(B/\\mathfrak q) \\otimes_A \\kappa(\\mathfrak p')$", "is an associated prime of $(B/\\mathfrak q) \\otimes_A \\kappa(\\mathfrak p')$ by", "Divisors, Lemma \\ref{divisors-lemma-minimal-support-in-ass}." ], "refs": [ "morphisms-lemma-locally-finite-type-characterize", "properties-lemma-finite-type-module", "algebra-remark-bourbaki", "divisors-lemma-associated-affine-open", "algebra-lemma-generic-flatness", "algebra-lemma-post-bourbaki", "algebra-lemma-generic-flatness", "algebra-lemma-flat-tor-zero", "divisors-lemma-minimal-support-in-ass" ], "ref_ids": [ 5198, 3002, 1563, 7856, 1015, 718, 1015, 532, 7862 ] } ], "ref_ids": [] }, { "id": 13809, "type": "theorem", "label": "more-morphisms-lemma-bad-case", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-bad-case", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{F}$ be a", "quasi-coherent $\\mathcal{O}_X$-module. Let $U \\subset X$ be an open", "subscheme. Assume", "\\begin{enumerate}", "\\item $f$ is of finite type,", "\\item $\\mathcal{F}$ is of finite type,", "\\item $Y$ is irreducible with generic point $\\eta$, and", "\\item $\\text{Ass}_{X_\\eta}(\\mathcal{F}_\\eta)$ is not contained in $U_\\eta$.", "\\end{enumerate}", "Then there exists a nonempty open subscheme $V \\subset Y$ such that", "for all $y \\in V$ the set $\\text{Ass}_{X_y}(\\mathcal{F}_y)$ is not", "contained in $U_y$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset X$ be the scheme theoretic support of $\\mathcal{F}$, see", "Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-support}.", "Then $Z_\\eta$ is the scheme theoretic support of $\\mathcal{F}_\\eta$", "(Morphisms, Lemma \\ref{morphisms-lemma-flat-pullback-support}).", "Hence the generic points of irreducible components of $Z_\\eta$", "are contained in $\\text{Ass}_{X_\\eta}(\\mathcal{F}_\\eta)$ by", "Divisors, Lemma \\ref{divisors-lemma-minimal-support-in-ass}.", "Hence we see that $Z_\\eta \\cap U_\\eta = \\emptyset$.", "Thus $T = Z \\setminus U$ is a closed subset of $Z$ with $T_\\eta = \\emptyset$.", "If we endow $T$ with the induced reduced scheme structure then", "$T \\to Y$ is a morphism of finite type. By", "Lemma \\ref{lemma-empty-generic-fibre}", "there is a nonempty open $V \\subset Y$ with $T_V = \\emptyset$.", "Then $V$ works." ], "refs": [ "morphisms-definition-scheme-theoretic-support", "morphisms-lemma-flat-pullback-support", "divisors-lemma-minimal-support-in-ass", "more-morphisms-lemma-empty-generic-fibre" ], "ref_ids": [ 5538, 5271, 7862, 13800 ] } ], "ref_ids": [] }, { "id": 13810, "type": "theorem", "label": "more-morphisms-lemma-good-case", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-good-case", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{F}$ be a", "quasi-coherent $\\mathcal{O}_X$-module. Let $U \\subset X$ be an open", "subscheme. Assume", "\\begin{enumerate}", "\\item $f$ is of finite type,", "\\item $\\mathcal{F}$ is of finite type,", "\\item $Y$ is irreducible with generic point $\\eta$, and", "\\item $\\text{Ass}_{X_\\eta}(\\mathcal{F}_\\eta) \\subset U_\\eta$.", "\\end{enumerate}", "Then there exists a nonempty open subscheme $V \\subset Y$ such that", "for all $y \\in V$ we have $\\text{Ass}_{X_y}(\\mathcal{F}_y) \\subset U_y$." ], "refs": [], "proofs": [ { "contents": [ "(This proof is the same as the proof of", "Lemma \\ref{lemma-scheme-theoretically-dense-generic-fibre}.", "We urge the reader to read that proof first.)", "Since the statement is about fibres it is clear that we may replace", "$Y$ by its reduction. Hence we may assume that $Y$ is integral, see", "Properties, Lemma \\ref{properties-lemma-characterize-integral}.", "We may also assume that $Y = \\Spec(A)$ is affine. Then $A$", "is a domain with fraction field $K$.", "\\medskip\\noindent", "As $f$ is of finite type we see that $X$ is quasi-compact.", "Write $X = X_1 \\cup \\ldots \\cup X_n$ for some affine opens $X_i$", "and set $\\mathcal{F}_i = \\mathcal{F}|_{X_i}$. By", "assumption the generic fibre of $U_i = X_i \\cap U$ contains", "$\\text{Ass}_{X_{i, \\eta}}(\\mathcal{F}_{i, \\eta})$.", "Thus it suffices to prove the result for the triples", "$(X_i, \\mathcal{F}_i, U_i)$,", "in other words we may assume that $X$ is affine.", "\\medskip\\noindent", "Write $X = \\Spec(B)$. Let $N$ be a finite $B$-module such that", "$\\mathcal{F} = \\widetilde{N}$.", "Note that $B_K$ is Noetherian as it is a", "finite type $K$-algebra. Hence $U_\\eta$ is quasi-compact. Thus we can", "find finitely many $g_1, \\ldots, g_m \\in B$ such that $D(g_j) \\subset U$", "and such that $U_\\eta = D(g_1)_\\eta \\cup \\ldots \\cup D(g_m)_\\eta$.", "Since $\\text{Ass}_{X_\\eta}(\\mathcal{F}_\\eta) \\subset U_\\eta$", "we see that $N_K \\to \\bigoplus_j (N_K)_{g_j}$ is injective. By", "Algebra, Lemma \\ref{algebra-lemma-when-injective-covering}", "this is equivalent to the injectivity of", "$N_K \\to \\bigoplus\\nolimits_{j = 1, \\ldots, m} N_K$,", "$n \\mapsto (g_1n, \\ldots, g_mn)$. Let $I$ and $M$ be the kernel and", "cokernel of this map over $A$, i.e., such that we have an exact sequence", "$$", "0 \\to I \\to N \\xrightarrow{(g_1, \\ldots, g_m)}", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} N \\to M \\to 0", "$$", "After replacing $A$ by $A_h$ for some nonzero $h$ we may assume that", "$B$ is a flat, finitely presented $A$-algebra and that", "both $M$ and $N$ are flat over $A$, see", "Algebra, Lemma \\ref{algebra-lemma-generic-flatness}.", "The flatness of $N$ over $A$ implies that $N$ is torsion free as an", "$A$-module, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-torsion-free}.", "Hence $N \\subset N_K$. By construction $I_K = 0$ which implies that $I = 0$", "(as $I \\subset N \\subset N_K$ is a subset of $I_K$). Hence now", "we have a short exact sequence", "$$", "0 \\to N \\xrightarrow{(g_1, \\ldots, g_m)}", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} N \\to M \\to 0", "$$", "with $M$ flat over $A$. Hence for every homomorphism $A \\to \\kappa$ where", "$\\kappa$ is a field, we obtain a short exact sequence", "$$", "0 \\to N \\otimes_A \\kappa \\xrightarrow{(g_1 \\otimes 1, \\ldots, g_m \\otimes 1)}", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} N \\otimes_A \\kappa \\to", "M \\otimes_A \\kappa \\to 0", "$$", "see", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}.", "Reversing the arguments above", "this means that $\\bigcup D(g_j \\otimes 1)$ contains", "$\\text{Ass}_{B \\otimes_A \\kappa}(N \\otimes_A \\kappa)$.", "As $\\bigcup D(g_j \\otimes 1) = \\bigcup D(g_j)_\\kappa \\subset U_\\kappa$", "we obtain that $U_\\kappa$ contains", "$\\text{Ass}_{X \\otimes \\kappa}(\\mathcal{F} \\otimes \\kappa)$", "which is what we wanted to prove." ], "refs": [ "more-morphisms-lemma-scheme-theoretically-dense-generic-fibre", "properties-lemma-characterize-integral", "algebra-lemma-when-injective-covering", "algebra-lemma-generic-flatness", "more-algebra-lemma-flat-torsion-free", "algebra-lemma-flat-tor-zero" ], "ref_ids": [ 13803, 2947, 416, 1015, 9919, 532 ] } ], "ref_ids": [] }, { "id": 13811, "type": "theorem", "label": "more-morphisms-lemma-base-change-assassin-in-U", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-assassin-in-U", "contents": [ "Let $f : X \\to S$ be a morphism which is locally of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module", "of finite type. Let $U \\subset X$ be an open subscheme.", "Let $g : S' \\to S$ be a morphism of schemes, let", "$f' : X' = X_{S'} \\to S'$ be the base change of $f$,", "let $g' : X' \\to X$ be the projection, set", "$\\mathcal{F}' = (g')^*\\mathcal{F}$, and set", "$U' = (g')^{-1}(U)$. Finally, let $s' \\in S'$ with image $s = g(s')$.", "In this case", "$$", "\\text{Ass}_{X_s}(\\mathcal{F}_s) \\subset U_s", "\\Leftrightarrow", "\\text{Ass}_{X'_{s'}}(\\mathcal{F}'_{s'}) \\subset U'_{s'}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from", "Divisors, Lemma \\ref{divisors-lemma-base-change-relative-assassin}.", "See also", "Divisors, Remark \\ref{divisors-remark-base-change-relative-assassin}." ], "refs": [ "divisors-lemma-base-change-relative-assassin", "divisors-remark-base-change-relative-assassin" ], "ref_ids": [ 7890, 8115 ] } ], "ref_ids": [] }, { "id": 13812, "type": "theorem", "label": "more-morphisms-lemma-relative-assassin-constructible", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-relative-assassin-constructible", "contents": [ "Let $f : X \\to Y$ be a morphism of finite presentation.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module", "of finite presentation. Let $U \\subset X$ be an open subscheme", "such that $U \\to Y$ is quasi-compact. Then the set", "$$", "E = \\{y \\in Y \\mid \\text{Ass}_{X_y}(\\mathcal{F}_y) \\subset U_y\\}", "$$", "is locally constructible in $Y$." ], "refs": [], "proofs": [ { "contents": [ "Let $y \\in Y$. We have to show that there exists an open neighbourhood", "$V$ of $y$ in $Y$ such that $E \\cap V$ is constructible in $V$. Thus we may", "assume that $Y$ is affine. Write $Y = \\Spec(A)$ and", "$A = \\colim A_i$ as a directed limit of finite type", "$\\mathbf{Z}$-algebras. By", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can find an $i$ and a morphism $f_i : X_i \\to \\Spec(A_i)$ of", "finite presentation whose base change to $Y$ recovers $f$.", "After possibly increasing $i$ we may assume there exists a", "quasi-coherent $\\mathcal{O}_{X_i}$-module $\\mathcal{F}_i$ of finite", "presentation whose pullback to $X$ is isomorphic to $\\mathcal{F}$, see", "Limits, Lemma \\ref{limits-lemma-descend-modules-finite-presentation}.", "After possibly increasing $i$ one more time we may assume there exists", "an open subscheme $U_i \\subset X_i$ whose inverse image in $X$", "is $U$, see", "Limits, Lemma \\ref{limits-lemma-descend-opens}.", "By", "Lemma \\ref{lemma-base-change-assassin-in-U}", "it suffices to prove the lemma for $f_i$. Thus we reduce to", "the case where $Y$ is the spectrum of a Noetherian ring.", "\\medskip\\noindent", "We will use the criterion of", "Topology, Lemma \\ref{topology-lemma-characterize-constructible-Noetherian}", "to prove that $E$ is constructible in case $Y$ is a Noetherian scheme.", "To see this let $Z \\subset Y$ be an irreducible closed subscheme.", "We have to show that $E \\cap Z$ either contains a nonempty open subset", "or is not dense in $Z$. This follows from", "Lemmas \\ref{lemma-bad-case} and", "\\ref{lemma-good-case}", "applied to the base change $(X, \\mathcal{F}, U) \\times_Y Z$ over $Z$." ], "refs": [ "limits-lemma-descend-finite-presentation", "limits-lemma-descend-modules-finite-presentation", "limits-lemma-descend-opens", "more-morphisms-lemma-base-change-assassin-in-U", "topology-lemma-characterize-constructible-Noetherian", "more-morphisms-lemma-bad-case", "more-morphisms-lemma-good-case" ], "ref_ids": [ 15077, 15078, 15041, 13811, 8269, 13809, 13810 ] } ], "ref_ids": [] }, { "id": 13813, "type": "theorem", "label": "more-morphisms-lemma-nonreduced-in-neighbourhood", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-nonreduced-in-neighbourhood", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with", "generic point $\\eta$ and $f$ of finite type. If $X_\\eta$ is nonreduced,", "then there exists a nonempty open $V \\subset Y$", "such that for all $y \\in V$ the fibre $X_y$ is nonreduced." ], "refs": [], "proofs": [ { "contents": [ "Let $Y' \\subset Y$ be the reduction of $Y$. Let $X' \\to Y'$", "be the base change of $f$. Note that $Y' \\to Y$", "induces a bijection on points and that $X' \\to X$ identifies fibres.", "Hence we may assume that $Y'$ is reduced, i.e., integral, see", "Properties, Lemma \\ref{properties-lemma-characterize-integral}.", "We may also replace $Y$ by an affine open. Hence we may assume that", "$Y = \\Spec(A)$ with $A$ a domain. Denote $K$ the", "fraction field of $A$.", "Pick an affine open $\\Spec(B) = U \\subset X$ and a section", "$h_\\eta \\in \\Gamma(U_\\eta, \\mathcal{O}_{U_\\eta}) = B_K$", "which is nonzero and nilpotent.", "After shrinking $Y$ we may assume that $h$ comes from", "$h \\in \\Gamma(U, \\mathcal{O}_U) = B$. After shrinking $Y$ a bit", "more we may assume that $h$ is nilpotent. Let", "$I = \\{b \\in B \\mid hb = 0\\}$ be the annihilator of $h$.", "Then $C = B/I$ is a finite type $A$-algebra whose generic fiber", "$(B/I)_K$ is nonzero (as $h_\\eta \\not = 0$). We apply", "generic flatness to $A \\to C$ and $A \\to B/hB$, see", "Algebra, Lemma \\ref{algebra-lemma-generic-flatness},", "and we obtain a $g \\in A$, $g \\not = 0$ such that $C_g$ is free as", "an $A_g$-module and $(B/hB)_g$ is flat as an $A_g$-module.", "Replace $Y$ by $D(g) \\subset Y$. Now we have the short exact sequence", "$$", "0 \\to C \\to B \\to B/hB \\to 0.", "$$", "with $B/hB$ flat over $A$ and with $C$ nonzero free as an $A$-module.", "It follows that for any homomorphism $A \\to \\kappa$ to a field", "the ring $C \\otimes_A \\kappa$ is nonzero and the sequence", "$$", "0 \\to C \\otimes_A \\kappa \\to B \\otimes_A \\kappa \\to", "B/hB \\otimes_A \\kappa \\to 0", "$$", "is exact, see", "Algebra, Lemma \\ref{algebra-lemma-flat-tor-zero}.", "Note that", "$B/hB \\otimes_A \\kappa = (B \\otimes_A \\kappa) / h(B \\otimes_A \\kappa)$", "by right exactness of tensor product. Thus we conclude that", "multiplication by $h$ is not zero on $B \\otimes_A \\kappa$.", "This clearly means that for any point $y \\in Y$ the element $h$", "restricts to a nonzero element of $U_y$, whence $X_y$ is nonreduced." ], "refs": [ "properties-lemma-characterize-integral", "algebra-lemma-generic-flatness", "algebra-lemma-flat-tor-zero" ], "ref_ids": [ 2947, 1015, 532 ] } ], "ref_ids": [] }, { "id": 13814, "type": "theorem", "label": "more-morphisms-lemma-base-change-fibres-geometrically-reduced", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-fibres-geometrically-reduced", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $g : Y' \\to Y$ be any morphism, and denote", "$f' : X' \\to Y'$ the base change of $f$.", "Then", "\\begin{align*}", "\\{y' \\in Y' \\mid X'_{y'}\\text{ is geometrically reduced}\\} \\\\", "= g^{-1}(\\{y \\in Y \\mid X_y\\text{ is geometrically reduced}\\}).", "\\end{align*}" ], "refs": [], "proofs": [ { "contents": [ "This comes down to the statement that for $y' \\in Y'$ with image", "$y \\in Y$ the fibre $X'_{y'} = X_y \\times_y y'$ is geometrically", "reduced over $\\kappa(y')$ if and only if $X_y$ is geometrically", "reduced over $\\kappa(y)$. This follows from", "Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced-upstairs}." ], "refs": [ "varieties-lemma-geometrically-reduced-upstairs" ], "ref_ids": [ 10910 ] } ], "ref_ids": [] }, { "id": 13815, "type": "theorem", "label": "more-morphisms-lemma-not-geometrically-reduced-in-neighbourhood", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-not-geometrically-reduced-in-neighbourhood", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with", "generic point $\\eta$ and $f$ of finite type. If $X_\\eta$ is not", "geometrically reduced, then there exists a nonempty open $V \\subset Y$", "such that for all $y \\in V$ the fibre $X_y$ is not geometrically reduced." ], "refs": [], "proofs": [ { "contents": [ "Apply", "Lemma \\ref{lemma-make-generic-fibre-geometrically-reduced}", "to get", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X_V \\ar[r] \\ar[d] & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & V \\ar[r] & Y", "}", "$$", "with all the properties mentioned in that lemma.", "Let $\\eta'$ be the generic point of $Y'$.", "Consider the morphism $X' \\to X_{Y'}$ (which is the reduction", "morphism) and the resulting morphism of generic fibres", "$X'_{\\eta'} \\to X_{\\eta'}$.", "Since $X'_{\\eta'}$ is geometrically reduced, and $X_\\eta$", "is not this cannot be an isomorphism, see", "Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced-upstairs}.", "Hence $X_{\\eta'}$ is nonreduced. Hence by", "Lemma \\ref{lemma-nonreduced-in-neighbourhood}", "the fibres of $X_{Y'} \\to Y'$ are nonreduced at all points $y' \\in V'$", "of a nonempty open $V' \\subset Y'$. Since $g : Y' \\to V$ is a homeomorphism", "Lemma \\ref{lemma-base-change-fibres-geometrically-reduced}", "proves that $g(V')$ is the open we are looking for." ], "refs": [ "more-morphisms-lemma-make-generic-fibre-geometrically-reduced", "varieties-lemma-geometrically-reduced-upstairs", "more-morphisms-lemma-nonreduced-in-neighbourhood", "more-morphisms-lemma-base-change-fibres-geometrically-reduced" ], "ref_ids": [ 13806, 10910, 13813, 13814 ] } ], "ref_ids": [] }, { "id": 13816, "type": "theorem", "label": "more-morphisms-lemma-geometrically-reduced-generic-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-geometrically-reduced-generic-fibre", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume", "\\begin{enumerate}", "\\item $Y$ is irreducible with generic point $\\eta$,", "\\item $X_\\eta$ is geometrically reduced, and", "\\item $f$ is of finite type.", "\\end{enumerate}", "Then there exists a nonempty open subscheme $V \\subset Y$", "such that $X_V \\to V$ has geometrically reduced fibres." ], "refs": [], "proofs": [ { "contents": [ "Let $Y' \\subset Y$ be the reduction of $Y$. Let $X' \\to Y'$", "be the base change of $f$. Note that $Y' \\to Y$", "induces a bijection on points and that $X' \\to X$ identifies fibres.", "Hence we may assume that $Y'$ is reduced, i.e., integral, see", "Properties, Lemma \\ref{properties-lemma-characterize-integral}.", "We may also replace $Y$ by an affine open. Hence we may assume that", "$Y = \\Spec(A)$ with $A$ a domain. Denote $K$ the", "fraction field of $A$. After shrinking $Y$ a bit we may also assume that", "$X \\to Y$ is flat and of finite presentation, see", "Morphisms, Proposition \\ref{morphisms-proposition-generic-flatness}.", "\\medskip\\noindent", "As $X_\\eta$ is geometrically reduced there exists an open dense", "subset $V \\subset X_\\eta$ such that $V \\to \\Spec(K)$ is smooth, see", "Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced-dense-smooth-open}.", "Let $U \\subset X$ be the set of points where $f$ is smooth. By", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-smooth}", "we see that $V \\subset U_\\eta$. Thus the generic fibre of $U$ is dense", "in the generic fibre of $X$. Since $X_\\eta$ is reduced, it follows", "that $U_\\eta$ is scheme theoretically dense in $X_\\eta$, see", "Morphisms, Lemma \\ref{morphisms-lemma-reduced-scheme-theoretically-dense}.", "We note that as $U \\to Y$ is smooth all the fibres of $U \\to Y$", "are geometrically reduced. Thus it suffices to show that, after", "shrinking $Y$, for all $y \\in Y$ the scheme $U_y$ is scheme theoretically", "dense in $X_y$, see", "Morphisms, Lemma \\ref{morphisms-lemma-reduced-subscheme-closure}.", "This follows from", "Lemma \\ref{lemma-scheme-theoretically-dense-generic-fibre}." ], "refs": [ "properties-lemma-characterize-integral", "morphisms-proposition-generic-flatness", "varieties-lemma-geometrically-reduced-dense-smooth-open", "morphisms-lemma-set-points-where-fibres-smooth", "morphisms-lemma-reduced-scheme-theoretically-dense", "morphisms-lemma-reduced-subscheme-closure", "more-morphisms-lemma-scheme-theoretically-dense-generic-fibre" ], "ref_ids": [ 2947, 5533, 11008, 5336, 5155, 5156, 13803 ] } ], "ref_ids": [] }, { "id": 13817, "type": "theorem", "label": "more-morphisms-lemma-geometrically-reduced-constructible", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-geometrically-reduced-constructible", "contents": [ "Let $f : X \\to Y$ be a morphism which is quasi-compact and", "locally of finite presentation. Then the set", "$$", "E = \\{y \\in Y \\mid X_y\\text{ is geometrically reduced}\\}", "$$", "is locally constructible in $Y$." ], "refs": [], "proofs": [ { "contents": [ "Let $y \\in Y$. We have to show that there exists an open neighbourhood", "$V$ of $y$ in $Y$ such that $E \\cap V$ is constructible in $V$. Thus we may", "assume that $Y$ is affine. Then $X$ is quasi-compact.", "Choose a finite affine open covering $X = U_1 \\cup \\ldots \\cup U_n$.", "Then the fibres of $U_i \\to Y$ at $y$ form an affine open covering", "of the fibre of $X \\to Y$ at $y$. Hence we may assume $X$ is affine", "as well. Write $Y = \\Spec(A)$.", "Write $A = \\colim A_i$ as a directed limit of finite type", "$\\mathbf{Z}$-algebras. By", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can find an $i$ and a morphism $f_i : X_i \\to \\Spec(A_i)$ of", "finite presentation whose base change to $Y$ recovers $f$. By", "Lemma \\ref{lemma-base-change-fibres-geometrically-reduced}", "it suffices to prove the lemma for $f_i$. Thus we reduce to", "the case where $Y$ is the spectrum of a Noetherian ring.", "\\medskip\\noindent", "We will use the criterion of", "Topology, Lemma \\ref{topology-lemma-characterize-constructible-Noetherian}", "to prove that $E$ is constructible in case $Y$ is a Noetherian scheme.", "To see this let $Z \\subset Y$ be an irreducible closed subscheme.", "We have to show that $E \\cap Z$ either contains a nonempty open subset", "or is not dense in $Z$. If $X_\\xi$ is geometrically reduced, then", "Lemma \\ref{lemma-geometrically-reduced-generic-fibre}", "(applied to the morphism $X_Z \\to Z$)", "implies that all fibres $X_y$ are geometrically reduced", "for a nonempty open $V \\subset Z$.", "If $X_\\xi$ is not geometrically reduced, then", "Lemma \\ref{lemma-not-geometrically-reduced-in-neighbourhood}", "(applied to the morphism $X_Z \\to Z$)", "implies that all fibres $X_y$ are geometrically reduced", "for a nonempty open $V \\subset Z$. Thus we win." ], "refs": [ "limits-lemma-descend-finite-presentation", "more-morphisms-lemma-base-change-fibres-geometrically-reduced", "topology-lemma-characterize-constructible-Noetherian", "more-morphisms-lemma-geometrically-reduced-generic-fibre", "more-morphisms-lemma-not-geometrically-reduced-in-neighbourhood" ], "ref_ids": [ 15077, 13814, 8269, 13816, 13815 ] } ], "ref_ids": [] }, { "id": 13818, "type": "theorem", "label": "more-morphisms-lemma-proper-flat-over-dvr-reduced-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-proper-flat-over-dvr-reduced-fibre", "contents": [ "Let $X \\to \\Spec(R)$ be a proper flat morphism where $R$ is a", "discrete valuation ring. If the special fibre is reduced, then", "both $X$ and the generic fibre $X_\\eta$ are reduced." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$ be a point in the generic fibre $X_\\eta$", "such that $\\mathcal{O}_{X_\\eta}$ is nonreduced.", "Then $\\mathcal{O}_{X, x}$ is nonreduced.", "Let $x \\leadsto x'$ be a specialization with $x'$", "in the special fibre; such a specialization exists", "as a proper morphism is closed. Consider the local", "ring $A = \\mathcal{O}_{X, x'}$. Let $\\pi \\in R$ be a uniformizer.", "If $a \\in A$ then there exists an $n \\geq 0$ and an element", "$a' \\in A$ such that $a = \\pi^n a'$ and $a' \\not \\in \\pi A$.", "This follows from Krull intersection theorem", "(Algebra, Lemma \\ref{algebra-lemma-intersect-powers-ideal-module-zero}).", "If $a$ is nilpotent, so is $a'$, because $\\pi$ is a nonzerodivisor", "by flatness of $A$ over $R$.", "But $a'$ maps to a nonzero element of the reduced ring", "$A/\\pi A = \\mathcal{O}_{X_s, x'}$.", "This is a contradiction unless $A$ is reduced, which", "is what we wanted to show." ], "refs": [ "algebra-lemma-intersect-powers-ideal-module-zero" ], "ref_ids": [ 627 ] } ], "ref_ids": [] }, { "id": 13819, "type": "theorem", "label": "more-morphisms-lemma-geometrically-reduced-open", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-geometrically-reduced-open", "contents": [ "Let $f : X \\to Y$ be a flat proper morphism of finite presentation.", "Then the set $\\{y \\in Y \\mid X_y\\text{ is geometrically reduced}\\}$", "is open in $Y$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $Y$ is affine. Then $Y$ is a cofiltered limit of affine", "schemes of finite type over $\\mathbf{Z}$.", "Hence we can assume $X \\to Y$ is the", "base change of $X_0 \\to Y_0$ where $Y_0$ is the spectrum of a finite", "type $\\mathbf{Z}$-algebra and $X_0 \\to Y_0$ is flat and proper.", "See Limits, Lemma \\ref{limits-lemma-descend-finite-presentation},", "\\ref{limits-lemma-descend-flat-finite-presentation}, and", "\\ref{limits-lemma-eventually-proper}. Since the formation of", "the set of points where the fibres are geometrically", "reduced commutes with base change", "(Lemma \\ref{lemma-base-change-fibres-geometrically-reduced}),", "we may assume the base is Noetherian.", "\\medskip\\noindent", "Assume $Y$ is Noetherian. The set is constructible by", "Lemma \\ref{lemma-geometrically-reduced-constructible}.", "Hence it suffices to show the set is stable under generalization", "(Topology, Lemma \\ref{topology-lemma-characterize-closed-Noetherian}). By", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian-specialization-dvr}", "we reduce to the case where $Y = \\Spec(R)$, $R$ is a discrete", "valuation ring, and the closed fibre $X_y$ is geometrically", "reduced. To show: the generic fibre $X_\\eta$ is geometrically reduced.", "\\medskip\\noindent", "If not then there exists a finite extension $L$ of the fraction", "field of $R$ such that $X_L$ is not reduced, see", "Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced}.", "There exists a discrete valuation ring", "$R' \\subset L$ with fraction field $L$ dominating $R$, see", "Algebra, Lemma \\ref{algebra-lemma-integral-closure-Dedekind}.", "After replacing $R$ by $R'$ we reduce to", "Lemma \\ref{lemma-proper-flat-over-dvr-reduced-fibre}." ], "refs": [ "limits-lemma-descend-finite-presentation", "limits-lemma-descend-flat-finite-presentation", "limits-lemma-eventually-proper", "more-morphisms-lemma-base-change-fibres-geometrically-reduced", "more-morphisms-lemma-geometrically-reduced-constructible", "topology-lemma-characterize-closed-Noetherian", "properties-lemma-locally-Noetherian-specialization-dvr", "varieties-lemma-geometrically-reduced", "algebra-lemma-integral-closure-Dedekind", "more-morphisms-lemma-proper-flat-over-dvr-reduced-fibre" ], "ref_ids": [ 15077, 15062, 15089, 13814, 13817, 8290, 2959, 10908, 1042, 13818 ] } ], "ref_ids": [] }, { "id": 13820, "type": "theorem", "label": "more-morphisms-lemma-irreducible-components-in-neighbourhood", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-irreducible-components-in-neighbourhood", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with", "generic point $\\eta$ and $f$ of finite type. If $X_\\eta$ has $n$", "irreducible components, then there exists a nonempty open $V \\subset Y$", "such that for all $y \\in V$ the fibre $X_y$ has at least $n$", "irreducible components." ], "refs": [], "proofs": [ { "contents": [ "As the question is purely topological we may replace $X$ and $Y$ by", "their reductions. In particular this implies that $Y$ is integral, see", "Properties, Lemma \\ref{properties-lemma-characterize-integral}.", "Let $X_\\eta = X_{1, \\eta} \\cup \\ldots \\cup X_{n, \\eta}$", "be the decomposition of $X_\\eta$ into irreducible components.", "Let $X_i \\subset X$ be the reduced closed subscheme whose generic", "fibre is $X_{i, \\eta}$. Note that $Z_{i, j} = X_i \\cap X_j$", "is a closed subset of $X_i$ whose generic fibre $Z_{i, j, \\eta}$", "is nowhere dense in $X_{i, \\eta}$. Hence after shrinking $Y$ we may", "assume that $Z_{i, j, y}$", "is nowhere dense in $X_{i, y}$ for every $y \\in Y$, see", "Lemma \\ref{lemma-nowhere-dense-generic-fibre}.", "After shrinking $Y$ some more we may assume that", "$X_y = \\bigcup X_{i, y}$ for $y \\in Y$, see", "Lemma \\ref{lemma-cover-generic-fibre-neighbourhood}.", "Moreover, after shrinking $Y$ we may assume that each $X_i \\to Y$", "is flat and of finite presentation, see", "Morphisms, Proposition \\ref{morphisms-proposition-generic-flatness}.", "The morphisms $X_i \\to Y$ are open, see", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}.", "Thus there exists an open neighbourhood $V$ of $\\eta$ which is contained", "in $f(X_i)$ for each $i$.", "For each $y \\in V$ the schemes $X_{i, y}$ are", "nonempty closed subsets of $X_y$, we have $X_y = \\bigcup X_{i, y}$", "and the intersections $Z_{i, j, y} = X_{i, y} \\cap X_{j, y}$", "are not dense in $X_{i, y}$. Clearly this implies that", "$X_y$ has at least $n$ irreducible components." ], "refs": [ "properties-lemma-characterize-integral", "more-morphisms-lemma-nowhere-dense-generic-fibre", "more-morphisms-lemma-cover-generic-fibre-neighbourhood", "morphisms-proposition-generic-flatness", "morphisms-lemma-fppf-open" ], "ref_ids": [ 2947, 13802, 13804, 5533, 5267 ] } ], "ref_ids": [] }, { "id": 13821, "type": "theorem", "label": "more-morphisms-lemma-base-change-fibres-geometrically-irreducible", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-fibres-geometrically-irreducible", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $g : Y' \\to Y$ be any morphism, and denote", "$f' : X' \\to Y'$ the base change of $f$.", "Then", "\\begin{align*}", "\\{y' \\in Y' \\mid X'_{y'}\\text{ is geometrically irreducible}\\} \\\\", "= g^{-1}(\\{y \\in Y \\mid X_y\\text{ is geometrically irreducible}\\}).", "\\end{align*}" ], "refs": [], "proofs": [ { "contents": [ "This comes down to the statement that for $y' \\in Y'$ with image", "$y \\in Y$ the fibre $X'_{y'} = X_y \\times_y y'$ is geometrically", "irreducible over $\\kappa(y')$ if and only if $X_y$ is geometrically", "irreducible over $\\kappa(y)$. This follows from", "Varieties,", "Lemma \\ref{varieties-lemma-geometrically-irreducible-check-after-extension}." ], "refs": [ "varieties-lemma-geometrically-irreducible-check-after-extension" ], "ref_ids": [ 10932 ] } ], "ref_ids": [] }, { "id": 13822, "type": "theorem", "label": "more-morphisms-lemma-base-change-fibres-nr-geometrically-irreducible-components", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-fibres-nr-geometrically-irreducible-components", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let", "$$", "n_{X/Y} : Y \\to \\{0, 1, 2, 3, \\ldots, \\infty\\}", "$$", "be the function which associates to $y \\in Y$ the number of irreducible", "components of $(X_y)_K$ where $K$ is a separably closed extension", "of $\\kappa(y)$. This is well defined and if $g : Y' \\to Y$ is a morphism", "then", "$$", "n_{X'/Y'} = n_{X/Y} \\circ g", "$$", "where $X' \\to Y'$ is the base change of $f$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $y' \\in Y'$ has image $y \\in Y$.", "Suppose $K \\supset \\kappa(y)$ and $K' \\supset \\kappa(y')$ are separably", "closed extensions. Then we may choose a commutative diagram", "$$", "\\xymatrix{", "K \\ar[r] & K'' & K' \\ar[l] \\\\", "\\kappa(y) \\ar[u] \\ar[rr] & & \\kappa(y') \\ar[u]", "}", "$$", "of fields. The result follows as the morphisms of schemes", "$$", "\\xymatrix{", "(X'_{y'})_{K'} &", "(X'_{y'})_{K''} = (X_y)_{K''} \\ar[l] \\ar[r] &", "(X_y)_K", "}", "$$", "induce bijections between irreducible components, see", "Varieties,", "Lemma \\ref{varieties-lemma-separably-closed-field-irreducible-components}." ], "refs": [ "varieties-lemma-separably-closed-field-irreducible-components" ], "ref_ids": [ 10937 ] } ], "ref_ids": [] }, { "id": 13823, "type": "theorem", "label": "more-morphisms-lemma-irreducible-polynomial-over-domain", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-irreducible-polynomial-over-domain", "contents": [ "Let $A$ be a domain with fraction field $K$.", "Let $P \\in A[x_1, \\ldots, x_n]$.", "Denote $\\overline{K}$ the algebraic closure of $K$.", "Assume $P$ is irreducible in $\\overline{K}[x_1, \\ldots, x_n]$.", "Then there exists a $f \\in A$ such that", "$P^\\varphi \\in \\kappa[x_1, \\ldots, x_n]$ is irreducible for all", "homomorphisms $\\varphi : A_f \\to \\kappa$ into fields." ], "refs": [], "proofs": [ { "contents": [ "There exists an automorphism $\\Psi$ of $A[x_1, \\ldots, x_n]$ over $A$", "such that $\\Psi(P) = ax_n^d +$ lower order terms in $x_n$ with", "$a \\not = 0$, see", "Algebra, Lemma \\ref{algebra-lemma-helper-polynomial}.", "We may replace $P$ by $\\Psi(P)$ and we may replace $A$ by $A_a$.", "Thus we may assume that $P$ is monic in $x_n$ of degree $d > 0$.", "For $i = 1, \\ldots, n - 1$ let $d_i$ be the degree of $P$ in $x_i$.", "Note that this implies that $P^\\varphi$ is monic of degree $d$ in $x_n$", "and has degree $\\leq d_i$ in $x_i$ for every homomorphism", "$\\varphi : A \\to \\kappa$ where $\\kappa$ is a field.", "Thus if $P^\\varphi$ is reducible, then we can write", "$$", "P^\\varphi = Q_1 Q_2", "$$", "with $Q_1, Q_2$ monic of degree $e_1, e_2 \\geq 0$ in $x_n$ with", "$e_1 + e_2 = d$ and having degree $\\leq d_i$ in $x_i$ for", "$i = 1, \\ldots, n - 1$. In other words we can write", "\\begin{equation}", "\\label{equation-factors}", "Q_j = x_n^{e_j} + \\sum\\nolimits_{0 \\leq l < e_j}", "\\left( \\sum\\nolimits_{L \\in \\mathcal{L}} a_{j, l, L} x^L \\right) x_n^l", "\\end{equation}", "where the sum is over the set $\\mathcal{L}$ of multi-indices $L$", "of the form $L = (l_1, \\ldots, l_{n - 1})$ with $0 \\leq l_i \\leq d_i$.", "For any $e_1, e_2 \\geq 0$ with $e_1 + e_2 = d$ we consider the $A$-algebra", "$$", "B_{e_1, e_2} =", "A[\\{a_{1, l, L}\\}_{0 \\leq l < e_1, L \\in \\mathcal{L}},", "\\{a_{2, l, L}\\}_{0 \\leq l < e_2, L \\in \\mathcal{L}}]/(\\text{relations})", "$$", "where the $(\\text{relations})$ is the ideal generated by the coefficients", "of the polynomial", "$$", "P - Q_1Q_2 \\in", "A[\\{a_{1, l, L}\\}_{0 \\leq l < e_1, L \\in \\mathcal{L}},", "\\{a_{2, l, L}\\}_{0 \\leq l < e_2, L \\in \\mathcal{L}}][x_1, \\ldots, x_n]", "$$", "with $Q_1$ and $Q_2$ defined as in (\\ref{equation-factors}). OK, and", "the assumption that $P$ is irreducible over $\\overline{K}$ implies that", "there does not exist any $A$-algebra homomorphism", "$B_{e_1, e_2} \\to \\overline{K}$. By the Hilbert Nullstellensatz, see", "Algebra, Theorem \\ref{algebra-theorem-nullstellensatz}", "this means that $B_{e_1, e_2} \\otimes_A K = 0$.", "As $B_{e_1, e_2}$ is a finitely generated $A$-algebra this signifies that", "we can find an $f_{e_1, e_2} \\in A$ such that", "$(B_{e_1, e_2})_{f_{e_1, e_2}} = 0$. By construction this means that", "if $\\varphi : A_{f_{e_1, e_2}} \\to \\kappa$ is a homomorphism to a field,", "then $P^\\varphi$ does not have a factorization $P^\\varphi = Q_1 Q_2$", "with $Q_1$ of degree $e_1$ in $x_n$ and $Q_2$ of degree $e_2$ in $x_n$.", "Thus taking", "$f = \\prod_{e1, e_2 \\geq 0, e_1 + e_2 = d} f_{e_1, e_2}$ we win." ], "refs": [ "algebra-lemma-helper-polynomial", "algebra-theorem-nullstellensatz" ], "ref_ids": [ 999, 316 ] } ], "ref_ids": [] }, { "id": 13824, "type": "theorem", "label": "more-morphisms-lemma-geom-irreducible-generic-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-geom-irreducible-generic-fibre", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume", "\\begin{enumerate}", "\\item $Y$ is irreducible with generic point $\\eta$,", "\\item $X_\\eta$ is geometrically irreducible, and", "\\item $f$ is of finite type.", "\\end{enumerate}", "Then there exists a nonempty open subscheme $V \\subset Y$", "such that $X_V \\to V$ has geometrically irreducible fibres." ], "refs": [], "proofs": [ { "contents": [ "[First proof of Lemma \\ref{lemma-geom-irreducible-generic-fibre}]", "We give two proofs of the lemma. These are essentially equivalent;", "the second is more self contained but a bit longer.", "Choose a diagram", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X_V \\ar[r] \\ar[d] & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & V \\ar[r] & Y", "}", "$$", "as in", "Lemma \\ref{lemma-make-generic-fibre-geometrically-reduced}.", "Note that the generic fibre of $f'$ is the reduction of the", "generic fibre of $f$ (see", "Lemma \\ref{lemma-reduction-generic-fibre})", "and hence is geometrically irreducible.", "Suppose that the lemma holds for the morphism $f'$. Then after shrinking", "$V$ all the fibres of $f'$ are geometrically irreducible.", "As $X' = (Y' \\times_V X_V)_{red}$ this implies that all the fibres", "of $Y' \\times_V X_V$ are geometrically irreducible. Hence by", "Lemma \\ref{lemma-base-change-fibres-geometrically-irreducible}", "all the fibres of $X_V \\to V$ are geometrically irreducible and", "we win. In this way we see that we may assume that the generic", "fibre is geometrically reduced as well as geometrically irreducible", "and we may assume $Y = \\Spec(A)$ with $A$ a domain.", "\\medskip\\noindent", "Let $x \\in X_\\eta$ be the generic point. As $X_\\eta$ is geometrically", "irreducible and reduced we see that $L = \\kappa(x)$ is a finitely generated", "extension of $K = \\kappa(\\eta)$ which is geometrically reduced and", "geometrically irreducible, see", "Varieties, Lemmas \\ref{varieties-lemma-geometrically-reduced-at-point} and", "\\ref{varieties-lemma-geometrically-irreducible-function-field}.", "In particular the field extension $K \\subset L$ is separable, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-separable-field-extensions}.", "Hence we can find $x_1, \\ldots, x_{r + 1} \\in L$ which generate $L$", "over $K$ and such that $x_1, \\ldots, x_r$ is a transcendence basis for", "$L$ over $K$, see", "Algebra, Lemma", "\\ref{algebra-lemma-generating-finitely-generated-separable-field-extensions}.", "Let $P \\in K(x_1, \\ldots, x_r)[T]$ be the minimal polynomial for", "$x_{r + 1}$. Clearing denominators we may assume that", "$P$ has coefficients in $A[x_1, \\ldots, x_r]$.", "Note that as $L$ is geometrically reduced and geometrically irreducible", "over $K$, the polynomial $P$ is irreducible in", "$\\overline{K}[x_1, \\ldots, x_r, T]$ where $\\overline{K}$ is the", "algebraic closure of $K$. Denote", "$$", "B' = A[x_1, \\ldots, x_{r + 1}]/(P(x_{r + 1}))", "$$", "and set $X' = \\Spec(B')$. By construction the fraction field of $B'$", "is isomorphic to $L = \\kappa(x)$ as $K$-extensions. Hence there exists an", "open $U \\subset X$, and open $U' \\subset X'$ and a $Y$-isomorphism", "$U \\to U'$, see", "Morphisms, Lemma \\ref{morphisms-lemma-common-open}.", "Here is a diagram:", "$$", "\\xymatrix{", "X \\ar[rd] &", "U \\ar[l] \\ar@{=}[r] \\ar[d] &", "U' \\ar[r] \\ar[d] &", "X' \\ar[ld] \\ar@{=}[r] & \\Spec(B') \\\\", "& Y \\ar@{=}[r] & Y &", "}", "$$", "Note that $U_\\eta \\subset X_\\eta$ and $U'_\\eta \\subset X'_\\eta$ are", "dense opens. Thus after shrinking $Y$ by applying", "Lemma \\ref{lemma-nowhere-dense-generic-fibre}", "we obtain that $U_y$ is dense in $X_y$ and $U'_y$ is dense in $X'_y$", "for all $y \\in Y$. Thus it suffices to prove the lemma for", "$X' \\to Y$ which is the content of", "Lemma \\ref{lemma-irreducible-polynomial-over-domain}." ], "refs": [ "more-morphisms-lemma-geom-irreducible-generic-fibre", "more-morphisms-lemma-make-generic-fibre-geometrically-reduced", "more-morphisms-lemma-reduction-generic-fibre", "more-morphisms-lemma-base-change-fibres-geometrically-irreducible", "varieties-lemma-geometrically-reduced-at-point", "varieties-lemma-geometrically-irreducible-function-field", "algebra-lemma-characterize-separable-field-extensions", "algebra-lemma-generating-finitely-generated-separable-field-extensions", "morphisms-lemma-common-open", "more-morphisms-lemma-nowhere-dense-generic-fibre", "more-morphisms-lemma-irreducible-polynomial-over-domain" ], "ref_ids": [ 13824, 13806, 13805, 13821, 10906, 10936, 569, 559, 5486, 13802, 13823 ] } ], "ref_ids": [] }, { "id": 13825, "type": "theorem", "label": "more-morphisms-lemma-nr-geom-irreducible-components-good", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-nr-geom-irreducible-components-good", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let", "$n_{X/Y}$ be the function on $Y$ counting the numbers of geometrically", "irreducible components of fibres of $f$ introduced in", "Lemma \\ref{lemma-base-change-fibres-nr-geometrically-irreducible-components}.", "Assume $f$ of finite type.", "Let $y \\in Y$ be a point. Then there exists a nonempty open", "$V \\subset \\overline{\\{y\\}}$ such that $n_{X/Y}|_V$ is constant." ], "refs": [ "more-morphisms-lemma-base-change-fibres-nr-geometrically-irreducible-components" ], "proofs": [ { "contents": [ "Let $Z$ be the reduced induced scheme structure on $\\overline{\\{y\\}}$.", "Let $f_Z : X_Z \\to Z$ be the base change of $f$. Clearly it suffices to prove", "the lemma for $f_Z$ and the generic point of $Z$. Hence we may assume that", "$Y$ is an integral scheme, see", "Properties, Lemma \\ref{properties-lemma-characterize-integral}.", "Our goal in this case is to produce a nonempty open $V \\subset Y$ such that", "$n_{X/Y}|_V$ is constant.", "\\medskip\\noindent", "We apply", "Lemma \\ref{lemma-make-components-generic-fibre-geometrically-irreducible}", "to $f : X \\to Y$ and we get $g : Y' \\to V \\subset Y$. As $g : Y' \\to V$ is", "surjective finite \\'etale, in particular open (see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-open}),", "it suffices to prove that there exists an open $V' \\subset Y'$", "such that $n_{X'/Y'}|_{V'}$ is constant, see", "Lemma \\ref{lemma-base-change-fibres-nr-geometrically-irreducible-components}.", "Thus we see that we may assume that all irreducible components of", "the generic fibre $X_\\eta$ are geometrically irreducible over $\\kappa(\\eta)$.", "\\medskip\\noindent", "At this point suppose that", "$X_\\eta = X_{1, \\eta} \\bigcup \\ldots \\bigcup X_{n, \\eta}$", "is the decomposition of the generic fibre into", "(geometrically) irreducible components.", "In particular $n_{X/Y}(\\eta) = n$.", "Let $X_i$ be the closure of", "$X_{i, \\eta}$ in $X$. After shrinking $Y$ we may assume that", "$X = \\bigcup X_i$, see", "Lemma \\ref{lemma-cover-generic-fibre-neighbourhood}.", "After shrinking $Y$ some more we see that each fibre of", "$f$ has at least $n$ irreducible components, see", "Lemma \\ref{lemma-irreducible-components-in-neighbourhood}.", "Hence $n_{X/Y}(y) \\geq n$ for all $y \\in Y$.", "After shrinking $Y$ some more we obtain that $X_{i, y}$", "is geometrically irreducible for each $i$ and all $y \\in Y$, see", "Lemma \\ref{lemma-geom-irreducible-generic-fibre}.", "Since $X_y = \\bigcup X_{i, y}$", "this shows that $n_{X/Y}(y) \\leq n$ and finishes the proof." ], "refs": [ "properties-lemma-characterize-integral", "more-morphisms-lemma-make-components-generic-fibre-geometrically-irreducible", "morphisms-lemma-etale-open", "more-morphisms-lemma-base-change-fibres-nr-geometrically-irreducible-components", "more-morphisms-lemma-cover-generic-fibre-neighbourhood", "more-morphisms-lemma-irreducible-components-in-neighbourhood", "more-morphisms-lemma-geom-irreducible-generic-fibre" ], "ref_ids": [ 2947, 13807, 5370, 13822, 13804, 13820, 13824 ] } ], "ref_ids": [ 13822 ] }, { "id": 13826, "type": "theorem", "label": "more-morphisms-lemma-nr-geom-irreducible-components-constructible", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-nr-geom-irreducible-components-constructible", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let", "$n_{X/Y}$ be the function on $Y$ counting the numbers of geometrically", "irreducible components of fibres of $f$ introduced in", "Lemma \\ref{lemma-base-change-fibres-nr-geometrically-irreducible-components}.", "Assume $f$ of finite presentation. Then the level sets", "$$", "E_n = \\{y \\in Y \\mid n_{X/Y}(y) = n\\}", "$$", "of $n_{X/Y}$ are locally constructible in $Y$." ], "refs": [ "more-morphisms-lemma-base-change-fibres-nr-geometrically-irreducible-components" ], "proofs": [ { "contents": [ "Fix $n$. Let $y \\in Y$. We have to show that there exists an open neighbourhood", "$V$ of $y$ in $Y$ such that $E_n \\cap V$ is constructible in $V$. Thus we may", "assume that $Y$ is affine. Write $Y = \\Spec(A)$ and", "$A = \\colim A_i$ as a directed limit of finite type", "$\\mathbf{Z}$-algebras. By", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can find an $i$ and a morphism $f_i : X_i \\to \\Spec(A_i)$ of", "finite presentation whose base change to $Y$ recovers $f$. By", "Lemma \\ref{lemma-base-change-fibres-nr-geometrically-irreducible-components}", "it suffices to prove the lemma for $f_i$. Thus we reduce to", "the case where $Y$ is the spectrum of a Noetherian ring.", "\\medskip\\noindent", "We will use the criterion of", "Topology, Lemma \\ref{topology-lemma-characterize-constructible-Noetherian}", "to prove that $E_n$ is constructible in case $Y$ is a Noetherian scheme.", "To see this let $Z \\subset Y$ be an irreducible closed subscheme.", "We have to show that $E_n \\cap Z$ either contains a nonempty open subset", "or is not dense in $Z$. Let $\\xi \\in Z$ be the generic point. Then", "Lemma \\ref{lemma-nr-geom-irreducible-components-good}", "shows that $n_{X/Y}$ is constant in a neighbourhood of $\\xi$ in $Z$.", "This clearly implies what we want." ], "refs": [ "limits-lemma-descend-finite-presentation", "more-morphisms-lemma-base-change-fibres-nr-geometrically-irreducible-components", "topology-lemma-characterize-constructible-Noetherian", "more-morphisms-lemma-nr-geom-irreducible-components-good" ], "ref_ids": [ 15077, 13822, 8269, 13825 ] } ], "ref_ids": [ 13822 ] }, { "id": 13827, "type": "theorem", "label": "more-morphisms-lemma-connected-components-in-neighbourhood", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-connected-components-in-neighbourhood", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with", "generic point $\\eta$ and $f$ of finite type. If $X_\\eta$ has $n$", "connected components, then there exists a nonempty open $V \\subset Y$", "such that for all $y \\in V$ the fibre $X_y$ has at least $n$", "connected components." ], "refs": [], "proofs": [ { "contents": [ "As the question is purely topological we may replace $X$ and $Y$ by", "their reductions. In particular this implies that $Y$ is integral, see", "Properties, Lemma \\ref{properties-lemma-characterize-integral}.", "Let $X_\\eta = X_{1, \\eta} \\cup \\ldots \\cup X_{n, \\eta}$", "be the decomposition of $X_\\eta$ into connected components.", "Let $X_i \\subset X$ be the reduced closed subscheme whose generic", "fibre is $X_{i, \\eta}$. Note that $Z_{i, j} = X_i \\cap X_j$", "is a closed subset of $X$ whose generic fibre $Z_{i, j, \\eta}$ is empty.", "Hence after shrinking $Y$ we may assume that $Z_{i, j} = \\emptyset$, see", "Lemma \\ref{lemma-empty-generic-fibre}.", "After shrinking $Y$ some more we may assume that", "$X_y = \\bigcup X_{i, y}$ for $y \\in Y$, see", "Lemma \\ref{lemma-cover-generic-fibre-neighbourhood}.", "Moreover, after shrinking $Y$ we may assume that each $X_i \\to Y$", "is flat and of finite presentation, see", "Morphisms, Proposition \\ref{morphisms-proposition-generic-flatness}.", "The morphisms $X_i \\to Y$ are open, see", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}.", "Thus there exists an open neighbourhood $V$ of $\\eta$ which is contained", "in $f(X_i)$ for each $i$.", "For each $y \\in V$ the schemes $X_{i, y}$ are", "nonempty closed subsets of $X_y$, we have $X_y = \\bigcup X_{i, y}$", "and the intersections $Z_{i, j, y} = X_{i, y} \\cap X_{j, y}$", "are empty! Clearly this implies that", "$X_y$ has at least $n$ connected components." ], "refs": [ "properties-lemma-characterize-integral", "more-morphisms-lemma-empty-generic-fibre", "more-morphisms-lemma-cover-generic-fibre-neighbourhood", "morphisms-proposition-generic-flatness", "morphisms-lemma-fppf-open" ], "ref_ids": [ 2947, 13800, 13804, 5533, 5267 ] } ], "ref_ids": [] }, { "id": 13828, "type": "theorem", "label": "more-morphisms-lemma-base-change-fibres-geometrically-connected", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-fibres-geometrically-connected", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $g : Y' \\to Y$ be any morphism, and denote", "$f' : X' \\to Y'$ the base change of $f$.", "Then", "\\begin{align*}", "\\{y' \\in Y' \\mid X'_{y'}\\text{ is geometrically connected}\\} \\\\", "= g^{-1}(\\{y \\in Y \\mid X_y\\text{ is geometrically connected}\\}).", "\\end{align*}" ], "refs": [], "proofs": [ { "contents": [ "This comes down to the statement that for $y' \\in Y'$ with image", "$y \\in Y$ the fibre $X'_{y'} = X_y \\times_y y'$ is geometrically", "connected over $\\kappa(y')$ if and only if $X_y$ is geometrically connected", "over $\\kappa(y)$. This follows from", "Varieties,", "Lemma \\ref{varieties-lemma-geometrically-connected-check-after-extension}." ], "refs": [ "varieties-lemma-geometrically-connected-check-after-extension" ], "ref_ids": [ 10915 ] } ], "ref_ids": [] }, { "id": 13829, "type": "theorem", "label": "more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let", "$$", "n_{X/Y} : Y \\to \\{0, 1, 2, 3, \\ldots, \\infty\\}", "$$", "be the function which associates to $y \\in Y$ the number of connected", "components of $(X_y)_K$ where $K$ is a separably closed extension", "of $\\kappa(y)$. This is well defined and if $g : Y' \\to Y$ is a morphism", "then", "$$", "n_{X'/Y'} = n_{X/Y} \\circ g", "$$", "where $X' \\to Y'$ is the base change of $f$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $y' \\in Y'$ has image $y \\in Y$.", "Suppose $K \\supset \\kappa(y)$ and $K' \\supset \\kappa(y')$ are separably", "closed extensions. Then we may choose a commutative diagram", "$$", "\\xymatrix{", "K \\ar[r] & K'' & K' \\ar[l] \\\\", "\\kappa(y) \\ar[u] \\ar[rr] & & \\kappa(y') \\ar[u]", "}", "$$", "of fields. The result follows as the morphisms of schemes", "$$", "\\xymatrix{", "(X'_{y'})_{K'} &", "(X'_{y'})_{K''} = (X_y)_{K''} \\ar[l] \\ar[r] &", "(X_y)_K", "}", "$$", "induce bijections between connected components, see", "Varieties,", "Lemma \\ref{varieties-lemma-separably-closed-field-connected-components}." ], "refs": [ "varieties-lemma-separably-closed-field-connected-components" ], "ref_ids": [ 10918 ] } ], "ref_ids": [] }, { "id": 13830, "type": "theorem", "label": "more-morphisms-lemma-geometrically-connected-generic-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-geometrically-connected-generic-fibre", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume", "\\begin{enumerate}", "\\item $Y$ is irreducible with generic point $\\eta$,", "\\item $X_\\eta$ is geometrically connected, and", "\\item $f$ is of finite type.", "\\end{enumerate}", "Then there exists a nonempty open subscheme $V \\subset Y$", "such that $X_V \\to V$ has geometrically connected fibres." ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X_V \\ar[r] \\ar[d] & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & V \\ar[r] & Y", "}", "$$", "as in", "Lemma \\ref{lemma-make-components-generic-fibre-geometrically-irreducible}.", "Note that the generic fibre of $f'$ is geometrically connected", "(for example by", "Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}).", "Suppose that the lemma holds for the morphism $f'$. This means that", "there exists a nonempty open $W \\subset Y'$ such that every fibre of", "$X' \\to Y'$ over $W$ is geometrically connected.", "Then, as $g$ is an open morphism by", "Morphisms, Lemma \\ref{morphisms-lemma-etale-open}", "all the fibres of $f$ at points of the nonempty open $V = g(W)$ are", "geometrically connected, see", "Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}.", "In this way we see that we may assume that the irreducible", "components of the generic fibre $X_\\eta$ are geometrically irreducible.", "\\medskip\\noindent", "Let $Y'$ be the reduction of $Y$, and set $X' = Y' \\times_Y X$.", "Then it suffices to prove the lemma for the morphism $X' \\to Y'$", "(for example by", "Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}", "once again). Since the generic fibre of $X' \\to Y'$ is the same as the", "generic fibre of $X \\to Y$ we see that we may assume that $Y$ is", "irreducible and reduced (i.e., integral, see", "Properties, Lemma \\ref{properties-lemma-characterize-integral})", "and that the irreducible", "components of the generic fibre $X_\\eta$ are geometrically irreducible.", "\\medskip\\noindent", "At this point suppose that", "$X_\\eta = X_{1, \\eta} \\bigcup \\ldots \\bigcup X_{n, \\eta}$", "is the decomposition of the generic fibre into", "(geometrically) irreducible components.", "Let $X_i$ be the closure of $X_{i, \\eta}$ in $X$.", "After shrinking $Y$ we may assume that", "$X = \\bigcup X_i$, see", "Lemma \\ref{lemma-cover-generic-fibre-neighbourhood}.", "Let $Z_{i, j} = X_i \\cap X_j$.", "Let", "$$", "\\{1, \\ldots, n\\} \\times \\{1, \\ldots, n\\} = I \\amalg J", "$$", "where $(i, j) \\in I$ if $Z_{i, j, \\eta} = \\emptyset$ and", "$(i, j) \\in J$ if $Z_{i, j, \\eta} \\not = \\emptyset$.", "After shrinking $Y$ we may assume that $Z_{i, j} = \\emptyset$", "for all $(i, j) \\in I$, see", "Lemma \\ref{lemma-empty-generic-fibre}.", "After shrinking $Y$ we obtain that $X_{i, y}$", "is geometrically irreducible for each $i$ and all $y \\in Y$, see", "Lemma \\ref{lemma-geom-irreducible-generic-fibre}.", "After shrinking $Y$ some more we achieve the situation where", "each $Z_{i, j} \\to Y$ is flat and of finite presentation for", "all $(i, j) \\in J$, see", "Morphisms, Proposition \\ref{morphisms-proposition-generic-flatness}.", "This means that $f(Z_{i, j}) \\subset Y$ is open, see", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}.", "We claim that", "$$", "V = \\bigcap\\nolimits_{(i, j) \\in J} f(Z_{i, j})", "$$", "works, i.e., that $X_y$ is geometrically connected for each", "$y \\in V$. Namely, the fact that $X_\\eta$ is connected implies that", "the equivalence relation generated by the pairs in $J$ has only", "one equivalence class. Now if $y \\in V$ and $K \\supset \\kappa(y)$", "is a separably closed extension, then the irreducible components", "of $(X_y)_K$ are the fibres $(X_{i, y})_K$. Moreover, we see by", "construction and $y \\in V$ that $(X_{i, y})_K$ meets $(X_{j, y})_K$", "if and only if $(i, j) \\in J$. Hence the remark on equivalence classes", "shows that $(X_y)_K$ is connected and we win." ], "refs": [ "more-morphisms-lemma-make-components-generic-fibre-geometrically-irreducible", "more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components", "morphisms-lemma-etale-open", "more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components", "more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components", "properties-lemma-characterize-integral", "more-morphisms-lemma-cover-generic-fibre-neighbourhood", "more-morphisms-lemma-empty-generic-fibre", "more-morphisms-lemma-geom-irreducible-generic-fibre", "morphisms-proposition-generic-flatness", "morphisms-lemma-fppf-open" ], "ref_ids": [ 13807, 13829, 5370, 13829, 13829, 2947, 13804, 13800, 13824, 5533, 5267 ] } ], "ref_ids": [] }, { "id": 13831, "type": "theorem", "label": "more-morphisms-lemma-nr-geom-connected-components-good", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-nr-geom-connected-components-good", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let", "$n_{X/Y}$ be the function on $Y$ counting the numbers of geometrically", "connected components of fibres of $f$ introduced in", "Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}.", "Assume $f$ of finite type.", "Let $y \\in Y$ be a point. Then there exists a nonempty open", "$V \\subset \\overline{\\{y\\}}$ such that $n_{X/Y}|_V$ is constant." ], "refs": [ "more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components" ], "proofs": [ { "contents": [ "Let $Z$ be the reduced induced scheme structure on $\\overline{\\{y\\}}$.", "Let $f_Z : X_Z \\to Z$ be the base change of $f$. Clearly it suffices to prove", "the lemma for $f_Z$ and the generic point of $Z$. Hence we may assume that", "$Y$ is an integral scheme, see", "Properties, Lemma \\ref{properties-lemma-characterize-integral}.", "Our goal in this case is to produce a nonempty open $V \\subset Y$ such that", "$n_{X/Y}|_V$ is constant.", "\\medskip\\noindent", "We apply", "Lemma \\ref{lemma-make-components-generic-fibre-geometrically-irreducible}", "to $f : X \\to Y$ and we get $g : Y' \\to V \\subset Y$. As $g : Y' \\to V$ is", "surjective finite \\'etale, in particular open (see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-open}),", "it suffices to prove that there exists an open $V' \\subset Y'$", "such that $n_{X'/Y'}|_{V'}$ is constant, see", "Lemma \\ref{lemma-base-change-fibres-nr-geometrically-irreducible-components}.", "Thus we see that we may assume that all irreducible components of", "the generic fibre $X_\\eta$ are geometrically irreducible over $\\kappa(\\eta)$.", "By", "Varieties, Lemma", "\\ref{varieties-lemma-irreducible-components-geometrically-irreducible}", "this implies that also the connected components of $X_\\eta$ are", "geometrically connected.", "\\medskip\\noindent", "At this point suppose that", "$X_\\eta = X_{1, \\eta} \\bigcup \\ldots \\bigcup X_{n, \\eta}$", "is the decomposition of the generic fibre into", "(geometrically) connected components.", "In particular $n_{X/Y}(\\eta) = n$.", "Let $X_i$ be the closure of", "$X_{i, \\eta}$ in $X$. After shrinking $Y$ we may assume that", "$X = \\bigcup X_i$, see", "Lemma \\ref{lemma-cover-generic-fibre-neighbourhood}.", "After shrinking $Y$ some more we see that each fibre of", "$f$ has at least $n$ connected components, see", "Lemma \\ref{lemma-connected-components-in-neighbourhood}.", "Hence $n_{X/Y}(y) \\geq n$ for all $y \\in Y$.", "After shrinking $Y$ some more we obtain that $X_{i, y}$", "is geometrically connected for each $i$ and all $y \\in Y$, see", "Lemma \\ref{lemma-geometrically-connected-generic-fibre}.", "Since $X_y = \\bigcup X_{i, y}$", "this shows that $n_{X/Y}(y) \\leq n$ and finishes the proof." ], "refs": [ "properties-lemma-characterize-integral", "more-morphisms-lemma-make-components-generic-fibre-geometrically-irreducible", "morphisms-lemma-etale-open", "more-morphisms-lemma-base-change-fibres-nr-geometrically-irreducible-components", "varieties-lemma-irreducible-components-geometrically-irreducible", "more-morphisms-lemma-cover-generic-fibre-neighbourhood", "more-morphisms-lemma-connected-components-in-neighbourhood", "more-morphisms-lemma-geometrically-connected-generic-fibre" ], "ref_ids": [ 2947, 13807, 5370, 13822, 10946, 13804, 13827, 13830 ] } ], "ref_ids": [ 13829 ] }, { "id": 13832, "type": "theorem", "label": "more-morphisms-lemma-nr-geom-connected-components-constructible", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-nr-geom-connected-components-constructible", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let", "$n_{X/Y}$ be the function on $Y$ counting the numbers of geometric", "connected components of fibres of $f$ introduced in", "Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}.", "Assume $f$ of finite presentation. Then the level sets", "$$", "E_n = \\{y \\in Y \\mid n_{X/Y}(y) = n\\}", "$$", "of $n_{X/Y}$ are locally constructible in $Y$." ], "refs": [ "more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components" ], "proofs": [ { "contents": [ "Fix $n$. Let $y \\in Y$. We have to show that there exists an open neighbourhood", "$V$ of $y$ in $Y$ such that $E_n \\cap V$ is constructible in $V$. Thus we may", "assume that $Y$ is affine. Write $Y = \\Spec(A)$ and", "$A = \\colim A_i$ as a directed limit of finite type", "$\\mathbf{Z}$-algebras. By", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can find an $i$ and a morphism $f_i : X_i \\to \\Spec(A_i)$ of", "finite presentation whose base change to $Y$ recovers $f$. By", "Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}", "it suffices to prove the lemma for $f_i$. Thus we reduce to", "the case where $Y$ is the spectrum of a Noetherian ring.", "\\medskip\\noindent", "We will use the criterion of", "Topology, Lemma \\ref{topology-lemma-characterize-constructible-Noetherian}", "to prove that $E_n$ is constructible in case $Y$ is a Noetherian scheme.", "To see this let $Z \\subset Y$ be an irreducible closed subscheme.", "We have to show that $E_n \\cap Z$ either contains a nonempty open subset", "or is not dense in $Z$. Let $\\xi \\in Z$ be the generic point. Then", "Lemma \\ref{lemma-nr-geom-connected-components-good}", "shows that $n_{X/Y}$ is constant in a neighbourhood of $\\xi$ in $Z$.", "This clearly implies what we want." ], "refs": [ "limits-lemma-descend-finite-presentation", "more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components", "topology-lemma-characterize-constructible-Noetherian", "more-morphisms-lemma-nr-geom-connected-components-good" ], "ref_ids": [ 15077, 13829, 8269, 13831 ] } ], "ref_ids": [ 13829 ] }, { "id": 13833, "type": "theorem", "label": "more-morphisms-lemma-connected-flat-over-dvr", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-connected-flat-over-dvr", "contents": [ "\\begin{slogan}", "A flat degeneration of a disconnected scheme is either disconnected", "or nonreduced.", "\\end{slogan}", "Let $f : X \\to S$ be a morphism of schemes.", "Assume that", "\\begin{enumerate}", "\\item $S$ is the spectrum of a discrete valuation ring,", "\\item $f$ is flat,", "\\item $X$ is connected,", "\\item the closed fibre $X_s$ is reduced.", "\\end{enumerate}", "Then the generic fibre $X_\\eta$ is connected." ], "refs": [], "proofs": [ { "contents": [ "Write $S = \\Spec(R)$ and let $\\pi \\in R$ be a uniformizer.", "To get a contradiction assume that $X_\\eta$ is disconnected.", "This means there exists a nontrivial idempotent", "$e \\in \\Gamma(X_\\eta, \\mathcal{O}_{X_\\eta})$.", "Let $U = \\Spec(A)$ be any affine open in $X$.", "Note that $\\pi$ is a nonzerodivisor on $A$ as $A$ is flat over $R$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-torsion-free}", "for example. Then $e|_{U_\\eta}$ corresponds to an element $e \\in A[1/\\pi]$.", "Let $z \\in A$ be an element such that $e = z/\\pi^n$ with $n \\geq 0$ minimal.", "Note that $z^2 = \\pi^nz$. This means that $z \\bmod \\pi A$ is nilpotent", "if $n > 0$. By assumption $A/\\pi A$ is reduced, and hence minimality of", "$n$ implies $n = 0$. Thus we conclude that $e \\in A$! In other words", "$e \\in \\Gamma(X, \\mathcal{O}_X)$. As $X$ is connected it follows", "that $e$ is a trivial idempotent which is a contradiction." ], "refs": [ "more-algebra-lemma-flat-torsion-free" ], "ref_ids": [ 9919 ] } ], "ref_ids": [] }, { "id": 13834, "type": "theorem", "label": "more-morphisms-lemma-base-change-connected-along-section", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-connected-along-section", "contents": [ "Let $f : X \\to Y$, $s : Y \\to X$ be as in", "Situation \\ref{situation-connected-along-section}.", "If $g : Y' \\to Y$ is any morphism, consider the base change diagram", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]^{f'} & X \\ar[d]_f \\\\", "Y' \\ar@/^1pc/[u]^{s'} \\ar[r]^g & Y \\ar@/_1pc/[u]_s", "}", "$$", "so that we obtain $(X')^0 \\subset X'$.", "Then $(X')^0 = (g')^{-1}(X^0)$." ], "refs": [], "proofs": [ { "contents": [ "Let $y' \\in Y'$ with image $y \\in Y$. We may think of", "$X^0_y$ as a closed subscheme of $X_y$, see for example", "Morphisms,", "Definition \\ref{morphisms-definition-scheme-structure-connected-component}.", "As $s(y) \\in X^0_y$ we conclude from", "Varieties, Lemma", "\\ref{varieties-lemma-geometrically-connected-if-connected-and-point}", "that $X_y^0$ is a geometrically connected scheme over $\\kappa(y)$.", "Hence $X_y^0 \\times_y y' \\to X'_{y'}$ is a connected closed subscheme", "which contains $s'(y')$. Thus $X_y^0 \\times_y y' \\subset (X'_{y'})^0$.", "The other inclusion $X_y^0 \\times_y y' \\supset (X'_{y'})^0$ is clear", "as the image of $(X'_{y'})^0$ in $X_y$ is a connected subset of $X_y$ which", "contains $s(y)$." ], "refs": [ "morphisms-definition-scheme-structure-connected-component", "varieties-lemma-geometrically-connected-if-connected-and-point" ], "ref_ids": [ 5558, 10926 ] } ], "ref_ids": [] }, { "id": 13835, "type": "theorem", "label": "more-morphisms-lemma-connected-along-section-good", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-connected-along-section-good", "contents": [ "Let $f : X \\to Y$, $s : Y \\to X$ be as in", "Situation \\ref{situation-connected-along-section}.", "Assume $f$ of finite type. Let $y \\in Y$ be a point.", "Then there exists a nonempty open $V \\subset \\overline{\\{y\\}}$ such that", "the inverse image of $X^0$ in the base change $X_V$ is open and closed in", "$X_V$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset Y$ be the induced reduced closed subscheme", "structure on $\\overline{\\{y\\}}$. Let $f_Z : X_Z \\to Z$ and $s_Z : Z \\to X_Z$", "be the base changes of $f$ and $s$. By", "Lemma \\ref{lemma-base-change-connected-along-section}", "we have $(X_Z)^0 = (X^0)_Z$. Hence it suffices to prove the lemma for", "the morphism $X_Z \\to Z$ and the point $x \\in X_Z$ which maps to the generic", "point of $Z$. In other words we have reduced the problem to the case", "where $Y$ is an integral scheme (see", "Properties, Lemma \\ref{properties-lemma-characterize-integral})", "with generic point $\\eta$. Our goal is to show that after shrinking", "$Y$ the subset $X^0$ becomes an open and closed subset of $X$.", "\\medskip\\noindent", "Note that the scheme $X_\\eta$ is of finite type over a field, hence Noetherian.", "Thus its connected components are open as well as closed. Hence we may write", "$X_\\eta = X_\\eta^0 \\amalg T_\\eta$ for some open and closed subset", "$T_\\eta$ of $X_\\eta$. Next, let $T \\subset X$ be the closure of $T_\\eta$", "and let $X^{00} \\subset X$ be the closure of $X_\\eta^0$. Note that", "$T_\\eta$, resp.\\ $X^0_\\eta$ is the generic fibre of $T$, resp.\\ $X^{00}$,", "see discussion preceding", "Lemma \\ref{lemma-cover-generic-fibre-neighbourhood}.", "Moreover, that lemma implies that after shrinking $Y$ we may assume that", "$X = X^{00} \\cup T$ (set theoretically).", "Note that $(T \\cap X^{00})_\\eta = T_\\eta \\cap X^0_\\eta = \\emptyset$.", "Hence after shrinking $Y$ we may assume that $T \\cap X^{00} = \\emptyset$, see", "Lemma \\ref{lemma-empty-generic-fibre}.", "In particular $X^{00}$ is open in $X$. Note that $X^0_\\eta$ is connected", "and has a rational point, namely $s(\\eta)$, hence it is geometrically", "connected, see", "Varieties,", "Lemma \\ref{varieties-lemma-geometrically-connected-if-connected-and-point}.", "Thus after shrinking $Y$ we may assume that all fibres of $X^{00} \\to Y$", "are geometrically connected, see", "Lemma \\ref{lemma-geometrically-connected-generic-fibre}.", "At this point it follows that the fibres $X^{00}_y$ are", "open, closed, and connected subsets of $X_y$ containing $\\sigma(y)$.", "It follows that $X^0 = X^{00}$ and we win." ], "refs": [ "properties-lemma-characterize-integral", "more-morphisms-lemma-cover-generic-fibre-neighbourhood", "more-morphisms-lemma-empty-generic-fibre", "varieties-lemma-geometrically-connected-if-connected-and-point", "more-morphisms-lemma-geometrically-connected-generic-fibre" ], "ref_ids": [ 2947, 13804, 13800, 10926, 13830 ] } ], "ref_ids": [] }, { "id": 13836, "type": "theorem", "label": "more-morphisms-lemma-connected-along-section-locally-constructible", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-connected-along-section-locally-constructible", "contents": [ "Let $f : X \\to Y$, $s : Y \\to X$ be as in", "Situation \\ref{situation-connected-along-section}.", "If $f$ is of finite presentation then $X^0$ is locally constructible", "in $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$. We have to show that there exists an open neighbourhood", "$U$ of $x$ such that $X^0 \\cap U$ is constructible in $U$.", "This reduces us to the case where $Y$ is affine.", "Write $Y = \\Spec(A)$ and $A = \\colim A_i$ as a directed", "limit of finite type $\\mathbf{Z}$-algebras. By", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can find an $i$ and a morphism $f_i : X_i \\to \\Spec(A_i)$ of", "finite presentation, endowed with a section $s_i : \\Spec(A_i) \\to X_i$", "whose base change to $Y$ recovers $f$ and the section $s$. By", "Lemma \\ref{lemma-base-change-connected-along-section}", "it suffices to prove the lemma for $f_i, s_i$. Thus we reduce to", "the case where $Y$ is the spectrum of a Noetherian ring.", "\\medskip\\noindent", "Assume $Y$ is a Noetherian affine scheme. Since $f$ is of finite presentation,", "i.e., of finite type, we see that $X$ is a Noetherian scheme too, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}.", "In order to prove the lemma in", "this case it suffices to show that for every irreducible closed subset", "$Z \\subset X$ the intersection $Z \\cap X^0$ either contains a nonempty", "open of $Z$ or is not dense in $Z$, see", "Topology, Lemma \\ref{topology-lemma-characterize-constructible-Noetherian}.", "Let $x \\in Z$ be the generic point, and let $y = f(x)$. By", "Lemma \\ref{lemma-connected-along-section-good}", "there exists a nonempty open subset $V \\subset \\overline{\\{y\\}}$ such", "that $X^0 \\cap X_V$ is open and closed in $X_V$. Since", "$f(Z) \\subset \\overline{\\{y\\}}$ and $f(x) = y \\in V$ we see that", "$W = f^{-1}(V) \\cap Z$ is a nonempty open subset of $Z$. It follows that", "$X^0 \\cap W$ is open and closed in $W$. Since $W$ is irreducible", "we see that $X^0 \\cap W$ is either empty or equal to $W$.", "This proves the lemma." ], "refs": [ "limits-lemma-descend-finite-presentation", "morphisms-lemma-finite-type-noetherian", "topology-lemma-characterize-constructible-Noetherian" ], "ref_ids": [ 15077, 5202, 8269 ] } ], "ref_ids": [] }, { "id": 13837, "type": "theorem", "label": "more-morphisms-lemma-connected-along-section-open-neighbourhood", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-connected-along-section-open-neighbourhood", "contents": [ "Let $f : X \\to Y$, $s : Y \\to X$ be as in", "Situation \\ref{situation-connected-along-section}.", "Let $y \\in Y$ be a point.", "Assume", "\\begin{enumerate}", "\\item $f$ is of finite presentation and flat, and", "\\item the fibre $X_y$ is geometrically reduced.", "\\end{enumerate}", "Then $X^0$ is a neighbourhood of $X^0_y$ in $X$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $Y$ with an affine open neighbourhood of $y$.", "Write $Y = \\Spec(A)$ and $A = \\colim A_i$ as a directed", "limit of finite type $\\mathbf{Z}$-algebras. By", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can find an $i$ and a morphism $f_i : X_i \\to \\Spec(A_i)$ of", "finite presentation, endowed with a section $s_i : \\Spec(A_i) \\to X_i$", "whose base change to $Y$ recovers $f$ and the section $s$.", "After possibly increasing $i$ we may also assume that $f_i$ is flat, see", "Limits, Lemma \\ref{limits-lemma-descend-flat-finite-presentation}.", "Let $y_i$ be the image of $y$ in $Y_i$. Note that", "$X_y = (X_{i, y_i}) \\times_{y_i} y$. Hence $X_{i, y_i}$ is geometrically", "reduced, see", "Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced-upstairs}.", "By", "Lemma \\ref{lemma-base-change-connected-along-section}", "it suffices to prove the lemma for the system $f_i, s_i, y_i \\in Y_i$.", "Thus we reduce to the case where $Y$ is the spectrum of a Noetherian ring.", "\\medskip\\noindent", "Assume $Y$ is the spectrum of a Noetherian ring.", "Since $f$ is of finite presentation,", "i.e., of finite type, we see that $X$ is a Noetherian scheme too, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}.", "Let $x \\in X^0$ be a point lying over $y$. By", "Topology, Lemma \\ref{topology-lemma-constructible-neighbourhood-Noetherian}", "it suffices to prove that for any irreducible closed $Z \\subset X$", "passing through $x$ the intersection $X^0 \\cap Z$ is dense in $Z$.", "In particular it suffices to prove that the generic point $x' \\in Z$", "is in $X^0$. By", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian-specialization-dvr}", "we can find a discrete valuation ring $R$ and a morphism", "$\\Spec(R) \\to X$ which maps the special point to $x$ and", "the generic point to $x'$. We are going to think of $\\Spec(R)$", "as a scheme over $Y$ via the composition $\\Spec(R) \\to X \\to Y$. By", "Lemma \\ref{lemma-base-change-connected-along-section}", "we have that $(X_R)^0$ is the inverse image of $X^0$.", "By construction we have a second section $t : \\Spec(R) \\to X_R$", "(besides the base change $s_R$ of $s$)", "of the structure morphism $X_R \\to \\Spec(R)$ such that", "$t(\\eta_R)$ is a point of $X_R$ which maps to $x'$ and", "$t(0_R)$ is a point of $X_R$ which maps to $x$. Note that", "$t(0_R)$ is in $(X_R)^0$ and that $t(\\eta_R) \\leadsto t(0_R)$.", "Thus it suffices to prove that this implies that $t(\\eta_R) \\in (X_R)^0$.", "Hence it suffices to prove the lemma in the case where $Y$", "is the spectrum of a discrete valuation ring and $y$ its closed point.", "\\medskip\\noindent", "Assume $Y$ is the spectrum of a discrete valuation ring and $y$ is its closed", "point. Our goal is to prove that $X^0$ is a neighbourhood of $X^0_y$.", "Note that $X^0_y$ is open and closed in $X_y$ as $X_y$ has finitely", "many irreducible components. Hence the complement $C = X_y \\setminus X_y^0$", "is closed in $X$. Thus $U = X \\setminus C$ is an open neighbourhood of", "$X^0_y$ and $U^0 = X^0$. Hence it suffices to prove the result for the", "morphism $U \\to Y$. In other words, we may assume that", "$X_y$ is connected. Suppose that $X$ is disconnected, say", "$X = X_1 \\amalg \\ldots \\amalg X_n$ is a decomposition into connected", "components. Then $s(Y)$ is completely contained in one of the $X_i$.", "Say $s(Y) \\subset X_1$. Then $X^0 \\subset X_1$. Hence we may replace", "$X$ by $X_1$ and assume that $X$ is connected. At this point", "Lemma \\ref{lemma-connected-flat-over-dvr}", "implies that $X_\\eta$ is connected, i.e., $X^0 = X$ and we win." ], "refs": [ "limits-lemma-descend-finite-presentation", "limits-lemma-descend-flat-finite-presentation", "varieties-lemma-geometrically-reduced-upstairs", "morphisms-lemma-finite-type-noetherian", "topology-lemma-constructible-neighbourhood-Noetherian", "properties-lemma-locally-Noetherian-specialization-dvr", "more-morphisms-lemma-connected-flat-over-dvr" ], "ref_ids": [ 15077, 15062, 10910, 5202, 8270, 2959, 13833 ] } ], "ref_ids": [] }, { "id": 13838, "type": "theorem", "label": "more-morphisms-lemma-connected-along-section-open", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-connected-along-section-open", "contents": [ "Let $f : X \\to Y$, $s : Y \\to X$ be as in", "Situation \\ref{situation-connected-along-section}.", "Assume", "\\begin{enumerate}", "\\item $f$ is of finite presentation and flat, and", "\\item all fibres of $f$ are geometrically reduced.", "\\end{enumerate}", "Then $X^0$ is open in $X$." ], "refs": [], "proofs": [ { "contents": [ "This is an immediate consequence of", "Lemma \\ref{lemma-connected-along-section-open-neighbourhood}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13839, "type": "theorem", "label": "more-morphisms-lemma-dimension-in-neighbourhood", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-dimension-in-neighbourhood", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with", "generic point $\\eta$ and $f$ of finite type. If $X_\\eta$ has dimension $n$,", "then there exists a nonempty open $V \\subset Y$", "such that for all $y \\in V$ the fibre $X_y$ has dimension $n$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z = \\{x \\in X \\mid \\dim_x(X_{f(x)}) > n \\}$. By", "Morphisms, Lemma \\ref{morphisms-lemma-openness-bounded-dimension-fibres}", "this is a closed subset of $X$. By assumption $Z_\\eta = \\emptyset$.", "Hence by", "Lemma \\ref{lemma-empty-generic-fibre}", "we may shrink $Y$ and assume that $Z = \\emptyset$. Let", "$Z' = \\{x \\in X \\mid \\dim_x(X_{f(x)}) > n - 1 \\} =", "\\{x \\in X \\mid \\dim_x(X_{f(x)}) = n\\}$. As before this is a closed subset", "of $X$. By assumption we have $Z'_\\eta \\not = \\emptyset$. Hence after", "shrinking $Y$ we may assume that $Z' \\to Y$ is surjective, see", "Lemma \\ref{lemma-nonempty-generic-fibre}.", "Hence we win." ], "refs": [ "morphisms-lemma-openness-bounded-dimension-fibres", "more-morphisms-lemma-empty-generic-fibre", "more-morphisms-lemma-nonempty-generic-fibre" ], "ref_ids": [ 5280, 13800, 13801 ] } ], "ref_ids": [] }, { "id": 13840, "type": "theorem", "label": "more-morphisms-lemma-base-change-dimension-fibres", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-dimension-fibres", "contents": [ "Let $f : X \\to Y$ be a morphism of finite type. Let", "$$", "n_{X/Y} : Y \\to \\{0, 1, 2, 3, \\ldots, \\infty\\}", "$$", "be the function which associates to $y \\in Y$ the dimension of $X_y$.", "If $g : Y' \\to Y$ is a morphism then", "$$", "n_{X'/Y'} = n_{X/Y} \\circ g", "$$", "where $X' \\to Y'$ is the base change of $f$." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}." ], "refs": [ "morphisms-lemma-dimension-fibre-after-base-change" ], "ref_ids": [ 5279 ] } ], "ref_ids": [] }, { "id": 13841, "type": "theorem", "label": "more-morphisms-lemma-dimension-fibres-constructible", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-dimension-fibres-constructible", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let", "$n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$", "introduced in", "Lemma \\ref{lemma-base-change-dimension-fibres}.", "Assume $f$ of finite presentation. Then the level sets", "$$", "E_n = \\{y \\in Y \\mid n_{X/Y}(y) = n\\}", "$$", "of $n_{X/Y}$ are locally constructible in $Y$." ], "refs": [ "more-morphisms-lemma-base-change-dimension-fibres" ], "proofs": [ { "contents": [ "Fix $n$. Let $y \\in Y$. We have to show that there exists an open neighbourhood", "$V$ of $y$ in $Y$ such that $E_n \\cap V$ is constructible in $V$. Thus we may", "assume that $Y$ is affine. Write $Y = \\Spec(A)$ and", "$A = \\colim A_i$ as a directed limit of finite type", "$\\mathbf{Z}$-algebras. By", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can find an $i$ and a morphism $f_i : X_i \\to \\Spec(A_i)$ of", "finite presentation whose base change to $Y$ recovers $f$. By", "Lemma \\ref{lemma-base-change-dimension-fibres}", "it suffices to prove the lemma for $f_i$. Thus we reduce to", "the case where $Y$ is the spectrum of a Noetherian ring.", "\\medskip\\noindent", "We will use the criterion of", "Topology, Lemma \\ref{topology-lemma-characterize-constructible-Noetherian}", "to prove that $E_n$ is constructible in case $Y$ is a Noetherian scheme.", "To see this let $Z \\subset Y$ be an irreducible closed subscheme.", "We have to show that $E_n \\cap Z$ either contains a nonempty open subset", "or is not dense in $Z$. Let $\\xi \\in Z$ be the generic point. Then", "Lemma \\ref{lemma-dimension-in-neighbourhood}", "shows that $n_{X/Y}$ is constant in a neighbourhood of $\\xi$ in $Z$.", "This implies what we want." ], "refs": [ "limits-lemma-descend-finite-presentation", "more-morphisms-lemma-base-change-dimension-fibres", "topology-lemma-characterize-constructible-Noetherian", "more-morphisms-lemma-dimension-in-neighbourhood" ], "ref_ids": [ 15077, 13840, 8269, 13839 ] } ], "ref_ids": [ 13840 ] }, { "id": 13842, "type": "theorem", "label": "more-morphisms-lemma-dimension-fibres-flat", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-dimension-fibres-flat", "contents": [ "Let $f : X \\to Y$ be a flat morphism of schemes of finite presentation. Let", "$n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$", "introduced in Lemma \\ref{lemma-base-change-dimension-fibres}.", "Then $n_{X/Y}$ is lower semi-continuous." ], "refs": [ "more-morphisms-lemma-base-change-dimension-fibres" ], "proofs": [ { "contents": [ "Let $W \\subset X$, $W = \\coprod_{d \\geq 0} U_d$ be the open constructed in", "Lemmas \\ref{lemma-flat-finite-presentation-CM-open} and", "\\ref{lemma-flat-finite-presentation-CM-pieces}.", "Let $y \\in Y$ be a point. If $n_{X/Y}(y) = \\dim(X_y) = n$, then", "$y$ is in the image of $U_n \\to Y$.", "By Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}", "we see that $f(U_n)$ is open in $Y$.", "Hence there is an open neighbourhoof of $y$ where", "$n_{X/Y}$ is $\\geq n$." ], "refs": [ "more-morphisms-lemma-flat-finite-presentation-CM-open", "more-morphisms-lemma-flat-finite-presentation-CM-pieces", "morphisms-lemma-fppf-open" ], "ref_ids": [ 13789, 13791, 5267 ] } ], "ref_ids": [ 13840 ] }, { "id": 13843, "type": "theorem", "label": "more-morphisms-lemma-dimension-fibres-proper", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-dimension-fibres-proper", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes. Let", "$n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$", "introduced in Lemma \\ref{lemma-base-change-dimension-fibres}.", "Then $n_{X/Y}$ is upper semi-continuous." ], "refs": [ "more-morphisms-lemma-base-change-dimension-fibres" ], "proofs": [ { "contents": [ "Let $Z_d = \\{x \\in X \\mid \\dim_x(X_{f(x)}) > d\\}$.", "Then $Z_d$ is a closed subset of $X$ by", "Morphisms, Lemma \\ref{morphisms-lemma-openness-bounded-dimension-fibres}.", "Since $f$ is proper $f(Z_d)$ is closed.", "Since $y \\in f(Z_d) \\Leftrightarrow n_{X/Y}(y) > d$", "we see that the lemma is true." ], "refs": [ "morphisms-lemma-openness-bounded-dimension-fibres" ], "ref_ids": [ 5280 ] } ], "ref_ids": [ 13840 ] }, { "id": 13844, "type": "theorem", "label": "more-morphisms-lemma-dimension-fibres-proper-flat", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-dimension-fibres-proper-flat", "contents": [ "Let $f : X \\to Y$ be a proper, flat morphism of schemes of finite presentation.", "Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$", "introduced in Lemma \\ref{lemma-base-change-dimension-fibres}.", "Then $n_{X/Y}$ is locally constant." ], "refs": [ "more-morphisms-lemma-base-change-dimension-fibres" ], "proofs": [ { "contents": [ "Immediate consequence of", "Lemmas \\ref{lemma-dimension-fibres-flat} and", "\\ref{lemma-dimension-fibres-proper}." ], "refs": [ "more-morphisms-lemma-dimension-fibres-flat", "more-morphisms-lemma-dimension-fibres-proper" ], "ref_ids": [ 13842, 13843 ] } ], "ref_ids": [ 13840 ] }, { "id": 13845, "type": "theorem", "label": "more-morphisms-lemma-amazing-bertini-lemma", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-amazing-bertini-lemma", "contents": [ "\\begin{reference}", "See pages 71 and 72 of \\cite{Jou}", "\\end{reference}", "Let $K/k$ be a geometrically irreducible and finitely generated", "field extension. Let $n \\geq 1$.", "Let $g_1, \\ldots, g_n \\in K$ be elements such that there", "exist $c_1, \\ldots, c_n \\in k$ such that the elements", "$$", "x_1, \\ldots, x_n, \\sum g_ix_i, \\sum c_ig_i \\in K(x_1, \\ldots, x_n)", "$$", "are algebraically independent over $k$. Then", "$K(x_1, \\ldots, x_n)$ is geometrically irreducible over", "$k(x_1, \\ldots, x_n, \\sum g_ix_i)$." ], "refs": [], "proofs": [ { "contents": [ "Let $c_1, \\ldots, c_n \\in k$ be as in the statement of the lemma.", "Write $\\xi = \\sum g_ix_i$ and $\\delta = \\sum c_ig_i$.", "For $a \\in k$ consider the automorphism $\\sigma_a$", "of $K(x_1, \\ldots, x_n)$ given by the identity on $K$ and the rules", "$$", "\\sigma_a(x_i) = x_i + a c_i", "$$", "Observe that $\\sigma_a(\\xi) = \\xi + a \\delta$ and $\\sigma_a(\\delta) = \\delta$.", "Consider the tower of fields", "$$", "K_0 = k(x_1, \\ldots, x_n) \\subset", "K_1 = K_0(\\xi) \\subset", "K_2 = K_0(\\xi, \\delta) \\subset K(x_1, \\ldots, x_n) = \\Omega", "$$", "Observe that $\\sigma_a(K_0) = K_0$ and $\\sigma_a(K_2) = K_2$.", "Let $\\theta \\in \\Omega$ be separable algebraic over $K_1$.", "We have to show $\\theta \\in K_1$, see Algebra, Lemma", "\\ref{algebra-lemma-geometrically-irreducible-separable-elements}.", "\\medskip\\noindent", "Denote $K'_2$ the separable algebraic closure of $K_2$ in $\\Omega$.", "Since $K'_2/K_2$ is finite (Algebra, Lemma", "\\ref{algebra-lemma-make-geometrically-irreducible}) and", "separable there are only a finite number of fields in", "between $K'_2$ and $K_2$ (Fields, Lemma \\ref{fields-lemma-primitive-element}).", "If $k$ is infinite\\footnote{We will deal with the finite field case in the", "last paragraph of the proof.}, then we can find distinct elements", "$a_1, a_2$ of $k$ such that", "$$", "K_2(\\sigma_{a_1}(\\theta)) = K_2(\\sigma_{a_2}(\\theta))", "$$", "as subfields of $\\Omega$. Write $\\theta_i = \\sigma_{a_i}(\\theta)$", "and $\\xi_i = \\sigma_{a_i}(\\xi) = \\xi + a_i \\delta$. Observe that", "$$", "K_2 = K_0(\\xi_1, \\xi_2)", "$$", "as we have $\\xi_i = \\xi + a_i \\delta$,", "$\\xi = (a_2 \\xi_1 - a_1 \\xi_2)/(a_2 - a_1)$, and", "$\\delta = (\\xi_1 - \\xi_2)/(a_1 - a_2)$.", "Since $K_2/K_0$ is purely transcendental of degree $2$ we conclude", "that $\\xi_1$ and $\\xi_2$ are algebraically indepedent over $K_0$.", "Since $\\theta_1$ is algebraic over $K_0(\\xi_1)$ we conclude that", "$\\xi_2$ is transcendental over $K_0(\\xi_1, \\theta_1)$.", "\\medskip\\noindent", "By assumption $K/k$ is geometrically irreducible. This implies", "that $K(x_1, \\ldots, x_n)/K_0$ is geometrically irreducible", "(Algebra, Lemma", "\\ref{algebra-lemma-geometrically-irreducible-base-change-transcendental}).", "This in turn implies that $K_0(\\xi_1, \\theta_1)/K_0$", "is geometrically irreducible as a subextension", "(Algebra, Lemma \\ref{algebra-lemma-subalgebra-geometrically-irreducible}).", "Since $\\xi_2$ is transcendental over $K_0(\\xi_1, \\theta_1)$", "we conclude that $K_0(\\xi_1, \\xi_2, \\theta_1)/K_0(\\xi_2)$", "is geometrically irreducible (Algebra, Lemma", "\\ref{algebra-lemma-geometrically-irreducible-add-transcendental}).", "By our choice of $a_1, a_2$ above we have", "$$", "K_0(\\xi_1, \\xi_2, \\theta_1) =", "K_2(\\sigma_{a_1}(\\theta)) =", "K_2(\\sigma_{a_2}(\\theta)) =", "K_0(\\xi_1, \\xi_2, \\theta_2)", "$$", "Since $\\theta_2$ is separably algebraic over $K_0(\\xi_2)$", "we conclude by Algebra, Lemma", "\\ref{algebra-lemma-geometrically-irreducible-separable-elements} again that", "$\\theta_2 \\in K_0(\\xi_2)$. Taking $\\sigma_{a_2}^{-1}$", "of this relation givens $\\theta \\in K_0(\\xi) = K_1$ as desired.", "\\medskip\\noindent", "This finishes the proof in case $k$ is infinite. If $k$ is finite,", "then we can choose a variable $t$ and consider the extension", "$K(t)/k(t)$ which is geometrically irreducible by", "Algebra, Lemma", "\\ref{algebra-lemma-geometrically-irreducible-base-change-transcendental}.", "Since it is still be true that", "$x_1, \\ldots, x_n, \\sum g_ix_i, \\sum c_ig_i$", "in $K(t, x_1, \\ldots, x_n)$ are algebraically independent over $k(t)$", "we conclude that $K(t, x_1, \\ldots, x_n)$", "is geometrically irreducible over", "$k(t, x_1, \\ldots, x_n, \\sum g_ix_i)$", "by the argument already given.", "Then using Algebra, Lemma", "\\ref{algebra-lemma-geometrically-irreducible-base-change-transcendental}", "once more finishes the job." ], "refs": [ "algebra-lemma-geometrically-irreducible-separable-elements", "algebra-lemma-make-geometrically-irreducible", "fields-lemma-primitive-element", "algebra-lemma-geometrically-irreducible-base-change-transcendental", "algebra-lemma-subalgebra-geometrically-irreducible", "algebra-lemma-geometrically-irreducible-add-transcendental", "algebra-lemma-geometrically-irreducible-separable-elements", "algebra-lemma-geometrically-irreducible-base-change-transcendental", "algebra-lemma-geometrically-irreducible-base-change-transcendental" ], "ref_ids": [ 597, 598, 4498, 595, 591, 596, 597, 595, 595 ] } ], "ref_ids": [] }, { "id": 13846, "type": "theorem", "label": "more-morphisms-lemma-algebraically-independent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-algebraically-independent", "contents": [ "Let $A$ be a domain of finite type over a field $k$. Let $n \\geq 2$.", "Let $g_1, \\ldots, g_n \\in A$ be elements such that $V(g_1, g_2)$", "has an irreducible component of dimension $\\dim(A) - 2$.", "Then there exist $c_1, \\ldots, c_n \\in k$ such that the elements", "$$", "x_1, \\ldots, x_n, \\sum g_ix_i, \\sum c_ig_i \\in", "\\text{Frac}(A)(x_1, \\ldots, x_n)", "$$", "are algebraically independent over $k$." ], "refs": [], "proofs": [ { "contents": [ "The algebraic independence over $k$ means that the morphism", "$$", "T = \\Spec(A[x_1, \\ldots, x_n])", "\\longrightarrow", "\\Spec(k[x_1, \\ldots, x_n, y, z]) = S", "$$", "given by $y = \\sum g_ix_i$ and $z = \\sum c_ig_i$ is dominant.", "Set $d = \\dim(A)$. If $T \\to S$ is not dominant, then the image", "has dimension $< n + 2$ and hence every irreducible component", "of every fibre has dimension $> d + n - (n + 2) = d - 2$, see", "Varieties, Lemma", "\\ref{varieties-lemma-dimension-fibres-locally-algebraic}.", "Choose a closed point $u \\in V(g_1, g_2)$ contained in an irreducible component", "of dimension $d - 2$ and in no other component of $V(g_1, g_2)$.", "Consider the closed point $t = (u, 1, 0, \\ldots 0)$", "of $T$ lying over $u$. Set $(c_1, \\ldots, c_n) = (0, 1, 0, \\ldots, 0)$.", "Then $t$ maps to the point $s = (1, 0, \\ldots, 0)$ of $S$.", "The fibre of $T \\to S$ over $s$ is cut out by", "$$", "x_1 - 1, x_2, \\ldots, x_n, \\sum x_ig_i, g_2", "$$", "and hence equivalently is cut out by", "$$", "x_1 - 1, x_2, \\ldots, x_n, g_1, g_2", "$$", "By our condition on $g_1, g_2$ this subscheme has an irreducible component", "of dimension $d - 2$." ], "refs": [ "varieties-lemma-dimension-fibres-locally-algebraic" ], "ref_ids": [ 10990 ] } ], "ref_ids": [] }, { "id": 13847, "type": "theorem", "label": "more-morphisms-lemma-bertini-irreducible", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-bertini-irreducible", "contents": [ "\\begin{reference}", "\\cite[Theorem 6.3 part 4)]{Jou}", "\\end{reference}", "In Varieties, Situation \\ref{varieties-situation-family-divisors} assume", "\\begin{enumerate}", "\\item $X$ is of finite type over $k$,", "\\item $X$ is geometrically irreducible over $k$,", "\\item there exist $v_1, v_2, v_3 \\in V$ and an irreducible component", "$Z$ of $H_{v_2} \\cap H_{v_3}$ such that $Z \\not \\subset H_{v_1}$ and", "$\\text{codim}(Z, X) = 2$, and", "\\item every irreducible component $Y$ of $\\bigcap_{v \\in V} H_v$", "has $\\text{codim}(Y, X) \\geq 2$.", "\\end{enumerate}", "Then for general $v \\in V \\otimes_k k'$", "the scheme $H_v$ is geometrically irreducible over $k'$." ], "refs": [], "proofs": [ { "contents": [ "In order for assumption (2) to hold, the elements $v_1, v_2, v_3$", "must be $k$-linearly independent in $V$ (small detail omitted).", "Thus we may choose a basis $v_1, \\ldots, v_r$ of $V$ incorporating", "these elements as the first $3$. Recall that", "$H_{univ} \\subset \\mathbf{A}^r_k \\times_k X$ is the ``universal divisor''.", "Consider the projection $q : H_{univ} \\to \\mathbf{A}^r_k$", "whose scheme theoretic fibres are the divisors $H_v$. By", "Lemma \\ref{lemma-geom-irreducible-generic-fibre}", "it suffices to show that the generic fibre of $q$ is geometrically", "irreducible.", "To prove this we may replace $X$ by its reduction, hence", "we may assume $X$ is an integral scheme of finite type over $k$.", "\\medskip\\noindent", "Let $U \\subset X$ be a nonempty affine open such that", "$\\mathcal{L}|_U \\cong \\mathcal{O}_U$. Write $U = \\Spec(A)$.", "Denote $f_i \\in A$ the element corresponding to section $\\psi(v_i)|_U$", "via the isomorphism $\\mathcal{L}|_U \\cong \\mathcal{O}_U$.", "Then $H_{univ} \\cap (\\mathbf{A}^r_k \\times_k U)$ is given by", "$$", "H_U = \\Spec(A[x_1, \\ldots, x_r]/(x_1f_1 + \\ldots + x_rf_r))", "$$", "By our choice of basis we see that $f_1$ cannot be zero because this", "would mean $v_1 = 0$ and hence $H_{v_1} = X$ which contradicts", "assumption (2). Hence $\\sum x_if_i$ is a nonzerodivisor in", "$A[x_1, \\ldots, x_r]$.", "It follows that every irreducible component of $H_U$ has dimension", "$d + r - 1$ where $d = \\dim(X) = \\dim(A)$. If $U' = U \\cap D(f_1)$", "then we see that", "$$", "H_{U'} =", "\\Spec(A_{f_1}[x_1, \\ldots, x_r]/(x_1f_1 + \\ldots + x_rf_r)) \\cong", "\\Spec(A_{f_1}[x_2, \\ldots x_r]) =", "\\mathbf{A}^{r - 1}_k \\times_k U'", "$$", "is irreducible. On the other hand, we have", "$$", "H_U \\setminus H_{U'} =", "\\Spec(A/(f_1)[x_1, \\ldots, x_r]/(x_2f_2 + \\ldots + x_rf_r))", "$$", "which has dimension at most $d + r - 2$. Namely, for $i \\not = 1$", "the scheme $(H_U \\setminus H_{U'}) \\times_U D(f_i)$ is either empty", "(if $f_i = 0$) or by the same argument as above isomorphic to an $r - 1$", "dimensional affine space over an open of $\\Spec(A/(f_1))$ and hence has", "dimension at most $d + r - 2$.", "On the other hand, $(H_U \\setminus H_{U'}) \\times_U V(f_2, \\ldots, f_r)$", "is an $r$ dimensional affine space over $\\Spec(A/(f_1, \\ldots, f_r))$", "and hence assumption (3) tells us this has dimension at most", "$d + r - 2$. We conclude that $H_U$ is irreducible for every $U$ as above.", "It follows that $H_{univ}$ is irreducible.", "\\medskip\\noindent", "Thus it suffices to show that the generic point of $H_{univ}$", "is geometrically irreducible over the generic point of $\\mathbf{A}^r_k$, see", "Varieties, Lemma", "\\ref{varieties-lemma-geometrically-irreducible-function-field}.", "Choose a nonempty affine open $U = \\Spec(A)$ of $X$", "contained in $X \\setminus H_{v_1}$ which meets the irreducible", "component $Z$ of $H_{v_2} \\cap H_{v_3}$ whose existence is averted in", "assumption (2). With notation as above we have to prove that", "the field extension", "$$", "\\text{Frac}(A[x_1, \\ldots, x_r]/(x_1f_1 + \\ldots + x_rf_r))/", "k(x_1, \\ldots , x_r)", "$$", "is geometrically irreducible. Observe that $f_1$ is invertible in $A$", "by our choice of $U$.", "Set $K = \\text{Frac}(A)$ equal to the fraction field of $A$.", "Eliminating the variable $x_1$ as above,", "we find that we have to show that the field extension", "$$", "K(x_2, \\ldots, x_r)/", "k(x_2, \\ldots, x_r, -\\sum\\nolimits_{i = 2, \\ldots, r} f_1^{-1}f_i x_i)", "$$", "is geometrically irreducible. By Lemma \\ref{lemma-amazing-bertini-lemma}", "it suffices to show that for some $c_2, \\ldots, c_r \\in k$ the elements", "$$", "x_2, \\ldots, x_r, \\sum\\nolimits_{i = 2, \\ldots, r} f_1^{-1}f_i x_i,", "\\sum\\nolimits_{i = 2, \\ldots, r} c_if_1^{-1}f_i", "$$", "are algebraically independent over $k$ in the fraction field of", "$A[x_2, \\ldots, x_r]$. This follows from", "Lemma \\ref{lemma-algebraically-independent}", "and the fact that $Z \\cap U$ is an irreducible", "component of $V(f_1^{-1}f_2, f_1^{-1}f_3) \\subset U$." ], "refs": [ "more-morphisms-lemma-geom-irreducible-generic-fibre", "varieties-lemma-geometrically-irreducible-function-field", "more-morphisms-lemma-amazing-bertini-lemma", "more-morphisms-lemma-algebraically-independent" ], "ref_ids": [ 13824, 10936, 13845, 13846 ] } ], "ref_ids": [] }, { "id": 13848, "type": "theorem", "label": "more-morphisms-lemma-diagonal-picard-flat-proper", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-diagonal-picard-flat-proper", "contents": [ "Let $f : X \\to S$ be a flat, proper morphism of finite presentation.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module.", "For a morphism $g : T \\to S$ consider the base change diagram", "$$", "\\xymatrix{", "X_T \\ar[d]_p \\ar[r]_q & X \\ar[d]^f \\\\", "T \\ar[r]^g & S", "}", "$$", "Assume $\\mathcal{O}_T \\to p_*\\mathcal{O}_{X_T}$ is an", "isomorphism for all $g : T \\to S$. Then there exists an", "immersion $j : Z \\to S$ of finite presentation such that", "a morphism $g : T \\to S$ factors through $Z$ if and only if", "there exists a finite locally free $\\mathcal{O}_T$-module $\\mathcal{N}$", "with $p^*\\mathcal{N} \\cong q^*\\mathcal{E}$." ], "refs": [], "proofs": [ { "contents": [ "Observe that the fibres $X_s$ of $f$ are connected by our assumption", "that $H^0(X_s, \\mathcal{O}_{X_s}) = \\kappa(s)$. Thus the rank of", "$\\mathcal{E}$ is constant on the fibres. Since $f$ is open", "(Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}) and closed", "we conclude that there is a decomposition $S = \\coprod S_r$", "of $S$ into open and closed subschemes such that $\\mathcal{E}$", "has constant rank $r$ on the inverse image of $S_r$.", "Thus we may assume $\\mathcal{E}$ has constant rank $r$.", "We will denote $\\mathcal{E}^\\vee = \\SheafHom(\\mathcal{E}, \\mathcal{O}_X)$", "the dual rank $r$ module.", "\\medskip\\noindent", "By cohomology and base change (more precisely by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image-general})", "we see that $E = Rf_*\\mathcal{E}$ is a perfect object of the", "derived category of $S$ and that its formation commutes with", "arbitrary change of base. Similarly for $E' = Rf_*\\mathcal{E}^\\vee$.", "Since there is never any cohomology in degrees $< 0$, we see that", "$E$ and $E'$ have (locally) tor-amplitude in $[0, b]$ for some $b$.", "Observe that for any $g : T \\to S$ we have", "$p_*(q^*\\mathcal{E}) = H^0(Lg^*E)$ and", "$p_*(q^*\\mathcal{E}^\\vee) = H^0(Lg^*E')$.", "Let $j : Z \\to S$ and $j' : Z' \\to S$ be immersions", "of finite presentation constructed in Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-locally-closed-where-H0-locally-free}", "for $E$ and $E'$ with $a = 0$ and $r = r$; these are roughly speaking", "characterized by the property that $H^0(Lj^*E)$ and $H^0((j')^*E')$", "are finite locally free modules compatible with pullback.", "\\medskip\\noindent", "Let $g : T \\to S$ be a morphism. If there exists an $\\mathcal{N}$", "as in the lemma, then, using the projection formula", "Cohomology, Lemma \\ref{cohomology-lemma-projection-formula},", "we see that the modules", "$$", "p_*(q^*\\mathcal{L}) \\cong", "p_*(p^*\\mathcal{N}) \\cong", "\\mathcal{N} \\otimes_{\\mathcal{O}_T} p_*\\mathcal{O}_{X_T} \\cong", "\\mathcal{N}\\quad\\text{and similarly }\\quad", "p_*(q^*\\mathcal{E}^\\vee) \\cong \\mathcal{N}^\\vee", "$$", "are finite locally free modules of rank $r$", "and remain so after any further base change $T' \\to T$.", "Hence in this case $T \\to S$ factors through $j$ and through $j'$.", "Thus we may replace $S$ by $Z \\times_S Z'$ and assume that", "$f_*\\mathcal{E}$ and $f_*\\mathcal{E}^\\vee$ are finite locally free", "$\\mathcal{O}_S$-modules of rank $r$", "whose formation commutes with arbitrary change of base", "(small detail omitted).", "\\medskip\\noindent", "In this sitation if $g : T \\to S$ be a morphism and there exists an", "$\\mathcal{N}$ as in the lemma, then the map (cup product in degree $0$)", "$$", "p_*(q^*\\mathcal{E})", "\\otimes_{\\mathcal{O}_T}", "p_*(q^*\\mathcal{E}^\\vee)", "\\longrightarrow \\mathcal{O}_T", "$$", "is a perfect pairing. Conversely, if this cup product map is a", "perfect pairing, then we see that locally on $T$ we may choose a", "basis of sections", "$\\sigma_1, \\ldots, \\sigma_r$ in $p_*(q^*\\mathcal{E})$", "and $\\tau_1, \\ldots, \\tau_r$ in $p_*(q^*\\mathcal{E}^\\vee)$", "whose products satisfy $\\sigma_i \\tau_j = \\delta_{ij}$.", "Thinking of $\\sigma_i$ as a section of $q^*\\mathcal{E}$ on $X_T$", "and $\\tau_j$ as a section of $q^*\\mathcal{E}^\\vee$ on $X_T$,", "we conclude that", "$$", "\\sigma_1, \\ldots, \\sigma_r :", "\\mathcal{O}_{X_T}^{\\oplus r}", "\\longrightarrow", "q^*\\mathcal{E}", "$$", "is an isomorphism with inverse given by", "$$", "\\tau_1, \\ldots, \\tau_r :", "q^*\\mathcal{E}", "\\longrightarrow", "\\mathcal{O}_{X_T}^{\\oplus r}", "$$", "In other words, we see that $p^*p_*q^*\\mathcal{E} \\cong q^*\\mathcal{E}$.", "But the condition that the cupproduct is nondegenerate picks", "out a retrocompact open subscheme (namely, the locus where a suitable", "determinant is nonzero) and the proof is complete." ], "refs": [ "morphisms-lemma-fppf-open", "perfect-lemma-flat-proper-perfect-direct-image-general", "perfect-lemma-locally-closed-where-H0-locally-free", "cohomology-lemma-projection-formula" ], "ref_ids": [ 5267, 7054, 7061, 2243 ] } ], "ref_ids": [] }, { "id": 13849, "type": "theorem", "label": "more-morphisms-lemma-trivial-on-fibres", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-trivial-on-fibres", "contents": [ "Let $f : X \\to S$ be a flat, proper morphism of finite presentation", "such that $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this remains", "true after arbitrary base change. Let $\\mathcal{E}$ be a finite", "locally free $\\mathcal{O}_X$-module. Assume", "\\begin{enumerate}", "\\item $\\mathcal{E}|_{X_s}$ is isomorphic to", "$\\mathcal{O}_{X_s}^{\\oplus r_s}$ for all $s \\in S$, and", "\\item $S$ is reduced.", "\\end{enumerate}", "Then $\\mathcal{E} = f^*\\mathcal{N}$ for some finite locally free", "$\\mathcal{O}_S$-module $\\mathcal{N}$." ], "refs": [], "proofs": [ { "contents": [ "Namely, in this case the locally closed immersion $j : Z \\to S$ of", "Lemma \\ref{lemma-diagonal-picard-flat-proper}", "is bijective and hence a", "closed immersion. But since $S$ is reduced, $j$ is an isomorphism." ], "refs": [ "more-morphisms-lemma-diagonal-picard-flat-proper" ], "ref_ids": [ 13848 ] } ], "ref_ids": [] }, { "id": 13850, "type": "theorem", "label": "more-morphisms-lemma-triviality-generic-fibre-valuation-ring", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-triviality-generic-fibre-valuation-ring", "contents": [ "Let $f : X \\to S$ be a proper flat morphism of finite presentation.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Assume", "\\begin{enumerate}", "\\item $S$ is the spectrum of a valuation ring,", "\\item $\\mathcal{L}$ is trivial on the generic fibre $X_\\eta$ of $f$,", "\\item the closed fibre $X_0$ of $f$ is integral,", "\\item $H^0(X_\\eta, \\mathcal{O}_{X_\\eta})$ is equal to the function field of $S$.", "\\end{enumerate}", "Then $\\mathcal{L}$ is trivial." ], "refs": [], "proofs": [ { "contents": [ "Write $S = \\Spec(A)$. We will first prove the lemma when $A$ is a", "discrete valuation ring (as this is the case most often used in practice).", "Let $\\pi \\in A$ be a uniformizer.", "Take a trivializing section $s \\in \\Gamma(X_\\eta, \\mathcal{L}_\\eta)$.", "After replacing $s$ by $\\pi^n s$ if necessary", "we can assume that $s \\in \\Gamma(X, \\mathcal{L})$.", "If $s|_{X_0} = 0$, then we see", "that $s$ is divisible by $\\pi$ (because $X_0$ is the scheme theoretic", "fibre and $X$ is flat over $A$). Thus we may assume that $s|_{X_0}$", "is nonzero. Then the zero locus $Z(s)$ of $s$ is contained in $X_0$", "but does not contain the generic point of $X_0$ (because $X_0$ is integral).", "This means that the $Z(s)$ has codimension $\\geq 2$ in $X$ which contradicts", "Divisors, Lemma \\ref{divisors-lemma-effective-Cartier-codimension-1}", "unless $Z(s) = \\emptyset$ as desired.", "\\medskip\\noindent", "Proof in the general case. Since the valuation ring $A$ is", "coherent (Algebra, Example \\ref{algebra-example-valuation-ring-coherent})", "we see that $H^0(X, \\mathcal{L})$ is a coherent $A$-module, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-cohomology-over-coherent-ring}.", "Equivalently, $H^0(X, \\mathcal{L})$ is a finitely presented", "$A$-module (Algebra, Lemma \\ref{algebra-lemma-coherent-ring}).", "Since $H^0(X, \\mathcal{L})$ is torsion free (by flatness of $X$ over $A$),", "we see from More on Algebra, Lemma", "\\ref{more-algebra-lemma-generalized-valuation-ring-modules}", "that $H^0(X, \\mathcal{L}) = A^{\\oplus n}$ for some $n$.", "By flat base change (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology})", "we have", "$$", "K = H^0(X_\\eta, \\mathcal{O}_{X_\\eta}) \\cong", "H^0(X_\\eta, \\mathcal{L}_\\eta) =", "H^0(X, \\mathcal{L}) \\otimes_A K", "$$", "where $K$ is the fraction field of $A$. Thus $n = 1$.", "Pick a generator $s \\in H^0(X, \\mathcal{L})$.", "Let $\\mathfrak m \\subset A$ be the maximal ideal.", "Then $\\kappa = A/\\mathfrak m = \\colim A/\\pi$ where", "this is a filtered colimit over nonzero $\\pi \\in \\mathfrak m$", "(here we use that $A$ is a valuation ring).", "Thus $X_0 = \\lim X \\times_S \\Spec(A/\\pi)$.", "If $s|_{X_0}$ is zero, then for some $\\pi$", "we see that $s$ restricts to zero on $X \\times_S \\Spec(A/\\pi)$, see", "Limits, Lemma \\ref{limits-lemma-descend-section}.", "But if this happens, then $\\pi^{-1} s$ is", "a global section of $\\mathcal{L}$ which contradicts", "the fact that $s$ is a generator of $H^0(X, \\mathcal{L})$.", "Thus $s|_{X_0}$ is not zero. Let $Z(s) \\subset X$ be the zero scheme of $s$.", "Since $s|_{X_0}$ is not zero and since $X_0$ is integral,", "we see that $Z(s)_0 \\subset X_0$ is an effective Cartier divisor.", "Since $f$ is proper and $S$ is local, every point of $Z(s)$", "specializes to a point of $Z(s)_0$. Thus by", "Divisors, Lemma \\ref{divisors-lemma-fibre-Cartier} part (3)", "we see that $Z(s)$ is a relative effective Cartier divisor,", "in particular $Z(s) \\to S$ is flat.", "Hence if $Z(s)$ were nonemtpy, then $Z(s)_\\eta$ would be nonempty", "which contradicts the fact that $s|_{X_\\eta}$ is a trivialization", "of $\\mathcal{L}_\\eta$. Thus $Z(s) = \\emptyset$ as desired." ], "refs": [ "divisors-lemma-effective-Cartier-codimension-1", "perfect-lemma-cohomology-over-coherent-ring", "algebra-lemma-coherent-ring", "more-algebra-lemma-generalized-valuation-ring-modules", "coherent-lemma-flat-base-change-cohomology", "divisors-lemma-fibre-Cartier" ], "ref_ids": [ 7947, 7070, 843, 10556, 3298, 7978 ] } ], "ref_ids": [] }, { "id": 13851, "type": "theorem", "label": "more-morphisms-lemma-get-a-closed", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-get-a-closed", "contents": [ "Let $f : X \\to S$ and $\\mathcal{E}$ be as in", "Lemma \\ref{lemma-diagonal-picard-flat-proper}", "and in addition assume $\\mathcal{E}$ is an invertible $\\mathcal{O}_X$-module.", "If moreover the geometric fibres of $f$ are", "integral, then $Z$ is closed in $S$." ], "refs": [ "more-morphisms-lemma-diagonal-picard-flat-proper" ], "proofs": [ { "contents": [ "Since $j : Z \\to S$ is of finite presentation, it suffices", "to show: for any morphism $g : \\Spec(A) \\to S$ where $A$ is a", "valuation ring with fraction field $K$ such that $g(\\Spec(K)) \\in j(Z)$", "we have $g(\\Spec(A)) \\subset j(Z)$. See", "Morphisms, Lemma \\ref{morphisms-lemma-reach-points-scheme-theoretic-image}.", "This follows from Lemma \\ref{lemma-triviality-generic-fibre-valuation-ring}", "and the characterization of $j : Z \\to S$ in", "Lemma \\ref{lemma-diagonal-picard-flat-proper}." ], "refs": [ "morphisms-lemma-reach-points-scheme-theoretic-image", "more-morphisms-lemma-triviality-generic-fibre-valuation-ring", "more-morphisms-lemma-diagonal-picard-flat-proper" ], "ref_ids": [ 5147, 13850, 13848 ] } ], "ref_ids": [ 13848 ] }, { "id": 13852, "type": "theorem", "label": "more-morphisms-lemma-H1-O-picard-flat-proper", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-H1-O-picard-flat-proper", "contents": [ "Consider a commutative diagram of schemes", "$$", "\\xymatrix{", "X' \\ar[rr] \\ar[dr]_{f'} & & X \\ar[dl]^f \\\\", "& S", "}", "$$", "with $f' : X' \\to S$ and $f : X \\to S$ satisfying the hypotheses of", "Lemma \\ref{lemma-diagonal-picard-flat-proper}.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module", "and let $\\mathcal{L}'$ be the pullback to $X'$. Let $Z \\subset S$,", "resp.\\ $Z' \\subset S$ be the locally closed subscheme constructed", "in Lemma \\ref{lemma-diagonal-picard-flat-proper}", "for $(f, \\mathcal{L})$, resp.\\ $(f', \\mathcal{L}')$", "so that $Z \\subset Z'$. If $s \\in Z$ and", "$$", "H^1(X_s, \\mathcal{O}) \\longrightarrow H^1(X'_s, \\mathcal{O})", "$$", "is injective, then $Z \\cap U = Z' \\cap U$ for some open neighbourhood", "$U$ of $s$." ], "refs": [ "more-morphisms-lemma-diagonal-picard-flat-proper", "more-morphisms-lemma-diagonal-picard-flat-proper" ], "proofs": [ { "contents": [ "We may replace $S$ by $Z'$. After shrinking $S$ to an affine open neighbourhood", "of $s$ we may assume that $\\mathcal{L}' = \\mathcal{O}_{X'}$.", "Let $E = Rf_*\\mathcal{L}$ and $E' = Rf'_*\\mathcal{L}' = Rf'_*\\mathcal{O}_{X'}$.", "These are perfect complexes whose formation commutes with arbitrary", "change of base (Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image-general}).", "In particular we see that", "$$", "E \\otimes_{\\mathcal{O}_S}^\\mathbf{L} \\kappa(s) =", "R\\Gamma(X_s, \\mathcal{L}_s) = R\\Gamma(X_s, \\mathcal{O}_{X_s})", "$$", "The second equality because $s \\in Z$. Set", "$h_i = \\dim_{\\kappa(s)} H^i(X_s, \\mathcal{O}_{X_s})$.", "After shrinking $S$ we can represent $E$ by a complex", "$$", "\\mathcal{O}_S \\to \\mathcal{O}_S^{\\oplus h_1} \\to", "\\mathcal{O}_S^{\\oplus h_2} \\to \\ldots", "$$", "see More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-perfect-from-residue-field}", "(strictly speaking this also uses", "Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-affine-compare-bounded} and", "\\ref{perfect-lemma-perfect-affine}). Similarly, we may assume $E'$", "is represented by a complex", "$$", "\\mathcal{O}_S \\to \\mathcal{O}_S^{\\oplus h'_1} \\to", "\\mathcal{O}_S^{\\oplus h'_2} \\to \\ldots", "$$", "where $h'_i = \\dim_{\\kappa(s)} H^i(X'_s, \\mathcal{O}_{X'_s})$.", "By functoriality of cohomology we have a map", "$$", "E \\longrightarrow E'", "$$", "in $D(\\mathcal{O}_S)$ whose formation commutes with change of base.", "Since the complex representing $E$ is a finite complex of finite free", "modules and since $S$ is affine, we can choose a map of complexes", "$$", "\\xymatrix{", "\\mathcal{O}_S \\ar[r]_d \\ar[d]_a &", "\\mathcal{O}_S^{\\oplus h_1} \\ar[r] \\ar[d]_b &", "\\mathcal{O}_S^{\\oplus h_2} \\ar[r] \\ar[d]_c & \\ldots \\\\", "\\mathcal{O}_S \\ar[r]^{d'} &", "\\mathcal{O}_S^{\\oplus h'_1} \\ar[r] &", "\\mathcal{O}_S^{\\oplus h'_2} \\ar[r] & \\ldots", "}", "$$", "representing the given map $E \\to E'$. Since $s \\in Z$ we see that", "the trivializing section of $\\mathcal{L}_s$ pulls back to a trivializing", "section of $\\mathcal{L}'_s = \\mathcal{O}_{X'_s}$. Thus", "$a \\otimes \\kappa(s)$ is an isomorphism, hence after shrinking $S$", "we see that $a$ is an isomorphism. Finally, we use the hypothesis", "that $H^1(X_s, \\mathcal{O}) \\to H^1(X'_s, \\mathcal{O})$", "is injective, to see that there exists a $h_1 \\times h_1$ minor of the", "matrix defining $b$ which maps to a nonzero", "element in $\\kappa(s)$. Hence after shrinking $S$ we may assume", "that $b$ is injective. However, since $\\mathcal{L}' = \\mathcal{O}_{X'}$", "we see that $d' = 0$. It follows that $d = 0$. In this way we see", "that the trivializing section of $\\mathcal{L}_s$ lifts to a section", "of $\\mathcal{L}$ over $X$. A straightforward topological argument (omitted)", "shows that this means that $\\mathcal{L}$ is trivial after possibly", "shrinking $S$ a bit further." ], "refs": [ "perfect-lemma-flat-proper-perfect-direct-image-general", "more-algebra-lemma-lift-perfect-from-residue-field", "perfect-lemma-affine-compare-bounded", "perfect-lemma-perfect-affine" ], "ref_ids": [ 7054, 10232, 6941, 6980 ] } ], "ref_ids": [ 13848, 13848 ] }, { "id": 13853, "type": "theorem", "label": "more-morphisms-lemma-H1-O-multiple-picard-flat-proper", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-H1-O-multiple-picard-flat-proper", "contents": [ "Consider $n$ commutative diagrams of schemes", "$$", "\\xymatrix{", "X_i \\ar[rr] \\ar[dr]_{f_i} & & X \\ar[dl]^f \\\\", "& S", "}", "$$", "with $f_i : X_i \\to S$ and $f : X \\to S$ satisfying the hypotheses of", "Lemma \\ref{lemma-diagonal-picard-flat-proper}.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module", "and let $\\mathcal{L}_i$ be the pullback to $X_i$. Let $Z \\subset S$,", "resp.\\ $Z_i \\subset S$ be the locally closed subscheme constructed", "in Lemma \\ref{lemma-diagonal-picard-flat-proper}", "for $(f, \\mathcal{L})$, resp.\\ $(f_i, \\mathcal{L}_i)$", "so that $Z \\subset \\bigcap_{i = 1, \\ldots, n} Z_i$. If $s \\in Z$ and", "$$", "H^1(X_s, \\mathcal{O}) \\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} H^1(X_{i, s}, \\mathcal{O})", "$$", "is injective, then $Z \\cap U = (\\bigcap_{i = 1, \\ldots, n} Z_i) \\cap U$", "(scheme theoretic intersection) for some open neighbourhood $U$ of $s$." ], "refs": [ "more-morphisms-lemma-diagonal-picard-flat-proper", "more-morphisms-lemma-diagonal-picard-flat-proper" ], "proofs": [ { "contents": [ "This lemma is a variant of Lemma \\ref{lemma-H1-O-picard-flat-proper}", "and we strongly urge the reader to read that proof first; this proof", "is basically a copy of that proof with minor modifications. It follows", "from the description of (scheme valued) points of $Z$ and the $Z_i$", "that $Z \\subset \\bigcap_{i = 1, \\ldots, n} Z_i$ where we take the", "scheme theoretic intersection. Thus we may replace $S$ by the scheme", "theoretic intersection $\\bigcap_{i = 1, \\ldots, n} Z_i$. After shrinking", "$S$ to an affine open neighbourhood of $s$ we may assume that", "$\\mathcal{L}_i = \\mathcal{O}_{X_i}$ for $i = 1, \\ldots, n$.", "Let $E = Rf_*\\mathcal{L}$ and", "$E_i = Rf_{i, *}\\mathcal{L}_i = Rf_{i, *}\\mathcal{O}_{X_i}$.", "These are perfect complexes whose formation commutes with arbitrary", "change of base (Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image-general}).", "In particular we see that", "$$", "E \\otimes_{\\mathcal{O}_S}^\\mathbf{L} \\kappa(s) =", "R\\Gamma(X_s, \\mathcal{L}_s) = R\\Gamma(X_s, \\mathcal{O}_{X_s})", "$$", "The second equality because $s \\in Z$. Set", "$h_j = \\dim_{\\kappa(s)} H^j(X_s, \\mathcal{O}_{X_s})$.", "After shrinking $S$ we can represent $E$ by a complex", "$$", "\\mathcal{O}_S \\to \\mathcal{O}_S^{\\oplus h_1} \\to", "\\mathcal{O}_S^{\\oplus h_2} \\to \\ldots", "$$", "see More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-perfect-from-residue-field}", "(strictly speaking this also uses", "Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-affine-compare-bounded} and", "\\ref{perfect-lemma-perfect-affine}). Similarly, we may assume $E_i$", "is represented by a complex", "$$", "\\mathcal{O}_S \\to \\mathcal{O}_S^{\\oplus h_{i, 1}} \\to", "\\mathcal{O}_S^{\\oplus h_{i, 2}} \\to \\ldots", "$$", "where $h_{i, j} = \\dim_{\\kappa(s)} H^j(X_{i, s}, \\mathcal{O}_{X_{i, s}})$.", "By functoriality of cohomology we have a map", "$$", "E \\longrightarrow E_i", "$$", "in $D(\\mathcal{O}_S)$ whose formation commutes with change of base.", "Since the complex representing $E$ is a finite complex of finite free", "modules and since $S$ is affine, we can choose a map of complexes", "$$", "\\xymatrix{", "\\mathcal{O}_S \\ar[r]_d \\ar[d]_{a_i} &", "\\mathcal{O}_S^{\\oplus h_1} \\ar[r] \\ar[d]_{b_i} &", "\\mathcal{O}_S^{\\oplus h_2} \\ar[r] \\ar[d]_{c_i} & \\ldots \\\\", "\\mathcal{O}_S \\ar[r]^{d_i} &", "\\mathcal{O}_S^{\\oplus h_{i, 1}} \\ar[r] &", "\\mathcal{O}_S^{\\oplus h_{i, 2}} \\ar[r] & \\ldots", "}", "$$", "representing the given map $E \\to E_i$. Since $s \\in Z$ we see that", "the trivializing section of $\\mathcal{L}_s$ pulls back to a trivializing", "section of $\\mathcal{L}_{i, s} = \\mathcal{O}_{X_{i, s}}$. Thus", "$a_i \\otimes \\kappa(s)$ is an isomorphism, hence after shrinking $S$", "we see that $a_i$ is an isomorphism. Finally, we use the hypothesis", "that $H^1(X_s, \\mathcal{O}) \\to", "\\bigoplus_{i = 1, \\ldots, n} H^1(X_{i, s}, \\mathcal{O})$", "is injective, to see that there exists a $h_1 \\times h_1$ minor of the", "matrix defining $\\oplus b_i$ which maps to a nonzero", "element in $\\kappa(s)$. Hence after shrinking $S$ we may assume that", "$(b_1, \\ldots, b_n) : \\mathcal{O}_S^{h_1}", "\\to \\bigoplus_{i = 1, \\ldots, n} \\mathcal{O}_S^{h_{i, 1}}$", "is injective. However, since $\\mathcal{L}_i = \\mathcal{O}_{X_i}$", "we see that $d_i = 0$ for $i = 1, \\ldots n$. It follows that $d = 0$", "because $(b_1, \\ldots, b_n) \\circ d = (\\oplus d_i) \\circ (a_1, \\ldots, a_n)$.", "In this way we see", "that the trivializing section of $\\mathcal{L}_s$ lifts to a section", "of $\\mathcal{L}$ over $X$. A straightforward topological argument (omitted)", "shows that this means that $\\mathcal{L}$ is trivial after possibly", "shrinking $S$ a bit further." ], "refs": [ "more-morphisms-lemma-H1-O-picard-flat-proper", "perfect-lemma-flat-proper-perfect-direct-image-general", "more-algebra-lemma-lift-perfect-from-residue-field", "perfect-lemma-affine-compare-bounded", "perfect-lemma-perfect-affine" ], "ref_ids": [ 13852, 7054, 10232, 6941, 6980 ] } ], "ref_ids": [ 13848, 13848 ] }, { "id": 13854, "type": "theorem", "label": "more-morphisms-lemma-pic-of-product", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pic-of-product", "contents": [ "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes", "satisfying the hypotheses of Lemma \\ref{lemma-diagonal-picard-flat-proper}.", "Let $\\sigma : S \\to X$ and $\\tau : S \\to Y$ be sections of", "$f$ and $g$. Let $s \\in S$.", "Let $\\mathcal{L}$ be an invertible sheaf on $X \\times_S Y$.", "If $(1 \\times \\tau)^*\\mathcal{L}$ on $X$, $(\\sigma \\times 1)^*\\mathcal{L}$", "on $Y$, and $\\mathcal{L}|_{(X \\times_S Y)_s}$ are trivial, then", "there is an open neighbourhood $U$ of $s$ such that", "$\\mathcal{L}$ is trivial over $(X \\times_S Y)_U$." ], "refs": [ "more-morphisms-lemma-diagonal-picard-flat-proper" ], "proofs": [ { "contents": [ "By K\\\"unneth (Varieties, Lemma \\ref{varieties-lemma-kunneth})", "the map", "$$", "H^1(X_s \\times_{\\Spec(\\kappa(s))} Y_s, \\mathcal{O}) \\to", "H^1(X_s, \\mathcal{O}) \\oplus H^1(Y_s, \\mathcal{O})", "$$", "is injective. Thus we may", "apply Lemma \\ref{lemma-H1-O-multiple-picard-flat-proper}", "to the two morphisms", "$$", "1 \\times \\tau : X \\to X \\times_S Y", "\\quad\\text{and}\\quad", "\\sigma \\times 1 : Y \\to X \\times_S Y", "$$", "to conclude." ], "refs": [ "varieties-lemma-kunneth", "more-morphisms-lemma-H1-O-multiple-picard-flat-proper" ], "ref_ids": [ 11023, 13853 ] } ], "ref_ids": [ 13848 ] }, { "id": 13855, "type": "theorem", "label": "more-morphisms-lemma-Noetherian-approximation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-Noetherian-approximation", "contents": [ "Let $f : X \\to S$ be a morphism of affine schemes, which is of finite", "presentation. Then there exists a cartesian diagram", "$$", "\\xymatrix{", "X_0 \\ar[d]_{f_0} & X \\ar[l]^g \\ar[d]^f \\\\", "S_0 & S \\ar[l]", "}", "$$", "such that", "\\begin{enumerate}", "\\item $X_0$, $S_0$ are affine schemes,", "\\item $S_0$ of finite type over $\\mathbf{Z}$,", "\\item $f_0$ is finite of finite type.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $S = \\Spec(A)$ and $X = \\Spec(B)$.", "As $f$ is of finite presentation we see that", "$B$ is of finite presentation as an $A$-algebra, see", "Morphisms,", "Lemma \\ref{morphisms-lemma-locally-finite-presentation-characterize}.", "Thus the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-limit-module-finite-presentation}." ], "refs": [ "morphisms-lemma-locally-finite-presentation-characterize", "algebra-lemma-limit-module-finite-presentation" ], "ref_ids": [ 5238, 1106 ] } ], "ref_ids": [] }, { "id": 13856, "type": "theorem", "label": "more-morphisms-lemma-Noetherian-approximation-module", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-Noetherian-approximation-module", "contents": [ "Let $f : X \\to S$ be a morphism of affine schemes, which is of finite", "presentation. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module", "of finite presentation. Then there exists a diagram as in", "Lemma \\ref{lemma-Noetherian-approximation}", "such that there exists a coherent $\\mathcal{O}_{X_0}$-module $\\mathcal{F}_0$", "with $g^*\\mathcal{F}_0 = \\mathcal{F}$." ], "refs": [ "more-morphisms-lemma-Noetherian-approximation" ], "proofs": [ { "contents": [ "Write $S = \\Spec(A)$, $X = \\Spec(B)$, and", "$\\mathcal{F} = \\widetilde{M}$. As $f$ is of finite presentation we see that", "$B$ is of finite presentation as an $A$-algebra, see", "Morphisms,", "Lemma \\ref{morphisms-lemma-locally-finite-presentation-characterize}.", "As $\\mathcal{F}$ is of finite presentation over $\\mathcal{O}_X$ we see that", "$M$ is of finite presentation as a $B$-module, see", "Properties, Lemma \\ref{properties-lemma-finite-presentation-module}.", "Thus the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-limit-module-finite-presentation}." ], "refs": [ "morphisms-lemma-locally-finite-presentation-characterize", "properties-lemma-finite-presentation-module", "algebra-lemma-limit-module-finite-presentation" ], "ref_ids": [ 5238, 3003, 1106 ] } ], "ref_ids": [ 13855 ] }, { "id": 13857, "type": "theorem", "label": "more-morphisms-lemma-Noetherian-approximation-flat-module", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-Noetherian-approximation-flat-module", "contents": [ "Let $f : X \\to S$ be a morphism of affine schemes, which is of finite", "presentation. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module", "of finite presentation and flat over $S$. Then we may choose a diagram as in", "Lemma \\ref{lemma-Noetherian-approximation-module}", "and sheaf $\\mathcal{F}_0$ such that in addition $\\mathcal{F}_0$", "is flat over $S_0$." ], "refs": [ "more-morphisms-lemma-Noetherian-approximation-module" ], "proofs": [ { "contents": [ "Write $S = \\Spec(A)$, $X = \\Spec(B)$, and", "$\\mathcal{F} = \\widetilde{M}$. As $f$ is of finite presentation we see that", "$B$ is of finite presentation as an $A$-algebra, see", "Morphisms,", "Lemma \\ref{morphisms-lemma-locally-finite-presentation-characterize}.", "As $\\mathcal{F}$ is of finite presentation over $\\mathcal{O}_X$ we see that", "$M$ is of finite presentation as a $B$-module, see", "Properties, Lemma \\ref{properties-lemma-finite-presentation-module}.", "As $\\mathcal{F}$ is flat over $S$ we see that $M$ is flat over $A$, see", "Morphisms, Lemma \\ref{morphisms-lemma-flat-module-characterize}.", "Thus the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}." ], "refs": [ "morphisms-lemma-locally-finite-presentation-characterize", "properties-lemma-finite-presentation-module", "morphisms-lemma-flat-module-characterize", "algebra-lemma-flat-finite-presentation-limit-flat" ], "ref_ids": [ 5238, 3003, 5259, 1389 ] } ], "ref_ids": [ 13856 ] }, { "id": 13858, "type": "theorem", "label": "more-morphisms-lemma-Noetherian-approximation-flat", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-Noetherian-approximation-flat", "contents": [ "Let $f : X \\to S$ be a morphism of affine schemes, which is of finite", "presentation and flat. Then there exists a diagram as in", "Lemma \\ref{lemma-Noetherian-approximation}", "such that in addition $f_0$ is flat." ], "refs": [ "more-morphisms-lemma-Noetherian-approximation" ], "proofs": [ { "contents": [ "This is a special case of", "Lemma \\ref{lemma-Noetherian-approximation-flat-module}." ], "refs": [ "more-morphisms-lemma-Noetherian-approximation-flat-module" ], "ref_ids": [ 13857 ] } ], "ref_ids": [ 13855 ] }, { "id": 13859, "type": "theorem", "label": "more-morphisms-lemma-Noetherian-approximation-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-Noetherian-approximation-smooth", "contents": [ "Let $f : X \\to S$ be a morphism of affine schemes, which is smooth.", "Then there exists a diagram as in", "Lemma \\ref{lemma-Noetherian-approximation}", "such that in addition $f_0$ is smooth." ], "refs": [ "more-morphisms-lemma-Noetherian-approximation" ], "proofs": [ { "contents": [ "Write $S = \\Spec(A)$, $X = \\Spec(B)$, and", "as $f$ is smooth we see that $B$ is smooth as an $A$-algebra, see", "Morphisms,", "Lemma \\ref{morphisms-lemma-smooth-characterize}.", "Hence the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-finite-presentation-fs-Noetherian}." ], "refs": [ "morphisms-lemma-smooth-characterize", "algebra-lemma-finite-presentation-fs-Noetherian" ], "ref_ids": [ 5324, 1213 ] } ], "ref_ids": [ 13855 ] }, { "id": 13860, "type": "theorem", "label": "more-morphisms-lemma-Noetherian-approximation-geometrically-reduced", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-Noetherian-approximation-geometrically-reduced", "contents": [ "Let $f : X \\to S$ be a morphism of affine schemes, which is", "of finite presentation with geometrically reduced fibres.", "Then there exists a diagram as in", "Lemma \\ref{lemma-Noetherian-approximation}", "such that in addition $f_0$ has geometrically reduced fibres." ], "refs": [ "more-morphisms-lemma-Noetherian-approximation" ], "proofs": [ { "contents": [ "Apply", "Lemma \\ref{lemma-Noetherian-approximation}", "to get a cartesian diagram", "$$", "\\xymatrix{", "X_0 \\ar[d]_{f_0} & X \\ar[l]^g \\ar[d]^f \\\\", "S_0 & S \\ar[l]_h", "}", "$$", "of affine schemes with $X_0 \\to S_0$ a finite type morphism of", "schemes of finite type over $\\mathbf{Z}$. By", "Lemma \\ref{lemma-geometrically-reduced-constructible}", "the set $E \\subset S_0$ of points where the fibre of", "$f_0$ is geometrically reduced is a constructible subset. By", "Lemma \\ref{lemma-base-change-fibres-geometrically-reduced}", "we have $h(S) \\subset E$. Write $S_0 = \\Spec(A_0)$ and", "$S = \\Spec(A)$. Write $A = \\colim_i A_i$ as a", "direct colimit of finite type $A_0$-algebras. By", "Limits, Lemma \\ref{limits-lemma-limit-contained-in-constructible}", "we see that $\\Spec(A_i) \\to S_0$ has image contained in $E$", "for some $i$. After replacing $S_0$ by $\\Spec(A_i)$ and", "$X_0$ by $X_0 \\times_{S_0} \\Spec(A_i)$ we see that", "all fibres of $f_0$ are geometrically reduced." ], "refs": [ "more-morphisms-lemma-Noetherian-approximation", "more-morphisms-lemma-geometrically-reduced-constructible", "more-morphisms-lemma-base-change-fibres-geometrically-reduced", "limits-lemma-limit-contained-in-constructible" ], "ref_ids": [ 13855, 13817, 13814, 15040 ] } ], "ref_ids": [ 13855 ] }, { "id": 13861, "type": "theorem", "label": "more-morphisms-lemma-Noetherian-approximation-geometrically-irreducible", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-Noetherian-approximation-geometrically-irreducible", "contents": [ "Let $f : X \\to S$ be a morphism of affine schemes, which is", "of finite presentation with geometrically irreducible fibres.", "Then there exists a diagram as in", "Lemma \\ref{lemma-Noetherian-approximation}", "such that in addition $f_0$ has geometrically irreducible fibres." ], "refs": [ "more-morphisms-lemma-Noetherian-approximation" ], "proofs": [ { "contents": [ "Apply", "Lemma \\ref{lemma-Noetherian-approximation}", "to get a cartesian diagram", "$$", "\\xymatrix{", "X_0 \\ar[d]_{f_0} & X \\ar[l]^g \\ar[d]^f \\\\", "S_0 & S \\ar[l]_h", "}", "$$", "of affine schemes with $X_0 \\to S_0$ a finite type morphism of", "schemes of finite type over $\\mathbf{Z}$. By", "Lemma \\ref{lemma-nr-geom-irreducible-components-constructible}", "the set $E \\subset S_0$ of points where the fibre of", "$f_0$ is geometrically irreducible is a constructible subset. By", "Lemma \\ref{lemma-base-change-fibres-geometrically-irreducible}", "we have $h(S) \\subset E$. Write $S_0 = \\Spec(A_0)$ and", "$S = \\Spec(A)$. Write $A = \\colim_i A_i$ as a", "direct colimit of finite type $A_0$-algebras. By", "Limits, Lemma \\ref{limits-lemma-limit-contained-in-constructible}", "we see that $\\Spec(A_i) \\to S_0$ has image contained in $E$", "for some $i$. After replacing $S_0$ by $\\Spec(A_i)$ and", "$X_0$ by $X_0 \\times_{S_0} \\Spec(A_i)$ we see that", "all fibres of $f_0$ are geometrically irreducible." ], "refs": [ "more-morphisms-lemma-Noetherian-approximation", "more-morphisms-lemma-nr-geom-irreducible-components-constructible", "more-morphisms-lemma-base-change-fibres-geometrically-irreducible", "limits-lemma-limit-contained-in-constructible" ], "ref_ids": [ 13855, 13826, 13821, 15040 ] } ], "ref_ids": [ 13855 ] }, { "id": 13862, "type": "theorem", "label": "more-morphisms-lemma-Noetherian-approximation-geometrically-connected", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-Noetherian-approximation-geometrically-connected", "contents": [ "Let $f : X \\to S$ be a morphism of affine schemes, which is", "of finite presentation with geometrically connected fibres.", "Then there exists a diagram as in", "Lemma \\ref{lemma-Noetherian-approximation}", "such that in addition $f_0$ has geometrically connected fibres." ], "refs": [ "more-morphisms-lemma-Noetherian-approximation" ], "proofs": [ { "contents": [ "Apply", "Lemma \\ref{lemma-Noetherian-approximation}", "to get a cartesian diagram", "$$", "\\xymatrix{", "X_0 \\ar[d]_{f_0} & X \\ar[l]^g \\ar[d]^f \\\\", "S_0 & S \\ar[l]_h", "}", "$$", "of affine schemes with $X_0 \\to S_0$ a finite type morphism of", "schemes of finite type over $\\mathbf{Z}$. By", "Lemma \\ref{lemma-nr-geom-connected-components-constructible}", "the set $E \\subset S_0$ of points where the fibre of", "$f_0$ is geometrically connected is a constructible subset. By", "Lemma \\ref{lemma-base-change-fibres-geometrically-connected}", "we have $h(S) \\subset E$. Write $S_0 = \\Spec(A_0)$ and", "$S = \\Spec(A)$. Write $A = \\colim_i A_i$ as a", "direct colimit of finite type $A_0$-algebras. By", "Limits, Lemma \\ref{limits-lemma-limit-contained-in-constructible}", "we see that $\\Spec(A_i) \\to S_0$ has image contained in $E$", "for some $i$. After replacing $S_0$ by $\\Spec(A_i)$ and", "$X_0$ by $X_0 \\times_{S_0} \\Spec(A_i)$ we see that", "all fibres of $f_0$ are geometrically connected." ], "refs": [ "more-morphisms-lemma-Noetherian-approximation", "more-morphisms-lemma-nr-geom-connected-components-constructible", "more-morphisms-lemma-base-change-fibres-geometrically-connected", "limits-lemma-limit-contained-in-constructible" ], "ref_ids": [ 13855, 13832, 13828, 15040 ] } ], "ref_ids": [ 13855 ] }, { "id": 13863, "type": "theorem", "label": "more-morphisms-lemma-Noetherian-approximation-dimension-d", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-Noetherian-approximation-dimension-d", "contents": [ "Let $d \\geq 0$ be an integer.", "Let $f : X \\to S$ be a morphism of affine schemes, which is", "of finite presentation all of whose fibres have dimension $d$.", "Then there exists a diagram as in", "Lemma \\ref{lemma-Noetherian-approximation}", "such that in addition all fibres of $f_0$ have dimension $d$." ], "refs": [ "more-morphisms-lemma-Noetherian-approximation" ], "proofs": [ { "contents": [ "Apply", "Lemma \\ref{lemma-Noetherian-approximation}", "to get a cartesian diagram", "$$", "\\xymatrix{", "X_0 \\ar[d]_{f_0} & X \\ar[l]^g \\ar[d]^f \\\\", "S_0 & S \\ar[l]_h", "}", "$$", "of affine schemes with $X_0 \\to S_0$ a finite type morphism of", "schemes of finite type over $\\mathbf{Z}$. By", "Lemma \\ref{lemma-dimension-fibres-constructible}", "the set $E \\subset S_0$ of points where the fibre of", "$f_0$ has dimension $d$ is a constructible subset. By", "Lemma \\ref{lemma-base-change-dimension-fibres}", "we have $h(S) \\subset E$. Write $S_0 = \\Spec(A_0)$ and", "$S = \\Spec(A)$. Write $A = \\colim_i A_i$ as a", "direct colimit of finite type $A_0$-algebras. By", "Limits, Lemma \\ref{limits-lemma-limit-contained-in-constructible}", "we see that $\\Spec(A_i) \\to S_0$ has image contained in $E$", "for some $i$. After replacing $S_0$ by $\\Spec(A_i)$ and", "$X_0$ by $X_0 \\times_{S_0} \\Spec(A_i)$ we see that", "all fibres of $f_0$ have dimension $d$." ], "refs": [ "more-morphisms-lemma-Noetherian-approximation", "more-morphisms-lemma-dimension-fibres-constructible", "more-morphisms-lemma-base-change-dimension-fibres", "limits-lemma-limit-contained-in-constructible" ], "ref_ids": [ 13855, 13841, 13840, 15040 ] } ], "ref_ids": [ 13855 ] }, { "id": 13864, "type": "theorem", "label": "more-morphisms-lemma-Noetherian-approximation-standard-syntomic", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-Noetherian-approximation-standard-syntomic", "contents": [ "Let $f : X \\to S$ be a morphism of affine schemes, which is", "standard syntomic (see", "Morphisms, Definition \\ref{morphisms-definition-syntomic}).", "Then there exists a diagram as in", "Lemma \\ref{lemma-Noetherian-approximation}", "such that in addition $f_0$ is standard syntomic." ], "refs": [ "morphisms-definition-syntomic", "more-morphisms-lemma-Noetherian-approximation" ], "proofs": [ { "contents": [ "This lemma is a copy of", "Algebra,", "Lemma \\ref{algebra-lemma-relative-global-complete-intersection-Noetherian}." ], "refs": [ "algebra-lemma-relative-global-complete-intersection-Noetherian" ], "ref_ids": [ 1182 ] } ], "ref_ids": [ 5560, 13855 ] }, { "id": 13865, "type": "theorem", "label": "more-morphisms-lemma-Noetherian-approximation-combine", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-Noetherian-approximation-combine", "contents": [ "(Noetherian approximation and combining properties.)", "Let $P$, $Q$ be properties of morphisms of schemes which are stable", "under base change. Let $f : X \\to S$ be a morphism of finite presentation", "of affine schemes. Assume we can find cartesian diagrams", "$$", "\\vcenter{", "\\xymatrix{", "X_1 \\ar[d]_{f_1} & X \\ar[l] \\ar[d]^f \\\\", "S_1 & S \\ar[l]", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "X_2 \\ar[d]_{f_2} & X \\ar[l] \\ar[d]^f \\\\", "S_2 & S \\ar[l]", "}", "}", "$$", "of affine schemes, with $S_1$, $S_2$ of finite type over $\\mathbf{Z}$", "and $f_1$, $f_2$ of finite type such that $f_1$ has property $P$", "and $f_2$ has property $Q$. Then we can find a cartesian diagram", "$$", "\\xymatrix{", "X_0 \\ar[d]_{f_0} & X \\ar[l] \\ar[d]^f \\\\", "S_0 & S \\ar[l]", "}", "$$", "of affine schemes with $S_0$ of finite type over $\\mathbf{Z}$", "and $f_0$ of finite type such that $f_0$ has both property $P$ and", "property $Q$." ], "refs": [], "proofs": [ { "contents": [ "The given pair of diagrams correspond to cocartesian diagrams of rings", "$$", "\\vcenter{", "\\xymatrix{", "B_1 \\ar[r] & B \\\\", "A_1 \\ar[u] \\ar[r] & A \\ar[u]", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "B_2 \\ar[r] & B \\\\", "A_2 \\ar[u] \\ar[r] & A \\ar[u]", "}", "}", "$$", "Let $A_0 \\subset A$ be a finite type $\\mathbf{Z}$-subalgebra of $A$", "containing the image of both $A_1 \\to A$ and $A_2 \\to A$. Such a subalgebra", "exists because by assumption both $A_1$ and $A_2$ are of finite type over", "$\\mathbf{Z}$. Note that the rings $B_{0, 1} = B_1 \\otimes_{A_1} A_0$", "and $B_{0, 2} = B_2 \\otimes_{A_2} A_0$ are finite type $A_0$-algebras", "with the property that", "$B_{0, 1} \\otimes_{A_0} A \\cong B \\cong B_{0, 2} \\otimes_{A_0} A$", "as $A$-algebras. As $A$ is the directed colimit of its finite type", "$A_0$-subalgebras, by", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we may assume after enlarging $A_0$ that there exists an isomorphism", "$B_{0, 1} \\cong B_{0, 2}$ as $A_0$-algebras. Since properties $P$ and $Q$", "are assumed stable under base change we conclude that setting", "$S_0 = \\Spec(A_0)$ and", "$$", "X_0 = X_1 \\times_{S_1} S_0 =", "\\Spec(B_{0, 1}) \\cong \\Spec(B_{0, 2}) = X_2 \\times_{S_2} S_0", "$$", "works." ], "refs": [ "limits-lemma-descend-finite-presentation" ], "ref_ids": [ 15077 ] } ], "ref_ids": [] }, { "id": 13866, "type": "theorem", "label": "more-morphisms-lemma-realize-prescribed-residue-field-extension-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-realize-prescribed-residue-field-extension-etale", "contents": [ "Let $S$ be a scheme.", "Let $s \\in S$.", "Let $\\kappa(s) \\subset k$ be a finite separable field extension.", "Then there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$", "such that the field extension $\\kappa(s) \\subset \\kappa(u)$ is", "isomorphic to $\\kappa(s) \\subset k$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $S$ is affine.", "In this case the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-make-etale-map-prescribed-residue-field}." ], "refs": [ "algebra-lemma-make-etale-map-prescribed-residue-field" ], "ref_ids": [ 1242 ] } ], "ref_ids": [] }, { "id": 13867, "type": "theorem", "label": "more-morphisms-lemma-etale-neighbourhoods-not-quite-filtered", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-etale-neighbourhoods-not-quite-filtered", "contents": [ "Let $S$ be a scheme, and let $s$ be a point of $S$.", "The category of \\'etale neighborhoods has the following properties:", "\\begin{enumerate}", "\\item Let $(U_i, u_i)_{i=1, 2}$ be two \\'etale neighborhoods of", "$s$ in $S$. Then there exists a third \\'etale neighborhood", "$(U, u)$ and morphisms", "$(U, u) \\to (U_i, u_i)$, $i = 1, 2$.", "\\item Let $h_1, h_2: (U, u) \\to (U', u')$ be two", "morphisms between \\'etale neighborhoods of $s$.", "Assume $h_1$, $h_2$ induce the same map $\\kappa(u') \\to \\kappa(u)$ of residue", "fields. Then there exist an \\'etale neighborhood $(U'', u'')$ and a morphism", "$h : (U'', u'') \\to (U, u)$", "which equalizes $h_1$ and $h_2$, i.e., such that", "$h_1 \\circ h = h_2 \\circ h$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "For part (1), consider the fibre product $U = U_1 \\times_S U_2$.", "It is \\'etale over both $U_1$ and $U_2$ because \\'etale morphisms are", "preserved under base change, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-etale}.", "There is a point of $U$ mapping to both $u_1$ and $u_2$ for example", "by the description of points of a fibre product in", "Schemes, Lemma \\ref{schemes-lemma-points-fibre-product}.", "For part (2), define $U''$ as the fibre product", "$$", "\\xymatrix{", "U'' \\ar[r] \\ar[d] & U \\ar[d]^{(h_1, h_2)} \\\\", "U' \\ar[r]^-\\Delta & U' \\times_S U'.", "}", "$$", "Since $h_1$ and $h_2$ induce the same map of residue fields", "$\\kappa(u') \\to \\kappa(u)$ there exists a point $u'' \\in U''$", "lying over $u'$ with $\\kappa(u'') = \\kappa(u')$.", "In particular $U'' \\not = \\emptyset$.", "Moreover, since $U'$ is \\'etale over $S$, so is the fibre product", "$U'\\times_S U'$ (see", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-etale} and", "\\ref{morphisms-lemma-composition-etale}).", "Hence the vertical arrow $(h_1, h_2)$ is \\'etale by", "Morphisms, Lemma \\ref{morphisms-lemma-etale-permanence}.", "Therefore $U''$ is \\'etale over $U'$ by base change, and hence also", "\\'etale over $S$ (because compositions of \\'etale morphisms are \\'etale).", "Thus $(U'', u'')$ is a solution to the problem." ], "refs": [ "morphisms-lemma-base-change-etale", "schemes-lemma-points-fibre-product", "morphisms-lemma-base-change-etale", "morphisms-lemma-composition-etale", "morphisms-lemma-etale-permanence" ], "ref_ids": [ 5361, 7693, 5361, 5360, 5375 ] } ], "ref_ids": [] }, { "id": 13868, "type": "theorem", "label": "more-morphisms-lemma-elementary-etale-neighbourhoods", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-elementary-etale-neighbourhoods", "contents": [ "Let $S$ be a scheme, and let $s$ be a point of $S$.", "The category of elementary \\'etale neighborhoods of $(S, s)$", "is cofiltered (see", "Categories, Definition \\ref{categories-definition-codirected})." ], "refs": [ "categories-definition-codirected" ], "proofs": [ { "contents": [ "This is immediate from the definitions and", "Lemma \\ref{lemma-etale-neighbourhoods-not-quite-filtered}." ], "refs": [ "more-morphisms-lemma-etale-neighbourhoods-not-quite-filtered" ], "ref_ids": [ 13867 ] } ], "ref_ids": [ 12364 ] }, { "id": 13869, "type": "theorem", "label": "more-morphisms-lemma-describe-henselization", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-describe-henselization", "contents": [ "Let $S$ be a scheme. Let $s \\in S$. Then we have", "$$", "\\mathcal{O}_{S, s}^h =", "\\colim_{(U, u)} \\mathcal{O}(U)", "$$", "where the colimit is over the filtered category which is opposite to the", "category of elementary \\'etale neighbourhoods $(U, u)$ of $(S, s)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\Spec(A) \\subset S$ be an affine neighbourhood of $s$.", "Let $\\mathfrak p \\subset A$ be the prime ideal corresponding to $s$.", "With these choices we have canonical isomorphisms", "$\\mathcal{O}_{S, s} = A_{\\mathfrak p}$ and $\\kappa(s) = \\kappa(\\mathfrak p)$.", "A cofinal system of elementary \\'etale neighbourhoods is given by those", "elementary \\'etale neighbourhoods $(U, u)$ such that $U$ is affine and", "$U \\to S$ factors through $\\Spec(A)$. In other words, we see that", "the right hand side is equal to $\\colim_{(B, \\mathfrak q)} B$", "where the colimit is over \\'etale $A$-algebras $B$ endowed with a prime", "$\\mathfrak q$ lying over $\\mathfrak p$ with", "$\\kappa(\\mathfrak p) = \\kappa(\\mathfrak q)$.", "Thus the lemma follows from", "Algebra, Lemma \\ref{algebra-lemma-henselization-different}." ], "refs": [ "algebra-lemma-henselization-different" ], "ref_ids": [ 1298 ] } ], "ref_ids": [] }, { "id": 13870, "type": "theorem", "label": "more-morphisms-lemma-lift-etale-neighbourhood-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-lift-etale-neighbourhood-fibre", "contents": [ "\\begin{slogan}", "Lift \\'etale neighbourhood of point on fibre to total space.", "\\end{slogan}", "Let $X \\to S$ be a morphism of schemes. Let $x \\in X$ with image $s \\in S$.", "Let $(V, v) \\to (X_s, x)$ be an \\'etale neighbourhood.", "Then there exists an \\'etale neighbourhood $(U, u) \\to (X, x)$", "such that there exists a morphism $(U_s, u) \\to (V, v)$", "of \\'etale neighbourhoods of $(X_s, x)$ which is an open immersion." ], "refs": [], "proofs": [ { "contents": [ "We may assume $X$, $V$, and $S$ affine. Say the morphism", "$X \\to S$ is given by $A \\to B$ the point $x$ by a prime", "$\\mathfrak q \\subset B$, the point $s$ by $\\mathfrak p = A \\cap \\mathfrak q$,", "and the morphism $V \\to X_s$ by $B \\otimes_A \\kappa(\\mathfrak p) \\to C$.", "Since $\\kappa(\\mathfrak p)$ is a localization of", "$A/\\mathfrak p$ there exists an $f \\in A$, $f \\not \\in \\mathfrak p$", "and an \\'etale ring map $B \\otimes_A (A/\\mathfrak p)_f \\to D$", "such that", "$$", "C = (B \\otimes_A \\kappa(\\mathfrak p))", "\\otimes_{B \\otimes_A (A/\\mathfrak p)_f} D", "$$", "See Algebra, Lemma \\ref{algebra-lemma-etale} part (9).", "After replacing $A$ by $A_f$ and $B$ by $B_f$ we may assume", "$D$ is \\'etale over $B \\otimes_A A/\\mathfrak p = B/\\mathfrak p B$.", "Then we can apply Algebra, Lemma \\ref{algebra-lemma-lift-etale}.", "This proves the lemma." ], "refs": [ "algebra-lemma-etale", "algebra-lemma-lift-etale" ], "ref_ids": [ 1231, 1238 ] } ], "ref_ids": [] }, { "id": 13871, "type": "theorem", "label": "more-morphisms-lemma-nr-minimal-primes", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-nr-minimal-primes", "contents": [ "Let $R = \\colim R_i$ be colimit of a directed system of rings", "whose transition maps are faithfully flat.", "Then the number of minimal primes of $R$", "taken as an element of $\\{0, 1, 2, \\ldots, \\infty\\}$", "is the supremum of the numbers of minimal primes of the $R_i$." ], "refs": [], "proofs": [ { "contents": [ "If $A \\to B$ is a flat ring map, then $\\Spec(B) \\to \\Spec(A)$", "maps minimal primes to minimal primes by going down", "(Algebra, Lemma \\ref{algebra-lemma-flat-going-down}).", "If $A \\to B$ is faithfully flat, then every minimal", "prime is the image of a minimal prime (by", "Algebra, Lemma \\ref{algebra-lemma-ff-rings} and", "\\ref{algebra-lemma-minimal-prime-image-minimal-prime}).", "Hence the number of minimal primes of $R_i$ is", "$\\geq$ the number of minimal primes of $R_{i'}$ if $i \\leq i'$.", "By Algebra, Lemma \\ref{algebra-lemma-colimit-faithfully-flat}", "each of the maps $R_i \\to R$ is", "faithfully flat and we also see that", "the number of minimal primes of $R$ is", "$\\geq$ the number of minimal primes of $R_i$.", "Finally, suppose that $\\mathfrak q_1, \\ldots, \\mathfrak q_n$", "are pairwise distinct minimal primes of $R$. Then we can", "find an $i$ such that $R_i \\cap \\mathfrak q_1, \\ldots, R_i \\cap \\mathfrak q_n$", "are pairwise distinct (as sets and hence as prime ideals).", "This implies the lemma." ], "refs": [ "algebra-lemma-flat-going-down", "algebra-lemma-ff-rings", "algebra-lemma-minimal-prime-image-minimal-prime", "algebra-lemma-colimit-faithfully-flat" ], "ref_ids": [ 539, 536, 447, 540 ] } ], "ref_ids": [] }, { "id": 13872, "type": "theorem", "label": "more-morphisms-lemma-nr-branches", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-nr-branches", "contents": [ "Let $X$ be a scheme and $x \\in X$ a point. Then", "\\begin{enumerate}", "\\item the number of branches of $X$ at $x$ is equal to", "the supremum of the number of irreducible components of $U$", "passing through $u$ taken over elementary \\'etale neighbourhoods", "$(U, u) \\to (X, x)$,", "\\item the number of geometric branches of $X$ at $x$ is equal to", "the supremum of the number of irreducible components of $U$", "passing through $u$ taken over \\'etale neighbourhoods", "$(U, u) \\to (X, x)$,", "\\item $X$ is unibranch at $x$ if and only if for every", "elementary \\'etale neighbourhood $(U, u) \\to (X, x)$ there", "is exactly one irreducible component of $U$ passing through $u$, and", "\\item $X$ is geometrically unibranch at $x$ if and only if for every", "\\'etale neighbourhood $(U, u) \\to (X, x)$ there", "is exactly one irreducible component of $U$ passing through $u$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Parts (3) and (4) follow from parts (1) and (2) via", "Properties, Lemma \\ref{properties-lemma-number-of-branches-1}.", "\\medskip\\noindent", "Proof of (1). Let $\\Spec(A)$ be an affine open neighbourhood", "of $x$ and let $\\mathfrak p \\subset A$ be the prime ideal", "corresponding to $x$. We may replace $X$ by $\\Spec(A)$ and", "it suffices to consider affine elementary \\'etale neighbourhoods", "$(U, u)$ in the supremum as they form a cofinal subsystem.", "Recall that the henselization $A_\\mathfrak p^h$", "is the colimit of the rings $B_\\mathfrak q$ over the category", "of pairs $(B, \\mathfrak q)$ where $B$ is an \\'etale $A$-algebra", "and $\\mathfrak q$ is a prime lying over $\\mathfrak p$ with", "$\\kappa(\\mathfrak q) = \\kappa(\\mathfrak p)$, see", "Algebra, Lemma \\ref{algebra-lemma-henselization-different}.", "These pairs $(B, \\mathfrak q)$ correspond exactly to", "the affine elementary \\'etale neighbourhoods $(U, u)$", "by the correspondence between rings and affine schemes.", "Observe that irreducible components of $\\Spec(B)$", "passing through $\\mathfrak q$ are exactly the minimal", "prime ideals of $B_\\mathfrak q$. The number of minimal", "primes of $A_\\mathfrak p^h$ is the number of branches", "of $X$ at $x$ by Properties, Definition", "\\ref{properties-definition-number-of-branches}.", "Observe that the transition", "maps $B_\\mathfrak q \\to B'_{\\mathfrak q'}$ in the system", "are all flat. Since a flat local ring map is faithfully flat", "(Algebra, Lemma \\ref{algebra-lemma-local-flat-ff})", "we see that the lemma follows", "from Lemma \\ref{lemma-nr-minimal-primes}.", "\\medskip\\noindent", "Proof of (2). The proof is the same as the proof of (1), except that we use", "Algebra, Lemma \\ref{algebra-lemma-strict-henselization-different}.", "There is a tiny difference: given a separable algebraic closure", "$\\kappa^{sep}$ of $\\kappa(x)$ for every \\'etale neighbourhood", "$(U, u)$ we can choose a $\\kappa(x)$-embedding", "$\\phi : \\kappa(u) \\to \\kappa^{sep}$", "because $\\kappa(u)/\\kappa(x)$ is finite separable", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-at-point}).", "Hence we can look at the supremum over all triples", "$(U, u, \\phi)$ where $(U, u) \\to (X, x)$ is an affine", "\\'etale neighbourhood and $\\phi : \\kappa(u) \\to \\kappa^{sep}$", "is a $\\kappa(x)$-embedding. These triples correspond", "exactly to the triples in", "Algebra, Lemma \\ref{algebra-lemma-strict-henselization-different}", "and the rest of the proof is exactly the same." ], "refs": [ "properties-lemma-number-of-branches-1", "algebra-lemma-henselization-different", "properties-definition-number-of-branches", "algebra-lemma-local-flat-ff", "more-morphisms-lemma-nr-minimal-primes", "algebra-lemma-strict-henselization-different", "morphisms-lemma-etale-at-point", "algebra-lemma-strict-henselization-different" ], "ref_ids": [ 3001, 1298, 3082, 537, 13871, 1304, 5372, 1304 ] } ], "ref_ids": [] }, { "id": 13873, "type": "theorem", "label": "more-morphisms-lemma-nr-branches-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-nr-branches-fibre", "contents": [ "Let $X \\to S$ be a morphism of schemes and $x \\in X$ a point with image $s$.", "Then", "\\begin{enumerate}", "\\item the number of branches of the fibre $X_s$ at $x$ is equal to", "the supremum of the number of irreducible components of the fibre $U_s$", "passing through $u$ taken over elementary \\'etale neighbourhoods", "$(U, u) \\to (X, x)$,", "\\item the number of geometric branches of the fibre $X_s$ at $x$ is equal to", "the supremum of the number of irreducible components of the fibre $U_s$", "passing through $u$ taken over \\'etale neighbourhoods", "$(U, u) \\to (X, x)$,", "\\item the fibre $X_s$ is unibranch at $x$ if and only if for every", "elementary \\'etale neighbourhood $(U, u) \\to (X, x)$ there is", "exactly one irreducible component of the fibre $U_s$ passing through $u$, and", "\\item $X$ is geometrically unibranch at $x$ if and only if for every", "\\'etale neighbourhood $(U, u) \\to (X, x)$ there", "is exactly one irreducible component of $U_s$ passing through $u$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-nr-branches} and", "\\ref{lemma-lift-etale-neighbourhood-fibre}." ], "refs": [ "more-morphisms-lemma-nr-branches", "more-morphisms-lemma-lift-etale-neighbourhood-fibre" ], "ref_ids": [ 13872, 13870 ] } ], "ref_ids": [] }, { "id": 13874, "type": "theorem", "label": "more-morphisms-lemma-number-of-branches-and-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-number-of-branches-and-smooth", "contents": [ "Let $X \\to S$ be a smooth morphism of schemes.", "Let $x \\in X$ with image $s \\in S$.", "Then", "\\begin{enumerate}", "\\item The number of geometric branches of $X$ at $x$", "is equal to the number of geometric branches of $S$ at $s$.", "\\item If $\\kappa(x)/\\kappa(s)$ is a purely inseparable\\footnote{In fact,", "it would suffice if $\\kappa(x)$ is geometrically irreducible over", "$\\kappa(s)$. If we ever need this we will add a detailed proof.}", "extension of fields, then number of branches of $X$ at $x$", "is equal to the number of branches of $S$ at $s$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from More on Algebra, Lemma", "\\ref{more-algebra-lemma-invariance-number-branches-smooth}", "and the definitions." ], "refs": [ "more-algebra-lemma-invariance-number-branches-smooth" ], "ref_ids": [ 10470 ] } ], "ref_ids": [] }, { "id": 13875, "type": "theorem", "label": "more-morphisms-lemma-slice-smooth-given-element", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-slice-smooth-given-element", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point with image $s \\in S$.", "Let $h \\in \\mathfrak m_x \\subset \\mathcal{O}_{X, x}$.", "Assume", "\\begin{enumerate}", "\\item $f$ is smooth at $x$, and", "\\item the image $\\text{d}\\overline{h}$ of $\\text{d}h$ in", "$$", "\\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) =", "\\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)", "$$", "is nonzero.", "\\end{enumerate}", "Then there exists an affine open neighbourhood $U \\subset X$ of $x$", "such that $h$ comes from $h \\in \\Gamma(U, \\mathcal{O}_U)$ and such", "that $D = V(h)$ is an effective Cartier divisor in $U$ with $x \\in D$ and", "$D \\to S$ smooth." ], "refs": [], "proofs": [ { "contents": [ "As $f$ is smooth at $x$ we may assume, after replacing $X$ by an open", "neighbourhood of $x$ that $f$ is smooth. In particular we see that", "$f$ is flat and locally of finite presentation. By", "Lemma \\ref{lemma-slice-given-element}", "we already know there exists an open neighbourhood $U \\subset X$ of $x$", "such that $h$ comes from $h \\in \\Gamma(U, \\mathcal{O}_U)$ and such", "that $D = V(h)$ is an effective Cartier divisor in $U$ with $x \\in D$ and", "$D \\to S$ flat and of finite presentation. By", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-relative-immersion}", "we have a short exact sequence", "$$", "\\mathcal{C}_{D/U} \\to i^*\\Omega_{U/S} \\to \\Omega_{D/S} \\to 0", "$$", "where $i : D \\to U$ is the closed immersion and $\\mathcal{C}_{D/U}$", "is the conormal sheaf of $D$ in $U$. As $D$ is an effective Cartier", "divisor cut out by $h \\in \\Gamma(U, \\mathcal{O}_U)$ we see that", "$\\mathcal{C}_{D/U} = h \\cdot \\mathcal{O}_S$. Since $U \\to S$ is smooth", "the sheaf $\\Omega_{U/S}$ is finite locally free, hence its pullback", "$i^*\\Omega_{U/S}$ is finite locally free also. The first arrow of", "the sequence maps the free generator $h$ to the section $\\text{d}h|_D$", "of $i^*\\Omega_{U/S}$ which has nonzero value in the fibre", "$\\Omega_{U/S, x} \\otimes \\kappa(x)$ by assumption. By right exactness", "of $\\otimes \\kappa(x)$ we conclude that", "$$", "\\dim_{\\kappa(x)} \\left( \\Omega_{D/S, x} \\otimes \\kappa(x) \\right)", "=", "\\dim_{\\kappa(x)} \\left( \\Omega_{U/S, x} \\otimes \\kappa(x) \\right) - 1.", "$$", "By", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-at-point}", "we see that $\\Omega_{U/S, x} \\otimes \\kappa(x)$ can be generated by", "at most $\\dim_x(U_s)$ elements. By the displayed formula we see that", "$\\Omega_{D/S, x} \\otimes \\kappa(x)$ can be generated by at most", "$\\dim_x(U_s) - 1$ elements. Note that", "$\\dim_x(D_s) = \\dim_x(U_s) - 1$ for example because", "$\\dim(\\mathcal{O}_{D_s, x}) = \\dim(\\mathcal{O}_{U_s, x}) - 1$ by", "Algebra, Lemma \\ref{algebra-lemma-one-equation}", "(also $D_s \\subset U_s$ is effective Cartier, see", "Divisors, Lemma \\ref{divisors-lemma-relative-Cartier})", "and then using", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}.", "Thus we conclude that $\\Omega_{D/S, x} \\otimes \\kappa(x)$ can be generated", "by at most $\\dim_x(D_s)$ elements and we conclude that $D \\to S$", "is smooth at $x$ by", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-at-point}", "again. After shrinking $U$ we get that $D \\to S$ is smooth and we win." ], "refs": [ "more-morphisms-lemma-slice-given-element", "morphisms-lemma-differentials-relative-immersion", "morphisms-lemma-smooth-at-point", "divisors-lemma-relative-Cartier", "morphisms-lemma-dimension-fibre-at-a-point", "morphisms-lemma-smooth-at-point" ], "ref_ids": [ 13794, 5319, 5335, 7972, 5277, 5335 ] } ], "ref_ids": [] }, { "id": 13876, "type": "theorem", "label": "more-morphisms-lemma-slice-smooth-once", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-slice-smooth-once", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point with image $s \\in S$.", "Assume", "\\begin{enumerate}", "\\item $f$ is smooth at $x$, and", "\\item the map", "$$", "\\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x)", "\\longrightarrow", "\\Omega_{\\kappa(x)/\\kappa(s)}", "$$", "has a nonzero kernel.", "\\end{enumerate}", "Then there exists an affine open neighbourhood $U \\subset X$ of $x$", "and an effective Cartier divisor $D \\subset U$ containing $x$ such that", "$D \\to S$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "Write $k = \\kappa(s)$ and $R = \\mathcal{O}_{X_s, x}$.", "Denote $\\mathfrak m$ the maximal ideal of $R$ and", "$\\kappa = R/\\mathfrak m$ so that $\\kappa = \\kappa(x)$.", "As formation of modules of differentials commutes with localization (see", "Algebra, Lemma \\ref{algebra-lemma-differentials-localize})", "we have $\\Omega_{X_s/s, x} = \\Omega_{R/k}$. By", "Algebra, Lemma \\ref{algebra-lemma-differential-seq}", "there is an exact sequence", "$$", "\\mathfrak m/\\mathfrak m^2 \\xrightarrow{\\text{d}}", "\\Omega_{R/k} \\otimes_R \\kappa \\to", "\\Omega_{\\kappa/k} \\to 0.", "$$", "Hence if (2) holds, there exists an element $\\overline{h} \\in \\mathfrak m$", "such that $\\text{d}\\overline{h}$ is nonzero. Choose a lift", "$h \\in \\mathcal{O}_{X, x}$ of $\\overline{h}$ and apply", "Lemma \\ref{lemma-slice-smooth-given-element}." ], "refs": [ "algebra-lemma-differentials-localize", "algebra-lemma-differential-seq", "more-morphisms-lemma-slice-smooth-given-element" ], "ref_ids": [ 1134, 1135, 13875 ] } ], "ref_ids": [] }, { "id": 13877, "type": "theorem", "label": "more-morphisms-lemma-slice-smooth-once-separable-residue-field-extension", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-slice-smooth-once-separable-residue-field-extension", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point with image $s \\in S$.", "Assume", "\\begin{enumerate}", "\\item $f$ is smooth at $x$,", "\\item the residue field extension $\\kappa(s) \\subset \\kappa(x)$", "is separable, and", "\\item $x$ is not a generic point of $X_s$.", "\\end{enumerate}", "Then there exists an affine open neighbourhood $U \\subset X$ of $x$", "and an effective Cartier divisor $D \\subset U$ containing $x$ such that", "$D \\to S$ is smooth." ], "refs": [], "proofs": [ { "contents": [ "Write $k = \\kappa(s)$ and $R = \\mathcal{O}_{X_s, x}$.", "Denote $\\mathfrak m$ the maximal ideal of $R$ and", "$\\kappa = R/\\mathfrak m$ so that $\\kappa = \\kappa(x)$.", "As formation of modules of differentials commutes with localization (see", "Algebra, Lemma \\ref{algebra-lemma-differentials-localize})", "we have $\\Omega_{X_s/s, x} = \\Omega_{R/k}$. By assumption (2) and", "Algebra, Lemma \\ref{algebra-lemma-computation-differential}", "the map", "$$", "\\text{d} :", "\\mathfrak m/\\mathfrak m^2", "\\longrightarrow", "\\Omega_{R/k} \\otimes_R \\kappa(\\mathfrak m)", "$$", "is injective. Assumption (3) implies that", "$\\mathfrak m/\\mathfrak m^2 \\not = 0$.", "Thus there exists an element $\\overline{h} \\in \\mathfrak m$", "such that $\\text{d}\\overline{h}$ is nonzero. Choose a lift", "$h \\in \\mathcal{O}_{X, x}$ of $\\overline{h}$ and apply", "Lemma \\ref{lemma-slice-smooth-given-element}." ], "refs": [ "algebra-lemma-differentials-localize", "algebra-lemma-computation-differential", "more-morphisms-lemma-slice-smooth-given-element" ], "ref_ids": [ 1134, 1224, 13875 ] } ], "ref_ids": [] }, { "id": 13878, "type": "theorem", "label": "more-morphisms-lemma-slice-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-slice-smooth", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ be a point with image $s \\in S$.", "Assume", "\\begin{enumerate}", "\\item $f$ is smooth at $x$, and", "\\item $x$ is a closed point of $X_s$ and $\\kappa(s) \\subset \\kappa(x)$", "is separable.", "\\end{enumerate}", "Then there exists an immersion $Z \\to X$ containing $x$ such that", "\\begin{enumerate}", "\\item $Z \\to S$ is \\'etale, and", "\\item $Z_s = \\{x\\}$ set theoretically.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We may and do replace $S$ by an affine open neighbourhood of $s$.", "We may and do replace $X$ by an affine open neighbourhood of $x$", "such that $X \\to S$ is smooth.", "We will prove the lemma for smooth morphisms of affines", "by induction on $d = \\dim_x(X_s)$.", "\\medskip\\noindent", "The case $d = 0$. In this case we show that we may take $Z$ to be", "an open neighbourhood of $x$. Namely, if $d = 0$, then $X \\to S$", "is quasi-finite at $x$, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}.", "Hence there exists an affine open neighbourhood $U \\subset X$ such", "that $U \\to S$ is quasi-finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-points-open}.", "Thus after replacing $X$ by $U$ we see that", "$X$ is quasi-finite and smooth over $S$, hence", "smooth of relative dimension $0$ over $S$, hence", "\\'etale over $S$. Moreover, the fibre $X_s$ is a finite", "discrete set. Hence after replacing $X$ by a further affine open neighbourhood", "of $X$ we see that $f^{-1}(\\{s\\}) = \\{x\\}$ (because the topology", "on $X_s$ is induced from the topology on $X$, see", "Schemes, Lemma \\ref{schemes-lemma-fibre-topological}).", "This proves the lemma in this case.", "\\medskip\\noindent", "Next, assume $d > 0$. Note that because $x$ is a closed point of its", "fibre the extension $\\kappa(s) \\subset \\kappa(x)$ is finite (by the", "Hilbert Nullstellensatz, see", "Morphisms, Lemma \\ref{morphisms-lemma-closed-point-fibre-locally-finite-type}).", "Thus we see $\\Omega_{\\kappa(x)/\\kappa(s)} = 0$ as this holds for", "algebraic separable field extensions.", "Thus we may apply", "Lemma \\ref{lemma-slice-smooth-once}", "to find a diagram", "$$", "\\xymatrix{", "D \\ar[r] \\ar[rrd] & U \\ar[r] \\ar[rd] & X \\ar[d] \\\\", "& & S", "}", "$$", "with $x \\in D$. Note that", "$\\dim_x(D_s) = \\dim_x(X_s) - 1$ for example because", "$\\dim(\\mathcal{O}_{D_s, x}) = \\dim(\\mathcal{O}_{X_s, x}) - 1$ by", "Algebra, Lemma \\ref{algebra-lemma-one-equation}", "(also $D_s \\subset X_s$ is effective Cartier, see", "Divisors, Lemma \\ref{divisors-lemma-relative-Cartier})", "and then using", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}.", "Thus the morphism $D \\to S$ is smooth with", "$\\dim_x(D_s) = \\dim_x(X_s) - 1 = d - 1$. By induction hypothesis", "we can find an immersion $Z \\to D$ as desired, which finishes the proof." ], "refs": [ "morphisms-lemma-locally-quasi-finite-rel-dimension-0", "morphisms-lemma-quasi-finite-points-open", "schemes-lemma-fibre-topological", "morphisms-lemma-closed-point-fibre-locally-finite-type", "more-morphisms-lemma-slice-smooth-once", "divisors-lemma-relative-Cartier", "morphisms-lemma-dimension-fibre-at-a-point" ], "ref_ids": [ 5287, 5521, 7696, 5223, 13876, 7972, 5277 ] } ], "ref_ids": [] }, { "id": 13879, "type": "theorem", "label": "more-morphisms-lemma-etale-nbhd-dominates-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-etale-nbhd-dominates-smooth", "contents": [ "Let $f : X \\to S$ be a smooth morphism of schemes.", "Let $s \\in S$ be a point in the image of $f$.", "Then there exists an \\'etale neighbourhood $(S', s') \\to (S, s)$", "and a $S$-morphism $S' \\to X$." ], "refs": [], "proofs": [ { "contents": [ "[First proof of Lemma \\ref{lemma-etale-nbhd-dominates-smooth}]", "By assumption $X_s \\not = \\emptyset$. By", "Varieties, Lemma \\ref{varieties-lemma-smooth-separable-closed-points-dense}", "there exists a closed point $x \\in X_s$ such that $\\kappa(x)$", "is a finite separable field extension of $\\kappa(s)$.", "Hence by", "Lemma \\ref{lemma-slice-smooth}", "there exists an immersion $Z \\to X$ such that $Z \\to S$ is \\'etale and such", "that $x \\in Z$. Take $(S' , s') = (Z, x)$." ], "refs": [ "more-morphisms-lemma-etale-nbhd-dominates-smooth", "varieties-lemma-smooth-separable-closed-points-dense", "more-morphisms-lemma-slice-smooth" ], "ref_ids": [ 13879, 11007, 13878 ] } ], "ref_ids": [] }, { "id": 13880, "type": "theorem", "label": "more-morphisms-lemma-etale-dominates-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-etale-dominates-smooth", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{U} = \\{S_i \\to S\\}_{i \\in I}$ be a smooth", "covering of $S$, see", "Topologies, Definition \\ref{topologies-definition-smooth-covering}.", "Then there exists an \\'etale covering $\\mathcal{V} = \\{T_j \\to S\\}_{j \\in J}$", "(see", "Topologies, Definition \\ref{topologies-definition-etale-covering})", "which refines (see", "Sites, Definition \\ref{sites-definition-morphism-coverings})", "$\\mathcal{U}$." ], "refs": [ "topologies-definition-smooth-covering", "topologies-definition-etale-covering", "sites-definition-morphism-coverings" ], "proofs": [ { "contents": [ "For every $s \\in S$ there exists an $i \\in I$ such that $s$ is in", "the image of $S_i \\to S$. By", "Lemma \\ref{lemma-etale-nbhd-dominates-smooth}", "we can find an \\'etale morphism $g_s : T_s \\to S$ such that $s \\in g_s(T_s)$", "and such that $g_s$ factors through $S_i \\to S$. Hence", "$\\{T_s \\to S\\}$ is an \\'etale covering of $S$ that refines $\\mathcal{U}$." ], "refs": [ "more-morphisms-lemma-etale-nbhd-dominates-smooth" ], "ref_ids": [ 13879 ] } ], "ref_ids": [ 12531, 12526, 8656 ] }, { "id": 13881, "type": "theorem", "label": "more-morphisms-lemma-cover-smooth-by-special", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-cover-smooth-by-special", "contents": [ "Let $f : X \\to S$ be a smooth morphism of schemes. Then there exists an", "\\'etale covering $\\{U_i \\to X\\}_{i \\in I}$ such that $U_i \\to S$", "factors as $U_i \\to V_i \\to S$ where $V_i \\to S$ is \\'etale and", "$U_i \\to V_i$ is a smooth morphism of affine schemes, which", "has a section, and has geometrically connected fibres." ], "refs": [], "proofs": [ { "contents": [ "Let $s \\in S$. By", "Varieties, Lemma \\ref{varieties-lemma-smooth-separable-closed-points-dense}", "the set of closed points $x \\in X_s$ such that $\\kappa(x)/\\kappa(s)$", "is separable is dense in $X_s$. Thus it suffices to construct an", "\\'etale morphism $U \\to X$ with $x$ in the image", "such that $U \\to S$ factors in the manner described in the lemma.", "To do this, choose an immersion $Z \\to X$ passing through $x$", "such that $Z \\to S$ is \\'etale (Lemma \\ref{lemma-slice-smooth}).", "After replacing $S$ by $Z$ and $X$ by $Z \\times_S X$", "we see that we may assume $X \\to S$ has a section $\\sigma : S \\to X$", "with $\\sigma(s) = x$. Then we can first replace $S$ by an affine", "open neighbourhood of $s$ and next replace $X$ by an affine open", "neighbourhood of $x$. Then finally, we consider the subset", "$X^0 \\subset X$ of Section \\ref{section-connected-components}.", "By Lemmas \\ref{lemma-connected-along-section-open} and", "\\ref{lemma-connected-along-section-locally-constructible}", "this is a retrocompact open subscheme containing $\\sigma$", "such that the fibres $X^0 \\to S$ are geometrically connected.", "If $X^0$ is not affine, then we choose an affine open $U \\subset X^0$", "containing $x$. Since $X^0 \\to S$ is smooth, the image of $U$", "is open. Choose an affine open neighbourhood $V \\subset S$ of $s$", "contained in $\\sigma^{-1}(U)$ and in the image of $U \\to S$.", "Finally, the reader sees that $U \\cap f^{-1}(V) \\to V$", "has all the desired properties. For example $U \\cap f^{-1}(V)$", "is equal to $U \\times_S V$ is affine as a fibre product of affine", "schemes. Also, the geometric fibres of $U \\cap f^{-1}(V) \\to V$ are", "nonempty open subschemes of the irreducible fibres of $X^0 \\to S$", "and hence connected. Some details omitted." ], "refs": [ "varieties-lemma-smooth-separable-closed-points-dense", "more-morphisms-lemma-slice-smooth" ], "ref_ids": [ 11007, 13878 ] } ], "ref_ids": [] }, { "id": 13882, "type": "theorem", "label": "more-morphisms-lemma-etale-local-structure", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-etale-local-structure", "contents": [ "Let $S$ be a scheme. Let $Y \\to X$ be a closed immersion of schemes", "smooth over $S$. For every $y \\in Y$ there exist integers", "$0 \\leq m, n$ and a commutative diagram", "$$", "\\xymatrix{", "Y \\ar[d] &", "V \\ar[l] \\ar[d] \\ar[r] &", "\\mathbf{A}^m_S", "\\ar[d]^{(a_1, \\ldots, a_m) \\mapsto (a_1, \\ldots, a_m, 0 \\ldots, 0)} \\\\", "X &", "U \\ar[l] \\ar[r]^-\\pi &", "\\mathbf{A}^{m + n}_S", "}", "$$", "where $U \\subset X$ is open, $V = Y \\cap U$,", "$\\pi$ is \\'etale, $V = \\pi^{-1}(\\mathbf{A}^m_S)$, and $y \\in V$." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$ hence we may replace $X$ by", "an open neighbourhood of $y$. Since $Y \\to X$ is a regular immersion", "by Divisors, Lemma", "\\ref{divisors-lemma-immersion-smooth-into-smooth-regular-immersion}", "we may assume $X = \\Spec(A)$ is affine and there exists a regular sequence", "$f_1, \\ldots, f_n \\in A$ such that $Y = V(f_1, \\ldots, f_n)$.", "After shrinking $X$ (and hence $Y$) further", "we may assume there exists an \\'etale morphism $Y \\to \\mathbf{A}^m_S$, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-etale-over-affine-space}.", "Let $\\overline{g}_1, \\ldots, \\overline{g}_m$ in $\\mathcal{O}_Y(Y)$", "be the coordinate functions of this \\'etale morphism.", "Choose lifts $g_1, \\ldots, g_m \\in A$ of these functions", "and consider the morphism", "$$", "(g_1, \\ldots, g_m, f_1, \\ldots, f_n) :", "X", "\\longrightarrow", "\\mathbf{A}^{m + n}_S", "$$", "over $S$. This is a morphism of schemes locally of finite presentation", "over $S$ and hence is locally of finite presentation", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-permanence}).", "The restriction of this morphism to", "$\\mathbf{A}^m_S \\subset \\mathbf{A}^{m + n}_S$", "is \\'etale by construction. Thus, in order to show that", "$X \\to \\mathbf{A}^{m + n}_S$ is \\'etale at $y$", "it suffices to show that $X \\to \\mathbf{A}^{m + n}_S$ is flat at $y$,", "see Morphisms, Lemma \\ref{morphisms-lemma-etale-at-point}.", "Let $s \\in S$ be the image of $y$. It suffices to check that", "$X_s \\to \\mathbf{A}^{m + n}_s$ is flat at $y$, see", "Theorem \\ref{theorem-criterion-flatness-fibre}.", "Let $z \\in \\mathbf{A}^{m + n}_s$ be the image of $y$.", "The local ring map", "$$", "\\mathcal{O}_{\\mathbf{A}^{m + n}_s, z}", "\\longrightarrow", "\\mathcal{O}_{X_s, y}", "$$", "is flat by Algebra, Lemma \\ref{algebra-lemma-CM-over-regular-flat}.", "Namely, schemes smooth over fields are regular and regular rings", "are Cohen-Macaulay, see Varieties, Lemma \\ref{varieties-lemma-smooth-regular}", "and Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}.", "Thus both source and target are regular local rings (and hence CM).", "The source and target have the same dimension: namely, we have", "$\\dim(\\mathcal{O}_{Y_s, y}) = \\dim(\\mathcal{O}_{\\mathbf{A}^m_s, z})$", "by More on Algebra, Lemma \\ref{more-algebra-lemma-dimension-etale-extension},", "we have $\\dim(\\mathcal{O}_{\\mathbf{A}^{m + n}_s, z}) = n +", "\\dim(\\mathcal{O}_{\\mathbf{A}^m_s, z})$, and we have", "$\\dim(\\mathcal{O}_{X_s, y}) = n + \\dim(\\mathcal{O}_{Y_s, y})$", "because $\\mathcal{O}_{Y_s, y}$ is the quotient of", "$\\mathcal{O}_{X_s, y}$ by the regular sequence $f_1, \\ldots, f_n$", "of length $n$ (see", "Divisors, Remark \\ref{divisors-remark-relative-regular-immersion-elements}).", "Finally, the fibre ring of the displayed arrow is finite over $\\kappa(z)$", "since $Y_s \\to \\mathbf{A}^m_s$ is \\'etale at $y$.", "This finishes the proof." ], "refs": [ "divisors-lemma-immersion-smooth-into-smooth-regular-immersion", "morphisms-lemma-smooth-etale-over-affine-space", "morphisms-lemma-finite-presentation-permanence", "morphisms-lemma-etale-at-point", "more-morphisms-theorem-criterion-flatness-fibre", "algebra-lemma-CM-over-regular-flat", "varieties-lemma-smooth-regular", "algebra-lemma-regular-ring-CM", "more-algebra-lemma-dimension-etale-extension", "divisors-remark-relative-regular-immersion-elements" ], "ref_ids": [ 8007, 5377, 5247, 5372, 13672, 1107, 11004, 941, 10052, 8122 ] } ], "ref_ids": [] }, { "id": 13883, "type": "theorem", "label": "more-morphisms-lemma-map-approximation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-map-approximation", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $X$, $Y$ be", "schemes locally of finite", "type over $S$. Let $x \\in X$ and $y \\in Y$ be points lying over the", "same point $s \\in S$. Assume $\\mathcal{O}_{S, s}$ is a G-ring.", "Assume further we are given a local $\\mathcal{O}_{S, s}$-algebra map", "$$", "\\varphi : \\mathcal{O}_{Y, y} \\longrightarrow \\mathcal{O}_{X, x}^\\wedge", "$$", "For every $N \\geq 1$", "there exists an elementary \\'etale neighbourhood", "$(U, u) \\to (X, x)$ and an $S$-morphism", "$f : U \\to Y$ mapping $u$ to $y$ such that the diagram", "$$", "\\xymatrix{", "\\mathcal{O}_{X, x}^\\wedge \\ar[r] &", "\\mathcal{O}_{U, u}^\\wedge \\\\", "\\mathcal{O}_{Y, y} \\ar[r]^{f^\\sharp_u} \\ar[u]^\\varphi &", "\\mathcal{O}_{U, u} \\ar[u]", "}", "$$", "commutes modulo $\\mathfrak m_u^N$." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $X$ hence we may assume $X$, $Y$, $S$ are affine.", "Say $S = \\Spec(R)$, $X = \\Spec(A)$, $Y = \\Spec(B)$.", "Write $B = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$.", "Let $\\mathfrak p \\subset A$ be the prime ideal corresponding to $x$.", "The local ring $\\mathcal{O}_{X, x} = A_\\mathfrak p$ is a G-ring by", "More on Algebra, Proposition", "\\ref{more-algebra-proposition-finite-type-over-G-ring}.", "The map $\\varphi$ is a map", "$$", "B_\\mathfrak q^\\wedge \\longrightarrow A_\\mathfrak p^\\wedge", "$$", "where $\\mathfrak q \\subset B$ is the prime corresponding to $y$.", "Let $a_1, \\ldots, a_n \\in A_\\mathfrak p^\\wedge$ be the images", "of $x_1, \\ldots, x_n$ via", "$R[x_1, \\ldots, x_n] \\to B \\to B_\\mathfrak q^\\wedge \\to A_\\mathfrak p^\\wedge$.", "Then we can apply Smoothing Ring Maps, Lemma", "\\ref{smoothing-lemma-approximation-property-variant}", "to get an \\'etale ring map $A \\to A'$ and a prime ideal", "$\\mathfrak p' \\subset A'$ and $b_1, \\ldots, b_n \\in A'$ such that", "$\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$,", "$a_i - b_i \\in (\\mathfrak p')^N(A'_{\\mathfrak p'})^\\wedge$, and", "$f_j(b_1, \\ldots, b_n) = 0$ for $j = 1, \\ldots, n$.", "This determines an $R$-algebra map $B \\to A'$ by sending the", "class of $x_i$ to $b_i \\in A'$. This finishes the proof", "by taking $U = \\Spec(A') \\to \\Spec(B)$ as the morphism $f$", "and $u = \\mathfrak p'$." ], "refs": [ "more-algebra-proposition-finite-type-over-G-ring", "smoothing-lemma-approximation-property-variant" ], "ref_ids": [ 10581, 5643 ] } ], "ref_ids": [] }, { "id": 13884, "type": "theorem", "label": "more-morphisms-lemma-isomorphism-approximation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-isomorphism-approximation", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $X$, $Y$ be", "schemes locally of finite", "type over $S$. Let $x \\in X$ and $y \\in Y$ be points lying over the", "same point $s \\in S$. Assume $\\mathcal{O}_{S, s}$ is a G-ring.", "Assume we have an $\\mathcal{O}_{S, s}$-algebra isomorphism", "$$", "\\varphi : \\mathcal{O}_{Y, y}^\\wedge \\longrightarrow \\mathcal{O}_{X, x}^\\wedge", "$$", "between the complete local rings. Then for every $N \\geq 1$", "there exists morphisms", "$$", "(X, x) \\leftarrow (U, u) \\rightarrow (Y, y)", "$$", "of pointed schemes over $S$ such that both arrows define elementary", "\\'etale neighbourhoods and such that the diagram", "$$", "\\xymatrix{", "& \\mathcal{O}_{U, u}^\\wedge \\\\", "\\mathcal{O}_{Y, y}^\\wedge \\ar[rr]^\\varphi \\ar[ru] & &", "\\mathcal{O}_{X, x}^\\wedge \\ar[lu]", "}", "$$", "commutes modulo $\\mathfrak m_u^N$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $N \\geq 2$. Apply Lemma \\ref{lemma-map-approximation} to get", "$(U, u) \\to (X, x)$ and $f : (U, u) \\to (Y, y)$.", "We claim that $f$ is \\'etale at $u$ which will finish the proof.", "In fact, we will show that the induced map", "$\\mathcal{O}_{Y, y}^\\wedge \\to \\mathcal{O}_{U, u}^\\wedge$", "is an isomorphism. Having proved this,", "Lemma \\ref{lemma-lifting-along-artinian-at-point}", "will show that $f$ is smooth at $u$ and of course", "$f$ is unramified at $u$ as well, so", "Morphisms, Lemma \\ref{morphisms-lemma-etale-smooth-unramified}", "tells us $f$ is \\'etale at $u$.", "For a local ring $(R, \\mathfrak m)$ we set", "$\\text{Gr}_\\mathfrak m(R) =", "\\bigoplus_{n \\geq 0} \\mathfrak m^n/\\mathfrak m^{n + 1}$.", "To prove the claim we look at the induced diagram", "of graded rings", "$$", "\\xymatrix{", "& \\text{Gr}_{\\mathfrak m_u}(\\mathcal{O}_{U, u}) \\\\", "\\text{Gr}_{\\mathfrak m_y}(\\mathcal{O}_{Y, y}) \\ar[rr]^\\varphi \\ar[ru] & &", "\\text{Gr}_{\\mathfrak m_x}(\\mathcal{O}_{X, x}) \\ar[lu]", "}", "$$", "Since $N \\geq 2$ this diagram is actually commutative as the", "displayed graded algebras are generated in degree $1$!", "By assumption the lower arrow is an isomorphism.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-flat-unramified}", "(for example) the map", "$\\mathcal{O}_{X, x}^\\wedge \\to \\mathcal{O}_{U, u}^\\wedge$", "is an isomorphism and hence the north-west arrow", "in the diagram is an isomorphism. We conclude that", "$f$ induces an isomorphism", "$\\text{Gr}_{\\mathfrak m_x}(\\mathcal{O}_{X, x}) \\to", "\\text{Gr}_{\\mathfrak m_y}(\\mathcal{O}_{U, u})$.", "Using induction and the short exact sequences", "$$", "0 \\to \\text{Gr}^n_{\\mathfrak m}(R) \\to R/\\mathfrak m^{n + 1} \\to", "R/\\mathfrak m^n \\to 0", "$$", "for both local rings we conclude (from the snake lemma)", "that $f$ induces isomorphisms", "$\\mathcal{O}_{Y, y}/\\mathfrak m_y^n \\to \\mathcal{O}_{U, u}/\\mathfrak m_u^n$", "for all $n$ which is what we wanted to show." ], "refs": [ "more-morphisms-lemma-map-approximation", "more-morphisms-lemma-lifting-along-artinian-at-point", "morphisms-lemma-etale-smooth-unramified", "more-algebra-lemma-flat-unramified" ], "ref_ids": [ 13883, 13741, 5362, 10050 ] } ], "ref_ids": [] }, { "id": 13885, "type": "theorem", "label": "more-morphisms-lemma-relative-map-approximation-pre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-relative-map-approximation-pre", "contents": [ "Let $X \\to S$, $Y \\to S$, $x$, $s$, $y$, $t$, $\\sigma$, $y_\\sigma$, and", "$\\varphi$ be given as follows: we have morphisms of schemes", "$$", "\\vcenter{", "\\xymatrix{", "X \\ar[d] & Y \\ar[d] \\\\", "S & T", "}", "}", "\\quad\\text{with points}\\quad", "\\vcenter{", "\\xymatrix{", "x \\ar[d] & y \\ar[d] \\\\", "s & t", "}", "}", "$$", "Here $S$ is locally Noetherian and $T$ is of finite type over $\\mathbf{Z}$.", "The morphisms $X \\to S$ and $Y \\to T$ are locally of finite type.", "The local ring $\\mathcal{O}_{S, s}$ is a G-ring. The map", "$$", "\\sigma : \\mathcal{O}_{T, t} \\longrightarrow \\mathcal{O}_{S, s}^\\wedge", "$$", "is a local homomorphism. Set", "$Y_\\sigma = Y \\times_{T, \\sigma} \\Spec(\\mathcal{O}_{S, s}^\\wedge)$.", "Next, $y_\\sigma$ is a point of $Y_\\sigma$ mapping to $y$ and", "the closed point of $\\Spec(\\mathcal{O}_{S, s}^\\wedge)$. Finally", "$$", "\\varphi :", "\\mathcal{O}_{X, x}^\\wedge", "\\longrightarrow", "\\mathcal{O}_{Y_\\sigma, y_\\sigma}^\\wedge", "$$", "is an isomorphism of $\\mathcal{O}_{S, s}^\\wedge$-algebras.", "In this situation there exists a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] &", "W \\ar[l] \\ar[rd] \\ar[rr] & &", "Y \\times_{T, \\tau} V \\ar[r] \\ar[ld] & Y \\ar[d] \\\\", "S & &", "V \\ar[ll] \\ar[rr]^\\tau & &", "T", "}", "$$", "of schemes and points $w \\in W$, $v \\in V$ such that", "\\begin{enumerate}", "\\item $(V, v) \\to (S, s)$ is an elementary \\'etale neighbourhood,", "\\item $(W, w) \\to (X, x)$ is an elementary \\'etale neighbourhood, and", "\\item $\\tau(v) = t$.", "\\end{enumerate}", "Let $y_\\tau \\in Y \\times_T V$ correspond to $y_\\sigma$", "via the identification $(Y_\\sigma)_s = (Y \\times_T V)_v$.", "Then", "\\begin{enumerate}", "\\item[(4)] $(W, w) \\to (Y \\times_{T, \\tau} V, y_\\tau)$ is an elementary", "\\'etale neighbourhood.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Denote $X_\\sigma = X \\times_S \\Spec(\\mathcal{O}_{S, s}^\\wedge)$", "and $x_\\sigma \\in X_\\sigma$ the unique point lying over $x$.", "Observe that $\\mathcal{O}_{S, s}^\\wedge$ is a G-ring by", "More on Algebra, Proposition", "\\ref{more-algebra-proposition-Noetherian-complete-G-ring}.", "By Lemma \\ref{lemma-isomorphism-approximation}", "we can choose", "$$", "(X_\\sigma, x_\\sigma) \\leftarrow (U, u) \\rightarrow (Y_\\sigma, y_\\sigma)", "$$", "where both arrows are elementary \\'etale neighbourhoods.", "\\medskip\\noindent", "After replacing $S$ by an open neighbourhood of $s$, we may", "assume $S = \\Spec(R)$ is affine. Since $\\mathcal{O}_{S, s}$", "is a G-ring by Smoothing Ring Maps, Theorem \\ref{smoothing-theorem-popescu}", "the ring $\\mathcal{O}_{S, s}^\\wedge$ is a filtered colimit of smooth", "$R$-algebras. Thus we can write", "$$", "\\Spec(\\mathcal{O}_{S, s}^\\wedge) = \\lim S_i", "$$", "as a directed limit of affine schemes $S_i$ smooth over $S$.", "Denote $s_i \\in S_i$ the image of the closed point of", "$\\Spec(\\mathcal{O}_{S, s}^\\wedge)$. Observe that $\\kappa(s) = \\kappa(s_i)$.", "Set $X_i = X \\times_S S_i$ and denote $x_i \\in X_i$ the unique", "point mapping to $x$. Note that $\\kappa(x) = \\kappa(x_i)$.", "Since $T$ is of finite type over $\\mathbf{Z}$ by Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}", "we can choose an $i$ and a morphism $\\sigma_i : (S_i, s_i) \\to (T, t)$", "of pointed schemes whose composition with", "$\\Spec(\\mathcal{O}_{S, s}) \\to S_i$ is equal to $\\sigma$.", "Set $Y_i = Y \\times_T S_i$ and denote $y_i$ the", "image of $y_\\sigma$. Note that $\\kappa(y_i) = \\kappa(y_\\sigma)$.", "By Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "we can choose an $i$ and a diagram", "$$", "\\xymatrix{", "X_i \\ar[rd] &", "U_i \\ar[l] \\ar[d] \\ar[r] &", "Y_i \\ar[ld] \\\\", "& S_i", "}", "$$", "whose base change to $\\Spec(\\mathcal{O}_{S, s}^\\wedge)$", "recovers $X_\\sigma \\leftarrow U \\rightarrow Y_\\sigma$.", "By Limits, Lemma \\ref{limits-lemma-descend-etale}", "after increasing $i$ we may assume the morphisms", "$X_i \\leftarrow U_i \\rightarrow Y_i$ are \\'etale.", "Let $u_i \\in U_i$ be the image of $u$. Then $u_i \\mapsto x_i$", "hence", "$\\kappa(x) = \\kappa(x_\\sigma) = \\kappa(u) \\supset \\kappa(u_i) \\supset", "\\kappa(x_i) = \\kappa(x)$ and we see that $\\kappa(u_i) = \\kappa(x_i)$.", "Hence $(X_i, x_i) \\leftarrow (U_i, u_i)$ is an elementary", "\\'etale neighbourhood. Since also $\\kappa(y_i) = \\kappa(y_\\sigma) = \\kappa(u)$", "we see that also $(U_i, u_i) \\to (Y_i, y_i)$ is an elementary", "\\'etale neighbourhood.", "\\medskip\\noindent", "At this point we have constructed a diagram", "$$", "\\xymatrix{", "X \\ar[d] &", "X \\times_S S_i \\ar[l] \\ar[rd] &", "U_i \\ar[l] \\ar[r] \\ar[d] &", "Y \\times_T S_i \\ar[r] \\ar[ld] &", "Y \\ar[d] \\\\", "S & &", "S_i \\ar[ll] \\ar[rr] & &", "T", "}", "$$", "as in the statement of the lemma, except that $S_i \\to S$ is smooth.", "By Lemma \\ref{lemma-slice-smooth} and after shrinking $S_i$", "we can assume there exists a closed subscheme $V \\subset S_i$ passing", "through $s_i$ such that $V \\to S$ is \\'etale.", "Setting $W$ equal to the scheme theoretic inverse image of $V$", "in $U_i$ we conclude." ], "refs": [ "more-algebra-proposition-Noetherian-complete-G-ring", "more-morphisms-lemma-isomorphism-approximation", "smoothing-theorem-popescu", "limits-proposition-characterize-locally-finite-presentation", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-etale", "more-morphisms-lemma-slice-smooth" ], "ref_ids": [ 10580, 13884, 5605, 15127, 15077, 15065, 13878 ] } ], "ref_ids": [] }, { "id": 13886, "type": "theorem", "label": "more-morphisms-lemma-relative-map-approximation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-relative-map-approximation", "contents": [ "Consider a diagram", "$$", "\\vcenter{", "\\xymatrix{", "X \\ar[d] & Y \\ar[d] \\\\", "S & T \\ar[l]", "}", "}", "\\quad\\text{with points}\\quad", "\\vcenter{", "\\xymatrix{", "x \\ar[d] & y \\ar[d] \\\\", "s & t \\ar[l]", "}", "}", "$$", "where $S$ be a locally Noetherian scheme and the morphisms are", "locally of finite type. Assume $\\mathcal{O}_{S, s}$ is a G-ring.", "Assume further we are given a local $\\mathcal{O}_{S, s}$-algebra map", "$$", "\\sigma : \\mathcal{O}_{T, t} \\longrightarrow \\mathcal{O}_{S, s}^\\wedge", "$$", "and a local $\\mathcal{O}_{S, s}$-algebra map", "$$", "\\varphi :", "\\mathcal{O}_{X, x}", "\\longrightarrow", "\\mathcal{O}_{Y_\\sigma, y_\\sigma}^\\wedge", "$$", "where $Y_\\sigma = Y \\times_{T, \\sigma} \\Spec(\\mathcal{O}_{S, s}^\\wedge)$", "and $y_\\sigma$ is the unique point of $Y_\\sigma$ lying over $y$.", "For every $N \\geq 1$ there exists a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] & X \\times_S V \\ar[l] \\ar[rd] &", "W \\ar[l]^-f \\ar[r] \\ar[d] &", "Y \\times_{T, \\tau} V \\ar[r] \\ar[ld] & Y \\ar[d] \\\\", "S & & V \\ar[ll] \\ar[rr]^\\tau & & T", "}", "$$", "of schemes over $S$ and points $w \\in W$, $v \\in V$ such that", "\\begin{enumerate}", "\\item $v \\mapsto s$, $\\tau(v) = t$, $f(w) = (x, v)$, and $w \\mapsto (y, v)$,", "\\item $(V, v) \\to (S, s)$ is an elementary \\'etale neighbourhood,", "\\item the diagram", "$$", "\\xymatrix{", "\\mathcal{O}_{S, s}^\\wedge \\ar[r] & \\mathcal{O}_{V, v}^\\wedge \\\\", "\\mathcal{O}_{T, t} \\ar[r]^{\\tau^\\sharp_v} \\ar[u]_\\sigma &", "\\mathcal{O}_{V, v} \\ar[u]", "}", "$$", "commutes module $\\mathfrak m_v^N$,", "\\item $(W, w) \\to (Y \\times_{T, \\tau} V, (y, v))$ is an", "elementary \\'etale neighbourhood,", "\\item the diagram", "$$", "\\xymatrix{", "\\mathcal{O}_{X, x} \\ar[r]_\\varphi &", "\\mathcal{O}_{Y_\\sigma, y_\\sigma}^\\wedge \\ar[r] &", "\\mathcal{O}_{Y_\\sigma, y_\\sigma}/\\mathfrak m_{y_\\sigma}^N \\ar@{=}[r] &", "\\mathcal{O}_{Y \\times_{T, \\tau} V, (y, v)}/\\mathfrak m_{(y, v)}^N", "\\ar[d]_{\\cong} \\\\", "\\mathcal{O}_{X, x} \\ar[r] \\ar@{=}[u] &", "\\mathcal{O}_{X \\times_S V, (x, v)} \\ar[r]^{f^\\sharp_w} &", "\\mathcal{O}_{W, w} \\ar[r] &", "\\mathcal{O}_{W, w}/\\mathfrak m_w^N", "}", "$$", "commutes. The equality comes from the fact that", "$Y_\\sigma$ and $Y \\times_{T, \\tau} V$ are canonically isomorphic over", "$\\mathcal{O}_{V, v}/\\mathfrak m_v^N = \\mathcal{O}_{S, s}/\\mathfrak m_s^N$", "by parts (2) and (3).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "After replacing $X$, $S$, $T$, $Y$ by affine open subschemes we", "may assume the diagram in the statement of the lemma comes", "from applying $\\Spec$ to a diagram", "$$", "\\vcenter{", "\\xymatrix{", "A & B \\\\", "R \\ar[u] \\ar[r] & C \\ar[u]", "}", "}", "\\quad\\text{with primes}\\quad", "\\vcenter{", "\\xymatrix{", "\\mathfrak p_A & \\mathfrak p_B \\\\", "\\mathfrak p_R \\ar@{-}[u] \\ar@{-}[r] & \\mathfrak p_C \\ar@{-}[u]", "}", "}", "$$", "of Noetherian rings and finite type ring maps.", "In this proof every ring $E$ will be a Noetherian $R$-algebra endowed with", "a prime ideal $\\mathfrak p_E$ lying over $\\mathfrak p_R$ and all ring maps", "will be $R$-algebra maps compatible with the given primes. Moreover,", "if we write $E^\\wedge$ we mean the completion of the localization", "of $E$ at $\\mathfrak p_E$. We will also use without further mention", "that an \\'etale ring map $E_1 \\to E_2$ such that", "$\\kappa(\\mathfrak p_{E_1}) = \\kappa(\\mathfrak p_{E_2})$ induces", "an isomorphism $E_1^\\wedge = E_2^\\wedge$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-unramified}.", "\\medskip\\noindent", "With this notation $\\sigma$ and $\\varphi$ correspond to ring maps", "$$", "\\sigma : C \\to R^\\wedge", "\\quad\\text{and}\\quad", "\\varphi : A \\longrightarrow (B \\otimes_{C, \\sigma} R^\\wedge)^\\wedge", "$$", "Here is a picture", "$$", "\\xymatrix{", "A \\ar@/^1em/[rrr]^\\varphi &", "B \\ar[r] &", "B \\otimes_{C, \\sigma} R^\\wedge \\ar[r] &", "(B \\otimes_{C, \\sigma} R^\\wedge)^\\wedge \\\\", "R \\ar[r] \\ar[u] &", "C \\ar[r]^\\sigma \\ar[u] & R^\\wedge \\ar[u] \\ar[ru]", "}", "$$", "Observe that $R^\\wedge$ is a G-ring by", "More on Algebra, Proposition", "\\ref{more-algebra-proposition-Noetherian-complete-G-ring}.", "Thus $B \\otimes_{C, \\sigma} R^\\wedge$ is a G-ring by", "More on Algebra, Proposition", "\\ref{more-algebra-proposition-finite-type-over-G-ring}.", "By Lemma \\ref{lemma-map-approximation} (translated into algebra)", "there exists an \\'etale ring map $B \\otimes_{C, \\sigma} R^\\wedge \\to B'$", "inducing an isomorphism", "$\\kappa(\\mathfrak p_{B \\otimes_{C, \\sigma} R^\\wedge})", "\\to \\kappa(\\mathfrak p_{B'})$", "and an $R$-algebra map $A \\to B'$ such that the composition", "$$", "A \\to B' \\to (B')^\\wedge = (B \\otimes_{C, \\sigma} R^\\wedge)^\\wedge", "$$", "is the same as $\\varphi$ modulo", "$(\\mathfrak p_{(B \\otimes_{C, \\sigma} R^\\wedge)^\\wedge})^N$.", "Thus we may replace $\\varphi$ by this composition because the", "only way $\\varphi$ enters the conclusion is via the commutativity", "requirement in part (5) of the statement of the lemma.", "Picture:", "$$", "\\xymatrix{", "& & B' \\ar[r] & (B')^\\wedge \\ar@{=}[d] \\\\", "A \\ar[rru] &", "B \\ar[r] &", "B \\otimes_{C, \\sigma} R^\\wedge \\ar[r] \\ar[u] &", "(B \\otimes_{C, \\sigma} R^\\wedge)^\\wedge \\\\", "R \\ar[r] \\ar[u] &", "C \\ar[r]^\\sigma \\ar[u] & R^\\wedge \\ar[u] \\ar[ru]", "}", "$$", "Next, we use that $R^\\wedge$ is a filtered colimit of smooth", "$R$-algebras (Smoothing Ring Maps, Theorem \\ref{smoothing-theorem-popescu})", "because $R_{\\mathfrak p_R}$ is a G-ring by assumption. Since $C$ is of", "finite presentation over $R$ we get a factorization", "$$", "C \\to R' \\to R^\\wedge", "$$", "for some $R \\to R'$ smooth, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-finite-presentation}.", "After increasing $R'$ we may assume there exists", "an \\'etale $B \\otimes_C R'$-algebra $B''$ whose base change", "to $B \\otimes_{C, \\sigma} R^\\wedge$ is $B'$, see", "Algebra, Lemma \\ref{algebra-lemma-etale}.", "Then $B'$ is the filtered colimit of these $B''$ and we", "conclude that after increasing $R'$ we may assume there is", "an $R$-algebra map $A \\to B''$ such that $A \\to B'' \\to B'$ is", "the previously constructed map (same reference as above). Picture", "$$", "\\xymatrix{", "& & B'' \\ar[r] & B' \\ar[r] & (B')^\\wedge \\ar@{=}[d] \\\\", "A \\ar[rru] &", "B \\ar[r] &", "B \\otimes_C R' \\ar[r] \\ar[u] &", "B \\otimes_{C, \\sigma} R^\\wedge \\ar[r] \\ar[u] &", "(B \\otimes_{C, \\sigma} R^\\wedge)^\\wedge \\\\", "R \\ar[r] \\ar[u] &", "C \\ar[r] \\ar[u] &", "R' \\ar[r] \\ar[u] &", "R^\\wedge \\ar[u] \\ar[ru]", "}", "$$", "and", "$$", "B' = B'' \\otimes_{(B \\otimes_C R')} (B \\otimes_{C, \\sigma} R^\\wedge)", "$$", "This means that we may replace $C$ by $R'$, $\\sigma : C \\to R^\\wedge$ by", "$R' \\to R^\\wedge$, and $B$ by $B''$ so that we simplify to the diagram", "$$", "\\xymatrix{", "A \\ar[r] &", "B \\ar[r] &", "B \\otimes_{C, \\sigma} R^\\wedge \\\\", "R \\ar[r] \\ar[u] &", "C \\ar[r]^\\sigma \\ar[u] & R^\\wedge \\ar[u]", "}", "$$", "with $\\varphi$ equal to the composition of the horizontal arrows", "followed by the canonical map from $B \\otimes_{C, \\sigma} R^\\wedge$", "to its completion.", "The final step in the proof is to apply Lemma \\ref{lemma-map-approximation}", "(or its proof)", "one more time to $\\Spec(C)$ and $\\Spec(R)$ over $\\Spec(R)$ and the map", "$C \\to R^\\wedge$. The lemma produces a ring map $C \\to D$", "such that $R \\to D$ is \\'etale, such that", "$\\kappa(\\mathfrak p_R) = \\kappa(\\mathfrak p_D)$, and such that", "$$", "C \\to D \\to D^\\wedge = R^\\wedge", "$$", "is equal to $\\sigma : C \\to R^\\wedge$ modulo $(\\mathfrak p_{R^\\wedge})^N$.", "Then we can take", "$$", "V = \\Spec(D)", "\\quad\\text{and}\\quad", "W = \\Spec(B \\otimes_C D)", "$$", "as our solution to the problem posed by the lemma. Namely the diagram", "$$", "\\xymatrix{", "A \\ar[r] &", "B \\otimes_{C, \\sigma} R^\\wedge \\ar[r] &", "B \\otimes_{C, \\sigma} R^\\wedge/(\\mathfrak p_{R^\\wedge})^N \\ar@{=}[r] &", "B \\otimes_C D/(\\mathfrak p_D)^N \\\\", "A \\ar@{=}[u] \\ar[r] &", "A \\otimes_R D \\ar[r] &", "B \\otimes_R D \\ar[r] &", "B \\otimes_C D/(\\mathfrak p_D)^N \\ar@{=}[u]", "}", "$$", "commutes because $C \\to D \\to D^\\wedge = R^\\wedge$ is equal to", "$\\sigma$ modulo $(\\mathfrak p_{R^\\wedge})^N$. This proves part (5)", "and the other properties are immediate from the construction." ], "refs": [ "more-algebra-lemma-flat-unramified", "more-algebra-proposition-Noetherian-complete-G-ring", "more-algebra-proposition-finite-type-over-G-ring", "more-morphisms-lemma-map-approximation", "smoothing-theorem-popescu", "algebra-lemma-characterize-finite-presentation", "algebra-lemma-etale", "more-morphisms-lemma-map-approximation" ], "ref_ids": [ 10050, 10580, 10581, 13883, 5605, 1092, 1231, 13883 ] } ], "ref_ids": [] }, { "id": 13887, "type": "theorem", "label": "more-morphisms-lemma-control-agreement", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-control-agreement", "contents": [ "Let $T \\to S$ be finite type morphisms of Noetherian schemes.", "Let $t \\in T$ map to $s \\in S$ and let", "$\\sigma : \\mathcal{O}_{T, t} \\to \\mathcal{O}_{S, s}^\\wedge$", "be a local $\\mathcal{O}_{S, s}$-algebra map. For every $N \\geq 1$", "there exists a finite type morphism $(T', t') \\to (T, t)$", "such that $\\sigma$ factors through", "$\\mathcal{O}_{T, t} \\to \\mathcal{O}_{T', t'}$", "and such that for every local $\\mathcal{O}_{S, s}$-algebra map", "$\\sigma' : \\mathcal{O}_{T, t} \\to \\mathcal{O}_{S, s}^\\wedge$", "which factors through $\\mathcal{O}_{T, t} \\to \\mathcal{O}_{T', t'}$", "the maps $\\sigma$ and $\\sigma'$ agree modulo $\\mathfrak m_s^N$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $S$ and $T$ are affine. Say $S = \\Spec(R)$ and", "$T = \\Spec(C)$. Let $c_1, \\ldots, c_n \\in C$ be generators", "of $C$ as an $R$-algebra. Let $\\mathfrak p \\subset R$ be the", "prime ideal corresponding to $s$. Say $\\mathfrak p = (f_1, \\ldots, f_m)$.", "After replacing $R$ by a principal localization", "(to clear denominators in $R_\\mathfrak p$)", "we may assume there exist", "$r_1, \\ldots, r_n \\in R$ and $a_{i, I} \\in \\mathcal{O}_{S, s}^\\wedge$", "where $I = (i_1, \\ldots, i_m)$ with $\\sum i_j = N$ such that", "$$", "\\sigma(c_i) = r_i + \\sum\\nolimits_I a_{i, I} f_1^{i_1} \\ldots f_m^{i_m}", "$$", "in $\\mathcal{O}_{S, s}^\\wedge$. Then we consider", "$$", "C' = C[t_{i, I}]/", "\\left(c_i - r_i - \\sum\\nolimits_I t_{i, I} f_1^{i_1} \\ldots f_m^{i_m}\\right)", "$$", "with $\\mathfrak p' = \\mathfrak pC' + (t_{i, I})$ and factorization", "of $\\sigma : C \\to \\mathcal{O}_{S, s}^\\wedge$ through $C'$", "given by sending $t_{i, I}$ to $a_{i, I}$.", "Taking $T' = \\Spec(C')$ works because any $\\sigma'$ as in the statement", "of the lemma will send $c_i$ to $r_i$ modulo the maximal ideal", "to the power $N$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13888, "type": "theorem", "label": "more-morphisms-lemma-control-graded", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-control-graded", "contents": [ "Let $Y \\to T \\to S$ be finite type morphisms of Noetherian schemes.", "Let $t \\in T$ map to $s \\in S$ and let", "$\\sigma : \\mathcal{O}_{T, t} \\to \\mathcal{O}_{S, s}^\\wedge$", "be a local $\\mathcal{O}_{S, s}$-algebra map.", "There exists a finite type morphism $(T', t') \\to (T, t)$", "such that $\\sigma$ factors through", "$\\mathcal{O}_{T, t} \\to \\mathcal{O}_{T', t'}$", "and such that for every local $\\mathcal{O}_{S, s}$-algebra map", "$\\sigma' : \\mathcal{O}_{T, t} \\to \\mathcal{O}_{S, s}^\\wedge$", "which factors through $\\mathcal{O}_{T, t} \\to \\mathcal{O}_{T', t'}$", "the closed immersions", "$$", "Y \\times_{T, \\sigma} \\Spec(\\mathcal{O}_{S, s}^\\wedge) = Y_\\sigma", "\\longleftarrow Y_t \\longrightarrow", "Y_{\\sigma'} =", "Y \\times_{T, \\sigma'} \\Spec(\\mathcal{O}_{S, s}^\\wedge)", "$$", "have isomorphic conormal algebras." ], "refs": [], "proofs": [ { "contents": [ "A useful observation is that $\\kappa(s) = \\kappa(t)$ by the existence of", "$\\sigma$. Observe that the statement makes sense as the fibres of $Y_\\sigma$", "and $Y_{\\sigma'}$ over $s \\in \\Spec(\\mathcal{O}_{S, s}^\\wedge)$", "are both canonically isomorphic to $Y_t$. We will think of the property", "``$\\sigma'$ factors through $\\mathcal{O}_{T, t} \\to \\mathcal{O}_{T', t'}$''", "as a constraint on $\\sigma'$. If we have several such constraints,", "say given by $(T'_i, t'_i) \\to (T, t)$, $i = 1, \\ldots, n$", "then we can combined them by considering", "$(T'_1 \\times_T \\ldots \\times_T T'_n, (t'_1, \\ldots, t'_n)) \\to", "(T, t)$. We will use this without further mention in the following.", "\\medskip\\noindent", "By Lemma \\ref{lemma-control-agreement} we can assume that any $\\sigma'$", "as in the statement of the lemma is the same as $\\sigma$ modulo", "$\\mathfrak m_s^2$. Note that the conormal algebra of $Y_t$ in", "$Y_\\sigma$ is just the quasi-coherent graded $\\mathcal{O}_{Y_t}$-algebra", "$$", "\\bigoplus\\nolimits_{n \\geq 0}", "\\mathfrak m_s^n\\mathcal{O}_{Y_\\sigma}/", "\\mathfrak m_s^{n + 1}\\mathcal{O}_{Y_\\sigma}", "$$", "and similarly for $Y_{\\sigma'}$. Since $\\sigma$ and $\\sigma'$", "agree modulo $\\mathfrak m_s^2$ we see that these two algebras", "are the same in degrees $0$ and $1$. On the other hand, these", "conormal algebras are generated in degree $1$ over degree $0$.", "Hence if there is an isomorphism extending the isomorphism", "just constructed in degrees $0$ and $1$, then it is unique.", "\\medskip\\noindent", "We may assume $S$ and $T$ are affine. Let $Y = Y_1 \\cup \\ldots \\cup Y_n$", "be an affine open covering. If we can construct $(T_i', t'_i) \\to (T, t)$", "as in the lemma such that the desired isomorphism (see previous paragraph)", "exists for $Y_i \\to T \\to S$ and $\\sigma$, then these glue by", "uniqueness to prove the result for $Y \\to T$. Thus we may assume $Y$ is affine.", "\\medskip\\noindent", "Write $S = \\Spec(R)$, $T = \\Spec(C)$, and $Y = \\Spec(B)$.", "Choose a presentation $B = C[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$.", "Denote $R^\\wedge = \\mathcal{O}_{S, s}^\\wedge$.", "Let $a_{kj} \\in R^\\wedge[x_1, \\ldots, x_n]$ be polynomials", "such that", "$$", "\\sum\\nolimits_{j = 1, \\ldots, m} a_{kj}\\sigma(f_j) = 0,\\quad", "\\text{for }k = 1, \\ldots, K", "$$", "is a set of generators for the module of relations among", "the $\\sigma(f_j) \\in R^\\wedge[x_1, \\ldots, x_n]$.", "Thus we have an exact sequence", "\\begin{equation}", "\\label{equation-resolution}", "R^\\wedge[x_1, \\ldots, x_n]^{\\oplus K} \\to", "R^\\wedge[x_1, \\ldots, x_n]^{\\oplus m} \\to", "R^\\wedge[x_1, \\ldots, x_n] \\to B \\otimes_{C, \\sigma} R^\\wedge \\to 0", "\\end{equation}", "Let $c$ be an integer which works in the Artin-Rees lemma for", "both the first and the second map in this sequence and the ideal", "$\\mathfrak m_{R^\\wedge}R^\\wedge[x_1, \\ldots, x_n]$ as defined in", "More on Algebra, Section \\ref{more-algebra-section-artin-rees}.", "Write", "$$", "a_{kj} = \\sum\\nolimits_{I \\in \\Omega} a_{kj, I} x^I", "\\quad\\text{and}\\quad", "f_j = \\sum\\nolimits_{I \\in \\Omega} f_{j, I} x^I", "$$", "in multiindex notation where $a_{kj, I} \\in R^\\wedge$, $f_{j, I} \\in C$,", "and $\\Omega$ a finite set of multiindices. Then we see that", "$$", "\\sum\\nolimits_{j = 1, \\ldots, m,\\ I, I' \\in \\Omega,\\ I + I' = I''}", "a_{kj, I} \\sigma(f_{j, I'}) = 0,\\quad", "I''\\text{ a multiindex}", "$$", "in $R^\\wedge$. Thus we take", "$$", "C' = C[t_{jk, I}]/", "\\left(", "\\sum\\nolimits_{j = 1, \\ldots, m,\\ I, I' \\in \\Omega,\\ I + I' = I''}", "t_{kj, I} f_{j, I'},\\ I''\\text{ a multiindex}\\right)", "$$", "Then $\\sigma$ factors through a map $\\tilde\\sigma : C' \\to R^\\wedge$", "sending $t_{kj, I}$ to $a_{jk, I}$.", "Thus $T' = \\Spec(C')$ comes with a point $t' \\in T'$ such that", "$\\sigma$ factors through $\\mathcal{O}_{T, t} \\to \\mathcal{O}_{T', t'}$.", "Let $t_{kj} = \\sum t_{kj, I} x^I$ in $C'[x_1, \\ldots, x_n]$.", "Then we see that we have a complex", "\\begin{equation}", "\\label{equation-resolution-new}", "C'[x_1, \\ldots, x_n]^{\\oplus K} \\to", "C'[x_1, \\ldots, x_n]^{\\oplus m} \\to", "C'[x_1, \\ldots, x_n] \\to B \\otimes_C C' \\to 0", "\\end{equation}", "which is exact at $C'[x_1, \\ldots, x_n]$ and whose base change", "by $\\tilde\\sigma$ gives (\\ref{equation-resolution}).", "\\medskip\\noindent", "By Lemma \\ref{lemma-control-agreement}", "we can find a further morphism $(T'', t'') \\to (T', t')$", "such that $\\tilde\\sigma$", "factors through $\\mathcal{O}_{T', t'} \\to \\mathcal{O}_{T'', t''}$", "and such that if $\\sigma' : C \\to R^\\wedge$", "factors through $\\mathcal{O}_{T'', t''}$, then the induced map", "$\\tilde \\sigma' : C' \\to R^\\wedge$", "agrees modulo $\\mathfrak m_s^{c + 1}$ with $\\tilde \\sigma$.", "Thus if $\\sigma'$ is such a map, then we obtain a complex", "$$", "R^\\wedge[x_1, \\ldots, x_n]^{\\oplus K} \\to", "R^\\wedge[x_1, \\ldots, x_n]^{\\oplus m} \\to", "R^\\wedge[x_1, \\ldots, x_n] \\to B \\otimes_{C, \\sigma'} R^\\wedge \\to 0", "$$", "over $R^\\wedge[x_1, \\ldots, x_n]$ by applying $\\tilde\\sigma'$", "to the polynomials $t_{kj}$ and $f_j$. In other words, this", "is the base change of the complex (\\ref{equation-resolution-new})", "by $\\tilde\\sigma'$. The matrices defining this complex", "are congruent modulo $\\mathfrak m_s^{c + 1}$ to the matrices", "defining the complex (\\ref{equation-resolution}) because", "$\\tilde \\sigma$ and $\\tilde \\sigma'$ are congruent modulo", "$\\mathfrak m_s^{c + 1}$. Since (\\ref{equation-resolution}) is exact,", "we can apply", "More on Algebra, Lemma \\ref{more-algebra-lemma-approximate-complex-graded}", "to conclude that", "$$", "\\text{Gr}_{\\mathfrak m_s}(B \\otimes_{C, \\sigma'} R^\\wedge)", "\\cong", "\\text{Gr}_{\\mathfrak m_s}(B \\otimes_{C, \\sigma} R^\\wedge)", "$$", "as desired." ], "refs": [ "more-morphisms-lemma-control-agreement", "more-morphisms-lemma-control-agreement", "more-algebra-lemma-approximate-complex-graded" ], "ref_ids": [ 13887, 13887, 9812 ] } ], "ref_ids": [] }, { "id": 13889, "type": "theorem", "label": "more-morphisms-lemma-relative-isomorphism-approximation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-relative-isomorphism-approximation", "contents": [ "With notation an assumptions as in Lemma \\ref{lemma-relative-map-approximation}", "assume that $\\varphi$ induces an isomorphism on completions.", "Then we can choose our diagram such that $f$ is \\'etale." ], "refs": [ "more-morphisms-lemma-relative-map-approximation" ], "proofs": [ { "contents": [ "We may assume $N \\geq 2$ and we may replace $(T, t)$ with $(T', t')$ as in", "Lemma \\ref{lemma-control-graded}. Since $(V, v) \\to (S, s)$ is an elementary", "\\'etale neighbourhood, so is $(X \\times_S V, (x, v)) \\to (X, x)$.", "Thus $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X \\times_S V, (x, v)}$", "induces an isomorphism on completions by", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-unramified}.", "We claim $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{W, w}$ induces", "an isomorphism on completions. Having proved this,", "Lemma \\ref{lemma-lifting-along-artinian-at-point}", "will show that $f$ is smooth at $w$ and of course", "$f$ is unramified at $u$ as well, so", "Morphisms, Lemma \\ref{morphisms-lemma-etale-smooth-unramified}", "tells us $f$ is \\'etale at $w$.", "\\medskip\\noindent", "First we use the commutativity in part (5) of", "Lemma \\ref{lemma-relative-map-approximation}", "to see that for $i \\leq N$ there is a commutative diagram", "$$", "\\xymatrix{", "\\text{Gr}^i_{\\mathfrak m_x}(\\mathcal{O}_{X, x})", "\\ar[r]_-\\varphi &", "\\text{Gr}^i_{\\mathfrak m_{y_\\sigma}}(\\mathcal{O}_{Y_\\sigma, y_\\sigma}^\\wedge)", "\\ar@{=}[r] &", "\\text{Gr}^i_{\\mathfrak m_{(y, v)}}(\\mathcal{O}_{Y \\times_{T, \\tau} V, (y, v)})", "\\ar[d]_{\\cong} \\\\", "\\text{Gr}^i_{\\mathfrak m_x}(\\mathcal{O}_{X, x})", "\\ar[r]^-{\\cong} \\ar@{=}[u] &", "\\text{Gr}^i_{\\mathfrak m_{(x, v)}}(\\mathcal{O}_{X \\times_S V, (x, v)})", "\\ar[r]^{f^\\sharp_w} &", "\\text{Gr}^i_{\\mathfrak m_w}(\\mathcal{O}_{W, w})", "}", "$$", "This implies that $f^\\sharp_w$ defines an isomorphism", "$\\kappa(x) \\to \\kappa(w)$ on residue fields and an isomorphism", "$\\mathfrak m_x/\\mathfrak m_x^2 \\to \\mathfrak m_w/\\mathfrak m_w^2$", "on cotangent spaces. Hence $f^\\sharp_w$ defines a surjection", "$\\mathcal{O}_{X, x}^\\wedge \\to \\mathcal{O}_{W, w}^\\wedge$", "on complete local rings.", "\\medskip\\noindent", "By Lemma \\ref{lemma-control-graded} there is an isomorphism of", "$\\text{Gr}_{\\mathfrak m_s}(\\mathcal{O}_{(Y \\times_{T, \\tau} V, (y, v)})$", "with", "$\\text{Gr}_{\\mathfrak m_s}(\\mathcal{O}_{Y_\\sigma, y_\\sigma})$.", "This follows by taking stalks of the isomorphism of conormal", "sheaves at the point $y$. Since our local rings are Noetherian", "taking associated graded with respect to $\\mathfrak m_s$", "commutes with completion because completion with respect to an ideal", "is an exact functor on finite modules over Noetherian rings.", "This produces the right vertical isomorphism in the diagram of graded rings", "$$", "\\xymatrix{", "\\text{Gr}_{\\mathfrak m_s}(\\mathcal{O}_{W, w}^\\wedge) &", "\\text{Gr}_{\\mathfrak m_s}", "(\\mathcal{O}_{(Y \\times_{T, \\tau} V, (y, v)}^\\wedge) \\ar[l] \\\\", "\\text{Gr}_{\\mathfrak m_s}(\\mathcal{O}_{X, x}^\\wedge)", "\\ar[r]^\\varphi \\ar[u] &", "\\text{Gr}_{\\mathfrak m_s}(\\mathcal{O}_{Y_\\sigma, y_\\sigma}^\\wedge)", "\\ar[u]_{\\cong}", "}", "$$", "We do not claim the diagram commutes. By the result of the previous", "paragraph the left arrow is surjective. The other three arrows", "are isomorphisms. It follows that the left arrow is a surjective map", "between isomorphic Noetherian rings. Hence it is an isomorphism", "by Algebra, Lemma \\ref{algebra-lemma-surjective-endo-noetherian-ring-is-iso}", "(you can argue this directly using Hilbert functions as well).", "In particular $\\mathcal{O}_{X, x}^\\wedge \\to \\mathcal{O}_{W, w}^\\wedge$", "must be injective as well as surjective which finishes the proof." ], "refs": [ "more-morphisms-lemma-control-graded", "more-algebra-lemma-flat-unramified", "more-morphisms-lemma-lifting-along-artinian-at-point", "morphisms-lemma-etale-smooth-unramified", "more-morphisms-lemma-relative-map-approximation", "more-morphisms-lemma-control-graded", "algebra-lemma-surjective-endo-noetherian-ring-is-iso" ], "ref_ids": [ 13888, 10050, 13741, 5362, 13886, 13888, 457 ] } ], "ref_ids": [ 13886 ] }, { "id": 13890, "type": "theorem", "label": "more-morphisms-lemma-dominate-etale-neighbourhood-finite-flat", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-dominate-etale-neighbourhood-finite-flat", "contents": [ "Let $S$ be a scheme. Let $s \\in S$.", "Let $f : (U, u) \\to (S, s)$ be an \\'etale neighbourhood.", "There exists an affine open neighbourhood $s \\in V \\subset S$", "and a surjective, finite locally free morphism $\\pi : T \\to V$", "such that for every $t \\in \\pi^{-1}(s)$ there exists an", "open neighbourhood $t \\in W_t \\subset T$ and a commutative", "diagram", "$$", "\\xymatrix{", "T \\ar[d]^\\pi & W_t \\ar[l] \\ar[rr]_{h_t} \\ar[rd] & & U \\ar[dl] \\\\", "V \\ar[rr] & & S", "}", "$$", "with $h_t(t) = u$." ], "refs": [], "proofs": [ { "contents": [ "The problem is local on $S$ hence we may replace $S$ by any", "open neighbourhood of $s$.", "We may also replace $U$ by an open neighbourhood of $u$.", "Hence, by Morphisms, Lemma \\ref{morphisms-lemma-etale-locally-standard-etale}", "we may assume that", "$U \\to S$ is a standard \\'etale morphism of affine schemes.", "In this case the lemma (with $V = S$) follows from", "Algebra, Lemma \\ref{algebra-lemma-standard-etale-finite-flat-Zariski}." ], "refs": [ "morphisms-lemma-etale-locally-standard-etale", "algebra-lemma-standard-etale-finite-flat-Zariski" ], "ref_ids": [ 5371, 1243 ] } ], "ref_ids": [] }, { "id": 13891, "type": "theorem", "label": "more-morphisms-lemma-dominate-etale-affine-finite-flat", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-dominate-etale-affine-finite-flat", "contents": [ "Let $f : U \\to S$ be a surjective \\'etale morphism of affine schemes.", "There exists a surjective, finite locally free morphism", "$\\pi : T \\to S$ and a finite open covering", "$T = T_1 \\cup \\ldots \\cup T_n$ such that each", "$T_i \\to S$ factors through $U \\to S$. Diagram:", "$$", "\\xymatrix{", "& \\coprod T_i \\ar[rd] \\ar[ld] & \\\\", "T \\ar[rd]^\\pi & & U \\ar[ld]_f \\\\", "& S &", "}", "$$", "where the south-west arrow is a Zariski-covering." ], "refs": [], "proofs": [ { "contents": [ "This is a restatement of", "Algebra, Lemma \\ref{algebra-lemma-etale-finite-flat-zariski}." ], "refs": [ "algebra-lemma-etale-finite-flat-zariski" ], "ref_ids": [ 1244 ] } ], "ref_ids": [] }, { "id": 13892, "type": "theorem", "label": "more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$. Set $s = f(x)$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite type, and", "\\item $x \\in X_s$ is isolated\\footnote{In the presence of (1)", "this means that $f$ is", "quasi-finite at $x$, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}.}.", "\\end{enumerate}", "Then there exist", "\\begin{enumerate}", "\\item[(a)] an elementary \\'etale neighbourhood $(U, u) \\to (S, s)$,", "\\item[(b)] an open subscheme $V \\subset X_U$", "(see \\ref{equation-basic-diagram})", "\\end{enumerate}", "such that", "\\begin{enumerate}", "\\item[(\\romannumeral1)] $V \\to U$ is a finite morphism,", "\\item[(\\romannumeral2)] there is a unique point $v$ of $V$", "mapping to $u$ in $U$, and", "\\item[(\\romannumeral3)] the point $v$ maps to $x$", "under the morphism $X_U \\to X$, inducing $\\kappa(x) = \\kappa(v)$.", "\\end{enumerate}", "Moreover, for any elementary \\'etale neighbourhood $(U', u') \\to (U, u)$", "setting $V' = U' \\times_U V \\subset X_{U'}$ the triple $(U', u', V')$", "satisfies the properties", "(\\romannumeral1), (\\romannumeral2), and (\\romannumeral3) as well." ], "refs": [ "morphisms-lemma-quasi-finite-at-point-characterize" ], "proofs": [ { "contents": [ "Let $Y \\subset X$, $W \\subset S$ be affine opens such that", "$f(Y) \\subset W$ and such that $x \\in Y$. Note that $x$ is", "also an isolated point of the fibre of the morphism $f|_Y : Y \\to W$.", "If we can prove the theorem for $f|_Y : Y \\to W$, then the", "theorem follows for $f$. Hence we reduce to the case where", "$f$ is a morphism of affine schemes. This case is", "Algebra, Lemma \\ref{algebra-lemma-etale-makes-quasi-finite-finite-one-prime}." ], "refs": [ "algebra-lemma-etale-makes-quasi-finite-finite-one-prime" ], "ref_ids": [ 1246 ] } ], "ref_ids": [ 5226 ] }, { "id": 13893, "type": "theorem", "label": "more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite type, and", "\\item $x_i \\in X_s$ is isolated for $i = 1, \\ldots, n$.", "\\end{enumerate}", "Then there exist", "\\begin{enumerate}", "\\item[(a)] an elementary \\'etale neighbourhood $(U, u) \\to (S, s)$,", "\\item[(b)] for each $i$ an open subscheme $V_i \\subset X_U$,", "\\end{enumerate}", "such that for each $i$ we have", "\\begin{enumerate}", "\\item[(\\romannumeral1)] $V_i \\to U$ is a finite morphism,", "\\item[(\\romannumeral2)] there is a unique point $v_i$ of $V_i$", "mapping to $u$ in $U$, and", "\\item[(\\romannumeral3)] the point $v_i$ maps to $x_i$ in $X$ and", "$\\kappa(x_i) = \\kappa(v_i)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will use induction on $n$.", "Namely, suppose $(U, u) \\to (S, s)$ and $V_i \\subset X_U$,", "$i = 1, \\ldots, n - 1$ work for $x_1, \\ldots, x_{n - 1}$. Since", "$\\kappa(s) = \\kappa(u)$ the fibre $(X_U)_u = X_s$. Hence there", "exists a unique point $x'_n \\in X_u \\subset X_U$ corresponding to", "$x_n \\in X_s$. Also $x'_n$ is isolated in $X_u$. Hence by", "Lemma \\ref{lemma-etale-makes-quasi-finite-finite-at-point} there", "exists an elementary \\'etale neighbourhood $(U', u') \\to (U, u)$", "and an open $V_n \\subset X_{U'}$ which works for $x'_n$ and hence", "for $x_n$.", "By the final assertion of", "Lemma \\ref{lemma-etale-makes-quasi-finite-finite-at-point}", "the open subschemes $V'_i = U'\\times_U V_i$ for $i = 1, \\ldots, n - 1$ still", "work with respect to $x_1, \\ldots, x_{n - 1}$. Hence we win." ], "refs": [ "more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point", "more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point" ], "ref_ids": [ 13892, 13892 ] } ], "ref_ids": [] }, { "id": 13894, "type": "theorem", "label": "more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points-var", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points-var", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$.", "Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite type, and", "\\item $x_i \\in X_s$ is isolated for $i = 1, \\ldots, n$.", "\\end{enumerate}", "Then there exist", "\\begin{enumerate}", "\\item[(a)] an \\'etale neighbourhood $(U, u) \\to (S, s)$,", "\\item[(b)] for each $i$ an integer $m_i$ and", "open subschemes $V_{i, j} \\subset X_U$, $j = 1, \\ldots, m_i$", "\\end{enumerate}", "such that we have", "\\begin{enumerate}", "\\item[(\\romannumeral1)] each $V_{i, j} \\to U$ is a finite morphism,", "\\item[(\\romannumeral2)] there is a unique point $v_{i, j}$ of $V_{i, j}$", "mapping to $u$ in $U$ with $\\kappa(u) \\subset \\kappa(v_{i, j})$", "finite purely inseparable,", "\\item[(\\romannumeral4)] if $v_{i, j} = v_{i', j'}$, then $i = i'$ and", "$j = j'$, and", "\\item[(\\romannumeral3)] the points $v_{i, j}$ map to $x_i$ in $X$ and", "no other points of $(X_U)_u$ map to $x_i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This proof is a variant of the proof of", "Algebra, Lemma \\ref{algebra-lemma-etale-makes-quasi-finite-finite-variant}", "in the language of schemes.", "By Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}", "the morphism $f$ is quasi-finite at each of the points $x_i$.", "Hence $\\kappa(s) \\subset \\kappa(x_i)$ is finite for each $i$", "(Morphisms, Lemma \\ref{morphisms-lemma-residue-field-quasi-finite}).", "For each $i$, let $\\kappa(s) \\subset L_i \\subset \\kappa(x_i)$", "be the subfield such that $L_i/\\kappa(s)$ is separable, and", "$\\kappa(x_i)/L_i$ is purely inseparable. Choose a finite Galois", "extension $\\kappa(s) \\subset L$ such that there exist", "$\\kappa(s)$-embeddings $L_i \\to L$ for $i = 1, \\ldots, n$.", "Choose an \\'etale neighbourhood $(U, u) \\to (S, s)$ such that", "$L \\cong \\kappa(u)$ as $\\kappa(s)$-extensions", "(Lemma \\ref{lemma-realize-prescribed-residue-field-extension-etale}).", "\\medskip\\noindent", "Let $y_{i, j}$, $j = 1, \\ldots, m_i$ be the points of $X_U$", "lying over $x_i \\in X$ and $u \\in U$. By", "Schemes, Lemma \\ref{schemes-lemma-points-fibre-product}", "these points $y_{i, j}$ correspond exactly to the primes in the rings", "$\\kappa(u) \\otimes_{\\kappa(s)} \\kappa(x_i)$. This also", "explains why there are finitely many; in fact", "$m_i = [L_i : \\kappa(s)]$ but we do not need this.", "By our choice of", "$L$ (and elementary field theory)", "we see that $\\kappa(u) \\subset \\kappa(y_{i, j})$ is", "finite purely inseparable for each pair $i, j$.", "Also, by Morphisms, Lemma \\ref{morphisms-lemma-base-change-quasi-finite}", "for example, the morphism", "$X_U \\to U$ is quasi-finite at the points $y_{i, j}$ for", "all $i, j$.", "\\medskip\\noindent", "Apply Lemma \\ref{lemma-etale-makes-quasi-finite-finite-multiple-points}", "to the morphism $X_U \\to U$, the point $u \\in U$", "and the points $y_{i, j} \\in (X_U)_u$. This gives an \\'etale neighbourhood", "$(U', u') \\to (U, u)$ with $\\kappa(u) = \\kappa(u')$ and", "opens $V_{i, j} \\subset X_{U'}$ with the properties", "(\\romannumeral1), (\\romannumeral2), and (\\romannumeral3)", "of that lemma. We claim that the \\'etale neighbourhood", "$(U', u') \\to (S, s)$ and the opens $V_{i, j} \\subset X_{U'}$", "are a solution to the problem posed by the lemma.", "We omit the verifications." ], "refs": [ "algebra-lemma-etale-makes-quasi-finite-finite-variant", "morphisms-lemma-quasi-finite-at-point-characterize", "morphisms-lemma-residue-field-quasi-finite", "more-morphisms-lemma-realize-prescribed-residue-field-extension-etale", "schemes-lemma-points-fibre-product", "morphisms-lemma-base-change-quasi-finite", "more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points" ], "ref_ids": [ 1248, 5226, 5225, 13866, 7693, 5233, 13893 ] } ], "ref_ids": [] }, { "id": 13895, "type": "theorem", "label": "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $s \\in S$. Let $x_1, \\ldots, x_n \\in X_s$. Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite type,", "\\item $f$ is separated, and", "\\item $x_1, \\ldots, x_n$ are pairwise distinct isolated points of $X_s$.", "\\end{enumerate}", "Then there exists an elementary \\'etale neighbourhood $(U, u) \\to (S, s)$", "and a decomposition", "$$", "U \\times_S X = W \\amalg V_1 \\amalg \\ldots \\amalg V_n", "$$", "into open and closed subschemes such that the morphisms", "$V_i \\to U$ are finite, the fibres of $V_i \\to U$ over $u$ are", "singletons $\\{v_i\\}$, each $v_i$ maps to $x_i$ with", "$\\kappa(x_i) = \\kappa(v_i)$, and the fibre of $W \\to U$", "over $u$ contains no points mapping to any of the $x_i$." ], "refs": [], "proofs": [ { "contents": [ "Choose $(U, u) \\to (S, s)$ and $V_i \\subset X_U$ as in", "Lemma \\ref{lemma-etale-makes-quasi-finite-finite-multiple-points}.", "Since $X_U \\to U$ is separated", "(Schemes, Lemma \\ref{schemes-lemma-separated-permanence})", "and $V_i \\to U$ is finite hence proper", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-proper})", "we see that $V_i \\subset X_U$ is closed by", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}.", "Hence $V_i \\cap V_j$ is a closed subset of $V_i$ which", "does not contain $v_i$. Hence the image of $V_i \\cap V_j$", "in $U$ is a closed set (because $V_i \\to U$ proper) not", "containing $u$. After shrinking $U$ we may therefore assume", "that $V_i \\cap V_j = \\emptyset$ for all $i, j$. This gives the", "decomposition as in the lemma." ], "refs": [ "more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points", "schemes-lemma-separated-permanence", "morphisms-lemma-finite-proper", "morphisms-lemma-image-proper-scheme-closed" ], "ref_ids": [ 13893, 7714, 5445, 5411 ] } ], "ref_ids": [] }, { "id": 13896, "type": "theorem", "label": "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $s \\in S$. Let $x_1, \\ldots, x_n \\in X_s$. Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite type,", "\\item $f$ is separated, and", "\\item $x_1, \\ldots, x_n$ are pairwise distinct isolated points of $X_s$.", "\\end{enumerate}", "Then there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$", "and a decomposition", "$$", "U \\times_S X =", "W \\amalg", "\\ \\coprod\\nolimits_{i = 1, \\ldots, n}", "\\ \\coprod\\nolimits_{j = 1, \\ldots, m_i}", "V_{i, j}", "$$", "into open and closed subschemes such that the morphisms", "$V_{i, j} \\to U$ are finite, the fibres of $V_{i, j} \\to U$ over $u$ are", "singletons $\\{v_{i, j}\\}$, each $v_{i, j}$ maps to $x_i$,", "$\\kappa(u) \\subset \\kappa(v_{i, j})$ is purely inseparable,", "and the fibre of $W \\to U$ over $u$ contains no points mapping", "to any of the $x_i$." ], "refs": [], "proofs": [ { "contents": [ "This is proved in exactly the same way as the proof of", "Lemma \\ref{lemma-etale-splits-off-quasi-finite-part-technical} except that it", "uses Lemma \\ref{lemma-etale-makes-quasi-finite-finite-multiple-points-var}", "instead of Lemma \\ref{lemma-etale-makes-quasi-finite-finite-multiple-points}." ], "refs": [ "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical", "more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points-var", "more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points" ], "ref_ids": [ 13895, 13894, 13893 ] } ], "ref_ids": [] }, { "id": 13897, "type": "theorem", "label": "more-morphisms-lemma-etale-splits-off-quasi-finite-part", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-etale-splits-off-quasi-finite-part", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $s \\in S$. Assume that", "\\begin{enumerate}", "\\item $f$ is locally of finite type,", "\\item $f$ is separated, and", "\\item $X_s$ has at most finitely many isolated points.", "\\end{enumerate}", "Then there exists an elementary \\'etale neighbourhood $(U, u) \\to (S, s)$", "and a decomposition", "$$", "U \\times_S X = W \\amalg V", "$$", "into open and closed subschemes such that the morphism", "$V \\to U$ is finite, and the fibre $W_u$ of the", "morphism $W \\to U$ contains no isolated points.", "In particular, if $f^{-1}(s)$ is a finite set, then $W_u = \\emptyset$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from", "Lemma \\ref{lemma-etale-splits-off-quasi-finite-part-technical}", "by choosing $x_1, \\ldots, x_n$ the complete set of", "isolated points of $X_s$ and setting $V = \\bigcup V_i$." ], "refs": [ "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical" ], "ref_ids": [ 13895 ] } ], "ref_ids": [] }, { "id": 13898, "type": "theorem", "label": "more-morphisms-lemma-etale-makes-integral-split", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-etale-makes-integral-split", "contents": [ "Let $R \\to S$ be an integral ring map. Let $\\mathfrak p \\subset R$ be a prime", "ideal. Assume", "\\begin{enumerate}", "\\item there are finitely many primes $\\mathfrak q_1, \\ldots, \\mathfrak q_n$", "lying over $\\mathfrak p$, and", "\\item for each $i$ the maximal separable subextension", "$\\kappa(\\mathfrak q)/\\kappa(\\mathfrak q_i)_{sep}/\\kappa(\\mathfrak p)$", "(Fields, Lemma \\ref{fields-lemma-separable-first})", "is finite over $\\kappa(\\mathfrak p)$.", "\\end{enumerate}", "Then there exists an \\'etale ring map $R \\to R'$ and a prime", "$\\mathfrak p'$ lying over $\\mathfrak p$ such that", "$$", "S \\otimes_R R' = A_1 \\times \\ldots \\times A_m", "$$", "with $R' \\to A_j$ integral having a unique prime $\\mathfrak r_j$", "over $\\mathfrak p'$ such that $\\kappa(\\mathfrak r_j)/\\kappa(\\mathfrak p')$", "is purely inseparable." ], "refs": [ "fields-lemma-separable-first" ], "proofs": [ { "contents": [ "[First proof]", "This proof uses", "Algebra, Lemma \\ref{algebra-lemma-etale-makes-quasi-finite-finite-variant}.", "Namely, choose a generator $\\theta_i \\in \\kappa(\\mathfrak q_i)_{sep}$", "of this field over $\\kappa(\\mathfrak p)$", "(Fields, Lemma \\ref{fields-lemma-primitive-element}).", "The spectrum of the fibre ring $S \\otimes_R \\kappa(\\mathfrak p)$", "is finite discrete with points corresponding to", "$\\mathfrak q_1, \\ldots, \\mathfrak q_n$.", "By the Chinese remainder theorem", "(Algebra, Lemma \\ref{algebra-lemma-chinese-remainder})", "we see that $S \\otimes_R \\kappa(\\mathfrak p) \\to \\prod \\kappa(\\mathfrak q_i)$", "is surjective. Hence after replacing $R$ by $R_g$ for some", "$g \\in R$, $g \\not \\in \\mathfrak p$ we may assume that", "$(0, \\ldots, 0, \\theta_i, 0, \\ldots, 0) \\in \\prod \\kappa(\\mathfrak q_i)$", "is the image of some $x_i \\in S$.", "Let $S' \\subset S$ be the $R$-subalgebra generated by our $x_i$.", "Since $\\Spec(S) \\to \\Spec(S')$ is surjective", "(Algebra, Lemma \\ref{algebra-lemma-integral-overring-surjective})", "we conclude that", "$\\mathfrak q_i' = S' \\cap \\mathfrak q_i$ are the primes of", "$S'$ over $\\mathfrak p$. By our choice of $x_i$ we conclude", "these primes are distinct that and", "$\\kappa(\\mathfrak q'_i)_{sep} = \\kappa(\\mathfrak q_i)_{sep}$.", "In particular the field extensions", "$\\kappa(\\mathfrak q_i)/\\kappa(\\mathfrak q'_i)$ are purely", "inseparable.", "Since $R \\to S'$ is finite we may apply", "Algebra, Lemma \\ref{algebra-lemma-etale-makes-quasi-finite-finite-variant}.", "and we get $R \\to R'$ and $\\mathfrak p'$ and a decomposition", "$$", "S' \\otimes_R R' = A'_1 \\times \\ldots \\times A'_m \\times B'", "$$", "with $R' \\to A'_j$ integral having a unique prime", "$\\mathfrak r'_j$ over $\\mathfrak p'$ such that", "$\\kappa(\\mathfrak r'_j)/\\kappa(\\mathfrak p')$", "is purely inseparable and such that $B'$ does not have a prime", "lying over $\\mathfrak p'$. Since $R' \\to B'$ is finite", "(as $R \\to S'$ is finite) we can", "after localizing $R'$ at some $g' \\in R'$, $g' \\not \\in \\mathfrak p'$", "assume that $B' = 0$. Via the map $S' \\otimes_R R' \\to S \\otimes_R R'$", "we get the corresponding decomposition for $S$." ], "refs": [ "algebra-lemma-etale-makes-quasi-finite-finite-variant", "fields-lemma-primitive-element", "algebra-lemma-chinese-remainder", "algebra-lemma-integral-overring-surjective", "algebra-lemma-etale-makes-quasi-finite-finite-variant" ], "ref_ids": [ 1248, 4498, 380, 495, 1248 ] } ], "ref_ids": [ 4482 ] }, { "id": 13899, "type": "theorem", "label": "more-morphisms-lemma-finite-type-separated", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-finite-type-separated", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume $f$ is of finite type and separated.", "Let $S'$ be the normalization of $S$ in $X$, see", "Morphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}.", "Picture:", "$$", "\\xymatrix{", "X \\ar[rd]_f \\ar[rr]_{f'} & & S' \\ar[ld]^\\nu \\\\", "& S &", "}", "$$", "Then there exists an open subscheme $U' \\subset S'$ such that", "\\begin{enumerate}", "\\item $(f')^{-1}(U') \\to U'$ is an isomorphism, and", "\\item $(f')^{-1}(U') \\subset X$ is the set of points at which", "$f$ is quasi-finite.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-normalization-X-in-Y" ], "proofs": [ { "contents": [ "By Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-points-open}", "the subset $U \\subset X$ of points where $f$ is quasi-finite is open.", "The lemma is equivalent to", "\\begin{enumerate}", "\\item[(a)] $U' = f'(U) \\subset S'$ is open,", "\\item[(b)] $U = f^{-1}(U')$, and", "\\item[(c)] $U \\to U'$ is an isomorphism.", "\\end{enumerate}", "Let $x \\in U$ be arbitrary. We claim there exists an open", "neighbourhood $f'(x) \\in V \\subset S'$ such that $(f')^{-1}V \\to V$ is an", "isomorphism. We first prove the claim implies the lemma.", "Namely, then $(f')^{-1}V \\cong V$ is both locally of finite", "type over $S$ (as an open subscheme of $X$) and for $v \\in V$ the residue", "field extension $\\kappa(v) \\supset \\kappa(\\nu(v))$ is algebraic (as", "$V \\subset S'$ and $S'$ is integral over $S$). Hence the fibres", "of $V \\to S$ are discrete (Morphisms, Lemma", "\\ref{morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre})", "and $(f')^{-1}V \\to S$ is locally quasi-finite", "(Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-fibres}).", "This implies $(f')^{-1}V \\subset U$ and $V \\subset U'$. Since $x$ was", "arbitrary we see that (a), (b), and (c) are true.", "\\medskip\\noindent", "Let $s = f(x)$. Let $(T, t) \\to (S, s)$ be an elementary \\'etale", "neighbourhood. Denote by a subscript ${}_T$ the base change to $T$.", "Let $y = (x, t) \\in X_T$ be the unique point in", "the fibre $X_t$ lying over $x$. Note that $U_T \\subset X_T$", "is the set of points where $f_T$ is quasi-finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-quasi-finite}.", "Note that", "$$", "X_T \\xrightarrow{f'_T} S'_T \\xrightarrow{\\nu_T} T", "$$", "is the normalization of $T$ in $X_T$, see", "Lemma \\ref{lemma-normalization-smooth-localization}.", "Suppose that the claim holds for $y \\in U_T \\subset X_T \\to S'_T \\to T$, i.e.,", "suppose that we can find an open neighbourhood", "$f'_T(y) \\in V' \\subset S'_T$ such that $(f'_T)^{-1}V' \\to V'$ is an", "isomorphism. The morphism $S'_T \\to S'$ is \\'etale hence the image", "$V \\subset S'$ of $V'$ is open. Observe that $f'(x) \\in V$ as $f'_T(y) \\in V'$.", "Observe that", "$$", "\\xymatrix{", "(f'_T)^{-1}V' \\ar[r] \\ar[d] & (f')^{-1}(V) \\ar[d] \\\\", "V' \\ar[r] & V", "}", "$$", "is a fibre square (as $S'_T \\times_{S'} X = X_T$).", "Since the left vertical arrow is an isomorphism", "and $\\{V' \\to V\\}$ is a \\'etale covering, we conclude that the right vertical", "arrow is an isomorphism by", "Descent, Lemma \\ref{descent-lemma-descending-property-isomorphism}.", "In other words, the claim holds for $x \\in U \\subset X \\to S' \\to S$.", "\\medskip\\noindent", "By the result of the previous paragraph we may replace $S$ by an", "elementary \\'etale neighbourhood of $s = f(x)$ in order to prove the claim.", "Thus we may assume there is a decomposition", "$$", "X = V \\amalg W", "$$", "into open and closed subschemes where $V \\to S$ is finite and $x \\in V$,", "see Lemma \\ref{lemma-etale-splits-off-quasi-finite-part-technical}.", "Since $X$ is a disjoint union of $V$ and $W$ over $S$ and since", "$V \\to S$ is finite we see that the", "normalization of $S$ in $X$ is the morphism", "$$", "X = V \\amalg W \\longrightarrow V \\amalg W' \\longrightarrow S", "$$", "where $W'$ is the normalization of $S$ in $W$, see", "Morphisms, Lemmas \\ref{morphisms-lemma-normalization-in-disjoint-union},", "\\ref{morphisms-lemma-finite-integral}, and", "\\ref{morphisms-lemma-normalization-in-integral}.", "The claim follows and we win." ], "refs": [ "morphisms-lemma-quasi-finite-points-open", "morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre", "morphisms-lemma-locally-quasi-finite-fibres", "morphisms-lemma-base-change-quasi-finite", "more-morphisms-lemma-normalization-smooth-localization", "descent-lemma-descending-property-isomorphism", "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical", "morphisms-lemma-normalization-in-disjoint-union", "morphisms-lemma-finite-integral", "morphisms-lemma-normalization-in-integral" ], "ref_ids": [ 5521, 5222, 5228, 5233, 13774, 14682, 13895, 5505, 5438, 5507 ] } ], "ref_ids": [ 5591 ] }, { "id": 13900, "type": "theorem", "label": "more-morphisms-lemma-quasi-finite-separated-quasi-affine", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-quasi-finite-separated-quasi-affine", "contents": [ "\\begin{slogan}", "Quasi-finite, separated morphisms are quasi-affine", "\\end{slogan}", "Let $f : X \\to S$ be a morphism of schemes.", "Assume $f$ is quasi-finite and separated.", "Let $S'$ be the normalization of $S$ in $X$, see", "Morphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}.", "Picture:", "$$", "\\xymatrix{", "X \\ar[rd]_f \\ar[rr]_{f'} & & S' \\ar[ld]^\\nu \\\\", "& S &", "}", "$$", "Then $f'$ is a quasi-compact open immersion and $\\nu$ is integral.", "In particular $f$ is quasi-affine." ], "refs": [ "morphisms-definition-normalization-X-in-Y" ], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-finite-type-separated}. Namely, by", "that lemma there exists an open subscheme $U' \\subset S'$ such that", "$(f')^{-1}(U') = X$ and $X \\to U'$ is an isomorphism. In other", "words, $f'$ is an open immersion. Note that $f'$ is quasi-compact as", "$f$ is quasi-compact and $\\nu : S' \\to S$ is separated", "(Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}).", "It follows that $f$ is quasi-affine by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-quasi-affine}." ], "refs": [ "more-morphisms-lemma-finite-type-separated", "schemes-lemma-quasi-compact-permanence", "morphisms-lemma-characterize-quasi-affine" ], "ref_ids": [ 13899, 7716, 5185 ] } ], "ref_ids": [ 5591 ] }, { "id": 13901, "type": "theorem", "label": "more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "contents": [ "\\begin{reference}", "\\cite[IV Corollary 18.12.13]{EGA}", "\\end{reference}", "Let $f : X \\to S$ be a morphism of schemes.", "Assume $f$ is quasi-finite and separated and assume that", "$S$ is quasi-compact and quasi-separated. Then there exists", "a factorization", "$$", "\\xymatrix{", "X \\ar[rd]_f \\ar[rr]_j & & T \\ar[ld]^\\pi \\\\", "& S &", "}", "$$", "where $j$ is a quasi-compact open immersion and $\\pi$ is finite." ], "refs": [], "proofs": [ { "contents": [ "Let $X \\to S' \\to S$ be as in the conclusion of", "Lemma \\ref{lemma-quasi-finite-separated-quasi-affine}.", "By", "Properties, Lemma", "\\ref{properties-lemma-integral-algebra-directed-colimit-finite}", "we can write", "$\\nu_*\\mathcal{O}_{S'} = \\colim_{i \\in I} \\mathcal{A}_i$ as a", "directed colimit of finite quasi-coherent $\\mathcal{O}_X$-algebras", "$\\mathcal{A}_i \\subset \\nu_*\\mathcal{O}_{S'}$. Then", "$\\pi_i : T_i = \\underline{\\Spec}_S(\\mathcal{A}_i) \\to S$", "is a finite morphism for each $i$.", "Note that the transition morphisms $T_{i'} \\to T_i$ are affine", "and that $S' = \\lim T_i$.", "\\medskip\\noindent", "By Limits, Lemma \\ref{limits-lemma-descend-opens}", "there exists an $i$ and a quasi-compact open", "$U_i \\subset T_i$ whose inverse image in $S'$ equals", "$f'(X)$. For $i' \\geq i$ let $U_{i'}$ be the inverse image", "of $U_i$ in $T_{i'}$. Then $X \\cong f'(X) = \\lim_{i' \\geq i} U_{i'}$, see", "Limits, Lemma \\ref{limits-lemma-directed-inverse-system-has-limit}.", "By Limits, Lemma \\ref{limits-lemma-finite-type-eventually-closed} we see that", "$X \\to U_{i'}$ is a closed immersion for some $i' \\geq i$.", "(In fact $X \\cong U_{i'}$ for sufficiently", "large $i'$ but we don't need this.) Hence $X \\to T_{i'}$ is an immersion. By", "Morphisms, Lemma \\ref{morphisms-lemma-factor-quasi-compact-immersion}", "we can factor this as $X \\to T \\to T_{i'}$ where the first arrow", "is an open immersion and the second a closed immersion. Thus we win." ], "refs": [ "more-morphisms-lemma-quasi-finite-separated-quasi-affine", "properties-lemma-integral-algebra-directed-colimit-finite", "limits-lemma-descend-opens", "limits-lemma-directed-inverse-system-has-limit", "limits-lemma-finite-type-eventually-closed", "morphisms-lemma-factor-quasi-compact-immersion" ], "ref_ids": [ 13900, 3030, 15041, 15027, 15046, 5133 ] } ], "ref_ids": [] }, { "id": 13902, "type": "theorem", "label": "more-morphisms-lemma-quasi-finite-separated-pass-through-finite-addendum", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-quasi-finite-separated-pass-through-finite-addendum", "contents": [ "With notation and hypotheses as in", "Lemma \\ref{lemma-quasi-finite-separated-pass-through-finite}.", "Assume moreover that $f$ is locally of finite presentation. Then we can", "choose the factorization such that $T$ is finite and of", "finite presentation over $S$." ], "refs": [ "more-morphisms-lemma-quasi-finite-separated-pass-through-finite" ], "proofs": [ { "contents": [ "By Limits, Lemma", "\\ref{limits-lemma-finite-in-finite-and-finite-presentation} we can write", "$T = \\lim T_i$ where all $T_i$ are finite and of finite presentation", "over $Y$ and the transition morphisms $T_{i'} \\to T_i$ are closed", "immersions. By", "Limits, Lemma \\ref{limits-lemma-descend-opens}", "there exists an $i$ and an open subscheme $U_i \\subset T_i$ whose inverse", "image in $T$ is $X$. By", "Limits, Lemma", "\\ref{limits-lemma-finite-type-eventually-closed}", "we see that $X \\cong U_i$ for large enough $i$.", "Replacing $T$ by $T_i$ finishes the proof." ], "refs": [ "limits-lemma-finite-in-finite-and-finite-presentation", "limits-lemma-descend-opens", "limits-lemma-finite-type-eventually-closed" ], "ref_ids": [ 15076, 15041, 15046 ] } ], "ref_ids": [ 13901 ] }, { "id": 13903, "type": "theorem", "label": "more-morphisms-lemma-characterize-finite", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-characterize-finite", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item $f$ is finite,", "\\item $f$ is proper with finite fibres,", "\\item $f$ is proper and locally quasi-finite,", "\\item $f$ is universally closed, separated, locally of finite type", "and has finite fibres.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have (1) implies (2) by", "Morphisms, Lemmas \\ref{morphisms-lemma-finite-proper},", "\\ref{morphisms-lemma-quasi-finite},", "and \\ref{morphisms-lemma-finite-quasi-finite}.", "We have (2) implies (3) by Morphisms, Lemma \\ref{morphisms-lemma-finite-fibre}.", "We have (3) implies (4) by the definition of proper morphisms and", "Morphisms, Lemmas \\ref{morphisms-lemma-quasi-finite-locally-quasi-compact} and", "\\ref{morphisms-lemma-quasi-finite}.", "\\medskip\\noindent", "Assume (4). Pick $s \\in S$. By", "Morphisms, Lemma \\ref{morphisms-lemma-finite-fibre} we", "see that all the finitely many points of $X_s$ are isolated in $X_s$.", "Choose an elementary \\'etale neighbourhood $(U, u) \\to (S, s)$", "and decomposition $X_U = V \\amalg W$ as in", "Lemma \\ref{lemma-etale-splits-off-quasi-finite-part}.", "Note that $W_u = \\emptyset$ because all points of $X_s$ are isolated.", "Since $f$ is universally closed we see that", "the image of $W$ in $U$ is a closed set not containing $u$.", "After shrinking $U$ we may assume that $W = \\emptyset$.", "In other words we see that $X_U = V$ is finite over $U$.", "Since $s \\in S$ was arbitrary", "this means there exists a family $\\{U_i \\to S\\}$", "of \\'etale morphisms whose images cover $S$ such that", "the base changes $X_{U_i} \\to U_i$ are finite.", "Note that $\\{U_i \\to S\\}$ is an \\'etale covering,", "see Topologies, Definition \\ref{topologies-definition-etale-covering}.", "Hence it is an fpqc covering, see", "Topologies,", "Lemma \\ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc}.", "Hence we conclude $f$ is finite by", "Descent, Lemma \\ref{descent-lemma-descending-property-finite}." ], "refs": [ "morphisms-lemma-finite-proper", "morphisms-lemma-quasi-finite", "morphisms-lemma-finite-quasi-finite", "morphisms-lemma-finite-fibre", "morphisms-lemma-quasi-finite-locally-quasi-compact", "morphisms-lemma-quasi-finite", "morphisms-lemma-finite-fibre", "more-morphisms-lemma-etale-splits-off-quasi-finite-part", "topologies-definition-etale-covering", "topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc", "descent-lemma-descending-property-finite" ], "ref_ids": [ 5445, 5230, 5444, 5227, 5229, 5230, 5227, 13897, 12526, 12497, 14688 ] } ], "ref_ids": [] }, { "id": 13904, "type": "theorem", "label": "more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $s \\in S$.", "Assume that $f$ is proper and $f^{-1}(\\{s\\})$ is a finite set.", "Then there exists an open neighbourhood $V \\subset S$ of $s$", "such that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite." ], "refs": [], "proofs": [ { "contents": [ "The morphism $f$ is quasi-finite at all the points of $f^{-1}(\\{s\\})$", "by Morphisms, Lemma \\ref{morphisms-lemma-finite-fibre}.", "By Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-points-open} the", "set of points at which $f$ is quasi-finite is an open $U \\subset X$.", "Let $Z = X \\setminus U$. Then $s \\not \\in f(Z)$. Since $f$ is proper", "the set $f(Z) \\subset S$ is closed. Choose any open neighbourhood", "$V \\subset S$ of $s$ with $f(Z) \\cap V = \\emptyset$. Then", "$f^{-1}(V) \\to V$ is locally quasi-finite and proper.", "Hence it is quasi-finite", "(Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-locally-quasi-compact}),", "hence has finite fibres", "(Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite}), hence", "is finite by Lemma \\ref{lemma-characterize-finite}." ], "refs": [ "morphisms-lemma-finite-fibre", "morphisms-lemma-quasi-finite-points-open", "morphisms-lemma-quasi-finite-locally-quasi-compact", "morphisms-lemma-quasi-finite", "more-morphisms-lemma-characterize-finite" ], "ref_ids": [ 5227, 5521, 5229, 5230, 13903 ] } ], "ref_ids": [] }, { "id": 13905, "type": "theorem", "label": "more-morphisms-lemma-flat-proper-family-cannot-collapse-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-proper-family-cannot-collapse-fibre", "contents": [ "Consider a commutative diagram of schemes", "$$", "\\xymatrix{", "X \\ar[rr]_h \\ar[rd]_f & & Y \\ar[ld]^g \\\\", "& S", "}", "$$", "Let $s \\in S$. Assume", "\\begin{enumerate}", "\\item $X \\to S$ is a proper morphism,", "\\item $Y \\to S$ is separated and locally of finite type, and", "\\item the image of $X_s \\to Y_s$ is finite.", "\\end{enumerate}", "Then there is an open", "subspace $U \\subset S$ containing $s$ such that $X_U \\to Y_U$", "factors through a closed subscheme $Z \\subset Y_U$ finite over $U$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z \\subset Y$ be the scheme theoretic image of $h$, see", "Morphisms, Section \\ref{morphisms-section-scheme-theoretic-image}.", "By Morphisms, Lemma \\ref{morphisms-lemma-scheme-theoretic-image-is-proper}", "the morphism $X \\to Z$ is surjective and $Z \\to S$ is proper.", "Thus $X_s \\to Z_s$ is surjective. We see that either", "(3) implies $Z_s$ is finite.", "Hence $Z \\to S$ is finite in an open neighbourhood of $s$ by", "Lemma \\ref{lemma-proper-finite-fibre-finite-in-neighbourhood}." ], "refs": [ "morphisms-lemma-scheme-theoretic-image-is-proper", "more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood" ], "ref_ids": [ 5414, 13904 ] } ], "ref_ids": [] }, { "id": 13906, "type": "theorem", "label": "more-morphisms-lemma-separated-locally-quasi-finite-over-affine", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-separated-locally-quasi-finite-over-affine", "contents": [ "Let $f : X \\to Y$ be a separated, locally quasi-finite morphism", "with $Y$ affine. Then every finite set of points of $X$ is contained", "in an open affine of $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $x_1, \\ldots, x_n \\in X$. Choose a quasi-compact open", "$U \\subset X$ with $x_i \\in U$. Then $U \\to Y$ is quasi-affine by", "Lemma \\ref{lemma-quasi-finite-separated-quasi-affine}.", "Hence there exists an affine open $V \\subset U$ containing", "$x_1, \\ldots, x_n$ by", "Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-affine}." ], "refs": [ "more-morphisms-lemma-quasi-finite-separated-quasi-affine", "properties-lemma-ample-finite-set-in-affine" ], "ref_ids": [ 13900, 3062 ] } ], "ref_ids": [] }, { "id": 13907, "type": "theorem", "label": "more-morphisms-lemma-quasi-finite-finite-over-dense-open", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-quasi-finite-finite-over-dense-open", "contents": [ "Let $f : Y \\to X$ be a quasi-finite morphism.", "There exists a dense open $U \\subset X$ such that", "$f|_{f^{-1}(U)} : f^{-1}(U) \\to U$ is finite." ], "refs": [], "proofs": [ { "contents": [ "If $U_i \\subset X$, $i \\in I$ is a collection of opens such that the", "restrictions $f|_{f^{-1}(U_i)} : f^{-1}(U_i) \\to U_i$ are finite,", "then with $U = \\bigcup U_i$ the restriction $f|_{f^{-1}(U)} : f^{-1}(U) \\to U$", "is finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-local}.", "Thus the problem is local on $X$ and we may assume that $X$ is affine.", "\\medskip\\noindent", "Assume $X$ is affine.", "Write $Y = \\bigcup_{j = 1, \\ldots, m} V_j$ with $V_j$ affine.", "This is possible since $f$ is quasi-finite and hence", "in particular quasi-compact. Each $V_j \\to X$ is quasi-finite", "and separated. Let $\\eta \\in X$ be a generic point of an irreducible", "component of $X$. We see from", "Morphisms, Lemmas", "\\ref{morphisms-lemma-quasi-finite} and \\ref{morphisms-lemma-generically-finite}", "that there exists an open neighbourhood $\\eta \\in U_\\eta$ such that", "$f^{-1}(U_\\eta) \\cap V_j \\to U_\\eta$ is finite. We may choose $U_\\eta$ such", "that it works for each $j = 1, \\ldots, m$.", "Note that the collection of generic points of $X$ is dense in $X$.", "Thus we see there exists a dense open $W = \\bigcup_\\eta U_\\eta$", "such that each $f^{-1}(W) \\cap V_j \\to W$ is finite.", "It suffices to show that there exists a dense open $U \\subset W$", "such that $f|_{f^{-1}(U)} : f^{-1}(U) \\to U$ is finite.", "Thus we may replace $X$ by an affine open subscheme of $W$ and", "assume that each $V_j \\to X$ is finite.", "\\medskip\\noindent", "Assume $X$ is affine, $Y = \\bigcup_{j = 1, \\ldots, m} V_j$ with $V_j$ affine,", "and the restrictions $f|_{V_j} : V_j \\to X$ are finite.", "Set", "$$", "\\Delta_{ij} =", "\\Big(\\overline{V_i \\cap V_j} \\setminus V_i \\cap V_j\\Big) \\cap V_j.", "$$", "This is a nowhere dense closed subset of $V_j$ because it is the boundary", "of the open subset $V_i \\cap V_j$ in $V_j$. By", "Morphisms, Lemma \\ref{morphisms-lemma-image-nowhere-dense-finite}", "the image $f(\\Delta_{ij})$ is a nowhere dense closed subset of $X$. By", "Topology, Lemma \\ref{topology-lemma-nowhere-dense}", "the union $T = \\bigcup f(\\Delta_{ij})$ is a nowhere dense closed", "subset of $X$. Thus $U = X \\setminus T$ is a dense open subset of $X$.", "We claim that $f|_{f^{-1}(U)} : f^{-1}(U) \\to U$ is finite.", "To see this let $U' \\subset U$ be an affine open.", "Set $Y' = f^{-1}(U') = U' \\times_X Y$,", "$V_j' = Y' \\cap V_j = U' \\times_X V_j$. Consider the restriction", "$$", "f' = f|_{Y'} : Y' \\longrightarrow U'", "$$", "of $f$. This morphism now has the property that", "$Y' = \\bigcup_{j = 1, \\ldots, m} V'_j$ is an affine open covering,", "each $V'_j \\to U'$ is finite, and $V_i' \\cap V_j'$ is (open and) closed", "both in $V'_i$ and $V'_j$. Hence $V_i' \\cap V_j'$ is affine, and the map", "$$", "\\mathcal{O}(V'_i) \\otimes_{\\mathbf{Z}} \\mathcal{O}(V'_j)", "\\longrightarrow", "\\mathcal{O}(V'_i \\cap V'_j)", "$$", "is surjective. This implies that $Y'$ is separated, see", "Schemes, Lemma \\ref{schemes-lemma-characterize-separated}.", "Finally, consider the commutative diagram", "$$", "\\xymatrix{", "\\coprod_{j = 1, \\ldots, m} V'_j \\ar[rd] \\ar[rr] & & Y' \\ar[ld] \\\\", "& U' &", "}", "$$", "The south-east arrow is finite, hence proper, the horizontal arrow is", "surjective, and the south-west arrow is separated. Hence by", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-is-proper}", "we conclude that $Y' \\to U'$ is proper. Since it is also quasi-finite,", "we see that it is finite by Lemma \\ref{lemma-characterize-finite},", "and we win." ], "refs": [ "morphisms-lemma-finite-local", "morphisms-lemma-quasi-finite", "morphisms-lemma-generically-finite", "morphisms-lemma-image-nowhere-dense-finite", "topology-lemma-nowhere-dense", "schemes-lemma-characterize-separated", "morphisms-lemma-image-proper-is-proper", "more-morphisms-lemma-characterize-finite" ], "ref_ids": [ 5437, 5230, 5487, 5476, 8294, 7710, 5413, 13903 ] } ], "ref_ids": [] }, { "id": 13908, "type": "theorem", "label": "more-morphisms-lemma-stratify-flat-fp-lqf-universally-bounded", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-stratify-flat-fp-lqf-universally-bounded", "contents": [ "Let $f : X \\to S$ be flat, locally of finite presentation, separated,", "locally quasi-finite with universally bounded fibres. Then there exist", "closed subsets", "$$", "\\emptyset = Z_{-1} \\subset Z_0 \\subset Z_1 \\subset Z_2 \\subset", "\\ldots \\subset Z_n = S", "$$", "such that with $S_r = Z_r \\setminus Z_{r - 1}$ the stratification", "$S = \\coprod_{r = 0, \\ldots, n} S_r$ is characterized by the following", "universal property: Given $g : T \\to S$ the projection", "$X \\times_S T \\to T$ is finite locally", "free of degree $r$ if and only if $g(T) \\subset S_r$ (set theoretically)." ], "refs": [], "proofs": [ { "contents": [ "Let $n$ be an integer bounding the degree of the fibres of $X \\to S$.", "By Morphisms, Lemma \\ref{morphisms-lemma-base-change-universally-bounded}", "we see that any base change has degrees of fibres bounded by $n$ also.", "In particular, all the integers $r$ that occur in the statement of the lemma", "will be $\\leq n$. We will prove the lemma by induction on $n$. The base", "case is $n = 0$ which is obvious.", "\\medskip\\noindent", "We claim the set of points $s \\in S$", "with $\\deg_{\\kappa(s)}(X_s) = n$ is an open subset $S_n \\subset S$", "and that $X \\times_S S_n \\to S_n$ is finite locally free of degree $n$.", "Namely, suppose that $s \\in S$ is such a point. Choose an elementary", "\\'etale morphism $(U, u) \\to (S, s)$ and a decomposition", "$U \\times_S X = W \\amalg V$ as in", "Lemma \\ref{lemma-etale-splits-off-quasi-finite-part}.", "Since $V \\to U$ is finite, flat, and locally of finite presentation,", "we see that $V \\to U$ is finite locally free, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}.", "After shrinking $U$ to a smaller neighbourhood of $u$", "we may assume $V \\to U$ is finite locally free of some degree $d$, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-locally-free}.", "As $u \\mapsto s$ and $W_u = \\emptyset$ we see that $d = n$. Since $n$", "is the maximum degree of a fibre we see that $W = \\emptyset$!", "Thus $U \\times_S X \\to U$ is finite locally free of degree $n$.", "By Descent, Lemma \\ref{descent-lemma-descending-property-finite-locally-free}", "we conclude that $X \\to S$ is finite locally free of degree $n$", "over $\\Im(U \\to S)$ which is an open neighbourhood of $s$", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-open}).", "This proves the claim.", "\\medskip\\noindent", "Let $S' = S \\setminus S_n$ endowed with the reduced induced scheme", "structure and set $X' = X \\times_S S'$. Note that the degrees of fibres", "of $X' \\to S'$ are universally bounded by $n - 1$. By induction we find a", "stratification $S' = S_0 \\amalg \\ldots \\amalg S_{n - 1}$ adapted", "to the morphism $X' \\to S'$. We claim that $S = \\coprod_{r = 0, \\ldots, n} S_r$", "works for the morphism $X \\to S$. Let $g : T \\to S$ be a morphism of schemes", "and assume that $X \\times_S T \\to T$ is finite locally free of degree $r$.", "As remarked above this implies that $r \\leq n$. If $r = n$, then it is", "clear that $T \\to S$ factors through $S_n$. If $r < n$, then", "$g(T) \\subset S' = S \\setminus S_d$ (set theoretically) hence", "$T_{red} \\to S$ factors through $S'$, see", "Schemes, Lemma \\ref{schemes-lemma-map-into-reduction}.", "Note that $X \\times_S T_{red} \\to T_{red}$ is also", "finite locally free of degree $r$ as a base change.", "By the universal property of the stratification", "$S' = \\coprod_{r = 0, \\ldots, n - 1} S_r$ we see that $g(T) = g(T_{red})$", "is contained in $S_r$.", "Conversely, suppose that we have $g : T \\to S$ such that", "$g(T) \\subset S_r$ (set theoretically).", "If $r = n$, then $g$ factors through $S_n$ and", "it is clear that $X \\times_S T \\to T$", "is finite locally free of degree $n$ as a base change.", "If $r < n$, then $X \\times_S T \\to T$ is a morphism which is", "separated, flat, and locally of finite presentation, such that", "the restriction to $T_{red}$ is finite locally free of degree $r$.", "Since $T_{red} \\to T$ is a universal homeomorphism, we conclude", "that $X \\times_S T_{red} \\to X \\times_S T$ is a universal homeomorphism", "too and hence $X \\times_S T \\to T$ is universally closed (as this", "is true for the finite morphism $X \\times_S T_{red} \\to T_{red}$).", "It follows that $X \\times_S T \\to T$ is finite, for example by", "Lemma \\ref{lemma-characterize-finite}. Then we can use", "Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}", "to see that $X \\times_S T \\to T$ is finite locally free.", "Finally, the degree is $r$ as all the fibres have degree $r$." ], "refs": [ "morphisms-lemma-base-change-universally-bounded", "more-morphisms-lemma-etale-splits-off-quasi-finite-part", "morphisms-lemma-finite-flat", "morphisms-lemma-finite-locally-free", "descent-lemma-descending-property-finite-locally-free", "morphisms-lemma-etale-open", "schemes-lemma-map-into-reduction", "more-morphisms-lemma-characterize-finite", "morphisms-lemma-finite-flat" ], "ref_ids": [ 5527, 13897, 5471, 5474, 14695, 5370, 7682, 13903, 5471 ] } ], "ref_ids": [] }, { "id": 13909, "type": "theorem", "label": "more-morphisms-lemma-stratify-flat-fp-qf", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-stratify-flat-fp-qf", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat, locally of", "finite presentation, separated, and quasi-finite. Then there exist", "closed subsets", "$$", "\\emptyset = Z_{-1} \\subset Z_0 \\subset Z_1 \\subset Z_2 \\subset", "\\ldots \\subset S", "$$", "such that with $S_r = Z_r \\setminus Z_{r - 1}$ the stratification", "$S = \\coprod S_r$ is characterized by the following universal property:", "Given a morphism $g : T \\to S$ the projection $X \\times_S T \\to T$ is", "finite locally free of degree $r$ if and only if $g(T) \\subset S_r$", "(set theoretically). Moreover, the inclusion maps $S_r \\to S$ are", "quasi-compact." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $S$, hence we may assume that $S$ is affine.", "By Morphisms, Lemma", "\\ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded}", "the fibres of $f$ are universally bounded in this case.", "Hence the existence of the stratification follows from", "Lemma \\ref{lemma-stratify-flat-fp-lqf-universally-bounded}.", "\\medskip\\noindent", "We will show that $U_r = S \\setminus Z_r \\to S$ is quasi-compact for", "each $r \\geq 0$. This will prove the final statement by elementary topology.", "Since a composition of quasi-compact maps is quasi-compact", "it suffices to prove that $U_r \\to U_{r - 1}$ is quasi-compact.", "Choose an affine open $W \\subset U_{r - 1}$. Write $W = \\Spec(A)$.", "Then $Z_r \\cap W = V(I)$ for some ideal $I \\subset A$", "and $X \\times_S \\Spec(A/I) \\to \\Spec(A/I)$ is finite locally", "free of degree $r$. Note that $A/I = \\colim A/I_i$ where $I_i \\subset I$", "runs through the finitely generated ideals. By", "Limits, Lemma \\ref{limits-lemma-descend-finite-locally-free}", "we see that $X \\times_S \\Spec(A/I_i) \\to \\Spec(A/I_i)$", "is finite locally free of degree $r$ for some $i$. (This uses", "that $X \\to S$ is of finite presentation, as it is locally of finite", "presentation, separated, and quasi-compact.)", "Hence $\\Spec(A/I_i) \\to \\Spec(A) = W$ factors (set theoretically)", "through $Z_r \\cap W$. It follows that $Z_r \\cap W = V(I_i)$ is the zero", "set of a finite subset of elements of $A$. This means that", "$W \\setminus Z_r$ is a finite union of standard opens, hence quasi-compact,", "as desired." ], "refs": [ "morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded", "more-morphisms-lemma-stratify-flat-fp-lqf-universally-bounded", "limits-lemma-descend-finite-locally-free" ], "ref_ids": [ 5531, 13908, 15063 ] } ], "ref_ids": [] }, { "id": 13910, "type": "theorem", "label": "more-morphisms-lemma-stratify-flat-fp-lqf", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-stratify-flat-fp-lqf", "contents": [ "Let $f : X \\to S$ be a flat, locally of finite presentation, separated, and", "locally quasi-finite morphism of schemes. Then there", "exist open subschemes", "$$", "S = U_0 \\supset U_1 \\supset U_2 \\supset \\ldots", "$$", "such that a morphism $\\Spec(k) \\to S$ factors through $U_d$ if and", "only if $X \\times_S \\Spec(k)$ has degree $\\geq d$ over $k$." ], "refs": [], "proofs": [ { "contents": [ "The statement simply means that the collection of points where the degree", "of the fibre is $\\geq d$ is open. Thus we can work locally on $S$ and", "assume $S$ is affine. In this case, for every $W \\subset X$ quasi-compact", "open, the set of points $U_d(W)$ where the fibres of $W \\to S$ have", "degree $\\geq d$ is open by Lemma \\ref{lemma-stratify-flat-fp-qf}.", "Since $U_d = \\bigcup_W U_d(W)$ the result follows." ], "refs": [ "more-morphisms-lemma-stratify-flat-fp-qf" ], "ref_ids": [ 13909 ] } ], "ref_ids": [] }, { "id": 13911, "type": "theorem", "label": "more-morphisms-lemma-go-down-with-annihilators", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-go-down-with-annihilators", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat, locally of", "finite presentation, and locally quasi-finite. Let", "$g \\in \\Gamma(X, \\mathcal{O}_X)$ nonzero. Then there exist", "an open $V \\subset X$ such that $g|_V \\not = 0$, an open", "$U \\subset S$ fitting into a commutative diagram", "$$", "\\xymatrix{", "V \\ar[r] \\ar[d]_\\pi & X \\ar[d]^f \\\\", "U \\ar[r] & S,", "}", "$$", "a quasi-coherent subsheaf $\\mathcal{F} \\subset \\mathcal{O}_U$, an integer", "$r > 0$, and an injective $\\mathcal{O}_U$-module map", "$\\mathcal{F}^{\\oplus r} \\to \\pi_*\\mathcal{O}_V$", "whose image contains $g|_V$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $X$ and $S$ affine. We obtain a filtration", "$\\emptyset = Z_{-1} \\subset Z_0 \\subset Z_1 \\subset Z_2 \\subset \\ldots", "\\subset Z_n = S$ as in", "Lemmas \\ref{lemma-stratify-flat-fp-lqf-universally-bounded} and", "\\ref{lemma-stratify-flat-fp-qf}.", "Let $T \\subset X$ be the scheme theoretic support of the finite", "$\\mathcal{O}_X$-module $\\Im(g : \\mathcal{O}_X \\to \\mathcal{O}_X)$.", "Note that $T$ is the support of $g$ as a section of $\\mathcal{O}_X$", "(Modules, Definition \\ref{modules-definition-support}) and", "for any open $V \\subset X$ we have $g|_V \\not = 0$ if and only if", "$V \\cap T \\not = \\emptyset$.", "Let $r$ be the smallest integer such that $f(T) \\subset Z_r$", "set theoretically. Let $\\xi \\in T$ be a generic point of an irreducible", "component of $T$ such that $f(\\xi) \\not \\in Z_{r - 1}$ (and hence", "$f(\\xi) \\in Z_r$). We may replace $S$ by an affine neighbourhood of", "$f(\\xi)$ contained in $S \\setminus Z_{r - 1}$. Write $S = \\Spec(A)$", "and let $I = (a_1, \\ldots, a_m) \\subset A$ be a finitely generated ideal", "such that $V(I) = Z_r$ (set theoretically, see", "Algebra, Lemma \\ref{algebra-lemma-qc-open}).", "Since the support of $g$ is contained in $f^{-1}V(I)$ by our choice of $r$", "we see that there exists an integer $N$ such that", "$a_j^N g = 0$ for $j = 1, \\ldots, m$. Replacing $a_j$ by $a_j^r$", "we may assume that $Ig = 0$. For any $A$-module $M$ write", "$M[I]$ for the $I$-torsion of $M$, i.e., $M[I] = \\{m \\in M \\mid Im = 0\\}$.", "Write $X = \\Spec(B)$, so $g \\in B[I]$. Since $A \\to B$ is flat we", "see that", "$$", "B[I] = A[I] \\otimes_A B \\cong A[I] \\otimes_{A/I} B/IB", "$$", "By our choice of $Z_r$, the $A/I$-module $B/IB$ is", "finite locally free of rank $r$. Hence after replacing $S$ by", "a smaller affine open neighbourhood of $f(\\xi)$ we may assume", "that $B/IB \\cong (A/IA)^{\\oplus r}$ as $A/I$-modules.", "Choose a map $\\psi : A^{\\oplus r} \\to B$ which reduces modulo $I$ to the", "isomorphism of the previous sentence. Then we see that", "the induced map", "$$", "A[I]^{\\oplus r} \\longrightarrow B[I]", "$$", "is an isomorphism. The lemma follows by taking $\\mathcal{F}$ the", "quasi-coherent sheaf associated to the $A$-module $A[I]$ and", "the map $\\mathcal{F}^{\\oplus r} \\to \\pi_*\\mathcal{O}_V$ the", "one corresponding to $A[I]^{\\oplus r} \\subset A^{\\oplus r} \\to B$." ], "refs": [ "more-morphisms-lemma-stratify-flat-fp-lqf-universally-bounded", "more-morphisms-lemma-stratify-flat-fp-qf", "modules-definition-support", "algebra-lemma-qc-open" ], "ref_ids": [ 13908, 13909, 13334, 432 ] } ], "ref_ids": [] }, { "id": 13912, "type": "theorem", "label": "more-morphisms-lemma-there-is-a-scheme-integral-over", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-there-is-a-scheme-integral-over", "contents": [ "Let $U \\to X$ be a surjective \\'etale morphism of schemes. Assume $X$", "is quasi-compact and quasi-separated. Then there exists a surjective", "integral morphism $Y \\to X$, such that for", "every $y \\in Y$ there is an open neighbourhood $V \\subset Y$", "such that $V \\to X$ factors through $U$. In fact, we may assume", "$Y \\to X$ is finite and of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is quasi-compact, there exist finitely many affine opens", "$U_i \\subset U$ such that $U' = \\coprod U_i \\to X$ is surjective.", "After replacing $U$ by $U'$, we see that we may assume $U$ is affine.", "In particular $U \\to X$ is separated", "(Schemes, Lemma \\ref{schemes-lemma-affine-separated}).", "Then there exists an integer $d$ bounding the degree of the geometric", "fibres of $U \\to X$ (see Morphisms, Lemma", "\\ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded}).", "We will prove the lemma by induction on $d$ for all quasi-compact", "and separated schemes $U$ mapping surjective and \\'etale onto $X$.", "If $d = 1$, then $U = X$ and the result holds with $Y = U$.", "Assume $d > 1$.", "\\medskip\\noindent", "We apply Lemma \\ref{lemma-quasi-finite-separated-quasi-affine}", "and we obtain a factorization", "$$", "\\xymatrix{", "U \\ar[rr]_j \\ar[rd] & & Y \\ar[ld]^\\pi \\\\", "& X", "}", "$$", "with $\\pi$ integral and $j$ a quasi-compact open immersion. We may and do", "assume that $j(U)$ is scheme theoretically dense in $Y$. Note that", "$$", "U \\times_X Y = U \\amalg W", "$$", "where the first summand is the image of $U \\to U \\times_X Y$", "(which is closed by", "Schemes, Lemma \\ref{schemes-lemma-semi-diagonal}", "and open because it is \\'etale as a morphism between schemes \\'etale over $Y$)", "and the second summand is the (open and closed) complement.", "The image $V \\subset Y$ of $W$ is an open subscheme containing", "$Y \\setminus U$.", "\\medskip\\noindent", "The \\'etale morphism $W \\to Y$ has geometric fibres of cardinality $< d$.", "Namely, this is clear for geometric points of $U \\subset Y$ by inspection.", "Since $U \\subset Y$ is dense, it holds for all geometric points of $Y$", "for example by Lemma", "\\ref{lemma-stratify-flat-fp-lqf-universally-bounded}", "(the degree of the fibres of a quasi-compact separated \\'etale morphism", "does not go up under specialization). Thus we may apply the induction", "hypothesis to $W \\to V$ and find a surjective integral morphism", "$Z \\to V$ with $Z$ a scheme, which Zariski locally factors through $W$.", "Choose a factorization $Z \\to Z' \\to Y$ with $Z' \\to Y$ integral and", "$Z \\to Z'$ open immersion", "(Lemma \\ref{lemma-quasi-finite-separated-quasi-affine}).", "After replacing $Z'$ by the scheme theoretic closure of $Z$ in $Z'$", "we may assume that $Z$ is scheme theoretically dense in $Z'$.", "After doing this we have $Z' \\times_Y V = Z$. Finally,", "let $T \\subset Y$ be the induced reduced closed subscheme structure", "on $Y \\setminus V$. Consider the morphism", "$$", "Z' \\amalg T \\longrightarrow X", "$$", "This is a surjective integral morphism by construction.", "Since $T \\subset U$ it is clear that the morphism $T \\to X$", "factors through $U$. On the other hand, let $z \\in Z'$", "be a point. If $z \\not \\in Z$, then $z$ maps to a point of", "$Y \\setminus V \\subset U$ and we find a neighbourhood of $z$", "on which the morphism factors through $U$.", "If $z \\in Z$, then we have a neighbourhood $\\Omega \\subset Z$", "which factors through $W \\subset U \\times_X Y$ and hence through $U$.", "This proves existence.", "\\medskip\\noindent", "Assume we have found $Y \\to X$ integral and surjective which Zariski", "locally factors through $U$. Choose a finite affine open covering", "$Y = \\bigcup V_j$ such that $V_j \\to X$ factors through $U$. We can", "write $Y = \\lim Y_i$ with $Y_i \\to X$ finite and of finite", "presentation, see Limits, Lemma", "\\ref{limits-lemma-integral-limit-finite-and-finite-presentation}.", "For large enough $i$ we can find affine opens $V_{i, j} \\subset Y_i$", "whose inverse image in $Y$ recovers $V_j$, see", "Limits, Lemma \\ref{limits-lemma-descend-opens}.", "For even larger $i$ the morphisms $V_j \\to U$ over $X$ come", "from morphisms $V_{i, j} \\to U$ over $X$, see", "Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}.", "This finishes the proof." ], "refs": [ "schemes-lemma-affine-separated", "morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded", "more-morphisms-lemma-quasi-finite-separated-quasi-affine", "schemes-lemma-semi-diagonal", "more-morphisms-lemma-stratify-flat-fp-lqf-universally-bounded", "more-morphisms-lemma-quasi-finite-separated-quasi-affine", "limits-lemma-integral-limit-finite-and-finite-presentation", "limits-lemma-descend-opens", "limits-proposition-characterize-locally-finite-presentation" ], "ref_ids": [ 7717, 5531, 13900, 7712, 13908, 13900, 15056, 15041, 15127 ] } ], "ref_ids": [] }, { "id": 13913, "type": "theorem", "label": "more-morphisms-lemma-descent-connected-fibres", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-descent-connected-fibres", "contents": [ "Consider a diagram of morphisms of schemes", "$$", "\\xymatrix{", "Z \\ar[r]_{\\sigma} \\ar[rd] & X \\ar[d] \\\\", "& Y", "}", "$$", "an a point $y \\in Y$. Assume", "\\begin{enumerate}", "\\item $X \\to Y$ is of finite presentation and flat,", "\\item $Z \\to Y$ is finite locally free,", "\\item $Z_y \\not = \\emptyset$,", "\\item all fibres of $X \\to Y$ are geometrically reduced, and", "\\item $X_y$ is geometrically connected over $\\kappa(y)$.", "\\end{enumerate}", "Then there exists a quasi-compact open $X^0 \\subset X$ such that $X^0_y = X_y$", "and such that all nonempty fibres of $X^0 \\to Y$ are geometrically connected." ], "refs": [], "proofs": [ { "contents": [ "In this proof we will use that flat, finite presentation, finite locally", "free are properties that are preserved under base change and composition.", "We will also use that a finite locally free morphism is both open and", "closed. You can find these facts as", "Morphisms, Lemmas", "\\ref{morphisms-lemma-base-change-flat},", "\\ref{morphisms-lemma-base-change-finite-presentation},", "\\ref{morphisms-lemma-base-change-finite-locally-free},", "\\ref{morphisms-lemma-composition-flat},", "\\ref{morphisms-lemma-composition-finite-presentation},", "\\ref{morphisms-lemma-composition-finite-locally-free},", "\\ref{morphisms-lemma-fppf-open}, and", "\\ref{morphisms-lemma-finite-proper}.", "\\medskip\\noindent", "Note that $X_Z \\to Z$ is flat morphism of finite presentation", "which has a section $s$ coming from $\\sigma$. Let $X_Z^0$ denote", "the subset of $X_Z$ defined in", "Situation \\ref{situation-connected-along-section}.", "By", "Lemma \\ref{lemma-connected-along-section-open}", "it is an open subset of $X_Z$.", "\\medskip\\noindent", "The pullback $X_{Z \\times_Y Z}$ of $X$ to $Z \\times_Y Z$ comes equipped", "with two sections $s_0, s_1$, namely the base changes of $s$ by", "$\\text{pr}_0, \\text{pr}_1 : Z \\times_Y Z \\to Z$. The construction of", "Situation \\ref{situation-connected-along-section}", "gives two subsets $(X_{Z \\times_Y Z})_{s_0}^0$ and", "$(X_{Z \\times_Y Z})_{s_1}^0$. By", "Lemma \\ref{lemma-base-change-connected-along-section}", "these are the inverse images of $X_Z^0$ under the morphisms", "$1_X \\times \\text{pr}_0, 1_X \\times \\text{pr}_1 : X_{Z \\times_Y Z} \\to X_Z$.", "In particular these subsets are open.", "\\medskip\\noindent", "Let $(Z \\times_Y Z)_y = \\{z_1, \\ldots, z_n\\}$.", "As $X_y$ is geometrically connected, we see that the fibres of", "$(X_{Z \\times_Y Z})_{s_0}^0$ and $(X_{Z \\times_Y Z})_{s_1}^0$", "over each $z_i$ agree (being equal to the whole fibre). Another", "way to say this is that", "$$", "s_0(z_i) \\in (X_{Z \\times_Y Z})_{s_1}^0", "\\quad\\text{and}\\quad", "s_1(z_i) \\in (X_{Z \\times_Y Z})_{s_0}^0.", "$$", "Since the sets $(X_{Z \\times_Y Z})_{s_0}^0$ and $(X_{Z \\times_Y Z})_{s_1}^0$", "are open in $X_{Z \\times_Y Z}$ there exists an open neighbourhood", "$W \\subset Z \\times_Y Z$ of $(Z \\times_Y Z)_y$ such that", "$$", "s_0(W) \\subset (X_{Z \\times_Y Z})_{s_1}^0", "\\quad\\text{and}\\quad", "s_1(W) \\subset (X_{Z \\times_Y Z})_{s_0}^0.", "$$", "Then it follows directly from the construction in", "Situation \\ref{situation-connected-along-section}", "that", "$$", "p^{-1}(W) \\cap (X_{Z \\times_Y Z})_{s_0}^0", "=", "p^{-1}(W) \\cap (X_{Z \\times_Y Z})_{s_1}^0", "$$", "where $p : X_{Z \\times_Y Z} \\to Z \\times_W Z$ is the projection.", "Because $Z \\times_Y Z \\to Y$ is finite locally free, hence open and closed,", "there exists an affine open neighbourhood $V \\subset Y$ of $y$ such that", "$q^{-1}(V) \\subset W$, where $q : Z \\times_Y Z \\to Y$ is the", "structure morphism. To prove the lemma we may replace $Y$ by $V$.", "After we do this we see that $X_Z^0 \\subset Y_Z$ is an open such that", "$$", "(1_X \\times \\text{pr}_0)^{-1}(X_Z^0) =", "(1_X \\times \\text{pr}_1)^{-1}(X_Z^0).", "$$", "This means that the image $X^0 \\subset X$ of $X_Z^0$ is an open such", "that $(X_Z \\to X)^{-1}(X^0) = X_Z^0$, see", "Descent, Lemma \\ref{descent-lemma-open-fpqc-covering}.", "Finally, $X^0$ is quasi-compact because $X_Z^0$ is quasi-compact", "by Lemma \\ref{lemma-connected-along-section-locally-constructible}", "(use that at this point $Y$ is affine, hence $X$ is quasi-compact and", "quasi-separated, hence locally constructible is the same as constructible", "and in particular quasi-compact; details omitted).", "In this way we see that $X^0$ has all the desired properties." ], "refs": [ "morphisms-lemma-base-change-flat", "morphisms-lemma-base-change-finite-presentation", "morphisms-lemma-base-change-finite-locally-free", "morphisms-lemma-composition-flat", "morphisms-lemma-composition-finite-presentation", "morphisms-lemma-composition-finite-locally-free", "morphisms-lemma-fppf-open", "morphisms-lemma-finite-proper", "descent-lemma-open-fpqc-covering" ], "ref_ids": [ 5265, 5240, 5473, 5263, 5239, 5472, 5267, 5445, 14637 ] } ], "ref_ids": [] }, { "id": 13914, "type": "theorem", "label": "more-morphisms-lemma-fibre-geometrically-connected-reduced", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-fibre-geometrically-connected-reduced", "contents": [ "Let $h : Y \\to S$ be a morphism of schemes.", "Let $s \\in S$ be a point.", "Let $T \\subset Y_s$ be an open subscheme.", "Assume", "\\begin{enumerate}", "\\item $h$ is flat and of finite presentation,", "\\item all fibres of $h$ are geometrically reduced, and", "\\item $T$ is geometrically connected over $\\kappa(s)$.", "\\end{enumerate}", "Then we can find an affine elementary \\'etale neighbourhood", "$(S', s') \\to (S, s)$", "and a quasi-compact open $V \\subset Y_{S'}$ such that", "\\begin{enumerate}", "\\item[(a)] all fibres of $V \\to S'$ are geometrically connected,", "\\item[(b)] $V_{s'} = T \\times_s s'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The problem is clearly local on $S$, hence we may replace $S$ by an", "affine open neighbourhood of $s$.", "The topology on $Y_s$ is induced from the topology on $Y$, see", "Schemes, Lemma \\ref{schemes-lemma-fibre-topological}.", "Hence we can find a quasi-compact open $V \\subset Y$ such that $V_s = T$.", "The restriction of $h$ to $V$ is quasi-compact (as $S$ affine and $V$", "quasi-compact), quasi-separated, locally of finite presentation, and", "flat hence flat of finite presentation.", "Thus after replacing $Y$ by $V$ we may assume, in addition", "to (1) and (2) that $Y_s = T$ and $S$ affine.", "\\medskip\\noindent", "Pick a closed point $y \\in Y_s$ such that $h$ is Cohen-Macaulay at $y$, see", "Lemma \\ref{lemma-flat-finite-presentation-CM-open}.", "By", "Lemma \\ref{lemma-slice-CM}", "there exists a diagram", "$$", "\\xymatrix{", "Z \\ar[r] \\ar[rd] & Y \\ar[d] \\\\", "& S", "}", "$$", "such that $Z \\to S$ is flat, locally of finite presentation, locally", "quasi-finite with $Z_s = \\{y\\}$. Apply", "Lemma \\ref{lemma-etale-makes-quasi-finite-finite-at-point}", "to find an elementary neighbourhood $(S', s') \\to (S, s)$ and an open", "$Z' \\subset Z_{S'} = S' \\times_S Z$ with $Z' \\to S'$ finite with a unique", "point $z' \\in Z'$ lying over $s$. Note that $Z' \\to S'$ is also", "locally of finite presentation and flat (as an open of the base change", "of $Z \\to S$), hence $Z' \\to S'$ is finite locally free, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}.", "Note that $Y_{S'} \\to S'$ is flat and of finite presentation", "with geometrically reduced fibres as a base change of $h$.", "Also $Y_{s'} = Y_s$ is geometrically connected.", "Apply Lemma \\ref{lemma-descent-connected-fibres}", "to $Z' \\to Y_{S'}$ over $S'$ to get $V \\subset Y_{S'}$ quasi-compact open", "satisfying (2) whose fibres over $S'$ are either empty or", "geometrically connected. As $V \\to S'$ is open", "(Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}), after replacing", "$S'$ by an affine open neighbourhood of $s'$", "we may assume $V \\to S'$ is surjective, whence (1) holds." ], "refs": [ "schemes-lemma-fibre-topological", "more-morphisms-lemma-flat-finite-presentation-CM-open", "more-morphisms-lemma-slice-CM", "more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point", "morphisms-lemma-finite-flat", "more-morphisms-lemma-descent-connected-fibres", "morphisms-lemma-fppf-open" ], "ref_ids": [ 7696, 13789, 13797, 13892, 5471, 13913, 5267 ] } ], "ref_ids": [] }, { "id": 13915, "type": "theorem", "label": "more-morphisms-lemma-cover-by-geometrically-connected", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-cover-by-geometrically-connected", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is", "locally of finite presentation and flat with geometrically", "reduced fibres. Then there", "exists an \\'etale covering $\\{X_i \\to X\\}_{i \\in I}$", "such that $X_i \\to S$ factors as $X_i \\to S_i \\to S$", "where $S_i \\to S$ is \\'etale and $X_i \\to S_i$ is", "flat of finite presentation with geometrically connected", "and geometrically reduced fibres." ], "refs": [], "proofs": [ { "contents": [ "Pick a point $x \\in X$ with image $s \\in S$. We will produce", "a diagram", "$$", "\\xymatrix{", "X' \\ar[r] \\ar[rd] & S' \\times_S X \\ar[r] \\ar[d] & X \\ar[d] \\\\", "& S' \\ar[r] & S", "}", "$$", "and points $s' \\in S'$, $x' \\in X'$, $y \\in S' \\times_S X$", "such that $x'$ maps to $x$, $(S', s') \\to (S, s)$", "is an \\'etale neighbourhood, $(X', x') \\to (S' \\times_S X, y)$", "is an \\'etale neighbourhood\\footnote{The proof actually gives", "an open $X' \\subset S' \\times_S X$.}, and", "$X' \\to S'$ has geometrically", "connected fibres. If we can do this for every $x \\in X$, then", "the lemma follows (with members of the covering given by the", "collection of \\'etale morphisms $X' \\to X$ so produced).", "The first step is the replace $X$ and $S$ by affine open neighbourhoods", "of $x$ and $s$ which reduces us to the case that $X$ and $S$ are affine", "(and hence $f$ of finite presentation).", "\\medskip\\noindent", "Choose a separable algebraic extension $\\overline{k}$ of $\\kappa(s)$.", "Denote $X_{\\overline{k}}$ the base change of $X_s$.", "Choose a point $\\overline{x}$ in $X_{\\overline{k}}$ mapping to $x \\in X_s$.", "Choose a connected quasi-compact open neighbourhood", "$\\overline{V} \\subset X_{\\overline{k}}$", "of $\\overline{x}$. (This is possible because any scheme", "locally of finite type over a field is locally connected", "as a locally Noetherian topological space.)", "By Varieties, Lemma \\ref{varieties-lemma-Galois-action-quasi-compact-open}", "we can find a finite separable extension $k'/\\kappa(s)$", "and a quasi-compact open $V' \\subset X_{k'}$ whose", "base change is $\\overline{V}$. In particular $V'$ is", "geometrically connected over $k'$, see", "Varieties, Lemma \\ref{varieties-lemma-characterize-geometrically-connected}. By", "Lemma \\ref{lemma-realize-prescribed-residue-field-extension-etale}", "we can find an \\'etale neighbourhood $(S', s') \\to (S, s)$", "such that $\\kappa(s')$ is isomorphic to $k'$ as an extension", "of $\\kappa(s)$.", "Denote $x' \\in (S' \\times_S X)_{s'} = X_{k'}$ the image of $\\overline{x}$.", "Thus after replacing $(S, s)$ by $(S', s')$ and $(X, x)$ by", "$(S' \\times_S X, x')$ we reduce to the case handled in the next", "paragrah.", "\\medskip\\noindent", "Assume there is a quasi-compact open $V \\subset X_s$", "which contains $x$ and is geometrically irreducible.", "Then we can apply Lemma \\ref{lemma-fibre-geometrically-connected-reduced}", "to find an affine \\'etale neighbourhood $(S', s') \\to (S, s)$", "and a quasi-compact open $X' \\subset S' \\times_S X$ such that", "$X' \\to S'$ has geometrically connected fibres", "and such that $X'$ contains a point mapping to $x$.", "This finishes the proof." ], "refs": [ "varieties-lemma-Galois-action-quasi-compact-open", "varieties-lemma-characterize-geometrically-connected", "more-morphisms-lemma-realize-prescribed-residue-field-extension-etale", "more-morphisms-lemma-fibre-geometrically-connected-reduced" ], "ref_ids": [ 10921, 10919, 13866, 13914 ] } ], "ref_ids": [] }, { "id": 13916, "type": "theorem", "label": "more-morphisms-lemma-normal-morphism-irreducible", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-normal-morphism-irreducible", "contents": [ "Let $h : Y \\to S$ be a morphism of schemes.", "Let $s \\in S$ be a point.", "Let $T \\subset Y_s$ be an open subscheme.", "Assume", "\\begin{enumerate}", "\\item $h$ is of finite presentation,", "\\item $h$ is normal, and", "\\item $T$ is geometrically irreducible over $\\kappa(s)$.", "\\end{enumerate}", "Then we can find an affine elementary \\'etale neighbourhood", "$(S', s') \\to (S, s)$ and a quasi-compact open $V \\subset Y_{S'}$ such that", "\\begin{enumerate}", "\\item[(a)] all fibres of $V \\to S'$ are geometrically integral,", "\\item[(b)] $V_{s'} = T \\times_s s'$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Apply", "Lemma \\ref{lemma-fibre-geometrically-connected-reduced}", "to find an affine elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and", "a quasi-compact open $V \\subset Y_{S'}$ such that all fibres of", "$V \\to S'$ are geometrically connected and $V_{s'} = T \\times_s s'$.", "As $V$ is an open of the base change of $h$ all fibres of $V \\to S'$", "are geometrically normal, see Lemma \\ref{lemma-normal}.", "In particular, they are geometrically reduced. To finish the proof", "we have to show they are geometrically irreducible. But, if $t \\in S'$", "then $V_t$ is of finite type over $\\kappa(t)$ and hence", "$V_t \\times_{\\kappa(t)} \\overline{\\kappa(t)}$ is of finite type", "over $\\overline{\\kappa(t)}$ hence Noetherian. By choice of $S' \\to S$", "the scheme $V_t \\times_{\\kappa(t)} \\overline{\\kappa(t)}$ is connected.", "Hence $V_t \\times_{\\kappa(t)} \\overline{\\kappa(t)}$ is irreducible by", "Properties, Lemma \\ref{properties-lemma-normal-Noetherian}", "and we win." ], "refs": [ "more-morphisms-lemma-fibre-geometrically-connected-reduced", "more-morphisms-lemma-normal", "properties-lemma-normal-Noetherian" ], "ref_ids": [ 13914, 13777, 2970 ] } ], "ref_ids": [] }, { "id": 13917, "type": "theorem", "label": "more-morphisms-lemma-local-structure-finite-type", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-local-structure-finite-type", "contents": [ "Let $f : X \\to S$ be a morphism. Let $x \\in X$ and set $s = f(x)$.", "Assume that $f$ is locally of finite type and that $n = \\dim_x(X_s)$.", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "X \\ar[dd] & X' \\ar[l]^g \\ar[d]^\\pi & x \\ar@{|->}[dd] &", "x' \\ar@{|->}[l] \\ar@{|->}[d] \\\\", "& Y \\ar[d]^h & & y \\ar@{|->}[d] \\\\", "S \\ar@{=}[r] & S & s & s \\ar@{=}[l]", "}", "$$", "and a point $x' \\in X'$ with $g(x') = x$ such that with $y = \\pi(x')$", "we have", "\\begin{enumerate}", "\\item $h : Y \\to S$ is smooth of relative dimension $n$,", "\\item $g : (X', x') \\to (X, x)$ is an elementary \\'etale neighbourhood,", "\\item $\\pi$ is finite, and $\\pi^{-1}(\\{y\\}) = \\{x'\\}$, and", "\\item $\\kappa(y)$ is a purely transcendental extension of $\\kappa(s)$.", "\\end{enumerate}", "Moreover, if $f$ is locally of finite presentation then $\\pi$ is", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "The problem is local on $X$ and $S$, hence we may assume that $X$ and", "$S$ are affine. By", "Algebra, Lemma \\ref{algebra-lemma-refined-quasi-finite-over-polynomial-algebra}", "after replacing $X$ by a standard open neighbourhood of $x$ in $X$", "we may assume there is a factorization", "$$", "\\xymatrix{", "X \\ar[r]^\\pi & \\mathbf{A}^n_S \\ar[r] & S", "}", "$$", "such that $\\pi$ is quasi-finite and such that $\\kappa(\\pi(x))$", "is purely transcendental over $\\kappa(s)$. By", "Lemma \\ref{lemma-etale-makes-quasi-finite-finite-at-point}", "there exists an elementary \\'etale neighbourhood", "$$", "(Y, y) \\to (\\mathbf{A}^n_S, \\pi(x))", "$$", "and an open $X' \\subset X \\times_{\\mathbf{A}^n_S} Y$ which contains a", "unique point $x'$ lying over $y$ such that $X' \\to Y$ is finite.", "This proves (1) -- (4) hold. For the final assertion, use", "Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-permanence}." ], "refs": [ "algebra-lemma-refined-quasi-finite-over-polynomial-algebra", "more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point", "morphisms-lemma-finite-presentation-permanence" ], "ref_ids": [ 1072, 13892, 5247 ] } ], "ref_ids": [] }, { "id": 13918, "type": "theorem", "label": "more-morphisms-lemma-local-local-structure-finite-type", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-local-local-structure-finite-type", "contents": [ "\\begin{slogan}", "A morphism of finite type is, in \\'etale neighbourhoods, finite over a", "smooth morphism.", "\\end{slogan}", "Let $f : X \\to S$ be a morphism. Let $x \\in X$ and set $s = f(x)$.", "Assume that $f$ is locally of finite type and that $n = \\dim_x(X_s)$.", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "X \\ar[dd] & X' \\ar[l]^g \\ar[d]^\\pi & x \\ar@{|->}[dd] &", "x' \\ar@{|->}[l] \\ar@{|->}[d] \\\\", "& Y' \\ar[d]^h & & y' \\ar@{|->}[d] \\\\", "S & S' \\ar[l]_e & s & s' \\ar@{|->}[l]", "}", "$$", "and a point $x' \\in X'$ with $g(x') = x$ such that with $y' = \\pi(x')$,", "$s' = h(y')$ we have", "\\begin{enumerate}", "\\item $h : Y' \\to S'$ is smooth of relative dimension $n$,", "\\item all fibres of $Y' \\to S'$ are geometrically integral,", "\\item $g : (X', x') \\to (X, x)$ is an elementary \\'etale neighbourhood,", "\\item $\\pi$ is finite, and $\\pi^{-1}(\\{y'\\}) = \\{x'\\}$,", "\\item $\\kappa(y')$ is a purely transcendental extension of $\\kappa(s')$, and", "\\item $e : (S', s') \\to (S, s)$ is an elementary \\'etale neighbourhood.", "\\end{enumerate}", "Moreover, if $f$ is locally of finite presentation, then $\\pi$ is", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "The question is local on $S$, hence we may replace $S$ by an affine open", "neighbourhood of $s$. Next, we apply", "Lemma \\ref{lemma-local-structure-finite-type}", "to get a commutative diagram", "$$", "\\xymatrix{", "X \\ar[dd] & X' \\ar[l]^g \\ar[d]^\\pi & x \\ar@{|->}[dd] &", "x' \\ar@{|->}[l] \\ar@{|->}[d] \\\\", "& Y \\ar[d]^h & & y \\ar@{|->}[d] \\\\", "S \\ar@{=}[r] & S & s & s \\ar@{=}[l]", "}", "$$", "where $h$ is smooth of relative dimension $n$ and $\\kappa(y)$ is", "a purely transcendental extension of $\\kappa(s)$. Since the question is", "local on $X$ also, we may replace $Y$ by an affine neighbourhood of $y$", "(and $X'$ by the inverse image of this under $\\pi$). As $S$ is affine", "this guarantees that $Y \\to S$ is quasi-compact, separated and smooth,", "in particular of finite presentation.", "Let $T$ be the connected component of $Y_s$ containing $y$.", "As $Y_s$ is Noetherian we see that $T$ is open.", "We also see that $T$ is geometrically connected over $\\kappa(s)$ by", "Varieties,", "Lemma \\ref{varieties-lemma-geometrically-connected-if-connected-and-point}.", "Since $T$ is also smooth over $\\kappa(s)$ it is geometrically normal, see", "Varieties, Lemma \\ref{varieties-lemma-smooth-geometrically-normal}.", "We conclude that $T$ is geometrically irreducible over $\\kappa(s)$ (as a", "connected Noetherian normal scheme is irreducible, see", "Properties, Lemma \\ref{properties-lemma-normal-Noetherian}).", "Finally, note that the smooth morphism $h$ is normal by", "Lemma \\ref{lemma-smooth-normal}.", "At this point we have verified all assumption of", "Lemma \\ref{lemma-normal-morphism-irreducible}", "hold for the morphism $h : Y \\to S$ and open $T \\subset Y_s$.", "As a result of applying", "Lemma \\ref{lemma-normal-morphism-irreducible}", "we obtain $e : S' \\to S$, $s' \\in S'$, $Y'$ as", "in the commutative diagram", "$$", "\\xymatrix{", "X \\ar[dd] & X' \\ar[l]^g \\ar[d]^\\pi & X' \\times_Y Y' \\ar[l] \\ar[d] &", "x \\ar@{|->}[dd] & x' \\ar@{|->}[l] \\ar@{|->}[d] &", "(x', s') \\ar@{|->}[l] \\ar@{|->}[d] \\\\", "& Y \\ar[d]^h & Y' \\ar[d] \\ar[l] & & y \\ar@{|->}[d] &", "(y, s') \\ar@{|->}[l] \\ar@{|->}[d] \\\\", "S \\ar@{=}[r] & S & S' \\ar[l]_e & s & s \\ar@{=}[l] & s' \\ar@{|->}[l]", "}", "$$", "where $e : (S', s') \\to (S, s)$ is an elementary \\'etale neighbourhood,", "and where $Y' \\subset Y_{S'}$ is an open neighbourhood all of whose fibres", "over $S'$ are geometrically irreducible, such that $Y'_{s'} = T$ via", "the identification $Y_s = Y_{S', s'}$. Let $(y, s') \\in Y'$ be the point", "corresponding to $y \\in T$; this is also the unique point of $Y \\times_S S'$", "lying over $y$ with residue field equal to $\\kappa(y)$ which maps to $s'$", "in $S'$. Similarly, let $(x', s') \\in X' \\times_Y Y' \\subset X' \\times_S S'$", "be the unique point over $x'$ with residue field equal to $\\kappa(x')$", "lying over $s'$. Then the outer part of this diagram is a solution to the", "problem posed in the lemma. Some minor details omitted." ], "refs": [ "more-morphisms-lemma-local-structure-finite-type", "varieties-lemma-geometrically-connected-if-connected-and-point", "varieties-lemma-smooth-geometrically-normal", "properties-lemma-normal-Noetherian", "more-morphisms-lemma-smooth-normal", "more-morphisms-lemma-normal-morphism-irreducible", "more-morphisms-lemma-normal-morphism-irreducible" ], "ref_ids": [ 13917, 10926, 11005, 2970, 13778, 13916, 13916 ] } ], "ref_ids": [] }, { "id": 13919, "type": "theorem", "label": "more-morphisms-lemma-local-local-structure-finite-type-affine", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-local-local-structure-finite-type-affine", "contents": [ "Assumption and notation as in", "Lemma \\ref{lemma-local-local-structure-finite-type}.", "In addition to properties (1) -- (6) we may also arrange it so that", "\\begin{enumerate}", "\\item[(7)] $S'$, $Y'$, $X'$ are affine.", "\\end{enumerate}" ], "refs": [ "more-morphisms-lemma-local-local-structure-finite-type" ], "proofs": [ { "contents": [ "Note that if $Y'$ is affine, then $X'$ is affine as $\\pi$ is finite.", "Choose an affine open neighbourhood $U' \\subset S'$ of $s'$.", "Choose an affine open neighbourhood $V' \\subset h^{-1}(U')$ of $y'$.", "Let $W' = h(V')$. This is an open neighbourhood of $s'$ in $S'$, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-open},", "contained in $U'$. Choose an affine open neighbourhood $U'' \\subset W'$", "of $s'$. Then $h^{-1}(U'') \\cap V'$ is affine because it is equal to", "$U'' \\times_{U'} V'$. By construction", "$h^{-1}(U'') \\cap V' \\to U''$ is a surjective smooth morphism whose", "fibres are (nonempty) open subschemes of geometrically integral fibres", "of $Y' \\to S'$, and hence geometrically integral. Thus we may replace", "$S'$ by $U''$ and $Y'$ by $h^{-1}(U'') \\cap V'$." ], "refs": [ "morphisms-lemma-smooth-open" ], "ref_ids": [ 5332 ] } ], "ref_ids": [ 13918 ] }, { "id": 13920, "type": "theorem", "label": "more-morphisms-lemma-finite-morphism-single-point-in-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-finite-morphism-single-point-in-fibre", "contents": [ "Let $\\pi : X \\to Y$ be a finite morphism.", "Let $x \\in X$ with $y = \\pi(x)$ such that $\\pi^{-1}(\\{y\\}) = \\{x\\}$.", "Then", "\\begin{enumerate}", "\\item For every neighbourhood $U \\subset X$ of $x$ in $X$, there", "exists a neighbourhood $V \\subset Y$ of $y$ such that", "$\\pi^{-1}(V) \\subset U$.", "\\item The ring map $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$", "is finite.", "\\item If $\\pi$ is of finite presentation, then", "$\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ is of finite presentation.", "\\item For any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "we have $\\mathcal{F}_x = \\pi_*\\mathcal{F}_y$ as", "$\\mathcal{O}_{Y, y}$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first assertion is purely topological; use that", "$\\pi$ is a continuous and closed map such that $\\pi^{-1}(\\{y\\}) = \\{x\\}$.", "To prove the second and third parts we may assume", "$X = \\Spec(B)$ and $Y = \\Spec(A)$. Then", "$A \\to B$ is a finite ring map and $y$ corresponds to a prime", "$\\mathfrak p$ of $A$ such that there exists a unique prime $\\mathfrak q$ of", "$B$ lying over $\\mathfrak p$. Then", "$B_{\\mathfrak q} = B_{\\mathfrak p}$, see", "Algebra, Lemma \\ref{algebra-lemma-unique-prime-over-localize-below}.", "In other words, the map $A_{\\mathfrak p} \\to B_{\\mathfrak q}$", "is equal to the map $A_{\\mathfrak p} \\to B_{\\mathfrak p}$ you get", "from localizing $A \\to B$ at $\\mathfrak p$.", "Thus (2) and (3) follow from simple properties of localization", "(some details omitted). For the final statement, suppose that", "$\\mathcal{F} = \\widetilde M$ for some $B$-module $M$.", "Then $\\mathcal{F} = M_{\\mathfrak q}$ and", "$\\pi_*\\mathcal{F}_y = M_{\\mathfrak p}$. By the above these", "localizations agree. Alternatively you can use part (1) and", "the definition of stalks to see that $\\mathcal{F}_x = \\pi_*\\mathcal{F}_y$", "directly." ], "refs": [ "algebra-lemma-unique-prime-over-localize-below" ], "ref_ids": [ 556 ] } ], "ref_ids": [] }, { "id": 13921, "type": "theorem", "label": "more-morphisms-lemma-dominate-fppf-etale-locally", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-dominate-fppf-etale-locally", "contents": [ "Let $S$ be a scheme. Let $\\{S_i \\to S\\}_{i \\in I}$ be an fppf covering.", "Then there exist", "\\begin{enumerate}", "\\item an \\'etale covering $\\{S'_a \\to S\\}$,", "\\item surjective finite locally free morphisms $V_a \\to S'_a$,", "\\end{enumerate}", "such that the fppf covering $\\{V_a \\to S\\}$ refines the given", "covering $\\{S_i \\to S\\}$." ], "refs": [], "proofs": [ { "contents": [ "We may assume that each $S_i \\to S$ is locally quasi-finite, see", "Lemma \\ref{lemma-qf-fp-flat-dominates-fppf}.", "\\medskip\\noindent", "Fix a point $s \\in S$. Pick an $i \\in I$ and a point", "$s_i \\in S_i$ mapping to $s$. Choose an elementary \\'etale neighbourhood", "$(S', s) \\to (S, s)$ such that there exists an open", "$$", "S_i \\times_S S' \\supset V", "$$", "which contains a unique point $v \\in V$ mapping to $s \\in S'$", "and such that $V \\to S'$ is finite, see", "Lemma \\ref{lemma-etale-makes-quasi-finite-finite-at-point}.", "Then $V \\to S'$ is finite locally free, because it is finite", "and because $S_i \\times_S S' \\to S'$ is flat and locally of finite presentation", "as a base change of the morphism $S_i \\to S$, see", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-finite-presentation},", "\\ref{morphisms-lemma-base-change-flat}, and", "\\ref{morphisms-lemma-finite-flat}.", "Hence $V \\to S'$ is open, and after shrinking $S'$", "we may assume that $V \\to S'$ is surjective finite locally free.", "Since we can do this for every point of $S$ we conclude that", "$\\{S_i \\to S\\}$ can be refined by a covering of the form", "$\\{V_a \\to S\\}_{a \\in A}$ where each $V_a \\to S$ factors as", "$V_a \\to S'_a \\to S$ with $S'_a \\to S$ \\'etale and $V_a \\to S'_a$ surjective", "finite locally free." ], "refs": [ "more-morphisms-lemma-qf-fp-flat-dominates-fppf", "more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point", "morphisms-lemma-base-change-finite-presentation", "morphisms-lemma-base-change-flat", "morphisms-lemma-finite-flat" ], "ref_ids": [ 13799, 13892, 5240, 5265, 5471 ] } ], "ref_ids": [] }, { "id": 13922, "type": "theorem", "label": "more-morphisms-lemma-dominate-fppf", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-dominate-fppf", "contents": [ "Let $S$ be a scheme. Let $\\{S_i \\to S\\}_{i \\in I}$ be an fppf covering.", "Then there exist", "\\begin{enumerate}", "\\item a Zariski open covering $S = \\bigcup U_j$,", "\\item surjective finite locally free morphisms $W_j \\to U_j$,", "\\item Zariski open coverings $W_j = \\bigcup_k W_{j, k}$,", "\\item surjective finite locally free morphisms $T_{j, k} \\to W_{j, k}$", "\\end{enumerate}", "such that the fppf covering $\\{T_{j, k} \\to S\\}$ refines the given", "covering $\\{S_i \\to S\\}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{V_a \\to S\\}_{a \\in A}$ be the fppf covering found in", "Lemma \\ref{lemma-dominate-fppf-etale-locally}.", "In other words, this covering refines", "$\\{S_i \\to S\\}$", "and each $V_a \\to S$ factors as", "$V_a \\to S'_a \\to S$ with $S'_a \\to S$ \\'etale and $V_a \\to S'_a$", "surjective finite locally free.", "\\medskip\\noindent", "By", "Remark \\ref{remark-topologies}", "there exists a Zariski open covering $S = \\bigcup U_j$,", "for each $j$ a finite locally free, surjective morphism", "$W_j \\to U_j$, and for each $j$ a Zariski open covering", "$\\{W_{j, k} \\to W_j\\}$ such that the family", "$\\{W_{j, k} \\to S\\}$ refines the \\'etale covering", "$\\{S'_a \\to S\\}$, i.e., for each pair $j, k$ there exists", "an $a(j, k)$ and a factorization $W_{j, k} \\to S'_a \\to S$", "of the morphism $W_{j, k} \\to S$. Set", "$T_{j, k} = W_{j, k} \\times_{S'_a} V_a$ and everything is clear." ], "refs": [ "more-morphisms-lemma-dominate-fppf-etale-locally", "more-morphisms-remark-topologies" ], "ref_ids": [ 13921, 14133 ] } ], "ref_ids": [] }, { "id": 13923, "type": "theorem", "label": "more-morphisms-lemma-extend-integral-surjective-morphisms", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-extend-integral-surjective-morphisms", "contents": [ "Let $S$ be a scheme. If $U \\subset S$ is open and $V \\to U$ is a surjective", "integral morphism, then there exists a surjective integral", "morphism $\\overline{V} \\to S$ with $\\overline{V} \\times_S U$", "isomorphic to $V$ as schemes over $U$." ], "refs": [], "proofs": [ { "contents": [ "Let $V' \\to S$ be the normalization of $S$ in $U$, see", "Morphisms, Section \\ref{morphisms-section-normalization-X-in-Y}.", "By construction $V' \\to S$ is integral.", "By Morphisms, Lemmas \\ref{morphisms-lemma-normalization-localization} and", "\\ref{morphisms-lemma-normalization-in-integral} we see that", "the inverse image of $U$ in $V'$ is $V$. Let $Z$ be the reduced", "induced scheme structure on $S \\setminus U$. Then", "$\\overline{V} = V' \\amalg Z$ works." ], "refs": [ "morphisms-lemma-normalization-localization", "morphisms-lemma-normalization-in-integral" ], "ref_ids": [ 5501, 5507 ] } ], "ref_ids": [] }, { "id": 13924, "type": "theorem", "label": "more-morphisms-lemma-extend-finite-surjective-morphisms", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-extend-finite-surjective-morphisms", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "If $U \\subset S$ is a quasi-compact open", "and $V \\to U$ is a surjective finite morphism, then there exists a", "surjective finite morphism $\\overline{V} \\to S$ with $\\overline{V} \\times_S U$", "isomorphic to $V$ as schemes over $U$." ], "refs": [], "proofs": [ { "contents": [ "By Zariski's Main Theorem", "(Lemma \\ref{lemma-quasi-finite-separated-pass-through-finite})", "we can assume $V$ is a quasi-compact open in a scheme $V'$", "finite over $S$. After replacing $V'$ by the scheme theoretic image", "of $V$ we may assume that $V$ is dense in $V'$.", "It follows that $V' \\times_S U = V$ because $V \\to V' \\times_S U$", "is closed as $V$ is finite over $U$. Let $Z$ be the reduced", "induced scheme structure on $S \\setminus U$. Then", "$\\overline{V} = V' \\amalg Z$ works." ], "refs": [ "more-morphisms-lemma-quasi-finite-separated-pass-through-finite" ], "ref_ids": [ 13901 ] } ], "ref_ids": [] }, { "id": 13925, "type": "theorem", "label": "more-morphisms-lemma-dominate-fppf-integral", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-dominate-fppf-integral", "contents": [ "Let $S$ be a scheme. Let $\\{S_i \\to S\\}_{i \\in I}$ be an fppf covering.", "Then there exists a surjective integral morphism $S' \\to S$ and an", "open covering $S' = \\bigcup U'_\\alpha$ such that for each $\\alpha$ the", "morphism $U'_\\alpha \\to S$ factors through $S_i \\to S$ for some $i$." ], "refs": [], "proofs": [ { "contents": [ "Choose $S = \\bigcup U_j$, $W_j \\to U_j$, $W_j = \\bigcup W_{j, k}$, and", "$T_{j, k} \\to W_{j, k}$ as in Lemma \\ref{lemma-dominate-fppf}.", "By Lemma \\ref{lemma-extend-integral-surjective-morphisms}", "we can extend $W_j \\to U_j$ to a surjective integral", "morphism $\\overline{W}_j \\to S$.", "After this we can extend $T_{j, k} \\to W_{j, k}$ to a surjective", "integral morphism $\\overline{T}_{j, k} \\to \\overline{W}_j$.", "We set $\\overline{T}_j$ equal to the product of all the schemes", "$\\overline{T}_{j, k}$ over $\\overline{W}_j$", "(Limits, Lemma \\ref{limits-lemma-infinite-product}).", "Then we set $S'$ equal to the product of all the schemes", "$\\overline{T}_j$ over $S$.", "If $x \\in S'$, then there is a $j$ such that the image of $x$ in", "$S$ lies in $U_j$. Hence there is a $k$ such that the image of", "$x$ under the projection $S' \\to \\overline{W}_j$ lies in $W_{j, k}$.", "Hence under the projection $S' \\to \\overline{T}_j \\to \\overline{T}_{j, k}$", "the point $x$ ends up in $T_{j, k}$. And $T_{j, k} \\to S$", "factors through $S_i$ for some $i$.", "Finally, the morphism $S' \\to S$ is integral and surjective", "by Limits, Lemmas \\ref{limits-lemma-infinite-product-integral} and", "\\ref{limits-lemma-infinite-product-surjective}." ], "refs": [ "more-morphisms-lemma-dominate-fppf", "more-morphisms-lemma-extend-integral-surjective-morphisms", "limits-lemma-infinite-product", "limits-lemma-infinite-product-integral", "limits-lemma-infinite-product-surjective" ], "ref_ids": [ 13922, 13923, 15029, 15031, 15030 ] } ], "ref_ids": [] }, { "id": 13926, "type": "theorem", "label": "more-morphisms-lemma-dominate-fppf-finite", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-dominate-fppf-finite", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $\\{S_i \\to S\\}_{i \\in I}$ be an fppf covering.", "Then there exists a surjective finite morphism $S' \\to S$", "of finite presentation and an", "open covering $S' = \\bigcup U'_\\alpha$ such that for each $\\alpha$ the", "morphism $U'_\\alpha \\to S$ factors through $S_i \\to S$ for some $i$." ], "refs": [], "proofs": [ { "contents": [ "Let $Y \\to X$ be the integral surjective morphism found in", "Lemma \\ref{lemma-dominate-fppf-integral}.", "Choose a finite affine open covering $Y = \\bigcup V_j$", "such that $V_j \\to X$ factors through $S_{i(j)}$.", "We can write $Y = \\lim Y_\\lambda$ with", "$Y_\\lambda \\to X$ finite and of finite presentation, see", "Limits, Lemma \\ref{limits-lemma-integral-limit-finite-and-finite-presentation}.", "For large enough $\\lambda$ we can find affine opens", "$V_{\\lambda, j} \\subset Y_\\lambda$", "whose inverse image in $Y$ recovers $V_j$, see", "Limits, Lemma \\ref{limits-lemma-descend-opens}.", "For even larger $\\lambda$ the morphisms $V_j \\to S_{i(j)}$", "over $X$ come from morphisms $V_{\\lambda, j} \\to S_{i(j)}$ over", "$X$, see", "Limits, Proposition", "\\ref{limits-proposition-characterize-locally-finite-presentation}.", "Setting $S' = Y_\\lambda$ for this $\\lambda$ finishes the proof." ], "refs": [ "more-morphisms-lemma-dominate-fppf-integral", "limits-lemma-integral-limit-finite-and-finite-presentation", "limits-lemma-descend-opens", "limits-proposition-characterize-locally-finite-presentation" ], "ref_ids": [ 13925, 15056, 15041, 15127 ] } ], "ref_ids": [] }, { "id": 13927, "type": "theorem", "label": "more-morphisms-lemma-fppf-ph", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-fppf-ph", "contents": [ "An fppf covering of schemes is a ph covering." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{T_i \\to T\\}$ be an fppf covering of schemes, see", "Topologies, Definition \\ref{topologies-definition-fppf-covering}.", "Observe that $T_i \\to T$ is locally of finite type.", "Let $U \\subset T$ be an affine open.", "It suffices to show that $\\{T_i \\times_T U \\to U\\}$", "can be refined by a standard ph covering, see", "Topologies, Definition \\ref{topologies-definition-ph-covering}.", "This follows immediately from Lemma \\ref{lemma-dominate-fppf-finite}", "and the fact that a finite morphism is proper", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-proper})." ], "refs": [ "topologies-definition-fppf-covering", "topologies-definition-ph-covering", "more-morphisms-lemma-dominate-fppf-finite", "morphisms-lemma-finite-proper" ], "ref_ids": [ 12539, 12544, 13926, 5445 ] } ], "ref_ids": [] }, { "id": 13928, "type": "theorem", "label": "more-morphisms-lemma-quasi-projective", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-quasi-projective", "contents": [ "Let $S$ be a scheme which has an ample invertible sheaf.", "Let $f : X \\to S$ be a morphism of schemes. The following are", "equivalent", "\\begin{enumerate}", "\\item $X \\to S$ is quasi-projective,", "\\item $X \\to S$ is H-quasi-projective,", "\\item there exists a quasi-compact open immersion $X \\to X'$ of schemes", "over $S$ with $X' \\to S$ projective,", "\\item $X \\to S$ is of finite type and $X$ has an ample invertible", "sheaf, and", "\\item $X \\to S$ is of finite type and there exists an", "$f$-very ample invertible sheaf.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is", "Morphisms, Lemma \\ref{morphisms-lemma-H-quasi-projective-quasi-projective}.", "The implication (1) $\\Rightarrow$ (2) is", "Morphisms, Lemma", "\\ref{morphisms-lemma-projective-over-quasi-projective-is-H-projective}.", "The implication (2) $\\Rightarrow$ (3) is", "Morphisms, Lemma \\ref{morphisms-lemma-H-quasi-projective-open-H-projective}", "\\medskip\\noindent", "Assume $X \\subset X'$ is as in (3). In particular $X \\to S$ is", "of finite type. By", "Morphisms, Lemma \\ref{morphisms-lemma-H-quasi-projective-open-H-projective}", "the morphism $X \\to S$ is H-projective.", "Thus there exists a quasi-compact immersion $i : X \\to \\mathbf{P}^n_S$.", "Hence $\\mathcal{L} = i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1)$", "is $f$-very ample. As $X \\to S$ is quasi-compact we conclude from", "Morphisms, Lemma \\ref{morphisms-lemma-ample-very-ample}", "that $\\mathcal{L}$ is $f$-ample. Thus $X \\to S$ is quasi-projective", "by definition.", "\\medskip\\noindent", "The implication (4) $\\Rightarrow$ (2) is", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-projective-finite-type-over-S}.", "\\medskip\\noindent", "Assume the equivalent conditions (1), (2), (3) hold.", "Choose an immersion $i : X \\to \\mathbf{P}^n_S$ over $S$.", "Let $\\mathcal{L}$ be an ample invertible sheaf on $S$. To finish the", "proof we will show that", "$\\mathcal{N} =", "f^*\\mathcal{L} \\otimes_{\\mathcal{O}_X} i^*\\mathcal{O}_{\\mathbf{P}^n_X}(1)$", "is ample on $X$. By", "Properties, Lemma \\ref{properties-lemma-ample-on-locally-closed}", "we reduce to the case $X = \\mathbf{P}^n_S$. Let", "$s \\in \\Gamma(S, \\mathcal{L}^{\\otimes d})$ be a section", "such that the corresponding open $S_s$ is affine.", "Say $S_s = \\Spec(A)$.", "Recall that $\\mathbf{P}^n_S$ is the projective bundle", "associated to $\\mathcal{O}_S T_0 \\oplus \\ldots \\oplus \\mathcal{O}_S T_n$, see", "Constructions, Lemma \\ref{constructions-lemma-projective-space-bundle}", "and its proof.", "Let $s_i \\in \\Gamma(\\mathbf{P}^n_S, \\mathcal{O}(1))$", "be the global section corresponding to the section $T_i$", "of $\\mathcal{O}_S T_0 \\oplus \\ldots \\oplus \\mathcal{O}_S T_n$.", "Then we see that $X_{f^*s \\otimes s_i^{\\otimes n}}$ is affine", "because it is equal to $\\Spec(A[T_0/T_i, \\ldots, T_n/T_i])$.", "This proves that $\\mathcal{N}$ is ample by definition.", "\\medskip\\noindent", "The equivalence of (1) and (5) follows from", "Morphisms, Lemmas \\ref{morphisms-lemma-ample-very-ample} and", "\\ref{morphisms-lemma-finite-type-ample-very-ample}." ], "refs": [ "morphisms-lemma-H-quasi-projective-quasi-projective", "morphisms-lemma-projective-over-quasi-projective-is-H-projective", "morphisms-lemma-H-quasi-projective-open-H-projective", "morphisms-lemma-H-quasi-projective-open-H-projective", "morphisms-lemma-ample-very-ample", "morphisms-lemma-quasi-projective-finite-type-over-S", "properties-lemma-ample-on-locally-closed", "constructions-lemma-projective-space-bundle", "morphisms-lemma-ample-very-ample", "morphisms-lemma-finite-type-ample-very-ample" ], "ref_ids": [ 5402, 5433, 5428, 5428, 5387, 5393, 3051, 12652, 5387, 5395 ] } ], "ref_ids": [] }, { "id": 13929, "type": "theorem", "label": "more-morphisms-lemma-category-quasi-projective", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-category-quasi-projective", "contents": [ "Let $S$ be a scheme which has an ample invertible sheaf.", "Let $\\text{QP}_S$ be the full subcategory of the", "category of schemes over $S$ satisfying the equivalent", "conditions of Lemma \\ref{lemma-quasi-projective}.", "\\begin{enumerate}", "\\item if $S' \\to S$ is a morphism of schemes and $S'$ has", "an ample invertible sheaf, then base change determines", "a functor $\\text{QP}_S \\to \\text{QP}_{S'}$,", "\\item if $X \\in \\text{QP}_S$ and $Y \\in \\text{QP}_X$, then $Y \\in \\text{QP}_S$,", "\\item the category $\\text{QP}_S$ is closed under fibre products,", "\\item the category $\\text{QP}_S$ is closed under", "finite disjoint unions,", "\\item if $X \\to S$ is projective, then $X \\in \\text{QP}_S$,", "\\item if $X \\to S$ is quasi-affine of finite type, then", "$X$ is in $\\text{QP}_S$,", "\\item if $X \\to S$ is quasi-finite and separated, then", "$X \\in \\text{QP}_S$,", "\\item if $X \\to S$ is a quasi-compact immersion, then", "$X \\in \\text{QP}_S$,", "\\item add more here.", "\\end{enumerate}" ], "refs": [ "more-morphisms-lemma-quasi-projective" ], "proofs": [ { "contents": [ "Part (1) follows from Morphisms, Lemma", "\\ref{morphisms-lemma-base-change-quasi-projective}.", "\\medskip\\noindent", "Part (2) follows from the fourth characterization of", "Lemma \\ref{lemma-quasi-projective}.", "\\medskip\\noindent", "If $X \\to S$ and $Y \\to S$ are quasi-projective, then", "$X \\times_S Y \\to Y$ is quasi-projective by", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-quasi-projective}.", "Hence (3) follows from (2).", "\\medskip\\noindent", "If $X = Y \\amalg Z$ is a disjoint union of schemes", "and $\\mathcal{L}$ is an invertible $\\mathcal{O}_X$-module", "such that $\\mathcal{L}|_Y$ and $\\mathcal{L}|_Z$ are ample, then", "$\\mathcal{L}$ is ample (details omitted). Thus", "part (4) follows from the fourth characterization of", "Lemma \\ref{lemma-quasi-projective}.", "\\medskip\\noindent", "Part (5) follows from", "Morphisms, Lemma \\ref{morphisms-lemma-projective-quasi-projective}.", "\\medskip\\noindent", "Part (6) follows from", "Morphisms, Lemma", "\\ref{morphisms-lemma-quasi-affine-finite-type-quasi-projective}.", "\\medskip\\noindent", "Part (7) follows from part (6) and", "Lemma \\ref{lemma-quasi-finite-separated-quasi-affine}.", "\\medskip\\noindent", "Part (8) follows from part (7) and", "Morphisms, Lemma \\ref{morphisms-lemma-immersion-locally-quasi-finite}." ], "refs": [ "morphisms-lemma-base-change-quasi-projective", "more-morphisms-lemma-quasi-projective", "morphisms-lemma-base-change-quasi-projective", "more-morphisms-lemma-quasi-projective", "morphisms-lemma-projective-quasi-projective", "morphisms-lemma-quasi-affine-finite-type-quasi-projective", "more-morphisms-lemma-quasi-finite-separated-quasi-affine", "morphisms-lemma-immersion-locally-quasi-finite" ], "ref_ids": [ 5399, 13928, 5399, 13928, 5427, 5404, 13900, 5236 ] } ], "ref_ids": [ 13928 ] }, { "id": 13930, "type": "theorem", "label": "more-morphisms-lemma-integral-over-quasi-affine", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-integral-over-quasi-affine", "contents": [ "Let $X$ be a quasi-affine scheme. Let $f : U \\to X$ be an integral", "morphism. Then $U$ is quasi-affine and the diagram", "$$", "\\xymatrix{", "U \\ar[r] \\ar[d] & \\Spec(\\Gamma(U, \\mathcal{O}_U)) \\ar[d] \\\\", "X \\ar[r] & \\Spec(\\Gamma(X, \\mathcal{O}_X))", "}", "$$", "is cartesian." ], "refs": [], "proofs": [ { "contents": [ "The scheme $U$ is quasi-affine because integral morphisms are affine,", "affine morphisms are quasi-affine, a scheme is quasi-affine if and only if", "the structure morphism to $\\Spec(\\mathbf{Z})$ is quasi-affine, and", "compositions of quasi-affine morphisms are quasi-affine.", "The first two statements follow immediately from the definition", "and the third is", "Morphisms, Lemma \\ref{morphisms-lemma-composition-quasi-affine}.", "Set $U' =", "X \\times_{\\Spec(\\Gamma(X, \\mathcal{O}_X))} \\Spec(\\Gamma(U, \\mathcal{O}_U))$", "and consider the extended diagram", "$$", "\\xymatrix{", "U \\ar[r]_j \\ar[rd] & U' \\ar[d] \\ar[r] &", "\\Spec(\\Gamma(U, \\mathcal{O}_U)) \\ar[d] \\\\", "& X \\ar[r] & \\Spec(\\Gamma(X, \\mathcal{O}_X))", "}", "$$", "The morphism $j$ is closed by", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}", "combined with the fact that an integral morphism is universally closed", "(Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed})", "and the fact that the vertical arrows are in the diagram are separated.", "On the other hand, $j$ is open because the horizontal", "arrows in the diagram of the lemma are open by", "Properties, Lemma \\ref{properties-lemma-quasi-affine}.", "Thus $j$ identifies $U$ with an open and closed subscheme of $U'$.", "If $U \\not = U'$ then $U$ isn't dense in $U'$ and a fortiori", "not dense in the spectrum of $\\Gamma(U, \\mathcal{O}_U)$.", "However, the scheme theoretic image of", "$U$ in $\\Spec(\\Gamma(U, \\mathcal{O}_U))$ is $\\Spec(\\Gamma(U, \\mathcal{O}_U))$", "because any ideal in $\\Gamma(U, \\mathcal{O}_U)$", "cutting out a closed subscheme through which $U$", "factors would have to be zero.", "Hence $U$ is dense in $\\Spec(\\Gamma(U, \\mathcal{O}_U))$ for example by", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}.", "Thus $U = U'$ and we win." ], "refs": [ "morphisms-lemma-composition-quasi-affine", "morphisms-lemma-image-proper-scheme-closed", "morphisms-lemma-integral-universally-closed", "properties-lemma-quasi-affine", "morphisms-lemma-quasi-compact-scheme-theoretic-image" ], "ref_ids": [ 5186, 5411, 5441, 3009, 5146 ] } ], "ref_ids": [] }, { "id": 13931, "type": "theorem", "label": "more-morphisms-lemma-projective", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-projective", "contents": [ "Let $S$ be a scheme which has an ample invertible sheaf.", "Let $f : X \\to S$ be a morphism of schemes. The following are", "equivalent", "\\begin{enumerate}", "\\item $X \\to S$ is projective,", "\\item $X \\to S$ is H-projective,", "\\item $X \\to S$ is quasi-projective and proper,", "\\item $X \\to S$ is H-quasi-projective and proper,", "\\item $X \\to S$ is proper and $X$ has an ample invertible sheaf,", "\\item $X \\to S$ is proper and there exists an $f$-ample invertible sheaf,", "\\item $X \\to S$ is proper and there exists an $f$-very ample invertible sheaf,", "\\item there is a quasi-coherent graded $\\mathcal{O}_S$-algebra $\\mathcal{A}$", "generated by $\\mathcal{A}_1$ over $\\mathcal{A}_0$ with $\\mathcal{A}_1$ a", "finite type $\\mathcal{O}_S$-module such that", "$X = \\underline{\\text{Proj}}_S(\\mathcal{A})$. ", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe first that in each case the morphism $f$ is proper, see", "Morphisms, Lemmas \\ref{morphisms-lemma-H-projective} and", "\\ref{morphisms-lemma-locally-projective-proper}.", "Hence it suffices to prove the equivalence of the notions in", "case $f$ is a proper morphism. We will use this without further", "mention in the following.", "\\medskip\\noindent", "The equivalences (1) $\\Leftrightarrow$ (3) and", "(2) $\\Leftrightarrow$ (4) are", "Morphisms, Lemma \\ref{morphisms-lemma-projective-is-quasi-projective-proper}.", "\\medskip\\noindent", "The implication (2) $\\Rightarrow$ (1) is", "Morphisms, Lemma \\ref{morphisms-lemma-H-projective}.", "\\medskip\\noindent", "The implications (1) $\\Rightarrow$ (2) and (3) $\\Rightarrow$ (4) are", "Morphisms, Lemma", "\\ref{morphisms-lemma-projective-over-quasi-projective-is-H-projective}.", "\\medskip\\noindent", "The implication (1) $\\Rightarrow$ (7) is immediate from", "Morphisms, Definitions \\ref{morphisms-definition-projective} and", "\\ref{morphisms-definition-very-ample}.", "\\medskip\\noindent", "The conditions (3) and (6) are equivalent by", "Morphisms, Definition \\ref{morphisms-definition-quasi-projective}.", "\\medskip\\noindent", "Thus (1) -- (4), (6) are equivalent and imply (7). By", "Lemma \\ref{lemma-quasi-projective}", "conditions (3), (5), and (7) are equivalent.", "Thus we see that (1) -- (7) are equivalent.", "\\medskip\\noindent", "By Divisors, Lemma \\ref{divisors-lemma-relative-proj-projective}", "we see that (8) implies (1). Conversely, if (2) holds, then", "we can choose a closed immersion", "$$", "i :", "X", "\\longrightarrow", "\\mathbf{P}^n_S = \\underline{\\text{Proj}}_S(\\mathcal{O}_S[T_0, \\ldots, T_n]).", "$$", "See Constructions, Lemma \\ref{constructions-lemma-projective-space-bundle}", "for the equality. By", "Divisors, Lemma \\ref{divisors-lemma-closed-subscheme-proj}", "we see that $X$ is the relative Proj of a quasi-coherent graded quotient", "algebra $\\mathcal{A}$ of $\\mathcal{O}_S[T_0, \\ldots, T_n]$.", "Then $\\mathcal{A}$ satisfies the conditions of (8)." ], "refs": [ "morphisms-lemma-H-projective", "morphisms-lemma-locally-projective-proper", "morphisms-lemma-projective-is-quasi-projective-proper", "morphisms-lemma-H-projective", "morphisms-lemma-projective-over-quasi-projective-is-H-projective", "morphisms-definition-projective", "morphisms-definition-very-ample", "morphisms-definition-quasi-projective", "more-morphisms-lemma-quasi-projective", "divisors-lemma-relative-proj-projective", "constructions-lemma-projective-space-bundle", "divisors-lemma-closed-subscheme-proj" ], "ref_ids": [ 5420, 5422, 5430, 5420, 5433, 5572, 5569, 5570, 13928, 8044, 12652, 8047 ] } ], "ref_ids": [] }, { "id": 13932, "type": "theorem", "label": "more-morphisms-lemma-category-projective", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-category-projective", "contents": [ "Let $S$ be a scheme which has an ample invertible sheaf.", "Let $\\text{P}_S$ be the full subcategory of the", "category of schemes over $S$ satisfying the equivalent", "conditions of Lemma \\ref{lemma-projective}.", "\\begin{enumerate}", "\\item if $S' \\to S$ is a morphism of schemes and $S'$ has", "an ample invertible sheaf, then base change determines", "a functor $\\text{P}_S \\to \\text{P}_{S'}$,", "\\item if $X \\in \\text{P}_S$ and $Y \\in \\text{P}_X$, then $Y \\in \\text{P}_S$,", "\\item the category $\\text{P}_S$ is closed under fibre products,", "\\item the category $\\text{P}_S$ is closed under", "finite disjoint unions,", "\\item if $X \\to S$ is finite, then $X$ is in $\\text{P}_S$,", "\\item add more here.", "\\end{enumerate}" ], "refs": [ "more-morphisms-lemma-projective" ], "proofs": [ { "contents": [ "Part (1) follows from Morphisms, Lemma", "\\ref{morphisms-lemma-base-change-projective}.", "\\medskip\\noindent", "Part (2) follows from the fifth characterization of", "Lemma \\ref{lemma-projective} and the fact that compositions", "of proper morphisms are proper", "(Morphisms, Lemma \\ref{morphisms-lemma-composition-proper}).", "\\medskip\\noindent", "If $X \\to S$ and $Y \\to S$ are projective, then", "$X \\times_S Y \\to Y$ is projective by", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-projective}.", "Hence (3) follows from (2).", "\\medskip\\noindent", "If $X = Y \\amalg Z$ is a disjoint union of schemes", "and $\\mathcal{L}$ is an invertible $\\mathcal{O}_X$-module", "such that $\\mathcal{L}|_Y$ and $\\mathcal{L}|_Z$ are ample, then", "$\\mathcal{L}$ is ample (details omitted). Thus", "part (4) follows from the fifth characterization of", "Lemma \\ref{lemma-projective}.", "\\medskip\\noindent", "Part (5) follows from", "Morphisms, Lemma \\ref{morphisms-lemma-finite-projective}." ], "refs": [ "morphisms-lemma-base-change-projective", "more-morphisms-lemma-projective", "morphisms-lemma-composition-proper", "morphisms-lemma-base-change-projective", "more-morphisms-lemma-projective", "morphisms-lemma-finite-projective" ], "ref_ids": [ 5426, 13931, 5408, 5426, 13931, 5450 ] } ], "ref_ids": [ 13931 ] }, { "id": 13933, "type": "theorem", "label": "more-morphisms-lemma-ample-in-neighbourhood", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-ample-in-neighbourhood", "contents": [ "\\begin{reference}", "\\cite[IV Corollary 9.6.4]{EGA}", "\\end{reference}", "Let $f : X \\to Y$ be a proper morphism of schemes.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $y \\in Y$ be a point such that $\\mathcal{L}_y$ is ample", "on $X_y$. Then there is an open neighbourhood $V \\subset Y$", "of $y$ such that $\\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $Y$ is affine. Then we find a directed set $I$", "and an inverse system of morphisms $X_i \\to Y_i$ of schemes", "with $Y_i$ of finite type over $\\mathbf{Z}$, with affine", "transition morphisms $X_i \\to X_{i'}$ and $Y_i \\to Y_{i'}$,", "with $X_i \\to Y_i$ proper, such that $X \\to Y = \\lim (X_i \\to Y_i)$.", "See Limits, Lemma", "\\ref{limits-lemma-proper-limit-of-proper-finite-presentation-noetherian}.", "After shrinking $I$ we can assume we have a compatible system of", "invertible $\\mathcal{O}_{X_i}$-modules $\\mathcal{L}_i$", "pulling back to $\\mathcal{L}$, see", "Limits, Lemma \\ref{limits-lemma-descend-invertible-modules}.", "Let $y_i \\in Y_i$ be the image of $y$.", "Then $\\kappa(y) = \\colim \\kappa(y_i)$.", "Hence for some $i$ we have $\\mathcal{L}_{i, y_i}$", "is ample on $X_{i, y_i}$ by", "Limits, Lemma \\ref{limits-lemma-limit-ample}.", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-ample-in-neighbourhood}", "we find an open neigbourhood", "$V_i \\subset Y_i$ of $y_i$ such that", "$\\mathcal{L}_i$ restricted to $f_i^{-1}(V_i)$", "is ample relative to $V_i$.", "Letting $V \\subset Y$ be the inverse image of", "$V_i$ finishes the proof (hints: use", "Morphisms, Lemma \\ref{morphisms-lemma-ample-base-change} and", "the fact that $X \\to Y \\times_{Y_i} X_i$ is affine", "and the fact that the pullback of an", "ample invertible sheaf by an affine morphism is ample by", "Morphisms, Lemma \\ref{morphisms-lemma-pullback-ample-tensor-relatively-ample})." ], "refs": [ "limits-lemma-proper-limit-of-proper-finite-presentation-noetherian", "limits-lemma-descend-invertible-modules", "limits-lemma-limit-ample", "coherent-lemma-ample-in-neighbourhood", "morphisms-lemma-ample-base-change", "morphisms-lemma-pullback-ample-tensor-relatively-ample" ], "ref_ids": [ 15091, 15079, 15045, 3368, 5385, 5383 ] } ], "ref_ids": [] }, { "id": 13934, "type": "theorem", "label": "more-morphisms-lemma-apply-proj-spec", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-apply-proj-spec", "contents": [ "Let $R$ be a ring. Let $P$ be a proper scheme over $R$ and let", "$\\mathcal{L}$ be an ample invertible $\\mathcal{O}_P$-module.", "Set $A = \\bigoplus_{m \\geq 0} \\Gamma(P, \\mathcal{L}^{\\otimes m})$.", "Then $P = \\text{Proj}(A)$ and diagram (\\ref{equation-proj-and-spec})", "becomes the diagram", "$$", "\\xymatrix{", "\\underline{\\Spec}_P \\left(", "\\bigoplus\\nolimits_{m \\in \\mathbf{Z}} \\mathcal{L}^{\\otimes m}", "\\right)", "\\ar[r] \\ar@{=}[d] &", "L =", "\\underline{\\Spec}_P \\left(", "\\bigoplus\\nolimits_{m \\geq 0} \\mathcal{L}^{\\otimes m}", "\\right) \\ar[d]^\\sigma \\ar[r]_-\\pi & P \\ar[d] \\\\", "U \\ar[r] & X \\ar[r] & Z", "}", "$$", "having the properties explained above." ], "refs": [], "proofs": [ { "contents": [ "We have $P = \\text{Proj}(A)$ by", "Morphisms, Lemma \\ref{morphisms-lemma-proper-ample-is-proj}.", "Moreover, by Properties, Lemma \\ref{properties-lemma-ample-gcd-is-one}", "via this identification we have $\\mathcal{O}_P(m) = \\mathcal{L}^{\\otimes m}$", "for all $m \\in \\mathbf{Z}$." ], "refs": [ "morphisms-lemma-proper-ample-is-proj", "properties-lemma-ample-gcd-is-one" ], "ref_ids": [ 5434, 3056 ] } ], "ref_ids": [] }, { "id": 13935, "type": "theorem", "label": "more-morphisms-lemma-locally-principal-vertical", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-locally-principal-vertical", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $Z \\subset X$ be a closed subscheme.", "Let $s \\in S$.", "Assume", "\\begin{enumerate}", "\\item $S$ is irreducible with generic point $\\eta$,", "\\item $X$ is irreducible,", "\\item $f$ is dominant,", "\\item $f$ is locally of finite type,", "\\item $\\dim(X_s) \\leq \\dim(X_\\eta)$,", "\\item $Z$ is locally principal in $X$, and", "\\item $Z_\\eta = \\emptyset$.", "\\end{enumerate}", "Then the fibre $Z_s$ is (set theoretically) a union of", "irreducible components of $X_s$." ], "refs": [], "proofs": [ { "contents": [ "Let $X_{red}$ denote the reduction of $X$. Then $Z \\cap X_{red}$ is", "a locally principal closed subscheme of $X_{red}$, see", "Divisors, Lemma \\ref{divisors-lemma-pullback-locally-principal}.", "Hence we may assume that $X$ is reduced. In other words $X$ is integral, see", "Properties, Lemma \\ref{properties-lemma-characterize-integral}.", "In this case the morphism $X \\to S$ factors through $S_{red}$, see", "Schemes, Lemma \\ref{schemes-lemma-map-into-reduction}.", "Thus we may replace $S$ by $S_{red}$ and assume that $S$ is integral too.", "\\medskip\\noindent", "The assertion that $f$ is dominant signifies that the generic point of $X$", "is mapped to $\\eta$, see", "Morphisms,", "Lemma \\ref{morphisms-lemma-dominant-finite-number-irreducible-components}.", "Moreover, the scheme $X_\\eta$ is an integral scheme which is locally of", "finite type over the field $\\kappa(\\eta)$. Hence", "$d = \\dim(X_\\eta) \\geq 0$ is equal to $\\dim_\\xi(X_\\eta)$ for", "every point $\\xi$ of $X_\\eta$, see", "Algebra, Lemmas \\ref{algebra-lemma-dimension-spell-it-out} and", "\\ref{algebra-lemma-dimension-at-a-point-finite-type-over-field}.", "In view of", "Morphisms, Lemma \\ref{morphisms-lemma-openness-bounded-dimension-fibres}", "and condition (5) we conclude that $\\dim_x(X_s) = d$", "for every $x \\in X_s$.", "\\medskip\\noindent", "In the Noetherian case the assertion can be proved as follows.", "If the lemma does not holds there exists $x \\in Z_s$ which is a generic", "point of an irreducible component of $Z_s$ but not a generic point", "of any irreducible component of $X_s$. Then we see that", "$\\dim_x(Z_s) \\leq d - 1$, because $\\dim_x(X_s) = d$ and in a neighbourhood", "of $x$ in $X_s$ the closed subscheme $Z_s$ does not contain any of the", "irreducible components of $X_s$. Hence after replacing $X$ by an", "open neighbourhood of $x$ we may assume that", "$\\dim_z(Z_{f(z)}) \\leq d - 1$ for all $z \\in Z$, see", "Morphisms, Lemma \\ref{morphisms-lemma-openness-bounded-dimension-fibres}.", "Let $\\xi' \\in Z$ be a generic point of an irreducible component of $Z$", "and set $s' = f(\\xi)$. As $Z \\not = X$ is locally principal we see that", "$\\dim(\\mathcal{O}_{X, \\xi}) = 1$, see", "Algebra, Lemma \\ref{algebra-lemma-minimal-over-1}", "(this is where we use $X$ is Noetherian).", "Let $\\xi \\in X$ be the generic point of $X$ and", "let $\\xi_1$ be a generic point of any irreducible component", "of $X_{s'}$ which contains $\\xi'$. Then we see that we have", "the specializations", "$$", "\\xi \\leadsto \\xi_1 \\leadsto \\xi'.", "$$", "As $\\dim(\\mathcal{O}_{X, \\xi}) = 1$ one of the two specializations", "has to be an equality.", "By assumption $s' \\not = \\eta$, hence the first specialization", "is not an equality.", "Hence $\\xi' = \\xi_1$ is a generic point of an irreducible component of", "$X_{s'}$. Applying", "Morphisms, Lemma \\ref{morphisms-lemma-openness-bounded-dimension-fibres}", "one more time this implies", "$\\dim_{\\xi'}(Z_{s'}) = \\dim_{\\xi'}(X_{s'}) \\geq \\dim(X_\\eta) = d$", "which gives the desired contradiction.", "\\medskip\\noindent", "In the general case we reduce to the Noetherian case as follows.", "If the lemma is false then there exists a point", "$x \\in X$ lying over $s$ such that $x$ is a generic point of an", "irreducible component of $Z_s$, but", "not a generic point of any of the irreducible components of $X_s$.", "Let $U \\subset S$ be an affine neighbourhood of $s$ and let", "$V \\subset X$ be an affine neighbourhood of $x$ with $f(V) \\subset U$.", "Write $U = \\Spec(A)$ and $V = \\Spec(B)$ so that $f|_V$", "is given by a ring map $A \\to B$. Let $\\mathfrak q \\subset B$,", "resp.\\ $\\mathfrak p \\subset A$ be the prime corresponding to $x$, resp.\\ $s$.", "After possibly shrinking $V$ we may assume $Z \\cap V$ is cut out by", "some element $g \\in B$. Denote $K$ the fraction field of $A$.", "What we know at this point is the following:", "\\begin{enumerate}", "\\item $A \\subset B$ is a finitely generated extension of domains,", "\\item the element $g \\otimes 1$ is invertible in $B \\otimes_A K$,", "\\item $d = \\dim(B \\otimes_A K) = \\dim(B \\otimes_A \\kappa(\\mathfrak p))$,", "\\item $g \\otimes 1$ is not a unit of $B \\otimes_A \\kappa(\\mathfrak p)$, and", "\\item $g \\otimes 1$ is not in any of the minimal primes of", "$B \\otimes_A \\kappa(\\mathfrak p)$.", "\\end{enumerate}", "We are seeking a contradiction.", "\\medskip\\noindent", "Pick elements $x_1, \\ldots, x_n \\in B$ which generate $B$ over $A$.", "For a finitely generated $\\mathbf{Z}$-algebra $A_0 \\subset A$", "let $B_0 \\subset B$ be the $A_0$-subalgebra generated by", "$x_1, \\ldots, x_n$, denote $K_0$ the fraction field of $A_0$, and set", "$\\mathfrak p_0 = A_0 \\cap \\mathfrak p$.", "We claim that when $A_0$ is large enough then (1) -- (5) also hold for", "the system $(A_0 \\subset B_0, g, \\mathfrak p_0)$.", "\\medskip\\noindent", "We prove each of the conditions in turn. Part (1) holds by construction.", "For part (2) write $(g \\otimes 1) h = 1$ for some", "$h \\otimes 1/a \\in B \\otimes_A K$. Write", "$g = \\sum a_I x^I$, $h = \\sum a'_I x^I$ (multi-index notation)", "for some coefficients $a_I, a'_I \\in A$. As soon as $A_0$ contains", "$a$ and the $a_I, a'_I$ then (2) holds because", "$B_0 \\otimes_{A_0} K_0 \\subset B \\otimes_A K$ (as localizations of the", "injective map $B_0 \\to B$).", "To achieve (3) consider the exact sequence", "$$", "0 \\to I \\to A[X_1, \\ldots, X_n] \\to B \\to 0", "$$", "which defines $I$ where the second map sends $X_i$ to $x_i$. Since $\\otimes$", "is right exact we see that $I \\otimes_A K$, respectively", "$I \\otimes_A \\kappa(\\mathfrak p)$ is the kernel of the surjection", "$K[X_1, \\ldots, X_n] \\to B \\otimes_A K$, respectively", "$\\kappa(\\mathfrak p)[X_1, \\ldots, X_n] \\to B \\otimes_A \\kappa(\\mathfrak p)$.", "As a polynomial ring over a field is Noetherian", "there exist finitely many elements $h_j \\in I$, $j = 1, \\ldots, m$", "which generate $I \\otimes_A K$ and $I \\otimes_A \\kappa(\\mathfrak p)$.", "Write $h_j = \\sum a_{j, I}X^I$. As soon as", "$A_0$ contains all $a_{j, I}$ we get to the situation where", "$$", "B_0 \\otimes_{A_0} K_0 \\otimes_{K_0} K = B \\otimes_A K", "\\quad\\text{and}\\quad", "B_0 \\otimes_{A_0} \\kappa(\\mathfrak p_0)", "\\otimes_{\\kappa(\\mathfrak p_0)} \\kappa(\\mathfrak p)", "=", "B \\otimes_A \\kappa(\\mathfrak p).", "$$", "By either", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}", "or", "Algebra, Lemma \\ref{algebra-lemma-dimension-preserved-field-extension}", "we see that the dimension equalities of (3) are satisfied.", "Part (4) is immediate. As", "$B_0 \\otimes_{A_0} \\kappa(\\mathfrak p_0) \\subset", "B \\otimes_A \\kappa(\\mathfrak p)$ each minimal prime of", "$B_0 \\otimes_{A_0} \\kappa(\\mathfrak p_0)$ lies under a minimal", "prime of $B \\otimes_A \\kappa(\\mathfrak p)$ by", "Algebra, Lemma \\ref{algebra-lemma-image-dense-generic-points}.", "This implies that (5) holds.", "In this way we reduce the problem to the Noetherian case which we", "have dealt with above." ], "refs": [ "divisors-lemma-pullback-locally-principal", "properties-lemma-characterize-integral", "schemes-lemma-map-into-reduction", "morphisms-lemma-dominant-finite-number-irreducible-components", "algebra-lemma-dimension-spell-it-out", "algebra-lemma-dimension-at-a-point-finite-type-over-field", "morphisms-lemma-openness-bounded-dimension-fibres", "morphisms-lemma-openness-bounded-dimension-fibres", "algebra-lemma-minimal-over-1", "morphisms-lemma-openness-bounded-dimension-fibres", "morphisms-lemma-dimension-fibre-after-base-change", "algebra-lemma-dimension-preserved-field-extension", "algebra-lemma-image-dense-generic-points" ], "ref_ids": [ 7935, 2947, 7682, 5161, 994, 995, 5280, 5280, 683, 5280, 5279, 1009, 446 ] } ], "ref_ids": [] }, { "id": 13936, "type": "theorem", "label": "more-morphisms-lemma-horizontal", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-horizontal", "contents": [ "Let $A \\to B$ be a local homomorphism of local rings, and", "$g \\in \\mathfrak m_B$. Assume", "\\begin{enumerate}", "\\item $A$ and $B$ are domains and $A \\subset B$,", "\\item $B$ is essentially of finite type over $A$,", "\\item $g$ is not contained in any minimal prime over $\\mathfrak m_AB$, and", "\\item $\\dim(B/\\mathfrak m_AB) +", "\\text{trdeg}_{\\kappa(\\mathfrak m_A)}(\\kappa(\\mathfrak m_B)) =", "\\text{trdeg}_A(B)$.", "\\end{enumerate}", "Then $A \\subset B/gB$, i.e., the generic point of $\\Spec(A)$", "is in the image of the morphism $\\Spec(B/gB) \\to \\Spec(A)$." ], "refs": [], "proofs": [ { "contents": [ "Note that the two assertions are equivalent by", "Algebra, Lemma \\ref{algebra-lemma-image-dense-generic-points}.", "To start the proof let $C$ be an $A$-algebra of finite type", "and $\\mathfrak q$ a prime of $C$ such that $B = C_{\\mathfrak q}$.", "Of course we may assume that $C$ is a domain and that $g \\in C$.", "After replacing $C$ by a localization we see that", "$\\dim(C/\\mathfrak m_AC) = \\dim(B/\\mathfrak m_AB) +", "\\text{trdeg}_{\\kappa(\\mathfrak m_A)}(\\kappa(\\mathfrak m_B))$, see", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}.", "Setting $K$ equal to the fraction field of $A$", "we see by the same reference that", "$\\dim(C \\otimes_A K) = \\text{trdeg}_A(B)$. Hence assumption", "(4) means that the generic and closed fibres of the morphism", "$\\Spec(C) \\to \\Spec(A)$ have the same dimension.", "\\medskip\\noindent", "Suppose that the lemma is false. Then $(B/gB) \\otimes_A K = 0$.", "This means that $g \\otimes 1$ is invertible in $B \\otimes_A K", "= C_{\\mathfrak q} \\otimes_A K$. As $C_{\\mathfrak q}$ is a limit", "of principal localizations we conclude that $g \\otimes 1$", "is invertible in $C_h \\otimes_A K$ for some", "$h \\in C$, $h \\not \\in \\mathfrak q$. Thus after replacing $C$", "by $C_h$ we may assume that $(C/gC) \\otimes_A K = 0$.", "We do one more replacement of $C$ to make sure that the minimal", "primes of $C/\\mathfrak m_AC$ correspond one-to-one with the minimal", "primes of $B/\\mathfrak m_AB$. At this point we apply", "Lemma \\ref{lemma-locally-principal-vertical}", "to $X = \\Spec(C) \\to \\Spec(A) = S$ and the locally closed", "subscheme $Z = \\Spec(C/gC)$. Since $Z_K = \\emptyset$ we see that", "$Z \\otimes \\kappa(\\mathfrak m_A)$ has to contain an irreducible", "component of", "$X \\otimes \\kappa(\\mathfrak m_A) = \\Spec(C/\\mathfrak m_AC)$.", "But this contradicts the assumption that $g$ is not contained", "in any prime minimal over $\\mathfrak m_AB$. The lemma follows." ], "refs": [ "algebra-lemma-image-dense-generic-points", "morphisms-lemma-dimension-fibre-at-a-point", "more-morphisms-lemma-locally-principal-vertical" ], "ref_ids": [ 446, 5277, 13935 ] } ], "ref_ids": [] }, { "id": 13937, "type": "theorem", "label": "more-morphisms-lemma-equality-dimensions", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-equality-dimensions", "contents": [ "Let $A \\to B$ be a local homomorphism of local rings. Assume", "\\begin{enumerate}", "\\item $A$ and $B$ are domains and $A \\subset B$,", "\\item $B$ is essentially of finite type over $A$, and", "\\item $B$ is flat over $A$.", "\\end{enumerate}", "Then we have", "$$", "\\dim(B/\\mathfrak m_AB) +", "\\text{trdeg}_{\\kappa(\\mathfrak m_A)}(\\kappa(\\mathfrak m_B)) =", "\\text{trdeg}_A(B).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let $C$ be an $A$-algebra of finite type and $\\mathfrak q$ a prime of $C$", "such that $B = C_{\\mathfrak q}$. We may assume $C$ is a domain.", "We have", "$\\dim_{\\mathfrak q}(C/\\mathfrak m_AC) = \\dim(B/\\mathfrak m_AB) +", "\\text{trdeg}_{\\kappa(\\mathfrak m_A)}(\\kappa(\\mathfrak m_B))$, see", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-at-a-point}.", "Setting $K$ equal to the fraction field of $A$", "we see by the same reference that", "$\\dim(C \\otimes_A K) = \\text{trdeg}_A(B)$.", "Thus we are really trying to prove that", "$\\dim_{\\mathfrak q}(C/\\mathfrak m_AC) = \\dim(C \\otimes_A K)$.", "Choose a valuation ring $A'$ in $K$ dominating $A$, see", "Algebra, Lemma \\ref{algebra-lemma-dominate}.", "Set $C' = C \\otimes_A A'$.", "Choose a prime $\\mathfrak q'$ of $C'$ lying over $\\mathfrak q$; such a", "prime exists because", "$$", "C'/\\mathfrak m_{A'}C' =", "C/\\mathfrak m_AC \\otimes_{\\kappa(\\mathfrak m_A)} \\kappa(\\mathfrak m_{A'})", "$$", "which proves that $C/\\mathfrak m_AC \\to C'/\\mathfrak m_{A'}C'$ is faithfully", "flat. This also proves that", "$\\dim_{\\mathfrak q}(C/\\mathfrak m_AC) =", "\\dim_{\\mathfrak q'}(C'/\\mathfrak m_{A'}C')$, see", "Algebra,", "Lemma \\ref{algebra-lemma-dimension-at-a-point-preserved-field-extension}.", "Note that $B' = C'_{\\mathfrak q'}$ is a localization of $B \\otimes_A A'$.", "Hence $B'$ is flat over $A'$. The generic fibre $B' \\otimes_{A'} K$", "is a localization of $B \\otimes_A K$. Hence $B'$ is a domain.", "If we prove the lemma for $A' \\subset B'$, then we get the equality", "$\\dim_{\\mathfrak q'}(C'/\\mathfrak m_{A'}C') = \\dim(C' \\otimes_{A'} K)$", "which implies the desired equality", "$\\dim_{\\mathfrak q}(C/\\mathfrak m_AC) = \\dim(C \\otimes_A K)$", "by what was said above. This reduces the", "lemma to the case where $A$ is a valuation ring.", "\\medskip\\noindent", "Let $A \\subset B$ be as in the lemma with $A$ a valuation ring.", "As before write $B = C_{\\mathfrak q}$ for some domain $C$ of finite", "type over $A$. By", "Algebra,", "Lemma \\ref{algebra-lemma-finite-type-domain-over-valuation-ring-dim-fibres}", "we obtain $\\dim(C/\\mathfrak m_AC) = \\dim(C \\otimes_A K)$ and we win." ], "refs": [ "morphisms-lemma-dimension-fibre-at-a-point", "algebra-lemma-dominate", "algebra-lemma-dimension-at-a-point-preserved-field-extension", "algebra-lemma-finite-type-domain-over-valuation-ring-dim-fibres" ], "ref_ids": [ 5277, 608, 1010, 1078 ] } ], "ref_ids": [] }, { "id": 13938, "type": "theorem", "label": "more-morphisms-lemma-closed-point-nearby-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-closed-point-nearby-fibre", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\leadsto x'$ be a specialization of points in $X$.", "Set $s = f(x)$ and $s' = f(x')$.", "Assume", "\\begin{enumerate}", "\\item $x'$ is a closed point of $X_{s'}$, and", "\\item $f$ is locally of finite type.", "\\end{enumerate}", "Then the set", "$$", "\\{x_1 \\in X", "\\text{ such that }", "f(x_1) = s", "\\text{ and }", "x_1\\text{ is closed in }X_s", "\\text{ and }", "x \\leadsto x_1 \\leadsto x'", "\\}", "$$", "is dense in the closure of $x$ in $X_s$." ], "refs": [], "proofs": [ { "contents": [ "We apply", "Schemes, Lemma \\ref{schemes-lemma-points-specialize}", "to the specialization $x \\leadsto x'$.", "This produces a morphism $\\varphi : \\Spec(B) \\to X$", "where $B$ is a valuation ring such that $\\varphi$ maps the", "generic point to $x$ and the closed point to $x'$. We may also", "assume that $\\kappa(x)$ is the fraction field of $B$.", "Let $A = B \\cap \\kappa(s)$. Note that this is a valuation ring (see", "Algebra, Lemma \\ref{algebra-lemma-valuation-ring-cap-field})", "which dominates the image of $\\mathcal{O}_{S, s'} \\to \\kappa(s)$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "\\Spec(B) \\ar[rd] \\ar[r] &", "X_A \\ar[d] \\ar[r] & X \\ar[d] \\\\", "& \\Spec(A) \\ar[r] & S", "}", "$$", "The generic (resp.\\ closed) point of $B$ maps to a point $x_A$", "(resp.\\ $x'_A$) of $X_A$ lying over the generic (resp.\\ closed)", "point of $\\Spec(A)$. Note that $x'_A$ is a closed point", "of the special fibre of $X_A$ by", "Morphisms,", "Lemma \\ref{morphisms-lemma-base-change-closed-point-fibre-locally-finite-type}.", "Note that the generic fibre of $X_A \\to \\Spec(A)$ is isomorphic", "to $X_s$. Thus we have reduced the lemma to the case where $S$ is", "the spectrum of a valuation ring, $s = \\eta \\in S$ is the generic point, and", "$s' \\in S$ is the closed point.", "\\medskip\\noindent", "We will prove the lemma by induction on $\\dim_x(X_\\eta)$.", "If $\\dim_x(X_\\eta) = 0$, then there are no other points of $X_\\eta$", "specializing to $x$ and $x$ is closed in its fibre, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize},", "and the result holds. Assume $\\dim_x(X_\\eta) > 0$.", "\\medskip\\noindent", "Let $X' \\subset X$ be the reduced induced scheme structure on", "the irreducible closed subscheme $\\overline{\\{x\\}}$ of $X$, see", "Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme}.", "To prove the lemma we may replace $X$ by $X'$ as this only decreases", "$\\dim_x(X_\\eta)$. Hence we may also assume that $X$ is an integral scheme", "and that $x$ is its generic point. In addition, we may replace $X$ by an", "affine neighbourhood of $x'$. Thus we have $X = \\Spec(B)$ where", "$A \\subset B$ is a finite type extension of domains. Note that in", "this case $\\dim_x(X_\\eta) = \\dim(X_\\eta) = \\dim(X_{s'})$, and that in fact", "$X_{s'}$ is equidimensional, see", "Algebra,", "Lemma \\ref{algebra-lemma-finite-type-domain-over-valuation-ring-dim-fibres}.", "\\medskip\\noindent", "Let $W \\subset X_\\eta$ be a proper closed subset (this is the", "subset we want to ``avoid''). As $X_s$ is of finite type over a field", "we see that $W$ has finitely many irreducible components", "$W = W_1 \\cup \\ldots \\cup W_n$. Let", "$\\mathfrak q_j \\subset B$, $j = 1, \\ldots, r$", "be the corresponding prime ideals. Let $\\mathfrak q \\subset B$", "be the maximal ideal corresponding to the point $x'$.", "Let $\\mathfrak p_1, \\ldots, \\mathfrak p_s \\subset B$ be the", "minimal primes lying over $\\mathfrak m_AB$. There are finitely", "many as these correspond to the irreducible components of the", "Noetherian scheme $X_{s'}$. Moreover, each of these irreducible", "components has dimension $> 0$ (see above) hence we see that", "$\\mathfrak p_i \\not = \\mathfrak q$ for all $i$.", "Now, pick an element $g \\in \\mathfrak q$ such that", "$g \\not \\in \\mathfrak q_j$ for all $j$ and $g \\not \\in \\mathfrak p_i$", "for all $i$, see", "Algebra, Lemma \\ref{algebra-lemma-silly}.", "Denote $Z \\subset X$ the locally principal closed subscheme defined by $g$.", "Let $Z_\\eta = Z_{1, \\eta} \\cup \\ldots \\cup Z_{n, \\eta}$, $n \\geq 0$", "be the decomposition of the generic fibre of $Z$ into irreducible", "components (finitely many as the generic fibre is Noetherian).", "Denote $Z_i \\subset X$ the closure of $Z_{i, \\eta}$.", "After replacing $X$ by a smaller affine neighbourhood", "we may assume that $x' \\in Z_i$ for each $i = 1, \\ldots, n$.", "By construction $Z \\cap X_{s'}$ does not contain any irreducible", "component of $X_{s'}$. Hence by", "Lemma \\ref{lemma-locally-principal-vertical}", "we conclude that $Z_\\eta \\not = \\emptyset$! In other words", "$n \\geq 1$. Letting $x_1 \\in Z_1$ be the generic point we see", "that $x_1 \\leadsto x'$ and $f(x_1) = \\eta$.", "Also, by construction $Z_{1, \\eta} \\cap W_j \\subset W_j$", "is a proper closed subset. Hence every irreducible component of", "$Z_{1, \\eta} \\cap W_j$ has codimension $\\geq 2$ in $X_\\eta$", "whereas $\\text{codim}(Z_{1, \\eta}, X_\\eta) = 1$ by", "Algebra, Lemma \\ref{algebra-lemma-minimal-over-1}.", "Thus $W \\cap Z_{1, \\eta}$ is a proper closed subset.", "At this point we see that the induction hypothesis applies to", "$Z_1 \\to S$ and the specialization $x_1 \\leadsto x'$.", "This produces a closed point $x_2$ of $Z_{1, \\eta}$ not contained", "in $W$ which specializes to $x'$. Thus we obtain", "$x \\leadsto x_2 \\leadsto x'$, the point $x_2$ is closed in $X_\\eta$,", "and $x_2 \\not \\in W$ as desired." ], "refs": [ "schemes-lemma-points-specialize", "algebra-lemma-valuation-ring-cap-field", "morphisms-lemma-base-change-closed-point-fibre-locally-finite-type", "morphisms-lemma-quasi-finite-at-point-characterize", "schemes-definition-reduced-induced-scheme", "algebra-lemma-finite-type-domain-over-valuation-ring-dim-fibres", "algebra-lemma-silly", "more-morphisms-lemma-locally-principal-vertical", "algebra-lemma-minimal-over-1" ], "ref_ids": [ 7704, 612, 5224, 5226, 7745, 1078, 378, 13935, 683 ] } ], "ref_ids": [] }, { "id": 13939, "type": "theorem", "label": "more-morphisms-lemma-quasi-finite-quasi-section-meeting-nearby-open", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-quasi-finite-quasi-section-meeting-nearby-open", "contents": [ "Let $\\varphi : A \\to B$ be a local ring map of local rings.", "Let $V \\subset \\Spec(B)$ be an open subscheme", "which contains at least one prime not lying over $\\mathfrak m_A$.", "Assume $A$ is Noetherian, $\\varphi$ essentially of finite type, and", "$A/\\mathfrak m_A \\subset B/\\mathfrak m_B$ is finite.", "Then there exists a $\\mathfrak q \\in V$,", "$\\mathfrak m_A \\not = \\mathfrak q \\cap A$ such that", "$A \\to B/\\mathfrak q$ is the localization of a quasi-finite ring map." ], "refs": [], "proofs": [ { "contents": [ "Since $A$ is Noetherian and $A \\to B$ is essentially of finite type,", "we know that $B$ is Noetherian too. By", "Properties, Lemma \\ref{properties-lemma-complement-closed-point-Jacobson}", "the topological space $\\Spec(B) \\setminus \\{\\mathfrak m_B\\}$", "is Jacobson. Hence we can choose a closed point $\\mathfrak q$", "which is contained in the nonempty open", "$$", "V \\setminus \\{\\mathfrak q \\subset B \\mid \\mathfrak m_A = \\mathfrak q \\cap A\\}.", "$$", "(Nonempty by assumption, open because $\\{\\mathfrak m_A\\}$ is a closed", "subset of $\\Spec(A)$.)", "Then $\\Spec(B/\\mathfrak q)$ has two points, namely $\\mathfrak m_B$", "and $\\mathfrak q$ and $\\mathfrak q$ does not lie over $\\mathfrak m_A$.", "Write $B/\\mathfrak q = C_{\\mathfrak m}$ for some finite type $A$-algebra", "$C$ and prime ideal $\\mathfrak m$. Then $A \\to C$ is quasi-finite at", "$\\mathfrak m$ by", "Algebra, Lemma \\ref{algebra-lemma-isolated-point-fibre} (2).", "Hence by", "Algebra, Lemma \\ref{algebra-lemma-quasi-finite-open}", "we see that after replacing $C$ by a principal localization the ring", "map $A \\to C$ is quasi-finite." ], "refs": [ "properties-lemma-complement-closed-point-Jacobson", "algebra-lemma-isolated-point-fibre", "algebra-lemma-quasi-finite-open" ], "ref_ids": [ 2965, 1049, 1066 ] } ], "ref_ids": [] }, { "id": 13940, "type": "theorem", "label": "more-morphisms-lemma-quasi-finite-quasi-section-meeting-nearby-open-X", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-quasi-finite-quasi-section-meeting-nearby-open-X", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$ with image $s \\in S$.", "Let $U \\subset X$ be an open subscheme.", "Assume $f$ locally of finite type, $S$ locally Noetherian, $x$ a closed", "point of $X_s$, and assume there exists a point $x' \\in U$ with", "$x' \\leadsto x$ and $f(x') \\not = s$. Then there exists a closed", "subscheme $Z \\subset X$ such that (a) $x \\in Z$, (b) $f|_Z : Z \\to S$ is", "quasi-finite at $x$, and (c) there exists a $z \\in Z$, $z \\in U$,", "$z \\leadsto x$ and $f(z) \\not = s$." ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of", "Lemma \\ref{lemma-quasi-finite-quasi-section-meeting-nearby-open}.", "Namely, set $A = \\mathcal{O}_{S, s}$ and $B = \\mathcal{O}_{X, x}$.", "Denote $V \\subset \\Spec(B)$ the inverse image of $U$.", "The ring map $f^\\sharp : A \\to B$ is essentially of finite type.", "By assumption there exists at least one point of $V$ which does not", "map to the closed point of $\\Spec(A)$. Hence all the assumptions of", "Lemma \\ref{lemma-quasi-finite-quasi-section-meeting-nearby-open}", "hold and we obtain a prime $\\mathfrak q \\subset B$ which does not", "lie over $\\mathfrak m_A$ and such that $A \\to B/\\mathfrak q$ is", "the localization of a quasi-finite ring map. Let $z \\in X$ be the", "image of the point $\\mathfrak q$ under the canonical", "morphism $\\Spec(B) \\to X$. Set $Z = \\overline{\\{z\\}}$", "with the induced reduced scheme structure. As $z \\leadsto x$", "we see that $x \\in Z$ and $\\mathcal{O}_{Z, x} = B/\\mathfrak q$.", "By construction $Z \\to S$ is quasi-finite at $x$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13941, "type": "theorem", "label": "more-morphisms-lemma-change-hypotheses", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-change-hypotheses", "contents": [ "Suppose that $f : X \\to S$ is locally of finite type, $S$ locally Noetherian,", "$x \\in X$ a closed point of its fibre $X_s$, and $U \\subset X$ an open", "subscheme such that $U \\cap X_s = \\emptyset$ and $x \\in \\overline{U}$, then", "the conclusions of", "Lemma \\ref{lemma-quasi-finite-quasi-section-meeting-nearby-open-X}", "hold." ], "refs": [], "proofs": [ { "contents": [ "Namely, we can reduce this to the cited lemma as follows: First we", "replace $X$ and $S$ by affine neighbourhoods of $x$ and $s$. Then $X$ is", "Noetherian, in particular $U$ is quasi-compact (see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}", "and", "Topology, Lemmas \\ref{topology-lemma-Noetherian} and", "\\ref{topology-lemma-Noetherian-quasi-compact}).", "Hence there exists a specialization $x' \\leadsto x$ with $x' \\in U$ (see", "Morphisms, Lemma \\ref{morphisms-lemma-reach-points-scheme-theoretic-image}).", "Note that $f(x') \\not = s$. Thus we see all hypotheses of the lemma", "are satisfied and we win." ], "refs": [ "morphisms-lemma-finite-type-noetherian", "topology-lemma-Noetherian", "topology-lemma-Noetherian-quasi-compact", "morphisms-lemma-reach-points-scheme-theoretic-image" ], "ref_ids": [ 5202, 8220, 8239, 5147 ] } ], "ref_ids": [] }, { "id": 13942, "type": "theorem", "label": "more-morphisms-lemma-stein-universally-closed", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-stein-universally-closed", "contents": [ "Let $S$ be a scheme. Let $f : X \\to S$ be a universally closed and", "quasi-separated morphism. There exists a factorization", "$$", "\\xymatrix{", "X \\ar[rr]_{f'} \\ar[rd]_f & & S' \\ar[dl]^\\pi \\\\", "& S &", "}", "$$", "with the following properties:", "\\begin{enumerate}", "\\item the morphism $f'$ is universally closed, quasi-compact, quasi-separated,", "and surjective,", "\\item the morphism $\\pi : S' \\to S$ is integral,", "\\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{S'}$,", "\\item we have $S' = \\underline{\\Spec}_S(f_*\\mathcal{O}_X)$, and", "\\item $S'$ is the normalization of $S$ in $X$, see", "Morphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}.", "\\end{enumerate}", "Formation of the factorization $f = \\pi \\circ f'$ commutes", "with flat base change." ], "refs": [ "morphisms-definition-normalization-X-in-Y" ], "proofs": [ { "contents": [ "By Morphisms, Lemma \\ref{morphisms-lemma-universally-closed-quasi-compact}", "the morphism $f$ is quasi-compact. Hence the normalization $S'$ of $S$ in", "$X$ is defined (Morphisms, Definition", "\\ref{morphisms-definition-normalization-X-in-Y})", "and we have the factorization $X \\to S' \\to S$. By", "Morphisms, Lemma \\ref{morphisms-lemma-normalization-in-universally-closed}", "we have (2), (4), and (5). The morphism $f'$ is universally closed by", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}.", "It is quasi-compact by", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}", "and quasi-separated by", "Schemes, Lemma \\ref{schemes-lemma-compose-after-separated}.", "\\medskip\\noindent", "To show the remaining statements we may assume the base scheme $S$ is affine,", "say $S = \\Spec(R)$. Then $S' = \\Spec(A)$ with", "$A = \\Gamma(X, \\mathcal{O}_X)$ an integral $R$-algebra.", "Thus it is clear that $f'_*\\mathcal{O}_X$", "is $\\mathcal{O}_{S'}$ (because $f'_*\\mathcal{O}_X$ is quasi-coherent,", "by", "Schemes, Lemma", "\\ref{schemes-lemma-push-forward-quasi-coherent},", "and hence equal to $\\widetilde{A}$). This proves (3).", "\\medskip\\noindent", "Let us show that $f'$ is surjective. As $f'$ is universally closed (see above)", "the image of $f'$ is a closed subset", "$V(I) \\subset S' = \\Spec(A)$. Pick $h \\in I$. Then", "$h|_X = f^\\sharp(h)$ is a global section of the structure sheaf of", "$X$ which vanishes at every point. As $X$ is quasi-compact this means", "that $h|_X$ is a nilpotent section, i.e., $h^n|X = 0$ for some $n > 0$.", "But $A = \\Gamma(X, \\mathcal{O}_X)$, hence $h^n = 0$.", "In other words $I$ is contained in the Jacobson radical ideal of $A$", "and we conclude that $V(I) = S'$ as desired." ], "refs": [ "morphisms-lemma-universally-closed-quasi-compact", "morphisms-definition-normalization-X-in-Y", "morphisms-lemma-normalization-in-universally-closed", "morphisms-lemma-image-proper-scheme-closed", "schemes-lemma-quasi-compact-permanence", "schemes-lemma-compose-after-separated", "schemes-lemma-push-forward-quasi-coherent" ], "ref_ids": [ 5412, 5591, 5506, 5411, 7716, 7715, 7730 ] } ], "ref_ids": [ 5591 ] }, { "id": 13943, "type": "theorem", "label": "more-morphisms-lemma-stein-universally-closed-residue-fields", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-stein-universally-closed-residue-fields", "contents": [ "In Lemma \\ref{lemma-stein-universally-closed} assume in addition that", "$f$ is locally of finite type. Then for $s \\in S$ the fibre", "$\\pi^{-1}(\\{s\\}) = \\{s_1, \\ldots, s_n\\}$ is finite and the field extensions", "$\\kappa(s_i)/\\kappa(s)$ are finite." ], "refs": [ "more-morphisms-lemma-stein-universally-closed" ], "proofs": [ { "contents": [ "Recall that there are no specializations among the points of $\\pi^{-1}(\\{s\\})$,", "see Algebra, Lemma \\ref{algebra-lemma-integral-no-inclusion}.", "As $f'$ is surjective, we find that $|X_s| \\to \\pi^{-1}(\\{s\\})$ is surjective.", "Observe that $X_s$ is a quasi-separated scheme of finite type", "over a field (quasi-compactness was shown in the proof of the", "referenced lemma). Thus $X_s$ is Noetherian", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}).", "A topological argument (omitted) now shows that $\\pi^{-1}(\\{s\\})$ is finite.", "For each $i$ we can pick a finite type point $x_i \\in X_s$ mapping to $s_i$", "(Morphisms, Lemma \\ref{morphisms-lemma-enough-finite-type-points}).", "We conclude that $\\kappa(s_i)/\\kappa(s)$ is finite:", "$x_i$ can be represented by a morphism $\\Spec(k_i) \\to X_s$", "of finite type (by our definition of finite type points)", "and hence $\\Spec(k_i) \\to s = \\Spec(\\kappa(s))$ is of finite type", "(as a composition of finite type morphisms),", "hence $k_i/\\kappa(s)$ is finite (Morphisms, Lemma", "\\ref{morphisms-lemma-point-finite-type})." ], "refs": [ "algebra-lemma-integral-no-inclusion", "morphisms-lemma-finite-type-noetherian", "morphisms-lemma-enough-finite-type-points", "morphisms-lemma-point-finite-type" ], "ref_ids": [ 498, 5202, 5210, 5205 ] } ], "ref_ids": [ 13942 ] }, { "id": 13944, "type": "theorem", "label": "more-morphisms-lemma-characterize-geometrically-connected-fibres", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-characterize-geometrically-connected-fibres", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $s \\in S$. Then $X_s$ is geometrically connected, if and", "only if for every \\'etale neighbourhood $(U, u) \\to (S, s)$", "the base change $X_U \\to U$ has connected fibre $X_u$." ], "refs": [], "proofs": [ { "contents": [ "If $X_s$ is geometrically connected, then any base change of it is connected.", "On the other hand, suppose that $X_s$ is not geometrically connected.", "Then by", "Varieties, Lemma", "\\ref{varieties-lemma-characterize-geometrically-disconnected}", "we see that $X_s \\times_{\\Spec(\\kappa(s)} \\Spec(k)$ is", "disconnected for some", "finite separable field extension $\\kappa(s) \\subset k$. By", "Lemma \\ref{lemma-realize-prescribed-residue-field-extension-etale}", "there exists an affine \\'etale neighbourhood $(U, u) \\to (S, s)$ such that", "$\\kappa(s) \\subset \\kappa(u)$ is identified with $\\kappa(s) \\subset k$.", "In this case $X_u$ is disconnected." ], "refs": [ "varieties-lemma-characterize-geometrically-disconnected", "more-morphisms-lemma-realize-prescribed-residue-field-extension-etale" ], "ref_ids": [ 10923, 13866 ] } ], "ref_ids": [] }, { "id": 13945, "type": "theorem", "label": "more-morphisms-lemma-geometrically-connected-fibres-towards-normal", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-geometrically-connected-fibres-towards-normal", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Assume", "\\begin{enumerate}", "\\item $f$ is proper,", "\\item $S$ is integral with generic point $\\xi$,", "\\item $S$ is normal,", "\\item $X$ is reduced,", "\\item every generic point of an irreducible component of $X$ maps to $\\xi$,", "\\item we have $H^0(X_\\xi, \\mathcal{O}) = \\kappa(\\xi)$.", "\\end{enumerate}", "Then $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and $f$", "has geometrically connected fibres." ], "refs": [], "proofs": [ { "contents": [ "Apply Theorem \\ref{theorem-stein-factorization-general} to get a", "factorization $X \\to S' \\to S$. It is enough to show that $S' = S$.", "This will follow from Morphisms, Lemma", "\\ref{morphisms-lemma-finite-birational-over-normal}.", "Namely, $S'$ is reduced because $X$ is reduced", "(Morphisms, Lemma \\ref{morphisms-lemma-normalization-in-reduced}).", "The morphism $S' \\to S$ is integral by the theorem cited above.", "Every generic point of $S'$ lies over $\\xi$ by", "Morphisms, Lemma \\ref{morphisms-lemma-normalization-generic}", "and assumption (5). On the other hand, since $S'$ is the relative", "spectrum of $f_*\\mathcal{O}_X$ we see that the scheme theoretic fibre", "$S'_\\xi$ is the spectrum of $H^0(X_\\xi, \\mathcal{O})$ which is", "equal to $\\kappa(\\xi)$ by assumption. Hence $S'$ is an integral", "scheme with function field equal to the function field of $S$.", "This finishes the proof." ], "refs": [ "more-morphisms-theorem-stein-factorization-general", "morphisms-lemma-finite-birational-over-normal", "morphisms-lemma-normalization-in-reduced", "morphisms-lemma-normalization-generic" ], "ref_ids": [ 13675, 5518, 5503, 5504 ] } ], "ref_ids": [] }, { "id": 13946, "type": "theorem", "label": "more-morphisms-lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous", "contents": [ "Let $X \\to S$ be a flat proper morphism of finite presentation. Let", "$n_{X/S}$ be the function on $S$ counting the numbers of geometric", "connected components of fibres of $f$ introduced in", "Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}.", "Then $n_{X/S}$ is lower semi-continuous." ], "refs": [ "more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components" ], "proofs": [ { "contents": [ "Let $s \\in S$. Set $n = n_{X/S}(s)$. Note that $n < \\infty$ as the geometric", "fibre of $X \\to S$ at $s$ is a proper scheme over a field, hence Noetherian,", "hence has a finite number of connected components. We have to find an open", "neighbourhood $V$ of $s$ such that $n_{X/S}|_V \\geq n$.", "Let $X \\to S' \\to S$ be the Stein factorization as in", "Theorem \\ref{theorem-stein-factorization-general}.", "By Lemma \\ref{lemma-stein-universally-closed-residue-fields}", "there are finitely many points $s'_1, \\ldots, s'_m \\in S'$ lying over $s$", "and the extensions $\\kappa(s'_i)/\\kappa(s)$ are finite.", "Then Lemma \\ref{lemma-etale-makes-integral-split}", "tells us that after replacing $S$ by an \\'etale neighbourhood", "of $s$ we may assume $S' = V_1 \\amalg \\ldots \\amalg V_m$ as a scheme", "with $s'_i \\in V_i$ and $\\kappa(s'_i)/\\kappa(s)$ purely inseparable.", "Then the schemes $X_{s_i'}$ are geometrically connected over", "$\\kappa(s)$, hence $m = n$. The schemes", "$X_i = (f')^{-1}(V_i)$, $i = 1, \\ldots, n$", "are flat and of finite presentation over $S$. Hence the image of $X_i \\to S$", "is open (Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}).", "Thus in a neighbourhood of $s$ we see that $n_{X/S}$ is", "at least $n$." ], "refs": [ "more-morphisms-theorem-stein-factorization-general", "more-morphisms-lemma-stein-universally-closed-residue-fields", "more-morphisms-lemma-etale-makes-integral-split", "morphisms-lemma-fppf-open" ], "ref_ids": [ 13675, 13943, 13898, 5267 ] } ], "ref_ids": [ 13829 ] }, { "id": 13947, "type": "theorem", "label": "more-morphisms-lemma-proper-flat-geom-red", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-proper-flat-geom-red", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Assume", "\\begin{enumerate}", "\\item $f$ is proper, flat, and of finite presentation, and", "\\item the geometric fibres of $f$ are reduced.", "\\end{enumerate}", "Then the function $n_{X/S} : S \\to \\mathbf{Z}$", "counting the numbers of geometric connected components", "of fibres of $f$ is locally constant." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous}", "the function $n_{X/S}$ is lower semincontinuous.", "For $s \\in S$ consider the $\\kappa(s)$-algebra", "$$", "A = H^0(X_s, \\mathcal{O}_{X_s})", "$$", "By Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}", "and the fact that $X_s$ is geometrically reduced", "$A$ is finite product of finite separable extensions of $\\kappa(s)$.", "Hence $A \\otimes_{\\kappa(s)} \\kappa(\\overline{s})$ is a product", "of $\\beta_0(s) = \\dim_{\\kappa(s)} H^0(E \\otimes^\\mathbf{L} \\kappa(s))$", "copies of $\\kappa(\\overline{s})$. Thus", "$X_{\\overline{s}}$ has $\\beta_0(s) = \\dim_{\\kappa(s)} A$", "connected components. In other words, we have $n_{X/S} = \\beta_0$", "as functions on $S$. Thus $n_{X/S}$ is upper semi-continuous by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-jump-loci-geometric}.", "This finishes the proof." ], "refs": [ "more-morphisms-lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous", "varieties-lemma-proper-geometrically-reduced-global-sections", "perfect-lemma-jump-loci-geometric" ], "ref_ids": [ 13946, 10948, 7062 ] } ], "ref_ids": [] }, { "id": 13948, "type": "theorem", "label": "more-morphisms-lemma-split-off-proper-part-henselian", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-split-off-proper-part-henselian", "contents": [ "\\begin{reference}", "A reference for the case of an adic Noetherian base is", "\\cite[III, Proposition 5.5.1]{EGA}", "\\end{reference}", "Let $(A, I)$ be a henselian pair. Let $X \\to \\Spec(A)$", "be separated and of finite type. Set $X_0 = X \\times_{\\Spec(A)} \\Spec(A/I)$.", "Let $Y \\subset X_0$ be an open and closed subscheme such that", "$Y \\to \\Spec(A/I)$ is proper. Then there exists an open and closed", "subscheme $W \\subset X$ which is proper over $A$ with", "$W \\times_{\\Spec(A)} \\Spec(A/I) = Y$." ], "refs": [], "proofs": [ { "contents": [ "We will denote $T \\mapsto T_0$ the base change by $\\Spec(A/I) \\to \\Spec(A)$.", "By Chow's lemma (in the form of", "Limits, Lemma \\ref{limits-lemma-chow-finite-type})", "there exists a surjective proper morphism $\\varphi : X' \\to X$ such", "that $X'$ admits an immersion into $\\mathbf{P}^n_A$.", "Set $Y' = \\varphi^{-1}(Y)$. This is an open and closed subscheme", "of $X'_0$. Suppose the lemma holds for $(X', Y')$. Let $W' \\subset X'$", "be the open and closed subscheme proper over $A$ such that $Y' = W'_0$.", "By Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}", "$W = \\varphi(W') \\subset X$ and", "$Q = \\varphi(X' \\setminus W') \\subset X$ are closed subsets and by", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-is-proper}", "$W$ is proper over $A$. The image of $W \\cap Q$ in $\\Spec(A)$ is closed.", "Since $(A, I)$ is henselian, if $W \\cap Q$ is nonempty, then we", "find that $W \\cap Q$ has a point lying over $\\Spec(A/I)$.", "This is impossible as $W'_0 = Y' = \\varphi^{-1}(Y)$.", "We conclude that $W$ is an open and closed subscheme", "of $X$ proper over $A$ with $W_0 = Y$.", "Thus we reduce to the case described in the next paragraph.", "\\medskip\\noindent", "Assume there exists an immersion $j : X \\to \\mathbf{P}^n_A$ over $A$.", "Let $\\overline{X}$ be the scheme theoretic image of $j$.", "Since $j$ is a quasi-compact morphism", "(Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence})", "we see that $j : X \\to \\overline{X}$ is an open immersion", "(Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-immersion}).", "Hence the base change $j_0 : X_0 \\to \\overline{X}_0$", "is an open immersion as well.", "Thus $j_0(Y) \\subset \\overline{X}_0$ is open.", "It is also closed by Morphisms, Lemma", "\\ref{morphisms-lemma-image-proper-scheme-closed}.", "Suppose that the lemma holds for $(\\overline{X}, j_0(Y))$.", "Let $\\overline{W} \\subset \\overline{X}$ be the", "corresponding open and closed subscheme proper over $A$", "such that $j_0(Y) = \\overline{W}_0$.", "Then $T = \\overline{W} \\setminus j(X)$ is closed in $\\overline{W}$,", "hence has closed image in $\\Spec(A)$ by properness of $\\overline{W}$", "over $A$. Since $(A, I)$ is henselian, we find that if $T$", "is nonempty, then there is a point of $T$ mapping into $\\Spec(A/I)$.", "This is impossible because $j_0(Y) = \\overline{W}_0$ is contained in $j(X)$.", "Hence $\\overline{W}$ is contained in $j(X)$ and we can", "set $W \\subset X$ equal to the unique open and closed", "subscheme mapping isomorphically to $\\overline{W}$ via $j$.", "Thus we reduce to the case described in the next paragraph.", "\\medskip\\noindent", "Assume $X \\subset \\mathbf{P}^n_A$ is a closed subscheme.", "Then $X \\to \\Spec(A)$ is a proper morphism.", "Let $Z = X_0 \\setminus Y$. This is an open and closed", "subscheme of $X_0$ and $X_0 = Y \\amalg Z$.", "Let $X \\to X' \\to \\Spec(A)$ be the Stein factorization as in", "Theorem \\ref{theorem-stein-factorization-general}.", "Let $Y' \\subset X'_0$ and $Z' \\subset X'_0$ be the images of", "$Y$ and $Z$.", "Since the fibres of $X \\to Z$ are geometrically connected,", "we see that $Y' \\cap Z' = \\emptyset$.", "Hence $X'_0 = Y' \\amalg Z'$ as $X \\to X'$ is surjective.", "Since $X' \\to \\Spec(A)$ is integral, we see that", "$X'$ is the spectrum of an $A$-algebra integral over $A$.", "Recall that open and closed subsets of spectra correspond", "$1$-to-$1$ with idempotents in the corresponding ring, see", "Algebra, Lemma \\ref{algebra-lemma-disjoint-decomposition}.", "Hence by", "More on Algebra, Lemma \\ref{more-algebra-lemma-characterize-henselian-pair}", "we see that we may write $X' = W' \\amalg V'$", "with $W'$ and $V'$ open and closed and", "with $Y' = W'_0$ and $Z' = V'_0$.", "Let $W$ be the inverse image in $X$", "to finish the proof." ], "refs": [ "limits-lemma-chow-finite-type", "morphisms-lemma-image-proper-scheme-closed", "morphisms-lemma-image-proper-is-proper", "schemes-lemma-quasi-compact-permanence", "morphisms-lemma-quasi-compact-immersion", "morphisms-lemma-image-proper-scheme-closed", "more-morphisms-theorem-stein-factorization-general", "algebra-lemma-disjoint-decomposition", "more-algebra-lemma-characterize-henselian-pair" ], "ref_ids": [ 15087, 5411, 5413, 7716, 5154, 5411, 13675, 405, 9861 ] } ], "ref_ids": [] }, { "id": 13949, "type": "theorem", "label": "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "contents": [ "Let $S$ be a scheme.", "Let $\\{X_i \\to S\\}_{i\\in I}$ be an fppf covering, see", "Topologies, Definition \\ref{topologies-definition-fppf-covering}.", "Let $(V_i/X_i, \\varphi_{ij})$ be a descent datum", "relative to $\\{X_i \\to S\\}$. If each morphism", "$V_i \\to X_i$ is separated and locally quasi-finite,", "then the descent datum is effective." ], "refs": [ "topologies-definition-fppf-covering" ], "proofs": [ { "contents": [ "Being separated and being locally quasi-finite", "are properties of morphisms of schemes", "which are preserved under any base change, see", "Schemes, Lemma \\ref{schemes-lemma-separated-permanence} and", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-quasi-finite}.", "Hence Descent, Lemma \\ref{descent-lemma-descending-types-morphisms}", "applies and it suffices to prove the statement of the lemma", "in case the fppf-covering is given by a single", "$\\{X \\to S\\}$ flat surjective morphism of finite presentation of affines.", "Say $X = \\Spec(A)$ and $S = \\Spec(R)$ so", "that $R \\to A$ is a faithfully flat ring map.", "Let $(V, \\varphi)$ be a descent datum relative to $X$ over $S$", "and assume that $\\pi : V \\to X$ is separated and", "locally quasi-finite.", "\\medskip\\noindent", "Let $W^1 \\subset V$ be any affine open.", "Consider $W = \\text{pr}_1(\\varphi(W^1 \\times_S X)) \\subset V$.", "Here is a picture", "$$", "\\xymatrix{", "W^1 \\times_S X \\ar[rrrrr] \\ar[ddd] \\ar[rd]", "& & & & &", "\\varphi(W^1 \\times_S X) \\ar[ddd] \\ar[ld] \\\\", "& V \\times_S X \\ar[rrr]^\\varphi \\ar[rd] \\ar[dd]", "& & &", "X \\times_S V \\ar[ld] \\ar[dd] & \\\\", "& &", "X \\times_S X \\ar[r]^1 \\ar[d]_{\\text{pr}_0}", "&", "X \\times_S X \\ar[d]^{\\text{pr}_1}", "& & \\\\", "W^1 \\ar[r] &", "V \\ar[r] &", "X &", "X &", "V \\ar[l] &", "W \\ar[l]", "}", "$$", "Ok, and now since $X \\to S$ is flat and of finite presentation it", "is universally open (Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}).", "Hence we conclude that $W$ is open. Moreover, it is", "also clearly the case that $W$ is quasi-compact, and", "$W^1 \\subset W$. Moreover, we note that", "$\\varphi(W \\times_S X) = X \\times_S W$ by the cocycle", "condition for $\\varphi$. Hence we obtain a new descent datum", "$(W, \\varphi')$ by restricting $\\varphi$ to $W \\times_S X$.", "Note that the morphism $W \\to X$ is quasi-compact, separated", "and locally quasi-finite. This implies that it is", "separated and quasi-finite by definition. Hence it is quasi-affine by", "Lemma \\ref{lemma-quasi-finite-separated-quasi-affine}.", "Thus by", "Descent, Lemma \\ref{descent-lemma-quasi-affine}", "we see that the descent datum", "$(W, \\varphi')$ is effective.", "\\medskip\\noindent", "In other words, we find that there exists an open covering", "$V = \\bigcup W_i$ by quasi-compact opens $W_i$ which are", "stable for the descent morphism $\\varphi$.", "Moreover, for each such quasi-compact open $W \\subset V$", "the corresponding descent data $(W, \\varphi')$ is effective.", "This means the original descent datum is effective by glueing the", "schemes obtained from descending the opens $W_i$, see", "Descent, Lemma \\ref{descent-lemma-effective-for-fpqc-is-local-upstairs}." ], "refs": [ "schemes-lemma-separated-permanence", "morphisms-lemma-base-change-quasi-finite", "descent-lemma-descending-types-morphisms", "morphisms-lemma-fppf-open", "more-morphisms-lemma-quasi-finite-separated-quasi-affine", "descent-lemma-quasi-affine", "descent-lemma-effective-for-fpqc-is-local-upstairs" ], "ref_ids": [ 7714, 5233, 14747, 5267, 13900, 14750, 14746 ] } ], "ref_ids": [ 12539 ] }, { "id": 13950, "type": "theorem", "label": "more-morphisms-lemma-relative-finite-presentation-characterize", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-relative-finite-presentation-characterize", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The following", "are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is of finite presentation relative to $S$,", "\\item for every affine opens $U \\subset X$, $V \\subset S$", "with $f(U) \\subset V$ the $\\mathcal{O}_X(U)$-module $\\mathcal{F}(U)$", "is finitely presented relative to $\\mathcal{O}_S(V)$.", "\\end{enumerate}", "Moreover, if this is true, then for every open subschemes", "$U \\subset X$ and $V \\subset S$ with $f(U) \\subset V$", "the restriction $\\mathcal{F}|_U$ is of finite presentation relative to $V$." ], "refs": [], "proofs": [ { "contents": [ "The final statement is clear from the equivalence of (1) and (2).", "It is also clear that (2) implies (1). Assume (1) holds.", "Let $S = \\bigcup V_i$ and $f^{-1}(V_i) = \\bigcup U_{ij}$ be", "affine open coverings as in", "Definition \\ref{definition-relatively-finitely-presented-sheaf}.", "Let $U \\subset X$ and $V \\subset S$ be as in (2).", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-glue-relative-finite-presentation}", "it suffices to find a standard open covering $U = \\bigcup U_k$ of $U$", "such that $\\mathcal{F}(U_k)$ is finitely presented relative to", "$\\mathcal{O}_S(V)$. In other words, for every $u \\in U$ it suffices", "to find a standard affine open $u \\in U' \\subset U$ such that", "$\\mathcal{F}(U')$ is finitely presented relative to $\\mathcal{O}_S(V)$.", "Pick $i$ such that $f(u) \\in V_i$ and then pick $j$ such that", "$u \\in U_{ij}$. By", "Schemes, Lemma \\ref{schemes-lemma-standard-open-two-affines}", "we can find $v \\in V' \\subset V \\cap V_i$ which is standard affine", "open in $V'$ and $V_i$. Then $f^{-1}V' \\cap U$, resp.\\ $f^{-1}V' \\cap U_{ij}$", "are standard affine opens of $U$, resp.\\ $U_{ij}$.", "Applying the lemma again we can find", "$u \\in U' \\subset f^{-1}V' \\cap U \\cap U_{ij}$ which is standard affine", "open in both $f^{-1}V' \\cap U$ and $f^{-1}V' \\cap U_{ij}$.", "Thus $U'$ is also a standard affine open of $U$ and $U_{ij}$.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-localize-relative-finite-presentation}", "the assumption that $\\mathcal{F}(U_{ij})$ is finitely presented", "relative to $\\mathcal{O}_S(V_i)$ implies that", "$\\mathcal{F}(U')$ is finitely presented relative to $\\mathcal{O}_S(V_i)$.", "Since $\\mathcal{O}_X(U') =", "\\mathcal{O}_X(U') \\otimes_{\\mathcal{O}_S(V_i)} \\mathcal{O}_S(V')$", "we see from More on Algebra, Lemma", "\\ref{more-algebra-lemma-base-change-relative-finite-presentation}", "that $\\mathcal{F}(U')$ is finitely presented relative to $\\mathcal{O}_S(V')$.", "Applying More on Algebra, Lemma", "\\ref{more-algebra-lemma-localize-relative-finite-presentation}", "again we conclude that", "$\\mathcal{F}(U')$ is finitely presented relative to $\\mathcal{O}_S(V)$.", "This finishes the proof." ], "refs": [ "more-morphisms-definition-relatively-finitely-presented-sheaf", "more-algebra-lemma-glue-relative-finite-presentation", "schemes-lemma-standard-open-two-affines", "more-algebra-lemma-localize-relative-finite-presentation", "more-algebra-lemma-base-change-relative-finite-presentation", "more-algebra-lemma-localize-relative-finite-presentation" ], "ref_ids": [ 14117, 10262, 7675, 10258, 10259, 10258 ] } ], "ref_ids": [] }, { "id": 13951, "type": "theorem", "label": "more-morphisms-lemma-relative-finite-presentation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-relative-finite-presentation", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite", "type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item If $f$ is locally of finite presentation, then $\\mathcal{F}$", "is of finite presentation relative to $S$ if and only if $\\mathcal{F}$", "is of finite presentation.", "\\item The morphism $f$ is locally of finite presentation if and only", "if $\\mathcal{O}_X$ is of finite presentation relative to $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from the definitions, see", "discussion following", "More on Algebra, Definition", "\\ref{more-algebra-definition-relatively-finitely-presented}." ], "refs": [ "more-algebra-definition-relatively-finitely-presented" ], "ref_ids": [ 10629 ] } ], "ref_ids": [] }, { "id": 13952, "type": "theorem", "label": "more-morphisms-lemma-finite-morphism-relative-finite-presentation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-finite-morphism-relative-finite-presentation", "contents": [ "Let $\\pi : X \\to Y$ be a finite morphism of schemes locally of finite", "type over a base scheme $S$. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is of finite presentation", "relative to $S$ if and only if $\\pi_*\\mathcal{F}$ is of finite presentation", "relative to $S$." ], "refs": [], "proofs": [ { "contents": [ "Translation of the result of", "More on Algebra, Lemma \\ref{more-algebra-lemma-finite-extension}", "into the language of schemes." ], "refs": [ "more-algebra-lemma-finite-extension" ], "ref_ids": [ 10257 ] } ], "ref_ids": [] }, { "id": 13953, "type": "theorem", "label": "more-morphisms-lemma-base-change-relative-finite-presentation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-relative-finite-presentation", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite", "type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $S' \\to S$ be a morphism of schemes, set $X' = X \\times_S S'$", "and denote $\\mathcal{F}'$ the pullback of $\\mathcal{F}$ to $X'$.", "If $\\mathcal{F}$ is of finite presentation relative to $S$, then", "$\\mathcal{F}'$ is of finite presentation relative to $S'$." ], "refs": [], "proofs": [ { "contents": [ "Translation of the result of", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-base-change-relative-finite-presentation}", "into the language of schemes." ], "refs": [ "more-algebra-lemma-base-change-relative-finite-presentation" ], "ref_ids": [ 10259 ] } ], "ref_ids": [] }, { "id": 13954, "type": "theorem", "label": "more-morphisms-lemma-pull-relative-finite-presentation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pull-relative-finite-presentation", "contents": [ "Let $X \\to Y \\to S$ be morphisms of schemes which are locally of finite", "type. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.", "If $f : X \\to Y$ is locally of finite presentation and", "$\\mathcal{G}$ of finite presentation relative to $S$, then", "$f^*\\mathcal{G}$ is of finite presentation relative to $S$." ], "refs": [], "proofs": [ { "contents": [ "Translation of the result of", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-pull-relative-finite-presentation}", "into the language of schemes." ], "refs": [ "more-algebra-lemma-pull-relative-finite-presentation" ], "ref_ids": [ 10260 ] } ], "ref_ids": [] }, { "id": 13955, "type": "theorem", "label": "more-morphisms-lemma-composition-relative-finite-presentation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-composition-relative-finite-presentation", "contents": [ "Let $X \\to Y \\to S$ be morphisms of schemes which are locally of finite", "type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "If $Y \\to S$ is locally of finite presentation and $\\mathcal{F}$", "is of finite presentation relative to $Y$, then $\\mathcal{F}$", "is of finite presentation relative to $S$." ], "refs": [], "proofs": [ { "contents": [ "Translation of the result of", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-composition-relative-finite-presentation}", "into the language of schemes." ], "refs": [ "more-algebra-lemma-composition-relative-finite-presentation" ], "ref_ids": [ 10261 ] } ], "ref_ids": [] }, { "id": 13956, "type": "theorem", "label": "more-morphisms-lemma-ses-relatively-finite-presentation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-ses-relatively-finite-presentation", "contents": [ "Let $X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $0 \\to \\mathcal{F}' \\to \\mathcal{F} \\to \\mathcal{F}'' \\to 0$", "be a short exact sequence of quasi-coherent $\\mathcal{O}_X$-modules.", "\\begin{enumerate}", "\\item If $\\mathcal{F}', \\mathcal{F}''$ are finitely presented relative to", "$S$, then so is $\\mathcal{F}$.", "\\item If $\\mathcal{F}'$ is a finite type $\\mathcal{O}_X$-module", "and $\\mathcal{F}$ is finitely presented relative to $S$, then", "$\\mathcal{F}''$ is finitely presented relative to $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Translation of the result of", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-ses-relatively-finite-presentation}", "into the language of schemes." ], "refs": [ "more-algebra-lemma-ses-relatively-finite-presentation" ], "ref_ids": [ 10263 ] } ], "ref_ids": [] }, { "id": 13957, "type": "theorem", "label": "more-morphisms-lemma-sum-relatively-finite-presentation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-sum-relatively-finite-presentation", "contents": [ "\\begin{slogan}", "A direct summand of a module inherits the property of being finitely", "presented relative to a base.", "\\end{slogan}", "Let $X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $\\mathcal{F}, \\mathcal{F}'$ be quasi-coherent $\\mathcal{O}_X$-modules.", "If $\\mathcal{F} \\oplus \\mathcal{F}'$ is finitely presented relative to $S$,", "then so are $\\mathcal{F}$ and $\\mathcal{F}'$." ], "refs": [], "proofs": [ { "contents": [ "Translation of the result of", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-sum-relatively-finite-presentation}", "into the language of schemes." ], "refs": [ "more-algebra-lemma-sum-relatively-finite-presentation" ], "ref_ids": [ 10264 ] } ], "ref_ids": [] }, { "id": 13958, "type": "theorem", "label": "more-morphisms-lemma-relatively-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-relatively-pseudo-coherent", "contents": [ "Let $X \\to S$ be a finite type morphism of affine schemes.", "Let $E$ be an object of $D(\\mathcal{O}_X)$.", "Let $m \\in \\mathbf{Z}$.", "The following are equivalent", "\\begin{enumerate}", "\\item for some closed immersion $i : X \\to \\mathbf{A}^n_S$", "the object $Ri_*E$ of $D(\\mathcal{O}_{\\mathbf{A}^n_S})$", "is $m$-pseudo-coherent, and", "\\item for all closed immersions $i : X \\to \\mathbf{A}^n_S$", "the object $Ri_*E$ of $D(\\mathcal{O}_{\\mathbf{A}^n_S})$", "is $m$-pseudo-coherent.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Say $S = \\Spec(R)$ and $X = \\Spec(A)$. Let $i$ correspond to the surjection", "$\\alpha : R[x_1, \\ldots, x_n] \\to A$ and let $X \\to \\mathbf{A}^m_S$", "correspond to $\\beta : R[y_1, \\ldots, y_m] \\to A$.", "Choose $f_j \\in R[x_1, \\ldots, x_n]$ with $\\alpha(f_j) = \\beta(y_j)$", "and $g_i \\in R[y_1, \\ldots, y_m]$ with $\\beta(g_i) = \\alpha(x_i)$.", "Then we get a commutative diagram", "$$", "\\xymatrix{", "R[x_1, \\ldots, x_n, y_1, \\ldots, y_m]", "\\ar[d]^{x_i \\mapsto g_i} \\ar[rr]_-{y_j \\mapsto f_j} & &", "R[x_1, \\ldots, x_n] \\ar[d] \\\\", "R[y_1, \\ldots, y_m] \\ar[rr] & & A", "}", "$$", "corresponding to the commutative diagram of closed immersions", "$$", "\\xymatrix{", "\\mathbf{A}^{n + m}_S & \\mathbf{A}^n_S \\ar[l] \\\\", "\\mathbf{A}^m_S \\ar[u] & X \\ar[u] \\ar[l]", "}", "$$", "Thus it suffices to show that under a closed immersion", "$$", "f : \\mathbf{A}^m_S \\to \\mathbf{A}^{n + m}_S", "$$", "an object $E$ of $D(\\mathcal{O}_{\\mathbf{A}^m_S})$ is", "$m$-pseudo-coherent if and only if $Rf_*E$ is $m$-pseudo-coherent.", "This follows from", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-closed-push-pseudo-coherent}", "and the fact that $f_*\\mathcal{O}_{\\mathbf{A}^m_S}$ is", "a pseudo-coherent $\\mathcal{O}_{\\mathbf{A}^{n + m}_S}$-module.", "The pseudo-coherence of $f_*\\mathcal{O}_{\\mathbf{A}^m_S}$ is", "straightforward to prove directly, but it also follows from", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-pseudo-coherent-affine}", "and", "More on Algebra, Lemma \\ref{more-algebra-lemma-relatively-pseudo-coherent}." ], "refs": [ "perfect-lemma-closed-push-pseudo-coherent", "perfect-lemma-pseudo-coherent-affine", "more-algebra-lemma-relatively-pseudo-coherent" ], "ref_ids": [ 6994, 6975, 10267 ] } ], "ref_ids": [] }, { "id": 13959, "type": "theorem", "label": "more-morphisms-lemma-relative-pseudo-coherence", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-relative-pseudo-coherence", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "If $E$ in $D(\\mathcal{O}_X)$ is $m$-pseudo-coherent relative to $S$,", "then $H^i(E)$ is a quasi-coherent $\\mathcal{O}_X$-module for $i > m$.", "If $E$ is pseudo-coherent relative to $S$, then $E$ is an object of", "$D_\\QCoh(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine open covering $S = \\bigcup V_i$ and", "for each $i$ an affine open covering $f^{-1}(V_i) = \\bigcup U_{ij}$", "such that the equivalent conditions of", "Lemma \\ref{lemma-relatively-pseudo-coherent}", "are satisfied for each of the pairs $(U_{ij} \\to V_i, E|_{U_{ij}})$.", "Since being quasi-coherent is local on $X$, we may assume", "that there exists an closed immersion $i : X \\to \\mathbf{A}^n_S$", "such that $Ri_*E$ is $m$-pseudo-coherent on $\\mathbf{A}^n_S$.", "By Derived Categories of Schemes, Lemma \\ref{perfect-lemma-pseudo-coherent}", "this means that $H^q(Ri_*E)$ is quasi-coherent for $q > m$.", "Since $i_*$ is an exact functor, we have $i_*H^q(E) = H^q(Ri_*E)$", "is quasi-coherent on $\\mathbf{A}^n_S$.", "By Morphisms, Lemma \\ref{morphisms-lemma-i-star-equivalence}", "this implies that $H^q(E)$ is quasi-coherent as desired", "(strictly speaking it implies there exists some quasi-coherent", "$\\mathcal{O}_X$-module $\\mathcal{F}$ such that", "$i_*\\mathcal{F} = i_*H^q(E)$ and then", "Modules, Lemma \\ref{modules-lemma-i-star-equivalence}", "tells us that $\\mathcal{F} \\cong H^q(E)$ hence the result)." ], "refs": [ "more-morphisms-lemma-relatively-pseudo-coherent", "perfect-lemma-pseudo-coherent", "morphisms-lemma-i-star-equivalence", "modules-lemma-i-star-equivalence" ], "ref_ids": [ 13958, 6974, 5136, 13260 ] } ], "ref_ids": [] }, { "id": 13960, "type": "theorem", "label": "more-morphisms-lemma-localize-relative-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-localize-relative-pseudo-coherent", "contents": [ "Let $S$ be an affine scheme. Let $V \\subset S$ be a standard open.", "Let $X \\to V$ be a finite type morphism of affine schemes.", "Let $U \\subset X$ be an affine open. Let $E$ be an object of", "$D(\\mathcal{O}_X)$. If the equivalent conditions of", "Lemma \\ref{lemma-relatively-pseudo-coherent}", "are satisfied for the pair $(X \\to V, E)$, then", "the equivalent conditions of", "Lemma \\ref{lemma-relatively-pseudo-coherent}", "are satisfied for the pair $(U \\to S, E|_U)$." ], "refs": [ "more-morphisms-lemma-relatively-pseudo-coherent", "more-morphisms-lemma-relatively-pseudo-coherent" ], "proofs": [ { "contents": [ "Write $S = \\Spec(R)$, $V = D(f)$, $X = \\Spec(A)$, and $U = D(g)$.", "Assume the equivalent conditions of", "Lemma \\ref{lemma-relatively-pseudo-coherent}", "are satisfied for the pair $(X \\to V, E)$.", "\\medskip\\noindent", "Choose $R_f[x_1, \\ldots, x_n] \\to A$ surjective. Write", "$R_f = R[x_0]/(fx_0 - 1)$. Then $R[x_0, x_1, \\ldots, x_n] \\to A$", "is surjective, and $R_f[x_1, \\ldots, x_n]$ is pseudo-coherent as", "an $R[x_0, \\ldots, x_n]$-module. Thus we have", "$$", "X \\to \\mathbf{A}^n_V \\to \\mathbf{A}^{n + 1}_S", "$$", "and we can apply", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-closed-push-pseudo-coherent}", "to conclude that the pushforward $E'$ of $E$ to $\\mathbf{A}^{n + 1}_S$", "is $m$-pseudo-coherent.", "\\medskip\\noindent", "Choose an element $g' \\in R[x_0, x_1, \\ldots, x_n]$ which maps to", "$g \\in A$. Consider the surjection", "$R[x_0, \\ldots, x_{n + 1}] \\to R[x_0, \\ldots, x_n, 1/g']$.", "We obtain", "$$", "\\xymatrix{", "X \\ar[d] & U \\ar[d] \\ar[l] \\ar[dr] \\\\", "\\mathbf{A}^{n + 1}_S & D(g')\\ar[l] \\ar[r] & \\mathbf{A}^{n + 2}_S", "}", "$$", "where the lower left arrow is an open immersion and the lower right arrow is", "a closed immersion. We conclude as before that the pushforward of", "$E'|_{D(g')}$ to $\\mathbf{A}^{n + 2}_S$ is $m$-pseudo-coherent.", "Since this is also the pushforward of $E|_U$ to $\\mathbf{A}^{n + 2}_S$", "we conclude the lemma is true." ], "refs": [ "more-morphisms-lemma-relatively-pseudo-coherent", "perfect-lemma-closed-push-pseudo-coherent" ], "ref_ids": [ 13958, 6994 ] } ], "ref_ids": [ 13958, 13958 ] }, { "id": 13961, "type": "theorem", "label": "more-morphisms-lemma-glue-relative-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-glue-relative-pseudo-coherent", "contents": [ "Let $X \\to S$ be a finite type morphism of affine schemes. Let $E$ be an", "object of $D(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$.", "Let $X = \\bigcup U_i$ be a standard affine open covering.", "The following are equivalent", "\\begin{enumerate}", "\\item the equivalent conditions of", "Lemma \\ref{lemma-relatively-pseudo-coherent}", "hold for the pairs $(U_i \\to S, E|_{U_i})$,", "\\item the equivalent conditions of", "Lemma \\ref{lemma-relatively-pseudo-coherent}", "hold for the pair $(X \\to S, E)$.", "\\end{enumerate}" ], "refs": [ "more-morphisms-lemma-relatively-pseudo-coherent", "more-morphisms-lemma-relatively-pseudo-coherent" ], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is", "Lemma \\ref{lemma-localize-relative-pseudo-coherent}.", "Assume (1). Say $S = \\Spec(R)$ and $X = \\Spec(A)$ and", "$U_i = D(f_i)$. Write $1 = \\sum f_ig_i$ in $A$.", "Consider the surjections", "$$", "R[x_i, y_i, z_i] \\to R[x_i, y_i, z_i]/(\\sum y_iz_i - 1) \\to A.", "$$", "which sends $y_i$ to $f_i$ and $z_i$ to $g_i$. Note that", "$R[x_i, y_i, z_i]/(\\sum y_iz_i - 1)$ is pseudo-coherent as an", "$R[x_i, y_i, z_i]$-module. Thus it suffices to prove that", "the pushforward of $E$ to $T = \\Spec(R[x_i, y_i, z_i]/(\\sum y_iz_i - 1))$", "is $m$-pseudo-coherent, see", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-closed-push-pseudo-coherent}.", "For each $i_0$ it suffices to prove the restriction of this", "pushforward to", "$W_{i_0} = \\Spec(R[x_i, y_i, z_i, 1/y_{i_0}]/(\\sum y_iz_i - 1))$", "is $m$-pseudo-coherent. Note that there is a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] & U_{i_0} \\ar[l] \\ar[d] \\\\", "T & W_{i_0} \\ar[l]", "}", "$$", "which implies that the pushforward of $E$ to $T$ restricted to $W_{i_0}$", "is the pushforward of $E|_{U_{i_0}}$ to $W_{i_0}$. Since", "$R[x_i, y_i, z_i, 1/y_{i_0}]/(\\sum y_iz_i - 1)$ is isomorphic", "to a polynomial ring over $R$ this proves what we want." ], "refs": [ "more-morphisms-lemma-localize-relative-pseudo-coherent", "perfect-lemma-closed-push-pseudo-coherent" ], "ref_ids": [ 13960, 6994 ] } ], "ref_ids": [ 13958, 13958 ] }, { "id": 13962, "type": "theorem", "label": "more-morphisms-lemma-relative-pseudo-coherence-characterize", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-relative-pseudo-coherence-characterize", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $E$ be an object of $D(\\mathcal{O}_X)$.", "Fix $m \\in \\mathbf{Z}$. The following are equivalent", "\\begin{enumerate}", "\\item $E$ is $m$-pseudo-coherent relative to $S$,", "\\item for every affine opens $U \\subset X$ and $V \\subset S$", "with $f(U) \\subset V$ the equivalent conditions of", "Lemma \\ref{lemma-relatively-pseudo-coherent}", "are satisfied for the pair $(U \\to V, E|_U)$.", "\\end{enumerate}", "Moreover, if this is true, then for every open subschemes", "$U \\subset X$ and $V \\subset S$ with $f(U) \\subset V$", "the restriction $E|_U$ is $m$-pseudo-coherent relative to $V$." ], "refs": [ "more-morphisms-lemma-relatively-pseudo-coherent" ], "proofs": [ { "contents": [ "The final statement is clear from the equivalence of (1) and (2).", "It is also clear that (2) implies (1). Assume (1) holds.", "Let $S = \\bigcup V_i$ and $f^{-1}(V_i) = \\bigcup U_{ij}$ be", "affine open coverings as in", "Definition \\ref{definition-relative-pseudo-coherence}.", "Let $U \\subset X$ and $V \\subset S$ be as in (2).", "By Lemma \\ref{lemma-glue-relative-pseudo-coherent}", "it suffices to find a standard open covering $U = \\bigcup U_k$ of $U$", "such that the equivalent conditions of", "Lemma \\ref{lemma-relatively-pseudo-coherent}", "are satisfied for the pairs $(U_k \\to V, E|_{U_k})$.", "In other words, for every $u \\in U$ it suffices", "to find a standard affine open $u \\in U' \\subset U$ such that", "the equivalent conditions of", "Lemma \\ref{lemma-relatively-pseudo-coherent}", "are satisfied for the pair $(U' \\to V, E|_{U'})$.", "Pick $i$ such that $f(u) \\in V_i$ and then pick $j$ such that", "$u \\in U_{ij}$. By", "Schemes, Lemma \\ref{schemes-lemma-standard-open-two-affines}", "we can find $v \\in V' \\subset V \\cap V_i$ which is standard affine", "open in $V'$ and $V_i$. Then $f^{-1}V' \\cap U$, resp.\\ $f^{-1}V' \\cap U_{ij}$", "are standard affine opens of $U$, resp.\\ $U_{ij}$.", "Applying the lemma again we can find", "$u \\in U' \\subset f^{-1}V' \\cap U \\cap U_{ij}$ which is standard affine", "open in both $f^{-1}V' \\cap U$ and $f^{-1}V' \\cap U_{ij}$.", "Thus $U'$ is also a standard affine open of $U$ and $U_{ij}$.", "By Lemma \\ref{lemma-localize-relative-pseudo-coherent}", "the assumption that the equivalent conditions of", "Lemma \\ref{lemma-relatively-pseudo-coherent}", "are satisfied for the pair $(U_{ij} \\to V_i, E|_{U_{ij}})$", "implies that the equivalent conditions of", "Lemma \\ref{lemma-relatively-pseudo-coherent}", "are satisfied for the pair $(U' \\to V, E|_{U'})$." ], "refs": [ "more-morphisms-definition-relative-pseudo-coherence", "more-morphisms-lemma-glue-relative-pseudo-coherent", "more-morphisms-lemma-relatively-pseudo-coherent", "more-morphisms-lemma-relatively-pseudo-coherent", "schemes-lemma-standard-open-two-affines", "more-morphisms-lemma-localize-relative-pseudo-coherent", "more-morphisms-lemma-relatively-pseudo-coherent", "more-morphisms-lemma-relatively-pseudo-coherent" ], "ref_ids": [ 14118, 13961, 13958, 13958, 7675, 13960, 13958, 13958 ] } ], "ref_ids": [ 13958 ] }, { "id": 13963, "type": "theorem", "label": "more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$.", "Fix $m \\in \\mathbf{Z}$. The following are equivalent", "\\begin{enumerate}", "\\item $E$ is $m$-pseudo-coherent relative to $S$,", "\\item there exists an affine open covering $S = \\bigcup V_i$ and", "for each $i$ an affine open covering $f^{-1}(V_i) = \\bigcup U_{ij}$", "such that the complex of $\\mathcal{O}_X(U_{ij})$-modules", "$R\\Gamma(U_{ij}, E)$ is $m$-pseudo-coherent relative to", "$\\mathcal{O}_S(V_i)$, and", "\\item for every affine opens $U \\subset X$ and $V \\subset S$", "with $f(U) \\subset V$ the complex of $\\mathcal{O}_X(U)$-modules", "$R\\Gamma(U, E)$ is $m$-pseudo-coherent relative to $\\mathcal{O}_S(V)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $U$ and $V$ be as in (2) and choose a closed immersion", "$i : U \\to \\mathbf{A}^n_V$. A formal argument, using", "Lemma \\ref{lemma-relative-pseudo-coherence-characterize}, shows it", "suffices to prove that", "$Ri_*(E|_U)$ is $m$-pseudo-coherent if and only if $R\\Gamma(U, E)$", "is $m$-pseudo-coherent relative to $\\mathcal{O}_S(V)$.", "Say $U = \\Spec(A)$, $V = \\Spec(R)$, and", "$\\mathbf{A}^n_V = \\Spec(R[x_1, \\ldots, x_n]$. By the remarks", "preceding the lemma, $E|_U$ is quasi-isomorphic to the", "complex of quasi-coherent sheaves on $U$ associated to the object", "$R\\Gamma(U, E)$ of $D(A)$. Note that", "$R\\Gamma(U, E) = R\\Gamma(\\mathbf{A}^n_V, Ri_*(E|_U))$ as $i$ is a", "closed immersion (and hence $i_*$ is exact). Thus $Ri_*E$", "is associated to $R\\Gamma(U, E)$ viewed as an object of", "$D(R[x_1, \\ldots, x_n])$. We conclude as $m$-pseudo-coherence", "of $Ri_*(E|_U)$ is equivalent to $m$-pseudo-coherence of", "$R\\Gamma(U, E)$ in $D(R[x_1, \\ldots, x_n])$ by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-pseudo-coherent-affine}", "which is equivalent to $R\\Gamma(U, E)$ is $m$-pseudo-coherent", "relative to $R = \\mathcal{O}_S(V)$ by definition." ], "refs": [ "more-morphisms-lemma-relative-pseudo-coherence-characterize", "perfect-lemma-pseudo-coherent-affine" ], "ref_ids": [ 13962, 6975 ] } ], "ref_ids": [] }, { "id": 13964, "type": "theorem", "label": "more-morphisms-lemma-closed-morphism-relative-pseudo-coherence", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-closed-morphism-relative-pseudo-coherence", "contents": [ "Let $i : X \\to Y$ morphism of schemes locally of finite type over a", "base scheme $S$. Assume that $i$ induces a homeomorphism of $X$ with a closed", "subset of $Y$. Let $E$ be an object of $D(\\mathcal{O}_X)$.", "Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if", "$Ri_*E$ is $m$-pseudo-coherent relative to $S$." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms, Lemma \\ref{morphisms-lemma-homeomorphism-affine}", "the morphism $i$ is affine. Thus we may assume $S$, $Y$, and $X$ are affine.", "Say $S = \\Spec(R)$, $Y = \\Spec(A)$, and $X = \\Spec(B)$.", "The condition means that $A/\\text{rad}(A) \\to B/\\text{rad}(B)$ is", "surjective; here $\\text{rad}(A)$ and $\\text{rad}(B)$ denote", "the Jacobson radical of $A$ and $B$.", "As $B$ is of finite type over $A$, we can find", "$b_1, \\ldots, b_m \\in \\text{rad}(B)$ which generate $B$ as an", "$A$-algebra. Say $b_j^N = 0$ for all $j$. Consider the diagram of", "rings", "$$", "\\xymatrix{", "B & R[x_i, y_j]/(y_j^N) \\ar[l] & R[x_i, y_j] \\ar[l] \\\\", "A \\ar[u] & R[x_i] \\ar[l] \\ar[u] \\ar[ru]", "}", "$$", "which translates into a diagram", "$$", "\\xymatrix{", "X \\ar[d] \\ar[r] & T \\ar[d] \\ar[r] & \\mathbf{A}^{n + m}_S \\ar[ld] \\\\", "Y \\ar[r] & \\mathbf{A}^n_S", "}", "$$", "of affine schemes. By Lemma \\ref{lemma-relative-pseudo-coherence-characterize}", "we see that $E$ is $m$-pseudo-coherent relative to $S$ if and only if its", "pushforward to $\\mathbf{A}^{n + m}_S$ is $m$-pseudo-coherent. ", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-closed-push-pseudo-coherent}", "we see that this is true if and only if its pushforward to $T$ is", "$m$-pseudo-coherent. The same lemma shows that this holds if and only", "if the pushforward to $\\mathbf{A}^n_S$ is $m$-pseudo-coherent.", "Again by", "Lemma \\ref{lemma-relative-pseudo-coherence-characterize}", "this holds if and only if $Ri_*E$ is $m$-pseudo-coherent relative to $S$." ], "refs": [ "morphisms-lemma-homeomorphism-affine", "more-morphisms-lemma-relative-pseudo-coherence-characterize", "perfect-lemma-closed-push-pseudo-coherent", "more-morphisms-lemma-relative-pseudo-coherence-characterize" ], "ref_ids": [ 5453, 13962, 6994, 13962 ] } ], "ref_ids": [] }, { "id": 13965, "type": "theorem", "label": "more-morphisms-lemma-finite-morphism-relative-pseudo-coherence", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-finite-morphism-relative-pseudo-coherence", "contents": [ "Let $\\pi : X \\to Y$ be a finite morphism of schemes locally of finite", "type over a base scheme $S$. Let $E$ be an object of", "$D_\\QCoh(\\mathcal{O}_X)$. Then $E$ is $m$-pseudo-coherent", "relative to $S$ if and only if $R\\pi_*E$ is $m$-pseudo-coherent", "relative to $S$." ], "refs": [], "proofs": [ { "contents": [ "Translation of the result of", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-finite-extension-pseudo-coherent}", "into the language of schemes. Observe that $R\\pi_*$ indeed", "maps $D_\\QCoh(\\mathcal{O}_X)$ into $D_\\QCoh(\\mathcal{O}_Y)$", "by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-direct-image}.", "To do the translation use", "Lemma \\ref{lemma-relative-pseudo-coherence-characterize}." ], "refs": [ "more-algebra-lemma-finite-extension-pseudo-coherent", "perfect-lemma-quasi-coherence-direct-image", "more-morphisms-lemma-relative-pseudo-coherence-characterize" ], "ref_ids": [ 10268, 6946, 13962 ] } ], "ref_ids": [] }, { "id": 13966, "type": "theorem", "label": "more-morphisms-lemma-cone-relatively-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-cone-relatively-pseudo-coherent", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $(E, E', E'')$ be a distinguished triangle of", "$D(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item If $E$ is $(m + 1)$-pseudo-coherent relative to $S$ and", "$E'$ is $m$-pseudo-coherent relative to $S$ then $E''$ is", "$m$-pseudo-coherent relative to $S$.", "\\item If $E, E''$ are $m$-pseudo-coherent relative to $S$,", "then $E'$ is $m$-pseudo-coherent relative to $S$.", "\\item If $E'$ is $(m + 1)$-pseudo-coherent relative to $S$", "and $E''$ is $m$-pseudo-coherent relative to $S$, then", "$E$ is $(m + 1)$-pseudo-coherent relative to $S$.", "\\end{enumerate}", "Moreover, if two out of three of $E, E', E''$ are pseudo-coherent", "relative to $S$, the so is the third." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemma \\ref{lemma-relative-pseudo-coherence-characterize} and", "Cohomology, Lemma \\ref{cohomology-lemma-cone-pseudo-coherent}." ], "refs": [ "more-morphisms-lemma-relative-pseudo-coherence-characterize", "cohomology-lemma-cone-pseudo-coherent" ], "ref_ids": [ 13962, 2207 ] } ], "ref_ids": [] }, { "id": 13967, "type": "theorem", "label": "more-morphisms-lemma-rel-n-pseudo-module", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-rel-n-pseudo-module", "contents": [ "Let $X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Then", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is $m$-pseudo-coherent relative to $S$ for all $m > 0$,", "\\item $\\mathcal{F}$ is $0$-pseudo-coherent relative to $S$ if and only if", "$\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module,", "\\item $\\mathcal{F}$ is $(-1)$-pseudo-coherent relative to $S$ if and only if", "$\\mathcal{F}$ is quasi-coherent and finitely presented relative to $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is immediate from the definition. To see part (3)", "we may work locally on $X$ (both properties are local). Thus we", "may assume $X$ and $S$ are affine. Choose a closed immersion", "$i : X \\to \\mathbf{A}^n_S$. Then we see that $\\mathcal{F}$ is", "$(-1)$-pseudo-coherent relative to $S$ if and only if $i_*\\mathcal{F}$", "is $(-1)$-pseudo-coherent, which is true if and only if $i_*\\mathcal{F}$", "is an $\\mathcal{O}_{\\mathbf{A}^n_S}$-module of finite presentation, see", "Cohomology, Lemma \\ref{cohomology-lemma-finite-cohomology}.", "A module of finite presentation is quasi-coherent, see", "Modules, Lemma \\ref{modules-lemma-finite-presentation-quasi-coherent}.", "By Morphisms, Lemma \\ref{morphisms-lemma-i-star-equivalence}", "we see that $\\mathcal{F}$ is quasi-coherent if and only if $i_*\\mathcal{F}$", "is quasi-coherent. Having said this part (3) follows. The proof of (2)", "is similar but less involved." ], "refs": [ "cohomology-lemma-finite-cohomology", "modules-lemma-finite-presentation-quasi-coherent", "morphisms-lemma-i-star-equivalence" ], "ref_ids": [ 2212, 13248, 5136 ] } ], "ref_ids": [] }, { "id": 13968, "type": "theorem", "label": "more-morphisms-lemma-summands-relative-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-summands-relative-pseudo-coherent", "contents": [ "Let $X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $m \\in \\mathbf{Z}$. Let $E, K$ be objects of $D(\\mathcal{O}_X)$.", "If $E \\oplus K$ is $m$-pseudo-coherent relative to $S$ so are $E$ and $K$." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Cohomology, Lemma \\ref{cohomology-lemma-summands-pseudo-coherent}", "and the definitions." ], "refs": [ "cohomology-lemma-summands-pseudo-coherent" ], "ref_ids": [ 2209 ] } ], "ref_ids": [] }, { "id": 13969, "type": "theorem", "label": "more-morphisms-lemma-complex-relative-pseudo-coherent-modules", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-complex-relative-pseudo-coherent-modules", "contents": [ "Let $X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $m \\in \\mathbf{Z}$. Let $\\mathcal{F}^\\bullet$ be a (locally) bounded", "above complex of $\\mathcal{O}_X$-modules such that $\\mathcal{F}^i$ is", "$(m - i)$-pseudo-coherent relative to $S$ for all $i$. Then", "$\\mathcal{F}^\\bullet$ is $m$-pseudo-coherent relative to $S$." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Cohomology, Lemma \\ref{cohomology-lemma-complex-pseudo-coherent-modules}", "and the definitions." ], "refs": [ "cohomology-lemma-complex-pseudo-coherent-modules" ], "ref_ids": [ 2210 ] } ], "ref_ids": [] }, { "id": 13970, "type": "theorem", "label": "more-morphisms-lemma-cohomology-relative-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-cohomology-relative-pseudo-coherent", "contents": [ "Let $X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $m \\in \\mathbf{Z}$. Let $E$ be an object of $D(\\mathcal{O}_X)$.", "If $E$ is (locally) bounded above and $H^i(E)$ is $(m - i)$-pseudo-coherent", "relative to $S$ for all $i$, then $E$ is $m$-pseudo-coherent relative to $S$." ], "refs": [], "proofs": [ { "contents": [ "Follows from", "Cohomology, Lemma \\ref{cohomology-lemma-cohomology-pseudo-coherent}", "and the definitions." ], "refs": [ "cohomology-lemma-cohomology-pseudo-coherent" ], "ref_ids": [ 2211 ] } ], "ref_ids": [] }, { "id": 13971, "type": "theorem", "label": "more-morphisms-lemma-base-change-relative-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-relative-pseudo-coherent", "contents": [ "Let $X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $m \\in \\mathbf{Z}$. Let $E$ be an object of $D(\\mathcal{O}_X)$", "which is $m$-pseudo-coherent relative to $S$. Let $S' \\to S$ be a", "morphism of schemes. Set $X' = X \\times_S S'$ and denote $E'$", "the derived pullback of $E$ to $X'$. If $S'$ and $X$ are", "Tor independent over $S$, then $E'$", "is $m$-pseudo-coherent relative to $S'$." ], "refs": [], "proofs": [ { "contents": [ "The problem is local on $X$ and $X'$ hence we may assume $X$, $S$, $S'$,", "and $X'$ are affine. Choose a closed immersion $i : X \\to \\mathbf{A}^n_S$", "and denote $i' : X' \\to \\mathbf{A}^n_{S'}$ the base change to $S'$.", "Denote $g : X' \\to X$ and $g' : \\mathbf{A}^n_{S'} \\to \\mathbf{A}^n_S$", "the projections, so $E' = Lg^*E$. Since $X$ and $S'$ are tor-independent", "over $S$, the base change map", "(Cohomology, Remark \\ref{cohomology-remark-base-change})", "induces an isomorphism", "$$", "Ri'_*(Lg^*E) = L(g')^*Ri_*E", "$$", "Namely, for a point $x' \\in X'$ lying over $x \\in X$ the base change", "map on stalks at $x'$ is the map", "$$", "E_x \\otimes_{\\mathcal{O}_{\\mathbf{A}^n_S, x}}^\\mathbf{L}", "\\mathcal{O}_{\\mathbf{A}^n_{S'}, x'}", "\\longrightarrow", "E_x \\otimes_{\\mathcal{O}_{X, x}}^\\mathbf{L}", "\\mathcal{O}_{X', x'}", "$$", "coming from the closed immersions $i$ and $i'$. Note that the source", "is quasi-isomorphic to a localization of", "$E_x \\otimes_{\\mathcal{O}_{S, s}}^\\mathbf{L} \\mathcal{O}_{S', s'}$", "which is isomorphic to the target as", "$\\mathcal{O}_{X', x'}$ is isomorphic to (the same) localization of", "$\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{S, s}}^\\mathbf{L}", "\\mathcal{O}_{S', s'}$ by assumption. We conclude the lemma holds", "by an application of", "Cohomology, Lemma \\ref{cohomology-lemma-pseudo-coherent-pullback}." ], "refs": [ "cohomology-remark-base-change", "cohomology-lemma-pseudo-coherent-pullback" ], "ref_ids": [ 2269, 2206 ] } ], "ref_ids": [] }, { "id": 13972, "type": "theorem", "label": "more-morphisms-lemma-pull-relative-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pull-relative-pseudo-coherent", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes locally of finite type", "over a base $S$. Let $m \\in \\mathbf{Z}$. Let $E$ be an object of", "$D(\\mathcal{O}_Y)$. Assume", "\\begin{enumerate}", "\\item $\\mathcal{O}_X$ is pseudo-coherent relative to $Y$\\footnote{This", "means $f$ is pseudo-coherent, see", "Definition \\ref{definition-pseudo-coherent}.}, and", "\\item $E$ is $m$-pseudo-coherent relative to $S$.", "\\end{enumerate}", "Then $Lf^*E$ is $m$-pseudo-coherent relative to $S$." ], "refs": [ "more-morphisms-definition-pseudo-coherent" ], "proofs": [ { "contents": [ "The problem is local on $X$. Thus we may assume $X$, $Y$, and $S$ are affine.", "Arguing as in the proof of More on Algebra, Lemma", "\\ref{more-algebra-lemma-pull-relative-pseudo-coherent}", "we can find a commutative diagram", "$$", "\\xymatrix{", "X \\ar[r]_i \\ar[d]_f &", "\\mathbf{A}^d_Y \\ar[r]_j \\ar[ld]^p &", "\\mathbf{A}^{n + d}_S \\ar[ld] \\\\", "Y \\ar[r] &", "\\mathbf{A}^n_S", "}", "$$", "Observe that", "$$", "Ri_* Lf^*E = Ri_* Li^* Lp^*E =", "Lp^*E \\otimes_{\\mathcal{O}_{\\mathbf{A}_Y^n}}^\\mathbf{L} Ri_*\\mathcal{O}_X", "$$", "by Cohomology, Lemma", "\\ref{cohomology-lemma-projection-formula-closed-immersion}.", "By assumption and the fact that $Y$ is affine, we can represent", "$Ri_*\\mathcal{O}_X = i_*\\mathcal{O}_X$ by a complexes of finite free", "$\\mathcal{O}_{\\mathbf{A}_Y^n}$-modules $\\mathcal{F}^\\bullet$, with", "$\\mathcal{F}^q = 0$ for $q > 0$", "(details omitted; use Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-pseudo-coherent-affine}", "and", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-rel-n-pseudo-module}).", "By assumption $E$ is bounded above, say $H^q(E) = 0$ for $q > a$.", "Represent $E$ by a complex $\\mathcal{E}^\\bullet$ of $\\mathcal{O}_Y$-modules", "with $\\mathcal{E}^q = 0$ for $q > a$. Then the derived tensor product above", "is represented by $\\text{Tot}(p^*\\mathcal{E}^\\bullet", "\\otimes_{\\mathcal{O}_{\\mathbf{A}_Y^n}} \\mathcal{F}^\\bullet)$.", "\\medskip\\noindent", "Since $j$ is a closed immersion, the functor $j_*$ is exact and", "$Rj_*$ is computed by applying $j_*$ to any representating complex", "of sheaves. Thus we have to show that", "$j_*\\text{Tot}(p^*\\mathcal{E}^\\bullet \\otimes_{\\mathcal{O}_{\\mathbf{A}_Y^n}}", "\\mathcal{F}^\\bullet)$ is $m$-pseudo-coherent", "as a complex of $\\mathcal{O}_{\\mathbf{A}^{n + m}_S}$-modules.", "Note that", "$\\text{Tot}(p^*\\mathcal{E}^\\bullet \\otimes_{\\mathcal{O}_{\\mathbf{A}_Y^n}}", "\\mathcal{F}^\\bullet)$ has a filtration by", "subcomplexes with successive quotients the complexes", "$p^*\\mathcal{E}^\\bullet", "\\otimes_{\\mathcal{O}_{\\mathbf{A}_Y^n}} \\mathcal{F}^q[-q]$.", "Note that for $q \\ll 0$ the complexes", "$p^*\\mathcal{E}^\\bullet \\otimes_{\\mathcal{O}_{\\mathbf{A}_Y^n}}", "\\mathcal{F}^q[-q]$", "have zero cohomology in degrees $\\leq m$ and hence are $m$-pseudo-coherent.", "Hence, applying", "Lemma \\ref{lemma-cone-relatively-pseudo-coherent}", "and induction, it suffices to show that", "$p^*\\mathcal{E}^\\bullet \\otimes_{\\mathcal{O}_{\\mathbf{A}_Y^n}}", "\\mathcal{F}^q[-q]$", "is pseudo-coherent relative to $S$ for all $q$. Note that $\\mathcal{F}^q = 0$", "for $q > 0$. Since also $\\mathcal{F}^q$ is finite free this", "reduces to proving that $p^*\\mathcal{E}^\\bullet$ is", "$m$-pseudo-coherent relative to $S$ which follows from", "Lemma \\ref{lemma-base-change-relative-pseudo-coherent}", "for instance." ], "refs": [ "more-algebra-lemma-pull-relative-pseudo-coherent", "cohomology-lemma-projection-formula-closed-immersion", "perfect-lemma-pseudo-coherent-affine", "more-algebra-lemma-rel-n-pseudo-module", "more-morphisms-lemma-cone-relatively-pseudo-coherent", "more-morphisms-lemma-base-change-relative-pseudo-coherent" ], "ref_ids": [ 10276, 2245, 6975, 10270, 13966, 13971 ] } ], "ref_ids": [ 14119 ] }, { "id": 13973, "type": "theorem", "label": "more-morphisms-lemma-composition-relative-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-composition-relative-pseudo-coherent", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes locally of finite type", "over a base $S$. Let $m \\in \\mathbf{Z}$. Let $E$ be an object of", "$D(\\mathcal{O}_X)$. Assume $\\mathcal{O}_Y$ is pseudo-coherent relative", "to $S$\\footnote{This means $Y \\to S$ is pseudo-coherent, see", "Definition \\ref{definition-pseudo-coherent}.}.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $E$ is $m$-pseudo-coherent relative to $Y$, and", "\\item $E$ is $m$-pseudo-coherent relative to $S$.", "\\end{enumerate}" ], "refs": [ "more-morphisms-definition-pseudo-coherent" ], "proofs": [ { "contents": [ "The question is local on $X$, hence we may assume $X$, $Y$, and $S$ are affine.", "Arguing as in the proof of More on Algebra, Lemma", "\\ref{more-algebra-lemma-pull-relative-pseudo-coherent}", "we can find a commutative diagram", "$$", "\\xymatrix{", "X \\ar[r]_i \\ar[d]_f &", "\\mathbf{A}^m_Y \\ar[r]_j \\ar[ld]^p &", "\\mathbf{A}^{n + m}_S \\ar[ld] \\\\", "Y \\ar[r] &", "\\mathbf{A}^n_S", "}", "$$", "The assumption that $\\mathcal{O}_Y$ is pseudo-coherent relative to $S$", "implies that $\\mathcal{O}_{\\mathbf{A}^m_Y}$ is pseudo-coherent relative", "to $\\mathbf{A}^m_S$ (by flat base change; this can be seen by using", "for example Lemma \\ref{lemma-base-change-relative-pseudo-coherent}).", "This in turn implies that $j_*\\mathcal{O}_{\\mathbf{A}^n_Y}$ is", "pseudo-coherent as an", "$\\mathcal{O}_{\\mathbf{A}^{n + m}_S}$-module. Then the equivalence of", "the lemma follows from", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-closed-push-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-pull-relative-pseudo-coherent", "more-morphisms-lemma-base-change-relative-pseudo-coherent", "perfect-lemma-closed-push-pseudo-coherent" ], "ref_ids": [ 10276, 13971, 6994 ] } ], "ref_ids": [ 14119 ] }, { "id": 13974, "type": "theorem", "label": "more-morphisms-lemma-check-relative-pseudo-coherence-on-charts", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-check-relative-pseudo-coherence-on-charts", "contents": [ "Let", "$$", "\\xymatrix{", "X \\ar[rd] \\ar[rr]_i & & P \\ar[ld] \\\\", "& S", "}", "$$", "be a commutative diagram of schemes. Assume $i$ is a closed immersion", "and $P \\to S$ flat and locally of finite presentation. Let $E$", "be an object of $D(\\mathcal{O}_X)$. Then the following", "are equivalent", "\\begin{enumerate}", "\\item $E$ is $m$-pseudo-coherent relative to $S$,", "\\item $Ri_*E$ is $m$-pseudo-coherent relative to $S$, and", "\\item $Ri_*E$ is $m$-pseudo-coherent on $P$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is", "Lemma \\ref{lemma-finite-morphism-relative-pseudo-coherence}.", "The equivalence of (2) and (3) follows from", "Lemma", "\\ref{lemma-composition-relative-pseudo-coherent}", "applied to $\\text{id} : P \\to P$", "provided we can show that $\\mathcal{O}_P$ is", "pseudo-coherent relative to $S$. This follows from", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-flat-finite-presentation-perfect}", "and the definitions." ], "refs": [ "more-morphisms-lemma-finite-morphism-relative-pseudo-coherence", "more-morphisms-lemma-composition-relative-pseudo-coherent", "more-algebra-lemma-flat-finite-presentation-perfect" ], "ref_ids": [ 13965, 13973, 10283 ] } ], "ref_ids": [] }, { "id": 13975, "type": "theorem", "label": "more-morphisms-lemma-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pseudo-coherent", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent", "\\begin{enumerate}", "\\item there exist an affine open covering $S = \\bigcup V_j$ and for each $j$", "an affine open covering $f^{-1}(V_j) = \\bigcup U_{ji}$ such that", "$\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_{ij})$ is a pseudo-coherent", "ring map,", "\\item for every pair of affine opens $U \\subset X$, $V \\subset S$", "such that $f(U) \\subset V$ the ring map", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is pseudo-coherent, and", "\\item $f$ is locally of finite type and $\\mathcal{O}_X$", "is pseudo-coherent relative to $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "To see the equivalence of (1) and (2) it suffices to check conditions", "(1)(a), (b), (c) of", "Morphisms, Definition \\ref{morphisms-definition-property-local}", "for the property of being a pseudo-coherent ring map.", "These properties follow (using localization is flat) from", "More on Algebra, Lemmas", "\\ref{more-algebra-lemma-base-change-relative-pseudo-coherent},", "\\ref{more-algebra-lemma-localize-relative-pseudo-coherent}, and", "\\ref{more-algebra-lemma-glue-relative-pseudo-coherent}.", "\\medskip\\noindent", "If (1) holds, then $f$ is locally of finite type as a pseudo-coherent", "ring map is of finite type by definition. Moreover, (1) implies", "via Lemma \\ref{lemma-qcoh-relative-pseudo-coherence-characterize}", "and the definitions that $\\mathcal{O}_X$ is pseudo-coherent", "relative to $S$. Conversely, if (3) holds, then we see", "that for every $U$ and $V$ as in (2) the ring", "$\\mathcal{O}_X(U)$ is of finite type over $\\mathcal{O}_S(V)$", "and $\\mathcal{O}_X(U)$ is as a module", "pseudo-coherent relative to $\\mathcal{O}_S(V)$, see", "Lemmas \\ref{lemma-relative-pseudo-coherence-characterize} and", "\\ref{lemma-qcoh-relative-pseudo-coherence-characterize}.", "This is the definition of a pseudo-coherent ring map, hence", "(2) and (1) hold." ], "refs": [ "morphisms-definition-property-local", "more-algebra-lemma-base-change-relative-pseudo-coherent", "more-algebra-lemma-localize-relative-pseudo-coherent", "more-algebra-lemma-glue-relative-pseudo-coherent", "more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize", "more-morphisms-lemma-relative-pseudo-coherence-characterize", "more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize" ], "ref_ids": [ 5547, 10275, 10274, 10279, 13963, 13962, 13963 ] } ], "ref_ids": [] }, { "id": 13976, "type": "theorem", "label": "more-morphisms-lemma-flat-base-change-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-base-change-pseudo-coherent", "contents": [ "A flat base change of a pseudo-coherent morphism is pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "This translates into the following algebra result:", "Let $A \\to B$ be a pseudo-coherent ring map.", "Let $A \\to A'$ be flat. Then $A' \\to B \\otimes_A A'$ is", "pseudo-coherent. This follows from the more general", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-base-change-relative-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-base-change-relative-pseudo-coherent" ], "ref_ids": [ 10275 ] } ], "ref_ids": [] }, { "id": 13977, "type": "theorem", "label": "more-morphisms-lemma-composition-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-composition-pseudo-coherent", "contents": [ "A composition of pseudo-coherent morphisms of schemes is", "pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "This translates into the following algebra result:", "If $A \\to B \\to C$ are composable pseudo-coherent ring maps", "then $A \\to C$ is pseudo-coherent. This follows from either", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-pull-relative-pseudo-coherent}", "or", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-composition-relative-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-pull-relative-pseudo-coherent", "more-algebra-lemma-composition-relative-pseudo-coherent" ], "ref_ids": [ 10276, 10278 ] } ], "ref_ids": [] }, { "id": 13978, "type": "theorem", "label": "more-morphisms-lemma-pseudo-coherent-finite-presentation", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pseudo-coherent-finite-presentation", "contents": [ "A pseudo-coherent morphism is locally of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 13979, "type": "theorem", "label": "more-morphisms-lemma-flat-finite-presentation-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-finite-presentation-pseudo-coherent", "contents": [ "A flat morphism which is locally of finite presentation is pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that a flat ring map of finite presentation is", "pseudo-coherent (and even perfect), see", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-flat-finite-presentation-perfect}." ], "refs": [ "more-algebra-lemma-flat-finite-presentation-perfect" ], "ref_ids": [ 10283 ] } ], "ref_ids": [] }, { "id": 13980, "type": "theorem", "label": "more-morphisms-lemma-permanence-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-permanence-pseudo-coherent", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes pseudo-coherent", "over a base scheme $S$. Then $f$ is pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "This translates into the following algebra result:", "If $R \\to A \\to B$ are composable ring maps", "and $R \\to A$, $R \\to B$ pseudo-coherent, then", "$R \\to B$ is pseudo-coherent. This follows from", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-composition-relative-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-composition-relative-pseudo-coherent" ], "ref_ids": [ 10278 ] } ], "ref_ids": [] }, { "id": 13981, "type": "theorem", "label": "more-morphisms-lemma-finite-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-finite-pseudo-coherent", "contents": [ "Let $f : X \\to S$ be a finite morphism of schemes.", "Then $f$ is pseudo-coherent if and only if $f_*\\mathcal{O}_X$", "is pseudo-coherent as an $\\mathcal{O}_S$-module." ], "refs": [], "proofs": [ { "contents": [ "Translated into algebra this lemma says the following: If $R \\to A$", "is a finite ring map, then $R \\to A$ is pseudo-coherent as a ring map", "(which means by definition that $A$ as an $A$-module is", "pseudo-coherent relative to $R$) if and only if", "$A$ is pseudo-coherent as an $R$-module. This follows from", "the more general", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-finite-extension-pseudo-coherent}." ], "refs": [ "more-algebra-lemma-finite-extension-pseudo-coherent" ], "ref_ids": [ 10268 ] } ], "ref_ids": [] }, { "id": 13982, "type": "theorem", "label": "more-morphisms-lemma-Noetherian-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-Noetherian-pseudo-coherent", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "If $S$ is locally Noetherian, then $f$ is pseudo-coherent if", "and only if $f$ is locally of finite type." ], "refs": [], "proofs": [ { "contents": [ "This translates into the following algebra result:", "If $R \\to A$ is a finite type ring map with $R$ Noetherian, then", "$R \\to A$ is pseudo-coherent if and only if $R \\to A$ is of finite type.", "To see this, note that a pseudo-coherent ring map is of finite type by", "definition. Conversely, if $R \\to A$ is of finite type, then", "we can write $A = R[x_1, \\ldots, x_n]/I$ and it follows from", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-Noetherian-pseudo-coherent}", "that $A$ is pseudo-coherent as an $R[x_1, \\ldots, x_n]$-module, i.e.,", "$R \\to A$ is a pseudo-coherent ring map." ], "refs": [ "more-algebra-lemma-Noetherian-pseudo-coherent" ], "ref_ids": [ 10160 ] } ], "ref_ids": [] }, { "id": 13983, "type": "theorem", "label": "more-morphisms-lemma-descending-property-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-descending-property-pseudo-coherent", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is pseudo-coherent''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion of", "Descent, Lemma \\ref{descent-lemma-descending-properties-morphisms}", "to prove this. By", "Definition \\ref{definition-pseudo-coherent}", "being pseudo-coherent is Zariski local on the base. By", "Lemma \\ref{lemma-flat-base-change-pseudo-coherent}", "being pseudo-coherent is preserved under flat base change.", "The final hypothesis (3) of", "Descent, Lemma \\ref{descent-lemma-descending-properties-morphisms}", "translates into the following algebra statement:", "Let $A \\to B$ be a faithfully flat ring map.", "Let $C = A[x_1, \\ldots, x_n]/I$ be an $A$-algebra.", "If $C \\otimes_A B$ is pseudo-coherent as an $B[x_1, \\ldots, x_n]$-module,", "then $C$ is pseudo-coherent as a $A[x_1, \\ldots, x_n]$-module.", "This is", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-descent-pseudo-coherent}." ], "refs": [ "descent-lemma-descending-properties-morphisms", "more-morphisms-definition-pseudo-coherent", "more-morphisms-lemma-flat-base-change-pseudo-coherent", "descent-lemma-descending-properties-morphisms", "more-algebra-lemma-flat-descent-pseudo-coherent" ], "ref_ids": [ 14665, 14119, 13976, 14665, 10158 ] } ], "ref_ids": [] }, { "id": 13984, "type": "theorem", "label": "more-morphisms-lemma-quotient-of-flat-finitely-presented", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-quotient-of-flat-finitely-presented", "contents": [ "Let $A \\to B$ be a flat ring map of finite presentation.", "Let $I \\subset B$ be an ideal. Then $A \\to B/I$ is pseudo-coherent", "if and only if $I$ is pseudo-coherent as a $B$-module." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $B = A[x_1, \\ldots, x_n]/J$.", "Note that $B$ is pseudo-coherent as an $A[x_1, \\ldots, x_n]$-module", "because $A \\to B$ is a pseudo-coherent ring map by", "Lemma \\ref{lemma-flat-finite-presentation-pseudo-coherent}.", "Note that $A \\to B/I$ is pseudo-coherent if and only if", "$B/I$ is pseudo-coherent as an $A[x_1, \\ldots, x_n]$-module. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-finite-push-pseudo-coherent}", "we see this is equivalent to the condition that $B/I$ is", "pseudo-coherent as an $B$-module. This proves the lemma as the", "short exact sequence $0 \\to I \\to B \\to B/I \\to 0$", "shows that $I$ is pseudo-coherent if and only if $B/I$ is (see", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-two-out-of-three-pseudo-coherent})." ], "refs": [ "more-morphisms-lemma-flat-finite-presentation-pseudo-coherent", "more-algebra-lemma-finite-push-pseudo-coherent", "more-algebra-lemma-two-out-of-three-pseudo-coherent" ], "ref_ids": [ 13979, 10154, 10149 ] } ], "ref_ids": [] }, { "id": 13985, "type": "theorem", "label": "more-morphisms-lemma-pseudo-coherent-syntomic-local-source", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pseudo-coherent-syntomic-local-source", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is pseudo-coherent''", "is syntomic local on the source." ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion of", "Descent, Lemma \\ref{descent-lemma-properties-morphisms-local-source}", "to prove this. It follows from", "Lemmas \\ref{lemma-flat-finite-presentation-pseudo-coherent} and", "\\ref{lemma-composition-pseudo-coherent}", "that being pseudo-coherent is preserved under precomposing with", "flat morphisms locally of finite presentation, in particular under", "precomposing with syntomic morphisms (see", "Morphisms, Lemmas \\ref{morphisms-lemma-syntomic-flat} and", "\\ref{morphisms-lemma-syntomic-locally-finite-presentation}).", "It is clear from", "Definition \\ref{definition-pseudo-coherent}", "that being pseudo-coherent is", "Zariski local on the source and target.", "Hence, according to the aforementioned", "Descent, Lemma \\ref{descent-lemma-properties-morphisms-local-source}", "it suffices to prove the following: Suppose $X' \\to X \\to Y$ are", "morphisms of affine schemes with $X' \\to X$ syntomic and $X' \\to Y$", "pseudo-coherent. Then $X \\to Y$ is pseudo-coherent.", "To see this, note that in any case $X \\to Y$ is of finite presentation by", "Descent, Lemma \\ref{descent-lemma-flat-finitely-presented-permanence-algebra}.", "Choose a closed immersion $X \\to \\mathbf{A}^n_Y$. By", "Algebra, Lemma \\ref{algebra-lemma-lift-syntomic}", "we can find an affine open covering $X' = \\bigcup_{i = 1, \\ldots, n} X'_i$", "and syntomic morphisms $W_i \\to \\mathbf{A}^n_Y$ lifting the morphisms", "$X'_i \\to X$, i.e., such that there are fibre product diagrams", "$$", "\\xymatrix{", "X'_i \\ar[d] \\ar[r] & W_i \\ar[d] \\\\", "X \\ar[r] & \\mathbf{A}^n_Y", "}", "$$", "After replacing $X'$ by $\\coprod X'_i$ and setting $W = \\coprod W_i$", "we obtain a fibre product diagram", "$$", "\\xymatrix{", "X' \\ar[d] \\ar[r] & W \\ar[d]^h \\\\", "X \\ar[r] & \\mathbf{A}^n_Y", "}", "$$", "with $W \\to \\mathbf{A}^n_Y$ flat and of finite presentation and", "$X' \\to Y$ still pseudo-coherent. Since", "$W \\to \\mathbf{A}^n_Y$ is open (see", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open})", "and $X' \\to X$ is surjective we can find", "$f \\in \\Gamma(\\mathbf{A}^n_Y, \\mathcal{O})$ such that", "$X \\subset D(f) \\subset \\Im(h)$. Write", "$Y = \\Spec(R)$, $X = \\Spec(A)$, $X' = \\Spec(A')$", "and $W = \\Spec(B)$, $A = R[x_1, \\ldots, x_n]/I$ and", "$A' = B/IB$. Then $R \\to A'$ is pseudo-coherent. Picture", "$$", "\\xymatrix{", "A' = B/IB & B \\ar[l] \\\\", "A = R[x_1, \\ldots, x_n]/I \\ar[u] & R[x_1, \\ldots, x_n] \\ar[l] \\ar[u]", "}", "$$", "By", "Lemma \\ref{lemma-quotient-of-flat-finitely-presented}", "we see that $IB$ is pseudo-coherent as a $B$-module.", "The ring map $R[x_1, \\ldots, x_n]_f \\to B_f$ is faithfully flat by", "our choice of $f$ above. This implies that", "$I_f \\subset R[x_1, \\ldots, x_n]_f$", "is pseudo-coherent, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-descent-pseudo-coherent}.", "Applying", "Lemma \\ref{lemma-quotient-of-flat-finitely-presented}", "one more time we see that $R \\to A$ is pseudo-coherent." ], "refs": [ "descent-lemma-properties-morphisms-local-source", "more-morphisms-lemma-flat-finite-presentation-pseudo-coherent", "more-morphisms-lemma-composition-pseudo-coherent", "morphisms-lemma-syntomic-flat", "morphisms-lemma-syntomic-locally-finite-presentation", "more-morphisms-definition-pseudo-coherent", "descent-lemma-properties-morphisms-local-source", "descent-lemma-flat-finitely-presented-permanence-algebra", "algebra-lemma-lift-syntomic", "morphisms-lemma-fppf-open", "more-morphisms-lemma-quotient-of-flat-finitely-presented", "more-algebra-lemma-flat-descent-pseudo-coherent", "more-morphisms-lemma-quotient-of-flat-finitely-presented" ], "ref_ids": [ 14707, 13979, 13977, 5294, 5293, 14119, 14707, 14640, 1188, 5267, 13984, 10158, 13984 ] } ], "ref_ids": [] }, { "id": 13986, "type": "theorem", "label": "more-morphisms-lemma-pseudo-coherent-fppf-local-source", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pseudo-coherent-fppf-local-source", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is pseudo-coherent''", "is fppf local on the source." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\{g_i : X_i \\to X\\}$ be an fppf covering such that each composition", "$f \\circ g_i$ is pseudo-coherent. According to", "Lemma \\ref{lemma-dominate-fppf}", "there exist", "\\begin{enumerate}", "\\item a Zariski open covering $X = \\bigcup U_j$,", "\\item surjective finite locally free morphisms $W_j \\to U_j$,", "\\item Zariski open coverings $W_j = \\bigcup_k W_{j, k}$,", "\\item surjective finite locally free morphisms $T_{j, k} \\to W_{j, k}$", "\\end{enumerate}", "such that the fppf covering $\\{h_{j, k} : T_{j, k} \\to X\\}$ refines the given", "covering $\\{X_i \\to X\\}$. Denote $\\psi_{j, k} : T_{j, k} \\to X_{\\alpha(j, k)}$", "the morphisms that witness the fact that $\\{T_{j, k} \\to X\\}$ refines", "the given covering $\\{X_i \\to X\\}$. Note that $T_{j, k} \\to X$ is a flat,", "locally finitely presented morphism, so both $X_i$ and $T_{j, k}$ are", "pseudo-coherent over $X$ by", "Lemma \\ref{lemma-flat-finite-presentation-pseudo-coherent}.", "Hence $\\psi_{j, k} : T_{j, k} \\to X_i$ is pseudo-coherent, see", "Lemma \\ref{lemma-permanence-pseudo-coherent}.", "Hence $T_{j, k} \\to S$ is pseudo coherent as the composition", "of $\\psi_{j, k}$ and $f \\circ g_{\\alpha(j, k)}$, see", "Lemma \\ref{lemma-composition-pseudo-coherent}.", "Thus we see we have reduced the lemma to the case of", "a Zariski open covering (which is OK) and the case of a covering", "given by a single surjective finite locally free morphism which we", "deal with in the following paragraph.", "\\medskip\\noindent", "Assume that $X' \\to X \\to S$ is a sequence of morphisms of schemes", "with $X' \\to X$ surjective finite locally free and $X' \\to Y$ pseudo-coherent.", "Our goal is to show that $X \\to S$ is pseudo-coherent.", "Note that by", "Descent, Lemma \\ref{descent-lemma-flat-finitely-presented-permanence}", "the morphism $X \\to S$ is locally of finite presentation.", "It is clear that the problem reduces to the case that $X'$, $X$ and $S$", "are affine and $X' \\to X$ is free of some rank $r > 0$. The corresponding", "algebra problem is the following: Suppose $R \\to A \\to A'$ are ring maps", "such that $R \\to A'$ is pseudo-coherent, $R \\to A$ is of finite presentation,", "and $A' \\cong A^{\\oplus r}$ as an $A$-module. Goal: Show $R \\to A$ is", "pseudo-coherent. The assumption that $R \\to A'$ is pseudo-coherent", "means that $A'$ as an $A'$-module is pseudo-coherent relative to $R$. By", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-finite-extension-pseudo-coherent}", "this implies that $A'$ as an $A$-module is pseudo-coherent relative to $R$.", "Since $A' \\cong A^{\\oplus r}$ as an $A$-module we see that", "$A$ as an $A$-module is pseudo-coherent relative to $R$, see", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-summands-relative-pseudo-coherent}.", "This by definition means that $R \\to A$ is pseudo-coherent and we win." ], "refs": [ "more-morphisms-lemma-dominate-fppf", "more-morphisms-lemma-flat-finite-presentation-pseudo-coherent", "more-morphisms-lemma-permanence-pseudo-coherent", "more-morphisms-lemma-composition-pseudo-coherent", "descent-lemma-flat-finitely-presented-permanence", "more-algebra-lemma-finite-extension-pseudo-coherent", "more-algebra-lemma-summands-relative-pseudo-coherent" ], "ref_ids": [ 13922, 13979, 13980, 13977, 14642, 10268, 10271 ] } ], "ref_ids": [] }, { "id": 13987, "type": "theorem", "label": "more-morphisms-lemma-perfect", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-perfect", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "The following are equivalent", "\\begin{enumerate}", "\\item there exist an affine open covering $S = \\bigcup V_j$ and for each $j$", "an affine open covering $f^{-1}(V_j) = \\bigcup U_{ji}$ such that", "$\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_{ij})$ is a perfect", "ring map, and", "\\item for every pair of affine opens $U \\subset X$, $V \\subset S$", "such that $f(U) \\subset V$ the ring map", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is perfect.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and let $U, V$ be as in (2).", "It follows from", "Lemma \\ref{lemma-pseudo-coherent}", "that $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is pseudo-coherent.", "Hence it suffices to prove that the property of a ring map", "being \"of finite tor dimension\" satisfies", "conditions (1)(a), (b), (c) of", "Morphisms, Definition \\ref{morphisms-definition-property-local}.", "These properties follow from", "More on Algebra,", "Lemmas \\ref{more-algebra-lemma-flat-push-tor-amplitude},", "\\ref{more-algebra-lemma-flat-base-change-finite-tor-dimension}, and", "\\ref{more-algebra-lemma-glue-tor-amplitude}.", "Some details omitted." ], "refs": [ "more-morphisms-lemma-pseudo-coherent", "morphisms-definition-property-local", "more-algebra-lemma-flat-push-tor-amplitude", "more-algebra-lemma-flat-base-change-finite-tor-dimension", "more-algebra-lemma-glue-tor-amplitude" ], "ref_ids": [ 13975, 5547, 10178, 10181, 10183 ] } ], "ref_ids": [] }, { "id": 13988, "type": "theorem", "label": "more-morphisms-lemma-flat-base-change-perfect", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-base-change-perfect", "contents": [ "A flat base change of a perfect morphism is perfect." ], "refs": [], "proofs": [ { "contents": [ "This translates into the following algebra result:", "Let $A \\to B$ be a perfect ring map.", "Let $A \\to A'$ be flat. Then $A' \\to B \\otimes_A A'$ is", "perfect. This result for pseudo-coherent ring maps we have seen in", "Lemma \\ref{lemma-flat-base-change-pseudo-coherent}.", "The corresponding fact for finite tor dimension follows from", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-flat-base-change-finite-tor-dimension}." ], "refs": [ "more-morphisms-lemma-flat-base-change-pseudo-coherent", "more-algebra-lemma-flat-base-change-finite-tor-dimension" ], "ref_ids": [ 13976, 10181 ] } ], "ref_ids": [] }, { "id": 13989, "type": "theorem", "label": "more-morphisms-lemma-composition-perfect", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-composition-perfect", "contents": [ "A composition of perfect morphisms of schemes is perfect." ], "refs": [], "proofs": [ { "contents": [ "This translates into the following algebra result:", "If $A \\to B \\to C$ are composable perfect ring maps", "then $A \\to C$ is perfect. We have seen this is the case for", "pseudo-coherent in", "Lemma \\ref{lemma-composition-pseudo-coherent}", "and its proof. By assumption there exist integers $n$, $m$ such", "that $B$ has tor dimension $\\leq n$ over $A$ and $C$ has tor dimension", "$\\leq m$ over $B$. Then for any $A$-module $M$ we have", "$$", "M \\otimes_A^{\\mathbf{L}} C =", "(M \\otimes_A^{\\mathbf{L}} B) \\otimes_B^{\\mathbf{L}} C", "$$", "and the spectral sequence of", "More on Algebra, Example \\ref{more-algebra-example-tor}", "shows that $\\text{Tor}^A_p(M, C) = 0$ for $p > n + m$ as desired." ], "refs": [ "more-morphisms-lemma-composition-pseudo-coherent" ], "ref_ids": [ 13977 ] } ], "ref_ids": [] }, { "id": 13990, "type": "theorem", "label": "more-morphisms-lemma-flat-finite-presentation-perfect", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-finite-presentation-perfect", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is flat and perfect, and", "\\item $f$ is flat and locally of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (2) $\\Rightarrow$ (1) is", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-flat-finite-presentation-perfect}.", "The converse follows from the fact that a pseudo-coherent morphism", "is locally of finite presentation, see", "Lemma \\ref{lemma-pseudo-coherent-finite-presentation}." ], "refs": [ "more-algebra-lemma-flat-finite-presentation-perfect", "more-morphisms-lemma-pseudo-coherent-finite-presentation" ], "ref_ids": [ 10283, 13978 ] } ], "ref_ids": [] }, { "id": 13991, "type": "theorem", "label": "more-morphisms-lemma-regular-target-perfect", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-regular-target-perfect", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume $S$ is regular and $f$ is locally of finite type.", "Then $f$ is perfect." ], "refs": [], "proofs": [ { "contents": [ "See", "More on Algebra, Lemma \\ref{more-algebra-lemma-regular-perfect-ring-map}." ], "refs": [ "more-algebra-lemma-regular-perfect-ring-map" ], "ref_ids": [ 10284 ] } ], "ref_ids": [] }, { "id": 13992, "type": "theorem", "label": "more-morphisms-lemma-regular-immersion-perfect", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-regular-immersion-perfect", "contents": [ "A regular immersion of schemes is perfect.", "A Koszul-regular immersion of schemes is perfect." ], "refs": [], "proofs": [ { "contents": [ "Since a regular immersion is a Koszul-regular immersion, see", "Divisors, Lemma \\ref{divisors-lemma-regular-quasi-regular-immersion},", "it suffices to prove the second statement. This translates into the", "following algebraic statement: Suppose that $I \\subset A$ is an", "ideal generated by a Koszul-regular sequence $f_1, \\ldots, f_r$ of $A$.", "Then $A \\to A/I$ is a perfect ring map. Since $A \\to A/I$ is surjective", "this is a presentation of $A/I$ by a polynomial algebra over $A$.", "Hence it suffices to see that $A/I$ is pseudo-coherent as an $A$-module", "and has finite tor dimension. By definition of a Koszul sequence", "the Koszul complex $K(A, f_1, \\ldots, f_r)$ is a finite free resolution", "of $A/I$. Hence $A/I$ is a perfect complex of $A$-modules and we win." ], "refs": [ "divisors-lemma-regular-quasi-regular-immersion" ], "ref_ids": [ 7989 ] } ], "ref_ids": [] }, { "id": 13993, "type": "theorem", "label": "more-morphisms-lemma-perfect-permanence", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-perfect-permanence", "contents": [ "Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes. Assume $Y \\to S$", "smooth and $X \\to S$ perfect. Then $f : X \\to Y$ is perfect." ], "refs": [], "proofs": [ { "contents": [ "We can factor $f$ as the composition", "$$", "X \\longrightarrow X \\times_S Y \\longrightarrow Y", "$$", "where the first morphism is the map $i = (1, f)$ and the second", "morphism is the projection. Since $Y \\to S$ is flat, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-flat},", "we see that $X \\times_S Y \\to Y$ is perfect by", "Lemma \\ref{lemma-flat-base-change-perfect}.", "As $Y \\to S$ is smooth, also $X \\times_S Y \\to X$ is smooth, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-smooth}.", "Hence $i$ is a section of a smooth morphism, therefore $i$ is", "a regular immersion, see", "Divisors, Lemma \\ref{divisors-lemma-section-smooth-regular-immersion}.", "This implies that $i$ is perfect, see", "Lemma \\ref{lemma-regular-immersion-perfect}.", "We conclude that $f$ is perfect because the composition of perfect", "morphisms is perfect, see", "Lemma \\ref{lemma-composition-perfect}." ], "refs": [ "morphisms-lemma-smooth-flat", "more-morphisms-lemma-flat-base-change-perfect", "morphisms-lemma-base-change-smooth", "more-morphisms-lemma-regular-immersion-perfect", "more-morphisms-lemma-composition-perfect" ], "ref_ids": [ 5331, 13988, 5327, 13992, 13989 ] } ], "ref_ids": [] }, { "id": 13994, "type": "theorem", "label": "more-morphisms-lemma-descending-property-perfect", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-descending-property-perfect", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is perfect''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion of", "Descent, Lemma \\ref{descent-lemma-descending-properties-morphisms}", "to prove this. By", "Definition \\ref{definition-perfect}", "being perfect is Zariski local on the base. By", "Lemma \\ref{lemma-flat-base-change-perfect}", "being perfect is preserved under flat base change.", "The final hypothesis (3) of", "Descent, Lemma \\ref{descent-lemma-descending-properties-morphisms}", "translates into the following algebra statement:", "Let $A \\to B$ be a faithfully flat ring map.", "Let $C = A[x_1, \\ldots, x_n]/I$ be an $A$-algebra.", "If $C \\otimes_A B$ is perfect as an $B[x_1, \\ldots, x_n]$-module,", "then $C$ is perfect as a $A[x_1, \\ldots, x_n]$-module.", "This is", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-descent-perfect}." ], "refs": [ "descent-lemma-descending-properties-morphisms", "more-morphisms-definition-perfect", "more-morphisms-lemma-flat-base-change-perfect", "descent-lemma-descending-properties-morphisms", "more-algebra-lemma-flat-descent-perfect" ], "ref_ids": [ 14665, 14120, 13988, 14665, 10222 ] } ], "ref_ids": [] }, { "id": 13995, "type": "theorem", "label": "more-morphisms-lemma-check-perfect-stalks", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-check-perfect-stalks", "contents": [ "Let $f : X \\to S$ be a pseudo-coherent morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is perfect,", "\\item $\\mathcal{O}_X$ locally has finite tor dimension as a", "sheaf of $f^{-1}\\mathcal{O}_S$-modules, and", "\\item for all $x \\in X$ the ring $\\mathcal{O}_{X, x}$ has finite tor", "dimension as an $\\mathcal{O}_{S, f(x)}$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The problem is local on $X$ and $S$. Hence we may assume that", "$X = \\Spec(B)$, $S = \\Spec(A)$ and $f$ corresponds to a pseudo-coherent", "ring map $A \\to B$.", "\\medskip\\noindent", "If (1) holds, then $B$ has finite tor dimension $d$ as $A$-module. Then", "$B_\\mathfrak q$ has tor dimension $d$ as an $A_\\mathfrak p$-module for all", "primes $\\mathfrak q \\subset B$ with $\\mathfrak p = A \\cap \\mathfrak q$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-tor-amplitude-localization}.", "Then $\\mathcal{O}_X$ has tor dimension $d$ as a ", "sheaf of $f^{-1}\\mathcal{O}_S$-modules by", "Cohomology, Lemma \\ref{cohomology-lemma-tor-amplitude-stalk}.", "Thus (1) implies (2).", "\\medskip\\noindent", "By Cohomology, Lemma \\ref{cohomology-lemma-tor-amplitude-stalk} (2) implies", "(3).", "\\medskip\\noindent", "Assume (3). We cannot use", "More on Algebra, Lemma \\ref{more-algebra-lemma-tor-amplitude-localization}", "to conclude as we are not given that the tor dimension of", "$B_\\mathfrak q$ over $A_\\mathfrak p$ is bounded independent of $\\mathfrak q$.", "Choose a presentation $A[x_1, \\ldots, x_n] \\to B$. Then $B$ is", "pseudo-coherent as a $A[x_1, \\ldots, x_n]$-module. Let", "$\\mathfrak q \\subset A[x_1, \\ldots, x_n]$ be a prime ideal", "lying over $\\mathfrak p \\subset A$. Then either $B_\\mathfrak q$ is zero", "or by assumption it has finite tor dimension as an", "$A_\\mathfrak p$-module. Since the fibres of $A \\to A[x_1, \\ldots, x_n]$", "have finite global dimension, we can apply", "More on Algebra, Lemma \\ref{more-algebra-lemma-perfect-over-polynomial-ring}", "to $A_\\mathfrak p \\to A[x_1, \\ldots, x_n]_\\mathfrak q$", "to see that $B_\\mathfrak q$ is a perfect", "$A[x_1, \\ldots, x_n]_\\mathfrak q$-module. Hence", "$B$ is a perfect $A[x_1, \\ldots, x_n]$-module by", "More on Algebra, Lemma \\ref{more-algebra-lemma-check-perfect-stalks}.", "Thus $A \\to B$ is a perfect ring map by definition." ], "refs": [ "more-algebra-lemma-tor-amplitude-localization", "cohomology-lemma-tor-amplitude-stalk", "cohomology-lemma-tor-amplitude-stalk", "more-algebra-lemma-tor-amplitude-localization", "more-algebra-lemma-perfect-over-polynomial-ring", "more-algebra-lemma-check-perfect-stalks" ], "ref_ids": [ 10182, 2217, 2217, 10182, 10245, 10243 ] } ], "ref_ids": [] }, { "id": 13996, "type": "theorem", "label": "more-morphisms-lemma-perfect-closed-immersion-perfect-direct-image", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-perfect-closed-immersion-perfect-direct-image", "contents": [ "Let $i : Z \\to X$ be a perfect closed immersion of schemes.", "Then $i_*\\mathcal{O}_X$ is a perfect $\\mathcal{O}_X$-module, i.e.,", "it is a perfect object of $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "This is more or less immediate from the definition. Namely, let", "$U = \\Spec(A)$ be an affine open of $X$. Then $i^{-1}(U) = \\Spec(A/I)$", "for some ideal $I \\subset A$ and $A/I$ has a finite resolution by", "finite projective $A$-modules by", "More on Algebra, Lemma \\ref{more-algebra-lemma-perfect-ring-map}.", "Hence $i_*\\mathcal{O}_Z|_U$ can be represented by a finite", "length complex of finite locally free $\\mathcal{O}_U$-modules.", "This is what we had to show, see", "Cohomology, Section \\ref{cohomology-section-perfect}." ], "refs": [ "more-algebra-lemma-perfect-ring-map" ], "ref_ids": [ 10281 ] } ], "ref_ids": [] }, { "id": 13997, "type": "theorem", "label": "more-morphisms-lemma-perfect-proper-perfect-direct-image", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-perfect-proper-perfect-direct-image", "contents": [ "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a perfect proper", "morphism of schemes. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Then", "$Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$." ], "refs": [], "proofs": [ { "contents": [ "We claim that Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-perfect-direct-image} applies.", "Conditions (1) and (2) are immediate. Condition (3) is local", "on $X$. Thus we may assume $X$ and $S$ affine and $E$", "represented by a strictly perfect complex of $\\mathcal{O}_X$-modules.", "Thus it suffices to show that $\\mathcal{O}_X$ has finite", "tor dimension as a sheaf of $f^{-1}\\mathcal{O}_S$-modules.", "This is equivalent to being perfect by", "Lemma \\ref{lemma-check-perfect-stalks}." ], "refs": [ "perfect-lemma-perfect-direct-image", "more-morphisms-lemma-check-perfect-stalks" ], "ref_ids": [ 7043, 13995 ] } ], "ref_ids": [] }, { "id": 13998, "type": "theorem", "label": "more-morphisms-lemma-perfect-fppf-local-source", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-perfect-fppf-local-source", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is perfect''", "is fppf local on the source." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{g_i : X_i \\to X\\}_{i \\in I}$ be an fppf covering of schemes and let", "$f : X \\to S$ be a morphism such that each $f \\circ g_i$ is", "perfect. By", "Lemma \\ref{lemma-pseudo-coherent-fppf-local-source}", "we conclude that $f$ is pseudo-coherent.", "Hence by", "Lemma \\ref{lemma-check-perfect-stalks}", "it suffices to check that $\\mathcal{O}_{X, x}$ is an", "$\\mathcal{O}_{S, f(x)}$-module of finite tor dimension for all $x \\in X$.", "Pick $i \\in I$ and $x_i \\in X_i$ mapping to $x$. Then we see that", "$\\mathcal{O}_{X_i, x_i}$ has finite tor dimension over", "$\\mathcal{O}_{S, f(x)}$ and that", "$\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X_i, x_i}$ is faithfully flat.", "The desired conclusion follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-descent-tor-amplitude}." ], "refs": [ "more-morphisms-lemma-pseudo-coherent-fppf-local-source", "more-morphisms-lemma-check-perfect-stalks", "more-algebra-lemma-flat-descent-tor-amplitude" ], "ref_ids": [ 13986, 13995, 10184 ] } ], "ref_ids": [] }, { "id": 13999, "type": "theorem", "label": "more-morphisms-lemma-factor-regular-immersion", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-factor-regular-immersion", "contents": [ "Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of schemes.", "Assume", "\\begin{enumerate}", "\\item $X$ is locally Noetherian,", "\\item $j \\circ i$ is a regular immersion, and", "\\item $i$ is perfect.", "\\end{enumerate}", "Then $i$ and $j$ are regular immersions." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ (and hence $Y$) is locally Noetherian all 4 types of regular", "immersions agree, and moreover we may check whether a morphism is a", "regular immersion on the level of local rings, see", "Divisors, Lemma \\ref{divisors-lemma-Noetherian-scheme-regular-ideal}.", "Thus the result follows from", "Divided Power Algebra, Lemma \\ref{dpa-lemma-regular-sequence}." ], "refs": [ "divisors-lemma-Noetherian-scheme-regular-ideal", "dpa-lemma-regular-sequence" ], "ref_ids": [ 7988, 1674 ] } ], "ref_ids": [] }, { "id": 14000, "type": "theorem", "label": "more-morphisms-lemma-koszul-independence-factorization", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-koszul-independence-factorization", "contents": [ "Let $S$ be a scheme. Let $U$, $P$, $P'$ be schemes over $S$.", "Let $u \\in U$. Let $i : U \\to P$, $i' : U \\to P'$ be immersions over $S$.", "Assume $P$ and $P'$ smooth over $S$. Then the following are equivalent", "\\begin{enumerate}", "\\item $i$ is a Koszul-regular immersion in a neighbourhood of $x$, and", "\\item $i'$ is a Koszul-regular immersion in a neighbourhood of $x$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $i$ is a Koszul-regular immersion in a neighbourhood of $x$.", "Consider the morphism $j = (i, i') : U \\to P \\times_S P' = P''$.", "Since $P'' = P \\times_S P' \\to P$ is smooth, it follows from", "Divisors, Lemma \\ref{divisors-lemma-lift-regular-immersion-to-smooth}", "that $j$ is a Koszul-regular immersion, whereupon it follows from", "Divisors, Lemma \\ref{divisors-lemma-push-regular-immersion-thru-smooth}", "that $i'$ is a Koszul-regular immersion." ], "refs": [ "divisors-lemma-lift-regular-immersion-to-smooth", "divisors-lemma-push-regular-immersion-thru-smooth" ], "ref_ids": [ 8005, 8008 ] } ], "ref_ids": [] }, { "id": 14001, "type": "theorem", "label": "more-morphisms-lemma-lci", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-lci", "contents": [ "Let $f : X \\to S$ be a local complete intersection morphism.", "Let $P$ be a scheme smooth over $S$. Let $U \\subset X$ be an open subscheme", "and $i : U \\to P$ an immersion of schemes over $S$.", "Then $i$ is a Koszul-regular immersion." ], "refs": [], "proofs": [ { "contents": [ "This is the defining property of a local complete intersection", "morphism. See discussion above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14002, "type": "theorem", "label": "more-morphisms-lemma-lci-properties", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-lci-properties", "contents": [ "Let $f : X \\to S$ be a local complete intersection morphism.", "Then", "\\begin{enumerate}", "\\item $f$ is locally of finite presentation,", "\\item $f$ is pseudo-coherent, and", "\\item $f$ is perfect.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since a perfect morphism is pseudo-coherent", "(because a perfect ring map is pseudo-coherent)", "and a pseudo-coherent morphism is locally of finite presentation", "(because a pseudo-coherent ring map is of finite presentation)", "it suffices to prove the last statement. Being perfect is a local", "property, hence we may assume that $f$ factors as $\\pi \\circ i$ where", "$\\pi$ is smooth and $i$ is a Koszul-regular immersion.", "A Koszul-regular immersion is perfect, see", "Lemma \\ref{lemma-regular-immersion-perfect}.", "A smooth morphism is perfect as it is flat and locally of finite", "presentation, see", "Lemma \\ref{lemma-flat-finite-presentation-perfect}.", "Finally a composition of perfect morphisms is perfect, see", "Lemma \\ref{lemma-composition-perfect}." ], "refs": [ "more-morphisms-lemma-regular-immersion-perfect", "more-morphisms-lemma-flat-finite-presentation-perfect", "more-morphisms-lemma-composition-perfect" ], "ref_ids": [ 13992, 13990, 13989 ] } ], "ref_ids": [] }, { "id": 14003, "type": "theorem", "label": "more-morphisms-lemma-affine-lci", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-affine-lci", "contents": [ "Let $f : X = \\Spec(B) \\to S = \\Spec(A)$ be a morphism of affine schemes.", "Then $f$ is a local complete intersection morphism if and only if", "$A \\to B$ is a local complete intersection homomorphism, see", "More on Algebra, Definition", "\\ref{more-algebra-definition-local-complete-intersection}." ], "refs": [ "more-algebra-definition-local-complete-intersection" ], "proofs": [ { "contents": [ "Follows immediately from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 10609 ] }, { "id": 14004, "type": "theorem", "label": "more-morphisms-lemma-flat-base-change-lci", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-base-change-lci", "contents": [ "A flat base change of a local complete intersection morphism is a", "local complete intersection morphism." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: This is true because a base change of a smooth morphism", "is smooth and a flat base change of a Koszul-regular immersion is a", "Koszul-regular immersion, see", "Divisors, Lemma \\ref{divisors-lemma-regular-immersion-noetherian}." ], "refs": [ "divisors-lemma-regular-immersion-noetherian" ], "ref_ids": [ 7990 ] } ], "ref_ids": [] }, { "id": 14005, "type": "theorem", "label": "more-morphisms-lemma-composition-lci", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-composition-lci", "contents": [ "A composition of local complete intersection morphisms", "is a local complete intersection morphism." ], "refs": [], "proofs": [ { "contents": [ "Let $g : Y \\to S$ and $f : X \\to Y$ be local complete intersection", "morphisms. Let $x \\in X$ and set $y = f(x)$. Choose an open neighbourhood", "$V \\subset Y$ of $y$ and a factorization $g|_V = \\pi \\circ i$ for some", "Koszul-regular immersion $i : V \\to P$ and smooth morphism $\\pi : P \\to S$.", "Next choose an open neighbourhood $U$ of $x \\in X$ and a factorization", "$f|_U = \\pi' \\circ i'$ for some Koszul-regular immersion $i' : U \\to P'$", "and smooth morphism $\\pi' : P' \\to Y$. In fact, we may assume that", "$P' = \\mathbf{A}^n_V$, see discussion preceding and following", "Definition \\ref{definition-lci}. Picture:", "$$", "\\xymatrix{", "X \\ar[d] & U \\ar[l] \\ar[r]_-{i'} & P' = \\mathbf{A}^n_V \\ar[d] \\\\", "Y \\ar[d] & & V \\ar[ll] \\ar[r]_i & P \\ar[d] \\\\", "S & & & S \\ar[lll]", "}", "$$", "Set $P'' = \\mathbf{A}^n_P$. Then $U \\to P' \\to P''$ is a", "Koszul-regular immersion as a composition", "of Koszul-regular immersions, namely $i'$ and the flat base change of", "$i$ via $P'' \\to P$, see", "Divisors,", "Lemma \\ref{divisors-lemma-regular-immersion-noetherian}", "and", "Divisors, Lemma \\ref{divisors-lemma-composition-regular-immersion}.", "Also $P'' \\to P \\to S$ is smooth as a composition of smooth morphisms,", "see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-smooth}.", "Hence we conclude that $X \\to S$ is Koszul at $x$ as desired." ], "refs": [ "more-morphisms-definition-lci", "divisors-lemma-regular-immersion-noetherian", "divisors-lemma-composition-regular-immersion", "morphisms-lemma-composition-smooth" ], "ref_ids": [ 14121, 7990, 7994, 5326 ] } ], "ref_ids": [] }, { "id": 14006, "type": "theorem", "label": "more-morphisms-lemma-flat-lci", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-lci", "contents": [ "\\begin{slogan}", "A morphism is flat and lci if and only if it is syntomic.", "\\end{slogan}", "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is flat and a local complete intersection morphism, and", "\\item $f$ is syntomic.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Working affine locally this is", "More on Algebra, Lemma \\ref{more-algebra-lemma-syntomic-lci}.", "We also give a more geometric proof.", "\\medskip\\noindent", "Assume (2). By", "Morphisms, Lemma \\ref{morphisms-lemma-syntomic-locally-standard-syntomic}", "for every point $x$ of $X$ there exist affine open neighbourhoods", "$U$ of $x$ and $V$ of $f(x)$ such that $f|_U : U \\to V$ is standard syntomic.", "This means that $U = \\Spec(R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c))", "\\to V = \\Spec(R)$ where $R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ is a", "relative global complete intersection over $R$. By", "Algebra,", "Lemma \\ref{algebra-lemma-relative-global-complete-intersection-conormal}", "the sequence $f_1, \\ldots, f_c$ is a regular sequence in each local", "ring $R[x_1, \\ldots, x_n]_{\\mathfrak q}$ for every prime", "$\\mathfrak q \\supset (f_1, \\ldots, f_c)$. Consider the Koszul complex", "$K_\\bullet = K_\\bullet(R[x_1, \\ldots, x_n], f_1, \\ldots, f_c)$", "with homology groups $H_i = H_i(K_\\bullet)$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-regular-koszul-regular}", "we see that $(H_i)_{\\mathfrak q} = 0$, $i > 0$ for every $\\mathfrak q$", "as above. On the other hand, by", "More on Algebra, Lemma \\ref{more-algebra-lemma-homotopy-koszul}", "we see that $H_i$ is annihilated by $(f_1, \\ldots, f_c)$. Hence we", "see that $H_i = 0$, $i > 0$ and $f_1, \\ldots, f_c$ is a Koszul-regular", "sequence. This proves that $U \\to V$ factors as a Koszul-regular", "immersion $U \\to \\mathbf{A}^n_V$ followed by a smooth morphism as desired.", "\\medskip\\noindent", "Assume (1). Then $f$ is a flat and locally of finite presentation", "(Lemma \\ref{lemma-lci-properties}).", "Hence, according to", "Morphisms, Lemma \\ref{morphisms-lemma-syntomic-locally-standard-syntomic}", "it suffices to show that the local rings $\\mathcal{O}_{X_s, x}$", "are local complete intersection rings. Choose, locally on $X$, a factorization", "$f = \\pi \\circ i$ for some Koszul-regular immersion $i : X \\to P$", "and smooth morphism $\\pi : P \\to S$. Note that $X \\to P$ is", "a relative quasi-regular immersion over $S$, see", "Divisors, Definition \\ref{divisors-definition-relative-H1-regular-immersion}.", "Hence according to", "Divisors,", "Lemma \\ref{divisors-lemma-relative-regular-immersion-flat-in-neighbourhood}", "we see that $X \\to P$ is a regular immersion and the same remains true", "after any base change. Thus each fibre is a regular immersion, whence", "all the local rings of all the fibres of $X$ are local complete intersections." ], "refs": [ "more-algebra-lemma-syntomic-lci", "morphisms-lemma-syntomic-locally-standard-syntomic", "algebra-lemma-relative-global-complete-intersection-conormal", "more-algebra-lemma-regular-koszul-regular", "more-algebra-lemma-homotopy-koszul", "more-morphisms-lemma-lci-properties", "morphisms-lemma-syntomic-locally-standard-syntomic", "divisors-definition-relative-H1-regular-immersion", "divisors-lemma-relative-regular-immersion-flat-in-neighbourhood" ], "ref_ids": [ 10002, 5297, 1183, 9973, 9960, 14002, 5297, 8100, 8001 ] } ], "ref_ids": [] }, { "id": 14007, "type": "theorem", "label": "more-morphisms-lemma-regular-immersion-lci", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-regular-immersion-lci", "contents": [ "A regular immersion of schemes is a local complete intersection morphism.", "A Koszul-regular immersion of schemes is a local complete intersection", "morphism." ], "refs": [], "proofs": [ { "contents": [ "Since a regular immersion is a Koszul-regular immersion, see", "Divisors, Lemma \\ref{divisors-lemma-regular-quasi-regular-immersion},", "it suffices to prove the second statement. The second statement", "follows immediately from the definition." ], "refs": [ "divisors-lemma-regular-quasi-regular-immersion" ], "ref_ids": [ 7989 ] } ], "ref_ids": [] }, { "id": 14008, "type": "theorem", "label": "more-morphisms-lemma-lci-permanence", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-lci-permanence", "contents": [ "Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes. Assume $Y \\to S$", "smooth and $X \\to S$ is a local complete intersection morphism.", "Then $f : X \\to Y$ is a local complete intersection morphism." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14009, "type": "theorem", "label": "more-morphisms-lemma-morphism-regular-schemes-is-lci", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-morphism-regular-schemes-is-lci", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. If $f$ is locally", "of finite type and $X$ and $Y$ are regular, then", "$f$ is a local complete intersection morphism." ], "refs": [], "proofs": [ { "contents": [ "We may assume there is a factorization $X \\to \\mathbf{A}^n_Y \\to Y$", "where the first arrow is an immersion.", "As $Y$ is regular also $\\mathbf{A}^n_Y$ is regular by", "Algebra, Lemma \\ref{algebra-lemma-regular-goes-up}.", "Hence $X \\to \\mathbf{A}^n_Y$ is a regular immersion by", "Divisors, Lemma \\ref{divisors-lemma-immersion-regular-regular-immersion}." ], "refs": [ "algebra-lemma-regular-goes-up", "divisors-lemma-immersion-regular-regular-immersion" ], "ref_ids": [ 1369, 7998 ] } ], "ref_ids": [] }, { "id": 14010, "type": "theorem", "label": "more-morphisms-lemma-lci-avramov", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-lci-avramov", "contents": [ "Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes. Assume", "\\begin{enumerate}", "\\item $S$ is locally Noetherian,", "\\item $Y \\to S$ is locally of finite type,", "\\item $f : X \\to Y$ is perfect,", "\\item $X \\to S$ is a local complete intersection morphism.", "\\end{enumerate}", "Then $X \\to Y$ is a local complete intersection morphism", "and $Y \\to S$ is Koszul at $f(x)$ for all $x \\in X$." ], "refs": [], "proofs": [ { "contents": [ "In the course of this proof all schemes will be locally Noetherian", "and all rings will be Noetherian. We will use without further mention", "that regular sequences and Koszul regular sequences agree in this", "setting, see More on Algebra, Lemma", "\\ref{more-algebra-lemma-noetherian-finite-all-equivalent}.", "Moreover, whether an ideal (resp.\\ ideal sheaf) is regular", "may be checked on local rings (resp.\\ stalks), see", "Algebra, Lemma \\ref{algebra-lemma-regular-sequence-in-neighbourhood}", "(resp.\\ Divisors, Lemma \\ref{divisors-lemma-Noetherian-scheme-regular-ideal})", "\\medskip\\noindent", "The question is local. Hence we may assume $S$, $X$, $Y$ are", "affine. In this situation we may choose a commutative diagram", "$$", "\\xymatrix{", "\\mathbf{A}^{n + m}_S \\ar[d] & X \\ar[l] \\ar[d] \\\\", "\\mathbf{A}^n_S \\ar[d] & Y \\ar[l] \\ar[ld] \\\\", "S", "}", "$$", "whose horizontal arrows are closed immersions. Let $x \\in X$ be a", "point and consider the corresponding commutative diagram of local", "rings", "$$", "\\xymatrix{", "J \\ar[r] &", "\\mathcal{O}_{\\mathbf{A}^{n + m}_S, x} \\ar[r] &", "\\mathcal{O}_{X, x} \\\\", "I \\ar[r] \\ar[u] &", "\\mathcal{O}_{\\mathbf{A}^n_S, f(x)} \\ar[r] \\ar[u] &", "\\mathcal{O}_{Y, f(x)} \\ar[u]", "}", "$$", "where $J$ and $I$ are the kernels of the horizontal arrows.", "Since $X \\to S$ is a local complete intersection morphism, the", "ideal $J$ is generated by a regular sequence. Since $X \\to Y$ is", "perfect the ring $\\mathcal{O}_{X, x}$ has finite tor dimension over", "$\\mathcal{O}_{Y, f(x)}$. Hence we may apply", "Divided Power Algebra, Lemma \\ref{dpa-lemma-perfect-map-ci}", "to conclude that $I$ and $J/I$ are generated by regular sequences.", "By our initial remarks, this finishes the proof." ], "refs": [ "more-algebra-lemma-noetherian-finite-all-equivalent", "algebra-lemma-regular-sequence-in-neighbourhood", "divisors-lemma-Noetherian-scheme-regular-ideal", "dpa-lemma-perfect-map-ci" ], "ref_ids": [ 9978, 741, 7988, 1675 ] } ], "ref_ids": [] }, { "id": 14011, "type": "theorem", "label": "more-morphisms-lemma-lci-to-regular", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-lci-to-regular", "contents": [ "Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes. Assume", "$S$ is locally Noetherian, $Y \\to S$ is locally of finite type,", "$Y$ is regular, and $X \\to S$ is a local complete intersection morphism.", "Then $f : X \\to Y$ is a local complete intersection morphism", "and $Y \\to S$ is Koszul at $f(x)$ for all $x \\in X$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-lci-avramov}", "in view of Lemma \\ref{lemma-regular-target-perfect}", "(and Morphisms, Lemma \\ref{morphisms-lemma-permanence-finite-type})." ], "refs": [ "more-morphisms-lemma-lci-avramov", "more-morphisms-lemma-regular-target-perfect", "morphisms-lemma-permanence-finite-type" ], "ref_ids": [ 14010, 13991, 5204 ] } ], "ref_ids": [] }, { "id": 14012, "type": "theorem", "label": "more-morphisms-lemma-perfect-conormal-free-lci", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-perfect-conormal-free-lci", "contents": [ "Let $i : X \\to Y$ be an immersion. If", "\\begin{enumerate}", "\\item $i$ is perfect,", "\\item $Y$ is locally Noetherian, and", "\\item the conormal sheaf $\\mathcal{C}_{Z/X}$ is finite locally free,", "\\end{enumerate}", "then $i$ is a regular immersion." ], "refs": [], "proofs": [ { "contents": [ "Translated into algebra, this is", "Divided Power Algebra, Proposition \\ref{dpa-proposition-regular-ideal}." ], "refs": [ "dpa-proposition-regular-ideal" ], "ref_ids": [ 1695 ] } ], "ref_ids": [] }, { "id": 14013, "type": "theorem", "label": "more-morphisms-lemma-lci-NL", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-lci-NL", "contents": [ "Let $f : X \\to Y$ be a local complete intersection homomorphism.", "Then the naive cotangent complex $\\NL_{X/Y}$ is a perfect object", "of $D(\\mathcal{O}_X)$ of tor-amplitude in $[-1, 0]$." ], "refs": [], "proofs": [ { "contents": [ "Translated into algebra this is", "More on Algebra, Lemma \\ref{more-algebra-lemma-lci-NL}.", "To do the translation use", "Lemmas \\ref{lemma-affine-lci} and", "\\ref{lemma-NL-affine} as well as", "Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-affine-compare-bounded},", "\\ref{perfect-lemma-tor-dimension-affine} and", "\\ref{perfect-lemma-perfect-affine}." ], "refs": [ "more-algebra-lemma-lci-NL", "more-morphisms-lemma-affine-lci", "more-morphisms-lemma-NL-affine", "perfect-lemma-affine-compare-bounded", "perfect-lemma-tor-dimension-affine", "perfect-lemma-perfect-affine" ], "ref_ids": [ 10311, 14003, 13745, 6941, 6977, 6980 ] } ], "ref_ids": [] }, { "id": 14014, "type": "theorem", "label": "more-morphisms-lemma-perfect-NL-lci", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-perfect-NL-lci", "contents": [ "Let $f : X \\to Y$ be a perfect morphism of locally Noetherian schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is a local complete intersection morphism,", "\\item $\\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and", "\\item $\\NL_{X/Y}$ is perfect with tor-amplitude in $[-1, 0]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Translated into algebra this is", "Divided Power Algebra, Lemma \\ref{dpa-lemma-perfect-NL-lci}.", "To do the translation use", "Lemmas \\ref{lemma-affine-lci} and", "\\ref{lemma-NL-affine} as well as", "Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-affine-compare-bounded},", "\\ref{perfect-lemma-tor-dimension-affine} and", "\\ref{perfect-lemma-perfect-affine}." ], "refs": [ "dpa-lemma-perfect-NL-lci", "more-morphisms-lemma-affine-lci", "more-morphisms-lemma-NL-affine", "perfect-lemma-affine-compare-bounded", "perfect-lemma-tor-dimension-affine", "perfect-lemma-perfect-affine" ], "ref_ids": [ 1691, 14003, 13745, 6941, 6977, 6980 ] } ], "ref_ids": [] }, { "id": 14015, "type": "theorem", "label": "more-morphisms-lemma-flat-fp-NL-lci", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-flat-fp-NL-lci", "contents": [ "Let $f : X \\to Y$ be a flat morphism of finite presentation.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is a local complete intersection morphism,", "\\item $f$ is syntomic,", "\\item $\\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and", "\\item $\\NL_{X/Y}$ is perfect with tor-amplitude in $[-1, 0]$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Translated into algebra this is", "Divided Power Algebra, Lemma \\ref{dpa-lemma-flat-fp-NL-lci}.", "To do the translation use", "Lemmas \\ref{lemma-affine-lci} and", "\\ref{lemma-NL-affine} as well as", "Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-affine-compare-bounded},", "\\ref{perfect-lemma-tor-dimension-affine} and", "\\ref{perfect-lemma-perfect-affine}." ], "refs": [ "dpa-lemma-flat-fp-NL-lci", "more-morphisms-lemma-affine-lci", "more-morphisms-lemma-NL-affine", "perfect-lemma-affine-compare-bounded", "perfect-lemma-tor-dimension-affine", "perfect-lemma-perfect-affine" ], "ref_ids": [ 1692, 14003, 13745, 6941, 6977, 6980 ] } ], "ref_ids": [] }, { "id": 14016, "type": "theorem", "label": "more-morphisms-lemma-smooth-diagonal-perfect", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-smooth-diagonal-perfect", "contents": [ "Let $f : X \\to Y$ be a finite type morphism of locally Noetherian schemes.", "Denote $\\Delta : X \\to X \\times_Y X$ the diagonal morphism.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is smooth,", "\\item $f$ is flat and $\\Delta : X \\to X \\times_Y X$ is a regular immersion,", "\\item $f$ is flat and $\\Delta : X \\to X \\times_Y X$ is a", "local complete intersection morphism,", "\\item $f$ is flat and $\\Delta : X \\to X \\times_Y X$ is perfect.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Then $f$ is flat by", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-flat}.", "The projections $X \\times_Y X \\to X$ are smooth by", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-smooth}. Hence the diagonal", "is a section to a smooth morphism and hence a regular immersion, see", "Divisors, Lemma \\ref{divisors-lemma-section-smooth-regular-immersion}.", "Hence (1) $\\Rightarrow$ (2).", "The implication (2) $\\Rightarrow$ (3) is", "Lemma \\ref{lemma-regular-immersion-lci}.", "The implication (3) $\\Rightarrow$ (4) is", "Lemma \\ref{lemma-lci-properties}.", "The interesting implication (4) $\\Rightarrow$ (1) follows immediately", "from Divided Power Algebra, Lemma \\ref{dpa-lemma-perfect-diagonal}." ], "refs": [ "morphisms-lemma-smooth-flat", "morphisms-lemma-base-change-smooth", "more-morphisms-lemma-regular-immersion-lci", "more-morphisms-lemma-lci-properties", "dpa-lemma-perfect-diagonal" ], "ref_ids": [ 5331, 5327, 14007, 14002, 1688 ] } ], "ref_ids": [] }, { "id": 14017, "type": "theorem", "label": "more-morphisms-lemma-descending-property-lci", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-descending-property-lci", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is a local complete intersection", "morphism'' is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\{S_i \\to S\\}$ be an fpqc covering of $S$.", "Assume that each base change $f_i : X_i \\to S_i$ of $f$ is", "a local complete intersection morphism.", "Note that this implies in particular that $f$ is locally of finite", "type, see", "Lemma \\ref{lemma-lci-properties}", "and", "Descent, Lemma \\ref{descent-lemma-descending-property-locally-finite-type}.", "Let $x \\in X$. Choose an open neighbourhood $U$ of $x$ and", "an immersion $j : U \\to \\mathbf{A}^n_S$ over $S$ (see", "discussion preceding", "Definition \\ref{definition-lci}).", "We have to show that $j$ is a Koszul-regular immersion.", "Since $f_i$ is a local complete intersection morphism, we see", "that the base change $j_i : U \\times_S S_i \\to \\mathbf{A}^n_{S_i}$", "is a Koszul-regular immersion, see", "Lemma \\ref{lemma-lci}.", "Because $\\{\\mathbf{A}^n_{S_i} \\to \\mathbf{A}^n_S\\}$ is a", "fpqc covering we see from", "Descent, Lemma \\ref{descent-lemma-descending-property-regular-immersion}", "that $j$ is a Koszul-regular immersion as desired." ], "refs": [ "more-morphisms-lemma-lci-properties", "descent-lemma-descending-property-locally-finite-type", "more-morphisms-definition-lci", "more-morphisms-lemma-lci", "descent-lemma-descending-property-regular-immersion" ], "ref_ids": [ 14002, 14675, 14121, 14001, 14697 ] } ], "ref_ids": [] }, { "id": 14018, "type": "theorem", "label": "more-morphisms-lemma-lci-syntomic-local-source", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-lci-syntomic-local-source", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is a local complete intersection", "morphism'' is syntomic local on the source." ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion of", "Descent, Lemma \\ref{descent-lemma-properties-morphisms-local-source}", "to prove this. It follows from", "Lemmas \\ref{lemma-flat-lci} and", "\\ref{lemma-composition-lci}", "that being a local complete intersection morphism is preserved under", "precomposing with syntomic morphisms. It is clear from", "Definition \\ref{definition-lci}", "that being a local complete intersection morphism is Zariski local on the", "source and target. Hence, according to the aforementioned", "Descent, Lemma \\ref{descent-lemma-properties-morphisms-local-source}", "it suffices to prove the following: Suppose $X' \\to X \\to Y$ are", "morphisms of affine schemes with $X' \\to X$ syntomic and $X' \\to Y$", "a local complete intersection morphism. Then $X \\to Y$ is a local complete", "intersection morphism. To see this, note that in any case $X \\to Y$ is of", "finite presentation by", "Descent, Lemma \\ref{descent-lemma-flat-finitely-presented-permanence-algebra}.", "Choose a closed immersion $X \\to \\mathbf{A}^n_Y$. By", "Algebra, Lemma \\ref{algebra-lemma-lift-syntomic}", "we can find an affine open covering $X' = \\bigcup_{i = 1, \\ldots, n} X'_i$", "and syntomic morphisms $W_i \\to \\mathbf{A}^n_Y$ lifting the morphisms", "$X'_i \\to X$, i.e., such that there are fibre product diagrams", "$$", "\\xymatrix{", "X'_i \\ar[d] \\ar[r] & W_i \\ar[d] \\\\", "X \\ar[r] & \\mathbf{A}^n_Y", "}", "$$", "After replacing $X'$ by $\\coprod X'_i$ and setting $W = \\coprod W_i$", "we obtain a fibre product diagram of affine schemes", "$$", "\\xymatrix{", "X' \\ar[d] \\ar[r] & W \\ar[d]^h \\\\", "X \\ar[r] & \\mathbf{A}^n_Y", "}", "$$", "with $h : W \\to \\mathbf{A}^n_Y$ syntomic and $X' \\to Y$ still a local complete", "intersection morphism. Since $W \\to \\mathbf{A}^n_Y$ is open (see", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open})", "and $X' \\to X$ is surjective we see that $X$ is contained in the image", "of $W \\to \\mathbf{A}^n_Y$. Choose a closed immersion", "$W \\to \\mathbf{A}^{n + m}_Y$ over $\\mathbf{A}^n_Y$. Now the diagram looks like", "$$", "\\xymatrix{", "X' \\ar[d] \\ar[r] & W \\ar[d]^h \\ar[r] & \\mathbf{A}^{n + m}_Y \\ar[ld] \\\\", "X \\ar[r] & \\mathbf{A}^n_Y", "}", "$$", "Because $h$ is syntomic and hence a local complete intersection morphism (see", "above) the morphism $W \\to \\mathbf{A}^{n + m}_Y$ is a Koszul-regular immersion.", "Because $X' \\to Y$ is a local complete intersection morphism the morphism", "$X' \\to \\mathbf{A}^{n + m}_Y$ is a Koszul-regular immersion. We conclude from", "Divisors, Lemma \\ref{divisors-lemma-permanence-regular-immersion}", "that $X' \\to W$ is a Koszul-regular immersion. Hence, since being", "a Koszul-regular immersion is fpqc local on the target (see", "Descent, Lemma \\ref{descent-lemma-descending-property-regular-immersion})", "we conclude that $X \\to \\mathbf{A}^n_Y$ is a Koszul-regular immersion", "which is what we had to show." ], "refs": [ "descent-lemma-properties-morphisms-local-source", "more-morphisms-lemma-flat-lci", "more-morphisms-lemma-composition-lci", "more-morphisms-definition-lci", "descent-lemma-properties-morphisms-local-source", "descent-lemma-flat-finitely-presented-permanence-algebra", "algebra-lemma-lift-syntomic", "morphisms-lemma-fppf-open", "divisors-lemma-permanence-regular-immersion", "descent-lemma-descending-property-regular-immersion" ], "ref_ids": [ 14707, 14006, 14005, 14121, 14707, 14640, 1188, 5267, 7995, 14697 ] } ], "ref_ids": [] }, { "id": 14019, "type": "theorem", "label": "more-morphisms-lemma-base-change-lci-fibres", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-lci-fibres", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$.", "Assume both $X$ and $Y$ are flat and locally of finite presentation over $S$.", "Then the set", "$$", "\\{x \\in X \\mid f\\text{ Koszul at }x\\}.", "$$", "is open in $X$ and its formation commutes with arbitrary base change", "$S' \\to S$." ], "refs": [], "proofs": [ { "contents": [ "The set is open by definition (see", "Definition \\ref{definition-lci}).", "Let $S' \\to S$ be a morphism of schemes. Set $X' = S' \\times_S X$,", "$Y' = S' \\times_S Y$, and denote $f' : X' \\to Y'$ the base change of $f$.", "Let $x' \\in X'$ be a point such that $f'$ is Koszul at $x'$. Denote", "$s' \\in S'$, $x \\in X$, $y' \\in Y'$ , $y \\in Y$, $s \\in S$ the image", "of $x'$. Note that $f$ is locally of finite presentation, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-permanence}.", "Hence we may choose an affine neighbourhood $U \\subset X$ of", "$x$ and an immersion $i : U \\to \\mathbf{A}^n_Y$. Denote $U' = S' \\times_S U$", "and $i' : U' \\to \\mathbf{A}^n_{Y'}$ the base change of $i$.", "The assumption that $f'$ is Koszul at $x'$ implies that $i'$ is a", "Koszul-regular immersion in a neighbourhood of $x'$, see", "Lemma \\ref{lemma-lci}.", "The scheme $X'$ is flat and locally of finite", "presentation over $S'$ as a base change of $X$ (see", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-flat} and", "\\ref{morphisms-lemma-base-change-finite-presentation}).", "Hence $i'$ is a relative $H_1$-regular immersion over $S'$", "in a neighbourhood of $x'$ (see", "Divisors, Definition \\ref{divisors-definition-relative-H1-regular-immersion}).", "Thus the base change $i'_{s'} : U'_{s'} \\to \\mathbf{A}^n_{Y'_{s'}}$ is", "a $H_1$-regular immersion in an open neighbourhood of $x'$, see", "Divisors, Lemma \\ref{divisors-lemma-relative-regular-immersion}", "and the discussion following", "Divisors, Definition \\ref{divisors-definition-relative-H1-regular-immersion}.", "Since $s' = \\Spec(\\kappa(s')) \\to \\Spec(\\kappa(s)) = s$", "is a surjective flat universally open morphism (see", "Morphisms, Lemma \\ref{morphisms-lemma-scheme-over-field-universally-open})", "we conclude that the base change $i_s : U_s \\to \\mathbf{A}^n_{Y_s}$ is an", "$H_1$-regular immersion in a neighbourhood of $x$, see", "Descent, Lemma \\ref{descent-lemma-descending-property-regular-immersion}.", "Finally, note that $\\mathbf{A}^n_Y$ is flat and locally of finite", "presentation over $S$, hence", "Divisors, Lemma \\ref{divisors-lemma-fibre-quasi-regular}", "implies that $i$ is a (Koszul-)regular immersion in a neighbourhood of $x$", "as desired." ], "refs": [ "more-morphisms-definition-lci", "morphisms-lemma-finite-presentation-permanence", "more-morphisms-lemma-lci", "morphisms-lemma-base-change-flat", "morphisms-lemma-base-change-finite-presentation", "divisors-definition-relative-H1-regular-immersion", "divisors-lemma-relative-regular-immersion", "divisors-definition-relative-H1-regular-immersion", "morphisms-lemma-scheme-over-field-universally-open", "descent-lemma-descending-property-regular-immersion", "divisors-lemma-fibre-quasi-regular" ], "ref_ids": [ 14121, 5247, 14001, 5265, 5240, 8100, 7999, 8100, 5254, 14697, 8003 ] } ], "ref_ids": [] }, { "id": 14020, "type": "theorem", "label": "more-morphisms-lemma-unramified-lci", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-unramified-lci", "contents": [ "Let $f : X \\to Y$ be a local complete intersection morphism of schemes.", "Then $f$ is unramified if and only if $f$ is formally unramified and in", "this case the conormal sheaf $\\mathcal{C}_{X/Y}$ is finite locally free", "on $X$." ], "refs": [], "proofs": [ { "contents": [ "The first assertion follows immediately from", "Lemma \\ref{lemma-unramified-formally-unramified}", "and the fact that a local complete intersection morphism is locally", "of finite type. To compute the conormal sheaf of $f$ we choose, locally", "on $X$, a factorization of $f$ as $f = p \\circ i$ where $i : X \\to V$", "is a Koszul-regular immersion and $V \\to Y$ is smooth. By", "Lemma \\ref{lemma-two-unramified-morphisms-formally-smooth}", "we see that $\\mathcal{C}_{X/Y}$ is a locally direct summand of", "$\\mathcal{C}_{X/V}$ which is finite locally free as $i$ is a Koszul-regular", "(hence quasi-regular) immersion, see", "Divisors, Lemma \\ref{divisors-lemma-quasi-regular-immersion}." ], "refs": [ "more-morphisms-lemma-unramified-formally-unramified", "more-morphisms-lemma-two-unramified-morphisms-formally-smooth", "divisors-lemma-quasi-regular-immersion" ], "ref_ids": [ 13696, 13740, 7992 ] } ], "ref_ids": [] }, { "id": 14021, "type": "theorem", "label": "more-morphisms-lemma-transitivity-conormal-lci", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-transitivity-conormal-lci", "contents": [ "Let $Z \\to Y \\to X$ be formally unramified morphisms of schemes.", "Assume that $Z \\to Y$ is a local complete intersection morphism.", "The exact sequence", "$$", "0 \\to i^*\\mathcal{C}_{Y/X} \\to", "\\mathcal{C}_{Z/X} \\to", "\\mathcal{C}_{Z/Y} \\to 0", "$$", "of", "Lemma \\ref{lemma-transitivity-conormal}", "is short exact." ], "refs": [ "more-morphisms-lemma-transitivity-conormal" ], "proofs": [ { "contents": [ "The question is local on $Z$ hence we may assume there exists a factorization", "$Z \\to \\mathbf{A}^n_Y \\to Y$ of the morphism $Z \\to Y$. Then we get a", "commutative diagram", "$$", "\\xymatrix{", "Z \\ar[r]_{i'} \\ar@{=}[d] &", "\\mathbf{A}^n_Y \\ar[r] \\ar[d] &", "\\mathbf{A}^n_X \\ar[d] \\\\", "Z \\ar[r]^i & Y \\ar[r] & X", "}", "$$", "As $Z \\to Y$ is a local complete intersection morphism, we see that", "$Z \\to \\mathbf{A}^n_Y$ is a Koszul-regular immersion. Hence by", "Divisors, Lemma \\ref{divisors-lemma-transitivity-conormal-quasi-regular}", "the sequence", "$$", "0 \\to (i')^*\\mathcal{C}_{\\mathbf{A}^n_Y/\\mathbf{A}^n_X} \\to", "\\mathcal{C}_{Z/\\mathbf{A}^n_X} \\to", "\\mathcal{C}_{Z/\\mathbf{A}^n_Y} \\to 0", "$$", "is exact and locally split. Note that", "$i^*\\mathcal{C}_{Y/X} = (i')^*\\mathcal{C}_{\\mathbf{A}^n_Y/\\mathbf{A}^n_X}$", "by", "Lemma \\ref{lemma-universal-thickening-fibre-product-flat}", "and note that the diagram", "$$", "\\xymatrix{", "(i')^*\\mathcal{C}_{\\mathbf{A}^n_Y/\\mathbf{A}^n_X} \\ar[r] &", "\\mathcal{C}_{Z/\\mathbf{A}^n_X} \\\\", "i^*\\mathcal{C}_{Y/X} \\ar[u]^{\\cong} \\ar[r] & \\mathcal{C}_{Z/X} \\ar[u]", "}", "$$", "is commutative. Hence the lower horizontal arrow is a locally split", "injection. This proves the lemma." ], "refs": [ "divisors-lemma-transitivity-conormal-quasi-regular", "more-morphisms-lemma-universal-thickening-fibre-product-flat" ], "ref_ids": [ 7993, 13702 ] } ], "ref_ids": [ 13707 ] }, { "id": 14022, "type": "theorem", "label": "more-morphisms-lemma-check-weakly-etale-stalks", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-check-weakly-etale-stalks", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent", "\\begin{enumerate}", "\\item $X \\to Y$ is weakly \\'etale, and", "\\item for every $x \\in X$ the ring map", "$\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is weakly \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Observe that under both assumptions (1) and (2) the morphism $f$ is flat.", "Thus we may assume $f$ is flat. Let $x \\in X$ with image $y = f(x)$ in $Y$.", "There are canonical maps of rings", "$$", "\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{Y, y}} \\mathcal{O}_{X, x}", "\\longrightarrow", "\\mathcal{O}_{X \\times_Y X, \\Delta_{X/Y}(x)}", "\\longrightarrow", "\\mathcal{O}_{X, x}", "$$", "where the first map is a localization (hence flat) and the second map is a", "surjection (hence an epimorphism of rings).", "Condition (1) means that for all $x$ the second arrow is flat.", "Condition (2) is that for all $x$ the composition is flat.", "These conditions are equivalent by", "Algebra, Lemma \\ref{algebra-lemma-composition-flat} and", "More on Algebra, Lemma \\ref{more-algebra-lemma-key}." ], "refs": [ "algebra-lemma-composition-flat", "more-algebra-lemma-key" ], "ref_ids": [ 524, 10439 ] } ], "ref_ids": [] }, { "id": 14023, "type": "theorem", "label": "more-morphisms-lemma-key", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-key", "contents": [ "Let $X \\to Y$ be a morphism of schemes such that", "$X \\to X \\times_Y X$ is flat. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "If $\\mathcal{F}$ is flat over $Y$, then $\\mathcal{F}$ is flat over $X$." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$ with image $y = f(x)$ in $Y$.", "Since $X \\to X \\times_Y X$ is flat, we see that", "$\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{Y, y}} \\mathcal{O}_{X, x} \\to", "\\mathcal{O}_{X, x}$ is flat. Hence the result follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-key}", "and the definitions." ], "refs": [ "more-algebra-lemma-key" ], "ref_ids": [ 10439 ] } ], "ref_ids": [] }, { "id": 14024, "type": "theorem", "label": "more-morphisms-lemma-weakly-etale-characterize", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-weakly-etale-characterize", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent", "\\begin{enumerate}", "\\item The morphism $f$ is weakly \\'etale.", "\\item For every affine opens $U \\subset X$, $V \\subset S$", "with $f(U) \\subset V$ the ring map", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is weakly \\'etale.", "\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$", "and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$", "is weakly \\'etale.", "\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$", "and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such", "that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is", "of weakly \\'etale, for all $j\\in J, i\\in I_j$.", "\\end{enumerate}", "Moreover, if $f$ is weakly \\'etale then for", "any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$", "the restriction $f|_U : U \\to V$ is weakly-\\'etale." ], "refs": [], "proofs": [ { "contents": [ "Suppose given open subschemes $U \\subset X$, $V \\subset S$ with", "$f(U) \\subset V$. Then $U \\times_V U \\subset X \\times_Y X$ is open", "(Schemes, Lemma \\ref{schemes-lemma-open-fibre-product})", "and the diagonal $\\Delta_{U/V}$ of $f|_U : U \\to V$ is", "the restriction $\\Delta_{X/Y}|_U : U \\to U \\times_V U$.", "Since flatness is a local property of morphisms of schemes", "(Morphisms, Lemma \\ref{morphisms-lemma-flat-characterize})", "the final statement of the lemma is follows", "as well as the equivalence of (1) and (3).", "If $X$ and $Y$ are affine, then $X \\to Y$ is weakly \\'etale", "if and only if $\\mathcal{O}_Y(Y) \\to \\mathcal{O}_X(X)$ is", "weakly \\'etale (use again", "Morphisms, Lemma \\ref{morphisms-lemma-flat-characterize}).", "Thus (1) and (3) are also equivalent to (2) and (4)." ], "refs": [ "schemes-lemma-open-fibre-product", "morphisms-lemma-flat-characterize", "morphisms-lemma-flat-characterize" ], "ref_ids": [ 7691, 5260, 5260 ] } ], "ref_ids": [] }, { "id": 14025, "type": "theorem", "label": "more-morphisms-lemma-composition-weakly-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-composition-weakly-etale", "contents": [ "Let $X \\to Y \\to Z$ be morphisms of schemes.", "\\begin{enumerate}", "\\item If $X \\to X \\times_Y X$ and $Y \\to Y \\times_Z Y$ are flat,", "then $X \\to X \\times_Z X$ is flat.", "\\item If $X \\to Y$ and $Y \\to Z$ are weakly \\'etale, then", "$X \\to Z$ is weakly \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from the factorization", "$$", "X \\to X \\times_Y X \\to X \\times_Z X", "$$", "of the diagonal of $X$ over $Z$, the fact that", "$$", "X \\times_Y X = (X \\times_Z X) \\times_{(Y \\times_Z Y)} Y,", "$$", "the fact that a base change of a flat morphism is flat, and", "the fact that the composition of flat morphisms is flat", "(Morphisms, Lemmas \\ref{morphisms-lemma-base-change-flat} and", "\\ref{morphisms-lemma-composition-flat}).", "Part (2) follows from part (1) and the fact (just used)", "that the composition of flat morphisms is flat." ], "refs": [ "morphisms-lemma-base-change-flat", "morphisms-lemma-composition-flat" ], "ref_ids": [ 5265, 5263 ] } ], "ref_ids": [] }, { "id": 14026, "type": "theorem", "label": "more-morphisms-lemma-base-change-weakly-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-weakly-etale", "contents": [ "Let $X \\to Y$ and $Y' \\to Y$ be morphisms of schemes and let", "$X' = Y' \\times_Y X$ be the base change of $X$.", "\\begin{enumerate}", "\\item If $X \\to X \\times_Y X$ is flat, then $X' \\to X' \\times_{Y'} X'$", "is flat.", "\\item If $X \\to Y$ is weakly \\'etale, then $X' \\to Y'$ is weakly \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume $X \\to X \\times_Y X$ is flat. The morphism $X' \\to X' \\times_{Y'} X'$", "is the base change of $X \\to X \\times_Y X$ by $Y' \\to Y$. Hence it", "is flat by Morphisms, Lemmas \\ref{morphisms-lemma-base-change-flat}.", "This proves (1). Part (2) follows from (1) and the fact (just used)", "that the base change of a flat morphism is flat." ], "refs": [ "morphisms-lemma-base-change-flat" ], "ref_ids": [ 5265 ] } ], "ref_ids": [] }, { "id": 14027, "type": "theorem", "label": "more-morphisms-lemma-go-down", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-go-down", "contents": [ "Let $X \\to Y \\to Z$ be morphisms of schemes. Assume that $X \\to Y$ is", "flat and surjective and that $X \\to X \\times_Z X$ is flat.", "Then $Y \\to Y \\times_Z Y$ is flat." ], "refs": [], "proofs": [ { "contents": [ "Consider the commutative diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d] & X \\times_Z X \\ar[d] \\\\", "Y \\ar[r] & Y \\times_Z Y", "}", "$$", "The top horizontal arrow is flat and the vertical arrows are flat.", "Hence $X$ is flat over $Y \\times_Z Y$. By", "Morphisms, Lemma \\ref{morphisms-lemma-flat-permanence}", "we see that $Y$ is flat over $Y \\times_Z Y$." ], "refs": [ "morphisms-lemma-flat-permanence" ], "ref_ids": [ 5270 ] } ], "ref_ids": [] }, { "id": 14028, "type": "theorem", "label": "more-morphisms-lemma-weakly-etale-formally-unramified", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-weakly-etale-formally-unramified", "contents": [ "Let $f : X \\to Y$ be a weakly \\'etale morphism of schemes.", "Then $f$ is formally unramified, i.e., $\\Omega_{X/Y} = 0$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $f$ is formally unramified if and only if $\\Omega_{X/Y} = 0$ by", "Lemma \\ref{lemma-formally-unramified-differentials}.", "Via Lemma \\ref{lemma-weakly-etale-characterize} and", "Morphisms, Lemma \\ref{morphisms-lemma-differentials-affine}", "this follows from the case of rings which is", "More on Algebra, Lemma \\ref{more-algebra-lemma-formally-unramified}." ], "refs": [ "more-morphisms-lemma-formally-unramified-differentials", "more-morphisms-lemma-weakly-etale-characterize", "morphisms-lemma-differentials-affine", "more-algebra-lemma-formally-unramified" ], "ref_ids": [ 13695, 14024, 5310, 10448 ] } ], "ref_ids": [] }, { "id": 14029, "type": "theorem", "label": "more-morphisms-lemma-when-weakly-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-when-weakly-etale", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Then $X \\to Y$ is weakly \\'etale", "in each of the following cases", "\\begin{enumerate}", "\\item $X \\to Y$ is a flat monomorphism,", "\\item $X \\to Y$ is an open immersion,", "\\item $X \\to Y$ is flat and unramified,", "\\item $X \\to Y$ is \\'etale.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (1) holds, then $\\Delta_{X/Y}$ is an isomorphism", "(Schemes, Lemma \\ref{schemes-lemma-monomorphism}), hence certainly", "$f$ is weakly \\'etale. Case (2) is a special case of (1).", "The diagonal of an unramified morphism is an open immersion", "(Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism}),", "hence flat. Thus a flat unramified morphism is weakly \\'etale.", "An \\'etale morphism is flat and unramified", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-smooth-unramified}),", "hence (4) follows from (3)." ], "refs": [ "schemes-lemma-monomorphism", "morphisms-lemma-diagonal-unramified-morphism", "morphisms-lemma-etale-smooth-unramified" ], "ref_ids": [ 7721, 5354, 5362 ] } ], "ref_ids": [] }, { "id": 14030, "type": "theorem", "label": "more-morphisms-lemma-reduced-goes-up", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-reduced-goes-up", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "If $Y$ is reduced and $f$ weakly \\'etale, then $X$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "Via Lemma \\ref{lemma-weakly-etale-characterize}", "this follows from the case of rings which is", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-absolutely-flat-over-absolutely-flat}." ], "refs": [ "more-morphisms-lemma-weakly-etale-characterize", "more-algebra-lemma-absolutely-flat-over-absolutely-flat" ], "ref_ids": [ 14024, 10444 ] } ], "ref_ids": [] }, { "id": 14031, "type": "theorem", "label": "more-morphisms-lemma-weakly-etale-strictly-henselian-local-rings", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-weakly-etale-strictly-henselian-local-rings", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is weakly \\'etale, and", "\\item for $x \\in X$ the local ring map", "$\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ induces an isomorphism", "on strict henselizations.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$ be a point with image $y = f(x)$ in $Y$.", "Choose a separable algebraic closure $\\kappa^{sep}$ of $\\kappa(x)$.", "Let $\\mathcal{O}_{X, x}^{sh}$ be the strict henselization", "corresponding to $\\kappa^{sep}$ and $\\mathcal{O}_{Y, y}^{sh}$", "the strict henselization relative to the separable algebraic", "closure of $\\kappa(y)$ in $\\kappa^{sep}$.", "Consider the commutative diagram", "$$", "\\xymatrix{", "\\mathcal{O}_{X, x} \\ar[r] & \\mathcal{O}_{X, x}^{sh} \\\\", "\\mathcal{O}_{Y, y} \\ar[u] \\ar[r] & \\mathcal{O}_{Y, y}^{sh} \\ar[u]", "}", "$$", "local homomorphisms of local rings, see", "Algebra, Lemma \\ref{algebra-lemma-strictly-henselian-functorial}.", "Since the strict henselization is a filtered colimit of \\'etale", "ring maps, More on Algebra, Lemma \\ref{more-algebra-lemma-when-weakly-etale}", "shows the horizontal maps are weakly \\'etale.", "Moreover, the horizontal maps are faithfully flat by", "More on Algebra, Lemma \\ref{more-algebra-lemma-dumb-properties-henselization}.", "\\medskip\\noindent", "Assume $f$ weakly \\'etale. By Lemma \\ref{lemma-check-weakly-etale-stalks}", "the left vertical arrow is weakly \\'etale. By", "More on Algebra, Lemmas \\ref{more-algebra-lemma-composition-weakly-etale} and", "\\ref{more-algebra-lemma-weakly-etale-permanence}", "the right vertical arrow is weakly \\'etale. By", "More on Algebra, Theorem \\ref{more-algebra-theorem-olivier}", "we conclude the right vertical map is an isomorphism.", "\\medskip\\noindent", "Assume $\\mathcal{O}_{Y, y}^{sh} \\to \\mathcal{O}_{X, x}^{sh}$ is an isomorphism.", "Then $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}^{sh}$ is weakly \\'etale.", "Since $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X, x}^{sh}$ is faithfully", "flat we conclude that $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$", "is weakly \\'etale by", "More on Algebra, Lemma \\ref{more-algebra-lemma-go-down}.", "Thus (2) implies (1) by Lemma \\ref{lemma-check-weakly-etale-stalks}." ], "refs": [ "algebra-lemma-strictly-henselian-functorial", "more-algebra-lemma-when-weakly-etale", "more-algebra-lemma-dumb-properties-henselization", "more-morphisms-lemma-check-weakly-etale-stalks", "more-algebra-lemma-composition-weakly-etale", "more-algebra-lemma-weakly-etale-permanence", "more-algebra-theorem-olivier", "more-algebra-lemma-go-down", "more-morphisms-lemma-check-weakly-etale-stalks" ], "ref_ids": [ 1303, 10450, 10055, 14022, 10445, 10447, 9805, 10446, 14022 ] } ], "ref_ids": [] }, { "id": 14032, "type": "theorem", "label": "more-morphisms-lemma-normal-goes-up", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-normal-goes-up", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. If $Y$ is a normal scheme", "and $f$ weakly \\'etale, then $X$ is a normal scheme." ], "refs": [], "proofs": [ { "contents": [ "By More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-normal}", "a scheme $S$ is normal if and only if for all $s \\in S$", "the strict henselization of $\\mathcal{O}_{S, s}$ is a normal domain.", "Hence the lemma follows from", "Lemma \\ref{lemma-weakly-etale-strictly-henselian-local-rings}." ], "refs": [ "more-algebra-lemma-henselization-normal", "more-morphisms-lemma-weakly-etale-strictly-henselian-local-rings" ], "ref_ids": [ 10060, 14031 ] } ], "ref_ids": [] }, { "id": 14033, "type": "theorem", "label": "more-morphisms-lemma-weakly-etale-permanence", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-weakly-etale-permanence", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$.", "If $X$, $Y$ are weakly \\'etale over $S$, then $f$ is weakly \\'etale." ], "refs": [], "proofs": [ { "contents": [ "We will use Morphisms, Lemmas \\ref{morphisms-lemma-base-change-flat} and", "\\ref{morphisms-lemma-composition-flat} without further mention.", "Write $X \\to Y$ as the composition $X \\to X \\times_S Y \\to Y$.", "The second morphism is flat as the base change of the flat morphism", "$X \\to S$. The first is the base change of the flat morphism", "$Y \\to Y \\times_S Y$ by the morphism $X \\times_S Y \\to Y \\times_S Y$,", "hence flat. Thus $X \\to Y$ is flat. The morphism", "$X \\times_Y X \\to X \\times_S X$ is an immersion.", "Thus Lemma \\ref{lemma-key} implies, that since", "$X$ is flat over $X \\times_S X$ it follows that $X$ is", "flat over $X \\times_Y X$." ], "refs": [ "morphisms-lemma-base-change-flat", "morphisms-lemma-composition-flat", "more-morphisms-lemma-key" ], "ref_ids": [ 5265, 5263, 14023 ] } ], "ref_ids": [] }, { "id": 14034, "type": "theorem", "label": "more-morphisms-lemma-weakly-etale-universal-homeomorphism", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-weakly-etale-universal-homeomorphism", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. If $f$ is weakly \\'etale and", "a universal homeomorphism, it is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is a universal homeomorphism, the diagonal", "$\\Delta : X \\to X \\times_Y X$ is a surjective closed immersion by", "Morphisms, Lemmas \\ref{morphisms-lemma-homeomorphism-affine} and", "\\ref{morphisms-lemma-universally-injective}. Since $\\Delta$ is also", "flat, we see that $\\Delta$ must be an isomorphism by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-flat-closed-immersions}.", "In other words, $f$ is a monomorphism", "(Schemes, Lemma \\ref{schemes-lemma-monomorphism}).", "Since $f$ is a universal homeomorphism it is certainly", "quasi-compact. Hence by Descent, Lemma", "\\ref{descent-lemma-flat-surjective-quasi-compact-monomorphism-isomorphism}", "we find that $f$ is an isomorphism." ], "refs": [ "morphisms-lemma-homeomorphism-affine", "morphisms-lemma-universally-injective", "morphisms-lemma-characterize-flat-closed-immersions", "schemes-lemma-monomorphism", "descent-lemma-flat-surjective-quasi-compact-monomorphism-isomorphism" ], "ref_ids": [ 5453, 5167, 5274, 7721, 14699 ] } ], "ref_ids": [] }, { "id": 14035, "type": "theorem", "label": "more-morphisms-lemma-relative-frobenius-weakly-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-relative-frobenius-weakly-etale", "contents": [ "Let $U \\to X$ be a weakly \\'etale morphism of schemes where $X$ is a scheme", "in characteristic $p$. Then the relative Frobenius", "$F_{U/X} : U \\to U \\times_{X, F_X} X$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "The morphism $F_{U/X}$ is a universal homeomorphism by", "Varieties, Lemma \\ref{varieties-lemma-relative-frobenius}.", "The morphism $F_{U/X}$ is weakly \\'etale as a morphism", "between schemes weakly \\'etale over $X$ by", "Lemma \\ref{lemma-weakly-etale-permanence}. Hence $F_{U/X}$", "is an isomorphism by Lemma \\ref{lemma-weakly-etale-universal-homeomorphism}." ], "refs": [ "varieties-lemma-relative-frobenius", "more-morphisms-lemma-weakly-etale-permanence", "more-morphisms-lemma-weakly-etale-universal-homeomorphism" ], "ref_ids": [ 11050, 14033, 14034 ] } ], "ref_ids": [] }, { "id": 14036, "type": "theorem", "label": "more-morphisms-lemma-normalized-base-change-with-reduced-fibre-over-curve", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-normalized-base-change-with-reduced-fibre-over-curve", "contents": [ "Let $f : X \\to S$ be a flat, finite type morphism of schemes.", "Assume $S$ is Nagata, integral with function field $K$, and", "regular of dimension $1$. Then there exists a finite extension $L/K$", "such that in the diagram", "$$", "\\xymatrix{", "Y \\ar[rd]_g \\ar[r]_-\\nu & X \\times_S T \\ar[d] \\ar[r] & X \\ar[d]_f \\\\", "& T \\ar[r] & S", "}", "$$", "the morphism $g$ is smooth at all generic points of fibres. Here", "$T$ is the normalization of $S$ in $\\Spec(L)$ and $\\nu : Y \\to X \\times_S T$", "is the normalization." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite affine open covering $S = \\bigcup \\Spec(A_i)$.", "Then $K$ is equal to the fraction field of $A_i$ for all $i$.", "Let $X_i = X \\times_S \\Spec(A_i)$.", "Choose $L_i/K$ as in", "Theorem \\ref{theorem-normalized-base-change-with-reduced-fibre}", "for the morphism $X_i \\to \\Spec(A_i)$.", "Let $B_i \\subset L_i$ be the integral closure of $A_i$ and", "let $Y_i$ be the normalized base change of $X$ to $B_i$.", "Let $L/K$ be a finite extension dominating each $L_i$.", "Let $T_i \\subset T$ be the inverse image of $\\Spec(A_i)$.", "For each $i$ we get a commutative diagram", "$$", "\\xymatrix{", "g^{-1}(T_i) \\ar[r] \\ar[d] & Y_i \\ar[r] \\ar[d] & X \\times_S \\Spec(A_i) \\ar[d] \\\\", "T_i \\ar[r] & \\Spec(B_i) \\ar[r] & \\Spec(A_i)", "}", "$$", "and in fact the left hand square is a normalized base change", "as discussed at the beginning of the section. In the proof", "of Theorem \\ref{theorem-normalized-base-change-with-reduced-fibre}", "we have seen that the smooth locus of $Y \\to T$ contains the", "inverse image in $g^{-1}(T_i)$ of the set of points", "where $Y_i$ is smooth over $B_i$. This proves the lemma." ], "refs": [ "more-morphisms-theorem-normalized-base-change-with-reduced-fibre", "more-morphisms-theorem-normalized-base-change-with-reduced-fibre" ], "ref_ids": [ 13676, 13676 ] } ], "ref_ids": [] }, { "id": 14037, "type": "theorem", "label": "more-morphisms-lemma-normalized-base-change-with-reduced-fibre-separable", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-normalized-base-change-with-reduced-fibre-separable", "contents": [ "Let $A$ be a Dedekind ring with fraction field $K$.", "Let $X$ be a scheme flat and of finite type over $A$.", "Assume $A$ is a Nagata ring and that for every generic point", "$\\eta$ of an irreducible component of $X$ the field", "extension $K \\subset \\kappa(\\eta)$ is separable.", "Then there exists a finite separable extension $K \\subset L$ such that", "the normalized base change $Y$ is smooth over $\\Spec(B)$", "at all generic points of all fibres." ], "refs": [], "proofs": [ { "contents": [ "This is proved in exactly the same manner as", "Theorem \\ref{theorem-normalized-base-change-with-reduced-fibre}", "with a few minor modifications. The most important change", "is to use More on Algebra, Lemma", "\\ref{more-algebra-lemma-epp-essentially-finite-type-separable}", "instead of More on Algebra, Proposition", "\\ref{more-algebra-proposition-epp-essentially-finite-type}.", "During the proof we will repeatedly use that formation of the set of points", "where a (flat, finitely presented) morphism like $X \\to \\Spec(A)$ is", "smooth commutes with base change, see", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-smooth}.", "\\medskip\\noindent", "Since $X$ is flat over $A$ every generic point $\\eta$ of $X$ maps to the", "generic point of $\\Spec(A)$.", "After replacing $X$ by its reduction we may assume $X$ is reduced.", "In this case $X_K$ is geometrically reduced over $K$", "by Varieties, Lemma \\ref{varieties-lemma-generic-points-geometrically-reduced}.", "Hence $X_K \\to \\Spec(K)$ is smooth on a dense open by", "Varieties, Lemma \\ref{varieties-lemma-geometrically-reduced-dense-smooth-open}.", "Thus the smooth locus $U \\subset X$ of the morphism $X \\to \\Spec(A)$", "is open (by Morphisms, Definition \\ref{morphisms-definition-smooth})", "and is dense in the generic fibre. This reduces us to the situation", "of the following paragraph.", "\\medskip\\noindent", "Assume $X$ is normal and the smooth locus $U \\subset X$ of $X \\to \\Spec(A)$", "is dense in the generic fibre. This implies that $U$ is dense in all but", "finitely many fibres, see Lemma \\ref{lemma-nowhere-dense-generic-fibre}.", "Let $x_1, \\ldots, x_r \\in X \\setminus U$ be the finitely many generic", "points of irreducible components of $X \\setminus U$ which are moreover", "generic points of irreducible components of fibres of $X \\to \\Spec(A)$.", "Set $\\mathcal{O}_i = \\mathcal{O}_{X, x_i}$. Observe that the fraction", "field of $\\mathcal{O}_i$ is the residue field of a generic point of $X$.", "Let $A_i$ be the localization of $A$ at the maximal ideal corresponding", "to the image of $x_i$ in $\\Spec(A)$. We may apply More on Algebra, Lemma", "\\ref{more-algebra-lemma-epp-essentially-finite-type-separable}", "and we find finite separable extensions $K \\subset K_i$ which are", "solutions for $A_i \\to \\mathcal{O}_i$. Let $K \\subset L$ be a finite", "separable extension dominating all of the extensions $K \\subset K_i$.", "Then $K \\subset L$ is still a solution for $A_i \\to \\mathcal{O}_i$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-formally-smooth-goes-up}.", "\\medskip\\noindent", "Consider the diagram (\\ref{equation-normalized-base-change})", "with the extension $L/K$ we just produced. Note that $U_B \\subset X_B$", "is smooth over $B$, hence normal (for example use", "Algebra, Lemma \\ref{algebra-lemma-normal-goes-up}).", "Thus $Y \\to X_B$ is an isomorphism over $U_B$.", "Let $y \\in Y$ be a generic point of an irreducible", "component of a fibre of $Y \\to \\Spec(B)$ lying over the maximal", "ideal $\\mathfrak m \\subset B$. Assume that $y \\not \\in U_B$.", "Then $y$ maps to one of the points $x_i$. It follows that", "$\\mathcal{O}_{Y, y}$ is a local ring of the integral closure", "of $\\mathcal{O}_i$ in $R(X) \\otimes_K L$ (details omitted).", "Hence because $K \\subset L$ is a solution for", "$A_i \\to \\mathcal{O}_i$ we see that", "$B_\\mathfrak m \\to \\mathcal{O}_{Y, y}$ is formally smooth", "(this is the definition of being a \"solution\").", "In other words, $\\mathfrak m\\mathcal{O}_{Y, y} = \\mathfrak m_y$", "and the residue field extension is separable. Hence the local ring", "of the fibre at $y$ is $\\kappa(y)$.", "This implies the fibre is smooth over $\\kappa(\\mathfrak m)$", "at $y$ for example by Algebra, Lemma \\ref{algebra-lemma-separable-smooth}.", "This finishes the proof." ], "refs": [ "more-morphisms-theorem-normalized-base-change-with-reduced-fibre", "more-algebra-lemma-epp-essentially-finite-type-separable", "more-algebra-proposition-epp-essentially-finite-type", "morphisms-lemma-set-points-where-fibres-smooth", "varieties-lemma-generic-points-geometrically-reduced", "varieties-lemma-geometrically-reduced-dense-smooth-open", "morphisms-definition-smooth", "more-morphisms-lemma-nowhere-dense-generic-fibre", "more-algebra-lemma-epp-essentially-finite-type-separable", "more-algebra-lemma-formally-smooth-goes-up", "algebra-lemma-normal-goes-up", "algebra-lemma-separable-smooth" ], "ref_ids": [ 13676, 10532, 10592, 5336, 10912, 11008, 5564, 13802, 10532, 10508, 1368, 1225 ] } ], "ref_ids": [] }, { "id": 14038, "type": "theorem", "label": "more-morphisms-lemma-normalized-base-change-with-reduced-fibre-over-curve-separable", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-normalized-base-change-with-reduced-fibre-over-curve-separable", "contents": [ "Let $f : X \\to S$ be a flat, finite type morphism of schemes.", "Assume $S$ is Nagata, integral with function field $K$, and", "regular of dimension $1$. Assume the field extensions $K \\subset \\kappa(\\eta)$", "are separable for every generic point $\\eta$ of an irreducible", "component of $X$. Then there exists a finite separable extension $L/K$", "such that in the diagram", "$$", "\\xymatrix{", "Y \\ar[rd]_g \\ar[r]_-\\nu & X \\times_S T \\ar[d] \\ar[r] & X \\ar[d]_f \\\\", "& T \\ar[r] & S", "}", "$$", "the morphism $g$ is smooth at all generic points of fibres. Here", "$T$ is the normalization of $S$ in $\\Spec(L)$ and $\\nu : Y \\to X \\times_S T$", "is the normalization." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-normalized-base-change-with-reduced-fibre-separable}", "in exactly the same manner that", "Lemma \\ref{lemma-normalized-base-change-with-reduced-fibre-over-curve}", "follows from", "Theorem \\ref{theorem-normalized-base-change-with-reduced-fibre}." ], "refs": [ "more-morphisms-lemma-normalized-base-change-with-reduced-fibre-separable", "more-morphisms-lemma-normalized-base-change-with-reduced-fibre-over-curve", "more-morphisms-theorem-normalized-base-change-with-reduced-fibre" ], "ref_ids": [ 14037, 14036, 13676 ] } ], "ref_ids": [] }, { "id": 14039, "type": "theorem", "label": "more-morphisms-lemma-ind-quasi-affine-alternative-definition", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-ind-quasi-affine-alternative-definition", "contents": [ "For a morphism of schemes $f : X \\to Y$, the following are equivalent:", "\\begin{enumerate}", "\\item $f$ is ind-quasi-affine,", "\\item for every affine open subscheme $V \\subset Y$ and", "every quasi-compact open subscheme $U \\subset f^{-1}(V)$,", "the induced morphism $U \\to V$ is quasi-affine. ", "\\item", "for some cover $\\{ V_j \\}_{j \\in J}$ of $Y$ by", "quasi-compact and quasi-separated open subschemes", "$V_j \\subset Y$, every $j \\in J$, and every quasi-compact", "open subscheme $U \\subset f^{-1}(V_j)$, the induced morphism", "$U \\to V_j$ is quasi-affine.", "\\item for every quasi-compact and quasi-separated open subscheme", "$V \\subset Y$ and every quasi-compact open subscheme", "$U \\subset f^{-1}(V)$, the induced morphism $U \\to V$ is quasi-affine.", "\\end{enumerate}", "In particular, the property of being an ind-quasi-affine morphism", "is Zariski local on the base." ], "refs": [], "proofs": [ { "contents": [ "The equivalence (1) $\\Leftrightarrow$ (2)", "follows from the definitions and", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-quasi-affine}.", "For (2) $\\Rightarrow$ (4), let $U$ and $V$ be as in (4). By", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}, the", "induced morphism $U \\to V$ is quasi-compact. Thus, for every affine", "open $V' \\subset V$, the fiber product $V' \\times_V U$ is quasi-compact,", "so, by (2), the induced map $V' \\times_V U \\to V'$ is quasi-affine.", "Thus, $U \\to V$ is also quasi-affine by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-quasi-affine}.", "This argument also gives (3) $\\Rightarrow$ (4): indeed, keeping the", "same notation, those affine opens $V' \\subset V$ that lie in one", "of the $V_j$ cover $V$, so one needs to argue that the", "quasi-compact map $V' \\times_V U \\to V'$ is quasi-affine.", "However, by (3), the composition $V' \\times_V U \\to V' \\to V_j$", "is quasi-affine and, by", "Schemes, Lemma \\ref{schemes-lemma-compose-after-separated}, the map", "$V' \\to V_j$ is quasi-separated. Thus, $V' \\times_V U \\to V'$", "is quasi-affine by", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-affine-permanence}.", "The final implications (4) $\\Rightarrow$ (2) and (4) $\\Rightarrow$ (3)", "are evident." ], "refs": [ "morphisms-lemma-characterize-quasi-affine", "schemes-lemma-quasi-compact-permanence", "morphisms-lemma-characterize-quasi-affine", "schemes-lemma-compose-after-separated", "morphisms-lemma-quasi-affine-permanence" ], "ref_ids": [ 5185, 7716, 5185, 7715, 5190 ] } ], "ref_ids": [] }, { "id": 14040, "type": "theorem", "label": "more-morphisms-lemma-ind-quasi-affine-composition", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-ind-quasi-affine-composition", "contents": [ "The property of being an ind-quasi-affine morphism is stable under composition." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be ind-quasi-affine morphisms.", "Let $V \\subset Z$ and $U \\subset f^{-1}(g^{-1}(V))$ be quasi-compact", "opens such that $V$ is also quasi-separated. The image $f(U)$ is a", "quasi-compact subset of $g^{-1}(V)$, so it is contained in some", "quasi-compact open $W \\subset g^{-1}(V)$ (a union of finitely many affines).", "We obtain a factorization $U \\to W \\to V$. The map $W \\to V$ is quasi-affine", "by Lemma \\ref{lemma-ind-quasi-affine-alternative-definition}, so, in", "particular, $W$ is quasi-separated. Then, by", "Lemma \\ref{lemma-ind-quasi-affine-alternative-definition} again, $U \\to W$", "is quasi-affine as well. Consequently, by Morphisms,", "Lemma \\ref{morphisms-lemma-composition-quasi-affine}, the composition", "$U \\to V$ is also quasi-affine, and it remains to apply", "Lemma \\ref{lemma-ind-quasi-affine-alternative-definition} once more." ], "refs": [ "more-morphisms-lemma-ind-quasi-affine-alternative-definition", "more-morphisms-lemma-ind-quasi-affine-alternative-definition", "morphisms-lemma-composition-quasi-affine", "more-morphisms-lemma-ind-quasi-affine-alternative-definition" ], "ref_ids": [ 14039, 14039, 5186, 14039 ] } ], "ref_ids": [] }, { "id": 14041, "type": "theorem", "label": "more-morphisms-lemma-ind-quasi-affine-examples", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-ind-quasi-affine-examples", "contents": [ "Any quasi-affine morphism is ind-quasi-affine.", "Any immersion is ind-quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "The first assertion is immediate from the definitions.", "In particular, affine morphisms, such as closed immersions,", "are ind-quasi-affine. Thus, by", "Lemma \\ref{lemma-ind-quasi-affine-composition}, it remains", "to show that an open immersion is ind-quasi-affine.", "This, however, is immediate from the definitions." ], "refs": [ "more-morphisms-lemma-ind-quasi-affine-composition" ], "ref_ids": [ 14040 ] } ], "ref_ids": [] }, { "id": 14042, "type": "theorem", "label": "more-morphisms-lemma-ind-quasi-affine-permanence", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-ind-quasi-affine-permanence", "contents": [ "If $f : X \\to Y$ and $g : Y \\to Z$ are morphisms of schemes", "such that $g \\circ f$ is ind-quasi-affine, then $f$ is ind-quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-ind-quasi-affine-alternative-definition}, we may", "work Zariski locally on $Z$ and then on $Y$, so we lose no generality", "by assuming that $Z$, and then also $Y$, is affine. Then any quasi-compact", "open of $X$ is quasi-affine, so", "Lemma \\ref{lemma-ind-quasi-affine-alternative-definition} gives the claim." ], "refs": [ "more-morphisms-lemma-ind-quasi-affine-alternative-definition", "more-morphisms-lemma-ind-quasi-affine-alternative-definition" ], "ref_ids": [ 14039, 14039 ] } ], "ref_ids": [] }, { "id": 14043, "type": "theorem", "label": "more-morphisms-lemma-base-change-ind-quasi-affine", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-base-change-ind-quasi-affine", "contents": [ "The property of being ind-quasi-affine is stable under base change." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be an ind-quasi-affine morphism.", "For checking that every base change of $f$ is ind-quasi-affine, by", "Lemma \\ref{lemma-ind-quasi-affine-alternative-definition}, we may work", "Zariski locally on $Y$, so we assume that $Y$ is affine.", "Furthermore, we may also assume that in the base change morphism", "$Z \\to Y$ the scheme $Z$ is affine, too. The base change", "$X \\times_Y Z \\to X$ is an affine morphism, so, by", "Lemmas \\ref{lemma-ind-quasi-affine-composition} and", "\\ref{lemma-ind-quasi-affine-examples},", "the map $X \\times_Y Z \\to Y$ is ind-quasi-affine. Then, by", "Lemma \\ref{lemma-ind-quasi-affine-permanence}, the", "base change $X \\times_Y Z \\to Z$ is ind-quasi-affine, as desired." ], "refs": [ "more-morphisms-lemma-ind-quasi-affine-alternative-definition", "more-morphisms-lemma-ind-quasi-affine-composition", "more-morphisms-lemma-ind-quasi-affine-examples", "more-morphisms-lemma-ind-quasi-affine-permanence" ], "ref_ids": [ 14039, 14040, 14041, 14042 ] } ], "ref_ids": [] }, { "id": 14044, "type": "theorem", "label": "more-morphisms-lemma-descending-property-ind-quasi-affine", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-descending-property-ind-quasi-affine", "contents": [ "The property of being ind-quasi-affine is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "The stability of ind-quasi-affineness under base change", "supplied by Lemma \\ref{lemma-base-change-ind-quasi-affine}", "gives one direction. For the other, let $f : X \\to Y$", "be a morphism of schemes and let $\\{g_i : Y_i \\to Y\\}$", "be an fpqc covering such that the base change $f_i : X_i \\to Y_i$", "is ind-quasi-affine for all $i$. We need to show $f$ is ind-quasi-affine.", "\\medskip\\noindent", "By Lemma \\ref{lemma-ind-quasi-affine-alternative-definition}, we may work", "Zariski locally on $Y$, so we assume that $Y$ is affine.", "Then we use stability under base change ensured by", "Lemma \\ref{lemma-base-change-ind-quasi-affine} to refine the cover", "and assume that it is given by a single affine, faithfully flat morphism", "$g : Y' \\to Y$. For any quasi-compact open $U \\subset X$, its", "$Y'$-base change $U \\times_Y Y' \\subset X \\times_Y Y'$ is also quasi-compact.", "It remains to observe that, by", "Descent, Lemma \\ref{descent-lemma-descending-property-quasi-affine},", "the map $U \\to Y$ is quasi-affine if and only if so is $U \\times_Y Y' \\to Y'$." ], "refs": [ "more-morphisms-lemma-base-change-ind-quasi-affine", "more-morphisms-lemma-ind-quasi-affine-alternative-definition", "more-morphisms-lemma-base-change-ind-quasi-affine", "descent-lemma-descending-property-quasi-affine" ], "ref_ids": [ 14043, 14039, 14043, 14685 ] } ], "ref_ids": [] }, { "id": 14045, "type": "theorem", "label": "more-morphisms-lemma-etale-separated-ind-quasi-affine", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-etale-separated-ind-quasi-affine", "contents": [ "A separated locally quasi-finite morphism of schemes is ind-quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a separated locally quasi-finite morphism of schemes.", "Let $V \\subset Y$ be affine and $U \\subset f^{-1}(V)$ quasi-compact", "open. We have to show $U$ is quasi-affine. Since $U \\to V$ is a", "separated quasi-finite morphism of schemes, this follows from", "Zariski's Main Theorem. See", "Lemma \\ref{lemma-quasi-finite-separated-quasi-affine}." ], "refs": [ "more-morphisms-lemma-quasi-finite-separated-quasi-affine" ], "ref_ids": [ 13900 ] } ], "ref_ids": [] }, { "id": 14046, "type": "theorem", "label": "more-morphisms-lemma-prepare-pushout-along-closed-immersion-and-integral", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-prepare-pushout-along-closed-immersion-and-integral", "contents": [ "In Situation \\ref{situation-pushout-along-closed-immersion-and-integral}", "then for $y \\in Y$ there exist affine opens $U \\subset X$ and", "$V \\subset Y$ with $i^{-1}(U) = j^{-1}(V)$ and $y \\in V$." ], "refs": [], "proofs": [ { "contents": [ "Let $y \\in Y$. Choose an affine open $U \\subset X$", "such that $j^{-1}(\\{y\\}) \\subset i^{-1}(U)$ (possible by assumption).", "Choose an affine open $V \\subset Y$ neighbourhood of $y$", "such that $j^{-1}(V) \\subset i^{-1}(U)$.", "This is possible because $j : Z \\to Y$ is a closed morphism", "(Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed}) and", "$i^{-1}(U)$ contains the fibre over $y$.", "Since $j$ is integral, the scheme theoretic fibre $Z_y$", "is the spectrum of an algebra integral over a field.", "By Limits, Lemma \\ref{limits-lemma-ample-profinite-set-in-principal-affine}", "we can find an $\\overline{f} \\in \\Gamma(i^{-1}(U), \\mathcal{O}_{i^{-1}(U)})$", "such that $Z_y \\subset D(\\overline{f}) \\subset j^{-1}(V)$.", "Since $i|_{i^{-1}(U)} : i^{-1}(U) \\to U$ is a closed immersion", "of affines, we can choose an $f \\in \\Gamma(U, \\mathcal{O}_U)$", "whose restriction to $i^{-1}(U)$ is $\\overline{f}$.", "After replacing $U$ by the principal open $D(f) \\subset U$", "we find affine opens $y \\in V \\subset Y$ and $U \\subset X$ with", "$$", "j^{-1}(\\{y\\}) \\subset i^{-1}(U) \\subset j^{-1}(V)", "$$", "Now we (in some sense) repeat the argument. Namely, we choose", "$g \\in \\Gamma(V, \\mathcal{O}_V)$ such that $y \\in D(g)$ and", "$j^{-1}(D(g)) \\subset i^{-1}(U)$ (possible by the same argument", "as above). Then we can pick $f \\in \\Gamma(U, \\mathcal{O}_U)$", "whose restriction to $i^{-1}(U)$ is the pullback of $g$", "by $i^{-1}(U) \\to V$ (again possible by the same reason as above).", "Then we finally have affine opens $y \\in V' = D(g) \\subset V \\subset Y$", "and $U' = D(f) \\subset U \\subset X$ with $j^{-1}(V') = i^{-1}(V')$." ], "refs": [ "morphisms-lemma-integral-universally-closed", "limits-lemma-ample-profinite-set-in-principal-affine" ], "ref_ids": [ 5441, 15086 ] } ], "ref_ids": [] }, { "id": 14047, "type": "theorem", "label": "more-morphisms-lemma-pushout-separated", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pushout-separated", "contents": [ "In Situation \\ref{situation-pushout-along-closed-immersion-and-integral}.", "If $X$ and $Y$ are separated, then the pushout $Y \\amalg_Z X$", "(Proposition \\ref{proposition-pushout-along-closed-immersion-and-integral})", "is separated. Same with ``separated over $S$'', ``quasi-separated'', and", "``quasi-separated over $S$''." ], "refs": [ "more-morphisms-proposition-pushout-along-closed-immersion-and-integral" ], "proofs": [ { "contents": [ "The morphism $Y \\amalg X \\to Y \\amalg_Z X$ is surjective", "and universall closed. Thus we may apply", "Morphisms, Lemma \\ref{morphisms-lemma-image-universally-closed-separated}." ], "refs": [ "morphisms-lemma-image-universally-closed-separated" ], "ref_ids": [ 5415 ] } ], "ref_ids": [ 14103 ] }, { "id": 14048, "type": "theorem", "label": "more-morphisms-lemma-pushout-finite-type", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pushout-finite-type", "contents": [ "In Situation \\ref{situation-pushout-along-closed-immersion-and-integral}", "assume $S$ is a locally Noetherian scheme and $X$, $Y$, and $Z$", "are locally of finite type over $S$. Then the pushout $Y \\amalg_Z X$", "(Proposition \\ref{proposition-pushout-along-closed-immersion-and-integral})", "is locally of finite type over $S$." ], "refs": [ "more-morphisms-proposition-pushout-along-closed-immersion-and-integral" ], "proofs": [ { "contents": [ "Looking on affine opens we recover the result of", "More on Algebra, Lemma \\ref{more-algebra-lemma-fibre-product-finite-type}." ], "refs": [ "more-algebra-lemma-fibre-product-finite-type" ], "ref_ids": [ 9814 ] } ], "ref_ids": [ 14103 ] }, { "id": 14049, "type": "theorem", "label": "more-morphisms-lemma-pushout-functor", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pushout-functor", "contents": [ "In Situation \\ref{situation-pushout-along-closed-immersion-and-integral}", "suppose given a commutative diagram", "$$", "\\xymatrix{", "Y' \\ar[d]^g & Z' \\ar[l]^{j'} \\ar[r]_{i'} \\ar[d]^h & X' \\ar[d]^f \\\\", "Y & Z \\ar[l] \\ar[r] & X", "}", "$$", "with cartesian squares and $f, g, h$ separated and locally quasi-finite. Then", "\\begin{enumerate}", "\\item the pushouts $Y \\amalg_Z X$ and $Y' \\amalg_{Z'} X'$ exist,", "\\item $Y' \\amalg_{Z'} X' \\to Y \\amalg_Z X$ is", "separated and locally quasi-finite, and", "\\item the squares", "$$", "\\xymatrix{", "Y' \\ar[r] \\ar[d] & Y' \\amalg_{Z'} X' \\ar[d] & X' \\ar[l] \\ar[d] \\\\", "Y \\ar[r] & Y \\amalg_Z X & X \\ar[l]", "}", "$$", "are cartesian.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The pushout $Y \\amalg_Z X$ exists by", "Proposition \\ref{proposition-pushout-along-closed-immersion-and-integral}.", "To see that the pushout $Y' \\amalg_{Z'} X'$ exists, we check", "condition (3) of", "Situation \\ref{situation-pushout-along-closed-immersion-and-integral}", "holds for $(X', Y', Z', i', j')$.", "Namely, let $y' \\in Y'$ and denote $y \\in Y$ the image.", "Choose $U \\subset X$ affine open with $i(j^{-1}(y)) \\subset U$.", "Choose a quasi-compact open $U' \\subset X'$ contained in", "$f^{-1}(U)$ containing the quasi-compact subset $i'((j')^{-1}(\\{y'\\}))$.", "By Lemma \\ref{lemma-etale-separated-ind-quasi-affine}", "we see that $U'$ is quasi-affine. Since $Z'_{y'}$ is the spectrum", "of an algebra integral over a field, we can apply", "Limits, Lemma \\ref{limits-lemma-ample-profinite-set-in-principal-affine}", "and we find there exists an affine open subscheme of $U'$ containing", "$i'((j')^{-1}(\\{y'\\}))$ as desired.", "\\medskip\\noindent", "Having verified existence we check the other assertions.", "Affine locally we are exactly in the situation of More on Algebra, Lemma", "\\ref{more-algebra-lemma-properties-algebras-over-fibre-product}", "with $B \\to D$ and $A' \\to C'$ locally quasi-finite\\footnote{To be precise", "$X, Y, Z, Y \\amalg_Z X, X', Y', Z', Y' \\amalg_{Z'} X'$", "correspond to $A', B, A, B', C', D, C, D'$.}.", "In particular, the morphism $Y' \\amalg_{Z'} X' \\to Y \\amalg_Z X$ is locally", "of finite type. The squares in of the diagram are cartesian by", "More on Algebra, Lemma \\ref{more-algebra-lemma-module-over-fibre-product}.", "Since being locally quasi-finite can be checked on fibres", "(Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize})", "we conclude that $Y' \\amalg_{Z'} X' \\to Y \\amalg_Z X$", "is locally quasi-finite.", "\\medskip\\noindent", "We still have to check $Y' \\amalg_{Z'} X' \\to Y \\amalg_Z X$ is separated.", "Observe that $Y' \\amalg X' \\to Y' \\amalg_{Z'} X'$ is universally closed", "and surjective by ", "Proposition \\ref{proposition-pushout-along-closed-immersion-and-integral}.", "Since also the morphism $Y' \\amalg X' \\to Y \\amalg_Z X$ is separated", "(as it factors as $Y' \\amalg X' \\to Y \\amalg X \\to Y \\amalg_Z X$)", "we conclude by", "Morphisms, Lemma \\ref{morphisms-lemma-image-universally-closed-separated}." ], "refs": [ "more-morphisms-proposition-pushout-along-closed-immersion-and-integral", "more-morphisms-lemma-etale-separated-ind-quasi-affine", "limits-lemma-ample-profinite-set-in-principal-affine", "more-algebra-lemma-properties-algebras-over-fibre-product", "more-algebra-lemma-module-over-fibre-product", "morphisms-lemma-quasi-finite-at-point-characterize", "more-morphisms-proposition-pushout-along-closed-immersion-and-integral", "morphisms-lemma-image-universally-closed-separated" ], "ref_ids": [ 14103, 14045, 15086, 9831, 9820, 5226, 14103, 5415 ] } ], "ref_ids": [] }, { "id": 14050, "type": "theorem", "label": "more-morphisms-lemma-pushout-functor-equivalence-flat", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pushout-functor-equivalence-flat", "contents": [ "In Situation \\ref{situation-pushout-along-closed-immersion-and-integral}", "the category of schemes flat, separated, and locally quasi-finite", "over the pushout $Y \\amalg_Z X$ is equivalent to the category of", "$(X', Y', Z', i', j', f, g, h)$ as in Lemma \\ref{lemma-pushout-functor}", "with $f, g, h$ flat. Similarly with ``flat'' replaced with", "``\\'etale''." ], "refs": [ "more-morphisms-lemma-pushout-functor" ], "proofs": [ { "contents": [ "If we start with $(X', Y', Z', i', j', f, g, h)$", "as in Lemma \\ref{lemma-pushout-functor} with $f, g, h$ flat", "or \\'etale, then $Y' \\amalg_{Z'} X' \\to Y \\amalg_Z X$ is flat", "or \\'etale by More on Algebra, Lemma", "\\ref{more-algebra-lemma-properties-algebras-over-fibre-product}.", "\\medskip\\noindent", "For the converse, let", "$W \\to Y \\amalg_Z X$ be a separated and locally quasi-finite morphism.", "Set $X' = W \\times_{Y \\amalg_Z X} X$, $Y' = W \\times_{Y \\amalg_Z X} Y$, and", "$Z' = W \\times_{Y \\amalg_Z X} Z$ with obvious morphisms $i', j', f, g, h$.", "Form the pushout $Y' \\amalg_{Z'} X'$. We obtain a morphism", "$$", "Y' \\amalg_{Z'} X' \\longrightarrow W", "$$", "of schemes over $Y \\amalg_X Z$ by the universal property of the", "pushout. If we do not assume that $W \\to Y \\amalg_Z X$ is flat,", "then in general this morphism won't be an isomorphism.", "(In fact, More on Algebra, Lemma", "\\ref{more-algebra-lemma-module-over-fibre-product-bis}", "shows the displayed arrow is a closed immersion but not an isomorphism", "in general.) However, if $W \\to Y \\times_Z X$ is flat, then", "it is an isomorphism by More on Algebra, Lemma", "\\ref{more-algebra-lemma-properties-algebras-over-fibre-product}." ], "refs": [ "more-morphisms-lemma-pushout-functor", "more-algebra-lemma-properties-algebras-over-fibre-product", "more-algebra-lemma-module-over-fibre-product-bis", "more-algebra-lemma-properties-algebras-over-fibre-product" ], "ref_ids": [ 14049, 9831, 9821, 9831 ] } ], "ref_ids": [ 14049 ] }, { "id": 14051, "type": "theorem", "label": "more-morphisms-lemma-pushout-along-closed-immersions", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pushout-along-closed-immersions", "contents": [ "Let $i : Z \\to X$ and $j : Z \\to Y$ be closed immersions of schemes.", "Then the pushout $Y \\amalg_Z X$ exists in the category of schemes. Picture", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[d]_j & X \\ar[d]^a \\\\", "Y \\ar[r]^-b & Y \\amalg_Z X", "}", "$$", "The diagram is a fibre square, the morphisms $a$ and $b$", "are closed immersions, and there is a short exact sequence", "$$", "0 \\to \\mathcal{O}_{Y \\amalg_Z X} \\to", "a_*\\mathcal{O}_X \\oplus b_*\\mathcal{O}_Y \\to", "c_*\\mathcal{O}_Z \\to 0", "$$", "where $c = a \\circ i = b \\circ j$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of", "Proposition \\ref{proposition-pushout-along-closed-immersion-and-integral}.", "Observe that hypothesis (3) in", "Situation \\ref{situation-pushout-along-closed-immersion-and-integral}", "is immediate because", "the fibres of $j$ are singletons. Finally, reverse the roles of the arrows", "to conclude that both $a$ and $b$ are closed immersions." ], "refs": [ "more-morphisms-proposition-pushout-along-closed-immersion-and-integral" ], "ref_ids": [ 14103 ] } ], "ref_ids": [] }, { "id": 14052, "type": "theorem", "label": "more-morphisms-lemma-pushout-along-closed-immersions-properties-above", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-pushout-along-closed-immersions-properties-above", "contents": [ "Let $i : Z \\to X$ and $j : Z \\to Y$ be closed immersions of schemes.", "Let $f : X' \\to X$ and $g : Y' \\to Y$ be morphisms of schemes and let", "$\\varphi : X' \\times_{X, i} Z \\to Y' \\times_{Y, j} Z$", "be an isomorphism of schemes over $Z$. Consider the morphism", "$$", "h :", "X' \\amalg_{X' \\times_{X, i} Z, \\varphi} Y'", "\\longrightarrow", "X \\amalg_Z Y", "$$", "Then we have", "\\begin{enumerate}", "\\item $h$ is locally of finite type if and only if $f$ and $g$ are", "locally of finite type,", "\\item $h$ is flat if and only if $f$ and $g$ are flat,", "\\item $h$ is flat and locally of finite presentation if and only if", "$f$ and $g$ are flat and locally of finite presentation,", "\\item $h$ is smooth if and only if $f$ and $g$ are smooth,", "\\item $h$ is \\'etale if and only if $f$ and $g$ are \\'etale, and", "\\item add more here as needed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We know that the pushouts exist by", "Lemma \\ref{lemma-pushout-along-closed-immersions}.", "In particular we get the morphism $h$.", "Hence we may replace all schemes in sight by", "affine schemes. In this case the assertions of the lemma", "are equivalent to the corresponding assertions of", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-properties-algebras-over-fibre-product}." ], "refs": [ "more-morphisms-lemma-pushout-along-closed-immersions", "more-algebra-lemma-properties-algebras-over-fibre-product" ], "ref_ids": [ 14051, 9831 ] } ], "ref_ids": [] }, { "id": 14053, "type": "theorem", "label": "more-morphisms-lemma-hom-from-finite-free-into-affine", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-hom-from-finite-free-into-affine", "contents": [ "Let $Z \\to S$ and $X \\to S$ be morphisms of affine schemes.", "Assume $\\Gamma(Z, \\mathcal{O}_Z)$ is a finite free", "$\\Gamma(S, \\mathcal{O}_S)$-module. Then $\\mathit{Mor}_S(Z, X)$", "is representable by an affine scheme over $S$." ], "refs": [], "proofs": [ { "contents": [ "Write $S = \\Spec(R)$. Choose a basis $\\{e_1, \\ldots, e_m\\}$", "for $\\Gamma(Z, \\mathcal{O}_Z)$ over $R$. Choose a presentation", "$$", "\\Gamma(X, \\mathcal{O}_X) = R[\\{x_i\\}_{i \\in I}]/(\\{f_k\\}_{k \\in K}).", "$$", "We will denote $\\overline{x}_i$ the image of $x_i$ in this quotient.", "Write", "$$", "P = R[\\{a_{ij}\\}_{i \\in I, 1 \\leq j \\leq m}].", "$$", "Consider the $R$-algebra map", "$$", "\\Psi :", "R[\\{x_i\\}_{i \\in I}]", "\\longrightarrow", "P \\otimes_R \\Gamma(Z, \\mathcal{O}_Z), \\quad", "x_i \\longmapsto \\sum\\nolimits_j a_{ij} \\otimes e_j.", "$$", "Write $\\Psi(f_k) = \\sum c_{kj} \\otimes e_j$ with $c_{kj} \\in P$.", "Finally, denote $J \\subset P$ the ideal generated by the elements", "$c_{kj}$, $k \\in K$, $1 \\leq j \\leq m$. We claim that", "$W = \\Spec(P/J)$ represents the functor $\\mathit{Mor}_S(Z, X)$.", "\\medskip\\noindent", "First, note that by construction $P/J$ is an $R$-algebra, hence", "a morphism $W \\to S$. Second, by construction the map", "$\\Psi$ factors through $\\Gamma(X, \\mathcal{O}_X)$, hence we obtain", "an $P/J$-algebra homomorphism", "$$", "P/J \\otimes_R \\Gamma(X, \\mathcal{O}_X)", "\\longrightarrow", "P/J \\otimes_R \\Gamma(Z, \\mathcal{O}_Z)", "$$", "which determines a morphism", "$b_{univ} : W \\times_S Z \\to W \\times_S X$.", "By the Yoneda lemma $b_{univ}$ determines a", "transformation of functors $W \\to \\mathit{Mor}_S(Z, X)$ which we", "claim is an isomorphism. To show that it is an isomorphism it suffices", "to show that it induces a bijection of sets", "$W(T) \\to \\mathit{Mor}_S(Z, X)(T)$ over any affine", "scheme $T$.", "\\medskip\\noindent", "Suppose $T = \\Spec(R')$ is an affine scheme over $S$", "and $b \\in \\mathit{Mor}_S(Z, X)(T)$. The structure morphism $T \\to S$", "defines an $R$-algebra structure on $R'$ and $b$ defines an $R'$-algebra map", "$$", "b^\\sharp :", "R' \\otimes_R \\Gamma(X, \\mathcal{O}_X)", "\\longrightarrow", "R' \\otimes_R \\Gamma(Z, \\mathcal{O}_Z).", "$$", "In particular we can write", "$b^\\sharp(1 \\otimes \\overline{x}_i) = \\sum \\alpha_{ij} \\otimes e_j$", "for some $\\alpha_{ij} \\in R'$. This corresponds to an $R$-algebra map", "$P \\to R'$ determined by the rule $a_{ij} \\mapsto \\alpha_{ij}$. This", "map factors through the quotient $P/J$ by the construction of the ideal", "$J$ to give a map $P/J \\to R'$. This in turn corresponds to a morphism", "$T \\to W$ such that $b$ is the pullback of $b_{univ}$.", "Some details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14054, "type": "theorem", "label": "more-morphisms-lemma-hom-from-finite-locally-free-into-affine", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-hom-from-finite-locally-free-into-affine", "contents": [ "Let $Z \\to S$ and $X \\to S$ be morphisms of schemes.", "If $Z \\to S$ is finite locally free and $X \\to S$ is affine,", "then $\\mathit{Mor}_S(Z, X)$ is representable by a scheme", "affine over $S$." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine open covering $S = \\bigcup U_i$ such that", "$\\Gamma(Z \\times_S U_i, \\mathcal{O}_{Z \\times_S U_i})$ is", "finite free over $\\mathcal{O}_S(U_i)$. Let $F_i \\subset \\mathit{Mor}_S(Z, X)$", "be the subfunctor which assigns to $T/S$ the empty set if", "$T \\to S$ does not factor through $U_i$ and $\\mathit{Mor}_S(Z, X)(T)$", "otherwise. Then the collection of these subfunctors satisfy the conditions", "(2)(a), (2)(b), (2)(c) of", "Schemes, Lemma \\ref{schemes-lemma-glue-functors} which proves the lemma.", "Condition (2)(a) follows from", "Lemma \\ref{lemma-hom-from-finite-free-into-affine}", "and the other two follow from straightforward arguments." ], "refs": [ "schemes-lemma-glue-functors", "more-morphisms-lemma-hom-from-finite-free-into-affine" ], "ref_ids": [ 7688, 14053 ] } ], "ref_ids": [] }, { "id": 14055, "type": "theorem", "label": "more-morphisms-lemma-hom-from-finite-locally-free-representable", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-hom-from-finite-locally-free-representable", "contents": [ "Let $Z \\to S$ and $X \\to S$ be morphisms of schemes.", "Assume", "\\begin{enumerate}", "\\item $Z \\to S$ is finite locally free, and", "\\item for all $(s, x_1, \\ldots, x_d)$ where $s \\in S$ and", "$x_1, \\ldots, x_d \\in X_s$ there exists an affine open $U \\subset X$", "with $x_1, \\ldots, x_d \\in U$.", "\\end{enumerate}", "Then $\\mathit{Mor}_S(Z, X)$ is representable by a scheme." ], "refs": [], "proofs": [ { "contents": [ "Consider the set $I$ of pairs $(U, V)$ where $U \\subset X$ and $V \\subset S$", "are affine open and $U \\to S$ factors through $V$. For $i \\in I$ denote", "$(U_i, V_i)$ the corresponding pair. Set", "$F_i = \\mathit{Mor}_{V_i}(Z_{V_i}, U_i)$.", "It is immediate that $F_i$ is a subfunctor of $\\mathit{Mor}_S(Z, X)$.", "Then we claim that conditions", "(2)(a), (2)(b), (2)(c) of", "Schemes, Lemma \\ref{schemes-lemma-glue-functors} which proves the lemma.", "\\medskip\\noindent", "Condition (2)(a) follows from", "Lemma \\ref{lemma-hom-from-finite-locally-free-into-affine}.", "\\medskip\\noindent", "To check condition (2)(b) consider $T/S$ and $b \\in \\mathit{Mor}_S(Z, X)$.", "Thinking of $b$ as a morphism $T \\times_S Z \\to X$ we find an open", "$b^{-1}(U_i) \\subset T \\times_S Z$. Clearly, $b \\in F_i(T)$", "if and only if $b^{-1}(U_i) = T \\times_S Z$. Since the projection", "$p : T \\times_S Z \\to T$ is finite hence closed, the set", "$U_{i, b} \\subset T$ of points $t \\in T$ with", "$p^{-1}(\\{t\\}) \\subset b^{-1}(U_i)$ is open.", "Then $f : T' \\to T$ factors through $U_{i, b}$ if and only", "if $b \\circ f \\in F_i(T')$ and we are done checking (2)(b).", "\\medskip\\noindent", "Finally, we check condition (2)(c) and this is where our condition", "on $X \\to S$ is used. Namely, consider", "$T/S$ and $b \\in \\mathit{Mor}_S(Z, X)$.", "It suffices to prove that every $t \\in T$", "is contained in one of the opens $U_{i, b}$ defined", "in the previous paragraph.", "This is equivalent to the condition that", "$b(p^{-1}(\\{t\\})) \\subset U_i$ for some $i$", "where $p : T \\times_S Z \\to T$ is the projection and", "$b : T \\times_S Z \\to X$ is the given morphism.", "Since $p$ is finite, the set $b(p^{-1}(\\{t\\})) \\subset X$", "is finite and contained in the fibre of $X \\to S$ over", "the image $s$ of $t$ in $S$.", "Thus our condition on $X \\to S$ exactly shows a", "suitable pair exists." ], "refs": [ "schemes-lemma-glue-functors", "more-morphisms-lemma-hom-from-finite-locally-free-into-affine" ], "ref_ids": [ 7688, 14054 ] } ], "ref_ids": [] }, { "id": 14056, "type": "theorem", "label": "more-morphisms-lemma-hom-from-finite-locally-free-separated-lqf", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-hom-from-finite-locally-free-separated-lqf", "contents": [ "Let $Z \\to S$ and $X \\to S$ be morphisms of schemes.", "Assume $Z \\to S$ is finite locally free and $X \\to S$", "is separated and locally quasi-finite.", "Then $\\mathit{Mor}_S(Z, X)$ is representable by a scheme." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemmas \\ref{lemma-hom-from-finite-locally-free-representable} and", "\\ref{lemma-separated-locally-quasi-finite-over-affine}." ], "refs": [ "more-morphisms-lemma-hom-from-finite-locally-free-representable", "more-morphisms-lemma-separated-locally-quasi-finite-over-affine" ], "ref_ids": [ 14055, 13906 ] } ], "ref_ids": [] }, { "id": 14057, "type": "theorem", "label": "more-morphisms-lemma-case-of-tor-independence", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-case-of-tor-independence", "contents": [ "Consider a commutative diagram of schemes", "$$", "\\xymatrix{", "Z' \\ar[d] \\ar[r] & Y' \\ar[d] \\\\", "X' \\ar[r] & S'", "}", "$$", "Let $S \\to S'$ be a morphism. Denote by $X$ and $Y$ the base", "changes of $X'$ and $Y'$ to $S$.", "Assume $Y' \\to S'$ and $Z' \\to X'$ are flat.", "Then $X \\times_S Y$ and $Z'$ are Tor independent over $X' \\times_{S'} Y'$." ], "refs": [], "proofs": [ { "contents": [ "The question is local, hence we may assume all schemes are affine", "(some details omitted). Observe that", "$$", "\\xymatrix{", "X \\times_S Y \\ar[r] \\ar[d] & X' \\times_{S'} Y' \\ar[d] \\\\", "X \\ar[r] & X'", "}", "$$", "is cartesian with flat vertical arrows.", "Write $X = \\Spec(A)$, $X' = \\Spec(A')$,", "$X' \\times_{S'} Y' = \\Spec(B')$. Then", "$X \\times_S Y = \\Spec(A \\otimes_{A'} B')$.", "Write $Z' = \\Spec(C')$. We have to show", "$$", "\\text{Tor}_p^{B'}(A \\otimes_{A'} B', C') = 0,", "\\quad\\text{for } p > 0", "$$", "Since $A' \\to B'$ is flat", "we have $A \\otimes_{A'} B' = A \\otimes_{A'}^\\mathbf{L} B'$.", "Hence", "$$", "(A \\otimes_{A'} B') \\otimes_{B'}^\\mathbf{L} C' =", "(A \\otimes_{A'}^\\mathbf{L} B') \\otimes_{B'}^\\mathbf{L} C' =", "A \\otimes_{A'}^\\mathbf{L} C' =", "A \\otimes_{A'} C'", "$$", "The second equality by More on Algebra, Lemma", "\\ref{more-algebra-lemma-double-base-change}.", "The last equality because $A' \\to C'$ is flat. This proves the lemma." ], "refs": [ "more-algebra-lemma-double-base-change" ], "ref_ids": [ 10138 ] } ], "ref_ids": [] }, { "id": 14058, "type": "theorem", "label": "more-morphisms-lemma-derived-chow", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-derived-chow", "contents": [ "Let $A$ be a ring. Let $X$ be a separated scheme of finite presentation", "over $A$. Let $x \\in X$. Then there exist", "an open neighbourhood $U \\subset X$ of $x$,", "an $n \\geq 0$,", "an open $V \\subset \\mathbf{P}^n_A$,", "a closed subscheme $Z \\subset X \\times_A \\mathbf{P}^n_A$,", "a point $z \\in Z$, and", "an object $E$ in $D(\\mathcal{O}_{X \\times_A \\mathbf{P}^n_A})$ such that", "\\begin{enumerate}", "\\item $Z \\to X \\times_A \\mathbf{P}^n_A$ is of finite presentation,", "\\item $b : Z \\to X$ is an isomorphism over $U$ and $b(z) = x$,", "\\item $c : Z \\to \\mathbf{P}^n_A$ is a closed immersion over $V$,", "\\item $b^{-1}(U) = c^{-1}(V)$, in particular $c(z) \\in V$,", "\\item $E|_{X \\times_A V} \\cong", "(b, c)_*\\mathcal{O}_Z|_{X \\times_A V}$,", "\\item $E$ is pseudo-coherent and supported on $Z$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We can find a finite type $\\mathbf{Z}$-subalgebra $A' \\subset A$", "and a scheme $X'$ separated and of finite presentation over $A'$", "whose base change to $A$ is $X$. See", "Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation} and", "\\ref{limits-lemma-descend-separated-finite-presentation}.", "Let $x' \\in X'$ be the image of $x$.", "If we can prove the lemma for $x' \\in X'/A'$, then", "the lemma follows for $x \\in X/A$.", "Namely, if $U', n', V', Z', z', E'$ provide the solution", "for $x' \\in X'/A'$, then we can let", "$U \\subset X$ be the inverse image of $U'$,", "let $n = n'$,", "let $V \\subset \\mathbf{P}^n_A$ be the inverse image of $V'$,", "let $Z \\subset X \\times \\mathbf{P}^n$ be", "the scheme theoretic inverse image of $Z'$,", "let $z \\in Z$ be the unique point mapping to $x$, and", "let $E$ be the derived pullback of $E'$.", "Observe that $E$ is pseudo-coherent by", "Cohomology, Lemma \\ref{cohomology-lemma-pseudo-coherent-pullback}.", "It only remains to check (5). To see this", "set $W = b^{-1}(U) = c^{-1}(V)$ and $W' = (b')^{-1}(U) = (c')^{-1}(V')$ ", "and consider the cartesian square", "$$", "\\xymatrix{", "W \\ar[d]_{(b, c)} \\ar[r] & W' \\ar[d]^{(b', c')} \\\\", "X \\times_A V \\ar[r] & X' \\times_{A'} V'", "}", "$$", "By Lemma \\ref{lemma-case-of-tor-independence} the schemes", "$X \\times_A V$ and $W'$ are Tor independent over $X' \\times_{A'} V'$.", "Hence the derived pullback of", "$(b', c')_*\\mathcal{O}_{W'}$ to $X \\times_A V$", "is $(b, c)_*\\mathcal{O}_W$ by", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-compare-base-change}.", "This also uses that $R(b', c')_*\\mathcal{O}_{Z'} = (b', c')_*\\mathcal{O}_{Z'}$", "because $(b', c')$ is a closed immersion and simiarly for", "$(b, c)_*\\mathcal{O}_Z$.", "Since $E'|_{U' \\times_{A'} V'} =", "(b', c')_*\\mathcal{O}_{W'}$ we obtain", "$E|_{U \\times_A V} = (b, c)_*\\mathcal{O}_W$", "and (5) holds.", "This reduces us to the situation described in the next", "paragraph.", "\\medskip\\noindent", "Assume $A$ is of finite type over $\\mathbf{Z}$.", "Choose an affine open neighbourhood $U \\subset X$ of $x$.", "Then $U$ is of finite type over $A$.", "Choose a closed immersion $U \\to \\mathbf{A}^n_A$ and denote", "$j : U \\to \\mathbf{P}^n_A$ the immersion we get by composing", "with the open immersion $\\mathbf{A}^n_A \\to \\mathbf{P}^n_A$.", "Let $Z$ be the scheme theoretic closure of", "$$", "(\\text{id}_U, j) : U \\longrightarrow X \\times_A \\mathbf{P}^n_A", "$$", "Since the projection $X \\times \\mathbf{P}^n \\to X$ is separated,", "we conclude from Morphisms, Lemma", "\\ref{morphisms-lemma-scheme-theoretic-image-of-partial-section}", "that $b : Z \\to X$ is an isomorphism over $U$.", "Let $z \\in Z$ be the unique point lying over $x$.", "\\medskip\\noindent", "Let $Y \\subset \\mathbf{P}^n_A$ be the scheme theoretic", "closure of $j$. Then it is clear that $Z \\subset X \\times_A Y$", "is the scheme theoretic closure of", "$(\\text{id}_U, j) : U \\to X \\times_A Y$.", "As $X$ is separated, the morphism", "$X \\times_A Y \\to Y$ is separated as well.", "Hence we see that $Z \\to Y$ is an isomorphism over", "the open subscheme $j(U) \\subset Y$ by the same lemma we used above.", "Choose $V \\subset \\mathbf{P}^n_A$ open with $V \\cap Y = j(U)$.", "Then we see that (3) and (4) hold.", "\\medskip\\noindent", "Because $A$ is Noetherian we see that $X$ and $X \\times_A \\mathbf{P}^n_A$", "are Noetherian schemes. Hence we can take $E = (b, c)_*\\mathcal{O}_Z$", "in this case, see Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-identify-pseudo-coherent-noetherian}.", "This finishes the proof." ], "refs": [ "limits-lemma-descend-finite-presentation", "limits-lemma-descend-separated-finite-presentation", "cohomology-lemma-pseudo-coherent-pullback", "more-morphisms-lemma-case-of-tor-independence", "perfect-lemma-compare-base-change", "perfect-lemma-identify-pseudo-coherent-noetherian" ], "ref_ids": [ 15077, 15061, 2206, 14057, 7028, 6976 ] } ], "ref_ids": [] }, { "id": 14059, "type": "theorem", "label": "more-morphisms-lemma-compute-Fourier-Mukai-for-derived-chow", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-compute-Fourier-Mukai-for-derived-chow", "contents": [ "Let $A$, $x \\in X$, and", "$U, n, V, Z, z, E$ be as in Lemma \\ref{lemma-derived-chow}.", "For any $K \\in D_\\QCoh(\\mathcal{O}_X)$ we have", "$$", "Rq_*(Lp^*K \\otimes^\\mathbf{L} E)|_V = R(U \\to V)_*K|_U", "$$", "where $p : X \\times_A \\mathbf{P}^n_A \\to X$ and", "$q : X \\times_A \\mathbf{P}^n_A \\to \\mathbf{P}^n_A$ are", "the projections and where the morphism $U \\to V$ is", "the finitely presented closed immersion $c \\circ (b|_U)^{-1}$." ], "refs": [ "more-morphisms-lemma-derived-chow" ], "proofs": [ { "contents": [ "Since $b^{-1}(U) = c^{-1}(V)$ and since $c$ is a closed immersion", "over $V$, we see that $c \\circ (b|_U)^{-1}$ is a closed immersion.", "It is of finite presentation because $U$ and $V$ are of finite", "presentation over $A$, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-permanence}.", "First we have", "$$", "Rq_*(Lp^*K \\otimes^\\mathbf{L} E)|_V =", "Rq'_*\\left((Lp^*K \\otimes^\\mathbf{L} E)|_{X \\times_A V}\\right)", "$$", "where $q' : X \\times_A V \\to V$ is the projection because", "formation of total direct image commutes with localization.", "Set $W = b^{-1}(U) = c^{-1}(V)$ and denote $i : W \\to X \\times_A V$", "the closed immersion $i = (b, c)|_W$. Then", "$$", "Rq'_*\\left((Lp^*K \\otimes^\\mathbf{L} E)|_{X \\times_A V}\\right) =", "Rq'_*(Lp^*K|_{X \\times_A V} \\otimes^\\mathbf{L} i_*\\mathcal{O}_W)", "$$", "by property (5). Since $i$ is a closed immersion we have", "$i_*\\mathcal{O}_W = Ri_*\\mathcal{O}_W$.", "Using", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-cohomology-base-change}", "we can rewrite this as", "$$", "Rq'_* Ri_* Li^* Lp^*K|_{X \\times_A V} =", "R(q' \\circ i)_* Lb^*K|_W =", "R(U \\to V)_* K|_U", "$$", "which is what we want." ], "refs": [ "morphisms-lemma-finite-presentation-permanence", "perfect-lemma-cohomology-base-change" ], "ref_ids": [ 5247, 7025 ] } ], "ref_ids": [ 14058 ] }, { "id": 14060, "type": "theorem", "label": "more-morphisms-lemma-characterize-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-characterize-pseudo-coherent", "contents": [ "Let $A$ be a ring. Let $X$ be a scheme separated and", "of finite presentation over $A$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$.", "If $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent", "in $D(A)$ for every pseudo-coherent $E$ in $D(\\mathcal{O}_X)$,", "then $K$ is pseudo-coherent relative to $A$." ], "refs": [], "proofs": [ { "contents": [ "Assume $K \\in D_\\QCoh(\\mathcal{O}_X)$ and", "$R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent", "in $D(A)$ for every pseudo-coherent $E$ in $D(\\mathcal{O}_X)$.", "Let $x \\in X$. We will show that $K$ is pseudo-coherent relative to $A$", "in a neighbourhood of $x$ and this will prove the lemma.", "\\medskip\\noindent", "Choose $U, n, V, Z, z, E$ as in Lemma \\ref{lemma-derived-chow}.", "Denote $p : X \\times \\mathbf{P}^n \\to X$ and", "$q : X \\times \\mathbf{P}^n \\to \\mathbf{P}^n_A$", "the projections.", "Then for any $i \\in \\mathbf{Z}$ we have", "\\begin{align*}", "& R\\Gamma(\\mathbf{P}^n_A,", "Rq_*(Lp^*K \\otimes^\\mathbf{L} E)", "\\otimes^\\mathbf{L}", "\\mathcal{O}_{\\mathbf{P}^n_A}(i)) \\\\", "& =", "R\\Gamma(X \\times \\mathbf{P}^n,", "Lp^*K \\otimes^\\mathbf{L} E \\otimes^\\mathbf{L}", "Lq^*\\mathcal{O}_{\\mathbf{P}^n_A}(i)) \\\\", "& =", "R\\Gamma(X,", "K \\otimes^\\mathbf{L} Rq_*(E \\otimes^\\mathbf{L}", "Lq^*\\mathcal{O}_{\\mathbf{P}^n_A}(i)))", "\\end{align*}", "by ", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-cohomology-base-change}.", "By", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-flat-proper-pseudo-coherent-direct-image-general}", "the complex $Rq_*(E \\otimes^\\mathbf{L} Lq^*\\mathcal{O}_{\\mathbf{P}^n_A}(i))$", "is pseudo-coherent on $X$. Hence the assumption tells us the expression", "in the displayed formula is a pseudo-coherent object of $D(A)$.", "By", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-pseudo-coherent-on-projective-space}", "we conclude that $Rq_*(Lp^*K \\otimes^\\mathbf{L} E)$ is", "pseudo-coherent on $\\mathbf{P}^n_A$.", "By Lemma \\ref{lemma-compute-Fourier-Mukai-for-derived-chow}", "we have", "$$", "Rq_*(Lp^*K \\otimes^\\mathbf{L} E)|_{X \\times_A V} =", "R(U \\to V)_*K|_U", "$$", "Since $U \\to V$ is a closed immersion into an open subscheme of", "$\\mathbf{P}^n_A$ this means $K|_U$ is pseudo-coherent relative to $A$", "by Lemma \\ref{lemma-check-relative-pseudo-coherence-on-charts}." ], "refs": [ "more-morphisms-lemma-derived-chow", "perfect-lemma-cohomology-base-change", "perfect-lemma-flat-proper-pseudo-coherent-direct-image-general", "perfect-lemma-pseudo-coherent-on-projective-space", "more-morphisms-lemma-compute-Fourier-Mukai-for-derived-chow", "more-morphisms-lemma-check-relative-pseudo-coherence-on-charts" ], "ref_ids": [ 14058, 7025, 7055, 7075, 14059, 13974 ] } ], "ref_ids": [] }, { "id": 14061, "type": "theorem", "label": "more-morphisms-lemma-characterize-pseudo-coh-improved", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-characterize-pseudo-coh-improved", "contents": [ "Let $A$ be a ring. Let $X$ be a scheme separated and", "of finite presentation over $A$. Let ", "$K \\in D_\\QCoh(\\mathcal{O}_X).$ If ", "$R \\Gamma (X, E \\otimes ^{\\mathbf{L}} K)$ is", "pseudo-coherent in $D(A)$ for every perfect ", "$E \\in D(\\mathcal{O}_X)$, then $K$ is pseudo-coherent", "relative to $A$." ], "refs": [], "proofs": [ { "contents": [ "In view of Lemma \\ref{lemma-characterize-pseudo-coherent}, it suffices", "to show $R \\Gamma (X, E \\otimes ^{\\mathbf{L}} K)$ is", "pseudo-coherent in $D(A)$ for every pseudo-coherent ", "$E \\in D(\\mathcal{O}_X)$. By Derived Categories of Schemes,", "Proposition \\ref{perfect-proposition-detecting-bounded-above}", "it follows that $K \\in D^-_\\QCoh (\\mathcal{O}_X)$. Now the", "result follows by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-perfect-enough}." ], "refs": [ "more-morphisms-lemma-characterize-pseudo-coherent", "perfect-proposition-detecting-bounded-above", "perfect-lemma-perfect-enough" ], "ref_ids": [ 14060, 7113, 7076 ] } ], "ref_ids": [] }, { "id": 14062, "type": "theorem", "label": "more-morphisms-lemma-characterize-relatively-perfect", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-characterize-relatively-perfect", "contents": [ "Let $A$ be a ring. Let $X$ be a scheme separated, of", "finite presentation, and flat over $A$. Let ", "$K \\in D_\\QCoh(\\mathcal{O}_X).$ If ", "$R \\Gamma (X, E \\otimes^\\mathbf{L} K)$ is perfect in", "$D(A)$ for every perfect $E \\in D(\\mathcal{O}_X)$, then $K$ is", "$\\Spec(A)$-perfect." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-characterize-pseudo-coh-improved},", "$K$ is pseudo-coherent relative to $A$. By Lemma", "\\ref{lemma-check-relative-pseudo-coherence-on-charts},", "$K$ is pseudo-coherent in $D( \\mathcal{O}_X)$.", "By Derived Categories of Schemes, Proposition", "\\ref{perfect-proposition-detecting-bounded-below}", "we see that $K$ is in $D^-(\\mathcal{O}_X)$.", "Let $\\mathfrak{p}$ be a prime ideal of $A$ and denote", "$i : Y \\to X$ the inclusion of the scheme theoretic", "fibre over $\\mathfrak{p}$, i.e., $Y$ is a scheme over $\\kappa(\\mathfrak p)$.", "By Derived Categories of Schemes, Lemma \\ref{perfect-lemma-bounded-on-fibres},", "we will be done if we can show $Li^*(K)$ is bounded below. ", "Let $G \\in D_{perf} (\\mathcal{O}_X)$ be a perfect", "complex which generates $D_\\QCoh (\\mathcal{O}_X)$,", "see Derived Categories of Schemes, Theorem", "\\ref{perfect-theorem-bondal-van-den-Bergh}.", "We have", "\\begin{align*}", "R\\Hom _{\\mathcal{O}_Y}(Li^*(G), Li^*(K))", "& =", "R\\Gamma(Y, Li^*(G ^\\vee \\otimes ^\\mathbf{L} K)) \\\\", "& =", "R\\Gamma(X, G^\\vee \\otimes ^{\\mathbf{L}} K)", "\\otimes^\\mathbf{L}_A \\kappa(\\mathfrak{p})", "\\end{align*}", "The first equality uses that $Li^*$ preserves perfect objects and duals", "and Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}; we omit", "some details. The second equality follows from", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-compare-base-change}", "as $X$ is flat over $A$. It follows from our hypothesis that this is a", "perfect object of $D(\\kappa(\\mathfrak{p}))$. The object", "$Li^*(G) \\in D_{perf}(\\mathcal{O}_Y)$ generates $D_\\QCoh(\\mathcal{O}_Y)$ by", "Derived Categories of Schemes, Remark \\ref{perfect-remark-pullback-generator}.", "Hence Derived Categories of Schemes, Proposition", "\\ref{perfect-proposition-detecting-bounded-below}", "now implies that $Li^*(K)$ is bounded below and we win." ], "refs": [ "more-morphisms-lemma-characterize-pseudo-coh-improved", "more-morphisms-lemma-check-relative-pseudo-coherence-on-charts", "perfect-proposition-detecting-bounded-below", "perfect-lemma-bounded-on-fibres", "perfect-theorem-bondal-van-den-Bergh", "cohomology-lemma-dual-perfect-complex", "perfect-lemma-compare-base-change", "perfect-remark-pullback-generator", "perfect-proposition-detecting-bounded-below" ], "ref_ids": [ 14061, 13974, 7114, 7085, 6935, 2233, 7028, 7128, 7114 ] } ], "ref_ids": [] }, { "id": 14063, "type": "theorem", "label": "more-morphisms-lemma-relative-pseudo-coherent-descends-fppf", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-relative-pseudo-coherent-descends-fppf", "contents": [ "Let $X \\to S$ be locally of finite type.", "Let $\\{f_i : X_i \\to X\\}$ be an fppf covering of schemes.", "Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$.", "Then $E$ is $m$-pseudo-coherent relative to $S$", "if and only if each $Lf_i^*E$ is $m$-pseudo-coherent relative to $S$." ], "refs": [], "proofs": [ { "contents": [ "Assume $E$ is $m$-pseudo-coherent relative to $S$.", "The morphisms $f_i$ are pseudo-coherent by", "Lemma \\ref{lemma-flat-finite-presentation-pseudo-coherent}.", "Hence $Lf_i^*E$ is $m$-pseudo-coherent relative to $S$", "by Lemma \\ref{lemma-pull-relative-pseudo-coherent}.", "\\medskip\\noindent", "Conversely, assume that $Lf_i^*E$ is $m$-pseudo-coherent relative to $S$", "for each $i$. Pick $S = \\bigcup U_j$, $W_j \\to U_j$,", "$W_j = \\bigcup W_{j, k}$, $T_{j, k} \\to W_{j, k}$, and", "morphisms $\\alpha_{j, k} : T_{j, k} \\to X_{i(j, k)}$ over $S$ as in", "Lemma \\ref{lemma-dominate-fppf}.", "Since the morphism $T_{j, K} \\to S$ is flat and of finite presentation,", "we see that $\\alpha_{j, k}$ is pseudo-coherent by", "Lemma \\ref{lemma-permanence-pseudo-coherent}.", "Hence", "$$", "L\\alpha_{j, k}^*Lf_{i(j, k)}^*E = L(T_{i, k} \\to S)^*E", "$$", "is $m$-pseudo-coherent relative to $S$ by", "Lemma \\ref{lemma-pull-relative-pseudo-coherent}.", "Now we want to descend this property", "through the coverings $\\{T_{j, k} \\to W_{j, k}\\}$,", "$W_j = \\bigcup W_{j, k}$, $\\{W_j \\to U_j\\}$, and $S = \\bigcup U_j$.", "Since for Zariski coverings the result is true", "(by the definition of $m$-pseudo-coherence relative to $S$),", "this means we may assume we have a single surjective", "finite locally free morphism $\\pi : Y \\to X$", "such that $L\\pi^*E$ is pseudo-coherent relative to $S$.", "In this case $R\\pi_*L\\pi^*E$ is pseudo-coherent relative to $S$", "by Lemma \\ref{lemma-finite-morphism-relative-pseudo-coherence}", "(this is the first time we use that $E$ has quasi-coherent cohomology sheaves).", "We have", "$R\\pi_*L\\pi^*E = E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\pi_*\\mathcal{O}_Y$", "for example by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-cohomology-base-change}", "and locally on $X$ the map $\\mathcal{O}_X \\to \\pi_*\\mathcal{O}_Y$", "is the inclusion of a direct summand. Hence we conclude by", "Lemma \\ref{lemma-summands-relative-pseudo-coherent}." ], "refs": [ "more-morphisms-lemma-flat-finite-presentation-pseudo-coherent", "more-morphisms-lemma-pull-relative-pseudo-coherent", "more-morphisms-lemma-dominate-fppf", "more-morphisms-lemma-permanence-pseudo-coherent", "more-morphisms-lemma-pull-relative-pseudo-coherent", "more-morphisms-lemma-finite-morphism-relative-pseudo-coherence", "perfect-lemma-cohomology-base-change", "more-morphisms-lemma-summands-relative-pseudo-coherent" ], "ref_ids": [ 13979, 13972, 13922, 13980, 13972, 13965, 7025, 13968 ] } ], "ref_ids": [] }, { "id": 14064, "type": "theorem", "label": "more-morphisms-lemma-relative-pseudo-coherent-post-compose", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-relative-pseudo-coherent-post-compose", "contents": [ "Let $X \\to T \\to S$ be morphisms of schemes. Assume $T \\to S$", "is flat and locally of finite presentation and $X \\to T$", "locally of finite type. Let $E \\in D(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$.", "Then $E$ is $m$-pseudo-coherent relative to $S$", "if and only if $E$ is $m$-pseudo-coherent relative to $T$." ], "refs": [], "proofs": [ { "contents": [ "Locally on $X$ we can choose a closed immersion $i : X \\to \\mathbf{A}^n_T$.", "Then $\\mathbf{A}^n_T \\to S$ is flat and locally of finite presentation.", "Thus we may", "apply Lemma \\ref{lemma-composition-relative-pseudo-coherent}", "to see the equivalence holds." ], "refs": [ "more-morphisms-lemma-composition-relative-pseudo-coherent" ], "ref_ids": [ 13973 ] } ], "ref_ids": [] }, { "id": 14065, "type": "theorem", "label": "more-morphisms-lemma-relative-pseudo-coherent-descends-fppf-base", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-relative-pseudo-coherent-descends-fppf-base", "contents": [ "Let $f : X \\to S$ be locally of finite type.", "Let $\\{S_i \\to S\\}$ be an fppf covering of schemes.", "Denote $f_i : X_i \\to S_i$ the base change of $f$", "and $g_i : X_i \\to X$ the projection.", "Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$.", "Then $E$ is $m$-pseudo-coherent relative to $S$", "if and only if each $Lg_i^*E$ is $m$-pseudo-coherent relative to $S_i$." ], "refs": [], "proofs": [ { "contents": [ "This follows formally from", "Lemmas \\ref{lemma-relative-pseudo-coherent-descends-fppf} and", "\\ref{lemma-relative-pseudo-coherent-post-compose}.", "Namely, if $E$ is $m$-pseudo-coherent relative to $S$,", "then $Lg_i^*E$ is $m$-pseudo-coherent relative to $S$ (by the first lemma),", "hence $Lg_i^*E$ is $m$-pseudo-coherent relative to $S_i$ (by the second).", "Conversely, if", "$Lg_i^*E$ is $m$-pseudo-coherent relative to $S_i$, then", "$Lg_i^*E$ is $m$-pseudo-coherent relative to $S$ (by the second lemma),", "hence $E$ is $m$-pseudo-coherent relative to $S$ (by the first lemma)." ], "refs": [ "more-morphisms-lemma-relative-pseudo-coherent-descends-fppf", "more-morphisms-lemma-relative-pseudo-coherent-post-compose" ], "ref_ids": [ 14063, 14064 ] } ], "ref_ids": [] }, { "id": 14066, "type": "theorem", "label": "more-morphisms-lemma-thickening-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-thickening-pseudo-coherent", "contents": [ "Let $i : X \\to X'$ be a finite order thickening of schemes. Let", "$K' \\in D(\\mathcal{O}_{X'})$ be an object such that", "$K = Li^*K'$ is pseudo-coherent. Then $K'$ is pseudo-coherent." ], "refs": [], "proofs": [ { "contents": [ "We first prove $K'$ has quasi-coherent cohomology sheaves.", "To do this, we may reduce to the case of a first order thickening, see", "Section \\ref{section-thickenings}. Let $\\mathcal{I} \\subset \\mathcal{O}_{X'}$", "be the quasi-coherent sheaf of ideals cutting out $X$.", "Tensoring the short exact sequence", "$$", "0 \\to \\mathcal{I} \\to \\mathcal{O}_{X'} \\to i_*\\mathcal{O}_X \\to 0", "$$", "with $K'$ we obtain a distinguished triangle", "$$", "K' \\otimes_{\\mathcal{O}_{X'}}^\\mathbf{L} \\mathcal{I}", "\\to K' \\to", "K' \\otimes_{\\mathcal{O}_{X'}}^\\mathbf{L} i_*\\mathcal{O}_X", "\\to", "(K' \\otimes_{\\mathcal{O}_{X'}}^\\mathbf{L} \\mathcal{I})[1]", "$$", "Since $i_* = Ri_*$ and since we may view $\\mathcal{I}$", "as a quasi-coherent $\\mathcal{O}_X$-module (as we have a first", "order thickening) we may rewrite this as", "$$", "i_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{I})", "\\to K' \\to", "i_*K \\to", "i_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{I})[1]", "$$", "Please use Cohomology, Lemma", "\\ref{cohomology-lemma-projection-formula-closed-immersion}", "to identify the terms. Since $K$ is in", "$D_\\QCoh(\\mathcal{O}_X)$ we conclude that", "$K'$ is in $D_\\QCoh(\\mathcal{O}_{X'})$; this uses", "Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-pseudo-coherent},", "\\ref{perfect-lemma-quasi-coherence-tensor-product}, and", "\\ref{perfect-lemma-quasi-coherence-direct-image}.", "\\medskip\\noindent", "Assume $K'$ is in $D_\\QCoh(\\mathcal{O}_{X'})$.", "The question is local on $X'$ hence we may assume $X'$ is affine.", "Say $X' = \\Spec(A')$ and $X = \\Spec(A)$ with $A = A'/I$ and $I$ nilpotent.", "Then $K'$ comes from an object $M' \\in D(A')$, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded}.", "Thus $M = M' \\otimes_{A'}^\\mathbf{L} A$ is a pseudo-coherent", "object of $D(A)$ by Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-pseudo-coherent-affine} and our assumption on $K$.", "Hence we can represent $M$", "by a bounded above complex of finite free $A$-modules $E^\\bullet$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-pseudo-coherent}.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-complex-projectives}", "we conclude that $M'$ is pseudo-coherent as desired." ], "refs": [ "cohomology-lemma-projection-formula-closed-immersion", "perfect-lemma-pseudo-coherent", "perfect-lemma-quasi-coherence-tensor-product", "perfect-lemma-quasi-coherence-direct-image", "perfect-lemma-affine-compare-bounded", "perfect-lemma-pseudo-coherent-affine", "more-algebra-lemma-pseudo-coherent", "more-algebra-lemma-lift-complex-projectives" ], "ref_ids": [ 2245, 6974, 6945, 6946, 6941, 6975, 10148, 10229 ] } ], "ref_ids": [] }, { "id": 14067, "type": "theorem", "label": "more-morphisms-lemma-thickening-relatively-perfect", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-thickening-relatively-perfect", "contents": [ "Consider a cartesian diagram", "$$", "\\xymatrix{", "X \\ar[r]_i \\ar[d]_f & X' \\ar[d]^{f'} \\\\", "Y \\ar[r]^j & Y'", "}", "$$", "of schemes. Assume $X' \\to Y'$ is flat and locally", "of finite presentation and $Y \\to Y'$ is a finite order thickening.", "Let $E' \\in D(\\mathcal{O}_{X'})$. If $E = Li^*(E')$ is $Y$-perfect,", "then $E'$ is $Y'$-perfect." ], "refs": [], "proofs": [ { "contents": [ "Recall that being $Y$-perfect for $E$ means $E$ is", "pseudo-coherent and locally has finite tor dimension as a complex", "of $f^{-1}\\mathcal{O}_Y$-modules", "(Derived Categories of Schemes,", "Definition \\ref{perfect-definition-relatively-perfect}).", "By Lemma \\ref{lemma-thickening-pseudo-coherent}", "we find that $E'$ is pseudo-coherent.", "In particular, $E'$ is in $D_\\QCoh(\\mathcal{O}_{X'})$, see", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-pseudo-coherent}.", "To prove that $E'$ locally has finite tor dimension", "we may work locally on $X'$. Hence we may assume", "$X'$, $S'$, $X$, $S$ are affine, say given by", "rings $A'$, $R'$, $A$, $R$.", "Then we reduce to the commutative algebra version by", "Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-affine-locally-rel-perfect}.", "The commutative algebra version in More on Algebra, Lemma", "\\ref{more-algebra-lemma-thickening-relatively-perfect}." ], "refs": [ "perfect-definition-relatively-perfect", "more-morphisms-lemma-thickening-pseudo-coherent", "perfect-lemma-pseudo-coherent", "perfect-lemma-affine-locally-rel-perfect", "more-algebra-lemma-thickening-relatively-perfect" ], "ref_ids": [ 7119, 14066, 6974, 7077, 10294 ] } ], "ref_ids": [] }, { "id": 14068, "type": "theorem", "label": "more-morphisms-lemma-henselian-relatively-perfect", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-henselian-relatively-perfect", "contents": [ "Let $(R, I)$ be a pair consisting of a ring and an ideal $I$", "contained in the Jacobson radical. Set $S = \\Spec(R)$ and $S_0 = \\Spec(R/I)$.", "Let $f : X \\to S$ be proper, flat, and of finite presentation.", "Denote $X_0 = S_0 \\times_S X$. Let $E \\in D(\\mathcal{O}_X)$", "be pseudo-coherent. If the derived restriction $E_0$ of $E$", "to $X_0$ is $S_0$-perfect, then $E$ is $S$-perfect." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite affine open covering $X = U_1 \\cup \\ldots \\cup U_n$.", "For each $i$ we can choose a closed immersion $U_i \\to \\mathbf{A}^{d_i}_S$.", "Set $U_{i, 0} = S_0 \\times_S U_i$.", "For each $i$ the complex $E_0|_{U_{i, 0}}$ has tor amplitude", "in $[a_i, b_i]$ for some $a_i, b_i \\in \\mathbf{Z}$.", "Let $x \\in X$ be a point.", "We will show that the tor amplitude of", "$E_x$ over $R$ is in $[a_i - d_i, b_i]$ for some $i$.", "This will finish the proof as the tor amplitude can be", "read off from the stalks by", "Cohomology, Lemma \\ref{cohomology-lemma-tor-amplitude-stalk}.", "\\medskip\\noindent", "Since $f$ is proper $f(\\overline{\\{x\\}})$ is a closed subset of $S$.", "Since $I$ is contained in the Jacobson radical, we see that", "$f(\\overline{\\{x\\}})$ meeting the closed subset $S_0 \\subset S$.", "Hence there is a specialization $x \\leadsto x_0$ with $x_0 \\in X_0$.", "Pick an $i$ with $x_0 \\in U_i$, so $x_0 \\in U_{i, 0}$.", "We will fix $i$ for the rest of the proof.", "Write $U_i = \\Spec(A)$. Then $A$ is a flat, finitely presented", "$R$-algebra which is a quotient of a polynomial $R$-algebra in", "$d_i$-variables. The restriction $E|_{U_i}$ corresponds", "(by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded} and", "\\ref{perfect-lemma-pseudo-coherent-affine})", "to a pseudo-coherent object $K$ of $D(A)$.", "Observe that $E_0$ corresponds to $K \\otimes_A^\\mathbf{L} A/IA$.", "Let $\\mathfrak q \\subset \\mathfrak q_0 \\subset A$ be the prime", "ideals corresponding to $x \\leadsto x_0$.", "Then $E_x = K_{\\mathfrak q}$ and $K_{\\mathfrak q}$", "is a localization of $K_{\\mathfrak q_0}$. Hence it suffices", "to show that $K_{\\mathfrak q_0}$ has tor amplitude in", "$[a_i - d_i, b_i]$ as a complex of $R$-modules.", "Let $I \\subset \\mathfrak p_0 \\subset R$ be the prime", "ideal corresponding to $f(x_0)$.", "Then we have", "\\begin{align*}", "K \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p_0)", "& =", "(K \\otimes_R^\\mathbf{L} R/I) \\otimes_{R/I}^\\mathbf{L}", "\\kappa(\\mathfrak p_0) \\\\", "& =", "(K \\otimes_A^\\mathbf{L} A/IA) \\otimes_{R/I}^\\mathbf{L} \\kappa(\\mathfrak p_0)", "\\end{align*}", "the second equality because $R \\to A$ is flat.", "By our choice of $a_i, b_i$ this complex has cohomology", "only in degrees in the interval $[a_i, b_i]$.", "Thus we may finally apply", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-from-fibre-relatively-perfect}", "to $R \\to A$, $\\mathfrak q_0$, $\\mathfrak p_0$ and $K$", "to conclude." ], "refs": [ "cohomology-lemma-tor-amplitude-stalk", "perfect-lemma-affine-compare-bounded", "perfect-lemma-pseudo-coherent-affine", "more-algebra-lemma-lift-from-fibre-relatively-perfect" ], "ref_ids": [ 2217, 6941, 6975, 10295 ] } ], "ref_ids": [] }, { "id": 14069, "type": "theorem", "label": "more-morphisms-lemma-check-h1-fibre-zero", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-check-h1-fibre-zero", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes. Let $y \\in Y$", "be a point with $\\dim(X_y) \\leq 1$. If", "\\begin{enumerate}", "\\item $R^1f_*\\mathcal{O}_X = 0$, or more generally", "\\item there is a morphism $g : Y' \\to Y$ such that $y$ is in the image", "of $g$ and such that $R'f'_*\\mathcal{O}_{X'} = 0$ where $f' : X' \\to Y'$", "is the base change of $f$ by $g$.", "\\end{enumerate}", "Then $H^1(X_y, \\mathcal{O}_{X_y}) = 0$." ], "refs": [], "proofs": [ { "contents": [ "To prove the lemma we may replace $Y$ by an open neighbourhood of $y$.", "Thus we may assume $Y$ is affine", "and that all fibres of $f$ have dimension $\\leq 1$, see", "Morphisms, Lemma \\ref{morphisms-lemma-openness-bounded-dimension-fibres}.", "In this case $R^1f_*\\mathcal{O}_X$ is a quasi-coherent $\\mathcal{O}_Y$-module", "of finite type and its formation commutes with arbitrary base change, see", "Limits, Lemmas \\ref{limits-lemma-proper-top-cohomology-finite-type} and", "\\ref{limits-lemma-higher-direct-images-zero-above-dimension-fibre}.", "The lemma follows immediately." ], "refs": [ "morphisms-lemma-openness-bounded-dimension-fibres", "limits-lemma-proper-top-cohomology-finite-type", "limits-lemma-higher-direct-images-zero-above-dimension-fibre" ], "ref_ids": [ 5280, 15110, 15109 ] } ], "ref_ids": [] }, { "id": 14070, "type": "theorem", "label": "more-morphisms-lemma-h1-fibre-zero", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-h1-fibre-zero", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes. Let $y \\in Y$", "be a point with $\\dim(X_y) \\leq 1$ and $H^1(X_y, \\mathcal{O}_{X_y}) = 0$.", "Then there is an open neighbourhood $V \\subset Y$ of $y$ such that", "$R^1f_*\\mathcal{O}_X|_V = 0$", "and the same is true after base change by any $Y' \\to V$." ], "refs": [], "proofs": [ { "contents": [ "To prove the lemma we may replace $Y$ by an open neighbourhood of $y$.", "Thus we may assume $Y$ is affine and that all fibres of $f$ have", "dimension $\\leq 1$, see", "Morphisms, Lemma \\ref{morphisms-lemma-openness-bounded-dimension-fibres}.", "In this case $R^1f_*\\mathcal{O}_X$ is a quasi-coherent $\\mathcal{O}_Y$-module", "of finite type and its formation commutes with arbitrary base change, see", "Limits, Lemmas \\ref{limits-lemma-proper-top-cohomology-finite-type} and", "\\ref{limits-lemma-higher-direct-images-zero-above-dimension-fibre}.", "Say $Y = \\Spec(A)$, $y$ corresponds to the prime $\\mathfrak p \\subset A$, and", "$R^1f_*\\mathcal{O}_X$ corresponds to the finite $A$-module $M$.", "Then $H^1(X_y, \\mathcal{O}_{X_y}) = 0$ means that", "$\\mathfrak pM_\\mathfrak p = M_\\mathfrak p$ by the statement", "on base change. By Nakayama's lemma", "we conclude $M_\\mathfrak p = 0$. Since $M$ is finite, we find", "an $f \\in A$, $f \\not \\in \\mathfrak p$ such that $M_f = 0$.", "Thus taking $V$ the principal open $D(f)$ we obtain the desired result." ], "refs": [ "morphisms-lemma-openness-bounded-dimension-fibres", "limits-lemma-proper-top-cohomology-finite-type", "limits-lemma-higher-direct-images-zero-above-dimension-fibre" ], "ref_ids": [ 5280, 15110, 15109 ] } ], "ref_ids": [] }, { "id": 14071, "type": "theorem", "label": "more-morphisms-lemma-globally-generated-vanishing", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-globally-generated-vanishing", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes such", "that $\\dim(X_y) \\leq 1$ and $H^1(X_y, \\mathcal{O}_{X_y}) = 0$", "for all $y \\in Y$. Let $\\mathcal{F}$ be quasi-coherent on $X$. Then", "\\begin{enumerate}", "\\item $R^pf_*\\mathcal{F} = 0$ for $p > 1$, and", "\\item $R^1f_*\\mathcal{F} = 0$ if there is a surjection", "$f^*\\mathcal{G} \\to \\mathcal{F}$ with $\\mathcal{G}$ quasi-coherent", "on $Y$.", "\\end{enumerate}", "If $Y$ is affine, then we also have", "\\begin{enumerate}", "\\item[(3)] $H^p(X, \\mathcal{F}) = 0$ for $p \\not \\in \\{0, 1\\}$, and", "\\item[(4)] $H^1(X, \\mathcal{F}) = 0$ if $\\mathcal{F}$ is globally generated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The vanishing in (1) is Limits, Lemma", "\\ref{limits-lemma-higher-direct-images-zero-above-dimension-fibre}.", "To prove (2) we may work locally on $Y$ and assume $Y$ is affine.", "Then $R^1f_*\\mathcal{F}$ is the quasi-coherent module on $Y$", "associated to the module $H^1(X, \\mathcal{F})$.", "Here we use that $Y$ is affine, quasi-coherence of higher direct", "images (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images}), and", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images-application}.", "Since $Y$ is affine, the quasi-coherent module $\\mathcal{G}$", "is globally generated, and hence so is $f^*\\mathcal{G}$", "and $\\mathcal{F}$.", "In this way we see that (4) implies (2).", "Part (3) follows from (1) as well as the remarks on", "quasi-coherence of direct images just made. Thus", "all that remains is the prove (4).", "If $\\mathcal{F}$ is globally generated, then there is a surjection", "$\\bigoplus_{i \\in I} \\mathcal{O}_X \\to \\mathcal{F}$. By part (1)", "and the long exact sequence of cohomology this", "induces a surjection on $H^1$. Since $H^1(X, \\mathcal{O}_X) = 0$", "because $R^1f_*\\mathcal{O}_X = 0$ by", "Lemma \\ref{lemma-h1-fibre-zero}, and", "since $H^1(X, -)$ commutes with direct sums", "(Cohomology, Lemma \\ref{cohomology-lemma-quasi-separated-cohomology-colimit})", "we conclude." ], "refs": [ "limits-lemma-higher-direct-images-zero-above-dimension-fibre", "coherent-lemma-quasi-coherence-higher-direct-images", "coherent-lemma-quasi-coherence-higher-direct-images-application", "more-morphisms-lemma-h1-fibre-zero", "cohomology-lemma-quasi-separated-cohomology-colimit" ], "ref_ids": [ 15109, 3295, 3296, 14070, 2082 ] } ], "ref_ids": [] }, { "id": 14072, "type": "theorem", "label": "more-morphisms-lemma-h1-fibre-zero-check-h0-kappa", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-h1-fibre-zero-check-h0-kappa", "contents": [ "Let $f : X \\to Y$ be a proper morphism of schemes. Assume", "\\begin{enumerate}", "\\item for all $y \\in Y$ we have $\\dim(X_y) \\leq 1$ and", "$H^1(X_y, \\mathcal{O}_{X_y}) = 0$, and", "\\item $\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$ is surjective.", "\\end{enumerate}", "Then $\\mathcal{O}_{Y'} \\to f'_*\\mathcal{O}_{X'}$ is surjective", "for any base change $f' : X' \\to Y'$ of $f$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $Y$ and $Y'$ affine. Then we can choose a closed", "immersion $Y' \\to Y''$ with $Y'' \\to Y$ a flat morphism of affines.", "By flat base change", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology})", "we see that the result holds for $X'' \\to Y''$.", "Thus we may assume $Y'$ is a closed subscheme of $Y$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_Y$ be the ideal cutting out $Y'$.", "Then there is a short exact sequence", "$$", "0 \\to \\mathcal{I}\\mathcal{O}_X \\to", "\\mathcal{O}_X \\to \\mathcal{O}_{X'} \\to 0", "$$", "where we view $\\mathcal{O}_{X'}$ as a quasi-coherent module on $X$.", "By Lemma \\ref{lemma-globally-generated-vanishing}", "we have $H^1(X, \\mathcal{I}\\mathcal{O}_X) = 0$.", "It follows that", "$$", "H^0(Y, \\mathcal{O}_Y) \\to", "H^0(Y, f_*\\mathcal{O}_X) = H^0(X, \\mathcal{O}_X) \\to", "H^0(X, \\mathcal{O}_{X'})", "$$", "is surjective as desired. The first arrow is surjective as $Y$", "is affine and since we assumed $\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$", "is surjective and the second by the long exact sequence of", "cohomology associated to the short exact sequence above and", "the vanishing just proved." ], "refs": [ "coherent-lemma-flat-base-change-cohomology", "more-morphisms-lemma-globally-generated-vanishing" ], "ref_ids": [ 3298, 14071 ] } ], "ref_ids": [] }, { "id": 14073, "type": "theorem", "label": "more-morphisms-lemma-h1-fibre-zero-h0-kappa", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-h1-fibre-zero-h0-kappa", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\", "& S", "}", "$$", "of morphisms of schemes. Let $s \\in S$ be a point. Assume", "\\begin{enumerate}", "\\item $X \\to S$ is locally of finite presentation and flat at", "points of $X_s$,", "\\item $f$ is proper,", "\\item the fibres of $f_s : X_s \\to Y_s$ have dimension $\\leq 1$", "and $R^1f_{s, *}\\mathcal{O}_{X_s} = 0$,", "\\item $\\mathcal{O}_{Y_s} \\to f_{s, *}\\mathcal{O}_{X_s}$ is surjective.", "\\end{enumerate}", "Then there is an open $Y_s \\subset V \\subset Y$ such that", "(a) $f^{-1}(V)$ is flat over $S$,", "(b) $\\dim(X_y) \\leq 1$ for $y \\in V$,", "(c) $R^1f_*\\mathcal{O}_X|_V = 0$,", "(d) $\\mathcal{O}_V \\to f_*\\mathcal{O}_X|_V$", "is surjective,", "and (b), (c), and (d) remain true after base change by any $Y' \\to V$." ], "refs": [], "proofs": [ { "contents": [ "Let $y \\in Y$ be a point over $s$.", "It suffices to find an open neighbourhood of $y$", "with the desired properties. As a first step, we replace $Y$", "by the open $V$ found in Lemma \\ref{lemma-h1-fibre-zero} so that", "$R^1f_*\\mathcal{O}_X$ is zero universally (the hypothesis of", "the lemma holds by Lemma \\ref{lemma-check-h1-fibre-zero}).", "We also shrink $Y$ so that all fibres of $f$ have dimension $\\leq 1$", "(use Morphisms, Lemma", "\\ref{morphisms-lemma-openness-bounded-dimension-fibres}", "and properness of $f$). Thus we may assume we have (b) and (c)", "with $V = Y$ and after any base change $Y' \\to Y$.", "Thus by Lemma \\ref{lemma-h1-fibre-zero-check-h0-kappa}", "it now suffices to show (d) over $Y$.", "We may still shrink $Y$ further; for example, we may and do", "assume $Y$ and $S$ are affine.", "\\medskip\\noindent", "By Theorem \\ref{theorem-openness-flatness}", "there is an open subset $U \\subset X$ where $X \\to S$", "is flat which contains $X_s$ by hypothesis.", "Then $f(X \\setminus U)$ is a closed subset", "not containing $y$. Thus after shrinking $Y$", "we may assume $X$ is flat over $S$.", "\\medskip\\noindent", "Say $S = \\Spec(R)$. Choose a closed immersion $Y \\to Y'$ where $Y'$", "is the spectrum of a polynomial ring $R[x_e; e \\in E]$ on a set $E$.", "Denote $f' : X \\to Y'$ the composition of $f$ with $Y \\to Y'$.", "Then the hypotheses (1) -- (4) as well as (b) and (c)", "hold for $f'$ and $s$. If we we show $\\mathcal{O}_{Y'} \\to f'_*\\mathcal{O}_X$", "is surjective in an open neighbourhood of $y$, then the same is true for", "$\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$. Thus we may assume $Y$ is", "the spectrum of $R[x_e; e \\in E]$.", "\\medskip\\noindent", "At this point $X$ and $Y$ are flat over $S$. Then $Y_s$ and $X$", "are tor independent over $Y$. We urge the reader to find their own", "proof, but it also follows from", "Lemma \\ref{lemma-case-of-tor-independence} applied to the", "square with corners $X, Y, S, S$ and its base change by $s \\to S$.", "Hence", "$$", "Rf_{s, *}\\mathcal{O}_{X_s} = L(Y_s \\to Y)^*Rf_*\\mathcal{O}_X", "$$", "by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-compare-base-change}.", "Because of the vanishing already established this implies", "$f_{s, *}\\mathcal{O}_{X_s} = (Y_s \\to Y)^*f_*\\mathcal{O}_X$.", "We conclude that $\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$ is a map of", "quasi-coherent $\\mathcal{O}_Y$-modules whose pullback", "to $Y_s$ is surjective. We claim", "$f_*\\mathcal{O}_X$ is a finite type $\\mathcal{O}_Y$-module.", "If true, then the cokernel $\\mathcal{F}$ of", "$\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$", "is a finite type quasi-coherent $\\mathcal{O}_Y$-module", "such that $\\mathcal{F}_y \\otimes \\kappa(y) = 0$.", "By Nakayama's lemma (Algebra, Lemma \\ref{algebra-lemma-NAK})", "we have $\\mathcal{F}_y = 0$. Thus $\\mathcal{F}$ is zero", "in an open neighbourhood of $y$", "(Modules, Lemma \\ref{modules-lemma-finite-type-stalk-zero})", "and the proof is complete.", "\\medskip\\noindent", "Proof of the claim.", "For a finite subset $E' \\subset E$ set $Y' = \\Spec(R[x_e; e \\in E'])$.", "For large enough $E'$ the morphism $f' : X \\to Y \\to Y'$ is proper, see", "Limits, Lemma \\ref{limits-lemma-finite-type-eventually-proper}.", "We fix $E'$ and $Y'$ in the following.", "Write $R = \\colim R_i$ as the colimit of its finite type", "$\\mathbf{Z}$-subalgebras. Set $S_i = \\Spec(R_i)$", "and $Y'_i = \\Spec(R_i[x_e; e \\in E'])$.", "For $i$ large enough we can find a diagram", "$$", "\\xymatrix{", "X \\ar[d] \\ar[r]_{f'} & Y' \\ar[d] \\ar[r] & S \\ar[d] \\\\", "X_i \\ar[r]^{f'_i} & Y'_i \\ar[r] & S_i", "}", "$$", "with cartesian squares such that $X_i$ is flat over $S_i$ and", "$X_i \\to Y'_i$ is proper. See", "Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation},", "\\ref{limits-lemma-descend-flat-finite-presentation}, and", "\\ref{limits-lemma-eventually-proper}.", "The same argument as above shows $Y'$ and $X_i$", "are tor independent over $Y'_i$ and hence", "$$", "R\\Gamma(X, \\mathcal{O}_X) =", "R\\Gamma(X_i, \\mathcal{O}_{X_i})", "\\otimes^\\mathbf{L}_{R_i[x_e; e \\in E']}", "R[x_e; e \\in E']", "$$", "by the same reference as above. By Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-over-affine-cohomology-finite}", "the complex $R\\Gamma(X_i, \\mathcal{O}_{X_i})$ is pseudo-coherent", "in the derived category of the Noetherian ring $R_i[x_e; e \\in E']$ (see", "More on Algebra, Lemma \\ref{more-algebra-lemma-Noetherian-pseudo-coherent}).", "Hence $R\\Gamma(X, \\mathcal{O}_X)$ is pseudo-coherent in the derived", "category of $R[x_e; e \\in E']$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-pull-pseudo-coherent}.", "Since the only nonvanishing cohomology module", "is $H^0(X, \\mathcal{O}_X)$ we conclude it is a finite", "$R[x_e; e \\in E']$-module, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-n-pseudo-module}.", "This concludes the proof." ], "refs": [ "more-morphisms-lemma-h1-fibre-zero", "more-morphisms-lemma-check-h1-fibre-zero", "morphisms-lemma-openness-bounded-dimension-fibres", "more-morphisms-lemma-h1-fibre-zero-check-h0-kappa", "more-morphisms-theorem-openness-flatness", "more-morphisms-lemma-case-of-tor-independence", "perfect-lemma-compare-base-change", "algebra-lemma-NAK", "modules-lemma-finite-type-stalk-zero", "limits-lemma-finite-type-eventually-proper", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-flat-finite-presentation", "limits-lemma-eventually-proper", "coherent-lemma-proper-over-affine-cohomology-finite", "more-algebra-lemma-Noetherian-pseudo-coherent", "more-algebra-lemma-pull-pseudo-coherent", "more-algebra-lemma-n-pseudo-module" ], "ref_ids": [ 14070, 14069, 5280, 14072, 13670, 14057, 7028, 401, 13239, 15092, 15077, 15062, 15089, 3355, 10160, 10155, 10147 ] } ], "ref_ids": [] }, { "id": 14074, "type": "theorem", "label": "more-morphisms-lemma-h1-fibre-zero-h0-flat", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-h1-fibre-zero-h0-flat", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\", "& S", "}", "$$", "of morphisms of schemes. Assume $X \\to S$", "is flat, $f$ is proper, $\\dim(X_y) \\leq 1$ for $y \\in Y$, and", "$R^1f_*\\mathcal{O}_X = 0$. Then $f_*\\mathcal{O}_X$", "is $S$-flat and formation of $f_*\\mathcal{O}_X$ commutes", "with arbitrary base change $S' \\to S$." ], "refs": [], "proofs": [ { "contents": [ "We may assume $Y$ and $S$ are affine, say $S = \\Spec(A)$.", "To show the quasi-coherent $\\mathcal{O}_Y$-module", "$f_*\\mathcal{O}_X$ is flat relative to $S$ it suffices", "to show that $H^0(X, \\mathcal{O}_X)$ is flat over $A$", "(some details omitted).", "By Lemma \\ref{lemma-globally-generated-vanishing} we have", "$H^1(X, \\mathcal{O}_X \\otimes_A M) = 0$ for every $A$-module $M$.", "Since also $\\mathcal{O}_X$ is flat over $A$ we deduce the functor", "$M \\mapsto H^0(X, \\mathcal{O}_X \\otimes_A M)$ is exact.", "Moreover, this functor commutes with direct sums by", "Cohomology, Lemma \\ref{cohomology-lemma-quasi-separated-cohomology-colimit}.", "Then it is an exercise to see that", "$H^0(X, \\mathcal{O}_X \\otimes_A M) = M \\otimes_A H^0(X, \\mathcal{O}_X)$", "functorially in $M$ and this gives the desired flatness.", "Finally, if $S' \\to S$ is a morphism of affines given by", "the ring map $A \\to A'$, then in the affine case just discussed", "we see that", "$$", "H^0(X \\times_S S', \\mathcal{O}_{X \\times_S S'}) =", "H^0(X, \\mathcal{O}_X \\otimes_A A') = H^0(X, \\mathcal{O}_X) \\otimes_A A'", "$$", "This shows that formation of $f_*\\mathcal{O}_X$ commutes", "with any base change $S' \\to S$. Some details omitted." ], "refs": [ "more-morphisms-lemma-globally-generated-vanishing", "cohomology-lemma-quasi-separated-cohomology-colimit" ], "ref_ids": [ 14071, 2082 ] } ], "ref_ids": [] }, { "id": 14075, "type": "theorem", "label": "more-morphisms-lemma-h1-fibre-zero-isom", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-h1-fibre-zero-isom", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\", "& S", "}", "$$", "of morphisms of schemes. Let $s \\in S$ be a point. Assume", "\\begin{enumerate}", "\\item $X \\to S$ is locally of finite presentation and flat at", "points of $X_s$,", "\\item $Y \\to S$ is locally of finite presentation,", "\\item $f$ is proper,", "\\item the fibres of $f_s : X_s \\to Y_s$ have dimension $\\leq 1$", "and $R^1f_{s, *}\\mathcal{O}_{X_s} = 0$,", "\\item $\\mathcal{O}_{Y_s} \\to f_{s, *}\\mathcal{O}_{X_s}$ is an isomorphism.", "\\end{enumerate}", "Then there is an open $Y_s \\subset V \\subset Y$ such that", "(a) $V$ is flat over $S$,", "(b) $f^{-1}(V)$ is flat over $S$,", "(c) $\\dim(X_y) \\leq 1$ for $y \\in V$,", "(d) $R^1f_*\\mathcal{O}_X|_V = 0$,", "(e) $\\mathcal{O}_V \\to f_*\\mathcal{O}_X|_V$", "is an isomorphism, and (a) -- (e)", "remain true after base change of $f^{-1}(V) \\to V$ by any $S' \\to S$." ], "refs": [], "proofs": [ { "contents": [ "Let $y \\in Y_s$. We may always replace $Y$ by an open neighbourhood of $y$.", "Thus we may assume $Y$ and $S$ affine. We may also assume that", "$X$ is flat over $S$, $\\dim(X_y) \\leq 1$ for $y \\in Y$,", "$R^1f_*\\mathcal{O}_X = 0$ universally, and that", "$\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$ is surjective, see", "Lemma \\ref{lemma-h1-fibre-zero-h0-kappa}. (We won't use all of this.)", "\\medskip\\noindent", "Assume $S$ and $Y$ affine.", "Write $S = \\lim S_i$ as a cofiltered of affine Noetherian schemes $S_i$.", "By Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}", "there exists an element $0 \\in I$ and a diagram", "$$", "\\xymatrix{", "X_0 \\ar[rr]_{f_0} \\ar[rd] & & Y_0 \\ar[ld] \\\\", "& S_0", "}", "$$", "of finite type morphisms of schemes whose base change to $S$", "is the diagram of the lemma. After increasing $0$ we may assume $Y_0$", "is affine and $X_0 \\to S_0$ proper, see", "Limits, Lemmas \\ref{limits-lemma-eventually-proper} and", "\\ref{limits-lemma-limit-affine}.", "Let $s_0 \\in S_0$ be the image of $s$.", "As $Y_s$ is affine, we see that", "$R^1f_{s, *}\\mathcal{O}_{X_s} = 0$ is equivalent", "to $H^1(X_s, \\mathcal{O}_{X_s}) = 0$.", "Since $X_s$ is the base change of $X_{0, s_0}$ by", "the faithfully flat map $\\kappa(s_0) \\to \\kappa(s)$", "we see that $H^1(X_{0, s_0}, \\mathcal{O}_{X_{0, s_0}}) = 0$", "and hence $R^1f_{0, *}\\mathcal{O}_{X_{0, s_0}} = 0$.", "Similarly, as $\\mathcal{O}_{Y_s} \\to f_{s, *}\\mathcal{O}_{X_s}$", "is an isomorphism, so is", "$\\mathcal{O}_{Y_{0, s_0}} \\to f_{0, *}\\mathcal{O}_{X_{0, s_0}}$.", "Since the dimensions of the fibres of $X_s \\to Y_s$ are at most $1$,", "the same is true for the morphism $X_{0, s_0} \\to Y_{0, s_0}$.", "Finally, since $X \\to S$ is flat, after increasing $0$", "we may assume $X_0$ is flat over $S_0$, see", "Limits, Lemma \\ref{limits-lemma-descend-flat-finite-presentation}.", "Thus it suffices to prove the lemma for", "$X_0 \\to Y_0 \\to S_0$ and the point $s_0$.", "\\medskip\\noindent", "Combining the reduction arguments above we reduce to the case where", "$S$ and $Y$ affine, $S$ Noetherian, the fibres of $f$ have dimension", "$\\leq 1$, and $R^1f_*\\mathcal{O}_X = 0$ universally.", "Let $y \\in Y_s$ be a point. Claim:", "$$", "\\mathcal{O}_{Y, y} \\longrightarrow (f_*\\mathcal{O}_X)_y", "$$", "is an isomorphism. The claim implies the lemma. Namely, since", "$f_*\\mathcal{O}_X$ is coherent (Cohomology of Schemes, Proposition", "\\ref{coherent-proposition-proper-pushforward-coherent})", "the claim means we can replace $Y$ by an open neighbourhood of $y$", "and obtain an isomorphism $\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$.", "Then we conclude that $Y$ is flat over $S$", "by Lemma \\ref{lemma-h1-fibre-zero-h0-flat}.", "Finally, the isomorphism $\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$", "remains an isomorphism after any base change $S' \\to S$", "by the final statement of Lemma \\ref{lemma-h1-fibre-zero-h0-flat}.", "\\medskip\\noindent", "Proof of the claim. We already know that", "$\\mathcal{O}_{Y, y} \\longrightarrow (f_*\\mathcal{O}_X)_y$", "is surjective (Lemma \\ref{lemma-h1-fibre-zero-h0-kappa})", "and that $(f_*\\mathcal{O}_X)_y$ is", "$\\mathcal{O}_{S, s}$-flat (Lemma \\ref{lemma-h1-fibre-zero-h0-flat})", "and that the induced map", "$$", "\\mathcal{O}_{Y_s, y} = \\mathcal{O}_{Y, y}/\\mathfrak m_s\\mathcal{O}_{Y, y}", "\\longrightarrow", "(f_*\\mathcal{O}_X)_y/\\mathfrak m_s (f_*\\mathcal{O}_X)_y", "\\to", "(f_{s, *}\\mathcal{O}_{X_s})_y", "$$", "is injective by the assumption in the lemma. Then it follows from", "Algebra, Lemma \\ref{algebra-lemma-mod-injective}", "that $\\mathcal{O}_{Y, y} \\longrightarrow (f_*\\mathcal{O}_X)_y$", "is injective as desired." ], "refs": [ "more-morphisms-lemma-h1-fibre-zero-h0-kappa", "limits-lemma-descend-finite-presentation", "limits-lemma-eventually-proper", "limits-lemma-limit-affine", "limits-lemma-descend-flat-finite-presentation", "coherent-proposition-proper-pushforward-coherent", "more-morphisms-lemma-h1-fibre-zero-h0-flat", "more-morphisms-lemma-h1-fibre-zero-h0-flat", "more-morphisms-lemma-h1-fibre-zero-h0-kappa", "more-morphisms-lemma-h1-fibre-zero-h0-flat", "algebra-lemma-mod-injective" ], "ref_ids": [ 14073, 15077, 15089, 15043, 15062, 3401, 14074, 14074, 14073, 14074, 883 ] } ], "ref_ids": [] }, { "id": 14076, "type": "theorem", "label": "more-morphisms-lemma-bijection-on-Pic", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-bijection-on-Pic", "contents": [ "Let $f : X \\to Y$ be a proper morphism of Noetherian schemes", "such that $f_*\\mathcal{O}_X = \\mathcal{O}_Y$, such that", "the fibres of $f$ have dimension $\\leq 1$, and such that", "$H^1(X_y, \\mathcal{O}_{X_y}) = 0$ for $y \\in Y$.", "Then $f^* : \\Pic(Y) \\to \\Pic(X)$ is a bijection onto", "the subgroup of $\\mathcal{L} \\in \\Pic(X)$ with", "$\\mathcal{L}|_{X_y} \\cong \\mathcal{O}_{X_y}$", "for all $y \\in Y$." ], "refs": [], "proofs": [ { "contents": [ "By the projection formula", "(Cohomology, Lemma \\ref{cohomology-lemma-projection-formula})", "we see that $f_*f^*\\mathcal{N} \\cong \\mathcal{N}$ for", "$\\mathcal{N} \\in \\Pic(Y)$.", "We claim that for $\\mathcal{L} \\in \\Pic(X)$ with", "$\\mathcal{L}|_{X_y} \\cong \\mathcal{O}_{X_y}$", "for all $y \\in Y$ we have $\\mathcal{N} = f_*\\mathcal{L}$", "is invertible and $\\mathcal{L} \\cong f^*\\mathcal{N}$.", "This will finish the proof.", "\\medskip\\noindent", "The $\\mathcal{O}_Y$-module $\\mathcal{N} = f_*\\mathcal{L}$", "is coherent by Cohomology of Schemes, Proposition", "\\ref{coherent-proposition-proper-pushforward-coherent}.", "Thus to see that it is an invertible $\\mathcal{O}_Y$-module,", "it suffices to check on stalks (Algebra, Lemma", "\\ref{algebra-lemma-finite-projective}).", "Since the map from a Noetherian local ring to its completion", "is faithfully flat, it suffices to check the completion", "$(f_*\\mathcal{L})_y^\\wedge$ is free (see", "Algebra, Section \\ref{algebra-section-completion-noetherian} and", "Lemma \\ref{algebra-lemma-finite-projective-descends}).", "For this we will use the theorem of formal functions as formulated in", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-formal-functions-stalk}.", "Since $f_*\\mathcal{O}_X = \\mathcal{O}_Y$ and hence", "$(f_*\\mathcal{O}_X)_y^\\wedge \\cong \\mathcal{O}_{Y, y}^\\wedge$,", "it suffices to show that $\\mathcal{L}|_{X_n} \\cong \\mathcal{O}_{X_n}$", "for each $n$ (compatibly for varying $n$.", "By Lemma \\ref{lemma-picard-group-first-order-thickening}", "we have an exact sequence", "$$", "H^1(X_y, \\mathfrak m_y^n\\mathcal{O}_X/\\mathfrak m_y^{n + 1}\\mathcal{O}_X)", "\\to \\Pic(X_{n + 1}) \\to \\Pic(X_n)", "$$", "with notation as in the theorem on formal functions.", "Observe that we have a surjection", "$$", "\\mathcal{O}_{X_y}^{\\oplus r_n} \\cong", "\\mathfrak m_y^n/\\mathfrak m_y^{n + 1} \\otimes_{\\kappa(y)} \\mathcal{O}_{X_y}", "\\longrightarrow", "\\mathfrak m_y^n\\mathcal{O}_X/\\mathfrak m_y^{n + 1}\\mathcal{O}_X", "$$", "for some integers $r_n \\geq 0$.", "Since $\\dim(X_y) \\leq 1$ this surjection induces a surjection", "on first cohomology groups (by the vanishing of cohomology in degrees $\\geq 2$", "coming from Cohomology, Proposition", "\\ref{cohomology-proposition-vanishing-Noetherian}).", "Hence the $H^1$ in the sequence is zero and the transition maps", "$\\Pic(X_{n + 1}) \\to \\Pic(X_n)$ are injective as desired.", "\\medskip\\noindent", "We still have to show that $f^*\\mathcal{N} \\cong \\mathcal{L}$.", "This is proved by the same method and we omit the details." ], "refs": [ "cohomology-lemma-projection-formula", "coherent-proposition-proper-pushforward-coherent", "algebra-lemma-finite-projective", "algebra-lemma-finite-projective-descends", "coherent-lemma-formal-functions-stalk", "more-morphisms-lemma-picard-group-first-order-thickening", "cohomology-proposition-vanishing-Noetherian" ], "ref_ids": [ 2243, 3401, 795, 798, 3362, 13686, 2246 ] } ], "ref_ids": [] }, { "id": 14077, "type": "theorem", "label": "more-morphisms-lemma-improve-stratification", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-improve-stratification", "contents": [ "Let $X$ be a scheme. Let $X = \\coprod_{i \\in I} X_i$ be a finite", "affine stratification. There exists an affine stratification", "with index set $\\{0, \\ldots, n\\}$ where $n$ is the length of $I$." ], "refs": [], "proofs": [ { "contents": [ "Recall that we have a partial ordering on $I$ such that", "the closure of $X_i$ is contained in $\\bigcup_{j \\leq i} X_j$", "for all $i \\in I$.", "Let $I' \\subset I$ be the set of maximal indices of $I$.", "If $i \\in I'$, then $X_i$ is open in $X$ because the union of the", "closures of the other strata is the complement of $X_i$.", "Let $U = \\bigcup_{i \\in I'} X_i$ viewed as an open subscheme of $X$", "so that $U_{red} = \\coprod_{i \\in I'} X_i$ as schemes.", "Then $U$ is an affine scheme by", "Schemes, Lemma \\ref{schemes-lemma-disjoint-union-affines}", "and Lemma \\ref{lemma-thickening-affine-scheme}. The morphism $U \\to X$", "is affine as each $X_i \\to X$, $i \\in I'$ is affine by the same reasoning", "using Lemma \\ref{lemma-thicken-property-morphisms}.", "The complement $Z = X \\setminus U$ endowed with the", "reduced induced scheme structure has the affine", "stratification $Z = \\bigcup_{i \\in I \\setminus I'} X_i$.", "Here we use that a morphism of schemes $T \\to Z$ is affine", "if and only if the composition $T \\to X$ is affine; this follows", "from Morphisms, Lemmas \\ref{morphisms-lemma-closed-immersion-affine},", "\\ref{morphisms-lemma-composition-affine}, and", "\\ref{morphisms-lemma-affine-permanence}.", "Observe that the partially ordered set $I \\setminus I'$", "has length exactly one less than the length of $I$.", "Hence by induction we find that $Z$ has an affine", "stratification $Z = Z_0 \\amalg \\ldots \\amalg Z_{n - 1}$", "with index set $\\{1, \\ldots, n\\}$. Setting $Z_n = U$", "we obtain the desired stratification of $X$." ], "refs": [ "schemes-lemma-disjoint-union-affines", "more-morphisms-lemma-thickening-affine-scheme", "more-morphisms-lemma-thicken-property-morphisms", "morphisms-lemma-closed-immersion-affine", "morphisms-lemma-composition-affine", "morphisms-lemma-affine-permanence" ], "ref_ids": [ 7659, 13678, 13682, 5177, 5175, 5179 ] } ], "ref_ids": [] }, { "id": 14078, "type": "theorem", "label": "more-morphisms-lemma-qc-affine-stratification", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-qc-affine-stratification", "contents": [ "Let $X$ be a scheme. The following are equivalent", "\\begin{enumerate}", "\\item $X$ has a finite affine stratification, and", "\\item $X$ is quasi-compact and quasi-separated.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $X = \\bigcup X_i$ be a finite affine stratification. Since each $X_i$", "is affine hence quasi-compact, we conclude that $X$ is quasi-compact.", "Let $U, V \\subset X$ be affine open. Then $U \\cap X_i$ and $V \\cap X_i$", "are affine open in $X_i$ since $X_i \\to X$ is an affine morphism. Hence", "$U \\cap V \\cap X_i$ is an affine open of the affine scheme $X_i$ (see", "Schemes, Lemma \\ref{schemes-lemma-characterize-separated} for example).", "Therefore $U \\cap V = \\coprod U \\cap V \\cap X_i$ is quasi-compact", "as a finite union of affine strata. We conclude that $X$ is", "quasi-separated by Schemes, Lemma", "\\ref{schemes-lemma-characterize-quasi-separated}.", "\\medskip\\noindent", "Assume $X$ is quasi-compact and quasi-separated. We may use the", "induction principle of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}", "to prove the assertion that $X$ has a finite affine stratification.", "If $X$ is empty, then it has an empty affine stratification.", "If $X$ is nonempty affine then it has an affine stratification", "with one stratum. Next, asssume $X = U \\cup V$ where $U$ is quasi-compact open,", "$V$ is affine open, and we have a finite affine stratifications", "$U = \\bigcup_{i \\in I} U_i$ and $U \\cap V = \\coprod_{j \\in J} W_j$.", "Denote $Z = X \\setminus V$ and $Z' = X \\setminus U$.", "Note that $Z$ is closed in $U$ and $Z'$ is closed in $V$.", "Observe that $U_i \\cap Z$ and $U_i \\cap W_j = U_i \\times_U W_j$ are affine", "schemes affine over $U$. (Hints: use that $U_i \\times_U W_j \\to W_j$ is", "affine as a base change of $U_i \\to U$, hence $U_i \\cap W_j$ is affine,", "hence $U_i \\cap W_j \\to U_i$ is affine, hence $U_i \\cap W_j \\to U$ is affine.)", "It follows that", "$$", "U = \\coprod\\nolimits_{i \\in I} (U_i \\cap Z) \\amalg", "\\coprod\\nolimits_{(i, j) \\in I \\times J} (U_i \\cap W_j)", "$$", "is a finite affine stratification with partial ordering on", "$I \\amalg I \\times J$ given by $i' \\leq (i, j) \\Leftrightarrow i' \\leq i$", "and $(i', j') \\leq (i, j) \\Leftrightarrow i' \\leq i$ and $j' \\leq j$.", "Observe that $(U_i \\cap Z) \\times_X V = \\emptyset$ and", "$(U_i \\cap W_j) \\times_X V = U_i \\cap W_j$ are affine.", "Hence the morphisms $U_i \\cap Z \\to X$ and $U_i \\cap W_j \\to X$", "are affine because we can check affineness of a morphism", "locally on the target", "(Morphisms, Lemma \\ref{morphisms-lemma-characterize-affine})", "and we have affineness over both $U$ and $V$.", "To finish the proof we take the stratification above", "and we add one additional stratum, namely $Z'$,", "whose index we add as a minimal element to the partially ordered set." ], "refs": [ "schemes-lemma-characterize-separated", "schemes-lemma-characterize-quasi-separated", "coherent-lemma-induction-principle", "morphisms-lemma-characterize-affine" ], "ref_ids": [ 7710, 7709, 3291, 5172 ] } ], "ref_ids": [] }, { "id": 14079, "type": "theorem", "label": "more-morphisms-lemma-affine-stratification-number-bound", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-affine-stratification-number-bound", "contents": [ "Let $X$ be a separated scheme which has an open covering by", "$n + 1$ affines. Then the affine stratification number of $X$", "is at most $n$." ], "refs": [], "proofs": [ { "contents": [ "Say $X = U_0 \\cup \\ldots \\cup U_n$ is an affine open covering.", "Set", "$$", "X_i = (U_i \\cup \\ldots \\cup U_n) \\setminus", "(U_{i + 1} \\cup \\ldots \\cup U_n)", "$$", "Then $X_i$ is affine as a closed subscheme of $U_i$. The morphism $X_i \\to X$", "is affine by Morphisms, Lemma \\ref{morphisms-lemma-affine-permanence}.", "Finally, we have $\\overline{X_i} \\subset X_i \\cup X_{i - 1} \\cup \\ldots X_0$." ], "refs": [ "morphisms-lemma-affine-permanence" ], "ref_ids": [ 5179 ] } ], "ref_ids": [] }, { "id": 14080, "type": "theorem", "label": "more-morphisms-lemma-affine-stratification-number-bound-Noetherian", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-affine-stratification-number-bound-Noetherian", "contents": [ "Let $X$ be a Noetherian scheme of dimension $\\infty > d \\geq 0$.", "Then the affine stratification number of $X$ is at most $d$." ], "refs": [], "proofs": [ { "contents": [ "By induction on $d$. If $d = 0$, then $X$ is affine, see", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian-dimension-0}.", "Assume $d > 0$. Let $\\eta_1, \\ldots, \\eta_n$ be the generic points", "of the irreducible components of $X$ (Properties, Lemma", "\\ref{properties-lemma-Noetherian-irreducible-components}).", "We can cover $X$ by affine opens containing $\\eta_1, \\ldots, \\eta_n$, see", "Properties, Lemma \\ref{properties-lemma-point-and-maximal-points-affine}.", "Since $X$ is quasi-compact we can find a finite affine open covering", "$X = \\bigcup_{j = 1, \\ldots, m} U_j$ with", "$\\eta_1, \\ldots, \\eta_n \\in U_j$ for all $j = 1, \\ldots, m$.", "Choose an affine open $U \\subset U_1 \\cap \\ldots \\cap U_m$", "containing $\\eta_1, \\ldots, \\eta_n$ (possible by the lemma", "already quoted). Then the morphism $U \\to X$ is affine", "because $U \\to U_j$ is affine for all $j$, see", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-affine}.", "Let $Z = X \\setminus U$.", "By construction $\\dim(Z) < \\dim(X)$.", "By induction hypothesis we can find an", "affine stratification $Z = \\bigcup_{i \\in \\{0, \\ldots, n\\}} Z_i$", "of $Z$ with $n \\leq \\dim(Z)$.", "Setting $U = X_{n + 1}$ and $X_i = Z_i$ for $i \\leq n$", "we conclude." ], "refs": [ "properties-lemma-locally-Noetherian-dimension-0", "properties-lemma-Noetherian-irreducible-components", "properties-lemma-point-and-maximal-points-affine", "morphisms-lemma-characterize-affine" ], "ref_ids": [ 2981, 2956, 3061, 5172 ] } ], "ref_ids": [] }, { "id": 14081, "type": "theorem", "label": "more-morphisms-lemma-test-universally-open", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-test-universally-open", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally open,", "\\item for every morphism $S' \\to S$ which is locally of finite presentation", "the base change $X_{S'} \\to S'$ is open, and", "\\item for every $n$ the morphism", "$\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$", "is open.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2) and (2) implies (3).", "Let us prove that (3) implies (1).", "Suppose that the base change $X_T \\to T$ is not open", "for some morphism of schemes $g : T \\to S$.", "Then we can find some affine opens", "$V \\subset S$, $U \\subset X$, $W \\subset T$", "with $f(U) \\subset V$ and $g(W) \\subset V$", "such that $U \\times_V W \\to W$ is not open.", "If we can show that this implies", "$\\mathbf{A}^n \\times U \\to \\mathbf{A}^n \\times V$", "is not open, then $\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$", "is not open and the proof is complete. This reduces us", "to the result proved in the next paragraph.", "\\medskip\\noindent", "Let $A \\to B$ be a ring map such that $A' \\to B' = A' \\otimes_A B$", "does not induce an open map of spectra for some $A$-algebra $A'$.", "As the principal opens give a basis for the topology of $\\Spec(B')$", "we conclude that the image of $D(g)$ in $\\Spec(A')$", "is not open for some $g \\in B'$. Write", "$g = \\sum_{i = 1, \\ldots, n} a'_i \\otimes b_i$", "for some $n$, $a'_i \\in A'$, and $b_i \\in B$.", "Consider the element $h = \\sum_{i = 1, \\ldots, n} x_i b_i$", "in $B[x_1, \\ldots, x_n]$. Assume that $D(h)$ maps to", "an open subset under the morphism", "$$", "\\Spec(B[x_1, \\ldots, x_n]) \\longrightarrow \\Spec(A[x_1, \\ldots, x_n])", "$$", "in order to get a contradiction. Then $D(h)$ would map surjectively", "onto a quasi-compact open $U \\subset \\Spec(A[x_1, \\ldots, x_n])$.", "Let $A[x_1, \\ldots, x_n] \\to A'$ be the $A$-algebra homomorphism", "sending $x_i$ to $a'_i$. This also induces a $B$-algebra", "homomorphism $B[x_1, \\ldots, x_n] \\to B'$ sending $h$ to $g$.", "Since", "$$", "\\xymatrix{", "\\Spec(B[x_1, \\ldots, x_n]) \\ar[d] &", "\\Spec(B') \\ar[l] \\ar[d] \\\\", "\\Spec(A[x_1, \\ldots, x_n]) &", "\\Spec(A') \\ar[l]", "}", "$$", "is cartesian the image of $D(g)$ in $\\Spec(A')$ is equal to the", "inverse image of $U$ in $\\Spec(A')$ and hence open which is", "the desired contradiction." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14082, "type": "theorem", "label": "more-morphisms-lemma-quasi-finite-Noetherian-universally-open", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-quasi-finite-Noetherian-universally-open", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. If", "\\begin{enumerate}", "\\item $f$ is locally quasi-finite,", "\\item $Y$ is unibranch and locally Noetherian, and", "\\item every irreducible component of $X$ dominates", "an irreducible component of $Y$,", "\\end{enumerate}", "then $f$ is universally open." ], "refs": [], "proofs": [ { "contents": [ "For any $n$ the scheme $\\mathbf{A}^n \\times Y$ is unibranch", "by Lemma \\ref{lemma-number-of-branches-and-smooth}. Hence", "the hypotheses of the lemma hold for the morphisms", "$\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times Y$ for all $n$.", "By Lemma \\ref{lemma-test-universally-open}", "it suffices to prove $f$ is open. By Morphisms, Lemma", "\\ref{morphisms-lemma-locally-finite-presentation-universally-open}", "it suffices to show that generalizations lift along $f$.", "Suppose that $y' \\leadsto y$ is a specialization of points in", "$Y$ and $x \\in X$ is a point mapping to $y$.", "As in Lemma \\ref{lemma-etale-makes-quasi-finite-finite-at-point}", "choose a diagram", "$$", "\\xymatrix{", "u \\ar[d] & U \\ar[d] \\ar[r] & X \\ar[d] \\\\", "v & V \\ar[r] & Y", "}", "$$", "where $(V, v) \\to (Y, y)$ is an elementary \\'etale neighbourhood,", "$U \\to V$ is finite, $u$ is the unique point of $U$ mapping to $v$,", "$U \\subset V \\times_Y X$ is open, and $v \\mapsto y$ and $u \\mapsto x$.", "Let $E$ be an irreducible component of $U$ passing through $u$", "(there is at least one of these). Since $U \\to X$ is \\'etale, $E$", "maps to an irreducible component of $X$,", "which in turn dominates an irreducible component of $Y$ (by assumption).", "Since $U \\to V$ is finite hence closed, we conclude that", "the image $E' \\subset V$ of $E$ is an irreducible closed subset", "passing through $v$ which dominates an irreducible component of $Y$.", "Since $V \\to Y$ is \\'etale $E'$ must be an irreducible component", "of $V$ passing through $v$.", "Since $Y$ is unibranch we see that $E'$ is the unique irreducible", "component of $V$ passing through $v$ (Lemma \\ref{lemma-nr-branches}).", "Since $V$ is locally Noetherian we may after shrinking $V$", "assume that $E' = V$ (equality of sets).", "\\medskip\\noindent", "Since $V \\to Y$ is \\'etale we can find a specialization", "$v' \\leadsto v$ whose image is $y' \\leadsto y$.", "By the above we can find $u' \\in U$ mapping to $v'$.", "Then $u' \\leadsto u$ because $u$ is the only point of", "$U$ mapping to $v$ and $U \\to V$ is closed.", "Then finally the image $x' \\in X$ of $u'$ is a point", "specializing to $x$ and mapping to $y'$ and the proof is complete." ], "refs": [ "more-morphisms-lemma-number-of-branches-and-smooth", "more-morphisms-lemma-test-universally-open", "morphisms-lemma-locally-finite-presentation-universally-open", "more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point", "more-morphisms-lemma-nr-branches" ], "ref_ids": [ 13874, 14081, 5252, 13892, 13872 ] } ], "ref_ids": [] }, { "id": 14083, "type": "theorem", "label": "more-morphisms-lemma-characterize-universally-open-finite", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-characterize-universally-open-finite", "contents": [ "Let $A \\to B$ be a ring map. Say $B$ is generated as an $A$-module by", "$b_1, \\ldots, b_d \\in B$. Set $h = \\sum x_ib_i \\in B[x_1, \\ldots, x_d]$.", "Then $\\Spec(B) \\to \\Spec(A)$ is universally open if and only if the image of", "$D(h)$ in $\\Spec(A[x_1, \\ldots, x_d])$ is open." ], "refs": [], "proofs": [ { "contents": [ "If $\\Spec(B) \\to \\Spec(A)$ is universally open, then of course", "the image of $D(h)$ is open. Conversely, assume the image $U$ of", "$D(h)$ is open. Let $A \\to A'$ be a ring map. It suffices to show", "that the image of any principal open $D(g) \\subset \\Spec(A' \\otimes_A B)$", "in $\\Spec(A')$ is open. We may write", "$g = \\sum_{i = 1, \\ldots, d} a'_i \\otimes b_i$ for some $a'_i \\in A'$.", "Let $A[x_1, \\ldots, x_n] \\to A'$ be the $A$-algebra homomorphism", "sending $x_i$ to $a'_i$. This also induces a $B$-algebra", "homomorphism $B[x_1, \\ldots, x_n] \\to A' \\otimes_A B$ sending $h$ to $g$.", "Since", "$$", "\\xymatrix{", "\\Spec(B[x_1, \\ldots, x_n]) \\ar[d] &", "\\Spec(B') \\ar[l] \\ar[d] \\\\", "\\Spec(A[x_1, \\ldots, x_n]) &", "\\Spec(A') \\ar[l]", "}", "$$", "is cartesian the image of $D(g)$ in $\\Spec(A')$ is equal to the", "inverse image of $U$ in $\\Spec(A')$ and hence open." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14084, "type": "theorem", "label": "more-morphisms-lemma-descend-quasi-finite-universally-open", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-descend-quasi-finite-universally-open", "contents": [ "Let $S = \\lim S_i$ be a limit of a directed system of schemes", "with affine transition morphisms.", "Let $0 \\in I$ and let $f_0 : X_0 \\to Y_0$ be a morphism of schemes over $S_0$.", "Assume $S_0$, $X_0$, $Y_0$ are quasi-compact and quasi-separated.", "Let $f_i : X_i \\to Y_i$ be the base change of $f_0$ to $S_i$ and", "let $f : X \\to Y$ be the base change of $f_0$ to $S$.", "If", "\\begin{enumerate}", "\\item $f$ is locally quasi-finite and universally open, and", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then there exists an $i \\geq 0$ such that $f_i$ is locally quasi-finite", "and universally open." ], "refs": [], "proofs": [ { "contents": [ "By Limits, Lemma \\ref{limits-lemma-descend-quasi-finite} after increasing", "$0$ we may assume $f_0$ is locally quasi-finite. Let $x \\in X$.", "By \\'etale localization of quasi-finite", "morphisms we can find a diagram", "$$", "\\xymatrix{", "X \\ar[d] & U \\ar[l] \\ar[d] \\\\", "Y & V \\ar[l]", "}", "$$", "where $V \\to Y$ is \\'etale, $U \\subset X_V$ is open, $U \\to V$ is finite,", "and $x$ is in the image of $U \\to X$, see", "Lemma \\ref{lemma-etale-makes-quasi-finite-finite-at-point}.", "After shrinking $V$ we may assume $V$ and $U$ are affine.", "Since $X$ is quasi-compact, it follows, by taking a finite disjoint", "union of such $V$ and $U$, that we can make a diagram as above", "such that $U \\to X$ is surjective. By", "Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation},", "\\ref{limits-lemma-descend-opens},", "\\ref{limits-lemma-descend-surjective},", "\\ref{limits-lemma-descend-finite-finite-presentation},", "\\ref{limits-lemma-descend-etale}, and", "\\ref{limits-lemma-limit-affine} after possibly increasing $0$", "we may assume we have a diagram", "$$", "\\xymatrix{", "X_0 \\ar[d] & U_0 \\ar[l] \\ar[d] \\\\", "Y_0 & V_0 \\ar[l]", "}", "$$", "where $V_0$ is affine, $V_0 \\to Y_0$ is \\'etale,", "$U_0 \\subset (X_0)_{V_0}$ is open, $U_0 \\to V_0$ is finite,", "and $U_0 \\to X_0$ is surjective. Since $V_i \\to Y_i$ is \\'etale", "and hence universally open, follows that it suffices", "to prove that $U_i \\to V_i$ is universally open for large", "enough $i$. This reduces us to the case discussed in the next", "paragraph.", "\\medskip\\noindent", "Let $A = \\colim A_i$ be a filtered colimit of rings.", "Let $A_0 \\to B_0$ be a ring map. Set $B = A \\otimes_{A_0} B_0$ and", "$B_i = A_i \\otimes_{A_0} B_0$. Assume $A_0 \\to B_0$ is finite,", "of finite presentation, and $A \\to B$ is universally open.", "We have to show that $A_i \\to B_i$ is universally open for $i$ large enough.", "Pick $b_{0, 1}, \\ldots, b_{0, d} \\in B_0$ which generate $B_0$", "as an $A_0$-module. Set $h_0 = \\sum_{j = 1, \\ldots, d} x_jb_{0, j}$", "in $B_0[x_1, \\ldots, x_d]$. Denote $h$, resp.\\ $h_i$ the image of $h_0$", "in $B[x_1, \\ldots, x_d]$, resp.\\ $B_i[x_1, \\ldots, x_d]$.", "The image $U$ of $D(h)$ in $\\Spec(A[x_1, \\ldots, x_d])$ is open", "as $A \\to B$ is universally open. Of course $U$ is quasi-compact", "as the image of an affine scheme. For $i$ large enough there", "is a quasi-compact open $U_i \\subset \\Spec(A_i[x_1, \\ldots, x_d])$", "whose inverse image in $\\Spec(A[x_1, \\ldots, x_d])$ is $U$, see", "Limits, Lemma \\ref{limits-lemma-descend-opens}.", "After increasing $i$ we may assume that $D(h_i)$ maps", "into $U_i$; this follows from the same lemma by considering the", "pullback of $U_i$ in $D(h_i)$. Finally, for $i$ even larger", "the morphism of schemes $D(h_i) \\to U_i$ will be surjective", "by an application of the already used", "Limits, Lemma \\ref{limits-lemma-descend-surjective}.", "We conclude $A_i \\to B_i$ is universally open by", "Lemma \\ref{lemma-characterize-universally-open-finite}." ], "refs": [ "limits-lemma-descend-quasi-finite", "more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-opens", "limits-lemma-descend-surjective", "limits-lemma-descend-finite-finite-presentation", "limits-lemma-descend-etale", "limits-lemma-limit-affine", "limits-lemma-descend-opens", "limits-lemma-descend-surjective", "more-morphisms-lemma-characterize-universally-open-finite" ], "ref_ids": [ 15105, 13892, 15077, 15041, 15069, 15058, 15065, 15043, 15041, 15069, 14083 ] } ], "ref_ids": [] }, { "id": 14085, "type": "theorem", "label": "more-morphisms-lemma-count-geometric-fibres", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-count-geometric-fibres", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism. Then", "\\begin{enumerate}", "\\item the functions $n_{X/Y}$ of", "Lemmas \\ref{lemma-base-change-fibres-nr-geometrically-irreducible-components}", "and \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}", "agree,", "\\item if $X$ is quasi-compact, then $n_{X/Y}$ attains a maximum $d < \\infty$.", "\\end{enumerate}" ], "refs": [ "more-morphisms-lemma-base-change-fibres-nr-geometrically-irreducible-components", "more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components" ], "proofs": [ { "contents": [ "Agreement of the functions is immediate from the fact that the", "(geometric) fibres of a locally quasi-finite morphism are discrete, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-fibres}.", "Boundedness follows from", "Morphisms, Lemmas \\ref{morphisms-lemma-characterize-universally-bounded} and", "\\ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded}." ], "refs": [ "morphisms-lemma-locally-quasi-finite-fibres", "morphisms-lemma-characterize-universally-bounded", "morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded" ], "ref_ids": [ 5228, 5524, 5531 ] } ], "ref_ids": [ 13822, 13829 ] }, { "id": 14086, "type": "theorem", "label": "more-morphisms-lemma-count-geometric-fibres-universally-open", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-count-geometric-fibres-universally-open", "contents": [ "Let $f : X \\to Y$ be a separated, locally quasi-finite, and universally open", "morphism of schemes. Let $n_{X/Y}$ be as in", "Lemma \\ref{lemma-count-geometric-fibres}.", "If $n_{X/Y}(y) \\geq d$ for some $y \\in Y$ and $d \\geq 0$,", "then $n_{X/Y} \\geq d$ in an open neighbourhood of $y$." ], "refs": [ "more-morphisms-lemma-count-geometric-fibres" ], "proofs": [ { "contents": [ "The question is local on $Y$ hence we may assume $Y$ affine.", "Let $K$ be an algebraic closure of the residue field $\\kappa(y)$.", "Our assumption is that $(X_y)_K$ has $\\geq d$ connected components.", "Then for a suitable quasi-compact open $X' \\subset X$", "the scheme $(X'_y)_K$ has $\\geq d$ connected components; details omitted.", "After replacing $X$ by $X'$ we may assume $X$ is quasi-compact.", "Then $f$ is quasi-finite. Let $x_1, \\ldots, x_n$ be the points of $X$", "lying over $y$. Apply", "Lemma \\ref{lemma-etale-splits-off-quasi-finite-part-technical-variant}", "to get an \\'etale neighbourhood $(U, u) \\to (Y, y)$ and a decomposition", "$$", "U \\times_Y X =", "W \\amalg", "\\ \\coprod\\nolimits_{i = 1, \\ldots, n}", "\\ \\coprod\\nolimits_{j = 1, \\ldots, m_i}", "V_{i, j}", "$$", "as in locus citatus. Observe that $n_{X/Y}(y) = \\sum_i m_i$", "in this situation; some details omitted.", "Since $f$ is universally open, we see", "that $V_{i, j} \\to U$ is open for all $i, j$. Hence after shrinking", "$U$ we may assume $V_{i, j} \\to U$ is surjective", "for all $i, j$. This proves that", "$n_{U \\times_Y X/U} \\geq \\sum_i m_i = n_{X/Y}(y) \\geq d$.", "Since the construction of $n_{X/Y}$ is compatible with", "base change the proof is complete." ], "refs": [ "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant" ], "ref_ids": [ 13896 ] } ], "ref_ids": [ 14085 ] }, { "id": 14087, "type": "theorem", "label": "more-morphisms-lemma-large-open", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-large-open", "contents": [ "Let $f : X \\to Y$ be a separated, locally quasi-finite, and universally open", "morphism of schemes. Let $n_{X/Y}$ be as in", "Lemma \\ref{lemma-count-geometric-fibres}.", "If $n_{X/Y}$ attains a maximum $d < \\infty$, then the set", "$$", "Y_d = \\{y \\in Y \\mid n_{X/Y}(y) = d\\}", "$$", "is open in $Y$ and the morphism $f^{-1}(Y_d) \\to Y_d$ is finite." ], "refs": [ "more-morphisms-lemma-count-geometric-fibres" ], "proofs": [ { "contents": [ "The openness of $Y_d$ is immediate from", "Lemma \\ref{lemma-count-geometric-fibres-universally-open}.", "To prove finiteness over $Y_d$ we redo the argument of the", "proof of that lemma. Namely, let $y \\in Y_d$. Then there are", "at most $d$ points of $X$ lying over $y$. Say", "$x_1, \\ldots, x_n$ are the points of $X$", "lying over $y$. Apply", "Lemma \\ref{lemma-etale-splits-off-quasi-finite-part-technical-variant}", "to get an \\'etale neighbourhood $(U, u) \\to (Y, y)$ and a decomposition", "$$", "U \\times_Y X =", "W \\amalg", "\\ \\coprod\\nolimits_{i = 1, \\ldots, n}", "\\ \\coprod\\nolimits_{j = 1, \\ldots, m_i}", "V_{i, j}", "$$", "as in locus citatus. Observe that $d = n_{X/Y}(y) = \\sum_i m_i$", "in this situation; some details omitted.", "Since $f$ is universally open, we see", "that $V_{i, j} \\to U$ is open for all $i, j$. Hence after shrinking", "$U$ we may assume $V_{i, j} \\to U$ is surjective", "for all $i, j$ and we may assume $U$ maps into $W$.", "This proves that $n_{U \\times_Y X/U} \\geq \\sum_i m_i = d$.", "Since the construction of $n_{X/Y}$ is compatible with", "base change we know that $n_{U \\times_Y X/U} = d$. This means that", "$W$ has to be empty and we conclude that $U \\times_Y X \\to U$ is finite.", "By Descent, Lemma \\ref{descent-lemma-descending-property-finite}", "this implies that $X \\to Y$ is finite over", "the image of the open morphism $U \\to Y$. In other words,", "we see that $f$ is finite over an open neighbourhood of $y$", "as desired." ], "refs": [ "more-morphisms-lemma-count-geometric-fibres-universally-open", "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant", "descent-lemma-descending-property-finite" ], "ref_ids": [ 14086, 13896, 14688 ] } ], "ref_ids": [ 14085 ] }, { "id": 14088, "type": "theorem", "label": "more-morphisms-lemma-weighting-check-after-etale-base-change", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-weighting-check-after-etale-base-change", "contents": [ "Given a cartesian square", "$$", "\\xymatrix{", "U \\ar[d]_\\pi & U' \\ar[l]^h \\ar[d]^{\\pi'} \\\\", "V & V' \\ar[l]_g", "}", "$$", "with $\\pi$ locally quasi-finite with finite fibres", "and a function $w : U \\to \\mathbf{Z}$", "we have $(\\int_\\pi w) \\circ g = \\int_{\\pi'} (w \\circ h)$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the second description of $\\int_\\pi w$", "given above. To prove it from the definition, you use that if", "$E/F$ is a finite extension of fields and $F'/F$ is another field extension,", "then writing $(E \\otimes_F F')_{red} = \\prod E'_i$ as a product of fields", "finite over $F'$, we have", "$$", "[E : F]_s = \\sum [E'_i : F']_s", "$$", "To prove this equality pick an algebraically closed field", "extension $\\Omega/F'$ and observe that", "\\begin{align*}", "[E : F]_s", "& =", "|\\Mor_F(E, \\Omega)| \\\\", "& =", "|\\Mor_{F'}(E \\otimes_F F', \\Omega)| \\\\", "& =", "|\\Mor_{F'}((E \\otimes_F F')_{red}, \\Omega)| \\\\", "& =", "\\sum |\\Mor_{F'}(E'_i, \\Omega)| \\\\", "& =", "\\sum [E'_i : F']_s", "\\end{align*}", "where we have used Fields, Lemma \\ref{fields-lemma-separable-degree}." ], "refs": [ "fields-lemma-separable-degree" ], "ref_ids": [ 4483 ] } ], "ref_ids": [] }, { "id": 14089, "type": "theorem", "label": "more-morphisms-lemma-weighting-base-change", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-weighting-base-change", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism.", "Let $w : X \\to \\mathbf{Z}$ be a weighting. Let $f' : X' \\to Y'$", "be the base change of $f$ by a morphism $Y' \\to Y$. Then the", "composition $w' : X' \\to \\mathbf{Z}$ of $w$ and the projection $X' \\to X$", "is a weighting of $f'$." ], "refs": [], "proofs": [ { "contents": [ "Consider a diagram", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} & U' \\ar[l]^{h'} \\ar[d]^{\\pi'} \\\\", "Y' & V' \\ar[l]_{g'}", "}", "$$", "as in Definition \\ref{definition-weighting} for the morphism $f'$.", "For any $v' \\in V'$ we have to show that $\\int_{\\pi'} (w' \\circ h')$ is", "constant in an open neighbourhood of $v'$. By", "Lemma \\ref{lemma-weighting-check-after-etale-base-change}", "(and the fact that \\'etale morphisms are open)", "we may replace $V'$ by any \\'etale neighbourhood of $v'$.", "After replacing $V'$ by an \\'etale neighbourhood of $v'$", "we may assume that $U' = U'_1 \\amalg \\ldots \\amalg U'_n$", "where each $U'_i$ has a unique point $u'_i$ lying over $v'$", "such that $\\kappa(u'_i)/\\kappa(v')$ is purely inseparable,", "see Lemma \\ref{lemma-etale-splits-off-quasi-finite-part-technical-variant}.", "Clearly, it suffices to prove that $\\int_{U'_i \\to V'} w'|_{U'_i}$", "is constant in a neighbourhood of $v'$.", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "We have $v' \\in V'$ and there is a unique point $u'$ of $U'$", "lying over $v'$ with $\\kappa(u')/\\kappa(v')$ purely inseparable.", "Denote $x \\in X$ and $y \\in Y$ the image of $u'$ and $v'$.", "We can find an \\'etale neighbourhood $(V, v) \\to (Y, y)$", "and an open $U \\subset X_V$ such that $\\pi : U \\to V$ is finite", "and such that there is a unique point $u \\in U$ lying over $v$", "which maps to $x \\in X$ via the projection $h : U \\to X$", "such that moreover $\\kappa(u)/\\kappa(v)$ is", "purely inseparable. This is possible by the lemma used above.", "Consider the morphism", "$$", "U'' = U \\times_X U' \\longrightarrow V \\times_Y V' = V''", "$$", "Since $u$ and $u'$ both map to $x \\in X$ there is a point", "$u'' \\in U''$ mapping to $(u, u')$. Denote $v'' \\in V''$", "the image of $u''$. After replacing $V', v'$ by $V'', v''$ ", "we may assume that the composition $V' \\to Y' \\to Y$ factors", "through a map of \\'etale neighbourhoods $(V', v') \\to (V, v)$", "such that the induced morphism $X'_{V'} = X_{V'} \\to X_V$ sends", "$u'$ to $u$. Inside the base change $X'_{V'} = X_{V'}$ we have", "two open subschemes, namely $U'$ and the inverse image $U_{V'}$ of", "$U \\subset X_V$. By construction both contain a unique point lying", "over $v'$, namely $u'$ for both of them.", "Thus after shrinking $V'$ we may assume these open subsets", "are the same; namely, $U' \\setminus (U' \\cap U_{V'})$ and", "$U_{V'} \\setminus (U' \\cap U_{V'})$ have", "a closed image in $V'$ and these images do not contain $v'$.", "Thus $U' = U_{V'}$ and we find a cartesian diagram as in", "Lemma \\ref{lemma-weighting-check-after-etale-base-change}.", "Since $\\int_\\pi (w \\circ h)$ is locally constant", "by assumption we conclude." ], "refs": [ "more-morphisms-definition-weighting", "more-morphisms-lemma-weighting-check-after-etale-base-change", "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant", "more-morphisms-lemma-weighting-check-after-etale-base-change" ], "ref_ids": [ 14126, 14088, 13896, 14088 ] } ], "ref_ids": [] }, { "id": 14090, "type": "theorem", "label": "more-morphisms-lemma-weighting-universally-open", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-weighting-universally-open", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism.", "Let $w : X \\to \\mathbf{Z}$ be a weighting. If $w(x) > 0$", "for all $x \\in X$, then $f$ is universally open." ], "refs": [], "proofs": [ { "contents": [ "Since the property is preserved by base change, see", "Lemma \\ref{lemma-weighting-base-change}, it suffices", "to prove that $f$ is open. Since we may also replace", "$X$ by any open of $X$, it suffices to prove that $f(X)$", "is open. Let $y \\in f(X)$. Choose $x \\in X$ with $f(x) = y$.", "It suffices to prove that $f(X)$ contains an open neighbourhood", "of $y$ and it suffices to do so after replacing $Y$ by an", "\\'etale neighbourhood of $y$. By \\'etale localization of", "quasi-finite morphisms, see Section \\ref{section-etale-localization},", "we may assume", "there is an open neighbourhood $U \\subset X$ of $x$", "such that $\\pi = f|_U : U \\to Y$ is finite. Then", "$\\int_\\pi w|_U$ is locally constant and has positive value at $y$.", "Hence $\\pi(U)$ contains an open neighbourhood of $y$", "and the proof is complete." ], "refs": [ "more-morphisms-lemma-weighting-base-change" ], "ref_ids": [ 14089 ] } ], "ref_ids": [] }, { "id": 14091, "type": "theorem", "label": "more-morphisms-lemma-weighting-flat-quasi-finite", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-weighting-flat-quasi-finite", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume $f$ is", "locally quasi-finite, locally of finite presentation, and flat.", "Then there is a positive weighting $w : X \\to \\mathbf{Z}_{> 0}$ of $f$", "given by the rule that sends $x \\in X$ lying over $y \\in Y$ to", "$$", "w(x) =", "\\text{length}_{\\mathcal{O}_{X, x}}", "(\\mathcal{O}_{X, x}/\\mathfrak m_y \\mathcal{O}_{X, x})", "[\\kappa(x) : \\kappa(y)]_i", "$$", "where $[\\kappa' : \\kappa]_i$ is the inseparable degree", "(Fields, Definition \\ref{fields-definition-insep-degree})." ], "refs": [ "fields-definition-insep-degree" ], "proofs": [ { "contents": [ "Consider a diagram as in Definition \\ref{definition-weighting}.", "Let $u \\in U$ with images $x, y, v$ in $X, Y, V$. Then we claim that", "$$", "\\text{length}_{\\mathcal{O}_{X, x}}", "(\\mathcal{O}_{X, x}/\\mathfrak m_y \\mathcal{O}_{X, x}) =", "\\text{length}_{\\mathcal{O}_{U, u}}", "(\\mathcal{O}_{U, u}/\\mathfrak m_v \\mathcal{O}_{U, u})", "$$", "and", "$$", "[\\kappa(x) : \\kappa(y)]_i =", "[\\kappa(u) : \\kappa(v)]_i", "$$", "The first equality follows as $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{U, u}$", "is a flat local homomorphism such that", "$\\mathfrak m_y \\mathcal{O}_{U, u} = \\mathfrak m_v \\mathcal{O}_{U, u}$", "and $\\mathfrak m_x \\mathcal{O}_{U, u} = \\mathfrak m_u$", "(because $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{V, v}$ and", "$\\mathcal{O}_{X, x} \\to \\mathcal{O}_{U, u}$ are unramified)", "and hence the equality by Algebra, Lemma \\ref{algebra-lemma-pullback-module}.", "The second equality follows because $\\kappa(v)/\\kappa(y)$ is a finite", "separable extension and $\\kappa(u)$ is a factor of", "$\\kappa(x) \\otimes_{\\kappa(y)} \\kappa(v)$ and hence the inseparable", "degree is unchanged. Having said this, we see that formation", "of the function in the lemma commutes with \\'etale base change.", "This reduces the problem to the discussion of the next paragraph.", "\\medskip\\noindent", "Assume that $f$ is a finite, flat morphism of finite presentation.", "We have to show that $\\int_f w$ is locally constant on $Y$.", "In fact, $f$ is finite locally free", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-flat})", "and we will show that $\\int_f w$ is equal to the degree of $f$", "(which is a locally constant function on $Y$). Namely,", "for $y \\in Y$ we see that", "\\begin{align*}", "(\\textstyle{\\int}_f w)(y)", "& =", "\\sum\\nolimits_{f(x) = y}", "\\text{length}_{\\mathcal{O}_{X, x}}", "(\\mathcal{O}_{X, x}/\\mathfrak m_y \\mathcal{O}_{X, x})", "[\\kappa(x) : \\kappa(y)]_i", "[\\kappa(x) : \\kappa(y)]_s \\\\", "& =", "\\sum\\nolimits_{f(x) = y}", "\\text{length}_{\\mathcal{O}_{X, x}}", "(\\mathcal{O}_{X, x}/\\mathfrak m_y \\mathcal{O}_{X, x})", "[\\kappa(x) : \\kappa(y)] \\\\", "& =", "\\text{length}_{\\mathcal{O}_{Y, y}}((f_*\\mathcal{O}_X)_y/", "\\mathfrak m_y (f_*\\mathcal{O}_X)_y)", "\\end{align*}", "Last equality by Algebra, Lemma \\ref{algebra-lemma-pushdown-module}.", "The final number is the rank of $f_*\\mathcal{O}_X$ at $y$ as desired." ], "refs": [ "more-morphisms-definition-weighting", "algebra-lemma-pullback-module", "morphisms-lemma-finite-flat", "algebra-lemma-pushdown-module" ], "ref_ids": [ 14126, 640, 5471, 639 ] } ], "ref_ids": [ 4540 ] }, { "id": 14092, "type": "theorem", "label": "more-morphisms-lemma-weighting-quasi-finite-Noetherian", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-weighting-quasi-finite-Noetherian", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Assume", "\\begin{enumerate}", "\\item $f$ is locally quasi-finite, and", "\\item $Y$ is unibranch and locally Noetherian.", "\\end{enumerate}", "Then there is a weighting $w : X \\to \\mathbf{Z}_{\\geq 0}$ given by", "the rule that sends $x \\in X$ lying over $y \\in Y$ to the", "``generic separable degree''", "of $\\mathcal{O}_{X, x}^{sh}$ over $\\mathcal{O}_{Y, y}^{sh}$." ], "refs": [], "proofs": [ { "contents": [ "It follows from Algebra, Lemmas", "\\ref{algebra-lemma-quasi-finite-strict-henselization}", "and \\ref{algebra-lemma-characterize-henselian}", "that $\\mathcal{O}_{Y, y}^{sh} \\to \\mathcal{O}_{X, x}^{sh}$", "is finite. Since $Y$ is unibranch", "there is a unique minimal prime $\\mathfrak p$ in", "$\\mathcal{O}_{Y, y}^{sh}$. Write", "$$", "(\\kappa(\\mathfrak p) \\otimes_{\\mathcal{O}_{Y, y}^{sh}}", "\\mathcal{O}_{X, x}^{sh})_{red} =", "\\prod K_i", "$$", "as a finite product of fields.", "We set $w(x) = \\sum [K_i : \\kappa(\\mathfrak p)]_s$.", "\\medskip\\noindent", "Since this definition is clearly insensitive to \\'etale localization,", "in order to show that $w$ is a weighting we reduce to showing that if", "$f$ is a finite morphism, then $\\int_f w$ is locally constant.", "Observe that the value of $\\int_f w$ in a generic point $\\eta$", "of $Y$ is just the number of points of the geometric fibre", "$X_{\\overline{\\eta}}$ of $X \\to Y$ over $\\eta$. Moreover, since", "$Y$ is unibranch a point $y$ of $Y$ is the specialization of a unique", "generic point $\\eta$. Hence it suffices to show that $(\\int_f w)(y)$", "is equal to the number of points of $X_{\\overline{\\eta}}$.", "After passing to an affine neighbourhood of $y$ we may assume", "$X \\to Y$ is given by a finite ring map $A \\to B$. Suppose", "$\\mathcal{O}_{Y, y}^{sh}$ is constructed using a map", "$\\kappa(y) \\to k$ into an algebraically closed field $k$.", "Then", "$$", "\\mathcal{O}_{Y, y}^{sh} \\otimes_A B =", "\\prod\\nolimits_{f(x) = y}", "\\prod\\nolimits_{\\varphi \\in \\Mor_{\\kappa(y)}(\\kappa(x), k)}", "\\mathcal{O}_{X, x}^{sh}", "$$", "by the references on strict henselization given above.", "Observe that the minimal prime $\\mathfrak p$ of $\\mathcal{O}_{Y, y}^{sh}$", "maps to the prime of $A$ corresponding to $\\eta$. Hence we see that", "the desired equality holds because the number of points of a geometric", "fibre is unchanged by a field extension." ], "refs": [ "algebra-lemma-quasi-finite-strict-henselization", "algebra-lemma-characterize-henselian" ], "ref_ids": [ 1306, 1276 ] } ], "ref_ids": [] }, { "id": 14093, "type": "theorem", "label": "more-morphisms-lemma-jumps-w", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-jumps-w", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism.", "Let $w : X \\to \\mathbf{Z}$ be a weighting of $f$. Then", "the level sets of the function $w$ are locally constructible in $X$." ], "refs": [], "proofs": [ { "contents": [ "The question is Zariski local on $X$ and $Y$, hence we may assume", "$X$ and $Y$ are affine. (Observe that the restriction of a weighting", "to an open is a weighting of the restriction of $f$ to that open.)", "By Lemma \\ref{lemma-weighting-base-change} after base change we still", "have a weighthing. If we can find a surjective morphism", "$Y' \\to Y$ of finite presentation such that the level sets of", "$w$ pull back to locally constructible subsets of $X' = Y' \\times_Y X$,", "then we conclude by Morphisms, Theorem \\ref{morphisms-theorem-chevalley}.", "We may choose an immersion $X \\to T$ where $T \\to Y$ is finite, see", "Lemma \\ref{lemma-quasi-finite-separated-pass-through-finite}.", "By Morphisms, Lemma \\ref{morphisms-lemma-massage-finite}", "after replacing $Y$ by $Y'$ surjective finite locally free over $Y$,", "replacing $X$ by $Y' \\times_Y X$ and $T$ by a scheme finite locally", "free over $Y'$ containing $Y' \\times_Y T$ as a closed subscheme,", "we may assume $T$ is finite locally free over $Y$,", "contains closed subschemes $T_i$ mapping isomorphically to $Y$", "such that $T = \\bigcup_{i = 1, \\ldots, n} T_i$ (set theoretically).", "Observe that for $I \\subset \\{1, \\ldots, n\\}$ the intersection", "$\\bigcap_{i \\in I} T_i$ is a constructible closed subset of $T$", "(and hence maps to a constructible closed subset of $Y$).", "\\medskip\\noindent", "For a disjoint union decomposition", "$\\{1, \\ldots, n\\} = I_1 \\amalg \\ldots \\amalg I_r$ with nonempty parts", "consider the subset $Y_{I_1, \\ldots, I_r} \\subset Y$", "consisting of points $y \\in Y$ such that $T_y = \\{x_1, \\ldots, x_r\\}$", "consists of exactly $r$ points with $x_j \\in T_i \\Leftrightarrow i \\in I_j$.", "This is a constructible partition of $Y$, hence we may", "assume that $Y = Y_{I_1, \\ldots, I_r}$ for some", "disjoint union decomposition", "$\\{1, \\ldots, n\\} = I_1 \\amalg \\ldots \\amalg I_r$.", "(You can base change by an affine scheme $Y'$ of finite presentation", "over $Y$ such that the image of $Y' \\to Y$ is exactly", "$Y_{I_1, \\ldots, I_r}$, see Algebra, Lemma", "\\ref{algebra-lemma-constructible-is-image}.)", "Thus $T = T(1) \\amalg \\ldots \\amalg T(r)$ with $T(j) = \\bigcap_{i \\in I_j} T_i$", "is a decomposition of $T$ into disjoint closed (and hence open) subsets.", "Intersecting with the locally closed subscheme $X$ we obtain an analogous", "decomposition $X = X(1) \\amalg \\ldots \\amalg X(r)$ into open and closed parts.", "The morphism $X(j) \\to Y$ an immersion.", "Since $w$ is a weigthing, it follows that $w|_{X(j)}$", "is locally constant\\footnote{In fact, if $f : X \\to Y$ is an", "immersion and $w$ is a weighting of $f$, then $f$ restricts to an", "open map on the locus where $w$ is nonzero.} and we conclude." ], "refs": [ "more-morphisms-lemma-weighting-base-change", "morphisms-theorem-chevalley", "more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "morphisms-lemma-massage-finite", "algebra-lemma-constructible-is-image" ], "ref_ids": [ 14089, 5123, 13901, 5475, 435 ] } ], "ref_ids": [] }, { "id": 14094, "type": "theorem", "label": "more-morphisms-lemma-jumps-int-w", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-jumps-int-w", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism of finite", "presentation. Let $w : X \\to \\mathbf{Z}$ be a weighting of $f$. Then", "the level sets of the function $\\int_f w$ are locally constructible in $Y$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-weighting-check-after-etale-base-change}", "formation of the function $\\int_f w$ commutes with arbitrary", "base change and by Lemma \\ref{lemma-weighting-base-change}", "after base change we still have a weighthing.", "This means that if we can find $Y' \\to Y$", "surjective and of finite presentation, then it", "suffices to prove the result after base change to $Y'$, see", "Morphisms, Theorem \\ref{morphisms-theorem-chevalley}.", "\\medskip\\noindent", "The question is local on $Y$ hence we may assume $Y$ is affine.", "Then $X$ is quasi-compact and quasi-separated", "(as $f$ is of finite presentation). Suppose that $X = U \\cup V$", "are quasi-compact open. Then we have", "$$", "\\textstyle{\\int}_f w =", "\\textstyle{\\int}_{f|_U} w|_U +", "\\textstyle{\\int}_{f|_V} w|_V -", "\\textstyle{\\int}_{f|_{U \\cap V}} w|_{U \\cap V}", "$$", "Thus if we know the result for $w|_U$, $w|_V$, $w|_{U \\cap V}$", "then we know the result for $w$. By the induction principle", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle})", "it suffices to prove the lemma when $X$ is affine.", "\\medskip\\noindent", "Assume $X$ and $Y$ are affine. We may choose an open immersion", "$X \\to T$ where $T \\to Y$ is finite, see", "Lemma \\ref{lemma-quasi-finite-separated-pass-through-finite}.", "Because we may still base change with a suitable $Y' \\to Y$", "we can use Morphisms, Lemma \\ref{morphisms-lemma-massage-finite}", "to reduce to the case where", "all residue field extensions induced by the morphism $T \\to Y$", "(and a foriori induced by $X \\to Y$) are trivial.", "In this situation $\\int_f w$ is just taking the sums", "of the values of $w$ in fibres. The level sets of $w$", "are locally constructible in $X$ (Lemma \\ref{lemma-jumps-w}).", "The function $w$ only takes a finite number of values by", "Properties, Lemma", "\\ref{properties-lemma-stratification-locally-finite-constructible}.", "Hence we conclude by", "Morphisms, Theorem \\ref{morphisms-theorem-chevalley}", "and some elementary arguments on sums of integers." ], "refs": [ "more-morphisms-lemma-weighting-check-after-etale-base-change", "more-morphisms-lemma-weighting-base-change", "morphisms-theorem-chevalley", "coherent-lemma-induction-principle", "more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "morphisms-lemma-massage-finite", "more-morphisms-lemma-jumps-w", "properties-lemma-stratification-locally-finite-constructible", "morphisms-theorem-chevalley" ], "ref_ids": [ 14088, 14089, 5123, 3291, 13901, 5475, 14093, 2944, 5123 ] } ], "ref_ids": [] }, { "id": 14095, "type": "theorem", "label": "more-morphisms-lemma-semicontinuous-w", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-semicontinuous-w", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism.", "Let $w : X \\to \\mathbf{Z}_{> 0}$ be a positive weighting of $f$.", "Then $w$ is upper semi-continuous." ], "refs": [], "proofs": [ { "contents": [ "Let $x \\in X$ with image $y \\in Y$. Choose an \\'etale neighbourhood", "$(V, v) \\to (Y, y)$ and an open $U \\subset X_V$ such that", "$\\pi : U \\to V$ is finite and there is a unique point $u \\in U$", "mapping to $v$ with $\\kappa(u)/\\kappa(v)$ purely inseparable.", "See Lemma \\ref{lemma-etale-makes-quasi-finite-finite-multiple-points-var}.", "Then $(\\int_\\pi w|_U)(v) = w(u)$.", "It follows from Definition \\ref{definition-weighting}", "that after replacing $V$ by a neighbourhood of $v$ we", "we have $w|_U(u') \\leq w|_U(u) = w(x)$ for all $u' \\in U$.", "Namely, $w|_U(u')$ occurs as a summand in the expression", "for $(\\int_\\pi w|_U)(\\pi(u'))$.", "This proves the lemma because the \\'etale morphism", "$U \\to X$ is open." ], "refs": [ "more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points-var", "more-morphisms-definition-weighting" ], "ref_ids": [ 13894, 14126 ] } ], "ref_ids": [] }, { "id": 14096, "type": "theorem", "label": "more-morphisms-lemma-semicontinuous-int-w", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-semicontinuous-int-w", "contents": [ "Let $f : X \\to Y$ be a separated, locally quasi-finite morphism", "with finite fibres.", "Let $w : X \\to \\mathbf{Z}_{> 0}$ be a positive weighting of $f$.", "Then $\\int_f w$ is lower semi-continuous." ], "refs": [], "proofs": [ { "contents": [ "Let $y \\in Y$. Let $x_1, \\ldots, x_r \\in X$ be the points lying over $y$.", "Apply", "Lemma \\ref{lemma-etale-splits-off-quasi-finite-part-technical-variant}", "to get an \\'etale neighbourhood $(U, u) \\to (Y, y)$ and a decomposition", "$$", "U \\times_Y X =", "W \\amalg", "\\ \\coprod\\nolimits_{i = 1, \\ldots, n}", "\\ \\coprod\\nolimits_{j = 1, \\ldots, m_i}", "V_{i, j}", "$$", "as in locus citatus. Observe that $(\\int_f w)(y) = \\sum w(v_{i, j})$", "where $w(v_{i, j}) = w(x_i)$. Since $\\int_{V_{i, j} \\to U} w|_{V_{i, j}}$", "is locally constant by definition, we may after shrinking $U$", "assume these functions are constant with value $w(v_{i, j})$.", "We conclude that", "$$", "\\textstyle{\\int}_{U \\times_Y X \\to U} w|_{U \\times_Y X} =", "\\textstyle{\\int}_{W \\to U} w|_W +", "\\sum \\textstyle{\\int}_{V_{i, j} \\to U} w|_{V_{i, j}} =", "\\textstyle{\\int}_{W \\to U} w|_W + (\\int_f w)(y)", "$$", "This is $\\geq (\\int_f w)(y)$ and we conclude because $U \\to Y$", "is open and formation of the integral commutes with base change", "(Lemma \\ref{lemma-weighting-check-after-etale-base-change})." ], "refs": [ "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant", "more-morphisms-lemma-weighting-check-after-etale-base-change" ], "ref_ids": [ 13896, 14088 ] } ], "ref_ids": [] }, { "id": 14097, "type": "theorem", "label": "more-morphisms-lemma-max-int-w", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-max-int-w", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism", "with $X$ quasi-compact. Let $w : X \\to \\mathbf{Z}$ be a weighting of $f$.", "Then $\\int_f w$ attains its maximum." ], "refs": [], "proofs": [ { "contents": [ "It follows from Lemma \\ref{lemma-jumps-w} and", "Properties, Lemma", "\\ref{properties-lemma-stratification-locally-finite-constructible}", "that $w$ only takes a finite number of values on $X$.", "It follows from Morphisms, Lemma", "\\ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded}", "that $X \\to Y$ has bounded geometric fibres.", "This shows that $\\int_f w$ is bounded." ], "refs": [ "more-morphisms-lemma-jumps-w", "properties-lemma-stratification-locally-finite-constructible", "morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded" ], "ref_ids": [ 14093, 2944, 5531 ] } ], "ref_ids": [] }, { "id": 14098, "type": "theorem", "label": "more-morphisms-lemma-max-int-finite", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-max-int-finite", "contents": [ "Let $f : X \\to Y$ be a separated, locally quasi-finite morphism.", "Let $w : X \\to \\mathbf{Z}_{> 0}$ be a positive weighting of $f$.", "Assume $\\int_w f$ attains its maximum $d$ and let $Y_d \\subset Y$", "be the open set of points $y$ with $(\\int_f w)(y) = d$. Then", "the morphism $f^{-1}(Y_d) \\to Y_d$ is finite." ], "refs": [], "proofs": [ { "contents": [ "Observe that $Y_d$ is open by Lemma \\ref{lemma-semicontinuous-int-w}.", "Let $y \\in Y_d$. Say $x_1, \\ldots, x_n$ are the points of $X$", "lying over $y$. Apply", "Lemma \\ref{lemma-etale-splits-off-quasi-finite-part-technical-variant}", "to get an \\'etale neighbourhood $(U, u) \\to (Y, y)$ and a decomposition", "$$", "U \\times_Y X =", "W \\amalg", "\\ \\coprod\\nolimits_{i = 1, \\ldots, n}", "\\ \\coprod\\nolimits_{j = 1, \\ldots, m_i}", "V_{i, j}", "$$", "as in locus citatus. Observe that $d = \\sum w(v_{i, j})$ where", "$w(v_{i, j}) = w(x_i)$. Since $\\int_{V_{i, j} \\to U} w|_{V_{i, j}}$", "is locally constant by definition, we may after shrinking $U$", "assume these functions are constant with value $w(v_{i, j})$.", "We conclude that", "$$", "\\textstyle{\\int}_{U \\times_Y X \\to U} w|_{U \\times_Y X} =", "\\textstyle{\\int}_{W \\to U} w|_W +", "\\sum \\textstyle{\\int}_{V_{i, j} \\to U} w|_{V_{i, j}} =", "\\textstyle{\\int}_{W \\to U} w|_W + (\\int_f w)(y)", "$$", "This is $\\geq (\\int_f w)(y) = d$ and we conclude that $W$", "must be the emptyset. Thus $U \\times_Y X \\to U$ is finite.", "By Descent, Lemma \\ref{descent-lemma-descending-property-finite}", "this implies that $X \\to Y$ is finite over", "the image of the open morphism $U \\to Y$. In other words,", "we see that $f$ is finite over an open neighbourhood of $y$", "as desired." ], "refs": [ "more-morphisms-lemma-semicontinuous-int-w", "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant", "descent-lemma-descending-property-finite" ], "ref_ids": [ 14096, 13896, 14688 ] } ], "ref_ids": [] }, { "id": 14099, "type": "theorem", "label": "more-morphisms-lemma-open-and-closed-in-finite", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-open-and-closed-in-finite", "contents": [ "Let $A \\to B$ be a ring map which is finite and of finite presentation.", "There exists a finitely presented ring map $A \\to A_{univ}$", "and an idempotent $e_{univ} \\in B \\otimes_A A_{univ}$", "such that for any ring map $A \\to A'$ and idempotent $e \\in B \\otimes_A A'$", "there is a ring map $A_{univ} \\to A'$ mapping $e_{univ}$ to $e$." ], "refs": [], "proofs": [ { "contents": [ "Choose $b_1, \\ldots, b_n \\in B$ generating $B$ as an $A$-module.", "For each $i$ choose a monic $P_i \\in A[x]$ such that $P_i(b_i) = 0$", "in $B$, see Algebra, Lemma \\ref{algebra-lemma-finite-is-integral}.", "Thus $B$ is a quotient of the finite free $A$-algebra", "$B' = A[x_1, \\ldots, x_n]/(P_1(x_1), \\ldots, P_n(x_n))$.", "Let $J \\subset B'$ be the kernel of the surjection $B' \\to B$.", "Then $J =(f_1, \\ldots, f_m)$ is finitely generated as $B$", "is a finitely generated $A$-algebra, see", "Algebra, Lemma \\ref{algebra-lemma-compose-finite-type}.", "Choose an $A$-basis $b'_1, \\ldots, b'_N$ of $B'$.", "Consider the algebra", "$$", "A_{univ} = A[z_1, \\ldots, z_N, y_1, \\ldots, y_m]/I", "$$", "where $I$ is the ideal generated by the coefficients in", "$A[z_1, \\ldots, z_n, y_1, \\ldots, y_m]$", "of the basis elements $b'_1, \\ldots, b'_N$ of the expresssion", "$$", "(\\sum z_j b'_j)^2 - \\sum z_j b'_j + \\sum y_k f_k", "$$", "in $B'[z_1, \\ldots, z_N, y_1, \\ldots, y_m]$. By construction", "the element $\\sum z_j b'_j$ maps to an idempotent $e_{univ}$ in the", "algebra $B \\otimes_A A_{univ}$. Moreover, if $e \\in B \\otimes_A A'$", "is an idempotent, then we can lift $e$ to an element of the form", "$\\sum b'_j \\otimes a'_j$ in $B' \\otimes_A A'$ and we can find", "$a''_k \\in A'$ such that", "$$", "(\\sum b'_j \\otimes a'_j)^2 - \\sum b'_j \\otimes a'_j + \\sum f_k \\otimes a''_k", "$$", "is zero in $B' \\otimes_A A'$. Hence we get an $A$-algebra map", "$A_{univ} \\to A$ sending $z_j$ to $a'_j$ and $y_k$ to $a''_k$", "mapping $e_{univ}$ to $e$. This finishes the proof." ], "refs": [ "algebra-lemma-finite-is-integral", "algebra-lemma-compose-finite-type" ], "ref_ids": [ 482, 333 ] } ], "ref_ids": [] }, { "id": 14100, "type": "theorem", "label": "more-morphisms-lemma-open-and-closed-in-quasi-finite", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-open-and-closed-in-quasi-finite", "contents": [ "Let $X \\to Y$ be a morphism of affine schemes which is quasi-finite and", "of finite presentation. There exists a morphism $Y_{univ} \\to Y$", "of finite presentation and an open subscheme", "$U_{univ} \\subset Y_{univ} \\times_Y X$ such that", "$U_{univ} \\to Y_{univ}$ is finite with the following property:", "given any morphism $Y' \\to Y$ of affine schemes", "and an open subscheme $U' \\subset Y' \\times_Y X$", "such that $U' \\to Y'$ is finite, there exists a morphism", "$Y' \\to Y_{univ}$ such that the inverse image of $U_{univ}$ is $U'$." ], "refs": [], "proofs": [ { "contents": [ "Recall that a finite type morphism is quasi-finite if and only", "if it has relative dimension $0$, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}.", "By Lemma \\ref{lemma-Noetherian-approximation-dimension-d}", "applied with $d = 0$ we reduce to the case where", "$X$ and $Y$ are Noetherian. We may choose an open immersion", "$X \\to X'$ such that $X' \\to Y$ is finite, see", "Algebra, Lemma \\ref{algebra-lemma-quasi-finite-open-integral-closure}.", "Note that if we have $Y' \\to Y$ and $U'$ as in (2), then", "$$", "U' \\to Y' \\times_Y X \\to Y' \\times_Y X'", "$$", "is open immersion between schemes finite over $Y'$ and hence", "is closed as well. We conclude that $U'$ corresponds to an", "idempotent in", "$$", "\\Gamma(Y', \\mathcal{O}_{Y'})", "\\otimes_{\\Gamma(Y, \\mathcal{O}_Y)}", "\\Gamma(X', \\mathcal{O}_{X'})", "$$", "whose corresponding open and closed subset is contained in", "the open $Y' \\times_Y X$. Let $Y'_{univ} \\to Y$ and idempotent", "$$", "e'_{univ} \\in", "\\Gamma(Y_{univ}, \\mathcal{O}_{Y_{univ}})", "\\otimes_{\\Gamma(Y, \\mathcal{O}_Y)}", "\\Gamma(X', \\mathcal{O}_{X'})", "$$", "be the pair constructed in Lemma \\ref{lemma-open-and-closed-in-finite}", "for the ring map $\\Gamma(Y, \\mathcal{O}_Y) \\to \\Gamma(X', \\mathcal{O}_{X'})$", "(here we use that $Y$ is Noetherian to see that $X'$ is of finite presentation", "over $Y$). Let $U'_{univ} \\subset Y'_{univ} \\times_Y X'$ be the corresponding", "open and closed subscheme. Then we see that", "$$", "U'_{univ} \\setminus Y'_{univ} \\times_Y X", "$$", "is a closed subset of $U'_{univ}$ and hence has closed image", "$T \\subset Y'_{univ}$. If we set $Y_{univ} = Y'_{univ} \\setminus T$", "and $U_{univ}$ the restriction of $U'_{univ}$ to", "$Y_{univ} \\times_Y X$, then we see that the lemma is true." ], "refs": [ "morphisms-lemma-locally-quasi-finite-rel-dimension-0", "more-morphisms-lemma-Noetherian-approximation-dimension-d", "algebra-lemma-quasi-finite-open-integral-closure", "more-morphisms-lemma-open-and-closed-in-finite" ], "ref_ids": [ 5287, 13863, 1067, 14099 ] } ], "ref_ids": [] }, { "id": 14101, "type": "theorem", "label": "more-morphisms-lemma-descend-weighting", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-descend-weighting", "contents": [ "Let $Y = \\lim Y_i$ be a directed limit of affine schemes. Let $0 \\in I$", "and let $f_0 : X_0 \\to Y_0$ be a morphism of affine schemes which is", "quasi-finite and of finite presentation. Let $f : X \\to Y$", "and $f_i : X_i \\to Y_i$ for $i \\geq 0$ be the base changes of $f_0$.", "If $w : X \\to \\mathbf{Z}$ is a weighting of $f$, then for sufficiently", "large $i$ there exists a weighting $w_i : X_i \\to \\mathbf{Z}$", "of $f_i$ whose pullback to $X$ is $w$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-jumps-w} the level sets of $w$ are constructible subsets", "$E_k$ of $X$. This implies the function $w$ only takes a finite number", "of values by Properties, Lemma", "\\ref{properties-lemma-stratification-locally-finite-constructible}.", "Thus there exists an $i$ such that $E_k$ descends to a construcible", "subset $E_{i, k}$ in $X_i$ for all $k$; moreover, we may assume", "$X_i = \\coprod E_{i, k}$. This follows as the topological space", "of $X$ is the limit in the category of topological spaces", "of the spectral spaces $X_i$ along a directed system with", "spectral transition maps. See", "Limits, Section \\ref{limits-section-descent}", "and", "Topology, Section \\ref{topology-section-limits-spectral}.", "We define $w_i : X_i \\to \\mathbf{Z}$ such that its level", "sets are the constructible sets $E_{i, k}$.", "\\medskip\\noindent", "Choose $Y_{i, univ} \\to Y_i$ and", "$U_{i, univ} \\subset Y_{i, univ} \\times_{Y_i} X_i$", "as in Lemma \\ref{lemma-open-and-closed-in-quasi-finite}.", "By the universal property of the construction, in order to", "show that $w_i$ is a weighting, it would suffice to show", "that", "$$", "\\tau_i = \\textstyle{\\int}_{U_{i, univ} \\to Y_{i, univ}} w_i|_{U_{i, univ}}", "$$", "is locally constant on $Y_{i, univ}$. By Lemma \\ref{lemma-jumps-int-w}", "this function has constructible level sets but it may", "not (yet) be locally constant. Set", "$Y_{univ} = Y_{i, univ} \\times_{Y_i} Y$", "and let $U_{univ} \\subset Y_{univ} \\times_Y X$", "be the inverse image of $U_{i, univ}$.", "Then, since the pullback of $w$ to $Y_{univ} \\times_Y X$", "is a weighting for $Y_{univ} \\times_Y X \\to Y_{univ}$", "(Lemma \\ref{lemma-weighting-base-change})", "we do have that", "$$", "\\tau = \\textstyle{\\int}_{U_{univ} \\to Y_{univ}} w_i|_{U_{univ}}", "$$", "is locally constant on $Y_{univ}$. Thus the level sets of", "$\\tau$ are open and closed. Finally, we have", "$Y_{univ} = \\lim_{i' \\geq i} Y_{i', univ}$", "and the level sets of $\\tau$ are the inverse limits of the", "level sets of $\\tau_{i'}$ (similarly defined).", "Hence the references above imply that for sufficiently", "large $i'$ the level sets of $\\tau_{i'}$ are open as well.", "For such an index $i'$ we conclude that $w_{i'}$", "is a weighting of $f_{i'}$ as desired." ], "refs": [ "more-morphisms-lemma-jumps-w", "properties-lemma-stratification-locally-finite-constructible", "more-morphisms-lemma-open-and-closed-in-quasi-finite", "more-morphisms-lemma-jumps-int-w", "more-morphisms-lemma-weighting-base-change" ], "ref_ids": [ 14093, 2944, 14100, 14094, 14089 ] } ], "ref_ids": [] }, { "id": 14102, "type": "theorem", "label": "more-morphisms-lemma-affineness-of-large-open", "categories": [ "more-morphisms" ], "title": "more-morphisms-lemma-affineness-of-large-open", "contents": [ "Let $f : X \\to Y$ be a morphism of affine schemes which is", "quasi-finite and of finite presentation.", "Let $w : X \\to \\mathbf{Z}_{> 0}$ be a postive weighting of $f$.", "Let $d < \\infty$ be the maximum value of $\\int_f w$. The open", "$$", "Y_d = \\{y \\in Y \\mid (\\textstyle{\\int}_f w)(y) = d \\}", "$$", "of $Y$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\int_f w$ attains its maximum by Lemma \\ref{lemma-max-int-w}.", "The set $Y_d$ is open by Lemma \\ref{lemma-semicontinuous-int-w}.", "Thus the statement of the lemma makes sense.", "\\medskip\\noindent", "Reduction to the Noetherian case; please skip this paragraph.", "Recall that a finite type morphism is quasi-finite if and only", "if it has relative dimension $0$, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}.", "By Lemma \\ref{lemma-Noetherian-approximation-dimension-d}", "applied with $d = 0$ we can find a quasi-finite morphism $f_0 : X_0 \\to Y_0$", "of affine Noetherian schemes and a morphism $Y \\to Y_0$ such that $f$", "is the base change of $f_0$. Then we can write $Y = \\lim Y_i$ as a directed", "limit of affine schemes of finite type over $Y_0$, see", "Algebra, Lemma \\ref{algebra-lemma-ring-colimit-fp}.", "By Lemma \\ref{lemma-descend-weighting}", "we can find an $i$ such that our weighting $w$", "descends to a weighting $w_i$ of the base change $f_i : X_i \\to Y_i$", "of $f_0$. Now if the lemma holds for $f_i, w_i$, then it implies", "the lemma for $f$ as formation of $\\int_f w$ commutes with base", "change, see Lemma \\ref{lemma-weighting-check-after-etale-base-change}.", "\\medskip\\noindent", "Assume $X$ and $Y$ Noetherian. Let $X' \\to Y'$ be the base change of $f$", "by a morphism $g : Y' \\to Y$. The formation of $\\int_f w$ and hence", "the open $Y_d$ commute with base change. If $g$ is finite and surjective, then", "$Y'_d \\to Y_d$ is finite and surjective. In this case proving that", "$Y_d$ is affine is equivalent to showing that $Y'_d$ is affine, see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-image-affine-finite-morphism-affine-Noetherian}.", "\\medskip\\noindent", "We may choose an immersion $X \\to T$ with $T$ finite over $Y$, see Lemma", "\\ref{lemma-quasi-finite-separated-pass-through-finite}.", "We are going to apply Morphisms, Lemma \\ref{morphisms-lemma-massage-finite}", "to the finite morphism $T \\to Y$. This lemma tells us that", "there is a finite surjective morphism $Y' \\to Y$ such that", "$Y' \\times_Y T$ is a closed subscheme of a scheme $T'$ finite over $Y'$", "which has a special form.", "By the discussion in the first paragraph, we may replace $Y$ by $Y'$,", "$T$ by $T'$, and $X$ by $Y' \\times_Y X$.", "Thus we may assume there is an immersion $X \\to T$ (not necessarily", "open or closed) and closed subschemes", "$T_i \\subset T$, $i = 1, \\ldots, n$ where", "\\begin{enumerate}", "\\item $T \\to Y$ is finite (and locally free),", "\\item $T_i \\to Y$ is an isomorphism, and", "\\item $T = \\bigcup_{i = 1, \\ldots, n} T_i$ set theoretically.", "\\end{enumerate}", "Let $Y' = \\coprod Y_k$ be the disjoint union of the irreducible", "components of $Y$ (viewed as integral closed subschemes of $Y$).", "Then we may base change once more by $Y' \\to Y$; here we", "are using that $Y$ is Noetherian. Thus we may in", "addition assume $Y$ is integral and Noetherian.", "\\medskip\\noindent", "We also may and do assume that $T_i \\not = T_j$ if $i \\not = j$ by", "removing repeats. Since $Y$ and hence all $T_i$ are integral, this", "means that if $T_i$ and $T_j$ intersect, then they intersect in a", "closed subset which maps to a proper closed subset of $Y$.", "\\medskip\\noindent", "Observe that $V_i = X \\cap T_i$ is a locally closed subset", "which is in addition a closed subscheme of $X$ hence affine.", "Let $\\eta \\in Y$ and $\\eta_i \\in T_i$ be the generic points.", "If $\\eta \\not \\in Y_d$, then $Y_d = \\emptyset$ and we're done.", "Assume $\\eta \\in Y_d$. Denote $I \\in \\{1, \\ldots, n\\}$", "the subset of indices $i$ such that $\\eta_i \\in V_i$.", "For $i \\in I$ the locally closed subset $V_i \\subset T_i$", "contains the generic point of the irreducible space $T_i$", "and hence is open. On the other hand, since $f$ is open", "(Lemma \\ref{lemma-weighting-universally-open}),", "for any $x \\in X$ we can find an $i \\in I$ and a specialization", "$\\eta_i \\leadsto x$. It follows that $x \\in T_i$ and hence", "$x \\in V_i$. In other words, we see that $X = \\bigcup_{i \\in I} V_i$", "set theoretically.", "We claim that $Y_d = \\bigcap_{i \\in I} \\Im(V_i \\to Y)$; this will", "finish the proof as the intersection of affine opens", "$\\Im(V_i \\to Y)$ of $Y$ is affine.", "\\medskip\\noindent", "For $y \\in Y$ let $f^{-1}(\\{y\\}) = \\{x_1, \\ldots, x_r\\}$ in $X$.", "For each $i \\in I$ there is at most one $j(i) \\in \\{1, \\ldots, x_r\\}$", "such that $\\eta_i \\leadsto x_{j(i)}$. In fact, $j(i)$ exists and is", "equal to $j$ if and only if $x_j \\in V_i$. If $i \\in I$ is such that", "$j = j(i)$ exists, then $V_i \\to Y$ is an isomorphism in a neighbourhood", "of $x_j \\mapsto y$. Hence $\\bigcup_{i \\in I,\\ j(i) = j} V_i \\to Y$", "is finite after replacing source and target by neighbourhoods of", "$x_j \\mapsto y$. Thus the definition of a weighting tells us that", "$w(x_j) = \\sum_{i \\in I,\\ j(i) = j} w(\\eta_i)$.", "Thus we see that", "$$", "(\\textstyle{\\int}_f w)(\\eta) =", "\\sum\\nolimits_{i \\in I} w(\\eta_i) \\geq", "\\sum\\nolimits_{j(i)\\text{ exists}} w(\\eta_i) =", "\\sum\\nolimits_j w(x_j) = (\\textstyle{\\int}_f w)(y)", "$$", "Thus equality holds if and only if $y$ is contained in", "$\\bigcap_{i \\in I} \\Im(V_i \\to Y)$ which is what we wanted to show." ], "refs": [ "more-morphisms-lemma-max-int-w", "more-morphisms-lemma-semicontinuous-int-w", "morphisms-lemma-locally-quasi-finite-rel-dimension-0", "more-morphisms-lemma-Noetherian-approximation-dimension-d", "algebra-lemma-ring-colimit-fp", "more-morphisms-lemma-descend-weighting", "more-morphisms-lemma-weighting-check-after-etale-base-change", "coherent-lemma-image-affine-finite-morphism-affine-Noetherian", "more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "morphisms-lemma-massage-finite", "more-morphisms-lemma-weighting-universally-open" ], "ref_ids": [ 14097, 14096, 5287, 13863, 1091, 14101, 14088, 3337, 13901, 5475, 14090 ] } ], "ref_ids": [] }, { "id": 14103, "type": "theorem", "label": "more-morphisms-proposition-pushout-along-closed-immersion-and-integral", "categories": [ "more-morphisms" ], "title": "more-morphisms-proposition-pushout-along-closed-immersion-and-integral", "contents": [ "\\begin{reference}", "\\cite[Theorem 7.1 part iii]{Ferrand-Conducteur}", "\\end{reference}", "In Situation \\ref{situation-pushout-along-closed-immersion-and-integral}", "the pushout $Y \\amalg_Z X$ exists in the category of schemes. Picture", "$$", "\\xymatrix{", "Z \\ar[r]_i \\ar[d]_j & X \\ar[d]^a \\\\", "Y \\ar[r]^-b & Y \\amalg_Z X", "}", "$$", "The diagram is a fibre square, the morphism $a$ is integral,", "the morphism $b$ is a closed immersion, and", "$$", "\\mathcal{O}_{Y \\amalg_Z X} =", "b_*\\mathcal{O}_Y \\times_{c_*\\mathcal{O}_Z} a_*\\mathcal{O}_X", "$$", "as sheaves of rings where $c = a \\circ i = b \\circ j$." ], "refs": [], "proofs": [ { "contents": [ "As a topological space we set $Y \\amalg_Z X$ equal to the pushout of", "the diagram in the category of topological spaces (Topology, Section", "\\ref{topology-section-colimits}). This is just the pushout", "of the underlying sets (Topology, Lemma \\ref{topology-lemma-colimits})", "endowed with the quotient topology.", "On $Y \\amalg_Z X$ we have the maps of sheaves of rings", "$$", "b_*\\mathcal{O}_Y \\longrightarrow c_*\\mathcal{O}_Z", "\\longleftarrow a_*\\mathcal{O}_X", "$$", "and we can define", "$$", "\\mathcal{O}_{Y \\amalg_Z X} =", "b_*\\mathcal{O}_Y \\times_{c_*\\mathcal{O}_Z} a_*\\mathcal{O}_X", "$$", "as the fibre product in the category of sheaves of rings. To prove that we", "obtain a scheme we have to show that every point has an", "affine open neighbourhood. This is clear for points not in the image of $c$", "as the image of $c$ is a closed subset whose complement is", "isomorphic as a ringed space to $(Y \\setminus j(Z)) \\amalg (X \\setminus i(Z))$.", "\\medskip\\noindent", "A point in the image of $c$ corresponds to a unique $y \\in Y$", "in the image of $j$. By", "Lemma \\ref{lemma-prepare-pushout-along-closed-immersion-and-integral}", "we find affine opens $U \\subset X$ and $V \\subset Y$ with", "$y \\in V$ and $i^{-1}(U) = j^{-1}(V)$.", "Since the construction of the first paragraph is clearly compatible", "with restriction to compatible open subschemes, to prove that it", "produces a scheme we may assume $X$, $Y$, and $Z$ are affine.", "\\medskip\\noindent", "If $X = \\Spec(A)$, $Y = \\Spec(B)$, and $Z = \\Spec(C)$ are affine, then", "More on Algebra, Lemma \\ref{more-algebra-lemma-points-of-fibre-product}", "shows that $Y \\amalg_Z X = \\Spec(B \\times_C A)$ as topological spaces.", "To finish the proof that $Y \\times_Z X$ is a scheme, it suffices to show", "that on $\\Spec(B \\times_C A)$ the structure sheaf is the fibre product", "of the pushforwards. This follows by applying", "More on Algebra, Lemma \\ref{more-algebra-lemma-diagram-localize}", "to principal affine opens of $\\Spec(B \\times_C A)$.", "\\medskip\\noindent", "The discussion above shows the scheme $Y \\amalg_X Z$", "has an affine open covering $Y \\amalg_X Z = \\bigcup W_i$", "such that $U_i = a^{-1}(W_i)$, $V_i = b^{-1}(W_i)$, and", "$\\Omega_i = c^{-1}(W_i)$ are affine open in $X$, $Y$, and $Z$.", "Thus $a$ and $b$ are affine.", "Moreover, if $A_i$, $B_i$, $C_i$ are the rings corresponding to", "$U_i$, $V_i$, $\\Omega_i$, then $A_i \\to C_i$ is surjective and", "$W_i$ corresponds to $A_i \\times_{C_i} B_i$ which surjects onto", "$B_i$. Hence $b$ is a closed immersion.", "The ring map $A_i \\times_{C_i} B_i \\to A_i$ is integral by", "More on Algebra, Lemma \\ref{more-algebra-lemma-fibre-product-integral}", "hence $a$ is integral. The diagram is cartesian because", "$$", "C_i \\cong B_i \\otimes_{B_i \\times_{C_i} A_i} A_i", "$$", "This follows as $B_i \\times_{C_i} A_i \\to B_i$", "and $A_i \\to C_i$ are surjective maps whose kernels are the same.", "\\medskip\\noindent", "Finally, we can apply Lemmas \\ref{lemma-basic-example-pushout} and", "\\ref{lemma-pushout-fpqc-local} to conclude our construction is a pushout", "in the category of schemes." ], "refs": [ "topology-lemma-colimits", "more-morphisms-lemma-prepare-pushout-along-closed-immersion-and-integral", "more-algebra-lemma-points-of-fibre-product", "more-algebra-lemma-diagram-localize", "more-algebra-lemma-fibre-product-integral", "more-morphisms-lemma-pushout-fpqc-local" ], "ref_ids": [ 8337, 14046, 9818, 9816, 9819, 13761 ] } ], "ref_ids": [] }, { "id": 14104, "type": "theorem", "label": "more-morphisms-proposition-vanishing-affine-stratification-number", "categories": [ "more-morphisms" ], "title": "more-morphisms-proposition-vanishing-affine-stratification-number", "contents": [ "Let $X$ be a nonempty quasi-compact and quasi-separated scheme with", "affine stratification number $n$. Then $H^p(X, \\mathcal{F}) = 0$, $p > n$", "for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "We will prove this by induction on the affine stratification number $n$.", "If $n = 0$, then $X$ is affine and the result is", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.", "Assume $n > 0$. By Definition \\ref{definition-affine-stratification-number}", "there is an affine scheme $U$ and an affine open immersion $j : U \\to X$", "such that the complement $Z$ has affine stratification number $n - 1$.", "As $U$ and $j$ are affine we have $H^p(X, j_*(\\mathcal{F}|_U)) = 0$", "for $p > 0$, see Cohomology of Schemes, Lemmas", "\\ref{coherent-lemma-relative-affine-cohomology} and", "\\ref{coherent-lemma-relative-affine-vanishing}.", "Denote $\\mathcal{K}$ and $\\mathcal{Q}$ the kernel and cokernel", "of the map $\\mathcal{F} \\to j_*(\\mathcal{F}|_U)$. Thus we obtain", "an exact sequence", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{F} \\to j_*(\\mathcal{F}|_U)", "\\to \\mathcal{Q} \\to 0", "$$", "of quasi-coherent $\\mathcal{O}_X$-modules (see", "Schemes, Section \\ref{schemes-section-quasi-coherent}).", "A standard argument, breaking our exact sequence into short", "exact sequences and using the long exact cohomology sequence,", "shows it suffices to prove $H^p(X, \\mathcal{K}) = 0$", "and $H^p(X, \\mathcal{Q}) = 0$ for $p \\geq n$.", "Since $\\mathcal{F} \\to j_*(\\mathcal{F}|_U)$ restricts", "to an isomorphism over $U$, we see that $\\mathcal{K}$", "and $\\mathcal{Q}$ are supported on $Z$.", "By Properties, Lemma", "\\ref{properties-lemma-quasi-coherent-colimit-finite-type}", "we can write these modules as the filtered colimits of", "their finite type quasi-coherent submodules.", "Using the fact that cohomology of sheaves on $X$ commutes", "with filtered colimits, see", "Cohomology, Lemma \\ref{cohomology-lemma-quasi-separated-cohomology-colimit},", "we conclude it suffices to show that if $\\mathcal{G}$", "is a finite type quasi-coherent module whose support", "is contained in $Z$, then $H^p(X, \\mathcal{G}) = 0$ for $p \\geq n$.", "Let $Z' \\subset X$ be the scheme theoretic support of", "$\\mathcal{G} \\oplus \\mathcal{O}_Z$; we may and do think", "of $\\mathcal{G}$ as a quasi-coherent module on $Z'$, see", "Morphisms, Section \\ref{morphisms-section-support}.", "Then $Z'$ and $Z$ have the same underlying topological space", "and hence the same affine stratification number, namely $n - 1$.", "Hence $H^p(X, \\mathcal{G}) = H^p(Z', \\mathcal{G})$", "(equality by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-relative-affine-cohomology})", "vanishes for $p \\geq n$ by induction hypothesis." ], "refs": [ "coherent-lemma-quasi-coherent-affine-cohomology-zero", "more-morphisms-definition-affine-stratification-number", "coherent-lemma-relative-affine-cohomology", "coherent-lemma-relative-affine-vanishing", "properties-lemma-quasi-coherent-colimit-finite-type", "cohomology-lemma-quasi-separated-cohomology-colimit", "coherent-lemma-relative-affine-cohomology" ], "ref_ids": [ 3282, 14125, 3284, 3283, 3020, 2082, 3284 ] } ], "ref_ids": [] }, { "id": 14105, "type": "theorem", "label": "more-morphisms-proposition-asn-weighting", "categories": [ "more-morphisms" ], "title": "more-morphisms-proposition-asn-weighting", "contents": [ "Let $f : X \\to Y$ be a surjective quasi-finite morphism of schemes.", "Let $w : X \\to \\mathbf{Z}_{> 0}$ be a positive weighting of $f$.", "Assume $X$ affine and $Y$ separated\\footnote{It suffices if the", "diagonal of $Y$ is affine.}. Then the affine stratification", "number of $Y$ is at most the number of distinct values of $\\int_f w$." ], "refs": [], "proofs": [ { "contents": [ "Note that since $Y$ is separated, the morphism $X \\to Y$ is affine", "(Morphisms, Lemma \\ref{morphisms-lemma-affine-permanence}).", "The function $\\int_f w$ attains its maximum $d$ by", "Lemma \\ref{lemma-max-int-w}. We will use induction on $d$.", "Consider the open subscheme $Y_d = \\{y \\in Y \\mid (\\int_f w)(y) = d\\}$", "of $Y$ and recall that $f^{-1}(Y_d) \\to Y_d$ is finite, see", "Lemma \\ref{lemma-max-int-finite}.", "By Lemma \\ref{lemma-affineness-of-large-open}", "for every affine open $W \\subset Y$ we have that $Y_d \\cap W$ is affine", "(this uses that $W \\times_Y X$ is affine, being affine over $X$).", "Hence $Y_d \\to Y$ is an affine morphism of schemes. We", "conclude that $f^{-1}(Y_d) = Y_d \\times_Y X$ is", "an affine scheme being affine over $X$.", "Then $f^{-1}(Y_d) \\to Y_d$ is surjective and", "hence $Y_d$ is affine by Limits, Lemma \\ref{limits-lemma-affine}.", "Set $X' = X \\setminus f^{-1}(Y_d)$ and $Y' = Y \\setminus Y_d$", "viewed as closed subschemes of $X$ and $Y$.", "Since $X'$ is closed in $X$ it is affine. Since", "$Y'$ is closed in $Y$ it is separated.", "The morphism $f' : X' \\to Y'$ is surjective and", "$w$ induces a weighting $w'$ of $f'$, see", "Lemma \\ref{lemma-weighting-base-change}.", "By induction $Y'$ has an affine stratification of", "length $\\leq$ the number of distinct values of", "$\\int_{f'} w'$ and the proof is complete." ], "refs": [ "morphisms-lemma-affine-permanence", "more-morphisms-lemma-max-int-w", "more-morphisms-lemma-max-int-finite", "more-morphisms-lemma-affineness-of-large-open", "limits-lemma-affine", "more-morphisms-lemma-weighting-base-change" ], "ref_ids": [ 5179, 14097, 14098, 14102, 15082, 14089 ] } ], "ref_ids": [] }, { "id": 14138, "type": "theorem", "label": "sites-modules-lemma-limits-colimits-abelian-presheaves", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-limits-colimits-abelian-presheaves", "contents": [ "Let $\\mathcal{C}$ be a category.", "\\begin{enumerate}", "\\item All limits and colimits exist in $\\textit{PAb}(\\mathcal{C})$.", "\\item All limits and colimits commute with taking sections over objects of", "$\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\to \\textit{PAb}(\\mathcal{C})$, $i \\mapsto \\mathcal{F}_i$", "be a diagram. We can simply define abelian presheaves", "$L$ and $C$ by the rules", "$$", "L : U \\longmapsto \\lim_i \\mathcal{F}_i(U)", "$$", "and", "$$", "C : U \\longmapsto \\colim_i \\mathcal{F}_i(U).", "$$", "It is clear that there are maps of abelian presheaves $L \\to \\mathcal{F}_i$", "and $\\mathcal{F}_i \\to C$, by using the corresponding maps on groups of", "sections over each $U$. It is straightforward to check that $L$ and $C$ endowed", "with these maps are the limit and colimit of the diagram in", "$\\textit{PAb}(\\mathcal{C})$. This proves (1) and (2). Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14139, "type": "theorem", "label": "sites-modules-lemma-abelian-abelian", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-abelian-abelian", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "be a morphism of abelian sheaves on $\\mathcal{C}$.", "\\begin{enumerate}", "\\item The category $\\textit{Ab}(\\mathcal{C})$ is an abelian category.", "\\item The kernel $\\Ker(\\varphi)$ of $\\varphi$ is the same as the", "kernel of $\\varphi$ as a morphism of presheaves.", "\\item The morphism $\\varphi$ is injective", "(Homology, Definition \\ref{homology-definition-injective-surjective})", "if and only if $\\varphi$ is injective as a map of presheaves", "(Sites, Definition \\ref{sites-definition-presheaves-injective-surjective}),", "if and only if $\\varphi$ is injective as a map of sheaves", "(Sites, Definition \\ref{sites-definition-sheaves-injective-surjective}).", "\\item The cokernel $\\Coker(\\varphi)$ of $\\varphi$ is the sheafification", "of the cokernel of $\\varphi$ as a morphism of presheaves.", "\\item The morphism $\\varphi$ is surjective", "(Homology, Definition \\ref{homology-definition-injective-surjective})", "if and only if $\\varphi$ is surjective as a map of sheaves", "(Sites, Definition \\ref{sites-definition-sheaves-injective-surjective}).", "\\item A complex of abelian sheaves", "$$", "\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}", "$$", "is exact at $\\mathcal{G}$ if and only if for all", "$U \\in \\Ob(\\mathcal{C})$ and all $s \\in \\mathcal{G}(U)$", "mapping to zero in $\\mathcal{H}(U)$ there exists a covering", "$\\{U_i \\to U\\}_{i \\in I}$ in $\\mathcal{C}$ such that each", "$s|_{U_i}$ is in the image of $\\mathcal{F}(U_i) \\to \\mathcal{G}(U_i)$.", "\\end{enumerate}" ], "refs": [ "homology-definition-injective-surjective", "sites-definition-presheaves-injective-surjective", "sites-definition-sheaves-injective-surjective", "homology-definition-injective-surjective", "sites-definition-sheaves-injective-surjective" ], "proofs": [ { "contents": [ "We claim that Homology, Lemma \\ref{homology-lemma-adjoint-get-abelian}", "applies to the categories $\\mathcal{A} = \\textit{Ab}(\\mathcal{C})$", "and $\\mathcal{B} = \\textit{PAb}(\\mathcal{C})$, and the functors", "$a : \\mathcal{A} \\to \\mathcal{B}$ (inclusion), and", "$b : \\mathcal{B} \\to \\mathcal{A}$ (sheafification).", "Let us check the assumptions of", "Homology, Lemma \\ref{homology-lemma-adjoint-get-abelian}.", "Assumption (1) is that $\\mathcal{A}$, $\\mathcal{B}$ are additive categories,", "$a$, $b$ are additive functors, and $a$ is right adjoint to $b$.", "The first two statements are clear and adjointness is", "Sites, Section \\ref{sites-section-sheaves-algebraic-structures} ($\\epsilon$).", "Assumption (2) says that $\\textit{PAb}(\\mathcal{C})$ is abelian", "which we saw in Section \\ref{section-abelian-pre-sheaves} and", "that sheafification is left exact, which is", "Sites, Section \\ref{sites-section-sheaves-algebraic-structures} ($\\zeta$).", "The final assumption is that $ba \\cong \\text{id}_\\mathcal{A}$ which is", "Sites, Section \\ref{sites-section-sheaves-algebraic-structures} ($\\delta$).", "Hence Homology, Lemma \\ref{homology-lemma-adjoint-get-abelian}", "applies and we conclude that $\\textit{Ab}(\\mathcal{C})$ is abelian.", "\\medskip\\noindent", "In the proof of Homology, Lemma \\ref{homology-lemma-adjoint-get-abelian}", "it is shown that $\\Ker(\\varphi)$ and $\\Coker(\\varphi)$", "are equal to the sheafification of the kernel and cokernel of $\\varphi$", "as a morphism of abelian presheaves. This proves (4). Since the kernel", "is a equalizer (i.e., a limit) and since sheafification commutes with", "finite limits, we conclude that (2) holds.", "\\medskip\\noindent", "Statement (2) implies (3). Statement (4) implies (5) by our description", "of sheafification. The characterization of exactness in (6) follows from", "(2) and (5), and the fact that the sequence is exact if and only if", "$\\Im(\\mathcal{F} \\to \\mathcal{G}) =", "\\Ker(\\mathcal{G} \\to \\mathcal{H})$." ], "refs": [ "homology-lemma-adjoint-get-abelian", "homology-lemma-adjoint-get-abelian", "homology-lemma-adjoint-get-abelian", "homology-lemma-adjoint-get-abelian" ], "ref_ids": [ 12036, 12036, 12036, 12036 ] } ], "ref_ids": [ 12138, 8648, 8660, 12138, 8660 ] }, { "id": 14140, "type": "theorem", "label": "sites-modules-lemma-limits-colimits-abelian-sheaves", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-limits-colimits-abelian-sheaves", "contents": [ "Let $\\mathcal{C}$ be a site.", "\\begin{enumerate}", "\\item All limits and colimits exist in $\\textit{Ab}(\\mathcal{C})$.", "\\item Limits are the same as the corresponding limits of abelian presheaves", "over $\\mathcal{C}$ (i.e., commute with taking sections over objects of", "$\\mathcal{C}$).", "\\item Finite direct sums are the same as the corresponding finite direct sums", "in the category of abelian pre-sheaves over $\\mathcal{C}$.", "\\item A colimit is the sheafification of the corresponding colimit in", "the category of abelian presheaves.", "\\item Filtered colimits are exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-limits-colimits-abelian-presheaves} limits and colimits", "of abelian presheaves exist, and are described by taking limits and colimits", "on the level of sections over objects.", "\\medskip\\noindent", "Let $\\mathcal{I} \\to \\textit{Ab}(\\mathcal{C})$, $i \\mapsto \\mathcal{F}_i$", "be a diagram. Let $\\lim_i \\mathcal{F}_i$ be the limit of the diagram", "as an abelian presheaf. By Sites, Lemma \\ref{sites-lemma-limit-sheaf}", "this is an abelian sheaf. Then it is quite easy to see that", "$\\lim_i \\mathcal{F}_i$ is the limit of the diagram in", "$\\textit{Ab}(\\mathcal{C})$. This proves limits exist and (2) holds.", "\\medskip\\noindent", "By Categories, Lemma \\ref{categories-lemma-adjoint-exact}, and because", "sheafification is left adjoint to the inclusion functor we see that", "$\\colim_i \\mathcal{F}$ exists and is the sheafification of the colimit", "in $\\textit{PAb}(\\mathcal{C})$. This proves colimits exist and (4) holds.", "\\medskip\\noindent", "Finite direct sums are the same thing as finite products in any abelian", "category. Hence (3) follows from (2).", "\\medskip\\noindent", "Proof of (5). The statement means that given a system", "$0 \\to \\mathcal{F}_i \\to \\mathcal{G}_i \\to \\mathcal{H}_i \\to 0$", "of exact sequences of abelian sheaves over a directed set $I$ the sequence", "$0 \\to \\colim \\mathcal{F}_i \\to \\colim \\mathcal{G}_i \\to", "\\colim \\mathcal{H}_i \\to 0$ is exact as well. A formal argument using", "Homology, Lemma \\ref{homology-lemma-check-exactness} and the", "definition of colimits shows that the sequence", "$\\colim \\mathcal{F}_i \\to \\colim \\mathcal{G}_i \\to \\colim \\mathcal{H}_i \\to 0$", "is exact. Note that $\\colim \\mathcal{F}_i \\to \\colim \\mathcal{G}_i$", "is the sheafification of the map of presheaf colimits which is", "injective as each of the maps $\\mathcal{F}_i \\to \\mathcal{G}_i$ is", "injective. Since sheafification is exact we conclude." ], "refs": [ "sites-modules-lemma-limits-colimits-abelian-presheaves", "sites-lemma-limit-sheaf", "categories-lemma-adjoint-exact", "homology-lemma-check-exactness" ], "ref_ids": [ 14138, 8508, 12249, 12019 ] } ], "ref_ids": [] }, { "id": 14141, "type": "theorem", "label": "sites-modules-lemma-obvious-adjointness", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-obvious-adjointness", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{G}$, $\\mathcal{F}$ be a presheaves of sets.", "Let $\\mathcal{A}$ be an abelian presheaf.", "Let $U$ be an object of $\\mathcal{C}$. Then", "we have", "\\begin{align*}", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(h_U, \\mathcal{F})", "& =", "\\mathcal{F}(U), \\\\", "\\Mor_{\\textit{PAb}(\\mathcal{C})}(\\mathbf{Z}_\\mathcal{G}, \\mathcal{A})", "& =", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(\\mathcal{G}, \\mathcal{A}), \\\\", "\\Mor_{\\textit{PAb}(\\mathcal{C})}(\\mathbf{Z}_U, \\mathcal{A})", "& =", "\\mathcal{A}(U).", "\\end{align*}", "All of these equalities are functorial." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14142, "type": "theorem", "label": "sites-modules-lemma-coproduct-sum-free-abelian-presheaf", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-coproduct-sum-free-abelian-presheaf", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $I$ be a set. For each $i \\in I$ let", "$\\mathcal{G}_i$ be a presheaf of sets.", "Then", "$$", "\\mathbf{Z}_{\\coprod_i \\mathcal{G}_i}", "=", "\\bigoplus\\nolimits_{i \\in I} \\mathbf{Z}_{\\mathcal{G}_i}", "$$", "in $\\textit{PAb}(\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14143, "type": "theorem", "label": "sites-modules-lemma-obvious-adjointness-sheaves", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-obvious-adjointness-sheaves", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{G}$, $\\mathcal{F}$ be a sheaves of sets.", "Let $\\mathcal{A}$ be an abelian sheaf.", "Let $U$ be an object of $\\mathcal{C}$. Then", "we have", "\\begin{align*}", "\\Mor_{\\Sh(\\mathcal{C})}(h_U^\\#, \\mathcal{F})", "& =", "\\mathcal{F}(U), \\\\", "\\Mor_{\\textit{Ab}(\\mathcal{C})}(\\mathbf{Z}_\\mathcal{G}^\\#,", "\\mathcal{A})", "& =", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{G}, \\mathcal{A}), \\\\", "\\Mor_{\\textit{Ab}(\\mathcal{C})}(\\mathbf{Z}_U^\\#, \\mathcal{A})", "& =", "\\mathcal{A}(U).", "\\end{align*}", "All of these equalities are functorial." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14144, "type": "theorem", "label": "sites-modules-lemma-may-sheafify-before-abelianize", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-may-sheafify-before-abelianize", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{G}$ be a presheaf of sets.", "Then $\\mathbf{Z}_\\mathcal{G}^\\# = (\\mathbf{Z}_{\\mathcal{G}^\\#})^\\#$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14145, "type": "theorem", "label": "sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites", "contents": [ "Let $(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi. There exists a factorization", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\ar[rr]_{(f, f^\\sharp)}", "\\ar[d]_{(g, g^\\sharp)}", "& &", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\ar[d]^{(e, e^\\sharp)}", "\\\\", "(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})", "\\ar[rr]^{(h, h^\\sharp)} & &", "(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'})", "}", "$$", "where", "\\begin{enumerate}", "\\item $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$", "is an equivalence of topoi induced by a special cocontinuous functor", "$\\mathcal{C} \\to \\mathcal{C}'$ (see", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}),", "\\item $e : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{D}')$", "is an equivalence of topoi induced by a special cocontinuous functor", "$\\mathcal{D} \\to \\mathcal{D}'$ (see", "Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}),", "\\item $\\mathcal{O}_{\\mathcal{C}'} = g_*\\mathcal{O}_\\mathcal{C}$", "and $g^\\sharp$ is the obvious map,", "\\item $\\mathcal{O}_{\\mathcal{D}'} = e_*\\mathcal{O}_\\mathcal{D}$", "and $e^\\sharp$ is the obvious map,", "\\item the sites $\\mathcal{C}'$ and $\\mathcal{D}'$ have final objects", "and fibre products (i.e., all finite limits),", "\\item $h$ is a morphism of sites induced by a continuous functor", "$u : \\mathcal{D}' \\to \\mathcal{C}'$ which commutes with all finite limits", "(i.e., it satisfies the assumptions of", "Sites, Proposition \\ref{sites-proposition-get-morphism}), and", "\\item given any set of sheaves $\\mathcal{F}_i$ (resp.\\ $\\mathcal{G}_j$)", "on $\\mathcal{C}$ (resp.\\ $\\mathcal{D}$) we may assume each of these is", "a representable sheaf on $\\mathcal{C}'$ (resp.\\ $\\mathcal{D}'$).", "\\end{enumerate}", "Moreover, if $(f, f^\\sharp)$ is an equivalence of ringed topoi,", "then we can choose the diagram such that", "$\\mathcal{C}' = \\mathcal{D}'$,", "$\\mathcal{O}_{\\mathcal{C}'} = \\mathcal{O}_{\\mathcal{D}'}$", "and $(h, h^\\sharp)$ is the identity." ], "refs": [ "sites-definition-special-cocontinuous-functor", "sites-definition-special-cocontinuous-functor", "sites-proposition-get-morphism" ], "proofs": [ { "contents": [ "This follows from", "Sites, Lemma \\ref{sites-lemma-morphism-topoi-comes-from-morphism-sites},", "and", "Sites, Remarks", "\\ref{sites-remark-morphism-topoi-comes-from-morphism-sites} and", "\\ref{sites-remark-equivalence-topoi-comes-from-morphism-sites}.", "You just have to carry along the sheaves of rings. Some details omitted." ], "refs": [ "sites-lemma-morphism-topoi-comes-from-morphism-sites", "sites-remark-morphism-topoi-comes-from-morphism-sites", "sites-remark-equivalence-topoi-comes-from-morphism-sites" ], "ref_ids": [ 8582, 8715, 8716 ] } ], "ref_ids": [ 8672, 8672, 8641 ] }, { "id": 14146, "type": "theorem", "label": "sites-modules-lemma-adjointness-tensor-restrict-presheaves", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-adjointness-tensor-restrict-presheaves", "contents": [ "With $\\mathcal{C}$, $\\mathcal{O}_1 \\to \\mathcal{O}_2$, $\\mathcal{F}$ and", "$\\mathcal{G}$ as above there exists a canonical bijection", "$$", "\\Hom_{\\mathcal{O}_1}(\\mathcal{G}, \\mathcal{F}_{\\mathcal{O}_1})", "=", "\\Hom_{\\mathcal{O}_2}(", "\\mathcal{O}_2 \\otimes_{p, \\mathcal{O}_1} \\mathcal{G},", "\\mathcal{F}", ")", "$$", "In other words, the restriction and change of rings functors defined", "above are adjoint to each other." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that for a ring map", "$A \\to B$ the restriction functor and the change", "of ring functor are adjoint to each other." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14147, "type": "theorem", "label": "sites-modules-lemma-sheafification-presheaf-modules", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-sheafification-presheaf-modules", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$.", "Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.", "Let $\\mathcal{O}^\\#$ be the sheafification of $\\mathcal{O}$ as a presheaf", "of rings, see Sites, Section \\ref{sites-section-sheaves-algebraic-structures}.", "Let $\\mathcal{F}^\\#$ be the sheafification of $\\mathcal{F}$", "as a presheaf of abelian groups. There exists a unique map of", "sheaves of sets", "$$", "\\mathcal{O}^\\# \\times \\mathcal{F}^\\#", "\\longrightarrow", "\\mathcal{F}^\\#", "$$", "which makes the diagram", "$$", "\\xymatrix{", "\\mathcal{O} \\times \\mathcal{F} \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[d] \\\\", "\\mathcal{O}^\\# \\times \\mathcal{F}^\\# \\ar[r] &", "\\mathcal{F}^\\#", "}", "$$", "commute and which makes $\\mathcal{F}^\\#$ into a sheaf", "of $\\mathcal{O}^\\#$-modules. In addition, if $\\mathcal{G}$", "is a sheaf of $\\mathcal{O}^\\#$-modules, then any morphism", "of presheaves of $\\mathcal{O}$-modules $\\mathcal{F} \\to \\mathcal{G}$", "(into the restriction of $\\mathcal{G}$ to a $\\mathcal{O}$-module)", "factors uniquely as $\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$", "where $\\mathcal{F}^\\# \\to \\mathcal{G}$ is a morphism of", "$\\mathcal{O}^\\#$-modules." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14148, "type": "theorem", "label": "sites-modules-lemma-sheafification-exact", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-sheafification-exact", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$", "The sheafification functor", "$$", "\\textit{PMod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O}^\\#), \\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}^\\#", "$$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "This is true because it holds for sheafification", "$\\textit{PAb}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{C})$.", "See the discussion in Section \\ref{section-abelian-sheaves}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14149, "type": "theorem", "label": "sites-modules-lemma-adjointness-tensor-restrict", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-adjointness-tensor-restrict", "contents": [ "With $X$, $\\mathcal{O}_1$, $\\mathcal{O}_2$, $\\mathcal{F}$ and", "$\\mathcal{G}$ as above there exists a canonical bijection", "$$", "\\Hom_{\\mathcal{O}_1}(\\mathcal{G}, \\mathcal{F}_{\\mathcal{O}_1})", "=", "\\Hom_{\\mathcal{O}_2}(", "\\mathcal{O}_2 \\otimes_{\\mathcal{O}_1} \\mathcal{G},", "\\mathcal{F}", ")", "$$", "In other words, the restriction and change of rings functors", "are adjoint to each other." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-adjointness-tensor-restrict-presheaves}", "and the fact that", "$\\Hom_{\\mathcal{O}_2}(", "\\mathcal{O}_2 \\otimes_{\\mathcal{O}_1} \\mathcal{G},", "\\mathcal{F}", ")", "=", "\\Hom_{\\mathcal{O}_2}(", "\\mathcal{O}_2 \\otimes_{p, \\mathcal{O}_1} \\mathcal{G},", "\\mathcal{F}", ")$", "because $\\mathcal{F}$ is a sheaf." ], "refs": [ "sites-modules-lemma-adjointness-tensor-restrict-presheaves" ], "ref_ids": [ 14146 ] } ], "ref_ids": [] }, { "id": 14150, "type": "theorem", "label": "sites-modules-lemma-epimorphism-modules", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-epimorphism-modules", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{O} \\to \\mathcal{O}'$ be an epimorphism of sheaves of rings.", "Let $\\mathcal{G}_1, \\mathcal{G}_2$ be $\\mathcal{O}'$-modules.", "Then", "$$", "\\Hom_{\\mathcal{O}'}(\\mathcal{G}_1, \\mathcal{G}_2) =", "\\Hom_\\mathcal{O}(\\mathcal{G}_1, \\mathcal{G}_2).", "$$", "In other words, the restriction functor", "$\\textit{Mod}(\\mathcal{O}') \\to \\textit{Mod}(\\mathcal{O})$ is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "This is the sheaf version of", "Algebra, Lemma \\ref{algebra-lemma-epimorphism-modules}", "and is proved in exactly the same way." ], "refs": [ "algebra-lemma-epimorphism-modules" ], "ref_ids": [ 959 ] } ], "ref_ids": [] }, { "id": 14151, "type": "theorem", "label": "sites-modules-lemma-pushforward-module", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-pushforward-module", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "be a morphism of topoi.", "Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.", "There is a natural map of sheaves of sets", "$$", "f_*\\mathcal{O} \\times f_*\\mathcal{F}", "\\longrightarrow", "f_*\\mathcal{F}", "$$", "which turns $f_*\\mathcal{F}$ into a sheaf of $f_*\\mathcal{O}$-modules.", "This construction is functorial in $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Denote $\\mu : \\mathcal{O} \\times \\mathcal{F} \\to \\mathcal{F}$ the", "multiplication map. Recall that $f_*$ (on sheaves of sets) is left exact", "and hence commutes with products. Hence $f_*\\mu$ is a map as", "indicated. This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14152, "type": "theorem", "label": "sites-modules-lemma-pullback-module", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-pullback-module", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "be a morphism of topoi.", "Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{D}$.", "Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules.", "There is a natural map of sheaves of sets", "$$", "f^{-1}\\mathcal{O} \\times f^{-1}\\mathcal{G}", "\\longrightarrow", "f^{-1}\\mathcal{G}", "$$", "which turns $f^{-1}\\mathcal{G}$ into a sheaf of $f^{-1}\\mathcal{O}$-modules.", "This construction is functorial in $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Denote $\\mu : \\mathcal{O} \\times \\mathcal{G} \\to \\mathcal{G}$ the", "multiplication map. Recall that $f^{-1}$ (on sheaves of sets) is exact", "and hence commutes with products. Hence $f^{-1}\\mu$ is a map as", "indicated. This proves the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14153, "type": "theorem", "label": "sites-modules-lemma-adjoint-push-pull-modules", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-adjoint-push-pull-modules", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "be a morphism of topoi.", "Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{D}$.", "Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules.", "Let $\\mathcal{F}$ be a sheaf of $f^{-1}\\mathcal{O}$-modules.", "Then", "$$", "\\Mor_{\\textit{Mod}(f^{-1}\\mathcal{O})}(f^{-1}\\mathcal{G}, \\mathcal{F})", "=", "\\Mor_{\\textit{Mod}(\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).", "$$", "Here we use", "Lemmas \\ref{lemma-pullback-module}", "and \\ref{lemma-pushforward-module}, and we think of", "$f_*\\mathcal{F}$ as an $\\mathcal{O}$-module by restriction via", "$\\mathcal{O} \\to f_*f^{-1}\\mathcal{O}$." ], "refs": [ "sites-modules-lemma-pullback-module", "sites-modules-lemma-pushforward-module" ], "proofs": [ { "contents": [ "First we note that we have", "$$", "\\Mor_{\\textit{Ab}(\\mathcal{C})}(f^{-1}\\mathcal{G}, \\mathcal{F})", "=", "\\Mor_{\\textit{Ab}(\\mathcal{D})}(\\mathcal{G}, f_*\\mathcal{F}).", "$$", "by Sites,", "Proposition \\ref{sites-proposition-functoriality-algebraic-structures-topoi}.", "Suppose that $\\alpha : f^{-1}\\mathcal{G} \\to \\mathcal{F}$ and", "$\\beta : \\mathcal{G} \\to f_*\\mathcal{F}$ are morphisms of abelian", "sheaves which correspond via the formula above. We have to show that", "$\\alpha$ is $f^{-1}\\mathcal{O}$-linear if and only if $\\beta$", "is $\\mathcal{O}$-linear. For example, suppose $\\alpha$ is", "$f^{-1}\\mathcal{O}$-linear, then clearly $f_*\\alpha$ is", "$f_*f^{-1}\\mathcal{O}$-linear, and hence (as restriction is a functor)", "is $\\mathcal{O}$-linear. Hence it suffices to prove that the", "adjunction map $\\mathcal{G} \\to f_*f^{-1}\\mathcal{G}$ is", "$\\mathcal{O}$-linear. Using that both $f_*$ and $f^{-1}$ commute", "with products (on sheaves of sets) this comes down to showing that", "$$", "\\xymatrix{", "\\mathcal{O} \\times \\mathcal{G} \\ar[r] \\ar[d] &", "f_*f^{-1}(\\mathcal{O} \\times \\mathcal{G}) \\ar[d] \\\\", "\\mathcal{G} \\ar[r] & f_*f^{-1}\\mathcal{G}", "}", "$$", "is commutative. This holds because the adjunction mapping", "$\\text{id}_{\\Sh(\\mathcal{D})} \\to f_*f^{-1}$ is a", "transformation of functors. We omit the proof of the implication", "$\\beta$ linear $\\Rightarrow$ $\\alpha$ linear." ], "refs": [ "sites-proposition-functoriality-algebraic-structures-topoi" ], "ref_ids": [ 8644 ] } ], "ref_ids": [ 14152, 14151 ] }, { "id": 14154, "type": "theorem", "label": "sites-modules-lemma-adjoint-pull-push-modules", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-adjoint-pull-push-modules", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "be a morphism of topoi.", "Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.", "Let $\\mathcal{G}$ be a sheaf of $f_*\\mathcal{O}$-modules.", "Then", "$$", "\\Mor_{\\textit{Mod}(\\mathcal{O})}(", "\\mathcal{O} \\otimes_{f^{-1}f_*\\mathcal{O}} f^{-1}\\mathcal{G}, \\mathcal{F})", "=", "\\Mor_{\\textit{Mod}(f_*\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).", "$$", "Here we use", "Lemmas \\ref{lemma-pullback-module}", "and \\ref{lemma-pushforward-module}, and we use", "the canonical map $f^{-1}f_*\\mathcal{O} \\to \\mathcal{O}$", "in the definition of the tensor product." ], "refs": [ "sites-modules-lemma-pullback-module", "sites-modules-lemma-pushforward-module" ], "proofs": [ { "contents": [ "Note that we have", "$$", "\\Mor_{\\textit{Mod}(\\mathcal{O})}(", "\\mathcal{O} \\otimes_{f^{-1}f_*\\mathcal{O}} f^{-1}\\mathcal{G}, \\mathcal{F})", "=", "\\Mor_{\\textit{Mod}(f^{-1}f_*\\mathcal{O})}(", "f^{-1}\\mathcal{G}, \\mathcal{F}_{f^{-1}f_*\\mathcal{O}})", "$$", "by Lemma \\ref{lemma-adjointness-tensor-restrict}. Hence the result follows", "from Lemma \\ref{lemma-adjoint-push-pull-modules}." ], "refs": [ "sites-modules-lemma-adjointness-tensor-restrict", "sites-modules-lemma-adjoint-push-pull-modules" ], "ref_ids": [ 14149, 14153 ] } ], "ref_ids": [ 14152, 14151 ] }, { "id": 14155, "type": "theorem", "label": "sites-modules-lemma-adjoint-pullback-pushforward-modules", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-adjoint-pullback-pushforward-modules", "contents": [ "Let", "$(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi or ringed sites.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{C}$-modules.", "Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_\\mathcal{D}$-modules.", "There is a canonical bijection", "$$", "\\Hom_{\\mathcal{O}_\\mathcal{C}}(f^*\\mathcal{G}, \\mathcal{F})", "=", "\\Hom_{\\mathcal{O}_\\mathcal{D}}(\\mathcal{G}, f_*\\mathcal{F}).", "$$", "In other words: the functor $f^*$ is the left adjoint to", "$f_*$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the work we did before:", "\\begin{eqnarray*}", "\\Hom_{\\mathcal{O}_\\mathcal{C}}(f^*\\mathcal{G}, \\mathcal{F})", "& = &", "\\Mor_{\\textit{Mod}(\\mathcal{O}_\\mathcal{C})}(", "\\mathcal{O}_\\mathcal{C}", "\\otimes_{f^{-1}\\mathcal{O}_\\mathcal{D}} f^{-1}\\mathcal{G},", "\\mathcal{F}) \\\\", "& = &", "\\Mor_{\\textit{Mod}(f^{-1}\\mathcal{O}_\\mathcal{D})}(", "f^{-1}\\mathcal{G}, \\mathcal{F}_{f^{-1}\\mathcal{O}_\\mathcal{D}}) \\\\", "& = &", "\\Hom_{\\mathcal{O}_\\mathcal{D}}(\\mathcal{G}, f_*\\mathcal{F}).", "\\end{eqnarray*}", "Here we use Lemmas \\ref{lemma-adjointness-tensor-restrict}", "and \\ref{lemma-adjoint-push-pull-modules}." ], "refs": [ "sites-modules-lemma-adjointness-tensor-restrict", "sites-modules-lemma-adjoint-push-pull-modules" ], "ref_ids": [ 14149, 14153 ] } ], "ref_ids": [] }, { "id": 14156, "type": "theorem", "label": "sites-modules-lemma-push-pull-composition-modules", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-push-pull-composition-modules", "contents": [ "$(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}_1), \\mathcal{O}_1)", "\\to (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2)$ and", "$(g, g^\\sharp) :", "(\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\to", "(\\Sh(\\mathcal{C}_3), \\mathcal{O}_3)$", "be morphisms of ringed topoi.", "There are canonical isomorphisms of functors", "$(g \\circ f)_* \\cong g_* \\circ f_*$ and", "$(g \\circ f)^* \\cong f^* \\circ g^*$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14157, "type": "theorem", "label": "sites-modules-lemma-abelian", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-abelian", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.", "The category $\\textit{Mod}(\\mathcal{O})$ is an abelian category.", "The forgetful functor", "$\\textit{Mod}(\\mathcal{O}) \\to \\textit{Ab}(\\mathcal{C})$", "is exact, hence kernels, cokernels and exactness of", "$\\mathcal{O}$-modules, correspond to the corresponding notions", "for abelian sheaves." ], "refs": [], "proofs": [ { "contents": [ "Above we have seen that $\\textit{Mod}(\\mathcal{O})$ is an additive", "category, with kernels and cokernels", "and that $\\textit{Mod}(\\mathcal{O}) \\to \\textit{Ab}(\\mathcal{C})$", "preserves kernels and cokernels.", "By Homology, Definition \\ref{homology-definition-abelian-category}", "we have to show that image and coimage agree. This is clear", "because it is true in $\\textit{Ab}(\\mathcal{C})$. The lemma follows." ], "refs": [ "homology-definition-abelian-category" ], "ref_ids": [ 12137 ] } ], "ref_ids": [] }, { "id": 14158, "type": "theorem", "label": "sites-modules-lemma-limits-colimits", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-limits-colimits", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.", "All limits and colimits exist in $\\textit{Mod}(\\mathcal{O})$", "and the forgetful functor", "$\\textit{Mod}(\\mathcal{O}) \\to \\textit{Ab}(\\mathcal{C})$", "commutes with them. Moreover, filtered colimits are exact." ], "refs": [], "proofs": [ { "contents": [ "The final statement follows from the first as filtered colimits are", "exact in $\\textit{Ab}(\\mathcal{C})$ by", "Lemma \\ref{lemma-limits-colimits-abelian-sheaves}.", "Let $\\mathcal{I} \\to \\textit{Mod}(\\mathcal{C})$, $i \\mapsto \\mathcal{F}_i$", "be a diagram. Let $\\lim_i \\mathcal{F}_i$ be the limit of the diagram", "in $\\textit{Ab}(\\mathcal{C})$. By the description of this limit in", "Lemma \\ref{lemma-limits-colimits-abelian-sheaves} we see immediately that", "there exists a multiplication", "$$", "\\mathcal{O} \\times \\lim_i \\mathcal{F}_i", "\\longrightarrow", "\\lim_i \\mathcal{F}_i", "$$", "which turns $\\lim_i \\mathcal{F}_i$ into a sheaf of", "$\\mathcal{O}$-modules. It is easy to see that this is the", "limit of the diagram in $\\textit{Mod}(\\mathcal{C})$. Let", "$\\colim_i \\mathcal{F}_i$ be the colimit of the diagram", "in $\\textit{PAb}(\\mathcal{C})$. By the description of this colimit", "in the proof of Lemma \\ref{lemma-limits-colimits-abelian-presheaves}", "we see immediately that there exists a multiplication", "$$", "\\mathcal{O} \\times \\colim_i \\mathcal{F}_i", "\\longrightarrow", "\\colim_i \\mathcal{F}_i", "$$", "which turns $\\colim_i \\mathcal{F}_i$ into a presheaf of", "$\\mathcal{O}$-modules. Applying sheafification we get a", "sheaf of $\\mathcal{O}$-modules $(\\colim_i \\mathcal{F}_i)^\\#$,", "see Lemma \\ref{lemma-sheafification-presheaf-modules}.", "It is easy to see that $(\\colim_i \\mathcal{F}_i)^\\#$", "is the colimit of the diagram in $\\textit{Mod}(\\mathcal{O})$, and", "by Lemma \\ref{lemma-limits-colimits-abelian-sheaves}", "forgetting the $\\mathcal{O}$-module structure is", "the colimit in $\\textit{Ab}(\\mathcal{C})$." ], "refs": [ "sites-modules-lemma-limits-colimits-abelian-sheaves", "sites-modules-lemma-limits-colimits-abelian-sheaves", "sites-modules-lemma-limits-colimits-abelian-presheaves", "sites-modules-lemma-sheafification-presheaf-modules", "sites-modules-lemma-limits-colimits-abelian-sheaves" ], "ref_ids": [ 14140, 14140, 14138, 14147, 14140 ] } ], "ref_ids": [] }, { "id": 14159, "type": "theorem", "label": "sites-modules-lemma-exactness-pushforward-pullback", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-exactness-pushforward-pullback", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi.", "\\begin{enumerate}", "\\item The functor $f_*$ is left exact. In fact it commutes with", "all limits.", "\\item The functor $f^*$ is right exact. In fact it commutes", "with all colimits.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is true because $(f^*, f_*)$ is an adjoint pair", "of functors, see", "Lemma \\ref{lemma-adjoint-pullback-pushforward-modules}.", "See Categories, Section \\ref{categories-section-adjoint}." ], "refs": [ "sites-modules-lemma-adjoint-pullback-pushforward-modules" ], "ref_ids": [ 14155 ] } ], "ref_ids": [] }, { "id": 14160, "type": "theorem", "label": "sites-modules-lemma-check-exactness-stalks", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-check-exactness-stalks", "contents": [ "Let $\\mathcal{C}$ be a site. If $\\{p_i\\}_{i \\in I}$ is a conservative", "family of points, then we may check exactness of a sequence of abelian", "sheaves on the stalks at the points $p_i$, $i \\in I$.", "If $\\mathcal{C}$ has enough points, then", "exactness of a sequence of abelian sheaves may", "be checked on stalks." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from", "Sites, Lemma \\ref{sites-lemma-exactness-stalks}." ], "refs": [ "sites-lemma-exactness-stalks" ], "ref_ids": [ 8609 ] } ], "ref_ids": [] }, { "id": 14161, "type": "theorem", "label": "sites-modules-lemma-reflect-surjections", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-reflect-surjections", "contents": [ "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be", "a morphism of topoi. The following are equivalent:", "\\begin{enumerate}", "\\item $f^{-1}f_*\\mathcal{F} \\to \\mathcal{F}$ is surjective for", "all $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C})$, and", "\\item $f_* : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{D})$", "reflects surjections.", "\\end{enumerate}", "In this case the functor", "$f_* : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{D})$", "is faithful." ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Suppose that $a : \\mathcal{F} \\to \\mathcal{F}'$", "is a map of abelian sheaves on $\\mathcal{C}$ such that $f_*a$ is surjective.", "As $f^{-1}$ is exact this implies that", "$f^{-1}f_*a : f^{-1}f_*\\mathcal{F} \\to f^{-1}f_*\\mathcal{F}'$", "is surjective. Combined with (1) this implies that $a$ is surjective.", "This means that (2) holds.", "\\medskip\\noindent", "Assume (2). Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}$.", "We have to show that the map $f^{-1}f_*\\mathcal{F} \\to \\mathcal{F}$ is", "surjective. By (2) it suffices to show that", "$f_*f^{-1}f_*\\mathcal{F} \\to f_*\\mathcal{F}$ is surjective.", "And this is true because there is a canonical map", "$f_*\\mathcal{F} \\to f_*f^{-1}f_*\\mathcal{F}$ which is a one-sided inverse.", "\\medskip\\noindent", "We omit the proof of the final assertion." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14162, "type": "theorem", "label": "sites-modules-lemma-exactness", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-exactness", "contents": [ "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be", "a morphism of topoi. Assume at least one of the following properties", "holds", "\\begin{enumerate}", "\\item $f_*$ transforms surjections of sheaves of sets into surjections,", "\\item $f_*$ transforms surjections of abelian sheaves into surjections,", "\\item $f_*$ commutes with coequalizers on sheaves of sets,", "\\item $f_*$ commutes with pushouts on sheaves of sets,", "\\end{enumerate}", "Then $f_* : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{D})$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "Since $f_* : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{D})$", "is a right adjoint we already know that it transforms a short exact sequence", "$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "of abelian sheaves on $\\mathcal{C}$ into an exact sequence", "$$", "0 \\to f_*\\mathcal{F}_1 \\to f_*\\mathcal{F}_2 \\to f_*\\mathcal{F}_3", "$$", "see", "Categories, Sections \\ref{categories-section-exact-functor} and", "\\ref{categories-section-adjoint}", "and", "Homology, Section \\ref{homology-section-functors}. Hence it suffices to", "prove that the map $f_*\\mathcal{F}_2 \\to f_*\\mathcal{F}_3$ is surjective.", "If (1), (2) holds, then this is clear from the definitions. By", "Sites, Lemma \\ref{sites-lemma-exactness-properties}", "we see that either (3) or (4) formally implies (1), hence in these cases", "we are done also." ], "refs": [ "sites-lemma-exactness-properties" ], "ref_ids": [ 8618 ] } ], "ref_ids": [] }, { "id": 14163, "type": "theorem", "label": "sites-modules-lemma-morphism-ringed-sites-almost-cocontinuous", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-morphism-ringed-sites-almost-cocontinuous", "contents": [ "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites", "associated to the continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$.", "Assume $u$ is almost cocontinuous. Then", "\\begin{enumerate}", "\\item $f_* : \\textit{Ab}(\\mathcal{D}) \\to \\textit{Ab}(\\mathcal{C})$ is exact.", "\\item if $f^\\sharp : f^{-1}\\mathcal{O}_\\mathcal{C} \\to \\mathcal{O}_\\mathcal{D}$", "is given so that $f$ becomes a morphism of ringed sites, then", "$f_* : \\textit{Mod}(\\mathcal{O}_\\mathcal{D}) \\to", "\\textit{Mod}(\\mathcal{O}_\\mathcal{C})$ is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (2) follows from part (1) by", "Lemma \\ref{lemma-limits-colimits}.", "Part (1) follows from", "Sites, Lemmas", "\\ref{sites-lemma-morphism-of-sites-almost-cocontinuous} and", "\\ref{sites-lemma-exactness-properties}." ], "refs": [ "sites-modules-lemma-limits-colimits", "sites-lemma-morphism-of-sites-almost-cocontinuous", "sites-lemma-exactness-properties" ], "ref_ids": [ 14158, 8624, 8618 ] } ], "ref_ids": [] }, { "id": 14164, "type": "theorem", "label": "sites-modules-lemma-g-shriek-adjoint", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-g-shriek-adjoint", "contents": [ "The functor $g_{p!}$ is a left adjoint to the functor $u^p$.", "The functor $g_!$ is a left adjoint to the functor $g^{-1}$.", "In other words the formulas", "\\begin{align*}", "\\Mor_{\\textit{PAb}(\\mathcal{C})}(\\mathcal{F}, u^p\\mathcal{G})", "& =", "\\Mor_{\\textit{PAb}(\\mathcal{D})}(g_{p!}\\mathcal{F}, \\mathcal{G}), \\\\", "\\Mor_{\\textit{Ab}(\\mathcal{C})}(\\mathcal{F}, g^{-1}\\mathcal{G})", "& =", "\\Mor_{\\textit{Ab}(\\mathcal{D})}(g_!\\mathcal{F}, \\mathcal{G})", "\\end{align*}", "hold bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "The second formula follows formally from the first, since if", "$\\mathcal{F}$ and $\\mathcal{G}$ are abelian sheaves then", "\\begin{align*}", "\\Mor_{\\textit{Ab}(\\mathcal{C})}(\\mathcal{F}, g^{-1}\\mathcal{G})", "& =", "\\Mor_{\\textit{PAb}(\\mathcal{D})}(g_{p!}\\mathcal{F}, \\mathcal{G}) \\\\", "& =", "\\Mor_{\\textit{Ab}(\\mathcal{D})}(g_!\\mathcal{F}, \\mathcal{G})", "\\end{align*}", "by the universal property of sheafification.", "\\medskip\\noindent", "To prove the first formula, let $\\mathcal{F}$, $\\mathcal{G}$ be abelian", "presheaves. To prove the lemma we will construct maps from the group on the", "left to the group on the right and omit the verification that these are", "mutually inverse.", "\\medskip\\noindent", "Note that there is a canonical map of abelian presheaves", "$\\mathcal{F} \\to u^pg_{p!}\\mathcal{F}$ which on sections over $U$ is the", "natural map", "$\\mathcal{F}(U) \\to \\colim_{u(U) \\to u(U')} \\mathcal{F}(U')$, see", "Sites, Lemma \\ref{sites-lemma-recover}.", "Given a map $\\alpha : g_{p!}\\mathcal{F} \\to \\mathcal{G}$", "we get $u^p\\alpha : u^pg_{p!}\\mathcal{F} \\to u^p\\mathcal{G}$.", "which we can precompose by the map $\\mathcal{F} \\to u^pg_{p!}\\mathcal{F}$.", "\\medskip\\noindent", "Note that there is a canonical map of abelian presheaves", "$g_{p!}u^p\\mathcal{G} \\to \\mathcal{G}$ which on sections over", "$V$ is the natural map", "$\\colim_{V \\to u(U)} \\mathcal{G}(u(U)) \\to \\mathcal{G}(V)$.", "It maps a section $s \\in u(U)$ in the summand corresponding to", "$t : V \\to u(U)$ to $t^*s \\in \\mathcal{G}(V)$.", "Hence, given a map $\\beta : \\mathcal{F} \\to u^p\\mathcal{G}$", "we get a map $g_{p!}\\beta : g_{p!}\\mathcal{F} \\to g_{p!}u^p\\mathcal{G}$", "which we can postcompose with the map $g_{p!}u^p\\mathcal{G} \\to \\mathcal{G}$", "above." ], "refs": [ "sites-lemma-recover" ], "ref_ids": [ 8499 ] } ], "ref_ids": [] }, { "id": 14165, "type": "theorem", "label": "sites-modules-lemma-exactness-lower-shriek", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-exactness-lower-shriek", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "Assume that", "\\begin{enumerate}", "\\item[(a)] $u$ is cocontinuous,", "\\item[(b)] $u$ is continuous, and", "\\item[(c)] fibre products and equalizers exist in $\\mathcal{C}$ and", "$u$ commutes with them.", "\\end{enumerate}", "In this case the functor", "$g_! : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{D})$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "Compare with", "Sites, Lemma \\ref{sites-lemma-preserve-equalizers}.", "Assume (a), (b), and (c).", "We already know that $g_!$ is right exact as it is a left adjoint, see", "Categories, Lemma \\ref{categories-lemma-exact-adjoint} and", "Homology, Section \\ref{homology-section-functors}.", "We have $g_! = (g_{p!}\\ )^\\#$. We have to show that", "$g_!$ transforms injective maps of abelian sheaves into injective maps", "of abelian presheaves.", "Recall that sheafification of abelian presheaves is exact, see", "Lemma \\ref{lemma-limits-colimits-abelian-sheaves}.", "Thus it suffices to show that $g_{p!}$ transforms injective maps of", "abelian presheaves into injective maps of abelian presheaves.", "To do this it suffices that colimits over the categories", "$(\\mathcal{I}_V^u)^{opp}$ of", "Sites, Section \\ref{sites-section-functoriality-PSh}", "transform injective maps between diagrams into injections.", "This follows from", "Sites, Lemma \\ref{sites-lemma-almost-directed}", "and", "Algebra, Lemma \\ref{algebra-lemma-almost-directed-colimit-exact}." ], "refs": [ "sites-lemma-preserve-equalizers", "categories-lemma-exact-adjoint", "sites-modules-lemma-limits-colimits-abelian-sheaves", "sites-lemma-almost-directed", "algebra-lemma-almost-directed-colimit-exact" ], "ref_ids": [ 8546, 12250, 14140, 8497, 344 ] } ], "ref_ids": [] }, { "id": 14166, "type": "theorem", "label": "sites-modules-lemma-back-and-forth", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-back-and-forth", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "Assume that", "\\begin{enumerate}", "\\item[(a)] $u$ is cocontinuous,", "\\item[(b)] $u$ is continuous, and", "\\item[(c)] $u$ is fully faithful.", "\\end{enumerate}", "For $g_!, g^{-1}, g_*$ as above the canonical maps", "$\\mathcal{F} \\to g^{-1}g_!\\mathcal{F}$ and", "$g^{-1}g_*\\mathcal{F} \\to \\mathcal{F}$", "are isomorphisms", "for all abelian sheaves $\\mathcal{F}$ on $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "The map $g^{-1}g_*\\mathcal{F} \\to \\mathcal{F}$ is an isomorphism", "by Sites, Lemma \\ref{sites-lemma-back-and-forth} and the fact that", "pullback and pushforward of abelian sheaves agrees with", "pullback and pushforward on the underlying sheaves of sets.", "\\medskip\\noindent", "Pick $U \\in \\Ob(\\mathcal{C})$. We will show that", "$g^{-1}g_!\\mathcal{F}(U) = \\mathcal{F}(U)$. First, note that", "$g^{-1}g_!\\mathcal{F}(U) = g_!\\mathcal{F}(u(U))$. Hence it suffices", "to show that $g_!\\mathcal{F}(u(U)) = \\mathcal{F}(U)$.", "We know that $g_!\\mathcal{F}$ is the (abelian) sheaf associated", "to the presheaf $g_{p!}\\mathcal{F}$ which is defined by the rule", "$$", "V \\longmapsto \\colim_{V \\to u(U')} \\mathcal{F}(U')", "$$", "with colimit taken in $\\textit{Ab}$. If $V = u(U)$, then, as $u$ is", "fully faithful this colimit is over $U \\to U'$. Hence we conclude", "that $g_{p!}\\mathcal{F}(u(U) = \\mathcal{F}(U)$.", "Since $u$ is cocontinuous and continuous any covering of $u(U)$ in", "$\\mathcal{D}$ can be refined by a covering (!) $\\{u(U_i) \\to u(U)\\}$", "of $\\mathcal{D}$ where $\\{U_i \\to U\\}$ is a covering in $\\mathcal{C}$.", "This implies that $(g_{p!}\\mathcal{F})^+(u(U)) = \\mathcal{F}(U)$ also,", "since in the colimit defining the value of $(g_{p!}\\mathcal{F})^+$", "on $u(U)$ we may restrict to the cofinal system of coverings", "$\\{u(U_i) \\to u(U)\\}$ as above. Hence we see that", "$(g_{p!}\\mathcal{F})^+(u(U)) = \\mathcal{F}(U)$ for all objects $U$", "of $\\mathcal{C}$ as well. Repeating this argument one more time", "gives the equality $(g_{p!}\\mathcal{F})^\\#(u(U)) = \\mathcal{F}(U)$", "for all objects $U$ of $\\mathcal{C}$. This produces the desired", "equality $g^{-1}g_!\\mathcal{F} = \\mathcal{F}$." ], "refs": [ "sites-lemma-back-and-forth" ], "ref_ids": [ 8547 ] } ], "ref_ids": [] }, { "id": 14167, "type": "theorem", "label": "sites-modules-lemma-have-left-adjoint", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-have-left-adjoint", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let", "$g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be the morphism of topoi", "associated to a continuous and cocontinuous functor", "$u : \\mathcal{C} \\to \\mathcal{D}$.", "\\begin{enumerate}", "\\item If $u$ has a left adjoint $w$, then $g_!$ agrees with $g_!^{\\Sh}$", "on underlying sheaves of sets and $g_!$ is exact.", "\\item If in addition $w$ is cocontinuous, then $g_! = h^{-1}$ and", "$g^{-1} = h_*$ where", "$h : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ is the morphism of topoi", "associated to $w$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This Lemma is the analogue of", "Sites, Lemma \\ref{sites-lemma-have-left-adjoint}.", "From Sites, Lemma \\ref{sites-lemma-adjoint-functors} we see that the categories", "$\\mathcal{I}_V^u$ have an initial object. Thus the underlying set of a", "colimit of a system of abelian groups over $(\\mathcal{I}_V^u)^{opp}$", "is the colimit of the underlying sets. Whence the agreement", "of $g_!^{\\Sh}$ and $g_!$ by our construction of $g_!$ in", "Definition \\ref{definition-g-shriek}.", "The exactness and (2) follow immediately from the corresponding statements of", "Sites, Lemma \\ref{sites-lemma-have-left-adjoint}." ], "refs": [ "sites-lemma-have-left-adjoint", "sites-lemma-adjoint-functors", "sites-modules-definition-g-shriek", "sites-lemma-have-left-adjoint" ], "ref_ids": [ 8551, 8538, 14285, 8551 ] } ], "ref_ids": [] }, { "id": 14168, "type": "theorem", "label": "sites-modules-lemma-global-pullback", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-global-pullback", "contents": [ "Let", "$(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{D}$-module.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is free then $f^*\\mathcal{F}$ is free.", "\\item If $\\mathcal{F}$ is finite free then $f^*\\mathcal{F}$ is finite free.", "\\item If $\\mathcal{F}$ is generated by global sections", "then $f^*\\mathcal{F}$ is generated by global sections.", "\\item Given $r \\geq 0$ if $\\mathcal{F}$ is generated by $r$ global", "sections, then $f^*\\mathcal{F}$ is generated by $r$ global sections.", "\\item If $\\mathcal{F}$ is generated by finitely many global sections", "then $f^*\\mathcal{F}$ is generated by finitely many global sections.", "\\item If $\\mathcal{F}$ has a global presentation then", "$f^*\\mathcal{F}$ has a global presentation.", "\\item If $\\mathcal{F}$ has a finite global presentation", "then $f^*\\mathcal{F}$ has a finite global presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is true because $f^*$ commutes with arbitrary colimits", "(Lemma \\ref{lemma-exactness-pushforward-pullback})", "and $f^*\\mathcal{O}_\\mathcal{D} = \\mathcal{O}_\\mathcal{C}$." ], "refs": [ "sites-modules-lemma-exactness-pushforward-pullback" ], "ref_ids": [ 14159 ] } ], "ref_ids": [] }, { "id": 14169, "type": "theorem", "label": "sites-modules-lemma-extension-by-zero", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-extension-by-zero", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $U \\in \\Ob(\\mathcal{C})$.", "The restriction functor", "$j_U^* : \\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_U)$", "has a left adjoint", "$j_{U!} : \\textit{Mod}(\\mathcal{O}_U) \\to \\textit{Mod}(\\mathcal{O})$.", "So", "$$", "\\Mor_{\\textit{Mod}(\\mathcal{O}_U)}(\\mathcal{G}, j_U^*\\mathcal{F})", "=", "\\Mor_{\\textit{Mod}(\\mathcal{O})}(j_{U!}\\mathcal{G}, \\mathcal{F})", "$$", "for $\\mathcal{F} \\in \\Ob(\\textit{Mod}(\\mathcal{O}))$", "and $\\mathcal{G} \\in \\Ob(\\textit{Mod}(\\mathcal{O}_U))$.", "Moreover, the extension by zero $j_{U!}\\mathcal{G}$ of $\\mathcal{G}$", "is the sheaf associated to the presheaf", "$$", "V", "\\longmapsto", "\\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{G}(V \\xrightarrow{\\varphi} U)", "$$", "with obvious restriction mappings and an obvious $\\mathcal{O}$-module", "structure." ], "refs": [], "proofs": [ { "contents": [ "The $\\mathcal{O}$-module structure on the presheaf is defined as", "follows. If $f \\in \\mathcal{O}(V)$ and", "$s \\in \\mathcal{G}(V \\xrightarrow{\\varphi} U)$, then", "we define $f \\cdot s = fs$ where", "$f \\in \\mathcal{O}_U(\\varphi : V \\to U) = \\mathcal{O}(V)$", "(because $\\mathcal{O}_U$ is the restriction of $\\mathcal{O}$ to", "$\\mathcal{C}/U$).", "\\medskip\\noindent", "Similarly, let $\\alpha : \\mathcal{G} \\to \\mathcal{F}|_U$ be a", "morphism of $\\mathcal{O}_U$-modules. In this case we can define", "a map from the presheaf of the lemma into $\\mathcal{F}$ by mapping", "$$", "\\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{G}(V \\xrightarrow{\\varphi} U)", "\\longrightarrow", "\\mathcal{F}(V)", "$$", "by the rule that $s \\in \\mathcal{G}(V \\xrightarrow{\\varphi} U)$", "maps to $\\alpha(s) \\in \\mathcal{F}(V)$. It is clear that this is", "$\\mathcal{O}$-linear, and hence induces a morphism of", "$\\mathcal{O}$-modules $\\alpha' : j_{U!}\\mathcal{G} \\to \\mathcal{F}$", "by the properties of sheafification of modules", "(Lemma \\ref{lemma-sheafification-presheaf-modules}).", "\\medskip\\noindent", "Conversely, let $\\beta : j_{U!}\\mathcal{G} \\to \\mathcal{F}$", "by a map of $\\mathcal{O}$-modules.", "Recall from Sites, Section \\ref{sites-section-localize}", "that there exists an extension by the empty set", "$j^{Sh}_{U!} : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})$", "on sheaves of sets which is left adjoint to $j_U^{-1}$.", "Moreover, $j^{Sh}_{U!}\\mathcal{G}$ is the sheaf associated to the presheaf", "$$", "V", "\\longmapsto", "\\coprod\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{G}(V \\xrightarrow{\\varphi} U)", "$$", "Hence there is a natural map", "$j^{Sh}_{U!}\\mathcal{G} \\to j_{U!}\\mathcal{G}$ of sheaves of sets.", "Hence precomposing $\\beta$ by this map we get a map of sheaves of sets", "$j^{Sh}_{U!}\\mathcal{G} \\to \\mathcal{F}$ which by adjunction corresponds", "to a map of sheaves of sets $\\beta' : \\mathcal{G} \\to \\mathcal{F}|_U$.", "We claim that $\\beta'$ is $\\mathcal{O}_U$-linear. Namely, suppose", "that $\\varphi : V \\to U$ is an object of $\\mathcal{C}/U$ and that", "$s, s' \\in \\mathcal{G}(\\varphi : V \\to U)$, and", "$f \\in \\mathcal{O}(V) = \\mathcal{O}_U(\\varphi : V \\to U)$.", "Then by the discussion above we see that", "$\\beta'(s + s')$, resp.\\ $\\beta'(fs)$ in $\\mathcal{F}|_U(\\varphi : V \\to U)$", "correspond to $\\beta(s + s')$, resp.\\ $\\beta(fs)$ in", "$\\mathcal{F}(V)$. Since $\\beta$ is a homomorphism we conclude.", "\\medskip\\noindent", "To conclude the proof of the lemma we have to show that the constructions", "$\\alpha \\mapsto \\alpha'$ and $\\beta \\mapsto \\beta'$ are mutually inverse.", "We omit the verifications." ], "refs": [ "sites-modules-lemma-sheafification-presheaf-modules" ], "ref_ids": [ 14147 ] } ], "ref_ids": [] }, { "id": 14170, "type": "theorem", "label": "sites-modules-lemma-extension-by-zero-exact", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-extension-by-zero-exact", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $U \\in \\Ob(\\mathcal{C})$.", "The functor", "$j_{U!} : \\textit{Mod}(\\mathcal{O}_U) \\to \\textit{Mod}(\\mathcal{O})$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "Since $j_{U!}$ is a left adjoint to $j_U^*$ we see that it is right exact", "(see", "Categories, Lemma \\ref{categories-lemma-exact-adjoint}", "and", "Homology, Section \\ref{homology-section-functors}).", "Hence it suffices to show that if $\\mathcal{G}_1 \\to \\mathcal{G}_2$", "is an injective map of $\\mathcal{O}_U$-modules, then", "$j_{U!}\\mathcal{G}_1 \\to j_{U!}\\mathcal{G}_2$ is injective.", "The map on sections of presheaves over an object $V$", "(as in Lemma \\ref{lemma-extension-by-zero}) is the map", "$$", "\\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{G}_1(V \\xrightarrow{\\varphi} U)", "\\longrightarrow", "\\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{G}_2(V \\xrightarrow{\\varphi} U)", "$$", "which is injective by assumption. Since sheafification is exact by", "Lemma \\ref{lemma-sheafification-exact}", "we conclude $j_{U!}\\mathcal{G}_1 \\to j_{U!}\\mathcal{G}_2$ is injective and", "we win." ], "refs": [ "categories-lemma-exact-adjoint", "sites-modules-lemma-extension-by-zero", "sites-modules-lemma-sheafification-exact" ], "ref_ids": [ 12250, 14169, 14148 ] } ], "ref_ids": [] }, { "id": 14171, "type": "theorem", "label": "sites-modules-lemma-j-shriek-reflects-exactness", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-j-shriek-reflects-exactness", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $U \\in \\Ob(\\mathcal{C})$. A complex of $\\mathcal{O}_U$-modules", "$\\mathcal{G}_1 \\to \\mathcal{G}_2 \\to \\mathcal{G}_3$ is exact", "if and only if", "$j_{U!}\\mathcal{G}_1 \\to j_{U!}\\mathcal{G}_2 \\to j_{U!}\\mathcal{G}_3$", "is exact as a sequence of $\\mathcal{O}$-modules." ], "refs": [], "proofs": [ { "contents": [ "We already know that $j_{U!}$ is exact, see", "Lemma \\ref{lemma-extension-by-zero-exact}.", "Thus it suffices to show that", "$j_{U!} : \\textit{Mod}(\\mathcal{O}_U) \\to \\textit{Mod}(\\mathcal{O})$", "reflects injections and surjections.", "\\medskip\\noindent", "For every $\\mathcal{G}$ in $\\textit{Mod}(\\mathcal{O}_U)$", "we have the unit $\\mathcal{G} \\to j_U^*j_{U!}\\mathcal{G}$", "of the adjunction. We claim this map is an injection of sheaves.", "Namely, looking at the construction of Lemma \\ref{lemma-extension-by-zero}", "we see that this map is the sheafification of the rule sending the object", "$V/U$ of $\\mathcal{C}/U$ to the injective map", "$$", "\\mathcal{G}(V/U) \\longrightarrow", "\\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{G}(V \\xrightarrow{\\varphi} U)", "$$", "given by the inclusion of the summand corresponding to the structure", "morphism $V \\to U$. Since sheafification is exact the claim follows.", "Some details omitted.", "\\medskip\\noindent", "If $\\mathcal{G} \\to \\mathcal{G}'$ is a map of $\\mathcal{O}_U$-modules with", "$j_{U!}\\mathcal{G} \\to j_{U!}\\mathcal{G}'$ injective,", "then $j_U^*j_{U!}\\mathcal{G} \\to j_U^*j_{U!}\\mathcal{G}'$ is injective", "(restriction is exact), hence", "$\\mathcal{G} \\to j_U^*j_{U!}\\mathcal{G}'$ is injective, hence", "$\\mathcal{G} \\to \\mathcal{G}'$ is injective.", "We conclude that $j_{U!}$ reflects injections.", "\\medskip\\noindent", "Let $a : \\mathcal{G} \\to \\mathcal{G}'$ be a map of $\\mathcal{O}_U$-modules", "such that $j_{U!}\\mathcal{G} \\to j_{U!}\\mathcal{G}'$ is surjective.", "Let $\\mathcal{H}$ be the cokernel of $a$.", "Then $j_{U!}\\mathcal{H} = 0$ as $j_{U!}$ is exact.", "By the above the map $\\mathcal{H} \\to j^*_U j_{U!}\\mathcal{H}$", "is injective. Hence $\\mathcal{H} = 0$ as desired." ], "refs": [ "sites-modules-lemma-extension-by-zero-exact", "sites-modules-lemma-extension-by-zero" ], "ref_ids": [ 14170, 14169 ] } ], "ref_ids": [] }, { "id": 14172, "type": "theorem", "label": "sites-modules-lemma-relocalize", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-relocalize", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $f : V \\to U$ be a morphism of $\\mathcal{C}$.", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}/V), \\mathcal{O}_V)", "\\ar[rd]_{(j_V, j_V^\\sharp)} \\ar[rr]_{(j, j^\\sharp)} & &", "(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)", "\\ar[ld]^{(j_U, j_U^\\sharp)} \\\\", "& (\\Sh(\\mathcal{C}), \\mathcal{O}) &", "}", "$$", "of ringed topoi. Here $(j, j^\\sharp)$ is the localization morphism", "associated to the object $V/U$ of the ringed site", "$(\\mathcal{C}/V, \\mathcal{O}_V)$." ], "refs": [], "proofs": [ { "contents": [ "The only thing to check is that", "$j_V^\\sharp = j^\\sharp \\circ j^{-1}(j_U^\\sharp)$,", "since everything else follows directly from", "Sites, Lemma \\ref{sites-lemma-relocalize} and", "Sites, Equation (\\ref{sites-equation-relocalize}).", "We omit the verification of the equality." ], "refs": [ "sites-lemma-relocalize" ], "ref_ids": [ 8559 ] } ], "ref_ids": [] }, { "id": 14173, "type": "theorem", "label": "sites-modules-lemma-restrict-back", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-restrict-back", "contents": [ "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$.", "Assume that every $X$ in $\\mathcal{C}$ has at most", "one morphism to $U$. Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}/U$.", "The canonical maps $\\mathcal{F} \\to j_U^{-1}j_{U!}\\mathcal{F}$", "and $j_U^{-1}j_{U*}\\mathcal{F} \\to \\mathcal{F}$ are", "isomorphisms." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-back-and-forth}", "because the assumption on $U$ is equivalent to the fully faithfulness", "of the localization functor $\\mathcal{C}/U \\to \\mathcal{C}$." ], "refs": [ "sites-modules-lemma-back-and-forth" ], "ref_ids": [ 14166 ] } ], "ref_ids": [] }, { "id": 14174, "type": "theorem", "label": "sites-modules-lemma-localize-morphism-ringed-sites", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-localize-morphism-ringed-sites", "contents": [ "Let", "$(f, f^\\sharp) :", "(\\mathcal{C}, \\mathcal{O})", "\\longrightarrow", "(\\mathcal{D}, \\mathcal{O}')$", "be a morphism of ringed sites where $f$ is given by the continuous", "functor $u : \\mathcal{D} \\to \\mathcal{C}$.", "Let $V$ be an object of $\\mathcal{D}$ and set $U = u(V)$.", "Then there is a canonical map of sheaves of rings $(f')^\\sharp$", "such that the diagram of", "Sites, Lemma \\ref{sites-lemma-localize-morphism}", "is turned into a commutative diagram of ringed topoi", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)", "\\ar[rr]_{(j_U, j_U^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} & &", "(\\Sh(\\mathcal{C}), \\mathcal{O})", "\\ar[d]^{(f, f^\\sharp)} \\\\", "(\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V)", "\\ar[rr]^{(j_V, j_V^\\sharp)} & &", "(\\Sh(\\mathcal{D}), \\mathcal{O}').", "}", "$$", "Moreover, in this situation we have $f'_*j_U^{-1} = j_V^{-1}f_*$", "and $f'_*j_U^* = j_V^*f_*$." ], "refs": [ "sites-lemma-localize-morphism" ], "proofs": [ { "contents": [ "Just take $(f')^\\sharp$ to be", "$$", "(f')^{-1}\\mathcal{O}'_V =", "(f')^{-1}j_V^{-1}\\mathcal{O}' =", "j_U^{-1}f^{-1}\\mathcal{O}' \\xrightarrow{j_U^{-1}f^\\sharp}", "j_U^{-1}\\mathcal{O} = \\mathcal{O}_U", "$$", "and everything else follows from", "Sites, Lemma \\ref{sites-lemma-localize-morphism}.", "(Note that $j^{-1} = j^*$ on sheaves of modules if $j$ is a localization", "morphism, hence the first equality of functors implies the second.)" ], "refs": [ "sites-lemma-localize-morphism" ], "ref_ids": [ 8571 ] } ], "ref_ids": [ 8571 ] }, { "id": 14175, "type": "theorem", "label": "sites-modules-lemma-relocalize-morphism-ringed-sites", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-relocalize-morphism-ringed-sites", "contents": [ "Let", "$(f, f^\\sharp) :", "(\\mathcal{C}, \\mathcal{O})", "\\longrightarrow", "(\\mathcal{D}, \\mathcal{O}')$", "be a morphism of ringed sites where $f$ is given by the continuous", "functor $u : \\mathcal{D} \\to \\mathcal{C}$.", "Let $V \\in \\Ob(\\mathcal{D})$, $U \\in \\Ob(\\mathcal{C})$", "and $c : U \\to u(V)$ a morphism of $\\mathcal{C}$.", "There exists a commutative diagram of ringed topoi", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)", "\\ar[rr]_{(j_U, j_U^\\sharp)} \\ar[d]_{(f_c, f_c^\\sharp)} & &", "(\\Sh(\\mathcal{C}), \\mathcal{O}) \\ar[d]^{(f, f^\\sharp)} \\\\", "(\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V)", "\\ar[rr]^{(j_V, j_V^\\sharp)} & &", "(\\Sh(\\mathcal{D}), \\mathcal{O}').", "}", "$$", "The morphism $(f_c, f_c^\\sharp)$", "is equal to the composition of the morphism", "$$", "(f', (f')^\\sharp) :", "(\\Sh(\\mathcal{C}/u(V)), \\mathcal{O}_{u(V)})", "\\longrightarrow", "(\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V)", "$$", "of", "Lemma \\ref{lemma-localize-morphism-ringed-sites}", "and the morphism", "$$", "(j, j^\\sharp) :", "(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)", "\\to", "(\\Sh(\\mathcal{C}/u(V)), \\mathcal{O}_{u(V)})", "$$", "of", "Lemma \\ref{lemma-relocalize}.", "Given any morphisms $b : V' \\to V$, $a : U' \\to U$ and", "$c' : U' \\to u(V')$ such that", "$$", "\\xymatrix{", "U' \\ar[r]_-{c'} \\ar[d]_a & u(V') \\ar[d]^{u(b)} \\\\", "U \\ar[r]^-c & u(V)", "}", "$$", "commutes, then the following diagram of ringed topoi", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}/U'), \\mathcal{O}_{U'})", "\\ar[rr]_{(j_{U'/U}, j_{U'/U}^\\sharp)} \\ar[d]_{(f_{c'}, f_{c'}^\\sharp)} & &", "(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)", "\\ar[d]^{(f_c, f_c^\\sharp)} \\\\", "(\\Sh(\\mathcal{D}/V'), \\mathcal{O}'_{V'})", "\\ar[rr]^{(j_{V'/V}, j_{V'/V}^\\sharp)} & &", "(\\Sh(\\mathcal{D}/V), \\mathcal{O}'_{V'})", "}", "$$", "commutes." ], "refs": [ "sites-modules-lemma-localize-morphism-ringed-sites", "sites-modules-lemma-relocalize" ], "proofs": [ { "contents": [ "On the level of morphisms of topoi this is", "Sites, Lemma \\ref{sites-lemma-relocalize-morphism}.", "To check that the diagrams commute as morphisms of ringed topoi use", "Lemmas \\ref{lemma-relocalize} and", "\\ref{lemma-localize-morphism-ringed-sites}", "exactly as in the proof of", "Sites, Lemma \\ref{sites-lemma-relocalize-morphism}." ], "refs": [ "sites-lemma-relocalize-morphism", "sites-modules-lemma-relocalize", "sites-modules-lemma-localize-morphism-ringed-sites", "sites-lemma-relocalize-morphism" ], "ref_ids": [ 8573, 14172, 14174, 8573 ] } ], "ref_ids": [ 14174, 14172 ] }, { "id": 14176, "type": "theorem", "label": "sites-modules-lemma-localize-ringed-topos", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-localize-ringed-topos", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.", "Let $\\mathcal{F} \\in \\Sh(\\mathcal{C})$ be a sheaf.", "For a sheaf $\\mathcal{H}$ on $\\mathcal{C}$ denote", "$\\mathcal{H}_\\mathcal{F}$ the sheaf $\\mathcal{H} \\times \\mathcal{F}$", "seen as an object of the category $\\Sh(\\mathcal{C})/\\mathcal{F}$.", "The pair", "$(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})$", "is a ringed topos and there is a canonical morphism of ringed topoi", "$$", "(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) :", "(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})", "\\longrightarrow", "(\\Sh(\\mathcal{C}), \\mathcal{O})", "$$", "which is a localization as in", "Section \\ref{section-localize}", "such that", "\\begin{enumerate}", "\\item the functor $j_\\mathcal{F}^{-1}$ is the functor", "$\\mathcal{H} \\mapsto \\mathcal{H}_\\mathcal{F}$,", "\\item the functor $j_\\mathcal{F}^*$ is the functor", "$\\mathcal{H} \\mapsto \\mathcal{H}_\\mathcal{F}$,", "\\item the functor $j_{\\mathcal{F}!}$ on sheaves of sets is the forgetful", "functor $\\mathcal{G}/\\mathcal{F} \\mapsto \\mathcal{G}$,", "\\item the functor $j_{\\mathcal{F}!}$ on sheaves of modules associates", "to the $\\mathcal{O}_\\mathcal{F}$-module", "$\\varphi : \\mathcal{G} \\to \\mathcal{F}$ the $\\mathcal{O}$-module", "which is the sheafification of the presheaf", "$$", "V \\longmapsto", "\\bigoplus\\nolimits_{s \\in \\mathcal{F}(V)}", "\\{\\sigma \\in \\mathcal{G}(V) \\mid \\varphi(\\sigma) = s \\}", "$$", "for $V \\in \\Ob(\\mathcal{C})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By", "Sites, Lemma \\ref{sites-lemma-localize-topos}", "we see that $\\Sh(\\mathcal{C})/\\mathcal{F}$ is a topos", "and that (1) and (3) are true. In particular this shows that", "$j_\\mathcal{F}^{-1}\\mathcal{O} = \\mathcal{O}_\\mathcal{F}$", "and shows that $\\mathcal{O}_\\mathcal{F}$ is a sheaf of rings.", "Thus we may choose the map $j_\\mathcal{F}^\\sharp$ to be the identity,", "in particular we see that (2) is true.", "Moreover, the proof of", "Sites, Lemma \\ref{sites-lemma-localize-topos}", "shows that we may assume $\\mathcal{C}$ is a site with all finite limits", "and a subcanonical topology and that $\\mathcal{F} = h_U$ for some object", "$U$ of $\\mathcal{C}$.", "Then (4) follows from the description of $j_{\\mathcal{F}!}$ in", "Lemma \\ref{lemma-extension-by-zero}.", "Alternatively one could show directly that the functor described", "in (4) is a left adjoint to $j_\\mathcal{F}^*$." ], "refs": [ "sites-lemma-localize-topos", "sites-lemma-localize-topos", "sites-modules-lemma-extension-by-zero" ], "ref_ids": [ 8583, 8583, 14169 ] } ], "ref_ids": [] }, { "id": 14177, "type": "theorem", "label": "sites-modules-lemma-localize-compare", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-localize-compare", "contents": [ "With", "$(\\Sh(\\mathcal{C}), \\mathcal{O})$ and", "$\\mathcal{F} \\in \\Sh(\\mathcal{C})$ as in", "Lemma \\ref{lemma-localize-ringed-topos}.", "If $\\mathcal{F} = h_U^\\#$ for some object $U$ of $\\mathcal{C}$", "then via the identification", "$\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$ of", "Sites, Lemma \\ref{sites-lemma-essential-image-j-shriek}", "we have", "\\begin{enumerate}", "\\item canonically $\\mathcal{O}_U = \\mathcal{O}_\\mathcal{F}$, and", "\\item with these identifications", "we have $(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) = (j_U, j_U^\\sharp)$.", "\\end{enumerate}" ], "refs": [ "sites-modules-lemma-localize-ringed-topos", "sites-lemma-essential-image-j-shriek" ], "proofs": [ { "contents": [ "The assertion for underlying topoi is", "Sites, Lemma \\ref{sites-lemma-localize-compare}.", "Note that $\\mathcal{O}_U$ is the restriction of $\\mathcal{O}$", "which by", "Sites, Lemma \\ref{sites-lemma-compute-j-shriek-restrict}", "corresponds to $\\mathcal{O} \\times h_U^\\#$ under the equivalence of", "Sites, Lemma \\ref{sites-lemma-essential-image-j-shriek}.", "By definition of $\\mathcal{O}_\\mathcal{F}$ we get (1).", "What's left is to prove that $j_\\mathcal{F}^\\sharp = j_U^\\sharp$", "under this identification. We omit the verification." ], "refs": [ "sites-lemma-localize-compare", "sites-lemma-compute-j-shriek-restrict", "sites-lemma-essential-image-j-shriek" ], "ref_ids": [ 8586, 8558, 8555 ] } ], "ref_ids": [ 14176, 8555 ] }, { "id": 14178, "type": "theorem", "label": "sites-modules-lemma-relocalize-ringed-topos", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-relocalize-ringed-topos", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.", "If $s : \\mathcal{G} \\to \\mathcal{F}$ is a morphism of sheaves", "on $\\mathcal{C}$ then there exists a natural commutative diagram of", "morphisms of ringed topoi", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C})/\\mathcal{G}, \\mathcal{O}_\\mathcal{G})", "\\ar[rd]_{(j_\\mathcal{G}, j_\\mathcal{G}^\\sharp)} \\ar[rr]_{(j, j^\\sharp)} & &", "(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})", "\\ar[ld]^{(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp)} \\\\", "& (\\Sh(\\mathcal{C}), \\mathcal{O}) &", "}", "$$", "where $(j, j^\\sharp)$ is the localization morphism of the ringed topos", "$(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})$", "at the object $\\mathcal{G}/\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "All assertions follow from", "Sites, Lemma \\ref{sites-lemma-relocalize-topos}", "except the assertion that", "$j_\\mathcal{G}^\\sharp = j^\\sharp \\circ j^{-1}(j_\\mathcal{F}^\\sharp)$.", "We omit the verification." ], "refs": [ "sites-lemma-relocalize-topos" ], "ref_ids": [ 8587 ] } ], "ref_ids": [] }, { "id": 14179, "type": "theorem", "label": "sites-modules-lemma-relocalize-compare", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-relocalize-compare", "contents": [ "With $(\\Sh(\\mathcal{C}), \\mathcal{O})$,", "$s : \\mathcal{G} \\to \\mathcal{F}$ as in", "Lemma \\ref{lemma-relocalize-ringed-topos}.", "If there exist a morphism $f : V \\to U$ of $\\mathcal{C}$", "such that $\\mathcal{G} = h_V^\\#$ and $\\mathcal{F} = h_U^\\#$", "and $s$ is induced by $f$, then the", "diagrams of", "Lemma \\ref{lemma-relocalize}", "and", "Lemma \\ref{lemma-relocalize-ringed-topos}", "agree via the identifications", "$(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) = (j_U, j_U^\\sharp)$", "and", "$(j_\\mathcal{G}, j_\\mathcal{G}^\\sharp) = (j_V, j_V^\\sharp)$", "of", "Lemma \\ref{lemma-localize-compare}." ], "refs": [ "sites-modules-lemma-relocalize-ringed-topos", "sites-modules-lemma-relocalize", "sites-modules-lemma-relocalize-ringed-topos", "sites-modules-lemma-localize-compare" ], "proofs": [ { "contents": [ "All assertions follow from", "Sites, Lemma \\ref{sites-lemma-relocalize-compare}", "except for the assertion that the two maps $j^\\sharp$", "agree. This holds since in both cases the map", "$j^\\sharp$ is simply the identity. Some details omitted." ], "refs": [ "sites-lemma-relocalize-compare" ], "ref_ids": [ 8588 ] } ], "ref_ids": [ 14178, 14172, 14178, 14177 ] }, { "id": 14180, "type": "theorem", "label": "sites-modules-lemma-localize-morphism-ringed-topoi", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-localize-morphism-ringed-topoi", "contents": [ "Let", "$$", "f :", "(\\Sh(\\mathcal{C}), \\mathcal{O})", "\\longrightarrow", "(\\Sh(\\mathcal{D}), \\mathcal{O}')", "$$", "be a morphism of ringed topoi. Let $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$.", "Set $\\mathcal{F} = f^{-1}\\mathcal{G}$.", "Then there exists a commutative diagram of ringed topoi", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})", "\\ar[rr]_{(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp)}", "\\ar[d]_{(f', (f')^\\sharp)} & &", "(\\Sh(\\mathcal{C}), \\mathcal{O}) \\ar[d]^{(f, f^\\sharp)} \\\\", "(\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_\\mathcal{G})", "\\ar[rr]^{(j_\\mathcal{G}, j_\\mathcal{G}^\\sharp)} & &", "(\\Sh(\\mathcal{D}), \\mathcal{O}')", "}", "$$", "We have $f'_*j_\\mathcal{F}^{-1} = j_\\mathcal{G}^{-1}f_*$", "and $f'_*j_\\mathcal{F}^* = j_\\mathcal{G}^*f_*$. Moreover, the", "morphism $f'$ is characterized by the rule", "$$", "(f')^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} \\mathcal{G})", "=", "(f^{-1}\\mathcal{H} \\xrightarrow{f^{-1}\\varphi} \\mathcal{F}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "By", "Sites, Lemma \\ref{sites-lemma-localize-morphism-topoi}", "we have the diagram of underlying topoi, the", "equality $f'_*j_\\mathcal{F}^{-1} = j_\\mathcal{G}^{-1}f_*$, and", "the description of $(f')^{-1}$.", "To define $(f')^\\sharp$ we use the map", "$$", "(f')^\\sharp :", "\\mathcal{O}'_\\mathcal{G} =", "j_\\mathcal{G}^{-1} \\mathcal{O}'", "\\xrightarrow{j_\\mathcal{G}^{-1}f^\\sharp}", "j_\\mathcal{G}^{-1} f_*\\mathcal{O} =", "f'_* j_\\mathcal{F}^{-1}\\mathcal{O} =", "f'_* \\mathcal{O}_\\mathcal{F}", "$$", "or equivalently the map", "$$", "(f')^\\sharp :", "(f')^{-1}\\mathcal{O}'_\\mathcal{G} =", "(f')^{-1}j_\\mathcal{G}^{-1} \\mathcal{O}' =", "j_\\mathcal{F}^{-1}f^{-1}\\mathcal{O}'", "\\xrightarrow{j_\\mathcal{F}^{-1}f^\\sharp}", "j_\\mathcal{F}^{-1} \\mathcal{O} =", "\\mathcal{O}_\\mathcal{F}.", "$$", "We omit the verification that these two maps are indeed adjoint", "to each other. The second construction of $(f')^\\sharp$ shows that", "the diagram commutes in the $2$-category of ringed topoi (as the", "maps $j_\\mathcal{F}^\\sharp$ and $j_\\mathcal{G}^\\sharp$ are identities).", "Finally, the equality $f'_*j_\\mathcal{F}^* = j_\\mathcal{G}^*f_*$", "follows from the equality", "$f'_*j_\\mathcal{F}^{-1} = j_\\mathcal{G}^{-1}f_*$", "and the fact that pullbacks of sheaves of modules and sheaves of sets agree,", "see", "Lemma \\ref{lemma-localize-ringed-topos}." ], "refs": [ "sites-lemma-localize-morphism-topoi", "sites-modules-lemma-localize-ringed-topos" ], "ref_ids": [ 8589, 14176 ] } ], "ref_ids": [] }, { "id": 14181, "type": "theorem", "label": "sites-modules-lemma-localize-morphism-compare", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-localize-morphism-compare", "contents": [ "Let", "$$", "f :", "(\\Sh(\\mathcal{C}), \\mathcal{O})", "\\longrightarrow", "(\\Sh(\\mathcal{D}), \\mathcal{O}')", "$$", "be a morphism of ringed topoi.", "Let $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$.", "Set $\\mathcal{F} = f^{-1}\\mathcal{G}$.", "If $f$ is given by a continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$", "and $\\mathcal{G} = h_V^\\#$, then the commutative diagrams of", "Lemma \\ref{lemma-localize-morphism-ringed-sites}", "and", "Lemma \\ref{lemma-localize-morphism-ringed-topoi}", "agree via the identifications of", "Lemma \\ref{lemma-localize-compare}." ], "refs": [ "sites-modules-lemma-localize-morphism-ringed-sites", "sites-modules-lemma-localize-morphism-ringed-topoi", "sites-modules-lemma-localize-compare" ], "proofs": [ { "contents": [ "At the level of morphisms of topoi this is", "Sites, Lemma \\ref{sites-lemma-localize-morphism-compare}.", "This works also on the level of morphisms of ringed topoi since", "the formulas defining $(f')^\\sharp$ in the proofs of", "Lemma \\ref{lemma-localize-morphism-ringed-sites}", "and", "Lemma \\ref{lemma-localize-morphism-ringed-topoi}", "agree." ], "refs": [ "sites-lemma-localize-morphism-compare", "sites-modules-lemma-localize-morphism-ringed-sites", "sites-modules-lemma-localize-morphism-ringed-topoi" ], "ref_ids": [ 8590, 14174, 14180 ] } ], "ref_ids": [ 14174, 14180, 14177 ] }, { "id": 14182, "type": "theorem", "label": "sites-modules-lemma-relocalize-morphism-ringed-topoi", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-relocalize-morphism-ringed-topoi", "contents": [ "Let", "$(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}), \\mathcal{O})", "\\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}')$", "be a morphism of ringed topoi.", "Let $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$,", "let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$,", "and let $s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ a morphism of sheaves.", "There exists a commutative diagram of ringed topoi", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})", "\\ar[rr]_{(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp)}", "\\ar[d]_{(f_c, f_c^\\sharp)} & &", "(\\Sh(\\mathcal{C}), \\mathcal{O})", "\\ar[d]^{(f, f^\\sharp)} \\\\", "(\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_\\mathcal{G})", "\\ar[rr]^{(j_\\mathcal{G}, j_\\mathcal{G}^\\sharp)} & &", "(\\Sh(\\mathcal{D}), \\mathcal{O}').", "}", "$$", "The morphism $(f_s, f_s^\\sharp)$", "is equal to the composition of the morphism", "$$", "(f', (f')^\\sharp) :", "(\\Sh(\\mathcal{C})/f^{-1}\\mathcal{G}, \\mathcal{O}_{f^{-1}\\mathcal{G}})", "\\longrightarrow", "(\\Sh(\\mathcal{D})/{\\mathcal{G}}, \\mathcal{O}'_\\mathcal{G})", "$$", "of", "Lemma \\ref{lemma-localize-morphism-ringed-topoi}", "and the morphism", "$$", "(j, j^\\sharp) :", "(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})", "\\to", "(\\Sh(\\mathcal{C})/f^{-1}\\mathcal{G}, \\mathcal{O}_{f^{-1}\\mathcal{G}})", "$$", "of", "Lemma \\ref{lemma-relocalize-ringed-topos}.", "Given any morphisms $b : \\mathcal{G}' \\to \\mathcal{G}$,", "$a : \\mathcal{F}' \\to \\mathcal{F}$, and", "$s' : \\mathcal{F}' \\to f^{-1}\\mathcal{G}'$ such that", "$$", "\\xymatrix{", "\\mathcal{F}' \\ar[r]_-{s'} \\ar[d]_a &", "f^{-1}\\mathcal{G}' \\ar[d]^{f^{-1}b} \\\\", "\\mathcal{F} \\ar[r]^-s &", "f^{-1}\\mathcal{G}", "}", "$$", "commutes, then the following diagram of ringed topoi", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C})/\\mathcal{F}', \\mathcal{O}_{\\mathcal{F}'})", "\\ar[rr]_{(j_{\\mathcal{F}'/\\mathcal{F}}, j_{\\mathcal{F}'/\\mathcal{F}}^\\sharp)}", "\\ar[d]_{(f_{s'}, f_{s'}^\\sharp)} & &", "(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})", "\\ar[d]^{(f_s, f_s^\\sharp)} \\\\", "(\\Sh(\\mathcal{D})/\\mathcal{G}', \\mathcal{O}'_{\\mathcal{G}'})", "\\ar[rr]^{(j_{\\mathcal{G}'/\\mathcal{G}}, j_{\\mathcal{G}'/\\mathcal{G}}^\\sharp)}", "& &", "(\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_{\\mathcal{G}'})", "}", "$$", "commutes." ], "refs": [ "sites-modules-lemma-localize-morphism-ringed-topoi", "sites-modules-lemma-relocalize-ringed-topos" ], "proofs": [ { "contents": [ "On the level of morphisms of topoi this is", "Sites, Lemma \\ref{sites-lemma-relocalize-morphism-topoi}.", "To check that the diagrams commute as morphisms of ringed topoi use", "the commutative diagrams of", "Lemmas \\ref{lemma-relocalize-ringed-topos} and", "\\ref{lemma-localize-morphism-ringed-topoi}." ], "refs": [ "sites-lemma-relocalize-morphism-topoi", "sites-modules-lemma-relocalize-ringed-topos", "sites-modules-lemma-localize-morphism-ringed-topoi" ], "ref_ids": [ 8591, 14178, 14180 ] } ], "ref_ids": [ 14180, 14178 ] }, { "id": 14183, "type": "theorem", "label": "sites-modules-lemma-relocalize-morphism-compare", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-relocalize-morphism-compare", "contents": [ "Let", "$(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}), \\mathcal{O})", "\\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}')$,", "$s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ be as in", "Lemma \\ref{lemma-relocalize-morphism-ringed-topoi}.", "If $f$ is given by a continuous functor", "$u : \\mathcal{D} \\to \\mathcal{C}$", "and $\\mathcal{G} = h_V^\\#$,", "$\\mathcal{F} = h_U^\\#$ and $s$ comes from a morphism", "$c : U \\to u(V)$, then", "the commutative diagrams of", "Lemma \\ref{lemma-relocalize-morphism-ringed-sites}", "and", "Lemma \\ref{lemma-relocalize-morphism-ringed-topoi}", "agree via the identifications of", "Lemma \\ref{lemma-localize-compare}." ], "refs": [ "sites-modules-lemma-relocalize-morphism-ringed-topoi", "sites-modules-lemma-relocalize-morphism-ringed-sites", "sites-modules-lemma-relocalize-morphism-ringed-topoi", "sites-modules-lemma-localize-compare" ], "proofs": [ { "contents": [ "This is formal using", "Lemmas \\ref{lemma-relocalize-compare} and", "\\ref{lemma-localize-morphism-compare}." ], "refs": [ "sites-modules-lemma-relocalize-compare", "sites-modules-lemma-localize-morphism-compare" ], "ref_ids": [ 14179, 14181 ] } ], "ref_ids": [ 14182, 14175, 14182, 14177 ] }, { "id": 14184, "type": "theorem", "label": "sites-modules-lemma-special-locally-free", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-special-locally-free", "contents": [ "Any of the properties (1) -- (8) of Definition \\ref{definition-site-local}", "is intrinsic (see discussion in Section \\ref{section-intrinsic})." ], "refs": [ "sites-modules-definition-site-local" ], "proofs": [ { "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a special cocontinuous functor.", "Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules on $\\mathcal{C}$.", "Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "be the equivalence of topoi associated to $u$.", "Set $\\mathcal{O}' = g_*\\mathcal{O}$, and let", "$g^\\sharp : \\mathcal{O}' \\to g_*\\mathcal{O}$ be the identity.", "Finally, set $\\mathcal{F}' = g_*\\mathcal{F}$.", "Let $\\mathcal{P}_l$ be one of the properties (1) -- (7) listed in", "Definition \\ref{definition-site-local}.", "(We will discuss the coherent case at the end of the proof.)", "Let $\\mathcal{P}_g$ denote the corresponding property listed in", "Definition \\ref{definition-global}. We have already seen that", "$\\mathcal{P}_g$ is intrinsic.", "We have to show that", "$\\mathcal{P}_l(\\mathcal{C}, \\mathcal{O}, \\mathcal{F})$", "holds if and only if", "$\\mathcal{P}_l(\\mathcal{D}, \\mathcal{O}', \\mathcal{F}')$", "holds.", "\\medskip\\noindent", "Assume that $\\mathcal{F}$ has $\\mathcal{P}_l$.", "Let $V$ be an object of $\\mathcal{D}$.", "One of the properties of a special cocontinuous functor is that there exists", "a covering $\\{u(U_i) \\to V\\}_{i \\in I}$ in the site $\\mathcal{D}$.", "By assumption, for each $i$ there exists a covering", "$\\{U_{ij} \\to U_i\\}_{j \\in J_i}$ in $\\mathcal{C}$ such that", "each restriction $\\mathcal{F}|_{U_{ij}}$ is $\\mathcal{P}_g$. By", "Sites, Lemma \\ref{sites-lemma-localize-special-cocontinuous}", "we have commutative diagrams of ringed topoi", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}/U_{ij}), \\mathcal{O}_{U_{ij}}) \\ar[r] \\ar[d] &", "(\\Sh(\\mathcal{C}), \\mathcal{O}) \\ar[d] \\\\", "(\\Sh(\\mathcal{D}/u(U_{ij})), \\mathcal{O}'_{u(U_{ij})}) \\ar[r] &", "(\\Sh(\\mathcal{D}), \\mathcal{O}')", "}", "$$", "where the vertical arrows are equivalences. Hence we conclude that", "$\\mathcal{F}'|_{u(U_{ij})}$ has property $\\mathcal{P}_g$ also.", "And moreover, $\\{u(U_{ij}) \\to V\\}_{i \\in I, j \\in J_i}$ is a", "covering of the site $\\mathcal{D}$. Hence $\\mathcal{F}'$ has", "property $\\mathcal{P}_l$.", "\\medskip\\noindent", "Assume that $\\mathcal{F}'$ has $\\mathcal{P}_l$.", "Let $U$ be an object of $\\mathcal{C}$.", "By assumption, there exists a covering", "$\\{V_i \\to u(U)\\}_{i \\in I}$ such that $\\mathcal{F}'|_{V_i}$", "has property $\\mathcal{P}_g$. Because $u$ is cocontinuous we", "can refine this covering by a family $\\{u(U_j) \\to u(U)\\}_{j \\in J}$", "where $\\{U_j \\to U\\}_{j \\in J}$ is a covering in $\\mathcal{C}$.", "Say the refinement is given by $\\alpha : J \\to I$ and", "$u(U_j) \\to V_{\\alpha(j)}$.", "Restricting is transitive, i.e.,", "$(\\mathcal{F}'|_{V_{\\alpha(j)}})|_{u(U_j)} = \\mathcal{F}'|_{u(U_j)}$.", "Hence by Lemma \\ref{lemma-global-pullback} we see that", "$\\mathcal{F}'|_{u(U_j)}$ has property $\\mathcal{P}_g$.", "Hence the diagram", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}/U_j), \\mathcal{O}_{U_j}) \\ar[r] \\ar[d] &", "(\\Sh(\\mathcal{C}), \\mathcal{O}) \\ar[d] \\\\", "(\\Sh(\\mathcal{D}/u(U_j)), \\mathcal{O}'_{u(U_j)})", "\\ar[r] &", "(\\Sh(\\mathcal{D}), \\mathcal{O}')", "}", "$$", "where the vertical arrows are equivalences shows that $\\mathcal{F}|_{U_j}$", "has property $\\mathcal{P}_g$ also. Thus $\\mathcal{F}$ has", "property $\\mathcal{P}_l$ as desired.", "\\medskip\\noindent", "Finally, we prove the lemma in case", "$\\mathcal{P}_l = coherent$\\footnote{The mechanics of this", "are a bit awkward, and we suggest the reader skip this part of the proof.}.", "Assume $\\mathcal{F}$ is coherent. This implies that $\\mathcal{F}$", "is of finite type and hence $\\mathcal{F}'$ is of finite type also by the", "first part of the proof. Let $V$ be an object of $\\mathcal{D}$ and let", "$s_1, \\ldots, s_n \\in \\mathcal{F}'(V)$. We have to show that the kernel", "$\\mathcal{K}'$ of", "$\\bigoplus_{j = 1, \\ldots, n} \\mathcal{O}_V \\to \\mathcal{F}'|_V$", "is of finite type on $\\mathcal{D}/V$. This means we have to show that", "for any $V'/V$ there exists a covering $\\{V'_i \\to V'\\}$ such that", "$\\mathcal{F}'|_{V'_i}$ is generated by finitely many sections.", "Replacing $V$ by $V'$ (and restricting the sections $s_j$ to $V'$)", "we reduce to the case where $V' = V$. Since $u$ is a special", "cocontinuous functor, there exists a covering $\\{u(U_i) \\to V\\}_{i \\in I}$", "in the site $\\mathcal{D}$. Using the isomorphism of topoi", "$\\Sh(\\mathcal{C}/U_i) = \\Sh(\\mathcal{D}/u(U_i))$", "we see that $\\mathcal{K}'|_{u(U_i)}$ corresponds to the kernel", "$\\mathcal{K}_i$ of a map", "$\\bigoplus_{j = 1, \\ldots, n} \\mathcal{O}_{U_i} \\to \\mathcal{F}|_{U_i}$.", "Since $\\mathcal{F}$ is coherent we see that $\\mathcal{K}_i$", "is of finite type. Hence we conclude (by the first part of the proof again)", "that $\\mathcal{K}|_{u(U_i)}$ is of finite type. Thus there exist coverings", "$\\{V_{il} \\to u(U_i)\\}$ such that $\\mathcal{K}|_{V_{il}}$ is generated", "by finitely many global sections. Since", "$\\{V_{il} \\to V\\}$ is a covering of $\\mathcal{D}$ we conclude that", "$\\mathcal{K}$ is of finite type as desired.", "\\medskip\\noindent", "Assume $\\mathcal{F}'$ is coherent. This implies that $\\mathcal{F}'$", "is of finite type and hence $\\mathcal{F}$ is of finite type also by the", "first part of the proof. Let $U$ be an object of $\\mathcal{C}$, and let", "$s_1, \\ldots, s_n \\in \\mathcal{F}(U)$. We have to show that the kernel", "$\\mathcal{K}$ of", "$\\bigoplus_{j = 1, \\ldots, n} \\mathcal{O}_U \\to \\mathcal{F}|_U$", "is of finite type on $\\mathcal{C}/U$. Using the isomorphism of topoi", "$\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{D}/u(U))$", "we see that $\\mathcal{K}|_U$ corresponds to the kernel", "$\\mathcal{K}'$ of a map", "$\\bigoplus_{j = 1, \\ldots, n} \\mathcal{O}_{u(U)} \\to \\mathcal{F}'|_{u(U)}$.", "As $\\mathcal{F}'$ is coherent, we see that $\\mathcal{K}'$ is of finite", "type. Hence, by the first part of the proof again, we conclude", "that $\\mathcal{K}$ is of finite type." ], "refs": [ "sites-modules-definition-site-local", "sites-modules-definition-global", "sites-lemma-localize-special-cocontinuous", "sites-modules-lemma-global-pullback" ], "ref_ids": [ 14289, 14286, 8579, 14168 ] } ], "ref_ids": [ 14289 ] }, { "id": 14185, "type": "theorem", "label": "sites-modules-lemma-local-final-object", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-local-final-object", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$", "be a ringed topos. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module.", "Assume that the site $\\mathcal{C}$ has a final object $X$.", "Then", "\\begin{enumerate}", "\\item The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is locally free,", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that", "each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is a locally free", "$\\mathcal{O}_{X_i}$-module, and", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that", "each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is a free", "$\\mathcal{O}_{X_i}$-module.", "\\end{enumerate}", "\\item The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is finite locally free,", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$", "such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$", "is a finite locally free $\\mathcal{O}_{X_i}$-module, and", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$", "such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$", "is a finite free $\\mathcal{O}_{X_i}$-module.", "\\end{enumerate}", "\\item The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is locally generated by sections,", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$", "such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$", "is an $\\mathcal{O}_{X_i}$-module locally generated by sections, and", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$", "such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$", "is an $\\mathcal{O}_{X_i}$-module globally generated by sections.", "\\end{enumerate}", "\\item Given $r \\geq 0$, the following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is locally generated by $r$ sections,", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$", "such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$", "is an $\\mathcal{O}_{X_i}$-module locally generated by $r$ sections, and", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$", "such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$", "is an $\\mathcal{O}_{X_i}$-module globally generated by $r$ sections.", "\\end{enumerate}", "\\item The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is of finite type,", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$", "such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$", "is an $\\mathcal{O}_{X_i}$-module of finite type, and", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$", "such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$", "is an $\\mathcal{O}_{X_i}$-module globally generated by finitely many sections.", "\\end{enumerate}", "\\item The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is quasi-coherent,", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$", "such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$", "is a quasi-coherent $\\mathcal{O}_{X_i}$-module, and", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$", "such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$", "is an $\\mathcal{O}_{X_i}$-module which has a global presentation.", "\\end{enumerate}", "\\item The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is of finite presentation,", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$", "such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$", "is an $\\mathcal{O}_{X_i}$-module of finite presentation, and", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$", "such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$", "is an $\\mathcal{O}_{X_i}$-module has a finite global presentation.", "\\end{enumerate}", "\\item The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is coherent, and", "\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$", "such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$", "is a coherent $\\mathcal{O}_{X_i}$-module.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In each case we have (a) $\\Rightarrow (b)$. In each of the cases (1) - (6)", "condition (b) implies condition (c) by axiom (2) of a site", "(see Sites, Definition \\ref{sites-definition-site})", "and the definition of the local types of modules.", "Suppose $\\{X_i \\to X\\}$ is a covering.", "Then for every object $U$ of $\\mathcal{C}$ we get an", "induced covering $\\{X_i \\times_X U \\to U\\}$. Moreover, the global", "property for $\\mathcal{F}|_{\\mathcal{C}/X_i}$ in part (c) implies", "the corresponding global property for", "$\\mathcal{F}|_{\\mathcal{C}/X_i \\times_X U}$ by", "Lemma \\ref{lemma-global-pullback}, hence the sheaf has property (a)", "by definition. We omit the proof of (b) $\\Rightarrow$ (a) in case (7)." ], "refs": [ "sites-definition-site", "sites-modules-lemma-global-pullback" ], "ref_ids": [ 8652, 14168 ] } ], "ref_ids": [] }, { "id": 14186, "type": "theorem", "label": "sites-modules-lemma-local-pullback", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-local-pullback", "contents": [ "Let", "$(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{D}$-module.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is locally free then $f^*\\mathcal{F}$ is locally free.", "\\item If $\\mathcal{F}$ is finite locally free then $f^*\\mathcal{F}$ is", "finite locally free.", "\\item If $\\mathcal{F}$ is locally generated by sections", "then $f^*\\mathcal{F}$ is locally generated by sections.", "\\item If $\\mathcal{F}$ is locally generated by $r$ sections", "then $f^*\\mathcal{F}$ is locally generated by $r$ sections.", "\\item If $\\mathcal{F}$ is of finite type", "then $f^*\\mathcal{F}$ is of finite type.", "\\item If $\\mathcal{F}$ is quasi-coherent then", "$f^*\\mathcal{F}$ is quasi-coherent.", "\\item If $\\mathcal{F}$ is of finite presentation", "then $f^*\\mathcal{F}$ is of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "According to the discussion in Section \\ref{section-intrinsic}", "we need only check preservation under pullback for a morphism of ringed sites", "$(f, f^\\sharp) :", "(\\mathcal{C}, \\mathcal{O}_\\mathcal{C})", "\\to", "(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$", "such that $f$ is given by a left exact, continuous functor", "$u : \\mathcal{D} \\to \\mathcal{C}$ between sites which have", "all finite limits.", "Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_\\mathcal{D}$-modules", "which has one of the properties (1) -- (6) of", "Definition \\ref{definition-site-local}.", "We know $\\mathcal{D}$ has a final object $Y$ and $X = u(Y)$", "is a final object for $\\mathcal{C}$. By assumption we have", "a covering $\\{Y_i \\to Y\\}$ such that $\\mathcal{G}|_{\\mathcal{D}/Y_i}$", "has the corresponding global property. Set $X_i = u(Y_i)$ so", "that $\\{X_i \\to X\\}$ is a covering in $\\mathcal{C}$.", "We get a commutative diagram of morphisms ringed sites", "$$", "\\xymatrix{", "(\\mathcal{C}/X_i, \\mathcal{O}_\\mathcal{C}|_{X_i}) \\ar[r] \\ar[d] &", "(\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\ar[d] \\\\", "(\\mathcal{D}/Y_i, \\mathcal{O}_\\mathcal{D}|_{Y_i}) \\ar[r] &", "(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})", "}", "$$", "by Sites, Lemma \\ref{sites-lemma-localize-morphism-strong}.", "Hence by Lemma \\ref{lemma-global-pullback}", "that $f^*\\mathcal{G}|_{X_i}$ has the corresponding global", "property. Hence we conclude that $\\mathcal{G}$ has the local", "property we started out with by Lemma \\ref{lemma-local-final-object}." ], "refs": [ "sites-modules-definition-site-local", "sites-lemma-localize-morphism-strong", "sites-modules-lemma-global-pullback", "sites-modules-lemma-local-final-object" ], "ref_ids": [ 14289, 8572, 14168, 14185 ] } ], "ref_ids": [] }, { "id": 14187, "type": "theorem", "label": "sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\theta : \\mathcal{G} \\to \\mathcal{F}$ be a surjective", "$\\mathcal{O}$-module map with $\\mathcal{F}$ of finite presentation", "and $\\mathcal{G}$ of finite type. Then $\\Ker(\\theta)$ is of finite type." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: See Modules, Lemma", "\\ref{modules-lemma-kernel-surjection-finite-free-onto-finite-presentation}." ], "refs": [ "modules-lemma-kernel-surjection-finite-free-onto-finite-presentation" ], "ref_ids": [ 13249 ] } ], "ref_ids": [] }, { "id": 14188, "type": "theorem", "label": "sites-modules-lemma-i-star-equivalence", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-i-star-equivalence", "contents": [ "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$", "be a morphism of ringed topoi. Assume $i$ is a closed immersion of topoi", "and $i^\\sharp : \\mathcal{O}' \\to i_*\\mathcal{O}$ is surjective.", "Denote $\\mathcal{I} \\subset \\mathcal{O}'$ the kernel of $i^\\sharp$.", "The functor", "$$", "i_* :", "\\textit{Mod}(\\mathcal{O})", "\\longrightarrow", "\\textit{Mod}(\\mathcal{O}')", "$$", "is exact, fully faithful, with essential image those", "$\\mathcal{O}'$-modules $\\mathcal{G}$ such that $\\mathcal{I}\\mathcal{G} = 0$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-exactness} and", "Sites, Lemma \\ref{sites-lemma-closed-immersion}", "we see that $i_*$ is exact. From the fact that", "$i_*$ is fully faithful on sheaves of sets, and the fact that", "$i^\\sharp$ is surjective it follows that $i_*$ is fully faithful", "as a functor $\\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}')$.", "Namely, suppose that $\\alpha : i_*\\mathcal{F}_1 \\to i_*\\mathcal{F}_2$", "is an $\\mathcal{O}'$-module map. By the fully faithfulness of $i_*$", "we obtain a map $\\beta : \\mathcal{F}_1 \\to \\mathcal{F}_2$ of sheaves", "of sets. To prove $\\beta$ is a map of modules we have to show", "that", "$$", "\\xymatrix{", "\\mathcal{O} \\times \\mathcal{F}_1 \\ar[r] \\ar[d] &", "\\mathcal{F}_1 \\ar[d] \\\\", "\\mathcal{O} \\times \\mathcal{F}_2 \\ar[r] &", "\\mathcal{F}_2", "}", "$$", "commutes. It suffices to prove commutativity after applying $i_*$.", "Consider", "$$", "\\xymatrix{", "\\mathcal{O}' \\times i_*\\mathcal{F}_1 \\ar[r] \\ar[d] &", "i_*\\mathcal{O} \\times i_*\\mathcal{F}_1 \\ar[r] \\ar[d] &", "i_*\\mathcal{F}_1 \\ar[d] \\\\", "\\mathcal{O}' \\times i_*\\mathcal{F}_2 \\ar[r] &", "i_*\\mathcal{O} \\times i_*\\mathcal{F}_2 \\ar[r] &", "i_*\\mathcal{F}_2", "}", "$$", "We know the outer rectangle commutes. Since $i^\\sharp$ is surjective", "we conclude.", "\\medskip\\noindent", "To finish the proof we have to prove the statement on the essential", "image of $i_*$. It is clear that $i_*\\mathcal{F}$ is annihilated by", "$\\mathcal{I}$ for any $\\mathcal{O}$-module $\\mathcal{F}$. Conversely,", "let $\\mathcal{G}$ be a $\\mathcal{O}'$-module with", "$\\mathcal{I}\\mathcal{G} = 0$. By definition of a closed subtopos", "there exists a subsheaf $\\mathcal{U}$ of the final object of", "$\\mathcal{D}$ such that the essential image of $i_*$ on sheaves of sets", "is the class of sheaves of sets $\\mathcal{H}$ such that", "$\\mathcal{H} \\times \\mathcal{U} \\to \\mathcal{U}$ is an isomorphism.", "In particular, $i_*\\mathcal{O} \\times \\mathcal{U} = \\mathcal{U}$.", "This implies that", "$\\mathcal{I} \\times \\mathcal{U} = \\mathcal{O} \\times \\mathcal{U}$.", "Hence our module $\\mathcal{G}$ satisfies", "$\\mathcal{G} \\times \\mathcal{U} = \\{0\\} \\times \\mathcal{U} = \\mathcal{U}$", "(because the zero module is isomorphic to the final object of sheaves", "of sets). Thus there exists a sheaf of sets $\\mathcal{F}$ on $\\mathcal{C}$", "with $i_*\\mathcal{F} = \\mathcal{G}$. Since $i_*$ is fully faithful on sheaves", "of sets, we see that in order to define the", "addition $\\mathcal{F} \\times \\mathcal{F} \\to \\mathcal{F}$ and the", "multiplication $\\mathcal{O} \\times \\mathcal{F} \\to \\mathcal{F}$", "it suffices to use the addition", "$$", "\\mathcal{G} \\times \\mathcal{G} \\longrightarrow \\mathcal{G}", "$$", "(given to us as $\\mathcal{G}$ is a $\\mathcal{O}'$-module)", "and the multiplication", "$$", "i_*\\mathcal{O} \\times \\mathcal{G} \\to \\mathcal{G}", "$$", "which is given to us as we have the multiplication by", "$\\mathcal{O}'$ which annihilates $\\mathcal{I}$ by assumption", "and $i_*\\mathcal{O} = \\mathcal{O}'/\\mathcal{I}$. By construction", "$\\mathcal{G}$ is isomorphic to the pushforward of the $\\mathcal{O}$-module", "$\\mathcal{F}$ so constructed." ], "refs": [ "sites-modules-lemma-exactness", "sites-lemma-closed-immersion" ], "ref_ids": [ 14162, 8627 ] } ], "ref_ids": [] }, { "id": 14189, "type": "theorem", "label": "sites-modules-lemma-tensor-product-pullback", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-tensor-product-pullback", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be", "a morphism of ringed topoi. Let $\\mathcal{F}$, $\\mathcal{G}$", "be $\\mathcal{O}_\\mathcal{D}$-modules. Then", "$f^*(\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{D}} \\mathcal{G})", "= f^*\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{C}} f^*\\mathcal{G}$", "functorially in $\\mathcal{F}$, $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "For a sheaf $\\mathcal{H}$ of $\\mathcal{O}_\\mathcal{C}$ modules we", "have", "\\begin{align*}", "\\Hom_{\\mathcal{O}_\\mathcal{C}}(", "f^*(\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}), \\mathcal{H})", "& =", "\\Hom_{\\mathcal{O}_\\mathcal{D}}(", "\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}, f_*\\mathcal{H}) \\\\", "& =", "\\text{Bilin}_{\\mathcal{O}_\\mathcal{D}}(", "\\mathcal{F} \\times \\mathcal{G}, f_*\\mathcal{H}) \\\\", "& =", "\\text{Bilin}_{f^{-1}\\mathcal{O}_\\mathcal{D}}(", "f^{-1}\\mathcal{F} \\times f^{-1}\\mathcal{G}, \\mathcal{H}) \\\\", "& =", "\\Hom_{f^{-1}\\mathcal{O}_\\mathcal{D}}(", "f^{-1}\\mathcal{F} \\otimes_{f^{-1}\\mathcal{O}_\\mathcal{D}} f^{-1}\\mathcal{G},", "\\mathcal{H}) \\\\", "& =", "\\Hom_{\\mathcal{O}_\\mathcal{C}}(", "f^*\\mathcal{F} \\otimes_{f^*\\mathcal{O}_\\mathcal{D}} f^*\\mathcal{G},", "\\mathcal{H})", "\\end{align*}", "The interesting ``$=$'' in this sequence of equalities is the", "third equality. It follows from the definition and adjointness of", "$f_*$ and $f^{-1}$ (as discussed in previous sections) in a", "straightforward manner." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14190, "type": "theorem", "label": "sites-modules-lemma-tensor-product-permanence", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-tensor-product-permanence", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of $\\mathcal{O}$-modules.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$, $\\mathcal{G}$ are locally free,", "so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$.", "\\item If $\\mathcal{F}$, $\\mathcal{G}$ are finite locally free,", "so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$.", "\\item If $\\mathcal{F}$, $\\mathcal{G}$ are locally generated", "by sections, so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$.", "\\item If $\\mathcal{F}$, $\\mathcal{G}$ are of finite type,", "so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$.", "\\item If $\\mathcal{F}$, $\\mathcal{G}$ are quasi-coherent,", "so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$.", "\\item If $\\mathcal{F}$, $\\mathcal{G}$ are of finite presentation,", "so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$.", "\\item If $\\mathcal{F}$ is of finite presentation and $\\mathcal{G}$ is coherent,", "then $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$ is coherent.", "\\item If $\\mathcal{F}$, $\\mathcal{G}$ are coherent,", "so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Compare with", "Sheaves of Modules, Lemma \\ref{modules-lemma-tensor-product-permanence}." ], "refs": [ "modules-lemma-tensor-product-permanence" ], "ref_ids": [ 13271 ] } ], "ref_ids": [] }, { "id": 14191, "type": "theorem", "label": "sites-modules-lemma-internal-hom", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-internal-hom", "contents": [ "If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings,", "$\\mathcal{F}$ is a presheaf of $\\mathcal{O}$-modules, and", "$\\mathcal{G}$ is a sheaf of $\\mathcal{O}$-modules, then", "$\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$", "is a sheaf of $\\mathcal{O}$-modules." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hints: Note first that", "$\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})", "= \\SheafHom_\\mathcal{O}(\\mathcal{F}^\\#, \\mathcal{G})$, which reduces", "the question to the case where both $\\mathcal{F}$ and $\\mathcal{G}$", "are sheaves. The result for sheaves of sets is", "Sites, Lemma \\ref{sites-lemma-glue-maps}." ], "refs": [ "sites-lemma-glue-maps" ], "ref_ids": [ 8561 ] } ], "ref_ids": [] }, { "id": 14192, "type": "theorem", "label": "sites-modules-lemma-internal-hom-restriction", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-internal-hom-restriction", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{F}, \\mathcal{G}$ be sheaves of $\\mathcal{O}$-modules.", "Then formation of $\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$", "commutes with restriction to $U$ for $U \\in \\Ob(\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "Immediate from the definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14193, "type": "theorem", "label": "sites-modules-lemma-internal-hom-commute-limits", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-internal-hom-commute-limits", "contents": [ "Internal hom and (co)limits.", "Let $\\mathcal{C}$ be a category and let $\\mathcal{O}$ be a presheaf of rings.", "\\begin{enumerate}", "\\item For any presheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ the functor", "$$", "\\textit{PMod}(\\mathcal{O}) \\longrightarrow \\textit{PMod}(\\mathcal{O})", ", \\quad", "\\mathcal{G} \\longmapsto \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})", "$$", "commutes with arbitrary limits.", "\\item For any presheaf of $\\mathcal{O}$-modules $\\mathcal{G}$ the functor", "$$", "\\textit{PMod}(\\mathcal{O}) \\longrightarrow \\textit{PMod}(\\mathcal{O})^{opp}", ", \\quad", "\\mathcal{F} \\longmapsto \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})", "$$", "commutes with arbitrary colimits, in a formula", "$$", "\\SheafHom_\\mathcal{O}(\\colim_i \\mathcal{F}_i, \\mathcal{G})", "=", "\\lim_i \\SheafHom_\\mathcal{O}(\\mathcal{F}_i, \\mathcal{G}).", "$$", "\\end{enumerate}", "Suppose that $\\mathcal{C}$ is a site, and $\\mathcal{O}$ is a sheaf of rings.", "\\begin{enumerate}", "\\item[(3)] For any sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ the functor", "$$", "\\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O})", ", \\quad", "\\mathcal{G} \\longmapsto \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})", "$$", "commutes with arbitrary limits.", "\\item[(4)] For any sheaf of $\\mathcal{O}$-modules $\\mathcal{G}$ the functor", "$$", "\\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O})^{opp}", ", \\quad", "\\mathcal{F} \\longmapsto \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})", "$$", "commutes with arbitrary colimits, in a formula", "$$", "\\SheafHom_\\mathcal{O}(\\colim_i \\mathcal{F}_i, \\mathcal{G})", "=", "\\lim_i \\SheafHom_\\mathcal{O}(\\mathcal{F}_i, \\mathcal{G}).", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\to \\textit{PMod}(\\mathcal{O})$, $i \\mapsto \\mathcal{G}_i$", "be a diagram. Let $U$ be an object of the category $\\mathcal{C}$.", "As $j_U^*$ is both a left and a right adjoint we see that", "$\\lim_i j_U^*\\mathcal{G}_i = j_U^* \\lim_i \\mathcal{G}_i$.", "Hence we have", "\\begin{align*}", "\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\lim_i \\mathcal{G}_i)(U)", "& =", "\\Hom_{\\mathcal{O}_U}(\\mathcal{F}|_U, \\lim_i \\mathcal{G}_i|_U) \\\\", "& =", "\\lim_i \\Hom_{\\mathcal{O}_U}(\\mathcal{F}|_U, \\mathcal{G}_i|_U) \\\\", "& = \\lim_i \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G}_i)(U)", "\\end{align*}", "by definition of a limit. This proves (1). Part (2) is proved in exactly the", "same way. Part (3) follows from (1) because the limit of a diagram of sheaves", "is the same as the limit in the category of presheaves.", "Finally, (4) follow because, in the formula we have", "$$", "\\Mor_{\\textit{Mod}(\\mathcal{O})}(", "\\colim_i \\mathcal{F}_i, \\mathcal{G})", "=", "\\Mor_{\\textit{PMod}(\\mathcal{O})}(", "\\colim^{PSh}_i \\mathcal{F}_i, \\mathcal{G})", "$$", "as the colimit $\\colim_i \\mathcal{F}_i$ is the sheafification of", "the colimit $\\colim^{PSh}_i \\mathcal{F}_i$ in", "$\\textit{PMod}(\\mathcal{O})$. Hence (4) follows from (2)", "(by the remark on limits above again)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14194, "type": "theorem", "label": "sites-modules-lemma-internal-hom-adjoint-tensor", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-internal-hom-adjoint-tensor", "contents": [ "Let $\\mathcal{C}$ be a category. Let $\\mathcal{O}$ be a presheaf of", "rings.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$, $\\mathcal{G}$, $\\mathcal{H}$ be", "presheaves of $\\mathcal{O}$-modules. There is a canonical isomorphism", "$$", "\\SheafHom_\\mathcal{O}", "(\\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G}, \\mathcal{H})", "\\longrightarrow", "\\SheafHom_\\mathcal{O}", "(\\mathcal{F}, \\SheafHom_\\mathcal{O}(\\mathcal{G}, \\mathcal{H}))", "$$", "which is functorial in all three entries (sheaf Hom in", "all three spots). In particular,", "$$", "\\Mor_{\\textit{PMod}(\\mathcal{O})}(", "\\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G}, \\mathcal{H})", "=", "\\Mor_{\\textit{PMod}(\\mathcal{O})}(", "\\mathcal{F}, \\SheafHom_\\mathcal{O}(\\mathcal{G}, \\mathcal{H}))", "$$", "\\item", "Suppose that $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings,", "and $\\mathcal{F}$, $\\mathcal{G}$, $\\mathcal{H}$ are sheaves of", "$\\mathcal{O}$-modules. There is a canonical isomorphism", "$$", "\\SheafHom_\\mathcal{O}", "(\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}, \\mathcal{H})", "\\longrightarrow", "\\SheafHom_\\mathcal{O}", "(\\mathcal{F}, \\SheafHom_\\mathcal{O}(\\mathcal{G}, \\mathcal{H}))", "$$", "which is functorial in all three entries (sheaf Hom in", "all three spots). In particular,", "$$", "\\Mor_{\\textit{Mod}(\\mathcal{O})}(", "\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}, \\mathcal{H})", "=", "\\Mor_{\\textit{Mod}(\\mathcal{O})}(", "\\mathcal{F}, \\SheafHom_\\mathcal{O}(\\mathcal{G}, \\mathcal{H}))", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is the analogue of", "Algebra, Lemma \\ref{algebra-lemma-hom-from-tensor-product}.", "The proof is the same, and is omitted." ], "refs": [ "algebra-lemma-hom-from-tensor-product" ], "ref_ids": [ 362 ] } ], "ref_ids": [] }, { "id": 14195, "type": "theorem", "label": "sites-modules-lemma-tensor-commute-colimits", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-tensor-commute-colimits", "contents": [ "Tensor product and colimits.", "Let $\\mathcal{C}$ be a category and let $\\mathcal{O}$ be a presheaf of rings.", "\\begin{enumerate}", "\\item For any presheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ the functor", "$$", "\\textit{PMod}(\\mathcal{O}) \\longrightarrow \\textit{PMod}(\\mathcal{O})", ", \\quad", "\\mathcal{G} \\longmapsto \\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G}", "$$", "commutes with arbitrary colimits.", "\\item", "Suppose that $\\mathcal{C}$ is a site, and $\\mathcal{O}$ is a sheaf of rings.", "For any sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ the functor", "$$", "\\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O})", ", \\quad", "\\mathcal{G} \\longmapsto \\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}", "$$", "commutes with arbitrary colimits.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is because tensor product is adjoint to internal hom according", "to Lemma \\ref{lemma-internal-hom-adjoint-tensor}.", "See Categories, Lemma \\ref{categories-lemma-adjoint-exact}." ], "refs": [ "sites-modules-lemma-internal-hom-adjoint-tensor", "categories-lemma-adjoint-exact" ], "ref_ids": [ 14194, 12249 ] } ], "ref_ids": [] }, { "id": 14196, "type": "theorem", "label": "sites-modules-lemma-adjoint-hom-restrict", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-adjoint-hom-restrict", "contents": [ "Let $\\mathcal{C}$ be a category, resp.\\ a site", "Let $\\mathcal{O} \\to \\mathcal{O}'$ be a map of presheaves, resp.\\ sheaves", "of rings. Then", "$$", "\\Hom_\\mathcal{O}(\\mathcal{G}, \\mathcal{F}) =", "\\Hom_{\\mathcal{O}'}(\\mathcal{G},", "\\SheafHom_\\mathcal{O}(\\mathcal{O}', \\mathcal{F}))", "$$", "for any $\\mathcal{O}'$-module $\\mathcal{G}$ and $\\mathcal{O}$-module", "$\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "This is the analogue of", "Algebra, Lemma \\ref{algebra-lemma-adjoint-hom-restrict}.", "The proof is the same, and is omitted." ], "refs": [ "algebra-lemma-adjoint-hom-restrict" ], "ref_ids": [ 375 ] } ], "ref_ids": [] }, { "id": 14197, "type": "theorem", "label": "sites-modules-lemma-j-shriek-and-tensor", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-j-shriek-and-tensor", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $U \\in \\Ob(\\mathcal{C})$.", "For $\\mathcal{G}$ in $\\textit{Mod}(\\mathcal{O}_U)$", "and $\\mathcal{F}$ in $\\textit{Mod}(\\mathcal{O})$", "we have $j_{U!}\\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F} =", "j_{U!}(\\mathcal{G} \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U)$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{H}$ be an object of $\\textit{Mod}(\\mathcal{O})$.", "Then", "\\begin{align*}", "\\Hom_\\mathcal{O}(j_{U!}(\\mathcal{G} \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U),", "\\mathcal{H})", "& =", "\\Hom_{\\mathcal{O}_U}(\\mathcal{G} \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U,", "\\mathcal{H}|_U) \\\\", "& =", "\\Hom_{\\mathcal{O}_U}(\\mathcal{G},", "\\SheafHom_{\\mathcal{O}_U}(\\mathcal{F}|_U, \\mathcal{H}|_U)) \\\\", "& =", "\\Hom_{\\mathcal{O}_U}(\\mathcal{G},", "\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{H})|_U) \\\\", "& =", "\\Hom_\\mathcal{O}(j_{U!}\\mathcal{G},", "\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{H})) \\\\", "& =", "\\Hom_\\mathcal{O}(j_{U!}\\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F},", "\\mathcal{H})", "\\end{align*}", "The first equality because $j_{U!}$ is a left adjoint to restriction", "of modules.", "The second by Lemma \\ref{lemma-internal-hom-adjoint-tensor}.", "The third by Lemma \\ref{lemma-internal-hom-restriction}.", "The fourth because $j_{U!}$ is a left adjoint to restriction", "of modules.", "The fifth by Lemma \\ref{lemma-internal-hom-adjoint-tensor}.", "The lemma follows from this and the Yoneda lemma." ], "refs": [ "sites-modules-lemma-internal-hom-adjoint-tensor", "sites-modules-lemma-internal-hom-restriction", "sites-modules-lemma-internal-hom-adjoint-tensor" ], "ref_ids": [ 14194, 14192, 14194 ] } ], "ref_ids": [] }, { "id": 14198, "type": "theorem", "label": "sites-modules-lemma-flatness-presheaves", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-flatness-presheaves", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{O}$ be a presheaf of rings.", "Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.", "If each $\\mathcal{F}(U)$ is a flat $\\mathcal{O}(U)$-module,", "then $\\mathcal{F}$ is flat." ], "refs": [], "proofs": [ { "contents": [ "This is immediate from the definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14199, "type": "theorem", "label": "sites-modules-lemma-flatness-sheafification", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-flatness-sheafification", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{O}$ be a presheaf of rings.", "Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.", "If $\\mathcal{F}$ is a flat $\\mathcal{O}$-module, then", "$\\mathcal{F}^\\#$ is a flat $\\mathcal{O}^\\#$-module." ], "refs": [], "proofs": [ { "contents": [ "Omitted. (Hint: Sheafification is exact.)" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14200, "type": "theorem", "label": "sites-modules-lemma-colimits-flat", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-colimits-flat", "contents": [ "Colimits and tensor product.", "\\begin{enumerate}", "\\item A filtered colimit of flat presheaves of modules", "is flat. A direct sum of flat presheaves of modules is flat.", "\\item A filtered colimit of flat sheaves of modules is flat.", "A direct sum of flat sheaves of modules is flat.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from Lemma \\ref{lemma-tensor-commute-colimits} and", "Algebra, Lemma \\ref{algebra-lemma-directed-colimit-exact}", "by looking at sections over objects.", "To see part (2), use Lemma \\ref{lemma-tensor-commute-colimits} and", "the fact that a filtered colimit of exact", "complexes is an exact complex (this uses that sheafification is exact", "and commutes with colimits). Some details omitted." ], "refs": [ "sites-modules-lemma-tensor-commute-colimits", "algebra-lemma-directed-colimit-exact", "sites-modules-lemma-tensor-commute-colimits" ], "ref_ids": [ 14195, 343, 14195 ] } ], "ref_ids": [] }, { "id": 14201, "type": "theorem", "label": "sites-modules-lemma-restriction-flat", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-restriction-flat", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $U$ be an object of $\\mathcal{C}$.", "If $\\mathcal{F}$ is a flat $\\mathcal{O}$-module, then", "$\\mathcal{F}|_U$ is a flat $\\mathcal{O}_U$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{G}_1 \\to \\mathcal{G}_2 \\to \\mathcal{G}_3$ be", "an exact complex of $\\mathcal{O}_U$-modules.", "Since $j_{U!}$ is exact (Lemma \\ref{lemma-extension-by-zero-exact})", "and $\\mathcal{F}$ is flat as an $\\mathcal{O}$-modules", "then we see that the complex made up of the modules", "$$", "j_{U!}(\\mathcal{G}_i \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U) =", "j_{U!}\\mathcal{G}_i \\otimes_\\mathcal{O} \\mathcal{F}", "$$", "(Lemma \\ref{lemma-j-shriek-and-tensor}) is exact. We conclude that", "$\\mathcal{G}_1 \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U \\to", "\\mathcal{G}_2 \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U \\to", "\\mathcal{G}_3 \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U$", "is exact by", "Lemma \\ref{lemma-j-shriek-reflects-exactness}." ], "refs": [ "sites-modules-lemma-extension-by-zero-exact", "sites-modules-lemma-j-shriek-and-tensor", "sites-modules-lemma-j-shriek-reflects-exactness" ], "ref_ids": [ 14170, 14197, 14171 ] } ], "ref_ids": [] }, { "id": 14202, "type": "theorem", "label": "sites-modules-lemma-j-shriek-flat", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-j-shriek-flat", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{O}$ be a presheaf of rings.", "Let $U$ be an object of $\\mathcal{C}$.", "Consider the functor $j_U : \\mathcal{C}/U \\to \\mathcal{C}$.", "\\begin{enumerate}", "\\item The presheaf of $\\mathcal{O}$-modules", "$j_{U!}\\mathcal{O}_U$ (see", "Remark \\ref{remark-localize-presheaves})", "is flat.", "\\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings,", "$j_{U!}\\mathcal{O}_U$ is a flat sheaf of $\\mathcal{O}$-modules.", "\\end{enumerate}" ], "refs": [ "sites-modules-remark-localize-presheaves" ], "proofs": [ { "contents": [ "Proof of (1). By the discussion in", "Remark \\ref{remark-localize-presheaves}", "we see that", "$$", "j_{U!}\\mathcal{O}_U(V)", "=", "\\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{O}(V)", "$$", "which is a flat $\\mathcal{O}(V)$-module. Hence (1) follows from", "Lemma \\ref{lemma-flatness-presheaves}.", "Then (2) follows as $j_{U!}\\mathcal{O}_U = (j_{U!}\\mathcal{O}_U)^\\#$", "(the first $j_{U!}$ on sheaves, the second on presheaves)", "and Lemma \\ref{lemma-flatness-sheafification}." ], "refs": [ "sites-modules-remark-localize-presheaves", "sites-modules-lemma-flatness-presheaves", "sites-modules-lemma-flatness-sheafification" ], "ref_ids": [ 14308, 14198, 14199 ] } ], "ref_ids": [ 14308 ] }, { "id": 14203, "type": "theorem", "label": "sites-modules-lemma-module-quotient-flat", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-module-quotient-flat", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{O}$ be a presheaf of rings.", "\\begin{enumerate}", "\\item Any presheaf of $\\mathcal{O}$-modules is a quotient of", "a direct sum $\\bigoplus j_{U_i!}\\mathcal{O}_{U_i}$.", "\\item Any presheaf of $\\mathcal{O}$-modules is a quotient of", "a flat presheaf of $\\mathcal{O}$-modules.", "\\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings,", "then any sheaf of $\\mathcal{O}$-modules is a quotient of", "a direct sum $\\bigoplus j_{U_i!}\\mathcal{O}_{U_i}$.", "\\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings,", "then any sheaf of $\\mathcal{O}$-modules is a quotient of", "a flat sheaf of $\\mathcal{O}$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). For every object $U$ of $\\mathcal{C}$ and every", "$s \\in \\mathcal{F}(U)$ we get a morphism", "$j_{U!}\\mathcal{O}_U \\to \\mathcal{F}$, namely the adjoint to", "the morphism $\\mathcal{O}_U \\to \\mathcal{F}|_U$, $1 \\mapsto s$.", "Clearly the map", "$$", "\\bigoplus\\nolimits_{(U, s)} j_{U!}\\mathcal{O}_U", "\\longrightarrow", "\\mathcal{F}", "$$", "is surjective. The source is flat by combining Lemmas", "\\ref{lemma-colimits-flat} and \\ref{lemma-j-shriek-flat}", "which proves (2). The sheaf case follows from this either by", "sheafifying or repeating the same argument." ], "refs": [ "sites-modules-lemma-colimits-flat", "sites-modules-lemma-j-shriek-flat" ], "ref_ids": [ 14200, 14202 ] } ], "ref_ids": [] }, { "id": 14204, "type": "theorem", "label": "sites-modules-lemma-flat-tor-zero", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-flat-tor-zero", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{O}$ be a presheaf of rings.", "Let", "$$", "0 \\to \\mathcal{F}'' \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0", "$$", "be a short exact sequence of presheaves of $\\mathcal{O}$-modules.", "Let $\\mathcal{G}$ be a presheaf of $\\mathcal{O}$-modules.", "\\begin{enumerate}", "\\item If $\\mathcal{F}$ is a flat presheaf of modules, then", "the sequence", "$$", "0 \\to", "\\mathcal{F}'' \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to", "\\mathcal{F}' \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to", "\\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to 0", "$$", "is exact.", "\\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$,", "$\\mathcal{F}$, $\\mathcal{F}'$, $\\mathcal{F}''$, and", "$\\mathcal{G}$ are sheaves, and $\\mathcal{F}$ is flat", "as a sheaf of modules, then the sequence", "$$", "0 \\to", "\\mathcal{F}'' \\otimes_\\mathcal{O} \\mathcal{G} \\to", "\\mathcal{F}' \\otimes_\\mathcal{O} \\mathcal{G} \\to", "\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G} \\to 0", "$$", "is exact.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose a flat presheaf of $\\mathcal{O}$-modules $\\mathcal{G}'$", "which surjects onto $\\mathcal{G}$. This is possible by", "Lemma \\ref{lemma-module-quotient-flat}. Let", "$\\mathcal{G}'' = \\Ker(\\mathcal{G}' \\to \\mathcal{G})$.", "The lemma follows by applying the snake lemma to the following", "diagram", "$$", "\\begin{matrix}", " & & 0 & & 0 & & 0 & & \\\\", " & & \\uparrow & & \\uparrow & & \\uparrow & & \\\\", " & & \\mathcal{F}'' \\otimes_{p, \\mathcal{O}} \\mathcal{G} & \\to &", " \\mathcal{F}' \\otimes_{p, \\mathcal{O}} \\mathcal{G} & \\to &", " \\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G} & \\to & 0 \\\\", " & & \\uparrow & & \\uparrow & & \\uparrow & & \\\\", "0 & \\to & \\mathcal{F}'' \\otimes_{p, \\mathcal{O}} \\mathcal{G}' & \\to &", " \\mathcal{F}' \\otimes_{p, \\mathcal{O}} \\mathcal{G}' & \\to &", " \\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G}' & \\to & 0 \\\\", " & & \\uparrow & & \\uparrow & & \\uparrow & & \\\\", " & & \\mathcal{F}'' \\otimes_{p, \\mathcal{O}} \\mathcal{G}'' & \\to &", " \\mathcal{F}' \\otimes_{p, \\mathcal{O}} \\mathcal{G}'' & \\to &", " \\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G}'' & \\to & 0 \\\\", " & & & & & & \\uparrow & & \\\\", " & & & & & & 0 & &", "\\end{matrix}", "$$", "with exact rows and columns. The middle row is exact because tensoring", "with the flat module $\\mathcal{G}'$ is exact. The proof in the case", "of sheaves is exactly the same." ], "refs": [ "sites-modules-lemma-module-quotient-flat" ], "ref_ids": [ 14203 ] } ], "ref_ids": [] }, { "id": 14205, "type": "theorem", "label": "sites-modules-lemma-flat-ses", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-flat-ses", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{O}$ be a presheaf of rings.", "Let", "$$", "0 \\to", "\\mathcal{F}_2 \\to", "\\mathcal{F}_1 \\to", "\\mathcal{F}_0 \\to 0", "$$", "be a short exact sequence of presheaves of $\\mathcal{O}$-modules.", "\\begin{enumerate}", "\\item If $\\mathcal{F}_2$ and $\\mathcal{F}_0$ are flat so is", "$\\mathcal{F}_1$.", "\\item If $\\mathcal{F}_1$ and $\\mathcal{F}_0$ are flat so is", "$\\mathcal{F}_2$.", "\\end{enumerate}", "If $\\mathcal{C}$ is a site and $\\mathcal{O}$ is a", "sheaf of rings then the same result holds in $\\textit{Mod}(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{G}^\\bullet$ be an arbitrary exact complex of presheaves", "of $\\mathcal{O}$-modules. Assume that $\\mathcal{F}_0$ is flat.", "By Lemma \\ref{lemma-flat-tor-zero} we see that", "$$", "0 \\to", "\\mathcal{G}^\\bullet \\otimes_{p, \\mathcal{O}} \\mathcal{F}_2 \\to", "\\mathcal{G}^\\bullet \\otimes_{p, \\mathcal{O}} \\mathcal{F}_1 \\to", "\\mathcal{G}^\\bullet \\otimes_{p, \\mathcal{O}} \\mathcal{F}_0 \\to 0", "$$", "is a short exact sequence of complexes of presheaves of", "$\\mathcal{O}$-modules. Hence (1) and (2) follow from the snake lemma.", "The case of sheaves of modules is proved in the same way." ], "refs": [ "sites-modules-lemma-flat-tor-zero" ], "ref_ids": [ 14204 ] } ], "ref_ids": [] }, { "id": 14206, "type": "theorem", "label": "sites-modules-lemma-flat-resolution-of-flat", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-flat-resolution-of-flat", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{O}$ be a presheaf of rings.", "Let", "$$", "\\ldots \\to", "\\mathcal{F}_2 \\to", "\\mathcal{F}_1 \\to", "\\mathcal{F}_0 \\to", "\\mathcal{Q} \\to 0", "$$", "be an exact complex of presheaves of $\\mathcal{O}$-modules.", "If $\\mathcal{Q}$ and all $\\mathcal{F}_i$ are flat $\\mathcal{O}$-modules,", "then for any presheaf $\\mathcal{G}$ of $\\mathcal{O}$-modules the", "complex", "$$", "\\ldots \\to", "\\mathcal{F}_2 \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to", "\\mathcal{F}_1 \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to", "\\mathcal{F}_0 \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to", "\\mathcal{Q} \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to 0", "$$", "is exact also. If $\\mathcal{C}$ is a site and $\\mathcal{O}$ is a", "sheaf of rings then the same result holds $\\textit{Mod}(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Lemma \\ref{lemma-flat-tor-zero} by splitting the complex", "into short exact sequences and using Lemma \\ref{lemma-flat-ses} to", "prove inductively that $\\Im(\\mathcal{F}_{i + 1} \\to \\mathcal{F}_i)$", "is flat." ], "refs": [ "sites-modules-lemma-flat-tor-zero", "sites-modules-lemma-flat-ses" ], "ref_ids": [ 14204, 14205 ] } ], "ref_ids": [] }, { "id": 14207, "type": "theorem", "label": "sites-modules-lemma-tensor-flats", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-tensor-flats", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. If $\\mathcal{G}$ and", "$\\mathcal{F}$ are flat $\\mathcal{O}$-modules, then", "$\\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F}$ is a flat $\\mathcal{O}$-module." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$$", "(\\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F}) \\otimes_\\mathcal{O} \\mathcal{H}", "=", "\\mathcal{G} \\otimes_\\mathcal{O} (\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{H})", "$$", "and a composition of exact functors is exact." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14208, "type": "theorem", "label": "sites-modules-lemma-flat-change-of-rings", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-flat-change-of-rings", "contents": [ "Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves", "of rings on a site $\\mathcal{C}$. If $\\mathcal{G}$ is a", "flat $\\mathcal{O}_1$-module, then", "$\\mathcal{G} \\otimes_{\\mathcal{O}_1} \\mathcal{O}_2$", "is a flat $\\mathcal{O}_2$-module." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$$", "(\\mathcal{G} \\otimes_{\\mathcal{O}_1} \\mathcal{O}_2)", "\\otimes_{\\mathcal{O}_2} \\mathcal{H}", "=", "\\mathcal{G} \\otimes_{\\mathcal{O}_1} \\mathcal{F}", "$$", "(as sheaves of abelian groups for example)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14209, "type": "theorem", "label": "sites-modules-lemma-flat-eq", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-flat-eq", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$ be an", "$\\mathcal{O}$-module. The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a flat $\\mathcal{O}$-module.", "\\item Let $U$ be an object of $\\mathcal{C}$ and let", "$$", "\\mathcal{O}_U \\xrightarrow{(f_1, \\ldots, f_n)}", "\\mathcal{O}_U^{\\oplus n} \\xrightarrow{(s_1, \\ldots, s_n)}", "\\mathcal{F}|_U", "$$", "be a complex of $\\mathcal{O}_U$-modules. Then there exists a covering", "$\\{U_i \\to U\\}$ and for each $i$ a factorization", "$$", "\\mathcal{O}_{U_i}^{\\oplus n}", "\\xrightarrow{B_i}", "\\mathcal{O}_{U_i}^{\\oplus l_i} \\xrightarrow{(t_{i1}, \\ldots, t_{il_i})}", "\\mathcal{F}|_{U_i}", "$$", "of $(s_1, \\ldots, s_n)|_{U_i}$ such that", "$B_i \\circ (f_1, \\ldots, f_n)|_{U_i} = 0$.", "\\item Let $U$ be an object of $\\mathcal{C}$ and let", "$$", "\\mathcal{O}_U^{\\oplus m} \\xrightarrow{A}", "\\mathcal{O}_U^{\\oplus n} \\xrightarrow{(s_1, \\ldots, s_n)}", "\\mathcal{F}|_U", "$$", "be a complex of $\\mathcal{O}_U$-modules. Then there exists a covering", "$\\{U_i \\to U\\}$ and for each $i$ a factorization", "$$", "\\mathcal{O}_{U_i}^{\\oplus n}", "\\xrightarrow{B_i}", "\\mathcal{O}_{U_i}^{\\oplus l_i} \\xrightarrow{(t_{i1}, \\ldots, t_{il_i})}", "\\mathcal{F}|_{U_i}", "$$", "of $(s_1, \\ldots, s_n)|_{U_i}$ such that", "$B_i \\circ A|_{U_i} = 0$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Let $\\mathcal{I} \\subset \\mathcal{O}_U$ be the sheaf of ideals", "generated by $f_1, \\ldots, f_n$. Then $\\sum f_j \\otimes s_j$ is", "a section of $\\mathcal{I} \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U$", "which maps to zero in $\\mathcal{F}|_U$. As $\\mathcal{F}|_U$ is flat", "(Lemma \\ref{lemma-restriction-flat}) the map", "$\\mathcal{I} \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U \\to \\mathcal{F}|_U$", "is injective. Since $\\mathcal{I} \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U$", "is the sheaf associated to the presheaf tensor product, we see", "there exists a covering $\\{U_i \\to U\\}$ such", "that $\\sum f_j|_{U_i} \\otimes s_j|_{U_i}$ is zero in", "$\\mathcal{I}(U_i) \\otimes_{\\mathcal{O}(U_i)} \\mathcal{F}(U_i)$.", "Unwinding the definitions using Algebra, Lemma \\ref{algebra-lemma-relations}", "we find $t_{i1}, \\ldots, t_{i l_i} \\in \\mathcal{F}(U_i)$ and", "$a_{ijk} \\in \\mathcal{O}(U_i)$", "such that $\\sum_j a_{ijk}f_j|_{U_i} = 0$ and", "$s_j|_{U_i} = \\sum_k a_{ijk}t_{ik}$.", "Thus (2) holds.", "\\medskip\\noindent", "Assume (2). Let $U$, $n$, $m$, $A$ and $s_1, \\ldots, s_n$ as in (3) be given.", "Observe that $A$ has $m$ columns. We will prove the assertion of (3)", "is true by induction on $m$. For the base case $m = 0$ we", "can use the factorization through the zero sheaf (in other words", "$l_i = 0$). Let $(f_1, \\ldots, f_n)$ be the last column of $A$", "and apply (2). This gives new diagrams", "$$", "\\mathcal{O}_{U_i}^{\\oplus m} \\xrightarrow{B_i \\circ A|_{U_i}}", "\\mathcal{O}_{U_i}^{\\oplus l_i} \\xrightarrow{(t_{i1}, \\ldots, t_{il_i})}", "\\mathcal{F}|_{U_i}", "$$", "but the first column of $A_i = B_i \\circ A|_{U_i}$ is zero.", "Hence we can apply the induction hypothesis to", "$U_i$, $l_i$, $m - 1$, the matrix consisting of", "the first $m - 1$ columns of $A_i$, and $t_{i1}, \\ldots, t_{il_i}$", "to get coverings $\\{U_{ij} \\to U_j\\}$ and factorizations", "$$", "\\mathcal{O}_{U_{ij}}^{\\oplus l_i}", "\\xrightarrow{C_{ij}}", "\\mathcal{O}_{U_{ij}}^{\\oplus k_{ij}}", "\\xrightarrow{(v_{ij1}, \\ldots, v_{ij k_{ij}})}", "\\mathcal{F}|_{U_{ij}}", "$$", "of $(t_{i1}, \\ldots, t_{il_i})|_{U_{ij}}$ such that", "$C_i \\circ B_i|_{U_{ij}} \\circ A|_{U_{ij}} = 0$.", "Then $\\{U_{ij} \\to U\\}$ is a covering and we get the", "desired factorizations", "using $B_{ij} = C_i \\circ B_i|_{U_{ij}}$ and", "$v_{ija}$. In this way we see that (2) implies (3).", "\\medskip\\noindent", "Assume (3). Let $\\mathcal{G} \\to \\mathcal{H}$ be an injective homomorphism", "of $\\mathcal{O}$-modules. We have to show that", "$\\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F} \\to", "\\mathcal{H} \\otimes_\\mathcal{O} \\mathcal{F}$", "is injective. Let $U$ be an object of $\\mathcal{C}$", "and let $s \\in (\\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F})(U)$", "be a section which maps to zero in", "$\\mathcal{H} \\otimes_\\mathcal{O} \\mathcal{F}$.", "We have to show that $s$ is zero. Since", "$\\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F}$", "is a sheaf, it suffices to find a covering $\\{U_i \\to U\\}_{i \\in I}$", "of $\\mathcal{C}$ such that $s|_{U_i}$ is zero for all $i \\in I$.", "Hence we may always replace $U$ by the members of a covering.", "In particular, since $\\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F}$", "is the sheafification of $\\mathcal{G} \\otimes_{p, \\mathcal{O}} \\mathcal{F}$", "we may assume that $s$ is the image of $s' \\in ", "\\mathcal{G}(U) \\otimes_{\\mathcal{O}(U)} \\mathcal{F}(U)$.", "Arguing similarly for $\\mathcal{H} \\otimes_\\mathcal{O} \\mathcal{F}$", "we may assume that $s'$ maps to zero in", "$\\mathcal{H}(U) \\otimes_{\\mathcal{O}(U)} \\mathcal{F}(U)$.", "Write $\\mathcal{F}(U) = \\colim M_\\alpha$ as a filtered colimit of finitely", "presented $\\mathcal{O}(U)$-modules $M_\\alpha$", "(Algebra, Lemma \\ref{algebra-lemma-module-colimit-fp}).", "Since tensor product commutes with filtered colimits", "(Algebra, Lemma \\ref{algebra-lemma-tensor-products-commute-with-limits})", "we can choose an $\\alpha$ such that $s'$", "comes from some $s'' \\in \\mathcal{G}(U) \\otimes_{\\mathcal{O}(U)} M_\\alpha$", "and such that $s''$ maps to zero in", "$\\mathcal{H}(U) \\otimes_{\\mathcal{O}(U)} M_\\alpha$.", "Fix $\\alpha$ and $s''$.", "Choose a presentation", "$$", "\\mathcal{O}(U)^{\\oplus m} \\xrightarrow{A} \\mathcal{O}(U)^{\\oplus n}", "\\to M_\\alpha \\to 0", "$$", "We apply (3) to the corresponding complex of $\\mathcal{O}_U$-modules", "$$", "\\mathcal{O}_U^{\\oplus m} \\xrightarrow{A}", "\\mathcal{O}_U^{\\oplus n} \\xrightarrow{(s_1, \\ldots, s_n)}", "\\mathcal{F}|_U", "$$", "After replacing $U$ by the members of the covering $U_i$", "we find that the map", "$$", "M_\\alpha \\to \\mathcal{F}(U)", "$$", "factors through a free module $\\mathcal{O}(U)^{\\oplus l}$ for some $l$.", "Since $\\mathcal{G}(U) \\to \\mathcal{H}(U)$ is injective", "we conclude that", "$$", "\\mathcal{G}(U) \\otimes_{\\mathcal{O}(U)} \\mathcal{O}(U)^{\\oplus l}", "\\to", "\\mathcal{H}(U) \\otimes_{\\mathcal{O}(U)} \\mathcal{O}(U)^{\\oplus l}", "$$", "is injective too. Hence as $s''$ maps to zero in the module on", "the right, it also maps to zero in the module on the left, i.e.,", "$s$ is zero as desired." ], "refs": [ "sites-modules-lemma-restriction-flat", "algebra-lemma-relations", "algebra-lemma-module-colimit-fp", "algebra-lemma-tensor-products-commute-with-limits" ], "ref_ids": [ 14201, 956, 355, 363 ] } ], "ref_ids": [] }, { "id": 14210, "type": "theorem", "label": "sites-modules-lemma-flat-over-thickening", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-flat-over-thickening", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}' \\to \\mathcal{O}$", "be a surjection of sheaves of rings whose kernel $\\mathcal{I}$ is", "an ideal of square zero. Let $\\mathcal{F}'$ be an $\\mathcal{O}'$-module", "and set $\\mathcal{F} = \\mathcal{F}'/\\mathcal{I}\\mathcal{F}'$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}'$ is a flat $\\mathcal{O}'$-module, and", "\\item $\\mathcal{F}$ is a flat $\\mathcal{O}$-module and", "$\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{F}'$", "is injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If (1) holds, then", "$\\mathcal{F} = \\mathcal{F}' \\otimes_{\\mathcal{O}'} \\mathcal{O}$ is", "flat over $\\mathcal{O}$ by", "Lemma \\ref{lemma-flat-change-of-rings}", "and we see the map", "$\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{F}'$", "is injective by applying $- \\otimes_{\\mathcal{O}'} \\mathcal{F}'$", "to the exact sequence", "$0 \\to \\mathcal{I} \\to \\mathcal{O}' \\to \\mathcal{O} \\to 0$, see", "Lemma \\ref{lemma-flat-tor-zero}.", "Assume (2). In the rest of the proof we will use without further mention", "that $\\mathcal{K} \\otimes_{\\mathcal{O}'} \\mathcal{F}' =", "\\mathcal{K} \\otimes_\\mathcal{O} \\mathcal{F}$", "for any $\\mathcal{O}'$-module $\\mathcal{K}$ annihilated by $\\mathcal{I}$.", "Let $\\alpha : \\mathcal{G}' \\to \\mathcal{H}'$", "be an injective map of $\\mathcal{O}'$-modules. Let", "$\\mathcal{G} \\subset \\mathcal{G}'$, resp.\\ $\\mathcal{H} \\subset \\mathcal{H}'$", "be the subsheaf of sections annihilated by $\\mathcal{I}$.", "Consider the diagram", "$$", "\\xymatrix{", "\\mathcal{G} \\otimes_{\\mathcal{O}'} \\mathcal{F}' \\ar[r] \\ar[d] &", "\\mathcal{G}' \\otimes_{\\mathcal{O}'} \\mathcal{F}' \\ar[r] \\ar[d] &", "\\mathcal{G}'/\\mathcal{G} \\otimes_{\\mathcal{O}'} \\mathcal{F}' \\ar[r] \\ar[d] &", "0 \\\\", "\\mathcal{H} \\otimes_{\\mathcal{O}'} \\mathcal{F}' \\ar[r] &", "\\mathcal{H}' \\otimes_{\\mathcal{O}'} \\mathcal{F}' \\ar[r] &", "\\mathcal{H}'/\\mathcal{H} \\otimes_{\\mathcal{O}'} \\mathcal{F}' \\ar[r] & 0", "}", "$$", "Note that $\\mathcal{G}'/\\mathcal{G}$ and $\\mathcal{H}'/\\mathcal{H}$", "are annihilated by $\\mathcal{I}$ and that", "$\\mathcal{G}'/\\mathcal{G} \\to \\mathcal{H}'/\\mathcal{H}$ is injective.", "Thus the right vertical arrow is injective as $\\mathcal{F}$ is flat", "over $\\mathcal{O}$. The same is true for the left vertical arrow.", "Hence the middle vertical arrow is injective and $\\mathcal{F}'$ is flat." ], "refs": [ "sites-modules-lemma-flat-change-of-rings", "sites-modules-lemma-flat-tor-zero" ], "ref_ids": [ 14208, 14204 ] } ], "ref_ids": [] }, { "id": 14211, "type": "theorem", "label": "sites-modules-lemma-left-dual-module", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-left-dual-module", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$ be a", "$\\mathcal{O}$-module. Let $\\mathcal{G}, \\eta, \\epsilon$", "be a left dual of $\\mathcal{F}$ in the monoidal category of", "$\\mathcal{O}$-modules, see", "Categories, Definition \\ref{categories-definition-dual}. Then", "\\begin{enumerate}", "\\item for every object $U$ of $\\mathcal{C}$", "there exists a covering $\\{U_i \\to U\\}$ such that $\\mathcal{F}|_{U_i}$", "is a direct summand of a finite free $\\mathcal{O}|_{U_i}$-module,", "\\item the map", "$e : \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{O}) \\to \\mathcal{G}$", "sending a local section $\\lambda$ to $(\\lambda \\otimes 1)(\\eta)$", "is an isomorphism,", "\\item we have $\\epsilon(f, g) = e^{-1}(g)(f)$ for local sections", "$f$ and $g$ of $\\mathcal{F}$ and $\\mathcal{G}$.", "\\end{enumerate}" ], "refs": [ "categories-definition-dual" ], "proofs": [ { "contents": [ "The assumptions mean that", "$$", "\\mathcal{F} \\xrightarrow{\\eta \\otimes 1}", "\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}", "\\otimes_\\mathcal{O} \\mathcal{F}", "\\xrightarrow{1 \\otimes \\epsilon} \\mathcal{F}", "\\quad\\text{and}\\quad", "\\mathcal{G} \\xrightarrow{1 \\otimes \\eta}", "\\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F}", "\\otimes_\\mathcal{O} \\mathcal{G}", "\\xrightarrow{\\epsilon \\otimes 1} \\mathcal{G}", "$$", "are the identity map. Let $U$ be an object of $\\mathcal{C}$.", "After replacing $U$ by the members of a covering of $U$,", "we can find a finite number of sections $f_1, \\ldots, f_n$", "and $g_1, \\ldots, g_n$ of", "$\\mathcal{F}$ and $\\mathcal{G}$ over $U$ such that", "$\\eta(1) = \\sum f_i g_i$. Denote", "$$", "\\mathcal{O}_U^{\\oplus n} \\to \\mathcal{F}|_U", "$$", "the map sending the $i$th basis vector to $f_i$. Then we", "can factor the map $\\eta|_U$ over a map", "$\\tilde \\eta : \\mathcal{O}_U \\to", "\\mathcal{O}_U^{\\oplus n} \\otimes_{\\mathcal{O}_U} \\mathcal{G}|_U$.", "We obtain a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{F}|_U", "\\ar[rr]_-{\\eta \\otimes 1} \\ar[rrd]_-{\\tilde \\eta \\otimes 1} & &", "\\mathcal{F}|_U \\otimes \\mathcal{G}|_U \\otimes \\mathcal{F}|_U", "\\ar[r]_-{1 \\otimes \\epsilon} &", "\\mathcal{F}|_U \\\\", "& &", "\\mathcal{O}_U^{\\oplus n} \\otimes \\mathcal{G}|_U \\otimes \\mathcal{F}|_U", "\\ar[u] \\ar[r]^-{1 \\otimes \\epsilon} &", "\\mathcal{O}_U^{\\oplus n} \\ar[u]", "}", "$$", "This shows that the identity on $\\mathcal{F}|_U$", "factors through a finite free $\\mathcal{O}_U$-module.", "This proves (1). Part (2) follows from", "Categories, Lemma \\ref{categories-lemma-left-dual} and its proof.", "Part (3) follows from the first equality of the proof.", "You can also deduce (2) and (3) from the uniqueness of left duals", "(Categories, Remark \\ref{categories-remark-left-dual-adjoint})", "and the construction of the left dual in", "Example \\ref{example-dual}." ], "refs": [ "categories-lemma-left-dual", "categories-remark-left-dual-adjoint" ], "ref_ids": [ 12325, 12430 ] } ], "ref_ids": [ 12407 ] }, { "id": 14212, "type": "theorem", "label": "sites-modules-lemma-flat-locally-finite-presentation", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-flat-locally-finite-presentation", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$", "be locally of finite presentation and flat. Then given an object", "$U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$ such that", "$\\mathcal{F}|_{U_i}$ is a direct summand of a finite free", "$\\mathcal{O}_{U_i}$-module." ], "refs": [], "proofs": [ { "contents": [ "Choose an object $U$ of $\\mathcal{C}$.", "After replacing $U$ by the members of a covering, we may", "assume there exists a presentation", "$$", "\\mathcal{O}_U^{\\oplus r} \\to", "\\mathcal{O}_U^{\\oplus n} \\to \\mathcal{F}|_U \\to 0", "$$", "By Lemma \\ref{lemma-flat-eq} we may, after replacing $U$", "by the members of a covering, assume there exists a factorization", "$$", "\\mathcal{O}_U^{\\oplus n} \\to", "\\mathcal{O}_U^{\\oplus n_1} \\to \\mathcal{F}|_U", "$$", "such that the composition", "$\\mathcal{O}_U^{\\oplus r} \\to \\mathcal{O}_U^{\\oplus n}", "\\to \\mathcal{O}_U^{\\oplus n_r}$ is zero.", "This means that the surjection $\\mathcal{O}_U^{\\oplus n_r} \\to \\mathcal{F}|_U$", "has a section and we win." ], "refs": [ "sites-modules-lemma-flat-eq" ], "ref_ids": [ 14209 ] } ], "ref_ids": [] }, { "id": 14213, "type": "theorem", "label": "sites-modules-lemma-covering-gives-surjection", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-covering-gives-surjection", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\{U_i \\to U\\}$", "be a covering of $\\mathcal{C}$. Then the sequence", "$$", "\\bigoplus j_{U_i \\times_U U_j!}\\mathcal{O}_{U_i \\times_U U_j} \\to", "\\bigoplus j_{U_i!}\\mathcal{O}_{U_i} \\to j_!\\mathcal{O}_U \\to 0", "$$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "For any $\\mathcal{O}$-module $\\mathcal{F}$ the functor", "$\\Hom_\\mathcal{O}(-, \\mathcal{F})$ turns our sequence into the exact sequence", "$0 \\to \\mathcal{F}(U) \\to \\prod \\mathcal{F}(U_i) \\to", "\\prod \\mathcal{F}(U_i \\times_U U_j)$, see", "(\\ref{equation-map-lower-shriek-OU-into-module}). The lemma follows from", "this and", "Homology, Lemma \\ref{homology-lemma-check-exactness}." ], "refs": [ "homology-lemma-check-exactness" ], "ref_ids": [ 12019 ] } ], "ref_ids": [] }, { "id": 14214, "type": "theorem", "label": "sites-modules-lemma-silly-quasi-compact", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-silly-quasi-compact", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be covering of $\\mathcal{C}$.", "If $U$ is quasi-compact, then there exist a finite subset", "$I' \\subset I$ such that the sequence", "$$", "\\bigoplus\\nolimits_{i, i' \\in I'}", "j_{U_i \\times_U U_{i'}!}\\mathcal{O}_{U_i \\times_U U_{i'}} \\to", "\\bigoplus\\nolimits_{i \\in I'}", "j_{U_i!}\\mathcal{O}_{U_i} \\to", "j_!\\mathcal{O}_U \\to 0", "$$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "This lemma is immediate from Lemma \\ref{lemma-covering-gives-surjection}", "if $U$ satisfies condition (3) of", "Sites, Lemma \\ref{sites-lemma-conclude-quasi-compact}.", "We urge the reader to skip the proof in the general case.", "By definition there exists a covering", "$\\mathcal{V} = \\{V_j \\to U\\}_{j \\in J}$ and a morphism", "$\\mathcal{V} \\to \\mathcal{U}$ of families of maps with fixed target", "given by $\\text{id} : U \\to U$, $\\alpha : J \\to I$, and", "$f_j : V_j \\to U_{\\alpha(j)}$", "(see Sites, Definition \\ref{sites-definition-morphism-coverings})", "such that the image $I' \\subset I$ of $\\alpha$ is finite.", "By Homology, Lemma \\ref{homology-lemma-check-exactness}", "it suffices to show that for any sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$", "the functor $\\Hom_\\mathcal{O}(-, \\mathcal{F})$ turns the sequence", "of the lemma into an exact sequence. By", "(\\ref{equation-map-lower-shriek-OU-into-module})", "we obtain the usual sequence", "$$", "0 \\to", "\\mathcal{F}(U) \\to", "\\prod\\nolimits_{i \\in I'} \\mathcal{F}(U_i) \\to", "\\prod\\nolimits_{i, i' \\in I'} \\mathcal{F}(U_i \\times_U U_{i'})", "$$", "This is an exact sequence by", "Sites, Lemma \\ref{sites-lemma-compare-sheaf-condition}", "applied to the family of maps $\\{U_i \\to U\\}_{i \\in I'}$", "which is refined by the covering $\\mathcal{V}$." ], "refs": [ "sites-modules-lemma-covering-gives-surjection", "sites-lemma-conclude-quasi-compact", "sites-definition-morphism-coverings", "homology-lemma-check-exactness", "sites-lemma-compare-sheaf-condition" ], "ref_ids": [ 14213, 8529, 8656, 12019, 8505 ] } ], "ref_ids": [] }, { "id": 14215, "type": "theorem", "label": "sites-modules-lemma-sections-over-quasi-compact", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-sections-over-quasi-compact", "contents": [ "Let $\\mathcal{C}$ be a site. Let $W$ be a quasi-compact", "object of $\\mathcal{C}$.", "\\begin{enumerate}", "\\item The functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}(W)$ commutes with coproducts.", "\\item Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$. The functor", "$\\textit{Mod}(\\mathcal{O}) \\to \\textit{Ab}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}(W)$", "commutes with direct sums.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Taking sections over $W$ commutes with filtered colimits", "with injective transition maps by", "Sites, Lemma \\ref{sites-lemma-directed-colimits-sections}.", "If $\\mathcal{F}_i$ is a family of sheaves of sets", "indexed by a set $I$. Then $\\coprod \\mathcal{F}_i$ is the filtered", "colimit over the partially ordered set of finite subsets", "$E \\subset I$ of the coproducts", "$\\mathcal{F}_E = \\coprod_{i \\in E} \\mathcal{F}_i$.", "Since the transition maps are injective we conclude.", "\\medskip\\noindent", "Proof of (2).", "Let $\\mathcal{F}_i$ be a family of sheaves of $\\mathcal{O}$-modules", "indexed by a set $I$. Then $\\bigoplus \\mathcal{F}_i$ is the filtered", "colimit over the partially ordered set of finite subsets", "$E \\subset I$ of the direct sums", "$\\mathcal{F}_E = \\bigoplus_{i \\in E} \\mathcal{F}_i$.", "A filtered colimit of abelian sheaves can be computed in the", "category of sheaves of sets. Moreover, for $E \\subset E'$ the transition map", "$\\mathcal{F}_E \\to \\mathcal{F}_{E'}$ is injective (as sheafification", "is exact and the injectivity is clear on underlying presheaves).", "Hence it suffices to show the result for a finite index set by", "Sites, Lemma \\ref{sites-lemma-directed-colimits-sections}.", "The finite case is dealt with in", "Lemma \\ref{lemma-limits-colimits-abelian-sheaves}", "(it holds over any object of $\\mathcal{C}$)." ], "refs": [ "sites-lemma-directed-colimits-sections", "sites-lemma-directed-colimits-sections", "sites-modules-lemma-limits-colimits-abelian-sheaves" ], "ref_ids": [ 8531, 8531, 14140 ] } ], "ref_ids": [] }, { "id": 14216, "type": "theorem", "label": "sites-modules-lemma-quasi-compact-hom-from", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-quasi-compact-hom-from", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be a quasi-compact", "object of $\\mathcal{C}$. Then the functor", "$\\Hom_\\mathcal{O}(j_!\\mathcal{O}_U, -)$ commutes with direct sums." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$\\Hom_\\mathcal{O}(j_!\\mathcal{O}_U, \\mathcal{F}) = \\mathcal{F}(U)$", "by (\\ref{equation-map-lower-shriek-OU-into-module})", "and because the functor $\\mathcal{F} \\mapsto \\mathcal{F}(U)$", "commutes with direct sums by", "Lemma \\ref{lemma-sections-over-quasi-compact}." ], "refs": [ "sites-modules-lemma-sections-over-quasi-compact" ], "ref_ids": [ 14215 ] } ], "ref_ids": [] }, { "id": 14217, "type": "theorem", "label": "sites-modules-lemma-module-quotient-direct-sum", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-module-quotient-direct-sum", "contents": [ "In Situation \\ref{situation-quasi-compact-objects} assume", "(\\ref{item-enough}) holds.", "\\begin{enumerate}", "\\item Every sheaf of sets is the target of a surjective map", "whose source is a coproduct $\\coprod h_{U_i}^\\#$ with $U_i$ in $\\mathcal{B}$.", "\\item If $\\mathcal{O}$ is a sheaf of rings, then every $\\mathcal{O}$-module", "is a quotient of a direct sum $\\bigoplus\\nolimits j_{U_i!}\\mathcal{O}_{U_i}$", "with $U_i$ in $\\mathcal{B}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) follows from Sites, Lemmas", "\\ref{sites-lemma-sheaf-coequalizer-representable} and", "\\ref{sites-lemma-covering-surjective-after-sheafification}.", "Part (2) follows from Lemmas \\ref{lemma-module-quotient-flat} and", "\\ref{lemma-covering-gives-surjection}." ], "refs": [ "sites-lemma-sheaf-coequalizer-representable", "sites-lemma-covering-surjective-after-sheafification", "sites-modules-lemma-module-quotient-flat", "sites-modules-lemma-covering-gives-surjection" ], "ref_ids": [ 8520, 8519, 14203, 14213 ] } ], "ref_ids": [] }, { "id": 14218, "type": "theorem", "label": "sites-modules-lemma-module-filtered-colimit-constructibles", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-module-filtered-colimit-constructibles", "contents": [ "In Situation \\ref{situation-quasi-compact-objects} assume", "(\\ref{item-enough}) and (\\ref{item-enough-qc}) hold.", "\\begin{enumerate}", "\\item Every sheaf of sets is a filtered colimit of sheaves of the form", "\\begin{equation}", "\\label{equation-towards-constructible-sets}", "\\text{Coequalizer}\\left(", "\\xymatrix{", "\\coprod\\nolimits_{j = 1, \\ldots, m} h_{V_j}^\\#", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\coprod\\nolimits_{i = 1, \\ldots, n} h_{U_i}^\\#", "}", "\\right)", "\\end{equation}", "with $U_i$ and $V_j$ in $\\mathcal{B}$.", "\\item If $\\mathcal{O}$ is a sheaf of rings, then every $\\mathcal{O}$-module", "is a filtered colimit of sheaves of the form", "\\begin{equation}", "\\label{equation-towards-constructible}", "\\Coker\\left(", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} j_{V_j!}\\mathcal{O}_{V_j}", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\mathcal{O}_{U_i}", "\\right)", "\\end{equation}", "with $U_i$ and $V_j$ in $\\mathcal{B}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). By Lemma \\ref{lemma-module-quotient-direct-sum}", "every sheaf of sets $\\mathcal{F}$ is the target of a surjection", "whose source is a coprod $\\mathcal{F}_0$", "of sheaves the form $h_{U}^\\#$ with $U \\in \\mathcal{B}$.", "Applying this to $\\mathcal{F}_0 \\times_\\mathcal{F} \\mathcal{F}_0$", "we find that $\\mathcal{F}$ is a coequalizer of a pair of maps", "$$", "\\xymatrix{", "\\coprod\\nolimits_{j \\in J} h_{V_j}^\\#", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\coprod\\nolimits_{i \\in I} h_{U_i}^\\#", "}", "$$", "for some index sets $I$, $J$ and $V_j$ and $U_i$ in $\\mathcal{B}$.", "For every finite subset $J' \\subset J$ there is a finite subset", "$I' \\subset I$ such that the coproduct over $j \\in J'$ maps into", "the coprod over $i \\in I'$ via both maps, see", "Sites, Lemma \\ref{sites-lemma-directed-colimits-sections}.", "(Details omitted; hint: an infinite coproduct is the filtered colimit", "of the finite sub-coproducts.)", "Thus our sheaf is the colimit of the cokernels of these maps", "between finite coproducts.", "\\medskip\\noindent", "Proof of (2).", "By Lemma \\ref{lemma-module-quotient-direct-sum}", "every module is a quotient of a direct sum of modules of the form", "$j_{U!}\\mathcal{O}_U$ with $U \\in \\mathcal{B}$. Thus every module", "is a cokernel", "$$", "\\Coker\\left(", "\\bigoplus\\nolimits_{j \\in J} j_{V_j!}\\mathcal{O}_{V_j}", "\\longrightarrow", "\\bigoplus\\nolimits_{i \\in I} j_{U_i!}\\mathcal{O}_{U_i}", "\\right)", "$$", "for some index sets $I$, $J$ and $V_j$ and $U_i$ in $\\mathcal{B}$.", "For every finite subset $J' \\subset J$ there is a finite subset", "$I' \\subset I$ such that the direct sum over $j \\in J'$ maps into", "the direct sum over $i \\in I'$, see", "Lemma \\ref{lemma-quasi-compact-hom-from}.", "Thus our module is the colimit of the cokernels of these maps", "between finite direct sums." ], "refs": [ "sites-modules-lemma-module-quotient-direct-sum", "sites-lemma-directed-colimits-sections", "sites-modules-lemma-module-quotient-direct-sum", "sites-modules-lemma-quasi-compact-hom-from" ], "ref_ids": [ 14217, 8531, 14217, 14216 ] } ], "ref_ids": [] }, { "id": 14219, "type": "theorem", "label": "sites-modules-lemma-cokernel-map-towards-constructibles", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-cokernel-map-towards-constructibles", "contents": [ "In Situation \\ref{situation-quasi-compact-objects} assume", "(\\ref{item-enough}) and (\\ref{item-enough-qc}) hold.", "Let $\\mathcal{O}$ be a sheaf of rings.", "Then a cokernel of a map between modules as in", "(\\ref{equation-towards-constructible}) is another module as", "in (\\ref{equation-towards-constructible})." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F} = \\Coker(\\bigoplus j_{V_j!}\\mathcal{O}_{V_j} \\to", "\\bigoplus j_{U_i!}\\mathcal{O}_{U_i})$", "as in (\\ref{equation-towards-constructible}). It suffices to show", "that the cokernel of a map $\\varphi : j_{W!}\\mathcal{O}_W \\to \\mathcal{F}$", "with $W \\in \\mathcal{B}$ is another module of the same type.", "The map $\\varphi$ corresponds to $s \\in \\mathcal{F}(W)$.", "Since $\\bigoplus j_{U_i!}\\mathcal{O}_{U_i} \\to \\mathcal{F}$ is", "surjective, by (\\ref{item-enough}) we may choose a covering", "$\\{W_k \\to W\\}_{k \\in K}$ with $W_k \\in \\mathcal{B}$", "such that $s|_{W_k}$ is the image of some section", "$s_k$ of $\\bigoplus j_{U_i!}\\mathcal{O}_{U_i})$.", "By (\\ref{item-enough-qc}) the object $W$ is quasi-compact.", "By Lemma \\ref{lemma-silly-quasi-compact}", "there is a finite subset $K' \\subset K$ such that", "$\\bigoplus_{k \\in K'} j_{W_k!}\\mathcal{O}_{W_k} \\to j_{W!}\\mathcal{O}_W$", "is surjective. We conclude that $\\Coker(\\varphi)$ is equal to", "$$", "\\Coker\\left(", "\\bigoplus\\nolimits_{k \\in K'} j_{W_k!}\\mathcal{O}_{W_k} \\oplus", "\\bigoplus j_{V_j!}\\mathcal{O}_{V_j}", "\\longrightarrow", "\\bigoplus j_{U_i!}\\mathcal{O}_{U_i}", "\\right)", "$$", "where the map $\\bigoplus_{k \\in K'} j_{W_k!}\\mathcal{O}_{W_k}", "\\to \\bigoplus j_{U_i!}\\mathcal{O}_{U_i}$ corresponds to", "$\\sum_{k \\in K'} s_k$. This finishes the proof." ], "refs": [ "sites-modules-lemma-silly-quasi-compact" ], "ref_ids": [ 14214 ] } ], "ref_ids": [] }, { "id": 14220, "type": "theorem", "label": "sites-modules-lemma-change-presentation-towards-constructibles", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-change-presentation-towards-constructibles", "contents": [ "In Situation \\ref{situation-quasi-compact-objects} assume", "(\\ref{item-enough}), (\\ref{item-enough-qc}), and (\\ref{item-enough-qc-qs})", "hold. Let $\\mathcal{O}$ be a sheaf of rings. Assume given a map", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} j_{V_j!}\\mathcal{O}_{V_j}", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\mathcal{O}_{U_i}", "$$", "with $U_i$ and $V_j$ in $\\mathcal{B}$, and coverings", "$\\{U_{ik} \\to U_i\\}_{k \\in K_i}$ with $U_{ik} \\in \\mathcal{B}$.", "Then there exist finite subsets $K'_i \\subset K_i$ and", "a finite set $L$ of $W_l \\in \\mathcal{B}$ and a commutative diagram", "$$", "\\xymatrix{", "\\bigoplus_{l \\in L} j_{W_l!}\\mathcal{O}_{W_l} \\ar[d] \\ar[r] &", "\\bigoplus_{i = 1, \\ldots, n} \\bigoplus_{k \\in K'_i}", "j_{U_{ik}!}\\mathcal{O}_{U_{ik}} \\ar[d] \\\\", "\\bigoplus_{j = 1, \\ldots, m} j_{V_j!}\\mathcal{O}_{V_j} \\ar[r] &", "\\bigoplus_{i = 1, \\ldots, n} j_{U_i!}\\mathcal{O}_{U_i}", "}", "$$", "inducing an isomorphism on cokernels of the horizontal maps." ], "refs": [], "proofs": [ { "contents": [ "Since $U_i$ is quasi-compact, we may choose finite subsets $K'_i \\subset K_i$", "as in Lemma \\ref{lemma-silly-quasi-compact}.", "Then since", "$\\bigoplus_{i = 1, \\ldots, n}", "\\bigoplus_{k \\in K'_i} j_{U_{ik}!}\\mathcal{O}_{U_{ik}} \\to", "\\bigoplus_{i = 1, \\ldots, n} j_{U_i!}\\mathcal{O}_{U_i}$ is surjective,", "we can find coverings $\\{V_{jm} \\to V_j\\}_{m \\in M_j}$ with", "$V_{jm} \\in \\mathcal{B}$ such that we can find a commutative diagram", "$$", "\\xymatrix{", "\\bigoplus_{j = 1, \\ldots, m} \\bigoplus_{m \\in M_j}", "j_{V_{jm}!}\\mathcal{O}_{V_{jm}} \\ar[d] \\ar[r] &", "\\bigoplus_{i = 1, \\ldots n} \\bigoplus_{k \\in K'_i}", "j_{U_{ik}!}\\mathcal{O}_{U_{ik}} \\ar[d] \\\\", "\\bigoplus_{j = 1, \\ldots, m} j_{V_j!}\\mathcal{O}_{V_j} \\ar[r] &", "\\bigoplus_{i = 1, \\ldots, n} j_{U_i!}\\mathcal{O}_{U_i}", "}", "$$", "Since $V_j$ is quasi-compact, we can choose finite subsets", "$M'_j \\subset M_j$ as in Lemma \\ref{lemma-silly-quasi-compact}.", "Set", "$$", "L = \\left(\\coprod\\nolimits_{i = 1, \\ldots, n} K'_i \\times K'_i \\right)", "\\coprod", "\\left(\\coprod\\nolimits_{j = 1, \\ldots, m} M'_j\\right)", "$$", "and for $l = (k, k') \\in K'_i \\times K'_i \\subset L$ set", "$W_l = U_{ik} \\times_{U_i} U_{ik'}$ and for", "$l = m \\in M'_j \\subset L$ set $W_l = V_{jm}$.", "Since we have the exact sequences of Lemma \\ref{lemma-silly-quasi-compact}", "for the families $\\{U_{ik} \\to U_i\\}_{k \\in K'_i}$", "we conclude that we get a diagram as in the statement of the lemma", "(details omitted), except that it is not yet clear that $W_l \\in \\mathcal{B}$.", "However, since $W_l$ is quasi-compact for all $l \\in L$", "we do another application of Lemma \\ref{lemma-silly-quasi-compact}", "and find finite families of maps $\\{W_{lt} \\to W_l\\}_{t \\in T_l}$", "with $W_{lt} \\in \\mathcal{B}$ such that", "$\\bigoplus j_{W_{lt}!}\\mathcal{O}_{W_{lt}} \\to j_{W_l!}\\mathcal{O}_{W_l}$", "is surjective. Then we replace $L$ by $\\coprod_{l \\in L} T_l$", "and everything is clear." ], "refs": [ "sites-modules-lemma-silly-quasi-compact", "sites-modules-lemma-silly-quasi-compact", "sites-modules-lemma-silly-quasi-compact", "sites-modules-lemma-silly-quasi-compact" ], "ref_ids": [ 14214, 14214, 14214, 14214 ] } ], "ref_ids": [] }, { "id": 14221, "type": "theorem", "label": "sites-modules-lemma-extension-towards-constructibles", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-extension-towards-constructibles", "contents": [ "In Situation \\ref{situation-quasi-compact-objects} assume", "(\\ref{item-enough}), (\\ref{item-enough-qc}), and (\\ref{item-enough-qc-qs})", "hold. Let $\\mathcal{O}$ be a sheaf of rings.", "Then an extension of modules as in (\\ref{equation-towards-constructible})", "is another module as in (\\ref{equation-towards-constructible})." ], "refs": [], "proofs": [ { "contents": [ "Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$", "be a short exact sequence of $\\mathcal{O}$-modules with", "$\\mathcal{F}_1$ and $\\mathcal{F}_3$ as in", "(\\ref{equation-towards-constructible}). Choose presentations", "$$", "\\bigoplus A_{V_j} \\to \\bigoplus A_{U_i} \\to \\mathcal{F}_1 \\to 0", "\\quad\\text{and}\\quad", "\\bigoplus A_{T_j} \\to \\bigoplus A_{W_i} \\to \\mathcal{F}_3 \\to 0", "$$", "In this proof the direct sums are always finite, and", "we write $A_U = j_{U!}\\mathcal{O}_U$ for $U \\in \\mathcal{B}$.", "Since $\\mathcal{F}_2 \\to \\mathcal{F}_3$ is surjective, we can", "choose coverings $\\{W_{ik} \\to W_i\\}$ with $W_{ik} \\in \\mathcal{B}$", "such that $A_{W_{ik}} \\to \\mathcal{F}_3$ lifts to a map", "$A_{W_{ik}} \\to \\mathcal{F}_2$.", "By Lemma \\ref{lemma-change-presentation-towards-constructibles}", "we may replace our collection $\\{W_i\\}$ by a finite subcollection", "of the collection $\\{W_{ik}\\}$ and assume the map", "$\\bigoplus A_{W_i} \\to \\mathcal{F}_3$ lifts to a map into $\\mathcal{F}_2$.", "Consider the kernel", "$$", "\\mathcal{K}_2 = \\Ker(\\bigoplus A_{U_i} \\oplus \\bigoplus A_{W_i}", "\\longrightarrow \\mathcal{F}_2)", "$$", "By the snake lemma this kernel surjects onto", "$\\mathcal{K}_3 = \\Ker(\\bigoplus A_{W_i} \\to \\mathcal{F}_3)$.", "Thus, arguing as above, after replacing each $T_j$ by a finite family of", "elements of $\\mathcal{B}$ (permissible by", "Lemma \\ref{lemma-silly-quasi-compact})", "we may assume there is a map", "$\\bigoplus A_{T_j} \\to \\mathcal{K}_2$ lifting the given map", "$\\bigoplus A_{T_j} \\to \\mathcal{K}_3$.", "Then $\\bigoplus A_{V_j} \\oplus \\bigoplus A_{T_j} \\to \\mathcal{K}_2$", "is surjective which finishes the proof." ], "refs": [ "sites-modules-lemma-change-presentation-towards-constructibles", "sites-modules-lemma-silly-quasi-compact" ], "ref_ids": [ 14220, 14214 ] } ], "ref_ids": [] }, { "id": 14222, "type": "theorem", "label": "sites-modules-lemma-towards-constructible-when-serre-subcategory", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-towards-constructible-when-serre-subcategory", "contents": [ "In Situation \\ref{situation-quasi-compact-objects} assume", "(\\ref{item-enough}), (\\ref{item-enough-qc}), and (\\ref{item-enough-qc-qs})", "hold. Let $\\mathcal{O}$ be a sheaf of rings.", "Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$ be the full", "subcategory of modules isomorphic to a cokernel as in", "(\\ref{equation-towards-constructible}).", "If the kernel of every map of $\\mathcal{O}$-modules of the form", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} j_{V_j!}\\mathcal{O}_{V_j}", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\mathcal{O}_{U_i}", "$$", "with $U_i$ and $V_j$ in $\\mathcal{B}$, is in $\\mathcal{A}$, then", "$\\mathcal{A}$ is weak Serre subcategory of $\\textit{Mod}(\\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion of", "Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}.", "By the results of", "Lemmas \\ref{lemma-cokernel-map-towards-constructibles} and", "\\ref{lemma-extension-towards-constructibles}", "it suffices to see that the kernel of a map $\\mathcal{F} \\to \\mathcal{G}$", "between objects of $\\mathcal{A}$ is in $\\mathcal{A}$. To prove this", "choose presentations", "$$", "\\bigoplus A_{V_j} \\to \\bigoplus A_{U_i} \\to \\mathcal{F} \\to 0", "\\quad\\text{and}\\quad", "\\bigoplus A_{T_j} \\to \\bigoplus A_{W_i} \\to \\mathcal{G} \\to 0", "$$", "In this proof the direct sums are always finite, and", "we write $A_U = j_{U!}\\mathcal{O}_U$ for $U \\in \\mathcal{B}$.", "Using Lemmas \\ref{lemma-covering-gives-surjection} and", "\\ref{lemma-change-presentation-towards-constructibles}", "and arguing as in the proof of", "Lemma \\ref{lemma-extension-towards-constructibles}", "we may assume that the map $\\mathcal{F} \\to \\mathcal{G}$", "lifts to a map of presentations", "$$", "\\xymatrix{", "\\bigoplus A_{V_j} \\ar[r] \\ar[d] &", "\\bigoplus A_{U_i} \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[r] \\ar[d] & 0 \\\\", "\\bigoplus A_{T_j} \\ar[r] &", "\\bigoplus A_{W_i} \\ar[r] &", "\\mathcal{G} \\ar[r] & 0", "}", "$$", "Then we see that", "$$", "\\Ker(\\mathcal{F} \\to \\mathcal{G}) =", "\\Coker\\left(\\bigoplus A_{V_j} \\to", "\\Ker\\left(", "\\bigoplus A_{T_j} \\oplus \\bigoplus A_{U_i} \\to \\bigoplus A_{W_i}\\right)\\right)", "$$", "and the lemma follows from the assumption and", "Lemma \\ref{lemma-cokernel-map-towards-constructibles}." ], "refs": [ "homology-lemma-characterize-weak-serre-subcategory", "sites-modules-lemma-cokernel-map-towards-constructibles", "sites-modules-lemma-extension-towards-constructibles", "sites-modules-lemma-covering-gives-surjection", "sites-modules-lemma-change-presentation-towards-constructibles", "sites-modules-lemma-extension-towards-constructibles", "sites-modules-lemma-cokernel-map-towards-constructibles" ], "ref_ids": [ 12046, 14219, 14221, 14213, 14220, 14221, 14219 ] } ], "ref_ids": [] }, { "id": 14223, "type": "theorem", "label": "sites-modules-lemma-flat-pullback-exact", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-flat-pullback-exact", "contents": [ "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$", "be a morphism of ringed topoi. Then", "$$", "f^{-1} : \\textit{Ab}(\\mathcal{C}') \\longrightarrow \\textit{Ab}(\\mathcal{C}),", "\\quad", "\\mathcal{F} \\longmapsto f^{-1}\\mathcal{F}", "$$", "is exact. If", "$(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}), \\mathcal{O})", "\\to", "(\\Sh(\\mathcal{C}'), \\mathcal{O}')$", "is a flat morphism of ringed topoi then", "$$", "f^* : \\textit{Mod}(\\mathcal{O}') \\longrightarrow \\textit{Mod}(\\mathcal{O}),", "\\quad", "\\mathcal{F} \\longmapsto f^*\\mathcal{F}", "$$", "is exact." ], "refs": [], "proofs": [ { "contents": [ "Given an abelian sheaf $\\mathcal{G}$ on $\\mathcal{C}'$", "the underlying sheaf of sets of $f^{-1}\\mathcal{G}$ is the same", "as $f^{-1}$ of the underlying sheaf of sets of $\\mathcal{G}$, see", "Sites, Section \\ref{sites-section-sheaves-algebraic-structures}.", "Hence the exactness of $f^{-1}$ for sheaves of sets (required in the", "definition of a morphism of topoi, see", "Sites, Definition \\ref{sites-definition-topos})", "implies the exactness of $f^{-1}$ as a functor on abelian sheaves.", "\\medskip\\noindent", "To see the statement on modules recall that $f^*\\mathcal{F}$ is defined", "as the tensor product", "$f^{-1}\\mathcal{F} \\otimes_{f^{-1}\\mathcal{O}', f^\\sharp} \\mathcal{O}$.", "Hence $f^*$ is a composition of functors both of which are exact." ], "refs": [ "sites-definition-topos" ], "ref_ids": [ 8667 ] } ], "ref_ids": [] }, { "id": 14224, "type": "theorem", "label": "sites-modules-lemma-invertible", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-invertible", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{L}$", "be an $\\mathcal{O}$-module. The following are equivalent:", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is invertible, and", "\\item there exists an $\\mathcal{O}$-module $\\mathcal{N}$", "such that", "$\\mathcal{L} \\otimes_\\mathcal{O} \\mathcal{N} \\cong \\mathcal{O}$.", "\\end{enumerate}", "In this case we have", "\\begin{enumerate}", "\\item[(a)] $\\mathcal{L}$ is a flat $\\mathcal{O}$-module of finite presentation,", "\\item[(b)] for every object $U$ of $\\mathcal{C}$ there exists a", "covering $U\\{U_i \\to U\\}$ such that $\\mathcal{L}|_{U_i}$", "is a direct summand of a finite free module, and", "\\item[(c)] the module $\\mathcal{N}$ in (2) is isomorphic to", "$\\SheafHom_\\mathcal{O}(\\mathcal{L}, \\mathcal{O})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Then the functor $- \\otimes_\\mathcal{O} \\mathcal{L}$", "is essentially surjective, hence there exists an $\\mathcal{O}$-module", "$\\mathcal{N}$ as in (2). If (2) holds, then the functor", "$- \\otimes_\\mathcal{O} \\mathcal{N}$ is a quasi-inverse", "to the functor $- \\otimes_\\mathcal{O} \\mathcal{L}$ and", "we see that (1) holds.", "\\medskip\\noindent", "Assume (1) and (2) hold. Since $- \\otimes_\\mathcal{O} \\mathcal{L}$ is an", "equivalence, it is exact, and hence $\\mathcal{L}$ is flat. Denote", "$\\psi : \\mathcal{L} \\otimes_\\mathcal{O} \\mathcal{N} \\to \\mathcal{O}$", "the given isomorphism. Let $U$ be an object of $\\mathcal{C}$.", "We will show that the restriction $\\mathcal{L}$ to the members", "of a covering of $U$ is a direct summand of a free module, which will", "certainly imply that $\\mathcal{L}$ is of finite presentation.", "By construction of $\\otimes$ we may assume (after replacing", "$U$ by the members of a covering) that there exists", "an integer $n \\geq 1$ and sections $x_i \\in \\mathcal{L}(U)$,", "$y_i \\in \\mathcal{N}(U)$ such that $\\psi(\\sum x_i \\otimes y_i) = 1$.", "Consider the isomorphisms", "$$", "\\mathcal{L}|_U \\to", "\\mathcal{L}|_U \\otimes_{\\mathcal{O}_U}", "\\mathcal{L}|_U \\otimes_{\\mathcal{O}_U} \\mathcal{N}|_U \\to \\mathcal{L}|_U", "$$", "where the first arrow sends $x$ to $\\sum x_i \\otimes x \\otimes y_i$", "and the second arrow sends $x \\otimes x' \\otimes y$ to $\\psi(x' \\otimes y)x$.", "We conclude that $x \\mapsto \\sum \\psi(x \\otimes y_i)x_i$ is", "an automorphism of $\\mathcal{L}|_U$. This automorphism factors as", "$$", "\\mathcal{L}|_U \\to \\mathcal{O}_U^{\\oplus n} \\to \\mathcal{L}|_U", "$$", "where the first arrow is given by", "$x \\mapsto (\\psi(x \\otimes y_1), \\ldots, \\psi(x \\otimes y_n))$", "and the second arrow by $(a_1, \\ldots, a_n) \\mapsto \\sum a_i x_i$.", "In this way we conclude that $\\mathcal{L}|_U$ is a direct summand", "of a finite free $\\mathcal{O}_U$-module.", "\\medskip\\noindent", "Assume (1) and (2) hold. Consider the evaluation map", "$$", "\\mathcal{L} \\otimes_\\mathcal{O}", "\\SheafHom_\\mathcal{O}(\\mathcal{L}, \\mathcal{O}_X)", "\\longrightarrow \\mathcal{O}_X", "$$", "To finish the proof of the lemma we will show this is an isomorphism.", "By Lemma \\ref{lemma-internal-hom-adjoint-tensor} we have", "$$", "\\Hom_\\mathcal{O}(\\mathcal{O}, \\mathcal{O}) =", "\\Hom_\\mathcal{O}", "(\\mathcal{N} \\otimes_\\mathcal{O} \\mathcal{L}, \\mathcal{O})", "\\longrightarrow", "\\Hom_\\mathcal{O}", "(\\mathcal{N}, \\SheafHom_\\mathcal{O}(\\mathcal{L}, \\mathcal{O}))", "$$", "The image of $1$ gives a morphism", "$\\mathcal{N} \\to \\SheafHom_\\mathcal{O}(\\mathcal{L}, \\mathcal{O})$.", "Tensoring with $\\mathcal{L}$ we obtain", "$$", "\\mathcal{O} = \\mathcal{L} \\otimes_\\mathcal{O} \\mathcal{N}", "\\longrightarrow", "\\mathcal{L} \\otimes_\\mathcal{O} \\SheafHom_\\mathcal{O}(\\mathcal{L}, \\mathcal{O})", "$$", "This map is the inverse to the evaluation map; computation omitted." ], "refs": [ "sites-modules-lemma-internal-hom-adjoint-tensor" ], "ref_ids": [ 14194 ] } ], "ref_ids": [] }, { "id": 14225, "type": "theorem", "label": "sites-modules-lemma-pullback-invertible", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-pullback-invertible", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a", "morphism of ringed topoi. The pullback $f^*\\mathcal{L}$ of an", "invertible $\\mathcal{O}_\\mathcal{D}$-module is invertible." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-invertible}", "there exists an $\\mathcal{O}_\\mathcal{D}$-module $\\mathcal{N}$ such that", "$\\mathcal{L} \\otimes_{\\mathcal{O}_\\mathcal{D}} \\mathcal{N} \\cong", "\\mathcal{O}_\\mathcal{D}$. Pulling back we get", "$f^*\\mathcal{L} \\otimes_{\\mathcal{O}_\\mathcal{C}} f^*\\mathcal{N} \\cong", "\\mathcal{O}_\\mathcal{C}$", "by Lemma \\ref{lemma-tensor-product-pullback}.", "Thus $f^*\\mathcal{L}$ is invertible by Lemma \\ref{lemma-invertible}." ], "refs": [ "sites-modules-lemma-invertible", "sites-modules-lemma-tensor-product-pullback", "sites-modules-lemma-invertible" ], "ref_ids": [ 14224, 14189, 14224 ] } ], "ref_ids": [] }, { "id": 14226, "type": "theorem", "label": "sites-modules-lemma-constructions-invertible", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-constructions-invertible", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed space.", "\\begin{enumerate}", "\\item If $\\mathcal{L}$, $\\mathcal{N}$ are invertible", "$\\mathcal{O}$-modules, then so is", "$\\mathcal{L} \\otimes_\\mathcal{O} \\mathcal{N}$.", "\\item If $\\mathcal{L}$ is an invertible", "$\\mathcal{O}$-module, then so is", "$\\SheafHom_\\mathcal{O}(\\mathcal{L}, \\mathcal{O})$ and the evaluation map", "$\\mathcal{L} \\otimes_\\mathcal{O}", "\\SheafHom_\\mathcal{O}(\\mathcal{L}, \\mathcal{O}) \\to \\mathcal{O}$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is clear from the definition and part (2) follows from", "Lemma \\ref{lemma-invertible} and its proof." ], "refs": [ "sites-modules-lemma-invertible" ], "ref_ids": [ 14224 ] } ], "ref_ids": [] }, { "id": 14227, "type": "theorem", "label": "sites-modules-lemma-pic-set", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-pic-set", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed space.", "There exists a set of invertible modules $\\{\\mathcal{L}_i\\}_{i \\in I}$", "such that each invertible module on $(\\mathcal{C}, \\mathcal{O})$", "is isomorphic to exactly one of the $\\mathcal{L}_i$." ], "refs": [], "proofs": [ { "contents": [ "Omitted, but see Sheaves of Modules, Lemma \\ref{modules-lemma-pic-set}." ], "refs": [ "modules-lemma-pic-set" ], "ref_ids": [ 13304 ] } ], "ref_ids": [] }, { "id": 14228, "type": "theorem", "label": "sites-modules-lemma-universal-module", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-universal-module", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings. The functor", "$$", "\\textit{Mod}(\\mathcal{O}_2) \\longrightarrow \\textit{Ab}, \\quad", "\\mathcal{F} \\longmapsto \\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})", "$$", "is representable." ], "refs": [], "proofs": [ { "contents": [ "This is proved in exactly the same way as the analogous statement in algebra.", "During this proof, for any sheaf of sets $\\mathcal{F}$ on $\\mathcal{C}$,", "let us denote $\\mathcal{O}_2[\\mathcal{F}]$ the sheafification of the", "presheaf $U \\mapsto \\mathcal{O}_2(U)[\\mathcal{F}(U)]$ where this denotes", "the free $\\mathcal{O}_2(U)$-module on the set $\\mathcal{F}(U)$.", "For $s \\in \\mathcal{F}(U)$ we denote $[s]$ the corresponding section", "of $\\mathcal{O}_2[\\mathcal{F}]$ over $U$. If $\\mathcal{F}$ is a sheaf of", "$\\mathcal{O}_2$-modules, then there is a canonical map", "$$", "c : \\mathcal{O}_2[\\mathcal{F}] \\longrightarrow \\mathcal{F}", "$$", "which on the presheaf level is given by the rule", "$\\sum f_s[s] \\mapsto \\sum f_s s$. We will employ the short hand", "$[s] \\mapsto s$ to", "describe this map and similarly for other maps below. Consider", "the map of $\\mathcal{O}_2$-modules", "\\begin{equation}", "\\label{equation-define-module-differentials}", "\\begin{matrix}", "\\mathcal{O}_2[\\mathcal{O}_2 \\times \\mathcal{O}_2] \\oplus", "\\mathcal{O}_2[\\mathcal{O}_2 \\times \\mathcal{O}_2] \\oplus", "\\mathcal{O}_2[\\mathcal{O}_1] &", "\\longrightarrow &", "\\mathcal{O}_2[\\mathcal{O}_2] \\\\", "[(a, b)] \\oplus [(f, g)] \\oplus [h] & \\longmapsto & [a + b] - [a] - [b] + \\\\", "& & [fg] - g[f] - f[g] + \\\\", "& & [\\varphi(h)]", "\\end{matrix}", "\\end{equation}", "with short hand notation as above. Set $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$", "equal to the cokernel of this map. Then it is clear that there exists", "a map of sheaves of sets", "$$", "\\text{d} : \\mathcal{O}_2 \\longrightarrow \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}", "$$", "mapping a local section $f$ to the image of $[f]$ in", "$\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$. By construction $\\text{d}$", "is a $\\mathcal{O}_1$-derivation. Next, let $\\mathcal{F}$", "be a sheaf of $\\mathcal{O}_2$-modules and let", "$D : \\mathcal{O}_2 \\to \\mathcal{F}$ be a $\\mathcal{O}_1$-derivation.", "Then we can consider the $\\mathcal{O}_2$-linear map", "$\\mathcal{O}_2[\\mathcal{O}_2] \\to \\mathcal{F}$ which sends $[g]$ to $D(g)$.", "It follows from the definition of a derivation that this map annihilates", "sections in the image of the map (\\ref{equation-define-module-differentials})", "and hence defines a map", "$$", "\\alpha_D : \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\longrightarrow \\mathcal{F}", "$$", "Since it is clear that $D = \\alpha_D \\circ \\text{d}$ the lemma is proved." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14229, "type": "theorem", "label": "sites-modules-lemma-differentials-sheafify", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-differentials-sheafify", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of presheaves of rings. Then", "$\\Omega_{\\mathcal{O}_2^\\#/\\mathcal{O}_1^\\#}$ is the sheaf associated to the", "presheaf $U \\mapsto \\Omega_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}$." ], "refs": [], "proofs": [ { "contents": [ "Consider the map (\\ref{equation-define-module-differentials}). There is", "a similar map of presheaves whose value on $U \\in \\Ob(\\mathcal{C})$ is", "$$", "\\mathcal{O}_2(U)[\\mathcal{O}_2(U) \\times \\mathcal{O}_2(U)] \\oplus", "\\mathcal{O}_2(U)[\\mathcal{O}_2(U) \\times \\mathcal{O}_2(U)] \\oplus", "\\mathcal{O}_2(U)[\\mathcal{O}_1(U)]", "\\longrightarrow", "\\mathcal{O}_2(U)[\\mathcal{O}_2(U)]", "$$", "The cokernel of this map has value $\\Omega_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}$", "over $U$ by the construction of the module of differentials in ", "Algebra, Definition \\ref{algebra-definition-differentials}.", "On the other hand, the sheaves in (\\ref{equation-define-module-differentials})", "are the sheafifications of the presheaves above. Thus the result follows", "as sheafification is exact." ], "refs": [ "algebra-definition-differentials" ], "ref_ids": [ 1526 ] } ], "ref_ids": [] }, { "id": 14230, "type": "theorem", "label": "sites-modules-lemma-pullback-differentials", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-pullback-differentials", "contents": [ "Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi.", "Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings on $\\mathcal{C}$.", "Then there is a canonical identification", "$f^{-1}\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} =", "\\Omega_{f^{-1}\\mathcal{O}_2/f^{-1}\\mathcal{O}_1}$", "compatible with universal derivations." ], "refs": [], "proofs": [ { "contents": [ "This holds because the sheaf $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$", "is the cokernel of the map (\\ref{equation-define-module-differentials})", "and a similar statement holds for", "$\\Omega_{f^{-1}\\mathcal{O}_2/f^{-1}\\mathcal{O}_1}$,", "because the functor $f^{-1}$ is exact, and because", "$f^{-1}(\\mathcal{O}_2[\\mathcal{O}_2]) =", "f^{-1}\\mathcal{O}_2[f^{-1}\\mathcal{O}_2]$,", "$f^{-1}(\\mathcal{O}_2[\\mathcal{O}_2 \\times \\mathcal{O}_2]) =", "f^{-1}\\mathcal{O}_2[f^{-1}\\mathcal{O}_2 \\times f^{-1}\\mathcal{O}_2]$, and", "$f^{-1}(\\mathcal{O}_2[\\mathcal{O}_1]) =", "f^{-1}\\mathcal{O}_2[f^{-1}\\mathcal{O}_1]$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14231, "type": "theorem", "label": "sites-modules-lemma-localize-differentials", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-localize-differentials", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings. For any object $U$ of $\\mathcal{C}$", "there is a canonical isomorphism", "$$", "\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}|_U =", "\\Omega_{(\\mathcal{O}_2|_U)/(\\mathcal{O}_1|_U)}", "$$", "compatible with universal derivations." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-pullback-differentials}." ], "refs": [ "sites-modules-lemma-pullback-differentials" ], "ref_ids": [ 14230 ] } ], "ref_ids": [] }, { "id": 14232, "type": "theorem", "label": "sites-modules-lemma-functoriality-differentials", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-functoriality-differentials", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$$", "\\xymatrix{", "\\mathcal{O}_2 \\ar[r]_\\varphi & \\mathcal{O}_2' \\\\", "\\mathcal{O}_1 \\ar[r] \\ar[u] & \\mathcal{O}'_1 \\ar[u]", "}", "$$", "be a commutative diagram of sheaves of rings on $\\mathcal{C}$. The map", "$\\mathcal{O}_2 \\to \\mathcal{O}'_2$ composed with the map", "$\\text{d} : \\mathcal{O}'_2 \\to \\Omega_{\\mathcal{O}'_2/\\mathcal{O}'_1}$", "is a $\\mathcal{O}_1$-derivation. Hence we obtain a canonical map of", "$\\mathcal{O}_2$-modules", "$\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\to", "\\Omega_{\\mathcal{O}'_2/\\mathcal{O}'_1}$.", "It is uniquely characterized by the property that", "$\\text{d}(f)$ mapsto $\\text{d}(\\varphi(f))$", "for any local section $f$ of $\\mathcal{O}_2$.", "In this way $\\Omega_{-/-}$ becomes a functor on the category", "of arrows of sheaves of rings." ], "refs": [], "proofs": [ { "contents": [ "This lemma proves itself." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14233, "type": "theorem", "label": "sites-modules-lemma-differential-seq", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-differential-seq", "contents": [ "In Lemma \\ref{lemma-functoriality-differentials} suppose that", "$\\mathcal{O}_2 \\to \\mathcal{O}'_2$ is surjective with kernel", "$\\mathcal{I} \\subset \\mathcal{O}_2$ and assume that", "$\\mathcal{O}_1 = \\mathcal{O}'_1$. Then there is a canonical exact", "sequence of $\\mathcal{O}'_2$-modules", "$$", "\\mathcal{I}/\\mathcal{I}^2", "\\longrightarrow", "\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\otimes_{\\mathcal{O}_2} \\mathcal{O}'_2", "\\longrightarrow", "\\Omega_{\\mathcal{O}'_2/\\mathcal{O}_1}", "\\longrightarrow", "0", "$$", "The leftmost map is characterized by the rule that a local section", "$f$ of $\\mathcal{I}$ maps to $\\text{d}f \\otimes 1$." ], "refs": [ "sites-modules-lemma-functoriality-differentials" ], "proofs": [ { "contents": [ "For a local section $f$ of $\\mathcal{I}$ denote $\\overline{f}$ the image of", "$f$ in $\\mathcal{I}/\\mathcal{I}^2$. To show that the map", "$\\overline{f} \\mapsto \\text{d}f \\otimes 1$ is well defined we just have to", "check that $\\text{d} f_1f_2 \\otimes 1 = 0$ if $f_1, f_2$ are local sections", "of $\\mathcal{I}$. And this is clear from the Leibniz rule", "$\\text{d} f_1f_2 \\otimes 1 =", "(f_1 \\text{d}f_2 + f_2 \\text{d} f_1 )\\otimes 1 =", "\\text{d}f_2 \\otimes f_1 + \\text{d}f_2 \\otimes f_1 = 0$.", "A similar computation show this map is", "$\\mathcal{O}'_2 = \\mathcal{O}_2/\\mathcal{I}$-linear. The map on the right", "is the one from Lemma \\ref{lemma-functoriality-differentials}.", "\\medskip\\noindent", "To see that the sequence is exact, we argue as follows. Let", "$\\mathcal{O}''_2 \\subset \\mathcal{O}'_2$ be the presheaf of", "$\\mathcal{O}_1$-algebras whose value on $U$ is the image of", "$\\mathcal{O}_2(U) \\to \\mathcal{O}'_2(U)$. By", "Algebra, Lemma \\ref{algebra-lemma-differential-seq}", "the sequences", "$$", "\\mathcal{I}(U)/\\mathcal{I}(U)^2", "\\longrightarrow", "\\Omega_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}", "\\otimes_{\\mathcal{O}_2(U)} \\mathcal{O}''_2(U)", "\\longrightarrow", "\\Omega_{\\mathcal{O}''_2(U)/\\mathcal{O}_1(U)}", "\\longrightarrow", "0", "$$", "are exact for all objects $U$ of $\\mathcal{C}$. Since sheafification is", "exact this gives an exact sequence of sheaves of", "$(\\mathcal{O}'_2)^\\#$-modules.", "By Lemma \\ref{lemma-differentials-sheafify}", "and the fact that $(\\mathcal{O}''_2)^\\# = \\mathcal{O}'_2$", "we conclude." ], "refs": [ "sites-modules-lemma-functoriality-differentials", "algebra-lemma-differential-seq", "sites-modules-lemma-differentials-sheafify" ], "ref_ids": [ 14232, 1135, 14229 ] } ], "ref_ids": [ 14232 ] }, { "id": 14234, "type": "theorem", "label": "sites-modules-lemma-double-structure-gives-derivation", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-double-structure-gives-derivation", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings.", "Consider a short exact sequence", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{A} \\to \\mathcal{O}_2 \\to 0", "$$", "Here $\\mathcal{A}$ is a sheaf of $\\mathcal{O}_1$-algebras,", "$\\pi : \\mathcal{A} \\to \\mathcal{O}_2$ is a surjection", "of sheaves of $\\mathcal{O}_1$-algebras, and", "$\\mathcal{F} = \\Ker(\\pi)$ is its kernel. Assume $\\mathcal{F}$ an ideal", "sheaf with square zero in $\\mathcal{A}$. So $\\mathcal{F}$", "has a natural structure of an $\\mathcal{O}_2$-module.", "A section $s : \\mathcal{O}_2 \\to \\mathcal{A}$ of $\\pi$", "is a $\\mathcal{O}_1$-algebra map such that $\\pi \\circ s = \\text{id}$.", "Given any section $s : \\mathcal{O}_2 \\to \\mathcal{F}$", "of $\\pi$ and any $\\varphi$-derivation $D : \\mathcal{O}_1 \\to \\mathcal{F}$", "the map", "$$", "s + D : \\mathcal{O}_1 \\to \\mathcal{A}", "$$", "is a section of $\\pi$ and every section $s'$ is of the form $s + D$", "for a unique $\\varphi$-derivation $D$." ], "refs": [], "proofs": [ { "contents": [ "Recall that the $\\mathcal{O}_2$-module structure on $\\mathcal{F}$", "is given by $h \\tau = \\tilde h \\tau$ (multiplication in $\\mathcal{A}$)", "where $h$ is a local section of $\\mathcal{O}_2$, and", "$\\tilde h$ is a local lift of $h$ to a local", "section of $\\mathcal{A}$, and $\\tau$ is a local section of $\\mathcal{F}$.", "In particular, given $s$, we may use $\\tilde h = s(h)$.", "To verify that $s + D$ is a homomorphism of sheaves of rings we", "compute", "\\begin{eqnarray*}", "(s + D)(ab) & = & s(ab) + D(ab) \\\\", "& = & s(a)s(b) + aD(b) + D(a)b \\\\", "& = & s(a) s(b) + s(a)D(b) + D(a)s(b) \\\\", "& = & (s(a) + D(a))(s(b) + D(b))", "\\end{eqnarray*}", "by the Leibniz rule. In the same manner one shows", "$s + D$ is a $\\mathcal{O}_1$-algebra map because $D$ is", "an $\\mathcal{O}_1$-derivation. Conversely, given $s'$ we set", "$D = s' - s$. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14235, "type": "theorem", "label": "sites-modules-lemma-functoriality-differentials-ringed-topoi", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-functoriality-differentials-ringed-topoi", "contents": [ "Let", "$X = (\\Sh(\\mathcal{C}_X), \\mathcal{O}_X)$,", "$Y = (\\Sh(\\mathcal{C}_Y), \\mathcal{O}_Y)$,", "$X' = (\\Sh(\\mathcal{C}_{X'}), \\mathcal{O}_{X'})$, and", "$Y' = (\\Sh(\\mathcal{C}_{Y'}), \\mathcal{O}_{Y'})$ be ringed topoi.", "Let", "$$", "\\xymatrix{", "X' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\", "Y' \\ar[r] & Y", "}", "$$", "be a commutative diagram of morphisms of ringed topoi. The map", "$f^\\sharp : \\mathcal{O}_X \\to f_*\\mathcal{O}_{X'}$ composed with the map", "$f_*\\text{d}_{X'/Y'} : f_*\\mathcal{O}_{X'} \\to f_*\\Omega_{X'/Y'}$ is a", "$Y$-derivation. Hence we obtain a canonical map of $\\mathcal{O}_X$-modules", "$\\Omega_{X/Y} \\to f_*\\Omega_{X'/Y'}$, and by", "adjointness of $f_*$ and $f^*$ a", "canonical $\\mathcal{O}_{X'}$-module homomorphism", "$$", "c_f : f^*\\Omega_{X/Y} \\longrightarrow \\Omega_{X'/Y'}.", "$$", "It is uniquely characterized by the property that", "$f^*\\text{d}_{X/Y}(t)$ mapsto $\\text{d}_{X'/Y'}(f^* t)$", "for any local section $t$ of $\\mathcal{O}_X$." ], "refs": [], "proofs": [ { "contents": [ "This is clear except for the last assertion. Let us explain the meaning of", "this. Let $U \\in \\Ob(\\mathcal{C}_X)$ and let $t \\in \\mathcal{O}_X(U)$.", "This is what it means for $t$ to be a local section of $\\mathcal{O}_X$.", "Now, we may think of $t$ as a map of sheaves of sets", "$t : h_U^\\# \\to \\mathcal{O}_X$. Then", "$f^{-1}t : f^{-1}h_U^\\# \\to f^{-1}\\mathcal{O}_X$. By $f^*t$ we mean", "the composition", "$$", "\\xymatrix{", "f^{-1}h_U^\\# \\ar[rr]^{f^{-1}t} \\ar@/^4ex/[rrrr]^{f^*t} & &", "f^{-1}\\mathcal{O}_X \\ar[rr]^{f^\\sharp} & &", "\\mathcal{O}_{X'}", "}", "$$", "Note that $\\text{d}_{X/Y}(t) \\in \\Omega_{X/Y}(U)$. Hence we may think of", "$\\text{d}_{X/Y}(t)$ as a map $\\text{d}_{X/Y}(t) : h_U^\\# \\to \\Omega_{X/Y}$.", "Then $f^{-1}\\text{d}_{X/Y}(t) : f^{-1}h_U^\\# \\to f^{-1}\\Omega_{X/Y}$.", "By $f^*\\text{d}_{X/Y}(t)$ we mean the composition", "$$", "\\xymatrix{", "f^{-1}h_U^\\#", "\\ar[rr]^{f^{-1}\\text{d}_{X/Y}(t)}", "\\ar@/^4ex/[rrrr]^{f^*\\text{d}_{X/Y}(t)} & &", "f^{-1}\\Omega_{X/Y} \\ar[rr]^{1 \\otimes \\text{id}} & &", "f^*\\Omega_{X/Y}", "}", "$$", "OK, and now the statement of the lemma means that we have", "$$", "c_f \\circ f^*t = f^*\\text{d}_{X/Y}(t)", "$$", "as maps from $f^{-1}h_U^\\#$ to $\\Omega_{X'/Y'}$. We omit the verification", "that this property holds for $c_f$ as defined in the lemma. (Hint: The first", "map $c'_f : \\Omega_{X/Y} \\to f_*\\Omega_{X'/Y'}$ satisfies", "$c'_f(\\text{d}_{X/Y}(t)) = f_*\\text{d}_{X'/Y'}(f^\\sharp(t))$ as sections of", "$f_*\\Omega_{X'/Y'}$ over $U$, and you have to", "turn this into the equality above by using adjunction.)", "The reason that this uniquely characterizes $c_f$ is that the images", "of $f^*\\text{d}_{X/Y}(t)$ generate the $\\mathcal{O}_{X'}$-module", "$f^*\\Omega_{X/Y}$ simply because the local sections $\\text{d}_{X/Y}(t)$", "generate the $\\mathcal{O}_X$-module $\\Omega_{X/Y}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14236, "type": "theorem", "label": "sites-modules-lemma-composition-differential-operators", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-composition-differential-operators", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings.", "Let $\\mathcal{E}, \\mathcal{F}, \\mathcal{G}$ be sheaves of", "$\\mathcal{O}_2$-modules.", "If $D : \\mathcal{E} \\to \\mathcal{F}$ and $D' : \\mathcal{F} \\to \\mathcal{G}$", "are differential operators of order $k$ and $k'$, then $D' \\circ D$ is a", "differential operator of order $k + k'$." ], "refs": [], "proofs": [ { "contents": [ "Let $g$ be a local section of $\\mathcal{O}_2$.", "Then the map which sends a local section $x$ of $\\mathcal{E}$ to", "$$", "D'(D(gx)) - gD'(D(x)) = D'(D(gx)) - D'(gD(x)) + D'(gD(x)) - gD'(D(x))", "$$", "is a sum of two compositions of differential operators of lower order.", "Hence the lemma follows by induction on $k + k'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14237, "type": "theorem", "label": "sites-modules-lemma-module-principal-parts", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-module-principal-parts", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_2$-modules.", "Let $k \\geq 0$. There exists a sheaf of $\\mathcal{O}_2$-modules", "$\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F})$", "and a canonical isomorphism", "$$", "\\text{Diff}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}, \\mathcal{G}) =", "\\Hom_{\\mathcal{O}_2}(", "\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}), \\mathcal{G})", "$$", "functorial in the $\\mathcal{O}_2$-module $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "The existence follows from general category theoretic arguments", "(insert future reference here), but we will also give a direct", "construction as this construction will be useful in the future proofs.", "We will freely use the notation introduced in the proof of", "Lemma \\ref{lemma-universal-module}.", "Given any differential operator $D : \\mathcal{F} \\to \\mathcal{G}$", "we obtain an $\\mathcal{O}_2$-linear map", "$L_D : \\mathcal{O}_2[\\mathcal{F}] \\to \\mathcal{G}$", "sending $[m]$ to $D(m)$. If $D$ has order $0$", "then $L_D$ annihilates the local sections", "$$", "[m + m'] - [m] - [m'],\\quad", "g_0[m] - [g_0m]", "$$", "where $g_0$ is a local section of $\\mathcal{O}_2$ and $m, m'$", "are local sections of $\\mathcal{F}$. If $D$ has order $1$, then $L_D$", "annihilates the local sections", "$$", "[m + m' - [m] - [m'],\\quad", "f[m] - [fm], \\quad", "g_0g_1[m] - g_0[g_1m] - g_1[g_0m] + [g_1g_0m]", "$$", "where $f$ is a local section of $\\mathcal{O}_1$,", "$g_0, g_1$ are local sections of $\\mathcal{O}_2$, and", "$m, m'$ are local sections of $\\mathcal{F}$.", "If $D$ has order $k$, then $L_D$ annihilates the local sections", "$[m + m'] - [m] - [m']$, $f[m] - [fm]$, and the local sections", "$$", "g_0g_1\\ldots g_k[m] - \\sum g_0 \\ldots \\hat g_i \\ldots g_k[g_im] + \\ldots", "+(-1)^{k + 1}[g_0\\ldots g_km]", "$$", "Conversely, if $L : \\mathcal{O}_2[\\mathcal{F}] \\to \\mathcal{G}$ is an", "$\\mathcal{O}_2$-linear map annihilating all the local sections", "listed in the previous sentence, then $m \\mapsto L([m])$ is a", "differential operator of order $k$. Thus we see that", "$\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F})$", "is the quotient of $\\mathcal{O}_2[\\mathcal{F}]$", "by the $\\mathcal{O}_2$-submodule generated by these local sections." ], "refs": [ "sites-modules-lemma-universal-module" ], "ref_ids": [ 14228 ] } ], "ref_ids": [] }, { "id": 14238, "type": "theorem", "label": "sites-modules-lemma-differential-operators-sheafify", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-differential-operators-sheafify", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of presheaves of rings. Let $\\mathcal{F}$ be a presheaf", "of $\\mathcal{O}_2$-modules. Then", "$\\mathcal{P}^k_{\\mathcal{O}_2^\\#/\\mathcal{O}_1^\\#}(\\mathcal{F}^\\#)$", "is the sheaf associated to the presheaf", "$U \\mapsto P^k_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}(\\mathcal{F}(U))$." ], "refs": [], "proofs": [ { "contents": [ "This can be proved in exactly the same way as is done for the sheaf", "of differentials in Lemma \\ref{lemma-differentials-sheafify}.", "Perhaps a more pleasing approach is to use the universal property", "of Lemma \\ref{lemma-module-principal-parts} directly to see the equality.", "We omit the details." ], "refs": [ "sites-modules-lemma-differentials-sheafify", "sites-modules-lemma-module-principal-parts" ], "ref_ids": [ 14229, 14237 ] } ], "ref_ids": [] }, { "id": 14239, "type": "theorem", "label": "sites-modules-lemma-sequence-of-principal-parts", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-sequence-of-principal-parts", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings. Let $\\mathcal{F}$ be a sheaf", "of $\\mathcal{O}_2$-modules. There is a", "canonical short exact sequence", "$$", "0 \\to", "\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\otimes_{\\mathcal{O}_2} \\mathcal{F} \\to", "\\mathcal{P}^1_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}) \\to", "\\mathcal{F} \\to 0", "$$", "functorial in $\\mathcal{F}$ called the {\\it sequence of principal parts}." ], "refs": [], "proofs": [ { "contents": [ "Follows from the commutative algebra version", "(Algebra, Lemma \\ref{algebra-lemma-sequence-of-principal-parts})", "and Lemmas \\ref{lemma-differentials-sheafify} and", "\\ref{lemma-differential-operators-sheafify}." ], "refs": [ "algebra-lemma-sequence-of-principal-parts", "sites-modules-lemma-differentials-sheafify", "sites-modules-lemma-differential-operators-sheafify" ], "ref_ids": [ 1146, 14229, 14238 ] } ], "ref_ids": [] }, { "id": 14240, "type": "theorem", "label": "sites-modules-lemma-NL-up-to-qis", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-NL-up-to-qis", "contents": [ "In the situation above there is a canonical isomorphism", "$\\NL(\\alpha) = \\NL_{\\mathcal{B}/\\mathcal{A}}$ in $D(\\mathcal{B})$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $\\NL_{\\mathcal{B}/\\mathcal{A}} = \\NL(\\text{id}_\\mathcal{B})$.", "Thus it suffices to show that given two maps", "$\\alpha_i : \\mathcal{E}_i \\to \\mathcal{B}$ as above, there is a", "canonical quasi-isomorphism $\\NL(\\alpha_1) = \\NL(\\alpha_2)$ in $D(\\mathcal{B})$.", "To see this set $\\mathcal{E} = \\mathcal{E}_1 \\amalg \\mathcal{E}_2$ and", "$\\alpha = \\alpha_1 \\amalg \\alpha_2 : \\mathcal{E} \\to \\mathcal{B}$.", "Set", "$\\mathcal{J}_i = \\Ker(\\mathcal{A}[\\mathcal{E}_i] \\to \\mathcal{B})$", "and", "$\\mathcal{J} = \\Ker(\\mathcal{A}[\\mathcal{E}] \\to \\mathcal{B})$.", "We obtain maps $\\mathcal{A}[\\mathcal{E}_i] \\to \\mathcal{A}[\\mathcal{E}]$", "which send $\\mathcal{J}_i$ into $\\mathcal{J}$.", "Thus we obtain canonical maps of complexes", "$$", "\\NL(\\alpha_i) \\longrightarrow \\NL(\\alpha)", "$$", "and it suffices to show these maps are quasi-isomorphism. To see this", "we argue as follows. First, observe that", "$H^0(\\NL(\\alpha_i)) = \\Omega_{\\mathcal{B}/\\mathcal{A}}$ and", "$H^0(\\NL(\\alpha)) = \\Omega_{\\mathcal{B}/\\mathcal{A}}$ by", "Lemma \\ref{lemma-differential-seq}", "hence the map is an isomorphism on cohomology sheaves in degree $0$.", "Similarly, we claim that $H^{-1}(\\NL(\\alpha_i))$ and $H^{-1}(\\NL(\\alpha))$", "are the sheaves associated to the presheaf", "$U \\mapsto H_1(L_{\\mathcal{B}(U)/\\mathcal{A}(U)})$ where", "$H_1(L_{-/-})$ is as in", "Algebra, Definition \\ref{algebra-definition-naive-cotangent-complex}.", "If the claim holds, then the proof is finished.", "\\medskip\\noindent", "Proof of the claim. Let $\\alpha : \\mathcal{E} \\to \\mathcal{B}$", "be as above. Let $\\mathcal{B}' \\subset \\mathcal{B}$ be the subpresheaf", "of $\\mathcal{A}$-algebras whose value on $U$ is the image of", "$\\mathcal{A}(U)[\\mathcal{E}(U)] \\to \\mathcal{B}(U)$. Let $\\mathcal{I}'$", "be the presheaf whose value on $U$ is the kernel of", "$\\mathcal{A}(U)[\\mathcal{E}(U)] \\to \\mathcal{B}(U)$. Then $\\mathcal{I}$", "is the sheafification of $\\mathcal{I}'$ and $\\mathcal{B}$ is", "the sheafification of $\\mathcal{B}'$. Similarly,", "$H^{-1}(\\NL(\\alpha))$ is the sheafification of the presheaf", "$$", "U \\longmapsto", "\\Ker(\\mathcal{I}'(U)/\\mathcal{I}'(U)^2 \\to", "\\Omega_{\\mathcal{A}(U)[\\mathcal{E}(U)]/\\mathcal{A}(U)}", "\\otimes_{\\mathcal{A}(U)[\\mathcal{E}(U)]} \\mathcal{B}'(U))", "$$", "by Lemma \\ref{lemma-differentials-sheafify}.", "By Algebra, Lemma \\ref{algebra-lemma-NL-homotopy} we conclude", "$H^{-1}(\\NL(\\alpha))$ is the sheaf associated to the presheaf", "$U \\mapsto H_1(L_{\\mathcal{B}'(U)/\\mathcal{A}(U)})$. Thus we have", "to show that the maps", "$H_1(L_{\\mathcal{B}'(U)/\\mathcal{A}(U)}) \\to", "H_1(L_{\\mathcal{B}(U)/\\mathcal{A}(U)})$ induce an isomorphism", "$\\mathcal{H}'_1 \\to \\mathcal{H}_1$ of sheafifications.", "\\medskip\\noindent", "Injectivity of $\\mathcal{H}'_1 \\to \\mathcal{H}_1$. Let", "$f \\in H_1(L_{\\mathcal{B}'(U)/\\mathcal{A}(U)})$ map to zero", "in $\\mathcal{H}_1(U)$. To show: $f$ maps to zero in", "$\\mathcal{H}'_1(U)$. The assumption means there is a covering", "$\\{U_i \\to U\\}$ such that $f$ maps to zero in", "$H_1(L_{\\mathcal{B}(U_i)/\\mathcal{A}(U_i)})$ for all $i$.", "Replace $U$ by $U_i$ to get to the point where $f$ maps to zero", "in $H_1(L_{\\mathcal{B}(U)/\\mathcal{A}(U)})$.", "By Algebra, Lemma \\ref{algebra-lemma-colimits-NL}", "we can find a finitely generated subalgebra", "$\\mathcal{B}'(U) \\subset B \\subset \\mathcal{B}(U)$ such", "that $f$ maps to zero in $H_1(L_{B/\\mathcal{A}(U)})$.", "Since $\\mathcal{B} = (\\mathcal{B}')^\\#$ we can find a covering", "$\\{U_i \\to U\\}$ such that $B \\to \\mathcal{B}(U_i)$ factors", "through $\\mathcal{B}'(U_i)$. Hence $f$ maps to zero in", "$H_1(L_{\\mathcal{B}'(U_i)/\\mathcal{A}(U_i)})$ as desired.", "\\medskip\\noindent", "The surjectivity of $\\mathcal{H}'_1 \\to \\mathcal{H}_1$ is proved", "in exactly the same way." ], "refs": [ "sites-modules-lemma-differential-seq", "algebra-definition-naive-cotangent-complex", "sites-modules-lemma-differentials-sheafify", "algebra-lemma-NL-homotopy", "algebra-lemma-colimits-NL" ], "ref_ids": [ 14233, 1529, 14229, 1151, 1157 ] } ], "ref_ids": [] }, { "id": 14241, "type": "theorem", "label": "sites-modules-lemma-pullback-NL", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-pullback-NL", "contents": [ "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be morphism of topoi.", "Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings", "on $\\mathcal{D}$. Then $f^{-1}\\NL_{\\mathcal{B}/\\mathcal{A}} =", "\\NL_{f^{-1}\\mathcal{B}/f^{-1}\\mathcal{A}}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Use Lemma \\ref{lemma-pullback-differentials}." ], "refs": [ "sites-modules-lemma-pullback-differentials" ], "ref_ids": [ 14230 ] } ], "ref_ids": [] }, { "id": 14242, "type": "theorem", "label": "sites-modules-lemma-stalk-exact", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-stalk-exact", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $p$ be a point of $\\mathcal{C}$.", "\\begin{enumerate}", "\\item We have $(\\mathcal{F}^\\#)_p = \\mathcal{F}_p$", "for any presheaf of sets on $\\mathcal{C}$.", "\\item The stalk functor", "$\\Sh(\\mathcal{C}) \\to \\textit{Sets}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_p$ is exact (see", "Categories, Definition \\ref{categories-definition-exact})", "and commutes with arbitrary colimits.", "\\item The stalk functor", "$\\textit{PSh}(\\mathcal{C}) \\to \\textit{Sets}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_p$ is exact (see", "Categories, Definition \\ref{categories-definition-exact})", "and commutes with arbitrary colimits.", "\\end{enumerate}" ], "refs": [ "categories-definition-exact", "categories-definition-exact" ], "proofs": [ { "contents": [ "By", "Sites, Lemma \\ref{sites-lemma-point-pushforward-sheaf}", "we have (1).", "By", "Sites, Lemmas \\ref{sites-lemma-adjoint-point-push-stalk}", "we see that", "$\\textit{PSh}(\\mathcal{C}) \\to \\textit{Sets}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_p$ is a left adjoint,", "and by", "Sites, Lemma \\ref{sites-lemma-point-pushforward-sheaf}", "we see the same thing for", "$\\textit{Sh}(\\mathcal{C}) \\to \\textit{Sets}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_p$.", "Hence the stalk functor commutes with arbitrary colimits (see", "Categories, Lemma \\ref{categories-lemma-adjoint-exact}).", "It follows from the definition of a point of a site, see", "Sites, Definition \\ref{sites-definition-point}", "that $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_p$", "is exact. Since sheafification is exact", "(Sites, Lemma \\ref{sites-lemma-sheafification-exact})", "it follows that $\\textit{PSh}(\\mathcal{C}) \\to \\textit{Sets}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_p$", "is exact." ], "refs": [ "sites-lemma-point-pushforward-sheaf", "sites-lemma-adjoint-point-push-stalk", "sites-lemma-point-pushforward-sheaf", "categories-lemma-adjoint-exact", "sites-definition-point", "sites-lemma-sheafification-exact" ], "ref_ids": [ 8595, 8594, 8595, 12249, 8675, 8515 ] } ], "ref_ids": [ 12370, 12370 ] }, { "id": 14243, "type": "theorem", "label": "sites-modules-lemma-stalk-exact-abelian", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-stalk-exact-abelian", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $p$ be a point of $\\mathcal{C}$.", "\\begin{enumerate}", "\\item The functor", "$\\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_p$ is exact.", "\\item The stalk functor", "$\\textit{PAb}(\\mathcal{C}) \\to \\textit{Ab}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_p$", "is exact.", "\\item For $\\mathcal{F} \\in \\Ob(\\textit{PAb}(\\mathcal{C}))$", "we have $\\mathcal{F}_p = \\mathcal{F}^\\#_p$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is formal from the results of", "Lemma \\ref{lemma-stalk-exact}", "and the construction of the stalk functor above." ], "refs": [ "sites-modules-lemma-stalk-exact" ], "ref_ids": [ 14242 ] } ], "ref_ids": [] }, { "id": 14244, "type": "theorem", "label": "sites-modules-lemma-stalk-exact-modules", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-stalk-exact-modules", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $p$ be a point of $\\mathcal{C}$.", "\\begin{enumerate}", "\\item The functor", "$\\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_p)$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_p$ is exact.", "\\item The stalk functor", "$\\textit{PMod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_p)$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_p$", "is exact.", "\\item For $\\mathcal{F} \\in \\Ob(\\textit{PMod}(\\mathcal{O}))$", "we have $\\mathcal{F}_p = \\mathcal{F}^\\#_p$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is formal from the results of", "Lemma \\ref{lemma-stalk-exact-abelian},", "the construction of the stalk functor above, and", "Lemma \\ref{lemma-abelian}." ], "refs": [ "sites-modules-lemma-stalk-exact-abelian", "sites-modules-lemma-abelian" ], "ref_ids": [ 14243, 14157 ] } ], "ref_ids": [] }, { "id": 14245, "type": "theorem", "label": "sites-modules-lemma-pullback-stalk", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-pullback-stalk", "contents": [ "Let", "$(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi or ringed sites.", "Let $p$ be a point of $\\mathcal{C}$ or $\\Sh(\\mathcal{C})$", "and set $q = f \\circ p$. Then", "$$", "(f^*\\mathcal{F})_p =", "\\mathcal{F}_q \\otimes_{\\mathcal{O}_{\\mathcal{D}, q}}", "\\mathcal{O}_{\\mathcal{C}, p}", "$$", "for any $\\mathcal{O}_\\mathcal{D}$-module $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "We have", "$$", "f^*\\mathcal{F} =", "f^{-1}\\mathcal{F} \\otimes_{f^{-1}\\mathcal{O}_\\mathcal{D}}", "\\mathcal{O}_\\mathcal{C}", "$$", "by definition. Since taking stalks at $p$ (i.e., applying", "$p^{-1}$) commutes with $\\otimes$ by", "Lemma \\ref{lemma-tensor-product-pullback}", "we win by the relation between the stalk of pullbacks at $p$", "and stalks at $q$ explained in", "Sites, Lemma \\ref{sites-lemma-point-morphism-sites} or", "Sites, Lemma \\ref{sites-lemma-point-morphism-topoi}." ], "refs": [ "sites-modules-lemma-tensor-product-pullback", "sites-lemma-point-morphism-sites", "sites-lemma-point-morphism-topoi" ], "ref_ids": [ 14189, 8603, 8604 ] } ], "ref_ids": [] }, { "id": 14246, "type": "theorem", "label": "sites-modules-lemma-skyscraper-exact", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-skyscraper-exact", "contents": [ "Let $\\mathcal{C}$ be a site. Let $p$ be a point of", "$\\mathcal{C}$ or of its associated topos.", "\\begin{enumerate}", "\\item The functor $p_* : \\textit{Ab} \\to \\textit{Ab}(\\mathcal{C})$,", "$A \\mapsto p_*A$ is exact.", "\\item There is a functorial direct sum decomposition", "$$", "p^{-1}p_*A = A \\oplus I(A)", "$$", "for $A \\in \\Ob(\\textit{Ab})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By", "Sites, Lemma \\ref{sites-lemma-stalk-skyscraper}", "there are functorial maps $A \\to p^{-1}p_*A \\to A$ whose composition", "equals $\\text{id}_A$. Hence a functorial direct sum decomposition", "as in (2) with $I(A)$ the kernel of the adjunction map", "$p^{-1}p_*A \\to A$. The functor $p_*$ is left exact by", "Lemma \\ref{lemma-exactness-pushforward-pullback}.", "The functor $p_*$ transforms surjections into surjections by", "Sites, Lemma \\ref{sites-lemma-skyscraper-functor-exact}.", "Hence (1) holds." ], "refs": [ "sites-lemma-stalk-skyscraper", "sites-modules-lemma-exactness-pushforward-pullback", "sites-lemma-skyscraper-functor-exact" ], "ref_ids": [ 8598, 14159, 8599 ] } ], "ref_ids": [] }, { "id": 14247, "type": "theorem", "label": "sites-modules-lemma-skyscraper-modules-exact", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-skyscraper-modules-exact", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.", "Let $p$ be a point of the topos $\\Sh(\\mathcal{C})$.", "\\begin{enumerate}", "\\item The functor", "$p_* : \\textit{Mod}(\\mathcal{O}_p) \\to \\textit{Mod}(\\mathcal{O})$,", "$M \\mapsto p_*M$ is exact.", "\\item The canonical surjection $p^{-1}p_*M \\to M$ is $\\mathcal{O}_p$-linear.", "\\item The functorial direct sum decomposition", "$p^{-1}p_*M = M \\oplus I(M)$ of Lemma \\ref{lemma-skyscraper-exact}", "is {\\bf not} $\\mathcal{O}_p$-linear in general.", "\\end{enumerate}" ], "refs": [ "sites-modules-lemma-skyscraper-exact" ], "proofs": [ { "contents": [ "Part (1) and surjectivity in (2)", "follow immediately from the corresponding result for abelian", "sheaves in", "Lemma \\ref{lemma-skyscraper-exact}.", "Since $p^{-1}\\mathcal{O} = \\mathcal{O}_p$", "we have $p^{-1} = p^*$ and hence $p^{-1}p_*M \\to M$", "is the same as the counit $p^*p_*M \\to M$", "of the adjunction for modules, whence linear.", "\\medskip\\noindent", "Proof of (3). Suppose that $G$ is a group. Consider the topos", "$G\\textit{-Sets} = \\Sh(\\mathcal{T}_G)$", "and the point $p : \\textit{Sets} \\to G\\textit{-Sets}$.", "See Sites, Section \\ref{sites-section-example-sheaf-G-sets} and", "Example \\ref{sites-example-point-G-sets}.", "Here $p^{-1}$ is the functor forgetting", "about the $G$-action. And $p_*$ is the right adjoint", "of the forgetful functor, sending $M$ to $\\text{Map}(G, M)$.", "The maps in the direct sum decomposition are the maps", "$$", "M \\to \\text{Map}(G, M) \\to M", "$$", "where the first sends $m \\in M$ to the constant map with value $m$ and", "where the second map is evaluation at the identity element $1$ of $G$.", "Next, suppose that $R$ is a ring endowed with an action of $G$.", "This determines a sheaf of rings $\\mathcal{O}$ on $\\mathcal{T}_G$.", "The category of $\\mathcal{O}$-modules is the category of $R$-modules $M$", "endowed with an action of $G$ compatible with the action on $R$.", "The $R$-module structure on $\\text{Map}(G, M)$ is given by", "$$", "( r f ) (\\sigma) = \\sigma(r) f(\\sigma)", "$$", "for $r \\in R$ and $f \\in \\text{Map}(G, M)$. This is true because it is the", "unique $G$-invariant $R$-module strucure compatible with evaluation at $1$.", "The reader observes that in general the image of $M \\to \\text{Map}(G, M)$", "is not an $R$-submodule (for example take $M = R$ and assume the", "$G$-action is nontrivial), which concludes the proof." ], "refs": [ "sites-modules-lemma-skyscraper-exact" ], "ref_ids": [ 14246 ] } ], "ref_ids": [ 14246 ] }, { "id": 14248, "type": "theorem", "label": "sites-modules-lemma-stalk-j-shriek", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-stalk-j-shriek", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $p$ be a point of $\\mathcal{C}$. Let $U$ be an object of $\\mathcal{C}$.", "For $\\mathcal{G}$ in $\\textit{Mod}(\\mathcal{O}_U)$ we have", "$$", "(j_{U!}\\mathcal{G})_p =", "\\bigoplus\\nolimits_q \\mathcal{G}_q", "$$", "where the coproduct is over the points $q$ of $\\mathcal{C}/U$", "lying over $p$, see", "Sites, Lemma \\ref{sites-lemma-points-above-point}." ], "refs": [ "sites-lemma-points-above-point" ], "proofs": [ { "contents": [ "We use the description of $j_{U!}\\mathcal{G}$ as the sheaf associated", "to the presheaf", "$V \\mapsto", "\\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{G}(V/_\\varphi U)$", "of", "Lemma \\ref{lemma-extension-by-zero}.", "The stalk of $j_{U!}\\mathcal{G}$ at $p$ is equal to the", "stalk of this presheaf, see", "Lemma \\ref{lemma-stalk-exact-modules}.", "Let $u : \\mathcal{C} \\to \\textit{Sets}$ be the functor corresponding", "to $p$ (see Sites, Section \\ref{sites-section-points}).", "Hence we see that", "$$", "(j_{U!}\\mathcal{G})_p = \\colim_{(V, y)}", "\\bigoplus\\nolimits_{\\varphi : V \\to U} \\mathcal{G}(V/_\\varphi U)", "$$", "where the colimit is taken in the category of abelian groups.", "To a quadruple $(V, y, \\varphi, s)$ occurring in this colimit, we can assign", "$x = u(\\varphi)(y) \\in u(U)$. Hence we obtain", "$$", "(j_{U!}\\mathcal{G})_p =", "\\bigoplus\\nolimits_{x \\in u(U)}", "\\colim_{(\\varphi : V \\to U, y), \\ u(\\varphi)(y) = x} \\mathcal{G}(V/_\\varphi U).", "$$", "This is equal to the expression of the lemma by the description", "of the points $q$ lying over $x$ in", "Sites, Lemma \\ref{sites-lemma-points-above-point}." ], "refs": [ "sites-modules-lemma-extension-by-zero", "sites-modules-lemma-stalk-exact-modules", "sites-lemma-points-above-point" ], "ref_ids": [ 14169, 14244, 8606 ] } ], "ref_ids": [ 8606 ] }, { "id": 14249, "type": "theorem", "label": "sites-modules-lemma-pullback-flat", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-pullback-flat", "contents": [ "\\begin{reference}", "\\cite[Expos\\'e V, Corollary 1.7.1]{SGA4}", "\\end{reference}", "Let", "$(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi or ringed sites.", "Then $f^*\\mathcal{F}$ is a flat $\\mathcal{O}_\\mathcal{C}$-module", "whenever $\\mathcal{F}$ is a flat $\\mathcal{O}_\\mathcal{D}$-module." ], "refs": [], "proofs": [ { "contents": [ "Choose a diagram as in", "Lemma \\ref{lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites}.", "Recall that being a flat module is intrinsic", "(see Section \\ref{section-intrinsic} and", "Definition \\ref{definition-flat}).", "Hence it suffices to prove the lemma for", "the morphism $(h, h^\\sharp) :", "(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})", "\\to", "(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'})$.", "In other words, we may assume that", "our sites $\\mathcal{C}$ and $\\mathcal{D}$", "have all finite limits and that $f$ is a morphism", "of sites induced by a continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$", "which commutes with finite limits.", "\\medskip\\noindent", "Recall that $f^*\\mathcal{F} =", "\\mathcal{O}_\\mathcal{C} \\otimes_{f^{-1}\\mathcal{O}_\\mathcal{D}}", "f^{-1}\\mathcal{F}$ (Definition \\ref{definition-pushforward}).", "By Lemma \\ref{lemma-flat-change-of-rings} it suffices to", "prove that $f^{-1}\\mathcal{F}$ is a flat", "$f^{-1}\\mathcal{O}_\\mathcal{D}$-module. Combined with", "the previous paragraph this reduces us to the situation", "of the next paragraph.", "\\medskip\\noindent", "Assume $\\mathcal{C}$ and $\\mathcal{D}$ are sites which", "have all finite limits and that $u : \\mathcal{D} \\to \\mathcal{C}$", "is a continuous functor which commutes with finite limits.", "Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{D}$", "and let $\\mathcal{F}$ be a flat $\\mathcal{O}$-module.", "Then $u$ defines a morphism of sites $f : \\mathcal{C} \\to \\mathcal{D}$", "(Sites, Proposition \\ref{sites-proposition-get-morphism}).", "To show: $f^{-1}\\mathcal{F}$ is a flat $f^{-1}\\mathcal{O}$-module.", "Let $U$ be an object of $\\mathcal{C}$ and let", "$$", "f^{-1}\\mathcal{O}|_U \\xrightarrow{(f_1, \\ldots, f_n)}", "f^{-1}\\mathcal{O}|_U^{\\oplus n} \\xrightarrow{(s_1, \\ldots, s_n)}", "f^{-1}\\mathcal{F}|_U", "$$", "be a complex of $f^{-1}\\mathcal{O}|_U$-modules.", "Our goal is to construct a factorization of", "$(s_1, \\ldots, s_n)$ on the members of a covering of $U$", "as in Lemma \\ref{lemma-flat-eq} part (2).", "Consider the elements $s_a \\in f^{-1}\\mathcal{F}(U)$", "and $f_a \\in f^{-1}\\mathcal{O}(U)$.", "Since $f^{-1}\\mathcal{F}$, resp.\\ $f^{-1}\\mathcal{O}$", "is the sheafification of $u_p\\mathcal{F}$ we may,", "after replacing $U$ by the members of a covering,", "assume that $s_a$ is the image of an element $s'_a \\in u_p\\mathcal{F}(U)$ and", "$f_a$ is the image of an element $f'_a \\in u_p\\mathcal{O}(U)$.", "Then after another replacement of $U$ by the members of a covering", "we may assume that $\\sum f'_as'_a$ is zero in $u_p\\mathcal{F}(U)$.", "Recall that the category $(\\mathcal{I}_U^u)^{opp}$ is directed", "(Sites, Lemma \\ref{sites-lemma-directed})", "and that $u_p\\mathcal{F}(U) = \\colim_{(\\mathcal{I}_U^u)^{opp}} \\mathcal{F}(V)$", "and $u_p\\mathcal{O}(U) = \\colim_{(\\mathcal{I}_U^u)^{opp}} \\mathcal{O}(V)$.", "Hence we may assume there is a pair $(V, \\phi) \\in \\Ob(\\mathcal{I}_U^u)$", "where $V$ is an object of $\\mathcal{D}$", "and $\\phi$ is a morphism $\\phi : U \\to u(V)$ of $\\mathcal{D}$", "and elements $s''_a \\in \\mathcal{F}(V)$ and $f''_a \\in \\mathcal{O}(V)$", "whose images in $u_p\\mathcal{F}(U)$ and $u_p\\mathcal{O}(U)$", "are equal to $s'_a$ and $f'_a$ and such that", "$\\sum f''_a s''_a = 0$ in $\\mathcal{F}(V)$.", "Then we obtain a complex", "$$", "\\mathcal{O}|_V \\xrightarrow{(f''_1, \\ldots, f''_n)}", "\\mathcal{O}|_V^{\\oplus n} \\xrightarrow{(s''_1, \\ldots, s''_n)}", "\\mathcal{F}|_V", "$$", "and we can apply the other direction of Lemma \\ref{lemma-flat-eq}", "to see there exists a covering $\\{V_i \\to V\\}$ of $\\mathcal{D}$", "and for each $i$ a factorization", "$$", "\\mathcal{O}|_{V_i}^{\\oplus n}", "\\xrightarrow{B''_i}", "\\mathcal{O}|_{V_i}^{\\oplus l_i} \\xrightarrow{(t''_{i1}, \\ldots, t''_{il_i})}", "\\mathcal{F}|_{V_i}", "$$", "of $(s''_1, \\ldots, s''_n)|_{V_i}$ such that", "$B_i \\circ (f''_1, \\ldots, f''_n)|_{V_i} = 0$.", "Set $U_i = U \\times_{\\phi, u(V)} u(V_i)$, denote", "$B_i \\in \\text{Mat}(l_i \\times n, f^{-1}\\mathcal{O}(U_i))$", "the image of $B''_i$, and denote", "$t_{ij} \\in f^{-1}\\mathcal{F}(U_i)$ the image of", "$t''_{ij}$. Then we get a factorization", "$$", "f^{-1}\\mathcal{O}|_{U_i}^{\\oplus n}", "\\xrightarrow{B_i}", "f^{-1}\\mathcal{O}|_{U_i}^{\\oplus l_i}", "\\xrightarrow{(t_{i1}, \\ldots, t_{il_i})}", "\\mathcal{F}|_{U_i}", "$$", "of $(s_1, \\ldots, s_n)|_{U_i}$ such that", "$B_i \\circ (f_1, \\ldots, f_n)|_{U_i} = 0$.", "This finishes the proof." ], "refs": [ "sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites", "sites-modules-definition-flat", "sites-modules-definition-pushforward", "sites-modules-lemma-flat-change-of-rings", "sites-proposition-get-morphism", "sites-modules-lemma-flat-eq", "sites-lemma-directed", "sites-modules-lemma-flat-eq" ], "ref_ids": [ 14145, 14290, 14284, 14208, 8641, 14209, 8498, 14209 ] } ], "ref_ids": [] }, { "id": 14250, "type": "theorem", "label": "sites-modules-lemma-stalk-flat", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-stalk-flat", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $p$ be a point of $\\mathcal{C}$.", "If $\\mathcal{F}$ is a flat $\\mathcal{O}$-module, then", "$\\mathcal{F}_p$ is a flat $\\mathcal{O}_p$-module." ], "refs": [], "proofs": [ { "contents": [ "In Section \\ref{section-skyscraper} we have seen", "that we can think of $p$ as a morphism of ringed topoi", "$$", "(p, \\text{id}_{\\mathcal{O}_p}) :", "(\\Sh(pt), \\mathcal{O}_p)", "\\longrightarrow", "(\\Sh(\\mathcal{C}), \\mathcal{O}).", "$$", "such that the pullback functor", "$p^* : \\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_p)$", "equals the stalk functor. Thus the lemma follows from", "Lemma \\ref{lemma-pullback-flat}." ], "refs": [ "sites-modules-lemma-pullback-flat" ], "ref_ids": [ 14249 ] } ], "ref_ids": [] }, { "id": 14251, "type": "theorem", "label": "sites-modules-lemma-check-flat-stalks", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-check-flat-stalks", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.", "Let $\\{p_i\\}_{i \\in I}$ be a conservative family of points of $\\mathcal{C}$.", "Then $\\mathcal{F}$ is flat if and only if $\\mathcal{F}_{p_i}$ is", "a flat $\\mathcal{O}_{p_i}$-module for all $i \\in I$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-stalk-flat}", "we see one of the implications.", "For the converse, use that", "$(\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G})_p =", "\\mathcal{F}_p \\otimes_{\\mathcal{O}_p} \\mathcal{G}_p$", "by", "Lemma \\ref{lemma-tensor-product-pullback} (as taking stalks at $p$", "is given by $p^{-1}$) and", "Lemma \\ref{lemma-check-exactness-stalks}." ], "refs": [ "sites-modules-lemma-stalk-flat", "sites-modules-lemma-tensor-product-pullback", "sites-modules-lemma-check-exactness-stalks" ], "ref_ids": [ 14250, 14189, 14160 ] } ], "ref_ids": [] }, { "id": 14252, "type": "theorem", "label": "sites-modules-lemma-pullback-ses", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-pullback-ses", "contents": [ "Let", "$f : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}'), \\mathcal{O})$", "be a morphism of ringed topoi. Let", "$0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0$", "be a short exact sequence of $\\mathcal{O}$-modules with $\\mathcal{H}$", "a flat $\\mathcal{O}$-module. Then the sequence", "$0 \\to f^*\\mathcal{F} \\to f^*\\mathcal{G} \\to f^*\\mathcal{H} \\to 0$", "is exact as well." ], "refs": [], "proofs": [ { "contents": [ "Since $f^{-1}$ is exact we have the short exact sequence", "$0 \\to f^{-1}\\mathcal{F} \\to f^{-1}\\mathcal{G} \\to f^{-1}\\mathcal{H} \\to 0$", "of $f^{-1}\\mathcal{O}$-modules. By", "Lemma \\ref{lemma-pullback-flat}", "the $f^{-1}\\mathcal{O}$-module $f^{-1}\\mathcal{H}$ is flat. By", "Lemma \\ref{lemma-flat-tor-zero} this implies that tensoring the sequence", "over $f^{-1}\\mathcal{O}$ with $\\mathcal{O}'$ the sequence", "remains exact. Since", "$f^*\\mathcal{F} = f^{-1}\\mathcal{F} \\otimes_{f^{-1}\\mathcal{O}} \\mathcal{O}'$", "and similarly for $\\mathcal{G}$ and $\\mathcal{H}$ we conclude." ], "refs": [ "sites-modules-lemma-pullback-flat", "sites-modules-lemma-flat-tor-zero" ], "ref_ids": [ 14249, 14204 ] } ], "ref_ids": [] }, { "id": 14253, "type": "theorem", "label": "sites-modules-lemma-locally-ringed", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-locally-ringed", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. The following", "are equivalent", "\\begin{enumerate}", "\\item For every object $U$ of $\\mathcal{C}$ and $f \\in \\mathcal{O}(U)$", "there exists a covering $\\{U_j \\to U\\}$ such that for each $j$", "either $f|_{U_j}$ is invertible or $(1 - f)|_{U_j}$ is invertible.", "\\item For $U \\in \\Ob(\\mathcal{C})$, $n \\geq 1$, and", "$f_1, \\ldots, f_n \\in \\mathcal{O}(U)$ which generate the unit ideal", "in $\\mathcal{O}(U)$ there exists a covering $\\{U_j \\to U\\}$", "such that for each $j$ there exists an $i$ such that $f_i|_{U_j}$", "is invertible.", "\\item The map of sheaves of sets", "$$", "(\\mathcal{O} \\times \\mathcal{O})", "\\amalg", "(\\mathcal{O} \\times \\mathcal{O})", "\\longrightarrow", "\\mathcal{O} \\times \\mathcal{O}", "$$", "which maps $(f, a)$ in the first component to $(f, af)$ and", "$(f, b)$ in the second component to $(f, b(1 - f))$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (2) implies (1). To show that (1) implies (2) we argue by", "induction on $n$. The first case is $n = 2$ (since $n = 1$ is trivial).", "In this case we have $a_1f_1 + a_2f_2 = 1$ for some", "$a_1, a_2 \\in \\mathcal{O}(U)$. By assumption we can find a covering", "$\\{U_j \\to U\\}$ such that for each $j$", "either $a_1f_1|_{U_j}$ is invertible or $a_2f_2|_{U_j}$ is invertible.", "Hence either $f_1|_{U_j}$ is invertible or $f_2|_{U_j}$ is invertible", "as desired. For $n > 2$ we have", "$a_1f_1 + \\ldots + a_nf_n = 1$ for some $a_1, \\ldots, a_n \\in \\mathcal{O}(U)$.", "By the case $n = 2$ we see that we have some covering $\\{U_j \\to U\\}_{j \\in J}$", "such that for each $j$ either $f_n|_{U_j}$ is invertible or", "$a_1f_1 + \\ldots + a_{n - 1}f_{n - 1}|_{U_j}$ is invertible.", "Say the first case happens for $j \\in J_n$. Set $J' = J \\setminus J_n$.", "By induction hypothesis, for each $j \\in J'$ we can find a covering", "$\\{U_{jk} \\to U_j\\}_{k \\in K_j}$ such that for each $k \\in K_j$ there", "exists an $i \\in \\{1, \\ldots, n - 1\\}$ such that", "$f_i|_{U_{jk}}$ is invertible. By the axioms of a site the family of", "morphisms", "$\\{U_j \\to U\\}_{j \\in J_n} \\cup \\{U_{jk} \\to U\\}_{j \\in J', k \\in K_j}$", "is a covering which has the desired property.", "\\medskip\\noindent", "Assume (1). To see that the map in (3) is surjective, let", "$(f, c)$ be a section of $\\mathcal{O} \\times \\mathcal{O}$ over $U$.", "By assumption there exists a covering $\\{U_j \\to U\\}$ such that", "for each $j$ either $f$ or $1 - f$ restricts to an invertible section.", "In the first case we can take $a = c|_{U_j} (f|_{U_j})^{-1}$, and", "in the second case we can take $b = c|_{U_j} (1 - f|_{U_j})^{-1}$.", "Hence $(f, c)$ is in the image of the map on each of the members.", "Conversely, assume (3) holds. For any $U$ and $f \\in \\mathcal{O}(U)$", "there exists a covering $\\{U_j \\to U\\}$ of $U$ such that the", "section $(f, 1)|_{U_j}$ is in the image of the map in (3) on sections", "over $U_j$. This means precisely that either $f$ or $1 - f$ restricts", "to an invertible section over $U_j$, and we see that (1) holds." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14254, "type": "theorem", "label": "sites-modules-lemma-locally-ringed-stalk", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-locally-ringed-stalk", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Consider the following conditions", "\\begin{enumerate}", "\\item For every object $U$ of $\\mathcal{C}$ and $f \\in \\mathcal{O}(U)$", "there exists a covering $\\{U_j \\to U\\}$ such that for each $j$", "either $f|_{U_j}$ is invertible or $(1 - f)|_{U_j}$ is invertible.", "\\item For every point $p$ of $\\mathcal{C}$ the stalk $\\mathcal{O}_p$", "is either the zero ring or a local ring.", "\\end{enumerate}", "We always have (1) $\\Rightarrow$ (2). If $\\mathcal{C}$ has enough points", "then (1) and (2) are equivalent." ], "refs": [], "proofs": [ { "contents": [ "Assume (1). Let $p$ be a point of $\\mathcal{C}$ given by a functor", "$u : \\mathcal{C} \\to \\textit{Sets}$. Let", "$f_p \\in \\mathcal{O}_p$. Since $\\mathcal{O}_p$ is computed by", "Sites, Equation (\\ref{sites-equation-stalk})", "we may represent $f_p$ by a triple", "$(U, x, f)$ where $x \\in U(U)$ and $f \\in \\mathcal{O}(U)$.", "By assumption there exists a covering $\\{U_i \\to U\\}$", "such that for each $i$ either $f$ or $1 - f$ is invertible", "on $U_i$. Because $u$ defines a point of the site we see that", "for some $i$ there exists an $x_i \\in u(U_i)$ which maps to", "$x \\in u(U)$. By the discussion surrounding", "Sites, Equation (\\ref{sites-equation-stalk})", "we see that $(U, x, f)$ and $(U_i, x_i, f|_{U_i})$ define the", "same element of $\\mathcal{O}_p$. Hence we conclude that", "either $f_p$ or $1 - f_p$ is invertible. Thus", "$\\mathcal{O}_p$ is a ring such that for every element $a$", "either $a$ or $1 - a$ is invertible. This means that $\\mathcal{O}_p$", "is either zero or a local ring, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-local-ring}.", "\\medskip\\noindent", "Assume (2) and assume that $\\mathcal{C}$ has enough points.", "Consider the map of sheaves of sets", "$$", "\\mathcal{O} \\times \\mathcal{O} \\amalg \\mathcal{O} \\times \\mathcal{O}", "\\longrightarrow", "\\mathcal{O} \\times \\mathcal{O}", "$$", "of", "Lemma \\ref{lemma-locally-ringed} part (3). For any local ring $R$", "the corresponding map", "$(R \\times R) \\amalg (R \\times R) \\to R \\times R$", "is surjective, see for example", "Algebra, Lemma \\ref{algebra-lemma-characterize-local-ring}.", "Since each $\\mathcal{O}_p$ is a local ring or zero the map is", "surjective on stalks. Hence, by our assumption that $\\mathcal{C}$", "has enough points it is surjective and we win." ], "refs": [ "algebra-lemma-characterize-local-ring", "sites-modules-lemma-locally-ringed", "algebra-lemma-characterize-local-ring" ], "ref_ids": [ 397, 14253, 397 ] } ], "ref_ids": [] }, { "id": 14255, "type": "theorem", "label": "sites-modules-lemma-ringed-stalk-not-zero", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-ringed-stalk-not-zero", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Consider the statements", "\\begin{enumerate}", "\\item (\\ref{equation-one-is-never-zero}) is an isomorphism, and", "\\item for every point $p$ of $\\mathcal{C}$ the stalk $\\mathcal{O}_p$", "is not the zero ring.", "\\end{enumerate}", "We always have (1) $\\Rightarrow$ (2) and if $\\mathcal{C}$ has enough points", "then (1) $\\Leftrightarrow$ (2)." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14256, "type": "theorem", "label": "sites-modules-lemma-locally-ringed-intrinsic", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-locally-ringed-intrinsic", "contents": [ "Being a locally ringed site is an intrinsic property.", "More precisely,", "\\begin{enumerate}", "\\item if $f : \\Sh(\\mathcal{C}') \\to \\Sh(\\mathcal{C})$", "is a morphism of topoi and $(\\mathcal{C}, \\mathcal{O})$ is", "a locally ringed site, then $(\\mathcal{C}', f^{-1}\\mathcal{O})$", "is a locally ringed site, and", "\\item if", "$(f, f^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}')", "\\to (\\Sh(\\mathcal{C}), \\mathcal{O})$", "is an equivalence of ringed topoi, then", "$(\\mathcal{C}, \\mathcal{O})$ is locally ringed if and only if", "$(\\mathcal{C}', \\mathcal{O}')$", "is locally ringed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (2) follows from (1). To prove (1) note that", "as $f^{-1}$ is exact we have $f^{-1}* = *$,", "$f^{-1}\\emptyset^\\# = \\emptyset^\\#$, and $f^{-1}$ commutes with", "products, equalizers and transforms isomorphisms and surjections", "into isomorphisms and surjections. Thus $f^{-1}$ transforms the", "isomorphism (\\ref{equation-one-is-never-zero}) into", "its analogue for $f^{-1}\\mathcal{O}$ and transforms the surjection of", "Lemma \\ref{lemma-locally-ringed} part (3)", "into the corresponding surjection for $f^{-1}\\mathcal{O}$." ], "refs": [ "sites-modules-lemma-locally-ringed" ], "ref_ids": [ 14253 ] } ], "ref_ids": [] }, { "id": 14257, "type": "theorem", "label": "sites-modules-lemma-invertible-is-locally-free-rank-1", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-invertible-is-locally-free-rank-1", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. Any locally free", "$\\mathcal{O}$-module of rank $1$ is invertible.", "If $(\\mathcal{C}, \\mathcal{O})$ is locally ringed, then", "the converse holds as well (but in general this is not the case)." ], "refs": [], "proofs": [ { "contents": [ "Assume $\\mathcal{L}$ is locally free of rank $1$ and consider the", "evaluation map", "$$", "\\mathcal{L} \\otimes_\\mathcal{O}", "\\SheafHom_\\mathcal{O}(\\mathcal{L}, \\mathcal{O})", "\\longrightarrow \\mathcal{O}", "$$", "Given any object $U$ of $\\mathcal{C}$ and restricting to the members", "of a covering trivializing $\\mathcal{L}$, we see", "that this map is an isomorphism (details omitted).", "Hence $\\mathcal{L}$ is invertible by Lemma \\ref{lemma-invertible}.", "\\medskip\\noindent", "Assume $(\\Sh(\\mathcal{C}), \\mathcal{O})$ is locally ringed.", "Let $U$ be an object of $\\mathcal{C}$.", "In the proof of Lemma \\ref{lemma-invertible}", "we have seen that there exists a covering $\\{U_i \\to U\\}$", "such that $\\mathcal{L}|_{\\mathcal{C}/U_i}$ is a direct summand", "of a finite free $\\mathcal{O}_{U_i}$-module. After replacing", "$U$ by $U_i$, let", "$p : \\mathcal{O}_U^{\\oplus r} \\to \\mathcal{O}_U^{\\oplus r}$", "be a projector whose image is isomorphic to $\\mathcal{L}|_{\\mathcal{C}/U}$.", "Then $p$ corresponds to a matrix", "$$", "P = (p_{ij}) \\in \\text{Mat}(r \\times r, \\mathcal{O}(U))", "$$", "which is a projector: $P^2 = P$. Set $A = \\mathcal{O}(U)$", "so that $P \\in \\text{Mat}(r \\times r, A)$.", "By Algebra, Lemma \\ref{algebra-lemma-finite-projective}", "the image of $P$ is a finite locally free module $M$ over $A$.", "Hence there are $f_1, \\ldots, f_t \\in A$ generating the unit", "ideal, such that $M_{f_i}$ is finite free. By", "Lemma \\ref{lemma-locally-ringed} after replacing $U$ by the members of an open", "covering, we may assume that $M$ is free. This means that", "$\\mathcal{L}|_U$ is free (details omitted). Of course, since", "$\\mathcal{L}$ is invertible, this is", "only possible if the rank of $\\mathcal{L}|_U$ is $1$", "and the proof is complete." ], "refs": [ "sites-modules-lemma-invertible", "sites-modules-lemma-invertible", "algebra-lemma-finite-projective", "sites-modules-lemma-locally-ringed" ], "ref_ids": [ 14224, 14224, 795, 14253 ] } ], "ref_ids": [] }, { "id": 14258, "type": "theorem", "label": "sites-modules-lemma-locally-ringed-morphism", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-locally-ringed-morphism", "contents": [ "Let", "$(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi.", "Consider the following conditions", "\\begin{enumerate}", "\\item The diagram of sheaves", "$$", "\\xymatrix{", "f^{-1}(\\mathcal{O}^*_\\mathcal{D}) \\ar[r]_-{f^\\sharp} \\ar[d] &", "\\mathcal{O}^*_\\mathcal{C} \\ar[d] \\\\", "f^{-1}(\\mathcal{O}_\\mathcal{D}) \\ar[r]^-{f^\\sharp} &", "\\mathcal{O}_\\mathcal{C}", "}", "$$", "is cartesian.", "\\item For any point $p$ of $\\mathcal{C}$, setting $q = f \\circ p$,", "the diagram", "$$", "\\xymatrix{", "\\mathcal{O}^*_{\\mathcal{D}, q} \\ar[r] \\ar[d] &", "\\mathcal{O}^*_{\\mathcal{C}, p} \\ar[d] \\\\", "\\mathcal{O}_{\\mathcal{D}, q} \\ar[r] &", "\\mathcal{O}_{\\mathcal{C}, p}", "}", "$$", "of sets is cartesian.", "\\end{enumerate}", "We always have (1) $\\Rightarrow$ (2). If $\\mathcal{C}$ has enough points", "then (1) and (2) are equivalent. If", "$(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$", "and", "$(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "are locally ringed topoi then (2) is equivalent to", "\\begin{enumerate}", "\\item[(3)] For any point $p$ of $\\mathcal{C}$, setting $q = f \\circ p$,", "the ring map $\\mathcal{O}_{\\mathcal{D}, q} \\to \\mathcal{O}_{\\mathcal{C}, p}$", "is a local ring map.", "\\end{enumerate}", "In fact, properties (2), or (3) for a conservative", "family of points implies (1)." ], "refs": [], "proofs": [ { "contents": [ "This lemma proves itself, in other words, it follows by unwinding the", "definitions." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14259, "type": "theorem", "label": "sites-modules-lemma-composition-morphisms-locally-ringed-topoi", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-composition-morphisms-locally-ringed-topoi", "contents": [ "Let", "$(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}_1), \\mathcal{O}_1)", "\\to (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2)$ and", "$(g, g^\\sharp) :", "(\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\to", "(\\Sh(\\mathcal{C}_3), \\mathcal{O}_3)$", "be morphisms of locally ringed topoi. Then the composition", "$(g, g^\\sharp) \\circ (f, f^\\sharp)$ (see", "Definition \\ref{definition-ringed-topos})", "is also a morphism of locally ringed topoi." ], "refs": [ "sites-modules-definition-ringed-topos" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 14280 ] }, { "id": 14260, "type": "theorem", "label": "sites-modules-lemma-locally-ringed-intrinsic-morphism", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-locally-ringed-intrinsic-morphism", "contents": [ "If $f : \\Sh(\\mathcal{C}') \\to \\Sh(\\mathcal{C})$", "is a morphism of topoi. If $\\mathcal{O}$ is a sheaf of rings", "on $\\mathcal{C}$, then", "$$", "f^{-1}(\\mathcal{O}^*) = (f^{-1}\\mathcal{O})^*.", "$$", "In particular, if $\\mathcal{O}$ turns $\\mathcal{C}$ into a locally", "ringed site, then setting $f^\\sharp = \\text{id}$", "the morphism of ringed topoi", "$$", "(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}'), f^{-1}\\mathcal{O})", "\\to", "(\\Sh(\\mathcal{C}, \\mathcal{O})", "$$", "is a morphism of locally ringed topoi." ], "refs": [], "proofs": [ { "contents": [ "Note that the diagram", "$$", "\\xymatrix{", "\\mathcal{O}^* \\ar[rr] \\ar[d]_{u \\mapsto (u, u^{-1})} & &", "{*} \\ar[d]^{1} \\\\", "\\mathcal{O} \\times \\mathcal{O} \\ar[rr]^-{(a, b) \\mapsto ab} & &", "\\mathcal{O}", "}", "$$", "is cartesian. Since $f^{-1}$ is exact we conclude that", "$$", "\\xymatrix{", "f^{-1}(\\mathcal{O}^*)", "\\ar[d]_{u \\mapsto (u, u^{-1})} \\ar[rr] & &", "{*} \\ar[d]^{1} \\\\", "f^{-1}\\mathcal{O} \\times f^{-1}\\mathcal{O} \\ar[rr]^-{(a, b) \\mapsto ab} & &", "f^{-1}\\mathcal{O}", "}", "$$", "is cartesian which implies the first assertion. For the second,", "note that $(\\mathcal{C}', f^{-1}\\mathcal{O})$ is a locally ringed site", "by", "Lemma \\ref{lemma-locally-ringed-intrinsic}", "so that the assertion makes sense. Now the first part implies that", "the morphism is a morphism of locally ringed topoi." ], "refs": [ "sites-modules-lemma-locally-ringed-intrinsic" ], "ref_ids": [ 14256 ] } ], "ref_ids": [] }, { "id": 14261, "type": "theorem", "label": "sites-modules-lemma-localize-morphism-locally-ringed-topoi", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-localize-morphism-locally-ringed-topoi", "contents": [ "Localization of locally ringed sites and topoi.", "\\begin{enumerate}", "\\item Let $(\\mathcal{C}, \\mathcal{O})$ be a locally ringed site.", "Let $U$ be an object of $\\mathcal{C}$. Then the localization", "$(\\mathcal{C}/U, \\mathcal{O}_U)$ is a locally ringed site, and", "the localization morphism", "$$", "(j_U, j_U^\\sharp) :", "(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)", "\\to", "(\\Sh(\\mathcal{C}), \\mathcal{O})", "$$", "is a morphism of locally ringed topoi.", "\\item Let $(\\mathcal{C}, \\mathcal{O})$ be a locally ringed site.", "Let $f : V \\to U$ be a morphism of $\\mathcal{C}$.", "Then the morphism", "$$", "(j, j^\\sharp) :", "(\\Sh(\\mathcal{C}/V), \\mathcal{O}_V)", "\\to", "(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)", "$$", "of", "Lemma \\ref{lemma-relocalize}", "is a morphism of locally ringed topoi.", "\\item Let", "$(f, f^\\sharp) :", "(\\mathcal{C}, \\mathcal{O})", "\\longrightarrow", "(\\mathcal{D}, \\mathcal{O}')$", "be a morphism of locally ringed sites where $f$ is given by the continuous", "functor $u : \\mathcal{D} \\to \\mathcal{C}$. Let $V$ be an object of", "$\\mathcal{D}$ and let $U = u(V)$. Then the morphism", "$$", "(f', (f')^\\sharp) :", "(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)", "\\to", "(\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V)", "$$", "of", "Lemma \\ref{lemma-localize-morphism-ringed-sites}", "is a morphism of locally ringed sites.", "\\item Let", "$(f, f^\\sharp) :", "(\\mathcal{C}, \\mathcal{O})", "\\longrightarrow", "(\\mathcal{D}, \\mathcal{O}')$", "be a morphism of locally ringed sites where $f$ is given by the continuous", "functor $u : \\mathcal{D} \\to \\mathcal{C}$. Let $V \\in \\Ob(\\mathcal{D})$,", "$U \\in \\Ob(\\mathcal{C})$, and $c : U \\to u(V)$. Then the morphism", "$$", "(f_c, (f_c)^\\sharp) :", "(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)", "\\to", "(\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V)", "$$", "of", "Lemma \\ref{lemma-relocalize-morphism-ringed-sites}", "is a morphism of locally ringed topoi.", "\\item Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a locally", "ringed topos. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.", "Then the localization", "$(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})$", "is a locally ringed topos and the localization morphism", "$$", "(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) :", "(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})", "\\to", "(\\Sh(\\mathcal{C}), \\mathcal{O})", "$$", "is a morphism of locally ringed topoi.", "\\item Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a locally", "ringed topos. Let $s : \\mathcal{G} \\to \\mathcal{F}$ be a map of sheaves", "on $\\mathcal{C}$. Then the morphism", "$$", "(j, j^\\sharp) :", "(\\Sh(\\mathcal{C})/\\mathcal{G}, \\mathcal{O}_\\mathcal{G})", "\\longrightarrow", "(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})", "$$", "of", "Lemma \\ref{lemma-relocalize-ringed-topos}", "is a morphism of locally ringed topoi.", "\\item Let", "$f :", "(\\Sh(\\mathcal{C}), \\mathcal{O})", "\\longrightarrow", "(\\Sh(\\mathcal{D}), \\mathcal{O}')$", "be a morphism of locally ringed topoi. Let $\\mathcal{G}$ be a sheaf", "on $\\mathcal{D}$. Set $\\mathcal{F} = f^{-1}\\mathcal{G}$.", "Then the morphism", "$$", "(f', (f')^\\sharp) :", "(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})", "\\longrightarrow", "(\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_\\mathcal{G})", "$$", "of", "Lemma \\ref{lemma-localize-morphism-ringed-topoi}", "is a morphism of locally ringed topoi.", "\\item Let", "$f :", "(\\Sh(\\mathcal{C}), \\mathcal{O})", "\\longrightarrow", "(\\Sh(\\mathcal{D}), \\mathcal{O}')$", "be a morphism of locally ringed topoi. Let $\\mathcal{G}$ be a sheaf", "on $\\mathcal{D}$, let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$, and", "let $s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ be a morphism of sheaves.", "Then the morphism", "$$", "(f_s, (f_s)^\\sharp) :", "(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})", "\\longrightarrow", "(\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_\\mathcal{G})", "$$", "of", "Lemma \\ref{lemma-relocalize-morphism-ringed-topoi}", "is a morphism of locally ringed topoi.", "\\end{enumerate}" ], "refs": [ "sites-modules-lemma-relocalize", "sites-modules-lemma-localize-morphism-ringed-sites", "sites-modules-lemma-relocalize-morphism-ringed-sites", "sites-modules-lemma-relocalize-ringed-topos", "sites-modules-lemma-localize-morphism-ringed-topoi", "sites-modules-lemma-relocalize-morphism-ringed-topoi" ], "proofs": [ { "contents": [ "Part (1) is clear since $\\mathcal{O}_U$ is just the", "restriction of $\\mathcal{O}$, so", "Lemmas \\ref{lemma-locally-ringed-intrinsic} and", "\\ref{lemma-locally-ringed-intrinsic-morphism}", "apply. Part (2) is clear as the morphism $(j, j^\\sharp)$", "is actually a localization of a locally ringed site so (1) applies.", "Part (3) is clear also since $(f')^\\sharp$ is just the", "restriction of $f^\\sharp$ to the topos", "$\\Sh(\\mathcal{C})/\\mathcal{F}$, see proof of", "Lemma \\ref{lemma-localize-morphism-ringed-topoi}", "(hence the diagram of", "Definition \\ref{definition-morphism-locally-ringed-topoi}", "for the morphism $f'$ is just the restriction of the corresponding", "diagram for $f$, and restriction is an exact functor).", "Part (4) follows formally on combining (2) and (3).", "Parts (5), (6), (7), and (8) follow from their counterparts", "(1), (2), (3), and (4) by enlarging the sites as in", "Lemma \\ref{lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites}", "and translating everything in terms of sites and morphisms of sites using", "the comparisons of", "Lemmas \\ref{lemma-localize-compare},", "\\ref{lemma-relocalize-compare},", "\\ref{lemma-localize-morphism-compare}, and", "\\ref{lemma-relocalize-morphism-compare}.", "(Alternatively one could use the same arguments as in the proofs", "of (1), (2), (3), and (4) to prove (5), (6), (7), and (8) directly.)" ], "refs": [ "sites-modules-lemma-locally-ringed-intrinsic", "sites-modules-lemma-locally-ringed-intrinsic-morphism", "sites-modules-lemma-localize-morphism-ringed-topoi", "sites-modules-definition-morphism-locally-ringed-topoi", "sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites", "sites-modules-lemma-localize-compare", "sites-modules-lemma-relocalize-compare", "sites-modules-lemma-localize-morphism-compare", "sites-modules-lemma-relocalize-morphism-compare" ], "ref_ids": [ 14256, 14260, 14180, 14304, 14145, 14177, 14179, 14181, 14183 ] } ], "ref_ids": [ 14172, 14174, 14175, 14178, 14180, 14182 ] }, { "id": 14262, "type": "theorem", "label": "sites-modules-lemma-lower-shriek-modules", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-lower-shriek-modules", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous", "functor between sites. Denote", "$g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ the associated", "morphism of topoi. Let $\\mathcal{O}_\\mathcal{D}$ be a sheaf of rings", "on $\\mathcal{D}$. Set", "$\\mathcal{O}_\\mathcal{C} = g^{-1}\\mathcal{O}_\\mathcal{D}$.", "Hence $g$ becomes a morphism of ringed topoi with $g^* = g^{-1}$.", "In this case there exists a functor", "$$", "g_! :", "\\textit{Mod}(\\mathcal{O}_\\mathcal{C})", "\\longrightarrow", "\\textit{Mod}(\\mathcal{O}_\\mathcal{D})", "$$", "which is left adjoint to $g^*$." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be an object of $\\mathcal{C}$. For any", "$\\mathcal{O}_\\mathcal{D}$-module $\\mathcal{G}$ we have", "\\begin{align*}", "\\Hom_{\\mathcal{O}_\\mathcal{C}}(j_{U!}\\mathcal{O}_U, g^{-1}\\mathcal{G})", "& =", "g^{-1}\\mathcal{G}(U) \\\\", "& =", "\\mathcal{G}(u(U)) \\\\", "& =", "\\Hom_{\\mathcal{O}_\\mathcal{D}}(j_{u(U)!}\\mathcal{O}_{u(U)}, \\mathcal{G})", "\\end{align*}", "because $g^{-1}$ is described by restriction, see", "Sites, Lemma \\ref{sites-lemma-when-shriek}.", "Of course a similar formula holds a direct sum of modules", "of the form $j_{U!}\\mathcal{O}_U$. By", "Homology, Lemma \\ref{homology-lemma-partially-defined-adjoint}", "and", "Lemma \\ref{lemma-module-quotient-flat}", "we see that $g_!$ exists." ], "refs": [ "sites-lemma-when-shriek", "homology-lemma-partially-defined-adjoint", "sites-modules-lemma-module-quotient-flat" ], "ref_ids": [ 8545, 12119, 14203 ] } ], "ref_ids": [] }, { "id": 14263, "type": "theorem", "label": "sites-modules-lemma-special-square-cocontinuous", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-special-square-cocontinuous", "contents": [ "Assume given a commutative diagram", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})", "\\ar[r]_{(g', (g')^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} &", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^{(f, f^\\sharp)} \\\\", "(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^{(g, g^\\sharp)} &", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})", "}", "$$", "of ringed topoi. Assume", "\\begin{enumerate}", "\\item $f$, $f'$, $g$, and $g'$ correspond to cocontinuous functors", "$u$, $u'$, $v$, and $v'$ as in", "Sites, Lemma \\ref{sites-lemma-cocontinuous-morphism-topoi},", "\\item $v \\circ u' = u \\circ v'$,", "\\item $v$ and $v'$ are continuous as well as cocontinuous,", "\\item for any object $V'$ of $\\mathcal{D}'$ the functor", "${}^{u'}_{V'}\\mathcal{I} \\to {}^{\\ \\ \\ u}_{v(V')}\\mathcal{I}$", "given by $v$ is cofinal, and", "\\item $g^{-1}\\mathcal{O}_{\\mathcal{D}} = \\mathcal{O}_{\\mathcal{D}'}$", "and $(g')^{-1}\\mathcal{O}_{\\mathcal{C}} = \\mathcal{O}_{\\mathcal{C}'}$.", "\\end{enumerate}", "Then we have $f'_* \\circ (g')^* = g^* \\circ f_*$ and", "$g'_! \\circ (f')^{-1} = f^{-1} \\circ g_!$ on modules." ], "refs": [ "sites-lemma-cocontinuous-morphism-topoi" ], "proofs": [ { "contents": [ "We have $(g')^*\\mathcal{F} = (g')^{-1}\\mathcal{F}$ and", "$g^*\\mathcal{G} = g^{-1}\\mathcal{G}$ because of condition (5).", "Thus the first equality follows immediately from the corresponding", "equality in Sites, Lemma \\ref{sites-lemma-special-square-cocontinuous}.", "Since the left adjoint functors $g_!$ and $g'_!$ to $g^*$ and $(g')^*$", "exist by Lemma \\ref{lemma-lower-shriek-modules}", "we see that the second equality follows by", "uniqueness of adjoint functors." ], "refs": [ "sites-lemma-special-square-cocontinuous", "sites-modules-lemma-lower-shriek-modules" ], "ref_ids": [ 8576, 14262 ] } ], "ref_ids": [ 8543 ] }, { "id": 14264, "type": "theorem", "label": "sites-modules-lemma-special-square-continuous", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-special-square-continuous", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})", "\\ar[r]_{(g', (g')^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} &", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^{(f, f^\\sharp)} \\\\", "(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^{(g, g^\\sharp)} &", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})", "}", "$$", "of ringed topoi and suppose we have functors", "$$", "\\xymatrix{", "\\mathcal{C}' \\ar[r]_{v'} &", "\\mathcal{C} \\\\", "\\mathcal{D}' \\ar[r]^v \\ar[u]^{u'} &", "\\mathcal{D} \\ar[u]_u", "}", "$$", "such that (with notation as in", "Sites, Sections \\ref{sites-section-morphism-sites} and", "\\ref{sites-section-cocontinuous-morphism-topoi}) we have", "\\begin{enumerate}", "\\item $u$ and $u'$ are continuous and give rise to the morphisms", "$f$ and $f'$,", "\\item $v$ and $v'$ are cocontinuous giving rise to the morphisms $g$ and $g'$,", "\\item $u \\circ v = v' \\circ u'$,", "\\item $v$ and $v'$ are continuous as well as cocontinuous, and", "\\item $g^{-1}\\mathcal{O}_{\\mathcal{D}} = \\mathcal{O}_{\\mathcal{D}'}$", "and $(g')^{-1}\\mathcal{O}_{\\mathcal{C}} = \\mathcal{O}_{\\mathcal{C}'}$.", "\\end{enumerate}", "Then $f'_* \\circ (g')^* = g^* \\circ f_*$ and", "$g'_! \\circ (f')^{-1} = f^{-1} \\circ g_!$ on modules." ], "refs": [], "proofs": [ { "contents": [ "We have $(g')^*\\mathcal{F} = (g')^{-1}\\mathcal{F}$ and", "$g^*\\mathcal{G} = g^{-1}\\mathcal{G}$ because of condition (5).", "Thus the first equality follows immediately from the corresponding", "equality in Sites, Lemma \\ref{sites-lemma-special-square-continuous}.", "Since the left adjoint functors $g_!$ and $g'_!$ to $g^*$ and $(g')^*$", "exist by Lemma \\ref{lemma-lower-shriek-modules}", "we see that the second equality follows by", "uniqueness of adjoint functors." ], "refs": [ "sites-lemma-special-square-continuous", "sites-modules-lemma-lower-shriek-modules" ], "ref_ids": [ 8577, 14262 ] } ], "ref_ids": [] }, { "id": 14265, "type": "theorem", "label": "sites-modules-lemma-constant-exact", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-constant-exact", "contents": [ "Let $\\mathcal{C}$ be a site. If $0 \\to A \\to B \\to C \\to 0$", "is a short exact sequence of abelian groups, then", "$0 \\to \\underline{A} \\to \\underline{B} \\to \\underline{C} \\to 0$", "is an exact sequence of abelian sheaves and in fact it is even", "exact as a sequence of abelian presheaves." ], "refs": [], "proofs": [ { "contents": [ "Since sheafification is exact it is clear that", "$0 \\to \\underline{A} \\to \\underline{B} \\to \\underline{C} \\to 0$", "is an exact sequence of abelian sheaves. Thus", "$0 \\to \\underline{A} \\to \\underline{B} \\to \\underline{C}$", "is an exact sequence of abelian presheaves. To see that", "$\\underline{B} \\to \\underline{C}$ is surjective, pick a", "set theoretical section $s : C \\to B$. This induces a", "section $\\underline{s} : \\underline{C} \\to \\underline{B}$", "of sheaves of sets left inverse to the surjection", "$\\underline{B} \\to \\underline{C}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14266, "type": "theorem", "label": "sites-modules-lemma-tensor-with-finitely-presented", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-tensor-with-finitely-presented", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a ring and let $M$", "and $Q$ be $\\Lambda$-modules. If $Q$ is a finitely presented", "$\\Lambda$-module, then we have", "$\\underline{M \\otimes_\\Lambda Q}(U) = \\underline{M}(U) \\otimes_\\Lambda Q$", "for all $U \\in \\Ob(\\mathcal{C})$." ], "refs": [], "proofs": [ { "contents": [ "Choose a presentation $\\Lambda^{\\oplus m} \\to \\Lambda^{\\oplus n} \\to Q \\to 0$.", "This gives an exact sequence", "$M^{\\oplus m} \\to M^{\\oplus n} \\to M \\otimes Q \\to 0$.", "By Lemma \\ref{lemma-constant-exact} we obtain an exact sequence", "$$", "\\underline{M}(U)^{\\oplus m} \\to", "\\underline{M}(U)^{\\oplus n} \\to", "\\underline{M \\otimes Q}(U) \\to 0", "$$", "which proves the lemma. (Note that taking sections over $U$ always", "commutes with finite direct sums, but not arbitrary direct sums.)" ], "refs": [ "sites-modules-lemma-constant-exact" ], "ref_ids": [ 14265 ] } ], "ref_ids": [] }, { "id": 14267, "type": "theorem", "label": "sites-modules-lemma-flat-sections", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-flat-sections", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a coherent ring.", "Let $M$ be a flat $\\Lambda$-module. For $U \\in \\Ob(\\mathcal{C})$ the", "module $\\underline{M}(U)$ is a flat $\\Lambda$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $I \\subset \\Lambda$ be a finitely generated ideal.", "By Algebra, Lemma \\ref{algebra-lemma-flat} it suffices to show that", "$\\underline{M}(U) \\otimes_\\Lambda I \\to \\underline{M}(U)$", "is injective. As $\\Lambda$ is coherent $I$ is finitely presented as", "a $\\Lambda$-module. By Lemma \\ref{lemma-tensor-with-finitely-presented}", "we see that $\\underline{M}(U) \\otimes I = \\underline{M \\otimes I}$.", "Since $M$ is flat the map $M \\otimes I \\to M$ is injective,", "whence $\\underline{M \\otimes I} \\to \\underline{M}$ is injective." ], "refs": [ "algebra-lemma-flat", "sites-modules-lemma-tensor-with-finitely-presented" ], "ref_ids": [ 525, 14266 ] } ], "ref_ids": [] }, { "id": 14268, "type": "theorem", "label": "sites-modules-lemma-completion-flat", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-completion-flat", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a Noetherian ring.", "Let $I \\subset \\Lambda$ be an ideal. The sheaf", "$\\underline{\\Lambda}^\\wedge = \\lim \\underline{\\Lambda/I^n}$", "is a flat $\\underline{\\Lambda}$-algebra.", "Moreover we have canonical identifications", "$$", "\\underline{\\Lambda}/I\\underline{\\Lambda} =", "\\underline{\\Lambda}/\\underline{I} =", "\\underline{\\Lambda}^\\wedge/I\\underline{\\Lambda}^\\wedge =", "\\underline{\\Lambda}^\\wedge/\\underline{I} \\cdot \\underline{\\Lambda}^\\wedge =", "\\underline{\\Lambda}^\\wedge/\\underline{I}^\\wedge =", "\\underline{\\Lambda/I}", "$$", "where $\\underline{I}^\\wedge = \\lim \\underline{I/I^n}$." ], "refs": [], "proofs": [ { "contents": [ "To prove $\\underline{\\Lambda}^\\wedge$ is flat, it suffices to show that", "$\\underline{\\Lambda}^\\wedge(U)$ is flat as a $\\Lambda$-module for each", "$U \\in \\Ob(\\mathcal{C})$, see", "Lemmas \\ref{lemma-flatness-presheaves} and", "\\ref{lemma-flatness-sheafification}.", "By Lemma \\ref{lemma-flat-sections} we see that", "$$", "\\underline{\\Lambda}^\\wedge(U) = \\lim \\underline{\\Lambda/I^n}(U)", "$$", "is a limit of a system of flat $\\Lambda/I^n$-modules.", "By Lemma \\ref{lemma-constant-exact} we see that the transition maps", "are surjective. We conclude by", "More on Algebra, Lemma \\ref{more-algebra-lemma-limit-flat}.", "\\medskip\\noindent", "To see the equalities, note that", "$\\underline{\\Lambda}(U)/I\\underline{\\Lambda}(U) = \\underline{\\Lambda/I}(U)$", "by Lemma \\ref{lemma-tensor-with-finitely-presented}.", "It follows that $\\underline{\\Lambda}/I\\underline{\\Lambda} =", "\\underline{\\Lambda}/\\underline{I} = \\underline{\\Lambda/I}$. The system", "of short exact sequences", "$$", "0 \\to \\underline{I/I^n}(U) \\to \\underline{\\Lambda/I^n}(U) \\to", "\\underline{\\Lambda/I}(U) \\to 0", "$$", "has surjective transition maps, hence gives a short exact sequence", "$$", "0 \\to \\lim \\underline{I/I^n}(U) \\to \\lim \\underline{\\Lambda/I^n}(U) \\to", "\\lim \\underline{\\Lambda/I}(U) \\to 0", "$$", "see Homology, Lemma \\ref{homology-lemma-Mittag-Leffler}. Thus we see that", "$\\underline{\\Lambda}^\\wedge/\\underline{I}^\\wedge =", "\\underline{\\Lambda/I}$. Since", "$$", "I \\underline{\\Lambda}^\\wedge \\subset", "\\underline{I} \\cdot \\underline{\\Lambda}^\\wedge \\subset", "\\underline{I}^\\wedge", "$$", "it suffices to show that", "$I \\underline{\\Lambda}^\\wedge(U) = \\underline{I}^\\wedge(U)$", "for all $U$. Choose generators $I = (f_1, \\ldots, f_r)$. This gives a", "short exact sequence $0 \\to K \\to \\Lambda^{\\oplus r} \\to I \\to 0$.", "We obtain short exact sequences", "$$", "0 \\to \\underline{(K \\cap I^n)/I^nK}(U) \\to", "\\underline{(\\Lambda/I^n)^{\\oplus r}}(U) \\to", "\\underline{I/I^n}(U) \\to 0", "$$", "By Artin-Rees (Algebra, Lemma \\ref{algebra-lemma-Artin-Rees})", "the system of modules on the left hand side has ML. (It is", "zero as a pro-object.) Thus we see that", "$(\\underline{\\Lambda}^\\wedge)^{\\oplus r}(U) \\to \\underline{I}^\\wedge(U)$", "is surjective by", "Homology, Lemma \\ref{homology-lemma-Mittag-Leffler}", "which is what we wanted to show." ], "refs": [ "sites-modules-lemma-flatness-presheaves", "sites-modules-lemma-flatness-sheafification", "sites-modules-lemma-flat-sections", "sites-modules-lemma-constant-exact", "more-algebra-lemma-limit-flat", "sites-modules-lemma-tensor-with-finitely-presented", "homology-lemma-Mittag-Leffler", "algebra-lemma-Artin-Rees", "homology-lemma-Mittag-Leffler" ], "ref_ids": [ 14198, 14199, 14267, 14265, 9955, 14266, 12124, 625, 12124 ] } ], "ref_ids": [] }, { "id": 14269, "type": "theorem", "label": "sites-modules-lemma-locally-constant-finite-type", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-locally-constant-finite-type", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a ring and let $M$ be a", "$\\Lambda$-module. Assume $\\Sh(\\mathcal{C})$ is not the empty topos. Then", "\\begin{enumerate}", "\\item $\\underline{M}$ is a finite type sheaf of $\\underline{\\Lambda}$-modules", "if and only if $M$ is a finite $\\Lambda$-module, and", "\\item $\\underline{M}$ is a finitely presented sheaf of", "$\\underline{\\Lambda}$-modules if and only if $M$ is a ", "finitely presented $\\Lambda$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). If $M$ is generated by $x_1, \\ldots, x_r$ then", "$x_1, \\ldots, x_r$ define global sections of $\\underline{M}$ which", "generate it, hence $\\underline{M}$ is of finite type. Conversely, assume", "$\\underline{M}$ is of finite type. Let $U \\in \\mathcal{C}$ be an object", "which is not sheaf theoretically empty", "(Sites, Definition \\ref{sites-definition-empty}).", "Such an object exists as we assumed $\\Sh(\\mathcal{C})$ is not", "the empty topos. Then there exists a covering $\\{U_i \\to U\\}$", "and finitely many sections $s_{ij} \\in \\underline{M}(U_i)$ generating", "$\\underline{M}|_{U_i}$. After refining the covering we may assume", "that $s_{ij}$ come from elements $x_{ij}$ of $M$. Then", "$x_{ij}$ define global sections of $\\underline{M}$ whose restriction", "to $U$ generate $\\underline{M}$.", "\\medskip\\noindent", "Assume there exist elements $x_1, \\ldots, x_r$ of $M$ which define", "global sections of $\\underline{M}$ generating $\\underline{M}$", "as a sheaf of $\\underline{\\Lambda}$-modules. We will show that", "$x_1, \\ldots, x_r$ generate $M$ as a $\\Lambda$-module.", "Let $x \\in M$. We can find a covering $\\{U_i \\to U\\}_{i \\in I}$", "and $f_{i, j} \\in \\underline{\\Lambda}(U_i)$ such that", "$x|_{U_i} = \\sum f_{i, j} x_j|_{U_i}$. After refining the covering", "we may assume $f_{i, j} \\in \\Lambda$. Since $U$ is not", "sheaf theoretically empty we see that $I \\not = \\emptyset$.", "Thus we can pick $i \\in I$ and we see that $x = \\sum f_{i, j}x_j$", "in $M$ as desired.", "\\medskip\\noindent", "Proof of (2). Assume $\\underline{M}$ is a $\\underline{\\Lambda}$-module", "of finite presentation. By (1) we see that $M$ is of finite type.", "Choose generators $x_1, \\ldots, x_r$ of $M$ as a $\\Lambda$-module.", "This determines a short exact sequence", "$0 \\to K \\to \\Lambda^{\\oplus r} \\to M \\to 0$ which", "turns into a short exact sequence", "$$", "0 \\to \\underline{K} \\to \\underline{\\Lambda}^{\\oplus r} \\to \\underline{M} \\to 0", "$$", "by Lemma \\ref{lemma-constant-exact}. By", "Lemma \\ref{lemma-kernel-surjection-finite-onto-finite-presentation}", "we see that $\\underline{K}$ is of finite type. Hence $K$ is a", "finite $\\Lambda$-module by (1). Thus $M$ is a $\\Lambda$-module", "of finite presentation." ], "refs": [ "sites-definition-empty", "sites-modules-lemma-constant-exact", "sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation" ], "ref_ids": [ 8681, 14265, 14187 ] } ], "ref_ids": [] }, { "id": 14270, "type": "theorem", "label": "sites-modules-lemma-pullback-locally-constant", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-pullback-locally-constant", "contents": [ "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be a morphism of topoi.", "If $\\mathcal{G}$ is a locally constant sheaf of", "sets, groups, abelian groups, rings, modules over a fixed ring $\\Lambda$, etc", "on $\\mathcal{D}$, the same is true for $f^{-1}\\mathcal{G}$", "on $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14271, "type": "theorem", "label": "sites-modules-lemma-morphism-locally-constant", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-morphism-locally-constant", "contents": [ "Let $\\mathcal{C}$ be a site with a final object $X$.", "\\begin{enumerate}", "\\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map", "of locally constant sheaves of sets on $\\mathcal{C}$.", "If $\\mathcal{F}$ is finite locally constant, there exists a", "covering $\\{U_i \\to X\\}$ such that", "$\\varphi|_{U_i}$ is the map of constant sheaves associated to", "a map of sets.", "\\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map", "of locally constant sheaves of abelian groups on $\\mathcal{C}$.", "If $\\mathcal{F}$ is finite locally constant, there exists a", "covering $\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is the map of", "constant abelian sheaves associated to a map of abelian groups.", "\\item Let $\\Lambda$ be a ring.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map", "of locally constant sheaves of $\\Lambda$-modules on $\\mathcal{C}$.", "If $\\mathcal{F}$ is of finite type, then there exists a covering", "$\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is the map of constant", "sheaves of $\\Lambda$-modules associated to a map of $\\Lambda$-modules.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14272, "type": "theorem", "label": "sites-modules-lemma-locally-constant", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-locally-constant", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a ring.", "Let $M$, $N$ be $\\Lambda$-modules.", "Let $\\mathcal{F}, \\mathcal{G}$ be a locally constant sheaves of", "$\\Lambda$-modules.", "\\begin{enumerate}", "\\item If $M$ is of finite presentation, then", "$$", "\\underline{\\Hom_\\Lambda(M, N)} =", "\\SheafHom_{\\underline{\\Lambda}}(\\underline{M}, \\underline{N})", "$$", "\\item If $M$ and $N$ are both of finite presentation, then", "$$", "\\underline{\\text{Isom}_\\Lambda(M, N)} =", "\\mathit{Isom}_{\\underline{\\Lambda}}(\\underline{M}, \\underline{N})", "$$", "\\item If $\\mathcal{F}$ is of finite presentation, then", "$\\SheafHom_{\\underline{\\Lambda}}(\\mathcal{F}, \\mathcal{G})$", "is a locally constant sheaf of $\\Lambda$-modules.", "\\item If $\\mathcal{F}$ and $\\mathcal{G}$ are both of finite presentation, then", "$\\mathit{Isom}_{\\underline{\\Lambda}}(\\mathcal{F}, \\mathcal{G})$", "is a locally constant sheaf of sets.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Set $E = \\Hom_\\Lambda(M, N)$. We want to show the canonical map", "$$", "\\underline{E}", "\\longrightarrow", "\\SheafHom_{\\underline{\\Lambda}}(\\underline{M}, \\underline{N})", "$$", "is an isomorphism. The module $M$ has a presentation", "$\\Lambda^{\\oplus s} \\to \\Lambda^{\\oplus t} \\to M \\to 0$.", "Then $E$ sits in an exact sequence", "$$", "0 \\to E \\to \\Hom_\\Lambda(\\Lambda^{\\oplus t}, N) \\to", "\\Hom_\\Lambda(\\Lambda^{\\oplus s}, N)", "$$", "and we have similarly", "$$", "0 \\to", "\\SheafHom_{\\underline{\\Lambda}}(\\underline{M}, \\underline{N}) \\to", "\\SheafHom_{\\underline{\\Lambda}}(\\underline{\\Lambda^{\\oplus t}}, \\underline{N})", "\\to", "\\SheafHom_{\\underline{\\Lambda}}(\\underline{\\Lambda^{\\oplus s}}, \\underline{N})", "$$", "This reduces the question to the case where $M$ is a finite free module", "where the result is clear.", "\\medskip\\noindent", "Proof of (3). The question is local on $\\mathcal{C}$, hence we may assume", "$\\mathcal{F} = \\underline{M}$ and $\\mathcal{G} = \\underline{N}$", "for some $\\Lambda$-modules $M$ and $N$.", "By Lemma \\ref{lemma-locally-constant-finite-type}", "the module $M$ is of finite presentation. Thus the result follows from (1).", "\\medskip\\noindent", "Parts (2) and (4) follow from parts (1) and (3) and the", "fact that $\\mathit{Isom}$ can be viewed as the subsheaf of sections of", "$\\SheafHom_{\\underline{\\Lambda}}(\\mathcal{F}, \\mathcal{G})$", "which have an inverse in", "$\\SheafHom_{\\underline{\\Lambda}}(\\mathcal{G}, \\mathcal{F})$." ], "refs": [ "sites-modules-lemma-locally-constant-finite-type" ], "ref_ids": [ 14269 ] } ], "ref_ids": [] }, { "id": 14273, "type": "theorem", "label": "sites-modules-lemma-kernel-finite-locally-constant", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-kernel-finite-locally-constant", "contents": [ "Let $\\mathcal{C}$ be a site.", "\\begin{enumerate}", "\\item The category of finite locally constant sheaves of sets", "is closed under finite limits and colimits inside $\\Sh(\\mathcal{C})$.", "\\item The category of finite locally constant abelian sheaves is a", "weak Serre subcategory of $\\textit{Ab}(\\mathcal{C})$.", "\\item Let $\\Lambda$ be a Noetherian ring. The category of", "finite type, locally constant sheaves of $\\Lambda$-modules on", "$\\mathcal{C}$ is a weak Serre subcategory of", "$\\textit{Mod}(\\mathcal{C}, \\Lambda)$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). We may work locally on $\\mathcal{C}$. Hence by", "Lemma \\ref{lemma-morphism-locally-constant} we may assume", "we are given a finite diagram of", "finite sets such that our diagram of sheaves is the associated", "diagram of constant sheaves. Then we just take the limit or colimit", "in the category of sets and take the associated constant sheaf.", "Some details omitted.", "\\medskip\\noindent", "To prove (2) and (3) we use the criterion of", "Homology, Lemma \\ref{homology-lemma-characterize-weak-serre-subcategory}.", "Existence of kernels and cokernels is argued in the same way", "as above. Of course, the reason for using", "a Noetherian ring in (3) is to assure us that the kernel of a map", "of finite $\\Lambda$-modules is a finite $\\Lambda$-module.", "To see that the category is closed under extensions", "(in the case of sheaves $\\Lambda$-modules), assume given", "an extension of sheaves of $\\Lambda$-modules", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{E} \\to \\mathcal{G} \\to 0", "$$", "on $\\mathcal{C}$ with $\\mathcal{F}$, $\\mathcal{G}$", "finite type and locally constant. Localizing on $\\mathcal{C}$", "we may assume $\\mathcal{F}$ and $\\mathcal{G}$ are constant, i.e., we", "get", "$$", "0 \\to \\underline{M} \\to \\mathcal{E} \\to \\underline{N} \\to 0", "$$", "for some $\\Lambda$-modules $M, N$. Choose generators $y_1, \\ldots, y_m$", "of $N$, so that we get a short exact sequence", "$0 \\to K \\to \\Lambda^{\\oplus m} \\to N \\to 0$ of $\\Lambda$-modules.", "Localizing further we may assume $y_j$ lifts to a section", "$s_j$ of $\\mathcal{E}$. Thus we see that $\\mathcal{E}$ is a", "pushout as in the following diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\underline{K} \\ar[d] \\ar[r] &", "\\underline{\\Lambda^{\\oplus m}} \\ar[d] \\ar[r] &", "\\underline{N} \\ar[d] \\ar[r] & 0 \\\\", "0 \\ar[r] &", "\\underline{M} \\ar[r] &", "\\mathcal{E} \\ar[r] &", "\\underline{N} \\ar[r] & 0", "}", "$$", "By Lemma \\ref{lemma-morphism-locally-constant} again (and the fact that", "$K$ is a finite $\\Lambda$-module as $\\Lambda$ is Noetherian) we see that", "the map $\\underline{K} \\to \\underline{M}$ is locally constant, hence", "we conclude." ], "refs": [ "sites-modules-lemma-morphism-locally-constant", "homology-lemma-characterize-weak-serre-subcategory", "sites-modules-lemma-morphism-locally-constant" ], "ref_ids": [ 14271, 12046, 14271 ] } ], "ref_ids": [] }, { "id": 14274, "type": "theorem", "label": "sites-modules-lemma-tensor-product-locally-constant", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-tensor-product-locally-constant", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a ring.", "The tensor product of two locally constant sheaves of $\\Lambda$-modules", "on $\\mathcal{C}$ is a locally constant sheaf of $\\Lambda$-modules." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14275, "type": "theorem", "label": "sites-modules-lemma-simple-invert", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-simple-invert", "contents": [ "In the situation above the map to the sheafification", "$$", "\\mathcal{O} \\longrightarrow (\\mathcal{S}^{-1}\\mathcal{O})^\\#", "$$", "is a homomorphism of sheaves of rings with the following", "universal property: for any homomorphism of sheaves of rings", "$\\mathcal{O} \\to \\mathcal{A}$ such that each local section", "of $\\mathcal{S}$ maps to an invertible section of $\\mathcal{A}$", "there exists a unique factorization", "$(\\mathcal{S}^{-1}\\mathcal{O})^\\# \\to \\mathcal{A}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14276, "type": "theorem", "label": "sites-modules-lemma-simple-invert-module", "categories": [ "sites-modules" ], "title": "sites-modules-lemma-simple-invert-module", "contents": [ "In the situation above the map to the sheafification", "$$", "\\mathcal{F} \\longrightarrow (\\mathcal{S}^{-1}\\mathcal{F})^\\#", "$$", "has the following universal property: for any homomorphism", "of $\\mathcal{O}$-modules $\\mathcal{F} \\to \\mathcal{G}$ such", "that each local section of $\\mathcal{S}$ acts invertibly on $\\mathcal{G}$", "there exists a unique factorization", "$(\\mathcal{S}^{-1}\\mathcal{F})^\\# \\to \\mathcal{G}$.", "Moreover we have", "$$", "(\\mathcal{S}^{-1}\\mathcal{F})^\\#", "=", "(\\mathcal{S}^{-1}\\mathcal{O})^\\# \\otimes_\\mathcal{O} \\mathcal{F}", "$$", "as sheaves of $(\\mathcal{S}^{-1}\\mathcal{O})^\\#$-modules." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14313, "type": "theorem", "label": "derham-lemma-base-change-de-rham", "categories": [ "derham" ], "title": "derham-lemma-base-change-de-rham", "contents": [ "Let", "$$", "\\xymatrix{", "X' \\ar[r]_f \\ar[d] & X \\ar[d] \\\\", "S' \\ar[r] & S", "}", "$$", "be a cartesian diagram of schemes. Then the maps discussed", "above induce isomorphisms", "$f^*\\Omega^p_{X/S} \\to \\Omega^p_{X'/S'}$." ], "refs": [], "proofs": [ { "contents": [ "Combine Morphisms, Lemma \\ref{morphisms-lemma-base-change-differentials}", "with the fact that formation of exterior power commutes with base change." ], "refs": [ "morphisms-lemma-base-change-differentials" ], "ref_ids": [ 5314 ] } ], "ref_ids": [] }, { "id": 14314, "type": "theorem", "label": "derham-lemma-etale", "categories": [ "derham" ], "title": "derham-lemma-etale", "contents": [ "Consider a commutative diagram of schemes", "$$", "\\xymatrix{", "X' \\ar[r]_f \\ar[d] & X \\ar[d] \\\\", "S' \\ar[r] & S", "}", "$$", "If $X' \\to X$ and $S' \\to S$ are \\'etale, then the maps discussed", "above induce isomorphisms", "$f^*\\Omega^p_{X/S} \\to \\Omega^p_{X'/S'}$." ], "refs": [], "proofs": [ { "contents": [ "We have $\\Omega_{S'/S} = 0$ and $\\Omega_{X'/X} = 0$, see for example", "Morphisms, Lemma \\ref{morphisms-lemma-etale-at-point}. Then by", "the short exact sequences of Morphisms, Lemmas", "\\ref{morphisms-lemma-triangle-differentials} and", "\\ref{morphisms-lemma-triangle-differentials-smooth}", "we see that $\\Omega_{X'/S'} = \\Omega_{X'/S} = f^*\\Omega_{X/S}$.", "Taking exterior powers we conclude." ], "refs": [ "morphisms-lemma-etale-at-point", "morphisms-lemma-triangle-differentials", "morphisms-lemma-triangle-differentials-smooth" ], "ref_ids": [ 5372, 5313, 5337 ] } ], "ref_ids": [] }, { "id": 14315, "type": "theorem", "label": "derham-lemma-de-rham-affine", "categories": [ "derham" ], "title": "derham-lemma-de-rham-affine", "contents": [ "Let $X \\to S$ be a morphism of affine schemes given by the ring map", "$R \\to A$. Then $R\\Gamma(X, \\Omega^\\bullet_{X/S}) = \\Omega^\\bullet_{A/R}$", "in $D(R)$ and $H^i_{dR}(X/S) = H^i(\\Omega^\\bullet_{A/R})$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}", "and Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})." ], "refs": [ "coherent-lemma-quasi-coherent-affine-cohomology-zero", "derived-lemma-leray-acyclicity" ], "ref_ids": [ 3282, 1844 ] } ], "ref_ids": [] }, { "id": 14316, "type": "theorem", "label": "derham-lemma-quasi-coherence-relative", "categories": [ "derham" ], "title": "derham-lemma-quasi-coherence-relative", "contents": [ "Let $p : X \\to S$ be a morphism of schemes. If $p$ is quasi-compact", "and quasi-separated, then $Rp_*\\Omega^\\bullet_{X/S}$ is an object", "of $D_\\QCoh(\\mathcal{O}_S)$." ], "refs": [], "proofs": [ { "contents": [ "There is a spectral sequence with first page", "$E_1^{a, b} = R^ap_*\\Omega^q_{X/S}$ converging to $Rp_*\\Omega^\\bullet_{X/S}$", "(see Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}).", "Hence by Homology, Lemma \\ref{homology-lemma-first-quadrant-ss}", "it suffices to show that $R^ap_*\\Omega^q_{X/S}$ is quasi-coherent.", "This follows from Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images}." ], "refs": [ "derived-lemma-two-ss-complex-functor", "homology-lemma-first-quadrant-ss", "coherent-lemma-quasi-coherence-higher-direct-images" ], "ref_ids": [ 1871, 12105, 3295 ] } ], "ref_ids": [] }, { "id": 14317, "type": "theorem", "label": "derham-lemma-coherence-relative", "categories": [ "derham" ], "title": "derham-lemma-coherence-relative", "contents": [ "Let $p : X \\to S$ be a proper morphism of schemes with $S$ locally", "Noetherian. Then $Rp_*\\Omega^\\bullet_{X/S}$ is an object", "of $D_{\\textit{Coh}}(\\mathcal{O}_S)$." ], "refs": [], "proofs": [ { "contents": [ "In this case by Morphisms, Lemma \\ref{morphisms-lemma-finite-type-differentials}", "the modules $\\Omega^i_{X/S}$ are coherent. Hence we can use exactly the", "same argument as in the proof of Lemma \\ref{lemma-quasi-coherence-relative}", "using Cohomology of Schemes, Proposition", "\\ref{coherent-proposition-proper-pushforward-coherent}." ], "refs": [ "morphisms-lemma-finite-type-differentials", "derham-lemma-quasi-coherence-relative", "coherent-proposition-proper-pushforward-coherent" ], "ref_ids": [ 5316, 14316, 3401 ] } ], "ref_ids": [] }, { "id": 14318, "type": "theorem", "label": "derham-lemma-finite-de-Rham", "categories": [ "derham" ], "title": "derham-lemma-finite-de-Rham", "contents": [ "Let $A$ be a Noetherian ring. Let $X$ be a proper scheme over $S = \\Spec(A)$.", "Then $H^i_{dR}(X/S)$ is a finite $A$-module for all $i$." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-coherence-relative}." ], "refs": [ "derham-lemma-coherence-relative" ], "ref_ids": [ 14317 ] } ], "ref_ids": [] }, { "id": 14319, "type": "theorem", "label": "derham-lemma-proper-smooth-de-Rham", "categories": [ "derham" ], "title": "derham-lemma-proper-smooth-de-Rham", "contents": [ "Let $f : X \\to S$ be a proper smooth morphism of schemes. Then", "$Rf_*\\Omega^p_{X/S}$, $p \\geq 0$ and $Rf_*\\Omega^\\bullet_{X/S}$ are", "perfect objects of $D(\\mathcal{O}_S)$ whose formation commutes", "with arbitrary change of base." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is smooth the modules $\\Omega^p_{X/S}$ are finite locally", "free $\\mathcal{O}_X$-modules, see Morphisms, Lemma", "\\ref{morphisms-lemma-smooth-omega-finite-locally-free}. Their", "formation commutes with arbitrary change of base by", "Lemma \\ref{lemma-base-change-de-rham}. Hence", "$Rf_*\\Omega^p_{X/S}$ is a perfect object of $D(\\mathcal{O}_S)$", "whose formation commutes with abitrary base change, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-perfect-direct-image-general}.", "This proves the first assertion of the lemma.", "\\medskip\\noindent", "To prove that $Rf_*\\Omega^\\bullet_{X/S}$ is perfect on $S$ we may work", "locally on $S$. Thus we may assume $S$ is quasi-compact. This means", "we may assume that $\\Omega^n_{X/S}$ is zero for $n$ large enough.", "For every $p \\geq 0$ we claim that", "$Rf_*\\sigma_{\\geq p}\\Omega^\\bullet_{X/S}$ is a", "perfect object of $D(\\mathcal{O}_S)$ whose formation commutes", "with arbitrary change of base. By the above we see that", "this is true for $p \\gg 0$. Suppose the claim holds for $p$", "and consider the distinguished triangle", "$$", "\\sigma_{\\geq p}\\Omega^\\bullet_{X/S} \\to", "\\sigma_{\\geq p - 1}\\Omega^\\bullet_{X/S} \\to", "\\Omega^{p - 1}_{X/S}[-(p - 1)] \\to", "(\\sigma_{\\geq p}\\Omega^\\bullet_{X/S})[1]", "$$", "in $D(f^{-1}\\mathcal{O}_S)$.", "Applying the exact functor $Rf_*$ we obtain a distinguished triangle", "in $D(\\mathcal{O}_S)$.", "Since we have the 2-out-of-3 property for being perfect", "(Cohomology, Lemma \\ref{cohomology-lemma-two-out-of-three-perfect})", "we conclude $Rf_*\\sigma_{\\geq p - 1}\\Omega^\\bullet_{X/S}$ is a", "perfect object of $D(\\mathcal{O}_S)$. Similarly for the", "commutation with arbitrary base change." ], "refs": [ "morphisms-lemma-smooth-omega-finite-locally-free", "derham-lemma-base-change-de-rham", "perfect-lemma-flat-proper-perfect-direct-image-general", "cohomology-lemma-two-out-of-three-perfect" ], "ref_ids": [ 5334, 14313, 7054, 2226 ] } ], "ref_ids": [] }, { "id": 14320, "type": "theorem", "label": "derham-lemma-cup-product-graded-commutative", "categories": [ "derham" ], "title": "derham-lemma-cup-product-graded-commutative", "contents": [ "Let $p : X \\to S$ be a morphism of schemes.", "The cup product on $H^*_{dR}(X/S)$ is associative and", "graded commutative." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Cohomology, Lemmas \\ref{cohomology-lemma-cup-product-associative} and", "\\ref{cohomology-lemma-cup-product-commutative}", "and the fact that $\\wedge$ is associative and graded commutative." ], "refs": [ "cohomology-lemma-cup-product-associative", "cohomology-lemma-cup-product-commutative" ], "ref_ids": [ 2131, 2132 ] } ], "ref_ids": [] }, { "id": 14321, "type": "theorem", "label": "derham-lemma-cup-product-hodge-graded-commutative", "categories": [ "derham" ], "title": "derham-lemma-cup-product-hodge-graded-commutative", "contents": [ "Let $p : X \\to S$ be a morphism of schemes.", "The cup product on $H^*_{Hodge}(X/S)$ is associative and graded commutative." ], "refs": [], "proofs": [ { "contents": [ "The proof is identical to the proof of", "Lemma \\ref{lemma-cup-product-graded-commutative}." ], "refs": [ "derham-lemma-cup-product-graded-commutative" ], "ref_ids": [ 14320 ] } ], "ref_ids": [] }, { "id": 14322, "type": "theorem", "label": "derham-lemma-de-rham-complex-product", "categories": [ "derham" ], "title": "derham-lemma-de-rham-complex-product", "contents": [ "In the situation above there is a canonical isomorphism", "$$", "\\text{Tot}(\\Omega^\\bullet_{X/S} \\boxtimes \\Omega^\\bullet_{Y/S})", "\\longrightarrow", "\\Omega^\\bullet_{X \\times_S Y/S}", "$$", "of complexes of $f^{-1}\\mathcal{O}_S$-modules." ], "refs": [], "proofs": [ { "contents": [ "We know that", "$", "\\Omega_{X \\times_S Y/S} = p^*\\Omega_{X/S} \\oplus q^*\\Omega_{Y/S}", "$", "by Morphisms, Lemma \\ref{morphisms-lemma-differential-product}.", "Taking exterior powers we obtain", "$$", "\\Omega^n_{X \\times_S Y/S} =", "\\bigoplus\\nolimits_{i + j = n}", "p^*\\Omega^i_{X/S} \\otimes_{\\mathcal{O}_{X \\times_S Y}} q^*\\Omega^j_{Y/S} =", "\\bigoplus\\nolimits_{i + j = n}", "\\Omega^i_{X/S} \\boxtimes \\Omega^j_{Y/S}", "$$", "by elementary properties of exterior powers. These identifications determine", "isomorphisms between the terms of the complexes on the left and the right", "of the arrow in the lemma. We omit the verification that these maps", "are compatible with differentials." ], "refs": [ "morphisms-lemma-differential-product" ], "ref_ids": [ 5315 ] } ], "ref_ids": [] }, { "id": 14323, "type": "theorem", "label": "derham-lemma-kunneth-de-rham", "categories": [ "derham" ], "title": "derham-lemma-kunneth-de-rham", "contents": [ "Assume $X$ and $Y$ are smooth, quasi-compact, with affine diagonal over", "$S = \\Spec(A)$. Then the map", "$$", "R\\Gamma(X, \\Omega^\\bullet_{X/S})", "\\otimes_A^\\mathbf{L}", "R\\Gamma(Y, \\Omega^\\bullet_{Y/S})", "\\longrightarrow", "R\\Gamma(X \\times_S Y, \\Omega^\\bullet_{X \\times_S Y/S})", "$$", "is an isomorphism in $D(A)$." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms, Lemma \\ref{morphisms-lemma-smooth-omega-finite-locally-free}", "the sheaves $\\Omega^n_{X/S}$ and $\\Omega^m_{Y/S}$ are finite locally free", "$\\mathcal{O}_X$ and $\\mathcal{O}_Y$-modules. On the other hand, $X$ and $Y$", "are flat over $S$ (Morphisms, Lemma \\ref{morphisms-lemma-smooth-flat})", "and hence we find that $\\Omega^n_{X/S}$ and $\\Omega^m_{Y/S}$ are flat over $S$.", "Also, observe that $\\Omega^\\bullet_{X/S}$ is a locally bounded. Thus", "the result by Lemma \\ref{lemma-de-rham-complex-product} and", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-kunneth-special}." ], "refs": [ "morphisms-lemma-smooth-omega-finite-locally-free", "morphisms-lemma-smooth-flat", "derham-lemma-de-rham-complex-product", "perfect-lemma-kunneth-special" ], "ref_ids": [ 5334, 5331, 14322, 7036 ] } ], "ref_ids": [] }, { "id": 14324, "type": "theorem", "label": "derham-lemma-kunneth-de-rham-relative", "categories": [ "derham" ], "title": "derham-lemma-kunneth-de-rham-relative", "contents": [ "Assume $X \\to S$ and $Y \\to S$ are smooth and quasi-compact", "and the morphisms $X \\to X \\times_S X$ and $Y \\to Y \\times_S Y$ are affine.", "Then the relative cup product", "$$", "Ra_*\\Omega^\\bullet_{X/S}", "\\otimes_{\\mathcal{O}_S}^\\mathbf{L}", "Rb_*\\Omega^\\bullet_{Y/S}", "\\longrightarrow", "Rf_*\\Omega^\\bullet_{X \\times_S Y/S}", "$$", "is an isomorphism in $D(\\mathcal{O}_S)$." ], "refs": [], "proofs": [ { "contents": [ "Immediate consequence of Lemma \\ref{lemma-kunneth-de-rham}." ], "refs": [ "derham-lemma-kunneth-de-rham" ], "ref_ids": [ 14323 ] } ], "ref_ids": [] }, { "id": 14325, "type": "theorem", "label": "derham-lemma-pullback-c1", "categories": [ "derham" ], "title": "derham-lemma-pullback-c1", "contents": [ "Given a commutative diagram", "$$", "\\xymatrix{", "X' \\ar[r]_f \\ar[d] & X \\ar[d] \\\\", "S' \\ar[r] & S", "}", "$$", "of schemes the diagrams", "$$", "\\xymatrix{", "\\Pic(X') \\ar[d]_{c_1^{dR}} &", "\\Pic(X) \\ar[d]^{c_1^{dR}} \\ar[l]^{f^*} \\\\", "H^2_{dR}(X'/S') &", "H^2_{dR}(X/S) \\ar[l]_{f^*}", "}", "\\quad", "\\xymatrix{", "\\Pic(X') \\ar[d]_{c_1^{Hodge}} &", "\\Pic(X) \\ar[d]^{c_1^{Hodge}} \\ar[l]^{f^*} \\\\", "H^1(X', \\Omega^1_{X'/S'}) &", "H^1(X, \\Omega^1_{X/S}) \\ar[l]_{f^*}", "}", "$$", "commute." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14326, "type": "theorem", "label": "derham-lemma-the-complex-for-L-star", "categories": [ "derham" ], "title": "derham-lemma-the-complex-for-L-star", "contents": [ "With notation as above, there is a short exact sequence of complexes", "$$", "0 \\to \\Omega^\\bullet_{X/S} \\to", "\\Omega^\\bullet_{L^\\star/S, 0} \\to", "\\Omega^\\bullet_{X/S}[-1] \\to 0", "$$" ], "refs": [], "proofs": [ { "contents": [ "We have constructed the map", "$\\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{L^\\star/S, 0}$ above.", "\\medskip\\noindent", "Construction of", "$\\text{Res} : \\Omega^\\bullet_{L^\\star/S, 0} \\to \\Omega^\\bullet_{X/S}[-1]$.", "Let $U \\subset X$ be an open and let $s \\in \\mathcal{L}(U)$", "and $s' \\in \\mathcal{L}^{\\otimes -1}(U)$ be sections such that", "$s' s = 1$. Then $s$ gives an invertible section of the sheaf of", "algebras $(L^\\star \\to X)_*\\mathcal{O}_{L^\\star}$ over $U$", "with inverse $s' = s^{-1}$. Then we can consider the $1$-form", "$\\text{d}\\log(s) = s' \\text{d}(s)$ which is an element of", "$\\Omega^1_{L^\\star/S, 0}(U)$ by our construction of the grading on", "$\\Omega^1_{L^\\star/S}$. Our computations on affines given below", "will show that $1$ and $\\text{d}\\log(s)$ freely generate", "$\\Omega^\\bullet_{L^\\star/S, 0}|_U$ as a right module over", "$\\Omega^\\bullet_{X/S}|_U$.", "Thus we can define $\\text{Res}$ over $U$ by the rule", "$$", "\\text{Res}(\\omega' + \\text{d}\\log(s) \\wedge \\omega) = \\omega", "$$", "for all $\\omega', \\omega \\in \\Omega^\\bullet_{X/S}(U)$. This", "map is independent of the choice of local generator $s$ and hence", "glues to give a global map. Namely, another choice of $s$", "would be of the form $gs$ for some invertible $g \\in \\mathcal{O}_X(U)$", "and we would get $\\text{d}\\log(gs) = g^{-1}\\text{d}(g) + \\text{d}\\log(s)$", "from which the independence easily follows.", "Finally, observe that our rule for $\\text{Res}$", "is compatible with differentials", "as $\\text{d}(\\omega' + \\text{d}\\log(s) \\wedge \\omega) =", "\\text{d}(\\omega') - \\text{d}\\log(s) \\wedge \\text{d}(\\omega)$", "and because the differential on $\\Omega^\\bullet_{X/S}[-1]$", "sends $\\omega'$ to $-\\text{d}(\\omega')$ by our sign convention in", "Homology, Definition \\ref{homology-definition-shift-cochain}.", "\\medskip\\noindent", "Local computation. We can cover $X$ by affine opens $U \\subset X$", "such that $\\mathcal{L}|_U \\cong \\mathcal{O}_U$ which moreover map", "into an affine open $V \\subset S$. Write $U = \\Spec(A)$, $V = \\Spec(R)$", "and choose a generator $s$ of $\\mathcal{L}$. We find that we have", "$$", "L^\\star \\times_X U = \\Spec(A[s, s^{-1}])", "$$", "Computing differentials we see that", "$$", "\\Omega^1_{A[s, s^{-1}]/R} =", "A[s, s^{-1}] \\otimes_A \\Omega^1_{A/R} \\oplus A[s, s^{-1}] \\text{d}\\log(s)", "$$", "and therefore taking exterior powers we obtain", "$$", "\\Omega^p_{A[s, s^{-1}]/R} =", "A[s, s^{-1}] \\otimes_A \\Omega^p_{A/R}", "\\oplus", "A[s, s^{-1}] \\text{d}\\log(s) \\otimes_A \\Omega^{p - 1}_{A/R}", "$$", "Taking degree $0$ parts we find", "$$", "\\Omega^p_{A[s, s^{-1}]/R, 0} =", "\\Omega^p_{A/R} \\oplus \\text{d}\\log(s) \\otimes_A \\Omega^{p - 1}_{A/R}", "$$", "and the proof of the lemma is complete." ], "refs": [ "homology-definition-shift-cochain" ], "ref_ids": [ 12158 ] } ], "ref_ids": [] }, { "id": 14327, "type": "theorem", "label": "derham-lemma-the-complex-for-L-star-gives-chern-class", "categories": [ "derham" ], "title": "derham-lemma-the-complex-for-L-star-gives-chern-class", "contents": [ "The ``boundary'' map", "$\\delta : \\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}[2]$", "in $D(X, f^{-1}\\mathcal{O}_S)$ coming from", "the short exact sequence in Lemma \\ref{lemma-the-complex-for-L-star}", "is the map of Remark \\ref{remark-cup-product-as-a-map}", "for $\\xi = c_1^{dR}(\\mathcal{L})$." ], "refs": [ "derham-lemma-the-complex-for-L-star", "derham-remark-cup-product-as-a-map" ], "proofs": [ { "contents": [ "To be precise we consider the shift", "$$", "0 \\to \\Omega^\\bullet_{X/S}[1] \\to", "\\Omega^\\bullet_{L^\\star/S, 0}[1] \\to", "\\Omega^\\bullet_{X/S} \\to 0", "$$", "of the short exact sequence of Lemma \\ref{lemma-the-complex-for-L-star}.", "As the degree zero part of a grading on", "$(L^\\star \\to X)_*\\Omega^\\bullet_{L^\\star/S}$", "we see that $\\Omega^\\bullet_{L^\\star/S, 0}$ is a differential", "graded $\\mathcal{O}_X$-algebra and that the map", "$\\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{L^\\star/S, 0}$", "is a homomorphism of differential graded $\\mathcal{O}_X$-algebras.", "Hence we may view $\\Omega^\\bullet_{X/S}[1] \\to", "\\Omega^\\bullet_{L^\\star/S, 0}[1]$ as a map of right differential graded", "$\\Omega^\\bullet_{X/S}$-modules on $X$. The map", "$\\text{Res} : \\Omega^\\bullet_{L^\\star/S, 0}[1] \\to \\Omega^\\bullet_{X/S}$", "is a map of right differential graded $\\Omega^\\bullet_{X/S}$-modules", "since it is locally defined by the rule", "$\\text{Res}(\\omega' + \\text{d}\\log(s) \\wedge \\omega) = \\omega$, see", "proof of Lemma \\ref{lemma-the-complex-for-L-star}.", "Thus by the discussion in", "Differential Graded Sheaves, Section \\ref{sdga-section-misc}", "we see that $\\delta$ comes from a map", "$\\delta' : \\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}[2]$", "in the derived category $D(\\Omega^\\bullet_{X/S}, \\text{d})$", "of right differential graded modules over the de Rham complex.", "The uniqueness averted in Remark \\ref{remark-cup-product-as-a-map}", "shows it suffices to prove that $\\delta(1) = c_1^{dR}(\\mathcal{L})$.", "\\medskip\\noindent", "We claim that there is a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{O}_X^* \\ar[r] \\ar[d]_{\\text{d}\\log} &", "E \\ar[r] \\ar[d] &", "\\underline{\\mathbf{Z}} \\ar[d] \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "\\Omega^\\bullet_{X/S}[1] \\ar[r] &", "\\Omega^\\bullet_{L^\\star/S, 0}[1] \\ar[r] &", "\\Omega^\\bullet_{X/S} \\ar[r] &", "0", "}", "$$", "where the top row is a short exact sequence of abelian sheaves whose", "boundary map sends $1$ to the class of $\\mathcal{L}$ in", "$H^1(X, \\mathcal{O}_X^*)$. It suffices to prove the claim", "by the compatibility of boundary maps with maps between short", "exact sequences. We define $E$ as the sheafification of the rule", "$$", "U \\longmapsto \\{(s, n) \\mid", "n \\in \\mathbf{Z},\\ s \\in \\mathcal{L}^{\\otimes n}(U)\\text{ generator}\\}", "$$", "with group structure given by $(s, n) \\cdot (t, m) = (s \\otimes t, n + m)$.", "The middle vertical map sends $(s, n)$ to $\\text{d}\\log(s)$. This produces", "a map of short exact sequences", "because the map $Res : \\Omega^1_{L^\\star/S, 0} \\to \\mathcal{O}_X$", "constructed in the proof of Lemma \\ref{lemma-the-complex-for-L-star} sends", "$\\text{d}\\log(s)$ to $1$ if $s$ is a local generator of $\\mathcal{L}$.", "To calculate the boundary of $1$ in the top row, choose local trivializations", "$s_i$ of $\\mathcal{L}$ over opens $U_i$ as in", "Section \\ref{section-first-chern-class}. On the overlaps", "$U_{i_0i_1} = U_{i_0} \\cap U_{i_1}$", "we have an invertible function $f_{i_0i_1}$ such that", "$f_{i_0i_1} = s_{i_1}|_{U_{i_0i_1}} s_{i_0}|_{U_{i_0i_1}}^{-1}$", "and the cohomology class of $\\mathcal{L}$ is given by the {\\v C}ech cocycle", "$\\{f_{i_0i_1}\\}$. Then of course we have", "$$", "(f_{i_0i_1}, 0) = (s_{i_1}, 1)|_{U_{i_0i_1}} \\cdot", "(s_{i_0}, 1)|_{U_{i_0i_1}}^{-1}", "$$", "as sections of $E$ which finishes the proof." ], "refs": [ "derham-lemma-the-complex-for-L-star", "derham-lemma-the-complex-for-L-star", "derham-remark-cup-product-as-a-map", "derham-lemma-the-complex-for-L-star" ], "ref_ids": [ 14326, 14326, 14382, 14326 ] } ], "ref_ids": [ 14326, 14382 ] }, { "id": 14328, "type": "theorem", "label": "derham-lemma-push-omega-a", "categories": [ "derham" ], "title": "derham-lemma-push-omega-a", "contents": [ "With notation as above we have", "\\begin{enumerate}", "\\item $\\Omega^p_{L^\\star/S, n} =", "\\Omega^p_{L^\\star/S, 0} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}$", "for all $n \\in \\mathbf{Z}$ as quasi-coherent $\\mathcal{O}_X$-modules,", "\\item $\\Omega^\\bullet_{X/S} = \\Omega^\\bullet_{L/X, 0}$", "as complexes, and", "\\item for $n > 0$ and $p \\geq 0$ we have", "$\\Omega^p_{L/X, n} = \\Omega^p_{L^\\star/S, n}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "In each case there is a globally defined canonical map which", "is an isomorphism by local calculations which we omit." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14329, "type": "theorem", "label": "derham-lemma-line-bundle-characteristic-zero", "categories": [ "derham" ], "title": "derham-lemma-line-bundle-characteristic-zero", "contents": [ "In the situation above, assume there is a morphism $S \\to \\Spec(\\mathbf{Q})$.", "Then $\\Omega^\\bullet_{X/S} \\to \\pi_*\\Omega^\\bullet_{L/S}$ is a", "quasi-isomorphism and $H_{dR}^*(X/S) = H_{dR}^*(L/S)$." ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a $\\mathbf{Q}$-algebra. Let $A$ be an $R$-algebra.", "The affine local statement is that the map", "$$", "\\Omega^\\bullet_{A/R} \\longrightarrow \\Omega^\\bullet_{A[t]/R}", "$$", "is a quasi-isomorphism of complexes of $R$-modules. In fact it is a", "homotopy equivalence with homotopy inverse given by the map sending", "$g \\omega + g' \\text{d}t \\wedge \\omega'$ to $g(0)\\omega$ for", "$g, g' \\in A[t]$ and $\\omega, \\omega' \\in \\Omega^\\bullet_{A/R}$.", "The homotopy sends $g \\omega + g' \\text{d}t \\wedge \\omega'$", "to $(\\int g') \\omega'$ were $\\int g' \\in A[t]$ is the polynomial", "with vanishing constant term whose derivative with respect to $t$", "is $g'$. Of course, here we use that $R$ contains $\\mathbf{Q}$", "as $\\int t^n = (1/n)t^{n + 1}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14330, "type": "theorem", "label": "derham-lemma-euler-sequence", "categories": [ "derham" ], "title": "derham-lemma-euler-sequence", "contents": [ "There exists a short exact sequence", "$$", "0 \\to \\Omega \\to \\mathcal{O}(-1)^{\\oplus n + 1} \\to \\mathcal{O} \\to 0", "$$" ], "refs": [], "proofs": [ { "contents": [ "To explain this, we recall that", "$\\mathbf{P}^n_A = \\text{Proj}(A[T_0, \\ldots, T_n])$,", "and we write symbolically", "$$", "\\mathcal{O}(-1)^{\\oplus n + 1} =", "\\bigoplus\\nolimits_{j = 0, \\ldots, n} \\mathcal{O}(-1) \\text{d}T_j", "$$", "The first arrow", "$$", "\\Omega \\to", "\\bigoplus\\nolimits_{j = 0, \\ldots, n} \\mathcal{O}(-1) \\text{d}T_j", "$$", "in the short exact sequence above", "is given on each of the standard opens", "$D_+(T_i) = \\Spec(A[T_0/T_i, \\ldots, T_n/T_i])$", "mentioned above by the rule", "$$", "\\sum\\nolimits_{j \\not = i} g_j \\text{d}(T_j/T_i)", "\\longmapsto", "\\sum\\nolimits_{j \\not = i} g_j/T_i \\text{d}T_j", "- (\\sum\\nolimits_{j \\not = i} g_jT_j/T_i^2) \\text{d}T_i", "$$", "This makes sense because $1/T_i$ is a section of $\\mathcal{O}(-1)$", "over $D_+(T_i)$. The map", "$$", "\\bigoplus\\nolimits_{j = 0, \\ldots, n} \\mathcal{O}(-1) \\text{d}T_j", "\\to", "\\mathcal{O}", "$$", "is given by sending $\\text{d}T_j$ to $T_j$, more precisely, on", "$D_+(T_i)$ we send the section $\\sum g_j \\text{d}T_j$ to", "$\\sum T_jg_j$. We omit the verification that this produces", "a short exact sequence." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14331, "type": "theorem", "label": "derham-lemma-twisted-hodge-cohomology-projective-space", "categories": [ "derham" ], "title": "derham-lemma-twisted-hodge-cohomology-projective-space", "contents": [ "In the situation above we have the following cohomology groups", "\\begin{enumerate}", "\\item $H^q(\\mathbf{P}^n_A, \\Omega^p) = 0$", "unless $0 \\leq p = q \\leq n$,", "\\item for $0 \\leq p \\leq n$ the $A$-module", "$H^p(\\mathbf{P}^n_A, \\Omega^p)$ free of rank $1$.", "\\item for $q > 0$, $k > 0$, and $p$ arbitrary we have", "$H^q(\\mathbf{P}^n_A, \\Omega^p(k)) = 0$, and", "\\item add more here.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We are going to use the results of Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cohomology-projective-space-over-ring}", "without further mention. In particular, the statements are true", "for $H^q(\\mathbf{P}^n_A, \\mathcal{O}(k))$.", "\\medskip\\noindent", "Proof for $p = 1$. Consider the short exact sequence", "$$", "0 \\to \\Omega \\to \\mathcal{O}(-1)^{\\oplus n + 1} \\to \\mathcal{O} \\to 0", "$$", "of Lemma \\ref{lemma-euler-sequence}. Since $\\mathcal{O}(-1)$ has", "vanishing cohomology in all degrees, this gives that", "$H^q(\\mathbf{P}^n_A, \\Omega)$ is zero except in degree $1$", "where it is freely generated by the boundary of $1$ in", "$H^0(\\mathbf{P}^n_A, \\mathcal{O})$.", "\\medskip\\noindent", "Assume $p > 1$. Let us think of the short exact sequence", "above as defining a $2$ step filtration on $\\mathcal{O}(-1)^{\\oplus n + 1}$.", "The induced filtration on $\\wedge^p\\mathcal{O}(-1)^{\\oplus n + 1}$ looks", "like this", "$$", "0 \\to \\Omega^p \\to \\wedge^p\\left(\\mathcal{O}(-1)^{\\oplus n + 1}\\right)", "\\to \\Omega^{p - 1} \\to 0", "$$", "Observe that $\\wedge^p\\mathcal{O}(-1)^{\\oplus n + 1}$ is isomorphic", "to a direct sum of $n + 1$ choose $p$ copies of $\\mathcal{O}(-p)$", "and hence has vanishing cohomology in all degrees.", "By induction hypothesis, this shows that $H^q(\\mathbf{P}^n_A, \\Omega^p)$", "is zero unless $q = p$ and $H^p(\\mathbf{P}^n_A, \\Omega^p)$ is free", "of rank $1$ with generator the boundary of the generator in", "$H^{p - 1}(\\mathbf{P}^n_A, \\Omega^{p - 1})$.", "\\medskip\\noindent", "Let $k > 0$. Observe that $\\Omega^n = \\mathcal{O}(-n - 1)$ for example", "by the short exact sequence above for $p = n + 1$.", "Hence $\\Omega^n(k)$ has vanishing cohomology in positive degrees.", "Using the short exact sequences", "$$", "0 \\to \\Omega^p(k) \\to \\wedge^p\\left(\\mathcal{O}(-1)^{\\oplus n + 1}\\right)(k)", "\\to \\Omega^{p - 1}(k) \\to 0", "$$", "and {\\it descending} induction on $p$ we get the vanishing of", "cohomology of $\\Omega^p(k)$ in positive degrees for all $p$." ], "refs": [ "coherent-lemma-cohomology-projective-space-over-ring", "derham-lemma-euler-sequence" ], "ref_ids": [ 3304, 14330 ] } ], "ref_ids": [] }, { "id": 14332, "type": "theorem", "label": "derham-lemma-hodge-cohomology-projective-space", "categories": [ "derham" ], "title": "derham-lemma-hodge-cohomology-projective-space", "contents": [ "We have $H^q(\\mathbf{P}^n_A, \\Omega^p) = 0$", "unless $0 \\leq p = q \\leq n$. For $0 \\leq p \\leq n$ the $A$-module", "$H^p(\\mathbf{P}^n_A, \\Omega^p)$ free of rank $1$ with basis element", "$c_1^{Hodge}(\\mathcal{O}(1))^p$." ], "refs": [], "proofs": [ { "contents": [ "We have the vanishing and and freeness by", "Lemma \\ref{lemma-twisted-hodge-cohomology-projective-space}.", "For $p = 0$ it is certainly true that", "$1 \\in H^0(\\mathbf{P}^n_A, \\mathcal{O})$ is a generator.", "\\medskip\\noindent", "Proof for $p = 1$. Consider the short exact sequence", "$$", "0 \\to \\Omega \\to \\mathcal{O}(-1)^{\\oplus n + 1} \\to \\mathcal{O} \\to 0", "$$", "of Lemma \\ref{lemma-euler-sequence}. In the proof of", "Lemma \\ref{lemma-twisted-hodge-cohomology-projective-space}", "we have seen that the generator of $H^1(\\mathbf{P}^n_A, \\Omega)$", "is the boundary $\\xi$ of $1 \\in H^0(\\mathbf{P}^n_A, \\mathcal{O})$.", "As in the proof of Lemma \\ref{lemma-euler-sequence} we will identify", "$\\mathcal{O}(-1)^{\\oplus n + 1}$ with", "$\\bigoplus_{j = 0, \\ldots, n} \\mathcal{O}(-1)\\text{d}T_j$.", "Consider the open covering", "$$", "\\mathcal{U} : ", "\\mathbf{P}^n_A =", "\\bigcup\\nolimits_{i = 0, \\ldots, n} D_{+}(T_i)", "$$", "We can lift the restriction of the global section $1$ of $\\mathcal{O}$", "to $U_i = D_+(T_i)$ by the section $T_i^{-1} \\text{d}T_i$ of", "$\\bigoplus \\mathcal{O}(-1)\\text{d}T_j$ over $U_i$. Thus the cocyle", "representing $\\xi$ is given by", "$$", "T_{i_1}^{-1} \\text{d}T_{i_1} - T_{i_0}^{-1} \\text{d}T_{i_0} =", "\\text{d}\\log(T_{i_1}/T_{i_0}) \\in \\Omega(U_{i_0i_1})", "$$", "On the other hand, for each $i$ the section $T_i$ is a trivializing", "section of $\\mathcal{O}(1)$ over $U_i$. Hence we see that", "$f_{i_0i_1} = T_{i_1}/T_{i_0} \\in \\mathcal{O}^*(U_{i_0i_1})$", "is the cocycle representing $\\mathcal{O}(1)$ in $\\Pic(\\mathbf{P}^n_A)$,", "see Section \\ref{section-first-chern-class}.", "Hence $c_1^{Hodge}(\\mathcal{O}(1))$", "is given by the cocycle $\\text{d}\\log(T_{i_1}/T_{i_0})$", "which agrees with what we got for $\\xi$ above.", "\\medskip\\noindent", "Proof for general $p$ by induction. The base cases $p = 0, 1$ were handled", "above. Assume $p > 1$. In the proof of", "Lemma \\ref{lemma-twisted-hodge-cohomology-projective-space}", "we have seen that the generator of $H^p(\\mathbf{P}^n_A, \\Omega^p)$", "is the boundary of $c_1^{Hodge}(\\mathcal{O}(1))^{p - 1}$", "in the long exact cohomology sequence associated to", "$$", "0 \\to \\Omega^p \\to \\wedge^p\\left(\\mathcal{O}(-1)^{\\oplus n + 1}\\right)", "\\to \\Omega^{p - 1} \\to 0", "$$", "By the calculation in Section \\ref{section-first-chern-class}", "the cohomology class $c_1^{Hodge}(\\mathcal{O}(1))^{p - 1}$", "is, up to a sign, represented by the cocycle with terms", "$$", "\\beta_{i_0 \\ldots i_{p - 1}} =", "\\text{d}\\log(T_{i_1}/T_{i_0}) \\wedge", "\\text{d}\\log(T_{i_2}/T_{i_1}) \\wedge \\ldots \\wedge", "\\text{d}\\log(T_{i_{p - 1}}/T_{i_{p - 2}})", "$$", "in $\\Omega^{p - 1}(U_{i_0 \\ldots i_{p - 1}})$. These", "$\\beta_{i_0 \\ldots i_{p - 1}}$ can be lifted to the sections", "$\\tilde \\beta_{i_0 \\ldots i_{p -1}} =", "T_{i_0}^{-1}\\text{d}T_{i_0} \\wedge \\beta_{i_0 \\ldots i_{p - 1}}$", "of $\\wedge^p(\\bigoplus \\mathcal{O}(-1) \\text{d}T_j)$ over", "$U_{i_0 \\ldots i_{p - 1}}$. We conclude that the generator of", "$H^p(\\mathbf{P}^n_A, \\Omega^p)$ is given by the cocycle whose", "components are", "\\begin{align*}", "\\sum\\nolimits_{a = 0}^p (-1)^a", "\\tilde \\beta_{i_0 \\ldots \\hat{i_a} \\ldots i_p}", "& =", "T_{i_1}^{-1}\\text{d}T_{i_1} \\wedge \\beta_{i_1 \\ldots i_p}", "+ \\sum\\nolimits_{a = 1}^p (-1)^a", "T_{i_0}^{-1}\\text{d}T_{i_0} \\wedge", "\\beta_{i_0 \\ldots \\hat{i_a} \\ldots i_p} \\\\", "& =", "(T_{i_1}^{-1}\\text{d}T_{i_1} - T_{i_0}^{-1}\\text{d}T_{i_0}) \\wedge", "\\beta_{i_1 \\ldots i_p} +", "T_{i_0}^{-1}\\text{d}T_{i_0} \\wedge \\text{d}(\\beta)_{i_0 \\ldots i_p} \\\\", "& =", "\\text{d}\\log(T_{i_1}/T_{i_0}) \\wedge \\beta_{i_1 \\ldots i_p}", "\\end{align*}", "viewed as a section of $\\Omega^p$ over $U_{i_0 \\ldots i_p}$.", "This is up to sign the same as the cocycle representing", "$c_1^{Hodge}(\\mathcal{O}(1))^p$ and the proof is complete." ], "refs": [ "derham-lemma-twisted-hodge-cohomology-projective-space", "derham-lemma-euler-sequence", "derham-lemma-twisted-hodge-cohomology-projective-space", "derham-lemma-euler-sequence", "derham-lemma-twisted-hodge-cohomology-projective-space" ], "ref_ids": [ 14331, 14330, 14331, 14330, 14331 ] } ], "ref_ids": [] }, { "id": 14333, "type": "theorem", "label": "derham-lemma-de-rham-cohomology-projective-space", "categories": [ "derham" ], "title": "derham-lemma-de-rham-cohomology-projective-space", "contents": [ "For $0 \\leq i \\leq n$ the de Rham cohomology", "$H^{2i}_{dR}(\\mathbf{P}^n_A/A)$ is a free $A$-module of rank $1$", "with basis element $c_1^{dR}(\\mathcal{O}(1))^i$.", "In all other degrees the de Rham cohomology of $\\mathbf{P}^n_A$", "over $A$ is zero." ], "refs": [], "proofs": [ { "contents": [ "Consider the Hodge-to-de Rham spectral sequence of", "Section \\ref{section-hodge-to-de-rham}.", "By the computation of the Hodge cohomology of $\\mathbf{P}^n_A$ over $A$", "done in Lemma \\ref{lemma-hodge-cohomology-projective-space}", "we see that the spectral sequence degenerates on the $E_1$ page.", "In this way we see that $H^{2i}_{dR}(\\mathbf{P}^n_A/A)$ is a free", "$A$-module of rank $1$ for $0 \\leq i \\leq n$ and zero else.", "Observe that $c_1^{dR}(\\mathcal{O}(1))^i \\in H^{2i}_{dR}(\\mathbf{P}^n_A/A)$", "for $i = 0, \\ldots, n$ and that for $i = n$ this element is the", "image of $c_1^{Hodge}(\\mathcal{L})^n$ by the map of complexes", "$$", "\\Omega^n_{\\mathbf{P}^n_A/A}[-n]", "\\longrightarrow", "\\Omega^\\bullet_{\\mathbf{P}^n_A/A}", "$$", "This follows for example from the discussion in Remark \\ref{remark-truncations}", "or from the explicit description of cocycles representing these classes in", "Section \\ref{section-first-chern-class}.", "The spectral sequence shows that the induced map", "$$", "H^n(\\mathbf{P}^n_A, \\Omega^n_{\\mathbf{P}^n_A/A}) \\longrightarrow", "H^{2n}_{dR}(\\mathbf{P}^n_A/A)", "$$", "is an isomorphism and since $c_1^{Hodge}(\\mathcal{L})^n$ is a generator of", "of the source (Lemma \\ref{lemma-hodge-cohomology-projective-space}),", "we conclude that $c_1^{dR}(\\mathcal{L})^n$ is a generator", "of the target. By the $A$-bilinearity of the cup products,", "it follows that also $c_1^{dR}(\\mathcal{L})^i$", "is a generator of $H^{2i}_{dR}(\\mathbf{P}^n_A/A)$ for", "$0 \\leq i \\leq n$." ], "refs": [ "derham-lemma-hodge-cohomology-projective-space", "derham-remark-truncations", "derham-lemma-hodge-cohomology-projective-space" ], "ref_ids": [ 14332, 14383, 14332 ] } ], "ref_ids": [] }, { "id": 14334, "type": "theorem", "label": "derham-lemma-spectral-sequence-smooth", "categories": [ "derham" ], "title": "derham-lemma-spectral-sequence-smooth", "contents": [ "Let $f : X \\to Y$ be a quasi-compact, quasi-separated, and smooth", "morphism of schemes over a base scheme $S$. There is a bounded spectral", "sequence with first page", "$$", "E_1^{p, q} =", "H^q(\\Omega^p_{Y/S} \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*\\Omega^\\bullet_{X/Y})", "$$", "converging to $R^{p + q}f_*\\Omega^\\bullet_{X/S}$." ], "refs": [], "proofs": [ { "contents": [ "Consider $\\Omega^\\bullet_{X/S}$ as a filtered complex with the", "filtration introduced above. The spectral sequence is the", "spectral sequence of Cohomology, Lemma", "\\ref{cohomology-lemma-relative-spectral-sequence-filtered-object}.", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-cohomology-de-rham-base-change} we have", "$$", "Rf_*\\text{gr}^k\\Omega^\\bullet_{X/S} =", "\\Omega^k_{Y/S}[-k] \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*\\Omega^\\bullet_{X/Y}", "$$", "and thus we conclude." ], "refs": [ "cohomology-lemma-relative-spectral-sequence-filtered-object", "perfect-lemma-cohomology-de-rham-base-change" ], "ref_ids": [ 2125, 7033 ] } ], "ref_ids": [] }, { "id": 14335, "type": "theorem", "label": "derham-lemma-relative-global-generation-on-fibres", "categories": [ "derham" ], "title": "derham-lemma-relative-global-generation-on-fibres", "contents": [ "Let $f : X \\to Y$ be a smooth proper morphism of schemes.", "Let $N$ and $n_1, \\ldots, n_N \\geq 0$ be integers and let", "$\\xi_i \\in H^{n_i}_{dR}(X/Y)$, $1 \\leq i \\leq N$.", "Assume for all points $y \\in Y$ the images of $\\xi_1, \\ldots, \\xi_N$", "in $H^*_{dR}(X_y/y)$ form a basis over $\\kappa(y)$. Then the map", "$$", "\\bigoplus\\nolimits_{i = 1}^N \\mathcal{O}_Y[-n_i]", "\\longrightarrow", "Rf_*\\Omega^\\bullet_{X/Y}", "$$", "associated to $\\xi_1, \\ldots, \\xi_N$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-proper-smooth-de-Rham}", "$Rf_*\\Omega^\\bullet_{X/Y}$ is a perfect object of $D(\\mathcal{O}_Y)$", "whose formation commutes with arbitrary base change.", "Thus the map of the lemma is a map $a : K \\to L$", "between perfect objects of $D(\\mathcal{O}_Y)$", "whose derived restriction to any point is an isomorphism", "by our assumption on fibres. Then the cone $C$ on $a$ is a perfect", "object of $D(\\mathcal{O}_Y)$ (Cohomology, Lemma", "\\ref{cohomology-lemma-two-out-of-three-perfect}) whose", "derived restriction to any point is zero. It follows that $C$", "is zero by More on Algebra, Lemma", "\\ref{more-algebra-lemma-lift-perfect-from-residue-field}", "and $a$ is an isomorphism. (This also uses Derived Categories of Schemes,", "Lemmas \\ref{perfect-lemma-affine-compare-bounded} and", "\\ref{perfect-lemma-perfect-affine} to translate into algebra.)" ], "refs": [ "derham-lemma-proper-smooth-de-Rham", "cohomology-lemma-two-out-of-three-perfect", "more-algebra-lemma-lift-perfect-from-residue-field", "perfect-lemma-affine-compare-bounded", "perfect-lemma-perfect-affine" ], "ref_ids": [ 14319, 2226, 10232, 6941, 6980 ] } ], "ref_ids": [] }, { "id": 14336, "type": "theorem", "label": "derham-lemma-global-generation-on-fibres", "categories": [ "derham" ], "title": "derham-lemma-global-generation-on-fibres", "contents": [ "Let $f : X \\to Y$ be a smooth proper morphism of schemes over a base $S$.", "Assume", "\\begin{enumerate}", "\\item $Y$ and $S$ are affine, and", "\\item there exist integers $N$ and $n_1, \\ldots, n_N \\geq 0$ and", "$\\xi_i \\in H^{n_i}_{dR}(X/S)$, $1 \\leq i \\leq N$ such that", "for all points $y \\in Y$ the images of $\\xi_1, \\ldots, \\xi_N$", "in $H^*_{dR}(X_y/y)$ form a basis over $\\kappa(y)$.", "\\end{enumerate}", "Then the map", "$$", "\\bigoplus\\nolimits_{i = 1}^N H^*_{dR}(Y/S) \\longrightarrow", "H^*_{dR}(X/S), \\quad", "(a_1, \\ldots, a_N) \\longmapsto \\sum \\xi_i \\cup f^*a_i", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Say $Y = \\Spec(A)$ and $S = \\Spec(R)$.", "In this case $\\Omega^\\bullet_{A/R}$ computes", "$R\\Gamma(Y, \\Omega^\\bullet_{Y/S})$ by Lemma \\ref{lemma-de-rham-affine}.", "Choose a finite affine open covering $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$.", "Consider the complex", "$$", "K^\\bullet =", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\Omega_{X/S}^\\bullet))", "$$", "as in", "Cohomology, Section \\ref{cohomology-section-cech-cohomology-of-complexes}.", "Let us collect some facts about this complex most of which", "can be found in the reference just given:", "\\begin{enumerate}", "\\item $K^\\bullet$ is a complex of $R$-modules whose terms are", "$A$-modules,", "\\item $K^\\bullet$ represents $R\\Gamma(X, \\Omega^\\bullet_{X/S})$ in $D(R)$", "(Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero} and", "Cohomology, Lemma \\ref{cohomology-lemma-cech-complex-complex-computes}),", "\\item there is a natural map $\\Omega^\\bullet_{A/R} \\to K^\\bullet$", "of complexes of $R$-modules which is $A$-linear on terms and", "induces the pullback map $H^*_{dR}(Y/S) \\to H^*_{dR}(X/S)$", "on cohomology,", "\\item $K^\\bullet$ has a multiplication denoted $\\wedge$", "which turns it into a differential graded $R$-algebra,", "\\item the multiplication on $K^\\bullet$", "induces the cup product on $H^*_{dR}(X/S)$", "(Cohomology, Section \\ref{cohomology-section-cup-product}),", "\\item the filtration $F$ on $\\Omega^*_{X/S}$ induces a filtration", "$$", "K^\\bullet =", "F^0K^\\bullet \\supset F^1K^\\bullet \\supset F^2K^\\bullet \\supset \\ldots", "$$", "by subcomplexes on $K^\\bullet$ such that", "\\begin{enumerate}", "\\item $F^kK^n \\subset K^n$ is an $A$-submmodule,", "\\item $F^kK^\\bullet \\wedge F^lK^\\bullet \\subset F^{k + l}K^\\bullet$,", "\\item $\\text{gr}^kK^\\bullet$ is a complex of $A$-modules,", "\\item $\\text{gr}^0K^\\bullet =", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\Omega_{X/Y}^\\bullet))$", "and represents $R\\Gamma(X, \\Omega^\\bullet_{X/Y})$ in $D(A)$,", "\\item multiplication induces an isomorphism", "$\\Omega^k_{A/R}[-k] \\otimes_A \\text{gr}^0K^\\bullet \\to \\text{gr}^kK^\\bullet$", "\\end{enumerate}", "\\end{enumerate}", "We omit the detailed proofs of these statements; please see discussion", "leading up to the construction of the spectral sequence in", "Lemma \\ref{lemma-spectral-sequence-smooth}.", "\\medskip\\noindent", "For every $i = 1, \\ldots, N$ we choose a cocycle $x_i \\in K^{n_i}$", "representing $\\xi_i$. Next, we look at the map of complexes", "$$", "\\tilde x :", "M^\\bullet = \\bigoplus\\nolimits_{i = 1, \\ldots, N}", "\\Omega^\\bullet_{A/R}[-n_i]", "\\longrightarrow", "K^\\bullet", "$$", "which sends $\\omega$ in the $i$th summand to $x_i \\wedge \\omega$.", "All that remains is to show that this map is a quasi-isomorphism.", "We endow $M^\\bullet$ with the structure of a filtered complex", "by the rule", "$$", "F^kM^\\bullet =", "\\bigoplus\\nolimits_{i = 1, \\ldots, N}", "(\\sigma_{\\geq k}\\Omega^\\bullet_{A/R})[-n_i]", "$$", "With this choice the map $\\tilde x$ is a morphism of filtered complexes.", "Observe that $\\text{gr}^0M^\\bullet = \\bigoplus A[-n_i]$", "and multiplication induces an isomorphism", "$\\Omega^k_{A/R}[-k] \\otimes_A \\text{gr}^0M^\\bullet \\to \\text{gr}^kM^\\bullet$.", "By construction and Lemma \\ref{lemma-relative-global-generation-on-fibres}", "we see that", "$$", "\\text{gr}^0\\tilde x :", "\\text{gr}^0M^\\bullet \\longrightarrow", "\\text{gr}^0K^\\bullet", "$$", "is an isomorphism in $D(A)$. It follows that for all $k \\geq 0$", "we obtain isomorphisms", "$$", "\\text{gr}^k \\tilde x :", "\\text{gr}^kM^\\bullet = \\Omega^k_{A/R}[-k] \\otimes_A \\text{gr}^0M^\\bullet", "\\longrightarrow", "\\Omega^k_{A/R}[-k] \\otimes_A \\text{gr}^0K^\\bullet =", "\\text{gr}^kK^\\bullet", "$$", "in $D(A)$. Namely, the complex", "$\\text{gr}^0K^\\bullet =", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\Omega_{X/Y}^\\bullet))$", "is K-flat as a complex of $A$-modules by Derived Categories of Schemes,", "Lemma \\ref{perfect-lemma-K-flat}.", "Hence the tensor product on the right hand side is the", "derived tensor product as is true by inspection on the left hand side.", "Finally, taking the derived tensor product", "$\\Omega^k_{A/R}[-k] \\otimes_A^\\mathbf{L} -$ is a functor on $D(A)$", "and therefore sends isomorphisms to isomorphisms.", "Arguing by induction on $k$ we deduce that", "$$", "\\tilde x : M^\\bullet/F^kM^\\bullet \\to K^\\bullet/F^kK^\\bullet", "$$", "is an isomorphism in $D(R)$ since we have the short exact sequences", "$$", "0 \\to F^kM^\\bullet/F^{k + 1}M^\\bullet \\to", "M^\\bullet/F^{k + 1}M^\\bullet \\to", "\\text{gr}^kM^\\bullet \\to 0", "$$", "and similarly for $K^\\bullet$. This proves that $\\tilde x$ is a", "quasi-isomorphism as the filtrations are finite in any given degree." ], "refs": [ "derham-lemma-de-rham-affine", "coherent-lemma-quasi-coherent-affine-cohomology-zero", "cohomology-lemma-cech-complex-complex-computes", "derham-lemma-spectral-sequence-smooth", "derham-lemma-relative-global-generation-on-fibres", "perfect-lemma-K-flat" ], "ref_ids": [ 14315, 3282, 2099, 14334, 14335, 7034 ] } ], "ref_ids": [] }, { "id": 14337, "type": "theorem", "label": "derham-lemma-log-complex", "categories": [ "derham" ], "title": "derham-lemma-log-complex", "contents": [ "Let $X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an", "effective Cartier divisor.", "Assume the de Rham complex of log poles is defined for $Y \\subset X$ over $S$.", "There is a canonical short exact sequence", "of complexes", "$$", "0 \\to \\Omega^\\bullet_{X/S} \\to", "\\Omega^\\bullet_{X/S}(\\log Y) \\to", "\\Omega^\\bullet_{Y/S}[-1] \\to 0", "$$" ], "refs": [], "proofs": [ { "contents": [ "Our assumption is that for every $y \\in Y$ and local equation", "$f \\in \\mathcal{O}_{X, y}$ of $Y$ we have", "$$", "\\Omega_{X/S, y} = \\mathcal{O}_{X, y}\\text{d}f \\oplus M", "\\quad\\text{and}\\quad", "\\Omega^p_{X/S, y} = \\wedge^{p - 1}(M)\\text{d}f \\oplus \\wedge^p(M)", "$$", "for some module $M$ with $f$-torsion free exterior powers $\\wedge^p(M)$.", "It follows that", "$$", "\\Omega^p_{Y/S, y} = \\wedge^p(M/fM) = \\wedge^p(M)/f\\wedge^p(M)", "$$", "Below we will tacitly use these facts.", "In particular the sheaves $\\Omega^p_{X/S}$ have no nonzero local", "sections supported on $Y$ and we have a canonical inclusion", "$$", "\\Omega^p_{X/S} \\subset \\Omega^p_{X/S}(Y)", "$$", "see More on Flatness, Section \\ref{flat-section-eta}. Let $U = \\Spec(A)$", "be an affine open subscheme such that $Y \\cap U = V(f)$ for some", "nonzerodivisor $f \\in A$. Let us consider the $\\mathcal{O}_U$-submodule", "of $\\Omega^p_{X/S}(Y)|_U$ generated by", "$\\Omega^p_{X/S}|_U$ and $\\text{d}\\log(f) \\wedge \\Omega^{p - 1}_{X/S}$", "where $\\text{d}\\log(f) = f^{-1}\\text{d}(f)$.", "This is independent of the choice of $f$ as another generator of the", "ideal of $Y$ on $U$ is equal to $uf$ for a unit $u \\in A$ and we get", "$$", "\\text{d}\\log(uf) - \\text{d}\\log(f) = \\text{d}\\log(u) = u^{-1}\\text{d}u", "$$", "which is a section of $\\Omega_{X/S}$ over $U$. These local", "sheaves glue to give a quasi-coherent submodule", "$$", "\\Omega^p_{X/S} \\subset \\Omega^p_{X/S}(\\log Y) \\subset \\Omega^p_{X/S}(Y)", "$$", "Let us agree to think of $\\Omega^p_{Y/S}$ as a quasi-coherent", "$\\mathcal{O}_X$-module. There is a unique surjective", "$\\mathcal{O}_X$-linear map", "$$", "\\text{Res} : \\Omega^p_{X/S}(\\log Y) \\to \\Omega^{p - 1}_Y", "$$", "defined by the rule", "$$", "\\text{Res}(\\eta' + \\text{d}\\log(f) \\wedge \\eta) = \\eta|_{Y \\cap U}", "$$", "for all opens $U$ as above and all", "$\\eta' \\in \\Omega^p_{X/S}(U)$ and $\\eta \\in \\Omega^{p - 1}_{X/S}(U)$.", "If a form $\\eta$ over $U$ restricts to zero on $Y \\cap U$, then", "$\\eta = \\text{d}f \\wedge \\eta' + f\\eta''$ for some forms $\\eta'$ and $\\eta''$", "over $U$. We conclude that", "we have a short exact sequence", "$$", "0 \\to \\Omega^p_{X/S} \\to \\Omega^p_{X/S}(\\log Y) \\to \\Omega^{p - 1}_{Y/S} \\to 0", "$$", "for all $p$. We still have to define the differentials", "$\\Omega^p_{X/S}(\\log Y) \\to \\Omega^{p + 1}_{X/S}(\\log Y)$.", "On the subsheaf $\\Omega^p_{X/S}$ we use the differential of", "the de Rham complex of $X$ over $S$. Finally, we define", "$\\text{d}(\\text{d}\\log(f) \\wedge \\eta) = -\\text{d}\\log(f) \\wedge \\text{d}\\eta$.", "The sign is forced on us by the Leibniz rule (on $\\Omega^\\bullet_{X/S}$)", "and it is compatible with the differential on $\\Omega^\\bullet_{Y/S}[-1]$", "which is after all $-\\text{d}_{Y/S}$ by our sign convention in", "Homology, Definition \\ref{homology-definition-shift-cochain}.", "In this way we obtain a short exact", "sequence of complexes as stated in the lemma." ], "refs": [ "homology-definition-shift-cochain" ], "ref_ids": [ 12158 ] } ], "ref_ids": [] }, { "id": 14338, "type": "theorem", "label": "derham-lemma-multiplication-log", "categories": [ "derham" ], "title": "derham-lemma-multiplication-log", "contents": [ "Let $p : X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an", "effective Cartier divisor. Assume the de Rham complex of log poles", "is defined for $Y \\subset X$ over $S$.", "\\begin{enumerate}", "\\item The maps", "$\\wedge : \\Omega^p_{X/S} \\times \\Omega^q_{X/S} \\to \\Omega^{p + q}_{X/S}$", "extend uniquely to $\\mathcal{O}_X$-bilinear maps", "$$", "\\wedge : \\Omega^p_{X/S}(\\log Y) \\times \\Omega^q_{X/S}(\\log Y)", "\\to \\Omega^{p + q}_{X/S}(\\log Y)", "$$", "satisfying the Leibniz rule", "$", "\\text{d}(\\omega \\wedge \\eta) = \\text{d}(\\omega) \\wedge \\eta +", "(-1)^{\\deg(\\omega)} \\omega \\wedge \\text{d}(\\eta)$,", "\\item with multiplication as in (1) the map", "$\\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}(\\log(Y)$", "is a homomorphism of differential graded $\\mathcal{O}_S$-algebras,", "\\item via the maps in (1) we have $\\Omega^p_{X/S}(\\log Y) =", "\\wedge^p(\\Omega^1_{X/S}(\\log Y))$, and", "\\item the map", "$\\text{Res} : \\Omega^\\bullet_{X/S}(\\log Y) \\to \\Omega^\\bullet_{Y/S}[-1]$", "satisfies", "$$", "\\text{Res}(\\omega \\wedge \\eta) =", "\\text{Res}(\\omega) \\wedge \\eta|_Y +", "(-1)^{\\deg(\\omega)} \\omega|_Y \\wedge \\text{Res}(\\eta)", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows by direct calcuation from the local construction", "of the complex in the proof of Lemma \\ref{lemma-log-complex}.", "Details omitted." ], "refs": [ "derham-lemma-log-complex" ], "ref_ids": [ 14337 ] } ], "ref_ids": [] }, { "id": 14339, "type": "theorem", "label": "derham-lemma-gysin-via-log-complex", "categories": [ "derham" ], "title": "derham-lemma-gysin-via-log-complex", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an effective", "Cartier divisor. Assume the de Rham complex of log poles is defined for", "$Y \\subset X$ over $S$. Denote", "$$", "\\delta : \\Omega^\\bullet_{Y/S} \\to \\Omega^\\bullet_{X/S}[2]", "$$", "in $D(X, f^{-1}\\mathcal{O}_S)$ the ``boundary'' map coming from the", "short exact sequence in Lemma \\ref{lemma-log-complex}. Denote", "$$", "\\xi' : \\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}[2]", "$$", "in $D(X, f^{-1}\\mathcal{O}_S)$ the map of", "Remark \\ref{remark-cup-product-as-a-map}", "corresponding to $\\xi = c_1^{dR}(\\mathcal{O}_X(-Y))$. Denote", "$$", "\\zeta' : \\Omega^\\bullet_{Y/S} \\to \\Omega^\\bullet_{Y/S}[2]", "$$", "in $D(Y, f|_Y^{-1}\\mathcal{O}_S)$ the map of", "Remark \\ref{remark-cup-product-as-a-map} corresponding to", "$\\zeta = c_1^{dR}(\\mathcal{O}_X(-Y)|_Y)$. Then the diagram", "$$", "\\xymatrix{", "\\Omega^\\bullet_{X/S} \\ar[d]_{\\xi'} \\ar[r] &", "\\Omega^\\bullet_{Y/S} \\ar[d]^{\\zeta'} \\ar[ld]_\\delta \\\\", "\\Omega^\\bullet_{X/S}[2] \\ar[r] &", "\\Omega^\\bullet_{Y/S}[2]", "}", "$$", "is commutative in $D(X, f^{-1}\\mathcal{O}_S)$." ], "refs": [ "derham-lemma-log-complex", "derham-remark-cup-product-as-a-map", "derham-remark-cup-product-as-a-map" ], "proofs": [ { "contents": [ "More precisely, we define $\\delta$ as the boundary map corresponding to the", "shifted short exact sequence", "$$", "0 \\to \\Omega^\\bullet_{X/S}[1] \\to", "\\Omega^\\bullet_{X/S}(\\log Y)[1] \\to", "\\Omega^\\bullet_{Y/S} \\to 0", "$$", "It suffices to prove each triangle commutes. Set", "$\\mathcal{L} = \\mathcal{O}_X(-Y)$. Denote $\\pi : L \\to X$ the line bundle", "with $\\pi_*\\mathcal{O}_L = \\bigoplus_{n \\geq 0} \\mathcal{L}^{\\otimes n}$.", "\\medskip\\noindent", "Commutativity of the upper left triangle.", "By Lemma \\ref{lemma-the-complex-for-L-star-gives-chern-class}", "the map $\\xi'$ is the boundary map of the triangle given in", "Lemma \\ref{lemma-the-complex-for-L-star}.", "By functoriality it suffices to prove there exists a morphism of", "short exact sequences", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Omega^\\bullet_{X/S}[1] \\ar[r] \\ar[d] &", "\\Omega^\\bullet_{L^\\star/S, 0}[1] \\ar[r] \\ar[d] &", "\\Omega^\\bullet_{X/S} \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\Omega^\\bullet_{X/S}[1] \\ar[r] &", "\\Omega^\\bullet_{X/S}(\\log Y)[1] \\ar[r] &", "\\Omega^\\bullet_{Y/S} \\ar[r] &", "0", "}", "$$", "where the left and right vertical arrows are the obvious ones.", "We can define the middle vertical arrow by the rule", "$$", "\\omega' + \\text{d}\\log(s) \\wedge \\omega \\longmapsto", "\\omega' + \\text{d}\\log(f) \\wedge \\omega", "$$", "where $\\omega', \\omega$ are local sections of $\\Omega^\\bullet_{X/S}$", "and where $s$ is a local generator of $\\mathcal{L}$ and", "$f \\in \\mathcal{O}_X(-Y)$ is the corresponding section of the ideal", "sheaf of $Y$ in $X$. Since the constructions of the maps in", "Lemmas \\ref{lemma-the-complex-for-L-star} and \\ref{lemma-log-complex}", "match exactly, this works.", "\\medskip\\noindent", "Commutativity of the lower right triangle. Denote", "$\\overline{L}$ the restriction of $L$ to $Y$.", "By Lemma \\ref{lemma-the-complex-for-L-star-gives-chern-class}", "the map $\\zeta'$ is the boundary map of the triangle given in", "Lemma \\ref{lemma-the-complex-for-L-star} using the line bundle", "$\\overline{L}$ on $Y$.", "By functoriality it suffices to prove there exists a morphism of", "short exact sequences", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\Omega^\\bullet_{X/S}[1] \\ar[r] \\ar[d] &", "\\Omega^\\bullet_{X/S}(\\log Y)[1] \\ar[r] \\ar[d] &", "\\Omega^\\bullet_{Y/S} \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\Omega^\\bullet_{Y/S}[1] \\ar[r] &", "\\Omega^\\bullet_{\\overline{L}^\\star/S, 0}[1] \\ar[r] &", "\\Omega^\\bullet_{Y/S} \\ar[r] &", "0 \\\\", "}", "$$", "where the left and right vertical arrows are the obvious ones.", "We can define the middle vertical arrow by the rule", "$$", "\\omega' + \\text{d}\\log(f) \\wedge \\omega \\longmapsto", "\\omega'|_Y + \\text{d}\\log(\\overline{s}) \\wedge \\omega|_Y", "$$", "where $\\omega', \\omega$ are local sections of $\\Omega^\\bullet_{X/S}$", "and where $f$ is a local generator of $\\mathcal{O}_X(-Y)$ viewed as", "a function on $X$ and where $\\overline{s}$ is $f|_Y$ viewed as a", "section of $\\mathcal{L}|_Y = \\mathcal{O}_X(-Y)|_Y$.", "Since the constructions of the maps in", "Lemmas \\ref{lemma-the-complex-for-L-star} and \\ref{lemma-log-complex}", "match exactly, this works." ], "refs": [ "derham-lemma-the-complex-for-L-star-gives-chern-class", "derham-lemma-the-complex-for-L-star", "derham-lemma-the-complex-for-L-star", "derham-lemma-log-complex", "derham-lemma-the-complex-for-L-star-gives-chern-class", "derham-lemma-the-complex-for-L-star", "derham-lemma-the-complex-for-L-star", "derham-lemma-log-complex" ], "ref_ids": [ 14327, 14326, 14326, 14337, 14327, 14326, 14326, 14337 ] } ], "ref_ids": [ 14337, 14382, 14382 ] }, { "id": 14340, "type": "theorem", "label": "derham-lemma-log-complex-consequence", "categories": [ "derham" ], "title": "derham-lemma-log-complex-consequence", "contents": [ "Let $X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an effective", "Cartier divisor. Assume the de Rham complex of log poles is defined for", "$Y \\subset X$ over $S$. Let $b \\in H^m_{dR}(X/S)$ be a de Rham cohomology", "class whose restriction to $Y$ is zero. Then", "$c_1^{dR}(\\mathcal{O}_X(Y)) \\cup b = 0$ in $H^{m + 2}_{dR}(X/S)$." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from Lemma \\ref{lemma-gysin-via-log-complex}.", "Namely, we have", "$$", "c_1^{dR}(\\mathcal{O}_X(Y)) \\cup b =", "-c_1^{dR}(\\mathcal{O}_X(-Y)) \\cup b = -\\xi'(b) = -\\delta(b|_Y) = 0", "$$", "as desired. For the second equality, see", "Remark \\ref{remark-cup-product-as-a-map}." ], "refs": [ "derham-lemma-gysin-via-log-complex", "derham-remark-cup-product-as-a-map" ], "ref_ids": [ 14339, 14382 ] } ], "ref_ids": [] }, { "id": 14341, "type": "theorem", "label": "derham-lemma-check-log-smooth", "categories": [ "derham" ], "title": "derham-lemma-check-log-smooth", "contents": [ "Let $X \\to T \\to S$ be morphisms of schemes. Let $Y \\subset X$ be an effective", "Cartier divisor. If both $X \\to T$ and $Y \\to T$ are smooth, then", "the de Rham complex of log poles is defined for $Y \\subset X$ over $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $y \\in Y$ be a point.", "By More on Morphisms, Lemma \\ref{more-morphisms-lemma-etale-local-structure}", "there exists an integer $0 \\geq m$ and a commutative diagram", "$$", "\\xymatrix{", "Y \\ar[d] &", "V \\ar[l] \\ar[d] \\ar[r] &", "\\mathbf{A}^m_T", "\\ar[d]^{(a_1, \\ldots, a_m) \\mapsto (a_1, \\ldots, a_m, 0)} \\\\", "X &", "U \\ar[l] \\ar[r]^-\\pi &", "\\mathbf{A}^{m + 1}_T", "}", "$$", "where $U \\subset X$ is open, $V = Y \\cap U$,", "$\\pi$ is \\'etale, $V = \\pi^{-1}(\\mathbf{A}^m_T)$, and $y \\in V$.", "Denote $z \\in \\mathbf{A}^m_T$ the image of $y$. Then we have", "$$", "\\Omega^p_{X/S, y} = \\Omega^p_{\\mathbf{A}^{m + 1}_T/S, z}", "\\otimes_{\\mathcal{O}_{\\mathbf{A}^{m + 1}_T, z}} \\mathcal{O}_{X, x}", "$$", "by Lemma \\ref{lemma-etale}. Denote $x_1, \\ldots, x_{m + 1}$", "the coordinate functions on $\\mathbf{A}^{m + 1}_T$.", "Since the conditions (1) and (2) in Definition \\ref{definition-local-product}", "do not depend on the choice of the local coordinate,", "it suffices to check the conditions (1) and (2) when $f$ is the", "image of $x_{m + 1}$ by the flat local ring homomorphism", "$\\mathcal{O}_{\\mathbf{A}^{m + 1}_T, z} \\to \\mathcal{O}_{X, x}$.", "In this way we see that it suffices to check conditions (1) and (2)", "for $\\mathbf{A}^m_T \\subset \\mathbf{A}^{m + 1}_T$ and the point $z$.", "To prove this case we may assume $S = \\Spec(A)$ and $T = \\Spec(B)$", "are affine. Let $A \\to B$ be the ring map corresponding to the morphism", "$T \\to S$ and set $P = B[x_1, \\ldots, x_{m + 1}]$ so that", "$\\mathbf{A}^{m + 1}_T = \\Spec(B)$. We have", "$$", "\\Omega_{P/A} = \\Omega_{B/A} \\otimes_B P \\oplus", "\\bigoplus\\nolimits_{j = 1, \\ldots, m} P \\text{d}x_j \\oplus", "P \\text{d}x_{m + 1}", "$$", "Hence the map $P \\to \\Omega_{P/A}$, $g \\mapsto g \\text{d}x_{m + 1}$", "is a split injection and $x_{m + 1}$ is a nonzerodivisor on", "$\\Omega^p_{P/A}$ for all $p \\geq 0$. Localizing at the prime ideal", "corresponding to $z$ finishes the proof." ], "refs": [ "more-morphisms-lemma-etale-local-structure", "derham-lemma-etale", "derham-definition-local-product" ], "ref_ids": [ 13882, 14314, 14379 ] } ], "ref_ids": [] }, { "id": 14342, "type": "theorem", "label": "derham-lemma-comparison", "categories": [ "derham" ], "title": "derham-lemma-comparison", "contents": [ "For $a \\geq 0$ we have", "\\begin{enumerate}", "\\item the map", "$\\Omega^a_{X/S} \\to b_*\\Omega^a_{L/S}$ is an isomorphism,", "\\item the map $\\Omega^a_{Z/S} \\to p_*\\Omega^a_{P/S}$ is an isomorphism,", "and", "\\item the map $Rb_*\\Omega^a_{L/S} \\to i_*Rp_*\\Omega^a_{P/S}$ is an isomorphism", "on cohomology sheaves in degree $\\geq 1$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us first prove part (2). Since", "$P = Z \\times_S \\mathbf{P}^{n - 1}_S$", "we see that", "$$", "\\Omega^a_{P/S} = \\bigoplus\\nolimits_{a = r + s}", "\\text{pr}_1^*\\Omega^r_{Z/S} \\otimes", "\\text{pr}_2^*\\Omega^s_{\\mathbf{P}^{n - 1}_S/S}", "$$", "Recalling that $p = \\text{pr}_1$ by the projection formula", "(Cohomology, Lemma \\ref{cohomology-lemma-projection-formula})", "we obtain", "$$", "p_*\\Omega^a_{P/S} = \\bigoplus\\nolimits_{a = r + s}", "\\Omega^r_{Z/S} \\otimes", "\\text{pr}_{1, *}\\text{pr}_2^*\\Omega^s_{\\mathbf{P}^{n - 1}_S/S}", "$$", "By the calculations in Section \\ref{section-projective-space}", "and in particular in", "the proof of Lemma \\ref{lemma-hodge-cohomology-projective-space}", "we have $\\text{pr}_{1, *}\\text{pr}_2^*\\Omega^s_{\\mathbf{P}^{n - 1}_S/S} = 0$", "except if $s = 0$ in which case we get", "$\\text{pr}_{1, *}\\mathcal{O}_P = \\mathcal{O}_Z$.", "This proves (2).", "\\medskip\\noindent", "By the material in Section \\ref{section-line-bundle} and in particular", "Lemma \\ref{lemma-push-omega-a} we have", "$\\pi_*\\Omega^a_{L/S} = \\Omega^a_{P/S} \\oplus", "\\bigoplus_{k \\geq 1} \\Omega^a_{L/S, k}$.", "Since the composition $\\pi \\circ 0$ in the diagram above", "is the identity morphism on $P$ to prove part (3) it suffices to show that", "$\\Omega^a_{L/S, k}$ has vanishing higher cohomology for $k > 0$.", "By Lemmas \\ref{lemma-the-complex-for-L-star} and \\ref{lemma-push-omega-a}", "there are short exact sequences", "$$", "0 \\to \\Omega^a_{P/S} \\otimes \\mathcal{O}_P(k)", "\\to \\Omega^a_{L/S, k} \\to", "\\Omega^{a - 1}_{P/S} \\otimes \\mathcal{O}_P(k) \\to 0", "$$", "where $\\Omega^{a - 1}_{P/S} = 0$ if $a = 0$. Since", "$P = Z \\times_S \\mathbf{P}^{n - 1}_S$ we have", "$$", "\\Omega^a_{P/S} = \\bigoplus\\nolimits_{i + j = a}", "\\Omega^i_{Z/S} \\boxtimes \\Omega^j_{\\mathbf{P}^{n - 1}_S/S}", "$$", "by Lemma \\ref{lemma-de-rham-complex-product}.", "Since $\\Omega^i_{Z/S}$ is free of finite rank", "we see that it suffices to show that the higher cohomology of", "$\\mathcal{O}_Z \\boxtimes \\Omega^j_{\\mathbf{P}^{n - 1}_S/S}(k)$", "is zero for $k > 0$. This follows from", "Lemma \\ref{lemma-twisted-hodge-cohomology-projective-space}", "applied to $P = Z \\times_S \\mathbf{P}^{n - 1}_S = \\mathbf{P}^{n - 1}_Z$", "and the proof of (3) is complete.", "\\medskip\\noindent", "We still have to prove (1). If $n = 1$, then we are blowing", "up an effective Cartier divisor and $b$ is an isomorphism", "and we have (1). If $n > 1$, then the composition", "$$", "\\Gamma(X, \\Omega^a_{X/S})", "\\to", "\\Gamma(L, \\Omega^a_{L/S})", "\\to", "\\Gamma(L \\setminus E, \\Omega^a_{L/S})", "=", "\\Gamma(X \\setminus Z, \\Omega^a_{X/S})", "$$", "is an isomorphism as $\\Omega^a_{X/S}$ is finite free", "(small detail omitted). Thus the only way (1) can fail is if", "there are nonzero elements of $\\Gamma(L, \\Omega^a_{L/S})$ which vanish", "outside of $E = 0(P)$. Since $L$ is a line bundle over $P$", "with zero section $0 : P \\to L$, it suffices to show that", "on a line bundle there are no nonzero sections of a sheaf", "of differentials which vanish identically outside the zero section.", "The reader sees this is true either (preferably) by a local caculation", "or by using that $\\Omega_{L/S, k} \\subset \\Omega_{L^\\star/S, k}$", "(see references above)." ], "refs": [ "cohomology-lemma-projection-formula", "derham-lemma-hodge-cohomology-projective-space", "derham-lemma-push-omega-a", "derham-lemma-the-complex-for-L-star", "derham-lemma-push-omega-a", "derham-lemma-de-rham-complex-product", "derham-lemma-twisted-hodge-cohomology-projective-space" ], "ref_ids": [ 2243, 14332, 14328, 14326, 14328, 14322, 14331 ] } ], "ref_ids": [] }, { "id": 14343, "type": "theorem", "label": "derham-lemma-comparison-bis", "categories": [ "derham" ], "title": "derham-lemma-comparison-bis", "contents": [ "For $a \\geq 0$ there are canonical maps", "$$", "b^*\\Omega^a_{X/S} \\longrightarrow", "\\Omega^a_{L/S} \\longrightarrow", "b^*\\Omega^a_{X/S} \\otimes_{\\mathcal{O}_L} \\mathcal{O}_L((n - 1)E)", "$$", "whose composition is induced by the inclusion", "$\\mathcal{O}_L \\subset \\mathcal{O}_L((n - 1)E)$." ], "refs": [], "proofs": [ { "contents": [ "The first arrow in the displayed formula is", "discussed in Section \\ref{section-de-rham-complex}.", "To get the second arrow we have to show that if we view", "a local section of $\\Omega^a_{L/S}$ as a ``meromorphic section''", "of $b^*\\Omega^a_{X/S}$, then it has a pole of order at most", "$n - 1$ along $E$. To see this we work on affine local charts", "on $L$. Namely, recall that $L$ is covered by the spectra of the", "affine blowup algebras $A[\\frac{I}{y_i}]$ where $I = A_{+}$", "is the ideal generated by $y_1, \\ldots, y_n$. See", "Algebra, Section \\ref{algebra-section-blow-up} and", "Divisors, Lemma \\ref{divisors-lemma-blowing-up-affine}.", "By symmetry it is enough to work on the", "chart corresponding to $i = 1$. Then", "$$", "A[\\frac{I}{y_1}] = R[x_1, \\ldots, x_m, y_1, t_2, \\ldots, t_n]", "$$", "where $t_i = y_i/y_1$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-blowup-regular-sequence}.", "Thus the module $\\Omega^1_{L/S}$ is over the corresponding", "affine open freely generated by", "$\\text{d}x_1, \\ldots, \\text{d}x_m$, $\\text{d}y_1$, and", "$\\text{d}t_1, \\ldots, \\text{d}t_n$.", "Of course, the first $m + 1$ of these generators come from", "$b^*\\Omega^1_{X/S}$ and for the remaining $n - 1$ we have", "$$", "\\text{d}t_j =", "\\text{d}\\frac{y_j}{y_1} =", "\\frac{1}{y_1}\\text{d}y_j - \\frac{y_j}{y_1^2}\\text{d}y_1 =", "\\frac{\\text{d}y_j - t_j \\text{d}y_1}{y_1}", "$$", "which has a pole of order $1$ along $E$ since $E$ is cut out by $y_1$", "on this chart. Since the wedges of $a$ of these elements give a basis", "of $\\Omega^a_{L/S}$ over this chart, and since there are at most", "$n - 1$ of the $\\text{d}t_j$ involved this finishes the proof." ], "refs": [ "divisors-lemma-blowing-up-affine", "more-algebra-lemma-blowup-regular-sequence" ], "ref_ids": [ 8052, 9990 ] } ], "ref_ids": [] }, { "id": 14344, "type": "theorem", "label": "derham-lemma-blowup-twist-same-cohomology", "categories": [ "derham" ], "title": "derham-lemma-blowup-twist-same-cohomology", "contents": [ "Let $E = 0(P)$ be the exceptional divisor of the blowing up $b$.", "For any locally free $\\mathcal{O}_X$-module $\\mathcal{E}$ and", "$0 \\leq i \\leq n - 1$ the map", "$$", "\\mathcal{E}", "\\longrightarrow", "Rb_*(b^*\\mathcal{E} \\otimes_{\\mathcal{O}_L} \\mathcal{O}_L(iE))", "$$", "is an isomorphism in $D(\\mathcal{O}_X)$." ], "refs": [], "proofs": [ { "contents": [ "By the projection formula it is enough to show this for", "$\\mathcal{E} = \\mathcal{O}_X$, see Cohomology, Lemma", "\\ref{cohomology-lemma-projection-formula}.", "Since $X$ is affine it suffices to show that the maps", "$$", "H^0(X, \\mathcal{O}_X) \\to", "H^0(L, \\mathcal{O}_L) \\to", "H^0(L, \\mathcal{O}_L(iE))", "$$", "are isomorphisms and that $H^j(X, \\mathcal{O}_L(iE)) = 0$", "for $j > 0$ and $0 \\leq i \\leq n - 1$, see Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images-application}.", "Since $\\pi$ is affine, we can compute global sections and", "cohomology after taking $\\pi_*$, see Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-relative-affine-cohomology}. If $n = 1$, then", "$L \\to X$ is an isomorphism and $i = 0$ hence the first statement holds.", "If $n > 1$, then we consider the composition", "$$", "H^0(X, \\mathcal{O}_X) \\to H^0(L, \\mathcal{O}_L) \\to", "H^0(L, \\mathcal{O}_L(iE)) \\to H^0(L \\setminus E, \\mathcal{O}_L) =", "H^0(X \\setminus Z, \\mathcal{O}_X)", "$$", "Since", "$H^0(X \\setminus Z, \\mathcal{O}_X) = H^0(X, \\mathcal{O}_X)$ in this", "case as $Z$ has codimension $n \\geq 2$ in $X$ (details omitted) we conclude", "the first statement holds. For the second, recall that", "$\\mathcal{O}_L(E) = \\mathcal{O}_L(-1)$, see Divisors, Lemma", "\\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}.", "Hence we have", "$$", "\\pi_*\\mathcal{O}_L(iE) =", "\\pi_*\\mathcal{O}_L(-i) =", "\\bigoplus\\nolimits_{k \\geq -i} \\mathcal{O}_P(k)", "$$", "as discussed in", "More on Morphisms, Section \\ref{more-morphisms-section-proj-spec}.", "Thus we conclude by the vanishing of the cohomology of twists", "of the structure sheaf on $P = \\mathbf{P}^{n - 1}_Z$", "shown in Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cohomology-projective-space-over-ring}." ], "refs": [ "cohomology-lemma-projection-formula", "coherent-lemma-quasi-coherence-higher-direct-images-application", "coherent-lemma-relative-affine-cohomology", "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "coherent-lemma-cohomology-projective-space-over-ring" ], "ref_ids": [ 2243, 3296, 3284, 8054, 3304 ] } ], "ref_ids": [] }, { "id": 14345, "type": "theorem", "label": "derham-lemma-blowup", "categories": [ "derham" ], "title": "derham-lemma-blowup", "contents": [ "Let $S$ be a scheme. Let $Z \\to X$ be a closed immersion of schemes", "smooth over $S$. Let $b : X' \\to X$ be the blowing up of $Z$ with", "exceptional divisor $E \\subset X'$. Then $X'$ and $E$ are smooth", "over $S$. The morphism $p : E \\to Z$ is canonically isomorphic", "to the projective space bundle", "$$", "\\mathbf{P}(\\mathcal{I}/\\mathcal{I}^2) \\longrightarrow Z", "$$", "where $\\mathcal{I} \\subset \\mathcal{O}_X$ is the ideal sheaf", "of $Z$. The relative $\\mathcal{O}_E(1)$ coming from the projective", "space bundle structure is isomorphic to the restriction of", "$\\mathcal{O}_{X'}(-E)$ to $E$." ], "refs": [], "proofs": [ { "contents": [ "By Divisors, Lemma", "\\ref{divisors-lemma-immersion-smooth-into-smooth-regular-immersion}", "the immersion $Z \\to X$ is a regular immmersion, hence", "the ideal sheaf $\\mathcal{I}$ is of finite type, hence $b$ is a projective", "morphism with relatively ample invertible sheaf", "$\\mathcal{O}_{X'}(1) = \\mathcal{O}_{X'}(-E)$, see", "Divisors, Lemmas", "\\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor} and", "\\ref{divisors-lemma-blowing-up-projective}.", "The canonical map $\\mathcal{I} \\to b_*\\mathcal{O}_{X'}(1)$", "gives a closed immersion", "$$", "X' \\longrightarrow", "\\mathbf{P}\\left(\\bigoplus\\nolimits_{n \\geq 0}", "\\text{Sym}^n_{\\mathcal{O}_X}(\\mathcal{I})\\right)", "$$", "by the very construction of the blowup. The restriction of this morphism", "to $E$ gives a canonical map", "$$", "E \\longrightarrow", "\\mathbf{P}\\left(\\bigoplus\\nolimits_{n \\geq 0}", "\\text{Sym}^n_{\\mathcal{O}_Z}(\\mathcal{I}/\\mathcal{I}^2)\\right)", "$$", "over $Z$. Since $\\mathcal{I}/\\mathcal{I}^2$ is finite locally free", "if this canonical map is an isomorphism, then the final part of the", "lemma holds. Having said all of this, now the question is \\'etale", "local on $X$. Namely, blowing up commutes with flat base change by", "Divisors, Lemma \\ref{divisors-lemma-flat-base-change-blowing-up}", "and we can check smoothness after precomposing with a surjective", "\\'etale morphism. Thus by the \\'etale local structure of a", "closed immersion of schemes over $S$ given in More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-etale-local-structure}", "this reduces to the situation discussed in", "Section \\ref{section-calculations}." ], "refs": [ "divisors-lemma-immersion-smooth-into-smooth-regular-immersion", "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "divisors-lemma-blowing-up-projective", "divisors-lemma-flat-base-change-blowing-up", "more-morphisms-lemma-etale-local-structure" ], "ref_ids": [ 8007, 8054, 8063, 8053, 13882 ] } ], "ref_ids": [] }, { "id": 14346, "type": "theorem", "label": "derham-lemma-comparison-general", "categories": [ "derham" ], "title": "derham-lemma-comparison-general", "contents": [ "With notation as in Lemma \\ref{lemma-blowup} for $a \\geq 0$ we have", "\\begin{enumerate}", "\\item the map", "$\\Omega^a_{X/S} \\to b_*\\Omega^a_{X'/S}$ is an isomorphism,", "\\item the map $\\Omega^a_{Z/S} \\to p_*\\Omega^a_{E/S}$ is an isomorphism,", "\\item the map $Rb_*\\Omega^a_{X'/S} \\to i_*Rp_*\\Omega^a_{E/S}$ is an isomorphism", "on cohomology sheaves in degree $\\geq 1$.", "\\end{enumerate}" ], "refs": [ "derham-lemma-blowup" ], "proofs": [ { "contents": [ "Let $\\epsilon : X_1 \\to X$ be a surjective \\'etale morphism. Denote", "$i_1 : Z_1 \\to X_1$, $b_1 : X'_1 \\to X_1$, $E_1 \\subset X'_1$, and", "$p_1 : E_1 \\to Z_1$ the base changes of the objects considered in", "Lemma \\ref{lemma-blowup}. Observe that $i_1$ is a closed immersion", "of schemes smooth over $S$ and that $b_1$ is the blowing up with center", "$Z_1$ by Divisors, Lemma \\ref{divisors-lemma-flat-base-change-blowing-up}.", "Suppose that we can prove (1), (2), and (3)", "for the morphisms $b_1$, $p_1$, and $i_1$. Then by", "Lemma \\ref{lemma-etale} we obtain that the pullback by $\\epsilon$", "of the maps in (1), (2), and (3) are isomorphisms. As $\\epsilon$", "is a surjective flat morphism we conclude.", "Thus working \\'etale locally, by", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-etale-local-structure},", "we may assume we are in the situation discussed in", "Section \\ref{section-calculations}. In this case the lemma", "is the same as Lemma \\ref{lemma-comparison}." ], "refs": [ "derham-lemma-blowup", "divisors-lemma-flat-base-change-blowing-up", "derham-lemma-etale", "more-morphisms-lemma-etale-local-structure", "derham-lemma-comparison" ], "ref_ids": [ 14345, 8053, 14314, 13882, 14342 ] } ], "ref_ids": [ 14345 ] }, { "id": 14347, "type": "theorem", "label": "derham-lemma-distinguished-triangle-blowup", "categories": [ "derham" ], "title": "derham-lemma-distinguished-triangle-blowup", "contents": [ "With notation as in Lemma \\ref{lemma-blowup} and denoting $f : X \\to S$", "the structure morphism there is a canonical", "distinguished triangle", "$$", "\\Omega^\\bullet_{X/S} \\to", "Rb_*(\\Omega^\\bullet_{X'/S}) \\oplus i_*\\Omega^\\bullet_{Z/S} \\to", "i_*Rp_*(\\Omega^\\bullet_{E/S}) \\to", "\\Omega^\\bullet_{X/S}[1]", "$$", "in $D(X, f^{-1}\\mathcal{O}_S)$ where the four maps", "$$", "\\begin{matrix}", "\\Omega^\\bullet_{X/S} & \\to & Rb_*(\\Omega^\\bullet_{X'/S}), \\\\", "\\Omega^\\bullet_{X/S} & \\to & i_*\\Omega^\\bullet_{Z/S}, \\\\", "Rb_*(\\Omega^\\bullet_{X'/S}) & \\to & i_*Rp_*(\\Omega^\\bullet_{E/S}), \\\\", "i_*\\Omega^\\bullet_{Z/S} & \\to & i_*Rp_*(\\Omega^\\bullet_{E/S})", "\\end{matrix}", "$$", "are the canonical ones (Section \\ref{section-de-rham-complex}),", "except with sign reversed for one of them." ], "refs": [ "derham-lemma-blowup" ], "proofs": [ { "contents": [ "Choose a distinguished triangle", "$$", "C \\to Rb_*\\Omega^\\bullet_{X'/S} \\oplus i_*\\Omega^\\bullet_{Z/S}", "\\to i_*Rp_*\\Omega^\\bullet_{E/S} \\to C[1]", "$$", "in $D(X, f^{-1}\\mathcal{O}_S)$. It suffices to show that", "$\\Omega^\\bullet_{X/S}$ is isomorphic to $C$ in a manner compatible", "with the canonical maps. By the axioms of triangulated categories", "there exists a map of distinguished triangles", "$$", "\\xymatrix{", "C' \\ar[r] \\ar[d] &", "b_*\\Omega^\\bullet_{X'/S} \\oplus i_*\\Omega^\\bullet_{Z/S} \\ar[r] \\ar[d] &", "i_*p_*\\Omega^\\bullet_{E/S} \\ar[r] \\ar[d] &", "C'[1] \\ar[d] \\\\", "C \\ar[r] &", "Rb_*\\Omega^\\bullet_{X'/S} \\oplus i_*\\Omega^\\bullet_{Z/S} \\ar[r] &", "i_*Rp_*\\Omega^\\bullet_{E/S} \\ar[r] &", "C[1]", "}", "$$", "By Lemma \\ref{lemma-comparison-general} part (3) and", "Derived Categories, Proposition \\ref{derived-proposition-9} we conclude that", "$C' \\to C$ is an isomorphism. By Lemma \\ref{lemma-comparison-general} part (2)", "the map $i_*\\Omega^\\bullet_{Z/S} \\to i_*p_*\\Omega^\\bullet_{E/S}$", "is an isomorphism. Thus $C' = b_*\\Omega^\\bullet_{X'/S}$", "in the derived category. Finally we use Lemma \\ref{lemma-comparison-general}", "part (1) tells us this is equal to $\\Omega^\\bullet_{X/S}$.", "We omit the verification this is compatible with the canonical maps." ], "refs": [ "derham-lemma-comparison-general", "derived-proposition-9", "derham-lemma-comparison-general", "derham-lemma-comparison-general" ], "ref_ids": [ 14346, 1958, 14346, 14346 ] } ], "ref_ids": [ 14345 ] }, { "id": 14348, "type": "theorem", "label": "derham-lemma-ext-zero", "categories": [ "derham" ], "title": "derham-lemma-ext-zero", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes which is regular of", "codimension $c$. Then $\\Ext^q_{\\mathcal{O}_X}(i_*\\mathcal{F}, \\mathcal{E}) = 0$", "for $q < c$ for $\\mathcal{E}$ locally free on $X$ and $\\mathcal{F}$", "any $\\mathcal{O}_Z$-module." ], "refs": [], "proofs": [ { "contents": [ "By the local to global spectral sequence of $\\Ext$ it suffices", "to prove this affine locally on $X$. See", "Cohomology, Section \\ref{cohomology-section-ext}.", "Thus we may assume $X = \\Spec(A)$", "and there exists a regular sequence $f_1, \\ldots, f_c$ in $A$", "such that $Z = \\Spec(A/(f_1, \\ldots, f_c))$. We may assume $c \\geq 1$.", "Then we see that $f_1 : \\mathcal{E} \\to \\mathcal{E}$", "is injective. Since $i_*\\mathcal{F}$ is annihilated by $f_1$", "this shows that the lemma holds for $i = 0$ and that we have", "a surjection", "$$", "\\Ext^{q - 1}_{\\mathcal{O}_X}(i_*\\mathcal{F}, \\mathcal{E}/f_1\\mathcal{E})", "\\longrightarrow", "\\Ext^q_{\\mathcal{O}_X}(i_*\\mathcal{F}, \\mathcal{E})", "$$", "Thus it suffices to show that the source of this arrow is zero.", "Next we repeat this argument: if $c \\geq 2$ the map", "$f_2 : \\mathcal{E}/f_1\\mathcal{E} \\to \\mathcal{E}/f_1\\mathcal{E}$", "is injective. Since $i_*\\mathcal{F}$ is annihilated by $f_2$", "this shows that the lemma holds for $q = 1$ and that we have a", "surjection", "$$", "\\Ext^{q - 2}_{\\mathcal{O}_X}(i_*\\mathcal{F},", "\\mathcal{E}/f_1\\mathcal{E} + f_2\\mathcal{E})", "\\longrightarrow", "\\Ext^{q - 1}_{\\mathcal{O}_X}(i_*\\mathcal{F}, \\mathcal{E}/f_1\\mathcal{E})", "$$", "Continuing in this fashion the lemma is proved." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14349, "type": "theorem", "label": "derham-lemma-splitting-on-omega-a", "categories": [ "derham" ], "title": "derham-lemma-splitting-on-omega-a", "contents": [ "With notation as in Lemma \\ref{lemma-blowup} for $a \\geq 0$", "there is a unique arrow", "$Rb_*\\Omega^a_{X'/S} \\to \\Omega^a_{X/S}$ in $D(\\mathcal{O}_X)$", "whose composition with $\\Omega^a_{X/S} \\to Rb_*\\Omega^a_{X'/S}$", "is the identity on $\\Omega^a_{X/S}$." ], "refs": [ "derham-lemma-blowup" ], "proofs": [ { "contents": [ "We may decompose $X$ into open and closed subschemes", "having fixed relative dimension to $S$, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-omega-finite-locally-free}.", "Since the derived category $D(X, f^{-1}\\mathcal{O})_S)$ correspondingly", "decomposes as a product of categories, we may assume $X$ has", "fixed relative dimension $N$ over $S$. We may decompose", "$Z = \\coprod Z_m$ into open and closed subschemes of relative", "dimension $m \\geq 0$ over $S$. The restriction $i_m : Z_m \\to X$ of", "$i$ to $Z_m$ is a regular immersion of codimension $N - m$, see Divisors, Lemma", "\\ref{divisors-lemma-immersion-smooth-into-smooth-regular-immersion}.", "Let $E = \\coprod E_m$ be the corresponding decomposition, i.e.,", "we set $E_m = p^{-1}(Z_m)$. We claim that there are natural maps", "$$", "b^*\\Omega^a_{X/S} \\to \\Omega^a_{X'/S} \\to", "b^*\\Omega^a_{X/S} \\otimes_{\\mathcal{O}_{X'}}", "\\mathcal{O}_{X'}(\\sum (N - m - 1)E_m)", "$$", "whose composition is induced by the inclusion", "$\\mathcal{O}_{X'} \\to \\mathcal{O}_{X'}(\\sum (N - m - 1)E_m)$.", "Namely, in order to prove this, it suffices to show that the", "cokernel of the first arrow is locally on $X'$ annihilated by", "a local equation of the effective Cartier divisor $\\sum (N - m - 1)E_m$.", "To see this in turn we can work \\'etale locally on $X$ as in the", "proof of Lemma \\ref{lemma-comparison-general} and apply", "Lemma \\ref{lemma-comparison-bis}.", "Computing \\'etale locally using Lemma \\ref{lemma-blowup-twist-same-cohomology}", "we see that the induced composition", "$$", "\\Omega^a_{X/S} \\to Rb_*\\Omega^a_{X'/S} \\to", "Rb_*\\left(b^*\\Omega^a_{X/S} \\otimes_{\\mathcal{O}_{X'}}", "\\mathcal{O}_{X'}(\\sum (N - m - 1)E_m)\\right)", "$$", "is an isomorphism in $D(\\mathcal{O}_X)$", "which is how we obtain the existence of the map in the lemma.", "\\medskip\\noindent", "For uniqueness, it suffices to show that there are no nonzero maps from", "$\\tau_{\\geq 1}Rb_*\\Omega_{X'/S}$ to $\\Omega^a_{X/S}$ in $D(\\mathcal{O}_X)$.", "For this it suffices in turn to show that there are no nonzero maps", "from $R^qb_*\\Omega_{X'/s}[-q]$ to $\\Omega^a_{X/S}$ in $D(\\mathcal{O}_X)$", "for $q \\geq 1$ (details omitted). By", "Lemma \\ref{lemma-comparison-general}", "we see that $R^qb_*\\Omega_{X'/s} \\cong i_*R^qp_*\\Omega^a_{E/S}$", "is the pushforward of a module on $Z = \\coprod Z_m$.", "Moreover, observe that the restriction of $R^qp_*\\Omega^a_{E/S}$", "to $Z_m$ is nonzero only for $q < N - m$. Namely, the fibres of", "$E_m \\to Z_m$ have dimension $N - m - 1$ and we can apply Limits, Lemma", "\\ref{limits-lemma-higher-direct-images-zero-above-dimension-fibre}.", "Thus the desired vanishing follows from Lemma \\ref{lemma-ext-zero}." ], "refs": [ "morphisms-lemma-smooth-omega-finite-locally-free", "divisors-lemma-immersion-smooth-into-smooth-regular-immersion", "derham-lemma-comparison-general", "derham-lemma-comparison-bis", "derham-lemma-blowup-twist-same-cohomology", "derham-lemma-comparison-general", "limits-lemma-higher-direct-images-zero-above-dimension-fibre", "derham-lemma-ext-zero" ], "ref_ids": [ 5334, 8007, 14346, 14343, 14344, 14346, 15109, 14348 ] } ], "ref_ids": [ 14345 ] }, { "id": 14350, "type": "theorem", "label": "derham-lemma-funny-map", "categories": [ "derham" ], "title": "derham-lemma-funny-map", "contents": [ "Let $R$ be a ring and consider a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "K^0 \\ar[r] &", "L^0 \\ar[r] &", "M^0 \\ar[r] & 0 \\\\", "& & L^{-1} \\ar[u]_\\partial \\ar@{=}[r] &", "M^{-1} \\ar[u]", "}", "$$", "of $R$-modules with exact top row and $M^0$ and $M^{-1}$", "finite free of the same rank. Then there are canonical maps", "$$", "\\wedge^i(H^0(L^\\bullet)) \\longrightarrow \\wedge^i(K^0) \\otimes_R \\det(M^\\bullet)", "$$", "whose composition with $\\wedge^i(K^0) \\to \\wedge^i(H^0(L^\\bullet))$", "is equal to multiplication with $\\delta(M^\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "Say $M^0$ and $M^{-1}$ are free of rank $n$. For every $i \\geq 0$", "there is a canonical surjection", "$$", "\\pi_i :", "\\wedge^{n + i}(L^0)", "\\longrightarrow", "\\wedge^i(K^0) \\otimes \\wedge^n(M^0)", "$$", "whose kernel is the submodule generated by wedges", "$l_1 \\wedge \\ldots \\wedge l_{n + i}$ such that $> i$ of the", "$l_j$ are in $K^0$. On the other hand, the exact sequence", "$$", "L^{-1} \\to L^0 \\to H^0(L^\\bullet) \\to 0", "$$", "similarly produces canonical maps", "$$", "\\wedge^i(H^0(L^\\bullet)) \\otimes \\wedge^n(L^{-1})", "\\longrightarrow", "\\wedge^{n + i}(L^0)", "$$", "by sending $\\eta \\otimes \\theta$ to $\\tilde \\eta \\wedge \\partial(\\theta)$", "where $\\tilde \\eta \\in \\wedge^i(L^0)$ is a lift of $\\eta$.", "The composition of these two maps, combined with the identification", "$\\wedge^n(L^{-1}) = \\wedge^n(M^{-1})$ gives a map", "$$", "\\wedge^i(H^0(L^\\bullet)) \\otimes \\wedge^n(M^{-1})", "\\longrightarrow", "\\wedge^i(K^0) \\otimes \\wedge^n(M^0)", "$$", "Since $\\det(M^\\bullet) = \\wedge^n(M^0) \\otimes", "(\\wedge^n(M^{-1}))^{\\otimes -1}$ this produces a map as", "in the statement of the lemma.", "If $\\eta$ is the image of $\\omega \\in \\wedge^i(K^0)$, then we see", "that $\\theta \\otimes \\eta$ is mapped to", "$\\pi_i(\\omega \\wedge \\partial(\\theta)) = \\omega \\otimes \\overline{\\theta}$ in", "$\\wedge^i(K^0) \\otimes \\wedge^n(M^0)$ where $\\overline{\\theta}$", "is the image of $\\theta$ in $\\wedge^n(M^0)$. Since", "$\\delta(M^\\bullet)$ is simply the determinant of the map", "$M^{-1} \\to M^0$ this proves the last statement." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14351, "type": "theorem", "label": "derham-lemma-Garel-upstairs", "categories": [ "derham" ], "title": "derham-lemma-Garel-upstairs", "contents": [ "There exists a unique rule that to every locally quasi-finite syntomic", "morphism of schemes $f : Y \\to X$ assigns $\\mathcal{O}_Y$-module maps", "$$", "c^p_{Y/X} :", "\\Omega^p_{Y/\\mathbf{Z}}", "\\longrightarrow", "f^*\\Omega^p_{X/\\mathbf{Z}} \\otimes_{\\mathcal{O}_Y} \\det(\\NL_{Y/X})", "$$", "satisfying the following two properties", "\\begin{enumerate}", "\\item the composition with", "$f^*\\Omega^p_{X/\\mathbf{Z}} \\to \\Omega^p_{Y/\\mathbf{Z}}$", "is multiplication by $\\delta(\\NL_{Y/X})$, and", "\\item the rule is compatible with restriction to opens and with", "base change.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This proof is very similar to the proof of", "Discriminants, Proposition \\ref{discriminant-proposition-tate-map}", "and we suggest the reader look at that proof first.", "We fix $p \\geq 0$ throughout the proof.", "\\medskip\\noindent", "Let us reformulate the statement. Consider the category", "$\\mathcal{C}$ whose objects, denoted $Y/X$, are locally quasi-finite syntomic", "morphism $f : Y \\to X$ of schemes and whose morphisms", "$b/a : Y'/X' \\to Y/X$ are commutative diagrams", "$$", "\\xymatrix{", "Y' \\ar[d]_{f'} \\ar[r]_b & Y \\ar[d]^f \\\\", "X' \\ar[r]^a & X", "}", "$$", "which induce an isomorphism of $Y'$ with an open subscheme of", "$X' \\times_X Y$. The lemma means that for every object", "$Y/X$ of $\\mathcal{C}$ we have maps $c^p_{Y/X}$ with property (1)", "and for every morphism $b/a : Y'/X' \\to Y/X$ of $\\mathcal{C}$ we have", "$b^*c^p_{Y/X} = c^p_{Y'/X'}$ via the identifications", "$b^*\\det(\\NL_{Y/X}) = \\det(\\NL_{Y'/X'})$", "(Discriminants, Section \\ref{discriminant-section-tate-map})", "and $b^*\\Omega^p_{Y/X} = \\Omega^p_{Y'/X'}$", "(Lemma \\ref{lemma-base-change-de-rham}).", "\\medskip\\noindent", "Given $Y/X$ in $\\mathcal{C}$ and $y \\in Y$ we can find", "an affine open $V \\subset Y$ and $U \\subset X$ with $f(V) \\subset U$", "such that there exists some maps", "$$", "\\Omega^p_{Y/\\mathbf{Z}}|_V", "\\longrightarrow", "\\left(", "f^*\\Omega^p_{X/\\mathbf{Z}} \\otimes_{\\mathcal{O}_Y} \\det(\\NL_{Y/X})", "\\right)|_V", "$$", "with property (1). This follows", "from picking affine opens as in", "Discriminants, Lemma \\ref{discriminant-lemma-syntomic-quasi-finite} part (5)", "and Remark \\ref{remark-local-description}.", "If $\\Omega^p_{X/\\mathbf{Z}}$ is finite locally free and", "annihilator of the section $\\delta(\\NL_{Y/X})$ is zero, then", "these local maps are unique and automatically glue!", "\\medskip\\noindent", "Let $\\mathcal{C}_{nice} \\subset \\mathcal{C}$ denote the full subcategory", "of $Y/X$ such that", "\\begin{enumerate}", "\\item $X$ is of finite type over $\\mathbf{Z}$,", "\\item $\\Omega_{X/\\mathbf{Z}}$ is locally free, and", "\\item the annihilator of $\\delta(\\NL_{Y/X})$ is zero.", "\\end{enumerate}", "By the remarks in the previous paragraph, we see that for any", "object $Y/X$ of $\\mathcal{C}_{nice}$ we have a unique map", "$c^p_{Y/X}$ satisfying condition (1). If $b/a : Y'/X' \\to Y/X$", "is a morphism of $\\mathcal{C}_{nice}$, then", "$b^*c^p_{Y/X}$ is equal to $c^p_{Y'/X'}$ because", "$b^*\\delta(\\NL_{Y/X}) = \\delta(\\NL_{Y'/X'})$ (see", "Discriminants, Section \\ref{discriminant-section-tate-map}).", "In other words, we have solved the problem", "on the full subcategory $\\mathcal{C}_{nice}$. For $Y/X$ in $\\mathcal{C}_{nice}$", "we continue to denote $c^p_{Y/X}$ the solution we've just found.", "\\medskip\\noindent", "Consider morphisms", "$$", "Y_1/X_1 \\xleftarrow{b_1/a_1} Y/X \\xrightarrow{b_2/a_2} Y_2/X_2", "$$", "in $\\mathcal{C}$ such that $Y_1/X_1$ and $Y_2/X_2$ are objects", "of $\\mathcal{C}_{nice}$.", "{\\bf Claim.} $b_1^*c^p_{Y_1/X_1} = b_2^*c^p_{Y_2/X_2}$.", "We will first show that the claim implies the lemma", "and then we will prove the claim.", "\\medskip\\noindent", "Let $d, n \\geq 1$ and consider the locally", "quasi-finite syntomic morphism $Y_{n, d} \\to X_{n, d}$", "constructed in Discriminants, Example", "\\ref{discriminant-example-universal-quasi-finite-syntomic}.", "Then $Y_{n, d}$ and $Y_{n, d}$ are irreducible schemes of finite type and", "smooth over $\\mathbf{Z}$. Namely, $X_{n, d}$ is a spectrum of a", "polynomial ring over $\\mathbf{Z}$ and $Y_{n, d}$ is an open subscheme", "of such. The morphism $Y_{n, d} \\to X_{n, d}$ is locally quasi-finite syntomic", "and \\'etale over a dense open, see Discriminants, Lemma", "\\ref{discriminant-lemma-universal-quasi-finite-syntomic-etale}.", "Thus $\\delta(\\NL_{Y_{n, d}/X_{n, d}})$ is nonzero: for example we have", "the local description of $\\delta(\\NL_{Y/X})$ in", "Discriminants, Remark \\ref{discriminant-remark-local-description-delta}", "and we have the local description of \\'etale morphisms in", "Morphisms, Lemma \\ref{morphisms-lemma-etale-at-point} part (8).", "Now a nonzero section of an invertible module over an irreducible", "regular scheme has vanishing annihilator. Thus", "$Y_{n, d}/X_{n, d}$ is an object of $\\mathcal{C}_{nice}$.", "\\medskip\\noindent", "Let $Y/X$ be an arbitrary object of $\\mathcal{C}$. Let $y \\in Y$.", "By Discriminants, Lemma \\ref{discriminant-lemma-locally-comes-from-universal}", "we can find $n, d \\geq 1$ and morphisms", "$$", "Y/X \\leftarrow V/U \\xrightarrow{b/a} Y_{n, d}/X_{n, d}", "$$", "of $\\mathcal{C}$ such that $V \\subset Y$ and $U \\subset X$ are open.", "Thus we can pullback the canonical morphism $c^p_{Y_{n, d}/X_{n, d}}$", "constructed above by $b$ to $V$. The claim guarantees these local", "isomorphisms glue! Thus we get a well defined global maps", "$c^p_{Y/X}$ with property (1).", "If $b/a : Y'/X' \\to Y/X$ is a morphism of $\\mathcal{C}$, then", "the claim also implies that the similarly constructed map", "$c^p_{Y'/X'}$ is the pullback by $b$ of the locally constructed", "map $c^p_{Y/X}$. Thus it remains to prove the claim.", "\\medskip\\noindent", "In the rest of the proof we prove the claim. We may pick a point", "$y \\in Y$ and prove the maps agree in an open neighbourhood of $y$.", "Thus we may replace $Y_1$, $Y_2$ by open neighbourhoods of the", "image of $y$ in $Y_1$ and $Y_2$. Thus we may assume", "$Y, X, Y_1, X_1, Y_2, X_2$ are affine.", "We may write $X = \\lim X_\\lambda$ as a cofiltered limit of affine", "schemes of finite type over $X_1 \\times X_2$. For each $\\lambda$", "we get", "$$", "Y_1 \\times_{X_1} X_\\lambda", "\\quad\\text{and}\\quad", "X_\\lambda \\times_{X_2} Y_2", "$$", "If we take limits we obtain", "$$", "\\lim Y_1 \\times_{X_1} X_\\lambda =", "Y_1 \\times_{X_1} X \\supset Y \\subset", "X \\times_{X_2} Y_2 = \\lim X_\\lambda \\times_{X_2} Y_2", "$$", "By Limits, Lemma \\ref{limits-lemma-descend-opens}", "we can find a $\\lambda$ and opens", "$V_{1, \\lambda} \\subset Y_1 \\times_{X_1} X_\\lambda$ and", "$V_{2, \\lambda} \\subset X_\\lambda \\times_{X_2} Y_2$", "whose base change to $X$ recovers $Y$ (on both sides).", "After increasing $\\lambda$ we may assume", "there is an isomorphism", "$V_{1, \\lambda} \\to V_{2, \\lambda}$ whose base change to $X$ is the", "identity on $Y$, see", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}.", "Then we have the commutative diagram", "$$", "\\xymatrix{", "& Y/X \\ar[d] \\ar[ld]_{b_1/a_1} \\ar[rd]^{b_2/a_2} \\\\", "Y_1/X_1 & V_{1, \\lambda}/X_\\lambda \\ar[l] \\ar[r] & Y_2/X_2", "}", "$$", "Thus it suffices to prove the claim for the lower row", "of the diagram and we reduce to the case discussed in the", "next paragraph.", "\\medskip\\noindent", "Assume $Y, X, Y_1, X_1, Y_2, X_2$ are affine of finite type over $\\mathbf{Z}$.", "Write $X = \\Spec(A)$, $X_i = \\Spec(A_i)$. The ring map $A_1 \\to A$ corresponding", "to $X \\to X_1$ is of finite type and hence we may choose a surjection", "$A_1[x_1, \\ldots, x_n] \\to A$. Similarly, we may choose a surjection", "$A_2[y_1, \\ldots, y_m] \\to A$. Set $X'_1 = \\Spec(A_1[x_1, \\ldots, x_n])$", "and $X'_2 = \\Spec(A_2[y_1, \\ldots, y_m])$. Observe that", "$\\Omega_{X'_1/\\mathbf{Z}}$ is the direct sum of the pullback of", "$\\Omega_{X_1/\\mathbf{Z}}$ and a finite free module.", "Similarly for $X'_2$. Set $Y'_1 = Y_1 \\times_{X_1} X'_1$ and", "$Y'_2 = Y_2 \\times_{X_2} X'_2$. We get the following diagram", "$$", "Y_1/X_1 \\leftarrow", "Y'_1/X'_1 \\leftarrow", "Y/X", "\\rightarrow Y'_2/X'_2", "\\rightarrow Y_2/X_2", "$$", "Since $X'_1 \\to X_1$ and $X'_2 \\to X_2$ are flat, the same is true", "for $Y'_1 \\to Y_1$ and $Y'_2 \\to Y_2$. It follows easily that the", "annihilators of $\\delta(\\NL_{Y'_1/X'_1})$ and $\\delta(\\NL_{Y'_2/X'_2})$", "are zero.", "Hence $Y'_1/X'_1$ and $Y'_2/X'_2$ are in $\\mathcal{C}_{nice}$.", "Thus the outer morphisms in the displayed diagram are morphisms", "of $\\mathcal{C}_{nice}$ for which we know the desired compatibilities.", "Thus it suffices to prove the claim for", "$Y'_1/X'_1 \\leftarrow Y/X \\rightarrow Y'_2/X'_2$. This reduces us", "to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $Y, X, Y_1, X_1, Y_2, X_2$ are affine of finite type over", "$\\mathbf{Z}$ and $X \\to X_1$ and $X \\to X_2$ are closed immersions.", "Consider the open embeddings", "$Y_1 \\times_{X_1} X \\supset Y \\subset X \\times_{X_2} Y_2$.", "There is an open neighbourhood $V \\subset Y$ of $y$ which is a", "standard open of both $Y_1 \\times_{X_1} X$ and $X \\times_{X_2} Y_2$.", "This follows from Schemes, Lemma \\ref{schemes-lemma-standard-open-two-affines}", "applied to the scheme obtained by glueing $Y_1 \\times_{X_1} X$ and", "$X \\times_{X_2} Y_2$ along $Y$; details omitted.", "Since $X \\times_{X_2} Y_2$ is a closed subscheme of $Y_2$", "we can find a standard open $V_2 \\subset Y_2$ such that", "$V_2 \\times_{X_2} X = V$. Similarly, we can find a standard open", "$V_1 \\subset Y_1$ such that $V_1 \\times_{X_1} X = V$.", "After replacing $Y, Y_1, Y_2$ by $V, V_1, V_2$ we reduce to the", "case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $Y, X, Y_1, X_1, Y_2, X_2$ are affine of finite type over", "$\\mathbf{Z}$ and $X \\to X_1$ and $X \\to X_2$ are closed immersions", "and $Y_1 \\times_{X_1} X = Y = X \\times_{X_2} Y_2$.", "Write $X = \\Spec(A)$, $X_i = \\Spec(A_i)$, $Y = \\Spec(B)$,", "$Y_i = \\Spec(B_i)$. Then we can consider the affine schemes", "$$", "X' = \\Spec(A_1 \\times_A A_2) = \\Spec(A')", "\\quad\\text{and}\\quad", "Y' = \\Spec(B_1 \\times_B B_2) = \\Spec(B')", "$$", "Observe that $X' = X_1 \\amalg_X X_2$ and $Y' = Y_1 \\amalg_Y Y_2$, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-basic-example-pushout}.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-fibre-product-finite-type}", "the rings $A'$ and $B'$ are of finite type over $\\mathbf{Z}$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-module-over-fibre-product}", "we have $B' \\otimes_A A_1 = B_1$ and $B' \\times_A A_2 = B_2$.", "In particular a fibre of $Y' \\to X'$ over a point of", "$X' = X_1 \\amalg_X X_2$ is always equal to either a fibre of $Y_1 \\to X_1$", "or a fibre of $Y_2 \\to X_2$. By More on Algebra, Lemma", "\\ref{more-algebra-lemma-flat-module-over-fibre-product}", "the ring map $A' \\to B'$ is flat. Thus by Discriminants, Lemma", "\\ref{discriminant-lemma-syntomic-quasi-finite} part (3)", "we conclude that $Y'/X'$ is an object of $\\mathcal{C}$.", "Consider now the commutative diagram", "$$", "\\xymatrix{", "& Y/X \\ar[ld]_{b_1/a_1} \\ar[rd]^{b_2/a_2} \\\\", "Y_1/X_1 \\ar[rd] & & Y_2/X_2 \\ar[ld] \\\\", "& Y'/X'", "}", "$$", "Now we would be done if $Y'/X'$ is an object of $\\mathcal{C}_{nice}$,", "but this is almost never the case. Namely, then pulling back $c^p_{Y'/X'}$", "around the two sides of the square, we would obtain the desired conclusion.", "To get around the problem that $Y'/X'$ is not in $\\mathcal{C}_{nice}$", "we note the arguments above show that, after possibly shrinking all", "of the schemes $X, Y, X_1, Y_1, X_2, Y_2, X', Y'$ we can find some", "$n, d \\geq 1$, and extend the diagram like so:", "$$", "\\xymatrix{", "& Y/X \\ar[ld]_{b_1/a_1} \\ar[rd]^{b_2/a_2} \\\\", "Y_1/X_1 \\ar[rd] & & Y_2/X_2 \\ar[ld] \\\\", "& Y'/X' \\ar[d] \\\\", "& Y_{n, d}/X_{n, d}", "}", "$$", "and then we can use the already given argument by pulling", "back from $c^p_{Y_{n, d}/X_{n, d}}$. This finishes the proof." ], "refs": [ "discriminant-proposition-tate-map", "derham-lemma-base-change-de-rham", "discriminant-lemma-syntomic-quasi-finite", "derham-remark-local-description", "discriminant-lemma-universal-quasi-finite-syntomic-etale", "discriminant-remark-local-description-delta", "morphisms-lemma-etale-at-point", "discriminant-lemma-locally-comes-from-universal", "limits-lemma-descend-opens", "limits-lemma-descend-finite-presentation", "schemes-lemma-standard-open-two-affines", "more-algebra-lemma-fibre-product-finite-type", "more-algebra-lemma-module-over-fibre-product", "more-algebra-lemma-flat-module-over-fibre-product", "discriminant-lemma-syntomic-quasi-finite" ], "ref_ids": [ 15002, 14313, 14981, 14390, 14985, 15010, 5372, 14986, 15041, 15077, 7675, 9814, 9820, 9824, 14981 ] } ], "ref_ids": [] }, { "id": 14352, "type": "theorem", "label": "derham-lemma-Garel", "categories": [ "derham" ], "title": "derham-lemma-Garel", "contents": [ "There exists a unique rule that to every finite syntomic", "morphism of schemes $f : Y \\to X$ assigns $\\mathcal{O}_X$-module maps", "$$", "\\Theta^p_{Y/X} :", "f_*\\Omega^p_{Y/\\mathbf{Z}}", "\\longrightarrow", "\\Omega^p_{X/\\mathbf{Z}}", "$$", "satisfying the following properties", "\\begin{enumerate}", "\\item the composition with", "$\\Omega^p_{X/\\mathbf{Z}} \\otimes_{\\mathcal{O}_X} f_*\\mathcal{O}_Y", "\\to f_*\\Omega^p_{Y/\\mathbf{Z}}$ is equal to", "$\\text{id} \\otimes \\text{Trace}_f$", "where $\\text{Trace}_f : f_*\\mathcal{O}_Y \\to \\mathcal{O}_X$", "is the map from", "Discriminants, Section \\ref{discriminant-section-discriminant},", "\\item the rule is compatible with base change.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "First, assume that $X$ is locally Noetherian. By", "Lemma \\ref{lemma-Garel-upstairs} we have a canonical map", "$$", "c^p_{Y/X} : \\Omega_{Y/S}^p", "\\longrightarrow", "f^*\\Omega_{X/S}^p \\otimes_{\\mathcal{O}_Y} \\det(\\NL_{Y/X})", "$$", "By Discriminants, Proposition \\ref{discriminant-proposition-tate-map}", "we have a canonical isomorphism", "$$", "c_{Y/X} : \\det(\\NL_{Y/X}) \\to \\omega_{Y/X}", "$$", "mapping $\\delta(\\NL_{Y/X})$ to $\\tau_{Y/X}$. Combined these maps give", "$$", "c^p_{Y/X} \\otimes c_{Y/X} :", "\\Omega_{Y/S}^p", "\\longrightarrow", "f^*\\Omega_{X/S}^p \\otimes_{\\mathcal{O}_Y} \\omega_{Y/X}", "$$", "By Discriminants, Section \\ref{discriminant-section-finite-morphisms}", "this is the same thing as a map", "$$", "\\Theta_{Y/X}^p :", "f_*\\Omega_{Y/S}^p", "\\longrightarrow", "\\Omega_{X/S}^p", "$$", "Recall that the relationship between $c^p_{Y/X} \\otimes c_{Y/X}$", "and $\\Theta_{Y/X}^p$ uses the evaluation map", "$f_*\\omega_{Y/X} \\to \\mathcal{O}_X$", "which sends $\\tau_{Y/X}$ to $\\text{Trace}_f(1)$, see", "Discriminants, Section \\ref{discriminant-section-finite-morphisms}.", "Hence property (1) holds. Property (2) holds for base changes by", "$X' \\to X$ with $X'$ locally Noetherian because both $c^p_{Y/X}$ and", "$c_{Y/X}$ are compatible with such base changes. For $f : Y \\to X$", "finite syntomic and $X$ locally Noetherian,", "we will continue to denote $\\Theta^p_{Y/X}$ the solution we've just found.", "\\medskip\\noindent", "Uniqueness. Suppose that we have a finite syntomic morphism", "$f: Y \\to X$ such that $X$ is smooth over $\\Spec(\\mathbf{Z})$", "and $f$ is \\'etale over a dense open of $X$. We claim that", "in this case $\\Theta^p_{Y/X}$ is uniquely determined by property (1).", "Namely, consider the maps", "$$", "\\Omega^p_{X/\\mathbf{Z}} \\otimes_{\\mathcal{O}_X} f_*\\mathcal{O}_Y \\to", "f_*\\Omega^p_{Y/\\mathbf{Z}} \\to", "\\Omega^p_{X/\\mathbf{Z}}", "$$", "The sheaf $\\Omega^p_{X/\\mathbf{Z}}$ is torsion free (by the assumed", "smoothness), hence it suffices to check that the restriction of", "$\\Theta^p_{Y/X}$ is uniquely determined over the dense open over", "which $f$ is \\'etale, i.e., we may assume $f$ is \\'etale.", "However, if $f$ is \\'etale, then", "$f^*\\Omega_{X/\\mathbf{Z}} = \\Omega_{Y/\\mathbf{Z}}$", "hence the first arrow in the displayed equation is an isomorphism.", "Since we've pinned down the composition, this guarantees uniqueness.", "\\medskip\\noindent", "Let $f : Y \\to X$ be a finite syntomic morphism of locally Noetherian schemes.", "Let $x \\in X$. By Discriminants, Lemma", "\\ref{discriminant-lemma-locally-comes-from-universal-finite}", "we can find $d \\geq 1$ and a commutative diagram", "$$", "\\xymatrix{", "Y \\ar[d] &", "V \\ar[d] \\ar[l] \\ar[r] &", "V_d \\ar[d] \\\\", "X &", "U \\ar[l] \\ar[r] &", "U_d", "}", "$$", "such that $x \\in U \\subset X$ is open, $V = f^{-1}(U)$", "and $V = U \\times_{U_d} V_d$. Thus $\\Theta^p_{Y/X}|_V$", "is the pullback of the map $\\Theta^p_{V_d/U_d}$.", "However, by the discussion on uniqueness above and", "Discriminants, Lemmas", "\\ref{discriminant-lemma-universal-finite-syntomic-smooth} and", "\\ref{discriminant-lemma-universal-finite-syntomic-etale}", "the map $\\Theta^p_{V_d/U_d}$ is uniquely determined", "by the requirement (1). Hence uniqueness holds.", "\\medskip\\noindent", "At this point we know that we have existence and uniqueness", "for all finite syntomic morphisms $Y \\to X$ with $X$ locally Noetherian.", "We could now give an argument similar to the proof of", "Lemma \\ref{lemma-Garel-upstairs} to extend to general $X$.", "However, instead it possible to directly use absolute Noetherian approximation", "to finish the proof. Namely, to construct $\\Theta^p_{Y/X}$", "it suffices to do so Zariski locally on $X$ (provided we also", "show the uniqueness). Hence we may assume $X$ is affine (small", "detail omitted). Then we can write $X = \\lim_{i \\in I} X_i$", "as the limit over a directed set $I$ of Noetherian affine schemes.", "By Algebra, Lemma \\ref{algebra-lemma-colimit-category-fp-algebras}", "we can find $0 \\in I$ and a finitely", "presented morphism of affines $f_0 : Y_0 \\to X_0$ whose base change to", "$X$ is $Y \\to X$. After increasing $0$ we may assume $Y_0 \\to X_0$", "is finite and syntomic, see", "Algebra, Lemma \\ref{algebra-lemma-colimit-lci} and", "\\ref{algebra-lemma-colimit-finite}. For $i \\geq 0$ also the", "base change $f_i : Y_i = Y_0 \\times_{X_0} X_i \\to X_i$ is finite syntomic.", "Then", "$$", "\\Gamma(X, f_*\\Omega^p_{Y/\\mathbf{Z}}) =", "\\Gamma(Y, \\Omega^p_{Y/\\mathbf{Z}}) =", "\\colim_{i \\geq 0} \\Gamma(Y_i, \\Omega^p_{Y_i/\\mathbf{Z}}) =", "\\colim_{i \\geq 0} \\Gamma(X_i, f_{i, *}\\Omega^p_{Y_i/\\mathbf{Z}})", "$$", "Hence we can (and are forced to) define $\\Theta^p_{Y/X}$ as the colimit", "of the maps $\\Theta^p_{Y_i/X_i}$. This map is compatible with any", "cartesian diagram", "$$", "\\xymatrix{", "Y' \\ar[r] \\ar[d] & Y \\ar[d] \\\\", "X' \\ar[r] & X", "}", "$$", "with $X'$ affine as we know this for the case of Noetherian affine schemes", "by the arguments given above (small detail omitted; hint: if we also", "write $X' = \\lim_{j \\in J} X'_j$ then for every $i \\in I$ there is a $j \\in J$", "and a morphism $X'_j \\to X_i$ compatible with the morphism $X' \\to X$).", "This finishes the proof." ], "refs": [ "derham-lemma-Garel-upstairs", "discriminant-proposition-tate-map", "discriminant-lemma-locally-comes-from-universal-finite", "discriminant-lemma-universal-finite-syntomic-smooth", "discriminant-lemma-universal-finite-syntomic-etale", "derham-lemma-Garel-upstairs", "algebra-lemma-colimit-category-fp-algebras", "algebra-lemma-colimit-lci", "algebra-lemma-colimit-finite" ], "ref_ids": [ 14351, 15002, 14991, 14989, 14990, 14351, 1097, 1397, 1391 ] } ], "ref_ids": [] }, { "id": 14353, "type": "theorem", "label": "derham-lemma-Garel-map-frobenius-smooth-char-p", "categories": [ "derham" ], "title": "derham-lemma-Garel-map-frobenius-smooth-char-p", "contents": [ "Let $p$ be a prime number. Let $X \\to S$ be a smooth morphism", "of relative dimension $d$ of schemes in characteristic $p$.", "The relative Frobenius $F_{X/S} : X \\to X^{(p)}$ of $X/S$", "(Varieties, Definition \\ref{varieties-definition-relative-frobenius})", "is finite syntomic and the corresponding map", "$$", "\\Theta_{X/X^{(p)}} :", "F_{X/S, *}\\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X^{(p)}/S}", "$$", "is zero in all degrees except in degree $d$ where it defines a", "surjection." ], "refs": [ "varieties-definition-relative-frobenius" ], "proofs": [ { "contents": [ "Observe that $F_{X/S}$ is a finite morphism by", "Varieties, Lemma \\ref{varieties-lemma-relative-frobenius-finite}.", "To prove that $F_{X/S}$ is flat, it suffices to show that", "the morphism $F_{X/S, s} : X_s \\to X^{(p)}_s$ between fibres", "is flat for all $s \\in S$, see More on Morphisms, Theorem", "\\ref{more-morphisms-theorem-criterion-flatness-fibre}.", "Flatness of $X_s \\to X^{(p)}_s$ follows from", "Algebra, Lemma \\ref{algebra-lemma-CM-over-regular-flat}", "(and the finiteness already shown).", "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-lci-permanence}", "the morphism $F_{X/S}$ is a local complete intersection morphism.", "Hence $F_{X/S}$ is finite syntomic (see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-flat-lci}).", "\\medskip\\noindent", "For every point $x \\in X$ we may choose a commutative diagram", "$$", "\\xymatrix{", "X \\ar[d] & U \\ar[l] \\ar[d]_\\pi \\\\", "S & \\mathbf{A}^d_S \\ar[l]", "}", "$$", "where $\\pi$ is \\'etale and $x \\in U$ is open in $X$, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-etale-over-affine-space}.", "Observe that", "$\\mathbf{A}^d_S \\to \\mathbf{A}^d_S$, $(x_1, \\ldots, x_d) \\mapsto", "(x_1^p, \\ldots, x_d^p)$ is the relative Frobenius for $\\mathcal{A}^d_S$", "over $S$. The commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_\\pi \\ar[r]_{F_{X/S}} & U^{(p)} \\ar[d]^{\\pi^{(p)}} \\\\", "\\mathbf{A}^d_S \\ar[r]^{x_i \\mapsto x_i^p} & \\mathbf{A}^d_S", "}", "$$", "of", "Varieties, Lemma \\ref{varieties-lemma-relative-frobenius-endomorphism-identity}", "for $\\pi : U \\to \\mathbf{A}^d_S$ is cartesian by", "\\'Etale Morphisms, Lemma", "\\ref{etale-lemma-relative-frobenius-etale}.", "Since the construction of $\\Theta$ is compatible with base change", "and since $\\Omega_{U/S} = \\pi^*\\Omega_{\\mathbf{A}^d_S/S}$", "(Lemma \\ref{lemma-etale})", "we conclude that it suffices to show the lemma for", "$\\mathbf{A}^d_S$.", "\\medskip\\noindent", "Let $A$ be a ring of characteristic $p$. Consider the unique $A$-algebra", "homomorphism $A[y_1, \\ldots, y_d] \\to A[x_1, \\ldots, x_d]$", "sending $y_i$ to $x_i^p$. The arguments above", "reduce us to computing the map", "$$", "\\Theta^i : \\Omega^i_{A[x_1, \\ldots, x_d]/A} \\to", "\\Omega^i_{A[y_1, \\ldots, y_d]/A}", "$$", "We urge the reader to do the computation in this case for themselves.", "As in Example \\ref{example-Garel} we may reduce this to computing", "a formula for $\\Theta^i$ in the universal case", "$$", "\\mathbf{Z}[y_1, \\ldots, y_d] \\to \\mathbf{Z}[x_1, \\ldots, x_d],\\quad", "y_i \\mapsto x_i^p", "$$", "In turn, we can find the formula for $\\Theta^i$ by computing in the complex", "case, i.e., for the $\\mathbf{C}$-algebra map", "$$", "\\mathbf{C}[y_1, \\ldots, y_d] \\to \\mathbf{C}[x_1, \\ldots, x_d],\\quad", "y_i \\mapsto x_i^p", "$$", "We may even invert $x_1, \\ldots, x_d$ and $y_1, \\ldots, y_d$.", "In this case, we have $\\text{d}x_i = p^{-1} x_i^{- p + 1}\\text{d}y_i$.", "Hence we see that", "\\begin{align*}", "\\Theta^i(", "x_1^{e_1} \\ldots x_d^{e_d} \\text{d}x_1 \\wedge \\ldots \\wedge \\text{d}x_i)", "& =", "p^{-i} \\Theta^i(", "x_1^{e_1 - p + 1} \\ldots x_i^{e_i - p + 1} x_{i + 1}^{e_{i + 1}} \\ldots", "x_d^{e_d} \\text{d}y_1 \\wedge \\ldots \\wedge \\text{d}y_i ) \\\\", "& =", "p^{-i} \\text{Trace}(x_1^{e_1 - p + 1} \\ldots x_i^{e_i - p + 1}", "x_{i + 1}^{e_{i + 1}} \\ldots x_d^{e_d})", "\\text{d}y_1 \\wedge \\ldots \\wedge \\text{d}y_i", "\\end{align*}", "by the properties of $\\Theta^i$. An elementary computation shows", "that the trace in the expression above is zero unless", "$e_1, \\ldots, e_i$ are congruent to $-1$ modulo $p$", "and $e_{i + 1}, \\ldots, e_d$ are divisible by $p$.", "Moreover, in this case we obtain", "$$", "p^{d - i} y_1^{(e_1 - p + 1)/p} \\ldots y_i^{(e_i - p + 1)/p}", "y_{i + 1}^{e_{i + 1}/p} \\ldots y_d^{e_d/p}", "\\text{d}y_1 \\wedge \\ldots \\wedge \\text{d}y_i", "$$", "We conclude that we get zero in characteristic $p$ unless $d = i$", "and in this case we get every possible $d$-form." ], "refs": [ "varieties-lemma-relative-frobenius-finite", "more-morphisms-theorem-criterion-flatness-fibre", "algebra-lemma-CM-over-regular-flat", "more-morphisms-lemma-lci-permanence", "more-morphisms-lemma-flat-lci", "morphisms-lemma-smooth-etale-over-affine-space", "varieties-lemma-relative-frobenius-endomorphism-identity", "etale-lemma-relative-frobenius-etale", "derham-lemma-etale" ], "ref_ids": [ 11052, 13672, 1107, 14008, 14006, 5377, 11049, 10708, 14314 ] } ], "ref_ids": [ 11156 ] }, { "id": 14354, "type": "theorem", "label": "derham-lemma-duality-hodge", "categories": [ "derham" ], "title": "derham-lemma-duality-hodge", "contents": [ "Let $k$ be a field. Let $X$ be a nonempty smooth proper scheme over $k$", "equidimensional of dimension $d$. There exists a $k$-linear map", "$$", "t : H^d(X, \\Omega^d_{X/k}) \\longrightarrow k", "$$", "unique up to precomposing by multiplication by a unit of", "$H^0(X, \\mathcal{O}_X)$ with the following property: for all $p, q$ the pairing", "$$", "H^q(X, \\Omega^p_{X/k}) \\times H^{d - q}(X, \\Omega^{d - p}_{X/k})", "\\longrightarrow", "k, \\quad", "(\\xi, \\xi') \\longmapsto t(\\xi \\cup \\xi')", "$$", "is perfect." ], "refs": [], "proofs": [ { "contents": [ "By Duality for Schemes, Lemma \\ref{duality-lemma-duality-proper-over-field}", "we have $\\omega_X^\\bullet = \\Omega^d_{X/k}[d]$.", "Since $\\Omega_{X/k}$ is locally free of rank $d$", "(Morphisms, Lemma \\ref{morphisms-lemma-smooth-omega-finite-locally-free})", "we have", "$$", "\\Omega^d_{X/k} \\otimes_{\\mathcal{O}_X} (\\Omega^p_{X/k})^\\vee", "\\cong", "\\Omega^{d - p}_{X/k}", "$$", "Thus we obtain a $k$-linear map $t : H^d(X, \\Omega^d_{X/k}) \\to k$", "such that the statement is true by Duality for Schemes, Lemma", "\\ref{duality-lemma-duality-proper-over-field-perfect}.", "In particular the pairing", "$H^0(X, \\mathcal{O}_X) \\times H^d(X, \\Omega^d_{X/k}) \\to k$", "is perfect, which implies that any $k$-linear map", "$t' : H^d(X, \\Omega^d_{X/k}) \\to k$ is of the form", "$\\xi \\mapsto t(g\\xi)$ for some $g \\in H^0(X, \\mathcal{O}_X)$.", "Of course, in order for $t'$ to still produce a duality", "between $H^0(X, \\mathcal{O}_X)$ and $H^d(X, \\Omega^d_{X/k})$", "we need $g$ to be a unit. Denote $\\langle -, - \\rangle_{p, q}$", "the pairing constructed using $t$ and denote $\\langle -, - \\rangle'_{p, q}$", "the pairing constructed using $t'$. Clearly we have", "$$", "\\langle \\xi, \\xi' \\rangle'_{p, q} =", "\\langle g\\xi, \\xi' \\rangle_{p, q}", "$$", "for $\\xi \\in H^q(X, \\Omega^p_{X/k})$ and", "$\\xi' \\in H^{d - q}(X, \\Omega^{d - p}_{X/k})$. Since $g$ is a unit, i.e.,", "invertible, we see that using $t'$ instead of $t$ we still get perfect", "pairings for all $p, q$." ], "refs": [ "duality-lemma-duality-proper-over-field", "morphisms-lemma-smooth-omega-finite-locally-free", "duality-lemma-duality-proper-over-field-perfect" ], "ref_ids": [ 13606, 5334, 13607 ] } ], "ref_ids": [] }, { "id": 14355, "type": "theorem", "label": "derham-lemma-bottom-part-degenerates", "categories": [ "derham" ], "title": "derham-lemma-bottom-part-degenerates", "contents": [ "Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. The map", "$$", "\\text{d} : H^0(X, \\mathcal{O}_X) \\to H^0(X, \\Omega^1_{X/k})", "$$", "is zero." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is smooth over $k$ it is geometrically reduced over $k$, see", "Varieties, Lemma \\ref{varieties-lemma-smooth-geometrically-normal}.", "Hence $H^0(X, \\mathcal{O}_X) = \\prod k_i$", "is a finite product of finite separable", "field extensions $k_i/k$, see Varieties, Lemma", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}.", "It follows that $\\Omega_{H^0(X, \\mathcal{O}_X)/k} = \\prod \\Omega_{k_i/k} = 0$", "(see for example Algebra, Lemma", "\\ref{algebra-lemma-characterize-separable-algebraic-field-extensions}).", "Since the map of the lemma factors as", "$$", "H^0(X, \\mathcal{O}_X) \\to", "\\Omega_{H^0(X, \\mathcal{O}_X)/k} \\to", "H^0(X, \\Omega_{X/k})", "$$", "by functoriality of the de Rham complex", "(see Section \\ref{section-de-rham-complex}), we conclude." ], "refs": [ "varieties-lemma-smooth-geometrically-normal", "varieties-lemma-proper-geometrically-reduced-global-sections", "algebra-lemma-characterize-separable-algebraic-field-extensions" ], "ref_ids": [ 11005, 10948, 1314 ] } ], "ref_ids": [] }, { "id": 14356, "type": "theorem", "label": "derham-lemma-top-part-degenerates", "categories": [ "derham" ], "title": "derham-lemma-top-part-degenerates", "contents": [ "Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$", "equidimensional of dimension $d$. The map", "$$", "\\text{d} : H^d(X, \\Omega^{d - 1}_{X/k}) \\to H^d(X, \\Omega^d_{X/k})", "$$", "is zero." ], "refs": [], "proofs": [ { "contents": [ "It is tempting to think this follows from a combination of", "Lemmas \\ref{lemma-bottom-part-degenerates} and \\ref{lemma-duality-hodge}.", "However this doesn't work because the maps $\\mathcal{O}_X \\to \\Omega^1_{X/k}$", "and $\\Omega^{d - 1}_{X/k} \\to \\Omega^d_{X/k}$ are not $\\mathcal{O}_X$-linear", "and hence we cannot use the functoriality discussed in", "Duality for Schemes, Remark", "\\ref{duality-remark-coherent-duality-proper-over-field}", "to conclude the map in Lemma \\ref{lemma-bottom-part-degenerates}", "is dual to the one in this lemma.", "\\medskip\\noindent", "We may replace $X$ by a connected component of $X$. Hence we may assume", "$X$ is irreducible. By", "Varieties, Lemmas \\ref{varieties-lemma-smooth-geometrically-normal} and", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections}", "we see that $k' = H^0(X, \\mathcal{O}_X)$ $X$ is a finite separable", "extension $k'/k$. Since $\\Omega_{k'/k} = 0$", "(see for example Algebra, Lemma", "\\ref{algebra-lemma-characterize-separable-algebraic-field-extensions})", "we see that $\\Omega_{X/k} = \\Omega_{X/k'}$", "(see Morphisms, Lemma \\ref{morphisms-lemma-triangle-differentials}).", "Thus we may replace $k$ by $k'$ and assume that $H^0(X, \\mathcal{O}_X) = k$.", "\\medskip\\noindent", "Assume $H^0(X, \\mathcal{O}_X) = k$. We conclude that", "$\\dim H^d(X, \\Omega^d_{X/k}) = 1$ by Lemma \\ref{lemma-duality-hodge}.", "Assume first that the characteristic of $k$ is a prime number $p$.", "Denote $F_{X/k} : X \\to X^{(p)}$ the relative Frobenius of $X$ over $k$;", "please keep in mind the facts proved about this morphism in", "Lemma \\ref{lemma-Garel-map-frobenius-smooth-char-p}.", "Consider the commutative diagram", "$$", "\\xymatrix{", "H^d(X, \\Omega^{d - 1}_{X/k}) \\ar[d] \\ar[r] &", "H^d(X^{(p)}, F_{X/k, *}\\Omega^{d - 1}_{X/k}) \\ar[d] \\ar[r]_{\\Theta^{d - 1}} &", "H^d(X^{(p)}, \\Omega^{d - 1}_{X^{(p)}/k}) \\ar[d] \\\\", "H^d(X, \\Omega^d_{X/k}) \\ar[r] &", "H^d(X^{(p)}, F_{X/k, *}\\Omega^d_{X/k}) \\ar[r]^{\\Theta^d} &", "H^d(X^{(p)}, \\Omega^d_{X^{(p)}/k})", "}", "$$", "The left two horizontal arrows are isomorphisms as $F_{X/k}$ is finite, see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology}.", "The right square commutes as $\\Theta_{X^{(p)}/X}$ is a morphism of", "complexes and $\\Theta^{d - 1}$ is zero. Thus it suffices to show that", "$\\Theta^d$ is nonzero (because the dimension of the source of the map", "$\\Theta^d$ is $1$ by the discussion above). However, we know that", "$$", "\\Theta^d : F_{X/k, *}\\Omega^d_{X/k} \\to \\Omega^d_{X^{(p)}/k}", "$$", "is surjective and hence surjective after applying the right exact", "functor $H^d(X^{(p)}, -)$ (right exactness by the vanishing of cohomology", "beyond $d$ as follows from", "Cohomology, Proposition \\ref{cohomology-proposition-vanishing-Noetherian}).", "Finally, $H^d(X^{(d)}, \\Omega^d_{X^{(d)}/k})$ is nonzero for example because", "it is dual to $H^0(X^{(d)}, \\mathcal{O}_{X^{(p)}})$ by", "Lemma \\ref{lemma-duality-hodge} applied to $X^{(p)}$ over $k$.", "This finishes the proof in this case.", "\\medskip\\noindent", "Finally, assume the characteristic of $k$ is $0$.", "We can write $k$ as the filtered colimit of its finite type", "$\\mathbf{Z}$-subalgebras $R$. For one of these we can find a", "cartesian diagram of schemes", "$$", "\\xymatrix{", "X \\ar[d] \\ar[r] & Y \\ar[d] \\\\", "\\Spec(k) \\ar[r] & \\Spec(R)", "}", "$$", "such that $Y \\to \\Spec(R)$ is smooth of relative dimension $d$ and proper.", "See Limits, Lemmas \\ref{limits-lemma-descend-finite-presentation},", "\\ref{limits-lemma-descend-smooth}, \\ref{limits-lemma-descend-dimension-d}, and", "\\ref{limits-lemma-eventually-proper}.", "The modules $M^{i, j} = H^j(Y, \\Omega^i_{Y/R})$ are finite $R$-modules, see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-over-affine-cohomology-finite}.", "Thus after replacing $R$ by a localization we may assume all of these", "modules are finite free. We have", "$M^{i, j} \\otimes_R k = H^j(X, \\Omega^i_{X/k})$", "by flat base change (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-flat-base-change-cohomology}).", "Thus it suffices to show that $M^{d - 1, d} \\to M^{d, d}$", "is zero. This is a map of finite free modules over a domain,", "hence it suffices to find a dense set of primes $\\mathfrak p \\subset R$", "such that after tensoring with $\\kappa(\\mathfrak p)$ we get zero.", "Since $R$ is of finite type over $\\mathbf{Z}$, we can take", "the collection of primes $\\mathfrak p$ whose residue field", "has positive characteristic (details omitted). Observe that", "$$", "M^{d - 1, d} \\otimes_R \\kappa(\\mathfrak p) =", "H^d(Y_{\\kappa(\\mathfrak p)},", "\\Omega^{d - 1}_{Y_{\\kappa(\\mathfrak p)}/\\kappa(\\mathfrak p)})", "$$", "for example by Limits, Lemma", "\\ref{limits-lemma-higher-direct-images-zero-above-dimension-fibre}.", "Similarly for $M^{d, d}$. Thus we see that", "$M^{d - 1, d} \\otimes_R \\kappa(\\mathfrak p) \\to", "M^{d, d} \\otimes_R \\kappa(\\mathfrak p)$", "is zero by the case of positive characteristic handled above." ], "refs": [ "derham-lemma-bottom-part-degenerates", "derham-lemma-duality-hodge", "duality-remark-coherent-duality-proper-over-field", "derham-lemma-bottom-part-degenerates", "varieties-lemma-smooth-geometrically-normal", "varieties-lemma-proper-geometrically-reduced-global-sections", "algebra-lemma-characterize-separable-algebraic-field-extensions", "morphisms-lemma-triangle-differentials", "derham-lemma-duality-hodge", "derham-lemma-Garel-map-frobenius-smooth-char-p", "coherent-lemma-relative-affine-cohomology", "cohomology-proposition-vanishing-Noetherian", "derham-lemma-duality-hodge", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-smooth", "limits-lemma-descend-dimension-d", "limits-lemma-eventually-proper", "coherent-lemma-proper-over-affine-cohomology-finite", "coherent-lemma-flat-base-change-cohomology", "limits-lemma-higher-direct-images-zero-above-dimension-fibre" ], "ref_ids": [ 14355, 14354, 13659, 14355, 11005, 10948, 1314, 5313, 14354, 14353, 3284, 2246, 14354, 15077, 15064, 15106, 15089, 3355, 3298, 15109 ] } ], "ref_ids": [] }, { "id": 14357, "type": "theorem", "label": "derham-lemma-chern-classes", "categories": [ "derham" ], "title": "derham-lemma-chern-classes", "contents": [ "There is a unique rule which assigns to every quasi-compact and", "quasi-separated scheme $X$ a total Chern class", "$$", "c^{dR} :", "K_0(\\textit{Vect}(X))", "\\longrightarrow", "\\prod\\nolimits_{i \\geq 0} H^{2i}_{dR}(X/\\mathbf{Z})", "$$", "with the following properties", "\\begin{enumerate}", "\\item we have $c^{dR}(\\alpha + \\beta) = c^{dR}(\\alpha) c^{dR}(\\beta)$", "for $\\alpha, \\beta \\in K_0(\\textit{Vect}(X))$,", "\\item if $f : X \\to X'$ is a morphism of quasi-compact and", "quasi-separated schemes, then $c^{dR}(f^*\\alpha) = f^*c^{dR}(\\alpha)$,", "\\item given $\\mathcal{L} \\in \\Pic(X)$ we have", "$c^{dR}([\\mathcal{L}]) = 1 + c_1^{dR}(\\mathcal{L})$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We will apply Weil Cohomology Theories, Proposition", "\\ref{weil-proposition-chern-class} to get this.", "\\medskip\\noindent", "Let $\\mathcal{C}$ be the category of all quasi-compact and quasi-separated", "schemes. This certainly satisfies conditions", "(1), (2), and (3) (a), (b), and (c) of Weil Cohomology Theories,", "Section \\ref{weil-section-chern}.", "\\medskip\\noindent", "As our contravariant functor $A$ from $\\mathcal{C}$ to the", "category of graded algebras will send $X$ to", "$A(X) = \\bigoplus_{i \\geq 0} H_{dR}^{2i}(X/\\mathbf{Z})$", "endowed with its cup product.", "Functoriality is discussed in Section \\ref{section-de-rham-cohomology}", "and the cup product in Section \\ref{section-cup-product}.", "For the additive maps $c_1^A$ we take $c_1^{dR}$ constructed", "in Section \\ref{section-first-chern-class}.", "\\medskip\\noindent", "In fact, we obtain commutative algebras by", "Lemma \\ref{lemma-cup-product-graded-commutative}", "which shows we have axiom (1) for $A$.", "\\medskip\\noindent", "To check axiom (2) for $A$ it suffices to check that", "$H^*_{dR}(X \\coprod Y/\\mathbf{Z}) = H^*_{dR}(X/\\mathbf{Z}) \\times", "H^*_{dR}(Y/\\mathbf{Z})$.", "This is a consequence of the fact that de Rham cohomology", "is constructed by taking the cohomology of a sheaf of differential", "graded algebras (in the Zariski topology).", "\\medskip\\noindent", "Axiom (3) for $A$ is just the statement that taking first Chern", "classes of invertible modules is compatible with pullbacks.", "This follows from the more general Lemma \\ref{lemma-pullback-c1}.", "\\medskip\\noindent", "Axiom (4) for $A$ is the projective space bundle formula which", "we proved in Proposition \\ref{proposition-projective-space-bundle-formula}.", "\\medskip\\noindent", "Axiom (5). Let $X$ be a quasi-compact and quasi-separated scheme and", "let $\\mathcal{E} \\to \\mathcal{F}$ be a surjection of finite locally free", "$\\mathcal{O}_X$-modules of ranks $r + 1$ and $r$. Denote", "$i : P' = \\mathbf{P}(\\mathcal{F}) \\to \\mathbf{P}(\\mathcal{E}) = P$ the", "corresponding incusion morphism. This is a morphism of smooth projective", "schemes over $X$ which exhibits $P'$ as an effective Cartier divisor on $P$.", "Thus by Lemma \\ref{lemma-check-log-smooth} the complex of log poles", "for $P' \\subset P$ over $\\mathbf{Z}$ is defined.", "Hence for $a \\in A(P)$ with $i^*a = 0$ we have", "$a \\cup c_1^A(\\mathcal{O}_P(P')) = 0$ by", "Lemma \\ref{lemma-log-complex-consequence}.", "This finishes the proof." ], "refs": [ "weil-proposition-chern-class", "derham-lemma-cup-product-graded-commutative", "derham-lemma-pullback-c1", "derham-proposition-projective-space-bundle-formula", "derham-lemma-check-log-smooth", "derham-lemma-log-complex-consequence" ], "ref_ids": [ 5111, 14320, 14325, 14372, 14341, 14340 ] } ], "ref_ids": [] }, { "id": 14358, "type": "theorem", "label": "derham-lemma-chern-character", "categories": [ "derham" ], "title": "derham-lemma-chern-character", "contents": [ "There is a unique rule which assigns to every quasi-compact and quasi-separated", "scheme $X$ over $\\mathbf{Q}$ a ``chern character''", "$$", "ch^{dR} : K_0(\\textit{Vect}(X)) \\longrightarrow", "\\prod\\nolimits_{i \\geq 0} H_{dR}^{2i}(X/\\mathbf{Q})", "$$", "with the following properties", "\\begin{enumerate}", "\\item $ch^{dR}$ is a ring map for all $X$,", "\\item if $f : X' \\to X$ is a morphism of quasi-compact and quasi-separated", "schemes over $\\mathbf{Q}$, then $f^* \\circ ch^{dR} = ch^{dR} \\circ f^*$, and", "\\item given $\\mathcal{L} \\in \\Pic(X)$", "we have $ch^{dR}([\\mathcal{L}]) = \\exp(c_1^{dR}(\\mathcal{L}))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Exactly as in the proof of Lemma \\ref{lemma-chern-classes}", "one shows that the category of quasi-compact and quasi-separated", "schemes over $\\mathbf{Q}$ together with the functor", "$A^*(X) = \\bigoplus_{i \\geq 0} H_{dR}^{2i}(X/\\mathbf{Q})$", "satisfy the axioms of", "Weil Cohomology Theories, Section \\ref{weil-section-chern}.", "Moreover, in this case $A(X)$ is a $\\mathbf{Q}$-algebra for", "all $X$. Hence the lemma follows from", "Weil Cohomology Theories, Proposition", "\\ref{weil-proposition-chern-character}." ], "refs": [ "derham-lemma-chern-classes", "weil-proposition-chern-character" ], "ref_ids": [ 14357, 5112 ] } ], "ref_ids": [] }, { "id": 14359, "type": "theorem", "label": "derham-lemma-gysin-differential", "categories": [ "derham" ], "title": "derham-lemma-gysin-differential", "contents": [ "The gysin map (\\ref{equation-gysin}) is compatible with the de Rham", "differentials on $\\Omega^\\bullet_{X/S}$ and $\\Omega^\\bullet_{Z/S}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from an almost trivial calculation once", "we correctly interpret this. First, we recall that the functor", "$\\mathcal{H}^c_Z$ computed on the category of $\\mathcal{O}_X$-modules", "agrees with the similarly defined functor on the category of abelian", "sheaves on $X$, see", "Cohomology, Lemma \\ref{cohomology-lemma-sections-support-abelian-unbounded}.", "Hence, the differential $\\text{d} : \\Omega^p_{X/S} \\to \\Omega^{p + 1}_{X/S}$", "induces a map", "$\\mathcal{H}^c_Z(\\Omega^p_{X/S}) \\to \\mathcal{H}^c_Z(\\Omega^{p + 1}_{X/S})$.", "Moreover, the formation of the extended alternating {\\v C}ech complex in", "Derived Categories of Schemes, Remark \\ref{perfect-remark-support-c-equations}", "works on the category of abelian sheaves. The map", "$$", "\\Coker\\left(\\bigoplus \\mathcal{F}_{1 \\ldots \\hat i \\ldots c} \\to", "\\mathcal{F}_{1 \\ldots c}\\right)", "\\longrightarrow", "i_*\\mathcal{H}^c_Z(\\mathcal{F})", "$$", "used in the construction of $c_{f_1, \\ldots, f_c}$ in", "Derived Categories of Schemes, Remark", "\\ref{perfect-remark-supported-map-c-equations}", "is well defined and", "functorial on the category of all abelian sheaves on $X$.", "Hence we see that the lemma follows from the equality", "$$", "\\text{d}\\left(", "\\frac{\\tilde \\omega \\wedge \\text{d}f_1 \\wedge \\ldots \\wedge", "\\text{d}f_c}{f_1 \\ldots f_c}\\right) =", "\\frac{\\text{d}(\\tilde \\omega) \\wedge", "\\text{d}f_1 \\wedge \\ldots \\wedge \\text{d}f_c}{f_1 \\ldots f_c}", "$$", "which is clear." ], "refs": [ "cohomology-lemma-sections-support-abelian-unbounded", "perfect-remark-support-c-equations", "perfect-remark-supported-map-c-equations" ], "ref_ids": [ 2157, 7122, 7124 ] } ], "ref_ids": [] }, { "id": 14360, "type": "theorem", "label": "derham-lemma-gysin-global", "categories": [ "derham" ], "title": "derham-lemma-gysin-global", "contents": [ "Let $X \\to S$ be a morphism of schemes. Let $Z \\to X$ be a closed immersion", "of finite presentation whose conormal sheaf $\\mathcal{C}_{Z/X}$ is", "locally free of rank $c$. Then there is a canonical map", "$$", "\\gamma^p : \\Omega^p_{Z/S} \\to \\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S})", "$$", "which is locally given by the maps $\\gamma^p_{f_1, \\ldots, f_c}$", "of Remark \\ref{remark-gysin-equations}." ], "refs": [ "derham-remark-gysin-equations" ], "proofs": [ { "contents": [ "The assumptions imply that given $x \\in Z \\subset X$ there exists an", "open neighbourhood $U$ of $x$ such that $Z$ is cut out by $c$", "elements $f_1, \\ldots, f_c \\in \\mathcal{O}_X(U)$. Thus", "it suffices to show that given $f_1, \\ldots, f_c$ and", "$g_1, \\ldots, g_c$ in $\\mathcal{O}_X(U)$ cutting out $Z \\cap U$,", "the maps $\\gamma^p_{f_1, \\ldots, f_c}$", "and $\\gamma^p_{g_1, \\ldots, g_c}$ are the same. To do this, after shrinking", "$U$ we may assume $g_j = \\sum a_{ji} f_i$ for some", "$a_{ji} \\in \\mathcal{O}_X(U)$. Then we have", "$c_{f_1, \\ldots, f_c} = \\det(a_{ji}) c_{g_1, \\ldots, g_c}$ by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-supported-map-determinant}.", "On the other hand we have", "$$", "\\text{d}(g_1) \\wedge \\ldots \\wedge \\text{d}(g_c) \\equiv", "\\det(a_{ji}) \\text{d}(f_1) \\wedge \\ldots \\wedge \\text{d}(f_c)", "\\bmod (f_1, \\ldots, f_c)\\Omega^c_{X/S}", "$$", "Combining these relations, a straightforward calculation gives the", "desired equality." ], "refs": [ "perfect-lemma-supported-map-determinant" ], "ref_ids": [ 6961 ] } ], "ref_ids": [ 14393 ] }, { "id": 14361, "type": "theorem", "label": "derham-lemma-gysin-differential-global", "categories": [ "derham" ], "title": "derham-lemma-gysin-differential-global", "contents": [ "Let $X \\to S$ and $i : Z \\to X$ be as in Lemma \\ref{lemma-gysin-global}.", "The gysin map $\\gamma^p$ is compatible with the de Rham", "differentials on $\\Omega^\\bullet_{X/S}$ and $\\Omega^\\bullet_{Z/S}$." ], "refs": [ "derham-lemma-gysin-global" ], "proofs": [ { "contents": [ "We may check this locally and then it follows from", "Lemma \\ref{lemma-gysin-differential}." ], "refs": [ "derham-lemma-gysin-differential" ], "ref_ids": [ 14359 ] } ], "ref_ids": [ 14360 ] }, { "id": 14362, "type": "theorem", "label": "derham-lemma-gysin-projection", "categories": [ "derham" ], "title": "derham-lemma-gysin-projection", "contents": [ "Let $X \\to S$ and $i : Z \\to X$ be as in Lemma \\ref{lemma-gysin-global}.", "Given $\\alpha \\in H^q(X, \\Omega^p_{X/S})$ we have", "$\\gamma^p(\\alpha|_Z) = i^{-1}\\alpha \\wedge \\gamma^0(1)$ in", "$H^q(Z, \\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S}))$.", "Please see proof for notation." ], "refs": [ "derham-lemma-gysin-global" ], "proofs": [ { "contents": [ "The restriction $\\alpha|_Z$ is the element of $H^q(Z, \\Omega^p_{Z/S})$", "given by functoriality for Hodge cohomology. Applying functoriality", "for cohomology using", "$\\gamma^p : \\Omega^p_{Z/S} \\to \\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S})$", "we get get $\\gamma^p(\\alpha|_Z)$ in", "$H^q(Z, \\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S}))$.", "This explains the left hand side of the formula.", "\\medskip\\noindent", "To explain the right hand side, we first pullback by the map", "of ringed spaces $i : (Z, i^{-1}\\mathcal{O}_X) \\to (X, \\mathcal{O}_X)$", "to get the element $i^{-1}\\alpha \\in H^q(Z, i^{-1}\\Omega^p_{X/S})$.", "Let $\\gamma^0(1) \\in H^0(Z, \\mathcal{H}_Z^c(\\Omega^c_{X/S}))$", "be the image of $1 \\in H^0(Z, \\mathcal{O}_Z) = H^0(Z, \\Omega^0_{Z/S})$", "by $\\gamma^0$. Using cup product we obtain an element", "$$", "i^{-1}\\alpha \\cup \\gamma^0(1)", "\\in", "H^{q + c}(Z, ", "i^{-1}\\Omega^p_{X/S} \\otimes_{i^{-1}\\mathcal{O}_X}", "\\mathcal{H}^c_Z(\\Omega^c_{X/S}))", "$$", "Using Cohomology, Remark \\ref{cohomology-remark-support-cup-product}", "and wedge product there are canonical maps", "$$", "i^{-1}\\Omega^p_{X/S} \\otimes_{i^{-1}\\mathcal{O}_X}^\\mathbf{L}", "R\\mathcal{H}_Z(\\Omega^c_{X/S}) \\to", "R\\mathcal{H}_Z(\\Omega^p_{X/S} \\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "\\Omega^c_{X/S}) \\to", "R\\mathcal{H}_Z(\\Omega^{p + c}_{X/S})", "$$", "By Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-supported-trivial-vanishing}", "the objects $R\\mathcal{H}_Z(\\Omega^j_{X/S})$ have vanishing", "cohomology sheaves in degrees $> c$. Hence on cohomology", "sheaves in degree $c$ we obtain a map", "$$", "i^{-1}\\Omega^p_{X/S} \\otimes_{i^{-1}\\mathcal{O}_X}", "\\mathcal{H}^c_Z(\\Omega^c_{X/S}) \\longrightarrow", "\\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S})", "$$", "The expression $i^{-1}\\alpha \\wedge \\gamma^0(1)$ is the image", "of the cup product $i^{-1}\\alpha \\cup \\gamma^0(1)$ by the", "functoriality of cohomology.", "\\medskip\\noindent", "Having explained the content of the formula in this manner, by", "general properties of cup products", "(Cohomology, Section \\ref{cohomology-section-cup-product}),", "it now suffices to prove that the diagram", "$$", "\\xymatrix{", "i^{-1}\\Omega^p_X \\otimes \\Omega^0_Z \\ar[rr]_{\\text{id} \\otimes \\gamma^0}", "\\ar[d] & &", "i^{-1}\\Omega^p_X \\otimes \\mathcal{H}^c_Z(\\Omega^c_X) \\ar[d]^\\wedge \\\\", "\\Omega^p_Z \\otimes \\Omega^0_Z \\ar[r]^\\wedge &", "\\Omega^p_Z \\ar[r]^{\\gamma^p} &", "\\mathcal{H}^c_Z(\\Omega^{p + c}_X)", "}", "$$", "is commutative in the category of sheaves on $Z$ (with obvious abuse of", "notation). This boils down to a simple computation for the maps", "$\\gamma^j_{f_1, \\ldots, f_c}$ which we omit; in fact these maps", "are chosen exactly such that this works and such that $1$ maps to", "$\\frac{\\text{d}f_1 \\wedge \\ldots \\wedge \\text{d}f_c}{f_1 \\ldots f_c}$." ], "refs": [ "cohomology-remark-support-cup-product", "perfect-lemma-supported-trivial-vanishing" ], "ref_ids": [ 2275, 6959 ] } ], "ref_ids": [ 14360 ] }, { "id": 14363, "type": "theorem", "label": "derham-lemma-gysin-transverse", "categories": [ "derham" ], "title": "derham-lemma-gysin-transverse", "contents": [ "Let $c \\geq 0$ be a integer. Let", "$$", "\\xymatrix{", "Z' \\ar[d]_h \\ar[r] & X' \\ar[d]_g \\ar[r] & S' \\ar[d] \\\\", "Z \\ar[r] & X \\ar[r] & S", "}", "$$", "be a commutative diagram of schemes.", "Assume", "\\begin{enumerate}", "\\item $Z \\to X$ and $Z' \\to X'$", "satisfy the assumptions of Lemma \\ref{lemma-gysin-global},", "\\item the left square in the diagram is cartesian, and", "\\item $h^*\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z'/X'}$", "(Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial})", "is an isomorphism.", "\\end{enumerate}", "Then the diagram", "$$", "\\xymatrix{", "h^*\\Omega^p_{Z/S} \\ar[rr]_-{h^{-1}\\gamma^p} \\ar[d] & &", "\\mathcal{O}_{X'}|_{Z'} \\otimes_{h^{-1}\\mathcal{O}_X|_Z}", "h^{-1}\\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S}) \\ar[d] \\\\", "\\Omega^p_{Z'/S'} \\ar[rr]^{\\gamma^p} & &", "\\mathcal{H}^c_{Z'}(\\Omega^{p + c}_{X'/S'})", "}", "$$", "is commutative. The left vertical arrow is functoriality of modules of", "differentials and the right vertical arrow uses", "Cohomology, Remark \\ref{cohomology-remark-support-functorial}." ], "refs": [ "derham-lemma-gysin-global", "morphisms-lemma-conormal-functorial", "cohomology-remark-support-functorial" ], "proofs": [ { "contents": [ "More precisely, consider the composition", "\\begin{align*}", "\\mathcal{O}_{X'}|_{Z'} \\otimes_{h^{-1}\\mathcal{O}_X|_Z}^\\mathbf{L}", "h^{-1}R\\mathcal{H}_Z(\\Omega^{p + c}_{X/S})", "& \\to", "R\\mathcal{H}_{Z'}(Lg^*\\Omega^{p + c}_{X/S}) \\\\", "& \\to", "R\\mathcal{H}_{Z'}(g^*\\Omega^{p + c}_{X/S}) \\\\", "& \\to", "R\\mathcal{H}_{Z'}(\\Omega^{p + c}_{X'/S'})", "\\end{align*}", "where the first arrow is given by", "Cohomology, Remark \\ref{cohomology-remark-support-functorial}", "and the last one by functoriality of differentials.", "Since we have the vanishing of cohomology sheaves in degrees $> c$", "by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-supported-trivial-vanishing}", "this induces the right vertical arrow.", "We can check the commutativity locally.", "Thus we may assume $Z$ is cut out by", "$f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$.", "Then $Z'$ is cut out by $f'_i = g^\\sharp(f_i)$.", "The maps $c_{f_1, \\ldots, f_c}$ and $c_{f'_1, \\ldots, f'_c}$", "fit into the commutative diagram", "$$", "\\xymatrix{", "h^*i^*\\Omega^p_{X/S} \\ar[rr]_-{h^{-1}c_{f_1, \\ldots, f_c}} \\ar[d] & &", "\\mathcal{O}_{X'}|_{Z'} \\otimes_{h^{-1}\\mathcal{O}_X|_Z}", "h^{-1}\\mathcal{H}^c_Z(\\Omega^p_{X/S}) \\ar[d] \\\\", "(i')^*\\Omega^p_{X'/S'} \\ar[rr]^{c_{f'_1, \\ldots, f'_c}} & &", "\\mathcal{H}^c_{Z'}(\\Omega^p_{X'/S'})", "}", "$$", "See Derived Categories of Schemes, Remark", "\\ref{perfect-remark-supported-functorial}.", "Recall given a $p$-form $\\omega$ on $Z$ we define", "$\\gamma^p(\\omega)$ by choosing (locally on $X$ and $Z$)", "a $p$-form $\\tilde \\omega$ on $X$ lifting $\\omega$ and taking", "$\\gamma^p(\\omega) =", "c_{f_1, \\ldots, f_c}(\\tilde \\omega) \\wedge", "\\text{d}f_1 \\wedge \\ldots \\wedge \\text{d}f_c$.", "Since the form $\\text{d}f_1 \\wedge \\ldots \\wedge \\text{d}f_c$", "pulls back to", "$\\text{d}f'_1 \\wedge \\ldots \\wedge \\text{d}f'_c$ we conclude." ], "refs": [ "cohomology-remark-support-functorial", "perfect-lemma-supported-trivial-vanishing", "perfect-remark-supported-functorial" ], "ref_ids": [ 2277, 6959, 7125 ] } ], "ref_ids": [ 14360, 5304, 2277 ] }, { "id": 14364, "type": "theorem", "label": "derham-lemma-gysin-differential-hodge", "categories": [ "derham" ], "title": "derham-lemma-gysin-differential-hodge", "contents": [ "Let $X \\to S$ and $i : Z \\to X$ be as in Lemma \\ref{lemma-gysin-global}.", "Assume $X \\to S$ is smooth and $Z \\to X$ Koszul regular.", "The gysin maps $\\gamma^{p, q}$ are compatible with the de Rham", "differentials on $\\Omega^\\bullet_{X/S}$ and $\\Omega^\\bullet_{Z/S}$." ], "refs": [ "derham-lemma-gysin-global" ], "proofs": [ { "contents": [ "This follows immediately from", "Lemma \\ref{lemma-gysin-differential-global}." ], "refs": [ "derham-lemma-gysin-differential-global" ], "ref_ids": [ 14361 ] } ], "ref_ids": [ 14360 ] }, { "id": 14365, "type": "theorem", "label": "derham-lemma-gysin-projection-global", "categories": [ "derham" ], "title": "derham-lemma-gysin-projection-global", "contents": [ "Let $X \\to S$, $i : Z \\to X$, and $c \\geq 0$ be as in", "Lemma \\ref{lemma-gysin-global}. Assume $X \\to S$ smooth and", "$Z \\to X$ Koszul regular. Given $\\alpha \\in H^q(X, \\Omega^p_{X/S})$ we have", "$\\gamma^{p, q}(\\alpha|_Z) = \\alpha \\cup \\gamma^{0, 0}(1)$ in", "$H^{q + c}(X, \\Omega^{p + c}_{X/S})$ with $\\gamma^{a, b}$ as in", "Remark \\ref{remark-how-to-use}." ], "refs": [ "derham-lemma-gysin-global", "derham-remark-how-to-use" ], "proofs": [ { "contents": [ "This lemma follows from Lemma \\ref{lemma-gysin-projection}", "and Cohomology, Lemma \\ref{cohomology-lemma-support-cup-product}.", "We suggest the reader skip over the more detailed discussion below.", "\\medskip\\noindent", "We will use without further mention that", "$R\\mathcal{H}_Z(\\Omega^j_{X/S}) = \\mathcal{H}^c_Z(\\Omega^j_{X/S})[-c]$", "for all $j$ as pointed out in Remark \\ref{remark-how-to-use}.", "We will also silently use the identifications", "$H^{q + c}_Z(X, \\Omega^j_{X/S}) = H^{q + c}(Z, R\\mathcal{H}_Z(\\Omega^j_{X/S}) =", "H^q(Z, \\mathcal{H}^c_Z(\\Omega^j_{X/S}))$, see", "Cohomology, Lemma \\ref{cohomology-lemma-local-to-global-sections-with-support}", "for the first one. With these identifications", "\\begin{enumerate}", "\\item $\\gamma^0(1) \\in H^c_Z(X, \\Omega^c_{X/S})$ maps to $\\gamma^{0, 0}(1)$", "in $H^c(X, \\Omega^c_{X/S})$,", "\\item the right hand side $i^{-1}\\alpha \\wedge \\gamma^0(1)$", "of the equality in Lemma \\ref{lemma-gysin-projection}", "is the (image by wedge product of the) cup product of", "Cohomology, Remark \\ref{cohomology-remark-support-cup-product-global} ", "of the elements $\\alpha$ and $\\gamma^0(1)$, in other words, the constructions", "in the proof of Lemma \\ref{lemma-gysin-projection} and in", "Cohomology, Remark \\ref{cohomology-remark-support-cup-product-global} match,", "\\item by Cohomology, Lemma \\ref{cohomology-lemma-support-cup-product}", "this maps to $\\alpha \\cup \\gamma^{0, 0}(1)$ in", "$H^{q + c}(X, \\Omega^p_{X/S} \\otimes \\Omega^c_{X/S})$, and", "\\item the left hand side $\\gamma^p(\\alpha|_Z)$ of the equality in", "Lemma \\ref{lemma-gysin-projection} maps to", "$\\gamma^{p, q}(\\alpha|_Z)$.", "\\end{enumerate}", "This finishes the proof." ], "refs": [ "derham-lemma-gysin-projection", "cohomology-lemma-support-cup-product", "derham-remark-how-to-use", "cohomology-lemma-local-to-global-sections-with-support", "derham-lemma-gysin-projection", "cohomology-remark-support-cup-product-global", "derham-lemma-gysin-projection", "cohomology-remark-support-cup-product-global", "cohomology-lemma-support-cup-product", "derham-lemma-gysin-projection" ], "ref_ids": [ 14362, 2158, 14394, 2153, 14362, 2276, 14362, 2276, 2158, 14362 ] } ], "ref_ids": [ 14360, 14394 ] }, { "id": 14366, "type": "theorem", "label": "derham-lemma-gysin-transverse-global", "categories": [ "derham" ], "title": "derham-lemma-gysin-transverse-global", "contents": [ "Let $c \\geq 0$ and", "$$", "\\xymatrix{", "Z' \\ar[d]_h \\ar[r] & X' \\ar[d]_g \\ar[r] & S' \\ar[d] \\\\", "Z \\ar[r] & X \\ar[r] & S", "}", "$$", "satisfy the assumptions of Lemma \\ref{lemma-gysin-transverse} and assume", "in addition that $X \\to S$ and $X' \\to S'$ are smooth and that", "$Z \\to X$ and $Z' \\to X'$ are Koszul regular immersions.", "Then the diagram", "$$", "\\xymatrix{", "H^q(Z, \\Omega^p_{Z/S}) \\ar[rr]_-{\\gamma^{p, q}} \\ar[d] & &", "H^{q + c}(X, \\Omega^{p + c}_{X/S}) \\ar[d] \\\\", "H^q(Z', \\Omega^p_{Z'/S'}) \\ar[rr]^{\\gamma^{p, q}} & &", "H^{q + c}(X', \\Omega^{p + c}_{X'/S'})", "}", "$$", "is commutative where $\\gamma^{p, q}$ is as in Remark \\ref{remark-how-to-use}." ], "refs": [ "derham-lemma-gysin-transverse", "derham-remark-how-to-use" ], "proofs": [ { "contents": [ "This follows on combining Lemma \\ref{lemma-gysin-transverse}", "and Cohomology, Lemma \\ref{cohomology-lemma-support-functorial}." ], "refs": [ "derham-lemma-gysin-transverse", "cohomology-lemma-support-functorial" ], "ref_ids": [ 14363, 2159 ] } ], "ref_ids": [ 14363, 14394 ] }, { "id": 14367, "type": "theorem", "label": "derham-lemma-class-of-a-point", "categories": [ "derham" ], "title": "derham-lemma-class-of-a-point", "contents": [ "Let $k$ be a field. Let $X$ be an irreducible smooth proper scheme over $k$", "of dimension $d$. Let $Z \\subset X$ be the reduced closed subscheme consisting", "of a single $k$-rational point $x$. Then the image of", "$1 \\in k = H^0(Z, \\mathcal{O}_Z) = H^0(Z, \\Omega^0_{Z/k})$", "by the map $H^0(Z, \\Omega^0_{Z/k}) \\to H^d(X, \\Omega^d_{X/k})$", "of Remark \\ref{remark-how-to-use} is nonzero." ], "refs": [ "derham-remark-how-to-use" ], "proofs": [ { "contents": [ "The map $\\gamma^0 : \\mathcal{O}_Z \\to", "\\mathcal{H}^d_Z(\\Omega^d_{X/k}) = R\\mathcal{H}_Z(\\Omega^d_{X/k})[d]$", "is adjoint to a map", "$$", "g^0 : i_*\\mathcal{O}_Z \\longrightarrow \\Omega^d_{X/k}[d]", "$$", "in $D(\\mathcal{O}_X)$. Recall that $\\Omega^d_{X/k} = \\omega_X$ is a", "dualizing sheaf for $X/k$, see", "Duality for Schemes, Lemma \\ref{duality-lemma-duality-proper-over-field}.", "Hence the $k$-linear dual of the map in the statement", "of the lemma is the map", "$$", "H^0(X, \\mathcal{O}_X) \\to \\Ext^d_X(i_*\\mathcal{O}_Z, \\omega_X)", "$$", "which sends $1$ to $g^0$. Thus it suffices to show that $g^0$ is nonzero.", "This we may do in any neighbourhood $U$ of the point $x$. Choose $U$", "such that there exist $f_1, \\ldots, f_d \\in \\mathcal{O}_X(U)$", "vanishing only at $x$ and generating the maximal ideal", "$\\mathfrak m_x \\subset \\mathcal{O}_{X, x}$. We may assume", "assume $U = \\Spec(R)$ is affine. Looking over the", "construction of $\\gamma^0$ we find that our extension is given by", "$$", "k \\to", "(R \\to \\bigoplus\\nolimits_{i_0} R_{f_{i_0}} \\to", "\\bigoplus\\nolimits_{i_0 < i_1} R_{f_{i_0}f_{i_1}} \\to", "\\ldots \\to R_{f_1\\ldots f_r})[d] \\to R[d]", "$$", "where $1$ maps to $1/f_1 \\ldots f_c$ under the first map.", "This is nonzero because $1/f_1 \\ldots f_c$ is a nonzero element", "of local cohomology group $H^d_{(f_1, \\ldots, f_d)}(R)$ in this case," ], "refs": [ "duality-lemma-duality-proper-over-field" ], "ref_ids": [ 13606 ] } ], "ref_ids": [ 14394 ] }, { "id": 14368, "type": "theorem", "label": "derham-lemma-relative-bottom-part-degenerates", "categories": [ "derham" ], "title": "derham-lemma-relative-bottom-part-degenerates", "contents": [ "In Situation \\ref{situation-relative-duality} the psuhforward", "$f_*\\mathcal{O}_X$ is a finite \\'etale $\\mathcal{O}_S$-algebra", "and locally on $S$ we have $Rf_*\\mathcal{O}_X = f_*\\mathcal{O}_X \\oplus P$", "in $D(\\mathcal{O}_S)$ with $P$ perfect of tor amplitude in $[1, \\infty)$.", "The map $\\text{d} : f_*\\mathcal{O}_X \\to f_*\\Omega_{X/S}$ is zero." ], "refs": [], "proofs": [ { "contents": [ "The first part of the statement follows from", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-proper-flat-geom-red}.", "Setting $S' = \\underline{\\Spec}_S(f_*\\mathcal{O}_X)$ we get a factorization", "$X \\to S' \\to S$ (this is the Stein factorization, see", "More on Morphisms, Section \\ref{more-morphisms-section-stein-factorization},", "although we don't need this)", "and we see that $\\Omega_{X/S} = \\Omega_{X/S'}$ for example by", "Morphisms, Lemma \\ref{morphisms-lemma-triangle-differentials} and", "\\ref{morphisms-lemma-etale-at-point}. This of course implies that", "$\\text{d} : f_*\\mathcal{O}_X \\to f_*\\Omega_{X/S}$ is zero." ], "refs": [ "perfect-lemma-proper-flat-geom-red", "morphisms-lemma-triangle-differentials", "morphisms-lemma-etale-at-point" ], "ref_ids": [ 7069, 5313, 5372 ] } ], "ref_ids": [] }, { "id": 14369, "type": "theorem", "label": "derham-lemma-relative-duality-hodge", "categories": [ "derham" ], "title": "derham-lemma-relative-duality-hodge", "contents": [ "In Situation \\ref{situation-relative-duality} there exists an", "$\\mathcal{O}_S$-module map", "$$", "t : Rf_*\\Omega^d_{X/S}[d] \\longrightarrow \\mathcal{O}_S", "$$", "unique up to precomposing by multiplication by a unit of", "$H^0(X, \\mathcal{O}_X)$ with the following property: for all $p$ the pairing", "$$", "Rf_*\\Omega^p_{X/S}", "\\otimes_{\\mathcal{O}_S}^\\mathbf{L}", "Rf_*\\Omega^{d - p}_{X/S}[d]", "\\longrightarrow", "\\mathcal{O}_S", "$$", "given by the relative cup product composed with $t$", "is a perfect pairing of perfect complexes on $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\omega^\\bullet_{X/S}$ be the relative dualizing complex of $X$ over $S$ as", "in Duality for Schemes, Remark \\ref{duality-remark-relative-dualizing-complex}", "and let $Rf_*\\omega_{X/S}^\\bullet \\to \\mathcal{O}_S$ be its trace map. By", "Duality for Schemes, Lemma \\ref{duality-lemma-smooth-proper}", "there exists an isomorphism $\\omega^\\bullet_{X/S} \\cong \\Omega^d_{X/S}[d]$", "and using this isomorphism we obtain $t$. The complexes $Rf_*\\Omega^p_{X/S}$", "are perfect by Lemma \\ref{lemma-proper-smooth-de-Rham}.", "Since $\\Omega^p_{X/S}$ is locally free and since", "$\\Omega^p_{X/S} \\otimes_{\\mathcal{O}_X} \\Omega^{d - p}_{X/S} \\to", "\\Omega^d_{X/S}$ exhibits an isomorphism $\\Omega^p_{X/S} \\cong", "\\SheafHom_{\\mathcal{O}_X}(\\Omega^{d - p}_{X/S}, \\Omega^d_{X/S})$", "we see that the pairing induced by the relative cup product is perfect by", "Duality for Schemes, Remark", "\\ref{duality-remark-relative-dualizing-complex-relative-cup-product}.", "\\medskip\\noindent", "Uniqueness of $t$. Choose a distinguished triangle", "$f_*\\mathcal{O}_X \\to Rf_*\\mathcal{O}_X \\to P \\to f_*\\mathcal{O}_X[1]$.", "By Lemma \\ref{lemma-relative-bottom-part-degenerates}", "the object $P$ is perfect of tor amplitude in $[1, \\infty)$", "and the triangle is locally on $S$ split.", "Thus $R\\SheafHom_{\\mathcal{O}_X}(P, \\mathcal{O}_X)$ is perfect", "of tor amplitude in $(-\\infty, -1]$. Hence duality (above) shows that", "locally on $S$ we have", "$$", "Rf_*\\Omega^d_{X/S}[d] \\cong", "R\\SheafHom_{\\mathcal{O}_S}(f_*\\mathcal{O}_X, \\mathcal{O}_S)", "\\oplus R\\SheafHom_{\\mathcal{O}_X}(P, \\mathcal{O}_X)", "$$", "This shows that $R^df_*\\Omega^d_{X/S}$ is finite locally free and", "that we obtain a perfect $\\mathcal{O}_S$-bilinear pairing", "$$", "f_*\\mathcal{O}_X \\times R^df_*\\Omega^d_{X/S} \\longrightarrow \\mathcal{O}_S", "$$", "using $t$.", "This implies that any $\\mathcal{O}_S$-linear map", "$t' : R^df_*\\Omega^d_{X/S} \\to \\mathcal{O}_S$ is of the form", "$t' = t \\circ g$ for some", "$g \\in \\Gamma(S, f_*\\mathcal{O}_X) = \\Gamma(X, \\mathcal{O}_X)$.", "In order for $t'$ to still determine a perfect pairing $g$ will have", "to be a unit. This finishes the proof." ], "refs": [ "duality-remark-relative-dualizing-complex", "duality-lemma-smooth-proper", "derham-lemma-proper-smooth-de-Rham", "duality-remark-relative-dualizing-complex-relative-cup-product", "derham-lemma-relative-bottom-part-degenerates" ], "ref_ids": [ 13649, 13550, 14319, 13650, 14368 ] } ], "ref_ids": [] }, { "id": 14370, "type": "theorem", "label": "derham-lemma-relative-top-part-degenerates", "categories": [ "derham" ], "title": "derham-lemma-relative-top-part-degenerates", "contents": [ "In Situation \\ref{situation-relative-duality} the map", "$\\text{d} : R^df_*\\Omega^{d - 1}_{X/S} \\to R^df_*\\Omega^d_{X/S}$", "is zero." ], "refs": [], "proofs": [ { "contents": [ "[Proof in case $S$ is reduced]", "Assume $S$ is reduced. Observe that", "$\\text{d} : R^df_*\\Omega^{d - 1}_{X/S} \\to R^df_*\\Omega^d_{X/S}$", "is an $\\mathcal{O}_S$-linear map of (quasi-coherent) $\\mathcal{O}_S$-modules.", "The $\\mathcal{O}_S$-module $R^df_*\\Omega^d_{X/S}$ is finite locally free", "(as the dual of the finite locally free $\\mathcal{O}_S$-module", "$f_*\\mathcal{O}_X$ by Lemmas", "\\ref{lemma-relative-duality-hodge} and", "\\ref{lemma-relative-bottom-part-degenerates}).", "Since $S$ is reduced it suffices to show that", "the stalk of $\\text{d}$ in every generic point $\\eta \\in S$", "is zero; this follows by looking at sections over affine opens,", "using that the target of $\\text{d}$ is locally free, and", "Algebra, Lemma \\ref{algebra-lemma-reduced-ring-sub-product-fields} part (2).", "Since $S$ is reduced we have $\\mathcal{O}_{S, \\eta} = \\kappa(\\eta)$, see", "Algebra, Lemma \\ref{algebra-lemma-minimal-prime-reduced-ring}.", "Thus $\\text{d}_\\eta$ is identified with the map", "$$", "\\text{d} :", "H^d(X_\\eta, \\Omega^{d - 1}_{X_\\eta/\\kappa(\\eta)})", "\\longrightarrow", "H^d(X_\\eta, \\Omega^d_{X_\\eta/\\kappa(\\eta)})", "$$", "which is zero by Lemma \\ref{lemma-top-part-degenerates}." ], "refs": [ "derham-lemma-relative-duality-hodge", "derham-lemma-relative-bottom-part-degenerates", "algebra-lemma-reduced-ring-sub-product-fields", "algebra-lemma-minimal-prime-reduced-ring", "derham-lemma-top-part-degenerates" ], "ref_ids": [ 14369, 14368, 419, 418, 14356 ] } ], "ref_ids": [] }, { "id": 14371, "type": "theorem", "label": "derham-proposition-global-generation-on-fibres", "categories": [ "derham" ], "title": "derham-proposition-global-generation-on-fibres", "contents": [ "Let $f : X \\to Y$ be a smooth proper morphism of schemes over a base $S$.", "Let $N$ and $n_1, \\ldots, n_N \\geq 0$ be integers and let", "$\\xi_i \\in H^{n_i}_{dR}(X/S)$, $1 \\leq i \\leq N$.", "Assume for all points $y \\in Y$ the images of $\\xi_1, \\ldots, \\xi_N$", "in $H^*_{dR}(X_y/y)$ form a basis over $\\kappa(y)$. The map", "$$", "\\tilde \\xi = \\bigoplus \\tilde \\xi_i[-n_i] :", "\\bigoplus \\Omega^\\bullet_{Y/S}[-n_i]", "\\longrightarrow", "Rf_*\\Omega^\\bullet_{X/S}", "$$", "(see proof) is an isomorphism in $D(Y, (Y \\to S)^{-1}\\mathcal{O}_S)$ and", "correspondingly the map", "$$", "\\bigoplus\\nolimits_{i = 1}^N H^*_{dR}(Y/S) \\longrightarrow", "H^*_{dR}(X/S), \\quad", "(a_1, \\ldots, a_N) \\longmapsto \\sum \\xi_i \\cup f^*a_i", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Denote $p : X \\to S$ and $q : Y \\to S$ be the structure morphisms.", "Let $\\xi'_i : \\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}[n_i]$", "be the map of Remark \\ref{remark-cup-product-as-a-map} corresponding", "to $\\xi_i$. Denote", "$$", "\\tilde \\xi_i :", "\\Omega^\\bullet_{Y/S} \\to Rf_*\\Omega^\\bullet_{X/S}[n_i]", "$$", "the composition of $\\xi'_i$ with the canonical map", "$\\Omega^\\bullet_{Y/S} \\to Rf_*\\Omega^\\bullet_{X/S}$.", "Using", "$$", "R\\Gamma(Y, Rf_*\\Omega^\\bullet_{X/S}) = R\\Gamma(X, \\Omega^\\bullet_{X/S})", "$$", "on cohomology $\\tilde \\xi_i$ is the map $\\eta \\mapsto \\xi_i \\cup f^*\\eta$", "from $H^m_{dR}(Y/S)$ to $H^{m + n}_{dR}(X/S)$.", "Further, since the formation of $\\xi'_i$ commutes with", "restrictions to opens, so does the formation of $\\tilde \\xi_i$", "commute with restriction to opens.", "\\medskip\\noindent", "Thus we can consider the map", "$$", "\\tilde \\xi = \\bigoplus \\tilde \\xi_i[-n_i] :", "\\bigoplus \\Omega^\\bullet_{Y/S}[-n_i]", "\\longrightarrow", "Rf_*\\Omega^\\bullet_{X/S}", "$$", "To prove the lemma it suffices to show that this is an isomorphism in", "$D(Y, q^{-1}\\mathcal{O}_S)$. If we could show $\\tilde \\xi$", "comes from a map of filtered complexes (with suitable filtrations),", "then we could appeal to the spectral sequence of", "Lemma \\ref{lemma-spectral-sequence-smooth} to finish the proof.", "This takes more work than is necessary and instead our approach", "will be to reduce to the affine case (whose proof does in some sense", "use the spectral sequence).", "\\medskip\\noindent", "Indeed, if $Y' \\subset Y$ is is any open with inverse image", "$X' \\subset X$, then $\\tilde \\xi|_{X'}$ induces the map", "$$", "\\bigoplus\\nolimits_{i = 1}^N H^*_{dR}(Y'/S) \\longrightarrow", "H^*_{dR}(X'/S), \\quad", "(a_1, \\ldots, a_N) \\longmapsto \\sum \\xi_i|_{X'} \\cup f^*a_i", "$$", "on cohomology over $Y'$, see discussion above.", "Thus it suffices to find a basis for the topology", "on $Y$ such that the proposition holds for the members of the basis", "(in particular we can forget about the map $\\tilde \\xi$ when", "we do this). This reduces us to the case where $Y$ and $S$", "are affine which is handled by Lemma \\ref{lemma-global-generation-on-fibres}", "and the proof is complete." ], "refs": [ "derham-remark-cup-product-as-a-map", "derham-lemma-spectral-sequence-smooth", "derham-lemma-global-generation-on-fibres" ], "ref_ids": [ 14382, 14334, 14336 ] } ], "ref_ids": [] }, { "id": 14372, "type": "theorem", "label": "derham-proposition-projective-space-bundle-formula", "categories": [ "derham" ], "title": "derham-proposition-projective-space-bundle-formula", "contents": [ "Let $X \\to S$ be a morphism of schemes. Let $\\mathcal{E}$ be a locally", "free $\\mathcal{O}_X$-module of constant rank $r$. Consider the morphism", "$p : P = \\mathbf{P}(\\mathcal{E}) \\to X$.", "Then the map", "$$", "\\bigoplus\\nolimits_{i = 0, \\ldots, r - 1} H^*_{dR}(X/S)", "\\longrightarrow", "H^*_{dR}(P/S)", "$$", "given by the rule", "$$", "(a_0, \\ldots, a_{r - 1}) \\longmapsto", "\\sum\\nolimits_{i = 0, \\ldots, r - 1} c_1^{dR}(\\mathcal{O}_P(1))^i \\cup p^*(a_i)", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine open $\\Spec(A) \\subset X$ such that $\\mathcal{E}$ restricts", "to the trivial locally free module $\\mathcal{O}_{\\Spec(A)}^{\\oplus r}$.", "Then $P \\times_X \\Spec(A) = \\mathbf{P}^{r - 1}_A$. Thus we see that", "$p$ is proper and smooth, see Section \\ref{section-projective-space}.", "Moreover, the classes $c_1^{dR}(\\mathcal{O}_P(1))^i$, $i = 0, 1, \\ldots, r - 1$", "restricted to a fibre $X_y = \\mathbf{P}^{r - 1}_y$ freely generate the", "de Rham cohomology $H^*_{dR}(X_y/y)$ over $\\kappa(y)$, see", "Lemma \\ref{lemma-de-rham-cohomology-projective-space}. Thus we've verified the", "conditions of Proposition \\ref{proposition-global-generation-on-fibres}", "and we win." ], "refs": [ "derham-lemma-de-rham-cohomology-projective-space", "derham-proposition-global-generation-on-fibres" ], "ref_ids": [ 14333, 14371 ] } ], "ref_ids": [] }, { "id": 14373, "type": "theorem", "label": "derham-proposition-blowup-split", "categories": [ "derham" ], "title": "derham-proposition-blowup-split", "contents": [ "With notation as in Lemma \\ref{lemma-blowup} the map", "$\\Omega^\\bullet_{X/S} \\to Rb_*\\Omega^\\bullet_{X'/S}$", "has a splitting in $D(X, (X \\to S)^{-1}\\mathcal{O}_S)$." ], "refs": [ "derham-lemma-blowup" ], "proofs": [ { "contents": [ "Consider the triangle constructed in", "Lemma \\ref{lemma-distinguished-triangle-blowup}.", "We claim that the map", "$$", "Rb_*(\\Omega^\\bullet_{X'/S}) \\oplus i_*\\Omega^\\bullet_{Z/S} \\to", "i_*Rp_*(\\Omega^\\bullet_{E/S})", "$$", "has a splitting whose image contains the summand $i_*\\Omega^\\bullet_{Z/S}$.", "By Derived Categories, Lemma \\ref{derived-lemma-split} this will show that", "the first arrow of the triangle has a splitting which vanishes on", "the summand $i_*\\Omega^\\bullet_{Z/S}$ which proves the lemma.", "We will prove the claim by decomposing $Rp_*\\Omega^\\bullet_{E/S}$", "into a direct sum where the first piece corresponds to", "$\\Omega^\\bullet_{Z/S}$ and the second piece can be lifted", "through $Rb_*\\Omega^\\bullet_{X'/S}$.", "\\medskip\\noindent", "Proof of the claim. We may decompose $X$ into open and closed subschemes", "having fixed relative dimension to $S$, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-omega-finite-locally-free}.", "Since the derived category $D(X, f^{-1}\\mathcal{O})_S)$ correspondingly", "decomposes as a product of categories, we may assume $X$ has", "fixed relative dimension $N$ over $S$. We may decompose", "$Z = \\coprod Z_m$ into open and closed subschemes of relative", "dimension $m \\geq 0$ over $S$. The restriction $i_m : Z_m \\to X$ of", "$i$ to $Z_m$ is a regular immersion of codimension $N - m$, see Divisors, Lemma", "\\ref{divisors-lemma-immersion-smooth-into-smooth-regular-immersion}.", "Let $E = \\coprod E_m$ be the corresponding decomposition, i.e.,", "we set $E_m = p^{-1}(Z_m)$. If $p_m : E_m \\to Z_m$ denotes the", "restriction of $p$ to $E_m$, then we have a canonical isomorphism", "$$", "\\tilde \\xi_m :", "\\bigoplus\\nolimits_{t = 0, \\ldots, N - m - 1}", "\\Omega^\\bullet_{Z_m/S}[-2t]", "\\longrightarrow", "Rp_{m, *}\\Omega^\\bullet_{E_m/S}", "$$", "in $D(Z_m, (Z_m \\to S)^{-1}\\mathcal{O}_S)$", "where in degree $0$ we have the canonical map", "$\\Omega^\\bullet_{Z_m/S} \\to Rp_{m, *}\\Omega^\\bullet_{E_m/S}$.", "See Remark \\ref{remark-projective-space-bundle-formula}.", "Thus we have an isomorphism", "$$", "\\tilde \\xi :", "\\bigoplus\\nolimits_m", "\\bigoplus\\nolimits_{t = 0, \\ldots, N - m - 1}", "\\Omega^\\bullet_{Z_m/S}[-2t]", "\\longrightarrow", "Rp_*(\\Omega^\\bullet_{E/S})", "$$", "in $D(Z, (Z \\to S)^{-1}\\mathcal{O}_S)$", "whose restriction to the summand", "$\\Omega^\\bullet_{Z/S} = \\bigoplus \\Omega^\\bullet_{Z_m/S}$ of the source", "is the canonical map $\\Omega^\\bullet_{Z/S} \\to Rp_*(\\Omega^\\bullet_{E/S})$.", "Consider the subcomplexes $M_m$ and $K_m$ of the complex", "$\\bigoplus\\nolimits_{t = 0, \\ldots, N - m - 1} \\Omega^\\bullet_{Z_m/S}[-2t]$", "introduced in Remark \\ref{remark-projective-space-bundle-formula}.", "We set", "$$", "M = \\bigoplus M_m", "\\quad\\text{and}\\quad", "K = \\bigoplus K_m", "$$", "We have $M = K[-2]$ and by construction the map", "$$", "c_{E/Z} \\oplus \\tilde \\xi|_M :", "\\Omega^\\bullet_{Z/S} \\oplus M", "\\longrightarrow", "Rp_*(\\Omega^\\bullet_{E/S})", "$$", "is an isomorphism (see remark referenced above).", "\\medskip\\noindent", "Consider the map", "$$", "\\delta : \\Omega^\\bullet_{E/S}[-2] \\longrightarrow \\Omega^\\bullet_{X'/S}", "$$", "in $D(X', (X' \\to S)^{-1}\\mathcal{O}_S)$ of", "Lemma \\ref{lemma-gysin-via-log-complex}", "with the property that the composition", "$$", "\\Omega^\\bullet_{E/S}[-2] \\longrightarrow \\Omega^\\bullet_{X'/S}", "\\longrightarrow", "\\Omega^\\bullet_{E/S}", "$$", "is the map $\\theta'$ of Remark \\ref{remark-cup-product-as-a-map} for", "$c_1^{dR}(\\mathcal{O}_{X'}(-E))|_E) = c_1^{dR}(\\mathcal{O}_E(1))$.", "The final assertion of Remark \\ref{remark-projective-space-bundle-formula}", "tells us that the diagram", "$$", "\\xymatrix{", "K[-2] \\ar[d]_{(\\tilde \\xi|_K)[-2]} \\ar[r]_{\\text{id}} &", "M \\ar[d]^{\\tilde x|_M} \\\\", "Rp_*\\Omega^\\bullet_{E/S}[-2] \\ar[r]^-{Rp_*\\theta'} &", "Rp_*\\Omega^\\bullet_{E/S}", "}", "$$", "commutes. Thus we see that we can obtain the desired splitting of", "the claim as the map", "\\begin{align*}", "Rp_*(\\Omega^\\bullet_{E/S})", "& \\xrightarrow{(c_{E/Z} \\oplus \\tilde \\xi|_M)^{-1}}", "\\Omega^\\bullet_{Z/S} \\oplus M \\\\", "& \\xrightarrow{\\text{id} \\oplus \\text{id}^{-1}}", "\\Omega^\\bullet_{Z/S} \\oplus K[-2] \\\\", "& \\xrightarrow{\\text{id} \\oplus (\\tilde \\xi|_K)[-2]}", "\\Omega^\\bullet_{Z/S} \\oplus Rp_*\\Omega^\\bullet_{E/S}[-2] \\\\", "& \\xrightarrow{\\text{id} \\oplus Rb_*\\delta}", "\\Omega^\\bullet_{Z/S} \\oplus Rb_*\\Omega^\\bullet_{X'/S}", "\\end{align*}", "The relationship between $\\theta'$ and $\\delta$ stated above", "together with the commutative diagram involving $\\theta'$, $\\tilde \\xi|_K$,", "and $\\tilde \\xi|_M$ above are exactly what's needed to", "show that this is a section to the canonical map", "$\\Omega^\\bullet_{Z/S} \\oplus Rb_*(\\Omega^\\bullet_{X'/S}) \\to", "Rp_*(\\Omega^\\bullet_{E/S})$ and the proof of the claim is complete." ], "refs": [ "derham-lemma-distinguished-triangle-blowup", "derived-lemma-split", "morphisms-lemma-smooth-omega-finite-locally-free", "divisors-lemma-immersion-smooth-into-smooth-regular-immersion", "derham-remark-projective-space-bundle-formula", "derham-remark-projective-space-bundle-formula", "derham-lemma-gysin-via-log-complex", "derham-remark-cup-product-as-a-map", "derham-remark-projective-space-bundle-formula" ], "ref_ids": [ 14347, 1766, 5334, 8007, 14387, 14387, 14339, 14382, 14387 ] } ], "ref_ids": [ 14345 ] }, { "id": 14374, "type": "theorem", "label": "derham-proposition-Garel", "categories": [ "derham" ], "title": "derham-proposition-Garel", "contents": [ "\\begin{reference}", "\\cite{Garel}", "\\end{reference}", "Let $f : Y \\to X$ be a finite syntomic morphism of schemes.", "The maps $\\Theta^p_{Y/X}$ of Lemma \\ref{lemma-Garel} define a map of complexes", "$$", "\\Theta_{Y/X} :", "f_*\\Omega^\\bullet_{Y/\\mathbf{Z}}", "\\longrightarrow", "\\Omega^\\bullet_{X/\\mathbf{Z}}", "$$", "with the following properties", "\\begin{enumerate}", "\\item in degree $0$ we get", "$\\text{Trace}_f : f_*\\mathcal{O}_Y \\to \\mathcal{O}_X$, see", "Discriminants, Section \\ref{discriminant-section-discriminant},", "\\item we have", "$\\Theta_{Y/X}(\\omega \\wedge \\eta) = \\omega \\wedge \\Theta_{Y/X}(\\eta)$", "for $\\omega$ in $\\Omega^\\bullet_{X/\\mathbf{Z}}$ and $\\eta$", "in $f_*\\Omega^\\bullet_{Y/\\mathbf{Z}}$,", "\\item if $f$ is a morphism over a base scheme $S$, then", "$\\Theta_{Y/X}$ induces a map of complexes", "$f_*\\Omega^\\bullet_{Y/S} \\to \\Omega^\\bullet_{X/S}$.", "\\end{enumerate}" ], "refs": [ "derham-lemma-Garel" ], "proofs": [ { "contents": [ "By Discriminants, Lemma", "\\ref{discriminant-lemma-locally-comes-from-universal-finite}", "for every $x \\in X$ we can find $d \\geq 1$ and a commutative diagram", "$$", "\\xymatrix{", "Y \\ar[d] &", "V \\ar[d] \\ar[l] \\ar[r] &", "V_d \\ar[d] \\ar[r] &", "Y_d = \\Spec(B_d) \\ar[d] \\\\", "X &", "U \\ar[l] \\ar[r] &", "U_d \\ar[r] &", "X_d = \\Spec(A_d)", "}", "$$", "such that $x \\in U \\subset X$ is affine open, $V = f^{-1}(U)$", "and $V = U \\times_{U_d} V_d$. Write $U = \\Spec(A)$ and $V = \\Spec(B)$", "and observe that $B = A \\otimes_{A_d} B_d$ and recall that", "$B_d = A_d e_1 \\oplus \\ldots \\oplus A_d e_d$. Suppose we have", "$a_1, \\ldots, a_r \\in A$ and $b_1, \\ldots, b_s \\in B$.", "We may write $b_j = \\sum a_{j, l} e_d$ with $a_{j, l} \\in A$.", "Set $N = r + sd$ and consider the factorizations", "$$", "\\xymatrix{", "V \\ar[r] \\ar[d] &", "V' = \\mathbf{A}^N \\times V_d \\ar[r] \\ar[d] &", "V_d \\ar[d] \\\\", "U \\ar[r]&", "U' = \\mathbf{A}^N \\times U_d \\ar[r] &", "U_d", "}", "$$", "Here the horizontal lower right arrow is given by the morphism", "$U \\to U_d$ (from the earlier diagram) and the morphism", "$U \\to \\mathbf{A}^N$ given by $a_1, \\ldots, a_r, a_{1, 1}, \\ldots, a_{s, d}$.", "Then we see that the functions $a_1, \\ldots, a_r$ are in the image of", "$\\Gamma(U', \\mathcal{O}_{U'}) \\to \\Gamma(U, \\mathcal{O}_U)$", "and the functions $b_1, \\ldots, b_s$ are in the image of", "$\\Gamma(V', \\mathcal{O}_{V'}) \\to \\Gamma(V, \\mathcal{O}_V)$.", "In this way we see that for any finite collection of elements\\footnote{After", "all these elements will be finite sums of elements of the form", "$a_0 \\text{d}a_1 \\wedge \\ldots \\wedge \\text{d}a_i$ with", "$a_0, \\ldots, a_i \\in A$ or finite sums of elements of the form", "$b_0 \\text{d}b_1 \\wedge \\ldots \\wedge \\text{d}b_j$ with", "$b_0, \\ldots, b_j \\in B$.} of the groups", "$$", "\\Gamma(V, \\Omega^i_{Y/\\mathbf{Z}}),\\quad i = 0, 1, 2, \\ldots", "\\quad\\text{and}\\quad", "\\Gamma(U, \\Omega^j_{X/\\mathbf{Z}}),\\quad j = 0, 1, 2, \\ldots", "$$", "we can find a factorizations $V \\to V' \\to V_d$ and", "$U \\to U' \\to U_d$ with $V' = \\mathbf{A}^N \\times V_d$ and", "$U' = \\mathbf{A}^N \\times U_d$ as above", "such that these sections are the pullbacks of sections from", "$$", "\\Gamma(V', \\Omega^i_{V'/\\mathbf{Z}}),\\quad i = 0, 1, 2, \\ldots", "\\quad\\text{and}\\quad", "\\Gamma(U', \\Omega^j_{U'/\\mathbf{Z}}),\\quad j = 0, 1, 2, \\ldots", "$$", "The upshot of this is that to check", "$\\text{d} \\circ \\Theta_{Y/X} = \\Theta_{Y/X} \\circ \\text{d}$", "it suffices to check this is true for $\\Theta_{V'/U'}$.", "Similarly, for property (2) of the lemma.", "\\medskip\\noindent", "By Discriminants, Lemmas", "\\ref{discriminant-lemma-universal-finite-syntomic-smooth} and", "\\ref{discriminant-lemma-universal-finite-syntomic-etale}", "the scheme $U_d$ is smooth and the morphism $V_d \\to U_d$", "is \\'etale over a dense open of $U_d$.", "Hence the same is true for the morphism", "$V' \\to U'$. Since $\\Omega_{U'/\\mathbf{Z}}$ is locally free and hence", "$\\Omega^p_{U'/\\mathbf{Z}}$ is torsion", "free, it suffices to check the desired relations", "after restricting to the open over which $V'$ is finite \\'etale.", "Then we may check the relations after a surjective \\'etale", "base change. Hence we may split the finite \\'etale cover", "and assume we are looking at a morphism of the form", "$$", "\\coprod\\nolimits_{i = 1, \\ldots, d} W \\longrightarrow W", "$$", "with $W$ smooth over $\\mathbf{Z}$.", "In this case any local properties of our construction are trivial to check", "(provided they are true). This finishes the proof of (1) and (2).", "\\medskip\\noindent", "Finally, we observe that (3) follows from (2) because $\\Omega_{Y/S}$", "is the quotient of $\\Omega_{Y/\\mathbf{Z}}$ by the submodule", "generated by pullbacks of local sections of $\\Omega_{S/\\mathbf{Z}}$." ], "refs": [ "discriminant-lemma-locally-comes-from-universal-finite", "discriminant-lemma-universal-finite-syntomic-smooth", "discriminant-lemma-universal-finite-syntomic-etale" ], "ref_ids": [ 14991, 14989, 14990 ] } ], "ref_ids": [ 14352 ] }, { "id": 14375, "type": "theorem", "label": "derham-proposition-poincare-duality", "categories": [ "derham" ], "title": "derham-proposition-poincare-duality", "contents": [ "Let $k$ be a field. Let $X$ be a nonempty smooth proper scheme over $k$", "equidimensional of dimension $d$. There exists a $k$-linear map", "$$", "t : H^{2d}_{dR}(X/k) \\longrightarrow k", "$$", "unique up to precomposing by multiplication by a unit of", "$H^0(X, \\mathcal{O}_X)$ with the following property: for all $i$ the pairing", "$$", "H^i_{dR}(X/k) \\times H_{dR}^{2d - i}(X/k)", "\\longrightarrow", "k, \\quad", "(\\xi, \\xi') \\longmapsto t(\\xi \\cup \\xi')", "$$", "is perfect." ], "refs": [], "proofs": [ { "contents": [ "By the Hodge-to-de Rham spectral sequence", "(Section \\ref{section-hodge-to-de-rham}), the vanishing", "of $\\Omega^i_{X/k}$ for $i > d$, the vanishing in", "Cohomology, Proposition \\ref{cohomology-proposition-vanishing-Noetherian}", "and the results of Lemmas \\ref{lemma-bottom-part-degenerates} and", "\\ref{lemma-top-part-degenerates}", "we see that $H^0_{dR}(X/k) = H^0(X, \\mathcal{O}_X)$", "and $H^d(X, \\Omega^d_{X/k}) = H_{dR}^{2d}(X/k)$.", "More precisesly, these identifications come from the maps", "of complexes", "$$", "\\Omega^\\bullet_{X/k} \\to \\mathcal{O}_X[0]", "\\quad\\text{and}\\quad", "\\Omega^d_{X/k}[-d] \\to \\Omega^\\bullet_{X/k}", "$$", "Let us choose $t : H_{dR}^{2d}(X/k) \\to k$ which via this identification", "corresponds to a $t$ as in Lemma \\ref{lemma-duality-hodge}.", "Then in any case we see that the pairing displayed in the lemma", "is perfect for $i = 0$.", "\\medskip\\noindent", "Denote $\\underline{k}$ the constant sheaf with value $k$ on $X$.", "Let us abbreviate $\\Omega^\\bullet = \\Omega^\\bullet_{X/k}$.", "Consider the map (\\ref{equation-wedge}) which in our situation reads", "$$", "\\wedge :", "\\text{Tot}(\\Omega^\\bullet \\otimes_{\\underline{k}} \\Omega^\\bullet)", "\\longrightarrow", "\\Omega^\\bullet", "$$", "For every integer $p = 0, 1, \\ldots, d$ this map", "annihilates the subcomplex", "$\\text{Tot}(\\sigma_{> p} \\Omega^\\bullet \\otimes_{\\underline{k}}", "\\sigma_{\\geq d - p} \\Omega^\\bullet)$ for degree reasons.", "Hence we find that the restriction of $\\wedge$ to the subcomplex", "$\\text{Tot}(\\Omega^\\bullet \\otimes_{\\underline{k}}", "\\sigma_{\\geq d - p}\\Omega^\\bullet)$ factors through a map of complexes", "$$", "\\gamma_p :", "\\text{Tot}(\\sigma_{\\leq p} \\Omega^\\bullet \\otimes_{\\underline{k}}", "\\sigma_{\\geq d - p} \\Omega^\\bullet)", "\\longrightarrow", "\\Omega^\\bullet", "$$", "Using the same procedure as in Section \\ref{section-cup-product} we obtain", "cup products", "$$", "H^i(X, \\sigma_{\\leq p} \\Omega^\\bullet) \\times", "H^{2d - i}(X, \\sigma_{\\geq d - p}\\Omega^\\bullet)", "\\longrightarrow", "H_{dR}^{2d}(X, \\Omega^\\bullet)", "$$", "We will prove by induction on $p$ that these cup products via $t$", "induce perfect pairings between $H^i(X, \\sigma_{\\leq p} \\Omega^\\bullet)$", "and $H^{2d - i}(X, \\sigma_{\\geq d - p}\\Omega^\\bullet)$. For $p = d$", "this is the assertion of the proposition.", "\\medskip\\noindent", "The base case is $p = 0$. In this case we simply obtain the pairing", "between $H^i(X, \\mathcal{O}_X)$ and $H^{d - i}(X, \\Omega^d)$ of", "Lemma \\ref{lemma-duality-hodge} and the result is true.", "\\medskip\\noindent", "Induction step. Say we know the result is true for $p$. Then", "we consider the distinguished triangle", "$$", "\\Omega^{p + 1}[-p - 1] \\to", "\\sigma_{\\leq p + 1}\\Omega^\\bullet \\to", "\\sigma_{\\leq p}\\Omega^\\bullet \\to", "\\Omega^{p + 1}[-p]", "$$", "and the distinguished triangle", "$$", "\\sigma_{\\geq d - p}\\Omega^\\bullet \\to", "\\sigma_{\\geq d - p - 1}\\Omega^\\bullet \\to", "\\Omega^{d - p - 1}[-d + p + 1] \\to", "(\\sigma_{\\geq d - p}\\Omega^\\bullet)[1]", "$$", "Observe that both are distinguished triangles in the homotopy category", "of complexes of sheaves of $\\underline{k}$-modules; in particular the", "maps $\\sigma_{\\leq p}\\Omega^\\bullet \\to \\Omega^{p + 1}[-p]$ and", "$\\Omega^{d - p - 1}[-d + p + 1] \\to (\\sigma_{\\geq d - p}\\Omega^\\bullet)[1]$", "are given by actual maps of complexes, namely using the differential", "$\\Omega^p \\to \\Omega^{p + 1}$ and the differential", "$\\Omega^{d - p - 1} \\to \\Omega^{d - p}$.", "Consider the long exact cohomology sequences associated to these", "distinguished triangles", "$$", "\\xymatrix{", "H^{i - 1}(X, \\sigma_{\\leq p}\\Omega^\\bullet) \\ar[d]_a \\\\", "H^i(X, \\Omega^{p + 1}[-p - 1]) \\ar[d]_b \\\\", "H^i(X, \\sigma_{\\leq p + 1}\\Omega^\\bullet) \\ar[d]_c \\\\", "H^i(X, \\sigma_{\\leq p}\\Omega^\\bullet) \\ar[d]_d \\\\", "H^{i + 1}(X, \\Omega^{p + 1}[-p - 1])", "}", "\\quad\\quad", "\\xymatrix{", "H^{2d - i + 1}(X, \\sigma_{\\geq d - p}\\Omega^\\bullet) \\\\", "H^{2d - i}(X, \\Omega^{d - p - 1}[-d + p + 1]) \\ar[u]_{a'} \\\\", "H^{2d - i}(X, \\sigma_{\\geq d - p - 1}\\Omega^\\bullet) \\ar[u]_{b'} \\\\", "H^{2d - i}(X, \\sigma_{\\geq d - p}\\Omega^\\bullet) \\ar[u]_{c'} \\\\", "H^{2d - i - 1}(X, \\Omega^{d - p - 1}[-d + p + 1]) \\ar[u]_{d'}", "}", "$$", "By induction and Lemma \\ref{lemma-duality-hodge}", "we know that the pairings constructed above between the", "$k$-vectorspaces on the first, second, fourth, and fifth", "rows are perfect. By the $5$-lemma, in order to show that", "the pairing between the cohomology groups in the middle row", "is perfect, it suffices to show that the pairs", "$(a, a')$, $(b, b')$, $(c, c')$, and $(d, d')$", "are compatible with the given pairings (see below).", "\\medskip\\noindent", "Let us prove this for the pair $(c, c')$. Here we observe simply", "that we have a commutative diagram", "$$", "\\xymatrix{", "\\text{Tot}(\\sigma_{\\leq p} \\Omega^\\bullet \\otimes_{\\underline{k}}", "\\sigma_{\\geq d - p} \\Omega^\\bullet) \\ar[d]_{\\gamma_p} &", "\\text{Tot}(\\sigma_{\\leq p + 1} \\Omega^\\bullet \\otimes_{\\underline{k}}", "\\sigma_{\\geq d - p} \\Omega^\\bullet) \\ar[l] \\ar[d] \\\\", "\\Omega^\\bullet &", "\\text{Tot}(\\sigma_{\\leq p + 1} \\Omega^\\bullet \\otimes_{\\underline{k}}", "\\sigma_{\\geq d - p - 1} \\Omega^\\bullet) \\ar[l]_-{\\gamma_{p + 1}}", "}", "$$", "Hence if we have $\\alpha \\in H^i(X, \\sigma_{\\leq p + 1}\\Omega^\\bullet)$", "and $\\beta \\in H^{2d - i}(X, \\sigma_{\\geq d - p}\\Omega^\\bullet)$", "then we get", "$\\gamma_p(\\alpha \\cup c'(\\beta)) = \\gamma_{p + 1}(c(\\alpha) \\cup \\beta)$", "by functoriality of the cup product.", "\\medskip\\noindent", "Similarly for the pair $(b, b')$ we use the commutative diagram", "$$", "\\xymatrix{", "\\text{Tot}(\\sigma_{\\leq p + 1} \\Omega^\\bullet \\otimes_{\\underline{k}}", "\\sigma_{\\geq d - p - 1} \\Omega^\\bullet) \\ar[d]_{\\gamma_{p + 1}} &", "\\text{Tot}(\\Omega^{p + 1}[-p - 1] \\otimes_{\\underline{k}}", "\\sigma_{\\geq d - p - 1} \\Omega^\\bullet) \\ar[l] \\ar[d] \\\\", "\\Omega^\\bullet &", "\\Omega^{p + 1}[-p - 1]", "\\otimes_{\\underline{k}}", "\\Omega^{d - p - 1}[-d + p + 1] \\ar[l]_-\\wedge", "}", "$$", "and argue in the same manner.", "\\medskip\\noindent", "For the pair $(d, d')$ we use the commutative diagram", "$$", "\\xymatrix{", "\\Omega^{p + 1}[-p] \\otimes_{\\underline{k}}", "\\Omega^{d - p - 1}[-d + p] \\ar[d] &", "\\text{Tot}(\\sigma_{\\leq p}\\Omega^\\bullet \\otimes_{\\underline{k}}", "\\Omega^{d - p - 1}[-d + p]) \\ar[l] \\ar[d] \\\\", "\\Omega^\\bullet &", "\\text{Tot}(\\sigma_{\\leq p}\\Omega^\\bullet \\otimes_{\\underline{k}}", "\\sigma_{\\geq d - p}\\Omega^\\bullet) \\ar[l]", "}", "$$", "and we look at cohomology classes in", "$H^i(X, \\sigma_{\\leq p}\\Omega^\\bullet)$ and", "$H^{2d - i}(X, \\Omega^{d - p - 1}[-d + p])$.", "Changing $i$ to $i - 1$ we get the result for the pair $(a, a')$", "thereby finishing the proof that our pairings are perfect.", "\\medskip\\noindent", "We omit the argument showing the uniqueness of $t$ up to", "precomposing by multiplication by a unit in $H^0(X, \\mathcal{O}_X)$." ], "refs": [ "cohomology-proposition-vanishing-Noetherian", "derham-lemma-bottom-part-degenerates", "derham-lemma-top-part-degenerates", "derham-lemma-duality-hodge", "derham-lemma-duality-hodge", "derham-lemma-duality-hodge" ], "ref_ids": [ 2246, 14355, 14356, 14354, 14354, 14354 ] } ], "ref_ids": [] }, { "id": 14376, "type": "theorem", "label": "derham-proposition-de-rham-is-weil", "categories": [ "derham" ], "title": "derham-proposition-de-rham-is-weil", "contents": [ "Let $k$ be a field of characteristic zero. The functor that", "sends a smooth projective scheme $X$ over $k$ to $H_{dR}^*(X/k)$", "is a Weil cohomology theory in the sense of", "Weil Cohomology Theories, Definition", "\\ref{weil-definition-weil-cohomology-theory}." ], "refs": [ "weil-definition-weil-cohomology-theory" ], "proofs": [ { "contents": [ "In the discussion above we showed that our data (D0), (D1), (D2')", "satisfies axioms (A1) -- (A9) of Weil Cohomology Theories, Section", "\\ref{weil-section-c1}. Hence we conclude by", "Weil Cohomology Theories, Proposition \\ref{weil-proposition-get-weil}.", "\\medskip\\noindent", "Please don't read what follows. In the proof of the assertions we also used", "Lemmas \\ref{lemma-proper-smooth-de-Rham},", "\\ref{lemma-pullback-c1},", "\\ref{lemma-log-complex-consequence},", "\\ref{lemma-kunneth-de-rham},", "\\ref{lemma-bottom-part-degenerates}, and", "\\ref{lemma-de-rham-cohomology-projective-space},", "Propositions", "\\ref{proposition-projective-space-bundle-formula},", "\\ref{proposition-blowup-split}, and", "\\ref{proposition-Garel},", "Weil Cohomology Theories, Lemmas", "\\ref{weil-lemma-check-over-extension},", "\\ref{weil-lemma-chern-classes},", "\\ref{weil-lemma-cycle-classes}, and", "\\ref{weil-lemma-poincare-duality},", "Weil Cohomology Theories, Remark \\ref{weil-remark-trace},", "Varieties, Lemmas", "\\ref{varieties-lemma-proper-geometrically-reduced-global-sections} and", "\\ref{varieties-lemma-ample-positive},", "Intersection Theory, Section \\ref{intersection-section-projection} and", "Lemma \\ref{intersection-lemma-projection-generically-finite},", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-lci-permanence},", "Algebra, Lemma \\ref{algebra-lemma-CM-over-regular-flat}, and", "Chow Homology, Lemma", "\\ref{chow-lemma-degrees-and-numerical-intersections}." ], "refs": [ "weil-proposition-get-weil", "derham-lemma-proper-smooth-de-Rham", "derham-lemma-pullback-c1", "derham-lemma-log-complex-consequence", "derham-lemma-kunneth-de-rham", "derham-lemma-bottom-part-degenerates", "derham-lemma-de-rham-cohomology-projective-space", "derham-proposition-projective-space-bundle-formula", "derham-proposition-blowup-split", "derham-proposition-Garel", "weil-lemma-check-over-extension", "weil-lemma-chern-classes", "weil-lemma-cycle-classes", "weil-lemma-poincare-duality", "weil-remark-trace", "varieties-lemma-proper-geometrically-reduced-global-sections", "varieties-lemma-ample-positive", "intersection-lemma-projection-generically-finite", "more-morphisms-lemma-lci-permanence", "algebra-lemma-CM-over-regular-flat", "chow-lemma-degrees-and-numerical-intersections" ], "ref_ids": [ 5113, 14319, 14325, 14340, 14323, 14355, 14333, 14372, 14373, 14374, 5108, 5093, 5094, 5097, 5122, 10948, 11128, 11990, 14008, 1107, 5760 ] } ], "ref_ids": [ 5116 ] }, { "id": 14377, "type": "theorem", "label": "derham-proposition-relative-poincare-duality", "categories": [ "derham" ], "title": "derham-proposition-relative-poincare-duality", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to S$", "be a proper smooth morphism of schemes all of whose fibres are nonempty", "and equidimensional of dimension $d$. There exists an", "$\\mathcal{O}_S$-module map", "$$", "t : R^{2d}f_*\\Omega^\\bullet_{X/S} \\longrightarrow \\mathcal{O}_S", "$$", "unique up to precomposing by multiplication by a unit of", "$H^0(X, \\mathcal{O}_X)$ with the following property: the pairing", "$$", "Rf_*\\Omega^\\bullet_{X/S}", "\\otimes_{\\mathcal{O}_S}^\\mathbf{L}", "Rf_*\\Omega^\\bullet_{X/S}[2d]", "\\longrightarrow", "\\mathcal{O}_S, \\quad", "(\\xi, \\xi') \\longmapsto t(\\xi \\cup \\xi')", "$$", "is a perfect pairing of perfect complexes on $S$." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Proposition \\ref{proposition-poincare-duality}.", "\\medskip\\noindent", "By the relative Hodge-to-de Rham spectral sequence", "$$", "E_1^{p, q} = R^qf_*\\Omega^p_{X/S} \\Rightarrow R^{p + q}f_*\\Omega^\\bullet_{X/S}", "$$", "(Section \\ref{section-hodge-to-de-rham}), the vanishing", "of $\\Omega^i_{X/S}$ for $i > d$, the vanishing in for example Limits, Lemma", "\\ref{limits-lemma-higher-direct-images-zero-above-dimension-fibre}", "and the results of Lemmas \\ref{lemma-relative-bottom-part-degenerates} and", "\\ref{lemma-relative-top-part-degenerates}", "we see that $R^0f_*\\Omega_{X/S} = R^0f_*\\mathcal{O}_X$", "and $R^df_*\\Omega^d_{X/S} = R^{2d}f_*\\Omega^\\bullet_{X/S}$.", "More precisesly, these identifications come from the maps", "of complexes", "$$", "\\Omega^\\bullet_{X/S} \\to \\mathcal{O}_X[0]", "\\quad\\text{and}\\quad", "\\Omega^d_{X/S}[-d] \\to \\Omega^\\bullet_{X/S}", "$$", "Let us choose $t : R^{2d}f_*\\Omega_{X/S} \\to \\mathcal{O}_S$", "which via this identification corresponds to a $t$ as in", "Lemma \\ref{lemma-relative-duality-hodge}.", "\\medskip\\noindent", "Let us abbreviate $\\Omega^\\bullet = \\Omega^\\bullet_{X/S}$.", "Consider the map (\\ref{equation-wedge}) which in our situation reads", "$$", "\\wedge :", "\\text{Tot}(\\Omega^\\bullet \\otimes_{f^{-1}\\mathcal{O}_S} \\Omega^\\bullet)", "\\longrightarrow", "\\Omega^\\bullet", "$$", "For every integer $p = 0, 1, \\ldots, d$ this map annihilates the subcomplex", "$\\text{Tot}(\\sigma_{> p} \\Omega^\\bullet \\otimes_{f^{-1}\\mathcal{O}_S}", "\\sigma_{\\geq d - p} \\Omega^\\bullet)$ for degree reasons.", "Hence we find that the restriction of $\\wedge$ to the subcomplex", "$\\text{Tot}(\\Omega^\\bullet \\otimes_{f^{-1}\\mathcal{O}_S}", "\\geq_{d - p}\\Omega^\\bullet)$ factors through a map of complexes", "$$", "\\gamma_p :", "\\text{Tot}(\\sigma_{\\leq p} \\Omega^\\bullet \\otimes_{f^{-1}\\mathcal{O}_S}", "\\sigma_{\\geq d - p} \\Omega^\\bullet)", "\\longrightarrow", "\\Omega^\\bullet", "$$", "Using the same procedure as in Section \\ref{section-cup-product} we obtain", "relative cup products", "$$", "Rf_*\\sigma_{\\leq p} \\Omega^\\bullet", "\\otimes_{\\mathcal{O}_S}^\\mathbf{L}", "Rf_*\\sigma_{\\geq d - p}\\Omega^\\bullet", "\\longrightarrow", "Rf_*\\Omega^\\bullet", "$$", "We will prove by induction on $p$ that these cup products via $t$", "induce perfect pairings between $Rf_*\\sigma_{\\leq p} \\Omega^\\bullet$", "and $Rf_*\\sigma_{\\geq d - p}\\Omega^\\bullet[2d]$. For $p = d$", "this is the assertion of the proposition.", "\\medskip\\noindent", "The base case is $p = 0$. In this case we have", "$$", "Rf_*\\sigma_{\\leq p}\\Omega^\\bullet = Rf_*\\mathcal{O}_X", "\\quad\\text{and}\\quad", "Rf_*\\sigma_{\\geq d - p}\\Omega^\\bullet[2d] = Rf_*(\\Omega^d[-d])[2d] =", "Rf_*\\Omega^d[d]", "$$", "In this case we simply obtain the pairing", "between $Rf_*\\mathcal{O}_X$ and $Rf_*\\Omega^d[d]$ of", "Lemma \\ref{lemma-relative-duality-hodge} and the result is true.", "\\medskip\\noindent", "Induction step. Say we know the result is true for $p$. Then", "we consider the distinguished triangle", "$$", "\\Omega^{p + 1}[-p - 1] \\to", "\\sigma_{\\leq p + 1}\\Omega^\\bullet \\to", "\\sigma_{\\leq p}\\Omega^\\bullet \\to", "\\Omega^{p + 1}[-p]", "$$", "and the distinguished triangle", "$$", "\\sigma_{\\geq d - p}\\Omega^\\bullet \\to", "\\sigma_{\\geq d - p - 1}\\Omega^\\bullet \\to", "\\Omega^{d - p - 1}[-d + p + 1] \\to", "(\\sigma_{\\geq d - p}\\Omega^\\bullet)[1]", "$$", "Observe that both are distinguished triangles in the homotopy category", "of complexes of sheaves of $f^{-1}\\mathcal{O}_S$-modules; in particular the", "maps $\\sigma_{\\leq p}\\Omega^\\bullet \\to \\Omega^{p + 1}[-p]$ and", "$\\Omega^{d - p - 1}[-d + p + 1] \\to (\\sigma_{\\geq d - p}\\Omega^\\bullet)[1]$", "are given by actual maps of complexes, namely using the differential", "$\\Omega^p \\to \\Omega^{p + 1}$ and the differential", "$\\Omega^{d - p - 1} \\to \\Omega^{d - p}$.", "Consider the distinguished triangles associated gotten from these", "distinguished triangles by applying $Rf_*$", "$$", "\\xymatrix{", "Rf_*\\sigma_{\\leq p}\\Omega^\\bullet \\ar[d]_a \\\\", "Rf_*\\Omega^{p + 1}[-p - 1] \\ar[d]_b \\\\", "Rf_*\\sigma_{\\leq p + 1}\\Omega^\\bullet \\ar[d]_c \\\\", "Rf_*\\sigma_{\\leq p}\\Omega^\\bullet \\ar[d]_d \\\\", "Rf_*\\Omega^{p + 1}[-p - 1]", "}", "\\quad\\quad", "\\xymatrix{", "Rf_*\\sigma_{\\geq d - p}\\Omega^\\bullet \\\\", "Rf_*\\Omega^{d - p - 1}[-d + p + 1] \\ar[u]_{a'} \\\\", "Rf_*\\sigma_{\\geq d - p - 1}\\Omega^\\bullet \\ar[u]_{b'} \\\\", "Rf_*\\sigma_{\\geq d - p}\\Omega^\\bullet \\ar[u]_{c'} \\\\", "Rf_*\\Omega^{d - p - 1}[-d + p + 1] \\ar[u]_{d'}", "}", "$$", "We will show below that the pairs $(a, a')$, $(b, b')$, $(c, c')$, and", "$(d, d')$ are compatible with the given pairings. This means we obtain a", "map from the distinguished triangle on the left to the distuiguished triangle", "obtained by applying $R\\SheafHom(-, \\mathcal{O}_S)$ to the distinguished", "triangle on the right. By induction and Lemma \\ref{lemma-duality-hodge}", "we know that the pairings constructed above between the", "complexes on the first, second, fourth, and fifth", "rows are perfect, i.e., determine isomorphisms after taking duals.", "By Derived Categories, Lemma \\ref{derived-lemma-third-isomorphism-triangle}", "we conclude the pairing between the complexes in the middle row", "is perfect as desired.", "\\medskip\\noindent", "Let $e : K \\to K'$ and $e' : M' \\to M$ be maps of objects", "of $D(\\mathcal{O}_S)$ and let", "$K \\otimes_{\\mathcal{O}_S}^\\mathbf{L} M \\to \\mathcal{O}_S$ and", "$K' \\otimes_{\\mathcal{O}_S}^\\mathbf{L} M' \\to \\mathcal{O}_S$", "be pairings. Then we say these pairings are compatible if the", "diagram", "$$", "\\xymatrix{", "K' \\otimes_{\\mathcal{O}_S}^\\mathbf{L} M' \\ar[d] &", "K \\otimes_{\\mathcal{O}_S}^\\mathbf{L} M'", "\\ar[l]^{e \\otimes 1} \\ar[d]^{1 \\otimes e'} \\\\", "\\mathcal{O}_S &", "K \\otimes_{\\mathcal{O}_S}^\\mathbf{L} M \\ar[l]", "}", "$$", "commutes. This indeed means that the diagram", "$$", "\\xymatrix{", "K \\ar[r] \\ar[d]_e & R\\SheafHom(M, \\mathcal{O}_S) \\ar[d]^{R\\SheafHom(e', -)} \\\\", "K' \\ar[r] & R\\SheafHom(M', \\mathcal{O}_S)", "}", "$$", "commutes and hence is sufficient for our purposes.", "\\medskip\\noindent", "Let us prove this for the pair $(c, c')$. Here we observe simply", "that we have a commutative diagram", "$$", "\\xymatrix{", "\\text{Tot}(\\sigma_{\\leq p} \\Omega^\\bullet \\otimes_{f^{-1}\\mathcal{O}_S}", "\\sigma_{\\geq d - p} \\Omega^\\bullet) \\ar[d]_{\\gamma_p} &", "\\text{Tot}(\\sigma_{\\leq p + 1} \\Omega^\\bullet \\otimes_{f^{-1}\\mathcal{O}_S}", "\\sigma_{\\geq d - p} \\Omega^\\bullet) \\ar[l] \\ar[d] \\\\", "\\Omega^\\bullet &", "\\text{Tot}(\\sigma_{\\leq p + 1} \\Omega^\\bullet \\otimes_{f^{-1}\\mathcal{O}_S}", "\\sigma_{\\geq d - p - 1} \\Omega^\\bullet) \\ar[l]_-{\\gamma_{p + 1}}", "}", "$$", "By functoriality of the cup product we obtain commutativity of the", "desired diagram.", "\\medskip\\noindent", "Similarly for the pair $(b, b')$ we use the commutative diagram", "$$", "\\xymatrix{", "\\text{Tot}(\\sigma_{\\leq p + 1} \\Omega^\\bullet \\otimes_{f^{-1}\\mathcal{O}_S}", "\\sigma_{\\geq d - p - 1} \\Omega^\\bullet) \\ar[d]_{\\gamma_{p + 1}} &", "\\text{Tot}(\\Omega^{p + 1}[-p - 1] \\otimes_{f^{-1}\\mathcal{O}_S}", "\\sigma_{\\geq d - p - 1} \\Omega^\\bullet) \\ar[l] \\ar[d] \\\\", "\\Omega^\\bullet &", "\\Omega^{p + 1}[-p - 1]", "\\otimes_{f^{-1}\\mathcal{O}_S}", "\\Omega^{d - p - 1}[-d + p + 1] \\ar[l]_-\\wedge", "}", "$$", "\\medskip\\noindent", "For the pairs $(d, d')$ and $(a, a')$ we use the commutative diagram", "$$", "\\xymatrix{", "\\Omega^{p + 1}[-p] \\otimes_{f^{-1}\\mathcal{O}_S}", "\\Omega^{d - p - 1}[-d + p] \\ar[d] &", "\\text{Tot}(\\sigma_{\\leq p}\\Omega^\\bullet \\otimes_{f^{-1}\\mathcal{O}_S}", "\\Omega^{d - p - 1}[-d + p]) \\ar[l] \\ar[d] \\\\", "\\Omega^\\bullet &", "\\text{Tot}(\\sigma_{\\leq p}\\Omega^\\bullet \\otimes_{f^{-1}\\mathcal{O}_S}", "\\sigma_{\\geq d - p}\\Omega^\\bullet) \\ar[l]", "}", "$$", "\\medskip\\noindent", "We omit the argument showing the uniqueness of $t$ up to", "precomposing by multiplication by a unit in $H^0(X, \\mathcal{O}_X)$." ], "refs": [ "derham-proposition-poincare-duality", "limits-lemma-higher-direct-images-zero-above-dimension-fibre", "derham-lemma-relative-bottom-part-degenerates", "derham-lemma-relative-top-part-degenerates", "derham-lemma-relative-duality-hodge", "derham-lemma-relative-duality-hodge", "derham-lemma-duality-hodge", "derived-lemma-third-isomorphism-triangle" ], "ref_ids": [ 14375, 15109, 14368, 14370, 14369, 14369, 14354, 1759 ] } ], "ref_ids": [] }, { "id": 14395, "type": "theorem", "label": "trace-theorem-baffling", "categories": [ "trace" ], "title": "trace-theorem-baffling", "contents": [ "Let $X$ be a scheme in characteristic $p > 0$. Then the absolute frobenius", "induces (by pullback) the trivial map on cohomology, i.e., for all", "integers $j\\geq 0$,", "$$", "F_X^* : H^j (X, \\underline{\\mathbf{Z}/n\\mathbf{Z}}) \\longrightarrow H^j (X,", "\\underline{\\mathbf{Z}/n\\mathbf{Z}})", "$$", "is the identity." ], "refs": [], "proofs": [ { "contents": [ "[Proof of Theorem \\ref{theorem-baffling}]", "We need to verify the existence of a functorial isomorphism as above. For an", "\\'etale morphism $\\varphi : U \\to X$, consider the diagram", "$$", "\\xymatrix{", "U \\ar@{-->}[rd] \\ar@/^1pc/[rrd]^{F_U}", "\\ar@/_1pc/[rdd]_\\varphi \\\\", "& {U \\times_{\\varphi, X, F_X} X} \\ar[r]_-{\\text{pr}_1}", "\\ar[d]^{\\text{pr}_2} & U \\ar[d]^\\varphi \\\\", "& X \\ar[r]^{F_X} & X.", "}", "$$", "The dotted arrow is an \\'etale morphism and a universal", "homeomorphism, so it is an isomorphism. See", "\\'Etale Morphisms, Lemma \\ref{etale-lemma-relative-frobenius-etale}." ], "refs": [ "trace-theorem-baffling", "etale-lemma-relative-frobenius-etale" ], "ref_ids": [ 14395, 10708 ] } ], "ref_ids": [] }, { "id": 14396, "type": "theorem", "label": "trace-theorem-geometric-arithmetic-inverse", "categories": [ "trace" ], "title": "trace-theorem-geometric-arithmetic-inverse", "contents": [ "Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Then for all", "$j\\geq 0$, $\\text{frob}_k$ acts on the cohomology group $H^j(X_{\\bar k},", "\\mathcal{F}|_{X_{\\bar k}})$ as the inverse of the map $\\pi_X^*$." ], "refs": [], "proofs": [ { "contents": [ "The composition $X_{\\bar k} \\xrightarrow{\\Spec(\\text{frob}_k)} X_{\\bar k}", "\\xrightarrow{\\pi_X} X_{\\bar k}$ is equal to $F_{X_{\\bar k}}^f$, hence the", "result follows from the baffling theorem suitably generalized to nontrivial", "coefficients. Note that the previous composition commutes in the sense that", "$F_{X_{\\bar k}}^f = \\pi_X \\circ \\Spec(\\text{frob}_k) =", "\\Spec(\\text{frob}_k) \\circ \\pi_X$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14397, "type": "theorem", "label": "trace-theorem-trace", "categories": [ "trace" ], "title": "trace-theorem-trace", "contents": [ "Let $X$ be a projective curve over a finite field $k$, $\\Lambda$ a finite ring", "and $K \\in D_{ctf}(X, \\Lambda)$. Then the global and local Lefschetz numbers", "of $K$ are equal, i.e.,", "\\begin{equation}", "\\label{equation-trace-formula}", "\\text{Tr}(\\pi^*_X |_{R\\Gamma(X_{\\bar k}, K)})", "=", "\\sum\\nolimits_{x\\in X(k)} \\text{Tr}(\\pi_X |_{K_{\\bar x}})", "\\end{equation}", "in $\\Lambda^\\natural$." ], "refs": [], "proofs": [ { "contents": [ "See discussion below." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14398, "type": "theorem", "label": "trace-theorem-weil-trace-formula", "categories": [ "trace" ], "title": "trace-theorem-weil-trace-formula", "contents": [ "Let $C$ be a nonsingular projective curve over an algebraically closed field", "$k$, and $\\varphi : C \\to C$ a $k$-endomorphism of $C$ distinct from the", "identity. Let $V(\\varphi) = \\Delta_C \\cdot \\Gamma_\\varphi$, where $\\Delta_C$ is", "the diagonal, $\\Gamma_\\varphi$ is the graph of $\\varphi$, and the intersection", "number is taken on $C \\times C$. Let $J = \\underline{\\Picardfunctor}^0_{C/k}$", "be the jacobian of $C$ and denote $\\varphi^* : J \\to J$ the action induced by", "$\\varphi$ by taking pullbacks. Then", "$$", "V(\\varphi) = 1 - \\text{Tr}_J(\\varphi^*) + \\deg \\varphi.", "$$" ], "refs": [], "proofs": [ { "contents": [ "The number $V(\\varphi)$ is the number of fixed points of $\\varphi$, it is equal", "to", "$$", "V(\\varphi) =", "\\sum\\nolimits_{c \\in |C| : \\varphi(c) = c} m_{\\text{Fix}(\\varphi)} (c)", "$$", "where $m_{\\text{Fix}(\\varphi)} (c)$ is the multiplicity of $c$ as a fixed point", "of $\\varphi$, namely the order or vanishing of the image of a local uniformizer", "under $\\varphi - \\text{id}_C$. Proofs of this theorem can be found in", "\\cite{Lang} and \\cite{Weil}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14399, "type": "theorem", "label": "trace-theorem-trace-formula-again", "categories": [ "trace" ], "title": "trace-theorem-trace-formula-again", "contents": [ "Let $k$ be a finite field and $X$ a finite type, separated scheme of dimension", "at most 1 over $k$. Let $\\Lambda$ be a finite ring whose cardinality is prime", "to that of $k$, and $K\\in D_{ctf}(X, \\Lambda)$. Then", "\\begin{equation}", "\\label{equation-trace-formula-again}", "\\text{Tr}(\\pi_X^* |_{R\\Gamma_c(X_{\\bar k}, K)})", "=", "\\sum\\nolimits_{x\\in X(k)}", "\\text{Tr}(\\pi_x |_{K_{\\bar x}})", "\\end{equation}", "in $\\Lambda^{\\natural}$." ], "refs": [], "proofs": [ { "contents": [ "[Proof of Theorem \\ref{theorem-trace-formula-again}]", "The proof proceeds in a number of steps.", "\\medskip\\noindent", "Step 1. {\\it Let $j : \\mathcal{U}\\hookrightarrow X$ be an open immersion with", "complement $Y = X - \\mathcal{U}$ and $i : Y \\hookrightarrow X$. Then", "$T''(X, K) = T''(\\mathcal{U}, j^{-1} K)+ T''(Y, i^{-1}K)$ and", "$T'(X, K) = T'(\\mathcal{U}, j^{-1} K)+ T'(Y, i^{-1}K)$.}", "\\medskip\\noindent", "This is clear for $T'$. For $T''$ use the exact sequence", "$$", "0\\to j_!j^{-1} K \\to K \\to i_* i^{-1} K \\to 0", "$$", "to get a filtration on $K$. This gives rise to an object", "$\\widetilde K \\in DF(X, \\Lambda)$", "whose graded pieces are $j_!j^{-1}K$ and $i_*i^{-1}K$,", "both of which lie in $D_{ctf}(X, \\Lambda)$. Then, by filtered derived", "abstract nonsense (INSERT REFERENCE),", "$R\\Gamma_c(X_{\\bar k}, K)\\in DF_{perf}(\\Lambda)$,", "and it comes equipped with $\\pi_x^*$ in", "$DF_{perf}(\\Lambda)$.", "By the discussion of traces on filtered complexes (INSERT REFERENCE) we get", "\\begin{eqnarray*}", "\\text{Tr}(\\pi_X^* |_{R\\Gamma_c(X_{\\bar k}, K)})", "& = & \\text{Tr}(\\pi_X^* |_{R\\Gamma_c(X_{\\bar k}, j_!j^{-1}K)}) +", "\\text{Tr}(\\pi_X^* |_{R\\Gamma_c(X_{\\bar k}, i_*i^{-1}K)}) \\\\", "& = & T''(U, i^{-1}K) + T''(Y, i^{-1}K).", "\\end{eqnarray*}", "\\noindent", "Step 2. {\\it The theorem holds if $\\dim X\\leq 0$. }", "\\medskip\\noindent", "Indeed, in that case", "$$", "R\\Gamma_c(X_{\\bar k}, K) = R\\Gamma(X_{\\bar k}, K) = \\Gamma(X_{\\bar k}, K) =", "\\bigoplus\\nolimits_{\\bar x\\in X_{\\bar k}} K_{\\bar x} \\leftarrow \\pi_X*.", "$$", "Since the fixed points of $\\pi_X: X_{\\bar k}\\to X_{\\bar k}$ are exactly the", "points $\\bar x\\in X_{\\bar k}$ which lie over a $k$-rational point $x\\in X(k)$", "we get", "$$", "\\text{Tr}\\big(\\pi_X^*|_{R\\Gamma_c(X_{\\bar k}, K)}\\big) =", "\\sum\\nolimits_{x\\in X(k)}\\text{Tr}(\\pi_{\\bar x}|_{K_{\\bar x}}).", "$$", "\\medskip\\noindent", "Step 3. {\\it It suffices to prove the equality", "$T'(\\mathcal{U}, \\mathcal{F}) = T''(\\mathcal{U}, \\mathcal{F})$", "in the case where", "\\begin{itemize}", "\\item $\\mathcal{U}$ is a smooth irreducible affine curve over $k$,", "\\item $\\mathcal{U}(k) = \\emptyset$,", "\\item $K=\\mathcal{F}$ is a finite locally constant sheaf of $\\Lambda$-modules", "on $\\mathcal{U}$ whose stalk(s) are finite projective $\\Lambda$-modules, and", "\\item $\\Lambda$ is killed by a power of a prime $\\ell$ and $\\ell \\in k^*$.", "\\end{itemize}", "}", "\\medskip\\noindent", "Indeed, because of Step 2, we can throw out any finite set of points. But we", "have only finitely many rational points, so we may assume there are", "none\\footnote{At this point, there should be an evil laugh in the background.}.", "We may assume that $\\mathcal{U}$ is smooth irreducible and affine by passing to", "irreducible components and throwing away the bad points if necessary. The", "assumptions of $\\mathcal{F}$ come from unwinding the definition of", "$D_{ctf}(X, \\Lambda)$ and those on $\\Lambda$ from considering its primary", "decomposition.", "\\medskip\\noindent", "For the remainder of the proof, we consider the situation", "$$", "\\xymatrix{", "\\mathcal{V} \\ar[d]_f \\ar[r] & Y \\ar[d]^{\\bar f} \\\\", "\\mathcal{U} \\ar[r] & X", "}", "$$", "where $\\mathcal{U}$ is as above, $f$ is a finite \\'etale Galois covering,", "$\\mathcal{V}$ is connected and the horizontal arrows are projective", "completions. Denoting $G=\\text{Aut}(\\mathcal{V}|\\mathcal{U})$, we also assume", "(as we may) that $f^{-1}\\mathcal{F} =\\underline M$ is constant, where the", "module $M = \\Gamma(\\mathcal{V}, f^{-1}\\mathcal{F})$ is a $\\Lambda[G]$-module", "which is finite and projective over $\\Lambda$. This corresponds to the trivial", "monoid extension", "$$", "1\\to G\\to \\Gamma = G \\times \\mathbf{N}\\to \\mathbf{N}\\to 1.", "$$", "In that context, using the reductions above, we need to show that", "$T''(\\mathcal{U}, \\mathcal{F}) = 0$.", "\\medskip\\noindent", "Step 4. {\\it There is a natural action of $G$ on $f_*f^{-1}\\mathcal{F}$ and", "the trace map $f_*f^{-1}\\mathcal{F}\\to \\mathcal{F}$ defines an isomorphism}", "$$", "(f_*f^{-1}\\mathcal{F})\\otimes_{\\Lambda[G]} \\Lambda =", "(f_*f^{-1}\\mathcal{F})_G \\cong \\mathcal{F}.", "$$", "\\medskip\\noindent", "To prove this, simply unwind everything at a geometric point.", "\\medskip\\noindent", "Step 5. {\\it Let $A = \\mathbf{Z}/\\ell^n \\mathbf{Z}$ with $n\\gg 0$. Then", "$f_*f^{-1}\\mathcal{F} \\cong (f_*\\underline A)", "\\otimes_{\\underline A} \\underline M$ with diagonal $G$-action.}", "\\medskip\\noindent", "Step 6. {\\it There is a canonical isomorphism", "$(f_*\\underline A \\otimes_{\\underline A} \\underline M)", "\\otimes_{\\Lambda[G]} \\underline \\Lambda \\cong \\mathcal{F}$.", "}", "\\medskip\\noindent", "In fact, this is a derived tensor product, because of the projectivity", "assumption on $\\mathcal{F}$.", "\\medskip\\noindent", "Step 7. {\\it There is a canonical isomorphism", "$$", "R\\Gamma_c(\\mathcal{U}_{\\bar k}, \\mathcal{F})", "= (R\\Gamma_c(\\mathcal{U}_{\\bar k}, f_*A)\\otimes_A^\\mathbf{L}", "M)\\otimes_{\\Lambda[G]}^\\mathbf{L} \\Lambda,", "$$", "compatible with the action of $\\pi^*_\\mathcal{U}$.", "}", "\\medskip\\noindent", "This comes from the universal coefficient theorem, i.e., the fact that", "$R\\Gamma_c$ commutes with $\\otimes^\\mathbf{L}$, and the flatness of", "$\\mathcal{F}$ as a $\\Lambda$-module.", "\\medskip\\noindent", "We have", "\\begin{eqnarray*}", "\\text{Tr}(", "\\pi_\\mathcal{U}^* |_{R\\Gamma_c(\\mathcal{U}_{\\bar k}, \\mathcal{F})})", "& = &", "{\\sum_{g \\in G}}'", "\\text{Tr}_{\\Lambda}^{Z_g}", "\\left(", "(g, \\pi_\\mathcal{U}^*)", "|_{R\\Gamma_c(\\mathcal{U}_{\\bar k}, f_*A)\\otimes_A^\\mathbf{L} M}", "\\right) \\\\", "& = &", "{\\sum_{g\\in G}}'", "\\text{Tr}_A^{Z_g}", "(", "(g, \\pi_\\mathcal{U}^*) |_{R\\Gamma_c(\\mathcal{U}_{\\bar k}, f_*A)}", ")", "\\cdot", "\\text{Tr}_\\Lambda(g|_M)", "\\end{eqnarray*}", "where $\\Gamma$ acts on $R\\Gamma_c(\\mathcal{U}_{\\bar k}, \\mathcal{F})$ by $G$", "and $(e, 1)$ acts via $\\pi_\\mathcal{U}^*$. So the monoidal extension is given", "by $\\Gamma = G \\times \\mathbf{N} \\to \\mathbf{N}$, $\\gamma \\mapsto 1$. The first", "equality follows from Lemma \\ref{lemma-trivial-trace} and the second from", "Lemma \\ref{lemma-weak-trace}.", "\\medskip\\noindent", "Step 8. {\\it It suffices to show that", "$\\text{Tr}_A^{Z_g}((g, \\pi_\\mathcal{U}^*)", "|_{R\\Gamma_c(\\mathcal{U}_{\\bar k}, f_*A)}) \\in A$", "maps to zero in $\\Lambda$.", "}", "\\medskip\\noindent", "Recall that", "\\begin{eqnarray*}", "\\# Z_g \\cdot \\text{Tr}_A^{Z_g}((g, \\pi_\\mathcal{U}^*)", "|_{R\\Gamma_c(\\mathcal{U}_{\\bar k}, f_*A)})", "& = & \\text{Tr}_A((g, \\pi_\\mathcal{U}^*)", "|_{R\\Gamma_c(\\mathcal{U}_{\\bar k}, f_*A)})\\\\", "& = &", "\\text{Tr}_A((g^{-1}\\pi_\\mathcal{V})^* |_{R\\Gamma_c(\\mathcal{V}_{\\bar k}, A)}).", "\\end{eqnarray*}", "The first equality is", "Lemma \\ref{lemma-gamma-z-gamma-trace},", "the second is the Leray", "spectral sequence, using the finiteness of $f$ and the fact that we are only", "taking traces over $A$. Now since $A=\\mathbf{Z}/\\ell^n\\mathbf{Z}$ with", "$n \\gg 0$ and $\\# Z_g = \\ell^a$ for some (fixed) $a$,", "it suffices to show the following result.", "\\medskip\\noindent", "Step 9. {\\it We have", "$\\text{Tr}_A((g^{-1}\\pi_\\mathcal{V})^* |_{R\\Gamma_c(\\mathcal{V}, A)}) = 0$", "in $A$.}", "\\medskip\\noindent", "By additivity again, we have", "\\begin{eqnarray*}", "& \\text{Tr}_A((g^{-1}\\pi_\\mathcal{V})^* |_{R\\Gamma_c(\\mathcal{V}_{\\bar k} A)})", "+", "\\text{Tr}_A((g^{-1}\\pi_\\mathcal{V})^*", "|_{R\\Gamma_c(Y-\\mathcal {V})_{\\bar k}, A)}) \\\\", "& =", "\\text{Tr}_A((g^{-1}\\pi_Y)^* |_{R\\Gamma(Y_{\\bar k}, A)})", "\\end{eqnarray*}", "The latter trace is the number of fixed points of $g^{-1}\\pi_Y$ on $Y$, by", "Weil's trace formula", "Theorem \\ref{theorem-weil-trace-formula}.", "Moreover, by the 0-dimensional case already proven in step 2,", "$$", "\\text{Tr}_A((g^{-1}\\pi_\\mathcal{V})^*|_{R\\Gamma_c(Y-\\mathcal{V})_{\\bar k}, A)})", "$$", "is the number of fixed points of $g^{-1}\\pi_Y$ on $(Y-\\mathcal{V})_{\\bar k}$.", "Therefore,", "$$", "\\text{Tr}_A((g^{-1}\\pi_\\mathcal{V})^* |_{R\\Gamma_c(\\mathcal{V}_{\\bar k}, A)})", "$$", "is the number of fixed points of $g^{-1}\\pi_Y$ on $\\mathcal{V}_{\\bar k}$. But", "there are no such points: if $\\bar y\\in Y_{\\bar k}$ is fixed under", "$g^{-1}\\pi_Y$, then $\\bar f(\\bar y) \\in X_{\\bar k}$ is fixed under $\\pi_X$. But", "$\\mathcal{U}$ has no $k$-rational point, so we must have $\\bar f(\\bar y)\\in", "(X-\\mathcal{U})_{\\bar k}$ and so $\\bar y\\notin \\mathcal{V}_{\\bar k}$, a", "contradiction.", "This finishes the proof." ], "refs": [ "trace-theorem-trace-formula-again", "trace-lemma-trivial-trace", "trace-lemma-weak-trace", "trace-lemma-gamma-z-gamma-trace", "trace-theorem-weil-trace-formula" ], "ref_ids": [ 14399, 14428, 14427, 14426, 14398 ] } ], "ref_ids": [] }, { "id": 14400, "type": "theorem", "label": "trace-theorem-A", "categories": [ "trace" ], "title": "trace-theorem-A", "contents": [ "Let $X$ be a scheme of finite type over a finite field $k$. Let $\\Lambda$ be a", "finite ring of order prime to the characteristic of $k$ and $\\mathcal{F}$ a", "constructible flat $\\Lambda$-module on $X_\\etale$. Then", "$$", "L(X, \\mathcal{F}) =", "\\det(1 - \\pi_X^*\\ T |_{R\\Gamma_c(X_{\\bar k}, \\mathcal{F})})^{-1}", "\\in \\Lambda[[T]].", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14401, "type": "theorem", "label": "trace-theorem-B", "categories": [ "trace" ], "title": "trace-theorem-B", "contents": [ "Let $X$ be a scheme of finite type over a finite field $k$, and $\\mathcal{F}$ a", "$\\mathbf{Q}_\\ell$-sheaf on $X$. Then", "$$", "L(X, \\mathcal{F}) =", "\\prod\\nolimits_i", "\\det(1 - \\pi_X^*T |_{H_c^i(X_{\\bar k} , \\mathcal{F})})^{(-1)^{i + 1}}", "\\in \\mathbf{Q}_\\ell[[T]].", "$$" ], "refs": [], "proofs": [ { "contents": [ "This is sketched below." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14402, "type": "theorem", "label": "trace-theorem-D", "categories": [ "trace" ], "title": "trace-theorem-D", "contents": [ "Let $X/k$ be as above, let $\\Lambda$ be a finite ring with $\\#\\Lambda \\in k^*$", "and $K\\in D_{ctf}(X, \\Lambda)$. Then $R\\Gamma_c(X_{\\bar k}, K)\\in", "D_{perf}(\\Lambda)$ and", "$$", "\\sum_{x\\in X(k)}\\text{Tr}\\left(\\pi_x |_{K_{\\bar x}}\\right) =", "\\text{Tr}\\left(\\pi_X^* |_{R\\Gamma_c(X_{\\bar k}, K )}\\right).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Note that we have already proved this (REFERENCE) when $\\dim X \\leq 1$. The", "general case follows easily from that case together with the proper base change", "theorem." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14403, "type": "theorem", "label": "trace-theorem-C", "categories": [ "trace" ], "title": "trace-theorem-C", "contents": [ "Let $X$ be a separated scheme of finite type over a finite field $k$ and", "$\\mathcal{F}$ be a $\\mathbf{Q}_\\ell$-sheaf on $X$. Then", "$\\dim_{\\mathbf{Q}_\\ell}H_c^i(X_{\\bar k}, \\mathcal{F})$ is finite for all $i$,", "and is nonzero for $0\\leq i \\leq 2 \\dim X$ only. Furthermore, we have", "$$", "\\sum_{x\\in X(k)} \\text{Tr}\\left(\\pi_x |_{\\mathcal{F}_{\\bar x}}\\right) =", "\\sum_i (-1)^i\\text{Tr}\\left(\\pi_X^* |_{H_c^i(X_{\\bar k}, \\mathcal{F})}\\right).", "$$" ], "refs": [], "proofs": [ { "contents": [ "We explain how to deduce this from Theorem \\ref{theorem-D}.", "We first use some \\'etale cohomology arguments to reduce the proof", "to an algebraic statement which we subsequently prove.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be as in the theorem. We can write", "$\\mathcal{F}$ as", "$\\mathcal{F}'\\otimes \\mathbf{Q}_\\ell$ where $\\mathcal{F}' =", "\\left\\{\\mathcal{F}'_n\\right\\}$ is a $\\mathbf{Z}_\\ell$-sheaf without torsion,", "i.e., $\\ell : \\mathcal{F}'\\to \\mathcal{F}'$ has trivial kernel in the", "category of $\\mathbf{Z}_\\ell$-sheaves. Then each $\\mathcal{F}_n'$ is a flat", "constructible $\\mathbf{Z}/\\ell^n\\mathbf{Z}$-module on $X_\\etale$, so", "$\\mathcal{F}'_n \\in D_{ctf}(X, \\mathbf{Z}/\\ell^n\\mathbf{Z})$ and", "$\\mathcal{F}_{n+1}'", "\\otimes^{\\mathbf{L}}_{\\mathbf{Z}/\\ell^{n+1}\\mathbf{Z}}", "\\mathbf{Z}/\\ell^n\\mathbf{Z} = \\mathcal{F}_n'$.", "Note that the last equality holds also", "for standard (non-derived) tensor product, since $\\mathcal{F}'_n$ is flat", "(it is the same equality). Therefore,", "\\begin{enumerate}", "\\item", "the complex $K_n = R\\Gamma_c\\left(X_{\\bar k}, \\mathcal{F}_n'\\right)$ is perfect,", "and it is endowed with an endomorphism $\\pi_n : K_n\\to K_n$ in", "$D(\\mathbf{Z}/\\ell^n\\mathbf{Z})$,", "\\item", "there are identifications", "$$", "K_{n+1}", "\\otimes^{\\mathbf{L}}_{\\mathbf{Z}/\\ell^{n+1}\\mathbf{Z}}", "\\mathbf{Z}/\\ell^n\\mathbf{Z}", "=", "K_n", "$$", "in $D_{perf}(\\mathbf{Z}/\\ell^n\\mathbf{Z})$, compatible with the endomorphisms", "$\\pi_{n+1}$ and $\\pi_n$ (see \\cite[Rapport 4.12]{SGA4.5}),", "\\item", "the equality $\\text{Tr}\\left(\\pi_X^* |_{K_n}\\right) =", "\\sum_{x\\in X(k)} \\text{Tr}\\left(\\pi_x |_{(\\mathcal{F}'_n)_{\\bar x}}\\right)$", "holds, and", "\\item", "for each $x\\in X(k)$, the elements", "$\\text{Tr}(\\pi_x |_{\\mathcal{F}'_{n, \\bar x}}) \\in \\mathbf{Z}/\\ell^n\\mathbf{Z}$", "form an element of", "$\\mathbf{Z}_\\ell$ which is equal to", "$\\text{Tr}(\\pi_x |_{\\mathcal{F}_{\\bar x}}) \\in \\mathbf{Q}_\\ell$.", "\\end{enumerate}", "It thus suffices to prove the following algebra lemma." ], "refs": [ "trace-theorem-D" ], "ref_ids": [ 14402 ] } ], "ref_ids": [] }, { "id": 14404, "type": "theorem", "label": "trace-theorem-fundamental-group", "categories": [ "trace" ], "title": "trace-theorem-fundamental-group", "contents": [ "Let $X$ be a connected scheme.", "\\begin{enumerate}", "\\item There is a topology on $\\pi_1(X, \\overline{x})$ such that the open", "subgroups form a fundamental system of open nbhds of $e\\in \\pi_1(X, \\overline", "x)$.", "\\item With topology of (1) the group", "$\\pi_1(X, \\overline{x})$ is a profinite group.", "\\item The functor", "$$", "\\begin{matrix}", "\\text{ schemes finite } \\atop \\text{ \\'etale over }X & \\to &", "\\text{ finite discrete continuous } \\atop \\pi_1(X, \\overline{x})\\text{-sets}\\\\", "Y / X& \\mapsto & F_{\\overline{x}}(Y) \\text{ with its natural action}", "\\end{matrix}", "$$", "is an equivalence of categories.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See \\cite{SGA1}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14405, "type": "theorem", "label": "trace-theorem-weil-II", "categories": [ "trace" ], "title": "trace-theorem-weil-II", "contents": [ "For a sheaf", "$\\mathcal{F}_\\rho$ with $\\rho$ satisfying the conclusions of the conjecture", "above then the eigenvalues of $\\pi_X^*$ on $H_c^i(X_{\\overline{k}},", "\\mathcal{F}_{\\rho})$ are algebraic numbers $\\alpha$ with absolute values", "$$", "|\\alpha|=q^{w/2}, \\text{ for }w\\in \\mathbf{Z},\\ w\\leq i", "$$", "Moreover, if $X$ smooth and proj. then $w = i$." ], "refs": [], "proofs": [ { "contents": [ "See \\cite{WeilII}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14406, "type": "theorem", "label": "trace-theorem-drinfeld-make-rho", "categories": [ "trace" ], "title": "trace-theorem-drinfeld-make-rho", "contents": [ "Given an eigenform $f$ with values in", "$\\overline{\\mathbf{Q}}_l$ and eigenvalues", "$u_v\\in \\overline{\\mathbf{Z}}_l^*$ then there exists", "$$", "\\rho : \\pi_1(X)\\to \\text{GL}_2(E)", "$$", "continuous, absolutely irreducible where", "$E$ is a finite extension of $\\mathbf{Q}_\\ell$ contained in", "$\\overline{\\mathbf{Q}}_l$ such that", "$t_v = \\text{Tr}(\\rho(F_v))$, and", "$u_v = q_v^{-1}\\det\\left(\\rho(F_v)\\right)$ for all places $v$." ], "refs": [], "proofs": [ { "contents": [ "See \\cite{D0}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14407, "type": "theorem", "label": "trace-theorem-drinfeld-make-f", "categories": [ "trace" ], "title": "trace-theorem-drinfeld-make-f", "contents": [ "Suppose $\\mathbf{Q}_l \\subset E$ finite, and", "$$", "\\rho : \\pi_1(X)\\to \\text{GL}_2(E)", "$$", "absolutely irreducible, continuous. Then there exists an eigenform $f$ with", "values in $\\overline{\\mathbf{Q}}_l$ whose eigenvalues $t_v$, $u_v$", "satisfy the equalities", "$t_v = \\text{Tr}(\\rho(F_v))$ and $u_v = q_v^{-1}\\det(\\rho(F_v))$." ], "refs": [], "proofs": [ { "contents": [ "See \\cite{D1}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14408, "type": "theorem", "label": "trace-theorem-conjecture-n-2", "categories": [ "trace" ], "title": "trace-theorem-conjecture-n-2", "contents": [ "The Conjecture holds if $n\\leq 2$." ], "refs": [], "proofs": [ { "contents": [ "See \\cite{dJ-conjecture}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14409, "type": "theorem", "label": "trace-theorem-conjecture-l-bigger-2n", "categories": [ "trace" ], "title": "trace-theorem-conjecture-l-bigger-2n", "contents": [ "Conjecture holds if $l > 2n$ modulo some unproven things." ], "refs": [], "proofs": [ { "contents": [ "See \\cite{Gaitsgory}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14410, "type": "theorem", "label": "trace-theorem-deformation-rings", "categories": [ "trace" ], "title": "trace-theorem-deformation-rings", "contents": [ "(See \\cite[Theorem 3.5]{dJ-conjecture})", "Suppose", "$$", "\\rho_0: \\pi_1(X)\\to \\text{GL}_n(\\mathbf{F}_l)", "$$", "is a continuous, $l\\neq p$. Assume", "\\begin{enumerate}", "\\item Conj. holds for $X$,", "\\item $\\rho_0 |_{\\pi_1(X_{\\overline{k}})}$ abs. irred., and", "\\item $l$ does not divide $n$.", "\\end{enumerate}", "Then the universal deformation ring $R_{\\text{univ}}$ of $\\rho_0$ is", "finite flat over $\\mathbf{Z}_l$." ], "refs": [], "proofs": [ { "contents": [ "[Sketch]", "Write", "$$", "\\varphi_*(\\overline{\\mathbf{Q}_l}) =", "\\oplus_{\\pi \\in \\widehat{G}} \\mathcal{F}_{\\pi}", "$$", "where $\\widehat{G}$ is the set of isomorphism", "classes of irred representations of", "$G$ over $\\overline{\\mathbf{Q}}_l$. For $\\pi \\in \\widehat{G}$", "let $\\chi_{\\pi}: G \\to \\overline{\\mathbf{Q}}_l$", "be the character of $\\pi$. Then", "$$", "H^*(Y_{\\overline{k}}, \\overline{\\mathbf{Q}}_l) =", "\\oplus_{\\pi\\in \\widehat{G}}", "H^*(Y_{\\overline{k}}, \\overline{\\mathbf{Q}}_l)_\\pi", "=_{(\\varphi\\text{ finite })}", "\\oplus_{\\pi\\in \\widehat{G}}", "H^*(X_{\\overline{k}}, \\mathcal{F}_\\pi)", "$$", "If $\\pi\\neq 1$ then we have", "$$", "H^0(X_{\\overline{k}}, \\mathcal{F}_\\pi) =", "H^2(X_{\\overline{k}}, \\mathcal{F}_\\pi) = 0,\\quad", "\\dim H^1(X_{\\overline{k}}, \\mathcal{F}_\\pi) = (2g_X - 2)d_\\pi^2", "$$", "(can get this from trace formula for acting on ...) and we see that", "$$", "|\\sum_{x \\in X(k_n)} \\chi_\\pi(\\mathcal{F}_x)| \\leq", "(2g_X - 2) d_\\pi^2\\sqrt{q^n}", "$$", "Write $1_C = \\sum_\\pi a_\\pi \\chi_\\pi$, then", "$a_\\pi = \\langle 1_C, \\chi_\\pi\\rangle$, and", "$a_1 = \\langle 1_C, \\chi_1\\rangle = \\frac{\\# C}{\\# G}$ where", "$$", "\\langle f, h\\rangle = \\frac{1}{\\# G}\\sum_{g \\in G} f(g)\\overline{h(g)}", "$$", "Thus we have the relation", "$$", "\\frac{\\# C}{\\# G} = ||1_C||^2 = \\sum|a_\\pi|^2", "$$", "Final step:", "\\begin{align*}", "\\#\\left\\{x \\in X(k_n) \\mid F_x \\in C\\right\\}", "& =", "\\sum_{x \\in X(k_n)} 1_C(x) \\\\", "& =", "\\sum_{x \\in X(k_n)} \\sum_\\pi a_\\pi \\chi_\\pi(F_x) \\\\", "& =", "\\underbrace{\\frac{\\# C}{\\# G} \\# X(k_n)}_{", "\\text{term for }\\pi = 1}", "+", "\\underbrace{\\sum_{\\pi\\neq 1}a_\\pi\\sum_{x\\in X(k_n)}\\chi_\\pi(F_x)}_{", "\\text{ error term (to be bounded by }E)}", "\\end{align*}", "We can bound the error term by", "\\begin{align*}", "|E|", "& \\leq", "\\sum_{\\pi \\in \\widehat{G}, \\atop \\pi \\neq 1}", "|a_\\pi| (2g - 2) d_\\pi^2 \\sqrt{q^n} \\\\", "& \\leq", "\\sum_{\\pi \\neq 1} \\frac{\\# C}{\\# G} (2g_X - 2) d_\\pi^3 \\sqrt{q^n}", "\\end{align*}", "By Weil's conjecture, $\\# X(k_n)\\sim q^n$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14411, "type": "theorem", "label": "trace-lemma-baffling", "categories": [ "trace" ], "title": "trace-lemma-baffling", "contents": [ "Let $X$ be a scheme and $g : X \\to X$ a morphism. Assume that for all", "$\\varphi : U \\to X$ \\'etale, there is an isomorphism", "$$", "\\xymatrix{", "U \\ar[rd]_\\varphi \\ar[rr]^-\\sim & & {U", "\\times_{\\varphi, X, g} X} \\ar[ld]^{\\text{pr}_2} \\\\", "& X", "}", "$$", "functorial in $U$. Then $g$ induces the identity on cohomology (for any sheaf)." ], "refs": [], "proofs": [ { "contents": [ "The proof is formal and without difficulty." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14412, "type": "theorem", "label": "trace-lemma-sheaf-over-finite-field-has-frobenius-descent", "categories": [ "trace" ], "title": "trace-lemma-sheaf-over-finite-field-has-frobenius-descent", "contents": [ "Let $\\mathcal{F}$ be a sheaf on $X_\\etale$.", "Then there are canonical isomorphisms", "$\\pi_X^{-1} \\mathcal{F} \\cong \\mathcal{F}$ and", "$\\mathcal{F} \\cong {\\pi_X}_*\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi : U \\to X$ be \\'etale. Recall that", "${\\pi_X}_* \\mathcal{F} (U) = \\mathcal{F} (U \\times_{\\varphi, X, \\pi_X} X)$.", "Since $\\pi_X = F_X^f$, it follows from the proof of", "Theorem \\ref{theorem-baffling} that there is a functorial isomorphism", "$$", "\\xymatrix{", "U \\ar[rd]_{\\varphi} \\ar[rr]_-{\\gamma_U}", "& & U \\times_{\\varphi, X, \\pi_X} X \\ar[ld]^{\\text{pr}_2} \\\\", "& X", "}", "$$", "where $\\gamma_U = (\\varphi, F_U^f)$. Now we define an", "isomorphism", "$$", "\\mathcal{F} (U) \\longrightarrow {\\pi_X}_* \\mathcal{F} (U) =", "\\mathcal{F} (U \\times_{\\varphi, X, \\pi_X} X)", "$$", "by taking the restriction map of $\\mathcal{F}$ along $\\gamma_U^{-1}$.", "The other isomorphism is analogous." ], "refs": [ "trace-theorem-baffling" ], "ref_ids": [ 14395 ] } ], "ref_ids": [] }, { "id": 14413, "type": "theorem", "label": "trace-lemma-two-actions-agree", "categories": [ "trace" ], "title": "trace-lemma-two-actions-agree", "contents": [ "In the situation above denote $\\alpha : X \\to \\Spec(k)$ the structure morphism.", "Consider the stalk $(R^j\\alpha_*\\mathcal{F})_{\\Spec(\\bar k)}$ endowed with its", "natural Galois action as in", "\\'Etale Cohomology, Section \\ref{etale-cohomology-section-galois-action-stalks}.", "Then the identification", "$$", "(R^j\\alpha_*\\mathcal{F})_{\\Spec(\\bar k)} \\cong H^j (X_{\\bar k},", "\\mathcal{F}|_{X_{\\bar k}})", "$$", "from", "\\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-higher-direct-images}", "is an isomorphism of $G_k$-modules." ], "refs": [ "etale-cohomology-theorem-higher-direct-images" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 6385 ] }, { "id": 14414, "type": "theorem", "label": "trace-lemma-when-in-bounded", "categories": [ "trace" ], "title": "trace-lemma-when-in-bounded", "contents": [ "An object $E$ of $D(\\mathcal{A})$ is contained in $D^+(\\mathcal{A})$ if and", "only if $H^i(E) =0 $ for all $i \\ll 0$. Similar statements hold for $D^-$ and", "$D^+$." ], "refs": [], "proofs": [ { "contents": [ "Hint: use truncation functors. See", "Derived Categories, Lemma \\ref{derived-lemma-complex-cohomology-bounded}." ], "refs": [ "derived-lemma-complex-cohomology-bounded" ], "ref_ids": [ 1812 ] } ], "ref_ids": [] }, { "id": 14415, "type": "theorem", "label": "trace-lemma-derived-categories", "categories": [ "trace" ], "title": "trace-lemma-derived-categories", "contents": [ "Morphisms between objects in the derived category.", "\\begin{enumerate}", "\\item", "Let $I^\\bullet \\in \\text{Comp}^+(\\mathcal{A})$ with $I^n$ injective for all", "$n \\in \\mathbf{Z}$. Then", "$$", "\\Hom_{D(\\mathcal{A})}(K^\\bullet, I^\\bullet)", "=", "\\Hom_{K(\\mathcal{A})}(K^\\bullet, I^\\bullet).", "$$", "\\item", "Let $P^\\bullet \\in \\text{Comp}^-(\\mathcal{A})$ with $P^n$ is projective for all", "$n \\in \\mathbf{Z}$. Then", "$$", "\\Hom_{D(\\mathcal{A})}(P^\\bullet, K^\\bullet)", "=", "\\Hom_{K(\\mathcal{A})}(P^\\bullet, K^\\bullet).", "$$", "\\item", "If $\\mathcal{A}$ has enough injectives and $\\mathcal{I} \\subset \\mathcal{A}$", "is the additive subcategory of injectives, then", "$", "D^+(\\mathcal{A})\\cong K^+(\\mathcal{I})", "$", "(as triangulated categories).", "\\item", "If $\\mathcal{A}$ has enough projectives and $\\mathcal{P} \\subset \\mathcal{A}$", "is the additive subcategory of projectives, then", "$", "D^-(\\mathcal{A}) \\cong K^-(\\mathcal{P}).", "$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14416, "type": "theorem", "label": "trace-lemma-filtered-derived-category", "categories": [ "trace" ], "title": "trace-lemma-filtered-derived-category", "contents": [ "If $\\mathcal{A}$ has enough injectives, then $DF^+(\\mathcal{A}) \\cong", "K^+(\\mathcal{I})$, where $\\mathcal{I}$ is the full additive subcategory of", "$\\text{Fil}^f(\\mathcal{A})$ consisting of filtered injective objects.", "Similarly, if $\\mathcal{A}$ has enough projectives, then $DF^-(\\mathcal{A})", "\\cong K^+(\\mathcal{P})$, where $\\mathcal P$ is the full additive subcategory of", "$\\text{Fil}^f(\\mathcal{A})$ consisting of filtered projective objects." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14417, "type": "theorem", "label": "trace-lemma-additivity", "categories": [ "trace" ], "title": "trace-lemma-additivity", "contents": [ "Let $K\\in DF_{\\text{perf}}(\\Lambda)$ and $f\\in", "\\text{End}_{DF}(K)$. Then", "$$", "\\text{Tr}(f|_K) =", "\\sum\\nolimits_{p\\in \\mathbf{Z}} \\text{Tr}(f|_{\\text{gr}^p K}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Proposition \\ref{proposition-trace-well-defined}, we may assume we have", "a bounded", "complex $P^\\bullet$ of filtered finite projectives of", "$\\text{Fil}^f(\\text{Mod}_\\Lambda)$ and a map $f^\\bullet : P^\\bullet\\to", "P^\\bullet$ in $\\text{Comp}(\\text{Fil}^f(\\text{Mod}_\\Lambda))$. So the lemma", "follows from the following result, which proof is left to the reader." ], "refs": [ "trace-proposition-trace-well-defined" ], "ref_ids": [ 14437 ] } ], "ref_ids": [] }, { "id": 14418, "type": "theorem", "label": "trace-lemma-additive-filtered-finite-projective", "categories": [ "trace" ], "title": "trace-lemma-additive-filtered-finite-projective", "contents": [ "Let $P \\in \\text{Fil}^f(\\text{Mod}_\\Lambda)$ be filtered finite projective, and", "$f : P \\to P$ an endomorphism in $\\text{Fil}^f(\\text{Mod}_\\Lambda)$. Then", "$$", "\\text{Tr}(f|_P) =", "\\sum\\nolimits_p \\text{Tr}(f|_{\\text{gr}^p(P)}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14419, "type": "theorem", "label": "trace-lemma-characterize-perfect", "categories": [ "trace" ], "title": "trace-lemma-characterize-perfect", "contents": [ "Let $\\Lambda$ be a left Noetherian ring and $K\\in D(\\Lambda)$. Then $K$ is", "perfect if and only if the two following conditions hold:", "\\begin{enumerate}", "\\item", "$K$ has finite $\\text{Tor}$-dimension, and", "\\item", "for all $i \\in \\mathbf{Z}$, $H^i(K)$ is a finite $\\Lambda$-module.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "See More on Algebra, Lemma \\ref{more-algebra-lemma-perfect}", "for the proof in the commutative case." ], "refs": [ "more-algebra-lemma-perfect" ], "ref_ids": [ 10212 ] } ], "ref_ids": [] }, { "id": 14420, "type": "theorem", "label": "trace-lemma-weil-mod", "categories": [ "trace" ], "title": "trace-lemma-weil-mod", "contents": [ "Consider the situation of", "Theorem \\ref{theorem-weil-trace-formula}", "and let $\\ell$ be a prime number invertible in $k$. Then", "$$", "\\sum\\nolimits_{i = 0}^2", "(-1)^i", "\\text{Tr}(\\varphi^* |_{H^i (C, \\underline{\\mathbf{Z}/\\ell^n \\mathbf{Z}})})", "=", "V(\\varphi) \\mod \\ell^n.", "$$" ], "refs": [ "trace-theorem-weil-trace-formula" ], "proofs": [ { "contents": [ "[Sketch of proof]", "Observe first that the assumption makes sense because $H^i(C,", "\\underline{\\mathbf{Z}/\\ell^n \\mathbf{Z}})$ is a free $\\mathbf{Z}/\\ell^n", "\\mathbf{Z}$-module for all $i$. The trace of $\\varphi^*$ on the 0th degree", "cohomology is 1. The choice of a primitive $\\ell^n$th root of unity in $k$", "gives an isomorphism", "$$", "H^i(C, \\underline{\\mathbf{Z}/\\ell^n \\mathbf{Z}}) \\cong H^i(C, \\mu_{\\ell^n})", "$$", "compatibly with the action of the geometric Frobenius. On the other hand,", "$H^1(C, \\mu_{\\ell^n}) = J[\\ell^n]$. Therefore,", "\\begin{eqnarray*}", "\\text{Tr}(\\varphi^* |_{H^1 (C, \\underline{\\mathbf{Z}/\\ell^n \\mathbf{Z}})})) & =", "& \\text{Tr}_J (\\varphi^*) \\mod \\ell^n \\\\", "& = & \\text{Tr}_{\\mathbf{Z}/\\ell^n \\mathbf{Z}} (\\varphi^* : J[\\ell^n] \\to", "J[\\ell^n]).", "\\end{eqnarray*}", "Moreover, $H^2(C, \\mu_{\\ell^n}) = \\Pic(C)/\\ell^n\\Pic(C) \\cong", "\\mathbf{Z}/\\ell^n \\mathbf{Z}$ where $\\varphi^*$ is multiplication by $\\deg", "\\varphi$. Hence", "$$", "\\text{Tr} (\\varphi^*|_{H^2 (C, \\underline{\\mathbf{Z}/\\ell^n \\mathbf{Z}})}) =", "\\deg \\varphi.", "$$", "Thus we have", "$$", "\\sum_{i = 0}^2 (-1)^i", "\\text{Tr}(\\varphi^* |_{H^i (C, \\underline{\\mathbf{Z}/\\ell^n \\mathbf{Z}})}) =", "1 - \\text{Tr}_J(\\varphi^*) + \\deg \\varphi \\mod \\ell^n", "$$", "and the corollary follows from Theorem \\ref{theorem-weil-trace-formula}." ], "refs": [ "trace-theorem-weil-trace-formula" ], "ref_ids": [ 14398 ] } ], "ref_ids": [ 14398 ] }, { "id": 14421, "type": "theorem", "label": "trace-lemma-epsilon", "categories": [ "trace" ], "title": "trace-lemma-epsilon", "contents": [ "Let $e\\in G$ denote the neutral element. The map", "$$", "\\begin{matrix}", "\\Lambda[G] & \\longrightarrow & \\Lambda^{\\natural}\\\\", "\\sum \\lambda_g\\cdot g & \\longmapsto & \\lambda_e", "\\end{matrix}", "$$", "factors through $\\Lambda[G]^\\natural$. We denote", "$\\varepsilon : \\Lambda[G]^\\natural\\to \\Lambda^\\natural$ the induced map." ], "refs": [], "proofs": [ { "contents": [ "We have to show the map annihilates commutators. One has", "$$", "\\left(\\sum\\lambda_g g\\right)\\left(\\sum\\mu_g g\\right)-\\left(\\sum \\mu_g", "g\\right)\\left(\\sum\\lambda_g g\\right)", "= \\sum_g\\left(\\sum_{g_1g_2=g}", "\\lambda_{g_1}\\mu_{g_2}-\\mu_{g_1}\\lambda_{g_2}\\right)g", "$$", "The coefficient of $e$ is", "$$", "\\sum_g\\left(\\lambda_g\\mu_{g^{-1}}-\\mu_g\\lambda_{g^{-1}}\\right) =", "\\sum_g\\left(\\lambda_g\\mu_{g^{-1}}-\\mu_{g^{-1}}\\lambda_g\\right)", "$$", "which is a sum of commutators, hence it zero in $\\Lambda^\\natural$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14422, "type": "theorem", "label": "trace-lemma-lambda-trace", "categories": [ "trace" ], "title": "trace-lemma-lambda-trace", "contents": [ "Let $f : P\\to P$ be an endomorphism of the finite projective", "$\\Lambda[G]$-module $P$. Then", "$$", "\\text{Tr}_{\\Lambda}(f; P) = \\# G \\cdot \\text{Tr}_\\Lambda^G(f; P).", "$$" ], "refs": [], "proofs": [ { "contents": [ "By additivity, reduce to the case $P = \\Lambda[G]$.", "In that case, $f$ is given by", "right multiplication by some element $\\sum\\lambda_g\\cdot g$ of $\\Lambda[G]$. In", "the basis $(g)_{g \\in G}$, the matrix of $f$ has coefficient", "$\\lambda_{g_2^{-1}g_1}$ in the $(g_1, g_2)$ position. In particular, all", "diagonal coefficients are $\\lambda_e$, and there are $\\# G$ such coefficients." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14423, "type": "theorem", "label": "trace-lemma-A-module-structure", "categories": [ "trace" ], "title": "trace-lemma-A-module-structure", "contents": [ "The map $A\\to \\Lambda$ defines an $A$-module structure on $\\Lambda^\\natural$." ], "refs": [], "proofs": [ { "contents": [ "This is clear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14424, "type": "theorem", "label": "trace-lemma-diagonal-action-projective-module", "categories": [ "trace" ], "title": "trace-lemma-diagonal-action-projective-module", "contents": [ "Let $P$ be a finite projective $A[G]$-module and $M$ a $\\Lambda[G]$-module,", "finite projective as a $\\Lambda$-module. Then $P \\otimes_A M$ is a finite", "projective $\\Lambda[G]$-module, for the structure induced by the diagonal", "action of $G$." ], "refs": [], "proofs": [ { "contents": [ "For any $\\Lambda[G]$-module $N$ one has", "$$", "\\Hom_{\\Lambda[G]}\\left(P \\otimes_A M, N\\right)= \\Hom_{A[G]}\\left(P,", "\\Hom_{\\Lambda}(M, N)\\right)", "$$", "where the $G$-action on $\\Hom_{\\Lambda}(M, N)$ is given by $(g\\cdot", "\\varphi)(m) = g \\varphi (g^{-1} m) $. Now it suffices to observe that the", "right-hand side is a composition of exact functors, because of the projectivity", "of $P$ and $M$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14425, "type": "theorem", "label": "trace-lemma-multiplicative-trace", "categories": [ "trace" ], "title": "trace-lemma-multiplicative-trace", "contents": [ "With assumptions as in", "Lemma \\ref{lemma-diagonal-action-projective-module},", "let", "$u\\in \\text{End}_{A[G]}(P)$ and $v\\in \\text{End}_{\\Lambda[G]}(M)$. Then", "$$", "\\text{Tr}_\\Lambda^G \\left(u \\otimes v; P \\otimes_A M\\right) = \\text{Tr}_A^G(u;", "P)\\cdot \\text{Tr}_\\Lambda(v;M).", "$$" ], "refs": [ "trace-lemma-diagonal-action-projective-module" ], "proofs": [ { "contents": [ "[Sketch of proof]", "Reduce to the case $P=A[G]$. In that case, $u$ is right multiplication by some", "element $a = \\sum a_gg$ of $A[G]$, which we write $u = R_a$. There is an", "isomorphism of $\\Lambda[G]$-modules", "$$", "\\begin{matrix}", "\\varphi : & A[G]\\otimes_A M & \\cong & \\left(A[G]\\otimes_A M\\right)'\\\\", "& g \\otimes m & \\longmapsto & g \\otimes g^{-1}m", "\\end{matrix}", "$$", "where $\\left(A[G]\\otimes_A M\\right)'$ has the module structure given by the", "left $G$-action, together with the $\\Lambda$-linearity on $M$. This transport", "of structure changes $u \\otimes v$ into $\\sum_ga_gR_g \\otimes g^{-1}v$. In other", "words,", "$$", "\\varphi \\circ (u \\otimes v) \\circ \\varphi^{-1}", "=", "\\sum_ga_gR_g \\otimes g^{-1}v.", "$$", "Working out explicitly both sides of the equation, we have to show", "$$", "\\text{Tr}_\\Lambda^G\\left(\\sum_g a_gR_g \\otimes g^{-1}v\\right) = a_e\\cdot", "\\text{Tr}_\\Lambda(v; M).", "$$", "This is done by showing that", "$$", "\\text{Tr}_\\Lambda^G\\left(a_gR_g \\otimes g^{-1}v\\right) =", "\\left\\{", "\\begin{matrix}", "0 & \\text{ if } g\\neq e\\\\", "a_e\\text{Tr}_\\Lambda\\left(v; M\\right) & \\text{ if }g = e", "\\end{matrix}", "\\right.", "$$", "by reducing to $M=\\Lambda$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 14424 ] }, { "id": 14426, "type": "theorem", "label": "trace-lemma-gamma-z-gamma-trace", "categories": [ "trace" ], "title": "trace-lemma-gamma-z-gamma-trace", "contents": [ "Let $P$ be a $\\Lambda[\\Gamma]$-module, finite and projective as a", "$\\Lambda[G]$-module, and $\\gamma \\in \\Gamma$. Then", "$$", "\\text{Tr}_{\\Lambda}(\\gamma, P) =", "\\# Z_\\gamma \\cdot \\text{Tr}_\\Lambda^{Z_\\gamma}\\left(\\gamma, P\\right).", "$$" ], "refs": [], "proofs": [ { "contents": [ "This follows readily from Lemma \\ref{lemma-lambda-trace}." ], "refs": [ "trace-lemma-lambda-trace" ], "ref_ids": [ 14422 ] } ], "ref_ids": [] }, { "id": 14427, "type": "theorem", "label": "trace-lemma-weak-trace", "categories": [ "trace" ], "title": "trace-lemma-weak-trace", "contents": [ "Let $P$ be an $A[\\Gamma]$-module, finite projective as $A[G]$-module. Let $M$", "be a $\\Lambda[\\Gamma]$-module, finite projective as a $\\Lambda$-module. Then", "$$", "\\text{Tr}_{\\Lambda}^{Z_\\gamma}(\\gamma, P \\otimes_A M) =", "\\text{Tr}_A^{Z_\\gamma}(\\gamma, P)\\cdot \\text{Tr}_\\Lambda(\\gamma, M).", "$$" ], "refs": [], "proofs": [ { "contents": [ "This follows directly from Lemma \\ref{lemma-multiplicative-trace}." ], "refs": [ "trace-lemma-multiplicative-trace" ], "ref_ids": [ 14425 ] } ], "ref_ids": [] }, { "id": 14428, "type": "theorem", "label": "trace-lemma-trivial-trace", "categories": [ "trace" ], "title": "trace-lemma-trivial-trace", "contents": [ "Let $P$ be a $\\Lambda[\\Gamma]$-module, finite projective as", "$\\Lambda[G]$-module. Then the coinvariants", "$P_G = \\Lambda \\otimes_{\\Lambda[G]} P$", "form a finite projective $\\Lambda$-module, endowed with an action of", "$\\Gamma/G = \\mathbf{N}$. Moreover, we have", "$$", "\\text{Tr}_\\Lambda(1; P_G) =", "\\sum\\nolimits'_{\\gamma \\mapsto 1} \\text{Tr}_\\Lambda^{Z_\\gamma}(\\gamma, P)", "$$", "where $\\sum_{\\gamma\\mapsto 1}'$ means taking the sum over the $G$-conjugacy", "classes in $\\Gamma$." ], "refs": [], "proofs": [ { "contents": [ "[Sketch of proof]", "We first prove this after multiplying by $\\# G$.", "$$", "\\# G\\cdot \\text{Tr}_\\Lambda(1; P_G)", "= \\text{Tr}_\\Lambda(\\sum\\nolimits_{\\gamma\\mapsto 1} \\gamma, P_G)", "= \\text{Tr}_\\Lambda(\\sum\\nolimits_{\\gamma\\mapsto 1} \\gamma, P)", "$$", "where the second equality follows by considering the commutative triangle", "$$", "\\xymatrix{", "P^G \\ar[rd]_a & & P_G \\ar[ll]^c \\\\", "& P \\ar[ur]_b", "}", "$$", "where $a$ is the canonical inclusion, $b$ the canonical surjection and $c =", "\\sum_{\\gamma \\mapsto 1} \\gamma$. Then we have", "$$", "(\\sum\\nolimits_{\\gamma \\mapsto 1} \\gamma) |_P = a \\circ c \\circ b", "\\quad\\text{and}\\quad", "(\\sum\\nolimits_{\\gamma \\mapsto 1} \\gamma) |_{P_G} = b \\circ a \\circ c", "$$", "hence they have the same trace. We then have", "$$", "\\# G\\cdot \\text{Tr}_\\Lambda(1; P_G)", "=", "{\\sum_{\\gamma\\mapsto 1}}'", "\\frac{\\# G}{\\# Z_\\gamma}\\text{Tr}_\\Lambda(\\gamma, P)", "= \\# G{\\sum_{\\gamma\\mapsto 1}}' \\text{Tr}_\\Lambda^{Z_\\gamma}(\\gamma, P).", "$$", "To finish the proof, reduce to case $\\Lambda$ torsion-free by some universality", "argument. See \\cite{SGA4.5} for details." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14429, "type": "theorem", "label": "trace-lemma-eventually-constant", "categories": [ "trace" ], "title": "trace-lemma-eventually-constant", "contents": [ "Let $\\{\\mathcal{G}_n\\}_{n\\geq 1}$ be an inverse system of constructible", "$\\mathbf{Z}/\\ell^n\\mathbf{Z}$-modules.", "Suppose that for all $k\\geq 1$, the maps", "$$", "\\mathcal{G}_{n+1}/\\ell^k \\mathcal{G}_{n+1}\\to \\mathcal{G}_n /\\ell^k", "\\mathcal{G}_n", "$$", "are isomorphisms for all $n\\gg 0$ (where the bound possibly depends on $k$).", "In other words, assume that the system", "$\\{\\mathcal{G}_n/\\ell^k\\mathcal{G}_n\\}_{n\\geq 1}$", "is eventually constant, and call $\\mathcal{F}_k$ the corresponding sheaf.", "Then the system $\\left\\{\\mathcal{F}_k\\right\\}_{k\\geq 1}$ forms a", "$\\mathbf{Z}_\\ell$-sheaf on $X$." ], "refs": [], "proofs": [ { "contents": [ "The proof is obvious." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14430, "type": "theorem", "label": "trace-lemma-l-adic-abelian", "categories": [ "trace" ], "title": "trace-lemma-l-adic-abelian", "contents": [ "The category of $\\mathbf{Z}_\\ell$-sheaves on $X$ is abelian." ], "refs": [], "proofs": [ { "contents": [ "Let", "$\\Phi = \\left\\{\\varphi_n\\right\\}_{n\\geq 1} :", "\\left\\{\\mathcal{F}_n\\right\\}", "\\to", "\\left\\{\\mathcal{G}_n\\right\\}$", "be a morphism of $\\mathbf{Z}_\\ell$-sheaves. Set", "$$", "\\Coker(\\Phi) =", "\\left\\{", "\\Coker\\left(\\mathcal{F}_n \\xrightarrow{\\varphi_n} \\mathcal{G}_n\\right)", "\\right\\}_{n\\geq 1}", "$$", "and $\\Ker(\\Phi)$ is the result of", "Lemma \\ref{lemma-eventually-constant}", "applied to the inverse system", "$$", "\\left\\{", "\\bigcap_{m\\geq n}", "\\Im\\left(\\Ker(\\varphi_m) \\to \\Ker(\\varphi_n)\\right)", "\\right\\}_{n \\geq 1}.", "$$", "That this defines an abelian category is left to the reader." ], "refs": [ "trace-lemma-eventually-constant" ], "ref_ids": [ 14429 ] } ], "ref_ids": [] }, { "id": 14431, "type": "theorem", "label": "trace-lemma-piece-together", "categories": [ "trace" ], "title": "trace-lemma-piece-together", "contents": [ "Suppose we have", "$K_n\\in D_{perf}(\\mathbf{Z}/\\ell^n\\mathbf{Z})$, $\\pi_n : K_n\\to K_n$", "and isomorphisms", "$\\varphi_n :", "K_{n+1} \\otimes^\\mathbf{L}_{\\mathbf{Z}/\\ell^{n+1}\\mathbf{Z}}", "\\mathbf{Z}/\\ell^n\\mathbf{Z}", "\\to K_n$", "compatible with $\\pi_{n+1}$ and $\\pi_n$. Then", "\\begin{enumerate}", "\\item", "the elements $t_n = \\text{Tr}(\\pi_n |_{K_n})\\in \\mathbf{Z}/\\ell^n\\mathbf{Z}$", "form an element $t_\\infty = \\{t_n\\}$ of $\\mathbf{Z}_\\ell$,", "\\item", "the $\\mathbf{Z}_\\ell$-module $H_\\infty^i = \\lim_n H^i(k_n)$ is finite and", "is nonzero for finitely many $i$ only, and", "\\item", "the operators $H^i(\\pi_n): H^i(K_n)\\to H^i(K_n)$ are compatible and define", "$\\pi_\\infty^i : H_\\infty^i\\to H_\\infty^i$ satisfying", "$$", "\\sum (-1)^i \\text{Tr}(", "\\pi_\\infty^i |_{H_\\infty^i \\otimes_{\\mathbf{Z}_\\ell}\\mathbf{Q}_\\ell}) =", "t_\\infty.", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $\\mathbf{Z}/\\ell^n\\mathbf{Z}$ is a local ring and $K_n$ is perfect, each", "$K_n$ can be represented by a finite complex $K_n^\\bullet$ of finite free", "$\\mathbf{Z}/\\ell^n \\mathbf{Z}$-modules such that the map $K_n^p \\to K_n^{p+1}$", "has image contained in $\\ell K_n^{p+1}$. It is a fact that such a complex is", "unique up to isomorphism. Moreover $\\pi_n$ can be represented by a morphism of", "complexes $\\pi_n^\\bullet : K_n^\\bullet\\to K_n^\\bullet$ (which is unique up to", "homotopy). By the same token the isomorphism", "$\\varphi_n : K_{n+1} \\otimes_{\\mathbf{Z}/\\ell^{n+1}\\mathbf{Z}}^{\\mathbf{L}}", "\\mathbf{Z}/\\ell^n\\mathbf{Z}\\to K_n$ is represented by a map of complexes", "$$", "\\varphi_n^\\bullet :", "K_{n+1}^\\bullet", "\\otimes_{\\mathbf{Z}/\\ell^{n+1}\\mathbf{Z}}", "\\mathbf{Z}/\\ell^n\\mathbf{Z} \\to K_n^\\bullet.", "$$", "In fact, $\\varphi_n^\\bullet$ is an isomorphism of complexes, thus we see that", "\\begin{itemize}", "\\item", "there exist $a, b\\in \\mathbf{Z}$ independent of $n$ such that $K_n^i = 0$ for", "all $i\\notin[a, b]$, and", "\\item", "the rank of $K_n^i$ is independent of $n$.", "\\end{itemize}", "Therefore, the module $K_\\infty^i = \\lim_n \\{K_n^i, \\varphi_n^i\\}$ is a", "finite free $\\mathbf{Z}_\\ell$-module and $K_\\infty^\\bullet$ is a finite complex", "of finite free $\\mathbf{Z}_\\ell$-modules. By induction on the number of nonzero", "terms, one can prove that $H^i\\left(K_\\infty^\\bullet\\right) = \\lim_n", "H^i\\left(K_n^\\bullet\\right)$ (this is not true for unbounded complexes). We", "conclude that $H_\\infty^i = H^i\\left(K_\\infty^\\bullet\\right)$ is a finite", "$\\mathbf{Z}_\\ell$-module. This proves {\\it ii}. To prove the remainder of the", "lemma, we need to overcome the possible noncommutativity of the diagrams", "$$", "\\xymatrix{", "{K_{n+1}^\\bullet} \\ar[d]_{\\pi_{n+1}^\\bullet} \\ar[r]^{\\varphi_n^\\bullet} &", "{K_n^\\bullet} \\ar[d]^{\\pi_n^\\bullet} \\\\", "{K_{n+1}^\\bullet} \\ar[r]_{\\varphi_n^\\bullet} & {K_n^\\bullet.}", "}", "$$", "However, this diagram does commute in the derived category, hence it commutes", "up to homotopy. We inductively replace $\\pi_n^\\bullet$ for $n\\geq 2$ by", "homotopic maps of complexes making these diagrams commute. Namely, if $h^i :", "K_{n+1}^i \\to K_n^{i-1}$ is a homotopy, i.e.,", "$$", "\\pi_n^\\bullet \\circ \\varphi_n^\\bullet -", "\\varphi_n^\\bullet \\circ \\pi_{n + 1}^\\bullet = dh + hd,", "$$", "then we choose $\\tilde h^i : K_{n+1}^i\\to K_{n+1}^{i-1}$ lifting $h^i$. This is", "possible because $K_{n+1}^i$ free and $K_{n+1}^{i-1}\\to K_n^{i-1}$ is", "surjective. Then replace $\\pi_n^\\bullet$ by $\\tilde\\pi_n^\\bullet$ defined by", "$$", "\\tilde\\pi_{n+1}^\\bullet = \\pi_{n+1}^\\bullet + d\\tilde h+\\tilde hd.", "$$", "With this choice of $\\{\\pi_n^\\bullet\\}$, the above diagrams commute, and the", "maps fit together to define an endomorphism $\\pi_\\infty^\\bullet =", "\\lim_n\\pi_n^\\bullet$ of $K_\\infty^\\bullet$. Then part {\\it i} is clear:", "the elements $t_n = \\sum(-1)^i \\text{Tr}\\left(\\pi_n^i |_{K_n^i}\\right)$", "fit into an element $t_\\infty$ of $\\mathbf{Z}_\\ell$. Moreover", "\\begin{align*}", "t_\\infty", "& =", "\\sum (-1)^i \\text{Tr}_{\\mathbf{Z}_\\ell}(\\pi_\\infty^i |_{K_\\infty^i}) \\\\", "& =", "\\sum (-1)^i \\text{Tr}_{\\mathbf{Q}_\\ell}(", "\\pi_\\infty^i |_{K_\\infty^i \\otimes_{\\mathbf{Z}_\\ell}\\mathbf{Q}_\\ell}) \\\\", "& =", "\\sum (-1)^i \\text{Tr}(", "\\pi_\\infty |_{H^i(K_\\infty^\\bullet \\otimes \\mathbf{Q}_\\ell)})", "\\end{align*}", "where the last equality follows from the fact that $\\mathbf{Q}_\\ell$ is a", "field, so the complex $K_\\infty^\\bullet \\otimes \\mathbf{Q}_\\ell$ is", "quasi-isomorphic to its cohomology", "$H^i(K_\\infty^\\bullet \\otimes \\mathbf{Q}_\\ell)$. The latter is also equal to", "$H^i(K_\\infty^\\bullet)\\otimes_{\\mathbf{Z}}\\mathbf{Q}_\\ell = H_\\infty^i \\otimes", "\\mathbf{Q}_\\ell$, which finishes the proof of the lemma, and also that of", "Theorem \\ref{theorem-C}." ], "refs": [ "trace-theorem-C" ], "ref_ids": [ 14403 ] } ], "ref_ids": [] }, { "id": 14432, "type": "theorem", "label": "trace-lemma-count-points-projective", "categories": [ "trace" ], "title": "trace-lemma-count-points-projective", "contents": [ "Let $X$ be a smooth, projective, geometrically irreducible", "curve over a finite field $k$. Then", "\\begin{enumerate}", "\\item the $L$-function $L(X, \\mathbf{Q}_\\ell)$ is a rational function,", "\\item the eigenvalues $\\alpha_1, \\ldots, \\alpha_{2g}$ of $\\pi_X^*$ on", "$H^1(X_{\\bar k}, \\mathbf{Q}_\\ell)$ are algebraic integers", "independent of $\\ell$,", "\\item the number of rational points of $X$ on $k_n$, where $[k_n : k] = n$, is", "$$", "\\# X(k_n) = 1 - \\sum\\nolimits_{i = 1}^{2g}\\alpha_i^n + q^n,", "$$", "\\item for each $i$, $|\\alpha_i| < q$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (3) is Theorem \\ref{theorem-C} applied to $\\mathcal{F} =", "\\underline{\\mathbf{Q}_\\ell}$ on $X \\otimes k_n$. For part (4), use the", "following result." ], "refs": [ "trace-theorem-C" ], "ref_ids": [ 14403 ] } ], "ref_ids": [] }, { "id": 14433, "type": "theorem", "label": "trace-lemma-identify-h2c", "categories": [ "trace" ], "title": "trace-lemma-identify-h2c", "contents": [ "There is a canonical isomorphism", "$$", "H_c^2(X_{\\overline{k}}, \\mathcal{F}_\\rho)=(M)_{\\pi_1(X_{\\overline{k}},", "\\overline\\eta)}(-1)", "$$", "as $\\text{Gal}(k^{^{sep}}/k)$-modules." ], "refs": [], "proofs": [ { "contents": [ "[Proof of Lemma \\ref{lemma-identify-h2c}]", "Let $Y\\to^{\\varphi}X$ be the finite \\'etale Galois covering", "corresponding to $\\Ker(\\rho) \\subset \\pi_1(X, \\overline\\eta)$. So", "$$", "\\text{Aut}(Y/X)=Ind(\\rho)", "$$", "is Galois group. Then $\\varphi^*\\mathcal{F}_\\rho =\\underline M_Y$ and", "$$", "\\varphi_*\\varphi^*\\mathcal{F}_\\rho\\to \\mathcal{F}_\\rho", "$$", "which gives", "\\begin{align*}", "& H_c^2(X_{\\overline{k}}, \\varphi_*\\varphi^*\\mathcal{F}_\\rho) \\to", "H_c^2(X_{\\overline{k}}, \\mathcal{F}_\\rho)\\\\", "& =H_c^2(Y_{\\overline{k}}, \\varphi^*\\mathcal{F}_\\rho)\\\\", "& =H_c^2(Y_{\\overline{k}}, \\underline M) = \\oplus_{\\text{irred.", "comp. of } \\atop Y_{\\overline{k}}}M", "\\end{align*}", "$$", "\\Im(\\rho) \\to H_c^2(Y_{\\overline{k}}, \\underline M) =", "\\oplus_{\\text{irred. comp. of } \\atop Y_{\\overline{k}}}", "M \\to_{\\Im(\\rho) \\text{equivalent}} H_c^2(X_{\\overline{k}},", "\\mathcal{F}_{\\rho}) \\to^{\\text{trivial }", "\\Im(\\rho) \\atop \\text{action}}", "$$", "irreducible curve $C/\\overline{k}$, $H_c^2(C, \\underline M)=M$.", "\\medskip\\noindent", "Since", "$$", "{\\text{set of irreducible } \\atop \\text{components of }Y_k} =", "\\frac{Im(\\rho)}{Im(\\rho|_{\\pi_1(X_{\\overline{k}}, \\overline \\eta)})}", "$$", "We conclude that $H_c^2(X_{\\overline{k}}, \\mathcal{F}_\\rho)$ is a", "quotient of $M_{\\pi_1(X_{\\overline{k}}, \\overline \\eta)}$. On the other hand,", "there is a surjection", "$$", "\\mathcal{F}_\\rho\\to \\mathcal{F}'' = {\\text{ sheaf on }", "X\\text{ associated to } \\atop (M)_{\\pi_1(X_{\\overline{k}}, \\overline", "\\eta)}\\leftarrow\\pi_1(X, \\overline \\eta)}", "$$", "$$", "H_c^2(X_{\\overline{k}}, \\mathcal{F}_\\rho)\\to", "M_{\\pi_1(X_{\\overline{k}}, \\overline\\eta)}", "$$", "The twist in Galois action comes from the fact that", "$H_c^2(X_{\\overline{k}}, \\mu_n)=^{\\text{can}} \\mathbf{Z}/n\\mathbf{Z}$." ], "refs": [ "trace-lemma-identify-h2c" ], "ref_ids": [ 14433 ] } ], "ref_ids": [] }, { "id": 14434, "type": "theorem", "label": "trace-lemma-eigenvalues-algebraic", "categories": [ "trace" ], "title": "trace-lemma-eigenvalues-algebraic", "contents": [ "Algebraicity of eigenvalues.", "If $\\Lambda$ is a field then the eigenvalues $t_v$ for $f\\in", "C(\\Lambda)$ are algebraic over the prime subfield", "$\\mathbf{F} \\subset \\Lambda$." ], "refs": [], "proofs": [ { "contents": [ "Follows from Proposition \\ref{proposition-cusp-forms-finite}." ], "refs": [ "trace-proposition-cusp-forms-finite" ], "ref_ids": [ 14441 ] } ], "ref_ids": [] }, { "id": 14435, "type": "theorem", "label": "trace-lemma-switch-l", "categories": [ "trace" ], "title": "trace-lemma-switch-l", "contents": [ "Switching $l$. Let $E$ be a number field.", "Start with", "$$", "\\rho : \\pi_1(X)\\to SL_2(E_\\lambda)", "$$", "absolutely irreducible continuous, where $\\lambda$ is a place of $E$", "not lying above $p$. Then for any second place $\\lambda'$ of $E$", "not lying above $p$ there exists a finite extension $E'_{\\lambda'}$", "and a absolutely irreducible continuous representation", "$$", "\\rho': \\pi_1(X)\\to SL_2(E'_{\\lambda'})", "$$", "which is compatible with $\\rho$ in the sense that the characteristic", "polynomials of all Frobenii are the same." ], "refs": [], "proofs": [ { "contents": [ "To prove the switching lemma use", "Theorem \\ref{theorem-drinfeld-make-f}", "to obtain $f\\in C(\\overline{\\mathbf{Q}}_l)$ eigenform ass. to $\\rho$.", "Next, use", "Proposition \\ref{proposition-cusp-forms-finite}", "to see that we may choose $f\\in C(E')$ with $E \\subset E'$ finite.", "Next we may complete $E'$ to see that we get", "$f\\in C(E'_{\\lambda'})$ eigenform with", "$E'_{\\lambda'}$ a finite extension of $E_{\\lambda'}$.", "And finally we use", "Theorem \\ref{theorem-drinfeld-make-rho}", "to obtain", "$\\rho': \\pi_1(X) \\to SL_2(E_{\\lambda'}')$ abs. irred. and continuous", "after perhaps enlarging $E'_{\\lambda'}$ a bit again." ], "refs": [ "trace-theorem-drinfeld-make-f", "trace-proposition-cusp-forms-finite", "trace-theorem-drinfeld-make-rho" ], "ref_ids": [ 14407, 14441, 14406 ] } ], "ref_ids": [] }, { "id": 14436, "type": "theorem", "label": "trace-proposition-compare-filtered-graded", "categories": [ "trace" ], "title": "trace-proposition-compare-filtered-graded", "contents": [ "In the situation above, we have", "$$", "\\text{gr}^p \\circ RT = RT \\circ \\text{gr}^p", "$$", "where the $RT$ on the left is the filtered derived functor while the one on the", "right is the total derived functor. That is, there is a commuting diagram", "$$", "\\xymatrix{", "DF^+(\\mathcal{A}) \\ar[r]^{RT} \\ar[d]_{\\text{gr}^p} & DF^+(\\mathcal{B})", "\\ar[d]^{\\text{gr}^p}\\\\", "D^+(\\mathcal{A}) \\ar[r]^{RT} & D^+(\\mathcal{B}).}", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14437, "type": "theorem", "label": "trace-proposition-trace-well-defined", "categories": [ "trace" ], "title": "trace-proposition-trace-well-defined", "contents": [ "Let $K\\in D_{perf}(\\Lambda)$ and $f\\in \\text{End}_{D(\\Lambda)}(K)$. Then the", "trace $\\text{Tr}(f)\\in \\Lambda^\\natural$ is well defined." ], "refs": [], "proofs": [ { "contents": [ "We will use Derived Categories, Lemma", "\\ref{derived-lemma-morphisms-from-projective-complex}", "without further mention in this proof.", "Let $P^\\bullet$ be a bounded complex of finite projective $\\Lambda$-modules", "and let $\\alpha : P^\\bullet \\to K$ be an isomorphism in $D(\\Lambda)$. Then", "$\\alpha^{-1}\\circ f\\circ \\alpha$ corresponds to a morphism of complexes", "$f^\\bullet : P^\\bullet \\to P^\\bullet$ well defined up to homotopy.", "Set", "$$", "\\text{Tr}(f) = \\sum_i (-1)^i \\text{Tr}(f^i : P^i \\to P^i) \\in \\Lambda^\\natural.", "$$", "Given $P^\\bullet$ and $\\alpha$, this is independent of the choice of", "$f^\\bullet$. Namely, any other choice is of the form", "$\\tilde{f}^\\bullet = f^\\bullet + dh +hd$ for some", "$h^i : P^i \\to P^{i-1}(i\\in \\mathbf{Z})$. But", "\\begin{eqnarray*}", "\\text{Tr}(dh) & = & \\sum_i (-1)^i \\text{Tr}(P^i\\xrightarrow{dh} P^i) \\\\", "& = & \\sum_i (-1)^i \\text{Tr}(P^{i-1}\\xrightarrow{hd} P^{i-1}) \\\\", "& = & -\\sum_i (-1)^{i-1}\\text{Tr}(P^{i-1}\\xrightarrow{hd} P^{i-1}) \\\\", "& = & - \\text{Tr}(hd)", "\\end{eqnarray*}", "and so $\\sum_i (-1)^i \\text{Tr} ((dh+hd)|_{P^i})=0$.", "Furthermore, this is independent of the choice of $(P^\\bullet , \\alpha)$:", "suppose $(Q^\\bullet, \\beta)$ is another choice. The compositions", "$$", "Q^\\bullet \\xrightarrow{\\beta} K \\xrightarrow{\\alpha^{-1}} P^\\bullet", "\\quad\\text{and}\\quad", "P^\\bullet \\xrightarrow{\\alpha} K \\xrightarrow{\\beta^{-1}} Q^\\bullet", "$$", "are representable by morphisms of complexes $\\gamma_1^\\bullet$ and", "$\\gamma_2^\\bullet$ respectively, such that $\\gamma_1^\\bullet \\circ", "\\gamma_2^\\bullet$ is homotopic to the identity. Thus, the morphism of complexes", "$\\gamma_2^\\bullet\\circ f^\\bullet\\circ \\gamma_1^\\bullet : Q^\\bullet\\to Q^\\bullet$", "represents the morphism $\\beta^{-1}\\circ f\\circ\\beta$ in $D(\\Lambda)$. Now", "\\begin{eqnarray*}", "\\text{Tr}(\\gamma_2^\\bullet\\circ f^\\bullet\\circ\\gamma_1^\\bullet|_{Q^\\bullet}) &", "= & \\text{Tr}(\\gamma_1^\\bullet \\circ\\gamma_2^\\bullet \\circ", "f^\\bullet|_{P^\\bullet})\\\\", "& = & \\text{Tr}(f^\\bullet|_{P^\\bullet})", "\\end{eqnarray*}", "by the fact that $\\gamma_1^\\bullet \\circ \\gamma_2^\\bullet$ is homotopic to the", "identity and the independence of the choice of $f^\\bullet$ we saw above." ], "refs": [ "derived-lemma-morphisms-from-projective-complex" ], "ref_ids": [ 1862 ] } ], "ref_ids": [] }, { "id": 14438, "type": "theorem", "label": "trace-proposition-projective-curve-constructible-cohomology", "categories": [ "trace" ], "title": "trace-proposition-projective-curve-constructible-cohomology", "contents": [ "Let $X$ be a projective curve over a field $k$, $\\Lambda$ a finite ring and", "$K\\in D_{ctf}(X, \\Lambda)$. Then $R\\Gamma(X_{\\bar k}, K)\\in", "D_{perf}(\\Lambda)$." ], "refs": [], "proofs": [ { "contents": [ "[Sketch of proof.]", "The first step is to show:", "\\begin{enumerate}", "\\item[(1)]", "{\\it The cohomology of $R\\Gamma(X_{\\bar k}, K)$ is bounded.}", "\\end{enumerate}", "Consider the spectral sequence", "$$", "H^i(X_{\\bar k}, \\underline H^j(K))", "\\Rightarrow", "H^{i+j} (R\\Gamma(X_{\\bar k}, K)).", "$$", "Since $K$ is bounded and $\\Lambda$ is finite, the sheaves $\\underline H^j(K)$", "are torsion. Moreover, $X_{\\bar k}$ has finite cohomological dimension, so the", "left-hand side is nonzero for finitely many $i$ and $j$ only. Therefore, so is", "the right-hand side.", "\\begin{enumerate}", "\\item[(2)]", "{\\it The cohomology groups $H^{i+j} (R\\Gamma(X_{\\bar k}, K))$ are finite.}", "\\end{enumerate}", "Since the sheaves $\\underline H^j(K)$ are constructible, the groups", "$H^i(X_{\\bar k}, \\underline H^j(K))$ are finite", "(\\'Etale Cohomology, Section \\ref{etale-cohomology-section-vanishing-torsion})", "so it follows by the spectral sequence again.", "\\begin{enumerate}", "\\item[(3)]", "{\\it $R\\Gamma(X_{\\bar k}, K)$ has finite $\\text{Tor}$-dimension.}", "\\end{enumerate}", "Let $N$ be a right $\\Lambda$-module (in fact, since $\\Lambda$ is finite, it", "suffices to assume that $N$ is finite). By the projection formula (change of", "module),", "$$", "N \\otimes^\\mathbf{L}_\\Lambda R \\Gamma(X_{\\bar k}, K) = R\\Gamma(X_{\\bar k},", "N \\otimes^\\mathbf{L}_\\Lambda K).", "$$", "Therefore,", "$$", "H^i (N \\otimes^\\mathbf{L}_\\Lambda R\\Gamma(X_{\\bar k}, K)) = H^i(R\\Gamma(X_{\\bar", "k}, N \\otimes_{\\Lambda}^\\mathbf{L} K)).", "$$", "Now consider the spectral sequence", "$$", "H^i (X_{\\bar k}, \\underline H^j (N \\otimes_{\\Lambda}^\\mathbf{L} K))", "\\Rightarrow", "H^{i+j}(R\\Gamma(X_{\\bar k}, N \\otimes_{\\Lambda}^\\mathbf{L} K)).", "$$", "Since $K$ has finite $\\text{Tor}$-dimension, $\\underline H^j", "(N \\otimes_{\\Lambda}^\\mathbf{L} K)$ vanishes universally for $j$ small enough,", "and the left-hand side vanishes whenever $i < 0$. Therefore $R\\Gamma(X_{\\bar", "k}, K)$ has finite $\\text{Tor}$-dimension, as claimed. So it is a perfect", "complex by Lemma \\ref{lemma-characterize-perfect}." ], "refs": [ "trace-lemma-characterize-perfect" ], "ref_ids": [ 14419 ] } ], "ref_ids": [] }, { "id": 14439, "type": "theorem", "label": "trace-proposition-integral-normal-fundamental-group", "categories": [ "trace" ], "title": "trace-proposition-integral-normal-fundamental-group", "contents": [ "Let $X$ be an integral normal Noetherian scheme. Let", "$\\overline y\\to X$ be an algebraic geometric point lying", "over the generic point $\\eta\\in X$. Then", "$$", "\\pi_x(X, \\overline \\eta) = Gal(M/\\kappa(\\eta))", "$$", "($\\kappa(\\eta)$, function field of $X$) where", "$$", "\\kappa(\\overline \\eta)\\supset M\\supset \\kappa(\\eta) = k(X)", "$$", "is the max sub-extension such that for every finite sub extension", "$M\\supset L\\supset \\kappa(\\eta)$ the normalization of $X$ in $L$ is finite", "\\'etale over $X$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14440, "type": "theorem", "label": "trace-proposition-curve-kpi1", "categories": [ "trace" ], "title": "trace-proposition-curve-kpi1", "contents": [ "Let $X/k$ as before but $X_{\\overline{k}}\\neq \\mathbf{P}^1_{\\overline{k}}$", "The functors", "$", "(M, \\rho)\\mapsto H_c^{2-i}(X_{\\overline{k}}, \\mathcal{F}_\\rho)", "$", "are the left derived functor of", "$(M, \\rho)\\mapsto H_c^2(X_{\\overline{k}}, \\mathcal{F}_\\rho)$", "so", "$$", "H_c^{2-i}(X_{\\overline{k}}, \\mathcal{F}_\\rho) =", "H_i(\\pi_1(X_{\\overline{k}}, \\overline \\eta), M)(-1)", "$$", "Moreover, there is a derived version, namely", "$$", "R\\Gamma_c(X_{\\overline{k}}, \\mathcal{F}_\\rho)", "=", "LH_0(\\pi_1(X_{\\overline{k}}, \\overline \\eta), M(-1))", "=", "M(-1)", "\\otimes_{\\Lambda[[\\pi_1(X_{\\overline{k}}, \\overline \\eta)]]}^\\mathbf{L}", "\\Lambda", "$$", "in $D(\\Lambda[[\\widehat{\\mathbf{Z}}]])$.", "Similarly, the functors", "$(M, \\rho)\\mapsto H^i(X_{\\overline{k}}, \\mathcal{F}_\\rho)$", "are the right derived functor of", "$(M, \\rho)\\mapsto M^{\\pi_1(X_{\\overline{k}}, \\overline \\eta)}$", "so", "$$", "H^i(X_{\\overline{k}}, \\mathcal{F}_\\rho) =", "H^i(\\pi_1(X_{\\overline{k}}, \\overline \\eta), M)", "$$", "Moreover, in this case there is a derived version too." ], "refs": [], "proofs": [ { "contents": [ "(Idea) Show both sides are universal $\\delta$-functors." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14441, "type": "theorem", "label": "trace-proposition-cusp-forms-finite", "categories": [ "trace" ], "title": "trace-proposition-cusp-forms-finite", "contents": [ "If $\\Lambda$ is Noetherian then $C(\\Lambda)$ is a", "finitely generated $\\Lambda$-module. Moreover, if $\\Lambda$ is a field with", "prime subfield $\\mathbf{F} \\subset \\Lambda$ then", "$$", "C(\\Lambda)=(C(\\mathbf{F}))\\otimes_{\\mathbf{F}}\\Lambda", "$$", "compatibly with $T_v$ acting." ], "refs": [], "proofs": [ { "contents": [ "See \\cite[Proposition 4.7]{dJ-conjecture}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14442, "type": "theorem", "label": "trace-proposition-finite-set-frobenii-generate-topologically", "categories": [ "trace" ], "title": "trace-proposition-finite-set-frobenii-generate-topologically", "contents": [ "There exists a finite set $x_1, \\ldots, x_n$ of closed points of $X$", "such that set of {\\bf all} frobenius elements corresponding to these", "points topologically generate $\\pi_1(X)$." ], "refs": [], "proofs": [ { "contents": [ "Pick $N\\gg 0$ and let", "$$", "\\{x_1, \\ldots, x_n\\} =", "{\\text{ set of all closed points of}", "\\atop X \\text{ of degree} \\leq N\\text{ over } k}", "$$", "Let $\\Gamma\\subset \\pi_1(X)$ be as in the variant statement for these", "points. Assume $\\Gamma \\neq \\pi_1(X)$. Then we can pick a normal open", "subgroup $U$ of $\\pi_1(X)$ containing $\\Gamma$ with", "$U \\neq \\pi_1(X)$. By R.H. for $X$ our set of points will have some", "$x_{i_1}$ of degree $N$, some $x_{i_2}$ of degree $N - 1$. This shows", "$\\deg : \\Gamma \\to \\widehat{\\mathbf{Z}}$ is surjective", "and so the same holds for $U$. This exactly", "means if $Y \\to X$ is the finite \\'etale Galois covering", "corresponding to $U$, then $Y_{\\overline{k}}$ irreducible.", "Set $G = \\text{Aut}(Y/X)$. Picture", "$$", "Y \\to^G X,\\quad G = \\pi_1(X)/U", "$$", "By construction all points of $X$ of degree $\\leq N$, split", "completely in $Y$. So, in particular", "$$", "\\# Y(k_N)\\geq (\\# G)\\# X(k_N)", "$$", "Use R.H. on both sides. So you get", "$$", "q^N+1+2g_Yq^{N/2}\\geq \\# G\\# X(k_N)\\geq \\#", "G(q^N+1-2g_Xq^{N/2})", "$$", "Since $2g_Y-2 = (\\# G)(2g_X-2)$, this means", "$$", "q^N + 1 + (\\# G)(2g_X - 1) + 1)q^{N/2}\\geq", "\\# G (q^N + 1 - 2g_Xq^{N/2})", "$$", "Thus we see that $G$ has to be the trivial group if $N$ is large enough." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14478, "type": "theorem", "label": "sheaves-lemma-product-presheaves", "categories": [ "sheaves" ], "title": "sheaves-lemma-product-presheaves", "contents": [ "Let $X$ be a topological space. The category of presheaves of sets", "on $X$ has products (see", "Categories, Definition \\ref{categories-definition-product}).", "Moreover, the set of", "sections of the product $\\mathcal{F} \\times \\mathcal{G}$", "over an open $U$ is the product of the sets of sections of", "$\\mathcal{F}$ and $\\mathcal{G}$ over $U$." ], "refs": [ "categories-definition-product" ], "proofs": [ { "contents": [ "Namely, suppose $\\mathcal{F}$ and $\\mathcal{G}$ are", "presheaves of sets on the topological space $X$.", "Consider the rule $U \\mapsto \\mathcal{F}(U) \\times \\mathcal{G}(U)$,", "denoted $\\mathcal{F} \\times \\mathcal{G}$. If $V \\subset U \\subset X$", "are open then define the restriction mapping", "$$", "(\\mathcal{F} \\times \\mathcal{G})(U)", "\\longrightarrow", "(\\mathcal{F} \\times \\mathcal{G})(V)", "$$", "by mapping $(s, t) \\mapsto (s|_V, t|_V)$. Then it is immediately", "clear that $\\mathcal{F} \\times \\mathcal{G}$ is a presheaf.", "Also, there are projection maps", "$p : \\mathcal{F} \\times \\mathcal{G} \\to \\mathcal{F}$", "and", "$q : \\mathcal{F} \\times \\mathcal{G} \\to \\mathcal{G}$.", "We leave it to the reader to show that", "for any third presheaf $\\mathcal{H}$ we have", "$\\Mor(\\mathcal{H}, \\mathcal{F} \\times \\mathcal{G})", "= \\Mor(\\mathcal{H}, \\mathcal{F}) \\times", "\\Mor(\\mathcal{H}, \\mathcal{G})$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12358 ] }, { "id": 14479, "type": "theorem", "label": "sheaves-lemma-abelian-presheaves", "categories": [ "sheaves" ], "title": "sheaves-lemma-abelian-presheaves", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{F}$ be a presheaf of sets.", "Consider the following types of structure on $\\mathcal{F}$:", "\\begin{enumerate}", "\\item For every open $U$ the structure of an abelian group", "on $\\mathcal{F}(U)$ such that all restriction maps are", "abelian group homomorphisms.", "\\item A map of presheaves", "$+ : \\mathcal{F} \\times \\mathcal{F} \\to \\mathcal{F}$,", "a map of presheaves $- : \\mathcal{F} \\to \\mathcal{F}$", "and a map $0 : * \\to \\mathcal{F}$", "(see Example \\ref{example-singleton-presheaf})", "satisfying all the axioms of $+, -, 0$ in a usual", "abelian group.", "\\item A map of presheaves", "$+ : \\mathcal{F} \\times \\mathcal{F} \\to \\mathcal{F}$,", "a map of presheaves $- : \\mathcal{F} \\to \\mathcal{F}$", "and a map $0 : * \\to \\mathcal{F}$", "such that for each open $U \\subset X$ the quadruple", "$(\\mathcal{F}(U), +, -, 0)$ is an abelian group,", "\\item A map of presheaves $+ : \\mathcal{F} \\times \\mathcal{F}", "\\to \\mathcal{F}$ such that for every open $U \\subset X$", "the map $+ : \\mathcal{F}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$", "defines the structure of an abelian group.", "\\end{enumerate}", "There are natural bijections between the collections of", "types of data (1) - (4) above." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14480, "type": "theorem", "label": "sheaves-lemma-adjointness-tensor-restrict-presheaves", "categories": [ "sheaves" ], "title": "sheaves-lemma-adjointness-tensor-restrict-presheaves", "contents": [ "With $X$, $\\mathcal{O}_1$, $\\mathcal{O}_2$, $\\mathcal{F}$ and", "$\\mathcal{G}$ as above there exists a canonical bijection", "$$", "\\Hom_{\\mathcal{O}_1}(\\mathcal{G}, \\mathcal{F}_{\\mathcal{O}_1})", "=", "\\Hom_{\\mathcal{O}_2}(", "\\mathcal{O}_2 \\otimes_{p, \\mathcal{O}_1} \\mathcal{G},", "\\mathcal{F}", ")", "$$", "In other words, the restriction and change of rings functors", "are adjoint to each other." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that for a ring map", "$A \\to B$ the restriction functor and the change", "of ring functor are adjoint to each other." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14481, "type": "theorem", "label": "sheaves-lemma-sheaves-structure", "categories": [ "sheaves" ], "title": "sheaves-lemma-sheaves-structure", "contents": [ "Suppose the category $\\mathcal{C}$ and", "the functor $F : \\mathcal{C} \\to \\textit{Sets}$", "have the following properties:", "\\begin{enumerate}", "\\item $F$ is faithful,", "\\item $\\mathcal{C}$ has limits and $F$ commutes with them, and", "\\item the functor $F$ reflects isomorphisms.", "\\end{enumerate}", "Let $X$ be a topological space. Let $\\mathcal{F}$", "be a presheaf with values in $\\mathcal{C}$.", "Then $\\mathcal{F}$ is a sheaf if and only if the", "underlying presheaf of sets is a sheaf." ], "refs": [], "proofs": [ { "contents": [ "Assume that $\\mathcal{F}$ is a sheaf. Then", "$\\mathcal{F}(U)$ is the equalizer of the diagram", "above and by assumption we see $F(\\mathcal{F}(U))$", "is the equalizer of the corresponding diagram", "of sets. Hence $F(\\mathcal{F})$ is a sheaf of sets.", "\\medskip\\noindent", "Assume that $F(\\mathcal{F})$ is a sheaf.", "Let $E \\in \\Ob(\\mathcal{C})$ be the", "equalizer of the two parallel arrows in", "Definition \\ref{definition-sheaf-values-in-category}.", "We get a canonical morphism $\\mathcal{F}(U) \\to E$,", "simply because $\\mathcal{F}$ is a presheaf.", "By assumption, the induced map $F(\\mathcal{F}(U)) \\to F(E)$", "is an isomorphism, because $F(E)$ is the equalizer", "of the corresponding diagram of sets. Hence we", "see $\\mathcal{F}(U) \\to E$ is an isomorphism", "by condition (3) of the lemma." ], "refs": [ "sheaves-definition-sheaf-values-in-category" ], "ref_ids": [ 14568 ] } ], "ref_ids": [] }, { "id": 14482, "type": "theorem", "label": "sheaves-lemma-sheaf-subset-stalks", "categories": [ "sheaves" ], "title": "sheaves-lemma-sheaf-subset-stalks", "contents": [ "Let $\\mathcal{F}$ be a sheaf of sets on the topological space $X$.", "For every open $U \\subset X$ the map", "$$", "\\mathcal{F}(U)", "\\longrightarrow", "\\prod\\nolimits_{x \\in U} \\mathcal{F}_x", "$$", "is injective." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $s, s' \\in \\mathcal{F}(U)$ map to the same element", "in every stalk $\\mathcal{F}_x$ for all $x \\in U$. This means that", "for every $x \\in U$, there exists an open $V^x \\subset U$,", "$x \\in V^x$ such that $s|_{V^x} = s'|_{V^x}$. But then", "$U = \\bigcup_{x \\in U} V^x$ is an open covering. Thus by the", "uniqueness in the sheaf condition we see that $s = s'$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14483, "type": "theorem", "label": "sheaves-lemma-stalk-abelian-presheaf", "categories": [ "sheaves" ], "title": "sheaves-lemma-stalk-abelian-presheaf", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{F}$ be a presheaf", "of abelian groups on $X$. There exists a unique structure of an", "abelian group on $\\mathcal{F}_x$ such that for every", "$U \\subset X$ open, $x\\in U$ the map $\\mathcal{F}(U) \\to \\mathcal{F}_x$", "is a group homomorphism. Moreover,", "$$", "\\mathcal{F}_x", "=", "\\colim_{x\\in U} \\mathcal{F}(U)", "$$", "holds in the category of abelian groups." ], "refs": [], "proofs": [ { "contents": [ "We define addition of a pair of elements", "$(U, s)$ and $(V, t)$ as the pair $(U \\cap V, s|_{U\\cap V} +", "t|_{U \\cap V})$. The rest is easy to check." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14484, "type": "theorem", "label": "sheaves-lemma-stalk-presheaf-values-in-category", "categories": [ "sheaves" ], "title": "sheaves-lemma-stalk-presheaf-values-in-category", "contents": [ "Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{C} \\to \\textit{Sets}$", "be a functor. Assume that", "\\begin{enumerate}", "\\item $F$ is faithful, and", "\\item directed colimits exist in $\\mathcal{C}$ and $F$ commutes with", "them.", "\\end{enumerate}", "Let $X$ be a topological space. Let $x \\in X$. Let $\\mathcal{F}$", "be a presheaf with values in $\\mathcal{C}$.", "Then", "$$", "\\mathcal{F}_x = \\colim_{x\\in U} \\mathcal{F}(U)", "$$", "exists in $\\mathcal{C}$. Its underlying set is equal to the", "stalk of the underlying presheaf of sets of $\\mathcal{F}$.", "Furthermore, the construction $\\mathcal{F} \\mapsto \\mathcal{F}_x$", "is a functor from the category of presheaves with values in", "$\\mathcal{C}$ to $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14485, "type": "theorem", "label": "sheaves-lemma-stalk-module", "categories": [ "sheaves" ], "title": "sheaves-lemma-stalk-module", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{O}$ be a presheaf of rings on $X$.", "Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.", "Let $x \\in X$.", "The canonical map $\\mathcal{O}_x \\times \\mathcal{F}_x", "\\to \\mathcal{F}_x$ coming from the multiplication map", "$\\mathcal{O} \\times \\mathcal{F} \\to \\mathcal{F}$ defines", "a $\\mathcal{O}_x$-module structure on the abelian group", "$\\mathcal{F}_x$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14486, "type": "theorem", "label": "sheaves-lemma-stalk-tensor-presheaf-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-stalk-tensor-presheaf-modules", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{O} \\to \\mathcal{O}'$ be a morphism of", "presheaves of rings on $X$.", "Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.", "Let $x \\in X$. We have", "$$", "\\mathcal{F}_x \\otimes_{\\mathcal{O}_x} \\mathcal{O}'_x", "=", "(\\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{O}')_x", "$$", "as $\\mathcal{O}'_x$-modules." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14487, "type": "theorem", "label": "sheaves-lemma-list-algebraic-structures", "categories": [ "sheaves" ], "title": "sheaves-lemma-list-algebraic-structures", "contents": [ "The following categories, endowed with the obvious forgetful", "functor, define types of algebraic structures:", "\\begin{enumerate}", "\\item The category of pointed sets.", "\\item The category of abelian groups.", "\\item The category of groups.", "\\item The category of monoids.", "\\item The category of rings.", "\\item The category of $R$-modules for a fixed ring $R$.", "\\item The category of Lie algebras over a fixed field.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14488, "type": "theorem", "label": "sheaves-lemma-properties-algebraic-structures", "categories": [ "sheaves" ], "title": "sheaves-lemma-properties-algebraic-structures", "contents": [ "Let $(\\mathcal{C}, F)$ be a type of algebraic structure.", "\\begin{enumerate}", "\\item $\\mathcal{C}$ has a final object $0$ and $F(0) = \\{ * \\}$.", "\\item $\\mathcal{C}$ has products and $F(\\prod A_i) = \\prod F(A_i)$.", "\\item $\\mathcal{C}$ has fibre products and", "$F(A \\times_B C) = F(A)\\times_{F(B)}F(C)$.", "\\item $\\mathcal{C}$ has equalizers, and if $E \\to A$", "is the equalizer of $a, b : A \\to B$, then", "$F(E) \\to F(A)$ is the equalizer of $F(a), F(b) : F(A) \\to F(B)$.", "\\item $A \\to B$ is a monomorphism if and only if", "$F(A) \\to F(B)$ is injective.", "\\item if $F(a) : F(A) \\to F(B)$ is surjective, then", "$a$ is an epimorphism.", "\\item given $A_1 \\to A_2 \\to A_3 \\to \\ldots$, then", "$\\colim A_i$ exists and $F(\\colim A_i) = \\colim F(A_i)$,", "and more generally for any filtered colimit.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. The only interesting statement is (5) which", "follows because $A \\to B$ is a monomorphism if and only if", "$A \\to A \\times_B A$ is an isomorphism, and then applying", "the fact that $F$ reflects isomorphisms." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14489, "type": "theorem", "label": "sheaves-lemma-image-contained-in", "categories": [ "sheaves" ], "title": "sheaves-lemma-image-contained-in", "contents": [ "Let $(\\mathcal{C}, F)$ be a type of algebraic structure.", "Suppose that $A, B, C \\in \\Ob(\\mathcal{C})$.", "Let $f : A \\to B$ and $g : C \\to B$ be morphisms of", "$\\mathcal{C}$. If $F(g)$ is injective, and", "$\\Im(F(f)) \\subset \\Im(F(g))$, then", "$f$ factors as $f = g \\circ t$ for some morphism", "$t : A \\to C$." ], "refs": [], "proofs": [ { "contents": [ "Consider $A \\times_B C$. The assumptions imply that", "$F(A \\times_B C) = F(A) \\times_{F(B)} F(C) = F(A)$.", "Hence $A = A \\times_B C$ because $F$ reflects isomorphisms.", "The result follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14490, "type": "theorem", "label": "sheaves-lemma-points-exactness", "categories": [ "sheaves" ], "title": "sheaves-lemma-points-exactness", "contents": [ "Let $X$ be a topological space. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "be a morphism of sheaves of sets on $X$.", "\\begin{enumerate}", "\\item The map $\\varphi$ is a monomorphism in the category of sheaves", "if and only if for all $x \\in X$ the map", "$\\varphi_x : \\mathcal{F}_x \\to \\mathcal{G}_x$", "is injective.", "\\item The map $\\varphi$ is an epimorphism in the category of sheaves", "if and only if for all $x \\in X$ the map", "$\\varphi_x : \\mathcal{F}_x \\to \\mathcal{G}_x$", "is surjective.", "\\item The map $\\varphi$ is an isomorphism in the category of sheaves", "if and only if for all $x \\in X$ the map", "$\\varphi_x : \\mathcal{F}_x \\to \\mathcal{G}_x$", "is bijective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14491, "type": "theorem", "label": "sheaves-lemma-characterize-epi-mono", "categories": [ "sheaves" ], "title": "sheaves-lemma-characterize-epi-mono", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item Epimorphisms (resp.\\ monomorphisms) in the category of", "presheaves are exactly the surjective (resp.\\ injective) maps", "of presheaves.", "\\item Epimorphisms (resp.\\ monomorphisms) in the category of", "sheaves are exactly the surjective (resp.\\ injective) maps", "of sheaves, and are exactly those maps with are surjective", "(resp.\\ injective) on all the stalks.", "\\item The sheafification of a surjective (resp.\\ injective)", "morphism of presheaves of sets is surjective (resp.\\ injective).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14492, "type": "theorem", "label": "sheaves-lemma-check-homomorphism-stalks", "categories": [ "sheaves" ], "title": "sheaves-lemma-check-homomorphism-stalks", "contents": [ "let $X$ be a topological space.", "Let $(\\mathcal{C}, F)$ be a type of algebraic structure.", "Suppose that $\\mathcal{F}$, $\\mathcal{G}$ are sheaves on $X$", "with values in $\\mathcal{C}$.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "be a map of the underlying sheaves of sets.", "If for all points $x \\in X$ the map", "$\\mathcal{F}_x \\to \\mathcal{G}_x$", "is a morphism of algebraic structures,", "then $\\varphi$ is a morphism of sheaves of algebraic structures." ], "refs": [], "proofs": [ { "contents": [ "Let $U$ be an open subset of $X$. Consider the diagram of", "(underlying) sets", "$$", "\\xymatrix{", "\\mathcal{F}(U) \\ar[r] \\ar[d] &", "\\prod_{x \\in U} \\mathcal{F}_x \\ar[d] \\\\", "\\mathcal{G}(U) \\ar[r] &", "\\prod_{x \\in U} \\mathcal{G}_x", "}", "$$", "By assumption, and previous results, all but the left vertical", "arrow are morphisms of algebraic structures. In addition the", "bottom horizontal arrow is injective,", "see Lemma \\ref{lemma-sheaf-subset-stalks}.", "Hence we conclude by Lemma \\ref{lemma-image-contained-in},", "see also Example \\ref{example-application-lemma-image-contained-in}" ], "refs": [ "sheaves-lemma-sheaf-subset-stalks", "sheaves-lemma-image-contained-in" ], "ref_ids": [ 14482, 14489 ] } ], "ref_ids": [] }, { "id": 14493, "type": "theorem", "label": "sheaves-lemma-sheafification-sheaf", "categories": [ "sheaves" ], "title": "sheaves-lemma-sheafification-sheaf", "contents": [ "The presheaf $\\mathcal{F}^{\\#}$ is a sheaf." ], "refs": [], "proofs": [ { "contents": [ "It is probably better for the reader to find their own explanation", "of this than to read the proof here. In fact the lemma is true", "for the same reason as why the presheaf of continuous", "function is a sheaf, see Example \\ref{example-basic-continuous-maps}", "(and this analogy can be made precise using the ``espace \\'etal\\'e'').", "\\medskip\\noindent", "Anyway, let $U = \\bigcup U_i$ be an open covering.", "Suppose that $s_i = (s_{i, u})_{u \\in U_i} \\in \\mathcal{F}^{\\#}(U_i)$", "such that $s_i$ and $s_j$ agree over $U_i \\cap U_j$.", "Because $\\Pi(\\mathcal{F})$ is a sheaf,", "we find an element $s = (s_u)_{u\\in U}$ in $\\prod_{u\\in U} \\mathcal{F}_u$", "restricting to $s_i$ on $U_i$. We have to check property $(*)$.", "Pick $u \\in U$. Then $u \\in U_i$ for some $i$. Hence by $(*)$ for $s_i$,", "there exists a $V$ open, $u \\in V \\subset U_i$", "and a $\\sigma \\in \\mathcal{F}(V)$", "such that $s_{i, v} = (V, \\sigma)$ in $\\mathcal{F}_v$", "for all $v \\in V$. Since $s_{i, v} = s_v$ we get $(*)$ for $s$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14494, "type": "theorem", "label": "sheaves-lemma-stalk-sheafification", "categories": [ "sheaves" ], "title": "sheaves-lemma-stalk-sheafification", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{F}$ be a presheaf of sets on $X$.", "Let $x \\in X$. Then $\\mathcal{F}_x = \\mathcal{F}^\\#_x$." ], "refs": [], "proofs": [ { "contents": [ "The map $\\mathcal{F}_x \\to \\mathcal{F}^\\#_x$", "is injective, since already the map", "$\\mathcal{F}_x \\to \\Pi(\\mathcal{F})_x$ is injective.", "Namely, there is a canonical map $\\Pi(\\mathcal{F})_x \\to \\mathcal{F}_x$", "which is a left inverse to the map $\\mathcal{F}_x \\to \\Pi(\\mathcal{F})_x$,", "see Example \\ref{example-sheaf-product-pointwise-stalk}.", "To show that it is surjective, suppose that", "$\\overline{s} \\in \\mathcal{F}^\\#_x$.", "We can find an open neighbourhood $U$ of $x$ such that", "$\\overline{s}$ is the equivalence class of $(U, s)$", "with $s \\in \\mathcal{F}^\\#(U)$.", "By definition, this means there exists an open neighbourhood", "$V \\subset U$ of $x$ and a section $\\sigma \\in \\mathcal{F}(V)$", "such that $s|_V$ is the image of $\\sigma$ in $\\Pi(\\mathcal{F})(V)$.", "Clearly the class of $(V, \\sigma)$ defines an element of", "$\\mathcal{F}_x$ mapping to $\\overline{s}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14495, "type": "theorem", "label": "sheaves-lemma-sheafify-universal", "categories": [ "sheaves" ], "title": "sheaves-lemma-sheafify-universal", "contents": [ "Let $\\mathcal{F}$ be a presheaf of sets on $X$.", "Any map $\\mathcal{F} \\to \\mathcal{G}$ into a sheaf of sets", "factors uniquely as", "$\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Clearly, there is a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{F} \\ar[r] \\ar[d] &", "\\mathcal{F}^\\# \\ar[r] \\ar[d] &", "\\Pi(\\mathcal{F}) \\ar[d] \\\\", "\\mathcal{G} \\ar[r] &", "\\mathcal{G}^\\# \\ar[r] &", "\\Pi(\\mathcal{G}) \\\\", "}", "$$", "So it suffices to prove that $\\mathcal{G} = \\mathcal{G}^\\#$.", "To see this it suffices to prove, for every point $x \\in X$ the", "map $\\mathcal{G}_x \\to \\mathcal{G}^\\#_x$ is bijective, by", "Lemma \\ref{lemma-points-exactness}. And this is", "Lemma \\ref{lemma-stalk-sheafification} above." ], "refs": [ "sheaves-lemma-points-exactness", "sheaves-lemma-stalk-sheafification" ], "ref_ids": [ 14490, 14494 ] } ], "ref_ids": [] }, { "id": 14496, "type": "theorem", "label": "sheaves-lemma-separated-presheaf-into-sheaf", "categories": [ "sheaves" ], "title": "sheaves-lemma-separated-presheaf-into-sheaf", "contents": [ "Let $X$ be a topological space.", "A presheaf $\\mathcal{F}$ is separated (see", "Definition \\ref{definition-separated}) if and only if", "the canonical map $\\mathcal{F} \\to \\mathcal{F}^\\#$ is injective." ], "refs": [ "sheaves-definition-separated" ], "proofs": [ { "contents": [ "This is clear from the construction of $\\mathcal{F}^\\#$ in this", "section." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 14570 ] }, { "id": 14497, "type": "theorem", "label": "sheaves-lemma-diagram-fibre-product", "categories": [ "sheaves" ], "title": "sheaves-lemma-diagram-fibre-product", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{F}$ be", "a presheaf of sets on $X$. Let $U \\subset X$ be open.", "There is a canonical fibre product diagram", "$$", "\\xymatrix{", "\\mathcal{F}^\\#(U) \\ar[d] \\ar[r] &", "\\Pi(\\mathcal{F})(U) \\ar[d] \\\\", "\\prod_{x \\in U} \\mathcal{F}_x", "\\ar[r] &", "\\prod_{x \\in U} \\Pi(\\mathcal{F})_x", "}", "$$", "where the maps are the following:", "\\begin{enumerate}", "\\item The left vertical map has components", "$\\mathcal{F}^\\#(U) \\to \\mathcal{F}^\\#_x = \\mathcal{F}_x$", "where the equality is Lemma \\ref{lemma-stalk-sheafification}.", "\\item The top horizontal map comes from the", "map of presheaves $\\mathcal{F} \\to \\Pi(\\mathcal{F})$ described", "in Section \\ref{section-sheafification}.", "\\item The right vertical map has obvious component", "maps $\\Pi(\\mathcal{F})(U) \\to \\Pi(\\mathcal{F})_x$.", "\\item The bottom horizontal map has components", "$\\mathcal{F}_x \\to \\Pi(\\mathcal{F})_x$", "which come from the map of presheaves", "$\\mathcal{F} \\to \\Pi(\\mathcal{F})$ described", "in Section \\ref{section-sheafification}.", "\\end{enumerate}" ], "refs": [ "sheaves-lemma-stalk-sheafification" ], "proofs": [ { "contents": [ "It is clear that the diagram commutes. We have to show", "it is a fibre product diagram. The bottom horizontal arrow", "is injective since all the maps $\\mathcal{F}_x \\to \\Pi(\\mathcal{F})_x$", "are injective (see beginning proof of", "Lemma \\ref{lemma-stalk-sheafification}).", "A section $s \\in \\Pi(\\mathcal{F})(U)$ is in $\\mathcal{F}^\\#$ if and", "only if $(*)$ holds. But $(*)$ says that around every point", "the section $s$ comes from a section of $\\mathcal{F}$. By definition", "of the stalk functors, this is equivalent to saying that", "the value of $s$ in every stalk $\\Pi(\\mathcal{F})_x$ comes", "from an element of the stalk $\\mathcal{F}_x$. Hence the lemma." ], "refs": [ "sheaves-lemma-stalk-sheafification" ], "ref_ids": [ 14494 ] } ], "ref_ids": [ 14494 ] }, { "id": 14498, "type": "theorem", "label": "sheaves-lemma-sheafify-abelian-presheaf", "categories": [ "sheaves" ], "title": "sheaves-lemma-sheafify-abelian-presheaf", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{F}$ be an abelian presheaf on $X$.", "Then there exists a unique structure of", "abelian sheaf on $\\mathcal{F}^\\#$ such that", "$\\mathcal{F} \\to \\mathcal{F}^\\#$ is a morphism", "of abelian presheaves. Moreover, the following adjointness", "property holds", "$$", "\\Mor_{\\textit{PAb}(X)}(\\mathcal{F}, i(\\mathcal{G}))", "=", "\\Mor_{\\textit{Ab}(X)}(\\mathcal{F}^\\#, \\mathcal{G}).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Recall the sheaf of sets $\\Pi(\\mathcal{F})$ defined in", "Section \\ref{section-sheafification}. All the stalks", "$\\mathcal{F}_x$ are abelian groups, see", "Lemma \\ref{lemma-stalk-abelian-presheaf}.", "Hence $\\Pi(\\mathcal{F})$ is a sheaf of abelian groups by", "Example \\ref{example-sheaf-product-pointwise-algebraic-structure}.", "Also, it is clear that the map $\\mathcal{F} \\to \\Pi(\\mathcal{F})$", "is a morphism of abelian presheaves. If we show that", "condition $(*)$ of Section \\ref{section-sheafification} defines a subgroup", "of $\\Pi(\\mathcal{F})(U)$ for all open subsets $U \\subset X$,", "then $\\mathcal{F}^\\#$ canonically inherits the structure of abelian sheaf.", "This is quite easy to do by hand, and we leave it to the", "reader to find a good simple argument. The argument we use here,", "which generalizes to presheaves of algebraic structures is the following:", "Lemma \\ref{lemma-diagram-fibre-product} show that", "$\\mathcal{F}^\\#(U)$ is the fibre product of a diagram", "of abelian groups. Thus $\\mathcal{F}^\\#$ is an abelian", "subgroup as desired.", "\\medskip\\noindent", "Note that at this point $\\mathcal{F}^\\#_x$ is an abelian", "group by Lemma \\ref{lemma-stalk-abelian-presheaf}", "and that $\\mathcal{F}_x \\to \\mathcal{F}^\\#_x$ is a", "bijection (Lemma \\ref{lemma-stalk-sheafification})", "and a homomorphism of abelian groups. Hence", "$\\mathcal{F}_x \\to \\mathcal{F}^\\#_x$ is an isomorphism", "of abelian groups. This will be used below without further mention.", "\\medskip\\noindent", "To prove the adjointness property we use the adjointness", "property of sheafification of presheaves of sets. For example", "if $\\psi : \\mathcal{F} \\to i(\\mathcal{G})$ is morphism of presheaves", "then we obtain a morphism of sheaves", "$\\psi' : \\mathcal{F}^\\# \\to \\mathcal{G}$. What we have to do is to check", "that this is a morphism of abelian sheaves. We may do this", "for example by noting that it is true on stalks,", "by Lemma \\ref{lemma-stalk-sheafification}, and then using", "Lemma \\ref{lemma-check-homomorphism-stalks} above." ], "refs": [ "sheaves-lemma-stalk-abelian-presheaf", "sheaves-lemma-diagram-fibre-product", "sheaves-lemma-stalk-abelian-presheaf", "sheaves-lemma-stalk-sheafification", "sheaves-lemma-stalk-sheafification", "sheaves-lemma-check-homomorphism-stalks" ], "ref_ids": [ 14483, 14497, 14483, 14494, 14494, 14492 ] } ], "ref_ids": [] }, { "id": 14499, "type": "theorem", "label": "sheaves-lemma-sheafify-presheaf-structures", "categories": [ "sheaves" ], "title": "sheaves-lemma-sheafify-presheaf-structures", "contents": [ "Let $X$ be a topological space.", "Let $(\\mathcal{C}, F)$ be a type of algebraic structure.", "Let $\\mathcal{F}$ be a presheaf with values in $\\mathcal{C}$", "on $X$. Then there exists a sheaf $\\mathcal{F}^\\#$ with values", "in $\\mathcal{C}$ and a morphism $\\mathcal{F} \\to \\mathcal{F}^\\#$", "of presheaves with values in $\\mathcal{C}$ with the", "following properties:", "\\begin{enumerate}", "\\item The map $\\mathcal{F} \\to \\mathcal{F}^\\#$ identifies", "the underlying sheaf of sets of $\\mathcal{F}^\\#$ with", "the sheafification of the underlying presheaf of sets of $\\mathcal{F}$.", "\\item For any morphism $\\mathcal{F} \\to \\mathcal{G}$, where", "$\\mathcal{G}$ is a sheaf with values in $\\mathcal{C}$ there exists", "a unique factorization $\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The proof is the same as the proof of", "Lemma \\ref{lemma-sheafify-abelian-presheaf},", "with repeated application of", "Lemma \\ref{lemma-image-contained-in} (see also", "Example \\ref{example-application-lemma-image-contained-in}).", "The main idea however, is to define $\\mathcal{F}^\\#(U)$", "as the fibre product in $\\mathcal{C}$ of the diagram", "$$", "\\xymatrix{", " &", "\\Pi(\\mathcal{F})(U) \\ar[d] \\\\", "\\prod_{x \\in U} \\mathcal{F}_x", "\\ar[r] &", "\\prod_{x \\in U} \\Pi(\\mathcal{F})_x", "}", "$$", "compare Lemma \\ref{lemma-diagram-fibre-product}." ], "refs": [ "sheaves-lemma-sheafify-abelian-presheaf", "sheaves-lemma-image-contained-in", "sheaves-lemma-diagram-fibre-product" ], "ref_ids": [ 14498, 14489, 14497 ] } ], "ref_ids": [] }, { "id": 14500, "type": "theorem", "label": "sheaves-lemma-sheafification-presheaf-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-sheafification-presheaf-modules", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{O}$ be a presheaf of rings on $X$.", "Let $\\mathcal{F}$ be a presheaf $\\mathcal{O}$-modules.", "Let $\\mathcal{O}^\\#$ be the sheafification of $\\mathcal{O}$.", "Let $\\mathcal{F}^\\#$ be the sheafification of $\\mathcal{F}$", "as a presheaf of abelian groups. There exists a map of", "sheaves of sets", "$$", "\\mathcal{O}^\\# \\times \\mathcal{F}^\\#", "\\longrightarrow", "\\mathcal{F}^\\#", "$$", "which makes the diagram", "$$", "\\xymatrix{", "\\mathcal{O} \\times \\mathcal{F} \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[d] \\\\", "\\mathcal{O}^\\# \\times \\mathcal{F}^\\# \\ar[r] &", "\\mathcal{F}^\\#", "}", "$$", "commute and which makes $\\mathcal{F}^\\#$ into a sheaf", "of $\\mathcal{O}^\\#$-modules. In addition, if $\\mathcal{G}$", "is a sheaf of $\\mathcal{O}^\\#$-modules, then any morphism", "of presheaves of $\\mathcal{O}$-modules $\\mathcal{F} \\to \\mathcal{G}$", "(into the restriction of $\\mathcal{G}$ to a $\\mathcal{O}$-module)", "factors uniquely as $\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$", "where $\\mathcal{F}^\\# \\to \\mathcal{G}$ is a morphism of", "$\\mathcal{O}^\\#$-modules." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14501, "type": "theorem", "label": "sheaves-lemma-adjointness-tensor-restrict", "categories": [ "sheaves" ], "title": "sheaves-lemma-adjointness-tensor-restrict", "contents": [ "With $X$, $\\mathcal{O}_1$, $\\mathcal{O}_2$, $\\mathcal{F}$ and", "$\\mathcal{G}$ as above there exists a canonical bijection", "$$", "\\Hom_{\\mathcal{O}_1}(\\mathcal{G}, \\mathcal{F}_{\\mathcal{O}_1})", "=", "\\Hom_{\\mathcal{O}_2}(", "\\mathcal{O}_2 \\otimes_{\\mathcal{O}_1} \\mathcal{G},", "\\mathcal{F}", ")", "$$", "In other words, the restriction and change of rings functors", "are adjoint to each other." ], "refs": [], "proofs": [ { "contents": [ "This follows from", "Lemma \\ref{lemma-adjointness-tensor-restrict-presheaves}", "and the fact that", "$\\Hom_{\\mathcal{O}_2}(", "\\mathcal{O}_2 \\otimes_{\\mathcal{O}_1} \\mathcal{G},", "\\mathcal{F}", ")", "=", "\\Hom_{\\mathcal{O}_2}(", "\\mathcal{O}_2 \\otimes_{p, \\mathcal{O}_1} \\mathcal{G},", "\\mathcal{F}", ")$", "because $\\mathcal{F}$ is a sheaf." ], "refs": [ "sheaves-lemma-adjointness-tensor-restrict-presheaves" ], "ref_ids": [ 14480 ] } ], "ref_ids": [] }, { "id": 14502, "type": "theorem", "label": "sheaves-lemma-stalk-tensor-sheaf-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-stalk-tensor-sheaf-modules", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{O} \\to \\mathcal{O}'$ be a morphism of", "sheaves of rings on $X$.", "Let $\\mathcal{F}$ be a sheaf $\\mathcal{O}$-modules.", "Let $x \\in X$. We have", "$$", "\\mathcal{F}_x \\otimes_{\\mathcal{O}_x} \\mathcal{O}'_x", "=", "(\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{O}')_x", "$$", "as $\\mathcal{O}'_x$-modules." ], "refs": [], "proofs": [ { "contents": [ "Follows directly from Lemma \\ref{lemma-stalk-tensor-presheaf-modules}", "and the fact that taking stalks commutes with sheafification." ], "refs": [ "sheaves-lemma-stalk-tensor-presheaf-modules" ], "ref_ids": [ 14486 ] } ], "ref_ids": [] }, { "id": 14503, "type": "theorem", "label": "sheaves-lemma-pushforward-sheaf", "categories": [ "sheaves" ], "title": "sheaves-lemma-pushforward-sheaf", "contents": [ "Let $f : X \\to Y$ be a continuous map.", "Let $\\mathcal{F}$ be a sheaf of sets on $X$.", "Then $f_*\\mathcal{F}$ is a sheaf on $Y$." ], "refs": [], "proofs": [ { "contents": [ "This immediately follows from the fact that", "if $V = \\bigcup V_j$ is an open covering in $Y$,", "then $f^{-1}(V) = \\bigcup f^{-1}(V_j)$ is an open covering in $X$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14504, "type": "theorem", "label": "sheaves-lemma-pushforward-composition", "categories": [ "sheaves" ], "title": "sheaves-lemma-pushforward-composition", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be continuous maps", "of topological spaces. The functors $(g \\circ f)_*$", "and $g_* \\circ f_*$ are equal (on both presheaves", "and sheaves of sets)." ], "refs": [], "proofs": [ { "contents": [ "This is because $(g \\circ f)_*\\mathcal{F}(W) =", "\\mathcal{F}((g \\circ f)^{-1}W)$ and", "$(g_* \\circ f_*)\\mathcal{F}(W) = \\mathcal{F}(f^{-1} g^{-1} W)$", "and $(g \\circ f)^{-1}W = f^{-1} g^{-1} W$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14505, "type": "theorem", "label": "sheaves-lemma-pullback-presheaves", "categories": [ "sheaves" ], "title": "sheaves-lemma-pullback-presheaves", "contents": [ "Let $f : X \\to Y$ be a continuous map.", "There exists a functor", "$f_p : \\textit{PSh}(Y) \\to \\textit{PSh}(X)$", "which is left adjoint to $f_*$. For a presheaf", "$\\mathcal{G}$ it is determined by the rule", "$$", "f_p\\mathcal{G}(U) = \\colim_{f(U) \\subset V} \\mathcal{G}(V)", "$$", "where the colimit is over the collection of open neighbourhoods", "$V$ of $f(U)$ in $Y$. The colimits are over", "directed partially ordered sets.", "(The restriction mappings of $f_p\\mathcal{G}$ are explained in the proof.)" ], "refs": [], "proofs": [ { "contents": [ "The colimit is over the partially ordered set consisting of open subsets", "$V \\subset Y$ which contain $f(U)$ with ordering by reverse inclusion.", "This is a directed partially ordered set, since if $V, V'$ are in it then", "so is $V \\cap V'$. Furthermore, if", "$U_1 \\subset U_2$, then every open neighbourhood of $f(U_2)$", "is an open neighbourhood of $f(U_1)$. Hence the system defining", "$f_p\\mathcal{G}(U_2)$ is a subsystem of the one defining", "$f_p\\mathcal{G}(U_1)$ and we obtain a restriction map (for", "example by applying the generalities in Categories,", "Lemma \\ref{categories-lemma-functorial-colimit}).", "\\medskip\\noindent", "Note that the construction of the colimit is clearly functorial", "in $\\mathcal{G}$, and similarly for the restriction mappings.", "Hence we have defined $f_p$ as a functor.", "\\medskip\\noindent", "A small useful remark is that there exists", "a canonical map $\\mathcal{G}(U) \\to f_p\\mathcal{G}(f^{-1}(U))$,", "because the system of open neighbourhoods", "of $f(f^{-1}(U))$ contains the element $U$. This is compatible", "with restriction mappings. In other words, there is a", "canonical map $i_\\mathcal{G} : \\mathcal{G} \\to f_* f_p \\mathcal{G}$.", "\\medskip\\noindent", "Let $\\mathcal{F}$ be a presheaf of sets on $X$.", "Suppose that $\\psi : f_p\\mathcal{G} \\to \\mathcal{F}$", "is a map of presheaves of sets. The corresponding map", "$\\mathcal{G} \\to f_*\\mathcal{F}$ is the map", "$f_*\\psi \\circ i_\\mathcal{G} :", "\\mathcal{G} \\to f_* f_p \\mathcal{G} \\to f_* \\mathcal{F}$.", "\\medskip\\noindent", "Another small useful remark is that there exists a", "canonical map $c_\\mathcal{F} : f_p f_* \\mathcal{F} \\to \\mathcal{F}$.", "Namely, let $U \\subset X$ open.", "For every open neighbourhood $V \\supset f(U)$ in $Y$", "there exists a map", "$f_*\\mathcal{F}(V) = \\mathcal{F}(f^{-1}(V))\\to \\mathcal{F}(U)$,", "namely the restriction map on $\\mathcal{F}$. And this is compatible", "with the restriction mappings between values of $\\mathcal{F}$", "on $f^{-1}$ of varying opens containing $f(U)$. Thus we obtain", "a canonical map $f_p f_* \\mathcal{F}(U) \\to \\mathcal{F}(U)$.", "Another trivial verification shows that these maps are compatible", "with restriction maps and define a map $c_\\mathcal{F}$", "of presheaves of sets.", "\\medskip\\noindent", "Suppose that $\\varphi : \\mathcal{G} \\to f_*\\mathcal{F}$", "is a map of presheaves of sets. Consider $f_p\\varphi :", "f_p \\mathcal{G} \\to f_p f_* \\mathcal{F}$.", "Postcomposing with $c_\\mathcal{F}$ gives the desired map", "$c_\\mathcal{F} \\circ f_p\\varphi : f_p\\mathcal{G} \\to \\mathcal{F}$.", "We omit the verification that this construction is inverse", "to the construction in the other direction given above." ], "refs": [ "categories-lemma-functorial-colimit" ], "ref_ids": [ 12210 ] } ], "ref_ids": [] }, { "id": 14506, "type": "theorem", "label": "sheaves-lemma-stalk-pullback-presheaf", "categories": [ "sheaves" ], "title": "sheaves-lemma-stalk-pullback-presheaf", "contents": [ "Let $f : X \\to Y$ be a continuous map.", "Let $x \\in X$. Let $\\mathcal{G}$ be a presheaf of sets on $Y$.", "There is a canonical bijection of stalks", "$(f_p\\mathcal{G})_x = \\mathcal{G}_{f(x)}$." ], "refs": [], "proofs": [ { "contents": [ "This you can see as follows", "\\begin{eqnarray*}", "(f_p\\mathcal{G})_x", "& = &", "\\colim_{x \\in U} f_p\\mathcal{G}(U) \\\\", "& = &", "\\colim_{x \\in U} \\colim_{f(U) \\subset V} \\mathcal{G}(V) \\\\", "& = &", "\\colim_{f(x) \\in V} \\mathcal{G}(V) \\\\", "& = &", "\\mathcal{G}_{f(x)}", "\\end{eqnarray*}", "Here we have used", "Categories, Lemma \\ref{categories-lemma-colimits-commute},", "and the fact that any $V$ open in $Y$ containing $f(x)$", "occurs in the third description above. Details omitted." ], "refs": [ "categories-lemma-colimits-commute" ], "ref_ids": [ 12212 ] } ], "ref_ids": [] }, { "id": 14507, "type": "theorem", "label": "sheaves-lemma-stalk-pullback", "categories": [ "sheaves" ], "title": "sheaves-lemma-stalk-pullback", "contents": [ "Let $x \\in X$. Let $\\mathcal{G}$ be a sheaf of sets on $Y$.", "There is a canonical bijection of stalks", "$(f^{-1}\\mathcal{G})_x = \\mathcal{G}_{f(x)}$." ], "refs": [], "proofs": [ { "contents": [ "This is a combination of Lemmas \\ref{lemma-stalk-sheafification}", "and \\ref{lemma-stalk-pullback-presheaf}." ], "refs": [ "sheaves-lemma-stalk-sheafification", "sheaves-lemma-stalk-pullback-presheaf" ], "ref_ids": [ 14494, 14506 ] } ], "ref_ids": [] }, { "id": 14508, "type": "theorem", "label": "sheaves-lemma-pullback-composition", "categories": [ "sheaves" ], "title": "sheaves-lemma-pullback-composition", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be continuous maps", "of topological spaces. The functors $(g \\circ f)^{-1}$", "and $f^{-1} \\circ g^{-1}$ are canonically isomorphic.", "Similarly $(g \\circ f)_p \\cong f_p \\circ g_p$ on", "presheaves." ], "refs": [], "proofs": [ { "contents": [ "To see this", "use that adjoint functors are unique up to unique isomorphism,", "and Lemma \\ref{lemma-pushforward-composition}." ], "refs": [ "sheaves-lemma-pushforward-composition" ], "ref_ids": [ 14504 ] } ], "ref_ids": [] }, { "id": 14509, "type": "theorem", "label": "sheaves-lemma-f-map", "categories": [ "sheaves" ], "title": "sheaves-lemma-f-map", "contents": [ "Let $f : X \\to Y$ be a continuous map.", "Let $\\mathcal{F}$ be a sheaf of sets on $X$ and", "let $\\mathcal{G}$ be a sheaf of sets on $Y$.", "There are canonical bijections between the following three sets:", "\\begin{enumerate}", "\\item The set of maps $\\mathcal{G} \\to f_*\\mathcal{F}$.", "\\item The set of maps $f^{-1}\\mathcal{G} \\to \\mathcal{F}$.", "\\item The set of $f$-maps $\\xi : \\mathcal{G} \\to \\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We leave the easy verification to the reader." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14510, "type": "theorem", "label": "sheaves-lemma-compose-f-maps-stalks", "categories": [ "sheaves" ], "title": "sheaves-lemma-compose-f-maps-stalks", "contents": [ "Suppose that $f : X \\to Y$ and $g : Y \\to Z$ are continuous", "maps of topological spaces. Suppose that $\\mathcal{F}$ is", "a sheaf on $X$, $\\mathcal{G}$ is a sheaf on $Y$, and", "$\\mathcal{H}$ is a sheaf on $Z$.", "Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be an $f$-map.", "Let $\\psi : \\mathcal{H} \\to \\mathcal{G}$ be an $g$-map.", "Let $x \\in X$ be a point. The map on stalks", "$(\\varphi \\circ \\psi)_x : \\mathcal{H}_{g(f(x))}", "\\to \\mathcal{F}_x$ is the composition", "$$", "\\mathcal{H}_{g(f(x))}", "\\xrightarrow{\\psi_{f(x)}}", "\\mathcal{G}_{f(x)}", "\\xrightarrow{\\varphi_x}", "\\mathcal{F}_x", "$$" ], "refs": [], "proofs": [ { "contents": [ "Immediate from Definition \\ref{definition-composition-f-maps}", "and the definition of the map on stalks above." ], "refs": [ "sheaves-definition-composition-f-maps" ], "ref_ids": [ 14574 ] } ], "ref_ids": [] }, { "id": 14511, "type": "theorem", "label": "sheaves-lemma-pullback-abelian-stalk", "categories": [ "sheaves" ], "title": "sheaves-lemma-pullback-abelian-stalk", "contents": [ "Let $f : X \\to Y$ be a continuous map.", "\\begin{enumerate}", "\\item Let $\\mathcal{G}$ be an abelian presheaf on $Y$.", "Let $x \\in X$. The bijection", "$\\mathcal{G}_{f(x)} \\to (f_p\\mathcal{G})_x$ of", "Lemma \\ref{lemma-stalk-pullback-presheaf} is an isomorphism of abelian groups.", "\\item Let $\\mathcal{G}$ be an abelian sheaf on $Y$.", "Let $x \\in X$. The bijection", "$\\mathcal{G}_{f(x)} \\to (f^{-1}\\mathcal{G})_x$ of", "Lemma \\ref{lemma-stalk-pullback} is an isomorphism of abelian groups.", "\\end{enumerate}" ], "refs": [ "sheaves-lemma-stalk-pullback-presheaf", "sheaves-lemma-stalk-pullback" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 14506, 14507 ] }, { "id": 14512, "type": "theorem", "label": "sheaves-lemma-f-map-sets-algebraic-structures", "categories": [ "sheaves" ], "title": "sheaves-lemma-f-map-sets-algebraic-structures", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Suppose given sheaves of algebraic structures", "$\\mathcal{F}$ on $X$, $\\mathcal{G}$ on $Y$. Let", "$\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be an $f$-map", "of underlying sheaves of sets. If for every $V \\subset Y$ open the", "map of sets $\\varphi_V : \\mathcal{G}(V) \\to \\mathcal{F}(f^{-1}V)$", "is the effect of a morphism in $\\mathcal{C}$ on underlying sets,", "then $\\varphi$ comes from a unique $f$-morphism between", "sheaves of algebraic structures." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14513, "type": "theorem", "label": "sheaves-lemma-pushforward-presheaf-module", "categories": [ "sheaves" ], "title": "sheaves-lemma-pushforward-presheaf-module", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Let $\\mathcal{O}$ be a presheaf of rings on $X$. Let", "$\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.", "There is a natural map of underlying presheaves of sets", "$$", "f_*\\mathcal{O} \\times f_*\\mathcal{F}", "\\longrightarrow", "f_*\\mathcal{F}", "$$", "which turns $f_*\\mathcal{F}$ into a presheaf of", "$f_*\\mathcal{O}$-modules. This construction is", "functorial in $\\mathcal{F}$." ], "refs": [], "proofs": [ { "contents": [ "Let $V \\subset Y$ is open. We define the map of the lemma", "to be the map", "$$", "f_*\\mathcal{O}(V) \\times f_*\\mathcal{F}(V)", "=", "\\mathcal{O}(f^{-1}V) \\times \\mathcal{F}(f^{-1}V)", "\\to", "\\mathcal{F}(f^{-1}V)", "=", "f_*\\mathcal{F}(V).", "$$", "Here the arrow in the middle is the multiplication map on $X$.", "We leave it to the reader to see this is compatible with", "restriction mappings and defines a structure of", "$f_*\\mathcal{O}$-module on $f_*\\mathcal{F}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14514, "type": "theorem", "label": "sheaves-lemma-pullback-presheaf-module", "categories": [ "sheaves" ], "title": "sheaves-lemma-pullback-presheaf-module", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Let $\\mathcal{O}$ be a presheaf of rings on $Y$. Let", "$\\mathcal{G}$ be a presheaf of $\\mathcal{O}$-modules.", "There is a natural map of underlying presheaves of sets", "$$", "f_p\\mathcal{O} \\times f_p\\mathcal{G}", "\\longrightarrow", "f_p\\mathcal{G}", "$$", "which turns $f_p\\mathcal{G}$ into a presheaf of $f_p\\mathcal{O}$-modules.", "This construction is functorial in $\\mathcal{G}$." ], "refs": [], "proofs": [ { "contents": [ "Let $U \\subset X$ is open. We define the map of the lemma", "to be the map", "\\begin{eqnarray*}", "f_p\\mathcal{O}(U) \\times f_p\\mathcal{G}(U)", "& = &", "\\colim_{f(U) \\subset V} \\mathcal{O}(V)", "\\times", "\\colim_{f(U) \\subset V} \\mathcal{G}(V) \\\\", "& = &", "\\colim_{f(U) \\subset V} (\\mathcal{O}(V)\\times \\mathcal{G}(V)) \\\\", "& \\to &", "\\colim_{f(U) \\subset V} \\mathcal{G}(V) \\\\", "& = &", "f_p\\mathcal{G}(U).", "\\end{eqnarray*}", "Here the arrow in the middle is the multiplication map on $Y$.", "The second equality holds because directed colimits commute", "with finite limits, see", "Categories, Lemma \\ref{categories-lemma-directed-commutes}.", "We leave it to the reader to see this is compatible with", "restriction mappings and defines a structure of", "$f_p\\mathcal{O}$-module on $f_p\\mathcal{G}$." ], "refs": [ "categories-lemma-directed-commutes" ], "ref_ids": [ 12228 ] } ], "ref_ids": [] }, { "id": 14515, "type": "theorem", "label": "sheaves-lemma-adjoint-push-pull-presheaves-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-adjoint-push-pull-presheaves-modules", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Let $\\mathcal{O}$ be a presheaf of rings on $Y$.", "Let $\\mathcal{G}$ be a presheaf of $\\mathcal{O}$-modules.", "Let $\\mathcal{F}$ be a presheaf of $f_p\\mathcal{O}$-modules.", "Then", "$$", "\\Mor_{\\textit{PMod}(f_p\\mathcal{O})}(f_p\\mathcal{G}, \\mathcal{F})", "=", "\\Mor_{\\textit{PMod}(\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).", "$$", "Here we use", "Lemmas \\ref{lemma-pullback-presheaf-module}", "and \\ref{lemma-pushforward-presheaf-module}, and we think of", "$f_*\\mathcal{F}$ as an $\\mathcal{O}$-module via the map", "$i_\\mathcal{O} : \\mathcal{O} \\to f_*f_p\\mathcal{O}$", "(defined first in the proof of Lemma \\ref{lemma-pullback-presheaves})." ], "refs": [ "sheaves-lemma-pullback-presheaf-module", "sheaves-lemma-pushforward-presheaf-module", "sheaves-lemma-pullback-presheaves" ], "proofs": [ { "contents": [ "Note that we have", "$$", "\\Mor_{\\textit{PAb}(X)}(f_p\\mathcal{G}, \\mathcal{F})", "=", "\\Mor_{\\textit{PAb}(Y)}(\\mathcal{G}, f_*\\mathcal{F}).", "$$", "according to Section \\ref{section-abelian-presheaves-functorial}.", "So what we have to prove is that under this correspondence, the", "subsets of module maps correspond. In addition, the correspondence", "is determined by the rule", "$$", "(\\psi : f_p\\mathcal{G} \\to \\mathcal{F})", "\\longmapsto", "(f_*\\psi \\circ i_\\mathcal{G} :", "\\mathcal{G} \\to f_* \\mathcal{F})", "$$", "and in the other direction by the rule", "$$", "(\\varphi : \\mathcal{G} \\to f_* \\mathcal{F})", "\\longmapsto", "(c_\\mathcal{F} \\circ f_p\\varphi : f_p\\mathcal{G} \\to \\mathcal{F})", "$$", "where $i_\\mathcal{G}$ and $c_\\mathcal{F}$ are as in", "Section \\ref{section-abelian-presheaves-functorial}.", "Hence, using the functoriality of $f_*$ and $f_p$ we see that", "it suffices to check that the maps", "$i_\\mathcal{G} : \\mathcal{G} \\to f_* f_p \\mathcal{G}$ and", "$c_\\mathcal{F} : f_p f_* \\mathcal{F} \\to \\mathcal{F}$", "are compatible with module structures, which we leave to the reader." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 14514, 14513, 14505 ] }, { "id": 14516, "type": "theorem", "label": "sheaves-lemma-adjoint-pull-push-presheaves-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-adjoint-pull-push-presheaves-modules", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Let $\\mathcal{O}$ be a presheaf of rings on $X$.", "Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.", "Let $\\mathcal{G}$ be a presheaf of $f_*\\mathcal{O}$-modules.", "Then", "$$", "\\Mor_{\\textit{PMod}(\\mathcal{O})}(", "\\mathcal{O} \\otimes_{p, f_pf_*\\mathcal{O}} f_p\\mathcal{G}, \\mathcal{F})", "=", "\\Mor_{\\textit{PMod}(f_*\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).", "$$", "Here we use", "Lemmas \\ref{lemma-pullback-presheaf-module}", "and \\ref{lemma-pushforward-presheaf-module}, and we use", "the map $c_\\mathcal{O} : f_pf_*\\mathcal{O} \\to \\mathcal{O}$", "in the definition of the tensor product." ], "refs": [ "sheaves-lemma-pullback-presheaf-module", "sheaves-lemma-pushforward-presheaf-module" ], "proofs": [ { "contents": [ "This follows from the equalities", "\\begin{eqnarray*}", "\\Mor_{\\textit{PMod}(\\mathcal{O})}(", "\\mathcal{O} \\otimes_{p, f_pf_*\\mathcal{O}} f_p\\mathcal{G}, \\mathcal{F})", "& = &", "\\Mor_{\\textit{PMod}(f_pf_*\\mathcal{O})}(", "f_p\\mathcal{G}, \\mathcal{F}_{f_pf_*\\mathcal{O}}) \\\\", "& = &", "\\Mor_{\\textit{PMod}(f_*\\mathcal{O})}(\\mathcal{G},", "f_*(\\mathcal{F}_{f_pf_*\\mathcal{O}})) \\\\", "& = &", "\\Mor_{\\textit{PMod}(f_*\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).", "\\end{eqnarray*}", "The first equality is", "Lemma \\ref{lemma-adjointness-tensor-restrict-presheaves}.", "The second equality is", "Lemma \\ref{lemma-adjoint-push-pull-presheaves-modules}.", "The third equality is given by the equality", "$f_*(\\mathcal{F}_{f_pf_*\\mathcal{O}}) = f_*\\mathcal{F}$", "of abelian sheaves which is $f_*\\mathcal{O}$-linear. Namely,", "$\\text{id}_{f_*\\mathcal{O}}$ corresponds to $c_\\mathcal{O}$", "under the adjunction described in the proof of", "Lemma \\ref{lemma-pullback-presheaves}", "and thus", "$\\text{id}_{f_*\\mathcal{O}} = f_*c_\\mathcal{O} \\circ i_{f_*\\mathcal{O}}$." ], "refs": [ "sheaves-lemma-adjointness-tensor-restrict-presheaves", "sheaves-lemma-adjoint-push-pull-presheaves-modules", "sheaves-lemma-pullback-presheaves" ], "ref_ids": [ 14480, 14515, 14505 ] } ], "ref_ids": [ 14514, 14513 ] }, { "id": 14517, "type": "theorem", "label": "sheaves-lemma-pushforward-module", "categories": [ "sheaves" ], "title": "sheaves-lemma-pushforward-module", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Let $\\mathcal{O}$ be a sheaf of rings on $X$. Let", "$\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.", "The pushforward $f_*\\mathcal{F}$, as defined in", "Lemma \\ref{lemma-pushforward-presheaf-module}", "is a sheaf of $f_*\\mathcal{O}$-modules." ], "refs": [ "sheaves-lemma-pushforward-presheaf-module" ], "proofs": [ { "contents": [ "Obvious from the definition and Lemma \\ref{lemma-pushforward-sheaf}." ], "refs": [ "sheaves-lemma-pushforward-sheaf" ], "ref_ids": [ 14503 ] } ], "ref_ids": [ 14513 ] }, { "id": 14518, "type": "theorem", "label": "sheaves-lemma-pullback-module", "categories": [ "sheaves" ], "title": "sheaves-lemma-pullback-module", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Let $\\mathcal{O}$ be a sheaf of rings on $Y$. Let", "$\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules.", "There is a natural map of underlying presheaves of sets", "$$", "f^{-1}\\mathcal{O} \\times f^{-1}\\mathcal{G}", "\\longrightarrow", "f^{-1}\\mathcal{G}", "$$", "which turns $f^{-1}\\mathcal{G}$ into a", "sheaf of $f^{-1}\\mathcal{O}$-modules." ], "refs": [], "proofs": [ { "contents": [ "Recall that $f^{-1}$ is defined as the composition of the", "functor $f_p$ and sheafification. Thus the lemma", "is a combination of Lemma \\ref{lemma-pullback-presheaf-module}", "and Lemma \\ref{lemma-sheafification-presheaf-modules}." ], "refs": [ "sheaves-lemma-pullback-presheaf-module", "sheaves-lemma-sheafification-presheaf-modules" ], "ref_ids": [ 14514, 14500 ] } ], "ref_ids": [] }, { "id": 14519, "type": "theorem", "label": "sheaves-lemma-adjoint-push-pull-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-adjoint-push-pull-modules", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Let $\\mathcal{O}$ be a sheaf of rings on $Y$.", "Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules.", "Let $\\mathcal{F}$ be a sheaf of $f^{-1}\\mathcal{O}$-modules.", "Then", "$$", "\\Mor_{\\textit{Mod}(f^{-1}\\mathcal{O})}(f^{-1}\\mathcal{G}, \\mathcal{F})", "=", "\\Mor_{\\textit{Mod}(\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).", "$$", "Here we use", "Lemmas \\ref{lemma-pullback-module}", "and \\ref{lemma-pushforward-module}, and we think of", "$f_*\\mathcal{F}$ as an $\\mathcal{O}$-module by restriction via", "$\\mathcal{O} \\to f_*f^{-1}\\mathcal{O}$." ], "refs": [ "sheaves-lemma-pullback-module", "sheaves-lemma-pushforward-module" ], "proofs": [ { "contents": [ "Argue by the equalities", "\\begin{eqnarray*}", "\\Mor_{\\textit{Mod}(f^{-1}\\mathcal{O})}(f^{-1}\\mathcal{G}, \\mathcal{F})", "& = &", "\\Mor_{\\textit{Mod}(f_p\\mathcal{O})}(f_p\\mathcal{G}, \\mathcal{F}) \\\\", "& = &", "\\Mor_{\\textit{Mod}(\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).", "\\end{eqnarray*}", "where the second is", "Lemmas \\ref{lemma-adjoint-push-pull-presheaves-modules}", "and the first is by Lemma \\ref{lemma-sheafification-presheaf-modules}." ], "refs": [ "sheaves-lemma-adjoint-push-pull-presheaves-modules", "sheaves-lemma-sheafification-presheaf-modules" ], "ref_ids": [ 14515, 14500 ] } ], "ref_ids": [ 14518, 14517 ] }, { "id": 14520, "type": "theorem", "label": "sheaves-lemma-adjoint-pull-push-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-adjoint-pull-push-modules", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Let $\\mathcal{O}$ be a sheaf of rings on $X$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.", "Let $\\mathcal{G}$ be a sheaf of $f_*\\mathcal{O}$-modules.", "Then", "$$", "\\Mor_{\\textit{Mod}(\\mathcal{O})}(", "\\mathcal{O} \\otimes_{f^{-1}f_*\\mathcal{O}} f^{-1}\\mathcal{G}, \\mathcal{F})", "=", "\\Mor_{\\textit{Mod}(f_*\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).", "$$", "Here we use", "Lemmas \\ref{lemma-pullback-module}", "and \\ref{lemma-pushforward-module}, and we use", "the canonical map $f^{-1}f_*\\mathcal{O} \\to \\mathcal{O}$", "in the definition of the tensor product." ], "refs": [ "sheaves-lemma-pullback-module", "sheaves-lemma-pushforward-module" ], "proofs": [ { "contents": [ "This follows from the equalities", "\\begin{eqnarray*}", "\\Mor_{\\textit{Mod}(\\mathcal{O})}(", "\\mathcal{O} \\otimes_{f^{-1}f_*\\mathcal{O}} f^{-1}\\mathcal{G}, \\mathcal{F})", "& = &", "\\Mor_{\\textit{Mod}(f^{-1}f_*\\mathcal{O})}(", "f^{-1}\\mathcal{G}, \\mathcal{F}_{f^{-1}f_*\\mathcal{O}}) \\\\", "& = &", "\\Mor_{\\textit{Mod}(f_*\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).", "\\end{eqnarray*}", "which are a combination of", "Lemma \\ref{lemma-adjointness-tensor-restrict}", "and \\ref{lemma-adjoint-push-pull-modules}." ], "refs": [ "sheaves-lemma-adjointness-tensor-restrict", "sheaves-lemma-adjoint-push-pull-modules" ], "ref_ids": [ 14501, 14519 ] } ], "ref_ids": [ 14518, 14517 ] }, { "id": 14521, "type": "theorem", "label": "sheaves-lemma-adjoint-pullback-pushforward-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-adjoint-pullback-pushforward-modules", "contents": [ "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules.", "There is a canonical bijection", "$$", "\\Hom_{\\mathcal{O}_X}(f^*\\mathcal{G}, \\mathcal{F})", "=", "\\Hom_{\\mathcal{O}_Y}(\\mathcal{G}, f_*\\mathcal{F}).", "$$", "In other words: the functor $f^*$ is the left adjoint to", "$f_*$." ], "refs": [], "proofs": [ { "contents": [ "This follows from the work we did before:", "\\begin{eqnarray*}", "\\Hom_{\\mathcal{O}_X}(f^*\\mathcal{G}, \\mathcal{F})", "& = &", "\\Mor_{\\textit{Mod}(\\mathcal{O}_X)}(", "\\mathcal{O}_X \\otimes_{f^{-1}\\mathcal{O}_Y} f^{-1}\\mathcal{G}, \\mathcal{F}) \\\\", "& = &", "\\Mor_{\\textit{Mod}(f^{-1}\\mathcal{O}_Y)}(", "f^{-1}\\mathcal{G}, \\mathcal{F}_{f^{-1}\\mathcal{O}_Y}) \\\\", "& = &", "\\Hom_{\\mathcal{O}_Y}(\\mathcal{G}, f_*\\mathcal{F}).", "\\end{eqnarray*}", "Here we use Lemmas \\ref{lemma-adjointness-tensor-restrict}", "and \\ref{lemma-adjoint-push-pull-modules}." ], "refs": [ "sheaves-lemma-adjointness-tensor-restrict", "sheaves-lemma-adjoint-push-pull-modules" ], "ref_ids": [ 14501, 14519 ] } ], "ref_ids": [] }, { "id": 14522, "type": "theorem", "label": "sheaves-lemma-push-pull-composition-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-push-pull-composition-modules", "contents": [ "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces.", "The functors $(g \\circ f)_*$ and $g_* \\circ f_*$ are equal.", "There is a canonical isomorphism of functors", "$(g \\circ f)^* \\cong f^* \\circ g^*$." ], "refs": [], "proofs": [ { "contents": [ "The result on pushforwards is a consequence of Lemma", "\\ref{lemma-pushforward-composition} and our definitions.", "The result on pullbacks follows from this by the same", "argument as in the proof of Lemma \\ref{lemma-pullback-composition}." ], "refs": [ "sheaves-lemma-pushforward-composition", "sheaves-lemma-pullback-composition" ], "ref_ids": [ 14504, 14508 ] } ], "ref_ids": [] }, { "id": 14523, "type": "theorem", "label": "sheaves-lemma-stalk-pullback-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-stalk-pullback-modules", "contents": [ "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces.", "Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules.", "Let $x \\in X$. Then", "$$", "(f^*\\mathcal{G})_x =", "\\mathcal{G}_{f(x)}", "\\otimes_{\\mathcal{O}_{Y, f(x)}}", "\\mathcal{O}_{X, x}", "$$", "as $\\mathcal{O}_{X, x}$-modules where the tensor product on the right", "uses $f^\\sharp_x : \\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$." ], "refs": [], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-stalk-tensor-sheaf-modules}", "and the identification of the stalks of pullback sheaves", "at $x$ with the corresponding stalks at $f(x)$. See the", "formulae in Section \\ref{section-presheaves-structures-functorial}", "for example." ], "refs": [ "sheaves-lemma-stalk-tensor-sheaf-modules" ], "ref_ids": [ 14502 ] } ], "ref_ids": [] }, { "id": 14524, "type": "theorem", "label": "sheaves-lemma-skyscraper-stalks", "categories": [ "sheaves" ], "title": "sheaves-lemma-skyscraper-stalks", "contents": [ "Let $X$ be a topological space, $x \\in X$ a point, and", "$A$ a set. For any point $x' \\in X$ the stalk of the", "skyscraper sheaf at $x$ with value $A$ at $x'$ is", "$$", "(i_{x, *}A)_{x'} =", "\\left\\{", "\\begin{matrix}", "A & \\text{if} & x' \\in \\overline{\\{x\\}} \\\\", "\\{*\\} & \\text{if} & x' \\not\\in \\overline{\\{x\\}}", "\\end{matrix}", "\\right.", "$$", "A similar description holds for the case of", "abelian groups, algebraic structures and", "sheaves of modules." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14525, "type": "theorem", "label": "sheaves-lemma-stalk-skyscraper-adjoint", "categories": [ "sheaves" ], "title": "sheaves-lemma-stalk-skyscraper-adjoint", "contents": [ "Let $X$ be a topological space, and let $x \\in X$ a point.", "The functors $\\mathcal{F} \\mapsto \\mathcal{F}_x$ and", "$A \\mapsto i_{x, *}A$ are adjoint. In a formula", "$$", "\\Mor_{\\textit{Sets}}(\\mathcal{F}_x, A)", "=", "\\Mor_{\\Sh(X)}(\\mathcal{F}, i_{x, *}A).", "$$", "A similar statement holds for the case of", "abelian groups, algebraic structures. In the case of", "sheaves of modules we have", "$$", "\\Hom_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x, A)", "=", "\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, i_{x, *}A).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: The stalk functor can be", "seen as the pullback functor for the morphism $i_x : \\{x\\} \\to X$.", "Then the adjointness follows from adjointness of", "$i_x^{-1}$ and $i_{x, *}$ (resp.\\ $i_x^*$ and $i_{x, *}$", "in the case of sheaves of modules)." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14526, "type": "theorem", "label": "sheaves-lemma-directed-colimits-sections", "categories": [ "sheaves" ], "title": "sheaves-lemma-directed-colimits-sections", "contents": [ "Let $X$ be a topological space. Let $I$ be a directed set.", "Let $(\\mathcal{F}_i, \\varphi_{ii'})$ be a system of sheaves of sets", "over $I$, see", "Categories, Section \\ref{categories-section-posets-limits}.", "Let $U \\subset X$ be an open subset.", "Consider the canonical map", "$$", "\\Psi :", "\\colim_i \\mathcal{F}_i(U)", "\\longrightarrow", "\\left(\\colim_i \\mathcal{F}_i\\right)(U)", "$$", "\\begin{enumerate}", "\\item If all the transition maps are injective then", "$\\Psi$ is injective for any open $U$.", "\\item If $U$ is quasi-compact, then $\\Psi$ is injective.", "\\item If $U$ is quasi-compact and all the transition maps are injective", "then $\\Psi$ is an isomorphism.", "\\item If $U$ has a cofinal system of open coverings", "$\\mathcal{U} : U = \\bigcup_{j\\in J} U_j$ with", "$J$ finite and $U_j \\cap U_{j'}$ quasi-compact", "for all $j, j' \\in J$, then $\\Psi$ is bijective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume all the transition maps are injective. In this case the presheaf", "$\\mathcal{F}' : V \\mapsto \\colim_i \\mathcal{F}_i(V)$ is", "separated (see Definition \\ref{definition-separated}).", "By the discussion above we have", "$(\\mathcal{F}')^\\# = \\colim_i \\mathcal{F}_i$.", "By Lemma \\ref{lemma-separated-presheaf-into-sheaf} we see that", "$\\mathcal{F}' \\to (\\mathcal{F}')^\\#$ is injective. This proves (1).", "\\medskip\\noindent", "Assume $U$ is quasi-compact.", "Suppose that $s \\in \\mathcal{F}_i(U)$ and", "$s' \\in \\mathcal{F}_{i'}(U)$ give rise to elements on", "the left hand side which have the same image under $\\Psi$.", "Since $U$ is quasi-compact this means there exists", "a finite open covering $U = \\bigcup_{j = 1, \\ldots, m} U_j$", "and for each $j$ an index $i_j \\in I$, $i_j \\geq i$, $i_j \\geq i'$", "such that $\\varphi_{ii_j}(s) = \\varphi_{i'i_j}(s')$.", "Let $i''\\in I$ be $\\geq$ than all of the $i_j$.", "We conclude that $\\varphi_{ii''}(s)$ and $\\varphi_{i'i''}(s)$", "agree on the opens $U_j$ for all $j$ and hence that", "$\\varphi_{ii''}(s) = \\varphi_{i'i''}(s)$. This proves (2).", "\\medskip\\noindent", "Assume $U$ is quasi-compact and all transition maps injective.", "Let $s$ be an element of the target of $\\Psi$.", "Since $U$ is quasi-compact", "there exists a finite open covering $U = \\bigcup_{j = 1, \\ldots, m} U_j$,", "for each $j$ an index $i_j \\in I$ and $s_j \\in \\mathcal{F}_{i_j}(U_j)$", "such that $s|_{U_j}$ comes from $s_j$ for all $j$.", "Pick $i \\in I$ which is $\\geq$ than all of the $i_j$.", "By (1) the sections $\\varphi_{i_ji}(s_j)$ agree over the", "overlaps $U_j \\cap U_{j'}$. Hence they glue to a section", "$s' \\in \\mathcal{F}_i(U)$ which maps to $s$ under $\\Psi$.", "This proves (3).", "\\medskip\\noindent", "Assume the hypothesis of (4). In particular we see that", "$U$ is quasi-compact and hence by (2) we have injectivity of $\\Psi$.", "Let $s$ be an element of the target of $\\Psi$.", "By assumption there exists a finite open covering", "$U = \\bigcup_{j = 1, \\ldots, m} U_j$, with $U_j \\cap U_{j'}$", "quasi-compact for all $j, j' \\in J$ and", "for each $j$ an index $i_j \\in I$ and $s_j \\in \\mathcal{F}_{i_j}(U_j)$", "such that $s|_{U_j}$ is the image of $s_j$ for all $j$.", "Since $U_j \\cap U_{j'}$ is quasi-compact we can apply (2)", "and we see that there exists an $i_{jj'} \\in I$,", "$i_{jj'} \\geq i_j$, $i_{jj'} \\geq i_{j'}$ such that", "$\\varphi_{i_ji_{jj'}}(s_j)$ and $\\varphi_{i_{j'}i_{jj'}}(s_{j'})$", "agree over $U_j \\cap U_{j'}$. Choose an index $i \\in I$", "wich is bigger or equal than all the $i_{jj'}$. Then we see that", "the sections $\\varphi_{i_ji}(s_j)$ of $\\mathcal{F}_i$ glue", "to a section of $\\mathcal{F}_i$ over $U$. This section is mapped", "to the element $s$ as desired." ], "refs": [ "sheaves-definition-separated", "sheaves-lemma-separated-presheaf-into-sheaf" ], "ref_ids": [ 14570, 14496 ] } ], "ref_ids": [] }, { "id": 14527, "type": "theorem", "label": "sheaves-lemma-compute-pullback-to-limit", "categories": [ "sheaves" ], "title": "sheaves-lemma-compute-pullback-to-limit", "contents": [ "In the situation described above, let $i \\in \\Ob(\\mathcal{I})$ and let", "$\\mathcal{G}$ be a sheaf on $X_i$. For $U_i \\subset X_i$", "quasi-compact open we have", "$$", "p_i^{-1}\\mathcal{G}(p_i^{-1}(U_i)) =", "\\colim_{a : j \\to i} f_a^{-1}\\mathcal{G}(f_a^{-1}(U_i))", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let us prove the canonical map", "$\\colim_{a : j \\to i} f_a^{-1}\\mathcal{G}(f_a^{-1}(U_i)) \\to", "p_i^{-1}\\mathcal{G}(p_i^{-1}(U_i))$ is injective.", "Let $s, s'$ be sections of $f_a^{-1}\\mathcal{G}$", "over $f_a^{-1}(U_i)$ for some $a : j \\to i$. For", "$b : k \\to j$ let $Z_k \\subset f_{a \\circ b}^{-1}(U_i)$", "be the closed subset of points $x$ such that the image of", "$s$ and $s'$ in the stalk $(f_{a \\circ b}^{-1}\\mathcal{G})_x$", "are different. If $Z_k$ is nonempty for all $b : k \\to j$, then by", "Topology, Lemma \\ref{topology-lemma-inverse-limit-spectral-spaces-nonempty}", "we see that $\\lim_{b : k \\to j} Z_k$ is nonempty too.", "Then for $x \\in \\lim_{b : k \\to j} Z_k \\subset X$", "(observe that $\\mathcal{I}/j \\to \\mathcal{I}$ is initial)", "we see that the image of $s$ and $s'$ in the stalk of", "$p_i^{-1}\\mathcal{G}$ at $x$ are different too since", "$(p_i^{-1}\\mathcal{G})_x = (f_{b \\circ a}^{-1}\\mathcal{G})_{p_k(x)}$", "for all $b : k \\to j$ as above. Thus if the images of $s$ and $s'$", "in $p_i^{-1}\\mathcal{G}(p_i^{-1}(U_i))$ are the same, then $Z_k$", "is empty for some $b : k \\to j$. This proves injectivity.", "\\medskip\\noindent", "Surjectivity. Let $s$ be a section of $p_i^{-1}\\mathcal{G}$ over", "$p_i^{-1}(U_i)$. By", "Topology, Lemma \\ref{topology-lemma-directed-inverse-limit-spectral-spaces}", "the set $p_i^{-1}(U_i)$ is a quasi-compact open of the spectral space $X$.", "By construction of the pullback sheaf, we can find an open covering", "$p_i^{-1}(U_i) = \\bigcup_{l \\in L} W_l$,", "opens $V_{l, i} \\subset X_i$,", "sections $s_{l, i} \\in \\mathcal{G}(V_{l, i})$", "such that $p_i(W_l) \\subset V_{l, i}$ and $p_i^{-1}s_{l, i}|_{W_l} = s|_{W_l}$.", "Because $X$ and $X_i$ are spectral and $p_i^{-1}(U_i)$ is quasi-compact open,", "we may assume $L$ is finite and $W_l$ and $V_{l, i}$ quasi-compact open", "for all $l$. Then we can apply", "Topology, Lemma \\ref{topology-lemma-descend-opens}", "to find $a : j \\to i$ and open covering", "$f_a^{-1}(U_i) = \\bigcup_{l \\in L} W_{l, j}$ by quasi-compact", "opens whose pullback to $X$ is the covering", "$p_i^{-1}(U_i) = \\bigcup_{l \\in L} W_l$", "and such that moreover $W_{l, j} \\subset f_a^{-1}(V_{l, i})$.", "Write $s_{l, j}$ the restriction of the pullback of $s_{l, i}$", "by $f_a$ to $W_{l, j}$. Then we see that $s_{l, j}$ and $s_{l', j}$", "restrict to elements of $(f_a^{-1}\\mathcal{G})(W_{l, j} \\cap W_{l', j})$", "which pullback to the same element $(p_i^{-1}\\mathcal{G})(W_l \\cap W_{l'})$,", "namely, the restriction of $s$. Hence by injectivity, we can find", "$b : k \\to j$ such that the sections $f_b^{-1}s_{l, j}$", "glue to a section over $f_{a \\circ b}^{-1}(U_i)$ as desired." ], "refs": [ "topology-lemma-inverse-limit-spectral-spaces-nonempty", "topology-lemma-directed-inverse-limit-spectral-spaces", "topology-lemma-descend-opens" ], "ref_ids": [ 8318, 8321, 8322 ] } ], "ref_ids": [] }, { "id": 14528, "type": "theorem", "label": "sheaves-lemma-descend-opens", "categories": [ "sheaves" ], "title": "sheaves-lemma-descend-opens", "contents": [ "In the situation described above, let $i \\in \\Ob(\\mathcal{I})$ and let", "$U_i \\subset X_i$ be a quasi-compact open. Then", "$$", "\\colim_{a : j \\to i} \\mathcal{F}_j(f_a^{-1}(U_i)) = \\mathcal{F}(p_i^{-1}(U_i))", "$$" ], "refs": [], "proofs": [ { "contents": [ "Recall that $p_i^{-1}(U_i)$ is a quasi-compact open of the spectral space", "$X$, see", "Topology, Lemma \\ref{topology-lemma-directed-inverse-limit-spectral-spaces}.", "Hence Lemma \\ref{lemma-directed-colimits-sections} applies and we have", "$$", "\\mathcal{F}(p_i^{-1}(U_i)) =", "\\colim_{a : j \\to i} p_j^{-1}\\mathcal{F}_j(p_i^{-1}(U_i)).", "$$", "A formal argument shows that", "$$", "\\colim_{a : j \\to i} \\mathcal{F}_j(f_a^{-1}(U_i)) =", "\\colim_{a : j \\to i} \\colim_{b : k \\to j}", "f_b^{-1}\\mathcal{F}_j(f_{a \\circ b}^{-1}(U_i))", "$$", "Thus it suffices to show that", "$$", "p_j^{-1}\\mathcal{F}_j(p_i^{-1}(U_i)) =", "\\colim_{b : k \\to j} f_b^{-1}\\mathcal{F}_j(f_{a \\circ b}^{-1}(U_i))", "$$", "This is Lemma \\ref{lemma-compute-pullback-to-limit}", "applied to $\\mathcal{F}_j$ and the quasi-compact open $f_a^{-1}(U_i)$." ], "refs": [ "topology-lemma-directed-inverse-limit-spectral-spaces", "sheaves-lemma-directed-colimits-sections", "sheaves-lemma-compute-pullback-to-limit" ], "ref_ids": [ 8321, 14526, 14527 ] } ], "ref_ids": [] }, { "id": 14529, "type": "theorem", "label": "sheaves-lemma-cofinal-systems-coverings", "categories": [ "sheaves" ], "title": "sheaves-lemma-cofinal-systems-coverings", "contents": [ "With notation as above.", "For each $U \\in \\mathcal{B}$, let $C(U) \\subset \\text{Cov}_\\mathcal{B}(U)$", "be a cofinal system. For each $U \\in \\mathcal{B}$, and each", "$\\mathcal{U} : U = \\bigcup U_i$ in $C(U)$, let coverings", "$\\mathcal{U}_{ij} : U_i \\cap U_j = \\bigcup U_{ijk}$,", "$U_{ijk} \\in \\mathcal{B}$ be given.", "Let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{B}$.", "The following are equivalent", "\\begin{enumerate}", "\\item The presheaf $\\mathcal{F}$ is a sheaf on $\\mathcal{B}$.", "\\item For every $U \\in \\mathcal{B}$ and every covering", "$\\mathcal{U} : U = \\bigcup U_i$ in $C(U)$ the sheaf condition", "$(**)$ holds (for the given coverings $\\mathcal{U}_{ij}$).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have to show that (2) implies (1).", "Suppose that $U \\in \\mathcal{B}$, and that", "$\\mathcal{U} : U = \\bigcup_{i\\in I} U_i$ is an arbitrary covering", "by elements of $\\mathcal{B}$. Because the system $C(U)$ is cofinal", "we can find an element $\\mathcal{V} : U = \\bigcup_{j \\in J} V_j$", "in $C(U)$ which refines $\\mathcal{U}$. This means there exists", "a map $\\alpha : J \\to I$ such that $V_j \\subset U_{\\alpha(j)}$.", "\\medskip\\noindent", "Note that if $s, s' \\in \\mathcal{F}(U)$ are sections such", "that $s|_{U_i} = s'|_{U_i}$, then", "$$", "s|_{V_j}", "= (s|_{U_{\\alpha(j)}})|_{V_j}", "= (s'|_{U_{\\alpha(j)}})|_{V_j}", "= s'|_{V_j}", "$$", "for all $j$. Hence by the uniqueness in $(**)$", "for the covering $\\mathcal{V}$ we conclude that $s = s'$.", "Thus we have proved the uniqueness part of $(**)$", "for our arbitrary covering $\\mathcal{U}$.", "\\medskip\\noindent", "Suppose furthermore that $U_i \\cap U_{i'} = \\bigcup_{k \\in I_{ii'}} U_{ii'k}$", "are arbitrary coverings by $U_{ii'k} \\in \\mathcal{B}$.", "Let us try to prove the existence part of $(**)$ for the system", "$(\\mathcal{U}, \\mathcal{U}_{ij})$. Thus let $s_i \\in \\mathcal{F}(U_i)$", "and suppose we have", "$$", "s_i|_{U_{ijk}} = s_{i'}|_{U_{ii'k}}", "$$", "for all $i, i', k$. Set $t_j = s_{\\alpha(j)}|_{V_j}$, where $\\mathcal{V}$", "and $\\alpha$ are as above.", "\\medskip\\noindent", "There is one small kink in the argument here. Namely, let", "$\\mathcal{V}_{jj'} : V_j \\cap V_{j'} = \\bigcup_{l \\in J_{jj'}} V_{jj'l}$", "be the covering given to us by the statement of the lemma.", "It is not a priori clear that", "$$", "t_j|_{V_{jj'l}} = t_{j'}|_{V_{jj'l}}", "$$", "for all $j, j', l$. To see this, note that we do have", "$$", "t_j|_W = t_{j'}|_W \\text{ for all } W \\in \\mathcal{B},", "W \\subset V_{jj'l} \\cap U_{\\alpha(j)\\alpha(j')k}", "$$", "for all $k \\in I_{\\alpha(j)\\alpha(j')}$, by our assumption on", "the family of elements $s_i$. And since", "$V_j \\cap V_{j'} \\subset U_{\\alpha(j)} \\cap U_{\\alpha(j')}$", "we see that $t_j|_{V_{jj'l}}$ and $t_{j'}|_{V_{jj'l}}$", "agree on the members of a covering of $V_{jj'l}$ by", "elements of $\\mathcal{B}$. Hence by the uniqueness part proved above", "we finally deduce the desired equality of", "$t_j|_{V_{jj'l}}$ and $t_{j'}|_{V_{jj'l}}$.", "Then we get the existence of an element $t \\in \\mathcal{F}(U)$", "by property $(**)$ for $(\\mathcal{V}, \\mathcal{V}_{jj'})$.", "\\medskip\\noindent", "Again there is a small snag. We know that $t$ restricts to $t_j$ on $V_j$", "but we do not yet know that $t$ restricts to $s_i$ on $U_i$. To conclude", "this note that the sets $U_i \\cap V_j$, $j \\in J$ cover $U_i$. Hence", "also the sets $U_{i \\alpha(j) k} \\cap V_j$, $j\\in J$, $k \\in I_{i\\alpha(j)}$", "cover $U_i$. We leave it to the reader to see that $t$ and $s_i$ restrict", "to the same section of $\\mathcal{F}$ on any $W \\in \\mathcal{B}$", "which is contained in one of the open sets", "$U_{i \\alpha(j) k} \\cap V_j$, $j\\in J$, $k \\in I_{i\\alpha(j)}$.", "Hence by the uniqueness part seen above we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14530, "type": "theorem", "label": "sheaves-lemma-cofinal-systems-coverings-standard-case", "categories": [ "sheaves" ], "title": "sheaves-lemma-cofinal-systems-coverings-standard-case", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the", "topology on $X$. Assume that for every triple $U, U', U'' \\in \\mathcal{B}$", "with $U' \\subset U$ and $U'' \\subset U$ we have $U' \\cap U'' \\in \\mathcal{B}$.", "For each $U \\in \\mathcal{B}$, let $C(U) \\subset \\text{Cov}_\\mathcal{B}(U)$", "be a cofinal system.", "Let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{B}$.", "The following are equivalent", "\\begin{enumerate}", "\\item The presheaf $\\mathcal{F}$ is a sheaf on $\\mathcal{B}$.", "\\item For every $U \\in \\mathcal{B}$ and every covering", "$\\mathcal{U} : U = \\bigcup U_i$ in $C(U)$ and for every", "family of sections $s_i \\in \\mathcal{F}(U_i)$ such", "that $s_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}$ there", "exists a unique section $s \\in \\mathcal{F}(U)$ which", "restricts to $s_i$ on $U_i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of", "Lemma \\ref{lemma-cofinal-systems-coverings} above", "in the special case where the coverings $\\mathcal{U}_{ij}$", "each consist of a single element. But also this case is much", "easier and is an easy exercise to do directly." ], "refs": [ "sheaves-lemma-cofinal-systems-coverings" ], "ref_ids": [ 14529 ] } ], "ref_ids": [] }, { "id": 14531, "type": "theorem", "label": "sheaves-lemma-condition-star-sections", "categories": [ "sheaves" ], "title": "sheaves-lemma-condition-star-sections", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{B}$ be a basis for the topology on $X$.", "Let $U \\in \\mathcal{B}$.", "Let $\\mathcal{F}$ be a sheaf of sets on $\\mathcal{B}$.", "The map", "$$", "\\mathcal{F}(U) \\to \\prod\\nolimits_{x \\in U} \\mathcal{F}_x", "$$", "identifies $\\mathcal{F}(U)$ with the elements $(s_x)_{x\\in U}$", "with the property", "\\begin{itemize}", "\\item[(*)] For any $x \\in U$ there exists a $V \\in \\mathcal{B}$,", "with $x \\in V \\subset U$ and a section $\\sigma \\in \\mathcal{F}(V)$", "such that for all $y \\in V$ we have $s_y = (V, \\sigma)$ in $\\mathcal{F}_y$.", "\\end{itemize}" ], "refs": [], "proofs": [ { "contents": [ "First note that the map", "$\\mathcal{F}(U) \\to \\prod\\nolimits_{x \\in U} \\mathcal{F}_x$", "is injective by the uniqueness in the sheaf condition", "of Definition \\ref{definition-sheaf-basis}. Let $(s_x)$ be", "any element on the right hand side which satisfies $(*)$.", "Clearly this means we can find a covering $U = \\bigcup U_i$,", "$U_i \\in \\mathcal{B}$ such that $(s_x)_{x \\in U_i}$ comes from", "certain $\\sigma_i \\in \\mathcal{F}(U_i)$. For every $y \\in U_i \\cap U_j$", "the sections $\\sigma_i$ and $\\sigma_j$ agree in the stalk", "$\\mathcal{F}_y$. Hence there exists an element $V_{ijy} \\in \\mathcal{B}$,", "$y \\in V_{ijy}$ such that $\\sigma_i|_{V_{ijy}} = \\sigma_j|_{V_{ijy}}$.", "Thus the sheaf condition $(**)$ of Definition \\ref{definition-sheaf-basis}", "applies to the system of $\\sigma_i$ and we obtain a section", "$s \\in \\mathcal{F}(U)$ with the desired property." ], "refs": [ "sheaves-definition-sheaf-basis", "sheaves-definition-sheaf-basis" ], "ref_ids": [ 14580, 14580 ] } ], "ref_ids": [] }, { "id": 14532, "type": "theorem", "label": "sheaves-lemma-extend-off-basis", "categories": [ "sheaves" ], "title": "sheaves-lemma-extend-off-basis", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{B}$ be a basis for the topology on $X$.", "Let $\\mathcal{F}$ be a sheaf of sets on $\\mathcal{B}$.", "There exists a unique sheaf of sets $\\mathcal{F}^{ext}$", "on $X$ such that $\\mathcal{F}^{ext}(U) = \\mathcal{F}(U)$", "for all $U \\in \\mathcal{B}$ compatibly with the restriction", "mappings." ], "refs": [], "proofs": [ { "contents": [ "We first construct a presheaf $\\mathcal{F}^{ext}$ with the", "desired property. Namely, for an arbitrary open $U \\subset X$ we", "define $\\mathcal{F}^{ext}(U)$ as the set of elements", "$(s_x)_{x \\in U}$ such that $(*)$ of", "Lemma \\ref{lemma-condition-star-sections} holds.", "It is clear that there are restriction mappings", "that turn $\\mathcal{F}^{ext}$ into a presheaf of sets.", "Also, by Lemma \\ref{lemma-condition-star-sections} we", "see that $\\mathcal{F}(U) = \\mathcal{F}^{ext}(U)$", "whenever $U$ is an element of the basis $\\mathcal{B}$.", "To see $\\mathcal{F}^{ext}$ is a sheaf one may", "argue as in the proof of Lemma \\ref{lemma-sheafification-sheaf}." ], "refs": [ "sheaves-lemma-condition-star-sections", "sheaves-lemma-condition-star-sections", "sheaves-lemma-sheafification-sheaf" ], "ref_ids": [ 14531, 14531, 14493 ] } ], "ref_ids": [] }, { "id": 14533, "type": "theorem", "label": "sheaves-lemma-restrict-basis-equivalence", "categories": [ "sheaves" ], "title": "sheaves-lemma-restrict-basis-equivalence", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{B}$ be a basis for the topology on $X$.", "Denote $\\Sh(\\mathcal{B})$ the category of", "sheaves on $\\mathcal{B}$.", "There is an equivalence of categories", "$$", "\\Sh(X) \\longrightarrow \\Sh(\\mathcal{B})", "$$", "which assigns to a sheaf on $X$ its restriction to", "the members of $\\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "The inverse functor in given in Lemma \\ref{lemma-extend-off-basis} above.", "Checking the obvious functorialities is left to the", "reader." ], "refs": [ "sheaves-lemma-extend-off-basis" ], "ref_ids": [ 14532 ] } ], "ref_ids": [] }, { "id": 14534, "type": "theorem", "label": "sheaves-lemma-extend-off-basis-structures", "categories": [ "sheaves" ], "title": "sheaves-lemma-extend-off-basis-structures", "contents": [ "Let $X$ be a topological space. Let $(\\mathcal{C}, F)$ be", "a type of algebraic structure.", "Let $\\mathcal{B}$ be a basis for the topology on $X$.", "Let $\\mathcal{F}$ be a sheaf with values in $\\mathcal{C}$", "on $\\mathcal{B}$.", "There exists a unique sheaf $\\mathcal{F}^{ext}$ with values in $\\mathcal{C}$", "on $X$ such that $\\mathcal{F}^{ext}(U) = \\mathcal{F}(U)$", "for all $U \\in \\mathcal{B}$ compatibly with the restriction", "mappings." ], "refs": [], "proofs": [ { "contents": [ "By the conditions imposed on the pair $(\\mathcal{C}, F)$ it", "suffices to come up with a presheaf $\\mathcal{F}^{ext}$", "which does the correct thing on the level of underlying", "presheaves of sets. Thus our first task is to construct", "a suitable object $\\mathcal{F}^{ext}(U)$ for all open $U \\subset X$.", "We could do this by imitating", "Lemma \\ref{lemma-diagram-fibre-product} in the setting", "of presheaves on $\\mathcal{B}$. However, a slightly different method", "(but basically equivalent) is the following:", "Define it as the directed colimit", "$$", "\\mathcal{F}^{ext}(U)", ":=", "\\colim_\\mathcal{U} FIB(\\mathcal{U})", "$$", "over all coverings", "$\\mathcal{U} : U = \\bigcup_{i\\in I} U_i$ by $U_i \\in \\mathcal{B}$", "of the fibre product", "$$", "\\xymatrix{", "FIB(\\mathcal{U}) \\ar[r] \\ar[d] &", "\\prod\\nolimits_{x\\in U} \\mathcal{F}_x \\ar[d] \\\\", "\\prod\\nolimits_{i\\in I} \\mathcal{F}(U_i) \\ar[r] &", "\\prod\\nolimits_{i \\in I} \\prod\\nolimits_{x\\in U_i} \\mathcal{F}_x", "}", "$$", "By the usual arguments, see Lemma \\ref{lemma-image-contained-in}", "and Example \\ref{example-application-lemma-image-contained-in}", "it suffices to show that this construction on underlying", "sets is the same as the definition using $(**)$ above.", "Details left to the reader." ], "refs": [ "sheaves-lemma-diagram-fibre-product", "sheaves-lemma-image-contained-in" ], "ref_ids": [ 14497, 14489 ] } ], "ref_ids": [] }, { "id": 14535, "type": "theorem", "label": "sheaves-lemma-restrict-basis-equivalence-structures", "categories": [ "sheaves" ], "title": "sheaves-lemma-restrict-basis-equivalence-structures", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{B}$ be a basis for the topology on $X$.", "Let $(\\mathcal{C}, F)$ be a type of algebraic structure.", "Denote $\\Sh(\\mathcal{B}, \\mathcal{C})$ the category of", "sheaves with values in $\\mathcal{C}$ on $\\mathcal{B}$.", "There is an equivalence of categories", "$$", "\\Sh(X, \\mathcal{C})", "\\longrightarrow", "\\Sh(\\mathcal{B}, \\mathcal{C})", "$$", "which assigns to a sheaf on $X$ its restriction to", "the members of $\\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "The inverse functor in given in", "Lemma \\ref{lemma-extend-off-basis-structures} above.", "Checking the obvious functorialities is left to the", "reader." ], "refs": [ "sheaves-lemma-extend-off-basis-structures" ], "ref_ids": [ 14534 ] } ], "ref_ids": [] }, { "id": 14536, "type": "theorem", "label": "sheaves-lemma-extend-off-basis-module", "categories": [ "sheaves" ], "title": "sheaves-lemma-extend-off-basis-module", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{B}$ be a basis for the topology on $X$.", "Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{B}$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules", "on $\\mathcal{B}$. Let $\\mathcal{O}^{ext}$ be the sheaf", "of rings on $X$ extending $\\mathcal{O}$ and let", "$\\mathcal{F}^{ext}$ be the abelian sheaf on $X$ extending", "$\\mathcal{F}$, see Lemma \\ref{lemma-extend-off-basis-structures}.", "There exists a canonical map", "$$", "\\mathcal{O}^{ext} \\times \\mathcal{F}^{ext}", "\\longrightarrow", "\\mathcal{F}^{ext}", "$$", "which agrees with the given map over elements of $\\mathcal{B}$", "and which endows $\\mathcal{F}^{ext}$ with the structure", "of an $\\mathcal{O}^{ext}$-module." ], "refs": [ "sheaves-lemma-extend-off-basis-structures" ], "proofs": [ { "contents": [ "It suffices to construct the multiplication map", "on the level of presheaves of sets. Perhaps the easiest", "way to see this is to prove directly that if", "$(f_x)_{x \\in U}$, $f_x \\in \\mathcal{O}_x$", "and", "$(m_x)_{x \\in U}$, $m_x \\in \\mathcal{F}_x$", "satisfy $(*)$, then the element", "$(f_xm_x)_{x \\in U}$ also satisfies $(*)$.", "Then we get the desired result, because in the proof", "of Lemma \\ref{lemma-extend-off-basis} we construct the extension", "in terms of families of elements of stalks satisfying $(*)$." ], "refs": [ "sheaves-lemma-extend-off-basis" ], "ref_ids": [ 14532 ] } ], "ref_ids": [ 14534 ] }, { "id": 14537, "type": "theorem", "label": "sheaves-lemma-restrict-basis-equivalence-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-restrict-basis-equivalence-modules", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{B}$ be a basis for the topology on $X$.", "Let $\\mathcal{O}$ be a sheaf of rings on $X$.", "Denote $\\textit{Mod}(\\mathcal{O}|_\\mathcal{B})$ the category of", "sheaves of $\\mathcal{O}|_\\mathcal{B}$-modules on $\\mathcal{B}$.", "There is an equivalence of categories", "$$", "\\textit{Mod}(\\mathcal{O})", "\\longrightarrow", "\\textit{Mod}(\\mathcal{O}|_\\mathcal{B})", "$$", "which assigns to a sheaf of $\\mathcal{O}$-modules on $X$ its restriction to", "the members of $\\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "The inverse functor in given in", "Lemma \\ref{lemma-extend-off-basis-module} above.", "Checking the obvious functorialities is left to the reader." ], "refs": [ "sheaves-lemma-extend-off-basis-module" ], "ref_ids": [ 14536 ] } ], "ref_ids": [] }, { "id": 14538, "type": "theorem", "label": "sheaves-lemma-f-map-basis-below-structures", "categories": [ "sheaves" ], "title": "sheaves-lemma-f-map-basis-below-structures", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Let $(\\mathcal{C}, F)$ be a type of algebraic structures.", "Let $\\mathcal{F}$ be a sheaf with values in $\\mathcal{C}$ on $X$.", "Let $\\mathcal{G}$ be a sheaf with values in $\\mathcal{C}$ on $Y$.", "Let $\\mathcal{B}$ be a basis for the topology on $Y$.", "Suppose given for every $V \\in \\mathcal{B}$ a morphism", "$$", "\\varphi_V :", "\\mathcal{G}(V)", "\\longrightarrow", "\\mathcal{F}(f^{-1}V)", "$$", "of $\\mathcal{C}$ compatible with restriction mappings.", "Then there is a unique $f$-map (see Definition \\ref{definition-f-map}", "and discussion of $f$-maps in", "Section \\ref{section-presheaves-structures-functorial})", "$\\varphi : \\mathcal{G} \\to \\mathcal{F}$", "recovering $\\varphi_V$ for $V \\in \\mathcal{B}$." ], "refs": [ "sheaves-definition-f-map" ], "proofs": [ { "contents": [ "This is trivial because the collection of maps", "amounts to a morphism between the restrictions", "of $\\mathcal{G}$ and $f_*\\mathcal{F}$ to $\\mathcal{B}$.", "By Lemma \\ref{lemma-restrict-basis-equivalence-structures}", "this is the same as giving a morphism from $\\mathcal{G}$", "to $f_*\\mathcal{F}$, which by Lemma \\ref{lemma-f-map}", "is the same as an $f$-map. See also", "Lemma \\ref{lemma-f-map-sets-algebraic-structures}", "and the discussion preceding it", "for how to deal with the case of sheaves of algebraic structures." ], "refs": [ "sheaves-lemma-restrict-basis-equivalence-structures", "sheaves-lemma-f-map", "sheaves-lemma-f-map-sets-algebraic-structures" ], "ref_ids": [ 14535, 14509, 14512 ] } ], "ref_ids": [ 14573 ] }, { "id": 14539, "type": "theorem", "label": "sheaves-lemma-f-map-basis-below-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-f-map-basis-below-modules", "contents": [ "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules.", "Let $\\mathcal{B}$ be a basis for the topology on $Y$.", "Suppose given for every $V \\in \\mathcal{B}$ a", "$\\mathcal{O}_Y(V)$-module map", "$$", "\\varphi_V :", "\\mathcal{G}(V)", "\\longrightarrow", "\\mathcal{F}(f^{-1}V)", "$$", "(where $\\mathcal{F}(f^{-1}V)$ has a module structure using", "$f^\\sharp_V : \\mathcal{O}_Y(V) \\to \\mathcal{O}_X(f^{-1}V)$)", "compatible with restriction mappings.", "Then there is a unique $f$-map (see discussion of $f$-maps in", "Section \\ref{section-ringed-spaces-functoriality-modules})", "$\\varphi : \\mathcal{G} \\to \\mathcal{F}$", "recovering $\\varphi_V$ for $V \\in \\mathcal{B}$." ], "refs": [], "proofs": [ { "contents": [ "Same as the proof of the corresponding lemma", "for sheaves of algebraic structures above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14540, "type": "theorem", "label": "sheaves-lemma-f-map-basis-above-and-below-structures", "categories": [ "sheaves" ], "title": "sheaves-lemma-f-map-basis-above-and-below-structures", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "Let $(\\mathcal{C}, F)$ be a type of algebraic structures.", "Let $\\mathcal{F}$ be a sheaf with values in $\\mathcal{C}$ on $X$.", "Let $\\mathcal{G}$ be a sheaf with values in $\\mathcal{C}$ on $Y$.", "Let $\\mathcal{B}_Y$ be a basis for the topology on $Y$.", "Let $\\mathcal{B}_X$ be a basis for the topology on $X$.", "Suppose given for every $V \\in \\mathcal{B}_Y$, and", "$U \\in \\mathcal{B}_X$ such that $f(U) \\subset V$ a morphism", "$$", "\\varphi_V^U :", "\\mathcal{G}(V)", "\\longrightarrow", "\\mathcal{F}(U)", "$$", "of $\\mathcal{C}$ compatible with restriction mappings.", "Then there is a unique $f$-map (see", "Definition \\ref{definition-f-map} and the discussion", "of $f$-maps in Section \\ref{section-presheaves-structures-functorial})", "$\\varphi : \\mathcal{G} \\to \\mathcal{F}$", "recovering $\\varphi_V^U$ as the composition", "$$", "\\mathcal{G}(V) \\xrightarrow{\\varphi_V}", "\\mathcal{F}(f^{-1}(V)) \\xrightarrow{\\text{restr.}}", "\\mathcal{F}(U)", "$$", "for every pair $(U, V)$ as above." ], "refs": [ "sheaves-definition-f-map" ], "proofs": [ { "contents": [ "Let us first proves this for sheaves of sets.", "Fix $V \\subset Y$ open. Pick $s \\in \\mathcal{G}(V)$.", "We are going to construct an element", "$\\varphi_V(s) \\in \\mathcal{F}(f^{-1}V)$.", "We can define a value $\\varphi(s)_x$ in the stalk $\\mathcal{F}_x$", "for every $x \\in f^{-1}V$ by picking a $U \\in \\mathcal{B}_X$", "with $x \\in U \\subset f^{-1}V$ and setting $\\varphi(s)_x$", "equal to the equivalence class of $(U, \\varphi_V^U(s))$", "in the stalk. Clearly, the family $(\\varphi(s)_x)_{x \\in f^{-1}V}$", "satisfies condition $(*)$ because the maps $\\varphi_V^U$", "for varying $U$", "are compatible with restrictions in the sheaf $\\mathcal{F}$.", "Thus, by the proof of Lemma \\ref{lemma-extend-off-basis}", "we see that $(\\varphi(s)_x)_{x \\in f^{-1}V}$ corresponds", "to a unique element $\\varphi_V(s)$ of $\\mathcal{F}(f^{-1}V)$.", "Thus we have defined a set map", "$\\varphi_V : \\mathcal{G}(V) \\to \\mathcal{F}(f^{-1}V)$.", "The compatibility between $\\varphi_V$ and $\\varphi_V^U$", "follows from Lemma \\ref{lemma-condition-star-sections}.", "\\medskip\\noindent", "We leave it to the reader to show that the construction", "of $\\varphi_V$ is compatible with restriction mappings as we vary", "$v \\in \\mathcal{B}_Y$. Thus we may apply Lemma", "\\ref{lemma-f-map-basis-below-structures} above to", "``glue'' them to the desired $f$-map.", "\\medskip\\noindent", "Finally, we note that the map of sheaves of sets so constructed", "satisfies the property that the map on stalks", "$$", "\\mathcal{G}_{f(x)} \\longrightarrow \\mathcal{F}_x", "$$", "is the colimit of the system of maps $\\varphi_V^U$ as", "$V \\in \\mathcal{B}_Y$ varies over those elements that", "contain $f(x)$ and $U \\in \\mathcal{B}_X$ varies over those elements that", "contain $x$. In particular, if $\\mathcal{G}$ and $\\mathcal{F}$", "are the underlying sheaves of sets of sheaves of algebraic structures,", "then we see that the maps on stalks is a morphism of algebraic", "structures. Hence we conclude that the associated map of", "sheaves of underlying sets $f^{-1}\\mathcal{G} \\to \\mathcal{F}$", "satisfies the assumptions of", "Lemma \\ref{lemma-f-map-sets-algebraic-structures}.", "We conclude that $f^{-1}\\mathcal{G} \\to \\mathcal{F}$", "is a morphism of sheaves with values in $\\mathcal{C}$.", "And by adjointness this means that $\\varphi$ is", "an $f$-map of sheaves of algebraic structures." ], "refs": [ "sheaves-lemma-extend-off-basis", "sheaves-lemma-condition-star-sections", "sheaves-lemma-f-map-basis-below-structures", "sheaves-lemma-f-map-sets-algebraic-structures" ], "ref_ids": [ 14532, 14531, 14538, 14512 ] } ], "ref_ids": [ 14573 ] }, { "id": 14541, "type": "theorem", "label": "sheaves-lemma-f-map-basis-above-and-below-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-f-map-basis-above-and-below-modules", "contents": [ "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules.", "Let $\\mathcal{B}_Y$ be a basis for the topology on $Y$.", "Let $\\mathcal{B}_X$ be a basis for the topology on $X$.", "Suppose given for every $V \\in \\mathcal{B}_Y$, and", "$U \\in \\mathcal{B}_X$ such that $f(U) \\subset V$ a", "$\\mathcal{O}_Y(V)$-module map", "$$", "\\varphi_V^U :", "\\mathcal{G}(V)", "\\longrightarrow", "\\mathcal{F}(U)", "$$", "compatible with restriction mappings. Here the", "$\\mathcal{O}_Y(V)$-module structure on $\\mathcal{F}(U)$", "comes from the $\\mathcal{O}_X(U)$-module structure", "via the map $f^\\sharp_V : \\mathcal{O}_Y(V)", "\\to \\mathcal{O}_X(f^{-1}V) \\to \\mathcal{O}_X(U)$.", "Then there is a unique $f$-map of sheaves of modules (see", "Definition \\ref{definition-f-map} and the discussion", "of $f$-maps in Section \\ref{section-ringed-spaces-functoriality-modules})", "$\\varphi : \\mathcal{G} \\to \\mathcal{F}$", "recovering $\\varphi_V^U$ as the composition", "$$", "\\mathcal{G}(V) \\xrightarrow{\\varphi_V}", "\\mathcal{F}(f^{-1}(V)) \\xrightarrow{\\text{restr.}}", "\\mathcal{F}(U)", "$$", "for every pair $(U, V)$ as above." ], "refs": [ "sheaves-definition-f-map" ], "proofs": [ { "contents": [ "Similar to the above and omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 14573 ] }, { "id": 14542, "type": "theorem", "label": "sheaves-lemma-j-pullback", "categories": [ "sheaves" ], "title": "sheaves-lemma-j-pullback", "contents": [ "Let $X$ be a topological space.", "Let $j : U \\to X$ be the inclusion of an open subset $U$ into $X$.", "\\begin{enumerate}", "\\item Let $\\mathcal{G}$ be a presheaf of sets on $X$.", "The presheaf $j_p\\mathcal{G}$", "(see Section \\ref{section-presheaves-functorial}) is given by the rule", "$V \\mapsto \\mathcal{G}(V)$ for $V \\subset U$ open.", "\\item Let $\\mathcal{G}$ be a sheaf of sets on $X$.", "The sheaf $j^{-1}\\mathcal{G}$ is given by the rule", "$V \\mapsto \\mathcal{G}(V)$ for $V \\subset U$ open.", "\\item For any point $u \\in U$ and any sheaf $\\mathcal{G}$ on $X$", "we have a canonical identification of stalks", "$$", "j^{-1}\\mathcal{G}_u = (\\mathcal{G}|_U)_u = \\mathcal{G}_u.", "$$", "\\item On the category of presheaves of $U$ we have $j_pj_* = \\text{id}$.", "\\item On the category of sheaves of $U$ we have $j^{-1}j_* = \\text{id}$.", "\\end{enumerate}", "The same description holds for (pre)sheaves of abelian groups,", "(pre)sheaves of algebraic structures, and (pre)sheaves of modules." ], "refs": [], "proofs": [ { "contents": [ "The colimit in the definition of $j_p\\mathcal{G}(V)$", "is over collection of all $W \\subset X$ open such that $V \\subset W$", "ordered by reverse inclusion.", "Hence this has a largest element, namely $V$. This proves (1).", "And (2) follows because the assignment $V \\mapsto \\mathcal{G}(V)$", "for $V \\subset U$ open is clearly a sheaf if $\\mathcal{G}$ is a", "sheaf. Assertion (3) follows from (2) since the collection", "of open neighbourhoods of $u$ which are contained in $U$ is cofinal", "in the collection of all open neighbourhoods of $u$ in $X$.", "Parts (4) and (5) follow by computing", "$j^{-1}j_*\\mathcal{F}(V) = j_*\\mathcal{F}(V) = \\mathcal{F}(V)$.", "\\medskip\\noindent", "The exact same arguments work for (pre)sheaves of abelian groups", "and (pre)sheaves of algebraic structures." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14543, "type": "theorem", "label": "sheaves-lemma-j-shriek", "categories": [ "sheaves" ], "title": "sheaves-lemma-j-shriek", "contents": [ "Let $X$ be a topological space.", "Let $j : U \\to X$ be the inclusion of an open subset.", "\\begin{enumerate}", "\\item The functor $j_{p!}$ is a left adjoint to the", "restriction functor $j_p$ (see Lemma \\ref{lemma-j-pullback}).", "\\item The functor $j_!$ is a left adjoint to restriction,", "in a formula", "$$", "\\Mor_{\\Sh(X)}(j_!\\mathcal{F}, \\mathcal{G})", "=", "\\Mor_{\\Sh(U)}(\\mathcal{F}, j^{-1}\\mathcal{G})", "=", "\\Mor_{\\Sh(U)}(\\mathcal{F}, \\mathcal{G}|_U)", "$$", "bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$.", "\\item Let $\\mathcal{F}$ be a sheaf of sets on $U$.", "The stalks of the sheaf $j_!\\mathcal{F}$ are described", "as follows", "$$", "j_{!}\\mathcal{F}_x =", "\\left\\{", "\\begin{matrix}", "\\emptyset & \\text{if} & x \\not \\in U \\\\", "\\mathcal{F}_x & \\text{if} & x \\in U", "\\end{matrix}", "\\right.", "$$", "\\item On the category of presheaves of $U$ we have $j_pj_{p!} = \\text{id}$.", "\\item On the category of sheaves of $U$ we have $j^{-1}j_! = \\text{id}$.", "\\end{enumerate}" ], "refs": [ "sheaves-lemma-j-pullback" ], "proofs": [ { "contents": [ "To map $j_{p!}\\mathcal{F}$ into $\\mathcal{G}$", "it is enough to map $\\mathcal{F}(V) \\to \\mathcal{G}(V)$", "whenever $V \\subset U$ compatibly with restriction", "mappings. And by Lemma \\ref{lemma-j-pullback}", "the same description holds for maps", "$\\mathcal{F} \\to \\mathcal{G}|_U$.", "The adjointness of $j_!$ and restriction follows", "from this and the properties of sheafification.", "The identification of stalks is obvious from the", "definition of the extension by the empty set", "and the definition of a stalk.", "Statements (4) and (5) follow by computing the", "value of the sheaf on any open of $U$." ], "refs": [ "sheaves-lemma-j-pullback" ], "ref_ids": [ 14542 ] } ], "ref_ids": [ 14542 ] }, { "id": 14544, "type": "theorem", "label": "sheaves-lemma-j-shriek-abelian", "categories": [ "sheaves" ], "title": "sheaves-lemma-j-shriek-abelian", "contents": [ "Let $X$ be a topological space.", "Let $j : U \\to X$ be the inclusion of an open subset.", "Consider the functors of restriction and extension", "by $0$ for abelian (pre)sheaves.", "\\begin{enumerate}", "\\item The functor $j_{p!}$ is a left adjoint to the", "restriction functor $j_p$ (see Lemma \\ref{lemma-j-pullback}).", "\\item The functor $j_!$ is a left adjoint to restriction,", "in a formula", "$$", "\\Mor_{\\textit{Ab}(X)}(j_!\\mathcal{F}, \\mathcal{G})", "=", "\\Mor_{\\textit{Ab}(U)}(\\mathcal{F}, j^{-1}\\mathcal{G})", "=", "\\Mor_{\\textit{Ab}(U)}(\\mathcal{F}, \\mathcal{G}|_U)", "$$", "bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$.", "\\item Let $\\mathcal{F}$ be an abelian sheaf on $U$.", "The stalks of the sheaf $j_!\\mathcal{F}$ are described", "as follows", "$$", "j_{!}\\mathcal{F}_x =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & x \\not \\in U \\\\", "\\mathcal{F}_x & \\text{if} & x \\in U", "\\end{matrix}", "\\right.", "$$", "\\item On the category of abelian presheaves of $U$", "we have $j_pj_{p!} = \\text{id}$.", "\\item On the category of abelian sheaves of $U$", "we have $j^{-1}j_! = \\text{id}$.", "\\end{enumerate}" ], "refs": [ "sheaves-lemma-j-pullback" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 14542 ] }, { "id": 14545, "type": "theorem", "label": "sheaves-lemma-j-shriek-structures", "categories": [ "sheaves" ], "title": "sheaves-lemma-j-shriek-structures", "contents": [ "Let $X$ be a topological space.", "Let $j : U \\to X$ be the inclusion of an open subset.", "Let $(\\mathcal{C}, F)$ be a type of algebraic structure", "such that $\\mathcal{C}$ has an initial object $e$.", "Consider the functors of restriction and extension", "by $e$ for (pre)sheaves of algebraic structure defined above.", "\\begin{enumerate}", "\\item The functor $j_{p!}$ is a left adjoint to the", "restriction functor $j_p$ (see Lemma \\ref{lemma-j-pullback}).", "\\item The functor $j_!$ is a left adjoint to restriction,", "in a formula", "$$", "\\Mor_{\\Sh(X, \\mathcal{C})}(j_!\\mathcal{F}, \\mathcal{G})", "=", "\\Mor_{\\Sh(U, \\mathcal{C})}(\\mathcal{F}, j^{-1}\\mathcal{G})", "=", "\\Mor_{\\Sh(U, \\mathcal{C})}(\\mathcal{F}, \\mathcal{G}|_U)", "$$", "bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$.", "\\item Let $\\mathcal{F}$ be a sheaf on $U$.", "The stalks of the sheaf $j_!\\mathcal{F}$ are described", "as follows", "$$", "j_{!}\\mathcal{F}_x =", "\\left\\{", "\\begin{matrix}", "e & \\text{if} & x \\not \\in U \\\\", "\\mathcal{F}_x & \\text{if} & x \\in U", "\\end{matrix}", "\\right.", "$$", "\\item On the category of presheaves of algebraic structures on $U$", "we have $j_pj_{p!} = \\text{id}$.", "\\item On the category of sheaves of algebraic structures on $U$", "we have $j^{-1}j_! = \\text{id}$.", "\\end{enumerate}" ], "refs": [ "sheaves-lemma-j-pullback" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 14542 ] }, { "id": 14546, "type": "theorem", "label": "sheaves-lemma-j-shriek-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-j-shriek-modules", "contents": [ "Let $(X, \\mathcal{O})$ be a ringed space.", "Let $j : (U, \\mathcal{O}|_U) \\to (X, \\mathcal{O})$", "be an open subspace.", "Consider the functors of restriction and extension", "by $0$ for (pre)sheaves of modules defined above.", "\\begin{enumerate}", "\\item The functor $j_{p!}$ is a left adjoint to restriction,", "in a formula", "$$", "\\Mor_{\\textit{PMod}(\\mathcal{O})}(j_{p!}\\mathcal{F}, \\mathcal{G})", "=", "\\Mor_{\\textit{PMod}(\\mathcal{O}|_U)}(\\mathcal{F}, \\mathcal{G}|_U)", "$$", "bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$.", "\\item The functor $j_!$ is a left adjoint to restriction,", "in a formula", "$$", "\\Mor_{\\textit{Mod}(\\mathcal{O})}(j_!\\mathcal{F}, \\mathcal{G})", "=", "\\Mor_{\\textit{Mod}(\\mathcal{O}|_U)}(\\mathcal{F}, \\mathcal{G}|_U)", "$$", "bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$.", "\\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules on $U$.", "The stalks of the sheaf $j_!\\mathcal{F}$ are described", "as follows", "$$", "j_{!}\\mathcal{F}_x =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & x \\not \\in U \\\\", "\\mathcal{F}_x & \\text{if} & x \\in U", "\\end{matrix}", "\\right.", "$$", "\\item On the category of sheaves of $\\mathcal{O}|_U$-modules on $U$", "we have $j^{-1}j_! = \\text{id}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14547, "type": "theorem", "label": "sheaves-lemma-equivalence-categories-open", "categories": [ "sheaves" ], "title": "sheaves-lemma-equivalence-categories-open", "contents": [ "Let $X$ be a topological space.", "Let $j : U \\to X$ be the inclusion of an open subset.", "The functor", "$$", "j_! : \\Sh(U) \\longrightarrow \\Sh(X)", "$$", "is fully faithful. Its essential image consists exactly", "of those sheaves $\\mathcal{G}$ such that", "$\\mathcal{G}_x = \\emptyset$ for all $x \\in X \\setminus U$." ], "refs": [], "proofs": [ { "contents": [ "Fully faithfulness follows formally from $j^{-1} j_! = \\text{id}$.", "We have seen that any sheaf in the image of the functor has", "the property on the stalks mentioned in the lemma. Conversely, suppose", "that $\\mathcal{G}$ has the indicated property.", "Then it is easy to check that", "$$", "j_! j^{-1} \\mathcal{G} \\to \\mathcal{G}", "$$", "is an isomorphism on all stalks and hence an isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14548, "type": "theorem", "label": "sheaves-lemma-equivalence-categories-open-abelian", "categories": [ "sheaves" ], "title": "sheaves-lemma-equivalence-categories-open-abelian", "contents": [ "Let $X$ be a topological space.", "Let $j : U \\to X$ be the inclusion of an open subset.", "The functor", "$$", "j_! : \\textit{Ab}(U) \\longrightarrow \\textit{Ab}(X)", "$$", "is fully faithful. Its essential image consists exactly", "of those sheaves $\\mathcal{G}$ such that", "$\\mathcal{G}_x = 0$ for all $x \\in X \\setminus U$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14549, "type": "theorem", "label": "sheaves-lemma-equivalence-categories-open-structures", "categories": [ "sheaves" ], "title": "sheaves-lemma-equivalence-categories-open-structures", "contents": [ "Let $X$ be a topological space.", "Let $j : U \\to X$ be the inclusion of an open subset.", "Let $(\\mathcal{C}, F)$ be a type of algebraic structure", "such that $\\mathcal{C}$ has an initial object $e$.", "The functor", "$$", "j_! : \\Sh(U, \\mathcal{C}) \\longrightarrow \\Sh(X, \\mathcal{C})", "$$", "is fully faithful. Its essential image consists exactly", "of those sheaves $\\mathcal{G}$ such that", "$\\mathcal{G}_x = e$ for all $x \\in X \\setminus U$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14550, "type": "theorem", "label": "sheaves-lemma-equivalence-categories-open-modules", "categories": [ "sheaves" ], "title": "sheaves-lemma-equivalence-categories-open-modules", "contents": [ "Let $(X, \\mathcal{O})$ be a ringed space.", "Let $j : (U, \\mathcal{O}|_U) \\to (X, \\mathcal{O})$", "be an open subspace.", "The functor", "$$", "j_! : \\textit{Mod}(\\mathcal{O}|_U) \\longrightarrow \\textit{Mod}(\\mathcal{O})", "$$", "is fully faithful. Its essential image consists exactly", "of those sheaves $\\mathcal{G}$ such that", "$\\mathcal{G}_x = 0$ for all $x \\in X \\setminus U$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14551, "type": "theorem", "label": "sheaves-lemma-stalks-closed-pushforward", "categories": [ "sheaves" ], "title": "sheaves-lemma-stalks-closed-pushforward", "contents": [ "Let $X$ be a topological space.", "Let $i : Z \\to X$ be the inclusion of a closed subset $Z$ into $X$.", "Let $\\mathcal{F}$ be a sheaf of sets on $Z$.", "The stalks of $i_*\\mathcal{F}$ are described as follows", "$$", "i_*\\mathcal{F}_x =", "\\left\\{", "\\begin{matrix}", "\\{*\\} & \\text{if} & x \\not \\in Z \\\\", "\\mathcal{F}_x & \\text{if} & x \\in Z", "\\end{matrix}", "\\right.", "$$", "where $\\{*\\}$ denotes a singleton set. Moreover,", "$i^{-1}i_* = \\text{id}$ on the category of sheaves", "of sets on $Z$. Moreover, the same holds for abelian", "sheaves on $Z$, resp.\\ sheaves of algebraic structures on $Z$", "where $\\{*\\}$ has to be replaced by $0$, resp.\\ a", "final object of the category of algebraic structures." ], "refs": [], "proofs": [ { "contents": [ "If $x \\not \\in Z$, then there exist arbitrarily small open", "neighbourhoods $U$ of $x$ which do not meet $Z$.", "Because $\\mathcal{F}$ is a sheaf", "we have $\\mathcal{F}(i^{-1}(U)) = \\{*\\}$ for any such $U$,", "see Remark \\ref{remark-confusion}. This proves the first case.", "The second case comes from the fact that for $z \\in Z$", "any open neighbourhood of $z$ is of the form $Z \\cap U$ for", "some open $U$ of $X$. For the statement that", "$i^{-1}i_* = \\text{id}$ consider the canonical map", "$i^{-1}i_*\\mathcal{F} \\to \\mathcal{F}$. This is an isomorphism", "on stalks (see above) and hence an isomorphism.", "\\medskip\\noindent", "For sheaves of abelian groups, and sheaves of algebraic structures", "you argue in the same manner." ], "refs": [ "sheaves-remark-confusion" ], "ref_ids": [ 14586 ] } ], "ref_ids": [] }, { "id": 14552, "type": "theorem", "label": "sheaves-lemma-equivalence-categories-closed", "categories": [ "sheaves" ], "title": "sheaves-lemma-equivalence-categories-closed", "contents": [ "Let $X$ be a topological space.", "Let $i : Z \\to X$ be the inclusion of a closed subset.", "The functor", "$$", "i_* : \\Sh(Z) \\longrightarrow \\Sh(X)", "$$", "is fully faithful. Its essential image consists exactly", "of those sheaves $\\mathcal{G}$ such that", "$\\mathcal{G}_x = \\{*\\}$ for all $x \\in X \\setminus Z$." ], "refs": [], "proofs": [ { "contents": [ "Fully faithfulness follows formally from $i^{-1} i_* = \\text{id}$.", "We have seen that any sheaf in the image of the functor has", "the property on the stalks mentioned in the lemma. Conversely, suppose", "that $\\mathcal{G}$ has the indicated property.", "Then it is easy to check that", "$$", "\\mathcal{G} \\to i_* i^{-1} \\mathcal{G}", "$$", "is an isomorphism on all stalks and hence an isomorphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14553, "type": "theorem", "label": "sheaves-lemma-equivalence-categories-closed-abelian", "categories": [ "sheaves" ], "title": "sheaves-lemma-equivalence-categories-closed-abelian", "contents": [ "Let $X$ be a topological space.", "Let $i : Z \\to X$ be the inclusion of a closed subset.", "The functor", "$$", "i_* : \\textit{Ab}(Z) \\longrightarrow \\textit{Ab}(X)", "$$", "is fully faithful. Its essential image consists exactly", "of those sheaves $\\mathcal{G}$ such that", "$\\mathcal{G}_x = 0$ for all $x \\in X \\setminus Z$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14554, "type": "theorem", "label": "sheaves-lemma-equivalence-categories-closed-structures", "categories": [ "sheaves" ], "title": "sheaves-lemma-equivalence-categories-closed-structures", "contents": [ "Let $X$ be a topological space.", "Let $i : Z \\to X$ be the inclusion of a closed subset.", "Let $(\\mathcal{C}, F)$ be a type of algebraic structure", "with final object $0$. The functor", "$$", "i_* : \\Sh(Z, \\mathcal{C}) \\longrightarrow \\Sh(X, \\mathcal{C})", "$$", "is fully faithful. Its essential image consists exactly", "of those sheaves $\\mathcal{G}$ such that", "$\\mathcal{G}_x = 0$ for all $x \\in X \\setminus Z$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14555, "type": "theorem", "label": "sheaves-lemma-glue-maps", "categories": [ "sheaves" ], "title": "sheaves-lemma-glue-maps", "contents": [ "Let $X$ be a topological space.", "Let $X = \\bigcup U_i$ be an open covering.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of sets on $X$.", "Given a collection", "$$", "\\varphi_i :", "\\mathcal{F}|_{U_i}", "\\longrightarrow", "\\mathcal{G}|_{U_i}", "$$", "of maps of sheaves such that for all $i, j \\in I$ the maps", "$\\varphi_i, \\varphi_j$ restrict to the same map", "$\\mathcal{F}|_{U_i \\cap U_j} \\to \\mathcal{G}|_{U_i \\cap U_j}$", "then there exists a unique map of sheaves", "$$", "\\varphi :", "\\mathcal{F}", "\\longrightarrow", "\\mathcal{G}", "$$", "whose restriction to each $U_i$ agrees with $\\varphi_i$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14556, "type": "theorem", "label": "sheaves-lemma-glue-sheaves", "categories": [ "sheaves" ], "title": "sheaves-lemma-glue-sheaves", "contents": [ "Let $X$ be a topological space.", "Let $X = \\bigcup_{i\\in I} U_i$ be an open covering.", "Given any glueing data $(\\mathcal{F}_i, \\varphi_{ij})$", "for sheaves of sets with respect to the covering $X = \\bigcup U_i$", "there exists a sheaf of sets $\\mathcal{F}$ on $X$", "together with isomorphisms", "$$", "\\varphi_i : \\mathcal{F}|_{U_i} \\to \\mathcal{F}_i", "$$", "such that the diagrams", "$$", "\\xymatrix{", "\\mathcal{F}|_{U_i \\cap U_j} \\ar[r]_{\\varphi_i} \\ar[d]_{\\text{id}} &", "\\mathcal{F}_i|_{U_i \\cap U_j} \\ar[d]^{\\varphi_{ij}} \\\\", "\\mathcal{F}|_{U_i \\cap U_j} \\ar[r]^{\\varphi_j} &", "\\mathcal{F}_j|_{U_i \\cap U_j}", "}", "$$", "are commutative." ], "refs": [], "proofs": [ { "contents": [ "First proof. In this proof we give a formula for the set of sections", "of $\\mathcal{F}$ over an open $W \\subset X$. Namely, we define", "$$", "\\mathcal{F}(W) =", "\\{", "(s_i)_{i \\in I} \\mid", "s_i \\in \\mathcal{F}_i(W \\cap U_i),", "\\varphi_{ij}(s_i|_{W \\cap U_i \\cap U_j}) = s_j|_{W \\cap U_i \\cap U_j}", "\\}.", "$$", "Restriction mappings for $W' \\subset W$ are defined by the restricting", "each of the $s_i$ to $W' \\cap U_i$. The sheaf condition for $\\mathcal{F}$", "follows immediately from the sheaf condition for each of the", "$\\mathcal{F}_i$.", "\\medskip\\noindent", "We still have to prove that $\\mathcal{F}|_{U_i}$ maps", "isomorphically to $\\mathcal{F}_i$. Let $W \\subset U_i$.", "In this case the condition in the definition of", "$\\mathcal{F}(W)$ implies that $s_j = \\varphi_{ij}(s_i|_{W \\cap U_j})$.", "And the commutativity of the diagrams in the definition", "of a glueing data assures that we may start with {\\it any}", "section $s \\in \\mathcal{F}_i(W)$ and obtain a compatible", "collection by setting $s_i = s$ and $s_j = \\varphi_{ij}(s_i|_{W \\cap U_j})$.", "\\medskip\\noindent", "Second proof (sketch). Let $\\mathcal{B}$ be the set of opens $U \\subset X$", "such that $U \\subset U_i$ for some $i \\in I$. Then $\\mathcal{B}$", "is a base for the topology on $X$. For $U \\in \\mathcal{B}$ we pick", "$i \\in I$ with $U \\subset U_i$ and we set $\\mathcal{F}(U) = \\mathcal{F}_i(U)$.", "Using the isomorphisms $\\varphi_{ij}$ we see that this prescription", "is ``independent of the choice of $i$''. Using the restriction mappings", "of $\\mathcal{F}_i$ we find that $\\mathcal{F}$ is a sheaf on $\\mathcal{B}$.", "Finally, use Lemma \\ref{lemma-extend-off-basis} to extend $\\mathcal{F}$", "to a unique sheaf $\\mathcal{F}$ on $X$." ], "refs": [ "sheaves-lemma-extend-off-basis" ], "ref_ids": [ 14532 ] } ], "ref_ids": [] }, { "id": 14557, "type": "theorem", "label": "sheaves-lemma-glue-sheaves-structures", "categories": [ "sheaves" ], "title": "sheaves-lemma-glue-sheaves-structures", "contents": [ "Let $X$ be a topological space.", "Let $X = \\bigcup U_i$ be an open covering.", "Let $(\\mathcal{F}_i, \\varphi_{ij})$ be a glueing data", "of sheaves of abelian groups, resp.\\ sheaves of algebraic structures,", "resp.\\ sheaves of $\\mathcal{O}$-modules for some sheaf of rings", "$\\mathcal{O}$ on $X$. Then the construction in the proof of", "Lemma \\ref{lemma-glue-sheaves} above leads to a sheaf", "of abelian groups, resp.\\ sheaf of algebraic structures,", "resp.\\ sheaf of $\\mathcal{O}$-modules." ], "refs": [ "sheaves-lemma-glue-sheaves" ], "proofs": [ { "contents": [ "This is true because in the construction the set of sections", "$\\mathcal{F}(W)$ over an open $W$ is given as the", "equalizer of the maps", "$$", "\\xymatrix{", "\\prod_{i \\in I} \\mathcal{F}_i(W \\cap U_i)", "\\ar@<1ex>[r]", "\\ar@<-1ex>[r]", "&", "\\prod_{i, j\\in I} \\mathcal{F}_i(W \\cap U_i \\cap U_j)", "}", "$$", "And in each of the cases envisioned this equalizer gives", "an object in the relevant category whose underlying set is", "the object considered in the cited lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 14556 ] }, { "id": 14558, "type": "theorem", "label": "sheaves-lemma-mapping-property-glue", "categories": [ "sheaves" ], "title": "sheaves-lemma-mapping-property-glue", "contents": [ "Let $X$ be a topological space.", "Let $X = \\bigcup_{i\\in I} U_i$ be an open covering.", "The functor which associates to a sheaf of", "sets $\\mathcal{F}$ the following collection of", "glueing data", "$$", "(\\mathcal{F}|_{U_i},", "(\\mathcal{F}|_{U_i})|_{U_i \\cap U_j}", "\\to", "(\\mathcal{F}|_{U_j})|_{U_i \\cap U_j}", ")", "$$", "with respect to the covering $X = \\bigcup U_i$", "defines an equivalence of categories between", "$\\Sh(X)$ and the category of glueing", "data. A similar statement holds for", "abelian sheaves, resp.\\ sheaves of algebraic structures,", "resp.\\ sheaves of $\\mathcal{O}$-modules." ], "refs": [], "proofs": [ { "contents": [ "The functor is fully faithful by", "Lemma \\ref{lemma-glue-maps}", "and essentially surjective (via an explicitly given quasi-inverse functor) by", "Lemma \\ref{lemma-glue-sheaves}." ], "refs": [ "sheaves-lemma-glue-maps", "sheaves-lemma-glue-sheaves" ], "ref_ids": [ 14555, 14556 ] } ], "ref_ids": [] }, { "id": 14590, "type": "theorem", "label": "descent-theorem-descent", "categories": [ "descent" ], "title": "descent-theorem-descent", "contents": [ "The following conditions are equivalent.", "\\begin{enumerate}", "\\item[(a)] The morphism $f$ is a descent morphism for modules.", "\\item[(b)] The morphism $f$ is an effective descent morphism for modules.", "\\item[(c)] The morphism $f$ is universally injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (b) implies (a). We now check that (a) implies (c). If $f$ is ", "not universally injective, we can find $M \\in \\text{Mod}_R$ such that the map", "$1_M \\otimes f: M \\to M \\otimes_R S$ has nontrivial kernel $N$.", "The natural projection $M \\to M/N$ is not an isomorphism, but its image in ", "$DD_{S/R}$ is an isomorphism.", "Hence $f^*$ is not fully faithful.", "\\medskip\\noindent", "We finally check that (c) implies (b). By Lemma \\ref{lemma-descent-lemma}, for", "$(M, \\theta) \\in DD_{S/R}$,", "the natural map $f^* f_*(M,\\theta) \\to M$ is an isomorphism of $S$-modules. On ", "the other hand, for $M_0 \\in \\text{Mod}_R$,", "we may tensor (\\ref{equation-equalizer-S}) with $M_0$ over $R$ to obtain an ", "equalizer sequence,", "so $M_0 \\to f_* f^* M$ is an isomorphism. Consequently, $f_*$ and $f^*$ are ", "quasi-inverse functors, proving the claim." ], "refs": [ "descent-lemma-descent-lemma" ], "ref_ids": [ 14607 ] } ], "ref_ids": [] }, { "id": 14591, "type": "theorem", "label": "descent-theorem-descend-module-properties", "categories": [ "descent" ], "title": "descent-theorem-descend-module-properties", "contents": [ "If $M \\otimes_R S$ has one of the following properties as an $S$-module", "\\begin{enumerate}", "\\item[(a)]", "finitely generated;", "\\item[(b)]", "finitely presented;", "\\item[(c)]", "flat;", "\\item[(d)]", "faithfully flat;", "\\item[(e)]", "finite projective;", "\\end{enumerate}", "then so does $M$ as an $R$-module (and conversely)." ], "refs": [], "proofs": [ { "contents": [ "To prove (a), choose a finite set $\\{n_i\\}$ of generators of $M \\otimes_R S$", "in $\\text{Mod}_S$. Write each $n_i$ as $\\sum_j m_{ij} \\otimes s_{ij}$ with", "$m_{ij} \\in M$ and $s_{ij} \\in S$. Let $F$ be the finite free $R$-module with", "basis $e_{ij}$ and let $F \\to M$ be the $R$-module map sending $e_{ij}$ to", "$m_{ij}$. Then $F \\otimes_R S\\to M \\otimes_R S$ is surjective, so", "$\\Coker(F \\to M) \\otimes_R S$ is zero and hence $\\Coker(F \\to M)$", "is zero. This proves (a).", "\\medskip\\noindent", "To see (b) assume $M \\otimes_R S$ is finitely presented. Then $M$ is finitely", "generated by (a). Choose a surjection $R^{\\oplus n} \\to M$ with kernel $K$.", "Then $K \\otimes_R S \\to S^{\\oplus r} \\to M \\otimes_R S \\to 0$ is exact.", "By Algebra, Lemma \\ref{algebra-lemma-extension}", "the kernel of $S^{\\oplus r} \\to M \\otimes_R S$", "is a finite $S$-module. Thus we can find finitely many elements", "$k_1, \\ldots, k_t \\in K$ such that the images of $k_i \\otimes 1$ in", "$S^{\\oplus r}$ generate the kernel of $S^{\\oplus r} \\to M \\otimes_R S$.", "Let $K' \\subset K$ be the submodule generated by $k_1, \\ldots, k_t$.", "Then $M' = R^{\\oplus r}/K'$ is a finitely presented $R$-module", "with a morphism $M' \\to M$ such that $M' \\otimes_R S \\to M \\otimes_R S$", "is an isomorphism. Thus $M' \\cong M$ as desired.", "\\medskip\\noindent", "To prove (c), let $0 \\to M' \\to M'' \\to M \\to 0$ be a short exact sequence in ", "$\\text{Mod}_R$. Since $\\bullet \\otimes_R S$ is a right exact functor,", "$M'' \\otimes_R S \\to M \\otimes_R S$ is surjective. So by", "Lemma \\ref{lemma-C-is-faithful} the map", "$C(M \\otimes_R S) \\to C(M'' \\otimes_R S)$ is injective.", "If $M \\otimes_R S$ is flat, then", "Lemma \\ref{lemma-flat-to-injective} shows", "$C(M \\otimes_R S)$ is an injective object of $\\text{Mod}_S$, so the injection", "$C(M \\otimes_R S) \\to C(M'' \\otimes_R S)$", "is split in $\\text{Mod}_S$ and hence also in $\\text{Mod}_R$.", "Since $C(M \\otimes_R S) \\to C(M)$ is a split surjection by ", "Lemma \\ref{lemma-split-surjection}, it follows that ", "$C(M) \\to C(M'')$ is a split injection in $\\text{Mod}_R$. That is, the sequence", "$$", "0 \\to C(M) \\to C(M'') \\to C(M') \\to 0", "$$", "is split exact. ", "For $N \\in \\text{Mod}_R$, by (\\ref{equation-adjunction}) we see that ", "$$", "0 \\to C(M \\otimes_R N) \\to C(M'' \\otimes_R N) \\to C(M' \\otimes_R N) \\to 0", "$$", "is split exact. By Lemma \\ref{lemma-C-is-faithful}, ", "$$", "0 \\to M' \\otimes_R N \\to M'' \\otimes_R N \\to M \\otimes_R N \\to 0", "$$", "is exact. This implies $M$ is flat over $R$. Namely, taking", "$M'$ a free module surjecting onto $M$ we conclude that", "$\\text{Tor}_1^R(M, N) = 0$ for all modules $N$ and we can use", "Algebra, Lemma \\ref{algebra-lemma-characterize-flat}.", "This proves (c).", "\\medskip\\noindent", "To deduce (d) from (c), note that if $N \\in \\text{Mod}_R$ and $M \\otimes_R N$ ", "is zero,", "then $M \\otimes_R S \\otimes_S (N \\otimes_R S) \\cong (M \\otimes_R N) \\otimes_R ", "S$ is zero,", "so $N \\otimes_R S$ is zero and hence $N$ is zero.", "\\medskip\\noindent", "To deduce (e) at this point, it suffices to recall that $M$ is finitely ", "generated and projective if and only if it is finitely presented and flat.", "See Algebra, Lemma \\ref{algebra-lemma-finite-projective}." ], "refs": [ "algebra-lemma-extension", "descent-lemma-C-is-faithful", "descent-lemma-flat-to-injective", "descent-lemma-split-surjection", "descent-lemma-C-is-faithful", "algebra-lemma-characterize-flat", "algebra-lemma-finite-projective" ], "ref_ids": [ 330, 14602, 14608, 14603, 14602, 786, 795 ] } ], "ref_ids": [] }, { "id": 14592, "type": "theorem", "label": "descent-theorem-descend-algebra-properties", "categories": [ "descent" ], "title": "descent-theorem-descend-algebra-properties", "contents": [ "If $A \\otimes_R S$ has one of the following properties as an $S$-algebra", "\\begin{enumerate}", "\\item[(a)]", "of finite type;", "\\item[(b)]", "of finite presentation;", "\\item[(c)]", "formally unramified;", "\\item[(d)]", "unramified;", "\\item[(e)]", "\\'etale;", "\\end{enumerate}", "then so does $A$ as an $R$-algebra (and of course conversely)." ], "refs": [], "proofs": [ { "contents": [ "To prove (a), choose a finite set $\\{x_i\\}$ of generators of $A \\otimes_R S$", "over $S$. Write each $x_i$ as $\\sum_j y_{ij} \\otimes s_{ij}$ with", "$y_{ij} \\in A$ and $s_{ij} \\in S$. Let $F$ be the polynomial $R$-algebra", "on variables $e_{ij}$ and let $F \\to M$ be the $R$-algebra map sending", "$e_{ij}$ to $y_{ij}$. Then $F \\otimes_R S\\to A \\otimes_R S$ is surjective, so", "$\\Coker(F \\to A) \\otimes_R S$ is zero and hence $\\Coker(F \\to A)$", "is zero. This proves (a).", "\\medskip\\noindent", "To see (b) assume $A \\otimes_R S$ is a finitely presented $S$-algebra.", "Then $A$ is finite type over $R$ by (a). Choose a surjection", "$R[x_1, \\ldots, x_n] \\to A$ with kernel $I$.", "Then $I \\otimes_R S \\to S[x_1, \\ldots, x_n] \\to A \\otimes_R S \\to 0$ is exact.", "By Algebra, Lemma \\ref{algebra-lemma-finite-presentation-independent}", "the kernel of $S[x_1, \\ldots, x_n] \\to A \\otimes_R S$", "is a finitely generated ideal. Thus we can find finitely many elements", "$y_1, \\ldots, y_t \\in I$ such that the images of $y_i \\otimes 1$ in", "$S[x_1, \\ldots, x_n]$ generate the kernel of", "$S[x_1, \\ldots, x_n] \\to A \\otimes_R S$.", "Let $I' \\subset I$ be the ideal generated by $y_1, \\ldots, y_t$.", "Then $A' = R[x_1, \\ldots, x_n]/I'$ is a finitely presented $R$-algebra", "with a morphism $A' \\to A$ such that $A' \\otimes_R S \\to A \\otimes_R S$", "is an isomorphism. Thus $A' \\cong A$ as desired.", "\\medskip\\noindent", "To prove (c), recall that $A$ is formally unramified over $R$ if and only", "if the module of relative differentials $\\Omega_{A/R}$ vanishes, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-formally-unramified} or", "\\cite[Proposition~17.2.1]{EGA4}.", "Since $\\Omega_{(A \\otimes_R S)/S} = \\Omega_{A/R} \\otimes_R S$,", "the vanishing descends by Theorem \\ref{theorem-descent}.", "\\medskip\\noindent", "To deduce (d) from the previous cases, recall that $A$ is unramified", "over $R$ if and only if $A$ is formally unramified and of finite type", "over $R$, see", "Algebra, Lemma \\ref{algebra-lemma-formally-unramified-unramified}.", "\\medskip\\noindent", "To prove (e), recall that by", "Algebra, Lemma \\ref{algebra-lemma-etale-flat-unramified-finite-presentation}", "or \\cite[Th\\'eor\\`eme~17.6.1]{EGA4} the algebra", "$A$ is \\'etale over $R$ if and only if", "$A$ is flat, unramified, and of finite presentation over $R$." ], "refs": [ "algebra-lemma-finite-presentation-independent", "algebra-lemma-characterize-formally-unramified", "descent-theorem-descent", "algebra-lemma-formally-unramified-unramified", "algebra-lemma-etale-flat-unramified-finite-presentation" ], "ref_ids": [ 334, 1254, 14590, 1265, 1271 ] } ], "ref_ids": [] }, { "id": 14593, "type": "theorem", "label": "descent-lemma-refine-descent-datum", "categories": [ "descent" ], "title": "descent-lemma-refine-descent-datum", "contents": [ "Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ and", "$\\mathcal{V} = \\{V_j \\to V\\}_{j \\in J}$", "be families of morphisms of schemes with fixed target.", "Let $(g, \\alpha : I \\to J, (g_i)) : \\mathcal{U} \\to \\mathcal{V}$", "be a morphism of families of maps with fixed target, see", "Sites, Definition \\ref{sites-definition-morphism-coverings}.", "Let $(\\mathcal{F}_j, \\varphi_{jj'})$ be a descent", "datum for quasi-coherent sheaves with respect to the", "family $\\{V_j \\to V\\}_{j \\in J}$. Then", "\\begin{enumerate}", "\\item The system", "$$", "\\left(g_i^*\\mathcal{F}_{\\alpha(i)},", "(g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')}\\right)", "$$", "is a descent datum with respect to the family $\\{U_i \\to U\\}_{i \\in I}$.", "\\item This construction is functorial in the descent datum", "$(\\mathcal{F}_j, \\varphi_{jj'})$.", "\\item Given a second morphism $(g', \\alpha' : I \\to J, (g'_i))$", "of families of maps with fixed target with $g = g'$", "there exists a functorial isomorphism of descent data", "$$", "(g_i^*\\mathcal{F}_{\\alpha(i)},", "(g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')})", "\\cong", "((g'_i)^*\\mathcal{F}_{\\alpha'(i)},", "(g'_i \\times g'_{i'})^*\\varphi_{\\alpha'(i)\\alpha'(i')}).", "$$", "\\end{enumerate}" ], "refs": [ "sites-definition-morphism-coverings" ], "proofs": [ { "contents": [ "Omitted. Hint: The maps", "$g_i^*\\mathcal{F}_{\\alpha(i)} \\to (g'_i)^*\\mathcal{F}_{\\alpha'(i)}$", "which give the isomorphism of descent data in part (3)", "are the pullbacks of the maps $\\varphi_{\\alpha(i)\\alpha'(i)}$ by the", "morphisms $(g_i, g'_i) : U_i \\to V_{\\alpha(i)} \\times_V V_{\\alpha'(i)}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 8656 ] }, { "id": 14594, "type": "theorem", "label": "descent-lemma-zariski-descent-effective", "categories": [ "descent" ], "title": "descent-lemma-zariski-descent-effective", "contents": [ "Let $S$ be a scheme.", "Let $S = \\bigcup U_i$ be an open covering.", "Any descent datum on quasi-coherent sheaves", "for the family $\\mathcal{U} = \\{U_i \\to S\\}$ is", "effective. Moreover, the functor from the category of", "quasi-coherent $\\mathcal{O}_S$-modules to the category", "of descent data with respect to $\\mathcal{U}$ is fully faithful." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from", "Sheaves, Section \\ref{sheaves-section-glueing-sheaves}", "and the fact that being quasi-coherent is a local property, see", "Modules, Definition \\ref{modules-definition-quasi-coherent}." ], "refs": [ "modules-definition-quasi-coherent" ], "ref_ids": [ 13337 ] } ], "ref_ids": [] }, { "id": 14595, "type": "theorem", "label": "descent-lemma-descent-datum-cosimplicial", "categories": [ "descent" ], "title": "descent-lemma-descent-datum-cosimplicial", "contents": [ "Let $R \\to A$ be a ring map.", "Given a descent datum $(N, \\varphi)$ we can associate to it a", "cosimplicial $(A/R)_\\bullet$-module $N_\\bullet$\\footnote{We should really", "write $(N, \\varphi)_\\bullet$.} by the", "rules $N_n = N_{n, n}$ and given $\\beta : [n] \\to [m]$", "setting we define", "$$", "N_\\bullet(\\beta) = (\\varphi^m_{\\beta(n)m}) \\circ N_{\\beta, n} :", "N_{n, n} \\longrightarrow N_{m, m}.", "$$", "This procedure is functorial in the descent datum." ], "refs": [], "proofs": [ { "contents": [ "Here are the first few maps", "where $\\varphi(n \\otimes 1) = \\sum \\alpha_i \\otimes x_i$", "$$", "\\begin{matrix}", "\\delta^1_0 & : & N & \\to & A \\otimes N & n & \\mapsto & 1 \\otimes n \\\\", "\\delta^1_1 & : & N & \\to & A \\otimes N & n & \\mapsto &", "\\sum \\alpha_i \\otimes x_i\\\\", "\\sigma^0_0 & : & A \\otimes N & \\to & N & a_0 \\otimes n & \\mapsto & a_0n \\\\", "\\delta^2_0 & : & A \\otimes N & \\to & A \\otimes A \\otimes N &", "a_0 \\otimes n & \\mapsto & 1 \\otimes a_0 \\otimes n \\\\", "\\delta^2_1 & : & A \\otimes N & \\to & A \\otimes A \\otimes N &", "a_0 \\otimes n & \\mapsto & a_0 \\otimes 1 \\otimes n \\\\", "\\delta^2_2 & : & A \\otimes N & \\to & A \\otimes A \\otimes N &", "a_0 \\otimes n & \\mapsto & \\sum a_0 \\otimes \\alpha_i \\otimes x_i \\\\", "\\sigma^1_0 & : & A \\otimes A \\otimes N & \\to & A \\otimes N &", "a_0 \\otimes a_1 \\otimes n & \\mapsto & a_0a_1 \\otimes n \\\\", "\\sigma^1_1 & : & A \\otimes A \\otimes N & \\to & A \\otimes N &", "a_0 \\otimes a_1 \\otimes n & \\mapsto & a_0 \\otimes a_1n", "\\end{matrix}", "$$", "with notation as in", "Simplicial, Section \\ref{simplicial-section-cosimplicial-object}.", "We first verify the two properties $\\sigma^0_0 \\circ \\delta^1_0 = \\text{id}$", "and $\\sigma^0_0 \\circ \\delta^1_1 = \\text{id}$.", "The first one, $\\sigma^0_0 \\circ \\delta^1_0 = \\text{id}$, is clear from", "the explicit description of the morphisms above.", "To prove the second relation we have to use the cocycle condition", "(because it does not holds for an arbitrary isomorphism", "$\\varphi : N \\otimes_R A \\to A \\otimes_R N$). Write", "$p = \\sigma^0_0 \\circ \\delta^1_1 : N \\to N$. By the description of the", "maps above we deduce that $p$ is also equal to", "$$", "p = \\varphi \\otimes \\text{id} :", "N = (N \\otimes_R A) \\otimes_{(A \\otimes_R A)} A", "\\longrightarrow", "(A \\otimes_R N) \\otimes_{(A \\otimes_R A)} A = N", "$$", "Since $\\varphi$ is an isomorphism we see that $p$ is an isomorphism.", "Write $\\varphi(n \\otimes 1) = \\sum \\alpha_i \\otimes x_i$ for certain", "$\\alpha_i \\in A$ and $x_i \\in N$. Then $p(n) = \\sum \\alpha_ix_i$.", "Next, write", "$\\varphi(x_i \\otimes 1) = \\sum \\alpha_{ij} \\otimes y_j$ for", "certain $\\alpha_{ij} \\in A$ and $y_j \\in N$. Then the cocycle condition", "says that", "$$", "\\sum \\alpha_i \\otimes \\alpha_{ij} \\otimes y_j", "=", "\\sum \\alpha_i \\otimes 1 \\otimes x_i.", "$$", "This means that $p(n) = \\sum \\alpha_ix_i = \\sum \\alpha_i\\alpha_{ij}y_j =", "\\sum \\alpha_i p(x_i) = p(p(n))$. Thus $p$ is a projector, and since it is", "an isomorphism it is the identity.", "\\medskip\\noindent", "To prove fully that $N_\\bullet$ is a cosimplicial module we have to check", "all 5 types of relations of", "Simplicial, Remark \\ref{simplicial-remark-relations-cosimplicial}.", "The relations on composing $\\sigma$'s are obvious.", "The relations on composing $\\delta$'s come down to the", "cocycle condition for $\\varphi$.", "In exactly the same way as above one checks the relations", "$\\sigma_j \\circ \\delta_j = \\sigma_j \\circ \\delta_{j + 1} = \\text{id}$.", "Finally, the other relations on compositions of $\\delta$'s and $\\sigma$'s", "hold for any $\\varphi$ whatsoever." ], "refs": [ "simplicial-remark-relations-cosimplicial" ], "ref_ids": [ 14933 ] } ], "ref_ids": [] }, { "id": 14596, "type": "theorem", "label": "descent-lemma-canonical-descent-datum-cosimplicial", "categories": [ "descent" ], "title": "descent-lemma-canonical-descent-datum-cosimplicial", "contents": [ "Let $R \\to A$ be a ring map.", "Let $M$ be an $R$-module. The cosimplicial", "$(A/R)_\\bullet$-module associated to the canonical descent", "datum is isomorphic to the cosimplicial module $(A/R)_\\bullet \\otimes_R M$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14597, "type": "theorem", "label": "descent-lemma-with-section-exact", "categories": [ "descent" ], "title": "descent-lemma-with-section-exact", "contents": [ "Suppose that $R \\to A$ has a section.", "Then for any $R$-module $M$ the extended cochain complex", "(\\ref{equation-extended-complex}) is exact." ], "refs": [], "proofs": [ { "contents": [ "By", "Simplicial, Lemma \\ref{simplicial-lemma-push-outs-simplicial-object-w-section}", "the map $R \\to (A/R)_\\bullet$ is a homotopy equivalence", "of cosimplicial $R$-algebras", "(here $R$ denotes the constant cosimplicial $R$-algebra).", "Hence $M \\to (A/R)_\\bullet \\otimes_R M$ is", "a homotopy equivalence in the category of cosimplicial", "$R$-modules, because $\\otimes_R M$ is a", "functor from the category of $R$-algebras to the category", "of $R$-modules, see", "Simplicial, Lemma \\ref{simplicial-lemma-functorial-homotopy}.", "This implies that the induced map of associated", "complexes is a homotopy equivalence, see", "Simplicial, Lemma \\ref{simplicial-lemma-homotopy-s-Q}.", "Since the complex associated to the constant cosimplicial", "$R$-module $M$ is the complex", "$$", "\\xymatrix{", "M \\ar[r]^0 & M \\ar[r]^1 & M \\ar[r]^0 & M \\ar[r]^1 & M \\ldots", "}", "$$", "we win (since the extended version simply puts an extra $M$ at", "the beginning)." ], "refs": [ "simplicial-lemma-functorial-homotopy", "simplicial-lemma-homotopy-s-Q" ], "ref_ids": [ 14879, 14881 ] } ], "ref_ids": [] }, { "id": 14598, "type": "theorem", "label": "descent-lemma-ff-exact", "categories": [ "descent" ], "title": "descent-lemma-ff-exact", "contents": [ "Suppose that $R \\to A$ is faithfully flat, see", "Algebra, Definition \\ref{algebra-definition-flat}.", "Then for any $R$-module $M$ the extended cochain complex", "(\\ref{equation-extended-complex}) is exact." ], "refs": [ "algebra-definition-flat" ], "proofs": [ { "contents": [ "Suppose we can show there exists a faithfully flat ring map", "$R \\to R'$ such that the result holds for the ring map", "$R' \\to A' = R' \\otimes_R A$. Then the result follows for", "$R \\to A$. Namely, for any $R$-module $M$ the cosimplicial", "module $(M \\otimes_R R') \\otimes_{R'} (A'/R')_\\bullet$ is", "just the cosimplicial module $R' \\otimes_R (M \\otimes_R (A/R)_\\bullet)$.", "Hence the vanishing of cohomology of the complex associated to", "$(M \\otimes_R R') \\otimes_{R'} (A'/R')_\\bullet$ implies the", "vanishing of the cohomology of the complex associated to", "$M \\otimes_R (A/R)_\\bullet$ by faithful flatness of $R \\to R'$.", "Similarly for the vanishing of cohomology groups in degrees", "$-1$ and $0$ of the extended complex (proof omitted).", "\\medskip\\noindent", "But we have such a faithful flat extension. Namely $R' = A$ works", "because the ring map $R' = A \\to A' = A \\otimes_R A$ has a section", "$a \\otimes a' \\mapsto aa'$ and", "Lemma \\ref{lemma-with-section-exact}", "applies." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 1456 ] }, { "id": 14599, "type": "theorem", "label": "descent-lemma-recognize-effective", "categories": [ "descent" ], "title": "descent-lemma-recognize-effective", "contents": [ "Let $R \\to A$ be a faithfully flat ring map.", "Let $(N, \\varphi)$ be a descent datum.", "Then $(N, \\varphi)$ is effective if and only if the canonical", "map", "$$", "A \\otimes_R H^0(s(N_\\bullet)) \\longrightarrow N", "$$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "If $(N, \\varphi)$ is effective, then we may write $N = A \\otimes_R M$", "with $\\varphi = can$. It follows that $H^0(s(N_\\bullet)) = M$ by", "Lemmas \\ref{lemma-canonical-descent-datum-cosimplicial}", "and \\ref{lemma-ff-exact}. Conversely, suppose the map of the lemma", "is an isomorphism. In this case set $M = H^0(s(N_\\bullet))$.", "This is an $R$-submodule of $N$,", "namely $M = \\{n \\in N \\mid 1 \\otimes n = \\varphi(n \\otimes 1)\\}$.", "The only thing to check is that via the isomorphism", "$A \\otimes_R M \\to N$", "the canonical descent data agrees with $\\varphi$.", "We omit the verification." ], "refs": [ "descent-lemma-canonical-descent-datum-cosimplicial", "descent-lemma-ff-exact" ], "ref_ids": [ 14596, 14598 ] } ], "ref_ids": [] }, { "id": 14600, "type": "theorem", "label": "descent-lemma-descent-descends", "categories": [ "descent" ], "title": "descent-lemma-descent-descends", "contents": [ "Let $R \\to A$ be a faithfully flat ring map, and let $R \\to R'$", "be faithfully flat. Set $A' = R' \\otimes_R A$. If all descent data", "for $R' \\to A'$ are effective, then so are all descent data for $R \\to A$." ], "refs": [], "proofs": [ { "contents": [ "Let $(N, \\varphi)$ be a descent datum for $R \\to A$.", "Set $N' = R' \\otimes_R N = A' \\otimes_A N$, and denote", "$\\varphi' = \\text{id}_{R'} \\otimes \\varphi$ the base change", "of the descent datum $\\varphi$. Then $(N', \\varphi')$ is", "a descent datum for $R' \\to A'$ and", "$H^0(s(N'_\\bullet)) = R' \\otimes_R H^0(s(N_\\bullet))$.", "Moreover, the map", "$A' \\otimes_{R'} H^0(s(N'_\\bullet)) \\to N'$ is identified", "with the base change of the $A$-module map", "$A \\otimes_R H^0(s(N)) \\to N$ via the faithfully flat map", "$A \\to A'$. Hence we conclude by Lemma \\ref{lemma-recognize-effective}." ], "refs": [ "descent-lemma-recognize-effective" ], "ref_ids": [ 14599 ] } ], "ref_ids": [] }, { "id": 14601, "type": "theorem", "label": "descent-lemma-faithfully-flat-universally-injective", "categories": [ "descent" ], "title": "descent-lemma-faithfully-flat-universally-injective", "contents": [ "Any faithfully flat ring map is universally injective." ], "refs": [], "proofs": [ { "contents": [ "This is a reformulation of Algebra, Lemma", "\\ref{algebra-lemma-faithfully-flat-universally-injective}." ], "refs": [ "algebra-lemma-faithfully-flat-universally-injective" ], "ref_ids": [ 814 ] } ], "ref_ids": [] }, { "id": 14602, "type": "theorem", "label": "descent-lemma-C-is-faithful", "categories": [ "descent" ], "title": "descent-lemma-C-is-faithful", "contents": [ "For a ring $R$, the functor $C : \\text{Mod}_R \\to \\text{Mod}_R$ is", "exact and reflects injections and surjections." ], "refs": [], "proofs": [ { "contents": [ "Exactness is More on Algebra, Lemma \\ref{more-algebra-lemma-vee-exact}", "and the other properties follow from this, see", "Remark \\ref{remark-reflects}." ], "refs": [ "more-algebra-lemma-vee-exact", "descent-remark-reflects" ], "ref_ids": [ 10114, 14785 ] } ], "ref_ids": [] }, { "id": 14603, "type": "theorem", "label": "descent-lemma-split-surjection", "categories": [ "descent" ], "title": "descent-lemma-split-surjection", "contents": [ "Let $R$ be a ring. A morphism $f: M \\to N$ in $\\text{Mod}_R$ is universally", "injective if and only if $C(f): C(N) \\to C(M)$ is a split surjection." ], "refs": [], "proofs": [ { "contents": [ "By (\\ref{equation-adjunction}), for any $P \\in \\text{Mod}_R$ we have a ", "commutative diagram", "$$", "\\xymatrix@C=9pc{", "\\Hom_R( P, C(N)) \\ar[r]_{\\Hom_R(P,C(f))} \\ar[d]^{\\cong} &", "\\Hom_R(P,C(M)) \\ar[d]^{\\cong} \\\\", "C(P \\otimes_R N ) \\ar[r]^{C(1_{P} \\otimes f)} & C(P \\otimes_R M ).", "}", "$$", "If $f$ is universally injective, then $1_{C(M)} \\otimes f: C(M) \\otimes_R M \\to ", "C(M) \\otimes_R N$ is injective,", "so both rows in the above diagram are surjective for $P = C(M)$. We may thus ", "lift", "$1_{C(M)} \\in \\Hom_R(C(M), C(M))$ to some $g \\in \\Hom_R(C(N), C(M))$ splitting ", "$C(f)$.", "Conversely, if $C(f)$ is a split surjection, then ", "both rows in the above diagram are surjective,", "so by Lemma \\ref{lemma-C-is-faithful}, $1_{P} \\otimes f$ is injective." ], "refs": [ "descent-lemma-C-is-faithful" ], "ref_ids": [ 14602 ] } ], "ref_ids": [] }, { "id": 14604, "type": "theorem", "label": "descent-lemma-equalizer-M", "categories": [ "descent" ], "title": "descent-lemma-equalizer-M", "contents": [ "For $(M,\\theta) \\in DD_{S/R}$, the diagram", "\\begin{equation}", "\\label{equation-equalizer-M}", "\\xymatrix@C=8pc{", "M \\ar[r]^{\\theta \\circ (1_M \\otimes \\delta_0^1)} &", "M \\otimes_{S, \\delta_1^1} S_2", "\\ar@<1ex>[r]^{(\\theta \\otimes \\delta_2^2) \\circ (1_M \\otimes \\delta^2_0)}", "\\ar@<-1ex>[r]_{1_{M \\otimes S_2} \\otimes \\delta^2_1} & ", "M \\otimes_{S, \\delta_{12}^1} S_3", "}", "\\end{equation}", "is a split equalizer." ], "refs": [], "proofs": [ { "contents": [ "Define the ring homomorphisms $\\sigma^0_0: S_2 \\to S_1$ and $\\sigma_0^1, ", "\\sigma_1^1: S_3 \\to S_2$ by the formulas", "\\begin{align*}", "\\sigma^0_0 (a_0 \\otimes a_1) & = a_0a_1 \\\\", "\\sigma^1_0 (a_0 \\otimes a_1 \\otimes a_2) & = a_0a_1 \\otimes a_2 \\\\", "\\sigma^1_1 (a_0 \\otimes a_1 \\otimes a_2) & = a_0 \\otimes a_1a_2.", "\\end{align*}", "We then take the auxiliary morphisms to be ", "$1_M \\otimes \\sigma_0^0: M \\otimes_{S, \\delta_1^1} S_2 \\to M$", "and $1_M \\otimes \\sigma_0^1: M \\otimes_{S,\\delta_{12}^1} S_3 \\to M \\otimes_{S, ", "\\delta_1^1} S_2$.", "Of the compatibilities required in (\\ref{equation-split-equalizer-conditions}), ", "the first follows from tensoring the cocycle condition", "(\\ref{equation-cocycle-condition}) with $\\sigma_1^1$", "and the others are immediate." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14605, "type": "theorem", "label": "descent-lemma-equalizer-CM", "categories": [ "descent" ], "title": "descent-lemma-equalizer-CM", "contents": [ "For $(M, \\theta) \\in DD_{S/R}$, the diagram", "\\begin{equation}", "\\label{equation-coequalizer-CM}", "\\xymatrix@C=8pc{", "C(M \\otimes_{S, \\delta_{12}^1} S_3)", "\\ar@<1ex>[r]^{C((\\theta \\otimes \\delta_2^2) \\circ (1_M \\otimes \\delta^2_0))}", "\\ar@<-1ex>[r]_{C(1_{M \\otimes S_2} \\otimes \\delta^2_1)} &", "C(M \\otimes_{S, \\delta_1^1} S_2 )", "\\ar[r]^{C(\\theta \\circ (1_M \\otimes \\delta_0^1))} & C(M).", "}", "\\end{equation}", "obtained by applying $C$ to (\\ref{equation-equalizer-M}) is a split", "coequalizer." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14606, "type": "theorem", "label": "descent-lemma-equalizer-S", "categories": [ "descent" ], "title": "descent-lemma-equalizer-S", "contents": [ "The diagram", "\\begin{equation}", "\\label{equation-equalizer-S}", "\\xymatrix@C=8pc{", "S_1 \\ar[r]^{\\delta^1_1} &", "S_2 \\ar@<1ex>[r]^{\\delta^2_2} \\ar@<-1ex>[r]_{\\delta^2_1} & ", "S_3", "}", "\\end{equation}", "is a split equalizer." ], "refs": [], "proofs": [ { "contents": [ "In Lemma \\ref{lemma-equalizer-M}, take $(M, \\theta) = f^*(S)$." ], "refs": [ "descent-lemma-equalizer-M" ], "ref_ids": [ 14604 ] } ], "ref_ids": [] }, { "id": 14607, "type": "theorem", "label": "descent-lemma-descent-lemma", "categories": [ "descent" ], "title": "descent-lemma-descent-lemma", "contents": [ "If $f$ is universally injective, then the diagram", "\\begin{equation}", "\\label{equation-equalizer-f2}", "\\xymatrix@C=8pc{", "f_*(M, \\theta) \\otimes_R S", "\\ar[r]^{\\theta \\circ (1_M \\otimes \\delta_0^1)} &", "M \\otimes_{S, \\delta_1^1} S_2 ", "\\ar@<1ex>[r]^{(\\theta \\otimes \\delta_2^2) \\circ (1_M \\otimes \\delta^2_0)}", "\\ar@<-1ex>[r]_{1_{M \\otimes S_2} \\otimes \\delta^2_1} &", "M \\otimes_{S, \\delta_{12}^1} S_3", "}", "\\end{equation}", "obtained by tensoring (\\ref{equation-equalizer-f}) over $R$ with $S$ is an ", "equalizer." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemma \\ref{lemma-split-surjection} and", "Remark \\ref{remark-functorial-splitting},", "the map $C(1_N \\otimes f): C(N \\otimes_R S) \\to C(N)$ can be split functorially ", "in $N$. This gives the upper vertical arrows in the commutative diagram", "$$", "\\xymatrix@C=8pc{", "C(M \\otimes_{S, \\delta_1^1} S_2)", "\\ar@<1ex>^{C(\\theta \\circ (1_M \\otimes \\delta_0^1))}[r]", "\\ar@<-1ex>_{C(1_M \\otimes \\delta_1^1)}[r] \\ar[d] &", "C(M) \\ar[r]\\ar[d] & C(f_*(M,\\theta)) \\ar@{-->}[d] \\\\", "C(M \\otimes_{S,\\delta_{12}^1} S_3)", "\\ar@<1ex>^{C((\\theta \\otimes \\delta_2^2) \\circ (1_M \\otimes \\delta^2_0))}[r]", "\\ar@<-1ex>_{C(1_{M \\otimes S_2} \\otimes \\delta^2_1)}[r] \\ar[d] &", "C(M \\otimes_{S, \\delta_1^1} S_2 )", "\\ar[r]^{C(\\theta \\circ (1_M \\otimes \\delta_0^1))}", "\\ar[d]^{C(1_M \\otimes \\delta_1^1)} &", "C(M) \\ar[d] \\ar@{=}[dl] \\\\", "C(M \\otimes_{S, \\delta_1^1} S_2)", "\\ar@<1ex>[r]^{C(\\theta \\circ (1_M \\otimes \\delta_0^1))}", "\\ar@<-1ex>[r]_{C(1_M \\otimes \\delta_1^1)} &", "C(M) \\ar[r] &", "C(f_*(M,\\theta))", "}", "$$", "in which the compositions along the columns are identity morphisms.", "The second row is the coequalizer diagram", "(\\ref{equation-coequalizer-CM}); this produces the dashed arrow.", "From the top right square, we obtain auxiliary morphisms $C(f_*(M,\\theta)) \\to ", "C(M)$ ", "and $C(M) \\to C(M\\otimes_{S,\\delta_1^1} S_2)$ which imply that the first row is ", "a split coequalizer diagram.", "By Remark \\ref{remark-adjunction}, we may tensor with $S$ inside $C$ to obtain ", "the split coequalizer diagram", "$$", "\\xymatrix@C=8pc{", "C(M \\otimes_{S,\\delta_2^2 \\circ \\delta_1^1} S_3)", "\\ar@<1ex>^{C((\\theta \\otimes \\delta_2^2) \\circ (1_M \\otimes \\delta^2_0))}[r] ", "\\ar@<-1ex>_{C(1_{M \\otimes S_2} \\otimes \\delta^2_1)}[r] &", "C(M \\otimes_{S, \\delta_1^1} S_2 )", "\\ar[r]^{C(\\theta \\circ (1_M \\otimes \\delta_0^1))} &", "C(f_*(M,\\theta) \\otimes_R S).", "}", "$$", "By Lemma \\ref{lemma-C-is-faithful}, we conclude", "(\\ref{equation-equalizer-f2}) must also be an equalizer." ], "refs": [ "descent-lemma-split-surjection", "descent-remark-functorial-splitting", "descent-remark-adjunction", "descent-lemma-C-is-faithful" ], "ref_ids": [ 14603, 14787, 14786, 14602 ] } ], "ref_ids": [] }, { "id": 14608, "type": "theorem", "label": "descent-lemma-flat-to-injective", "categories": [ "descent" ], "title": "descent-lemma-flat-to-injective", "contents": [ "If $M \\in \\text{Mod}_R$ is flat, then $C(M)$ is an injective $R$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $0 \\to N \\to P \\to Q \\to 0$ be an exact sequence in $\\text{Mod}_R$. Since ", "$M$ is flat,", "$$", "0 \\to N \\otimes_R M \\to P \\otimes_R M \\to Q \\otimes_R M \\to 0", "$$", "is exact.", "By Lemma \\ref{lemma-C-is-faithful},", "$$", "0 \\to C(Q \\otimes_R M) \\to C(P \\otimes_R M) \\to C(N \\otimes_R M) \\to 0", "$$", "is exact. By (\\ref{equation-adjunction}), this last sequence can be rewritten", "as", "$$", "0 \\to \\Hom_R(Q, C(M)) \\to \\Hom_R(P, C(M)) \\to \\Hom_R(N, C(M)) \\to 0.", "$$", "Hence $C(M)$ is an injective object of $\\text{Mod}_R$." ], "refs": [ "descent-lemma-C-is-faithful" ], "ref_ids": [ 14602 ] } ], "ref_ids": [] }, { "id": 14609, "type": "theorem", "label": "descent-lemma-standard-fpqc-covering", "categories": [ "descent" ], "title": "descent-lemma-standard-fpqc-covering", "contents": [ "Let $S$ be an affine scheme.", "Let $\\mathcal{U} = \\{f_i : U_i \\to S\\}_{i = 1, \\ldots, n}$", "be a standard fpqc covering of $S$, see", "Topologies, Definition \\ref{topologies-definition-standard-fpqc}.", "Any descent datum on quasi-coherent sheaves", "for $\\mathcal{U} = \\{U_i \\to S\\}$ is effective.", "Moreover, the functor from the category of", "quasi-coherent $\\mathcal{O}_S$-modules to the category", "of descent data with respect to $\\mathcal{U}$ is fully faithful." ], "refs": [ "topologies-definition-standard-fpqc" ], "proofs": [ { "contents": [ "This is a restatement of Proposition \\ref{proposition-descent-module}", "in terms of schemes. First, note that a descent datum $\\xi$", "for quasi-coherent sheaves with respect to $\\mathcal{U}$", "is exactly the same as a descent datum $\\xi'$ for quasi-coherent sheaves", "with respect to the covering", "$\\mathcal{U}' = \\{\\coprod_{i = 1, \\ldots, n} U_i \\to S\\}$.", "Moreover, effectivity for $\\xi$ is the same as effectivity for $\\xi'$.", "Hence we may assume $n = 1$, i.e., $\\mathcal{U} = \\{U \\to S\\}$", "where $U$ and $S$ are affine. In this case descent data", "correspond to descent data on modules with respect to the ring map", "$$", "\\Gamma(S, \\mathcal{O})", "\\longrightarrow", "\\Gamma(U, \\mathcal{O}).", "$$", "Since $U \\to S$ is surjective and flat, we see that this ring map", "is faithfully flat. In other words,", "Proposition \\ref{proposition-descent-module} applies and we win." ], "refs": [ "descent-proposition-descent-module", "descent-proposition-descent-module" ], "ref_ids": [ 14752, 14752 ] } ], "ref_ids": [ 12548 ] }, { "id": 14610, "type": "theorem", "label": "descent-lemma-galois-descent", "categories": [ "descent" ], "title": "descent-lemma-galois-descent", "contents": [ "Let $k'/k$ be a (finite) Galois extension with Galois group $G$.", "Let $X$ be a scheme over $k$. The category of quasi-coherent", "$\\mathcal{O}_X$-modules is equivalent to the category of systems", "$(\\mathcal{F}, (\\varphi_\\sigma)_{\\sigma \\in G})$ where", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a quasi-coherent module on $X_{k'}$,", "\\item $\\varphi_\\sigma : \\mathcal{F} \\to f_\\sigma^*\\mathcal{F}$", "is an isomorphism of modules,", "\\item $\\varphi_{\\sigma\\tau} = f_\\sigma^*\\varphi_\\tau \\circ \\varphi_\\sigma$", "for all $\\sigma, \\tau \\in G$.", "\\end{enumerate}", "Here $f_\\sigma = \\text{id}_X \\times \\Spec(\\sigma) : X_{k'} \\to X_{k'}$." ], "refs": [], "proofs": [ { "contents": [ "As seen above a datum $(\\mathcal{F}, (\\varphi_\\sigma)_{\\sigma \\in G})$", "as in the lemma is the same thing as a descent datum for the", "fpqc covering $\\{X_{k'} \\to X\\}$. Thus the lemma follows from", "Proposition \\ref{proposition-fpqc-descent-quasi-coherent}." ], "refs": [ "descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 14753 ] } ], "ref_ids": [] }, { "id": 14611, "type": "theorem", "label": "descent-lemma-galois-descent-more-general", "categories": [ "descent" ], "title": "descent-lemma-galois-descent-more-general", "contents": [ "Let $X \\to Y$, $G$, and $f_\\sigma : X \\to X$ be as above.", "The category of quasi-coherent", "$\\mathcal{O}_Y$-modules is equivalent to the category of systems", "$(\\mathcal{F}, (\\varphi_\\sigma)_{\\sigma \\in G})$ where", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module,", "\\item $\\varphi_\\sigma : \\mathcal{F} \\to f_\\sigma^*\\mathcal{F}$", "is an isomorphism of modules,", "\\item $\\varphi_{\\sigma\\tau} = f_\\sigma^*\\varphi_\\tau \\circ \\varphi_\\sigma$", "for all $\\sigma, \\tau \\in G$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Since $X \\to Y$ is surjective finite \\'etale $\\{X \\to Y\\}$ is", "an fpqc covering. Since", "$G \\times X \\to X \\times_Y X$, $(\\sigma, x) \\mapsto (x, f_\\sigma(x))$", "is an isomorphism, we see that", "$G \\times G \\times X \\to X \\times_Y X \\times_Y X$,", "$(\\sigma, \\tau, x) \\mapsto (x, f_\\sigma(x), f_{\\sigma\\tau}(x))$", "is an isomorphism too. Using these identifications, the category of", "data as in the lemma is the same as the category of descent data", "for quasi-coherent sheaves for the covering $\\{x \\to Y\\}$.", "Thus the lemma follows from", "Proposition \\ref{proposition-fpqc-descent-quasi-coherent}." ], "refs": [ "descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 14753 ] } ], "ref_ids": [] }, { "id": 14612, "type": "theorem", "label": "descent-lemma-finite-type-descends", "categories": [ "descent" ], "title": "descent-lemma-finite-type-descends", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that", "each $f_i^*\\mathcal{F}$ is a finite type $\\mathcal{O}_{X_i}$-module.", "Then $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "Omitted. For the affine case, see", "Algebra, Lemma \\ref{algebra-lemma-descend-properties-modules}." ], "refs": [ "algebra-lemma-descend-properties-modules" ], "ref_ids": [ 819 ] } ], "ref_ids": [] }, { "id": 14613, "type": "theorem", "label": "descent-lemma-finite-type-descends-fppf", "categories": [ "descent" ], "title": "descent-lemma-finite-type-descends-fppf", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of", "locally ringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_Y$-modules.", "If", "\\begin{enumerate}", "\\item $f$ is open as a map of topological spaces,", "\\item $f$ is surjective and flat, and", "\\item $f^*\\mathcal{F}$ is of finite type,", "\\end{enumerate}", "then $\\mathcal{F}$ is of finite type." ], "refs": [], "proofs": [ { "contents": [ "Let $y \\in Y$ be a point. Choose a point $x \\in X$ mapping to $y$.", "Choose an open $x \\in U \\subset X$ and elements $s_1, \\ldots, s_n$", "of $f^*\\mathcal{F}(U)$ which generate $f^*\\mathcal{F}$ over $U$.", "Since $f^*\\mathcal{F} =", "f^{-1}\\mathcal{F} \\otimes_{f^{-1}\\mathcal{O}_Y} \\mathcal{O}_X$", "we can after shrinking $U$ assume $s_i = \\sum t_{ij} \\otimes a_{ij}$", "with $t_{ij} \\in f^{-1}\\mathcal{F}(U)$ and $a_{ij} \\in \\mathcal{O}_X(U)$.", "After shrinking $U$ further we may assume that $t_{ij}$ comes from", "a section $s_{ij} \\in \\mathcal{F}(V)$ for some $V \\subset Y$ open", "with $f(U) \\subset V$. Let $N$ be the number of sections $s_{ij}$ and", "consider the map", "$$", "\\sigma = (s_{ij}) : \\mathcal{O}_V^{\\oplus N} \\to \\mathcal{F}|_V", "$$", "By our choice of the sections we see that $f^*\\sigma|_U$ is surjective.", "Hence for every $u \\in U$ the map", "$$", "\\sigma_{f(u)} \\otimes_{\\mathcal{O}_{Y, f(u)}} \\mathcal{O}_{X, u} :", "\\mathcal{O}_{X, u}^{\\oplus N}", "\\longrightarrow", "\\mathcal{F}_{f(u)} \\otimes_{\\mathcal{O}_{Y, f(u)}} \\mathcal{O}_{X, u}", "$$", "is surjective. As $f$ is flat, the local ring map", "$\\mathcal{O}_{Y, f(u)} \\to \\mathcal{O}_{X, u}$ is flat, hence", "faithfully flat (Algebra, Lemma \\ref{algebra-lemma-local-flat-ff}).", "Hence $\\sigma_{f(u)}$ is surjective. Since $f$ is open, $f(U)$ is", "an open neighbourhood of $y$ and the proof is done." ], "refs": [ "algebra-lemma-local-flat-ff" ], "ref_ids": [ 537 ] } ], "ref_ids": [] }, { "id": 14614, "type": "theorem", "label": "descent-lemma-finite-presentation-descends", "categories": [ "descent" ], "title": "descent-lemma-finite-presentation-descends", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that", "each $f_i^*\\mathcal{F}$ is an $\\mathcal{O}_{X_i}$-module of finite", "presentation. Then $\\mathcal{F}$ is an $\\mathcal{O}_X$-module", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Omitted. For the affine case, see", "Algebra, Lemma \\ref{algebra-lemma-descend-properties-modules}." ], "refs": [ "algebra-lemma-descend-properties-modules" ], "ref_ids": [ 819 ] } ], "ref_ids": [] }, { "id": 14615, "type": "theorem", "label": "descent-lemma-locally-generated-by-r-sections-descends", "categories": [ "descent" ], "title": "descent-lemma-locally-generated-by-r-sections-descends", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that", "each $f_i^*\\mathcal{F}$ is locally generated by $r$ sections as an", "$\\mathcal{O}_{X_i}$-module. Then $\\mathcal{F}$ is locally generated by", "$r$ sections as an $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-type-descends} we see that $\\mathcal{F}$", "is of finite type. Hence Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK}) implies that $\\mathcal{F}$", "is generated by $r$ sections in the neighbourhood of a point $x \\in X$", "if and only if $\\dim_{\\kappa(x)} \\mathcal{F}_x \\otimes \\kappa(x) \\leq r$.", "Choose an $i$ and a point $x_i \\in X_i$ mapping to $x$. Then", "$\\dim_{\\kappa(x)} \\mathcal{F}_x \\otimes \\kappa(x) = ", "\\dim_{\\kappa(x_i)} (f_i^*\\mathcal{F})_{x_i} \\otimes \\kappa(x_i)$", "which is $\\leq r$ as $f_i^*\\mathcal{F}$ is locally generated by $r$", "sections." ], "refs": [ "descent-lemma-finite-type-descends", "algebra-lemma-NAK" ], "ref_ids": [ 14612, 401 ] } ], "ref_ids": [] }, { "id": 14616, "type": "theorem", "label": "descent-lemma-flat-descends", "categories": [ "descent" ], "title": "descent-lemma-flat-descends", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that", "each $f_i^*\\mathcal{F}$ is a flat $\\mathcal{O}_{X_i}$-module.", "Then $\\mathcal{F}$ is a flat $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "Omitted. For the affine case, see", "Algebra, Lemma \\ref{algebra-lemma-descend-properties-modules}." ], "refs": [ "algebra-lemma-descend-properties-modules" ], "ref_ids": [ 819 ] } ], "ref_ids": [] }, { "id": 14617, "type": "theorem", "label": "descent-lemma-finite-locally-free-descends", "categories": [ "descent" ], "title": "descent-lemma-finite-locally-free-descends", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that", "each $f_i^*\\mathcal{F}$ is a finite locally free $\\mathcal{O}_{X_i}$-module.", "Then $\\mathcal{F}$ is a finite locally free $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "This follows from the fact that a quasi-coherent sheaf is finite locally", "free if and only if it is of finite presentation and flat, see", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}.", "Namely, if each $f_i^*\\mathcal{F}$ is flat and of finite presentation,", "then so is $\\mathcal{F}$ by", "Lemmas \\ref{lemma-flat-descends} and", "\\ref{lemma-finite-presentation-descends}." ], "refs": [ "algebra-lemma-finite-projective", "descent-lemma-flat-descends", "descent-lemma-finite-presentation-descends" ], "ref_ids": [ 795, 14616, 14614 ] } ], "ref_ids": [] }, { "id": 14618, "type": "theorem", "label": "descent-lemma-locally-projective-descends", "categories": [ "descent" ], "title": "descent-lemma-locally-projective-descends", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that", "each $f_i^*\\mathcal{F}$ is a locally projective $\\mathcal{O}_{X_i}$-module.", "Then $\\mathcal{F}$ is a locally projective $\\mathcal{O}_X$-module." ], "refs": [], "proofs": [ { "contents": [ "Omitted. For Zariski coverings this is", "Properties, Lemma \\ref{properties-lemma-locally-projective}.", "For the affine case this is", "Algebra, Theorem \\ref{algebra-theorem-ffdescent-projectivity}." ], "refs": [ "properties-lemma-locally-projective", "algebra-theorem-ffdescent-projectivity" ], "ref_ids": [ 3016, 324 ] } ], "ref_ids": [] }, { "id": 14619, "type": "theorem", "label": "descent-lemma-finite-over-finite-module", "categories": [ "descent" ], "title": "descent-lemma-finite-over-finite-module", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume $f$ is a finite morphism.", "Then $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type", "if and only if $f_*\\mathcal{F}$ is an $\\mathcal{O}_Y$-module of finite", "type." ], "refs": [], "proofs": [ { "contents": [ "As $f$ is finite it is affine. This reduces us to the case where", "$f$ is the morphism $\\Spec(B) \\to \\Spec(A)$ given", "by a finite ring map $A \\to B$.", "Moreover, then $\\mathcal{F} = \\widetilde{M}$ is the sheaf of modules", "associated to the $B$-module $M$.", "Note that $M$ is finite as a $B$-module if and only if", "$M$ is finite as an $A$-module, see", "Algebra, Lemma \\ref{algebra-lemma-finite-module-over-finite-extension}.", "Combined with", "Properties, Lemma \\ref{properties-lemma-finite-type-module}", "this proves the lemma." ], "refs": [ "algebra-lemma-finite-module-over-finite-extension", "properties-lemma-finite-type-module" ], "ref_ids": [ 336, 3002 ] } ], "ref_ids": [] }, { "id": 14620, "type": "theorem", "label": "descent-lemma-finite-finitely-presented-module", "categories": [ "descent" ], "title": "descent-lemma-finite-finitely-presented-module", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume $f$ is finite and of finite presentation.", "Then $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation", "if and only if $f_*\\mathcal{F}$ is an $\\mathcal{O}_Y$-module of finite", "presentation." ], "refs": [], "proofs": [ { "contents": [ "As $f$ is finite it is affine. This reduces us to the case where", "$f$ is the morphism $\\Spec(B) \\to \\Spec(A)$ given", "by a finite and finitely presented ring map $A \\to B$.", "Moreover, then $\\mathcal{F} = \\widetilde{M}$ is the sheaf of modules", "associated to the $B$-module $M$.", "Note that $M$ is finitely presented as a $B$-module if and only if", "$M$ is finitely presented as an $A$-module, see", "Algebra, Lemma \\ref{algebra-lemma-finite-finitely-presented-extension}.", "Combined with", "Properties, Lemma \\ref{properties-lemma-finite-presentation-module}", "this proves the lemma." ], "refs": [ "algebra-lemma-finite-finitely-presented-extension", "properties-lemma-finite-presentation-module" ], "ref_ids": [ 501, 3003 ] } ], "ref_ids": [] }, { "id": 14621, "type": "theorem", "label": "descent-lemma-sheaf-condition-holds", "categories": [ "descent" ], "title": "descent-lemma-sheaf-condition-holds", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module.", "Let $\\tau \\in \\{Zariski, \\linebreak[0] fpqc, \\linebreak[0] fppf, \\linebreak[0]", "\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$.", "The functor defined in (\\ref{equation-quasi-coherent-presheaf})", "satisfies the sheaf condition with respect to any $\\tau$-covering", "$\\{T_i \\to T\\}_{i \\in I}$ of any scheme $T$ over $S$." ], "refs": [], "proofs": [ { "contents": [ "For $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0] \\etale,", "\\linebreak[0] smooth, \\linebreak[0] syntomic\\}$ a $\\tau$-covering", "is also a fpqc-covering, see the results in", "Topologies, Lemmas", "\\ref{topologies-lemma-zariski-etale},", "\\ref{topologies-lemma-zariski-etale-smooth},", "\\ref{topologies-lemma-zariski-etale-smooth-syntomic},", "\\ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf}, and", "\\ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc}.", "Hence it suffices to prove the theorem", "for a fpqc covering. Assume that $\\{f_i : T_i \\to T\\}_{i \\in I}$", "is an fpqc covering where $f : T \\to S$ is given. Suppose that", "we have a family of sections $s_i \\in \\Gamma(T_i , f_i^*f^*\\mathcal{F})$", "such that $s_i|_{T_i \\times_T T_j} = s_j|_{T_i \\times_T T_j}$.", "We have to find the correspond section $s \\in \\Gamma(T, f^*\\mathcal{F})$.", "We can reinterpret the $s_i$ as a family of maps", "$\\varphi_i : f_i^*\\mathcal{O}_T = \\mathcal{O}_{T_i} \\to f_i^*f^*\\mathcal{F}$", "compatible with the canonical descent data associated to the", "quasi-coherent sheaves $\\mathcal{O}_T$ and $f^*\\mathcal{F}$ on $T$.", "Hence by Proposition \\ref{proposition-fpqc-descent-quasi-coherent}", "we see that we may (uniquely) descend", "these to a map $\\mathcal{O}_T \\to f^*\\mathcal{F}$ which gives", "us our section $s$." ], "refs": [ "topologies-lemma-zariski-etale", "topologies-lemma-zariski-etale-smooth", "topologies-lemma-zariski-etale-smooth-syntomic", "topologies-lemma-zariski-etale-smooth-syntomic-fppf", "topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc", "descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 12445, 12459, 12465, 12471, 12497, 14753 ] } ], "ref_ids": [] }, { "id": 14622, "type": "theorem", "label": "descent-lemma-compare-sites", "categories": [ "descent" ], "title": "descent-lemma-compare-sites", "contents": [ "Let $S$ be a scheme. Denote", "$$", "\\begin{matrix}", "\\text{id}_{\\tau, Zar} & : & (\\Sch/S)_\\tau \\to S_{Zar}, &", "\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\} \\\\", "\\text{id}_{\\tau, \\etale} & : &", "(\\Sch/S)_\\tau \\to S_\\etale, &", "\\tau \\in \\{\\etale, smooth, syntomic, fppf\\} \\\\", "\\text{id}_{small, \\etale, Zar} & : & S_\\etale \\to S_{Zar},", "\\end{matrix}", "$$", "the morphisms of ringed sites of", "Remark \\ref{remark-change-topologies-ringed}.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_S$-modules", "which we view a sheaf of $\\mathcal{O}$-modules on $S_{Zar}$. Then", "\\begin{enumerate}", "\\item $(\\text{id}_{\\tau, Zar})^*\\mathcal{F}$ is the $\\tau$-sheafification", "of the Zariski sheaf", "$$", "(f : T \\to S) \\longmapsto \\Gamma(T, f^*\\mathcal{F})", "$$", "on $(\\Sch/S)_\\tau$, and", "\\item $(\\text{id}_{small, \\etale, Zar})^*\\mathcal{F}$ is the", "\\'etale sheafification of the Zariski sheaf", "$$", "(f : T \\to S) \\longmapsto \\Gamma(T, f^*\\mathcal{F})", "$$", "on $S_\\etale$.", "\\end{enumerate}", "Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules", "on $S_\\etale$. Then", "\\begin{enumerate}", "\\item[(3)] $(\\text{id}_{\\tau, \\etale})^*\\mathcal{G}$ is the", "$\\tau$-sheafification of the \\'etale sheaf", "$$", "(f : T \\to S) \\longmapsto \\Gamma(T, f_{small}^*\\mathcal{G})", "$$", "where $f_{small} : T_\\etale \\to S_\\etale$", "is the morphism of ringed small \\'etale sites of", "Remark \\ref{remark-change-topologies-ringed}.", "\\end{enumerate}" ], "refs": [ "descent-remark-change-topologies-ringed", "descent-remark-change-topologies-ringed" ], "proofs": [ { "contents": [ "Proof of (1). We first note that the result is true when $\\tau = Zar$", "because in that case we have the morphism of topoi", "$i_f : \\Sh(T_{Zar}) \\to \\Sh((\\Sch/S)_{Zar})$", "such that $\\text{id}_{\\tau, Zar} \\circ i_f = f_{small}$ as morphisms", "$T_{Zar} \\to S_{Zar}$, see", "Topologies, Lemmas \\ref{topologies-lemma-put-in-T} and", "\\ref{topologies-lemma-morphism-big-small}.", "Since pullback is transitive (see", "Modules on Sites,", "Lemma \\ref{sites-modules-lemma-push-pull-composition-modules})", "we see that", "$i_f^*(\\text{id}_{\\tau, Zar})^*\\mathcal{F} = f_{small}^*\\mathcal{F}$", "as desired. Hence, by the remark preceding this lemma we see that", "$(\\text{id}_{\\tau, Zar})^*\\mathcal{F}$ is the $\\tau$-sheafification of", "the presheaf $T \\mapsto \\Gamma(T, f^*\\mathcal{F})$.", "\\medskip\\noindent", "The proof of (3) is exactly the same as the proof of (1), except that it", "uses", "Topologies, Lemmas \\ref{topologies-lemma-put-in-T-etale} and", "\\ref{topologies-lemma-morphism-big-small-etale}.", "We omit the proof of (2)." ], "refs": [ "topologies-lemma-put-in-T", "topologies-lemma-morphism-big-small", "sites-modules-lemma-push-pull-composition-modules", "topologies-lemma-put-in-T-etale", "topologies-lemma-morphism-big-small-etale" ], "ref_ids": [ 12438, 12441, 14156, 12452, 12455 ] } ], "ref_ids": [ 14792, 14792 ] }, { "id": 14623, "type": "theorem", "label": "descent-lemma-quasi-coherent-gives-quasi-coherent", "categories": [ "descent" ], "title": "descent-lemma-quasi-coherent-gives-quasi-coherent", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module.", "Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0]", "\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$.", "\\begin{enumerate}", "\\item The sheaf $\\mathcal{F}^a$ is a quasi-coherent", "$\\mathcal{O}$-module on $(\\Sch/S)_\\tau$, as defined in", "Modules on Sites, Definition \\ref{sites-modules-definition-site-local}.", "\\item If $\\tau = \\etale$ (resp.\\ $\\tau = Zariski$), then the sheaf", "$\\mathcal{F}^a$ is a quasi-coherent $\\mathcal{O}$-module on", "$S_\\etale$ (resp.\\ $S_{Zar}$) as defined in", "Modules on Sites, Definition \\ref{sites-modules-definition-site-local}.", "\\end{enumerate}" ], "refs": [ "sites-modules-definition-site-local", "sites-modules-definition-site-local" ], "proofs": [ { "contents": [ "Let $\\{S_i \\to S\\}$ be a Zariski covering such that we have exact sequences", "$$", "\\bigoplus\\nolimits_{k \\in K_i} \\mathcal{O}_{S_i} \\longrightarrow", "\\bigoplus\\nolimits_{j \\in J_i} \\mathcal{O}_{S_i} \\longrightarrow", "\\mathcal{F} \\longrightarrow 0", "$$", "for some index sets $K_i$ and $J_i$. This is possible by the definition", "of a quasi-coherent sheaf on a ringed space", "(See Modules, Definition \\ref{modules-definition-quasi-coherent}).", "\\medskip\\noindent", "Proof of (1). Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0]", "\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$.", "It is clear that $\\mathcal{F}^a|_{(\\Sch/S_i)_\\tau}$ also", "sits in an exact sequence", "$$", "\\bigoplus\\nolimits_{k \\in K_i} \\mathcal{O}|_{(\\Sch/S_i)_\\tau}", "\\longrightarrow", "\\bigoplus\\nolimits_{j \\in J_i} \\mathcal{O}|_{(\\Sch/S_i)_\\tau}", "\\longrightarrow", "\\mathcal{F}^a|_{(\\Sch/S_i)_\\tau} \\longrightarrow 0", "$$", "Hence $\\mathcal{F}^a$ is quasi-coherent by Modules on Sites,", "Lemma \\ref{sites-modules-lemma-local-final-object}.", "\\medskip\\noindent", "Proof of (2). Let $\\tau = \\etale$.", "It is clear that $\\mathcal{F}^a|_{(S_i)_\\etale}$ also sits", "in an exact sequence", "$$", "\\bigoplus\\nolimits_{k \\in K_i} \\mathcal{O}|_{(S_i)_\\etale}", "\\longrightarrow", "\\bigoplus\\nolimits_{j \\in J_i} \\mathcal{O}|_{(S_i)_\\etale}", "\\longrightarrow", "\\mathcal{F}^a|_{(S_i)_\\etale} \\longrightarrow 0", "$$", "Hence $\\mathcal{F}^a$ is quasi-coherent by Modules on Sites,", "Lemma \\ref{sites-modules-lemma-local-final-object}.", "The case $\\tau = Zariski$ is similar (actually, it is really", "tautological since the corresponding ringed topoi agree)." ], "refs": [ "modules-definition-quasi-coherent", "sites-modules-lemma-local-final-object", "sites-modules-lemma-local-final-object" ], "ref_ids": [ 13337, 14185, 14185 ] } ], "ref_ids": [ 14289, 14289 ] }, { "id": 14624, "type": "theorem", "label": "descent-lemma-standard-covering-Cech", "categories": [ "descent" ], "title": "descent-lemma-standard-covering-Cech", "contents": [ "Let $S$ be a scheme. Let", "\\begin{enumerate}", "\\item[(a)] $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0]", "\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$", "and $\\mathcal{C} = (\\Sch/S)_\\tau$, or", "\\item[(b)] let $\\tau = \\etale$ and $\\mathcal{C} = S_\\etale$, or", "\\item[(c)] let $\\tau = Zariski$ and $\\mathcal{C} = S_{Zar}$.", "\\end{enumerate}", "Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}$.", "Let $U \\in \\Ob(\\mathcal{C})$ be affine.", "Let $\\mathcal{U} = \\{U_i \\to U\\}_{i = 1, \\ldots, n}$ be a standard affine", "$\\tau$-covering in $\\mathcal{C}$. Then", "\\begin{enumerate}", "\\item $\\mathcal{V} = \\{\\coprod_{i = 1, \\ldots, n} U_i \\to U\\}$ is a", "$\\tau$-covering of $U$,", "\\item $\\mathcal{U}$ is a refinement of $\\mathcal{V}$, and", "\\item the induced map on {\\v C}ech complexes", "(Cohomology on Sites,", "Equation (\\ref{sites-cohomology-equation-map-cech-complexes}))", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{F})", "\\longrightarrow", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})", "$$", "is an isomorphism of complexes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This follows because", "$$", "(\\coprod\\nolimits_{i_0 = 1, \\ldots, n} U_{i_0}) \\times_U", "\\ldots \\times_U", "(\\coprod\\nolimits_{i_p = 1, \\ldots, n} U_{i_p})", "=", "\\coprod\\nolimits_{i_0, \\ldots, i_p \\in \\{1, \\ldots, n\\}}", "U_{i_0} \\times_U \\ldots \\times_U U_{i_p}", "$$", "and the fact that $\\mathcal{F}(\\coprod_a V_a) = \\prod_a \\mathcal{F}(V_a)$", "since disjoint unions are $\\tau$-coverings." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14625, "type": "theorem", "label": "descent-lemma-standard-covering-Cech-quasi-coherent", "categories": [ "descent" ], "title": "descent-lemma-standard-covering-Cech-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $S$.", "Let $\\tau$, $\\mathcal{C}$, $U$, $\\mathcal{U}$ be as in", "Lemma \\ref{lemma-standard-covering-Cech}. Then there is an isomorphism", "of complexes", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^a)", "\\cong", "s((A/R)_\\bullet \\otimes_R M)", "$$", "(see Section \\ref{section-descent-modules})", "where $R = \\Gamma(U, \\mathcal{O}_U)$, $M = \\Gamma(U, \\mathcal{F}^a)$", "and $R \\to A$ is a faithfully flat ring map. In particular", "$$", "\\check{H}^p(\\mathcal{U}, \\mathcal{F}^a) = 0", "$$", "for all $p \\geq 1$." ], "refs": [ "descent-lemma-standard-covering-Cech" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-standard-covering-Cech} we see that", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^a)$", "is isomorphic to $\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{F}^a)$", "where $\\mathcal{V} = \\{V \\to U\\}$ with $V = \\coprod_{i = 1, \\ldots n} U_i$", "affine also. Set $A = \\Gamma(V, \\mathcal{O}_V)$. Since $\\{V \\to U\\}$", "is a $\\tau$-covering we see that $R \\to A$ is faithfully flat.", "On the other hand, by definition of $\\mathcal{F}^a$ we have", "that the degree $p$ term $\\check{\\mathcal{C}}^p(\\mathcal{V}, \\mathcal{F}^a)$", "is", "$$", "\\Gamma(V \\times_U \\ldots \\times_U V, \\mathcal{F}^a)", "=", "\\Gamma(\\Spec(A \\otimes_R \\ldots \\otimes_R A), \\mathcal{F}^a)", "=", "A \\otimes_R \\ldots \\otimes_R A \\otimes_R M", "$$", "We omit the verification that the maps of the {\\v C}ech complex agree with", "the maps in the complex $s((A/R)_\\bullet \\otimes_R M)$. The vanishing", "of cohomology is Lemma \\ref{lemma-ff-exact}." ], "refs": [ "descent-lemma-standard-covering-Cech", "descent-lemma-ff-exact" ], "ref_ids": [ 14624, 14598 ] } ], "ref_ids": [ 14624 ] }, { "id": 14626, "type": "theorem", "label": "descent-lemma-equivalence-quasi-coherent-properties", "categories": [ "descent" ], "title": "descent-lemma-equivalence-quasi-coherent-properties", "contents": [ "Let $S$ be a scheme.", "Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0]", "\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$.", "Let $\\mathcal{P}$ be one of the properties of modules\\footnote{The list is:", "free, finite free, generated by global sections,", "generated by $r$ global sections, generated by finitely many global sections,", "having a global presentation, having a global finite presentation,", "locally free, finite locally free, locally generated by sections,", "locally generated by $r$ sections, finite type, of finite presentation,", "coherent, or flat.} defined in", "Modules on Sites, Definitions \\ref{sites-modules-definition-global},", "\\ref{sites-modules-definition-site-local}, and", "\\ref{sites-modules-definition-flat}.", "The equivalences of categories", "$$", "\\QCoh(\\mathcal{O}_S)", "\\longrightarrow", "\\QCoh((\\Sch/S)_\\tau, \\mathcal{O})", "\\quad\\text{and}\\quad", "\\QCoh(\\mathcal{O}_S)", "\\longrightarrow", "\\QCoh(S_\\tau, \\mathcal{O})", "$$", "defined by the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ seen in", "Proposition \\ref{proposition-equivalence-quasi-coherent}", "have the property", "$$", "\\mathcal{F}\\text{ has }\\mathcal{P}", "\\Leftrightarrow", "\\mathcal{F}^a\\text{ has }\\mathcal{P}\\text{ as an }\\mathcal{O}\\text{-module}", "$$", "except (possibly) when $\\mathcal{P}$ is ``locally free'' or ``coherent''.", "If $\\mathcal{P}=$``coherent'' the equivalence", "holds for $\\QCoh(\\mathcal{O}_S) \\to \\QCoh(S_\\tau, \\mathcal{O})$", "when $S$ is locally Noetherian and $\\tau$ is Zariski or \\'etale." ], "refs": [ "sites-modules-definition-global", "sites-modules-definition-site-local", "sites-modules-definition-flat", "descent-proposition-equivalence-quasi-coherent" ], "proofs": [ { "contents": [ "This is immediate for the global properties, i.e., those defined in", "Modules on Sites, Definition \\ref{sites-modules-definition-global}.", "For the local properties we can use", "Modules on Sites, Lemma \\ref{sites-modules-lemma-local-final-object}", "to translate ``$\\mathcal{F}^a$ has $\\mathcal{P}$'' into a property", "on the members of a covering of $X$. Hence the result follows from", "Lemmas \\ref{lemma-finite-type-descends},", "\\ref{lemma-finite-presentation-descends},", "\\ref{lemma-locally-generated-by-r-sections-descends},", "\\ref{lemma-flat-descends}, and", "\\ref{lemma-finite-locally-free-descends}.", "Being coherent for a quasi-coherent module is the same as being", "of finite type over a locally Noetherian scheme (see", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-coherent-Noetherian})", "hence this reduces", "to the case of finite type modules (details omitted)." ], "refs": [ "sites-modules-definition-global", "sites-modules-lemma-local-final-object", "descent-lemma-finite-type-descends", "descent-lemma-finite-presentation-descends", "descent-lemma-locally-generated-by-r-sections-descends", "descent-lemma-flat-descends", "descent-lemma-finite-locally-free-descends", "coherent-lemma-coherent-Noetherian" ], "ref_ids": [ 14286, 14185, 14612, 14614, 14615, 14616, 14617, 3308 ] } ], "ref_ids": [ 14286, 14289, 14290, 14755 ] }, { "id": 14627, "type": "theorem", "label": "descent-lemma-equivalence-quasi-coherent-limits", "categories": [ "descent" ], "title": "descent-lemma-equivalence-quasi-coherent-limits", "contents": [ "Let $S$ be a scheme.", "Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0]", "\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$.", "The functors", "$$", "\\QCoh(\\mathcal{O}_S)", "\\longrightarrow", "\\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O})", "\\quad\\text{and}\\quad", "\\QCoh(\\mathcal{O}_S)", "\\longrightarrow", "\\textit{Mod}(S_\\tau, \\mathcal{O})", "$$", "defined by the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ seen in", "Proposition \\ref{proposition-equivalence-quasi-coherent}", "are", "\\begin{enumerate}", "\\item fully faithful,", "\\item compatible with direct sums,", "\\item compatible with colimits,", "\\item right exact,", "\\item exact as a functor", "$\\QCoh(\\mathcal{O}_S) \\to \\textit{Mod}(S_\\etale, \\mathcal{O})$,", "\\item {\\bf not} exact as a functor", "$\\QCoh(\\mathcal{O}_S) \\to", "\\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O})$", "in general,", "\\item given two quasi-coherent $\\mathcal{O}_S$-modules", "$\\mathcal{F}$, $\\mathcal{G}$ we have", "$(\\mathcal{F} \\otimes_{\\mathcal{O}_S} \\mathcal{G})^a =", "\\mathcal{F}^a \\otimes_\\mathcal{O} \\mathcal{G}^a$,", "\\item given two quasi-coherent $\\mathcal{O}_S$-modules", "$\\mathcal{F}$, $\\mathcal{G}$ such that $\\mathcal{F}$", "is of finite presentation we have", "$(\\SheafHom_{\\mathcal{O}_S}(\\mathcal{F}, \\mathcal{G}))^a =", "\\SheafHom_\\mathcal{O}(\\mathcal{F}^a, \\mathcal{G}^a)$, and", "\\item given a short exact sequence", "$0 \\to \\mathcal{F}_1^a \\to \\mathcal{E} \\to \\mathcal{F}_2^a \\to 0$", "of $\\mathcal{O}$-modules then $\\mathcal{E}$ is", "quasi-coherent\\footnote{Warning: This is misleading. See part (6).}, i.e.,", "$\\mathcal{E}$ is in the essential image of the functor.", "\\end{enumerate}" ], "refs": [ "descent-proposition-equivalence-quasi-coherent" ], "proofs": [ { "contents": [ "Part (1) we saw in", "Proposition \\ref{proposition-equivalence-quasi-coherent}.", "\\medskip\\noindent", "We have seen in", "Schemes, Section \\ref{schemes-section-quasi-coherent}", "that a colimit of quasi-coherent sheaves on a scheme is a quasi-coherent", "sheaf. Moreover, in", "Remark \\ref{remark-change-topologies-ringed-sites}", "we saw that $\\mathcal{F} \\mapsto \\mathcal{F}^a$ is the pullback functor", "for a morphism of ringed sites, hence commutes with all colimits, see", "Modules on Sites, Lemma", "\\ref{sites-modules-lemma-exactness-pushforward-pullback}.", "Thus (3) and its special case (3) hold.", "\\medskip\\noindent", "This also shows that the functor is right exact (i.e., commutes with", "finite colimits), hence (4).", "\\medskip\\noindent", "The functor $\\QCoh(\\mathcal{O}_S) \\to", "\\QCoh(S_\\etale, \\mathcal{O})$,", "$\\mathcal{F} \\mapsto \\mathcal{F}^a$", "is left exact because an \\'etale morphism is flat, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-flat}.", "This proves (5).", "\\medskip\\noindent", "To see (6), suppose that $S = \\Spec(\\mathbf{Z})$.", "Then $2 : \\mathcal{O}_S \\to \\mathcal{O}_S$ is injective but the associated", "map of $\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$ isn't", "injective because $2 : \\mathbf{F}_2 \\to \\mathbf{F}_2$ isn't injective", "and $\\Spec(\\mathbf{F}_2)$ is an object of $(\\Sch/S)_\\tau$.", "\\medskip\\noindent", "We omit the proofs of (7) and (8).", "\\medskip\\noindent", "Let $0 \\to \\mathcal{F}_1^a \\to \\mathcal{E} \\to \\mathcal{F}_2^a \\to 0$", "be a short exact sequence of $\\mathcal{O}$-modules with $\\mathcal{F}_1$", "and $\\mathcal{F}_2$ quasi-coherent on $S$. Consider the restriction", "$$", "0 \\to \\mathcal{F}_1 \\to \\mathcal{E}|_{S_{Zar}} \\to \\mathcal{F}_2", "$$", "to $S_{Zar}$. By", "Proposition \\ref{proposition-same-cohomology-quasi-coherent}", "we see that on any affine $U \\subset S$ we have", "$H^1(U, \\mathcal{F}_1^a) = H^1(U, \\mathcal{F}_1) = 0$.", "Hence the sequence above is also exact on the right. By", "Schemes, Section \\ref{schemes-section-quasi-coherent}", "we conclude that $\\mathcal{F} = \\mathcal{E}|_{S_{Zar}}$ is", "quasi-coherent. Thus we obtain a commutative diagram", "$$", "\\xymatrix{", "& \\mathcal{F}_1^a \\ar[r] \\ar[d] &", "\\mathcal{F}^a \\ar[r] \\ar[d] &", "\\mathcal{F}_2^a \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "\\mathcal{F}_1^a \\ar[r] &", "\\mathcal{E} \\ar[r] &", "\\mathcal{F}_2^a \\ar[r] & 0", "}", "$$", "To finish the proof it suffices to show that the top row is also", "right exact. To do this, denote once more $U = \\Spec(A) \\subset S$", "an affine open of $S$. We have seen above that", "$0 \\to \\mathcal{F}_1(U) \\to \\mathcal{E}(U) \\to \\mathcal{F}_2(U) \\to 0$", "is exact. For any affine scheme $V/U$,", "$V = \\Spec(B)$ the map $\\mathcal{F}_1^a(V) \\to \\mathcal{E}(V)$", "is injective. We have $\\mathcal{F}_1^a(V) = \\mathcal{F}_1(U) \\otimes_A B$", "by definition. The injection", "$\\mathcal{F}_1^a(V) \\to \\mathcal{E}(V)$ factors as", "$$", "\\mathcal{F}_1(U) \\otimes_A B \\to", "\\mathcal{E}(U) \\otimes_A B \\to \\mathcal{E}(U)", "$$", "Considering $A$-algebras $B$ of the form $B = A \\oplus M$", "we see that $\\mathcal{F}_1(U) \\to \\mathcal{E}(U)$ is", "universally injective (see", "Algebra, Definition \\ref{algebra-definition-universally-injective}).", "Since $\\mathcal{E}(U) = \\mathcal{F}(U)$ we conclude that", "$\\mathcal{F}_1 \\to \\mathcal{F}$ remains injective after any base change,", "or equivalently that $\\mathcal{F}_1^a \\to \\mathcal{F}^a$ is injective." ], "refs": [ "descent-proposition-equivalence-quasi-coherent", "descent-remark-change-topologies-ringed-sites", "sites-modules-lemma-exactness-pushforward-pullback", "morphisms-lemma-etale-flat", "descent-proposition-same-cohomology-quasi-coherent", "algebra-definition-universally-injective" ], "ref_ids": [ 14755, 14793, 14159, 5369, 14754, 1493 ] } ], "ref_ids": [ 14755 ] }, { "id": 14628, "type": "theorem", "label": "descent-lemma-higher-direct-images-small-etale", "categories": [ "descent" ], "title": "descent-lemma-higher-direct-images-small-etale", "contents": [ "Let $f : T \\to S$ be a quasi-compact and quasi-separated morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $T$. For either the \\'etale", "or Zariski topology, there are canonical isomorphisms", "$R^if_{small, *}(\\mathcal{F}^a) = (R^if_*\\mathcal{F})^a$." ], "refs": [], "proofs": [ { "contents": [ "We prove this for the \\'etale topology; we omit the proof in the case", "of the Zariski topology. By Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images}", "the sheaves $R^if_*\\mathcal{F}$ are quasi-coherent so that the assertion", "makes sense. The sheaf $R^if_{small, *}\\mathcal{F}^a$ is the sheaf associated", "to the presheaf", "$$", "U \\longmapsto H^i(U \\times_S T, \\mathcal{F}^a)", "$$", "where $g : U \\to S$ is an object of $S_\\etale$, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-higher-direct-images}.", "By our conventions the right hand side is the \\'etale", "cohomology of the restriction of $\\mathcal{F}^a$ to the localization", "$T_\\etale/U \\times_S T$ which equals", "$(U \\times_S T)_\\etale$. By", "Proposition \\ref{proposition-same-cohomology-quasi-coherent}", "this is presheaf the same as the presheaf", "$$", "U \\longmapsto", "H^i(U \\times_S T, (g')^*\\mathcal{F}),", "$$", "where $g' : U \\times_S T \\to T$ is the projection. If $U$ is affine", "then this is the same as $H^0(U, R^if'_*(g')^*\\mathcal{F})$, see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images-application}.", "By", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}", "this is equal to $H^0(U, g^*R^if_*\\mathcal{F})$ which is the value", "of $(R^if_*\\mathcal{F})^a$ on $U$.", "Thus the values of the sheaves of modules", "$R^if_{small, *}(\\mathcal{F}^a)$ and $(R^if_*\\mathcal{F})^a$", "on every affine object of $S_\\etale$ are canonically isomorphic", "which implies they are canonically isomorphic." ], "refs": [ "coherent-lemma-quasi-coherence-higher-direct-images", "sites-cohomology-lemma-higher-direct-images", "descent-proposition-same-cohomology-quasi-coherent", "coherent-lemma-quasi-coherence-higher-direct-images-application", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 3295, 4189, 14754, 3296, 3298 ] } ], "ref_ids": [] }, { "id": 14629, "type": "theorem", "label": "descent-lemma-cohomology-parasitic", "categories": [ "descent" ], "title": "descent-lemma-cohomology-parasitic", "contents": [ "Let $S$ be a scheme. Let $\\tau \\in \\{Zar, \\etale, smooth,", "syntomic, fppf\\}$. Let $\\mathcal{G}$ be a presheaf of", "$\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$.", "\\begin{enumerate}", "\\item If $\\mathcal{G}$ is parasitic for the $\\tau$-topology, then", "$H^p_\\tau(U, \\mathcal{G}) = 0$ for every $U$ open in $S$,", "resp.\\ \\'etale over $S$,", "resp.\\ smooth over $S$,", "resp.\\ syntomic over $S$,", "resp.\\ flat and locally of finite presentation over $S$.", "\\item If $\\mathcal{G}$ is parasitic then $H^p_\\tau(U, \\mathcal{G}) = 0$", "for every $U$ flat over $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof in case $\\tau = fppf$; the other cases are proved in the", "exact same way. The assumption means that $\\mathcal{G}(U) = 0$ for any", "$U \\to S$ flat and locally of finite presentation. Apply", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cech-vanish-collection}", "to the subset $\\mathcal{B} \\subset \\Ob((\\Sch/S)_{fppf})$ consisting", "of $U \\to S$ flat and locally of finite presentation and the collection", "$\\text{Cov}$ of all fppf coverings of elements of $\\mathcal{B}$." ], "refs": [ "sites-cohomology-lemma-cech-vanish-collection" ], "ref_ids": [ 4205 ] } ], "ref_ids": [] }, { "id": 14630, "type": "theorem", "label": "descent-lemma-direct-image-parasitic", "categories": [ "descent" ], "title": "descent-lemma-direct-image-parasitic", "contents": [ "Let $f : T \\to S$ be a morphism of schemes. For any parasitic", "$\\mathcal{O}$-module on $(\\Sch/T)_\\tau$ the pushforward", "$f_*\\mathcal{F}$ and the higher direct images $R^if_*\\mathcal{F}$", "are parasitic $\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $R^if_*\\mathcal{F}$ is the sheaf associated to the", "presheaf", "$$", "U \\mapsto H^i((\\Sch/U \\times_S T)_\\tau, \\mathcal{F})", "$$", "see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-higher-direct-images}.", "If $U \\to S$ is flat, then $U \\times_S T \\to T$ is flat as a base change.", "Hence the displayed group is zero by", "Lemma \\ref{lemma-cohomology-parasitic}.", "If $\\{U_i \\to U\\}$ is a $\\tau$-covering then", "$U_i \\times_S T \\to T$ is also flat.", "Hence it is clear that the sheafification of the displayed", "presheaf is zero on schemes $U$ flat over $S$." ], "refs": [ "sites-cohomology-lemma-higher-direct-images", "descent-lemma-cohomology-parasitic" ], "ref_ids": [ 4189, 14629 ] } ], "ref_ids": [] }, { "id": 14631, "type": "theorem", "label": "descent-lemma-quasi-coherent-and-flat-base-change", "categories": [ "descent" ], "title": "descent-lemma-quasi-coherent-and-flat-base-change", "contents": [ "Let $S$ be a scheme. Let $\\tau \\in \\{Zar, \\etale\\}$.", "Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules on", "$(\\Sch/S)_{fppf}$ such that", "\\begin{enumerate}", "\\item $\\mathcal{G}|_{S_\\tau}$ is quasi-coherent, and", "\\item for every flat, locally finitely presented morphism", "$g : U \\to S$ the canonical map", "$g_{\\tau, small}^*(\\mathcal{G}|_{S_\\tau}) \\to \\mathcal{G}|_{U_\\tau}$", "is an isomorphism.", "\\end{enumerate}", "Then $H^p(U, \\mathcal{G}) = H^p(U, \\mathcal{G}|_{U_\\tau})$", "for every $U$ flat and locally of finite presentation over $S$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{F}$ be the pullback of $\\mathcal{G}|_{S_\\tau}$", "to the big fppf site $(\\Sch/S)_{fppf}$. Note that $\\mathcal{F}$", "is quasi-coherent. There is a canonical", "comparison map $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ which by", "assumptions (1) and (2) induces an isomorphism", "$\\mathcal{F}|_{U_\\tau} \\to \\mathcal{G}|_{U_\\tau}$", "for all $g : U \\to S$ flat and locally of finite presentation.", "Hence in the short exact sequences", "$$", "0 \\to \\Ker(\\varphi) \\to \\mathcal{F} \\to \\Im(\\varphi) \\to 0", "$$", "and", "$$", "0 \\to \\Im(\\varphi) \\to \\mathcal{G} \\to \\Coker(\\varphi) \\to 0", "$$", "the sheaves $\\Ker(\\varphi)$ and $\\Coker(\\varphi)$ are", "parasitic for the fppf topology. By", "Lemma \\ref{lemma-cohomology-parasitic}", "we conclude that $H^p(U, \\mathcal{F}) \\to H^p(U, \\mathcal{G})$", "is an isomorphism for $g : U \\to S$ flat and locally of finite presentation.", "Since the result holds for $\\mathcal{F}$ by", "Proposition \\ref{proposition-same-cohomology-quasi-coherent}", "we win." ], "refs": [ "descent-lemma-cohomology-parasitic", "descent-proposition-same-cohomology-quasi-coherent" ], "ref_ids": [ 14629, 14754 ] } ], "ref_ids": [] }, { "id": 14632, "type": "theorem", "label": "descent-lemma-equiv-fibre-product", "categories": [ "descent" ], "title": "descent-lemma-equiv-fibre-product", "contents": [ "For a scheme $X$ denote $|X|$ the underlying set.", "Let $f : X \\to S$ be a morphism of schemes.", "Then", "$$", "|X \\times_S X| \\to |X| \\times_{|S|} |X|", "$$", "is surjective." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from the description of points on the", "fibre product in Schemes, Lemma \\ref{schemes-lemma-points-fibre-product}." ], "refs": [ "schemes-lemma-points-fibre-product" ], "ref_ids": [ 7693 ] } ], "ref_ids": [] }, { "id": 14633, "type": "theorem", "label": "descent-lemma-universal-effective-epimorphism-affine", "categories": [ "descent" ], "title": "descent-lemma-universal-effective-epimorphism-affine", "contents": [ "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms of affine schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item for any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ we have", "$$", "\\Gamma(X, \\mathcal{F}) =", "\\text{Equalizer}\\left(", "\\xymatrix{", "\\prod\\nolimits_{i \\in I} \\Gamma(X_i, f_i^*\\mathcal{F})", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\prod\\nolimits_{i, j \\in I}", "\\Gamma(X_i \\times_X X_j, (f_i \\times f_j)^*\\mathcal{F})", "}", "\\right)", "$$", "\\item $\\{f_i : X_i \\to X\\}_{i \\in I}$ is a universal effective epimorphism", "(Sites, Definition \\ref{sites-definition-universal-effective-epimorphisms})", "in the category of affine schemes.", "\\end{enumerate}" ], "refs": [ "sites-definition-universal-effective-epimorphisms" ], "proofs": [ { "contents": [ "Assume (2) holds and let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. Consider the scheme", "(Constructions, Section \\ref{constructions-section-spec})", "$$", "X' = \\underline{\\Spec}_X(\\mathcal{O}_X \\oplus \\mathcal{F})", "$$", "where $\\mathcal{O}_X \\oplus \\mathcal{F}$ is an", "$\\mathcal{O}_X$-algebra with multiplication", "$(f, s)(f', s') = (ff', fs' + f's)$.", "If $s_i \\in \\Gamma(X_i, f_i^*\\mathcal{F})$ is a section,", "then $s_i$ determines a unique element of", "$$", "\\Gamma(X' \\times_X X_i, \\mathcal{O}_{X' \\times_X X_i}) =", "\\Gamma(X_i, \\mathcal{O}_{X_i}) \\oplus \\Gamma(X_i, f_i^*\\mathcal{F})", "$$", "Proof of equality omitted.", "If $(s_i)_{i \\in I}$ is in the equalizer of (1), then, using the equality", "$$", "\\Mor(T, \\mathbf{A}^1_\\mathbf{Z}) = \\Gamma(T, \\mathcal{O}_T)", "$$", "which holds for any scheme $T$, we see that these sections define", "a family of morphisms $h_i : X' \\times_X X_i \\to \\mathbf{A}^1_\\mathbf{Z}$ with", "$h_i \\circ \\text{pr}_1 = h_j \\circ \\text{pr}_2$ as morphisms", "$(X' \\times_X X_i) \\times_{X'} (X' \\times_X X_j) \\to \\mathbf{A}^1_\\mathbf{Z}$.", "Since we've assume (2) we obtain a morphism", "$h : X' \\to \\mathbf{A}^1_\\mathbf{Z}$ compatible with the morphisms $h_i$", "which in turn determines", "an element $s \\in \\Gamma(X, \\mathcal{F})$.", "We omit the verification that $s$ maps to $s_i$ in", "$\\Gamma(X_i, f_i^*\\mathcal{F})$.", "\\medskip\\noindent", "Assume (1). Let $T$ be an affine scheme and let $h_i : X_i \\to T$", "be a family of morphisms such that", "$h_i \\circ \\text{pr}_1 = h_j \\circ \\text{pr}_2$ on", "$X_i \\times_X X_j$ for all $i, j \\in I$. Then", "$$", "\\prod h_i^\\sharp :", "\\Gamma(T, \\mathcal{O}_T)", "\\to", "\\prod \\Gamma(X_i, \\mathcal{O}_{X_i})", "$$", "maps into the equalizer and we find that we get a ring map", "$\\Gamma(T, \\mathcal{O}_T) \\to \\Gamma(X, \\mathcal{O}_X)$", "by the assumption of the lemma for $\\mathcal{F} = \\mathcal{O}_X$.", "This ring map corresponds to a morphism $h : X \\to T$ such", "that $h_i = h \\circ f_i$. Hence our family is an effective", "epimorphism.", "\\medskip\\noindent", "Let $p : Y \\to X$ be a morphism of affines. We will show", "the base changes $g_i : Y_i \\to Y$ of $f_i$ form an effective epimorphism", "by applying the result of the previous paragraph.", "Namely, if $\\mathcal{G}$ is a quasi-coherent $\\mathcal{O}_Y$-module, then", "$$", "\\Gamma(Y, \\mathcal{G}) = \\Gamma(X, p_*\\mathcal{G}),\\quad", "\\Gamma(Y_i, g_i^*\\mathcal{G}) = \\Gamma(X, f_i^*p_*\\mathcal{G}),", "$$", "and", "$$", "\\Gamma(Y_i \\times_Y Y_j, (g_i \\times g_j)^*\\mathcal{G}) =", "\\Gamma(X, (f_i \\times f_j)^*p_*\\mathcal{G})", "$$", "by the trivial base change formula", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-affine-base-change}).", "Thus we see property (1) lemma holds for the family $g_i$." ], "refs": [ "coherent-lemma-affine-base-change" ], "ref_ids": [ 3297 ] } ], "ref_ids": [ 8661 ] }, { "id": 14634, "type": "theorem", "label": "descent-lemma-universal-effective-epimorphism-surjective", "categories": [ "descent" ], "title": "descent-lemma-universal-effective-epimorphism-surjective", "contents": [ "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms of schemes.", "\\begin{enumerate}", "\\item If the family is universal effective", "epimorphism in the category of schemes, then $\\coprod f_i$ is surjective.", "\\item If $X$ and $X_i$ are affine and the family is a universal effective", "epimorphism in the category of affine schemes, then", "$\\coprod f_i$ is surjective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: perform base change by $\\Spec(\\kappa(x)) \\to X$", "to see that any $x \\in X$ has to be in the image." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14635, "type": "theorem", "label": "descent-lemma-check-universal-effective-epimorphism-affine", "categories": [ "descent" ], "title": "descent-lemma-check-universal-effective-epimorphism-affine", "contents": [ "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms of schemes.", "If for every morphism $Y \\to X$ with $Y$ affine the family of base changes", "$g_i : Y_i \\to Y$ forms an effective epimorphism, then", "the family of $f_i$ forms a universally effective epimorphism", "in the category of schemes." ], "refs": [], "proofs": [ { "contents": [ "Let $Y \\to X$ be a morphism of schemes. We have to show that", "the base changes $g_i : Y_i \\to Y$ form an effective epimorphism.", "To do this, assume given a scheme $T$ and morphisms $h_i : Y_i \\to T$", "with $h_i \\circ \\text{pr}_1 = h_j \\circ \\text{pr}_2$ on", "$Y_i \\times_Y Y_j$.", "Choose an affine open covering $Y = \\bigcup V_\\alpha$.", "Set $V_{\\alpha, i}$ equal to the inverse image of", "$V_\\alpha$ in $Y_i$. Then we see that", "$V_{\\alpha, i} \\to V_\\alpha$ is the base change of", "$f_i$ by $V_\\alpha \\to X$. Thus by assumption", "the family of restrictions $h_i|_{V_{\\alpha, i}}$", "come from a morphism of schemes $h_\\alpha : V_\\alpha \\to T$.", "We leave it to the reader to show that these agree", "on overlaps and define the desired morphism $Y \\to T$.", "See discussion in Schemes, Section \\ref{schemes-section-glueing-schemes}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14636, "type": "theorem", "label": "descent-lemma-universal-effective-epimorphism", "categories": [ "descent" ], "title": "descent-lemma-universal-effective-epimorphism", "contents": [ "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms of affine", "schemes. Assume the equivalent assumption of", "Lemma \\ref{lemma-universal-effective-epimorphism-affine} hold", "and that moreover for any morphism of affines $Y \\to X$ the map", "$$", "\\coprod X_i \\times_X Y \\longrightarrow Y", "$$", "is a submersive map of topological spaces", "(Topology, Definition \\ref{topology-definition-submersive}).", "Then our family of morphisms is a universal effective epimorphism", "in the category of schemes." ], "refs": [ "descent-lemma-universal-effective-epimorphism-affine", "topology-definition-submersive" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-check-universal-effective-epimorphism-affine}", "it suffices to base change our family of morphisms", "by $Y \\to X$ with $Y$ affine. Set $Y_i = X_i \\times_X Y$.", "Let $T$ be a scheme and let $h_i : Y_i \\to Y$ be a family of morphisms", "such that $h_i \\circ \\text{pr}_1 = h_j \\circ \\text{pr}_2$", "on $Y_i \\times_Y Y_j$. Note that $Y$ as a set is the coequalizer", "of the two maps from $\\coprod Y_i \\times_Y Y_j$ to $\\coprod Y_i$.", "Namely, surjectivity by the affine case of", "Lemma \\ref{lemma-universal-effective-epimorphism-surjective}", "and injectivity by Lemma \\ref{lemma-equiv-fibre-product}.", "Hence there is a set map of underlying sets $h : Y \\to T$", "compatible with the maps $h_i$. By the second condition of", "the lemma we see that $h$ is continuous!", "Thus if $y \\in Y$ and $U \\subset T$ is an affine open", "neighbourhood of $h(y)$, then we can find an affine open", "$V \\subset Y$ such that $h(V) \\subset U$.", "Setting $V_i = Y_i \\times_Y V = X_i \\times_X V$", "we can use the result proved in", "Lemma \\ref{lemma-universal-effective-epimorphism-affine}", "to see that $h|_V : V \\to U \\subset T$ comes from a unique", "morphism of affine schemes $h_V : V \\to U$ agreeing with $h_i|_{V_i}$", "as morphisms of schemes for all $i$. Glueing these $h_V$", "(see Schemes, Section \\ref{schemes-section-glueing-schemes})", "gives a morphism $Y \\to T$ as desired." ], "refs": [ "descent-lemma-check-universal-effective-epimorphism-affine", "descent-lemma-universal-effective-epimorphism-surjective", "descent-lemma-equiv-fibre-product", "descent-lemma-universal-effective-epimorphism-affine" ], "ref_ids": [ 14635, 14634, 14632, 14633 ] } ], "ref_ids": [ 14633, 8349 ] }, { "id": 14637, "type": "theorem", "label": "descent-lemma-open-fpqc-covering", "categories": [ "descent" ], "title": "descent-lemma-open-fpqc-covering", "contents": [ "Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a fpqc covering.", "Suppose that for each $i$ we have an open subset $W_i \\subset T_i$", "such that for all $i, j \\in I$ we have", "$\\text{pr}_0^{-1}(W_i) = \\text{pr}_1^{-1}(W_j)$ as open", "subsets of $T_i \\times_T T_j$. Then there exists a unique open subset", "$W \\subset T$ such that $W_i = f_i^{-1}(W)$ for each $i$." ], "refs": [], "proofs": [ { "contents": [ "Apply", "Lemma \\ref{lemma-equiv-fibre-product}", "to the map $\\coprod_{i \\in I} T_i \\to T$.", "It implies there exists a subset $W \\subset T$ such that", "$W_i = f_i^{-1}(W)$ for each $i$, namely $W = \\bigcup f_i(W_i)$.", "To see that $W$ is open we may work Zariski locally on $T$.", "Hence we may assume that $T$ is affine. Using the definition", "of a fpqc covering, this reduces us to the case where", "$\\{f_i : T_i \\to T\\}$ is a standard fpqc covering. In this case we", "may apply", "Morphisms, Lemma \\ref{morphisms-lemma-fpqc-quotient-topology}", "to the morphism", "$\\coprod T_i \\to T$ to conclude that $W$ is open." ], "refs": [ "descent-lemma-equiv-fibre-product", "morphisms-lemma-fpqc-quotient-topology" ], "ref_ids": [ 14632, 5269 ] } ], "ref_ids": [] }, { "id": 14638, "type": "theorem", "label": "descent-lemma-fpqc-universal-effective-epimorphisms", "categories": [ "descent" ], "title": "descent-lemma-fpqc-universal-effective-epimorphisms", "contents": [ "Let $\\{T_i \\to T\\}$ be an fpqc covering, see", "Topologies, Definition \\ref{topologies-definition-fpqc-covering}.", "Then $\\{T_i \\to T\\}$ is a universal effective epimorphism", "in the category of schemes, see", "Sites, Definition \\ref{sites-definition-universal-effective-epimorphisms}.", "In other words, every representable functor on the category of schemes", "satisfies the sheaf condition for the fpqc topology, see", "Topologies, Definition \\ref{topologies-definition-sheaf-property-fpqc}." ], "refs": [ "topologies-definition-fpqc-covering", "sites-definition-universal-effective-epimorphisms", "topologies-definition-sheaf-property-fpqc" ], "proofs": [ { "contents": [ "Let $S$ be a scheme. We have to show the following:", "Given morphisms $\\varphi_i : T_i \\to S$", "such that $\\varphi_i|_{T_i \\times_T T_j} = \\varphi_j|_{T_i \\times_T T_j}$", "there exists a unique morphism $T \\to S$ which restricts", "to $\\varphi_i$ on each $T_i$.", "In other words, we have to show that the functor", "$h_S = \\Mor_{\\Sch}( - , S)$ satisfies", "the sheaf property for the fpqc topology.", "\\medskip\\noindent", "If $\\{T_i \\to T\\}$ is a Zariski covering, then this follows from", "Schemes, Lemma \\ref{schemes-lemma-glue}.", "Thus Topologies, Lemma \\ref{topologies-lemma-sheaf-property-fpqc}", "reduces us to the case of a covering $\\{X \\to Y\\}$", "given by a single surjective flat morphism of affines.", "\\medskip\\noindent", "First proof. By Lemma \\ref{lemma-sheaf-condition-holds}", "we have the sheaf condition for quasi-coherent modules", "for $\\{X \\to Y\\}$. By Lemma \\ref{lemma-open-fpqc-covering}", "the morphism $X \\to Y$ is universally submersive.", "Hence we may apply Lemma \\ref{lemma-universal-effective-epimorphism}", "to see that $\\{X \\to Y\\}$ is a universal effective epimorphism.", "\\medskip\\noindent", "Second proof. Let $R \\to A$ be the faithfully flat ring map", "corresponding to our surjective flat morphism $\\pi : X \\to Y$.", "Let $f : X \\to S$ be a morphism", "such that $f \\circ \\text{pr}_1 = f \\circ \\text{pr}_2$", "as morphisms $X \\times_Y X = \\Spec(A \\otimes_R A) \\to S$.", "By Lemma \\ref{lemma-equiv-fibre-product} we see that", "as a map on the underlying", "sets $f$ is of the form $f = g \\circ \\pi$ for some", "(set theoretic) map $g : \\Spec(R) \\to S$.", "By Morphisms, Lemma \\ref{morphisms-lemma-fpqc-quotient-topology}", "and the fact that $f$ is continuous we see that $g$", "is continuous.", "\\medskip\\noindent", "Pick $y \\in Y = \\Spec(R)$.", "Choose $U \\subset S$ affine open containing $g(y)$.", "Say $U = \\Spec(B)$.", "By the above we may choose an $r \\in R$ such that", "$y \\in D(r) \\subset g^{-1}(U)$.", "The restriction of $f$ to $\\pi^{-1}(D(r))$ into $U$", "corresponds to a ring map $B \\to A_r$. The two induced", "ring maps $B \\to A_r \\otimes_{R_r} A_r = (A \\otimes_R A)_r$ are equal", "by assumption on $f$.", "Note that $R_r \\to A_r$ is faithfully flat.", "By Lemma \\ref{lemma-ff-exact} the equalizer of", "the two arrows $A_r \\to A_r \\otimes_{R_r} A_r$ is $R_r$.", "We conclude that $B \\to A_r$ factors uniquely through a map $B \\to R_r$.", "This map in turn gives a morphism of schemes $D(r) \\to U \\to S$,", "see Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}.", "\\medskip\\noindent", "What have we proved so far? We have shown that for any prime", "$\\mathfrak p \\subset R$, there exists a standard affine open", "$D(r) \\subset \\Spec(R)$ such that the morphism", "$f|_{\\pi^{-1}(D(r))} : \\pi^{-1}(D(r)) \\to S$ factors uniquely", "through some morphism of schemes $D(r) \\to S$. We omit the", "verification that these morphisms glue to the desired", "morphism $\\Spec(R) \\to S$." ], "refs": [ "schemes-lemma-glue", "topologies-lemma-sheaf-property-fpqc", "descent-lemma-sheaf-condition-holds", "descent-lemma-open-fpqc-covering", "descent-lemma-universal-effective-epimorphism", "descent-lemma-equiv-fibre-product", "morphisms-lemma-fpqc-quotient-topology", "descent-lemma-ff-exact", "schemes-lemma-morphism-into-affine" ], "ref_ids": [ 7686, 12502, 14621, 14637, 14636, 14632, 5269, 14598, 7655 ] } ], "ref_ids": [ 12547, 8661, 12549 ] }, { "id": 14639, "type": "theorem", "label": "descent-lemma-coequalizer-fpqc-local", "categories": [ "descent" ], "title": "descent-lemma-coequalizer-fpqc-local", "contents": [ "Consider schemes $X, Y, Z$ and morphisms $a, b : X \\to Y$ and", "a morphism $c : Y \\to Z$ with $c \\circ a = c \\circ b$. Set", "$d = c \\circ a = c \\circ b$. If there exists an", "fpqc covering $\\{Z_i \\to Z\\}$ such that", "\\begin{enumerate}", "\\item for all $i$ the morphism $Y \\times_{c, Z} Z_i \\to Z_i$", "is the coequalizer of $(a, 1) : X \\times_{d, Z} Z_i \\to Y \\times_{c, Z} Z_i$", "and $(b, 1) : X \\times_{d, Z} Z_i \\to Y \\times_{c, Z} Z_i$, and", "\\item for all $i$ and $i'$ the morphism", "$Y \\times_{c, Z} (Z_i \\times_Z Z_{i'}) \\to (Z_i \\times_Z Z_{i'})$", "is the coequalizer of", "$(a, 1) : X \\times_{d, Z} (Z_i \\times_Z Z_{i'}) \\to", "Y \\times_{c, Z} (Z_i \\times_Z Z_{i'})$ and", "$(b, 1) : X \\times_{d, Z} (Z_i \\times_Z Z_{i'}) \\to", "Y \\times_{c, Z} (Z_i \\times_Z Z_{i'})$", "\\end{enumerate}", "then $c$ is the coequalizer of $a$ and $b$." ], "refs": [], "proofs": [ { "contents": [ "Namely, for a scheme $T$ a morphism $Z \\to T$ is the same thing as", "a collection of morphism $Z_i \\to T$ which agree on overlaps by", "Lemma \\ref{lemma-fpqc-universal-effective-epimorphisms}." ], "refs": [ "descent-lemma-fpqc-universal-effective-epimorphisms" ], "ref_ids": [ 14638 ] } ], "ref_ids": [] }, { "id": 14640, "type": "theorem", "label": "descent-lemma-flat-finitely-presented-permanence-algebra", "categories": [ "descent" ], "title": "descent-lemma-flat-finitely-presented-permanence-algebra", "contents": [ "Let $R \\to A \\to B$ be ring maps.", "Assume $R \\to B$ is of finite presentation and", "$A \\to B$ faithfully flat and of finite presentation.", "Then $R \\to A$ is of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Consider the algebra $C = B \\otimes_A B$ together with the", "pair of maps $p, q : B \\to C$ given by $p(b) = b \\otimes 1$", "and $q(b) = 1 \\otimes b$. Of course the two compositions", "$A \\to B \\to C$ are the same. Note that as", "$p : B \\to C$ is flat and of finite presentation (base change of", "$A \\to B$), the ring map $R \\to C$ is of finite presentation", "(as the composite of $R \\to B \\to C$).", "\\medskip\\noindent", "We are going to use the criterion", "Algebra, Lemma \\ref{algebra-lemma-characterize-finite-presentation}", "to show that $R \\to A$ is of finite presentation.", "Let $S$ be any $R$-algebra, and suppose that", "$S = \\colim_{\\lambda \\in \\Lambda} S_\\lambda$ is written", "as a directed colimit of $R$-algebras.", "Let $A \\to S$ be an $R$-algebra homomorphism. We have to", "show that $A \\to S$ factors through one of the $S_\\lambda$.", "Consider the rings $B' = S \\otimes_A B$ and", "$C' = S \\otimes_A C = B' \\otimes_S B'$.", "As $B$ is faithfully flat of finite presentation over $A$, also $B'$", "is faithfully flat of finite presentation over $S$.", "By Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}", "part (2) applied to the pair $(S \\to B', B')$ and the system $(S_\\lambda)$", "there exists a $\\lambda_0 \\in \\Lambda$", "and a flat, finitely presented $S_{\\lambda_0}$-algebra", "$B_{\\lambda_0}$ such that $B' = S \\otimes_{S_{\\lambda_0}} B_{\\lambda_0}$.", "For $\\lambda \\geq \\lambda_0$ set", "$B_\\lambda = S_\\lambda \\otimes_{S_{\\lambda_0}} B_{\\lambda_0}$ and", "$C_\\lambda = B_\\lambda \\otimes_{S_\\lambda} B_\\lambda$.", "\\medskip\\noindent", "We interrupt the flow of the argument to show that $S_\\lambda \\to B_\\lambda$", "is faithfully flat for $\\lambda$ large enough. (This should really", "be a separate lemma somewhere else, maybe in the chapter on limits.)", "Since $\\Spec(B_{\\lambda_0}) \\to \\Spec(S_{\\lambda_0})$ is", "flat and of finite presentation it is open (see Morphisms,", "Lemma \\ref{morphisms-lemma-fppf-open}).", "Let $I \\subset S_{\\lambda_0}$ be an ideal such that", "$V(I) \\subset \\Spec(S_{\\lambda_0})$ is the complement", "of the image. Note that formation of the image commutes", "with base change. Hence, since $\\Spec(B') \\to \\Spec(S)$", "is surjective, and $B' = B_{\\lambda_0} \\otimes_{S_{\\lambda_0}} S$", "we see that $IS = S$. Thus for some $\\lambda \\geq \\lambda_0$ we", "have $IS_{\\lambda} = S_\\lambda$. For this and all greater", "$\\lambda$ the morphism", "$\\Spec(B_\\lambda) \\to \\Spec(S_\\lambda)$ is surjective.", "\\medskip\\noindent", "By analogy with the notation in the first paragraph of the proof denote", "$p_\\lambda, q_\\lambda : B_\\lambda \\to C_\\lambda$ the two canonical maps.", "Then $B' = \\colim_{\\lambda \\geq \\lambda_0} B_\\lambda$", "and $C' = \\colim_{\\lambda \\geq \\lambda_0} C_\\lambda$.", "Since $B$ and $C$ are finitely presented over $R$ there exist", "(by Algebra, Lemma \\ref{algebra-lemma-characterize-finite-presentation}", "applied several times)", "a $\\lambda \\geq \\lambda_0$ and an $R$-algebra maps", "$B \\to B_\\lambda$, $C \\to C_\\lambda$ such that", "the diagram", "$$", "\\xymatrix{", "C \\ar[rr] & &", "C_\\lambda \\\\", "B \\ar[rr]", "\\ar@<1ex>[u]^-p", "\\ar@<-1ex>[u]_-q", "& &", "B_\\lambda", "\\ar@<1ex>[u]^-{p_\\lambda}", "\\ar@<-1ex>[u]_-{q_\\lambda}", "}", "$$", "is commutative. OK, and this means that $A \\to B \\to B_\\lambda$", "maps into the equalizer of $p_\\lambda$ and $q_\\lambda$.", "By Lemma \\ref{lemma-ff-exact} we", "see that $S_\\lambda$ is the equalizer of $p_\\lambda$ and $q_\\lambda$.", "Thus we get the desired ring map $A \\to S_\\lambda$ and we win." ], "refs": [ "algebra-lemma-characterize-finite-presentation", "algebra-lemma-flat-finite-presentation-limit-flat", "morphisms-lemma-fppf-open", "algebra-lemma-characterize-finite-presentation", "descent-lemma-ff-exact" ], "ref_ids": [ 1092, 1389, 5267, 1092, 14598 ] } ], "ref_ids": [] }, { "id": 14641, "type": "theorem", "label": "descent-lemma-finite-type-local-source-fppf-algebra", "categories": [ "descent" ], "title": "descent-lemma-finite-type-local-source-fppf-algebra", "contents": [ "Let $R \\to A \\to B$ be ring maps.", "Assume $R \\to B$ is of finite type and", "$A \\to B$ faithfully flat and of finite presentation.", "Then $R \\to A$ is of finite type." ], "refs": [], "proofs": [ { "contents": [ "By", "Algebra, Lemma \\ref{algebra-lemma-descend-faithfully-flat-finite-presentation}", "there exists a commutative diagram", "$$", "\\xymatrix{", "R \\ar[r] \\ar@{=}[d] &", "A_0 \\ar[d] \\ar[r] &", "B_0 \\ar[d] \\\\", "R \\ar[r] & A \\ar[r] & B", "}", "$$", "with $R \\to A_0$ of finite presentation,", "$A_0 \\to B_0$ faithfully flat of finite presentation", "and $B = A \\otimes_{A_0} B_0$. Since $R \\to B$ is of finite", "type by assumption, we may add some elements to $A_0$ and assume", "that the map $B_0 \\to B$ is surjective!", "In this case, since $A_0 \\to B_0$ is faithfully flat, we see", "that as", "$$", "(A_0 \\to A) \\otimes_{A_0} B_0 \\cong (B_0 \\to B)", "$$", "is surjective, also $A_0 \\to A$ is surjective. Hence we win." ], "refs": [ "algebra-lemma-descend-faithfully-flat-finite-presentation" ], "ref_ids": [ 1390 ] } ], "ref_ids": [] }, { "id": 14642, "type": "theorem", "label": "descent-lemma-flat-finitely-presented-permanence", "categories": [ "descent" ], "title": "descent-lemma-flat-finitely-presented-permanence", "contents": [ "\\begin{reference}", "\\cite[IV, 17.7.5 (i) and (ii)]{EGA}.", "\\end{reference}", "Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & &", "Y \\ar[dl]^q \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes. Assume that $f$ is", "surjective, flat and locally of finite presentation and assume", "that $p$ is locally of finite presentation (resp.\\ locally of finite type).", "Then $q$ is locally of finite presentation (resp.\\ locally of finite type)." ], "refs": [], "proofs": [ { "contents": [ "The problem is local on $S$ and $Y$. Hence we may assume that", "$S$ and $Y$ are affine. Since $f$ is flat and locally of finite", "presentation, we see that $f$ is open", "(Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}).", "Hence, since $Y$ is quasi-compact, there exist finitely many affine opens", "$X_i \\subset X$ such that $Y = \\bigcup f(X_i)$.", "Clearly we may replace $X$ by $\\coprod X_i$, and hence we", "may assume $X$ is affine as well.", "In this case the lemma is equivalent to", "Lemma \\ref{lemma-flat-finitely-presented-permanence-algebra}", "(resp. Lemma \\ref{lemma-finite-type-local-source-fppf-algebra})", "above." ], "refs": [ "morphisms-lemma-fppf-open", "descent-lemma-flat-finitely-presented-permanence-algebra", "descent-lemma-finite-type-local-source-fppf-algebra" ], "ref_ids": [ 5267, 14640, 14641 ] } ], "ref_ids": [] }, { "id": 14643, "type": "theorem", "label": "descent-lemma-syntomic-smooth-etale-permanence", "categories": [ "descent" ], "title": "descent-lemma-syntomic-smooth-etale-permanence", "contents": [ "Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & &", "Y \\ar[dl]^q \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes. Assume that", "\\begin{enumerate}", "\\item $f$ is surjective, and syntomic (resp.\\ smooth, resp.\\ \\'etale),", "\\item $p$ is syntomic (resp.\\ smooth, resp.\\ \\'etale).", "\\end{enumerate}", "Then $q$ is syntomic (resp.\\ smooth, resp.\\ \\'etale)." ], "refs": [], "proofs": [ { "contents": [ "Combine Morphisms, Lemmas", "\\ref{morphisms-lemma-syntomic-permanence},", "\\ref{morphisms-lemma-smooth-permanence}, and", "\\ref{morphisms-lemma-etale-permanence-two}", "with Lemma \\ref{lemma-flat-finitely-presented-permanence} above." ], "refs": [ "morphisms-lemma-syntomic-permanence", "morphisms-lemma-smooth-permanence", "morphisms-lemma-etale-permanence-two", "descent-lemma-flat-finitely-presented-permanence" ], "ref_ids": [ 5302, 5340, 5376, 14642 ] } ], "ref_ids": [] }, { "id": 14644, "type": "theorem", "label": "descent-lemma-smooth-permanence", "categories": [ "descent" ], "title": "descent-lemma-smooth-permanence", "contents": [ "Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & &", "Y \\ar[dl]^q \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes. Assume that", "\\begin{enumerate}", "\\item $f$ is surjective, flat, and locally of finite presentation,", "\\item $p$ is smooth (resp.\\ \\'etale).", "\\end{enumerate}", "Then $q$ is smooth (resp.\\ \\'etale)." ], "refs": [], "proofs": [ { "contents": [ "Assume (1) and that $p$ is smooth. By", "Lemma \\ref{lemma-flat-finitely-presented-permanence}", "we see that $q$ is locally of finite presentation.", "By", "Morphisms, Lemma \\ref{morphisms-lemma-flat-permanence}", "we see that $q$ is flat.", "Hence now it suffices to show that the fibres of $q$ are smooth, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-flat-smooth-fibres}.", "Apply", "Varieties, Lemma \\ref{varieties-lemma-flat-under-smooth}", "to the flat surjective morphisms $X_s \\to Y_s$ for $s \\in S$ to", "conclude. We omit the proof of the \\'etale case." ], "refs": [ "descent-lemma-flat-finitely-presented-permanence", "morphisms-lemma-flat-permanence", "morphisms-lemma-smooth-flat-smooth-fibres", "varieties-lemma-flat-under-smooth" ], "ref_ids": [ 14642, 5270, 5325, 11010 ] } ], "ref_ids": [] }, { "id": 14645, "type": "theorem", "label": "descent-lemma-syntomic-permanence", "categories": [ "descent" ], "title": "descent-lemma-syntomic-permanence", "contents": [ "Let", "$$", "\\xymatrix{", "X \\ar[rr]_f \\ar[rd]_p & &", "Y \\ar[dl]^q \\\\", "& S", "}", "$$", "be a commutative diagram of morphisms of schemes. Assume that", "\\begin{enumerate}", "\\item $f$ is surjective, flat, and locally of finite presentation,", "\\item $p$ is syntomic.", "\\end{enumerate}", "Then both $q$ and $f$ are syntomic." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-flat-finitely-presented-permanence} we see that $q$", "is of finite presentation. By", "Morphisms, Lemma \\ref{morphisms-lemma-flat-permanence}", "we see that $q$ is flat.", "By Morphisms, Lemma \\ref{morphisms-lemma-syntomic-locally-standard-syntomic}", "it now suffices to show that the local rings of the fibres of", "$Y \\to S$ and the fibres of $X \\to Y$ are local complete intersection", "rings. To do this we may take the fibre of $X \\to Y \\to S$ at", "a point $s \\in S$, i.e., we may assume $S$ is the spectrum of a", "field. Pick a point $x \\in X$ with image $y \\in Y$ and", "consider the ring map", "$$", "\\mathcal{O}_{Y, y} \\longrightarrow \\mathcal{O}_{X, x}", "$$", "This is a flat local homomorphism of local Noetherian rings.", "The local ring $\\mathcal{O}_{X, x}$ is a complete intersection.", "Thus may use Avramov's result, see", "Divided Power Algebra, Lemma \\ref{dpa-lemma-avramov},", "to conclude that both $\\mathcal{O}_{Y, y}$ and", "$\\mathcal{O}_{X, x}/\\mathfrak m_y\\mathcal{O}_{X, x}$ are", "complete intersection rings." ], "refs": [ "descent-lemma-flat-finitely-presented-permanence", "morphisms-lemma-flat-permanence", "morphisms-lemma-syntomic-locally-standard-syntomic", "dpa-lemma-avramov" ], "ref_ids": [ 14642, 5270, 5297, 1682 ] } ], "ref_ids": [] }, { "id": 14646, "type": "theorem", "label": "descent-lemma-curiosity", "categories": [ "descent" ], "title": "descent-lemma-curiosity", "contents": [ "Let $X \\to Y \\to Z$ be morphism of schemes.", "Let $P$ be one of the following properties of morphisms of schemes:", "flat, locally finite type, locally finite presentation.", "Assume that $X \\to Z$ has $P$ and that $\\{X \\to Y\\}$", "can be refined by an fppf covering of $Y$. Then $Y \\to Z$ is $P$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\Spec(C) \\subset Z$ be an affine open and let", "$\\Spec(B) \\subset Y$ be an affine open which maps into", "$\\Spec(C)$. The assumption on $X \\to Y$ implies we can", "find a standard affine fppf covering $\\{\\Spec(B_j) \\to \\Spec(B)\\}$", "and lifts $x_j : \\Spec(B_j) \\to X$. Since $\\Spec(B_j)$", "is quasi-compact we can find finitely many affine opens", "$\\Spec(A_i) \\subset X$ lying over $\\Spec(B)$", "such that the image of each $x_j$", "is contained in the union $\\bigcup \\Spec(A_i)$. Hence after", "replacing each $\\Spec(B_j)$ by a standard affine Zariski coverings", "of itself we may assume we have a", "standard affine fppf covering $\\{\\Spec(B_i) \\to \\Spec(B)\\}$", "such that each $\\Spec(B_i) \\to Y$ factors through an affine", "open $\\Spec(A_i) \\subset X$ lying over $\\Spec(B)$.", "In other words, we have ring maps $C \\to B \\to A_i \\to B_i$ for each $i$.", "Note that we can also consider", "$$", "C \\to B \\to A = \\prod A_i \\to B' = \\prod B_i", "$$", "and that the ring map $B \\to \\prod B_i$ is faithfully flat and", "of finite presentation.", "\\medskip\\noindent", "The case $P = flat$. In this case we know that $C \\to A$ is flat", "and we have to prove that $C \\to B$ is flat. Suppose that", "$N \\to N' \\to N''$ is an exact sequence of $C$-modules. We want to", "show that $N \\otimes_C B \\to N' \\otimes_C B \\to N'' \\otimes_C B$", "is exact. Let $H$ be its cohomology and let $H'$ be the cohomology", "of $N \\otimes_C B' \\to N' \\otimes_C B' \\to N'' \\otimes_C B'$. As", "$B \\to B'$ is flat we know that $H' = H \\otimes_B B'$. On the other hand", "$N \\otimes_C A \\to N' \\otimes_C A \\to N'' \\otimes_C A$", "is exact hence has zero cohomology. Hence the map", "$H \\to H'$ is zero (as it factors through the zero module).", "Thus $H' = 0$. As $B \\to B'$ is faithfully flat we conclude that", "$H = 0$ as desired.", "\\medskip\\noindent", "The case $P = locally\\ finite\\ type$.", "In this case we know that $C \\to A$ is of finite type and", "we have to prove that $C \\to B$ is of finite type.", "Because $B \\to B'$ is of finite presentation (hence of finite type)", "we see that $A \\to B'$ is of finite type, see", "Algebra, Lemma \\ref{algebra-lemma-compose-finite-type}.", "Therefore $C \\to B'$ is of finite type and we conclude by", "Lemma \\ref{lemma-finite-type-local-source-fppf-algebra}.", "\\medskip\\noindent", "The case $P = locally\\ finite\\ presentation$.", "In this case we know that $C \\to A$ is of finite presentation and", "we have to prove that $C \\to B$ is of finite presentation.", "Because $B \\to B'$ is of finite presentation and $B \\to A$", "of finite type we see that $A \\to B'$ is of finite presentation, see", "Algebra, Lemma \\ref{algebra-lemma-compose-finite-type}.", "Therefore $C \\to B'$ is of finite presentation and we conclude by", "Lemma \\ref{lemma-flat-finitely-presented-permanence-algebra}." ], "refs": [ "algebra-lemma-compose-finite-type", "descent-lemma-finite-type-local-source-fppf-algebra", "algebra-lemma-compose-finite-type", "descent-lemma-flat-finitely-presented-permanence-algebra" ], "ref_ids": [ 333, 14641, 333, 14640 ] } ], "ref_ids": [] }, { "id": 14647, "type": "theorem", "label": "descent-lemma-descending-properties", "categories": [ "descent" ], "title": "descent-lemma-descending-properties", "contents": [ "Let $\\mathcal{P}$ be a property of schemes.", "Let $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0]", "\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$.", "Assume that", "\\begin{enumerate}", "\\item the property is local in the Zariski topology,", "\\item for any morphism of affine schemes $S' \\to S$", "which is flat, flat of finite presentation,", "\\'etale, smooth or syntomic depending on whether $\\tau$ is", "fpqc, fppf, \\'etale, smooth, or syntomic,", "property $\\mathcal{P}$ holds for $S'$ if property $\\mathcal{P}$", "holds for $S$, and", "\\item for any surjective morphism of affine schemes $S' \\to S$", "which is flat, flat of finite presentation,", "\\'etale, smooth or syntomic depending on whether $\\tau$ is", "fpqc, fppf, \\'etale, smooth, or syntomic,", "property $\\mathcal{P}$ holds for $S$ if property $\\mathcal{P}$", "holds for $S'$.", "\\end{enumerate}", "Then $\\mathcal{P}$ is $\\tau$ local on the base." ], "refs": [], "proofs": [ { "contents": [ "This follows almost immediately from the definition of", "a $\\tau$-covering, see", "Topologies, Definition", "\\ref{topologies-definition-fpqc-covering}", "\\ref{topologies-definition-fppf-covering}", "\\ref{topologies-definition-etale-covering}", "\\ref{topologies-definition-smooth-covering}, or", "\\ref{topologies-definition-syntomic-covering}", "and Topologies, Lemma", "\\ref{topologies-lemma-fpqc-affine},", "\\ref{topologies-lemma-fppf-affine},", "\\ref{topologies-lemma-etale-affine},", "\\ref{topologies-lemma-smooth-affine}, or", "\\ref{topologies-lemma-syntomic-affine}.", "Details omitted." ], "refs": [ "topologies-definition-fpqc-covering", "topologies-definition-fppf-covering", "topologies-definition-etale-covering", "topologies-definition-smooth-covering", "topologies-definition-syntomic-covering", "topologies-lemma-fpqc-affine", "topologies-lemma-fppf-affine", "topologies-lemma-etale-affine", "topologies-lemma-smooth-affine", "topologies-lemma-syntomic-affine" ], "ref_ids": [ 12547, 12539, 12526, 12531, 12535, 12499, 12473, 12447, 12461, 12467 ] } ], "ref_ids": [] }, { "id": 14648, "type": "theorem", "label": "descent-lemma-Noetherian-local-fppf", "categories": [ "descent" ], "title": "descent-lemma-Noetherian-local-fppf", "contents": [ "The property $\\mathcal{P}(S) =$``$S$ is locally Noetherian'' is local", "in the fppf topology." ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-descending-properties}.", "First we note that ``being locally Noetherian'' is local", "in the Zariski topology. This is clear from the definition,", "see Properties, Definition \\ref{properties-definition-noetherian}.", "Next, we show that if $S' \\to S$ is a flat, finitely presented", "morphism of affines and $S$ is locally Noetherian, then $S'$ is", "locally Noetherian. This is", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}.", "Finally, we have to show that if $S' \\to S$ is a surjective", "flat, finitely presented morphism of affines and $S'$ is", "locally Noetherian, then $S$ is locally Noetherian. This follows from", "Algebra, Lemma \\ref{algebra-lemma-descent-Noetherian}.", "Thus (1), (2) and (3) of Lemma \\ref{lemma-descending-properties} hold", "and we win." ], "refs": [ "descent-lemma-descending-properties", "properties-definition-noetherian", "morphisms-lemma-finite-type-noetherian", "algebra-lemma-descent-Noetherian", "descent-lemma-descending-properties" ], "ref_ids": [ 14647, 3071, 5202, 1370, 14647 ] } ], "ref_ids": [] }, { "id": 14649, "type": "theorem", "label": "descent-lemma-Jacobson-local-fppf", "categories": [ "descent" ], "title": "descent-lemma-Jacobson-local-fppf", "contents": [ "The property $\\mathcal{P}(S) =$``$S$ is Jacobson'' is local", "in the fppf topology." ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-descending-properties}.", "First we note that ``being Jacobson'' is local", "in the Zariski topology. This is", "Properties, Lemma \\ref{properties-lemma-locally-jacobson}.", "Next, we show that if $S' \\to S$ is a flat, finitely presented", "morphism of affines and $S$ is Jacobson, then $S'$ is", "Jacobson. This is", "Morphisms, Lemma \\ref{morphisms-lemma-Jacobson-universally-Jacobson}.", "Finally, we have to show that if $f : S' \\to S$ is a surjective", "flat, finitely presented morphism of affines and $S'$ is", "Jacobson, then $S$ is Jacobson. Say $S = \\Spec(A)$ and", "$S' = \\Spec(B)$ and $S' \\to S$ given by $A \\to B$.", "Then $A \\to B$ is finitely presented and faithfully flat.", "Moreover, the ring $B$ is Jacobson, see", "Properties, Lemma \\ref{properties-lemma-locally-jacobson}.", "\\medskip\\noindent", "By Algebra, Lemma \\ref{algebra-lemma-fppf-fpqf} there exists a diagram", "$$", "\\xymatrix{", "B \\ar[rr] & & B' \\\\", "& A \\ar[ru] \\ar[lu] &", "}", "$$", "with $A \\to B'$ finitely presented, faithfully flat and quasi-finite.", "In particular, $B \\to B'$ is finite type, and we see from", "Algebra, Proposition \\ref{algebra-proposition-Jacobson-permanence}", "that $B'$ is Jacobson. Hence we may assume that $A \\to B$ is quasi-finite", "as well as faithfully flat and of finite presentation.", "\\medskip\\noindent", "Assume $A$ is not Jacobson to get a contradiction.", "According to Algebra, Lemma \\ref{algebra-lemma-characterize-jacobson}", "there exists a nonmaximal prime $\\mathfrak p \\subset A$ and", "an element $f \\in A$, $f \\not \\in \\mathfrak p$ such that", "$V(\\mathfrak p) \\cap D(f) = \\{\\mathfrak p\\}$.", "\\medskip\\noindent", "This leads to a contradiction as follows. First let", "$\\mathfrak p \\subset \\mathfrak m$ be a maximal ideal of $A$.", "Pick a prime $\\mathfrak m' \\subset B$ lying over $\\mathfrak m$", "(exists because $A \\to B$ is faithfully flat, see", "Algebra, Lemma \\ref{algebra-lemma-ff-rings}).", "As $A \\to B$ is flat, by going down see", "Algebra, Lemma \\ref{algebra-lemma-flat-going-down},", "we can find a prime $\\mathfrak q \\subset \\mathfrak m'$ lying over", "$\\mathfrak p$. In particular we see that $\\mathfrak q$ is not", "maximal. Hence according to", "Algebra, Lemma \\ref{algebra-lemma-characterize-jacobson} again", "the set $V(\\mathfrak q) \\cap D(f)$ is infinite", "(here we finally use that $B$ is Jacobson).", "All points of $V(\\mathfrak q) \\cap D(f)$ map to", "$V(\\mathfrak p) \\cap D(f) = \\{\\mathfrak p\\}$. Hence the", "fibre over $\\mathfrak p$ is infinite. This contradicts the", "fact that $A \\to B$ is quasi-finite (see", "Algebra, Lemma \\ref{algebra-lemma-quasi-finite}", "or more explicitly", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite}).", "Thus the lemma is proved." ], "refs": [ "descent-lemma-descending-properties", "properties-lemma-locally-jacobson", "morphisms-lemma-Jacobson-universally-Jacobson", "properties-lemma-locally-jacobson", "algebra-lemma-fppf-fpqf", "algebra-proposition-Jacobson-permanence", "algebra-lemma-characterize-jacobson", "algebra-lemma-ff-rings", "algebra-lemma-flat-going-down", "algebra-lemma-characterize-jacobson", "algebra-lemma-quasi-finite", "morphisms-lemma-quasi-finite" ], "ref_ids": [ 14647, 2964, 5212, 2964, 1398, 1405, 470, 536, 539, 470, 1050, 5230 ] } ], "ref_ids": [] }, { "id": 14650, "type": "theorem", "label": "descent-lemma-locally-finite-nr-irred-local-fppf", "categories": [ "descent" ], "title": "descent-lemma-locally-finite-nr-irred-local-fppf", "contents": [ "The property $\\mathcal{P}(S) =$``every quasi-compact open of $S$", "has a finite number of irreducible components'' is local", "in the fppf topology." ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-descending-properties}. First we note that", "$\\mathcal{P}$ is local in the Zariski topology.", "Next, we show that if $T \\to S$ is a flat, finitely presented", "morphism of affines and $S$ has a finite number of irreducible", "components, then so does $T$. Namely, since $T \\to S$ is flat,", "the generic points of $T$ map to the generic points of $S$, see", "Morphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat}.", "Hence it suffices to show that for $s \\in S$ the fibre $T_s$", "has a finite number of generic points. Note that $T_s$ is an", "affine scheme of finite type over $\\kappa(s)$, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-type}.", "Hence $T_s$ is Noetherian and has a finite number of irreducible components", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian} and", "Properties, Lemma \\ref{properties-lemma-Noetherian-irreducible-components}).", "Finally, we have to show that if $T \\to S$ is a surjective", "flat, finitely presented morphism of affines and $T$ has a finite", "number of irreducible components, then so does $S$. In this case", "the arguments above show that every generic point of $S$ is the", "image of a generic point of $T$ and the result is clear.", "Thus (1), (2) and (3) of Lemma \\ref{lemma-descending-properties} hold", "and we win." ], "refs": [ "descent-lemma-descending-properties", "morphisms-lemma-generalizations-lift-flat", "morphisms-lemma-base-change-finite-type", "morphisms-lemma-finite-type-noetherian", "properties-lemma-Noetherian-irreducible-components", "descent-lemma-descending-properties" ], "ref_ids": [ 14647, 5266, 5200, 5202, 2956, 14647 ] } ], "ref_ids": [] }, { "id": 14651, "type": "theorem", "label": "descent-lemma-Sk-local-syntomic", "categories": [ "descent" ], "title": "descent-lemma-Sk-local-syntomic", "contents": [ "The property $\\mathcal{P}(S) =$``$S$ is locally Noetherian and $(S_k)$''", "is local in the syntomic topology." ], "refs": [], "proofs": [ { "contents": [ "We will check (1), (2) and (3) of Lemma \\ref{lemma-descending-properties}.", "As a syntomic morphism is flat of finite presentation", "(Morphisms, Lemmas \\ref{morphisms-lemma-syntomic-flat}", "and \\ref{morphisms-lemma-syntomic-locally-finite-presentation})", "we have already checked this for ``being locally Noetherian'' in the proof", "of Lemma \\ref{lemma-Noetherian-local-fppf}.", "We will use this without further mention in the proof.", "First we note that $\\mathcal{P}$ is local in the Zariski topology.", "This is clear from the definition,", "see Cohomology of Schemes, Definition \\ref{coherent-definition-depth}.", "Next, we show that if $S' \\to S$ is a syntomic morphism of affines", "and $S$ has $\\mathcal{P}$, then $S'$ has $\\mathcal{P}$. This", "is Algebra, Lemma \\ref{algebra-lemma-Sk-goes-up}", "(use", "Morphisms, Lemma \\ref{morphisms-lemma-syntomic-characterize}", "and", "Algebra, Definition \\ref{algebra-definition-lci} and", "Lemma \\ref{algebra-lemma-lci-CM}).", "Finally, we show that if $S' \\to S$ is a surjective", "syntomic morphism of affines and $S'$ has $\\mathcal{P}$,", "then $S$ has $\\mathcal{P}$. This is", "Algebra, Lemma \\ref{algebra-lemma-descent-Sk}.", "Thus (1), (2) and (3) of Lemma \\ref{lemma-descending-properties} hold", "and we win." ], "refs": [ "descent-lemma-descending-properties", "morphisms-lemma-syntomic-flat", "morphisms-lemma-syntomic-locally-finite-presentation", "descent-lemma-Noetherian-local-fppf", "coherent-definition-depth", "algebra-lemma-Sk-goes-up", "morphisms-lemma-syntomic-characterize", "algebra-definition-lci", "algebra-lemma-lci-CM", "algebra-lemma-descent-Sk", "descent-lemma-descending-properties" ], "ref_ids": [ 14647, 5294, 5293, 14648, 3403, 1363, 5289, 1532, 1166, 1374, 14647 ] } ], "ref_ids": [] }, { "id": 14652, "type": "theorem", "label": "descent-lemma-CM-local-syntomic", "categories": [ "descent" ], "title": "descent-lemma-CM-local-syntomic", "contents": [ "The property $\\mathcal{P}(S) =$``$S$ is Cohen-Macaulay''", "is local in the syntomic topology." ], "refs": [], "proofs": [ { "contents": [ "This is clear from Lemma \\ref{lemma-Sk-local-syntomic}", "above since a scheme is Cohen-Macaulay if and only if", "it is locally Noetherian and $(S_k)$ for all $k \\geq 0$, see", "Properties, Lemma \\ref{properties-lemma-scheme-CM-iff-all-Sk}." ], "refs": [ "descent-lemma-Sk-local-syntomic", "properties-lemma-scheme-CM-iff-all-Sk" ], "ref_ids": [ 14651, 2987 ] } ], "ref_ids": [] }, { "id": 14653, "type": "theorem", "label": "descent-lemma-reduced-local-smooth", "categories": [ "descent" ], "title": "descent-lemma-reduced-local-smooth", "contents": [ "The property $\\mathcal{P}(S) =$``$S$ is reduced'' is local in the smooth", "topology." ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-descending-properties}.", "First we note that ``being reduced'' is local", "in the Zariski topology. This is clear from the definition,", "see Schemes, Definition \\ref{schemes-definition-reduced}.", "Next, we show that if $S' \\to S$ is a smooth morphism of affines", "and $S$ is reduced, then $S'$ is reduced. This is", "Algebra, Lemma \\ref{algebra-lemma-reduced-goes-up}.", "Finally, we show that if $S' \\to S$ is a surjective", "smooth morphism of affines", "and $S'$ is reduced, then $S$ is reduced. This is", "Algebra, Lemma \\ref{algebra-lemma-descent-reduced}.", "Thus (1), (2) and (3) of Lemma \\ref{lemma-descending-properties} hold", "and we win." ], "refs": [ "descent-lemma-descending-properties", "schemes-definition-reduced", "algebra-lemma-reduced-goes-up", "algebra-lemma-descent-reduced", "descent-lemma-descending-properties" ], "ref_ids": [ 14647, 7744, 1366, 1371, 14647 ] } ], "ref_ids": [] }, { "id": 14654, "type": "theorem", "label": "descent-lemma-normal-local-smooth", "categories": [ "descent" ], "title": "descent-lemma-normal-local-smooth", "contents": [ "\\begin{slogan}", "Normality is local in the smooth topology.", "\\end{slogan}", "The property $\\mathcal{P}(S) =$``$S$ is normal'' is local in the smooth", "topology." ], "refs": [], "proofs": [ { "contents": [ "We will use Lemma \\ref{lemma-descending-properties}.", "First we show ``being normal'' is local", "in the Zariski topology. This is clear from the definition,", "see Properties, Definition \\ref{properties-definition-normal}.", "Next, we show that if $S' \\to S$ is a smooth morphism of affines", "and $S$ is normal, then $S'$ is normal. This is", "Algebra, Lemma \\ref{algebra-lemma-normal-goes-up}.", "Finally, we show that if $S' \\to S$ is a surjective", "smooth morphism of affines", "and $S'$ is normal, then $S$ is normal. This is", "Algebra, Lemma \\ref{algebra-lemma-descent-normal}.", "Thus (1), (2) and (3) of Lemma \\ref{lemma-descending-properties} hold", "and we win." ], "refs": [ "descent-lemma-descending-properties", "properties-definition-normal", "algebra-lemma-normal-goes-up", "algebra-lemma-descent-normal", "descent-lemma-descending-properties" ], "ref_ids": [ 14647, 3073, 1368, 1372, 14647 ] } ], "ref_ids": [] }, { "id": 14655, "type": "theorem", "label": "descent-lemma-Rk-local-smooth", "categories": [ "descent" ], "title": "descent-lemma-Rk-local-smooth", "contents": [ "The property $\\mathcal{P}(S) =$``$S$ is locally Noetherian and $(R_k)$''", "is local in the smooth topology." ], "refs": [], "proofs": [ { "contents": [ "We will check (1), (2) and (3) of Lemma \\ref{lemma-descending-properties}.", "As a smooth morphism is flat of finite presentation", "(Morphisms, Lemmas \\ref{morphisms-lemma-smooth-flat}", "and \\ref{morphisms-lemma-smooth-locally-finite-presentation})", "we have already checked this for ``being locally Noetherian'' in the proof", "of Lemma \\ref{lemma-Noetherian-local-fppf}.", "We will use this without further mention in the proof.", "First we note that $\\mathcal{P}$ is local in the Zariski topology.", "This is clear from the definition,", "see Properties, Definition \\ref{properties-definition-Rk}.", "Next, we show that if $S' \\to S$ is a smooth morphism of affines", "and $S$ has $\\mathcal{P}$, then $S'$ has $\\mathcal{P}$. This", "is Algebra, Lemmas \\ref{algebra-lemma-Rk-goes-up}", "(use Morphisms, Lemma \\ref{morphisms-lemma-smooth-characterize},", "Algebra, Lemmas \\ref{algebra-lemma-base-change-smooth}", "and \\ref{algebra-lemma-characterize-smooth-over-field}).", "Finally, we show that if $S' \\to S$ is a surjective", "smooth morphism of affines and $S'$ has $\\mathcal{P}$,", "then $S$ has $\\mathcal{P}$. This is", "Algebra, Lemma \\ref{algebra-lemma-descent-Rk}.", "Thus (1), (2) and (3) of Lemma \\ref{lemma-descending-properties} hold", "and we win." ], "refs": [ "descent-lemma-descending-properties", "morphisms-lemma-smooth-flat", "morphisms-lemma-smooth-locally-finite-presentation", "descent-lemma-Noetherian-local-fppf", "properties-definition-Rk", "algebra-lemma-Rk-goes-up", "morphisms-lemma-smooth-characterize", "algebra-lemma-base-change-smooth", "algebra-lemma-characterize-smooth-over-field", "algebra-lemma-descent-Rk", "descent-lemma-descending-properties" ], "ref_ids": [ 14647, 5331, 5330, 14648, 3078, 1364, 5324, 1191, 1223, 1375, 14647 ] } ], "ref_ids": [] }, { "id": 14656, "type": "theorem", "label": "descent-lemma-regular-local-smooth", "categories": [ "descent" ], "title": "descent-lemma-regular-local-smooth", "contents": [ "The property $\\mathcal{P}(S) =$``$S$ is regular''", "is local in the smooth topology." ], "refs": [], "proofs": [ { "contents": [ "This is clear from Lemma \\ref{lemma-Rk-local-smooth}", "above since a locally Noetherian scheme is regular if and only if", "it is locally Noetherian and $(R_k)$ for all $k \\geq 0$." ], "refs": [ "descent-lemma-Rk-local-smooth" ], "ref_ids": [ 14655 ] } ], "ref_ids": [] }, { "id": 14657, "type": "theorem", "label": "descent-lemma-Nagata-local-smooth", "categories": [ "descent" ], "title": "descent-lemma-Nagata-local-smooth", "contents": [ "The property $\\mathcal{P}(S) =$``$S$ is Nagata''", "is local in the smooth topology." ], "refs": [], "proofs": [ { "contents": [ "We will check (1), (2) and (3) of Lemma \\ref{lemma-descending-properties}.", "First we note that being Nagata is local in the Zariski topology.", "This is Properties, Lemma \\ref{properties-lemma-locally-nagata}.", "Next, we show that if $S' \\to S$ is a smooth morphism of affines", "and $S$ is Nagata, then $S'$ is Nagata. This", "is Morphisms, Lemma \\ref{morphisms-lemma-finite-type-nagata}.", "Finally, we show that if $S' \\to S$ is a surjective", "smooth morphism of affines and $S'$ is Nagata,", "then $S$ is Nagata. This is", "Algebra, Lemma \\ref{algebra-lemma-descent-nagata}.", "Thus (1), (2) and (3) of Lemma \\ref{lemma-descending-properties} hold", "and we win." ], "refs": [ "descent-lemma-descending-properties", "properties-lemma-locally-nagata", "morphisms-lemma-finite-type-nagata", "algebra-lemma-descent-nagata", "descent-lemma-descending-properties" ], "ref_ids": [ 14647, 2995, 5218, 1376, 14647 ] } ], "ref_ids": [] }, { "id": 14658, "type": "theorem", "label": "descent-lemma-descend-reduced", "categories": [ "descent" ], "title": "descent-lemma-descend-reduced", "contents": [ "If $f : X \\to Y$ is a flat and surjective morphism of schemes", "and $X$ is reduced, then $Y$ is reduced." ], "refs": [], "proofs": [ { "contents": [ "The result follows by looking at local rings", "(Schemes, Definition \\ref{schemes-definition-reduced})", "and", "Algebra, Lemma \\ref{algebra-lemma-descent-reduced}." ], "refs": [ "schemes-definition-reduced", "algebra-lemma-descent-reduced" ], "ref_ids": [ 7744, 1371 ] } ], "ref_ids": [] }, { "id": 14659, "type": "theorem", "label": "descent-lemma-descend-regular", "categories": [ "descent" ], "title": "descent-lemma-descend-regular", "contents": [ "Let $f : X \\to Y$ be a morphism of algebraic spaces.", "If $f$ is locally of finite presentation, flat, and surjective and", "$X$ is regular, then $Y$ is regular." ], "refs": [], "proofs": [ { "contents": [ "This lemma reduces to the following algebra statement: If $A \\to B$ is", "a faithfully flat, finitely presented ring homomorphism with $B$ Noetherian", "and regular, then $A$ is Noetherian and regular. We see that", "$A$ is Noetherian by", "Algebra, Lemma \\ref{algebra-lemma-descent-Noetherian}", "and regular by", "Algebra, Lemma \\ref{algebra-lemma-flat-under-regular}." ], "refs": [ "algebra-lemma-descent-Noetherian", "algebra-lemma-flat-under-regular" ], "ref_ids": [ 1370, 981 ] } ], "ref_ids": [] }, { "id": 14660, "type": "theorem", "label": "descent-lemma-dimension-at-point-local", "categories": [ "descent" ], "title": "descent-lemma-dimension-at-point-local", "contents": [ "Let $f : U \\to V$ be an \\'etale morphism of schemes.", "Let $u \\in U$ and $v = f(u)$. Then $\\dim_u(U) = \\dim_v(V)$." ], "refs": [], "proofs": [ { "contents": [ "In the statement $\\dim_u(U)$ is the dimension of $U$ at $u$ as defined in", "Topology, Definition \\ref{topology-definition-Krull}", "as the minimum of the Krull dimensions of open neighbourhoods of $u$ in $U$.", "Similarly for $\\dim_v(V)$.", "\\medskip\\noindent", "Let us show that $\\dim_v(V) \\geq \\dim_u(U)$.", "Let $V'$ be an open neighbourhood of $v$ in $V$.", "Then there exists an open neighbourhood $U'$ of $u$ in $U$", "contained in $f^{-1}(V')$ such that $\\dim_u(U) = \\dim(U')$. Suppose that", "$Z_0 \\subset Z_1 \\subset \\ldots \\subset Z_n$ is a chain of irreducible", "closed subschemes of $U'$. If $\\xi_i \\in Z_i$ is the generic point", "then we have specializations", "$\\xi_n \\leadsto \\xi_{n - 1} \\leadsto \\ldots \\leadsto \\xi_0$.", "This gives specializations", "$f(\\xi_n) \\leadsto f(\\xi_{n - 1}) \\leadsto \\ldots \\leadsto f(\\xi_0)$", "in $V'$. Note that $f(\\xi_j) \\not = f(\\xi_i)$ if $i \\not = j$ as", "the fibres of $f$ are discrete (see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}).", "Hence we see that $\\dim(V') \\geq n$. The inequality", "$\\dim_v(V) \\geq \\dim_u(U)$ follows formally.", "\\medskip\\noindent", "Let us show that $\\dim_u(U) \\geq \\dim_v(V)$.", "Let $U'$ be an open neighbourhood of $u$ in $U$.", "Note that $V' = f(U')$ is an open neighbourhood of $v$ by", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}.", "Hence $\\dim(V') \\geq \\dim_v(V)$. Pick a chain", "$Z_0 \\subset Z_1 \\subset \\ldots \\subset Z_n$ of irreducible", "closed subschemes of $V'$. Let $\\xi_i \\in Z_i$ be the generic point,", "so we have specializations", "$\\xi_n \\leadsto \\xi_{n - 1} \\leadsto \\ldots \\leadsto \\xi_0$.", "Since $\\xi_0 \\in f(U')$ we can find a point $\\eta_0 \\in U'$", "with $f(\\eta_0) = \\xi_0$. Consider the map of local rings", "$$", "\\mathcal{O}_{V', \\xi_0} \\longrightarrow \\mathcal{O}_{U', \\eta_0}", "$$", "which is a flat local ring map by", "Morphisms, Lemma \\ref{morphisms-lemma-etale-flat}.", "Note that the points $\\xi_i$ correspond to primes of the ring on the left by", "Schemes, Lemma \\ref{schemes-lemma-specialize-points}.", "Hence by going down (see", "Algebra, Section \\ref{algebra-section-going-up})", "for the displayed ring map we can find a sequence of specializations", "$\\eta_n \\leadsto \\eta_{n - 1} \\leadsto \\ldots \\leadsto \\eta_0$", "in $U'$ mapping to the sequence", "$\\xi_n \\leadsto \\xi_{n - 1} \\leadsto \\ldots \\leadsto \\xi_0$", "under $f$. This implies that $\\dim_u(U) \\geq \\dim_v(V)$." ], "refs": [ "topology-definition-Krull", "morphisms-lemma-etale-over-field", "morphisms-lemma-fppf-open", "morphisms-lemma-etale-flat", "schemes-lemma-specialize-points" ], "ref_ids": [ 8356, 5364, 5267, 5369, 7684 ] } ], "ref_ids": [] }, { "id": 14661, "type": "theorem", "label": "descent-lemma-dimension-local-ring-local", "categories": [ "descent" ], "title": "descent-lemma-dimension-local-ring-local", "contents": [ "Let $f : U \\to V$ be an \\'etale morphism of schemes.", "Let $u \\in U$ and $v = f(u)$. Then", "$\\dim(\\mathcal{O}_{U, u}) = \\dim(\\mathcal{O}_{V, v})$." ], "refs": [], "proofs": [ { "contents": [ "The algebraic statement we are asked to prove is the following:", "If $A \\to B$ is an \\'etale ring map and $\\mathfrak q$ is a prime of", "$B$ lying over $\\mathfrak p \\subset A$, then", "$\\dim(A_{\\mathfrak p}) = \\dim(B_{\\mathfrak q})$.", "This is", "More on Algebra, Lemma \\ref{more-algebra-lemma-dimension-etale-extension}." ], "refs": [ "more-algebra-lemma-dimension-etale-extension" ], "ref_ids": [ 10052 ] } ], "ref_ids": [] }, { "id": 14662, "type": "theorem", "label": "descent-lemma-regular-local-ring-local", "categories": [ "descent" ], "title": "descent-lemma-regular-local-ring-local", "contents": [ "Let $f : U \\to V$ be an \\'etale morphism of schemes.", "Let $u \\in U$ and $v = f(u)$. Then", "$\\mathcal{O}_{U, u}$ is a regular local ring if and only if", "$\\mathcal{O}_{V, v}$ is a regular local ring." ], "refs": [], "proofs": [ { "contents": [ "The algebraic statement we are asked to prove is the following:", "If $A \\to B$ is an \\'etale ring map and $\\mathfrak q$ is a prime of", "$B$ lying over $\\mathfrak p \\subset A$, then", "$A_{\\mathfrak p}$ is regular if and only if $B_{\\mathfrak q}$ is regular.", "This is More on Algebra, Lemma", "\\ref{more-algebra-lemma-regular-etale-extension}." ], "refs": [ "more-algebra-lemma-regular-etale-extension" ], "ref_ids": [ 10053 ] } ], "ref_ids": [] }, { "id": 14663, "type": "theorem", "label": "descent-lemma-pullback-property-local-target", "categories": [ "descent" ], "title": "descent-lemma-pullback-property-local-target", "contents": [ "Let $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale, Zariski\\}$.", "Let $\\mathcal{P}$ be a property of morphisms which is $\\tau$ local", "on the target. Let $f : X \\to Y$ have property $\\mathcal{P}$.", "For any morphism $Y' \\to Y$ which is", "flat, resp.\\ flat and locally of finite presentation, resp.\\ syntomic,", "resp.\\ \\'etale, resp.\\ an open immersion, the base change", "$f' : Y' \\times_Y X \\to Y'$ of $f$ has property $\\mathcal{P}$." ], "refs": [], "proofs": [ { "contents": [ "This is true because we can fit $Y' \\to Y$ into a family of", "morphisms which forms a $\\tau$-covering." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14664, "type": "theorem", "label": "descent-lemma-largest-open-of-the-base", "categories": [ "descent" ], "title": "descent-lemma-largest-open-of-the-base", "contents": [ "Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale\\}$.", "Let $\\mathcal{P}$ be a property of morphisms which is $\\tau$ local", "on the target. For any morphism of schemes $f : X \\to Y$ there exists", "a largest open $W(f) \\subset Y$ such that the restriction", "$X_{W(f)} \\to W(f)$ has $\\mathcal{P}$. Moreover,", "\\begin{enumerate}", "\\item if $g : Y' \\to Y$ is flat and locally of finite presentation,", "syntomic, smooth, or \\'etale and the base change $f' : X_{Y'} \\to Y'$", "has $\\mathcal{P}$, then $g(Y') \\subset W(f)$,", "\\item if $g : Y' \\to Y$ is flat and locally of finite presentation,", "syntomic, smooth, or \\'etale, then $W(f') = g^{-1}(W(f))$, and", "\\item if $\\{g_i : Y_i \\to Y\\}$ is a $\\tau$-covering, then", "$g_i^{-1}(W(f)) = W(f_i)$, where $f_i$ is the base change of $f$", "by $Y_i \\to Y$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Consider the union $W$ of the images $g(Y') \\subset Y$ of", "morphisms $g : Y' \\to Y$ with the properties:", "\\begin{enumerate}", "\\item $g$ is flat and locally of finite presentation, syntomic,", "smooth, or \\'etale, and", "\\item the base change $Y' \\times_{g, Y} X \\to Y'$ has property", "$\\mathcal{P}$.", "\\end{enumerate}", "Since such a morphism $g$ is open (see", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open})", "we see that $W \\subset Y$ is an open subset of $Y$. Since $\\mathcal{P}$", "is local in the $\\tau$ topology the restriction $X_W \\to W$ has property", "$\\mathcal{P}$ because we are given a covering $\\{Y' \\to W\\}$ of $W$ such that", "the pullbacks have $\\mathcal{P}$. This proves the existence and proves", "that $W(f)$ has property (1). To see property (2) note that", "$W(f') \\supset g^{-1}(W(f))$ because $\\mathcal{P}$ is stable under", "base change by flat and locally of finite presentation,", "syntomic, smooth, or \\'etale morphisms, see", "Lemma \\ref{lemma-pullback-property-local-target}.", "On the other hand, if $Y'' \\subset Y'$ is an open such that", "$X_{Y''} \\to Y''$ has property $\\mathcal{P}$, then $Y'' \\to Y$ factors", "through $W$ by construction, i.e., $Y'' \\subset g^{-1}(W(f))$. This", "proves (2). Assertion (3) follows from (2) because each morphism", "$Y_i \\to Y$ is flat and locally of finite presentation, syntomic,", "smooth, or \\'etale by our definition of a $\\tau$-covering." ], "refs": [ "morphisms-lemma-fppf-open", "descent-lemma-pullback-property-local-target" ], "ref_ids": [ 5267, 14663 ] } ], "ref_ids": [] }, { "id": 14665, "type": "theorem", "label": "descent-lemma-descending-properties-morphisms", "categories": [ "descent" ], "title": "descent-lemma-descending-properties-morphisms", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes over a base.", "Let $\\tau \\in \\{fpqc, fppf, \\etale, smooth, syntomic\\}$.", "Assume that", "\\begin{enumerate}", "\\item the property is preserved under", "flat, flat and locally of finite presentation, \\'etale, smooth, or syntomic", "base change depending on whether $\\tau$ is fpqc, fppf, \\'etale, smooth, or", "syntomic (compare with", "Schemes, Definition \\ref{schemes-definition-preserved-by-base-change}),", "\\item the property is Zariski local on the base.", "\\item for any surjective morphism of affine schemes $S' \\to S$", "which is flat, flat of finite presentation,", "\\'etale, smooth or syntomic depending on whether $\\tau$ is", "fpqc, fppf, \\'etale, smooth, or syntomic,", "and any morphism of schemes $f : X \\to S$ property", "$\\mathcal{P}$ holds for $f$ if property $\\mathcal{P}$", "holds for the base change $f' : X' = S' \\times_S X \\to S'$.", "\\end{enumerate}", "Then $\\mathcal{P}$ is $\\tau$ local on the base." ], "refs": [ "schemes-definition-preserved-by-base-change" ], "proofs": [ { "contents": [ "This follows almost immediately from the definition of", "a $\\tau$-covering, see", "Topologies, Definition", "\\ref{topologies-definition-fpqc-covering}", "\\ref{topologies-definition-fppf-covering}", "\\ref{topologies-definition-etale-covering}", "\\ref{topologies-definition-smooth-covering}, or", "\\ref{topologies-definition-syntomic-covering}", "and Topologies, Lemma", "\\ref{topologies-lemma-fpqc-affine},", "\\ref{topologies-lemma-fppf-affine},", "\\ref{topologies-lemma-etale-affine},", "\\ref{topologies-lemma-smooth-affine}, or", "\\ref{topologies-lemma-syntomic-affine}.", "Details omitted." ], "refs": [ "topologies-definition-fpqc-covering", "topologies-definition-fppf-covering", "topologies-definition-etale-covering", "topologies-definition-smooth-covering", "topologies-definition-syntomic-covering", "topologies-lemma-fpqc-affine", "topologies-lemma-fppf-affine", "topologies-lemma-etale-affine", "topologies-lemma-smooth-affine", "topologies-lemma-syntomic-affine" ], "ref_ids": [ 12547, 12539, 12526, 12531, 12535, 12499, 12473, 12447, 12461, 12467 ] } ], "ref_ids": [ 7751 ] }, { "id": 14666, "type": "theorem", "label": "descent-lemma-descending-property-quasi-compact", "categories": [ "descent" ], "title": "descent-lemma-descending-property-quasi-compact", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is quasi-compact''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "A base change of a quasi-compact morphism is quasi-compact, see", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-preserved-base-change}.", "Being quasi-compact is Zariski local on the base, see", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-affine}.", "Finally, let", "$S' \\to S$ be a flat surjective morphism of affine schemes,", "and let $f : X \\to S$ be a morphism. Assume that the base change", "$f' : X' \\to S'$ is quasi-compact. Then $X'$ is quasi-compact,", "and $X' \\to X$ is surjective. Hence $X$ is quasi-compact.", "This implies that $f$ is quasi-compact.", "Therefore Lemma \\ref{lemma-descending-properties-morphisms} applies and we win." ], "refs": [ "schemes-lemma-quasi-compact-preserved-base-change", "schemes-lemma-quasi-compact-affine", "descent-lemma-descending-properties-morphisms" ], "ref_ids": [ 7698, 7697, 14665 ] } ], "ref_ids": [] }, { "id": 14667, "type": "theorem", "label": "descent-lemma-descending-property-quasi-separated", "categories": [ "descent" ], "title": "descent-lemma-descending-property-quasi-separated", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is quasi-separated''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Any base change of a quasi-separated morphism is quasi-separated, see", "Schemes, Lemma \\ref{schemes-lemma-separated-permanence}.", "Being quasi-separated is Zariski local on the base (from the", "definition or by", "Schemes, Lemma \\ref{schemes-lemma-characterize-quasi-separated}).", "Finally, let", "$S' \\to S$ be a flat surjective morphism of affine schemes,", "and let $f : X \\to S$ be a morphism. Assume that the base change", "$f' : X' \\to S'$ is quasi-separated. This means that", "$\\Delta' : X' \\to X'\\times_{S'} X'$ is quasi-compact.", "Note that $\\Delta'$ is the base change of $\\Delta : X \\to X \\times_S X$", "via $S' \\to S$. By Lemma \\ref{lemma-descending-property-quasi-compact}", "this implies $\\Delta$ is quasi-compact, and hence $f$ is", "quasi-separated.", "Therefore Lemma \\ref{lemma-descending-properties-morphisms} applies and we win." ], "refs": [ "schemes-lemma-separated-permanence", "schemes-lemma-characterize-quasi-separated", "descent-lemma-descending-property-quasi-compact", "descent-lemma-descending-properties-morphisms" ], "ref_ids": [ 7714, 7709, 14666, 14665 ] } ], "ref_ids": [] }, { "id": 14668, "type": "theorem", "label": "descent-lemma-descending-property-universally-closed", "categories": [ "descent" ], "title": "descent-lemma-descending-property-universally-closed", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is universally closed''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "A base change of a universally closed morphism is universally closed", "by definition.", "Being universally closed is Zariski local on the base (from the", "definition or by", "Morphisms, Lemma", "\\ref{morphisms-lemma-universally-closed-local-on-the-base}).", "Finally, let", "$S' \\to S$ be a flat surjective morphism of affine schemes,", "and let $f : X \\to S$ be a morphism. Assume that the base change", "$f' : X' \\to S'$ is universally closed. Let $T \\to S$ be any morphism.", "Consider the diagram", "$$", "\\xymatrix{", "X' \\ar[d] &", "S' \\times_S T \\times_S X \\ar[d] \\ar[r] \\ar[l] &", "T \\times_S X \\ar[d] \\\\", "S' &", "S' \\times_S T \\ar[r] \\ar[l] &", "T", "}", "$$", "in which both squares are cartesian.", "Thus the assumption implies that the middle vertical", "arrow is closed. The right horizontal arrows are flat, quasi-compact", "and surjective (as base changes of $S' \\to S$).", "Hence a subset of $T$ is closed if and only if its inverse", "image in $S' \\times_S T$ is closed, see Morphisms,", "Lemma \\ref{morphisms-lemma-fpqc-quotient-topology}.", "An easy diagram chase shows that the right vertical", "arrow is closed too, and we conclude $X \\to S$ is", "universally closed.", "Therefore Lemma \\ref{lemma-descending-properties-morphisms} applies and we win." ], "refs": [ "morphisms-lemma-universally-closed-local-on-the-base", "morphisms-lemma-fpqc-quotient-topology", "descent-lemma-descending-properties-morphisms" ], "ref_ids": [ 5406, 5269, 14665 ] } ], "ref_ids": [] }, { "id": 14669, "type": "theorem", "label": "descent-lemma-descending-property-universally-open", "categories": [ "descent" ], "title": "descent-lemma-descending-property-universally-open", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is universally open''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "The proof is the same as the proof of", "Lemma \\ref{lemma-descending-property-universally-closed}." ], "refs": [ "descent-lemma-descending-property-universally-closed" ], "ref_ids": [ 14668 ] } ], "ref_ids": [] }, { "id": 14670, "type": "theorem", "label": "descent-lemma-descending-property-universally-submersive", "categories": [ "descent" ], "title": "descent-lemma-descending-property-universally-submersive", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is universally submersive''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "The proof is the same as the proof of", "Lemma \\ref{lemma-descending-property-universally-closed}", "using that a quasi-compact flat surjective morphism is", "universally submersive by", "Morphisms, Lemma \\ref{morphisms-lemma-fpqc-quotient-topology}." ], "refs": [ "descent-lemma-descending-property-universally-closed", "morphisms-lemma-fpqc-quotient-topology" ], "ref_ids": [ 14668, 5269 ] } ], "ref_ids": [] }, { "id": 14671, "type": "theorem", "label": "descent-lemma-descending-property-separated", "categories": [ "descent" ], "title": "descent-lemma-descending-property-separated", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is separated''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "A base change of a separated morphism is separated, see", "Schemes, Lemma \\ref{schemes-lemma-separated-permanence}.", "Being separated is Zariski local on the base (from the", "definition or by", "Schemes, Lemma \\ref{schemes-lemma-characterize-separated}).", "Finally, let", "$S' \\to S$ be a flat surjective morphism of affine schemes,", "and let $f : X \\to S$ be a morphism. Assume that the base change", "$f' : X' \\to S'$ is separated. This means that", "$\\Delta' : X' \\to X'\\times_{S'} X'$ is a closed immersion,", "hence universally closed.", "Note that $\\Delta'$ is the base change of $\\Delta : X \\to X \\times_S X$", "via $S' \\to S$. By Lemma \\ref{lemma-descending-property-universally-closed}", "this implies $\\Delta$ is universally closed. Since it is", "an immersion", "(Schemes, Lemma \\ref{schemes-lemma-diagonal-immersion})", "we conclude $\\Delta$ is a closed immersion.", "Hence $f$ is separated.", "Therefore Lemma \\ref{lemma-descending-properties-morphisms} applies and we win." ], "refs": [ "schemes-lemma-separated-permanence", "schemes-lemma-characterize-separated", "descent-lemma-descending-property-universally-closed", "schemes-lemma-diagonal-immersion", "descent-lemma-descending-properties-morphisms" ], "ref_ids": [ 7714, 7710, 14668, 7707, 14665 ] } ], "ref_ids": [] }, { "id": 14672, "type": "theorem", "label": "descent-lemma-descending-property-surjective", "categories": [ "descent" ], "title": "descent-lemma-descending-property-surjective", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is surjective''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "This is clear." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14673, "type": "theorem", "label": "descent-lemma-descending-property-universally-injective", "categories": [ "descent" ], "title": "descent-lemma-descending-property-universally-injective", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is universally injective''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "A base change of a universally injective morphism is universally", "injective (this is formal). Being universally injective is Zariski", "local on the base; this is clear from the definition.", "Finally, let", "$S' \\to S$ be a flat surjective morphism of affine schemes,", "and let $f : X \\to S$ be a morphism. Assume that the base change", "$f' : X' \\to S'$ is universally injective. Let $K$ be a field, and let", "$a, b : \\Spec(K) \\to X$ be two morphisms such that", "$f \\circ a = f \\circ b$. As $S' \\to S$ is surjective and", "by the discussion in Schemes,", "Section \\ref{schemes-section-points} there exists a field", "extension $K \\subset K'$ and a morphism $\\Spec(K')", "\\to S'$ such that the following solid diagram commutes", "$$", "\\xymatrix{", "\\Spec(K') \\ar[rrd] \\ar@{-->}[rd]_{a', b'} \\ar[dd] \\\\", " &", "X' \\ar[r] \\ar[d] &", "S' \\ar[d] \\\\", "\\Spec(K) \\ar[r]^{a, b} &", "X \\ar[r] &", "S", "}", "$$", "As the square is cartesian we get the two dotted arrows $a'$, $b'$ making the", "diagram commute. Since $X' \\to S'$ is universally injective we get $a' = b'$,", "by", "Morphisms, Lemma \\ref{morphisms-lemma-universally-injective}.", "Clearly this forces $a = b$ (by the discussion in Schemes,", "Section \\ref{schemes-section-points}).", "Therefore Lemma \\ref{lemma-descending-properties-morphisms} applies and we win.", "\\medskip\\noindent", "An alternative proof would be to use the characterization of a universally", "injective morphism as one whose diagonal is surjective, see", "Morphisms, Lemma \\ref{morphisms-lemma-universally-injective}.", "The lemma then follows from the fact that", "the property of being surjective is fpqc local on the base, see", "Lemma \\ref{lemma-descending-property-surjective}.", "(Hint: use that the base change of the diagonal is the diagonal", "of the base change.)" ], "refs": [ "morphisms-lemma-universally-injective", "descent-lemma-descending-properties-morphisms", "morphisms-lemma-universally-injective", "descent-lemma-descending-property-surjective" ], "ref_ids": [ 5167, 14665, 5167, 14672 ] } ], "ref_ids": [] }, { "id": 14674, "type": "theorem", "label": "descent-lemma-descending-property-universal-homeomorphism", "categories": [ "descent" ], "title": "descent-lemma-descending-property-universal-homeomorphism", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is a universal homeomorphism''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "This can be proved in exactly the same manner as", "Lemma \\ref{lemma-descending-property-universally-closed}.", "Alternatively, one can use that", "a map of topological spaces is a homeomorphism if and only if", "it is injective, surjective, and open. Thus", "a universal homeomorphism is the same thing as a", "surjective, universally injective, and universally open morphism.", "Thus the lemma follows from", "Lemmas \\ref{lemma-descending-property-surjective},", "\\ref{lemma-descending-property-universally-injective}, and", "\\ref{lemma-descending-property-universally-open}." ], "refs": [ "descent-lemma-descending-property-universally-closed", "descent-lemma-descending-property-surjective", "descent-lemma-descending-property-universally-injective", "descent-lemma-descending-property-universally-open" ], "ref_ids": [ 14668, 14672, 14673, 14669 ] } ], "ref_ids": [] }, { "id": 14675, "type": "theorem", "label": "descent-lemma-descending-property-locally-finite-type", "categories": [ "descent" ], "title": "descent-lemma-descending-property-locally-finite-type", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is locally of finite type''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Being locally of finite type is preserved under base change, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-type}.", "Being locally of finite type is Zariski local on the base, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-finite-type-characterize}.", "Finally, let", "$S' \\to S$ be a flat surjective morphism of affine schemes,", "and let $f : X \\to S$ be a morphism. Assume that the base change", "$f' : X' \\to S'$ is locally of finite type.", "Let $U \\subset X$ be an affine open. Then $U' = S' \\times_S U$", "is affine and of finite type over $S'$. Write", "$S = \\Spec(R)$,", "$S' = \\Spec(R')$,", "$U = \\Spec(A)$, and", "$U' = \\Spec(A')$.", "We know that $R \\to R'$ is faithfully flat,", "$A' = R' \\otimes_R A$ and $R' \\to A'$ is of finite type.", "We have to show that $R \\to A$ is of finite type.", "This is the result of", "Algebra, Lemma \\ref{algebra-lemma-finite-type-descends}.", "It follows that $f$ is locally of finite type.", "Therefore Lemma \\ref{lemma-descending-properties-morphisms} applies and we win." ], "refs": [ "morphisms-lemma-base-change-finite-type", "morphisms-lemma-locally-finite-type-characterize", "algebra-lemma-finite-type-descends", "descent-lemma-descending-properties-morphisms" ], "ref_ids": [ 5200, 5198, 1079, 14665 ] } ], "ref_ids": [] }, { "id": 14676, "type": "theorem", "label": "descent-lemma-descending-property-locally-finite-presentation", "categories": [ "descent" ], "title": "descent-lemma-descending-property-locally-finite-presentation", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is locally of finite presentation''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Being locally of finite presentation is preserved under base change, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-presentation}.", "Being locally of finite type is Zariski local on the base, see Morphisms,", "Lemma \\ref{morphisms-lemma-locally-finite-presentation-characterize}.", "Finally, let", "$S' \\to S$ be a flat surjective morphism of affine schemes,", "and let $f : X \\to S$ be a morphism. Assume that the base change", "$f' : X' \\to S'$ is locally of finite presentation.", "Let $U \\subset X$ be an affine open. Then $U' = S' \\times_S U$", "is affine and of finite type over $S'$. Write", "$S = \\Spec(R)$,", "$S' = \\Spec(R')$,", "$U = \\Spec(A)$, and", "$U' = \\Spec(A')$.", "We know that $R \\to R'$ is faithfully flat,", "$A' = R' \\otimes_R A$ and $R' \\to A'$ is of finite presentation.", "We have to show that $R \\to A$ is of finite presentation.", "This is the result of", "Algebra, Lemma \\ref{algebra-lemma-finite-presentation-descends}.", "It follows that $f$ is locally of finite presentation.", "Therefore Lemma \\ref{lemma-descending-properties-morphisms} applies and we win." ], "refs": [ "morphisms-lemma-base-change-finite-presentation", "morphisms-lemma-locally-finite-presentation-characterize", "algebra-lemma-finite-presentation-descends", "descent-lemma-descending-properties-morphisms" ], "ref_ids": [ 5240, 5238, 1080, 14665 ] } ], "ref_ids": [] }, { "id": 14677, "type": "theorem", "label": "descent-lemma-descending-property-finite-type", "categories": [ "descent" ], "title": "descent-lemma-descending-property-finite-type", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is of finite type''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-descending-property-quasi-compact}", "and \\ref{lemma-descending-property-locally-finite-type}." ], "refs": [ "descent-lemma-descending-property-quasi-compact", "descent-lemma-descending-property-locally-finite-type" ], "ref_ids": [ 14666, 14675 ] } ], "ref_ids": [] }, { "id": 14678, "type": "theorem", "label": "descent-lemma-descending-property-finite-presentation", "categories": [ "descent" ], "title": "descent-lemma-descending-property-finite-presentation", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is of finite presentation''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-descending-property-quasi-compact},", "\\ref{lemma-descending-property-quasi-separated} and", "\\ref{lemma-descending-property-locally-finite-presentation}." ], "refs": [ "descent-lemma-descending-property-quasi-compact", "descent-lemma-descending-property-quasi-separated", "descent-lemma-descending-property-locally-finite-presentation" ], "ref_ids": [ 14666, 14667, 14676 ] } ], "ref_ids": [] }, { "id": 14679, "type": "theorem", "label": "descent-lemma-descending-property-proper", "categories": [ "descent" ], "title": "descent-lemma-descending-property-proper", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is proper''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "The lemma follows by combining", "Lemmas \\ref{lemma-descending-property-universally-closed},", "\\ref{lemma-descending-property-separated}", "and \\ref{lemma-descending-property-finite-type}." ], "refs": [ "descent-lemma-descending-property-universally-closed", "descent-lemma-descending-property-separated", "descent-lemma-descending-property-finite-type" ], "ref_ids": [ 14668, 14671, 14677 ] } ], "ref_ids": [] }, { "id": 14680, "type": "theorem", "label": "descent-lemma-descending-property-flat", "categories": [ "descent" ], "title": "descent-lemma-descending-property-flat", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is flat''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Being flat is preserved under arbitrary base change, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat}.", "Being flat is Zariski local on the base by definition.", "Finally, let", "$S' \\to S$ be a flat surjective morphism of affine schemes,", "and let $f : X \\to S$ be a morphism. Assume that the base change", "$f' : X' \\to S'$ is flat.", "Let $U \\subset X$ be an affine open. Then $U' = S' \\times_S U$", "is affine. Write", "$S = \\Spec(R)$,", "$S' = \\Spec(R')$,", "$U = \\Spec(A)$, and", "$U' = \\Spec(A')$.", "We know that $R \\to R'$ is faithfully flat,", "$A' = R' \\otimes_R A$ and $R' \\to A'$ is flat.", "Goal: Show that $R \\to A$ is flat.", "This follows immediately from", "Algebra, Lemma \\ref{algebra-lemma-flatness-descends}.", "Hence $f$ is flat.", "Therefore Lemma \\ref{lemma-descending-properties-morphisms} applies and we win." ], "refs": [ "morphisms-lemma-base-change-flat", "algebra-lemma-flatness-descends", "descent-lemma-descending-properties-morphisms" ], "ref_ids": [ 5265, 528, 14665 ] } ], "ref_ids": [] }, { "id": 14681, "type": "theorem", "label": "descent-lemma-descending-property-open-immersion", "categories": [ "descent" ], "title": "descent-lemma-descending-property-open-immersion", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is an open immersion''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "The property of being an open immersion is stable under base change,", "see Schemes, Lemma \\ref{schemes-lemma-base-change-immersion}.", "The property of being an open immersion is Zariski local on the base", "(this is obvious).", "\\medskip\\noindent", "Let $S' \\to S$ be a flat surjective morphism of affine schemes,", "and let $f : X \\to S$ be a morphism. Assume that the base change", "$f' : X' \\to S'$ is an open immersion. We claim that $f$ is an", "open immersion.", "Then $f'$ is universally open, and universally injective.", "Hence we conclude that $f$ is universally open by", "Lemma \\ref{lemma-descending-property-universally-open}, and", "universally injective by", "Lemma \\ref{lemma-descending-property-universally-injective}.", "In particular $f(X) \\subset S$ is open. If for every affine", "open $U \\subset f(X)$ we can prove that $f^{-1}(U) \\to U$", "is an isomorphism, then $f$ is an open immersion and we're done.", "If $U' \\subset S'$ denotes the inverse image of $U$,", "then $U' \\to U$ is a faithfully flat morphism of affines and", "$(f')^{-1}(U') \\to U'$ is an isomorphism (as $f'(X')$ contains $U'$", "by our choice of $U$). Thus we reduce to the case discussed", "in the next paragraph.", "\\medskip\\noindent", "Let $S' \\to S$ be a flat surjective morphism of affine schemes,", "let $f : X \\to S$ be a morphism, and assume that the base change", "$f' : X' \\to S'$ is an isomorphism. We have to show that $f$ is an", "isomorphism also. It is clear that $f$ is surjective, universally injective,", "and universally open (see arguments above for the last two).", "Hence $f$ is bijective, i.e., $f$ is a homeomorphism.", "Thus $f$ is affine by", "Morphisms, Lemma \\ref{morphisms-lemma-homeomorphism-affine}.", "Since", "$$", "\\mathcal{O}(S') \\to", "\\mathcal{O}(X') =", "\\mathcal{O}(S') \\otimes_{\\mathcal{O}(S)} \\mathcal{O}(X)", "$$", "is an isomorphism and since $\\mathcal{O}(S) \\to \\mathcal{O}(S')$", "is faithfully flat this implies that $\\mathcal{O}(S) \\to \\mathcal{O}(X)$", "is an isomorphism. Thus $f$ is an isomorphism. This finishes the proof of", "the claim above.", "Therefore Lemma \\ref{lemma-descending-properties-morphisms} applies and we win." ], "refs": [ "schemes-lemma-base-change-immersion", "descent-lemma-descending-property-universally-open", "descent-lemma-descending-property-universally-injective", "morphisms-lemma-homeomorphism-affine", "descent-lemma-descending-properties-morphisms" ], "ref_ids": [ 7695, 14669, 14673, 5453, 14665 ] } ], "ref_ids": [] }, { "id": 14682, "type": "theorem", "label": "descent-lemma-descending-property-isomorphism", "categories": [ "descent" ], "title": "descent-lemma-descending-property-isomorphism", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is an isomorphism''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-descending-property-surjective}", "and \\ref{lemma-descending-property-open-immersion}." ], "refs": [ "descent-lemma-descending-property-surjective", "descent-lemma-descending-property-open-immersion" ], "ref_ids": [ 14672, 14681 ] } ], "ref_ids": [] }, { "id": 14683, "type": "theorem", "label": "descent-lemma-descending-property-affine", "categories": [ "descent" ], "title": "descent-lemma-descending-property-affine", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is affine''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "A base change of an affine morphism is affine, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-affine}.", "Being affine is Zariski local on the base, see", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-affine}.", "Finally, let", "$g : S' \\to S$ be a flat surjective morphism of affine schemes,", "and let $f : X \\to S$ be a morphism. Assume that the base change", "$f' : X' \\to S'$ is affine. In other words, $X'$ is affine, say", "$X' = \\Spec(A')$. Also write $S = \\Spec(R)$", "and $S' = \\Spec(R')$. We have to show that $X$ is affine.", "\\medskip\\noindent", "By Lemmas \\ref{lemma-descending-property-quasi-compact}", "and \\ref{lemma-descending-property-separated} we see that", "$X \\to S$ is separated and quasi-compact. Thus", "$f_*\\mathcal{O}_X$ is a quasi-coherent sheaf of $\\mathcal{O}_S$-algebras,", "see Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}.", "Hence $f_*\\mathcal{O}_X = \\widetilde{A}$ for some $R$-algebra $A$.", "In fact $A = \\Gamma(X, \\mathcal{O}_X)$ of course.", "Also, by flat base change", "(see for example", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology})", "we have $g^*f_*\\mathcal{O}_X = f'_*\\mathcal{O}_{X'}$.", "In other words, we have $A' = R' \\otimes_R A$.", "Consider the canonical morphism", "$$", "X \\longrightarrow \\Spec(A)", "$$", "over $S$ from Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}.", "By the above the base change of this morphism to $S'$ is an isomorphism.", "Hence it is an isomorphism by", "Lemma \\ref{lemma-descending-property-isomorphism}.", "Therefore Lemma \\ref{lemma-descending-properties-morphisms} applies and we win." ], "refs": [ "morphisms-lemma-base-change-affine", "morphisms-lemma-characterize-affine", "descent-lemma-descending-property-quasi-compact", "descent-lemma-descending-property-separated", "schemes-lemma-push-forward-quasi-coherent", "coherent-lemma-flat-base-change-cohomology", "schemes-lemma-morphism-into-affine", "descent-lemma-descending-property-isomorphism", "descent-lemma-descending-properties-morphisms" ], "ref_ids": [ 5176, 5172, 14666, 14671, 7730, 3298, 7655, 14682, 14665 ] } ], "ref_ids": [] }, { "id": 14684, "type": "theorem", "label": "descent-lemma-descending-property-closed-immersion", "categories": [ "descent" ], "title": "descent-lemma-descending-property-closed-immersion", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is a closed immersion''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\{Y_i \\to Y\\}$ be an fpqc covering.", "Assume that each $f_i : Y_i \\times_Y X \\to Y_i$", "is a closed immersion.", "This implies that each $f_i$ is affine, see", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-affine}.", "By Lemma \\ref{lemma-descending-property-affine}", "we conclude that $f$ is affine. It remains to show that", "$\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$ is surjective.", "For every $y \\in Y$ there exists an $i$ and a point", "$y_i \\in Y_i$ mapping to $y$.", "By Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}", "the sheaf $f_{i, *}(\\mathcal{O}_{Y_i \\times_Y X})$", "is the pullback of $f_*\\mathcal{O}_X$.", "By assumption it is a quotient of $\\mathcal{O}_{Y_i}$.", "Hence we see that", "$$", "\\Big(", "\\mathcal{O}_{Y, y} \\longrightarrow (f_*\\mathcal{O}_X)_y", "\\Big)", "\\otimes_{\\mathcal{O}_{Y, y}} \\mathcal{O}_{Y_i, y_i}", "$$", "is surjective. Since $\\mathcal{O}_{Y_i, y_i}$ is faithfully", "flat over $\\mathcal{O}_{Y, y}$ this implies the surjectivity", "of $\\mathcal{O}_{Y, y} \\longrightarrow (f_*\\mathcal{O}_X)_y$ as", "desired." ], "refs": [ "morphisms-lemma-closed-immersion-affine", "descent-lemma-descending-property-affine", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 5177, 14683, 3298 ] } ], "ref_ids": [] }, { "id": 14685, "type": "theorem", "label": "descent-lemma-descending-property-quasi-affine", "categories": [ "descent" ], "title": "descent-lemma-descending-property-quasi-affine", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is quasi-affine''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\{g_i : Y_i \\to Y\\}$ be an fpqc covering.", "Assume that each $f_i : Y_i \\times_Y X \\to Y_i$", "is quasi-affine.", "This implies that each $f_i$ is quasi-compact and separated.", "By Lemmas \\ref{lemma-descending-property-quasi-compact}", "and \\ref{lemma-descending-property-separated}", "this implies that $f$ is quasi-compact and separated.", "Consider the sheaf of $\\mathcal{O}_Y$-algebras", "$\\mathcal{A} = f_*\\mathcal{O}_X$.", "By Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}", "it is a quasi-coherent $\\mathcal{O}_Y$-algebra.", "Consider the canonical morphism", "$$", "j : X \\longrightarrow \\underline{\\Spec}_Y(\\mathcal{A})", "$$", "see Constructions, Lemma \\ref{constructions-lemma-canonical-morphism}.", "By flat base change", "(see for example", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology})", "we have $g_i^*f_*\\mathcal{O}_X = f_{i, *}\\mathcal{O}_{X'}$", "where $g_i : Y_i \\to Y$ are the given flat", "maps. Hence the base change $j_i$ of $j$ by $g_i$ is the canonical", "morphism of Constructions, Lemma \\ref{constructions-lemma-canonical-morphism}", "for the morphism $f_i$. By assumption and", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-quasi-affine}", "all of these", "morphisms $j_i$ are quasi-compact open immersions. Hence, by", "Lemmas \\ref{lemma-descending-property-quasi-compact} and", "\\ref{lemma-descending-property-open-immersion} we", "see that $j$ is a quasi-compact open immersion.", "Hence by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-quasi-affine}", "again we conclude that $f$ is quasi-affine." ], "refs": [ "descent-lemma-descending-property-quasi-compact", "descent-lemma-descending-property-separated", "schemes-lemma-push-forward-quasi-coherent", "constructions-lemma-canonical-morphism", "coherent-lemma-flat-base-change-cohomology", "constructions-lemma-canonical-morphism", "morphisms-lemma-characterize-quasi-affine", "descent-lemma-descending-property-quasi-compact", "descent-lemma-descending-property-open-immersion", "morphisms-lemma-characterize-quasi-affine" ], "ref_ids": [ 14666, 14671, 7730, 12591, 3298, 12591, 5185, 14666, 14681, 5185 ] } ], "ref_ids": [] }, { "id": 14686, "type": "theorem", "label": "descent-lemma-descending-property-quasi-compact-immersion", "categories": [ "descent" ], "title": "descent-lemma-descending-property-quasi-compact-immersion", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is a quasi-compact immersion''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\{Y_i \\to Y\\}$ be an fpqc covering.", "Write $X_i = Y_i \\times_Y X$ and $f_i : X_i \\to Y_i$", "the base change of $f$. Also denote", "$q_i : Y_i \\to Y$ the given flat morphisms.", "Assume each $f_i$ is a quasi-compact immersion.", "By Schemes, Lemma \\ref{schemes-lemma-immersions-monomorphisms}", "each $f_i$ is separated.", "By Lemmas \\ref{lemma-descending-property-quasi-compact} and", "\\ref{lemma-descending-property-separated}", "this implies that $f$ is quasi-compact and separated.", "Let $X \\to Z \\to Y$ be the factorization of $f$ through its", "scheme theoretic image. By", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}", "the closed subscheme $Z \\subset Y$ is cut out by the", "quasi-coherent sheaf of ideals", "$\\mathcal{I} = \\Ker(\\mathcal{O}_Y \\to f_*\\mathcal{O}_X)$", "as $f$ is quasi-compact. By flat base change", "(see for example", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology};", "here we use $f$ is separated)", "we see $f_{i, *}\\mathcal{O}_{X_i}$ is the pullback $q_i^*f_*\\mathcal{O}_X$.", "Hence $Y_i \\times_Y Z$ is cut out by the", "quasi-coherent sheaf of ideals $q_i^*\\mathcal{I} =", "\\Ker(\\mathcal{O}_{Y_i} \\to f_{i, *}\\mathcal{O}_{X_i})$.", "By Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-immersion}", "the morphisms $X_i \\to Y_i \\times_Y Z$", "are open immersions. Hence by", "Lemma \\ref{lemma-descending-property-open-immersion}", "we see that $X \\to Z$ is an open immersion and", "hence $f$ is a immersion as desired", "(we already saw it was quasi-compact)." ], "refs": [ "schemes-lemma-immersions-monomorphisms", "descent-lemma-descending-property-quasi-compact", "descent-lemma-descending-property-separated", "morphisms-lemma-quasi-compact-scheme-theoretic-image", "coherent-lemma-flat-base-change-cohomology", "morphisms-lemma-quasi-compact-immersion", "descent-lemma-descending-property-open-immersion" ], "ref_ids": [ 7727, 14666, 14671, 5146, 3298, 5154, 14681 ] } ], "ref_ids": [] }, { "id": 14687, "type": "theorem", "label": "descent-lemma-descending-property-integral", "categories": [ "descent" ], "title": "descent-lemma-descending-property-integral", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is integral''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "An integral morphism is the same thing as an affine,", "universally closed morphism. See", "Morphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed}.", "Hence the lemma follows on combining", "Lemmas \\ref{lemma-descending-property-universally-closed}", "and \\ref{lemma-descending-property-affine}." ], "refs": [ "morphisms-lemma-integral-universally-closed", "descent-lemma-descending-property-universally-closed", "descent-lemma-descending-property-affine" ], "ref_ids": [ 5441, 14668, 14683 ] } ], "ref_ids": [] }, { "id": 14688, "type": "theorem", "label": "descent-lemma-descending-property-finite", "categories": [ "descent" ], "title": "descent-lemma-descending-property-finite", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is finite''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "An finite morphism is the same thing as an integral", "morphism which is locally of finite type. See", "Morphisms, Lemma \\ref{morphisms-lemma-finite-integral}.", "Hence the lemma follows on combining", "Lemmas \\ref{lemma-descending-property-locally-finite-type}", "and \\ref{lemma-descending-property-integral}." ], "refs": [ "morphisms-lemma-finite-integral", "descent-lemma-descending-property-locally-finite-type", "descent-lemma-descending-property-integral" ], "ref_ids": [ 5438, 14675, 14687 ] } ], "ref_ids": [] }, { "id": 14689, "type": "theorem", "label": "descent-lemma-descending-property-quasi-finite", "categories": [ "descent" ], "title": "descent-lemma-descending-property-quasi-finite", "contents": [ "The properties", "$\\mathcal{P}(f) =$``$f$ is locally quasi-finite''", "and", "$\\mathcal{P}(f) =$``$f$ is quasi-finite''", "are fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be a morphism of schemes, and let $\\{S_i \\to S\\}$", "be an fpqc covering such that each base change", "$f_i : X_i \\to S_i$ is locally quasi-finite.", "We have already seen", "(Lemma \\ref{lemma-descending-property-locally-finite-type})", "that ``locally of finite type'' is fpqc local", "on the base, and hence we see that $f$ is locally of finite type.", "Then it follows from", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-quasi-finite}", "that $f$ is locally quasi-finite. The quasi-finite case follows", "as we have already seen that ``quasi-compact'' is fpqc local on the base", "(Lemma \\ref{lemma-descending-property-quasi-compact})." ], "refs": [ "descent-lemma-descending-property-locally-finite-type", "morphisms-lemma-base-change-quasi-finite", "descent-lemma-descending-property-quasi-compact" ], "ref_ids": [ 14675, 5233, 14666 ] } ], "ref_ids": [] }, { "id": 14690, "type": "theorem", "label": "descent-lemma-descending-property-relative-dimension-d", "categories": [ "descent" ], "title": "descent-lemma-descending-property-relative-dimension-d", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is locally of finite type", "of relative dimension $d$'' is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the fact that being locally of finite", "type is fpqc local on the base and", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}." ], "refs": [ "morphisms-lemma-dimension-fibre-after-base-change" ], "ref_ids": [ 5279 ] } ], "ref_ids": [] }, { "id": 14691, "type": "theorem", "label": "descent-lemma-descending-property-syntomic", "categories": [ "descent" ], "title": "descent-lemma-descending-property-syntomic", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is syntomic''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "A morphism is syntomic if and only if it is locally of finite presentation,", "flat, and has locally complete intersections as fibres. We have seen", "already that being flat and locally of finite presentation are", "fpqc local on the base (Lemmas", "\\ref{lemma-descending-property-flat}, and", "\\ref{lemma-descending-property-locally-finite-presentation}).", "Hence the result follows for syntomic from", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-lci}." ], "refs": [ "descent-lemma-descending-property-flat", "descent-lemma-descending-property-locally-finite-presentation", "morphisms-lemma-set-points-where-fibres-lci" ], "ref_ids": [ 14680, 14676, 5299 ] } ], "ref_ids": [] }, { "id": 14692, "type": "theorem", "label": "descent-lemma-descending-property-smooth", "categories": [ "descent" ], "title": "descent-lemma-descending-property-smooth", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is smooth''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "A morphism is smooth if and only if it is locally of finite presentation,", "flat, and has smooth fibres. We have seen", "already that being flat and locally of finite presentation are", "fpqc local on the base (Lemmas", "\\ref{lemma-descending-property-flat}, and", "\\ref{lemma-descending-property-locally-finite-presentation}).", "Hence the result follows for smooth from", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-smooth}." ], "refs": [ "descent-lemma-descending-property-flat", "descent-lemma-descending-property-locally-finite-presentation", "morphisms-lemma-set-points-where-fibres-smooth" ], "ref_ids": [ 14680, 14676, 5336 ] } ], "ref_ids": [] }, { "id": 14693, "type": "theorem", "label": "descent-lemma-descending-property-unramified", "categories": [ "descent" ], "title": "descent-lemma-descending-property-unramified", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is unramified''", "is fpqc local on the base.", "The property $\\mathcal{P}(f) =$``$f$ is G-unramified''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "A morphism is unramified (resp.\\ G-unramified) if and only if it is", "locally of finite type (resp.\\ finite presentation)", "and its diagonal morphism is an open immersion (see", "Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism}).", "We have seen already that being locally of finite type", "(resp.\\ locally of finite presentation) and an open immersion is", "fpqc local on the base (Lemmas", "\\ref{lemma-descending-property-locally-finite-presentation},", "\\ref{lemma-descending-property-locally-finite-type}, and", "\\ref{lemma-descending-property-open-immersion}).", "Hence the result follows formally." ], "refs": [ "morphisms-lemma-diagonal-unramified-morphism", "descent-lemma-descending-property-locally-finite-presentation", "descent-lemma-descending-property-locally-finite-type", "descent-lemma-descending-property-open-immersion" ], "ref_ids": [ 5354, 14676, 14675, 14681 ] } ], "ref_ids": [] }, { "id": 14694, "type": "theorem", "label": "descent-lemma-descending-property-etale", "categories": [ "descent" ], "title": "descent-lemma-descending-property-etale", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is \\'etale''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "A morphism is \\'etale if and only if it flat and G-unramified.", "See Morphisms, Lemma \\ref{morphisms-lemma-flat-unramified-etale}.", "We have seen already that being flat and G-unramified", "are fpqc local on the base (Lemmas", "\\ref{lemma-descending-property-flat}, and", "\\ref{lemma-descending-property-unramified}).", "Hence the result follows." ], "refs": [ "morphisms-lemma-flat-unramified-etale", "descent-lemma-descending-property-flat", "descent-lemma-descending-property-unramified" ], "ref_ids": [ 5373, 14680, 14693 ] } ], "ref_ids": [] }, { "id": 14695, "type": "theorem", "label": "descent-lemma-descending-property-finite-locally-free", "categories": [ "descent" ], "title": "descent-lemma-descending-property-finite-locally-free", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is finite locally free''", "is fpqc local on the base.", "Let $d \\geq 0$.", "The property $\\mathcal{P}(f) =$``$f$ is finite locally free of degree $d$''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Being finite locally free is equivalent to being", "finite, flat and locally of finite presentation", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}).", "Hence this follows from Lemmas", "\\ref{lemma-descending-property-finite},", "\\ref{lemma-descending-property-flat}, and", "\\ref{lemma-descending-property-locally-finite-presentation}.", "If $f : Z \\to U$ is finite locally free, and $\\{U_i \\to U\\}$ is a surjective", "family of morphisms such that each pullback $Z \\times_U U_i \\to U_i$ has", "degree $d$, then $Z \\to U$ has degree $d$, for example because we", "can read off the degree in a point $u \\in U$ from the fibre", "$(f_*\\mathcal{O}_Z)_u \\otimes_{\\mathcal{O}_{U, u}} \\kappa(u)$." ], "refs": [ "morphisms-lemma-finite-flat", "descent-lemma-descending-property-finite", "descent-lemma-descending-property-flat", "descent-lemma-descending-property-locally-finite-presentation" ], "ref_ids": [ 5471, 14688, 14680, 14676 ] } ], "ref_ids": [] }, { "id": 14696, "type": "theorem", "label": "descent-lemma-descending-property-monomorphism", "categories": [ "descent" ], "title": "descent-lemma-descending-property-monomorphism", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is a monomorphism''", "is fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\{S_i \\to S\\}$ be an fpqc covering, and assume", "each of the base changes $f_i : X_i \\to S_i$ of $f$ is", "a monomorphism. Let $a, b : T \\to X$ be two morphisms", "such that $f \\circ a = f \\circ b$. We have to show that $a = b$.", "Since $f_i$ is a monomorphism we see that $a_i = b_i$, where", "$a_i, b_i : S_i \\times_S T \\to X_i$ are", "the base changes. In particular the compositions", "$S_i \\times_S T \\to T \\to X$ are equal.", "Since $\\coprod S_i \\times_S T \\to T$", "is an epimorphism (see", "e.g.\\ Lemma \\ref{lemma-fpqc-universal-effective-epimorphisms})", "we conclude $a = b$." ], "refs": [ "descent-lemma-fpqc-universal-effective-epimorphisms" ], "ref_ids": [ 14638 ] } ], "ref_ids": [] }, { "id": 14697, "type": "theorem", "label": "descent-lemma-descending-property-regular-immersion", "categories": [ "descent" ], "title": "descent-lemma-descending-property-regular-immersion", "contents": [ "The properties", "\\begin{enumerate}", "\\item[] $\\mathcal{P}(f) =$``$f$ is a Koszul-regular immersion'',", "\\item[] $\\mathcal{P}(f) =$``$f$ is an $H_1$-regular immersion'', and", "\\item[] $\\mathcal{P}(f) =$``$f$ is a quasi-regular immersion''", "\\end{enumerate}", "are fpqc local on the base." ], "refs": [], "proofs": [ { "contents": [ "We will use the criterion of", "Lemma \\ref{lemma-descending-properties-morphisms}", "to prove this. By", "Divisors, Definition \\ref{divisors-definition-regular-immersion}", "being a Koszul-regular (resp.\\ $H_1$-regular, quasi-regular)", "immersion is Zariski local on the base. By", "Divisors, Lemma \\ref{divisors-lemma-flat-base-change-regular-immersion}", "being a Koszul-regular (resp.\\ $H_1$-regular, quasi-regular)", "immersion is preserved under flat base change.", "The final hypothesis (3) of", "Lemma \\ref{lemma-descending-properties-morphisms}", "translates into the following algebra statement:", "Let $A \\to B$ be a faithfully flat ring map. Let $I \\subset A$ be an ideal.", "If $IB$ is locally on $\\Spec(B)$ generated by a Koszul-regular", "(resp.\\ $H_1$-regular, quasi-regular) sequence in $B$, then $I \\subset A$", "is locally on $\\Spec(A)$ generated by a Koszul-regular", "(resp.\\ $H_1$-regular, quasi-regular) sequence in $A$. This is", "More on Algebra, Lemma \\ref{more-algebra-lemma-flat-descent-regular-ideal}." ], "refs": [ "descent-lemma-descending-properties-morphisms", "divisors-definition-regular-immersion", "divisors-lemma-flat-base-change-regular-immersion", "descent-lemma-descending-properties-morphisms", "more-algebra-lemma-flat-descent-regular-ideal" ], "ref_ids": [ 14665, 8099, 7991, 14665, 9997 ] } ], "ref_ids": [] }, { "id": 14698, "type": "theorem", "label": "descent-lemma-descending-fppf-property-immersion", "categories": [ "descent" ], "title": "descent-lemma-descending-fppf-property-immersion", "contents": [ "The property $\\mathcal{P}(f) =$``$f$ is an immersion''", "is fppf local on the base." ], "refs": [], "proofs": [ { "contents": [ "The property of being an immersion is stable under base change,", "see Schemes, Lemma \\ref{schemes-lemma-base-change-immersion}.", "The property of being an immersion is Zariski local on the base.", "Finally, let", "$\\pi : S' \\to S$ be a surjective morphism of affine schemes,", "which is flat and locally of finite presentation.", "Note that $\\pi : S' \\to S$ is open by", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}.", "Let $f : X \\to S$ be a morphism.", "Assume that the base change $f' : X' \\to S'$ is an immersion.", "In particular we see that $f'(X') = \\pi^{-1}(f(X))$ is locally closed.", "Hence by Topology, Lemma \\ref{topology-lemma-open-morphism-quotient-topology}", "we see that $f(X) \\subset S$", "is locally closed. Let $Z \\subset S$ be", "the closed subset $Z = \\overline{f(X)} \\setminus f(X)$.", "By Topology, Lemma \\ref{topology-lemma-open-morphism-quotient-topology}", "again we see that $f'(X')$ is closed in $S' \\setminus Z'$.", "Hence we may apply Lemma \\ref{lemma-descending-property-closed-immersion}", "to the fpqc covering $\\{S' \\setminus Z' \\to S \\setminus Z\\}$", "and conclude that $f : X \\to S \\setminus Z$ is a closed", "immersion. In other words, $f$ is an immersion.", "Therefore Lemma \\ref{lemma-descending-properties-morphisms} applies and we win." ], "refs": [ "schemes-lemma-base-change-immersion", "morphisms-lemma-fppf-open", "topology-lemma-open-morphism-quotient-topology", "topology-lemma-open-morphism-quotient-topology", "descent-lemma-descending-property-closed-immersion", "descent-lemma-descending-properties-morphisms" ], "ref_ids": [ 7695, 5267, 8203, 8203, 14684, 14665 ] } ], "ref_ids": [] }, { "id": 14699, "type": "theorem", "label": "descent-lemma-flat-surjective-quasi-compact-monomorphism-isomorphism", "categories": [ "descent" ], "title": "descent-lemma-flat-surjective-quasi-compact-monomorphism-isomorphism", "contents": [ "Let $f : X \\to Y$ be a flat, quasi-compact, surjective monomorphism.", "Then f is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "As $f$ is a flat, quasi-compact, surjective morphism", "we see $\\{X \\to Y\\}$ is an fpqc covering of $Y$.", "The diagonal $\\Delta : X \\to X \\times_Y X$ is an isomorphism.", "This implies that the base change of $f$ by $f$ is an", "isomorphism. Hence we see $f$ is an isomorphism by", "Lemma \\ref{lemma-descending-property-isomorphism}." ], "refs": [ "descent-lemma-descending-property-isomorphism" ], "ref_ids": [ 14682 ] } ], "ref_ids": [] }, { "id": 14700, "type": "theorem", "label": "descent-lemma-universally-injective-etale-open-immersion", "categories": [ "descent" ], "title": "descent-lemma-universally-injective-etale-open-immersion", "contents": [ "A universally injective \\'etale morphism is an open immersion." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "Let $f : X \\to Y$ be an \\'etale morphism which is universally injective.", "Then $f$ is open", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-open})", "hence we can replace $Y$ by $f(X)$ and we may assume that $f$ is surjective.", "Then $f$ is bijective and open hence a homeomorphism. Hence $f$ is", "quasi-compact. Thus by", "Lemma \\ref{lemma-flat-surjective-quasi-compact-monomorphism-isomorphism}", "it suffices to show that $f$ is a monomorphism. As $X \\to Y$ is \\'etale", "the morphism $\\Delta_{X/Y} : X \\to X \\times_Y X$ is an open immersion by", "Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism}", "(and", "Morphisms, Lemma \\ref{morphisms-lemma-flat-unramified-etale}).", "As $f$ is universally injective $\\Delta_{X/Y}$ is also surjective, see", "Morphisms, Lemma \\ref{morphisms-lemma-universally-injective}.", "Hence $\\Delta_{X/Y}$ is an isomorphism, i.e., $X \\to Y$ is a monomorphism." ], "refs": [ "morphisms-lemma-etale-open", "descent-lemma-flat-surjective-quasi-compact-monomorphism-isomorphism", "morphisms-lemma-diagonal-unramified-morphism", "morphisms-lemma-flat-unramified-etale", "morphisms-lemma-universally-injective" ], "ref_ids": [ 5370, 14699, 5354, 5373, 5167 ] } ], "ref_ids": [] }, { "id": 14701, "type": "theorem", "label": "descent-lemma-flat-universally-injective", "categories": [ "descent" ], "title": "descent-lemma-flat-universally-injective", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $X^0$ denote the set", "of generic points of irreducible components of $X$. If", "\\begin{enumerate}", "\\item $f$ is flat and separated,", "\\item for $\\xi \\in X^0$ we have $\\kappa(f(\\xi)) = \\kappa(\\xi)$, and", "\\item if $\\xi, \\xi' \\in X^0$, $\\xi \\not = \\xi'$, then $f(\\xi) \\not = f(\\xi')$,", "\\end{enumerate}", "then $f$ is universally injective." ], "refs": [], "proofs": [ { "contents": [ "We have to show that $\\Delta : X \\to X \\times_Y X$ is surjective, see", "Morphisms, Lemma \\ref{morphisms-lemma-universally-injective}.", "As $X \\to Y$ is separated, the image of $\\Delta$ is closed.", "Thus if $\\Delta$ is not surjective, we can find a generic point", "$\\eta \\in X \\times_S X$ of an irreducible component of $X \\times_S X$", "which is not in the image of $\\Delta$. The projection", "$\\text{pr}_1 : X \\times_Y X \\to X$", "is flat as a base change of the flat morphism $X \\to Y$, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat}.", "Hence generalizations lift along $\\text{pr}_1$, see", "Morphisms, Lemma \\ref{morphisms-lemma-generalizations-lift-flat}.", "We conclude that $\\xi = \\text{pr}_1(\\eta) \\in X^0$.", "However, assumptions (2) and (3) guarantee that the scheme", "$(X \\times_Y X)_{f(\\xi)}$ has at most one point for every $\\xi \\in X^0$.", "In other words, we have $\\Delta(\\xi) = \\eta$ a contradiction." ], "refs": [ "morphisms-lemma-universally-injective", "morphisms-lemma-base-change-flat", "morphisms-lemma-generalizations-lift-flat" ], "ref_ids": [ 5167, 5265, 5266 ] } ], "ref_ids": [] }, { "id": 14702, "type": "theorem", "label": "descent-lemma-characterize-open-immersion", "categories": [ "descent" ], "title": "descent-lemma-characterize-open-immersion", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $X^0$ denote the set", "of generic points of irreducible components of $X$. If", "\\begin{enumerate}", "\\item $f$ is \\'etale and separated,", "\\item for $\\xi \\in X^0$ we have $\\kappa(f(\\xi)) = \\kappa(\\xi)$, and", "\\item if $\\xi, \\xi' \\in X^0$, $\\xi \\not = \\xi'$, then $f(\\xi) \\not = f(\\xi')$,", "\\end{enumerate}", "then $f$ is an open immersion." ], "refs": [], "proofs": [ { "contents": [ "Immediate from Lemmas \\ref{lemma-flat-universally-injective} and", "\\ref{lemma-universally-injective-etale-open-immersion}." ], "refs": [ "descent-lemma-flat-universally-injective", "descent-lemma-universally-injective-etale-open-immersion" ], "ref_ids": [ 14701, 14700 ] } ], "ref_ids": [] }, { "id": 14703, "type": "theorem", "label": "descent-lemma-descending-property-proper-over-base", "categories": [ "descent" ], "title": "descent-lemma-descending-property-proper-over-base", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes which is locally of finite type.", "Let $Z$ be a closed subset of $X$. If there exists an fpqc covering", "$\\{Y_i \\to Y\\}$ such that the inverse image $Z_i \\subset Y_i \\times_Y X$", "is proper over $Y_i$", "(Cohomology of Schemes, Definition \\ref{coherent-definition-proper-over-base})", "then $Z$ is proper over $Y$." ], "refs": [ "coherent-definition-proper-over-base" ], "proofs": [ { "contents": [ "Endow $Z$ with the reduced induced closed subscheme structure, see", "Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme}.", "For every $i$ the base change $Y_i \\times_Y Z$ is a closed subscheme", "of $Y_i \\times_Y X$ whose underlying closed subset is $Z_i$.", "By definition (via", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-closed-proper-over-base})", "we conclude that the projections $Y_i \\times_Y Z \\to Y_i$ are proper", "morphisms. Hence $Z \\to Y$ is a proper morphism by", "Lemma \\ref{lemma-descending-property-proper}.", "Thus $Z$ is proper over $Y$ by definition." ], "refs": [ "schemes-definition-reduced-induced-scheme", "coherent-lemma-closed-proper-over-base", "descent-lemma-descending-property-proper" ], "ref_ids": [ 7745, 3386, 14679 ] } ], "ref_ids": [ 3405 ] }, { "id": 14704, "type": "theorem", "label": "descent-lemma-descending-property-ample", "categories": [ "descent" ], "title": "descent-lemma-descending-property-ample", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $\\{g_i : S_i \\to S\\}_{i \\in I}$ be an fpqc covering.", "Let $f_i : X_i \\to S_i$ be the base change of $f$ and let $\\mathcal{L}_i$", "be the pullback of $\\mathcal{L}$ to $X_i$.", "The following are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{L}$ is ample on $X/S$, and", "\\item $\\mathcal{L}_i$ is ample on $X_i/S_i$", "for every $i \\in I$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) follows from", "Morphisms, Lemma \\ref{morphisms-lemma-ample-base-change}.", "Assume $\\mathcal{L}_i$ is ample on $X_i/S_i$ for every $i \\in I$.", "By Morphisms, Definition \\ref{morphisms-definition-relatively-ample}", "this implies that $X_i \\to S_i$ is quasi-compact and by", "Morphisms, Lemma \\ref{morphisms-lemma-relatively-ample-separated}", "this implies $X_i \\to S$ is separated.", "Hence $f$ is quasi-compact and separated by", "Lemmas \\ref{lemma-descending-property-quasi-compact} and", "\\ref{lemma-descending-property-separated}.", "\\medskip\\noindent", "This means that", "$\\mathcal{A} = \\bigoplus_{d \\geq 0} f_*\\mathcal{L}^{\\otimes d}$", "is a quasi-coherent graded $\\mathcal{O}_S$-algebra", "(Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}).", "Moreover, the formation of $\\mathcal{A}$ commutes with flat", "base change by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}.", "In particular, if we set", "$\\mathcal{A}_i = \\bigoplus_{d \\geq 0} f_{i, *}\\mathcal{L}_i^{\\otimes d}$", "then we have $\\mathcal{A}_i = g_i^*\\mathcal{A}$.", "It follows that the natural maps", "$\\psi_d : f^*\\mathcal{A}_d \\to \\mathcal{L}^{\\otimes d}$", "of $\\mathcal{O}_X$", "pullback to give the natural maps", "$\\psi_{i, d} : f_i^*(\\mathcal{A}_i)_d \\to \\mathcal{L}_i^{\\otimes d}$", "of $\\mathcal{O}_{X_i}$-modules. Since $\\mathcal{L}_i$ is ample on $X_i/S_i$", "we see that for any point $x_i \\in X_i$, there exists a $d \\geq 1$", "such that $f_i^*(\\mathcal{A}_i)_d \\to \\mathcal{L}_i^{\\otimes d}$", "is surjective on stalks at $x_i$. This follows either directly", "from the definition of a relatively ample module or from", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-relatively-ample}.", "If $x \\in X$, then we can choose an $i$ and an $x_i \\in X_i$", "mapping to $x$. Since $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X_i, x_i}$", "is flat hence faithfully flat, we conclude that for every $x \\in X$", "there exists a $d \\geq 1$ such that", "$f^*\\mathcal{A}_d \\to \\mathcal{L}^{\\otimes d}$", "is surjective on stalks at $x$.", "This implies that the open subset $U(\\psi) \\subset X$ of", "Constructions, Lemma", "\\ref{constructions-lemma-invertible-map-into-relative-proj}", "corresponding to the map", "$\\psi : f^*\\mathcal{A} \\to \\bigoplus_{d \\geq 0} \\mathcal{L}^{\\otimes d}$", "of graded $\\mathcal{O}_X$-algebras", "is equal to $X$. Consider the corresponding morphism", "$$", "r_{\\mathcal{L}, \\psi} : X \\longrightarrow \\underline{\\text{Proj}}_S(\\mathcal{A})", "$$", "It is clear from the above that the base change of", "$r_{\\mathcal{L}, \\psi}$ to $S_i$ is the morphism", "$r_{\\mathcal{L}_i, \\psi_i}$ which is an open immersion by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-relatively-ample}.", "Hence $r_{\\mathcal{L}, \\psi}$ is an open immersion", "by Lemma \\ref{lemma-descending-property-open-immersion}", "and we conclude $\\mathcal{L}$ is ample on $X/S$ by", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-relatively-ample}." ], "refs": [ "morphisms-lemma-ample-base-change", "morphisms-definition-relatively-ample", "morphisms-lemma-relatively-ample-separated", "descent-lemma-descending-property-quasi-compact", "descent-lemma-descending-property-separated", "schemes-lemma-push-forward-quasi-coherent", "coherent-lemma-flat-base-change-cohomology", "morphisms-lemma-characterize-relatively-ample", "constructions-lemma-invertible-map-into-relative-proj", "morphisms-lemma-characterize-relatively-ample", "descent-lemma-descending-property-open-immersion", "morphisms-lemma-characterize-relatively-ample" ], "ref_ids": [ 5385, 5568, 5379, 14666, 14671, 7730, 3298, 5380, 12649, 5380, 14681, 5380 ] } ], "ref_ids": [] }, { "id": 14705, "type": "theorem", "label": "descent-lemma-precompose-property-local-source", "categories": [ "descent" ], "title": "descent-lemma-precompose-property-local-source", "contents": [ "Let $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale, Zariski\\}$.", "Let $\\mathcal{P}$ be a property of morphisms which is $\\tau$ local", "on the source. Let $f : X \\to Y$ have property $\\mathcal{P}$.", "For any morphism $a : X' \\to X$ which is", "flat, resp.\\ flat and locally of finite presentation, resp.\\ syntomic,", "resp.\\ \\'etale, resp.\\ an open immersion, the composition", "$f \\circ a : X' \\to Y$ has property $\\mathcal{P}$." ], "refs": [], "proofs": [ { "contents": [ "This is true because we can fit $X' \\to X$ into a family of", "morphisms which forms a $\\tau$-covering." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14706, "type": "theorem", "label": "descent-lemma-largest-open-of-the-source", "categories": [ "descent" ], "title": "descent-lemma-largest-open-of-the-source", "contents": [ "Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale\\}$.", "Let $\\mathcal{P}$ be a property of morphisms which is $\\tau$ local", "on the source. For any morphism of schemes $f : X \\to Y$ there exists", "a largest open $W(f) \\subset X$ such that the restriction", "$f|_{W(f)} : W(f) \\to Y$ has $\\mathcal{P}$. Moreover,", "if $g : X' \\to X$ is flat and locally of finite presentation,", "syntomic, smooth, or \\'etale and $f' = f \\circ g : X' \\to Y$, then", "$g^{-1}(W(f)) = W(f')$." ], "refs": [], "proofs": [ { "contents": [ "Consider the union $W$ of the images $g(X') \\subset X$ of", "morphisms $g : X' \\to X$ with the properties:", "\\begin{enumerate}", "\\item $g$ is flat and locally of finite presentation, syntomic,", "smooth, or \\'etale, and", "\\item the composition $X' \\to X \\to Y$ has property $\\mathcal{P}$.", "\\end{enumerate}", "Since such a morphism $g$ is open (see", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open})", "we see that $W \\subset X$ is an open subset of $X$. Since $\\mathcal{P}$", "is local in the $\\tau$ topology the restriction $f|_W : W \\to Y$ has property", "$\\mathcal{P}$ because we are given a $\\tau$ covering $\\{X' \\to W\\}$ of $W$", "such that the pullbacks have $\\mathcal{P}$. This proves the existence of $W(f)$.", "The compatibility stated in the last sentence follows immediately", "from the construction of $W(f)$." ], "refs": [ "morphisms-lemma-fppf-open" ], "ref_ids": [ 5267 ] } ], "ref_ids": [] }, { "id": 14707, "type": "theorem", "label": "descent-lemma-properties-morphisms-local-source", "categories": [ "descent" ], "title": "descent-lemma-properties-morphisms-local-source", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes.", "Let $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0]", "\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$.", "Assume that", "\\begin{enumerate}", "\\item the property is preserved under precomposing with", "flat, flat locally of finite presentation, \\'etale, smooth or syntomic morphisms", "depending on whether $\\tau$ is fpqc, fppf, \\'etale, smooth, or syntomic,", "\\item the property is Zariski local on the source,", "\\item the property is Zariski local on the target,", "\\item for any morphism of affine schemes $X \\to Y$, and", "any surjective morphism of affine schemes $X' \\to X$", "which is flat, flat of finite presentation,", "\\'etale, smooth or syntomic depending on whether $\\tau$ is", "fpqc, fppf, \\'etale, smooth, or syntomic, property", "$\\mathcal{P}$ holds for $f$ if property $\\mathcal{P}$", "holds for the composition $f' : X' \\to Y$.", "\\end{enumerate}", "Then $\\mathcal{P}$ is $\\tau$ local on the source." ], "refs": [], "proofs": [ { "contents": [ "This follows almost immediately from the definition of", "a $\\tau$-covering, see", "Topologies, Definition", "\\ref{topologies-definition-fpqc-covering}", "\\ref{topologies-definition-fppf-covering}", "\\ref{topologies-definition-etale-covering}", "\\ref{topologies-definition-smooth-covering}, or", "\\ref{topologies-definition-syntomic-covering}", "and Topologies, Lemma", "\\ref{topologies-lemma-fpqc-affine},", "\\ref{topologies-lemma-fppf-affine},", "\\ref{topologies-lemma-etale-affine},", "\\ref{topologies-lemma-smooth-affine}, or", "\\ref{topologies-lemma-syntomic-affine}.", "Details omitted. (Hint: Use locality on the source and target to", "reduce the verification of property $\\mathcal{P}$ to the case of", "a morphism between affines. Then apply (1) and (4).)" ], "refs": [ "topologies-definition-fpqc-covering", "topologies-definition-fppf-covering", "topologies-definition-etale-covering", "topologies-definition-smooth-covering", "topologies-definition-syntomic-covering", "topologies-lemma-fpqc-affine", "topologies-lemma-fppf-affine", "topologies-lemma-etale-affine", "topologies-lemma-smooth-affine", "topologies-lemma-syntomic-affine" ], "ref_ids": [ 12547, 12539, 12526, 12531, 12535, 12499, 12473, 12447, 12461, 12467 ] } ], "ref_ids": [] }, { "id": 14708, "type": "theorem", "label": "descent-lemma-flat-fpqc-local-source", "categories": [ "descent" ], "title": "descent-lemma-flat-fpqc-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is flat'' is fpqc local on the source." ], "refs": [], "proofs": [ { "contents": [ "Since flatness is defined in terms of the maps of local rings", "(Morphisms, Definition \\ref{morphisms-definition-flat})", "what has to be shown is the following", "algebraic fact: Suppose $A \\to B \\to C$ are local homomorphisms of local", "rings, and assume $B \\to C$ is flat. Then $A \\to B$ is", "flat if and only if $A \\to C$ is flat.", "If $A \\to B$ is flat, then $A \\to C$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-composition-flat}.", "Conversely, assume $A \\to C$ is flat.", "Note that $B \\to C$ is faithfully", "flat, see", "Algebra, Lemma \\ref{algebra-lemma-local-flat-ff}.", "Hence $A \\to B$ is flat by", "Algebra, Lemma \\ref{algebra-lemma-flat-permanence}.", "(Also see Morphisms, Lemma \\ref{morphisms-lemma-flat-permanence}", "for a direct proof.)" ], "refs": [ "morphisms-definition-flat", "algebra-lemma-composition-flat", "algebra-lemma-local-flat-ff", "algebra-lemma-flat-permanence", "morphisms-lemma-flat-permanence" ], "ref_ids": [ 5557, 524, 537, 530, 5270 ] } ], "ref_ids": [] }, { "id": 14709, "type": "theorem", "label": "descent-lemma-injective-local-rings-fpqc-local-source", "categories": [ "descent" ], "title": "descent-lemma-injective-local-rings-fpqc-local-source", "contents": [ "Then property", "$\\mathcal{P}(f : X \\to Y)=$``for every $x \\in X$ the map of local", "rings $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is injective''", "is fpqc local on the source." ], "refs": [], "proofs": [ { "contents": [ "Omitted. This is just a (probably misguided) attempt to be playful." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14710, "type": "theorem", "label": "descent-lemma-locally-finite-presentation-fppf-local-source", "categories": [ "descent" ], "title": "descent-lemma-locally-finite-presentation-fppf-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is locally of finite presentation''", "is fppf local on the source." ], "refs": [], "proofs": [ { "contents": [ "Being locally of finite presentation is Zariski local on the source", "and the target, see Morphisms,", "Lemma \\ref{morphisms-lemma-locally-finite-presentation-characterize}.", "It is a property which is preserved under composition, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-presentation}.", "This proves", "(1), (2) and (3) of Lemma \\ref{lemma-properties-morphisms-local-source}.", "The final condition (4) is", "Lemma \\ref{lemma-flat-finitely-presented-permanence-algebra}. Hence we win." ], "refs": [ "morphisms-lemma-locally-finite-presentation-characterize", "morphisms-lemma-composition-finite-presentation", "descent-lemma-properties-morphisms-local-source", "descent-lemma-flat-finitely-presented-permanence-algebra" ], "ref_ids": [ 5238, 5239, 14707, 14640 ] } ], "ref_ids": [] }, { "id": 14711, "type": "theorem", "label": "descent-lemma-locally-finite-type-fppf-local-source", "categories": [ "descent" ], "title": "descent-lemma-locally-finite-type-fppf-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is locally of finite type''", "is fppf local on the source." ], "refs": [], "proofs": [ { "contents": [ "Being locally of finite type is Zariski local on the source", "and the target, see Morphisms,", "Lemma \\ref{morphisms-lemma-locally-finite-type-characterize}.", "It is a property which is preserved under composition, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-type}, and", "a flat morphism locally of finite presentation is locally of finite type, see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-presentation-finite-type}.", "This proves", "(1), (2) and (3) of Lemma \\ref{lemma-properties-morphisms-local-source}.", "The final condition (4) is", "Lemma \\ref{lemma-finite-type-local-source-fppf-algebra}. Hence we win." ], "refs": [ "morphisms-lemma-locally-finite-type-characterize", "morphisms-lemma-composition-finite-type", "morphisms-lemma-finite-presentation-finite-type", "descent-lemma-properties-morphisms-local-source", "descent-lemma-finite-type-local-source-fppf-algebra" ], "ref_ids": [ 5198, 5199, 5244, 14707, 14641 ] } ], "ref_ids": [] }, { "id": 14712, "type": "theorem", "label": "descent-lemma-open-fppf-local-source", "categories": [ "descent" ], "title": "descent-lemma-open-fppf-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is open''", "is fppf local on the source." ], "refs": [], "proofs": [ { "contents": [ "Being an open morphism is clearly Zariski local on the source and the target.", "It is a property which is preserved under composition, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-open}, and", "a flat morphism of finite presentation is open, see", "Morphisms, Lemma \\ref{morphisms-lemma-fppf-open}", "This proves", "(1), (2) and (3) of Lemma \\ref{lemma-properties-morphisms-local-source}.", "The final condition (4) follows from", "Morphisms, Lemma \\ref{morphisms-lemma-fpqc-quotient-topology}.", "Hence we win." ], "refs": [ "morphisms-lemma-composition-open", "morphisms-lemma-fppf-open", "descent-lemma-properties-morphisms-local-source", "morphisms-lemma-fpqc-quotient-topology" ], "ref_ids": [ 5253, 5267, 14707, 5269 ] } ], "ref_ids": [] }, { "id": 14713, "type": "theorem", "label": "descent-lemma-universally-open-fppf-local-source", "categories": [ "descent" ], "title": "descent-lemma-universally-open-fppf-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is universally open''", "is fppf local on the source." ], "refs": [], "proofs": [ { "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\{X_i \\to X\\}_{i \\in I}$ be an fppf covering.", "Denote $f_i : X_i \\to X$ the compositions.", "We have to show that $f$ is universally open if and only if", "each $f_i$ is universally open. If $f$ is universally open,", "then also each $f_i$ is universally open since the maps", "$X_i \\to X$ are universally open and compositions", "of universally open morphisms are universally open", "(Morphisms, Lemmas \\ref{morphisms-lemma-fppf-open}", "and \\ref{morphisms-lemma-composition-open}).", "Conversely, assume each $f_i$ is universally open.", "Let $Y' \\to Y$ be a morphism of schemes.", "Denote $X' = Y' \\times_Y X$ and $X'_i = Y' \\times_Y X_i$.", "Note that $\\{X_i' \\to X'\\}_{i \\in I}$ is an fppf covering also.", "The morphisms $f'_i : X_i' \\to Y'$ are open by assumption.", "Hence by the Lemma \\ref{lemma-open-fppf-local-source}", "above we conclude that $f' : X' \\to Y'$ is open as desired." ], "refs": [ "morphisms-lemma-fppf-open", "morphisms-lemma-composition-open", "descent-lemma-open-fppf-local-source" ], "ref_ids": [ 5267, 5253, 14712 ] } ], "ref_ids": [] }, { "id": 14714, "type": "theorem", "label": "descent-lemma-syntomic-syntomic-local-source", "categories": [ "descent" ], "title": "descent-lemma-syntomic-syntomic-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is syntomic''", "is syntomic local on the source." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-properties-morphisms-local-source} with", "Morphisms, Lemma \\ref{morphisms-lemma-syntomic-characterize}", "(local for Zariski on source and target),", "Morphisms, Lemma \\ref{morphisms-lemma-composition-syntomic} (pre-composing),", "and Lemma \\ref{lemma-syntomic-smooth-etale-permanence} (part (4))." ], "refs": [ "descent-lemma-properties-morphisms-local-source", "morphisms-lemma-syntomic-characterize", "morphisms-lemma-composition-syntomic", "descent-lemma-syntomic-smooth-etale-permanence" ], "ref_ids": [ 14707, 5289, 5290, 14643 ] } ], "ref_ids": [] }, { "id": 14715, "type": "theorem", "label": "descent-lemma-smooth-smooth-local-source", "categories": [ "descent" ], "title": "descent-lemma-smooth-smooth-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is smooth''", "is smooth local on the source." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-properties-morphisms-local-source} with", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-characterize}", "(local for Zariski on source and target),", "Morphisms, Lemma \\ref{morphisms-lemma-composition-smooth} (pre-composing), and", "Lemma \\ref{lemma-syntomic-smooth-etale-permanence} (part (4))." ], "refs": [ "descent-lemma-properties-morphisms-local-source", "morphisms-lemma-smooth-characterize", "morphisms-lemma-composition-smooth", "descent-lemma-syntomic-smooth-etale-permanence" ], "ref_ids": [ 14707, 5324, 5326, 14643 ] } ], "ref_ids": [] }, { "id": 14716, "type": "theorem", "label": "descent-lemma-etale-etale-local-source", "categories": [ "descent" ], "title": "descent-lemma-etale-etale-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is \\'etale''", "is \\'etale local on the source." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemma \\ref{lemma-properties-morphisms-local-source} with", "Morphisms, Lemma \\ref{morphisms-lemma-etale-characterize}", "(local for Zariski on source and target),", "Morphisms, Lemma \\ref{morphisms-lemma-composition-etale} (pre-composing), and", "Lemma \\ref{lemma-syntomic-smooth-etale-permanence} (part (4))." ], "refs": [ "descent-lemma-properties-morphisms-local-source", "morphisms-lemma-etale-characterize", "morphisms-lemma-composition-etale", "descent-lemma-syntomic-smooth-etale-permanence" ], "ref_ids": [ 14707, 5359, 5360, 14643 ] } ], "ref_ids": [] }, { "id": 14717, "type": "theorem", "label": "descent-lemma-locally-quasi-finite-etale-local-source", "categories": [ "descent" ], "title": "descent-lemma-locally-quasi-finite-etale-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is locally quasi-finite''", "is \\'etale local on the source." ], "refs": [], "proofs": [ { "contents": [ "We are going to use", "Lemma \\ref{lemma-properties-morphisms-local-source}.", "By", "Morphisms, Lemma", "\\ref{morphisms-lemma-locally-quasi-finite-characterize}", "the property of being locally quasi-finite is local for Zariski on source", "and target. By", "Morphisms, Lemmas", "\\ref{morphisms-lemma-composition-quasi-finite} and", "\\ref{morphisms-lemma-etale-locally-quasi-finite}", "we see the precomposition", "of a locally quasi-finite morphism by an \\'etale morphism is locally", "quasi-finite. Finally, suppose that $X \\to Y$ is a morphism of affine schemes", "and that $X' \\to X$ is a surjective \\'etale morphism of affine schemes", "such that $X' \\to Y$ is locally quasi-finite. Then $X' \\to Y$ is of finite", "type, and by", "Lemma \\ref{lemma-finite-type-local-source-fppf-algebra}", "we see that $X \\to Y$ is of finite type also.", "Moreover, by assumption $X' \\to Y$ has finite fibres, and hence $X \\to Y$", "has finite fibres also. We conclude that $X \\to Y$ is quasi-finite by", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite}.", "This proves the last assumption of", "Lemma \\ref{lemma-properties-morphisms-local-source}", "and finishes the proof." ], "refs": [ "descent-lemma-properties-morphisms-local-source", "morphisms-lemma-locally-quasi-finite-characterize", "morphisms-lemma-composition-quasi-finite", "morphisms-lemma-etale-locally-quasi-finite", "descent-lemma-finite-type-local-source-fppf-algebra", "morphisms-lemma-quasi-finite", "descent-lemma-properties-morphisms-local-source" ], "ref_ids": [ 14707, 5231, 5232, 5363, 14641, 5230, 14707 ] } ], "ref_ids": [] }, { "id": 14718, "type": "theorem", "label": "descent-lemma-unramified-etale-local-source", "categories": [ "descent" ], "title": "descent-lemma-unramified-etale-local-source", "contents": [ "The property $\\mathcal{P}(f)=$``$f$ is unramified''", "is \\'etale local on the source.", "The property $\\mathcal{P}(f)=$``$f$ is G-unramified''", "is \\'etale local on the source." ], "refs": [], "proofs": [ { "contents": [ "We are going to use", "Lemma \\ref{lemma-properties-morphisms-local-source}.", "By", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-characterize}", "the property of being unramified (resp.\\ G-unramified)", "is local for Zariski on source and target. By", "Morphisms, Lemmas \\ref{morphisms-lemma-composition-unramified} and", "\\ref{morphisms-lemma-etale-smooth-unramified}", "we see the precomposition", "of an unramified (resp.\\ G-unramified) morphism by an \\'etale morphism is", "unramified (resp.\\ G-unramified).", "Finally, suppose that $X \\to Y$ is a morphism of affine schemes", "and that $f : X' \\to X$ is a surjective \\'etale morphism of affine schemes", "such that $X' \\to Y$ is unramified (resp.\\ G-unramified).", "Then $X' \\to Y$ is of finite type (resp.\\ finite presentation), and by", "Lemma \\ref{lemma-finite-type-local-source-fppf-algebra}", "(resp.\\ Lemma \\ref{lemma-flat-finitely-presented-permanence-algebra})", "we see that $X \\to Y$ is of finite type (resp.\\ finite presentation) also. By", "Morphisms, Lemma \\ref{morphisms-lemma-triangle-differentials-smooth}", "we have a short exact sequence", "$$", "0 \\to f^*\\Omega_{X/Y} \\to \\Omega_{X'/Y} \\to \\Omega_{X'/X} \\to 0.", "$$", "As $X' \\to Y$ is unramified we see that the middle term is zero.", "Hence, as $f$ is faithfully flat we see that $\\Omega_{X/Y} = 0$.", "Hence $X \\to Y$ is unramified (resp.\\ G-unramified), see", "Morphisms, Lemma \\ref{morphisms-lemma-unramified-omega-zero}.", "This proves the last assumption of", "Lemma \\ref{lemma-properties-morphisms-local-source}", "and finishes the proof." ], "refs": [ "descent-lemma-properties-morphisms-local-source", "morphisms-lemma-unramified-characterize", "morphisms-lemma-composition-unramified", "morphisms-lemma-etale-smooth-unramified", "descent-lemma-finite-type-local-source-fppf-algebra", "descent-lemma-flat-finitely-presented-permanence-algebra", "morphisms-lemma-triangle-differentials-smooth", "morphisms-lemma-unramified-omega-zero", "descent-lemma-properties-morphisms-local-source" ], "ref_ids": [ 14707, 5344, 5345, 5362, 14641, 14640, 5337, 5343, 14707 ] } ], "ref_ids": [] }, { "id": 14719, "type": "theorem", "label": "descent-lemma-local-source-target-implies", "categories": [ "descent" ], "title": "descent-lemma-local-source-target-implies", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes which is", "\\'etale local on source-and-target. Then", "\\begin{enumerate}", "\\item $\\mathcal{P}$ is \\'etale local on the source,", "\\item $\\mathcal{P}$ is \\'etale local on the target,", "\\item $\\mathcal{P}$ is stable under postcomposing with \\'etale morphisms:", "if $f : X \\to Y$ has $\\mathcal{P}$ and $g : Y \\to Z$ is \\'etale, then", "$g \\circ f$ has $\\mathcal{P}$, and", "\\item $\\mathcal{P}$ has a permanence property: given $f : X \\to Y$ and", "$g : Y \\to Z$ \\'etale such that $g \\circ f$ has $\\mathcal{P}$, then", "$f$ has $\\mathcal{P}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We write everything out completely.", "\\medskip\\noindent", "Proof of (1). Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\{X_i \\to X\\}_{i \\in I}$ be an \\'etale covering of $X$. If each composition", "$h_i : X_i \\to Y$ has $\\mathcal{P}$, then for each $x \\in X$ we can find", "an $i \\in I$ and a point $x_i \\in X_i$ mapping to $x$. Then", "$(X_i, x_i) \\to (X, x)$ is an \\'etale morphism of germs, and", "$\\text{id}_Y : Y \\to Y$ is an \\'etale morphism, and $h_i$ is as in part (3) of", "Definition \\ref{definition-local-source-target}.", "Thus we see that $f$ has $\\mathcal{P}$.", "Conversely, if $f$ has $\\mathcal{P}$ then each $X_i \\to Y$ has", "$\\mathcal{P}$ by", "Definition \\ref{definition-local-source-target} part (1).", "\\medskip\\noindent", "Proof of (2). Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\{Y_i \\to Y\\}_{i \\in I}$ be an \\'etale covering of $Y$.", "Write $X_i = Y_i \\times_Y X$ and $h_i : X_i \\to Y_i$ for the base change", "of $f$. If each $h_i : X_i \\to Y_i$ has $\\mathcal{P}$, then for each", "$x \\in X$ we pick an $i \\in I$ and a point $x_i \\in X_i$ mapping to $x$.", "Then $(X_i, x_i) \\to (X, x)$ is an \\'etale morphism of germs, $Y_i \\to Y$ is", "\\'etale, and $h_i$ is as in part (3) of", "Definition \\ref{definition-local-source-target}.", "Thus we see that $f$ has $\\mathcal{P}$.", "Conversely, if $f$ has $\\mathcal{P}$, then each $X_i \\to Y_i$ has", "$\\mathcal{P}$ by", "Definition \\ref{definition-local-source-target} part (2).", "\\medskip\\noindent", "Proof of (3). Assume $f : X \\to Y$ has $\\mathcal{P}$ and $g : Y \\to Z$ is", "\\'etale. For every $x \\in X$ we can think of $(X, x) \\to (X, x)$ as an", "\\'etale morphism of germs, $Y \\to Z$ is an \\'etale morphism, and $h = f$ is as", "in part (3) of", "Definition \\ref{definition-local-source-target}.", "Thus we see that $g \\circ f$ has $\\mathcal{P}$.", "\\medskip\\noindent", "Proof of (4). Let $f : X \\to Y$ be a morphism and $g : Y \\to Z$ \\'etale", "such that $g \\circ f$ has $\\mathcal{P}$. Then by", "Definition \\ref{definition-local-source-target} part (2)", "we see that $\\text{pr}_Y : Y \\times_Z X \\to Y$ has $\\mathcal{P}$. But", "the morphism $(f, 1) : X \\to Y \\times_Z X$ is \\'etale as a section to the", "\\'etale projection $\\text{pr}_X : Y \\times_Z X \\to X$, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-permanence}.", "Hence $f = \\text{pr}_Y \\circ (f, 1)$ has $\\mathcal{P}$ by", "Definition \\ref{definition-local-source-target} part (1)." ], "refs": [ "descent-definition-local-source-target", "descent-definition-local-source-target", "descent-definition-local-source-target", "descent-definition-local-source-target", "descent-definition-local-source-target", "descent-definition-local-source-target", "morphisms-lemma-etale-permanence", "descent-definition-local-source-target" ], "ref_ids": [ 14774, 14774, 14774, 14774, 14774, 14774, 5375, 14774 ] } ], "ref_ids": [] }, { "id": 14720, "type": "theorem", "label": "descent-lemma-local-source-target-characterize", "categories": [ "descent" ], "title": "descent-lemma-local-source-target-characterize", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes which is", "\\'etale local on source-and-target. Let $f : X \\to Y$ be a morphism", "of schemes. The following are equivalent:", "\\begin{enumerate}", "\\item[(a)] $f$ has property $\\mathcal{P}$,", "\\item[(b)] for every $x \\in X$ there exists an \\'etale morphism of germs", "$a : (U, u) \\to (X, x)$, an \\'etale morphism $b : V \\to Y$, and", "a morphism $h : U \\to V$ such that $f \\circ a = b \\circ h$ and", "$h$ has $\\mathcal{P}$,", "\\item[(c)]", "for any commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "with $a$, $b$ \\'etale the morphism $h$ has $\\mathcal{P}$,", "\\item[(d)] for some diagram as in (c)", "with $a : U \\to X$ surjective $h$ has $\\mathcal{P}$,", "\\item[(e)] there exists an \\'etale covering $\\{Y_i \\to Y\\}_{i \\in I}$ such", "that each base change $Y_i \\times_Y X \\to Y_i$ has $\\mathcal{P}$,", "\\item[(f)] there exists an \\'etale covering $\\{X_i \\to X\\}_{i \\in I}$ such", "that each composition $X_i \\to Y$ has $\\mathcal{P}$,", "\\item[(g)] there exists an \\'etale covering $\\{Y_i \\to Y\\}_{i \\in I}$ and", "for each $i \\in I$ an \\'etale covering", "$\\{X_{ij} \\to Y_i \\times_Y X\\}_{j \\in J_i}$ such that each morphism", "$X_{ij} \\to Y_i$ has $\\mathcal{P}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (a) and (b) is part of", "Definition \\ref{definition-local-source-target}.", "The equivalence of (a) and (e) is", "Lemma \\ref{lemma-local-source-target-implies} part (2).", "The equivalence of (a) and (f) is", "Lemma \\ref{lemma-local-source-target-implies} part (1).", "As (a) is now equivalent to (e) and (f) it follows that", "(a) equivalent to (g).", "\\medskip\\noindent", "It is clear that (c) implies (a). If (a) holds, then for any", "diagram as in (c) the morphism $f \\circ a$ has $\\mathcal{P}$ by", "Definition \\ref{definition-local-source-target} part (1),", "whereupon $h$ has $\\mathcal{P}$ by", "Lemma \\ref{lemma-local-source-target-implies} part (4).", "Thus (a) and (c) are equivalent. It is clear that (c) implies (d).", "To see that (d) implies (a) assume we have a diagram as in (c)", "with $a : U \\to X$ surjective and $h$ having $\\mathcal{P}$.", "Then $b \\circ h$ has $\\mathcal{P}$ by", "Lemma \\ref{lemma-local-source-target-implies} part (3).", "Since $\\{a : U \\to X\\}$ is an \\'etale covering we conclude that", "$f$ has $\\mathcal{P}$ by", "Lemma \\ref{lemma-local-source-target-implies} part (1)." ], "refs": [ "descent-definition-local-source-target", "descent-lemma-local-source-target-implies", "descent-lemma-local-source-target-implies", "descent-definition-local-source-target", "descent-lemma-local-source-target-implies", "descent-lemma-local-source-target-implies", "descent-lemma-local-source-target-implies" ], "ref_ids": [ 14774, 14719, 14719, 14774, 14719, 14719, 14719 ] } ], "ref_ids": [] }, { "id": 14721, "type": "theorem", "label": "descent-lemma-etale-local-source-target", "categories": [ "descent" ], "title": "descent-lemma-etale-local-source-target", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes.", "Assume", "\\begin{enumerate}", "\\item $\\mathcal{P}$ is \\'etale local on the source,", "\\item $\\mathcal{P}$ is \\'etale local on the target, and", "\\item $\\mathcal{P}$ is stable under postcomposing with open immersions:", "if $f : X \\to Y$ has $\\mathcal{P}$ and $Y \\subset Z$ is an open", "subscheme then $X \\to Z$ has $\\mathcal{P}$.", "\\end{enumerate}", "Then $\\mathcal{P}$ is \\'etale local on the source-and-target." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes which", "satisfies conditions (1), (2) and (3) of the lemma. By", "Lemma \\ref{lemma-precompose-property-local-source}", "we see that $\\mathcal{P}$ is stable under precomposing with", "\\'etale morphisms. By", "Lemma \\ref{lemma-pullback-property-local-target}", "we see that $\\mathcal{P}$ is stable under \\'etale base change.", "Hence it suffices to prove part (3) of", "Definition \\ref{definition-local-source-target}", "holds.", "\\medskip\\noindent", "More precisely, suppose that $f : X \\to Y$ is a morphism", "of schemes which satisfies", "Definition \\ref{definition-local-source-target} part (3)(b).", "In other words, for every $x \\in X$ there exists an \\'etale", "morphism $a_x : U_x \\to X$, a point $u_x \\in U_x$ mapping to $x$,", "an \\'etale morphism $b_x : V_x \\to Y$, and a morphism $h_x : U_x \\to V_x$", "such that $f \\circ a_x = b_x \\circ h_x$ and $h_x$ has $\\mathcal{P}$.", "The proof of the lemma is complete once we show that $f$ has $\\mathcal{P}$.", "Set $U = \\coprod U_x$, $a = \\coprod a_x$, $V = \\coprod V_x$,", "$b = \\coprod b_x$, and $h = \\coprod h_x$. We obtain a", "commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "with $a$, $b$ \\'etale, $a$ surjective. Note that $h$ has $\\mathcal{P}$", "as each $h_x$ does and $\\mathcal{P}$ is \\'etale local on the target.", "Because $a$ is surjective and $\\mathcal{P}$ is \\'etale local on the source,", "it suffices to prove that $b \\circ h$ has $\\mathcal{P}$.", "This reduces the lemma to proving that $\\mathcal{P}$ is stable under", "postcomposing with an \\'etale morphism.", "\\medskip\\noindent", "During the rest of the proof we let $f : X \\to Y$ be a", "morphism with property $\\mathcal{P}$ and $g : Y \\to Z$ is an \\'etale", "morphism. Consider the following statements:", "\\begin{enumerate}", "\\item[(-)] With no additional assumptions $g \\circ f$", "has property $\\mathcal{P}$.", "\\item[(A)] Whenever $Z$ is affine", "$g \\circ f$ has property $\\mathcal{P}$.", "\\item[(AA)] Whenever $X$ and $Z$ are affine", "$g \\circ f$ has property $\\mathcal{P}$.", "\\item[(AAA)] Whenever $X$, $Y$, and $Z$ are affine", "$g \\circ f$ has property $\\mathcal{P}$.", "\\end{enumerate}", "Once we have proved (-) the proof of the lemma will be complete.", "\\medskip\\noindent", "Claim 1: (AAA) $\\Rightarrow$ (AA).", "Namely, let $f : X \\to Y$, $g : Y \\to Z$ be as above with $X$, $Z$ affine.", "As $X$ is affine hence quasi-compact we can find finitely many", "affine open $Y_i \\subset Y$, $i = 1, \\ldots, n$ such that", "$X = \\bigcup_{i = 1, \\ldots, n} f^{-1}(Y_i)$. Set $X_i = f^{-1}(Y_i)$. By", "Lemma \\ref{lemma-pullback-property-local-target}", "each of the morphisms $X_i \\to Y_i$ has $\\mathcal{P}$.", "Hence $\\coprod_{i = 1, \\ldots, n} X_i \\to \\coprod_{i = 1, \\ldots, n} Y_i$", "has $\\mathcal{P}$ as $\\mathcal{P}$ is \\'etale local on the target.", "By (AAA) applied to", "$\\coprod_{i = 1, \\ldots, n} X_i \\to \\coprod_{i = 1, \\ldots, n} Y_i$", "and the \\'etale morphism $\\coprod_{i = 1, \\ldots, n} Y_i \\to Z$", "we see that $\\coprod_{i = 1, \\ldots, n} X_i \\to Z$ has $\\mathcal{P}$.", "Now $\\{\\coprod_{i = 1, \\ldots, n} X_i \\to X\\}$ is an \\'etale", "covering, hence as $\\mathcal{P}$ is \\'etale local on the source", "we conclude that $X \\to Z$ has $\\mathcal{P}$ as desired.", "\\medskip\\noindent", "Claim 2: (AAA) $\\Rightarrow$ (A).", "Namely, let $f : X \\to Y$, $g : Y \\to Z$ be as above with $Z$ affine.", "Choose an affine open covering $X = \\bigcup X_i$.", "As $\\mathcal{P}$ is \\'etale local on the source we see that", "each $f|_{X_i} : X_i \\to Y$ has $\\mathcal{P}$.", "By (AA), which follows from (AAA) according to Claim 1, we see that", "$X_i \\to Z$ has $\\mathcal{P}$ for each $i$.", "Since $\\{X_i \\to X\\}$ is an \\'etale covering and $\\mathcal{P}$ is \\'etale", "local on the source we conclude that", "$X \\to Z$ has $\\mathcal{P}$.", "\\medskip\\noindent", "Claim 3: (AAA) $\\Rightarrow$ (-).", "Namely, let $f : X \\to Y$, $g : Y \\to Z$ be as above.", "Choose an affine open covering $Z = \\bigcup Z_i$.", "Set $Y_i = g^{-1}(Z_i)$ and $X_i = f^{-1}(Y_i)$. By", "Lemma \\ref{lemma-pullback-property-local-target}", "each of the morphisms $X_i \\to Y_i$ has $\\mathcal{P}$.", "By (A), which follows from (AAA) according to Claim 2, we see that", "$X_i \\to Z_i$ has $\\mathcal{P}$ for each $i$.", "Since $\\mathcal{P}$ is local on the target and $X_i = (g \\circ f)^{-1}(Z_i)$", "we conclude that $X \\to Z$ has $\\mathcal{P}$.", "\\medskip\\noindent", "Thus to prove the lemma it suffices to prove (AAA).", "Let $f : X \\to Y$ and $g : Y \\to Z$ be as above $X, Y, Z$ affine.", "Note that an \\'etale morphism of affines has universally bounded fibres, see", "Morphisms,", "Lemma \\ref{morphisms-lemma-etale-locally-quasi-finite} and", "Lemma \\ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded}.", "Hence we can do induction on the integer $n$ bounding the degree of the fibres", "of $Y \\to Z$. See", "Morphisms, Lemma \\ref{morphisms-lemma-etale-universally-bounded}", "for a description of this integer in the case of an \\'etale morphism.", "If $n = 1$, then $Y \\to Z$ is an open immersion, see", "Lemma \\ref{lemma-universally-injective-etale-open-immersion},", "and the result follows from assumption (3) of the lemma. Assume $n > 1$.", "\\medskip\\noindent", "Consider the following commutative diagram", "$$", "\\xymatrix{", "X \\times_Z Y \\ar[d] \\ar[r]_{f_Y} &", "Y \\times_Z Y \\ar[d] \\ar[r]_-{\\text{pr}} &", "Y \\ar[d] \\\\", "X \\ar[r]^f &", "Y \\ar[r]^g &", "Z", "}", "$$", "Note that we have a decomposition into open and closed", "subschemes $Y \\times_Z Y = \\Delta_{Y/Z}(Y) \\amalg Y'$, see", "Morphisms, Lemma \\ref{morphisms-lemma-diagonal-unramified-morphism}.", "As a base change the degrees of the fibres of the second projection", "$\\text{pr} : Y \\times_Z Y \\to Y$ are bounded by $n$, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-universally-bounded}.", "On the other hand, $\\text{pr}|_{\\Delta(Y)} : \\Delta(Y) \\to Y$ is", "an isomorphism and every fibre has exactly one point.", "Thus, on applying", "Morphisms, Lemma \\ref{morphisms-lemma-etale-universally-bounded}", "we conclude the degrees of the fibres of the restriction", "$\\text{pr}|_{Y'} : Y' \\to Y$ are bounded by $n - 1$.", "Set $X' = f_Y^{-1}(Y')$. Picture", "$$", "\\xymatrix{", "X \\amalg X' \\ar@{=}[d] \\ar[r]_-{f \\amalg f'} &", "\\Delta(Y) \\amalg Y' \\ar@{=}[d] \\ar[r] &", "Y \\ar@{=}[d] \\\\", "X \\times_Z Y \\ar[r]^{f_Y} &", "Y \\times_Z Y \\ar[r]^-{\\text{pr}} &", "Y", "}", "$$", "As $\\mathcal{P}$ is \\'etale local on the target and hence stable under", "\\'etale base change (see", "Lemma \\ref{lemma-pullback-property-local-target})", "we see that $f_Y$ has $\\mathcal{P}$.", "Hence, as $\\mathcal{P}$ is \\'etale local on the source,", "$f' = f_Y|_{X'}$ has $\\mathcal{P}$. By induction hypothesis", "we see that $X' \\to Y$ has $\\mathcal{P}$.", "As $\\mathcal{P}$ is local on the source, and", "$\\{X \\to X \\times_Z Y, X' \\to X \\times_Y Z\\}$ is an \\'etale covering,", "we conclude that $\\text{pr} \\circ f_Y$ has $\\mathcal{P}$.", "Note that $g \\circ f$ can be viewed as a morphism", "$g \\circ f : X \\to g(Y)$. As $\\text{pr} \\circ f_Y$ is the pullback of", "$g \\circ f : X \\to g(Y)$ via the \\'etale covering $\\{Y \\to g(Y)\\}$,", "and as $\\mathcal{P}$ is \\'etale local on the target, we conclude that", "$g \\circ f : X \\to g(Y)$ has property $\\mathcal{P}$. Finally, applying", "assumption (3) of the lemma once more we conclude that", "$g \\circ f : X \\to Z$ has property $\\mathcal{P}$." ], "refs": [ "descent-lemma-precompose-property-local-source", "descent-lemma-pullback-property-local-target", "descent-definition-local-source-target", "descent-definition-local-source-target", "descent-lemma-pullback-property-local-target", "descent-lemma-pullback-property-local-target", "morphisms-lemma-etale-locally-quasi-finite", "morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded", "morphisms-lemma-etale-universally-bounded", "descent-lemma-universally-injective-etale-open-immersion", "morphisms-lemma-diagonal-unramified-morphism", "morphisms-lemma-base-change-universally-bounded", "morphisms-lemma-etale-universally-bounded", "descent-lemma-pullback-property-local-target" ], "ref_ids": [ 14705, 14663, 14774, 14774, 14663, 14663, 5363, 5531, 5530, 14700, 5354, 5527, 5530, 14663 ] } ], "ref_ids": [] }, { "id": 14722, "type": "theorem", "label": "descent-lemma-etale-etale-local-source-target", "categories": [ "descent" ], "title": "descent-lemma-etale-etale-local-source-target", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes which", "is \\'etale local on the source-and-target.", "Given a commutative diagram of schemes", "$$", "\\vcenter{", "\\xymatrix{", "X' \\ar[d]_{g'} \\ar[r]_{f'} & Y' \\ar[d]^g \\\\", "X \\ar[r]^f & Y", "}", "}", "\\quad\\text{with points}\\quad", "\\vcenter{", "\\xymatrix{", "x' \\ar[d] \\ar[r] & y' \\ar[d] \\\\", "x \\ar[r] & y", "}", "}", "$$", "such that $g'$ is \\'etale at $x'$ and $g$ is \\'etale at $y'$, then", "$x \\in W(f) \\Leftrightarrow x' \\in W(f')$", "where $W(-)$ is as in Lemma \\ref{lemma-largest-open-of-the-source}." ], "refs": [ "descent-lemma-largest-open-of-the-source" ], "proofs": [ { "contents": [ "Lemma \\ref{lemma-largest-open-of-the-source} applies since", "$\\mathcal{P}$ is \\'etale local on the source by", "Lemma \\ref{lemma-local-source-target-implies}.", "\\medskip\\noindent", "Assume $x \\in W(f)$. Let $U' \\subset X'$ and $V' \\subset Y'$", "be open neighbourhoods of $x'$ and $y'$ such that $f'(U') \\subset V'$,", "$g'(U') \\subset W(f)$ and $g'|_{U'}$ and $g|_{V'}$ are \\'etale.", "Then $f \\circ g'|_{U'} = g \\circ f'|_{U'}$", "has $\\mathcal{P}$ by property (1) of", "Definition \\ref{definition-local-source-target}.", "Then $f'|_{U'} : U' \\to V'$ has property $\\mathcal{P}$", "by (4) of Lemma \\ref{lemma-local-source-target-implies}.", "Then by (3) of Lemma \\ref{lemma-local-source-target-implies}", "we conclude that $f'_{U'} : U' \\to Y'$ has $\\mathcal{P}$.", "Hence $U' \\subset W(f')$ by definition. Hence $x' \\in W(f')$.", "\\medskip\\noindent", "Assume $x' \\in W(f')$. Let $U' \\subset X'$ and $V' \\subset Y'$", "be open neighbourhoods of $x'$ and $y'$ such that $f'(U') \\subset V'$,", "$U' \\subset W(f')$ and $g'|_{U'}$ and $g|_{V'}$ are \\'etale.", "Then $U' \\to Y'$ has $\\mathcal{P}$ by definition of $W(f')$.", "Then $U' \\to V'$ has $\\mathcal{P}$ by (4) of", "Lemma \\ref{lemma-local-source-target-implies}.", "Then $U' \\to Y$ has $\\mathcal{P}$ by (3) of", "Lemma \\ref{lemma-local-source-target-implies}.", "Let $U \\subset X$ be the image of the \\'etale (hence open)", "morphism $g'|_U' : U' \\to X$. Then $\\{U' \\to U\\}$", "is an \\'etale covering and we conclude that", "$U \\to Y$ has $\\mathcal{P}$ by (1) of", "Lemma \\ref{lemma-local-source-target-implies}.", "Thus $U \\subset W(f)$ by definition. Hence $x \\in W(f)$." ], "refs": [ "descent-lemma-largest-open-of-the-source", "descent-lemma-local-source-target-implies", "descent-definition-local-source-target", "descent-lemma-local-source-target-implies", "descent-lemma-local-source-target-implies", "descent-lemma-local-source-target-implies", "descent-lemma-local-source-target-implies", "descent-lemma-local-source-target-implies" ], "ref_ids": [ 14706, 14719, 14774, 14719, 14719, 14719, 14719, 14719 ] } ], "ref_ids": [ 14706 ] }, { "id": 14723, "type": "theorem", "label": "descent-lemma-orbits", "categories": [ "descent" ], "title": "descent-lemma-orbits", "contents": [ "Let $k$ be a field. Let $n \\geq 2$. For $1 \\leq i, j \\leq n$ with", "$i \\not = j$ and $d \\geq 0$ denote $T_{i, j, d}$ the automorphism", "of $\\mathbf{A}^n_k$ given in coordinates by", "$$", "(x_1, \\ldots, x_n) \\longmapsto", "(x_1, \\ldots, x_{i - 1}, x_i + x_j^d, x_{i + 1}, \\ldots, x_n)", "$$", "Let $W \\subset \\mathbf{A}^n_k$ be a nonempty open subscheme", "such that $T_{i, j, d}(W) = W$ for all $i, j, d$ as above.", "Then either $W = \\mathbf{A}^n_k$ or the characteristic of $k$", "is $p > 0$ and $\\mathbf{A}^n_k \\setminus W$ is a finite set", "of closed points whose coordinates are algebraic over $\\mathbf{F}_p$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $k$ by any extension field in order to prove this.", "Let $Z$ be an irreducible component of $\\mathbf{A}^n_k \\setminus W$.", "Assume $\\dim(Z) \\geq 1$, to get a contradiction.", "Then there exists an extension field $k'/k$ and a $k'$-valued", "point $\\xi = (\\xi_1, \\ldots, \\xi_n) \\in (k')^n$ of", "$Z_{k'} \\subset \\mathbf{A}^n_{k'}$", "such that at least one of $x_1, \\ldots, x_n$ is transcendental over the", "prime field. Claim: the orbit of $\\xi$ under the group generated by", "the transformations $T_{i, j, d}$ is Zariski", "dense in $\\mathbf{A}^n_{k'}$. The claim will give the desired contradiction.", "\\medskip\\noindent", "If the characteristic of $k'$ is zero, then already the operators", "$T_{i, j, 0}$ will be enough since these transform $\\xi$ into", "the points", "$$", "(\\xi_1 + a_1, \\ldots, \\xi_n + a_n)", "$$", "for arbitrary $(a_1, \\ldots, a_n) \\in \\mathbf{Z}_{\\geq 0}^n$.", "If the characteristic is $p > 0$, we may assume after renumbering", "that $\\xi_n$ is transcendental over $\\mathbf{F}_p$. By", "successively applying the operators $T_{i, n, d}$ for", "$i < n$ we see the orbit of $\\xi$ contains the elements", "$$", "(\\xi_1 + P_1(\\xi_n), \\ldots, \\xi_{n - 1} + P_{n - 1}(\\xi_n), \\xi_n)", "$$", "for arbitrary $(P_1, \\ldots, P_{n - 1}) \\in \\mathbf{F}_p[t]$.", "Thus the Zariski closure of the orbit contains the coordinate", "hyperplane $x_n = \\xi_n$. Repeating the argument with a different", "coordinate, we conclude that the Zariski closure contains", "$x_i = \\xi_i + P(\\xi_n)$ for any $P \\in \\mathbf{F}_p[t]$", "such that $\\xi_i + P(\\xi_n)$ is transcendental over $\\mathbf{F}_p$.", "Since there are infinitely many such $P$ the claim follows.", "\\medskip\\noindent", "Of course the argument in the preceding paragraph also applies", "if $Z = \\{z\\}$ has dimension $0$ and the coordinates of $z$", "in $\\kappa(z)$ are not algebraic over $\\mathbf{F}_p$. The lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14724, "type": "theorem", "label": "descent-lemma-etale-tau-local-source-target", "categories": [ "descent" ], "title": "descent-lemma-etale-tau-local-source-target", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes. Assume", "\\begin{enumerate}", "\\item $\\mathcal{P}$ is \\'etale local on the source,", "\\item $\\mathcal{P}$ is smooth local on the target,", "\\item $\\mathcal{P}$ is stable under postcomposing with open immersions:", "if $f : X \\to Y$ has $\\mathcal{P}$ and $Y \\subset Z$ is an open", "subscheme then $X \\to Z$ has $\\mathcal{P}$.", "\\end{enumerate}", "Given a commutative diagram of schemes", "$$", "\\vcenter{", "\\xymatrix{", "X' \\ar[d]_{g'} \\ar[r]_{f'} & Y' \\ar[d]^g \\\\", "X \\ar[r]^f & Y", "}", "}", "\\quad\\text{with points}\\quad", "\\vcenter{", "\\xymatrix{", "x' \\ar[d] \\ar[r] & y' \\ar[d] \\\\", "x \\ar[r] & y", "}", "}", "$$", "such that $g$ is smooth $y'$ and $X' \\to X \\times_Y Y'$ is \\'etale", "at $x'$, then $x \\in W(f) \\Leftrightarrow x' \\in W(f')$", "where $W(-)$ is as in Lemma \\ref{lemma-largest-open-of-the-source}." ], "refs": [ "descent-lemma-largest-open-of-the-source" ], "proofs": [ { "contents": [ "Since $\\mathcal{P}$ is \\'etale local on the source we see", "that $x \\in W(f)$ if and only if the image of $x$ in", "$X \\times_Y Y'$ is in $W(X \\times_Y Y' \\to Y')$. Hence we", "may assume the diagram in the lemma is cartesian.", "\\medskip\\noindent", "Assume $x \\in W(f)$. Since $\\mathcal{P}$ is smooth local on the target", "we see that $(g')^{-1}W(f) = W(f) \\times_Y Y' \\to Y'$ has $\\mathcal{P}$.", "Hence $(g')^{-1}W(f) \\subset W(f')$. We conclude $x' \\in W(f')$.", "\\medskip\\noindent", "Assume $x' \\in W(f')$.", "For any open neighbourhood $V' \\subset Y'$ of $y'$ we may replace", "$Y'$ by $V'$ and $X'$ by $U' = (f')^{-1}V'$ because $V' \\to Y'$ is smooth", "and hence the base change $W(f') \\cap U' \\to V'$ of $W(f') \\to Y'$", "has property $\\mathcal{P}$. Thus we may assume there exists", "an \\'etale morphism $Y' \\to \\mathbf{A}^n_Y$ over $Y$, see", "Morphisms, Lemma \\ref{morphisms-lemma-smooth-etale-over-affine-space}.", "Picture", "$$", "\\xymatrix{", "X' \\ar[r] \\ar[d] & Y' \\ar[d] \\\\", "\\mathbf{A}^n_X \\ar[r]_{f_n} \\ar[d] & \\mathbf{A}^n_Y \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "By Lemma \\ref{lemma-etale-local-source-target}", "(and because \\'etale coverings are smooth coverings)", "we see that $\\mathcal{P}$ is \\'etale local on the source-and-target.", "By Lemma \\ref{lemma-etale-etale-local-source-target}", "we see that $W(f')$ is the inverse image of", "the open $W(f_n) \\subset \\mathbf{A}^n_X$. In particular", "$W(f_n)$ contains a point lying over $x$.", "After replacing $X$ by the image of $W(f_n)$ (which is open)", "we may assume $W(f_n) \\to X$ is surjective.", "Claim: $W(f_n) = \\mathbf{A}^n_X$.", "The claim implies $f$ has $\\mathcal{P}$ as", "$\\mathcal{P}$ is local in the smooth topology", "and $\\{\\mathbf{A}^n_Y \\to Y\\}$ is a smooth covering.", "\\medskip\\noindent", "Essentially, the claim follows as $W(f_n) \\subset \\mathbf{A}^n_X$ is a", "``translation invariant'' open which meets every fibre of", "$\\mathbf{A}^n_X \\to X$. However, to produce an argument along these lines", "one has to do \\'etale localization on $Y$ to produce enough translations", "and it becomes a bit annoying. Instead we use the automorphisms", "of Lemma \\ref{lemma-orbits} and \\'etale morphisms of affine spaces.", "We may assume $n \\geq 2$. Namely, if $n = 0$, then we are done.", "If $n = 1$, then we consider the diagram", "$$", "\\xymatrix{", "\\mathbf{A}^2_X \\ar[r]_{f_2} \\ar[d]_p & \\mathbf{A}^2_Y \\ar[d] \\\\", "\\mathbf{A}^1_X \\ar[r]^{f_1} & \\mathbf{A}^1_Y", "}", "$$", "We have $p^{-1}(W(f_1)) \\subset W(f_2)$ (see first paragraph", "of the proof). Thus $W(f_2) \\to X$ is still surjective", "and we may work with $f_2$. Assume $n \\geq 2$.", "\\medskip\\noindent", "For any $1 \\leq i, j \\leq n$ with $i \\not = j$ and $d \\geq 0$", "denote $T_{i, j, d}$ the automorphism of $\\mathbf{A}^n$ defined", "in Lemma \\ref{lemma-orbits}. Then we get a commutative diagram", "$$", "\\xymatrix{", "\\mathbf{A}^n_X \\ar[r]_{f_n} \\ar[d]_{T_{i, j, d}} &", "\\mathbf{A}^n_Y \\ar[d]^{T_{i, j, d}} \\\\", "\\mathbf{A}^n_X \\ar[r]^{f_n} & \\mathbf{A}^n_Y", "}", "$$", "whose vertical arrows are isomorphisms. We conclude that", "$T_{i, j, d}(W(f_n)) = W(f_n)$. Applying Lemma \\ref{lemma-orbits}", "we conclude for any $x \\in X$ the fibre $W(f_n)_x \\subset \\mathbf{A}^n_x$ is", "either $\\mathbf{A}^n_x$ (this is what we want) or $\\kappa(x)$", "has characteristic $p > 0$ and $W(f_n)_x$", "is the complement of a finite set $Z_x \\subset \\mathbf{A}^n_x$", "of closed points. The second possibility cannot occur. Namely,", "consider the morphism $T_p : \\mathbf{A}^n \\to \\mathbf{A}^n$ given by", "$$", "(x_1, \\ldots, x_n) \\mapsto (x_1 - x_1^p, \\ldots, x_n - x_n^p)", "$$", "As above we get a commutative diagram", "$$", "\\xymatrix{", "\\mathbf{A}^n_X \\ar[r]_{f_n} \\ar[d]_{T_p} &", "\\mathbf{A}^n_Y \\ar[d]^{T_p} \\\\", "\\mathbf{A}^n_X \\ar[r]^{f_n} & \\mathbf{A}^n_Y", "}", "$$", "The morphism $T_p : \\mathbf{A}^n_X \\to \\mathbf{A}^n_X$", "is \\'etale at every point lying over $x$", "and the morphism $T_p : \\mathbf{A}^n_Y \\to \\mathbf{A}^n_Y$", "is \\'etale at every point lying over the image of $x$ in $Y$.", "(Details omitted; hint: compute the derivatives.)", "We conclude that", "$$", "T_p^{-1}(W) \\cap \\mathbf{A}^n_x = W \\cap \\mathbf{A}^n_x", "$$", "by Lemma \\ref{lemma-etale-etale-local-source-target}", "(we've already seen $\\mathcal{P}$ is", "\\'etale local on the source-and-target).", "Since $T_p : \\mathbf{A}^n_x \\to \\mathbf{A}^n_x$ is finite \\'etale", "of degree $p^n > 1$ we see that if $Z_x$ is not empty then it contains", "$T_p^{-1}(Z_x)$ which is bigger. This contradiction finishes", "the proof." ], "refs": [ "morphisms-lemma-smooth-etale-over-affine-space", "descent-lemma-etale-local-source-target", "descent-lemma-etale-etale-local-source-target", "descent-lemma-orbits", "descent-lemma-orbits", "descent-lemma-orbits", "descent-lemma-etale-etale-local-source-target" ], "ref_ids": [ 5377, 14721, 14722, 14723, 14723, 14723, 14722 ] } ], "ref_ids": [ 14706 ] }, { "id": 14725, "type": "theorem", "label": "descent-lemma-local-source-target-global-implies-local", "categories": [ "descent" ], "title": "descent-lemma-local-source-target-global-implies-local", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes", "which is \\'etale local on the source-and-target.", "Consider the property $\\mathcal{Q}$ of", "morphisms of germs defined by the rule", "$$", "\\mathcal{Q}((X, x) \\to (S, s))", "\\Leftrightarrow", "\\text{there exists a representative }U \\to S", "\\text{ which has }\\mathcal{P}", "$$", "Then $\\mathcal{Q}$ is \\'etale local on the source-and-target as in", "Definition \\ref{definition-local-source-target-at-point}." ], "refs": [ "descent-definition-local-source-target-at-point" ], "proofs": [ { "contents": [ "If a morphism of germs $(X, x) \\to (S, s)$ has $\\mathcal{Q}$,", "then there are arbitrarily small neighbourhoods", "$U \\subset X$ of $x$ and $V \\subset S$ of $s$", "such that a representative $U \\to V$ of $(X, x) \\to (S, s)$ has $\\mathcal{P}$.", "This follows from Lemma \\ref{lemma-local-source-target-implies}. Let", "$$", "\\xymatrix{", "(U', u') \\ar[r]_{h'} \\ar[d]_a & (V', v') \\ar[d]^b \\\\", "(U, u) \\ar[r]^h & (V, v)", "}", "$$", "be as in Definition \\ref{definition-local-source-target-at-point}.", "Choose $U_1 \\subset U$ and a representative $h_1 : U_1 \\to V$ of $h$.", "Choose $V'_1 \\subset V'$ and an \\'etale representative $b_1 : V'_1 \\to V$", "of $b$ (Definition \\ref{definition-etale-morphism-germs}).", "Choose $U'_1 \\subset U'$ and representatives $a_1 : U'_1 \\to U_1$", "and $h'_1 : U'_1 \\to V'_1$ of $a$ and $h'$ with $a_1$ \\'etale.", "After shrinking $U'_1$ we may assume $h_1 \\circ a_1 = b_1 \\circ h'_1$.", "By the initial remark of the proof, we are trying to show", "$u' \\in W(h'_1) \\Leftrightarrow u \\in W(h_1)$ where $W(-)$ is as", "in Lemma \\ref{lemma-largest-open-of-the-source}.", "Thus the lemma follows from Lemma \\ref{lemma-etale-etale-local-source-target}." ], "refs": [ "descent-lemma-local-source-target-implies", "descent-definition-local-source-target-at-point", "descent-definition-etale-morphism-germs", "descent-lemma-largest-open-of-the-source", "descent-lemma-etale-etale-local-source-target" ], "ref_ids": [ 14719, 14775, 14770, 14706, 14722 ] } ], "ref_ids": [ 14775 ] }, { "id": 14726, "type": "theorem", "label": "descent-lemma-local-source-target-local-implies-global", "categories": [ "descent" ], "title": "descent-lemma-local-source-target-local-implies-global", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes which is", "\\'etale local on source-and-target. Let $Q$ be the associated property", "of morphisms of germs, see", "Lemma \\ref{lemma-local-source-target-global-implies-local}.", "Let $f : X \\to Y$ be a morphism", "of schemes. The following are equivalent:", "\\begin{enumerate}", "\\item $f$ has property $\\mathcal{P}$, and", "\\item for every $x \\in X$ the morphism of germs $(X, x) \\to (Y, f(x))$", "has property $\\mathcal{Q}$.", "\\end{enumerate}" ], "refs": [ "descent-lemma-local-source-target-global-implies-local" ], "proofs": [ { "contents": [ "The implication (1) $\\Rightarrow$ (2) is direct from the definitions.", "The implication (2) $\\Rightarrow$ (1) also follows from part (3) of", "Definition \\ref{definition-local-source-target}." ], "refs": [ "descent-definition-local-source-target" ], "ref_ids": [ 14774 ] } ], "ref_ids": [ 14725 ] }, { "id": 14727, "type": "theorem", "label": "descent-lemma-flat-at-point", "categories": [ "descent" ], "title": "descent-lemma-flat-at-point", "contents": [ "The property of morphisms of germs", "$$", "\\mathcal{P}((X, x) \\to (S, s)) =", "\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}\\text{ is flat}", "$$", "is \\'etale local on the source-and-target." ], "refs": [], "proofs": [ { "contents": [ "Given a diagram as in", "Definition \\ref{definition-local-source-target-at-point}", "we obtain the following diagram of local homomorphisms of local rings", "$$", "\\xymatrix{", "\\mathcal{O}_{U', u'} & \\mathcal{O}_{V', v'} \\ar[l] \\\\", "\\mathcal{O}_{U, u} \\ar[u] & \\mathcal{O}_{V, v} \\ar[l] \\ar[u]", "}", "$$", "Note that the vertical arrows are localizations of \\'etale ring maps,", "in particular they are essentially of finite presentation, flat,", "and unramified (see", "Algebra, Section \\ref{algebra-section-etale}).", "In particular the vertical maps are faithfully flat, see", "Algebra, Lemma \\ref{algebra-lemma-local-flat-ff}.", "Now, if the upper horizontal arrow is flat, then the lower horizontal", "arrow is flat by an application of", "Algebra, Lemma \\ref{algebra-lemma-flat-permanence}", "with $R = \\mathcal{O}_{V, v}$, $S = \\mathcal{O}_{U, u}$ and", "$M = \\mathcal{O}_{U', u'}$. If the lower horizontal arrow is", "flat, then the ring map", "$$", "\\mathcal{O}_{V', v'} \\otimes_{\\mathcal{O}_{V, v}} \\mathcal{O}_{U, u}", "\\longleftarrow", "\\mathcal{O}_{V', v'}", "$$", "is flat by", "Algebra, Lemma \\ref{algebra-lemma-flat-base-change}.", "And the ring map", "$$", "\\mathcal{O}_{U', u'}", "\\longleftarrow", "\\mathcal{O}_{V', v'} \\otimes_{\\mathcal{O}_{V, v}} \\mathcal{O}_{U, u}", "$$", "is a localization of a map between \\'etale ring extensions of", "$\\mathcal{O}_{U, u}$, hence flat by", "Algebra, Lemma \\ref{algebra-lemma-map-between-etale}." ], "refs": [ "descent-definition-local-source-target-at-point", "algebra-lemma-local-flat-ff", "algebra-lemma-flat-permanence", "algebra-lemma-flat-base-change", "algebra-lemma-map-between-etale" ], "ref_ids": [ 14775, 537, 530, 527, 1236 ] } ], "ref_ids": [] }, { "id": 14728, "type": "theorem", "label": "descent-lemma-etale-on-fiber", "categories": [ "descent" ], "title": "descent-lemma-etale-on-fiber", "contents": [ "Consider a commutative diagram of morphisms of schemes", "$$", "\\xymatrix{", "U' \\ar[r] \\ar[d] & V' \\ar[d] \\\\", "U \\ar[r] & V", "}", "$$", "with \\'etale vertical arrows and a point $v' \\in V'$ mapping to $v \\in V$.", "Then the morphism of fibres $U'_{v'} \\to U_v$ is \\'etale." ], "refs": [], "proofs": [ { "contents": [ "Note that $U'_v \\to U_v$ is \\'etale as a base change of the \\'etale", "morphism $U' \\to U$. The scheme $U'_v$ is a scheme over $V'_v$. By", "Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}", "the scheme $V'_v$ is a disjoint union of spectra", "of finite separable field extensions of $\\kappa(v)$.", "One of these is $v' = \\Spec(\\kappa(v'))$. Hence", "$U'_{v'}$ is an open and closed subscheme of $U'_v$ and it follows", "that $U'_{v'} \\to U'_v \\to U_v$ is \\'etale (as a composition of an", "open immersion and an \\'etale morphism, see", "Morphisms, Section \\ref{morphisms-section-etale})." ], "refs": [ "morphisms-lemma-etale-over-field" ], "ref_ids": [ 5364 ] } ], "ref_ids": [] }, { "id": 14729, "type": "theorem", "label": "descent-lemma-dimension-local-ring-fibre", "categories": [ "descent" ], "title": "descent-lemma-dimension-local-ring-fibre", "contents": [ "Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$.", "The property of morphisms of germs", "$$", "\\mathcal{P}_d((X, x) \\to (S, s)) =", "\\text{the local ring }", "\\mathcal{O}_{X_s, x}", "\\text{ of the fibre has dimension }d", "$$", "is \\'etale local on the source-and-target." ], "refs": [], "proofs": [ { "contents": [ "Given a diagram as in", "Definition \\ref{definition-local-source-target-at-point}", "we obtain an \\'etale morphism of fibres", "$U'_{v'} \\to U_v$ mapping $u'$ to $u$, see", "Lemma \\ref{lemma-etale-on-fiber}.", "Hence the result follows from", "Lemma \\ref{lemma-dimension-local-ring-local}." ], "refs": [ "descent-definition-local-source-target-at-point", "descent-lemma-etale-on-fiber", "descent-lemma-dimension-local-ring-local" ], "ref_ids": [ 14775, 14728, 14661 ] } ], "ref_ids": [] }, { "id": 14730, "type": "theorem", "label": "descent-lemma-transcendence-degree-at-point", "categories": [ "descent" ], "title": "descent-lemma-transcendence-degree-at-point", "contents": [ "Let $r \\in \\{0, 1, 2, \\ldots, \\infty\\}$.", "The property of morphisms of germs", "$$", "\\mathcal{P}_r((X, x) \\to (S, s))", "\\Leftrightarrow", "\\text{trdeg}_{\\kappa(s)} \\kappa(x) = r", "$$", "is \\'etale local on the source-and-target." ], "refs": [], "proofs": [ { "contents": [ "Given a diagram as in", "Definition \\ref{definition-local-source-target-at-point}", "we obtain the following diagram of local homomorphisms of local rings", "$$", "\\xymatrix{", "\\mathcal{O}_{U', u'} & \\mathcal{O}_{V', v'} \\ar[l] \\\\", "\\mathcal{O}_{U, u} \\ar[u] & \\mathcal{O}_{V, v} \\ar[l] \\ar[u]", "}", "$$", "Note that the vertical arrows are localizations of \\'etale ring maps,", "in particular they are unramified (see", "Algebra, Section \\ref{algebra-section-etale}).", "Hence $\\kappa(u) \\subset \\kappa(u')$ and $\\kappa(v) \\subset \\kappa(v')$", "are finite separable field extensions.", "Thus we have", "$\\text{trdeg}_{\\kappa(v)} \\kappa(u) = \\text{trdeg}_{\\kappa(v')} \\kappa(u)$", "which proves the lemma." ], "refs": [ "descent-definition-local-source-target-at-point" ], "ref_ids": [ 14775 ] } ], "ref_ids": [] }, { "id": 14731, "type": "theorem", "label": "descent-lemma-dimension-at-point", "categories": [ "descent" ], "title": "descent-lemma-dimension-at-point", "contents": [ "Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$.", "The property of morphisms of germs", "$$", "\\mathcal{P}_d((X, x) \\to (S, s))", "\\Leftrightarrow", "\\dim_x (X_s) = d", "$$", "is \\'etale local on the source-and-target." ], "refs": [], "proofs": [ { "contents": [ "Given a diagram as in", "Definition \\ref{definition-local-source-target-at-point}", "we obtain an \\'etale morphism of fibres", "$U'_{v'} \\to U_v$ mapping $u'$ to $u$, see", "Lemma \\ref{lemma-etale-on-fiber}.", "Hence now the equality $\\dim_u(U_v) = \\dim_{u'}(U'_{v'})$ follows from", "Lemma \\ref{lemma-dimension-at-point-local}." ], "refs": [ "descent-definition-local-source-target-at-point", "descent-lemma-etale-on-fiber", "descent-lemma-dimension-at-point-local" ], "ref_ids": [ 14775, 14728, 14660 ] } ], "ref_ids": [] }, { "id": 14732, "type": "theorem", "label": "descent-lemma-family-is-one", "categories": [ "descent" ], "title": "descent-lemma-family-is-one", "contents": [ "Let $S$ be a scheme.", "Let $\\{X_i \\to S\\}_{i \\in I}$ be a family of morphisms with target $S$.", "Set $X = \\coprod_{i \\in I} X_i$, and consider it as an $S$-scheme.", "There is a canonical equivalence of categories", "$$", "\\begin{matrix}", "\\text{category of descent data } \\\\", "\\text{relative to the family } \\{X_i \\to S\\}_{i \\in I}", "\\end{matrix}", "\\longrightarrow", "\\begin{matrix}", "\\text{ category of descent data} \\\\", "\\text{ relative to } X/S", "\\end{matrix}", "$$", "which maps $(V_i, \\varphi_{ij})$ to $(V, \\varphi)$ with", "$V = \\coprod_{i\\in I} V_i$ and $\\varphi = \\coprod \\varphi_{ij}$." ], "refs": [], "proofs": [ { "contents": [ "Observe that $X \\times_S X = \\coprod_{ij} X_i \\times_S X_j$", "and similarly for higher fibre products.", "Giving a morphism $V \\to X$ is exactly the same as", "giving a family $V_i \\to X_i$. And giving a descent datum", "$\\varphi$ is exactly the same as giving a family $\\varphi_{ij}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14733, "type": "theorem", "label": "descent-lemma-pullback", "categories": [ "descent" ], "title": "descent-lemma-pullback", "contents": [ "Pullback of descent data for schemes over schemes.", "\\begin{enumerate}", "\\item Let", "$$", "\\xymatrix{", "X' \\ar[r]_f \\ar[d]_{a'} & X \\ar[d]^a \\\\", "S' \\ar[r]^h & S", "}", "$$", "be a commutative diagram of morphisms of schemes.", "The construction", "$$", "(V \\to X, \\varphi) \\longmapsto f^*(V \\to X, \\varphi) = (V' \\to X', \\varphi')", "$$", "where $V' = X' \\times_X V$ and where", "$\\varphi'$ is defined as the composition", "$$", "\\xymatrix{", "V' \\times_{S'} X' \\ar@{=}[r] &", "(X' \\times_X V) \\times_{S'} X' \\ar@{=}[r] &", "(X' \\times_{S'} X') \\times_{X \\times_S X} (V \\times_S X)", "\\ar[d]^{\\text{id} \\times \\varphi} \\\\", "X' \\times_{S'} V' \\ar@{=}[r] &", "X' \\times_{S'} (X' \\times_X V) &", "(X' \\times_{S'} X') \\times_{X \\times_S X} (X \\times_S V) \\ar@{=}[l]", "}", "$$", "defines a functor from the category of descent data", "relative to $X \\to S$ to the category of descent data", "relative to $X' \\to S'$.", "\\item Given two morphisms $f_i : X' \\to X$, $i = 0, 1$ making the", "diagram commute the functors $f_0^*$ and $f_1^*$ are", "canonically isomorphic.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We omit the proof of (1), but we remark that the morphism", "$\\varphi'$ is the morphism $(f \\times f)^*\\varphi$ in the", "notation introduced in Remark \\ref{remark-easier}.", "For (2) we indicate which morphism", "$f_0^*V \\to f_1^*V$ gives the functorial isomorphism. Namely,", "since $f_0$ and $f_1$ both fit into the commutative diagram", "we see there is a unique morphism $r : X' \\to X \\times_S X$", "with $f_i = \\text{pr}_i \\circ r$. Then we take", "\\begin{eqnarray*}", "f_0^*V & = &", "X' \\times_{f_0, X} V \\\\", "& = &", "X' \\times_{\\text{pr}_0 \\circ r, X} V \\\\", "& = &", "X' \\times_{r, X \\times_S X} (X \\times_S X) \\times_{\\text{pr}_0, X} V \\\\", "& \\xrightarrow{\\varphi} &", "X' \\times_{r, X \\times_S X} (X \\times_S X) \\times_{\\text{pr}_1, X} V \\\\", "& = &", "X' \\times_{\\text{pr}_1 \\circ r, X} V \\\\", "& = &", "X' \\times_{f_1, X} V \\\\", "& = & f_1^*V", "\\end{eqnarray*}", "We omit the verification that this works." ], "refs": [ "descent-remark-easier" ], "ref_ids": [ 14800 ] } ], "ref_ids": [] }, { "id": 14734, "type": "theorem", "label": "descent-lemma-pullback-family", "categories": [ "descent" ], "title": "descent-lemma-pullback-family", "contents": [ "Let $\\mathcal{U} = \\{U_i \\to S'\\}_{i \\in I}$ and", "$\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$ be families of morphisms with", "fixed target. Let $\\alpha : I \\to J$, $h : S' \\to S$ and", "$g_i : U_i \\to V_{\\alpha(i)}$ be a morphism of families", "of maps with fixed target, see", "Sites, Definition \\ref{sites-definition-morphism-coverings}.", "\\begin{enumerate}", "\\item Let $(Y_j, \\varphi_{jj'})$ be a descent datum relative to the", "family $\\{V_j \\to S'\\}$. The system", "$$", "\\left(", "g_i^*Y_{\\alpha(i)},", "(g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')}", "\\right)", "$$", "(with notation as in Remark \\ref{remark-easier-family})", "is a descent datum relative to $\\mathcal{V}$.", "\\item This construction defines a functor between descent data relative", "to $\\mathcal{U}$ and descent data relative to $\\mathcal{V}$.", "\\item Given a second $\\alpha' : I \\to J$, $h' : S' \\to S$ and", "$g'_i : U_i \\to V_{\\alpha'(i)}$ morphism of families", "of maps with fixed target, then if $h = h'$ the two resulting functors", "between descent data are canonically isomorphic.", "\\item These functors agree, via Lemma \\ref{lemma-family-is-one},", "with the pullback functors constructed in Lemma \\ref{lemma-pullback}.", "\\end{enumerate}" ], "refs": [ "sites-definition-morphism-coverings", "descent-remark-easier-family", "descent-lemma-family-is-one", "descent-lemma-pullback" ], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-pullback} via the", "correspondence of Lemma \\ref{lemma-family-is-one}." ], "refs": [ "descent-lemma-pullback", "descent-lemma-family-is-one" ], "ref_ids": [ 14733, 14732 ] } ], "ref_ids": [ 8656, 14801, 14732, 14733 ] }, { "id": 14735, "type": "theorem", "label": "descent-lemma-surjective-flat-epi", "categories": [ "descent" ], "title": "descent-lemma-surjective-flat-epi", "contents": [ "A surjective and flat morphism is an epimorphism in the", "category of schemes." ], "refs": [], "proofs": [ { "contents": [ "Suppose we have $h : X' \\to X$ surjective and flat and", "$a, b : X \\to Y$ morphisms such that $a \\circ h = b \\circ h$.", "As $h$ is surjective we see that $a$ and $b$ agree on underlying", "topological spaces. Pick $x' \\in X'$ and set $x = h(x')$ and", "$y = a(x) = b(x)$. Consider the local ring maps", "$$", "a^\\sharp_x, b^\\sharp_x : \\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}", "$$", "These become equal when composed with", "the flat local homomorphism", "$h^\\sharp_{x'} : \\mathcal{O}_{X, x} \\to \\mathcal{O}_{X', x'}$.", "Since a flat local homomorphism is faithfully flat", "(Algebra, Lemma \\ref{algebra-lemma-local-flat-ff})", "we conclude that $h^\\sharp_{x'}$ is injective.", "Hence $a^\\sharp_x = b^\\sharp_x$ which implies $a = b$ as desired." ], "refs": [ "algebra-lemma-local-flat-ff" ], "ref_ids": [ 537 ] } ], "ref_ids": [] }, { "id": 14736, "type": "theorem", "label": "descent-lemma-ff-base-change-faithful", "categories": [ "descent" ], "title": "descent-lemma-ff-base-change-faithful", "contents": [ "Let $h : S' \\to S$ be a surjective, flat morphism of", "schemes. The base change functor", "$$", "\\Sch/S \\longrightarrow \\Sch/S', \\quad", "X \\longmapsto S' \\times_S X", "$$", "is faithful." ], "refs": [], "proofs": [ { "contents": [ "Let $X_1$, $X_2$ be schemes over $S$.", "Let $\\alpha, \\beta : X_2 \\to X_1$ be morphisms over $S$.", "If $\\alpha$, $\\beta$ base change to the same morphism then", "we get a commutative diagram as follows", "$$", "\\xymatrix{", "X_2 \\ar[d]^\\alpha &", "S' \\times_S X_2 \\ar[l] \\ar[d] \\ar[r] &", "X_2 \\ar[d]^\\beta \\\\", "X_1 &", "S' \\times_S X_1 \\ar[l] \\ar[r] &", "X_1", "}", "$$", "Hence it suffices to show that $S' \\times_S X_2 \\to X_2$", "is an epimorphism. As the base change of a surjective and", "flat morphism it is surjective and flat (see", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-surjective}", "and \\ref{morphisms-lemma-base-change-flat}). Hence the lemma follows", "from Lemma \\ref{lemma-surjective-flat-epi}." ], "refs": [ "morphisms-lemma-base-change-surjective", "morphisms-lemma-base-change-flat", "descent-lemma-surjective-flat-epi" ], "ref_ids": [ 5165, 5265, 14735 ] } ], "ref_ids": [] }, { "id": 14737, "type": "theorem", "label": "descent-lemma-faithful", "categories": [ "descent" ], "title": "descent-lemma-faithful", "contents": [ "In the situation of Lemma \\ref{lemma-pullback}", "assume that $f : X' \\to X$ is surjective", "and flat. Then the pullback functor is faithful." ], "refs": [ "descent-lemma-pullback" ], "proofs": [ { "contents": [ "Let $(V_i, \\varphi_i)$, $i = 1, 2$ be descent data for $X \\to S$.", "Let $\\alpha, \\beta : V_1 \\to V_2$ be morphisms of descent data.", "Suppose that $f^*\\alpha = f^*\\beta$. Our task is to show that", "$\\alpha = \\beta$. Note that $\\alpha$, $\\beta$ are morphisms", "of schemes over $X$, and that $f^*\\alpha$, $f^*\\beta$ are", "simply the base changes of $\\alpha$, $\\beta$ to morphisms over", "$X'$. Hence the lemma follows from Lemma \\ref{lemma-ff-base-change-faithful}." ], "refs": [ "descent-lemma-ff-base-change-faithful" ], "ref_ids": [ 14736 ] } ], "ref_ids": [ 14733 ] }, { "id": 14738, "type": "theorem", "label": "descent-lemma-fully-faithful", "categories": [ "descent" ], "title": "descent-lemma-fully-faithful", "contents": [ "In the situation of Lemma \\ref{lemma-pullback}", "assume", "\\begin{enumerate}", "\\item $\\{f : X' \\to X\\}$ is an fpqc covering (for example if $f$ is", "surjective, flat, and quasi-compact), and", "\\item $S = S'$.", "\\end{enumerate}", "Then the pullback functor is fully faithful." ], "refs": [ "descent-lemma-pullback" ], "proofs": [ { "contents": [ "Assumption (1) implies that $f$ is surjective and flat.", "Hence the pullback functor is faithful by", "Lemma \\ref{lemma-faithful}.", "Let $(V, \\varphi)$ and $(W, \\psi)$ be two descent data relative", "to $X \\to S$. Set $(V', \\varphi') = f^*(V, \\varphi)$ and", "$(W', \\psi') = f^*(W, \\psi)$.", "Let $\\alpha' : V' \\to W'$ be a morphism of descent data for $X'$ over $S$.", "We have to show there exists a morphism $\\alpha : V \\to W$ of", "descent data for $X$ over $S$ whose pullback is $\\alpha'$.", "\\medskip\\noindent", "Recall that $V'$ is the base change of $V$ by $f$ and that", "$\\varphi'$ is the base change of $\\varphi$ by $f \\times f$", "(see Remark \\ref{remark-easier}).", "By assumption the diagram", "$$", "\\xymatrix{", "V' \\times_S X' \\ar[r]_{\\varphi'} \\ar[d]_{\\alpha' \\times \\text{id}} &", "X' \\times_S V' \\ar[d]^{\\text{id} \\times \\alpha'} \\\\", "W' \\times_S X' \\ar[r]^{\\psi'} &", "X' \\times_S W'", "}", "$$", "commutes. We claim the two compositions", "$$", "\\xymatrix{", "V' \\times_V V' \\ar[r]^-{\\text{pr}_i} &", "V' \\ar[r]^{\\alpha'} &", "W' \\ar[r] &", "W", "}", ", \\quad i = 0, 1", "$$", "are the same. The reader is advised to prove this themselves rather", "than read the rest of this paragraph. (Please email if you find a", "nice clean argument.)", "Let $v_0, v_1$ be points of $V'$ which map to the same point $v \\in V$.", "Let $x_i \\in X'$ be the image of $v_i$, and let", "$x$ be the point of $X$ which is the image of $v$ in $X$. In other words,", "$v_i = (x_i, v)$ in $V' = X' \\times_X V$. Write", "$\\varphi(v, x) = (x, v')$ for some point $v'$ of $V$.", "This is possible because $\\varphi$ is", "a morphism over $X \\times_S X$. Denote", "$v_i' = (x_i, v')$ which is a point of $V'$.", "Then a calculation (using the definition of $\\varphi'$)", "shows that $\\varphi'(v_i, x_j) = (x_i, v'_j)$. Denote", "$w_i = \\alpha'(v_i)$ and $w'_i = \\alpha'(v_i')$.", "Now we may write $w_i = (x_i, u_i)$ for some point $u_i$ of $W$,", "and $w_i' = (x_i, u'_i)$ for some point $u_i'$ of $W$.", "The claim is equivalent to the assertion: $u_0 = u_1$.", "A formal calculation using the definition of $\\psi'$", "(see Lemma \\ref{lemma-pullback}) shows", "that the commutativity of the diagram displayed above says that", "$$", "((x_i, x_j), \\psi(u_i, x)) = ((x_i, x_j), (x, u'_j))", "$$", "as points of", "$(X' \\times_S X') \\times_{X \\times_S X} (X \\times_S W)$", "for all $i, j \\in \\{0, 1\\}$. This shows that $\\psi(u_0, x) = \\psi(u_1, x)$", "and hence $u_0 = u_1$ by taking $\\psi^{-1}$.", "This proves the claim because the argument above was formal", "and we can take scheme points (in other words, we may", "take $(v_0, v_1) = \\text{id}_{V' \\times_V V'}$).", "\\medskip\\noindent", "At this point we can use", "Lemma \\ref{lemma-fpqc-universal-effective-epimorphisms}.", "Namely, $\\{V' \\to V\\}$ is a fpqc covering as", "the base change of the morphism $f : X' \\to X$.", "Hence, by", "Lemma \\ref{lemma-fpqc-universal-effective-epimorphisms}", "the morphism $\\alpha' : V' \\to W' \\to W$ factors through", "a unique morphism $\\alpha : V \\to W$ whose base change is", "necessarily $\\alpha'$. Finally, we see the diagram", "$$", "\\xymatrix{", "V \\times_S X \\ar[r]_{\\varphi} \\ar[d]_{\\alpha \\times \\text{id}} &", "X \\times_S V \\ar[d]^{\\text{id} \\times \\alpha} \\\\", "W \\times_S X \\ar[r]^{\\psi} & X \\times_S W", "}", "$$", "commutes because its base change to $X' \\times_S X'$", "commutes and the morphism $X' \\times_S X' \\to X \\times_S X$", "is surjective and flat (use Lemma \\ref{lemma-ff-base-change-faithful}).", "Hence $\\alpha$ is a morphism of descent data", "$(V, \\varphi) \\to (W, \\psi)$ as desired." ], "refs": [ "descent-lemma-faithful", "descent-remark-easier", "descent-lemma-pullback", "descent-lemma-fpqc-universal-effective-epimorphisms", "descent-lemma-fpqc-universal-effective-epimorphisms", "descent-lemma-ff-base-change-faithful" ], "ref_ids": [ 14737, 14800, 14733, 14638, 14638, 14736 ] } ], "ref_ids": [ 14733 ] }, { "id": 14739, "type": "theorem", "label": "descent-lemma-pullback-selfmap", "categories": [ "descent" ], "title": "descent-lemma-pullback-selfmap", "contents": [ "Let $X \\to S$ be a morphism of schemes.", "Let $f : X \\to X$ be a selfmap of $X$ over $S$.", "In this case pullback by $f$ is isomorphic to the", "identity functor on the category of descent data", "relative to $X \\to S$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from Lemma \\ref{lemma-pullback} since it tells us that", "$f^* \\cong \\text{id}^*$." ], "refs": [ "descent-lemma-pullback" ], "ref_ids": [ 14733 ] } ], "ref_ids": [] }, { "id": 14740, "type": "theorem", "label": "descent-lemma-morphism-with-section-equivalence", "categories": [ "descent" ], "title": "descent-lemma-morphism-with-section-equivalence", "contents": [ "Let $f : X' \\to X$ be a morphism of schemes over a base scheme $S$.", "Assume there exists a morphism $g : X \\to X'$ over $S$, for example", "if $f$ has a section. Then the pullback functor", "of Lemma \\ref{lemma-pullback} defines an equivalence of", "categories between the category of descent data relative to", "$X/S$ and $X'/S$." ], "refs": [ "descent-lemma-pullback" ], "proofs": [ { "contents": [ "Let $g : X \\to X'$ be a morphism over $S$.", "Lemma \\ref{lemma-pullback-selfmap} above shows that the functors", "$f^* \\circ g^* = (g \\circ f)^*$ and $g^* \\circ f^* = (f \\circ g)^*$", "are isomorphic", "to the respective identity functors as desired." ], "refs": [ "descent-lemma-pullback-selfmap" ], "ref_ids": [ 14739 ] } ], "ref_ids": [ 14733 ] }, { "id": 14741, "type": "theorem", "label": "descent-lemma-morphism-source-faithfully-flat", "categories": [ "descent" ], "title": "descent-lemma-morphism-source-faithfully-flat", "contents": [ "Let $f : X \\to X'$ be a morphism of schemes over a base scheme $S$.", "Assume $X \\to S$ is surjective and flat. Then the pullback functor", "of Lemma \\ref{lemma-pullback} is a faithful functor", "from the category of descent data relative to $X'/S$ to the", "category of descent data relative to $X/S$." ], "refs": [ "descent-lemma-pullback" ], "proofs": [ { "contents": [ "We may factor $X \\to X'$ as $X \\to X \\times_S X' \\to X'$.", "The first morphism has a section, hence induces an equivalence of", "categories of descent data by", "Lemma \\ref{lemma-morphism-with-section-equivalence}.", "The second morphism is surjective and flat, hence induces a", "faithful functor by Lemma \\ref{lemma-faithful}." ], "refs": [ "descent-lemma-faithful" ], "ref_ids": [ 14737 ] } ], "ref_ids": [ 14733 ] }, { "id": 14742, "type": "theorem", "label": "descent-lemma-morphism-source-fpqc-covering", "categories": [ "descent" ], "title": "descent-lemma-morphism-source-fpqc-covering", "contents": [ "Let $f : X \\to X'$ be a morphism of schemes over a base scheme $S$.", "Assume $\\{X \\to S\\}$ is an fpqc covering (for example if $f$ is", "surjective, flat and quasi-compact).", "Then the pullback functor of Lemma \\ref{lemma-pullback} is a", "fully faithful functor from the category of descent data relative", "to $X'/S$ to the category of descent data relative to $X/S$." ], "refs": [ "descent-lemma-pullback" ], "proofs": [ { "contents": [ "We may factor $X \\to X'$ as $X \\to X \\times_S X' \\to X'$.", "The first morphism has a section, hence induces an equivalence of", "categories of descent data by", "Lemma \\ref{lemma-morphism-with-section-equivalence}.", "The second morphism is an fpqc covering", "hence induces a fully faithful functor by Lemma \\ref{lemma-fully-faithful}." ], "refs": [ "descent-lemma-fully-faithful" ], "ref_ids": [ 14738 ] } ], "ref_ids": [ 14733 ] }, { "id": 14743, "type": "theorem", "label": "descent-lemma-fpqc-refinement-coverings-fully-faithful", "categories": [ "descent" ], "title": "descent-lemma-fpqc-refinement-coverings-fully-faithful", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{U} = \\{U_i \\to S\\}_{i \\in I}$, and", "$\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$,", "be families of morphisms with target $S$.", "Let $\\alpha : I \\to J$, $\\text{id} : S \\to S$ and", "$g_i : U_i \\to V_{\\alpha(i)}$ be a morphism of families", "of maps with fixed target, see", "Sites, Definition \\ref{sites-definition-morphism-coverings}.", "Assume that for each $j \\in J$ the family", "$\\{g_i : U_i \\to V_j\\}_{\\alpha(i) = j}$ is an fpqc", "covering of $V_j$. Then the pullback functor", "$$", "\\text{descent data relative to }", "\\mathcal{V}", "\\longrightarrow", "\\text{descent data relative to }", "\\mathcal{U}", "$$", "of Lemma \\ref{lemma-pullback-family} is fully faithful." ], "refs": [ "sites-definition-morphism-coverings", "descent-lemma-pullback-family" ], "proofs": [ { "contents": [ "Consider the morphism of schemes", "$$", "g :", "X = \\coprod\\nolimits_{i \\in I} U_i", "\\longrightarrow", "Y = \\coprod\\nolimits_{j \\in J} V_j", "$$", "over $S$ which on the $i$th component maps into the $\\alpha(i)$th component", "via the morphism $g_{\\alpha(i)}$. We claim that $\\{g : X \\to Y\\}$", "is an fpqc covering of schemes. Namely, by", "Topologies, Lemma \\ref{topologies-lemma-disjoint-union-is-fpqc-covering}", "for each $j$ the morphism $\\{\\coprod_{\\alpha(i) = j} U_i \\to V_j\\}$ is an", "fpqc covering. Thus for every affine open $V \\subset V_j$", "(which we may think of as an affine open of $Y$)", "we can find finitely many affine opens", "$W_1, \\ldots, W_n \\subset \\coprod_{\\alpha(i) = j} U_i$", "(which we may think of as affine opens of $X$)", "such that $V = \\bigcup_{i = 1, \\ldots, n} g(W_i)$.", "This provides enough affine opens of $Y$ which can be covered by finitely", "many affine opens of $X$ so that", "Topologies, Lemma \\ref{topologies-lemma-recognize-fpqc-covering} part (3)", "applies, and the claim follows. Let us write $DD(X/S)$,", "resp.\\ $DD(\\mathcal{U})$ for the category of descent data with respect", "to $X/S$, resp.\\ $\\mathcal{U}$, and similarly for $Y/S$ and $\\mathcal{V}$.", "Consider the diagram", "$$", "\\xymatrix{", "DD(Y/S) \\ar[r] & DD(X/S) \\\\", "DD(\\mathcal{V}) \\ar[u]^{\\text{Lemma }\\ref{lemma-family-is-one}} \\ar[r] &", "DD(\\mathcal{U}) \\ar[u]_{\\text{Lemma }\\ref{lemma-family-is-one}}", "}", "$$", "This diagram is commutative, see the proof of", "Lemma \\ref{lemma-pullback-family}.", "The vertical arrows are equivalences. Hence the lemma follows from", "Lemma \\ref{lemma-fully-faithful} which shows the top horizontal arrow", "of the diagram is fully faithful." ], "refs": [ "topologies-lemma-disjoint-union-is-fpqc-covering", "topologies-lemma-recognize-fpqc-covering", "descent-lemma-family-is-one", "descent-lemma-family-is-one", "descent-lemma-pullback-family", "descent-lemma-fully-faithful" ], "ref_ids": [ 12494, 12493, 14732, 14732, 14734, 14738 ] } ], "ref_ids": [ 8656, 14734 ] }, { "id": 14744, "type": "theorem", "label": "descent-lemma-Zariski-refinement-coverings-equivalence", "categories": [ "descent" ], "title": "descent-lemma-Zariski-refinement-coverings-equivalence", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{U} = \\{U_i \\to S\\}_{i \\in I}$, and", "$\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$,", "be families of morphisms with target $S$.", "Let $\\alpha : I \\to J$, $\\text{id} : S \\to S$ and", "$g_i : U_i \\to V_{\\alpha(i)}$ be a morphism of families", "of maps with fixed target, see", "Sites, Definition \\ref{sites-definition-morphism-coverings}.", "Assume that for each $j \\in J$ the family", "$\\{g_i : U_i \\to V_j\\}_{\\alpha(i) = j}$ is a Zariski covering (see", "Topologies, Definition \\ref{topologies-definition-zariski-covering})", "of $V_j$. Then the pullback functor", "$$", "\\text{descent data relative to }", "\\mathcal{V}", "\\longrightarrow", "\\text{descent data relative to }", "\\mathcal{U}", "$$", "of Lemma \\ref{lemma-pullback-family} is an equivalence of categories.", "In particular, the category of schemes over $S$", "is equivalent to the category", "of descent data relative to any Zariski covering of $S$." ], "refs": [ "sites-definition-morphism-coverings", "topologies-definition-zariski-covering", "descent-lemma-pullback-family" ], "proofs": [ { "contents": [ "The functor is faithful and fully faithful by", "Lemma \\ref{lemma-fpqc-refinement-coverings-fully-faithful}.", "Let us indicate how to prove that it is essentially surjective.", "Let $(X_i, \\varphi_{ii'})$ be a descent datum relative to $\\mathcal{U}$.", "Fix $j \\in J$ and set $I_j = \\{i \\in I \\mid \\alpha(i) = j\\}$.", "For $i, i' \\in I_j$ note that there is a canonical morphism", "$$", "c_{ii'} : U_i \\times_{g_i, V_j, g_{i'}} U_{i'} \\to U_i \\times_S U_{i'}.", "$$", "Hence we can pullback $\\varphi_{ii'}$ by this morphism", "and set $\\psi_{ii'} = c_{ii'}^*\\varphi_{ii'}$ for $i, i' \\in I_j$.", "In this way we obtain a descent datum $(X_i, \\psi_{ii'})$", "relative to the Zariski covering", "$\\{g_i : U_i \\to V_j\\}_{i \\in I_j}$.", "Note that $\\psi_{ii'}$ is an isomorphism from the open", "$X_{i, U_i \\times_{V_j} U_{i'}}$ of $X_i$ to the corresponding", "open of $X_{i'}$. It follows from", "Schemes, Section \\ref{schemes-section-glueing-schemes}", "that we may glue $(X_i, \\psi_{ii'})$ into a scheme", "$Y_j$ over $V_j$. Moreover, the morphisms $\\varphi_{ii'}$", "for $i \\in I_j$ and $i' \\in I_{j'}$ glue to a morphism", "$\\varphi_{jj'} : Y_j \\times_S V_{j'} \\to V_j \\times_S Y_{j'}$", "satisfying the cocycle condition (details omitted).", "Hence we obtain the desired descent datum", "$(Y_j, \\varphi_{jj'})$ relative to $\\mathcal{V}$." ], "refs": [ "descent-lemma-fpqc-refinement-coverings-fully-faithful" ], "ref_ids": [ 14743 ] } ], "ref_ids": [ 8656, 12521, 14734 ] }, { "id": 14745, "type": "theorem", "label": "descent-lemma-refine-coverings-fully-faithful", "categories": [ "descent" ], "title": "descent-lemma-refine-coverings-fully-faithful", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{U} = \\{U_i \\to S\\}_{i \\in I}$, and", "$\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$,", "be fpqc-coverings of $S$.", "If $\\mathcal{U}$ is a refinement of $\\mathcal{V}$,", "then the pullback functor", "$$", "\\text{descent data relative to }", "\\mathcal{V}", "\\longrightarrow", "\\text{descent data relative to }", "\\mathcal{U}", "$$", "is fully faithful.", "In particular, the category of schemes over $S$", "is identified with a full subcategory of the category", "of descent data relative to any fpqc-covering of $S$." ], "refs": [], "proofs": [ { "contents": [ "Consider the fpqc-covering", "$\\mathcal{W} = \\{U_i \\times_S V_j \\to S\\}_{(i, j) \\in I \\times J}$ of $S$.", "It is a refinement of both $\\mathcal{U}$ and $\\mathcal{V}$.", "Hence we have a $2$-commutative diagram of functors and categories", "$$", "\\xymatrix{", "DD(\\mathcal{V}) \\ar[rd] \\ar[rr] & & DD(\\mathcal{U}) \\ar[ld] \\\\", "& DD(\\mathcal{W}) &", "}", "$$", "Notation as in the proof of", "Lemma \\ref{lemma-fpqc-refinement-coverings-fully-faithful} and", "commutativity by Lemma \\ref{lemma-pullback-family} part (3).", "Hence clearly it suffices to prove the functors", "$DD(\\mathcal{V}) \\to DD(\\mathcal{W})$ and", "$DD(\\mathcal{U}) \\to DD(\\mathcal{W})$ are fully faithful.", "This follows from", "Lemma \\ref{lemma-fpqc-refinement-coverings-fully-faithful}", "as desired." ], "refs": [ "descent-lemma-fpqc-refinement-coverings-fully-faithful", "descent-lemma-pullback-family", "descent-lemma-fpqc-refinement-coverings-fully-faithful" ], "ref_ids": [ 14743, 14734, 14743 ] } ], "ref_ids": [] }, { "id": 14746, "type": "theorem", "label": "descent-lemma-effective-for-fpqc-is-local-upstairs", "categories": [ "descent" ], "title": "descent-lemma-effective-for-fpqc-is-local-upstairs", "contents": [ "Let $X \\to S$ be a surjective, quasi-compact, flat morphism of", "schemes. Let $(V, \\varphi)$ be a descent datum relative to $X/S$.", "Suppose that for all $v \\in V$ there exists an open subscheme", "$v \\in W \\subset V$ such that $\\varphi(W \\times_S X) \\subset X \\times_S W$", "and such that the descent datum $(W, \\varphi|_{W \\times_S X})$", "is effective. Then $(V, \\varphi)$ is effective." ], "refs": [], "proofs": [ { "contents": [ "Let $V = \\bigcup W_i$ be an open covering with", "$\\varphi(W_i \\times_S X) \\subset X \\times_S W_i$", "and such that the descent datum $(W_i, \\varphi|_{W_i \\times_S X})$", "is effective. Let $U_i \\to S$ be a scheme and let", "$\\alpha_i : (X \\times_S U_i, can) \\to (W_i, \\varphi|_{W_i \\times_S X})$", "be an isomorphism of descent data. For each pair of indices", "$(i, j)$ consider the open", "$\\alpha_i^{-1}(W_i \\cap W_j) \\subset X \\times_S U_i$.", "Because everything is compatible with descent data", "and since $\\{X \\to S\\}$ is an fpqc covering, we", "may apply Lemma \\ref{lemma-open-fpqc-covering}", "to find an open $V_{ij} \\subset V_j$ such that", "$\\alpha_i^{-1}(W_i \\cap W_j) = X \\times_S V_{ij}$.", "Now the identity morphism on $W_i \\cap W_j$ is", "compatible with descent data, hence comes from a", "unique morphism $\\varphi_{ij} : U_{ij} \\to U_{ji}$ over $S$", "(see Remark \\ref{remark-morphisms-of-schemes-satisfy-fpqc-descent}).", "Then $(U_i, U_{ij}, \\varphi_{ij})$ is a glueing", "data as in Schemes, Section \\ref{schemes-section-glueing-schemes}", "(proof omitted). Thus we may assume there is a scheme $U$ over $S$", "such that $U_i \\subset U$ is open, $U_{ij} = U_i \\cap U_j$ and", "$\\varphi_{ij} = \\text{id}_{U_i \\cap U_j}$, see", "Schemes, Lemma \\ref{schemes-lemma-glue}.", "Pulling back to $X$ we can use the $\\alpha_i$ to", "get the desired isomorphism $\\alpha : X \\times_S U \\to V$." ], "refs": [ "descent-lemma-open-fpqc-covering", "descent-remark-morphisms-of-schemes-satisfy-fpqc-descent", "schemes-lemma-glue" ], "ref_ids": [ 14637, 14802, 7686 ] } ], "ref_ids": [] }, { "id": 14747, "type": "theorem", "label": "descent-lemma-descending-types-morphisms", "categories": [ "descent" ], "title": "descent-lemma-descending-types-morphisms", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes over a base.", "Let $\\tau \\in \\{fpqc, fppf, \\etale, smooth, syntomic\\}$.", "Suppose that", "\\begin{enumerate}", "\\item $\\mathcal{P}$ is stable under any base change", "(see Schemes, Definition \\ref{schemes-definition-preserved-by-base-change}),", "\\item if $Y_j \\to V_j$, $j = 1, \\ldots, m$ have $\\mathcal{P}$,", "then so does $\\coprod Y_j \\to \\coprod V_j$, and", "\\item for any surjective morphism of affines", "$X \\to S$ which is flat, flat of finite presentation,", "\\'etale, smooth or syntomic depending on whether $\\tau$ is", "fpqc, fppf, \\'etale, smooth, or syntomic,", "any descent datum $(V, \\varphi)$ relative", "to $X$ over $S$ such that $\\mathcal{P}$ holds for", "$V \\to X$ is effective.", "\\end{enumerate}", "Then morphisms of type $\\mathcal{P}$ satisfy descent for $\\tau$-coverings." ], "refs": [ "schemes-definition-preserved-by-base-change" ], "proofs": [ { "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{U} = \\{\\varphi_i : U_i \\to S\\}_{i \\in I}$", "be a $\\tau$-covering of $S$.", "Let $(X_i, \\varphi_{ii'})$ be a descent datum relative to", "$\\mathcal{U}$ and assume that each morphism $X_i \\to U_i$ has property", "$\\mathcal{P}$. We have to show there exists a scheme $X \\to S$ such that", "$(X_i, \\varphi_{ii'}) \\cong (U_i \\times_S X, can)$.", "\\medskip\\noindent", "Before we start the proof proper we remark that for any", "family of morphisms $\\mathcal{V} : \\{V_j \\to S\\}$ and any", "morphism of families $\\mathcal{V} \\to \\mathcal{U}$, if we pullback", "the descent datum $(X_i, \\varphi_{ii'})$ to a descent datum", "$(Y_j, \\varphi_{jj'})$ over $\\mathcal{V}$, then each of the", "morphisms $Y_j \\to V_j$ has property $\\mathcal{P}$ also.", "This is true because of assumption (1) that $\\mathcal{P}$ is stable", "under any base change and the definition of pullback", "(see Definition \\ref{definition-pullback-functor-family}).", "We will use this without further mention.", "\\medskip\\noindent", "First, let us prove the lemma when $S$ is affine.", "By Topologies, Lemma", "\\ref{topologies-lemma-fpqc-affine},", "\\ref{topologies-lemma-fppf-affine},", "\\ref{topologies-lemma-etale-affine},", "\\ref{topologies-lemma-smooth-affine}, or", "\\ref{topologies-lemma-syntomic-affine}", "there exists a standard $\\tau$-covering", "$\\mathcal{V} : \\{V_j \\to S\\}_{j = 1, \\ldots, m}$", "which refines $\\mathcal{U}$. The pullback functor", "$DD(\\mathcal{U}) \\to DD(\\mathcal{V})$", "between categories of descent data is fully faithful", "by Lemma \\ref{lemma-refine-coverings-fully-faithful}.", "Hence it suffices to prove that the descent datum over", "the standard $\\tau$-covering $\\mathcal{V}$ is effective.", "By assumption (2) we see that $\\coprod Y_j \\to \\coprod V_j$", "has property $\\mathcal{P}$.", "By Lemma \\ref{lemma-family-is-one} this reduces us to the covering", "$\\{\\coprod_{j = 1, \\ldots, m} V_j \\to S\\}$ for which we have", "assumed the result in assumption (3) of the lemma.", "Hence the lemma holds when $S$ is affine.", "\\medskip\\noindent", "Assume $S$ is general. Let $V \\subset S$ be an affine open.", "By the properties of site the family", "$\\mathcal{U}_V = \\{V \\times_S U_i \\to V\\}_{i \\in I}$ is a", "$\\tau$-covering of $V$. Denote", "$(X_i, \\varphi_{ii'})_V$ the restriction (or pullback) of", "the given descent datum to $\\mathcal{U}_V$.", "Hence by what we just saw we obtain a scheme $X_V$ over $V$", "whose canonical descent datum with respect to", "$\\mathcal{U}_V$ is isomorphic to $(X_i, \\varphi_{ii'})_V$.", "Suppose that $V' \\subset V$ is an affine open of $V$.", "Then both $X_{V'}$ and $V' \\times_V X_V$ have canonical", "descent data isomorphic to $(X_i, \\varphi_{ii'})_{V'}$.", "Hence, by Lemma \\ref{lemma-refine-coverings-fully-faithful}", "again we obtain a canonical morphism", "$\\rho^V_{V'} : X_{V'} \\to X_V$ over $S$ which identifies", "$X_{V'}$ with the inverse image of $V'$ in $X_V$.", "We omit the verification that given affine opens", "$V'' \\subset V' \\subset V$ of $S$ we have", "$\\rho^V_{V''} = \\rho^V_{V'} \\circ \\rho^{V'}_{V''}$.", "\\medskip\\noindent", "By Constructions, Lemma \\ref{constructions-lemma-relative-glueing} the data", "$(X_V, \\rho^V_{V'})$ glue to a scheme $X \\to S$.", "Moreover, we are given isomorphisms $V \\times_S X \\to X_V$", "which recover the maps $\\rho^V_{V'}$. Unwinding the construction", "of the schemes $X_V$ we obtain isomorphisms", "$$", "V \\times_S U_i \\times_S X", "\\longrightarrow", "V \\times_S X_i", "$$", "compatible with the maps $\\varphi_{ii'}$ and compatible with", "restricting to smaller affine opens in $X$. This implies that", "the canonical descent datum on $U_i \\times_S X$ is isomorphic", "to the given descent datum and we win." ], "refs": [ "descent-definition-pullback-functor-family", "topologies-lemma-fpqc-affine", "topologies-lemma-fppf-affine", "topologies-lemma-etale-affine", "topologies-lemma-smooth-affine", "topologies-lemma-syntomic-affine", "descent-lemma-refine-coverings-fully-faithful", "descent-lemma-family-is-one", "descent-lemma-refine-coverings-fully-faithful", "constructions-lemma-relative-glueing" ], "ref_ids": [ 14779, 12499, 12473, 12447, 12461, 12467, 14745, 14732, 14745, 12581 ] } ], "ref_ids": [ 7751 ] }, { "id": 14748, "type": "theorem", "label": "descent-lemma-affine", "categories": [ "descent" ], "title": "descent-lemma-affine", "contents": [ "Let $S$ be a scheme.", "Let $\\{X_i \\to S\\}_{i\\in I}$ be an fpqc covering, see", "Topologies, Definition \\ref{topologies-definition-fpqc-covering}.", "Let $(V_i/X_i, \\varphi_{ij})$ be a descent datum", "relative to $\\{X_i \\to S\\}$. If each morphism", "$V_i \\to X_i$ is affine, then the descent datum is", "effective." ], "refs": [ "topologies-definition-fpqc-covering" ], "proofs": [ { "contents": [ "Being affine is a property of morphisms of schemes", "which is local on the base and preserved under any base change, see", "Morphisms, Lemmas \\ref{morphisms-lemma-characterize-affine} and", "\\ref{morphisms-lemma-base-change-affine}.", "Hence Lemma \\ref{lemma-descending-types-morphisms} applies", "and it suffices to prove the statement of the lemma", "in case the fpqc-covering is given by a single", "$\\{X \\to S\\}$ flat surjective morphism of affines.", "Say $X = \\Spec(A)$ and $S = \\Spec(R)$ so", "that $R \\to A$ is a faithfully flat ring map.", "Let $(V, \\varphi)$ be a descent datum relative to $X$ over $S$", "and assume that $V \\to X$ is affine.", "Then $V \\to X$ being affine implies that $V = \\Spec(B)$", "for some $A$-algebra $B$ (see", "Morphisms, Definition \\ref{morphisms-definition-affine}).", "The isomorphism $\\varphi$ corresponds to an isomorphism", "of rings", "$$", "\\varphi^\\sharp :", "B \\otimes_R A \\longleftarrow A \\otimes_R B", "$$", "as $A \\otimes_R A$-algebras. The cocycle condition on $\\varphi$", "says that", "$$", "\\xymatrix{", "B \\otimes_R A \\otimes_R A & &", "A \\otimes_R A \\otimes_R B \\ar[ll] \\ar[ld]\\\\", "& A \\otimes_R B \\otimes_R A \\ar[lu] &", "}", "$$", "is commutative. Inverting these arrows we see that we have a", "descent datum for modules with respect to $R \\to A$ as in", "Definition \\ref{definition-descent-datum-modules}.", "Hence we may apply Proposition \\ref{proposition-descent-module}", "to obtain an $R$-module", "$C = \\Ker(B \\to A \\otimes_R B)$", "and an isomorphism $A \\otimes_R C \\cong B$", "respecting descent data. Given any pair $c, c' \\in C$", "the product $cc'$ in $B$ lies in $C$ since the", "map $\\varphi$ is an algebra homomorphism. Hence", "$C$ is an $R$-algebra whose base change to $A$ is", "isomorphic to $B$ compatibly with descent data.", "Applying $\\Spec$ we obtain a scheme", "$U$ over $S$ such that $(V, \\varphi) \\cong (X \\times_S U, can)$", "as desired." ], "refs": [ "morphisms-lemma-characterize-affine", "morphisms-lemma-base-change-affine", "descent-lemma-descending-types-morphisms", "morphisms-definition-affine", "descent-definition-descent-datum-modules", "descent-proposition-descent-module" ], "ref_ids": [ 5172, 5176, 14747, 5544, 14759, 14752 ] } ], "ref_ids": [ 12547 ] }, { "id": 14749, "type": "theorem", "label": "descent-lemma-closed-immersion", "categories": [ "descent" ], "title": "descent-lemma-closed-immersion", "contents": [ "Let $S$ be a scheme.", "Let $\\{X_i \\to S\\}_{i\\in I}$ be an fpqc covering, see", "Topologies, Definition \\ref{topologies-definition-fpqc-covering}.", "Let $(V_i/X_i, \\varphi_{ij})$ be a descent datum", "relative to $\\{X_i \\to S\\}$. If each morphism", "$V_i \\to X_i$ is a closed immersion, then the descent datum is", "effective." ], "refs": [ "topologies-definition-fpqc-covering" ], "proofs": [ { "contents": [ "This is true because a closed immersion is an affine morphism", "(Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-affine}),", "and hence Lemma \\ref{lemma-affine} applies." ], "refs": [ "morphisms-lemma-closed-immersion-affine", "descent-lemma-affine" ], "ref_ids": [ 5177, 14748 ] } ], "ref_ids": [ 12547 ] }, { "id": 14750, "type": "theorem", "label": "descent-lemma-quasi-affine", "categories": [ "descent" ], "title": "descent-lemma-quasi-affine", "contents": [ "Let $S$ be a scheme.", "Let $\\{X_i \\to S\\}_{i\\in I}$ be an fpqc covering, see", "Topologies, Definition \\ref{topologies-definition-fpqc-covering}.", "Let $(V_i/X_i, \\varphi_{ij})$ be a descent datum", "relative to $\\{X_i \\to S\\}$. If each morphism", "$V_i \\to X_i$ is quasi-affine, then the descent datum is", "effective." ], "refs": [ "topologies-definition-fpqc-covering" ], "proofs": [ { "contents": [ "Being quasi-affine is a property of morphisms of schemes", "which is preserved under any base change, see", "Morphisms, Lemmas \\ref{morphisms-lemma-characterize-quasi-affine} and", "\\ref{morphisms-lemma-base-change-quasi-affine}.", "Hence Lemma \\ref{lemma-descending-types-morphisms} applies", "and it suffices to prove the statement of the lemma", "in case the fpqc-covering is given by a single", "$\\{X \\to S\\}$ flat surjective morphism of affines.", "Say $X = \\Spec(A)$ and $S = \\Spec(R)$ so", "that $R \\to A$ is a faithfully flat ring map.", "Let $(V, \\varphi)$ be a descent datum relative to $X$ over $S$", "and assume that $\\pi : V \\to X$ is quasi-affine.", "\\medskip\\noindent", "According to Morphisms, Lemma \\ref{morphisms-lemma-characterize-quasi-affine}", "this means that", "$$", "V \\longrightarrow \\underline{\\Spec}_X(\\pi_*\\mathcal{O}_V) = W", "$$", "is a quasi-compact open immersion of schemes over $X$.", "The projections $\\text{pr}_i : X \\times_S X \\to X$ are flat", "and hence we have", "$$", "\\text{pr}_0^*\\pi_*\\mathcal{O}_V =", "(\\pi \\times \\text{id}_X)_*\\mathcal{O}_{V \\times_S X}, \\quad", "\\text{pr}_1^*\\pi_*\\mathcal{O}_V =", "(\\text{id}_X \\times \\pi)_*\\mathcal{O}_{X \\times_S V}", "$$", "by flat base change", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}).", "Thus the isomorphism $\\varphi : V \\times_S X \\to X \\times_S V$ (which", "is an isomorphism over $X \\times_S X$) induces an isomorphism", "of quasi-coherent sheaves of algebras", "$$", "\\varphi^\\sharp :", "\\text{pr}_0^*\\pi_*\\mathcal{O}_V", "\\longrightarrow", "\\text{pr}_1^*\\pi_*\\mathcal{O}_V", "$$", "on $X \\times_S X$.", "The cocycle condition for $\\varphi$ implies the cocycle condition", "for $\\varphi^\\sharp$. Another way to say this is that it produces", "a descent datum $\\varphi'$ on the affine scheme $W$ relative to", "$X$ over $S$, which moreover has the property that the morphism", "$V \\to W$ is a morphism of descent data.", "Hence by Lemma \\ref{lemma-affine}", "(or by effectivity of descent for quasi-coherent", "algebras) we obtain a scheme $U' \\to S$ with an isomorphism", "$(W, \\varphi') \\cong (X \\times_S U', can)$ of descent data.", "We note in passing that $U'$ is affine by", "Lemma \\ref{lemma-descending-property-affine}.", "\\medskip\\noindent", "And now we can think of $V$ as a (quasi-compact)", "open $V \\subset X \\times_S U'$ with the property that", "it is stable under the descent datum", "$$", "can : X \\times_S U' \\times_S X \\to X \\times_S X \\times_S U',", "(x_0, u', x_1) \\mapsto (x_0, x_1, u').", "$$", "In other words $(x_0, u') \\in V \\Rightarrow (x_1, u') \\in V$", "for any $x_0, x_1, u'$ mapping to the same point of $S$.", "Because $X \\to S$ is surjective we immediately find that", "$V$ is the inverse image of a subset $U \\subset U'$ under", "the morphism $X \\times_S U' \\to U'$.", "Because $X \\to S$ is quasi-compact, flat and surjective", "also $X \\times_S U' \\to U'$ is quasi-compact flat and surjective.", "Hence by Morphisms, Lemma \\ref{morphisms-lemma-fpqc-quotient-topology}", "this subset $U \\subset U'$ is open and we win." ], "refs": [ "morphisms-lemma-characterize-quasi-affine", "morphisms-lemma-base-change-quasi-affine", "descent-lemma-descending-types-morphisms", "morphisms-lemma-characterize-quasi-affine", "coherent-lemma-flat-base-change-cohomology", "descent-lemma-affine", "descent-lemma-descending-property-affine", "morphisms-lemma-fpqc-quotient-topology" ], "ref_ids": [ 5185, 5187, 14747, 5185, 3298, 14748, 14683, 5269 ] } ], "ref_ids": [ 12547 ] }, { "id": 14751, "type": "theorem", "label": "descent-lemma-descent-data-sheaves", "categories": [ "descent" ], "title": "descent-lemma-descent-data-sheaves", "contents": [ "Let $\\tau \\in \\{Zariski, fppf, \\etale, smooth, syntomic\\}$\\footnote{The", "fact that fpqc is missing is not a typo. See discussion", "in Topologies, Section \\ref{topologies-section-fpqc}.}.", "Let $\\Sch_\\tau$ be a big $\\tau$-site.", "Let $S \\in \\Ob(\\Sch_\\tau)$.", "Let $\\{S_i \\to S\\}_{i \\in I}$ be a covering in the", "site $(\\Sch/S)_\\tau$. There is an equivalence of", "categories", "$$", "\\left\\{", "\\begin{matrix}", "\\text{descent data }(X_i, \\varphi_{ii'})\\text{ such that}\\\\", "\\text{each }X_i \\in \\Ob((\\Sch/S)_\\tau)", "\\end{matrix}", "\\right\\}", "\\leftrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{sheaves }F\\text{ on }(\\Sch/S)_\\tau\\text{ such that}\\\\", "\\text{each }h_{S_i} \\times F\\text{ is representable}", "\\end{matrix}", "\\right\\}.", "$$", "Moreover,", "\\begin{enumerate}", "\\item the objects representing $h_{S_i} \\times F$ on the right hand side", "correspond to the schemes $X_i$ on the left hand side, and", "\\item the sheaf $F$ is representable if and only if the", "corresponding descent datum $(X_i, \\varphi_{ii'})$ is effective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have seen in Section \\ref{section-fpqc-universal-effective-epimorphisms}", "that representable presheaves are sheaves on the site $(\\Sch/S)_\\tau$.", "Moreover, the Yoneda lemma (Categories, Lemma \\ref{categories-lemma-yoneda})", "guarantees that maps between representable", "sheaves correspond one to one with maps between the representing objects.", "We will use these remarks without further mention during the proof.", "\\medskip\\noindent", "Let us construct the functor from right to left.", "Let $F$ be a sheaf on $(\\Sch/S)_\\tau$ such that each", "$h_{S_i} \\times F$ is representable. In this case let $X_i$", "be a representing object in $(\\Sch/S)_\\tau$.", "It comes equipped with a morphism $X_i \\to S_i$.", "Then both $X_i \\times_S S_{i'}$ and $S_i \\times_S X_{i'}$", "represent the sheaf $h_{S_i} \\times F \\times h_{S_{i'}}$", "and hence we obtain an isomorphism", "$$", "\\varphi_{ii'} : X_i \\times_S S_{i'} \\to S_i \\times_S X_{i'}", "$$", "It is straightforward to see that the maps $\\varphi_{ii'}$", "are morphisms over $S_i \\times_S S_{i'}$ and satisfy the", "cocycle condition. The functor from right to left is given", "by this construction $F \\mapsto (X_i, \\varphi_{ii'})$.", "\\medskip\\noindent", "Let us construct a functor from left to right.", "For each $i$ denote $F_i$ the sheaf $h_{X_i}$.", "The isomorphisms $\\varphi_{ii'}$ give isomorphisms", "$$", "\\varphi_{ii'} :", "F_i \\times h_{S_{i'}}", "\\longrightarrow", "h_{S_i} \\times F_{i'}", "$$", "over $h_{S_i} \\times h_{S_{i'}}$.", "Set $F$ equal to the coequalizer in the following diagram", "$$", "\\xymatrix{", "\\coprod_{i, i'} F_i \\times h_{S_{i'}}", "\\ar@<1ex>[rr]^-{\\text{pr}_0}", "\\ar@<-1ex>[rr]_-{\\text{pr}_1 \\circ \\varphi_{ii'}}", "& &", "\\coprod_i F_i \\ar[r]", "&", "F", "}", "$$", "The cocycle condition guarantees that $h_{S_i} \\times F$ is", "isomorphic to $F_i$ and hence representable.", "The functor from left to right is given", "by this construction $(X_i, \\varphi_{ii'}) \\mapsto F$.", "\\medskip\\noindent", "We omit the verification that these constructions", "are mutually quasi-inverse functors. The final statements", "(1) and (2) follow from the constructions." ], "refs": [ "categories-lemma-yoneda" ], "ref_ids": [ 12203 ] } ], "ref_ids": [] }, { "id": 14752, "type": "theorem", "label": "descent-proposition-descent-module", "categories": [ "descent" ], "title": "descent-proposition-descent-module", "contents": [ "\\begin{slogan}", "Effective descent for modules along faithfully flat ring maps.", "\\end{slogan}", "Let $R \\to A$ be a faithfully flat ring map.", "Then", "\\begin{enumerate}", "\\item any descent datum on modules with respect to $R \\to A$", "is effective,", "\\item the functor $M \\mapsto (A \\otimes_R M, can)$ from $R$-modules", "to the category of descent data is an equivalence, and", "\\item the inverse functor is given by $(N, \\varphi) \\mapsto H^0(s(N_\\bullet))$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We only prove (1) and omit the proofs of (2) and (3).", "As $R \\to A$ is faithfully flat, there exists a faithfully flat", "base change $R \\to R'$ such that $R' \\to A' = R' \\otimes_R A$ has", "a section (namely take $R' = A$ as in the proof of", "Lemma \\ref{lemma-ff-exact}). Hence, using", "Lemma \\ref{lemma-descent-descends}", "we may assume that $R \\to A$ has a section, say $\\sigma : A \\to R$.", "Let $(N, \\varphi)$ be a descent datum relative to $R \\to A$.", "Set", "$$", "M = H^0(s(N_\\bullet)) = \\{n \\in N \\mid 1 \\otimes n = \\varphi(n \\otimes 1)\\}", "\\subset", "N", "$$", "By Lemma \\ref{lemma-recognize-effective} it suffices to show that", "$A \\otimes_R M \\to N$ is an isomorphism.", "\\medskip\\noindent", "Take an element $n \\in N$. Write", "$\\varphi(n \\otimes 1) = \\sum a_i \\otimes x_i$ for certain", "$a_i \\in A$ and $x_i \\in N$. By Lemma \\ref{lemma-descent-datum-cosimplicial}", "we have $n = \\sum a_i x_i$ in $N$ (because", "$\\sigma^0_0 \\circ \\delta^1_1 = \\text{id}$ in any cosimplicial object).", "Next, write $\\varphi(x_i \\otimes 1) = \\sum a_{ij} \\otimes y_j$ for", "certain $a_{ij} \\in A$ and $y_j \\in N$.", "The cocycle condition means that", "$$", "\\sum a_i \\otimes a_{ij} \\otimes y_j = \\sum a_i \\otimes 1 \\otimes x_i", "$$", "in $A \\otimes_R A \\otimes_R N$. We conclude two things from this.", "First, by applying $\\sigma$ to the first $A$ we conclude that", "$\\sum \\sigma(a_i) \\varphi(x_i \\otimes 1) = \\sum \\sigma(a_i) \\otimes x_i$", "which means that $\\sum \\sigma(a_i) x_i \\in M$. Next, by applying", "$\\sigma$ to the middle $A$ and multiplying out we conclude that", "$\\sum_i a_i (\\sum_j \\sigma(a_{ij}) y_j) = \\sum a_i x_i = n$. Hence", "by the first conclusion we see that $A \\otimes_R M \\to N$ is", "surjective. Finally, suppose that $m_i \\in M$ and", "$\\sum a_i m_i = 0$. Then we see by applying $\\varphi$ to", "$\\sum a_im_i \\otimes 1$ that $\\sum a_i \\otimes m_i = 0$.", "In other words $A \\otimes_R M \\to N$ is injective and we win." ], "refs": [ "descent-lemma-ff-exact", "descent-lemma-descent-descends", "descent-lemma-recognize-effective", "descent-lemma-descent-datum-cosimplicial" ], "ref_ids": [ 14598, 14600, 14599, 14595 ] } ], "ref_ids": [] }, { "id": 14753, "type": "theorem", "label": "descent-proposition-fpqc-descent-quasi-coherent", "categories": [ "descent" ], "title": "descent-proposition-fpqc-descent-quasi-coherent", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{U} = \\{\\varphi_i : U_i \\to S\\}$ be an fpqc covering, see", "Topologies, Definition \\ref{topologies-definition-fpqc-covering}.", "Any descent datum on quasi-coherent sheaves", "for $\\mathcal{U} = \\{U_i \\to S\\}$ is effective.", "Moreover, the functor from the category of", "quasi-coherent $\\mathcal{O}_S$-modules to the category", "of descent data with respect to $\\mathcal{U}$ is fully faithful." ], "refs": [ "topologies-definition-fpqc-covering" ], "proofs": [ { "contents": [ "Let $S = \\bigcup_{j \\in J} V_j$ be an affine open covering.", "For $j, j' \\in J$ we denote $V_{jj'} = V_j \\cap V_{j'}$ the intersection", "(which need not be affine). For $V \\subset S$ open we denote", "$\\mathcal{U}_V = \\{V \\times_S U_i \\to V\\}_{i \\in I}$ which is a", "fpqc-covering (Topologies, Lemma \\ref{topologies-lemma-fpqc}).", "By definition of an fpqc covering, we can find for each $j \\in J$ a", "finite set $K_j$, a map $\\underline{i} : K_j \\to I$,", "affine opens $U_{\\underline{i}(k), k} \\subset U_{\\underline{i}(k)}$,", "$k \\in K_j$ such that", "$\\mathcal{V}_j = \\{U_{\\underline{i}(k), k} \\to V_j\\}_{k \\in K_j}$ is", "a standard fpqc covering of $V_j$. And of course, $\\mathcal{V}_j$", "is a refinement of $\\mathcal{U}_{V_j}$. Picture", "$$", "\\xymatrix{", "\\mathcal{V}_j \\ar[r] \\ar@{~>}[d] &", "\\mathcal{U}_{V_j} \\ar[r] \\ar@{~>}[d] &", "\\mathcal{U} \\ar@{~>}[d] \\\\", "V_j \\ar@{=}[r] & V_j \\ar[r] & S", "}", "$$", "where the top horizontal arrows are morphisms of families of", "morphisms with fixed target (see", "Sites, Definition \\ref{sites-definition-morphism-coverings}).", "\\medskip\\noindent", "To prove the proposition you show successively the", "faithfulness, fullness, and essential surjectivity of the", "functor from quasi-coherent sheaves to descent data.", "\\medskip\\noindent", "Faithfulness. Let $\\mathcal{F}$, $\\mathcal{G}$ be quasi-coherent", "sheaves on $S$ and let $a, b : \\mathcal{F} \\to \\mathcal{G}$ be", "homomorphisms of $\\mathcal{O}_S$-modules.", "Suppose $\\varphi_i^*(a) = \\varphi^*(b)$ for all $i$.", "Pick $s \\in S$. Then $s = \\varphi_i(u)$ for some $i \\in I$ and", "$u \\in U_i$. Since $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{U_i, u}$", "is flat, hence faithfully flat", "(Algebra, Lemma \\ref{algebra-lemma-local-flat-ff}) we see", "that $a_s = b_s : \\mathcal{F}_s \\to \\mathcal{G}_s$. Hence $a = b$.", "\\medskip\\noindent", "Fully faithfulness. Let $\\mathcal{F}$, $\\mathcal{G}$ be quasi-coherent", "sheaves on $S$ and let", "$a_i : \\varphi_i^*\\mathcal{F} \\to \\varphi_i^*\\mathcal{G}$ be", "homomorphisms of $\\mathcal{O}_{U_i}$-modules such that", "$\\text{pr}_0^*a_i = \\text{pr}_1^*a_j$ on $U_i \\times_U U_j$.", "We can pull back these morphisms to get morphisms", "$$", "a_k :", "\\mathcal{F}|_{U_{\\underline{i}(k), k}}", "\\longrightarrow", "\\mathcal{G}|_{U_{\\underline{i}(k), k}}", "$$", "$k \\in K_j$ with notation as above. Moreover,", "Lemma \\ref{lemma-refine-descent-datum} assures us", "that these define a morphism between (canonical) descent data on", "$\\mathcal{V}_j$. Hence, by", "Lemma \\ref{lemma-standard-fpqc-covering}, we get correspondingly", "unique morphisms $a_j : \\mathcal{F}|_{V_j} \\to \\mathcal{G}|_{V_j}$.", "To see that $a_j|_{V_{jj'}} = a_{j'}|_{V_{jj'}}$ we use that", "both $a_j$ and $a_{j'}$ agree with the pullback of the morphism", "$(a_i)_{i \\in I}$ of (canonical) descent data to any covering", "refining both $\\mathcal{V}_{j, V_{jj'}}$ and", "$\\mathcal{V}_{j', V_{jj'}}$, and using the faithfulness already", "shown. For example the covering", "$\\mathcal{V}_{jj'} =", "\\{V_k \\times_S V_{k'} \\to V_{jj'}\\}_{k \\in K_j, k' \\in K_{j'}}$", "will do.", "\\medskip\\noindent", "Essential surjectivity. Let $\\xi = (\\mathcal{F}_i, \\varphi_{ii'})$", "be a descent datum for quasi-coherent sheaves relative to the covering", "$\\mathcal{U}$. Pull back this descent datum to get descent data", "$\\xi_j$ for quasi-coherent sheaves relative to the coverings", "$\\mathcal{V}_j$ of $V_j$. By Lemma \\ref{lemma-standard-fpqc-covering}", "once again there exist", "quasi-coherent sheaves $\\mathcal{F}_j$ on $V_j$ whose associated", "canonical descent datum is isomorphic to $\\xi_j$. By fully faithfulness", "(proved above) we see there are isomorphisms", "$$", "\\phi_{jj'} :", "\\mathcal{F}_j|_{V_{jj'}}", "\\longrightarrow", "\\mathcal{F}_{j'}|_{V_{jj'}}", "$$", "corresponding to the isomorphism of descent data between the pullback", "of $\\xi_j$ and $\\xi_{j'}$ to $\\mathcal{V}_{jj'}$. To see that these", "maps $\\phi_{jj'}$ satisfy the cocycle condition we use faithfulness", "(proved above) over the triple intersections $V_{jj'j''}$. Hence, by", "Lemma \\ref{lemma-zariski-descent-effective}", "we see that the sheaves $\\mathcal{F}_j$", "glue to a quasi-coherent sheaf $\\mathcal{F}$ as desired.", "We still have to verify that the canonical descent datum relative to", "$\\mathcal{U}$ associated to $\\mathcal{F}$ is isomorphic to the descent", "datum we started out with. This verification is omitted." ], "refs": [ "topologies-lemma-fpqc", "sites-definition-morphism-coverings", "algebra-lemma-local-flat-ff", "descent-lemma-refine-descent-datum", "descent-lemma-standard-fpqc-covering", "descent-lemma-standard-fpqc-covering", "descent-lemma-zariski-descent-effective" ], "ref_ids": [ 12498, 8656, 537, 14593, 14609, 14609, 14594 ] } ], "ref_ids": [ 12547 ] }, { "id": 14754, "type": "theorem", "label": "descent-proposition-same-cohomology-quasi-coherent", "categories": [ "descent" ], "title": "descent-proposition-same-cohomology-quasi-coherent", "contents": [ "\\begin{slogan}", "Cohomology of quasi-coherent sheaves is the same no matter which", "topology you use.", "\\end{slogan}", "Let $S$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $S$.", "Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0]", "\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$.", "\\begin{enumerate}", "\\item There is a canonical isomorphism", "$$", "H^q(S, \\mathcal{F}) = H^q((\\Sch/S)_\\tau, \\mathcal{F}^a).", "$$", "\\item There are canonical isomorphisms", "$$", "H^q(S, \\mathcal{F}) =", "H^q(S_{Zar}, \\mathcal{F}^a) =", "H^q(S_\\etale, \\mathcal{F}^a).", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The result for $q = 0$ is clear from the definition of $\\mathcal{F}^a$.", "Let $\\mathcal{C} = (\\Sch/S)_\\tau$, or $\\mathcal{C} = S_\\etale$,", "or $\\mathcal{C} = S_{Zar}$.", "\\medskip\\noindent", "We are going to apply", "Cohomology on Sites,", "Lemma \\ref{sites-cohomology-lemma-cech-vanish-collection}", "with $\\mathcal{F} = \\mathcal{F}^a$,", "$\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ the set of affine schemes", "in $\\mathcal{C}$, and $\\text{Cov} \\subset \\text{Cov}_\\mathcal{C}$ the", "set of standard affine $\\tau$-coverings. Assumption (3) of", "the lemma is satisfied by", "Lemma \\ref{lemma-standard-covering-Cech-quasi-coherent}.", "Hence we conclude that $H^p(U, \\mathcal{F}^a) = 0$ for every", "affine object $U$ of $\\mathcal{C}$.", "\\medskip\\noindent", "Next, let $U \\in \\Ob(\\mathcal{C})$ be any separated object.", "Denote $f : U \\to S$ the structure morphism.", "Let $U = \\bigcup U_i$ be an affine open covering.", "We may also think of this as a $\\tau$-covering", "$\\mathcal{U} = \\{U_i \\to U\\}$ of $U$ in $\\mathcal{C}$.", "Note that", "$U_{i_0} \\times_U \\ldots \\times_U U_{i_p} =", "U_{i_0} \\cap \\ldots \\cap U_{i_p}$ is affine as we assumed $U$ separated.", "By", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-cech-spectral-sequence-application}", "and the result above we see that", "$$", "H^p(U, \\mathcal{F}^a) = \\check{H}^p(\\mathcal{U}, \\mathcal{F}^a)", "= H^p(U, f^*\\mathcal{F})", "$$", "the last equality by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cech-cohomology-quasi-coherent}.", "In particular, if $S$ is separated we can take $U = S$ and", "$f = \\text{id}_S$ and the proposition is proved.", "We suggest the reader skip the rest of the proof (or rewrite it", "to give a clearer exposition).", "\\medskip\\noindent", "Choose an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ on $S$.", "Choose an injective resolution $\\mathcal{F}^a \\to \\mathcal{J}^\\bullet$", "on $\\mathcal{C}$. Denote $\\mathcal{J}^n|_S$ the restriction of $\\mathcal{J}^n$", "to opens of $S$; this is a sheaf on the topological space $S$ as open", "coverings are $\\tau$-coverings. We get a complex", "$$", "0 \\to \\mathcal{F} \\to \\mathcal{J}^0|_S \\to \\mathcal{J}^1|_S \\to \\ldots", "$$", "which is exact since its sections over any affine open $U \\subset S$", "is exact (by the vanishing of $H^p(U, \\mathcal{F}^a)$, $p > 0$ seen", "above). Hence by", "Derived Categories, Lemma \\ref{derived-lemma-morphisms-lift}", "there exists map of complexes", "$\\mathcal{J}^\\bullet|_S \\to \\mathcal{I}^\\bullet$ which in particular", "induces a map", "$$", "R\\Gamma(\\mathcal{C}, \\mathcal{F}^a)", "=", "\\Gamma(S, \\mathcal{J}^\\bullet)", "\\longrightarrow", "\\Gamma(S, \\mathcal{I}^\\bullet)", "=", "R\\Gamma(S, \\mathcal{F}).", "$$", "Taking cohomology gives the map", "$H^n(\\mathcal{C}, \\mathcal{F}^a) \\to H^n(S, \\mathcal{F})$ which", "we have to prove is an isomorphism.", "Let $\\mathcal{U} : S = \\bigcup U_i$ be an affine open covering", "which we may think of as a $\\tau$-covering also.", "By the above we get a map of double complexes", "$$", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{J})", "=", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{J}|_S)", "\\longrightarrow", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}).", "$$", "This map induces a map of spectral sequences", "$$", "{}^\\tau\\! E_2^{p, q} = \\check{H}^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F}^a))", "\\longrightarrow", "E_2^{p, q} = \\check{H}^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F}))", "$$", "The first spectral sequence converges to", "$H^{p + q}(\\mathcal{C}, \\mathcal{F})$ and the second to", "$H^{p + q}(S, \\mathcal{F})$. On the other hand, we have seen", "that the induced maps ${}^\\tau\\! E_2^{p, q} \\to E_2^{p, q}$ are", "bijections (as all the intersections are separated being opens in affines).", "Whence also the maps $H^n(\\mathcal{C}, \\mathcal{F}^a) \\to H^n(S, \\mathcal{F})$", "are isomorphisms, and we win." ], "refs": [ "sites-cohomology-lemma-cech-vanish-collection", "descent-lemma-standard-covering-Cech-quasi-coherent", "sites-cohomology-lemma-cech-spectral-sequence-application", "coherent-lemma-cech-cohomology-quasi-coherent", "derived-lemma-morphisms-lift" ], "ref_ids": [ 4205, 14625, 4203, 3286, 1853 ] } ], "ref_ids": [] }, { "id": 14755, "type": "theorem", "label": "descent-proposition-equivalence-quasi-coherent", "categories": [ "descent" ], "title": "descent-proposition-equivalence-quasi-coherent", "contents": [ "Let $S$ be a scheme.", "Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0]", "\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$.", "\\begin{enumerate}", "\\item The functor $\\mathcal{F} \\mapsto \\mathcal{F}^a$", "defines an equivalence of categories", "$$", "\\QCoh(\\mathcal{O}_S)", "\\longrightarrow", "\\QCoh((\\Sch/S)_\\tau, \\mathcal{O})", "$$", "between the category of quasi-coherent sheaves on $S$ and the category", "of quasi-coherent $\\mathcal{O}$-modules on the big $\\tau$ site of $S$.", "\\item Let $\\tau = \\etale$, or $\\tau = Zariski$.", "The functor $\\mathcal{F} \\mapsto \\mathcal{F}^a$", "defines an equivalence of categories", "$$", "\\QCoh(\\mathcal{O}_S)", "\\longrightarrow", "\\QCoh(S_\\tau, \\mathcal{O})", "$$", "between the category of quasi-coherent sheaves on $S$ and the category", "of quasi-coherent $\\mathcal{O}$-modules on the small $\\tau$ site of $S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We have seen in Lemma \\ref{lemma-quasi-coherent-gives-quasi-coherent}", "that the functor is well defined.", "It is straightforward to show that the functor is fully faithful (we omit", "the verification). To finish the proof we will show that a", "quasi-coherent $\\mathcal{O}$-module on $(\\Sch/S)_\\tau$ gives", "rise to a descent datum for quasi-coherent sheaves relative to a", "$\\tau$-covering of $S$. Having produced this descent datum we will appeal", "to Proposition \\ref{proposition-fpqc-descent-quasi-coherent} to get the", "corresponding quasi-coherent sheaf on $S$.", "\\medskip\\noindent", "Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}$-modules on", "the big $\\tau$ site of $S$. By", "Modules on Sites, Definition \\ref{sites-modules-definition-site-local}", "there exists a $\\tau$-covering $\\{S_i \\to S\\}_{i \\in I}$ of $S$", "such that each of the restrictions", "$\\mathcal{G}|_{(\\Sch/S_i)_\\tau}$ has a global presentation", "$$", "\\bigoplus\\nolimits_{k \\in K_i} \\mathcal{O}|_{(\\Sch/S_i)_\\tau}", "\\longrightarrow", "\\bigoplus\\nolimits_{j \\in J_i} \\mathcal{O}|_{(\\Sch/S_i)_\\tau}", "\\longrightarrow", "\\mathcal{G}|_{(\\Sch/S_i)_\\tau} \\longrightarrow 0", "$$", "for some index sets $J_i$ and $K_i$. We claim that this implies", "that $\\mathcal{G}|_{(\\Sch/S_i)_\\tau}$ is $\\mathcal{F}_i^a$", "for some quasi-coherent sheaf $\\mathcal{F}_i$ on $S_i$. Namely,", "this is clear for the direct sums", "$\\bigoplus\\nolimits_{k \\in K_i} \\mathcal{O}|_{(\\Sch/S_i)_\\tau}$", "and", "$\\bigoplus\\nolimits_{j \\in J_i} \\mathcal{O}|_{(\\Sch/S_i)_\\tau}$.", "Hence we see that $\\mathcal{G}|_{(\\Sch/S_i)_\\tau}$ is a", "cokernel of a map $\\varphi : \\mathcal{K}_i^a \\to \\mathcal{L}_i^a$", "for some quasi-coherent sheaves $\\mathcal{K}_i$, $\\mathcal{L}_i$", "on $S_i$. By the fully faithfulness of $(\\ )^a$ we see that", "$\\varphi = \\phi^a$ for some map of quasi-coherent sheaves", "$\\phi : \\mathcal{K}_i \\to \\mathcal{L}_i$ on $S_i$. Then it is", "clear that", "$\\mathcal{G}|_{(\\Sch/S_i)_\\tau} \\cong \\Coker(\\phi)^a$", "as claimed.", "\\medskip\\noindent", "Since $\\mathcal{G}$ lives on all of the category", "$(\\Sch/S)_\\tau$ we see that", "$$", "(\\text{pr}_0^*\\mathcal{F}_i)^a", "\\cong", "\\mathcal{G}|_{(\\Sch/(S_i \\times_S S_j))_\\tau}", "\\cong", "(\\text{pr}_1^*\\mathcal{F})^a", "$$", "as $\\mathcal{O}$-modules on $(\\Sch/(S_i \\times_S S_j))_\\tau$.", "Hence, using fully faithfulness again we get canonical isomorphisms", "$$", "\\phi_{ij} :", "\\text{pr}_0^*\\mathcal{F}_i", "\\longrightarrow", "\\text{pr}_1^*\\mathcal{F}_j", "$$", "of quasi-coherent modules over $S_i \\times_S S_j$. We omit the verification", "that these satisfy the cocycle condition. Since they do we see by", "effectivity of descent for quasi-coherent sheaves and the covering", "$\\{S_i \\to S\\}$ (Proposition \\ref{proposition-fpqc-descent-quasi-coherent})", "that there exists a quasi-coherent sheaf $\\mathcal{F}$ on $S$", "with $\\mathcal{F}|_{S_i} \\cong \\mathcal{F}_i$ compatible", "with the given descent data. In other words we are given", "$\\mathcal{O}$-module isomorphisms", "$$", "\\phi_i :", "\\mathcal{F}^a|_{(\\Sch/S_i)_\\tau}", "\\longrightarrow", "\\mathcal{G}|_{(\\Sch/S_i)_\\tau}", "$$", "which agree over $S_i \\times_S S_j$. Hence, since", "$\\SheafHom_\\mathcal{O}(\\mathcal{F}^a, \\mathcal{G})$ is", "a sheaf (Modules on Sites, Lemma \\ref{sites-modules-lemma-internal-hom}),", "we conclude that", "there is a morphism of $\\mathcal{O}$-modules $\\mathcal{F}^a \\to \\mathcal{G}$", "recovering the isomorphisms $\\phi_i$ above. Hence this is an isomorphism", "and we win.", "\\medskip\\noindent", "The case of the sites $S_\\etale$ and $S_{Zar}$ is proved in the", "exact same manner." ], "refs": [ "descent-lemma-quasi-coherent-gives-quasi-coherent", "descent-proposition-fpqc-descent-quasi-coherent", "sites-modules-definition-site-local", "descent-proposition-fpqc-descent-quasi-coherent", "sites-modules-lemma-internal-hom" ], "ref_ids": [ 14623, 14753, 14289, 14753, 14191 ] } ], "ref_ids": [] }, { "id": 14756, "type": "theorem", "label": "descent-proposition-equivalence-quasi-coherent-functorial", "categories": [ "descent" ], "title": "descent-proposition-equivalence-quasi-coherent-functorial", "contents": [ "Let $f : T \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item The equivalences of categories of", "Proposition \\ref{proposition-equivalence-quasi-coherent}", "are compatible with pullback.", "More precisely, we have $f^*(\\mathcal{G}^a) = (f^*\\mathcal{G})^a$", "for any quasi-coherent sheaf $\\mathcal{G}$ on $S$.", "\\item The equivalences of categories of", "Proposition \\ref{proposition-equivalence-quasi-coherent} part (1)", "are {\\bf not} compatible with pushforward in general.", "\\item If $f$ is quasi-compact and quasi-separated, and", "$\\tau \\in \\{Zariski, \\etale\\}$ then $f_*$ and $f_{small, *}$", "preserve quasi-coherent sheaves and the diagram", "$$", "\\xymatrix{", "\\QCoh(\\mathcal{O}_T)", "\\ar[rr]_{f_*} \\ar[d]_{\\mathcal{F} \\mapsto \\mathcal{F}^a} & &", "\\QCoh(\\mathcal{O}_S)", "\\ar[d]^{\\mathcal{G} \\mapsto \\mathcal{G}^a} \\\\", "\\QCoh(T_\\tau, \\mathcal{O}) \\ar[rr]^{f_{small, *}} & &", "\\QCoh(S_\\tau, \\mathcal{O})", "}", "$$", "is commutative, i.e., $f_{small, *}(\\mathcal{F}^a) = (f_*\\mathcal{F})^a$.", "\\end{enumerate}" ], "refs": [ "descent-proposition-equivalence-quasi-coherent", "descent-proposition-equivalence-quasi-coherent" ], "proofs": [ { "contents": [ "Part (1) follows from the discussion in", "Remark \\ref{remark-change-topologies-ringed-sites}.", "Part (2) is just a warning, and can be explained in the following way:", "First the statement cannot be made precise since $f_*$ does not", "transform quasi-coherent sheaves into quasi-coherent sheaves in general.", "Even if this is the case for $f$ (and any base change of $f$), then the", "compatibility over the big sites would mean that formation of $f_*\\mathcal{F}$", "commutes with any base change, which does not hold in general.", "An explicit example is the quasi-compact open immersion", "$j : X = \\mathbf{A}^2_k \\setminus \\{0\\} \\to \\mathbf{A}^2_k = Y$", "where $k$ is a field. We have $j_*\\mathcal{O}_X = \\mathcal{O}_Y$", "but after base change to $\\Spec(k)$ by the $0$ map", "we see that the pushforward is zero.", "\\medskip\\noindent", "Let us prove (3) in case $\\tau = \\etale$. Note that $f$, and any", "base change of $f$, transforms quasi-coherent sheaves", "into quasi-coherent sheaves, see", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}.", "The equality $f_{small, *}(\\mathcal{F}^a) = (f_*\\mathcal{F})^a$", "means that for any \\'etale morphism $g : U \\to S$ we have", "$\\Gamma(U, g^*f_*\\mathcal{F}) = \\Gamma(U \\times_S T, (g')^*\\mathcal{F})$", "where $g' : U \\times_S T \\to T$ is the projection. This is true by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-flat-base-change-cohomology}." ], "refs": [ "descent-remark-change-topologies-ringed-sites", "schemes-lemma-push-forward-quasi-coherent", "coherent-lemma-flat-base-change-cohomology" ], "ref_ids": [ 14793, 7730, 3298 ] } ], "ref_ids": [ 14755, 14755 ] }, { "id": 14804, "type": "theorem", "label": "simplicial-theorem-dold-kan", "categories": [ "simplicial" ], "title": "simplicial-theorem-dold-kan", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "The functor $N$ induces an equivalence of", "categories", "$$", "N :", "\\text{Simp}(\\mathcal{A})", "\\longrightarrow", "\\text{Ch}_{\\geq 0}(\\mathcal{A})", "$$" ], "refs": [], "proofs": [ { "contents": [ "We will describe a functor in the reverse direction", "inspired by the construction of Lemma \\ref{lemma-extension}", "(except that we throw in a sign to get the boundaries", "right). Let $A_\\bullet$ be a chain complex with boundary maps", "$d_{A, n} : A_n \\to A_{n - 1}$. For each $n \\geq 0$ denote", "$$", "I_n =", "\\Big\\{", "\\alpha : [n] \\to \\{0, 1, 2, \\ldots\\}", "\\mid", "\\Im(\\alpha) = [k]\\text{ for some }k", "\\Big\\}.", "$$", "For $\\alpha \\in I_n$ we denote $k(\\alpha)$ the unique", "integer such that $\\Im(\\alpha) = [k]$.", "We define a simplicial object $S(A_\\bullet)$ as follows:", "\\begin{enumerate}", "\\item $S(A_\\bullet)_n = \\bigoplus_{\\alpha \\in I_n} A_{k(\\alpha)}$, which", "we will write as", "$\\bigoplus_{\\alpha \\in I_n} A_{k(\\alpha)} \\cdot \\alpha$", "to suggest thinking of ``$\\alpha$'' as a basis vector for the", "summand corresponding to it,", "\\item given $\\varphi : [m] \\to [n]$ we define", "$S(A_\\bullet)(\\varphi)$ by its restriction to", "the direct summand $A_{k(\\alpha)} \\cdot \\alpha$", "of $S(A_\\bullet)_n$ as follows", "\\begin{enumerate}", "\\item $\\alpha \\circ \\varphi \\not \\in I_m$ then we set it equal to zero,", "\\item $\\alpha \\circ \\varphi \\in I_m$ but $k(\\alpha \\circ \\varphi)$", "not equal to either $k(\\alpha)$ or $k(\\alpha) - 1$ then we set it", "equal to zero as well,", "\\item if $\\alpha \\circ \\varphi \\in I_m$", "and $k(\\alpha \\circ \\varphi) = k(\\alpha)$ then we use", "the identity map to the summand", "$A_{k(\\alpha \\circ \\varphi)} \\cdot (\\alpha \\circ \\varphi)$", "of $S(A_\\bullet)_m$, and", "\\item if $\\alpha \\circ \\varphi \\in I_m$", "and $k(\\alpha \\circ \\varphi) = k(\\alpha) - 1$", "then we use $(-1)^{k(\\alpha)} d_{A, k(\\alpha)}$ to the summand", "$A_{k(\\alpha \\circ \\varphi)}\\cdot (\\alpha \\circ \\varphi)$", "of $S(A_\\bullet)_m$.", "\\end{enumerate}", "\\end{enumerate}", "Let us show that $S(A_\\bullet)$ is a simplicial object of $\\mathcal{A}$.", "To do this, assume we have maps $\\varphi : [m] \\to [n]$ and", "$\\psi : [n] \\to [p]$. We will show that", "$S(A_\\bullet)(\\varphi) \\circ S(A_\\bullet)(\\psi) =", "S(A_\\bullet)(\\psi \\circ \\varphi)$. Choose $\\beta \\in I_p$ and set", "$\\alpha = \\beta \\circ \\psi$ and $\\gamma = \\alpha \\circ \\varphi$", "viewed as maps $\\alpha : [n] \\to \\{0, 1, 2, \\ldots\\}$", "and $\\gamma : [m] \\to \\{0, 1, 2, \\ldots\\}$. Picture", "$$", "\\xymatrix{", "[m] \\ar[r]_\\varphi \\ar[d]_\\gamma &", "[n] \\ar[r]_\\psi \\ar[d]_\\alpha &", "[p] \\ar[d]^\\beta \\\\", "\\Im(\\gamma) \\ar[r] &", "\\Im(\\alpha) \\ar[r] &", "[k(\\beta)]", "}", "$$", "We will show that the restriction of the maps", "$S(A_\\bullet)(\\varphi) \\circ S(A_\\bullet)(\\psi)$ and", "$S(A_\\bullet)(\\psi \\circ \\varphi)$.", "to the summand $A_{k(\\beta)} \\cdot \\beta$ agree.", "There are several cases to consider", "\\begin{enumerate}", "\\item Say $\\alpha \\not \\in I_n$ so the restriction of", "$S(A_\\bullet)(\\psi)$ to $A_{k(\\beta)} \\cdot \\beta$ is zero.", "Then either $\\gamma \\not \\in I_m$ or we have", "$[k(\\gamma)] = \\Im(\\gamma) \\subset \\Im(\\alpha) \\subset [k(\\beta)]$", "and the subset $\\Im(\\alpha)$ of $[k(\\beta)]$ has a gap so", "$k(\\gamma) < k(\\beta) - 1$. In both cases we see that", "the restriction of", "$S(A_\\bullet)(\\psi \\circ \\varphi)$ to $A_{k(\\beta)} \\cdot \\beta$", "is zero as well.", "\\item Say $\\alpha \\in I_n$ and $k(\\alpha) < k(\\beta) - 1$ so the restriction of", "$S(A_\\bullet)(\\psi)$ to $A_{k(\\beta)} \\cdot \\beta$ is zero.", "Then either $\\gamma \\not \\in I_m$ or we have", "$[k(\\gamma)] \\subset [k(\\alpha)] \\subset [k(\\beta)]$", "and it follows that $k(\\gamma) < k(\\beta) - 1$. In both cases we see that", "the restriction of", "$S(A_\\bullet)(\\psi \\circ \\varphi)$ to $A_{k(\\beta)} \\cdot \\beta$", "is zero as well.", "\\item Say $\\alpha \\in I_n$ and $k(\\alpha) = k(\\beta)$ so the restriction of", "$S(A_\\bullet)(\\psi)$ to $A_{k(\\beta)} \\cdot \\beta$ is", "the identity map from $A_{k(\\beta)} \\cdot \\beta$ to", "$A_{k(\\alpha)} \\cdot \\alpha$. In this case because $\\Im(\\alpha) =[k(\\beta)]$", "the rule describing the restriction of $S(A_\\bullet)(\\psi \\circ \\varphi)$", "to the summand $A_{k(\\beta)} \\cdot \\beta$ is exactly the same", "as the rule describing the restriction of $S(A_\\bullet)(\\varphi)$", "to the summand $A_{k(\\alpha)} \\cdot \\alpha$ and hence agreement holds.", "\\item Say $\\alpha \\in I_n$ and $k(\\alpha) = k(\\beta) - 1$", "so the restriction of $S(A_\\bullet)(\\psi)$ to $A_{k(\\beta)} \\cdot \\beta$", "is given by $(-1)^{k(\\beta)}d_{A, k(\\beta)}$ to $A_{k(\\alpha)} \\cdot \\alpha$.", "Subcases", "\\begin{enumerate}", "\\item If $\\gamma \\not \\in I_m$, then both the restriction of", "$S(A_\\bullet)(\\psi \\circ \\varphi)$ to the summand $A_{k(\\beta)} \\cdot \\beta$", "and the restriction of $S(A_\\bullet)(\\varphi)$ to the summand", "$A_{k(\\alpha)} \\cdot \\alpha$ are zero and we get agreement.", "\\item If $\\gamma \\in I_m$ but $k(\\gamma) < k(\\alpha) - 1$, then", "again both restrictions are zero and we get agreement.", "\\item If $\\gamma \\in I_m$ and $k(\\gamma) = k(\\alpha)$ then", "$\\Im(\\gamma) = \\Im(\\alpha)$. In this case", "the restriction of $S(A_\\bullet)(\\psi \\circ \\varphi)$", "to the summand $A_{k(\\beta)} \\cdot \\beta$ is given by", "$(-1)^{k(\\beta)}d_{A, k(\\beta)}$ to $A_{k(\\gamma)} \\cdot \\gamma$", "and the restriction of $S(A_\\bullet)(\\varphi)$ to the summand", "$A_{k(\\alpha)} \\cdot \\alpha$ is the identity map", "$A_{k(\\alpha)} \\cdot \\alpha \\to A_{k(\\gamma)} \\cdot \\gamma$.", "Hence agreement holds.", "\\item Finally, if $\\gamma \\in I_m$ and $k(\\gamma) = k(\\alpha) - 1$ then", "the restriction of $S(A_\\bullet)(\\varphi)$ to the summand", "$A_{k(\\alpha)} \\cdot \\alpha$ is given by $(-1)^{k(\\alpha)} d_{A, k(\\alpha)}$", "as a map", "$A_{k(\\alpha)} \\cdot \\alpha \\to A_{k(\\beta)} \\cdot \\beta$.", "Since $A_\\bullet$ is a complex we see that the composition", "$A_{k(\\beta)} \\cdot \\beta \\to", "A_{k(\\alpha)} \\cdot \\alpha \\to A_{k(\\gamma)} \\cdot \\gamma$ is zero", "which matches what we get for the restriction of", "$S(A_\\bullet)(\\psi \\circ \\varphi)$", "to the summand $A_{k(\\beta)} \\cdot \\beta$", "because $k(\\gamma) = k(\\beta) - 2 < k(\\beta) - 1$.", "\\end{enumerate}", "\\end{enumerate}", "Thus $S(A_\\bullet)$ is a simplicial object of $\\mathcal{A}$.", "\\medskip\\noindent", "Let us construct an isomorphism $A_\\bullet \\to N(S(A_\\bullet))$", "functorial in $A_\\bullet$. Recall that", "$$", "S(A_\\bullet) = N(S(A_\\bullet)) \\oplus D(S(A_\\bullet))", "$$", "as chain complexes by Lemma \\ref{lemma-decompose-associated-complexes}.", "On the other hand it follows from Remark \\ref{remark-degenerate-subcomplex}", "and the construction of $S(A_\\bullet)$ that", "$$", "D(S(A_\\bullet))_n =", "\\bigoplus\\nolimits_{\\alpha \\in I_n,\\ k(\\alpha) < n} A_{k(\\alpha)} \\cdot \\alpha", "\\subset", "\\bigoplus\\nolimits_{\\alpha \\in I_n} A_{k(\\alpha)} \\cdot \\alpha", "$$", "However, if $\\alpha \\in I_n$ then we have", "$k(\\alpha) \\geq n \\Leftrightarrow \\alpha = \\text{id}_{[n]} : [n] \\to [n]$.", "Thus the summand $A_n \\cdot \\text{id}_{[n]}$ of $S(A_\\bullet)_n$", "is a complement to the summand $D(S(A_\\bullet))_n$.", "All the maps $d^n_i : S(A_\\bullet)_n \\to S(A_\\bullet)_n$", "restrict to zero on the summand $A_n \\cdot \\text{id}_{[n]}$", "except for $d^n_n$ which produces $(-1)^n d_{A, n}$", "from $A_n \\cdot \\text{id}_{[n]}$ to $A_{n - 1} \\cdot \\text{id}_{[n - 1]}$.", "We conclude that $A_n \\cdot \\text{id}_{[n]}$ must be", "equal to the summand $N(S(A_\\bullet))_n$ and moreover", "the restriction of the differential", "$d_n = \\sum (-1)^id^n_i : S(A_\\bullet)_n \\to S(A_\\bullet)_{n - 1}$", "to the summand $A_n \\cdot \\text{id}_{[n]}$ gives what we want!", "\\medskip\\noindent", "Finally, we have to show that $S \\circ N$ is isomorphic to the", "identity functor. Let $U$ be a simplicial object of $\\mathcal{A}$.", "Then we can define an obvious map", "$$", "S(N(U))_n = \\bigoplus\\nolimits_{\\alpha \\in I_n} N(U)_{k(\\alpha)} \\cdot \\alpha", "\\longrightarrow", "U_n", "$$", "by using $U(\\alpha) : N(U)_{k(\\alpha)} \\to U_n$ on the summand corresponding", "to $\\alpha$. By Definition \\ref{definition-split} this is an isomorphism.", "To finish the proof we have to show that this is compatible with", "the maps in the simplicial objects. Thus let $\\varphi : [m] \\to [n]$", "and let $\\alpha \\in I_n$. Set $\\beta = \\alpha \\circ \\varphi$.", "Picture", "$$", "\\xymatrix{", "[m] \\ar[r]_\\varphi \\ar[d]_\\beta &", "[n] \\ar[d]_\\alpha \\\\", "\\Im(\\beta) \\ar[r] &", "[k(\\alpha)]", "}", "$$", "There are several cases to consider", "\\begin{enumerate}", "\\item Say $\\beta \\not \\in I_m$. Then there exists an index", "$0 \\leq j < k(\\alpha)$ with $j \\not \\in \\Im(\\alpha \\circ \\varphi)$", "and hence we can choose a factorization", "$\\alpha \\circ \\varphi = \\delta^{k(\\alpha)}_j \\circ \\psi$", "for some $\\psi : [m] \\to [k(\\alpha) - 1]$.", "It follows that $U(\\varphi)$ is zero on the image of the summand", "$N(U)_{k(\\alpha)} \\cdot \\alpha$ because $U(\\varphi) \\circ U(\\alpha) =", "U(\\alpha \\circ \\varphi) = U(\\psi) \\circ d^{k(\\alpha)}_j$", "is zero on $N(U)_{k(\\alpha)}$ by construction of $N$.", "This matches our rule for $S(N(U))$ given above.", "\\item Say $\\beta \\in I_m$ and $k(\\beta) < k(\\alpha) - 1$. Here we", "argue exactly as in case (1) with $j = k(\\alpha) - 1$.", "\\item Say $\\beta \\in I_m$ and $k(\\beta) = k(\\alpha)$. Here", "the summand $N(U)_{k(\\alpha)} \\cdot \\alpha$ is mapped by the", "identity to the summand $N(U)_{k(\\beta)} \\cdot \\beta$.", "This is the same as the effect of $U(\\varphi)$ since in", "this case $U(\\varphi) \\circ U(\\alpha) = U(\\beta)$.", "\\item Say $\\beta \\in I_m$ and $k(\\beta) = k(\\alpha) - 1$.", "Here we use the differential $(-1)^{k(\\alpha)} d_{N(U), k(\\alpha)}$", "to map the summand $N(U)_{k(\\alpha)} \\cdot \\alpha$", "to the summand $N(U)_{k(\\beta)} \\cdot \\beta$.", "On the other hand, since $\\Im(\\beta) = [k(\\beta)]$ in this case", "we get $\\alpha \\circ \\varphi = \\delta^{k(\\alpha)}_{k(\\alpha)} \\circ \\beta$.", "Thus we see that $U(\\varphi)$ composed with the restriction of", "$U(\\alpha)$ to $N(U)_{k(\\alpha)}$ is equal to", "$U(\\beta)$ precomposed with $d^{k(\\alpha)}_{k(\\alpha)}$ restricted to", "$N(U)_{k(\\alpha)}$.", "Since $d_{N(U), k(\\alpha)} = \\sum (-1)^i d^{k(\\alpha)}_i$", "and since $d^{k(\\alpha)}_i$ restricts to zero on", "$N(U)_{k(\\alpha)}$ for $i < k(\\alpha)$", "we see that equality holds.", "\\end{enumerate}", "This finishes the proof of the theorem." ], "refs": [ "simplicial-lemma-extension", "simplicial-lemma-decompose-associated-complexes", "simplicial-remark-degenerate-subcomplex", "simplicial-definition-split" ], "ref_ids": [ 14859, 14866, 14940, 14924 ] } ], "ref_ids": [] }, { "id": 14805, "type": "theorem", "label": "simplicial-lemma-face-degeneracy", "categories": [ "simplicial" ], "title": "simplicial-lemma-face-degeneracy", "contents": [ "Any morphism in $\\Delta$ can be written as a composition", "of the morphisms $\\delta^n_j$ and $\\sigma^n_j$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\varphi : [n] \\to [m]$ be a morphism of $\\Delta$.", "If $j \\not \\in \\Im(\\varphi)$, then we can write", "$\\varphi$ as $\\delta^m_j \\circ \\psi$ for some morphism", "$\\psi : [n] \\to [m - 1]$. If $\\varphi(j) = \\varphi(j + 1)$", "then we can write $\\varphi$ as $\\psi \\circ \\sigma^{n - 1}_j$", "for some morphism $\\psi : [n - 1] \\to [m]$.", "The result follows because each replacement", "as above lowers $n + m$ and hence at some point", "$\\varphi$ is both injective and surjective, hence", "an identity morphism." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14806, "type": "theorem", "label": "simplicial-lemma-relations-face-degeneracy", "categories": [ "simplicial" ], "title": "simplicial-lemma-relations-face-degeneracy", "contents": [ "The morphisms $\\delta^n_j$ and $\\sigma^n_j$ satisfy the following relations.", "\\begin{enumerate}", "\\item If $0 \\leq i < j \\leq n + 1$, then", "$\\delta^{n + 1}_j \\circ \\delta^n_i =", "\\delta^{n + 1}_i \\circ \\delta^n_{j - 1}$.", "In other words the diagram", "$$", "\\xymatrix{", "& [n] \\ar[rd]^{\\delta^{n + 1}_j} & \\\\", "[n - 1] \\ar[ru]^{\\delta^n_i} \\ar[rd]_{\\delta^n_{j - 1}} & &", "[n + 1] \\\\", "& [n] \\ar[ru]_{\\delta^{n + 1}_i} &", "}", "$$", "commutes.", "\\item If $0 \\leq i < j \\leq n - 1$, then", "$\\sigma^{n - 1}_j \\circ \\delta^n_i =", "\\delta^{n - 1}_i \\circ \\sigma^{n - 2}_{j - 1}$.", "In other words the diagram", "$$", "\\xymatrix{", "& [n] \\ar[rd]^{\\sigma^{n - 1}_j} & \\\\", "[n - 1] \\ar[ru]^{\\delta^n_i} \\ar[rd]_{\\sigma^{n - 2}_{j - 1}} & &", "[n - 1] \\\\", "& [n - 2] \\ar[ru]_{\\delta^{n - 1}_i} &", "}", "$$", "commutes.", "\\item If $0 \\leq j \\leq n - 1$, then", "$\\sigma^{n - 1}_j \\circ \\delta^n_j = \\text{id}_{[n - 1]}$", "and", "$\\sigma^{n - 1}_j \\circ \\delta^n_{j + 1} = \\text{id}_{[n - 1]}$.", "In other words the diagram", "$$", "\\xymatrix{", "& [n] \\ar[rd]^{\\sigma^{n - 1}_j} & \\\\", "[n - 1]", "\\ar[ru]^{\\delta^n_j}", "\\ar[rd]_{\\delta^n_{j + 1}}", "\\ar[rr]^{\\text{id}_{[n - 1]}} & & [n - 1] \\\\", "& [n] \\ar[ru]_{\\sigma^{n - 1}_j} &", "}", "$$", "commutes.", "\\item If $0 < j + 1 < i \\leq n$, then", "$\\sigma^{n - 1}_j \\circ \\delta^n_i =", "\\delta^{n - 1}_{i - 1} \\circ \\sigma^{n - 2}_j$.", "In other words the diagram", "$$", "\\xymatrix{", "& [n] \\ar[rd]^{\\sigma^{n - 1}_j} & \\\\", "[n - 1] \\ar[ru]^{\\delta^n_i} \\ar[rd]_{\\sigma^{n - 2}_j} & &", "[n - 1] \\\\", "& [n - 2] \\ar[ru]_{\\delta^{n - 1}_{i - 1}} &", "}", "$$", "commutes.", "\\item If $0 \\leq i \\leq j \\leq n - 1$, then", "$\\sigma^{n - 1}_j \\circ \\sigma^n_i =", "\\sigma^{n - 1}_i \\circ \\sigma^n_{j + 1}$.", "In other words the diagram", "$$", "\\xymatrix{", "& [n] \\ar[rd]^{\\sigma^{n - 1}_j} & \\\\", "[n + 1] \\ar[ru]^{\\sigma^n_i} \\ar[rd]_{\\sigma^n_{j + 1}} & &", "[n - 1] \\\\", "& [n] \\ar[ru]_{\\sigma^{n - 1}_i} &", "}", "$$", "commutes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14807, "type": "theorem", "label": "simplicial-lemma-face-degeneracy-category", "categories": [ "simplicial" ], "title": "simplicial-lemma-face-degeneracy-category", "contents": [ "The category $\\Delta$ is the universal category", "with objects $[n]$, $n \\geq 0$ and morphisms", "$\\delta^n_j$ and $\\sigma^n_j$ such that (a) every morphism is", "a composition of these morphisms, (b) the relations", "listed in Lemma \\ref{lemma-relations-face-degeneracy} are satisfied,", "and (c) any relation among the morphisms is a consequence of", "those relations." ], "refs": [ "simplicial-lemma-relations-face-degeneracy" ], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 14806 ] }, { "id": 14808, "type": "theorem", "label": "simplicial-lemma-characterize-simplicial-object", "categories": [ "simplicial" ], "title": "simplicial-lemma-characterize-simplicial-object", "contents": [ "Let $\\mathcal{C}$ be a category.", "\\begin{enumerate}", "\\item Given a simplicial object $U$ in $\\mathcal{C}$", "we obtain a sequence of objects $U_n = U([n])$ endowed", "with the morphisms $d^n_j = U(\\delta^n_j) : U_n \\to U_{n-1}$ and", "$s^n_j = U(\\sigma^n_j) : U_n \\to U_{n + 1}$. These morphisms", "satisfy the opposites of the relations displayed in", "Lemma \\ref{lemma-relations-face-degeneracy}, namely", "\\begin{enumerate}", "\\item If $0 \\leq i < j \\leq n + 1$, then", "$d^n_i \\circ d^{n + 1}_j = d^n_{j - 1} \\circ d^{n + 1}_i$.", "\\item If $0 \\leq i < j \\leq n - 1$, then", "$d^n_i \\circ s^{n - 1}_j = s^{n - 2}_{j - 1} \\circ d^{n - 1}_i$.", "\\item If $0 \\leq j \\leq n - 1$, then", "$\\text{id} = d^n_j \\circ s^{n - 1}_j = d^n_{j + 1} \\circ s^{n - 1}_j$.", "\\item If $0 < j + 1 < i \\leq n$, then", "$d^n_i \\circ s^{n - 1}_j = s^{n - 2}_j \\circ d^{n - 1}_{i - 1}$.", "\\item If $0 \\leq i \\leq j \\leq n - 1$, then", "$s^n_i \\circ s^{n - 1}_j = s^n_{j + 1} \\circ s^{n - 1}_i$.", "\\end{enumerate}", "\\item Conversely, given a sequence of objects $U_n$ and morphisms", "$d^n_j$, $s^n_j$ satisfying (1)(a) -- (e) there exists a unique", "simplicial object $U$ in $\\mathcal{C}$ such that $U_n = U([n])$,", "$d^n_j = U(\\delta^n_j)$, and $s^n_j = U(\\sigma^n_j)$.", "\\item A morphism between simplicial objects $U$ and $U'$", "is given by a family of morphisms $U_n \\to U'_n$ commuting", "with the morphisms $d^n_j$ and $s^n_j$.", "\\end{enumerate}" ], "refs": [ "simplicial-lemma-relations-face-degeneracy" ], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-face-degeneracy-category}." ], "refs": [ "simplicial-lemma-face-degeneracy-category" ], "ref_ids": [ 14807 ] } ], "ref_ids": [ 14806 ] }, { "id": 14809, "type": "theorem", "label": "simplicial-lemma-si-injective", "categories": [ "simplicial" ], "title": "simplicial-lemma-si-injective", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $U$ be a simplicial object of $\\mathcal{C}$.", "Each of the morphisms $s^n_i : U_n \\to U_{n + 1}$", "has a left inverse. In particular $s^n_i$ is a monomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is true because $d_i^{n + 1} \\circ s^n_i = \\text{id}_{U_n}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14810, "type": "theorem", "label": "simplicial-lemma-characterize-cosimplicial-object", "categories": [ "simplicial" ], "title": "simplicial-lemma-characterize-cosimplicial-object", "contents": [ "Let $\\mathcal{C}$ be a category.", "\\begin{enumerate}", "\\item Given a cosimplicial object $U$ in $\\mathcal{C}$", "we obtain a sequence of objects $U_n = U([n])$ endowed", "with the morphisms $\\delta^n_j = U(\\delta^n_j) : U_{n - 1} \\to U_n$ and", "$\\sigma^n_j = U(\\sigma^n_j) : U_{n + 1} \\to U_n$. These morphisms", "satisfy the relations displayed in", "Lemma \\ref{lemma-relations-face-degeneracy}.", "\\item Conversely, given a sequence of objects $U_n$ and morphisms", "$\\delta^n_j$, $\\sigma^n_j$ satisfying these relations there exists a unique", "cosimplicial object $U$ in $\\mathcal{C}$ such that $U_n = U([n])$,", "$\\delta^n_j = U(\\delta^n_j)$, and $\\sigma^n_j = U(\\sigma^n_j)$.", "\\item A morphism between cosimplicial objects $U$ and $U'$", "is given by a family of morphisms $U_n \\to U'_n$ commuting", "with the morphisms $\\delta^n_j$ and $\\sigma^n_j$.", "\\end{enumerate}" ], "refs": [ "simplicial-lemma-relations-face-degeneracy" ], "proofs": [ { "contents": [ "This follows from Lemma \\ref{lemma-face-degeneracy-category}." ], "refs": [ "simplicial-lemma-face-degeneracy-category" ], "ref_ids": [ 14807 ] } ], "ref_ids": [ 14806 ] }, { "id": 14811, "type": "theorem", "label": "simplicial-lemma-di-injective", "categories": [ "simplicial" ], "title": "simplicial-lemma-di-injective", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $U$ be a cosimplicial object of $\\mathcal{C}$.", "Each of the morphisms $\\delta^n_i : U_{n - 1} \\to U_n$", "has a left inverse. In particular $\\delta^n_i$ is a monomorphism." ], "refs": [], "proofs": [ { "contents": [ "This is true because", "$\\sigma_i^{n - 1} \\circ \\delta^n_i = \\text{id}_{U_n}$", "for $j < n$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14812, "type": "theorem", "label": "simplicial-lemma-product", "categories": [ "simplicial" ], "title": "simplicial-lemma-product", "contents": [ "If $U$ and $V$ are simplicial objects in the category $\\mathcal{C}$,", "and if $U \\times V$ exists, then we have", "$$", "\\Mor(W, U \\times V) =", "\\Mor(W, U) \\times", "\\Mor(W, V)", "$$", "for any third simplicial object $W$ of $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14813, "type": "theorem", "label": "simplicial-lemma-fibre-product", "categories": [ "simplicial" ], "title": "simplicial-lemma-fibre-product", "contents": [ "If $U, V, W$ are simplicial objects in the category $\\mathcal{C}$,", "and if $a : V \\to U$, $b : W \\to U$ are morphisms", "and if $V \\times_U W$ exists, then we have", "$$", "\\Mor(T, V \\times_U W) =", "\\Mor(T, V) \\times_{\\Mor(T, U)}", "\\Mor(T, W)", "$$", "for any fourth simplicial object $T$ of $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14814, "type": "theorem", "label": "simplicial-lemma-push-out", "categories": [ "simplicial" ], "title": "simplicial-lemma-push-out", "contents": [ "If $U, V, W$ are simplicial objects in the category $\\mathcal{C}$,", "and if $a : U \\to V$, $b : U \\to W$ are morphisms", "and if $V\\amalg_U W$ exists, then we have", "$$", "\\Mor(V\\amalg_U W, T) =", "\\Mor(V, T) \\times_{\\Mor(U, T)}", "\\Mor(W, T)", "$$", "for any fourth simplicial object $T$ of $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14815, "type": "theorem", "label": "simplicial-lemma-product-cosimplicial-objects", "categories": [ "simplicial" ], "title": "simplicial-lemma-product-cosimplicial-objects", "contents": [ "If $U$ and $V$ are cosimplicial objects in the category $\\mathcal{C}$,", "and if $U \\times V$ exists, then we have", "$$", "\\Mor(W, U \\times V) =", "\\Mor(W, U) \\times", "\\Mor(W, V)", "$$", "for any third cosimplicial object $W$ of $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14816, "type": "theorem", "label": "simplicial-lemma-fibre-product-cosimplicial-objects", "categories": [ "simplicial" ], "title": "simplicial-lemma-fibre-product-cosimplicial-objects", "contents": [ "If $U, V, W$ are cosimplicial objects in the category $\\mathcal{C}$,", "and if $a : V \\to U$, $b : W \\to U$ are morphisms", "and if $V \\times_U W$ exists, then we have", "$$", "\\Mor(T, V \\times_U W) =", "\\Mor(T, V) \\times_{\\Mor(T, U)}", "\\Mor(T, W)", "$$", "for any fourth cosimplicial object $T$ of $\\mathcal{C}$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14817, "type": "theorem", "label": "simplicial-lemma-simplex-map", "categories": [ "simplicial" ], "title": "simplicial-lemma-simplex-map", "contents": [ "Let $U$ be a simplicial set. Let $n \\geq 0$ be an integer.", "There is a canonical bijection", "$$", "\\Mor(\\Delta[n], U)", "\\longrightarrow", "U_n", "$$", "which maps a morphism $\\varphi$ to the value of $\\varphi$", "on the unique nondegenerate $n$-simplex of $\\Delta[n]$." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14818, "type": "theorem", "label": "simplicial-lemma-product-degenerate", "categories": [ "simplicial" ], "title": "simplicial-lemma-product-degenerate", "contents": [ "Let $U$, $V$ be simplicial sets.", "Let $a, b \\geq 0$ be integers.", "Assume every $n$-simplex of $U$ is degenerate if $n > a$.", "Assume every $n$-simplex of $V$ is degenerate if $n > b$.", "Then every $n$-simplex of $U \\times V$ is degenerate", "if $n > a + b$." ], "refs": [], "proofs": [ { "contents": [ "Suppose $n > a + b$. Let $(u, v) \\in (U \\times V)_n = U_n \\times V_n$.", "By assumption, there exists a $\\alpha : [n] \\to [a]$ and a", "$u' \\in U_a$ and a $\\beta : [n] \\to [b]$ and a $v' \\in V_b$", "such that $u = U(\\alpha)(u')$ and $v = V(\\beta)(v')$. Because", "$n > a + b$, there exists an $0 \\leq i \\leq a + b$ such that", "$\\alpha(i) = \\alpha(i + 1)$ and", "$\\beta(i) = \\beta(i + 1)$. It follows immediately", "that $(u, v)$ is in the image of $s^{n - 1}_i$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14819, "type": "theorem", "label": "simplicial-lemma-check-product-with-simplicial-set", "categories": [ "simplicial" ], "title": "simplicial-lemma-check-product-with-simplicial-set", "contents": [ "Let $\\mathcal{C}$ be a category such that the coproduct of", "any two objects of $\\mathcal{C}$ exists. Let", "$U$ be a simplicial set. Let $V$ be a simplicial", "object of $\\mathcal{C}$. Assume that each $U_n$ is", "finite nonempty. The functor", "$W \\mapsto \\Mor_{\\text{Simp}(\\mathcal{C})}(U \\times V, W)$", "is canonically isomorphic to the functor which", "maps $W$ to the set in", "Equation (\\ref{equation-functor-product-with-simplicial-set})." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14820, "type": "theorem", "label": "simplicial-lemma-back-to-U", "categories": [ "simplicial" ], "title": "simplicial-lemma-back-to-U", "contents": [ "Let $\\mathcal{C}$ be a category such that the coproduct of", "any two objects of $\\mathcal{C}$ exists. Let us temporarily", "denote $\\textit{FSSets}$ the category of simplicial sets", "all of whose components are finite nonempty.", "\\begin{enumerate}", "\\item The rule $(U, V) \\mapsto U \\times V$", "defines a functor", "$\\textit{FSSets} \\times \\text{Simp}(\\mathcal{C})", "\\to \\text{Simp}(\\mathcal{C})$.", "\\item For every $U$, $V$ as above", "there is a canonical map of simplicial objects", "$$", "U \\times V \\longrightarrow V", "$$", "defined by taking the identity on each component of", "$(U \\times V)_n = \\coprod_u V_n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14821, "type": "theorem", "label": "simplicial-lemma-morphism-from-coproduct", "categories": [ "simplicial" ], "title": "simplicial-lemma-morphism-from-coproduct", "contents": [ "With $X$ and $k$ as above.", "For any simplicial object $V$ of", "$\\mathcal{C}$ we have the following", "canonical bijection", "$$", "\\Mor_{\\text{Simp}(\\mathcal{C})}(X \\times \\Delta[k], V)", "\\longrightarrow", "\\Mor_\\mathcal{C}(X, V_k).", "$$", "which maps $\\gamma$ to the restriction of the", "morphism $\\gamma_k$ to the component corresponding", "to $\\text{id}_{[k]}$.", "Similarly, for any $n \\geq k$, if $W$ is an", "$n$-truncated simplicial object", "of $\\mathcal{C}$, then we have", "$$", "\\Mor_{\\text{Simp}_n(\\mathcal{C})}(\\text{sk}_n(X \\times \\Delta[k]), W)", "=", "\\Mor_\\mathcal{C}(X, W_k).", "$$" ], "refs": [], "proofs": [ { "contents": [ "A morphism $\\gamma : X \\times \\Delta[k] \\to V$ is given by", "a family of morphisms $\\gamma_\\alpha : X \\to V_n$ where", "$\\alpha : [n] \\to [k]$. The morphisms have to satisfy the", "rules that for all $\\varphi : [m] \\to [n]$ the diagrams", "$$", "\\xymatrix{", "X \\ar[r]^{\\gamma_\\alpha} \\ar[d]^{\\text{id}_X} & V_n \\ar[d]^{V(\\varphi)} \\\\", "X \\ar[r]^{\\gamma_{\\alpha \\circ \\varphi}} & V_m", "}", "$$", "commute. Taking $\\alpha = \\text{id}_{[k]}$, we see that", "for any $\\varphi : [m] \\to [k]$ we have $\\gamma_\\varphi =", "V(\\varphi) \\circ \\gamma_{\\text{id}_{[k]}}$. Thus the morphism", "$\\gamma$ is determined by the value of $\\gamma$ on the", "component corresponding to $\\text{id}_{[k]}$. Conversely,", "given such a morphism $f : X \\to V_k$ we easily", "construct a morphism $\\gamma$ by putting", "$\\gamma_\\alpha = V(\\alpha) \\circ f$.", "\\medskip\\noindent", "The truncated case is similar, and left to the reader." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14822, "type": "theorem", "label": "simplicial-lemma-morphism-into-product", "categories": [ "simplicial" ], "title": "simplicial-lemma-morphism-into-product", "contents": [ "With $X$, $k$ and $U$ as above.", "\\begin{enumerate}", "\\item For any simplicial object $V$ of", "$\\mathcal{C}$ we have the following", "canonical bijection", "$$", "\\Mor_{\\text{Simp}(\\mathcal{C})}(V, U)", "\\longrightarrow", "\\Mor_\\mathcal{C}(V_k, X).", "$$", "wich maps $\\gamma$ to the morphism $\\gamma_k$ composed with", "the projection onto the factor corresponding to $\\text{id}_{[k]}$.", "\\item Similarly, if $W$ is an $k$-truncated simplicial object", "of $\\mathcal{C}$, then we have", "$$", "\\Mor_{\\text{Simp}_k(\\mathcal{C})}(W, \\text{sk}_k U)", "=", "\\Mor_\\mathcal{C}(W_k, X).", "$$", "\\item The object $U$ constructed above is an", "incarnation of $\\Hom(C[k], X)$ where $C[k]$ is the cosimplicial set from", "Example \\ref{example-simplex-cosimplicial-set}.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "We first prove (1).", "Suppose that $\\gamma : V \\to U$ is a morphism.", "This is given by a family of morphisms", "$\\gamma_{\\alpha} : V_n \\to X$ for $\\alpha : [k] \\to [n]$.", "The morphisms have to satisfy the", "rules that for all $\\varphi : [m] \\to [n]$ the diagrams", "$$", "\\xymatrix{", "X \\ar[d]^{\\text{id}_X} &", "V_n \\ar[d]^{V(\\varphi)}", "\\ar[l]^{\\gamma_{\\varphi \\circ \\alpha'}} \\\\", "X &", "V_m \\ar[l]_{\\gamma_{\\alpha'}}", "}", "$$", "commute for all $\\alpha' : [k] \\to [m]$.", "Taking $\\alpha' = \\text{id}_{[k]}$, we see that", "for any $\\varphi : [k] \\to [n]$ we have $\\gamma_\\varphi =", "\\gamma_{\\text{id}_{[k]}} \\circ V(\\varphi)$. Thus the morphism", "$\\gamma$ is determined by the component of $\\gamma_k$", "corresponding to $\\text{id}_{[k]}$. Conversely,", "given such a morphism $f : V_k \\to X$ we easily", "construct a morphism $\\gamma$ by putting", "$\\gamma_\\alpha = f \\circ V(\\alpha)$.", "\\medskip\\noindent", "The truncated case is similar, and left to the reader.", "\\medskip\\noindent", "Part (3) is immediate from the construction of $U$ and the", "fact that $C[k]_n = \\Mor([k], [n])$ which are the index sets", "used in the construction of $U_n$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14823, "type": "theorem", "label": "simplicial-lemma-exists-hom-0-from-simplicial-set", "categories": [ "simplicial" ], "title": "simplicial-lemma-exists-hom-0-from-simplicial-set", "contents": [ "Assume the category $\\mathcal{C}$", "has coproducts of any two objects and countable", "limits. Let $U$ be a simplicial set, with $U_n$ finite nonempty", "for all $n \\geq 0$.", "Let $V$ be a simplicial object of $\\mathcal{C}$.", "Then the functor", "\\begin{eqnarray*}", "\\mathcal{C}^{opp} & \\longrightarrow & \\textit{Sets} \\\\", "X", "& \\longmapsto &", "\\Mor_{\\text{Simp}(\\mathcal{C})}(X \\times U, V)", "\\end{eqnarray*}", "is representable." ], "refs": [], "proofs": [ { "contents": [ "A morphism from $X \\times U$ into $V$ is given by a collection", "of morphisms $f_u : X \\to V_n$ with $n \\geq 0$ and $u \\in U_n$.", "And such a collection actually defines a morphism if and only", "if for all $\\varphi : [m] \\to [n]$ all the diagrams", "$$", "\\xymatrix{", "X \\ar[r]^{f_u} \\ar[d]_{\\text{id}_X} & V_n \\ar[d]^{V(\\varphi)} \\\\", "X \\ar[r]^{f_{U(\\varphi)(u)}} & V_m", "}", "$$", "commute. Thus it is natural to introduce a category", "$\\mathcal{U}$ and a functor", "$\\mathcal{V} : \\mathcal{U}^{opp} \\to \\mathcal{C}$", "as follows:", "\\begin{enumerate}", "\\item The set of objects of $\\mathcal{U}$ is", "$\\coprod_{n \\geq 0} U_n$,", "\\item a morphism from $u' \\in U_m$ to $u \\in U_n$", "is a $\\varphi : [m] \\to [n]$ such that $U(\\varphi)(u) = u'$", "\\item for $u \\in U_n$ we set $\\mathcal{V}(u) = V_n$, and", "\\item for $\\varphi : [m] \\to [n]$ such that $U(\\varphi)(u) = u'$", "we set $\\mathcal{V}(\\varphi) = V(\\varphi) : V_n \\to V_m$.", "\\end{enumerate}", "At this point it is clear that our functor is nothing but the", "functor defining", "$$", "\\lim_{\\mathcal{U}^{opp}} \\mathcal{V}", "$$", "Thus if $\\mathcal{C}$ has countable limits then this limit", "and hence an object representing the functor of the lemma", "exist." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14824, "type": "theorem", "label": "simplicial-lemma-exists-hom-0-from-simplicial-set-finite", "categories": [ "simplicial" ], "title": "simplicial-lemma-exists-hom-0-from-simplicial-set-finite", "contents": [ "Assume the category $\\mathcal{C}$", "has coproducts of any two objects and finite", "limits. Let $U$ be a simplicial set, with $U_n$ finite nonempty", "for all $n \\geq 0$. Assume that all $n$-simplices", "of $U$ are degenerate for all $n \\gg 0$.", "Let $V$ be a simplicial object of $\\mathcal{C}$.", "Then the functor", "\\begin{eqnarray*}", "\\mathcal{C}^{opp} & \\longrightarrow & \\textit{Sets} \\\\", "X", "& \\longmapsto &", "\\Mor_{\\text{Simp}(\\mathcal{C})}(X \\times U, V)", "\\end{eqnarray*}", "is representable." ], "refs": [], "proofs": [ { "contents": [ "We have to show that the category $\\mathcal{U}$ described", "in the proof of Lemma \\ref{lemma-exists-hom-0-from-simplicial-set}", "has a finite subcategory $\\mathcal{U}'$ such that the limit", "of $\\mathcal{V}$ over $\\mathcal{U}'$ is the same as the", "limit of $\\mathcal{V}$ over $\\mathcal{U}$. We will use", "Categories, Lemma \\ref{categories-lemma-initial}.", "For $m > 0$ let $\\mathcal{U}_{\\leq m}$ denote the full", "subcategory with objects $\\coprod_{0 \\leq n \\leq m} U_m$.", "Let $m_0$ be an integer such that every $n$-simplex", "of the simplicial set $U$ is degenerate if $n > m_0$.", "For any $m \\geq m_0$ large enough, the subcategory", "$\\mathcal{U}_{\\leq m}$ satisfies property (1) of", "Categories, Definition \\ref{categories-definition-initial}.", "\\medskip\\noindent", "Suppose that $u \\in U_n$ and", "$u' \\in U_{n'}$ with $n, n' \\leq m_0$ and suppose that", "$\\varphi : [k] \\to [n]$, $\\varphi' : [k] \\to [n']$", "are morphisms such that $U(\\varphi)(u) = U(\\varphi')(u')$.", "A simple combinatorial argument shows that if $k > 2m_0$,", "then there exists an index $0 \\leq i \\leq 2m_0$ such that", "$\\varphi(i) =\\varphi(i + 1)$ and $\\varphi'(i) = \\varphi'(i + 1)$.", "(The pigeon hole principle would tell you this works if", "$k > m_0^2$ which is good enough for the argument below", "anyways.) Hence, if $k > 2m_0$, we may write", "$\\varphi = \\psi \\circ \\sigma^{k - 1}_i$ and", "$\\varphi' = \\psi' \\circ \\sigma^{k - 1}_i$ for some", "$\\psi : [k - 1] \\to [n]$ and some $\\psi' : [k - 1] \\to [n']$.", "Since $s^{k - 1}_i : U_{k - 1} \\to U_k$ is injective,", "see Lemma \\ref{lemma-si-injective}, we conclude that", "$U(\\psi)(u) = U(\\psi')(u')$ also. Continuing in this", "fashion we conclude that given morphisms", "$u \\to z$ and $u' \\to z$ of $\\mathcal{U}$", "with $u, u' \\in \\mathcal{U}_{\\leq m_0}$, there exists", "a commutative diagram", "$$", "\\xymatrix{", "u \\ar[rd] \\ar[rrd] & & \\\\", "& a \\ar[r] & z \\\\", "u' \\ar[ru] \\ar[rru]", "}", "$$", "with $a \\in \\mathcal{U}_{\\leq 2m_0}$.", "\\medskip\\noindent", "It is easy to deduce from this that the finite subcategory", "$\\mathcal{U}_{\\leq 2m_0}$ works. Namely, suppose given", "$x' \\in U_n$ and $x'' \\in U_{n'}$ with $n, n' \\leq 2m_0$ as well as", "morphisms $x' \\to x$ and $x'' \\to x$ of $\\mathcal{U}$", "with the same target. By our choice of $m_0$ we can", "find objects $u, u'$ of $\\mathcal{U}_{\\leq m_0}$ and", "morphisms $u \\to x'$, $u' \\to x''$.", "By the above we can find $a \\in \\mathcal{U}_{\\leq 2m_0}$", "and morphisms $u \\to a$, $u' \\to a$ such that", "$$", "\\xymatrix{", "u \\ar[rd] \\ar[rrd] \\ar[r] & x' \\ar[rd] & \\\\", "& a \\ar[r] & x \\\\", "u' \\ar[ru] \\ar[rru] \\ar[r] & x'' \\ar[ru] &", "}", "$$", "is commutative. Turning this diagram 90 degrees clockwise", "we get the desired diagram as in (2) of", "Categories, Definition \\ref{categories-definition-initial}." ], "refs": [ "simplicial-lemma-exists-hom-0-from-simplicial-set", "categories-lemma-initial", "categories-definition-initial", "simplicial-lemma-si-injective", "categories-definition-initial" ], "ref_ids": [ 14823, 12218, 12362, 14809, 12362 ] } ], "ref_ids": [] }, { "id": 14825, "type": "theorem", "label": "simplicial-lemma-exists-hom-from-simplicial-set-finite", "categories": [ "simplicial" ], "title": "simplicial-lemma-exists-hom-from-simplicial-set-finite", "contents": [ "Assume the category $\\mathcal{C}$", "has coproducts of any two objects and finite", "limits. Let $U$ be a simplicial set, with $U_n$ finite nonempty", "for all $n \\geq 0$. Assume that all $n$-simplices", "of $U$ are degenerate for all $n \\gg 0$.", "Let $V$ be a simplicial object of $\\mathcal{C}$.", "Then $\\Hom(U, V)$ exists, moreover", "we have the expected equalities", "$$", "\\Hom(U, V)_n = \\Hom(U \\times \\Delta[n], V)_0.", "$$" ], "refs": [], "proofs": [ { "contents": [ "We construct this simplicial object as follows.", "For $n \\geq 0$ let $\\Hom(U, V)_n$ denote", "the object of $\\mathcal{C}$ representing the", "functor", "$$", "X", "\\longmapsto", "\\Mor_{\\text{Simp}(\\mathcal{C})}(X \\times U \\times \\Delta[n], V)", "$$", "This exists by Lemma \\ref{lemma-exists-hom-0-from-simplicial-set-finite}", "because $U \\times \\Delta[n]$ is a simplicial set with finite", "sets of simplices and no nondegenerate simplices in high enough degree,", "see Lemma \\ref{lemma-product-degenerate}.", "For $\\varphi : [m] \\to [n]$ we obtain an induced map of simplicial", "sets $\\varphi : \\Delta[m] \\to \\Delta[n]$. Hence we obtain a morphism", "$X \\times U \\times \\Delta[m] \\to X \\times U \\times \\Delta[n]$", "functorial in $X$, and hence a transformation of functors,", "which in turn gives", "$$", "\\Hom(U, V)(\\varphi) :", "\\Hom(U, V)_n", "\\longrightarrow", "\\Hom(U, V)_m.", "$$", "Clearly this defines a contravariant functor", "$\\Hom(U, V)$ from", "$\\Delta$ into the category $\\mathcal{C}$.", "In other words, we have a simplicial object of $\\mathcal{C}$.", "\\medskip\\noindent", "We have to show that $\\Hom(U, V)$ satisfies the desired", "universal property", "$$", "\\Mor_{\\text{Simp}(\\mathcal{C})}(W, \\Hom(U, V))", "=", "\\Mor_{\\text{Simp}(\\mathcal{C})}(W \\times U, V)", "$$", "To see this, let $f : W \\to \\Hom(U, V)$ be given.", "We want to construct the element $f' : W \\times U \\to V$", "of the right hand side.", "By construction, each $f_n : W_n \\to \\Hom(U, V)_n$", "corresponds to a morphism", "$f_n : W_n \\times U \\times \\Delta[n] \\to V$. Further,", "for every morphism $\\varphi : [m] \\to [n]$ the", "diagram", "$$", "\\xymatrix{", "W_n \\times U \\times \\Delta[m]", "\\ar[rr]_{W(\\varphi)\\times \\text{id} \\times \\text{id}}", "\\ar[d]_{\\text{id} \\times \\text{id} \\times \\varphi} & &", "W_m \\times U \\times \\Delta[m] \\ar[d]^{f_m} \\\\", "W_n \\times U \\times \\Delta[n] \\ar[rr]^{f_n} & & V", "}", "$$", "is commutative. For $\\psi : [n] \\to [k]$ in $(\\Delta[n])_k$", "we denote $(f_n)_{k, \\psi} : W_n \\times U_k \\to V_k$", "the component of $(f_n)_k$ corresponding to the element", "$\\psi$. We define $f'_n : W_n \\times U_n \\to V_n$", "as $f'_n = (f_n)_{n, \\text{id}}$, in other words, as", "the restriction of", "$(f_n)_n : W_n \\times U_n \\times (\\Delta[n])_n \\to V_n$", "to $W_n \\times U_n \\times \\text{id}_{[n]}$.", "To see that the collection $(f'_n)$ defines a", "morphism of simplicial objects, we have to show", "for any $\\varphi : [m] \\to [n]$ that", "$V(\\varphi) \\circ f'_n =", "f'_m \\circ W(\\varphi) \\times U(\\varphi)$.", "The commutative diagram above says that", "$(f_n)_{m, \\varphi} : W_n \\times U_m \\to V_m$", "is equal to", "$(f_m)_{m, \\text{id}} \\circ W(\\varphi) :", "W_n \\times U_m \\to V_m$.", "But then the fact that $f_n$ is a morphism of simplicial", "objects implies that the diagram", "$$", "\\xymatrix{", "W_n \\times U_n \\times (\\Delta[n])_n", "\\ar[r]_-{(f_n)_n}", "\\ar[d]_{\\text{id} \\times U(\\varphi) \\times \\varphi}", "& V_n \\ar[d]^{V(\\varphi)} \\\\", "W_n \\times U_m \\times (\\Delta[n])_m \\ar[r]^-{(f_n)_m} & V_m", "}", "$$", "is commutative. And this implies that", "$(f_n)_{m, \\varphi} \\circ U(\\varphi)$ is", "equal to $V(\\varphi) \\circ (f_n)_{n, \\text{id}}$.", "Altogether we obtain", "$", "V(\\varphi) \\circ (f_n)_{n, \\text{id}}", "=", "(f_n)_{m, \\varphi} \\circ U(\\varphi)", "=", "(f_m)_{m, \\text{id}} \\circ W(\\varphi)\\circ U(\\varphi)", "=", "(f_m)_{m, \\text{id}} \\circ W(\\varphi)\\times U(\\varphi)", "$", "as desired.", "\\medskip\\noindent", "On the other hand, given a morphism", "$f' : W \\times U \\to V$ we define", "a morphism $f : W \\to \\Hom(U, V)$", "as follows. By Lemma \\ref{lemma-morphism-from-coproduct} the morphisms", "$\\text{id} : W_n \\to W_n$ corresponds to a unique", "morphism $c_n : W_n \\times \\Delta[n] \\to W$.", "Hence we can consider the composition", "$$", "W_n \\times \\Delta[n] \\times U", "\\xrightarrow{c_n}", "W \\times U", "\\xrightarrow{f'}", "V.", "$$", "By construction this corresponds to a unique morphism", "$f_n : W_n \\to \\Hom(U, V)_n$. We leave it to the reader", "to see that these define a morphism of simplicial sets as", "desired.", "\\medskip\\noindent", "We also leave it to the reader to see that", "$f \\mapsto f'$ and $f' \\mapsto f$ are mutually inverse", "operations." ], "refs": [ "simplicial-lemma-exists-hom-0-from-simplicial-set-finite", "simplicial-lemma-product-degenerate", "simplicial-lemma-morphism-from-coproduct" ], "ref_ids": [ 14824, 14818, 14821 ] } ], "ref_ids": [] }, { "id": 14826, "type": "theorem", "label": "simplicial-lemma-hom-from-coprod", "categories": [ "simplicial" ], "title": "simplicial-lemma-hom-from-coprod", "contents": [ "Assume the category $\\mathcal{C}$", "has coproducts of any two objects and finite", "limits. Let $a : U \\to V$, $b : U \\to W$", "be morphisms of simplicial sets.", "Assume $U_n, V_n, W_n$ finite nonempty for all $n \\geq 0$.", "Assume that all $n$-simplices of $U, V, W$", "are degenerate for all $n \\gg 0$.", "Let $T$ be a simplicial object of $\\mathcal{C}$.", "Then", "$$", "\\Hom(V, T) \\times_{\\Hom(U, T)} \\Hom(W, T)", "=", "\\Hom(V \\amalg_U W, T)", "$$", "In other words, the fibre product on the left hand", "side is represented by the Hom object on the right hand side." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-exists-hom-from-simplicial-set-finite}", "all the required $\\Hom$ objects exist and satisfy the", "correct functorial properties. Now we can identify", "the $n$th term on the left hand side as the object", "representing the functor that associates to $X$", "the first set of the following sequence of functorial", "equalities", "\\begin{align*}", "&", "\\Mor(X \\times \\Delta[n],", "\\Hom(V, T) \\times_{\\Hom(U, T)} \\Hom(W, T)) \\\\", "& =", "\\Mor(X \\times \\Delta[n], \\Hom(V, T))", "\\times_{\\Mor(X \\times \\Delta[n], \\Hom(U, T))}", "\\Mor(X \\times \\Delta[n], \\Hom(W, T)) \\\\", "& =", "\\Mor(X \\times \\Delta[n] \\times V, T)", "\\times_{\\Mor(X \\times \\Delta[n] \\times U, T)}", "\\Mor(X \\times \\Delta[n] \\times W, T) \\\\", "& =", "\\Mor(X \\times \\Delta[n] \\times (V \\amalg_U W), T))", "\\end{align*}", "Here we have used the fact that", "$$", "(X \\times \\Delta[n] \\times V)", "\\times_{X \\times \\Delta[n] \\times U}", "(X \\times \\Delta[n] \\times W)", "=", "X \\times \\Delta[n] \\times (V \\amalg_U W)", "$$", "which is easy to verify term by term. The result of the lemma", "follows as the last term in the displayed sequence of", "equalities corresponds to $\\Hom(V \\amalg_U W, T)_n$." ], "refs": [ "simplicial-lemma-exists-hom-from-simplicial-set-finite" ], "ref_ids": [ 14825 ] } ], "ref_ids": [] }, { "id": 14827, "type": "theorem", "label": "simplicial-lemma-splitting-simplicial-sets", "categories": [ "simplicial" ], "title": "simplicial-lemma-splitting-simplicial-sets", "contents": [ "Let $U$ be a simplicial set. Then $U$ has a unique splitting", "with $N(U_m)$ equal to the set of nondegenerate $m$-simplices." ], "refs": [], "proofs": [ { "contents": [ "From the definition it follows immediately, that if there is a", "splitting then $N(U_m)$ has the be the set of nondegenerate simplices.", "Let $x \\in U_n$. Suppose that there are surjections $\\varphi : [n] \\to [k]$", "and $\\psi : [n] \\to [l]$ and nondegenerate simplices", "$y \\in U_k$, $z \\in U_l$ such that $x = U(\\varphi)(y)$", "and $x = U(\\psi)(z)$. Choose a right inverse $\\xi : [l] \\to [n]$", "of $\\psi$, i.e., $\\psi \\circ \\xi = \\text{id}_{[l]}$.", "Then $z = U(\\xi)(x)$. Hence $z = U(\\xi)(x) = U(\\varphi \\circ \\xi)(y)$.", "Since $z$ is nondegenerate we conclude that $\\varphi \\circ \\xi :", "[l] \\to [k]$ is surjective, and hence $l \\geq k$. Similarly", "$k \\geq l$. Hence we see that $\\varphi \\circ \\xi : [l] \\to [k]$", "has to be the identity map for any choice of right inverse", "$\\xi$ of $\\psi$. This easily implies that $\\psi = \\varphi$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14828, "type": "theorem", "label": "simplicial-lemma-injective-map-simplicial-sets", "categories": [ "simplicial" ], "title": "simplicial-lemma-injective-map-simplicial-sets", "contents": [ "Let $f : U \\to V$ be a morphism of simplicial sets.", "Suppose that (a) the image of every nondegenerate simplex of", "$U$ is a nondegenerate simplex of $V$ and (b) the restriction", "of $f$ to a map from the set of nondegenerate simplices of $U$", "to the set of nondegenerate simplices of $V$ is injective.", "Then $f_n$ is injective for all $n$.", "Same holds with ``injective'' replaced by", "``surjective'' or ``bijective''." ], "refs": [], "proofs": [ { "contents": [ "Under hypothesis (a) we see that the map $f$ preserves", "the disjoint union decompositions of the splitting", "of Lemma \\ref{lemma-splitting-simplicial-sets}, in other words", "that we get commutative diagrams", "$$", "\\xymatrix{", "\\coprod\\nolimits_{\\varphi : [n] \\to [m]\\text{ surjective}}", "N(U_m)", "\\ar[r] \\ar[d] &", "U_n \\ar[d] \\\\", "\\coprod\\nolimits_{\\varphi : [n] \\to [m]\\text{ surjective}}", "N(V_m)", "\\ar[r] &", "V_n.", "}", "$$", "And then (b) clearly shows that the left vertical arrow is", "injective (resp.\\ surjective, resp.\\ bijective)." ], "refs": [ "simplicial-lemma-splitting-simplicial-sets" ], "ref_ids": [ 14827 ] } ], "ref_ids": [] }, { "id": 14829, "type": "theorem", "label": "simplicial-lemma-simplicial-set-n-skel-sub", "categories": [ "simplicial" ], "title": "simplicial-lemma-simplicial-set-n-skel-sub", "contents": [ "Let $U$ be a simplicial set.", "Let $n \\geq 0$ be an integer.", "The rule", "$$", "U'_m = \\bigcup\\nolimits_{\\varphi : [m] \\to [i], \\ i\\leq n} \\Im(U(\\varphi))", "$$", "defines a sub simplicial set $U' \\subset U$ with", "$U'_i = U_i$ for $i \\leq n$.", "Moreover, all $m$-simplices of $U'$ are degenerate for", "all $m > n$." ], "refs": [], "proofs": [ { "contents": [ "If $x \\in U_m$ and $x = U(\\varphi)(y)$", "for some $y \\in U_i$, $i \\leq n$ and some $\\varphi : [m] \\to [i]$", "then any image $U(\\psi)(x)$ for any $\\psi : [m'] \\to [m]$ is", "equal to $U(\\varphi \\circ \\psi)(y)$ and $\\varphi \\circ \\psi :", "[m'] \\to [i]$. Hence $U'$ is a simplicial set. By construction", "all simplices in dimension $n + 1$ and higher are degenerate." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14830, "type": "theorem", "label": "simplicial-lemma-splitting-simplicial-groups", "categories": [ "simplicial" ], "title": "simplicial-lemma-splitting-simplicial-groups", "contents": [ "Let $U$ be a simplicial abelian group.", "Then $U$ has a splitting obtained by taking $N(U_0) = U_0$ and", "for $m \\geq 1$ taking", "$$", "N(U_m) = \\bigcap\\nolimits_{i = 0}^{m - 1} \\Ker(d^m_i).", "$$", "Moreover, this splitting is functorial on the category", "of simplicial abelian groups." ], "refs": [], "proofs": [ { "contents": [ "By induction on $n$ we will show that the choice of $N(U_m)$", "in the lemma guarantees that (\\ref{equation-splitting}) is", "an isomorphism for $m \\leq n$. This is clear for $n = 0$.", "In the rest of this proof we are going to", "drop the superscripts from the maps $d_i$ and $s_i$ in order", "to improve readability. We will also repeatedly use the relations", "from Remark \\ref{remark-relations}.", "\\medskip\\noindent", "First we make a general remark.", "For $0 \\leq i \\leq m$ and $z \\in U_m$ we have", "$d_i(s_i(z)) = z$. Hence we can write", "any $x \\in U_{m + 1}$ uniquely as", "$x = x' + x''$ with $d_i(x') = 0$", "and $x'' \\in \\Im(s_i)$", "by taking $x' = (x - s_i(d_i(x)))$ and", "$x'' = s_i(d_i(x))$. Moreover, the element", "$z \\in U_m$ such that $x'' = s_i(z)$", "is unique because $s_i$ is injective.", "\\medskip\\noindent", "Here is a procedure for decomposing", "any $x \\in U_{n + 1}$.", "First, write $x = x_0 + s_0(z_0)$ with $d_0(x_0) = 0$.", "Next, write $x_0 = x_1 + s_1(z_1)$ with", "$d_n(x_1) = 0$. Continue like this to get", "\\begin{eqnarray*}", "x & = & x_0 + s_0(z_0), \\\\", "x_0 & = & x_1 + s_1(z_1), \\\\", "x_1 & = & x_2 + s_2(z_2), \\\\", "\\ldots & \\ldots & \\ldots \\\\", "x_{n - 1} & = & x_n + s_n(z_n)", "\\end{eqnarray*}", "where $d_i(x_i) = 0$ for all $i = n, \\ldots, 0$.", "By our general remark above all of the $x_i$", "and $z_i$ are determined uniquely by $x$.", "We claim that", "$x_i \\in", "\\Ker(d_0) \\cap", "\\Ker(d_1) \\cap", "\\ldots \\cap", "\\Ker(d_i)$", "and", "$z_i \\in", "\\Ker(d_0) \\cap", "\\ldots \\cap", "\\Ker(d_{i - 1})$", "for $i = n, \\ldots, 0$.", "Here and in the following", "an empty intersection of kernels indicates", "the whole space; i.e.,", "the notation", "$z_0 \\in \\Ker(d_0) \\cap", "\\ldots \\cap", "\\Ker(d_{i - 1})$", "when $i = 0$ means $z_0 \\in U_n$ with no restriction.", "\\medskip\\noindent", "We prove this by ascending induction on $i$.", "It is clear for $i = 0$ by construction of $x_0$ and $z_0$.", "Let us prove it for $0 < i \\leq n$ assuming the result for $i - 1$.", "First of all we have $d_i(x_i) = 0$ by construction.", "So pick a $j$ with $0 \\leq j < i$. We have", "$d_j(x_{i - 1}) = 0$ by induction. Hence", "$$", "0 = d_j(x_{i - 1})", "= d_j(x_i) + d_j(s_i(z_i))", "= d_j(x_i) + s_{i - 1}(d_j(z_i)).", "$$", "The last equality by the relations of Remark \\ref{remark-relations}.", "These relations also imply that", "$d_{i - 1}(d_j(x_i)) = d_j(d_i(x_i)) = 0$", "because $d_i(x_i)= 0$ by construction.", "Then the uniqueness in the general remark above shows the equality", "$0 = x' + x'' = d_j(x_i) + s_{i - 1}(d_j(z_i))$", "can only hold if both terms are zero. We conclude that", "$d_j(x_i) = 0$ and by injectivity of $s_{i - 1}$ we also", "conclude that $d_j(z_i) = 0$. This proves the claim.", "\\medskip\\noindent", "The claim implies we can uniquely write", "$$", "x = s_0(z_0) + s_1(z_1) + \\ldots + s_n(z_n) + x_0", "$$", "with $x_0 \\in N(U_{n + 1})$ and", "$z_i \\in \\Ker(d_0) \\cap \\ldots \\cap \\Ker(d_{i - 1})$.", "We can reformulate this as saying that we have found a direct", "sum decomposition", "$$", "U_{n + 1}", "=", "N(U_{n + 1})", "\\oplus", "\\bigoplus\\nolimits_{i = 0}^{i = n}", "s_i\\Big(\\Ker(d_0) \\cap \\ldots \\cap \\Ker(d_{i - 1})\\Big)", "$$", "with the property that", "$$", "\\Ker(d_0) \\cap \\ldots \\cap \\Ker(d_j)", "=", "N(U_{n + 1}) \\oplus", "\\bigoplus\\nolimits_{i = j + 1}^{i = n}", "s_i\\Big(\\Ker(d_n) \\cap \\ldots \\cap \\Ker(d_{i - 1})\\Big)", "$$", "for $j = 0, \\ldots, n$.", "The result follows from this statement as follows.", "Each of the $z_i$ in the expression for $x$", "can be written uniquely as", "$$", "z_i = s_i(z'_{i, i}) + \\ldots + s_{n - 1}(z'_{i, n - 1}) + z_{i, 0}", "$$", "with $z_{i, 0} \\in N(U_n)$ and", "$z'_{i, j} \\in \\Ker(d_0) \\cap \\ldots \\cap \\Ker(d_{j - 1})$.", "The first few steps in the decomposition of $z_i$ are zero because", "$z_i$ already is in the kernel of $d_0, \\ldots, d_i$.", "This in turn uniquely gives", "$$", "x = x_0 + s_0(z_{0, 0}) + s_1(z_{1, 0}) + \\ldots + s_n(z_{n, 0}) +", "\\sum\\nolimits_{0 \\leq i \\leq j \\leq n - 1} s_i(s_j(z'_{i, j})).", "$$", "Continuing in this fashion we see that we in the end obtain", "a decomposition of $x$ as a sum of terms", "of the form", "$$", "s_{i_1} s_{i_2} \\ldots s_{i_k} (z)", "$$", "with $0 \\leq i_1 \\leq i_2 \\leq \\ldots \\leq i_k \\leq n - k + 1$ and", "$z \\in N(U_{n + 1 - k})$. This is exactly the required", "decomposition, because any surjective map $[n + 1] \\to [n + 1 - k]$", "can be uniquely expressed in the form", "$$", "\\sigma^{n - k}_{i_k} \\ldots \\sigma^{n - 1}_{i_2} \\sigma^n_{i_1}", "$$", "with $0 \\leq i_1 \\leq i_2 \\leq \\ldots \\leq i_k \\leq n - k + 1$." ], "refs": [ "simplicial-remark-relations", "simplicial-remark-relations" ], "ref_ids": [ 14932, 14932 ] } ], "ref_ids": [] }, { "id": 14831, "type": "theorem", "label": "simplicial-lemma-splitting-abelian-category", "categories": [ "simplicial" ], "title": "simplicial-lemma-splitting-abelian-category", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $U$ be a simplicial object in $\\mathcal{A}$.", "Then $U$ has a splitting obtained by taking $N(U_0) = U_0$ and", "for $m \\geq 1$ taking", "$$", "N(U_m) = \\bigcap\\nolimits_{i = 0}^{m - 1} \\Ker(d^m_i).", "$$", "Moreover, this splitting is functorial on the category of", "simplicial objects of $\\mathcal{A}$." ], "refs": [], "proofs": [ { "contents": [ "For any object $A$ of $\\mathcal{A}$ we obtain", "a simplicial abelian group $\\Mor_\\mathcal{A}(A, U)$.", "Each of these are canonically split by Lemma", "\\ref{lemma-splitting-simplicial-groups}. Moreover,", "$$", "N(\\Mor_\\mathcal{A}(A, U_m)) =", "\\bigcap\\nolimits_{i = 0}^{m - 1} \\Ker(d^m_i) =", "\\Mor_\\mathcal{A}(A, N(U_m)).", "$$", "Hence we see that the morphism (\\ref{equation-splitting})", "becomes an isomorphism after applying the functor", "$\\Mor_\\mathcal{A}(A, -)$ for any object of $\\mathcal{A}$.", "Hence it is an isomorphism by the Yoneda lemma." ], "refs": [ "simplicial-lemma-splitting-simplicial-groups" ], "ref_ids": [ 14830 ] } ], "ref_ids": [] }, { "id": 14832, "type": "theorem", "label": "simplicial-lemma-injective-map-simplicial-abelian", "categories": [ "simplicial" ], "title": "simplicial-lemma-injective-map-simplicial-abelian", "contents": [ "\\begin{slogan}", "The Dold-Kan normalization functor reflects", "injectivity, surjectivity, and isomorphy.", "\\end{slogan}", "Let $\\mathcal{A}$ be an abelian category.", "Let $f : U \\to V$ be a morphism of", "simplicial objects of $\\mathcal{A}$.", "If the induced morphisms $N(f)_i : N(U)_i \\to N(V)_i$", "are injective for all $i$, then $f_i$ is", "injective for all $i$. Same holds with ``injective'' replaced", "with ``surjective'', or ``isomorphism''." ], "refs": [], "proofs": [ { "contents": [ "This is clear from Lemma \\ref{lemma-splitting-abelian-category}", "and the definition of a splitting." ], "refs": [ "simplicial-lemma-splitting-abelian-category" ], "ref_ids": [ 14831 ] } ], "ref_ids": [] }, { "id": 14833, "type": "theorem", "label": "simplicial-lemma-N-d-in-N", "categories": [ "simplicial" ], "title": "simplicial-lemma-N-d-in-N", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $U$ be a simplicial object in $\\mathcal{A}$.", "Let $N(U_m)$ as in Lemma \\ref{lemma-splitting-abelian-category} above.", "Then $d^m_m(N(U_m)) \\subset N(U_{m - 1})$." ], "refs": [ "simplicial-lemma-splitting-abelian-category" ], "proofs": [ { "contents": [ "For $j = 0, \\ldots, m - 2$ we have", "$d^{m - 1}_j d^m_m = d^{m - 1}_{m - 1} d^m_j$", "by the relations in Remark \\ref{remark-relations}.", "The result follows." ], "refs": [ "simplicial-remark-relations" ], "ref_ids": [ 14932 ] } ], "ref_ids": [ 14831 ] }, { "id": 14834, "type": "theorem", "label": "simplicial-lemma-simplicial-abelian-n-skel-sub", "categories": [ "simplicial" ], "title": "simplicial-lemma-simplicial-abelian-n-skel-sub", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $U$ be a simplicial object of $\\mathcal{A}$.", "Let $n \\geq 0$ be an integer.", "The rule", "$$", "U'_m = \\sum\\nolimits_{\\varphi : [m] \\to [i], \\ i\\leq n} \\Im(U(\\varphi))", "$$", "defines a sub simplicial object $U' \\subset U$ with $U'_i = U_i$", "for $i \\leq n$.", "Moreover, $N(U'_m) = 0$ for all $m > n$." ], "refs": [], "proofs": [ { "contents": [ "Pick $m$, $i \\leq n$ and some $\\varphi : [m] \\to [i]$.", "The image under $U(\\psi)$ of $\\Im(U(\\varphi))$", "for any $\\psi : [m'] \\to [m]$ is", "equal to the image of $U(\\varphi \\circ \\psi)$ and", "$\\varphi \\circ \\psi : [m'] \\to [i]$.", "Hence $U'$ is a simplicial object.", "Pick $m > n$. We have to show $N(U'_m) = 0$.", "By definition of $N(U_m)$ and $N(U'_m)$ we have", "$N(U'_m) = U'_m \\cap N(U_m)$ (intersection of subobjects).", "Since $U$ is split by Lemma \\ref{lemma-splitting-abelian-category},", "it suffices to show that $U'_m$ is contained in the sum", "$$", "\\sum\\nolimits_{\\varphi : [m] \\to [m']\\text{ surjective}, \\ m' < m}", "\\Im(U(\\varphi)|_{N(U_{m'})}).", "$$", "By the splitting each $U_{m'}$ is the sum of images of", "$N(U_{m''})$ via $U(\\psi)$ for surjective maps", "$\\psi : [m'] \\to [m'']$. Hence the displayed sum above", "is the same as", "$$", "\\sum\\nolimits_{\\varphi : [m] \\to [m']\\text{ surjective}, \\ m' < m}", "\\Im(U(\\varphi)).", "$$", "Clearly $U'_m$ is contained in this by the simple fact that", "any $\\varphi : [m] \\to [i]$, $i \\leq n$ occurring in the definition", "of $U'_m$ may be factored as", "$[m] \\to [m'] \\to [i]$ with $[m] \\to [m']$ surjective", "and $m' < m$ as in the last displayed sum above." ], "refs": [ "simplicial-lemma-splitting-abelian-category" ], "ref_ids": [ 14831 ] } ], "ref_ids": [] }, { "id": 14835, "type": "theorem", "label": "simplicial-lemma-existence-cosk", "categories": [ "simplicial" ], "title": "simplicial-lemma-existence-cosk", "contents": [ "If the category $\\mathcal{C}$ has finite limits, then", "$\\text{cosk}_m$ functors exist for all $m$. Moreover,", "for any $m$-truncated simplicial object $U$ the", "simplicial object $\\text{cosk}_mU$ is described", "by the formula", "$$", "(\\text{cosk}_mU)_n = \\lim_{(\\Delta/[n])_{\\leq m}^{opp}} U(n)", "$$", "and for $\\varphi : [n] \\to [n']$ the map", "$\\text{cosk}_mU(\\varphi)$ comes from the", "identification $U(n') \\circ \\overline{\\varphi} = U(n)$ above", "via Categories, Lemma \\ref{categories-lemma-functorial-limit}." ], "refs": [ "categories-lemma-functorial-limit" ], "proofs": [ { "contents": [ "During the proof of this lemma we denote $\\text{cosk}_mU$ the", "simplicial object with $(\\text{cosk}_mU)_n$ equal to", "$\\lim_{(\\Delta/[n])_{\\leq m}^{opp}} U(n)$.", "We will conclude at the end of the proof that it does", "satisfy the required mapping property.", "\\medskip\\noindent", "Suppose that $V$ is a simplicial object.", "A morphism $\\gamma : V \\to \\text{cosk}_mU$ is given by a sequence", "of morphisms $\\gamma_n : V_n \\to (\\text{cosk}_mU)_n$.", "By definition of a limit, this is given by a", "collection of morphisms $\\gamma(\\alpha) : V_n \\to U_k$", "where $\\alpha$ ranges over all $\\alpha : [k] \\to [n]$", "with $k \\leq m$. These morphisms then also satisfy", "the rules that", "$$", "\\xymatrix{", "V_n \\ar[r]_{\\gamma(\\alpha)} & U_k \\\\", "V_{n'} \\ar[r]^{\\gamma(\\alpha')} \\ar[u]^{V(\\varphi)} & U_{k'} \\ar[u]_{U(\\psi)}", "}", "$$", "are commutative, given any $0 \\leq k, k' \\leq m$, $0 \\leq n, n'$", "and any $\\psi : [k] \\to [k']$, $\\varphi : [n] \\to [n']$,", "$\\alpha : [k] \\to [n]$ and $\\alpha' : [k'] \\to [n']$ in $\\Delta$", "such that $\\varphi \\circ \\alpha = \\alpha' \\circ \\psi$.", "Taking $n = k$, $\\varphi = \\alpha'$, and $\\alpha = \\psi = \\text{id}_{[k]}$", "we deduce that $\\gamma(\\alpha') = \\gamma(\\text{id}_{[k]}) \\circ V(\\alpha')$.", "In other words, the morphisms $\\gamma(\\text{id}_{[k]})$, $k \\leq m$", "determine the morphism $\\gamma$. And it is easy to see that these", "morphisms form a morphism $\\text{sk}_m V \\to U$.", "\\medskip\\noindent", "Conversely, given a morphism $\\gamma : \\text{sk}_m V \\to U$,", "we obtain a family of morphisms $\\gamma(\\alpha)$", "where $\\alpha$ ranges over all $\\alpha : [k] \\to [n]$", "with $k \\leq m$ by setting $\\gamma(\\alpha) =", "\\gamma(\\text{id}_{[k]}) \\circ V(\\alpha)$. These morphisms", "satisfy all the displayed commutativity restraints pictured", "above, and hence give rise to a morphism $V \\to \\text{cosk}_m U$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12211 ] }, { "id": 14836, "type": "theorem", "label": "simplicial-lemma-trivial-cosk", "categories": [ "simplicial" ], "title": "simplicial-lemma-trivial-cosk", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $U$ be an $m$-truncated simplicial object of $\\mathcal{C}$.", "For $n \\leq m$ the limit $\\lim_{(\\Delta/[n])_{\\leq m}^{opp}} U(n)$", "exists and is canonically isomorphic to $U_n$." ], "refs": [], "proofs": [ { "contents": [ "This is true because the category $(\\Delta/[n])_{\\leq m}$", "has an final object in this case, namely the identity", "map $[n] \\to [n]$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14837, "type": "theorem", "label": "simplicial-lemma-recover-cosk", "categories": [ "simplicial" ], "title": "simplicial-lemma-recover-cosk", "contents": [ "Let $\\mathcal{C}$ be a category with finite limits.", "Let $U$ be an $n$-truncated simplicial object of $\\mathcal{C}$.", "The morphism $\\text{sk}_n \\text{cosk}_n U \\to U$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-existence-cosk} and \\ref{lemma-trivial-cosk}." ], "refs": [ "simplicial-lemma-existence-cosk", "simplicial-lemma-trivial-cosk" ], "ref_ids": [ 14835, 14836 ] } ], "ref_ids": [] }, { "id": 14838, "type": "theorem", "label": "simplicial-lemma-formula-limit", "categories": [ "simplicial" ], "title": "simplicial-lemma-formula-limit", "contents": [ "Let $n$ be an integer $\\geq 1$.", "Let $U$ be a $n$-truncated simplicial object of $\\mathcal{C}$.", "Consider the contravariant functor from $\\mathcal{C}$ to", "$\\textit{Sets}$ which associates to an object $T$ the set", "$$", "\\{ (f_0, \\ldots, f_{n + 1}) \\in \\Mor_\\mathcal{C}(T, U_n)", "\\mid", "d^n_{j - 1} \\circ f_i = d^n_i \\circ f_j", "\\ \\forall\\ 0\\leq i < j\\leq n + 1\\}", "$$", "If this functor is representable by some object $U_{n + 1}$", "of $\\mathcal{C}$, then", "$$", "U_{n + 1} = \\lim_{(\\Delta/[n + 1])_{\\leq n}^{opp}} U(n)", "$$" ], "refs": [], "proofs": [ { "contents": [ "The limit, if it exists, represents the functor", "that associates to an object $T$ the set", "$$", "\\{", "(f_\\alpha)_{\\alpha : [k] \\to [n + 1], k \\leq n}", "\\mid", "f_{\\alpha \\circ \\psi} = U(\\psi) \\circ f_\\alpha\\ \\forall", "\\ \\psi : [k'] \\to [k], \\alpha : [k] \\to [n + 1]", "\\}.", "$$", "In fact we will show this functor is isomorphic to the", "one displayed in the lemma. The map in one direction", "is given by the rule", "$$", "(f_\\alpha)_{\\alpha}", "\\longmapsto", "(f_{\\delta^{n + 1}_0}, \\ldots, f_{\\delta^{n + 1}_{n + 1}}).", "$$", "This satisfies the conditions of the lemma because", "$$", "d^n_{j - 1} \\circ f_{\\delta^{n + 1}_i} =", "f_{\\delta^{n + 1}_i \\circ \\delta^n_{j - 1}} =", "f_{\\delta^{n + 1}_j \\circ \\delta^n_i} =", "d^n_i \\circ f_{\\delta^{n + 1}_j}", "$$", "by the relations we recalled above the lemma. To construct a map", "in the other direction we have to associate to a system", "$(f_0, \\ldots, f_{n + 1})$ as in the displayed formula", "of the lemma a system of maps $f_\\alpha$. Let $\\alpha : [k] \\to [n + 1]$", "be given. Since $k \\leq n$ the map $\\alpha$ is not surjective.", "Hence we can write $\\alpha = \\delta^{n + 1}_i \\circ \\psi$", "for some $0 \\leq i \\leq n + 1$ and some", "$\\psi : [k] \\to [n]$. We have no choice but to define", "$$", "f_\\alpha = U(\\psi) \\circ f_i.", "$$", "Of course we have to check that this is independent of the", "choice of the pair $(i, \\psi)$. First, observe that given $i$", "there is a unique $\\psi$ which works. Second, suppose that $(j, \\phi)$ is", "another pair. Then $i \\not = j$ and we may assume $i < j$. Since", "both $i, j$ are not in the image of $\\alpha$ we may actually", "write $\\alpha = \\delta^{n + 1}_{i, j} \\circ \\xi$ and then", "we see that $\\psi = \\delta^n_{j - 1} \\circ \\xi$ and", "$\\phi = \\delta^n_i \\circ \\xi$. Thus", "\\begin{eqnarray*}", "U(\\psi) \\circ f_i & = & U(\\delta^n_{j - 1} \\circ \\xi) \\circ f_i \\\\", "& = & U(\\xi) \\circ d^n_{j - 1} \\circ f_i \\\\", "& = & U(\\xi) \\circ d^n_i \\circ f_j \\\\", "& = & U(\\delta^n_i \\circ \\xi) \\circ f_j \\\\", "& = & U(\\phi) \\circ f_j", "\\end{eqnarray*}", "as desired. We still have to verify that the maps", "$f_\\alpha$ so defined satisfy the rules of a system", "of maps $(f_\\alpha)_\\alpha$. To see this suppose that", "$\\psi : [k'] \\to [k]$, $\\alpha : [k] \\to [n + 1]$ with", "$k, k' \\leq n$. Set $\\alpha' = \\alpha \\circ \\psi$.", "Choose $i$ not in the image of $\\alpha$. Then clearly", "$i$ is not in the image of $\\alpha'$ also. Write", "$\\alpha = \\delta^{n + 1}_i \\circ \\phi$ (we cannot use the letter $\\psi$ here", "because we've already used it). Then obviously", "$\\alpha' = \\delta^{n + 1}_i \\circ \\phi \\circ \\psi$. By construction above", "we then have", "$$", "U(\\psi) \\circ f_\\alpha = U(\\psi) \\circ U(\\phi) \\circ f_i", "= U(\\phi \\circ \\psi) \\circ f_i = f_{\\alpha \\circ \\psi} = f_{\\alpha'}", "$$", "as desired. We leave to the reader the pleasant task of verifying", "that our constructions are mutually inverse bijections, and are", "functorial in $T$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14839, "type": "theorem", "label": "simplicial-lemma-work-out", "categories": [ "simplicial" ], "title": "simplicial-lemma-work-out", "contents": [ "Let $n$ be an integer $\\geq 1$. Let $U$ be a $n$-truncated", "simplicial object of $\\mathcal{C}$. Consider the", "contravariant functor from $\\mathcal{C}$ to $\\textit{Sets}$", "which associates to an object $T$ the set", "$$", "\\{ (f_0, \\ldots, f_{n + 1}) \\in \\Mor_\\mathcal{C}(T, U_n)", "\\mid", "d^n_{j - 1} \\circ f_i = d^n_i \\circ f_j", "\\ \\forall\\ 0\\leq i < j\\leq n + 1\\}", "$$", "If this functor is representable by some object $U_{n + 1}$", "of $\\mathcal{C}$, then there exists an $(n + 1)$-truncated", "simplicial object $\\tilde U$, with $\\text{sk}_n \\tilde U = U$", "and $\\tilde U_{n + 1} = U_{n + 1}$ such that the following", "adjointness holds", "$$", "\\Mor_{\\text{Simp}_{n + 1}(\\mathcal{C})}(V, \\tilde U)", "=", "\\Mor_{\\text{Simp}_n(\\mathcal{C})}(\\text{sk}_nV, U)", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-trivial-cosk} there are identifications", "$$", "U_i = \\lim_{(\\Delta/[i])_{\\leq n}^{opp}} U(i)", "$$", "for $0 \\leq i \\leq n$. By Lemma \\ref{lemma-formula-limit}", "we have", "$$", "U_{n + 1} = \\lim_{(\\Delta/[n + 1])_{\\leq n}^{opp}} U(n).", "$$", "Thus we may define for any $\\varphi : [i] \\to [j]$", "with $i, j \\leq n + 1$ the corresponding map", "$\\tilde U(\\varphi) : \\tilde U_j \\to \\tilde U_i$ exactly as", "in Lemma \\ref{lemma-existence-cosk}. This defines", "an $(n + 1)$-truncated simplicial object $\\tilde U$", "with $\\text{sk}_n \\tilde U = U$.", "\\medskip\\noindent", "To see the adjointness we argue as follows. Given any element", "$\\gamma : \\text{sk}_n V \\to U$ of the right hand side of the formula", "consider the morphisms", "$f_i = \\gamma_n \\circ d^{n + 1}_i : V_{n + 1} \\to V_n \\to U_n$.", "These clearly satisfy the relations $d^n_{j - 1} \\circ f_i = d^n_i \\circ f_j$", "and hence define a unique morphism $V_{n + 1} \\to U_{n + 1}$", "by our choice of $U_{n + 1}$.", "Conversely, given a morphism $\\gamma' : V \\to \\tilde U$", "of the left hand side we can simply restrict to", "$\\Delta_{\\leq n}$ to get an element of the right hand side.", "We leave it to the reader to show these are mutually inverse", "constructions." ], "refs": [ "simplicial-lemma-trivial-cosk", "simplicial-lemma-formula-limit", "simplicial-lemma-existence-cosk" ], "ref_ids": [ 14836, 14838, 14835 ] } ], "ref_ids": [] }, { "id": 14840, "type": "theorem", "label": "simplicial-lemma-cosk-up", "categories": [ "simplicial" ], "title": "simplicial-lemma-cosk-up", "contents": [ "Let $\\mathcal{C}$ be a category which has finite limits.", "\\begin{enumerate}", "\\item For every $n$ the functor $\\text{sk}_n : \\text{Simp}(\\mathcal{C})", "\\to \\text{Simp}_n(\\mathcal{C})$ has a right adjoint $\\text{cosk}_n$.", "\\item For every $n' \\geq n$ the functor", "$\\text{sk}_n : \\text{Simp}_{n'}(\\mathcal{C}) \\to \\text{Simp}_n(\\mathcal{C})$", "has a right adjoint, namely $\\text{sk}_{n'}\\text{cosk}_n$.", "\\item For every $m \\geq n \\geq 0$ and every $n$-truncated simplicial", "object $U$ of $\\mathcal{C}$ we have", "$\\text{cosk}_m \\text{sk}_m \\text{cosk}_n U = \\text{cosk}_n U$.", "\\item If $U$ is a simplicial object of $\\mathcal{C}$ such that", "the canonical map", "$U \\to \\text{cosk}_n \\text{sk}_nU$", "is an isomorphism for some $n \\geq 0$, then the canonical map", "$U \\to \\text{cosk}_m \\text{sk}_mU$", "is an isomorphism for all $m \\geq n$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The existence in (1) follows from Lemma \\ref{lemma-existence-cosk} above.", "Parts (2) and (3) follow from the discussion", "in Remark \\ref{remark-inductive-coskeleton}. After this (4) is obvious." ], "refs": [ "simplicial-lemma-existence-cosk", "simplicial-remark-inductive-coskeleton" ], "ref_ids": [ 14835, 14936 ] } ], "ref_ids": [] }, { "id": 14841, "type": "theorem", "label": "simplicial-lemma-cosk-product", "categories": [ "simplicial" ], "title": "simplicial-lemma-cosk-product", "contents": [ "Let $U$, $V$ be $n$-truncated simplicial objects of a", "category $\\mathcal{C}$. Then", "$$", "\\text{cosk}_n (U \\times V) = \\text{cosk}_nU \\times \\text{cosk}_nV", "$$", "whenever the left and right hand sides exist." ], "refs": [], "proofs": [ { "contents": [ "Let $W$ be a simplicial object. We have", "\\begin{eqnarray*}", "\\Mor(W, \\text{cosk}_n (U \\times V))", "& = &", "\\Mor(\\text{sk}_n W, U \\times V) \\\\", "& = &", "\\Mor(\\text{sk}_n W, U)", "\\times", "\\Mor(\\text{sk}_nW, V) \\\\", "& = &", "\\Mor(W, \\text{cosk}_n U)", "\\times", "\\Mor(W, \\text{cosk}_n V) \\\\", "& = &", "\\Mor(W, \\text{cosk}_n U \\times \\text{cosk}_n V)", "\\end{eqnarray*}", "The lemma follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14842, "type": "theorem", "label": "simplicial-lemma-cosk-fibre-product", "categories": [ "simplicial" ], "title": "simplicial-lemma-cosk-fibre-product", "contents": [ "Assume $\\mathcal{C}$ has fibre products.", "Let $U \\to V$ and $W \\to V$ be morphisms", "of $n$-truncated simplicial objects of the", "category $\\mathcal{C}$. Then", "$$", "\\text{cosk}_n (U \\times_V W)", "=", "\\text{cosk}_nU \\times_{\\text{cosk}_n V} \\text{cosk}_nW", "$$", "whenever the left and right hand side exist." ], "refs": [], "proofs": [ { "contents": [ "Omitted, but very similar to the proof of", "Lemma \\ref{lemma-cosk-product} above." ], "refs": [ "simplicial-lemma-cosk-product" ], "ref_ids": [ 14841 ] } ], "ref_ids": [] }, { "id": 14843, "type": "theorem", "label": "simplicial-lemma-cosk-above-object", "categories": [ "simplicial" ], "title": "simplicial-lemma-cosk-above-object", "contents": [ "Let $\\mathcal{C}$ be a category with finite limits.", "Let $X \\in \\Ob(\\mathcal{C})$.", "The functor $\\mathcal{C}/X \\to \\mathcal{C}$ commutes with", "the coskeleton functors $\\text{cosk}_k$ for $k \\geq 1$." ], "refs": [], "proofs": [ { "contents": [ "The statement means that if $U$ is a simplicial object of $\\mathcal{C}/X$", "which we can think of as a simplicial object of $\\mathcal{C}$ with a morphism", "towards the constant simplicial object $X$, then $\\text{cosk}_k U$", "computed in $\\mathcal{C}/X$ is the same as computed in $\\mathcal{C}$.", "This follows for example from", "Categories, Lemma \\ref{categories-lemma-connected-limit-over-X}", "because the categories $(\\Delta/[n])_{\\leq k}$ for $k \\geq 1$ and", "$n \\geq k + 1$ used in Lemma \\ref{lemma-existence-cosk}", "are connected. Observe that we do not need the categories", "for $n \\leq k$ by Lemma \\ref{lemma-trivial-cosk} or", "Lemma \\ref{lemma-recover-cosk}." ], "refs": [ "categories-lemma-connected-limit-over-X", "simplicial-lemma-existence-cosk", "simplicial-lemma-trivial-cosk", "simplicial-lemma-recover-cosk" ], "ref_ids": [ 12215, 14835, 14836, 14837 ] } ], "ref_ids": [] }, { "id": 14844, "type": "theorem", "label": "simplicial-lemma-simplex-cosk", "categories": [ "simplicial" ], "title": "simplicial-lemma-simplex-cosk", "contents": [ "The canonical map", "$\\Delta[n] \\to \\text{cosk}_1 \\text{sk}_1 \\Delta[n]$", "is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Consider a simplicial set $U$ and a morphism", "$f : U \\to \\Delta[n]$. This is a rule that associates", "to each $u \\in U_i$ a map $f_u : [i] \\to [n]$ in $\\Delta$.", "Furthermore, these maps should have the property that", "$f_u \\circ \\varphi = f_{U(\\varphi)(u)}$ for any", "$\\varphi : [j] \\to [i]$. Denote $\\epsilon^i_j : [0] \\to [i]$", "the map which maps $0$ to $j$. Denote $F : U_0 \\to [n]$", "the map $u \\mapsto f_u(0)$. Then we see that", "$$", "f_u(j) = F(\\epsilon^i_j(u))", "$$", "for all $0 \\leq j \\leq i$ and $u \\in U_i$.", "In particular, if we know the function $F$", "then we know the maps $f_u$ for all $u\\in U_i$ all $i$.", "Conversely, given a map $F : U_0 \\to [n]$,", "we can set for any $i$, and any $u \\in U_i$", "and any $0 \\leq j \\leq i$", "$$", "f_u(j) = F(\\epsilon^i_j(u))", "$$", "This does not in general define a morphism $f$ of simplicial sets", "as above. Namely, the condition is that all the maps $f_u$ are", "nondecreasing. This clearly is equivalent to the condition", "that $F(\\epsilon^i_j(u)) \\leq F(\\epsilon^i_{j'}(u))$", "whenever $0 \\leq j \\leq j' \\leq i$ and $u \\in U_i$. But in this", "case the morphisms", "$$", "\\epsilon^i_j, \\epsilon^i_{j'} : [0] \\to [i]", "$$", "both factor through the map", "$\\epsilon^i_{j, j'} : [1] \\to [i]$ defined by the rules", "$0 \\mapsto j$, $1 \\mapsto j'$.", "In other words, it is enough to check the inequalities for", "$i = 1$ and $u \\in X_1$. In other words, we have", "$$", "\\Mor(U, \\Delta[n])", "=", "\\Mor(\\text{sk}_1 U, \\text{sk}_1 \\Delta[n])", "$$", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14845, "type": "theorem", "label": "simplicial-lemma-augmentation-howto", "categories": [ "simplicial" ], "title": "simplicial-lemma-augmentation-howto", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $X \\in \\Ob(\\mathcal{C})$.", "Let $U$ be a simplicial object of $\\mathcal{C}$.", "To give an augmentation of $U$ towards $X$ is", "the same as giving a morphism $\\epsilon_0 : U_0 \\to X$", "such that $\\epsilon_0 \\circ d^1_0 = \\epsilon_0 \\circ d^1_1$." ], "refs": [], "proofs": [ { "contents": [ "Given a morphism $\\epsilon : U \\to X$", "we certainly obtain an $\\epsilon_0$ as in the lemma.", "Conversely, given $\\epsilon_0$ as in the lemma, define", "$\\epsilon_n : U_n \\to X$ by choosing any", "morphism $\\alpha : [0] \\to [n]$ and taking", "$\\epsilon_n = \\epsilon_0 \\circ U(\\alpha)$.", "Namely, if $\\beta : [0] \\to [n]$ is another", "choice, then there exists a morphism", "$\\gamma : [1] \\to [n]$ such that $\\alpha$", "and $\\beta$ both factor as $[0] \\to [1] \\to [n]$.", "Hence the condition on $\\epsilon_0$ shows that", "$\\epsilon_n$ is well defined. Then it is", "easy to show that $(\\epsilon_n) : U \\to X$", "is a morphism of simplicial objects." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14846, "type": "theorem", "label": "simplicial-lemma-cosk-minus-one", "categories": [ "simplicial" ], "title": "simplicial-lemma-cosk-minus-one", "contents": [ "Let $\\mathcal{C}$ be a category with fibred products.", "Let $f : Y\\to X$ be a morphism of $\\mathcal{C}$. Let $U$ be the", "simplicial object of $\\mathcal{C}$ whose $n$th term", "is the $(n + 1)$fold fibred product", "$Y \\times_X Y \\times_X \\ldots \\times_X Y$.", "See Example \\ref{example-fibre-products-simplicial-object}.", "For any simplicial object $V$ of $\\mathcal{C}$ we have", "\\begin{align*}", "\\Mor_{\\text{Simp}(\\mathcal{C})}(V, U)", "& =", "\\Mor_{\\text{Simp}_1(\\mathcal{C})}(\\text{sk}_1 V, \\text{sk}_1 U) \\\\", "& =", "\\{g_0 : V_0 \\to Y \\mid f \\circ g_0 \\circ d^1_0 = f \\circ g_0 \\circ d^1_1\\}", "\\end{align*}", "In particular we have $U = \\text{cosk}_1 \\text{sk}_1 U$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $g : \\text{sk}_1V \\to \\text{sk}_1U$ is a morphism of", "$1$-truncated simplicial objects. Then the diagram", "$$", "\\xymatrix{", "V_1 \\ar@<1ex>[r]^{d^1_0} \\ar@<-1ex>[r]_{d^1_1} \\ar[d]_{g_1} &", "V_0 \\ar[d]^{g_0} \\\\", "Y \\times_X Y \\ar@<1ex>[r]^{pr_1} \\ar@<-1ex>[r]_{pr_0} &", "Y \\ar[r] & X", "}", "$$", "is commutative, which proves that the relation shown in", "the lemma holds. We have to show that,", "conversely, given a morphism $g_0$ satisfying the relation", "$f \\circ g_0 \\circ d^1_0 = f \\circ g_0 \\circ d^1_1$", "we get a unique morphism of simplicial objects $g : V \\to U$.", "This is done as follows. For any $n \\geq 1$ let", "$g_{n, i} = g_0 \\circ V([0] \\to [n], 0 \\mapsto i) :", "V_n \\to Y$. The equality above implies that", "$f \\circ g_{n, i} = f \\circ g_{n, i + 1}$ because of", "the commutative diagram", "$$", "\\xymatrix{", "[0] \\ar[rd]_{0 \\mapsto 0} \\ar[rrrrrd]^{0 \\mapsto i} \\\\", "& [1] \\ar[rrrr]^{0 \\mapsto i, 1\\mapsto i + 1} & & & & [n] \\\\", "[0] \\ar[ru]^{0 \\mapsto 1} \\ar[rrrrru]_{0 \\mapsto i + 1}", "}", "$$", "Hence we get", "$(g_{n, 0}, \\ldots, g_{n, n}) : V_n \\to Y \\times_X\\ldots \\times_X Y = U_n$.", "We leave it to the reader to see that this is a morphism of simplicial", "objects. The last assertion of the lemma is equivalent to the", "first equality in the displayed formula of the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14847, "type": "theorem", "label": "simplicial-lemma-left-adjoint-exists", "categories": [ "simplicial" ], "title": "simplicial-lemma-left-adjoint-exists", "contents": [ "Let $\\mathcal{C}$ be a category which has finite colimits.", "The functors $i_{m!}$ exist for all $m$.", "Let $U$ be an $m$-truncated simplicial object of $\\mathcal{C}$.", "The simplicial object $i_{m!}U$", "is described by the formula", "$$", "(i_{m!}U)_n = \\colim_{([n]/\\Delta)_{\\leq m}^{opp}} U(n)", "$$", "and for $\\varphi : [n] \\to [n']$ the map", "$i_{m!}U(\\varphi)$ comes from the", "identification $U(n) \\circ \\underline{\\varphi} = U(n')$ above", "via Categories, Lemma \\ref{categories-lemma-functorial-colimit}." ], "refs": [ "categories-lemma-functorial-colimit" ], "proofs": [ { "contents": [ "In this proof we denote $i_{m!}U$ the simplicial object", "whose $n$th term is given by the displayed formula of the", "lemma. We will show it satisfies the adjointness property.", "\\medskip\\noindent", "Let $V$ be a simplicial object of $\\mathcal{C}$.", "Let $\\gamma : U \\to \\text{sk}_mV$ be given.", "A morphism", "$$", "\\colim_{([n]/\\Delta)_{\\leq m}^{opp}} U(n) \\to T", "$$", "is given by a compatible system of morphisms", "$f_\\alpha : U_k \\to T$ where $\\alpha : [n] \\to [k]$", "with $k \\leq m$. Certainly, we have such a system of", "morphisms by taking the compositions", "$$", "U_k \\xrightarrow{\\gamma_k} V_k \\xrightarrow{V(\\alpha)} V_n.", "$$", "Hence we get an induced morphism $(i_{m!}U)_n \\to V_n$.", "We leave it to the reader to see that these form a", "morphism of simplicial objects $\\gamma' : i_{m!}U \\to V$.", "\\medskip\\noindent", "Conversely, given a morphism $\\gamma' : i_{m!}U \\to V$ we obtain", "a morphism $\\gamma : U \\to \\text{sk}_m V$ by setting", "$\\gamma_i : U_i \\to V_i$ equal to the composition", "$$", "U_i", "\\xrightarrow{\\text{id}_{[i]}}", "\\colim_{([i]/\\Delta)_{\\leq m}^{opp}} U(i)", "\\xrightarrow{\\gamma'_i}", "V_i", "$$", "for $0 \\leq i \\leq n$. We leave it to the reader to see that", "this is the inverse of the construction above." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 12210 ] }, { "id": 14848, "type": "theorem", "label": "simplicial-lemma-recovering-U", "categories": [ "simplicial" ], "title": "simplicial-lemma-recovering-U", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $U$ be an $m$-truncated simplicial object of $\\mathcal{C}$.", "For any $n \\leq m$ the colimit", "$$", "\\colim_{([n]/\\Delta)_{\\leq m}^{opp}} U(n)", "$$", "exists and is equal to $U_n$." ], "refs": [], "proofs": [ { "contents": [ "This is so because the category $([n]/\\Delta)_{\\leq m}$", "has an initial object, namely $\\text{id} : [n] \\to [n]$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14849, "type": "theorem", "label": "simplicial-lemma-recovering-U-for-real", "categories": [ "simplicial" ], "title": "simplicial-lemma-recovering-U-for-real", "contents": [ "Let $\\mathcal{C}$ be a category which has finite colimits.", "Let $U$ be an $m$-truncated simplicial object of $\\mathcal{C}$.", "The map $U \\to \\text{sk}_m i_{m!}U$ is an isomorphism." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-left-adjoint-exists} and \\ref{lemma-recovering-U}." ], "refs": [ "simplicial-lemma-left-adjoint-exists", "simplicial-lemma-recovering-U" ], "ref_ids": [ 14847, 14848 ] } ], "ref_ids": [] }, { "id": 14850, "type": "theorem", "label": "simplicial-lemma-imshriek-sets", "categories": [ "simplicial" ], "title": "simplicial-lemma-imshriek-sets", "contents": [ "If $U$ is an $m$-truncated simplicial set and $n > m$", "then all $n$-simplices of $i_{m!}U$ are degenerate." ], "refs": [], "proofs": [ { "contents": [ "This can be seen from the construction of", "$i_{m!}U$ in Lemma \\ref{lemma-left-adjoint-exists},", "but we can also argue directly as follows.", "Write $V = i_{m!}U$. Let $V' \\subset V$ be the", "simplicial subset with $V'_i = V_i$ for $i \\leq m$", "and all $i$ simplices degenerate for $i > m$,", "see Lemma \\ref{lemma-simplicial-set-n-skel-sub}.", "By the adjunction formula,", "since $\\text{sk}_m V' = U$, there is an inverse to the", "injection $V' \\to V$. Hence $V' = V$." ], "refs": [ "simplicial-lemma-left-adjoint-exists", "simplicial-lemma-simplicial-set-n-skel-sub" ], "ref_ids": [ 14847, 14829 ] } ], "ref_ids": [] }, { "id": 14851, "type": "theorem", "label": "simplicial-lemma-n-skeleton-sets", "categories": [ "simplicial" ], "title": "simplicial-lemma-n-skeleton-sets", "contents": [ "Let $U$ be a simplicial set.", "Let $n \\geq 0$ be an integer.", "The morphism $i_{n!} \\text{sk}_n U \\to U$ identifies", "$i_{n!} \\text{sk}_n U$ with the simplicial set", "$U' \\subset U$ defined in Lemma \\ref{lemma-simplicial-set-n-skel-sub}." ], "refs": [ "simplicial-lemma-simplicial-set-n-skel-sub" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-imshriek-sets} the only", "nondegenerate simplices of $i_{n!} \\text{sk}_n U$", "are in degrees $\\leq n$. The map", "$i_{n!} \\text{sk}_n U \\to U$ is an isomorphism", "in degrees $\\leq n$. Combined we conclude", "that the map $i_{n!} \\text{sk}_n U \\to U$ maps", "nondegenerate simplices to nondegenerate simplices", "and no two nondegenerate simplices have the same image.", "Hence Lemma \\ref{lemma-injective-map-simplicial-sets} applies.", "Thus $i_{n!} \\text{sk}_n U \\to U$", "is injective. The result follows easily from this." ], "refs": [ "simplicial-lemma-imshriek-sets", "simplicial-lemma-injective-map-simplicial-sets" ], "ref_ids": [ 14850, 14828 ] } ], "ref_ids": [ 14829 ] }, { "id": 14852, "type": "theorem", "label": "simplicial-lemma-glue-simplex", "categories": [ "simplicial" ], "title": "simplicial-lemma-glue-simplex", "contents": [ "Let $U \\subset V$ be simplicial sets.", "Suppose $n \\geq 0$ and $x \\in V_n$, $x \\not \\in U_n$ are such that", "\\begin{enumerate}", "\\item $V_i = U_i$ for $i < n$,", "\\item $V_n = U_n \\cup \\{x\\}$,", "\\item any $z \\in V_j$, $z \\not \\in U_j$ for $j > n$", "is degenerate.", "\\end{enumerate}", "Let $\\Delta[n] \\to V$ be the unique morphism mapping the", "nondegenerate $n$-simplex of $\\Delta[n]$ to $x$.", "In this case the diagram", "$$", "\\xymatrix{", "\\Delta[n] \\ar[r] & V \\\\", "i_{(n - 1)!} \\text{sk}_{n - 1} \\Delta[n] \\ar[r] \\ar[u] & U \\ar[u]", "}", "$$", "is a pushout diagram." ], "refs": [], "proofs": [ { "contents": [ "Let us denote $\\partial \\Delta[n] = i_{(n - 1)!} \\text{sk}_{n - 1} \\Delta[n]$", "for convenience. There is a natural map", "$U \\amalg_{\\partial \\Delta[n]} \\Delta[n] \\to V$.", "We have to show that it is bijective in degree $j$", "for all $j$. This is clear for $j \\leq n$. Let $j > n$.", "The third condition means that any $z \\in V_j$, $z \\not \\in U_j$", "is a degenerate simplex, say $z = s^{j - 1}_i(z')$. Of course", "$z' \\not \\in U_{j - 1}$. By induction it follows that $z'$", "is a degeneracy of $x$. Thus we conclude that all $j$-simplices", "of $V$ are either in $U$ or degeneracies of $x$. This implies", "that the map $U \\amalg_{\\partial \\Delta[n]} \\Delta[n] \\to V$", "is surjective. Note that a nondegenerate simplex of", "$U \\amalg_{\\partial \\Delta[n]} \\Delta[n]$ is either", "the image of a nondegenerate simplex of $U$, or", "the image of the (unique) nondegenerate $n$-simplex", "of $\\Delta[n]$. Since clearly $x$ is nondegenerate we", "deduce that $U \\amalg_{\\partial \\Delta[n]} \\Delta[n] \\to V$", "maps nondegenerate simplices to nondegenerate simplices", "and is injective on nondegenerate simplices. Hence it is", "injective, by Lemma \\ref{lemma-injective-map-simplicial-sets}." ], "refs": [ "simplicial-lemma-injective-map-simplicial-sets" ], "ref_ids": [ 14828 ] } ], "ref_ids": [] }, { "id": 14853, "type": "theorem", "label": "simplicial-lemma-add-simplices", "categories": [ "simplicial" ], "title": "simplicial-lemma-add-simplices", "contents": [ "Let $U \\subset V$ be simplicial sets, with $U_n, V_n$", "finite nonempty for all $n$.", "Assume that $U$ and $V$ have finitely many nondegenerate simplices.", "Then there exists a sequence of sub simplicial sets", "$$", "U = W^0 \\subset W^1 \\subset W^2 \\subset \\ldots W^r = V", "$$", "such that Lemma \\ref{lemma-glue-simplex} applies to each of", "the inclusions $W^i \\subset W^{i + 1}$." ], "refs": [ "simplicial-lemma-glue-simplex" ], "proofs": [ { "contents": [ "Let $n$ be the smallest integer such that $V$ has a nondegenerate", "simplex that does not belong to $U$. Let $x \\in V_n$, $x\\not \\in U_n$", "be such a nondegenerate simplex. Let $W \\subset V$ be the set", "of elements which are either in $U$, or are a (repeated) degeneracy", "of $x$ (in other words, are of the form $V(\\varphi)(x)$", "with $\\varphi : [m] \\to [n]$ surjective). It is easy to see", "that $W$ is a simplicial set. The", "inclusion $U \\subset W$ satisfies the conditions of Lemma", "\\ref{lemma-glue-simplex}. Moreover the number of nondegenerate", "simplices of $V$ which are not contained in $W$ is exactly", "one less than the number of nondegenerate", "simplices of $V$ which are not contained in $U$.", "Hence we win by induction on this number." ], "refs": [ "simplicial-lemma-glue-simplex" ], "ref_ids": [ 14852 ] } ], "ref_ids": [ 14852 ] }, { "id": 14854, "type": "theorem", "label": "simplicial-lemma-imshriek-abelian", "categories": [ "simplicial" ], "title": "simplicial-lemma-imshriek-abelian", "contents": [ "Let $\\mathcal{A}$ be an abelian category", "Let $U$ be an $m$-truncated simplicial object of", "$\\mathcal{A}$. For $n > m$ we have $N(i_{m!}U)_n = 0$." ], "refs": [], "proofs": [ { "contents": [ "Write $V = i_{m!}U$. Let $V' \\subset V$ be the", "simplicial subobject of $V$ with $V'_i = V_i$ for $i \\leq m$", "and $N(V'_i) = 0$ for $i > m$,", "see Lemma \\ref{lemma-simplicial-abelian-n-skel-sub}.", "By the adjunction formula,", "since $\\text{sk}_m V' = U$, there is an inverse to the", "injection $V' \\to V$. Hence $V' = V$." ], "refs": [ "simplicial-lemma-simplicial-abelian-n-skel-sub" ], "ref_ids": [ 14834 ] } ], "ref_ids": [] }, { "id": 14855, "type": "theorem", "label": "simplicial-lemma-n-skeleton-abelian", "categories": [ "simplicial" ], "title": "simplicial-lemma-n-skeleton-abelian", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $U$ be a simplicial object of $\\mathcal{A}$.", "Let $n \\geq 0$ be an integer.", "The morphism $i_{n!} \\text{sk}_n U \\to U$ identifies", "$i_{n!} \\text{sk}_n U$ with the simplicial subobject", "$U' \\subset U$ defined in Lemma \\ref{lemma-simplicial-abelian-n-skel-sub}." ], "refs": [ "simplicial-lemma-simplicial-abelian-n-skel-sub" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-imshriek-abelian}", "we have $N(i_{n!} \\text{sk}_n U)_i = 0$", "for $i > n$. The map", "$i_{n!} \\text{sk}_n U \\to U$ is an isomorphism", "in degrees $\\leq n$, see Lemma \\ref{lemma-recovering-U-for-real}.", "Combined we conclude that the map $i_{n!} \\text{sk}_n U \\to U$", "induces injective maps $N(i_{n!} \\text{sk}_n U)_i \\to N(U)_i$", "for all $i$. Hence Lemma \\ref{lemma-injective-map-simplicial-abelian}", "applies. Thus $i_{n!} \\text{sk}_n U \\to U$", "is injective. The result follows easily from this." ], "refs": [ "simplicial-lemma-imshriek-abelian", "simplicial-lemma-recovering-U-for-real", "simplicial-lemma-injective-map-simplicial-abelian" ], "ref_ids": [ 14854, 14849, 14832 ] } ], "ref_ids": [ 14834 ] }, { "id": 14856, "type": "theorem", "label": "simplicial-lemma-cosk-shriek", "categories": [ "simplicial" ], "title": "simplicial-lemma-cosk-shriek", "contents": [ "Let $\\mathcal{C}$ be a category with finite coproducts", "and finite limits. Let $V$ be a simplicial object of $\\mathcal{C}$.", "In this case", "$$", "(\\text{cosk}_n \\text{sk}_n V)_{n + 1}", "=", "\\Hom(i_{n !}\\text{sk}_n \\Delta[n + 1], V)_0.", "$$" ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-morphism-from-coproduct}", "the object on the left represents the functor", "which assigns to $X$ the first set of the following", "equalities", "\\begin{eqnarray*}", "\\Mor(X \\times \\Delta[n + 1], \\text{cosk}_n \\text{sk}_n V)", "& = &", "\\Mor(X \\times \\text{sk}_n \\Delta[n + 1], \\text{sk}_n V) \\\\", "& = &", "\\Mor(X \\times i_{n !} \\text{sk}_n \\Delta[n + 1], V).", "\\end{eqnarray*}", "The object on the right in the formula of the lemma", "is represented by the functor which assigns to $X$", "the last set in the sequence of equalities.", "This proves the result.", "\\medskip\\noindent", "In the sequence of equalities we have used that", "$\\text{sk}_n (X \\times \\Delta[n + 1]) = X \\times \\text{sk}_n \\Delta[n + 1]$", "and that", "$i_{n!}(X \\times \\text{sk}_n \\Delta[n + 1]) =", "X \\times i_{n !} \\text{sk}_n \\Delta[n + 1]$.", "The first equality is obvious. For any (possibly truncated)", "simplicial object $W$", "of $\\mathcal{C}$ and any object $X$ of $\\mathcal{C}$", "denote temporarily $\\Mor_\\mathcal{C}(X, W)$ the", "(possibly truncated) simplicial set", "$[n] \\mapsto \\Mor_\\mathcal{C}(X, W_n)$. From the definitions", "it follows that $\\Mor(U \\times X, W) =", "\\Mor(U, \\Mor_\\mathcal{C}(X, W))$ for any", "(possibly truncated) simplicial set $U$. Hence", "\\begin{eqnarray*}", "\\Mor(X \\times i_{n !} \\text{sk}_n \\Delta[n + 1], W)", "& = &", "\\Mor(i_{n !} \\text{sk}_n \\Delta[n + 1], \\Mor_\\mathcal{C}(X, W)) \\\\", "& = &", "\\Mor(\\text{sk}_n \\Delta[n + 1],", "\\text{sk}_n\\Mor_\\mathcal{C}(X, W)) \\\\", "& = &", "\\Mor(X \\times \\text{sk}_n \\Delta[n + 1], \\text{sk}_nW) \\\\", "& = &", "\\Mor(i_{n!}(X \\times \\text{sk}_n \\Delta[n + 1]), W).", "\\end{eqnarray*}", "This proves the second equality used, and ends the proof of the lemma." ], "refs": [ "simplicial-lemma-morphism-from-coproduct" ], "ref_ids": [ 14821 ] } ], "ref_ids": [] }, { "id": 14857, "type": "theorem", "label": "simplicial-lemma-abelian", "categories": [ "simplicial" ], "title": "simplicial-lemma-abelian", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item The categories $\\text{Simp}(\\mathcal{A})$ and", "$\\text{CoSimp}(\\mathcal{A})$ are abelian.", "\\item A morphism of (co)simplicial objects", "$f : A \\to B$ is injective", "if and only if each $f_n : A_n \\to B_n$ is injective.", "\\item A morphism of (co)simplicial objects", "$f : A \\to B$ is surjective", "if and only if each $f_n : A_n \\to B_n$ is surjective.", "\\item A sequence of (co)simplicial objects", "$$", "A \\xrightarrow{f} B \\xrightarrow{g} C", "$$", "is exact at $B$ if and only if each sequence", "$$", "A_i \\xrightarrow{f_i} B_i \\xrightarrow{g_i} C_i", "$$", "is exact at $B_i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Pre-additivity is easy. A final object is", "given by $U_n = 0$ in all degrees.", "Existence of direct products we saw in", "Lemmas \\ref{lemma-product} and", "\\ref{lemma-product-cosimplicial-objects}.", "Kernels and cokernels are obtained by taking", "termwise kernels and cokernels." ], "refs": [ "simplicial-lemma-product", "simplicial-lemma-product-cosimplicial-objects" ], "ref_ids": [ 14812, 14815 ] } ], "ref_ids": [] }, { "id": 14858, "type": "theorem", "label": "simplicial-lemma-eilenberg-maclane-object", "categories": [ "simplicial" ], "title": "simplicial-lemma-eilenberg-maclane-object", "contents": [ "With $A$, $k$ and $U$ as above, so $U_i = 0$, $i < k$ and $U_k = A$.", "\\begin{enumerate}", "\\item Given a $k$-truncated simplicial object $V$", "we have", "$$", "\\Mor(U, V)", "=", "\\{ f : A \\to V_k \\mid d^k_i \\circ f = 0, \\ i = 0, \\ldots, k \\}", "$$", "and", "$$", "\\Mor(V, U)", "=", "\\{ f : V_k \\to A \\mid f \\circ s^{k - 1}_i = 0, \\ i = 0, \\ldots, k - 1 \\}.", "$$", "\\item The object $i_{k!} U$ has $n$th term equal to", "$\\bigoplus_\\alpha A$ where $\\alpha$ runs over all", "surjective morphisms $\\alpha : [n] \\to [k]$.", "\\item For any $\\varphi : [m] \\to [n]$ the map", "$i_{k!} U(\\varphi)$ is described as the mapping", "$\\bigoplus_\\alpha A \\to \\bigoplus_{\\alpha'} A$", "which maps to component corresponding to $\\alpha : [n] \\to [k]$", "to zero if $\\alpha \\circ \\varphi$ is not surjective and", "by the identity to the component corresponding to", "$\\alpha \\circ \\varphi$ if it is surjective.", "\\item The object $\\text{cosk}_k U$ has $n$th term equal to", "$\\bigoplus_\\beta A$, where $\\beta$ runs over all", "injective morphisms $\\beta : [k] \\to [n]$.", "\\item For any $\\varphi : [m] \\to [n]$ the map", "$\\text{cosk}_k U(\\varphi)$ is described as the mapping", "$\\bigoplus_\\beta A \\to \\bigoplus_{\\beta'} A$", "which maps to component corresponding to $\\beta : [k] \\to [n]$", "to zero if $\\beta$ does not factor through $\\varphi$ and", "by the identity to each of the components corresponding to", "$\\beta'$ such that $\\beta = \\varphi \\circ \\beta'$", "if it does.", "\\item The canonical map", "$", "c : i_{k !} U \\to \\text{cosk}_k U", "$", "in degree $n$ has $(\\alpha, \\beta)$ coefficient $A \\to A$", "equal to zero if $\\alpha \\circ \\beta$ is not the identity", "and equal to $\\text{id}_A$ if it is.", "\\item The canonical map", "$", "c : i_{k !} U \\to \\text{cosk}_k U", "$", "is injective.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The proof of (1) is left to the reader.", "\\medskip\\noindent", "Let us take the rules of (2) and (3)", "as the definition of a simplicial object, call it $\\tilde U$.", "We will show that it is an incarnation of $i_{k!}U$.", "This will prove (2), (3) at the same time. We have to show", "that given a morphism $f : U \\to \\text{sk}_kV$", "there exists a unique morphism $\\tilde f : \\tilde U \\to V$", "which recovers $f$ upon taking the $k$-skeleton.", "From (1) we see that $f$ corresponds with a morphism", "$f_k : A \\to V_k$ which maps into the kernel of", "$d^k_i$ for all $i$. For any surjective $\\alpha : [n] \\to [k]$", "we set $\\tilde f_\\alpha : A \\to V_n$ equal to the composition", "$\\tilde f_\\alpha = V(\\alpha) \\circ f_k : A \\to V_n$. We define", "$\\tilde f_n : \\tilde U_n \\to V_n$ as the sum of", "the $\\tilde f_\\alpha$ over $\\alpha : [n] \\to [k]$ surjective.", "Such a collection of $\\tilde f_\\alpha$ defines a morphism", "of simplicial objects if and only if", "for any $\\varphi : [m] \\to [n]$ the diagram", "$$", "\\xymatrix{", "\\bigoplus_{\\alpha : [n] \\to [k]\\text{ surjective}} A", "\\ar[r]_-{\\tilde f_n}", "\\ar[d]_{(3)} &", "V_n \\ar[d]^{V(\\varphi)} \\\\", "\\bigoplus_{\\alpha' : [m] \\to [k]\\text{ surjective}} A", "\\ar[r]^-{\\tilde f_m} &", "V_m", "}", "$$", "is commutative. Choosing $\\varphi = \\alpha$ shows our choice of", "$\\tilde f_\\alpha$ is uniquely determined by $f_k$.", "The commutativity in general may be checked for each summand", "of the left upper corner separately. It is clear for the", "summands corresponding to $\\alpha$ where", "$\\alpha \\circ \\varphi$ is surjective, because those get", "mapped by $\\text{id}_A$ to the summand with", "$\\alpha' = \\alpha \\circ \\varphi$, and we have", "$\\tilde f_{\\alpha'} = V(\\alpha') \\circ f_k =", "V(\\alpha \\circ \\varphi) \\circ f_k = V(\\varphi) \\circ \\tilde f_\\alpha$.", "For those where $\\alpha \\circ \\varphi$", "is not surjective, we have to show that $V(\\varphi) \\circ \\tilde f_\\alpha = 0$.", "By definition this is equal to", "$V(\\varphi) \\circ V(\\alpha) \\circ f_k = V(\\alpha \\circ \\varphi) \\circ f_k$.", "Since $\\alpha \\circ \\varphi$ is not surjective we can write it", "as $\\delta^k_i \\circ \\psi$, and we deduce that", "$V(\\varphi) \\circ V(\\alpha) \\circ f_k =", "V(\\psi) \\circ d^k_i \\circ f_k = 0$ see above.", "\\medskip\\noindent", "Let us take the rules of (4) and (5)", "as the definition of a simplicial object, call it $\\tilde U$.", "We will show that it is an incarnation of $\\text{cosk}_k U$.", "This will prove (4), (5) at the same time. The argument is completely dual", "to the proof of (2), (3) above, but we give it anyway.", "We have to show", "that given a morphism $f : \\text{sk}_kV \\to U$", "there exists a unique morphism $\\tilde f : V \\to \\tilde U$", "which recovers $f$ upon taking the $k$-skeleton.", "From (1) we see that $f$ corresponds with a morphism", "$f_k : V_k \\to A$ which is zero on the image of $s^{k - 1}_i$", "for all $i$. For any injective $\\beta : [k] \\to [n]$", "we set $\\tilde f_\\beta : V_n \\to A$ equal to the composition", "$\\tilde f_\\beta = f_k \\circ V(\\beta) : V_n \\to A$. We define", "$\\tilde f_n : V_n \\to \\tilde U_n$ as the sum of", "the $\\tilde f_\\beta$ over $\\beta : [k] \\to [n]$ injective.", "Such a collection of $\\tilde f_\\beta$ defines a morphism", "of simplicial objects if and only if", "for any $\\varphi : [m] \\to [n]$ the diagram", "$$", "\\xymatrix{", "V_n", "\\ar[d]_{V(\\varphi)}", "\\ar[r]_-{\\tilde f_n}", "&", "\\bigoplus_{\\beta : [k] \\to [n]\\text{ injective}} A", "\\ar[d]^{(5)}", "\\\\", "V_m", "\\ar[r]^-{\\tilde f_m}", "&", "\\bigoplus_{\\beta' : [k] \\to [m]\\text{ injective}} A", "}", "$$", "is commutative. Choosing $\\varphi = \\beta$ shows our choice of", "$\\tilde f_\\beta$ is uniquely determined by $f_k$.", "The commutativity in general may be checked for each summand", "of the right lower corner separately. It is clear for the", "summands corresponding to $\\beta'$ where", "$\\varphi \\circ \\beta'$ is injective, because these summands", "get mapped into by exactly the summand with", "$\\beta = \\varphi \\circ \\beta'$ and we have in that case", "$\\tilde f_{\\beta'} \\circ V(\\varphi) =", "f_k \\circ V(\\beta') \\circ V(\\varphi) =", "f_k \\circ V(\\beta) = \\tilde f_\\beta$. For those where", "$\\varphi \\circ \\beta'$ is not injective,", "we have to show that $\\tilde f_{\\beta'} \\circ V(\\varphi) = 0$.", "By definition this is equal to", "$f_k \\circ V(\\beta') \\circ V(\\varphi) =", "f_k \\circ V(\\varphi \\circ \\beta')$.", "Since $\\varphi \\circ \\beta'$ is not injective we can write it", "as $\\psi \\circ \\sigma^{k - 1}_i$, and we deduce that", "$f_k \\circ V(\\beta') \\circ V(\\varphi) =", "f_k \\circ s^{k - 1}_i \\circ V(\\psi) = 0$ see above.", "\\medskip\\noindent", "The composition $i_{k!}U \\to \\text{cosk}_kU$ is the", "unique map of simplicial objects which is", "the identity on $A = U_k = (i_{k!}U)_k = (\\text{cosk}_kU)_k$.", "Hence it suffices to check that the proposed rule defines", "a morphism of simplicial objects.", "To see this we have to show that", "for any $\\varphi : [m] \\to [n]$ the diagram", "$$", "\\xymatrix{", "\\bigoplus_{\\alpha : [n] \\to [k]\\text{ surjective}} A", "\\ar[d]_{(3)}", "\\ar[r]_{(6)}", "&", "\\bigoplus_{\\beta : [k] \\to [n]\\text{ injective}} A", "\\ar[d]^{(5)}", "\\\\", "\\bigoplus_{\\alpha' : [m] \\to [k]\\text{ surjective}} A", "\\ar[r]^{(6)}", "&", "\\bigoplus_{\\beta' : [k] \\to [m]\\text{ injective}} A", "}", "$$", "is commutative. Now we can think of this in terms of", "matrices filled with only $0$'s and $1$'s as follows:", "The matrix of (3) has a nonzero", "$(\\alpha', \\alpha)$ entry if and only if", "$\\alpha' = \\alpha \\circ \\varphi$. Likewise", "the matrix of (5) has a nonzero", "$(\\beta', \\beta)$ entry if and only if", "$\\beta = \\varphi \\circ \\beta'$. The upper matrix", "of (6) has a nonzero $(\\alpha, \\beta)$ entry if and only if", "$\\alpha \\circ \\beta = \\text{id}_{[k]}$. Similarly for the", "lower matrix of (6). The commutativity of the", "diagram then comes down to computing the", "$(\\alpha, \\beta')$ entry for both compositions", "and seeing they are equal. This comes down to the", "following equality", "$$", "\\# \\left\\{", "\\beta", "\\mid", "\\beta = \\varphi \\circ \\beta'", "\\text{ and }", "\\alpha \\circ \\beta = \\text{id}_{[k]}", "\\right\\}", "=", "\\# \\left\\{", "\\alpha'", "\\mid", "\\alpha' = \\alpha \\circ \\varphi", "\\text{ and }", "\\alpha' \\circ \\beta' = \\text{id}_{[k]}", "\\right\\}", "$$", "whose proof may safely be left to the reader.", "\\medskip\\noindent", "Finally, we prove (7). This follows directly from", "Lemmas \\ref{lemma-injective-map-simplicial-abelian},", "\\ref{lemma-recover-cosk}, \\ref{lemma-recovering-U-for-real}", "and \\ref{lemma-imshriek-abelian}." ], "refs": [ "simplicial-lemma-injective-map-simplicial-abelian", "simplicial-lemma-recover-cosk", "simplicial-lemma-recovering-U-for-real", "simplicial-lemma-imshriek-abelian" ], "ref_ids": [ 14832, 14837, 14849, 14854 ] } ], "ref_ids": [] }, { "id": 14859, "type": "theorem", "label": "simplicial-lemma-extension", "categories": [ "simplicial" ], "title": "simplicial-lemma-extension", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A$ be an object of $\\mathcal{A}$ and", "let $k$ be an integer $\\geq 0$. Consider the", "simplicial object $E$ defined by the following rules", "\\begin{enumerate}", "\\item $E_n = \\bigoplus_\\alpha A$, where the", "sum is over $\\alpha : [n] \\to [k + 1]$ whose", "image is either $[k]$ or $[k + 1]$.", "\\item Given $\\varphi : [m] \\to [n]$ the map", "$E_n \\to E_m$ maps the summand corresponding", "to $\\alpha$ via $\\text{id}_A$ to the summand", "corresponding to $\\alpha \\circ \\varphi$,", "provided $\\Im(\\alpha \\circ \\varphi)$", "is equal to $[k]$ or $[k + 1]$.", "\\end{enumerate}", "Then there exists a short exact sequence", "$$", "0 \\to K(A, k) \\to E \\to K(A, k + 1) \\to 0", "$$", "which is term by term split exact." ], "refs": [], "proofs": [ { "contents": [ "The maps $K(A, k)_n \\to E_n$ resp.\\ $E_n \\to K(A, k + 1)_n$", "are given by the inclusion of direct sums, resp.\\ projection", "of direct sums which is obvious from the inclusions of index", "sets. It is clear that these are maps of simplicial", "objects." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14860, "type": "theorem", "label": "simplicial-lemma-abelian-limit-skeleta", "categories": [ "simplicial" ], "title": "simplicial-lemma-abelian-limit-skeleta", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "For any simplicial object $V$ of $\\mathcal{A}$ we have", "$$", "V = \\colim_n i_{n!}\\text{sk}_n V", "$$", "where all the transition maps are injections." ], "refs": [], "proofs": [ { "contents": [ "This is true simply because each $V_m$ is", "equal to $(i_{n!}\\text{sk}_n V)_m$ as", "soon as $n \\geq m$. See also Lemma \\ref{lemma-n-skeleton-abelian}", "for the transition maps." ], "refs": [ "simplicial-lemma-n-skeleton-abelian" ], "ref_ids": [ 14855 ] } ], "ref_ids": [] }, { "id": 14861, "type": "theorem", "label": "simplicial-lemma-s-exact", "categories": [ "simplicial" ], "title": "simplicial-lemma-s-exact", "contents": [ "The functor $s$ is exact." ], "refs": [], "proofs": [ { "contents": [ "Clear from Lemma \\ref{lemma-abelian}." ], "refs": [ "simplicial-lemma-abelian" ], "ref_ids": [ 14857 ] } ], "ref_ids": [] }, { "id": 14862, "type": "theorem", "label": "simplicial-lemma-homology-extension", "categories": [ "simplicial" ], "title": "simplicial-lemma-homology-extension", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A$ be an object of $\\mathcal{A}$ and", "let $k$ be an integer. Let $E$ be the object", "described in Lemma \\ref{lemma-extension}.", "Then the complex $s(E)$ is acyclic." ], "refs": [ "simplicial-lemma-extension" ], "proofs": [ { "contents": [ "For a morphism $\\alpha : [n] \\to [k + 1]$", "we define $\\alpha' : [n + 1] \\to [k + 1]$ to be", "the map such that $\\alpha'|_{[n]} = \\alpha$ and", "$\\alpha'(n + 1) = k + 1$. Note that if the", "image of $\\alpha$ is $[k]$ or $[k + 1]$, then", "the image of $\\alpha'$ is $[k + 1]$.", "Consider the family of", "maps $h_n : E_n \\to E_{n + 1}$ which maps", "the summand corresponding to $\\alpha$ to", "the summand corresponding to $\\alpha'$ via", "the identity on $A$.", "Let us compute $d_{n + 1} \\circ h_n - h_{n - 1} \\circ d_n$.", "We will first do this in case the category $\\mathcal{A}$ is", "the category of abelian groups.", "Let us use the notation $x_\\alpha$ to indicate", "the element $x \\in A$ in the summand of $E_n$ corresponding", "to the map $\\alpha$ occurring in the index set.", "Let us also adopt the convention that", "$x_\\alpha$ designates the zero element of $E_n$", "whenever $\\Im(\\alpha)$ is not $[k]$ or $[k + 1]$.", "With these conventions we see that", "$$", "d_{n + 1}(h_n(x_\\alpha)) =", "\\sum\\nolimits_{i = 0}^{n + 1} (-1)^i x_{\\alpha' \\circ \\delta^{n + 1}_i}", "$$", "and", "$$", "h_{n - 1}(d_n(x_\\alpha)) =", "\\sum\\nolimits_{i = 0}^n (-1)^i x_{(\\alpha \\circ \\delta_i^n)'}", "$$", "It is easy to see that", "$\\alpha' \\circ \\delta^{n + 1}_i = (\\alpha \\circ \\delta_i^n)'$", "for $i = 0, \\ldots, n$. It is also easy to see that", "$\\alpha' \\circ \\delta^{n + 1}_{n + 1} = \\alpha$. Thus we", "see that", "$$", "(d_{n + 1} \\circ h_n - h_{n - 1} \\circ d_n)(x_\\alpha)", "=", "(-1)^{n + 1} x_\\alpha", "$$", "These identities continue to hold if $\\mathcal{A}$ is any abelian", "category because they hold in the simplicial abelian group", "$[n] \\mapsto \\Hom(A, E_n)$; details left to the reader.", "We conclude that the identity map on $E$ is", "homotopic to zero, with homotopy given by the", "system of maps $h'_n = (-1)^{n + 1}h_n : E_n \\to E_{n + 1}$.", "Hence we see that $E$ is acyclic, for", "example by Homology, Lemma \\ref{homology-lemma-map-homology-homotopy}." ], "refs": [ "homology-lemma-map-homology-homotopy" ], "ref_ids": [ 12056 ] } ], "ref_ids": [ 14859 ] }, { "id": 14863, "type": "theorem", "label": "simplicial-lemma-homology-eilenberg-maclane", "categories": [ "simplicial" ], "title": "simplicial-lemma-homology-eilenberg-maclane", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A$ be an object of $\\mathcal{A}$ and", "let $k$ be an integer. We have", "$H_i(s(K(A, k))) = A$ if $i = k$ and", "$0$ else." ], "refs": [], "proofs": [ { "contents": [ "First, let us prove this if $k = 0$.", "In this case we have $K(A, 0)_n = A$ for all $n$.", "Furthermore, all the maps in this simplicial abelian", "group are $\\text{id}_A$, in other words $K(A, 0)$", "is the constant simplicial object with value $A$.", "The boundary maps $d_n = \\sum_{i = 0}^n (-1)^i \\text{id}_A", "= 0$ if $n$ odd and $ = \\text{id}_A$ if $n$ is even.", "Thus $s(K(A, 0))$ looks like this", "$$", "\\ldots \\to A \\xrightarrow{0} A \\xrightarrow{1} A \\xrightarrow{0} A \\to 0", "$$", "and the result is clear.", "\\medskip\\noindent", "Next, we prove the result for all $k$ by induction.", "Given the result for $k$ consider the short exact sequence", "$$", "0 \\to K(A, k) \\to E \\to K(A, k + 1) \\to 0", "$$", "from Lemma \\ref{lemma-extension}.", "By Lemma \\ref{lemma-abelian} the associated sequence of", "chain complexes is exact.", "By Lemma \\ref{lemma-homology-extension} we see that", "$s(E)$ is acyclic. Hence the result for $k + 1$", "follows from the long exact sequence of homology,", "see Homology, Lemma \\ref{homology-lemma-long-exact-sequence-chain}." ], "refs": [ "simplicial-lemma-extension", "simplicial-lemma-abelian", "simplicial-lemma-homology-extension", "homology-lemma-long-exact-sequence-chain" ], "ref_ids": [ 14859, 14857, 14862, 12057 ] } ], "ref_ids": [] }, { "id": 14864, "type": "theorem", "label": "simplicial-lemma-map-associated-complexes", "categories": [ "simplicial" ], "title": "simplicial-lemma-map-associated-complexes", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $U$ be a simplicial object of $\\mathcal{A}$.", "The canonical map $N(U_n) \\to U_n$ gives rise to", "a morphism of complexes $N(U) \\to s(U)$." ], "refs": [], "proofs": [ { "contents": [ "This is clear because the differential", "on $s(U)_n = U_n$ is $\\sum (-1)^i d^n_i$ and", "the maps $d^n_i$, $i < n$ are zero on $N(U_n)$,", "whereas the restriction of $(-1)^nd^n_n$ is the boundary", "map of $N(U)$ by definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14865, "type": "theorem", "label": "simplicial-lemma-N-K", "categories": [ "simplicial" ], "title": "simplicial-lemma-N-K", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A$ be an object of $\\mathcal{A}$ and", "let $k$ be an integer. We have", "$N(K(A, k))_i = A$ if $i = k$ and", "$0$ else." ], "refs": [], "proofs": [ { "contents": [ "It is clear that $N(K(A, k))_i = 0$ when $i < k$", "because $K(A, k)_i = 0$ in that case.", "It is clear that $N(K(A, k))_k = A$ since", "$K(A, k)_{k - 1} = 0$ and $K(A, k)_k = A$.", "For $i > k$ we have $N(K(A, k))_i = 0$", "by Lemma \\ref{lemma-imshriek-abelian} and", "the definition of $K(A, k)$, see Definition", "\\ref{definition-eilenberg-maclane}." ], "refs": [ "simplicial-lemma-imshriek-abelian", "simplicial-definition-eilenberg-maclane" ], "ref_ids": [ 14854, 14926 ] } ], "ref_ids": [] }, { "id": 14866, "type": "theorem", "label": "simplicial-lemma-decompose-associated-complexes", "categories": [ "simplicial" ], "title": "simplicial-lemma-decompose-associated-complexes", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $U$ be a simplicial object of $\\mathcal{A}$.", "The canonical morphism of chain complexes", "$N(U) \\to s(U)$ is split. In fact,", "$$", "s(U) = N(U) \\oplus D(U)", "$$", "for some complex $D(U)$. The construction $U \\mapsto D(U)$", "is functorial." ], "refs": [], "proofs": [ { "contents": [ "Define $D(U)_n$ to be the image of", "$$", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]\\text{ surjective}, \\ m < n} N(U_m)", "\\xrightarrow{\\bigoplus U(\\varphi)}", "U_n", "$$", "which is a subobject of $U_n$ complementary to", "$N(U_n)$ according to Lemma \\ref{lemma-splitting-abelian-category} and", "Definition \\ref{definition-split}. We show that", "$D(U)$ is a subcomplex. Pick a surjective", "map $\\varphi : [n] \\to [m]$ with $m < n$ and consider", "the composition", "$$", "N(U_m) \\xrightarrow{U(\\varphi)} U_n \\xrightarrow{d_n} U_{n - 1}.", "$$", "This composition is the sum of the maps", "$$", "N(U_m) \\xrightarrow{U(\\varphi \\circ \\delta^n_i)} U_{n - 1}", "$$", "with sign $(-1)^i$, $i = 0, \\ldots, n$.", "\\medskip\\noindent", "First we will prove by ascending induction on $m$,", "$0 \\leq m < n - 1$ that all the maps $U(\\varphi \\circ \\delta^n_i)$", "map $N(U_m)$ into $D(U)_{n - 1}$. (The case $m = n - 1$ is treated below.)", "Whenever the map $\\varphi \\circ \\delta^n_i : [n - 1] \\to [m]$", "is surjective then the image of $N(U_m)$ under $U(\\varphi \\circ \\delta^n_i)$", "is contained in $D(U)_{n - 1}$ by definition.", "If $\\varphi \\circ \\delta^n_i : [n - 1] \\to [m]$ is not surjective,", "set $j = \\varphi(i)$ and observe that $i$ is the unique", "index whose image under $\\varphi$ is $j$. We may write", "$\\varphi \\circ \\delta^n_i = \\delta^m_j \\circ \\psi \\circ \\delta^n_i$", "for some $\\psi : [n - 1] \\to [m - 1]$. Hence", "$U(\\varphi \\circ \\delta^n_i) = U(\\psi \\circ \\delta^n_i) \\circ d^m_j$ which", "is zero on $N(U_m)$ unless $j = m$. If $j = m$, then", "$d^m_m(N(U_m)) \\subset N(U_{m - 1})$ and hence", "$U(\\varphi \\circ \\delta^n_i)(N(U_m)) \\subset", "U(\\psi \\circ \\delta^n_i)(N(U_{m - 1}))$ and we win", "by induction hypothesis.", "\\medskip\\noindent", "To finish proving that $D(U)$ is a subcomplex", "we still have to deal with the composition", "$$", "N(U_m) \\xrightarrow{U(\\varphi)} U_n \\xrightarrow{d_n} U_{n - 1}.", "$$", "in case $m = n - 1$. In this case $\\varphi = \\sigma^{n - 1}_j$", "for some $0 \\leq j \\leq n - 1$ and $U(\\varphi) = s^{n - 1}_j$.", "Thus the composition is given by the sum", "$$", "\\sum (-1)^i d^n_i \\circ s^{n - 1}_j", "$$", "Recall from Remark \\ref{remark-relations} that", "$d^n_j \\circ s^{n - 1}_j = d^n_{j + 1} \\circ s^{n - 1}_j = \\text{id}$", "and these drop out because the corresponding terms have opposite signs.", "The map $d^n_n \\circ s^{n - 1}_j$, if $j < n - 1$, is equal to", "$s^{n - 2}_j \\circ d^{n - 1}_{n - 1}$. Since", "$d^{n - 1}_{n - 1}$ maps $N(U_{n - 1})$ into $N(U_{n - 2})$,", "we see that the image $d^n_n ( s^{n - 1}_j (N(U_{n - 1}))$", "is contained in $s^{n - 2}_j(N(U_{n - 2}))$ which", "is contained in $D(U_{n - 1})$ by definition. For all", "other combinations of $(i, j)$ we have", "either $d^n_i \\circ s^{n - 1}_j = s^{n - 2}_{j - 1} \\circ d^{n - 1}_i$", "(if $i < j$), or", "$d^n_i \\circ s^{n - 1}_j = s^{n - 2}_j \\circ d^{n - 1}_{i - 1}$", "(if $n > i > j + 1$) and in these cases the map is zero because", "of the definition of $N(U_{n - 1})$." ], "refs": [ "simplicial-lemma-splitting-abelian-category", "simplicial-definition-split", "simplicial-remark-relations" ], "ref_ids": [ 14831, 14924, 14932 ] } ], "ref_ids": [] }, { "id": 14867, "type": "theorem", "label": "simplicial-lemma-N-exact", "categories": [ "simplicial" ], "title": "simplicial-lemma-N-exact", "contents": [ "The functor $N$ is exact." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-s-exact} and the functorial", "decomposition of Lemma \\ref{lemma-decompose-associated-complexes}." ], "refs": [ "simplicial-lemma-s-exact", "simplicial-lemma-decompose-associated-complexes" ], "ref_ids": [ 14861, 14866 ] } ], "ref_ids": [] }, { "id": 14868, "type": "theorem", "label": "simplicial-lemma-quasi-isomorphism", "categories": [ "simplicial" ], "title": "simplicial-lemma-quasi-isomorphism", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $V$ be a simplicial object of $\\mathcal{A}$.", "The canonical morphism of chain complexes", "$N(V) \\to s(V)$ is a quasi-isomorphism.", "In other words, the complex $D(V)$ of Lemma", "\\ref{lemma-decompose-associated-complexes} is acyclic." ], "refs": [ "simplicial-lemma-decompose-associated-complexes" ], "proofs": [ { "contents": [ "Note that the result holds for $K(A, k)$ for", "any object $A$ and any $k \\geq 0$, by Lemmas", "\\ref{lemma-homology-eilenberg-maclane} and \\ref{lemma-N-K}.", "Consider the hypothesis $IH_{n, m}$:", "for all $V$ such that $V_j = 0$ for", "$j \\leq m$ and all $i \\leq n$ the map", "$N(V) \\to s(V)$ induces an isomorphism", "$H_i(N(V)) \\to H_i(s(V))$.", "\\medskip\\noindent", "To start of the induction, note that $IH_{n, n}$", "is trivially true, because in that case $N(V)_n = 0$", "and $s(V)_n = 0$.", "\\medskip\\noindent", "Assume $IH_{n, m}$, with $m \\leq n$.", "Pick a simplicial object $V$ such that", "$V_j = 0$ for $j < m$. By Lemma \\ref{lemma-eilenberg-maclane-object}", "and Definition \\ref{definition-eilenberg-maclane}", "we have $K(V_m, m) = i_{m!} \\text{sk}_mV$.", "By Lemma \\ref{lemma-n-skeleton-abelian} the natural morphism", "$$", "K(V_m, m) = i_{m!} \\text{sk}_mV \\to V", "$$", "is injective. Thus we get a short exact sequence", "$$", "0 \\to K(V_m, m) \\to V \\to W \\to 0", "$$", "for some $W$ with $W_i = 0$ for $i = 0, \\ldots, m$. This short exact sequence", "induces a morphism of short exact sequence of associated complexes", "$$", "\\xymatrix{", "0 \\ar[r] &", "N(K(V_m, m)) \\ar[r] \\ar[d] &", "N(V) \\ar[r] \\ar[d] &", "N(W) \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "s(K(V_m, m)) \\ar[r] &", "s(V) \\ar[r] &", "s(W) \\ar[r] &", "0", "}", "$$", "see Lemmas \\ref{lemma-s-exact} and \\ref{lemma-N-exact}.", "Hence we deduce the result for $V$ from the result", "on the ends." ], "refs": [ "simplicial-lemma-homology-eilenberg-maclane", "simplicial-lemma-N-K", "simplicial-lemma-eilenberg-maclane-object", "simplicial-definition-eilenberg-maclane", "simplicial-lemma-n-skeleton-abelian", "simplicial-lemma-s-exact", "simplicial-lemma-N-exact" ], "ref_ids": [ 14863, 14865, 14858, 14926, 14855, 14861, 14867 ] } ], "ref_ids": [ 14866 ] }, { "id": 14869, "type": "theorem", "label": "simplicial-lemma-N-faithful", "categories": [ "simplicial" ], "title": "simplicial-lemma-N-faithful", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "The functor $N$ is faithful, and reflects", "isomorphisms, injections and surjections." ], "refs": [], "proofs": [ { "contents": [ "The faithfulness is immediate from the canonical", "splitting of Lemma \\ref{lemma-splitting-abelian-category}.", "The statement on reflecting injections, surjections, and", "isomorphisms follows from", "Lemma \\ref{lemma-injective-map-simplicial-abelian}." ], "refs": [ "simplicial-lemma-splitting-abelian-category", "simplicial-lemma-injective-map-simplicial-abelian" ], "ref_ids": [ 14831, 14832 ] } ], "ref_ids": [] }, { "id": 14870, "type": "theorem", "label": "simplicial-lemma-S-N", "categories": [ "simplicial" ], "title": "simplicial-lemma-S-N", "contents": [ "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.", "Let $N : \\mathcal{A} \\to \\mathcal{B}$, and", "$S : \\mathcal{B} \\to \\mathcal{A}$ be functors.", "Suppose that", "\\begin{enumerate}", "\\item the functors $S$ and $N$ are exact,", "\\item there is an isomorphism $g : N \\circ S \\to \\text{id}_\\mathcal{B}$", "to the identity functor of $\\mathcal{B}$,", "\\item $N$ is faithful, and", "\\item $S$ is essentially surjective.", "\\end{enumerate}", "Then $S$ and $N$ are quasi-inverse equivalences of categories." ], "refs": [], "proofs": [ { "contents": [ "It suffices to construct a functorial", "isomorphism $S(N(A)) \\cong A$. To do this", "choose $B$ and an isomorphism $f : A \\to S(B)$.", "Consider the map", "$$", "f^{-1} \\circ g_{S(B)} \\circ S(N(f)) :", "S(N(A)) \\to S(N(S(B))) \\to S(B) \\to A.", "$$", "It is easy to show this does not depend on", "the choice of $f, B$ and gives the desired", "isomorphism $S \\circ N \\to \\text{id}_\\mathcal{A}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14871, "type": "theorem", "label": "simplicial-lemma-dual-dold-kan", "categories": [ "simplicial" ], "title": "simplicial-lemma-dual-dold-kan", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item The functor", "$s : \\text{CoSimp}(\\mathcal{A}) \\to \\text{CoCh}_{\\geq 0}(\\mathcal{A})$", "is exact.", "\\item The maps $s(U)^n \\to Q(U)^n$ define a morphism", "of cochain complexes.", "\\item There exists a functorial direct sum decomposition", "$s(U) = D(U) \\oplus Q(U)$ in $\\text{CoCh}_{\\geq 0}(\\mathcal{A})$.", "\\item The functor $Q$ is exact.", "\\item The morphism of complexes $s(U) \\to Q(U)$ is a quasi-isomorphism.", "\\item The functor $U \\mapsto Q(U)^\\bullet$ defines", "an equivalence of categories", "$\\text{CoSimp}(\\mathcal{A}) \\to \\text{CoCh}_{\\geq 0}(\\mathcal{A})$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted. But the results are the exact dual statements to", "Lemmas \\ref{lemma-s-exact}, \\ref{lemma-map-associated-complexes},", "\\ref{lemma-decompose-associated-complexes},", "\\ref{lemma-N-exact}, \\ref{lemma-quasi-isomorphism}, and", "Theorem \\ref{theorem-dold-kan}." ], "refs": [ "simplicial-lemma-s-exact", "simplicial-lemma-map-associated-complexes", "simplicial-lemma-decompose-associated-complexes", "simplicial-lemma-N-exact", "simplicial-lemma-quasi-isomorphism", "simplicial-theorem-dold-kan" ], "ref_ids": [ 14861, 14864, 14866, 14867, 14868, 14804 ] } ], "ref_ids": [] }, { "id": 14872, "type": "theorem", "label": "simplicial-lemma-relations-homotopy", "categories": [ "simplicial" ], "title": "simplicial-lemma-relations-homotopy", "contents": [ "In the situation above, we have the following relations:", "\\begin{enumerate}", "\\item We have $h_{n, 0} = b_n$ and $h_{n, n + 1} = a_n$.", "\\item We have $d^n_j \\circ h_{n, i} = h_{n - 1, i - 1} \\circ d^n_j$", "for $i > j$.", "\\item We have $d^n_j \\circ h_{n, i} = h_{n - 1, i} \\circ d^n_j$", "for $i \\leq j$.", "\\item We have $s^n_j \\circ h_{n, i} = h_{n + 1, i + 1} \\circ s^n_j$", "for $i > j$.", "\\item We have $s^n_j \\circ h_{n, i} = h_{n + 1, i} \\circ s^n_j$", "for $i \\leq j$.", "\\end{enumerate}", "Conversely, given a system of maps $h_{n, i}$ satisfying the", "properties listed above, then these define a morphism", "$h$ which is a homotopy from $a$ to $b$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. You can prove the last statement using the fact,", "see Lemma \\ref{lemma-face-degeneracy-category} that", "to give a morphism of simplicial objects is the", "same as giving a sequence of morphisms $h_n$ commuting", "with all $d^n_j$ and $s^n_j$." ], "refs": [ "simplicial-lemma-face-degeneracy-category" ], "ref_ids": [ 14807 ] } ], "ref_ids": [] }, { "id": 14873, "type": "theorem", "label": "simplicial-lemma-contractible", "categories": [ "simplicial" ], "title": "simplicial-lemma-contractible", "contents": [ "Let $\\mathcal{C}$ be a category with finite coproducts.", "Let $U$ be a simplicial object of $\\mathcal{C}$.", "Consider the maps $e_1, e_0 : U \\to U \\times \\Delta[1]$,", "and $\\pi : U \\times \\Delta[1] \\to U$, see", "Lemma \\ref{lemma-back-to-U}.", "\\begin{enumerate}", "\\item We have $\\pi \\circ e_1 = \\pi \\circ e_0 = \\text{id}_U$, and", "\\item The morphisms $\\text{id}_{U \\times \\Delta[1]}$,", "and $e_0 \\circ \\pi$ are homotopic.", "\\item The morphisms $\\text{id}_{U \\times \\Delta[1]}$,", "and $e_1 \\circ \\pi$ are homotopic.", "\\end{enumerate}" ], "refs": [ "simplicial-lemma-back-to-U" ], "proofs": [ { "contents": [ "The first assertion is trivial.", "For the second, consider the map", "of simplicial sets", "$\\Delta[1] \\times \\Delta[1] \\longrightarrow \\Delta[1]$", "which in degree $n$ assigns to a pair $(\\beta_1, \\beta_2)$,", "$\\beta_i : [n] \\to [1]$ the morphism", "$\\beta : [n] \\to [1]$ defined by the rule", "$$", "\\beta(i) = \\max\\{\\beta_1(i), \\beta_2(i)\\}.", "$$", "It is a morphism of simplicial sets, because the action", "$\\Delta[1](\\varphi) : \\Delta[1]_n \\to \\Delta[1]_m$", "of $\\varphi : [m] \\to [n]$ is by precomposing.", "Clearly, using notation from Section \\ref{section-homotopy},", "we have $\\beta = \\beta_1$ if $\\beta_2 = \\alpha^n_0$", "and $\\beta = \\alpha^n_{n + 1}$ if $\\beta_2 = \\alpha^n_{n + 1}$.", "This implies easily that the induced morphism", "$$", "U \\times \\Delta[1] \\times \\Delta[1]", "\\longrightarrow", "U \\times \\Delta[1]", "$$", "of Lemma \\ref{lemma-back-to-U}", "is a homotopy from $\\text{id}_{U \\times \\Delta[1]}$ to $e_0 \\circ \\pi$.", "Similarly for $e_1 \\circ \\pi$ (use minimum instead of maximum)." ], "refs": [ "simplicial-lemma-back-to-U" ], "ref_ids": [ 14820 ] } ], "ref_ids": [ 14820 ] }, { "id": 14874, "type": "theorem", "label": "simplicial-lemma-fibre-products-simplicial-object-w-section", "categories": [ "simplicial" ], "title": "simplicial-lemma-fibre-products-simplicial-object-w-section", "contents": [ "Let $f : Y \\to X$ be a morphism of a category $\\mathcal{C}$ with", "fibre products. Assume $f$ has a section $s$. Consider the", "simplicial object $U$ constructed in", "Example \\ref{example-fibre-products-simplicial-object}", "starting with $f$. The morphism $U \\to U$ which in each degree", "is the self map $(s \\circ f)^{n + 1}$ of $Y \\times_X \\ldots \\times_X Y$", "given by $s \\circ f$ on each factor is homotopic to the identity on $U$.", "In particular, $U$ is homotopy equivalent to the constant", "simplicial object $X$." ], "refs": [], "proofs": [ { "contents": [ "Set $g^0 = \\text{id}_Y$ and $g^1 = s \\circ f$.", "We use the morphisms", "\\begin{eqnarray*}", "Y \\times_X \\ldots \\times_X Y \\times \\Mor([n], [1])", "& \\to &", "Y \\times_X \\ldots \\times_X Y \\\\", "(y_0, \\ldots, y_n) \\times \\alpha", "& \\mapsto &", "(g^{\\alpha(0)}(y_0), \\ldots, g^{\\alpha(n)}(y_n))", "\\end{eqnarray*}", "where we use the functor of points point of view to define the maps.", "Another way to say this is to say that", "$h_{n, 0} = \\text{id}$, $h_{n, n + 1} = (s \\circ f)^{n + 1}$ and", "$h_{n, i} = \\text{id}_Y^{i + 1} \\times (s \\circ f)^{n + 1 - i}$.", "We leave it to the reader to show that these satisfy the relations", "of Lemma \\ref{lemma-relations-homotopy}. Hence they define", "the desired homotopy. See also Remark \\ref{remark-homotopy-better}", "which shows that", "we do not need to assume anything else on the category $\\mathcal{C}$." ], "refs": [ "simplicial-lemma-relations-homotopy", "simplicial-remark-homotopy-better" ], "ref_ids": [ 14872, 14941 ] } ], "ref_ids": [] }, { "id": 14875, "type": "theorem", "label": "simplicial-lemma-products-homotopy", "categories": [ "simplicial" ], "title": "simplicial-lemma-products-homotopy", "contents": [ "Let $\\mathcal{C}$ be a category. Let $T$ be a set. For $t \\in T$", "let $X_t$, $Y_t$ be simplicial objects of $\\mathcal{C}$. Assume", "$X = \\prod_{t \\in T} X_t$ and $Y = \\prod_{t \\in T} Y_t$ exist.", "\\begin{enumerate}", "\\item If $X_t$ and $Y_t$ are homotopy equivalent for all $t \\in T$", "and $T$ is finite, then $X$ and $Y$ are homotopy equivalent.", "\\end{enumerate}", "For $t \\in T$ let $a_t, b_t : X_t \\to Y_t$ be morphisms.", "Set $a = \\prod a_t : X \\to Y$ and $b = \\prod b_t : X \\to Y$.", "\\begin{enumerate}", "\\item[(2)] If there exists a homotopy from $a_t$ to $b_t$ for", "all $t \\in T$, then there exists a homotopy from $a$ to $b$.", "\\item[(3)] If $T$ is finite and $a_t, b_t : X_t \\to Y_t$ for $t \\in T$", "are homotopic, then $a$ and $b$ are homotopic.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "If $h_t = (h_{t, n , i})$ is a homotopy from $a_t$ to $b_t$", "(see Remark \\ref{remark-homotopy-better}), then", "$h = (\\prod_t h_{t, n, i})$ is a homotopy from $\\prod a_t$ to $\\prod b_t$.", "This proves (2).", "\\medskip\\noindent", "Proof of (3). Choose $t \\in T$. There exists an integer $n \\geq 0$", "and a chain $a_t = a_{t, 0}, a_{t, 1}, \\ldots, a_{t, n} = b_t$ such that", "for every $1 \\leq i \\leq n$ either there is a homotopy from", "$a_{t, i - 1}$ to $a_{t, i}$ or there is a homotopy from", "$a_{t, i}$ to $a_{t, i - 1}$. If $n = 0$, then we pick another $t$.", "(We're done if $a_t = b_t$ for all $t \\in T$.) So assume $n > 0$.", "By Example \\ref{example-trivial-homotopy}", "there are is a homotopy from $b_{t'}$ to $b_{t'}$ for all", "$t' \\in T \\setminus \\{t\\}$. Thus by (2) there is a homotopy from", "$a_{t, n - 1} \\times \\prod_{t'} b_{t'}$ to $b$ or there is a", "homotopy from $b$ to $a_{t, n - 1} \\times \\prod_{t'} b_{t'}$.", "In this way we can decrease $n$ by $1$. This proves (3).", "\\medskip\\noindent", "Part (1) follows from part (3) and the definitions." ], "refs": [ "simplicial-remark-homotopy-better" ], "ref_ids": [ 14941 ] } ], "ref_ids": [] }, { "id": 14876, "type": "theorem", "label": "simplicial-lemma-homotopy-s-N", "categories": [ "simplicial" ], "title": "simplicial-lemma-homotopy-s-N", "contents": [ "Let $\\mathcal{A}$ be an additive category. Let $a, b : U \\to V$ be morphisms", "of simplicial objects of $\\mathcal{A}$. If $a$, $b$ are homotopic,", "then $s(a), s(b) : s(U) \\to s(V)$ are homotopic maps of chain complexes.", "If $\\mathcal{A}$ is abelian, then also $N(a), N(b) : N(U) \\to N(V)$ are", "homotopic maps of chain complexes." ], "refs": [], "proofs": [ { "contents": [ "We may choose a sequence $a = a_0, a_1, \\ldots, a_n = b$ of morphisms from", "$U$ to $V$ such that for each $i = 1, \\ldots, n$ either there is a", "homotopy from $a_i$ to $a_{i - 1}$ or there is a", "homotopy from $a_{i - 1}$ to $a_i$. The calculation above", "shows that in this case either", "$s(a_i)$ is homotopic to $s(a_{i - 1})$ as a map of chain complexes", "or $s(a_{i - 1})$ is homotopic to $s(a_i)$ as a map of chain complexes.", "Of course, these things are equivalent and moreover being homotopic", "is an equivalence relation on the set of maps of chain complexes, see", "Homology, Section \\ref{homology-section-complexes}.", "This proves that $s(a)$ and $s(b)$ are homotopic as maps", "of chain complexes.", "\\medskip\\noindent", "Next, we turn to $N(a)$ and $N(b)$. It follows from", "Lemma \\ref{lemma-decompose-associated-complexes} that", "$N(a)$, $N(b)$ are compositions", "$$", "N(U) \\to s(U) \\to s(V) \\to N(V)", "$$", "where we use $s(a)$, $s(b)$ in the middle. Hence the assertion", "follows from", "Homology, Lemma \\ref{homology-lemma-compose-homotopy}." ], "refs": [ "simplicial-lemma-decompose-associated-complexes", "homology-lemma-compose-homotopy" ], "ref_ids": [ 14866, 12054 ] } ], "ref_ids": [] }, { "id": 14877, "type": "theorem", "label": "simplicial-lemma-homotopy-equivalence-s-N", "categories": [ "simplicial" ], "title": "simplicial-lemma-homotopy-equivalence-s-N", "contents": [ "Let $\\mathcal{A}$ be an additive category. Let $a : U \\to V$ be a morphism", "of simplicial objects of $\\mathcal{A}$. If $a$ is a homotopy equivalence,", "then $s(a) : s(U) \\to s(V)$ is a homotopy equivalence of chain complexes.", "If in addition $\\mathcal{A}$ is abelian, then also", "$N(a) : N(U) \\to N(V)$ is a homotopy equivalence of chain complexes." ], "refs": [], "proofs": [ { "contents": [ "Omitted. See Lemma \\ref{lemma-homotopy-s-N} above." ], "refs": [ "simplicial-lemma-homotopy-s-N" ], "ref_ids": [ 14876 ] } ], "ref_ids": [] }, { "id": 14878, "type": "theorem", "label": "simplicial-lemma-compare-homotopies", "categories": [ "simplicial" ], "title": "simplicial-lemma-compare-homotopies", "contents": [ "Let $\\mathcal{C}$ be a category. Suppose that $U$ and $V$ are two", "cosimplicial objects of $\\mathcal{C}$. Let $a, b : U \\to V$ be morphisms", "of cosimplicial objects. Recall that $U$, $V$ correspond", "to simplicial objects $U'$, $V'$ of $\\mathcal{C}^{opp}$.", "Moreover $a, b$ correspond to morphisms $a', b' : V' \\to U'$.", "The following are equivalent", "\\begin{enumerate}", "\\item There exists a homotopy $h = \\{h_{n, \\alpha}\\}$ from", "$a$ to $b$ as in Remark \\ref{remark-homotopy-cosimplicial-better}.", "\\item There exists a homotopy $h = \\{h_{n, i}\\}$ from $a'$ to $b'$", "as in Remark \\ref{remark-homotopy-better}.", "\\end{enumerate}", "Thus $a$ is homotopic to $b$ as in", "Remark \\ref{remark-homotopy-cosimplicial-better}", "if and only if $a'$ is homotopic to $b'$ as in", "Remark \\ref{remark-homotopy-better}." ], "refs": [ "simplicial-remark-homotopy-cosimplicial-better", "simplicial-remark-homotopy-better", "simplicial-remark-homotopy-cosimplicial-better", "simplicial-remark-homotopy-better" ], "proofs": [ { "contents": [ "In case $\\mathcal{C}$ has finite products, then $\\mathcal{C}^{opp}$", "has finite coproducts and we may use", "Definitions \\ref{definition-homotopy-cosimplicial}", "and \\ref{definition-homotopy} instead of", "Remarks \\ref{remark-homotopy-cosimplicial-better} and", "\\ref{remark-homotopy-better}.", "In this case $h : U \\to \\Hom(\\Delta[1], V)$", "is the same as a morphism $h' : \\Hom(\\Delta[1], V)' \\to U'$.", "Since products and coproducts get switched too, it is immediate", "that $(\\Hom(\\Delta[1], V))' = V' \\times \\Delta[1]$.", "Moreover, the ``primed'' version of the morphisms", "$e_0, e_1 : \\Hom(\\Delta[1], V) \\to V$ are the morphisms", "$e_0, e_1 : V' \\to \\Delta[1] \\times V$.", "Thus $e_0 \\circ h = a$ translates into $h' \\circ e_0 = a'$ and", "similarly $e_1 \\circ h = b$ translates into $h' \\circ e_1 = b'$.", "This proves the lemma in this case.", "\\medskip\\noindent", "In the general case, one needs to translate the relations given", "by (\\ref{equation-property-homotopy-cosimplicial}) into the relations", "given in Lemma \\ref{lemma-relations-homotopy}. We omit the details.", "\\medskip\\noindent", "The final assertion is formal from the equivalence of (1) and (2)." ], "refs": [ "simplicial-definition-homotopy-cosimplicial", "simplicial-definition-homotopy", "simplicial-remark-homotopy-cosimplicial-better", "simplicial-remark-homotopy-better", "simplicial-lemma-relations-homotopy" ], "ref_ids": [ 14929, 14927, 14943, 14941, 14872 ] } ], "ref_ids": [ 14943, 14941, 14943, 14941 ] }, { "id": 14879, "type": "theorem", "label": "simplicial-lemma-functorial-homotopy", "categories": [ "simplicial" ], "title": "simplicial-lemma-functorial-homotopy", "contents": [ "\\begin{slogan}", "Functors preserve homotopic morphisms of (co)simplicial objects.", "\\end{slogan}", "Let $\\mathcal{C}, \\mathcal{C}', \\mathcal{D}, \\mathcal{D}'$ be categories.", "With terminology as in Remarks \\ref{remark-homotopy-cosimplicial-better} and", "\\ref{remark-homotopy-better}.", "\\begin{enumerate}", "\\item Let $a, b : U \\to V$ be morphisms of simplicial objects", "of $\\mathcal{D}$. Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be a covariant", "functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$", "are homotopic morphisms $F(U) \\to F(V)$ of simplicial objects.", "\\item Let $a, b : U \\to V$ be morphisms of cosimplicial objects", "of $\\mathcal{C}$. Let $F : \\mathcal{C} \\to \\mathcal{C}'$ be a covariant", "functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$", "are homotopic morphisms $F(U) \\to F(V)$ of cosimplicial objects.", "\\item Let $a, b : U \\to V$ be morphisms of simplicial objects of $\\mathcal{D}$.", "Let $F : \\mathcal{D} \\to \\mathcal{C}$ be a contravariant", "functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$", "are homotopic morphisms $F(V) \\to F(U)$ of cosimplicial objects.", "\\item Let $a, b : U \\to V$ be morphisms of cosimplicial objects of", "$\\mathcal{C}$.", "Let $F : \\mathcal{C} \\to \\mathcal{D}$ be a contravariant", "functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$", "are homotopic morphisms $F(V) \\to F(U)$ of simplicial objects.", "\\end{enumerate}" ], "refs": [ "simplicial-remark-homotopy-cosimplicial-better", "simplicial-remark-homotopy-better" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-compare-homotopies} above, we can", "turn $F$ into a covariant functor between a pair of", "categories, and we have to show that the functor preserves homotopic pairs", "of maps. This is explained in Remark \\ref{remark-homotopy-better}." ], "refs": [ "simplicial-lemma-compare-homotopies", "simplicial-remark-homotopy-better" ], "ref_ids": [ 14878, 14941 ] } ], "ref_ids": [ 14943, 14941 ] }, { "id": 14880, "type": "theorem", "label": "simplicial-lemma-push-outs-simplicial-object-w-section", "categories": [ "simplicial" ], "title": "simplicial-lemma-push-outs-simplicial-object-w-section", "contents": [ "Let $f : X \\to Y$ be a morphism of a category $\\mathcal{C}$ with", "pushouts. Assume there is a morphism $s : Y \\to X$ with", "$s \\circ f = \\text{id}_X$. Consider the cosimplicial object $U$ constructed in", "Example \\ref{example-push-outs-simplicial-object}", "starting with $f$. The morphism $U \\to U$ which in each degree", "is the self map of $Y \\amalg_X \\ldots \\amalg_X Y$", "given by $f \\circ s$ on each factor is homotopic to the identity on $U$.", "In particular, $U$ is homotopy equivalent to the constant", "cosimplicial object $X$." ], "refs": [], "proofs": [ { "contents": [ "This lemma is dual to", "Lemma \\ref{lemma-fibre-products-simplicial-object-w-section}.", "Hence this lemma follows on applying", "Lemma \\ref{lemma-compare-homotopies}." ], "refs": [ "simplicial-lemma-compare-homotopies" ], "ref_ids": [ 14878 ] } ], "ref_ids": [] }, { "id": 14881, "type": "theorem", "label": "simplicial-lemma-homotopy-s-Q", "categories": [ "simplicial" ], "title": "simplicial-lemma-homotopy-s-Q", "contents": [ "\\begin{slogan}", "The (cosimplicial) Dold-Kan functor carries homotopic maps to homotopic maps.", "\\end{slogan}", "Let $\\mathcal{A}$ be an additive category. Let $a, b : U \\to V$ be morphisms", "of cosimplicial objects of $\\mathcal{A}$. If $a$, $b$ are homotopic,", "then $s(a), s(b) : s(U) \\to s(V)$ are homotopic maps of cochain complexes.", "If in addition $\\mathcal{A}$ is abelian, then $Q(a), Q(b) : Q(U) \\to Q(V)$", "are homotopic maps of cochain complexes." ], "refs": [], "proofs": [ { "contents": [ "Let $(-)' : \\mathcal{A} \\to \\mathcal{A}^{opp}$", "be the contravariant functor $A \\mapsto A$. By", "Lemma \\ref{lemma-push-outs-simplicial-object-w-section}", "the maps $a'$ and $b'$ are homotopic.", "By Lemma \\ref{lemma-homotopy-s-N} we see that", "$s(a')$ and $s(b')$ are homotopic maps of chain", "complexes. Since $s(a') = (s(a))'$ and", "$s(b') = (s(b))'$ we conclude that also", "$s(a)$ and $s(b)$ are homotopic by applying the additive", "contravariant functor $(-)'' : \\mathcal{A}^{opp} \\to \\mathcal{A}$.", "The result for the $Q$-complexes follows in the same manner", "using that $Q(U)' = N(U')$." ], "refs": [ "simplicial-lemma-homotopy-s-N" ], "ref_ids": [ 14876 ] } ], "ref_ids": [] }, { "id": 14882, "type": "theorem", "label": "simplicial-lemma-homotopy-equivalence-s-Q", "categories": [ "simplicial" ], "title": "simplicial-lemma-homotopy-equivalence-s-Q", "contents": [ "Let $\\mathcal{A}$ be an additive category. Let $a : U \\to V$ be a morphism", "of cosimplicial objects of $\\mathcal{A}$. If $a$ is a homotopy equivalence,", "then $s(a) : s(U) \\to s(V)$ is a homotopy equivalence of chain complexes.", "If in addition $\\mathcal{A}$ is abelian, then also", "$Q(a) : Q(U) \\to Q(V)$ is a homotopy equivalence of chain complexes." ], "refs": [], "proofs": [ { "contents": [ "Omitted. See Lemma \\ref{lemma-homotopy-s-Q} above." ], "refs": [ "simplicial-lemma-homotopy-s-Q" ], "ref_ids": [ 14881 ] } ], "ref_ids": [] }, { "id": 14883, "type": "theorem", "label": "simplicial-lemma-represent-homotopy", "categories": [ "simplicial" ], "title": "simplicial-lemma-represent-homotopy", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $A$ be a chain complex.", "Consider the covariant functor", "$$", "B \\longmapsto", "\\{", "(a, b, h)", "\\mid", "a, b : A \\to B\\text{ and }h\\text{ a homotopy between }a, b", "\\}", "$$", "There exists a chain complex $\\diamond A$", "such that $\\Mor_{\\text{Ch}(\\mathcal{A})}(\\diamond A, -)$", "is isomorphic to the displayed functor.", "The construction $A \\mapsto \\diamond A$ is functorial." ], "refs": [], "proofs": [ { "contents": [ "We set $\\diamond A_n = A_n \\oplus A_n \\oplus A_{n - 1}$,", "and we define $d_{\\diamond A, n}$ by the matrix", "$$", "d_{\\diamond A, n}", "=", "\\left(", "\\begin{matrix}", "d_{A, n} & 0 & \\text{id}_{A_{n - 1}} \\\\", "0 & d_{A, n} & -\\text{id}_{A_{n - 1}} \\\\", "0 & 0 & -d_{A, n - 1}", "\\end{matrix}", "\\right) :", "A_n \\oplus A_n \\oplus A_{n - 1} \\to", "A_{n - 1} \\oplus A_{n - 1} \\oplus A_{n - 2}", "$$", "If $\\mathcal{A}$ is the category of abelian groups,", "and $(x, y, z) \\in A_n \\oplus A_n \\oplus A_{n - 1}$", "then $d_{\\diamond A, n}(x, y, z) = (d_n(x) + z, d_n(y) - z, -d_{n - 1}(z))$.", "It is easy to verify that $d^2 = 0$. Clearly,", "there are two maps $\\diamond a, \\diamond b : A \\to \\diamond A$", "(first summand and second summand),", "and a map $\\diamond A \\to A[-1]$ which give", "a short exact sequence", "$$", "0 \\to", "A \\oplus A \\to", "\\diamond A \\to", "A[-1] \\to", "0", "$$", "which is termwise split. Moreover, there is a sequence", "of maps $\\diamond h_n : A_n \\to \\diamond A_{n + 1}$, namely", "the identity from $A_n$ to the summand $A_n$ of $\\diamond A_{n + 1}$,", "such that $\\diamond h$ is a homotopy between $\\diamond a$ and $\\diamond b$.", "\\medskip\\noindent", "We conclude that any morphism $f : \\diamond A \\to B$", "gives rise to a triple $(a, b, h)$ by setting $a = f \\circ \\diamond a$,", "$b = f \\circ \\diamond b$ and $h_n = f_{n + 1} \\circ \\diamond h_n$.", "Conversely, given a triple $(a, b, h)$ we get a morphism", "$f : \\diamond A \\to B$ by taking", "$$", "f_n = (a_n, b_n, h_{n - 1}).", "$$", "To see that this is a morphism of chain complexes you have", "to do a calculation. We only do this in case $\\mathcal{A}$", "is the category of abelian groups:", "Say $(x, y, z) \\in \\diamond A_n = A_n \\oplus A_n \\oplus A_{n - 1}$.", "Then", "\\begin{eqnarray*}", "f_{n - 1}(d_n(x, y, z)) & = &", "f_{n - 1}(d_n(x) + z, d_n(y) - z, -d_{n - 1}(z)) \\\\", "& = &", "a_n(d_n(x)) + a_n(z) + b_n(d_n(y)) - b_n(z) - h_{n - 2}(d_{n - 1}(z))", "\\end{eqnarray*}", "and", "\\begin{eqnarray*}", "d_n(f_n(x, y, z) & = &", "d_n(a_n(x) + b_n(y) + h_{n - 1}(z)) \\\\", "& = &", "d_n(a_n(x)) + d_n(b_n(y)) + d_n(h_{n - 1}(z))", "\\end{eqnarray*}", "which are the same by definition of a homotopy." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14884, "type": "theorem", "label": "simplicial-lemma-map-into-diamond", "categories": [ "simplicial" ], "title": "simplicial-lemma-map-into-diamond", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let", "$$", "0 \\to A \\oplus A \\to B \\to C \\to 0", "$$", "be a short exact sequence of chain complexes of $\\mathcal{A}$.", "Suppose given in addition morphisms $s_n : C_n \\to B_n$", "splitting the associated short exact sequence in degree $n$.", "Let $\\delta(s) : C \\to (A \\oplus A)[-1] = A[-1] \\oplus A[-1]$", "be the associated morphism of complexes, see", "Homology, Lemma \\ref{homology-lemma-ses-termwise-split}.", "If $\\delta(s)$ factors through the morphism", "$(1, -1) : A[-1] \\to A[-1] \\oplus A[-1]$, then", "there is a unique morphism $B \\to \\diamond A$", "fitting into a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "A \\oplus A \\ar[d] \\ar[r] &", "B \\ar[r] \\ar[d] &", "C \\ar[d] \\ar[r] &", "0 \\\\", "0 \\ar[r] &", "A \\oplus A \\ar[r] &", "\\diamond A \\ar[r] &", "A[-1] \\ar[r] &", "0", "}", "$$", "where the vertical maps are compatible with the splittings", "$s_n$ and the splittings of $\\diamond A_n \\to A[-1]_n$", "as well." ], "refs": [ "homology-lemma-ses-termwise-split" ], "proofs": [ { "contents": [ "Denote $(p_n, q_n) : B_n \\to A_n \\oplus A_n$ the morphism $\\pi_n$", "of Homology, Lemma \\ref{homology-lemma-ses-termwise-split}.", "Also write $(a, b) : A \\oplus A \\to B$, and", "$r : B \\to C$ for the maps in the short exact sequence.", "Write the factorization of $\\delta(s)$ as", "$\\delta(s) = (1, -1) \\circ f$. This means that", "$p_{n - 1} \\circ d_{B, n} \\circ s_n = f_n$, and", "$q_{n - 1} \\circ d_{B, n} \\circ s_n = - f_n$, and", "Set $B_n \\to \\diamond A_n = A_n \\oplus A_n \\oplus A_{n - 1}$", "equal to $(p_n, q_n, f_n \\circ r_n)$.", "\\medskip\\noindent", "Now we have to check that this actually defines a morphism", "of complexes. We will only do this in the case of abelian groups.", "Pick $x \\in B_n$. Then $x = a_n(x_1) + b_n(x_2) + s_n(x_3)$", "and it suffices to show that our definition commutes", "with differential for each term separately. For the term", "$a_n(x_1)$ we have $(p_n, q_n, f_n \\circ r_n)(a_n(x_1)) =", "(x_1, 0, 0)$ and the result is obvious. Similarly for", "the term $b_n(x_2)$. For the term $s_n(x_3)$ we have", "\\begin{eqnarray*}", "(p_n, q_n, f_n \\circ r_n)(d_n(s_n(x_3))) & = &", "(p_n, q_n, f_n \\circ r_n)( \\\\", "& & \\ \\ \\ \\ \\ a_n(f_n(x_3)) - b_n(f_n(x_3)) + s_n(d_n(x_3))) \\\\", "& = &", "(f_n(x_3), -f_n(x_3), f_n(d_n(x_3)))", "\\end{eqnarray*}", "by definition of $f_n$. And", "\\begin{eqnarray*}", "d_n(p_n, q_n, f_n \\circ r_n)(s_n(x_3)) & = & d_n(0, 0, f_n(x_3)) \\\\", "& = &", "(f_n(x_3), - f_n(x_3), d_{A[-1], n}(f_n(x_3)))", "\\end{eqnarray*}", "The result follows as $f$ is a morphism of complexes." ], "refs": [ "homology-lemma-ses-termwise-split" ], "ref_ids": [ 12063 ] } ], "ref_ids": [ 12063 ] }, { "id": 14885, "type": "theorem", "label": "simplicial-lemma-backwards-homotopy", "categories": [ "simplicial" ], "title": "simplicial-lemma-backwards-homotopy", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $U$, $V$ be simplicial objects of $\\mathcal{A}$.", "Let $a, b : U \\to V$ be a pair of morphisms.", "Assume the corresponding maps of chain complexes", "$N(a), N(b) : N(U) \\to N(V)$ are homotopic by", "a homotopy $\\{N_n : N(U)_n \\to N(V)_{n + 1}\\}$.", "Then there exists a homotopy from $a$ to $b$ as in", "Definition \\ref{definition-homotopy}. Moreover, one can choose the", "homotopy $h : U \\times \\Delta[1] \\to V$ such that", "$N_n = N(h)_n$ where $N(h)$ is the homotopy coming", "from $h$ as in Section \\ref{section-homotopy-abelian}." ], "refs": [ "simplicial-definition-homotopy" ], "proofs": [ { "contents": [ "Let $(\\diamond N(U), \\diamond a, \\diamond b, \\diamond h)$", "be as in Lemma \\ref{lemma-represent-homotopy} and its proof. By that lemma", "there exists a morphism $\\diamond N(U) \\to N(V)$ representing", "the triple $(N(a), N(b), \\{N_n\\})$. We will show", "there exists a morphism", "$\\psi : N(U \\times \\Delta[1]) \\to \\diamond{N(U)}$", "such that $\\diamond a = \\psi \\circ N(e_0)$, and", "$\\diamond b = \\psi \\circ N(e_1)$. Moreover, we will", "show that the homotopy between $N(e_0), N(e_1) :", "N(U) \\to N(U \\times \\Delta[1])$ coming from", "(\\ref{equation-homotopy-to-homotopy}) and", "Lemma \\ref{lemma-homotopy-s-N} with", "$h = \\text{id}_{U \\times \\Delta[1]}$ is mapped via", "$\\psi$ to the canonical homotopy $\\diamond h$ between the two", "maps $\\diamond a, \\diamond b : N(U) \\to \\diamond{N(U)}$. Certainly this", "will imply the lemma.", "\\medskip\\noindent", "Note that $N : \\text{Simp}(\\mathcal{A}) \\to \\text{Ch}_{\\geq 0}(\\mathcal{A})$", "as a functor is a direct summand of the functor", "$s : \\text{Simp}(\\mathcal{A}) \\to \\text{Ch}_{\\geq 0}(\\mathcal{A})$.", "Also, the functor $\\diamond$ is compatible with direct sums.", "Thus it suffices instead to construct a morphism", "$\\Psi : s(U \\times \\Delta[1]) \\to \\diamond{s(U)}$ with the", "corresponding properties. This is what we do below.", "\\medskip\\noindent", "By Definition \\ref{definition-homotopy}", "the morphisms $e_0 : U \\to U \\times \\Delta[1]$", "and $e_1 : U \\to U \\times \\Delta[1]$ are homotopic", "with homotopy $\\text{id}_{U \\times \\Delta[1]}$.", "By Lemma \\ref{lemma-homotopy-s-N}", "we get an explicit homotopy", "$\\{h_n : s(U)_n \\to s(U \\times \\Delta[1])_{n + 1}\\}$", "between the morphisms", "of chain complexes $s(e_0) : s(U) \\to s(U \\times \\Delta[1])$", "and $s(e_1) : s(U) \\to s(U \\times \\Delta[1])$. By", "Lemma \\ref{lemma-map-into-diamond} above we get a corresponding morphism", "$$", "\\Phi : \\diamond{s(U)} \\to s(U \\times \\Delta[1])", "$$", "According to the construction, $\\Phi_n$ restricted to the summand", "$s(U)[-1]_n = s(U)_{n - 1}$ of $\\diamond{s(U)}_n$", "is equal to $h_{n - 1}$. And", "$$", "h_{n - 1} = \\sum\\nolimits_{i = 0}^{n - 1}", "(-1)^{i + 1} s^n_i \\cdot \\alpha^n_{i + 1} :", "U_{n - 1} \\to \\bigoplus\\nolimits_j U_n \\cdot \\alpha^n_j.", "$$", "with obvious notation.", "\\medskip\\noindent", "On the other hand, the morphisms $e_i : U \\to U \\times \\Delta[1]$ induce", "a morphism $(e_0, e_1) : U \\oplus U \\to U \\times \\Delta[1]$.", "Denote $W$ the cokernel. Note that, if we write", "$(U \\times \\Delta[1])_n = \\bigoplus_{\\alpha : [n] \\to [1]} U_n \\cdot \\alpha$,", "then we may identify", "$W_n = \\bigoplus_{i = 1}^n U_n \\cdot \\alpha^n_i$ with", "$\\alpha^n_i$ as in Section \\ref{section-homotopy}.", "We have a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "U \\oplus U \\ar[rd]_{(1, 1)} \\ar[r] &", "U \\times \\Delta[1] \\ar[d]^\\pi \\ar[r] &", "W \\ar[r] &", "0 \\\\", "& & U & &", "}", "$$", "This implies we have a similar commutative diagram after", "applying the functor $s$. Next, we choose the splittings", "$\\sigma_n : s(W)_n \\to s(U \\times \\Delta[1])_n$ by mapping", "the summand $U_n \\cdot \\alpha^n_i \\subset W_n$ via $(-1, 1)$", "to the summands $U_n \\cdot \\alpha^n_0 \\oplus U_n \\cdot \\alpha^n_i", "\\subset (U \\times \\Delta[1])_n$. Note that $s(\\pi)_n \\circ \\sigma_n = 0$.", "It follows that $(1, 1) \\circ \\delta(\\sigma)_n = 0$.", "Hence $\\delta(\\sigma)$ factors as in", "Lemma \\ref{lemma-map-into-diamond}. By that lemma", "we obtain a canonical morphism", "$\\Psi : s(U \\times \\Delta[1]) \\to \\diamond{s(U)}$.", "\\medskip\\noindent", "To compute $\\Psi$ we first compute the morphism", "$\\delta(\\sigma) : s(W) \\to s(U)[-1] \\oplus s(U)[-1]$.", "According to Homology, Lemma \\ref{homology-lemma-ses-termwise-split}", "and its proof,", "to do this we have compute", "$$", "d_{s(U \\times \\delta[1]), n} \\circ \\sigma_n", "-", "\\sigma_{n - 1} \\circ d_{s(W), n}", "$$", "and write it as a morphism into", "$U_{n - 1} \\cdot \\alpha^{n - 1}_0 \\oplus U_{n - 1} \\cdot \\alpha^{n - 1}_n$.", "We only do this in case $\\mathcal{A}$ is the category of abelian", "groups. We use the short hand notation $x_{\\alpha}$ for $x \\in U_n$", "to denote the element $x$ in the summand $U_n \\cdot \\alpha$", "of $(U \\times \\Delta[1])_n$. Recall that", "$$", "d_{s(U \\times \\delta[1]), n}", "=", "\\sum\\nolimits_{i = 0}^n (-1)^i d^n_i", "$$", "where $d^n_i$ maps the summand $U_n \\cdot \\alpha$", "to the summand $U_{n - 1} \\cdot (\\alpha \\circ \\delta^n_i)$", "via the morphism $d^n_i$ of the simplicial object $U$.", "In terms of the notation above this means", "$$", "d_{s(U \\times \\delta[1]), n}(x_\\alpha) =", "\\sum\\nolimits_{i = 0}^n (-1)^i (d^n_i(x))_{\\alpha \\circ \\delta^n_i}", "$$", "Starting with $x_\\alpha \\in W_n$, in other words $\\alpha = \\alpha^n_j$", "for some $j \\in \\{1, \\ldots, n\\}$, we see that", "$\\sigma_n(x_\\alpha) = x_\\alpha - x_{\\alpha^n_0}$ and", "hence", "$$", "(d_{s(U \\times \\delta[1]), n} \\circ \\sigma_n)(x_\\alpha)", "=", "\\sum\\nolimits_{i = 0}^n (-1)^i (d^n_i(x))_{\\alpha \\circ \\delta^n_i}", "-", "\\sum\\nolimits_{i = 0}^n (-1)^i (d^n_i(x))_{\\alpha^n_0 \\circ \\delta^n_i}", "$$", "To compute $d_{s(W), n}(x_\\alpha)$, we have to omit", "all terms where $\\alpha \\circ \\delta^n_i = \\alpha^{n - 1}_0, \\alpha^{n - 1}_n$.", "Hence we get", "$$", "\\begin{matrix}", "(\\sigma_{n - 1} \\circ d_{s(W), n})(x_\\alpha) = \\\\", "\\sum\\nolimits_{i = 0, \\ldots, n\\text{ and }", "\\alpha \\circ \\delta^n_i \\not = \\alpha^{n - 1}_0\\text{ or }\\alpha^{n - 1}_n}", "\\Big((-1)^i (d^n_i(x))_{\\alpha \\circ \\delta^n_i}", "-", "(-1)^i (d^n_i(x))_{\\alpha^{n - 1}_0}", "\\Big)", "\\end{matrix}", "$$", "Clearly the difference of the two terms is the sum", "$$", "\\sum\\nolimits_{i = 0, \\ldots, n\\text{ and }", "\\alpha \\circ \\delta^n_i = \\alpha^{n - 1}_0\\text{ or }\\alpha^{n - 1}_n}", "\\Big((-1)^i (d^n_i(x))_{\\alpha \\circ \\delta^n_i}", "-", "(-1)^i (d^n_i(x))_{\\alpha^{n - 1}_0}", "\\Big)", "$$", "Of course, if $\\alpha \\circ \\delta^n_i = \\alpha^{n - 1}_0$", "then the term drops out. Recall that $\\alpha = \\alpha^n_j$", "for some $j \\in \\{1, \\ldots, n\\}$. The only way", "$\\alpha^n_j \\circ \\delta^n_i = \\alpha^{n - 1}_n$", "is if $j = n$ and $i = n$. Thus we actually get", "$0$ unless $j = n$ and in that case we get", "$(-1)^n(d^n_n(x))_{\\alpha^{n - 1}_n} - (-1)^n(d^n_n(x))_{\\alpha^{n - 1}_0}$.", "In other words, we conclude the morphism", "$$", "\\delta(\\sigma)_n :", "W_n", "\\to", "(s(U)[-1] \\oplus s(U)[-1])_n = U_{n - 1} \\oplus U_{n - 1}", "$$", "is zero on all summands except $U_n \\cdot \\alpha^n_n$", "and on that summand it is equal to $((-1)^nd^n_n, -(-1)^nd^n_n)$.", "(Namely, the first summand of the two corresponds to the factor", "with $\\alpha^{n - 1}_n$ because that is the map $[n - 1] \\to [1]$", "which maps everybody to $0$, and hence corresponds to $e_0$.)", "\\medskip\\noindent", "We obtain a canonical diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "s(U) \\oplus s(U) \\ar[r] \\ar[d] &", "\\diamond{s(U)} \\ar[r] \\ar[d]^{\\Phi}&", "s(U)[-1] \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "s(U) \\oplus s(U) \\ar[r] \\ar[d] &", "s(U \\times \\Delta[1]) \\ar[r] \\ar[d]^\\Psi &", "s(W) \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "s(U) \\oplus s(U) \\ar[r] &", "\\diamond{s(U)} \\ar[r] &", "s(U)[-1] \\ar[r] &", "0", "}", "$$", "We claim that $\\Phi \\circ \\Psi$ is the identity.", "To see this it is enough to prove that the composition", "of $\\Phi$ and $\\delta(\\sigma)$ as a map", "$s(U)[-1] \\to s(W) \\to s(U)[-1] \\oplus s(U)[-1]$ is the", "identity in the first factor and minus identity in the second.", "By the computations above it is", "$((-1)^nd^n_0, -(-1)^nd^n_0) \\circ (-1)^n s^n_n = (1, -1)$", "as desired." ], "refs": [ "simplicial-lemma-represent-homotopy", "simplicial-lemma-homotopy-s-N", "simplicial-definition-homotopy", "simplicial-lemma-homotopy-s-N", "simplicial-lemma-map-into-diamond", "simplicial-lemma-map-into-diamond", "homology-lemma-ses-termwise-split" ], "ref_ids": [ 14883, 14876, 14927, 14876, 14884, 14884, 12063 ] } ], "ref_ids": [ 14927 ] }, { "id": 14886, "type": "theorem", "label": "simplicial-lemma-trivial-kan", "categories": [ "simplicial" ], "title": "simplicial-lemma-trivial-kan", "contents": [ "Let $f : X \\to Y$ be a trivial Kan fibration of simplicial sets.", "For any solid commutative diagram", "$$", "\\xymatrix{", "Z \\ar[r]_b \\ar[d] & X \\ar[d] \\\\", "W \\ar[r]^a \\ar@{-->}[ru] & Y", "}", "$$", "of simplicial sets with $Z \\to W$ (termwise) injective", "a dotted arrow exists making the diagram commute." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $Z \\not = W$. Let $n$ be the smallest integer such that", "$Z_n \\not = W_n$. Let $x \\in W_n$, $x \\not \\in Z_n$. Denote", "$Z' \\subset W$ the simplicial subset containing $Z$, $x$, and", "all degeneracies of $x$. Let", "$\\varphi : \\Delta[n] \\to Z'$ be the morphism corresponding to $x$", "(Lemma \\ref{lemma-simplex-map}).", "Then $\\varphi|_{\\partial \\Delta[n]}$ maps into $Z$", "as all the nondegenerate simplices of $\\partial \\Delta[n]$ end up in $Z$.", "By assumption we can extend $b \\circ \\varphi|_{\\partial \\Delta[n]}$", "to $\\beta : \\Delta[n] \\to X$.", "By Lemma \\ref{lemma-glue-simplex} the simplicial set $Z'$", "is the pushout of $\\Delta[n]$ and $Z$ along $\\partial \\Delta[n]$.", "Hence $b$ and $\\beta$ define a morphism $b' : Z' \\to X$.", "In other words, we have extended the morphism $b$ to a bigger", "simplicial subset of $Z$.", "\\medskip\\noindent", "The proof is finished by an application of Zorn's lemma (omitted)." ], "refs": [ "simplicial-lemma-simplex-map", "simplicial-lemma-glue-simplex" ], "ref_ids": [ 14817, 14852 ] } ], "ref_ids": [] }, { "id": 14887, "type": "theorem", "label": "simplicial-lemma-trivial-kan-base-change", "categories": [ "simplicial" ], "title": "simplicial-lemma-trivial-kan-base-change", "contents": [ "Let $f : X \\to Y$ be a trivial Kan fibration of simplicial sets.", "Let $Y' \\to Y$ be a morphism of simplicial sets.", "Then $X \\times_Y Y' \\to Y'$ is a trivial Kan fibration." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the functorial properties of the fibre", "product (Lemma \\ref{lemma-fibre-product}) and the definitions." ], "refs": [ "simplicial-lemma-fibre-product" ], "ref_ids": [ 14813 ] } ], "ref_ids": [] }, { "id": 14888, "type": "theorem", "label": "simplicial-lemma-trivial-kan-composition", "categories": [ "simplicial" ], "title": "simplicial-lemma-trivial-kan-composition", "contents": [ "The composition of two trivial Kan fibrations is a trivial Kan fibration." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14889, "type": "theorem", "label": "simplicial-lemma-limit-trivial-kan", "categories": [ "simplicial" ], "title": "simplicial-lemma-limit-trivial-kan", "contents": [ "Let $\\ldots \\to U^2 \\to U^1 \\to U^0$ be a sequence of trivial Kan", "fibrations. Let $U = \\lim U^t$ defined by taking $U_n = \\lim U_n^t$.", "Then $U \\to U^0$ is a trivial Kan fibration." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: use that for a countable sequence of surjections of sets", "the inverse limit is nonempty." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14890, "type": "theorem", "label": "simplicial-lemma-product-trivial-kan", "categories": [ "simplicial" ], "title": "simplicial-lemma-product-trivial-kan", "contents": [ "\\begin{slogan}", "Products of trivial Kan fibrations are trivial Kan fibrations.", "\\end{slogan}", "Let $X_i \\to Y_i$ be a set of trivial Kan fibrations. Then", "$\\prod X_i \\to \\prod Y_i$ is a trivial Kan fibration." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14891, "type": "theorem", "label": "simplicial-lemma-filtered-colimit-trivial-kan", "categories": [ "simplicial" ], "title": "simplicial-lemma-filtered-colimit-trivial-kan", "contents": [ "A filtered colimit of trivial Kan fibrations is a trivial Kan fibration." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: See description of filtered colimits of sets in", "Categories, Section \\ref{categories-section-directed-colimits}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14892, "type": "theorem", "label": "simplicial-lemma-trivial-kan-homotopy", "categories": [ "simplicial" ], "title": "simplicial-lemma-trivial-kan-homotopy", "contents": [ "Let $f : X \\to Y$ be a trivial Kan fibration of simplicial sets.", "Then $f$ is a homotopy equivalence." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-trivial-kan} we can choose an right inverse $g : Y \\to X$", "to $f$. Consider the diagram", "$$", "\\xymatrix{", "\\partial \\Delta[1] \\times X \\ar[d] \\ar[r] & X \\ar[d] \\\\", "\\Delta[1] \\times X \\ar[r] \\ar@{-->}[ru] & Y", "}", "$$", "Here the top horizontal arrow is given by $\\text{id}_X$ and $g \\circ f$", "where we use that", "$(\\partial \\Delta[1] \\times X)_n = X_n \\amalg X_n$ for all $n \\geq 0$.", "The bottom horizontal arrow is given by the map $\\Delta[1] \\to \\Delta[0]$", "and $f : X \\to Y$. The diagram commutes as $f \\circ g \\circ f = f$.", "By Lemma \\ref{lemma-trivial-kan} we can fill in the dotted arrow", "and we win." ], "refs": [ "simplicial-lemma-trivial-kan", "simplicial-lemma-trivial-kan" ], "ref_ids": [ 14886, 14886 ] } ], "ref_ids": [] }, { "id": 14893, "type": "theorem", "label": "simplicial-lemma-kan-base-change", "categories": [ "simplicial" ], "title": "simplicial-lemma-kan-base-change", "contents": [ "Let $f : X \\to Y$ be a Kan fibration of simplicial sets.", "Let $Y' \\to Y$ be a morphism of simplicial sets.", "Then $X \\times_Y Y' \\to Y'$ is a Kan fibration." ], "refs": [], "proofs": [ { "contents": [ "This follows immediately from the functorial properties of the fibre", "product (Lemma \\ref{lemma-fibre-product}) and the definitions." ], "refs": [ "simplicial-lemma-fibre-product" ], "ref_ids": [ 14813 ] } ], "ref_ids": [] }, { "id": 14894, "type": "theorem", "label": "simplicial-lemma-kan-composition", "categories": [ "simplicial" ], "title": "simplicial-lemma-kan-composition", "contents": [ "The composition of two Kan fibrations is a Kan fibration." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14895, "type": "theorem", "label": "simplicial-lemma-limit-kan", "categories": [ "simplicial" ], "title": "simplicial-lemma-limit-kan", "contents": [ "Let $\\ldots \\to U^2 \\to U^1 \\to U^0$ be a sequence of Kan", "fibrations. Let $U = \\lim U^t$ defined by taking $U_n = \\lim U_n^t$.", "Then $U \\to U^0$ is a Kan fibration." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: use that for a countable sequence of surjections of sets", "the inverse limit is nonempty." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14896, "type": "theorem", "label": "simplicial-lemma-product-kan", "categories": [ "simplicial" ], "title": "simplicial-lemma-product-kan", "contents": [ "Let $X_i \\to Y_i$ be a set of Kan fibrations. Then", "$\\prod X_i \\to \\prod Y_i$ is a Kan fibration." ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14897, "type": "theorem", "label": "simplicial-lemma-simplicial-group-kan", "categories": [ "simplicial" ], "title": "simplicial-lemma-simplicial-group-kan", "contents": [ "Let $X$ be a simplicial group. Then $X$ is a Kan complex." ], "refs": [], "proofs": [ { "contents": [ "The following proof is basically just a translation into English", "of the proof in the reference mentioned above.", "Using the terminology as explained in the introduction to this section,", "suppose $f : \\Lambda_k[n] \\to X$ is a morphism from a horn. Set", "$x_i = f(\\sigma_i) \\in X_{n - 1}$ for $i = 0, \\ldots, \\hat k, \\ldots, n$.", "This means that for $i < j$ we have $d_i x_j = d_{j - 1} x_i$", "whenever $i, j \\not = k$.", "We have to find an $x \\in X_n$ such that $x_i = d_ix$ for", "$i = 0, \\ldots, \\hat k, \\ldots, n$.", "\\medskip\\noindent", "We first prove there exists a $u \\in X_n$ such that $d_i u = x_i$", "for $i < k$. This is trivial for $k = 0$. If $k > 0$, one defines by", "induction an element $u^r \\in X_n$ such that $d_i u^r = x_i$ for", "$0 \\leq i \\leq r$. Start with $u^0 = s_0x_0$. If $r < k - 1$,", "we set", "$$", "y^r = s_{r + 1}((d_{r + 1}u^r)^{-1}x_{r + 1}),\\quad", "u^{r + 1} = u^r y^r.", "$$", "An easy calculation shows that $d_iy^r = 1$ (unit element of the group", "$X_{n - 1}$) for $i \\leq r$ and", "$d_{r + 1}y^r = (d_{r + 1}u^r)^{-1}x_{r + 1}$. It follows that", "$d_iu^{r + 1} = x_i$ for $i \\leq r + 1$. Finally, take $u = u^{k - 1}$", "to get $u$ as promised.", "\\medskip\\noindent", "Next we prove, by induction on the integer $r$, $0 \\leq r \\leq n - k$,", "there exists a $x^r \\in X_n$ such that", "$$", "d_i x^r = x_i\\quad\\text{for }i < k\\text{ and }i > n - r.", "$$", "Start with $x^0 = u$ for $r = 0$. Having defined $x^r$ for $r \\leq n - k - 1$", "we set", "$$", "z^r = s_{n - r - 1}((d_{n - r}x^r)^{-1}x_{n - r}),\\quad x^{r + 1} = x^rz^r", "$$", "A simple calculation, using the given relations, shows that", "$d_iz^r = 1$ for $i < k$ and $i > n - r$ and that", "$d_{n - r}(z^r) = (d_{n - r}x^r)^{-1}x_{n - r}$. It follows that", "$d_ix^{r + 1} = x_i$ for $i < k$ and $i > n - r - 1$. Finally, we", "take $x = x^{n - k}$ which finishes the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14898, "type": "theorem", "label": "simplicial-lemma-surjection-simplicial-abelian-groups-kan", "categories": [ "simplicial" ], "title": "simplicial-lemma-surjection-simplicial-abelian-groups-kan", "contents": [ "Let $f : X \\to Y$ be a homomorphism of simplicial abelian groups", "which is termwise surjective. Then $f$ is a Kan fibration of", "simplicial sets." ], "refs": [], "proofs": [ { "contents": [ "Consider a commutative solid diagram", "$$", "\\xymatrix{", "\\Lambda_k[n] \\ar[r]_a \\ar[d] & X \\ar[d] \\\\", "\\Delta[n] \\ar[r]^b \\ar@{-->}[ru] & Y", "}", "$$", "as in Definition \\ref{definition-kan}. The map $a$ corresponds", "to $x_0, \\ldots, \\hat x_k, \\ldots, x_n \\in X_{n - 1}$ satisfying", "$d_i x_j = d_{j - 1} x_i$ for $i < j$, $i, j \\not = k$.", "The map $b$ corresponds to an element $y \\in Y_n$ such", "that $d_iy = f(x_i)$ for $i \\not = k$. Our task is to produce an", "$x \\in X_n$ such that $d_ix = x_i$ for $i \\not = k$ and $f(x) = y$.", "\\medskip\\noindent", "Since $f$ is termwise surjective we can find $x \\in X_n$ with $f(x) = y$.", "Replace $y$ by $0 = y - f(x)$ and $x_i$ by $x_i - d_ix$ for $i \\not = k$.", "Then we see that we may assume $y = 0$. In particular $f(x_i) = 0$.", "In other words, we can replace $X$ by $\\Ker(f) \\subset X$", "and $Y$ by $0$. In this case the statement become", "Lemma \\ref{lemma-simplicial-group-kan}." ], "refs": [ "simplicial-definition-kan", "simplicial-lemma-simplicial-group-kan" ], "ref_ids": [ 14931, 14897 ] } ], "ref_ids": [] }, { "id": 14899, "type": "theorem", "label": "simplicial-lemma-qis-simplicial-abelian-groups", "categories": [ "simplicial" ], "title": "simplicial-lemma-qis-simplicial-abelian-groups", "contents": [ "Let $f : X \\to Y$ be a homomorphism of simplicial abelian groups", "which is termwise surjective and induces a quasi-isomorphism on", "associated chain complexes. Then $f$ is a trivial Kan fibration of", "simplicial sets." ], "refs": [], "proofs": [ { "contents": [ "Consider a commutative solid diagram", "$$", "\\xymatrix{", "\\partial \\Delta[n] \\ar[r]_a \\ar[d] & X \\ar[d] \\\\", "\\Delta[n] \\ar[r]^b \\ar@{-->}[ru] & Y", "}", "$$", "as in Definition \\ref{definition-trivial-kan}. The map $a$ corresponds", "to $x_0, \\ldots, x_n \\in X_{n - 1}$ satisfying $d_i x_j = d_{j - 1} x_i$", "for $i < j$. The map $b$ corresponds to an element $y \\in Y_n$ such", "that $d_iy = f(x_i)$. Our task is to produce an $x \\in X_n$ such that", "$d_ix = x_i$ and $f(x) = y$.", "\\medskip\\noindent", "Since $f$ is termwise surjective we can find $x \\in X_n$ with $f(x) = y$.", "Replace $y$ by $0 = y - f(x)$ and $x_i$ by $x_i - d_ix$. Then we", "see that we may assume $y = 0$. In particular $f(x_i) = 0$.", "In other words, we can replace $X$ by $\\Ker(f) \\subset X$", "and $Y$ by $0$. This works, because by", "Homology, Lemma \\ref{homology-lemma-long-exact-sequence-chain}", "the homology of the chain complex associated to $\\Ker(f)$", "is zero and hence $\\Ker(f) \\to 0$ induces a quasi-isomorphism", "on associated chain complexes.", "\\medskip\\noindent", "Since $X$ is a Kan complex (Lemma \\ref{lemma-simplicial-group-kan})", "we can find $x \\in X_n$ with $d_i x = x_i$ for $i = 0, \\ldots, n - 1$.", "After replacing $x_i$ by $x_i - d_ix$ for $i = 0, \\ldots, n$ we", "may assume that $x_0 = x_1 = \\ldots = x_{n - 1} = 0$.", "In this case we see that $d_i x_n = 0$ for $i = 0, \\ldots, n - 1$.", "Thus $x_n \\in N(X)_{n - 1}$ and lies in the kernel of the differential", "$N(X)_{n - 1} \\to N(X)_{n - 2}$. Here $N(X)$ is the normalized chain", "complex associated to $X$, see Section \\ref{section-complexes}.", "Since $N(X)$ is quasi-isomorphic to $s(X)$", "(Lemma \\ref{lemma-quasi-isomorphism})", "and thus acyclic we find $x \\in N(X_n)$ whose differential is $x_n$.", "This $x$ answers the question posed by the lemma and we are done." ], "refs": [ "simplicial-definition-trivial-kan", "homology-lemma-long-exact-sequence-chain", "simplicial-lemma-simplicial-group-kan", "simplicial-lemma-quasi-isomorphism" ], "ref_ids": [ 14930, 12057, 14897, 14868 ] } ], "ref_ids": [] }, { "id": 14900, "type": "theorem", "label": "simplicial-lemma-homotopy-equivalence", "categories": [ "simplicial" ], "title": "simplicial-lemma-homotopy-equivalence", "contents": [ "Let $f : X \\to Y$ be a map of simplicial abelian groups. If $f$ is a", "homotopy equivalence of simplicial sets, then $f$ induces a", "quasi-isomorphism of associated chain complexes." ], "refs": [], "proofs": [ { "contents": [ "In this proof we will write $H_n(Z) = H_n(s(Z)) = H_n(N(Z))$", "when $Z$ is a simplicial abelian group, with $s$ and $N$ as in", "Section \\ref{section-complexes}.", "Let $\\mathbf{Z}[X]$ denote the free abelian group on $X$ viewed as", "a simplicial set and similarly for $\\mathbf{Z}[Y]$. Consider the", "commutative diagram", "$$", "\\xymatrix{", "\\mathbf{Z}[X] \\ar[r]_g \\ar[d] &", "\\mathbf{Z}[Y] \\ar[d] \\\\", "X \\ar[r]^f & Y", "}", "$$", "of simplicial abelian groups. Since taking the free abelian group on a set", "is a functor, we see that the horizontal arrow is a homotopy equivalence of", "simplicial abelian groups, see Lemma \\ref{lemma-functorial-homotopy}.", "By Lemma \\ref{lemma-homotopy-equivalence-s-N}", "we see that $H_n(g) : H_n(\\mathbf{Z}[X]) \\to H_n(\\mathbf{Z}[Y])$ is bijective", "for all $n \\geq 0$.", "\\medskip\\noindent", "Let $\\xi \\in H_n(Y)$. By definition of $N(Y)$ we can represent", "$\\xi$ by an element $y \\in N(Y_n)$ whose boundary is zero.", "This means $y \\in Y_n$ with $d^n_0(y) = \\ldots = d^n_{n - 1}(y) = 0$", "because $y \\in N(Y_n)$ and $d^n_n(y) = 0$ because the boundary", "of $y$ is zero. Denote $0_n \\in Y_n$ the zero element. Then we see that", "$$", "\\tilde y = [y] - [0_n] \\in (\\mathbf{Z}[Y])_n", "$$", "is an element with $d^n_0(\\tilde y) = \\ldots = d^n_{n - 1}(\\tilde y) = 0$", "and $d^n_n(\\tilde y) = 0$. Thus $\\tilde y$ is in $N(\\mathbf{Z}[Y])_n$", "has boundary $0$, i.e., $\\tilde y$ determines a", "class $\\tilde \\xi \\in H_n(\\mathbf{Z}[Y])$ mapping to $\\xi$.", "Because $H_n(\\mathbf{Z}[X]) \\to H_n(\\mathbf{Z}[Y])$ is bijective", "we can lift $\\tilde \\xi$ to a class in $H_n(\\mathbf{Z}[X])$.", "Looking at the commutative diagram above", "we see that $\\xi$ is in the image of $H_n(X) \\to H_n(Y)$.", "\\medskip\\noindent", "Let $\\xi \\in H_n(X)$ be an element mapping to zero in $H_n(Y)$.", "Exactly as in the previous parapgraph we can represent", "$\\xi$ by an element $x \\in N(X_n)$ whose boundary is zero, i.e.,", "$d^n_0(x) = \\ldots = d^n_{n - 1}(x) = d^n_n(x) = 0$.", "In particular, we see that $[x] - [0_n]$ is an element of", "$N(\\mathbf{Z}[X])_n$ whose boundary is zero, whence defines", "a lift $\\tilde \\xi \\in H_n(\\mathbf{Z}[x])$ of $\\xi$.", "The fact that $\\xi$ maps to zero in $H_n(Y)$", "means there exists a $y \\in N(Y_{n + 1})$ whose boundary is $f_n(x)$.", "This means $d^{n + 1}_0(y) = \\ldots = d^{n + 1}_n(y) = 0$ and", "$d^{n + 1}_{n + 1}(y) = f(x)$.", "However, this means exactly that", "$z = [y] - [0_{n + 1}]$ is in $N(\\mathbf{Z}[y])_{n + 1}$ and", "$$", "g([x] - [0_n]) = [f(x)] - [0_n] = \\text{boundary of }z", "$$", "This proves that $\\tilde \\xi$ maps to zero in $H_n(\\mathbf{Z}[y])$.", "As $H_n(\\mathbf{Z}[X]) \\to H_n(\\mathbf{Z}[Y])$ is bijective", "we conclude $\\tilde \\xi = 0$ and hence $\\xi = 0$." ], "refs": [ "simplicial-lemma-functorial-homotopy", "simplicial-lemma-homotopy-equivalence-s-N" ], "ref_ids": [ 14879, 14877 ] } ], "ref_ids": [] }, { "id": 14901, "type": "theorem", "label": "simplicial-lemma-section", "categories": [ "simplicial" ], "title": "simplicial-lemma-section", "contents": [ "Let $f : V \\to U$ be a morphism of simplicial sets. Let $n \\geq 0$ be an", "integer. Assume", "\\begin{enumerate}", "\\item The map $f_i : V_i \\to U_i$ is a bijection for $i < n$.", "\\item The map $f_n : V_n \\to U_n$ is a surjection.", "\\item The canonical morphism $U \\to \\text{cosk}_n \\text{sk}_n U$", "is an isomorphism.", "\\item The canonical morphism $V \\to \\text{cosk}_n \\text{sk}_n V$", "is an isomorphism.", "\\end{enumerate}", "Then $f$ is a trivial Kan fibration." ], "refs": [], "proofs": [ { "contents": [ "Consider a solid diagram", "$$", "\\xymatrix{", "\\partial \\Delta[k] \\ar[r] \\ar[d] & V \\ar[d] \\\\", "\\Delta[k] \\ar[r] \\ar@{-->}[ru] & U", "}", "$$", "as in Definition \\ref{definition-trivial-kan}. Let $x \\in U_k$ be", "the $k$-simplex corresponding to the lower horizontal arrow.", "If $k \\leq n$ then the dotted arrow is the one corresponding to", "a lift $y \\in V_k$ of $x$; the diagram will commute as the other", "nondegenerate simplices of $\\Delta[k]$ are in degrees $< k$ where", "$f$ is an isomorphism. If $k > n$, then by conditions (3) and (4)", "we have (using adjointness of skeleton and coskeleton functors)", "$$", "\\Mor(\\Delta[k], U) =", "\\Mor(\\text{sk}_n\\Delta[k], \\text{sk}_nU) =", "\\Mor(\\text{sk}_n\\partial\\Delta[k], \\text{sk}_nU) =", "\\Mor(\\partial \\Delta[k], U)", "$$", "and similarly for $V$ because", "$\\text{sk}_n\\Delta[k] = \\text{sk}_n\\partial\\Delta[k]$ for $k > n$.", "Thus we obtain a unique dotted arrow fitting into the diagram in", "this case also." ], "refs": [ "simplicial-definition-trivial-kan" ], "ref_ids": [ 14930 ] } ], "ref_ids": [] }, { "id": 14902, "type": "theorem", "label": "simplicial-lemma-homotopy", "categories": [ "simplicial" ], "title": "simplicial-lemma-homotopy", "contents": [ "Let $f^0, f^1 : V \\to U$ be maps of simplicial sets.", "Let $n \\geq 0$ be an integer.", "Assume", "\\begin{enumerate}", "\\item The maps $f^j_i : V_i \\to U_i$, $j = 0, 1$ are equal for $i < n$.", "\\item The canonical morphism $U \\to \\text{cosk}_n \\text{sk}_n U$", "is an isomorphism.", "\\item The canonical morphism $V \\to \\text{cosk}_n \\text{sk}_n V$", "is an isomorphism.", "\\end{enumerate}", "Then $f^0$ is homotopic to $f^1$." ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "Let $W$ be the $n$-truncated simplicial set with $W_i = U_i$ for", "$i < n$ and $W_n = U_n / \\sim$ where $\\sim$ is the equivalence relation", "generated by $f^0(y) \\sim f^1(y)$ for $y \\in V_n$. This makes sense", "as the morphisms $U(\\varphi) : U_n \\to U_i$ corresponding to", "$\\varphi : [i] \\to [n]$ for $i < n$ factor through the quotient map", "$U_n \\to W_n$ because $f^0$ and $f^1$ are morphisms of simplicial sets and", "equal in degrees $< n$. Next, we upgrade $W$ to a simplicial set", "by taking $\\text{cosk}_n W$. By Lemma \\ref{lemma-section}", "the morphism $g : U \\to W$ is a trivial Kan fibration. Observe", "that $g \\circ f^0 = g \\circ f^1$ by construction and denote", "this morphism $f : V \\to W$. Consider the diagram", "$$", "\\xymatrix{", "\\partial \\Delta[1] \\times V \\ar[rr]_{f^0, f^1} \\ar[d] & & U \\ar[d] \\\\", "\\Delta[1] \\times V \\ar[rr]^f \\ar@{-->}[rru] & & W", "}", "$$", "By Lemma \\ref{lemma-trivial-kan} the dotted arrow exists and the proof is done." ], "refs": [ "simplicial-lemma-trivial-kan" ], "ref_ids": [ 14886 ] } ], "ref_ids": [] }, { "id": 14903, "type": "theorem", "label": "simplicial-lemma-cosk-minus-one-equivalence", "categories": [ "simplicial" ], "title": "simplicial-lemma-cosk-minus-one-equivalence", "contents": [ "Let $A$, $B$ be sets, and that $f : A \\to B$ is a map. Consider the simplicial", "set $U$ with $n$-simplices", "$$", "A \\times_B A \\times_B \\ldots \\times_B A\\ (n + 1 \\text{ factors)}.", "$$", "see Example \\ref{example-fibre-products-simplicial-object}.", "If $f$ is surjective, the morphism $U \\to B$", "where $B$ indicates the constant simplicial set with value $B$", "is a trivial Kan fibration." ], "refs": [], "proofs": [ { "contents": [ "Observe that $U$ fits into a cartesian square", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & \\text{cosk}_0(B) \\ar[d] \\\\", "B \\ar[r] & \\text{cosk}_0(A)", "}", "$$", "Since the right vertical arrow is a trivial Kan fibration by", "Lemma \\ref{lemma-section}, so is the left by", "Lemma \\ref{lemma-trivial-kan-base-change}." ], "refs": [ "simplicial-lemma-trivial-kan-base-change" ], "ref_ids": [ 14887 ] } ], "ref_ids": [] }, { "id": 14904, "type": "theorem", "label": "simplicial-lemma-godement", "categories": [ "simplicial" ], "title": "simplicial-lemma-godement", "contents": [ "In Example \\ref{example-godement} if", "$$", "1_Y = (d \\star 1_Y) \\circ s = (1_Y \\star d) \\circ s", "\\quad\\text{and}\\quad", "(s \\star 1) \\circ s = (1 \\star s) \\circ s", "$$", "then $X = (X_n, d^n_j, s^n_j)$ is a simplicial object in the category", "of endofunctors of $\\mathcal{C}$ and $d : X_0 = Y \\to \\text{id}_\\mathcal{C}$", "defines an augmentation." ], "refs": [], "proofs": [ { "contents": [ "To see that we obtain a simplicial object we have to check that the", "relations (1)(a) -- (e) of Lemma \\ref{lemma-characterize-simplicial-object}", "are satisfied. We will use the short hand notation", "$$", "1_a = 1_{X_{a - 1}} = 1_Y \\star \\ldots \\star 1_Y \\quad (a\\text{ factors})", "$$", "for $a \\geq 0$. With this notation we have", "$$", "d^n_j = 1_j \\star d \\star 1_{n - j}", "\\quad\\text{and}\\quad", "s^n_j = 1_j \\star s \\star 1_{n - j}", "$$", "We are repeatedly going to use the rule that for transformations", "of funtors $a, a', b, b'$ we have", "$(a' \\circ a) \\star (b' \\circ b) = (a' \\star b') \\circ (a \\star b)$", "provided that the $\\star$ and $\\circ$ compositions in this formula", "make sense, see", "Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats}.", "\\medskip\\noindent", "Condition (1)(a) always holds (no conditions needed on $d$ and $s$).", "Namely, let $0 \\leq i < j \\leq n + 1$. We have to show that", "$d^n_i \\circ d^{n + 1}_j = d^n_{j - 1} \\circ d^{n + 1}_i$, i.e.,", "$$", "(1_i \\star d \\star 1_{n - i}) \\circ", "(1_j \\star d \\star 1_{n + 1 - j}) =", "(1_{j - 1} \\star d \\star 1_{n + 1 - j}) \\circ", "(1_i \\star d \\star 1_{n + 1 - i})", "$$", "We can rewrite the left hand side as", "\\begin{align*}", "& (1_i \\star d \\star 1_{j - i - 1} \\star 1_{n + 1 - j}) \\circ", "(1_i \\star 1_1 \\star 1_{j - i - 1} \\star d \\star 1_{n + 1 - j}) \\\\", "& =", "1_i \\star", "\\left((d \\star 1_{j - i - 1}) \\circ", "(1_1 \\star 1_{j - i - 1} \\star d)\\right) \\star 1_{n + 1 - j} \\\\", "& =", "1_i \\star d \\star 1_{j - i - 1} \\star d \\star 1_{n + 1 - j} ", "\\end{align*}", "The second equality is true because $d \\circ 1_1 = d$ and", "$1_{j - i} \\circ (1_{j - i - 1} \\star d) = 1_{j - i - 1} \\star d$.", "A similar computation gives the same result for the right hand side.", "\\medskip\\noindent", "We check condition (1)(b). Let $0 \\leq i < j \\leq n - 1$. We have to show", "that $d^n_i \\circ s^{n - 1}_j = s^{n - 2}_{j - 1} \\circ d^{n - 1}_i$, i.e.,", "$$", "(1_i \\star d \\star 1_{n - i}) \\circ", "(1_j \\star s \\star 1_{n - 1 - j}) =", "(1_{j - 1} \\star s \\star 1_{n - 1 - j}) \\circ", "(1_i \\star d \\star 1_{n - 1 - i})", "$$", "By the same kind of calculus as in case (1)(a) both sides simplify", "to $1_i \\star d \\star 1_{j - i - 1} \\star s \\star 1_{n - j - 1}$.", "\\medskip\\noindent", "We check condition (1)(c). Let $0 \\leq j \\leq n - 1$. We have to show", "$\\text{id} = d^n_j \\circ s^{n - 1}_j = d^n_{j + 1} \\circ s^{n - 1}_j$, i.e.,", "$$", "1_n =", "(1_j \\star d \\star 1_{n - j}) \\circ", "(1_j \\star s \\star 1_{n - 1 - j}) =", "(1_{j + 1} \\star d \\star 1_{n - j - 1}) \\circ", "(1_j \\star s \\star 1_{n - 1 - j})", "$$", "This is easily seen to be implied by the first assumption of the lemma.", "\\medskip\\noindent", "We check condition (1)(d). Let $0 < j + 1 < i \\leq n$. We have to show", "$d^n_i \\circ s^{n - 1}_j = s^{n - 2}_j \\circ d^{n - 1}_{i - 1}$, i.e.,", "$$", "(1_i \\star d \\star 1_{n - i}) \\circ", "(1_j \\star s \\star 1_{n - 1 - j}) =", "(1_j \\star s \\star 1_{n - 2 - j}) \\circ", "(1_{i - 1} \\star d \\star 1_{n - i})", "$$", "By the same kind of calculus as in case (1)(a) both sides simplify", "to $1_j \\star s \\star 1_{i - j - 2} \\star d \\star 1_{n - i}$.", "\\medskip\\noindent", "We check condition (1)(e). Let $0 \\leq i \\leq j \\leq n - 1$. We", "have to show that", "$s^n_i \\circ s^{n - 1}_j = s^n_{j + 1} \\circ s^{n - 1}_i$, i.e.,", "$$", "(1_i \\star s \\star 1_{n - i}) \\circ", "(1_j \\star s \\star 1_{n - 1 - j}) =", "(1_{j + 1} \\star s \\star 1_{n - 1 - j}) \\circ", "(1_i \\star s \\star 1_{n - 1 - i})", "$$", "By the same kind of calculus as in case (1)(a) this reduces to", "$$", "(s \\star 1_{j - i + 1}) \\circ (1_{j - i} \\star s) =", "(1_{j - i + 1} \\star s) \\circ (s \\star 1_{j - i})", "$$", "If $j = i$ this is exactly one of the two assumptions of the lemma.", "For $j > i$ left and", "right hand side both reduce to the equality $s \\star 1_{j - i - 1} \\star s$", "by calculations similar to those we did in case (1)(a).", "\\medskip\\noindent", "Finally, in order to show that $d$ defines an augmentation we have", "to show that $d \\circ (1_1 \\star d) = d \\circ (d \\star 1_1)$ which is", "true because both sides are equal to $d \\star d$." ], "refs": [ "simplicial-lemma-characterize-simplicial-object", "categories-lemma-properties-2-cat-cats" ], "ref_ids": [ 14808, 12269 ] } ], "ref_ids": [] }, { "id": 14905, "type": "theorem", "label": "simplicial-lemma-godement-section", "categories": [ "simplicial" ], "title": "simplicial-lemma-godement-section", "contents": [ "Let $\\mathcal{A}$, $\\mathcal{B}$, $\\mathcal{C}$, $Y$, $d$, $s$, $F$, $G$", "be as in Example \\ref{example-godement-functorial}. Given a transformation", "of functors $h_0 : G \\circ F \\to G \\circ Y \\circ F$ such that", "$$", "1_{G \\circ F} = (1_G \\star d \\star 1_F) \\circ h_0", "$$", "Then there is a morphism $h : G \\circ F \\to G \\circ X \\circ F$", "of simplicial objects such that $\\epsilon \\circ h = \\text{id}$ where", "$\\epsilon : G \\circ X \\circ F \\to G \\circ F$ is the augmentation." ], "refs": [], "proofs": [ { "contents": [ "Denote $u_n : Y = X_0 \\to X_n$ the map of the simplicial object $X$", "corresponding to the unique morphism $[n] \\to [0]$ in $\\Delta$.", "Set $h_n : G \\circ F \\to G \\circ X_n \\circ F$ equal to", "$(1_G \\star u_n \\star 1_F) \\circ h_0$.", "\\medskip\\noindent", "For any simplicial object $X = (X_n)$ in any category", "$u =(u_n) : X_0 \\to X$ is a morphism from the constant simplicial object", "on $X_0$ to $X$. Hence $h$ is a", "morphism of simplicial objects because it is the composition of", "$1_G \\star u \\star 1_F$ and $h_0$.", "\\medskip\\noindent", "Let us check that $\\epsilon \\circ h = \\text{id}$. We compute", "$$", "\\epsilon_n \\circ (1_G \\star u_n \\star 1_F) \\circ h_0 =", "\\epsilon_0 \\circ h_0 = \\text{id}", "$$", "The first equality because $\\epsilon$ is a morphism of simplicial objects", "and the second equality because", "$\\epsilon_0 = (1_G \\star d \\star 1_F)$ and we can apply the", "assumption in the statement of the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14906, "type": "theorem", "label": "simplicial-lemma-godement-two-maps", "categories": [ "simplicial" ], "title": "simplicial-lemma-godement-two-maps", "contents": [ "Let $\\mathcal{A}$, $\\mathcal{B}$, $\\mathcal{C}$, $Y$, $d$, $s$, $F$, $G$", "be as in Example \\ref{example-godement-functorial}. Let", "$F' : \\mathcal{A} \\to \\mathcal{C}$ and", "$G' : \\mathcal{C} \\to \\mathcal{B}$ be two functors.", "Let $(a_n) : G \\circ X \\to G' \\circ X$ be a morphism of simplicial objects", "compatible via augmentations with $a : G \\to G'$.", "Let $(b_n) : X \\circ F \\to X \\circ F'$ be a morphism of simplicial objects", "compatible via augmentations with $b : F \\to F'$. Then the two maps", "$$", "a \\star (b_n), (a_n) \\star b : G \\circ X \\circ F \\to G' \\circ X \\circ F'", "$$", "are homotopic." ], "refs": [], "proofs": [ { "contents": [ "To show the morphisms are homotopic we construct morphisms", "$$", "h_{n, i} : G \\circ X_n \\circ F \\to G' \\circ X_n \\circ F'", "$$", "for $n \\geq 0$ and $0 \\leq i \\leq n + 1$", "satisfying the relations described in Lemma \\ref{lemma-relations-homotopy}.", "See also Remark \\ref{remark-homotopy-better}. To satisfy condition (1)", "of Lemma \\ref{lemma-relations-homotopy} we are forced to set", "$h_{n, 0} = a \\star b_n$ and $h_{n , n + 1} = a_n \\star b$.", "Thus a logical choice is", "$$", "h_{n , i} = a_{i - 1} \\star b_{n - i}", "$$", "for $1 \\leq i \\leq n$. Setting $a = a_{-1}$ and $b = b_{-1}$ we", "see the displayed formular holds for $0 \\leq i \\leq n + 1$.", "\\medskip\\noindent", "Recall that", "$$", "d^n_j = 1_G \\star 1_j \\star d \\star 1_{n - j} \\star 1_F", "$$", "on $G \\circ X \\circ F$ where we use", "the notation $1_a = 1_{Y \\circ \\ldots \\circ Y}$ introduced", "in the proof of Lemma \\ref{lemma-godement}. We are going to use below", "that we can rewrite this as", "\\begin{align*}", "d^n_j", "& =", "d^j_j \\star 1_{n - j} =", "d^{j + 1}_j \\star 1_{n - j} = \\ldots = d^{n - 1}_j \\star 1_1 \\\\", "& =", "1_j \\star d^{n - j}_0 = 1_{j - 1} \\star d^{n - j + 1}_1 = \\ldots =", "1_1 \\star d^{n - 1}_{j - 1}", "\\end{align*}", "Of course we have the analogous formulae for $d^n_j$ on $G' \\circ X \\circ F'$.", "\\medskip\\noindent", "We check condition (2) of Lemma \\ref{lemma-relations-homotopy}.", "Let $i > j$. We have to show", "$$", "d^n_j \\circ (a_{i - 1} \\star b_{n - i}) =", "(a_{i - 2} \\star b_{n - i}) \\circ d^n_j", "$$", "Since $i - 1 \\geq j$ we can use one of the possible descriptions of", "$d^n_j$ to rewrite the left hand side as", "$$", "(d^{i - 1}_j \\star 1_{n - i + 1}) \\circ (a_{i - 1} \\star b_{n - i}) =", "(d^{i - 1}_j \\circ a_{i - 1}) \\star b_{n - i} =", "(a_{i - 2} \\circ d^{i - 1}_j) \\star b_{n - i}", "$$", "Similarly the right hand side becomes", "$$", "(a_{i - 2} \\star b_{n - i}) \\circ (d^{i - 1}_j \\star 1_{n - i + 1}) =", "(a_{i - 2} \\circ d^{i - 1}_j) \\star b_{n - i}", "$$", "Thus we obtain the same result and (2) is checked.", "\\medskip\\noindent", "We check condition (3) of Lemma \\ref{lemma-relations-homotopy}.", "Let $i \\leq j$. We have to show", "$$", "d^n_j \\circ (a_{i - 1} \\star b_{n - i}) =", "(a_{i - 1} \\star b_{n - 1 - i}) \\circ d^n_j", "$$", "Since $j \\geq i$ we may rewrite the left hand side as", "$$", "(1_i \\star d^{n - i}_{j - i}) \\circ (a_{i - 1} \\star b_{n - i}) =", "a_{i - 1} \\star (b_{n - 1 - i} \\circ d^{n - i}_{j - i})", "$$", "A similar manipulation shows this agrees with the right hand side.", "\\medskip\\noindent", "Recall that", "$$", "s^n_j = 1_G \\star 1_j \\star s \\star 1_{n - j} \\star 1_F", "$$", "on $G \\circ X \\circ F$. We are going to use below", "that we can rewrite this as", "\\begin{align*}", "s^n_j", "& =", "s^j_j \\star 1_{n - j} = s^{j + 1}_j \\star 1_{n - j - 1} = \\ldots =", "s^{n - 1}_j \\star 1_1 \\\\", "& =", "1_j \\star s^{n - j}_0 = 1_{j - 1} \\star s^{n - j + 1}_1 = \\ldots =", "1_1 \\star s^{n - 1}_{j - 1}", "\\end{align*}", "Of course we have the analogous formulae for $s^n_j$ on $G' \\circ X \\circ F'$.", "\\medskip\\noindent", "We check condition (4) of Lemma \\ref{lemma-relations-homotopy}.", "Let $i > j$. We have to show", "$$", "s^n_j \\circ (a_{i - 1} \\star b_{n - i}) =", "(a_i \\star b_{n - i}) \\circ s^n_j", "$$", "Since $i - 1 \\geq j$ we can rewrite the left hand side as", "$$", "(s^{i - 1}_j \\star 1_{n - i + 1}) \\circ (a_{i - 1} \\star b_{n - i}) =", "(s^{i - 1}_j \\circ a_{i - 1}) \\star b_{n - i} =", "(a_i \\circ s^{i - 1}_j) \\star b_{n - i}", "$$", "Similarly the right hand side becomes", "$$", "(a_i \\star b_{n - i}) \\circ (s^{i - 1}_j \\star 1_{n - i + 1}) =", "(a_i \\circ s^{i - 1}_j) \\star b_{n - i}", "$$", "as desired.", "\\medskip\\noindent", "We check condition (5) of Lemma \\ref{lemma-relations-homotopy}.", "Let $i \\leq j$. We have to show", "$$", "s^n_j \\circ (a_{i - 1} \\star b_{n - i}) =", "(a_{i - 1} \\star b_{n + 1 - i}) \\circ s^n_j", "$$", "This equality holds because both sides evaluate to", "$a_{i - 1} \\star (s^{n - i}_{j - i} \\circ b_{n - i}) =", "a_{i - 1} \\star (b_{n + 1 - i} \\circ s^{n - i}_{j - i})$", "by exactly the same arguments as above." ], "refs": [ "simplicial-lemma-relations-homotopy", "simplicial-remark-homotopy-better", "simplicial-lemma-relations-homotopy", "simplicial-lemma-godement", "simplicial-lemma-relations-homotopy", "simplicial-lemma-relations-homotopy", "simplicial-lemma-relations-homotopy", "simplicial-lemma-relations-homotopy" ], "ref_ids": [ 14872, 14941, 14872, 14904, 14872, 14872, 14872, 14872 ] } ], "ref_ids": [] }, { "id": 14907, "type": "theorem", "label": "simplicial-lemma-godement-before-after", "categories": [ "simplicial" ], "title": "simplicial-lemma-godement-before-after", "contents": [ "Let $\\mathcal{C}$, $Y$, $d$, $s$ be as in Example \\ref{example-godement}", "satisfying the equations of Lemma \\ref{lemma-godement}. Let", "$f : \\text{id}_\\mathcal{C} \\to \\text{id}_\\mathcal{C}$ be an endomorphism", "of the identity functor. Then $f \\star 1_X, 1_X \\star f : X \\to X$", "are maps of simplicial objects compatible with $f$ via the augmentation", "$\\epsilon : X \\to \\text{id}_\\mathcal{C}$. Moreover, $f \\star 1_X$", "and $1_X \\star f$ are homotopic." ], "refs": [ "simplicial-lemma-godement" ], "proofs": [ { "contents": [ "The map $f \\star 1_X$ is the map with components", "$$", "X_n = \\text{id}_\\mathcal{C} \\circ X_n", "\\xrightarrow{f \\star 1_{X_n}}", "\\text{id}_\\mathcal{C} \\circ X_n = X_n", "$$", "For a transformation $a : F \\to G$ of endofunctors of $\\mathcal{C}$ we have", "$a \\circ (f \\star 1_F) = f \\star a = (f \\star 1_G) \\circ a$. Thus $f \\star 1_X$", "is indeed a morphism of simplicial objects. Similarly for $1_X \\star f$.", "\\medskip\\noindent", "To show the morphisms are homotopic we construct", "morphisms $h_{n, i} : X_n \\to X_n$ for $n \\geq 0$ and $0 \\leq i \\leq n + 1$", "satisfying the relations described in Lemma \\ref{lemma-relations-homotopy}.", "See also Remark \\ref{remark-homotopy-better}. It turns out we can take", "$$", "h_{n, i} = 1_i \\star f \\star 1_{n + 1 - i}", "$$", "where $1_i$ is the identity transformation on $Y \\circ \\ldots \\circ Y$", "as in the proof of Lemma \\ref{lemma-godement}. We have", "$h_{n, 0} = f \\star 1_{X_n}$ and $h_{n, n + 1} = 1_{X_n} \\star f$", "which checks the first condition. In checking the other conditions", "we use the comments made in the proof of", "Lemma \\ref{lemma-godement-two-maps} about the maps $d^n_j$ and $s^n_j$.", "\\medskip\\noindent", "We check condition (2) of Lemma \\ref{lemma-relations-homotopy}.", "Let $i > j$. We have to show", "$$", "d^n_j \\circ (1_i \\star f \\star 1_{n + 1 - i}) =", "(1_{i - 1} \\star f \\star 1_{n + 1 - i}) \\circ d^n_j", "$$", "Since $i - 1 \\geq j$ we can use one of the possible descriptions of", "$d^n_j$ to rewrite the left hand side as", "$$", "(d^{i - 1}_j \\star 1_{n - i + 1}) \\circ (1_i \\star f \\star 1_{n + 1 - i}) =", "d^{i - 1}_j \\star f \\star 1_{n + 1 - i}", "$$", "Similarly the right hand side becomes", "$$", "(1_{i - 1} \\star f \\star 1_{n + 1 - i}) \\circ", "(d^{i - 1}_j \\star 1_{n - i + 1}) =", "d^{i - 1}_j \\star f \\star 1_{n + 1 - i}", "$$", "Thus we obtain the same result and (2) is checked.", "\\medskip\\noindent", "The conditions (3), (4), and (5) of Lemma \\ref{lemma-relations-homotopy}", "are checked in exactly the same manner using the strategy of the proof", "of Lemma \\ref{lemma-godement-two-maps}. We omit the details\\footnote{When", "$f$ is invertible it suffices to prove that $(a_n) = 1_X$ and", "$(b_n) = f^{-1} \\star 1_X \\star f$ are homotopic.", "But this follows from Lemma \\ref{lemma-godement-two-maps}", "because in this case $a = b = 1_{\\text{id}_\\mathcal{C}}$.}." ], "refs": [ "simplicial-lemma-relations-homotopy", "simplicial-remark-homotopy-better", "simplicial-lemma-godement", "simplicial-lemma-godement-two-maps", "simplicial-lemma-relations-homotopy", "simplicial-lemma-relations-homotopy", "simplicial-lemma-godement-two-maps", "simplicial-lemma-godement-two-maps" ], "ref_ids": [ 14872, 14941, 14904, 14906, 14872, 14872, 14906, 14906 ] } ], "ref_ids": [ 14904 ] }, { "id": 14908, "type": "theorem", "label": "simplicial-lemma-standard-simplicial", "categories": [ "simplicial" ], "title": "simplicial-lemma-standard-simplicial", "contents": [ "In Situation \\ref{situation-adjoint-functors}", "the system $X = (X_n, d^n_j, s^n_j)$", "is a simplicial object of $\\text{Fun}(\\mathcal{A}, \\mathcal{A})$", "and $\\epsilon_0$ defines an augmentation $\\epsilon$ from $X$", "to the constant simplicial object with value $X_{-1} = \\text{id}_\\mathcal{A}$." ], "refs": [], "proofs": [ { "contents": [ "Consider $Y = U \\circ V : \\mathcal{A} \\to \\mathcal{A}$.", "We already have the transformation", "$d : Y = U \\circ V \\to \\text{id}_\\mathcal{A}$. Let us denote", "$$", "s = 1_U \\star \\eta \\star 1_V :", "Y = U \\circ \\text{id}_\\mathcal{S} \\circ V", "\\longrightarrow", "U \\circ V \\circ U \\circ V = Y \\circ Y", "$$", "This places us in the sitation of Example \\ref{example-godement}.", "It is immediate from the formulas that", "the $X, d^n_i, s^n_i$ constructed above and the", "$X, s^n_i, s^n_i$ constructed from $Y, d, s$", "in Example \\ref{example-godement} agree. Thus, according to", "Lemma \\ref{lemma-godement} it suffices to prove that", "$$", "1_Y = (d \\star 1_Y) \\circ s = (1_Y \\star d) \\circ s", "\\quad\\text{and}\\quad", "(s \\star 1) \\circ s = (1 \\star s) \\circ s", "$$", "The first equal sign translates into the equality", "$$", "1_U \\star 1_V = (d \\star 1_U \\star 1_V) \\circ (1_U \\star \\eta \\star 1_V)", "$$", "which holds if we have $1_U = (d \\star 1_U) \\circ (1_U \\star \\eta)$", "which in turn holds by (\\ref{equation-composition}). Similarly for the", "second equal sign. For the last equation we need to prove", "$$", "(1_U \\star \\eta \\star 1_V \\star 1_U \\star 1_V) \\circ", "(1_U \\star \\eta \\star 1_V) =", "(1_U \\star 1_V \\star 1_U \\star \\eta \\star 1_V) \\circ", "(1_U \\star \\eta \\star 1_V)", "$$", "For this it suffices to prove", "$(\\eta \\star 1_V \\star 1_U) \\circ \\eta = (1_V \\star 1_U \\star \\eta) \\circ \\eta$", "which is true because both sides are the same as $\\eta \\star \\eta$." ], "refs": [ "simplicial-lemma-godement" ], "ref_ids": [ 14904 ] } ], "ref_ids": [] }, { "id": 14909, "type": "theorem", "label": "simplicial-lemma-standard-simplicial-homotopy", "categories": [ "simplicial" ], "title": "simplicial-lemma-standard-simplicial-homotopy", "contents": [ "In Situation \\ref{situation-adjoint-functors} the maps", "$$", "1_V \\star \\epsilon : V \\circ X \\to V,", "\\quad\\text{and}\\quad", "\\epsilon \\star 1_U : X \\circ U \\to U", "$$", "are homotopy equivalences." ], "refs": [], "proofs": [ { "contents": [ "As in the proof of Lemma \\ref{lemma-standard-simplicial}", "we set $Y = U \\circ V$ so that we are", "in the sitation of Example \\ref{example-godement}.", "\\medskip\\noindent", "Proof of the first homotopy equivalence. By Lemma \\ref{lemma-godement-section}", "to construct a map", "$h : V \\to V \\circ X$ right inverse to $1_V \\star \\epsilon$ it", "suffices to construct a map $h_0 : V \\to V \\circ Y = V \\circ U \\circ V$", "such that $1_V = (1_V \\star d) \\circ h_0$. Of course we take", "$h_0 = \\eta \\star 1_V$ and the equality holds by (\\ref{equation-composition}).", "To finish the proof we need to show the two maps", "$$", "(1_V \\star \\epsilon) \\circ h, 1_V \\star \\text{id}_X :", "V \\circ X \\longrightarrow V \\circ X", "$$", "are homotopic. This follows immediately from", "Lemma \\ref{lemma-godement-two-maps} (with", "$G = G' = V$ and $F = F' = \\text{id}_\\mathcal{S}$).", "\\medskip\\noindent", "The proof of the second homotopy equivalence. By", "Lemma \\ref{lemma-godement-section} to construct a map", "$h : U \\to X \\circ U$ right inverse to $\\epsilon \\star 1_U$ it", "suffices to construct a map $h_0 : U \\to Y \\circ U = U \\circ V \\circ U$", "such that $1_U = (d \\star 1_U) \\circ h_0$. Of course we take", "$h_0 = 1_U \\star \\eta$ and the equality holds by (\\ref{equation-composition}).", "To finish the proof we need to show the two maps", "$$", "(\\epsilon \\star 1_U) \\circ h, \\text{id}_X \\star 1_U :", "X \\circ U \\longrightarrow X \\circ U", "$$", "are homotopic. This follows immediately from", "Lemma \\ref{lemma-godement-two-maps} (with", "$G = G' = \\text{id}_\\mathcal{A}$ and $F = F' = U$)." ], "refs": [ "simplicial-lemma-standard-simplicial", "simplicial-lemma-godement-two-maps", "simplicial-lemma-godement-two-maps" ], "ref_ids": [ 14908, 14906, 14906 ] } ], "ref_ids": [] }, { "id": 14944, "type": "theorem", "label": "discriminant-lemma-dominate-factorizations", "categories": [ "discriminant" ], "title": "discriminant-lemma-dominate-factorizations", "contents": [ "Let $A \\to B$ be a quasi-finite ring map. Given two factorizations", "$A \\to B' \\to B$ and $A \\to B'' \\to B$ with", "$A \\to B'$ and $A \\to B''$ finite and $\\Spec(B) \\to \\Spec(B')$", "and $\\Spec(B) \\to \\Spec(B'')$ open immersions, there exists", "an $A$-subalgebra $B''' \\subset B$ finite over $A$ such that", "$\\Spec(B) \\to \\Spec(B''')$ an open immersion and $B' \\to B$ and", "$B'' \\to B$ factor through $B'''$." ], "refs": [], "proofs": [ { "contents": [ "Let $B''' \\subset B$ be the $A$-subalgebra generated by the images", "of $B' \\to B$ and $B'' \\to B$. As $B'$ and $B''$ are each generated", "by finitely many elements integral over $A$, we see that $B'''$ is", "generated by finitely many elements integral over $A$ and we conclude", "that $B'''$ is finite over $A$", "(Algebra, Lemma \\ref{algebra-lemma-characterize-finite-in-terms-of-integral}).", "Consider the maps", "$$", "B = B' \\otimes_{B'} B \\to B''' \\otimes_{B'} B \\to B \\otimes_{B'} B = B", "$$", "The final equality holds because $\\Spec(B) \\to \\Spec(B')$ is an", "open immersion (and hence a monomorphism). The second arrow is injective", "as $B' \\to B$ is flat. Hence both arrows are isomorphisms.", "This means that", "$$", "\\xymatrix{", "\\Spec(B''') \\ar[d] & \\Spec(B) \\ar[d] \\ar[l] \\\\", "\\Spec(B') & \\Spec(B) \\ar[l]", "}", "$$", "is cartesian. Since the base change of an open immersion is an", "open immersion we conclude." ], "refs": [ "algebra-lemma-characterize-finite-in-terms-of-integral" ], "ref_ids": [ 484 ] } ], "ref_ids": [] }, { "id": 14945, "type": "theorem", "label": "discriminant-lemma-dualizing-well-defined", "categories": [ "discriminant" ], "title": "discriminant-lemma-dualizing-well-defined", "contents": [ "The module (\\ref{equation-dualizing}) is well defined, i.e.,", "independent of the choice of the factorization." ], "refs": [], "proofs": [ { "contents": [ "Let $B', B'', B'''$ be as in Lemma \\ref{lemma-dominate-factorizations}.", "We obtain a canonical map", "$$", "\\omega''' = \\Hom_A(B''', A) \\otimes_{B'''} B \\longrightarrow", "\\Hom_A(B', A) \\otimes_{B'} B = \\omega'", "$$", "and a similar one involving $B''$. If we show these maps are isomorphisms", "then the lemma is proved. Let $g \\in B'$ be an element such that", "$B'_g \\to B_g$ is an isomorphism and hence $B'_g \\to (B''')_g \\to B_g$", "are isomorphisms. It suffices to show that $(\\omega''')_g \\to \\omega'_g$", "is an isomorphism. The kernel and cokernel of the ring map $B' \\to B'''$", "are finite $A$-modules and $g$-power torsion.", "Hence they are annihilated by a power of $g$.", "This easily implies the result." ], "refs": [ "discriminant-lemma-dominate-factorizations" ], "ref_ids": [ 14944 ] } ], "ref_ids": [] }, { "id": 14946, "type": "theorem", "label": "discriminant-lemma-localize-dualizing", "categories": [ "discriminant" ], "title": "discriminant-lemma-localize-dualizing", "contents": [ "Let $A \\to B$ be a quasi-finite map of Noetherian rings.", "\\begin{enumerate}", "\\item If $A \\to B$ factors as $A \\to A_f \\to B$ for some $f \\in A$,", "then $\\omega_{B/A} = \\omega_{B/A_f}$.", "\\item If $g \\in B$, then $(\\omega_{B/A})_g = \\omega_{B_g/A}$.", "\\item If $f \\in A$, then $\\omega_{B_f/A_f} = (\\omega_{B/A})_f$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Say $A \\to B' \\to B$ is a factorization with $A \\to B'$ finite and", "$\\Spec(B) \\to \\Spec(B')$ an open immersion. In case (1) we may use", "the factorization $A_f \\to B'_f \\to B$ to compute $\\omega_{B/A_f}$", "and use Algebra, Lemma \\ref{algebra-lemma-hom-from-finitely-presented}.", "In case (2) use the factorization $A \\to B' \\to B_g$ to see the result.", "Part (3) follows from a combination of (1) and (2)." ], "refs": [ "algebra-lemma-hom-from-finitely-presented" ], "ref_ids": [ 353 ] } ], "ref_ids": [] }, { "id": 14947, "type": "theorem", "label": "discriminant-lemma-bc-map-dualizing", "categories": [ "discriminant" ], "title": "discriminant-lemma-bc-map-dualizing", "contents": [ "The base change map (\\ref{equation-bc-dualizing})", "is independent of the choice of the", "factorization $A \\to B' \\to B$. Given ring maps $A \\to A_1 \\to A_2$", "the composition of the base change maps for $A \\to A_1$ and $A_1 \\to A_2$", "is the base change map for $A \\to A_2$." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: argue in exactly the same way as in", "Lemma \\ref{lemma-dualizing-well-defined}", "using Lemma \\ref{lemma-dominate-factorizations}." ], "refs": [ "discriminant-lemma-dualizing-well-defined", "discriminant-lemma-dominate-factorizations" ], "ref_ids": [ 14945, 14944 ] } ], "ref_ids": [] }, { "id": 14948, "type": "theorem", "label": "discriminant-lemma-dualizing-flat-base-change", "categories": [ "discriminant" ], "title": "discriminant-lemma-dualizing-flat-base-change", "contents": [ "If $A \\to A_1$ is flat, then", "the base change map (\\ref{equation-bc-dualizing}) induces an isomorphism", "$\\omega_{B/A} \\otimes_B B_1 \\to \\omega_{B_1/A_1}$." ], "refs": [], "proofs": [ { "contents": [ "Assume that $A \\to A_1$ is flat. By construction of $\\omega_{B/A}$ we may", "assume that $A \\to B$ is finite. Then $\\omega_{B/A} = \\Hom_A(B, A)$ and", "$\\omega_{B_1/A_1} = \\Hom_{A_1}(B_1, A_1)$. Since $B_1 = B \\otimes_A A_1$", "the result follows from More on Algebra, Lemma", "\\ref{more-algebra-lemma-pseudo-coherence-and-base-change-ext}." ], "refs": [ "more-algebra-lemma-pseudo-coherence-and-base-change-ext" ], "ref_ids": [ 10165 ] } ], "ref_ids": [] }, { "id": 14949, "type": "theorem", "label": "discriminant-lemma-dualizing-composition", "categories": [ "discriminant" ], "title": "discriminant-lemma-dualizing-composition", "contents": [ "Let $A \\to B \\to C$ be quasi-finite homomorphisms of Noetherian rings.", "There is a canonical map", "$\\omega_{B/A} \\otimes_B \\omega_{C/B} \\to \\omega_{C/A}$." ], "refs": [], "proofs": [ { "contents": [ "Choose $A \\to B' \\to B$ with $A \\to B'$ finite such that", "$\\Spec(B) \\to \\Spec(B')$ is an open immersion. Then", "$B' \\to C$ is quasi-finite too. Choose $B' \\to C' \\to C$", "with $B' \\to C'$ finite and $\\Spec(C) \\to \\Spec(C')$ an", "open immersion. Then the source of the arrow is", "$$", "\\Hom_A(B', A) \\otimes_{B'} B \\otimes_B", "\\Hom_B(B \\otimes_{B'} C', B) \\otimes_{B \\otimes_{B'} C'} C", "$$", "which is equal to", "$$", "\\Hom_A(B', A) \\otimes_{B'}", "\\Hom_{B'}(C', B) \\otimes_{C'} C", "$$", "This indeed comes with a canonical map to", "$\\Hom_A(C', A) \\otimes_{C'} C = \\omega_{C/A}$", "coming from composition", "$\\Hom_A(B', A) \\times \\Hom_{B'}(C', B) \\to \\Hom_A(C', A)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14950, "type": "theorem", "label": "discriminant-lemma-dualizing-product", "categories": [ "discriminant" ], "title": "discriminant-lemma-dualizing-product", "contents": [ "Let $A \\to B$ and $A \\to C$ be quasi-finite maps of Noetherian rings.", "Then $\\omega_{B \\times C/A} = \\omega_{B/A} \\times \\omega_{C/A}$", "as modules over $B \\times C$." ], "refs": [], "proofs": [ { "contents": [ "Choose factorizations $A \\to B' \\to B$ and $A \\to C' \\to C$ such that", "$A \\to B'$ and $A \\to C'$ are finite and such that $\\Spec(B) \\to \\Spec(B')$", "and $\\Spec(C) \\to \\Spec(C')$ are open immersions. Then", "$A \\to B' \\times C' \\to B \\times C$ is a similar factorization.", "Using this factorization to compute $\\omega_{B \\times C/A}$", "gives the lemma." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14951, "type": "theorem", "label": "discriminant-lemma-dualizing-associated-primes", "categories": [ "discriminant" ], "title": "discriminant-lemma-dualizing-associated-primes", "contents": [ "Let $A \\to B$ be a quasi-finite homomorphism of Noetherian rings.", "Then $\\text{Ass}_B(\\omega_{B/A})$ is the set of primes of $B$", "lying over associated primes of $A$." ], "refs": [], "proofs": [ { "contents": [ "Choose a factorization $A \\to B' \\to B$ with $A \\to B'$ finite and", "$B' \\to B$ inducing an open immersion on spectra. As", "$\\omega_{B/A} = \\omega_{B'/A} \\otimes_{B'} B$ it suffices", "to prove the statement for $\\omega_{B'/A}$. Thus we may assume $A \\to B$", "is finite.", "\\medskip\\noindent", "Assume $\\mathfrak p \\in \\text{Ass}(A)$ and $\\mathfrak q$ is a prime", "of $B$ lying over $\\mathfrak p$. Let $x \\in A$ be an element whose", "annihilator is $\\mathfrak p$. Choose a nonzero $\\kappa(\\mathfrak p)$", "linear map $\\lambda : \\kappa(\\mathfrak q) \\to \\kappa(\\mathfrak p)$.", "Since $A/\\mathfrak p \\subset B/\\mathfrak q$ is a finite extension", "of rings, there is an $f \\in A$, $f \\not \\in \\mathfrak p$", "such that $f\\lambda$ maps $B/\\mathfrak q$ into $A/\\mathfrak p$.", "Hence we obtain a nonzero $A$-linear map", "$$", "B \\to B/\\mathfrak q \\to A/\\mathfrak p \\to A,\\quad", "b \\mapsto f\\lambda(b)x", "$$", "An easy computation shows that this element of $\\omega_{B/A}$", "has annihilator $\\mathfrak q$, whence", "$\\mathfrak q \\in \\text{Ass}(\\omega_{B/A})$.", "\\medskip\\noindent", "Conversely, suppose that $\\mathfrak q \\subset B$ is a prime ideal", "lying over a prime $\\mathfrak p \\subset A$ which is not an associated", "prime of $A$. We have to show that", "$\\mathfrak q \\not \\in \\text{Ass}_B(\\omega_{B/A})$.", "After replacing $A$ by $A_\\mathfrak p$ and $B$ by", "$B_\\mathfrak p$ we may assume that $\\mathfrak p$ is a maximal ideal", "of $A$. This is allowed by Lemma \\ref{lemma-dualizing-flat-base-change} and", "Algebra, Lemma \\ref{algebra-lemma-localize-ass}.", "Then there exists an $f \\in \\mathfrak m$", "which is a nonzerodivisor on $A$.", "Then $f$ is a nonzerodivisor on $\\omega_{B/A}$", "and hence $\\mathfrak q$ is not an associated prime of this module." ], "refs": [ "discriminant-lemma-dualizing-flat-base-change", "algebra-lemma-localize-ass" ], "ref_ids": [ 14948, 710 ] } ], "ref_ids": [] }, { "id": 14952, "type": "theorem", "label": "discriminant-lemma-dualizing-base-flat-flat", "categories": [ "discriminant" ], "title": "discriminant-lemma-dualizing-base-flat-flat", "contents": [ "Let $A \\to B$ be a flat quasi-finite homomorphism of Noetherian rings.", "Then $\\omega_{B/A}$ is a flat $A$-module." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q \\subset B$ be a prime lying over $\\mathfrak p \\subset A$.", "We will show that the localization $\\omega_{B/A, \\mathfrak q}$ is flat", "over $A_\\mathfrak p$.", "This suffices by Algebra, Lemma \\ref{algebra-lemma-flat-localization}.", "By", "Algebra, Lemma \\ref{algebra-lemma-etale-makes-quasi-finite-finite-one-prime}", "we can find an \\'etale ring map $A \\to A'$ and a prime", "ideal $\\mathfrak p' \\subset A'$ lying over $\\mathfrak p$", "such that $\\kappa(\\mathfrak p') = \\kappa(\\mathfrak p)$ and", "such that", "$$", "B' = B \\otimes_A A' = C \\times D", "$$", "with $A' \\to C$ finite and such that the unique prime $\\mathfrak q'$", "of $B \\otimes_A A'$ lying over $\\mathfrak q$ and $\\mathfrak p'$", "corresponds to a prime of $C$. By ", "Lemma \\ref{lemma-dualizing-flat-base-change}", "and Algebra, Lemma \\ref{algebra-lemma-base-change-flat-up-down}", "it suffices to show $\\omega_{B'/A', \\mathfrak q'}$", "is flat over $A'_{\\mathfrak p'}$.", "Since $\\omega_{B'/A'} = \\omega_{C/A'} \\times \\omega_{D/A'}$", "by Lemma \\ref{lemma-dualizing-product}", "this reduces us to the case where $B$ is finite flat over $A$.", "In this case $B$ is finite locally free as an $A$-module", "and $\\omega_{B/A} = \\Hom_A(B, A)$ is the dual finite", "locally free $A$-module." ], "refs": [ "algebra-lemma-flat-localization", "algebra-lemma-etale-makes-quasi-finite-finite-one-prime", "discriminant-lemma-dualizing-flat-base-change", "algebra-lemma-base-change-flat-up-down", "discriminant-lemma-dualizing-product" ], "ref_ids": [ 538, 1246, 14948, 898, 14950 ] } ], "ref_ids": [] }, { "id": 14953, "type": "theorem", "label": "discriminant-lemma-dualizing-base-change-of-flat", "categories": [ "discriminant" ], "title": "discriminant-lemma-dualizing-base-change-of-flat", "contents": [ "If $A \\to B$ is flat, then the base change map (\\ref{equation-bc-dualizing})", "induces an isomorphism $\\omega_{B/A} \\otimes_B B_1 \\to \\omega_{B_1/A_1}$." ], "refs": [], "proofs": [ { "contents": [ "If $A \\to B$ is finite flat, then $B$ is finite locally free as an $A$-module.", "In this case $\\omega_{B/A} = \\Hom_A(B, A)$ is the dual finite", "locally free $A$-module and formation of this module commutes", "with arbitrary base change which proves the lemma in this case.", "In the next paragraph we reduce the general (quasi-finite flat)", "case to the finite flat case just discussed.", "\\medskip\\noindent", "Let $\\mathfrak q_1 \\subset B_1$ be a prime. We will show that the", "localization of the map at the prime $\\mathfrak q_1$ is an isomorphism, which", "suffices by Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}.", "Let $\\mathfrak q \\subset B$ and $\\mathfrak p \\subset A$ be the prime", "ideals lying under $\\mathfrak q_1$. By", "Algebra, Lemma \\ref{algebra-lemma-etale-makes-quasi-finite-finite-one-prime}", "we can find an \\'etale ring map $A \\to A'$ and a prime", "ideal $\\mathfrak p' \\subset A'$ lying over $\\mathfrak p$", "such that $\\kappa(\\mathfrak p') = \\kappa(\\mathfrak p)$ and", "such that", "$$", "B' = B \\otimes_A A' = C \\times D", "$$", "with $A' \\to C$ finite and such that the unique prime $\\mathfrak q'$", "of $B \\otimes_A A'$ lying over $\\mathfrak q$ and $\\mathfrak p'$", "corresponds to a prime of $C$. Set $A'_1 = A' \\otimes_A A_1$ and", "consider the base change maps", "(\\ref{equation-bc-dualizing}) for the ring maps", "$A \\to A' \\to A'_1$ and $A \\to A_1 \\to A'_1$ as in the diagram", "$$", "\\xymatrix{", "\\omega_{B'/A'} \\otimes_{B'} B'_1 \\ar[r] & \\omega_{B'_1/A'_1} \\\\", "\\omega_{B/A} \\otimes_B B'_1 \\ar[r] \\ar[u] &", "\\omega_{B_1/A_1} \\otimes_{B_1} B'_1 \\ar[u]", "}", "$$", "where $B' = B \\otimes_A A'$, $B_1 = B \\otimes_A A_1$, and", "$B_1' = B \\otimes_A (A' \\otimes_A A_1)$. By", "Lemma \\ref{lemma-bc-map-dualizing} the diagram commutes. By", "Lemma \\ref{lemma-dualizing-flat-base-change}", "the vertical arrows are isomorphisms.", "As $B_1 \\to B'_1$ is \\'etale and hence flat it suffices", "to prove the top horizontal arrow is an isomorphism after localizing", "at a prime $\\mathfrak q'_1$ of $B'_1$ lying over $\\mathfrak q$", "(there is such a prime and use", "Algebra, Lemma \\ref{algebra-lemma-local-flat-ff}).", "Thus we may assume that $B = C \\times D$ with $A \\to C$", "finite and $\\mathfrak q$ corresponding to a prime of $C$.", "In this case the dualizing module $\\omega_{B/A}$ decomposes", "in a similar fashion (Lemma \\ref{lemma-dualizing-product})", "which reduces the question", "to the finite flat case $A \\to C$ handled above." ], "refs": [ "algebra-lemma-characterize-zero-local", "algebra-lemma-etale-makes-quasi-finite-finite-one-prime", "discriminant-lemma-bc-map-dualizing", "discriminant-lemma-dualizing-flat-base-change", "algebra-lemma-local-flat-ff", "discriminant-lemma-dualizing-product" ], "ref_ids": [ 410, 1246, 14947, 14948, 537, 14950 ] } ], "ref_ids": [] }, { "id": 14954, "type": "theorem", "label": "discriminant-lemma-compare-dualizing-algebraic", "categories": [ "discriminant" ], "title": "discriminant-lemma-compare-dualizing-algebraic", "contents": [ "Let $A \\to B$ be a quasi-finite homomorphism of Noetherian rings.", "Let $\\omega_{B/A}^\\bullet \\in D(B)$ be the algebraic relative dualizing", "complex discussed in Dualizing Complexes, Section", "\\ref{dualizing-section-relative-dualizing-complexes-Noetherian}.", "Then there is a (nonunique) isomorphism", "$\\omega_{B/A} = H^0(\\omega_{B/A}^\\bullet)$." ], "refs": [], "proofs": [ { "contents": [ "Choose a factorization $A \\to B' \\to B$", "where $A \\to B'$ is finite and $\\Spec(B') \\to \\Spec(B)$", "is an open immersion. Then", "$\\omega_{B/A}^\\bullet = \\omega_{B'/A}^\\bullet \\otimes_B^\\mathbf{L} B'$", "by Dualizing Complexes, Lemmas", "\\ref{dualizing-lemma-composition-shriek-algebraic} and", "\\ref{dualizing-lemma-upper-shriek-localize} and", "the definition of $\\omega_{B/A}^\\bullet$. Hence", "it suffices to show there is an isomorphism when $A \\to B$ is finite.", "In this case we can use", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-upper-shriek-finite}", "to see that $\\omega_{B/A}^\\bullet = R\\Hom(B, A)$ and hence", "$H^0(\\omega^\\bullet_{B/A}) = \\Hom_A(B, A)$ as desired." ], "refs": [ "dualizing-lemma-composition-shriek-algebraic", "dualizing-lemma-upper-shriek-localize", "dualizing-lemma-upper-shriek-finite" ], "ref_ids": [ 2900, 2902, 2901 ] } ], "ref_ids": [] }, { "id": 14955, "type": "theorem", "label": "discriminant-lemma-discriminant", "categories": [ "discriminant" ], "title": "discriminant-lemma-discriminant", "contents": [ "Let $\\pi : X \\to Y$ be a morphism of schemes which is finite locally", "free. Then $\\pi$ is \\'etale if and only if its discriminant is empty." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms, Lemma \\ref{morphisms-lemma-etale-flat-etale-fibres}", "it suffices to check that the fibres of $\\pi$ are \\'etale.", "Since the construction of the trace pairing commutes with base", "change we reduce to the following question: Let $k$ be a field", "and let $A$ be a finite dimensional $k$-algebra. Show that", "$A$ is \\'etale over $k$ if and only if the trace pairing", "$Q_{A/k} : A \\times A \\to k$, $(a, b) \\mapsto \\text{Trace}_{A/k}(ab)$", "is nondegenerate.", "\\medskip\\noindent", "Assume $Q_{A/k}$ is nondegenerate. If $a \\in A$ is a nilpotent element, then", "$ab$ is nilpotent for all $b \\in A$ and we conclude that $Q_{A/k}(a, -)$ is", "identically zero. Hence $A$ is reduced. Then we can write", "$A = K_1 \\times \\ldots \\times K_n$ as a product where each $K_i$", "is a field (see", "Algebra, Lemmas \\ref{algebra-lemma-finite-dimensional-algebra},", "\\ref{algebra-lemma-artinian-finite-length}, and", "\\ref{algebra-lemma-minimal-prime-reduced-ring}).", "In this case the quadratic", "space $(A, Q_{A/k})$ is the orthogonal direct sum of the spaces", "$(K_i, Q_{K_i/k})$. It follows from", "Fields, Lemma \\ref{fields-lemma-separable-trace-pairing}", "that each $K_i$ is separable over $k$. This means that $A$ is \\'etale", "over $k$ by Algebra, Lemma \\ref{algebra-lemma-etale-over-field}.", "The converse is proved by reading the argument backwards." ], "refs": [ "morphisms-lemma-etale-flat-etale-fibres", "algebra-lemma-finite-dimensional-algebra", "algebra-lemma-artinian-finite-length", "algebra-lemma-minimal-prime-reduced-ring", "fields-lemma-separable-trace-pairing", "algebra-lemma-etale-over-field" ], "ref_ids": [ 5365, 642, 646, 418, 4503, 1232 ] } ], "ref_ids": [] }, { "id": 14956, "type": "theorem", "label": "discriminant-lemma-trace-unique", "categories": [ "discriminant" ], "title": "discriminant-lemma-trace-unique", "contents": [ "Let $A \\to B$ be a flat quasi-finite map of Noetherian rings.", "Then there is at most one trace element in $\\omega_{B/A}$." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathfrak q \\subset B$ be a prime ideal lying over the prime", "$\\mathfrak p \\subset A$. By", "Algebra, Lemma \\ref{algebra-lemma-etale-makes-quasi-finite-finite-one-prime}", "we can find an \\'etale ring map $A \\to A_1$ and a prime", "ideal $\\mathfrak p_1 \\subset A_1$ lying over $\\mathfrak p$", "such that $\\kappa(\\mathfrak p_1) = \\kappa(\\mathfrak p)$ and", "such that", "$$", "B_1 = B \\otimes_A A_1 = C \\times D", "$$", "with $A_1 \\to C$ finite and such that the unique prime $\\mathfrak q_1$", "of $B \\otimes_A A_1$ lying over $\\mathfrak q$ and $\\mathfrak p_1$", "corresponds to a prime of $C$. Observe that", "$\\omega_{C/A_1} = \\omega_{B/A} \\otimes_B C$", "(combine Lemmas \\ref{lemma-dualizing-flat-base-change} and", "\\ref{lemma-dualizing-product}). Since the collection", "of ring maps $B \\to C$ obtained in this manner is a jointly", "injective family of flat maps and since the image of $\\tau_{B/A}$", "in $\\omega_{C/A_1}$ is prescribed the uniqueness follows." ], "refs": [ "algebra-lemma-etale-makes-quasi-finite-finite-one-prime", "discriminant-lemma-dualizing-flat-base-change", "discriminant-lemma-dualizing-product" ], "ref_ids": [ 1246, 14948, 14950 ] } ], "ref_ids": [] }, { "id": 14957, "type": "theorem", "label": "discriminant-lemma-finite-flat-trace", "categories": [ "discriminant" ], "title": "discriminant-lemma-finite-flat-trace", "contents": [ "Let $A \\to B$ be a finite flat map of Noetherian rings.", "Then $\\text{Trace}_{B/A} \\in \\omega_{B/A}$ is the trace element." ], "refs": [], "proofs": [ { "contents": [ "Suppose we have $A \\to A_1$ with $A_1$ Noetherian and", "a product decomposition $B \\otimes_A A_1 = C \\times D$ with $A_1 \\to C$", "finite. Of course in this case $A_1 \\to D$ is also finite.", "Set $B_1 = B \\otimes_A A_1$.", "Since the construction of traces commutes with base change", "we see that $\\text{Trace}_{B/A}$ maps to $\\text{Trace}_{B_1/A_1}$.", "Thus the proof is finished by noticing that", "$\\text{Trace}_{B_1/A_1} = (\\text{Trace}_{C/A_1}, \\text{Trace}_{D/A_1})$", "under the isomorphism", "$\\omega_{B_1/A_1} = \\omega_{C/A_1} \\times \\omega_{D/A_1}$", "of Lemma \\ref{lemma-dualizing-product}." ], "refs": [ "discriminant-lemma-dualizing-product" ], "ref_ids": [ 14950 ] } ], "ref_ids": [] }, { "id": 14958, "type": "theorem", "label": "discriminant-lemma-trace-base-change", "categories": [ "discriminant" ], "title": "discriminant-lemma-trace-base-change", "contents": [ "Let $A \\to B$ be a flat quasi-finite map of Noetherian rings.", "Let $\\tau \\in \\omega_{B/A}$ be a trace element.", "\\begin{enumerate}", "\\item If $A \\to A_1$ is a map with $A_1$ Noetherian, then with", "$B_1 = A_1 \\otimes_A B$ the image of $\\tau$ in $\\omega_{B_1/A_1}$ is a", "trace element.", "\\item If $A = R_f$, then $\\tau$ is a trace element in $\\omega_{B/R}$.", "\\item If $g \\in B$, then the image of $\\tau$ in $\\omega_{B_g/A}$", "is a trace element.", "\\item If $B = B_1 \\times B_2$, then $\\tau$ maps to a trace element", "in both $\\omega_{B_1/A}$ and $\\omega_{B_2/A}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is a formal consequence of the definition.", "\\medskip\\noindent", "Statement (2) makes sense because $\\omega_{B/R} = \\omega_{B/A}$", "by Lemma \\ref{lemma-localize-dualizing}. Denote $\\tau'$ the element", "$\\tau$ but viewed as an element of $\\omega_{B/R}$. To see that (2) is true", "suppose that we have $R \\to R_1$ with $R_1$ Noetherian and a product", "decomposition $B \\otimes_R R_1 = C \\times D$ with $R_1 \\to C$ finite.", "Then with $A_1 = (R_1)_f$ we see that $B \\otimes_A A_1 = C \\times D$.", "Since $R_1 \\to C$ is finite, a fortiori $A_1 \\to C$ is finite.", "Hence we can use the defining property of $\\tau$ to get the corresponding", "property of $\\tau'$.", "\\medskip\\noindent", "Statement (3) makes sense because $\\omega_{B_g/A} = (\\omega_{B/A})_g$", "by Lemma \\ref{lemma-localize-dualizing}. The proof is similar to the proof", "of (2). Suppose we have $A \\to A_1$ with $A_1$ Noetherian and", "a product decomposition $B_g \\otimes_A A_1 = C \\times D$ with $A_1 \\to C$", "finite. Set $B_1 = B \\otimes_A A_1$. Then", "$\\Spec(C) \\to \\Spec(B_1)$ is an open immersion as $B_g \\otimes_A A_1 = (B_1)_g$", "and the image is closed because $B_1 \\to C$ is finite", "(as $A_1 \\to C$ is finite).", "Thus we see that $B_1 = C \\times D_1$ and $D = (D_1)_g$. Then we can use", "the defining property of $\\tau$ to get the corresponding property", "for the image of $\\tau$ in $\\omega_{B_g/A}$.", "\\medskip\\noindent", "Statement (4) makes sense because", "$\\omega_{B/A} = \\omega_{B_1/A} \\times \\omega_{B_2/A}$ by", "Lemma \\ref{lemma-dualizing-product}.", "Suppose we have $A \\to A'$ with $A'$ Noetherian and", "a product decomposition $B \\otimes_A A' = C \\times D$ with $A' \\to C$", "finite. Then it is clear that we can refine this product", "decomposition into $B \\otimes_A A' = C_1 \\times C_2 \\times D_1 \\times D_2$", "with $A' \\to C_i$ finite such that $B_i \\otimes_A A' = C_i \\times D_i$.", "Then we can use the defining property of $\\tau$ to get the corresponding", "property for the image of $\\tau$ in $\\omega_{B_i/A}$. This uses the obvious", "fact that", "$\\text{Trace}_{C/A'} = (\\text{Trace}_{C_1/A'}, \\text{Trace}_{C_2/A'})$", "under the decomposition", "$\\omega_{C/A'} = \\omega_{C_1/A'} \\times \\omega_{C_2/A'}$." ], "refs": [ "discriminant-lemma-localize-dualizing", "discriminant-lemma-localize-dualizing", "discriminant-lemma-dualizing-product" ], "ref_ids": [ 14946, 14946, 14950 ] } ], "ref_ids": [] }, { "id": 14959, "type": "theorem", "label": "discriminant-lemma-glue-trace", "categories": [ "discriminant" ], "title": "discriminant-lemma-glue-trace", "contents": [ "Let $A \\to B$ be a flat quasi-finite map of Noetherian rings.", "Let $g_1, \\ldots, g_m \\in B$ be elements generating the unit ideal.", "Let $\\tau \\in \\omega_{B/A}$ be an element whose image in", "$\\omega_{B_{g_i}/A}$ is a trace element for $A \\to B_{g_i}$.", "Then $\\tau$ is a trace element." ], "refs": [], "proofs": [ { "contents": [ "Suppose we have $A \\to A_1$ with $A_1$ Noetherian and a product", "decomposition $B \\otimes_A A_1 = C \\times D$ with $A_1 \\to C$ finite.", "We have to show that the image of $\\tau$ in $\\omega_{C/A_1}$ is", "$\\text{Trace}_{C/A_1}$. Observe that $g_1, \\ldots, g_m$", "generate the unit ideal in $B_1 = B \\otimes_A A_1$ and that", "$\\tau$ maps to a trace element in $\\omega_{(B_1)_{g_i}/A_1}$", "by Lemma \\ref{lemma-trace-base-change}. Hence we may replace", "$A$ by $A_1$ and $B$ by $B_1$ to get to the situation as described", "in the next paragraph.", "\\medskip\\noindent", "Here we assume that $B = C \\times D$ with $A \\to C$ is finite.", "Let $\\tau_C$ be the image of $\\tau$ in $\\omega_{C/A}$.", "We have to prove that $\\tau_C = \\text{Trace}_{C/A}$ in $\\omega_{C/A}$.", "By the compatibility of trace elements with products", "(Lemma \\ref{lemma-trace-base-change})", "we see that $\\tau_C$ maps to a trace element in $\\omega_{C_{g_i}/A}$.", "Hence, after replacing $B$ by $C$ we may assume that $A \\to B$", "is finite flat.", "\\medskip\\noindent", "Assume $A \\to B$ is finite flat. In this case $\\text{Trace}_{B/A}$", "is a trace element by Lemma \\ref{lemma-finite-flat-trace}.", "Hence $\\text{Trace}_{B/A}$ maps to a trace element in", "$\\omega_{B_{g_i}/A}$ by Lemma \\ref{lemma-trace-base-change}.", "Since trace elements are unique (Lemma \\ref{lemma-trace-unique})", "we find that $\\text{Trace}_{B/A}$ and $\\tau$ map", "to the same elements in $\\omega_{B_{g_i}/A} = (\\omega_{B/A})_{g_i}$.", "As $g_1, \\ldots, g_m$ generate the unit ideal of $B$ the map", "$\\omega_{B/A} \\to \\prod \\omega_{B_{g_i}/A}$ is injective", "and we conclude that $\\tau_C = \\text{Trace}_{B/A}$ as desired." ], "refs": [ "discriminant-lemma-trace-base-change", "discriminant-lemma-trace-base-change", "discriminant-lemma-finite-flat-trace", "discriminant-lemma-trace-base-change", "discriminant-lemma-trace-unique" ], "ref_ids": [ 14958, 14958, 14957, 14958, 14956 ] } ], "ref_ids": [] }, { "id": 14960, "type": "theorem", "label": "discriminant-lemma-dualizing-tau", "categories": [ "discriminant" ], "title": "discriminant-lemma-dualizing-tau", "contents": [ "Let $A \\to B$ be a flat quasi-finite map of Noetherian rings.", "There exists a trace element $\\tau \\in \\omega_{B/A}$." ], "refs": [], "proofs": [ { "contents": [ "Choose a factorization $A \\to B' \\to B$ with $A \\to B'$ finite and", "$\\Spec(B) \\to \\Spec(B')$ an open immersion. Let $g_1, \\ldots, g_n \\in B'$", "be elements such that $\\Spec(B) = \\bigcup D(g_i)$ as opens of $\\Spec(B')$.", "Suppose that we can prove the existence of trace elements $\\tau_i$ for the", "quasi-finite flat ring maps $A \\to B_{g_i}$. Then for all $i, j$ the elements", "$\\tau_i$ and $\\tau_j$ map to trace elements of $\\omega_{B_{g_ig_j}/A}$", "by Lemma \\ref{lemma-trace-base-change}. By uniqueness of", "trace elements (Lemma \\ref{lemma-trace-unique}) they map to the same element.", "Hence the sheaf condition for the quasi-coherent module associated to", "$\\omega_{B/A}$ (see Algebra, Lemma \\ref{algebra-lemma-cover-module})", "produces an element $\\tau \\in \\omega_{B/A}$.", "Then $\\tau$ is a trace element by", "Lemma \\ref{lemma-glue-trace}.", "In this way we reduce to the case treated in the next paragraph.", "\\medskip\\noindent", "Assume we have $A \\to B'$ finite and $g \\in B'$ with $B = B'_g$ flat over $A$.", "It is our task to construct a trace element in", "$\\omega_{B/A} = \\Hom_A(B', A) \\otimes_{B'} B$.", "Choose a resolution $F_1 \\to F_0 \\to B' \\to 0$ of $B'$ by finite free", "$A$-modules $F_0$ and $F_1$. Then we have an exact sequence", "$$", "0 \\to \\Hom_A(B', A) \\to F_0^\\vee \\to F_1^\\vee", "$$", "where $F_i^\\vee = \\Hom_A(F_i, A)$ is the dual finite free module.", "Similarly we have the exact sequence", "$$", "0 \\to \\Hom_A(B', B') \\to F_0^\\vee \\otimes_A B' \\to F_1^\\vee \\otimes_A B'", "$$", "The idea of the construction of $\\tau$ is to use the diagram", "$$", "B' \\xrightarrow{\\mu} \\Hom_A(B', B')", "\\leftarrow \\Hom_A(B', A) \\otimes_A B'", "\\xrightarrow{ev} A", "$$", "where the first arrow sends $b' \\in B'$ to the $A$-linear operator", "given by multiplication by $b'$ and the last arrow is the evaluation map.", "The problem is that the middle arrow, which sends $\\lambda' \\otimes b'$", "to the map $b'' \\mapsto \\lambda'(b'')b'$, is not an isomorphism.", "If $B'$ is flat over $A$, the exact sequences above show that it", "is an isomorphism and the composition from left to right is the usual trace", "$\\text{Trace}_{B'/A}$. In the general case, we consider", "the diagram", "$$", "\\xymatrix{", "& \\Hom_A(B', A) \\otimes_A B' \\ar[r] \\ar[d] &", "\\Hom_A(B', A) \\otimes_A B'_g \\ar[d] \\\\", "B' \\ar[r]_-\\mu \\ar@{..>}[rru] \\ar@{..>}[ru]^\\psi &", "\\Hom_A(B', B') \\ar[r] &", "\\Ker(F_0^\\vee \\otimes_A B'_g \\to F_1^\\vee \\otimes_A B'_g)", "}", "$$", "By flatness of $A \\to B'_g$ we see that the right vertical arrow is an", "isomorphism. Hence we obtain the unadorned dotted arrow.", "Since $B'_g = \\colim \\frac{1}{g^n}B'$, since", "colimits commute with tensor products,", "and since $B'$ is a finitely presented $A$-module", "we can find an $n \\geq 0$ and a $B'$-linear (for right $B'$-module structure)", "map $\\psi : B' \\to \\Hom_A(B', A) \\otimes_A B'$", "whose composition with the left vertical arrow is $g^n\\mu$.", "Composing with $ev$ we obtain an element", "$ev \\circ \\psi \\in \\Hom_A(B', A)$. Then we set ", "$$", "\\tau = (ev \\circ \\psi) \\otimes g^{-n} \\in", "\\Hom_A(B', A) \\otimes_{B'} B'_g = \\omega_{B'_g/A} = \\omega_{B/A}", "$$", "We omit the easy verification that this element does not depend", "on the choice of $n$ and $\\psi$ above.", "\\medskip\\noindent", "Let us prove that $\\tau$ as constructed in the previous paragraph", "has the desired property in a special case. Namely, say", "$B' = C' \\times D'$ and $g = (f, h)$ where $A \\to C'$ flat, $D'_h$ is flat, and", "$f$ is a unit in $C'$.", "To show: $\\tau$ maps to $\\text{Trace}_{C'/A}$ in $\\omega_{C'/A}$.", "In this case we first choose $n_D$ and", "$\\psi_D : D' \\to \\Hom_A(D', A) \\otimes_A D'$ as above for the pair", "$(D', h)$ and we can let", "$\\psi_C : C' \\to \\Hom_A(C', A) \\otimes_A C' = \\Hom_A(C', C')$", "be the map seconding $c' \\in C'$ to multiplication by $c'$.", "Then we take $n = n_D$ and $\\psi = (f^{n_D} \\psi_C, \\psi_D)$", "and the desired compatibility is clear because", "$\\text{Trace}_{C'/A} = ev \\circ \\psi_C$ as remarked above.", "\\medskip\\noindent", "To prove the desired property in general, suppose given", "$A \\to A_1$ with $A_1$ Noetherian and a product decomposition", "$B'_g \\otimes_A A_1 = C \\times D$ with $A_1 \\to C$ finite.", "Set $B'_1 = B' \\otimes_A A_1$. Then $\\Spec(C) \\to \\Spec(B'_1)$", "is an open immersion as $B'_g \\otimes_A A_1 = (B'_1)_g$ and", "the image is closed as $B'_1 \\to C$ is finite (since $A_1 \\to C$", "is finite). Thus $B'_1 = C \\times D'$ and $D'_g = D$.", "We conclude that $B'_1 = C \\times D'$ and $g$ over $A_1$", "are as in the previous paragraph.", "Since formation of the displayed diagram above", "commutes with base change, the formation of $\\tau$ commutes", "with the base change $A \\to A_1$ (details omitted; use the", "resolution $F_1 \\otimes_A A_1 \\to F_0 \\otimes_A A_1 \\to B'_1 \\to 0$", "to see this). Thus the desired compatibility follows from the result", "of the previous paragraph." ], "refs": [ "discriminant-lemma-trace-base-change", "discriminant-lemma-trace-unique", "algebra-lemma-cover-module", "discriminant-lemma-glue-trace" ], "ref_ids": [ 14958, 14956, 413, 14959 ] } ], "ref_ids": [] }, { "id": 14961, "type": "theorem", "label": "discriminant-lemma-tau-nonzero", "categories": [ "discriminant" ], "title": "discriminant-lemma-tau-nonzero", "contents": [ "Let $k$ be a field and let $A$ be a finite $k$-algebra. Assume $A$", "is local with residue field $k'$. The following are equivalent", "\\begin{enumerate}", "\\item $\\text{Trace}_{A/k}$ is nonzero,", "\\item $\\tau_{A/k} \\in \\omega_{A/k}$ is nonzero, and", "\\item $k'/k$ is separable and $\\text{length}_A(A)$ is prime", "to the characteristic of $k$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Conditions (1) and (2) are equivalent by Lemma \\ref{lemma-finite-flat-trace}.", "Let $\\mathfrak m \\subset A$. Since $\\dim_k(A) < \\infty$ it is clear that", "$A$ has finite length over $A$. Choose a filtration", "$$", "A = I_0 \\supset \\mathfrak m = I_1 \\supset I_2 \\supset \\ldots I_n = 0", "$$", "by ideals such that $I_i/I_{i + 1} \\cong k'$ as $A$-modules. See", "Algebra, Lemma \\ref{algebra-lemma-simple-pieces} which also shows that", "$n = \\text{length}_A(A)$. If $a \\in \\mathfrak m$ then $aI_i \\subset I_{i + 1}$", "and it is immediate that $\\text{Trace}_{A/k}(a) = 0$.", "If $a \\not \\in \\mathfrak m$ with image $\\lambda \\in k'$, then", "we conclude", "$$", "\\text{Trace}_{A/k}(a) =", "\\sum\\nolimits_{i = 0, \\ldots, n - 1}", "\\text{Trace}_k(a : I_i/I_{i - 1} \\to I_i/I_{i - 1}) =", "n \\text{Trace}_{k'/k}(\\lambda)", "$$", "The proof of the lemma is finished by applying", "Fields, Lemma \\ref{fields-lemma-separable-trace-pairing}." ], "refs": [ "discriminant-lemma-finite-flat-trace", "algebra-lemma-simple-pieces", "fields-lemma-separable-trace-pairing" ], "ref_ids": [ 14957, 638, 4503 ] } ], "ref_ids": [] }, { "id": 14962, "type": "theorem", "label": "discriminant-lemma-noether-different-product", "categories": [ "discriminant" ], "title": "discriminant-lemma-noether-different-product", "contents": [ "Let $A \\to B_i$, $i = 1, 2$ be ring maps. Set $B = B_1 \\times B_2$.", "\\begin{enumerate}", "\\item The annihilator $J$ of $\\Ker(B \\otimes_A B \\to B)$ is $J_1 \\times J_2$", "where $J_i$ is the annihilator of $\\Ker(B_i \\otimes_A B_i \\to B_i)$.", "\\item The Noether different $\\mathfrak{D}$ of $B$ over $A$ is", "$\\mathfrak{D}_1 \\times \\mathfrak{D}_2$, where $\\mathfrak{D}_i$ is", "the Noether different of $B_i$ over $A$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14963, "type": "theorem", "label": "discriminant-lemma-noether-different-base-change", "categories": [ "discriminant" ], "title": "discriminant-lemma-noether-different-base-change", "contents": [ "Let $A \\to B$ be a finite type ring map. Let $A \\to A'$ be a flat ring map.", "Set $B' = B \\otimes_A A'$.", "\\begin{enumerate}", "\\item The annihilator $J'$ of $\\Ker(B' \\otimes_{A'} B' \\to B')$ is", "$J \\otimes_A A'$ where $J$ is the annihilator of $\\Ker(B \\otimes_A B \\to B)$.", "\\item The Noether different $\\mathfrak{D}'$ of $B'$ over $A'$ is", "$\\mathfrak{D}B'$, where $\\mathfrak{D}$ is", "the Noether different of $B$ over $A$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose generators $b_1, \\ldots, b_n$ of $B$ as an $A$-algebra.", "Then", "$$", "J = \\Ker(B \\otimes_A B \\xrightarrow{b_i \\otimes 1 - 1 \\otimes b_i}", "(B \\otimes_A B)^{\\oplus n})", "$$", "Hence we see that the formation of $J$ commutes with flat base change.", "The result on the Noether different follows immediately from this." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 14964, "type": "theorem", "label": "discriminant-lemma-noether-different-localization", "categories": [ "discriminant" ], "title": "discriminant-lemma-noether-different-localization", "contents": [ "Let $A \\to B' \\to B$ be ring maps with $A \\to B'$", "of finite type and $B' \\to B$ inducing an open immersion of spectra.", "\\begin{enumerate}", "\\item The annihilator $J$ of $\\Ker(B \\otimes_A B \\to B)$ is", "$J' \\otimes_{B'} B$ where $J'$ is the annihilator of", "$\\Ker(B' \\otimes_A B' \\to B')$.", "\\item The Noether different $\\mathfrak{D}$ of $B$ over $A$ is", "$\\mathfrak{D}'B$, where $\\mathfrak{D}'$ is", "the Noether different of $B'$ over $A$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Write $I = \\Ker(B \\otimes_A B \\to B)$ and $I' = \\Ker(B' \\otimes_A B' \\to B')$.", "As $\\Spec(B) \\to \\Spec(B')$ is an open immersion, it follows that", "$B = (B \\otimes_A B) \\otimes_{B' \\otimes_A B'} B'$. Thus we see that", "$I = I'(B \\otimes_A B)$. Since $I'$ is finitely generated and", "$B' \\otimes_A B' \\to B \\otimes_A B$ is flat, we conclude that", "$J = J'(B \\otimes_A B)$, see", "Algebra, Lemma \\ref{algebra-lemma-annihilator-flat-base-change}.", "Since the $B' \\otimes_A B'$-module structure of $J'$", "factors through $B' \\otimes_A B' \\to B'$ we conclude that (1) is true.", "Part (2) is a consequence of (1)." ], "refs": [ "algebra-lemma-annihilator-flat-base-change" ], "ref_ids": [ 542 ] } ], "ref_ids": [] }, { "id": 14965, "type": "theorem", "label": "discriminant-lemma-noether-pairing-compatibilities", "categories": [ "discriminant" ], "title": "discriminant-lemma-noether-pairing-compatibilities", "contents": [ "Let $A \\to B$ be a quasi-finite homomorphism of Noetherian rings.", "\\begin{enumerate}", "\\item If $A \\to A'$ is a flat map of Noetherian rings, then", "$$", "\\xymatrix{", "\\omega_{B/A} \\times J \\ar[r] \\ar[d] & B \\ar[d] \\\\", "\\omega_{B'/A'} \\times J' \\ar[r] & B'", "}", "$$", "is commutative where notation as in", "Lemma \\ref{lemma-noether-different-base-change}", "and horizontal arrows are given by", "(\\ref{equation-pairing-noether}).", "\\item If $B = B_1 \\times B_2$, then", "$$", "\\xymatrix{", "\\omega_{B/A} \\times J \\ar[r] \\ar[d] & B \\ar[d] \\\\", "\\omega_{B_i/A} \\times J_i \\ar[r] & B_i", "}", "$$", "is commutative for $i = 1, 2$ where notation as in", "Lemma \\ref{lemma-noether-different-product}", "and horizontal arrows are given by", "(\\ref{equation-pairing-noether}).", "\\end{enumerate}" ], "refs": [ "discriminant-lemma-noether-different-base-change", "discriminant-lemma-noether-different-product" ], "proofs": [ { "contents": [ "Because of the construction of the pairing in", "Remark \\ref{remark-construction-pairing}", "both (1) and (2) reduce to the case where $A \\to B$ is finite.", "Then (1) follows from the fact that the contraction map", "$\\Hom_A(M, A) \\otimes_A M \\otimes_A M \\to M$,", "$\\lambda \\otimes m \\otimes m' \\mapsto \\lambda(m)m'$", "commuted with base change. To see (2) use that", "$J = J_1 \\times J_2$ is contained in the summands", "$B_1 \\otimes_A B_1$ and $B_2 \\otimes_A B_2$", "of $B \\otimes_A B$." ], "refs": [ "discriminant-remark-construction-pairing" ], "ref_ids": [ 15008 ] } ], "ref_ids": [ 14963, 14962 ] }, { "id": 14966, "type": "theorem", "label": "discriminant-lemma-noether-pairing-flat-quasi-finite", "categories": [ "discriminant" ], "title": "discriminant-lemma-noether-pairing-flat-quasi-finite", "contents": [ "Let $A \\to B$ be a flat quasi-finite homomorphism of Noetherian rings.", "The pairing of Remark \\ref{remark-construction-pairing} induces an isomorphism", "$J \\to \\Hom_B(\\omega_{B/A}, B)$." ], "refs": [ "discriminant-remark-construction-pairing" ], "proofs": [ { "contents": [ "We first prove this when $A \\to B$ is finite and flat. In this case we can", "localize on $A$ and assume $B$ is finite free as an $A$-module. Let", "$b_1, \\ldots, b_n$ be a basis of $B$ as an $A$-module and denote", "$b_1^\\vee, \\ldots, b_n^\\vee$ the dual basis of $\\omega_{B/A}$. Note that", "$\\sum b_i \\otimes c_i \\in J$ maps to the element of $\\Hom_B(\\omega_{B/A}, B)$", "which sends $b_i^\\vee$ to $c_i$. Suppose $\\varphi : \\omega_{B/A} \\to B$", "is $B$-linear. Then we claim that $\\xi = \\sum b_i \\otimes \\varphi(b_i^\\vee)$", "is an element of $J$. Namely, the $B$-linearity of $\\varphi$", "exactly implies that $(b \\otimes 1)\\xi = (1 \\otimes b)\\xi$ for all $b \\in B$.", "Thus our map has an inverse and it is an isomorphism.", "\\medskip\\noindent", "Let $\\mathfrak q \\subset B$ be a prime lying over $\\mathfrak p \\subset A$.", "We will show that the localization", "$$", "J_\\mathfrak q", "\\longrightarrow", "\\Hom_B(\\omega_B/A, B)_\\mathfrak q", "$$", "is an isomorphism.", "This suffices by Algebra, Lemma \\ref{algebra-lemma-characterize-zero-local}.", "By", "Algebra, Lemma \\ref{algebra-lemma-etale-makes-quasi-finite-finite-one-prime}", "we can find an \\'etale ring map $A \\to A'$ and a prime", "ideal $\\mathfrak p' \\subset A'$ lying over $\\mathfrak p$", "such that $\\kappa(\\mathfrak p') = \\kappa(\\mathfrak p)$ and", "such that", "$$", "B' = B \\otimes_A A' = C \\times D", "$$", "with $A' \\to C$ finite and such that the unique prime $\\mathfrak q'$", "of $B \\otimes_A A'$ lying over $\\mathfrak q$ and $\\mathfrak p'$", "corresponds to a prime of $C$. Let $J'$ be the annihilator of", "$\\Ker(B' \\otimes_{A'} B' \\to B')$. By", "Lemmas \\ref{lemma-dualizing-flat-base-change},", "\\ref{lemma-noether-different-base-change}, and", "\\ref{lemma-noether-pairing-compatibilities}", "the map $J' \\to \\Hom_{B'}(\\omega_{B'/A'}, B')$", "is gotten by applying the functor $- \\otimes_B B'$", "to the map $J \\to \\Hom_B(\\omega_{B/A}, B)$.", "Since $B_\\mathfrak q \\to B'_{\\mathfrak q'}$ is faithfully flat", "it suffices to prove the result for $(A' \\to B', \\mathfrak q')$.", "By Lemmas \\ref{lemma-dualizing-product},", "\\ref{lemma-noether-different-product}, and", "\\ref{lemma-noether-pairing-compatibilities}", "this reduces us to the case proved in the first", "paragraph of the proof." ], "refs": [ "algebra-lemma-characterize-zero-local", "algebra-lemma-etale-makes-quasi-finite-finite-one-prime", "discriminant-lemma-dualizing-flat-base-change", "discriminant-lemma-noether-different-base-change", "discriminant-lemma-noether-pairing-compatibilities", "discriminant-lemma-dualizing-product", "discriminant-lemma-noether-different-product", "discriminant-lemma-noether-pairing-compatibilities" ], "ref_ids": [ 410, 1246, 14948, 14963, 14965, 14950, 14962, 14965 ] } ], "ref_ids": [ 15008 ] }, { "id": 14967, "type": "theorem", "label": "discriminant-lemma-noether-different-flat-quasi-finite", "categories": [ "discriminant" ], "title": "discriminant-lemma-noether-different-flat-quasi-finite", "contents": [ "Let $A \\to B$ be a flat quasi-finite homomorphism of Noetherian rings.", "The diagram", "$$", "\\xymatrix{", "J \\ar[rr] \\ar[rd]_\\mu & &", "\\Hom_B(\\omega_{B/A}, B) \\ar[ld]^{\\varphi \\mapsto \\varphi(\\tau_{B/A})} \\\\", "& B", "}", "$$", "commutes where the horizontal arrow is the isomorphism of", "Lemma \\ref{lemma-noether-pairing-flat-quasi-finite}.", "Hence the Noether different of $B$ over $A$", "is the image of the map $\\Hom_B(\\omega_{B/A}, B) \\to B$." ], "refs": [ "discriminant-lemma-noether-pairing-flat-quasi-finite" ], "proofs": [ { "contents": [ "Exactly as in the proof of Lemma \\ref{lemma-noether-pairing-flat-quasi-finite}", "this reduces to the case of a finite free map $A \\to B$.", "In this case $\\tau_{B/A} = \\text{Trace}_{B/A}$.", "Choose a basis $b_1, \\ldots, b_n$ of $B$ as an $A$-module.", "Let $\\xi = \\sum b_i \\otimes c_i \\in J$. Then $\\mu(\\xi) = \\sum b_i c_i$.", "On the other hand, the image of $\\xi$ in $\\Hom_B(\\omega_{B/A}, B)$", "sends $\\text{Trace}_{B/A}$ to $\\sum \\text{Trace}_{B/A}(b_i)c_i$.", "Thus we have to show", "$$", "\\sum b_ic_i = \\sum \\text{Trace}_{B/A}(b_i)c_i", "$$", "when $\\xi = \\sum b_i \\otimes c_i \\in J$. Write $b_i b_j = \\sum_k a_{ij}^k b_k$", "for some $a_{ij}^k \\in A$. Then the right hand side is", "$\\sum_{i, j} a_{ij}^j c_i$. On the other hand, $\\xi \\in J$ implies", "$$", "(b_j \\otimes 1)(\\sum\\nolimits_i b_i \\otimes c_i) =", "(1 \\otimes b_j)(\\sum\\nolimits_i b_i \\otimes c_i)", "$$", "which implies that $b_j c_i = \\sum_k a_{jk}^i c_k$. Thus the left hand side", "is $\\sum_{i, j} a_{ij}^i c_j$. Since $a_{ij}^k = a_{ji}^k$ the equality holds." ], "refs": [ "discriminant-lemma-noether-pairing-flat-quasi-finite" ], "ref_ids": [ 14966 ] } ], "ref_ids": [ 14966 ] }, { "id": 14968, "type": "theorem", "label": "discriminant-lemma-noether-different", "categories": [ "discriminant" ], "title": "discriminant-lemma-noether-different", "contents": [ "Let $A \\to B$ be a finite type ring map. Let $\\mathfrak{D} \\subset B$", "be the Noether different. Then $V(\\mathfrak{D})$ is the set of primes", "$\\mathfrak q \\subset B$ such that $A \\to B$ is not unramified at $\\mathfrak q$." ], "refs": [], "proofs": [ { "contents": [ "Assume $A \\to B$ is unramified at $\\mathfrak q$. After replacing", "$B$ by $B_g$ for some $g \\in B$, $g \\not \\in \\mathfrak q$ we may", "assume $A \\to B$ is unramified (Algebra, Definition", "\\ref{algebra-definition-unramified} and", "Lemma \\ref{lemma-noether-different-localization}).", "In this case $\\Omega_{B/A} = 0$. Hence if $I = \\Ker(B \\otimes_A B \\to B)$,", "then $I/I^2 = 0$ by", "Algebra, Lemma \\ref{algebra-lemma-differentials-diagonal}.", "Since $A \\to B$ is of finite type, we see that $I$ is finitely", "generated. Hence by Nakayama's lemma", "(Algebra, Lemma \\ref{algebra-lemma-NAK})", "there exists an element of the form $1 + i$", "annihilating $I$. It follows that $\\mathfrak{D} = B$.", "\\medskip\\noindent", "Conversely, assume that $\\mathfrak{D} \\not \\subset \\mathfrak q$.", "Then after replacing $B$ by a principal localization as above", "we may assume $\\mathfrak{D} = B$. This means there exists an", "element of the form $1 + i$ in the annihilator of $I$.", "Conversely this implies that $I/I^2 = \\Omega_{B/A}$ is zero", "and we conclude." ], "refs": [ "algebra-definition-unramified", "discriminant-lemma-noether-different-localization", "algebra-lemma-differentials-diagonal", "algebra-lemma-NAK" ], "ref_ids": [ 1544, 14964, 1139, 401 ] } ], "ref_ids": [] }, { "id": 14969, "type": "theorem", "label": "discriminant-lemma-base-change-kahler-different", "categories": [ "discriminant" ], "title": "discriminant-lemma-base-change-kahler-different", "contents": [ "Consider a cartesian diagram of schemes", "$$", "\\xymatrix{", "Y' \\ar[d]_{f'} \\ar[r] & Y \\ar[d]^f \\\\", "X' \\ar[r]^g & X", "}", "$$", "with $f$ locally of finite type. Let $R \\subset Y$, resp.\\ $R' \\subset Y'$", "be the closed subscheme cut out by the K\\\"ahler different of $f$, resp.\\ $f'$.", "Then $Y' \\to Y$ induces an isomorphism $R' \\to R \\times_Y Y'$." ], "refs": [], "proofs": [ { "contents": [ "This is true because $\\Omega_{Y'/X'}$ is the pullback of $\\Omega_{Y/X}$", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-differentials})", "and then we can apply", "More on Algebra, Lemma \\ref{more-algebra-lemma-fitting-ideal-basics}." ], "refs": [ "morphisms-lemma-base-change-differentials", "more-algebra-lemma-fitting-ideal-basics" ], "ref_ids": [ 5314, 9834 ] } ], "ref_ids": [] }, { "id": 14970, "type": "theorem", "label": "discriminant-lemma-kahler-different", "categories": [ "discriminant" ], "title": "discriminant-lemma-kahler-different", "contents": [ "Let $f : Y \\to X$ be a morphism of schemes which is locally of finite type.", "Let $R \\subset Y$ be the closed subscheme defined by", "the K\\\"ahler different. Then $R \\subset Y$ is exactly", "the set of points where $f$ is not unramified." ], "refs": [], "proofs": [ { "contents": [ "This is a copy of", "Divisors, Lemma \\ref{divisors-lemma-base-change-and-fitting-ideal-omega}." ], "refs": [ "divisors-lemma-base-change-and-fitting-ideal-omega" ], "ref_ids": [ 7900 ] } ], "ref_ids": [] }, { "id": 14971, "type": "theorem", "label": "discriminant-lemma-kahler-different-complete-intersection", "categories": [ "discriminant" ], "title": "discriminant-lemma-kahler-different-complete-intersection", "contents": [ "Let $A$ be a ring. Let $n \\geq 1$ and", "$f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$.", "Set $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$.", "The K\\\"ahler different of $B$ over $A$ is the ideal", "of $B$ generated by $\\det(\\partial f_i/\\partial x_j)$." ], "refs": [], "proofs": [ { "contents": [ "This is true because $\\Omega_{B/A}$ has a presentation", "$$", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} B f_i", "\\xrightarrow{\\text{d}}", "\\bigoplus\\nolimits_{j = 1, \\ldots, n} B \\text{d}x_j", "\\rightarrow \\Omega_{B/A} \\rightarrow 0", "$$", "by Algebra, Lemma \\ref{algebra-lemma-differential-seq}." ], "refs": [ "algebra-lemma-differential-seq" ], "ref_ids": [ 1135 ] } ], "ref_ids": [] }, { "id": 14972, "type": "theorem", "label": "discriminant-lemma-dedekind-different-ideal", "categories": [ "discriminant" ], "title": "discriminant-lemma-dedekind-different-ideal", "contents": [ "Assume the Dedekind different of $A \\to B$ is defined. Consider the statements", "\\begin{enumerate}", "\\item $A \\to B$ is flat,", "\\item $A$ is a normal ring,", "\\item $\\text{Trace}_{L/K}(B) \\subset A$,", "\\item $1 \\in \\mathcal{L}_{B/A}$, and", "\\item the Dedekind different $\\mathfrak{D}_{B/A}$ is an ideal of $B$.", "\\end{enumerate}", "Then we have (1) $\\Rightarrow$ (3), (2) $\\Rightarrow$ (3),", "(3) $\\Leftrightarrow$ (4), and (4) $\\Rightarrow$ (5)." ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (3) and (4) and the", "implication (4) $\\Rightarrow$ (5) are immediate.", "\\medskip\\noindent", "If $A \\to B$ is flat, then we see that $\\text{Trace}_{B/A} : B \\to A$ is", "defined and that $\\text{Trace}_{L/K}$ is the base change. Hence (3) holds.", "\\medskip\\noindent", "If $A$ is normal, then $A$ is a finite product of normal domains,", "hence we reduce to the case of a normal domain. Then $K$ is", "the fraction field of $A$ and $L = \\prod L_i$ is a finite product of", "finite separable field extensions of $K$. Then", "$\\text{Trace}_{L/K}(b) = \\sum \\text{Trace}_{L_i/K}(b_i)$", "where $b_i \\in L_i$ is the image of $b$.", "Since $b$ is integral over $A$ as $B$ is finite over $A$,", "these traces are in $A$. This is true because the", "minimal polynomial of $b_i$ over $K$ has coefficients in $A$", "(Algebra, Lemma \\ref{algebra-lemma-minimal-polynomial-normal-domain})", "and because $\\text{Trace}_{L_i/K}(b_i)$ is an", "integer multiple of one of these coefficients", "(Fields, Lemma \\ref{fields-lemma-trace-and-norm-from-minimal-polynomial})." ], "refs": [ "algebra-lemma-minimal-polynomial-normal-domain", "fields-lemma-trace-and-norm-from-minimal-polynomial" ], "ref_ids": [ 521, 4500 ] } ], "ref_ids": [] }, { "id": 14973, "type": "theorem", "label": "discriminant-lemma-dedekind-complementary-module", "categories": [ "discriminant" ], "title": "discriminant-lemma-dedekind-complementary-module", "contents": [ "If the Dedekind different of $A \\to B$ is defined, then", "there is a canonical isomorphism", "$\\mathcal{L}_{B/A} \\to \\omega_{B/A}$." ], "refs": [], "proofs": [ { "contents": [ "Recall that $\\omega_{B/A} = \\Hom_A(B, A)$ as $A \\to B$ is finite.", "We send $x \\in \\mathcal{L}_{B/A}$ to the map", "$b \\mapsto \\text{Trace}_{L/K}(bx)$.", "Conversely, given an $A$-linear map $\\varphi : B \\to A$", "we obtain a $K$-linear map $\\varphi_K : L \\to K$. Since $K \\to L$ is finite", "\\'etale, we see that the trace pairing is nondegenerate", "(Lemma \\ref{lemma-discriminant}) and hence there exists a $x \\in L$ such that", "$\\varphi_K(y) = \\text{Trace}_{L/K}(xy)$ for all $y \\in L$.", "Then $x \\in \\mathcal{L}_{B/A}$ maps to $\\varphi$ in $\\omega_{B/A}$." ], "refs": [ "discriminant-lemma-discriminant" ], "ref_ids": [ 14955 ] } ], "ref_ids": [] }, { "id": 14974, "type": "theorem", "label": "discriminant-lemma-flat-dedekind-complementary-module-trace", "categories": [ "discriminant" ], "title": "discriminant-lemma-flat-dedekind-complementary-module-trace", "contents": [ "If the Dedekind different of $A \\to B$ is defined and $A \\to B$ is flat, then", "\\begin{enumerate}", "\\item the canonical isomorphism $\\mathcal{L}_{B/A} \\to \\omega_{B/A}$", "sends $1 \\in \\mathcal{L}_{B/A}$ to the trace element", "$\\tau_{B/A} \\in \\omega_{B/A}$, and", "\\item the Dedekind different is", "$\\mathfrak{D}_{B/A} = \\{b \\in B \\mid b\\omega_{B/A} \\subset B\\tau_{B/A}\\}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The first assertion", "follows from the proof of Lemma \\ref{lemma-dedekind-different-ideal}", "and Lemma \\ref{lemma-finite-flat-trace}.", "The second assertion is immediate from the first and the", "definitions." ], "refs": [ "discriminant-lemma-dedekind-different-ideal", "discriminant-lemma-finite-flat-trace" ], "ref_ids": [ 14972, 14957 ] } ], "ref_ids": [] }, { "id": 14975, "type": "theorem", "label": "discriminant-lemma-flat-agree-dedekind", "categories": [ "discriminant" ], "title": "discriminant-lemma-flat-agree-dedekind", "contents": [ "Let $f : Y \\to X$ be a flat quasi-finite morphism of Noetherian schemes.", "Let $V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$", "be affine open subschemes with $f(V) \\subset U$.", "If the Dedekind different of $A \\to B$ is defined, then", "$$", "\\mathfrak{D}_f|_V = \\widetilde{\\mathfrak{D}_{B/A}}", "$$", "as coherent ideal sheaves on $V$." ], "refs": [], "proofs": [ { "contents": [ "This is clear from Lemmas \\ref{lemma-dedekind-different-ideal} and", "\\ref{lemma-flat-dedekind-complementary-module-trace}." ], "refs": [ "discriminant-lemma-dedekind-different-ideal", "discriminant-lemma-flat-dedekind-complementary-module-trace" ], "ref_ids": [ 14972, 14974 ] } ], "ref_ids": [] }, { "id": 14976, "type": "theorem", "label": "discriminant-lemma-flat-gorenstein-agree-noether", "categories": [ "discriminant" ], "title": "discriminant-lemma-flat-gorenstein-agree-noether", "contents": [ "Let $f : Y \\to X$ be a flat quasi-finite morphism of Noetherian schemes.", "Let $V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$", "be affine open subschemes with $f(V) \\subset U$.", "If $\\omega_{Y/X}|_V$ is invertible, i.e., if $\\omega_{B/A}$", "is an invertible $B$-module, then", "$$", "\\mathfrak{D}_f|_V = \\widetilde{\\mathfrak{D}}", "$$", "as coherent ideal sheaves on $V$ where", "$\\mathfrak{D} \\subset B$ is the Noether different of $B$ over $A$." ], "refs": [], "proofs": [ { "contents": [ "Consider the map", "$$", "\\SheafHom_{\\mathcal{O}_Y}(\\omega_{Y/X}, \\mathcal{O}_Y)", "\\longrightarrow", "\\mathcal{O}_Y,\\quad", "\\varphi \\longmapsto \\varphi(\\tau_{Y/X})", "$$", "The image of this map corresponds to the Noether different", "on affine opens, see Lemma \\ref{lemma-noether-different-flat-quasi-finite}.", "Hence the result follows from the elementary fact that given", "an invertible module $\\omega$ and a global section $\\tau$", "the image of", "$\\tau : \\SheafHom(\\omega, \\mathcal{O}) = \\omega^{\\otimes -1} \\to \\mathcal{O}$", "is the same as the annihilator of $\\Coker(\\tau : \\mathcal{O} \\to \\omega)$." ], "refs": [ "discriminant-lemma-noether-different-flat-quasi-finite" ], "ref_ids": [ 14967 ] } ], "ref_ids": [] }, { "id": 14977, "type": "theorem", "label": "discriminant-lemma-base-change-different", "categories": [ "discriminant" ], "title": "discriminant-lemma-base-change-different", "contents": [ "Consider a cartesian diagram of Noetherian schemes", "$$", "\\xymatrix{", "Y' \\ar[d]_{f'} \\ar[r] & Y \\ar[d]^f \\\\", "X' \\ar[r]^g & X", "}", "$$", "with $f$ flat and quasi-finite. Let $R \\subset Y$, resp.\\ $R' \\subset Y'$", "be the closed subscheme cut out by the different", "$\\mathfrak{D}_f$, resp.\\ $\\mathfrak{D}_{f'}$.", "Then $Y' \\to Y$ induces a bijective closed immersion $R' \\to R \\times_Y Y'$.", "If $g$ is flat or if $\\omega_{Y/X}$ is invertible, then", "$R' = R \\times_Y Y'$." ], "refs": [], "proofs": [ { "contents": [ "There is an immediate reduction to the case where $X$, $X'$, $Y$, $Y'$", "are affine. In other words, we have a cocartesian diagram of Noetherian", "rings", "$$", "\\xymatrix{", "B' & B \\ar[l] \\\\", "A' \\ar[u] & A \\ar[l] \\ar[u]", "}", "$$", "with $A \\to B$ flat and quasi-finite. The base change map", "$\\omega_{B/A} \\otimes_B B' \\to \\omega_{B'/A'}$ is an isomorphism", "(Lemma \\ref{lemma-dualizing-base-change-of-flat}) and maps", "the trace element $\\tau_{B/A}$ to the trace element $\\tau_{B'/A'}$", "(Lemma \\ref{lemma-trace-base-change}).", "Hence the finite $B$-module $Q = \\Coker(\\tau_{B/A} : B \\to \\omega_{B/A})$", "satisfies $Q \\otimes_B B' = \\Coker(\\tau_{B'/A'} : B' \\to \\omega_{B'/A'})$.", "Thus $\\mathfrak{D}_{B/A}B' \\subset \\mathfrak{D}_{B'/A'}$ which means", "we obtain the closed immersion $R' \\to R \\times_Y Y'$.", "Since $R = \\text{Supp}(Q)$ and $R' = \\text{Supp}(Q \\otimes_B B')$", "(Algebra, Lemma \\ref{algebra-lemma-support-closed})", "we see that $R' \\to R \\times_Y Y'$ is bijective by", "Algebra, Lemma \\ref{algebra-lemma-support-base-change}.", "The equality $\\mathfrak{D}_{B/A}B' = \\mathfrak{D}_{B'/A'}$ holds", "if $B \\to B'$ is flat, e.g., if $A \\to A'$ is flat, see", "Algebra, Lemma \\ref{algebra-lemma-annihilator-flat-base-change}.", "Finally, if $\\omega_{B/A}$ is invertible, then we can localize", "and assume $\\omega_{B/A} = B \\lambda$. Writing $\\tau_{B/A} = b\\lambda$", "we see that $Q = B/bB$ and $\\mathfrak{D}_{B/A} = bB$.", "The same reasoning over $B'$", "gives $\\mathfrak{D}_{B'/A'} = bB'$ and the lemma is proved." ], "refs": [ "discriminant-lemma-dualizing-base-change-of-flat", "discriminant-lemma-trace-base-change", "algebra-lemma-support-closed", "algebra-lemma-support-base-change", "algebra-lemma-annihilator-flat-base-change" ], "ref_ids": [ 14953, 14958, 543, 544, 542 ] } ], "ref_ids": [] }, { "id": 14978, "type": "theorem", "label": "discriminant-lemma-norm-different-in-discriminant", "categories": [ "discriminant" ], "title": "discriminant-lemma-norm-different-in-discriminant", "contents": [ "Let $f : Y \\to X$ be a finite flat morphism of Noetherian schemes.", "Then $\\text{Norm}_f : f_*\\mathcal{O}_Y \\to \\mathcal{O}_X$ maps", "$f_*\\mathfrak{D}_f$ into the ideal sheaf of the discriminant $D_f$." ], "refs": [], "proofs": [ { "contents": [ "The norm map is constructed in", "Divisors, Lemma \\ref{divisors-lemma-finite-locally-free-has-norm}", "and the discriminant of $f$ in Section \\ref{section-discriminant}.", "The question is affine local, hence we may assume $X = \\Spec(A)$,", "$Y = \\Spec(B)$ and $f$ given by a finite locally free ring map $A \\to B$.", "Localizing further we may assume $B$ is finite free as an $A$-module.", "Choose a basis $b_1, \\ldots, b_n \\in B$ for $B$ as an $A$-module.", "Denote $b_1^\\vee, \\ldots, b_n^\\vee$ the dual basis of", "$\\omega_{B/A} = \\Hom_A(B, A)$ as an $A$-module.", "Since the norm of $b$ is the determinant of $b : B \\to B$ as an", "$A$-linear map, we see that", "$\\text{Norm}_{B/A}(b) = \\det(b_i^\\vee(bb_j))$.", "The discriminant is the principal closed subscheme of $\\Spec(A)$", "defined by $\\det(\\text{Trace}_{B/A}(b_ib_j))$.", "If $b \\in \\mathfrak{D}_{B/A}$ then", "there exist $c_i \\in B$ such that", "$b \\cdot b_i^\\vee = c_i \\cdot \\text{Trace}_{B/A}$ where", "we use a dot to indicate the $B$-module structure on $\\omega_{B/A}$.", "Write $c_i = \\sum a_{il} b_l$.", "We have", "\\begin{align*}", "\\text{Norm}_{B/A}(b)", "& =", "\\det(b_i^\\vee(bb_j)) \\\\", "& =", "\\det( (b \\cdot b_i^\\vee)(b_j)) \\\\", "& =", "\\det((c_i \\cdot \\text{Trace}_{B/A})(b_j)) \\\\", "& =", "\\det(\\text{Trace}_{B/A}(c_ib_j)) \\\\", "& =", "\\det(a_{il}) \\det(\\text{Trace}_{B/A}(b_l b_j))", "\\end{align*}", "which proves the lemma." ], "refs": [ "divisors-lemma-finite-locally-free-has-norm" ], "ref_ids": [ 7968 ] } ], "ref_ids": [] }, { "id": 14979, "type": "theorem", "label": "discriminant-lemma-different-ramification", "categories": [ "discriminant" ], "title": "discriminant-lemma-different-ramification", "contents": [ "Let $f : Y \\to X$ be a flat quasi-finite morphism of Noetherian schemes.", "The closed subscheme $R \\subset Y$ defined by the different $\\mathfrak{D}_f$", "is exactly the set of points where $f$ is not \\'etale", "(equivalently not unramified)." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is of finite presentation and flat, we see that it is \\'etale", "at a point if and only if it is unramified at that point. Moreover, the", "formation of the locus of ramified points commutes with base change.", "See Morphisms, Section \\ref{morphisms-section-etale} and especially", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-etale}.", "By Lemma \\ref{lemma-base-change-different} the formation of $R$ commutes", "set theoretically with base change. Hence it suffices to prove the", "lemma when $X$ is the spectrum of a field. On the other hand, the", "construction of $(\\omega_{Y/X}, \\tau_{Y/X})$ is local on $Y$.", "Since $Y$ is a finite discrete space (being quasi-finite", "over a field), we may assume $Y$ has a unique point.", "\\medskip\\noindent", "Say $X = \\Spec(k)$ and $Y = \\Spec(B)$ where $k$ is a field and $B$ is", "a finite local $k$-algebra. If $Y \\to X$ is \\'etale, then", "$B$ is a finite separable extension of $k$, and the trace", "element $\\text{Trace}_{B/k}$ is a basis element of $\\omega_{B/k}$", "by Fields, Lemma \\ref{fields-lemma-separable-trace-pairing}.", "Thus $\\mathfrak{D}_{B/k} = B$ in this case.", "Conversely, if $\\mathfrak{D}_{B/k} = B$, then we see from", "Lemma \\ref{lemma-norm-different-in-discriminant}", "and the fact that the norm of $1$ equals $1$ that the", "discriminant is empty. Hence", "$Y \\to X$ is \\'etale by Lemma \\ref{lemma-discriminant}." ], "refs": [ "morphisms-lemma-set-points-where-fibres-etale", "discriminant-lemma-base-change-different", "fields-lemma-separable-trace-pairing", "discriminant-lemma-norm-different-in-discriminant", "discriminant-lemma-discriminant" ], "ref_ids": [ 5374, 14977, 4503, 14978, 14955 ] } ], "ref_ids": [] }, { "id": 14980, "type": "theorem", "label": "discriminant-lemma-norm-different-is-discriminant", "categories": [ "discriminant" ], "title": "discriminant-lemma-norm-different-is-discriminant", "contents": [ "Let $f : Y \\to X$ be a flat quasi-finite morphism of Noetherian schemes.", "Let $R \\subset Y$ be the closed subscheme defined by $\\mathfrak{D}_f$.", "\\begin{enumerate}", "\\item If $\\omega_{Y/X}$ is invertible,", "then $R$ is a locally principal closed subscheme of $Y$.", "\\item If $\\omega_{Y/X}$ is invertible and $f$ is finite, then", "the norm of $R$ is the discriminant $D_f$ of $f$.", "\\item If $\\omega_{Y/X}$ is invertible and $f$", "is \\'etale at the associated points of $Y$, then $R$", "is an effective Cartier divisor and there is an", "isomorphism $\\mathcal{O}_Y(R) = \\omega_{Y/X}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). We may work locally on $Y$, hence we may assume", "$\\omega_{Y/X}$ is free of rank $1$. Say $\\omega_{Y/X} = \\mathcal{O}_Y\\lambda$.", "Then we can write $\\tau_{Y/X} = h \\lambda$ and then we see that", "$R$ is defined by $h$, i.e., $R$ is locally principal.", "\\medskip\\noindent", "Proof of (2). We may assume $Y \\to X$ is given by a finite free ring", "map $A \\to B$ and that $\\omega_{B/A}$ is free of rank $1$ as $B$-module.", "Choose a $B$-basis element $\\lambda$ for $\\omega_{B/A}$ and write", "$\\text{Trace}_{B/A} = b \\cdot \\lambda$ for some $b \\in B$.", "Then $\\mathfrak{D}_{B/A} = (b)$ and $D_f$ is cut out by", "$\\det(\\text{Trace}_{B/A}(b_ib_j))$ where $b_1, \\ldots, b_n$ is a", "basis of $B$ as an $A$-module. Let $b_1^\\vee, \\ldots, b_n^\\vee$", "be the dual basis.", "Writing $b_i^\\vee = c_i \\cdot \\lambda$ we see that", "$c_1, \\ldots, c_n$ is a basis of $B$ as well.", "Hence with $c_i = \\sum a_{il}b_l$ we see that $\\det(a_{il})$", "is a unit in $A$. Clearly,", "$b \\cdot b_i^\\vee = c_i \\cdot \\text{Trace}_{B/A}$", "hence we conclude from the computation in the proof of", "Lemma \\ref{lemma-norm-different-in-discriminant}", "that $\\text{Norm}_{B/A}(b)$ is a unit times", "$\\det(\\text{Trace}_{B/A}(b_ib_j))$.", "\\medskip\\noindent", "Proof of (3). In the notation above we see from", "Lemma \\ref{lemma-different-ramification} and the assumption", "that $h$ does not vanish in", "the associated points of $Y$, which implies that $h$ is a nonzerodivisor.", "The canonical isomorphism sends $1$ to $\\tau_{Y/X}$, see", "Divisors, Lemma \\ref{divisors-lemma-characterize-OD}." ], "refs": [ "discriminant-lemma-norm-different-in-discriminant", "discriminant-lemma-different-ramification", "divisors-lemma-characterize-OD" ], "ref_ids": [ 14978, 14979, 7944 ] } ], "ref_ids": [] }, { "id": 14981, "type": "theorem", "label": "discriminant-lemma-syntomic-quasi-finite", "categories": [ "discriminant" ], "title": "discriminant-lemma-syntomic-quasi-finite", "contents": [ "Let $f : Y \\to X$ be a morphism of schemes. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is locally quasi-finite and syntomic,", "\\item $f$ is locally quasi-finite, flat, and a local complete intersection", "morphism,", "\\item $f$ is locally quasi-finite, flat, locally of finite presentation,", "and the fibres of $f$ are local complete intersections,", "\\item $f$ is locally quasi-finite and for every $y \\in Y$ there are", "affine opens $y \\in V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$", "with $f(V) \\subset U$ an integer $n$ and", "$h, f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$ such that", "$B = A[x_1, \\ldots, x_n, 1/h]/(f_1, \\ldots, f_n)$,", "\\item for every $y \\in Y$ there are affine opens", "$y \\in V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$", "with $f(V) \\subset U$ such that $A \\to B$ is a relative global complete", "intersection of the form $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$,", "\\item $f$ is locally quasi-finite, flat, locally of finite presentation,", "and $\\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and", "\\item $f$ is flat, locally of finite presentation,", "$\\NL_{X/Y}$ is perfect of rank $0$ with tor-amplitude in $[-1, 0]$,", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-flat-lci}.", "The equivalence of (1) and (3) is ", "Morphisms, Lemma \\ref{morphisms-lemma-syntomic-flat-fibres}.", "\\medskip\\noindent", "If $A \\to B$ is as in (4), then", "$B = A[x, x_1, \\ldots, x_n]/(xh - 1, f_1, \\ldots, f_n]$", "is a relative global complete intersection by see Algebra, Definition", "\\ref{algebra-definition-relative-global-complete-intersection}.", "Thus (4) implies (5).", "It is clear that (5) implies (4).", "\\medskip\\noindent", "Condition (5) implies (1): by", "Algebra, Lemma \\ref{algebra-lemma-relative-global-complete-intersection}", "a relative global complete intersection is syntomic and", "the definition of a relative global complete intersection", "guarantees that a relative global complete intersection on", "$n$ variables with $n$ equations is quasi-finite, see", "Algebra, Definition", "\\ref{algebra-definition-relative-global-complete-intersection} and", "Lemma \\ref{algebra-lemma-isolated-point-fibre}.", "\\medskip\\noindent", "Either Algebra, Lemma \\ref{algebra-lemma-syntomic} or", "Morphisms, Lemma \\ref{morphisms-lemma-syntomic-locally-standard-syntomic}", "shows that (1) implies (5).", "\\medskip\\noindent", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-flat-fp-NL-lci} shows that", "(6) is equivalent to (1). If the equivalent conditions (1) -- (6) hold,", "then we see that affine locally $Y \\to X$ is given by a relative global", "complete intersection $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$", "with the same number of variables as the number of", "equations. Using this presentation we see that", "$$", "\\NL_{B/A} =\\left(", "(f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} B \\text{d} x_i\\right)", "$$", "By Algebra, Lemma", "\\ref{algebra-lemma-relative-global-complete-intersection-conormal}", "the module $(f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2$", "is free with generators the congruence classes of the elements", "$f_1, \\ldots, f_n$. Thus $\\NL_{B/A}$ has rank $0$ and so does $\\NL_{Y/X}$.", "In this way we see that (1) -- (6) imply (7).", "\\medskip\\noindent", "Finally, assume (7). By", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-flat-fp-NL-lci}", "we see that $f$ is syntomic. Thus on suitable affine opens", "$f$ is given by a relative global complete intersection", "$A \\to B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$, see", "Morphisms, Lemma \\ref{morphisms-lemma-syntomic-locally-standard-syntomic}.", "Exactly as above we see that $\\NL_{B/A}$ is a perfect complex", "of rank $n - m$. Thus $n = m$ and we see that (5) holds.", "This finishes the proof." ], "refs": [ "more-morphisms-lemma-flat-lci", "morphisms-lemma-syntomic-flat-fibres", "algebra-definition-relative-global-complete-intersection", "algebra-lemma-relative-global-complete-intersection", "algebra-definition-relative-global-complete-intersection", "algebra-lemma-isolated-point-fibre", "algebra-lemma-syntomic", "morphisms-lemma-syntomic-locally-standard-syntomic", "more-morphisms-lemma-flat-fp-NL-lci", "algebra-lemma-relative-global-complete-intersection-conormal", "more-morphisms-lemma-flat-fp-NL-lci", "morphisms-lemma-syntomic-locally-standard-syntomic" ], "ref_ids": [ 14006, 5298, 1533, 1184, 1533, 1049, 1185, 5297, 14015, 1183, 14015, 5297 ] } ], "ref_ids": [] }, { "id": 14982, "type": "theorem", "label": "discriminant-lemma-characterize-invertible", "categories": [ "discriminant" ], "title": "discriminant-lemma-characterize-invertible", "contents": [ "Invertibility of the relative dualizing module.", "\\begin{enumerate}", "\\item If $A \\to B$ is a quasi-finite flat homomorphism of Noetherian rings,", "then $\\omega_{B/A}$ is an invertible $B$-module if and only if", "$\\omega_{B \\otimes_A \\kappa(\\mathfrak p)/\\kappa(\\mathfrak p)}$", "is an invertible $B \\otimes_A \\kappa(\\mathfrak p)$-module", "for all primes $\\mathfrak p \\subset A$.", "\\item If $Y \\to X$ is a quasi-finite flat morphism of", "Noetherian schemes, then $\\omega_{Y/X}$ is invertible", "if and only if $\\omega_{Y_x/x}$ is invertible for all $x \\in X$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). As $A \\to B$ is flat, the module", "$\\omega_{B/A}$ is $A$-flat, see Lemma \\ref{lemma-dualizing-base-flat-flat}.", "Thus $\\omega_{B/A}$ is an invertible $B$-module if and only if", "$\\omega_{B/A} \\otimes_A \\kappa(\\mathfrak p)$", "is an invertible $B \\otimes_A \\kappa(\\mathfrak p)$-module for", "every prime $\\mathfrak p \\subset A$, see More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-flat-and-free-at-point-fibre}.", "Still using that $A \\to B$ is flat, we have that", "formation of $\\omega_{B/A}$ commutes with base change, see", "Lemma \\ref{lemma-dualizing-base-change-of-flat}.", "Thus we see that invertibility of the relative dualizing module,", "in the presence of flatness, is equivalent to invertibility", "of the relative dualizing module for the maps", "$\\kappa(\\mathfrak p) \\to B \\otimes_A \\kappa(\\mathfrak p)$.", "\\medskip\\noindent", "Part (2) follows from (1) and the fact that affine locally", "the dualizing modules are given by their algebraic counterparts, see", "Remark \\ref{remark-relative-dualizing-for-quasi-finite}." ], "refs": [ "discriminant-lemma-dualizing-base-flat-flat", "more-morphisms-lemma-flat-and-free-at-point-fibre", "discriminant-lemma-dualizing-base-change-of-flat", "discriminant-remark-relative-dualizing-for-quasi-finite" ], "ref_ids": [ 14952, 13771, 14953, 15006 ] } ], "ref_ids": [] }, { "id": 14983, "type": "theorem", "label": "discriminant-lemma-dim-zero-global-complete-intersection-over-field", "categories": [ "discriminant" ], "title": "discriminant-lemma-dim-zero-global-complete-intersection-over-field", "contents": [ "Let $k$ be a field. Let $B = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$", "be a global complete intersection over $k$ of dimension $0$.", "Then $\\omega_{B/k}$ is invertible." ], "refs": [], "proofs": [ { "contents": [ "By Noether normalization, see", "Algebra, Lemma \\ref{algebra-lemma-Noether-normalization}", "we see that there exists a finite injection $k \\to B$, i.e.,", "$\\dim_k(B) < \\infty$. Hence $\\omega_{B/k} = \\Hom_k(B, k)$", "as a $B$-module.", "By Dualizing Complexes, Lemma \\ref{dualizing-lemma-dualizing-finite}", "we see that $R\\Hom(B, k)$ is a dualizing complex for $B$", "and by Dualizing Complexes, Lemma \\ref{dualizing-lemma-RHom-ext}", "we see that $R\\Hom(B, k)$ is equal to $\\omega_{B/k}$", "placed in degree $0$. Thus it suffices to show that", "$B$ is Gorenstein", "(Dualizing Complexes, Lemma \\ref{dualizing-lemma-gorenstein}).", "This is true by Dualizing Complexes, Lemma", "\\ref{dualizing-lemma-gorenstein-lci}." ], "refs": [ "algebra-lemma-Noether-normalization", "dualizing-lemma-dualizing-finite", "dualizing-lemma-RHom-ext", "dualizing-lemma-gorenstein", "dualizing-lemma-gorenstein-lci" ], "ref_ids": [ 1001, 2853, 2838, 2881, 2884 ] } ], "ref_ids": [] }, { "id": 14984, "type": "theorem", "label": "discriminant-lemma-dualizing-syntomic-quasi-finite", "categories": [ "discriminant" ], "title": "discriminant-lemma-dualizing-syntomic-quasi-finite", "contents": [ "Let $f : Y \\to X$ be a morphism of locally Noetherian schemes. If $f$", "satisfies the equivalent conditions of Lemma \\ref{lemma-syntomic-quasi-finite}", "then $\\omega_{Y/X}$ is an invertible $\\mathcal{O}_Y$-module." ], "refs": [ "discriminant-lemma-syntomic-quasi-finite" ], "proofs": [ { "contents": [ "We may assume $A \\to B$ is a relative global complete", "intersection of the form $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$", "and we have to show $\\omega_{B/A}$ is invertible.", "This follows in combining Lemmas \\ref{lemma-characterize-invertible} and", "\\ref{lemma-dim-zero-global-complete-intersection-over-field}." ], "refs": [ "discriminant-lemma-characterize-invertible", "discriminant-lemma-dim-zero-global-complete-intersection-over-field" ], "ref_ids": [ 14982, 14983 ] } ], "ref_ids": [ 14981 ] }, { "id": 14985, "type": "theorem", "label": "discriminant-lemma-universal-quasi-finite-syntomic-etale", "categories": [ "discriminant" ], "title": "discriminant-lemma-universal-quasi-finite-syntomic-etale", "contents": [ "With notation as in Example \\ref{example-universal-quasi-finite-syntomic}", "the schemes $X_{n, d}$ and $Y_{n, d}$ are regular and irreducible,", "the morphism $Y_{n, d} \\to X_{n, d}$ is locally quasi-finite and", "syntomic, and there is a dense open subscheme $V \\subset Y_{n, d}$", "such that $Y_{n, d} \\to X_{n, d}$ restricts to an \\'etale morphism", "$V \\to X_{n, d}$." ], "refs": [], "proofs": [ { "contents": [ "The scheme $X_{n, d}$ is the spectrum of the polynomial ring $A$.", "Hence $X_{n, d}$ is regular and irreducible. Since we can write", "$$", "f_i = a_{i, (0, \\ldots, 0)} +", "\\sum\\nolimits_{E \\in T, E \\not = (0, \\ldots, 0)} a_{i, E} x^E", "$$", "we see that the ring $B$ is isomorphic to the polynomial ring", "on $x_1, \\ldots, x_n$ and the elements $a_{i, E}$ with", "$E \\not = (0, \\ldots, 0)$. Hence $\\Spec(B)$ is an irreducible and", "regular scheme and so is the open $Y_{n, d}$. The morphism", "$Y_{n, d} \\to X_{n, d}$ is locally quasi-finite and syntomic by", "Lemma \\ref{lemma-syntomic-quasi-finite}. To find $V$ it suffices", "to find a single point where $Y_{n, d} \\to X_{n, d}$ is \\'etale", "(the locus of points where a morphism is \\'etale is open by", "definition). Thus it suffices to find a point of $X_{n, d}$", "where the fibre of $Y_{n, d} \\to X_{n, d}$ is nonempty and \\'etale, see", "Morphisms, Lemma \\ref{morphisms-lemma-etale-at-point}. We choose", "the point corresponding to the ring map $\\chi : A \\to \\mathbf{Q}$", "sending $f_i$ to $1 + x_i^d$. Then", "$$", "B \\otimes_{A, \\chi} \\mathbf{Q} =", "\\mathbf{Q}[x_1, \\ldots, x_n]/(x_1^d - 1, \\ldots, x_n^d - 1)", "$$", "which is a nonzero \\'etale algebra over $\\mathbf{Q}$." ], "refs": [ "discriminant-lemma-syntomic-quasi-finite", "morphisms-lemma-etale-at-point" ], "ref_ids": [ 14981, 5372 ] } ], "ref_ids": [] }, { "id": 14986, "type": "theorem", "label": "discriminant-lemma-locally-comes-from-universal", "categories": [ "discriminant" ], "title": "discriminant-lemma-locally-comes-from-universal", "contents": [ "Let $f : Y \\to X$ be a morphism of schemes. If $f$ satisfies the equivalent", "conditions of Lemma \\ref{lemma-syntomic-quasi-finite} then for every", "$y \\in Y$ there exist $n, d$ and a commutative diagram", "$$", "\\xymatrix{", "Y \\ar[d] &", "V \\ar[d] \\ar[l] \\ar[r] &", "Y_{n, d} \\ar[d] \\\\", "X & U \\ar[l] \\ar[r] &", "X_{n, d}", "}", "$$", "where $U \\subset X$ and $V \\subset Y$ are open, where $Y_{n, d} \\to X_{n, d}$", "is as in Example \\ref{example-universal-quasi-finite-syntomic}, and", "where the square on the right hand side is cartesian." ], "refs": [ "discriminant-lemma-syntomic-quasi-finite" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-syntomic-quasi-finite}", "we can choose $U$ and $V$ affine so that", "$U = \\Spec(R)$ and $V = \\Spec(S)$ with", "$S = R[y_1, \\ldots, y_n]/(g_1, \\ldots, g_n)$.", "With notation as in Example \\ref{example-universal-quasi-finite-syntomic}", "if we pick $d$ large enough, then we can write each $g_i$ as", "$g_i = \\sum_{E \\in T} g_{i, E}y^E$ with $g_{i, E} \\in R$.", "Then the map $A \\to R$ sending $a_{i, E}$ to $g_{i, E}$", "and the map $B \\to S$ sending $x_i \\to y_i$ give a cocartesian", "diagram of rings", "$$", "\\xymatrix{", "S & B \\ar[l] \\\\", "R \\ar[u] & A \\ar[l] \\ar[u]", "}", "$$", "which proves the lemma." ], "refs": [ "discriminant-lemma-syntomic-quasi-finite" ], "ref_ids": [ 14981 ] } ], "ref_ids": [ 14981 ] }, { "id": 14987, "type": "theorem", "label": "discriminant-lemma-syntomic-finite", "categories": [ "discriminant" ], "title": "discriminant-lemma-syntomic-finite", "contents": [ "Let $f : Y \\to X$ be a morphism of schemes. The following are equivalent", "\\begin{enumerate}", "\\item $f$ is finite and syntomic,", "\\item $f$ is finite, flat, and a local complete intersection morphism,", "\\item $f$ is finite, flat, locally of finite presentation,", "and the fibres of $f$ are local complete intersections,", "\\item $f$ is finite and for every $x \\in X$ there is an", "affine open $x \\in U = \\Spec(A) \\subset X$ an integer $n$", "and $f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$ such that", "$f^{-1}(U)$ is isomorphic to the spectrum of", "$A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$,", "\\item $f$ is finite, flat, locally of finite presentation,", "and $\\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and", "\\item $f$ is finite, flat, locally of finite presentation, and", "$\\NL_{X/Y}$ is perfect of rank $0$ with tor-amplitude in $[-1, 0]$,", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1), (2), (3), (5), and (6)", "and the implication (4) $\\Rightarrow$ (1) follow immediately", "from Lemma \\ref{lemma-syntomic-quasi-finite}. Assume the equivalent conditions", "(1), (2), (3), (5), (6) hold.", "Choose a point $x \\in X$ and an affine open $U = \\Spec(A)$", "of $x$ in $X$ and say $x$ corresponds to the prime ideal", "$\\mathfrak p \\subset A$. Write $f^{-1}(U) = \\Spec(B)$.", "Write $B = A[x_1, \\ldots, x_n]/I$. Since $\\NL_{B/A}$", "is perfect of tor-amplitude in $[-1, 0]$ by (6)", "we see that $I/I^2$ is a finite locally free $B$-module", "of rank $n$. Since $B_\\mathfrak p$ is semi-local we see that", "$(I/I^2)_\\mathfrak p$ is free of rank $n$, see", "Algebra, Lemma \\ref{algebra-lemma-locally-free-semi-local-free}.", "Thus after replacing $A$ by a principal localization at", "an element not in $\\mathfrak p$ we may assume $I/I^2$", "is a free $B$-module of rank $n$.", "Thus by Algebra, Lemma \\ref{algebra-lemma-huber}", "we can find a presentation of $B$ over $A$", "with the same number of variables as equations. In other words,", "we may assume $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$.", "This proves (4)." ], "refs": [ "discriminant-lemma-syntomic-quasi-finite", "algebra-lemma-locally-free-semi-local-free", "algebra-lemma-huber" ], "ref_ids": [ 14981, 799, 1178 ] } ], "ref_ids": [] }, { "id": 14988, "type": "theorem", "label": "discriminant-lemma-universal-finite-syntomic", "categories": [ "discriminant" ], "title": "discriminant-lemma-universal-finite-syntomic", "contents": [ "With notation as in Example \\ref{example-universal-finite-syntomic}", "there is an open subscheme $U_d \\subset X_d$ with the following property:", "a morphism of schemes $X \\to X_d$ factors through $U_d$ if and only", "if $Y_d \\times_{X_d} X \\to X$ is syntomic." ], "refs": [], "proofs": [ { "contents": [ "Recall that being syntomic is the same thing as being flat and", "a local complete intersection morphism, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-flat-lci}.", "The set $W_d \\subset Y_d$ of points where $\\pi_d$ is Koszul", "is open in $Y_d$ and its formation commutes with arbitrary base change, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-base-change-lci-fibres}.", "Since $\\pi_d$ is finite and hence closed, we see that", "$Z = \\pi_d(Y_d \\setminus W_d)$ is closed. Since clearly $U_d = X_d \\setminus Z$", "and since its formation commutes with base change we find that the lemma", "is true." ], "refs": [ "more-morphisms-lemma-flat-lci", "more-morphisms-lemma-base-change-lci-fibres" ], "ref_ids": [ 14006, 14019 ] } ], "ref_ids": [] }, { "id": 14989, "type": "theorem", "label": "discriminant-lemma-universal-finite-syntomic-smooth", "categories": [ "discriminant" ], "title": "discriminant-lemma-universal-finite-syntomic-smooth", "contents": [ "With notation as in Example \\ref{example-universal-finite-syntomic}", "and $U_d$ as in Lemma \\ref{lemma-universal-finite-syntomic}", "then $U_d$ is smooth over $\\Spec(\\mathbf{Z})$." ], "refs": [ "discriminant-lemma-universal-finite-syntomic" ], "proofs": [ { "contents": [ "Let us use More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-lifting-along-artinian-at-point}", "to show that $U_d \\to \\Spec(\\mathbf{Z})$ is smooth.", "Namely, suppose that $\\Spec(A) \\to U_d$ is a morphism", "and $A' \\to A$ is a small extension. Then $B = A \\otimes_{A_d} B_d$", "is a finite free $A$-algebra which is syntomic over $A$", "(by construction of $U_d$). By", "Smoothing Ring Maps, Proposition \\ref{smoothing-proposition-lift-smooth}", "there exists a syntomic ring map $A' \\to B'$ such that", "$B \\cong B' \\otimes_{A'} A$. Set $e'_1 = 1 \\in B'$. For $1 < i \\leq d$", "choose lifts $e'_i \\in B'$ of the elements", "$1 \\otimes e_i \\in A \\otimes_{A_d} B_d = B$. Then $e'_1, \\ldots, e'_d$", "is a basis for $B'$ over $A'$ (for example see Algebra, Lemma", "\\ref{algebra-lemma-local-artinian-basis-when-flat}).", "Thus we can write $e'_i e'_j = \\sum \\alpha_{ij}^l e'_l$ for unique", "elements $\\alpha_{ij}^l \\in A'$ which satisfy the relations", "$\\sum_l \\alpha_{ij}^l \\alpha_{lk}^m = \\sum_l \\alpha_{il}^m \\alpha _{jk}^l$", "and $\\alpha_{ij}^k = \\alpha_{ji}^k$ and $\\alpha_{i1}^j - \\delta_{ij}$", "in $A'$. This determines a morphism $\\Spec(A') \\to X_d$ by", "sending $a_{ij}^l \\in A_d$ to $\\alpha_{ij}^l \\in A'$. This morphism", "agrees with the given morphism $\\Spec(A) \\to U_d$. Since $\\Spec(A')$", "and $\\Spec(A)$ have the same underlying topological space, we see", "that we obtain the desired lift $\\Spec(A') \\to U_d$ and we", "conclude that $U_d$ is smooth over $\\mathbf{Z}$." ], "refs": [ "more-morphisms-lemma-lifting-along-artinian-at-point", "smoothing-proposition-lift-smooth", "algebra-lemma-local-artinian-basis-when-flat" ], "ref_ids": [ 13741, 5645, 900 ] } ], "ref_ids": [ 14988 ] }, { "id": 14990, "type": "theorem", "label": "discriminant-lemma-universal-finite-syntomic-etale", "categories": [ "discriminant" ], "title": "discriminant-lemma-universal-finite-syntomic-etale", "contents": [ "With notation as in Example \\ref{example-universal-finite-syntomic}", "consider the open subscheme $U'_d \\subset X_d$ over which", "$\\pi_d$ is \\'etale. Then $U'_d$ is a dense subset of the", "open $U_d$ of Lemma \\ref{lemma-universal-finite-syntomic}" ], "refs": [ "discriminant-lemma-universal-finite-syntomic" ], "proofs": [ { "contents": [ "By exactly the same reasoning as in the proof of", "Lemma \\ref{lemma-universal-finite-syntomic}, using", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-etale},", "there is a maximal open $U'_d \\subset X_d$ over which $\\pi_d$ is", "\\'etale. Moreover, since an \\'etale morphism is syntomic, we see", "that $U'_d \\subset U_d$. To finish the proof we have to show", "that $U'_d \\subset U_d$ is dense. Let $u : \\Spec(k) \\to U_d$ be a morphism", "where $k$ is a field. Let $B = k \\otimes_{A_d} B_d$ as in the", "proof of Lemma \\ref{lemma-universal-finite-syntomic-smooth}.", "We will show there is a local domain $A'$ with residue field $k$", "and a finite syntomic $A'$ algebra $B'$ with $B = k \\otimes_{A'} B'$", "whose generic fibre is \\'etale. Exactly as in the previous paragraph", "this will determine a morphism $\\Spec(A') \\to U_d$ which will map the", "generic point into $U'_d$ and the closed point to $u$, thereby", "finishing the proof.", "\\medskip\\noindent", "By Lemma \\ref{lemma-syntomic-finite} part (4) we can choose a presentation", "$B = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$.", "Let $d'$ be the maximum total degree of the polynomials $f_1, \\ldots, f_n$.", "Let $Y_{n, d'} \\to X_{n, d'}$ be as in", "Example \\ref{example-universal-quasi-finite-syntomic}.", "By construction there is a morphism $u' : \\Spec(k) \\to X_{n, d'}$", "such that", "$$", "\\Spec(B) \\cong Y_{n, d'} \\times_{X_{n, d'}, u'} \\Spec(k)", "$$", "Denote $A = \\mathcal{O}_{X_{n, d'}, u'}^h$ the henselization of the", "local ring of $X_{n, d'}$ at the image of $u'$. Then we can write", "$$", "Y_{n, d'} \\times_{X_{n, d'}} \\Spec(A) = Z \\amalg W", "$$", "with $Z \\to \\Spec(A)$ finite and $W \\to \\Spec(A)$ having empty", "closed fibre, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-henselian} part (13)", "or the discussion in More on Morphisms, Section", "\\ref{more-morphisms-section-etale-localization}.", "By Lemma \\ref{lemma-universal-quasi-finite-syntomic-etale}", "the local ring $A$ is regular (here we also use", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-regular})", "and the morphism $Z \\to \\Spec(A)$ is \\'etale over the generic point of", "$\\Spec(A)$ (because it is mapped to the generic point of $X_{d, n'}$).", "By construction $Z \\times_{\\Spec(A)} \\Spec(k) \\cong \\Spec(B)$.", "This proves what we want except that the map from", "residue field of $A$ to $k$ may not be an isomorphism.", "By Algebra, Lemma \\ref{algebra-lemma-flat-local-given-residue-field}", "there exists a flat local ring map $A \\to A'$ such that the residue", "field of $A'$ is $k$. If $A'$ isn't a domain, then we choose a", "minimal prime $\\mathfrak p \\subset A'$ (which lies over the", "unique minimal prime of $A$ by flatness) and we replace", "$A'$ by $A'/\\mathfrak p$. Set $B'$ equal to the unique $A'$-algebra", "such that $Z \\times_{\\Spec(A)} \\Spec(A') = \\Spec(B')$.", "This finishes the proof." ], "refs": [ "discriminant-lemma-universal-finite-syntomic", "morphisms-lemma-set-points-where-fibres-etale", "discriminant-lemma-universal-finite-syntomic-smooth", "discriminant-lemma-syntomic-finite", "algebra-lemma-characterize-henselian", "discriminant-lemma-universal-quasi-finite-syntomic-etale", "more-algebra-lemma-henselization-regular", "algebra-lemma-flat-local-given-residue-field" ], "ref_ids": [ 14988, 5374, 14989, 14987, 1276, 14985, 10064, 1324 ] } ], "ref_ids": [ 14988 ] }, { "id": 14991, "type": "theorem", "label": "discriminant-lemma-locally-comes-from-universal-finite", "categories": [ "discriminant" ], "title": "discriminant-lemma-locally-comes-from-universal-finite", "contents": [ "Let $f : Y \\to X$ be a morphism of schemes. If $f$ satisfies the equivalent", "conditions of Lemma \\ref{lemma-syntomic-finite} then for every", "$x \\in X$ there exist a $d$ and a commutative diagram", "$$", "\\xymatrix{", "Y \\ar[d] &", "V \\ar[d] \\ar[l] \\ar[r] &", "V_d \\ar[d] \\ar[r] &", "Y_d \\ar[d]^{\\pi_d}\\\\", "X &", "U \\ar[l] \\ar[r] &", "U_d \\ar[r] &", "X_d", "}", "$$", "with the following properties", "\\begin{enumerate}", "\\item $U \\subset X$ is open and $V = f^{-1}(U)$,", "\\item $\\pi_d : Y_d \\to X_d$ is as in", "Example \\ref{example-universal-finite-syntomic},", "\\item $U_d \\subset X_d$ is as in Lemma \\ref{lemma-universal-finite-syntomic}", "and $V_d = \\pi_d^{-1}(U_d) \\subset Y_d$,", "\\item where the middle square is cartesian.", "\\end{enumerate}" ], "refs": [ "discriminant-lemma-syntomic-finite", "discriminant-lemma-universal-finite-syntomic" ], "proofs": [ { "contents": [ "Choose an affine open neighbourhood $U = \\Spec(A) \\subset X$ of $x$.", "Write $V = f^{-1}(U) = \\Spec(B)$. Then $B$ is a finite locally free", "$A$-module and the inclusion $A \\subset B$ is a locally direct summand.", "Thus after shrinking $U$ we can choose a basis $1 = e_1, e_2, \\ldots, e_d$", "of $B$ as an $A$-module. Write", "$e_i e_j = \\sum \\alpha_{ij}^l e_l$ for unique", "elements $\\alpha_{ij}^l \\in A$ which satisfy the relations", "$\\sum_l \\alpha_{ij}^l \\alpha_{lk}^m = \\sum_l \\alpha_{il}^m \\alpha _{jk}^l$", "and $\\alpha_{ij}^k = \\alpha_{ji}^k$ and $\\alpha_{i1}^j - \\delta_{ij}$", "in $A$. This determines a morphism $\\Spec(A) \\to X_d$ by sending", "$a_{ij}^l \\in A_d$ to $\\alpha_{ij}^l \\in A$. By construction", "$V \\cong \\Spec(A) \\times_{X_d} Y_d$. By the definition of $U_d$", "we see that $\\Spec(A) \\to X_d$ factors through $U_d$. This", "finishes the proof." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 14987, 14988 ] }, { "id": 14992, "type": "theorem", "label": "discriminant-lemma-tate", "categories": [ "discriminant" ], "title": "discriminant-lemma-tate", "contents": [ "\\begin{reference}", "\\cite[Appendix]{Mazur-Roberts}", "\\end{reference}", "Let $A \\to P$ be a ring map. Let $f_1, \\ldots, f_n \\in P$ be a", "Koszul regular sequence. Assume $B = P/(f_1, \\ldots, f_n)$", "is flat over $A$. Let $g_1, \\ldots, g_n \\in P \\otimes_A B$", "be a Koszul regular sequence generating the kernel of the multiplication", "map $P \\otimes_A B \\to B$. Write $f_i \\otimes 1 = \\sum g_{ij} g_j$.", "Then the annihilator of $\\Ker(B \\otimes_A B \\to B)$ is a principal", "ideal generated by the image of $\\det(g_{ij})$." ], "refs": [], "proofs": [ { "contents": [ "The Koszul complex $K_\\bullet = K(P, f_1, \\ldots, f_n)$ is a resolution", "of $B$ by finite free $P$-modules. The Koszul complex", "$M_\\bullet = K(P \\otimes_A B, g_1, \\ldots, g_n)$ is a resolution", "of $B$ by finite free $P \\otimes_A B$-modules. There is a map of", "complexes", "$$", "K_\\bullet \\longrightarrow M_\\bullet", "$$", "which in degree $1$ is given by the matrix $(g_{ij})$ and", "in degree $n$ by $\\det(g_{ij})$. See", "More on Algebra, Lemma \\ref{more-algebra-lemma-functorial}.", "As $B$ is a flat $A$-module, we can view $M_\\bullet$ as a complex", "of flat $P$-modules (via $P \\to P \\otimes_A B$, $p \\mapsto p \\otimes 1$).", "Thus we may use both complexes to compute $\\text{Tor}_*^P(B, B)$ and", "it follows that the displayed map defines a quasi-isomorphism after tensoring", "with $B$. It is clear that $H_n(K_\\bullet \\otimes_P B) = B$.", "On the other hand, $H_n(M_\\bullet \\otimes_P B)$ is the kernel of", "$$", "B \\otimes_A B \\xrightarrow{g_1, \\ldots, g_n} (B \\otimes_A B)^{\\oplus n}", "$$", "Since $g_1, \\ldots, g_n$ generate the kernel of $B \\otimes_A B \\to B$", "this proves the lemma." ], "refs": [ "more-algebra-lemma-functorial" ], "ref_ids": [ 9957 ] } ], "ref_ids": [] }, { "id": 14993, "type": "theorem", "label": "discriminant-lemma-quasi-finite-complete-intersection", "categories": [ "discriminant" ], "title": "discriminant-lemma-quasi-finite-complete-intersection", "contents": [ "Let $A$ be a ring. Let $n \\geq 1$ and", "$h, f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$.", "Set $B = A[x_1, \\ldots, x_n, 1/h]/(f_1, \\ldots, f_n)$.", "Assume that $B$ is quasi-finite over $A$.", "Then", "\\begin{enumerate}", "\\item $B$ is flat over $A$ and $A \\to B$ is a relative local complete", "intersection,", "\\item the annihilator $J$ of $I = \\Ker(B \\otimes_A B \\to B)$", "is free of rank $1$ over $B$,", "\\item the Noether different of $B$ over $A$ is generated", "by $\\det(\\partial f_i/\\partial x_j)$ in $B$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Note that", "$B = A[x, x_1, \\ldots, x_n]/(xh - 1, f_1, \\ldots, f_n)$", "is a relative global complete intersection over $A$, see", "Algebra, Definition", "\\ref{algebra-definition-relative-global-complete-intersection}.", "By Algebra, Lemma \\ref{algebra-lemma-relative-global-complete-intersection}", "we see that $B$ is flat over $A$.", "\\medskip\\noindent", "Write $P' = A[x, x_1, \\ldots, x_n]$ and", "$P = P'/(xh - 1) = A[x_1, \\ldots, x_n, 1/g]$.", "Then we have $P' \\to P \\to B$.", "By More on Algebra, Lemma", "\\ref{more-algebra-lemma-relative-global-complete-intersection-koszul}", "we see that $xh - 1, f_1, \\ldots, f_n$ is a Koszul regular sequence", "in $P'$. Since $xh - 1$ is a Koszul regular sequence of length", "one in $P'$ (by the same lemma for example) we conclude that ", "$f_1, \\ldots, f_n$ is a Koszul regular sequence in $P$ by", "More on Algebra, Lemma \\ref{more-algebra-lemma-truncate-koszul-regular}.", "\\medskip\\noindent", "Let $g_i \\in P \\otimes_A B$ be the image of $x_i \\otimes 1 - 1 \\otimes x_i$.", "Let us use the short hand $y_i = x_i \\otimes 1$ and $z_i = 1 \\otimes x_i$", "in $A[x_1, \\ldots, x_n] \\otimes_A A[x_1, \\ldots, x_n]$", "so that $g_i$ is the image of $y_i - z_i$. For a polynomial", "$f \\in A[x_1, \\ldots, x_n]$ we write $f(y) = f \\otimes 1$", "and $f(z) = 1 \\otimes f$ in the above tensor product.", "Then we have", "$$", "P \\otimes_A B/(g_1, \\ldots, g_n) =", "\\frac{A[y_1, \\ldots, y_n, z_1, \\ldots, z_n, \\frac{1}{h(y)h(z)}]}", "{(f_1(z), \\ldots, f_n(z), y_1 - z_1, \\ldots, y_n - z_n)}", "$$", "which is clearly isomorphic to $B$. Hence by the same arguments", "as above we find that $f_1(z), \\ldots, f_n(z), y_1 - z_1, \\ldots, y_n - z_n$", "is a Koszul regular sequence in", "$A[y_1, \\ldots, y_n, z_1, \\ldots, z_n, \\frac{1}{h(y)h(z)}]$.", "The sequence $f_1(z), \\ldots, f_n(z)$ is a Koszul regular in", "$A[y_1, \\ldots, y_n, z_1, \\ldots, z_n, \\frac{1}{h(y)h(z)}]$", "by flatness of the map", "$$", "P \\longrightarrow A[y_1, \\ldots, y_n, z_1, \\ldots, z_n,", "\\textstyle{\\frac{1}{h(y)h(z)}}],\\quad x_i \\longmapsto z_i", "$$", "and More on Algebra, Lemma", "\\ref{more-algebra-lemma-koszul-regular-flat-base-change}.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-truncate-koszul-regular}", "we conclude that $g_1, \\ldots, g_n$ is a regular sequence", "in $P \\otimes_A B$.", "\\medskip\\noindent", "At this point we have verified all the assumptions of Lemma \\ref{lemma-tate}", "above with $P$, $f_1, \\ldots, f_n$, and $g_i \\in P \\otimes_A B$ as above.", "In particular the annihilator $J$ of $I$ is freely generated by one", "element $\\delta$ over $B$.", "Set $f_{ij} = \\partial f_i/\\partial x_j \\in A[x_1, \\ldots, x_n]$.", "An elementary computation shows that we can write", "$$", "f_i(y) =", "f_i(z_1 + g_1, \\ldots, z_n + g_n) =", "f_i(z) + \\sum\\nolimits_j f_{ij}(z) g_j +", "\\sum\\nolimits_{j, j'} F_{ijj'}g_jg_{j'}", "$$", "for some $F_{ijj'} \\in A[y_1, \\ldots, y_n, z_1, \\ldots, z_n]$.", "Taking the image in $P \\otimes_A B$ the terms $f_i(z)$ map to", "zero and we obtain", "$$", "f_i \\otimes 1 = \\sum\\nolimits_j", "\\left(1 \\otimes f_{ij} + \\sum\\nolimits_{j'} F_{ijj'}g_{j'}\\right)g_j", "$$", "Thus we conclude from Lemma \\ref{lemma-tate}", "that $\\delta = \\det(g_{ij})$ with", "$g_{ij} = 1 \\otimes f_{ij} + \\sum_{j'} F_{ijj'}g_{j'}$.", "Since $g_{j'}$ maps to zero in $B$, we conclude", "that the image of $\\det(\\partial f_i/\\partial x_j)$ in $B$", "generates the Noether different of $B$ over $A$." ], "refs": [ "algebra-definition-relative-global-complete-intersection", "algebra-lemma-relative-global-complete-intersection", "more-algebra-lemma-relative-global-complete-intersection-koszul", "more-algebra-lemma-truncate-koszul-regular", "more-algebra-lemma-koszul-regular-flat-base-change", "more-algebra-lemma-truncate-koszul-regular", "discriminant-lemma-tate", "discriminant-lemma-tate" ], "ref_ids": [ 1533, 1184, 10001, 9985, 9976, 9985, 14992, 14992 ] } ], "ref_ids": [] }, { "id": 14994, "type": "theorem", "label": "discriminant-lemma-different-syntomic-quasi-finite", "categories": [ "discriminant" ], "title": "discriminant-lemma-different-syntomic-quasi-finite", "contents": [ "Let $f : Y \\to X$ be a morphism of Noetherian schemes. If $f$", "satisfies the equivalent conditions of Lemma \\ref{lemma-syntomic-quasi-finite}", "then the different $\\mathfrak{D}_f$ of $f$ is the K\\\"ahler different", "of $f$." ], "refs": [ "discriminant-lemma-syntomic-quasi-finite" ], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-flat-gorenstein-agree-noether} and", "\\ref{lemma-dualizing-syntomic-quasi-finite}", "the different of $f$ affine locally is the same as the", "Noether different. Then the lemma follows from the", "computation of the Noether different and the K\\\"ahler", "different on standard affine pieces done in", "Lemmas \\ref{lemma-kahler-different-complete-intersection} and", "\\ref{lemma-quasi-finite-complete-intersection}." ], "refs": [ "discriminant-lemma-flat-gorenstein-agree-noether", "discriminant-lemma-dualizing-syntomic-quasi-finite", "discriminant-lemma-kahler-different-complete-intersection", "discriminant-lemma-quasi-finite-complete-intersection" ], "ref_ids": [ 14976, 14984, 14971, 14993 ] } ], "ref_ids": [ 14981 ] }, { "id": 14995, "type": "theorem", "label": "discriminant-lemma-different-quasi-finite-complete-intersection", "categories": [ "discriminant" ], "title": "discriminant-lemma-different-quasi-finite-complete-intersection", "contents": [ "Let $A$ be a ring. Let $n \\geq 1$ and", "$h, f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$.", "Set $B = A[x_1, \\ldots, x_n, 1/h]/(f_1, \\ldots, f_n)$.", "Assume that $B$ is quasi-finite over $A$.", "Then there is an isomorphism $B \\to \\omega_{B/A}$", "mapping $\\det(\\partial f_i/\\partial x_j)$ to $\\tau_{B/A}$." ], "refs": [], "proofs": [ { "contents": [ "Let $J$ be the annihilator of $\\Ker(B \\otimes_A B \\to B)$.", "By Lemma \\ref{lemma-quasi-finite-complete-intersection}", "the map $A \\to B$ is flat and", "$J$ is a free $B$-module with generator $\\xi$ mapping to", "$\\det(\\partial f_i/\\partial x_j)$ in $B$.", "Thus the lemma follows from", "Lemma \\ref{lemma-noether-different-flat-quasi-finite}", "and the fact (Lemma \\ref{lemma-dualizing-syntomic-quasi-finite})", "that $\\omega_{B/A}$ is an invertible $B$-module.", "(Warning: it is necessary to prove $\\omega_{B/A}$", "is invertible because a finite $B$-module $M$ such", "that $\\Hom_B(M, B) \\cong B$ need not be free.)" ], "refs": [ "discriminant-lemma-quasi-finite-complete-intersection", "discriminant-lemma-noether-different-flat-quasi-finite", "discriminant-lemma-dualizing-syntomic-quasi-finite" ], "ref_ids": [ 14993, 14967, 14984 ] } ], "ref_ids": [] }, { "id": 14996, "type": "theorem", "label": "discriminant-lemma-discriminant-quasi-finite-morphism-smooth", "categories": [ "discriminant" ], "title": "discriminant-lemma-discriminant-quasi-finite-morphism-smooth", "contents": [ "Let $S$ be a Noetherian scheme. Let $X$, $Y$ be smooth schemes", "of relative dimension $n$ over $S$. Let $f : Y \\to X$ be a", "quasi-finite morphism over $S$.", "Then $f$ is flat and the closed subscheme $R \\subset Y$", "cut out by the different of $f$ is the locally principal", "closed subscheme cut out by", "$$", "\\wedge^n(\\text{d}f) \\in", "\\Gamma(Y,", "(f^*\\Omega^n_{X/S})^{\\otimes -1} \\otimes_{\\mathcal{O}_Y} \\Omega^n_{Y/S})", "$$", "If $f$ is \\'etale at the associated points of $Y$, then $R$ is an", "effective Cartier divisor and", "$$", "f^*\\Omega^n_{X/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{O}(R) =", "\\Omega^n_{Y/S}", "$$", "as invertible sheaves on $Y$." ], "refs": [], "proofs": [ { "contents": [ "To prove that $f$ is flat, it suffices to prove $Y_s \\to X_s$", "is flat for all $s \\in S$ (More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-morphism-between-flat-Noetherian}).", "Flatness of $Y_s \\to X_s$ follows from", "Algebra, Lemma \\ref{algebra-lemma-CM-over-regular-flat}.", "By More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-lci-permanence}", "the morphism $f$ is a local complete intersection morphism.", "Thus the statement on the different follows from the", "corresponding statement on the K\\\"ahler different by", "Lemma \\ref{lemma-different-syntomic-quasi-finite}.", "Finally, since we have the exact sequence", "$$", "f^*\\Omega_{X/S} \\xrightarrow{\\text{d}f} \\Omega_{X/S} \\to \\Omega_{Y/X} \\to 0", "$$", "by Morphisms, Lemma \\ref{morphisms-lemma-triangle-differentials}", "and since $\\Omega_{X/S}$ and $\\Omega_{Y/S}$ are finite locally free", "of rank $n$ (Morphisms, Lemma", "\\ref{morphisms-lemma-smooth-omega-finite-locally-free}),", "the statement for the K\\\"ahler different is clear from the definition", "of the zeroth fitting ideal. If $f$ is \\'etale at the associated", "points of $Y$, then $\\wedge^n\\text{d}f$ does not vanish in", "the associated points of $Y$, which implies that the local equation", "of $R$ is a nonzerodivisor. Hence $R$ is an effective Cartier divisor.", "The canonical isomorphism sends $1$ to $\\wedge^n\\text{d}f$, see", "Divisors, Lemma \\ref{divisors-lemma-characterize-OD}." ], "refs": [ "more-morphisms-lemma-morphism-between-flat-Noetherian", "algebra-lemma-CM-over-regular-flat", "more-morphisms-lemma-lci-permanence", "discriminant-lemma-different-syntomic-quasi-finite", "morphisms-lemma-triangle-differentials", "morphisms-lemma-smooth-omega-finite-locally-free", "divisors-lemma-characterize-OD" ], "ref_ids": [ 13767, 1107, 14008, 14994, 5313, 5334, 7944 ] } ], "ref_ids": [] }, { "id": 14997, "type": "theorem", "label": "discriminant-lemma-explain-condition", "categories": [ "discriminant" ], "title": "discriminant-lemma-explain-condition", "contents": [ "Let $A \\to B$ be a map of Noetherian rings. Consider the conditions", "\\begin{enumerate}", "\\item nonzerodivisors of $A$ map to nonzerodivisors of $B$,", "\\item (1) holds and $Q(A) \\to Q(A) \\otimes_A B$ is flat,", "\\item $A \\to B_\\mathfrak q$ is flat for every", "$\\mathfrak q \\in \\text{Ass}(B)$,", "\\item (3) holds and $A \\to B_\\mathfrak q$ is flat for every $\\mathfrak q$", "lying over an element in $\\text{Ass}(A)$.", "\\end{enumerate}", "Then we have the following implications", "$$", "\\xymatrix{", "(1) & (2) \\ar@{=>}[l] \\ar@{=>}[d] \\\\", "(3) \\ar@{=>}[u] & (4) \\ar@{=>}[l]", "}", "$$", "If going up holds for $A \\to B$ then (2) and (4) are equivalent." ], "refs": [], "proofs": [ { "contents": [ "The horizontal implications in the diagram are trivial.", "Let $S \\subset A$ be the set of nonzerodivisors so that", "$Q(A) = S^{-1}A$ and $Q(A) \\otimes_A B = S^{-1}B$. Recall that", "$S = A \\setminus \\bigcup_{\\mathfrak p \\in \\text{Ass}(A)} \\mathfrak p$", "by Algebra, Lemma \\ref{algebra-lemma-ass-zero-divisors}.", "Let $\\mathfrak q \\subset B$ be a prime lying over $\\mathfrak p \\subset A$.", "\\medskip\\noindent", "Assume (2). If $\\mathfrak q \\in \\text{Ass}(B)$ then", "$\\mathfrak q$ consists of zerodivisors, hence (1) implies", "the same is true for $\\mathfrak p$. Hence", "$\\mathfrak p$ corresponds to a prime of $S^{-1}A$.", "Hence $A \\to B_\\mathfrak q$ is flat by our assumption (2).", "If $\\mathfrak q$ lies over an associated prime $\\mathfrak p$", "of $A$, then certainly $\\mathfrak p \\in \\Spec(S^{-1}A)$ and the", "same argument works.", "\\medskip\\noindent", "Assume (3). Let $f \\in A$ be a nonzerodivisor. If $f$ were a zerodivisor", "on $B$, then $f$ is contained in an associated prime $\\mathfrak q$", "of $B$. Since $A \\to B_\\mathfrak q$ is flat by assumption, we conclude that", "$\\mathfrak p$ is an associated prime of $A$ by", "Algebra, Lemma \\ref{algebra-lemma-bourbaki}. This would imply that", "$f$ is a zerodivisor on $A$, a contradiction.", "\\medskip\\noindent", "Assume (4) and going up for $A \\to B$. We already know (1) holds.", "If $\\mathfrak q$ corresponds to a prime of $S^{-1}B$ then $\\mathfrak p$", "is contained in an associated prime $\\mathfrak p'$ of $A$. By going up", "there exists a prime $\\mathfrak q'$ containing $\\mathfrak q$ and lying", "over $\\mathfrak p$. Then $A \\to B_{\\mathfrak q'}$ is flat by", "(4). Hence $A \\to B_{\\mathfrak q}$ is flat as a localization.", "Thus $A \\to S^{-1}B$ is flat and so is $S^{-1}A \\to S^{-1}B$, see", "Algebra, Lemma \\ref{algebra-lemma-flat-localization}." ], "refs": [ "algebra-lemma-ass-zero-divisors", "algebra-lemma-bourbaki", "algebra-lemma-flat-localization" ], "ref_ids": [ 704, 717, 538 ] } ], "ref_ids": [] }, { "id": 14998, "type": "theorem", "label": "discriminant-lemma-agree-dedekind", "categories": [ "discriminant" ], "title": "discriminant-lemma-agree-dedekind", "contents": [ "Assume the Dedekind different is defined for $A \\to B$.", "Set $X = \\Spec(A)$ and $Y = \\Spec(B)$. The generalization of", "Remark \\ref{remark-different-generalization}", "applies to the morphism $f : Y \\to X$ if and only if", "$1 \\in \\mathcal{L}_{B/A}$ (e.g., if $A$ is normal, see", "Lemma \\ref{lemma-dedekind-different-ideal}).", "In this case $\\mathfrak{D}_{B/A}$ is an ideal of $B$ and we have", "$$", "\\mathfrak{D}_f = \\widetilde{\\mathfrak{D}_{B/A}}", "$$", "as coherent ideal sheaves on $Y$." ], "refs": [ "discriminant-remark-different-generalization", "discriminant-lemma-dedekind-different-ideal" ], "proofs": [ { "contents": [ "As the Dedekind different for $A \\to B$ is defined we can apply", "Lemma \\ref{lemma-explain-condition} to see that", "$Y \\to X$ satisfies condition (1) of", "Remark \\ref{remark-different-generalization}.", "Recall that there is a canonical isomorphism", "$c : \\mathcal{L}_{B/A} \\to \\omega_{B/A}$, see", "Lemma \\ref{lemma-dedekind-complementary-module}.", "Let $K = Q(A)$ and $L = K \\otimes_A B$ as above.", "By construction the map $c$ fits into a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{L}_{B/A} \\ar[r] \\ar[d]_c & L \\ar[d] \\\\", "\\omega_{B/A} \\ar[r] & \\Hom_K(L, K)", "}", "$$", "where the right vertical arrow sends $x \\in L$ to the map", "$y \\mapsto \\text{Trace}_{L/K}(xy)$ and the lower horizontal", "arrow is the base change map (\\ref{equation-bc-dualizing}) for $\\omega_{B/A}$.", "We can factor the lower horizontal map as", "$$", "\\omega_{B/A} = \\Gamma(Y, \\omega_{Y/X})", "\\to \\Gamma(V, \\omega_{V/X}) \\to \\Hom_K(L, K)", "$$", "Since all associated points of $\\omega_{V/X}$", "map to associated primes of $A$", "(Lemma \\ref{lemma-dualizing-associated-primes})", "we see that the second map is injective.", "The element $\\tau_{V/X}$ maps to $\\text{Trace}_{L/K}$ in", "$\\Hom_K(L, K)$ by the very definition of trace elements", "(Definition \\ref{definition-trace-element}).", "Thus $\\tau$ as in condition (2) of", "Remark \\ref{remark-different-generalization}", "exists if and only if $1 \\in \\mathcal{L}_{B/A}$ and then", "$\\tau = c(1)$. In this case, by Lemma \\ref{lemma-dedekind-different-ideal}", "we see that $\\mathfrak{D}_{B/A} \\subset B$.", "Finally, the agreement of $\\mathfrak{D}_f$ with $\\mathfrak{D}_{B/A}$", "is immediate from the definitions and the fact $\\tau = c(1)$ seen above." ], "refs": [ "discriminant-lemma-explain-condition", "discriminant-remark-different-generalization", "discriminant-lemma-dedekind-complementary-module", "discriminant-lemma-dualizing-associated-primes", "discriminant-definition-trace-element", "discriminant-remark-different-generalization", "discriminant-lemma-dedekind-different-ideal" ], "ref_ids": [ 14997, 15011, 14973, 14951, 15003, 15011, 14972 ] } ], "ref_ids": [ 15011, 14972 ] }, { "id": 14999, "type": "theorem", "label": "discriminant-lemma-compare-dualizing", "categories": [ "discriminant" ], "title": "discriminant-lemma-compare-dualizing", "contents": [ "Let $f : Y \\to X$ be a quasi-finite separated morphism of Noetherian schemes.", "For every pair of affine opens $\\Spec(B) = V \\subset Y$,", "$\\Spec(A) = U \\subset X$ with $f(V) \\subset U$ there is an isomorphism", "$$", "H^0(V, f^!\\mathcal{O}_Y) = \\omega_{B/A}", "$$", "where $f^!$ is as in", "Duality for Schemes, Section \\ref{duality-section-upper-shriek}.", "These isomorphisms are compatible with restriction maps and define a canonical", "isomorphism $H^0(f^!\\mathcal{O}_X) = \\omega_{Y/X}$ with", "$\\omega_{Y/X}$ as in Remark \\ref{remark-relative-dualizing-for-quasi-finite}.", "Similarly, if $f : Y \\to X$ is a quasi-finite morphism of schemes of", "finite type over a Noetherian base $S$ endowed with a dualizing complex", "$\\omega_S^\\bullet$, then $H^0(f_{new}^!\\mathcal{O}_X) = \\omega_{Y/X}$." ], "refs": [ "discriminant-remark-relative-dualizing-for-quasi-finite" ], "proofs": [ { "contents": [ "By Zariski's main theorem we can choose a factorization $f = f' \\circ j$", "where $j : Y \\to Y'$ is an open immersion and $f' : Y' \\to X$ is a finite", "morphism, see More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite}.", "By our construction in", "Duality for Schemes, Lemma \\ref{duality-lemma-shriek-well-defined} we have", "$f^! = j^* \\circ a'$ where", "$a' : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_{Y'})$", "is the right adjoint to $Rf'_*$ of", "Duality for Schemes, Lemma \\ref{duality-lemma-twisted-inverse-image}.", "By Duality for Schemes, Lemma \\ref{duality-lemma-finite-twisted}", "we see that", "$\\Phi(a'(\\mathcal{O}_X)) = R\\SheafHom(f'_*\\mathcal{O}_{Y'}, \\mathcal{O}_X)$ in", "$D_\\QCoh^+(f'_*\\mathcal{O}_{Y'})$. In particular $a'(\\mathcal{O}_X)$ has", "vanishing cohomology sheaves in degrees $< 0$. The zeroth cohomology sheaf", "is determined by the isomorphism", "$$", "f'_*H^0(a'(\\mathcal{O}_X)) =", "\\SheafHom_{\\mathcal{O}_X}(f'_*\\mathcal{O}_{Y'}, \\mathcal{O}_X)", "$$", "as $f'_*\\mathcal{O}_{Y'}$-modules via the equivalence of", "Morphisms, Lemma \\ref{morphisms-lemma-affine-equivalence-modules}.", "Writing $(f')^{-1}U = V' = \\Spec(B')$, we obtain", "$$", "H^0(V', a'(\\mathcal{O}_X)) = \\Hom_A(B', A).", "$$", "As the zeroth cohomology sheaf of $a'(\\mathcal{O}_X)$", "is a quasi-coherent module we find that", "the restriction to $V$ is given by", "$\\omega_{B/A} = \\Hom_A(B', A) \\otimes_{B'} B$ as desired.", "\\medskip\\noindent", "The statement about restriction maps signifies that the restriction mappings", "of the quasi-coherent $\\mathcal{O}_{Y'}$-module $H^0(a'(\\mathcal{O}_X))$", "for opens in $Y'$ agrees with the maps defined in", "Lemma \\ref{lemma-localize-dualizing}", "for the modules $\\omega_{B/A}$ via the isomorphisms given above.", "This is clear.", "\\medskip\\noindent", "Let $f : Y \\to X$ be a quasi-finite morphism of schemes of finite type", "over a Noetherian base $S$ endowed with a dualizing complex $\\omega_S^\\bullet$.", "Consider opens $V \\subset Y$ and $U \\subset X$ with $f(V) \\subset U$", "and $V$ and $U$ separated over $S$. Denote $f|_V : V \\to U$ the restriction", "of $f$. By the discussion above and", "Duality for Schemes, Lemma \\ref{duality-lemma-duality-bootstrap}", "there are canonical isomorphisms", "$$", "H^0(f_{new}^!\\mathcal{O}_X)|_V = H^0((f|_V)^!\\mathcal{O}_U) = \\omega_{V/U} =", "\\omega_{Y/X}|_V", "$$", "We omit the verification that these isomorphisms glue to a global", "isomorphism $H^0(f_{new}^!\\mathcal{O}_X) \\to \\omega_{Y/X}$." ], "refs": [ "more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "duality-lemma-shriek-well-defined", "duality-lemma-twisted-inverse-image", "duality-lemma-finite-twisted", "morphisms-lemma-affine-equivalence-modules", "discriminant-lemma-localize-dualizing", "duality-lemma-duality-bootstrap" ], "ref_ids": [ 13901, 13551, 13503, 13532, 5174, 14946, 13575 ] } ], "ref_ids": [ 15006 ] }, { "id": 15000, "type": "theorem", "label": "discriminant-lemma-compare-trace", "categories": [ "discriminant" ], "title": "discriminant-lemma-compare-trace", "contents": [ "Let $f : Y \\to X$ be a finite flat morphism of Noetherian schemes.", "The map", "$$", "\\text{Trace}_f : f_*\\mathcal{O}_Y \\longrightarrow \\mathcal{O}_X", "$$", "of Section \\ref{section-discriminant}", "corresponds to a map $\\mathcal{O}_Y \\to f^!\\mathcal{O}_X$.", "Denote $\\tau_{Y/X} \\in H^0(Y, f^!\\mathcal{O}_X)$ the image of $1$.", "Via the isomorphism $H^0(f^!\\mathcal{O}_X) = \\omega_{X/Y}$ of", "Lemma \\ref{lemma-compare-dualizing}", "this agrees with the construction in", "Remark \\ref{remark-relative-dualizing-for-flat-quasi-finite}." ], "refs": [ "discriminant-lemma-compare-dualizing", "discriminant-remark-relative-dualizing-for-flat-quasi-finite" ], "proofs": [ { "contents": [ "Unwinding all the definitions, this is immediate from the fact", "that if $A \\to B$ is finite flat, then $\\tau_{B/A} = \\text{Trace}_{B/A}$", "(Lemma \\ref{lemma-finite-flat-trace}) and the compatibility", "of traces with localizations (Lemma \\ref{lemma-trace-base-change})." ], "refs": [ "discriminant-lemma-finite-flat-trace", "discriminant-lemma-trace-base-change" ], "ref_ids": [ 14957, 14958 ] } ], "ref_ids": [ 14999, 15007 ] }, { "id": 15001, "type": "theorem", "label": "discriminant-lemma-gorenstein-quasi-finite", "categories": [ "discriminant" ], "title": "discriminant-lemma-gorenstein-quasi-finite", "contents": [ "Let $f : Y \\to X$ be a quasi-finite morphism of Noetherian schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is Gorenstein,", "\\item $f$ is flat and the fibres of $f$ are Gorenstein,", "\\item $f$ is flat and $\\omega_{Y/X}$ is invertible", "(Remark \\ref{remark-relative-dualizing-for-quasi-finite}),", "\\item for every $y \\in Y$ there are affine opens", "$y \\in V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$", "with $f(V) \\subset U$ such that $A \\to B$ is flat", "and $\\omega_{B/A}$ is an invertible $B$-module.", "\\end{enumerate}" ], "refs": [ "discriminant-remark-relative-dualizing-for-quasi-finite" ], "proofs": [ { "contents": [ "Parts (1) and (2) are equivalent by definition. Parts (3) and (4)", "are equivalent by the construction of $\\omega_{Y/X}$ in", "Remark \\ref{remark-relative-dualizing-for-quasi-finite}.", "Thus we have to show that (1)-(2) is equivalent to (3)-(4).", "\\medskip\\noindent", "First proof. Working affine locally we can assume $f$ is a separated", "morphism and apply Lemma \\ref{lemma-compare-dualizing} to see that", "$\\omega_{Y/X}$ is the zeroth cohomology sheaf of $f^!\\mathcal{O}_X$.", "Under both assumptions $f$ is flat and quasi-finite, hence", "$f^!\\mathcal{O}_X$ is isomorphic to $\\omega_{Y/X}[0]$, see", "Duality for Schemes, Lemma \\ref{duality-lemma-flat-quasi-finite-shriek}. Hence", "the equivalence follows from", "Duality for Schemes, Lemma", "\\ref{duality-lemma-affine-flat-Noetherian-gorenstein}.", "\\medskip\\noindent", "Second proof. By Lemma \\ref{lemma-characterize-invertible},", "we see that it suffices to prove the equivalence of", "(2) and (3) when $X$ is the spectrum of a field $k$.", "Then $Y = \\Spec(B)$ where $B$ is a finite $k$-algebra.", "In this case $\\omega_{B/A} = \\omega_{B/k} = \\Hom_k(B, k)$", "placed in degree $0$ is a dualizing complex for $B$, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-dualizing-finite}.", "Thus the equivalence follows from", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-gorenstein}." ], "refs": [ "discriminant-remark-relative-dualizing-for-quasi-finite", "discriminant-lemma-compare-dualizing", "duality-lemma-flat-quasi-finite-shriek", "duality-lemma-affine-flat-Noetherian-gorenstein", "discriminant-lemma-characterize-invertible", "dualizing-lemma-dualizing-finite", "dualizing-lemma-gorenstein" ], "ref_ids": [ 15006, 14999, 13581, 13602, 14982, 2853, 2881 ] } ], "ref_ids": [ 15006 ] }, { "id": 15002, "type": "theorem", "label": "discriminant-proposition-tate-map", "categories": [ "discriminant" ], "title": "discriminant-proposition-tate-map", "contents": [ "There exists a unique rule that to every locally quasi-finite syntomic", "morphism of locally Noetherian schemes $Y \\to X$ assigns an isomorphism", "$$", "c_{Y/X} : \\det(\\NL_{Y/X}) \\longrightarrow \\omega_{Y/X}", "$$", "satisfying the following two properties", "\\begin{enumerate}", "\\item the section $\\delta(\\NL_{Y/X})$ is mapped to $\\tau_{Y/X}$, and", "\\item the rule is compatible with restriction to opens and with", "base change.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let us reformulate the statement of the proposition. Consider the category", "$\\mathcal{C}$ whose objects, denoted $Y/X$, are locally quasi-finite syntomic", "morphism $Y \\to X$ of locally Noetherian schemes and whose morphisms", "$b/a : Y'/X' \\to Y/X$ are commutative diagrams", "$$", "\\xymatrix{", "Y' \\ar[d] \\ar[r]_b & Y \\ar[d] \\\\", "X' \\ar[r]^a & X", "}", "$$", "which induce an isomorphism of $Y'$ with an open subscheme of", "$X' \\times_X Y$. The proposition means that for every object", "$Y/X$ of $\\mathcal{C}$ we have an isomorphism", "$c_{Y/X} : \\det(\\NL_{Y/X}) \\to \\omega_{Y/X}$", "with $c_{Y/X}(\\delta(\\NL_{Y/X})) = \\tau_{Y/X}$", "and for every morphism $b/a : Y'/X' \\to Y/X$ of $\\mathcal{C}$ we have", "$b^*c_{Y/X} = c_{Y'/X'}$ via the identifications", "$b^*\\det(\\NL_{Y/X}) = \\det(\\NL_{Y'/X'})$ and", "$b^*\\omega_{Y/X} = \\omega_{Y'/X'}$ described above.", "\\medskip\\noindent", "Given $Y/X$ in $\\mathcal{C}$ and $y \\in Y$ we can find", "an affine open $V \\subset Y$ and $U \\subset X$ with $f(V) \\subset U$", "such that there exists some isomorphism", "$$", "\\det(\\NL_{Y/X})|_V \\longrightarrow \\omega_{Y/X}|_V", "$$", "mapping $\\delta(\\NL_{Y/X})|_V$ to $\\tau_{Y/X}|_V$. This follows", "from picking affine opens as in", "Lemma \\ref{lemma-syntomic-quasi-finite} part (5), the affine", "local description of $\\delta(\\NL_{Y/X})$ in", "Remark \\ref{remark-local-description-delta}, and", "Lemma \\ref{lemma-different-quasi-finite-complete-intersection}.", "If the annihilator of the section $\\tau_{Y/X}$ is zero, then", "these local maps are unique and automatically glue. Hence if the annihilator", "of $\\tau_{Y/X}$ is zero, then there is a unique isomorphism", "$c_{Y/X} : \\det(\\NL_{Y/X}) \\to \\omega_{Y/X}$ with", "$c_{Y/X}(\\delta(\\NL_{Y/X})) = \\tau_{Y/X}$.", "If $b/a : Y'/X' \\to Y/X$ is a morphism of $\\mathcal{C}$", "and the annihilator of $\\tau_{Y'/X'}$ is zero as well,", "then $b^*c_{Y/X}$ is the unique isomorphism", "$c_{Y'/X'} : \\det(\\NL_{Y'/X'}) \\to \\omega_{Y'/X'}$ with", "$c_{Y'/X'}(\\delta(\\NL_{Y'/X'})) = \\tau_{Y'/X'}$.", "This follows formally from the fact that", "$b^*\\delta(\\NL_{Y/X}) = \\delta(\\NL_{Y'/X'})$ and", "$b^*\\tau_{Y/X} = \\tau_{Y'/X'}$.", "\\medskip\\noindent", "We can summarize the results of the previous paragraph as follows.", "Let $\\mathcal{C}_{nice} \\subset \\mathcal{C}$ denote the", "full subcategory of $Y/X$ such that the annihilator of", "$\\tau_{Y/X}$ is zero. Then we have solved the problem", "on $\\mathcal{C}_{nice}$. For $Y/X$ in $\\mathcal{C}_{nice}$", "we continue to denote $c_{Y/X}$ the solution we've just found.", "\\medskip\\noindent", "Consider morphisms", "$$", "Y_1/X_1 \\xleftarrow{b_1/a_1} Y/X \\xrightarrow{b_2/a_2} Y_2/X_2", "$$", "in $\\mathcal{C}$ such that $Y_1/X_1$ and $Y_2/X_2$ are objects", "of $\\mathcal{C}_{nice}$. {\\bf Claim.} $b_1^*c_{Y_1/X_1} = b_2^*c_{Y_2/X_2}$.", "We will first show that the claim implies the proposition", "and then we will prove the claim.", "\\medskip\\noindent", "Let $d, n \\geq 1$ and consider the locally", "quasi-finite syntomic morphism $Y_{n, d} \\to X_{n, d}$", "constructed in Example \\ref{example-universal-quasi-finite-syntomic}.", "Then $Y_{n, d}$ is an irreducible regular scheme and the", "morphism $Y_{n, d} \\to X_{n, d}$ is locally quasi-finite syntomic", "and \\'etale over a dense open, see", "Lemma \\ref{lemma-universal-quasi-finite-syntomic-etale}.", "Thus $\\tau_{Y_{n, d}/X_{n, d}}$ is nonzero for example by", "Lemma \\ref{lemma-different-ramification}. Now a nonzero section", "of an invertible module over an irreducible regular scheme", "has vanishing annihilator. Thus", "$Y_{n, d}/X_{n, d}$ is an object of $\\mathcal{C}_{nice}$.", "\\medskip\\noindent", "Let $Y/X$ be an arbitrary object of $\\mathcal{C}$. Let $y \\in Y$.", "By Lemma \\ref{lemma-locally-comes-from-universal} we can find", "$n, d \\geq 1$ and morphisms", "$$", "Y/X \\leftarrow V/U \\xrightarrow{b/a} Y_{n, d}/X_{n, d}", "$$", "of $\\mathcal{C}$ such that $V \\subset Y$ and $U \\subset X$ are open.", "Thus we can pullback the canonical morphism $c_{Y_{n, d}/X_{n, d}}$", "constructed above by $b$ to $V$. The claim guarantees these local", "isomorphisms glue! Thus we get a well defined global isomorphism", "$c_{Y/X} : \\det(\\NL_{Y/X}) \\to \\omega_{Y/X}$ with", "$c_{Y/X}(\\delta(\\NL_{Y/X})) = \\tau_{Y/X}$.", "If $b/a : Y'/X' \\to Y/X$ is a morphism of $\\mathcal{C}$, then", "the claim also implies that the similarly constructed map", "$c_{Y'/X'}$ is the pullback by $b$ of the locally constructed", "map $c_{Y/X}$. Thus it remains to prove the claim.", "\\medskip\\noindent", "In the rest of the proof we prove the claim. We may pick a point", "$y \\in Y$ and prove the maps agree in an open neighbourhood of $y$.", "Thus we may replace $Y_1$, $Y_2$ by open neighbourhoods of the", "image of $y$ in $Y_1$ and $Y_2$. Thus we may assume there are", "morphisms", "$$", "Y_{n_1, d_1}/X_{n_1, d_1} \\leftarrow Y_1/X_1", "\\quad\\text{and}\\quad", "Y_2/X_2 \\rightarrow Y_{n_2, d_2}/X_{n_2, d_2}", "$$", "These are morphisms of $\\mathcal{C}_{nice}$ for which we know the", "desired compatibilities. Thus we may replace", "$Y_1/X_1$ by $Y_{n_1, d_1}/X_{n_1, d_1}$ and", "$Y_2/X_2$ by $Y_{n_2, d_2}/X_{n_2, d_2}$. This reduces us to the", "case that $Y_1, X_1, Y_2, X_2$ are of finite type over $\\mathbf{Z}$.", "(The astute reader will realize that this step wouldn't have been", "necessary if we'd defined $\\mathcal{C}_{nice}$ to consist only", "of those objects $Y/X$ with $Y$ and $X$ of finite type over $\\mathbf{Z}$.)", "\\medskip\\noindent", "Assume $Y_1, X_1, Y_2, X_2$ are of finite type over $\\mathbf{Z}$.", "After replacing $Y, X, Y_1, X_1, Y_2, X_2$ by suitable open neighbourhoods", "of the image of $y$ we may assume $Y, X, Y_1, X_1, Y_2, X_2$ are affine.", "We may write $X = \\lim X_\\lambda$ as a cofiltered limit of affine", "schemes of finite type over $X_1 \\times X_2$. For each $\\lambda$", "we get", "$$", "Y_1 \\times_{X_1} X_\\lambda", "\\quad\\text{and}\\quad", "X_\\lambda \\times_{X_2} Y_2", "$$", "If we take limits we obtain", "$$", "\\lim Y_1 \\times_{X_1} X_\\lambda =", "Y_1 \\times_{X_1} X \\supset Y \\subset", "X \\times_{X_2} Y_2 = \\lim X_\\lambda \\times_{X_2} Y_2", "$$", "By Limits, Lemma \\ref{limits-lemma-descend-opens}", "we can find a $\\lambda$ and opens", "$V_{1, \\lambda} \\subset Y_1 \\times_{X_1} X_\\lambda$ and", "$V_{2, \\lambda} \\subset X_\\lambda \\times_{X_2} Y_2$", "whose base change to $X$ recovers $Y$ (on both sides).", "After increasing $\\lambda$ we may assume", "there is an isomorphism", "$V_{1, \\lambda} \\to V_{2, \\lambda}$ whose base change to $X$ is the", "identity on $Y$, see", "Limits, Lemma \\ref{limits-lemma-descend-finite-presentation}.", "Then we have the commutative diagram", "$$", "\\xymatrix{", "& Y/X \\ar[d] \\ar[ld]_{b_1/a_1} \\ar[rd]^{b_2/a_2} \\\\", "Y_1/X_1 & V_{1, \\lambda}/X_\\lambda \\ar[l] \\ar[r] & Y_2/X_2", "}", "$$", "Thus it suffices to prove the claim for the lower row", "of the diagram and we reduce to the case discussed in the", "next paragraph.", "\\medskip\\noindent", "Assume $Y, X, Y_1, X_1, Y_2, X_2$ are affine of finite type over $\\mathbf{Z}$.", "Write $X = \\Spec(A)$, $X_i = \\Spec(A_i)$. The ring map $A_1 \\to A$ corresponding", "to $X \\to X_1$ is of finite type and hence we may choose a surjection", "$A_1[x_1, \\ldots, x_n] \\to A$. Similarly, we may choose a surjection", "$A_2[y_1, \\ldots, y_m] \\to A$. Set $X'_1 = \\Spec(A_1[x_1, \\ldots, x_n])$", "and $X'_2 = \\Spec(A_2[y_1, \\ldots, y_m])$.", "Set $Y'_1 = Y_1 \\times_{X_1} X'_1$ and $Y'_2 = Y_2 \\times_{X_2} X'_2$.", "We get the following diagram", "$$", "Y_1/X_1 \\leftarrow", "Y'_1/X'_1 \\leftarrow", "Y/X", "\\rightarrow Y'_2/X'_2", "\\rightarrow Y_2/X_2", "$$", "Since $X'_1 \\to X_1$ and $X'_2 \\to X_2$ are flat, the same is true", "for $Y'_1 \\to Y_1$ and $Y'_2 \\to Y_2$. It follows easily that the", "annihilators of $\\tau_{Y'_1/X'_1}$ and $\\tau_{Y'_2/X'_2}$ are zero.", "Hence $Y'_1/X'_1$ and $Y'_2/X'_2$ are in $\\mathcal{C}_{nice}$.", "Thus the outer morphisms in the displayed diagram are morphisms", "of $\\mathcal{C}_{nice}$ for which we know the desired compatibilities.", "Thus it suffices to prove the claim for", "$Y'_1/X'_1 \\leftarrow Y/X \\rightarrow Y'_2/X'_2$. This reduces us", "to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $Y, X, Y_1, X_1, Y_2, X_2$ are affine of finite type over", "$\\mathbf{Z}$ and $X \\to X_1$ and $X \\to X_2$ are closed immersions.", "Consider the open embeddings", "$Y_1 \\times_{X_1} X \\supset Y \\subset X \\times_{X_2} Y_2$.", "There is an open neighbourhood $V \\subset Y$ of $y$ which is a", "standard open of both $Y_1 \\times_{X_1} X$ and $X \\times_{X_2} Y_2$.", "This follows from Schemes, Lemma \\ref{schemes-lemma-standard-open-two-affines}", "applied to the scheme obtained by glueing $Y_1 \\times_{X_1} X$ and", "$X \\times_{X_2} Y_2$ along $Y$; details omitted.", "Since $X \\times_{X_2} Y_2$ is a closed subscheme of $Y_2$", "we can find a standard open $V_2 \\subset Y_2$ such that", "$V_2 \\times_{X_2} X = V$. Similarly, we can find a standard open", "$V_1 \\subset Y_1$ such that $V_1 \\times_{X_1} X = V$.", "After replacing $Y, Y_1, Y_2$ by $V, V_1, V_2$ we reduce to the", "case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $Y, X, Y_1, X_1, Y_2, X_2$ are affine of finite type over", "$\\mathbf{Z}$ and $X \\to X_1$ and $X \\to X_2$ are closed immersions", "and $Y_1 \\times_{X_1} X = Y = X \\times_{X_2} Y_2$.", "Write $X = \\Spec(A)$, $X_i = \\Spec(A_i)$, $Y = \\Spec(B)$,", "$Y_i = \\Spec(B_i)$. Then we can consider the affine schemes", "$$", "X' = \\Spec(A_1 \\times_A A_2) = \\Spec(A')", "\\quad\\text{and}\\quad", "Y' = \\Spec(B_1 \\times_B B_2) = \\Spec(B')", "$$", "Observe that $X' = X_1 \\amalg_X X_2$ and $Y' = Y_1 \\amalg_Y Y_2$, see", "More on Morphisms, Lemma \\ref{more-morphisms-lemma-basic-example-pushout}.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-fibre-product-finite-type}", "the rings $A'$ and $B'$ are of finite type over $\\mathbf{Z}$. By", "More on Algebra, Lemma \\ref{more-algebra-lemma-module-over-fibre-product}", "we have $B' \\otimes_A A_1 = B_1$ and $B' \\times_A A_2 = B_2$.", "In particular a fibre of $Y' \\to X'$ over a point of", "$X' = X_1 \\amalg_X X_2$ is always equal to either a fibre of $Y_1 \\to X_1$", "or a fibre of $Y_2 \\to X_2$. By More on Algebra, Lemma", "\\ref{more-algebra-lemma-flat-module-over-fibre-product}", "the ring map $A' \\to B'$ is flat. Thus by", "Lemma \\ref{lemma-syntomic-quasi-finite} part (3)", "we conclude that $Y'/X'$ is an object of $\\mathcal{C}$.", "Consider now the commutative diagram", "$$", "\\xymatrix{", "& Y/X \\ar[ld]_{b_1/a_1} \\ar[rd]^{b_2/a_2} \\\\", "Y_1/X_1 \\ar[rd] & & Y_2/X_2 \\ar[ld] \\\\", "& Y'/X'", "}", "$$", "Now we would be done if $Y'/X'$ is an object of $\\mathcal{C}_{nice}$.", "Namely, then pulling back $c_{Y'/X'}$ around the two sides of the", "square, we would obtain the desired conclusion. Now, in fact, it", "is true that $Y'/X'$ is an object of", "$\\mathcal{C}_{nice}$\\footnote{Namely, the structure", "sheaf $\\mathcal{O}_{Y'}$ is a subsheaf of", "$(Y_1 \\to Y')_*\\mathcal{O}_{Y_1} \\times (Y_2 \\to Y')_*\\mathcal{O}_{Y_2}$.}.", "But it is amusing to note that we don't even need this.", "Namely, the arguments above show that,", "after possibly shrinking all of the schemes", "$X, Y, X_1, Y_1, X_2, Y_2, X', Y'$ we can find some", "$n, d \\geq 1$, and extend the diagram like so:", "$$", "\\xymatrix{", "& Y/X \\ar[ld]_{b_1/a_1} \\ar[rd]^{b_2/a_2} \\\\", "Y_1/X_1 \\ar[rd] & & Y_2/X_2 \\ar[ld] \\\\", "& Y'/X' \\ar[d] \\\\", "& Y_{n, d}/X_{n, d}", "}", "$$", "and then we can use the already given argument by pulling", "back from $c_{Y_{n, d}/X_{n, d}}$. This finishes the proof." ], "refs": [ "discriminant-lemma-syntomic-quasi-finite", "discriminant-remark-local-description-delta", "discriminant-lemma-different-quasi-finite-complete-intersection", "discriminant-lemma-universal-quasi-finite-syntomic-etale", "discriminant-lemma-different-ramification", "discriminant-lemma-locally-comes-from-universal", "limits-lemma-descend-opens", "limits-lemma-descend-finite-presentation", "schemes-lemma-standard-open-two-affines", "more-algebra-lemma-fibre-product-finite-type", "more-algebra-lemma-module-over-fibre-product", "more-algebra-lemma-flat-module-over-fibre-product", "discriminant-lemma-syntomic-quasi-finite" ], "ref_ids": [ 14981, 15010, 14995, 14985, 14979, 14986, 15041, 15077, 7675, 9814, 9820, 9824, 14981 ] } ], "ref_ids": [] }, { "id": 15014, "type": "theorem", "label": "stacks-limits-lemma-limit-preserving-objects", "categories": [ "stacks-limits" ], "title": "stacks-limits-lemma-limit-preserving-objects", "contents": [ "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a", "$1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "If $f$ is limit preserving (Definition \\ref{definition-limit-preserving}),", "then $f$ is limit preserving on objects (Criteria for Representability, Section", "\\ref{criteria-section-limit-preserving})." ], "refs": [ "stacks-limits-definition-limit-preserving" ], "proofs": [ { "contents": [ "If for every directed limit $U = \\lim U_i$ of affine schemes over $U$,", "the functor", "$$", "\\colim \\mathcal{X}_{U_i} \\longrightarrow", "(\\colim \\mathcal{Y}_{U_i}) \\times_{\\mathcal{Y}_U} \\mathcal{X}_U", "$$", "is essentially surjective, then $f$ is limit preserving on objects." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 15025 ] }, { "id": 15015, "type": "theorem", "label": "stacks-limits-lemma-base-change-limit-preserving", "categories": [ "stacks-limits" ], "title": "stacks-limits-lemma-base-change-limit-preserving", "contents": [ "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$", "be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "If $p : \\mathcal{X} \\to \\mathcal{Y}$ is limit preserving, then so", "is the base change", "$p' : \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$", "of $p$ by $q$." ], "refs": [], "proofs": [ { "contents": [ "This is formal. Let $U = \\lim_{i \\in I} U_i$ be the directed limit", "of affine schemes $U_i$ over $S$. For each $i$ we have", "$$", "(\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z})_{U_i} =", "\\mathcal{X}_{U_i} \\times_{\\mathcal{Y}_{U_i}} \\mathcal{Z}_{U_i}", "$$", "Filtered colimits commute with $2$-fibre products of categories", "(details omitted) hence if $p$ is limit preserving we get", "\\begin{align*}", "\\colim (\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z})_{U_i}", "& =", "\\colim \\mathcal{X}_{U_i} \\times_{\\colim \\mathcal{Y}_{U_i}}", "\\colim \\mathcal{Z}_{U_i} \\\\", "& =", "\\mathcal{X}_U \\times_{\\mathcal{Y}_U} \\colim \\mathcal{Y}_{U_i}", "\\times_{\\colim \\mathcal{Y}_{U_i}}", "\\colim \\mathcal{Z}_{U_i} \\\\", "& =", "\\mathcal{X}_U \\times_{\\mathcal{Y}_U} \\colim \\mathcal{Z}_{U_i} \\\\", "& =", "\\mathcal{X}_U \\times_{\\mathcal{Y}_U} \\mathcal{Z}_U \\times_{\\mathcal{Z}_U}", "\\colim \\mathcal{Z}_{U_i} \\\\", "& =", "(\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z})_U \\times_{\\mathcal{Z}_U}", "\\colim \\mathcal{Z}_{U_i}", "\\end{align*}", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 15016, "type": "theorem", "label": "stacks-limits-lemma-composition-limit-preserving", "categories": [ "stacks-limits" ], "title": "stacks-limits-lemma-composition-limit-preserving", "contents": [ "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Y} \\to \\mathcal{Z}$", "be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "If $p$ and $q$ are limit preserving, then so is the composition $q \\circ p$." ], "refs": [], "proofs": [ { "contents": [ "This is formal. Let $U = \\lim_{i \\in I} U_i$ be the directed limit", "of affine schemes $U_i$ over $S$. If $p$ and $q$ are limit preserving we get", "\\begin{align*}", "\\colim \\mathcal{X}_{U_i}", "& =", "\\mathcal{X}_U \\times_{\\mathcal{Y}_U} \\colim \\mathcal{Y}_{U_i} \\\\", "& =", "\\mathcal{X}_U \\times_{\\mathcal{Y}_U} \\mathcal{Y}_U", "\\times_{\\mathcal{Z}_U} \\colim \\mathcal{Z}_{U_i} \\\\", "& =", "\\mathcal{X}_U \\times_{\\mathcal{Z}_U} \\colim \\mathcal{Z}_{U_i}", "\\end{align*}", "as desired." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 15017, "type": "theorem", "label": "stacks-limits-lemma-representable-by-spaces-limit-preserving", "categories": [ "stacks-limits" ], "title": "stacks-limits-lemma-representable-by-spaces-limit-preserving", "contents": [ "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ is", "representable by algebraic spaces, then the following are equivalent:", "\\begin{enumerate}", "\\item $p$ is limit preserving,", "\\item $p$ is limit preserving on objects, and", "\\item $p$ is locally of finite presentation (see", "Algebraic Stacks,", "Definition \\ref{algebraic-definition-relative-representable-property}).", "\\end{enumerate}" ], "refs": [ "algebraic-definition-relative-representable-property" ], "proofs": [ { "contents": [ "In Criteria for Representability, Lemma", "\\ref{criteria-lemma-representable-by-spaces-limit-preserving}", "we have seen that (2) and (3) are equivalent.", "Thus it suffices to show that (1) and (2) are equivalent.", "One direction we saw in Lemma \\ref{lemma-limit-preserving-objects}.", "For the other direction, let $U = \\lim_{i \\in I} U_i$ be the directed limit", "of affine schemes $U_i$ over $S$. We have to show that", "$$", "\\colim \\mathcal{X}_{U_i} \\longrightarrow", "\\mathcal{X}_U \\times_{\\mathcal{Y}_U} \\colim \\mathcal{Y}_{U_i}", "$$", "is an equivalence. Since we are assuming (2) we know that it is essentially", "surjective. Hence we need to prove it is fully faithful.", "Since $p$ is faithful on fibre categories", "(Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids})", "we see that the functor is faithful. Let $x_i$ and $x'_i$ be objects", "in the fibre category of $\\mathcal{X}$ over $U_i$.", "The functor above sends $x_i$ to $(x_i|_U, p(x_i), can)$ where", "$can$ is the canonical isomorphism $p(x_i|_U) \\to p(x_i)|_U$.", "Thus we assume given a morphism", "$$", "(\\alpha, \\beta_i) : (x_i|_U, p(x_i), can) \\longrightarrow", "(x'_i|_U, p(x'_i), can)", "$$", "in the category of the right hand side of the first displayed arrow", "of this proof. Our task is to produce an $i' \\geq i$ and a morphism", "$x_i|_{U_{i'}} \\to x'_i|_{U_{i'}}$ which maps to", "$(\\alpha, \\beta_i|_{U_{i'}})$.", "\\medskip\\noindent", "Set $y_i = p(x_i)$ and $y'_i = p(x'_i)$.", "By (Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids})", "the functor", "$$", "X_{y_i} : (\\Sch/U_i)^{opp} \\to \\textit{Sets},\\quad", "V/U_i \\mapsto", "\\{(x, \\phi) \\mid x \\in \\Ob(\\mathcal{X}_V), \\phi : f(x) \\to y_i|V\\}/\\cong", "$$", "is an algebraic space over $U_i$ and the same is true for the", "analogously defined functor $X_{y'_i}$. Since (2) is equivalent to (3)", "we see that $X_{y'_i}$ is locally of finite presentation over $U_i$.", "Observe that $(x_i, \\text{id})$ and $(x'_i, \\text{id})$ define", "$U_i$-valued points of $X_{y_i}$ and $X_{y'_i}$.", "There is a transformation of functors", "$$", "\\beta_i : X_{y_i} \\to X_{y'_i},\\quad", "(x/V, \\phi) \\mapsto (x/V, \\beta_i|_V \\circ \\phi)", "$$", "in other words, this is a morphism of algebraic spaces over $U_i$.", "We claim that", "$$", "\\xymatrix{", "U \\ar[d] \\ar[rr] & & U_i \\ar[d]^{(x'_i, \\text{id})} \\\\", "U_i \\ar[r]^{(x_i, \\text{id})} & X_{y_i} \\ar[r]^{\\beta_i} & X_{y'_i}", "}", "$$", "commutes. Namely, this is equivalent to the condition that", "the pairs $(x_i|_U, \\beta_i|_U)$ and $(x'_i|_U, \\text{id})$", "as in the definition of the functor $X_{y'_i}$ are isomorphic.", "And the morphism $\\alpha : x_i|_U \\to x'_i|_U$ exactly produces", "such an isomorphism. Arguing backwards the reader sees that", "if we can find an $i' \\geq i$ such that the diagram", "$$", "\\xymatrix{", "U_{i'} \\ar[d] \\ar[rr] & & U_i \\ar[d]^{(x'_i, \\text{id})} \\\\", "U_i \\ar[r]^{(x_i, \\text{id})} & X_{y_i} \\ar[r]^{\\beta_i} & X_{y'_i}", "}", "$$", "commutes, then we obtain an isomorphism $x_i|_{U_{i'}} \\to x'_i|_{U_{i'}}$", "which is a solution to the problem posed in the preceding paragraph.", "However, the diagonal morphism", "$$", "\\Delta : X_{y'_i} \\to X_{y'_i} \\times_{U_i} X_{y'_i}", "$$", "is locally of finite presentation (Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-diagonal-morphism-finite-type})", "hence the fact that $U \\to U_i$ equalizes the two morphisms to $X_{y'_i}$,", "means that for some $i' \\geq i$ the morphism $U_{i'} \\to U_i$", "equalizes the two morphisms, see", "Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-characterize-locally-finite-presentation}." ], "refs": [ "criteria-lemma-representable-by-spaces-limit-preserving", "stacks-limits-lemma-limit-preserving-objects", "algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids", "algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids", "spaces-morphisms-lemma-diagonal-morphism-finite-type", "spaces-limits-proposition-characterize-locally-finite-presentation" ], "ref_ids": [ 3101, 15014, 8442, 8442, 4847, 4655 ] } ], "ref_ids": [ 8483 ] }, { "id": 15018, "type": "theorem", "label": "stacks-limits-lemma-limit-preserving-diagonal", "categories": [ "stacks-limits" ], "title": "stacks-limits-lemma-limit-preserving-diagonal", "contents": [ "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. The following are equivalent", "\\begin{enumerate}", "\\item the diagonal", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$", "is limit preserving, and", "\\item for every directed limit $U = \\lim U_i$ of affine schemes over $S$", "the functor", "$$", "\\colim \\mathcal{X}_{U_i} \\longrightarrow", "\\mathcal{X}_U \\times_{\\mathcal{Y}_U} \\colim \\mathcal{Y}_{U_i}", "$$", "is fully faithful.", "\\end{enumerate}", "In particular, if $p$ is limit preserving, then $\\Delta$ is too." ], "refs": [], "proofs": [ { "contents": [ "Let $U = \\lim U_i$ be a directed limit of affine schemes over $S$.", "We claim that the functor", "$$", "\\colim \\mathcal{X}_{U_i} \\longrightarrow", "\\mathcal{X}_U \\times_{\\mathcal{Y}_U} \\colim \\mathcal{Y}_{U_i}", "$$", "is fully faithful if and only if the functor", "$$", "\\colim \\mathcal{X}_{U_i} \\longrightarrow", "\\mathcal{X}_U \\times_{(\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X})_U}", "\\colim (\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X})_{U_i}", "$$", "is an equivalence. This will prove the lemma.", "Since", "$(\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X})_U =", "\\mathcal{X}_U \\times_{\\mathcal{Y}_U} \\mathcal{X}_U$", "and", "$(\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X})_{U_i} =", "\\mathcal{X}_{U_i} \\times_{\\mathcal{Y}_{U_i}} \\mathcal{X}_{U_i}$", "this is a purely category theoretic assertion which we discuss", "in the next paragraph.", "\\medskip\\noindent", "Let $\\mathcal{I}$ be a filtered index category.", "Let $(\\mathcal{C}_i)$ and $(\\mathcal{D}_i)$ be systems", "of groupoids over $\\mathcal{I}$.", "Let $p : (\\mathcal{C}_i) \\to (\\mathcal{D}_i)$ be a map", "of systems of groupoids over $\\mathcal{I}$.", "Suppose we have a functor $p : \\mathcal{C} \\to \\mathcal{D}$", "of groupoids and functors", "$f : \\colim \\mathcal{C}_i \\to \\mathcal{C}$ and", "$g : \\colim \\mathcal{D}_i \\to \\mathcal{D}$", "fitting into a commutative diagram", "$$", "\\xymatrix{", "\\colim \\mathcal{C}_i \\ar[d]_p \\ar[r]_f & \\mathcal{C} \\ar[d]^p \\\\", "\\colim \\mathcal{D}_i \\ar[r]^g & \\mathcal{D}", "}", "$$", "Then we claim that", "$$", "A : \\colim \\mathcal{C}_i \\longrightarrow", "\\mathcal{C} \\times_\\mathcal{D} \\colim \\mathcal{D}_i", "$$", "is fully faithful if and only if the functor", "$$", "B : \\colim \\mathcal{C}_i \\longrightarrow", "\\mathcal{C}", "\\times_{\\Delta, \\mathcal{C} \\times_\\mathcal{D} \\mathcal{C}, f \\times_g f}", "\\colim (\\mathcal{C}_i \\times_{\\mathcal{D}_i} \\mathcal{C}_i)", "$$", "is an equivalence. Set $\\mathcal{C}' = \\colim \\mathcal{C}_i$ and", "$\\mathcal{D}' = \\colim \\mathcal{D}_i$.", "Since $2$-fibre products commute with filtered colimits we see that", "$A$ and $B$ become the functors", "$$", "A' : \\mathcal{C}' \\to \\mathcal{C} \\times_\\mathcal{D} \\mathcal{D}'", "\\quad\\text{and}\\quad", "B' : \\mathcal{C}' \\longrightarrow", "\\mathcal{C}", "\\times_{\\Delta, \\mathcal{C} \\times_\\mathcal{D} \\mathcal{C}, f \\times_g f}", "(\\mathcal{C}' \\times_{\\mathcal{D}'} \\mathcal{C}')", "$$", "Thus it suffices to prove that if", "$$", "\\xymatrix{", "\\mathcal{C}' \\ar[d]_p \\ar[r]_f & \\mathcal{C} \\ar[d]^p \\\\", "\\mathcal{D}' \\ar[r]^g & \\mathcal{D}", "}", "$$", "is a commutative diagram of groupoids, then $A'$ is fully faithful", "if and only if $B'$ is an equivalence. This follows from", "Categories, Lemma \\ref{categories-lemma-fully-faithful-diagonal-equivalence}", "(with trivial, i.e., punctual, base category) because", "$$", "\\mathcal{C}", "\\times_{\\Delta, \\mathcal{C} \\times_\\mathcal{D} \\mathcal{C}, f \\times_g f}", "(\\mathcal{C}' \\times_{\\mathcal{D}'} \\mathcal{C}') =", "\\mathcal{C}'", "\\times_{A', \\mathcal{C} \\times_\\mathcal{D} \\mathcal{D}', A'}", "\\mathcal{C}'", "$$", "This finishes the proof." ], "refs": [ "categories-lemma-fully-faithful-diagonal-equivalence" ], "ref_ids": [ 12298 ] } ], "ref_ids": [] }, { "id": 15019, "type": "theorem", "label": "stacks-limits-lemma-locally-finite-presentation-limit-preserving", "categories": [ "stacks-limits" ], "title": "stacks-limits-lemma-locally-finite-presentation-limit-preserving", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack", "over $S$. If $\\mathcal{X} \\to S$ is locally of finite presentation,", "then $\\mathcal{X}$ is limit preserving in the sense of", "Artin's Axioms, Definition \\ref{artin-definition-limit-preserving}", "(equivalently: the morphism $\\mathcal{X} \\to S$ is limit preserving)." ], "refs": [ "artin-definition-limit-preserving" ], "proofs": [ { "contents": [ "Choose a surjective smooth morphism $U \\to \\mathcal{X}$ for some scheme $U$.", "Then $U \\to S$ is locally of finite presentation, see", "Morphisms of Stacks, Section", "\\ref{stacks-morphisms-section-finite-presentation}.", "We can write $\\mathcal{X} = [U/R]$ for some smooth groupoid in", "algebraic spaces $(U, R, s, t, c)$, see", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-stack-presentation}.", "Since $U$ is locally of finite presentation over $S$", "it follows that the algebraic space $R$ is", "locally of finite presentation over $S$.", "Recall that $[U/R]$ is the stack in groupoids over $(\\Sch/S)_{fppf}$", "obtained by stackyfying the category fibred in groupoids", "whose fibre category over $T$ is the groupoid $(U(T), R(T), s, t, c)$.", "Since $U$ and $R$ are limit preserving as functors", "(Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-characterize-locally-finite-presentation})", "this category fibred in groupoids is limit preserving.", "Thus it suffices to show that fppf stackyfication preserves", "the property of being limit preserving. This is true", "(hint: use Topologies, Lemma", "\\ref{topologies-lemma-limit-fppf-topology}).", "However, we give a direct proof below using that in this", "case we know what the stackyfication amounts to.", "\\medskip\\noindent", "Let $T = \\lim T_\\lambda$ be a directed limit of affine schemes over $S$.", "We have to show that the functor", "$$", "\\colim [U/R]_{T_\\lambda} \\longrightarrow [U/R]_T", "$$", "is an equivalence of categories. Let us show this functor is", "essentially surjective. Let $x \\in \\Ob([U/R]_T)$. In", "Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-quotient-stack-objects}", "the reader finds a description of the category $[U/R]_T$.", "In particular $x$ corresponds to an fppf covering", "$\\{T_i \\to T\\}_{i \\in I}$ and a $[U/R]$-descent datum", "$(u_i, r_{ij})$ relative to this covering.", "After refining this covering we may assume it is a standard", "fppf covering of the affine scheme $T$.", "By Topologies, Lemma", "\\ref{topologies-lemma-limit-fppf-topology}", "we may choose a $\\lambda$ and a standard fppf covering", "$\\{T_{\\lambda, i} \\to T_\\lambda\\}_{i \\in I}$ whose base change to $T$", "is equal to $\\{T_i \\to T\\}_{i \\in I}$.", "For each $i$, after increasing $\\lambda$, we can find", "a $u_{\\lambda, i} : T_{\\lambda, i} \\to U$ whose composition", "with $T_i \\to T_{\\lambda, i}$ is the given morphism $u_i$", "(this is where we use that $U$ is limit preserving).", "Similarly, for each $i, j$, after increasing $\\lambda$, we can find", "a $r_{\\lambda, ij} : T_{\\lambda, i} \\times_{T_\\lambda} T_{\\lambda, j} \\to R$", "whose composition with $T_{ij} \\to T_{\\lambda, ij}$ is the given morphism", "$r_{ij}$ (this is where we use that $R$ is limit preserving).", "After increasing $\\lambda$ we can further assume that", "$$", "s \\circ r_{\\lambda, ij} = u_{\\lambda, i} \\circ \\text{pr}_0", "\\quad\\text{and}\\quad", "t \\circ r_{\\lambda, ij} = u_{\\lambda, j} \\circ \\text{pr}_1,", "$$", "and", "$$", "c \\circ (r_{\\lambda, jk} \\circ \\text{pr}_{12}, r_{\\lambda, ij}", "\\circ \\text{pr}_{01}) = r_{\\lambda, ik} \\circ \\text{pr}_{02}.", "$$", "In other words, we may assume that $(u_{\\lambda, i}, r_{\\lambda, ij})$", "is a $[U/R]$-descent datum relative to the covering", "$\\{T_{\\lambda, i} \\to T_\\lambda\\}_{i \\in I}$.", "Then we obtain a corresponding object of $[U/R]$ over $T_\\lambda$", "whose pullback to $T$ is isomorphic to $x$ as desired.", "The proof of fully faithfulness works in exactly the same", "way using the description of morphisms", "in the fibre categories of $[U/T]$ given in", "Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-quotient-stack-objects}." ], "refs": [ "algebraic-lemma-stack-presentation", "spaces-limits-proposition-characterize-locally-finite-presentation", "topologies-lemma-limit-fppf-topology", "spaces-groupoids-lemma-quotient-stack-objects", "topologies-lemma-limit-fppf-topology", "spaces-groupoids-lemma-quotient-stack-objects" ], "ref_ids": [ 8474, 4655, 12518, 9328, 12518, 9328 ] } ], "ref_ids": [ 11420 ] }, { "id": 15020, "type": "theorem", "label": "stacks-limits-lemma-eventually-separated", "categories": [ "stacks-limits" ], "title": "stacks-limits-lemma-eventually-separated", "contents": [ "In Situation \\ref{situation-descent} assume that $\\mathcal{X}_0 \\to Y_0$", "is a morphism from algebraic stack to $Y_0$. Assume $\\mathcal{X}_0$", "is quasi-compact and quasi-separated.", "If $Y \\times_{Y_0} \\mathcal{X}_0 \\to Y$ is separated, then", "$Y_i \\times_{Y_0} \\mathcal{X}_0 \\to Y_i$ is separated for all", "sufficiently large $i \\in I$." ], "refs": [], "proofs": [ { "contents": [ "Write $\\mathcal{X} = Y \\times_{Y_0} \\mathcal{X}_0$ and", "$\\mathcal{X}_i = Y_i \\times_{Y_0} \\mathcal{X}_0$.", "Choose an affine scheme $U_0$ and a surjective smooth morphism", "$U_0 \\to \\mathcal{X}_0$. Set $U = Y \\times_{Y_0} U_0$", "and $U_i = Y_i \\times_{Y_0} U_0$. Then $U$ and $U_i$ are", "affine and $U \\to \\mathcal{X}$ and $U_i \\to \\mathcal{X}_i$", "are smooth and surjective. Set $R_0 = U_0 \\times_{\\mathcal{X}_0} U_0$.", "Set $R = Y \\times_{Y_0} R_0$ and $R_i = Y_i \\times_{Y_0} R_0$.", "Then $R = U \\times_\\mathcal{X} U$ and $R_i = U_i \\times_{\\mathcal{X}_i} U_i$.", "\\medskip\\noindent", "With this notation note that $\\mathcal{X} \\to Y$ is separated", "implies that $R \\to U \\times_Y U$ is proper as the base change", "of $\\mathcal{X} \\to \\mathcal{X} \\times_Y \\mathcal{X}$", "by $U \\times_Y U \\to \\mathcal{X} \\times_Y \\mathcal{X}$.", "Conversely, we see that $\\mathcal{X}_i \\to Y_i$ is separated", "if $R_i \\to U_i \\times_{Y_i} U_i$ is proper because", "$U_i \\times_{Y_i} U_i \\to \\mathcal{X}_i \\times_{Y_i} \\mathcal{X}_i$", "is surjective and smooth, see", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}.", "Observe that $R_0 \\to U_0 \\times_{Y_0} U_0$", "is locally of finite type and that $R_0$ is", "quasi-compact and quasi-separated.", "By Limits of Spaces, Lemma \\ref{spaces-limits-lemma-eventually-proper}", "we see that $R_i \\to U_i \\times_{Y_i} U_i$ is", "proper for large enough $i$ which finishes the proof." ], "refs": [ "stacks-properties-lemma-check-property-covering", "spaces-limits-lemma-eventually-proper" ], "ref_ids": [ 8859, 4596 ] } ], "ref_ids": [] }, { "id": 15021, "type": "theorem", "label": "stacks-limits-lemma-descend-a-stack-down", "categories": [ "stacks-limits" ], "title": "stacks-limits-lemma-descend-a-stack-down", "contents": [ "Let $I$ be a directed set. Let $(X_i, f_{ii'})$ be an inverse system", "of algebraic spaces over $I$. Assume", "\\begin{enumerate}", "\\item the morphisms $f_{ii'} : X_i \\to X_{i'}$ are affine,", "\\item the spaces $X_i$ are quasi-compact and quasi-separated.", "\\end{enumerate}", "Let $X = \\lim X_i$.", "If $\\mathcal{X}$ is an algebraic stack of finite presentation over $X$,", "then there exists an $i \\in I$ and an algebraic stack $\\mathcal{X}_i$", "of finite presentation over $X_i$ with", "$\\mathcal{X} \\cong \\mathcal{X}_i \\times_{X_i} X$ as", "algebraic stacks over $X$." ], "refs": [], "proofs": [ { "contents": [ "By Morphisms of Stacks, Definition", "\\ref{stacks-morphisms-definition-locally-finite-presentation}", "the morphism $\\mathcal{X} \\to X$ is", "quasi-compact, locally of finite presentation, and quasi-separated.", "Since $X$ is quasi-compact and $\\mathcal{X} \\to X$ is quasi-compact,", "we see that $\\mathcal{X}$ is quasi-compact", "(Morphisms of Stacks, Definition", "\\ref{stacks-morphisms-definition-quasi-compact}).", "Hence we can find an affine", "scheme $U$ and a surjective smooth morphism $U \\to \\mathcal{X}$", "(Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-quasi-compact-stack}).", "Set $R = U \\times_\\mathcal{X} U$. We obtain a smooth groupoid", "in algebraic spaces $(U, R, s, t, c)$ over $X$ such that", "$\\mathcal{X} = [U/R]$, see Algebraic Stacks, Lemma", "\\ref{algebraic-lemma-stack-presentation}.", "Since $\\mathcal{X} \\to X$ is quasi-separated and $X$ is quasi-separated", "we see that $\\mathcal{X}$ is quasi-separated (Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-composition-separated}).", "Thus $R \\to U \\times U$ is quasi-compact and quasi-separated", "(Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-fibre-product-after-map})", "and hence $R$ is a quasi-separated and quasi-compact algebraic space.", "On the other hand $U \\to X$ is locally of finite presentation", "and hence also $R \\to X$ is locally of finite presentation", "(because $s : R \\to U$ is smooth hence locally of finite presentation).", "Thus $(U, R, s, t, c)$ is a groupoid object in the category", "of algebraic spaces which are of finite presentation over $X$.", "By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-descend-finite-presentation}", "there exists an $i$ and a groupoid in algebraic spaces", "$(U_i, R_i, s_i, t_i, c_i)$ over $X_i$ whose pullback", "to $X$ is isomorphic to $(U, R, s, t, c)$.", "After increasing $i$ we may assume that", "$s_i$ and $t_i$ are smooth, see", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-descend-smooth}.", "The quotient stack $\\mathcal{X}_i = [U_i/R_i]$", "is an algebraic stack (Algebraic Stacks, Theorem", "\\ref{algebraic-theorem-smooth-groupoid-gives-algebraic-stack}).", "\\medskip\\noindent", "There is a morphism $[U/R] \\to [U_i/R_i]$, see", "Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-functorial}.", "We claim that combined with the morphisms", "$[U/R] \\to X$ and $[U_i/R_i] \\to X_i$", "(Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-quotient-stack-arrows})", "we obtain an isomorphism (i.e., equivalence)", "$$", "[U/R] \\longrightarrow [U_i/R_i] \\times_{X_i} X", "$$", "The corresponding map", "$$", "[U/_{\\!p}R] \\longrightarrow [U_i/_{\\!p}R_i] \\times_{X_i} X", "$$", "on the level of ``presheaves of groupoids'' as in", "Groupoids in Spaces, Equation (\\ref{spaces-groupoids-equation-quotient-stack})", "is an isomorphism. Thus the claim follows from the fact that", "stackification commutes with fibre products, see Stacks, Lemma", "\\ref{stacks-lemma-stackification-fibre-product-fibred-categories}." ], "refs": [ "stacks-morphisms-definition-locally-finite-presentation", "stacks-morphisms-definition-quasi-compact", "stacks-properties-lemma-quasi-compact-stack", "algebraic-lemma-stack-presentation", "stacks-morphisms-lemma-composition-separated", "stacks-morphisms-lemma-fibre-product-after-map", "spaces-limits-lemma-descend-finite-presentation", "spaces-limits-lemma-descend-smooth", "algebraic-theorem-smooth-groupoid-gives-algebraic-stack", "spaces-groupoids-lemma-quotient-stack-functorial", "spaces-groupoids-lemma-quotient-stack-arrows", "stacks-lemma-stackification-fibre-product-fibred-categories" ], "ref_ids": [ 7620, 7604, 8873, 8474, 7404, 7401, 4598, 4586, 8435, 9321, 9319, 8964 ] } ], "ref_ids": [] }, { "id": 15022, "type": "theorem", "label": "stacks-limits-lemma-finite-type-closed-in-finite-presentation", "categories": [ "stacks-limits" ], "title": "stacks-limits-lemma-finite-type-closed-in-finite-presentation", "contents": [ "Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack", "to an algebraic space. Assume:", "\\begin{enumerate}", "\\item $f$ is of finite type and quasi-separated,", "\\item $Y$ is quasi-compact and quasi-separated.", "\\end{enumerate}", "Then there exists a morphism of finite presentation", "$f' : \\mathcal{X}' \\to Y$ and a closed immersion", "$\\mathcal{X} \\to \\mathcal{X}'$ of", "algebraic stacks over $Y$." ], "refs": [], "proofs": [ { "contents": [ "Write $Y = \\lim_{i \\in I} Y_i$ as a limit of algebraic spaces", "over a directed set $I$ with affine transition morphisms and", "with $Y_i$ Noetherian, see", "Limits of Spaces, Proposition \\ref{spaces-limits-proposition-approximate}.", "We will use the material from", "Limits of Spaces, Section", "\\ref{spaces-limits-section-finite-type-quasi-separated}.", "\\medskip\\noindent", "Choose a presentation $\\mathcal{X} = [U/R]$.", "Denote $(U, R, s, t, c, e, i)$ the corresponding groupoid", "in algebraic spaces over $Y$. We may and do assume $U$ is affine.", "Then $U$, $R$, $R \\times_{s, U, t} R$ are quasi-separated", "algebraic spaces of finite type over $Y$. We have two morpisms", "$s, t : R \\to U$, three morphisms", "$c : R \\times_{s, U, t} R \\to R$,", "$\\text{pr}_1 : R \\times_{s, U, t} R \\to R$,", "$\\text{pr}_2 : R \\times_{s, U, t} R \\to R$,", "a morphism $e : U \\to R$, and finally a morphism $i : R \\to R$.", "These morphisms satisfy a list of axioms which are detailed", "in Groupoids, Section \\ref{groupoids-section-groupoids}.", "\\medskip\\noindent", "According to Limits of Spaces, Remark", "\\ref{spaces-limits-remark-finite-type-gives-well-defined-system}", "we can find an $i_0 \\in I$ and inverse systems", "\\begin{enumerate}", "\\item $(U_i)_{i \\geq i_0}$,", "\\item $(R_i)_{i \\geq i_0}$,", "\\item $(T_i)_{i \\geq i_0}$", "\\end{enumerate}", "over $(Y_i)_{i \\geq i_0}$ such that", "$U = \\lim_{i \\geq i_0} U_i$,", "$R = \\lim_{i \\geq i_0} R_i$, and", "$R \\times_{s, U, t} R = \\lim_{i \\geq i_0} T_i$", "and such that there exist morphisms of systems", "\\begin{enumerate}", "\\item $(s_i)_{i \\geq i_0} : (R_i)_{i \\geq i_0} \\to (U_i)_{i \\geq i_0}$,", "\\item $(t_i)_{i \\geq i_0} : (R_i)_{i \\geq i_0} \\to (U_i)_{i \\geq i_0}$,", "\\item $(c_i)_{i \\geq i_0} : (T_i)_{i \\geq i_0} \\to (R_i)_{i \\geq i_0}$,", "\\item $(p_i)_{i \\geq i_0} : (T_i)_{i \\geq i_0} \\to (R_i)_{i \\geq i_0}$,", "\\item $(q_i)_{i \\geq i_0} : (T_i)_{i \\geq i_0} \\to (R_i)_{i \\geq i_0}$,", "\\item $(e_i)_{i \\geq i_0} : (U_i)_{i \\geq i_0} \\to (R_i)_{i \\geq i_0}$,", "\\item $(i_i)_{i \\geq i_0} : (R_i)_{i \\geq i_0} \\to (R_i)_{i \\geq i_0}$", "\\end{enumerate}", "with", "$s = \\lim_{i \\geq i_0} s_i$,", "$t = \\lim_{i \\geq i_0} t_i$,", "$c = \\lim_{i \\geq i_0} c_i$,", "$\\text{pr}_1 = \\lim_{i \\geq i_0} p_i$,", "$\\text{pr}_2 = \\lim_{i \\geq i_0} q_i$,", "$e = \\lim_{i \\geq i_0} e_i$, and", "$i = \\lim_{i \\geq i_0} i_i$.", "By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-morphism-good-diagram-smooth}", "we see that we may assume that $s_i$ and $t_i$ are smooth", "(this may require increasing $i_0$).", "By Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-morphism-good-diagram-flat}", "we may assume that the maps", "$R \\to U \\times_{U_i, s_i} R_i$ given by $s$ and $R \\to R_i$ and", "$R \\to U \\times_{U_i, t_i} R_i$ given by $t$ and $R \\to R_i$", "are isomorphisms for all $i \\geq i_0$.", "By Limits of Spaces, Lemma \\ref{spaces-limits-lemma-good-diagram-fibre-product}", "we see that we may assume that the diagrams", "$$", "\\xymatrix{", "T_i \\ar[r]_{q_i} \\ar[d]_{p_i} & R_i \\ar[d]^{t_i} \\\\", "R_i \\ar[r]^{s_i} & U_i", "}", "$$", "are cartesian. The uniqueness of", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-morphism-good-diagram}", "then guarantees that for a sufficiently large $i$", "the relations between the morphisms $s, t, c, e, i$ mentioned above", "are satisfied by $s_i, t_i, c_i, e_i, i_i$. Fix such an $i$.", "\\medskip\\noindent", "It follows that $(U_i, R_i, s_i, t_i, c_i, e_i, i_i)$", "is a smooth groupoid in algebraic spaces over $Y_i$.", "Hence $\\mathcal{X}_i = [U_i/R_i]$ is an algebraic stack", "(Algebraic Stacks, Theorem", "\\ref{algebraic-theorem-smooth-groupoid-gives-algebraic-stack}).", "The morphism of groupoids", "$$", "(U, R, s, t, c, e, i) \\to (U_i, R_i, s_i, t_i, c_i, e_i, i_i)", "$$", "over $Y \\to Y_i$ determines a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{X} \\ar[d] \\ar[r] & \\mathcal{X}_i \\ar[d] \\\\", "Y \\ar[r] & Y_i", "}", "$$", "(Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-quotient-stack-functorial}).", "We claim that the morphism $\\mathcal{X} \\to Y \\times_{Y_i} \\mathcal{X}_i$", "is a closed immersion. The claim finishes the proof because", "the algebraic stack $\\mathcal{X}_i \\to Y_i$ is of finite presentation", "by construction. To prove the claim, note that the left diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & U_i \\ar[d] \\\\", "\\mathcal{X} \\ar[r] & \\mathcal{X}_i", "}", "\\quad\\quad", "\\xymatrix{", "U \\ar[d] \\ar[r] & Y \\times_{Y_i} U_i \\ar[d] \\\\", "\\mathcal{X} \\ar[r] & Y \\times_{Y_i} \\mathcal{X}_i", "}", "$$", "is cartesian by Groupoids in Spaces, Lemma", "\\ref{spaces-groupoids-lemma-criterion-fibre-product}", "and the results mentioned above.", "Hence the right commutative diagram is cartesian too.", "Then the desired result follows from the fact that", "$U \\to Y \\times_{Y_i} U_i$ is a closed immersion", "by construction of the inverse system $(U_i)$ in", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-limit-from-good-diagram},", "the fact that $Y \\times_{Y_i} U_i \\to Y \\times_{Y_i} \\mathcal{X}_i$", "is smooth and surjective, and Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-immersion-covering}." ], "refs": [ "spaces-limits-proposition-approximate", "spaces-limits-remark-finite-type-gives-well-defined-system", "spaces-limits-lemma-morphism-good-diagram-smooth", "spaces-limits-lemma-morphism-good-diagram-flat", "spaces-limits-lemma-good-diagram-fibre-product", "spaces-limits-lemma-morphism-good-diagram", "algebraic-theorem-smooth-groupoid-gives-algebraic-stack", "spaces-groupoids-lemma-quotient-stack-functorial", "spaces-groupoids-lemma-criterion-fibre-product", "spaces-limits-lemma-limit-from-good-diagram", "stacks-properties-lemma-check-immersion-covering" ], "ref_ids": [ 4656, 4665, 4652, 4651, 4654, 4650, 8435, 9321, 9331, 4649, 8884 ] } ], "ref_ids": [] }, { "id": 15023, "type": "theorem", "label": "stacks-limits-lemma-separated-closed-in-finite-presentation", "categories": [ "stacks-limits" ], "title": "stacks-limits-lemma-separated-closed-in-finite-presentation", "contents": [ "Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack", "to an algebraic space. Assume:", "\\begin{enumerate}", "\\item $f$ is of finite type and separated,", "\\item $Y$ is quasi-compact and quasi-separated.", "\\end{enumerate}", "Then there exists a separated morphism of finite presentation", "$f' : \\mathcal{X}' \\to Y$ and a closed immersion", "$\\mathcal{X} \\to \\mathcal{X}'$ of", "algebraic stacks over $Y$." ], "refs": [], "proofs": [ { "contents": [ "First we use exactly the same procedure as in the proof of", "Lemma \\ref{lemma-finite-type-closed-in-finite-presentation}", "(and we borrow its notation)", "to construct the embedding $\\mathcal{X} \\to \\mathcal{X}'$ as a morphism", "$\\mathcal{X} \\to \\mathcal{X}' = Y \\times_{Y_i} \\mathcal{X}_i$ with", "$\\mathcal{X}_i = [U_i/R_i]$.", "Thus it is enough to show that $\\mathcal{X}_i \\to Y_i$ is", "separated for sufficiently large $i$.", "In other words, it is enough to show that", "$\\mathcal{X}_i \\to \\mathcal{X}_i \\times_{Y_i} \\mathcal{X}_i$", "is proper for $i$ sufficiently large. Since the morphism", "$U_i \\times_{Y_i} U_i \\to \\mathcal{X}_i \\times_{Y_i} \\mathcal{X}_i$", "is surjective and smooth and since", "$R_i = \\mathcal{X}_i", "\\times_{\\mathcal{X}_i \\times_{Y_i} \\mathcal{X}_i} U_i \\times_{Y_i} U_i$", "it is enough to show that the morphism", "$(s_i, t_i) : R_i \\to U_i \\times_{Y_i} U_i$", "is proper for $i$ sufficiently large, see", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-check-property-covering}.", "We prove this in the next paragraph.", "\\medskip\\noindent", "Observe that $U \\times_Y U \\to Y$ is quasi-separated and of finite type.", "Hence we can use the construction of ", "Limits of Spaces, Remark", "\\ref{spaces-limits-remark-finite-type-gives-well-defined-system}", "to find an $i_1 \\in I$ and an inverse system $(V_i)_{i \\geq i_1}$", "with $U \\times_Y U = \\lim_{i \\geq i_1} V_i$.", "By Limits of Spaces, Lemma \\ref{spaces-limits-lemma-good-diagram-fibre-product}", "for $i$ sufficiently large the functoriality of the construction", "applied to the projections $U \\times_Y U \\to U$", "gives closed immersions", "$$", "V_i \\to U_i \\times_{Y_i} U_i", "$$", "(There is a small mismatch here because in truth we should replace", "$Y_i$ by the scheme theoretic image of $Y \\to Y_i$, but clearly this", "does not change the fibre product.)", "On the other hand, by Limits of Spaces, Lemma", "\\ref{spaces-limits-lemma-morphism-good-diagram-proper}", "the functoriality applied to the proper morphism", "$(s, t) : R \\to U \\times_Y U$ (here we use that $\\mathcal{X}$ is separated)", "leads to morphisms $R_i \\to V_i$ which are proper for", "large enough $i$.", "Composing these morphisms we obtain a proper morphisms", "$R_i \\to U_i \\times_{Y_i} U_i$ for all $i$ large enough.", "The functoriality of the construction of", "Limits of Spaces, Remark", "\\ref{spaces-limits-remark-finite-type-gives-well-defined-system}", "shows that this is the morphism is the same as $(s_i, t_i)$", "for large enough $i$ and the proof is complete." ], "refs": [ "stacks-limits-lemma-finite-type-closed-in-finite-presentation", "stacks-properties-lemma-check-property-covering", "spaces-limits-remark-finite-type-gives-well-defined-system", "spaces-limits-lemma-good-diagram-fibre-product", "spaces-limits-lemma-morphism-good-diagram-proper", "spaces-limits-remark-finite-type-gives-well-defined-system" ], "ref_ids": [ 15022, 8859, 4665, 4654, 4653, 4665 ] } ], "ref_ids": [] }, { "id": 15024, "type": "theorem", "label": "stacks-limits-proposition-characterize-locally-finite-presentation", "categories": [ "stacks-limits" ], "title": "stacks-limits-proposition-characterize-locally-finite-presentation", "contents": [ "\\begin{reference}", "This is a special case of \\cite[Lemma 2.3.15]{Emerton-Gee}", "\\end{reference}", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is limit preserving,", "\\item $f$ is limit preserving on objects, and", "\\item $f$ is locally of finite presentation.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Assume (3). Let $T = \\lim T_i$ be a directed limit of affine schemes.", "Consider the functor", "$$", "\\colim \\mathcal{X}_{T_i} \\longrightarrow", "\\mathcal{X}_T \\times_{\\mathcal{Y}_T} \\colim \\mathcal{Y}_{T_i}", "$$", "Let $(x, y_i, \\beta)$ be an object on the right hand side, i.e.,", "$x \\in \\Ob(\\mathcal{X}_T)$, $y_i \\in \\Ob(\\mathcal{Y}_{T_i})$, and", "$\\beta : f(x) \\to y_i|_T$ in $\\mathcal{Y}_T$.", "Then we can consider $(x, y_i, \\beta)$ as an object of the", "algebraic stack $\\mathcal{X}_{y_i} = \\mathcal{X} \\times_{\\mathcal{Y}, y_i} T_i$", "over $T$. Since $\\mathcal{X}_{y_i} \\to T_i$ is locally of finite presentation", "(as a base change of $f$) we see that it is limit preserving", "by Lemma \\ref{lemma-locally-finite-presentation-limit-preserving}.", "This means that $(x, y_i, \\beta)$ comes from an object over $T_{i'}$", "for some $i' \\geq i$ and unwinding the definitions we find that", "$(x, y_i, \\beta)$ is in the essential image of the displayed functor.", "In other words, the displayed functor is essentially surjective.", "Another formulation is that this means", "$f$ is limit preserving on objects.", "Now we apply this to the diagonal $\\Delta$ of $f$. Namely,", "by Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-diagonal-morphism-finite-type}", "the morphism $\\Delta$ is locally of finite presentation.", "Thus the argument above shows that $\\Delta$ is limit preserving on objects.", "By Lemma \\ref{lemma-representable-by-spaces-limit-preserving}", "this implies that $\\Delta$ is limit preserving.", "By Lemma \\ref{lemma-limit-preserving-diagonal}", "we conclude that the displayed functor above is fully faithful.", "Thus it is an equivalence (as we already proved essential surjectivity)", "and we conclude that (1) holds.", "\\medskip\\noindent", "The implication (1) $\\Rightarrow$ (2) is trivial. Assume (2).", "Choose a scheme $V$ and a surjective smooth morphism $V \\to \\mathcal{Y}$.", "By Criteria for Representability, Lemma", "\\ref{criteria-lemma-base-change-limit-preserving}", "the base change $\\mathcal{X} \\times_\\mathcal{Y} V \\to V$", "is limit preserving on objects.", "Choose a scheme $U$ and a surjective smooth morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$.", "Since a smooth morphism is locally of finite presentation,", "we see that $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ is", "limit preserving (first part of the proof).", "By Criteria for Representability, Lemma", "\\ref{criteria-lemma-composition-limit-preserving}", "we find that the composition $U \\to V$ is limit preserving", "on objects. We conclude that $U \\to V$ is locally of finite presentation, see", "Criteria for Representability, Lemma", "\\ref{criteria-lemma-representable-by-spaces-limit-preserving}.", "This is exactly the condition that $f$ is locally of finite presentation, see", "Morphisms of Stacks, Definition", "\\ref{stacks-morphisms-definition-locally-finite-presentation}." ], "refs": [ "stacks-limits-lemma-locally-finite-presentation-limit-preserving", "stacks-morphisms-lemma-diagonal-morphism-finite-type", "stacks-limits-lemma-representable-by-spaces-limit-preserving", "stacks-limits-lemma-limit-preserving-diagonal", "criteria-lemma-base-change-limit-preserving", "criteria-lemma-composition-limit-preserving", "criteria-lemma-representable-by-spaces-limit-preserving", "stacks-morphisms-definition-locally-finite-presentation" ], "ref_ids": [ 15019, 7505, 15017, 15018, 3099, 3100, 3101, 7620 ] } ], "ref_ids": [] }, { "id": 15026, "type": "theorem", "label": "limits-lemma-directed-inverse-system-affine-schemes-has-limit", "categories": [ "limits" ], "title": "limits-lemma-directed-inverse-system-affine-schemes-has-limit", "contents": [ "Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of", "schemes over $I$. If all the schemes $S_i$", "are affine, then the limit $S = \\lim_i S_i$ exists", "in the category of schemes.", "In fact $S$ is affine and $S = \\Spec(\\colim_i R_i)$", "with $R_i = \\Gamma(S_i, \\mathcal{O})$." ], "refs": [], "proofs": [ { "contents": [ "Just define $S = \\Spec(\\colim_i R_i)$.", "It follows from Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}", "that $S$ is the limit even in the category of locally ringed spaces." ], "refs": [ "schemes-lemma-morphism-into-affine" ], "ref_ids": [ 7655 ] } ], "ref_ids": [] }, { "id": 15027, "type": "theorem", "label": "limits-lemma-directed-inverse-system-has-limit", "categories": [ "limits" ], "title": "limits-lemma-directed-inverse-system-has-limit", "contents": [ "Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an", "inverse system of schemes over $I$. If all the morphisms", "$f_{ii'} : S_i \\to S_{i'}$ are affine, then the limit $S = \\lim_i S_i$ exists", "in the category of schemes. Moreover,", "\\begin{enumerate}", "\\item each of the morphisms $f_i : S \\to S_i$ is affine,", "\\item for an element $0 \\in I$ and any open subscheme $U_0 \\subset S_0$", "we have", "$$", "f_0^{-1}(U_0) = \\lim_{i \\geq 0} f_{i0}^{-1}(U_0)", "$$", "in the category of schemes.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose an element $0 \\in I$. Note that $I$ is nonempty as the limit is", "directed. For every $i \\geq 0$ consider the quasi-coherent sheaf of", "$\\mathcal{O}_{S_0}$-algebras $\\mathcal{A}_i = f_{i0, *}\\mathcal{O}_{S_i}$.", "Recall that $S_i = \\underline{\\Spec}_{S_0}(\\mathcal{A}_i)$,", "see Morphisms, Lemma \\ref{morphisms-lemma-characterize-affine}.", "Set $\\mathcal{A} = \\colim_{i \\geq 0} \\mathcal{A}_i$.", "This is a quasi-coherent sheaf of $\\mathcal{O}_{S_0}$-algebras,", "see Schemes, Section \\ref{schemes-section-quasi-coherent}.", "Set $S = \\underline{\\Spec}_{S_0}(\\mathcal{A})$.", "By Morphisms, Lemma \\ref{morphisms-lemma-affine-equivalence-algebras}", "we get for $i \\geq 0$ morphisms $f_i : S \\to S_i$ compatible with", "the transition morphisms. Note that the morphisms $f_i$ are", "affine by Morphisms, Lemma \\ref{morphisms-lemma-affine-permanence} for example.", "By Lemma \\ref{lemma-directed-inverse-system-affine-schemes-has-limit} above", "we see that for any affine open $U_0 \\subset S_0$ the", "inverse image $U = f_0^{-1}(U_0) \\subset S$ is the limit of the", "system of opens $U_i = f_{i0}^{-1}(U_0)$, $i \\geq 0$ in the", "category of schemes.", "\\medskip\\noindent", "Let $T$ be a scheme. Let $g_i : T \\to S_i$ be a compatible system", "of morphisms. To show that $S = \\lim_i S_i$ we have", "to prove there is a unique morphism $g : T \\to S$ with", "$g_i = f_i \\circ g$ for all $i \\in I$.", "For every $t \\in T$ there exists an affine open", "$U_0 \\subset S_0$ containing $g_0(t)$. Let $V \\subset g_0^{-1}(U_0)$", "be an affine open neighbourhood containing $t$.", "By the remarks above we obtain a unique morphism", "$g_V : V \\to U = f_0^{-1}(U_0)$ such that $f_i \\circ g_V = g_i|_{U_i}$", "for all $i$. The open sets $V \\subset T$ so constructed form", "a basis for the topology of $T$. The morphisms $g_V$ glue to a morphism", "$g : T \\to S$ because of the uniqueness property. This gives the", "desired morphism $g : T \\to S$.", "\\medskip\\noindent", "The final statement is clear from the construction of the limit above." ], "refs": [ "morphisms-lemma-characterize-affine", "morphisms-lemma-affine-equivalence-algebras", "morphisms-lemma-affine-permanence", "limits-lemma-directed-inverse-system-affine-schemes-has-limit" ], "ref_ids": [ 5172, 5173, 5179, 15026 ] } ], "ref_ids": [] }, { "id": 15028, "type": "theorem", "label": "limits-lemma-scheme-over-limit", "categories": [ "limits" ], "title": "limits-lemma-scheme-over-limit", "contents": [ "Let $I$ be a directed set.", "Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$.", "Assume all the morphisms $f_{ii'} : S_i \\to S_{i'}$ are affine,", "Let $S = \\lim_i S_i$. Let $0 \\in I$.", "Suppose that $T$ is a scheme over $S_0$.", "Then", "$$", "T \\times_{S_0} S = \\lim_{i \\geq 0} T \\times_{S_0} S_i", "$$" ], "refs": [], "proofs": [ { "contents": [ "The right hand side is a scheme by", "Lemma \\ref{lemma-directed-inverse-system-has-limit}.", "The equality is formal, see", "Categories, Lemma \\ref{categories-lemma-colimits-commute}." ], "refs": [ "limits-lemma-directed-inverse-system-has-limit", "categories-lemma-colimits-commute" ], "ref_ids": [ 15027, 12212 ] } ], "ref_ids": [] }, { "id": 15029, "type": "theorem", "label": "limits-lemma-infinite-product", "categories": [ "limits" ], "title": "limits-lemma-infinite-product", "contents": [ "\\begin{slogan}", "Infinite products of affine schemes exist and are affine.", "\\end{slogan}", "Let $S$ be a scheme. Let $I$ be a set and for each $i \\in I$", "let $f_i : T_i \\to S$ be an affine morphism. Then the", "product $T = \\prod T_i$ exists in the category of schemes", "over $S$. In fact, we have", "$$", "T = \\lim_{\\{i_1, \\ldots, i_n\\} \\subset I}", "T_{i_1} \\times_S \\ldots \\times_S T_{i_n}", "$$", "and the projection morphisms $T \\to T_{i_1} \\times_S \\ldots \\times_S T_{i_n}$", "are affine." ], "refs": [], "proofs": [ { "contents": [ "Omitted. Hint: Argue as in the discussion preceding the lemma", "and use Lemma \\ref{lemma-directed-inverse-system-has-limit}", "for existence of the limit." ], "refs": [ "limits-lemma-directed-inverse-system-has-limit" ], "ref_ids": [ 15027 ] } ], "ref_ids": [] }, { "id": 15030, "type": "theorem", "label": "limits-lemma-infinite-product-surjective", "categories": [ "limits" ], "title": "limits-lemma-infinite-product-surjective", "contents": [ "Let $S$ be a scheme. Let $I$ be a set and for each $i \\in I$", "let $f_i : T_i \\to S$ be a surjective affine morphism. Then the", "product $T = \\prod T_i$ in the category of schemes over $S$", "(Lemma \\ref{lemma-infinite-product})", "maps surjectively to $S$." ], "refs": [ "limits-lemma-infinite-product" ], "proofs": [ { "contents": [ "Let $s \\in S$. Choose $t_i \\in T_i$ mapping to $s$.", "Choose a huge field extension $K/\\kappa(s)$ such that", "$\\kappa(s_i)$ embeds into $K$ for each $i$. Then we get", "morphisms $\\Spec(K) \\to T_i$ with image $s_i$ agreeing", "as morphisms to $S$. Whence a morphism $\\Spec(K) \\to T$", "which proves there is a point of $T$ mapping to $s$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 15029 ] }, { "id": 15031, "type": "theorem", "label": "limits-lemma-infinite-product-integral", "categories": [ "limits" ], "title": "limits-lemma-infinite-product-integral", "contents": [ "Let $S$ be a scheme. Let $I$ be a set and for each $i \\in I$", "let $f_i : T_i \\to S$ be an integral morphism. Then the", "product $T = \\prod T_i$ in the category of schemes over $S$", "(Lemma \\ref{lemma-infinite-product})", "is integral over $S$." ], "refs": [ "limits-lemma-infinite-product" ], "proofs": [ { "contents": [ "Omitted. Hint: On affine pieces this reduces to the following", "algebra fact: if $A \\to B_i$ is integral for all $i$, then", "$A \\to \\otimes_A B_i$ is integral." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 15029 ] }, { "id": 15032, "type": "theorem", "label": "limits-lemma-inverse-limit-sets", "categories": [ "limits" ], "title": "limits-lemma-inverse-limit-sets", "contents": [ "Let $S = \\lim S_i$ be the limit of a directed inverse system", "of schemes with affine transition morphisms", "(Lemma \\ref{lemma-directed-inverse-system-has-limit}). Then", "$S_{set} = \\lim_i S_{i, set}$ where $S_{set}$", "indicates the underlying set of the scheme $S$." ], "refs": [ "limits-lemma-directed-inverse-system-has-limit" ], "proofs": [ { "contents": [ "Pick $i \\in I$. Take $U_i \\subset S_i$ an affine open.", "Denote $U_{i'} = f_{i'i}^{-1}(U_i)$ and $U = f_i^{-1}(U_i)$.", "Here $f_{i'i} : S_{i'} \\to S_i$ is the transtion morphism", "and $f_i : S \\to S_i$ is the projection.", "By Lemma \\ref{lemma-directed-inverse-system-has-limit}", "we have $U = \\lim_{i' \\geq i} U_i$.", "Suppose we can show that $U_{set} = \\lim_{i' \\geq i} U_{i', set}$. Then", "the lemma follows by a simple argument using an affine covering of $S_i$.", "Hence we may assume all $S_i$ and $S$ affine. This reduces us to the", "algebra question considered in the next paragraph.", "\\medskip\\noindent", "Suppose given a system of rings $(A_i, \\varphi_{ii'})$", "over $I$. Set $A = \\colim_i A_i$ with canonical maps $\\varphi_i : A_i \\to A$.", "Then", "$$", "\\Spec(A) = \\lim_i \\Spec(A_i)", "$$", "Namely, suppose that we are given primes $\\mathfrak p_i \\subset A_i$", "such that $\\mathfrak p_i = \\varphi_{ii'}^{-1}(\\mathfrak p_{i'})$", "for all $i' \\geq i$. Then we simply set", "$$", "\\mathfrak p =", "\\{x \\in A", "\\mid", "\\exists i, x_i \\in \\mathfrak p_i \\text{ with }\\varphi_i(x_i) = x\\}", "$$", "It is clear that this is an ideal and has the property that", "$\\varphi_i^{-1}(\\mathfrak p) = \\mathfrak p_i$. Then it follows", "easily that it is a prime ideal as well." ], "refs": [ "limits-lemma-directed-inverse-system-has-limit" ], "ref_ids": [ 15027 ] } ], "ref_ids": [ 15027 ] }, { "id": 15033, "type": "theorem", "label": "limits-lemma-inverse-limit-top", "categories": [ "limits" ], "title": "limits-lemma-inverse-limit-top", "contents": [ "\\begin{reference}", "\\cite[IV, Proposition 8.2.9]{EGA}", "\\end{reference}", "Let $S = \\lim S_i$ be the limit of a directed inverse system", "of schemes with affine transition morphisms", "(Lemma \\ref{lemma-directed-inverse-system-has-limit}). Then", "$S_{top} = \\lim_i S_{i, top}$ where $S_{top}$", "indicates the underlying topological space of the scheme $S$." ], "refs": [ "limits-lemma-directed-inverse-system-has-limit" ], "proofs": [ { "contents": [ "We will use the criterion of", "Topology, Lemma \\ref{topology-lemma-characterize-limit}.", "We have seen that $S_{set} = \\lim_i S_{i, set}$ in", "Lemma \\ref{lemma-inverse-limit-sets}.", "The maps $f_i : S \\to S_i$ are morphisms of schemes", "hence continuous. Thus $f_i^{-1}(U_i)$ is open for each", "open $U_i \\subset S_i$. Finally, ", "let $s \\in S$ and let $s \\in V \\subset S$ be an open", "neighbourhood. Choose $0 \\in I$ and choose an", "affine open neighbourhood $U_0 \\subset S_0$ of the image of $s$.", "Then $f_0^{-1}(U_0) = \\lim_{i \\geq 0} f_{i0}^{-1}(U_0)$, see", "Lemma \\ref{lemma-directed-inverse-system-has-limit}.", "Then $f_0^{-1}(U_0)$ and $f_{i0}^{-1}(U_0)$ are affine and", "$$", "\\mathcal{O}_S(f_0^{-1}(U_0)) =", "\\colim_{i \\geq 0} \\mathcal{O}_{S_i}(f_{i0}^{-1}(U_0))", "$$", "either by the proof of", "Lemma \\ref{lemma-directed-inverse-system-has-limit}", "or by", "Lemma \\ref{lemma-directed-inverse-system-affine-schemes-has-limit}.", "Choose $a \\in \\mathcal{O}_S(f_0^{-1}(U_0))$ such that", "$s \\in D(a) \\subset V$. This is possible because the principal", "opens form a basis for the topology on the affine scheme $f_0^{-1}(U_0)$.", "Then we can pick an $i \\geq 0$ and", "$a_i \\in \\mathcal{O}_{S_i}(f_{i0}^{-1}(U_0))$ mapping to $a$.", "It follows that $D(a_i) \\subset f_{i0}^{-1}(U_0) \\subset S_i$", "is an open subset whose inverse image in $S$ is $D(a)$.", "This finishes the proof." ], "refs": [ "topology-lemma-characterize-limit", "limits-lemma-inverse-limit-sets", "limits-lemma-directed-inverse-system-has-limit", "limits-lemma-directed-inverse-system-has-limit", "limits-lemma-directed-inverse-system-affine-schemes-has-limit" ], "ref_ids": [ 8250, 15032, 15027, 15027, 15026 ] } ], "ref_ids": [ 15027 ] }, { "id": 15034, "type": "theorem", "label": "limits-lemma-limit-nonempty", "categories": [ "limits" ], "title": "limits-lemma-limit-nonempty", "contents": [ "Let $S = \\lim S_i$ be the limit of a directed inverse system", "of schemes with affine transition morphisms", "(Lemma \\ref{lemma-directed-inverse-system-has-limit}).", "If all the schemes $S_i$ are nonempty and quasi-compact,", "then the limit $S = \\lim_i S_i$ is nonempty." ], "refs": [ "limits-lemma-directed-inverse-system-has-limit" ], "proofs": [ { "contents": [ "Choose $0 \\in I$. Note that $I$ is nonempty as the limit is directed.", "Choose an affine open covering $S_0 = \\bigcup_{j = 1, \\ldots, m} U_j$.", "Since $I$ is directed there exists a $j \\in \\{1, \\ldots, m\\}$", "such that $f_{i0}^{-1}(U_j) \\not = \\emptyset$ for all", "$i \\geq 0$. Hence $\\lim_{i \\geq 0} f_{i0}^{-1}(U_j)$ is not", "empty since a directed colimit of nonzero rings is nonzero", "(because $1 \\not = 0$). As $\\lim_{i \\geq 0} f_{i0}^{-1}(U_j)$", "is an open subscheme of the limit we win." ], "refs": [], "ref_ids": [] } ], "ref_ids": [ 15027 ] }, { "id": 15035, "type": "theorem", "label": "limits-lemma-inverse-limit-irreducibles", "categories": [ "limits" ], "title": "limits-lemma-inverse-limit-irreducibles", "contents": [ "Let $S = \\lim S_i$ be the limit of a directed inverse system", "of schemes with affine transition morphisms", "(Lemma \\ref{lemma-directed-inverse-system-has-limit}).", "Let $s \\in S$ with images $s_i \\in S_i$.", "Then", "\\begin{enumerate}", "\\item $s = \\lim s_i$ as schemes, i.e., $\\kappa(s) = \\colim \\kappa(s_i)$,", "\\item $\\overline{\\{s\\}} = \\lim \\overline{\\{s_i\\}}$ as sets, and", "\\item $\\overline{\\{s\\}} = \\lim \\overline{\\{s_i\\}}$ as schemes", "where $\\overline{\\{s\\}}$ and $\\overline{\\{s_i\\}}$ are", "endowed with the reduced induced scheme structure.", "\\end{enumerate}" ], "refs": [ "limits-lemma-directed-inverse-system-has-limit" ], "proofs": [ { "contents": [ "Choose $0 \\in I$ and an affine open covering", "$U_0 = \\bigcup_{j \\in J} U_{0, j}$.", "For $i \\geq 0$ let $U_{i, j} = f_{i, 0}^{-1}(U_{0, j})$", "and set $U_j = f_0^{-1}(U_{0, j})$.", "Here $f_{i'i} : S_{i'} \\to S_i$ is the transtion morphism", "and $f_i : S \\to S_i$ is the projection.", "For $j \\in J$ the following are equivalent:", "(a) $s \\in U_j$, (b) $s_0 \\in U_{0, j}$,", "(c) $s_i \\in U_{i, j}$ for all $i \\geq 0$.", "Let $J' \\subset J$ be the set of indices for which (a), (b), (c) are true.", "Then $\\overline{\\{s\\}} = \\bigcup_{j \\in J'} (\\overline{\\{s\\}} \\cap U_j)$", "and similarly for $\\overline{\\{s_i\\}}$ for $i \\geq 0$.", "Note that $\\overline{\\{s\\}} \\cap U_j$ is the closure of the", "set $\\{s\\}$ in the topological space $U_j$. Similarly for", "$\\overline{\\{s_i\\}} \\cap U_{i, j}$ for $i \\geq 0$.", "Hence it suffices to prove the lemma in the case", "$S$ and $S_i$ affine for all $i$. This reduces us to the", "algebra question considered in the next paragraph.", "\\medskip\\noindent", "Suppose given a system of rings $(A_i, \\varphi_{ii'})$", "over $I$. Set $A = \\colim_i A_i$ with canonical maps", "$\\varphi_i : A_i \\to A$. Let $\\mathfrak p \\subset A$ be a", "prime and set $\\mathfrak p_i = \\varphi_i^{-1}(\\mathfrak p)$.", "Then", "$$", "V(\\mathfrak p) = \\lim_i V(\\mathfrak p_i)", "$$", "This follows from Lemma \\ref{lemma-inverse-limit-sets}", "because $A/\\mathfrak p = \\colim A_i/\\mathfrak p_i$.", "This equality of rings also shows the final statement", "about reduced induced scheme structures holds true.", "The equality $\\kappa(\\mathfrak p) = \\colim \\kappa(\\mathfrak p_i)$", "follows from the statement as well." ], "refs": [ "limits-lemma-inverse-limit-sets" ], "ref_ids": [ 15032 ] } ], "ref_ids": [ 15027 ] }, { "id": 15036, "type": "theorem", "label": "limits-lemma-topology-limit", "categories": [ "limits" ], "title": "limits-lemma-topology-limit", "contents": [ "In Situation \\ref{situation-descent}.", "\\begin{enumerate}", "\\item We have $S_{set} = \\lim_i S_{i, set}$ where $S_{set}$", "indicates the underlying set of the scheme $S$.", "\\item We have $S_{top} = \\lim_i S_{i, top}$ where $S_{top}$", "indicates the underlying topological space of the scheme $S$.", "\\item If $s, s' \\in S$ and $s'$ is not a specialization of $s$", "then for some $i \\in I$ the image $s'_i \\in S_i$ of $s'$ is not", "a specialization of the image $s_i \\in S_i$ of $s$.", "\\item Add more easy facts on topology of $S$ here.", "(Requirement: whatever is added should be easy in the affine case.)", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Part (1) is a special case of Lemma \\ref{lemma-inverse-limit-sets}.", "\\medskip\\noindent", "Part (2) is a special case of Lemma \\ref{lemma-inverse-limit-top}.", "\\medskip\\noindent", "Part (3) is a special case of Lemma \\ref{lemma-inverse-limit-irreducibles}." ], "refs": [ "limits-lemma-inverse-limit-sets", "limits-lemma-inverse-limit-top", "limits-lemma-inverse-limit-irreducibles" ], "ref_ids": [ 15032, 15033, 15035 ] } ], "ref_ids": [] }, { "id": 15037, "type": "theorem", "label": "limits-lemma-descend-section", "categories": [ "limits" ], "title": "limits-lemma-descend-section", "contents": [ "In Situation \\ref{situation-descent}.", "Suppose that $\\mathcal{F}_0$ is a quasi-coherent sheaf on $S_0$.", "Set $\\mathcal{F}_i = f_{i0}^*\\mathcal{F}_0$ for $i \\geq 0$ and set", "$\\mathcal{F} = f_0^*\\mathcal{F}_0$.", "Then", "$$", "\\Gamma(S, \\mathcal{F}) = \\colim_{i \\geq 0} \\Gamma(S_i, \\mathcal{F}_i)", "$$" ], "refs": [], "proofs": [ { "contents": [ "Write $\\mathcal{A}_j = f_{i0, *} \\mathcal{O}_{S_i}$.", "This is a quasi-coherent sheaf of $\\mathcal{O}_{S_0}$-algebras", "(see Morphisms, Lemma \\ref{morphisms-lemma-affine-equivalence-algebras})", "and $S_i$ is the relative spectrum of $\\mathcal{A}_i$ over $S_0$.", "In the proof of Lemma \\ref{lemma-directed-inverse-system-has-limit}", "we constructed $S$ as the relative spectrum of", "$\\mathcal{A} = \\colim_{i \\geq 0} \\mathcal{A}_i$", "over $S_0$. Set", "$$", "\\mathcal{M}_i = \\mathcal{F}_0 \\otimes_{\\mathcal{O}_{S_0}} \\mathcal{A}_i", "$$", "and", "$$", "\\mathcal{M} = \\mathcal{F}_0 \\otimes_{\\mathcal{O}_{S_0}} \\mathcal{A}.", "$$", "Then we have $f_{i0, *} \\mathcal{F}_i = \\mathcal{M}_i$", "and $f_{0, *}\\mathcal{F} = \\mathcal{M}$. Since $\\mathcal{A}$", "is the colimit of the sheaves $\\mathcal{A}_i$ and since tensor", "product commutes with directed colimits, we conclude that", "$\\mathcal{M} = \\colim_{i \\geq 0} \\mathcal{M}_i$.", "Since $S_0$ is quasi-compact and quasi-separated we see that", "\\begin{eqnarray*}", "\\Gamma(S, \\mathcal{F})", "& = &", "\\Gamma(S_0, \\mathcal{M}) \\\\", "& = &", "\\Gamma(S_0, \\colim_{i \\geq 0} \\mathcal{M}_i) \\\\", "& = &", "\\colim_{i \\geq 0} \\Gamma(S_0, \\mathcal{M}_i) \\\\", "& = &", "\\colim_{i \\geq 0} \\Gamma(S_i, \\mathcal{F}_i)", "\\end{eqnarray*}", "see Sheaves, Lemma \\ref{sheaves-lemma-directed-colimits-sections} and", "Topology, Lemma \\ref{topology-lemma-topology-quasi-separated-scheme}", "for the middle equality." ], "refs": [ "morphisms-lemma-affine-equivalence-algebras", "limits-lemma-directed-inverse-system-has-limit", "sheaves-lemma-directed-colimits-sections", "topology-lemma-topology-quasi-separated-scheme" ], "ref_ids": [ 5173, 15027, 14526, 8333 ] } ], "ref_ids": [] }, { "id": 15038, "type": "theorem", "label": "limits-lemma-limit-closed-nonempty", "categories": [ "limits" ], "title": "limits-lemma-limit-closed-nonempty", "contents": [ "In Situation \\ref{situation-descent}.", "Suppose for each $i$ we are given a nonempty closed subset", "$Z_i \\subset S_i$ with $f_{ii'}(Z_i) \\subset Z_{i'}$.", "Then there exists a point $s \\in S$ with $f_i(s) \\in Z_i$ for", "all $i$." ], "refs": [], "proofs": [ { "contents": [ "Let $Z_i \\subset S_i$ also denote the reduced closed subscheme", "associated to $Z_i$, see Schemes,", "Definition \\ref{schemes-definition-reduced-induced-scheme}.", "A closed immersion is affine, and a composition of affine", "morphisms is affine (see", "Morphisms, Lemmas \\ref{morphisms-lemma-closed-immersion-affine}", "and \\ref{morphisms-lemma-composition-affine}), and hence $Z_i \\to S_{i'}$ is", "affine when $i \\geq i'$. We conclude that the morphism", "$f_{ii'} : Z_i \\to Z_{i'}$ is affine by", "Morphisms, Lemma \\ref{morphisms-lemma-affine-permanence}.", "Each of the schemes $Z_i$ is quasi-compact as a closed", "subscheme of a quasi-compact scheme. Hence we may apply", "Lemma \\ref{lemma-limit-nonempty} to see that", "$Z = \\lim_i Z_i$ is nonempty. Since there is a", "canonical morphism $Z \\to S$ we win." ], "refs": [ "schemes-definition-reduced-induced-scheme", "morphisms-lemma-closed-immersion-affine", "morphisms-lemma-composition-affine", "morphisms-lemma-affine-permanence", "limits-lemma-limit-nonempty" ], "ref_ids": [ 7745, 5177, 5175, 5179, 15034 ] } ], "ref_ids": [] }, { "id": 15039, "type": "theorem", "label": "limits-lemma-limit-fibre-product-empty", "categories": [ "limits" ], "title": "limits-lemma-limit-fibre-product-empty", "contents": [ "In Situation \\ref{situation-descent}.", "Suppose we are given an $i$ and a morphism $T \\to S_i$ such that", "\\begin{enumerate}", "\\item $T \\times_{S_i} S = \\emptyset$, and", "\\item $T$ is quasi-compact.", "\\end{enumerate}", "Then $T \\times_{S_i} S_{i'} = \\emptyset$ for all sufficiently large $i'$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-scheme-over-limit}", "we see that $T \\times_{S_i} S = \\lim_{i' \\geq i} T \\times_{S_i} S_{i'}$.", "Hence the result follows from", "Lemma \\ref{lemma-limit-nonempty}." ], "refs": [ "limits-lemma-scheme-over-limit", "limits-lemma-limit-nonempty" ], "ref_ids": [ 15028, 15034 ] } ], "ref_ids": [] }, { "id": 15040, "type": "theorem", "label": "limits-lemma-limit-contained-in-constructible", "categories": [ "limits" ], "title": "limits-lemma-limit-contained-in-constructible", "contents": [ "In Situation \\ref{situation-descent}.", "Suppose we are given an $i$ and a locally constructible subset", "$E \\subset S_i$ such that $f_i(S) \\subset E$.", "Then $f_{i'i}(S_{i'}) \\subset E$ for all sufficiently large $i'$." ], "refs": [], "proofs": [ { "contents": [ "Writing $S_i$ as a finite union of open affine subschemes reduces", "the question to the case that $S_i$ is affine and $E$ is constructible, see", "Lemma \\ref{lemma-directed-inverse-system-has-limit}", "and", "Properties, Lemma \\ref{properties-lemma-locally-constructible}.", "In this case the complement $S_i \\setminus E$ is constructible too.", "Hence there exists an affine scheme $T$ and a morphism $T \\to S_i$", "whose image is $S_i \\setminus E$, see", "Algebra, Lemma \\ref{algebra-lemma-constructible-is-image}.", "By", "Lemma \\ref{lemma-limit-fibre-product-empty}", "we see that $T \\times_{S_i} S_{i'}$ is empty for all sufficiently large", "$i'$, and hence $f_{i'i}(S_{i'}) \\subset E$ for all sufficiently large $i'$." ], "refs": [ "limits-lemma-directed-inverse-system-has-limit", "properties-lemma-locally-constructible", "algebra-lemma-constructible-is-image", "limits-lemma-limit-fibre-product-empty" ], "ref_ids": [ 15027, 2938, 435, 15039 ] } ], "ref_ids": [] }, { "id": 15041, "type": "theorem", "label": "limits-lemma-descend-opens", "categories": [ "limits" ], "title": "limits-lemma-descend-opens", "contents": [ "In Situation \\ref{situation-descent} we have the following:", "\\begin{enumerate}", "\\item Given any quasi-compact open $V \\subset S = \\lim_i S_i$", "there exists an $i \\in I$ and a quasi-compact open $V_i \\subset S_i$", "such that $f_i^{-1}(V_i) = V$.", "\\item Given $V_i \\subset S_i$ and $V_{i'} \\subset S_{i'}$", "quasi-compact opens such that $f_i^{-1}(V_i) = f_{i'}^{-1}(V_{i'})$", "there exists an index $i'' \\geq i, i'$ such that", "$f_{i''i}^{-1}(V_i) = f_{i''i'}^{-1}(V_{i'})$.", "\\item If $V_{1, i}, \\ldots, V_{n, i} \\subset S_i$ are quasi-compact", "opens and $S = f_i^{-1}(V_{1, i}) \\cup \\ldots \\cup f_i^{-1}(V_{n, i})$", "then $S_{i'} = f_{i'i}^{-1}(V_{1, i}) \\cup \\ldots \\cup f_{i'i}^{-1}(V_{n, i})$", "for some $i' \\geq i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Choose $i_0 \\in I$. Note that $I$ is nonempty as the limit is directed.", "For convenience we write $S_0 = S_{i_0}$ and $i_0 = 0$.", "Choose an affine open covering $S_0 = U_{1, 0} \\cup \\ldots \\cup U_{m, 0}$.", "Denote $U_{j, i} \\subset S_i$ the inverse image of $U_{j, 0}$", "under the transition morphism for $i \\geq 0$.", "Denote $U_j$ the inverse image of $U_{j, 0}$ in $S$.", "Note that $U_j = \\lim_i U_{j, i}$ is a limit of affine", "schemes.", "\\medskip\\noindent", "We first prove the uniqueness statement: Let", "$V_i \\subset S_i$ and $V_{i'} \\subset S_{i'}$", "quasi-compact opens such that $f_i^{-1}(V_i) = f_{i'}^{-1}(V_{i'})$.", "It suffices to show that $f_{i''i}^{-1}(V_i \\cap U_{j, i''})$ and", "$f_{i''i'}^{-1}(V_{i'} \\cap U_{j, i''})$ become equal", "for $i''$ large enough. Hence we reduce to the case", "of a limit of affine schemes. In this case write", "$S = \\Spec(R)$ and $S_i = \\Spec(R_i)$ for all $i \\in I$.", "We may write $V_i = S_i \\setminus V(h_1, \\ldots, h_m)$", "and $V_{i'} = S_{i'} \\setminus V(g_1, \\ldots, g_n)$.", "The assumption means that the ideals", "$\\sum g_jR$ and $\\sum h_jR$ have the same radical", "in $R$. This means that $g_j^N = \\sum a_{jj'}h_{j'}$ and", "$h_j^N = \\sum b_{jj'} g_{j'}$ for some $N \\gg 0$ and $a_{jj'}$", "and $b_{jj'}$ in $R$.", "Since $R = \\colim_i R_i$ we can chose an index", "$i'' \\geq i$ such that the equations", "$g_j^N = \\sum a_{jj'}h_{j'}$ and", "$h_j^N = \\sum b_{jj'} g_{j'}$ hold in $R_{i''}$ for some", "$a_{jj'}$ and $b_{jj'}$ in $R_{i''}$. This implies that", "the ideals $\\sum g_jR_{i''}$ and $\\sum h_jR_{i''}$ have the same radical", "in $R_{i''}$ as desired.", "\\medskip\\noindent", "We prove existence: If $S_0$ is affine, then $S_i = \\Spec(R_i)$ for all", "$i \\geq 0$ and $S = \\Spec(R)$ with $R = \\colim R_i$. Then", "$V = S \\setminus V(g_1, \\ldots, g_n)$ for some $g_1, \\ldots, g_n \\in R$.", "Choose any $i$ large enough so that each of the $g_j$ comes from an", "element $g_{j, i} \\in R_i$ and take", "$V_i = S_i \\setminus V(g_{1, i}, \\ldots, g_{n, i})$.", "If $S_0$ is general, then the opens $V \\cap U_j$", "are quasi-compact because $S$ is quasi-separated. Hence by the", "affine case we see that for each $j = 1, \\ldots, m$", "there exists an $i_j \\in I$ and a quasi-compact open", "$V_{i_j} \\subset U_{j, i_j}$ whose inverse image in $U_j$", "is $V \\cap U_j$. Set $i = \\max(i_1, \\ldots, i_m)$", "and let $V_i = \\bigcup f_{ii_j}^{-1}(V_{i_j})$.", "\\medskip\\noindent", "The statement on coverings follows from the uniqueness statement", "for the opens $V_{1, i} \\cup \\ldots \\cup V_{n, i}$ and $S_i$ of $S_i$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 15042, "type": "theorem", "label": "limits-lemma-limit-quasi-affine", "categories": [ "limits" ], "title": "limits-lemma-limit-quasi-affine", "contents": [ "In Situation \\ref{situation-descent} if $S$ is quasi-affine, then", "for some $i_0 \\in I$ the schemes $S_i$ for $i \\geq i_0$ are quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "Choose $i_0 \\in I$. Note that $I$ is nonempty as the limit is directed.", "For convenience we write $S_0 = S_{i_0}$ and $i_0 = 0$.", "Let $s \\in S$. We may choose an affine open", "$U_0 \\subset S_0$ containing $f_0(s)$. Since $S$ is quasi-affine", "we may choose an element $a \\in \\Gamma(S, \\mathcal{O}_S)$ such", "that $s \\in D(a) \\subset f_0^{-1}(U_0)$, and such that", "$D(a)$ is affine. By Lemma \\ref{lemma-descend-section}", "there exists an $i \\geq 0$ such that $a$", "comes from an element $a_i \\in \\Gamma(S_i, \\mathcal{O}_{S_i})$.", "For any index $j \\geq i$ we denote $a_j$", "the image of $a_i$ in the global sections of the", "structure sheaf of $S_j$.", "Consider the opens $D(a_j) \\subset S_j$", "and $U_j = f_{j0}^{-1}(U_0)$. Note that", "$U_j$ is affine and $D(a_j)$ is a quasi-compact open of $S_j$,", "see Properties, Lemma \\ref{properties-lemma-affine-cap-s-open}", "for example. Hence we may apply Lemma \\ref{lemma-descend-opens} to the opens", "$U_j$ and $U_j \\cup D(a_j)$ to conclude that", "$D(a_j) \\subset U_j$ for some $j \\geq i$.", "For such an index $j$ we see that $D(a_j) \\subset S_j$ is an affine open", "(because $D(a_j)$ is a standard affine open of the affine open $U_j$)", "containing the image $f_j(s)$.", "\\medskip\\noindent", "We conclude that for every $s \\in S$ there exist", "an index $i \\in I$, and a global section", "$a \\in \\Gamma(S_i, \\mathcal{O}_{S_i})$", "such that $D(a) \\subset S_i$ is an affine open", "containing $f_i(s)$. Because $S$ is quasi-compact we", "may choose a single index $i \\in I$ and global sections", "$a_1, \\ldots, a_m \\in \\Gamma(S_i, \\mathcal{O}_{S_i})$", "such that each $D(a_j) \\subset S_i$ is affine open", "and such that $f_i : S \\to S_i$ has image contained", "in the union $W_i = \\bigcup_{j = 1, \\ldots, m} D(a_j)$.", "For $i' \\geq i$ set $W_{i'} = f_{i'i}^{-1}(W_i)$.", "Since $f_i^{-1}(W_i)$ is all of $S$ we see", "(by Lemma \\ref{lemma-descend-opens} again)", "that for a suitable $i' \\geq i$ we", "have $S_{i'} = W_{i'}$. Thus we may replace $i$ by", "$i'$ and assume that $S_i = \\bigcup_{j = 1, \\ldots, m} D(a_j)$.", "This implies that $\\mathcal{O}_{S_i}$ is an ample invertible", "sheaf on $S_i$ (see Properties, Definition \\ref{properties-definition-ample})", "and hence that $S_i$ is quasi-affine, see", "Properties, Lemma \\ref{properties-lemma-quasi-affine-O-ample}.", "Hence we win." ], "refs": [ "properties-lemma-affine-cap-s-open", "limits-lemma-descend-opens", "limits-lemma-descend-opens", "properties-definition-ample", "properties-lemma-quasi-affine-O-ample" ], "ref_ids": [ 3042, 15041, 15041, 3088, 3053 ] } ], "ref_ids": [] }, { "id": 15043, "type": "theorem", "label": "limits-lemma-limit-affine", "categories": [ "limits" ], "title": "limits-lemma-limit-affine", "contents": [ "In Situation \\ref{situation-descent} if $S$ is affine,", "then for some $i_0 \\in I$ the schemes $S_i$ for $i \\geq i_0$", "are affine." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-limit-quasi-affine} we may assume that $S_0$ is", "quasi-affine for some $0 \\in I$. Set $R_0 = \\Gamma(S_0, \\mathcal{O}_{S_0})$.", "Then $S_0$ is a quasi-compact open of $T_0 = \\Spec(R_0)$. Denote", "$j_0 : S_0 \\to T_0$ the corresponding quasi-compact open immersion.", "For $i \\geq 0$ set $\\mathcal{A}_i = f_{i0, *}\\mathcal{O}_{S_i}$.", "Since $f_{i0}$ is affine we see that", "$S_i = \\underline{\\Spec}_{S_0}(\\mathcal{A}_i)$.", "Set $T_i = \\underline{\\Spec}_{T_0}(j_{0, *}\\mathcal{A}_i)$.", "Then $T_i \\to T_0$ is affine, hence $T_i$ is affine. Thus", "$T_i$ is the spectrum of", "$$", "R_i = \\Gamma(T_0, j_{0, *}\\mathcal{A}_i) = \\Gamma(S_0, \\mathcal{A}_i) =", "\\Gamma(S_i, \\mathcal{O}_{S_i}).", "$$", "Write $S = \\Spec(R)$. We have $R = \\colim_i R_i$", "by Lemma \\ref{lemma-descend-section}.", "Hence also $S = \\lim_i T_i$. As formation of the relative spectrum commutes", "with base change, the inverse image", "of the open $S_0 \\subset T_0$ in $T_i$ is $S_i$.", "Let $Z_0 = T_0 \\setminus S_0$ and let $Z_i \\subset T_i$", "be the inverse image of $Z_0$. As $S_i = T_i \\setminus Z_i$, it suffices", "to show that $Z_i$ is empty for some $i$. Assume $Z_i$ is nonempty for all", "$i$ to get a contradiction. By Lemma \\ref{lemma-limit-closed-nonempty}", "there exists a point $s$ of $S = \\lim T_i$ which maps to a point of $Z_i$", "for every $i$. But $S = \\lim_i S_i$, and hence we arrive at a contradiction", "by Lemma \\ref{lemma-topology-limit}." ], "refs": [ "limits-lemma-limit-quasi-affine", "limits-lemma-limit-closed-nonempty", "limits-lemma-topology-limit" ], "ref_ids": [ 15042, 15038, 15036 ] } ], "ref_ids": [] }, { "id": 15044, "type": "theorem", "label": "limits-lemma-limit-separated", "categories": [ "limits" ], "title": "limits-lemma-limit-separated", "contents": [ "In Situation \\ref{situation-descent} if $S$ is separated,", "then for some $i_0 \\in I$ the schemes $S_i$ for $i \\geq i_0$", "are separated." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite affine open covering", "$S_0 = U_{0, 1} \\cup \\ldots \\cup U_{0, m}$.", "Set $U_{i, j} \\subset S_i$ and $U_j \\subset S$", "equal to the inverse image of $U_{0, j}$.", "Note that $U_{i, j}$ and $U_j$ are affine. As $S$ is separated", "the intersections $U_{j_1} \\cap U_{j_2}$ are affine. Since", "$U_{j_1} \\cap U_{j_2} = \\lim_{i \\geq 0} U_{i, j_1} \\cap U_{i, j_2}$", "we see that $U_{i, j_1} \\cap U_{i, j_2}$ is affine for large $i$", "by Lemma \\ref{lemma-limit-affine}. To show that $S_i$ is separated", "for large $i$ it now suffices to show that", "$$", "\\mathcal{O}_{S_i}(U_{i, j_1})", "\\otimes_{\\mathcal{O}_S(S)}", "\\mathcal{O}_{S_i}(U_{i, j_2})", "\\longrightarrow", "\\mathcal{O}_{S_i}(U_{i, j_1} \\cap U_{i, j_2})", "$$", "is surjective for large $i$", "(Schemes, Lemma \\ref{schemes-lemma-characterize-separated}).", "\\medskip\\noindent", "To get rid of the annoying indices, assume we have affine opens", "$U, V \\subset S_0$ such that $U \\cap V$ is affine too.", "Let $U_i, V_i \\subset S_i$, resp.\\ $U, V \\subset S$ be the inverse images.", "We have to show that", "$\\mathcal{O}(U_i) \\otimes \\mathcal{O}(V_i) \\to", "\\mathcal{O}(U_i \\cap V_i)$", "is surjective for $i$ large enough and we know that", "$\\mathcal{O}(U) \\otimes \\mathcal{O}(V) \\to \\mathcal{O}(U \\cap V)$", "is surjective. Note that", "$\\mathcal{O}(U_0) \\otimes \\mathcal{O}(V_0) \\to", "\\mathcal{O}(U_0 \\cap V_0)$", "is of finite type, as the diagonal morphism $S_i \\to S_i \\times S_i$", "is an immersion (Schemes, Lemma \\ref{schemes-lemma-diagonal-immersion})", "hence locally of finite type", "(Morphisms, Lemmas \\ref{morphisms-lemma-locally-finite-type-characterize} and", "\\ref{morphisms-lemma-immersion-locally-finite-type}).", "Thus we can choose elements", "$f_{0, 1}, \\ldots, f_{0, n} \\in \\mathcal{O}(U_0 \\cap V_0)$", "which generate $\\mathcal{O}(U_0 \\cap V_0)$ over", "$\\mathcal{O}(U_0) \\otimes \\mathcal{O}(V_0)$.", "Observe that for $i \\geq 0$ the diagram of schemes", "$$", "\\xymatrix{", "U_i \\cap V_i \\ar[r] \\ar[d] & U_i \\ar[d] \\\\", "U_0 \\cap V_0 \\ar[r] & U_0", "}", "$$", "is cartesian. Thus we see that the images", "$f_{i, 1}, \\ldots, f_{i, n} \\in \\mathcal{O}(U_i \\cap V_i)$", "generate $\\mathcal{O}(U_i \\cap V_i)$ over", "$\\mathcal{O}(U_i) \\otimes \\mathcal{O}(V_0)$", "and a fortiori over", "$\\mathcal{O}(U_i) \\otimes \\mathcal{O}(V_i)$.", "By assumption the images $f_1, \\ldots, f_n \\in \\mathcal{O}(U \\otimes V)$", "are in the image of the map", "$\\mathcal{O}(U) \\otimes \\mathcal{O}(V) \\to \\mathcal{O}(U \\cap V)$.", "Since", "$\\mathcal{O}(U) \\otimes \\mathcal{O}(V) =", "\\colim \\mathcal{O}(U_i) \\otimes \\mathcal{O}(V_i)$", "we see that they are in the image of the map at some finite level", "and the lemma is proved." ], "refs": [ "limits-lemma-limit-affine", "schemes-lemma-characterize-separated", "schemes-lemma-diagonal-immersion", "morphisms-lemma-locally-finite-type-characterize", "morphisms-lemma-immersion-locally-finite-type" ], "ref_ids": [ 15043, 7710, 7707, 5198, 5201 ] } ], "ref_ids": [] }, { "id": 15045, "type": "theorem", "label": "limits-lemma-limit-ample", "categories": [ "limits" ], "title": "limits-lemma-limit-ample", "contents": [ "In Situation \\ref{situation-descent} let $\\mathcal{L}_0$ be an invertible", "sheaf of modules on $S_0$. If the pullback $\\mathcal{L}$ to $S$ is ample,", "then for some $i \\in I$ the pullback $\\mathcal{L}_i$ to $S_i$ is ample." ], "refs": [], "proofs": [ { "contents": [ "The assumption means there are finitely many sections", "$s_1, \\ldots, s_m \\in \\Gamma(S, \\mathcal{L})$ such that", "$S_{s_j}$ is affine and such that $S = \\bigcup S_{s_j}$, see", "Properties, Definition \\ref{properties-definition-ample}.", "By Lemma \\ref{lemma-descend-section} we can find an $i \\in I$", "and sections $s_{i, j} \\in \\Gamma(S_i, \\mathcal{L}_i)$ mapping to $s_j$.", "By Lemma \\ref{lemma-limit-affine} we may, after increasing $i$, assume", "that $(S_i)_{s_{i, j}}$ is affine for $j = 1, \\ldots, m$.", "By Lemma \\ref{lemma-descend-opens} we may, after increasing $i$ a", "last time, assume that $S_i = \\bigcup (S_i)_{s_{i, j}}$.", "Then $\\mathcal{L}_i$ is ample by definition." ], "refs": [ "properties-definition-ample", "limits-lemma-limit-affine", "limits-lemma-descend-opens" ], "ref_ids": [ 3088, 15043, 15041 ] } ], "ref_ids": [] }, { "id": 15046, "type": "theorem", "label": "limits-lemma-finite-type-eventually-closed", "categories": [ "limits" ], "title": "limits-lemma-finite-type-eventually-closed", "contents": [ "Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of", "schemes over $S$ with affine transition morphisms. Let $Y \\to X$", "be a morphism of schemes over $S$.", "\\begin{enumerate}", "\\item If $Y \\to X$ is a closed immersion, $X_i$ quasi-compact, and", "$Y$ locally of finite type over $S$, then $Y \\to X_i$ is a closed", "immersion for $i$ large enough.", "\\item If $Y \\to X$ is an immersion, $X_i$ quasi-separated, $Y \\to S$ locally", "of finite type, and $Y$ quasi-compact, then $Y \\to X_i$ is an", "immersion for $i$ large enough.", "\\item If $Y \\to X$ is an isomorphism, $X_i$ quasi-compact,", "$X_i \\to S$ locally of finite type, the transition morphisms", "$X_{i'} \\to X_i$ are closed immersions, and $Y \\to S$ is locally", "of finite presentation, then $Y \\to X_i$ is an isomorphism for $i$", "large enough.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Proof of (1). Choose $0 \\in I$ and a finite affine open covering", "$X_0 = U_{0, 1} \\cup \\ldots \\cup U_{0, m}$ with the property that", "$U_{0, j}$ maps into an affine open $W_j \\subset S$.", "Let $V_j \\subset Y$, resp.\\ $U_{i, j} \\subset X_i$, $i \\geq 0$,", "resp. $U_j \\subset X$ be the inverse image of $U_{0, j}$. It suffices", "to prove that $V_j \\to U_{i, j}$ is a closed immersion for $i$", "sufficiently large and we know that $V_j \\to U_j$ is a closed immersion.", "Thus we reduce to the following algebra fact: If $A = \\colim A_i$ is a", "directed colimit of $R$-algebras, $A \\to B$ is a surjection of $R$-algebras,", "and $B$ is a finitely generated $R$-algebra, then", "$A_i \\to B$ is surjective for $i$ sufficiently large.", "\\medskip\\noindent", "Proof of (2). Choose $0 \\in I$. Choose a quasi-compact open", "$X'_0 \\subset X_0$ such that $Y \\to X_0$ factors through $X'_0$.", "After replacing $X_i$ by the inverse image of $X'_0$ for $i \\geq 0$", "we may assume all $X_i'$ are quasi-compact and quasi-separated.", "Let $U \\subset X$ be a quasi-compact open such that $Y \\to X$ factors", "through a closed immersion $Y \\to U$ ($U$ exists as $Y$ is quasi-compact). By", "Lemma \\ref{lemma-descend-opens}", "we may assume that $U = \\lim U_i$ with $U_i \\subset X_i$ quasi-compact", "open. By part (1) we see that $Y \\to U_i$ is a closed immersion for some", "$i$. Thus (2) holds.", "\\medskip\\noindent", "Proof of (3). Working affine locally on $X_0$ for some $0 \\in I$ as in", "the proof of (1) we reduce to the following algebra fact: If $A = \\lim A_i$", "is a directed colimit of $R$-algebras with surjective transition", "maps and $A$ of finite presentation over $A_0$, then $A = A_i$ for", "some $i$. Namely, write $A = A_0/(f_1, \\ldots, f_n)$. Pick $i$ such", "that $f_1, \\ldots, f_n$ map to zero under the surjective map $A_0 \\to A_i$." ], "refs": [ "limits-lemma-descend-opens" ], "ref_ids": [ 15041 ] } ], "ref_ids": [] }, { "id": 15047, "type": "theorem", "label": "limits-lemma-eventually-separated", "categories": [ "limits" ], "title": "limits-lemma-eventually-separated", "contents": [ "Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed", "limit of schemes over $S$ with affine transition morphisms.", "Assume", "\\begin{enumerate}", "\\item $S$ quasi-separated,", "\\item $X_i$ quasi-compact and quasi-separated,", "\\item $X \\to S$ separated.", "\\end{enumerate}", "Then $X_i \\to S$ is separated for all $i$ large enough." ], "refs": [], "proofs": [ { "contents": [ "Let $0 \\in I$. Note that $I$ is nonempty as the limit is directed.", "As $X_0$ is quasi-compact we can find finitely many", "affine opens $U_1, \\ldots, U_n \\subset S$ such that", "$X_0 \\to S$ maps into $U_1 \\cup \\ldots \\cup U_n$.", "Denote $h_i : X_i \\to S$ the structure morphism.", "It suffices to check that for some $i \\geq 0$ the morphisms", "$h_i^{-1}(U_j) \\to U_j$ are separated for $j = 1, \\ldots, n$.", "Since $S$ is quasi-separated the morphisms $U_j \\to S$ are quasi-compact.", "Hence $h_i^{-1}(U_j)$ is quasi-compact and quasi-separated.", "In this way we reduce to the case $S$ affine. In this case we", "have to show that $X_i$ is separated and we know that $X$ is separated.", "Thus the lemma follows from Lemma \\ref{lemma-limit-separated}." ], "refs": [ "limits-lemma-limit-separated" ], "ref_ids": [ 15044 ] } ], "ref_ids": [] }, { "id": 15048, "type": "theorem", "label": "limits-lemma-eventually-affine", "categories": [ "limits" ], "title": "limits-lemma-eventually-affine", "contents": [ "Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of schemes", "over $S$ with affine transition morphisms. Assume", "\\begin{enumerate}", "\\item $S$ quasi-compact and quasi-separated,", "\\item $X_i$ quasi-compact and quasi-separated,", "\\item $X \\to S$ affine.", "\\end{enumerate}", "Then $X_i \\to S$ is affine for $i$ large enough." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite affine open covering $S = \\bigcup_{j = 1, \\ldots, n} V_j$.", "Denote $f : X \\to S$ and $f_i : X_i \\to S$ the structure morphisms.", "For each $j$ the scheme $f^{-1}(V_j) = \\lim_i f_i^{-1}(V_j)$", "is affine (as a finite morphism is affine by definition). Hence by", "Lemma \\ref{lemma-limit-affine} there exists an $i \\in I$ such that", "each $f_i^{-1}(V_j)$ is affine. In other words, $f_i : X_i \\to S$ is", "affine for $i$ large enough, see", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-affine}." ], "refs": [ "limits-lemma-limit-affine", "morphisms-lemma-characterize-affine" ], "ref_ids": [ 15043, 5172 ] } ], "ref_ids": [] }, { "id": 15049, "type": "theorem", "label": "limits-lemma-eventually-finite", "categories": [ "limits" ], "title": "limits-lemma-eventually-finite", "contents": [ "Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of schemes", "over $S$ with affine transition morphisms. Assume", "\\begin{enumerate}", "\\item $S$ quasi-compact and quasi-separated,", "\\item $X_i$ quasi-compact and quasi-separated,", "\\item the transition morphisms $X_{i'} \\to X_i$ are finite,", "\\item $X_i \\to S$ locally of finite type", "\\item $X \\to S$ integral.", "\\end{enumerate}", "Then $X_i \\to S$ is finite for $i$ large enough." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-eventually-affine}", "we may assume $X_i \\to S$ is affine for all $i$.", "Choose a finite affine open covering $S = \\bigcup_{j = 1, \\ldots, n} V_j$.", "Denote $f : X \\to S$ and $f_i : X_i \\to S$ the structure morphisms.", "It suffices to show that there exists an $i$ such that", "$f_i^{-1}(V_j)$ is finite over $V_j$ for $j = 1, \\ldots, m$", "(Morphisms, Lemma \\ref{morphisms-lemma-finite-local}).", "Namely, for $i' \\geq i$ the composition $X_{i'} \\to X_i \\to S$", "will be finite as a composition of finite morphisms", "(Morphisms, Lemma \\ref{morphisms-lemma-composition-finite}).", "This reduces us to the affine case: Let $R$ be a ring and", "$A = \\colim A_i$ with $R \\to A$ integral and $A_i \\to A_{i'}$", "finite for all $i \\leq i'$. Moreover $R \\to A_i$ is of finite type", "for all $i$. Goal: Show that $A_i$ is finite over $R$ for some $i$.", "To prove this choose an $i \\in I$ and pick generators", "$x_1, \\ldots, x_m \\in A_i$ of $A_i$ as an $R$-algebra.", "Since $A$ is integral over $R$ we can find monic polynomials", "$P_j \\in R[T]$ such that $P_j(x_j) = 0$ in $A$.", "Thus there exists an $i' \\geq i$ such that $P_j(x_j) = 0$ in $A_{i'}$", "for $j = 1, \\ldots, m$. Then the image $A'_i$ of $A_i$ in $A_{i'}$", "is finite over $R$ by", "Algebra, Lemma \\ref{algebra-lemma-characterize-finite-in-terms-of-integral}.", "Since $A'_i \\subset A_{i'}$ is finite too we conclude", "that $A_{i'}$ is finite over $R$ by", "Algebra, Lemma \\ref{algebra-lemma-finite-transitive}." ], "refs": [ "limits-lemma-eventually-affine", "morphisms-lemma-finite-local", "morphisms-lemma-composition-finite", "algebra-lemma-characterize-finite-in-terms-of-integral", "algebra-lemma-finite-transitive" ], "ref_ids": [ 15048, 5437, 5439, 484, 337 ] } ], "ref_ids": [] }, { "id": 15050, "type": "theorem", "label": "limits-lemma-eventually-closed-immersion", "categories": [ "limits" ], "title": "limits-lemma-eventually-closed-immersion", "contents": [ "Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of schemes", "over $S$ with affine transition morphisms. Assume", "\\begin{enumerate}", "\\item $S$ quasi-compact and quasi-separated,", "\\item $X_i$ quasi-compact and quasi-separated,", "\\item the transition morphisms $X_{i'} \\to X_i$ are closed immersions,", "\\item $X_i \\to S$ locally of finite type", "\\item $X \\to S$ a closed immersion.", "\\end{enumerate}", "Then $X_i \\to S$ is a closed immersion for $i$ large enough." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-eventually-affine}", "we may assume $X_i \\to S$ is affine for all $i$.", "Choose a finite affine open covering $S = \\bigcup_{j = 1, \\ldots, n} V_j$.", "Denote $f : X \\to S$ and $f_i : X_i \\to S$ the structure morphisms.", "It suffices to show that there exists an $i$ such that", "$f_i^{-1}(V_j)$ is a closed subscheme of $V_j$ for $j = 1, \\ldots, m$", "(Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion}).", "This reduces us to the affine case: Let $R$ be a ring and", "$A = \\colim A_i$ with $R \\to A$ surjective and $A_i \\to A_{i'}$", "surjective for all $i \\leq i'$. Moreover $R \\to A_i$ is of finite type", "for all $i$. Goal: Show that $R \\to A_i$ is surjective for some $i$.", "To prove this choose an $i \\in I$ and pick generators", "$x_1, \\ldots, x_m \\in A_i$ of $A_i$ as an $R$-algebra.", "Since $R \\to A$ is surjective we can find", "$r_j \\in R$ such that $r_j$ maps to $x_j$ in $A$.", "Thus there exists an $i' \\geq i$ such that $r_j$ maps to the image", "of $x_j$ in $A_{i'}$ for $j = 1, \\ldots, m$. Since $A_i \\to A_{i'}$", "is surjective this implies that $R \\to A_{i'}$ is surjective." ], "refs": [ "limits-lemma-eventually-affine", "morphisms-lemma-closed-immersion" ], "ref_ids": [ 15048, 5125 ] } ], "ref_ids": [] }, { "id": 15051, "type": "theorem", "label": "limits-lemma-quasi-affine-finite-type-over-Z", "categories": [ "limits" ], "title": "limits-lemma-quasi-affine-finite-type-over-Z", "contents": [ "Let $W$ be a quasi-affine scheme of finite type over", "$\\mathbf{Z}$. Suppose $W \\to \\Spec(R)$ is an", "open immersion into an affine scheme. There exists a", "finite type $\\mathbf{Z}$-algebra $A \\subset R$", "which induces an open immersion $W \\to \\Spec(A)$.", "Moreover, $R$ is the directed colimit of such subalgebras." ], "refs": [], "proofs": [ { "contents": [ "Choose an affine open covering $W = \\bigcup_{i = 1, \\ldots, n} W_i$", "such that each $W_i$ is a standard affine open in $\\Spec(R)$.", "In other words, if we write $W_i = \\Spec(R_i)$", "then $R_i = R_{f_i}$ for some $f_i \\in R$.", "Choose finitely many $x_{ij} \\in R_i$ which generate", "$R_i$ over $\\mathbf{Z}$.", "Pick an $N \\gg 0$ such that each $f_i^Nx_{ij}$ comes from an", "element of $R$, say $y_{ij} \\in R$.", "Set $A$ equal to the $\\mathbf{Z}$-algebra generated by", "the $f_i$ and the $y_{ij}$ and (optionally) finitely many", "additional elements of $R$. Then $A$ works. Details omitted." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 15052, "type": "theorem", "label": "limits-lemma-diagram", "categories": [ "limits" ], "title": "limits-lemma-diagram", "contents": [ "Suppose given a cartesian diagram of rings", "$$", "\\xymatrix{", "B \\ar[r]_s & R \\\\", "B'\\ar[u] \\ar[r] & R' \\ar[u]_t", "}", "$$", "Let $W' \\subset \\Spec(R')$ be an open of", "the form $W' = D(f_1) \\cup \\ldots \\cup D(f_n)$", "such that $t(f_i) = s(g_i)$ for some $g_i \\in B$", "and $B_{g_i} \\cong R_{s(g_i)}$. Then $B' \\to R'$", "induces an open immersion of $W'$ into $\\Spec(B')$." ], "refs": [], "proofs": [ { "contents": [ "Set $h_i = (g_i, f_i) \\in B'$. More on Algebra,", "Lemma \\ref{more-algebra-lemma-diagram-localize} shows that", "$(B')_{h_i} \\cong (R')_{f_i}$ as desired." ], "refs": [ "more-algebra-lemma-diagram-localize" ], "ref_ids": [ 9816 ] } ], "ref_ids": [] }, { "id": 15053, "type": "theorem", "label": "limits-lemma-approximate", "categories": [ "limits" ], "title": "limits-lemma-approximate", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme. Let $V \\subset S$", "be a quasi-compact open. Let $I$ be a directed set", "and let $(V_i, f_{ii'})$ be an inverse system of schemes over $I$", "with affine transition maps, with each $V_i$ of finite type over $\\mathbf{Z}$,", "and with $V = \\lim V_i$. Then there exist", "\\begin{enumerate}", "\\item a directed set $J$,", "\\item an inverse system of schemes $(S_j, g_{jj'})$ over $J$,", "\\item an order preserving map $\\alpha : J \\to I$,", "\\item open subschemes $V'_j \\subset S_j$, and", "\\item isomorphisms $V'_j \\to V_{\\alpha(j)}$", "\\end{enumerate}", "such that", "\\begin{enumerate}", "\\item the transition morphisms $g_{jj'} : S_j \\to S_{j'}$ are affine,", "\\item each $S_j$ is of finite type over $\\mathbf{Z}$,", "\\item $g_{jj'}^{-1}(V'_{j'}) = V'_j$,", "\\item $S = \\lim S_j$ and $V = \\lim V'_j$, and", "\\item the diagrams", "$$", "\\vcenter{", "\\xymatrix{", "V \\ar[d] \\ar[rd] \\\\", "V'_j \\ar[r] & V_{\\alpha(j)}", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "V'_j \\ar[r] \\ar[d] & V_{\\alpha(j)} \\ar[d] \\\\", "V'_{j'} \\ar[r] & V_{\\alpha(j')}", "}", "}", "$$", "are commutative.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Set $Z = S \\setminus V$. Choose affine opens $U_1, \\ldots, U_m \\subset S$", "such that $Z \\subset \\bigcup_{l = 1, \\ldots, m} U_l$. Consider the opens", "$$", "V \\subset V \\cup U_1 \\subset V \\cup U_1 \\cup U_2 \\subset", "\\ldots \\subset V \\cup \\bigcup\\nolimits_{l = 1, \\ldots, m} U_l = S", "$$", "If we can prove the lemma successively for each of the cases", "$$", "V \\cup U_1 \\cup \\ldots \\cup U_l", "\\subset", "V \\cup U_1 \\cup \\ldots \\cup U_{l + 1}", "$$", "then the lemma will follow for $V \\subset S$. In each case we are adding", "one affine open. Thus we may assume", "\\begin{enumerate}", "\\item $S = U \\cup V$,", "\\item $U$ affine open in $S$,", "\\item $V$ quasi-compact open in $S$, and", "\\item $V = \\lim_i V_i$ with $(V_i, f_{ii'})$", "an inverse system over a directed set $I$, each $f_{ii'}$", "affine and each $V_i$ of finite type over $\\mathbf{Z}$.", "\\end{enumerate}", "Denote $f_i : V \\to V_i$ the projections.", "Set $W = U \\cap V$. As $S$ is quasi-separated, this is a quasi-compact open", "of $V$. By Lemma \\ref{lemma-descend-opens}", "(and after shrinking $I$) we may assume that there exist", "opens $W_i \\subset V_i$ such that $f_{ii'}^{-1}(W_{i'}) = W_i$", "and such that $f_i^{-1}(W_i) = W$. Since $W$ is a quasi-compact open", "of $U$ it is quasi-affine. Hence we may assume (after shrinking $I$ again)", "that $W_i$ is quasi-affine for all $i$, see", "Lemma \\ref{lemma-limit-quasi-affine}.", "\\medskip\\noindent", "Write $U = \\Spec(B)$. Set $R = \\Gamma(W, \\mathcal{O}_W)$,", "and $R_i = \\Gamma(W_i, \\mathcal{O}_{W_i})$.", "By Lemma \\ref{lemma-descend-section} we have $R = \\colim_i R_i$.", "Now we have the maps of rings", "$$", "\\xymatrix{", "B \\ar[r]_s & R \\\\", "& R_i \\ar[u]_{t_i}", "}", "$$", "We set $B_i = \\{(b, r) \\in B \\times R_i \\mid s(b) = t_i(t)\\}$ so that we", "have a cartesian diagram", "$$", "\\xymatrix{", "B \\ar[r]_s & R \\\\", "B_i \\ar[u] \\ar[r] & R_i \\ar[u]_{t_i}", "}", "$$", "for each $i$. The transition maps $R_i \\to R_{i'}$ induce maps", "$B_i \\to B_{i'}$. It is clear that $B = \\colim_i B_i$.", "In the next paragraph we show that for all sufficiently large $i$", "the composition $W_i \\to \\Spec(R_i) \\to \\Spec(B_i)$ is an open immersion.", "\\medskip\\noindent", "As $W$ is a quasi-compact open of $U = \\Spec(B)$", "we can find a finitely many elements $g_l \\in B$, $l = 1, \\ldots, m$", "such that $D(g_l) \\subset W$ and such that", "$W = \\bigcup_{l = 1, \\ldots, m} D(g_l)$.", "Note that this implies $D(g_l) = W_{s(g_l)}$ as open subsets of $U$,", "where $W_{s(g_l)}$ denotes the largest open subset of $W$ on which", "$s(g_l)$ is invertible. Hence", "$$", "B_{g_l} =", "\\Gamma(D(g_l), \\mathcal{O}_U) =", "\\Gamma(W_{s(g_l)}, \\mathcal{O}_W) = R_{s(g_l)},", "$$", "where the last equality is", "Properties, Lemma \\ref{properties-lemma-invert-f-sections}.", "Since $W_{s(g_l)}$ is affine this also", "implies that $D(s(g_l)) = W_{s(g_l)}$ as open subsets of $\\Spec(R)$.", "Since $R = \\colim_i R_i$ we can (after shrinking $I$)", "assume there exist $g_{l, i} \\in R_i$ for all $i \\in I$ such that", "$s(g_l) = t_i(g_{l, i})$. Of course we choose the $g_{l, i}$", "such that $g_{l, i}$ maps to $g_{l, i'}$ under the transition maps", "$R_i \\to R_{i'}$. Then, by Lemma \\ref{lemma-descend-opens} we can", "(after shrinking $I$ again)", "assume the corresponding opens $D(g_{l, i}) \\subset \\Spec(R_i)$", "are contained in $W_i$ for $l = 1, \\ldots, m$ and cover $W_i$.", "We conclude that the morphism $W_i \\to \\Spec(R_i) \\to \\Spec(B_i)$", "is an open immersion, see Lemma \\ref{lemma-diagram}.", "\\medskip\\noindent", "By Lemma \\ref{lemma-quasi-affine-finite-type-over-Z}", "we can write $B_i$ as a directed colimit of subalgebras", "$A_{i, p} \\subset B_i$, $p \\in P_i$ each", "of finite type over $\\mathbf{Z}$ and such that $W_i$ is", "identified with an open subscheme of $\\Spec(A_{i, p})$.", "Let $S_{i, p}$ be the scheme obtained by glueing", "$V_i$ and $\\Spec(A_{i, p})$ along the open $W_i$, see", "Schemes, Section \\ref{schemes-section-glueing-schemes}.", "Here is the resulting commutative diagram of schemes:", "$$", "\\xymatrix{", "& & V \\ar[lld] \\ar[d] & W \\ar[l] \\ar[lld] \\ar[d] \\\\", "V_i \\ar[d] & W_i \\ar[l] \\ar[d] & S \\ar[lld] & U \\ar[lld] \\ar[l] \\\\", "S_{i, p} & \\Spec(A_{i, p}) \\ar[l]", "}", "$$", "The morphism $S \\to S_{i, p}$ arises because the upper right", "square is a pushout in the category of schemes.", "Note that $S_{i, p}$ is of finite type over $\\mathbf{Z}$ since", "it has a finite affine open covering whose members are", "spectra of finite type $\\mathbf{Z}$-algebras.", "We define a preorder on $J = \\coprod_{i \\in I} P_i$", "by the rule $(i', p') \\geq (i, p)$ if and only if", "$i' \\geq i$ and the map $B_i \\to B_{i'}$ maps $A_{i, p}$ into", "$A_{i', p'}$. This is exactly the condition needed to", "define a morphism $S_{i', p'} \\to S_{i, p}$: namely make a commutative", "diagram as above using the transition morphisms $V_{i'} \\to V_i$", "and $W_{i'} \\to W_i$ and", "the morphism $\\Spec(A_{i', p'}) \\to \\Spec(A_{i, p})$ induced", "by the ring map $A_{i, p} \\to A_{i', p'}$. The relevant commutativities", "have been built into the constructions.", "We claim that $S$ is the directed limit of the schemes $S_{i, p}$.", "Since by construction the schemes $V_i$ have limit $V$ this boils", "down to the fact that $B$ is the limit of the rings $A_{i, p}$", "which is true by construction. The map $\\alpha : J \\to I$ is given", "by the rule $j = (i, p) \\mapsto i$. The open subscheme $V'_j$ is", "just the image of $V_i \\to S_{i, p}$ above. The commutativity of", "the diagrams in (5) is clear from the construction.", "This finishes the proof of the lemma." ], "refs": [ "limits-lemma-descend-opens", "limits-lemma-limit-quasi-affine", "properties-lemma-invert-f-sections", "limits-lemma-descend-opens", "limits-lemma-diagram", "limits-lemma-quasi-affine-finite-type-over-Z" ], "ref_ids": [ 15041, 15042, 3004, 15041, 15052, 15051 ] } ], "ref_ids": [] }, { "id": 15054, "type": "theorem", "label": "limits-lemma-surjection-is-enough", "categories": [ "limits" ], "title": "limits-lemma-surjection-is-enough", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. If for every directed limit", "$T = \\lim_{i \\in I} T_i$ of affine schemes over $S$ the map", "$$", "\\colim \\Mor_S(T_i, X) \\longrightarrow \\Mor_S(T, X)", "$$", "is surjective, then $f$ is locally of finite presentation.", "In other words, in", "Proposition \\ref{proposition-characterize-locally-finite-presentation}", "parts (2) and (3) it suffices to check surjectivity of the map." ], "refs": [ "limits-proposition-characterize-locally-finite-presentation" ], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of the implication", "``(2) implies (1)'' in", "Proposition \\ref{proposition-characterize-locally-finite-presentation}.", "Choose any affine opens $U \\subset X$ and $V \\subset S$ such that", "$f(U) \\subset V$. We have to show that", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is of finite presentation.", "Let $(A_i, \\varphi_{ii'})$ be a directed system of", "$\\mathcal{O}_S(V)$-algebras. Set $A = \\colim_i A_i$.", "According to", "Algebra, Lemma \\ref{algebra-lemma-characterize-finite-presentation}", "it suffices to show that", "$$", "\\colim_i \\Hom_{\\mathcal{O}_S(V)}(\\mathcal{O}_X(U), A_i) \\to", "\\Hom_{\\mathcal{O}_S(V)}(\\mathcal{O}_X(U), A)", "$$", "is surjective. Consider the schemes $T_i = \\Spec(A_i)$. They", "form an inverse system of $V$-schemes over $I$", "with transition morphisms $f_{ii'} : T_i \\to T_{i'}$", "induced by the $\\mathcal{O}_S(V)$-algebra maps $\\varphi_{i'i}$.", "Set $T := \\Spec(A) = \\lim_i T_i$.", "The formula above becomes in terms of morphism sets of schemes", "$$", "\\colim_i \\Mor_V(T_i, U) \\to \\Mor_V(\\lim_i T_i, U)", "$$", "We first observe that", "$\\Mor_V(T_i, U) = \\Mor_S(T_i, U)$", "and", "$\\Mor_V(T, U) = \\Mor_S(T, U)$.", "Hence we have to show that", "$$", "\\colim_i \\Mor_S(T_i, U) \\to", "\\Mor_S(\\lim_i T_i, U)", "$$", "is surjective and we are given that", "$$", "\\colim_i \\Mor_S(T_i, X) \\to", "\\Mor_S(\\lim_i T_i, X)", "$$", "is surjective.", "Hence it suffices to prove that given a morphism $g_i : T_i \\to X$ over $S$", "such that the composition $T \\to T_i \\to X$ ends up in $U$ there exists some", "$i' \\geq i$ such that the composition $g_{i'} : T_{i'} \\to T_i \\to X$ ends up", "in $U$. Denote $Z_{i'} = g_{i'}^{-1}(X \\setminus U)$.", "Assume each $Z_{i'}$ is nonempty", "to get a contradiction. By Lemma \\ref{lemma-limit-closed-nonempty}", "there exists a point $t$ of $T$ which is mapped into $Z_{i'}$ for all", "$i' \\geq i$. Such a point is not mapped into $U$. A contradiction." ], "refs": [ "limits-proposition-characterize-locally-finite-presentation", "algebra-lemma-characterize-finite-presentation", "limits-lemma-limit-closed-nonempty" ], "ref_ids": [ 15127, 1092, 15038 ] } ], "ref_ids": [ 15127 ] }, { "id": 15055, "type": "theorem", "label": "limits-lemma-relative-approximation", "categories": [ "limits" ], "title": "limits-lemma-relative-approximation", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Assume that", "\\begin{enumerate}", "\\item $X$ is quasi-compact and quasi-separated, and", "\\item $S$ is quasi-separated.", "\\end{enumerate}", "Then $X = \\lim X_i$ is a limit of a directed system of schemes", "$X_i$ of finite presentation over $S$ with affine transition morphisms", "over $S$." ], "refs": [], "proofs": [ { "contents": [ "Since $f(X)$ is quasi-compact we may replace $S$ by a quasi-compact", "open containing $f(X)$. Hence we may assume $S$ is quasi-compact as well.", "Write $X = \\lim X_a$ and $S = \\lim S_b$ as in", "Proposition \\ref{proposition-approximate}, i.e., with $X_a$ and $S_b$", "of finite type over $\\mathbf{Z}$ and with affine transition morphisms.", "By Proposition \\ref{proposition-characterize-locally-finite-presentation}", "we find that for each $b$ there exists an $a$ and a morphism", "$f_{a, b} : X_a \\to S_b$ making the diagram", "$$", "\\xymatrix{", "X \\ar[d] \\ar[r] & S \\ar[d] \\\\", "X_a \\ar[r] & S_b", "}", "$$", "commute. Moreover the same proposition implies that, given a second", "triple $(a', b', f_{a', b'})$, there exists an $a'' \\geq a'$ such that", "the compositions $X_{a''} \\to X_a \\to S_b$ and", "$X_{a''} \\to X_{a'} \\to S_{b'} \\to S_b$ are equal.", "Consider the set of triples $(a, b, f_{a, b})$ endowed with the preorder", "$$", "(a, b, f_{a, b}) \\geq (a', b', f_{a', b'})", "\\Leftrightarrow", "a \\geq a',\\ b' \\geq b,\\text{ and }", "f_{a', b'} \\circ h_{a, a'} = g_{b', b} \\circ f_{a, b}", "$$", "where $h_{a, a'} : X_a \\to X_{a'}$ and $g_{b', b} : S_{b'} \\to S_b$", "are the transition morphisms. The remarks above show that this system", "is directed. It follows formally from the equalities", "$X = \\lim X_a$ and $S = \\lim S_b$ that", "$$", "X = \\lim_{(a, b, f_{a, b})} X_a \\times_{f_{a, b}, S_b} S.", "$$", "where the limit is over our directed system above. The transition morphisms", "$X_a \\times_{S_b} S \\to X_{a'} \\times_{S_{b'}} S$ are affine as", "the composition", "$$", "X_a \\times_{S_b} S \\to X_a \\times_{S_{b'}} S \\to X_{a'} \\times_{S_{b'}} S", "$$", "where the first morphism is a closed immersion (by", "Schemes, Lemma \\ref{schemes-lemma-fibre-product-after-map})", "and the second is a base change of an affine morphism", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-affine})", "and the composition of affine morphisms is affine", "(Morphisms, Lemma \\ref{morphisms-lemma-composition-affine}).", "The morphisms $f_{a, b}$ are of finite presentation", "(Morphisms, Lemmas", "\\ref{morphisms-lemma-noetherian-finite-type-finite-presentation} and", "\\ref{morphisms-lemma-finite-presentation-permanence})", "and hence the base changes $X_a \\times_{f_{a, b}, S_b} S \\to S$", "are of finite presentation", "(Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-presentation})." ], "refs": [ "limits-proposition-approximate", "limits-proposition-characterize-locally-finite-presentation", "schemes-lemma-fibre-product-after-map", "morphisms-lemma-base-change-affine", "morphisms-lemma-composition-affine", "morphisms-lemma-noetherian-finite-type-finite-presentation", "morphisms-lemma-finite-presentation-permanence", "morphisms-lemma-base-change-finite-presentation" ], "ref_ids": [ 15126, 15127, 7711, 5176, 5175, 5245, 5247, 5240 ] } ], "ref_ids": [] }, { "id": 15056, "type": "theorem", "label": "limits-lemma-integral-limit-finite-and-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-integral-limit-finite-and-finite-presentation", "contents": [ "Let $X \\to S$ be an integral morphism with $S$ quasi-compact and", "quasi-separated. Then $X = \\lim X_i$ with $X_i \\to S$ finite and", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Consider the sheaf $\\mathcal{A} = f_*\\mathcal{O}_X$.", "This is a quasi-coherent sheaf of $\\mathcal{O}_S$-algebras, see", "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}.", "Combining", "Properties, Lemma", "\\ref{properties-lemma-integral-algebra-directed-colimit-finite}", "we can write $\\mathcal{A} = \\colim_i \\mathcal{A}_i$ as a filtered", "colimit of finite and finitely presented $\\mathcal{O}_S$-algebras.", "Then", "$$", "X_i = \\underline{\\Spec}_S(\\mathcal{A}_i)", "\\longrightarrow", "S", "$$", "is a finite and finitely presented morphism of schemes. By construction", "$X = \\lim_i X_i$ which proves the lemma." ], "refs": [ "schemes-lemma-push-forward-quasi-coherent", "properties-lemma-integral-algebra-directed-colimit-finite" ], "ref_ids": [ 7730, 3030 ] } ], "ref_ids": [] }, { "id": 15057, "type": "theorem", "label": "limits-lemma-descend-affine-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-descend-affine-finite-presentation", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If $f$ is affine, then there exists an index $i \\geq 0$", "such that $f_i$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Let $Y_0 = \\bigcup_{j = 1, \\ldots, m} V_{j, 0}$ be a finite affine", "open covering. Set $U_{j, 0} = f_0^{-1}(V_{j, 0})$. For $i \\geq 0$", "we denote $V_{j, i}$ the inverse image of $V_{j, 0}$ in $Y_i$ and", "$U_{j, i} = f_i^{-1}(V_{j, i})$. Similarly we have", "$U_j = f^{-1}(V_j)$. Then $U_j = \\lim_{i \\geq 0} U_{j, i}$", "(see Lemma \\ref{lemma-directed-inverse-system-has-limit}).", "Since $U_j$ is affine by assumption we see that", "each $U_{j, i}$ is affine for $i$ large enough, see", "Lemma \\ref{lemma-limit-affine}. As there are finitely many $j$ we", "can pick an $i$ which works for all $j$. Thus $f_i$ is", "affine for $i$ large enough, see", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-affine}." ], "refs": [ "limits-lemma-directed-inverse-system-has-limit", "limits-lemma-limit-affine", "morphisms-lemma-characterize-affine" ], "ref_ids": [ 15027, 15043, 5172 ] } ], "ref_ids": [] }, { "id": 15058, "type": "theorem", "label": "limits-lemma-descend-finite-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-descend-finite-finite-presentation", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is a finite morphism, and", "\\item $f_0$ is locally of finite type,", "\\end{enumerate}", "then there exists an $i \\geq 0$ such that $f_i$ is finite." ], "refs": [], "proofs": [ { "contents": [ "A finite morphism is affine, see", "Morphisms, Definition \\ref{morphisms-definition-integral}.", "Hence by Lemma \\ref{lemma-descend-affine-finite-presentation} above", "after increasing $0$ we may assume that $f_0$ is affine.", "By writing $Y_0$ as a finite union of affines we reduce to proving", "the result when $X_0$ and $Y_0$ are affine and map", "into a common affine $W \\subset S_0$. The corresponding algebra", "statement follows from Algebra, Lemma \\ref{algebra-lemma-colimit-finite}." ], "refs": [ "morphisms-definition-integral", "limits-lemma-descend-affine-finite-presentation", "algebra-lemma-colimit-finite" ], "ref_ids": [ 5573, 15057, 1391 ] } ], "ref_ids": [] }, { "id": 15059, "type": "theorem", "label": "limits-lemma-descend-unramified", "categories": [ "limits" ], "title": "limits-lemma-descend-unramified", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is unramified, and", "\\item $f_0$ is locally of finite type,", "\\end{enumerate}", "then there exists an $i \\geq 0$ such that $f_i$ is unramified." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite affine open covering", "$Y_0 = \\bigcup_{j = 1, \\ldots, m} Y_{j, 0}$", "such that each $Y_{j, 0}$ maps into an affine open", "$S_{j, 0} \\subset S_0$. For each $j$ let", "$f_0^{-1}Y_{j, 0} = \\bigcup_{k = 1, \\ldots, n_j} X_{k, 0}$ be a", "finite affine open covering. Since the property of being unramified is", "local we see that it suffices to prove the lemma for the morphisms", "of affines $X_{k, i} \\to Y_{j, i} \\to S_{j, i}$ which are the base", "changes of $X_{k, 0} \\to Y_{j, 0} \\to S_{j, 0}$ to $S_i$.", "Thus we reduce to the case that $X_0, Y_0, S_0$ are affine", "\\medskip\\noindent", "In the affine case we reduce to the following algebra result.", "Suppose that $R = \\colim_{i \\in I} R_i$. For some $0 \\in I$", "suppose given an $R_0$-algebra map $A_i \\to B_i$ of finite type.", "If $R \\otimes_{R_0} A_0 \\to R \\otimes_{R_0} B_0$ is unramified, then", "for some $i \\geq 0$ the map", "$R_i \\otimes_{R_0} A_0 \\to R_i \\otimes_{R_0} B_0$ is unramified.", "This follows from Algebra,", "Lemma \\ref{algebra-lemma-colimit-unramified}." ], "refs": [ "algebra-lemma-colimit-unramified" ], "ref_ids": [ 1393 ] } ], "ref_ids": [] }, { "id": 15060, "type": "theorem", "label": "limits-lemma-descend-closed-immersion-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-descend-closed-immersion-finite-presentation", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is a closed immersion, and", "\\item $f_0$ is locally of finite type,", "\\end{enumerate}", "then there exists an $i \\geq 0$ such that $f_i$ is a closed immersion." ], "refs": [], "proofs": [ { "contents": [ "A closed immersion is affine, see", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-affine}.", "Hence by Lemma \\ref{lemma-descend-affine-finite-presentation} above", "after increasing $0$ we may assume that $f_0$ is affine.", "By writing $Y_0$ as a finite union of affines we reduce to proving", "the result when $X_0$ and $Y_0$ are affine and map", "into a common affine $W \\subset S_0$. The corresponding algebra", "statement is a consequence of", "Algebra, Lemma \\ref{algebra-lemma-colimit-surjective}." ], "refs": [ "morphisms-lemma-closed-immersion-affine", "limits-lemma-descend-affine-finite-presentation", "algebra-lemma-colimit-surjective" ], "ref_ids": [ 5177, 15057, 1392 ] } ], "ref_ids": [] }, { "id": 15061, "type": "theorem", "label": "limits-lemma-descend-separated-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-descend-separated-finite-presentation", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If $f$ is separated, then $f_i$ is separated for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-descend-closed-immersion-finite-presentation}", "to the diagonal morphism $\\Delta_{X_0/S_0} : X_0 \\to X_0 \\times_{S_0} X_0$.", "(This is permissible as diagonal morphisms are locally of finite type", "and the fibre product $X_0 \\times_{S_0} X_0$ is quasi-compact and", "quasi-separated, see", "Schemes, Lemma \\ref{schemes-lemma-diagonal-immersion},", "Morphisms, Lemma \\ref{morphisms-lemma-immersion-locally-finite-type}, and", "Schemes, Remark \\ref{schemes-remark-quasi-compact-and-quasi-separated}." ], "refs": [ "limits-lemma-descend-closed-immersion-finite-presentation", "schemes-lemma-diagonal-immersion", "morphisms-lemma-immersion-locally-finite-type", "schemes-remark-quasi-compact-and-quasi-separated" ], "ref_ids": [ 15060, 7707, 5201, 7763 ] } ], "ref_ids": [] }, { "id": 15062, "type": "theorem", "label": "limits-lemma-descend-flat-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-descend-flat-finite-presentation", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is flat,", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then $f_i$ is flat for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite affine open covering", "$Y_0 = \\bigcup_{j = 1, \\ldots, m} Y_{j, 0}$", "such that each $Y_{j, 0}$ maps into an affine open", "$S_{j, 0} \\subset S_0$. For each $j$ let", "$f_0^{-1}Y_{j, 0} = \\bigcup_{k = 1, \\ldots, n_j} X_{k, 0}$ be a", "finite affine open covering. Since the property of being flat is", "local we see that it suffices to prove the lemma for the morphisms", "of affines $X_{k, i} \\to Y_{j, i} \\to S_{j, i}$ which are the base", "changes of $X_{k, 0} \\to Y_{j, 0} \\to S_{j, 0}$ to $S_i$.", "Thus we reduce to the case that $X_0, Y_0, S_0$ are affine", "\\medskip\\noindent", "In the affine case we reduce to the following algebra result.", "Suppose that $R = \\colim_{i \\in I} R_i$. For some $0 \\in I$", "suppose given an $R_0$-algebra map $A_i \\to B_i$ of finite presentation.", "If $R \\otimes_{R_0} A_0 \\to R \\otimes_{R_0} B_0$ is flat, then", "for some $i \\geq 0$ the map", "$R_i \\otimes_{R_0} A_0 \\to R_i \\otimes_{R_0} B_0$ is flat.", "This follows from Algebra,", "Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat} part (3)." ], "refs": [ "algebra-lemma-flat-finite-presentation-limit-flat" ], "ref_ids": [ 1389 ] } ], "ref_ids": [] }, { "id": 15063, "type": "theorem", "label": "limits-lemma-descend-finite-locally-free", "categories": [ "limits" ], "title": "limits-lemma-descend-finite-locally-free", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is finite locally free (of degree $d$),", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then $f_i$ is finite locally free (of degree $d$) for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "By", "Lemmas \\ref{lemma-descend-flat-finite-presentation} and", "\\ref{lemma-descend-finite-finite-presentation}", "we find an $i$ such that $f_i$ is flat and finite.", "On the other hand, $f_i$ is locally of finite presentation.", "Hence $f_i$ is finite locally free by", "Morphisms, Lemma \\ref{morphisms-lemma-finite-flat}.", "If moreover $f$ is finite locally free of degree $d$,", "then the image of $Y \\to Y_i$ is contained in the", "open and closed locus $W_d \\subset Y_i$ over which $f_i$ has degree", "$d$. By", "Lemma \\ref{lemma-limit-contained-in-constructible}", "we see that for some $i' \\geq i$ the image of $Y_{i'} \\to Y_i$", "is contained in $W_d$.", "Then $f_{i'}$ will be finite locally free of degree $d$." ], "refs": [ "limits-lemma-descend-flat-finite-presentation", "limits-lemma-descend-finite-finite-presentation", "morphisms-lemma-finite-flat", "limits-lemma-limit-contained-in-constructible" ], "ref_ids": [ 15062, 15058, 5471, 15040 ] } ], "ref_ids": [] }, { "id": 15064, "type": "theorem", "label": "limits-lemma-descend-smooth", "categories": [ "limits" ], "title": "limits-lemma-descend-smooth", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is smooth,", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then $f_i$ is smooth for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Being smooth is local on the source and the target (Morphisms,", "Lemma \\ref{morphisms-lemma-smooth-characterize}) hence we may assume", "$S_0, X_0, Y_0$ affine (details omitted). The corresponding algebra fact is", "Algebra, Lemma \\ref{algebra-lemma-colimit-smooth}." ], "refs": [ "morphisms-lemma-smooth-characterize", "algebra-lemma-colimit-smooth" ], "ref_ids": [ 5324, 1396 ] } ], "ref_ids": [] }, { "id": 15065, "type": "theorem", "label": "limits-lemma-descend-etale", "categories": [ "limits" ], "title": "limits-lemma-descend-etale", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is \\'etale,", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then $f_i$ is \\'etale for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Being \\'etale is local on the source and the target (Morphisms,", "Lemma \\ref{morphisms-lemma-etale-characterize}) hence we may assume", "$S_0, X_0, Y_0$ affine (details omitted). The corresponding algebra fact is", "Algebra, Lemma \\ref{algebra-lemma-colimit-etale}." ], "refs": [ "morphisms-lemma-etale-characterize", "algebra-lemma-colimit-etale" ], "ref_ids": [ 5359, 1395 ] } ], "ref_ids": [] }, { "id": 15066, "type": "theorem", "label": "limits-lemma-descend-isomorphism", "categories": [ "limits" ], "title": "limits-lemma-descend-isomorphism", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is an isomorphism, and", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then $f_i$ is an isomorphism for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "By Lemmas \\ref{lemma-descend-etale} and", "\\ref{lemma-descend-closed-immersion-finite-presentation}", "we can find an $i$ such that $f_i$ is flat and a closed immersion.", "Then $f_i$ identifies $X_i$ with an open and closed subscheme of", "$Y_i$, see Morphisms, Lemma", "\\ref{morphisms-lemma-flat-closed-immersions-finite-presentation}.", "By assumption the image of $Y \\to Y_i$ maps into $f_i(X_i)$.", "Thus by Lemma \\ref{lemma-limit-contained-in-constructible}", "we find that $Y_{i'}$ maps into $f_i(X_i)$ for some $i' \\geq i$.", "It follows that $X_{i'} \\to Y_{i'}$ is surjective and we win." ], "refs": [ "limits-lemma-descend-etale", "limits-lemma-descend-closed-immersion-finite-presentation", "morphisms-lemma-flat-closed-immersions-finite-presentation", "limits-lemma-limit-contained-in-constructible" ], "ref_ids": [ 15065, 15060, 5275, 15040 ] } ], "ref_ids": [] }, { "id": 15067, "type": "theorem", "label": "limits-lemma-descend-open-immersion", "categories": [ "limits" ], "title": "limits-lemma-descend-open-immersion", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is an open immersion, and", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then $f_i$ is an open immersion for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-descend-etale} we can find an $i$ such that $f_i$", "is \\'etale. Then $V_i = f_i(X_i)$ is a quasi-compact open subscheme", "of $Y_i$ (Morphisms, Lemma \\ref{morphisms-lemma-etale-open}).", "let $V$ and $V_{i'}$ for $i' \\geq i$ be the inverse image of $V_i$", "in $Y$ and $Y_{i'}$. Then $f : X \\to V$ is an isomorphism", "(namely it is a surjective open immersion). Hence by", "Lemma \\ref{lemma-descend-isomorphism} we see that", "$X_{i'} \\to V_{i'}$ is an isomorphism", "for some $i' \\geq i$ as desired." ], "refs": [ "limits-lemma-descend-etale", "morphisms-lemma-etale-open", "limits-lemma-descend-isomorphism" ], "ref_ids": [ 15065, 5370, 15066 ] } ], "ref_ids": [] }, { "id": 15068, "type": "theorem", "label": "limits-lemma-descend-monomorphism", "categories": [ "limits" ], "title": "limits-lemma-descend-monomorphism", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is a monomorphism, and", "\\item $f_0$ is locally of finite type,", "\\end{enumerate}", "then $f_i$ is a monomorphism for some $i \\geq 0$." ], "refs": [], "proofs": [ { "contents": [ "Recall that a morphism of schemes $V \\to W$ is a monomorphism if and", "only if the diagonal $V \\to V \\times_W V$ is an isomorphism", "(Schemes, Lemma \\ref{schemes-lemma-monomorphism}).", "The morphism $X_0 \\to X_0 \\times_{Y_0} X_0$ is locally of finite", "presentation by", "Morphisms, Lemma \\ref{morphisms-lemma-diagonal-morphism-finite-type}.", "Since $X_0 \\times_{Y_0} X_0$ is quasi-compact and quasi-separated", "(Schemes, Remark \\ref{schemes-remark-quasi-compact-and-quasi-separated})", "we conclude from", "Lemma \\ref{lemma-descend-isomorphism}", "that $\\Delta_i : X_i \\to X_i \\times_{Y_i} X_i$ is an isomorphism for", "some $i \\geq 0$. For this $i$ the morphism $f_i$ is a monomorphism." ], "refs": [ "schemes-lemma-monomorphism", "morphisms-lemma-diagonal-morphism-finite-type", "schemes-remark-quasi-compact-and-quasi-separated", "limits-lemma-descend-isomorphism" ], "ref_ids": [ 7721, 5248, 7763, 15066 ] } ], "ref_ids": [] }, { "id": 15069, "type": "theorem", "label": "limits-lemma-descend-surjective", "categories": [ "limits" ], "title": "limits-lemma-descend-surjective", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is surjective, and", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then there exists an $i \\geq 0$ such that $f_i$ is surjective." ], "refs": [], "proofs": [ { "contents": [ "The morphism $f_0$ is of finite presentation.", "Hence $E = f_0(X_0)$ is a constructible subset of $Y_0$, see", "Morphisms, Lemma \\ref{morphisms-lemma-chevalley}.", "Since $f_i$ is the base change of $f_0$ by", "$Y_i \\to Y_0$ we see that the image of $f_i$ is the", "inverse image of $E$ in $Y_i$. Moreover, we know that", "$Y \\to Y_0$ maps into $E$. Hence we win by", "Lemma \\ref{lemma-limit-contained-in-constructible}." ], "refs": [ "morphisms-lemma-chevalley", "limits-lemma-limit-contained-in-constructible" ], "ref_ids": [ 5250, 15040 ] } ], "ref_ids": [] }, { "id": 15070, "type": "theorem", "label": "limits-lemma-descend-syntomic", "categories": [ "limits" ], "title": "limits-lemma-descend-syntomic", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is syntomic, and", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then there exists an $i \\geq 0$ such that $f_i$ is syntomic." ], "refs": [], "proofs": [ { "contents": [ "Choose a finite affine open covering", "$Y_0 = \\bigcup_{j = 1, \\ldots, m} Y_{j, 0}$", "such that each $Y_{j, 0}$ maps into an affine open", "$S_{j, 0} \\subset S_0$. For each $j$ let", "$f_0^{-1}Y_{j, 0} = \\bigcup_{k = 1, \\ldots, n_j} X_{k, 0}$ be a", "finite affine open covering. Since the property of being syntomic is", "local we see that it suffices to prove the lemma for the morphisms", "of affines $X_{k, i} \\to Y_{j, i} \\to S_{j, i}$ which are the base", "changes of $X_{k, 0} \\to Y_{j, 0} \\to S_{j, 0}$ to $S_i$.", "Thus we reduce to the case that $X_0, Y_0, S_0$ are affine", "\\medskip\\noindent", "In the affine case we reduce to the following algebra result.", "Suppose that $R = \\colim_{i \\in I} R_i$. For some $0 \\in I$", "suppose given an $R_0$-algebra map $A_i \\to B_i$ of finite presentation.", "If $R \\otimes_{R_0} A_0 \\to R \\otimes_{R_0} B_0$ is syntomic, then", "for some $i \\geq 0$ the map", "$R_i \\otimes_{R_0} A_0 \\to R_i \\otimes_{R_0} B_0$ is syntomic.", "This follows from Algebra,", "Lemma \\ref{algebra-lemma-colimit-syntomic}." ], "refs": [ "algebra-lemma-colimit-syntomic" ], "ref_ids": [ 1323 ] } ], "ref_ids": [] }, { "id": 15071, "type": "theorem", "label": "limits-lemma-locally-finite-type-in-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-locally-finite-type-in-finite-presentation", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume:", "\\begin{enumerate}", "\\item The morphism $f$ is locally of finite type.", "\\item The scheme $X$ is quasi-compact and quasi-separated.", "\\end{enumerate}", "Then there exists a morphism of finite presentation", "$f' : X' \\to S$ and an immersion $X \\to X'$ of schemes over $S$." ], "refs": [], "proofs": [ { "contents": [ "By", "Proposition \\ref{proposition-approximate}", "we can write", "$X = \\lim_i X_i$ with each $X_i$ of finite type over $\\mathbf{Z}$ and", "with transition morphisms $f_{ii'} : X_i \\to X_{i'}$ affine.", "Consider the commutative diagram", "$$", "\\xymatrix{", "X \\ar[r] \\ar[rd] & X_{i, S} \\ar[r] \\ar[d] & X_i \\ar[d] \\\\", "& S \\ar[r] & \\Spec(\\mathbf{Z})", "}", "$$", "Note that $X_i$ is of finite presentation over $\\Spec(\\mathbf{Z})$, see", "Morphisms,", "Lemma \\ref{morphisms-lemma-noetherian-finite-type-finite-presentation}.", "Hence the base change $X_{i, S} \\to S$ is of finite presentation by", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-presentation}.", "Thus it suffices to show that the arrow $X \\to X_{i, S}$ is an", "immersion for $i$ sufficiently large.", "\\medskip\\noindent", "To do this we choose a finite affine open covering", "$X = V_1 \\cup \\ldots \\cup V_n$ such that", "$f$ maps each $V_j$ into an affine open $U_j \\subset S$.", "Let $h_{j, a} \\in \\mathcal{O}_X(V_j)$ be a finite", "set of elements which generate $\\mathcal{O}_X(V_j)$ as", "an $\\mathcal{O}_S(U_j)$-algebra, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-finite-type-characterize}.", "By Lemmas \\ref{lemma-descend-opens} and \\ref{lemma-limit-affine}", "(after possibly shrinking $I$) we may assume that", "there exist affine open coverings", "$X_i = V_{1, i} \\cup \\ldots \\cup V_{n, i}$", "compatible with transition maps such that $V_j = \\lim_i V_{j, i}$.", "By Lemma \\ref{lemma-descend-section} we can choose $i$ so large that each", "$h_{j, a}$ comes from an element", "$h_{j, a, i} \\in \\mathcal{O}_{X_i}(V_{j, i})$.", "Thus the arrow in", "$$", "V_j \\longrightarrow U_j \\times_{\\Spec(\\mathbf{Z})} V_{j, i} =", "(V_{j, i})_{U_j} \\subset (V_{j, i})_S \\subset X_{i, S}", "$$", "is a closed immersion. Since $\\bigcup (V_{j, i})_{U_j}$ forms an open of", "$X_{i, S}$ and since the inverse image of $(V_{j, i})_{U_j}$ in $X$", "is $V_j$ it follows that $X \\to X_{i, S}$ is an immersion." ], "refs": [ "limits-proposition-approximate", "morphisms-lemma-noetherian-finite-type-finite-presentation", "morphisms-lemma-base-change-finite-presentation", "morphisms-lemma-locally-finite-type-characterize", "limits-lemma-descend-opens", "limits-lemma-limit-affine" ], "ref_ids": [ 15126, 5245, 5240, 5198, 15041, 15043 ] } ], "ref_ids": [] }, { "id": 15072, "type": "theorem", "label": "limits-lemma-finite-type-closed-in-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-finite-type-closed-in-finite-presentation", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume:", "\\begin{enumerate}", "\\item The morphism $f$ is of locally of finite type.", "\\item The scheme $X$ is quasi-compact and quasi-separated, and", "\\item The scheme $S$ is quasi-separated.", "\\end{enumerate}", "Then there exists a morphism of finite presentation", "$f' : X' \\to S$ and a closed immersion $X \\to X'$ of schemes over $S$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-locally-finite-type-in-finite-presentation} above", "there exists a morphism $Y \\to S$ of finite presentation and an", "immersion $i : X \\to Y$ of schemes over $S$.", "For every point $x \\in X$, there exists an affine open", "$V_x \\subset Y$ such that $i^{-1}(V_x) \\to V_x$ is a", "closed immersion. Since $X$ is quasi-compact we can find", "finitely may affine opens $V_1, \\ldots, V_n \\subset Y$", "such that $i(X) \\subset V_1 \\cup \\ldots \\cup V_n$ and", "$i^{-1}(V_j) \\to V_j$ is a closed immersion. In other words", "such that $i : X \\to X' = V_1 \\cup \\ldots \\cup V_n$ is a", "closed immersion of schemes over $S$.", "Since $S$ is quasi-separated and $Y$ is quasi-separated over $S$", "we deduce that $Y$ is quasi-separated, see", "Schemes, Lemma \\ref{schemes-lemma-separated-permanence}.", "Hence the open immersion $X' = V_1 \\cup \\ldots \\cup V_n \\to Y$", "is quasi-compact. This implies that", "$X' \\to Y$ is of finite presentation, see", "Morphisms,", "Lemma \\ref{morphisms-lemma-quasi-compact-open-immersion-finite-presentation}.", "We conclude since then $X' \\to Y \\to S$ is a composition of morphisms", "of finite presentation, and hence of finite presentation (see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-presentation})." ], "refs": [ "limits-lemma-locally-finite-type-in-finite-presentation", "schemes-lemma-separated-permanence", "morphisms-lemma-quasi-compact-open-immersion-finite-presentation", "morphisms-lemma-composition-finite-presentation" ], "ref_ids": [ 15071, 7714, 5242, 5239 ] } ], "ref_ids": [] }, { "id": 15073, "type": "theorem", "label": "limits-lemma-closed-is-limit-closed-and-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-closed-is-limit-closed-and-finite-presentation", "contents": [ "Let $X \\to Y$ be a closed immersion of schemes. Assume $Y$ quasi-compact and", "quasi-separated. Then $X$ can be written as a directed limit $X = \\lim X_i$", "of schemes over $Y$ where $X_i \\to Y$ is a closed immersion", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{I} \\subset \\mathcal{O}_Y$ be the quasi-coherent sheaf of", "ideals defining $X$ as a closed subscheme of $Y$. By", "Properties, Lemma \\ref{properties-lemma-quasi-coherent-colimit-finite-type}", "we can write $\\mathcal{I}$ as a directed colimit", "$\\mathcal{I} = \\colim_{i \\in I} \\mathcal{I}_i$ of its", "quasi-coherent sheaves of ideals of finite type.", "Let $X_i \\subset Y$ be the closed subscheme defined by $\\mathcal{I}_i$.", "These form an inverse system of schemes indexed by $I$.", "The transition morphisms $X_i \\to X_{i'}$ are affine because", "they are closed immersions. Each $X_i$ is quasi-compact and quasi-separated", "since it is a closed subscheme of $Y$ and $Y$ is quasi-compact and", "quasi-separated by our assumptions.", "We have $X = \\lim_i X_i$ as follows directly from the", "fact that $\\mathcal{I} = \\colim_{i \\in I} \\mathcal{I}_a$.", "Each of the morphisms $X_i \\to Y$ is of finite presentation, see", "Morphisms, Lemma \\ref{morphisms-lemma-closed-immersion-finite-presentation}." ], "refs": [ "properties-lemma-quasi-coherent-colimit-finite-type", "morphisms-lemma-closed-immersion-finite-presentation" ], "ref_ids": [ 3020, 5243 ] } ], "ref_ids": [] }, { "id": 15074, "type": "theorem", "label": "limits-lemma-finite-type-is-limit-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-finite-type-is-limit-finite-presentation", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Assume", "\\begin{enumerate}", "\\item The morphism $f$ is of locally of finite type.", "\\item The scheme $X$ is quasi-compact and quasi-separated, and", "\\item The scheme $S$ is quasi-separated.", "\\end{enumerate}", "Then $X = \\lim X_i$ where the $X_i \\to S$ are of", "finite presentation, the $X_i$ are quasi-compact and quasi-separated,", "and the transition morphisms $X_{i'} \\to X_i$ are closed immersions", "(which implies that $X \\to X_i$ are closed immersions for all $i$)." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-type-closed-in-finite-presentation}", "there is a closed immersion $X \\to Y$ with $Y \\to S$ of", "finite presentation. Then $Y$ is quasi-separated by", "Schemes, Lemma \\ref{schemes-lemma-separated-permanence}.", "Since $X$ is quasi-compact, we may assume", "$Y$ is quasi-compact by replacing $Y$ with a quasi-compact open", "containing $X$. We see that $X = \\lim X_i$ with $X_i \\to Y$ a closed", "immersion of finite presentation by", "Lemma \\ref{lemma-closed-is-limit-closed-and-finite-presentation}.", "The morphisms $X_i \\to S$ are of finite presentation by", "Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-presentation}." ], "refs": [ "limits-lemma-finite-type-closed-in-finite-presentation", "schemes-lemma-separated-permanence", "limits-lemma-closed-is-limit-closed-and-finite-presentation", "morphisms-lemma-composition-finite-presentation" ], "ref_ids": [ 15072, 7714, 15073, 5239 ] } ], "ref_ids": [] }, { "id": 15075, "type": "theorem", "label": "limits-lemma-finite-closed-in-finite-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-finite-closed-in-finite-finite-presentation", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Assume", "\\begin{enumerate}", "\\item $f$ is finite, and", "\\item $S$ is quasi-compact and quasi-separated.", "\\end{enumerate}", "Then there exists a morphism which is finite and of finite presentation", "$f' : X' \\to S$ and a closed immersion $X \\to X'$ of schemes over $S$." ], "refs": [], "proofs": [ { "contents": [ "We may write $X = \\lim X_i$ as in", "Lemma \\ref{lemma-finite-type-is-limit-finite-presentation}.", "Applying Lemma \\ref{lemma-eventually-finite} we see that $X_i \\to S$", "is finite for large enough $i$." ], "refs": [ "limits-lemma-finite-type-is-limit-finite-presentation", "limits-lemma-eventually-finite" ], "ref_ids": [ 15074, 15049 ] } ], "ref_ids": [] }, { "id": 15076, "type": "theorem", "label": "limits-lemma-finite-in-finite-and-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-finite-in-finite-and-finite-presentation", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Assume", "\\begin{enumerate}", "\\item $f$ is finite, and", "\\item $S$ quasi-compact and quasi-separated.", "\\end{enumerate}", "Then $X$ is a directed limit $X = \\lim X_i$", "where the transition maps are closed immersions and the objects", "$X_i$ are finite and of finite presentation over $S$." ], "refs": [], "proofs": [ { "contents": [ "We may write $X = \\lim X_i$ as in", "Lemma \\ref{lemma-finite-type-is-limit-finite-presentation}.", "Applying Lemma \\ref{lemma-eventually-finite} we see that $X_i \\to S$", "is finite for large enough $i$." ], "refs": [ "limits-lemma-finite-type-is-limit-finite-presentation", "limits-lemma-eventually-finite" ], "ref_ids": [ 15074, 15049 ] } ], "ref_ids": [] }, { "id": 15077, "type": "theorem", "label": "limits-lemma-descend-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-descend-finite-presentation", "contents": [ "Let $I$ be a directed set.", "Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$.", "Assume", "\\begin{enumerate}", "\\item the morphisms $f_{ii'} : S_i \\to S_{i'}$ are affine,", "\\item the schemes $S_i$ are quasi-compact and quasi-separated.", "\\end{enumerate}", "Let $S = \\lim_i S_i$. Then we have the following:", "\\begin{enumerate}", "\\item For any morphism of finite presentation $X \\to S$", "there exists an index $i \\in I$ and a morphism of finite", "presentation $X_i \\to S_i$ such that $X \\cong X_{i, S}$ as", "schemes over $S$.", "\\item Given an index $i \\in I$, schemes", "$X_i$, $Y_i$ of finite presentation over $S_i$, and a morphism", "$\\varphi : X_{i, S} \\to Y_{i, S}$ over $S$, there exists an index", "$i' \\geq i$ and a morphism", "$\\varphi_{i'} : X_{i, S_{i'}} \\to Y_{i, S_{i'}}$", "whose base change to $S$ is $\\varphi$.", "\\item Given an index $i \\in I$, schemes $X_i$, $Y_i$ of finite presentation", "over $S_i$ and a pair of morphisms $\\varphi_i, \\psi_i : X_i \\to Y_i$", "whose base changes $\\varphi_{i, S} = \\psi_{i, S}$ are equal,", "there exists an index $i' \\geq i$ such that", "$\\varphi_{i, S_{i'}} = \\psi_{i, S_{i'}}$.", "\\end{enumerate}", "In other words, the category of schemes of finite presentation over", "$S$ is the colimit over $I$ of the categories of schemes of finite", "presentation over $S_i$." ], "refs": [], "proofs": [ { "contents": [ "In case each of the schemes $S_i$ is affine, and we consider", "only affine schemes of finite presentation over $S_i$, resp.\\ $S$", "this lemma is equivalent to", "Algebra, Lemma \\ref{algebra-lemma-colimit-category-fp-algebras}.", "We claim that the affine case implies the lemma in general.", "\\medskip\\noindent", "Let us prove (3). Suppose given an index $i \\in I$, schemes", "$X_i$, $Y_i$ of finite presentation over $S_i$ and a pair of morphisms", "$\\varphi_i, \\psi_i : X_i \\to Y_i$. Assume that the base changes are", "equal: $\\varphi_{i, S} = \\psi_{i, S}$. We will use the notation", "$X_{i'} = X_{i, S_{i'}}$ and $Y_{i'} = Y_{i, S_{i'}}$ for", "$i' \\geq i$. We also set $X = X_{i, S}$ and $Y = Y_{i, S}$.", "Note that according to Lemma \\ref{lemma-scheme-over-limit} we have", "$X = \\lim_{i' \\geq i} X_{i'}$ and similarly for $Y$.", "Additionally we denote $\\varphi_{i'}$ and $\\psi_{i'}$", "(resp.\\ $\\varphi$ and $\\psi$)", "the base change of $\\varphi_i$ and $\\psi_i$ to $S_{i'}$", "(resp.\\ $S$). So our assumption means that $\\varphi = \\psi$.", "Since $Y_i$ and $X_i$ are of finite presentation", "over $S_i$, and since $S_i$ is quasi-compact and quasi-separated, also", "$X_i$ and $Y_i$ are quasi-compact and quasi-separated", "(see Morphisms,", "Lemma \\ref{morphisms-lemma-finite-presentation-quasi-compact-quasi-separated}).", "Hence we may choose a finite affine open covering", "$Y_i = \\bigcup V_{j, i}$ such that each $V_{j, i}$ maps into", "an affine open of $S$. As above, denote $V_{j, i'}$ the inverse", "image of $V_{j, i}$ in $Y_{i'}$ and $V_j$ the inverse image in $Y$.", "The immersions $V_{j, i'} \\to Y_{i'}$ are quasi-compact, and the inverse images", "$U_{j, i'} = \\varphi_i^{-1}(V_{j, i'})$ and", "$U_{j, i'}' = \\psi_i^{-1}(V_{j, i'})$", "are quasi-compact opens of $X_{i'}$. By assumption the inverse images of", "$V_j$ under $\\varphi$ and $\\psi$ in $X$ are equal.", "Hence by Lemma \\ref{lemma-descend-opens}", "there exists an index $i' \\geq i$ such that", "of $U_{j, i'} = U_{j, i'}'$ in $X_{i'}$.", "Choose an finite affine open covering", "$U_{j, i'} = U_{j, i'}' = \\bigcup W_{j, k, i'}$", "which induce coverings $U_{j, i''} = U_{j, i''}' = \\bigcup W_{j, k, i''}$", "for all $i'' \\geq i'$.", "By the affine case there exists", "an index $i''$ such that", "$\\varphi_{i''}|_{W_{j, k, i''}} = \\psi_{i''}|_{W_{j, k, i''}}$", "for all $j, k$. Then $i''$ is an index such that", "$\\varphi_{i''} = \\psi_{i''}$ and (3) is proved.", "\\medskip\\noindent", "Let us prove (2). Suppose given an index $i \\in I$, schemes", "$X_i$, $Y_i$ of finite presentation over $S_i$ and a morphism", "$\\varphi : X_{i, S} \\to Y_{i, S}$. We will use the notation", "$X_{i'} = X_{i, S_{i'}}$ and $Y_{i'} = Y_{i, S_{i'}}$ for", "$i' \\geq i$. We also set $X = X_{i, S}$ and $Y = Y_{i, S}$.", "Note that according to Lemma \\ref{lemma-scheme-over-limit} we have", "$X = \\lim_{i' \\geq i} X_{i'}$ and similarly for $Y$.", "Since $Y_i$ and $X_i$ are of finite presentation", "over $S_i$, and since $S_i$ is quasi-compact and quasi-separated, also", "$X_i$ and $Y_i$ are quasi-compact and quasi-separated", "(see Morphisms,", "Lemma \\ref{morphisms-lemma-finite-presentation-quasi-compact-quasi-separated}).", "Hence we may choose a finite affine open covering", "$Y_i = \\bigcup V_{j, i}$ such that each $V_{j, i}$ maps into", "an affine open of $S$. As above, denote $V_{j, i'}$ the inverse", "image of $V_{j, i}$ in $Y_{i'}$ and $V_j$ the inverse image in $Y$.", "The immersions $V_j \\to Y$ are quasi-compact, and the inverse images", "$U_j = \\varphi^{-1}(V_j)$ are quasi-compact opens of $X$.", "Hence by Lemma \\ref{lemma-descend-opens} there exists an index", "$i' \\geq i$ and quasi-compact opens $U_{j, i'}$ of $X_{i'}$", "whose inverse image in $X$ is $U_j$. Choose an finite affine open covering", "$U_{j, i'} = \\bigcup W_{j, k, i'}$ which induce affine open", "coverings $U_{j, i''} = \\bigcup W_{j, k, i''}$", "for all $i'' \\geq i'$ and an affine open covering", "$U_j = \\bigcup W_{j, k}$. By the affine case there exists", "an index $i''$ and morphisms", "$\\varphi_{j, k, i''} : W_{j, k, i''} \\to V_{j, i''}$", "such that", "$\\varphi|_{W_{j, k}} = \\varphi_{j, k, i'', S}$ for all $j, k$.", "By part (3) proved above, there is a further index $i''' \\geq i''$", "such that", "$$", "\\varphi_{j_1, k_1, i'', S_{i'''}}|_{W_{j_1, k_1, i'''} \\cap W_{j_2, k_2, i'''}}", "=", "\\varphi_{j_2, k_2, i'', S_{i'''}}|_{W_{j_1, k_1, i'''} \\cap W_{j_2, k_2, i'''}}", "$$", "for all $j_1, j_2, k_1, k_2$. Then $i'''$ is an index such that", "there exists a morphism $\\varphi_{i'''} : X_{i'''} \\to Y_{i'''}$", "whose base change to $S$ gives $\\varphi$. Hence (2) holds.", "\\medskip\\noindent", "Let us prove (1). Suppose given a scheme $X$ of finite presentation", "over $S$. Since $X$ is of finite presentation", "over $S$, and since $S$ is quasi-compact and quasi-separated, also", "$X$ is quasi-compact and quasi-separated", "(see Morphisms,", "Lemma \\ref{morphisms-lemma-finite-presentation-quasi-compact-quasi-separated}).", "Choose a finite affine open covering $X = \\bigcup U_j$", "such that each $U_j$ maps into an affine open $V_j \\subset S$.", "Denote $U_{j_1j_2} = U_{j_1} \\cap U_{j_2}$ and", "$U_{j_1j_2j_3} = U_{j_1} \\cap U_{j_2} \\cap U_{j_3}$.", "By Lemmas \\ref{lemma-descend-opens} and \\ref{lemma-limit-affine}", "we can find an index $i_1$ and affine opens $V_{j, i_1} \\subset S_{i_1}$", "such that each $V_j$ is the inverse of this in $S$.", "Let $V_{j, i}$ be the inverse image of $V_{j, i_1}$ in $S_i$ for", "$i \\geq i_1$. By the affine case we may find an index $i_2 \\geq i_1$ and", "affine schemes $U_{j, i_2} \\to V_{j, i_2}$ such", "that $U_j = S \\times_{S_{i_2}} U_{j, i_2}$ is the base change.", "Denote $U_{j, i} = S_i \\times_{S_{i_2}} U_{j, i_2}$ for $i \\geq i_2$.", "By Lemma \\ref{lemma-descend-opens} there exists an index", "$i_3 \\geq i_2$ and open subschemes", "$W_{j_1, j_2, i_3} \\subset U_{j_1, i_3}$", "whose base change to $S$ is equal to $U_{j_1j_2}$.", "Denote $W_{j_1, j_2, i} = S_i \\times_{S_{i_3}} W_{j_1, j_2, i_3}$", "for $i \\geq i_3$. By part (2) shown above there exists an index", "$i_4 \\geq i_3$ and morphisms", "$\\varphi_{j_1, j_2, i_4} : W_{j_1, j_2, i_4} \\to W_{j_2, j_1, i_4}$", "whose base change to $S$ gives the identity morphism", "$U_{j_1j_2} = U_{j_2j_1}$ for all $j_1, j_2$.", "For all $i \\geq i_4$ denote", "$\\varphi_{j_1, j_2, i} = \\text{id}_S \\times \\varphi_{j_1, j_2, i_4}$", "the base change. We claim that for some $i_5 \\geq i_4$ the system", "$((U_{j, i_5})_j, (W_{j_1, j_2, i_5})_{j_1, j_2},", "(\\varphi_{j_1, j_2, i_5})_{j_1, j_2})$ forms a glueing datum", "as in Schemes, Section \\ref{schemes-section-glueing-schemes}.", "In order to see this we have to verify that for $i$ large enough", "we have", "$$", "\\varphi_{j_1, j_2, i}^{-1}(W_{j_1, j_2, i} \\cap W_{j_1, j_3, i})", "=", "W_{j_1, j_2, i} \\cap W_{j_1, j_3, i}", "$$", "and that for large enough $i$ the cocycle condition holds.", "The first condition follows from Lemma \\ref{lemma-descend-opens}", "and the fact that $U_{j_2j_1j_3} = U_{j_1j_2j_3}$.", "The second from part (1) of the lemma proved above and the fact", "that the cocycle condition holds for the maps", "$\\text{id} : U_{j_1j_2} \\to U_{j_2j_1}$.", "Ok, so now we can use Schemes, Lemma \\ref{schemes-lemma-glue-schemes}", "to glue the system", "$((U_{j, i_5})_j, (W_{j_1, j_2, i_5})_{j_1, j_2},", "(\\varphi_{j_1, j_2, i_5})_{j_1, j_2})$ to get a scheme", "$X_{i_5} \\to S_{i_5}$. By construction the base change of", "$X_{i_5}$ to $S$ is formed by glueing the open affines", "$U_j$ along the opens $U_{j_1} \\leftarrow U_{j_1j_2} \\rightarrow U_{j_2}$.", "Hence $S \\times_{S_{i_5}} X_{i_5} \\cong X$ as desired." ], "refs": [ "algebra-lemma-colimit-category-fp-algebras", "limits-lemma-scheme-over-limit", "morphisms-lemma-finite-presentation-quasi-compact-quasi-separated", "limits-lemma-descend-opens", "limits-lemma-scheme-over-limit", "morphisms-lemma-finite-presentation-quasi-compact-quasi-separated", "limits-lemma-descend-opens", "morphisms-lemma-finite-presentation-quasi-compact-quasi-separated", "limits-lemma-descend-opens", "limits-lemma-limit-affine", "limits-lemma-descend-opens", "limits-lemma-descend-opens", "schemes-lemma-glue-schemes" ], "ref_ids": [ 1097, 15028, 5246, 15041, 15028, 5246, 15041, 5246, 15041, 15043, 15041, 15041, 7687 ] } ], "ref_ids": [] }, { "id": 15078, "type": "theorem", "label": "limits-lemma-descend-modules-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-descend-modules-finite-presentation", "contents": [ "Let $I$ be a directed set.", "Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$.", "Assume", "\\begin{enumerate}", "\\item all the morphisms $f_{ii'} : S_i \\to S_{i'}$ are affine,", "\\item all the schemes $S_i$ are quasi-compact and quasi-separated.", "\\end{enumerate}", "Let $S = \\lim_i S_i$. Then we have the following:", "\\begin{enumerate}", "\\item For any sheaf of $\\mathcal{O}_S$-modules", "$\\mathcal{F}$ of finite presentation there exists an index", "$i \\in I$ and a sheaf of $\\mathcal{O}_{S_i}$-modules of finite", "presentation $\\mathcal{F}_i$ such that", "$\\mathcal{F} \\cong f_i^*\\mathcal{F}_i$.", "\\item Suppose given an index $i \\in I$, sheaves", "of $\\mathcal{O}_{S_i}$-modules $\\mathcal{F}_i$, $\\mathcal{G}_i$", "of finite presentation and a morphism", "$\\varphi : f_i^*\\mathcal{F}_i \\to f_i^*\\mathcal{G}_i$ over $S$.", "Then there exists an index $i' \\geq i$ and a morphism", "$\\varphi_{i'} : f_{i'i}^*\\mathcal{F}_i \\to f_{i'i}^*\\mathcal{G}_i$", "whose base change to $S$ is $\\varphi$.", "\\item Suppose given an index $i \\in I$, sheaves of $\\mathcal{O}_{S_i}$-modules", "$\\mathcal{F}_i$, $\\mathcal{G}_i$ of finite presentation", "and a pair of morphisms $\\varphi_i, \\psi_i : \\mathcal{F}_i \\to \\mathcal{G}_i$.", "Assume that the base changes are equal: $f_i^*\\varphi_i = f_i^*\\psi_i$.", "Then there exists an index $i' \\geq i$ such that", "$f_{i'i}^*\\varphi_i = f_{i'i}^*\\psi_i$.", "\\end{enumerate}", "In other words, the category of modules", "of finite presentation over $S$ is the colimit over $I$", "of the categories modules of finite presentation over $S_i$." ], "refs": [], "proofs": [ { "contents": [ "We sketch two proofs, but we omit the details.", "\\medskip\\noindent", "First proof. If $S$ and $S_i$ are affine schemes, then this lemma", "is equivalent to", "Algebra, Lemma \\ref{algebra-lemma-colimit-category-fp-modules}.", "In the general case, use Zariski glueing to deduce it from the affine case.", "\\medskip\\noindent", "Second proof. We use", "\\begin{enumerate}", "\\item there is an equivalence of categories between quasi-coherent", "$\\mathcal{O}_S$-modules and vector bundles over $S$, see", "Constructions, Section \\ref{constructions-section-vector-bundle}, and", "\\item a vector bundle $\\mathbf{V}(\\mathcal{F}) \\to S$ is", "of finite presentation over $S$ if and only if $\\mathcal{F}$", "is an $\\mathcal{O}_S$-module of finite presentation.", "\\end{enumerate}", "Having said this, we can use Lemma \\ref{lemma-descend-finite-presentation}", "to show that the category of vector bundles of finite", "presentation over $S$ is the colimit over $I$", "of the categories of vector bundles over $S_i$." ], "refs": [ "algebra-lemma-colimit-category-fp-modules", "limits-lemma-descend-finite-presentation" ], "ref_ids": [ 1095, 15077 ] } ], "ref_ids": [] }, { "id": 15079, "type": "theorem", "label": "limits-lemma-descend-invertible-modules", "categories": [ "limits" ], "title": "limits-lemma-descend-invertible-modules", "contents": [ "Let $S = \\lim S_i$ be the limit of a directed system of quasi-compact and", "quasi-separated schemes $S_i$ with affine transition morphisms. Then", "\\begin{enumerate}", "\\item any finite locally free $\\mathcal{O}_S$-module is the pullback", "of a finite locally free $\\mathcal{O}_{S_i}$-module for some $i$,", "\\item any invertible $\\mathcal{O}_S$-module is the pullback of an invertible", "$\\mathcal{O}_{S_i}$-module for some $i$, and", "\\item any finite type quasi-coherent ideal $\\mathcal{I} \\subset \\mathcal{O}_S$", "is of the form $\\mathcal{I}_i \\cdot \\mathcal{O}_S$ for some $i$ and some", "finite type quasi-coherent ideal $\\mathcal{I}_i \\subset \\mathcal{O}_{S_i}$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_S$-module. Since", "finite locally free modules are of finite presentation we can find an $i$", "and an $\\mathcal{O}_{S_i}$-module $\\mathcal{E}_i$ of finite presentation", "such that $f_i^*\\mathcal{E}_i \\cong \\mathcal{E}$, see", "Lemma \\ref{lemma-descend-modules-finite-presentation}.", "After increasing $i$ we may assume $\\mathcal{E}_i$ is a flat", "$\\mathcal{O}_{S_i}$-module, see", "Algebra, Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat}.", "(Using this lemma is not necessary, but it is convenient.)", "Then $\\mathcal{E}_i$ is finite locally free by", "Algebra, Lemma \\ref{algebra-lemma-finite-projective}.", "\\medskip\\noindent", "If $\\mathcal{L}$ is an invertible $\\mathcal{O}_S$-module,", "then by the above we can find an $i$ and finite locally", "free $\\mathcal{O}_{S_i}$-modules $\\mathcal{L}_i$", "and $\\mathcal{N}_i$ pulling back to $\\mathcal{L}$ and", "$\\mathcal{L}^{\\otimes -1}$. After possible increasing", "$i$ we see that the map", "$\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes -1}", "\\to \\mathcal{O}_X$ descends to a map", "$\\mathcal{L}_i \\otimes_{\\mathcal{O}_{S_i}} \\mathcal{N}_i \\to", "\\mathcal{O}_{S_i}$. And after increasing $i$ further, we", "may assume it is an isomorphism. It follows that", "$\\mathcal{L}_i$ is an invertible module", "(Modules, Lemma \\ref{modules-lemma-invertible}) and", "the proof of (2) is complete.", "\\medskip\\noindent", "Given $\\mathcal{I}$ as in (3) we see that", "$\\mathcal{O}_S \\to \\mathcal{O}_S/\\mathcal{I}$", "is a map of finitely presented $\\mathcal{O}_S$-modules.", "Hence by Lemma \\ref{lemma-descend-modules-finite-presentation}", "this is the pullback of some", "map $\\mathcal{O}_{S_i} \\to \\mathcal{F}_i$ of finitely", "presented $\\mathcal{O}_{S_i}$-modules. After increasing $i$", "we may assume this map is surjective (details omitted; hint: use", "Algebra, Lemma \\ref{algebra-lemma-module-map-property-in-colimit}", "on affine open cover). Then the kernel of $\\mathcal{O}_{S_i} \\to \\mathcal{F}_i$", "is a finite type quasi-coherent ideal in $\\mathcal{O}_{S_i}$", "whose pullback gives $\\mathcal{I}$." ], "refs": [ "limits-lemma-descend-modules-finite-presentation", "algebra-lemma-flat-finite-presentation-limit-flat", "algebra-lemma-finite-projective", "modules-lemma-invertible", "limits-lemma-descend-modules-finite-presentation", "algebra-lemma-module-map-property-in-colimit" ], "ref_ids": [ 15078, 1389, 795, 13300, 15078, 1094 ] } ], "ref_ids": [] }, { "id": 15080, "type": "theorem", "label": "limits-lemma-descend-module-flat-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-descend-module-flat-finite-presentation", "contents": [ "With notation and assumptions as in", "Lemma \\ref{lemma-descend-finite-presentation}.", "Let $i \\in I$.", "Suppose that $\\varphi_i : X_i \\to Y_i$ is a morphism of schemes", "of finite presentation over $S_i$ and that $\\mathcal{F}_i$ is a", "quasi-coherent $\\mathcal{O}_{X_i}$-module of finite presentation.", "If the pullback of $\\mathcal{F}_i$ to $X_i \\times_{S_i} S$ is flat", "over $Y_i \\times_{S_i} S$, then there exists an index $i' \\geq i$", "such that the pullback of $\\mathcal{F}_i$ to $X_i \\times_{S_i} S_{i'}$", "is flat over $Y_i \\times_{S_i} S_{i'}$." ], "refs": [ "limits-lemma-descend-finite-presentation" ], "proofs": [ { "contents": [ "(This lemma is the analogue of", "Lemma \\ref{lemma-descend-flat-finite-presentation}", "for modules.)", "For $i' \\geq i$ denote $X_{i'} = S_{i'} \\times_{S_i} X_i$,", "$\\mathcal{F}_{i'} = (X_{i'} \\to X_i)^*\\mathcal{F}_i$ and similarly", "for $Y_{i'}$. Denote $\\varphi_{i'}$ the base change", "of $\\varphi_i$ to $S_{i'}$. Also set $X = S \\times_{S_i} X_i$,", "$Y =S \\times_{S_i} X_i$, $\\mathcal{F} = (X \\to X_i)^*\\mathcal{F}_i$", "and $\\varphi$ the base change of $\\varphi_i$ to $S$.", "Let $Y_i = \\bigcup_{j = 1, \\ldots, m} V_{j, i}$ be a finite affine open", "covering such that each $V_{j, i}$ maps into some affine open of $S_i$.", "For each $j = 1, \\ldots m$ let", "$\\varphi_i^{-1}(V_{j, i}) = \\bigcup_{k = 1, \\ldots, m(j)} U_{k, j, i}$", "be a finite affine open covering. For $i' \\geq i$ we denote", "$V_{j, i'}$ the inverse image of $V_{j, i}$ in $Y_{i'}$ and", "$U_{k, j, i'}$ the inverse image of $U_{k, j, i}$ in $X_{i'}$.", "Similarly we have $U_{k, j} \\subset X$ and $V_j \\subset Y$.", "Then $U_{k, j} = \\lim_{i' \\geq i} U_{k, j, i'}$", "and $V_j = \\lim_{i' \\geq i} V_j$", "(see Lemma \\ref{lemma-directed-inverse-system-has-limit}).", "Since $X_{i'} = \\bigcup_{k, j} U_{k, j, i'}$ is a finite open covering", "it suffices to prove the lemma for each of the morphisms", "$U_{k, j, i} \\to V_{j, i}$ and the sheaf $\\mathcal{F}_i|_{U_{k, j, i}}$.", "Hence we see that the lemma reduces to the case that $X_i$ and", "$Y_i$ are affine and map into an affine open of $S_i$, i.e., we", "may also assume that $S$ is affine.", "\\medskip\\noindent", "In the affine case we reduce to the following algebra result.", "Suppose that $R = \\colim_{i \\in I} R_i$. For some $i \\in I$", "suppose given a map $A_i \\to B_i$ of finitely presented $R_i$-algebras.", "Let $N_i$ be a finitely presented $B_i$-module.", "Then, if $R \\otimes_{R_i} N_i$ is flat over $R \\otimes_{R_i} A_i$,", "then for some $i' \\geq i$ the module", "$R_{i'} \\otimes_{R_i} N_i$ is flat over $R_{i'} \\otimes_{R_i} A$.", "This is exactly the result proved in", "Algebra,", "Lemma \\ref{algebra-lemma-flat-finite-presentation-limit-flat} part (3)." ], "refs": [ "limits-lemma-descend-flat-finite-presentation", "limits-lemma-directed-inverse-system-has-limit", "algebra-lemma-flat-finite-presentation-limit-flat" ], "ref_ids": [ 15062, 15027, 1389 ] } ], "ref_ids": [ 15077 ] }, { "id": 15081, "type": "theorem", "label": "limits-lemma-descend-finite-presentation-variant", "categories": [ "limits" ], "title": "limits-lemma-descend-finite-presentation-variant", "contents": [ "For a scheme $T$ denote $\\mathcal{C}_T$ the full subcategory of", "schemes $W$ over $T$ such that $W$ is quasi-compact and quasi-separated", "and such that the structure morphism $W \\to T$ is", "locally of finite presentation.", "Let $S = \\lim S_i$ be a directed limit of schemes with affine", "transition morphisms. Then there is an equivalence", "of categories", "$$", "\\colim \\mathcal{C}_{S_i} \\longrightarrow \\mathcal{C}_S", "$$", "given by the base change functors." ], "refs": [], "proofs": [ { "contents": [ "Fully faithfulness. Suppose we have $i \\in I$ and objects", "$X_i$, $Y_i$ of $\\mathcal{C}_{S_i}$. Denote", "$X = X_i \\times_{S_i} S$ and $Y = Y_i \\times_{S_i} S$.", "Suppose given a morphism $f : X \\to Y$ over $S$.", "We can choose a finite affine open covering", "$Y_i = V_{i, 1} \\cup \\ldots \\cup V_{i, m}$", "such that $V_{i, j} \\to Y_i \\to S_i$ maps into an affine", "open $W_{i, j}$ of $S_i$. Denote $Y = V_1 \\cup \\ldots \\cup V_m$ the", "induced affine open covering of $Y$.", "Since $f : X \\to Y$ is quasi-compact", "(Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence})", "after increasing $i$ we may assume that there is a", "finite open covering $X_i = U_{i, 1} \\cup \\ldots \\cup U_{i, m}$", "by quasi-compact opens such that the inverse image of", "$U_{i, j}$ in $Y$ is $f^{-1}(V_j)$, see", "Lemma \\ref{lemma-descend-opens}.", "By Lemma \\ref{lemma-descend-finite-presentation}", "applied to $f|_{f^{-1}(V_j)}$ over $W_j$ we may assume, after", "increasing $i$, that there is a morphism", "$f_{i, j} : V_{i, j} \\to U_{i, j}$ over $S$ whose base change", "to $S$ is $f|_{f^{-1}(V_j)}$.", "Increasing $i$ more we may assume $f_{i, j}$ and $f_{i, j'}$", "agree on the quasi-compact open $U_{i, j} \\cap U_{i, j'}$.", "Then we can glue these morphisms to get the", "desired morphism $f_i : X_i \\to Y_i$.", "This morphism is unique (up to increasing $i$)", "because this is true for the morphisms $f_{i, j}$.", "\\medskip\\noindent", "To show that the functor is essentially surjective we argue in", "exactly the same way. Namely, suppose that", "$X$ is an object of $\\mathcal{C}_S$. Pick $i \\in I$.", "We can choose a finite affine open covering", "$X = U_1 \\cup \\ldots \\cup U_m$ such that $U_j \\to X \\to S \\to S_i$", "factors through an affine open $W_{i, j} \\subset S_i$.", "Set $W_j = W_{i, j} \\times_{S_i} S$. This is an affine open of $S$.", "By Lemma \\ref{lemma-descend-finite-presentation},", "after increasing $i$, we may assume there exist", "$U_{i, j} \\to W_{i, j}$ of finite presentation", "whose base change to $W_j$ is $U_j$.", "After increasing $i$ we may assume there exist", "quasi-compact opens $U_{i, j, j'} \\subset U_{i, j}$", "whose base changes to $S$ are equal to $U_j \\cap U_{j'}$.", "Claim: after increasing $i$ we may assume the image of the morphism", "$U_{i, j, j'} \\to U_{i, j} \\to W_{i, j}$", "ends up in $W_{i, j} \\cap W_{i, j'}$.", "Namely, because the complement of $W_{i, j} \\cap W_{i, j'}$", "is closed in the affine scheme $W_{i, j}$ it is affine.", "Since $U_j \\cap U_{j'} = \\lim U_{i, j, j'}$ does map into", "$W_{i, j} \\cap W_{i, j'}$", "we can apply Lemma \\ref{lemma-limit-fibre-product-empty}", "to get the claim. Thus we can view both", "$$", "U_{i, j, j'} \\quad\\text{and}\\quad U_{i, j', j}", "$$", "as schemes over $W_{i, j'}$ whose base changes to $W_{j'}$", "recover $U_j \\cap U_{j'}$. Hence after increasing $i$, using", "Lemma \\ref{lemma-descend-finite-presentation},", "we may assume there are isomorphisms", "$U_{i, j, j'} \\to U_{i, j', j}$ over $W_{i, j'}$ and hence", "over $S_i$. Increasing $i$ further (details omitted)", "we may assume these isomorphisms", "satisfy the cocycle condition mentioned in", "Schemes, Section \\ref{schemes-section-glueing-schemes}.", "Applying Schemes, Lemma \\ref{schemes-lemma-glue}", "we obtain an object $X_i$ of $\\mathcal{C}_{S_i}$ whose", "base change to $S$ is isomorphic to $X$; we omit some", "of the verifications." ], "refs": [ "schemes-lemma-quasi-compact-permanence", "limits-lemma-descend-opens", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-finite-presentation", "limits-lemma-limit-fibre-product-empty", "limits-lemma-descend-finite-presentation", "schemes-lemma-glue" ], "ref_ids": [ 7716, 15041, 15077, 15077, 15039, 15077, 7686 ] } ], "ref_ids": [] }, { "id": 15082, "type": "theorem", "label": "limits-lemma-affine", "categories": [ "limits" ], "title": "limits-lemma-affine", "contents": [ "\\begin{slogan}", "A scheme, admitting a finite surjective map from an affine scheme, is affine.", "\\end{slogan}", "Let $f : X \\to S$ be a morphism of schemes.", "Assume that $f$ is surjective and finite, and assume that $X$ is affine.", "Then $S$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is surjective and $X$ is quasi-compact we see that $S$ is", "quasi-compact. Since $X$ is separated and $f$ is surjective and", "universally closed (Morphisms, Lemma", "\\ref{morphisms-lemma-integral-universally-closed}), we see that $S$", "is separated (Morphisms, Lemma", "\\ref{morphisms-lemma-image-universally-closed-separated}).", "\\medskip\\noindent", "By Lemma \\ref{lemma-finite-in-finite-and-finite-presentation}", "we can write $X = \\lim_a X_a$ with $X_a \\to S$ finite and of finite", "presentation. By Lemma \\ref{lemma-limit-affine} we see that $X_a$", "is affine for some $a \\in A$. Replacing $X$ by $X_a$ we may assume", "that $X \\to S$ is surjective, finite, of finite presentation and", "that $X$ is affine.", "\\medskip\\noindent", "By Proposition \\ref{proposition-approximate} we may write", "$S = \\lim_{i \\in I} S_i$ as", "a directed limits of schemes of finite type over $\\mathbf{Z}$.", "By Lemma \\ref{lemma-descend-finite-presentation} we can", "after shrinking $I$ assume there exist schemes $X_i \\to S_i$", "of finite presentation such that $X_{i'} = X_i \\times_S S_{i'}$", "for $i' \\geq i$ and such that $X = \\lim_i X_i$. By", "Lemma \\ref{lemma-descend-finite-finite-presentation} we may", "assume that $X_i \\to S_i$ is finite for all $i \\in I$ as well.", "By Lemma \\ref{lemma-limit-affine} once again we may assume that $X_i$ is", "affine for all $i \\in I$. Hence the result follows from the", "Noetherian case, see Cohomology of Schemes,", "Lemma \\ref{coherent-lemma-image-affine-finite-morphism-affine-Noetherian}." ], "refs": [ "morphisms-lemma-integral-universally-closed", "morphisms-lemma-image-universally-closed-separated", "limits-lemma-finite-in-finite-and-finite-presentation", "limits-lemma-limit-affine", "limits-proposition-approximate", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-finite-finite-presentation", "limits-lemma-limit-affine", "coherent-lemma-image-affine-finite-morphism-affine-Noetherian" ], "ref_ids": [ 5441, 5415, 15076, 15043, 15126, 15077, 15058, 15043, 3337 ] } ], "ref_ids": [] }, { "id": 15083, "type": "theorem", "label": "limits-lemma-affines-glued-in-closed-affine", "categories": [ "limits" ], "title": "limits-lemma-affines-glued-in-closed-affine", "contents": [ "Let $X$ be a scheme which is set theoretically the union of", "finitely many affine closed subschemes. Then $X$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Let $Z_i \\subset X$, $i = 1, \\ldots, n$ be affine closed subschemes such that", "$X = \\bigcup Z_i$ set theoretically. Then $\\coprod Z_i \\to X$ is surjective", "and integral with affine source. Hence $X$ is affine by", "Proposition \\ref{proposition-affine}." ], "refs": [ "limits-proposition-affine" ], "ref_ids": [ 15129 ] } ], "ref_ids": [] }, { "id": 15084, "type": "theorem", "label": "limits-lemma-ample-on-reduction", "categories": [ "limits" ], "title": "limits-lemma-ample-on-reduction", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes", "inducing a homeomorphism of underlying topological spaces.", "Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "Then $i^*\\mathcal{L}$ is ample on $Z$, if and only if $\\mathcal{L}$", "is ample on $X$." ], "refs": [], "proofs": [ { "contents": [ "If $\\mathcal{L}$ is ample, then $i^*\\mathcal{L}$ is ample for", "example by Morphisms, Lemma", "\\ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}.", "Assume $i^*\\mathcal{L}$ is ample. Then $Z$ is quasi-compact", "(Properties, Definition \\ref{properties-definition-ample})", "and separated", "(Properties, Lemma \\ref{properties-lemma-ample-separated}).", "Since $i$ is surjective, we see that $X$ is quasi-compact.", "Since $i$ is universally closed and surjective, we see that", "$X$ is separated (Morphisms, Lemma", "\\ref{morphisms-lemma-image-universally-closed-separated}).", "\\medskip\\noindent", "By Proposition \\ref{proposition-approximate} we can write", "$X = \\lim X_i$ as a directed limit of finite type schemes over $\\mathbf{Z}$", "with affine transition morphisms. We can find an $i$ and an invertible", "sheaf $\\mathcal{L}_i$ on $X_i$ whose pullback to $X$ is isomorphic to", "$\\mathcal{L}$, see Lemma \\ref{lemma-descend-modules-finite-presentation}.", "\\medskip\\noindent", "For each $i$ let $Z_i \\subset X_i$ be the scheme theoretic image", "of the morphism $Z \\to X$. If $\\Spec(A_i) \\subset X_i$ is an affine", "open subscheme with inverse image of $\\Spec(A)$ in $X$ and if", "$Z \\cap \\Spec(A)$ is defined by the ideal $I \\subset A$, then", "$Z_i \\cap \\Spec(A_i)$ is defined by the ideal $I_i \\subset A_i$", "which is the inverse image of $I$ in $A_i$ under the ring map", "$A_i \\to A$, see", "Morphisms, Example \\ref{morphisms-example-scheme-theoretic-image}.", "Since $\\colim A_i/I_i = A/I$ it follows that $\\lim Z_i = Z$.", "By Lemma \\ref{lemma-limit-ample} we see that $\\mathcal{L}_i|_{Z_i}$", "is ample for some $i$. Since $Z$ and hence $X$ maps into $Z_i$", "set theoretically, we see that $X_{i'} \\to X_i$ maps into $Z_i$", "set theoretically for some $i' \\geq i$, see", "Lemma \\ref{lemma-limit-contained-in-constructible}.", "(Observe that since $X_i$ is Noetherian, every closed subset", "of $X_i$ is constructible.) Let $T \\subset X_{i'}$", "be the scheme theoretic inverse image of $Z_i$ in $X_{i'}$.", "Observe that $\\mathcal{L}_{i'}|_T$ is the pullback", "of $\\mathcal{L}_i|_{Z_i}$ and hence ample by", "Morphisms, Lemma \\ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}", "and the fact that $T \\to Z_i$ is an affine morphism.", "Thus we see that $\\mathcal{L}_{i'}$ is ample on $X_{i'}$", "by Cohomology of Schemes, Lemma \\ref{coherent-lemma-ample-on-reduction}.", "Pulling back to $X$ (using the same lemma as above)", "we find that $\\mathcal{L}$ is ample." ], "refs": [ "morphisms-lemma-pullback-ample-tensor-relatively-ample", "properties-definition-ample", "properties-lemma-ample-separated", "morphisms-lemma-image-universally-closed-separated", "limits-proposition-approximate", "limits-lemma-descend-modules-finite-presentation", "limits-lemma-limit-ample", "limits-lemma-limit-contained-in-constructible", "morphisms-lemma-pullback-ample-tensor-relatively-ample", "coherent-lemma-ample-on-reduction" ], "ref_ids": [ 5383, 3088, 3046, 5415, 15126, 15078, 15045, 15040, 5383, 3350 ] } ], "ref_ids": [] }, { "id": 15085, "type": "theorem", "label": "limits-lemma-thickening-quasi-affine", "categories": [ "limits" ], "title": "limits-lemma-thickening-quasi-affine", "contents": [ "Let $i : Z \\to X$ be a closed immersion of schemes", "inducing a homeomorphism of underlying topological spaces.", "Then $X$ is quasi-affine if and only if $Z$ is quasi-affine." ], "refs": [], "proofs": [ { "contents": [ "Recall that a scheme is quasi-affine", "if and only if the structure sheaf is ample, see", "Properties, Lemma \\ref{properties-lemma-quasi-affine-O-ample}.", "Hence if $Z$ is quasi-affine, then $\\mathcal{O}_Z$ is ample,", "hence $\\mathcal{O}_X$ is ample by", "Lemma \\ref{lemma-ample-on-reduction}, hence", "$X$ is quasi-affine. A proof of the converse, which", "can also be seen in an elementary way, is gotten by", "reading the argument just given backwards." ], "refs": [ "properties-lemma-quasi-affine-O-ample", "limits-lemma-ample-on-reduction" ], "ref_ids": [ 3053, 15084 ] } ], "ref_ids": [] }, { "id": 15086, "type": "theorem", "label": "limits-lemma-ample-profinite-set-in-principal-affine", "categories": [ "limits" ], "title": "limits-lemma-ample-profinite-set-in-principal-affine", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{L}$ be an ample invertible sheaf on $X$.", "Assume we have morphisms of schemes", "$$", "\\Spec(k) \\leftarrow \\Spec(A) \\to W \\subset X", "$$", "where $k$ is a field, $A$ is an integral $k$-algebra, $W$ is open in $X$.", "Then there exists an $n > 0$ and a section", "$s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ such that", "$X_s$ is affine, $X_s \\subset W$, and $\\Spec(A) \\to W$ factors through $X_s$" ], "refs": [], "proofs": [ { "contents": [ "Since $\\Spec(A)$ is quasi-compact, we may replace $W$ by a quasi-compact", "open still containing the image of $\\Spec(A) \\to X$.", "Recall that $X$ is quasi-separated and quasi-compact by dint", "of having an ample invertible sheaf, see", "Properties, Definition \\ref{properties-definition-ample} and", "Lemma \\ref{properties-lemma-affine-s-opens-cover-quasi-separated}.", "By Proposition \\ref{proposition-approximate} we can", "write $X = \\lim X_i$ as a limit of a directed system", "of schemes of finite type over $\\mathbf{Z}$", "with affine transition morphisms.", "For some $i$ the ample invertible sheaf $\\mathcal{L}$ on $X$", "descends to an ample invertible sheaf $\\mathcal{L}_i$ on $X_i$", "and the open $W$ is the inverse image of a quasi-compact", "open $W_i \\subset X_i$, see", "Lemmas \\ref{lemma-limit-ample}, \\ref{lemma-descend-invertible-modules}, and", "\\ref{lemma-descend-opens}.", "We may replace $X, W, \\mathcal{L}$ by $X_i, W_i, \\mathcal{L}_i$", "and assume $X$ is of finite presentation over $\\mathbf{Z}$.", "Write $A = \\colim A_j$ as the colimit of its finite $k$-subalgebras.", "Then for some $j$ the morphism $\\Spec(A) \\to X$ factors through", "a morphism $\\Spec(A_j) \\to X$, see", "Proposition \\ref{proposition-characterize-locally-finite-presentation}.", "Since $\\Spec(A_j)$ is finite this reduces the lemma to", "Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-principal-affine}." ], "refs": [ "properties-definition-ample", "properties-lemma-affine-s-opens-cover-quasi-separated", "limits-proposition-approximate", "limits-lemma-limit-ample", "limits-lemma-descend-invertible-modules", "limits-lemma-descend-opens", "limits-proposition-characterize-locally-finite-presentation", "properties-lemma-ample-finite-set-in-principal-affine" ], "ref_ids": [ 3088, 3045, 15126, 15045, 15079, 15041, 15127, 3063 ] } ], "ref_ids": [] }, { "id": 15087, "type": "theorem", "label": "limits-lemma-chow-finite-type", "categories": [ "limits" ], "title": "limits-lemma-chow-finite-type", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $f : X \\to S$ be a separated morphism of finite type.", "Then there exists an $n \\geq 0$ and a diagram", "$$", "\\xymatrix{", "X \\ar[rd] & X' \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_S \\ar[dl] \\\\", "& S &", "}", "$$", "where $X' \\to \\mathbf{P}^n_S$ is an immersion, and", "$\\pi : X' \\to X$ is proper and surjective." ], "refs": [], "proofs": [ { "contents": [ "By Proposition \\ref{proposition-separated-closed-in-finite-presentation}", "we can find a closed immersion $X \\to Y$ where $Y$ is separated", "and of finite presentation over $S$. Clearly, if we prove the assertion", "for $Y$, then the result follows for $X$. Hence we may assume that", "$X$ is of finite presentation over $S$.", "\\medskip\\noindent", "Write $S = \\lim_i S_i$ as a directed limit of Noetherian schemes, see", "Proposition \\ref{proposition-approximate}. By", "Lemma \\ref{lemma-descend-finite-presentation} we can", "find an index $i \\in I$ and a scheme $X_i \\to S_i$ of finite presentation", "so that $X = S \\times_{S_i} X_i$.", "By Lemma \\ref{lemma-descend-separated-finite-presentation}", "we may assume that $X_i \\to S_i$ is separated.", "Clearly, if we prove the assertion for", "$X_i$ over $S_i$, then the assertion holds for $X$. The case", "$X_i \\to S_i$ is treated by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-chow-Noetherian}." ], "refs": [ "limits-proposition-separated-closed-in-finite-presentation", "limits-proposition-approximate", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-separated-finite-presentation", "coherent-lemma-chow-Noetherian" ], "ref_ids": [ 15128, 15126, 15077, 15061, 3354 ] } ], "ref_ids": [] }, { "id": 15088, "type": "theorem", "label": "limits-lemma-chow-EGA", "categories": [ "limits" ], "title": "limits-lemma-chow-EGA", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $f : X \\to S$ be a separated morphism of finite type.", "Assume that $X$ has finitely many irreducible components.", "Then there exists an $n \\geq 0$ and a diagram", "$$", "\\xymatrix{", "X \\ar[rd] & X' \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_S \\ar[dl] \\\\", "& S &", "}", "$$", "where $X' \\to \\mathbf{P}^n_S$ is an immersion, and", "$\\pi : X' \\to X$ is proper and surjective. Moreover, there exists", "an open dense subscheme $U \\subset X$ such that $\\pi^{-1}(U) \\to U$", "is an isomorphism of schemes." ], "refs": [], "proofs": [ { "contents": [ "Let $X = Z_1 \\cup \\ldots \\cup Z_n$ be the decomposition of $X$", "into irreducible components. Let $\\eta_j \\in Z_j$ be the generic point.", "\\medskip\\noindent", "There are (at least) two ways to proceed with the proof.", "The first is to redo the proof of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-chow-Noetherian}", "using the general", "Properties, Lemma \\ref{properties-lemma-point-and-maximal-points-affine}", "to find suitable affine opens in $X$. (This is the ``standard'' proof.)", "The second is to use absolute Noetherian approximation as in", "the proof of Lemma \\ref{lemma-chow-finite-type} above.", "This is what we will do here.", "\\medskip\\noindent", "By Proposition \\ref{proposition-separated-closed-in-finite-presentation}", "we can find a closed immersion $X \\to Y$ where $Y$ is separated", "and of finite presentation over $S$.", "Write $S = \\lim_i S_i$ as a directed limit of Noetherian schemes, see", "Proposition \\ref{proposition-approximate}. By", "Lemma \\ref{lemma-descend-finite-presentation} we can", "find an index $i \\in I$ and a scheme $Y_i \\to S_i$ of finite presentation", "so that $Y = S \\times_{S_i} Y_i$.", "By Lemma \\ref{lemma-descend-separated-finite-presentation}", "we may assume that $Y_i \\to S_i$ is separated.", "We have the following diagram", "$$", "\\xymatrix{", "\\eta_j \\in Z_j \\ar[r] & X \\ar[r] \\ar[rd] & Y \\ar[r] \\ar[d] & Y_i \\ar[d] \\\\", "& & S \\ar[r] & S_i", "}", "$$", "Denote $h : X \\to Y_i$ the composition.", "\\medskip\\noindent", "For $i' \\geq i$ write $Y_{i'} = S_{i'} \\times_{S_i} Y_i$.", "Then $Y = \\lim_{i' \\geq i} Y_{i'}$, see", "Lemma \\ref{lemma-scheme-over-limit}.", "Choose $j, j' \\in \\{1, \\ldots, n\\}$, $j \\not = j'$.", "Note that $\\eta_j$ is not a specialization of $\\eta_{j'}$.", "By Lemma \\ref{lemma-topology-limit}", "we can replace $i$ by a bigger index and assume", "that $h(\\eta_j)$ is not a specialization of $h(\\eta_{j'})$", "for all pairs $(j, j')$ as above.", "For such an index, let", "$Y' \\subset Y_i$ be the scheme theoretic image of", "$h : X \\to Y_i$, see", "Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-image}.", "The morphism $h$ is quasi-compact as the composition of the quasi-compact", "morphisms $X \\to Y$ and $Y \\to Y_i$ (which is affine).", "Hence by", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}", "the morphism $X \\to Y'$ is dominant. Thus the generic points", "of $Y'$ are all contained in the set", "$\\{h(\\eta_1), \\ldots, h(\\eta_n)\\}$, see", "Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-dominant}.", "Since none of the $h(\\eta_j)$ is the specialization of another", "we see that the points $h(\\eta_1), \\ldots, h(\\eta_n)$ are pairwise", "distinct and are each a generic point of $Y'$.", "\\medskip\\noindent", "We apply Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-chow-Noetherian} above to the morphism", "$Y' \\to S_i$. This gives a diagram", "$$", "\\xymatrix{", "Y' \\ar[rd] & Y^* \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_{S_i} \\ar[dl] \\\\", "& S_i &", "}", "$$", "such that $\\pi$ is proper and surjective and an isomorphism over", "a dense open subscheme $V \\subset Y'$. By our choice of $i$ above", "we know that $h(\\eta_1), \\ldots, h(\\eta_n) \\in V$. Consider", "the commutative diagram", "$$", "\\xymatrix{", "X' \\ar@{=}[r] &", "X \\times_{Y'} Y^* \\ar[r] \\ar[d] &", "Y^* \\ar[r] \\ar[d] &", "\\mathbf{P}^n_{S_i} \\ar[ddl] \\\\", "& X \\ar[r] \\ar[d] & Y' \\ar[d] & \\\\", "& S \\ar[r] & S_i &", "}", "$$", "Note that $X' \\to X$ is an isomorphism over the open subscheme", "$U = h^{-1}(V)$ which contains each of the $\\eta_j$ and hence is", "dense in $X$. We conclude $X \\leftarrow X' \\rightarrow \\mathbf{P}^n_S$", "is a solution to the problem posed in the lemma." ], "refs": [ "coherent-lemma-chow-Noetherian", "properties-lemma-point-and-maximal-points-affine", "limits-lemma-chow-finite-type", "limits-proposition-separated-closed-in-finite-presentation", "limits-proposition-approximate", "limits-lemma-descend-finite-presentation", "limits-lemma-descend-separated-finite-presentation", "limits-lemma-scheme-over-limit", "limits-lemma-topology-limit", "morphisms-definition-scheme-theoretic-image", "morphisms-lemma-quasi-compact-scheme-theoretic-image", "morphisms-lemma-quasi-compact-dominant", "coherent-lemma-chow-Noetherian" ], "ref_ids": [ 3354, 3061, 15087, 15128, 15126, 15077, 15061, 15028, 15036, 5539, 5146, 5159, 3354 ] } ], "ref_ids": [] }, { "id": 15089, "type": "theorem", "label": "limits-lemma-eventually-proper", "categories": [ "limits" ], "title": "limits-lemma-eventually-proper", "contents": [ "\\begin{slogan}", "If the base change of a scheme to a limit is proper, then", "already the base change is proper at a finite level.", "\\end{slogan}", "Assumptions and notation as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is proper, and", "\\item $f_0$ is locally of finite type,", "\\end{enumerate}", "then there exists an $i$ such that $f_i$ is proper." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-descend-separated-finite-presentation} we see that", "$f_i$ is separated for some $i \\geq 0$. Replacing", "$0$ by $i$ we may assume that $f_0$ is separated.", "Observe that $f_0$ is quasi-compact, see", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}.", "By Lemma \\ref{lemma-chow-finite-type} we can choose a diagram", "$$", "\\xymatrix{", "X_0 \\ar[rd] & X_0' \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_{Y_0} \\ar[dl] \\\\", "& Y_0 &", "}", "$$", "where $X_0' \\to \\mathbf{P}^n_{Y_0}$ is an immersion, and", "$\\pi : X_0' \\to X_0$ is proper and surjective. Introduce", "$X' = X_0' \\times_{Y_0} Y$ and $X_i' = X_0' \\times_{Y_0} Y_i$.", "By Morphisms, Lemmas \\ref{morphisms-lemma-composition-proper} and", "\\ref{morphisms-lemma-base-change-proper}", "we see that $X' \\to Y$ is proper. Hence $X' \\to \\mathbf{P}^n_Y$ is", "a closed immersion (Morphisms, Lemma", "\\ref{morphisms-lemma-image-proper-scheme-closed}). By", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-is-proper}", "it suffices to prove that $X'_i \\to Y_i$ is proper for some $i$.", "By Lemma \\ref{lemma-descend-closed-immersion-finite-presentation}", "we find that $X'_i \\to \\mathbf{P}^n_{Y_i}$ is", "a closed immersion for $i$ large enough. Then $X'_i \\to Y_i$", "is proper and we win." ], "refs": [ "limits-lemma-descend-separated-finite-presentation", "schemes-lemma-quasi-compact-permanence", "limits-lemma-chow-finite-type", "morphisms-lemma-composition-proper", "morphisms-lemma-base-change-proper", "morphisms-lemma-image-proper-scheme-closed", "morphisms-lemma-image-proper-is-proper", "limits-lemma-descend-closed-immersion-finite-presentation" ], "ref_ids": [ 15061, 7716, 15087, 5408, 5409, 5411, 5413, 15060 ] } ], "ref_ids": [] }, { "id": 15090, "type": "theorem", "label": "limits-lemma-proper-limit-of-proper-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-proper-limit-of-proper-finite-presentation", "contents": [ "Let $f : X \\to S$ be a proper morphism with $S$ quasi-compact and", "quasi-separated. Then $X = \\lim X_i$ is a directed limit of schemes", "$X_i$ proper and of finite presentation over $S$ such that", "all transition morphisms and the morphisms $X \\to X_i$ are closed", "immersions." ], "refs": [], "proofs": [ { "contents": [ "By Proposition \\ref{proposition-separated-closed-in-finite-presentation}", "we can find a closed immersion $X \\to Y$ with $Y$ separated and of", "finite presentation over $S$. By Lemma \\ref{lemma-chow-finite-type}", "we can find a diagram", "$$", "\\xymatrix{", "Y \\ar[rd] & Y' \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_S \\ar[dl] \\\\", "& S &", "}", "$$", "where $Y' \\to \\mathbf{P}^n_S$ is an immersion, and", "$\\pi : Y' \\to Y$ is proper and surjective. By", "Lemma \\ref{lemma-closed-is-limit-closed-and-finite-presentation}", "we can write $X = \\lim X_i$ with $X_i \\to Y$ a closed immersion of", "finite presentation. Denote $X'_i \\subset Y'$, resp.\\ $X' \\subset Y'$", "the scheme theoretic inverse image of $X_i \\subset Y$, resp.\\ $X \\subset Y$.", "Then $\\lim X'_i = X'$. Since $X' \\to S$ is proper", "(Morphisms, Lemmas \\ref{morphisms-lemma-composition-proper}), we see that", "$X' \\to \\mathbf{P}^n_S$ is a closed immersion (Morphisms, Lemma", "\\ref{morphisms-lemma-image-proper-scheme-closed}). Hence for $i$ large enough", "we find that $X'_i \\to \\mathbf{P}^n_S$ is a closed immersion by", "Lemma \\ref{lemma-eventually-closed-immersion}.", "Thus $X'_i$ is proper over $S$.", "For such $i$ the morphism $X_i \\to S$ is proper by", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-is-proper}." ], "refs": [ "limits-proposition-separated-closed-in-finite-presentation", "limits-lemma-chow-finite-type", "limits-lemma-closed-is-limit-closed-and-finite-presentation", "morphisms-lemma-composition-proper", "morphisms-lemma-image-proper-scheme-closed", "limits-lemma-eventually-closed-immersion", "morphisms-lemma-image-proper-is-proper" ], "ref_ids": [ 15128, 15087, 15073, 5408, 5411, 15050, 5413 ] } ], "ref_ids": [] }, { "id": 15091, "type": "theorem", "label": "limits-lemma-proper-limit-of-proper-finite-presentation-noetherian", "categories": [ "limits" ], "title": "limits-lemma-proper-limit-of-proper-finite-presentation-noetherian", "contents": [ "Let $f : X \\to S$ be a proper morphism with $S$ quasi-compact and", "quasi-separated. Then there exists a directed set $I$, an", "inverse system $(f_i : X_i \\to S_i)$ of morphisms of schemes over $I$,", "such that the transition morphisms $X_i \\to X_{i'}$ and $S_i \\to S_{i'}$", "are affine, such that $f_i$ is proper, such that $S_i$ is of finite", "type over $\\mathbf{Z}$, and such that", "$(X \\to S) = \\lim (X_i \\to S_i)$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-proper-limit-of-proper-finite-presentation}", "we can write $X = \\lim_{k \\in K} X_k$ with $X_k \\to S$ proper and", "of finite presentation. Next, by absolute Noetherian approximation", "(Proposition \\ref{proposition-approximate}) we can", "write $S = \\lim_{j \\in J} S_j$ with $S_j$ of finite type over $\\mathbf{Z}$.", "For each $k$ there exists a $j$ and a morphism $X_{k, j} \\to S_j$", "of finite presentation with $X_k \\cong S \\times_{S_j} X_{k, j}$", "as schemes over $S$, see", "Lemma \\ref{lemma-descend-finite-presentation}.", "After increasing $j$ we may assume $X_{k, j} \\to S_j$ is proper, see", "Lemma \\ref{lemma-eventually-proper}. The set $I$ will be consist", "of these pairs $(k, j)$ and the corresponding morphism is $X_{k, j} \\to S_j$.", "For every $k' \\geq k$ we can find a $j' \\geq j$ and a morphism", "$X_{j', k'} \\to X_{j, k}$ over $S_{j'} \\to S_j$ whose base change to $S$", "gives the morphism $X_{k'} \\to X_k$ (follows again from", "Lemma \\ref{lemma-descend-finite-presentation}).", "These morphisms form the transition morphisms of the system. Some details", "omitted." ], "refs": [ "limits-lemma-proper-limit-of-proper-finite-presentation", "limits-proposition-approximate", "limits-lemma-descend-finite-presentation", "limits-lemma-eventually-proper", "limits-lemma-descend-finite-presentation" ], "ref_ids": [ 15090, 15126, 15077, 15089, 15077 ] } ], "ref_ids": [] }, { "id": 15092, "type": "theorem", "label": "limits-lemma-finite-type-eventually-proper", "categories": [ "limits" ], "title": "limits-lemma-finite-type-eventually-proper", "contents": [ "Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of", "schemes over $S$ with affine transition morphisms. Let $Y \\to X$", "be a morphism of schemes over $S$.", "If $Y \\to X$ is proper, $X_i$ quasi-compact and quasi-separated, and", "$Y$ locally of finite type over $S$, then $Y \\to X_i$ is proper", "for $i$ large enough." ], "refs": [], "proofs": [ { "contents": [ "Choose a closed immersion $Y \\to Y'$ with $Y'$ proper and of finite", "presentation over $X$, see", "Lemma \\ref{lemma-proper-limit-of-proper-finite-presentation}.", "Then choose an $i$ and a proper morphism $Y'_i \\to X_i$", "such that $Y' = X \\times_{X_i} Y'_i$. This is possible by", "Lemmas \\ref{lemma-descend-finite-presentation} and", "\\ref{lemma-eventually-proper}. Then after replacing $i$", "by a larger index we have that $Y \\to Y'_i$ is a closed", "immersion, see Lemma \\ref{lemma-finite-type-eventually-closed}." ], "refs": [ "limits-lemma-proper-limit-of-proper-finite-presentation", "limits-lemma-descend-finite-presentation", "limits-lemma-eventually-proper", "limits-lemma-finite-type-eventually-closed" ], "ref_ids": [ 15090, 15077, 15089, 15046 ] } ], "ref_ids": [] }, { "id": 15093, "type": "theorem", "label": "limits-lemma-eventually-proper-support", "categories": [ "limits" ], "title": "limits-lemma-eventually-proper-support", "contents": [ "Assumptions and notation as in Situation \\ref{situation-descent-property}.", "Let $\\mathcal{F}_0$ be a quasi-coherent $\\mathcal{O}_{X_0}$-module.", "Denote $\\mathcal{F}$ and $\\mathcal{F}_i$ the pullbacks of", "$\\mathcal{F}_0$ to $X$ and $X_i$. Assume", "\\begin{enumerate}", "\\item $f_0$ is locally of finite type,", "\\item $\\mathcal{F}_0$ is of finite type,", "\\item the scheme theoretic support of $\\mathcal{F}$ is proper over $Y$.", "\\end{enumerate}", "Then the scheme theoretic support of $\\mathcal{F}_i$ is proper over $Y_i$", "for some $i$." ], "refs": [], "proofs": [ { "contents": [ "We may replace $X_0$ by the scheme theoretic support of $\\mathcal{F}_0$.", "By Morphisms, Lemma \\ref{morphisms-lemma-support-finite-type} this", "guarantees that $X_i$ is the support of $\\mathcal{F}_i$ and $X$ is the", "support of $\\mathcal{F}$. Then, if $Z \\subset X$ denotes the scheme", "theoretic support of $\\mathcal{F}$, we see that $Z \\to X$ is a universal", "homeomorphism. We conclude that $X \\to Y$ is proper as this is true for", "$Z \\to Y$ by assumption, see", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-is-proper}.", "By Lemma \\ref{lemma-eventually-proper} we see that $X_i \\to Y$ is proper", "for some $i$. Then it follows that the scheme theoretic support $Z_i$ of", "$\\mathcal{F}_i$ is proper over $Y$ by", "Morphisms, Lemmas \\ref{morphisms-lemma-closed-immersion-proper} and", "\\ref{morphisms-lemma-composition-proper}." ], "refs": [ "morphisms-lemma-support-finite-type", "morphisms-lemma-image-proper-is-proper", "limits-lemma-eventually-proper", "morphisms-lemma-closed-immersion-proper", "morphisms-lemma-composition-proper" ], "ref_ids": [ 5143, 5413, 15089, 5410, 5408 ] } ], "ref_ids": [] }, { "id": 15094, "type": "theorem", "label": "limits-lemma-separate", "categories": [ "limits" ], "title": "limits-lemma-separate", "contents": [ "Let $f : X \\to S$ be a quasi-compact morphism of schemes.", "Let $g : T \\to S$ be a morphism of schemes.", "Let $t \\in T$ be a point and $Z \\subset X_T$ be a closed", "subscheme such that $Z \\cap X_t = \\emptyset$.", "Then there exists an open neighbourhood", "$V \\subset T$ of $t$, a commutative diagram", "$$", "\\xymatrix{", "V \\ar[d] \\ar[r]_a & T' \\ar[d]^b \\\\", "T \\ar[r]^g & S,", "}", "$$", "and a closed subscheme $Z' \\subset X_{T'}$ such that", "\\begin{enumerate}", "\\item the morphism $b : T' \\to S$ is locally of finite presentation,", "\\item with $t' = a(t)$ we have $Z' \\cap X_{t'} = \\emptyset$, and", "\\item $Z \\cap X_V$ maps into $Z'$ via the morphism $X_V \\to X_{T'}$.", "\\end{enumerate}", "Moreover, we may assume $V$ and $T'$ are affine." ], "refs": [], "proofs": [ { "contents": [ "Let $s = g(t)$. During the proof we may always replace $T$ by an", "open neighbourhood of $t$. Hence we may also replace $S$ by an open", "neighbourhood of $s$. Thus we may and do assume that $T$ and $S$ are affine.", "Say $S = \\Spec(A)$, $T = \\Spec(B)$, $g$ is given by the", "ring map $A \\to B$, and $t$ correspond to the prime ideal", "$\\mathfrak q \\subset B$.", "\\medskip\\noindent", "As $X \\to S$ is quasi-compact and $S$ is affine we may write", "$X = \\bigcup_{i = 1, \\ldots, n} U_i$ as a finite union of affine opens.", "Write $U_i = \\Spec(C_i)$. In particular we have", "$X_T = \\bigcup_{i = 1, \\ldots, n} U_{i, T} =", "\\bigcup_{i = 1, \\ldots n} \\Spec(C_i \\otimes_A B)$.", "Let $I_i \\subset C_i \\otimes_A B$ be the ideal corresponding to the", "closed subscheme $Z \\cap U_{i, T}$. The condition that", "$Z \\cap X_t = \\emptyset$ signifies that $I_i$ generates the", "unit ideal in the ring", "$$", "C_i \\otimes_A \\kappa(\\mathfrak q) =", "(B \\setminus \\mathfrak q)^{-1}\\left(", "C_i \\otimes_A B/\\mathfrak q C_i \\otimes_A B \\right)", "$$", "Since $I_i (B \\setminus \\mathfrak q)^{-1}(C_i \\otimes_A B) =", "(B \\setminus \\mathfrak q)^{-1} I_i$ this means that $1 = x_i/g_i$", "for some $x_i \\in I_i$ and $g_i \\in B$, $g_i \\not \\in \\mathfrak q$.", "Thus, clearing denominators we can find a relation of the form", "$$", "x_i + \\sum\\nolimits_j f_{i, j}c_{i, j} = g_i", "$$", "with $x_i \\in I_i$, $f_{i, j} \\in \\mathfrak q$, $c_{i, j} \\in C_i \\otimes_A B$,", "and $g_i \\in B$, $g_i \\not \\in \\mathfrak q$. After replacing $B$ by", "$B_{g_1 \\ldots g_n}$, i.e., after replacing $T$ by a smaller affine", "neighbourhood of $t$, we may assume the equations read", "$$", "x_i + \\sum\\nolimits_j f_{i, j}c_{i, j} = 1", "$$", "with $x_i \\in I_i$, $f_{i, j} \\in \\mathfrak q$, $c_{i, j} \\in C_i \\otimes_A B$.", "\\medskip\\noindent", "To finish the argument write $B$ as a colimit of finitely presented", "$A$-algebras $B_\\lambda$ over a directed set $\\Lambda$.", "For each $\\lambda$ set", "$\\mathfrak q_\\lambda = (B_\\lambda \\to B)^{-1}(\\mathfrak q)$.", "For sufficiently large $\\lambda \\in \\Lambda$ we can find", "\\begin{enumerate}", "\\item an element", "$x_{i, \\lambda} \\in C_i \\otimes_A B_\\lambda$ which maps to $x_i$,", "\\item elements $f_{i, j, \\lambda} \\in \\mathfrak q_{i, \\lambda}$", "mapping to $f_{i, j}$, and", "\\item elements $c_{i, j, \\lambda} \\in C_i \\otimes_A B_\\lambda$", "mapping to $c_{i, j}$.", "\\end{enumerate}", "After increasing $\\lambda$ a bit more the equation", "$$", "x_{i, \\lambda} + \\sum\\nolimits_j f_{i, j, \\lambda}c_{i, j, \\lambda} = 1", "$$", "will hold. Fix such a $\\lambda$ and set $T' = \\Spec(B_\\lambda)$.", "Then $t' \\in T'$ is the point corresponding to the prime $\\mathfrak q_\\lambda$.", "Finally, let $Z' \\subset X_{T'}$ be the scheme theoretic image of", "$Z \\to X_T \\to X_{T'}$. As $X_T \\to X_{T'}$ is affine, we can compute $Z'$", "on the affine open pieces $U_{i, T'}$ as the closed subscheme associated", "to $\\Ker(C_i \\otimes_A B_\\lambda \\to C_i \\otimes_A B/I_i)$, see", "Morphisms, Example \\ref{morphisms-example-scheme-theoretic-image}.", "Hence $x_{i, \\lambda}$ is in the ideal defining $Z'$. Thus the last", "displayed equation shows that $Z' \\cap X_{t'}$ is empty." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 15095, "type": "theorem", "label": "limits-lemma-test-universally-closed", "categories": [ "limits" ], "title": "limits-lemma-test-universally-closed", "contents": [ "Let $f : X \\to S$ be a quasi-compact morphism of schemes.", "The following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally closed,", "\\item for every morphism $S' \\to S$ which is locally of finite presentation", "the base change $X_{S'} \\to S'$ is closed, and", "\\item for every $n$ the morphism", "$\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$", "is closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (1) implies (2). Let us prove that (2) implies (1).", "Suppose that the base change $X_T \\to T$ is not closed for some", "scheme $T$ over $S$. By", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-closed}", "this means that there exists some specialization $t_1 \\leadsto t$ in", "$T$ and a point $\\xi \\in X_T$ mapping to $t_1$ such that $\\xi$ does not", "specialize to a point in the fibre over $t$. Set", "$Z = \\overline{\\{\\xi\\}} \\subset X_T$. Then $Z \\cap X_t = \\emptyset$. Apply", "Lemma \\ref{lemma-separate}.", "We find an open neighbourhood $V \\subset T$ of $t$, a commutative diagram", "$$", "\\xymatrix{", "V \\ar[d] \\ar[r]_a & T' \\ar[d]^b \\\\", "T \\ar[r]^g & S,", "}", "$$", "and a closed subscheme $Z' \\subset X_{T'}$ such that", "\\begin{enumerate}", "\\item the morphism $b : T' \\to S$ is locally of finite presentation,", "\\item with $t' = a(t)$ we have $Z' \\cap X_{t'} = \\emptyset$, and", "\\item $Z \\cap X_V$ maps into $Z'$ via the morphism $X_V \\to X_{T'}$.", "\\end{enumerate}", "Clearly this means that $X_{T'} \\to T'$ maps the closed subset $Z'$", "to a subset of $T'$ which contains $a(t_1)$ but not $t' = a(t)$.", "Since $a(t_1) \\leadsto a(t) = t'$ we conclude that $X_{T'} \\to T'$", "is not closed. Hence we have shown that $X \\to S$ not universally closed", "implies that $X_{T'} \\to T'$ is not closed for some $T' \\to S$", "which is locally of finite presentation. In order words (2)", "implies (1).", "\\medskip\\noindent", "Assume that $\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$ is", "closed for every integer $n$. We want to prove that $X_T \\to T$ is", "closed for every scheme $T$ which is locally of finite presentation", "over $S$. We may of course assume that $T$ is affine and maps into", "an affine open $V$ of $S$ (since $X_T \\to T$ being a closed is local on $T$).", "In this case there exists a closed immersion $T \\to \\mathbf{A}^n \\times V$", "because $\\mathcal{O}_T(T)$ is a finitely presented", "$\\mathcal{O}_S(V)$-algebra, see", "Morphisms,", "Lemma \\ref{morphisms-lemma-locally-finite-presentation-characterize}.", "Then $T \\to \\mathbf{A}^n \\times S$ is a locally closed immersion.", "Hence we get a cartesian diagram", "$$", "\\xymatrix{", "X_T \\ar[d]_{f_T} \\ar[r] & \\mathbf{A}^n \\times X \\ar[d]^{f_n} \\\\", "T \\ar[r] & \\mathbf{A}^n \\times S", "}", "$$", "of schemes where the horizontal arrows are locally closed immersions.", "Hence any closed subset $Z \\subset X_T$ can be written as", "$X_T \\cap Z'$ for some closed subset $Z' \\subset \\mathbf{A}^n \\times X$.", "Then $f_T(Z) = T \\cap f_n(Z')$ and we see that if $f_n$ is closed, then", "also $f_T$ is closed." ], "refs": [ "schemes-lemma-quasi-compact-closed", "limits-lemma-separate", "morphisms-lemma-locally-finite-presentation-characterize" ], "ref_ids": [ 7702, 15094, 5238 ] } ], "ref_ids": [] }, { "id": 15096, "type": "theorem", "label": "limits-lemma-limited-base-change", "categories": [ "limits" ], "title": "limits-lemma-limited-base-change", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to S$ be a separated morphism of finite type.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is proper.", "\\item For any morphism $S' \\to S$ which is locally of finite type", "the base change $X_{S'} \\to S'$ is closed.", "\\item For every $n \\geq 0$ the morphism", "$\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$ is closed.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "[First proof]", "In view of the fact that a proper morphism is the same thing as", "a separated, finite type, and universally closed morphism, this", "lemma is a special case of Lemma \\ref{lemma-test-universally-closed}." ], "refs": [ "limits-lemma-test-universally-closed" ], "ref_ids": [ 15095 ] } ], "ref_ids": [] }, { "id": 15097, "type": "theorem", "label": "limits-lemma-reach-point-closure-Noetherian", "categories": [ "limits" ], "title": "limits-lemma-reach-point-closure-Noetherian", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume $f$ finite type and $Y$ locally Noetherian.", "Let $y \\in Y$ be a point in the closure of the image of $f$.", "Then there exists a commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d]^f \\\\", "\\Spec(A) \\ar[r] & Y", "}", "$$", "where $A$ is a discrete valuation ring and $K$ is its field of fractions", "mapping the closed point of $\\Spec(A)$ to $y$. Moreover, we can assume", "that the image point of $\\Spec(K) \\to X$ is a generic point $\\eta$", "of an irreducible component of $X$ and that $K = \\kappa(\\eta)$." ], "refs": [], "proofs": [ { "contents": [ "By the non-Noetherian version of this lemma", "(Morphisms, Lemma \\ref{morphisms-lemma-reach-points-scheme-theoretic-image})", "there exists a point $x \\in X$ such that $f(x)$ specializes to $y$.", "We may replace $x$ by any point specializing to $x$, hence we may", "assume that $x$ is a generic point of an irreducible component of $X$.", "This produces a ring map $\\mathcal{O}_{Y, y} \\to \\kappa(x)$", "(see Schemes, Section \\ref{schemes-section-points}).", "Let $R \\subset \\kappa(x)$ be the image. Then $R$ is Noetherian as a quotient", "of the Noetherian local ring $\\mathcal{O}_{Y, y}$.", "On the other hand, the extension $\\kappa(x)$ is", "a finitely generated extension of the fraction field of $R$", "as $f$ is of finite type.", "Thus there exists a discrete valuation ring $A \\subset \\kappa(x)$", "with fraction field $\\kappa(x)$ dominating $R$ by", "Algebra, Lemma \\ref{algebra-lemma-exists-dvr}. Then", "$$", "\\xymatrix{", "\\Spec(\\kappa(x)) \\ar[d] \\ar[rrr] & & & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] & \\Spec(R) \\ar[r] & \\Spec(\\mathcal{O}_{Y, y}) \\ar[r] & Y", "}", "$$", "gives the desired diagram." ], "refs": [ "morphisms-lemma-reach-points-scheme-theoretic-image", "algebra-lemma-exists-dvr" ], "ref_ids": [ 5147, 1028 ] } ], "ref_ids": [] }, { "id": 15098, "type": "theorem", "label": "limits-lemma-Noetherian-dvr-valuative-separation", "categories": [ "limits" ], "title": "limits-lemma-Noetherian-dvr-valuative-separation", "contents": [ "Let $S$ be a locally Noetherian scheme.", "Let $f : X \\to S$ be a morphism of schemes.", "Assume $f$ is locally of finite type.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is separated.", "\\item For any diagram (\\ref{equation-valuative}) there is at most", "one dotted arrow.", "\\item For all diagrams (\\ref{equation-valuative}) with $A$ a discrete", "valuation ring there is at most one dotted arrow.", "\\item For any irreducible component $X_0$ of $X$ with", "generic point $\\eta \\in X_0$, for any discrete valuation ring", "$A \\subset K = \\kappa(\\eta)$ with fraction field $K$ and any", "diagram (\\ref{equation-valuative}) such that", "the morphism $\\Spec(K) \\to X$ is the canonical one", "(see Schemes, Section \\ref{schemes-section-points})", "there is at most one dotted arrow.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Clearly (1) implies (2), (2) implies (3), and (3) implies (4). It", "remains to show (4) implies (1). Assume (4).", "We begin by reducing to $S$ affine. Being separated is a local", "on the base (see", "Schemes, Lemma \\ref{schemes-lemma-characterize-separated}).", "Hence, if we can show that whenever", "$X \\to S$ has (4) that the restriction $X_\\alpha \\to S_\\alpha$ has (4)", "where $S_\\alpha \\subset S$ is an (affine) open subset and $X_\\alpha :=", "f^{-1}(S_\\alpha)$, then we will be done. The", "generic points of the irreducible components of $X_\\alpha$ will be the", "generic points of irreducible components of $X$, since $X_\\alpha$ is", "open in $X$. Therefore, any two distinct dotted arrows in the diagram", "\\begin{equation}", "\\label{equation-valuative-alpha}", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X_\\alpha \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & S_\\alpha", "}", "\\end{equation}", "would then give two distinct arrows in diagram", "(\\ref{equation-valuative}) via the maps $X_\\alpha \\to X$ and", "$S_\\alpha \\to S$, which is a contradiction. Thus we have reduced", "to the case $S$ is affine. We remark that in the course of this", "reduction, we prove that if $X \\to S$ has (4) then the restriction $U", "\\to V$ has (4) for opens $U \\subset X$ and $V \\subset S$ with", "$f(U) \\subset V$.", "\\medskip\\noindent", "We next wish to reduce to the case $X \\to S$ is finite type. Assume", "that we know (4) implies (1) when $X$ is finite type. Since", "$S$ is Noetherian and $X$ is locally of finite type over $S$", "we see $X$ is locally Noetherian as well (see Morphisms,", "Lemma \\ref{morphisms-lemma-finite-type-noetherian}).", "Thus, $X \\to S$ is quasi-separated (see", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian-quasi-separated}),", "and therefore we may apply the valuative criterion to check whether $X$", "is separated (see", "Schemes, Lemma \\ref{schemes-lemma-valuative-criterion-separatedness}).", "Let $X = \\bigcup_\\alpha X_\\alpha$ be an affine open", "cover of $X$. Given any two dotted arrows, in a diagram", "(\\ref{equation-valuative}), the image of the closed points of", "$\\Spec A$ will", "fall in two sets $X_\\alpha$ and $X_\\beta$. Since $X_\\alpha \\cup", "X_\\beta$ is open, for topological reasons it must contain the image of", "$\\Spec(A)$ under both maps. Therefore, the two dotted arrows factor", "through $X_\\alpha \\cup X_\\beta \\to X$, which is a scheme of finite type over", "$S$. Since $X_\\alpha \\cup X_\\beta$ is an open subset of $X$, by our", "previous remark, $X_\\alpha \\cup X_\\beta$ satisfies (4), so by", "assumption, is separated. This implies the two given dotted", "arrows are the same. Therefore, we have reduced to $X \\to S$ is finite type.", "\\medskip\\noindent", "Assume $X \\to S$ of finite type and assume (4).", "Since $X \\to S$ is finite type, and $S$ is an affine Noetherian", "scheme, $X$ is also Noetherian (see", "Morphisms, Lemma \\ref{morphisms-lemma-finite-type-noetherian}).", "Therefore, $X \\to X \\times_S X$ will", "be a quasi-compact immersion of Noetherian schemes. We proceed by", "contradiction. Assume that $X \\to X \\times_S X$ is not closed. Then,", "there is some $y \\in X \\times_S X$ in the closure of the image that is", "not in the image. As $X$ is Noetherian it has finitely many irreducible", "components. Therefore, $y$ is in the closure of the image of one of", "the irreducible components $X_0 \\subset X$. Give $X_0$ the reduced", "induced structure. The composition $X_0 \\to X \\to X \\times_S X$", "factors through the closed subscheme $X_0 \\times_S X_0 \\subset X \\times_S X$.", "Denote the closure of $\\Delta(X_0)$ in $X_0 \\times_S X_0$", "by $\\bar X_0$ (again as a reduced closed subscheme). Thus $y \\in \\bar X_0$.", "Since $X_0 \\to X_0 \\times_S X_0$ is an immersion, the image of $X_0$", "will be open in $\\bar X_0$. Hence $X_0$ and $\\bar X_0$ are", "birational. Since $\\bar{X}_0$ is a closed subscheme of a", "Noetherian scheme, it is Noetherian. Thus, the local ring", "$\\mathcal O_{{\\bar X_0, y}}$ is a local Noetherian domain with fraction", "field $K$ equal to the function field of $X_0$. By the Krull-Akizuki", "theorem (see Algebra, Lemma \\ref{algebra-lemma-exists-dvr}), there exists a", "discrete valuation ring $A$ dominating $\\mathcal O_{{\\bar X_0, y}}$", "with fraction field $K$. This allows to construct a diagram:", "\\begin{equation}", "\\label{equation-valuative-generic}", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X_0 \\ar[d]^{\\Delta} \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ur]& X_0 \\times_S X_0 \\\\", "}", "\\end{equation}", "which sends $\\Spec K$ to the generic point of $\\Delta(X_0)$ and", "the closed point of $A$ to $y \\in X_0 \\times_S X_0$ (use the material in", "Schemes, Section \\ref{schemes-section-points} to construct the arrows).", "There cannot even exist", "a set theoretic dotted arrow, since $y$ is not in the image of", "$\\Delta$ by our choice of $y$. By categorical means, the existence of", "the dotted arrow in the above diagram is equivalent to the uniqueness", "of the dotted arrow in the following diagram:", "\\begin{equation}", "\\label{equation-valuative-nonexistent}", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X_0 \\ar[d]\\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ur] & S \\\\", "}", "\\end{equation}", "Therefore, we have non-uniqueness in this latter diagram by the", "nonexistence in the first. Therefore, $X_0$ does not satisfy", "uniqueness for discrete valuation rings, and since $X_0$ is an", "irreducible component of $X$, we have that $X \\to S$ does not satisfy", "(4). Therefore, we have shown (4) implies (1)." ], "refs": [ "schemes-lemma-characterize-separated", "morphisms-lemma-finite-type-noetherian", "properties-lemma-locally-Noetherian-quasi-separated", "schemes-lemma-valuative-criterion-separatedness", "morphisms-lemma-finite-type-noetherian", "algebra-lemma-exists-dvr" ], "ref_ids": [ 7710, 5202, 2953, 7720, 5202, 1028 ] } ], "ref_ids": [] }, { "id": 15099, "type": "theorem", "label": "limits-lemma-Noetherian-dvr-valuative-proper", "categories": [ "limits" ], "title": "limits-lemma-Noetherian-dvr-valuative-proper", "contents": [ "Let $S$ be a locally Noetherian scheme.", "Let $f : X \\to S$ be a morphism of finite type.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is proper.", "\\item For any diagram (\\ref{equation-valuative}) there exists exactly", "one dotted arrow.", "\\item For all diagrams (\\ref{equation-valuative}) with $A$ a discrete", "valuation ring there exists exactly one dotted arrow.", "\\item For any irreducible component $X_0$ of $X$ with", "generic point $\\eta \\in X_0$, for any discrete valuation ring", "$A \\subset K = \\kappa(\\eta)$ with fraction field $K$ and any", "diagram (\\ref{equation-valuative}) such that", "the morphism $\\Spec(K) \\to X$ is the canonical one", "(see", "Schemes, Section \\ref{schemes-section-points})", "there exists exactly one dotted arrow.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "(1) implies (2) implies (3) implies (4). We will now show (4) implies", "(1). As in the proof of Lemma \\ref{lemma-Noetherian-dvr-valuative-separation},", "we can reduce to the", "case $S$ is affine, since properness is local on the base, and if $X", "\\to S$ satisfies (4), then $X_\\alpha \\to S_\\alpha$ does as well for", "open $S_\\alpha \\subset S$ and $X_\\alpha = f^{-1}(S_\\alpha)$.", "\\medskip\\noindent", "Now $S$ is a Noetherian scheme, and so $X$ is as well, since $X \\to", "S$ is of finite type. Now we may use Chow's lemma", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-chow-Noetherian})", "to get a surjective, proper, birational", "$X' \\to X$ and an immersion $X' \\to \\mathbf{P}^n_S$. We wish to", "show $X \\to S$ is universally closed. As in the proof of Lemma", "\\ref{lemma-limited-base-change}, it is enough to check that", "$X' \\to \\mathbf{P}^n_S$ is a closed immersion.", "For the sake of contradiction, assume that $X' \\to", "\\mathbf{P}^n_S$ is not a closed immersion. Then there is some $y", "\\in \\mathbf{P}^n_S$ that is in the closure of the image of $X'$, but", "is not in the image. So $y$ is in the closure of the image of an", "irreducible component $X_0'$ of $X'$, but not in the image.", "Let $\\bar X_0' \\subset \\mathbf{P}^n_S$ be the closure of", "the image of $X_0'$. As $X' \\to \\mathbf{P}^n_S$ is an immersion", "of Noetherian schemes, the morphism $X'_0 \\to \\bar X_0'$ is", "open and dense. By", "Algebra, Lemma \\ref{algebra-lemma-exists-dvr}", "or", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian-specialization-dvr}", "we can find a discrete valuation ring $A$ dominating", "$\\mathcal{O}_{\\bar X_0', y}$ and with identical field", "of fractions $K$. It is clear that", "$K$ is the residue field at the generic point of $X_0'$.", "Thus the solid commutative diagram", "\\begin{equation}", "\\label{equation-solid}", "\\xymatrix{", "\\Spec K \\ar[r] \\ar[d] & X' \\ar [r] \\ar[d] &", "\\mathbf{P}^n_S \\ar[d] \\\\", "\\Spec A \\ar@{-->}[r] \\ar@{-->}[ru] \\ar[urr] & X \\ar[r] & S\\\\", "}", "\\end{equation}", "Note that the closed point of $A$ maps to $y \\in \\mathbf{P}^n_S$. By", "construction, there does not exist a set theoretic lift to $X'$.", "As $X' \\to X$ is birational, the image of $X'_0$ in $X$ is an", "irreducible component $X_0$ of $X$ and $K$ is also identified with", "the function field of $X_0$. Hence, as $X \\to S$ is assumed to satisfy (4),", "the dotted arrow $\\Spec(A) \\to X$ exists.", "Since $X' \\to X$ is proper, the dotted", "arrow lifts to the dotted arrow $\\Spec(A) \\to X'$ (use Schemes,", "Proposition \\ref{schemes-proposition-characterize-universally-closed}).", "We can compose this with the immersion $X' \\to \\mathbf{P}^n_S$ to obtain", "another morphism (not depicted in the diagram) from", "$\\Spec(A) \\to \\mathbf{P}^n_S$. Since $\\mathbf{P}^n_S$", "is proper over $S$, it satisfies (2), and so these two morphisms", "agree. This is a contradiction, for we have constructed the", "forbidden lift of our original map $\\Spec(A) \\to \\mathbf{P}^n_S$", "to $X'$." ], "refs": [ "limits-lemma-Noetherian-dvr-valuative-separation", "coherent-lemma-chow-Noetherian", "limits-lemma-limited-base-change", "algebra-lemma-exists-dvr", "properties-lemma-locally-Noetherian-specialization-dvr", "schemes-proposition-characterize-universally-closed" ], "ref_ids": [ 15098, 3354, 15096, 1028, 2959, 7733 ] } ], "ref_ids": [] }, { "id": 15100, "type": "theorem", "label": "limits-lemma-check-universally-closed-Noetherian", "categories": [ "limits" ], "title": "limits-lemma-check-universally-closed-Noetherian", "contents": [ "Let $f : X \\to S$ be a finite type morphism of schemes.", "Assume $S$ is locally Noetherian. Then the following are equivalent", "\\begin{enumerate}", "\\item $f$ is universally closed,", "\\item for every $n$ the morphism", "$\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$ is closed,", "\\item for any diagram (\\ref{equation-valuative}) there exists some", "dotted arrow,", "\\item for all diagrams (\\ref{equation-valuative}) with $A$ a discrete", "valuation ring there exists some dotted arrow.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "The equivalence of (1) and (2) is a special case of", "Lemma \\ref{lemma-test-universally-closed}.", "The equivalence of (1) and (3) is a special case of", "Schemes, Proposition \\ref{schemes-proposition-characterize-universally-closed}.", "Trivially (3) implies (4).", "Thus all we have to do is prove that (4) implies (2).", "We will prove that $\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$", "is closed by the criterion of", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-closed}.", "Pick $n$ and a specialization $z \\leadsto z'$ of points", "in $\\mathbf{A}^n \\times S$ and a point $y \\in \\mathbf{A}^n \\times X$", "lying over $z$. Note that $\\kappa(y)$ is a finitely generated field", "extension of $\\kappa(z)$ as $\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$", "is of finite type. Hence by", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian-specialization-dvr}", "or", "Algebra, Lemma \\ref{algebra-lemma-exists-dvr}", "implies that there exists a discrete valuation ring $A \\subset \\kappa(y)$", "with fraction field $\\kappa(z)$ dominating the image of", "$\\mathcal{O}_{\\mathbf{A}^n \\times S, z'}$ in $\\kappa(z)$.", "This gives a commutative diagram", "$$", "\\xymatrix{", "\\Spec(\\kappa(y)) \\ar[r] \\ar[d] &", "\\mathbf{A}^n \\times X \\ar[d] \\ar[r] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] & \\mathbf{A}^n \\times S \\ar[r] & S", "}", "$$", "Now property (4) implies that there exists a morphism", "$\\Spec(A) \\to X$ which fits into this diagram.", "Since we already have the morphism $\\Spec(A) \\to \\mathbf{A}^n$", "from the left lower horizontal arrow we also get a morphism", "$\\Spec(A) \\to \\mathbf{A}^n \\times X$ fitting into the", "left square. Thus the image $y' \\in \\mathbf{A}^n \\times X$", "of the closed point is a specialization of $y$ lying over $z'$.", "This proves that specializations lift along", "$\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$", "and we win." ], "refs": [ "limits-lemma-test-universally-closed", "schemes-proposition-characterize-universally-closed", "schemes-lemma-quasi-compact-closed", "properties-lemma-locally-Noetherian-specialization-dvr", "algebra-lemma-exists-dvr" ], "ref_ids": [ 15095, 7733, 7702, 2959, 1028 ] } ], "ref_ids": [] }, { "id": 15101, "type": "theorem", "label": "limits-lemma-refined-valuative-criterion-proper", "categories": [ "limits" ], "title": "limits-lemma-refined-valuative-criterion-proper", "contents": [ "Let $f : X \\to S$ and $h : U \\to X$ be morphisms of schemes.", "Assume that $S$ is locally Noetherian, that $f$ and $h$ are of finite type,", "that $f$ is separated, and that $h(U)$ is dense in $X$.", "If given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\", "\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & S", "}", "$$", "where $A$ is a discrete valuation ring with field of fractions $K$, there", "exists a dotted arrow making the diagram commute, then $f$ is proper." ], "refs": [], "proofs": [ { "contents": [ "There is an immediate reduction to the case where $S$ is affine.", "Then $U$ is quasi-compact.", "Let $U = U_1 \\cup \\ldots \\cup U_n$ be an affine open covering.", "We may replace $U$ by $U_1 \\amalg \\ldots \\amalg U_n$ without", "changing the assumptions, hence we may assume $U$ is affine.", "Thus we can find an open immersion $U \\to Y$ over $X$", "with $Y$ proper over $X$. (First put $U$ inside $\\mathbf{A}^n_X$", "using Morphisms, Lemma \\ref{morphisms-lemma-quasi-affine-finite-type-over-S}", "and then take the closure inside $\\mathbf{P}^n_X$, or you can directly", "use Morphisms, Lemma \\ref{morphisms-lemma-quasi-projective-open-projective}.)", "We can assume $U$ is dense in $Y$ (replace $Y$ by the scheme theoretic", "closure of $U$ if necessary, see Morphisms, Section", "\\ref{morphisms-section-scheme-theoretic-closure}).", "Note that $g : Y \\to X$ is surjective as the image is closed", "and contains the dense subset $h(U)$.", "We will show that $Y \\to S$ is proper. This will imply that", "$X \\to S$ is proper by", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-is-proper}", "thereby finishing the proof.", "To show that $Y \\to S$ is proper we will use", "part (4) of Lemma \\ref{lemma-Noetherian-dvr-valuative-proper}.", "To do this consider a diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_y \\ar[d] & Y \\ar[d]^{f \\circ g} \\\\", "\\Spec(A) \\ar[r] \\ar@{..>}[ru] & S", "}", "$$", "where $A$ is a discrete valuation ring with fraction field $K$", "and where $y : \\Spec(K) \\to Y$ is the inclusion of a generic point.", "We have to show there exists a unique dotted arrow.", "Uniqueness holds by the converse to the valuative criterion", "for separatedness", "(Schemes, Lemma \\ref{schemes-lemma-separated-implies-valuative})", "since $Y \\to S$ is separated as the", "composition of the separated morphisms $Y \\to X$ and $X \\to S$", "(Schemes, Lemma \\ref{schemes-lemma-separated-permanence}).", "Existence can be seen as follows.", "As $y$ is a generic point of $Y$, it is contained in $U$.", "By assumption of the lemma", "there exists a morphism $a : \\Spec(A) \\to X$ such that", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_y \\ar[d] & U \\ar[r] & X \\ar[d]^f \\\\", "\\Spec(A) \\ar[rr] \\ar[rru]^a & & S", "}", "$$", "is commutative. Then since $Y \\to X$ is proper, we can", "apply the valuative criterion for properness", "(Morphisms, Lemma \\ref{morphisms-lemma-characterize-proper})", "to find a morphism $b : \\Spec(A) \\to Y$ such that", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r]_y \\ar[d] & Y \\ar[d]^g \\\\", "\\Spec(A) \\ar[r]^a \\ar[ru]^b & X", "}", "$$", "is commutative. This finishes the proof since", "$b$ can serve as the dotted arrow above." ], "refs": [ "morphisms-lemma-quasi-affine-finite-type-over-S", "morphisms-lemma-quasi-projective-open-projective", "morphisms-lemma-image-proper-is-proper", "limits-lemma-Noetherian-dvr-valuative-proper", "schemes-lemma-separated-implies-valuative", "schemes-lemma-separated-permanence", "morphisms-lemma-characterize-proper" ], "ref_ids": [ 5392, 5429, 5413, 15099, 7719, 7714, 5416 ] } ], "ref_ids": [] }, { "id": 15102, "type": "theorem", "label": "limits-lemma-refined-valuative-criterion-separated", "categories": [ "limits" ], "title": "limits-lemma-refined-valuative-criterion-separated", "contents": [ "Let $f : X \\to S$ and $h : U \\to X$ be morphisms of schemes.", "Assume that $S$ is locally Noetherian, that $f$ is locally of finite type,", "that $h$ is of finite type, and that $h(U)$ is dense in $X$.", "If given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\", "\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & S", "}", "$$", "where $A$ is a discrete valuation ring with field of fractions $K$, there", "exists at most one dotted arrow making the diagram commute, then $f$ is", "separated." ], "refs": [], "proofs": [ { "contents": [ "We will apply Lemma \\ref{lemma-refined-valuative-criterion-proper}", "to the morphisms $U \\to X$ and $\\Delta : X \\to X \\times_S X$.", "We check the conditions. Observe that $\\Delta$ is quasi-compact by", "Properties, Lemma \\ref{properties-lemma-locally-Noetherian-quasi-separated}", "(and Schemes, Lemma \\ref{schemes-lemma-compose-after-separated}).", "Of course $\\Delta$ is locally of finite type and separated (true", "for any diagonal morphism).", "Finally, suppose given a commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^\\Delta \\\\", "\\Spec(A) \\ar[rr]^{(a, b)} \\ar@{-->}[rru] & & X \\times_S X", "}", "$$", "where $A$ is a discrete valuation ring with field of fractions $K$.", "Then $a$ and $b$ give two dotted arrows in the diagram of the lemma", "and have to be equal. Hence as dotted arrow we can use $a = b$", "which gives existence. This finishes the proof." ], "refs": [ "limits-lemma-refined-valuative-criterion-proper", "properties-lemma-locally-Noetherian-quasi-separated", "schemes-lemma-compose-after-separated" ], "ref_ids": [ 15101, 2953, 7715 ] } ], "ref_ids": [] }, { "id": 15103, "type": "theorem", "label": "limits-lemma-refined-valuative-criterion-universally-closed", "categories": [ "limits" ], "title": "limits-lemma-refined-valuative-criterion-universally-closed", "contents": [ "Let $f : X \\to S$ and $h : U \\to X$ be morphisms of schemes.", "Assume that $S$ is locally Noetherian, that $f$ and $h$ are of finite type, and", "that $h(U)$ is dense in $X$. If given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\", "\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & S", "}", "$$", "where $A$ is a discrete valuation ring with field of fractions $K$, there", "exists a unique dotted arrow making the diagram commute, then $f$ is proper." ], "refs": [], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-refined-valuative-criterion-separated} and", "\\ref{lemma-refined-valuative-criterion-proper}." ], "refs": [ "limits-lemma-refined-valuative-criterion-separated", "limits-lemma-refined-valuative-criterion-proper" ], "ref_ids": [ 15102, 15101 ] } ], "ref_ids": [] }, { "id": 15104, "type": "theorem", "label": "limits-lemma-limit-dimension", "categories": [ "limits" ], "title": "limits-lemma-limit-dimension", "contents": [ "Let $I$ be a directed set.", "Let $(f_i : X_i \\to S_i)$ be an inverse system of morphisms of schemes", "over $I$. Assume", "\\begin{enumerate}", "\\item all the morphisms $S_{i'} \\to S_i$ are affine,", "\\item all the schemes $S_i$ are quasi-compact and quasi-separated,", "\\item the morphisms $f_i$ are of finite type, and", "\\item the morphisms $X_{i'} \\to X_i \\times_{S_i} S_{i'}$ are closed", "immersions.", "\\end{enumerate}", "Let $f : X = \\lim_i X_i \\to S = \\lim_i S_i$ be the limit.", "Let $d \\geq 0$.", "If every fibre of $f$ has dimension $\\leq d$, then for some $i$", "every fibre of $f_i$ has dimension $\\leq d$." ], "refs": [], "proofs": [ { "contents": [ "For each $i$ let $U_i = \\{x \\in X_i \\mid \\dim_x((X_i)_{f_i(x)}) \\leq d\\}$.", "This is an open subset of $X_i$, see", "Morphisms, Lemma \\ref{morphisms-lemma-openness-bounded-dimension-fibres}.", "Set $Z_i = X_i \\setminus U_i$ (with reduced induced scheme structure).", "We have to show that $Z_i = \\emptyset$ for some $i$.", "If not, then $Z = \\lim Z_i \\not = \\emptyset$, see", "Lemma \\ref{lemma-limit-nonempty}.", "Say $z \\in Z$ is a point. Note that $Z \\subset X$ is a closed subscheme.", "Set $s = f(z)$. For each $i$ let $s_i \\in S_i$ be the image", "of $s$. We remark that $Z_s$ is the limit of the schemes $(Z_i)_{s_i}$", "and $Z_s$ is also the limit of the schemes $(Z_i)_{s_i}$ base", "changed to $\\kappa(s)$. Moreover, all the morphisms", "$$", "Z_s", "\\longrightarrow", "(Z_{i'})_{s_{i'}} \\times_{\\Spec(\\kappa(s_{i'}))} \\Spec(\\kappa(s))", "\\longrightarrow", "(Z_i)_{s_i} \\times_{\\Spec(\\kappa(s_i))} \\Spec(\\kappa(s))", "\\longrightarrow", "X_s", "$$", "are closed immersions by assumption (4). Hence $Z_s$ is the scheme", "theoretic intersection of the closed subschemes", "$(Z_i)_{s_i} \\times_{\\Spec(\\kappa(s_i))} \\Spec(\\kappa(s))$", "in $X_s$. Since all the irreducible components of the schemes", "$(Z_i)_{s_i} \\times_{\\Spec(\\kappa(s_i))} \\Spec(\\kappa(s))$", "have dimension $> d$ and contain $z$ we conclude that", "$Z_s$ contains an irreducible component of dimension $> d$ passing", "through $z$ which contradicts the fact that $Z_s \\subset X_s$ and", "$\\dim(X_s) \\leq d$." ], "refs": [ "morphisms-lemma-openness-bounded-dimension-fibres", "limits-lemma-limit-nonempty" ], "ref_ids": [ 5280, 15034 ] } ], "ref_ids": [] }, { "id": 15105, "type": "theorem", "label": "limits-lemma-descend-quasi-finite", "categories": [ "limits" ], "title": "limits-lemma-descend-quasi-finite", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ is a quasi-finite morphism, and", "\\item $f_0$ is locally of finite type,", "\\end{enumerate}", "then there exists an $i \\geq 0$ such that $f_i$ is quasi-finite." ], "refs": [], "proofs": [ { "contents": [ "Follows immediately from Lemma \\ref{lemma-limit-dimension}." ], "refs": [ "limits-lemma-limit-dimension" ], "ref_ids": [ 15104 ] } ], "ref_ids": [] }, { "id": 15106, "type": "theorem", "label": "limits-lemma-descend-dimension-d", "categories": [ "limits" ], "title": "limits-lemma-descend-dimension-d", "contents": [ "Notation and assumptions as in Situation \\ref{situation-descent-property}.", "If", "\\begin{enumerate}", "\\item $f$ has relative dimension $d$, and", "\\item $f_0$ is locally of finite presentation,", "\\end{enumerate}", "then there exists an $i \\geq 0$ such that $f_i$", "has relative dimension $d$." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-limit-dimension} we may assume all fibres", "of $f_0$ have dimension $\\leq d$. By Morphisms, Lemma", "\\ref{morphisms-lemma-openness-bounded-dimension-fibres-finite-presentation}", "the set $U_0 \\subset X_0$ of points $x \\in X_0$ such that", "the dimension of the fibre of $X_0 \\to Y_0$ at $x$ is $\\leq d - 1$", "is open and retrocompact in $X_0$. Hence the complement", "$E = X_0 \\setminus U_0$ is constructible.", "Moreover the image of $X \\to X_0$ is contained in $E$", "by Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}.", "Thus for $i \\gg 0$ we have that the image", "of $X_i \\to X_0$ is contained in $E$", "(Lemma \\ref{lemma-limit-contained-in-constructible}). Then all fibres", "of $X_i \\to Y_i$ have dimension $d$ by the aforementioned", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}." ], "refs": [ "limits-lemma-limit-dimension", "morphisms-lemma-openness-bounded-dimension-fibres-finite-presentation", "morphisms-lemma-dimension-fibre-after-base-change", "limits-lemma-limit-contained-in-constructible", "morphisms-lemma-dimension-fibre-after-base-change" ], "ref_ids": [ 15104, 5282, 5279, 15040, 5279 ] } ], "ref_ids": [] }, { "id": 15107, "type": "theorem", "label": "limits-lemma-approximate-given-relative-dimension", "categories": [ "limits" ], "title": "limits-lemma-approximate-given-relative-dimension", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "Let $f : X \\to S$ be a morphism of finite presentation.", "Let $d \\geq 0$ be an integer.", "If $Z \\subset X$ be a closed subscheme such that", "$\\dim(Z_s) \\leq d$ for all $s \\in S$, then there exists a", "closed subscheme $Z' \\subset X$ such that", "\\begin{enumerate}", "\\item $Z \\subset Z'$,", "\\item $Z' \\to X$ is of finite presentation, and", "\\item $\\dim(Z'_s) \\leq d$ for all $s \\in S$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "By", "Proposition \\ref{proposition-approximate}", "we can write $S = \\lim S_i$ as the limit of a directed inverse", "system of Noetherian schemes with affine transition maps. By", "Lemma \\ref{lemma-descend-finite-presentation}", "we may assume that there exist a system of morphisms", "$f_i : X_i \\to S_i$ of finite presentation such that", "$X_{i'} = X_i \\times_{S_i} S_{i'}$", "for all $i' \\geq i$ and such that $X = X_i \\times_{S_i} S$.", "Let $Z_i \\subset X_i$ be the scheme theoretic image of", "$Z \\to X \\to X_i$. Then for $i' \\geq i$ the morphism $X_{i'} \\to X_i$", "maps $Z_{i'}$ into $Z_i$ and the induced morphism", "$Z_{i'} \\to Z_i \\times_{S_i} S_{i'}$ is a closed immersion. By", "Lemma \\ref{lemma-limit-dimension}", "we see that the dimension of the fibres of $Z_i \\to S_i$", "all have dimension $\\leq d$ for a suitable $i \\in I$.", "Fix such an $i$ and set $Z' = Z_i \\times_{S_i} S \\subset X$.", "Since $S_i$ is Noetherian, we see that $X_i$ is Noetherian, and hence", "the morphism $Z_i \\to X_i$ is of finite presentation.", "Therefore also the base change $Z' \\to X$ is of finite presentation.", "Moreover, the fibres of $Z' \\to S$ are base changes of the fibres", "of $Z_i \\to S_i$ and hence have dimension $\\leq d$." ], "refs": [ "limits-proposition-approximate", "limits-lemma-descend-finite-presentation", "limits-lemma-limit-dimension" ], "ref_ids": [ 15126, 15077, 15104 ] } ], "ref_ids": [] }, { "id": 15108, "type": "theorem", "label": "limits-lemma-top-cohomology-functor", "categories": [ "limits" ], "title": "limits-lemma-top-cohomology-functor", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $d \\geq 0$. Assume", "\\begin{enumerate}", "\\item $X$ and $Y$ are quasi-compact and quasi-separated, and", "\\item $R^if_*\\mathcal{F} = 0$ for $i > d$ and", "every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$.", "\\end{enumerate}", "Then we have", "\\begin{enumerate}", "\\item[(a)] for any base change diagram", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "we have $R^if'_*\\mathcal{F}' = 0$ for $i > d$ and any quasi-coherent", "$\\mathcal{O}_{X'}$-module $\\mathcal{F}'$,", "\\item[(b)]", "$R^df'_*(\\mathcal{F}' \\otimes_{\\mathcal{O}_{X'}} (f')^*\\mathcal{G}') =", "R^df'_*\\mathcal{F}' \\otimes_{\\mathcal{O}_{Y'}} \\mathcal{G}'$", "for any quasi-coherent $\\mathcal{O}_{Y'}$-module $\\mathcal{G}'$,", "\\item[(c)] formation of $R^df'_*\\mathcal{F}'$ commutes with arbitrary", "further base change (see proof for explanation).", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "Before giving the proofs, we explain the meaning of (c). Suppose", "we have an additional cartesian square", "$$", "\\xymatrix{", "X'' \\ar[d]_{f''} \\ar[r]_{h'} &", "X' \\ar[d]_{f'} \\ar[r]_{g'} &", "X \\ar[d]^f \\\\", "Y'' \\ar[r]^h &", "Y' \\ar[r]^g &", "Y", "}", "$$", "tacked onto our given diagram. If (a) holds, then there is a canonical", "map $\\gamma : h^*R^df'_*\\mathcal{F}' \\to R^df''_*(h')^*\\mathcal{F}'$.", "Namely, $\\gamma$ is the map on degree $d$ cohomology sheaves", "induced by the composition", "$$", "Lh^*Rf'_*\\mathcal{F}' \\longrightarrow", "Rf''_*L(h')^*\\mathcal{F}' \\longrightarrow Rf''_*(h')^*\\mathcal{F}'", "$$", "Here the first arrow is the base change map", "(Cohomology, Remark \\ref{cohomology-remark-base-change}) and", "the second arrow complex from the canonical map", "$L(g')^*\\mathcal{F} \\to (g')^*\\mathcal{F}$.", "Similarly, since $Rf'_*\\mathcal{F}$ has no nonzero cohomology sheaves in", "degrees $> d$ by (a) we have", "$H^d(Lh^*Rf_*\\mathcal{F}') = h^*R^df_*\\mathcal{F}$.", "The content of (c) is that $\\gamma$ is an isomorphism.", "\\medskip\\noindent", "Having said this, we can check (a), (b), and (c) locally on $Y'$ and $Y''$.", "Suppose that $V \\subset Y$ is a quasi-compact open subscheme. Then we", "claim (1) and (2) hold for $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$.", "Namely, (1) is immediate and (2) follows because any quasi-coherent", "module on $f^{-1}(V)$ is the restriction of a quasi-coherent module", "on $X$ (Properties, Lemma \\ref{properties-lemma-extend-trivial}) and formation", "of higher direct images commutes with restriction to opens.", "Thus we may also work locally on $Y$. In other words, we may", "assume $Y''$, $Y'$, and $Y$ are affine schemes.", "\\medskip\\noindent", "Proof of (a) when $Y'$ and $Y$ are affine. In this case the morphisms", "$g$ and $g'$ are affine. Thus $g_* = Rg_*$ and $g'_* = Rg'_*$", "(Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-vanishing})", "and $g_*$ is identified with the restriction functor on modules", "(Schemes, Lemma \\ref{schemes-lemma-widetilde-pullback}).", "Then", "$$", "g_*(R^if'_*\\mathcal{F}') = H^i(Rg_*Rf'_*\\mathcal{F}') =", "H^i(Rf_*Rg'_*\\mathcal{F}') =", "H^i(Rf_*g'_*\\mathcal{F}') =", "Rf^i_*g'_*\\mathcal{F}'", "$$", "which is zero by assumption (2). Hence (a) by our description of $g_*$.", "\\medskip\\noindent", "Proof of (b) when $Y'$ is affine, say $Y' = \\Spec(R')$.", "By part (a) we have $H^{d + 1}(X', \\mathcal{F}') = 0$", "for any quasi-coherent $\\mathcal{O}_{X'}$-module $\\mathcal{F}'$, see", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images-application}.", "Consider the functor $F$ on $R'$-modules defined by the rule", "$$", "F(M) = H^d(X', \\mathcal{F}' \\otimes_{\\mathcal{O}_{X'}} (f')^*\\widetilde{M})", "$$", "By Cohomology, Lemma \\ref{cohomology-lemma-quasi-separated-cohomology-colimit}", "this functor commutes with direct sums (this is where we use that", "$X$ and hence $X'$ is quasi-compact and quasi-separated).", "On the other hand, if $M_1 \\to M_2 \\to M_3 \\to 0$ is an exact sequence,", "then", "$$", "\\mathcal{F}' \\otimes_{\\mathcal{O}_{X'}} (f')^*\\widetilde{M}_1 \\to", "\\mathcal{F}' \\otimes_{\\mathcal{O}_{X'}} (f')^*\\widetilde{M}_2 \\to", "\\mathcal{F}' \\otimes_{\\mathcal{O}_{X'}} (f')^*\\widetilde{M}_3 \\to 0", "$$", "is an exact sequence of quasi-coherent modules on $X'$", "and by the vanishing of higher cohomology given above we get an exact sequence", "$$", "F(M_1) \\to F(M_2) \\to F(M_3) \\to 0", "$$", "In other words, $F$ is right exact. Any right exact $R'$-linear functor", "$F : \\text{Mod}_{R'} \\to \\text{Mod}_{R'}$", "which commutes with direct sums is given by tensoring with an $R'$-module", "(omitted; left as exercise for the reader).", "Thus we obtain $F(M) = H^d(X', \\mathcal{F}') \\otimes_{R'} M$.", "Since $R^d(f')_*\\mathcal{F}'$ and", "$R^d(f')_*(\\mathcal{F}' \\otimes_{\\mathcal{O}_{X'}} (f')^*\\widetilde{M})$", "are quasi-coherent (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images}),", "the fact that $F(M) = H^d(X', \\mathcal{F}') \\otimes_{R'} M$", "translates into the statement given in (b).", "\\medskip\\noindent", "Proof of (c) when $Y'' \\to Y' \\to Y$ are morphisms of affine schemes.", "Say $Y'' = \\Spec(R'')$ and $Y' = \\Spec(R')$.", "Then we see that $R^df''_*(h')^*\\mathcal{F}'$", "is the quasi-coherent module on $Y'$ associated to the $R''$-module", "$H^d(X'', (h')^*\\mathcal{F}')$. Now $h' : X'' \\to X'$ is affine", "hence $H^d(X'', (h')^*\\mathcal{F}') = H^d(X, h'_*(h')^*\\mathcal{F}')$", "by the already used", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology}.", "We have", "$$", "h'_*(h')^*\\mathcal{F}' =", "\\mathcal{F}' \\otimes_{\\mathcal{O}_{X'}} (f')^*\\widetilde{R''}", "$$", "as the reader sees by checking on an affine open covering. Thus", "$H^d(X'', (h')^*\\mathcal{F}') = H^d(X', \\mathcal{F}') \\otimes_{R'} R''$", "by part (b) applied to $f'$ and the proof is complete." ], "refs": [ "cohomology-remark-base-change", "properties-lemma-extend-trivial", "coherent-lemma-relative-affine-vanishing", "schemes-lemma-widetilde-pullback", "coherent-lemma-quasi-coherence-higher-direct-images-application", "cohomology-lemma-quasi-separated-cohomology-colimit", "coherent-lemma-quasi-coherence-higher-direct-images", "coherent-lemma-relative-affine-cohomology" ], "ref_ids": [ 2269, 3018, 3283, 7662, 3296, 2082, 3295, 3284 ] } ], "ref_ids": [] }, { "id": 15109, "type": "theorem", "label": "limits-lemma-higher-direct-images-zero-above-dimension-fibre", "categories": [ "limits" ], "title": "limits-lemma-higher-direct-images-zero-above-dimension-fibre", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $y \\in Y$.", "Assume $f$ is proper and $\\dim(X_y) = d$.", "Then", "\\begin{enumerate}", "\\item for $\\mathcal{F} \\in \\QCoh(\\mathcal{O}_X)$", "we have $(R^if_*\\mathcal{F})_y = 0$ for all $i > d$,", "\\item there is an affine open neighbourhood $V \\subset Y$", "of $y$ such that $f^{-1}(V) \\to V$ and $d$ satisfy the", "assumptions and conclusions of Lemma \\ref{lemma-top-cohomology-functor}.", "\\end{enumerate}" ], "refs": [ "limits-lemma-top-cohomology-functor" ], "proofs": [ { "contents": [ "By Morphisms, Lemma", "\\ref{morphisms-lemma-openness-bounded-dimension-fibres}", "and the fact that $f$ is closed, we can find an affine open neighbourhood", "$V$ of $y$ such that the fibres over points of $V$ all have dimension", "$\\leq d$. Thus we may assume $X \\to Y$ is a proper morphism all", "of whose fibres have dimension $\\leq d$ with $Y$ affine.", "We will show that (2) holds, which will immediately imply (1)", "for all $y \\in Y$.", "\\medskip\\noindent", "By Lemma \\ref{lemma-proper-limit-of-proper-finite-presentation}", "we can write $X = \\lim X_i$ as a cofiltered limit with $X_i \\to Y$ proper", "and of finite presentation and such that both $X \\to X_i$", "and transition morphisms are closed immersions.", "For some $i$ we have that $X_i \\to Y$ has fibres of dimension $\\leq d$,", "see Lemma \\ref{lemma-limit-dimension}.", "For a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ we have", "$R^pf_*\\mathcal{F} = R^pf_{i, *}(X \\to X_i)_*\\mathcal{F}$ by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-vanishing}", "and Leray (Cohomology, Lemma \\ref{cohomology-lemma-relative-Leray}).", "Thus we may replace $X$ by $X_i$ and", "reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $Y$ is affine and $f : X \\to Y$ is proper and of finite presentation", "and all fibres have dimension $\\leq d$. It suffices to show that", "$H^p(X, \\mathcal{F}) = 0$ for $p > d$. Namely, by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images-application}", "we have $H^p(X, \\mathcal{F}) = H^0(Y, R^pf_*\\mathcal{F})$.", "On the other hand, $R^pf_*\\mathcal{F}$ is quasi-coherent on $Y$", "by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images},", "hence vanishing of global sections implies vanishing.", "Write $Y = \\lim_{i \\in I} Y_i$ as a cofiltered limit of affine schemes", "with $Y_i$ the spectrum of a Noetherian ring", "(for example a finite type $\\mathbf{Z}$-algebra).", "We can choose an element $0 \\in I$ and a finite type morphism", "$X_0 \\to Y_0$ such that $X \\cong Y \\times_{Y_0} X_0$, see", "Lemma \\ref{lemma-descend-finite-presentation}.", "After increasing $0$ we may assume $X_0 \\to Y_0$ is proper", "(Lemma \\ref{lemma-eventually-proper})", "and that the fibres of $X_0 \\to Y_0$ have dimension $\\leq d$", "(Lemma \\ref{lemma-limit-dimension}).", "Since $X \\to X_0$ is affine, we find that", "$H^p(X, \\mathcal{F}) = H^p(X_0, (X \\to X_0)_*\\mathcal{F})$ by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology}.", "This reduces us to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $Y$ is affine Noetherian and $f : X \\to Y$ is proper", "and all fibres have dimension $\\leq d$.", "In this case we can write $\\mathcal{F} = \\colim \\mathcal{F}_i$", "as a filtered colimit of coherent $\\mathcal{O}_X$-modules, see", "Properties, Lemma", "\\ref{properties-lemma-directed-colimit-finite-presentation}.", "Then $H^p(X, \\mathcal{F}) = \\colim H^p(X, \\mathcal{F}_i)$ by", "Cohomology, Lemma \\ref{cohomology-lemma-quasi-separated-cohomology-colimit}.", "Thus we may assume $\\mathcal{F}$ is coherent.", "In this case we see that $(R^pf_*\\mathcal{F})_y = 0$ for", "all $y \\in Y$ by Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-higher-direct-images-zero-above-dimension-fibre}.", "Thus $R^pf_*\\mathcal{F} = 0$ and therefore", "$H^p(X, \\mathcal{F}) = 0$ (see above) and we win." ], "refs": [ "morphisms-lemma-openness-bounded-dimension-fibres", "limits-lemma-proper-limit-of-proper-finite-presentation", "limits-lemma-limit-dimension", "coherent-lemma-relative-affine-vanishing", "cohomology-lemma-relative-Leray", "coherent-lemma-quasi-coherence-higher-direct-images-application", "coherent-lemma-quasi-coherence-higher-direct-images", "limits-lemma-descend-finite-presentation", "limits-lemma-eventually-proper", "limits-lemma-limit-dimension", "coherent-lemma-relative-affine-cohomology", "properties-lemma-directed-colimit-finite-presentation", "cohomology-lemma-quasi-separated-cohomology-colimit", "coherent-lemma-higher-direct-images-zero-above-dimension-fibre" ], "ref_ids": [ 5280, 15090, 15104, 3283, 2073, 3296, 3295, 15077, 15089, 15104, 3284, 3024, 2082, 3364 ] } ], "ref_ids": [ 15108 ] }, { "id": 15110, "type": "theorem", "label": "limits-lemma-proper-top-cohomology-finite-type", "categories": [ "limits" ], "title": "limits-lemma-proper-top-cohomology-finite-type", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $d \\geq 0$. Let $\\mathcal{F}$", "be an $\\mathcal{O}_X$-module. Assume", "\\begin{enumerate}", "\\item $f$ is a proper morphism all of whose fibres have dimension $\\leq d$,", "\\item $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module of finite type.", "\\end{enumerate}", "Then $R^df_*\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module", "of finite type." ], "refs": [], "proofs": [ { "contents": [ "The module $R^df_*\\mathcal{F}$ is quasi-coherent by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images}.", "The question is local on $Y$ hence we may assume $Y$ is affine.", "Say $Y = \\Spec(R)$. Then it suffices to prove that $H^d(X, \\mathcal{F})$", "is a finite $R$-module.", "\\medskip\\noindent", "By Lemma \\ref{lemma-proper-limit-of-proper-finite-presentation}", "we can write $X = \\lim X_i$ as a cofiltered limit with $X_i \\to Y$ proper", "and of finite presentation and such that both $X \\to X_i$", "and transition morphisms are closed immersions.", "For some $i$ we have that $X_i \\to Y$ has fibres of dimension $\\leq d$,", "see Lemma \\ref{lemma-limit-dimension}. We have", "$R^pf_*\\mathcal{F} = R^pf_{i, *}(X \\to X_i)_*\\mathcal{F}$ by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-vanishing}", "and Leray (Cohomology, Lemma \\ref{cohomology-lemma-relative-Leray}).", "Thus we may replace $X$ by $X_i$ and", "reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $Y$ is affine and $f : X \\to Y$ is proper and of finite presentation", "and all fibres have dimension $\\leq d$.", "We can write $\\mathcal{F}$ as a quotient of a finitely presented", "$\\mathcal{O}_X$-module $\\mathcal{F}'$, see", "Properties, Lemma", "\\ref{properties-lemma-finite-directed-colimit-surjective-maps}.", "The map $H^d(X, \\mathcal{F}') \\to H^d(X, \\mathcal{F})$ is", "surjective, as we have $H^{d + 1}(X, \\Ker(\\mathcal{F}' \\to \\mathcal{F})) = 0$", "by the vanishing of higher cohomology seen in", "Lemma \\ref{lemma-higher-direct-images-zero-above-dimension-fibre}", "(or its proof). Thus we reduce to the case discussed in the next paragraph.", "\\medskip\\noindent", "Assume $Y = \\Spec(R)$ is affine and $f : X \\to Y$", "is proper and of finite presentation", "and all fibres have dimension $\\leq d$ and $\\mathcal{F}$ is an", "$\\mathcal{O}_X$-module of finite presentation.", "Write $Y = \\lim_{i \\in I} Y_i$ as a cofiltered limit of affine schemes", "with $Y_i = \\Spec(R_i)$ the spectrum of a Noetherian ring", "(for example a finite type $\\mathbf{Z}$-algebra).", "We can choose an element $0 \\in I$ and a finite type morphism", "$X_0 \\to Y_0$ such that $X \\cong Y \\times_{Y_0} X_0$, see", "Lemma \\ref{lemma-descend-finite-presentation}.", "After increasing $0$ we may assume $X_0 \\to Y_0$ is proper", "(Lemma \\ref{lemma-eventually-proper})", "and that the fibres of $X_0 \\to Y_0$ have dimension $\\leq d$", "(Lemma \\ref{lemma-limit-dimension}).", "After increasing $0$ we can assume there is a coherent", "$\\mathcal{O}_{X_0}$-module $\\mathcal{F}_0$ which pulls", "back to $\\mathcal{F}$, see", "Lemma \\ref{lemma-descend-modules-finite-presentation}.", "By Lemma \\ref{lemma-top-cohomology-functor}", "we have", "$$", "H^d(X, \\mathcal{F}) = H^d(X_0, \\mathcal{F}_0) \\otimes_{R_0} R", "$$", "This finishes the proof because the cohomology module", "$H^d(X_0, \\mathcal{F}_0)$ is finite by", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-proper-over-affine-cohomology-finite}." ], "refs": [ "coherent-lemma-quasi-coherence-higher-direct-images", "limits-lemma-proper-limit-of-proper-finite-presentation", "limits-lemma-limit-dimension", "coherent-lemma-relative-affine-vanishing", "cohomology-lemma-relative-Leray", "properties-lemma-finite-directed-colimit-surjective-maps", "limits-lemma-higher-direct-images-zero-above-dimension-fibre", "limits-lemma-descend-finite-presentation", "limits-lemma-eventually-proper", "limits-lemma-limit-dimension", "limits-lemma-descend-modules-finite-presentation", "limits-lemma-top-cohomology-functor", "coherent-lemma-proper-over-affine-cohomology-finite" ], "ref_ids": [ 3295, 15090, 15104, 3283, 2073, 3025, 15109, 15077, 15089, 15104, 15078, 15108, 3355 ] } ], "ref_ids": [] }, { "id": 15111, "type": "theorem", "label": "limits-lemma-proper-top-cohomology-finite-presentation", "categories": [ "limits" ], "title": "limits-lemma-proper-top-cohomology-finite-presentation", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $d \\geq 0$.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Assume", "\\begin{enumerate}", "\\item $f$ is a proper morphism of finite presentation", "all of whose fibres have dimension $\\leq d$,", "\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation.", "\\end{enumerate}", "Then $R^df_*\\mathcal{F}$ is an $\\mathcal{O}_X$-module", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "The proof is exactly the same as the proof of", "Lemma \\ref{lemma-proper-top-cohomology-finite-type}", "except that the third paragraph can be skipped.", "We omit the details." ], "refs": [ "limits-lemma-proper-top-cohomology-finite-type" ], "ref_ids": [ 15110 ] } ], "ref_ids": [] }, { "id": 15112, "type": "theorem", "label": "limits-lemma-glueing-near-closed-point", "categories": [ "limits" ], "title": "limits-lemma-glueing-near-closed-point", "contents": [ "Let $S$ be a scheme. Let $s \\in S$ be a closed point such that", "$U = S \\setminus \\{s\\} \\to S$ is quasi-compact. With", "$V = \\Spec(\\mathcal{O}_{S, s}) \\setminus \\{s\\}$ there is", "an equivalence of categories", "$$", "\\left\\{", "\\begin{matrix}", "X \\to S\\text{ of finite presentation}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\vcenter{", "\\xymatrix{", "X' \\ar[d] & Y' \\ar[d] \\ar[l] \\ar[r] & Y \\ar[d] \\\\", "U & V \\ar[l] \\ar[r] & \\Spec(\\mathcal{O}_{S, s})", "}", "}", "\\right\\}", "$$", "where on the right hand side we consider commutative diagrams", "whose squares are cartesian and whose vertical arrows are", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Let $W \\subset S$ be an open neighbourhood of $s$. By", "glueing of relative schemes, see", "Constructions, Section \\ref{constructions-section-relative-glueing},", "the functor", "$$", "\\left\\{", "\\begin{matrix}", "X \\to S\\text{ of finite presentation}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\vcenter{", "\\xymatrix{", "X' \\ar[d] & Y' \\ar[d] \\ar[l] \\ar[r] & Y \\ar[d] \\\\", "U & W \\setminus \\{s\\} \\ar[l] \\ar[r] & W", "}", "}", "\\right\\}", "$$", "is an equivalence of categories. We have", "$\\mathcal{O}_{S, s} = \\colim \\mathcal{O}_W(W)$ where", "$W$ runs over the affine open neighbourhoods of $s$.", "Hence $\\Spec(\\mathcal{O}_{S, s}) = \\lim W$ where $W$", "runs over the affine open neighbourhoods of $s$.", "Thus the category of schemes of finite presentation", "over $\\Spec(\\mathcal{O}_{S, s})$ is the limit of the", "category of schemes of finite presentation over", "$W$ where $W$ runs over the affine open neighbourhoods", "of $s$, see", "Lemma \\ref{lemma-descend-finite-presentation}.", "For every affine open $s \\in W$ we see that $U \\cap W$", "is quasi-compact as $U \\to S$ is quasi-compact.", "Hence $V = \\lim W \\cap U = \\lim W \\setminus \\{s\\}$ is a limit of", "quasi-compact and quasi-separated schemes (see", "Lemma \\ref{lemma-directed-inverse-system-has-limit}).", "Thus also the category of schemes of finite presentation", "over $V$ is the limit of the", "categories of schemes of finite presentation over", "$W \\cap U$ where $W$ runs over the affine open neighbourhoods", "of $s$. The lemma follows formally from a combination", "of these results." ], "refs": [ "limits-lemma-descend-finite-presentation", "limits-lemma-directed-inverse-system-has-limit" ], "ref_ids": [ 15077, 15027 ] } ], "ref_ids": [] }, { "id": 15113, "type": "theorem", "label": "limits-lemma-glueing-near-closed-point-modules", "categories": [ "limits" ], "title": "limits-lemma-glueing-near-closed-point-modules", "contents": [ "Let $S$ be a scheme. Let $s \\in S$ be a closed point such that", "$U = S \\setminus \\{s\\} \\to S$ is quasi-compact. With", "$V = \\Spec(\\mathcal{O}_{S, s}) \\setminus \\{s\\}$ there is", "an equivalence of categories", "$$", "\\left\\{", "\\mathcal{O}_S\\text{-modules }\\mathcal{F}\\text{ of finite presentation}", "\\right\\}", "\\longrightarrow", "\\left\\{", "(\\mathcal{G}, \\mathcal{H}, \\alpha)", "\\right\\}", "$$", "where on the right hand side we consider triples", "consisting of a $\\mathcal{O}_U$-module $\\mathcal{G}$ of", "finite presentation, a $\\mathcal{O}_{\\Spec(\\mathcal{O}_{S, s})}$-module", "$\\mathcal{H}$ of finite presentation, and an isomorphism", "$\\alpha : \\mathcal{G}|_V \\to \\mathcal{H}|_V$ of", "$\\mathcal{O}_V$-modules." ], "refs": [], "proofs": [ { "contents": [ "You can either prove this by", "redoing the proof of Lemma \\ref{lemma-glueing-near-closed-point}", "using Lemma \\ref{lemma-descend-modules-finite-presentation}", "or you can deduce it from Lemma \\ref{lemma-glueing-near-closed-point}", "using the equivalence between quasi-coherent modules and", "``vector bundles'' from", "Constructions, Section \\ref{constructions-section-vector-bundle}.", "We omit the details." ], "refs": [ "limits-lemma-glueing-near-closed-point", "limits-lemma-descend-modules-finite-presentation", "limits-lemma-glueing-near-closed-point" ], "ref_ids": [ 15112, 15078, 15112 ] } ], "ref_ids": [] }, { "id": 15114, "type": "theorem", "label": "limits-lemma-glueing-near-point", "categories": [ "limits" ], "title": "limits-lemma-glueing-near-point", "contents": [ "Let $S$ be a scheme. Let $U \\subset S$ be a retrocompact open.", "Let $s \\in S$ be a point in the complement of $U$. With", "$V = \\Spec(\\mathcal{O}_{S, s}) \\cap U$ there is", "an equivalence of categories", "$$", "\\colim_{s \\in U' \\supset U\\text{ open}}", "\\left\\{", "\\vcenter{", "\\xymatrix{", "X \\ar[d] \\\\", "U'", "}", "}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\vcenter{", "\\xymatrix{", "X' \\ar[d] & Y' \\ar[d] \\ar[l] \\ar[r] & Y \\ar[d] \\\\", "U & V \\ar[l] \\ar[r] & \\Spec(\\mathcal{O}_{S, s})", "}", "}", "\\right\\}", "$$", "where on the left hand side the vertical arrow is of finite", "presentation and on the right hand side we consider commutative diagrams", "whose squares are cartesian and whose vertical arrows are", "of finite presentation." ], "refs": [], "proofs": [ { "contents": [ "Let $W \\subset S$ be an open neighbourhood of $s$. By", "glueing of relative schemes, see", "Constructions, Section \\ref{constructions-section-relative-glueing},", "the functor", "$$", "\\left\\{", "\\begin{matrix}", "X \\to U' = U \\cup W \\text{ of finite presentation}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\vcenter{", "\\xymatrix{", "X' \\ar[d] & Y' \\ar[d] \\ar[l] \\ar[r] & Y \\ar[d] \\\\", "U & W \\cap U \\ar[l] \\ar[r] & W", "}", "}", "\\right\\}", "$$", "is an equivalence of categories. We have", "$\\mathcal{O}_{S, s} = \\colim \\mathcal{O}_W(W)$ where", "$W$ runs over the affine open neighbourhoods of $s$.", "Hence $\\Spec(\\mathcal{O}_{S, s}) = \\lim W$ where $W$", "runs over the affine open neighbourhoods of $s$.", "Thus the category of schemes of finite presentation", "over $\\Spec(\\mathcal{O}_{S, s})$ is the limit of the", "category of schemes of finite presentation over", "$W$ where $W$ runs over the affine open neighbourhoods", "of $s$, see", "Lemma \\ref{lemma-descend-finite-presentation}.", "For every affine open $s \\in W$ we see that $U \\cap W$", "is quasi-compact as $U \\to S$ is quasi-compact.", "Hence $V = \\lim W \\cap U$ is a limit of", "quasi-compact and quasi-separated schemes (see", "Lemma \\ref{lemma-directed-inverse-system-has-limit}).", "Thus also the category of schemes of finite presentation", "over $V$ is the limit of the", "categories of schemes of finite presentation over", "$W \\cap U$ where $W$ runs over the affine open neighbourhoods", "of $s$. The lemma follows formally from a combination", "of these results." ], "refs": [ "limits-lemma-descend-finite-presentation", "limits-lemma-directed-inverse-system-has-limit" ], "ref_ids": [ 15077, 15027 ] } ], "ref_ids": [] }, { "id": 15115, "type": "theorem", "label": "limits-lemma-glueing-near-point-properties", "categories": [ "limits" ], "title": "limits-lemma-glueing-near-point-properties", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-glueing-near-point}.", "Let $U \\subset U' \\subset X$ be an open containing $s$.", "\\begin{enumerate}", "\\item Let $f' : X \\to U'$ correspond to $f : X' \\to U$ and", "$g : Y \\to \\Spec(\\mathcal{O}_{S, s})$", "via the equivalence. If $f$ and $g$ are separated, proper, finite, \\'etale,", "then after possibly shrinking $U'$ the morphism $f'$ has the same property.", "\\item Let $a : X_1 \\to X_2$", "be a morphism of schemes of finite presentation over $U'$", "with base change $a' : X'_1 \\to X'_2$ over $U$ and", "$b : Y_1 \\to Y_2$ over $\\Spec(\\mathcal{O}_{S, s})$.", "If $a'$ and $b$ are separated, proper, finite, \\'etale,", "then after possibly shrinking $U'$ the morphism $a$ has the same property.", "\\end{enumerate}" ], "refs": [ "limits-lemma-glueing-near-point" ], "proofs": [ { "contents": [ "Proof of (1). Recall that $\\Spec(\\mathcal{O}_{S, s})$ is the limit of the", "affine open neighbourhoods of $s$ in $S$. Since $g$ has the property", "in question, then the restriction of $f'$ to one of these", "affine open neighbourhoods does too, see", "Lemmas \\ref{lemma-descend-separated-finite-presentation},", "\\ref{lemma-eventually-proper},", "\\ref{lemma-descend-finite-finite-presentation}, and", "\\ref{lemma-descend-etale}.", "Since $f'$ has the given property over $U$ as $f$ does,", "we conclude as one can check the property locally on the base.", "\\medskip\\noindent", "Proof of (2). If we write $\\Spec(\\mathcal{O}_{S, s}) = \\lim W$", "where $W$ runs over the affine open neighbourhoods of $s$ in $S$,", "then we have $Y_i = \\lim W \\times_S X_i$. Thus we can use", "exactly the same arguments as in the proof of (1)." ], "refs": [ "limits-lemma-descend-separated-finite-presentation", "limits-lemma-eventually-proper", "limits-lemma-descend-finite-finite-presentation", "limits-lemma-descend-etale" ], "ref_ids": [ 15061, 15089, 15058, 15065 ] } ], "ref_ids": [ 15114 ] }, { "id": 15116, "type": "theorem", "label": "limits-lemma-glueing-near-multiple-closed-points", "categories": [ "limits" ], "title": "limits-lemma-glueing-near-multiple-closed-points", "contents": [ "Let $S$ be a scheme. Let $s_1, \\ldots, s_n \\in S$ be pairwise distinct", "closed points such that", "$U = S \\setminus \\{s_1, \\ldots, s_n\\} \\to S$ is quasi-compact. With", "$S_i = \\Spec(\\mathcal{O}_{S, s_i})$ and $U_i = S_i \\setminus \\{s_i\\}$", "there is an equivalence of categories", "$$", "FP_S \\longrightarrow", "FP_U \\times_{(FP_{U_1} \\times \\ldots \\times FP_{U_n})}", "(FP_{S_1} \\times \\ldots \\times FP_{S_n})", "$$", "where $FP_T$ is the category of schemes of finite presentation over", "the scheme $T$." ], "refs": [], "proofs": [ { "contents": [ "For $n = 1$ this is Lemma \\ref{lemma-glueing-near-closed-point}.", "For $n > 1$ the lemma can be proved in exactly the same way or it", "can be deduced from it. For example, suppose that $f_i : X_i \\to S_i$", "are objects of $FP_{S_i}$ and $f : X \\to U$ is an object", "of $FP_U$ and we're given isomorphisms $X_i \\times_{S_i} U_i = X \\times_U U_i$.", "By Lemma \\ref{lemma-glueing-near-closed-point} we can find", "a morphism $f' : X' \\to U' = S \\setminus \\{s_1, \\ldots, s_{n - 1}\\}$", "which is of finite presentation, which is isomorphic to", "$X_i$ over $S_i$, which is isomorphic to $X$ over $U$, and", "these isomorphisms are compatible with the given isomorphism", "$X_i \\times_{S_n} U_n = X \\times_U U_n$.", "Then we can apply induction to", "$f_i : X_i \\to S_i$, $i \\leq n - 1$,", "$f' : X' \\to U'$, and the induced", "isomorphisms $X_i \\times_{S_i} U_i = X' \\times_{U'} U_i$, $i \\leq n - 1$.", "This shows essential surjectivity. We omit the proof of", "fully faithfulness." ], "refs": [ "limits-lemma-glueing-near-closed-point", "limits-lemma-glueing-near-closed-point" ], "ref_ids": [ 15112, 15112 ] } ], "ref_ids": [] }, { "id": 15117, "type": "theorem", "label": "limits-lemma-modifications", "categories": [ "limits" ], "title": "limits-lemma-modifications", "contents": [ "Let $S$ be a scheme. Let $s \\in S$ be a closed point such that", "$U = S \\setminus \\{s\\} \\to S$ is quasi-compact. With", "$V = \\Spec(\\mathcal{O}_{S, s}) \\setminus \\{s\\}$ the base change functor", "$$", "\\left\\{", "\\begin{matrix}", "f : X \\to S\\text{ of finite presentation} \\\\", "f^{-1}(U) \\to U\\text{ is an isomorphism}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "g : Y \\to \\Spec(\\mathcal{O}_{S, s})\\text{ of finite presentation} \\\\", "g^{-1}(V) \\to V\\text{ is an isomorphism}", "\\end{matrix}", "\\right\\}", "$$", "is an equivalence of categories." ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-glueing-near-closed-point}." ], "refs": [ "limits-lemma-glueing-near-closed-point" ], "ref_ids": [ 15112 ] } ], "ref_ids": [] }, { "id": 15118, "type": "theorem", "label": "limits-lemma-modifications-properties", "categories": [ "limits" ], "title": "limits-lemma-modifications-properties", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-modifications}.", "Let $f : X \\to S$ correspond to $g : Y \\to \\Spec(\\mathcal{O}_{S, s})$", "via the equivalence. Then $f$ is separated, proper, finite, \\'etale", "and add more here if and only if $g$ is so." ], "refs": [ "limits-lemma-modifications" ], "proofs": [ { "contents": [ "The property of being separated, proper, integral, finite, etc", "is stable under base change. See", "Schemes, Lemma \\ref{schemes-lemma-separated-permanence}", "and", "Morphisms, Lemmas \\ref{morphisms-lemma-base-change-proper} and", "\\ref{morphisms-lemma-base-change-finite}.", "Hence if $f$ has the property, then so does $g$.", "The converse follows from Lemma \\ref{lemma-glueing-near-point-properties}", "but we also give a direct proof here.", "Namely, if $g$ has to property, then $f$ does in a neighbourhood of $s$ by", "Lemmas \\ref{lemma-descend-separated-finite-presentation},", "\\ref{lemma-eventually-proper},", "\\ref{lemma-descend-finite-finite-presentation}, and", "\\ref{lemma-descend-etale}.", "Since $f$ clearly has the given property over $S \\setminus \\{s\\}$", "we conclude as one can check the property locally on the base." ], "refs": [ "schemes-lemma-separated-permanence", "morphisms-lemma-base-change-proper", "morphisms-lemma-base-change-finite", "limits-lemma-glueing-near-point-properties", "limits-lemma-descend-separated-finite-presentation", "limits-lemma-eventually-proper", "limits-lemma-descend-finite-finite-presentation", "limits-lemma-descend-etale" ], "ref_ids": [ 7714, 5409, 5440, 15115, 15061, 15089, 15058, 15065 ] } ], "ref_ids": [ 15117 ] }, { "id": 15119, "type": "theorem", "label": "limits-lemma-good-diagram", "categories": [ "limits" ], "title": "limits-lemma-good-diagram", "contents": [ "In Situation \\ref{situation-limit-noetherian}.", "Let $X \\to S$ be quasi-separated and of finite type.", "Then there exists an $i \\in I$ and a diagram", "\\begin{equation}", "\\label{equation-good-diagram}", "\\vcenter{", "\\xymatrix{", "X \\ar[r] \\ar[d] & W \\ar[d] \\\\", "S \\ar[r] & S_i", "}", "}", "\\end{equation}", "such that $W \\to S_i$ is of finite type and such that", "the induced morphism $X \\to S \\times_{S_i} W$ is a closed", "immersion." ], "refs": [], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-finite-type-closed-in-finite-presentation}", "we can find a closed immersion $X \\to X'$", "over $S$ where $X'$ is a scheme of finite presentation over $S$.", "By Lemma \\ref{lemma-descend-finite-presentation}", "we can find an $i$ and a morphism of finite presentation", "$X'_i \\to S_i$ whose pull back is $X'$. Set $W = X'_i$." ], "refs": [ "limits-lemma-finite-type-closed-in-finite-presentation", "limits-lemma-descend-finite-presentation" ], "ref_ids": [ 15072, 15077 ] } ], "ref_ids": [] }, { "id": 15120, "type": "theorem", "label": "limits-lemma-limit-from-good-diagram", "categories": [ "limits" ], "title": "limits-lemma-limit-from-good-diagram", "contents": [ "In Situation \\ref{situation-limit-noetherian}.", "Let $X \\to S$ be quasi-separated and of finite type.", "Given $i \\in I$ and a diagram", "$$", "\\vcenter{", "\\xymatrix{", "X \\ar[r] \\ar[d] & W \\ar[d] \\\\", "S \\ar[r] & S_i", "}", "}", "$$", "as in (\\ref{equation-good-diagram}) for $i' \\geq i$ let", "$X_{i'}$ be the scheme theoretic image of $X \\to S_{i'} \\times_{S_i} W$.", "Then $X = \\lim_{i' \\geq i} X_{i'}$." ], "refs": [], "proofs": [ { "contents": [ "Since $X$ is quasi-compact and quasi-separated formation of the", "scheme theoretic image of $X \\to S_{i'} \\times_{S_i} W$", "commutes with restriction to open subschemes", "(Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}).", "Hence we may and do assume $W$ is affine and maps into an affine open", "$U_i$ of $S_i$. Let $U \\subset S$, $U_{i'} \\subset S_{i'}$", "be the inverse image of $U_i$. Then $U$, $U_{i'}$, ", "$S_{i'} \\times_{S_i} W = U_{i'} \\times_{U_i} W$, and", "$S \\times_{S_i} W = U \\times_{U_i} W$ are all affine.", "This implies $X$ is affine because $X \\to S \\times_{S_i} W$ is", "a closed immersion. This also shows the ring map", "$$", "\\mathcal{O}(U) \\otimes_{\\mathcal{O}(U_i)} \\mathcal{O}(W) \\to", "\\mathcal{O}(X)", "$$", "is surjective. Let $I$ be the kernel. Then we see that $X_{i'}$", "is the spectrum of the ring", "$$", "\\mathcal{O}(X_{i'}) =", "\\mathcal{O}(U_{i'}) \\otimes_{\\mathcal{O}(U_i)} \\mathcal{O}(W)/I_{i'}", "$$", "where $I_{i'}$ is the inverse image of the ideal $I$ (see", "Morphisms, Example \\ref{morphisms-example-scheme-theoretic-image}).", "Since $\\mathcal{O}(U) = \\colim \\mathcal{O}(U_{i'})$", "we see that $I = \\colim I_{i'}$ and we conclude", "that $\\colim \\mathcal{O}(X_{i'}) = \\mathcal{O}(X)$." ], "refs": [ "morphisms-lemma-quasi-compact-scheme-theoretic-image" ], "ref_ids": [ 5146 ] } ], "ref_ids": [] }, { "id": 15121, "type": "theorem", "label": "limits-lemma-morphism-good-diagram", "categories": [ "limits" ], "title": "limits-lemma-morphism-good-diagram", "contents": [ "In Situation \\ref{situation-limit-noetherian}.", "Let $f : X \\to Y$ be a morphism of schemes quasi-separated", "and of finite type over $S$. Let", "$$", "\\vcenter{", "\\xymatrix{", "X \\ar[r] \\ar[d] & W \\ar[d] \\\\", "S \\ar[r] & S_{i_1}", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "Y \\ar[r] \\ar[d] & V \\ar[d] \\\\", "S \\ar[r] & S_{i_2}", "}", "}", "$$", "be diagrams as in (\\ref{equation-good-diagram}). Let", "$X = \\lim_{i \\geq i_1} X_i$ and", "$Y = \\lim_{i \\geq i_2} Y_i$ be the corresponding", "limit descriptions as in Lemma \\ref{lemma-limit-from-good-diagram}.", "Then there exists an $i_0 \\geq \\max(i_1, i_2)$ and a morphism", "$$", "(f_i)_{i \\geq i_0} : (X_i)_{i \\geq i_0} \\to (Y_i)_{i \\geq i_0}", "$$", "of inverse systems over $(S_i)_{i \\geq i_0}$ such that", "such that $f = \\lim_{i \\geq i_0} f_i$.", "If $(g_i)_{i \\geq i_0} : (X_i)_{i \\geq i_0} \\to (Y_i)_{i \\geq i_0}$", "is a second morphism of inverse systems over $(S_i)_{i \\geq i_0}$ such that", "such that $f = \\lim_{i \\geq i_0} g_i$", "then $f_i = g_i$ for all $i \\gg i_0$." ], "refs": [ "limits-lemma-limit-from-good-diagram" ], "proofs": [ { "contents": [ "Since $V \\to S_{i_2}$ is of finite presentation and", "$X = \\lim_{i \\geq i_1} X_i$ we can appeal to Proposition", "\\ref{proposition-characterize-locally-finite-presentation}", "to find an $i_0 \\geq \\max(i_1, i_2)$ and a morphism $h : X_{i_0} \\to V$", "over $S_{i_2}$ such that $X \\to X_{i_0} \\to V$ is equal to $X \\to Y \\to V$.", "For $i \\geq i_0$ we get a commutative solid diagram", "$$", "\\xymatrix{", "X \\ar[d] \\ar[r] &", "X_i \\ar[r] \\ar@{..>}[d] \\ar@/_2pc/[dd] |!{[d];[ld]}\\hole &", "X_{i_0} \\ar[d]^h \\\\", "Y \\ar[r] \\ar[d] & Y_i \\ar[r] \\ar[d] & V \\ar[d] \\\\", "S \\ar[r] & S_i \\ar[r] & S_{i_0}", "}", "$$", "Since $X \\to X_i$ has scheme theoretically dense image", "and since $Y_i$ is the scheme theoretic image of", "$Y \\to S_i \\times_{S_{i_2}} V$", "we find that the morphism $X_i \\to S_i \\times_{S_{i_2}} V$", "induced by the diagram", "factors through $Y_i$ (Morphisms, Lemma \\ref{morphisms-lemma-factor-factor}).", "This proves existence.", "\\medskip\\noindent", "Uniqueness. Let $E_i \\subset X_i$ be the equalizer of $f_i$ and $g_i$", "for $i \\geq i_0$. By", "Schemes, Lemma \\ref{schemes-lemma-where-are-they-equal}", "$E_i$ is a locally closed subscheme of $X_i$.", "Since $X_i$ is a closed subscheme of $S_i \\times_{S_{i_0}} X_{i_0}$", "and similarly for $Y_i$ we see that", "$$", "E_i = X_i \\times_{(S_i \\times_{S_{i_0}} X_{i_0})} (S_i \\times_{S_{i_0}} E_{i_0})", "$$", "Thus to finish the proof it suffices to show that $X_i \\to X_{i_0}$", "factors through $E_{i_0}$ for some $i \\geq i_0$.", "To do this we will use that $X \\to X_{i_0}$ factors through $E_{i_0}$", "as both $f_{i_0}$ and $g_{i_0}$ are compatible with $f$.", "Since $X_i$ is Noetherian, we see that the underlying", "topological space $|E_{i_0}|$ is a constructible subset of $|X_{i_0}|$", "(Topology, Lemma \\ref{topology-lemma-constructible-Noetherian-space}).", "Hence $X_i \\to X_{i_0}$ factors through $E_{i_0}$ set theoretically", "for large enough $i$ by Lemma \\ref{lemma-limit-contained-in-constructible}.", "For such an $i$ the scheme theoretic inverse image", "$(X_i \\to X_{i_0})^{-1}(E_{i_0})$ is a closed subscheme of $X_i$", "through which $X$ factors and hence equal to $X_i$ since", "$X \\to X_i$ has scheme theoretically dense image by construction.", "This concludes the proof." ], "refs": [ "limits-proposition-characterize-locally-finite-presentation", "morphisms-lemma-factor-factor", "schemes-lemma-where-are-they-equal", "topology-lemma-constructible-Noetherian-space", "limits-lemma-limit-contained-in-constructible" ], "ref_ids": [ 15127, 5148, 7708, 8267, 15040 ] } ], "ref_ids": [ 15120 ] }, { "id": 15122, "type": "theorem", "label": "limits-lemma-morphism-good-diagram-flat", "categories": [ "limits" ], "title": "limits-lemma-morphism-good-diagram-flat", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}.", "If $f$ is flat and of finite presentation, then", "there exists an $i_3 \\geq i_0$ such that for $i \\geq i_3$ we have", "$f_i$ is flat, $X_i = Y_i \\times_{Y_{i_3}} X_{i_3}$, and", "$X = Y \\times_{Y_{i_3}} X_{i_3}$." ], "refs": [ "limits-lemma-morphism-good-diagram" ], "proofs": [ { "contents": [ "By Lemma \\ref{lemma-descend-finite-presentation}", "we can choose an $i \\geq i_2$ and a morphism", "$U \\to Y_i$ of finite presentation such that $X = Y \\times_{Y_i} U$", "(this is where we use that $f$ is of finite presentation).", "After increasing $i$ we may assume that $U \\to Y_i$ is flat, see", "Lemma \\ref{lemma-descend-flat-finite-presentation}.", "As discussed in Remark \\ref{remark-finite-type-gives-well-defined-system}", "we may and do replace the initial diagram used to define the system", "$(X_i)_{i \\geq i_1}$ by the system corresponding to", "$X \\to U \\to S_i$. Thus $X_{i'}$ for $i' \\geq i$ is defined as", "the scheme theoretic image of $X \\to S_{i'} \\times_{S_i} U$.", "\\medskip\\noindent", "Because $U \\to Y_i$ is flat (this is where we use that $f$ is flat),", "because $X = Y \\times_{Y_i} U$, and", "because the scheme theoretic image of $Y \\to Y_i$ is $Y_i$,", "we see that the scheme theoretic image of $X \\to U$ is $U$", "(Morphisms, Lemma", "\\ref{morphisms-lemma-flat-base-change-scheme-theoretic-image}).", "Observe that $Y_{i'} \\to S_{i'} \\times_{S_i} Y_i$ is a closed", "immersion for $i' \\geq i$ by construction of the system of $Y_j$.", "Then the same argument as above shows that the scheme theoretic image", "of $X \\to S_{i'} \\times_{S_i} U$", "is equal to the closed subscheme $Y_{i'} \\times_{Y_i} U$.", "Thus we see that $X_{i'} = Y_{i'} \\times_{Y_i} U$ for all $i' \\geq i$", "and hence the lemma holds with $i_3 = i$." ], "refs": [ "limits-lemma-descend-finite-presentation", "limits-lemma-descend-flat-finite-presentation", "limits-remark-finite-type-gives-well-defined-system", "morphisms-lemma-flat-base-change-scheme-theoretic-image" ], "ref_ids": [ 15077, 15062, 15133, 5273 ] } ], "ref_ids": [ 15121 ] }, { "id": 15123, "type": "theorem", "label": "limits-lemma-morphism-good-diagram-smooth", "categories": [ "limits" ], "title": "limits-lemma-morphism-good-diagram-smooth", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}.", "If $f$ is smooth, then there exists an $i_3 \\geq i_0$ such that for", "$i \\geq i_3$ we have $f_i$ is smooth." ], "refs": [ "limits-lemma-morphism-good-diagram" ], "proofs": [ { "contents": [ "Combine Lemmas \\ref{lemma-morphism-good-diagram-flat} and", "\\ref{lemma-descend-smooth}." ], "refs": [ "limits-lemma-morphism-good-diagram-flat", "limits-lemma-descend-smooth" ], "ref_ids": [ 15122, 15064 ] } ], "ref_ids": [ 15121 ] }, { "id": 15124, "type": "theorem", "label": "limits-lemma-morphism-good-diagram-proper", "categories": [ "limits" ], "title": "limits-lemma-morphism-good-diagram-proper", "contents": [ "Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}.", "If $f$ is proper, then there exists an $i_3 \\geq i_0$ such that for", "$i \\geq i_3$ we have $f_i$ is proper." ], "refs": [ "limits-lemma-morphism-good-diagram" ], "proofs": [ { "contents": [ "By the discussion in ", "Remark \\ref{remark-finite-type-gives-well-defined-system}", "the choice of $i_1$ and $W$ fitting into a diagram as in", "(\\ref{equation-good-diagram}) is immaterial for the truth of", "the lemma. Thus we choose $W$ as follows.", "First we choose a closed immersion $X \\to X'$", "with $X' \\to S$ proper and of finite presentation, see", "Lemma \\ref{lemma-proper-limit-of-proper-finite-presentation}.", "Then we choose an $i_3 \\geq i_2$ and a proper morphism $W \\to Y_{i_3}$", "such that $X' = Y \\times_{Y_{i_3}} W$. This is possible because", "$Y = \\lim_{i \\geq i_2} Y_i$ and", "Lemmas \\ref{lemma-descend-finite-presentation} and", "\\ref{lemma-eventually-proper}.", "With this choice of $W$ it is immediate from the construction that", "for $i \\geq i_3$ the scheme $X_i$ is a closed subscheme of", "$Y_i \\times_{Y_{i_3}} W \\subset S_i \\times_{S_{i_3}} W$", "and hence proper over $Y_i$." ], "refs": [ "limits-remark-finite-type-gives-well-defined-system", "limits-lemma-proper-limit-of-proper-finite-presentation", "limits-lemma-descend-finite-presentation", "limits-lemma-eventually-proper" ], "ref_ids": [ 15133, 15090, 15077, 15089 ] } ], "ref_ids": [ 15121 ] }, { "id": 15125, "type": "theorem", "label": "limits-lemma-good-diagram-fibre-product", "categories": [ "limits" ], "title": "limits-lemma-good-diagram-fibre-product", "contents": [ "In Situation \\ref{situation-limit-noetherian} suppose that we have a", "cartesian diagram", "$$", "\\xymatrix{", "X^1 \\ar[r]_p \\ar[d]_q & X^3 \\ar[d]^a \\\\", "X^2 \\ar[r]^b & X^4", "}", "$$", "of schemes quasi-separated and of finite type over $S$.", "For each $j = 1, 2, 3, 4$ choose $i_j \\in I$ and a diagram", "$$", "\\xymatrix{", "X^j \\ar[r] \\ar[d] & W^j \\ar[d] \\\\", "S \\ar[r] & S_{i_j}", "}", "$$", "as in (\\ref{equation-good-diagram}). Let", "$X^j = \\lim_{i \\geq i_j} X^j_i$ be the corresponding limit descriptions", "as in Lemma \\ref{lemma-morphism-good-diagram}.", "Let $(a_i)_{i \\geq i_5}$, $(b_i)_{i \\geq i_6}$, $(p_i)_{i \\geq i_7}$, and", "$(q_i)_{i \\geq i_8}$ be the corresponding morphisms of systems contructed", "in Lemma \\ref{lemma-morphism-good-diagram}. Then there exists an", "$i_9 \\geq \\max(i_5, i_6, i_7, i_8)$ such that for $i \\geq i_9$ we have", "$a_i \\circ p_i = b_i \\circ q_i$ and such that", "$$", "(q_i, p_i) : X^1_i \\longrightarrow X^2_i \\times_{b_i, X^4_i, a_i} X^3_i", "$$", "is a closed immersion.", "If $a$ and $b$ are flat and of finite presentation, then there exists an", "$i_{10} \\geq \\max(i_5, i_6, i_7, i_8, i_9)$ such that for $i \\geq i_{10}$", "the last displayed morphism is an isomorphism." ], "refs": [ "limits-lemma-morphism-good-diagram", "limits-lemma-morphism-good-diagram" ], "proofs": [ { "contents": [ "According to the discussion in", "Remark \\ref{remark-finite-type-gives-well-defined-system}", "the choice of $W^1$ fitting into a diagram as in", "(\\ref{equation-good-diagram}) is immaterial for the truth of", "the lemma. Thus we may choose $W^1 = W^2 \\times_{W^4} W^3$.", "Then it is immediate from the construction of $X^1_i$ that ", "$a_i \\circ p_i = b_i \\circ q_i$ and that", "$$", "(q_i, p_i) : X^1_i \\longrightarrow X^2_i \\times_{b_i, X^4_i, a_i} X^3_i", "$$", "is a closed immersion.", "\\medskip\\noindent", "If $a$ and $b$ are flat and of finite presentation, then so are", "$p$ and $q$ as base changes of $a$ and $b$. Thus we can apply", "Lemma \\ref{lemma-morphism-good-diagram-flat}", "to each of $a$, $b$, $p$, $q$, and $a \\circ p = b \\circ q$.", "It follows that there exists an $i_9 \\in I$ such that", "$$", "(q_i, p_i) : X^1_i \\to X^2_i \\times_{X^4_i} X^3_i", "$$", "is the base change of $(q_{i_9}, p_{i_9})$ by the morphism", "by the morphism $X^4_i \\to X^4_{i_9}$ for all $i \\geq i_9$.", "We conclude that $(q_i, p_i)$ is an isomorphism for all sufficiently", "large $i$ by Lemma \\ref{lemma-descend-isomorphism}." ], "refs": [ "limits-remark-finite-type-gives-well-defined-system", "limits-lemma-morphism-good-diagram-flat", "limits-lemma-descend-isomorphism" ], "ref_ids": [ 15133, 15122, 15066 ] } ], "ref_ids": [ 15121, 15121 ] }, { "id": 15126, "type": "theorem", "label": "limits-proposition-approximate", "categories": [ "limits" ], "title": "limits-proposition-approximate", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme.", "There exist a directed set $I$", "and an inverse system of schemes $(S_i, f_{ii'})$ over $I$", "such that", "\\begin{enumerate}", "\\item the transition morphisms $f_{ii'}$ are affine", "\\item each $S_i$ is of finite type over $\\mathbf{Z}$, and", "\\item $S = \\lim_i S_i$.", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "This is a special case of Lemma \\ref{lemma-approximate}", "with $V = \\emptyset$." ], "refs": [ "limits-lemma-approximate" ], "ref_ids": [ 15053 ] } ], "ref_ids": [] }, { "id": 15127, "type": "theorem", "label": "limits-proposition-characterize-locally-finite-presentation", "categories": [ "limits" ], "title": "limits-proposition-characterize-locally-finite-presentation", "contents": [ "\\begin{reference}", "\\cite[IV, Proposition 8.14.2]{EGA}", "\\end{reference}", "Let $f : X \\to S$ be a morphism of schemes.", "The following are equivalent:", "\\begin{enumerate}", "\\item The morphism $f$ is locally of finite presentation.", "\\item For any directed set $I$, and any", "inverse system $(T_i, f_{ii'})$ of $S$-schemes over $I$", "with each $T_i$ affine, we have", "$$", "\\Mor_S(\\lim_i T_i, X) =", "\\colim_i \\Mor_S(T_i, X)", "$$", "\\item For any directed set $I$, and any", "inverse system $(T_i, f_{ii'})$ of $S$-schemes over $I$", "with each $f_{ii'}$ affine and every $T_i$ quasi-compact and", "quasi-separated as a scheme, we have", "$$", "\\Mor_S(\\lim_i T_i, X) =", "\\colim_i \\Mor_S(T_i, X)", "$$", "\\end{enumerate}" ], "refs": [], "proofs": [ { "contents": [ "It is clear that (3) implies (2).", "\\medskip\\noindent", "Let us prove that (2) implies (1). Assume (2).", "Choose any affine opens $U \\subset X$ and $V \\subset S$ such that", "$f(U) \\subset V$. We have to show that", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is of finite presentation.", "Let $(A_i, \\varphi_{ii'})$ be a directed system of", "$\\mathcal{O}_S(V)$-algebras. Set $A = \\colim_i A_i$.", "According to", "Algebra, Lemma \\ref{algebra-lemma-characterize-finite-presentation}", "we have to show that", "$$", "\\Hom_{\\mathcal{O}_S(V)}(\\mathcal{O}_X(U), A) =", "\\colim_i \\Hom_{\\mathcal{O}_S(V)}(\\mathcal{O}_X(U), A_i)", "$$", "Consider the schemes $T_i = \\Spec(A_i)$. They", "form an inverse system of $V$-schemes over $I$", "with transition morphisms $f_{ii'} : T_i \\to T_{i'}$", "induced by the $\\mathcal{O}_S(V)$-algebra maps $\\varphi_{i'i}$.", "Set $T := \\Spec(A) = \\lim_i T_i$.", "The formula above becomes in terms of morphism sets of schemes", "$$", "\\Mor_V(\\lim_i T_i, U) =", "\\colim_i \\Mor_V(T_i, U).", "$$", "We first observe that", "$\\Mor_V(T_i, U) = \\Mor_S(T_i, U)$", "and", "$\\Mor_V(T, U) = \\Mor_S(T, U)$.", "Hence we have to show that", "$$", "\\Mor_S(\\lim_i T_i, U) =", "\\colim_i \\Mor_S(T_i, U)", "$$", "and we are given that", "$$", "\\Mor_S(\\lim_i T_i, X) =", "\\colim_i \\Mor_S(T_i, X).", "$$", "Hence it suffices to prove that given a morphism $g_i : T_i \\to X$ over $S$", "such that the composition $T \\to T_i \\to X$ ends up in $U$ there exists some", "$i' \\geq i$ such that the composition $g_{i'} : T_{i'} \\to T_i \\to X$ ends up", "in $U$. Denote $Z_{i'} = g_{i'}^{-1}(X \\setminus U)$.", "Assume each $Z_{i'}$ is nonempty", "to get a contradiction. By Lemma \\ref{lemma-limit-closed-nonempty}", "there exists a point $t$ of $T$ which is mapped into $Z_{i'}$ for all", "$i' \\geq i$. Such a point is not mapped into $U$. A contradiction.", "\\medskip\\noindent", "Finally, let us prove that (1) implies (3). Assume (1). Let an inverse directed", "system $(T_i, f_{ii'})$ of $S$-schemes be given. Assume the morphisms $f_{ii'}$", "are affine and each $T_i$ is quasi-compact and quasi-separated as a scheme. Let", "$T = \\lim_i T_i$. Denote $f_i : T \\to T_i$ the projection morphisms.", "We have to show:", "\\begin{enumerate}", "\\item[(a)] Given morphisms $g_i, g'_i : T_i \\to X$ over $S$ such that", "$g_i \\circ f_i = g'_i \\circ f_i$, then there exists an $i' \\geq i$", "such that $g_i \\circ f_{i'i} = g'_i \\circ f_{i'i}$.", "\\item[(b)] Given any morphism", "$g : T \\to X$ over $S$ there exists an $i \\in I$ and a morphism", "$g_i : T_i \\to X$ such that $g = f_i \\circ g_i$.", "\\end{enumerate}", "\\noindent", "First let us prove the uniqueness part (a). Let $g_i, g'_i : T_i \\to X$ be", "morphisms such that $g_i \\circ f_i = g'_i \\circ f_i$. For any $i' \\geq i$", "we set $g_{i'} = g_i \\circ f_{i'i}$ and $g'_{i'} = g'_i \\circ f_{i'i}$.", "We also set $g = g_i \\circ f_i = g'_i \\circ f_i$.", "Consider the morphism", "$(g_i, g'_i) : T_i \\to X \\times_S X$. Set", "$$", "W =", "\\bigcup\\nolimits_{U \\subset X\\text{ affine open},", "V \\subset S\\text{ affine open}, f(U) \\subset V}", "U \\times_V U.", "$$", "This is an open in $X \\times_S X$, with the property that the morphism", "$\\Delta_{X/S}$ factors through a closed immersion into $W$, see the proof", "of Schemes, Lemma \\ref{schemes-lemma-diagonal-immersion}.", "Note that the composition", "$(g_i, g'_i) \\circ f_i : T \\to X \\times_S X$ is a morphism into $W$", "because it factors through the diagonal by assumption.", "Set $Z_{i'} = (g_{i'}, g'_{i'})^{-1}(X \\times_S X \\setminus W)$.", "If each $Z_{i'}$ is nonempty, then by Lemma \\ref{lemma-limit-closed-nonempty}", "there exists a point $t \\in T$ which maps to $Z_{i'}$ for all", "$i' \\geq i$. This is a contradiction with the fact that $T$ maps into $W$.", "Hence we may increase $i$ and assume that $(g_i, g'_i) : T_i \\to X \\times_S X$", "is a morphism into $W$. By construction of $W$, and since $T_i$ is", "quasi-compact we can find a finite affine open covering", "$T_i = T_{1, i} \\cup \\ldots \\cup T_{n, i}$ such that", "$(g_i, g'_i)|_{T_{j, i}}$ is a morphism into $U \\times_V U$ for", "some pair $(U, V)$ as in the definition of $W$ above.", "Since it suffices to prove that $g_{i'}$ and $g'_{i'}$ agree", "on each of the $f_{i'i}^{-1}(T_{j, i})$ this reduces us to the affine case.", "The affine case follows from", "Algebra, Lemma \\ref{algebra-lemma-characterize-finite-presentation}", "and the fact that the ring map", "$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is of finite presentation", "(see Morphisms,", "Lemma \\ref{morphisms-lemma-locally-finite-presentation-characterize}).", "\\medskip\\noindent", "Finally, we prove the existence part (b).", "Let $g : T \\to X$ be a morphism of schemes over $S$.", "We can find a finite affine open covering", "$T = W_1 \\cup \\ldots \\cup W_n$ such that for", "each $j \\in \\{1, \\ldots, n\\}$ there exist affine opens", "$U_j \\subset X$ and $V_j \\subset S$ with $f(U_j) \\subset V_j$", "and $g(W_j) \\subset U_j$. By Lemmas \\ref{lemma-descend-opens}", "and \\ref{lemma-limit-affine}", "(after possibly shrinking $I$) we may assume that", "there exist affine open coverings $T_i = W_{1, i} \\cup \\ldots \\cup W_{n, i}$", "compatible with transition maps such that $W_j = \\lim_i W_{j, i}$.", "We apply Algebra, Lemma \\ref{algebra-lemma-characterize-finite-presentation}", "to the rings corresponding to the affine schemes $U_j$, $V_j$, $W_{j, i}$ and", "$W_j$ using that $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_j)$ is of finite", "presentation (see Morphisms,", "Lemma \\ref{morphisms-lemma-locally-finite-presentation-characterize}).", "Thus we can find for each $j$ an index $i_j \\in I$ and a morphism", "$g_{j, i_j} : W_{j, i_j} \\to X$ such that", "$g_{j, i_j} \\circ f_i|_{W_j} : W_j \\to W_{j, i} \\to X$", "equals $g|_{W_j}$. By part (a) proved above, using the quasi-compactness of", "$W_{j_1, i} \\cap W_{j_2, i}$ which follows as $T_i$ is quasi-separated,", "we can find an index $i' \\in I$ larger than all $i_j$ such that", "$$", "g_{j_1, i_{j_1}} \\circ f_{i'i_{j_1}}|_{W_{j_1, i'} \\cap W_{j_2, i'}} =", "g_{j_2, i_{j_2}} \\circ f_{i'i_{j_2}}|_{W_{j_1, i'} \\cap W_{j_2, i'}}", "$$", "for all $j_1, j_2 \\in \\{1, \\ldots, n\\}$. Hence the morphisms", "$g_{j, i_j} \\circ f_{i'i_j}|_{W_{j, i'}}$ glue to given the", "desired morphism $T_{i'} \\to X$." ], "refs": [ "algebra-lemma-characterize-finite-presentation", "limits-lemma-limit-closed-nonempty", "schemes-lemma-diagonal-immersion", "limits-lemma-limit-closed-nonempty", "algebra-lemma-characterize-finite-presentation", "morphisms-lemma-locally-finite-presentation-characterize", "limits-lemma-descend-opens", "limits-lemma-limit-affine", "algebra-lemma-characterize-finite-presentation", "morphisms-lemma-locally-finite-presentation-characterize" ], "ref_ids": [ 1092, 15038, 7707, 15038, 1092, 5238, 15041, 15043, 1092, 5238 ] } ], "ref_ids": [] }, { "id": 15128, "type": "theorem", "label": "limits-proposition-separated-closed-in-finite-presentation", "categories": [ "limits" ], "title": "limits-proposition-separated-closed-in-finite-presentation", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Assume", "\\begin{enumerate}", "\\item $f$ is of finite type and separated, and", "\\item $S$ is quasi-compact and quasi-separated.", "\\end{enumerate}", "Then there exists a separated morphism of finite presentation", "$f' : X' \\to S$ and a closed immersion $X \\to X'$ of schemes over $S$." ], "refs": [], "proofs": [ { "contents": [ "Apply Lemma \\ref{lemma-finite-type-is-limit-finite-presentation}", "and note that $X_i \\to S$ is separated for", "large $i$ by Lemma \\ref{lemma-eventually-separated} as we have", "assumed that $X \\to S$ is separated." ], "refs": [ "limits-lemma-finite-type-is-limit-finite-presentation", "limits-lemma-eventually-separated" ], "ref_ids": [ 15074, 15047 ] } ], "ref_ids": [] }, { "id": 15129, "type": "theorem", "label": "limits-proposition-affine", "categories": [ "limits" ], "title": "limits-proposition-affine", "contents": [ "\\begin{slogan}", "A scheme admitting a surjective integral map from an affine scheme is affine.", "\\end{slogan}", "Let $f : X \\to S$ be a morphism of schemes.", "Assume that $f$ is surjective and integral, and assume that $X$ is affine.", "Then $S$ is affine." ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is surjective and $X$ is quasi-compact we see that $S$ is", "quasi-compact. Since $X$ is separated and $f$ is surjective and", "universally closed (Morphisms, Lemma", "\\ref{morphisms-lemma-integral-universally-closed}), we see that $S$", "is separated (Morphisms, Lemma", "\\ref{morphisms-lemma-image-universally-closed-separated}).", "\\medskip\\noindent", "By Lemma \\ref{lemma-integral-limit-finite-and-finite-presentation}", "we can write $X = \\lim_i X_i$ with $X_i \\to S$ finite. By", "Lemma \\ref{lemma-limit-affine}", "we see that for $i$ sufficiently large the scheme $X_i$ is affine.", "Moreover, since $X \\to S$ factors through each $X_i$ we see that", "$X_i \\to S$ is surjective. Hence we conclude that $S$ is affine by", "Lemma \\ref{lemma-affine}." ], "refs": [ "morphisms-lemma-integral-universally-closed", "morphisms-lemma-image-universally-closed-separated", "limits-lemma-integral-limit-finite-and-finite-presentation", "limits-lemma-limit-affine", "limits-lemma-affine" ], "ref_ids": [ 5441, 5415, 15056, 15043, 15082 ] } ], "ref_ids": [] } ], "definitions": [ { "id": 8, "type": "definition", "label": "stacks-perfect-definition-derived", "categories": [ "stacks-perfect" ], "title": "stacks-perfect-definition-derived", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let", "$\\mathcal{M}_\\mathcal{X} \\subset \\textit{Mod}(\\mathcal{O}_\\mathcal{X})$", "denote the category of locally quasi-coherent", "$\\mathcal{O}_\\mathcal{X}$-modules with the flat base change property.", "Let $\\mathcal{P}_\\mathcal{X} \\subset \\mathcal{M}_\\mathcal{X}$", "be the full subcategory consisting of parasitic objects.", "We define the {\\it derived category of $\\mathcal{O}_\\mathcal{X}$-modules with", "quasi-coherent cohomology sheaves} as the Verdier quotient\\footnote{This", "definition is different from the one in the literature, see", "\\cite[6.3]{olsson_sheaves}, but it agrees with that definition", "by Lemma \\ref{lemma-derived-quasi-coherent}.}", "$$", "D_\\QCoh(\\mathcal{O}_\\mathcal{X}) =", "D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})/", "D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})", "$$" ], "refs": [ "stacks-perfect-lemma-derived-quasi-coherent" ], "ref_ids": [ 4 ] }, { "id": 278, "type": "definition", "label": "spaces-more-morphisms-definition-radicial", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-radicial", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. We say $f$ is {\\it radicial} if for any morphism", "$\\Spec(K) \\to Y$ where $K$ is a field the reduction", "$(\\Spec(K) \\times_Y X)_{red}$ is either empty or", "representable by the spectrum of a purely inseparable field extension of $K$." ], "refs": [], "ref_ids": [] }, { "id": 279, "type": "definition", "label": "spaces-more-morphisms-definition-conormal-sheaf", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-conormal-sheaf", "contents": [ "Let $i : Z \\to X$ be an immersion. The {\\it conormal sheaf", "$\\mathcal{C}_{Z/X}$ of $Z$ in $X$} or the {\\it conormal sheaf of $i$}", "is the quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{I}/\\mathcal{I}^2$", "described above." ], "refs": [], "ref_ids": [] }, { "id": 280, "type": "definition", "label": "spaces-more-morphisms-definition-conormal-algebra", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-conormal-algebra", "contents": [ "Let $i : Z \\to X$ be an immersion. The {\\it conormal algebra", "$\\mathcal{C}_{Z/X, *}$ of $Z$ in $X$} or the {\\it conormal algebra of $i$}", "is the quasi-coherent sheaf of graded $\\mathcal{O}_Z$-algebras", "$\\bigoplus_{n \\geq 0} \\mathcal{I}^n/\\mathcal{I}^{n + 1}$ described above." ], "refs": [], "ref_ids": [] }, { "id": 281, "type": "definition", "label": "spaces-more-morphisms-definition-normal-cone", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-normal-cone", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic spaces", "over $S$. The {\\it normal cone $C_ZX$} of $Z$ in $X$ is", "$$", "C_ZX = \\underline{\\Spec}_Z(\\mathcal{C}_{Z/X, *})", "$$", "see Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-relative-spec}. The", "{\\it normal bundle} of $Z$ in $X$ is the vector bundle", "$$", "N_ZX = \\underline{\\Spec}_Z(\\text{Sym}(\\mathcal{C}_{Z/X}))", "$$" ], "refs": [ "spaces-morphisms-definition-relative-spec" ], "ref_ids": [ 4999 ] }, { "id": 282, "type": "definition", "label": "spaces-more-morphisms-definition-sheaf-differentials", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-sheaf-differentials", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. The {\\it sheaf of differentials $\\Omega_{X/Y}$ of $X$ over $Y$}", "is sheaf of differentials", "(Modules on Sites,", "Definition \\ref{sites-modules-definition-sheaf-differentials})", "for the morphism of ringed topoi", "$$", "(f_{small}, f^\\sharp) :", "(X_\\etale, \\mathcal{O}_X)", "\\to", "(Y_\\etale, \\mathcal{O}_Y)", "$$", "of", "Properties of Spaces,", "Lemma \\ref{spaces-properties-lemma-morphism-ringed-topoi}.", "The {\\it universal $Y$-derivation} will be denoted", "$\\text{d}_{X/Y} : \\mathcal{O}_X \\to \\Omega_{X/Y}$." ], "refs": [ "sites-modules-definition-sheaf-differentials", "spaces-properties-lemma-morphism-ringed-topoi" ], "ref_ids": [ 14297, 11882 ] }, { "id": 283, "type": "definition", "label": "spaces-more-morphisms-definition-thickening", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-thickening", "contents": [ "Thickenings. Let $S$ be a scheme.", "\\begin{enumerate}", "\\item We say an algebraic space $X'$ is a {\\it thickening} of an algebraic", "space $X$ if $X$ is a closed subspace of $X'$ and the associated topological", "spaces are equal.", "\\item We say $X'$ is a {\\it first order thickening} of $X$ if", "$X$ is a closed subspace of $X'$ and the quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_{X'}$ defining $X$ has square zero.", "\\item Given two thickenings $X \\subset X'$ and $Y \\subset Y'$ a", "{\\it morphism of thickenings} is a morphism $f' : X' \\to Y'$ such that", "$f(X) \\subset Y$, i.e., such that $f'|_X$ factors through the closed", "subspace $Y$. In this situation we set $f = f'|_X : X \\to Y$ and we say", "that $(f, f') : (X \\subset X') \\to (Y \\subset Y')$ is a morphism of", "thickenings.", "\\item Let $B$ be an algebraic space. We similarly define", "{\\it thickenings over $B$}, and", "{\\it morphisms of thickenings over $B$}. This means that the spaces", "$X, X', Y, Y'$ above are algebraic spaces endowed with a structure", "morphism to $B$, and that the morphisms", "$X \\to X'$, $Y \\to Y'$ and $f' : X' \\to Y'$ are morphisms over $B$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 284, "type": "definition", "label": "spaces-more-morphisms-definition-first-order-infinitesimal-neighbourhood", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-first-order-infinitesimal-neighbourhood", "contents": [ "Let $i : Z \\to X$ be an immersion of algebraic spaces. The", "{\\it first order infinitesimal neighbourhood} of $Z$ in $X$ is", "the first order thickening $Z \\subset Z'$ over $X$ described above." ], "refs": [], "ref_ids": [] }, { "id": 285, "type": "definition", "label": "spaces-more-morphisms-definition-formally-smooth-etale-unramified", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-formally-smooth-etale-unramified", "contents": [ "Let $S$ be a scheme.", "Let $a : F \\to G$ be a transformation of functors", "$F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "Consider commutative solid diagrams of the form", "$$", "\\xymatrix{", "F \\ar[d]_a & T \\ar[d]^i \\ar[l] \\\\", "G & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "where $T$ and $T'$ are affine schemes and $i$ is a closed immersion", "defined by an ideal of square zero.", "\\begin{enumerate}", "\\item We say $a$ is {\\it formally smooth} if given any solid", "diagram as above there exists a dotted arrow making the diagram", "commute\\footnote{This is just one possible definition that one can", "make here. Another slightly weaker condition would be to require that", "the dotted arrow exists fppf locally on $T'$. This weaker notion", "has in some sense better formal properties.}.", "\\item We say $a$ is {\\it formally \\'etale} if given any solid", "diagram as above there exists exactly one dotted arrow making the diagram", "commute.", "\\item We say $a$ is {\\it formally unramified} if given any solid", "diagram as above there exists at most one dotted arrow making the diagram", "commute.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 286, "type": "definition", "label": "spaces-more-morphisms-definition-formally-unramified", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-formally-unramified", "contents": [ "Let $S$ be a scheme. A morphism $f : X \\to Y$ of algebraic spaces over $S$", "is said to be {\\it formally unramified} if it is formally unramified as a", "transformation of functors as in", "Definition \\ref{definition-formally-smooth-etale-unramified}." ], "refs": [ "spaces-more-morphisms-definition-formally-smooth-etale-unramified" ], "ref_ids": [ 285 ] }, { "id": 287, "type": "definition", "label": "spaces-more-morphisms-definition-universal-thickening", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-universal-thickening", "contents": [ "Let $S$ be a scheme.", "Let $h : Z \\to X$ be a formally unramified morphism of", "algebraic spaces over $S$.", "\\begin{enumerate}", "\\item The {\\it universal first order thickening} of $Z$ over $X$", "is the thickening $Z \\subset Z'$ constructed in", "Lemma \\ref{lemma-universal-thickening}.", "\\item The {\\it conormal sheaf of $Z$ over $X$} is the conormal sheaf", "of $Z$ in its universal first order thickening $Z'$ over $X$.", "\\end{enumerate}", "We often denote the conormal sheaf $\\mathcal{C}_{Z/X}$ in this situation." ], "refs": [ "spaces-more-morphisms-lemma-universal-thickening" ], "ref_ids": [ 78 ] }, { "id": 288, "type": "definition", "label": "spaces-more-morphisms-definition-formally-etale", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-formally-etale", "contents": [ "Let $S$ be a scheme. A morphism $f : X \\to Y$ of algebraic spaces over $S$", "is said to be {\\it formally \\'etale} if it is formally \\'etale as a", "transformation of functors as in", "Definition \\ref{definition-formally-smooth-etale-unramified}." ], "refs": [ "spaces-more-morphisms-definition-formally-smooth-etale-unramified" ], "ref_ids": [ 285 ] }, { "id": 289, "type": "definition", "label": "spaces-more-morphisms-definition-formally-smooth", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-formally-smooth", "contents": [ "Let $S$ be a scheme. A morphism $f : X \\to Y$ of algebraic spaces over $S$", "is said to be {\\it formally smooth} if it is formally smooth as a", "transformation of functors as in", "Definition \\ref{definition-formally-smooth-etale-unramified}." ], "refs": [ "spaces-more-morphisms-definition-formally-smooth-etale-unramified" ], "ref_ids": [ 285 ] }, { "id": 290, "type": "definition", "label": "spaces-more-morphisms-definition-netherlander", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-netherlander", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "The {\\it naive cotangent complex of $f$}", "is the complex defined in Modules on Sites, Definition", "\\ref{sites-modules-definition-cotangent-complex-morphism-ringed-topoi}", "for the morphism of ringed topoi $f_{small}$ between the", "small \\'etale sites of $X$ and $Y$, see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-morphism-ringed-topoi}.", "Notation: $\\NL_f$ or $\\NL_{X/Y}$." ], "refs": [ "sites-modules-definition-cotangent-complex-morphism-ringed-topoi", "spaces-properties-lemma-morphism-ringed-topoi" ], "ref_ids": [ 14301, 11882 ] }, { "id": 291, "type": "definition", "label": "spaces-more-morphisms-definition-module-flat-on-fibre", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-module-flat-on-fibre", "contents": [ "Let $S$ be a scheme. Let $X \\to Y \\to Z$ be morphisms of algebraic", "spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $x \\in |X|$ be a point and denote $z \\in |Z|$ its image.", "\\begin{enumerate}", "\\item We say {\\it the restriction of $\\mathcal{F}$ to its fibre over $z$", "is flat at $x$ over the fibre of $Y$ over $z$} if the equivalent conditions of", "Lemma \\ref{lemma-flat-on-fibres-at-point}", "are satisfied.", "\\item We say {\\it the fibre of $X$ over $z$ is flat at $x$ over the fibre of", "$Y$ over $z$} if the equivalent conditions of", "Lemma \\ref{lemma-flat-on-fibres-at-point}", "hold with $\\mathcal{F} = \\mathcal{O}_X$.", "\\item We say {\\it the fibre of $X$ over $z$ is flat over the fibre of $Y$", "over $z$} if for all $x \\in |X|$ lying over $z$ the fibre of $X$ over $z$", "is flat at $x$ over the fibre of $Y$ over $z$", "\\end{enumerate}" ], "refs": [ "spaces-more-morphisms-lemma-flat-on-fibres-at-point", "spaces-more-morphisms-lemma-flat-on-fibres-at-point" ], "ref_ids": [ 130, 130 ] }, { "id": 292, "type": "definition", "label": "spaces-more-morphisms-definition-CM", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-CM", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume the fibres of $f$ are locally Noetherian", "(Divisors on Spaces, Definition", "\\ref{spaces-divisors-definition-locally-Noetherian-fibre}).", "\\begin{enumerate}", "\\item Let $x \\in |X|$, and $y = f(x)$. We say that $f$ is", "{\\it Cohen-Macaulay at $x$} if $f$ is flat at $x$ and", "the equivalent conditions of", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-local-source-target-at-point}", "hold for the property $\\mathcal{P}$ described in", "Lemma \\ref{lemma-CM-local-ring-fibre}.", "\\item We say $f$ is a {\\it Cohen-Macaulay morphism} if $f$ is", "Cohen-Macaulay at every point of $X$.", "\\end{enumerate}" ], "refs": [ "spaces-divisors-definition-locally-Noetherian-fibre", "spaces-morphisms-lemma-local-source-target-at-point", "spaces-more-morphisms-lemma-CM-local-ring-fibre" ], "ref_ids": [ 13014, 4812, 140 ] }, { "id": 293, "type": "definition", "label": "spaces-more-morphisms-definition-gorenstein", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-gorenstein", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Assume the fibres of $f$ are locally Noetherian", "(Divisors on Spaces, Definition", "\\ref{spaces-divisors-definition-locally-Noetherian-fibre}).", "\\begin{enumerate}", "\\item Let $x \\in |X|$, and $y = f(x)$. We say that $f$ is", "{\\it Gorenstein at $x$} if $f$ is flat at $x$ and", "the equivalent conditions of", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-local-source-target-at-point}", "hold for the property $\\mathcal{P}$ described in", "Lemma \\ref{lemma-gorenstein-local-ring-fibre}.", "\\item We say $f$ is a {\\it Gorenstein morphism} if $f$ is", "Gorenstein at every point of $X$.", "\\end{enumerate}" ], "refs": [ "spaces-divisors-definition-locally-Noetherian-fibre", "spaces-morphisms-lemma-local-source-target-at-point", "spaces-more-morphisms-lemma-gorenstein-local-ring-fibre" ], "ref_ids": [ 13014, 4812, 147 ] }, { "id": 294, "type": "definition", "label": "spaces-more-morphisms-definition-geometrically-reduced-fibre", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-geometrically-reduced-fibre", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a", "morphism of algebraic spaces over $S$. Let $y \\in |Y|$.", "We say {\\it the fibre of $f : X \\to Y$ at $y$ is geometrically reduced}", "if the equivalent conditions of", "Lemma \\ref{lemma-geometrically-reduced-fibre} hold." ], "refs": [ "spaces-more-morphisms-lemma-geometrically-reduced-fibre" ], "ref_ids": [ 154 ] }, { "id": 295, "type": "definition", "label": "spaces-more-morphisms-definition-regular-immersion", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-regular-immersion", "contents": [ "Let $S$ be a scheme. Let $i : X \\to Y$ be a morphism of algebraic", "spaces over $S$.", "\\begin{enumerate}", "\\item We say $i$ is a {\\it Koszul-regular immersion} if $i$ is representable", "and the equivalent conditions of", "Lemma \\ref{lemma-representable-etale-local-target}", "hold with $\\mathcal{P}(f) =$``$f$ is a Koszul-regular immersion''.", "\\item We say $i$ is an {\\it $H_1$-regular immersion} if $i$ is representable", "and the equivalent conditions of", "Lemma \\ref{lemma-representable-etale-local-target}", "hold with $\\mathcal{P}(f) =$``$f$ is an $H_1$-regular immersion''.", "\\item We say $i$ is a {\\it quasi-regular immersion} if $i$ is representable", "and the equivalent conditions of", "Lemma \\ref{lemma-representable-etale-local-target}", "hold with $\\mathcal{P}(f) =$``$f$ is a quasi-regular immersion''.", "\\end{enumerate}" ], "refs": [ "spaces-more-morphisms-lemma-representable-etale-local-target", "spaces-more-morphisms-lemma-representable-etale-local-target", "spaces-more-morphisms-lemma-representable-etale-local-target" ], "ref_ids": [ 214, 214, 214 ] }, { "id": 296, "type": "definition", "label": "spaces-more-morphisms-definition-relative-pseudo-coherence", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-relative-pseudo-coherence", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "algebraic spaces over $S$ which is locally of finite type.", "Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. Let $\\mathcal{F}$ be a", "quasi-coherent $\\mathcal{O}_X$-module. Fix $m \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item We say $E$ is {\\it $m$-pseudo-coherent relative to $Y$}", "if the equivalent conditions of", "Lemma \\ref{lemma-qcoh-relative-pseudo-coherence-characterize} are satisfied.", "\\item We say $E$ is {\\it pseudo-coherent relative to $Y$}", "if $E$ is $m$-pseudo-coherent relative to $Y$ for all $m \\in \\mathbf{Z}$.", "\\item We say $\\mathcal{F}$ is {\\it $m$-pseudo-coherent relative to $Y$} if", "$\\mathcal{F}$ viewed as an object of $D_\\QCoh(\\mathcal{O}_X)$ is", "$m$-pseudo-coherent relative to $Y$.", "\\item We say $\\mathcal{F}$ is {\\it pseudo-coherent relative to $Y$} if", "$\\mathcal{F}$ viewed as an object of $D_\\QCoh(\\mathcal{O}_X)$ is", "pseudo-coherent relative to $Y$.", "\\end{enumerate}" ], "refs": [ "spaces-more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize" ], "ref_ids": [ 223 ] }, { "id": 297, "type": "definition", "label": "spaces-more-morphisms-definition-pseudo-coherent", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-pseudo-coherent", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item We say $f$ is {\\it pseudo-coherent} if the equivalent conditions of", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-local-source-target}", "hold with $\\mathcal{P} =$``pseudo-coherent''.", "\\item Let $x \\in |X|$. We say $f$ is {\\it pseudo-coherent at $x$} if", "there exists an open neighbourhood $X' \\subset X$ of $x$ such", "that $f|_{X'} : X' \\to Y$ is pseudo-coherent.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-lemma-local-source-target" ], "ref_ids": [ 4811 ] }, { "id": 298, "type": "definition", "label": "spaces-more-morphisms-definition-perfect", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-perfect", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item We say $f$ is {\\it perfect} if the equivalent conditions of", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-local-source-target}", "hold with $\\mathcal{P} =$``perfect''.", "\\item Let $x \\in |X|$. We say $f$ is {\\it perfect at $x$} if", "there exists an open neighbourhood $X' \\subset X$ of $x$ such", "that $f|_{X'} : X' \\to Y$ is perfect.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-lemma-local-source-target" ], "ref_ids": [ 4811 ] }, { "id": 299, "type": "definition", "label": "spaces-more-morphisms-definition-lci", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-lci", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item We say $f$ is a {\\it Koszul morphism}, or that $f$ is a", "{\\it local complete intersection morphism} if the equivalent conditions of", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-local-source-target}", "hold with $\\mathcal{P}(f) =$``$f$ is a local complete intersection morphism''.", "\\item Let $x \\in |X|$. We say $f$ is {\\it Koszul at $x$} if", "there exists an open neighbourhood $X' \\subset X$ of $x$ such", "that $f|_{X'} : X' \\to Y$ is a local complete intersection morphism.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-lemma-local-source-target" ], "ref_ids": [ 4811 ] }, { "id": 300, "type": "definition", "label": "spaces-more-morphisms-definition-relatively-perfect", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-relatively-perfect", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ which is flat and locally of finite presentation.", "An object $E$ of $D(\\mathcal{O}_X)$ is {\\it perfect relative to $Y$} or", "{\\it $Y$-perfect} if $E$ is pseudo-coherent", "(Cohomology on Sites, Definition", "\\ref{sites-cohomology-definition-pseudo-coherent}) and", "$E$ locally has finite tor dimension as an object of", "$D(f^{-1}\\mathcal{O}_Y)$", "(Cohomology on Sites, Definition", "\\ref{sites-cohomology-definition-tor-amplitude})." ], "refs": [ "sites-cohomology-definition-pseudo-coherent", "sites-cohomology-definition-tor-amplitude" ], "ref_ids": [ 4420, 4421 ] }, { "id": 301, "type": "definition", "label": "spaces-more-morphisms-definition-nodal-family", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-definition-nodal-family", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "We say $f$ is {\\it at-worst-nodal of relative dimension $1$}", "if the equivalent conditions of", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-local-source-target}", "hold with $\\mathcal{P} =$``at-worst-nodal of relative dimension $1$''." ], "refs": [ "spaces-morphisms-lemma-local-source-target" ], "ref_ids": [ 4811 ] }, { "id": 1432, "type": "definition", "label": "algebra-definition-module-finite-type", "categories": [ "algebra" ], "title": "algebra-definition-module-finite-type", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item We say $M$ is a {\\it finite $R$-module}, or a {\\it finitely generated", "$R$-module} if there exist $n \\in \\mathbf{N}$ and $x_1, \\ldots, x_n \\in M$", "such that every element of $M$ is a $R$-linear combination of the $x_i$.", "Equivalently, this means there exists a surjection", "$R^{\\oplus n} \\to M$ for some $n \\in \\mathbf{N}$.", "\\item We say $M$ is a {\\it finitely presented $R$-module} or an", "{\\it $R$-module of finite presentation} if there exist integers", "$n, m \\in \\mathbf{N}$ and an exact sequence", "$$", "R^{\\oplus m} \\longrightarrow R^{\\oplus n} \\longrightarrow M \\longrightarrow 0", "$$", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1433, "type": "definition", "label": "algebra-definition-finite-type", "categories": [ "algebra" ], "title": "algebra-definition-finite-type", "contents": [ "Let $R \\to S$ be a ring map.", "\\begin{enumerate}", "\\item We say $R \\to S$ is of {\\it finite type}, or that {\\it $S$ is a finite", "type $R$-algebra} if there exists an $n \\in \\mathbf{N}$ and an surjection", "of $R$-algebras $R[x_1, \\ldots, x_n] \\to S$.", "\\item We say $R \\to S$ is of {\\it finite presentation} if there", "exist integers $n, m \\in \\mathbf{N}$ and polynomials", "$f_1, \\ldots, f_m \\in R[x_1, \\ldots, x_n]$", "and an isomorphism of $R$-algebras", "$R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m) \\cong S$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1434, "type": "definition", "label": "algebra-definition-finite-ring-map", "categories": [ "algebra" ], "title": "algebra-definition-finite-ring-map", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. We say $\\varphi : R \\to S$ is", "{\\it finite} if $S$ is finite as an $R$-module." ], "refs": [], "ref_ids": [] }, { "id": 1435, "type": "definition", "label": "algebra-definition-directed-system", "categories": [ "algebra" ], "title": "algebra-definition-directed-system", "contents": [ "Let $(I, \\leq)$ be a preordered set.", "A {\\it system $(M_i, \\mu_{ij})$ of $R$-modules over $I$}", "consists of a family of $R$-modules $\\{M_i\\}_{i\\in I}$ indexed", "by $I$ and a family of $R$-module maps $\\{\\mu_{ij} : M_i \\to M_j\\}_{i \\leq j}$", "such that for all $i \\leq j \\leq k$", "$$", "\\mu_{ii} = \\text{id}_{M_i}\\quad", "\\mu_{ik} = \\mu_{jk}\\circ \\mu_{ij}", "$$", "We say $(M_i, \\mu_{ij})$ is a {\\it directed system} if $I$ is a directed set." ], "refs": [], "ref_ids": [] }, { "id": 1436, "type": "definition", "label": "algebra-definition-homomorphism-directed-systems", "categories": [ "algebra" ], "title": "algebra-definition-homomorphism-directed-systems", "contents": [ "Let $(M_i, \\mu_{ij})$, $(N_i, \\nu_{ij})$ be", "systems of $R$-modules over the same preordered set $I$.", "A {\\it homomorphism of systems} $\\Phi$ from $(M_i, \\mu_{ij})$ to", "$(N_i, \\nu_{ij})$ is by definition a family of $R$-module homomorphisms", "$\\phi_i : M_i \\to N_i$", "such that $\\phi_j \\circ \\mu_{ij} = \\nu_{ij} \\circ \\phi_i$", "for all $i \\leq j$." ], "refs": [], "ref_ids": [] }, { "id": 1437, "type": "definition", "label": "algebra-definition-multiplicative-subset", "categories": [ "algebra" ], "title": "algebra-definition-multiplicative-subset", "contents": [ "Let $R$ be a ring, $S$ a subset of $R$.", "We say $S$ is a {\\it multiplicative subset of $R$} if", "$1\\in S$ and $S$ is closed", "under multiplication, i.e., $s, s' \\in S \\Rightarrow ss' \\in S$." ], "refs": [], "ref_ids": [] }, { "id": 1438, "type": "definition", "label": "algebra-definition-localization", "categories": [ "algebra" ], "title": "algebra-definition-localization", "contents": [ "This ring is called the {\\it localization of $A$ with respect to $S$}." ], "refs": [], "ref_ids": [] }, { "id": 1439, "type": "definition", "label": "algebra-definition-localization-module", "categories": [ "algebra" ], "title": "algebra-definition-localization-module", "contents": [ "The $S^{-1}A$-module $S^{-1}M$ is called the {\\it localization} of $M$ at $S$." ], "refs": [], "ref_ids": [] }, { "id": 1440, "type": "definition", "label": "algebra-definition-relation", "categories": [ "algebra" ], "title": "algebra-definition-relation", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "Let $n \\geq 0$ and $x_i \\in M$ for $i = 1, \\ldots, n$.", "A {\\it relation} between $x_1, \\ldots, x_n$ in $M$ is a", "sequence of elements $f_1, \\ldots, f_n \\in R$ such that", "$\\sum_{i = 1, \\ldots, n} f_i x_i = 0$." ], "refs": [], "ref_ids": [] }, { "id": 1441, "type": "definition", "label": "algebra-definition-bilinear", "categories": [ "algebra" ], "title": "algebra-definition-bilinear", "contents": [ "Let $R$ be a ring, $M, N, P$ be three $R$-modules.", "A mapping $f : M \\times N \\to P$ (where $M \\times N$", "is viewed only as Cartesian product of two $R$-modules) is said to be", "{\\it $R$-bilinear} if for each $x \\in M$", "the mapping $y\\mapsto f(x, y)$ of $N$ into $P$ is $R$-linear, and for each", "$y\\in N$ the mapping $x\\mapsto f(x, y)$ is also $R$-linear." ], "refs": [], "ref_ids": [] }, { "id": 1442, "type": "definition", "label": "algebra-definition-bimodule", "categories": [ "algebra" ], "title": "algebra-definition-bimodule", "contents": [ "An abelian group $N$ is called an {\\it $(A, B)$-bimodule} if it is both an", "$A$-module and a $B$-module, and", "the actions $A \\to End(M)$ and $B \\to End(M)$", "are compatible in the sense that $(ax)b = a(xb)$ for all", "$a\\in A, b\\in B, x\\in N$. Usually we denote it as $_AN_B$." ], "refs": [], "ref_ids": [] }, { "id": 1443, "type": "definition", "label": "algebra-definition-base-change", "categories": [ "algebra" ], "title": "algebra-definition-base-change", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module.", "Let $R \\to R'$ be any ring map. The {\\it base change} of $\\varphi$", "by $R \\to R'$ is the ring map $R' \\to S \\otimes_R R'$. In this situation", "we often write $S' = S \\otimes_R R'$.", "The {\\it base change} of the $S$-module $M$ is the $S'$-module", "$M \\otimes_R R'$." ], "refs": [], "ref_ids": [] }, { "id": 1444, "type": "definition", "label": "algebra-definition-spectrum-ring", "categories": [ "algebra" ], "title": "algebra-definition-spectrum-ring", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item The {\\it spectrum} of $R$ is the set of prime ideals of $R$.", "It is usually denoted $\\Spec(R)$.", "\\item Given a subset $T \\subset R$ we let $V(T) \\subset \\Spec(R)$", "be the set of primes containing $T$, i.e., $V(T) = \\{ \\mathfrak p \\in", "\\Spec(R) \\mid \\forall f\\in T, f\\in \\mathfrak p\\}$.", "\\item Given an element $f \\in R$ we let $D(f) \\subset \\Spec(R)$", "be the set of primes not containing $f$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1445, "type": "definition", "label": "algebra-definition-Zariski-topology", "categories": [ "algebra" ], "title": "algebra-definition-Zariski-topology", "contents": [ "Let $R$ be a ring.", "The topology on $\\Spec(R)$ whose closed sets are the", "sets $V(T)$ is called the {\\it Zariski} topology. The open", "subsets $D(f)$ are called the {\\it standard opens} of $\\Spec(R)$." ], "refs": [], "ref_ids": [] }, { "id": 1446, "type": "definition", "label": "algebra-definition-local-ring", "categories": [ "algebra" ], "title": "algebra-definition-local-ring", "contents": [ "A {\\it local ring} is a ring with exactly one maximal ideal.", "The maximal ideal is often denoted $\\mathfrak m_R$ in this case.", "We often say ``let $(R, \\mathfrak m, \\kappa)$ be a local ring''", "to indicate that $R$ is local, $\\mathfrak m$ is its unique maximal", "ideal and $\\kappa = R/\\mathfrak m$ is its residue field.", "A {\\it local homomorphism of local rings} is a ring map", "$\\varphi : R \\to S$ such that $R$ and $S$ are local rings and such", "that $\\varphi(\\mathfrak m_R) \\subset \\mathfrak m_S$.", "If it is given that $R$ and $S$ are local rings, then the phrase", "``{\\it local ring map $\\varphi : R \\to S$}'' means that $\\varphi$", "is a local homomorphism of local rings." ], "refs": [], "ref_ids": [] }, { "id": 1447, "type": "definition", "label": "algebra-definition-oka-family", "categories": [ "algebra" ], "title": "algebra-definition-oka-family", "contents": [ "Let $R$ be a ring. Let $\\mathcal{F}$ be a set of ideals of $R$. We say", "$\\mathcal{F}$ is an {\\it Oka family} if $R \\in \\mathcal{F}$ and", "whenever $I \\subset R$ is an ideal and $(I : a), (I, a) \\in \\mathcal{F}$", "for some $a \\in R$, then $I \\in \\mathcal{F}$." ], "refs": [], "ref_ids": [] }, { "id": 1448, "type": "definition", "label": "algebra-definition-locally-nilpotent-ideal", "categories": [ "algebra" ], "title": "algebra-definition-locally-nilpotent-ideal", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal.", "We say $I$ is {\\it locally nilpotent} if for every", "$x \\in I$ there exists an $n \\in \\mathbf{N}$ such", "that $x^n = 0$. We say $I$ is {\\it nilpotent} if", "there exists an $n \\in \\mathbf{N}$ such that $I^n = 0$." ], "refs": [], "ref_ids": [] }, { "id": 1449, "type": "definition", "label": "algebra-definition-ring-jacobson", "categories": [ "algebra" ], "title": "algebra-definition-ring-jacobson", "contents": [ "Let $R$ be a ring. We say that $R$ is a", "{\\it Jacobson ring} if every radical", "ideal $I$ is the intersection of the", "maximal ideals containing it." ], "refs": [], "ref_ids": [] }, { "id": 1450, "type": "definition", "label": "algebra-definition-integral-ring-map", "categories": [ "algebra" ], "title": "algebra-definition-integral-ring-map", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "\\begin{enumerate}", "\\item An element $s \\in S$", "is {\\it integral over $R$} if there exists a monic", "polynomial $P(x) \\in R[x]$ such that", "$P^\\varphi(s) = 0$, where $P^\\varphi(x) \\in S[x]$", "is the image of $P$ under $\\varphi : R[x] \\to S[x]$.", "\\item The ring map $\\varphi$ is {\\it integral}", "if every $s \\in S$ is integral over $R$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1451, "type": "definition", "label": "algebra-definition-integral-closure", "categories": [ "algebra" ], "title": "algebra-definition-integral-closure", "contents": [ "Let $R \\to S$ be a ring map.", "The ring $S' \\subset S$ of elements integral over", "$R$, see Lemma \\ref{lemma-integral-closure-is-ring},", "is called the {\\it integral closure} of $R$", "in $S$. If $R \\subset S$ we say that $R$ is", "{\\it integrally closed} in $S$ if $R = S'$." ], "refs": [ "algebra-lemma-integral-closure-is-ring" ], "ref_ids": [ 486 ] }, { "id": 1452, "type": "definition", "label": "algebra-definition-domain-normal", "categories": [ "algebra" ], "title": "algebra-definition-domain-normal", "contents": [ "A domain $R$ is called {\\it normal} if it is integrally", "closed in its field of fractions." ], "refs": [], "ref_ids": [] }, { "id": 1453, "type": "definition", "label": "algebra-definition-almost-integral", "categories": [ "algebra" ], "title": "algebra-definition-almost-integral", "contents": [ "Let $R$ be a domain.", "\\begin{enumerate}", "\\item An element $g$ of the fraction", "field of $R$ is called {\\it almost integral over $R$}", "if there exists an element $r \\in R$, $r\\not = 0$", "such that $rg^n \\in R$ for all $n \\geq 0$.", "\\item The domain $R$ is called {\\it completely normal} if every", "almost integral element of the fraction field of $R$ is", "contained in $R$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1454, "type": "definition", "label": "algebra-definition-ring-normal", "categories": [ "algebra" ], "title": "algebra-definition-ring-normal", "contents": [ "A ring $R$ is called {\\it normal} if for every prime", "$\\mathfrak p \\subset R$ the localization $R_{\\mathfrak p}$ is", "a normal domain (see Definition \\ref{definition-domain-normal})." ], "refs": [ "algebra-definition-domain-normal" ], "ref_ids": [ 1452 ] }, { "id": 1455, "type": "definition", "label": "algebra-definition-integral-over-ideal", "categories": [ "algebra" ], "title": "algebra-definition-integral-over-ideal", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "Let $I \\subset R$ be an ideal.", "We say an element $g \\in S$ is", "{\\it integral over $I$} if", "there exists a monic", "polynomial $P = x^d + \\sum_{j < d} a_j x^j$", "with coefficients $a_j \\in I^{d-j}$ such", "that $P^\\varphi(g) = 0$ in $S$." ], "refs": [], "ref_ids": [] }, { "id": 1456, "type": "definition", "label": "algebra-definition-flat", "categories": [ "algebra" ], "title": "algebra-definition-flat", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item An $R$-module $M$ is called {\\it flat} if whenever", "$N_1 \\to N_2 \\to N_3$ is an exact sequence of $R$-modules", "the sequence $M \\otimes_R N_1 \\to M \\otimes_R N_2 \\to M \\otimes_R N_3$", "is exact as well.", "\\item An $R$-module $M$ is called {\\it faithfully flat} if the", "complex of $R$-modules", "$N_1 \\to N_2 \\to N_3$ is exact if and only if", "the sequence $M \\otimes_R N_1 \\to M \\otimes_R N_2 \\to M \\otimes_R N_3$", "is exact.", "\\item A ring map $R \\to S$ is called {\\it flat} if", "$S$ is flat as an $R$-module.", "\\item A ring map $R \\to S$ is called {\\it faithfully flat} if", "$S$ is faithfully flat as an $R$-module.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1457, "type": "definition", "label": "algebra-definition-support-module", "categories": [ "algebra" ], "title": "algebra-definition-support-module", "contents": [ "Let $R$ be a ring and let $M$ be an $R$-module.", "The {\\it support of $M$} is the set", "$$", "\\text{Supp}(M)", "=", "\\{", "\\mathfrak p \\in \\Spec(R)", "\\mid", "M_{\\mathfrak p} \\not = 0", "\\}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 1458, "type": "definition", "label": "algebra-definition-annihilator", "categories": [ "algebra" ], "title": "algebra-definition-annihilator", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item Given an element $m \\in M$ the {\\it annihilator of $m$}", "is the ideal", "$$", "\\text{Ann}_R(m) = \\text{Ann}(m) = \\{f \\in R \\mid fm = 0\\}.", "$$", "\\item The {\\it annihilator of $M$}", "is the ideal", "$$", "\\text{Ann}_R(M) = \\text{Ann}(M) = \\{f \\in R \\mid fm = 0\\ \\forall m \\in M\\}.", "$$", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1459, "type": "definition", "label": "algebra-definition-going-up-down", "categories": [ "algebra" ], "title": "algebra-definition-going-up-down", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "\\begin{enumerate}", "\\item We say a $\\varphi : R \\to S$ satisfies {\\it going up} if", "given primes $\\mathfrak p \\subset \\mathfrak p'$ in $R$", "and a prime $\\mathfrak q$ in $S$ lying over $\\mathfrak p$", "there exists a prime $\\mathfrak q'$ of $S$ such that", "(a) $\\mathfrak q \\subset \\mathfrak q'$, and (b)", "$\\mathfrak q'$ lies over $\\mathfrak p'$.", "\\item We say a $\\varphi : R \\to S$ satisfies {\\it going down} if", "given primes $\\mathfrak p \\subset \\mathfrak p'$ in $R$", "and a prime $\\mathfrak q'$ in $S$ lying over $\\mathfrak p'$", "there exists a prime $\\mathfrak q$ of $S$ such that", "(a) $\\mathfrak q \\subset \\mathfrak q'$, and (b)", "$\\mathfrak q$ lies over $\\mathfrak p$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1460, "type": "definition", "label": "algebra-definition-separable-field-extension", "categories": [ "algebra" ], "title": "algebra-definition-separable-field-extension", "contents": [ "Let $k \\subset K$ be a field extension.", "\\begin{enumerate}", "\\item We say $K$ is {\\it separably generated over $k$} if there exists", "a transcendence basis $\\{x_i; i \\in I\\}$ of $K/k$ such that the extension", "$k(x_i; i\\in I) \\subset K$ is a separable algebraic extension.", "\\item We say $K$ is {\\it separable over $k$} if for every subextension", "$k \\subset K' \\subset K$ with $K'$ finitely generated", "over $k$, the extension $k \\subset K'$ is separably generated.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1461, "type": "definition", "label": "algebra-definition-geometrically-reduced", "categories": [ "algebra" ], "title": "algebra-definition-geometrically-reduced", "contents": [ "Let $k$ be a field. Let $S$ be a $k$-algebra.", "We say $S$ is {\\it geometrically reduced over $k$}", "if for every field extension $k \\subset K$ the", "$K$-algebra $K \\otimes_k S$ is reduced." ], "refs": [], "ref_ids": [] }, { "id": 1462, "type": "definition", "label": "algebra-definition-perfect", "categories": [ "algebra" ], "title": "algebra-definition-perfect", "contents": [ "Let $k$ be a field. We say $k$ is {\\it perfect}", "if every field extension of $k$ is separable over $k$." ], "refs": [], "ref_ids": [] }, { "id": 1463, "type": "definition", "label": "algebra-definition-perfection", "categories": [ "algebra" ], "title": "algebra-definition-perfection", "contents": [ "Let $k$ be a field. The field extension $k \\subset k'$ of", "Lemma \\ref{lemma-perfection}", "is called the {\\it perfect closure} of $k$. Notation $k \\subset k^{perf}$." ], "refs": [ "algebra-lemma-perfection" ], "ref_ids": [ 574 ] }, { "id": 1464, "type": "definition", "label": "algebra-definition-geometrically-irreducible", "categories": [ "algebra" ], "title": "algebra-definition-geometrically-irreducible", "contents": [ "Let $k$ be a field.", "Let $S$ be a $k$-algebra.", "We say $S$ is {\\it geometrically irreducible over $k$}", "if for every field extension $k \\subset k'$ the spectrum of", "$S \\otimes_k k'$ is irreducible\\footnote{An irreducible space is nonempty.}." ], "refs": [], "ref_ids": [] }, { "id": 1465, "type": "definition", "label": "algebra-definition-geometrically-connected", "categories": [ "algebra" ], "title": "algebra-definition-geometrically-connected", "contents": [ "Let $k$ be a field.", "Let $S$ be a $k$-algebra.", "We say $S$ is {\\it geometrically connected over $k$}", "if for every field extension $k \\subset k'$ the spectrum", "of $S \\otimes_k k'$ is connected." ], "refs": [], "ref_ids": [] }, { "id": 1466, "type": "definition", "label": "algebra-definition-geometrically-integral", "categories": [ "algebra" ], "title": "algebra-definition-geometrically-integral", "contents": [ "Let $k$ be a field.", "Let $S$ be a $k$-algebra.", "We say $S$ is {\\it geometrically integral over $k$}", "if for every field extension $k \\subset k'$ the ring", "of $S \\otimes_k k'$ is a domain." ], "refs": [], "ref_ids": [] }, { "id": 1467, "type": "definition", "label": "algebra-definition-valuation-ring", "categories": [ "algebra" ], "title": "algebra-definition-valuation-ring", "contents": [ "Valuation rings.", "\\begin{enumerate}", "\\item Let $K$ be a field. Let $A$, $B$ be local rings contained", "in $K$. We say that $B$ {\\it dominates} $A$ if $A \\subset B$", "and $\\mathfrak m_A = A \\cap \\mathfrak m_B$.", "\\item Let $A$ be a ring. We say $A$ is a {\\it valuation ring}", "if $A$ is a local domain and if $A$ is maximal", "for the relation of domination among local rings contained in", "the fraction field of $A$.", "\\item Let $A$ be a valuation ring with fraction field $K$.", "If $R \\subset K$ is a subring of $K$, then we say $A$", "is {\\it centered} on $R$ if $R \\subset A$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1468, "type": "definition", "label": "algebra-definition-value-group", "categories": [ "algebra" ], "title": "algebra-definition-value-group", "contents": [ "Let $A$ be a valuation ring.", "\\begin{enumerate}", "\\item The totally ordered abelian group $(\\Gamma, \\geq)$ of", "Lemma \\ref{lemma-valuation-group} is called the", "{\\it value group} of the valuation ring $A$.", "\\item The map $v : A - \\{0\\} \\to \\Gamma$ and also $v : K^* \\to \\Gamma$ is", "called the {\\it valuation} associated to $A$.", "\\item The valuation ring $A$ is called a {\\it discrete valuation ring}", "if $\\Gamma \\cong \\mathbf{Z}$.", "\\end{enumerate}" ], "refs": [ "algebra-lemma-valuation-group" ], "ref_ids": [ 618 ] }, { "id": 1469, "type": "definition", "label": "algebra-definition-length", "categories": [ "algebra" ], "title": "algebra-definition-length", "contents": [ "Let $R$ be a ring. For any $R$-module $M$", "we define the {\\it length} of $M$ over $R$ by the", "formula", "$$", "\\text{length}_R(M)", "=", "\\sup", "\\{", "n", "\\mid", "\\exists\\ 0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M,", "\\text{ }M_i \\not = M_{i + 1}", "\\}.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 1470, "type": "definition", "label": "algebra-definition-simple-module", "categories": [ "algebra" ], "title": "algebra-definition-simple-module", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "We say $M$ is {\\it simple} if $M \\not = 0$ and", "every submodule of $M$ is either equal to $M$ or", "to $0$." ], "refs": [], "ref_ids": [] }, { "id": 1471, "type": "definition", "label": "algebra-definition-artinian", "categories": [ "algebra" ], "title": "algebra-definition-artinian", "contents": [ "A ring $R$ is {\\it Artinian} if it satisfies the", "descending chain condition for ideals." ], "refs": [], "ref_ids": [] }, { "id": 1472, "type": "definition", "label": "algebra-definition-essentially-finite-p-t", "categories": [ "algebra" ], "title": "algebra-definition-essentially-finite-p-t", "contents": [ "Let $R \\to S$ be a ring map.", "\\begin{enumerate}", "\\item We say that $R \\to S$ is {\\it essentially of finite type} if", "$S$ is the localization of an $R$-algebra of finite type.", "\\item We say that $R \\to S$ is {\\it essentially of finite presentation} if", "$S$ is the localization of an $R$-algebra of finite presentation.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1473, "type": "definition", "label": "algebra-definition-proj", "categories": [ "algebra" ], "title": "algebra-definition-proj", "contents": [ "Let $S$ be a graded ring.", "We define $\\text{Proj}(S)$ to be the set of homogeneous", "prime ideals $\\mathfrak p$ of $S$ such that", "$S_{+} \\not \\subset \\mathfrak p$.", "The set $\\text{Proj}(S)$ is a subset of $\\Spec(S)$", "and we endow it with the induced topology.", "The topological space $\\text{Proj}(S)$ is called the", "{\\it homogeneous spectrum} of the graded ring $S$." ], "refs": [], "ref_ids": [] }, { "id": 1474, "type": "definition", "label": "algebra-definition-numerical-polynomial", "categories": [ "algebra" ], "title": "algebra-definition-numerical-polynomial", "contents": [ "Let $A$ be an abelian group.", "We say that a function $f : n \\mapsto f(n) \\in A$", "defined for all sufficient large integers $n$ is a", "{\\it numerical polynomial} if there exists $r \\geq 0$,", "elements $a_0, \\ldots, a_r\\in A$ such that", "$$", "f(n) = \\sum\\nolimits_{i = 0}^r \\binom{n}{i} a_i", "$$", "for all $n \\gg 0$." ], "refs": [], "ref_ids": [] }, { "id": 1475, "type": "definition", "label": "algebra-definition-ideal-definition", "categories": [ "algebra" ], "title": "algebra-definition-ideal-definition", "contents": [ "Let $(R, \\mathfrak m)$ be a local Noetherian ring.", "An ideal $I \\subset R$ such that $\\sqrt{I} = \\mathfrak m$ is called", "{\\it an ideal of definition of $R$}." ], "refs": [], "ref_ids": [] }, { "id": 1476, "type": "definition", "label": "algebra-definition-hilbert-polynomial", "categories": [ "algebra" ], "title": "algebra-definition-hilbert-polynomial", "contents": [ "Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module.", "The {\\it Hilbert polynomial} of $M$ over $R$ is the element", "$P(t) \\in \\mathbf{Q}[t]$ such that $P(n) = \\varphi_M(n)$ for $n \\gg 0$." ], "refs": [], "ref_ids": [] }, { "id": 1477, "type": "definition", "label": "algebra-definition-d", "categories": [ "algebra" ], "title": "algebra-definition-d", "contents": [ "Let $R$ be a local Noetherian ring and $M$ a finite $R$-module.", "We denote {\\it $d(M)$} the element of $\\{-\\infty, 0, 1, 2, \\ldots \\}$", "defined as follows:", "\\begin{enumerate}", "\\item If $M = 0$ we set $d(M) = -\\infty$,", "\\item if $M \\not = 0$ then $d(M)$ is the degree of the numerical", "polynomial $\\chi_M$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1478, "type": "definition", "label": "algebra-definition-Krull", "categories": [ "algebra" ], "title": "algebra-definition-Krull", "contents": [ "The {\\it Krull dimension} of the ring $R$ is the", "Krull dimension of the topological space $\\Spec(R)$, see", "Topology, Definition \\ref{topology-definition-Krull}.", "In other words it is the supremum of the integers $n\\geq 0$", "such that there exists a chain of prime ideals of length $n$:", "$$", "\\mathfrak p_0", "\\subset", "\\mathfrak p_1", "\\subset", "\\ldots", "\\subset", "\\mathfrak p_n, \\quad", "\\mathfrak p_i \\not = \\mathfrak p_{i + 1}.", "$$" ], "refs": [ "topology-definition-Krull" ], "ref_ids": [ 8356 ] }, { "id": 1479, "type": "definition", "label": "algebra-definition-height", "categories": [ "algebra" ], "title": "algebra-definition-height", "contents": [ "The {\\it height} of a prime ideal $\\mathfrak p$ of", "a ring $R$ is the dimension of the local ring $R_{\\mathfrak p}$." ], "refs": [], "ref_ids": [] }, { "id": 1480, "type": "definition", "label": "algebra-definition-regular-local", "categories": [ "algebra" ], "title": "algebra-definition-regular-local", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $d$.", "\\begin{enumerate}", "\\item A {\\it system of parameters of $R$} is a sequence of elements", "$x_1, \\ldots, x_d \\in \\mathfrak m$ which generates an ideal of", "definition of $R$,", "\\item if there exist $x_1, \\ldots, x_d \\in \\mathfrak m$", "such that $\\mathfrak m = (x_1, \\ldots, x_d)$ then we call", "$R$ a {\\it regular local ring} and $x_1, \\ldots, x_d$ a {\\it regular", "system of parameters}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1481, "type": "definition", "label": "algebra-definition-associated", "categories": [ "algebra" ], "title": "algebra-definition-associated", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "A prime $\\mathfrak p$ of $R$ is {\\it associated} to $M$", "if there exists an element $m \\in M$ whose annihilator", "is $\\mathfrak p$.", "The set of all such primes is denoted $\\text{Ass}_R(M)$", "or $\\text{Ass}(M)$." ], "refs": [], "ref_ids": [] }, { "id": 1482, "type": "definition", "label": "algebra-definition-symbolic-power", "categories": [ "algebra" ], "title": "algebra-definition-symbolic-power", "contents": [ "Let $R$ be a ring. Let $\\mathfrak p$ be a prime ideal. For $n \\geq 0$ the", "$n$th {\\it symbolic power} of $\\mathfrak p$ is the ideal", "$\\mathfrak p^{(n)} = \\Ker(R \\to R_\\mathfrak p/\\mathfrak p^nR_\\mathfrak p)$." ], "refs": [], "ref_ids": [] }, { "id": 1483, "type": "definition", "label": "algebra-definition-relative-assassin", "categories": [ "algebra" ], "title": "algebra-definition-relative-assassin", "contents": [ "Let $R \\to S$ be a ring map. Let $N$ be an $S$-module.", "The {\\it relative assassin of $N$ over $S/R$} is the set", "$$", "\\text{Ass}_{S/R}(N)", "=", "\\{ \\mathfrak q \\subset S \\mid", "\\mathfrak q \\in \\text{Ass}_S(N \\otimes_R \\kappa(\\mathfrak p))", "\\text{ with }\\mathfrak p = R \\cap \\mathfrak q\\}.", "$$", "This is the set named $A$ in", "Lemma \\ref{lemma-compare-relative-assassins}." ], "refs": [ "algebra-lemma-compare-relative-assassins" ], "ref_ids": [ 716 ] }, { "id": 1484, "type": "definition", "label": "algebra-definition-weakly-associated", "categories": [ "algebra" ], "title": "algebra-definition-weakly-associated", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "A prime $\\mathfrak p$ of $R$ is {\\it weakly associated} to $M$", "if there exists an element $m \\in M$ such that $\\mathfrak p$ is minimal", "among the prime ideals containing the annihilator", "$\\text{Ann}(m) = \\{f \\in R \\mid fm = 0\\}$.", "The set of all such primes is denoted $\\text{WeakAss}_R(M)$", "or $\\text{WeakAss}(M)$." ], "refs": [], "ref_ids": [] }, { "id": 1485, "type": "definition", "label": "algebra-definition-embedded-primes", "categories": [ "algebra" ], "title": "algebra-definition-embedded-primes", "contents": [ "Let $R$ be a ring.", "Let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item The associated primes of $M$ which are", "not minimal among the associated primes of $M$ are called the", "{\\it embedded associated primes} of $M$.", "\\item The {\\it embedded primes of $R$}", "are the embedded associated primes of $R$ as an $R$-module.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1486, "type": "definition", "label": "algebra-definition-regular-sequence", "categories": [ "algebra" ], "title": "algebra-definition-regular-sequence", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module. A sequence of elements", "$f_1, \\ldots, f_r$ of $R$ is called an {\\it $M$-regular sequence}", "if the following conditions hold:", "\\begin{enumerate}", "\\item $f_i$ is a nonzerodivisor on", "$M/(f_1, \\ldots, f_{i - 1})M$", "for each $i = 1, \\ldots, r$, and", "\\item the module $M/(f_1, \\ldots, f_r)M$ is not zero.", "\\end{enumerate}", "If $I$ is an ideal of $R$ and $f_1, \\ldots, f_r \\in I$", "then we call $f_1, \\ldots, f_r$ a {\\it $M$-regular sequence", "in $I$}. If $M = R$, we call $f_1, \\ldots, f_r$ simply a", "{\\it regular sequence} (in $I$)." ], "refs": [], "ref_ids": [] }, { "id": 1487, "type": "definition", "label": "algebra-definition-quasi-regular-sequence", "categories": [ "algebra" ], "title": "algebra-definition-quasi-regular-sequence", "contents": [ "Let $R$ be a ring.", "Let $M$ be an $R$-module.", "A sequence of elements $f_1, \\ldots, f_c$ of $R$ is called", "{\\it $M$-quasi-regular} if (\\ref{equation-quasi-regular})", "is an isomorphism. If $M = R$, we call $f_1, \\ldots, f_c$ simply a", "{\\it quasi-regular sequence}." ], "refs": [], "ref_ids": [] }, { "id": 1488, "type": "definition", "label": "algebra-definition-blow-up", "categories": [ "algebra" ], "title": "algebra-definition-blow-up", "contents": [ "Let $R$ be a ring.", "Let $I \\subset R$ be an ideal.", "\\begin{enumerate}", "\\item The {\\it blowup algebra}, or the {\\it Rees algebra}, associated to", "the pair $(R, I)$ is the graded $R$-algebra", "$$", "\\text{Bl}_I(R) =", "\\bigoplus\\nolimits_{n \\geq 0} I^n =", "R \\oplus I \\oplus I^2 \\oplus \\ldots", "$$", "where the summand $I^n$ is placed in degree $n$.", "\\item Let $a \\in I$ be an element. Denote $a^{(1)}$ the element $a$", "seen as an element of degree $1$ in the Rees algebra. Then the", "{\\it affine blowup algebra} $R[\\frac{I}{a}]$ is the algebra", "$(\\text{Bl}_I(R))_{(a^{(1)})}$ constructed in Section \\ref{section-proj}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1489, "type": "definition", "label": "algebra-definition-finite-free-resolution", "categories": [ "algebra" ], "title": "algebra-definition-finite-free-resolution", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item A (left) {\\it resolution} $F_\\bullet \\to M$ of $M$ is an exact complex", "$$", "\\ldots \\to F_2 \\to F_1 \\to F_0 \\to M \\to 0", "$$", "of $R$-modules.", "\\item A {\\it resolution of $M$ by free $R$-modules} is a resolution", "$F_\\bullet \\to M$ where each $F_i$ is a free $R$-module.", "\\item A {\\it resolution of $M$ by finite free $R$-modules} is a resolution", "$F_\\bullet \\to M$ where each $F_i$ is a finite free $R$-module.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1490, "type": "definition", "label": "algebra-definition-depth", "categories": [ "algebra" ], "title": "algebra-definition-depth", "contents": [ "Let $R$ be a ring, and $I \\subset R$ an ideal. Let $M$ be a finite $R$-module.", "The {\\it $I$-depth} of $M$, denoted $\\text{depth}_I(M)$, is defined as follows:", "\\begin{enumerate}", "\\item if $IM \\not = M$, then $\\text{depth}_I(M)$ is the supremum in", "$\\{0, 1, 2, \\ldots, \\infty\\}$ of the lengths of $M$-regular sequences in $I$,", "\\item if $IM = M$ we set $\\text{depth}_I(M) = \\infty$.", "\\end{enumerate}", "If $(R, \\mathfrak m)$ is local we call $\\text{depth}_{\\mathfrak m}(M)$ simply", "the {\\it depth} of $M$." ], "refs": [], "ref_ids": [] }, { "id": 1491, "type": "definition", "label": "algebra-definition-projective", "categories": [ "algebra" ], "title": "algebra-definition-projective", "contents": [ "Let $R$ be a ring. An $R$-module $P$ is {\\it projective} if and only if", "the functor $\\Hom_R(P, -) : \\text{Mod}_R \\to \\text{Mod}_R$ is", "an exact functor." ], "refs": [], "ref_ids": [] }, { "id": 1492, "type": "definition", "label": "algebra-definition-locally-free", "categories": [ "algebra" ], "title": "algebra-definition-locally-free", "contents": [ "Let $R$ be a ring and $M$ an $R$-module.", "\\begin{enumerate}", "\\item We say that $M$ is {\\it locally free} if we can cover $\\Spec(R)$ by", "standard opens $D(f_i)$, $i \\in I$ such that $M_{f_i}$ is a free", "$R_{f_i}$-module for all $i \\in I$.", "\\item We say that $M$ is {\\it finite locally free} if we can choose", "the covering such that each $M_{f_i}$ is finite free.", "\\item We say that $M$ is {\\it finite locally free of rank $r$}", "if we can choose the covering such that each $M_{f_i}$ is isomorphic", "to $R_{f_i}^{\\oplus r}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1493, "type": "definition", "label": "algebra-definition-universally-injective", "categories": [ "algebra" ], "title": "algebra-definition-universally-injective", "contents": [ "Let $f: M \\to N$ be a map of $R$-modules. Then $f$ is called", "{\\it universally injective} if for every $R$-module $Q$, the map $f", "\\otimes_R \\text{id}_Q: M \\otimes_R Q \\to N \\otimes_R Q$", "is injective. A sequence $0 \\to M_1 \\to M_2 \\to M_3", "\\to 0$ of $R$-modules is called {\\it universally exact} if it is exact", "and $M_1 \\to M_2$ is universally injective." ], "refs": [], "ref_ids": [] }, { "id": 1494, "type": "definition", "label": "algebra-definition-devissage", "categories": [ "algebra" ], "title": "algebra-definition-devissage", "contents": [ "Let $M$ be an $R$-module. A {\\it direct sum d\\'evissage} of $M$ is a family", "of submodules $(M_{\\alpha})_{\\alpha \\in S}$, indexed by an ordinal $S$ and", "increasing (with respect to inclusion), such that:", "\\begin{enumerate}", "\\item[(0)] $M_0 = 0$;", "\\item[(1)] $M = \\bigcup_{\\alpha} M_{\\alpha}$;", "\\item[(2)] if $\\alpha \\in S$ is a limit ordinal, then $M_{\\alpha} =", "\\bigcup_{\\beta < \\alpha} M_{\\beta}$;", "\\item[(3)] if $\\alpha + 1 \\in S$, then $M_{\\alpha}$ is a direct summand of", "$M_{\\alpha + 1}$.", "\\end{enumerate}", "If moreover", "\\begin{enumerate}", "\\item[(4)] $M_{\\alpha + 1}/M_{\\alpha}$ is countably generated for", "$\\alpha + 1 \\in S$,", "\\end{enumerate}", "then $(M_{\\alpha})_{\\alpha \\in S}$ is called a {\\it Kaplansky d\\'evissage}", "of $M$." ], "refs": [], "ref_ids": [] }, { "id": 1495, "type": "definition", "label": "algebra-definition-ML-system", "categories": [ "algebra" ], "title": "algebra-definition-ML-system", "contents": [ "Let $(A_i, \\varphi_{ji})$ be a directed inverse system of sets over $I$. Then", "we say $(A_i, \\varphi_{ji})$ is {\\it Mittag-Leffler} if for", "each $i \\in I$, the family $\\varphi_{ji}(A_j) \\subset A_i$ for", "$j \\geq i$ stabilizes. Explicitly, this means that for each $i \\in I$, there", "exists $j \\geq i$ such that for $k \\geq j$ we have $\\varphi_{ki}(A_k) =", "\\varphi_{ji}( A_j)$. If $(A_i, \\varphi_{ji})$ is a directed inverse system", "of modules over a ring $R$, we say that it is Mittag-Leffler if the underlying", "inverse system of sets is Mittag-Leffler." ], "refs": [], "ref_ids": [] }, { "id": 1496, "type": "definition", "label": "algebra-definition-ML-inductive-system", "categories": [ "algebra" ], "title": "algebra-definition-ML-inductive-system", "contents": [ "Let $(M_i, f_{ij})$ be a directed system of $R$-modules. We say that", "$(M_i, f_{ij})$ is a {\\it Mittag-Leffler directed system of modules} if each", "$M_i$ is an $R$-module of finite presentation and if for every $R$-module $N$,", "the inverse system", "$$", "(\\Hom_R(M_i, N), \\Hom_R(f_{ij}, N))", "$$", "is Mittag-Leffler." ], "refs": [], "ref_ids": [] }, { "id": 1497, "type": "definition", "label": "algebra-definition-domination", "categories": [ "algebra" ], "title": "algebra-definition-domination", "contents": [ "Let $f: M \\to N$ and $g: M \\to M'$ be maps of $R$-modules.", "Then we say $g$ {\\it dominates} $f$ if for any $R$-module $Q$, we have $\\Ker(f", "\\otimes_R \\text{id}_Q) \\subset \\Ker(g \\otimes_R \\text{id}_Q)$." ], "refs": [], "ref_ids": [] }, { "id": 1498, "type": "definition", "label": "algebra-definition-mittag-leffler-module", "categories": [ "algebra" ], "title": "algebra-definition-mittag-leffler-module", "contents": [ "Let $M$ be an $R$-module. We say that $M$ is {\\it Mittag-Leffler} if the", "equivalent conditions of", "Proposition \\ref{proposition-ML-characterization}", "hold." ], "refs": [ "algebra-proposition-ML-characterization" ], "ref_ids": [ 1414 ] }, { "id": 1499, "type": "definition", "label": "algebra-definition-coherent", "categories": [ "algebra" ], "title": "algebra-definition-coherent", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item We say $M$ is a {\\it coherent module} if it is finitely generated", "and every finitely generated submodule of $M$ is finitely presented over", "$R$.", "\\item We say $R$ is a {\\it coherent ring} if it is coherent as a module", "over itself.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1500, "type": "definition", "label": "algebra-definition-complete", "categories": [ "algebra" ], "title": "algebra-definition-complete", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal.", "Let $M$ be an $R$-module. We say $M$ is {\\it $I$-adically complete}", "if the map", "$$", "M \\longrightarrow M^\\wedge = \\lim_n M/I^nM", "$$", "is an isomorphism\\footnote{This includes the condition that", "$\\bigcap I^nM = (0)$.}. We say $R$ is {\\it $I$-adically complete}", "if $R$ is $I$-adically complete as an $R$-module." ], "refs": [], "ref_ids": [] }, { "id": 1501, "type": "definition", "label": "algebra-definition-rank", "categories": [ "algebra" ], "title": "algebra-definition-rank", "contents": [ "Let $R$ be a ring. Suppose that $\\varphi : R^m \\to R^n$ is a map", "of finite free modules.", "\\begin{enumerate}", "\\item The {\\it rank} of $\\varphi$ is the maximal $r$ such that", "$\\wedge^r \\varphi : \\wedge^r R^m \\to \\wedge^r R^n$ is nonzero.", "\\item We let $I(\\varphi) \\subset R$ be the ideal generated by", "the $r \\times r$ minors of the matrix of $\\varphi$, where $r$", "is the rank as defined above.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1502, "type": "definition", "label": "algebra-definition-CM", "categories": [ "algebra" ], "title": "algebra-definition-CM", "contents": [ "Let $R$ be a Noetherian local ring.", "Let $M$ be a finite $R$-module.", "We say $M$ is {\\it Cohen-Macaulay}", "if $\\dim(\\text{Supp}(M)) = \\text{depth}(M)$." ], "refs": [], "ref_ids": [] }, { "id": 1503, "type": "definition", "label": "algebra-definition-maximal-CM", "categories": [ "algebra" ], "title": "algebra-definition-maximal-CM", "contents": [ "Let $R$ be a Noetherian local ring.", "A finite module $M$ over $R$ is called a {\\it maximal Cohen-Macaulay}", "module if $\\text{depth}(M) = \\dim(R)$." ], "refs": [], "ref_ids": [] }, { "id": 1504, "type": "definition", "label": "algebra-definition-module-CM", "categories": [ "algebra" ], "title": "algebra-definition-module-CM", "contents": [ "Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module.", "We say $M$ is {\\it Cohen-Macaulay} if $M_\\mathfrak p$ is a Cohen-Macaulay", "module over $R_\\mathfrak p$ for all primes $\\mathfrak p$ of $R$." ], "refs": [], "ref_ids": [] }, { "id": 1505, "type": "definition", "label": "algebra-definition-local-ring-CM", "categories": [ "algebra" ], "title": "algebra-definition-local-ring-CM", "contents": [ "A Noetherian local ring $R$ is called {\\it Cohen-Macaulay}", "if it is Cohen-Macaulay as a module over itself." ], "refs": [], "ref_ids": [] }, { "id": 1506, "type": "definition", "label": "algebra-definition-ring-CM", "categories": [ "algebra" ], "title": "algebra-definition-ring-CM", "contents": [ "A Noetherian ring $R$ is called {\\it Cohen-Macaulay} if all", "its local rings are Cohen-Macaulay." ], "refs": [], "ref_ids": [] }, { "id": 1507, "type": "definition", "label": "algebra-definition-catenary", "categories": [ "algebra" ], "title": "algebra-definition-catenary", "contents": [ "A ring $R$ is said to be {\\it catenary} if for any pair of prime ideals", "$\\mathfrak p \\subset \\mathfrak q$, all maximal chains of primes", "$\\mathfrak p = \\mathfrak p_0 \\subset \\mathfrak p_1 \\subset \\ldots \\subset", "\\mathfrak p_e = \\mathfrak q$ have the same (finite) length." ], "refs": [], "ref_ids": [] }, { "id": 1508, "type": "definition", "label": "algebra-definition-universally-catenary", "categories": [ "algebra" ], "title": "algebra-definition-universally-catenary", "contents": [ "A Noetherian ring $R$ is said to be {\\it universally catenary}", "if every $R$ algebra of finite type is catenary." ], "refs": [], "ref_ids": [] }, { "id": 1509, "type": "definition", "label": "algebra-definition-pure-ideal", "categories": [ "algebra" ], "title": "algebra-definition-pure-ideal", "contents": [ "Let $R$ be a ring. We say that $I \\subset R$ is {\\it pure}", "if the quotient ring $R/I$ is flat over $R$." ], "refs": [], "ref_ids": [] }, { "id": 1510, "type": "definition", "label": "algebra-definition-finite-proj-dim", "categories": [ "algebra" ], "title": "algebra-definition-finite-proj-dim", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module. We say $M$ has", "{\\it finite projective dimension} if it has a finite length", "resolution by projective $R$-modules. The minimal length of such a", "resolution is called the {\\it projective dimension}", "of $M$." ], "refs": [], "ref_ids": [] }, { "id": 1511, "type": "definition", "label": "algebra-definition-finite-gl-dim", "categories": [ "algebra" ], "title": "algebra-definition-finite-gl-dim", "contents": [ "Let $R$ be a ring. The ring", "$R$ is said to have {\\it finite global dimension}", "if there exists an integer $n$ such that", "every $R$-module has a resolution by", "projective $R$-modules of length at most $n$.", "The minimal such $n$ is then called the {\\it global dimension}", "of $R$." ], "refs": [], "ref_ids": [] }, { "id": 1512, "type": "definition", "label": "algebra-definition-regular", "categories": [ "algebra" ], "title": "algebra-definition-regular", "contents": [ "A Noetherian ring $R$ is said to be {\\it regular}", "if all the localizations $R_{\\mathfrak p}$ at primes are", "regular local rings." ], "refs": [], "ref_ids": [] }, { "id": 1513, "type": "definition", "label": "algebra-definition-fibre", "categories": [ "algebra" ], "title": "algebra-definition-fibre", "contents": [ "Suppose that $R \\to S$ is a ring map.", "Let $\\mathfrak q \\subset S$ be a prime lying", "over the prime $\\mathfrak p$ of $R$.", "The {\\it local ring of the fibre at $\\mathfrak q$}", "is the local ring", "$$", "S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}", "=", "(S/\\mathfrak pS)_{\\mathfrak q}", "=", "(S \\otimes_R \\kappa(\\mathfrak p))_{\\mathfrak q}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 1514, "type": "definition", "label": "algebra-definition-uniformizer", "categories": [ "algebra" ], "title": "algebra-definition-uniformizer", "contents": [ "Let $A$ be a discrete valuation ring. A {\\it uniformizer} is an element", "$\\pi \\in A$ which generates the maximal ideal of $A$." ], "refs": [], "ref_ids": [] }, { "id": 1515, "type": "definition", "label": "algebra-definition-irreducible-prime-element", "categories": [ "algebra" ], "title": "algebra-definition-irreducible-prime-element", "contents": [ "Let $R$ be a domain.", "\\begin{enumerate}", "\\item Elements $x, y \\in R$ are called {\\it associates} if", "there exists a unit $u \\in R^*$ such that $x = uy$.", "\\item An element $x \\in R$ is called {\\it irreducible}", "if it is nonzero, not a unit and whenever $x = yz$, $y, z \\in R$,", "then $y$ is either a unit or an associate of $x$.", "\\item An element $x \\in R$ is called {\\it prime} if the ideal", "generated by $x$ is a prime ideal.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1516, "type": "definition", "label": "algebra-definition-UFD", "categories": [ "algebra" ], "title": "algebra-definition-UFD", "contents": [ "A {\\it unique factorization domain}, abbreviated {\\it UFD},", "is a domain $R$ such that", "if $x \\in R$ is a nonzero, nonunit, then $x$ has a factorization", "into irreducibles, and if", "$$", "x = a_1 \\ldots a_m = b_1 \\ldots b_n", "$$", "are factorizations into irreducibles then $n = m$ and", "there exists a permutation $\\sigma : \\{1, \\ldots, n\\} \\to \\{1, \\ldots, n\\}$", "such that $a_i$ and $b_{\\sigma(i)}$ are associates." ], "refs": [], "ref_ids": [] }, { "id": 1517, "type": "definition", "label": "algebra-definition-PID", "categories": [ "algebra" ], "title": "algebra-definition-PID", "contents": [ "A {\\it principal ideal domain}, abbreviated {\\it PID},", "is a domain $R$ such that every ideal is a principal ideal." ], "refs": [], "ref_ids": [] }, { "id": 1518, "type": "definition", "label": "algebra-definition-dedekind-domain", "categories": [ "algebra" ], "title": "algebra-definition-dedekind-domain", "contents": [ "A {\\it Dedekind domain} is a domain $R$ such that every", "nonzero ideal $I \\subset R$ can be written as a product", "$$", "I = \\mathfrak p_1 \\ldots \\mathfrak p_r", "$$", "of nonzero prime ideals uniquely up to permutation of the $\\mathfrak p_i$." ], "refs": [], "ref_ids": [] }, { "id": 1519, "type": "definition", "label": "algebra-definition-ord", "categories": [ "algebra" ], "title": "algebra-definition-ord", "contents": [ "Suppose that $K$ is a field, and $R \\subset K$ is a", "local\\footnote{We could also define this when $R$ is only", "semi-local but this is probably never really what you want!}", "Noetherian subring of dimension $1$ with fraction field $K$.", "In this case we define the {\\it order of vanishing along $R$}", "$$", "\\text{ord}_R : K^* \\longrightarrow \\mathbf{Z}", "$$", "by the rule", "$$", "\\text{ord}_R(x) = \\text{length}_R(R/(x))", "$$", "if $x \\in R$ and we set", "$\\text{ord}_R(x/y) = \\text{ord}_R(x) - \\text{ord}_R(y)$", "for $x, y \\in R$ both nonzero." ], "refs": [], "ref_ids": [] }, { "id": 1520, "type": "definition", "label": "algebra-definition-lattice", "categories": [ "algebra" ], "title": "algebra-definition-lattice", "contents": [ "Let $R$ be a Noetherian local domain of dimension $1$ with", "fraction field $K$. Let $V$ be a finite dimensional $K$-vector space.", "A {\\it lattice in $V$} is a finite $R$-submodule $M \\subset V$ such", "that $V = K \\otimes_R M$." ], "refs": [], "ref_ids": [] }, { "id": 1521, "type": "definition", "label": "algebra-definition-distance", "categories": [ "algebra" ], "title": "algebra-definition-distance", "contents": [ "Let $R$ be a Noetherian local domain of dimension $1$ with", "fraction field $K$. Let $V$ be a finite dimensional $K$-vector space.", "Let $M$, $M'$ be two lattices in $V$. The {\\it distance between", "$M$ and $M'$} is the integer", "$$", "d(M, M') = \\text{length}_R(M/M \\cap M') - \\text{length}_R(M'/M \\cap M')", "$$", "of Lemma \\ref{lemma-compare-lattices} part (5)." ], "refs": [ "algebra-lemma-compare-lattices" ], "ref_ids": [ 1044 ] }, { "id": 1522, "type": "definition", "label": "algebra-definition-quasi-finite", "categories": [ "algebra" ], "title": "algebra-definition-quasi-finite", "contents": [ "Let $R \\to S$ be a finite type ring map.", "Let $\\mathfrak q \\subset S$ be a prime.", "\\begin{enumerate}", "\\item If the equivalent conditions of Lemma \\ref{lemma-isolated-point-fibre}", "are satisfied then we say $R \\to S$ is {\\it quasi-finite at $\\mathfrak q$}.", "\\item We say a ring map $A \\to B$ is {\\it quasi-finite}", "if it is of finite type and quasi-finite at all primes of $B$.", "\\end{enumerate}" ], "refs": [ "algebra-lemma-isolated-point-fibre" ], "ref_ids": [ 1049 ] }, { "id": 1523, "type": "definition", "label": "algebra-definition-strongly-transcendental", "categories": [ "algebra" ], "title": "algebra-definition-strongly-transcendental", "contents": [ "Given an inclusion of rings $R \\subset S$ and", "an element $x \\in S$ we say that $x$ is", "{\\it strongly transcendental over $R$} if", "whenever $u(a_0 + a_1 x + \\ldots + a_k x^k) = 0$", "with $u \\in S$ and $a_i \\in R$, then", "we have $ua_i = 0$ for all $i$." ], "refs": [], "ref_ids": [] }, { "id": 1524, "type": "definition", "label": "algebra-definition-relative-dimension", "categories": [ "algebra" ], "title": "algebra-definition-relative-dimension", "contents": [ "Suppose that $R \\to S$ is of finite type, and let", "$\\mathfrak q \\subset S$ be a prime lying over a prime", "$\\mathfrak p$ of $R$.", "We define the {\\it relative dimension", "of $S/R$ at $\\mathfrak q$}, denoted", "$\\dim_{\\mathfrak q}(S/R)$, to be the dimension", "of $\\Spec(S \\otimes_R \\kappa(\\mathfrak p))$", "at the point corresponding to $\\mathfrak q$. We let", "$\\dim(S/R)$ be the supremum of $\\dim_{\\mathfrak q}(S/R)$", "over all $\\mathfrak q$. This is called the", "{\\it relative dimension of} $S/R$." ], "refs": [], "ref_ids": [] }, { "id": 1525, "type": "definition", "label": "algebra-definition-derivation", "categories": [ "algebra" ], "title": "algebra-definition-derivation", "contents": [ "Let $\\varphi : R \\to S$ be a ring map and let $M$ be an $S$-module.", "A {\\it derivation}, or more precisely an", "{\\it $R$-derivation} into $M$ is a map $D : S \\to M$", "which is additive, annihilates elements of $\\varphi(R)$,", "and satisfies the {\\it Leibniz rule}: $D(ab) = aD(b) + bD(a)$." ], "refs": [], "ref_ids": [] }, { "id": 1526, "type": "definition", "label": "algebra-definition-differentials", "categories": [ "algebra" ], "title": "algebra-definition-differentials", "contents": [ "The pair $(\\Omega_{S/R}, \\text{d})$ is called the {\\it module", "of K\\\"ahler differentials} or the {\\it module of differentials}", "of $S$ over $R$." ], "refs": [], "ref_ids": [] }, { "id": 1527, "type": "definition", "label": "algebra-definition-differential-operators", "categories": [ "algebra" ], "title": "algebra-definition-differential-operators", "contents": [ "Let $R \\to S$ be a ring map. Let $M$, $N$ be $S$-modules.", "Let $k \\geq 0$ be an integer. We inductively define a", "{\\it differential operator $D : M \\to N$ of order $k$}", "to be an $R$-linear map such that for all $g \\in S$ the map", "$m \\mapsto D(gm) - gD(m)$ is a differential operator of", "order $k - 1$. For the base case $k = 0$ we define a", "differential operator of order $0$ to be an $S$-linear map." ], "refs": [], "ref_ids": [] }, { "id": 1528, "type": "definition", "label": "algebra-definition-module-principal-parts", "categories": [ "algebra" ], "title": "algebra-definition-module-principal-parts", "contents": [ "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. The module", "$P^k_{S/R}(M)$ constructed in Lemma \\ref{lemma-module-principal-parts}", "is called the {\\it module of principal parts of order $k$} of $M$." ], "refs": [ "algebra-lemma-module-principal-parts" ], "ref_ids": [ 1145 ] }, { "id": 1529, "type": "definition", "label": "algebra-definition-naive-cotangent-complex", "categories": [ "algebra" ], "title": "algebra-definition-naive-cotangent-complex", "contents": [ "Let $R \\to S$ be a ring map. The {\\it naive cotangent complex}", "$\\NL_{S/R}$ is the chain complex (\\ref{equation-naive-cotangent-complex})", "$$", "\\NL_{S/R} = \\left(I/I^2 \\longrightarrow \\Omega_{R[S]/R} \\otimes_{R[S]} S\\right)", "$$", "with $I/I^2$ placed in (homological) degree $1$ and", "$\\Omega_{R[S]/R} \\otimes_{R[S]} S$ placed in degree $0$. We will denote", "$H_1(L_{S/R}) = H_1(\\NL_{S/R})$\\footnote{This module is sometimes", "denoted $\\Gamma_{S/R}$ in the literature.} the homology in degree $1$." ], "refs": [], "ref_ids": [] }, { "id": 1530, "type": "definition", "label": "algebra-definition-lci-field", "categories": [ "algebra" ], "title": "algebra-definition-lci-field", "contents": [ "Let $k$ be a field.", "Let $S$ be a finite type $k$-algebra.", "\\begin{enumerate}", "\\item We say that $S$ is a {\\it global complete intersection over $k$}", "if there exists a presentation $S = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$", "such that $\\dim(S) = n - c$.", "\\item We say that $S$ is a {\\it local complete intersection over $k$}", "if there exists a covering $\\Spec(S) = \\bigcup D(g_i)$ such", "that each of the rings $S_{g_i}$ is a global complete intersection", "over $k$.", "\\end{enumerate}", "We will also use the convention that the zero ring is a global", "complete intersection over $k$." ], "refs": [], "ref_ids": [] }, { "id": 1531, "type": "definition", "label": "algebra-definition-lci-local-ring", "categories": [ "algebra" ], "title": "algebra-definition-lci-local-ring", "contents": [ "Let $k$ be a field. Let $S$ be a local $k$-algebra essentially of finite type", "over $k$. We say $S$ is a {\\it complete intersection (over $k$)}", "if there exists a local $k$-algebra $R$ and elements", "$f_1, \\ldots, f_c \\in \\mathfrak m_R$ such that", "\\begin{enumerate}", "\\item $R$ is essentially of finite type over $k$,", "\\item $R$ is a regular local ring,", "\\item $f_1, \\ldots, f_c$ form a regular sequence in $R$, and", "\\item $S \\cong R/(f_1, \\ldots, f_c)$ as $k$-algebras.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1532, "type": "definition", "label": "algebra-definition-lci", "categories": [ "algebra" ], "title": "algebra-definition-lci", "contents": [ "A ring map $R \\to S$ is called {\\it syntomic}, or we say $S$ is a", "{\\it flat local complete intersection over $R$}", "if it is flat, of finite presentation, and if all of its fibre rings", "$S \\otimes_R \\kappa(\\mathfrak p)$ are local complete intersections,", "see Definition \\ref{definition-lci-field}." ], "refs": [ "algebra-definition-lci-field" ], "ref_ids": [ 1530 ] }, { "id": 1533, "type": "definition", "label": "algebra-definition-relative-global-complete-intersection", "categories": [ "algebra" ], "title": "algebra-definition-relative-global-complete-intersection", "contents": [ "Let $R \\to S$ be a ring map. We say that $R \\to S$ is", "a {\\it relative global complete intersection} if there exists", "a presentation $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ and", "every nonempty fibre of $\\Spec(S) \\to \\Spec(R)$ has dimension $n - c$.", "We will say ``let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ be a relative", "global complete intersection'' to indicate this situation." ], "refs": [], "ref_ids": [] }, { "id": 1534, "type": "definition", "label": "algebra-definition-smooth", "categories": [ "algebra" ], "title": "algebra-definition-smooth", "contents": [ "A ring map $R \\to S$ is {\\it smooth} if it is of finite presentation", "and the naive cotangent complex $\\NL_{S/R}$ is quasi-isomorphic to a", "finite projective $S$-module placed in degree $0$." ], "refs": [], "ref_ids": [] }, { "id": 1535, "type": "definition", "label": "algebra-definition-standard-smooth", "categories": [ "algebra" ], "title": "algebra-definition-standard-smooth", "contents": [ "Let $R$ be a ring. Given integers $n \\geq c \\geq 0$ and", "$f_1, \\ldots, f_c \\in R[x_1, \\ldots, x_n]$ we say", "$$", "S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)", "$$", "is a {\\it standard smooth algebra over $R$} if the polynomial", "$$", "g =", "\\det", "\\left(", "\\begin{matrix}", "\\partial f_1/\\partial x_1 &", "\\partial f_2/\\partial x_1 &", "\\ldots &", "\\partial f_c/\\partial x_1 \\\\", "\\partial f_1/\\partial x_2 &", "\\partial f_2/\\partial x_2 &", "\\ldots &", "\\partial f_c/\\partial x_2 \\\\", "\\ldots & \\ldots & \\ldots & \\ldots \\\\", "\\partial f_1/\\partial x_c &", "\\partial f_2/\\partial x_c &", "\\ldots &", "\\partial f_c/\\partial x_c", "\\end{matrix}", "\\right)", "$$", "maps to an invertible element in $S$." ], "refs": [], "ref_ids": [] }, { "id": 1536, "type": "definition", "label": "algebra-definition-smooth-at-prime", "categories": [ "algebra" ], "title": "algebra-definition-smooth-at-prime", "contents": [ "Let $R \\to S$ be a ring map.", "Let $\\mathfrak q$ be a prime of $S$.", "We say $R \\to S$ is {\\it smooth at $\\mathfrak q$} if there", "exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such", "that $R \\to S_g$ is smooth." ], "refs": [], "ref_ids": [] }, { "id": 1537, "type": "definition", "label": "algebra-definition-formally-smooth", "categories": [ "algebra" ], "title": "algebra-definition-formally-smooth", "contents": [ "Let $R \\to S$ be a ring map.", "We say $S$ is {\\it formally smooth over $R$} if for every", "commutative solid diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar@{-->}[rd] & A/I \\\\", "R \\ar[r] \\ar[u] & A \\ar[u]", "}", "$$", "where $I \\subset A$ is an ideal of square zero, a dotted", "arrow exists which makes the diagram commute." ], "refs": [], "ref_ids": [] }, { "id": 1538, "type": "definition", "label": "algebra-definition-small-extension", "categories": [ "algebra" ], "title": "algebra-definition-small-extension", "contents": [ "Let $\\varphi : B' \\to B$ be a ring map.", "We say $\\varphi$ is a {\\it small extension} if", "$B'$ and $B$ are local Artinian rings, $\\varphi$ is surjective", "and $I = \\Ker(\\varphi)$ has length $1$ as a $B'$-module." ], "refs": [], "ref_ids": [] }, { "id": 1539, "type": "definition", "label": "algebra-definition-etale", "categories": [ "algebra" ], "title": "algebra-definition-etale", "contents": [ "Let $R \\to S$ be a ring map. We say $R \\to S$ is {\\it \\'etale} if it is", "of finite presentation and the naive cotangent complex", "$\\NL_{S/R}$ is quasi-isomorphic to zero. Given a prime $\\mathfrak q$", "of $S$ we say that $R \\to S$ is {\\it \\'etale at $\\mathfrak q$}", "if there exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that", "$R \\to S_g$ is \\'etale." ], "refs": [], "ref_ids": [] }, { "id": 1540, "type": "definition", "label": "algebra-definition-standard-etale", "categories": [ "algebra" ], "title": "algebra-definition-standard-etale", "contents": [ "Let $R$ be a ring. Let $g , f \\in R[x]$.", "Assume that $f$ is monic and the derivative $f'$ is invertible in", "the localization $R[x]_g/(f)$.", "In this case the ring map $R \\to R[x]_g/(f)$ is said to be", "{\\it standard \\'etale}." ], "refs": [], "ref_ids": [] }, { "id": 1541, "type": "definition", "label": "algebra-definition-formally-unramified", "categories": [ "algebra" ], "title": "algebra-definition-formally-unramified", "contents": [ "Let $R \\to S$ be a ring map.", "We say $S$ is {\\it formally unramified over $R$} if for every", "commutative solid diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar@{-->}[rd] & A/I \\\\", "R \\ar[r] \\ar[u] & A \\ar[u]", "}", "$$", "where $I \\subset A$ is an ideal of square zero, there exists", "at most one dotted arrow making the diagram commute." ], "refs": [], "ref_ids": [] }, { "id": 1542, "type": "definition", "label": "algebra-definition-universal-thickening", "categories": [ "algebra" ], "title": "algebra-definition-universal-thickening", "contents": [ "Let $R \\to S$ be a formally unramified ring map.", "\\begin{enumerate}", "\\item The {\\it universal first order thickening} of $S$ over $R$ is", "the surjection of $R$-algebras $S' \\to S$ of", "Lemma \\ref{lemma-universal-thickening}.", "\\item The {\\it conormal module} of $R \\to S$ is the kernel $I$ of the", "universal first order thickening $S' \\to S$, seen as an $S$-module.", "\\end{enumerate}", "We often denote the conormal module {\\it $C_{S/R}$} in this situation." ], "refs": [ "algebra-lemma-universal-thickening" ], "ref_ids": [ 1258 ] }, { "id": 1543, "type": "definition", "label": "algebra-definition-formally-etale", "categories": [ "algebra" ], "title": "algebra-definition-formally-etale", "contents": [ "Let $R \\to S$ be a ring map.", "We say $S$ is {\\it formally \\'etale over $R$} if for every", "commutative solid diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar@{-->}[rd] & A/I \\\\", "R \\ar[r] \\ar[u] & A \\ar[u]", "}", "$$", "where $I \\subset A$ is an ideal of square zero, there exists", "a unique dotted arrow making the diagram commute." ], "refs": [], "ref_ids": [] }, { "id": 1544, "type": "definition", "label": "algebra-definition-unramified", "categories": [ "algebra" ], "title": "algebra-definition-unramified", "contents": [ "Let $R \\to S$ be a ring map.", "\\begin{enumerate}", "\\item We say $R \\to S$ is {\\it unramified} if $R \\to S$ is of", "finite type and $\\Omega_{S/R} = 0$.", "\\item We say $R \\to S$ is {\\it G-unramified} if $R \\to S$ is of finite", "presentation and $\\Omega_{S/R} = 0$.", "\\item Given a prime $\\mathfrak q$ of $S$ we say that $S$ is", "{\\it unramified at $\\mathfrak q$} if there exists a", "$g \\in S$, $g \\not \\in \\mathfrak q$ such that $R \\to S_g$ is unramified.", "\\item Given a prime $\\mathfrak q$ of $S$ we say that $S$ is", "{\\it G-unramified at $\\mathfrak q$} if there exists a", "$g \\in S$, $g \\not \\in \\mathfrak q$ such that $R \\to S_g$ is G-unramified.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1545, "type": "definition", "label": "algebra-definition-henselian", "categories": [ "algebra" ], "title": "algebra-definition-henselian", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring.", "\\begin{enumerate}", "\\item We say $R$ is {\\it henselian} if for every monic $f \\in R[T]$ and", "every root $a_0 \\in \\kappa$ of $\\overline{f}$ such that", "$\\overline{f'}(a_0) \\not = 0$", "there exists an $a \\in R$ such that $f(a) = 0$ and", "$a_0 = \\overline{a}$.", "\\item We say $R$ is {\\it strictly henselian} if $R$ is henselian", "and its residue field is separably algebraically closed.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1546, "type": "definition", "label": "algebra-definition-henselization", "categories": [ "algebra" ], "title": "algebra-definition-henselization", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring.", "\\begin{enumerate}", "\\item The local ring map $R \\to R^h$ constructed in", "Lemma \\ref{lemma-henselization}", "is called the {\\it henselization} of $R$.", "\\item Given a separable algebraic closure $\\kappa \\subset \\kappa^{sep}$", "the local ring map $R \\to R^{sh}$ constructed in", "Lemma \\ref{lemma-strict-henselization}", "is called the", "{\\it strict henselization of $R$ with respect to", "$\\kappa \\subset \\kappa^{sep}$}.", "\\item A local ring map $R \\to R^{sh}$ is called a {\\it strict henselization}", "of $R$ if it is isomorphic to one of the local ring maps constructed in", "Lemma \\ref{lemma-strict-henselization}", "\\end{enumerate}" ], "refs": [ "algebra-lemma-henselization", "algebra-lemma-strict-henselization", "algebra-lemma-strict-henselization" ], "ref_ids": [ 1294, 1295, 1295 ] }, { "id": 1547, "type": "definition", "label": "algebra-definition-conditions", "categories": [ "algebra" ], "title": "algebra-definition-conditions", "contents": [ "Let $R$ be a Noetherian ring.", "Let $k \\geq 0$ be an integer.", "\\begin{enumerate}", "\\item We say $R$ has property {\\it $(R_k)$} if for every prime $\\mathfrak p$", "of height $\\leq k$ the local ring $R_{\\mathfrak p}$ is regular.", "We also say that $R$ is {\\it regular in codimension $\\leq k$}.", "\\item We say $R$ has property {\\it $(S_k)$} if for every prime $\\mathfrak p$", "the local ring $R_{\\mathfrak p}$ has depth at least", "$\\min\\{k, \\dim(R_{\\mathfrak p})\\}$.", "\\item Let $M$ be a finite $R$-module. We say $M$ has property $(S_k)$", "if for every prime $\\mathfrak p$ the module", "$M_{\\mathfrak p}$ has depth at least", "$\\min\\{k, \\dim(\\text{Supp}(M_{\\mathfrak p}))\\}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1548, "type": "definition", "label": "algebra-definition-complete-local-ring", "categories": [ "algebra" ], "title": "algebra-definition-complete-local-ring", "contents": [ "Let $(R, \\mathfrak m)$ be a local ring. We say $R$ is a", "{\\it complete local ring} if the canonical map", "$$", "R \\longrightarrow \\lim_n R/\\mathfrak m^n", "$$", "to the completion of $R$ with respect to $\\mathfrak m$ is an", "isomorphism\\footnote{This includes the condition", "that $\\bigcap \\mathfrak m^n = (0)$; in some texts this may be indicated", "by saying that $R$ is complete and separated. Warning: It can happen", "that the completion $\\lim_n R/\\mathfrak m^n$ of a local ring is", "non-complete, see", "Examples, Lemma \\ref{examples-lemma-noncomplete-completion}.", "This does not happen when $\\mathfrak m$ is finitely generated, see", "Lemma \\ref{lemma-hathat-finitely-generated} in which", "case the completion is Noetherian, see", "Lemma \\ref{lemma-completion-Noetherian}.}." ], "refs": [ "examples-lemma-noncomplete-completion", "algebra-lemma-hathat-finitely-generated", "algebra-lemma-completion-Noetherian" ], "ref_ids": [ 2510, 859, 873 ] }, { "id": 1549, "type": "definition", "label": "algebra-definition-coefficient-ring", "categories": [ "algebra" ], "title": "algebra-definition-coefficient-ring", "contents": [ "Let $(R, \\mathfrak m)$ be a complete local ring.", "A subring $\\Lambda \\subset R$ is", "called a {\\it coefficient ring} if the following conditions hold:", "\\begin{enumerate}", "\\item $\\Lambda$ is a complete local ring with maximal ideal", "$\\Lambda \\cap \\mathfrak m$,", "\\item the residue field of $\\Lambda$ maps isomorphically to the", "residue field of $R$, and", "\\item $\\Lambda \\cap \\mathfrak m = p\\Lambda$, where $p$ is the characteristic", "of the residue field of $R$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1550, "type": "definition", "label": "algebra-definition-cohen-ring", "categories": [ "algebra" ], "title": "algebra-definition-cohen-ring", "contents": [ "A {\\it Cohen ring} is a complete discrete valuation ring with", "uniformizer $p$ a prime number." ], "refs": [], "ref_ids": [] }, { "id": 1551, "type": "definition", "label": "algebra-definition-N", "categories": [ "algebra" ], "title": "algebra-definition-N", "contents": [ "\\begin{reference}", "\\cite[Chapter 0, Definition 23.1.1]{EGA}", "\\end{reference}", "Let $R$ be a domain with field of fractions $K$.", "\\begin{enumerate}", "\\item We say $R$ is {\\it N-1} if the integral closure of $R$ in $K$", "is a finite $R$-module.", "\\item We say $R$ is {\\it N-2} or {\\it Japanese} if for any finite", "extension $K \\subset L$ of fields the integral closure of $R$ in $L$", "is finite over $R$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1552, "type": "definition", "label": "algebra-definition-nagata", "categories": [ "algebra" ], "title": "algebra-definition-nagata", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item We say $R$ is {\\it universally Japanese} if for any finite", "type ring map $R \\to S$ with $S$ a domain we have that $S$ is N-2", "(i.e., Japanese).", "\\item We say that $R$ is a {\\it Nagata ring} if $R$ is Noetherian and", "for every prime ideal $\\mathfrak p$ the ring $R/\\mathfrak p$ is N-2.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1553, "type": "definition", "label": "algebra-definition-analytically-unramified", "categories": [ "algebra" ], "title": "algebra-definition-analytically-unramified", "contents": [ "Let $(R, \\mathfrak m)$ be a Noetherian local ring.", "We say $R$ is {\\it analytically unramified} if its completion", "$R^\\wedge = \\lim_n R/\\mathfrak m^n$ is reduced.", "A prime ideal $\\mathfrak p \\subset R$ is said to be", "{\\it analytically unramified} if $R/\\mathfrak p$ is analytically", "unramified." ], "refs": [], "ref_ids": [] }, { "id": 1554, "type": "definition", "label": "algebra-definition-geometrically-normal", "categories": [ "algebra" ], "title": "algebra-definition-geometrically-normal", "contents": [ "Let $k$ be a field.", "A $k$-algebra $R$ is called {\\it geometrically normal} over $k$ if", "the equivalent conditions of Lemma \\ref{lemma-geometrically-normal} hold." ], "refs": [ "algebra-lemma-geometrically-normal" ], "ref_ids": [ 1377 ] }, { "id": 1555, "type": "definition", "label": "algebra-definition-geometrically-regular", "categories": [ "algebra" ], "title": "algebra-definition-geometrically-regular", "contents": [ "Let $k$ be a field. Let $R$ be a Noetherian $k$-algebra.", "The $k$-algebra $R$ is called {\\it geometrically regular} over $k$ if", "the equivalent conditions of Lemma \\ref{lemma-geometrically-regular} hold." ], "refs": [ "algebra-lemma-geometrically-regular" ], "ref_ids": [ 1382 ] }, { "id": 1643, "type": "definition", "label": "moduli-curves-definition-deligne-mumford-smooth", "categories": [ "moduli-curves" ], "title": "moduli-curves-definition-deligne-mumford-smooth", "contents": [ "\\begin{reference}", "\\cite{DM}", "\\end{reference}", "We denote $\\mathcal{M}$ and we name it the", "{\\it moduli stack of smooth proper curves}", "the algebraic stack", "$\\Curvesstack^{smooth, h0}$ parametrizing families of curves", "introduced in Lemma \\ref{lemma-smooth-curves-h0}.", "For $g \\geq 0$ we denote $\\mathcal{M}_g$ and we name it the", "{\\it moduli stack of smooth proper curves of genus $g$}", "the algebraic stack introduced in", "Lemma \\ref{lemma-smooth-one-piece-per-genus}." ], "refs": [ "moduli-curves-lemma-smooth-curves-h0", "moduli-curves-lemma-smooth-one-piece-per-genus" ], "ref_ids": [ 1613, 1614 ] }, { "id": 1644, "type": "definition", "label": "moduli-curves-definition-relative-dualizing-sheaf", "categories": [ "moduli-curves" ], "title": "moduli-curves-definition-relative-dualizing-sheaf", "contents": [ "Let $f : X \\to S$ be a family of curves with Cohen-Macaulay fibres", "equidimensional of dimension $1$ (Lemma \\ref{lemma-CM-1-curves}).", "Then the $\\mathcal{O}_X$-module", "$$", "\\omega_{X/S} = H^{-1}(\\omega_{X/S}^\\bullet)", "$$", "studied in Lemma \\ref{lemma-CM-dualizing}", "is called the {\\it relative dualizing sheaf} of $f$." ], "refs": [ "moduli-curves-lemma-CM-1-curves", "moduli-curves-lemma-CM-dualizing" ], "ref_ids": [ 1595, 1619 ] }, { "id": 1645, "type": "definition", "label": "moduli-curves-definition-prestable", "categories": [ "moduli-curves" ], "title": "moduli-curves-definition-prestable", "contents": [ "Let $f : X \\to S$ be a family of curves. We say $f$ is a", "{\\it prestable family of curves} if", "\\begin{enumerate}", "\\item $f$ is at-worst-nodal of relative dimension $1$, and", "\\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this holds after", "any base change\\footnote{In fact, it suffices to require", "$f_*\\mathcal{O}_X = \\mathcal{O}_S$ because the Stein factorization", "of $f$ is \\'etale in this case, see", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-stein-factorization-etale}.", "The condition may also be replaced by asking the geometric", "fibres to be connected, see Lemma \\ref{lemma-geomredcon-in-h0-1}.}.", "\\end{enumerate}" ], "refs": [ "spaces-more-morphisms-lemma-stein-factorization-etale", "moduli-curves-lemma-geomredcon-in-h0-1" ], "ref_ids": [ 182, 1603 ] }, { "id": 1646, "type": "definition", "label": "moduli-curves-definition-semistable", "categories": [ "moduli-curves" ], "title": "moduli-curves-definition-semistable", "contents": [ "Let $f : X \\to S$ be a family of curves.", "We say $f$ is a {\\it semistable family of curves} if", "\\begin{enumerate}", "\\item $X \\to S$ is a prestable family of curves, and", "\\item $X_s$ has genus $\\geq 1$ and", "does not have a rational tail for all $s \\in S$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1647, "type": "definition", "label": "moduli-curves-definition-stable", "categories": [ "moduli-curves" ], "title": "moduli-curves-definition-stable", "contents": [ "Let $f : X \\to S$ be a family of curves.", "We say $f$ is a {\\it stable family of curves} if", "\\begin{enumerate}", "\\item $X \\to S$ is a prestable family of curves, and", "\\item $X_s$ has genus $\\geq 2$ and does not have a rational tails", "or bridges for all $s \\in S$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1648, "type": "definition", "label": "moduli-curves-definition-deligne-mumford", "categories": [ "moduli-curves" ], "title": "moduli-curves-definition-deligne-mumford", "contents": [ "\\begin{reference}", "\\cite{DM}", "\\end{reference}", "We denote $\\overline{\\mathcal{M}}$ and we name the", "{\\it moduli stack of stable curves} the algebraic stack", "$\\Curvesstack^{stable}$ parametrizing stable families of curves", "introduced in Lemma \\ref{lemma-stable-curves}.", "For $g \\geq 2$ we denote $\\overline{\\mathcal{M}}_g$ and we name the", "{\\it moduli stack of stable curves of genus $g$}", "the algebraic stack introduced in Lemma \\ref{lemma-stable-one-piece-per-genus}." ], "refs": [ "moduli-curves-lemma-stable-curves", "moduli-curves-lemma-stable-one-piece-per-genus" ], "ref_ids": [ 1629, 1630 ] }, { "id": 1696, "type": "definition", "label": "dpa-definition-divided-powers", "categories": [ "dpa" ], "title": "dpa-definition-divided-powers", "contents": [ "Let $A$ be a ring. Let $I$ be an ideal of $A$. A collection of maps", "$\\gamma_n : I \\to I$, $n > 0$ is called a {\\it divided power structure}", "on $I$ if for all $n \\geq 0$, $m > 0$, $x, y \\in I$, and $a \\in A$ we have", "\\begin{enumerate}", "\\item $\\gamma_1(x) = x$, we also set $\\gamma_0(x) = 1$,", "\\item $\\gamma_n(x)\\gamma_m(x) = \\frac{(n + m)!}{n! m!} \\gamma_{n + m}(x)$,", "\\item $\\gamma_n(ax) = a^n \\gamma_n(x)$,", "\\item $\\gamma_n(x + y) = \\sum_{i = 0, \\ldots, n} \\gamma_i(x)\\gamma_{n - i}(y)$,", "\\item $\\gamma_n(\\gamma_m(x)) = \\frac{(nm)!}{n! (m!)^n} \\gamma_{nm}(x)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1697, "type": "definition", "label": "dpa-definition-divided-power-ring", "categories": [ "dpa" ], "title": "dpa-definition-divided-power-ring", "contents": [ "A {\\it divided power ring} is a triple $(A, I, \\gamma)$ where", "$A$ is a ring, $I \\subset A$ is an ideal, and $\\gamma = (\\gamma_n)_{n \\geq 1}$", "is a divided power structure on $I$.", "A {\\it homomorphism of divided power rings}", "$\\varphi : (A, I, \\gamma) \\to (B, J, \\delta)$ is a ring homomorphism", "$\\varphi : A \\to B$ such that $\\varphi(I) \\subset J$ and such that", "$\\delta_n(\\varphi(x)) = \\varphi(\\gamma_n(x))$ for all $x \\in I$ and", "$n \\geq 1$." ], "refs": [], "ref_ids": [] }, { "id": 1698, "type": "definition", "label": "dpa-definition-extends", "categories": [ "dpa" ], "title": "dpa-definition-extends", "contents": [ "Given a divided power ring $(A, I, \\gamma)$ and a ring map", "$A \\to B$ we say $\\gamma$ {\\it extends} to $B$ if there exists a", "divided power structure $\\bar \\gamma$ on $IB$ such that", "$(A, I, \\gamma) \\to (B, IB, \\bar\\gamma)$ is a homomorphism of", "divided power rings." ], "refs": [], "ref_ids": [] }, { "id": 1699, "type": "definition", "label": "dpa-definition-divided-powers-graded", "categories": [ "dpa" ], "title": "dpa-definition-divided-powers-graded", "contents": [ "Let $R$ be a ring. Let $A = \\bigoplus_{d \\geq 0} A_d$ be a graded", "$R$-algebra which is strictly graded commutative. A collection of maps", "$\\gamma_n : A_{even, +} \\to A_{even, +}$ defined for all $n > 0$ is called", "a {\\it divided power structure} on $A$ if we have", "\\begin{enumerate}", "\\item $\\gamma_n(x) \\in A_{2nd}$ if $x \\in A_{2d}$,", "\\item $\\gamma_1(x) = x$ for any $x$, we also set $\\gamma_0(x) = 1$,", "\\item $\\gamma_n(x)\\gamma_m(x) = \\frac{(n + m)!}{n! m!} \\gamma_{n + m}(x)$,", "\\item $\\gamma_n(xy) = x^n \\gamma_n(y)$ for all $x \\in A_{even}$ and", "$y \\in A_{even, +}$,", "\\item $\\gamma_n(xy) = 0$ if $x, y \\in A_{odd}$ homogeneous and $n > 1$", "\\item if $x, y \\in A_{even, +}$ then", "$\\gamma_n(x + y) = \\sum_{i = 0, \\ldots, n} \\gamma_i(x)\\gamma_{n - i}(y)$,", "\\item $\\gamma_n(\\gamma_m(x)) =", "\\frac{(nm)!}{n! (m!)^n} \\gamma_{nm}(x)$ for $x \\in A_{even, +}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1700, "type": "definition", "label": "dpa-definition-divided-powers-dga", "categories": [ "dpa" ], "title": "dpa-definition-divided-powers-dga", "contents": [ "Let $R$ be a ring. Let $A = \\bigoplus_{d \\geq 0} A_d$ be a", "differential graded $R$-algebra which is strictly graded commutative.", "A divided power structure $\\gamma$ on $A$ is {\\it compatible with", "the differential graded structure} if", "$\\text{d}(\\gamma_n(x)) = \\text{d}(x) \\gamma_{n - 1}(x)$ for", "all $x \\in A_{even, +}$." ], "refs": [], "ref_ids": [] }, { "id": 1701, "type": "definition", "label": "dpa-definition-lci", "categories": [ "dpa" ], "title": "dpa-definition-lci", "contents": [ "Let $A$ be a Noetherian ring.", "\\begin{enumerate}", "\\item If $A$ is local, then we say $A$ is a {\\it complete intersection}", "if its completion is a complete intersection in the sense above.", "\\item In general we say $A$ is a {\\it local complete intersection}", "if all of its local rings are complete intersections.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1967, "type": "definition", "label": "derived-definition-triangle", "categories": [ "derived" ], "title": "derived-definition-triangle", "contents": [ "Let $\\mathcal{D}$ be an additive category.", "Let $[n] : \\mathcal{D} \\to \\mathcal{D}$, $E \\mapsto E[n]$", "be a collection of additive functors indexed by $n \\in \\mathbf{Z}$ such that", "$[n] \\circ [m] = [n + m]$ and $[0] = \\text{id}$ (equality as functors).", "In this situation we define a {\\it triangle} to be a sextuple", "$(X, Y, Z, f, g, h)$ where $X, Y, Z \\in \\Ob(\\mathcal{D})$ and", "$f : X \\to Y$, $g : Y \\to Z$ and $h : Z \\to X[1]$ are morphisms", "of $\\mathcal{D}$.", "A {\\it morphism of triangles}", "$(X, Y, Z, f, g, h) \\to (X', Y', Z', f', g', h')$", "is given by morphisms $a : X \\to X'$, $b : Y \\to Y'$ and $c : Z \\to Z'$", "of $\\mathcal{D}$ such that", "$b \\circ f = f' \\circ a$, $c \\circ g = g' \\circ b$ and", "$a[1] \\circ h = h' \\circ c$." ], "refs": [], "ref_ids": [] }, { "id": 1968, "type": "definition", "label": "derived-definition-triangulated-category", "categories": [ "derived" ], "title": "derived-definition-triangulated-category", "contents": [ "A {\\it triangulated category} consists of a triple", "$(\\mathcal{D}, \\{[n]\\}_{n\\in \\mathbf{Z}}, \\mathcal{T})$", "where", "\\begin{enumerate}", "\\item $\\mathcal{D}$ is an additive category,", "\\item $[n] : \\mathcal{D} \\to \\mathcal{D}$, $E \\mapsto E[n]$", "is a collection of additive functors indexed by $n \\in \\mathbf{Z}$ such that", "$[n] \\circ [m] = [n + m]$ and $[0] = \\text{id}$ (equality as functors), and", "\\item $\\mathcal{T}$ is a set of triangles called the", "{\\it distinguished triangles}", "\\end{enumerate}", "subject to the following conditions", "\\begin{enumerate}", "\\item[TR1] Any triangle isomorphic to a distinguished triangle is", "a distinguished triangle. Any triangle of the form", "$(X, X, 0, \\text{id}, 0, 0)$ is distinguished.", "For any morphism $f : X \\to Y$ of $\\mathcal{D}$ there exists a", "distinguished triangle of the form $(X, Y, Z, f, g, h)$.", "\\item[TR2] The triangle $(X, Y, Z, f, g, h)$ is distinguished", "if and only if the triangle $(Y, Z, X[1], g, h, -f[1])$ is.", "\\item[TR3] Given a solid diagram", "$$", "\\xymatrix{", "X \\ar[r]^f \\ar[d]^a &", "Y \\ar[r]^g \\ar[d]^b &", "Z \\ar[r]^h \\ar@{-->}[d] &", "X[1] \\ar[d]^{a[1]} \\\\", "X' \\ar[r]^{f'} &", "Y' \\ar[r]^{g'} &", "Z' \\ar[r]^{h'} &", "X'[1]", "}", "$$", "whose rows are distinguished triangles and which satisfies", "$b \\circ f = f' \\circ a$, there exists a morphism", "$c : Z \\to Z'$ such that $(a, b, c)$ is a morphism of triangles.", "\\item[TR4] Given objects $X$, $Y$, $Z$ of $\\mathcal{D}$, and morphisms", "$f : X \\to Y$, $g : Y \\to Z$, and distinguished triangles", "$(X, Y, Q_1, f, p_1, d_1)$,", "$(X, Z, Q_2, g \\circ f, p_2, d_2)$,", "and", "$(Y, Z, Q_3, g, p_3, d_3)$,", "there exist", "morphisms $a : Q_1 \\to Q_2$ and $b : Q_2 \\to Q_3$ such", "that", "\\begin{enumerate}", "\\item $(Q_1, Q_2, Q_3, a, b, p_1[1] \\circ d_3)$ is a", "distinguished triangle,", "\\item the triple $(\\text{id}_X, g, a)$ is", "a morphism of triangles", "$(X, Y, Q_1, f, p_1, d_1) \\to (X, Z, Q_2, g \\circ f, p_2, d_2)$, and", "\\item the triple $(f, \\text{id}_Z, b)$ is a morphism of triangles", "$(X, Z, Q_2, g \\circ f, p_2, d_2) \\to (Y, Z, Q_3, g, p_3, d_3)$.", "\\end{enumerate}", "\\end{enumerate}", "We will call $(\\mathcal{D}, [\\ ], \\mathcal{T})$ a", "{\\it pre-triangulated category} if TR1, TR2 and TR3", "hold.\\footnote{We use $[\\ ]$ as an abbreviation for the", "family $\\{[n]\\}_{n\\in \\mathbf{Z}}$.}" ], "refs": [], "ref_ids": [] }, { "id": 1969, "type": "definition", "label": "derived-definition-exact-functor-triangulated-categories", "categories": [ "derived" ], "title": "derived-definition-exact-functor-triangulated-categories", "contents": [ "Let $\\mathcal{D}$, $\\mathcal{D}'$ be pre-triangulated", "categories. An {\\it exact functor}, or a {\\it triangulated functor}", "from $\\mathcal{D}$ to $\\mathcal{D}'$ is a functor", "$F : \\mathcal{D} \\to \\mathcal{D}'$ together", "with given functorial isomorphisms $\\xi_X : F(X[1]) \\to F(X)[1]$", "such that for every distinguished triangle", "$(X, Y, Z, f, g, h)$ of $\\mathcal{D}$ the triangle", "$(F(X), F(Y), F(Z), F(f), F(g), \\xi_X \\circ F(h))$", "is a distinguished triangle of $\\mathcal{D}'$." ], "refs": [], "ref_ids": [] }, { "id": 1970, "type": "definition", "label": "derived-definition-triangulated-subcategory", "categories": [ "derived" ], "title": "derived-definition-triangulated-subcategory", "contents": [ "Let $(\\mathcal{D}, [\\ ], \\mathcal{T})$ be a pre-triangulated category.", "A {\\it pre-triangulated subcategory}\\footnote{This definition may be", "nonstandard. If $\\mathcal{D}'$ is a full subcategory then $\\mathcal{T}'$", "is the intersection of the set of triangles in $\\mathcal{D}'$ with", "$\\mathcal{T}$, see", "Lemma \\ref{lemma-triangulated-subcategory}.", "In this case we drop $\\mathcal{T}'$ from the notation.}", "is a pair $(\\mathcal{D}', \\mathcal{T}')$ such that", "\\begin{enumerate}", "\\item $\\mathcal{D}'$ is an additive subcategory of $\\mathcal{D}$", "which is preserved under $[1]$ and $[-1]$,", "\\item $\\mathcal{T}' \\subset \\mathcal{T}$ is a subset such that for every", "$(X, Y, Z, f, g, h) \\in \\mathcal{T}'$ we have", "$X, Y, Z \\in \\Ob(\\mathcal{D}')$ and", "$f, g, h \\in \\text{Arrows}(\\mathcal{D}')$, and", "\\item $(\\mathcal{D}', [\\ ], \\mathcal{T}')$ is a pre-triangulated", "category.", "\\end{enumerate}", "If $\\mathcal{D}$ is a triangulated category, then we say", "$(\\mathcal{D}', \\mathcal{T}')$ is a {\\it triangulated subcategory} if", "it is a pre-triangulated subcategory and", "$(\\mathcal{D}', [\\ ], \\mathcal{T}')$ is a triangulated category." ], "refs": [ "derived-lemma-triangulated-subcategory" ], "ref_ids": [ 1771 ] }, { "id": 1971, "type": "definition", "label": "derived-definition-homological", "categories": [ "derived" ], "title": "derived-definition-homological", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category.", "Let $\\mathcal{A}$ be an abelian category.", "An additive functor $H : \\mathcal{D} \\to \\mathcal{A}$ is called", "{\\it homological} if for every distinguished triangle", "$(X, Y, Z, f, g, h)$ the sequence", "$$", "H(X) \\to H(Y) \\to H(Z)", "$$", "is exact in the abelian category $\\mathcal{A}$. An additive functor", "$H : \\mathcal{D}^{opp} \\to \\mathcal{A}$ is called {\\it cohomological}", "if the corresponding functor $\\mathcal{D} \\to \\mathcal{A}^{opp}$ is", "homological." ], "refs": [], "ref_ids": [] }, { "id": 1972, "type": "definition", "label": "derived-definition-delta-functor", "categories": [ "derived" ], "title": "derived-definition-delta-functor", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $\\mathcal{D}$ be a triangulated category.", "A {\\it $\\delta$-functor from $\\mathcal{A}$ to $\\mathcal{D}$} is", "given by a functor $G : \\mathcal{A} \\to \\mathcal{D}$ and", "a rule which assigns to every short exact sequence", "$$", "0 \\to A \\xrightarrow{a} B \\xrightarrow{b} C \\to 0", "$$", "a morphism $\\delta = \\delta_{A \\to B \\to C} : G(C) \\to G(A)[1]$", "such that", "\\begin{enumerate}", "\\item the triangle", "$(G(A), G(B), G(C), G(a), G(b), \\delta_{A \\to B \\to C})$", "is a distinguished triangle of $\\mathcal{D}$", "for any short exact sequence as above, and", "\\item for every morphism $(A \\to B \\to C) \\to (A' \\to B' \\to C')$", "of short exact sequences the diagram", "$$", "\\xymatrix{", "G(C) \\ar[d] \\ar[rr]_{\\delta_{A \\to B \\to C}} & &", "G(A)[1] \\ar[d] \\\\", "G(C') \\ar[rr]^{\\delta_{A' \\to B' \\to C'}} & &", "G(A')[1]", "}", "$$", "is commutative.", "\\end{enumerate}", "In this situation we call", "$(G(A), G(B), G(C), G(a), G(b), \\delta_{A \\to B \\to C})$", "the {\\it image of the short exact sequence under the", "given $\\delta$-functor}." ], "refs": [], "ref_ids": [] }, { "id": 1973, "type": "definition", "label": "derived-definition-localization", "categories": [ "derived" ], "title": "derived-definition-localization", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category. We say a multiplicative", "system $S$ is {\\it compatible with the triangulated structure} if", "the following two conditions hold:", "\\begin{enumerate}", "\\item[MS5] For $s \\in S$ we have $s[n] \\in S$ for all $n \\in \\mathbf{Z}$.", "\\item[MS6] Given a solid commutative square", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d]^s &", "Y \\ar[r] \\ar[d]^{s'} &", "Z \\ar[r] \\ar@{-->}[d] &", "X[1] \\ar[d]^{s[1]} \\\\", "X' \\ar[r] &", "Y' \\ar[r] &", "Z' \\ar[r] &", "X'[1]", "}", "$$", "whose rows are distinguished triangles with $s, s' \\in S$", "there exists a morphism $s'' : Z \\to Z'$ in $S$ such that", "$(s, s', s'')$ is a morphism of triangles.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1974, "type": "definition", "label": "derived-definition-saturated", "categories": [ "derived" ], "title": "derived-definition-saturated", "contents": [ "Let $\\mathcal{D}$ be a pre-triangulated category. We say a full", "pre-triangulated subcategory $\\mathcal{D}'$ of $\\mathcal{D}$ is", "{\\it saturated} if whenever $X \\oplus Y$ is isomorphic to an object", "of $\\mathcal{D}'$ then both $X$ and $Y$ are isomorphic to objects", "of $\\mathcal{D}'$." ], "refs": [], "ref_ids": [] }, { "id": 1975, "type": "definition", "label": "derived-definition-kernel-category", "categories": [ "derived" ], "title": "derived-definition-kernel-category", "contents": [ "Let $\\mathcal{D}$ be a (pre-)triangulated category.", "\\begin{enumerate}", "\\item Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor.", "The {\\it kernel of $F$} is the strictly full saturated", "(pre-)triangulated subcategory described in", "Lemma \\ref{lemma-triangle-functor-kernel}.", "\\item Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor.", "The {\\it kernel of $H$} is the strictly full saturated", "(pre-)triangulated subcategory described in", "Lemma \\ref{lemma-homological-functor-kernel}.", "\\end{enumerate}", "These are sometimes denoted $\\Ker(F)$ or $\\Ker(H)$." ], "refs": [ "derived-lemma-triangle-functor-kernel", "derived-lemma-homological-functor-kernel" ], "ref_ids": [ 1784, 1785 ] }, { "id": 1976, "type": "definition", "label": "derived-definition-quotient-category", "categories": [ "derived" ], "title": "derived-definition-quotient-category", "contents": [ "Let $\\mathcal{D}$ be a triangulated category.", "Let $\\mathcal{B}$ be a full triangulated subcategory.", "We define the {\\it quotient category $\\mathcal{D}/\\mathcal{B}$}", "by the formula $\\mathcal{D}/\\mathcal{B} = S^{-1}\\mathcal{D}$, where", "$S$ is the multiplicative system of $\\mathcal{D}$ associated to", "$\\mathcal{B}$ via", "Lemma \\ref{lemma-construct-multiplicative-system}.", "The localization functor $Q : \\mathcal{D} \\to \\mathcal{D}/\\mathcal{B}$", "is called the {\\it quotient functor} in this case." ], "refs": [ "derived-lemma-construct-multiplicative-system" ], "ref_ids": [ 1787 ] }, { "id": 1977, "type": "definition", "label": "derived-definition-complexes-notation", "categories": [ "derived" ], "title": "derived-definition-complexes-notation", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "\\begin{enumerate}", "\\item We set $\\text{Comp}(\\mathcal{A}) = \\text{CoCh}(\\mathcal{A})$", "be the {\\it category of (cochain) complexes}.", "\\item A complex $K^\\bullet$ is said to be", "{\\it bounded below} if $K^n = 0$ for all $n \\ll 0$.", "\\item A complex $K^\\bullet$ is said to be", "{\\it bounded above} if $K^n = 0$ for all $n \\gg 0$.", "\\item A complex $K^\\bullet$ is said to be", "{\\it bounded} if $K^n = 0$ for all $|n| \\gg 0$.", "\\item We let", "$\\text{Comp}^{+}(\\mathcal{A})$, $\\text{Comp}^{-}(\\mathcal{A})$,", "resp.\\ $\\text{Comp}^b(\\mathcal{A})$ be the full subcategory", "of $\\text{Comp}(\\mathcal{A})$ whose objects are the complexes", "which are bounded below, bounded above, resp.\\ bounded.", "\\item We let $K(\\mathcal{A})$ be the category with the same objects", "as $\\text{Comp}(\\mathcal{A})$ but as morphisms homotopy classes of", "maps of complexes (see", "Homology, Lemma \\ref{homology-lemma-compose-homotopy-cochain}).", "\\item We let $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$,", "resp.\\ $K^b(\\mathcal{A})$ be the full subcategory of $K(\\mathcal{A})$", "whose objects are bounded below, bounded above, resp.\\ bounded", "complexes of $\\mathcal{A}$.", "\\end{enumerate}" ], "refs": [ "homology-lemma-compose-homotopy-cochain" ], "ref_ids": [ 12058 ] }, { "id": 1978, "type": "definition", "label": "derived-definition-cone", "categories": [ "derived" ], "title": "derived-definition-cone", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let $f : K^\\bullet \\to L^\\bullet$ be a morphism of", "complexes of $\\mathcal{A}$. The {\\it cone} of $f$", "is the complex $C(f)^\\bullet$ given by", "$C(f)^n = L^n \\oplus K^{n + 1}$ and", "differential", "$$", "d_{C(f)}^n =", "\\left(", "\\begin{matrix}", "d^n_L & f^{n + 1} \\\\", "0 & -d_K^{n + 1}", "\\end{matrix}", "\\right)", "$$", "It comes equipped with canonical morphisms of complexes", "$i : L^\\bullet \\to C(f)^\\bullet$ and $p : C(f)^\\bullet \\to K^\\bullet[1]$", "induced by the obvious maps $L^n \\to C(f)^n \\to K^{n + 1}$." ], "refs": [], "ref_ids": [] }, { "id": 1979, "type": "definition", "label": "derived-definition-termwise-split-map", "categories": [ "derived" ], "title": "derived-definition-termwise-split-map", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "A {\\it termwise split injection $\\alpha : A^\\bullet \\to B^\\bullet$}", "is a morphism of complexes such that each $A^n \\to B^n$", "is isomorphic to the inclusion of a direct summand.", "A {\\it termwise split surjection $\\beta : B^\\bullet \\to C^\\bullet$}", "is a morphism of complexes such that each $B^n \\to C^n$", "is isomorphic to the projection onto a direct summand." ], "refs": [], "ref_ids": [] }, { "id": 1980, "type": "definition", "label": "derived-definition-split-ses", "categories": [ "derived" ], "title": "derived-definition-split-ses", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "A {\\it termwise split exact sequence of complexes of $\\mathcal{A}$}", "is a complex of complexes", "$$", "0 \\to", "A^\\bullet \\xrightarrow{\\alpha}", "B^\\bullet \\xrightarrow{\\beta}", "C^\\bullet \\to 0", "$$", "together with given direct sum decompositions", "$B^n = A^n \\oplus C^n$", "compatible with $\\alpha^n$ and $\\beta^n$.", "We often write $s^n : C^n \\to B^n$ and $\\pi^n : B^n \\to A^n$", "for the maps induced by the direct sum decompositions.", "According to", "Homology, Lemma \\ref{homology-lemma-ses-termwise-split-cochain}", "we get an associated morphism of complexes", "$$", "\\delta : C^\\bullet \\longrightarrow A^\\bullet[1]", "$$", "which in degree $n$ is the map $\\pi^{n + 1} \\circ d_B^n \\circ s^n$.", "In other words", "$(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta)$", "forms a triangle", "$$", "A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to A^\\bullet[1]", "$$", "This will be the {\\it triangle associated to the termwise", "split sequence of complexes}." ], "refs": [ "homology-lemma-ses-termwise-split-cochain" ], "ref_ids": [ 12067 ] }, { "id": 1981, "type": "definition", "label": "derived-definition-distinguished-triangle", "categories": [ "derived" ], "title": "derived-definition-distinguished-triangle", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "A triangle $(X, Y, Z, f, g, h)$ of $K(\\mathcal{A})$ is", "called a {\\it distinguished triangle of $K(\\mathcal{A})$}", "if it is isomorphic to the triangle associated to", "a termwise split exact sequence of complexes, see Definition", "\\ref{definition-split-ses}.", "Same definition for $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, and", "$K^b(\\mathcal{A})$." ], "refs": [ "derived-definition-split-ses" ], "ref_ids": [ 1980 ] }, { "id": 1982, "type": "definition", "label": "derived-definition-unbounded-derived-category", "categories": [ "derived" ], "title": "derived-definition-unbounded-derived-category", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $\\text{Ac}(\\mathcal{A})$ and $\\text{Qis}(\\mathcal{A})$", "be as in", "Lemma \\ref{lemma-acyclic}.", "The {\\it derived category of $\\mathcal{A}$} is the triangulated", "category", "$$", "D(\\mathcal{A}) =", "K(\\mathcal{A})/\\text{Ac}(\\mathcal{A}) =", "\\text{Qis}(\\mathcal{A})^{-1} K(\\mathcal{A}).", "$$", "We denote $H^0 : D(\\mathcal{A}) \\to \\mathcal{A}$ the unique functor", "whose composition with the quotient functor gives back the functor", "$H^0$ defined above. Using", "Lemma \\ref{lemma-homological-functor-bounded}", "we introduce the strictly full saturated triangulated subcategories", "$D^{+}(\\mathcal{A}), D^{-}(\\mathcal{A}), D^b(\\mathcal{A})$", "whose sets of objects are", "$$", "\\begin{matrix}", "\\Ob(D^{+}(\\mathcal{A})) =", "\\{X \\in \\Ob(D(\\mathcal{A})) \\mid", "H^n(X) = 0\\text{ for all }n \\ll 0\\} \\\\", "\\Ob(D^{-}(\\mathcal{A})) =", "\\{X \\in \\Ob(D(\\mathcal{A})) \\mid", "H^n(X) = 0\\text{ for all }n \\gg 0\\} \\\\", "\\Ob(D^b(\\mathcal{A})) =", "\\{X \\in \\Ob(D(\\mathcal{A})) \\mid", "H^n(X) = 0\\text{ for all }|n| \\gg 0\\}", "\\end{matrix}", "$$", "The category $D^b(\\mathcal{A})$ is called the {\\it bounded derived", "category} of $\\mathcal{A}$." ], "refs": [ "derived-lemma-acyclic", "derived-lemma-homological-functor-bounded" ], "ref_ids": [ 1811, 1786 ] }, { "id": 1983, "type": "definition", "label": "derived-definition-finite-filtered", "categories": [ "derived" ], "title": "derived-definition-finite-filtered", "contents": [ "Let $\\mathcal{A}$ be an abelian category. The", "{\\it category of finite filtered objects of $\\mathcal{A}$}", "is the category of filtered objects", "$(A, F)$ of $\\mathcal{A}$ whose filtration $F$ is finite.", "We denote it $\\text{Fil}^f(\\mathcal{A})$." ], "refs": [], "ref_ids": [] }, { "id": 1984, "type": "definition", "label": "derived-definition-filtered-acyclic", "categories": [ "derived" ], "title": "derived-definition-filtered-acyclic", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item Let $\\alpha : K^\\bullet \\to L^\\bullet$ be a morphism of", "$K(\\text{Fil}^f(\\mathcal{A}))$. We say that", "$\\alpha$ is a {\\it filtered quasi-isomorphism} if", "the morphism $\\text{gr}(\\alpha)$ is a quasi-isomorphism.", "\\item Let $K^\\bullet$ be an object of $K(\\text{Fil}^f(\\mathcal{A}))$.", "We say that $K^\\bullet$ is {\\it filtered acyclic} if", "the complex $\\text{gr}(K^\\bullet)$ is acyclic.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1985, "type": "definition", "label": "derived-definition-filtered-derived", "categories": [ "derived" ], "title": "derived-definition-filtered-derived", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $\\text{FAc}(\\mathcal{A})$ and $\\text{FQis}(\\mathcal{A})$", "be as in", "Lemma \\ref{lemma-filtered-acyclic}.", "The {\\it filtered derived category of $\\mathcal{A}$}", "is the triangulated category", "$$", "DF(\\mathcal{A}) =", "K(\\text{Fil}^f(\\mathcal{A}))/\\text{FAc}(\\mathcal{A}) =", "\\text{FQis}(\\mathcal{A})^{-1} K(\\text{Fil}^f(\\mathcal{A})).", "$$" ], "refs": [ "derived-lemma-filtered-acyclic" ], "ref_ids": [ 1819 ] }, { "id": 1986, "type": "definition", "label": "derived-definition-filtered-derived-bounded", "categories": [ "derived" ], "title": "derived-definition-filtered-derived-bounded", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "The {\\it bounded filtered derived category} $DF^b(\\mathcal{A})$ is", "the full subcategory of $DF(\\mathcal{A})$ with objects those $X$", "such that $\\text{gr}(X) \\in D^b(\\mathcal{A})$.", "Similarly for the bounded below filtered derived category", "$DF^{+}(\\mathcal{A})$ and the bounded above filtered derived category", "$DF^{-}(\\mathcal{A})$." ], "refs": [], "ref_ids": [] }, { "id": 1987, "type": "definition", "label": "derived-definition-right-derived-functor-defined", "categories": [ "derived" ], "title": "derived-definition-right-derived-functor-defined", "contents": [ "Assumptions and notation as in", "Situation \\ref{situation-derived-functor}.", "Let $X \\in \\Ob(\\mathcal{D})$.", "\\begin{enumerate}", "\\item we say the {\\it right derived functor $RF$ is defined at}", "$X$ if the ind-object", "$$", "(X/S) \\longrightarrow \\mathcal{D}', \\quad", "(s : X \\to X') \\longmapsto F(X')", "$$", "is essentially constant\\footnote{For a discussion of when an ind-object", "or pro-object of a category is essentially constant we refer to", "Categories, Section \\ref{categories-section-essentially-constant}.};", "in this case the value", "$Y$ in $\\mathcal{D}'$ is called the {\\it value of $RF$ at $X$}.", "\\item we say the {\\it left derived functor $LF$ is defined at} $X$", "if the pro-object", "$$", "(S/X) \\longrightarrow \\mathcal{D}', \\quad", "(s: X' \\to X) \\longmapsto F(X')", "$$", "is essentially constant; in this case the value $Y$ in $\\mathcal{D}'$", "is called the {\\it value of $LF$ at $X$}.", "\\end{enumerate}", "By abuse of notation we often denote the values simply", "$RF(X)$ or $LF(X)$." ], "refs": [], "ref_ids": [] }, { "id": 1988, "type": "definition", "label": "derived-definition-everywhere-defined", "categories": [ "derived" ], "title": "derived-definition-everywhere-defined", "contents": [ "In", "Situation \\ref{situation-derived-functor}.", "We say $F$ is {\\it right derivable}, or that $RF$ {\\it everywhere defined}", "if $RF$ is defined at every object of $\\mathcal{D}$.", "We say $F$ is {\\it left derivable}, or that $LF$ {\\it everywhere defined}", "if $LF$ is defined at every object of $\\mathcal{D}$." ], "refs": [], "ref_ids": [] }, { "id": 1989, "type": "definition", "label": "derived-definition-computes", "categories": [ "derived" ], "title": "derived-definition-computes", "contents": [ "In", "Situation \\ref{situation-derived-functor}.", "\\begin{enumerate}", "\\item An object $X$ of $\\mathcal{D}$ {\\it computes} $RF$ if $RF$ is defined", "at $X$ and the canonical map $F(X) \\to RF(X)$ is an isomorphism.", "\\item An object $X$ of $\\mathcal{D}$ {\\it computes} $LF$ if $LF$ is defined", "at $X$ and the canonical map $LF(X) \\to F(X)$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1990, "type": "definition", "label": "derived-definition-derived-functor", "categories": [ "derived" ], "title": "derived-definition-derived-functor", "contents": [ "In", "Situation \\ref{situation-classical}.", "\\begin{enumerate}", "\\item The {\\it right derived functors of $F$} are the partial functors", "$RF$ associated to cases (1) and (2) of", "Situation \\ref{situation-classical}.", "\\item The {\\it left derived functors of $F$} are the partial functors", "$LF$ associated to cases (3) and (4) of", "Situation \\ref{situation-classical}.", "\\item An object $A$ of $\\mathcal{A}$ is said to be", "{\\it right acyclic for $F$}, or {\\it acyclic for $RF$}", "if $A[0]$ computes $RF$.", "\\item An object $A$ of $\\mathcal{A}$ is said to be", "{\\it left acyclic for $F$}, or {\\it acyclic for $LF$}", "if $A[0]$ computes $LF$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1991, "type": "definition", "label": "derived-definition-higher-derived-functors", "categories": [ "derived" ], "title": "derived-definition-higher-derived-functors", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor", "between abelian categories. Assume", "$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere", "defined. Let $i \\in \\mathbf{Z}$.", "The {\\it $i$th right derived functor $R^iF$ of $F$} is the functor", "$$", "R^iF = H^i \\circ RF :", "\\mathcal{A}", "\\longrightarrow", "\\mathcal{B}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 1992, "type": "definition", "label": "derived-definition-injective-resolution", "categories": [ "derived" ], "title": "derived-definition-injective-resolution", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A \\in \\Ob(\\mathcal{A})$.", "An {\\it injective resolution of $A$} is a complex", "$I^\\bullet$ together with a map $A \\to I^0$ such", "that:", "\\begin{enumerate}", "\\item We have $I^n = 0$ for $n < 0$.", "\\item Each $I^n$ is an injective object of $\\mathcal{A}$.", "\\item The map $A \\to I^0$ is an isomorphism onto $\\Ker(d^0)$.", "\\item We have $H^i(I^\\bullet) = 0$ for $i > 0$.", "\\end{enumerate}", "Hence $A[0] \\to I^\\bullet$ is a quasi-isomorphism.", "In other words the complex", "$$", "\\ldots \\to 0 \\to A \\to I^0 \\to I^1 \\to \\ldots", "$$", "is acyclic.", "Let $K^\\bullet$ be a complex in $\\mathcal{A}$.", "An {\\it injective resolution of $K^\\bullet$} is a complex", "$I^\\bullet$ together with a map $\\alpha : K^\\bullet \\to I^\\bullet$", "of complexes such that", "\\begin{enumerate}", "\\item We have $I^n = 0$ for $n \\ll 0$, i.e., $I^\\bullet$ is bounded below.", "\\item Each $I^n$ is an injective object of $\\mathcal{A}$.", "\\item The map $\\alpha : K^\\bullet \\to I^\\bullet$ is a", "quasi-isomorphism.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1993, "type": "definition", "label": "derived-definition-projective-resolution", "categories": [ "derived" ], "title": "derived-definition-projective-resolution", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A \\in \\Ob(\\mathcal{A})$.", "An {\\it projective resolution of $A$} is a complex", "$P^\\bullet$ together with a map $P^0 \\to A$ such", "that:", "\\begin{enumerate}", "\\item We have $P^n = 0$ for $n > 0$.", "\\item Each $P^n$ is an projective object of $\\mathcal{A}$.", "\\item The map $P^0 \\to A$ induces an isomorphism $\\Coker(d^{-1}) \\to A$.", "\\item We have $H^i(P^\\bullet) = 0$ for $i < 0$.", "\\end{enumerate}", "Hence $P^\\bullet \\to A[0]$ is a quasi-isomorphism.", "In other words the complex", "$$", "\\ldots \\to P^{-1} \\to P^0 \\to A \\to 0 \\to \\ldots", "$$", "is acyclic. Let $K^\\bullet$ be a complex in $\\mathcal{A}$.", "An {\\it projective resolution of $K^\\bullet$} is a complex", "$P^\\bullet$ together with a map $\\alpha : P^\\bullet \\to K^\\bullet$", "of complexes such that", "\\begin{enumerate}", "\\item We have $P^n = 0$ for $n \\gg 0$, i.e., $P^\\bullet$ is bounded above.", "\\item Each $P^n$ is an projective object of $\\mathcal{A}$.", "\\item The map $\\alpha : P^\\bullet \\to K^\\bullet$ is a", "quasi-isomorphism.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1994, "type": "definition", "label": "derived-definition-cartan-eilenberg", "categories": [ "derived" ], "title": "derived-definition-cartan-eilenberg", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $K^\\bullet$ be a bounded below complex.", "A {\\it Cartan-Eilenberg resolution} of $K^\\bullet$", "is given by a double complex $I^{\\bullet, \\bullet}$", "and a morphism of complexes $\\epsilon : K^\\bullet \\to I^{\\bullet, 0}$", "with the following properties:", "\\begin{enumerate}", "\\item There exists a $i \\ll 0$ such that $I^{p, q} = 0$ for all $p < i$", "and all $q$.", "\\item We have $I^{p, q} = 0$ if $q < 0$.", "\\item The complex $I^{p, \\bullet}$ is an injective resolution of $K^p$.", "\\item The complex $\\Ker(d_1^{p, \\bullet})$ is an injective resolution", "of $\\Ker(d_K^p)$.", "\\item The complex $\\Im(d_1^{p, \\bullet})$ is an injective resolution", "of $\\Im(d_K^p)$.", "\\item The complex $H^p_I(I^{\\bullet, \\bullet})$ is an injective resolution", "of $H^p(K^\\bullet)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1995, "type": "definition", "label": "derived-definition-localization-functor", "categories": [ "derived" ], "title": "derived-definition-localization-functor", "contents": [ "Let $\\mathcal{A}$ be an abelian category with enough injectives.", "A {\\it resolution functor}\\footnote{This is likely nonstandard terminology.}", "for $\\mathcal{A}$ is given by the following data:", "\\begin{enumerate}", "\\item for all $K^\\bullet \\in \\Ob(K^{+}(\\mathcal{A}))$ a", "bounded below complex of injectives $j(K^\\bullet)$, and", "\\item for all $K^\\bullet \\in \\Ob(K^{+}(\\mathcal{A}))$ a", "quasi-isomorphism $i_{K^\\bullet} : K^\\bullet \\to j(K^\\bullet)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 1996, "type": "definition", "label": "derived-definition-filtered-complexes-notation", "categories": [ "derived" ], "title": "derived-definition-filtered-complexes-notation", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "We say an object $I$ of $\\text{Fil}^f(\\mathcal{A})$", "is {\\it filtered injective} if each $\\text{gr}^p(I)$ is", "an injective object of $\\mathcal{A}$." ], "refs": [], "ref_ids": [] }, { "id": 1997, "type": "definition", "label": "derived-definition-ext", "categories": [ "derived" ], "title": "derived-definition-ext", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $i \\in \\mathbf{Z}$. Let", "$X, Y$ be objects of $D(\\mathcal{A})$. The {\\it $i$th extension group}", "of $X$ by $Y$ is the group", "$$", "\\Ext^i_\\mathcal{A}(X, Y) =", "\\Hom_{D(\\mathcal{A})}(X, Y[i]) =", "\\Hom_{D(\\mathcal{A})}(X[-i], Y).", "$$", "If $A, B \\in \\Ob(\\mathcal{A})$ we set", "$\\Ext^i_\\mathcal{A}(A, B) = \\text{Ext}^i_\\mathcal{A}(A[0], B[0])$." ], "refs": [], "ref_ids": [] }, { "id": 1998, "type": "definition", "label": "derived-definition-yoneda-extension", "categories": [ "derived" ], "title": "derived-definition-yoneda-extension", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A, B \\in \\Ob(\\mathcal{A})$.", "A degree $i$ {\\it Yoneda extension} of $B$ by $A$ is an exact sequence", "$$", "E : 0 \\to A \\to Z_{i - 1} \\to Z_{i - 2} \\to \\ldots \\to Z_0 \\to B \\to 0", "$$", "in $\\mathcal{A}$. We say two Yoneda extensions $E$ and $E'$ of the same degree", "are {\\it equivalent} if there exists a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] & A \\ar[r] & Z_{i - 1} \\ar[r] & \\ldots \\ar[r] &", "Z_0 \\ar[r] & B \\ar[r] & 0 \\\\", "0 \\ar[r] &", "A \\ar[r] \\ar[u]^{\\text{id}} \\ar[d]_{\\text{id}} &", "Z''_{i - 1} \\ar[r] \\ar[u] \\ar[d] &", "\\ldots \\ar[r] &", "Z''_0 \\ar[r] \\ar[u] \\ar[d] &", "B \\ar[r] \\ar[u]_{\\text{id}} \\ar[d]^{\\text{id}} & 0 \\\\", "0 \\ar[r] & A \\ar[r] & Z'_{i - 1} \\ar[r] & \\ldots \\ar[r] &", "Z'_0 \\ar[r] & B \\ar[r] & 0", "}", "$$", "where the middle row is a Yoneda extension as well." ], "refs": [], "ref_ids": [] }, { "id": 1999, "type": "definition", "label": "derived-definition-K-zero", "categories": [ "derived" ], "title": "derived-definition-K-zero", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. We denote $K_0(\\mathcal{D})$ the", "{\\it zeroth $K$-group of $\\mathcal{D}$}. It is the abelian group constructed", "as follows. Take the free abelian group on the objects on $\\mathcal{D}$", "and for every distinguished triangle $X \\to Y \\to Z$", "impose the relation $[Y] - [X] - [Z] = 0$." ], "refs": [], "ref_ids": [] }, { "id": 2000, "type": "definition", "label": "derived-definition-K-injective", "categories": [ "derived" ], "title": "derived-definition-K-injective", "contents": [ "Let $\\mathcal{A}$ be an abelian category. A complex $I^\\bullet$", "is {\\it K-injective} if for every acyclic complex $M^\\bullet$ we", "have $\\Hom_{K(\\mathcal{A})}(M^\\bullet, I^\\bullet) = 0$." ], "refs": [], "ref_ids": [] }, { "id": 2001, "type": "definition", "label": "derived-definition-derived-colimit", "categories": [ "derived" ], "title": "derived-definition-derived-colimit", "contents": [ "Let $\\mathcal{D}$ be a triangulated category.", "Let $(K_n, f_n)$ be a system of objects of $\\mathcal{D}$.", "We say an object $K$ is a {\\it derived colimit}, or a", "{\\it homotopy colimit} of the system $(K_n)$ if", "the direct sum $\\bigoplus K_n$ exists and there is a distinguished triangle", "$$", "\\bigoplus K_n \\to \\bigoplus K_n \\to K \\to \\bigoplus K_n[1]", "$$", "where the map $\\bigoplus K_n \\to \\bigoplus K_n$ is given", "by $1 - f_n$ in degree $n$. If this is the", "case, then we sometimes indicate this by the notation", "$K = \\text{hocolim} K_n$." ], "refs": [], "ref_ids": [] }, { "id": 2002, "type": "definition", "label": "derived-definition-derived-limit", "categories": [ "derived" ], "title": "derived-definition-derived-limit", "contents": [ "Let $\\mathcal{D}$ be a triangulated category.", "Let $(K_n, f_n)$ be an inverse system of objects of $\\mathcal{D}$.", "We say an object $K$ is a {\\it derived limit}, or a", "{\\it homotopy limit} of the system $(K_n)$ if", "the product $\\prod K_n$ exists and there is a distinguished triangle", "$$", "K \\to \\prod K_n \\to \\prod K_n \\to K[1]", "$$", "where the map $\\prod K_n \\to \\prod K_n$ is given", "by $(k_n) \\mapsto (k_n - f_{n+1}(k_{n + 1}))$. If this is the", "case, then we sometimes indicate this by the notation $K = R\\lim K_n$." ], "refs": [], "ref_ids": [] }, { "id": 2003, "type": "definition", "label": "derived-definition-generators", "categories": [ "derived" ], "title": "derived-definition-generators", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let $E$ be an object", "of $\\mathcal{D}$.", "\\begin{enumerate}", "\\item We say $E$ is a {\\it classical generator} of $\\mathcal{D}$", "if the smallest strictly full, saturated, triangulated subcategory", "of $\\mathcal{D}$ containing $E$ is equal to $\\mathcal{D}$, in", "other words, if $\\langle E \\rangle = \\mathcal{D}$.", "\\item We say $E$ is a {\\it strong generator} of $\\mathcal{D}$", "if $\\langle E \\rangle_n = \\mathcal{D}$ for some $n \\geq 1$.", "\\item We say $E$ is a {\\it weak generator} or a {\\it generator}", "of $\\mathcal{D}$", "if for any nonzero object $K$ of $\\mathcal{D}$ there exists", "an integer $n$ and a nonzero map $E \\to K[n]$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 2004, "type": "definition", "label": "derived-definition-compact-object", "categories": [ "derived" ], "title": "derived-definition-compact-object", "contents": [ "Let $\\mathcal{D}$ be an additive category with arbitrary direct", "sums. A {\\it compact object} of $\\mathcal{D}$ is an object $K$", "such that the map", "$$", "\\bigoplus\\nolimits_{i \\in I} \\Hom_{\\mathcal{D}}(K, E_i)", "\\longrightarrow", "\\Hom_{\\mathcal{D}}(K, \\bigoplus\\nolimits_{i \\in I} E_i)", "$$", "is bijective for any set $I$ and objects", "$E_i \\in \\Ob(\\mathcal{D})$ parametrized by $i \\in I$." ], "refs": [], "ref_ids": [] }, { "id": 2005, "type": "definition", "label": "derived-definition-compactly-generated", "categories": [ "derived" ], "title": "derived-definition-compactly-generated", "contents": [ "Let $\\mathcal{D}$ be a triangulated category with arbitrary direct", "sums. We say $\\mathcal{D}$ is {\\it compactly generated} if", "there exists a set $E_i$, $i \\in I$ of compact objects such that", "$\\bigoplus E_i$ generates $\\mathcal{D}$." ], "refs": [], "ref_ids": [] }, { "id": 2006, "type": "definition", "label": "derived-definition-orthogonal", "categories": [ "derived" ], "title": "derived-definition-orthogonal", "contents": [ "Let $\\mathcal{D}$ be an additive category. Let $\\mathcal{A} \\subset \\mathcal{D}$", "be a full subcategory. The {\\it right orthogonal} $\\mathcal{A}^\\perp$ of", "$\\mathcal{A}$ is the full subcategory consisting of the objects $X$ of", "$\\mathcal{D}$ such that $\\Hom(A, X) = 0$ for all $A \\in \\Ob(\\mathcal{A})$.", "The {\\it left orthogonal} ${}^\\perp\\mathcal{A}$ of", "$\\mathcal{A}$ is the full subcategory consisting of the objects $X$ of", "$\\mathcal{D}$ such that $\\Hom(X, A) = 0$ for all $A \\in \\Ob(\\mathcal{A})$." ], "refs": [], "ref_ids": [] }, { "id": 2007, "type": "definition", "label": "derived-definition-admissible", "categories": [ "derived" ], "title": "derived-definition-admissible", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. A {\\it right admissible}", "subcategory of $\\mathcal{D}$ is a strictly full triangulated subcategory", "satisfying the equivalent conditions of Lemma \\ref{lemma-right-adjoint}.", "A {\\it left admissible}", "subcategory of $\\mathcal{D}$ is a strictly full triangulated subcategory", "satisfying the equivalent conditions of Lemma \\ref{lemma-left-adjoint}.", "A {\\it two-sided admissible} subcategory is one which is both", "right and left admissible." ], "refs": [ "derived-lemma-right-adjoint", "derived-lemma-left-adjoint" ], "ref_ids": [ 1947, 1948 ] }, { "id": 2008, "type": "definition", "label": "derived-definition-postnikov-system", "categories": [ "derived" ], "title": "derived-definition-postnikov-system", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let", "$$", "X_n \\to X_{n - 1} \\to \\ldots \\to X_0", "$$", "be a complex in $\\mathcal{D}$. A {\\it Postnikov system} is defined", "inductively as follows.", "\\begin{enumerate}", "\\item If $n = 0$, then it is an isomorphism $Y_0 \\to X_0$.", "\\item If $n = 1$, then it is a choice of a distinguished triangle", "$$", "Y_1 \\to X_1 \\to Y_0 \\to Y_1[1]", "$$", "where $X_1 \\to Y_0$ composed with $Y_0 \\to X_0$ is the given morphism", "$X_1 \\to X_0$.", "\\item If $n > 1$, then it is a choice of a Postnikov system", "for $X_{n - 1} \\to \\ldots \\to X_0$ and a choice of a distinguished", "triangle", "$$", "Y_n \\to X_n \\to Y_{n - 1} \\to Y_n[1]", "$$", "where the morphism $X_n \\to Y_{n - 1}$ composed with", "$Y_{n - 1} \\to X_{n - 1}$ is the given morphism $X_n \\to X_{n - 1}$.", "\\end{enumerate}", "Given a morphism", "\\begin{equation}", "\\label{equation-map-complexes}", "\\vcenter{", "\\xymatrix{", "X_n \\ar[r] \\ar[d] &", "X_{n - 1} \\ar[r] \\ar[d] &", "\\ldots \\ar[r] &", "X_0 \\ar[d] \\\\", "X'_n \\ar[r] &", "X'_{n - 1} \\ar[r] &", "\\ldots \\ar[r] &", "X'_0", "}", "}", "\\end{equation}", "between complexes of the same length in $\\mathcal{D}$", "there is an obvious notion of a {\\it morphism of Postnikov systems}." ], "refs": [], "ref_ids": [] }, { "id": 2248, "type": "definition", "label": "cohomology-definition-torsor", "categories": [ "cohomology" ], "title": "cohomology-definition-torsor", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $X$.", "A {\\it torsor}, or more precisely a {\\it $\\mathcal{G}$-torsor}, is a sheaf", "of sets $\\mathcal{F}$ on $X$ endowed with an action", "$\\mathcal{G} \\times \\mathcal{F} \\to \\mathcal{F}$ such that", "\\begin{enumerate}", "\\item whenever $\\mathcal{F}(U)$ is nonempty the action", "$\\mathcal{G}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$", "is simply transitive, and", "\\item for every $x \\in X$ the stalk $\\mathcal{F}_x$ is nonempty.", "\\end{enumerate}", "A {\\it morphism of $\\mathcal{G}$-torsors} $\\mathcal{F} \\to \\mathcal{F}'$", "is simply a morphism of sheaves of sets compatible with the", "$\\mathcal{G}$-actions. The {\\it trivial $\\mathcal{G}$-torsor}", "is the sheaf $\\mathcal{G}$ endowed with the obvious left", "$\\mathcal{G}$-action." ], "refs": [], "ref_ids": [] }, { "id": 2249, "type": "definition", "label": "cohomology-definition-cech-complex", "categories": [ "cohomology" ], "title": "cohomology-definition-cech-complex", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.", "Let $\\mathcal{F}$ be an abelian presheaf on $X$.", "The complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "is the {\\it {\\v C}ech complex} associated to $\\mathcal{F}$ and the", "open covering $\\mathcal{U}$. Its cohomology groups", "$H^i(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}))$ are", "called the {\\it {\\v C}ech cohomology groups} associated to", "$\\mathcal{F}$ and the covering $\\mathcal{U}$.", "They are denoted $\\check H^i(\\mathcal{U}, \\mathcal{F})$." ], "refs": [], "ref_ids": [] }, { "id": 2250, "type": "definition", "label": "cohomology-definition-flasque", "categories": [ "cohomology" ], "title": "cohomology-definition-flasque", "contents": [ "Let $X$ be a topological space. We say a presheaf of sets", "$\\mathcal{F}$ is {\\it flasque} or {\\it flabby} if for every", "$U \\subset V$ open in $X$ the restriction map", "$\\mathcal{F}(V) \\to \\mathcal{F}(U)$ is surjective." ], "refs": [], "ref_ids": [] }, { "id": 2251, "type": "definition", "label": "cohomology-definition-alternating-cech-complex", "categories": [ "cohomology" ], "title": "cohomology-definition-alternating-cech-complex", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$", "be an open covering. Let $\\mathcal{F}$ be an abelian presheaf on $X$.", "The complex $\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "is the {\\it alternating {\\v C}ech complex} associated to $\\mathcal{F}$ and the", "open covering $\\mathcal{U}$." ], "refs": [], "ref_ids": [] }, { "id": 2252, "type": "definition", "label": "cohomology-definition-ordered-cech-complex", "categories": [ "cohomology" ], "title": "cohomology-definition-ordered-cech-complex", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.", "Assume given a total ordering on $I$.", "Let $\\mathcal{F}$ be an abelian presheaf on $X$.", "The complex $\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "is the {\\it ordered {\\v C}ech complex} associated to $\\mathcal{F}$, the", "open covering $\\mathcal{U}$ and the given total ordering on $I$." ], "refs": [], "ref_ids": [] }, { "id": 2253, "type": "definition", "label": "cohomology-definition-covering-locally-finite", "categories": [ "cohomology" ], "title": "cohomology-definition-covering-locally-finite", "contents": [ "Let $X$ be a topological space.", "An open covering $X = \\bigcup_{i \\in I} U_i$ is said to be", "{\\it locally finite} if for every $x \\in X$ there exists an open neighbourhood", "$W$ of $x$ such that $\\{i \\in I \\mid W \\cap U_i \\not = \\emptyset\\}$ is finite." ], "refs": [], "ref_ids": [] }, { "id": 2254, "type": "definition", "label": "cohomology-definition-K-flat", "categories": [ "cohomology" ], "title": "cohomology-definition-K-flat", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "A complex $\\mathcal{K}^\\bullet$ of $\\mathcal{O}_X$-modules is", "called {\\it K-flat} if for every acyclic complex $\\mathcal{F}^\\bullet$", "of $\\mathcal{O}_X$-modules the complex", "$$", "\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{K}^\\bullet)", "$$", "is acyclic." ], "refs": [], "ref_ids": [] }, { "id": 2255, "type": "definition", "label": "cohomology-definition-derived-tor", "categories": [ "cohomology" ], "title": "cohomology-definition-derived-tor", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}^\\bullet$ be an object of $D(\\mathcal{O}_X)$.", "The {\\it derived tensor product}", "$$", "- \\otimes_{\\mathcal{O}_X}^{\\mathbf{L}} \\mathcal{F}^\\bullet :", "D(\\mathcal{O}_X)", "\\longrightarrow", "D(\\mathcal{O}_X)", "$$", "is the exact functor of triangulated categories described above." ], "refs": [], "ref_ids": [] }, { "id": 2256, "type": "definition", "label": "cohomology-definition-tor", "categories": [ "cohomology" ], "title": "cohomology-definition-tor", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules.", "The {\\it Tor}'s of $\\mathcal{F}$ and $\\mathcal{G}$ are define by", "the formula", "$$", "\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) =", "H^{-p}(\\mathcal{F} \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G})", "$$", "with derived tensor product as defined above." ], "refs": [], "ref_ids": [] }, { "id": 2257, "type": "definition", "label": "cohomology-definition-strictly-perfect", "categories": [ "cohomology" ], "title": "cohomology-definition-strictly-perfect", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules.", "We say $\\mathcal{E}^\\bullet$ is {\\it strictly perfect}", "if $\\mathcal{E}^i$ is zero for all but finitely many $i$ and", "$\\mathcal{E}^i$ is a direct summand of a finite free", "$\\mathcal{O}_X$-module for all $i$." ], "refs": [], "ref_ids": [] }, { "id": 2258, "type": "definition", "label": "cohomology-definition-pseudo-coherent", "categories": [ "cohomology" ], "title": "cohomology-definition-pseudo-coherent", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{E}^\\bullet$", "be a complex of $\\mathcal{O}_X$-modules. Let $m \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item We say $\\mathcal{E}^\\bullet$ is {\\it $m$-pseudo-coherent}", "if there exists an open covering $X = \\bigcup U_i$ and for each $i$", "a morphism of complexes", "$\\alpha_i : \\mathcal{E}_i^\\bullet \\to \\mathcal{E}^\\bullet|_{U_i}$", "where $\\mathcal{E}_i$ is strictly perfect on $U_i$ and", "$H^j(\\alpha_i)$ is an isomorphism for $j > m$ and $H^m(\\alpha_i)$", "is surjective.", "\\item We say $\\mathcal{E}^\\bullet$ is {\\it pseudo-coherent}", "if it is $m$-pseudo-coherent for all $m$.", "\\item We say an object $E$ of $D(\\mathcal{O}_X)$ is", "{\\it $m$-pseudo-coherent} (resp.\\ {\\it pseudo-coherent})", "if and only if it can be represented by a $m$-pseudo-coherent", "(resp.\\ pseudo-coherent) complex of $\\mathcal{O}_X$-modules.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 2259, "type": "definition", "label": "cohomology-definition-tor-amplitude", "categories": [ "cohomology" ], "title": "cohomology-definition-tor-amplitude", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $E$ be an object of $D(\\mathcal{O}_X)$.", "Let $a, b \\in \\mathbf{Z}$ with $a \\leq b$.", "\\begin{enumerate}", "\\item We say $E$ has {\\it tor-amplitude in $[a, b]$}", "if $H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}) = 0$", "for all $\\mathcal{O}_X$-modules $\\mathcal{F}$ and all $i \\not \\in [a, b]$.", "\\item We say $E$ has {\\it finite tor dimension}", "if it has tor-amplitude in $[a, b]$ for some $a, b$.", "\\item We say $E$ {\\it locally has finite tor dimension}", "if there exists an open covering $X = \\bigcup U_i$ such that", "$E|_{U_i}$ has finite tor dimension for all $i$.", "\\end{enumerate}", "An $\\mathcal{O}_X$-module $\\mathcal{F}$ has {\\it tor dimension $\\leq d$}", "if $\\mathcal{F}[0]$ viewed as an object of $D(\\mathcal{O}_X)$ has", "tor-amplitude in $[-d, 0]$." ], "refs": [], "ref_ids": [] }, { "id": 2260, "type": "definition", "label": "cohomology-definition-perfect", "categories": [ "cohomology" ], "title": "cohomology-definition-perfect", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules.", "We say $\\mathcal{E}^\\bullet$ is {\\it perfect} if there exists", "an open covering $X = \\bigcup U_i$ such that for each $i$", "there exists a morphism of complexes", "$\\mathcal{E}_i^\\bullet \\to \\mathcal{E}^\\bullet|_{U_i}$", "which is a quasi-isomorphism with $\\mathcal{E}_i^\\bullet$", "a strictly perfect complex of $\\mathcal{O}_{U_i}$-modules.", "An object $E$ of $D(\\mathcal{O}_X)$ is {\\it perfect}", "if it can be represented by a perfect complex of $\\mathcal{O}_X$-modules." ], "refs": [], "ref_ids": [] }, { "id": 2288, "type": "definition", "label": "stacks-introduction-definition-smooth", "categories": [ "stacks-introduction" ], "title": "stacks-introduction-definition-smooth", "contents": [ "We say a morphism $S \\to \\mathcal{M}_{1, 1}$ is {\\it smooth} if for every", "morphism $S' \\to \\mathcal{M}_{1, 1}$ the projection morphism", "$$", "S \\times_{\\mathcal{M}_{1, 1}} S' \\longrightarrow S'", "$$", "is smooth." ], "refs": [], "ref_ids": [] }, { "id": 2289, "type": "definition", "label": "stacks-introduction-definition-algebraic-stack", "categories": [ "stacks-introduction" ], "title": "stacks-introduction-definition-algebraic-stack", "contents": [ "We say $\\mathcal{M}_{1, 1}$ is an {\\it algebraic stack} if and only if", "\\begin{enumerate}", "\\item We have descent for objects for the \\'etale topology on $\\Sch$.", "\\item The key fact holds.", "\\item there exists a surjective and smooth morphism", "$S \\to \\mathcal{M}_{1, 1}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 2434, "type": "definition", "label": "restricted-definition-rig-smooth-homomorphism", "categories": [ "restricted" ], "title": "restricted-definition-rig-smooth-homomorphism", "contents": [ "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.", "Let $B$ be an object of (\\ref{equation-C-prime}). We say", "$B$ is {\\it rig-smooth over $(A, I)$} if there exists an integer $c \\geq 0$", "such that $I^c$ annihilates $\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$ for every", "$B$-module $N$." ], "refs": [], "ref_ids": [] }, { "id": 2435, "type": "definition", "label": "restricted-definition-rig-etale-homomorphism", "categories": [ "restricted" ], "title": "restricted-definition-rig-etale-homomorphism", "contents": [ "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.", "Let $B$ be an object of (\\ref{equation-C-prime}). We say", "$B$ is {\\it rig-\\'etale over $(A, I)$} if there exists an integer", "$c \\geq 0$ such that for all $a \\in I^c$", "multiplication by $a$ on $\\NL_{B/A}^\\wedge$", "is zero in $D(B)$." ], "refs": [], "ref_ids": [] }, { "id": 2436, "type": "definition", "label": "restricted-definition-flat", "categories": [ "restricted" ], "title": "restricted-definition-flat", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally", "Noetherian formal algebraic spaces over $S$. We say $f$ is", "{\\it flat} if for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$", "representable by algebraic spaces and \\'etale, the morphism $U \\to V$", "corresponds to a flat map of adic Noetherian topological rings." ], "refs": [], "ref_ids": [] }, { "id": 2437, "type": "definition", "label": "restricted-definition-rig-closed", "categories": [ "restricted" ], "title": "restricted-definition-rig-closed", "contents": [ "Let $A$ be a Noetherian adic topological ring. Let", "$\\mathfrak q \\subset A$ be a prime ideal. We say", "$\\mathfrak q$ is {\\it rig-closed} if the equivalent", "conditions of Lemma \\ref{lemma-rig-point} are satisfied." ], "refs": [ "restricted-lemma-rig-point" ], "ref_ids": [ 2342 ] }, { "id": 2438, "type": "definition", "label": "restricted-definition-completed-principal-localization", "categories": [ "restricted" ], "title": "restricted-definition-completed-principal-localization", "contents": [ "Let $A$ be an adic topological ring which has a finitely generated ideal", "of definition. Let $f \\in A$. The {\\it completed principal localization}", "$A_{\\{f\\}}$ of $A$ is the completion of $A_f = A[1/f]$", "of the principal localization of $A$ at $f$ with respect to any", "ideal of definition of $A$." ], "refs": [], "ref_ids": [] }, { "id": 2439, "type": "definition", "label": "restricted-definition-naively-rig-flat", "categories": [ "restricted" ], "title": "restricted-definition-naively-rig-flat", "contents": [ "Let $\\varphi : A \\to B$ be a continuous ring homomorphism", "between adic Noetherian topological rings, i.e., $\\varphi$", "is an arrow of $\\textit{WAdm}^{Noeth}$. We say $\\varphi$ is", "{\\it naively rig-flat} if $\\varphi$ is adic, topologically", "of finite type, and satisfies the equivalent conditions of", "Lemma \\ref{lemma-naively-rig-flat-continuous}." ], "refs": [ "restricted-lemma-naively-rig-flat-continuous" ], "ref_ids": [ 2352 ] }, { "id": 2440, "type": "definition", "label": "restricted-definition-rig-flat-continuous-homomorphism", "categories": [ "restricted" ], "title": "restricted-definition-rig-flat-continuous-homomorphism", "contents": [ "Let $\\varphi : A \\to B$ be a continuous ring homomorphism between", "adic Noetherian topological rings, i.e., $\\varphi$ is an arrow of", "$\\textit{WAdm}^{Noeth}$. We say $\\varphi$ is {\\it rig-flat} if $\\varphi$", "is adic, topologically of finite type, and for all $f \\in A$ the induced map", "$$", "A_{\\{f\\}} \\longrightarrow B_{\\{f\\}}", "$$", "is naively rig-flat (Definition \\ref{definition-naively-rig-flat})." ], "refs": [ "restricted-definition-naively-rig-flat" ], "ref_ids": [ 2439 ] }, { "id": 2441, "type": "definition", "label": "restricted-definition-rig-flat", "categories": [ "restricted" ], "title": "restricted-definition-rig-flat", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally", "Noetherian formal algebraic spaces over $S$. We say $f$ is", "{\\it rig-flat} if for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$", "representable by algebraic spaces and \\'etale, the morphism $U \\to V$", "corresponds to a rig-flat map of adic Noetherian topological rings." ], "refs": [], "ref_ids": [] }, { "id": 2442, "type": "definition", "label": "restricted-definition-rig-smooth-continuous-homomorphism", "categories": [ "restricted" ], "title": "restricted-definition-rig-smooth-continuous-homomorphism", "contents": [ "Let $\\varphi : A \\to B$ be a continuous ring homomorphism", "between adic Noetherian topological rings, i.e., $\\varphi$", "is an arrow of $\\textit{WAdm}^{Noeth}$. We say", "$\\varphi$ is {\\it rig-smooth} if the equivalent conditions", "of Lemma \\ref{lemma-rig-smooth-continuous} hold." ], "refs": [ "restricted-lemma-rig-smooth-continuous" ], "ref_ids": [ 2362 ] }, { "id": 2443, "type": "definition", "label": "restricted-definition-rig-smooth", "categories": [ "restricted" ], "title": "restricted-definition-rig-smooth", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally", "Noetherian formal algebraic spaces over $S$. We say $f$ is", "{\\it rig-smooth} if for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$", "representable by algebraic spaces and \\'etale, the morphism $U \\to V$", "corresponds to a rig-smooth map of adic Noetherian topological rings." ], "refs": [], "ref_ids": [] }, { "id": 2444, "type": "definition", "label": "restricted-definition-rig-etale-continuous-homomorphism", "categories": [ "restricted" ], "title": "restricted-definition-rig-etale-continuous-homomorphism", "contents": [ "Let $\\varphi : A \\to B$ be a continuous ring homomorphism", "between adic Noetherian topological rings, i.e., $\\varphi$", "is an arrow of $\\textit{WAdm}^{Noeth}$. We say", "$\\varphi$ is {\\it rig-etale} if the equivalent conditions", "of Lemma \\ref{lemma-rig-etale-continuous} hold." ], "refs": [ "restricted-lemma-rig-etale-continuous" ], "ref_ids": [ 2372 ] }, { "id": 2445, "type": "definition", "label": "restricted-definition-rig-etale", "categories": [ "restricted" ], "title": "restricted-definition-rig-etale", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally", "Noetherian formal algebraic spaces over $S$. We say $f$ is", "{\\it rig-\\'etale} if for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[d] \\ar[r] & V \\ar[d] \\\\", "X \\ar[r] & Y", "}", "$$", "with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$", "representable by algebraic spaces and \\'etale, the morphism $U \\to V$", "corresponds to a rig-\\'etale map of adic Noetherian topological rings." ], "refs": [], "ref_ids": [] }, { "id": 2446, "type": "definition", "label": "restricted-definition-rig-surjective", "categories": [ "restricted" ], "title": "restricted-definition-rig-surjective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal", "algebraic spaces over $S$. Assume that $X$ and $Y$ are locally", "Noetherian and that $f$ is locally of finite type. We say", "$f$ is {\\it rig-surjective} if for every solid diagram", "$$", "\\xymatrix{", "\\text{Spf}(R') \\ar@{..>}[r] \\ar@{..>}[d] & X \\ar[d]^f \\\\", "\\text{Spf}(R) \\ar[r]^-p & Y", "}", "$$", "where $R$ is a complete discrete valuation ring and where", "$p$ is an adic morphism there exists an", "extension of complete discrete valuation rings $R \\subset R'$", "and a morphism $\\text{Spf}(R') \\to X$ making the displayed diagram commute." ], "refs": [], "ref_ids": [] }, { "id": 2447, "type": "definition", "label": "restricted-definition-formal-modification", "categories": [ "restricted" ], "title": "restricted-definition-formal-modification", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "locally Noetherian formal algebraic spaces over $S$. We say $f$ is a", "{\\it formal modification} if", "\\begin{enumerate}", "\\item $f$ is a proper morphism (Formal Spaces, Definition", "\\ref{formal-spaces-definition-proper}),", "\\item $f$ is rig-\\'etale,", "\\item $f$ is rig-surjective,", "\\item $\\Delta_f : X \\to X \\times_Y X$ is rig-surjective.", "\\end{enumerate}" ], "refs": [ "formal-spaces-definition-proper" ], "ref_ids": [ 3995 ] }, { "id": 2637, "type": "definition", "label": "bootstrap-definition-morphism-representable-by-spaces", "categories": [ "bootstrap" ], "title": "bootstrap-definition-morphism-representable-by-spaces", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F$, $G$ be presheaves on $\\Sch_{fppf}/S$.", "We say a morphism $a : F \\to G$ is", "{\\it representable by algebraic spaces}", "if for every $U \\in \\Ob((\\Sch/S)_{fppf})$ and", "any $\\xi : U \\to G$ the fiber product $U \\times_{\\xi, G} F$", "is an algebraic space." ], "refs": [], "ref_ids": [] }, { "id": 2638, "type": "definition", "label": "bootstrap-definition-property-transformation", "categories": [ "bootstrap" ], "title": "bootstrap-definition-property-transformation", "contents": [ "Let $S$ be a scheme. Let $a : F \\to G$ be a map of presheaves on", "$(\\Sch/S)_{fppf}$ which is representable by algebraic spaces.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which", "\\begin{enumerate}", "\\item is preserved under any base change, and", "\\item is fppf local on the base, see", "Descent on Spaces,", "Definition \\ref{spaces-descent-definition-property-morphisms-local}.", "\\end{enumerate}", "In this case we say that $a$ has {\\it property $\\mathcal{P}$} if for every", "scheme $U$ and $\\xi : U \\to G$ the resulting morphism of algebraic spaces", "$U \\times_G F \\to U$ has property $\\mathcal{P}$." ], "refs": [ "spaces-descent-definition-property-morphisms-local" ], "ref_ids": [ 9440 ] }, { "id": 2761, "type": "definition", "label": "spaces-perfect-definition-supported-on", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-definition-supported-on", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $E$ be an object of $D(\\mathcal{O}_X)$.", "Let $T \\subset |X|$ be a closed subset.", "We say $E$ is {\\it supported on $T$} if the", "cohomology sheaves $H^i(E)$ are supported on $T$." ], "refs": [], "ref_ids": [] }, { "id": 2762, "type": "definition", "label": "spaces-perfect-definition-derived-quasi-coherent", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-definition-derived-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The {\\it derived category of $\\mathcal{O}_X$-modules with", "quasi-coherent cohomology sheaves} is denoted", "$D_\\QCoh(\\mathcal{O}_X)$." ], "refs": [], "ref_ids": [] }, { "id": 2763, "type": "definition", "label": "spaces-perfect-definition-proper-over-base", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-definition-proper-over-base", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$", "which is locally of finite type.", "Let $T \\subset |X|$ be a closed subset.", "We say {\\it $T$ is proper over $Y$}", "if the equivalent conditions of Lemma \\ref{lemma-closed-proper-over-base}", "are satisfied." ], "refs": [ "spaces-perfect-lemma-closed-proper-over-base" ], "ref_ids": [ 2656 ] }, { "id": 2764, "type": "definition", "label": "spaces-perfect-definition-elementary-distinguished-square", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-definition-elementary-distinguished-square", "contents": [ "Let $S$ be a scheme. A commutative diagram", "$$", "\\xymatrix{", "U \\times_W V \\ar[r] \\ar[d] & V \\ar[d]^f \\\\", "U \\ar[r]^j & W", "}", "$$", "of algebraic spaces over $S$ is called an {\\it elementary distinguished square}", "if", "\\begin{enumerate}", "\\item $U$ is an open subspace of $W$ and $j$ is the inclusion morphism,", "\\item $f$ is \\'etale, and", "\\item setting $T = W \\setminus U$ (with reduced induced", "subspace structure) the morphism $f^{-1}(T) \\to T$ is an isomorphism.", "\\end{enumerate}", "We will indicate this by saying: ``Let $(U \\subset W, f : V \\to W)$", "be an elementary distinguished square.''" ], "refs": [], "ref_ids": [] }, { "id": 2765, "type": "definition", "label": "spaces-perfect-definition-approximation-holds", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-definition-approximation-holds", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Consider triples $(T, E, m)$ where", "\\begin{enumerate}", "\\item $T \\subset |X|$ is a closed subset,", "\\item $E$ is an object of $D_\\QCoh(\\mathcal{O}_X)$, and", "\\item $m \\in \\mathbf{Z}$.", "\\end{enumerate}", "We say {\\it approximation holds for the triple} $(T, E, m)$ if", "there exists a perfect object $P$ of $D(\\mathcal{O}_X)$ supported on $T$", "and a map $\\alpha : P \\to E$ which induces isomorphisms $H^i(P) \\to H^i(E)$", "for $i > m$ and a surjection $H^m(P) \\to H^m(E)$." ], "refs": [], "ref_ids": [] }, { "id": 2766, "type": "definition", "label": "spaces-perfect-definition-approximation", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-definition-approximation", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "We say {\\it approximation by perfect complexes holds}", "on $X$ if for any closed subset $T \\subset |X|$ such that", "the morphism $X \\setminus T \\to X$ is quasi-compact", "there exists an integer $r$ such that for every triple $(T, E, m)$ as in", "Definition \\ref{definition-approximation-holds} with", "\\begin{enumerate}", "\\item $E$ is $(m - r)$-pseudo-coherent, and", "\\item $H^i(E)$ is supported on $T$ for $i \\geq m - r$", "\\end{enumerate}", "approximation holds." ], "refs": [ "spaces-perfect-definition-approximation-holds" ], "ref_ids": [ 2765 ] }, { "id": 2767, "type": "definition", "label": "spaces-perfect-definition-tor-independent", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-definition-tor-independent", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $X$, $Y$ be algebraic spaces over $B$. We say $X$ and", "$Y$ are {\\it Tor independent over $B$} if and only if for every", "commutative diagram", "$$", "\\xymatrix{", "\\Spec(k) \\ar[d]_{\\overline{y}} \\ar[dr]_{\\overline{b}} \\ar[r]_-{\\overline{x}} &", "X \\ar[d] \\\\", "Y \\ar[r] & B", "}", "$$", "of geometric points the rings", "$\\mathcal{O}_{X, \\overline{x}}$ and $\\mathcal{O}_{Y, \\overline{y}}$", "are Tor independent over $\\mathcal{O}_{B, \\overline{b}}$ (see", "More on Algebra, Definition \\ref{more-algebra-definition-tor-independent})." ], "refs": [ "more-algebra-definition-tor-independent" ], "ref_ids": [ 10622 ] }, { "id": 2927, "type": "definition", "label": "dualizing-definition-essential", "categories": [ "dualizing" ], "title": "dualizing-definition-essential", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item An injection $A \\subset B$ of $\\mathcal{A}$ is {\\it essential},", "or we say that $B$ is an {\\it essential extension of} $A$,", "if every nonzero subobject $B' \\subset B$ has nonzero intersection with $A$.", "\\item A surjection $f : A \\to B$ of $\\mathcal{A}$ is {\\it essential}", "if for every proper subobject $A' \\subset A$ we have $f(A') \\not = B$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 2928, "type": "definition", "label": "dualizing-definition-projective-cover", "categories": [ "dualizing" ], "title": "dualizing-definition-projective-cover", "contents": [ "Let $R$ be a ring. A surjection $P \\to M$ of $R$-modules is said", "to be a {\\it projective cover}, or sometimes a {\\it projective envelope},", "if $P$ is a projective $R$-module and $P \\to M$ is an essential", "surjection." ], "refs": [], "ref_ids": [] }, { "id": 2929, "type": "definition", "label": "dualizing-definition-injective-hull", "categories": [ "dualizing" ], "title": "dualizing-definition-injective-hull", "contents": [ "Let $R$ be a ring. A injection $M \\to I$ of $R$-modules is said", "to be an {\\it injective hull} if $I$ is a injective $R$-module and", "$M \\to I$ is an essential injection." ], "refs": [], "ref_ids": [] }, { "id": 2930, "type": "definition", "label": "dualizing-definition-indecomposable", "categories": [ "dualizing" ], "title": "dualizing-definition-indecomposable", "contents": [ "An object $X$ of an additive category is called {\\it indecomposable}", "if it is nonzero and if $X = Y \\oplus Z$, then either $Y = 0$ or $Z = 0$." ], "refs": [], "ref_ids": [] }, { "id": 2931, "type": "definition", "label": "dualizing-definition-dualizing", "categories": [ "dualizing" ], "title": "dualizing-definition-dualizing", "contents": [ "Let $A$ be a Noetherian ring. A {\\it dualizing complex} is a", "complex of $A$-modules $\\omega_A^\\bullet$ such that", "\\begin{enumerate}", "\\item $\\omega_A^\\bullet$ has finite injective dimension,", "\\item $H^i(\\omega_A^\\bullet)$ is a finite $A$-module for all $i$, and", "\\item $A \\to R\\Hom_A(\\omega_A^\\bullet, \\omega_A^\\bullet)$", "is a quasi-isomorphism.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 2932, "type": "definition", "label": "dualizing-definition-gorenstein", "categories": [ "dualizing" ], "title": "dualizing-definition-gorenstein", "contents": [ "Gorenstein rings.", "\\begin{enumerate}", "\\item Let $A$ be a Noetherian local ring. We say $A$ is {\\it Gorenstein}", "if $A[0]$ is a dualizing complex for $A$.", "\\item Let $A$ be a Noetherian ring. We say $A$ is {\\it Gorenstein}", "if $A_\\mathfrak p$ is Gorenstein for every prime $\\mathfrak p$ of $A$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 2933, "type": "definition", "label": "dualizing-definition-relative-dualizing-complex", "categories": [ "dualizing" ], "title": "dualizing-definition-relative-dualizing-complex", "contents": [ "Let $R \\to A$ be a flat ring map of finite presentation.", "A {\\it relative dualizing complex} is an object $K \\in D(A)$ such that", "\\begin{enumerate}", "\\item $K$ is $R$-perfect (More on Algebra, Definition", "\\ref{more-algebra-definition-relatively-perfect}), and", "\\item $R\\Hom_{A \\otimes_R A}(A, K \\otimes_A^\\mathbf{L} (A \\otimes_R A))$", "is isomorphic to $A$.", "\\end{enumerate}" ], "refs": [ "more-algebra-definition-relatively-perfect" ], "ref_ids": [ 10632 ] }, { "id": 3068, "type": "definition", "label": "properties-definition-integral", "categories": [ "properties" ], "title": "properties-definition-integral", "contents": [ "Let $X$ be a scheme. We say $X$ is {\\it integral} if it is nonempty and", "for every nonempty affine open $\\Spec(R) = U \\subset X$ the ring $R$", "is an integral domain." ], "refs": [], "ref_ids": [] }, { "id": 3069, "type": "definition", "label": "properties-definition-property-local", "categories": [ "properties" ], "title": "properties-definition-property-local", "contents": [ "Let $P$ be a property of rings.", "We say that $P$ is {\\it local} if the following hold:", "\\begin{enumerate}", "\\item For any ring $R$, and any $f \\in R$ we have", "$P(R) \\Rightarrow P(R_f)$.", "\\item For any ring $R$, and $f_i \\in R$ such that", "$(f_1, \\ldots, f_n) = R$ then", "$\\forall i, P(R_{f_i}) \\Rightarrow P(R)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3070, "type": "definition", "label": "properties-definition-locally-P", "categories": [ "properties" ], "title": "properties-definition-locally-P", "contents": [ "Let $P$ be a property of rings. Let $X$ be a scheme.", "We say $X$ is {\\it locally $P$} if for any $x \\in X$", "there exists an affine open neighbourhood $U$ of $x$", "in $X$ such that $\\mathcal{O}_X(U)$ has property $P$." ], "refs": [], "ref_ids": [] }, { "id": 3071, "type": "definition", "label": "properties-definition-noetherian", "categories": [ "properties" ], "title": "properties-definition-noetherian", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item We say $X$ is {\\it locally Noetherian} if every", "$x \\in X$ has an affine open neighbourhood", "$\\Spec(R) = U \\subset X$ such that the ring $R$ is Noetherian.", "\\item We say $X$ is {\\it Noetherian} if $X$ is locally Noetherian", "and quasi-compact.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3072, "type": "definition", "label": "properties-definition-jacobson", "categories": [ "properties" ], "title": "properties-definition-jacobson", "contents": [ "A scheme $S$ is said to be {\\it Jacobson} if its underlying topological", "space is Jacobson." ], "refs": [], "ref_ids": [] }, { "id": 3073, "type": "definition", "label": "properties-definition-normal", "categories": [ "properties" ], "title": "properties-definition-normal", "contents": [ "A scheme $X$ is {\\it normal} if and only if for all $x \\in X$ the local ring", "$\\mathcal{O}_{X, x}$ is a normal domain." ], "refs": [], "ref_ids": [] }, { "id": 3074, "type": "definition", "label": "properties-definition-Cohen-Macaulay", "categories": [ "properties" ], "title": "properties-definition-Cohen-Macaulay", "contents": [ "Let $X$ be a scheme. We say $X$ is {\\it Cohen-Macaulay} if", "for every $x \\in X$ there exists an affine open neighbourhood", "$U \\subset X$ of $x$ such that the ring $\\mathcal{O}_X(U)$ is", "Noetherian and Cohen-Macaulay." ], "refs": [], "ref_ids": [] }, { "id": 3075, "type": "definition", "label": "properties-definition-regular", "categories": [ "properties" ], "title": "properties-definition-regular", "contents": [ "Let $X$ be a scheme. We say $X$ is {\\it regular}, or {\\it nonsingular} if", "for every $x \\in X$ there exists an affine open neighbourhood", "$U \\subset X$ of $x$ such that the ring $\\mathcal{O}_X(U)$ is", "Noetherian and regular." ], "refs": [], "ref_ids": [] }, { "id": 3076, "type": "definition", "label": "properties-definition-dimension", "categories": [ "properties" ], "title": "properties-definition-dimension", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item The {\\it dimension} of $X$ is just the dimension of $X$", "as a topological spaces, see", "Topology, Definition \\ref{topology-definition-Krull}.", "\\item For $x \\in X$ we denote $\\dim_x(X)$ the dimension of the underlying", "topological space of $X$ at $x$ as in", "Topology, Definition \\ref{topology-definition-Krull}.", "We say $\\dim_x(X)$ is the {\\it dimension of $X$ at $x$}.", "\\end{enumerate}" ], "refs": [ "topology-definition-Krull", "topology-definition-Krull" ], "ref_ids": [ 8356, 8356 ] }, { "id": 3077, "type": "definition", "label": "properties-definition-catenary", "categories": [ "properties" ], "title": "properties-definition-catenary", "contents": [ "Let $S$ be a scheme. We say $S$ is {\\it catenary} if the", "underlying topological space of $S$ is catenary." ], "refs": [], "ref_ids": [] }, { "id": 3078, "type": "definition", "label": "properties-definition-Rk", "categories": [ "properties" ], "title": "properties-definition-Rk", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $k \\geq 0$.", "\\begin{enumerate}", "\\item We say $X$ is {\\it regular in codimension $k$},", "or we say $X$ has property {\\it $(R_k)$} if for every $x \\in X$", "we have", "$$", "\\dim(\\mathcal{O}_{X, x}) \\leq k", "\\Rightarrow", "\\mathcal{O}_{X, x}\\text{ is regular}", "$$", "\\item We say $X$ has property {\\it $(S_k)$} if for every $x \\in X$ we have", "$\\text{depth}(\\mathcal{O}_{X, x}) \\geq \\min(k, \\dim(\\mathcal{O}_{X, x}))$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3079, "type": "definition", "label": "properties-definition-nagata", "categories": [ "properties" ], "title": "properties-definition-nagata", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item Assume $X$ integral. We say $X$ is {\\it Japanese}", "if for every $x \\in X$ there exists an", "affine open neighbourhood $x \\in U \\subset X$ such that the ring", "$\\mathcal{O}_X(U)$ is Japanese (see", "Algebra, Definition \\ref{algebra-definition-N}).", "\\item We say $X$ is {\\it universally Japanese} if for every $x \\in X$", "there exists an affine open neighbourhood $x \\in U \\subset X$ such that", "the ring $\\mathcal{O}_X(U)$ is universally Japanese (see", "Algebra, Definition \\ref{algebra-definition-nagata}).", "\\item We say $X$ is {\\it Nagata} if for every $x \\in X$ there exists an", "affine open neighbourhood $x \\in U \\subset X$ such that the ring", "$\\mathcal{O}_X(U)$ is Nagata (see", "Algebra, Definition \\ref{algebra-definition-nagata}).", "\\end{enumerate}" ], "refs": [ "algebra-definition-N", "algebra-definition-nagata", "algebra-definition-nagata" ], "ref_ids": [ 1551, 1552, 1552 ] }, { "id": 3080, "type": "definition", "label": "properties-definition-singular-locus", "categories": [ "properties" ], "title": "properties-definition-singular-locus", "contents": [ "Let $X$ be a locally Noetherian scheme. The {\\it regular locus}", "$\\text{Reg}(X)$ of $X$ is the set of $x \\in X$ such that $\\mathcal{O}_{X, x}$", "is a regular local ring. The {\\it singular locus} $\\text{Sing}(X)$ is the", "complement $X \\setminus \\text{Reg}(X)$, i.e., the set of points $x \\in X$", "such that $\\mathcal{O}_{X, x}$ is not a regular local ring." ], "refs": [], "ref_ids": [] }, { "id": 3081, "type": "definition", "label": "properties-definition-unibranch", "categories": [ "properties" ], "title": "properties-definition-unibranch", "contents": [ "\\begin{reference}", "\\cite[Chapter IV (6.15.1)]{EGA4}", "\\end{reference}", "Let $X$ be a scheme. Let $x \\in X$. We say $X$ is {\\it unibranch at $x$}", "if the local ring $\\mathcal{O}_{X, x}$ is unibranch. We say $X$ is", "{\\it geometrically unibranch at $x$}", "if the local ring $\\mathcal{O}_{X, x}$ is geometrically unibranch.", "We say $X$ is {\\it unibranch} if $X$ is unibranch at all of its points.", "We say $X$ is {\\it geometrically unibranch} if $X$ is", "geometrically unibranch at all of its points." ], "refs": [], "ref_ids": [] }, { "id": 3082, "type": "definition", "label": "properties-definition-number-of-branches", "categories": [ "properties" ], "title": "properties-definition-number-of-branches", "contents": [ "Let $X$ be a scheme. Let $x \\in X$. The {\\it number of branches of $X$", "at $x$} is the number of branches of the local ring $\\mathcal{O}_{X, x}$", "as defined in", "More on Algebra, Definition \\ref{more-algebra-definition-number-of-branches}.", "The {\\it number of geometric branches of $X$ at $x$} is the number of", "geometric branches of the local ring $\\mathcal{O}_{X, x}$ as defined in", "More on Algebra, Definition \\ref{more-algebra-definition-number-of-branches}." ], "refs": [ "more-algebra-definition-number-of-branches", "more-algebra-definition-number-of-branches" ], "ref_ids": [ 10638, 10638 ] }, { "id": 3083, "type": "definition", "label": "properties-definition-quasi-affine", "categories": [ "properties" ], "title": "properties-definition-quasi-affine", "contents": [ "A scheme $X$ is called {\\it quasi-affine} if it is quasi-compact", "and isomorphic to an open subscheme of an affine scheme." ], "refs": [], "ref_ids": [] }, { "id": 3084, "type": "definition", "label": "properties-definition-locally-projective", "categories": [ "properties" ], "title": "properties-definition-locally-projective", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module. We say $\\mathcal{F}$ is {\\it locally projective}", "if for every affine open $U \\subset X$ the $\\mathcal{O}_X(U)$-module", "$\\mathcal{F}(U)$ is projective." ], "refs": [], "ref_ids": [] }, { "id": 3085, "type": "definition", "label": "properties-definition-kappa-generated", "categories": [ "properties" ], "title": "properties-definition-kappa-generated", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\kappa$ be an infinite", "cardinal. We say a sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ is", "{\\it $\\kappa$-generated} if there exists an open covering", "$X = \\bigcup U_i$ such that $\\mathcal{F}|_{U_i}$ is generated by", "a subset $R_i \\subset \\mathcal{F}(U_i)$ whose cardinality is", "at most $\\kappa$." ], "refs": [], "ref_ids": [] }, { "id": 3086, "type": "definition", "label": "properties-definition-subsheaf-sections-annihilated-by-ideal", "categories": [ "properties" ], "title": "properties-definition-subsheaf-sections-annihilated-by-ideal", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals", "of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "The subsheaf $\\mathcal{F}' \\subset \\mathcal{F}$ defined in", "Lemma \\ref{lemma-sections-annihilated-by-ideal} above is called", "the {\\it subsheaf of sections annihilated by $\\mathcal{I}$}." ], "refs": [ "properties-lemma-sections-annihilated-by-ideal" ], "ref_ids": [ 3034 ] }, { "id": 3087, "type": "definition", "label": "properties-definition-subsheaf-sections-supported-on-closed", "categories": [ "properties" ], "title": "properties-definition-subsheaf-sections-supported-on-closed", "contents": [ "Let $X$ be a scheme.", "Let $T \\subset X$ be a closed subset whose complement", "is retrocompact in $X$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "The quasi-coherent subsheaf $\\mathcal{F}' \\subset \\mathcal{F}$ defined in", "Lemma \\ref{lemma-sections-supported-on-closed-subset} is called", "the {\\it subsheaf of sections supported on $T$}." ], "refs": [ "properties-lemma-sections-supported-on-closed-subset" ], "ref_ids": [ 3036 ] }, { "id": 3088, "type": "definition", "label": "properties-definition-ample", "categories": [ "properties" ], "title": "properties-definition-ample", "contents": [ "\\begin{reference}", "\\cite[II Definition 4.5.3]{EGA}", "\\end{reference}", "Let $X$ be a scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "We say $\\mathcal{L}$ is {\\it ample} if", "\\begin{enumerate}", "\\item $X$ is quasi-compact, and", "\\item for every $x \\in X$ there exists an $n \\geq 1$", "and $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ such", "that $x \\in X_s$ and $X_s$ is affine.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3145, "type": "definition", "label": "criteria-definition-algebraic", "categories": [ "criteria" ], "title": "criteria-definition-algebraic", "contents": [ "Let $S$ be a scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a", "$1$-morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$.", "We say that $F$ is {\\it algebraic} if for every scheme $T$ and every", "object $\\xi$ of $\\mathcal{Y}$ over $T$ the $2$-fibre product", "$$", "(\\Sch/T)_{fppf} \\times_{\\xi, \\mathcal{Y}} \\mathcal{X}", "$$", "is an algebraic stack over $S$." ], "refs": [], "ref_ids": [] }, { "id": 3403, "type": "definition", "label": "coherent-definition-depth", "categories": [ "coherent" ], "title": "coherent-definition-depth", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "Let $k \\geq 0$ be an integer.", "\\begin{enumerate}", "\\item We say $\\mathcal{F}$ has {\\it depth $k$ at a point}", "$x$ of $X$ if $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) = k$.", "\\item We say $X$ has {\\it depth $k$ at a point} $x$ of $X$ if", "$\\text{depth}(\\mathcal{O}_{X, x}) = k$.", "\\item We say $\\mathcal{F}$ has property {\\it $(S_k)$} if", "$$", "\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x)", "\\geq \\min(k, \\dim(\\text{Supp}(\\mathcal{F}_x)))", "$$", "for all $x \\in X$.", "\\item We say $X$ has property {\\it $(S_k)$} if $\\mathcal{O}_X$ has", "property $(S_k)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3404, "type": "definition", "label": "coherent-definition-Cohen-Macaulay", "categories": [ "coherent" ], "title": "coherent-definition-Cohen-Macaulay", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "We say $\\mathcal{F}$ is {\\it Cohen-Macaulay} if and only", "if $(S_k)$ holds for all $k \\geq 0$." ], "refs": [], "ref_ids": [] }, { "id": 3405, "type": "definition", "label": "coherent-definition-proper-over-base", "categories": [ "coherent" ], "title": "coherent-definition-proper-over-base", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $Z \\subset X$ be a closed subset.", "We say {\\it $Z$ is proper over $S$}", "if the equivalent conditions of Lemma \\ref{lemma-closed-proper-over-base}", "are satisfied." ], "refs": [ "coherent-lemma-closed-proper-over-base" ], "ref_ids": [ 3386 ] }, { "id": 3510, "type": "definition", "label": "formal-defos-definition-CLambda", "categories": [ "formal-defos" ], "title": "formal-defos-definition-CLambda", "contents": [ "Let $\\Lambda$ be a Noetherian ring and let $\\Lambda \\to k$ be a finite", "ring map where $k$ is a field. We define {\\it $\\mathcal{C}_\\Lambda$} to be", "the category with", "\\begin{enumerate}", "\\item objects are pairs $(A, \\varphi)$ where $A$ is an Artinian local", "$\\Lambda$-algebra and where $\\varphi : A/\\mathfrak m_A \\to k$ is a", "$\\Lambda$-algebra isomorphism, and", "\\item morphisms $f : (B, \\psi) \\to (A, \\varphi)$ are local $\\Lambda$-algebra", "homomorphisms such that $\\varphi \\circ (f \\bmod \\mathfrak m) = \\psi$.", "\\end{enumerate}", "We say we are in the {\\it classical case} if $\\Lambda$ is a Noetherian", "complete local ring and $k$ is its residue field." ], "refs": [], "ref_ids": [] }, { "id": 3511, "type": "definition", "label": "formal-defos-definition-small-extension", "categories": [ "formal-defos" ], "title": "formal-defos-definition-small-extension", "contents": [ "Let $f: B \\to A$ be a ring map in $\\mathcal{C}_\\Lambda$. We say $f$", "is a {\\it small extension} if it is surjective and $\\Ker(f)$ is a nonzero", "principal ideal which is annihilated by $\\mathfrak{m}_B$." ], "refs": [], "ref_ids": [] }, { "id": 3512, "type": "definition", "label": "formal-defos-definition-tangent-space-ring", "categories": [ "formal-defos" ], "title": "formal-defos-definition-tangent-space-ring", "contents": [ "Let $R \\to S$ be a local homomorphism of local rings. The", "{\\it relative cotangent space}\\footnote{Caution: We will see later", "that in our general setting the tangent", "space of an object $A \\in \\mathcal{C}_\\Lambda$ over $\\Lambda$ should", "not be defined simply as the $k$-linear dual of the relative", "cotangent space. In fact, the correct definition of the relative", "cotangent space is", "$\\Omega_{S/R} \\otimes_S S/\\mathfrak m_S$.} of $R$ over $S$ is the", "$S/\\mathfrak m_S$-vector space", "$\\mathfrak m_S/(\\mathfrak m_R S + \\mathfrak m_S^2)$." ], "refs": [], "ref_ids": [] }, { "id": 3513, "type": "definition", "label": "formal-defos-definition-essential-surjection", "categories": [ "formal-defos" ], "title": "formal-defos-definition-essential-surjection", "contents": [ "Let $f: B \\to A$ be a ring map in $\\mathcal{C}_\\Lambda$. We say $f$", "is an {\\it essential surjection} if it has the following properties:", "\\begin{enumerate}", "\\item $f$ is surjective.", "\\item If $g: C \\to B$ is a ring map in $\\mathcal{C}_\\Lambda$ such that", "$f \\circ g$ is surjective, then $g$ is surjective.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3514, "type": "definition", "label": "formal-defos-definition-completion-CLambda", "categories": [ "formal-defos" ], "title": "formal-defos-definition-completion-CLambda", "contents": [ "Let $\\Lambda$ be a Noetherian ring and let $\\Lambda \\to k$ be a finite", "ring map where $k$ is a field. We define {\\it $\\widehat{\\mathcal{C}}_\\Lambda$}", "to be the category with", "\\begin{enumerate}", "\\item objects are pairs $(R, \\varphi)$ where $R$ is a Noetherian complete", "local $\\Lambda$-algebra and where $\\varphi : R/\\mathfrak m_R \\to k$ is a", "$\\Lambda$-algebra isomorphism, and", "\\item morphisms $f : (S, \\psi) \\to (R, \\varphi)$ are local $\\Lambda$-algebra", "homomorphisms such that $\\varphi \\circ (f \\bmod \\mathfrak m) = \\psi$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3515, "type": "definition", "label": "formal-defos-definition-category-cofibred-groupoids", "categories": [ "formal-defos" ], "title": "formal-defos-definition-category-cofibred-groupoids", "contents": [ "Let $\\mathcal{C}$ be a category. A {\\it category cofibered in groupoids over", "$\\mathcal{C}$} is a category $\\mathcal{F}$ equipped with a functor", "$p: \\mathcal{F} \\to \\mathcal{C}$ such that $\\mathcal{F}^{opp}$ is a category", "fibered in groupoids over $\\mathcal{C}^{opp}$ via", "$p^{opp}: \\mathcal{F}^{opp} \\to \\mathcal{C}^{opp}$." ], "refs": [], "ref_ids": [] }, { "id": 3516, "type": "definition", "label": "formal-defos-definition-prorepresentable", "categories": [ "formal-defos" ], "title": "formal-defos-definition-prorepresentable", "contents": [ "Let $F : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a functor.", "We say $F$ is {\\it prorepresentable} if there exists an isomorphism", "$F \\cong \\underline{R}|_{\\mathcal{C}_\\Lambda}$ of functors for some", "$R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$." ], "refs": [], "ref_ids": [] }, { "id": 3517, "type": "definition", "label": "formal-defos-definition-predeformation-category", "categories": [ "formal-defos" ], "title": "formal-defos-definition-predeformation-category", "contents": [ "A {\\it predeformation category} $\\mathcal{F}$ is a category cofibered", "in groupoids over $\\mathcal{C}_\\Lambda$ such that $\\mathcal{F}(k)$ is", "equivalent to a category with a single object and a single morphism,", "i.e., $\\mathcal{F}(k)$ contains at least one object and there is a", "unique morphism between any two objects. A {\\it morphism of predeformation", "categories} is a morphism of categories cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$." ], "refs": [], "ref_ids": [] }, { "id": 3518, "type": "definition", "label": "formal-defos-definition-formal-objects", "categories": [ "formal-defos" ], "title": "formal-defos-definition-formal-objects", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$. The {\\it category $\\widehat{\\mathcal{F}}$ of formal", "objects of $\\mathcal{F}$} is the category with the following objects and", "morphisms.", "\\begin{enumerate}", "\\item A {\\it formal object $\\xi = (R, \\xi_n, f_n)$ of $\\mathcal{F}$}", "consists of an object $R$ of $\\widehat{\\mathcal{C}}_\\Lambda$, and a collection", "indexed by $n \\in \\mathbf{N}$ of objects $\\xi_n$ of", "$\\mathcal{F}(R/\\mathfrak m_R^n)$ and morphisms", "$f_n : \\xi_{n + 1} \\to \\xi_n$ lying over the projection", "$R/\\mathfrak m_R^{n + 1} \\to R/\\mathfrak m_R^n$.", "\\item Let $\\xi = (R, \\xi_n, f_n)$ and $\\eta = (S, \\eta_n, g_n)$ be", "formal objects of $\\mathcal{F}$. A {\\it morphism $a : \\xi \\to \\eta$ of", "formal objects} consists of a map $a_0 : R \\to S$ in", "$\\widehat{\\mathcal{C}}_\\Lambda$ and a collection $a_n : \\xi_n \\to \\eta_n$", "of morphisms of $\\mathcal{F}$ lying over", "$R/\\mathfrak m_R^n \\to S/\\mathfrak m_S^n$,", "such that for every $n$ the diagram", "$$", "\\xymatrix{", "\\xi_{n + 1} \\ar[r]_{f_n} \\ar[d]_{a_{n + 1}} & \\xi_n \\ar[d]^{a_n} \\\\", "\\eta_{n + 1} \\ar[r]^{g_n} & \\eta_n", "}", "$$", "commutes.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3519, "type": "definition", "label": "formal-defos-definition-completion", "categories": [ "formal-defos" ], "title": "formal-defos-definition-completion", "contents": [ "Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in", "groupoids. The category cofibered in groupoids", "$\\widehat{p} : \\widehat{\\mathcal F} \\to \\widehat{\\mathcal{C}}_\\Lambda$", "is called the {\\it completion of $\\mathcal{F}$}." ], "refs": [], "ref_ids": [] }, { "id": 3520, "type": "definition", "label": "formal-defos-definition-smooth-morphism", "categories": [ "formal-defos" ], "title": "formal-defos-definition-smooth-morphism", "contents": [ "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of categories", "cofibered in groupoids over $\\mathcal{C}_\\Lambda$. We say $\\varphi$ is", "{\\it smooth} if it satisfies the following condition: Let $B \\to A$ be", "a surjective ring map in $\\mathcal{C}_\\Lambda$. Let $y \\in", "\\Ob(\\mathcal{G}(B)), x \\in \\Ob(\\mathcal{F}(A))$, and $y", "\\to \\varphi(x)$ be a morphism lying over $B \\to A$. Then there", "exists $x' \\in \\Ob(\\mathcal{F}(B))$, a morphism $x' \\to x$", "lying over $B \\to A$, and a morphism $\\varphi(x') \\to y$ lying", "over $\\text{id}: B \\to B$, such that the diagram", "$$", "\\xymatrix{", "\\varphi(x') \\ar[r] \\ar[dr] & y \\ar[d] \\\\", "& \\varphi(x)", "}", "$$", "commutes." ], "refs": [], "ref_ids": [] }, { "id": 3521, "type": "definition", "label": "formal-defos-definition-versal", "categories": [ "formal-defos" ], "title": "formal-defos-definition-versal", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids. Let $\\xi$ be a formal", "object of $\\mathcal{F}$ lying over $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$.", "We say $\\xi$ is {\\it versal} if the corresponding morphism", "$\\underline{\\xi}: \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$", "of Remark \\ref{remark-formal-objects-yoneda} is smooth." ], "refs": [ "formal-defos-remark-formal-objects-yoneda" ], "ref_ids": [ 3553 ] }, { "id": 3522, "type": "definition", "label": "formal-defos-definition-cofibered-groupoid-projection-smooth", "categories": [ "formal-defos" ], "title": "formal-defos-definition-cofibered-groupoid-projection-smooth", "contents": [ "Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in", "groupoids. We say $\\mathcal{F}$ is {\\it smooth} or {\\it unobstructed}", "if its structure morphism $p$ is smooth", "in the sense of Definition \\ref{definition-smooth-morphism}." ], "refs": [ "formal-defos-definition-smooth-morphism" ], "ref_ids": [ 3520 ] }, { "id": 3523, "type": "definition", "label": "formal-defos-definition-S1-S2", "categories": [ "formal-defos" ], "title": "formal-defos-definition-S1-S2", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal", "C_\\Lambda$. We define {\\it conditions (S1) and (S2)}", "on $\\mathcal{F}$ as follows:", "\\begin{enumerate}", "\\item[(S1)] Every diagram in $\\mathcal{F}$", "$$", "\\vcenter{", "\\xymatrix{", " & x_2 \\ar[d] \\\\", "x_1 \\ar[r] & x", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", " & A_2 \\ar[d] \\\\", "A_1 \\ar[r] & A", "}", "}", "$$", "in $\\mathcal{C}_\\Lambda$ with $A_2 \\to A$ surjective can be completed", "to a commutative diagram", "$$", "\\vcenter{", "\\xymatrix{", "y \\ar[r] \\ar[d] & x_2 \\ar[d] \\\\", "x_1 \\ar[r] & x", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "A_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d] \\\\", "A_1 \\ar[r] & A.", "}", "}", "$$", "\\item[(S2)]", "The condition of (S1) holds for diagrams in $\\mathcal{F}$ lying over", "a diagram in $\\mathcal{C}_\\Lambda$ of the form", "$$", "\\xymatrix{", " & k[\\epsilon] \\ar[d] \\\\", "A \\ar[r] & k.", "}", "$$", "Moreover, if we have two commutative diagrams in $\\mathcal{F}$", "$$", "\\vcenter{", "\\xymatrix{", "y \\ar[r]_c \\ar[d]_a & x_\\epsilon \\ar[d]^e \\\\", "x \\ar[r]^d & x_0", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "y' \\ar[r]_{c'} \\ar[d]_{a'} & x_\\epsilon \\ar[d]^e \\\\", "x \\ar[r]^d & x_0", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", "A \\times_k k[\\epsilon] \\ar[r] \\ar[d] & k[\\epsilon] \\ar[d] \\\\", "A \\ar[r] & k", "}", "}", "$$", "then there exists a morphism $b : y \\to y'$ in", "$\\mathcal{F}(A \\times_k k[\\epsilon])$ such that $a = a' \\circ b$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3524, "type": "definition", "label": "formal-defos-definition-linear", "categories": [ "formal-defos" ], "title": "formal-defos-definition-linear", "contents": [ "Let $L: \\text{Mod}^{fg}_R \\to \\text{Mod}_R$,", "resp.\\ $L: \\text{Mod}_R \\to \\text{Mod}_R$", "be a functor. We say that $L$ is {\\it $R$-linear} if for every", "pair of objects $M, N$ of $\\text{Mod}^{fg}_R$, resp.\\ $\\text{Mod}_R$", "the map", "$$", "L : \\Hom_R(M, N) \\longrightarrow \\Hom_R(L(M), L(N))", "$$", "is a map of $R$-modules." ], "refs": [], "ref_ids": [] }, { "id": 3525, "type": "definition", "label": "formal-defos-definition-tangent-space-over-R", "categories": [ "formal-defos" ], "title": "formal-defos-definition-tangent-space-over-R", "contents": [ "Let $\\mathcal{C}$ be a category as in", "Lemma \\ref{lemma-tangent-space-functor}.", "Let $F : \\mathcal{C} \\to \\textit{Sets}$ be a functor such that", "$F(R)$ is a one element set. The {\\it tangent space $TF$ of $F$} is", "$F(R[\\epsilon])$." ], "refs": [ "formal-defos-lemma-tangent-space-functor" ], "ref_ids": [ 3449 ] }, { "id": 3526, "type": "definition", "label": "formal-defos-definition-tangent-space", "categories": [ "formal-defos" ], "title": "formal-defos-definition-tangent-space", "contents": [ "Let $\\mathcal{F}$ be a predeformation category.", "The {\\it tangent space $T \\mathcal{F}$ of $\\mathcal{F}$}", "is the set $\\overline{\\mathcal{F}}(k[\\epsilon])$", "of isomorphism classes of objects in the fiber category $\\mathcal", "F(k[\\epsilon])$." ], "refs": [], "ref_ids": [] }, { "id": 3527, "type": "definition", "label": "formal-defos-definition-differential", "categories": [ "formal-defos" ], "title": "formal-defos-definition-differential", "contents": [ "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism predeformation", "categories. The", "{\\it differential $d \\varphi : T \\mathcal{F} \\to T \\mathcal{G}$ of $\\varphi$}", "is the map obtained by evaluating the morphism of functors", "$\\overline{\\varphi}: \\overline{\\mathcal{F}} \\to \\overline{\\mathcal{G}}$", "at $A = k[\\epsilon]$." ], "refs": [], "ref_ids": [] }, { "id": 3528, "type": "definition", "label": "formal-defos-definition-minimal-versal", "categories": [ "formal-defos" ], "title": "formal-defos-definition-minimal-versal", "contents": [ "Let $\\mathcal{F}$ be a predeformation category.", "We say a versal formal object $\\xi$ of $\\mathcal{F}$ is", "{\\it minimal}\\footnote{This may be nonstandard terminology. Many", "authors tie this notion in with properties of tangent spaces.", "We will make the link in", "Section \\ref{section-miniversal-objects-existence}.}", "if for any morphism of formal objects", "$\\xi' \\to \\xi$ the underlying map on rings is surjective.", "Sometimes a minimal versal formal object is called {\\it miniversal}." ], "refs": [], "ref_ids": [] }, { "id": 3529, "type": "definition", "label": "formal-defos-definition-RS", "categories": [ "formal-defos" ], "title": "formal-defos-definition-RS", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal", "C_\\Lambda$. We say that $\\mathcal{F}$ satisfies {\\it condition (RS)}", "if for every diagram in $\\mathcal{F}$", "$$", "\\vcenter{", "\\xymatrix{", " & x_2 \\ar[d] \\\\", "x_1 \\ar[r] & x", "}", "}", "\\quad\\text{lying over}\\quad", "\\vcenter{", "\\xymatrix{", " & A_2 \\ar[d] \\\\", "A_1 \\ar[r] & A", "}", "}", "$$", "in $\\mathcal{C}_\\Lambda$ with $A_2 \\to A$ surjective, there exists a", "fiber product $x_1 \\times_x x_2$ in $\\mathcal{F}$ such that the diagram", "$$", "\\vcenter{", "\\xymatrix{", "x_1 \\times_x x_2 \\ar[r] \\ar[d] & x_2 \\ar[d] \\\\", "x_1 \\ar[r] & x", "}", "}", "\\quad\\text{lies over}\\quad", "\\vcenter{", "\\xymatrix{", "A_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d] \\\\", "A_1 \\ar[r] & A.", "}", "}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 3530, "type": "definition", "label": "formal-defos-definition-deformation-category", "categories": [ "formal-defos" ], "title": "formal-defos-definition-deformation-category", "contents": [ "A {\\it deformation category} is a predeformation category $\\mathcal{F}$", "satisfying (RS). A morphism of deformation categories is a morphism of", "categories over $\\mathcal{C}_\\Lambda$." ], "refs": [], "ref_ids": [] }, { "id": 3531, "type": "definition", "label": "formal-defos-definition-lifts", "categories": [ "formal-defos" ], "title": "formal-defos-definition-lifts", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$. Let $f: A' \\to A$ be a map in $\\mathcal{C}_\\Lambda$.", "Let $x \\in \\mathcal{F}(A)$. The category $\\textit{Lift}(x, f)$ of lifts of $x$", "along $f$ is the category with the following objects and", "morphisms.", "\\begin{enumerate}", "\\item Objects: A {\\it lift of $x$ along $f$} is a morphism $x' \\to x$", "lying over $f$.", "\\item Morphisms: A {\\it morphism of lifts} from $a_1 : x'_1 \\to x$ to", "$a_2 : x'_2 \\to x$ is a morphism $b : x'_1 \\to x'_2$ in", "$\\mathcal{F}(A')$ such that $a_2 = a_1 \\circ b$.", "\\end{enumerate}", "The set $\\text{Lift}(x, f)$ of lifts of $x$ along $f$ is the set of", "isomorphism classes of $\\textit{Lift}(x, f)$." ], "refs": [], "ref_ids": [] }, { "id": 3532, "type": "definition", "label": "formal-defos-definition-relative-infinitesimal-auts", "categories": [ "formal-defos" ], "title": "formal-defos-definition-relative-infinitesimal-auts", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal", "C_\\Lambda$. Let $x' \\to x$ be a morphism in $\\mathcal{F}$ lying over", "$A' \\to A$. The kernel", "$$", "\\text{Inf}(x'/x) = \\Ker(\\text{Aut}_{A'}(x') \\to \\text{Aut}_A(x))", "$$", "is the {\\it group of infinitesimal automorphisms of $x'$ over $x$}." ], "refs": [], "ref_ids": [] }, { "id": 3533, "type": "definition", "label": "formal-defos-definition-infinitesimal-auts", "categories": [ "formal-defos" ], "title": "formal-defos-definition-infinitesimal-auts", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal", "C_\\Lambda$. Let $x_0 \\in \\Ob(\\mathcal{F}(k))$. Assume a choice of", "pushforward $x_0 \\to x_0'$ of $x_0$ along the map", "$k \\to k[\\epsilon], a \\mapsto a$ has been made.", "Then there is a unique map $x'_0 \\to x_0$ such that", "$x_0 \\to x_0' \\to x_0$ is the identity on $x_0$.", "Then", "$$", "\\text{Inf}_{x_0}(\\mathcal F) = \\text{Inf}(x'_0/x_0)", "$$", "is the {\\it group of infinitesimal automorphisms of $x_0$}" ], "refs": [], "ref_ids": [] }, { "id": 3534, "type": "definition", "label": "formal-defos-definition-automorphism-functor", "categories": [ "formal-defos" ], "title": "formal-defos-definition-automorphism-functor", "contents": [ "Let $p : \\mathcal{F} \\to \\mathcal{C}$ be a category cofibered in groupoids", "over an arbitrary base category $\\mathcal{C}$. Assume a choice of pushforwards", "has been made. Let $x \\in \\Ob(\\mathcal{F})$ and let $U = p(x)$.", "Let $U/\\mathcal{C}$ denote the category of objects under $U$. The", "{\\it automorphism functor of $x$} is the functor", "$\\mathit{Aut}(x) : U/\\mathcal{C} \\to \\textit{Sets}$ sending an object", "$f : U \\to V$ to $\\text{Aut}_V(f_*x)$ and sending a morphism", "$$", "\\xymatrix{", "V' \\ar[rr] & & V\\\\", " & U \\ar[ul]^{f'} \\ar[ur]_f &", "}", "$$", "to the homomorphism", "$\\text{Aut}_{V'}(f'_*x) \\to \\text{Aut}_V(f_*x)$", "coming from the unique morphism $f'_*x \\to f_*x$ lying over", "$V' \\to V$ and compatible with $x \\to f'_*x$ and $x \\to f_*x$." ], "refs": [], "ref_ids": [] }, { "id": 3535, "type": "definition", "label": "formal-defos-definition-groupoid-in-functors", "categories": [ "formal-defos" ], "title": "formal-defos-definition-groupoid-in-functors", "contents": [ "Let $\\mathcal{C}$ be a category. The", "{\\it category of groupoids in functors on $\\mathcal{C}$}", "is the category with the following objects and morphisms.", "\\begin{enumerate}", "\\item Objects: A {\\it groupoid in functors on $\\mathcal{C}$} is a quintuple", "$(U, R, s, t, c)$ where $U, R : \\mathcal{C} \\to \\textit{Sets}$ are", "functors and $s, t : R \\to U$ and $c : R \\times_{s, U, t} R \\to R$", "are morphisms with the following property: For any object $T$ of $\\mathcal{C}$,", "the quintuple", "$$", "(U(T), R(T), s, t, c)", "$$", "is a groupoid category.", "\\item Morphisms: A {\\it morphism $(U, R, s, t, c) \\to (U', R', s', t', c')$ of", "groupoids in functors on $\\mathcal{C}$} consists of morphisms $U \\to U'$", "and $R \\to R'$ with the following property: For any object $T$ of", "$\\mathcal{C}$, the induced maps $U(T) \\to U'(T)$ and", "$R(T) \\to R'(T)$ define a functor between groupoid categories", "$$", "(U(T), R(T), s, t, c) \\to (U'(T), R'(T), s', t', c').", "$$", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3536, "type": "definition", "label": "formal-defos-definition-representable", "categories": [ "formal-defos" ], "title": "formal-defos-definition-representable", "contents": [ "Let $\\mathcal{C}$ be a category. A groupoid in functors on $\\mathcal{C}$ is", "{\\it representable} if it is isomorphic to one of the form", "$(\\underline{U}, \\underline{R}, s, t, c)$ where $U$ and $R$ are objects of", "$\\mathcal{C}$ and the pushout $R \\amalg_{s, U, t} R$ exists." ], "refs": [], "ref_ids": [] }, { "id": 3537, "type": "definition", "label": "formal-defos-definition-restricting-groupoids-in-functors", "categories": [ "formal-defos" ], "title": "formal-defos-definition-restricting-groupoids-in-functors", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\\mathcal{C}$.", "Let $\\mathcal{C}'$ be a subcategory of $\\mathcal{C}$. The", "{\\it restriction $(U, R, s, t, c)|_{\\mathcal{C}'}$ of $(U, R, s, t, c)$", "to $\\mathcal{C}'$} is the groupoid", "in functors on $\\mathcal{C}'$ given by $(U|_{\\mathcal{C}'}, R|_{\\mathcal", "C'}, s|_{\\mathcal{C}'}, t|_{\\mathcal{C}'}, c|_{\\mathcal{C}'})$." ], "refs": [], "ref_ids": [] }, { "id": 3538, "type": "definition", "label": "formal-defos-definition-quotient", "categories": [ "formal-defos" ], "title": "formal-defos-definition-quotient", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\\mathcal{C}$.", "\\begin{enumerate}", "\\item The assignment $T \\mapsto (U(T), R(T), s, t, c)$ determines a functor", "$\\mathcal{C} \\to \\textit{Groupoids}$. The {\\it quotient category", "cofibered in groupoids $[U/R] \\to \\mathcal{C}$} is the category", "cofibered in groupoids over $\\mathcal{C}$ associated to this functor (as in", "Remarks \\ref{remarks-cofibered-groupoids}", "(\\ref{item-construction-associated-cofibered-groupoid})).", "\\item The {\\it quotient morphism $U \\to [U/R]$} is the morphism of", "categories cofibered in groupoids over $\\mathcal{C}$ defined by the", "rules", "\\begin{enumerate}", "\\item $x \\in U(T)$ maps to the object $(T, x) \\in \\Ob([U/R](T))$, and", "\\item $x \\in U(T)$ and $f : T \\to T'$ give rise to the morphism", "$(f, \\text{id}_{U(f)(x)}): (T, x) \\to (T, U(f)(x))$ lying over", "$f : T \\to T'$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [ "formal-defos-remarks-cofibered-groupoids" ], "ref_ids": [ 3585 ] }, { "id": 3539, "type": "definition", "label": "formal-defos-definition-prorepresentable-groupoid-in-functors", "categories": [ "formal-defos" ], "title": "formal-defos-definition-prorepresentable-groupoid-in-functors", "contents": [ "A groupoid in functors on $\\mathcal{C}_\\Lambda$ is {\\it prorepresentable}", "if it is isomorphic to", "$(\\underline{R_0}, \\underline{R_1}, s, t, c)|_{\\mathcal{C}_\\Lambda}$", "for some representable groupoid in functors", "$(\\underline{R_0}, \\underline{R_1}, s, t, c)$ on the category", "$\\widehat{\\mathcal{C}}_\\Lambda$." ], "refs": [], "ref_ids": [] }, { "id": 3540, "type": "definition", "label": "formal-defos-definition-completion-groupoid-in-functors", "categories": [ "formal-defos" ], "title": "formal-defos-definition-completion-groupoid-in-functors", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}_\\Lambda$.", "The {\\it completion $(U, R, s, t, c)^{\\wedge}$ of $(U, R, s, t, c)$} is the", "groupoid in functors", "$(\\widehat{U}, \\widehat{R}, \\widehat{s}, \\widehat{t}, \\widehat{c})$", "on $\\widehat{\\mathcal{C}}_\\Lambda$ described above." ], "refs": [], "ref_ids": [] }, { "id": 3541, "type": "definition", "label": "formal-defos-definition-smooth-groupoid-in-functors", "categories": [ "formal-defos" ], "title": "formal-defos-definition-smooth-groupoid-in-functors", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}_\\Lambda$. We", "say $(U, R, s, t, c)$ is {\\it smooth} if $s, t: R \\to U$ are smooth." ], "refs": [], "ref_ids": [] }, { "id": 3542, "type": "definition", "label": "formal-defos-definition-presentation", "categories": [ "formal-defos" ], "title": "formal-defos-definition-presentation", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over a category", "$\\mathcal{C}$. Let $(U, R, s, t, c)$ be a groupoid in functors on", "$\\mathcal{C}$. A", "{\\it presentation of $\\mathcal{F}$ by $(U, R, s, t, c)$} is an equivalence", "$\\varphi : [U/R] \\to \\mathcal{F}$ of categories cofibered in groupoids", "over $\\mathcal{C}$." ], "refs": [], "ref_ids": [] }, { "id": 3543, "type": "definition", "label": "formal-defos-definition-minimal-groupoid-in-functors", "categories": [ "formal-defos" ], "title": "formal-defos-definition-minimal-groupoid-in-functors", "contents": [ "Let $(U, R, s, t, c)$ be a smooth prorepresentable groupoid in functors", "on $\\mathcal{C}_\\Lambda$.", "\\begin{enumerate}", "\\item We say $(U, R, s, t, c)$ is {\\it normalized} if the groupoid", "$(U(k[\\epsilon]), R(k[\\epsilon]), s, t, c)$ is totally disconnected,", "i.e., there are no morphisms between distinct objects.", "\\item We say $(U, R, s, t, c)$ is {\\it minimal} if the $U \\to [U/R]$", "is given by a minimal versal formal object of $[U/R]$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3637, "type": "definition", "label": "adequate-definition-module-valued-functor", "categories": [ "adequate" ], "title": "adequate-definition-module-valued-functor", "contents": [ "Let $A$ be a ring. A {\\it module-valued functor} is a functor", "$F : \\textit{Alg}_A \\to \\textit{Ab}$ such that", "\\begin{enumerate}", "\\item for every object $B$ of $\\textit{Alg}_A$ the group", "$F(B)$ is endowed with the structure of a $B$-module, and", "\\item for any morphism $B \\to B'$ of $\\textit{Alg}_A$ the map", "$F(B) \\to F(B')$ is $B$-linear.", "\\end{enumerate}", "A {\\it morphism of module-valued functors} is a transformation of", "functors $\\varphi : F \\to G$ such that $F(B) \\to G(B)$ is $B$-linear", "for all $B \\in \\Ob(\\textit{Alg}_A)$." ], "refs": [], "ref_ids": [] }, { "id": 3638, "type": "definition", "label": "adequate-definition-adequate-functor", "categories": [ "adequate" ], "title": "adequate-definition-adequate-functor", "contents": [ "Let $A$ be a ring. A module-valued functor $F$ on $\\textit{Alg}_A$ is", "called", "\\begin{enumerate}", "\\item {\\it adequate} if there exists a", "map of $A$-modules $M \\to N$ such that $F$ is isomorphic to", "$\\Ker(\\underline{M} \\to \\underline{N})$.", "\\item {\\it linearly adequate} if $F$ is isomorphic to the", "kernel of a map $\\underline{A^{\\oplus n}} \\to \\underline{A^{\\oplus m}}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3639, "type": "definition", "label": "adequate-definition-adequate", "categories": [ "adequate" ], "title": "adequate-definition-adequate", "contents": [ "A sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ on $(\\Sch/S)_\\tau$ is", "{\\it adequate} if there exists a $\\tau$-covering", "$\\{\\Spec(A_i) \\to S\\}_{i \\in I}$ such that $F_{\\mathcal{F}, A_i}$ is", "adequate for all $i \\in I$." ], "refs": [], "ref_ids": [] }, { "id": 3640, "type": "definition", "label": "adequate-definition-category-adequate-modules", "categories": [ "adequate" ], "title": "adequate-definition-category-adequate-modules", "contents": [ "Let $S$ be a scheme. The category of adequate $\\mathcal{O}$-modules on", "$(\\Sch/S)_\\tau$ is denoted {\\it $\\textit{Adeq}(\\mathcal{O})$} or", "{\\it $\\textit{Adeq}((\\Sch/S)_\\tau, \\mathcal{O})$}. If we want to think just", "about the abelian category of adequate modules without choosing a", "topology we simply write {\\it $\\textit{Adeq}(S)$}." ], "refs": [], "ref_ids": [] }, { "id": 3641, "type": "definition", "label": "adequate-definition-pure", "categories": [ "adequate" ], "title": "adequate-definition-pure", "contents": [ "Let $A$ be a ring.", "\\begin{enumerate}", "\\item An $A$-module $P$ is said to be {\\it pure projective}", "if for every universally exact sequence", "$0 \\to K \\to M \\to N \\to 0$ of $A$-module the sequence", "$0 \\to \\Hom_A(P, K) \\to \\Hom_A(P, M) \\to \\Hom_A(P, N) \\to 0$", "is exact.", "\\item An $A$-module $I$ is said to be {\\it pure injective}", "if for every universally exact sequence", "$0 \\to K \\to M \\to N \\to 0$ of $A$-module the sequence", "$0 \\to \\Hom_A(N, I) \\to \\Hom_A(M, I) \\to \\Hom_A(K, I) \\to 0$", "is exact.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3642, "type": "definition", "label": "adequate-definition-pure-resolution", "categories": [ "adequate" ], "title": "adequate-definition-pure-resolution", "contents": [ "Let $A$ be a ring. Let $M$ be an $A$-module.", "\\begin{enumerate}", "\\item A {\\it pure projective resolution} $P_\\bullet \\to M$", "is a universally exact sequence", "$$", "\\ldots \\to P_1 \\to P_0 \\to M \\to 0", "$$", "with each $P_i$ pure projective.", "\\item A {\\it pure injective resolution} $M \\to I^\\bullet$ is a universally", "exact sequence", "$$", "0 \\to M \\to I^0 \\to I^1 \\to \\ldots", "$$", "with each $I^i$ pure injective.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3643, "type": "definition", "label": "adequate-definition-pure-ext", "categories": [ "adequate" ], "title": "adequate-definition-pure-ext", "contents": [ "Let $A$ be a ring and let $M$, $N$ be $A$-modules.", "The $i$th {\\it pure extension module} $\\text{Pext}^i_A(M, N)$", "is the $i$th cohomology module of the complex", "$\\Hom_A(M, I^\\bullet)$ where $I^\\bullet$ is a pure injective", "resolution of $N$." ], "refs": [], "ref_ids": [] }, { "id": 3681, "type": "definition", "label": "spaces-topologies-definition-zariski-covering", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-definition-zariski-covering", "contents": [ "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.", "A {\\it Zariski covering of $X$} is a family of morphisms", "$\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$", "such that each $f_i$ is an open immersion", "and such that", "$$", "|X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|),", "$$", "i.e., the morphisms are jointly surjective." ], "refs": [], "ref_ids": [] }, { "id": 3682, "type": "definition", "label": "spaces-topologies-definition-etale-covering", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-definition-etale-covering", "contents": [ "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.", "An {\\it \\'etale covering of $X$} is a family of morphisms", "$\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$", "such that each $f_i$ is \\'etale", "and such that", "$$", "|X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|),", "$$", "i.e., the morphisms are jointly surjective." ], "refs": [], "ref_ids": [] }, { "id": 3683, "type": "definition", "label": "spaces-topologies-definition-big-etale-site", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-definition-big-etale-site", "contents": [ "Let $S$ be a scheme. A big \\'etale site {\\it $(\\textit{Spaces}/S)_\\etale$}", "is any site constructed as follows:", "\\begin{enumerate}", "\\item Choose a big \\'etale site $(\\Sch/S)_\\etale$ as in", "Topologies, Section \\ref{topologies-section-etale}.", "\\item As underlying category take the category $\\textit{Spaces}/S$", "of algebraic spaces over $S$ (see discussion in", "Section \\ref{section-procedure} why this is a set).", "\\item Choose any set of coverings as in", "Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the", "category $\\textit{Spaces}/S$ and the class of \\'etale coverings", "of Definition \\ref{definition-etale-covering}.", "\\end{enumerate}" ], "refs": [ "sets-lemma-coverings-site", "spaces-topologies-definition-etale-covering" ], "ref_ids": [ 8800, 3682 ] }, { "id": 3684, "type": "definition", "label": "spaces-topologies-definition-big-small-etale", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-definition-big-small-etale", "contents": [ "Let $S$ be a scheme. Let $(\\textit{Spaces}/S)_\\etale$ be as in", "Definition \\ref{definition-big-etale-site}.", "Let $X$ be an algebraic space over $S$, i.e., an object of", "$(\\textit{Spaces}/S)_\\etale$. Then the big \\'etale site", "{\\it $(\\textit{Spaces}/X)_\\etale$} of $X$", "is the localization of the site $(\\textit{Spaces}/S)_\\etale$", "at $X$ introduced in Sites, Section \\ref{sites-section-localize}." ], "refs": [ "spaces-topologies-definition-big-etale-site" ], "ref_ids": [ 3683 ] }, { "id": 3685, "type": "definition", "label": "spaces-topologies-definition-restriction-small-etale", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-definition-restriction-small-etale", "contents": [ "In the situation of Lemma \\ref{lemma-at-the-bottom-etale}", "the functor $i_X^{-1} = \\pi_{X, *}$ is often", "called the {\\it restriction to the small \\'etale site}, and for a sheaf", "$\\mathcal{F}$ on the big \\'etale site we often denote", "$\\mathcal{F}|_{X_\\etale}$ this restriction." ], "refs": [ "spaces-topologies-lemma-at-the-bottom-etale" ], "ref_ids": [ 3655 ] }, { "id": 3686, "type": "definition", "label": "spaces-topologies-definition-smooth-covering", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-definition-smooth-covering", "contents": [ "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.", "A {\\it smooth covering of $X$} is a family of morphisms", "$\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$", "such that each $f_i$ is smooth", "and such that", "$$", "|X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|),", "$$", "i.e., the morphisms are jointly surjective." ], "refs": [], "ref_ids": [] }, { "id": 3687, "type": "definition", "label": "spaces-topologies-definition-syntomic-covering", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-definition-syntomic-covering", "contents": [ "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.", "A {\\it syntomic covering of $X$} is a family of morphisms", "$\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$", "such that each $f_i$ is syntomic", "and such that", "$$", "|X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|),", "$$", "i.e., the morphisms are jointly surjective." ], "refs": [], "ref_ids": [] }, { "id": 3688, "type": "definition", "label": "spaces-topologies-definition-fppf-covering", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-definition-fppf-covering", "contents": [ "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.", "An {\\it fppf covering of $X$} is a family of morphisms", "$\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$", "such that each $f_i$ is flat and locally of finite presentation", "and such that", "$$", "|X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|),", "$$", "i.e., the morphisms are jointly surjective." ], "refs": [], "ref_ids": [] }, { "id": 3689, "type": "definition", "label": "spaces-topologies-definition-big-fppf-site", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-definition-big-fppf-site", "contents": [ "Let $S$ be a scheme. A big fppf site {\\it $(\\textit{Spaces}/S)_{fppf}$}", "is any site constructed as follows:", "\\begin{enumerate}", "\\item Choose a big fppf site $(\\Sch/S)_{fppf}$ as in", "Topologies, Section \\ref{topologies-section-fppf}.", "\\item As underlying category take the category $\\textit{Spaces}/S$", "of algebraic spaces over $S$ (see discussion in", "Section \\ref{section-procedure} why this is a set).", "\\item Choose any set of coverings as in", "Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the", "category $\\textit{Spaces}/S$ and the class of fppf coverings", "of Definition \\ref{definition-fppf-covering}.", "\\end{enumerate}" ], "refs": [ "sets-lemma-coverings-site", "spaces-topologies-definition-fppf-covering" ], "ref_ids": [ 8800, 3688 ] }, { "id": 3690, "type": "definition", "label": "spaces-topologies-definition-big-small-fppf", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-definition-big-small-fppf", "contents": [ "Let $S$ be a scheme. Let $(\\textit{Spaces}/S)_{fppf}$ be as in", "Definition \\ref{definition-big-fppf-site}.", "Let $X$ be an algebraic space over $S$, i.e., an object of", "$(\\textit{Spaces}/S)_{fppf}$. Then the big fppf site", "{\\it $(\\textit{Spaces}/X)_{fppf}$} of $X$", "is the localization of the site $(\\textit{Spaces}/S)_{fppf}$", "at $X$ introduced in Sites, Section \\ref{sites-section-localize}." ], "refs": [ "spaces-topologies-definition-big-fppf-site" ], "ref_ids": [ 3689 ] }, { "id": 3691, "type": "definition", "label": "spaces-topologies-definition-ph-covering", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-definition-ph-covering", "contents": [ "Let $S$ be a scheme and let $X$ be an algebraic space over $S$.", "A {\\it ph covering of $X$} is a family", "of morphisms $\\{X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$", "such that $f_i$ is locally of finite type and such that for every", "$U \\to X$ with $U$ affine there exists a standard ph covering", "$\\{U_j \\to U\\}_{j = 1, \\ldots, m}$ refining the family", "$\\{X_i \\times_X U \\to U\\}_{i \\in I}$." ], "refs": [], "ref_ids": [] }, { "id": 3692, "type": "definition", "label": "spaces-topologies-definition-big-ph-site", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-definition-big-ph-site", "contents": [ "Let $S$ be a scheme. A big ph site {\\it $(\\textit{Spaces}/S)_{ph}$}", "is any site constructed as follows:", "\\begin{enumerate}", "\\item Choose a big ph site $(\\Sch/S)_{ph}$ as in", "Topologies, Section \\ref{topologies-section-ph}.", "\\item As underlying category take the category $\\textit{Spaces}/S$", "of algebraic spaces over $S$ (see discussion in", "Section \\ref{section-procedure} why this is a set).", "\\item Choose any set of coverings as in", "Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the", "category $\\textit{Spaces}/S$ and the class of ph coverings", "of Definition \\ref{definition-ph-covering}.", "\\end{enumerate}" ], "refs": [ "sets-lemma-coverings-site", "spaces-topologies-definition-ph-covering" ], "ref_ids": [ 8800, 3691 ] }, { "id": 3693, "type": "definition", "label": "spaces-topologies-definition-big-small-ph", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-definition-big-small-ph", "contents": [ "Let $S$ be a scheme. Let $(\\textit{Spaces}/S)_{ph}$ be as in", "Definition \\ref{definition-big-ph-site}.", "Let $X$ be an algebraic space over $S$, i.e., an object of", "$(\\textit{Spaces}/S)_{ph}$. Then the big ph site", "{\\it $(\\textit{Spaces}/X)_{ph}$} of $X$", "is the localization of the site $(\\textit{Spaces}/S)_{ph}$", "at $X$ introduced in Sites, Section \\ref{sites-section-localize}." ], "refs": [ "spaces-topologies-definition-big-ph-site" ], "ref_ids": [ 3692 ] }, { "id": 3694, "type": "definition", "label": "spaces-topologies-definition-fpqc-covering", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-definition-fpqc-covering", "contents": [ "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.", "An {\\it fpqc covering of $X$} is a family of morphisms", "$\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces", "such that each $f_i$ is flat and such that for every affine scheme", "$Z$ and morphism $h : Z \\to X$ there exists a standard fpqc covering", "$\\{g_j : Z_j \\to Z\\}_{j = 1, \\ldots, m}$ which refines the family", "$\\{X_i \\times_X Z \\to Z\\}_{i \\in I}$." ], "refs": [], "ref_ids": [] }, { "id": 3830, "type": "definition", "label": "proetale-definition-w-local", "categories": [ "proetale" ], "title": "proetale-definition-w-local", "contents": [ "A spectral space $X$ is {\\it w-local} if the set of closed points $X_0$", "is closed and every point of $X$ specializes to a unique closed point.", "A continuous map $f : X \\to Y$ of w-local spaces is {\\it w-local}", "if it is spectral and maps any closed point of $X$ to a closed point of $Y$." ], "refs": [], "ref_ids": [] }, { "id": 3831, "type": "definition", "label": "proetale-definition-local-isomorphism", "categories": [ "proetale" ], "title": "proetale-definition-local-isomorphism", "contents": [ "Let $\\varphi : A \\to B$ be a ring map.", "\\begin{enumerate}", "\\item We say $A \\to B$ is a {\\it local isomorphism} if for every prime", "$\\mathfrak q \\subset B$ there exists a $g \\in B$, $g \\not \\in \\mathfrak q$", "such that $A \\to B_g$ induces an open immersion $\\Spec(B_g) \\to \\Spec(A)$.", "\\item We say $A \\to B$ {\\it identifies local rings} if for every prime", "$\\mathfrak q \\subset B$ the canonical map", "$A_{\\varphi^{-1}(\\mathfrak q)} \\to B_\\mathfrak q$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3832, "type": "definition", "label": "proetale-definition-ind-zariski", "categories": [ "proetale" ], "title": "proetale-definition-ind-zariski", "contents": [ "A ring map $A \\to B$ is said to be {\\it ind-Zariski} if $B$ can be written", "as a filtered colimit $B = \\colim B_i$ with each $A \\to B_i$ a local", "isomorphism." ], "refs": [], "ref_ids": [] }, { "id": 3833, "type": "definition", "label": "proetale-definition-ind-etale", "categories": [ "proetale" ], "title": "proetale-definition-ind-etale", "contents": [ "A ring map $A \\to B$ is said to be {\\it ind-\\'etale} if $B$ can be written", "as a filtered colimit of \\'etale $A$-algebras." ], "refs": [], "ref_ids": [] }, { "id": 3834, "type": "definition", "label": "proetale-definition-w-contractible", "categories": [ "proetale" ], "title": "proetale-definition-w-contractible", "contents": [ "Let $A$ be a ring. We say $A$ is {\\it w-contractible} if every", "faithfully flat weakly \\'etale ring map $A \\to B$ has a section." ], "refs": [], "ref_ids": [] }, { "id": 3835, "type": "definition", "label": "proetale-definition-fpqc-covering", "categories": [ "proetale" ], "title": "proetale-definition-fpqc-covering", "contents": [ "Let $T$ be a scheme. A {\\it pro-\\'etale covering of $T$} is a family", "of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes", "such that each $f_i$ is weakly-\\'etale and such that for every affine open", "$U \\subset T$ there exists $n \\geq 0$, a map", "$a : \\{1, \\ldots, n\\} \\to I$ and affine opens", "$V_j \\subset T_{a(j)}$, $j = 1, \\ldots, n$", "with $\\bigcup_{j = 1}^n f_{a(j)}(V_j) = U$." ], "refs": [], "ref_ids": [] }, { "id": 3836, "type": "definition", "label": "proetale-definition-standard-proetale", "categories": [ "proetale" ], "title": "proetale-definition-standard-proetale", "contents": [ "Let $T$ be an affine scheme. A {\\it standard pro-\\'etale covering}", "of $T$ is a family $\\{f_i : T_i \\to T\\}_{i = 1, \\ldots, n}$", "where each $T_j$ is affine, each $f_i$ is weakly \\'etale, and", "$T = \\bigcup f_i(T_i)$." ], "refs": [], "ref_ids": [] }, { "id": 3837, "type": "definition", "label": "proetale-definition-big-proetale-site", "categories": [ "proetale" ], "title": "proetale-definition-big-proetale-site", "contents": [ "A {\\it big pro-\\'etale site} is any site $\\Sch_\\proetale$ as in", "Sites, Definition \\ref{sites-definition-site} constructed as follows:", "\\begin{enumerate}", "\\item Choose any set of schemes $S_0$, and any set of pro-\\'etale coverings", "$\\text{Cov}_0$ among these schemes.", "\\item Change the function $Bound$ of", "Sets, Equation (\\ref{sets-equation-bound}) into", "$$", "Bound(\\kappa) = \\max\\{\\kappa^{2^{2^{2^\\kappa}}}, \\kappa^{\\aleph_0}, \\kappa^+\\}.", "$$", "\\item As underlying category take any category $\\Sch_\\alpha$", "constructed as in Sets, Lemma \\ref{sets-lemma-construct-category}", "starting with the set $S_0$ and the function $Bound$.", "\\item Choose any set of coverings as in", "Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the", "category $\\Sch_\\alpha$ and the class of pro-\\'etale coverings,", "and the set $\\text{Cov}_0$ chosen above.", "\\end{enumerate}" ], "refs": [ "sites-definition-site", "sets-lemma-construct-category", "sets-lemma-coverings-site" ], "ref_ids": [ 8652, 8789, 8800 ] }, { "id": 3838, "type": "definition", "label": "proetale-definition-big-small-proetale", "categories": [ "proetale" ], "title": "proetale-definition-big-small-proetale", "contents": [ "Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale", "site containing $S$.", "\\begin{enumerate}", "\\item The {\\it big pro-\\'etale site of $S$}, denoted", "$(\\Sch/S)_\\proetale$, is the site $\\Sch_\\proetale/S$", "introduced in Sites, Section \\ref{sites-section-localize}.", "\\item The {\\it small pro-\\'etale site of $S$}, which we denote", "$S_\\proetale$, is the full subcategory of $(\\Sch/S)_\\proetale$", "whose objects are those $U/S$ such that $U \\to S$ is weakly \\'etale.", "A covering of $S_\\proetale$ is any covering $\\{U_i \\to U\\}$ of", "$(\\Sch/S)_\\proetale$ with $U \\in \\Ob(S_\\proetale)$.", "\\item The {\\it big affine pro-\\'etale site of $S$}, denoted", "$(\\textit{Aff}/S)_\\proetale$, is the full subcategory of", "$(\\Sch/S)_\\proetale$ whose objects are affine $U/S$.", "A covering of $(\\textit{Aff}/S)_\\proetale$ is any covering", "$\\{U_i \\to U\\}$ of $(\\Sch/S)_\\proetale$ which is a", "standard pro-\\'etale covering.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3839, "type": "definition", "label": "proetale-definition-restriction-small-proetale", "categories": [ "proetale" ], "title": "proetale-definition-restriction-small-proetale", "contents": [ "In the situation of", "Lemma \\ref{lemma-at-the-bottom}", "the functor $i_S^{-1} = \\pi_{S, *}$ is often", "called the {\\it restriction to the small pro-\\'etale site}, and for a sheaf", "$\\mathcal{F}$ on the big pro-\\'etale site we denote", "$\\mathcal{F}|_{S_\\proetale}$ this restriction." ], "refs": [ "proetale-lemma-at-the-bottom" ], "ref_ids": [ 3751 ] }, { "id": 3840, "type": "definition", "label": "proetale-definition-extension-zero", "categories": [ "proetale" ], "title": "proetale-definition-extension-zero", "contents": [ "Let $j : U \\to X$ be a weakly \\'etale morphism of schemes.", "\\begin{enumerate}", "\\item The restriction functor", "$j^{-1} : \\Sh(X_\\proetale) \\to \\Sh(U_\\proetale)$", "has a left adjoint", "$j_!^{Sh} : \\Sh(X_\\proetale) \\to \\Sh(U_\\proetale)$.", "\\item The restriction functor", "$j^{-1} : \\textit{Ab}(X_\\proetale) \\to \\textit{Ab}(U_\\proetale)$", "has a left adjoint which is denoted", "$j_! : \\textit{Ab}(U_\\proetale) \\to \\textit{Ab}(X_\\proetale)$", "and called {\\it extension by zero}.", "\\item Let $\\Lambda$ be a ring. The functor", "$j^{-1} : \\textit{Mod}(X_\\proetale, \\Lambda) \\to", "\\textit{Mod}(U_\\proetale, \\Lambda)$", "has a left adjoint", "$j_! : \\textit{Mod}(U_\\proetale, \\Lambda) \\to", "\\textit{Mod}(X_\\proetale, \\Lambda)$", "and called {\\it extension by zero}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3841, "type": "definition", "label": "proetale-definition-constructible", "categories": [ "proetale" ], "title": "proetale-definition-constructible", "contents": [ "Let $X$ be a scheme.", "Let $\\Lambda$ be a Noetherian ring. A sheaf of $\\Lambda$-modules", "on $X_\\proetale$ is {\\it constructible} if for every affine open", "$U \\subset X$ there exists a finite decomposition", "of $U$ into constructible locally closed subschemes", "$U = \\coprod_i U_i$ such that", "$\\mathcal{F}|_{U_i}$ is of finite type and locally constant for all $i$." ], "refs": [], "ref_ids": [] }, { "id": 3842, "type": "definition", "label": "proetale-definition-adic", "categories": [ "proetale" ], "title": "proetale-definition-adic", "contents": [ "Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal.", "Let $X$ be a scheme. Let $\\mathcal{F}$ be a sheaf of $\\Lambda$-modules", "on $X_\\proetale$.", "\\begin{enumerate}", "\\item We say $\\mathcal{F}$ is a {\\it constructible $\\Lambda$-sheaf}", "if $\\mathcal{F} = \\lim \\mathcal{F}/I^n\\mathcal{F}$ and each", "$\\mathcal{F}/I^n\\mathcal{F}$ is a constructible sheaf of $\\Lambda/I^n$-modules.", "\\item If $\\mathcal{F}$ is a constructible $\\Lambda$-sheaf, then we say", "$\\mathcal{F}$ is {\\it lisse} if each $\\mathcal{F}/I^n\\mathcal{F}$ is", "locally constant.", "\\item We say $\\mathcal{F}$ is {\\it adic lisse}\\footnote{This may", "be nonstandard notation.} if there exists a", "$I$-adically complete $\\Lambda$-module $M$ with $M/IM$ finite", "such that $\\mathcal{F}$ is locally isomorphic to", "$$", "\\underline{M}^\\wedge = \\lim \\underline{M/I^nM}.", "$$", "\\item We say $\\mathcal{F}$ is", "{\\it adic constructible}\\footnote{This may be nonstandard notation.}", "if for every affine open $U \\subset X$", "there exists a decomposition $U = \\coprod U_i$ into", "constructible locally closed subschemes such that $\\mathcal{F}|_{U_i}$", "is adic lisse.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3843, "type": "definition", "label": "proetale-definition-Dbc", "categories": [ "proetale" ], "title": "proetale-definition-Dbc", "contents": [ "Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal.", "Let $X$ be a scheme. An object $K$ of $D(X_\\proetale, \\Lambda)$ is called", "{\\it constructible} if", "\\begin{enumerate}", "\\item $K$ is derived complete with respect to $I$,", "\\item $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$", "has constructible cohomology sheaves and locally has finite tor dimension.", "\\end{enumerate}", "We denote $D_{cons}(X, \\Lambda)$ the full subcategory of constructible", "$K$ in $D(X_\\proetale, \\Lambda)$." ], "refs": [], "ref_ids": [] }, { "id": 3844, "type": "definition", "label": "proetale-definition-adic-constructible", "categories": [ "proetale" ], "title": "proetale-definition-adic-constructible", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring and let", "$I \\subset \\Lambda$ be an ideal. Let $K \\in D(X_\\proetale, \\Lambda)$.", "\\begin{enumerate}", "\\item We say $K$ is {\\it adic lisse}\\footnote{This may be", "nonstandard notation} if there exists a finite complex of finite", "projective $\\Lambda^\\wedge$-modules $M^\\bullet$ such that", "$K$ is locally isomorphic to", "$$", "\\underline{M^a}^\\wedge \\to \\ldots \\to \\underline{M^b}^\\wedge", "$$", "\\item We say $K$ is {\\it adic constructible}\\footnote{This may be", "nonstandard notation.} if for every affine open $U \\subset X$", "there exists a decomposition $U = \\coprod U_i$ into", "constructible locally closed subschemes such that $K|_{U_i}$", "is adic lisse.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3974, "type": "definition", "label": "formal-spaces-definition-toplogy-tensor-product", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-toplogy-tensor-product", "contents": [ "Let $R$ be a topological ring. Let $M$ and $N$ be linearly", "topologized $R$-modules. The {\\it tensor product} of $M$ and $N$", "is the (usual) tensor product $M \\otimes_R N$ endowed", "with the linear topology defined by declaring", "$$", "\\Im(M_\\mu \\otimes_R N + M \\otimes_R N_\\nu \\longrightarrow M \\otimes_R N)", "$$", "to be a fundamental system of open submodules, where", "$M_\\mu \\subset M$ and $N_\\nu \\subset N$ run through fundamental", "systems of open submodules in $M$ and $N$.", "The {\\it completed tensor product}", "$$", "M \\widehat{\\otimes}_R N =", "\\lim M \\otimes_R N/(M_\\mu \\otimes_R N + M \\otimes_R N_\\nu) =", "\\lim M/M_\\mu \\otimes_R N/N_\\nu", "$$", "is the completion of the tensor product." ], "refs": [], "ref_ids": [] }, { "id": 3975, "type": "definition", "label": "formal-spaces-definition-weakly-admissible", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-weakly-admissible", "contents": [ "Let $A$ be a linearly topologized ring.", "\\begin{enumerate}", "\\item An element $f \\in A$ is called {\\it topologically nilpotent}", "if $f^n \\to 0$ as $n \\to \\infty$.", "\\item A {\\it weak ideal of definition} for $A$ is an open ideal", "$I \\subset A$ consisting entirely of topologically nilpotent elements.", "\\item We say $A$ is {\\it weakly pre-admissible} if $A$ has a weak", "ideal of definition.", "\\item We say $A$ is {\\it weakly admissible} if $A$ is weakly pre-admissible", "and complete\\footnote{By our conventions this includes separated.}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3976, "type": "definition", "label": "formal-spaces-definition-taut", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-taut", "contents": [ "Let $\\varphi : A \\to B$ be a continuous map of linearly topologized rings.", "We say $\\varphi$ is {\\it taut}\\footnote{This is nonstandard notation.", "The definition generalizes to modules, by saying a linearly topologized", "$A$-module $M$ is $A$-taut if for every open ideal $I \\subset A$ the closure", "of $IM$ in $M$ is open and these closures form a fundamental system of", "neighbourhoods of $0$ in $M$.}", "if for every open ideal $I \\subset A$ the closure of the ideal $\\varphi(I)B$", "is open and these closures form a fundamental system of open ideals." ], "refs": [], "ref_ids": [] }, { "id": 3977, "type": "definition", "label": "formal-spaces-definition-affine-formal-algebraic-space", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-affine-formal-algebraic-space", "contents": [ "Let $S$ be a scheme. We say a sheaf $X$ on $(\\Sch/S)_{fppf}$ is an", "{\\it affine formal algebraic space} if there exist", "\\begin{enumerate}", "\\item a directed set $\\Lambda$,", "\\item a system $(X_\\lambda, f_{\\lambda \\mu})$ over $\\Lambda$", "in $(\\Sch/S)_{fppf}$ where", "\\begin{enumerate}", "\\item each $X_\\lambda$ is affine,", "\\item each $f_{\\lambda \\mu} : X_\\lambda \\to X_\\mu$ is a thickening,", "\\end{enumerate}", "\\end{enumerate}", "such that", "$$", "X \\cong \\colim_{\\lambda \\in \\Lambda} X_\\lambda", "$$", "as fppf sheaves and $X$ satisfies a set theoretic condition", "(see Remark \\ref{remark-set-theoretic}). A", "{\\it morphism of affine formal algebraic spaces}", "over $S$ is a map of sheaves." ], "refs": [ "formal-spaces-remark-set-theoretic" ], "ref_ids": [ 4001 ] }, { "id": 3978, "type": "definition", "label": "formal-spaces-definition-types-affine-formal-algebraic-space", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-types-affine-formal-algebraic-space", "contents": [ "Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$.", "We say $X$ is {\\it McQuillan} if $X$ satisfies the equivalent conditions", "of Lemma \\ref{lemma-mcquillan-affine-formal-algebraic-space}. Let $A$", "be the weakly admissible topological ring associated to $X$. We say", "\\begin{enumerate}", "\\item $X$ is {\\it classical} if $X$ is McQuillan and $A$ is admissible,", "\\item $X$ is {\\it adic} if $X$ is McQuillan and $A$ is adic,", "\\item $X$ is {\\it adic*} if $X$ is McQuillan, $A$ is adic, and $A$", "has a finitely generated ideal of definition, and", "\\item $X$ is {\\it Noetherian} if $X$ is McQuillan and $A$ is", "both Noetherian and adic.", "\\end{enumerate}" ], "refs": [ "formal-spaces-lemma-mcquillan-affine-formal-algebraic-space" ], "ref_ids": [ 3870 ] }, { "id": 3979, "type": "definition", "label": "formal-spaces-definition-affine-formal-spectrum", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-affine-formal-spectrum", "contents": [ "Let $S$ be a scheme. Let $A$ be a weakly admissible topological ring over", "$S$, see Definition \\ref{definition-weakly-admissible}\\footnote{See", "More on Algebra, Definition", "\\ref{more-algebra-definition-topological-ring}", "for the classical case and see Remark \\ref{remark-mcquillan}", "for a discussion of differences.}.", "The {\\it formal spectrum} of $A$ is the affine formal algebraic space", "$$", "\\text{Spf}(A) = \\colim \\Spec(A/I)", "$$", "where the colimit is over the set of weak ideals of definition of $A$", "and taken in the category $\\Sh((\\Sch/S)_{fppf})$." ], "refs": [ "formal-spaces-definition-weakly-admissible", "more-algebra-definition-topological-ring", "formal-spaces-remark-mcquillan" ], "ref_ids": [ 3975, 10610, 3997 ] }, { "id": 3980, "type": "definition", "label": "formal-spaces-definition-countable", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-countable", "contents": [ "Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$.", "We say $X$ is {\\it countably indexed} if the equivalent conditions of", "Lemma \\ref{lemma-countable-affine-formal-algebraic-space} are satisfied." ], "refs": [ "formal-spaces-lemma-countable-affine-formal-algebraic-space" ], "ref_ids": [ 3873 ] }, { "id": 3981, "type": "definition", "label": "formal-spaces-definition-formal-algebraic-space", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-formal-algebraic-space", "contents": [ "Let $S$ be a scheme. We say a sheaf $X$ on $(\\Sch/S)_{fppf}$ is a", "{\\it formal algebraic space} if there exist a family of maps", "$\\{X_i \\to X\\}_{i \\in I}$ of sheaves such that", "\\begin{enumerate}", "\\item $X_i$ is an affine formal algebraic space,", "\\item $X_i \\to X$ is representable by algebraic spaces and \\'etale,", "\\item $\\coprod X_i \\to X$ is surjective as a map of sheaves", "\\end{enumerate}", "and $X$ satisfies a set theoretic condition", "(see Remark \\ref{remark-set-theoretic}). A", "{\\it morphism of formal algebraic spaces}", "over $S$ is a map of sheaves." ], "refs": [ "formal-spaces-remark-set-theoretic" ], "ref_ids": [ 4001 ] }, { "id": 3982, "type": "definition", "label": "formal-spaces-definition-completion", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-completion", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $T \\subset |X|$ be a closed subset. The formal algebraic space", "of Lemma \\ref{lemma-completion-is-formal-algebraic-space}", "is called the {\\it completion of $X$ along $T$}." ], "refs": [ "formal-spaces-lemma-completion-is-formal-algebraic-space" ], "ref_ids": [ 3884 ] }, { "id": 3983, "type": "definition", "label": "formal-spaces-definition-separated", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-separated", "contents": [ "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.", "We say", "\\begin{enumerate}", "\\item $X$ is {\\it quasi-separated} if the equivalent conditions of", "Lemma \\ref{lemma-characterize-quasi-separated} are satisfied.", "\\item $X$ is {\\it separated} if the equivalent conditions of", "Lemma \\ref{lemma-characterize-separated} are satisfied.", "\\end{enumerate}" ], "refs": [ "formal-spaces-lemma-characterize-quasi-separated", "formal-spaces-lemma-characterize-separated" ], "ref_ids": [ 3894, 3895 ] }, { "id": 3984, "type": "definition", "label": "formal-spaces-definition-quasi-compact", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-quasi-compact", "contents": [ "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.", "We say $X$ is {\\it quasi-compact} if the equivalent conditions of", "Lemma \\ref{lemma-characterize-quasi-compact} are satisfied." ], "refs": [ "formal-spaces-lemma-characterize-quasi-compact" ], "ref_ids": [ 3898 ] }, { "id": 3985, "type": "definition", "label": "formal-spaces-definition-quasi-compact-morphism", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-quasi-compact-morphism", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of", "formal algebraic spaces over $S$.", "We say $f$ is {\\it quasi-compact} if the equivalent conditions of", "Lemma \\ref{lemma-characterize-quasi-compact-morphism} are satisfied." ], "refs": [ "formal-spaces-lemma-characterize-quasi-compact-morphism" ], "ref_ids": [ 3899 ] }, { "id": 3986, "type": "definition", "label": "formal-spaces-definition-types-formal-algebraic-spaces", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-types-formal-algebraic-spaces", "contents": [ "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.", "We say $X$ is {\\it locally countably indexed},", "{\\it locally adic*}, or {\\it locally Noetherian}", "if the equivalent conditions of Lemma \\ref{lemma-type-local}", "hold for the corresponding property." ], "refs": [ "formal-spaces-lemma-type-local" ], "ref_ids": [ 3918 ] }, { "id": 3987, "type": "definition", "label": "formal-spaces-definition-adic-homomorphism", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-adic-homomorphism", "contents": [ "Let $A$ and $B$ be pre-adic topological rings. A ring homomorphism", "$\\varphi : A \\to B$ is {\\it adic}\\footnote{This may be nonstandard terminology.}", "if there exists an ideal of definition $I \\subset A$ such that", "the topology on $B$ is the $I$-adic topology." ], "refs": [], "ref_ids": [] }, { "id": 3988, "type": "definition", "label": "formal-spaces-definition-adic-morphism", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-adic-morphism", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic", "spaces over $S$. Assume $X$ and $Y$ are locally adic*. We say $f$ is", "an {\\it adic morphism} if $f$ is representable by algebraic spaces.", "See discussion above." ], "refs": [], "ref_ids": [] }, { "id": 3989, "type": "definition", "label": "formal-spaces-definition-finite-type", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-finite-type", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of formal algebraic", "spaces over $S$.", "\\begin{enumerate}", "\\item We say $f$ is {\\it locally of finite type}", "if $f$ is representable by algebraic spaces and is locally", "of finite type in the sense of", "Bootstrap, Definition \\ref{bootstrap-definition-property-transformation}.", "\\item We say $f$ is of {\\it finite type} if $f$ is locally of finite type and", "quasi-compact (Definition \\ref{definition-quasi-compact-morphism}).", "\\end{enumerate}" ], "refs": [ "bootstrap-definition-property-transformation", "formal-spaces-definition-quasi-compact-morphism" ], "ref_ids": [ 2638, 3985 ] }, { "id": 3990, "type": "definition", "label": "formal-spaces-definition-surjective", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-surjective", "contents": [ "Let $S$ be a scheme. A morphism $f : X \\to Y$ of formal algebraic spaces", "over $S$ is said to be {\\it surjective} if it induces a surjective morphism", "$X_{red} \\to Y_{red}$ on underlying reduced algebraic spaces." ], "refs": [], "ref_ids": [] }, { "id": 3991, "type": "definition", "label": "formal-spaces-definition-monomorphism", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-monomorphism", "contents": [ "Let $S$ be a scheme.", "A morphism of formal algebraic spaces over $S$ is called a", "{\\it monomorphism} if it is an injective map of sheaves." ], "refs": [], "ref_ids": [] }, { "id": 3992, "type": "definition", "label": "formal-spaces-definition-closed-immersion", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-closed-immersion", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of formal algebraic", "spaces over $S$. We say $f$ is a {\\it closed immersion}", "if $f$ is representable by algebraic spaces and a closed immersion", "in the sense of", "Bootstrap, Definition \\ref{bootstrap-definition-property-transformation}." ], "refs": [ "bootstrap-definition-property-transformation" ], "ref_ids": [ 2638 ] }, { "id": 3993, "type": "definition", "label": "formal-spaces-definition-topologically-finite-type", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-topologically-finite-type", "contents": [ "Let $A \\to B$ be a continuous map of topological rings", "(More on Algebra, Definition \\ref{more-algebra-definition-topological-ring}).", "We say $B$ is {\\it topologically of finite type over} $A$ if", "there exists an $A$-algebra map $A[x_1, \\ldots, x_n] \\to B$ whose", "image is dense in $B$." ], "refs": [ "more-algebra-definition-topological-ring" ], "ref_ids": [ 10610 ] }, { "id": 3994, "type": "definition", "label": "formal-spaces-definition-separated-morphism", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-separated-morphism", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$.", "Let $\\Delta_{X/Y} : X \\to X \\times_Y X$ be the diagonal morphism.", "\\begin{enumerate}", "\\item We say $f$ is {\\it separated} if $\\Delta_{X/Y}$ is a closed immersion.", "\\item We say $f$ is {\\it quasi-separated} if $\\Delta_{X/Y}$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 3995, "type": "definition", "label": "formal-spaces-definition-proper", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-proper", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of formal algebraic", "spaces over $S$. We say $f$ is {\\it proper}", "if $f$ is representable by algebraic spaces and is proper in the sense of", "Bootstrap, Definition \\ref{bootstrap-definition-property-transformation}." ], "refs": [ "bootstrap-definition-property-transformation" ], "ref_ids": [ 2638 ] }, { "id": 3996, "type": "definition", "label": "formal-spaces-definition-etale-sites", "categories": [ "formal-spaces" ], "title": "formal-spaces-definition-etale-sites", "contents": [ "Let $S$ be a scheme. Let $X$ be a formal algebraic space with", "reduction $X_{red}$ (Lemma \\ref{lemma-reduction-formal-algebraic-space}).", "\\begin{enumerate}", "\\item The {\\it small \\'etale site} $X_\\etale$ of $X$ is", "the site $X_{red, \\etale}$ of Properties of Spaces, Definition", "\\ref{spaces-properties-definition-etale-site}.", "\\item The site $X_{spaces, \\etale}$ is the site", "$X_{red, spaces, \\etale}$ of Properties of Spaces, Definition", "\\ref{spaces-properties-definition-spaces-etale-site}.", "\\item The site $X_{affine, \\etale}$ is the site", "$X_{red, affine, \\etale}$ of Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-alternative}.", "\\end{enumerate}" ], "refs": [ "formal-spaces-lemma-reduction-formal-algebraic-space", "spaces-properties-definition-etale-site", "spaces-properties-definition-spaces-etale-site", "spaces-properties-lemma-alternative" ], "ref_ids": [ 3879, 11934, 11935, 11863 ] }, { "id": 4141, "type": "definition", "label": "pione-definition-G-set-continuous", "categories": [ "pione" ], "title": "pione-definition-G-set-continuous", "contents": [ "Let $G$ be a topological group.", "A {\\it $G$-set}, sometimes called a {\\it discrete $G$-set},", "is a set $X$ endowed with a left action $a : G \\times X \\to X$", "such that $a$ is continuous when $X$ is given the discrete topology and", "$G \\times X$ the product topology.", "A {\\it morphism of $G$-sets} $f : X \\to Y$ is simply any $G$-equivariant", "map from $X$ to $Y$.", "The category of $G$-sets is denoted {\\it $G\\textit{-Sets}$}." ], "refs": [], "ref_ids": [] }, { "id": 4142, "type": "definition", "label": "pione-definition-galois-category", "categories": [ "pione" ], "title": "pione-definition-galois-category", "contents": [ "\\begin{reference}", "Different from the definition in \\cite[Expos\\'e V, Definition 5.1]{SGA1}.", "Compare with \\cite[Definition 7.2.1]{BS}.", "\\end{reference}", "Let $\\mathcal{C}$ be a category and let $F : \\mathcal{C} \\to \\textit{Sets}$", "be a functor. The pair $(\\mathcal{C}, F)$ is a {\\it Galois category} if", "\\begin{enumerate}", "\\item $\\mathcal{C}$ has finite limits and finite colimits,", "\\item", "\\label{item-connected-components}", "every object of $\\mathcal{C}$ is a finite (possibly empty)", "coproduct of connected objects,", "\\item $F(X)$ is finite for all $X \\in \\Ob(\\mathcal{C})$, and", "\\item $F$ reflects isomorphisms and is exact.", "\\end{enumerate}", "Here we say $X \\in \\Ob(\\mathcal{C})$ is connected if", "it is not initial and for any monomorphism $Y \\to X$", "either $Y$ is initial or $Y \\to X$ is an isomorphism." ], "refs": [], "ref_ids": [] }, { "id": 4143, "type": "definition", "label": "pione-definition-fundamental-group", "categories": [ "pione" ], "title": "pione-definition-fundamental-group", "contents": [ "Let $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point", "of $X$. The {\\it fundamental group} of $X$ with", "{\\it base point} $\\overline{x}$ is the group", "$$", "\\pi_1(X, \\overline{x}) = \\text{Aut}(F_{\\overline{x}})", "$$", "of automorphisms of the fibre functor", "$F_{\\overline{x}} : \\textit{F\\'Et}_X \\to \\textit{Sets}$", "endowed with its canonical profinite topology from", "Lemma \\ref{lemma-aut-inverse-limit}." ], "refs": [ "pione-lemma-aut-inverse-limit" ], "ref_ids": [ 4025 ] }, { "id": 4177, "type": "definition", "label": "stacks-cohomology-definition-flat-base-change", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-definition-flat-base-change", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack and let $\\mathcal{F}$ in", "$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.", "We say $\\mathcal{F}$ has the {\\it flat base change property}\\footnote{This", "may be nonstandard notation.}", "if and only if $c_\\varphi$ is an isomorphism whenever $f$ is flat." ], "refs": [], "ref_ids": [] }, { "id": 4178, "type": "definition", "label": "stacks-cohomology-definition-parasitic", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-definition-parasitic", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "A presheaf of $\\mathcal{O}_\\mathcal{X}$-modules $\\mathcal{F}$ is", "{\\it parasitic} if we have $\\mathcal{F}(x) = 0$ for any object $x$", "of $\\mathcal{X}$ which lies over a scheme $U$ such that the corresponding", "morphism $x : U \\to \\mathcal{X}$ is flat." ], "refs": [], "ref_ids": [] }, { "id": 4179, "type": "definition", "label": "stacks-cohomology-definition-lisse-etale", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-definition-lisse-etale", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "\\begin{enumerate}", "\\item The {\\it lisse-\\'etale site} of $\\mathcal{X}$ is the full subcategory", "$\\mathcal{X}_{lisse,\\etale}$\\footnote{In the literature the", "site is denoted $\\text{Lis-\\'et}(\\mathcal{X})$ or", "$\\text{Lis-Et}(\\mathcal{X})$ and the associated topos is denoted", "$\\mathcal{X}_{\\text{lis-\\'e}t}$ or $\\mathcal{X}_{\\text{lis-et}}$.", "In the Stacks project our convention is to name the site and", "denote the corresponding topos by $\\Sh(\\mathcal{C})$.} of $\\mathcal{X}$", "whose objects are those $x \\in \\Ob(\\mathcal{X})$ lying over a scheme $U$", "such that $x : U \\to \\mathcal{X}$ is smooth. A covering of", "$\\mathcal{X}_{lisse,\\etale}$ is a family of morphisms", "$\\{x_i \\to x\\}_{i \\in I}$ of $\\mathcal{X}_{lisse,\\etale}$", "which forms a covering of $\\mathcal{X}_\\etale$.", "\\item The {\\it flat-fppf site} of $\\mathcal{X}$ is the full subcategory", "$\\mathcal{X}_{flat,fppf}$ of $\\mathcal{X}$", "whose objects are those $x \\in \\Ob(\\mathcal{X})$ lying over a scheme $U$", "such that $x : U \\to \\mathcal{X}$ is flat. A covering of", "$\\mathcal{X}_{flat,fppf}$ is a family of morphisms", "$\\{x_i \\to x\\}_{i \\in I}$ of $\\mathcal{X}_{flat,fppf}$", "which forms a covering of $\\mathcal{X}_{fppf}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 4411, "type": "definition", "label": "sites-cohomology-definition-torsor", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-definition-torsor", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{G}$ be a sheaf of (possibly non-commutative)", "groups on $\\mathcal{C}$.", "A {\\it pseudo torsor}, or more precisely a", "{\\it pseudo $\\mathcal{G}$-torsor}, is a sheaf", "of sets $\\mathcal{F}$ on $\\mathcal{C}$ endowed with an action", "$\\mathcal{G} \\times \\mathcal{F} \\to \\mathcal{F}$ such that", "\\begin{enumerate}", "\\item whenever $\\mathcal{F}(U)$ is nonempty the action", "$\\mathcal{G}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$", "is simply transitive.", "\\end{enumerate}", "A {\\it morphism of pseudo $\\mathcal{G}$-torsors}", "$\\mathcal{F} \\to \\mathcal{F}'$", "is simply a morphism of sheaves of sets compatible with the", "$\\mathcal{G}$-actions.", "A {\\it torsor}, or more precisely a", "{\\it $\\mathcal{G}$-torsor}, is a pseudo $\\mathcal{G}$-torsor such that", "in addition", "\\begin{enumerate}", "\\item[(2)] for every $U \\in \\Ob(\\mathcal{C})$", "there exists a covering $\\{U_i \\to U\\}_{i \\in I}$ of $U$", "such that $\\mathcal{F}(U_i)$ is nonempty for all $i \\in I$.", "\\end{enumerate}", "A {\\it morphism of $\\mathcal{G}$-torsors} is simply a morphism of", "pseudo $\\mathcal{G}$-torsors.", "The {\\it trivial $\\mathcal{G}$-torsor}", "is the sheaf $\\mathcal{G}$ endowed with the obvious left", "$\\mathcal{G}$-action." ], "refs": [], "ref_ids": [] }, { "id": 4412, "type": "definition", "label": "sites-cohomology-definition-cech-complex", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-definition-cech-complex", "contents": [ "Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$", "be a family of morphisms with fixed target such that all fibre products", "$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$.", "Let $\\mathcal{F}$ be an abelian presheaf on $\\mathcal{C}$.", "The complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$", "is the {\\it {\\v C}ech complex} associated to $\\mathcal{F}$ and the", "family $\\mathcal{U}$. Its cohomology groups", "$H^i(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}))$ are", "called the {\\it {\\v C}ech cohomology groups} of $\\mathcal{F}$ with respect", "to $\\mathcal{U}$. They are denoted $\\check H^i(\\mathcal{U}, \\mathcal{F})$." ], "refs": [], "ref_ids": [] }, { "id": 4413, "type": "definition", "label": "sites-cohomology-definition-limp", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-definition-limp", "contents": [ "Let $\\mathcal{C}$ be a site.", "We say an abelian sheaf $\\mathcal{F}$ is", "{\\it totally acyclic}\\footnote{Although this terminology is is used in", "\\cite[Vbis, Proposition 1.3.10]{SGA4} this is probably nonstandard notation.", "In \\cite[V, Definition 4.1]{SGA4} this property is dubbed ``flasque'', but", "we cannot use this because it would clash with our definition", "of flasque sheaves on topological spaces. Please email", "\\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}", "if you have a better suggestion.}", "if for every sheaf of sets $K$ we have $H^p(K, \\mathcal{F}) = 0$", "for all $p \\geq 1$." ], "refs": [], "ref_ids": [] }, { "id": 4414, "type": "definition", "label": "sites-cohomology-definition-K-flat", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-definition-K-flat", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "A complex $\\mathcal{K}^\\bullet$ of $\\mathcal{O}$-modules is", "called {\\it K-flat} if for every acyclic complex $\\mathcal{F}^\\bullet$", "of $\\mathcal{O}$-modules the complex", "$$", "\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{K}^\\bullet)", "$$", "is acyclic." ], "refs": [], "ref_ids": [] }, { "id": 4415, "type": "definition", "label": "sites-cohomology-definition-derived-tor", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-definition-derived-tor", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{F}^\\bullet$ be an object of $D(\\mathcal{O})$.", "The {\\it derived tensor product}", "$$", "- \\otimes_\\mathcal{O}^{\\mathbf{L}} \\mathcal{F}^\\bullet :", "D(\\mathcal{O})", "\\longrightarrow", "D(\\mathcal{O})", "$$", "is the exact functor of triangulated categories described above." ], "refs": [], "ref_ids": [] }, { "id": 4416, "type": "definition", "label": "sites-cohomology-definition-tor", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-definition-tor", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}$-modules.", "The {\\it Tor}'s of $\\mathcal{F}$ and $\\mathcal{G}$ are defined by", "the formula", "$$", "\\text{Tor}_p^\\mathcal{O}(\\mathcal{F}, \\mathcal{G}) =", "H^{-p}(\\mathcal{F} \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{G})", "$$", "with derived tensor product as defined above." ], "refs": [], "ref_ids": [] }, { "id": 4417, "type": "definition", "label": "sites-cohomology-definition-covering-LC", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-definition-covering-LC", "contents": [ "Let $\\{f_i : X_i \\to X\\}$ be a family of morphisms with fixed target", "in the category $\\textit{LC}$. We say this family is a", "{\\it qc covering}\\footnote{This is nonstandard notation.", "We chose it to remind the reader of fpqc coverings of schemes.}", "if for every $x \\in X$ there exist $i_1, \\ldots, i_n \\in I$ and", "quasi-compact subsets $E_j \\subset X_{i_j}$ such that", "$\\bigcup f_{i_j}(E_j)$ is a neighbourhood of $x$." ], "refs": [], "ref_ids": [] }, { "id": 4418, "type": "definition", "label": "sites-cohomology-definition-simplicial-module", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-definition-simplicial-module", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A}_\\bullet$ be a simplicial", "sheaf of rings on $\\mathcal{C}$. A", "{\\it simplicial $\\mathcal{A}_\\bullet$-module} $\\mathcal{F}_\\bullet$", "(sometimes called a", "{\\it simplicial sheaf of $\\mathcal{A}_\\bullet$-modules})", "is a sheaf of modules over the sheaf of rings on $\\Delta \\times \\mathcal{C}$", "associated to $\\mathcal{A}_\\bullet$." ], "refs": [], "ref_ids": [] }, { "id": 4419, "type": "definition", "label": "sites-cohomology-definition-strictly-perfect", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-definition-strictly-perfect", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}$-modules.", "We say $\\mathcal{E}^\\bullet$ is {\\it strictly perfect}", "if $\\mathcal{E}^i$ is zero for all but finitely many $i$ and", "$\\mathcal{E}^i$ is a direct summand of a finite free", "$\\mathcal{O}$-module for all $i$." ], "refs": [], "ref_ids": [] }, { "id": 4420, "type": "definition", "label": "sites-cohomology-definition-pseudo-coherent", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-definition-pseudo-coherent", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{E}^\\bullet$", "be a complex of $\\mathcal{O}$-modules. Let $m \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item We say $\\mathcal{E}^\\bullet$ is {\\it $m$-pseudo-coherent}", "if for every object $U$ of $\\mathcal{C}$ there exists a covering", "$\\{U_i \\to U\\}$ and for each $i$ a morphism of complexes", "$\\alpha_i : \\mathcal{E}_i^\\bullet \\to \\mathcal{E}^\\bullet|_{U_i}$", "where $\\mathcal{E}_i$ is a strictly perfect complex of", "$\\mathcal{O}_{U_i}$-modules and $H^j(\\alpha_i)$ is an isomorphism", "for $j > m$ and $H^m(\\alpha_i)$ is surjective.", "\\item We say $\\mathcal{E}^\\bullet$ is {\\it pseudo-coherent}", "if it is $m$-pseudo-coherent for all $m$.", "\\item We say an object $E$ of $D(\\mathcal{O})$ is", "{\\it $m$-pseudo-coherent} (resp.\\ {\\it pseudo-coherent})", "if and only if it can be represented by a $m$-pseudo-coherent", "(resp.\\ pseudo-coherent) complex of $\\mathcal{O}$-modules.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 4421, "type": "definition", "label": "sites-cohomology-definition-tor-amplitude", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-definition-tor-amplitude", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $E$ be an object of $D(\\mathcal{O})$.", "Let $a, b \\in \\mathbf{Z}$ with $a \\leq b$.", "\\begin{enumerate}", "\\item We say $E$ has {\\it tor-amplitude in $[a, b]$}", "if $H^i(E \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{F}) = 0$", "for all $\\mathcal{O}$-modules $\\mathcal{F}$ and all $i \\not \\in [a, b]$.", "\\item We say $E$ has {\\it finite tor dimension}", "if it has tor-amplitude in $[a, b]$ for some $a, b$.", "\\item We say $E$ {\\it locally has finite tor dimension} if for any", "object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$", "such that $E|_{U_i}$ has finite tor dimension for all $i$.", "\\end{enumerate}", "An $\\mathcal{O}$-module $\\mathcal{F}$ has {\\it tor dimension $\\leq d$}", "if $\\mathcal{F}[0]$ viewed as an object of $D(\\mathcal{O})$ has", "tor-amplitude in $[-d, 0]$." ], "refs": [], "ref_ids": [] }, { "id": 4422, "type": "definition", "label": "sites-cohomology-definition-perfect", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-definition-perfect", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}$-modules.", "We say $\\mathcal{E}^\\bullet$ is {\\it perfect} if for every object $U$ of", "$\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$ such that for each $i$", "there exists a morphism of complexes", "$\\mathcal{E}_i^\\bullet \\to \\mathcal{E}^\\bullet|_{U_i}$", "which is a quasi-isomorphism with $\\mathcal{E}_i^\\bullet$", "strictly perfect.", "An object $E$ of $D(\\mathcal{O})$ is {\\it perfect}", "if it can be represented by a perfect complex of $\\mathcal{O}$-modules." ], "refs": [], "ref_ids": [] }, { "id": 4524, "type": "definition", "label": "fields-definition-field", "categories": [ "fields" ], "title": "fields-definition-field", "contents": [ "A {\\it field} is a nonzero ring where every nonzero element is invertible.", "Given a field a {\\it subfield} is a subring that is itself a field." ], "refs": [], "ref_ids": [] }, { "id": 4525, "type": "definition", "label": "fields-definition-domain", "categories": [ "fields" ], "title": "fields-definition-domain", "contents": [ "A {\\it domain} or an {\\it integral domain} is a nonzero ring where $0$", "is the only zerodivisor." ], "refs": [], "ref_ids": [] }, { "id": 4526, "type": "definition", "label": "fields-definition-characteristic", "categories": [ "fields" ], "title": "fields-definition-characteristic", "contents": [ "The {\\it characteristic} of a field $F$ is $0$ if", "$\\mathbf{Z} \\subset F$, or is a prime $p$ if $p = 0$ in $F$.", "The {\\it prime subfield of $F$} is the smallest subfield of $F$", "which is either $\\mathbf{Q} \\subset F$ if the characteristic is zero, or", "$\\mathbf{F}_p \\subset F$ if the characteristic is $p > 0$." ], "refs": [], "ref_ids": [] }, { "id": 4527, "type": "definition", "label": "fields-definition-extension", "categories": [ "fields" ], "title": "fields-definition-extension", "contents": [ "If $F$ is a field contained in a field $E$, then $E$ is said", "to be a {\\it field extension} of $F$. We shall write $E/F$ to indicate", "that $E$ is an extension of $F$." ], "refs": [], "ref_ids": [] }, { "id": 4528, "type": "definition", "label": "fields-definition-tower", "categories": [ "fields" ], "title": "fields-definition-tower", "contents": [ "A {\\it tower} of fields $E_n/E_{n - 1}/\\ldots/E_0$ consists of a sequence of", "extensions of fields", "$E_n/E_{n - 1}$, $E_{n - 1}/E_{n - 2}$, $\\ldots$, $E_1/E_0$." ], "refs": [], "ref_ids": [] }, { "id": 4529, "type": "definition", "label": "fields-definition-generated-by", "categories": [ "fields" ], "title": "fields-definition-generated-by", "contents": [ "Let $k$ be a field. If $F/k$ is an extension of fields and", "$S \\subset F$, we write $k(S)$ for the smallest subfield of $F$", "containing $k$ and $S$. We will say that $S$ {\\it generates the", "field extension} $k(S)/k$. If $S = \\{\\alpha\\}$ is a singleton, then we", "write $k(\\alpha)$ instead of $k(\\{\\alpha\\})$. We say $F/k$ is a", "{\\it finitely generated field extension} if there exists a", "finite subset $S \\subset F$ with $F = k(S)$." ], "refs": [], "ref_ids": [] }, { "id": 4530, "type": "definition", "label": "fields-definition-degree", "categories": [ "fields" ], "title": "fields-definition-degree", "contents": [ "Let $F/E$ be an extension of fields. The dimension of $F$ considered as an", "$E$-vector space is called the {\\it degree} of the extension and is", "denoted $[F : E]$. If $[F : E] < \\infty$ then $F$ is said to be a", "{\\it finite} extension of $E$." ], "refs": [], "ref_ids": [] }, { "id": 4531, "type": "definition", "label": "fields-definition-number-field", "categories": [ "fields" ], "title": "fields-definition-number-field", "contents": [ "A field $K$ is said to be a {\\it number field} if it has characteristic", "$0$ and the extension $\\mathbf{Q} \\subset K$ is finite." ], "refs": [], "ref_ids": [] }, { "id": 4532, "type": "definition", "label": "fields-definition-algebraic", "categories": [ "fields" ], "title": "fields-definition-algebraic", "contents": [ "Consider a field extension $F/E$. An element $\\alpha \\in F$ is said to be", "{\\it algebraic} over $E$ if $\\alpha$ is the root of some nonzero polynomial", "with coefficients in $E$. If all elements of $F$ are algebraic then $F$ is", "said to be an {\\it algebraic extension} of $E$." ], "refs": [], "ref_ids": [] }, { "id": 4533, "type": "definition", "label": "fields-definition-minimal-polynomial", "categories": [ "fields" ], "title": "fields-definition-minimal-polynomial", "contents": [ "The polynomial $P$ above is called the {\\it minimal polynomial}", "of $\\alpha$ over $k$." ], "refs": [], "ref_ids": [] }, { "id": 4534, "type": "definition", "label": "fields-definition-algebraically-closed", "categories": [ "fields" ], "title": "fields-definition-algebraically-closed", "contents": [ "A field $F$ is said to be {\\it algebraically closed} if every algebraic", "extension $E/F$ is trivial, i.e., $E = F$." ], "refs": [], "ref_ids": [] }, { "id": 4535, "type": "definition", "label": "fields-definition-algebraic-closure", "categories": [ "fields" ], "title": "fields-definition-algebraic-closure", "contents": [ "Let $F$ be a field. We say $F$ is {\\it algebraically closed} if every", "algebraic extension $E/F$ is trivial, i.e., $E = F$. An {\\it algebraic closure}", "of $F$ is a field $\\overline{F}$ containing $F$ such that:", "\\begin{enumerate}", "\\item $\\overline{F}$ is algebraic over $F$.", "\\item $\\overline{F}$ is algebraically closed.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 4536, "type": "definition", "label": "fields-definition-relatively-prime", "categories": [ "fields" ], "title": "fields-definition-relatively-prime", "contents": [ "If $k$ is any field, we say that two polynomials in $k[x]$ are", "{\\it relatively prime} if they generate the unit ideal in $k[x]$." ], "refs": [], "ref_ids": [] }, { "id": 4537, "type": "definition", "label": "fields-definition-separable", "categories": [ "fields" ], "title": "fields-definition-separable", "contents": [ "Let $F$ be a field. Let $K/F$ be an extension of fields.", "\\begin{enumerate}", "\\item We say an irreducible polynomial $P$ over $F$ is {\\it separable}", "if it is relatively prime to its derivative.", "\\item Given $\\alpha \\in K$ algebraic over $F$ we say $\\alpha$ is", "{\\it separable} over $F$ if its minimal polynomial is separable over $F$.", "\\item If $K$ is an algebraic extension of $F$, we say $K$ is", "{\\it separable}\\footnote{For nonalgebraic extensions", "this definition does not make sense and is not the correct one. We refer", "the reader to Algebra, Sections \\ref{algebra-section-separability} and", "\\ref{algebra-section-separability-continued}.}", "over $F$ if every element of $K$ is separable over $F$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 4538, "type": "definition", "label": "fields-definition-separable-degree", "categories": [ "fields" ], "title": "fields-definition-separable-degree", "contents": [ "Let $F$ be a field. Let $P$ be an irreducible polynomial over $F$.", "The {\\it separable degree} of $P$ is the cardinality of the", "set of roots of $P$ in any algebraic closure of $F$ (see discussion", "above). Notation $\\deg_s(P)$." ], "refs": [], "ref_ids": [] }, { "id": 4539, "type": "definition", "label": "fields-definition-purely-inseparable", "categories": [ "fields" ], "title": "fields-definition-purely-inseparable", "contents": [ "Let $F$ be a field of characteristic $p > 0$. Let $K/F$ be an extension.", "\\begin{enumerate}", "\\item An element $\\alpha \\in K$ is {\\it purely inseparable} over $F$", "if there exists a power $q$ of $p$ such that $\\alpha^q \\in F$.", "\\item The extension $K/F$ is said to be {\\it purely inseparable}", "if and only if every element of $K$ is purely inseparable over $F$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 4540, "type": "definition", "label": "fields-definition-insep-degree", "categories": [ "fields" ], "title": "fields-definition-insep-degree", "contents": [ "Let $E/F$ be an algebraic field extension. Let $E_{sep}$ be the subextension", "found in Lemma \\ref{lemma-separable-first}.", "\\begin{enumerate}", "\\item The integer $[E_{sep} : F]$ is called the {\\it separable", "degree} of the extension. Notation $[E : F]_s$.", "\\item The integer $[E : E_{sep}]$ is called the {\\it inseparable", "degree}, or the {\\it degree of inseparability} of the extension.", "Notation $[E : F]_i$.", "\\end{enumerate}" ], "refs": [ "fields-lemma-separable-first" ], "ref_ids": [ 4482 ] }, { "id": 4541, "type": "definition", "label": "fields-definition-normal", "categories": [ "fields" ], "title": "fields-definition-normal", "contents": [ "Let $E/F$ be an algebraic field extension. We say $E$ is {\\it normal}", "over $F$ if for all $\\alpha \\in E$ the minimal polynomial $P$", "of $\\alpha$ over $F$ splits completely into linear factors over $E$." ], "refs": [], "ref_ids": [] }, { "id": 4542, "type": "definition", "label": "fields-definition-automorphisms", "categories": [ "fields" ], "title": "fields-definition-automorphisms", "contents": [ "Let $E/F$ be an extension of fields. Then $\\text{Aut}(E/F)$ or", "$\\text{Aut}_F(E)$ denotes the automorphism group of $E$ as an object", "of the category of $F$-extensions. Elements of $\\text{Aut}(E/F)$", "are called {\\it automorphisms of $E$ over $F$} or", "{\\it automorphisms of $E/F$}." ], "refs": [], "ref_ids": [] }, { "id": 4543, "type": "definition", "label": "fields-definition-splitting-field", "categories": [ "fields" ], "title": "fields-definition-splitting-field", "contents": [ "Let $F$ be a field. Let $P \\in F[x]$ be a nonconstant polynomial.", "The field extension $E/F$ constructed in Lemma \\ref{lemma-splitting-field}", "is called the {\\it splitting field of $P$ over $F$}." ], "refs": [ "fields-lemma-splitting-field" ], "ref_ids": [ 4493 ] }, { "id": 4544, "type": "definition", "label": "fields-definition-normal-closure", "categories": [ "fields" ], "title": "fields-definition-normal-closure", "contents": [ "Let $E/F$ be a finite extension of fields. The field extension $K/E$", "constructed in Lemma \\ref{lemma-normal-closure}", "is called the {\\it normal closure $E$ over $F$}." ], "refs": [ "fields-lemma-normal-closure" ], "ref_ids": [ 4494 ] }, { "id": 4545, "type": "definition", "label": "fields-definition-trace-norm", "categories": [ "fields" ], "title": "fields-definition-trace-norm", "contents": [ "Let $L/K$ be a finite extension of fields. For $\\alpha \\in L$ we define", "the {\\it trace}", "$\\text{Trace}_{L/K}(\\alpha) = \\text{Trace}_K(\\alpha : L \\to L)$", "and the {\\it norm}", "$\\text{Norm}_{L/K}(\\alpha) = \\det_K(\\alpha : L \\to L)$." ], "refs": [], "ref_ids": [] }, { "id": 4546, "type": "definition", "label": "fields-definition-trace-pairing", "categories": [ "fields" ], "title": "fields-definition-trace-pairing", "contents": [ "Let $L/K$ be a finite extension of fields. The {\\it trace pairing}", "for $L/K$ is the symmetric $K$-bilinear form", "$$", "Q_{L/K} : L \\times L \\longrightarrow K,\\quad", "(\\alpha, \\beta) \\longmapsto \\text{Trace}_{L/K}(\\alpha\\beta)", "$$" ], "refs": [], "ref_ids": [] }, { "id": 4547, "type": "definition", "label": "fields-definition-discriminant", "categories": [ "fields" ], "title": "fields-definition-discriminant", "contents": [ "Let $L/K$ be a finite extension of fields. The", "{\\it discriminant of $L/K$} is the discriminant of", "the trace pairing $Q_{L/K}$." ], "refs": [], "ref_ids": [] }, { "id": 4548, "type": "definition", "label": "fields-definition-galois", "categories": [ "fields" ], "title": "fields-definition-galois", "contents": [ "A field extension $E/F$ is called {\\it Galois} if it is algebraic,", "separable, and normal." ], "refs": [], "ref_ids": [] }, { "id": 4549, "type": "definition", "label": "fields-definition-galois-group", "categories": [ "fields" ], "title": "fields-definition-galois-group", "contents": [ "If $E/F$ is a Galois extension, then the group $\\text{Aut}(E/F)$ is", "called the {\\it Galois group} and it is denoted $\\text{Gal}(E/F)$." ], "refs": [], "ref_ids": [] }, { "id": 4550, "type": "definition", "label": "fields-definition-transcendence", "categories": [ "fields" ], "title": "fields-definition-transcendence", "contents": [ "Let $k \\subset K$ be a field extension.", "\\begin{enumerate}", "\\item A collection of elements $\\{x_i\\}_{i \\in I}$ of $K$ is called", "{\\it algebraically independent} over $k$ if the map", "$$", "k[X_i; i\\in I] \\longrightarrow K", "$$", "which maps $X_i$ to $x_i$ is injective.", "\\item The field of fractions of a polynomial ring", "$k[x_i; i \\in I]$ is denoted $k(x_i; i\\in I)$.", "\\item A {\\it purely transcendental extension} of $k$ is any", "field extension $k \\subset K$ isomorphic to the field of", "fractions of a polynomial ring over $k$.", "\\item A {\\it transcendence basis} of $K/k$ is a", "collection of elements $\\{x_i\\}_{i \\in I}$ which are", "algebraically independent over $k$ and such that", "the extension $k(x_i; i\\in I) \\subset K$ is algebraic.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 4551, "type": "definition", "label": "fields-definition-transcendence-degree", "categories": [ "fields" ], "title": "fields-definition-transcendence-degree", "contents": [ "Let $k \\subset K$ be a field extension.", "The {\\it transcendence degree} of $K$ over $k$ is", "the cardinality of a transcendence basis of $K$ over $k$.", "It is denoted $\\text{trdeg}_k(K)$." ], "refs": [], "ref_ids": [] }, { "id": 4552, "type": "definition", "label": "fields-definition-algebraically-closed-in", "categories": [ "fields" ], "title": "fields-definition-algebraically-closed-in", "contents": [ "Let $k \\subset K$ be a field extension.", "\\begin{enumerate}", "\\item The {\\it algebraic closure of $k$ in $K$} is the subfield", "$k'$ of $K$ consisting of elements of $K$ which are algebraic over $k$.", "\\item We say $k$ is {\\it algebraically closed in $K$} if", "every element of $K$ which is algebraic over $k$ is", "contained in $k$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 4553, "type": "definition", "label": "fields-definition-compositum", "categories": [ "fields" ], "title": "fields-definition-compositum", "contents": [ "Consider a diagram", "\\begin{equation}", "\\label{equation-inside-omega}", "\\vcenter{", "\\xymatrix{", "L \\ar[r] & \\Omega \\\\", "k \\ar[r] \\ar[u] & K \\ar[u]", "}", "}", "\\end{equation}", "of field extensions. The {\\it compositum of $K$ and $L$ in $\\Omega$}", "written $KL$ is the smallest subfield of $\\Omega$ containing both", "$L$ and $K$." ], "refs": [], "ref_ids": [] }, { "id": 4554, "type": "definition", "label": "fields-definition-linearly-disjoint", "categories": [ "fields" ], "title": "fields-definition-linearly-disjoint", "contents": [ "Consider a diagram of fields as in (\\ref{equation-inside-omega}).", "We say that $K$ and $L$ are {\\it linearly disjoint over $k$ in $\\Omega$}", "if the map", "$$", "K \\otimes_k L \\longrightarrow KL,\\quad", "\\sum x_i \\otimes y_i \\longmapsto \\sum x_i y_i", "$$", "is injective." ], "refs": [], "ref_ids": [] }, { "id": 4555, "type": "definition", "label": "fields-definition-separable-algebraic", "categories": [ "fields" ], "title": "fields-definition-separable-algebraic", "contents": [ "Algebraic field extensions.", "\\begin{enumerate}", "\\item A field extension $k \\subset K$ is called {\\it algebraic}", "if every element of $K$ is algebraic over $k$.", "\\item An algebraic extension $k \\subset k'$ is called {\\it separable}", "if every $\\alpha \\in k'$ is separable over $k$.", "\\item An algebraic", "extension $k \\subset k'$ is called {\\it purely inseparable} if", "the characteristic of $k$ is $p > 0$ and for every element", "$\\alpha \\in k'$ there exists a power $q$ of $p$ such that", "$\\alpha^q \\in k$.", "\\item An algebraic extension $k \\subset k'$ is called {\\it normal}", "if for every $\\alpha \\in k'$ the minimal polynomial $P(T) \\in k[T]$", "of $\\alpha$ over $k$ splits completely into linear factors over $k'$.", "\\item An algebraic extension $k \\subset k'$ is called {\\it Galois}", "if it is separable and normal.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 4660, "type": "definition", "label": "spaces-limits-definition-locally-finite-presentation", "categories": [ "spaces-limits" ], "title": "spaces-limits-definition-locally-finite-presentation", "contents": [ "Let $S$ be a scheme.", "\\begin{enumerate}", "\\item A functor $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$", "is said to be {\\it limit preserving} or {\\it locally of finite presentation} if", "for every affine scheme $T$ over $S$ which is a limit $T = \\lim T_i$", "of a directed inverse system of affine schemes $T_i$ over $S$, we have", "$$", "F(T) = \\colim F(T_i).", "$$", "We sometimes say that $F$ is {\\it locally of finite presentation over $S$}.", "\\item Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.", "A transformation of functors $a : F \\to G$", "is {\\it limit preserving} or {\\it locally of finite presentation}", "if for every scheme $T$ over $S$ and every $y \\in G(T)$ the functor", "$$", "F_y : (\\Sch/T)_{fppf}^{opp} \\longrightarrow \\textit{Sets}, \\quad", "T'/T \\longmapsto \\{x \\in F(T') \\mid a(x) = y|_{T'}\\}", "$$", "is locally of finite presentation over $T$\\footnote{The characterization (2) in", "Lemma \\ref{lemma-characterize-relative-limit-preserving}", "may be easier to parse.}. We sometimes say that", "$F$ is {\\it relatively limit preserving} over $G$.", "\\end{enumerate}" ], "refs": [ "spaces-limits-lemma-characterize-relative-limit-preserving" ], "ref_ids": [ 4556 ] }, { "id": 4661, "type": "definition", "label": "spaces-limits-definition-subsheaf-sections-annihilated-by-ideal", "categories": [ "spaces-limits" ], "title": "spaces-limits-definition-subsheaf-sections-annihilated-by-ideal", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent", "sheaf of ideals of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "The subsheaf $\\mathcal{F}' \\subset \\mathcal{F}$ defined in", "Lemma \\ref{lemma-sections-annihilated-by-ideal} above is called", "the {\\it subsheaf of sections annihilated by $\\mathcal{I}$}." ], "refs": [ "spaces-limits-lemma-sections-annihilated-by-ideal" ], "ref_ids": [ 4622 ] }, { "id": 4662, "type": "definition", "label": "spaces-limits-definition-subsheaf-sections-supported-on-closed", "categories": [ "spaces-limits" ], "title": "spaces-limits-definition-subsheaf-sections-supported-on-closed", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $T \\subset |X|$ be a closed subset whose complement", "corresponds to an open subspace $U \\subset X$", "with quasi-compact inclusion morphism $U \\to X$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "The quasi-coherent subsheaf $\\mathcal{F}' \\subset \\mathcal{F}$ defined in", "Lemma \\ref{lemma-sections-supported-on-closed-subset} above is called", "the {\\it subsheaf of sections supported on $T$}." ], "refs": [ "spaces-limits-lemma-sections-supported-on-closed-subset" ], "ref_ids": [ 4624 ] }, { "id": 4696, "type": "definition", "label": "stacks-geometry-definition-versal-ring-at-x", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-definition-versal-ring-at-x", "contents": [ "In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$", "be a morphism, where $k$ is a finite type field over $S$.", "A {\\it versal ring to $\\mathcal{X}$ at $x_0$} is a complete", "Noetherian local $S$-algebra $A$ with residue field $k$", "such that there exists a versal formal object", "$(A, \\xi_n, f_n)$ as in Artin's Axioms, Definition", "\\ref{artin-definition-versal-formal-object}", "with $\\xi_1 \\cong x_0$ (a $2$-isomorphism)." ], "refs": [ "artin-definition-versal-formal-object" ], "ref_ids": [ 11421 ] }, { "id": 4697, "type": "definition", "label": "stacks-geometry-definition-multiplicity", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-definition-multiplicity", "contents": [ "Let $\\mathcal{X}$ be a locally Noetherian algebraic stack. Let", "$T \\subset |\\mathcal{X}|$ be an irreducible component.", "The {\\it multiplicity} of $T$ in $\\mathcal{X}$ is defined as", "$m_{T, \\mathcal{X}} = m_{T', U}$ where $f : U \\to \\mathcal{X}$", "is a smooth morphism from a scheme and $T' \\subset |U|$", "is an irreducible component with $f(T') \\subset T$." ], "refs": [], "ref_ids": [] }, { "id": 4698, "type": "definition", "label": "stacks-geometry-definition-formal-branches", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-definition-formal-branches", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack locally of finite type", "over a locally Noetherian scheme $S$. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$", "is a morphism where $k$ is a field of finite type over $S$.", "The {\\it formal branches of $\\mathcal{X}$ through $x_0$}", "is the set of irreducible components of $\\Spec(A)$", "for any choice of versal ring to $\\mathcal{X}$ at $x_0$", "identified for different choices of $A$ by the procedure described above." ], "refs": [], "ref_ids": [] }, { "id": 4699, "type": "definition", "label": "stacks-geometry-definition-multiplicity-formal-branches", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-definition-multiplicity-formal-branches", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack locally of finite type", "over a locally Noetherian scheme $S$. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$", "is a morphism where $k$ is a field of finite type over $S$.", "The {\\it multiplicity of a formal branch of $\\mathcal{X}$ through $x_0$}", "is the multiplicity of the corresponding irreducible component of", "$\\Spec(A)$ for any choice of versal ring to $\\mathcal{X}$ at $x_0$", "(see discussion above)." ], "refs": [], "ref_ids": [] }, { "id": 4700, "type": "definition", "label": "stacks-geometry-definition-relative-dimension", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-definition-relative-dimension", "contents": [ "If $f : T \\to \\mathcal{X}$ is a locally of finite type morphism from an", "algebraic space to an algebraic stack,", "and if $t \\in |T|$ is a point with image $x \\in | \\mathcal{X}|$, then we define", "{\\it the relative dimension} of $f$ at $t$, denoted", "$\\dim_t(T_x),$", "as follows:", "choose a morphism $\\Spec k \\to \\mathcal{X}$, with source the spectrum of", "a field, which represents $x$, and choose a point", "$t' \\in |T \\times_{\\mathcal{X}} \\Spec k|$", "mapping to $t$ under the projection to $|T|$", "(such a point $t'$ exists, by", "Properties of Stacks, Lemma \\ref{stacks-properties-lemma-points-cartesian});", "then", "$$", "\\dim_t(T_x) = \\dim_{t'}(T \\times_{\\mathcal{X}} \\Spec k ).", "$$" ], "refs": [ "stacks-properties-lemma-points-cartesian" ], "ref_ids": [ 8864 ] }, { "id": 4701, "type": "definition", "label": "stacks-geometry-definition-relative-dimension-for-stacks", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-definition-relative-dimension-for-stacks", "contents": [ "If $f : \\mathcal{T} \\to \\mathcal{X}$", "is a locally of finite type morphism between", "locally Noetherian algebraic stacks, and if", "$t \\in |\\mathcal{T}|$ is a point with image $x \\in |\\mathcal{X}|$, then", "we define the {\\it relative dimension} of $f$ at $t$, denoted", "$\\dim_t(\\mathcal{T}_x),$ as follows:", "choose a morphism $\\Spec k \\to \\mathcal{X}$, with source the spectrum of", "a field, which represents $x$, and choose a point", "$t' \\in |\\mathcal{T} \\times_{\\mathcal{X}} \\Spec k|$", "mapping to $t$ under the projection to $|\\mathcal{T}|$", "(such a point $t'$ exists, by", "Properties of Stacks, Lemma", "\\ref{stacks-properties-lemma-points-cartesian}; then", "$$", "\\dim_t(\\mathcal{T}_x) = \\dim_{t'}(\\mathcal{T} \\times_{\\mathcal{X}} \\Spec k ).", "$$" ], "refs": [ "stacks-properties-lemma-points-cartesian" ], "ref_ids": [ 8864 ] }, { "id": 4702, "type": "definition", "label": "stacks-geometry-definition-pseudo-catenary", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-definition-pseudo-catenary", "contents": [ "We say that a locally Noetherian algebraic stack $\\mathcal{X}$", "is {\\it pseudo-catenary} if there exists a smooth", "and surjective morphism $U \\to \\mathcal{X}$ whose source is", "a universally catenary scheme." ], "refs": [], "ref_ids": [] }, { "id": 4703, "type": "definition", "label": "stacks-geometry-definition-dimension-local-ring", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-definition-dimension-local-ring", "contents": [ "Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.", "Let $x \\in |\\mathcal{X}|$ be a finite type point.", "The {\\it dimension of the local ring of $\\mathcal{X}$ at $x$}", "is $d \\in \\mathbf{Z}$ if the equivalent conditions of", "Lemma \\ref{lemma-dimension-local-ring} are satisfied." ], "refs": [ "stacks-geometry-lemma-dimension-local-ring" ], "ref_ids": [ 4694 ] }, { "id": 4984, "type": "definition", "label": "spaces-morphisms-definition-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-separated", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\Delta_{X/Y} : X \\to X \\times_Y X$ be the diagonal morphism.", "\\begin{enumerate}", "\\item We say $f$ is {\\it separated} if $\\Delta_{X/Y}$ is a closed immersion.", "\\item We say $f$ is {\\it locally separated}\\footnote{In the literature", "this term often refers to quasi-separated and locally separated morphisms.}", "if $\\Delta_{X/Y}$ is an immersion.", "\\item We say $f$ is {\\it quasi-separated} if $\\Delta_{X/Y}$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 4985, "type": "definition", "label": "spaces-morphisms-definition-surjective", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-surjective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. We say $f$ is {\\it surjective}", "if the map $|f| : |X| \\to |Y|$ of associated topological spaces", "is surjective." ], "refs": [], "ref_ids": [] }, { "id": 4986, "type": "definition", "label": "spaces-morphisms-definition-open", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-open", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$.", "\\begin{enumerate}", "\\item We say $f$ is {\\it open} if the map of topological spaces", "$|f| : |X| \\to |Y|$ is open.", "\\item We say $f$ is {\\it universally open} if for every morphism", "of algebraic spaces $Z \\to Y$ the morphism of topological spaces", "$$", "|Z \\times_Y X| \\to |Z|", "$$", "is open, i.e., the base change $Z \\times_Y X \\to Z$ is open.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 4987, "type": "definition", "label": "spaces-morphisms-definition-submersive", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-submersive", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item We say $f$ is {\\it submersive}\\footnote{This is very different", "from the notion of a submersion of differential manifolds.}", "if the continuous map $|X| \\to |Y|$ is submersive, see", "Topology, Definition \\ref{topology-definition-submersive}.", "\\item We say $f$ is {\\it universally submersive} if for every", "morphism of algebraic spaces $Y' \\to Y$ the base change", "$Y' \\times_Y X \\to Y'$ is submersive.", "\\end{enumerate}" ], "refs": [ "topology-definition-submersive" ], "ref_ids": [ 8349 ] }, { "id": 4988, "type": "definition", "label": "spaces-morphisms-definition-quasi-compact", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-quasi-compact", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "We say $f$ is {\\it quasi-compact} if for every quasi-compact", "algebraic space $Z$ and morphism $Z \\to Y$ the fibre product", "$Z \\times_Y X$ is quasi-compact." ], "refs": [], "ref_ids": [] }, { "id": 4989, "type": "definition", "label": "spaces-morphisms-definition-closed", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-closed", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$.", "\\begin{enumerate}", "\\item We say $f$ is {\\it closed} if the map of topological", "spaces $|X| \\to |Y|$ is closed.", "\\item We say $f$ is {\\it universally closed} if for every morphism", "of algebraic spaces $Z \\to Y$ the morphism of topological spaces", "$$", "|Z \\times_Y X| \\to |Z|", "$$", "is closed, i.e., the base change $Z \\times_Y X \\to Z$ is closed.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 4990, "type": "definition", "label": "spaces-morphisms-definition-monomorphism", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-monomorphism", "contents": [ "Let $S$ be a scheme.", "A morphism of algebraic spaces over $S$ is called a {\\it monomorphism}", "if it is an injective map of sheaves, i.e., a monomorphism in the category", "of sheaves on $(\\Sch/S)_{fppf}$." ], "refs": [], "ref_ids": [] }, { "id": 4991, "type": "definition", "label": "spaces-morphisms-definition-inverse-image-closed-subspace", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-inverse-image-closed-subspace", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces", "over $S$. Let $Z \\subset X$ be a closed subspace. The", "{\\it inverse image $f^{-1}(Z)$ of the closed subspace $Z$}", "is the closed subspace $Z \\times_X Y$ of $Y$." ], "refs": [], "ref_ids": [] }, { "id": 4992, "type": "definition", "label": "spaces-morphisms-definition-scheme-theoretic-intersection-union", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-scheme-theoretic-intersection-union", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $Z, Y \\subset X$ be closed subspaces", "corresponding to quasi-coherent ideal sheaves", "$\\mathcal{I}, \\mathcal{J} \\subset \\mathcal{O}_X$.", "The {\\it scheme theoretic intersection} of $Z$ and $Y$", "is the closed subspace of $X$ cut out by $\\mathcal{I} + \\mathcal{J}$.", "Then {\\it scheme theoretic union} of $Z$ and $Y$", "is the closed subspace of $X$ cut out by", "$\\mathcal{I} \\cap \\mathcal{J}$." ], "refs": [], "ref_ids": [] }, { "id": 4993, "type": "definition", "label": "spaces-morphisms-definition-scheme-theoretic-support", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-scheme-theoretic-support", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "The {\\it scheme theoretic support of $\\mathcal{F}$} is the closed subspace", "$Z \\subset X$ constructed in Lemma \\ref{lemma-scheme-theoretic-support}." ], "refs": [ "spaces-morphisms-lemma-scheme-theoretic-support" ], "ref_ids": [ 4778 ] }, { "id": 4994, "type": "definition", "label": "spaces-morphisms-definition-scheme-theoretic-image", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-scheme-theoretic-image", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. The {\\it scheme theoretic image} of $f$ is the smallest closed", "subspace $Z \\subset Y$ through which $f$", "factors, see Lemma \\ref{lemma-scheme-theoretic-image} above." ], "refs": [ "spaces-morphisms-lemma-scheme-theoretic-image" ], "ref_ids": [ 4779 ] }, { "id": 4995, "type": "definition", "label": "spaces-morphisms-definition-scheme-theoretically-dense", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-scheme-theoretically-dense", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $U \\subset X$ be an open subspace.", "\\begin{enumerate}", "\\item The scheme theoretic image of the morphism $U \\to X$", "is called the {\\it scheme theoretic closure of $U$ in $X$}.", "\\item We say $U$ is {\\it scheme theoretically dense in $X$}", "if the equivalent conditions of", "Lemma \\ref{lemma-scheme-theoretically-dense} are satisfied.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-lemma-scheme-theoretically-dense" ], "ref_ids": [ 4786 ] }, { "id": 4996, "type": "definition", "label": "spaces-morphisms-definition-dominant", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-dominant", "contents": [ "Let $S$ be a scheme. A morphism $f : X \\to Y$ of algebraic spaces over $S$ is", "called {\\it dominant} if the image of $|f| : |X| \\to |Y|$ is dense in $|Y|$." ], "refs": [], "ref_ids": [] }, { "id": 4997, "type": "definition", "label": "spaces-morphisms-definition-universally-injective", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-universally-injective", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic", "spaces over $S$. We say $f$ is {\\it universally injective} if", "for every morphism $Y' \\to Y$ the induced map", "$|Y' \\times_Y X| \\to |Y'|$ is injective." ], "refs": [], "ref_ids": [] }, { "id": 4998, "type": "definition", "label": "spaces-morphisms-definition-affine", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-affine", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "We say $f$ is {\\it affine} if for every affine scheme $Z$ and", "morphism $Z \\to Y$ the algebraic space $X \\times_Y Z$ is representable", "by an affine scheme." ], "refs": [], "ref_ids": [] }, { "id": 4999, "type": "definition", "label": "spaces-morphisms-definition-relative-spec", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-relative-spec", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{A}$ be a quasi-coherent sheaf of", "$\\mathcal{O}_X$-algebras. The {\\it relative spectrum of $\\mathcal{A}$ over", "$X$}, or simply the {\\it spectrum of $\\mathcal{A}$ over $X$} is the", "affine morphism $\\underline{\\Spec}(\\mathcal{A}) \\to X$", "corresponding to $\\mathcal{A}$ under the equivalence of categories of", "Lemma \\ref{lemma-affine-equivalence-algebras}." ], "refs": [ "spaces-morphisms-lemma-affine-equivalence-algebras" ], "ref_ids": [ 4802 ] }, { "id": 5000, "type": "definition", "label": "spaces-morphisms-definition-quasi-affine", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-quasi-affine", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "We say $f$ is {\\it quasi-affine} if for every affine scheme $Z$ and", "morphism $Z \\to Y$ the algebraic space $X \\times_Y Z$ is representable", "by a quasi-affine scheme." ], "refs": [], "ref_ids": [] }, { "id": 5001, "type": "definition", "label": "spaces-morphisms-definition-P", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-P", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{P}$ be a property of morphisms of schemes", "which is \\'etale local on the source-and-target.", "We say a morphism $f : X \\to Y$ of algebraic spaces over $S$", "{\\it has property $\\mathcal{P}$} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target}", "hold." ], "refs": [ "spaces-morphisms-lemma-local-source-target" ], "ref_ids": [ 4811 ] }, { "id": 5002, "type": "definition", "label": "spaces-morphisms-definition-P-at-point", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-P-at-point", "contents": [ "Let $\\mathcal{Q}$ be a property of morphisms of germs", "of schemes which is \\'etale local on the source-and-target.", "Let $S$ be a scheme.", "Given a morphism $f : X \\to Y$ of algebraic spaces over $S$ and", "a point $x \\in |X|$ we say that $f$", "{\\it has property $\\mathcal{Q}$ at $x$} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target-at-point}", "hold." ], "refs": [ "spaces-morphisms-lemma-local-source-target-at-point" ], "ref_ids": [ 4812 ] }, { "id": 5003, "type": "definition", "label": "spaces-morphisms-definition-locally-finite-type", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-locally-finite-type", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item We say $f$", "{\\it locally of finite type} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target}", "hold with", "$\\mathcal{P} = \\text{locally of finite type}$.", "\\item Let $x \\in |X|$. We say $f$ is of {\\it finite type at $x$}", "if there exists an open neighbourhood $X' \\subset X$ of $x$ such", "that $f|_{X'} : X' \\to Y$ is locally of finite type.", "\\item We say $f$ is", "{\\it of finite type} if it is locally of finite type and quasi-compact.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-lemma-local-source-target" ], "ref_ids": [ 4811 ] }, { "id": 5004, "type": "definition", "label": "spaces-morphisms-definition-finite-type-point", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-finite-type-point", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "We say a point $x \\in |X|$ is a {\\it finite type point}\\footnote{This is a", "slight abuse of language as it would perhaps be more correct to say", "``locally finite type point''.} if the equivalent conditions of", "Lemma \\ref{lemma-point-finite-type}", "are satisfied. We denote $X_{\\text{ft-pts}}$ the set of finite type points", "of $X$." ], "refs": [ "spaces-morphisms-lemma-point-finite-type" ], "ref_ids": [ 4822 ] }, { "id": 5005, "type": "definition", "label": "spaces-morphisms-definition-locally-quasi-finite", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-locally-quasi-finite", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item We say $f$ is", "{\\it locally quasi-finite} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target} hold with", "$\\mathcal{P} = \\text{locally quasi-finite}$.", "\\item Let $x \\in |X|$. We say $f$ is {\\it quasi-finite at $x$}", "if there exists an open neighbourhood $X' \\subset X$ of $x$ such", "that $f|_{X'} : X' \\to Y$ is locally quasi-finite.", "\\item A morphism of algebraic spaces $f : X \\to Y$ is", "{\\it quasi-finite} if it is locally quasi-finite and quasi-compact.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-lemma-local-source-target" ], "ref_ids": [ 4811 ] }, { "id": 5006, "type": "definition", "label": "spaces-morphisms-definition-locally-finite-presentation", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-locally-finite-presentation", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item We say $f$ is {\\it locally of finite presentation} if", "the equivalent conditions of", "Lemma \\ref{lemma-local-source-target}", "hold with $\\mathcal{P} =$``locally of finite presentation''.", "\\item Let $x \\in |X|$. We say $f$ is of {\\it finite presentation at $x$}", "if there exists an open neighbourhood $X' \\subset X$ of $x$ such", "that $f|_{X'} : X' \\to Y$ is locally of finite", "presentation\\footnote{It seems awkward to use ``locally of finite presentation", "at $x$'', but the current terminology may be misleading in the sense that", "``of finite presentation at $x$'' does {\\bf not} mean that there is", "an open neighbourhood $X' \\subset X$ such that $f|_{X'}$ is of finite", "presentation.}.", "\\item A morphism of algebraic spaces $f : X \\to Y$ is", "{\\it of finite presentation}", "if it is locally of finite presentation, quasi-compact and", "quasi-separated.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-lemma-local-source-target" ], "ref_ids": [ 4811 ] }, { "id": 5007, "type": "definition", "label": "spaces-morphisms-definition-flat", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-flat", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item We say $f$ is {\\it flat} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target} with", "$\\mathcal{P} =$``flat''.", "\\item Let $x \\in |X|$. We say $f$ is {\\it flat at $x$} if the", "equivalent conditions of", "Lemma \\ref{lemma-local-source-target-at-point}", "holds with $\\mathcal{Q} =$``induced map local rings is flat''.", "\\end{enumerate}", "Note that the second part makes sense by", "Descent, Lemma \\ref{descent-lemma-flat-at-point}." ], "refs": [ "spaces-morphisms-lemma-local-source-target", "spaces-morphisms-lemma-local-source-target-at-point", "descent-lemma-flat-at-point" ], "ref_ids": [ 4811, 4812, 14727 ] }, { "id": 5008, "type": "definition", "label": "spaces-morphisms-definition-flat-module", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-flat-module", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "\\begin{enumerate}", "\\item Let $x \\in |X|$. We say $\\mathcal{F}$ is {\\it flat at $x$ over $Y$}", "if the equivalent conditions of", "Lemma \\ref{lemma-flat-at-point}", "hold.", "\\item We say $\\mathcal{F}$ is {\\it flat over $Y$} if $\\mathcal{F}$ is", "flat over $Y$ at all $x \\in |X|$.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-lemma-flat-at-point" ], "ref_ids": [ 4862 ] }, { "id": 5009, "type": "definition", "label": "spaces-morphisms-definition-dimension-fibre", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-dimension-fibre", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $x \\in |X|$.", "Let $d, r \\in \\{0, 1, 2, \\ldots, \\infty\\}$.", "\\begin{enumerate}", "\\item We say the", "{\\it dimension of the local ring of the fibre of $f$ at $x$} is $d$", "if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target-at-point}", "hold for the property", "$\\mathcal{P}_d$ described in", "Descent, Lemma \\ref{descent-lemma-dimension-local-ring-fibre}.", "\\item We say the", "{\\it transcendence degree of $x/f(x)$} is $r$", "if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target-at-point}", "hold for the property", "$\\mathcal{P}_r$ described in", "Descent, Lemma \\ref{descent-lemma-transcendence-degree-at-point}.", "\\item We say", "{\\it $f$ has relative dimension $d$ at $x$}", "if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target-at-point}", "hold for the property", "$\\mathcal{P}_d$ described in", "Descent, Lemma \\ref{descent-lemma-dimension-at-point}.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-lemma-local-source-target-at-point", "descent-lemma-dimension-local-ring-fibre", "spaces-morphisms-lemma-local-source-target-at-point", "descent-lemma-transcendence-degree-at-point", "spaces-morphisms-lemma-local-source-target-at-point", "descent-lemma-dimension-at-point" ], "ref_ids": [ 4812, 14729, 4812, 14730, 4812, 14731 ] }, { "id": 5010, "type": "definition", "label": "spaces-morphisms-definition-relative-dimension", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-relative-dimension", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $d \\in \\{0, 1, 2, \\ldots\\}$.", "\\begin{enumerate}", "\\item We say $f$ has {\\it relative dimension $\\leq d$} if", "$f$ has relative dimension $\\leq d$ at all $x \\in |X|$.", "\\item We say $f$ has {\\it relative dimension $d$} if", "$f$ has relative dimension $d$ at all $x \\in |X|$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5011, "type": "definition", "label": "spaces-morphisms-definition-syntomic", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-syntomic", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item We say $f$ is {\\it syntomic} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target}", "hold with $\\mathcal{P} =$``syntomic''.", "\\item Let $x \\in |X|$. We say $f$ is {\\it syntomic at $x$} if", "there exists an open neighbourhood $X' \\subset X$ of $x$ such", "that $f|_{X'} : X' \\to Y$ is syntomic.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-lemma-local-source-target" ], "ref_ids": [ 4811 ] }, { "id": 5012, "type": "definition", "label": "spaces-morphisms-definition-smooth", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-smooth", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item We say $f$ is {\\it smooth} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target} hold with", "$\\mathcal{P} =$``smooth''.", "\\item Let $x \\in |X|$. We say $f$ is {\\it smooth at $x$} if there exists", "an open neighbourhood $X' \\subset X$ of $x$ such that $f|_{X'} : X' \\to Y$", "is smooth.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-lemma-local-source-target" ], "ref_ids": [ 4811 ] }, { "id": 5013, "type": "definition", "label": "spaces-morphisms-definition-unramified", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-unramified", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item We say $f$ is {\\it unramified} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target}", "hold with $\\mathcal{P} = \\text{unramified}$.", "\\item Let $x \\in |X|$. We say $f$ is {\\it unramified at $x$} if there", "exists an open neighbourhood $X' \\subset X$ of $x$ such that", "$f|_{X'} : X' \\to Y$ is unramified.", "\\item We say $f$ is {\\it G-unramified} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target}", "hold with $\\mathcal{P} = \\text{G-unramified}$.", "\\item Let $x \\in |X|$. We say $f$ is {\\it G-unramified at $x$} if there", "exists an open neighbourhood $X' \\subset X$ of $x$ such that", "$f|_{X'} : X' \\to Y$ is G-unramified.", "\\end{enumerate}" ], "refs": [ "spaces-morphisms-lemma-local-source-target", "spaces-morphisms-lemma-local-source-target" ], "ref_ids": [ 4811, 4811 ] }, { "id": 5014, "type": "definition", "label": "spaces-morphisms-definition-etale", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-etale", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $x \\in |X|$. We say $f$ is {\\it \\'etale at $x$} if there", "exists an open neighbourhood $X' \\subset X$ of $x$ such that", "$f|_{X'} : X' \\to Y$ is \\'etale." ], "refs": [], "ref_ids": [] }, { "id": 5015, "type": "definition", "label": "spaces-morphisms-definition-proper", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-proper", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "We say $f$ is {\\it proper} if $f$ is separated, finite type, and", "universally closed." ], "refs": [], "ref_ids": [] }, { "id": 5016, "type": "definition", "label": "spaces-morphisms-definition-valuative-criterion", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-valuative-criterion", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "We say $f$ {\\it satisfies the uniqueness part of the valuative criterion}", "if given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$, there exists", "at most one dotted arrow (without requiring existence).", "We say $f$ {\\it satisfies the existence part of the valuative criterion}", "if given any solid diagram as above there exists an extension", "$K \\subset K'$ of fields, a valuation ring $A' \\subset K'$ dominating", "$A$ and a morphism $\\Spec(A') \\to X$ such that the following", "diagram commutes", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\", "\\Spec(A') \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] & Y", "}", "$$", "We say $f$ {\\it satisfies the valuative criterion}", "if $f$ satisfies both the existence and uniqueness part." ], "refs": [], "ref_ids": [] }, { "id": 5017, "type": "definition", "label": "spaces-morphisms-definition-integral", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-integral", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item We say that $f$ is {\\it integral} if for every affine scheme $Z$", "and morphisms $Z \\to Y$ the algebraic space $X \\times_Y Z$ is", "representable by an affine scheme integral over $Z$.", "\\item We say that $f$ is {\\it finite} if for every affine scheme $Z$", "and morphisms $Z \\to Y$ the algebraic space $X \\times_Y Z$ is", "representable by an affine scheme finite over $Z$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5018, "type": "definition", "label": "spaces-morphisms-definition-finite-locally-free", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-finite-locally-free", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "We say that $f$ is {\\it finite locally free} if $f$ is affine", "and $f_*\\mathcal{O}_X$ is a finite locally free $\\mathcal{O}_Y$-module.", "In this case we say $f$ is", "has {\\it rank} or {\\it degree} $d$", "if the sheaf $f_*\\mathcal{O}_X$ is finite locally free of rank $d$." ], "refs": [], "ref_ids": [] }, { "id": 5019, "type": "definition", "label": "spaces-morphisms-definition-rational-map", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-rational-map", "contents": [ "Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$.", "\\begin{enumerate}", "\\item Let $f : U \\to Y$, $g : V \\to Y$ be morphisms of algebraic spaces", "over $S$ defined on dense open subspaces $U$, $V$ of $X$. We say that $f$ is", "{\\it equivalent} to $g$ if $f|_W = g|_W$ for some dense open", "subspace $W \\subset U \\cap V$.", "\\item A {\\it rational map from $X$ to $Y$}", "is an equivalence class for the equivalence relation defined in (1).", "\\item Given morphisms $X \\to B$ and $Y \\to B$ of algebraic spaces over $S$", "we say that a rational map from $X$ to $Y$ is a", "{\\it $B$-rational map from $X$ to $Y$}", "if there exists a representative $f : U \\to Y$ of the equivalence", "class which is a morphism over $B$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5020, "type": "definition", "label": "spaces-morphisms-definition-rational-function", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-rational-function", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A", "{\\it rational function on $X$} is a rational map from $X$ to $\\mathbf{A}^1_S$." ], "refs": [], "ref_ids": [] }, { "id": 5021, "type": "definition", "label": "spaces-morphisms-definition-ring-of-rational-functions", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-ring-of-rational-functions", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "The {\\it ring of rational functions on $X$}", "is the ring $R(X)$ whose elements are rational functions with", "addition and multiplication as just described." ], "refs": [], "ref_ids": [] }, { "id": 5022, "type": "definition", "label": "spaces-morphisms-definition-domain-of-definition", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-domain-of-definition", "contents": [ "Let $S$ be a scheme. Let $\\varphi$ be a rational map between two", "algebraic spaces $X$ and $Y$ over $S$. We say", "$\\varphi$ is {\\it defined in a point $x \\in |X|$} if there exists a", "representative $(U, f)$ of $\\varphi$ with $x \\in |U|$. The", "{\\it domain of definition} of $\\varphi$ is the set of all points", "where $\\varphi$ is defined." ], "refs": [], "ref_ids": [] }, { "id": 5023, "type": "definition", "label": "spaces-morphisms-definition-dominant-rational", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-dominant-rational", "contents": [ "Let $S$ be a scheme. Let $X$ and $Y$ be algebraic spaces over $S$.", "Assume $|X|$ and $|Y|$ are irreducible. A rational map from $X$ to $Y$", "is called {\\it dominant} if any representative $f : U \\to Y$ is a dominant", "morphism in the sense of Definition \\ref{definition-dominant}." ], "refs": [ "spaces-morphisms-definition-dominant" ], "ref_ids": [ 4996 ] }, { "id": 5024, "type": "definition", "label": "spaces-morphisms-definition-birational-spaces", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-birational-spaces", "contents": [ "Let $S$ be a scheme. Let $X$ and $Y$ be algebraic spaces", "over $S$ with $|X|$ and $|Y|$ irreducible.", "We say $X$ and $Y$ are {\\it birational} if $X$ and $Y$ are isomorphic", "in the category of irreducible algebraic spaces over $S$", "and dominant rational maps." ], "refs": [], "ref_ids": [] }, { "id": 5025, "type": "definition", "label": "spaces-morphisms-definition-integral-closure", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-integral-closure", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{A}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras.", "The {\\it integral closure of $\\mathcal{O}_X$ in $\\mathcal{A}$} is the", "quasi-coherent $\\mathcal{O}_X$-subalgebra $\\mathcal{A}' \\subset \\mathcal{A}$", "constructed in Lemma \\ref{lemma-integral-closure} above." ], "refs": [ "spaces-morphisms-lemma-integral-closure" ], "ref_ids": [ 4957 ] }, { "id": 5026, "type": "definition", "label": "spaces-morphisms-definition-normalization-X-in-Y", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-normalization-X-in-Y", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated", "morphism of algebraic spaces over $S$. Let $\\mathcal{O}'$ be the integral", "closure of $\\mathcal{O}_X$ in $f_*\\mathcal{O}_Y$. The {\\it normalization of", "$X$ in $Y$} is the morphism of algebraic spaces", "$$", "\\nu : X' = \\underline{\\Spec}_X(\\mathcal{O}') \\to X", "$$", "over $S$. It comes equipped with a natural factorization", "$$", "Y \\xrightarrow{f'} X' \\xrightarrow{\\nu} X", "$$", "of the initial morphism $f$." ], "refs": [], "ref_ids": [] }, { "id": 5027, "type": "definition", "label": "spaces-morphisms-definition-normalization", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-normalization", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying the", "equivalent conditions of Lemma \\ref{lemma-prepare-normalization}.", "We define the {\\it normalization} of $X$ as the morphism", "$$", "\\nu : X^\\nu \\longrightarrow X", "$$", "constructed in Lemma \\ref{lemma-normalization}." ], "refs": [ "spaces-morphisms-lemma-prepare-normalization", "spaces-morphisms-lemma-normalization" ], "ref_ids": [ 4966, 4967 ] }, { "id": 5028, "type": "definition", "label": "spaces-morphisms-definition-universal-homeomorphism", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-definition-universal-homeomorphism", "contents": [ "Let $S$ be a scheme.", "A morphism $f : X \\to Y$ of algebraic spaces over $S$", "is called a {\\it universal homeomorphism}", "if and only if for every morphism of algebraic spaces $Z \\to Y$", "the base change $Z \\times_Y X \\to Z$ induces a homeomorphism", "$|Z \\times_Y X| \\to |Z|$." ], "refs": [], "ref_ids": [] }, { "id": 5114, "type": "definition", "label": "weil-definition-chow-group-motives", "categories": [ "weil" ], "title": "weil-definition-chow-group-motives", "contents": [ "Let $k$ be a base field. Let $M = (X, p, m)$ be a Chow motive over $k$.", "For $i \\in \\mathbf{Z}$ we define the {\\it $i$th Chow group of $M$}", "by the formula", "$$", "\\CH^i(M) = p\\left(\\CH^{i + m}(X) \\otimes \\mathbf{Q}\\right)", "$$" ], "refs": [], "ref_ids": [] }, { "id": 5115, "type": "definition", "label": "weil-definition-weil-cohomology-theory-classical", "categories": [ "weil" ], "title": "weil-definition-weil-cohomology-theory-classical", "contents": [ "Let $k$ be an algebraically closed field.", "Let $F$ be a field of characteristic $0$.", "A {\\it classical Weil cohomology theory} over $k$ with coefficients in $F$", "is given by data (D1), (D2), and (D3) satisfying", "Poincar\\'e duality, the K\\\"unneth formula, and compatibility", "with cycle classes, more precisely, satisfying (A), (B), and (C)." ], "refs": [], "ref_ids": [] }, { "id": 5116, "type": "definition", "label": "weil-definition-weil-cohomology-theory", "categories": [ "weil" ], "title": "weil-definition-weil-cohomology-theory", "contents": [ "Let $k$ be a field. Let $F$ be a field of characteristic $0$.", "A {\\it Weil cohomology theory} over $k$ with coefficients in $F$", "is given by data (D0), (D1), (D2), and (D3) satisfying", "Poincar\\'e duality, the K\\\"unneth formula, and compatibility", "with cycle classes, more precisely, satisfying axioms (A), (B), and (C)", "of Section \\ref{section-axioms}", "and in addition such that the equivalent conditions (1) and (2) of", "Lemma \\ref{lemma-H-0-separable} hold for every smooth projective $X$ over $k$." ], "refs": [ "weil-lemma-H-0-separable" ], "ref_ids": [ 5087 ] }, { "id": 5537, "type": "definition", "label": "morphisms-definition-scheme-theoretic-intersection-union", "categories": [ "morphisms" ], "title": "morphisms-definition-scheme-theoretic-intersection-union", "contents": [ "Let $X$ be a scheme. Let $Z, Y \\subset X$ be closed subschemes", "corresponding to quasi-coherent ideal sheaves", "$\\mathcal{I}, \\mathcal{J} \\subset \\mathcal{O}_X$.", "The {\\it scheme theoretic intersection} of $Z$ and $Y$", "is the closed subscheme of $X$ cut out by $\\mathcal{I} + \\mathcal{J}$.", "The {\\it scheme theoretic union} of $Z$ and $Y$", "is the closed subscheme of $X$ cut out by", "$\\mathcal{I} \\cap \\mathcal{J}$." ], "refs": [], "ref_ids": [] }, { "id": 5538, "type": "definition", "label": "morphisms-definition-scheme-theoretic-support", "categories": [ "morphisms" ], "title": "morphisms-definition-scheme-theoretic-support", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module of finite type. The {\\it scheme theoretic support", "of $\\mathcal{F}$} is the closed subscheme $Z \\subset X$ constructed in", "Lemma \\ref{lemma-scheme-theoretic-support}." ], "refs": [ "morphisms-lemma-scheme-theoretic-support" ], "ref_ids": [ 5144 ] }, { "id": 5539, "type": "definition", "label": "morphisms-definition-scheme-theoretic-image", "categories": [ "morphisms" ], "title": "morphisms-definition-scheme-theoretic-image", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. The {\\it scheme theoretic image}", "of $f$ is the smallest closed subscheme $Z \\subset Y$ through which $f$", "factors, see Lemma \\ref{lemma-scheme-theoretic-image} above." ], "refs": [ "morphisms-lemma-scheme-theoretic-image" ], "ref_ids": [ 5145 ] }, { "id": 5540, "type": "definition", "label": "morphisms-definition-scheme-theoretically-dense", "categories": [ "morphisms" ], "title": "morphisms-definition-scheme-theoretically-dense", "contents": [ "Let $X$ be a scheme. Let $U \\subset X$ be an open subscheme.", "\\begin{enumerate}", "\\item The scheme theoretic image of the morphism $U \\to X$", "is called the {\\it scheme theoretic closure of $U$ in $X$}.", "\\item We say $U$ is {\\it scheme theoretically dense in $X$}", "if for every open $V \\subset X$ the scheme theoretic closure", "of $U \\cap V$ in $V$ is equal to $V$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5541, "type": "definition", "label": "morphisms-definition-dominant", "categories": [ "morphisms" ], "title": "morphisms-definition-dominant", "contents": [ "A morphism $f : X \\to S$ of schemes is called {\\it dominant} if the", "image of $f$ is a dense subset of $S$." ], "refs": [], "ref_ids": [] }, { "id": 5542, "type": "definition", "label": "morphisms-definition-surjective", "categories": [ "morphisms" ], "title": "morphisms-definition-surjective", "contents": [ "A morphism of schemes is said to be {\\it surjective}", "if it is surjective on underlying topological", "spaces." ], "refs": [], "ref_ids": [] }, { "id": 5543, "type": "definition", "label": "morphisms-definition-universally-injective", "categories": [ "morphisms" ], "title": "morphisms-definition-universally-injective", "contents": [ "Let $f : X \\to S$ be a morphism.", "\\begin{enumerate}", "\\item We say that $f$ is {\\it universally injective} if and only", "if for any morphism of schemes $S' \\to S$ the base change", "$f' : X_{S'} \\to S'$ is injective (on underlying topological spaces).", "\\item We say $f$ is {\\it radicial} if $f$ is injective as a", "map of topological spaces, and for every $x \\in X$ the field", "extension $\\kappa(x) \\supset \\kappa(f(x))$ is purely inseparable.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5544, "type": "definition", "label": "morphisms-definition-affine", "categories": [ "morphisms" ], "title": "morphisms-definition-affine", "contents": [ "A morphism of schemes $f : X \\to S$ is called {\\it affine} if", "the inverse image of every affine open of $S$ is an affine", "open of $X$." ], "refs": [], "ref_ids": [] }, { "id": 5545, "type": "definition", "label": "morphisms-definition-family-ample-invertible-modules", "categories": [ "morphisms" ], "title": "morphisms-definition-family-ample-invertible-modules", "contents": [ "\\begin{reference}", "\\cite[II Definition 2.2.4]{SGA6}", "\\end{reference}", "Let $X$ be a scheme. Let $\\{\\mathcal{L}_i\\}_{i \\in I}$", "be a family of invertible $\\mathcal{O}_X$-modules. We say", "$\\{\\mathcal{L}_i\\}_{i \\in I}$ is an", "{\\it ample family of invertible modules on $X$} if", "\\begin{enumerate}", "\\item $X$ is quasi-compact, and", "\\item for every $x \\in X$ there exists an $i \\in I$, an $n \\geq 1$,", "and $s \\in \\Gamma(X, \\mathcal{L}_i^{\\otimes n})$ such", "that $x \\in X_s$ and $X_s$ is affine.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5546, "type": "definition", "label": "morphisms-definition-quasi-affine", "categories": [ "morphisms" ], "title": "morphisms-definition-quasi-affine", "contents": [ "A morphism of schemes $f : X \\to S$ is called {\\it quasi-affine} if the", "inverse image of every affine open of $S$ is a quasi-affine scheme." ], "refs": [], "ref_ids": [] }, { "id": 5547, "type": "definition", "label": "morphisms-definition-property-local", "categories": [ "morphisms" ], "title": "morphisms-definition-property-local", "contents": [ "Let $P$ be a property of ring maps.", "\\begin{enumerate}", "\\item We say that $P$ is {\\it local} if the following hold:", "\\begin{enumerate}", "\\item For any ring map $R \\to A$, and any $f \\in R$ we have", "$P(R \\to A) \\Rightarrow P(R_f \\to A_f)$.", "\\item For any rings $R$, $A$, any $f \\in R$, $a\\in A$, and any ring map", "$R_f \\to A$ we have $P(R_f \\to A) \\Rightarrow P(R \\to A_a)$.", "\\item For any ring map $R \\to A$, and $a_i \\in A$ such that", "$(a_1, \\ldots, a_n) = A$ then", "$\\forall i, P(R \\to A_{a_i}) \\Rightarrow P(R \\to A)$.", "\\end{enumerate}", "\\item We say that $P$ is {\\it stable under base change} if for any", "ring maps $R \\to A$, $R \\to R'$ we have", "$P(R \\to A) \\Rightarrow P(R' \\to R' \\otimes_R A)$.", "\\item We say that $P$ is {\\it stable under composition} if for any", "ring maps $A \\to B$, $B \\to C$ we have", "$P(A \\to B) \\wedge P(B \\to C) \\Rightarrow P(A \\to C)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5548, "type": "definition", "label": "morphisms-definition-locally-P", "categories": [ "morphisms" ], "title": "morphisms-definition-locally-P", "contents": [ "Let $P$ be a property of ring maps.", "Let $f : X \\to S$ be a morphism of schemes.", "We say $f$ is {\\it locally of type $P$} if for any $x \\in X$", "there exists an affine open neighbourhood $U$ of $x$", "in $X$ which maps into an affine open $V \\subset S$ such that", "the induced ring map $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$", "has property $P$." ], "refs": [], "ref_ids": [] }, { "id": 5549, "type": "definition", "label": "morphisms-definition-finite-type", "categories": [ "morphisms" ], "title": "morphisms-definition-finite-type", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item We say that $f$ is of {\\it finite type at $x \\in X$} if", "there exists an affine open neighbourhood $\\Spec(A) = U \\subset X$", "of $x$ and an affine open $\\Spec(R) = V \\subset S$", "with $f(U) \\subset V$ such that the induced ring map", "$R \\to A$ is of finite type.", "\\item We say that $f$ is {\\it locally of finite type} if it is", "of finite type at every point of $X$.", "\\item We say that $f$ is of {\\it finite type} if it is locally of", "finite type and quasi-compact.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5550, "type": "definition", "label": "morphisms-definition-finite-type-point", "categories": [ "morphisms" ], "title": "morphisms-definition-finite-type-point", "contents": [ "Let $S$ be a scheme.", "Let us say that a point $s$ of $S$ is a {\\it finite type point}", "if the canonical morphism $\\Spec(\\kappa(s)) \\to S$ is of finite type.", "We denote $S_{\\text{ft-pts}}$ the set of finite type points of $S$." ], "refs": [], "ref_ids": [] }, { "id": 5551, "type": "definition", "label": "morphisms-definition-universally-catenary", "categories": [ "morphisms" ], "title": "morphisms-definition-universally-catenary", "contents": [ "Let $S$ be a scheme. Assume $S$ is locally Noetherian.", "We say $S$ is {\\it universally catenary} if for every", "morphism $X \\to S$ locally of finite type the scheme $X$ is catenary." ], "refs": [], "ref_ids": [] }, { "id": 5552, "type": "definition", "label": "morphisms-definition-J", "categories": [ "morphisms" ], "title": "morphisms-definition-J", "contents": [ "Let $X$ be a locally Noetherian scheme. We say $X$ is {\\it J-2}", "if for every morphism $Y \\to X$ which is locally of finite type", "the regular locus $\\text{Reg}(Y)$ is open in $Y$." ], "refs": [], "ref_ids": [] }, { "id": 5553, "type": "definition", "label": "morphisms-definition-quasi-finite", "categories": [ "morphisms" ], "title": "morphisms-definition-quasi-finite", "contents": [ "\\begin{reference}", "\\cite[II Definition 6.2.3]{EGA}", "\\end{reference}", "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item We say that $f$ is {\\it quasi-finite at a point $x \\in X$}", "if there exist an affine neighbourhood $\\Spec(A) = U \\subset X$", "of $x$ and an affine open $\\Spec(R) = V \\subset S$ such that", "$f(U) \\subset V$, the ring map $R \\to A$ is of finite type,", "and $R \\to A$ is quasi-finite at the prime of $A$ corresponding to $x$", "(see above).", "\\item We say $f$ is {\\it locally quasi-finite} if $f$ is", "quasi-finite at every point $x$ of $X$.", "\\item We say that $f$ is {\\it quasi-finite} if $f$ is of finite type", "and every point $x$ is an isolated point of its fibre.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5554, "type": "definition", "label": "morphisms-definition-finite-presentation", "categories": [ "morphisms" ], "title": "morphisms-definition-finite-presentation", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item We say that $f$ is of {\\it finite presentation at $x \\in X$} if", "there exists an affine open neighbourhood $\\Spec(A) = U \\subset X$", "of $x$ and affine open $\\Spec(R) = V \\subset S$", "with $f(U) \\subset V$ such that the induced ring map", "$R \\to A$ is of finite presentation.", "\\item We say that $f$ is {\\it locally of finite presentation} if it is", "of finite presentation at every point of $X$.", "\\item We say that $f$ is of {\\it finite presentation} if it is locally of", "finite presentation, quasi-compact and quasi-separated.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5555, "type": "definition", "label": "morphisms-definition-open", "categories": [ "morphisms" ], "title": "morphisms-definition-open", "contents": [ "Let $f : X \\to S$ be a morphism.", "\\begin{enumerate}", "\\item We say $f$ is {\\it open} if the map on underlying", "topological spaces is open.", "\\item We say $f$ is {\\it universally open} if for any morphism of", "schemes $S' \\to S$ the base change $f' : X_{S'} \\to S'$ is open.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5556, "type": "definition", "label": "morphisms-definition-submersive", "categories": [ "morphisms" ], "title": "morphisms-definition-submersive", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "\\begin{enumerate}", "\\item We say $f$ is {\\it submersive}\\footnote{This is very different", "from the notion of a submersion of differential manifolds.}", "if the continuous map of underlying topological spaces is submersive, see", "Topology, Definition \\ref{topology-definition-submersive}.", "\\item We say $f$ is {\\it universally submersive} if for every", "morphism of schemes $Y' \\to Y$ the base change", "$Y' \\times_Y X \\to Y'$ is submersive.", "\\end{enumerate}" ], "refs": [ "topology-definition-submersive" ], "ref_ids": [ 8349 ] }, { "id": 5557, "type": "definition", "label": "morphisms-definition-flat", "categories": [ "morphisms" ], "title": "morphisms-definition-flat", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.", "\\begin{enumerate}", "\\item We say $f$ is {\\it flat at a point $x \\in X$} if the", "local ring $\\mathcal{O}_{X, x}$ is flat over the local ring", "$\\mathcal{O}_{S, f(x)}$.", "\\item We say that $\\mathcal{F}$ is {\\it flat over $S$ at a point $x \\in X$}", "if the stalk $\\mathcal{F}_x$ is a flat $\\mathcal{O}_{S, f(x)}$-module.", "\\item We say $f$ is {\\it flat} if $f$ is flat at every point of $X$.", "\\item We say that $\\mathcal{F}$ is {\\it flat over $S$} if", "$\\mathcal{F}$ is flat over $S$ at every point $x$ of $X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5558, "type": "definition", "label": "morphisms-definition-scheme-structure-connected-component", "categories": [ "morphisms" ], "title": "morphisms-definition-scheme-structure-connected-component", "contents": [ "Let $X$ be a scheme. Let $T \\subset X$ be a connected component.", "The {\\it canonical scheme structure on $T$} is the unique", "scheme structure on $T$ such that the closed immersion $T \\to X$", "is flat, see", "Lemma \\ref{lemma-characterize-flat-closed-immersions}." ], "refs": [ "morphisms-lemma-characterize-flat-closed-immersions" ], "ref_ids": [ 5274 ] }, { "id": 5559, "type": "definition", "label": "morphisms-definition-relative-dimension-d", "categories": [ "morphisms" ], "title": "morphisms-definition-relative-dimension-d", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Assume $f$ is locally of finite type.", "\\begin{enumerate}", "\\item We say $f$ is of {\\it relative dimension $\\leq d$ at $x$} if", "$\\dim_x(X_{f(x)}) \\leq d$.", "\\item We say $f$ is of {\\it relative dimension $\\leq d$} if", "$\\dim_x(X_{f(x)}) \\leq d$ for all $x \\in X$.", "\\item We say $f$ is of {\\it relative dimension $d$} if", "all nonempty fibres $X_s$ are equidimensional of dimension $d$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5560, "type": "definition", "label": "morphisms-definition-syntomic", "categories": [ "morphisms" ], "title": "morphisms-definition-syntomic", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item We say that $f$ is {\\it syntomic at $x \\in X$} if", "there exists an affine open neighbourhood $\\Spec(A) = U \\subset X$", "of $x$ and affine open $\\Spec(R) = V \\subset S$", "with $f(U) \\subset V$ such that the induced ring map", "$R \\to A$ is syntomic.", "\\item We say that $f$ is {\\it syntomic} if it is syntomic", "at every point of $X$.", "\\item If $S = \\Spec(k)$ and $f$ is syntomic, then we say that", "$X$ is a {\\it local complete intersection over $k$}.", "\\item A morphism of affine schemes $f : X \\to S$", "is called {\\it standard syntomic} if there exists a", "global relative complete intersection", "$R \\to R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ (see", "Algebra,", "Definition \\ref{algebra-definition-relative-global-complete-intersection})", "such that $X \\to S$ is isomorphic to", "$$", "\\Spec(R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)) \\to \\Spec(R).", "$$", "\\end{enumerate}" ], "refs": [ "algebra-definition-relative-global-complete-intersection" ], "ref_ids": [ 1533 ] }, { "id": 5561, "type": "definition", "label": "morphisms-definition-syntomic-relative-dimension", "categories": [ "morphisms" ], "title": "morphisms-definition-syntomic-relative-dimension", "contents": [ "Let $d \\geq 0$ be an integer. We say a morphism of schemes $f : X \\to S$", "is {\\it syntomic of relative dimension $d$} if $f$ is syntomic and", "the function $\\dim_x(X_{f(x)}) = d$ for all $x \\in X$." ], "refs": [], "ref_ids": [] }, { "id": 5562, "type": "definition", "label": "morphisms-definition-conormal-sheaf", "categories": [ "morphisms" ], "title": "morphisms-definition-conormal-sheaf", "contents": [ "Let $i : Z \\to X$ be an immersion. The {\\it conormal sheaf", "$\\mathcal{C}_{Z/X}$ of $Z$ in $X$} or the {\\it conormal sheaf of $i$}", "is the quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{I}/\\mathcal{I}^2$", "described above." ], "refs": [], "ref_ids": [] }, { "id": 5563, "type": "definition", "label": "morphisms-definition-sheaf-differentials", "categories": [ "morphisms" ], "title": "morphisms-definition-sheaf-differentials", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "The {\\it sheaf of differentials $\\Omega_{X/S}$ of $X$ over $S$} is", "the sheaf of differentials of $f$ viewed as a morphism of ringed spaces", "(Modules, Definition \\ref{modules-definition-differentials})", "equipped with its {\\it universal $S$-derivation}", "$$", "\\text{d}_{X/S} : \\mathcal{O}_X \\longrightarrow \\Omega_{X/S}.", "$$" ], "refs": [ "modules-definition-differentials" ], "ref_ids": [ 13355 ] }, { "id": 5564, "type": "definition", "label": "morphisms-definition-smooth", "categories": [ "morphisms" ], "title": "morphisms-definition-smooth", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item We say that $f$ is {\\it smooth at $x \\in X$} if", "there exists an affine open neighbourhood $\\Spec(A) = U \\subset X$", "of $x$ and affine open $\\Spec(R) = V \\subset S$", "with $f(U) \\subset V$ such that the induced ring map", "$R \\to A$ is smooth.", "\\item We say that $f$ is {\\it smooth} if it is smooth at every point of $X$.", "\\item A morphism of affine schemes $f : X \\to S$", "is called {\\it standard smooth} if there exists a standard smooth ring", "map $R \\to R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ (see", "Algebra, Definition \\ref{algebra-definition-standard-smooth})", "such that $X \\to S$ is isomorphic to", "$$", "\\Spec(R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)) \\to \\Spec(R).", "$$", "\\end{enumerate}" ], "refs": [ "algebra-definition-standard-smooth" ], "ref_ids": [ 1535 ] }, { "id": 5565, "type": "definition", "label": "morphisms-definition-smooth-relative-dimension", "categories": [ "morphisms" ], "title": "morphisms-definition-smooth-relative-dimension", "contents": [ "Let $d \\geq 0$ be an integer. We say a morphism of schemes $f : X \\to S$", "is {\\it smooth of relative dimension $d$} if $f$ is smooth and", "$\\Omega_{X/S}$ is finite locally free of constant rank $d$." ], "refs": [], "ref_ids": [] }, { "id": 5566, "type": "definition", "label": "morphisms-definition-unramified", "categories": [ "morphisms" ], "title": "morphisms-definition-unramified", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item We say that $f$ is {\\it unramified at $x \\in X$} if", "there exists an affine open neighbourhood $\\Spec(A) = U \\subset X$", "of $x$ and affine open $\\Spec(R) = V \\subset S$", "with $f(U) \\subset V$ such that the induced ring map", "$R \\to A$ is unramified.", "\\item We say that $f$ is {\\it G-unramified at $x \\in X$} if", "there exists an affine open neighbourhood $\\Spec(A) = U \\subset X$", "of $x$ and affine open $\\Spec(R) = V \\subset S$", "with $f(U) \\subset V$ such that the induced ring map", "$R \\to A$ is G-unramified.", "\\item We say that $f$ is {\\it unramified} if it is unramified", "at every point of $X$.", "\\item We say that $f$ is {\\it G-unramified} if it is G-unramified", "at every point of $X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5567, "type": "definition", "label": "morphisms-definition-etale", "categories": [ "morphisms" ], "title": "morphisms-definition-etale", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item We say that $f$ is {\\it \\'etale at $x \\in X$} if", "there exists an affine open neighbourhood $\\Spec(A) = U \\subset X$", "of $x$ and affine open $\\Spec(R) = V \\subset S$", "with $f(U) \\subset V$ such that the induced ring map", "$R \\to A$ is \\'etale.", "\\item We say that $f$ is {\\it \\'etale} if it is \\'etale at every point of $X$.", "\\item A morphism of affine schemes $f : X \\to S$ is called", "{\\it standard \\'etale} if $X \\to S$ is isomorphic to", "$$", "\\Spec(R[x]_h/(g)) \\to \\Spec(R)", "$$", "where $R \\to R[x]_h/(g)$ is a standard \\'etale ring map, see", "Algebra, Definition \\ref{algebra-definition-standard-etale},", "i.e., $g$ is monic and $g'$ invertible in $R[x]_h/(g)$.", "\\end{enumerate}" ], "refs": [ "algebra-definition-standard-etale" ], "ref_ids": [ 1540 ] }, { "id": 5568, "type": "definition", "label": "morphisms-definition-relatively-ample", "categories": [ "morphisms" ], "title": "morphisms-definition-relatively-ample", "contents": [ "\\begin{reference}", "\\cite[II Definition 4.6.1]{EGA}", "\\end{reference}", "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "We say $\\mathcal{L}$ is {\\it relatively ample}, or {\\it $f$-relatively ample},", "or {\\it ample on $X/S$}, or {\\it $f$-ample} if $f : X \\to S$", "is quasi-compact, and if for every affine open $V \\subset S$", "the restriction of $\\mathcal{L}$ to the open subscheme", "$f^{-1}(V)$ of $X$ is ample." ], "refs": [], "ref_ids": [] }, { "id": 5569, "type": "definition", "label": "morphisms-definition-very-ample", "categories": [ "morphisms" ], "title": "morphisms-definition-very-ample", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "We say $\\mathcal{L}$ is {\\it relatively very ample} or more", "precisely {\\it $f$-relatively very ample}, or", "{\\it very ample on $X/S$}, or {\\it $f$-very ample} if", "there exist a quasi-coherent $\\mathcal{O}_S$-module", "$\\mathcal{E}$ and an immersion $i : X \\to \\mathbf{P}(\\mathcal{E})$", "over $S$ such that", "$\\mathcal{L} \\cong i^*\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(1)$." ], "refs": [], "ref_ids": [] }, { "id": 5570, "type": "definition", "label": "morphisms-definition-quasi-projective", "categories": [ "morphisms" ], "title": "morphisms-definition-quasi-projective", "contents": [ "\\begin{reference}", "\\cite[II, Definition 5.3.1]{EGA} and \\cite[page 103]{H}", "\\end{reference}", "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item We say $f$ is {\\it quasi-projective} if $f$ is of finite type", "and there exists an $f$-relatively ample invertible $\\mathcal{O}_X$-module.", "\\item We say $f$ is {\\it H-quasi-projective} if there exists", "a quasi-compact immersion $X \\to \\mathbf{P}^n_S$ over $S$ for some", "$n$.\\footnote{This is not exactly the same as the definition in Hartshorne.", "Namely, the definition in Hartshorne (8th corrected printing, 1997) is that", "$f$ should be the composition of an open immersion followed by a H-projective", "morphism (see Definition \\ref{definition-projective}), which does not imply", "$f$ is quasi-compact. See", "Lemma \\ref{lemma-H-quasi-projective-open-H-projective} for", "the implication in the other direction.}", "\\item We say $f$ is {\\it locally quasi-projective} if there exists", "an open covering $S = \\bigcup V_j$ such that each $f^{-1}(V_j) \\to V_j$", "is quasi-projective.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-projective", "morphisms-lemma-H-quasi-projective-open-H-projective" ], "ref_ids": [ 5572, 5428 ] }, { "id": 5571, "type": "definition", "label": "morphisms-definition-proper", "categories": [ "morphisms" ], "title": "morphisms-definition-proper", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "We say $f$ is {\\it proper} if $f$ is separated, finite type, and", "universally closed." ], "refs": [], "ref_ids": [] }, { "id": 5572, "type": "definition", "label": "morphisms-definition-projective", "categories": [ "morphisms" ], "title": "morphisms-definition-projective", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item We say $f$ is {\\it projective} if $X$ is isomorphic as", "an $S$-scheme to a closed subscheme of a projective", "bundle $\\mathbf{P}(\\mathcal{E})$", "for some quasi-coherent, finite type $\\mathcal{O}_S$-module $\\mathcal{E}$.", "\\item We say $f$ is {\\it H-projective} if there exists an integer $n$ and", "a closed immersion $X \\to \\mathbf{P}^n_S$ over $S$.", "\\item We say $f$ is {\\it locally projective} if there exists an open", "covering $S = \\bigcup U_i$ such that each $f^{-1}(U_i) \\to U_i$ is", "projective.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5573, "type": "definition", "label": "morphisms-definition-integral", "categories": [ "morphisms" ], "title": "morphisms-definition-integral", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item We say that $f$ is {\\it integral} if $f$ is affine", "and if for every affine open $\\Spec(R) = V \\subset S$", "with inverse image $\\Spec(A) = f^{-1}(V) \\subset X$", "the associated ring map $R \\to A$ is integral.", "\\item We say that $f$ is {\\it finite} if $f$ is affine", "and if for every affine open $\\Spec(R) = V \\subset S$", "with inverse image $\\Spec(A) = f^{-1}(V) \\subset X$", "the associated ring map $R \\to A$ is finite.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5574, "type": "definition", "label": "morphisms-definition-universal-homeomorphism", "categories": [ "morphisms" ], "title": "morphisms-definition-universal-homeomorphism", "contents": [ "A morphisms $f : X \\to Y$ of schemes is called a {\\it universal homeomorphism}", "if the base change $f' : Y' \\times_Y X \\to Y'$ is a homeomorphism for", "every morphism $Y' \\to Y$." ], "refs": [], "ref_ids": [] }, { "id": 5575, "type": "definition", "label": "morphisms-definition-seminormal-ring", "categories": [ "morphisms" ], "title": "morphisms-definition-seminormal-ring", "contents": [ "Let $A$ be a ring.", "\\begin{enumerate}", "\\item We say $A$ is {\\it seminormal} if for all $x, y \\in A$", "with $x^3 = y^2$ there is a unique $a \\in A$ with", "$x = a^2$ and $y = a^3$.", "\\item We say $A$ is {\\it absolutely weakly normal} if", "(a) $A$ is seminormal and", "(b) for any prime number $p$ and $x, y \\in A$ with $p^px = y^p$", "there is a unique $a \\in A$ with $x = a^p$ and $y = pa$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5576, "type": "definition", "label": "morphisms-definition-seminormal", "categories": [ "morphisms" ], "title": "morphisms-definition-seminormal", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item We say $X$ is {\\it seminormal} if every $x \\in X$ has", "an affine open neighbourhood $\\Spec(R) = U \\subset X$", "such that the ring $R$ is seminormal.", "\\item We say $X$ is {\\it absolutely weakly normal} if every $x \\in X$ has", "an affine open neighbourhood $\\Spec(R) = U \\subset X$", "such that the ring $R$ is absolutely weakly normal.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5577, "type": "definition", "label": "morphisms-definition-seminormalization", "categories": [ "morphisms" ], "title": "morphisms-definition-seminormalization", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item The morphism $X^{sn} \\to X$ constructed in", "Lemma \\ref{lemma-seminormalization}", "is the {\\it seminormalization} of $X$.", "\\item The morphism $X^{awn} \\to X$ constructed in", "Lemma \\ref{lemma-seminormalization}", "is the {\\it absolute weak normalization} of $X$.", "\\end{enumerate}" ], "refs": [ "morphisms-lemma-seminormalization", "morphisms-lemma-seminormalization" ], "ref_ids": [ 5470, 5470 ] }, { "id": 5578, "type": "definition", "label": "morphisms-definition-finite-locally-free", "categories": [ "morphisms" ], "title": "morphisms-definition-finite-locally-free", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "We say $f$ is {\\it finite locally free} if $f$ is", "affine and $f_*\\mathcal{O}_X$ is a finite locally", "free $\\mathcal{O}_S$-module. In this case we say $f$ is", "has {\\it rank} or {\\it degree} $d$", "if the sheaf $f_*\\mathcal{O}_X$ is finite locally free of degree $d$." ], "refs": [], "ref_ids": [] }, { "id": 5579, "type": "definition", "label": "morphisms-definition-rational-map", "categories": [ "morphisms" ], "title": "morphisms-definition-rational-map", "contents": [ "Let $X$, $Y$ be schemes.", "\\begin{enumerate}", "\\item Let $f : U \\to Y$, $g : V \\to Y$ be morphisms of schemes defined", "on dense open subsets $U$, $V$ of $X$. We say that $f$ is", "{\\it equivalent} to $g$ if $f|_W = g|_W$ for some $W \\subset U \\cap V$", "dense open in $X$.", "\\item A {\\it rational map from $X$ to $Y$}", "is an equivalence class for the equivalence relation defined in (1).", "\\item If $X$, $Y$ are schemes over a base scheme $S$ we say that", "a rational map from $X$ to $Y$ is an {\\it $S$-rational map from $X$", "to $Y$} if there exists a representative $f : U \\to Y$ of the equivalence", "class which is an $S$-morphism.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5580, "type": "definition", "label": "morphisms-definition-rational-function", "categories": [ "morphisms" ], "title": "morphisms-definition-rational-function", "contents": [ "Let $X$ be a scheme. A {\\it rational function on $X$} is a rational map", "from $X$ to $\\mathbf{A}^1_{\\mathbf{Z}}$." ], "refs": [], "ref_ids": [] }, { "id": 5581, "type": "definition", "label": "morphisms-definition-ring-of-rational-functions", "categories": [ "morphisms" ], "title": "morphisms-definition-ring-of-rational-functions", "contents": [ "Let $X$ be a scheme. The {\\it ring of rational functions on $X$}", "is the ring $R(X)$ whose elements are rational functions with", "addition and multiplication as just described." ], "refs": [], "ref_ids": [] }, { "id": 5582, "type": "definition", "label": "morphisms-definition-function-field", "categories": [ "morphisms" ], "title": "morphisms-definition-function-field", "contents": [ "Let $X$ be an integral scheme.", "The {\\it function field}, or the {\\it field of rational functions}", "of $X$ is the field $R(X)$." ], "refs": [], "ref_ids": [] }, { "id": 5583, "type": "definition", "label": "morphisms-definition-domain-of-definition", "categories": [ "morphisms" ], "title": "morphisms-definition-domain-of-definition", "contents": [ "Let $\\varphi$ be a rational map between two schemes $X$ and $Y$. We say", "$\\varphi$ is {\\it defined in a point $x \\in X$} if there exists a", "representative $(U, f)$ of $\\varphi$ with $x \\in U$. The", "{\\it domain of definition} of $\\varphi$ is the set of all points", "where $\\varphi$ is defined." ], "refs": [], "ref_ids": [] }, { "id": 5584, "type": "definition", "label": "morphisms-definition-dominant-rational", "categories": [ "morphisms" ], "title": "morphisms-definition-dominant-rational", "contents": [ "Let $X$ and $Y$ be irreducible schemes. A rational map from $X$ to $Y$", "is called {\\it dominant} if any representative $f : U \\to Y$ is a dominant", "morphism of schemes." ], "refs": [], "ref_ids": [] }, { "id": 5585, "type": "definition", "label": "morphisms-definition-birational-schemes", "categories": [ "morphisms" ], "title": "morphisms-definition-birational-schemes", "contents": [ "Let $X$ and $Y$ be irreducible schemes.", "\\begin{enumerate}", "\\item We say $X$ and $Y$ are {\\it birational} if $X$ and $Y$ are isomorphic", "in the category of irreducible schemes and dominant rational maps.", "\\item Assume $X$ and $Y$ are schemes over a base scheme $S$.", "We say $X$ and $Y$ are {\\it $S$-birational} if $X$ and $Y$ are", "isomorphic in the category of irreducible schemes over $S$ and", "dominant $S$-rational maps.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5586, "type": "definition", "label": "morphisms-definition-birational", "categories": [ "morphisms" ], "title": "morphisms-definition-birational", "contents": [ "\\begin{reference}", "\\cite[(2.2.9)]{EGA1}", "\\end{reference}", "Let $X$, $Y$ be schemes. Assume $X$ and $Y$ have finitely many", "irreducible components. We say a morphism $f : X \\to Y$ is", "{\\it birational} if", "\\begin{enumerate}", "\\item $f$ induces a bijection between the set of generic points", "of irreducible components of $X$ and the set of generic points", "of the irreducible components of $Y$, and", "\\item for every generic point $\\eta \\in X$ of an irreducible component", "of $X$ the local ring map", "$\\mathcal{O}_{Y, f(\\eta)} \\to \\mathcal{O}_{X, \\eta}$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5587, "type": "definition", "label": "morphisms-definition-degree", "categories": [ "morphisms" ], "title": "morphisms-definition-degree", "contents": [ "Let $X$ and $Y$ be integral schemes.", "Let $f : X \\to Y$ be locally of finite type and dominant.", "Assume $[R(X) : R(Y)] < \\infty$, or any other of the equivalent", "conditions (1) -- (4) of Lemma \\ref{lemma-finite-degree}.", "Then the positive integer", "$$", "\\text{deg}(X/Y) = [R(X) : R(Y)]", "$$", "is called the {\\it degree of $X$ over $Y$}." ], "refs": [ "morphisms-lemma-finite-degree" ], "ref_ids": [ 5491 ] }, { "id": 5588, "type": "definition", "label": "morphisms-definition-modification", "categories": [ "morphisms" ], "title": "morphisms-definition-modification", "contents": [ "Let $X$ be an integral scheme. A {\\it modification of $X$}", "is a birational proper morphism $f : X' \\to X$ with $X'$", "integral." ], "refs": [], "ref_ids": [] }, { "id": 5589, "type": "definition", "label": "morphisms-definition-alteration", "categories": [ "morphisms" ], "title": "morphisms-definition-alteration", "contents": [ "\\begin{reference}", "\\cite[Definition 2.20]{alterations}", "\\end{reference}", "Let $X$ be an integral scheme. An {\\it alteration of $X$}", "is a proper dominant morphism $f : Y \\to X$ with $Y$ integral such", "that $f^{-1}(U) \\to U$ is finite for some nonempty open $U \\subset X$." ], "refs": [], "ref_ids": [] }, { "id": 5590, "type": "definition", "label": "morphisms-definition-integral-closure", "categories": [ "morphisms" ], "title": "morphisms-definition-integral-closure", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent sheaf", "of $\\mathcal{O}_X$-algebras. The {\\it integral closure of $\\mathcal{O}_X$", "in $\\mathcal{A}$} is the quasi-coherent $\\mathcal{O}_X$-subalgebra", "$\\mathcal{A}' \\subset \\mathcal{A}$ constructed in", "Lemma \\ref{lemma-integral-closure} above." ], "refs": [ "morphisms-lemma-integral-closure" ], "ref_ids": [ 5498 ] }, { "id": 5591, "type": "definition", "label": "morphisms-definition-normalization-X-in-Y", "categories": [ "morphisms" ], "title": "morphisms-definition-normalization-X-in-Y", "contents": [ "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes.", "Let $\\mathcal{O}'$ be the integral closure of $\\mathcal{O}_X$ in", "$f_*\\mathcal{O}_Y$. The {\\it normalization of $X$ in $Y$} is the", "scheme\\footnote{The scheme $X'$ need not be normal, for example if", "$Y = X$ and $f = \\text{id}_X$, then $X' = X$.}", "$$", "\\nu : X' = \\underline{\\Spec}_X(\\mathcal{O}') \\to X", "$$", "over $X$. It comes equipped with a natural factorization", "$$", "Y \\xrightarrow{f'} X' \\xrightarrow{\\nu} X", "$$", "of the initial morphism $f$." ], "refs": [], "ref_ids": [] }, { "id": 5592, "type": "definition", "label": "morphisms-definition-normalization", "categories": [ "morphisms" ], "title": "morphisms-definition-normalization", "contents": [ "Let $X$ be a scheme such that every quasi-compact open has", "finitely many irreducible components. We define the", "{\\it normalization} of $X$ as the morphism", "$$", "\\nu : X^\\nu \\longrightarrow X", "$$", "which is the normalization of $X$ in the morphism $f : Y \\to X$", "(\\ref{equation-generic-points}) constructed above." ], "refs": [], "ref_ids": [] }, { "id": 5593, "type": "definition", "label": "morphisms-definition-universally-bounded", "categories": [ "morphisms" ], "title": "morphisms-definition-universally-bounded", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "\\begin{enumerate}", "\\item We say the integer $n$ {\\it bounds the degrees of the fibres", "of $f$} if for all $y \\in Y$", "the fibre $X_y$ is a finite scheme over $\\kappa(y)$ whose", "degree over $\\kappa(y)$ is $\\leq n$.", "\\item We say the {\\it fibres of $f$ are universally bounded}\\footnote{This is", "probably nonstandard notation.}", "if there exists an integer $n$ which bounds the degrees of the fibres", "of $f$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5647, "type": "definition", "label": "smoothing-definition-singular-ideal", "categories": [ "smoothing" ], "title": "smoothing-definition-singular-ideal", "contents": [ "Let $R \\to A$ be a ring map. The {\\it singular ideal of $A$ over $R$},", "denoted $H_{A/R}$ is the unique radical ideal $H_{A/R} \\subset A$ with", "$$", "V(H_{A/R}) = \\{\\mathfrak q \\in \\Spec(A) \\mid R \\to A", "\\text{ not smooth at }\\mathfrak q\\}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 5648, "type": "definition", "label": "smoothing-definition-strictly-standard", "categories": [ "smoothing" ], "title": "smoothing-definition-strictly-standard", "contents": [ "Let $R \\to A$ be a ring map of finite presentation.", "We say an element $a \\in A$ is {\\it elementary standard in $A$ over $R$}", "if there exists a presentation", "$A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$", "and $0 \\leq c \\leq \\min(n, m)$ such that", "\\begin{equation}", "\\label{equation-elementary-standard-one}", "a = a' \\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}", "\\end{equation}", "for some $a' \\in A$ and", "\\begin{equation}", "\\label{equation-elementary-standard-two}", "a f_{c + j} \\in (f_1, \\ldots, f_c) + (f_1, \\ldots, f_m)^2", "\\end{equation}", "for $j = 1, \\ldots, m - c$. We say $a \\in A$ is", "{\\it strictly standard in $A$ over $R$} if there exists a presentation", "$A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$", "and $0 \\leq c \\leq \\min(n, m)$ such that", "\\begin{equation}", "\\label{equation-strictly-standard-one}", "a = \\sum\\nolimits_{I \\subset \\{1, \\ldots, n\\},\\ |I| = c}", "a_I \\det(\\partial f_j/\\partial x_i)_{j = 1, \\ldots, c,\\ i \\in I}", "\\end{equation}", "for some $a_I \\in A$ and", "\\begin{equation}", "\\label{equation-strictly-standard-two}", "a f_{c + j} \\in (f_1, \\ldots, f_c) + (f_1, \\ldots, f_m)^2", "\\end{equation}", "for $j = 1, \\ldots, m - c$." ], "refs": [], "ref_ids": [] }, { "id": 5903, "type": "definition", "label": "chow-definition-periodic-complex", "categories": [ "chow" ], "title": "chow-definition-periodic-complex", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item A {\\it $2$-periodic complex} over $R$ is given", "by a quadruple $(M, N, \\varphi, \\psi)$ consisting of", "$R$-modules $M$, $N$ and $R$-module maps $\\varphi : M \\to N$,", "$\\psi : N \\to M$ such that", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "M \\ar[r]^\\varphi &", "N \\ar[r]^\\psi &", "M \\ar[r]^\\varphi &", "N \\ar[r] & \\ldots", "}", "$$", "is a complex. In this setting we define the {\\it cohomology modules}", "of the complex to be the $R$-modules", "$$", "H^0(M, N, \\varphi, \\psi) = \\Ker(\\varphi)/\\Im(\\psi)", "\\quad\\text{and}\\quad", "H^1(M, N, \\varphi, \\psi) = \\Ker(\\psi)/\\Im(\\varphi).", "$$", "We say the $2$-periodic complex is {\\it exact} if the cohomology", "groups are zero.", "\\item A {\\it $(2, 1)$-periodic complex} over $R$ is given", "by a triple $(M, \\varphi, \\psi)$ consisting of an $R$-module $M$ and", "$R$-module maps $\\varphi : M \\to M$, $\\psi : M \\to M$", "such that", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "M \\ar[r]^\\varphi &", "M \\ar[r]^\\psi &", "M \\ar[r]^\\varphi &", "M \\ar[r] & \\ldots", "}", "$$", "is a complex. Since this is a special case of a $2$-periodic complex", "we have its {\\it cohomology modules} $H^0(M, \\varphi, \\psi)$,", "$H^1(M, \\varphi, \\psi)$ and a notion of exactness.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5904, "type": "definition", "label": "chow-definition-periodic-length", "categories": [ "chow" ], "title": "chow-definition-periodic-length", "contents": [ "Let $(M, N, \\varphi, \\psi)$ be a $2$-periodic complex", "over a ring $R$ whose cohomology modules have finite length.", "In this case we define the {\\it multiplicity} of $(M, N, \\varphi, \\psi)$", "to be the integer", "$$", "e_R(M, N, \\varphi, \\psi) =", "\\text{length}_R(H^0(M, N, \\varphi, \\psi))", "-", "\\text{length}_R(H^1(M, N, \\varphi, \\psi))", "$$", "In the case of a $(2, 1)$-periodic complex $(M, \\varphi, \\psi)$,", "we denote this by $e_R(M, \\varphi, \\psi)$ and we will sometimes call this", "the {\\it (additive) Herbrand quotient}." ], "refs": [], "ref_ids": [] }, { "id": 5905, "type": "definition", "label": "chow-definition-delta-dimension", "categories": [ "chow" ], "title": "chow-definition-delta-dimension", "contents": [ "Let $(S, \\delta)$ as in Situation \\ref{situation-setup}.", "For any scheme $X$ locally of finite type over $S$", "and any irreducible closed subset $Z \\subset X$ we define", "$$", "\\dim_\\delta(Z) = \\delta(\\xi)", "$$", "where $\\xi \\in Z$ is the generic point of $Z$.", "We will call this the {\\it $\\delta$-dimension of $Z$}.", "If $Z$ is a closed subscheme of $X$, then we define", "$\\dim_\\delta(Z)$ as the supremum of the $\\delta$-dimensions", "of its irreducible components." ], "refs": [], "ref_ids": [] }, { "id": 5906, "type": "definition", "label": "chow-definition-cycles", "categories": [ "chow" ], "title": "chow-definition-cycles", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $k \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item A {\\it cycle on $X$} is a formal sum", "$$", "\\alpha = \\sum n_Z [Z]", "$$", "where the sum is over integral closed subschemes $Z \\subset X$,", "each $n_Z \\in \\mathbf{Z}$, and the collection", "$\\{Z; n_Z \\not = 0\\}$ is locally finite", "(Topology, Definition \\ref{topology-definition-locally-finite}).", "\\item A {\\it $k$-cycle} on $X$ is a cycle", "$$", "\\alpha = \\sum n_Z [Z]", "$$", "where $n_Z \\not = 0 \\Rightarrow \\dim_\\delta(Z) = k$.", "\\item The abelian group of all $k$-cycles on $X$ is denoted $Z_k(X)$.", "\\end{enumerate}" ], "refs": [ "topology-definition-locally-finite" ], "ref_ids": [ 8376 ] }, { "id": 5907, "type": "definition", "label": "chow-definition-cycle-associated-to-closed-subscheme", "categories": [ "chow" ], "title": "chow-definition-cycle-associated-to-closed-subscheme", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $Z \\subset X$ be a closed subscheme.", "\\begin{enumerate}", "\\item For any irreducible component $Z' \\subset Z$ with generic point $\\xi$", "the integer", "$m_{Z', Z} = \\text{length}_{\\mathcal{O}_{X, \\xi}} \\mathcal{O}_{Z, \\xi}$", "(Lemma \\ref{lemma-multiplicity-finite})", "is called the {\\it multiplicity of $Z'$ in $Z$}.", "\\item Assume $\\dim_\\delta(Z) \\leq k$.", "The {\\it $k$-cycle associated to $Z$} is", "$$", "[Z]_k", "=", "\\sum m_{Z', Z}[Z']", "$$", "where the sum is over the irreducible components of $Z$", "of $\\delta$-dimension $k$. (This is a $k$-cycle by", "Divisors, Lemma \\ref{divisors-lemma-components-locally-finite}.)", "\\end{enumerate}" ], "refs": [ "chow-lemma-multiplicity-finite", "divisors-lemma-components-locally-finite" ], "ref_ids": [ 5668, 8022 ] }, { "id": 5908, "type": "definition", "label": "chow-definition-cycle-associated-to-coherent-sheaf", "categories": [ "chow" ], "title": "chow-definition-cycle-associated-to-coherent-sheaf", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item For any irreducible component $Z' \\subset \\text{Supp}(\\mathcal{F})$", "with generic point $\\xi$ the integer", "$m_{Z', \\mathcal{F}} = \\text{length}_{\\mathcal{O}_{X, \\xi}} \\mathcal{F}_\\xi$", "(Lemma \\ref{lemma-length-finite})", "is called the {\\it multiplicity of $Z'$ in $\\mathcal{F}$}.", "\\item Assume $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$.", "The {\\it $k$-cycle associated to $\\mathcal{F}$} is", "$$", "[\\mathcal{F}]_k", "=", "\\sum m_{Z', \\mathcal{F}}[Z']", "$$", "where the sum is over the irreducible components of", "$\\text{Supp}(\\mathcal{F})$ of $\\delta$-dimension $k$.", "(This is a $k$-cycle by Lemma \\ref{lemma-length-finite}.)", "\\end{enumerate}" ], "refs": [ "chow-lemma-length-finite", "chow-lemma-length-finite" ], "ref_ids": [ 5669, 5669 ] }, { "id": 5909, "type": "definition", "label": "chow-definition-proper-pushforward", "categories": [ "chow" ], "title": "chow-definition-proper-pushforward", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $f : X \\to Y$ be a morphism.", "Assume $f$ is proper.", "\\begin{enumerate}", "\\item Let $Z \\subset X$ be an integral closed subscheme", "with $\\dim_\\delta(Z) = k$. We define", "$$", "f_*[Z] =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & \\dim_\\delta(f(Z))< k, \\\\", "\\deg(Z/f(Z)) [f(Z)] & \\text{if} & \\dim_\\delta(f(Z)) = k.", "\\end{matrix}", "\\right.", "$$", "Here we think of $f(Z) \\subset Y$ as an integral closed subscheme.", "The degree of $Z$ over $f(Z)$ is finite if", "$\\dim_\\delta(f(Z)) = \\dim_\\delta(Z)$", "by Lemma \\ref{lemma-equal-dimension}.", "\\item Let $\\alpha = \\sum n_Z [Z]$ be a $k$-cycle on $X$. The", "{\\it pushforward} of $\\alpha$ as the sum", "$$", "f_* \\alpha = \\sum n_Z f_*[Z]", "$$", "where each $f_*[Z]$ is defined as above. The sum is locally finite", "by Lemma \\ref{lemma-quasi-compact-locally-finite} above.", "\\end{enumerate}" ], "refs": [ "chow-lemma-equal-dimension", "chow-lemma-quasi-compact-locally-finite" ], "ref_ids": [ 5672, 5673 ] }, { "id": 5910, "type": "definition", "label": "chow-definition-flat-pullback", "categories": [ "chow" ], "title": "chow-definition-flat-pullback", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$, $Y$ be locally of finite type over $S$.", "Let $f : X \\to Y$ be a morphism.", "Assume $f$ is flat of relative dimension $r$.", "\\begin{enumerate}", "\\item Let $Z \\subset Y$ be an integral closed subscheme of", "$\\delta$-dimension $k$. We define $f^*[Z]$ to be the", "$(k+r)$-cycle on $X$ to the scheme theoretic inverse image", "$$", "f^*[Z] = [f^{-1}(Z)]_{k+r}.", "$$", "This makes sense since $\\dim_\\delta(f^{-1}(Z)) = k + r$", "by Lemma \\ref{lemma-flat-inverse-image-dimension}.", "\\item Let $\\alpha = \\sum n_i [Z_i]$ be", "a $k$-cycle on $Y$. The {\\it flat pullback of $\\alpha$ by $f$}", "is the sum", "$$", "f^* \\alpha = \\sum n_i f^*[Z_i]", "$$", "where each $f^*[Z_i]$ is defined as above.", "The sum is locally finite by Lemma \\ref{lemma-inverse-image-locally-finite}.", "\\item We denote $f^* : Z_k(Y) \\to Z_{k + r}(X)$ the map of abelian", "groups so obtained.", "\\end{enumerate}" ], "refs": [ "chow-lemma-flat-inverse-image-dimension", "chow-lemma-inverse-image-locally-finite" ], "ref_ids": [ 5677, 5678 ] }, { "id": 5911, "type": "definition", "label": "chow-definition-principal-divisor", "categories": [ "chow" ], "title": "chow-definition-principal-divisor", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. Assume $X$ is", "integral with $\\dim_\\delta(X) = n$.", "Let $f \\in R(X)^*$. The {\\it principal divisor", "associated to $f$} is the $(n - 1)$-cycle", "$$", "\\text{div}(f) = \\text{div}_X(f) = \\sum \\text{ord}_Z(f) [Z]", "$$", "defined in Divisors, Definition \\ref{divisors-definition-principal-divisor}.", "This makes sense because prime divisors have $\\delta$-dimension $n - 1$ by", "Lemma \\ref{lemma-divisor-delta-dimension}." ], "refs": [ "divisors-definition-principal-divisor", "chow-lemma-divisor-delta-dimension" ], "ref_ids": [ 8108, 5684 ] }, { "id": 5912, "type": "definition", "label": "chow-definition-rational-equivalence", "categories": [ "chow" ], "title": "chow-definition-rational-equivalence", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be a scheme locally of finite type over $S$.", "Let $k \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item Given any locally finite collection $\\{W_j \\subset X\\}$", "of integral closed subschemes with $\\dim_\\delta(W_j) = k + 1$,", "and any $f_j \\in R(W_j)^*$ we may consider", "$$", "\\sum (i_j)_*\\text{div}(f_j) \\in Z_k(X)", "$$", "where $i_j : W_j \\to X$ is the inclusion morphism.", "This makes sense as the morphism", "$\\coprod i_j : \\coprod W_j \\to X$ is proper.", "\\item We say that $\\alpha \\in Z_k(X)$ is {\\it rationally equivalent to zero}", "if $\\alpha$ is a cycle of the form displayed above.", "\\item We say $\\alpha, \\beta \\in Z_k(X)$ are", "{\\it rationally equivalent} and we write $\\alpha \\sim_{rat} \\beta$", "if $\\alpha - \\beta$ is rationally equivalent to zero.", "\\item We define", "$$", "\\CH_k(X) = Z_k(X) / \\sim_{rat}", "$$", "to be the {\\it Chow group of $k$-cycles on $X$}. This is sometimes called", "the {\\it Chow group of $k$-cycles modulo rational equivalence on $X$}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5913, "type": "definition", "label": "chow-definition-divisor-invertible-sheaf", "categories": [ "chow" ], "title": "chow-definition-divisor-invertible-sheaf", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. Assume $X$ is", "integral and $n = \\dim_\\delta(X)$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item For any nonzero meromorphic section $s$ of $\\mathcal{L}$", "we define the {\\it Weil divisor associated to $s$} is the", "$(n - 1)$-cycle", "$$", "\\text{div}_\\mathcal{L}(s) =", "\\sum \\text{ord}_{Z, \\mathcal{L}}(s) [Z]", "$$", "defined in Divisors, Definition", "\\ref{divisors-definition-divisor-invertible-sheaf}.", "This makes sense because Weil divisors have $\\delta$-dimension $n - 1$", "by Lemma \\ref{lemma-divisor-delta-dimension}.", "\\item We define {\\it Weil divisor associated to $\\mathcal{L}$} as", "$$", "c_1(\\mathcal{L}) \\cap [X] =", "\\text{class of }\\text{div}_\\mathcal{L}(s) \\in \\CH_{n - 1}(X)", "$$", "where $s$ is any nonzero meromorphic section of $\\mathcal{L}$ over", "$X$. This is well defined by", "Divisors, Lemma \\ref{divisors-lemma-divisor-meromorphic-well-defined}.", "\\end{enumerate}" ], "refs": [ "divisors-definition-divisor-invertible-sheaf", "chow-lemma-divisor-delta-dimension", "divisors-lemma-divisor-meromorphic-well-defined" ], "ref_ids": [ 8111, 5684, 8026 ] }, { "id": 5914, "type": "definition", "label": "chow-definition-cap-c1", "categories": [ "chow" ], "title": "chow-definition-cap-c1", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "We define, for every integer $k$, an operation", "$$", "c_1(\\mathcal{L}) \\cap - :", "Z_{k + 1}(X) \\to \\CH_k(X)", "$$", "called {\\it intersection with the first Chern class of $\\mathcal{L}$}.", "\\begin{enumerate}", "\\item Given an integral closed subscheme $i : W \\to X$ with", "$\\dim_\\delta(W) = k + 1$ we define", "$$", "c_1(\\mathcal{L}) \\cap [W] = i_*(c_1({i^*\\mathcal{L}}) \\cap [W])", "$$", "where the right hand side is defined in", "Definition \\ref{definition-divisor-invertible-sheaf}.", "\\item For a general $(k + 1)$-cycle $\\alpha = \\sum n_i [W_i]$ we set", "$$", "c_1(\\mathcal{L}) \\cap \\alpha = \\sum n_i c_1(\\mathcal{L}) \\cap [W_i]", "$$", "\\end{enumerate}" ], "refs": [ "chow-definition-divisor-invertible-sheaf" ], "ref_ids": [ 5913 ] }, { "id": 5915, "type": "definition", "label": "chow-definition-gysin-homomorphism", "categories": [ "chow" ], "title": "chow-definition-gysin-homomorphism", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $(\\mathcal{L}, s)$ be a pair consisting of an invertible", "sheaf and a global section $s \\in \\Gamma(X, \\mathcal{L})$.", "Let $D = Z(s)$ be the zero scheme of $s$, and", "denote $i : D \\to X$ the closed immersion.", "We define, for every integer $k$, a {\\it Gysin homomorphism}", "$$", "i^* : Z_{k + 1}(X) \\to \\CH_k(D).", "$$", "by the following rules:", "\\begin{enumerate}", "\\item Given a integral closed subscheme $W \\subset X$ with", "$\\dim_\\delta(W) = k + 1$ we define", "\\begin{enumerate}", "\\item if $W \\not \\subset D$, then $i^*[W] = [D \\cap W]_k$ as a", "$k$-cycle on $D$, and", "\\item if $W \\subset D$, then", "$i^*[W] = i'_*(c_1(\\mathcal{L}|_W) \\cap [W])$,", "where $i' : W \\to D$ is the induced closed immersion.", "\\end{enumerate}", "\\item For a general $(k + 1)$-cycle $\\alpha = \\sum n_j[W_j]$", "we set", "$$", "i^*\\alpha = \\sum n_j i^*[W_j]", "$$", "\\item If $D$ is an effective Cartier divisor, then we denote", "$D \\cdot \\alpha = i_*i^*\\alpha$ the pushforward of the class $i^*\\alpha$", "to a class on $X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5916, "type": "definition", "label": "chow-definition-bivariant-class", "categories": [ "chow" ], "title": "chow-definition-bivariant-class", "contents": [ "\\begin{reference}", "Similar to \\cite[Definition 17.1]{F}", "\\end{reference}", "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$.", "Let $p \\in \\mathbf{Z}$.", "A {\\it bivariant class $c$ of degree $p$ for $f$} is given by a rule", "which assigns to every locally of finite type morphism $Y' \\to Y$", "and every $k$ a map", "$$", "c \\cap - : \\CH_k(Y') \\longrightarrow \\CH_{k - p}(X')", "$$", "where $X' = Y' \\times_Y X$, satisfying the following conditions", "\\begin{enumerate}", "\\item if $Y'' \\to Y'$ is a proper, then", "$c \\cap (Y'' \\to Y')_*\\alpha'' = (X'' \\to X')_*(c \\cap \\alpha'')$", "for all $\\alpha''$ on $Y''$ where $X'' = Y'' \\times_Y X$,", "\\item if $Y'' \\to Y'$ is flat locally of finite type of", "fixed relative dimension, then", "$c \\cap (Y'' \\to Y')^*\\alpha' = (X'' \\to X')^*(c \\cap \\alpha')$", "for all $\\alpha'$ on $Y'$, and", "\\item if $(\\mathcal{L}', s', i' : D' \\to Y')$ is as in", "Definition \\ref{definition-gysin-homomorphism}", "with pullback $(\\mathcal{N}', t', j' : E' \\to X')$ to $X'$,", "then we have $c \\cap (i')^*\\alpha' = (j')^*(c \\cap \\alpha')$", "for all $\\alpha'$ on $Y'$.", "\\end{enumerate}", "The collection of all bivariant classes of degree $p$ for $f$ is", "denoted $A^p(X \\to Y)$." ], "refs": [ "chow-definition-gysin-homomorphism" ], "ref_ids": [ 5915 ] }, { "id": 5917, "type": "definition", "label": "chow-definition-chow-cohomology", "categories": [ "chow" ], "title": "chow-definition-chow-cohomology", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. The {\\it Chow cohomology}", "of $X$ is the graded $\\mathbf{Z}$-algebra $A^*(X)$ whose degree", "$p$ component is $A^p(X \\to X)$." ], "refs": [], "ref_ids": [] }, { "id": 5918, "type": "definition", "label": "chow-definition-first-chern-class", "categories": [ "chow" ], "title": "chow-definition-first-chern-class", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$", "be locally of finite type over $S$. Let $\\mathcal{L}$ be an invertible", "$\\mathcal{O}_X$-module. The {\\it first Chern class}", "$c_1(\\mathcal{L}) \\in A^1(X)$ of $\\mathcal{L}$", "is the bivariant class of Lemma \\ref{lemma-cap-c1-bivariant}." ], "refs": [ "chow-lemma-cap-c1-bivariant" ], "ref_ids": [ 5735 ] }, { "id": 5919, "type": "definition", "label": "chow-definition-chern-classes", "categories": [ "chow" ], "title": "chow-definition-chern-classes", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Assume $X$ is integral and $n = \\dim_\\delta(X)$.", "Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$", "on $X$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$ be the projective space", "bundle associated to $\\mathcal{E}$.", "\\begin{enumerate}", "\\item By Lemma \\ref{lemma-chow-ring-projective-bundle} there are", "elements $c_i \\in \\CH_{n - i}(X)$, $i = 0, \\ldots, r$", "such that $c_0 = [X]$, and", "\\begin{equation}", "\\label{equation-chern-classes}", "\\sum\\nolimits_{i = 0}^r", "(-1)^i c_1(\\mathcal{O}_P(1))^i \\cap \\pi^*c_{r - i}", "= 0.", "\\end{equation}", "\\item With notation as above we set", "$c_i(\\mathcal{E}) \\cap [X] = c_i$", "as an element of $\\CH_{n - i}(X)$.", "We call these the {\\it Chern classes of $\\mathcal{E}$ on $X$}.", "\\item The {\\it total Chern class of $\\mathcal{E}$ on $X$}", "is the combination", "$$", "c({\\mathcal E}) \\cap [X] =", "c_0({\\mathcal E}) \\cap [X]", "+ c_1({\\mathcal E}) \\cap [X] + \\ldots", "+ c_r({\\mathcal E}) \\cap [X]", "$$", "which is an element of", "$\\CH_*(X) = \\bigoplus_{k \\in \\mathbf{Z}} \\CH_k(X)$.", "\\end{enumerate}" ], "refs": [ "chow-lemma-chow-ring-projective-bundle" ], "ref_ids": [ 5743 ] }, { "id": 5920, "type": "definition", "label": "chow-definition-cap-chern-classes", "categories": [ "chow" ], "title": "chow-definition-cap-chern-classes", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.", "We define, for every integer $k$ and any $0 \\leq j \\leq r$,", "an operation", "$$", "c_j(\\mathcal{E}) \\cap - : Z_k(X) \\to \\CH_{k - j}(X)", "$$", "called {\\it intersection with the $j$th Chern class of $\\mathcal{E}$}.", "\\begin{enumerate}", "\\item Given an integral closed subscheme $i : W \\to X$ of $\\delta$-dimension", "$k$ we define", "$$", "c_j(\\mathcal{E}) \\cap [W] = i_*(c_j({i^*\\mathcal{E}}) \\cap [W])", "\\in", "\\CH_{k - j}(X)", "$$", "where $c_j({i^*\\mathcal{E}}) \\cap [W]$ is as defined in", "Definition \\ref{definition-chern-classes}.", "\\item For a general $k$-cycle $\\alpha = \\sum n_i [W_i]$ we set", "$$", "c_j(\\mathcal{E}) \\cap \\alpha = \\sum n_i c_j(\\mathcal{E}) \\cap [W_i]", "$$", "\\end{enumerate}" ], "refs": [ "chow-definition-chern-classes" ], "ref_ids": [ 5919 ] }, { "id": 5921, "type": "definition", "label": "chow-definition-chern-classes-final", "categories": [ "chow" ], "title": "chow-definition-chern-classes-final", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module", "of rank $r$. For $i = 0, \\ldots, r$ the {\\it $i$th Chern class}", "of $\\mathcal{E}$ is the bivariant class", "$c_i(\\mathcal{E}) \\in A^i(X)$ of degree $i$", "constructed in Lemma \\ref{lemma-cap-cp-bivariant}. The", "{\\it total Chern class} of $\\mathcal{E}$ is the formal sum", "$$", "c(\\mathcal{E}) = ", "c_0(\\mathcal{E}) + c_1(\\mathcal{E}) + \\ldots + c_r(\\mathcal{E})", "$$", "which is viewed as a nonhomogeneous bivariant class on $X$." ], "refs": [ "chow-lemma-cap-cp-bivariant" ], "ref_ids": [ 5751 ] }, { "id": 5922, "type": "definition", "label": "chow-definition-degree-zero-cycle", "categories": [ "chow" ], "title": "chow-definition-degree-zero-cycle", "contents": [ "Let $k$ be a field (Example \\ref{example-field}). Let $p : X \\to \\Spec(k)$", "be proper. The {\\it degree of a zero cycle} on $X$ is given by proper", "pushforward", "$$", "p_* : \\CH_0(X) \\to \\CH_0(\\Spec(k))", "$$", "(Lemma \\ref{lemma-proper-pushforward-rational-equivalence})", "combined with the natural isomorphism $\\CH_0(\\Spec(k)) = \\mathbf{Z}$", "which maps $[\\Spec(k)]$ to $1$. Notation: $\\deg(\\alpha)$." ], "refs": [ "chow-lemma-proper-pushforward-rational-equivalence" ], "ref_ids": [ 5694 ] }, { "id": 5923, "type": "definition", "label": "chow-definition-defined-on-perfect", "categories": [ "chow" ], "title": "chow-definition-defined-on-perfect", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. Let $E \\in D(\\mathcal{O}_X)$", "be a perfect object. If $E$ is isomorphic in $D(\\mathcal{O}_X)$", "to a finite complex $\\mathcal{E}^\\bullet$ of finite locally free", "$\\mathcal{O}_X$-modules, then we say {\\it Chern classes of $E$ are defined}.", "If this is the case, then we define $c(E) = c(\\mathcal{E}^\\bullet) \\in A^*(X)$,", "$ch(E) = ch(\\mathcal{E}^\\bullet) \\in A^*(X) \\otimes \\mathbf{Q}$, and", "$P_p(E) = P_p(\\mathcal{E}^\\bullet) \\in A^p(X)$." ], "refs": [], "ref_ids": [] }, { "id": 5924, "type": "definition", "label": "chow-definition-localized-chern", "categories": [ "chow" ], "title": "chow-definition-localized-chern", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme", "locally of finite type over $S$. Let $i : Z \\to X$ be a closed immersion.", "Let $E \\in D(\\mathcal{O}_X)$ be a perfect object whose Chern classes", "are defined.", "\\begin{enumerate}", "\\item If the restriction $E|_{X \\setminus Z}$ is zero, then for all", "$p \\geq 0$ we define", "$$", "P_p(Z \\to X, E) \\in A^p(Z \\to X)", "$$", "by the construction given above and we define the", "{\\it localized Chern character} by the formula", "$$", "ch(Z \\to X, E) =", "\\sum\\nolimits_{p = 0, 1, 2, \\ldots} \\frac{P_p(Z \\to X, E)}{p!}", "\\quad\\text{in}\\quad A^*(Z \\to X) \\otimes \\mathbf{Q}", "$$", "\\item If the restriction $E|_{X \\setminus Z}$ is isomorphic to a", "finite locally free $\\mathcal{O}_{X \\setminus Z}$-module of rank $< p$", "sitting in cohomological degree $0$, then we define the", "{\\it localized $p$th Chern class} $c_p(Z \\to X, E)$ by the construction above.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5925, "type": "definition", "label": "chow-definition-lci-gysin", "categories": [ "chow" ], "title": "chow-definition-lci-gysin", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ be a local complete intersection morphism", "of schemes locally of finite type over $S$. We say", "{\\it the gysin map for $f$ exists} if we can write", "$f = g \\circ i$ with $g$ smooth and $i$ an immersion.", "In this case we define the", "{\\it gysin map} $f^! = i^! \\circ g^* \\in A^*(X \\to Y)$ as above." ], "refs": [], "ref_ids": [] }, { "id": 5926, "type": "definition", "label": "chow-definition-determinant", "categories": [ "chow" ], "title": "chow-definition-determinant", "contents": [ "Let $R$ be a local ring with maximal ideal $\\mathfrak m$ and", "residue field $\\kappa$. Let $M$ be a finite length $R$-module.", "Say $l = \\text{length}_R(M)$.", "\\begin{enumerate}", "\\item Given elements $x_1, \\ldots, x_r \\in M$ we denote", "$\\langle x_1, \\ldots, x_r \\rangle = Rx_1 + \\ldots + Rx_r$ the", "$R$-submodule of $M$ generated by $x_1, \\ldots, x_r$.", "\\item We will say an $l$-tuple of elements", "$(e_1, \\ldots, e_l)$ of $M$ is {\\it admissible} if", "$\\mathfrak m e_i \\subset \\langle e_1, \\ldots, e_{i - 1} \\rangle$", "for $i = 1, \\ldots, l$.", "\\item A {\\it symbol} $[e_1, \\ldots, e_l]$ will mean", "$(e_1, \\ldots, e_l)$ is an admissible $l$-tuple.", "\\item An {\\it admissible relation} between symbols is one of the following:", "\\begin{enumerate}", "\\item if $(e_1, \\ldots, e_l)$ is an admissible sequence and", "for some $1 \\leq a \\leq l$ we have", "$e_a \\in \\langle e_1, \\ldots, e_{a - 1}\\rangle$, then", "$[e_1, \\ldots, e_l] = 0$,", "\\item if $(e_1, \\ldots, e_l)$ is an admissible sequence and", "for some $1 \\leq a \\leq l$ we have $e_a = \\lambda e'_a + x$", "with $\\lambda \\in R^*$, and", "$x \\in \\langle e_1, \\ldots, e_{a - 1}\\rangle$, then", "$$", "[e_1, \\ldots, e_l] =", "\\overline{\\lambda} [e_1, \\ldots, e_{a - 1}, e'_a, e_{a + 1}, \\ldots, e_l]", "$$", "where $\\overline{\\lambda} \\in \\kappa^*$ is the image of $\\lambda$ in", "the residue field, and", "\\item if $(e_1, \\ldots, e_l)$ is an admissible sequence and", "$\\mathfrak m e_a \\subset \\langle e_1, \\ldots, e_{a - 2}\\rangle$ then", "$$", "[e_1, \\ldots, e_l] =", "- [e_1, \\ldots, e_{a - 2}, e_a, e_{a - 1}, e_{a + 1}, \\ldots, e_l].", "$$", "\\end{enumerate}", "\\item", "We define the {\\it determinant of the finite length $R$-module $M$} to be", "$$", "\\det\\nolimits_\\kappa(M) =", "\\left\\{", "\\frac{\\kappa\\text{-vector space generated by symbols}}", "{\\kappa\\text{-linear combinations of admissible relations}}", "\\right\\}", "$$", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 5927, "type": "definition", "label": "chow-definition-periodic-determinant", "categories": [ "chow" ], "title": "chow-definition-periodic-determinant", "contents": [ "Let $R$ be a local ring with residue field $\\kappa$.", "Let $(M, \\varphi, \\psi)$ be a $(2, 1)$-periodic complex over $R$.", "Assume that $M$ has finite length and that $(M, \\varphi, \\psi)$ is", "exact. The {\\it determinant of $(M, \\varphi, \\psi)$} is", "the element", "$$", "\\det\\nolimits_\\kappa(M, \\varphi, \\psi) \\in \\kappa^*", "$$", "such that the composition", "$$", "\\det\\nolimits_\\kappa(M)", "\\xrightarrow{\\gamma_\\psi \\circ \\sigma \\circ \\gamma_\\varphi^{-1}}", "\\det\\nolimits_\\kappa(M)", "$$", "is multiplication by", "$(-1)^{\\text{length}_R(I_\\varphi)\\text{length}_R(I_\\psi)}", "\\det\\nolimits_\\kappa(M, \\varphi, \\psi)$." ], "refs": [], "ref_ids": [] }, { "id": 5928, "type": "definition", "label": "chow-definition-symbol-M", "categories": [ "chow" ], "title": "chow-definition-symbol-M", "contents": [ "Let $A$ be a Noetherian local ring with residue field $\\kappa$.", "Let $a, b \\in A$.", "Let $M$ be a finite $A$-module of dimension $1$", "such that $a, b$ are nonzerodivisors on $M$.", "We define the {\\it symbol associated to $M, a, b$}", "to be the element", "$$", "d_M(a, b) =", "\\det\\nolimits_\\kappa(M/abM, a, b) \\in \\kappa^*", "$$" ], "refs": [], "ref_ids": [] }, { "id": 5929, "type": "definition", "label": "chow-definition-tame-symbol", "categories": [ "chow" ], "title": "chow-definition-tame-symbol", "contents": [ "Let $A$ be a Noetherian local domain of dimension $1$", "with residue field $\\kappa$.", "Let $K$ be the fraction field of $A$.", "We define the {\\it tame symbol} of $A$ to be the map", "$$", "K^* \\times K^* \\longrightarrow \\kappa^*,", "\\quad", "(x, y) \\longmapsto d_A(x, y)", "$$", "where $d_A(x, y)$ is extended to $K^* \\times K^*$ by the multiplicativity of", "Lemma \\ref{lemma-multiplicativity-symbol}." ], "refs": [ "chow-lemma-multiplicativity-symbol" ], "ref_ids": [ 5882 ] }, { "id": 6206, "type": "definition", "label": "flat-definition-one-step-devissage", "categories": [ "flat" ], "title": "flat-definition-one-step-devissage", "contents": [ "Let $S$ be a scheme.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.", "Let $s \\in S$ be a point.", "A {\\it one step d\\'evissage of $\\mathcal{F}/X/S$ over $s$}", "is given by morphisms of schemes over $S$", "$$", "\\xymatrix{", "X & Z \\ar[l]_i \\ar[r]^\\pi & Y", "}", "$$", "and a quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{G}$ of finite type", "such that", "\\begin{enumerate}", "\\item $X$, $S$, $Z$ and $Y$ are affine,", "\\item $i$ is a closed immersion of finite presentation,", "\\item $\\mathcal{F} \\cong i_*\\mathcal{G}$,", "\\item $\\pi$ is finite, and", "\\item the structure morphism $Y \\to S$ is smooth with", "geometrically irreducible fibres of", "dimension $\\dim(\\text{Supp}(\\mathcal{F}_s))$.", "\\end{enumerate}", "In this case we say $(Z, Y, i, \\pi, \\mathcal{G})$ is a one step", "d\\'evissage of $\\mathcal{F}/X/S$ over $s$." ], "refs": [], "ref_ids": [] }, { "id": 6207, "type": "definition", "label": "flat-definition-one-step-devissage-at-x", "categories": [ "flat" ], "title": "flat-definition-one-step-devissage-at-x", "contents": [ "Let $S$ be a scheme.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.", "Let $x \\in X$ be a point with image $s$ in $S$.", "A {\\it one step d\\'evissage of $\\mathcal{F}/X/S$ at $x$}", "is a system $(Z, Y, i, \\pi, \\mathcal{G}, z, y)$, where", "$(Z, Y, i, \\pi, \\mathcal{G})$ is a one step d\\'evissage of", "$\\mathcal{F}/X/S$ over $s$ and", "\\begin{enumerate}", "\\item $\\dim_x(\\text{Supp}(\\mathcal{F}_s)) = \\dim(\\text{Supp}(\\mathcal{F}_s))$,", "\\item $z \\in Z$ is a point with $i(z) = x$ and $\\pi(z) = y$,", "\\item we have $\\pi^{-1}(\\{y\\}) = \\{z\\}$,", "\\item the extension $\\kappa(s) \\subset \\kappa(y)$ is purely", "transcendental.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6208, "type": "definition", "label": "flat-definition-shrink", "categories": [ "flat" ], "title": "flat-definition-shrink", "contents": [ "Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in", "Definition \\ref{definition-one-step-devissage-at-x}.", "Let $(Z, Y, i, \\pi, \\mathcal{G}, z, y)$ be a one step d\\'evissage", "of $\\mathcal{F}/X/S$ at $x$. Let us define a", "{\\it standard shrinking} of this situation to be", "given by standard opens $S' \\subset S$, $X' \\subset X$, $Z' \\subset Z$,", "and $Y' \\subset Y$ such that $s \\in S'$, $x \\in X'$, $z \\in Z'$, and", "$y \\in Y'$ and such that", "$$", "(Z', Y', i|_{Z'}, \\pi|_{Z'}, \\mathcal{G}|_{Z'}, z, y)", "$$", "is a one step d\\'evissage of $\\mathcal{F}|_{X'}/X'/S'$ at $x$." ], "refs": [ "flat-definition-one-step-devissage-at-x" ], "ref_ids": [ 6207 ] }, { "id": 6209, "type": "definition", "label": "flat-definition-complete-devissage", "categories": [ "flat" ], "title": "flat-definition-complete-devissage", "contents": [ "Let $S$ be a scheme.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.", "Let $s \\in S$ be a point.", "A {\\it complete d\\'evissage of $\\mathcal{F}/X/S$ over $s$} is given by a", "diagram", "$$", "\\xymatrix{", "X & Z_1 \\ar[l]^{i_1} \\ar[d]^{\\pi_1} \\\\", "& Y_1 & Z_2 \\ar[l]^{i_2} \\ar[d]^{\\pi_2} \\\\", "& & Y_2 & Z_3 \\ar[l] \\ar[d] \\\\", "& & & ... & ... \\ar[l] \\ar[d] \\\\", "& & & & Y_n", "}", "$$", "of schemes over $S$, finite type quasi-coherent $\\mathcal{O}_{Z_k}$-modules", "$\\mathcal{G}_k$, and $\\mathcal{O}_{Y_k}$-module maps", "$$", "\\alpha_k :", "\\mathcal{O}_{Y_k}^{\\oplus r_k}", "\\longrightarrow", "\\pi_{k, *}\\mathcal{G}_k,", "\\quad", "k = 1, \\ldots, n", "$$", "satisfying the following properties:", "\\begin{enumerate}", "\\item $(Z_1, Y_1, i_1, \\pi_1, \\mathcal{G}_1)$ is a one step", "d\\'evissage of $\\mathcal{F}/X/S$ over $s$,", "\\item the map $\\alpha_k$ induces an isomorphism", "$$", "\\kappa(\\xi_k)^{\\oplus r_k} \\longrightarrow", "(\\pi_{k, *}\\mathcal{G}_k)_{\\xi_k}", "\\otimes_{\\mathcal{O}_{Y_k, \\xi_k}} \\kappa(\\xi_k)", "$$", "where $\\xi_k \\in (Y_k)_s$ is the unique generic point,", "\\item for $k = 2, \\ldots, n$ the system", "$(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k)$", "is a one step d\\'evissage of $\\Coker(\\alpha_{k - 1})/Y_{k - 1}/S$", "over $s$,", "\\item $\\Coker(\\alpha_n) = 0$.", "\\end{enumerate}", "In this case we say that", "$(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k)_{k = 1, \\ldots, n}$", "is a complete d\\'evissage of $\\mathcal{F}/X/S$ over $s$." ], "refs": [], "ref_ids": [] }, { "id": 6210, "type": "definition", "label": "flat-definition-complete-devissage-at-x", "categories": [ "flat" ], "title": "flat-definition-complete-devissage-at-x", "contents": [ "Let $S$ be a scheme.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.", "Let $x \\in X$ be a point with image $s \\in S$.", "A {\\it complete d\\'evissage of $\\mathcal{F}/X/S$ at $x$} is given by a", "system", "$$", "(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k, z_k, y_k)_{k = 1, \\ldots, n}", "$$", "such that $(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k)$ is a", "complete d\\'evissage of $\\mathcal{F}/X/S$ over $s$, and such that", "\\begin{enumerate}", "\\item $(Z_1, Y_1, i_1, \\pi_1, \\mathcal{G}_1, z_1, y_1)$ is a one step", "d\\'evissage of $\\mathcal{F}/X/S$ at $x$,", "\\item for $k = 2, \\ldots, n$ the system", "$(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, z_k, y_k)$", "is a one step d\\'evissage of $\\Coker(\\alpha_{k - 1})/Y_{k - 1}/S$", "at $y_{k - 1}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6211, "type": "definition", "label": "flat-definition-shrink-complete", "categories": [ "flat" ], "title": "flat-definition-shrink-complete", "contents": [ "Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in", "Definition \\ref{definition-complete-devissage-at-x}.", "Consider a complete d\\'evissage", "$(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k, z_k, y_k)_{k = 1, \\ldots, n}$", "of $\\mathcal{F}/X/S$ at $x$. Let us define a", "{\\it standard shrinking} of this situation to be", "given by standard opens $S' \\subset S$, $X' \\subset X$,", "$Z'_k \\subset Z_k$, and $Y'_k \\subset Y_k$ such that $s_k \\in S'$,", "$x_k \\in X'$, $z_k \\in Z'$, and $y_k \\in Y'$ and such that", "$$", "(Z'_k, Y'_k, i'_k, \\pi'_k,", "\\mathcal{G}'_k, \\alpha'_k, z_k, y_k)_{k = 1, \\ldots, n}", "$$", "is a one step d\\'evissage of $\\mathcal{F}'/X'/S'$ at $x$ where", "$\\mathcal{G}'_k = \\mathcal{G}_k|_{Z'_k}$ and", "$\\mathcal{F}' = \\mathcal{F}|_{X'}$." ], "refs": [ "flat-definition-complete-devissage-at-x" ], "ref_ids": [ 6210 ] }, { "id": 6212, "type": "definition", "label": "flat-definition-elementary-etale-neighbourhood", "categories": [ "flat" ], "title": "flat-definition-elementary-etale-neighbourhood", "contents": [ "Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime of $S$ lying over", "the prime $\\mathfrak p$ of $R$. A {\\it elementary \\'etale localization of", "the ring map $R \\to S$ at $\\mathfrak q$} is given by a commutative diagram", "of rings and accompanying primes", "$$", "\\xymatrix{", "S \\ar[r] & S' \\\\", "R \\ar[u] \\ar[r] & R' \\ar[u]", "}", "\\quad\\quad", "\\xymatrix{", "\\mathfrak q \\ar@{-}[r] & \\mathfrak q' \\\\", "\\mathfrak p \\ar@{-}[u] \\ar@{-}[r] & \\mathfrak p' \\ar@{-}[u]", "}", "$$", "such that $R \\to R'$ and $S \\to S'$ are \\'etale ring maps and", "$\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$ and", "$\\kappa(\\mathfrak q) = \\kappa(\\mathfrak q')$." ], "refs": [], "ref_ids": [] }, { "id": 6213, "type": "definition", "label": "flat-definition-complete-devissage-algebra", "categories": [ "flat" ], "title": "flat-definition-complete-devissage-algebra", "contents": [ "Let $R \\to S$ be a finite type ring map.", "Let $\\mathfrak r$ be a prime of $R$.", "Let $N$ be a finite $S$-module.", "A {\\it complete d\\'evissage of $N/S/R$ over $\\mathfrak r$}", "is given by $R$-algebra maps", "$$", "\\xymatrix{", "& A_1 & & A_2 & & ... & & A_n \\\\", "S \\ar[ru] & & B_1 \\ar[lu] \\ar[ru] & & ... \\ar[lu] \\ar[ru] & &", "... \\ar[lu] \\ar[ru] & & B_n \\ar[lu]", "}", "$$", "finite $A_i$-modules $M_i$ and $B_i$-module maps", "$\\alpha_i : B_i^{\\oplus r_i} \\to M_i$ such that", "\\begin{enumerate}", "\\item $S \\to A_1$ is surjective and of finite presentation,", "\\item $B_i \\to A_{i + 1}$ is surjective and of finite presentation,", "\\item $B_i \\to A_i$ is finite,", "\\item $R \\to B_i$ is smooth with geometrically irreducible fibres,", "\\item $N \\cong M_1$ as $S$-modules,", "\\item $\\Coker(\\alpha_i) \\cong M_{i + 1}$ as $B_i$-modules,", "\\item $\\alpha_i : \\kappa(\\mathfrak p_i)^{\\oplus r_i}", "\\to M_i \\otimes_{B_i} \\kappa(\\mathfrak p_i)$ is an isomorphism", "where $\\mathfrak p_i = \\mathfrak rB_i$, and", "\\item $\\Coker(\\alpha_n) = 0$.", "\\end{enumerate}", "In this situation we say that", "$(A_i, B_i, M_i, \\alpha_i)_{i = 1, \\ldots, n}$", "is a complete d\\'evissage of $N/S/R$ over $\\mathfrak r$." ], "refs": [], "ref_ids": [] }, { "id": 6214, "type": "definition", "label": "flat-definition-complete-devissage-at-x-algebra", "categories": [ "flat" ], "title": "flat-definition-complete-devissage-at-x-algebra", "contents": [ "Let $R \\to S$ be a finite type ring map.", "Let $\\mathfrak q$ be a prime of $S$ lying over the prime $\\mathfrak r$ of $R$.", "Let $N$ be a finite $S$-module.", "A {\\it complete d\\'evissage of $N/S/R$ at $\\mathfrak q$} is given by a", "complete d\\'evissage $(A_i, B_i, M_i, \\alpha_i)_{i = 1, \\ldots, n}$", "of $N/S/R$ over $\\mathfrak r$ and prime ideals $\\mathfrak q_i \\subset B_i$", "lying over $\\mathfrak r$ such that", "\\begin{enumerate}", "\\item $\\kappa(\\mathfrak r) \\subset \\kappa(\\mathfrak q_i)$ is purely", "transcendental,", "\\item there is a unique prime $\\mathfrak q'_i \\subset A_i$", "lying over $\\mathfrak q_i \\subset B_i$,", "\\item $\\mathfrak q = \\mathfrak q'_1 \\cap S$ and", "$\\mathfrak q_i = \\mathfrak q'_{i + 1} \\cap A_i$,", "\\item $R \\to B_i$ has relative dimension", "$\\dim_{\\mathfrak q_i}(\\text{Supp}(M_i \\otimes_R \\kappa(\\mathfrak r)))$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6215, "type": "definition", "label": "flat-definition-impurity", "categories": [ "flat" ], "title": "flat-definition-impurity", "contents": [ "In", "Situation \\ref{situation-pre-pure}", "we say a diagram (\\ref{equation-impurity}) defines an", "{\\it impurity of $\\mathcal{F}$ above $s$}", "if $\\xi \\in \\text{Ass}_{X_T/T}(\\mathcal{F}_T)$ and", "$\\overline{\\{\\xi\\}} \\cap X_t = \\emptyset$. We will indicate", "this by saying ``let $(g : T \\to S, t' \\leadsto t, \\xi)$ be", "an impurity of $\\mathcal{F}$ above $s$''." ], "refs": [], "ref_ids": [] }, { "id": 6216, "type": "definition", "label": "flat-definition-pure", "categories": [ "flat" ], "title": "flat-definition-pure", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is of finite type.", "Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item Let $s \\in S$. We say $\\mathcal{F}$ is {\\it pure along $X_s$}", "if there is no impurity $(g : T \\to S, t' \\leadsto t, \\xi)$", "of $\\mathcal{F}$ above $s$ with $(T, t) \\to (S, s)$ an", "elementary \\'etale neighbourhood.", "\\item We say $\\mathcal{F}$ is {\\it universally pure along $X_s$}", "if there does not exist any impurity of $\\mathcal{F}$ above $s$.", "\\item We say that $X$ is {\\it pure along $X_s$} if $\\mathcal{O}_X$", "is pure along $X_s$.", "\\item We say $\\mathcal{F}$ is {\\it universally $S$-pure}, or", "{\\it universally pure relative to $S$} if $\\mathcal{F}$ is universally", "pure along $X_s$ for every $s \\in S$.", "\\item We say $\\mathcal{F}$ is {\\it $S$-pure}, or", "{\\it pure relative to $S$} if $\\mathcal{F}$ is pure along $X_s$", "for every $s \\in S$.", "\\item We say that $X$ is {\\it $S$-pure} or {\\it pure relative to $S$}", "if $\\mathcal{O}_X$ is pure relative to $S$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6217, "type": "definition", "label": "flat-definition-flat-dimension-n", "categories": [ "flat" ], "title": "flat-definition-flat-dimension-n", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.", "Let $n \\geq 0$.", "We say {\\it $\\mathcal{F}$ is flat over $S$ in dimensions $\\geq n$}", "if the equivalent conditions of Lemma \\ref{lemma-pre-flat-dimension-n}", "are satisfied." ], "refs": [ "flat-lemma-pre-flat-dimension-n" ], "ref_ids": [ 6081 ] }, { "id": 6218, "type": "definition", "label": "flat-definition-flattening", "categories": [ "flat" ], "title": "flat-definition-flattening", "contents": [ "Let $X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "We say that the {\\it universal flattening of $\\mathcal{F}$ exists}", "if the functor $F_{flat}$ defined in Situation \\ref{situation-flat}", "is representable by a scheme $S'$ over $S$.", "We say that the {\\it universal flattening of $X$ exists}", "if the universal flattening of $\\mathcal{O}_X$ exists." ], "refs": [], "ref_ids": [] }, { "id": 6219, "type": "definition", "label": "flat-definition-flattening-stratification", "categories": [ "flat" ], "title": "flat-definition-flattening-stratification", "contents": [ "Let $X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "We say that $\\mathcal{F}$ has a {\\it flattening stratification}", "if the functor $F_{flat}$ defined in Situation \\ref{situation-flat}", "is representable by a monomorphism $S' \\to S$ associated", "to a stratification of $S$ by locally closed subschemes.", "We say that $X$ has a {\\it flattening stratification}", "if $\\mathcal{O}_X$ has a flattening stratification." ], "refs": [], "ref_ids": [] }, { "id": 6220, "type": "definition", "label": "flat-definition-h-covering", "categories": [ "flat" ], "title": "flat-definition-h-covering", "contents": [ "Let $T$ be a scheme. A {\\it h covering of $T$} is a family of morphisms", "$\\{f_i : T_i \\to T\\}_{i \\in I}$ such that each $f_i$ is", "locally of finite presentation and one of the equivalent conditions of", "Lemma \\ref{lemma-equivalence-h-v-locally-finite-presentation} is satisfied." ], "refs": [ "flat-lemma-equivalence-h-v-locally-finite-presentation" ], "ref_ids": [ 6139 ] }, { "id": 6221, "type": "definition", "label": "flat-definition-big-h-site", "categories": [ "flat" ], "title": "flat-definition-big-h-site", "contents": [ "A {\\it big h site} is any site $\\Sch_h$ as in", "Sites, Definition \\ref{sites-definition-site} constructed as follows:", "\\begin{enumerate}", "\\item Choose any set of schemes $S_0$, and any set of h coverings", "$\\text{Cov}_0$ among these schemes.", "\\item As underlying category take any category $\\Sch_\\alpha$", "constructed as in Sets, Lemma \\ref{sets-lemma-construct-category}", "starting with the set $S_0$.", "\\item Choose any set of coverings as in", "Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the", "category $\\Sch_\\alpha$ and the class of h coverings,", "and the set $\\text{Cov}_0$ chosen above.", "\\end{enumerate}" ], "refs": [ "sites-definition-site", "sets-lemma-construct-category", "sets-lemma-coverings-site" ], "ref_ids": [ 8652, 8789, 8800 ] }, { "id": 6222, "type": "definition", "label": "flat-definition-standard-h", "categories": [ "flat" ], "title": "flat-definition-standard-h", "contents": [ "Let $T$ be an affine scheme. A {\\it standard h covering} of $T$", "is a family $\\{f_i : T_i \\to T\\}_{i = 1, \\ldots, n}$ with each $T_i$", "affine, with $f_i$ of finite presentation satisfying either of the", "following equivalent conditions: (1) $\\{U_i \\to U\\}$ can be refined by", "a standard ph covering or (2) $\\{U_i \\to U\\}$ is a V covering." ], "refs": [], "ref_ids": [] }, { "id": 6223, "type": "definition", "label": "flat-definition-big-small-h", "categories": [ "flat" ], "title": "flat-definition-big-small-h", "contents": [ "Let $S$ be a scheme. Let $\\Sch_h$ be a big h site containing $S$.", "\\begin{enumerate}", "\\item The {\\it big h site of $S$}, denoted", "$(\\Sch/S)_h$, is the site $\\Sch_h/S$", "introduced in Sites, Section \\ref{sites-section-localize}.", "\\item The {\\it big affine h site of $S$}, denoted", "$(\\textit{Aff}/S)_h$, is the full subcategory of", "$(\\Sch/S)_h$ whose objects are affine $U/S$.", "A covering of $(\\textit{Aff}/S)_h$ is any covering", "$\\{U_i \\to U\\}$ of $(\\Sch/S)_h$ which is a standard h covering.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6353, "type": "definition", "label": "curves-definition-normal-projective-model", "categories": [ "curves" ], "title": "curves-definition-normal-projective-model", "contents": [ "Let $k$ be a field. Let $X$ be a curve.", "A {\\it nonsingular projective model of $X$}", "is a pair $(Y, \\varphi)$ where $Y$ is a nonsingular projective", "curve and $\\varphi : k(X) \\to k(Y)$ is an isomorphism", "of function fields." ], "refs": [], "ref_ids": [] }, { "id": 6354, "type": "definition", "label": "curves-definition-linear-series", "categories": [ "curves" ], "title": "curves-definition-linear-series", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$.", "Let $d \\geq 0$ and $r \\geq 0$.", "A {\\it linear series of degree $d$ and dimension $r$}", "is a pair $(\\mathcal{L}, V)$ where $\\mathcal{L}$ in an", "invertible $\\mathcal{O}_X$-module of degree $d$", "(Varieties, Definition \\ref{varieties-definition-degree-invertible-sheaf})", "and $V \\subset H^0(X, \\mathcal{L})$ is a $k$-subvector space", "of dimension $r + 1$. We will abbreviate this by saying", "$(\\mathcal{L}, V)$ is a {\\it $\\mathfrak g^r_d$} on $X$." ], "refs": [ "varieties-definition-degree-invertible-sheaf" ], "ref_ids": [ 11161 ] }, { "id": 6355, "type": "definition", "label": "curves-definition-genus", "categories": [ "curves" ], "title": "curves-definition-genus", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having", "dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$.", "Then the {\\it genus} of $X$ is $g = \\dim_k H^1(X, \\mathcal{O}_X)$." ], "refs": [], "ref_ids": [] }, { "id": 6356, "type": "definition", "label": "curves-definition-geometric-genus", "categories": [ "curves" ], "title": "curves-definition-geometric-genus", "contents": [ "Let $k$ be a field. Let $X$ be a geometrically irreducible", "curve over $k$. The {\\it geometric genus} of $X$ is the genus", "of a smooth projective model of $X$ possibly defined over", "an extension field of $k$ as in", "Lemma \\ref{lemma-smooth-models}." ], "refs": [ "curves-lemma-smooth-models" ], "ref_ids": [ 6245 ] }, { "id": 6357, "type": "definition", "label": "curves-definition-multicross", "categories": [ "curves" ], "title": "curves-definition-multicross", "contents": [ "Let $k$ be an algebraically closed field. Let $X$ be an algebraic", "$1$-dimensional $k$-scheme. Let $x \\in X$ be a closed point.", "We say $x$ defines a {\\it multicross singularity} if the completion", "$\\mathcal{O}_{X, x}^\\wedge$", "is isomorphic to (\\ref{equation-multicross}) for some $n \\geq 2$.", "We say $x$ is a {\\it node}, or an {\\it ordinary double point}, or", "{\\it defines a nodal singularity} if $n = 2$." ], "refs": [], "ref_ids": [] }, { "id": 6358, "type": "definition", "label": "curves-definition-nodal", "categories": [ "curves" ], "title": "curves-definition-nodal", "contents": [ "Let $k$ be a field. Let $X$ be a $1$-dimensional locally algebraic $k$-scheme.", "\\begin{enumerate}", "\\item We say a closed point $x \\in X$ is a {\\it node}, or an", "{\\it ordinary double point}, or {\\it defines a nodal singularity}", "if there exists an ordinary double point $\\overline{x} \\in X_{\\overline{k}}$", "mapping to $x$.", "\\item We say the {\\it singularities of $X$ are at-worst-nodal} if", "all closed points of $X$ are either in the smooth locus of", "the structure morphism $X \\to \\Spec(k)$ or are ordinary double points.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6359, "type": "definition", "label": "curves-definition-split-node", "categories": [ "curves" ], "title": "curves-definition-split-node", "contents": [ "Let $k$ be a field. Let $X$ be a $1$-dimensional algebraic $k$-scheme.", "Let $x \\in X$ be a closed point. We say $x$ is a {\\it split node}", "if $x$ is a node, $\\kappa(x) = k$, and the equivalent assertions of", "Remark \\ref{remark-trivial-quadratic-extension}", "hold for $A = \\mathcal{O}_{X, x}$." ], "refs": [ "curves-remark-trivial-quadratic-extension" ], "ref_ids": [ 6365 ] }, { "id": 6360, "type": "definition", "label": "curves-definition-nodal-family", "categories": [ "curves" ], "title": "curves-definition-nodal-family", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. We say $f$ is", "{\\it at-worst-nodal of relative dimension $1$} if $f$ satisfies", "the equivalent conditions of Lemma \\ref{lemma-nodal-family}." ], "refs": [ "curves-lemma-nodal-family" ], "ref_ids": [ 6317 ] }, { "id": 6710, "type": "definition", "label": "etale-cohomology-definition-etale-covering-initial", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-etale-covering-initial", "contents": [ "A family of morphisms $\\{ \\varphi_i : U_i \\to X\\}_{i \\in I}$ is", "called an {\\it \\'etale covering} if each $\\varphi_i$ is an \\'etale morphism", "and their images cover $X$, i.e.,", "$X = \\bigcup_{i \\in I} \\varphi_i(U_i)$." ], "refs": [], "ref_ids": [] }, { "id": 6711, "type": "definition", "label": "etale-cohomology-definition-presheaf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-presheaf", "contents": [ "Let $\\mathcal{C}$ be a category. A {\\it presheaf of sets} (respectively, an", "{\\it abelian presheaf}) on $\\mathcal{C}$ is a functor $\\mathcal{C}^{opp} \\to", "\\textit{Sets}$ (resp.\\ $\\textit{Ab}$)." ], "refs": [], "ref_ids": [] }, { "id": 6712, "type": "definition", "label": "etale-cohomology-definition-family-morphisms-fixed-target", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-family-morphisms-fixed-target", "contents": [ "Let $\\mathcal{C}$ be a category. A {\\it family of morphisms with fixed target}", "$\\mathcal{U} = \\{\\varphi_i : U_i \\to U\\}_{i\\in I}$ is the data of", "\\begin{enumerate}", "\\item an object $U \\in \\mathcal{C}$,", "\\item a set $I$ (possibly empty), and", "\\item for all $i\\in I$, a morphism $\\varphi_i : U_i \\to U$ of $\\mathcal{C}$", "with target $U$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6713, "type": "definition", "label": "etale-cohomology-definition-site", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-site", "contents": [ "A {\\it site}\\footnote{What we call a site is a called a category endowed with", "a pretopology in \\cite[Expos\\'e II, D\\'efinition 1.3]{SGA4}.", "In \\cite{ArtinTopologies} it is called a category with a Grothendieck", "topology.} consists of a category $\\mathcal{C}$ and a set", "$\\text{Cov}(\\mathcal{C})$ consisting of families of morphisms with fixed target", "called {\\it coverings}, such that", "\\begin{enumerate}", "\\item (isomorphism) if $\\varphi : V \\to U$ is an isomorphism in $\\mathcal{C}$,", "then $\\{\\varphi : V \\to U\\}$ is a covering,", "\\item (locality) if $\\{\\varphi_i : U_i \\to U\\}_{i\\in I}$ is a covering and", "for all $i \\in I$ we are given a covering", "$\\{\\psi_{ij} : U_{ij} \\to U_i \\}_{j\\in I_i}$, then", "$$", "\\{", "\\varphi_i \\circ \\psi_{ij} : U_{ij} \\to U", "\\}_{(i, j)\\in \\prod_{i\\in I} \\{i\\} \\times I_i}", "$$", "is also a covering, and", "\\item (base change) if $\\{U_i \\to U\\}_{i\\in I}$", "is a covering and $V \\to U$ is a morphism in $\\mathcal{C}$, then", "\\begin{enumerate}", "\\item for all $i \\in I$ the fibre product", "$U_i \\times_U V$ exists in $\\mathcal{C}$, and", "\\item $\\{U_i \\times_U V \\to V\\}_{i\\in I}$ is a covering.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6714, "type": "definition", "label": "etale-cohomology-definition-sheaf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-sheaf", "contents": [ "A presheaf $\\mathcal{F}$ of sets (resp. abelian presheaf) on a site", "$\\mathcal{C}$ is said to be a {\\it separated presheaf} if for all coverings", "$\\{\\varphi_i : U_i \\to U\\}_{i\\in I} \\in \\text{Cov} (\\mathcal{C})$", "the map", "$$", "\\mathcal{F}(U) \\longrightarrow \\prod\\nolimits_{i\\in I} \\mathcal{F}(U_i)", "$$", "is injective. Here the map is $s \\mapsto (s|_{U_i})_{i\\in I}$.", "The presheaf $\\mathcal{F}$ is a {\\it sheaf} if for all coverings", "$\\{\\varphi_i : U_i \\to U\\}_{i\\in I} \\in \\text{Cov} (\\mathcal{C})$, the", "diagram", "\\begin{equation}", "\\label{equation-sheaf-axiom}", "\\xymatrix{", "\\mathcal{F}(U) \\ar[r] &", "\\prod_{i\\in I} \\mathcal{F}(U_i) \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\prod_{i, j \\in I} \\mathcal{F}(U_i \\times_U U_j),", "}", "\\end{equation}", "where the first map is $s \\mapsto (s|_{U_i})_{i\\in I}$ and the two", "maps on the right are", "$(s_i)_{i\\in I} \\mapsto (s_i |_{U_i \\times_U U_j})$ and", "$(s_i)_{i\\in I} \\mapsto (s_j |_{U_i \\times_U U_j})$,", "is an equalizer diagram in the category of sets (resp.\\ abelian groups)." ], "refs": [], "ref_ids": [] }, { "id": 6715, "type": "definition", "label": "etale-cohomology-definition-category-sheaves", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-category-sheaves", "contents": [ "We denote $\\Sh(\\mathcal{C})$ (resp.\\ $\\textit{Ab}(\\mathcal{C})$)", "the full subcategory of $\\textit{PSh}(\\mathcal{C})$", "(resp.\\ $\\textit{PAb}(\\mathcal{C})$) whose objects are sheaves. This is the", "{\\it category of sheaves of sets} (resp.\\ {\\it abelian sheaves}) on", "$\\mathcal{C}$." ], "refs": [], "ref_ids": [] }, { "id": 6716, "type": "definition", "label": "etale-cohomology-definition-0-cech", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-0-cech", "contents": [ "Let $\\mathcal{F}$ be a presheaf on the site $\\mathcal{C}$ and", "$\\mathcal{U} = \\{U_i \\to U\\} \\in \\text{Cov} (\\mathcal{C})$.", "We define the {\\it zeroth {\\v C}ech cohomology group} of", "$\\mathcal{F}$ with respect to $\\mathcal{U}$ by", "$$", "\\check H^0 (\\mathcal{U}, \\mathcal{F}) =", "\\left\\{", "(s_i)_{i\\in I} \\in \\prod\\nolimits_{i\\in I }\\mathcal{F}(U_i)", "\\text{ such that }", "s_i|_{U_i \\times_U U_j} = s_j |_{U_i \\times_U U_j}", "\\right\\}.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 6717, "type": "definition", "label": "etale-cohomology-definition-fpqc-covering", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-fpqc-covering", "contents": [ "Let $T$ be a scheme. An {\\it fpqc covering} of $T$ is a family", "$\\{ \\varphi_i : T_i \\to T\\}_{i \\in I}$ such that", "\\begin{enumerate}", "\\item each $\\varphi_i$ is a flat morphism and", "$\\bigcup_{i\\in I} \\varphi_i(T_i) = T$, and", "\\item for each affine open $U \\subset T$ there exists a finite", "set $K$, a map $\\mathbf{i} : K \\to I$ and affine opens", "$U_{\\mathbf{i}(k)} \\subset T_{\\mathbf{i}(k)}$ such that", "$U = \\bigcup_{k \\in K} \\varphi_{\\mathbf{i}(k)}(U_{\\mathbf{i}(k)})$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6718, "type": "definition", "label": "etale-cohomology-definition-sheaf-property-fpqc", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-sheaf-property-fpqc", "contents": [ "Let $S$ be a scheme. The category of schemes over $S$ is denoted", "$\\Sch/S$. Consider a functor", "$\\mathcal{F} : (\\Sch/S)^{opp} \\to \\textit{Sets}$, in other words", "a presheaf of sets. We say $\\mathcal{F}$", "{\\it satisfies the sheaf property for the fpqc topology}", "if for every fpqc covering $\\{U_i \\to U\\}_{i \\in I}$ of schemes over $S$", "the diagram (\\ref{equation-sheaf-axiom}) is an equalizer diagram." ], "refs": [], "ref_ids": [] }, { "id": 6719, "type": "definition", "label": "etale-cohomology-definition-descent-datum", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-descent-datum", "contents": [ "Let $\\mathcal{U} = \\{ t_i : T_i \\to T\\}_{i \\in I}$ be a family of", "morphisms of schemes with fixed target. A {\\it descent datum} for", "quasi-coherent sheaves with respect to $\\mathcal{U}$ is a collection", "$((\\mathcal{F}_i)_{i \\in I}, (\\varphi_{ij})_{i, j \\in I})$ where", "\\begin{enumerate}", "\\item $\\mathcal{F}_i$ is a quasi-coherent sheaf on $T_i$, and", "\\item $\\varphi_{ij} : \\text{pr}_0^* \\mathcal{F}_i \\to", "\\text{pr}_1^* \\mathcal{F}_j$ is an isomorphism of modules", "on $T_i \\times_T T_j$,", "\\end{enumerate}", "such that the {\\it cocycle condition} holds: the diagrams", "$$", "\\xymatrix{", "\\text{pr}_0^*\\mathcal{F}_i \\ar[dr]_{\\text{pr}_{02}^*\\varphi_{ik}}", "\\ar[rr]^{\\text{pr}_{01}^*\\varphi_{ij}} & &", "\\text{pr}_1^*\\mathcal{F}_j \\ar[dl]^{\\text{pr}_{12}^*\\varphi_{jk}} \\\\", "& \\text{pr}_2^*\\mathcal{F}_k", "}", "$$", "commute on $T_i \\times_T T_j \\times_T T_k$.", "This descent datum is called {\\it effective} if there exist a quasi-coherent", "sheaf $\\mathcal{F}$ over $T$ and $\\mathcal{O}_{T_i}$-module isomorphisms", "$\\varphi_i : t_i^* \\mathcal{F} \\cong \\mathcal{F}_i$ compatible with", "the maps $\\varphi_{ij}$, namely", "$$", "\\varphi_{ij} = \\text{pr}_1^* (\\varphi_j) \\circ \\text{pr}_0^* (\\varphi_i)^{-1}.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 6720, "type": "definition", "label": "etale-cohomology-definition-descent-datum-modules", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-descent-datum-modules", "contents": [ "Let $A \\to B$ be a ring map and $N$ a $B$-module. A {\\it descent datum} for", "$N$ with respect to $A \\to B$ is an isomorphism", "$\\varphi : N \\otimes_A B \\cong B \\otimes_A N$ of $B \\otimes_A B$-modules such", "that the diagram of $B \\otimes_A B \\otimes_A B$-modules", "$$", "\\xymatrix{", "{N \\otimes_A B \\otimes_A B} \\ar[dr]_{\\varphi_{02}} \\ar[rr]^{\\varphi_{01}} & &", "{B \\otimes_A N \\otimes_A B} \\ar[dl]^{\\varphi_{12}} \\\\", "& {B \\otimes_A B \\otimes_A N}", "}", "$$", "commutes where $\\varphi_{01} = \\varphi \\otimes \\text{id}_B$ and similarly", "for $\\varphi_{12}$ and $\\varphi_{02}$." ], "refs": [], "ref_ids": [] }, { "id": 6721, "type": "definition", "label": "etale-cohomology-definition-effective-modules", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-effective-modules", "contents": [ "A descent datum $(N, \\varphi)$ is called {\\it effective} if there exists an", "$A$-module $M$ such that $(N, \\varphi) \\cong (B \\otimes_A M,", "\\varphi_\\text{can})$, with the obvious notion of isomorphism of descent data." ], "refs": [], "ref_ids": [] }, { "id": 6722, "type": "definition", "label": "etale-cohomology-definition-ringed-site", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-ringed-site", "contents": [ "Let $\\mathcal{C}$ be a {\\it ringed site}, i.e., a site endowed with a", "sheaf of rings $\\mathcal{O}$. A sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ on", "$\\mathcal{C}$ is called {\\it quasi-coherent} if for all", "$U \\in \\Ob(\\mathcal{C})$ there exists a covering", "$\\{U_i \\to U\\}_{i\\in I}$ of $\\mathcal{C}$ such that the restriction", "$\\mathcal{F}|_{\\mathcal{C}/U_i}$ is isomorphic to the cokernel of", "an $\\mathcal{O}$-linear map of free $\\mathcal{O}$-modules", "$$", "\\bigoplus\\nolimits_{k \\in K} \\mathcal{O}|_{\\mathcal{C}/U_i}", "\\longrightarrow", "\\bigoplus\\nolimits_{l \\in L} \\mathcal{O}|_{\\mathcal{C}/U_i}.", "$$", "The direct sum over $K$ is the sheaf associated to the presheaf", "$V \\mapsto \\bigoplus_{k \\in K} \\mathcal{O}(V)$ and similarly for the other." ], "refs": [], "ref_ids": [] }, { "id": 6723, "type": "definition", "label": "etale-cohomology-definition-cech-complex", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-cech-complex", "contents": [ "Let $\\mathcal{C}$ be a category,", "$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ a family of morphisms of $\\mathcal{C}$", "with fixed target, and $\\mathcal{F} \\in \\textit{PAb}(\\mathcal{C})$ an abelian", "presheaf. We define the {\\it {\\v C}ech complex}", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$ by", "$$", "\\prod_{i_0\\in I} \\mathcal{F}(U_{i_0}) \\to", "\\prod_{i_0, i_1\\in I} \\mathcal{F}(U_{i_0} \\times_U U_{i_1}) \\to", "\\prod_{i_0, i_1, i_2 \\in I}", "\\mathcal{F}(U_{i_0} \\times_U U_{i_1} \\times_U U_{i_2}) \\to \\ldots", "$$", "where the first term is in degree 0, and the maps are the usual ones. Again, it", "is essential to allow the case $i_0 = i_1$ etc. The", "{\\it {\\v C}ech cohomology groups} are defined by", "$$", "\\check{H}^p(\\mathcal{U}, \\mathcal{F}) =", "H^p(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})).", "$$" ], "refs": [], "ref_ids": [] }, { "id": 6724, "type": "definition", "label": "etale-cohomology-definition-free-abelian-presheaf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-free-abelian-presheaf", "contents": [ "Let $\\mathcal{C}$ be a category.", "Given a presheaf of sets $\\mathcal{G}$, we define the", "{\\it free abelian presheaf on $\\mathcal{G}$},", "denoted $\\mathbf{Z}_\\mathcal{G}$, by the rule", "$$", "\\mathbf{Z}_\\mathcal{G}(U)", "=", "\\mathbf{Z}[\\mathcal{G}(U)]", "$$", "for $U \\in \\Ob(\\mathcal{C})$", "with restriction maps induced by the restriction maps of $\\mathcal{G}$.", "In the special case $\\mathcal{G} = h_U$ we write simply", "$\\mathbf{Z}_U = \\mathbf{Z}_{h_U}$." ], "refs": [], "ref_ids": [] }, { "id": 6725, "type": "definition", "label": "etale-cohomology-definition-tau-covering", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-tau-covering", "contents": [ "(See", "Topologies, Definitions", "\\ref{topologies-definition-fppf-covering},", "\\ref{topologies-definition-syntomic-covering},", "\\ref{topologies-definition-smooth-covering},", "\\ref{topologies-definition-etale-covering}, and", "\\ref{topologies-definition-zariski-covering}.)", "Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$.", "A family of morphisms of schemes $\\{f_i : T_i \\to T\\}_{i \\in I}$ with fixed", "target is called a {\\it $\\tau$-covering} if and only if", "each $f_i$ is flat of finite presentation, syntomic, smooth, \\'etale,", "resp.\\ an open immersion, and we have $\\bigcup f_i(T_i) = T$." ], "refs": [ "topologies-definition-fppf-covering", "topologies-definition-syntomic-covering", "topologies-definition-smooth-covering", "topologies-definition-etale-covering", "topologies-definition-zariski-covering" ], "ref_ids": [ 12539, 12535, 12531, 12526, 12521 ] }, { "id": 6726, "type": "definition", "label": "etale-cohomology-definition-standard-tau", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-standard-tau", "contents": [ "(See", "Topologies, Definitions", "\\ref{topologies-definition-standard-fppf},", "\\ref{topologies-definition-standard-syntomic},", "\\ref{topologies-definition-standard-smooth},", "\\ref{topologies-definition-standard-etale}, and", "\\ref{topologies-definition-standard-Zariski}.)", "Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$.", "Let $T$ be an affine scheme.", "A {\\it standard $\\tau$-covering} of $T$ is a family", "$\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ with each $U_j$ is affine,", "and each $f_j$ flat and of finite presentation,", "standard syntomic, standard smooth, \\'etale, resp.\\ the immersion of a", "standard principal open in $T$ and $T = \\bigcup f_j(U_j)$." ], "refs": [ "topologies-definition-standard-fppf", "topologies-definition-standard-syntomic", "topologies-definition-standard-smooth", "topologies-definition-standard-etale", "topologies-definition-standard-Zariski" ], "ref_ids": [ 12540, 12536, 12532, 12527, 12522 ] }, { "id": 6727, "type": "definition", "label": "etale-cohomology-definition-tau-site", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-tau-site", "contents": [ "Let $S$ be a scheme.", "Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, \\linebreak[0] Zariski\\}$.", "\\begin{enumerate}", "\\item A {\\it big $\\tau$-site of $S$} is any of the sites", "$(\\Sch/S)_\\tau$ constructed as explained above and in more detail in", "Topologies, Definitions", "\\ref{topologies-definition-big-small-fppf},", "\\ref{topologies-definition-big-small-syntomic},", "\\ref{topologies-definition-big-small-smooth},", "\\ref{topologies-definition-big-small-etale}, and", "\\ref{topologies-definition-big-small-Zariski}.", "\\item If $\\tau \\in \\{\\etale, Zariski\\}$, then the", "{\\it small $\\tau$-site of $S$}", "is the full subcategory $S_\\tau$ of $(\\Sch/S)_\\tau$ whose objects", "are schemes $T$ over $S$ whose structure morphism $T \\to S$ is \\'etale,", "resp.\\ an open immersion. A covering in $S_\\tau$ is a covering", "$\\{U_i \\to U\\}$ in $(\\Sch/S)_\\tau$", "such that $U$ is an object of $S_\\tau$.", "\\end{enumerate}" ], "refs": [ "topologies-definition-big-small-fppf", "topologies-definition-big-small-syntomic", "topologies-definition-big-small-smooth", "topologies-definition-big-small-etale", "topologies-definition-big-small-Zariski" ], "ref_ids": [ 12542, 12538, 12534, 12529, 12524 ] }, { "id": 6728, "type": "definition", "label": "etale-cohomology-definition-etale-topos", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-etale-topos", "contents": [ "Let $S$ be a scheme.", "\\begin{enumerate}", "\\item The {\\it \\'etale topos}, or the {\\it small \\'etale topos}", "of $S$ is the category $\\Sh(S_\\etale)$ of sheaves of sets on", "the small \\'etale site of $S$.", "\\item The {\\it Zariski topos}, or the {\\it small Zariski topos}", "of $S$ is the category $\\Sh(S_{Zar})$ of sheaves of sets on the", "small Zariski site of $S$.", "\\item For $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$ a", "{\\it big $\\tau$-topos} is the category of sheaves of set on a", "big $\\tau$-topos of $S$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6729, "type": "definition", "label": "etale-cohomology-definition-additive-sheaf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-additive-sheaf", "contents": [ "On any of the sites $(\\Sch/S)_\\tau$ or $S_\\tau$ of", "Section \\ref{section-big-small}.", "\\begin{enumerate}", "\\item The sheaf $T \\mapsto \\Gamma(T, \\mathcal{O}_T)$ is denoted", "$\\mathcal{O}_S$, or $\\mathbf{G}_a$, or $\\mathbf{G}_{a, S}$ if we", "want to indicate the base scheme.", "\\item Similarly, the sheaf", "$T \\mapsto \\Gamma(T, \\mathcal{O}^*_T)$ is denoted $\\mathcal{O}_S^*$, or", "$\\mathbf{G}_m$, or $\\mathbf{G}_{m, S}$ if we want", "to indicate the base scheme.", "\\item The {\\it constant sheaf} $\\underline{\\mathbf{Z}/n\\mathbf{Z}}$ on any", "site is the sheafification of the constant presheaf", "$U \\mapsto \\mathbf{Z}/n\\mathbf{Z}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6730, "type": "definition", "label": "etale-cohomology-definition-structure-sheaf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-structure-sheaf", "contents": [ "Let $S$ be a scheme. The {\\it structure sheaf} of $S$ is the sheaf of rings", "$\\mathcal{O}_S$", "on any of the sites $S_{Zar}$, $S_\\etale$, or $(\\Sch/S)_\\tau$", "discussed above." ], "refs": [], "ref_ids": [] }, { "id": 6731, "type": "definition", "label": "etale-cohomology-definition-etale-morphism", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-etale-morphism", "contents": [ "A morphism of schemes is {\\it \\'etale} if it is smooth of relative dimension 0." ], "refs": [], "ref_ids": [] }, { "id": 6732, "type": "definition", "label": "etale-cohomology-definition-standard-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-standard-etale", "contents": [ "A ring map $A \\to B$ is called {\\it standard \\'etale} if", "$B \\cong \\left(A[t]/(f)\\right)_g$ with $f, g \\in A[t]$, with $f$ monic,", "and $\\text{d}f/\\text{d}t$ invertible in $B$." ], "refs": [], "ref_ids": [] }, { "id": 6733, "type": "definition", "label": "etale-cohomology-definition-etale-covering", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-etale-covering", "contents": [ "An {\\it \\'etale covering} of a scheme $U$ is a family of morphisms", "of schemes", "$\\{\\varphi_i : U_i \\to U\\}_{i \\in I}$ such that", "\\begin{enumerate}", "\\item each $\\varphi_i$ is an \\'etale morphism,", "\\item the $U_i$ cover $U$, i.e., $U = \\bigcup_{i\\in I}\\varphi_i(U_i)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6734, "type": "definition", "label": "etale-cohomology-definition-big-etale-site", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-big-etale-site", "contents": [ "(For more details see Section \\ref{section-big-small}, or", "Topologies, Section \\ref{topologies-section-etale}.)", "Let $S$ be a scheme.", "The {\\it big \\'etale site over $S$} is the site", "$(\\Sch/S)_\\etale$, see", "Definition \\ref{definition-tau-site}.", "The {\\it small \\'etale site over $S$} is the site $S_\\etale$, see", "Definition \\ref{definition-tau-site}.", "We define similarly the {\\it big} and {\\it small Zariski sites} on $S$,", "denoted $(\\Sch/S)_{Zar}$ and $S_{Zar}$." ], "refs": [ "etale-cohomology-definition-tau-site", "etale-cohomology-definition-tau-site" ], "ref_ids": [ 6727, 6727 ] }, { "id": 6735, "type": "definition", "label": "etale-cohomology-definition-geometric-point", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-geometric-point", "contents": [ "Let $S$ be a scheme.", "\\begin{enumerate}", "\\item A {\\it geometric point} of $S$ is a morphism", "$\\Spec(k) \\to S$ where $k$ is algebraically closed.", "Such a point is usually denoted $\\overline{s}$, i.e., by an overlined", "small case letter. We often use $\\overline{s}$ to denote the scheme", "$\\Spec(k)$ as well as the morphism, and we use $\\kappa(\\overline{s})$", "to denote $k$.", "\\item We say $\\overline{s}$ {\\it lies over} $s$", "to indicate that $s \\in S$ is the image of $\\overline{s}$.", "\\item An {\\it \\'etale neighborhood} of a geometric point $\\overline{s}$", "of $S$ is a commutative diagram", "$$", "\\xymatrix{", "& U \\ar[d]^\\varphi \\\\", "{\\overline{s}} \\ar[r]^{\\overline{s}} \\ar[ur]^{\\bar u} & S", "}", "$$", "where $\\varphi$ is an \\'etale morphism of schemes.", "We write $(U, \\overline{u}) \\to (S, \\overline{s})$.", "\\item A {\\it morphism of \\'etale neighborhoods}", "$(U, \\overline{u}) \\to (U', \\overline{u}')$", "is an $S$-morphism $h: U \\to U'$", "such that $\\overline{u}' = h \\circ \\overline{u}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6736, "type": "definition", "label": "etale-cohomology-definition-stalk", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-stalk", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{F}$ be a presheaf on $S_\\etale$.", "Let $\\overline{s}$ be a geometric point of $S$.", "The {\\it stalk} of $\\mathcal{F}$ at $\\overline{s}$ is", "$$", "\\mathcal{F}_{\\overline{s}}", "=", "\\colim_{(U, \\overline{u})} \\mathcal{F}(U)", "$$", "where $(U, \\overline{u})$ runs over all \\'etale", "neighborhoods of $\\overline{s}$ in $S$." ], "refs": [], "ref_ids": [] }, { "id": 6737, "type": "definition", "label": "etale-cohomology-definition-support", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-support", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{F}$ be an abelian sheaf on $S_\\etale$.", "\\begin{enumerate}", "\\item The {\\it support of $\\mathcal{F}$} is the set of", "points $s \\in S$ such that $\\mathcal{F}_{\\overline{s}} \\not = 0$", "for any (some) geometric point $\\overline{s}$ lying over $s$.", "\\item Let $\\sigma \\in \\mathcal{F}(U)$ be a section.", "The {\\it support of $\\sigma$} is the closed subset $U \\setminus W$, where", "$W \\subset U$ is the largest open subset of $U$ on which $\\sigma$", "restricts to zero (see", "Lemma \\ref{lemma-zero-over-image}).", "\\end{enumerate}" ], "refs": [ "etale-cohomology-lemma-zero-over-image" ], "ref_ids": [ 6429 ] }, { "id": 6738, "type": "definition", "label": "etale-cohomology-definition-henselian", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-henselian", "contents": [ "(See Algebra, Definition \\ref{algebra-definition-henselian}.)", "A local ring $(R, \\mathfrak m, \\kappa)$ is called", "{\\it henselian} if for all", "$f \\in R[T]$ monic, for all $a_0 \\in \\kappa$ such that", "$\\bar f(a_0) = 0$ and $\\bar f'(a_0) \\neq 0$, there exists", "an $a \\in R$ such that $f(a) = 0$ and $a \\bmod \\mathfrak m = a_0$." ], "refs": [ "algebra-definition-henselian" ], "ref_ids": [ 1545 ] }, { "id": 6739, "type": "definition", "label": "etale-cohomology-definition-strictly-henselian", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-strictly-henselian", "contents": [ "A local ring $R$ is called {\\it strictly henselian} if it is henselian and its", "residue field is separably closed." ], "refs": [], "ref_ids": [] }, { "id": 6740, "type": "definition", "label": "etale-cohomology-definition-etale-local-rings", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-etale-local-rings", "contents": [ "Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point of $S$", "lying over the point $s \\in S$.", "\\begin{enumerate}", "\\item The {\\it \\'etale local ring of $S$ at $\\overline{s}$}", "is the stalk of the structure sheaf $\\mathcal{O}_S$ on $S_\\etale$", "at $\\overline{s}$. We sometimes call this the", "{\\it strict henselization of $\\mathcal{O}_{S, s}$} relative", "to the geometric point $\\overline{s}$.", "Notation used:", "$\\mathcal{O}_{S, \\overline{s}} = \\mathcal{O}_{S, s}^{sh}$.", "\\item The {\\it henselization of $\\mathcal{O}_{S, s}$} is the", "henselization of the local ring of $S$ at $s$. See", "Algebra, Definition \\ref{algebra-definition-henselization},", "and", "Theorem \\ref{theorem-henselization}.", "Notation: $\\mathcal{O}_{S, s}^h$.", "\\item The {\\it strict henselization of $S$ at $\\overline{s}$}", "is the scheme $\\Spec(\\mathcal{O}_{S, s}^{sh})$.", "\\item The {\\it henselization of $S$ at $s$} is the scheme", "$\\Spec(\\mathcal{O}_{S, s}^h)$.", "\\end{enumerate}" ], "refs": [ "algebra-definition-henselization", "etale-cohomology-theorem-henselization" ], "ref_ids": [ 1546, 6380 ] }, { "id": 6741, "type": "definition", "label": "etale-cohomology-definition-direct-image-presheaf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-direct-image-presheaf", "contents": [ "Let $f: X\\to Y$ be a morphism of schemes.", "Let $\\mathcal{F} $ a presheaf of sets on $X_\\etale$.", "The {\\it direct image}, or {\\it pushforward} of $\\mathcal{F}$", "(under $f$) is", "$$", "f_*\\mathcal{F} : Y_\\etale^{opp} \\longrightarrow \\textit{Sets}, \\quad", "(V/Y) \\longmapsto \\mathcal{F}(X \\times_Y V/X).", "$$", "We sometimes write $f_* = f_{small, *}$ to distinguish from other", "direct image functors (such as usual Zariski pushforward or $f_{big, *}$)." ], "refs": [], "ref_ids": [] }, { "id": 6742, "type": "definition", "label": "etale-cohomology-definition-direct-image-sheaf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-direct-image-sheaf", "contents": [ "Let $f: X\\to Y$ be a morphism of schemes.", "Let $\\mathcal{F} $ a sheaf of sets on $X_\\etale$.", "The {\\it direct image}, or {\\it pushforward} of $\\mathcal{F}$", "(under $f$) is", "$$", "f_*\\mathcal{F} : Y_\\etale^{opp} \\longrightarrow \\textit{Sets}, \\quad", "(V/Y) \\longmapsto \\mathcal{F}(X \\times_Y V/X)", "$$", "which is a sheaf by", "Remark \\ref{remark-direct-image-sheaf}.", "We sometimes write $f_* = f_{small, *}$ to distinguish from other", "direct image functors (such as usual Zariski pushforward or $f_{big, *}$)." ], "refs": [ "etale-cohomology-remark-direct-image-sheaf" ], "ref_ids": [ 6779 ] }, { "id": 6743, "type": "definition", "label": "etale-cohomology-definition-higher-direct-images", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-higher-direct-images", "contents": [ "Let $f: X \\to Y$ be a morphism of schemes.", "The right derived functors $\\{R^pf_*\\}_{p \\geq 1}$ of", "$f_* : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$", "are called {\\it higher direct images}." ], "refs": [], "ref_ids": [] }, { "id": 6744, "type": "definition", "label": "etale-cohomology-definition-inverse-image", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-inverse-image", "contents": [ "Let $f: X\\to Y$ be a morphism of schemes. The {\\it inverse image}, or", "{\\it pullback}\\footnote{We use the notation $f^{-1}$ for pullbacks of", "sheaves of sets or sheaves of abelian groups, and we reserve $f^*$ for", "pullbacks of sheaves of modules via a morphism of ringed sites/topoi.}", "functors are the functors", "$$", "f^{-1} = f_{small}^{-1} :", "\\Sh(Y_\\etale)", "\\longrightarrow", "\\Sh(X_\\etale)", "$$", "and", "$$", "f^{-1} = f_{small}^{-1} :", "\\textit{Ab}(Y_\\etale)", "\\longrightarrow", "\\textit{Ab}(X_\\etale)", "$$", "which are left adjoint to $f_* = f_{small, *}$. Thus", "$f^{-1}$ thus characterized by the fact that", "$$", "\\Hom_{{\\Sh(X_\\etale)}} (f^{-1}\\mathcal{G}, \\mathcal{F})", "=", "\\Hom_{\\Sh(Y_\\etale)} (\\mathcal{G}, f_*\\mathcal{F})", "$$", "functorially, for any $\\mathcal{F} \\in \\Sh(X_\\etale)$ and", "$\\mathcal{G} \\in \\Sh(Y_\\etale)$. We similarly have", "$$", "\\Hom_{{\\textit{Ab}(X_\\etale)}} (f^{-1}\\mathcal{G}, \\mathcal{F})", "=", "\\Hom_{\\textit{Ab}(Y_\\etale)} (\\mathcal{G}, f_*\\mathcal{F})", "$$", "for $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ and", "$\\mathcal{G} \\in \\textit{Ab}(Y_\\etale)$." ], "refs": [], "ref_ids": [] }, { "id": 6745, "type": "definition", "label": "etale-cohomology-definition-inverse-system-sheaves", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-inverse-system-sheaves", "contents": [ "Let $I$ be a preordered set. Let $(X_i, f_{i'i})$ be an inverse", "system of schemes over $I$.", "A {\\it system $(\\mathcal{F}_i, \\varphi_{i'i})$ of sheaves", "on $(X_i, f_{i'i})$} is given by", "\\begin{enumerate}", "\\item a sheaf $\\mathcal{F}_i$ on $(X_i)_\\etale$ for all $i \\in I$,", "\\item for $i' \\geq i$ a map", "$\\varphi_{i'i} : f_{i'i}^{-1}\\mathcal{F}_i \\to \\mathcal{F}_{i'}$", "of sheaves on $(X_{i'})_\\etale$", "\\end{enumerate}", "such that $\\varphi_{i''i} = \\varphi_{i''i'} \\circ f_{i'' i'}^{-1}\\varphi_{i'i}$", "whenever $i'' \\geq i' \\geq i$." ], "refs": [], "ref_ids": [] }, { "id": 6746, "type": "definition", "label": "etale-cohomology-definition-algebraic-geometric-point", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-algebraic-geometric-point", "contents": [ "Let $S$ be a scheme.", "Let $\\overline{s}$ be a geometric point lying over the point $s$ of $S$.", "Let $\\kappa(s) \\subset \\kappa(s)^{sep} \\subset \\kappa(\\overline{s})$", "denote the separable algebraic closure of $\\kappa(s)$ in the algebraically", "closed field $\\kappa(\\overline{s})$.", "\\begin{enumerate}", "\\item In this situation the {\\it absolute Galois group} of $\\kappa(s)$", "is $\\text{Gal}(\\kappa(s)^{sep}/\\kappa(s))$. It is sometimes denoted", "$\\text{Gal}_{\\kappa(s)}$.", "\\item The geometric point $\\overline{s}$ is called", "{\\it algebraic} if $\\kappa(s) \\subset \\kappa(\\overline{s})$ is", "an algebraic closure of $\\kappa(s)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6747, "type": "definition", "label": "etale-cohomology-definition-G-module-continuous", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-G-module-continuous", "contents": [ "Let $G$ be a topological group.", "\\begin{enumerate}", "\\item A {\\it $G$-module}, sometimes called a {\\it discrete $G$-module},", "is an abelian group $M$ endowed with a left action $a : G \\times M \\to M$", "by group homomorphisms such that $a$ is continuous when $M$ is given the", "discrete topology.", "\\item A {\\it morphism of $G$-modules} $f : M \\to N$ is a", "$G$-equivariant homomorphism from $M$ to $N$.", "\\item The category of $G$-modules is denoted $\\text{Mod}_G$.", "\\end{enumerate}", "Let $R$ be a ring.", "\\begin{enumerate}", "\\item An {\\it $R\\text{-}G$-module} is an $R$-module $M$ endowed with", "a left action $a : G \\times M \\to M$ by $R$-linear maps such that $a$", "is continuous when $M$ is given the discrete topology.", "\\item A {\\it morphism of $R\\text{-}G$-modules} $f : M \\to N$ is a", "$G$-equivariant $R$-module map from $M$ to $N$.", "\\item The category of $R\\text{-}G$-modules is denoted $\\text{Mod}_{R, G}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6748, "type": "definition", "label": "etale-cohomology-definition-galois-cohomology", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-galois-cohomology", "contents": [ "Let $G$ be a topological group. Let $M$ be a discrete $G$-module", "with continuous $G$-action. In other words, $M$ is an object", "of the category $\\text{Mod}_G$ introduced in", "Definition \\ref{definition-G-module-continuous}.", "\\begin{enumerate}", "\\item The right derived functors $H^i(G, M)$ of $H^0(G, M)$ on the", "category $\\text{Mod}_G$ are called the", "{\\it continuous group cohomology groups} of $M$.", "\\item If $G$ is an abstract group endowed with the discrete topology", "then the $H^i(G, M)$ are called the {\\it group cohomology groups} of $M$.", "\\item If $G$ is a Galois group, then the groups $H^i(G, M)$ are called", "the {\\it Galois cohomology groups} of $M$.", "\\item If $G$ is the absolute Galois group of a field $K$, then the groups", "$H^i(G, M)$ are sometimes called the {\\it Galois cohomology groups of $K$", "with coefficients in $M$}. In this case we sometimes write", "$H^i(K, M)$ instead of $H^i(G, M)$.", "\\end{enumerate}" ], "refs": [ "etale-cohomology-definition-G-module-continuous" ], "ref_ids": [ 6747 ] }, { "id": 6749, "type": "definition", "label": "etale-cohomology-definition-brauer-equivalent", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-brauer-equivalent", "contents": [ "Two finite central simple algebras $A_1$ and $A_2$ over $K$ are called", "{\\it similar}, or {\\it equivalent} if there exist $m, n \\geq 1$", "such that $\\text{Mat}(n \\times n, A_1)", "\\cong \\text{Mat}(m \\times m, A_2)$. We write $A_1 \\sim A_2$." ], "refs": [], "ref_ids": [] }, { "id": 6750, "type": "definition", "label": "etale-cohomology-definition-brauer-group", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-brauer-group", "contents": [ "Let $K$ be a field. The {\\it Brauer group} of $K$ is the set $\\text{Br} (K)$", "of similarity classes of finite central simple algebras over $K$, endowed with", "the group law induced by tensor product (over $K$). The class of $A$ in", "$\\text{Br}(K)$ is denoted by $[A]$. The neutral element is", "$[K] = [\\text{Mat}(d \\times d, K)]$ for any $d \\geq 1$." ], "refs": [], "ref_ids": [] }, { "id": 6751, "type": "definition", "label": "etale-cohomology-definition-finite-locally-constant", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-finite-locally-constant", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$.", "\\begin{enumerate}", "\\item Let $E$ be a set. We say $\\mathcal{F}$ is the", "{\\it constant sheaf with value $E$} if $\\mathcal{F}$ is the", "sheafification of the presheaf $U \\mapsto E$.", "Notation: $\\underline{E}_X$ or $\\underline{E}$.", "\\item We say $\\mathcal{F}$ is a {\\it constant sheaf} if it is", "isomorphic to a sheaf as in (1).", "\\item We say $\\mathcal{F}$ is {\\it locally constant} if there exists a", "covering $\\{U_i \\to X\\}$ such that $\\mathcal{F}|_{U_i}$ is a constant sheaf.", "\\item We say that $\\mathcal{F}$ is {\\it finite locally constant} if it", "is locally constant and the values are finite sets.", "\\end{enumerate}", "Let $\\mathcal{F}$ be a sheaf of abelian groups on $X_\\etale$.", "\\begin{enumerate}", "\\item Let $A$ be an abelian group.", "We say $\\mathcal{F}$ is the {\\it constant sheaf with value $A$} if", "$\\mathcal{F}$ is the sheafification of the presheaf $U \\mapsto A$.", "Notation: $\\underline{A}_X$ or $\\underline{A}$.", "\\item We say $\\mathcal{F}$ is a {\\it constant sheaf} if it is isomorphic", "as an abelian sheaf to a sheaf as in (1).", "\\item We say $\\mathcal{F}$ is {\\it locally constant} if there exists a", "covering $\\{U_i \\to X\\}$ such that $\\mathcal{F}|_{U_i}$ is a constant sheaf.", "\\item We say that $\\mathcal{F}$ is {\\it finite locally constant} if it", "is locally constant and the values are finite abelian groups.", "\\end{enumerate}", "Let $\\Lambda$ be a ring. Let $\\mathcal{F}$ be a sheaf of $\\Lambda$-modules", "on $X_\\etale$.", "\\begin{enumerate}", "\\item Let $M$ be a $\\Lambda$-module.", "We say $\\mathcal{F}$ is the {\\it constant sheaf with value $M$} if", "$\\mathcal{F}$ is the sheafification of the presheaf $U \\mapsto M$.", "Notation: $\\underline{M}_X$ or $\\underline{M}$.", "\\item We say $\\mathcal{F}$ is a {\\it constant sheaf} if it is isomorphic", "as a sheaf of $\\Lambda$-modules to a sheaf as in (1).", "\\item We say $\\mathcal{F}$ is {\\it locally constant} if there exists a", "covering $\\{U_i \\to X\\}$ such that $\\mathcal{F}|_{U_i}$ is a constant sheaf.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6752, "type": "definition", "label": "etale-cohomology-definition-trace-map", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-trace-map", "contents": [ "Let $f : Y \\to X$ be a finite \\'etale morphism of schemes.", "The map $f_* f^{-1} \\to \\text{id}$ described above and explicitly below", "is called the {\\it trace}." ], "refs": [], "ref_ids": [] }, { "id": 6753, "type": "definition", "label": "etale-cohomology-definition-Cr", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-Cr", "contents": [ "A field $K$ is called {\\it $C_r$}", "if for every $0 < d^r < n$ and every $f \\in K[T_1,", "\\ldots, T_n]$ homogeneous of degree $d$, there exist $\\alpha = (\\alpha_1,", "\\ldots, \\alpha_n)$, $\\alpha_i \\in K$ not all zero, such that $f(\\alpha) = 0$.", "Such an $\\alpha$ is called a {\\it nontrivial solution} of $f$." ], "refs": [], "ref_ids": [] }, { "id": 6754, "type": "definition", "label": "etale-cohomology-definition-variety", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-variety", "contents": [ "Let $k$ be a field. A {\\it variety} is separated, integral scheme of", "finite type over $k$. A {\\it curve} is a variety of dimension $1$." ], "refs": [], "ref_ids": [] }, { "id": 6755, "type": "definition", "label": "etale-cohomology-definition-extension-zero", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-extension-zero", "contents": [ "Let $j : U \\to X$ be an \\'etale morphism of schemes.", "\\begin{enumerate}", "\\item The restriction functor", "$j^{-1} : \\Sh(X_\\etale) \\to \\Sh(U_\\etale)$", "has a left adjoint", "$j_!^{Sh} : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$.", "\\item The restriction functor", "$j^{-1} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(U_\\etale)$", "has a left adjoint which is denoted", "$j_! : \\textit{Ab}(U_\\etale) \\to \\textit{Ab}(X_\\etale)$", "and called {\\it extension by zero}.", "\\item Let $\\Lambda$ be a ring. The restriction functor", "$j^{-1} : \\textit{Mod}(X_\\etale, \\Lambda) \\to", "\\textit{Mod}(U_\\etale, \\Lambda)$", "has a left adjoint which is denoted", "$j_! : \\textit{Mod}(U_\\etale, \\Lambda) \\to", "\\textit{Mod}(X_\\etale, \\Lambda)$", "and called {\\it extension by zero}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6756, "type": "definition", "label": "etale-cohomology-definition-constructible", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-constructible", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item A sheaf of sets on $X_\\etale$ is {\\it constructible}", "if for every affine open $U \\subset X$ there exists a finite decomposition", "of $U$ into constructible locally closed subschemes $U = \\coprod_i U_i$", "such that $\\mathcal{F}|_{U_i}$ is finite locally constant for all $i$.", "\\item A sheaf of abelian groups on $X_\\etale$ is {\\it constructible}", "if for every affine open $U \\subset X$ there exists a finite decomposition", "of $U$ into constructible locally closed subschemes $U = \\coprod_i U_i$", "such that $\\mathcal{F}|_{U_i}$ is finite locally constant for all $i$.", "\\item Let $\\Lambda$ be a Noetherian ring. A sheaf of $\\Lambda$-modules", "on $X_\\etale$ is {\\it constructible} if for every affine open", "$U \\subset X$ there exists a finite decomposition", "of $U$ into constructible locally closed subschemes", "$U = \\coprod_i U_i$ such that", "$\\mathcal{F}|_{U_i}$ is of finite type and locally constant for all $i$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6757, "type": "definition", "label": "etale-cohomology-definition-c", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-c", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.", "We denote {\\it $D_c(X_\\etale, \\Lambda)$} the full subcategory", "of $D(X_\\etale, \\Lambda)$ of complexes whose cohomology sheaves", "are constructible sheaves of $\\Lambda$-modules." ], "refs": [], "ref_ids": [] }, { "id": 6758, "type": "definition", "label": "etale-cohomology-definition-ctf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-ctf", "contents": [ "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring. We denote", "{\\it $D_{ctf}(X_\\etale, \\Lambda)$} the full subcategory of", "$D_c(X_\\etale, \\Lambda)$", "consisting of objects having locally finite tor dimension." ], "refs": [], "ref_ids": [] }, { "id": 6759, "type": "definition", "label": "etale-cohomology-definition-cd", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-cd", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "The {\\it cohomological dimension of $X$} is the smallest", "element", "$$", "\\text{cd}(X) \\in \\{0, 1, 2, \\ldots\\} \\cup \\{\\infty\\}", "$$", "such that for any abelian torsion sheaf $\\mathcal{F}$", "on $X_\\etale$ we have $H^i_\\etale(X, \\mathcal{F}) = 0$", "for $i > \\text{cd}(X)$. If $X = \\Spec(A)$ we sometimes", "call this the cohomological dimension of $A$." ], "refs": [], "ref_ids": [] }, { "id": 6760, "type": "definition", "label": "etale-cohomology-definition-cd-f", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-definition-cd-f", "contents": [ "Let $f : X \\to Y$ be a quasi-compact and quasi-separated", "morphism of schemes.", "The {\\it cohomological dimension of $f$} is the smallest", "element", "$$", "\\text{cd}(f) \\in \\{0, 1, 2, \\ldots\\} \\cup \\{\\infty\\}", "$$", "such that for any abelian torsion sheaf $\\mathcal{F}$", "on $X_\\etale$ we have $R^if_*\\mathcal{F} = 0$", "for $i > \\text{cd}(f)$." ], "refs": [], "ref_ids": [] }, { "id": 6874, "type": "definition", "label": "equiv-definition-Serre-functor", "categories": [ "equiv" ], "title": "equiv-definition-Serre-functor", "contents": [ "Let $k$ be a field. Let $\\mathcal{T}$ be a $k$-linear", "triangulated category such that $\\dim_k \\Hom_\\mathcal{T}(X, Y) < \\infty$", "for all $X, Y \\in \\Ob(\\mathcal{T})$. We say {\\it a Serre functor", "exists} if the equivalent conditions of Lemma \\ref{lemma-Serre-functor-exists}", "are satisfied. In this case a {\\it Serre functor} is a $k$-linear equivalence", "$S : \\mathcal{T} \\to \\mathcal{T}$ endowed with $k$-linear isomorphisms", "$c_{X, Y} : \\Hom_\\mathcal{T}(X, Y) \\to \\Hom_\\mathcal{T}(Y, S(X))^\\vee$", "functorial in $X, Y \\in \\Ob(\\mathcal{T})$." ], "refs": [ "equiv-lemma-Serre-functor-exists" ], "ref_ids": [ 6802 ] }, { "id": 6875, "type": "definition", "label": "equiv-definition-fourier-mukai-functor", "categories": [ "equiv" ], "title": "equiv-definition-fourier-mukai-functor", "contents": [ "Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$.", "Let $K \\in D(\\mathcal{O}_{X \\times_S Y})$. The exact functor", "$$", "\\Phi_K : D(\\mathcal{O}_X) \\longrightarrow D(\\mathcal{O}_Y),\\quad", "M \\longmapsto R\\text{pr}_{2, *}(", "L\\text{pr}_1^*M \\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} K)", "$$", "of triangulated categories is called a {\\it Fourier-Mukai functor}", "and $K$ is called a {\\it Fourier-Mukai kernel} for this functor.", "Moreover,", "\\begin{enumerate}", "\\item if $\\Phi_K$ sends $D_\\QCoh(\\mathcal{O}_X)$ into $D_\\QCoh(\\mathcal{O}_Y)$", "then the resulting exact functor", "$\\Phi_K : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$", "is called a Fourier-Mukai functor,", "\\item if $\\Phi_K$ sends $D_{perf}(\\mathcal{O}_X)$ into", "$D_{perf}(\\mathcal{O}_Y)$ then the resulting exact functor", "$\\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$", "is called a Fourier-Mukai functor, and", "\\item if $X$ and $Y$ are Noetherian and $\\Phi_K$ sends", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ into $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$", "then the resulting exact functor", "$\\Phi_K : D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to", "D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$", "is called a Fourier-Mukai functor.", "Similarly for $D_{\\textit{Coh}}$, $D^+_{\\textit{Coh}}$, $D^-_{\\textit{Coh}}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6876, "type": "definition", "label": "equiv-definition-siblings", "categories": [ "equiv" ], "title": "equiv-definition-siblings", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{D}$ be a", "triangulated category. We say two exact functors of triangulated categories", "$$", "F, F' : D^b(\\mathcal{A}) \\longrightarrow \\mathcal{D}", "$$", "are {\\it siblings}, or we say $F'$ is a {\\it sibling} of $F$,", "if the following two conditions are satisfied", "\\begin{enumerate}", "\\item the functors $F \\circ i$ and $F' \\circ i$ are isomorphic", "where $i : \\mathcal{A} \\to D^b(\\mathcal{A})$ is the inclusion functor, and", "\\item $F(K) \\cong F'(K)$ for any $K$ in $D^b(\\mathcal{A})$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6877, "type": "definition", "label": "equiv-definition-siblings-geometric", "categories": [ "equiv" ], "title": "equiv-definition-siblings-geometric", "contents": [ "Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$.", "Recall that", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X) = D^b(\\textit{Coh}(\\mathcal{O}_X))$", "by Derived Categories of Schemes, Proposition \\ref{perfect-proposition-DCoh}.", "We say two $k$-linear exact functors", "$$", "F, F' :", "D^b_{\\textit{Coh}}(\\mathcal{O}_X) = D^b(\\textit{Coh}(\\mathcal{O}_X))", "\\longrightarrow", "D^b_{\\textit{Coh}}(\\mathcal{O}_Y)", "$$", "are {\\it siblings}, or we say $F'$ is a {\\it sibling} of $F$ if $F$ and $F'$", "are siblings in the sense of Definition \\ref{definition-siblings}", "with abelian category being $\\textit{Coh}(\\mathcal{O}_X)$.", "If $X$ is regular then", "$D_{perf}(\\mathcal{O}_X) = D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-perfect-on-noetherian}", "and we use the same terminology for $k$-linear exact functors", "$F, F' : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$." ], "refs": [ "perfect-proposition-DCoh", "equiv-definition-siblings", "perfect-lemma-perfect-on-noetherian" ], "ref_ids": [ 7110, 6876, 6987 ] }, { "id": 6878, "type": "definition", "label": "equiv-definition-relative-equivalence-kernel", "categories": [ "equiv" ], "title": "equiv-definition-relative-equivalence-kernel", "contents": [ "Let $S$ be a scheme. Let $X \\to S$ and $Y \\to S$ be smooth proper morphisms.", "An object $K \\in D_{perf}(\\mathcal{O}_{X \\times_S Y})$", "is said to be {\\it the Fourier-Mukai kernel of a relative equivalence", "from $X$ to $Y$ over $S$}", "if there exist an object $K' \\in D_{perf}(\\mathcal{O}_{X \\times_S Y})$", "such that", "$$", "\\Delta_{X/S, *}\\mathcal{O}_X \\cong", "R\\text{pr}_{13, *}(L\\text{pr}_{12}^*K", "\\otimes_{\\mathcal{O}_{X \\times_S Y \\times_S X}}^\\mathbf{L}", "L\\text{pr}_{23}^*K')", "$$", "in $D(\\mathcal{O}_{X \\times_S X})$ and", "$$", "\\Delta_{Y/S, *}\\mathcal{O}_Y \\cong", "R\\text{pr}_{13, *}(L\\text{pr}_{12}^*K'", "\\otimes_{\\mathcal{O}_{Y \\times_S X \\times_S Y}}^\\mathbf{L}", "L\\text{pr}_{23}^*K)", "$$", "in $D(\\mathcal{O}_{Y \\times_S Y})$. In other words, the isomorphism class", "of $K$ defines an invertible arrow in the category defined in", "Section \\ref{section-category-Fourier-Mukai-kernels}." ], "refs": [], "ref_ids": [] }, { "id": 6879, "type": "definition", "label": "equiv-definition-derived-equivalent", "categories": [ "equiv" ], "title": "equiv-definition-derived-equivalent", "contents": [ "Let $k$ be a field. Let $X$ and $Y$ be smooth projective schemes over $k$.", "We say $X$ and $Y$ are {\\it derived equivalent} if there exists a $k$-linear", "exact equivalence", "$D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$." ], "refs": [], "ref_ids": [] }, { "id": 6928, "type": "definition", "label": "stacks-more-morphisms-definition-thickening", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-definition-thickening", "contents": [ "Thickenings.", "\\begin{enumerate}", "\\item We say an algebraic stack $\\mathcal{X}'$ is a {\\it thickening}", "of an algebraic stack $\\mathcal{X}$ if $\\mathcal{X}$ is a closed substack", "of $\\mathcal{X}'$ and the associated topological spaces are equal.", "\\item Given two thickenings $\\mathcal{X} \\subset \\mathcal{X}'$ and", "$\\mathcal{Y} \\subset \\mathcal{Y}'$ a {\\it morphism of thickenings}", "is a morphism $f' : \\mathcal{X}' \\to \\mathcal{Y}'$ of algebraic stacks", "such that $f'|_\\mathcal{X}$ factors through the closed", "substack $\\mathcal{Y}$. In this situation we set", "$f = f'|_\\mathcal{X} : \\mathcal{X} \\to \\mathcal{Y}$ and we say that", "$(f, f') : (\\mathcal{X} \\subset \\mathcal{X}') \\to", "(\\mathcal{Y} \\subset \\mathcal{Y}')$ is a morphism of thickenings.", "\\item Let $\\mathcal{Z}$ be an algebraic stack. We similarly define", "{\\it thickenings over $\\mathcal{Z}$} and", "{\\it morphisms of thickenings over $\\mathcal{Z}$}.", "This means that the algebraic stacks", "$\\mathcal{X}'$ and $\\mathcal{Y}'$", "are algebraic stack endowed with a structure", "morphism to $\\mathcal{Z}$ and that $f'$ fits into a suitable", "$2$-commutative diagram of algebraic stacks.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6929, "type": "definition", "label": "stacks-more-morphisms-definition-first-order-thickening", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-definition-first-order-thickening", "contents": [ "We say an algebraic stack $\\mathcal{X}'$ is a {\\it first order thickening}", "of an algebraic stack $\\mathcal{X}$ if $\\mathcal{X}$ is a closed substack", "of $\\mathcal{X}'$ and $\\mathcal{X} \\to \\mathcal{X}'$ is a first order", "thickening in the sense of Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}." ], "refs": [], "ref_ids": [] }, { "id": 6930, "type": "definition", "label": "stacks-more-morphisms-definition-formally-smooth", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-definition-formally-smooth", "contents": [ "A morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks is said to be", "{\\it formally smooth} if it is formally smooth on objects as a", "$1$-morphism in categories fibred in groupoids as explained in", "Criteria for Representability, Section \\ref{criteria-section-formally-smooth}." ], "refs": [], "ref_ids": [] }, { "id": 6931, "type": "definition", "label": "stacks-more-morphisms-definition-categorical-quotient", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-definition-categorical-quotient", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let", "$f : \\mathcal{X} \\to Y$ be a morphism to an algebraic space $Y$.", "\\begin{enumerate}", "\\item We say $f$ is a {\\it categorical moduli space} if any morphism", "$\\mathcal{X} \\to W$ to an algebraic space $W$ factors uniquely through $f$.", "\\item We say $f$ is a {\\it uniform categorical moduli space}", "if for any flat morphism $Y' \\to Y$ of algebraic spaces the base change", "$f' : Y' \\times_Y \\mathcal{X} \\to Y'$ is a categorical moduli space.", "\\end{enumerate}", "Let $\\mathcal{C}$ be a full subcategory of the category of algebraic", "spaces.", "\\begin{enumerate}", "\\item[(3)] We say $f$ is a {\\it categorical moduli space in $\\mathcal{C}$}", "if $Y \\in \\Ob(\\mathcal{C})$ and any morphism $\\mathcal{X} \\to W$ with", "$W \\in \\Ob(\\mathcal{C})$ factors uniquely through $f$.", "\\item[(4)] We say is a {\\it uniform categorical moduli space in $\\mathcal{C}$}", "if $Y \\in \\Ob(\\mathcal{C})$ and for every flat morphism $Y' \\to Y$ in", "$\\mathcal{C}$ the base change $f' : Y' \\times_Y \\mathcal{X} \\to Y'$ is a", "categorical moduli space in $\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 6932, "type": "definition", "label": "stacks-more-morphisms-definition-well-nigh-affine", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-definition-well-nigh-affine", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. We say $\\mathcal{X}$", "is {\\it well-nigh affine} if there exists an affine scheme $U$", "and a surjective, flat, finite, and finitely presented morphism", "$U \\to \\mathcal{X}$." ], "refs": [], "ref_ids": [] }, { "id": 7115, "type": "definition", "label": "perfect-definition-supported-on", "categories": [ "perfect" ], "title": "perfect-definition-supported-on", "contents": [ "Let $X$ be a scheme. Let $E$ be an object of $D(\\mathcal{O}_X)$.", "Let $T \\subset X$ be a closed subset.", "We say $E$ is {\\it supported on $T$} if the", "cohomology sheaves $H^i(E)$ are supported on $T$." ], "refs": [], "ref_ids": [] }, { "id": 7116, "type": "definition", "label": "perfect-definition-approximation-holds", "categories": [ "perfect" ], "title": "perfect-definition-approximation-holds", "contents": [ "Let $X$ be a scheme. Consider triples $(T, E, m)$ where", "\\begin{enumerate}", "\\item $T \\subset X$ is a closed subset,", "\\item $E$ is an object of $D_\\QCoh(\\mathcal{O}_X)$, and", "\\item $m \\in \\mathbf{Z}$.", "\\end{enumerate}", "We say {\\it approximation holds for the triple} $(T, E, m)$ if", "there exists a perfect object $P$ of $D(\\mathcal{O}_X)$ supported on $T$", "and a map $\\alpha : P \\to E$ which induces isomorphisms $H^i(P) \\to H^i(E)$", "for $i > m$ and a surjection $H^m(P) \\to H^m(E)$." ], "refs": [], "ref_ids": [] }, { "id": 7117, "type": "definition", "label": "perfect-definition-approximation", "categories": [ "perfect" ], "title": "perfect-definition-approximation", "contents": [ "Let $X$ be a scheme. We say {\\it approximation by perfect complexes holds}", "on $X$ if for any closed subset $T \\subset X$ with $X \\setminus T$", "retro-compact in $X$ there exists an integer $r$ such that", "for every triple $(T, E, m)$ as in", "Definition \\ref{definition-approximation-holds} with", "\\begin{enumerate}", "\\item $E$ is $(m - r)$-pseudo-coherent, and", "\\item $H^i(E)$ is supported on $T$ for $i \\geq m - r$", "\\end{enumerate}", "approximation holds." ], "refs": [ "perfect-definition-approximation-holds" ], "ref_ids": [ 7116 ] }, { "id": 7118, "type": "definition", "label": "perfect-definition-tor-independent", "categories": [ "perfect" ], "title": "perfect-definition-tor-independent", "contents": [ "Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. We say $X$ and", "$Y$ are {\\it Tor independent over $S$} if for every $x \\in X$ and", "$y \\in Y$ mapping to the same point $s \\in S$ the rings", "$\\mathcal{O}_{X, x}$ and $\\mathcal{O}_{Y, y}$ are Tor independent", "over $\\mathcal{O}_{S, s}$ (see", "More on Algebra, Definition \\ref{more-algebra-definition-tor-independent})." ], "refs": [ "more-algebra-definition-tor-independent" ], "ref_ids": [ 10622 ] }, { "id": 7119, "type": "definition", "label": "perfect-definition-relatively-perfect", "categories": [ "perfect" ], "title": "perfect-definition-relatively-perfect", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is flat and", "locally of finite presentation. An object $E$ of $D(\\mathcal{O}_X)$ is", "{\\it perfect relative to $S$} or", "{\\it $S$-perfect} if $E$ is pseudo-coherent", "(Cohomology, Definition \\ref{cohomology-definition-pseudo-coherent}) and", "$E$ locally has finite tor dimension as an object of", "$D(f^{-1}\\mathcal{O}_S)$", "(Cohomology, Definition \\ref{cohomology-definition-tor-amplitude})." ], "refs": [ "cohomology-definition-pseudo-coherent", "cohomology-definition-tor-amplitude" ], "ref_ids": [ 2258, 2259 ] }, { "id": 7120, "type": "definition", "label": "perfect-definition-resolution-property", "categories": [ "perfect" ], "title": "perfect-definition-resolution-property", "contents": [ "Let $X$ be a scheme. We say $X$ has the {\\it resolution property}", "if every quasi-coherent $\\mathcal{O}_X$-module of finite type", "is the quotient of a finite locally free $\\mathcal{O}_X$-module." ], "refs": [], "ref_ids": [] }, { "id": 7121, "type": "definition", "label": "perfect-definition-K-group", "categories": [ "perfect" ], "title": "perfect-definition-K-group", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item We denote $K_0(X)$ the {\\it Grothendieck group of $X$}. It is the", "zeroth K-group of the strictly full, saturated, triangulated subcategory", "$D_{perf}(\\mathcal{O}_X)$ of $D(\\mathcal{O}_X)$ consisting of perfect objects.", "In a formula", "$$", "K_0(X) = K_0(D_{perf}(\\mathcal{O}_X))", "$$", "\\item If $X$ is locally Noetherian, then we denote $K'_0(X)$ the", "{\\it Grothendieck group of coherent sheaves on $X$}. It is the", "is the zeroth $K$-group of the abelian category", "of coherent $\\mathcal{O}_X$-modules. In a formula", "$$", "K'_0(X) = K_0(\\textit{Coh}(\\mathcal{O}_X))", "$$", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7201, "type": "definition", "label": "spaces-flat-definition-impurity", "categories": [ "spaces-flat" ], "title": "spaces-flat-definition-impurity", "contents": [ "In", "Situation \\ref{situation-pre-pure}", "we say a diagram (\\ref{equation-impurity}) defines an", "{\\it impurity of $\\mathcal{F}$ above $y$}", "if $\\xi \\in \\text{Ass}_{X_T/T}(\\mathcal{F}_T)$ and", "$t \\not \\in f_T(\\overline{\\{\\xi\\}})$. We will indicate", "this by saying ``let $(g : T \\to Y, t' \\leadsto t, \\xi)$ be", "an impurity of $\\mathcal{F}$ above $y$''." ], "refs": [], "ref_ids": [] }, { "id": 7202, "type": "definition", "label": "spaces-flat-definition-pure", "categories": [ "spaces-flat" ], "title": "spaces-flat-definition-pure", "contents": [ "In Situation \\ref{situation-pre-pure}.", "\\begin{enumerate}", "\\item We say $\\mathcal{F}$ is {\\it pure above $y$} if {\\bf none} of the", "equivalent conditions of Lemma \\ref{lemma-pure-along-X-y} hold.", "\\item We say $\\mathcal{F}$ is {\\it universally pure above $y$}", "if there does not exist any impurity of $\\mathcal{F}$ above $y$.", "\\item We say that $X$ is {\\it pure above $y$} if $\\mathcal{O}_X$", "is pure above $y$.", "\\item We say $\\mathcal{F}$ is {\\it universally $Y$-pure}, or", "{\\it universally pure relative to $Y$} if $\\mathcal{F}$ is universally", "pure above $y$ for every $y \\in |Y|$.", "\\item We say $\\mathcal{F}$ is {\\it $Y$-pure}, or", "{\\it pure relative to $Y$} if $\\mathcal{F}$ is pure above $y$", "for every $y \\in |Y|$.", "\\item We say that $X$ is {\\it $Y$-pure} or {\\it pure relative to $Y$}", "if $\\mathcal{O}_X$ is pure relative to $Y$.", "\\end{enumerate}" ], "refs": [ "spaces-flat-lemma-pure-along-X-y" ], "ref_ids": [ 7152 ] }, { "id": 7203, "type": "definition", "label": "spaces-flat-definition-flattening", "categories": [ "spaces-flat" ], "title": "spaces-flat-definition-flattening", "contents": [ "Let $S$ be a scheme.", "Let $X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "We say that the {\\it universal flattening of $\\mathcal{F}$ exists}", "if the functor $F_{flat}$ defined in Situation \\ref{situation-flat}", "is an algebraic space.", "We say that the {\\it universal flattening of $X$ exists}", "if the universal flattening of $\\mathcal{O}_X$ exists." ], "refs": [], "ref_ids": [] }, { "id": 7204, "type": "definition", "label": "spaces-flat-definition-flat-dimension-n", "categories": [ "spaces-flat" ], "title": "spaces-flat-definition-flat-dimension-n", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$ which is locally of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.", "Let $n \\geq 0$.", "We say {\\it $\\mathcal{F}$ is flat over $Y$ in dimensions $\\geq n$}", "if the equivalent conditions of Lemma \\ref{lemma-pre-flat-dimension-n}", "are satisfied." ], "refs": [ "spaces-flat-lemma-pre-flat-dimension-n" ], "ref_ids": [ 7186 ] }, { "id": 7281, "type": "definition", "label": "spaces-chow-definition-delta-dimension", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-delta-dimension", "contents": [ "In Situation \\ref{situation-setup} for any good $X/B$", "and any irreducible closed subset $T \\subset |X|$ we define", "$$", "\\dim_\\delta(T) = \\delta(\\xi)", "$$", "where $\\xi \\in T$ is the generic point of $T$.", "We will call this the {\\it $\\delta$-dimension of $T$}.", "If $T \\subset |X|$ is any closed subset, then we define", "$\\dim_\\delta(T)$ as the supremum of the $\\delta$-dimensions", "of the irreducible components of $T$.", "If $Z$ is a closed subspace of $X$, then we set", "$\\dim_\\delta(Z) = \\dim_\\delta(|Z|)$." ], "refs": [], "ref_ids": [] }, { "id": 7282, "type": "definition", "label": "spaces-chow-definition-cycles", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-cycles", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $k \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item A {\\it cycle on $X$} is a formal sum", "$$", "\\alpha = \\sum n_Z [Z]", "$$", "where the sum is over integral closed subspaces $Z \\subset X$,", "each $n_Z \\in \\mathbf{Z}$, and", "$\\{|Z|; n_Z \\not = 0\\}$ is a locally finite", "collection of subsets of $|X|$", "(Topology, Definition \\ref{topology-definition-locally-finite}).", "\\item A {\\it $k$-cycle} on $X$ is", "a cycle", "$$", "\\alpha = \\sum n_Z [Z]", "$$", "where $n_Z \\not = 0 \\Rightarrow \\dim_\\delta(Z) = k$.", "\\item The abelian group of all $k$-cycles on $X$ is denoted $Z_k(X)$.", "\\end{enumerate}" ], "refs": [ "topology-definition-locally-finite" ], "ref_ids": [ 8376 ] }, { "id": 7283, "type": "definition", "label": "spaces-chow-definition-length-at-x", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-length-at-x", "contents": [ "Let $S$ be a scheme and let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Let $x \\in |X|$. Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$.", "We say {\\it $\\mathcal{F}$ has length $d$ at $x$}", "if the equivalent conditions of Lemma \\ref{lemma-length}", "are satisfied." ], "refs": [ "spaces-chow-lemma-length" ], "ref_ids": [ 7207 ] }, { "id": 7284, "type": "definition", "label": "spaces-chow-definition-cycle-associated-to-closed-subscheme", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-cycle-associated-to-closed-subscheme", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $Y \\subset X$ be a closed subspace.", "\\begin{enumerate}", "\\item For an irreducible component $Z \\subset Y$ with generic point $\\xi$", "the length of $\\mathcal{O}_Y$ at $\\xi$", "(Definition \\ref{definition-length-at-x}) is called the", "{\\it multiplicity of $Z$ in $Y$}.", "By Lemma \\ref{lemma-length-finite} applied to $\\mathcal{O}_Y$", "on $Y$ this is a positive integer.", "\\item Assume $\\dim_\\delta(Y) \\leq k$.", "The {\\it $k$-cycle associated to $Y$} is", "$$", "[Y]_k = \\sum m_{Z, Y}[Z]", "$$", "where the sum is over the irreducible components $Z$ of $Y$", "of $\\delta$-dimension $k$ and $m_{Z, Y}$ is the multiplicity", "of $Z$ in $Y$.", "This is a $k$-cycle by Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-components-locally-finite}.", "\\end{enumerate}" ], "refs": [ "spaces-chow-definition-length-at-x", "spaces-chow-lemma-length-finite", "spaces-over-fields-lemma-components-locally-finite" ], "ref_ids": [ 7283, 7209, 12831 ] }, { "id": 7285, "type": "definition", "label": "spaces-chow-definition-cycle-associated-to-coherent-sheaf", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-cycle-associated-to-coherent-sheaf", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item For an integral closed subspace $Z \\subset X$ with generic point $\\xi$", "such that $|Z|$ is an irreducible component of $\\text{Supp}(\\mathcal{F})$", "the length of $\\mathcal{F}$ at $\\xi$ (Definition \\ref{definition-length-at-x})", "is called the {\\it multiplicity of $Z$ in $\\mathcal{F}$}.", "By Lemma \\ref{lemma-length-finite} this is a positive integer.", "\\item Assume $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$.", "The {\\it $k$-cycle associated to $\\mathcal{F}$} is", "$$", "[\\mathcal{F}]_k = \\sum m_{Z, \\mathcal{F}}[Z]", "$$", "where the sum is over the integral closed subspaces $Z \\subset X$", "corresponding to irreducible components of", "$\\text{Supp}(\\mathcal{F})$ of $\\delta$-dimension $k$", "and $m_{Z, \\mathcal{F}}$ is the multiplicity of $Z$ in $\\mathcal{F}$.", "This is a $k$-cycle by Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-components-locally-finite}.", "\\end{enumerate}" ], "refs": [ "spaces-chow-definition-length-at-x", "spaces-chow-lemma-length-finite", "spaces-over-fields-lemma-components-locally-finite" ], "ref_ids": [ 7283, 7209, 12831 ] }, { "id": 7286, "type": "definition", "label": "spaces-chow-definition-proper-pushforward", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-proper-pushforward", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Let $f : X \\to Y$ be a morphism over $B$.", "Assume $f$ is proper.", "\\begin{enumerate}", "\\item Let $Z \\subset X$ be an integral closed subspace", "with $\\dim_\\delta(Z) = k$. Let $Z' \\subset Y$ be the", "image of $Z$ as in Lemma \\ref{lemma-proper-image}.", "We define", "$$", "f_*[Z] =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & \\dim_\\delta(Z')< k, \\\\", "\\deg(Z/Z') [Z'] & \\text{if} & \\dim_\\delta(Z') = k.", "\\end{matrix}", "\\right.", "$$", "The degree of $Z$ over $Z'$ is defined and finite if", "$\\dim_\\delta(Z') = \\dim_\\delta(Z)$ by Lemma \\ref{lemma-equal-dimension} and", "Spaces over Fields, Definition \\ref{spaces-over-fields-definition-degree}.", "\\item Let $\\alpha = \\sum n_Z [Z]$ be a $k$-cycle on $X$. The", "{\\it pushforward} of $\\alpha$ as the sum", "$$", "f_* \\alpha = \\sum n_Z f_*[Z]", "$$", "where each $f_*[Z]$ is defined as above. The sum is locally finite", "by Lemma \\ref{lemma-quasi-compact-locally-finite} above.", "\\end{enumerate}" ], "refs": [ "spaces-chow-lemma-proper-image", "spaces-chow-lemma-equal-dimension", "spaces-over-fields-definition-degree", "spaces-chow-lemma-quasi-compact-locally-finite" ], "ref_ids": [ 7214, 7215, 12885, 7216 ] }, { "id": 7287, "type": "definition", "label": "spaces-chow-definition-flat-pullback", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-flat-pullback", "contents": [ "In Situation \\ref{situation-setup} let $X, Y/B$ be good.", "Let $f : X \\to Y$ be a morphism over $B$.", "Assume $f$ is flat of relative dimension $r$.", "\\begin{enumerate}", "\\item Let $Z \\subset Y$ be an integral closed subspace of", "$\\delta$-dimension $k$. We define $f^*[Z]$ to be the", "$(k+r)$-cycle on $X$ associated to the scheme theoretic inverse image", "$$", "f^*[Z] = [f^{-1}(Z)]_{k+r}.", "$$", "This makes sense since $\\dim_\\delta(f^{-1}(Z)) = k + r$", "by Lemma \\ref{lemma-flat-inverse-image-dimension}.", "\\item Let $\\alpha = \\sum n_i [Z_i]$ be", "a $k$-cycle on $Y$. The {\\it flat pullback of $\\alpha$ by $f$}", "is the sum", "$$", "f^* \\alpha = \\sum n_i f^*[Z_i]", "$$", "where each $f^*[Z_i]$ is defined as above.", "The sum is locally finite by Lemma \\ref{lemma-inverse-image-locally-finite}.", "\\item We denote $f^* : Z_k(Y) \\to Z_{k + r}(X)$ the map of abelian", "groups so obtained.", "\\end{enumerate}" ], "refs": [ "spaces-chow-lemma-flat-inverse-image-dimension", "spaces-chow-lemma-inverse-image-locally-finite" ], "ref_ids": [ 7219, 7220 ] }, { "id": 7288, "type": "definition", "label": "spaces-chow-definition-principal-divisor", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-principal-divisor", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good. Assume $X$ is", "integral with $\\dim_\\delta(X) = n$.", "Let $f \\in R(X)^*$. The {\\it principal divisor associated to $f$}", "is the $(n - 1)$-cycle", "$$", "\\text{div}(f) = \\text{div}_X(f) = \\sum \\text{ord}_Z(f) [Z]", "$$", "defined in Spaces over Fields, Definition", "\\ref{spaces-over-fields-definition-principal-divisor}.", "This makes sense because prime divisors have $\\delta$-dimension $n - 1$ by", "Lemma \\ref{lemma-divisor-delta-dimension}." ], "refs": [ "spaces-over-fields-definition-principal-divisor", "spaces-chow-lemma-divisor-delta-dimension" ], "ref_ids": [ 12888, 7227 ] }, { "id": 7289, "type": "definition", "label": "spaces-chow-definition-rational-equivalence", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-rational-equivalence", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $k \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item Given any locally finite collection $\\{W_j \\subset X\\}$", "of integral closed subspaces with $\\dim_\\delta(W_j) = k + 1$,", "and any $f_j \\in R(W_j)^*$ we may consider", "$$", "\\sum (i_j)_*\\text{div}(f_j) \\in Z_k(X)", "$$", "where $i_j : W_j \\to X$ is the inclusion morphism.", "This makes sense as the morphism", "$\\coprod i_j : \\coprod W_j \\to X$ is proper.", "\\item We say that $\\alpha \\in Z_k(X)$ is {\\it rationally equivalent to zero}", "if $\\alpha$ is a cycle of the form displayed above.", "\\item We say $\\alpha, \\beta \\in Z_k(X)$ are", "{\\it rationally equivalent} and we write $\\alpha \\sim_{rat} \\beta$", "if $\\alpha - \\beta$ is rationally equivalent to zero.", "\\item We define", "$$", "\\CH_k(X) = Z_k(X) / \\sim_{rat}", "$$", "to be the {\\it Chow group of $k$-cycles on $X$}. This is sometimes called", "the {\\it Chow group of $k$-cycles modulo rational equivalence on $X$}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7290, "type": "definition", "label": "spaces-chow-definition-divisor-invertible-sheaf", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-divisor-invertible-sheaf", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Assume $X$ is integral and $n = \\dim_\\delta(X)$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item For any nonzero meromorphic section $s$ of $\\mathcal{L}$", "we define the {\\it Weil divisor associated to $s$} is the", "$(n - 1)$-cycle", "$$", "\\text{div}_\\mathcal{L}(s) =", "\\sum \\text{ord}_{Z, \\mathcal{L}}(s) [Z]", "$$", "defined in Spaces over Fields, Definition", "\\ref{spaces-over-fields-definition-divisor-invertible-sheaf}.", "This makes sense because Weil divisors have $\\delta$-dimension $n - 1$", "by Lemma \\ref{lemma-divisor-delta-dimension}.", "\\item We define {\\it Weil divisor associated to $\\mathcal{L}$} as", "$$", "c_1(\\mathcal{L}) \\cap [X] =", "\\text{class of }\\text{div}_\\mathcal{L}(s) \\in \\CH_{n - 1}(X)", "$$", "where $s$ is any nonzero meromorphic section of $\\mathcal{L}$ over", "$X$. This is well defined by", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-divisor-meromorphic-well-defined}.", "\\end{enumerate}" ], "refs": [ "spaces-over-fields-definition-divisor-invertible-sheaf", "spaces-chow-lemma-divisor-delta-dimension", "spaces-over-fields-lemma-divisor-meromorphic-well-defined" ], "ref_ids": [ 12891, 7227, 12837 ] }, { "id": 7291, "type": "definition", "label": "spaces-chow-definition-cap-c1", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-cap-c1", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "We define, for every integer $k$, an operation", "$$", "c_1(\\mathcal{L}) \\cap - :", "Z_{k + 1}(X) \\to \\CH_k(X)", "$$", "called {\\it intersection with the first Chern class of $\\mathcal{L}$}.", "\\begin{enumerate}", "\\item Given an integral closed subspace $i : W \\to X$ with", "$\\dim_\\delta(W) = k + 1$ we define", "$$", "c_1(\\mathcal{L}) \\cap [W] = i_*(c_1({i^*\\mathcal{L}}) \\cap [W])", "$$", "where the right hand side is defined in", "Definition \\ref{definition-divisor-invertible-sheaf}.", "\\item For a general $(k + 1)$-cycle $\\alpha = \\sum n_i [W_i]$ we set", "$$", "c_1(\\mathcal{L}) \\cap \\alpha = \\sum n_i c_1(\\mathcal{L}) \\cap [W_i]", "$$", "\\end{enumerate}" ], "refs": [ "spaces-chow-definition-divisor-invertible-sheaf" ], "ref_ids": [ 7290 ] }, { "id": 7292, "type": "definition", "label": "spaces-chow-definition-gysin-homomorphism", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-gysin-homomorphism", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $(\\mathcal{L}, s)$ be a pair consisting of an invertible", "sheaf and a global section $s \\in \\Gamma(X, \\mathcal{L})$.", "Let $D = Z(s)$ be the vanishing locus of $s$, and", "denote $i : D \\to X$ the closed immersion.", "We define, for every integer $k$, a (refined) {\\it Gysin homomorphism}", "$$", "i^* : Z_{k + 1}(X) \\to \\CH_k(D).", "$$", "by the following rules:", "\\begin{enumerate}", "\\item Given a integral closed subspace $W \\subset X$ with", "$\\dim_\\delta(W) = k + 1$ we define", "\\begin{enumerate}", "\\item if $W \\not \\subset D$, then $i^*[W] = [D \\cap W]_k$ as a", "$k$-cycle on $D$, and", "\\item if $W \\subset D$, then", "$i^*[W] = i'_*(c_1(\\mathcal{L}|_W) \\cap [W])$,", "where $i' : W \\to D$ is the induced closed immersion.", "\\end{enumerate}", "\\item For a general $(k + 1)$-cycle $\\alpha = \\sum n_j[W_j]$", "we set", "$$", "i^*\\alpha = \\sum n_j i^*[W_j]", "$$", "\\item If $D$ is an effective Cartier divisor, then we denote", "$D \\cdot \\alpha = i_*i^*\\alpha$ the pushforward of", "the class to a class on $X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7293, "type": "definition", "label": "spaces-chow-definition-bivariant-class", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-bivariant-class", "contents": [ "\\begin{reference}", "Similar to \\cite[Definition 17.1]{F}", "\\end{reference}", "In Situation \\ref{situation-setup} let $f : X \\to Y$ be a morphism of", "good algebraic spaces over $B$. Let $p \\in \\mathbf{Z}$.", "A {\\it bivariant class $c$ of degree $p$ for $f$} is given by a rule", "which assigns to every morphism $Y' \\to Y$ of good algebraic spaces over $B$", "and every $k$ a map", "$$", "c \\cap - : \\CH_k(Y') \\longrightarrow \\CH_{k - p}(X')", "$$", "where $X' = Y' \\times_Y X$, satisfying the following conditions", "\\begin{enumerate}", "\\item if $Y'' \\to Y'$ is a proper morphism, then", "$c \\cap (Y'' \\to Y')_*\\alpha'' = (X'' \\to X')_*(c \\cap \\alpha'')$", "for all $\\alpha''$ on $Y''$,", "\\item if $Y'' \\to Y'$ a morphism of good algebraic spaces over $B$", "which is flat of relative dimension $r$, then", "$c \\cap (Y'' \\to Y')^*\\alpha' = (X'' \\to X')^*(c \\cap \\alpha')$", "for all $\\alpha'$ on $Y'$,", "\\item if $(\\mathcal{L}', s', i' : D' \\to Y')$ is as in", "Definition \\ref{definition-gysin-homomorphism}", "with pullback $(\\mathcal{N}', t', j' : E' \\to X')$ to $X'$,", "then we have $c \\cap (i')^*\\alpha' = (j')^*(c \\cap \\alpha')$", "for all $\\alpha'$ on $Y'$.", "\\end{enumerate}", "The collection of all bivariant classes of degree $p$ for $f$ is", "denoted $A^p(X \\to Y)$." ], "refs": [ "spaces-chow-definition-gysin-homomorphism" ], "ref_ids": [ 7292 ] }, { "id": 7294, "type": "definition", "label": "spaces-chow-definition-chow-cohomology", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-chow-cohomology", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good. The {\\it Chow cohomology}", "of $X$ is the graded $\\mathbf{Z}$-algebra $A^*(X)$ whose degree", "$p$ component is $A^p(X \\to X)$." ], "refs": [], "ref_ids": [] }, { "id": 7295, "type": "definition", "label": "spaces-chow-definition-chern-classes", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-chern-classes", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.", "For $i = 0, \\ldots, r$ the {\\it $i$th Chern class of $\\mathcal{E}$}", "is the bivariant class $c_i(\\mathcal{E}) \\in A^i(X)$ of degree $i$", "constructed in Lemma \\ref{lemma-segre-classes}.", "The {\\it total Chern class of $\\mathcal{E}$}", "is the formal sum", "$$", "c(\\mathcal{E}) =", "c_0(\\mathcal{E}) + c_1(\\mathcal{E}) + \\ldots + c_r(\\mathcal{E})", "$$", "which is viewed as a nonhomogeneous bivariant class on $X$." ], "refs": [ "spaces-chow-lemma-segre-classes" ], "ref_ids": [ 7268 ] }, { "id": 7296, "type": "definition", "label": "spaces-chow-definition-degree-zero-cycle", "categories": [ "spaces-chow" ], "title": "spaces-chow-definition-degree-zero-cycle", "contents": [ "Let $k$ be a field. Let $p : X \\to \\Spec(k)$ be a proper morphism of", "algebraic spaces. The {\\it degree of a zero cycle} on $X$ is given by", "proper pushforward", "$$", "p_* : \\CH_0(X) \\longrightarrow \\CH_0(\\Spec(k)) \\longrightarrow \\mathbf{Z}", "$$", "(Lemma \\ref{lemma-proper-pushforward-rational-equivalence})", "composed with the natural isomorphism $\\CH_0(\\Spec(k)) \\to \\mathbf{Z}$", "which maps $[\\Spec(k)]$ to $1$. Notation: $\\deg(\\alpha)$." ], "refs": [ "spaces-chow-lemma-proper-pushforward-rational-equivalence" ], "ref_ids": [ 7233 ] }, { "id": 7369, "type": "definition", "label": "sdga-definition-ga", "categories": [ "sdga" ], "title": "sdga-definition-ga", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. A", "{\\it sheaf of graded $\\mathcal{O}$-algebras}", "or a {\\it sheaf of graded algebras} on $(\\mathcal{C}, \\mathcal{O})$", "is given by a family $\\mathcal{A}^n$ indexed by $n \\in \\mathbf{Z}$", "of $\\mathcal{O}$-modules endowed with $\\mathcal{O}$-bilinear maps", "$$", "\\mathcal{A}^n \\times \\mathcal{A}^m \\to \\mathcal{A}^{n + m},\\quad", "(a, b) \\longmapsto ab", "$$", "called the multiplication maps with the following properties", "\\begin{enumerate}", "\\item multiplication is associative, and", "\\item there is a global section $1$ of $\\mathcal{A}^0$", "which is a two-sided identity for multiplication.", "\\end{enumerate}", "We often denote such a structure $\\mathcal{A}$.", "A {\\it homomorphism of graded $\\mathcal{O}$-algebras}", "$f : \\mathcal{A} \\to \\mathcal{B}$ is a family of maps", "$f^n : \\mathcal{A}^n \\to \\mathcal{B}^n$", "of $\\mathcal{O}$-modules compatible with the multiplication maps." ], "refs": [], "ref_ids": [] }, { "id": 7370, "type": "definition", "label": "sdga-definition-gm", "categories": [ "sdga" ], "title": "sdga-definition-gm", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$.", "A (right) {\\it graded $\\mathcal{A}$-module} or (right)", "{\\it graded module} over $\\mathcal{A}$", "is given by a family $\\mathcal{M}^n$ indexed by $n \\in \\mathbf{Z}$", "of $\\mathcal{O}$-modules endowed with", "$\\mathcal{O}$-bilinear maps", "$$", "\\mathcal{M}^n \\times \\mathcal{A}^m \\to \\mathcal{M}^{n + m},\\quad", "(x, a) \\longmapsto xa", "$$", "called the multiplication maps with the following properties", "\\begin{enumerate}", "\\item multiplication satisfies $(xa)a' = x(aa')$,", "\\item the identity section $1$ of $\\mathcal{A}^0$", "acts as the identity on $\\mathcal{M}^n$ for all $n$.", "\\end{enumerate}", "We often say ``let $\\mathcal{M}$ be a graded $\\mathcal{A}$-module''", "to indicate this situation.", "A {\\it homomorphism of graded $\\mathcal{A}$-modules}", "$f : \\mathcal{M} \\to \\mathcal{N}$ is a family of maps", "$f^n : \\mathcal{M}^n \\to \\mathcal{N}^n$", "of $\\mathcal{O}$-modules compatible with the multiplication maps.", "The category of (right) graded $\\mathcal{A}$-modules", "is denoted $\\text{Mod}_\\mathcal{A}$." ], "refs": [], "ref_ids": [] }, { "id": 7371, "type": "definition", "label": "sdga-definition-bimodule", "categories": [ "sdga" ], "title": "sdga-definition-bimodule", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$", "and $\\mathcal{B}$ be a sheaves of graded algebras on", "$(\\mathcal{C}, \\mathcal{O})$. A", "{\\it graded $(\\mathcal{A}, \\mathcal{B})$-bimodule}", "is given by a family $\\mathcal{M}^n$ indexed by $n \\in \\mathbf{Z}$", "of $\\mathcal{O}$-modules endowed with $\\mathcal{O}$-bilinear maps", "$$", "\\mathcal{M}^n \\times \\mathcal{B}^m \\to \\mathcal{M}^{n + m},\\quad", "(x, b) \\longmapsto xb", "$$", "and", "$$", "\\mathcal{A}^n \\times \\mathcal{M}^m \\to \\mathcal{M}^{n + m},\\quad", "(a, x) \\longmapsto ax", "$$", "called the multiplication maps with the following properties", "\\begin{enumerate}", "\\item multiplication satisfies $a(a'x) = (aa')x$ and", "$(xb)b' = x(bb')$,", "\\item $(ax)b = a(xb)$,", "\\item the identity section $1$ of $\\mathcal{A}^0$ acts as the", "identity by multiplication, and", "\\item the identity section $1$ of", "$\\mathcal{B}^0$ acts as the identity by multiplication.", "\\end{enumerate}", "We often denote such a structure $\\mathcal{M}$.", "A {\\it homomorphism of graded $(\\mathcal{A}, \\mathcal{B})$-bimodules}", "$f : \\mathcal{M} \\to \\mathcal{N}$ is a family of maps", "$f^n : \\mathcal{M}^n \\to \\mathcal{N}^n$", "of $\\mathcal{O}$-modules compatible with the multiplication maps." ], "refs": [], "ref_ids": [] }, { "id": 7372, "type": "definition", "label": "sdga-definition-dga", "categories": [ "sdga" ], "title": "sdga-definition-dga", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. A", "{\\it sheaf of differential graded $\\mathcal{O}$-algebras}", "or a {\\it sheaf of differential graded algebras} on", "$(\\mathcal{C}, \\mathcal{O})$ is a cochain complex", "$\\mathcal{A}^\\bullet$ of $\\mathcal{O}$-modules", "endowed with $\\mathcal{O}$-bilinear maps", "$$", "\\mathcal{A}^n \\times \\mathcal{A}^m \\to \\mathcal{A}^{n + m},\\quad", "(a, b) \\longmapsto ab", "$$", "called the multiplication maps with the following properties", "\\begin{enumerate}", "\\item multiplication is associative,", "\\item there is a global section $1$ of $\\mathcal{A}^0$", "which is a two-sided identity for multiplication,", "\\item for $U \\in \\Ob(\\mathcal{C})$, $a \\in \\mathcal{A}^n(U)$, and", "$b \\in \\mathcal{A}^m(U)$ we have", "$$", "\\text{d}^{n + m}(ab) = \\text{d}^n(a)b + (-1)^n a\\text{d}^m(b)", "$$", "\\end{enumerate}", "We often denote such a structure $(\\mathcal{A}, \\text{d})$.", "A {\\it homomorphism of differential graded $\\mathcal{O}$-algebras}", "from $(\\mathcal{A}, \\text{d})$ to $(\\mathcal{B}, \\text{d})$ is a map", "$f : \\mathcal{A}^\\bullet \\to \\mathcal{B}^\\bullet$ of complexes", "of $\\mathcal{O}$-modules compatible with the multiplication maps." ], "refs": [], "ref_ids": [] }, { "id": 7373, "type": "definition", "label": "sdga-definition-dgm", "categories": [ "sdga" ], "title": "sdga-definition-dgm", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$.", "A (right) {\\it differential graded $\\mathcal{A}$-module} or (right)", "{\\it differential graded module} over $\\mathcal{A}$", "is a cochain complex $\\mathcal{M}^\\bullet$ endowed with", "$\\mathcal{O}$-bilinear maps", "$$", "\\mathcal{M}^n \\times \\mathcal{A}^m \\to \\mathcal{M}^{n + m},\\quad", "(x, a) \\longmapsto xa", "$$", "called the multiplication maps with the following properties", "\\begin{enumerate}", "\\item multiplication satisfies $(xa)a' = x(aa')$,", "\\item the identity section $1$ of $\\mathcal{A}^0$", "acts as the identity on $\\mathcal{M}^n$ for all $n$,", "\\item for $U \\in \\Ob(\\mathcal{C})$, $x \\in \\mathcal{M}^n(U)$, and", "$a \\in \\mathcal{A}^m(U)$ we have", "$$", "\\text{d}^{n + m}(xa) = \\text{d}^n(x)a + (-1)^n x\\text{d}^m(a)", "$$", "\\end{enumerate}", "We often say ``let $\\mathcal{M}$ be a differential graded", "$\\mathcal{A}$-module'' to indicate this situation.", "A {\\it homomorphism of differential graded $\\mathcal{A}$-modules}", "from $\\mathcal{M}$ to $\\mathcal{N}$ is a map", "$f : \\mathcal{M}^\\bullet \\to \\mathcal{N}^\\bullet$ of complexes", "of $\\mathcal{O}$-modules compatible with the multiplication maps.", "The category of (right) differential graded $\\mathcal{A}$-modules", "is denoted $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$." ], "refs": [], "ref_ids": [] }, { "id": 7374, "type": "definition", "label": "sdga-definition-dg-bimodule", "categories": [ "sdga" ], "title": "sdga-definition-dg-bimodule", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$", "and $\\mathcal{B}$ be a sheaves of differential graded algebras on", "$(\\mathcal{C}, \\mathcal{O})$. A", "{\\it differential graded $(\\mathcal{A}, \\mathcal{B})$-bimodule}", "is given by a complex $\\mathcal{M}^\\bullet$", "of $\\mathcal{O}$-modules endowed with $\\mathcal{O}$-bilinear maps", "$$", "\\mathcal{M}^n \\times \\mathcal{B}^m \\to \\mathcal{M}^{n + m},\\quad", "(x, b) \\longmapsto xb", "$$", "and", "$$", "\\mathcal{A}^n \\times \\mathcal{M}^m \\to \\mathcal{M}^{n + m},\\quad", "(a, x) \\longmapsto ax", "$$", "called the multiplication maps with the following properties", "\\begin{enumerate}", "\\item multiplication satisfies $a(a'x) = (aa')x$ and", "$(xb)b' = x(bb')$,", "\\item $(ax)b = a(xb)$,", "\\item $\\text{d}(ax) = \\text{d}(a) x + (-1)^{\\deg(a)}a \\text{d}(x)$ and", "$\\text{d}(xb) = \\text{d}(x) b + (-1)^{\\deg(x)}x \\text{d}(b)$,", "\\item the identity section $1$ of $\\mathcal{A}^0$ acts as the", "identity by multiplication, and", "\\item the identity section $1$ of", "$\\mathcal{B}^0$ acts as the identity by multiplication.", "\\end{enumerate}", "We often denote such a structure $\\mathcal{M}$ and sometimes", "we write ${}_\\mathcal{A}\\mathcal{M}_\\mathcal{B}$.", "A {\\it homomorphism of differential graded", "$(\\mathcal{A}, \\mathcal{B})$-bimodules}", "$f : \\mathcal{M} \\to \\mathcal{N}$ is a map of complexes", "$f : \\mathcal{M}^\\bullet \\to \\mathcal{N}^\\bullet$", "of $\\mathcal{O}$-modules compatible with the multiplication maps." ], "refs": [], "ref_ids": [] }, { "id": 7375, "type": "definition", "label": "sdga-definition-homotopy", "categories": [ "sdga" ], "title": "sdga-definition-homotopy", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. Let", "$f, g : \\mathcal{M} \\to \\mathcal{N}$", "be homomorphisms of differential graded $\\mathcal{A}$-modules.", "A {\\it homotopy between $f$ and $g$} is a graded $\\mathcal{A}$-module", "map $h : \\mathcal{M} \\to \\mathcal{N}$ homogeneous of degree $-1$", "such that", "$$", "f - g = \\text{d}_\\mathcal{N} \\circ h + h \\circ \\text{d}_\\mathcal{M}", "$$", "If a homotopy exists, then we say $f$ and $g$ are {\\it homotopic}." ], "refs": [], "ref_ids": [] }, { "id": 7376, "type": "definition", "label": "sdga-definition-complexes-notation", "categories": [ "sdga" ], "title": "sdga-definition-complexes-notation", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$.", "The {\\it homotopy category}, denoted $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$,", "is the category whose objects are the objects of", "$\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ and whose morphisms are homotopy classes", "of homomorphisms of differential graded $\\mathcal{A}$-modules." ], "refs": [], "ref_ids": [] }, { "id": 7377, "type": "definition", "label": "sdga-definition-cone", "categories": [ "sdga" ], "title": "sdga-definition-cone", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$.", "Let $f : \\mathcal{K} \\to \\mathcal{L}$", "be a homomorphism of differential graded $\\mathcal{A}$-modules.", "The {\\it cone} of $f$ is the differential graded $\\mathcal{A}$-module", "$C(f)$ defined as follows:", "\\begin{enumerate}", "\\item the underlying complex of $\\mathcal{O}$-modules", "is the cone of the corresponding map", "$f : \\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet$ of", "complexes of $\\mathcal{A}$-modules, i.e., we have", "$C(f)^n = \\mathcal{L}^n \\oplus \\mathcal{K}^{n + 1}$ and", "differential", "$$", "d_{C(f)} =", "\\left(", "\\begin{matrix}", "\\text{d}_\\mathcal{L} & f \\\\", "0 & -\\text{d}_\\mathcal{K}", "\\end{matrix}", "\\right)", "$$", "\\item the multiplication map", "$$", "C(f)^n \\times \\mathcal{A}^m \\to C(f)^{n + m}", "$$", "is the direct sum of the multiplication map", "$\\mathcal{L}^n \\times \\mathcal{A}^m \\to \\mathcal{L}^{n + m}$", "and the multiplication map", "$\\mathcal{K}^{n + 1} \\times \\mathcal{A}^m \\to \\mathcal{K}^{n + 1 + m}$.", "\\end{enumerate}", "It comes equipped with canonical hommorphisms of differential graded", "$\\mathcal{A}$-modules $i : \\mathcal{L} \\to C(f)$", "and $p : C(f) \\to \\mathcal{K}[1]$ induced by the obvious maps." ], "refs": [], "ref_ids": [] }, { "id": 7378, "type": "definition", "label": "sdga-definition-graded-injective", "categories": [ "sdga" ], "title": "sdga-definition-graded-injective", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. A diffential graded $\\mathcal{A}$-module", "$\\mathcal{I}$ is said to be {\\it graded injective}\\footnote{This may be", "nonstandard terminology.} if $\\mathcal{M}$ viewed as a graded", "$\\mathcal{A}$-module is an injective object of the category", "$\\text{Mod}_\\mathcal{A}$ of graded $\\mathcal{A}$-modules." ], "refs": [], "ref_ids": [] }, { "id": 7379, "type": "definition", "label": "sdga-definition-K-injective", "categories": [ "sdga" ], "title": "sdga-definition-K-injective", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. A diffential graded $\\mathcal{A}$-module", "$\\mathcal{I}$ is {\\it K-injective} if for every acyclic", "differential graded $\\mathcal{M}$ we have ", "$$", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}, \\mathcal{I}) = 0", "$$" ], "refs": [], "ref_ids": [] }, { "id": 7380, "type": "definition", "label": "sdga-definition-derived-category", "categories": [ "sdga" ], "title": "sdga-definition-derived-category", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. Let $\\text{Qis}$ be as in", "Lemma \\ref{lemma-qis}. The", "{\\it derived category of $(\\mathcal{A}, \\text{d})$} is the triangulated", "category", "$$", "D(\\mathcal{A}, \\text{d}) = \\text{Qis}^{-1}K(\\text{Mod}_{(A, \\text{d})})", "$$", "discussed in more detail above." ], "refs": [ "sdga-lemma-qis" ], "ref_ids": [ 7344 ] }, { "id": 7381, "type": "definition", "label": "sdga-definition-pullback", "categories": [ "sdga" ], "title": "sdga-definition-pullback", "contents": [ "Derived tensor product and derived pullback.", "\\begin{enumerate}", "\\item Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$\\mathcal{A}$, $\\mathcal{B}$ be differential graded $\\mathcal{O}$-algebras.", "Let $\\mathcal{N}$ be a differential graded", "$(\\mathcal{A}, \\mathcal{B})$-bimodule.", "The functor $D(\\mathcal{A}, \\text{d}) \\to D(\\mathcal{B}, \\text{d})$", "constructed in Lemma \\ref{lemma-derived-tensor-product}", "is called the {\\it derived tensor product} and denoted", "$- \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}$.", "\\item Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi. Let $\\mathcal{A}$ be a differential", "graded $\\mathcal{O}_\\mathcal{C}$-algebra. Let $\\mathcal{B}$ be a", "differential graded $\\mathcal{O}_\\mathcal{D}$-algebra. Let", "$\\varphi : \\mathcal{B} \\to f_*\\mathcal{A}$ be a homomorphism", "of differential graded $\\mathcal{O}_\\mathcal{D}$-algebras.", "The functor $D(\\mathcal{B}, \\text{d}) \\to D(\\mathcal{A}, \\text{d})$", "constructed in Lemma \\ref{lemma-derived-tensor-product}", "is called {\\it derived pullback}", "and denote $Lf^*$.", "\\end{enumerate}" ], "refs": [ "sdga-lemma-derived-tensor-product", "sdga-lemma-derived-tensor-product" ], "ref_ids": [ 7351, 7351 ] }, { "id": 7382, "type": "definition", "label": "sdga-definition-pushforward", "categories": [ "sdga" ], "title": "sdga-definition-pushforward", "contents": [ "Derived internal hom and derived pushforward.", "\\begin{enumerate}", "\\item Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$\\mathcal{A}$, $\\mathcal{B}$ be differential graded $\\mathcal{O}$-algebras.", "Let $\\mathcal{N}$ be a differential graded", "$(\\mathcal{A}, \\mathcal{B})$-bimodule. The right derived extension", "$$", "R\\SheafHom_\\mathcal{B}(\\mathcal{N}, -) :", "D(\\mathcal{B}, \\text{d})", "\\longrightarrow", "D(\\mathcal{A}, \\text{d})", "$$", "of the internal hom functor $\\SheafHom_\\mathcal{B}^{dg}(\\mathcal{N}, -)$", "is called {\\it derived internal hom}.", "\\item Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi. Let $\\mathcal{A}$ be a differential", "graded $\\mathcal{O}_\\mathcal{C}$-algebra. Let $\\mathcal{B}$ be a", "differential graded $\\mathcal{O}_\\mathcal{D}$-algebra. Let", "$\\varphi : \\mathcal{B} \\to f_*\\mathcal{A}$ be a homomorphism", "of differential graded $\\mathcal{O}_\\mathcal{D}$-algebras.", "The right derived extension", "$$", "Rf_* :", "D(\\mathcal{A}, \\text{d})", "\\longrightarrow", "D(\\mathcal{B}, \\text{d})", "$$", "of the pushforward $f_*$ is called {\\it derived pushforward}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7601, "type": "definition", "label": "stacks-morphisms-definition-separated", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-separated", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "\\begin{enumerate}", "\\item We say $f$ is {\\it DM} if $\\Delta_f$ is unramified\\footnote{The", "letters DM stand for Deligne-Mumford. If $f$ is DM then given any scheme", "$T$ and any morphism $T \\to \\mathcal{Y}$ the fibre product", "$\\mathcal{X}_T = \\mathcal{X} \\times_\\mathcal{Y} T$", "is an algebraic stack over $T$ whose diagonal is unramified, i.e.,", "$\\mathcal{X}_T$ is DM. This implies $\\mathcal{X}_T$", "is a Deligne-Mumford stack, see Theorem \\ref{theorem-DM}.", "In other words a DM morphism is one whose ``fibres'' are Deligne-Mumford", "stacks. This hopefully at least motivates the terminology.}.", "\\item We say $f$ is {\\it quasi-DM} if $\\Delta_f$ is", "locally quasi-finite\\footnote{If $f$ is quasi-DM, then the", "``fibres'' $\\mathcal{X}_T$ of $\\mathcal{X} \\to \\mathcal{Y}$ are quasi-DM. An", "algebraic stack $\\mathcal{X}$ is quasi-DM exactly if there exists a", "scheme $U$ and a surjective flat morphism $U \\to \\mathcal{X}$ of finite", "presentation which is locally quasi-finite, see", "Theorem \\ref{theorem-quasi-DM}.", "Note the similarity to being Deligne-Mumford, which", "is defined in terms of having an \\'etale covering by a scheme.}.", "\\item We say $f$ is {\\it separated} if $\\Delta_f$ is proper.", "\\item We say $f$ is {\\it quasi-separated} if $\\Delta_f$", "is quasi-compact and quasi-separated.", "\\end{enumerate}" ], "refs": [ "stacks-morphisms-theorem-DM", "stacks-morphisms-theorem-quasi-DM" ], "ref_ids": [ 7389, 7388 ] }, { "id": 7602, "type": "definition", "label": "stacks-morphisms-definition-absolute-separated", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-absolute-separated", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack over the base scheme $S$.", "Denote $p : \\mathcal{X} \\to S$ the structure morphism.", "\\begin{enumerate}", "\\item We say $\\mathcal{X}$ is {\\it DM over $S$}", "if $p : \\mathcal{X} \\to S$ is DM.", "\\item We say $\\mathcal{X}$ is {\\it quasi-DM over $S$}", "if $p : \\mathcal{X} \\to S$ is quasi-DM.", "\\item We say $\\mathcal{X}$ is {\\it separated over $S$}", "if $p : \\mathcal{X} \\to S$ is separated.", "\\item We say $\\mathcal{X}$ is {\\it quasi-separated over $S$} if", "$p : \\mathcal{X} \\to S$ is quasi-separated.", "\\item We say $\\mathcal{X}$ is {\\it DM}", "if $\\mathcal{X}$ is DM\\footnote{Theorem \\ref{theorem-DM} shows", "that this is equivalent to $\\mathcal{X}$ being a Deligne-Mumford stack.}", "over $\\Spec(\\mathbf{Z})$.", "\\item We say $\\mathcal{X}$ is {\\it quasi-DM}", "if $\\mathcal{X}$ is quasi-DM over $\\Spec(\\mathbf{Z})$.", "\\item We say $\\mathcal{X}$ is {\\it separated} if $\\mathcal{X}$", "is separated over $\\Spec(\\mathbf{Z})$.", "\\item We say $\\mathcal{X}$ is {\\it quasi-separated} if $\\mathcal{X}$", "is quasi-separated over $\\Spec(\\mathbf{Z})$.", "\\end{enumerate}", "In the last 4 definitions we view $\\mathcal{X}$", "as an algebraic stack over $\\Spec(\\mathbf{Z})$", "via", "Algebraic Stacks, Definition \\ref{algebraic-definition-viewed-as}." ], "refs": [ "stacks-morphisms-theorem-DM", "algebraic-definition-viewed-as" ], "ref_ids": [ 7389, 8489 ] }, { "id": 7603, "type": "definition", "label": "stacks-morphisms-definition-isom", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-isom", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $Z$ be an algebraic space.", "\\begin{enumerate}", "\\item Let $x : Z \\to \\mathcal{X}$ be a morphism. We set", "$$", "\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x, x) =", "Z \\times_{x, \\mathcal{X}} \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}", "$$", "We endow it with the structure of a group algebraic space over $Z$", "by pulling back the composition law discussed in", "Remark \\ref{remark-inertia-is-group-in-spaces}.", "We will sometimes refer to $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x, x)$", "as the {\\it relative sheaf of automorphisms of $x$}.", "\\item Let $x_1, x_2 : Z \\to \\mathcal{X}$ be morphisms. Set", "$y_i = f \\circ x_i$. Let $\\alpha : y_1 \\to y_2$ be a $2$-morphism.", "Then $\\alpha$ determines a morphism", "$\\Delta^\\alpha : Z \\to Z \\times_{y_1, \\mathcal{Y}, y_2} Z$ and we set", "$$", "\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}^\\alpha(x_1, x_2) =", "(Z \\times_{x_1, \\mathcal{X}, x_2} Z)", "\\times_{Z \\times_{y_1, \\mathcal{Y}, y_2} Z, \\Delta^\\alpha} Z.", "$$", "We will sometimes refer to", "$\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}^\\alpha(x_1, x_2)$", "as the {\\it relative sheaf of isomorphisms from $x_1$ to $x_2$}.", "\\end{enumerate}", "If $\\mathcal{Y} = \\Spec(\\mathbf{Z})$ or more generally when $\\mathcal{Y}$", "is an algebraic space, then we use the notation", "$\\mathit{Isom}_\\mathcal{X}(x, x)$ and $\\mathit{Isom}_\\mathcal{X}(x_1, x_2)$", "and we use the terminology {\\it sheaf of automorphisms of $x$}", "and {\\it sheaf of isomorphisms from $x_1$ to $x_2$}." ], "refs": [ "stacks-morphisms-remark-inertia-is-group-in-spaces" ], "ref_ids": [ 7632 ] }, { "id": 7604, "type": "definition", "label": "stacks-morphisms-definition-quasi-compact", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-quasi-compact", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $f$ is {\\it quasi-compact} if for every quasi-compact", "algebraic stack $\\mathcal{Z}$ and morphism $\\mathcal{Z} \\to \\mathcal{Y}$", "the fibre product $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$", "is quasi-compact." ], "refs": [], "ref_ids": [] }, { "id": 7605, "type": "definition", "label": "stacks-morphisms-definition-noetherian", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-noetherian", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. We say $\\mathcal{X}$ is", "{\\it Noetherian} if $\\mathcal{X}$ is quasi-compact, quasi-separated", "and locally Noetherian." ], "refs": [], "ref_ids": [] }, { "id": 7606, "type": "definition", "label": "stacks-morphisms-definition-affine", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-affine", "contents": [ "A morphism of algebraic stacks is said to be {\\it affine}", "if it is representable and affine in the sense of", "Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}." ], "refs": [], "ref_ids": [] }, { "id": 7607, "type": "definition", "label": "stacks-morphisms-definition-integral", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-integral", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "\\begin{enumerate}", "\\item We say $f$ is {\\it integral} if $f$ is representable and integral", "in the sense of Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}.", "\\item We say $f$ is {\\it finite} if $f$ is representable and finite", "in the sense of Properties of Stacks, Section", "\\ref{stacks-properties-section-properties-morphisms}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7608, "type": "definition", "label": "stacks-morphisms-definition-open", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-open", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "\\begin{enumerate}", "\\item We say $f$ is {\\it open} if the map of topological", "spaces $|\\mathcal{X}| \\to |\\mathcal{Y}|$ is open.", "\\item We say $f$ is {\\it universally open} if for every morphism", "of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$", "the morphism of topological spaces", "$$", "|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|", "$$", "is open, i.e., the base change", "$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$ is open.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7609, "type": "definition", "label": "stacks-morphisms-definition-submersive", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-submersive", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "\\begin{enumerate}", "\\item We say $f$ is {\\it submersive}\\footnote{This is very different", "from the notion of a submersion of differential manifolds.}", "if the continuous map $|\\mathcal{X}| \\to |\\mathcal{Y}|$ is submersive, see", "Topology, Definition \\ref{topology-definition-submersive}.", "\\item We say $f$ is {\\it universally submersive} if for every", "morphism of algebraic stacks $\\mathcal{Y}' \\to \\mathcal{Y}$", "the base change $\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Y}'$", "is submersive.", "\\end{enumerate}" ], "refs": [ "topology-definition-submersive" ], "ref_ids": [ 8349 ] }, { "id": 7610, "type": "definition", "label": "stacks-morphisms-definition-closed", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-closed", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "\\begin{enumerate}", "\\item We say $f$ is {\\it closed} if the map of topological", "spaces $|\\mathcal{X}| \\to |\\mathcal{Y}|$ is closed.", "\\item We say $f$ is {\\it universally closed} if for every morphism", "of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$", "the morphism of topological spaces", "$$", "|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|", "$$", "is closed, i.e., the base change", "$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$ is closed.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7611, "type": "definition", "label": "stacks-morphisms-definition-universally-injective", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-universally-injective", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $f$ is {\\it universally injective} if for every morphism", "of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$ the map", "$$", "|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|", "$$", "is injective." ], "refs": [], "ref_ids": [] }, { "id": 7612, "type": "definition", "label": "stacks-morphisms-definition-universal-homeomorphism", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-universal-homeomorphism", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $f$ is a {\\it universal homeomorphism} if for every morphism", "of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$", "the map of topological spaces", "$$", "|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|", "$$", "is a homeomorphism." ], "refs": [], "ref_ids": [] }, { "id": 7613, "type": "definition", "label": "stacks-morphisms-definition-P", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-P", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces", "which is smooth local on the source-and-target.", "We say a morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks", "{\\it has property $\\mathcal{P}$} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target}", "hold." ], "refs": [ "stacks-morphisms-lemma-local-source-target" ], "ref_ids": [ 7458 ] }, { "id": 7614, "type": "definition", "label": "stacks-morphisms-definition-locally-finite-type", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-locally-finite-type", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "\\begin{enumerate}", "\\item We say $f$", "{\\it locally of finite type} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target}", "hold with", "$\\mathcal{P} = \\text{locally of finite type}$.", "\\item We say $f$ is", "{\\it of finite type} if it is locally of finite type and quasi-compact.", "\\end{enumerate}" ], "refs": [ "stacks-morphisms-lemma-local-source-target" ], "ref_ids": [ 7458 ] }, { "id": 7615, "type": "definition", "label": "stacks-morphisms-definition-finite-type-point", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-finite-type-point", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. We say a point $x \\in |\\mathcal{X}|$", "is a {\\it finite type point}\\footnote{This is a", "slight abuse of language as it would perhaps be more correct to say", "``locally finite type point''.} if the equivalent conditions of", "Lemma \\ref{lemma-point-finite-type}", "are satisfied. We denote $\\mathcal{X}_{\\text{ft-pts}}$", "the set of finite type points of $\\mathcal{X}$." ], "refs": [ "stacks-morphisms-lemma-point-finite-type" ], "ref_ids": [ 7466 ] }, { "id": 7616, "type": "definition", "label": "stacks-morphisms-definition-locally-quasi-finite", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-locally-quasi-finite", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $f$ is {\\it locally quasi-finite} if $f$ is quasi-DM, locally of", "finite type, and for every morphism $\\Spec(k) \\to \\mathcal{Y}$", "where $k$ is a field the space $|\\mathcal{X}_k|$ is discrete." ], "refs": [], "ref_ids": [] }, { "id": 7617, "type": "definition", "label": "stacks-morphisms-definition-quasi-finite", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-quasi-finite", "contents": [ "\\begin{reference}", "\\cite{rydh_approx}", "\\end{reference}", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $f$ is {\\it quasi-finite} if $f$ is locally quasi-finite", "(Definition \\ref{definition-locally-quasi-finite})", "and quasi-compact (Definition \\ref{definition-quasi-compact})." ], "refs": [ "stacks-morphisms-definition-locally-quasi-finite", "stacks-morphisms-definition-quasi-compact" ], "ref_ids": [ 7616, 7604 ] }, { "id": 7618, "type": "definition", "label": "stacks-morphisms-definition-flat", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-flat", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $f$ is {\\it flat} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target}", "hold with $\\mathcal{P} = \\text{flat}$." ], "refs": [ "stacks-morphisms-lemma-local-source-target" ], "ref_ids": [ 7458 ] }, { "id": 7619, "type": "definition", "label": "stacks-morphisms-definition-flat-at-point", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-flat-at-point", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $x \\in |\\mathcal{X}|$. We say $f$ is {\\it flat at $x$} if the", "equivalent conditions of Lemma \\ref{lemma-flat-at-point} hold." ], "refs": [ "stacks-morphisms-lemma-flat-at-point" ], "ref_ids": [ 7499 ] }, { "id": 7620, "type": "definition", "label": "stacks-morphisms-definition-locally-finite-presentation", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-locally-finite-presentation", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "\\begin{enumerate}", "\\item We say $f$", "{\\it locally of finite presentation} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target}", "hold with", "$\\mathcal{P} = \\text{locally of finite presentation}$.", "\\item We say $f$ is", "{\\it of finite presentation} if it is locally of finite presentation,", "quasi-compact, and quasi-separated.", "\\end{enumerate}" ], "refs": [ "stacks-morphisms-lemma-local-source-target" ], "ref_ids": [ 7458 ] }, { "id": 7621, "type": "definition", "label": "stacks-morphisms-definition-gerbe", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-gerbe", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $\\mathcal{X}$ is a {\\it gerbe over} $\\mathcal{Y}$ if", "$\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$ as stacks", "in groupoids over $(\\Sch/S)_{fppf}$, see", "Stacks, Definition \\ref{stacks-definition-gerbe-over-stack-in-groupoids}.", "We say an algebraic stack $\\mathcal{X}$ is a {\\it gerbe} if there exists", "a morphism $\\mathcal{X} \\to X$ where $X$ is an algebraic space which", "turns $\\mathcal{X}$ into a gerbe over $X$." ], "refs": [ "stacks-definition-gerbe-over-stack-in-groupoids" ], "ref_ids": [ 9004 ] }, { "id": 7622, "type": "definition", "label": "stacks-morphisms-definition-smooth", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-smooth", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $f$ is {\\it smooth} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target}", "hold with $\\mathcal{P} = \\text{smooth}$." ], "refs": [ "stacks-morphisms-lemma-local-source-target" ], "ref_ids": [ 7458 ] }, { "id": 7623, "type": "definition", "label": "stacks-morphisms-definition-etale-smooth-P", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-etale-smooth-P", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces", "which is \\'etale-smooth local on the source-and-target.", "We say a DM morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks", "{\\it has property $\\mathcal{P}$} if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target}", "hold." ], "refs": [ "stacks-morphisms-lemma-local-source-target" ], "ref_ids": [ 7458 ] }, { "id": 7624, "type": "definition", "label": "stacks-morphisms-definition-etale", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-etale", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $f$ is {\\it \\'etale} if $f$ is DM and the equivalent conditions of", "Lemma \\ref{lemma-etale-smooth-local-source-target}", "hold with $\\mathcal{P} = \\etale$." ], "refs": [ "stacks-morphisms-lemma-etale-smooth-local-source-target" ], "ref_ids": [ 7546 ] }, { "id": 7625, "type": "definition", "label": "stacks-morphisms-definition-unramified", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-unramified", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $f$ is {\\it unramified} if $f$ is DM and the equivalent conditions of", "Lemma \\ref{lemma-etale-smooth-local-source-target}", "hold with $\\mathcal{P} =$``unramified''." ], "refs": [ "stacks-morphisms-lemma-etale-smooth-local-source-target" ], "ref_ids": [ 7546 ] }, { "id": 7626, "type": "definition", "label": "stacks-morphisms-definition-proper", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-proper", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$", "be a morphism of algebraic stacks.", "We say $f$ is {\\it proper} if $f$ is separated, finite type, and", "universally closed." ], "refs": [], "ref_ids": [] }, { "id": 7627, "type": "definition", "label": "stacks-morphisms-definition-scheme-theoretic-image", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-scheme-theoretic-image", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "The {\\it scheme theoretic image} of $f$ is the smallest closed substack", "$\\mathcal{Z} \\subset \\mathcal{Y}$ through which $f$", "factors\\footnote{We will see in", "Lemma \\ref{lemma-scheme-theoretic-image-existence}", "that the scheme theoretic image always exists.}." ], "refs": [ "stacks-morphisms-lemma-scheme-theoretic-image-existence" ], "ref_ids": [ 7566 ] }, { "id": 7628, "type": "definition", "label": "stacks-morphisms-definition-fill-in-diagram", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-fill-in-diagram", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Consider a $2$-commutative solid diagram", "\\begin{equation}", "\\label{equation-diagram}", "\\vcenter{", "\\xymatrix{", "\\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\", "\\Spec(A) \\ar[r]^-y \\ar@{..>}[ru] & \\mathcal{Y}", "}", "}", "\\end{equation}", "where $A$ is a valuation ring with field of fractions $K$. Let", "$$", "\\gamma : y \\circ j \\longrightarrow f \\circ x", "$$", "be a $2$-morphism witnessing the $2$-commutativity of the diagram.", "(Notation as in Categories, Sections \\ref{categories-section-formal-cat-cat}", "and \\ref{categories-section-2-categories}.)", "Given (\\ref{equation-diagram}) and $\\gamma$", "a {\\it dotted arrow} is a triple $(a, \\alpha, \\beta)$ consisting of a", "morphism $a : \\Spec(A) \\to \\mathcal{X}$ and $2$-arrows", "$\\alpha : a \\circ j \\to x$, $\\beta : y \\to f \\circ a$", "such that", "$\\gamma = (\\text{id}_f \\star \\alpha) \\circ (\\beta \\star \\text{id}_j)$,", "in other words such that", "$$", "\\xymatrix{", "& f \\circ a \\circ j \\ar[rd]^{\\text{id}_f \\star \\alpha} \\\\", "y \\circ j \\ar[ru]^{\\beta \\star \\text{id}_j} \\ar[rr]^\\gamma & &", "f \\circ x", "}", "$$", "is commutative. A {\\it morphism of dotted arrows}", "$(a, \\alpha, \\beta) \\to (a', \\alpha', \\beta')$ is a", "$2$-arrow $\\theta : a \\to a'$ such that", "$\\alpha = \\alpha' \\circ (\\theta \\star \\text{id}_j)$ and", "$\\beta' = (\\text{id}_f \\star \\theta) \\circ \\beta$." ], "refs": [], "ref_ids": [] }, { "id": 7629, "type": "definition", "label": "stacks-morphisms-definition-uniqueness", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-uniqueness", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $f$ satisfies the {\\it uniqueness part of the valuative criterion}", "if for every diagram (\\ref{equation-diagram}) and $\\gamma$", "as in Definition \\ref{definition-fill-in-diagram}", "the category of dotted arrows is either empty or", "a setoid with exactly one isomorphism class." ], "refs": [ "stacks-morphisms-definition-fill-in-diagram" ], "ref_ids": [ 7628 ] }, { "id": 7630, "type": "definition", "label": "stacks-morphisms-definition-existence", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-existence", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $f$ satisfies the {\\it existence part of the valuative criterion}", "if for every diagram (\\ref{equation-diagram}) and $\\gamma$", "as in Definition \\ref{definition-fill-in-diagram}", "there exists an extension $K'/K$ of fields, a valuation ring $A' \\subset K'$", "dominating $A$ such that the category of dotted arrows for the", "outer rectangle of the diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar@/^2em/[rr]_{x'} \\ar[d]_{j'} &", "\\Spec(K) \\ar[d]_j \\ar[r]_-x &", "\\mathcal{X} \\ar[d]^f \\\\", "\\Spec(A') \\ar[r] \\ar@/_2em/[rr]^{y'} &", "\\Spec(A) \\ar[r]^-y &", "\\mathcal{Y}", "}", "$$", "with induced $2$-arrow $\\gamma' : y' \\circ j' \\to f \\circ x'$ is nonempty." ], "refs": [ "stacks-morphisms-definition-fill-in-diagram" ], "ref_ids": [ 7628 ] }, { "id": 7631, "type": "definition", "label": "stacks-morphisms-definition-lci", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-definition-lci", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $f$ is a {\\it local complete intersection morphism} or {\\it Koszul}", "if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target}", "hold with $\\mathcal{P} = \\text{local complete intersection}$." ], "refs": [ "stacks-morphisms-lemma-local-source-target" ], "ref_ids": [ 7458 ] }, { "id": 7734, "type": "definition", "label": "schemes-definition-locally-ringed-space", "categories": [ "schemes" ], "title": "schemes-definition-locally-ringed-space", "contents": [ "Locally ringed spaces.", "\\begin{enumerate}", "\\item A {\\it locally ringed space $(X, \\mathcal{O}_X)$}", "is a pair consisting of a", "topological space $X$ and a sheaf of rings $\\mathcal{O}_X$ all of whose stalks", "are local rings.", "\\item Given a locally ringed space $(X, \\mathcal{O}_X)$ we say that", "$\\mathcal{O}_{X, x}$ is the {\\it local ring of $X$ at $x$}.", "We denote $\\mathfrak{m}_{X, x}$ or simply $\\mathfrak{m}_x$", "the maximal ideal of $\\mathcal{O}_{X, x}$. Moreover, the", "{\\it residue field of $X$ at $x$} is the residue field", "$\\kappa(x) = \\mathcal{O}_{X, x}/\\mathfrak{m}_x$.", "\\item A {\\it morphism of locally ringed spaces}", "$(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "is a morphism of ringed spaces such that for all $x \\in X$", "the induced ring map $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is a", "local ring map.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7735, "type": "definition", "label": "schemes-definition-immersion-locally-ringed-spaces", "categories": [ "schemes" ], "title": "schemes-definition-immersion-locally-ringed-spaces", "contents": [ "Let $f : X \\to Y$ be a morphism of locally ringed spaces.", "We say that $f$ is an {\\it open immersion} if", "$f$ is a homeomorphism of $X$ onto an open subset", "of $Y$, and the map $f^{-1}\\mathcal{O}_Y \\to \\mathcal{O}_X$", "is an isomorphism." ], "refs": [], "ref_ids": [] }, { "id": 7736, "type": "definition", "label": "schemes-definition-open-subspace", "categories": [ "schemes" ], "title": "schemes-definition-open-subspace", "contents": [ "Let $X$ be a locally ringed space.", "Let $U \\subset X$ be an open subset.", "The locally ringed space $(U, \\mathcal{O}_U)$", "of Example \\ref{example-open-subspace} above", "is the {\\it open subspace of $X$ associated to $U$}." ], "refs": [], "ref_ids": [] }, { "id": 7737, "type": "definition", "label": "schemes-definition-closed-immersion-locally-ringed-spaces", "categories": [ "schemes" ], "title": "schemes-definition-closed-immersion-locally-ringed-spaces", "contents": [ "Let $i : Z \\to X$ be a morphism of locally ringed spaces.", "We say that $i$ is a {\\it closed immersion} if:", "\\begin{enumerate}", "\\item The map $i$ is a homeomorphism of $Z$ onto a closed subset of $X$.", "\\item The map $\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective;", "let $\\mathcal{I}$ denote the kernel.", "\\item The $\\mathcal{O}_X$-module $\\mathcal{I}$", "is locally generated by sections.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7738, "type": "definition", "label": "schemes-definition-closed-subspace", "categories": [ "schemes" ], "title": "schemes-definition-closed-subspace", "contents": [ "Let $X$ be a locally ringed space.", "Let $\\mathcal{I}$ be a sheaf of ideals on $X$", "which is locally generated by sections.", "The locally ringed space $(Z, \\mathcal{O}_Z)$", "of Example \\ref{example-closed-subspace} above", "is the {\\it closed subspace of $X$ associated to", "the sheaf of ideals $\\mathcal{I}$}." ], "refs": [], "ref_ids": [] }, { "id": 7739, "type": "definition", "label": "schemes-definition-standard-covering", "categories": [ "schemes" ], "title": "schemes-definition-standard-covering", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item A {\\it standard open covering} of $\\Spec(R)$", "is a covering $\\Spec(R) = \\bigcup_{i = 1}^n D(f_i)$,", "where $f_1, \\ldots, f_n \\in R$.", "\\item Suppose that $D(f) \\subset \\Spec(R)$ is a standard", "open. A {\\it standard open covering} of $D(f)$", "is a covering $D(f) = \\bigcup_{i = 1}^n D(g_i)$,", "where $g_1, \\ldots, g_n \\in R$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7740, "type": "definition", "label": "schemes-definition-structure-sheaf", "categories": [ "schemes" ], "title": "schemes-definition-structure-sheaf", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item The {\\it structure sheaf $\\mathcal{O}_{\\Spec(R)}$ of the", "spectrum of $R$} is the unique sheaf of rings $\\mathcal{O}_{\\Spec(R)}$", "which agrees with $\\widetilde R$ on the basis of standard opens.", "\\item The locally ringed space", "$(\\Spec(R), \\mathcal{O}_{\\Spec(R)})$ is called", "the {\\it spectrum} of $R$ and denoted $\\Spec(R)$.", "\\item The sheaf of $\\mathcal{O}_{\\Spec(R)}$-modules", "extending $\\widetilde M$ to all opens of $\\Spec(R)$", "is called the sheaf of $\\mathcal{O}_{\\Spec(R)}$-modules", "associated to $M$. This sheaf is denoted $\\widetilde M$ as", "well.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7741, "type": "definition", "label": "schemes-definition-affine-scheme", "categories": [ "schemes" ], "title": "schemes-definition-affine-scheme", "contents": [ "An {\\it affine scheme} is a locally ringed space isomorphic", "as a locally ringed space to $\\Spec(R)$ for some ring $R$.", "A {\\it morphism of affine schemes} is a morphism in the category", "of locally ringed spaces." ], "refs": [], "ref_ids": [] }, { "id": 7742, "type": "definition", "label": "schemes-definition-scheme", "categories": [ "schemes" ], "title": "schemes-definition-scheme", "contents": [ "\\begin{history}", "In \\cite{EGA1} what we call a scheme was called a ``pre-sch\\'ema'' and the", "name ``sch\\'ema'' was reserved for what is a separated scheme in the", "Stacks project. In the second edition \\cite{EGA1-second} the terminology", "was changed to the terminology that is now standard. However, one may", "occasionally encounter the terminology ``prescheme'', for example in", "\\cite{Murre-lectures}.", "\\end{history}", "A {\\it scheme} is a locally ringed space with the property that", "every point has an open neighbourhood which is an affine scheme.", "A {\\it morphism of schemes} is a morphism of locally", "ringed spaces. The category of schemes will be denoted", "$\\Sch$." ], "refs": [], "ref_ids": [] }, { "id": 7743, "type": "definition", "label": "schemes-definition-immersion", "categories": [ "schemes" ], "title": "schemes-definition-immersion", "contents": [ "Let $X$ be a scheme.", "\\begin{enumerate}", "\\item A morphism of schemes is called an {\\it open immersion}", "if it is an open immersion of locally ringed spaces (see", "Definition \\ref{definition-immersion-locally-ringed-spaces}).", "\\item An {\\it open subscheme} of $X$ is an open subspace of $X$", "in the sense of Definition \\ref{definition-open-subspace}; an open subscheme", "of $X$ is a scheme by Lemma \\ref{lemma-open-subspace-scheme}.", "\\item A morphism of schemes is called a {\\it closed immersion}", "if it is a closed immersion of locally ringed spaces (see", "Definition \\ref{definition-closed-immersion-locally-ringed-spaces}).", "\\item A {\\it closed subscheme} of $X$ is a closed subspace of $X$", "in the sense of Definition \\ref{definition-closed-subspace}; a closed subscheme", "is a scheme by Lemma \\ref{lemma-closed-subspace-scheme}.", "\\item A morphism of schemes $f : X \\to Y$ is called an {\\it immersion},", "or a {\\it locally closed immersion} if it can be factored as", "$j \\circ i$ where $i$ is a closed immersion and $j$ is an open", "immersion.", "\\end{enumerate}" ], "refs": [ "schemes-definition-immersion-locally-ringed-spaces", "schemes-definition-open-subspace", "schemes-lemma-open-subspace-scheme", "schemes-definition-closed-immersion-locally-ringed-spaces", "schemes-definition-closed-subspace", "schemes-lemma-closed-subspace-scheme" ], "ref_ids": [ 7735, 7736, 7669, 7737, 7738, 7670 ] }, { "id": 7744, "type": "definition", "label": "schemes-definition-reduced", "categories": [ "schemes" ], "title": "schemes-definition-reduced", "contents": [ "Let $X$ be a scheme. We say $X$ is {\\it reduced} if every local ring", "$\\mathcal{O}_{X, x}$ is reduced." ], "refs": [], "ref_ids": [] }, { "id": 7745, "type": "definition", "label": "schemes-definition-reduced-induced-scheme", "categories": [ "schemes" ], "title": "schemes-definition-reduced-induced-scheme", "contents": [ "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subset.", "A {\\it scheme structure on $Z$} is given by a closed subscheme $Z'$ of", "$X$ whose underlying set is equal to $Z$. We often say", "``let $(Z, \\mathcal{O}_Z)$ be a scheme structure on $Z$'' to", "indicate this. The {\\it reduced induced scheme structure}", "on $Z$ is the one constructed in Lemma \\ref{lemma-reduced-closed-subscheme}.", "The {\\it reduction $X_{red}$ of $X$} is the reduced induced scheme", "structure on $X$ itself." ], "refs": [ "schemes-lemma-reduced-closed-subscheme" ], "ref_ids": [ 7681 ] }, { "id": 7746, "type": "definition", "label": "schemes-definition-representable-functor", "categories": [ "schemes" ], "title": "schemes-definition-representable-functor", "contents": [ "(See Categories, Definition \\ref{categories-definition-representable-functor}.)", "Let $F$ be a contravariant functor from the category", "of schemes to the category of sets (as above).", "We say that $F$ is {\\it representable by a scheme}", "or {\\it representable} if there exists a scheme $X$", "such that $h_X \\cong F$." ], "refs": [ "categories-definition-representable-functor" ], "ref_ids": [ 12340 ] }, { "id": 7747, "type": "definition", "label": "schemes-definition-representable-by-open-immersions", "categories": [ "schemes" ], "title": "schemes-definition-representable-by-open-immersions", "contents": [ "Let $F$ be a contravariant functor on the category", "of schemes with values in sets.", "\\begin{enumerate}", "\\item We say that $F$ {\\it satisfies the sheaf property for the", "Zariski topology} if for every scheme $T$ and every open covering", "$T = \\bigcup_{i \\in I} U_i$, and for any collection of elements", "$\\xi_i \\in F(U_i)$ such that $\\xi_i|_{U_i \\cap U_j} =", "\\xi_j|_{U_i \\cap U_j}$ there exists a unique element", "$\\xi \\in F(T)$ such that $\\xi_i = \\xi|_{U_i}$ in $F(U_i)$.", "\\item A {\\it subfunctor $H \\subset F$} is a rule that associates", "to every scheme $T$ a subset $H(T) \\subset F(T)$ such that", "the maps $F(f) : F(T) \\to F(T')$ maps $H(T)$ into", "$H(T')$ for all morphisms of schemes $f : T' \\to T$.", "\\item Let $H \\subset F$ be a subfunctor. We say that", "$H \\subset F$ is {\\it representable by open immersions}", "if for all pairs $(T, \\xi)$, where $T$ is a scheme and $\\xi \\in F(T)$", "there exists an open subscheme $U_\\xi \\subset T$ with the following", "property:", "\\begin{itemize}", "\\item[(*)] A morphism $f : T' \\to T$ factors through $U_\\xi$ if and only", "if $f^*\\xi \\in H(T')$.", "\\end{itemize}", "\\item Let $I$ be a set. For each $i \\in I$ let $H_i \\subset F$", "be a subfunctor. We say that the collection $(H_i)_{i \\in I}$", "{\\it covers $F$} if and only if for every $\\xi \\in F(T)$", "there exists an open covering $T = \\bigcup U_i$ such that", "$\\xi|_{U_i} \\in H_i(U_i)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7748, "type": "definition", "label": "schemes-definition-fibre-product", "categories": [ "schemes" ], "title": "schemes-definition-fibre-product", "contents": [ "Given morphisms of schemes $f : X \\to S$ and $g : Y \\to S$", "the {\\it fibre product} is a scheme $X \\times_S Y$ together", "with projection morphisms $p : X \\times_S Y \\to X$", "and $q : X \\times_S Y \\to Y$ sitting into the following", "commutative diagram", "$$", "\\xymatrix{", "X \\times_S Y \\ar[r]_q \\ar[d]_p & Y \\ar[d]^g \\\\", "X \\ar[r]^f & S", "}", "$$", "which is universal among all diagrams of this sort,", "see Categories, Definition \\ref{categories-definition-fibre-products}." ], "refs": [ "categories-definition-fibre-products" ], "ref_ids": [ 12345 ] }, { "id": 7749, "type": "definition", "label": "schemes-definition-inverse-image-closed-subscheme", "categories": [ "schemes" ], "title": "schemes-definition-inverse-image-closed-subscheme", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. Let $Z \\subset Y$ be a", "closed subscheme of $Y$. The {\\it inverse image $f^{-1}(Z)$ of the", "closed subscheme $Z$} is the closed subscheme $Z \\times_Y X$ of", "$X$. See Lemma \\ref{lemma-fibre-product-immersion} above." ], "refs": [ "schemes-lemma-fibre-product-immersion" ], "ref_ids": [ 7694 ] }, { "id": 7750, "type": "definition", "label": "schemes-definition-base-change", "categories": [ "schemes" ], "title": "schemes-definition-base-change", "contents": [ "Let $S$ be a scheme.", "\\begin{enumerate}", "\\item We say $X$ is a {\\it scheme over $S$} to mean that $X$", "comes equipped with a morphism of schemes $X \\to S$.", "The morphism $X \\to S$ is sometimes called the", "{\\it structure morphism}.", "\\item If $R$ is a ring we say", "$X$ is a {\\it scheme over $R$} instead of", "$X$ is a scheme over $\\Spec(R)$.", "\\item A {\\it morphism $f : X \\to Y$ of schemes over $S$}", "is a morphism of schemes such that the composition", "$X \\to Y \\to S$ of $f$ with the structure morphism of $Y$ is", "equal to the structure morphism of $X$.", "\\item We denote $\\Mor_S(X, Y)$ the set of all morphisms", "from $X$ to $Y$ over $S$.", "\\item Let $X$ be a scheme over $S$. Let $S' \\to S$ be a", "morphism of schemes. The {\\it base change} of $X$", "is the scheme $X_{S'} = S' \\times_S X$ over $S'$.", "\\item Let $f : X \\to Y$ be a morphism of schemes over $S$. Let $S' \\to S$", "be a morphism of schemes. The {\\it base change} of $f$ is", "the induced morphism $f' : X_{S'} \\to Y_{S'}$ (namely the", "morphism $\\text{id}_{S'} \\times_{\\text{id}_S} f$).", "\\item Let $R$ be a ring. Let $X$ be a scheme over $R$.", "Let $R \\to R'$ be a ring map. The {\\it base change} $X_{R'}$", "is the scheme $\\Spec(R') \\times_{\\Spec(R)} X$", "over $R'$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7751, "type": "definition", "label": "schemes-definition-preserved-by-base-change", "categories": [ "schemes" ], "title": "schemes-definition-preserved-by-base-change", "contents": [ "Properties and base change.", "\\begin{enumerate}", "\\item Let $\\mathcal{P}$ be a property of schemes over a base.", "We say that $\\mathcal{P}$ is {\\it preserved under arbitrary base change},", "or simply that $\\mathcal{P}$ is {\\it preserved under base change}", "if whenever $X/S$", "has $\\mathcal{P}$, any base change $X_{S'}/S'$ has $\\mathcal{P}$.", "\\item Let $\\mathcal{P}$ be a property of morphisms of schemes over a base.", "We say that $\\mathcal{P}$ is {\\it preserved under arbitrary base change},", "or simply that {\\it preserved under base change} if whenever", "$f : X \\to Y$ over $S$ has $\\mathcal{P}$, any base change", "$f' : X_{S'} \\to Y_{S'}$ over $S'$ has $\\mathcal{P}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7752, "type": "definition", "label": "schemes-definition-fibre", "categories": [ "schemes" ], "title": "schemes-definition-fibre", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $s \\in S$ be a point.", "The {\\it scheme theoretic fibre $X_s$ of $f$ over $s$},", "or simply the {\\it fibre of $f$ over $s$},", "is the scheme fitting in the following fibre product diagram", "$$", "\\xymatrix{", "X_s = \\Spec(\\kappa(s)) \\times_S X \\ar[r] \\ar[d] &", "X \\ar[d] \\\\", "\\Spec(\\kappa(s)) \\ar[r] &", "S", "}", "$$", "We think of the fibre $X_s$ always as a scheme over $\\kappa(s)$." ], "refs": [], "ref_ids": [] }, { "id": 7753, "type": "definition", "label": "schemes-definition-quasi-compact", "categories": [ "schemes" ], "title": "schemes-definition-quasi-compact", "contents": [ "A morphism of schemes is called {\\it quasi-compact}", "if the underlying map of topological spaces is", "quasi-compact, see", "Topology, Definition \\ref{topology-definition-quasi-compact}." ], "refs": [ "topology-definition-quasi-compact" ], "ref_ids": [ 8360 ] }, { "id": 7754, "type": "definition", "label": "schemes-definition-universally-closed", "categories": [ "schemes" ], "title": "schemes-definition-universally-closed", "contents": [ "A morphism of schemes $f : X \\to S$ is said to be", "{\\it universally closed} if every base change", "$f' : X_{S'} \\to S'$ is closed." ], "refs": [], "ref_ids": [] }, { "id": 7755, "type": "definition", "label": "schemes-definition-valuative-criterion", "categories": [ "schemes" ], "title": "schemes-definition-valuative-criterion", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. We say $f$", "{\\it satisfies the existence part of the valuative criterion}", "if given any commutative solid diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & S", "}", "$$", "where $A$ is a valuation ring with field of fractions $K$, the", "dotted arrow exists. We say $f$ {\\it satisfies the uniqueness", "part of the valuative criterion} if there is at most one", "dotted arrow given any diagram as above (without requiring", "existence of course)." ], "refs": [], "ref_ids": [] }, { "id": 7756, "type": "definition", "label": "schemes-definition-separated", "categories": [ "schemes" ], "title": "schemes-definition-separated", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item We say $f$ is {\\it separated} if the diagonal morphism $\\Delta_{X/S}$", "is a closed immersion.", "\\item We say $f$ is {\\it quasi-separated} if the diagonal morphism", "$\\Delta_{X/S}$ is a quasi-compact morphism.", "\\item We say a scheme $Y$ is {\\it separated} if the morphism", "$Y \\to \\Spec(\\mathbf{Z})$ is separated.", "\\item We say a scheme $Y$ is {\\it quasi-separated} if the morphism", "$Y \\to \\Spec(\\mathbf{Z})$ is quasi-separated.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 7757, "type": "definition", "label": "schemes-definition-monomorphism", "categories": [ "schemes" ], "title": "schemes-definition-monomorphism", "contents": [ "A morphism of schemes is called a {\\it monomorphism} if it is", "a monomorphism in the category of schemes, see", "Categories, Definition \\ref{categories-definition-mono-epi}." ], "refs": [ "categories-definition-mono-epi" ], "ref_ids": [ 12355 ] }, { "id": 7809, "type": "definition", "label": "injectives-definition-small", "categories": [ "injectives" ], "title": "injectives-definition-small", "contents": [ "Let $\\mathcal{C}$ be a category, let $I \\subset \\text{Arrows}(\\mathcal{C})$,", "and let $\\alpha$ be an ordinal. An object $A$ of $\\mathcal{C}$ is said to", "be {\\it $\\alpha$-small with respect to $I$} if whenever $\\{B_\\beta\\}$ is", "a system over $\\alpha$ with transition maps in $I$, then", "the map (\\ref{equation-compare}) is an isomorphism." ], "refs": [], "ref_ids": [] }, { "id": 7810, "type": "definition", "label": "injectives-definition-grothendieck-conditions", "categories": [ "injectives" ], "title": "injectives-definition-grothendieck-conditions", "contents": [ "Let $\\mathcal{A}$ be an abelian category. We name some conditions", "\\begin{enumerate}", "\\item[AB3] $\\mathcal{A}$ has direct sums,", "\\item[AB4] $\\mathcal{A}$ has AB3 and direct sums are exact,", "\\item[AB5] $\\mathcal{A}$ has AB3 and filtered colimits are exact.", "\\end{enumerate}", "Here are the dual notions", "\\begin{enumerate}", "\\item[AB3*] $\\mathcal{A}$ has products,", "\\item[AB4*] $\\mathcal{A}$ has AB3* and products are exact,", "\\item[AB5*] $\\mathcal{A}$ has AB3* and filtered limits are exact.", "\\end{enumerate}", "We say an object $U$ of $\\mathcal{A}$ is a {\\it generator} if", "for every $N \\subset M$, $N \\not = M$ in $\\mathcal{A}$ there exists a morphism", "$U \\to M$ which does not factor through $N$.", "We say $\\mathcal{A}$ is a {\\it Grothendieck abelian category} if", "it has AB5 and a generator." ], "refs": [], "ref_ids": [] }, { "id": 7811, "type": "definition", "label": "injectives-definition-size", "categories": [ "injectives" ], "title": "injectives-definition-size", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category.", "Let $M$ be an object of $\\mathcal{A}$.", "The {\\it size} $|M|$ of $M$ is the cardinality of the set of subobjects", "of $M$." ], "refs": [], "ref_ids": [] }, { "id": 7849, "type": "definition", "label": "brauer-definition-finite", "categories": [ "brauer" ], "title": "brauer-definition-finite", "contents": [ "Let $A$ be a $k$-algebra. We say $A$ is {\\it finite} if $\\dim_k(A) < \\infty$.", "In this case we write $[A : k] = \\dim_k(A)$." ], "refs": [], "ref_ids": [] }, { "id": 7850, "type": "definition", "label": "brauer-definition-skew-field", "categories": [ "brauer" ], "title": "brauer-definition-skew-field", "contents": [ "A {\\it skew field} is a possibly noncommutative ring with an identity", "element $1$, with $1 \\not = 0$, in which every nonzero element", "has a multiplicative inverse." ], "refs": [], "ref_ids": [] }, { "id": 7851, "type": "definition", "label": "brauer-definition-simple", "categories": [ "brauer" ], "title": "brauer-definition-simple", "contents": [ "Let $A$ be a $k$-algebra.", "We say an $A$-module $M$ is {\\it simple} if it is nonzero and", "the only $A$-submodules are $0$ and $M$.", "We say $A$ is {\\it simple} if the only two-sided ideals of $A$ are", "$0$ and $A$." ], "refs": [], "ref_ids": [] }, { "id": 7852, "type": "definition", "label": "brauer-definition-central", "categories": [ "brauer" ], "title": "brauer-definition-central", "contents": [ "A $k$-algebra $A$ is {\\it central} if the center of $A$ is the image of", "$k \\to A$." ], "refs": [], "ref_ids": [] }, { "id": 7853, "type": "definition", "label": "brauer-definition-opposite", "categories": [ "brauer" ], "title": "brauer-definition-opposite", "contents": [ "Given a $k$-algebra $A$ we denote $A^{op}$ the $k$-algebra we get by", "reversing the order of multiplication in $A$. This is called the", "{\\it opposite algebra}." ], "refs": [], "ref_ids": [] }, { "id": 7854, "type": "definition", "label": "brauer-definition-brauer-group", "categories": [ "brauer" ], "title": "brauer-definition-brauer-group", "contents": [ "Let $k$ be a field. The {\\it Brauer group} of $k$ is the abelian group", "of similarity classes of finite central simple $k$-algebras defined", "above. Notation $\\text{Br}(k)$." ], "refs": [], "ref_ids": [] }, { "id": 7855, "type": "definition", "label": "brauer-definition-splitting", "categories": [ "brauer" ], "title": "brauer-definition-splitting", "contents": [ "Let $A$ be a finite central simple $k$-algebra.", "We say a field extension $k \\subset k'$ {\\it splits} $A$, or", "$k'$ is a {\\it splitting field} for $A$ if $A \\otimes_k k'$ is", "a matrix algebra over $k'$." ], "refs": [], "ref_ids": [] }, { "id": 8082, "type": "definition", "label": "divisors-definition-associated", "categories": [ "divisors" ], "title": "divisors-definition-associated", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "\\begin{enumerate}", "\\item We say $x \\in X$ is {\\it associated} to $\\mathcal{F}$", "if the maximal ideal", "$\\mathfrak m_x$ is associated to the $\\mathcal{O}_{X, x}$-module", "$\\mathcal{F}_x$.", "\\item We denote $\\text{Ass}(\\mathcal{F})$ or $\\text{Ass}_X(\\mathcal{F})$", "the set of associated points of $\\mathcal{F}$.", "\\item The {\\it associated points of $X$} are the associated", "points of $\\mathcal{O}_X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8083, "type": "definition", "label": "divisors-definition-embedded", "categories": [ "divisors" ], "title": "divisors-definition-embedded", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "\\begin{enumerate}", "\\item An {\\it embedded associated point} of $\\mathcal{F}$", "is an associated point which is not maximal among the", "associated points of $\\mathcal{F}$, i.e., it is the specialization", "of another associated point of $\\mathcal{F}$.", "\\item A point $x$ of $X$ is called an {\\it embedded point}", "if $x$ is an embedded associated point of $\\mathcal{O}_X$.", "\\item An {\\it embedded component} of $X$ is an irreducible", "closed subset $Z = \\overline{\\{x\\}}$ where $x$ is an embedded", "point of $X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8084, "type": "definition", "label": "divisors-definition-weakly-associated", "categories": [ "divisors" ], "title": "divisors-definition-weakly-associated", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "\\begin{enumerate}", "\\item We say $x \\in X$ is {\\it weakly associated} to $\\mathcal{F}$", "if the maximal ideal $\\mathfrak m_x$ is weakly associated to the", "$\\mathcal{O}_{X, x}$-module $\\mathcal{F}_x$.", "\\item We denote $\\text{WeakAss}(\\mathcal{F})$ the set of weakly associated", "points of $\\mathcal{F}$.", "\\item The {\\it weakly associated points of $X$} are the weakly associated", "points of $\\mathcal{O}_X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8085, "type": "definition", "label": "divisors-definition-relative-assassin", "categories": [ "divisors" ], "title": "divisors-definition-relative-assassin", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "The {\\it relative assassin of $\\mathcal{F}$ in $X$ over $S$}", "is the set", "$$", "\\text{Ass}_{X/S}(\\mathcal{F}) =", "\\bigcup\\nolimits_{s \\in S} \\text{Ass}_{X_s}(\\mathcal{F}_s)", "$$", "where $\\mathcal{F}_s = (X_s \\to X)^*\\mathcal{F}$ is the restriction", "of $\\mathcal{F}$ to the fibre of $f$ at $s$." ], "refs": [], "ref_ids": [] }, { "id": 8086, "type": "definition", "label": "divisors-definition-relative-weak-assassin", "categories": [ "divisors" ], "title": "divisors-definition-relative-weak-assassin", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "The {\\it relative weak assassin of $\\mathcal{F}$ in $X$ over $S$}", "is the set", "$$", "\\text{WeakAss}_{X/S}(\\mathcal{F}) =", "\\bigcup\\nolimits_{s \\in S} \\text{WeakAss}(\\mathcal{F}_s)", "$$", "where $\\mathcal{F}_s = (X_s \\to X)^*\\mathcal{F}$ is the restriction", "of $\\mathcal{F}$ to the fibre of $f$ at $s$." ], "refs": [], "ref_ids": [] }, { "id": 8087, "type": "definition", "label": "divisors-definition-torsion", "categories": [ "divisors" ], "title": "divisors-definition-torsion", "contents": [ "Let $X$ be an integral scheme. Let $\\mathcal{F}$ be a quasi-coherent", "$\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item We say a local section of $\\mathcal{F}$ is {\\it torsion}", "if it satisfies the equivalent conditions of Lemma \\ref{lemma-torsion-sections}.", "\\item We say $\\mathcal{F}$ is {\\it torsion free} if every torsion section", "of $\\mathcal{F}$ is $0$.", "\\end{enumerate}" ], "refs": [ "divisors-lemma-torsion-sections" ], "ref_ids": [ 7903 ] }, { "id": 8088, "type": "definition", "label": "divisors-definition-reflexive", "categories": [ "divisors" ], "title": "divisors-definition-reflexive", "contents": [ "Let $X$ be an integral locally Noetherian scheme. Let $\\mathcal{F}$", "be a coherent $\\mathcal{O}_X$-module. The {\\it reflexive hull}", "of $\\mathcal{F}$ is the $\\mathcal{O}_X$-module", "$$", "\\mathcal{F}^{**} = \\SheafHom_{\\mathcal{O}_X}(", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X), \\mathcal{O}_X)", "$$", "We say $\\mathcal{F}$ is {\\it reflexive} if the natural map", "$j : \\mathcal{F} \\longrightarrow \\mathcal{F}^{**}$", "is an isomorphism." ], "refs": [], "ref_ids": [] }, { "id": 8089, "type": "definition", "label": "divisors-definition-effective-Cartier-divisor", "categories": [ "divisors" ], "title": "divisors-definition-effective-Cartier-divisor", "contents": [ "Let $S$ be a scheme.", "\\begin{enumerate}", "\\item A {\\it locally principal closed subscheme} of $S$ is a closed subscheme", "whose sheaf of ideals is locally generated by a single element.", "\\item An {\\it effective Cartier divisor} on $S$ is a closed subscheme", "$D \\subset S$ whose ideal sheaf $\\mathcal{I}_D \\subset \\mathcal{O}_S$", "is an invertible $\\mathcal{O}_S$-module.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8090, "type": "definition", "label": "divisors-definition-sum-effective-Cartier-divisors", "categories": [ "divisors" ], "title": "divisors-definition-sum-effective-Cartier-divisors", "contents": [ "Let $S$ be a scheme. Given effective Cartier divisors", "$D_1$, $D_2$ on $S$ we set $D = D_1 + D_2$ equal to the", "closed subscheme of $S$ corresponding to the quasi-coherent", "sheaf of ideals", "$\\mathcal{I}_{D_1}\\mathcal{I}_{D_2} \\subset \\mathcal{O}_S$.", "We call this the {\\it sum of the effective Cartier divisors", "$D_1$ and $D_2$}." ], "refs": [], "ref_ids": [] }, { "id": 8091, "type": "definition", "label": "divisors-definition-pullback-effective-Cartier-divisor", "categories": [ "divisors" ], "title": "divisors-definition-pullback-effective-Cartier-divisor", "contents": [ "Let $f : S' \\to S$ be a morphism of schemes. Let $D \\subset S$", "be an effective Cartier divisor. We say the {\\it pullback of", "$D$ by $f$ is defined} if the closed subscheme $f^{-1}(D) \\subset S'$", "is an effective Cartier divisor. In this case we denote it either", "$f^*D$ or $f^{-1}(D)$ and we call it the", "{\\it pullback of the effective Cartier divisor}." ], "refs": [], "ref_ids": [] }, { "id": 8092, "type": "definition", "label": "divisors-definition-invertible-sheaf-effective-Cartier-divisor", "categories": [ "divisors" ], "title": "divisors-definition-invertible-sheaf-effective-Cartier-divisor", "contents": [ "Let $S$ be a scheme. Let $D \\subset S$ be an effective Cartier divisor", "with ideal sheaf $\\mathcal{I}_D$.", "\\begin{enumerate}", "\\item The {\\it invertible sheaf $\\mathcal{O}_S(D)$ associated to $D$}", "is defined by", "$$", "\\mathcal{O}_S(D) =", "\\SheafHom_{\\mathcal{O}_S}(\\mathcal{I}_D, \\mathcal{O}_S) =", "\\mathcal{I}_D^{\\otimes -1}.", "$$", "\\item The {\\it canonical section}, usually denoted $1$ or $1_D$, is the", "global section of $\\mathcal{O}_S(D)$ corresponding to", "the inclusion mapping $\\mathcal{I}_D \\to \\mathcal{O}_S$.", "\\item We write", "$\\mathcal{O}_S(-D) = \\mathcal{O}_S(D)^{\\otimes -1} = \\mathcal{I}_D$.", "\\item Given a second effective Cartier divisor $D' \\subset S$ we define", "$\\mathcal{O}_S(D - D') =", "\\mathcal{O}_S(D) \\otimes_{\\mathcal{O}_S} \\mathcal{O}_S(-D')$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8093, "type": "definition", "label": "divisors-definition-regular-section", "categories": [ "divisors" ], "title": "divisors-definition-regular-section", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a locally ringed space.", "Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "A global section $s \\in \\Gamma(X, \\mathcal{L})$ is called a", "{\\it regular section} if the map $\\mathcal{O}_X \\to \\mathcal{L}$,", "$f \\mapsto fs$ is injective." ], "refs": [], "ref_ids": [] }, { "id": 8094, "type": "definition", "label": "divisors-definition-zero-scheme-s", "categories": [ "divisors" ], "title": "divisors-definition-zero-scheme-s", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible sheaf.", "Let $s \\in \\Gamma(X, \\mathcal{L})$ be a global section.", "The {\\it zero scheme} of $s$ is the closed subscheme $Z(s) \\subset X$", "defined by the quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_X$ which is the image of the", "map $s : \\mathcal{L}^{\\otimes -1} \\to \\mathcal{O}_X$." ], "refs": [], "ref_ids": [] }, { "id": 8095, "type": "definition", "label": "divisors-definition-relative-effective-Cartier-divisor", "categories": [ "divisors" ], "title": "divisors-definition-relative-effective-Cartier-divisor", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "A {\\it relative effective Cartier divisor} on $X/S$ is an", "effective Cartier divisor $D \\subset X$ such that $D \\to S$", "is a flat morphism of schemes." ], "refs": [], "ref_ids": [] }, { "id": 8096, "type": "definition", "label": "divisors-definition-conormal-sheaf", "categories": [ "divisors" ], "title": "divisors-definition-conormal-sheaf", "contents": [ "Let $f : Z \\to X$ be an immersion. The {\\it conormal algebra", "$\\mathcal{C}_{Z/X, *}$ of $Z$ in $X$} or the {\\it conormal algebra of $f$}", "is the quasi-coherent sheaf of graded $\\mathcal{O}_Z$-algebras", "$\\bigoplus_{n \\geq 0} \\mathcal{I}^n/\\mathcal{I}^{n + 1}$ described above." ], "refs": [], "ref_ids": [] }, { "id": 8097, "type": "definition", "label": "divisors-definition-normal-cone", "categories": [ "divisors" ], "title": "divisors-definition-normal-cone", "contents": [ "Let $i : Z \\to X$ be an immersion of schemes.", "The {\\it normal cone $C_ZX$} of $Z$ in $X$ is", "$$", "C_ZX = \\underline{\\Spec}_Z(\\mathcal{C}_{Z/X, *})", "$$", "see", "Constructions,", "Definitions \\ref{constructions-definition-cone} and", "\\ref{constructions-definition-abstract-cone}. The {\\it normal bundle}", "of $Z$ in $X$ is the vector bundle", "$$", "N_ZX = \\underline{\\Spec}_Z(\\text{Sym}(\\mathcal{C}_{Z/X}))", "$$", "see", "Constructions,", "Definitions \\ref{constructions-definition-vector-bundle} and", "\\ref{constructions-definition-abstract-vector-bundle}." ], "refs": [ "constructions-definition-cone", "constructions-definition-abstract-cone", "constructions-definition-vector-bundle", "constructions-definition-abstract-vector-bundle" ], "ref_ids": [ 12659, 12660, 12657, 12658 ] }, { "id": 8098, "type": "definition", "label": "divisors-definition-regular-ideal-sheaf", "categories": [ "divisors" ], "title": "divisors-definition-regular-ideal-sheaf", "contents": [ "Let $X$ be a ringed space. Let $\\mathcal{J} \\subset \\mathcal{O}_X$", "be a sheaf of ideals.", "\\begin{enumerate}", "\\item We say $\\mathcal{J}$ is {\\it regular} if for every", "$x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists an open", "neighbourhood $x \\in U \\subset X$ and a regular sequence", "$f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ such that $\\mathcal{J}|_U$", "is generated by $f_1, \\ldots, f_r$.", "\\item We say $\\mathcal{J}$ is {\\it Koszul-regular} if for every", "$x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists an open", "neighbourhood $x \\in U \\subset X$ and a Koszul-regular sequence", "$f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ such that $\\mathcal{J}|_U$", "is generated by $f_1, \\ldots, f_r$.", "\\item We say $\\mathcal{J}$ is {\\it $H_1$-regular} if for every", "$x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists an open", "neighbourhood $x \\in U \\subset X$ and a $H_1$-regular sequence", "$f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ such that $\\mathcal{J}|_U$", "is generated by $f_1, \\ldots, f_r$.", "\\item We say $\\mathcal{J}$ is {\\it quasi-regular} if for every", "$x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists an open", "neighbourhood $x \\in U \\subset X$ and a quasi-regular sequence", "$f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ such that $\\mathcal{J}|_U$", "is generated by $f_1, \\ldots, f_r$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8099, "type": "definition", "label": "divisors-definition-regular-immersion", "categories": [ "divisors" ], "title": "divisors-definition-regular-immersion", "contents": [ "Let $i : Z \\to X$ be an immersion of schemes. Choose an open subscheme", "$U \\subset X$ such that $i$ identifies $Z$ with a closed", "subscheme of $U$ and denote $\\mathcal{I} \\subset \\mathcal{O}_U$", "the corresponding quasi-coherent sheaf of ideals.", "\\begin{enumerate}", "\\item We say $i$ is a {\\it regular immersion} if", "$\\mathcal{I}$ is regular.", "\\item We say $i$ is a {\\it Koszul-regular immersion} if", "$\\mathcal{I}$ is Koszul-regular.", "\\item We say $i$ is a {\\it $H_1$-regular immersion} if", "$\\mathcal{I}$ is $H_1$-regular.", "\\item We say $i$ is a {\\it quasi-regular immersion} if", "$\\mathcal{I}$ is quasi-regular.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8100, "type": "definition", "label": "divisors-definition-relative-H1-regular-immersion", "categories": [ "divisors" ], "title": "divisors-definition-relative-H1-regular-immersion", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $i : Z \\to X$ be an immersion.", "\\begin{enumerate}", "\\item We say $i$ is a {\\it relative quasi-regular immersion}", "if $Z \\to S$ is flat and $i$ is a quasi-regular immersion.", "\\item We say $i$ is a {\\it relative $H_1$-regular immersion}", "if $Z \\to S$ is flat and $i$ is an $H_1$-regular immersion.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8101, "type": "definition", "label": "divisors-definition-sheaf-meromorphic-functions", "categories": [ "divisors" ], "title": "divisors-definition-sheaf-meromorphic-functions", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a locally ringed space.", "The {\\it sheaf of meromorphic functions on $X$} is", "the sheaf {\\it $\\mathcal{K}_X$} associated to the presheaf", "displayed above. A {\\it meromorphic function} on $X$", "is a global section of $\\mathcal{K}_X$." ], "refs": [], "ref_ids": [] }, { "id": 8102, "type": "definition", "label": "divisors-definition-meromorphic-section", "categories": [ "divisors" ], "title": "divisors-definition-meromorphic-section", "contents": [ "Let $X$ be a locally ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "\\begin{enumerate}", "\\item We denote", "$\\mathcal{K}_X(\\mathcal{F})$", "the sheaf of $\\mathcal{K}_X$-modules which is", "the sheafification of the presheaf", "$U \\mapsto \\mathcal{S}(U)^{-1}\\mathcal{F}(U)$. Equivalently", "$\\mathcal{K}_X(\\mathcal{F}) =", "\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{K}_X$ (see above).", "\\item A {\\it meromorphic section of $\\mathcal{F}$}", "is a global section of $\\mathcal{K}_X(\\mathcal{F})$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8103, "type": "definition", "label": "divisors-definition-pullback-meromorphic-sections", "categories": [ "divisors" ], "title": "divisors-definition-pullback-meromorphic-sections", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism", "of locally ringed spaces. We say that {\\it pullbacks of meromorphic", "functions are defined for $f$} if for every pair of open", "$U \\subset X$, $V \\subset Y$ such that $f(U) \\subset V$, and any", "section $s \\in \\Gamma(V, \\mathcal{S}_Y)$ the pullback", "$f^\\sharp(s) \\in \\Gamma(U, \\mathcal{O}_X)$ is an element", "of $\\Gamma(U, \\mathcal{S}_X)$." ], "refs": [], "ref_ids": [] }, { "id": 8104, "type": "definition", "label": "divisors-definition-regular-meromorphic-section", "categories": [ "divisors" ], "title": "divisors-definition-regular-meromorphic-section", "contents": [ "Let $X$ be a locally ringed space.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "A meromorphic section $s$ of $\\mathcal{L}$ is said to be {\\it regular}", "if the induced map", "$\\mathcal{K}_X \\to \\mathcal{K}_X(\\mathcal{L})$", "is injective. In other words, $s$ is a regular", "section of the invertible $\\mathcal{K}_X$-module", "$\\mathcal{K}_X(\\mathcal{L})$, see", "Definition \\ref{definition-regular-section}." ], "refs": [], "ref_ids": [] }, { "id": 8105, "type": "definition", "label": "divisors-definition-regular-meromorphic-ideal-denominators", "categories": [ "divisors" ], "title": "divisors-definition-regular-meromorphic-ideal-denominators", "contents": [ "Let $X$ be a scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s$ be a regular meromorphic section of $\\mathcal{L}$.", "The sheaf of ideals $\\mathcal{I}$ constructed in", "Lemma \\ref{lemma-regular-meromorphic-ideal-denominators}", "is called the {\\it ideal sheaf of denominators of $s$}." ], "refs": [ "divisors-lemma-regular-meromorphic-ideal-denominators" ], "ref_ids": [ 8012 ] }, { "id": 8106, "type": "definition", "label": "divisors-definition-Weil-divisor", "categories": [ "divisors" ], "title": "divisors-definition-Weil-divisor", "contents": [ "Let $X$ be a locally Noetherian integral scheme.", "\\begin{enumerate}", "\\item A {\\it prime divisor} is an integral closed subscheme $Z \\subset X$", "of codimension $1$.", "\\item A {\\it Weil divisor} is a formal sum $D = \\sum n_Z Z$ where", "the sum is over prime divisors of $X$ and the collection", "$\\{Z \\mid n_Z \\not = 0\\}$ is locally finite", "(Topology, Definition \\ref{topology-definition-locally-finite}).", "\\end{enumerate}", "The group of all Weil divisors on $X$ is denoted $\\text{Div}(X)$." ], "refs": [ "topology-definition-locally-finite" ], "ref_ids": [ 8376 ] }, { "id": 8107, "type": "definition", "label": "divisors-definition-order-vanishing", "categories": [ "divisors" ], "title": "divisors-definition-order-vanishing", "contents": [ "Let $X$ be a locally Noetherian integral scheme. Let $f \\in R(X)^*$.", "For every prime divisor $Z \\subset X$ we define the", "{\\it order of vanishing of $f$ along $Z$} as the integer", "$$", "\\text{ord}_Z(f) = \\text{ord}_{\\mathcal{O}_{X, \\xi}}(f)", "$$", "where the right hand side is the notion of", "Algebra, Definition \\ref{algebra-definition-ord}", "and $\\xi$ is the generic point of $Z$." ], "refs": [ "algebra-definition-ord" ], "ref_ids": [ 1519 ] }, { "id": 8108, "type": "definition", "label": "divisors-definition-principal-divisor", "categories": [ "divisors" ], "title": "divisors-definition-principal-divisor", "contents": [ "Let $X$ be a locally Noetherian integral scheme. Let $f \\in R(X)^*$.", "The {\\it principal Weil divisor associated to $f$} is the Weil divisor", "$$", "\\text{div}(f) = \\text{div}_X(f) = \\sum \\text{ord}_Z(f) [Z]", "$$", "where the sum is over prime divisors and $\\text{ord}_Z(f)$ is as in", "Definition \\ref{definition-order-vanishing}. This makes sense", "by Lemma \\ref{lemma-divisor-locally-finite}." ], "refs": [ "divisors-definition-order-vanishing", "divisors-lemma-divisor-locally-finite" ], "ref_ids": [ 8107, 8023 ] }, { "id": 8109, "type": "definition", "label": "divisors-definition-class-group", "categories": [ "divisors" ], "title": "divisors-definition-class-group", "contents": [ "Let $X$ be a locally Noetherian integral scheme. The", "{\\it Weil divisor class group} of $X$ is the quotient of", "the group of Weil divisors by the subgroup of principal Weil divisors.", "Notation: $\\text{Cl}(X)$." ], "refs": [], "ref_ids": [] }, { "id": 8110, "type": "definition", "label": "divisors-definition-order-vanishing-meromorphic", "categories": [ "divisors" ], "title": "divisors-definition-order-vanishing-meromorphic", "contents": [ "Let $X$ be a locally Noetherian integral scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{K}_X(\\mathcal{L}))$", "be a regular meromorphic section of $\\mathcal{L}$.", "For every prime divisor $Z \\subset X$ we define the", "{\\it order of vanishing of $s$ along $Z$} as the integer", "$$", "\\text{ord}_{Z, \\mathcal{L}}(s)", "= \\text{ord}_{\\mathcal{O}_{X, \\xi}}(s/s_\\xi)", "$$", "where the right hand side is the notion of", "Algebra, Definition \\ref{algebra-definition-ord},", "$\\xi \\in Z$ is the generic point,", "and $s_\\xi \\in \\mathcal{L}_\\xi$ is a generator." ], "refs": [ "algebra-definition-ord" ], "ref_ids": [ 1519 ] }, { "id": 8111, "type": "definition", "label": "divisors-definition-divisor-invertible-sheaf", "categories": [ "divisors" ], "title": "divisors-definition-divisor-invertible-sheaf", "contents": [ "Let $X$ be a locally Noetherian integral scheme.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item For any nonzero meromorphic section $s$ of $\\mathcal{L}$", "we define the {\\it Weil divisor associated to $s$} as", "$$", "\\text{div}_\\mathcal{L}(s) =", "\\sum \\text{ord}_{Z, \\mathcal{L}}(s) [Z] \\in \\text{Div}(X)", "$$", "where the sum is over prime divisors.", "\\item We define {\\it Weil divisor class associated to $\\mathcal{L}$}", "as the image of $\\text{div}_\\mathcal{L}(s)$ in $\\text{Cl}(X)$", "where $s$ is any nonzero meromorphic section of $\\mathcal{L}$ over", "$X$. This is well defined by", "Lemma \\ref{lemma-divisor-meromorphic-well-defined}.", "\\end{enumerate}" ], "refs": [ "divisors-lemma-divisor-meromorphic-well-defined" ], "ref_ids": [ 8026 ] }, { "id": 8112, "type": "definition", "label": "divisors-definition-blow-up", "categories": [ "divisors" ], "title": "divisors-definition-blow-up", "contents": [ "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a", "quasi-coherent sheaf of ideals, and let $Z \\subset X$ be the closed subscheme", "corresponding to $\\mathcal{I}$, see", "Schemes, Definition \\ref{schemes-definition-immersion}.", "The {\\it blowing up of $X$ along $Z$}, or the", "{\\it blowing up of $X$ in the ideal sheaf $\\mathcal{I}$} is", "the morphism", "$$", "b :", "\\underline{\\text{Proj}}_X", "\\left(\\bigoplus\\nolimits_{n \\geq 0} \\mathcal{I}^n\\right)", "\\longrightarrow", "X", "$$", "The {\\it exceptional divisor} of the blowup is the inverse image", "$b^{-1}(Z)$. Sometimes $Z$ is called the {\\it center} of the blowup." ], "refs": [ "schemes-definition-immersion" ], "ref_ids": [ 7743 ] }, { "id": 8113, "type": "definition", "label": "divisors-definition-strict-transform", "categories": [ "divisors" ], "title": "divisors-definition-strict-transform", "contents": [ "With $Z \\subset S$ and $f : X \\to S$ as above.", "\\begin{enumerate}", "\\item Given a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "the {\\it strict transform} of $\\mathcal{F}$ with respect to the blowup", "of $S$ in $Z$ is the quotient $\\mathcal{F}'$ of $\\text{pr}_X^*\\mathcal{F}$", "by the submodule of sections supported on $\\text{pr}_{S'}^{-1}E$.", "\\item The {\\it strict transform} of $X$ is the closed subscheme", "$X' \\subset X \\times_S S'$ cut out by the quasi-coherent ideal of", "sections of $\\mathcal{O}_{X \\times_S S'}$ supported on $\\text{pr}_{S'}^{-1}E$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8114, "type": "definition", "label": "divisors-definition-admissible-blowup", "categories": [ "divisors" ], "title": "divisors-definition-admissible-blowup", "contents": [ "Let $X$ be a scheme. Let $U \\subset X$ be an open subscheme. A morphism", "$X' \\to X$ is called a {\\it $U$-admissible blowup} if there exists a", "closed immersion $Z \\to X$ of finite presentation with $Z$ disjoint from", "$U$ such that $X'$ is isomorphic to the blowup of $X$ in $Z$." ], "refs": [], "ref_ids": [] }, { "id": 8173, "type": "definition", "label": "spaces-definition-relative-representable-property", "categories": [ "spaces" ], "title": "spaces-definition-relative-representable-property", "contents": [ "With $S$, and $a : F \\to G$ representable as above.", "Let $\\mathcal{P}$ be a property of morphisms of schemes which", "\\begin{enumerate}", "\\item is preserved under any base change,", "see Schemes, Definition \\ref{schemes-definition-preserved-by-base-change},", "and", "\\item is fppf local on the base, see", "Descent, Definition \\ref{descent-definition-property-morphisms-local}.", "\\end{enumerate}", "In this case we say that $a$ has {\\it property $\\mathcal{P}$} if for every", "$U \\in \\Ob((\\Sch/S)_{fppf})$ and", "any $\\xi \\in G(U)$ the resulting morphism of schemes", "$V_\\xi \\to U$ has property $\\mathcal{P}$." ], "refs": [ "schemes-definition-preserved-by-base-change", "descent-definition-property-morphisms-local" ], "ref_ids": [ 7751, 14772 ] }, { "id": 8174, "type": "definition", "label": "spaces-definition-algebraic-space", "categories": [ "spaces" ], "title": "spaces-definition-algebraic-space", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "An {\\it algebraic space over $S$} is a presheaf", "$$", "F : (\\Sch/S)^{opp}_{fppf} \\longrightarrow \\textit{Sets}", "$$", "with the following properties", "\\begin{enumerate}", "\\item The presheaf $F$ is a sheaf.", "\\item The diagonal morphism $F \\to F \\times F$ is representable.", "\\item There exists a scheme $U \\in \\Ob((\\Sch/S)_{fppf})$", "and a map $h_U \\to F$ which is surjective, and \\'etale.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8175, "type": "definition", "label": "spaces-definition-morphism-algebraic-spaces", "categories": [ "spaces" ], "title": "spaces-definition-morphism-algebraic-spaces", "contents": [ "Let $F$, $F'$ be algebraic spaces over $S$.", "A {\\it morphism $f : F \\to F'$ of algebraic spaces over $S$}", "is a transformation of functors from $F$ to $F'$." ], "refs": [], "ref_ids": [] }, { "id": 8176, "type": "definition", "label": "spaces-definition-etale-equivalence-relation", "categories": [ "spaces" ], "title": "spaces-definition-etale-equivalence-relation", "contents": [ "Let $S$ be a scheme. Let $U$ be a scheme over $S$.", "An {\\it \\'etale equivalence relation} on $U$ over $S$", "is an equivalence relation $j : R \\to U \\times_S U$", "such that $s, t : R \\to U$ are \\'etale morphisms of schemes." ], "refs": [], "ref_ids": [] }, { "id": 8177, "type": "definition", "label": "spaces-definition-presentation", "categories": [ "spaces" ], "title": "spaces-definition-presentation", "contents": [ "Let $F$ be an algebraic space over $S$.", "A {\\it presentation} of $F$ is given by a scheme", "$U$ over $S$ and an \\'etale equivalence relation $R$ on $U$ over $S$, and", "a surjective \\'etale morphism $U \\to F$ such that $R = U \\times_F U$." ], "refs": [], "ref_ids": [] }, { "id": 8178, "type": "definition", "label": "spaces-definition-immersion", "categories": [ "spaces" ], "title": "spaces-definition-immersion", "contents": [ "Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme.", "Let $F$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item A morphism of algebraic spaces over $S$", "is called an {\\it open immersion} if it is representable, and an open immersion", "in the sense of Definition \\ref{definition-relative-representable-property}.", "\\item An {\\it open subspace} of $F$ is a subfunctor $F' \\subset F$", "such that $F'$ is an algebraic space and $F' \\to F$ is an", "open immersion.", "\\item A morphism of algebraic spaces over $S$", "is called a {\\it closed immersion} if it is representable, and a closed", "immersion in the sense of", "Definition \\ref{definition-relative-representable-property}.", "\\item A {\\it closed subspace} of $F$ is a subfunctor $F' \\subset F$", "such that $F'$ is an algebraic space and $F' \\to F$ is a", "closed immersion.", "\\item A morphism of algebraic spaces over $S$", "is called an {\\it immersion} if it is representable, and an immersion", "in the sense of Definition \\ref{definition-relative-representable-property}.", "\\item A {\\it locally closed subspace} of $F$ is a subfunctor $F' \\subset F$", "such that $F'$ is an algebraic space and $F' \\to F$ is an", "immersion.", "\\end{enumerate}" ], "refs": [ "spaces-definition-relative-representable-property", "spaces-definition-relative-representable-property", "spaces-definition-relative-representable-property" ], "ref_ids": [ 8173, 8173, 8173 ] }, { "id": 8179, "type": "definition", "label": "spaces-definition-Zariski-open-covering", "categories": [ "spaces" ], "title": "spaces-definition-Zariski-open-covering", "contents": [ "Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme.", "Let $F$ be an algebraic space over $S$.", "A {\\it Zariski covering} $\\{F_i \\subset F\\}_{i \\in I}$ of $F$", "is given by a set $I$ and a collection of open subspaces", "$F_i \\subset F$ such that $\\coprod F_i \\to F$ is a surjective", "map of sheaves." ], "refs": [], "ref_ids": [] }, { "id": 8180, "type": "definition", "label": "spaces-definition-small-Zariski-site", "categories": [ "spaces" ], "title": "spaces-definition-small-Zariski-site", "contents": [ "Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme. Let $F$ be an algebraic space over", "$S$. A {\\it small Zariski site $F_{Zar}$} of an algebraic space $F$ is one", "of the sites described above." ], "refs": [], "ref_ids": [] }, { "id": 8181, "type": "definition", "label": "spaces-definition-separated", "categories": [ "spaces" ], "title": "spaces-definition-separated", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $F$ be an algebraic space over $S$.", "Let $\\Delta : F \\to F \\times F$ be the diagonal morphism.", "\\begin{enumerate}", "\\item We say $F$ is {\\it separated over $S$} if $\\Delta$ is a closed immersion.", "\\item We say $F$ is {\\it locally separated over $S$}\\footnote{In the", "literature this often refers to quasi-separated and", "locally separated algebraic spaces.} if $\\Delta$ is an", "immersion.", "\\item We say $F$ is {\\it quasi-separated over $S$} if $\\Delta$ is quasi-compact.", "\\item We say $F$ is {\\it Zariski locally quasi-separated over $S$}\\footnote{This", "definition was suggested by B.\\ Conrad.} if there", "exists a Zariski covering $F = \\bigcup_{i \\in I} F_i$ such that", "each $F_i$ is quasi-separated.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8182, "type": "definition", "label": "spaces-definition-quotient", "categories": [ "spaces" ], "title": "spaces-definition-quotient", "contents": [ "Notation $U \\to S$, $G$, $R$ as in Lemma \\ref{lemma-quotient}.", "If the action of $G$ on $U$ satisfies $(*)$ we say $G$ {\\it acts freely}", "on the scheme $U$. In this case the algebraic space $U/R$ is denoted", "$U/G$ and is called the {\\it quotient of $U$ by $G$}." ], "refs": [ "spaces-lemma-quotient" ], "ref_ids": [ 8164 ] }, { "id": 8183, "type": "definition", "label": "spaces-definition-base-change", "categories": [ "spaces" ], "title": "spaces-definition-base-change", "contents": [ "Let $\\Sch_{fppf}$ be a big fppf site.", "Let $S \\to S'$ be a morphism of this site.", "\\begin{enumerate}", "\\item If $F'$ is an algebraic space over $S'$, then the", "{\\it base change of $F'$ to $S$} is the", "algebraic space $j^{-1}F'$ described in", "Lemma \\ref{lemma-change-base-scheme}. We denote it $F'_S$.", "\\item If $F$ is an algebraic space over $S$, then $F$", "{\\it viewed as an algebraic space over $S'$}", "is the algebraic space $j_!F$ over $S'$ described in", "Lemma \\ref{lemma-change-base-scheme}. We often simply denote this", "$F$; if not then we will write $j_!F$.", "\\end{enumerate}" ], "refs": [ "spaces-lemma-change-base-scheme", "spaces-lemma-change-base-scheme" ], "ref_ids": [ 8169, 8169 ] }, { "id": 8346, "type": "definition", "label": "topology-definition-separated", "categories": [ "topology" ], "title": "topology-definition-separated", "contents": [ "A continuous map $f : X \\to Y$ of topological spaces is called", "{\\it separated} if and only if the diagonal $\\Delta : X \\to X \\times_Y X$", "is a closed map." ], "refs": [], "ref_ids": [] }, { "id": 8347, "type": "definition", "label": "topology-definition-base", "categories": [ "topology" ], "title": "topology-definition-base", "contents": [ "Let $X$ be a topological space. A collection of subsets $\\mathcal{B}$ of $X$", "is called a {\\it base for the topology on $X$} or a {\\it basis for the", "topology on $X$} if the following conditions hold:", "\\begin{enumerate}", "\\item Every element $B \\in \\mathcal{B}$ is open in $X$.", "\\item For every open $U \\subset X$ and every $x \\in U$,", "there exists an element $B \\in \\mathcal{B}$ such that", "$x \\in B \\subset U$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8348, "type": "definition", "label": "topology-definition-subbase", "categories": [ "topology" ], "title": "topology-definition-subbase", "contents": [ "Let $X$ be a topological space. A collection of subsets $\\mathcal{B}$ of $X$", "is called a {\\it subbase for the topology on $X$} or a {\\it subbasis for the", "topology on $X$} if the finite intersections of", "elements of $\\mathcal{B}$ form a basis for the topology on $X$." ], "refs": [], "ref_ids": [] }, { "id": 8349, "type": "definition", "label": "topology-definition-submersive", "categories": [ "topology" ], "title": "topology-definition-submersive", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "\\begin{enumerate}", "\\item We say $f$ is a {\\it strict map of topological spaces}", "if the induced topology and the quotient topology on $f(X)$ agree", "(see discussion above).", "\\item We say $f$ is {\\it submersive}\\footnote{This is very different from", "the notion of a submersion between differential manifolds! It is probably", "a good idea to use ``strict and surjective'' in stead of ``submersive''.}", "if $f$ is surjective and strict.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8350, "type": "definition", "label": "topology-definition-connected-components", "categories": [ "topology" ], "title": "topology-definition-connected-components", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item We say $X$ is {\\it connected} if $X$ is not empty and whenever", "$X = T_1 \\amalg T_2$ with $T_i \\subset X$ open and closed, then either", "$T_1 = \\emptyset$ or $T_2 = \\emptyset$.", "\\item We say $T \\subset X$ is a {\\it connected component} of $X$ if", "$T$ is a maximal connected subset of $X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8351, "type": "definition", "label": "topology-definition-totally-disconnected", "categories": [ "topology" ], "title": "topology-definition-totally-disconnected", "contents": [ "A topological space is {\\it totally disconnected} if the connected components", "are all singletons." ], "refs": [], "ref_ids": [] }, { "id": 8352, "type": "definition", "label": "topology-definition-locally-connected", "categories": [ "topology" ], "title": "topology-definition-locally-connected", "contents": [ "A topological space $X$ is called {\\it locally connected} if", "every point $x \\in X$ has a fundamental system of connected neighbourhoods." ], "refs": [], "ref_ids": [] }, { "id": 8353, "type": "definition", "label": "topology-definition-irreducible-components", "categories": [ "topology" ], "title": "topology-definition-irreducible-components", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item We say $X$ is {\\it irreducible}, if $X$ is not empty, and whenever", "$X = Z_1 \\cup Z_2$ with $Z_i$ closed, we have $X = Z_1$ or $X = Z_2$.", "\\item We say $Z \\subset X$ is an {\\it irreducible component} of $X$", "if $Z$ is a maximal irreducible subset of $X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8354, "type": "definition", "label": "topology-definition-generic-point", "categories": [ "topology" ], "title": "topology-definition-generic-point", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item Let $Z \\subset X$ be an irreducible closed subset.", "A {\\it generic point} of $Z$ is a point $\\xi \\in Z$ such", "that $Z = \\overline{\\{\\xi\\}}$.", "\\item The space $X$ is called {\\it Kolmogorov}, if for every $x, x' \\in X$,", "$x \\not = x'$ there exists a closed subset of $X$ which contains", "exactly one of the two points.", "\\item The space $X$ is called {\\it quasi-sober} if every", "irreducible closed subset has a generic point.", "\\item The space $X$ is called {\\it sober} if every", "irreducible closed subset has a unique generic point.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8355, "type": "definition", "label": "topology-definition-noetherian", "categories": [ "topology" ], "title": "topology-definition-noetherian", "contents": [ "A topological space is called {\\it Noetherian}", "if the descending chain condition holds for", "closed subsets of $X$. A topological space is called", "{\\it locally Noetherian} if every point has a neighbourhood", "which is Noetherian." ], "refs": [], "ref_ids": [] }, { "id": 8356, "type": "definition", "label": "topology-definition-Krull", "categories": [ "topology" ], "title": "topology-definition-Krull", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item A {\\it chain of irreducible closed subsets} of $X$", "is a sequence $Z_0 \\subset Z_1 \\subset \\ldots \\subset Z_n \\subset X$", "with $Z_i$ closed irreducible and $Z_i \\not = Z_{i + 1}$ for", "$i = 0, \\ldots, n - 1$.", "\\item The {\\it length} of a chain", "$Z_0 \\subset Z_1 \\subset \\ldots \\subset Z_n \\subset X$", "of irreducible closed subsets of $X$ is the", "integer $n$.", "\\item The {\\it dimension} or more precisely the {\\it Krull dimension}", "$\\dim(X)$ of $X$ is the element of", "$\\{-\\infty, 0, 1, 2, 3, \\ldots, \\infty\\}$ defined by the formula:", "$$", "\\dim(X) =", "\\sup \\{\\text{lengths of chains of irreducible closed subsets}\\}", "$$", "Thus $\\dim(X) = -\\infty$ if and only if $X$ is the empty space.", "\\item Let $x \\in X$.", "The {\\it Krull dimension of $X$ at $x$} is defined as", "$$", "\\dim_x(X) = \\min \\{\\dim(U), x\\in U\\subset X\\text{ open}\\}", "$$", "the minimum of $\\dim(U)$ where $U$ runs over the open", "neighbourhoods of $x$ in $X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8357, "type": "definition", "label": "topology-definition-equidimensional", "categories": [ "topology" ], "title": "topology-definition-equidimensional", "contents": [ "Let $X$ be a topological space.", "We say that $X$ is {\\it equidimensional} if every irreducible", "component of $X$ has the same dimension." ], "refs": [], "ref_ids": [] }, { "id": 8358, "type": "definition", "label": "topology-definition-codimension", "categories": [ "topology" ], "title": "topology-definition-codimension", "contents": [ "Let $X$ be a topological space.", "Let $Y \\subset X$ be an irreducible closed subset.", "The {\\it codimension} of $Y$ in $X$ is the supremum of", "the lengths $e$ of chains", "$$", "Y = Y_0 \\subset Y_1 \\subset \\ldots \\subset Y_e \\subset X", "$$", "of irreducible closed subsets in $X$ starting with $Y$.", "We will denote this $\\text{codim}(Y, X)$." ], "refs": [], "ref_ids": [] }, { "id": 8359, "type": "definition", "label": "topology-definition-catenary", "categories": [ "topology" ], "title": "topology-definition-catenary", "contents": [ "Let $X$ be a topological space. We say $X$ is {\\it catenary} if", "for every pair of irreducible closed subsets $T \\subset T'$", "we have $\\text{codim}(T, T') < \\infty$ and every maximal chain", "of irreducible closed subsets", "$$", "T = T_0 \\subset T_1 \\subset \\ldots \\subset T_e = T'", "$$", "has the same length (equal to the codimension)." ], "refs": [], "ref_ids": [] }, { "id": 8360, "type": "definition", "label": "topology-definition-quasi-compact", "categories": [ "topology" ], "title": "topology-definition-quasi-compact", "contents": [ "Quasi-compactness.", "\\begin{enumerate}", "\\item We say that a topological space $X$ is {\\it quasi-compact}", "if every open covering of $X$ has a finite refinement.", "\\item We say that a continuous map $f : X \\to Y$ is {\\it quasi-compact}", "if the inverse image $f^{-1}(V)$ of every quasi-compact open $V \\subset Y$", "is quasi-compact.", "\\item We say a subset $Z \\subset X$ is {\\it retrocompact}", "if the inclusion map $Z \\to X$ is quasi-compact.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8361, "type": "definition", "label": "topology-definition-locally-quasi-compact", "categories": [ "topology" ], "title": "topology-definition-locally-quasi-compact", "contents": [ "A topological space $X$ is called", "{\\it locally quasi-compact}\\footnote{This may not be standard notation.", "Alternative notions used in the literature are: (1) Every point has some", "quasi-compact neighbourhood, and (2) Every point has a closed quasi-compact", "neighbourhood. A scheme has the property that every point has a fundamental", "system of open quasi-compact neighbourhoods.} if every", "point has a fundamental system of quasi-compact neighbourhoods." ], "refs": [], "ref_ids": [] }, { "id": 8362, "type": "definition", "label": "topology-definition-constructible", "categories": [ "topology" ], "title": "topology-definition-constructible", "contents": [ "Let $X$ be a topological space. Let $E \\subset X$ be a subset of $X$.", "\\begin{enumerate}", "\\item We say $E$ is {\\it constructible}\\footnote{In the second edition", "of EGA I \\cite{EGA1-second} this was called a ``globally constructible''", "set and a the terminology ``constructible'' was used for what we call a locally", "constructible set.}", "in $X$ if $E$ is a finite union", "of subsets of the form $U \\cap V^c$ where $U, V \\subset X$ are open and", "retrocompact in $X$.", "\\item We say $E$ is {\\it locally constructible} in $X$ if there exists an open", "covering $X = \\bigcup V_i$ such that each $E \\cap V_i$ is constructible", "in $V_i$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8363, "type": "definition", "label": "topology-definition-proper-map", "categories": [ "topology" ], "title": "topology-definition-proper-map", "contents": [ "Let $f : X\\to Y$ be a continuous map between topological spaces.", "\\begin{enumerate}", "\\item We say that the map $f$ is {\\it closed}", "iff the image of every closed subset is closed.", "\\item We say that the map $f$ is {\\it proper}\\footnote{This is the", "terminology used in \\cite{Bourbaki}. Usually this is what", "is called ``universally closed'' in the literature. Thus our notion", "of proper does not involve any separation conditions.} iff", "the map $Z \\times X\\to Z \\times Y$ is closed for any topological space", "$Z$.", "\\item We say that the map $f$ is {\\it quasi-proper} iff", "the inverse image $f^{-1}(V)$ of every quasi-compact subset $V \\subset Y$", "is quasi-compact.", "\\item We say that $f$ is {\\it universally closed} iff", "the map $f': Z \\times_Y X \\to Z$ is closed for any map $g: Z \\to Y$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8364, "type": "definition", "label": "topology-definition-space-jacobson", "categories": [ "topology" ], "title": "topology-definition-space-jacobson", "contents": [ "Let $X$ be a topological space.", "Let $X_0$ be the set of closed points of $X$.", "We say that $X$ is {\\it Jacobson} if every", "closed subset $Z \\subset X$ is the closure", "of $Z \\cap X_0$." ], "refs": [], "ref_ids": [] }, { "id": 8365, "type": "definition", "label": "topology-definition-specialization", "categories": [ "topology" ], "title": "topology-definition-specialization", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item If $x, x' \\in X$ then we say $x$ is a {\\it specialization} of $x'$,", "or $x'$ is a {\\it generalization} of $x$ if $x \\in \\overline{\\{x'\\}}$.", "Notation: $x' \\leadsto x$.", "\\item A subset $T \\subset X$ is {\\it stable under specialization}", "if for all $x' \\in T$ and every specialization $x' \\leadsto x$ we have", "$x \\in T$.", "\\item A subset $T \\subset X$ is {\\it stable under generalization}", "if for all $x \\in T$ and every generalization $x' \\leadsto x$ we have", "$x' \\in T$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8366, "type": "definition", "label": "topology-definition-lift-specializations", "categories": [ "topology" ], "title": "topology-definition-lift-specializations", "contents": [ "Let $f : X \\to Y$ be a continuous map of topological spaces.", "\\begin{enumerate}", "\\item We say that {\\it specializations lift along $f$} or that $f$ is", "{\\it specializing} if given $y' \\leadsto y$ in $Y$ and any $x'\\in X$ with", "$f(x') = y'$ there exists a specialization $x' \\leadsto x$ of $x'$ in $X$ such", "that $f(x) = y$.", "\\item We say that {\\it generalizations lift along $f$} or that $f$ is", "{\\it generalizing} if given $y' \\leadsto y$ in $Y$ and any $x\\in X$ with", "$f(x) = y$ there exists a generalization $x' \\leadsto x$ of $x$ in $X$ such", "that $f(x') = y'$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8367, "type": "definition", "label": "topology-definition-dimension-function", "categories": [ "topology" ], "title": "topology-definition-dimension-function", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item Let $x, y \\in X$, $x \\not = y$. Suppose $x \\leadsto y$, that", "is $y$ is a specialization of $x$.", "We say $y$ is an {\\it immediate specialization}", "of $x$ if there is no", "$z \\in X \\setminus \\{x, y\\}$ with $x \\leadsto z$ and $z \\leadsto y$.", "\\item A map $\\delta : X \\to \\mathbf{Z}$ is called a", "{\\it dimension function}\\footnote{This is likely nonstandard", "notation. This notion is usually introduced only for (locally) Noetherian", "schemes, in which case condition (a) is implied by (b).} if", "\\begin{enumerate}", "\\item whenever $x \\leadsto y$ and $x \\not = y$", "we have $\\delta(x) > \\delta(y)$, and", "\\item for every immediate specialization $x \\leadsto y$ in $X$", "we have $\\delta(x) = \\delta(y) + 1$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8368, "type": "definition", "label": "topology-definition-nowhere-dense", "categories": [ "topology" ], "title": "topology-definition-nowhere-dense", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item Given a subset $T \\subset X$ the {\\it interior} of $T$ is the", "largest open subset of $X$ contained in $T$.", "\\item A subset $T \\subset X$ is called {\\it nowhere dense} if the closure of", "$T$ has empty interior.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8369, "type": "definition", "label": "topology-definition-profinite", "categories": [ "topology" ], "title": "topology-definition-profinite", "contents": [ "A topological space is {\\it profinite} if it is homeomorphic to a limit", "of a diagram of finite discrete spaces." ], "refs": [], "ref_ids": [] }, { "id": 8370, "type": "definition", "label": "topology-definition-spectral-space", "categories": [ "topology" ], "title": "topology-definition-spectral-space", "contents": [ "A topological space $X$ is called {\\it spectral} if it is sober,", "quasi-compact, the intersection of two quasi-compact opens is", "quasi-compact, and the collection of quasi-compact opens forms a", "basis for the topology. A continuous map $f : X \\to Y$ of spectral", "spaces is called {\\it spectral} if the inverse image of a quasi-compact", "open is quasi-compact." ], "refs": [], "ref_ids": [] }, { "id": 8371, "type": "definition", "label": "topology-definition-extremally-disconnected", "categories": [ "topology" ], "title": "topology-definition-extremally-disconnected", "contents": [ "A topological space $X$ is called {\\it extremally disconnected}", "if the closure of every open subset of $X$ is open." ], "refs": [], "ref_ids": [] }, { "id": 8372, "type": "definition", "label": "topology-definition-isolated-point", "categories": [ "topology" ], "title": "topology-definition-isolated-point", "contents": [ "Let $X$ be a topological space. We say $x \\in X$ is an", "{\\it isolated point} of $X$ if $\\{x\\}$ is open in $X$." ], "refs": [], "ref_ids": [] }, { "id": 8373, "type": "definition", "label": "topology-definition-paritition", "categories": [ "topology" ], "title": "topology-definition-paritition", "contents": [ "Let $X$ be a topological space. A {\\it partition} of $X$ is a", "decomposition $X = \\coprod X_i$ into locally closed subsets $X_i$.", "The $X_i$ are called the {\\it parts} of the partition.", "Given two partitions of $X$ we say one {\\it refines} the other if", "the parts of one are unions of parts of the other." ], "refs": [], "ref_ids": [] }, { "id": 8374, "type": "definition", "label": "topology-definition-good-stratification", "categories": [ "topology" ], "title": "topology-definition-good-stratification", "contents": [ "Let $X$ be a topological space. A {\\it good stratification}", "of $X$ is a partition $X = \\coprod X_i$ such that for all", "$i, j \\in I$ we have", "$$", "X_i \\cap \\overline{X_j} \\not = \\emptyset", "\\Rightarrow", "X_i \\subset \\overline{X_j}.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 8375, "type": "definition", "label": "topology-definition-stratification", "categories": [ "topology" ], "title": "topology-definition-stratification", "contents": [ "Let $X$ be a topological space. A {\\it stratification} of $X$ is", "given by a partition $X = \\coprod_{i \\in I} X_i$ and a partial ordering", "on $I$ such that for each $j \\in I$ we have", "$$", "\\overline{X_j} \\subset \\bigcup\\nolimits_{i \\leq j} X_i", "$$", "The parts $X_i$ are called the {\\it strata} of the stratification." ], "refs": [], "ref_ids": [] }, { "id": 8376, "type": "definition", "label": "topology-definition-locally-finite", "categories": [ "topology" ], "title": "topology-definition-locally-finite", "contents": [ "Let $X$ be a topological space. Let $I$ be a set and for $i \\in I$", "let $E_i \\subset X$ be a subset. We say the collection $\\{E_i\\}_{i \\in I}$", "is {\\it locally finite} if for all $x \\in X$ there exists an open", "neighbourhood $U$ of $x$ such that", "$\\{i \\in I | E_i \\cap U \\not = \\emptyset\\}$ is finite." ], "refs": [], "ref_ids": [] }, { "id": 8377, "type": "definition", "label": "topology-definition-topological-group", "categories": [ "topology" ], "title": "topology-definition-topological-group", "contents": [ "A {\\it topological group} is a group $G$ endowed with a topology", "such that multiplication $G \\times G \\to G$, $(x, y) \\mapsto xy$ and", "inverse $G \\to G$, $x \\mapsto x^{-1}$ are continuous.", "A {\\it homomorphism of topological groups} is a homomorphism of groups", "which is continuous." ], "refs": [], "ref_ids": [] }, { "id": 8378, "type": "definition", "label": "topology-definition-profinite-group", "categories": [ "topology" ], "title": "topology-definition-profinite-group", "contents": [ "A topological group is called a {\\it profinite group} if it satisfies", "the equivalent conditions of Lemma \\ref{lemma-profinite-group}." ], "refs": [ "topology-lemma-profinite-group" ], "ref_ids": [ 8339 ] }, { "id": 8379, "type": "definition", "label": "topology-definition-topological-ring", "categories": [ "topology" ], "title": "topology-definition-topological-ring", "contents": [ "A {\\it topological ring} is a ring $R$ endowed with a topology", "such that addition $R \\times R \\to R$, $(x, y) \\mapsto x + y$ and", "multiplication $R \\times R \\to R$, $(x, y) \\mapsto xy$ are continuous.", "A {\\it homomorphism of topological rings} is a homomorphism of rings", "which is continuous." ], "refs": [], "ref_ids": [] }, { "id": 8380, "type": "definition", "label": "topology-definition-topological-module", "categories": [ "topology" ], "title": "topology-definition-topological-module", "contents": [ "Let $R$ be a topological ring. A {\\it topological module} is an $R$-module", "$M$ endowed with a topology such that addition $M \\times M \\to M$ and", "scalar multiplication $R \\times M \\to M$ are continuous.", "A {\\it homomorphism of topological modules} is a homomorphism of", "modules which is continuous." ], "refs": [], "ref_ids": [] }, { "id": 8421, "type": "definition", "label": "hypercovering-definition-SR", "categories": [ "hypercovering" ], "title": "hypercovering-definition-SR", "contents": [ "Let $\\mathcal{C}$ be a category. We denote $\\text{SR}(\\mathcal{C})$", "the category of {\\it semi-representable objects} defined as follows", "\\begin{enumerate}", "\\item objects are families of objects $\\{U_i\\}_{i \\in I}$, and", "\\item morphisms $\\{U_i\\}_{i \\in I} \\to \\{V_j\\}_{j \\in J}$ are given by", "a map $\\alpha : I \\to J$ and for each $i \\in I$", "a morphism $f_i : U_i \\to V_{\\alpha(i)}$ of $\\mathcal{C}$.", "\\end{enumerate}", "Let $X \\in \\Ob(\\mathcal{C})$ be an object of $\\mathcal{C}$.", "The category of {\\it semi-representable objects over $X$}", "is the category", "$\\text{SR}(\\mathcal{C}, X) = \\text{SR}(\\mathcal{C}/X)$." ], "refs": [], "ref_ids": [] }, { "id": 8422, "type": "definition", "label": "hypercovering-definition-SR-F", "categories": [ "hypercovering" ], "title": "hypercovering-definition-SR-F", "contents": [ "Let $\\mathcal{C}$ be a category.", "We denote $F$ the functor {\\it which associates a presheaf to a", "semi-representable object}. In a formula", "\\begin{eqnarray*}", "F : \\text{SR}(\\mathcal{C}) & \\longrightarrow & \\textit{PSh}(\\mathcal{C}) \\\\", "\\{U_i\\}_{i \\in I} & \\longmapsto & \\amalg_{i\\in I} h_{U_i}", "\\end{eqnarray*}", "where $h_U$ denotes the representable presheaf associated to", "the object $U$." ], "refs": [], "ref_ids": [] }, { "id": 8423, "type": "definition", "label": "hypercovering-definition-covering-SR", "categories": [ "hypercovering" ], "title": "hypercovering-definition-covering-SR", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$f = (\\alpha, f_i) : \\{U_i\\}_{i \\in I} \\to \\{V_j\\}_{j \\in J}$", "be a morphism in the category $\\text{SR}(\\mathcal{C})$.", "We say that $f$ is a {\\it covering} if for every $j \\in J$ the", "family of morphisms $\\{U_i \\to V_j\\}_{i \\in I, \\alpha(i) = j}$", "is a covering for the site $\\mathcal{C}$.", "Let $X$ be an object of $\\mathcal{C}$.", "A morphism $K \\to L$ in $\\text{SR}(\\mathcal{C}, X)$ is", "a {\\it covering} if its image in $\\text{SR}(\\mathcal{C})$ is", "a covering." ], "refs": [], "ref_ids": [] }, { "id": 8424, "type": "definition", "label": "hypercovering-definition-hypercovering", "categories": [ "hypercovering" ], "title": "hypercovering-definition-hypercovering", "contents": [ "Let $\\mathcal{C}$ be a site. Assume $\\mathcal{C}$ has fibre products.", "Let $X \\in \\Ob(\\mathcal{C})$ be an object of $\\mathcal{C}$.", "A {\\it hypercovering of $X$} is a simplicial object", "$K$ of $\\text{SR}(\\mathcal{C}, X)$ such that", "\\begin{enumerate}", "\\item The object $K_0$ is a covering of $X$ for the site $\\mathcal{C}$.", "\\item For every $n \\geq 0$ the canonical morphism", "$$", "K_{n + 1} \\longrightarrow (\\text{cosk}_n \\text{sk}_n K)_{n + 1}", "$$", "is a covering in the sense defined above.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8425, "type": "definition", "label": "hypercovering-definition-homology", "categories": [ "hypercovering" ], "title": "hypercovering-definition-homology", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $K$ be a simplicial object of $\\textit{PSh}(\\mathcal{C})$.", "By the above we get a simplicial object $\\mathbf{Z}_K^\\#$ of", "$\\textit{Ab}(\\mathcal{C})$. We can take its associated", "complex of abelian presheaves $s(\\mathbf{Z}_K^\\#)$, see", "Simplicial, Section \\ref{simplicial-section-complexes}.", "The {\\it homology of $K$} is the homology of the", "complex of abelian sheaves $s(\\mathbf{Z}_K^\\#)$." ], "refs": [], "ref_ids": [] }, { "id": 8426, "type": "definition", "label": "hypercovering-definition-hypercovering-variant", "categories": [ "hypercovering" ], "title": "hypercovering-definition-hypercovering-variant", "contents": [ "Let $\\mathcal{C}$ be a site. Assume $\\mathcal{C}$ has equalizers", "and fibre products. Let $\\mathcal{G}$ be a presheaf of sets.", "A {\\it hypercovering of $\\mathcal{G}$} is a simplicial object", "$K$ of $\\text{SR}(\\mathcal{C})$ endowed with an augmentation", "$F(K) \\to \\mathcal{G}$ such that", "\\begin{enumerate}", "\\item $F(K_0) \\to \\mathcal{G}$ becomes surjective", "after sheafification,", "\\item $F(K_1) \\to F(K_0) \\times_\\mathcal{G} F(K_0)$", "becomes surjective after sheafification, and", "\\item $F(K_{n + 1}) \\longrightarrow F((\\text{cosk}_n \\text{sk}_n K)_{n + 1})$", "for $n \\geq 1$ becomes surjective after sheafification.", "\\end{enumerate}", "We say that a simplicial object $K$ of $\\text{SR}(\\mathcal{C})$", "is a {\\it hypercovering} if $K$ is a hypercovering of the final", "object $*$ of $\\textit{PSh}(\\mathcal{C})$." ], "refs": [], "ref_ids": [] }, { "id": 8481, "type": "definition", "label": "algebraic-definition-representable-by-algebraic-space", "categories": [ "algebraic" ], "title": "algebraic-definition-representable-by-algebraic-space", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "A category fibred in groupoids $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$", "is called {\\it representable by an algebraic space over $S$}", "if there exists an algebraic space $F$ over $S$ and an equivalence", "$j : \\mathcal{X} \\to \\mathcal{S}_F$", "of categories over $(\\Sch/S)_{fppf}$." ], "refs": [], "ref_ids": [] }, { "id": 8482, "type": "definition", "label": "algebraic-definition-representable-by-algebraic-spaces", "categories": [ "algebraic" ], "title": "algebraic-definition-representable-by-algebraic-spaces", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "A $1$-morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of", "categories fibred in groupoids over $(\\Sch/S)_{fppf}$", "is called {\\it representable by algebraic spaces} if", "for any $U \\in \\Ob((\\Sch/S)_{fppf})$", "and any $y : (\\Sch/U)_{fppf} \\to \\mathcal{Y}$", "the category fibred in groupoids", "$$", "(\\Sch/U)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}", "$$", "over $(\\Sch/U)_{fppf}$", "is representable by an algebraic space over $U$." ], "refs": [], "ref_ids": [] }, { "id": 8483, "type": "definition", "label": "algebraic-definition-relative-representable-property", "categories": [ "algebraic" ], "title": "algebraic-definition-relative-representable-property", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism", "of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "Assume $f$ is representable by algebraic spaces.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which", "\\begin{enumerate}", "\\item is preserved under any base change, and", "\\item is fppf local on the base, see", "Descent on Spaces,", "Definition \\ref{spaces-descent-definition-property-morphisms-local}.", "\\end{enumerate}", "In this case we say that $f$ has {\\it property $\\mathcal{P}$} if for every", "$U \\in \\Ob((\\Sch/S)_{fppf})$ and", "any $y \\in \\mathcal{Y}_U$ the resulting morphism of algebraic spaces", "$f_y : F_y \\to U$, see", "diagram (\\ref{equation-representable-by-algebraic-spaces}),", "has property $\\mathcal{P}$." ], "refs": [ "spaces-descent-definition-property-morphisms-local" ], "ref_ids": [ 9440 ] }, { "id": 8484, "type": "definition", "label": "algebraic-definition-algebraic-stack", "categories": [ "algebraic" ], "title": "algebraic-definition-algebraic-stack", "contents": [ "Let $S$ be a base scheme contained in $\\Sch_{fppf}$.", "An {\\it algebraic stack over $S$} is a category", "$$", "p : \\mathcal{X} \\to (\\Sch/S)_{fppf}", "$$", "over $(\\Sch/S)_{fppf}$ with the following properties:", "\\begin{enumerate}", "\\item The category $\\mathcal{X}$ is a stack in groupoids over", "$(\\Sch/S)_{fppf}$.", "\\item The diagonal", "$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$", "is representable by algebraic spaces.", "\\item There exists a scheme $U \\in \\Ob((\\Sch/S)_{fppf})$", "and a $1$-morphism $(\\Sch/U)_{fppf} \\to \\mathcal{X}$", "which is surjective and smooth\\footnote{In future chapters we will denote", "this simply $U \\to \\mathcal{X}$ as is customary in the literature. Another", "good alternative would be to formulate this condition as the existence of a", "representable category fibred in groupoids $\\mathcal{U}$ and a surjective", "smooth $1$-morphism $\\mathcal{U} \\to \\mathcal{X}$.}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8485, "type": "definition", "label": "algebraic-definition-deligne-mumford", "categories": [ "algebraic" ], "title": "algebraic-definition-deligne-mumford", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "Let $\\mathcal{X}$ be an algebraic stack over $S$.", "We say $\\mathcal{X}$ is a {\\it Deligne-Mumford stack} if there exists", "a scheme $U$ and a surjective \\'etale morphism", "$(\\Sch/U)_{fppf} \\to \\mathcal{X}$." ], "refs": [], "ref_ids": [] }, { "id": 8486, "type": "definition", "label": "algebraic-definition-morphism-algebraic-stacks", "categories": [ "algebraic" ], "title": "algebraic-definition-morphism-algebraic-stacks", "contents": [ "Let $S$ be a scheme contained in $\\Sch_{fppf}$.", "The {\\it $2$-category of algebraic stacks over $S$} is the", "sub $2$-category of the $2$-category of categories fibred in", "groupoids over $(\\Sch/S)_{fppf}$ (see", "Categories,", "Definition \\ref{categories-definition-categories-fibred-in-groupoids-over-C})", "defined as follows:", "\\begin{enumerate}", "\\item Its objects are those categories fibred in groupoids", "over $(\\Sch/S)_{fppf}$ which are algebraic stacks over $S$.", "\\item Its $1$-morphisms $f : \\mathcal{X} \\to \\mathcal{Y}$ are", "any functors of categories over $(\\Sch/S)_{fppf}$, as in", "Categories, Definition \\ref{categories-definition-categories-over-C}.", "\\item Its $2$-morphisms are transformations between functors", "over $(\\Sch/S)_{fppf}$, as in", "Categories, Definition \\ref{categories-definition-categories-over-C}.", "\\end{enumerate}" ], "refs": [ "categories-definition-categories-fibred-in-groupoids-over-C", "categories-definition-categories-over-C", "categories-definition-categories-over-C" ], "ref_ids": [ 12393, 12385, 12385 ] }, { "id": 8487, "type": "definition", "label": "algebraic-definition-smooth-groupoid", "categories": [ "algebraic" ], "title": "algebraic-definition-smooth-groupoid", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "We say $(U, R, s, t, c)$ is a {\\it smooth groupoid}\\footnote{This terminology", "might be a bit confusing: it does not imply that $[U/R]$ is smooth", "over anything.}", "if $s, t : R \\to U$ are smooth morphisms of algebraic spaces." ], "refs": [], "ref_ids": [] }, { "id": 8488, "type": "definition", "label": "algebraic-definition-presentation", "categories": [ "algebraic" ], "title": "algebraic-definition-presentation", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack over $S$.", "A {\\it presentation} of $\\mathcal{X}$ is given by a smooth groupoid", "$(U, R, s, t, c)$ in algebraic spaces over $S$, and an", "equivalence $f : [U/R] \\to \\mathcal{X}$." ], "refs": [], "ref_ids": [] }, { "id": 8489, "type": "definition", "label": "algebraic-definition-viewed-as", "categories": [ "algebraic" ], "title": "algebraic-definition-viewed-as", "contents": [ "Let $\\Sch_{fppf}$ be a big fppf site.", "Let $S \\to S'$ be a morphism of this site.", "If $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$", "is an algebraic stack over $S$, then", "$\\mathcal{X}$ {\\it viewed as an algebraic stack over $S'$}", "is the algebraic stack", "$$", "\\mathcal{X} \\longrightarrow (\\Sch/S')_{fppf}", "$$", "gotten by applying construction A of", "Lemma \\ref{lemma-category-of-spaces-over-smaller-base-scheme}", "to $\\mathcal{X}$." ], "refs": [ "algebraic-lemma-category-of-spaces-over-smaller-base-scheme" ], "ref_ids": [ 8479 ] }, { "id": 8490, "type": "definition", "label": "algebraic-definition-change-of-base", "categories": [ "algebraic" ], "title": "algebraic-definition-change-of-base", "contents": [ "Let $\\Sch_{fppf}$ be a big fppf site.", "Let $S \\to S'$ be a morphism of this site.", "Let $\\mathcal{X}'$ be an algebraic stack over $S'$.", "The {\\it change of base of $\\mathcal{X}'$} is the", "algebraic space $\\mathcal{X}'_S$ over $S$ described above." ], "refs": [], "ref_ids": [] }, { "id": 8646, "type": "definition", "label": "sites-definition-presheaves-sets", "categories": [ "sites" ], "title": "sites-definition-presheaves-sets", "contents": [ "A {\\it presheaf of sets} on $\\mathcal{C}$ is a contravariant", "functor from $\\mathcal{C}$ to $\\textit{Sets}$. {\\it Morphisms", "of presheaves} are transformations of functors. The category", "of presheaves of sets is denoted $\\textit{PSh}(\\mathcal{C})$." ], "refs": [], "ref_ids": [] }, { "id": 8647, "type": "definition", "label": "sites-definition-presheaf", "categories": [ "sites" ], "title": "sites-definition-presheaf", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{A}$ be categories.", "A {\\it presheaf} $\\mathcal{F}$ on $\\mathcal{C}$", "with values in $\\mathcal{A}$ is a contravariant", "functor from $\\mathcal{C}$ to $\\mathcal{A}$,", "i.e., $\\mathcal{F} : \\mathcal{C}^{opp} \\to \\mathcal{A}$.", "A {\\it morphism} of presheaves $\\mathcal{F} \\to \\mathcal{G}$", "on $\\mathcal{C}$ with values in $\\mathcal{A}$ is a transformation", "of functors from $\\mathcal{F}$ to $\\mathcal{G}$." ], "refs": [], "ref_ids": [] }, { "id": 8648, "type": "definition", "label": "sites-definition-presheaves-injective-surjective", "categories": [ "sites" ], "title": "sites-definition-presheaves-injective-surjective", "contents": [ "Let $\\mathcal{C}$ be a category, and let $\\varphi : \\mathcal{F}", "\\to \\mathcal{G}$ be a map of presheaves of sets.", "\\begin{enumerate}", "\\item We say that $\\varphi$ is {\\it injective} if for every object", "$U$ of $\\mathcal{C}$ the map $\\varphi_U : \\mathcal{F}(U)", "\\to \\mathcal{G}(U)$ is injective.", "\\item We say that $\\varphi$ is {\\it surjective} if for every object", "$U$ of $\\mathcal{C}$ the map $\\varphi_U : \\mathcal{F}(U)", "\\to \\mathcal{G}(U)$ is surjective.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8649, "type": "definition", "label": "sites-definition-sub-presheaf", "categories": [ "sites" ], "title": "sites-definition-sub-presheaf", "contents": [ "We say $\\mathcal{F}$ is a {\\it subpresheaf} of $\\mathcal{G}$", "if for every object $U \\in \\Ob(\\mathcal{C})$ the set", "$\\mathcal{F}(U)$ is a subset of $\\mathcal{G}(U)$, compatibly", "with the restriction mappings." ], "refs": [], "ref_ids": [] }, { "id": 8650, "type": "definition", "label": "sites-definition-image", "categories": [ "sites" ], "title": "sites-definition-image", "contents": [ "Notation as in Lemma \\ref{lemma-image}. We", "say that $\\mathcal{G}'$ is the {\\it image of $\\varphi$}." ], "refs": [ "sites-lemma-image" ], "ref_ids": [ 8496 ] }, { "id": 8651, "type": "definition", "label": "sites-definition-family-morphisms-fixed-target", "categories": [ "sites" ], "title": "sites-definition-family-morphisms-fixed-target", "contents": [ "Let $\\mathcal{C}$ be a category, see", "Conventions, Section \\ref{conventions-section-categories}.", "A {\\it family of morphisms with fixed target} in $\\mathcal{C}$ is", "given by an object $U \\in \\Ob(\\mathcal{C})$, a set $I$ and", "for each $i\\in I$ a morphism $U_i \\to U$ of $\\mathcal{C}$ with target $U$.", "We use the notation $\\{U_i \\to U\\}_{i\\in I}$ to indicate this." ], "refs": [], "ref_ids": [] }, { "id": 8652, "type": "definition", "label": "sites-definition-site", "categories": [ "sites" ], "title": "sites-definition-site", "contents": [ "A {\\it site}\\footnote{This notation differs from that of \\cite{SGA4}, as", "explained in the introduction.} is given by a category $\\mathcal{C}$ and a set", "$\\text{Cov}(\\mathcal{C})$ of families of morphisms with fixed target", "$\\{U_i \\to U\\}_{i \\in I}$, called {\\it coverings of $\\mathcal{C}$},", "satisfying the following axioms", "\\begin{enumerate}", "\\item If $V \\to U$ is an isomorphism then $\\{V \\to U\\} \\in", "\\text{Cov}(\\mathcal{C})$.", "\\item If $\\{U_i \\to U\\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$ and for each", "$i$ we have $\\{V_{ij} \\to U_i\\}_{j\\in J_i} \\in \\text{Cov}(\\mathcal{C})$, then", "$\\{V_{ij} \\to U\\}_{i \\in I, j\\in J_i} \\in \\text{Cov}(\\mathcal{C})$.", "\\item If $\\{U_i \\to U\\}_{i\\in I}\\in \\text{Cov}(\\mathcal{C})$", "and $V \\to U$ is a morphism of $\\mathcal{C}$ then $U_i \\times_U V$", "exists for all $i$ and", "$\\{U_i \\times_U V \\to V \\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8653, "type": "definition", "label": "sites-definition-sheaf-sets", "categories": [ "sites" ], "title": "sites-definition-sheaf-sets", "contents": [ "Let $\\mathcal{C}$ be a site, and let $\\mathcal{F}$ be a presheaf of sets", "on $\\mathcal{C}$. We say $\\mathcal{F}$ is a {\\it sheaf} if", "for every covering $\\{U_i \\to U\\}_{i \\in I} \\in \\text{Cov}(\\mathcal{C})$", "the diagram", "\\begin{equation}", "\\label{equation-sheaf-condition}", "\\xymatrix{", "\\mathcal{F}(U) \\ar[r]", "&", "\\prod\\nolimits_{i\\in I}", "\\mathcal{F}(U_i)", "\\ar@<1ex>[r]^-{\\text{pr}_0^*} \\ar@<-1ex>[r]_-{\\text{pr}_1^*}", "&", "\\prod\\nolimits_{(i_0, i_1) \\in I \\times I}", "\\mathcal{F}(U_{i_0} \\times_U U_{i_1})", "}", "\\end{equation}", "represents the first arrow as the equalizer of $\\text{pr}_0^*$", "and $\\text{pr}_1^*$." ], "refs": [], "ref_ids": [] }, { "id": 8654, "type": "definition", "label": "sites-definition-category-sheaves-sets", "categories": [ "sites" ], "title": "sites-definition-category-sheaves-sets", "contents": [ "The category {\\it $\\Sh(\\mathcal{C})$}", "of sheaves of sets is the full subcategory of the category", "$\\textit{PSh}(\\mathcal{C})$ whose objects are the sheaves of sets." ], "refs": [], "ref_ids": [] }, { "id": 8655, "type": "definition", "label": "sites-definition-sheaf", "categories": [ "sites" ], "title": "sites-definition-sheaf", "contents": [ "Let $\\mathcal{C}$ be a site, let $\\mathcal{A}$ be a category", "and let $\\mathcal{F}$ be a presheaf on $\\mathcal{C}$ with values in", "$\\mathcal{A}$. We say that $\\mathcal{F}$ is a {\\it sheaf}", "if for all objects $X$ of $\\mathcal{A}$ the presheaf of sets", "$\\mathcal{F}_X$ (defined above) is a sheaf." ], "refs": [], "ref_ids": [] }, { "id": 8656, "type": "definition", "label": "sites-definition-morphism-coverings", "categories": [ "sites" ], "title": "sites-definition-morphism-coverings", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{U} = \\{U_i \\to U\\}_{i\\in I}$ be a family", "of morphisms of $\\mathcal{C}$ with fixed target.", "Let $\\mathcal{V} = \\{V_j \\to V\\}_{j\\in J}$ be another.", "\\begin{enumerate}", "\\item", "A {\\it morphism of families of maps with fixed target", "of $\\mathcal{C}$ from $\\mathcal{U}$ to $\\mathcal{V}$},", "or simply a {\\it morphism from $\\mathcal{U}$ to $\\mathcal{V}$}", "is given by a morphism $U \\to V$, a map of sets", "$\\alpha : I \\to J$ and for each $i\\in I$", "a morphism $U_i \\to V_{\\alpha(i)}$ such that the diagram", "$$", "\\xymatrix{", "U_i \\ar[r] \\ar[d]", "&", "V_{\\alpha(i)} \\ar[d]", "\\\\", "U \\ar[r]", "&", "V", "}", "$$", "is commutative.", "\\item In the special case that $U = V$ and $U \\to V$ is the identity", "we call $\\mathcal{U}$ a {\\it refinement} of the family $\\mathcal{V}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8657, "type": "definition", "label": "sites-definition-combinatorial-tautological", "categories": [ "sites" ], "title": "sites-definition-combinatorial-tautological", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{U} = \\{\\varphi_i : U_i \\to U\\}_{i\\in I}$, and", "$\\mathcal{V} = \\{\\psi_j : V_j \\to U\\}_{j\\in J}$ be two families of morphisms", "with fixed target.", "\\begin{enumerate}", "\\item We say $\\mathcal{U}$ and $\\mathcal{V}$ are", "{\\it combinatorially equivalent}", "if there exist maps", "$\\alpha : I \\to J$ and $\\beta : J\\to I$ such that", "$\\varphi_i = \\psi_{\\alpha(i)}$ and $\\psi_j = \\varphi_{\\beta(j)}$.", "\\item We say $\\mathcal{U}$ and $\\mathcal{V}$ are", "{\\it tautologically equivalent} if there exist maps", "$\\alpha : I \\to J$ and $\\beta : J\\to I$ and", "for all $i\\in I$ and $j \\in J$ commutative diagrams", "$$", "\\xymatrix{", "U_i \\ar[rd] \\ar[rr] & &", "V_{\\alpha(i)} \\ar[ld] & &", "V_j \\ar[rd] \\ar[rr] & &", "U_{\\beta(j)} \\ar[ld] \\\\", "&", "U & & & &", "U &", "}", "$$", "with isomorphisms as horizontal arrows.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8658, "type": "definition", "label": "sites-definition-separated", "categories": [ "sites" ], "title": "sites-definition-separated", "contents": [ "We say that a presheaf of sets $\\mathcal{F}$ on a site", "$\\mathcal{C}$ is {\\it separated} if, for all coverings", "of $\\{U_i \\rightarrow U\\}$, the map", "$\\mathcal{F}(U) \\to \\prod \\mathcal{F}(U_i)$ is injective." ], "refs": [], "ref_ids": [] }, { "id": 8659, "type": "definition", "label": "sites-definition-associated-sheaf", "categories": [ "sites" ], "title": "sites-definition-associated-sheaf", "contents": [ "Let $\\mathcal{C}$ be a site and let $\\mathcal{F}$ be a presheaf", "of sets on $\\mathcal{C}$. The sheaf $\\mathcal{F}^\\# := \\mathcal{F}^{++}$", "together with the canonical map $\\mathcal{F} \\to \\mathcal{F}^\\#$", "is called the {\\it sheaf associated to $\\mathcal{F}$}." ], "refs": [], "ref_ids": [] }, { "id": 8660, "type": "definition", "label": "sites-definition-sheaves-injective-surjective", "categories": [ "sites" ], "title": "sites-definition-sheaves-injective-surjective", "contents": [ "Let $\\mathcal{C}$ be a site, and let $\\varphi : \\mathcal{F}", "\\to \\mathcal{G}$ be a map of sheaves of sets.", "\\begin{enumerate}", "\\item We say that $\\varphi$ is {\\it injective} if for every object", "$U$ of $\\mathcal{C}$ the map $\\varphi : \\mathcal{F}(U)", "\\to \\mathcal{G}(U)$ is injective.", "\\item We say that $\\varphi$ is {\\it surjective} if for every object", "$U$ of $\\mathcal{C}$ and every section $s\\in \\mathcal{G}(U)$", "there exists a covering $\\{U_i \\to U\\}$ such that for", "all $i$ the restriction $s|_{U_i}$ is in the image of", "$\\varphi : \\mathcal{F}(U_i) \\to \\mathcal{G}(U_i)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8661, "type": "definition", "label": "sites-definition-universal-effective-epimorphisms", "categories": [ "sites" ], "title": "sites-definition-universal-effective-epimorphisms", "contents": [ "Let $\\mathcal{C}$ be a category. We say that a family $\\{U_i \\to U\\}_{i \\in I}$", "is an {\\it effective epimorphism} if all the morphisms $U_i \\to U$ are", "representable (see", "Categories, Definition \\ref{categories-definition-representable-morphism}),", "and for any $X\\in \\Ob(\\mathcal{C})$ the sequence", "$$", "\\xymatrix{", "\\Mor_\\mathcal{C}(U, X) \\ar[r]", "&", "\\prod\\nolimits_{i \\in I} \\Mor_\\mathcal{C}(U_i, X)", "\\ar@<1ex>[r] \\ar@<-1ex>[r]", "&", "\\prod\\nolimits_{(i, j) \\in I^2} \\Mor_\\mathcal{C}(U_i \\times_U U_j, X)", "}", "$$", "is an equalizer diagram. We say that a family $\\{U_i \\to U\\}$ is a", "{\\it universal effective epimorphism} if for any morphism $V \\to U$", "the base change $\\{U_i \\times_U V \\to V\\}$ is an effective epimorphism." ], "refs": [ "categories-definition-representable-morphism" ], "ref_ids": [ 12348 ] }, { "id": 8662, "type": "definition", "label": "sites-definition-weaker-than-canonical", "categories": [ "sites" ], "title": "sites-definition-weaker-than-canonical", "contents": [ "We say that the topology on a site $\\mathcal{C}$ is", "{\\it weaker than the canonical topology}, or that the topology is", "{\\it subcanonical} if all the coverings", "of $\\mathcal{C}$ are universal effective epimorphisms." ], "refs": [], "ref_ids": [] }, { "id": 8663, "type": "definition", "label": "sites-definition-representable-sheaf", "categories": [ "sites" ], "title": "sites-definition-representable-sheaf", "contents": [ "Let $\\mathcal{C}$ be a site whose topology is subcanonical.", "The Yoneda embedding $h$ (see", "Categories, Section \\ref{categories-section-opposite})", "presents $\\mathcal{C}$ as a full subcategory of the", "category of sheaves of $\\mathcal{C}$. In this case", "we call sheaves of the form $h_U$ with $U \\in \\Ob(\\mathcal{C})$", "{\\it representable sheaves} on $\\mathcal{C}$.", "Notation: Sometimes, the representable sheaf $h_U$ associated to $U$ is", "denoted {\\it $\\underline{U}$}." ], "refs": [], "ref_ids": [] }, { "id": 8664, "type": "definition", "label": "sites-definition-continuous", "categories": [ "sites" ], "title": "sites-definition-continuous", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "A functor $u : \\mathcal{C} \\to \\mathcal{D}$ is called", "{\\it continuous} if for every", "$\\{V_i \\to V\\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$", "we have the following", "\\begin{enumerate}", "\\item $\\{u(V_i) \\to u(V)\\}_{i\\in I}$ is in $\\text{Cov}(\\mathcal{D})$, and", "\\item for any morphism $T \\to V$ in $\\mathcal{C}$ the morphism", "$u(T \\times_V V_i) \\to u(T) \\times_{u(V)} u(V_i)$ is an isomorphism.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8665, "type": "definition", "label": "sites-definition-morphism-sites", "categories": [ "sites" ], "title": "sites-definition-morphism-sites", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "A {\\it morphism of sites} $f : \\mathcal{D} \\to \\mathcal{C}$", "is given by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$", "such that the functor $u_s$ is exact." ], "refs": [], "ref_ids": [] }, { "id": 8666, "type": "definition", "label": "sites-definition-composition-morphisms-sites", "categories": [ "sites" ], "title": "sites-definition-composition-morphisms-sites", "contents": [ "Let $\\mathcal{C}_i$, $i = 1, 2, 3$ be sites. Let", "$f : \\mathcal{C}_1 \\to \\mathcal{C}_2$ and", "$g : \\mathcal{C}_2 \\to \\mathcal{C}_3$ be morphisms of sites", "given by continuous functors $u : \\mathcal{C}_2 \\to \\mathcal{C}_1$", "and $v : \\mathcal{C}_3 \\to \\mathcal{C}_2$. The {\\it composition}", "$g \\circ f$ is the morphism of sites corresponding to the", "functor $u \\circ v$." ], "refs": [], "ref_ids": [] }, { "id": 8667, "type": "definition", "label": "sites-definition-topos", "categories": [ "sites" ], "title": "sites-definition-topos", "contents": [ "A {\\it topos} is the category $\\Sh(\\mathcal{C})$ of sheaves", "on a site $\\mathcal{C}$.", "\\begin{enumerate}", "\\item Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "A {\\it morphism of topoi} $f$ from $\\Sh(\\mathcal{D})$", "to $\\Sh(\\mathcal{C})$ is given by a pair of functors", "$f_* : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$", "and", "$f^{-1} : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "such that", "\\begin{enumerate}", "\\item we have", "$$", "\\Mor_{\\Sh(\\mathcal{D})}(f^{-1}\\mathcal{G}, \\mathcal{F})", "=", "\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{G}, f_*\\mathcal{F})", "$$", "bifunctorially, and", "\\item the functor $f^{-1}$ commutes with finite limits, i.e.,", "is left exact.", "\\end{enumerate}", "\\item Let $\\mathcal{C}$, $\\mathcal{D}$, $\\mathcal{E}$ be sites.", "Given morphisms of topoi", "$f :\\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ and", "$g :\\Sh(\\mathcal{E}) \\to \\Sh(\\mathcal{D})$ the", "{\\it composition $f\\circ g$} is the morphism of topoi defined", "by the functors", "$(f \\circ g)_* = f_* \\circ g_*$ and", "$(f \\circ g)^{-1} = g^{-1} \\circ f^{-1}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8668, "type": "definition", "label": "sites-definition-quasi-compact", "categories": [ "sites" ], "title": "sites-definition-quasi-compact", "contents": [ "Let $\\mathcal{C}$ be a site. An object $U$ of $\\mathcal{C}$ is", "{\\it quasi-compact} if given a covering $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$", "in $\\mathcal{C}$ there exists another covering", "$\\mathcal{V} = \\{V_j \\to U\\}_{j \\in J}$ and a morphism", "$\\mathcal{V} \\to \\mathcal{U}$ of families of maps with fixed target", "given by $\\text{id} : U \\to U$, $\\alpha : J \\to I$, and $V_j \\to U_{\\alpha(j)}$", "(see Definition \\ref{definition-morphism-coverings})", "such that the image of $\\alpha$ is a finite subset of $I$." ], "refs": [ "sites-definition-morphism-coverings" ], "ref_ids": [ 8656 ] }, { "id": 8669, "type": "definition", "label": "sites-definition-quasi-compact-topos", "categories": [ "sites" ], "title": "sites-definition-quasi-compact-topos", "contents": [ "An object $\\mathcal{F}$ of a topos $\\Sh(\\mathcal{C})$ is {\\it quasi-compact}", "if for any surjective map $\\coprod_{i \\in I} \\mathcal{F}_i \\to \\mathcal{F}$", "of $\\Sh(\\mathcal{C})$ there exists a finite subset $J \\subset I$ such", "that $\\coprod_{i \\in J} \\mathcal{F}_i \\to \\mathcal{F}$ is surjective.", "A topos $\\Sh(\\mathcal{C})$ is said to be {\\it quasi-compact}", "if its final object $*$ is a quasi-compact object." ], "refs": [], "ref_ids": [] }, { "id": 8670, "type": "definition", "label": "sites-definition-cocontinuous", "categories": [ "sites" ], "title": "sites-definition-cocontinuous", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "The functor $u$ is called {\\it cocontinuous}", "if for every $U \\in \\Ob(\\mathcal{C})$", "and every covering $\\{V_j \\to u(U)\\}_{j \\in J}$ of $\\mathcal{D}$", "there exists a covering", "$\\{U_i \\to U\\}_{i\\in I}$ of $\\mathcal{C}$", "such that the family of maps $\\{u(U_i) \\to u(U)\\}_{i \\in I}$", "refines the covering $\\{V_j \\to u(U)\\}_{j \\in J}$." ], "refs": [], "ref_ids": [] }, { "id": 8671, "type": "definition", "label": "sites-definition-localize", "categories": [ "sites" ], "title": "sites-definition-localize", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $U \\in \\Ob(\\mathcal{C})$.", "\\begin{enumerate}", "\\item The site $\\mathcal{C}/U$ is called the {\\it localization of", "the site $\\mathcal{C}$ at the object $U$}.", "\\item The morphism of topoi", "$j_U : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})$", "is called the {\\it localization morphism}.", "\\item The functor $j_{U*}$ is called the {\\it direct image functor}.", "\\item For a sheaf $\\mathcal{F}$ on $\\mathcal{C}$ the sheaf", "$j_U^{-1}\\mathcal{F}$ is called the {\\it restriction of $\\mathcal{F}$", "to $\\mathcal{C}/U$}.", "\\item For a sheaf $\\mathcal{G}$ on $\\mathcal{C}/U$", "the sheaf $j_{U!}\\mathcal{G}$ is called the", "{\\it extension of $\\mathcal{G}$ by the empty set}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8672, "type": "definition", "label": "sites-definition-special-cocontinuous-functor", "categories": [ "sites" ], "title": "sites-definition-special-cocontinuous-functor", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "A {\\it special cocontinuous functor $u$ from $\\mathcal{C}$ to $\\mathcal{D}$}", "is a cocontinuous functor $u : \\mathcal{C} \\to \\mathcal{D}$ satisfying", "the assumptions and conclusions of Lemma \\ref{lemma-equivalence}." ], "refs": [ "sites-lemma-equivalence" ], "ref_ids": [ 8578 ] }, { "id": 8673, "type": "definition", "label": "sites-definition-localize-topos", "categories": [ "sites" ], "title": "sites-definition-localize-topos", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.", "\\begin{enumerate}", "\\item The topos $\\Sh(\\mathcal{C})/\\mathcal{F}$", "is called the", "{\\it localization of the topos $\\Sh(\\mathcal{C})$ at $\\mathcal{F}$}.", "\\item The morphism of topoi", "$j_\\mathcal{F} :", "\\Sh(\\mathcal{C})/\\mathcal{F}", "\\to", "\\Sh(\\mathcal{C})$ of", "Lemma \\ref{lemma-localize-topos}", "is called the {\\it localization morphism}.", "\\end{enumerate}" ], "refs": [ "sites-lemma-localize-topos" ], "ref_ids": [ 8583 ] }, { "id": 8674, "type": "definition", "label": "sites-definition-point-topos", "categories": [ "sites" ], "title": "sites-definition-point-topos", "contents": [ "Let $\\mathcal{C}$ be a site.", "A {\\it point of the topos $\\Sh(\\mathcal{C})$}", "is a morphism of topoi $p$ from $\\Sh(pt)$ to", "$\\Sh(\\mathcal{C})$." ], "refs": [], "ref_ids": [] }, { "id": 8675, "type": "definition", "label": "sites-definition-point", "categories": [ "sites" ], "title": "sites-definition-point", "contents": [ "Let $\\mathcal{C}$ be a site. A {\\it point $p$ of the site", "$\\mathcal{C}$} is given by a functor $u : \\mathcal{C}", "\\to \\textit{Sets}$ such that", "\\begin{enumerate}", "\\item For every covering $\\{U_i \\to U\\}$ of $\\mathcal{C}$ the map", "$\\coprod u(U_i) \\to u(U)$ is surjective.", "\\item For every covering $\\{U_i \\to U\\}$ of $\\mathcal{C}$ and", "every morphism $V \\to U$ the maps", "$u(U_i \\times_U V) \\to u(U_i) \\times_{u(U)} u(V)$ are bijective.", "\\item The stalk functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$,", "$\\mathcal{F} \\mapsto \\mathcal{F}_p$ is left exact.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8676, "type": "definition", "label": "sites-definition-pushforward-point", "categories": [ "sites" ], "title": "sites-definition-pushforward-point", "contents": [ "Let $p$ be a point of the site $\\mathcal{C}$ given by the functor $u$.", "For a set $E$ we define $p_*E = u^sE$ the sheaf", "described in Lemma \\ref{lemma-point-pushforward-sheaf} above.", "We sometimes call this a {\\it skyscraper sheaf}." ], "refs": [ "sites-lemma-point-pushforward-sheaf" ], "ref_ids": [ 8595 ] }, { "id": 8677, "type": "definition", "label": "sites-definition-2-morphism-topoi", "categories": [ "sites" ], "title": "sites-definition-2-morphism-topoi", "contents": [ "Let $f, g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "be two morphisms of topoi. A {\\it 2-morphism from $f$ to $g$}", "is given by a transformation of functors $t : f_* \\to g_*$." ], "refs": [], "ref_ids": [] }, { "id": 8678, "type": "definition", "label": "sites-definition-morphism-points", "categories": [ "sites" ], "title": "sites-definition-morphism-points", "contents": [ "Let $\\mathcal{C}$ be a site. Let $p, p'$ be points of $\\mathcal{C}$", "given by functors $u, u' : \\mathcal{C} \\to \\textit{Sets}$.", "A {\\it morphism $f : p \\to p'$} is given by a transformation of", "functors", "$$", "f_u : u' \\to u.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 8679, "type": "definition", "label": "sites-definition-enough-points", "categories": [ "sites" ], "title": "sites-definition-enough-points", "contents": [ "Let $\\mathcal{C}$ be a site.", "\\begin{enumerate}", "\\item A family of points $\\{p_i\\}_{i\\in I}$ is called {\\it conservative}", "if every map of sheaves $\\phi : \\mathcal{F} \\to \\mathcal{G}$", "which is an isomorphism on all the fibres $\\mathcal{F}_{p_i}", "\\to \\mathcal{G}_{p_i}$ is an isomorphism.", "\\item We say that $\\mathcal{C}$ {\\it has enough points}", "if there exists a conservative family of points.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8680, "type": "definition", "label": "sites-definition-w-contractible", "categories": [ "sites" ], "title": "sites-definition-w-contractible", "contents": [ "Let $\\mathcal{C}$ be a site.", "\\begin{enumerate}", "\\item We say an object $U$ of $\\mathcal{C}$ is {\\it weakly contractible}", "if the equivalent conditions of Lemma \\ref{lemma-w-contractible} hold.", "\\item We say a site has {\\it enough weakly contractible objects}", "if every object $U$ of $\\mathcal{C}$ has a covering $\\{U_i \\to U\\}$", "with $U_i$ weakly contractible for all $i$.", "\\item More generally, if $P$ is a property of objects of $\\mathcal{C}$", "we say that $\\mathcal{C}$ has {\\it enough $P$ objects} if every object $U$ of", "$\\mathcal{C}$ has a covering $\\{U_i \\to U\\}$ such that $U_i$ has $P$", "for all $i$.", "\\end{enumerate}" ], "refs": [ "sites-lemma-w-contractible" ], "ref_ids": [ 8617 ] }, { "id": 8681, "type": "definition", "label": "sites-definition-empty", "categories": [ "sites" ], "title": "sites-definition-empty", "contents": [ "Let $\\mathcal{C}$ be a site. We say an object $U$ of $\\mathcal{C}$", "is {\\it sheaf theoretically empty} if $\\emptyset^\\# \\to h_U^\\#$", "is an isomorphism of sheaves." ], "refs": [], "ref_ids": [] }, { "id": 8682, "type": "definition", "label": "sites-definition-almost-cocontinuous", "categories": [ "sites" ], "title": "sites-definition-almost-cocontinuous", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "We say $u$ is {\\it almost cocontinuous} if for every", "object $U$ of $\\mathcal{C}$ and every covering", "$\\{V_j \\to u(U)\\}_{j \\in J}$ there exists a covering", "$\\{U_i \\to U\\}_{i \\in I}$ in $\\mathcal{C}$ such that", "for each $i$ in $I$ we have at least one of the following two conditions", "\\begin{enumerate}", "\\item $u(U_i)$ is sheaf theoretically empty, or", "\\item the morphism $u(U_i) \\to u(U)$ factors through $V_j$ for some $j \\in J$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8683, "type": "definition", "label": "sites-definition-embedding", "categories": [ "sites" ], "title": "sites-definition-embedding", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "A morphism of topoi $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$", "is called an {\\it embedding} if $f_*$ is fully faithful." ], "refs": [], "ref_ids": [] }, { "id": 8684, "type": "definition", "label": "sites-definition-subtopos", "categories": [ "sites" ], "title": "sites-definition-subtopos", "contents": [ "Let $\\mathcal{C}$ be a site. A strictly full subcategory", "$E \\subset \\Sh(\\mathcal{C})$ is a {\\it subtopos} if there", "exists an embedding of topoi $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$", "such that $E$ is equal to the essential image of the functor $f_*$." ], "refs": [], "ref_ids": [] }, { "id": 8685, "type": "definition", "label": "sites-definition-open-subtopos", "categories": [ "sites" ], "title": "sites-definition-open-subtopos", "contents": [ "Let $\\mathcal{C}$ be a site. A strictly full subcategory", "$E \\subset \\Sh(\\mathcal{C})$ is an {\\it open subtopos}", "if there exists a subsheaf $\\mathcal{F}$ of the final object", "of $\\Sh(\\mathcal{C})$ such that $E$ is the subtopos", "$\\Sh(\\mathcal{C})/\\mathcal{F}$ described in Lemma \\ref{lemma-open-subtopos}." ], "refs": [ "sites-lemma-open-subtopos" ], "ref_ids": [ 8625 ] }, { "id": 8686, "type": "definition", "label": "sites-definition-closed-subtopos", "categories": [ "sites" ], "title": "sites-definition-closed-subtopos", "contents": [ "Let $\\mathcal{C}$ be a site. A strictly full subcategory", "$E \\subset \\Sh(\\mathcal{C})$ is an {\\it closed subtopos}", "if there exists a subsheaf $\\mathcal{F}$ of the final object", "of $\\Sh(\\mathcal{C})$ such that $E$ is the subtopos", "described in Lemma \\ref{lemma-closed-subtopos}." ], "refs": [ "sites-lemma-closed-subtopos" ], "ref_ids": [ 8626 ] }, { "id": 8687, "type": "definition", "label": "sites-definition-immersion-topoi", "categories": [ "sites" ], "title": "sites-definition-immersion-topoi", "contents": [ "Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi.", "\\begin{enumerate}", "\\item We say $f$ is an {\\it open immersion} if $f$ is an embedding", "and the essential image of $f_*$ is an open subtopos.", "\\item We say $f$ is a {\\it closed immersion} if $f$ is an embedding", "and the essential image of $f_*$ is a closed subtopos.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8688, "type": "definition", "label": "sites-definition-pushforward-algebraic-structures", "categories": [ "sites" ], "title": "sites-definition-pushforward-algebraic-structures", "contents": [ "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites", "given by a functor $u : \\mathcal{C} \\to \\mathcal{D}$.", "We define the {\\it pushforward} functor for presheaves of algebraic structures", "by the rule $u^p\\mathcal{F}(U) = \\mathcal{F}(uU)$,", "and for sheaves of algebraic structures by the same rule, namely", "$f_*\\mathcal{F}(U) = \\mathcal{F}(uU)$." ], "refs": [], "ref_ids": [] }, { "id": 8689, "type": "definition", "label": "sites-definition-global-sections", "categories": [ "sites" ], "title": "sites-definition-global-sections", "contents": [ "The {\\it global sections} of a presheaf of sets $\\mathcal{F}$ over a", "site $\\mathcal{C}$ is the set", "$$", "\\Gamma(\\mathcal{C}, \\mathcal{F}) =", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(*, \\mathcal{F})", "$$", "where $*$ is the final object in the category of presheaves on", "$\\mathcal{C}$, i.e., the presheaf which associates to every object", "a singleton." ], "refs": [], "ref_ids": [] }, { "id": 8690, "type": "definition", "label": "sites-definition-sieve", "categories": [ "sites" ], "title": "sites-definition-sieve", "contents": [ "Let $\\mathcal{C}$ be a category. Let $U \\in \\Ob(\\mathcal{C})$.", "A {\\it sieve $S$ on $U$} is a subpresheaf $S \\subset h_U$." ], "refs": [], "ref_ids": [] }, { "id": 8691, "type": "definition", "label": "sites-definition-sieve-generated", "categories": [ "sites" ], "title": "sites-definition-sieve-generated", "contents": [ "Let $\\mathcal{C}$ be a category.", "Given a family of morphisms $\\{f_i : U_i \\to U\\}_{i\\in I}$", "of $\\mathcal{C}$ with target $U$ we say the sieve", "$S$ on $U$ described in Lemma \\ref{lemma-sieves-set}", "part (\\ref{item-sieve-generated}) is the {\\it sieve on $U$", "generated by the morphisms $f_i$}." ], "refs": [ "sites-lemma-sieves-set" ], "ref_ids": [ 8630 ] }, { "id": 8692, "type": "definition", "label": "sites-definition-pullback-sieve", "categories": [ "sites" ], "title": "sites-definition-pullback-sieve", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $f : V \\to U$ be a morphism of $\\mathcal{C}$.", "Let $S \\subset h_U$ be a sieve. We define the", "{\\it pullback of $S$ by $f$} to be the sieve", "$S \\times_U V$ of $V$ defined by the rule", "$$", "(\\alpha : T \\to V) \\in (S \\times_U V)(T)", "\\Leftrightarrow", "(f \\circ \\alpha : T \\to U) \\in S(T)", "$$" ], "refs": [], "ref_ids": [] }, { "id": 8693, "type": "definition", "label": "sites-definition-topology", "categories": [ "sites" ], "title": "sites-definition-topology", "contents": [ "Let $\\mathcal{C}$ be a category. A {\\it topology on $\\mathcal{C}$} is given", "by a rule which assigns to every $U \\in \\Ob(\\mathcal{C})$", "a subset $J(U)$ of the set of all sieves on $U$ satisfying", "the following conditions", "\\begin{enumerate}", "\\item For every morphism $f : V \\to U$ in $\\mathcal{C}$, and", "every element $S \\in J(U)$ the pullback $S \\times_U V$", "is an element of $J(V)$.", "\\item If $S$ and $S'$ are sieves on $U \\in \\Ob(\\mathcal{C})$,", "if $S \\in J(U)$, and if for all $f \\in S(V)$ the pullback", "$S' \\times_U V$ belongs to $J(V)$, then $S'$ belongs to $J(U)$.", "\\item For every $U \\in \\Ob(\\mathcal{C})$ the", "maximal sieve $S = h_U$ belongs to $J(U)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8694, "type": "definition", "label": "sites-definition-finer", "categories": [ "sites" ], "title": "sites-definition-finer", "contents": [ "Let $\\mathcal{C}$ be a category. Let $J$, $J'$ be", "two topologies on $\\mathcal{C}$. We say that $J$ is", "{\\it finer} or {\\it stronger} than $J'$ if and only if for every object", "$U$ of $\\mathcal{C}$ we have $J'(U) \\subset J(U)$.", "In this case we also say that $J'$ is", "{\\it coarser} or {\\it weaker} than $J$." ], "refs": [], "ref_ids": [] }, { "id": 8695, "type": "definition", "label": "sites-definition-sheaf-sets-topology", "categories": [ "sites" ], "title": "sites-definition-sheaf-sets-topology", "contents": [ "Let $\\mathcal{C}$ be a category endowed with a", "topology $J$. Let $\\mathcal{F}$ be a presheaf of sets", "on $\\mathcal{C}$.", "We say that $\\mathcal{F}$ is a", "{\\it sheaf} on $\\mathcal{C}$", "if for every $U \\in \\Ob(\\mathcal{C})$ and for", "every covering sieve $S$ of $U$ the canonical map", "$$", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(h_U, \\mathcal{F})", "\\longrightarrow", "\\Mor_{\\textit{PSh}(\\mathcal{C})}(S, \\mathcal{F})", "$$", "is bijective." ], "refs": [], "ref_ids": [] }, { "id": 8696, "type": "definition", "label": "sites-definition-canonical-topology", "categories": [ "sites" ], "title": "sites-definition-canonical-topology", "contents": [ "Let $\\mathcal{C}$ be a category.", "The finest topology on $\\mathcal{C}$ such that", "all representable presheaves are sheaves, see", "Lemma \\ref{lemma-topology-presheaves-sheaves},", "is called the {\\it canonical topology} of $\\mathcal{C}$." ], "refs": [ "sites-lemma-topology-presheaves-sheaves" ], "ref_ids": [ 8634 ] }, { "id": 8697, "type": "definition", "label": "sites-definition-topology-associated-site", "categories": [ "sites" ], "title": "sites-definition-topology-associated-site", "contents": [ "Let $\\mathcal{C}$ be a site with coverings $\\text{Cov}(\\mathcal{C})$.", "The {\\it topology associated to $\\mathcal{C}$} is the topology", "$J$ constructed in Lemma \\ref{lemma-site-gives-topology} above." ], "refs": [ "sites-lemma-site-gives-topology" ], "ref_ids": [ 8635 ] }, { "id": 8698, "type": "definition", "label": "sites-definition-presheaf-separated-topology", "categories": [ "sites" ], "title": "sites-definition-presheaf-separated-topology", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $J$ be a topology on $\\mathcal{C}$.", "We say that a presheaf of sets $\\mathcal{F}$", "is {\\it separated} if for every object $U$ and", "every covering sieve $S$ on $U$ the canonical map", "$\\mathcal{F}(U) \\to \\Mor_{\\textit{PSh}(\\mathcal{C})}(S, \\mathcal{F})$", "is injective." ], "refs": [], "ref_ids": [] }, { "id": 8699, "type": "definition", "label": "sites-definition-associated-sheaf-topology", "categories": [ "sites" ], "title": "sites-definition-associated-sheaf-topology", "contents": [ "Let $\\mathcal{C}$ be a category endowed with a topology $J$.", "Let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{C}$.", "The sheaf $\\mathcal{F}^\\# := LL\\mathcal{F}$", "together with the canonical map $\\mathcal{F} \\to \\mathcal{F}^\\#$", "is called the {\\it sheaf associated to $\\mathcal{F}$}." ], "refs": [], "ref_ids": [] }, { "id": 8700, "type": "definition", "label": "sites-definition-point-topology", "categories": [ "sites" ], "title": "sites-definition-point-topology", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $J$ be a topology on $\\mathcal{C}$.", "A {\\it point $p$} of the topology is given by a functor", "$u : \\mathcal{C} \\to \\textit{Sets}$ such that", "\\begin{enumerate}", "\\item For every covering sieve $S$ on $U$ the map", "$S_p \\to (h_U)_p$ is surjective.", "\\item The stalk functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$,", "$\\mathcal{F} \\to \\mathcal{F}_p$ is exact.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8846, "type": "definition", "label": "more-etale-definition-f-shriek-separated", "categories": [ "more-etale" ], "title": "more-etale-definition-f-shriek-separated", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes which is separated (!) and", "locally of finite type. Let $\\mathcal{F}$ be an abelian sheaf on", "$X_\\etale$. The subsheaf $f_!\\mathcal{F} \\subset f_*\\mathcal{F}$", "constructed in Lemma \\ref{lemma-f-shriek-separated} is called the", "{\\it direct image with compact support}." ], "refs": [ "more-etale-lemma-f-shriek-separated" ], "ref_ids": [ 8808 ] }, { "id": 8847, "type": "definition", "label": "more-etale-definition-compact-support", "categories": [ "more-etale" ], "title": "more-etale-definition-compact-support", "contents": [ "Let $X$ be a separated scheme locally of finite type over a field $k$.", "Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. We let", "$H^0_c(X, \\mathcal{F}) \\subset H^0(X, \\mathcal{F})$ be the", "set of sections whose support is proper over $k$. Elements of", "$H^0_c(X, \\mathcal{F})$ are called {\\it sections with compact support}." ], "refs": [], "ref_ids": [] }, { "id": 8848, "type": "definition", "label": "more-etale-definition-f-shriek-lqf", "categories": [ "more-etale" ], "title": "more-etale-definition-f-shriek-lqf", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes.", "We define the {\\it direct image with compact support} to be the", "functor", "$$", "f_! : \\textit{Ab}(X_\\etale) \\longrightarrow \\textit{Ab}(Y_\\etale)", "$$", "defined by the formula $f_!\\mathcal{F} = (f_{p!}\\mathcal{F})^\\#$,", "i.e., $f_!\\mathcal{F}$ is the sheafification of the presheaf", "$f_{p!}\\mathcal{F}$ constructed above." ], "refs": [], "ref_ids": [] }, { "id": 8913, "type": "definition", "label": "stacks-properties-definition-points", "categories": [ "stacks-properties" ], "title": "stacks-properties-definition-points", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "A {\\it point} of $\\mathcal{X}$ is an equivalence class of morphisms", "from spectra of fields into $\\mathcal{X}$.", "The set of points of $\\mathcal{X}$ is denoted $|\\mathcal{X}|$." ], "refs": [], "ref_ids": [] }, { "id": 8914, "type": "definition", "label": "stacks-properties-definition-topological-space", "categories": [ "stacks-properties" ], "title": "stacks-properties-definition-topological-space", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "The underlying {\\it topological space} of $\\mathcal{X}$ is the set of points", "$|\\mathcal{X}|$ endowed with the topology constructed in", "Lemma \\ref{lemma-topology-points}." ], "refs": [ "stacks-properties-lemma-topology-points" ], "ref_ids": [ 8867 ] }, { "id": 8915, "type": "definition", "label": "stacks-properties-definition-surjective", "categories": [ "stacks-properties" ], "title": "stacks-properties-definition-surjective", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $f$ is {\\it surjective} if the map", "$|f| : |\\mathcal{X}| \\to |\\mathcal{Y}|$ of associated topological spaces", "is surjective." ], "refs": [], "ref_ids": [] }, { "id": 8916, "type": "definition", "label": "stacks-properties-definition-quasi-compact", "categories": [ "stacks-properties" ], "title": "stacks-properties-definition-quasi-compact", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "We say $\\mathcal{X}$ is {\\it quasi-compact}", "if and only if $|\\mathcal{X}|$ is quasi-compact." ], "refs": [], "ref_ids": [] }, { "id": 8917, "type": "definition", "label": "stacks-properties-definition-type-property", "categories": [ "stacks-properties" ], "title": "stacks-properties-definition-type-property", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Let $\\mathcal{P}$ be a property of schemes which is", "local in the smooth topology.", "We say $\\mathcal{X}$ {\\it has property $\\mathcal{P}$}", "if any of the equivalent conditions of", "Lemma \\ref{lemma-type-property}", "hold." ], "refs": [ "stacks-properties-lemma-type-property" ], "ref_ids": [ 8875 ] }, { "id": 8918, "type": "definition", "label": "stacks-properties-definition-property-at-point", "categories": [ "stacks-properties" ], "title": "stacks-properties-definition-property-at-point", "contents": [ "Let $\\mathcal{P}$ be a property of germs of schemes which is", "smooth local. Let $\\mathcal{X}$ be an algebraic stack.", "Let $x \\in |\\mathcal{X}|$.", "We say $\\mathcal{X}$ {\\it has property $\\mathcal{P}$ at $x$}", "if any of the equivalent conditions of", "Lemma \\ref{lemma-local-source-target-at-point}", "holds." ], "refs": [ "stacks-properties-lemma-local-source-target-at-point" ], "ref_ids": [ 8876 ] }, { "id": 8919, "type": "definition", "label": "stacks-properties-definition-monomorphism", "categories": [ "stacks-properties" ], "title": "stacks-properties-definition-monomorphism", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "We say $f$ is a {\\it monomorphism}", "if it is representable by algebraic spaces and a monomorphism in the sense of", "Section \\ref{section-properties-morphisms}." ], "refs": [], "ref_ids": [] }, { "id": 8920, "type": "definition", "label": "stacks-properties-definition-immersion", "categories": [ "stacks-properties" ], "title": "stacks-properties-definition-immersion", "contents": [ "Immersions.", "\\begin{enumerate}", "\\item A morphism of algebraic stacks is called an {\\it open immersion}", "if it is representable, and an open immersion", "in the sense of", "Section \\ref{section-properties-morphisms}.", "\\item A morphism of algebraic stacks is called a {\\it closed immersion}", "if it is representable, and a closed immersion", "in the sense of", "Section \\ref{section-properties-morphisms}.", "\\item A morphism of algebraic stacks is called an {\\it immersion}", "if it is representable, and an immersion", "in the sense of", "Section \\ref{section-properties-morphisms}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8921, "type": "definition", "label": "stacks-properties-definition-substacks", "categories": [ "stacks-properties" ], "title": "stacks-properties-definition-substacks", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "\\begin{enumerate}", "\\item An {\\it open substack} of $\\mathcal{X}$ is a strictly full subcategory", "$\\mathcal{X}' \\subset \\mathcal{X}$ such that $\\mathcal{X}'$ is an algebraic", "stack and $\\mathcal{X}' \\to \\mathcal{X}$ is an open immersion.", "\\item A {\\it closed substack} of $\\mathcal{X}$ is a strictly full subcategory", "$\\mathcal{X}' \\subset \\mathcal{X}$ such that $\\mathcal{X}'$ is an algebraic", "stack and $\\mathcal{X}' \\to \\mathcal{X}$ is a closed immersion.", "\\item A {\\it locally closed substack} of $\\mathcal{X}$ is a strictly full", "subcategory $\\mathcal{X}' \\subset \\mathcal{X}$ such that $\\mathcal{X}'$", "is an algebraic stack and $\\mathcal{X}' \\to \\mathcal{X}$ is an immersion.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 8922, "type": "definition", "label": "stacks-properties-definition-reduced-induced-stack", "categories": [ "stacks-properties" ], "title": "stacks-properties-definition-reduced-induced-stack", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack.", "Let $Z \\subset |\\mathcal{X}|$ be a closed subset.", "An {\\it algebraic stack structure on $Z$} is given by a closed substack", "$\\mathcal{Z}$ of $\\mathcal{X}$ with $|\\mathcal{Z}|$ equal to $Z$.", "The {\\it reduced induced algebraic stack structure}", "on $Z$ is the one constructed in", "Lemma \\ref{lemma-reduced-closed-substack}.", "The {\\it reduction $\\mathcal{X}_{red}$ of $\\mathcal{X}$}", "is the reduced induced algebraic stack structure on $|\\mathcal{X}|$." ], "refs": [ "stacks-properties-lemma-reduced-closed-substack" ], "ref_ids": [ 8897 ] }, { "id": 8923, "type": "definition", "label": "stacks-properties-definition-residual-gerbe", "categories": [ "stacks-properties" ], "title": "stacks-properties-definition-residual-gerbe", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$.", "\\begin{enumerate}", "\\item We say the {\\it residual gerbe of $\\mathcal{X}$ at $x$ exists}", "if the equivalent conditions (1), (2), and (3) of", "Lemma \\ref{lemma-residual-gerbe}", "hold.", "\\item If the residual gerbe of $\\mathcal{X}$ at $x$ exists, then the", "{\\it residual gerbe of $\\mathcal{X}$ at $x$}\\footnote{This clashes with", "\\cite{LM-B} in spirit, but not in fact. Namely, in Chapter 11 they associate", "to any point on any quasi-separated algebraic stack a gerbe (not necessarily", "algebraic) which they call the residual gerbe. We will see in", "Morphisms of Stacks, Lemma", "\\ref{stacks-morphisms-lemma-every-point-residual-gerbe}", "that on a quasi-separated algebraic stack every point", "has a residual gerbe in our sense which is then equivalent to theirs. For", "more information on this topic see", "\\cite[Appendix B]{rydh_etale_devissage}.}", "is the strictly full", "subcategory $\\mathcal{Z}_x \\subset \\mathcal{X}$ constructed in", "Lemma \\ref{lemma-residual-gerbe}.", "\\end{enumerate}" ], "refs": [ "stacks-properties-lemma-residual-gerbe", "stacks-morphisms-lemma-every-point-residual-gerbe", "stacks-properties-lemma-residual-gerbe" ], "ref_ids": [ 8905, 7532, 8905 ] }, { "id": 8924, "type": "definition", "label": "stacks-properties-definition-dimension-at-point", "categories": [ "stacks-properties" ], "title": "stacks-properties-definition-dimension-at-point", "contents": [ "Let $\\mathcal{X}$ be a locally Noetherian algebraic stack over a scheme $S$.", "Let $x \\in |\\mathcal{X}|$ be a point of $\\mathcal{X}$.", "Let $[U/R] \\to \\mathcal{X}$ be a presentation", "(Algebraic Stacks, Definition \\ref{algebraic-definition-presentation})", "where $U$ is a scheme", "and let $u \\in U$ be a point that maps to $x$.", "We define the {\\it dimension of $\\mathcal{X}$ at $x$} to be", "the element $\\dim_x(\\mathcal{X}) \\in \\mathbf{Z} \\cup \\infty$", "such that ", "$$", "\\dim_x(\\mathcal{X}) = \\dim_u(U)-\\dim_{e(u)}(R_u).", "$$", "with notation as in Lemma \\ref{lemma-dimension-at-point-well-defined}." ], "refs": [ "algebraic-definition-presentation", "stacks-properties-lemma-dimension-at-point-well-defined" ], "ref_ids": [ 8488, 8911 ] }, { "id": 8925, "type": "definition", "label": "stacks-properties-definition-dimension", "categories": [ "stacks-properties" ], "title": "stacks-properties-definition-dimension", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X}$ be", "a locally Noetherian algebraic stack over $S$.", "The {\\it dimension} $\\dim(\\mathcal{X})$ of $\\mathcal{X}$ is defined to be", "$$", "\\dim(\\mathcal{X}) = \\sup\\nolimits_{x \\in |\\mathcal{X}|} \\dim_x(\\mathcal{X})", "$$" ], "refs": [], "ref_ids": [] }, { "id": 8926, "type": "definition", "label": "stacks-properties-definition-number-of-geometric-branches", "categories": [ "stacks-properties" ], "title": "stacks-properties-definition-number-of-geometric-branches", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$.", "\\begin{enumerate}", "\\item The {\\it number of geometric branches of $\\mathcal{X}$ at $x$}", "is either $n \\in \\mathbf{N}$ if the equivalent conditions of", "Lemma \\ref{lemma-local-source-target-at-point} hold for", "$\\mathcal{P}_n$ defined above, or else $\\infty$.", "\\item We say $\\mathcal{X}$ is {\\it geometrically unibranch at $x$}", "if the number of geometric branches of $\\mathcal{X}$ at $x$ is $1$.", "\\end{enumerate}" ], "refs": [ "stacks-properties-lemma-local-source-target-at-point" ], "ref_ids": [ 8876 ] }, { "id": 8992, "type": "definition", "label": "stacks-definition-mor-presheaf", "categories": [ "stacks" ], "title": "stacks-definition-mor-presheaf", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category,", "see Categories, Section \\ref{categories-section-fibred-categories}.", "Given an object $U$ of $\\mathcal{C}$ and objects", "$x$, $y$ of the fibre category, the {\\it presheaf", "of morphisms from $x$ to $y$} is the presheaf", "$$", "(f : V \\to U) \\longmapsto \\Mor_{\\mathcal{S}_V}(f^*x, f^*y)", "$$", "described above. It is denoted $\\mathit{Mor}(x, y)$.", "The subpresheaf $\\mathit{Isom}(x, y)$ whose values", "over $V$ is the set of isomorphisms", "$f^*x \\to f^*y$ in the fibre category $\\mathcal{S}_V$", "is called the {\\it presheaf of isomorphisms from $x$ to $y$}." ], "refs": [], "ref_ids": [] }, { "id": 8993, "type": "definition", "label": "stacks-definition-descent-data", "categories": [ "stacks" ], "title": "stacks-definition-descent-data", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category.", "Make a choice of pullbacks as in Categories,", "Definition \\ref{categories-definition-pullback-functor-fibred-category}.", "Let $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$", "be a family of morphisms of $\\mathcal{C}$. Assume all the fibre products", "$U_i \\times_U U_j$, and $U_i \\times_U U_j \\times_U U_k$ exist.", "\\begin{enumerate}", "\\item A {\\it descent datum $(X_i, \\varphi_{ij})$ in $\\mathcal{S}$", "relative to the family $\\{f_i : U_i \\to U\\}$} is given by an object $X_i$", "of $\\mathcal{S}_{U_i}$ for each $i \\in I$, an isomorphism", "$\\varphi_{ij} : \\text{pr}_0^*X_i \\to \\text{pr}_1^*X_j$", "in $\\mathcal{S}_{U_i \\times_U U_j}$ for each pair $(i, j) \\in I^2$", "such that for every triple of indices $(i, j, k) \\in I^3$ the", "diagram", "$$", "\\xymatrix{", "\\text{pr}_0^*X_i \\ar[rd]_{\\text{pr}_{01}^*\\varphi_{ij}}", "\\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} & &", "\\text{pr}_2^*X_k \\\\", "& \\text{pr}_1^*X_j \\ar[ru]_{\\text{pr}_{12}^*\\varphi_{jk}} &", "}", "$$", "in the category $\\mathcal{S}_{U_i \\times_U U_j \\times_U U_k}$", "commutes. This is called the {\\it cocycle condition}.", "\\item A {\\it morphism $\\psi : (X_i, \\varphi_{ij}) \\to", "(X'_i, \\varphi'_{ij})$ of descent data} is given", "by a family $\\psi = (\\psi_i)_{i\\in I}$ of morphisms", "$\\psi_i : X_i \\to X'_i$ in $\\mathcal{S}_{U_i}$", "such that all the diagrams", "$$", "\\xymatrix{", "\\text{pr}_0^*X_i \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\text{pr}_0^*\\psi_i}", "& \\text{pr}_1^*X_j \\ar[d]^{\\text{pr}_1^*\\psi_j} \\\\", "\\text{pr}_0^*X'_i \\ar[r]^{\\varphi'_{ij}} &", "\\text{pr}_1^*X'_j \\\\", "}", "$$", "in the categories $\\mathcal{S}_{U_i \\times_U U_j}$ commute.", "\\item The category of descent data relative to", "$\\mathcal{U}$ is denoted $DD(\\mathcal{U})$.", "\\end{enumerate}" ], "refs": [ "categories-definition-pullback-functor-fibred-category" ], "ref_ids": [ 12389 ] }, { "id": 8994, "type": "definition", "label": "stacks-definition-pullback-functor", "categories": [ "stacks" ], "title": "stacks-definition-pullback-functor", "contents": [ "With $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$,", "$\\mathcal{V} = \\{V_j \\to V\\}_{j \\in J}$,", "$\\alpha : I \\to J$, $h : U \\to V$,", "and $g_i : U_i \\to V_{\\alpha(i)}$ as in Lemma \\ref{lemma-pullback}", "the functor", "$$", "(Y_j, \\varphi_{jj'}) \\longmapsto", "(g_i^*Y_{\\alpha(i)}, (g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')})", "$$", "constructed in that lemma", "is called the {\\it pullback functor} on descent data." ], "refs": [ "stacks-lemma-pullback" ], "ref_ids": [ 8937 ] }, { "id": 8995, "type": "definition", "label": "stacks-definition-effective-descent-datum", "categories": [ "stacks" ], "title": "stacks-definition-effective-descent-datum", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category.", "Make a choice of pullbacks as in Categories,", "Definition \\ref{categories-definition-pullback-functor-fibred-category}.", "Let $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ be a family of morphisms", "with target $U$. Assume all the fibre products", "$U_i \\times_U U_j$ and $U_i \\times_U U_j \\times_U U_k$ exist.", "\\begin{enumerate}", "\\item Given an object $X$ of $\\mathcal{S}_U$ the {\\it trivial descent datum}", "is the descent datum $(X, \\text{id}_X)$ with respect to the family", "$\\{\\text{id}_U : U \\to U\\}$.", "\\item Given an object $X$ of $\\mathcal{S}_U$", "we have a {\\it canonical descent datum} on the family of", "objects $f_i^*X$ by pulling back the trivial", "descent datum $(X, \\text{id}_X)$ via the", "obvious map $\\{f_i : U_i \\to U\\} \\to \\{\\text{id}_U : U \\to U\\}$.", "We denote this descent datum $(f_i^*X, can)$.", "\\item A descent datum $(X_i, \\varphi_{ij})$", "relative to $\\{f_i : U_i \\to U\\}$ is called {\\it effective}", "if there exists an object $X$ of $\\mathcal{S}_U$ such that", "$(X_i, \\varphi_{ij})$ is isomorphic to $(f_i^*X, can)$.", "\\end{enumerate}" ], "refs": [ "categories-definition-pullback-functor-fibred-category" ], "ref_ids": [ 12389 ] }, { "id": 8996, "type": "definition", "label": "stacks-definition-stack", "categories": [ "stacks" ], "title": "stacks-definition-stack", "contents": [ "Let $\\mathcal{C}$ be a site. A {\\it stack} over $\\mathcal{C}$", "is a category $p : \\mathcal{S} \\to \\mathcal{C}$ over $\\mathcal{C}$ which", "satisfies the following conditions:", "\\begin{enumerate}", "\\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category, see", "Categories, Definition \\ref{categories-definition-fibred-category},", "\\item for any $U \\in \\Ob(\\mathcal{C})$ and any $x, y \\in \\mathcal{S}_U$", "the presheaf $\\mathit{Mor}(x, y)$ (see", "Definition \\ref{definition-mor-presheaf}) is a sheaf on", "the site $\\mathcal{C}/U$, and", "\\item for any covering $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$", "of the site $\\mathcal{C}$, any descent datum in $\\mathcal{S}$", "relative to $\\mathcal{U}$ is effective.", "\\end{enumerate}" ], "refs": [ "categories-definition-fibred-category", "stacks-definition-mor-presheaf" ], "ref_ids": [ 12388, 8992 ] }, { "id": 8997, "type": "definition", "label": "stacks-definition-stacks-over-C", "categories": [ "stacks" ], "title": "stacks-definition-stacks-over-C", "contents": [ "Let $\\mathcal{C}$ be a site.", "The {\\it $2$-category of stacks over $\\mathcal{C}$}", "is the sub $2$-category of the $2$-category of fibred categories", "over $\\mathcal{C}$ (see", "Categories, Definition \\ref{categories-definition-fibred-categories-over-C})", "defined as follows:", "\\begin{enumerate}", "\\item Its objects will be stacks $p : \\mathcal{S} \\to \\mathcal{C}$.", "\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that", "$p' \\circ G = p$ and such that $G$ maps strongly cartesian", "morphisms to strongly cartesian morphisms.", "\\item Its $2$-morphisms $t : G \\to H$ for", "$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be morphisms of functors", "such that $p'(t_x) = \\text{id}_{p(x)}$", "for all $x \\in \\Ob(\\mathcal{S})$.", "\\end{enumerate}" ], "refs": [ "categories-definition-fibred-categories-over-C" ], "ref_ids": [ 12390 ] }, { "id": 8998, "type": "definition", "label": "stacks-definition-stack-in-groupoids", "categories": [ "stacks" ], "title": "stacks-definition-stack-in-groupoids", "contents": [ "A {\\it stack in groupoids} over a site $\\mathcal{C}$ is a", "category $p : \\mathcal{S} \\to \\mathcal{C}$ over $\\mathcal{C}$", "such that", "\\begin{enumerate}", "\\item $p : \\mathcal{S} \\to \\mathcal{C}$ is fibred", "in groupoids over $\\mathcal{C}$ (see", "Categories, Definition \\ref{categories-definition-fibred-groupoids}),", "\\item for all $U \\in \\Ob(\\mathcal{C})$,", "for all $x, y\\in \\Ob(\\mathcal{S}_U)$ the presheaf", "$\\mathit{Isom}(x, y)$ is a sheaf on the site $\\mathcal{C}/U$, and", "\\item for all coverings $\\mathcal{U} = \\{U_i \\to U\\}$ in $\\mathcal{C}$,", "all descent data $(x_i, \\phi_{ij})$ for $\\mathcal{U}$ are effective.", "\\end{enumerate}" ], "refs": [ "categories-definition-fibred-groupoids" ], "ref_ids": [ 12392 ] }, { "id": 8999, "type": "definition", "label": "stacks-definition-stacks-in-groupoids-over-C", "categories": [ "stacks" ], "title": "stacks-definition-stacks-in-groupoids-over-C", "contents": [ "Let $\\mathcal{C}$ be a site.", "The {\\it $2$-category of stacks in groupoids over $\\mathcal{C}$}", "is the sub $2$-category of the $2$-category of stacks", "over $\\mathcal{C}$ (see Definition \\ref{definition-stacks-over-C})", "defined as follows:", "\\begin{enumerate}", "\\item Its objects will be stacks in groupoids", "$p : \\mathcal{S} \\to \\mathcal{C}$.", "\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that", "$p' \\circ G = p$. (Since every morphism is strongly cartesian", "every functor preserves them.)", "\\item Its $2$-morphisms $t : G \\to H$ for", "$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be morphisms of functors", "such that $p'(t_x) = \\text{id}_{p(x)}$", "for all $x \\in \\Ob(\\mathcal{S})$.", "\\end{enumerate}" ], "refs": [ "stacks-definition-stacks-over-C" ], "ref_ids": [ 8997 ] }, { "id": 9000, "type": "definition", "label": "stacks-definition-stack-in-sets", "categories": [ "stacks" ], "title": "stacks-definition-stack-in-sets", "contents": [ "Let $\\mathcal{C}$ be a site.", "\\begin{enumerate}", "\\item A {\\it stack in setoids} over $\\mathcal{C}$", "is a stack over $\\mathcal{C}$ all of whose fibre categories are", "setoids.", "\\item A {\\it stack in sets}, or a {\\it stack in discrete categories}", "is a stack over $\\mathcal{C}$ all of whose fibre categories are discrete.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9001, "type": "definition", "label": "stacks-definition-stacks-in-setoids-over-C", "categories": [ "stacks" ], "title": "stacks-definition-stacks-in-setoids-over-C", "contents": [ "Let $\\mathcal{C}$ be a site.", "The {\\it $2$-category of stacks in setoids over $\\mathcal{C}$}", "is the sub $2$-category of the $2$-category of stacks", "over $\\mathcal{C}$ (see Definition \\ref{definition-stacks-over-C})", "defined as follows:", "\\begin{enumerate}", "\\item Its objects will be stacks in setoids", "$p : \\mathcal{S} \\to \\mathcal{C}$.", "\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that", "$p' \\circ G = p$. (Since every morphism is strongly cartesian", "every functor preserves them.)", "\\item Its $2$-morphisms $t : G \\to H$ for", "$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be morphisms of functors", "such that $p'(t_x) = \\text{id}_{p(x)}$", "for all $x \\in \\Ob(\\mathcal{S})$.", "\\end{enumerate}" ], "refs": [ "stacks-definition-stacks-over-C" ], "ref_ids": [ 8997 ] }, { "id": 9002, "type": "definition", "label": "stacks-definition-topology-inherited", "categories": [ "stacks" ], "title": "stacks-definition-topology-inherited", "contents": [ "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a", "fibred category. We say $(\\mathcal{S}, \\text{Cov}(\\mathcal{S}))$ as in", "Lemma \\ref{lemma-topology-inherited}", "is the {\\it structure of site on $\\mathcal{S}$ inherited from $\\mathcal{C}$}.", "We sometimes indicate this by saying that", "{\\it $\\mathcal{S}$ is endowed with the topology inherited from $\\mathcal{C}$}." ], "refs": [ "stacks-lemma-topology-inherited" ], "ref_ids": [ 8969 ] }, { "id": 9003, "type": "definition", "label": "stacks-definition-gerbe", "categories": [ "stacks" ], "title": "stacks-definition-gerbe", "contents": [ "A {\\it gerbe} over a site $\\mathcal{C}$ is a category", "$p : \\mathcal{S} \\to \\mathcal{C}$ over $\\mathcal{C}$ such that", "\\begin{enumerate}", "\\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a stack", "in groupoids over $\\mathcal{C}$ (see", "Definition \\ref{definition-stack-in-groupoids}),", "\\item for $U \\in \\Ob(\\mathcal{C})$ there exists", "a covering $\\{U_i \\to U\\}$ in $\\mathcal{C}$ such that", "$\\mathcal{S}_{U_i}$ is nonempty, and", "\\item for $U \\in \\Ob(\\mathcal{C})$ and", "$x, y \\in \\Ob(\\mathcal{S}_U)$ there exists", "a covering $\\{U_i \\to U\\}$ in $\\mathcal{C}$ such that", "$x|_{U_i} \\cong y|_{U_i}$ in $\\mathcal{S}_{U_i}$.", "\\end{enumerate}" ], "refs": [ "stacks-definition-stack-in-groupoids" ], "ref_ids": [ 8998 ] }, { "id": 9004, "type": "definition", "label": "stacks-definition-gerbe-over-stack-in-groupoids", "categories": [ "stacks" ], "title": "stacks-definition-gerbe-over-stack-in-groupoids", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{X}$", "and $\\mathcal{Y}$ be stacks in groupoids over $\\mathcal{C}$.", "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "over $\\mathcal{C}$. We say $\\mathcal{X}$ is a {\\it gerbe over} $\\mathcal{Y}$", "if the equivalent conditions of", "Lemma \\ref{lemma-when-gerbe}", "are satisfied." ], "refs": [ "stacks-lemma-when-gerbe" ], "ref_ids": [ 8975 ] }, { "id": 9005, "type": "definition", "label": "stacks-definition-pushforward-stack", "categories": [ "stacks" ], "title": "stacks-definition-pushforward-stack", "contents": [ "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites", "given by the continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$.", "Let $\\mathcal{S}$ be a fibred category over $\\mathcal{D}$.", "In this setting we write {\\it $f_*\\mathcal{S}$} for the fibred", "category $u^p\\mathcal{S}$ defined above. We say that", "$f_*\\mathcal{S}$ is the {\\it pushforward of $\\mathcal{S}$ along $f$}." ], "refs": [], "ref_ids": [] }, { "id": 9006, "type": "definition", "label": "stacks-definition-pullback-stack", "categories": [ "stacks" ], "title": "stacks-definition-pullback-stack", "contents": [ "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites", "given by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$", "satisfying the hypotheses and conclusions of", "Sites, Proposition \\ref{sites-proposition-get-morphism}.", "Let $\\mathcal{S}$ be a stack over $\\mathcal{C}$.", "In this setting we write {\\it $f^{-1}\\mathcal{S}$} for the stackification", "of the fibred category $u_p\\mathcal{S}$ over $\\mathcal{D}$ constructed", "above. We say that $f^{-1}\\mathcal{S}$ is the", "{\\it pullback of $\\mathcal{S}$ along $f$}." ], "refs": [ "sites-proposition-get-morphism" ], "ref_ids": [ 8641 ] }, { "id": 9147, "type": "definition", "label": "spaces-simplicial-definition-cartesian-sheaf", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-definition-cartesian-sheaf", "contents": [ "In Situation \\ref{situation-simplicial-site}.", "\\begin{enumerate}", "\\item A sheaf $\\mathcal{F}$ of sets or of abelian groups on", "$\\mathcal{C}$ is {\\it cartesian} if the maps", "$\\mathcal{F}(\\varphi) : f_\\varphi^{-1}\\mathcal{F}_m \\to \\mathcal{F}_n$", "are isomorphisms for all $\\varphi : [m] \\to [n]$.", "\\item If $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}_{total}$,", "then a sheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules is", "{\\it cartesian} if the maps $f_\\varphi^*\\mathcal{F}_m \\to \\mathcal{F}_n$", "are isomorphisms for all $\\varphi : [m] \\to [n]$.", "\\item An object $K$ of $D(\\mathcal{C}_{total})$ is {\\it cartesian} if the maps", "$f_\\varphi^{-1}K_m \\to K_n$", "are isomorphisms for all $\\varphi : [m] \\to [n]$.", "\\item If $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}_{total}$, then", "an object $K$ of $D(\\mathcal{O})$ is {\\it cartesian} if the maps", "$Lf_\\varphi^*K_m \\to K_n$", "are isomorphisms for all $\\varphi : [m] \\to [n]$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9148, "type": "definition", "label": "spaces-simplicial-definition-cartesian-derived", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-definition-cartesian-derived", "contents": [ "In Situation \\ref{situation-simplicial-site}. A", "{\\it simplicial system of the derived category}", "consists of the following data", "\\begin{enumerate}", "\\item for every $n$ an object $K_n$ of $D(\\mathcal{C}_n)$,", "\\item for every $\\varphi : [m] \\to [n]$ a map", "$K_\\varphi : f_\\varphi^{-1}K_m \\to K_n$ in $D(\\mathcal{C}_n)$", "\\end{enumerate}", "subject to the condition that", "$$", "K_{\\varphi \\circ \\psi} = K_\\varphi \\circ f_\\varphi^{-1}K_\\psi :", "f_{\\varphi \\circ \\psi}^{-1}K_l = f_\\varphi^{-1} f_\\psi^{-1}K_l", "\\longrightarrow", "K_n", "$$", "for any morphisms $\\varphi : [m] \\to [n]$ and $\\psi : [l] \\to [m]$ of $\\Delta$.", "We say the simplicial system is {\\it cartesian} if the maps $K_\\varphi$", "are isomorphisms for all $\\varphi$.", "Given two simplicial systems of the derived category", "there is an obvious notion of a", "{\\it morphism of simplicial systems of the derived category}." ], "refs": [], "ref_ids": [] }, { "id": 9149, "type": "definition", "label": "spaces-simplicial-definition-cartesian-derived-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-definition-cartesian-derived-modules", "contents": [ "In Situation \\ref{situation-simplicial-site}. Let $\\mathcal{O}$", "be a sheaf of rings on $\\mathcal{C}_{total}$. A", "{\\it simplicial system of the derived category of modules}", "consists of the following data", "\\begin{enumerate}", "\\item for every $n$ an object $K_n$ of $D(\\mathcal{O}_n)$,", "\\item for every $\\varphi : [m] \\to [n]$ a map", "$K_\\varphi : Lf_\\varphi^*K_m \\to K_n$ in $D(\\mathcal{O}_n)$", "\\end{enumerate}", "subject to the condition that", "$$", "K_{\\varphi \\circ \\psi} = K_\\varphi \\circ Lf_\\varphi^*K_\\psi :", "Lf_{\\varphi \\circ \\psi}^*K_l = Lf_\\varphi^* Lf_\\psi^*K_l", "\\longrightarrow", "K_n", "$$", "for any morphisms $\\varphi : [m] \\to [n]$ and $\\psi : [l] \\to [m]$ of $\\Delta$.", "We say the simplicial system is {\\it cartesian} if the maps $K_\\varphi$", "are isomorphisms for all $\\varphi$.", "Given two simplicial systems of the derived category", "there is an obvious notion of a", "{\\it morphism of simplicial systems of the derived category of modules}." ], "refs": [], "ref_ids": [] }, { "id": 9150, "type": "definition", "label": "spaces-simplicial-definition-cartesian-morphism", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-definition-cartesian-morphism", "contents": [ "Let $a : Y \\to X$ be a morphism of simplicial schemes.", "We say $a$ is {\\it cartesian}, or that {\\it $Y$ is cartesian over $X$},", "if for every morphism $\\varphi : [n] \\to [m]$ of $\\Delta$ the corresponding", "diagram", "$$", "\\xymatrix{", "Y_m \\ar[r]_a \\ar[d]_{Y(\\varphi)} & X_m \\ar[d]^{X(\\varphi)}\\\\", "Y_n \\ar[r]^{a} & X_n", "}", "$$", "is a fibre square in the category of schemes." ], "refs": [], "ref_ids": [] }, { "id": 9151, "type": "definition", "label": "spaces-simplicial-definition-fibre-products-simplicial-scheme", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-definition-fibre-products-simplicial-scheme", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. The {\\it simplicial scheme", "associated to $f$}, denoted $(X/S)_\\bullet$, is the functor", "$\\Delta^{opp} \\to \\Sch$, $[n] \\mapsto X \\times_S \\ldots \\times_S X$", "described in", "Simplicial, Example \\ref{simplicial-example-fibre-products-simplicial-object}." ], "refs": [], "ref_ids": [] }, { "id": 9184, "type": "definition", "label": "examples-stacks-definition-hilbert-d-stack", "categories": [ "examples-stacks" ], "title": "examples-stacks-definition-hilbert-d-stack", "contents": [ "We will denote $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$", "the {\\it degree $d$ finite Hilbert stack of $\\mathcal{X}$ over $\\mathcal{Y}$}", "constructed above. If $\\mathcal{Y} = S$ we write", "$\\mathcal{H}_d(\\mathcal{X}) = \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$.", "If $\\mathcal{X} = \\mathcal{Y} = S$ we denote it $\\mathcal{H}_d$." ], "refs": [], "ref_ids": [] }, { "id": 9269, "type": "definition", "label": "models-definition-type", "categories": [ "models" ], "title": "models-definition-type", "contents": [ "A {\\it numerical type} $T$ is given by", "$$", "n, m_i, a_{ij}, w_i, g_i", "$$", "where $n \\geq 1$ is an integer and $m_i$, $a_{ij}$, $w_i$, $g_i$", "are integers for $1 \\leq i, j \\leq n$ subject to the following conditions", "\\begin{enumerate}", "\\item $m_i > 0$, $w_i > 0$, $g_i \\geq 0$,", "\\item the matrix $A = (a_{ij})$ is symmetric and $a_{ij} \\geq 0$", "for $i \\not = j$,", "\\item there is no proper nonempty subset $I \\subset \\{1, \\ldots, n\\}$", "such that $a_{ij} = 0$ for $i \\in I$, $j \\not \\in I$,", "\\item for each $i$ we have $\\sum_j a_{ij}m_j = 0$, and", "\\item $w_i | a_{ij}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9270, "type": "definition", "label": "models-definition-type-equivalent", "categories": [ "models" ], "title": "models-definition-type-equivalent", "contents": [ "We say two numerical types $n, m_i, a_{ij}, w_i, g_i$ and", "$n', m'_i, a'_{ij}, w'_i, g'_i$ are {\\it equivalent types} if", "there exists a permutation $\\sigma$ of $\\{1, \\ldots, n\\}$", "such that $m_i = m'_{\\sigma(i)}$, $a_{ij} = a'_{\\sigma(i)\\sigma(j)}$,", "$w_i = w'_{\\sigma(i)}$, and $g_i = g'_{\\sigma(i)}$." ], "refs": [], "ref_ids": [] }, { "id": 9271, "type": "definition", "label": "models-definition-genus", "categories": [ "models" ], "title": "models-definition-genus", "contents": [ "We say $n, m_i, a_{ij}, w_i, g_i$ is a {\\it numerical type of genus $g$}", "if $g = 1 + \\sum m_i(w_i(g_i - 1) - \\frac{1}{2} a_{ii})$ is the integer", "from Lemma \\ref{lemma-genus}." ], "refs": [ "models-lemma-genus" ], "ref_ids": [ 9197 ] }, { "id": 9272, "type": "definition", "label": "models-definition-type-minus-one", "categories": [ "models" ], "title": "models-definition-type-minus-one", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type.", "We say $i$ is a {\\it $(-1)$-index} if $g_i = 0$ and $a_{ii} = -w_i$." ], "refs": [], "ref_ids": [] }, { "id": 9273, "type": "definition", "label": "models-definition-top-genus", "categories": [ "models" ], "title": "models-definition-top-genus", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. The", "{\\it topological genus of $T$} is the nonnegative integer", "$g_{top} = 1 - n + e$ from Lemma \\ref{lemma-top-genus}." ], "refs": [ "models-lemma-top-genus" ], "ref_ids": [ 9202 ] }, { "id": 9274, "type": "definition", "label": "models-definition-type-minimal", "categories": [ "models" ], "title": "models-definition-type-minimal", "contents": [ "We say the numerical type $n, m_i, a_{ij}, w_i, g_i$ of genus $g$", "is {\\it minimal} if there does not exist an $i$", "with $g_i = 0$ and $a_{ii} = -w_i$, in other words, if there", "does not exist a $(-1)$-index." ], "refs": [], "ref_ids": [] }, { "id": 9275, "type": "definition", "label": "models-definition-type-minus-two", "categories": [ "models" ], "title": "models-definition-type-minus-two", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$.", "We say $i$ is a {\\it $(-2)$-index} if $g_i = 0$ and $a_{ii} = -2w_i$." ], "refs": [], "ref_ids": [] }, { "id": 9276, "type": "definition", "label": "models-definition-picard-group", "categories": [ "models" ], "title": "models-definition-picard-group", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. The", "{\\it Picard group of $T$} is the cokernel of the matrix", "$(a_{ij}/w_i)$, more precisely", "$$", "\\Pic(T) =", "\\Coker\\left(", "\\mathbf{Z}^{\\oplus n} \\to \\mathbf{Z}^{\\oplus n},\\quad", "e_i", "\\mapsto", "\\sum \\frac{a_{ij}}{w_j}e_j", "\\right)", "$$", "where $e_i$ denotes the $i$th standard basis vector for $\\mathbf{Z}^{\\oplus n}$." ], "refs": [], "ref_ids": [] }, { "id": 9277, "type": "definition", "label": "models-definition-minimal-model", "categories": [ "models" ], "title": "models-definition-minimal-model", "contents": [ "Let $C$ be a smooth projective curve over $K$ with", "$H^0(C, \\mathcal{O}_C) = K$. A {\\it minimal model}", "will be a regular, proper model $X$ for $C$ such that", "$X$ does not contain an exceptional curve of the first kind", "(Resolution of Surfaces, Section \\ref{resolve-section-minus-one})." ], "refs": [], "ref_ids": [] }, { "id": 9278, "type": "definition", "label": "models-definition-numerical-type-model", "categories": [ "models" ], "title": "models-definition-numerical-type-model", "contents": [ "In Situation \\ref{situation-regular-model} the", "{\\it numerical type associated to $X$} is the numerical", "type described in Lemma \\ref{lemma-numerical-type-of-model}." ], "refs": [ "models-lemma-numerical-type-of-model" ], "ref_ids": [ 9249 ] }, { "id": 9279, "type": "definition", "label": "models-definition-semistable", "categories": [ "models" ], "title": "models-definition-semistable", "contents": [ "Let $R$ be a discrete valuation ring with fraction field $K$.", "Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$.", "We say that $C$ has {\\it semistable reduction} if the equivalent", "conditions of Lemma \\ref{lemma-semistable} are satisfied." ], "refs": [ "models-lemma-semistable" ], "ref_ids": [ 9264 ] }, { "id": 9280, "type": "definition", "label": "models-definition-good", "categories": [ "models" ], "title": "models-definition-good", "contents": [ "Let $R$ be a discrete valuation ring with fraction field $K$.", "Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$.", "We say that $C$ has {\\it good reduction} if the equivalent", "conditions of Lemma \\ref{lemma-good} are satisfied." ], "refs": [ "models-lemma-good" ], "ref_ids": [ 9265 ] }, { "id": 9339, "type": "definition", "label": "spaces-groupoids-definition-equivalence-relation", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-equivalence-relation", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $U$ be an algebraic space over $B$.", "\\begin{enumerate}", "\\item A {\\it pre-relation} on $U$ over $B$ is any morphism", "$j : R \\to U \\times_B U$ of algebraic spaces over $B$.", "In this case we set", "$t = \\text{pr}_0 \\circ j$ and $s = \\text{pr}_1 \\circ j$, so", "that $j = (t, s)$.", "\\item A {\\it relation} on $U$ over $B$ is a monomorphism", "$j : R \\to U \\times_B U$ of algebraic spaces over $B$.", "\\item A {\\it pre-equivalence relation} is a pre-relation", "$j : R \\to U \\times_B U$ such that the image of", "$j : R(T) \\to U(T) \\times U(T)$ is an equivalence relation for", "all schemes $T$ over $B$.", "\\item We say a morphism $R \\to U \\times_B U$ of algebraic spaces over $B$", "is an {\\it equivalence relation on $U$ over $B$}", "if and only if for every $T$ over $B$ the $T$-valued", "points of $R$ define an equivalence relation", "on the set of $T$-valued points of $U$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9340, "type": "definition", "label": "spaces-groupoids-definition-restrict-relation", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-restrict-relation", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $U$ be an algebraic space over $B$.", "Let $j : R \\to U \\times_B U$ be a pre-relation.", "Let $g : U' \\to U$ be a morphism of algebraic spaces over $B$.", "The pre-relation $j' : R' \\to U' \\times_B U'$ of", "Lemma \\ref{lemma-restrict-relation} is called", "the {\\it restriction}, or {\\it pullback} of the pre-relation $j$ to $U'$.", "In this situation we sometimes write $R' = R|_{U'}$." ], "refs": [ "spaces-groupoids-lemma-restrict-relation" ], "ref_ids": [ 9286 ] }, { "id": 9341, "type": "definition", "label": "spaces-groupoids-definition-group-space", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-group-space", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "\\begin{enumerate}", "\\item A {\\it group algebraic space over $B$} is a pair $(G, m)$, where", "$G$ is an algebraic space over $B$ and $m : G \\times_B G \\to G$ is", "a morphism of algebraic spaces over $B$ with the following property:", "For every scheme $T$ over $B$ the pair $(G(T), m)$ is a group.", "\\item A {\\it morphism $\\psi : (G, m) \\to (G', m')$ of", "group algebraic spaces over $B$}", "is a morphism $\\psi : G \\to G'$ of algebraic spaces over $B$ such that for", "every $T/B$ the induced map $\\psi : G(T) \\to G'(T)$ is a homomorphism", "of groups.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9342, "type": "definition", "label": "spaces-groupoids-definition-action-group-space", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-action-group-space", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(G, m)$ be a group algebraic space over $B$.", "Let $X$ be an algebraic space over $B$.", "\\begin{enumerate}", "\\item An {\\it action of $G$ on the algebraic space $X/B$} is", "a morphism $a : G \\times_B X \\to X$ over $B$ such that", "for every scheme $T$ over $B$ the map $a : G(T) \\times X(T) \\to X(T)$", "defines the structure of a $G(T)$-set on $X(T)$.", "\\item Suppose that $X$, $Y$ are algebraic spaces over $B$ each endowed", "with an action of $G$. An {\\it equivariant} or more precisely", "a {\\it $G$-equivariant} morphism $\\psi : X \\to Y$", "is a morphism of algebraic spaces over $B$ such", "that for every $T$ over $B$ the map $\\psi : X(T) \\to Y(T)$ is", "a morphism of $G(T)$-sets.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9343, "type": "definition", "label": "spaces-groupoids-definition-free-action", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-free-action", "contents": [ "Let $B \\to S$, $G \\to B$, and $X \\to B$ as in", "Definition \\ref{definition-action-group-space}.", "Let $a : G \\times_B X \\to X$ be an action of $G$ on $X/B$.", "We say the action is {\\it free} if for every scheme $T$ over $B$", "the action $a : G(T) \\times X(T) \\to X(T)$ is a free action of", "the group $G(T)$ on the set $X(T)$." ], "refs": [ "spaces-groupoids-definition-action-group-space" ], "ref_ids": [ 9342 ] }, { "id": 9344, "type": "definition", "label": "spaces-groupoids-definition-pseudo-torsor", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-pseudo-torsor", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.", "Let $(G, m)$ be a group algebraic space over $B$.", "Let $X$ be an algebraic space over $B$, and let", "$a : G \\times_B X \\to X$ be an action of $G$ on $X$.", "\\begin{enumerate}", "\\item We say $X$ is a {\\it pseudo $G$-torsor} or that $X$ is", "{\\it formally principally homogeneous under $G$} if the induced", "morphism $G \\times_B X \\to X \\times_B X$,", "$(g, x) \\mapsto (a(g, x), x)$ is an isomorphism.", "\\item A pseudo $G$-torsor $X$ is called {\\it trivial} if there exists", "an $G$-equivariant isomorphism $G \\to X$ over $B$ where $G$ acts on", "$G$ by left multiplication.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9345, "type": "definition", "label": "spaces-groupoids-definition-principal-homogeneous-space", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-principal-homogeneous-space", "contents": [ "Let $S$ be a scheme.", "Let $B$ be an algebraic space over $S$.", "Let $(G, m)$ be a group algebraic space over $B$.", "Let $X$ be a pseudo $G$-torsor over $B$.", "\\begin{enumerate}", "\\item We say $X$ is a", "{\\it principal homogeneous space}, or more precisely a", "{\\it principal homogeneous $G$-space over $B$}", "if there exists a fpqc covering\\footnote{The default type of torsor in", "Groupoids, Definition \\ref{groupoids-definition-principal-homogeneous-space}", "is a pseudo torsor which is trivial on an fpqc covering.", "Since $G$, as an algebraic space, can be seen a sheaf of groups", "there already is a notion of a $G$-torsor which corresponds", "to fppf-torsor, see", "Lemma \\ref{lemma-torsor}.", "Hence we use ``principal homogeneous space'' for a pseudo torsor which", "is fpqc locally trivial, and we try to avoid using the word torsor in", "this situation.}", "$\\{B_i \\to B\\}_{i \\in I}$ such that each", "$X_{B_i} \\to B_i$ has a section (i.e., is a trivial pseudo $G_{B_i}$-torsor).", "\\item Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.", "We say $X$ is a {\\it $G$-torsor in the $\\tau$ topology}, or a", "{\\it $\\tau$ $G$-torsor}, or simply a {\\it $\\tau$ torsor}", "if there exists a $\\tau$ covering $\\{B_i \\to B\\}_{i \\in I}$", "such that each $X_{B_i} \\to B_i$ has a section.", "\\item If $X$ is a principal homogeneous $G$-space over $B$,", "then we say that it is", "{\\it quasi-isotrivial} if it is a torsor for the \\'etale topology.", "\\item If $X$ is a principal homogeneous $G$-space over $B$,", "then we say that it is", "{\\it locally trivial} if it is a torsor for the Zariski topology.", "\\end{enumerate}" ], "refs": [ "groupoids-definition-principal-homogeneous-space", "spaces-groupoids-lemma-torsor" ], "ref_ids": [ 9679, 9294 ] }, { "id": 9346, "type": "definition", "label": "spaces-groupoids-definition-equivariant-module", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-equivariant-module", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(G, m)$ be a group algebraic space over $B$, and", "let $a : G \\times_B X \\to X$ be an action of $G$", "on the algebraic space $X$ over $B$.", "An {\\it $G$-equivariant quasi-coherent $\\mathcal{O}_X$-module},", "or simply a {\\it equivariant quasi-coherent $\\mathcal{O}_X$-module},", "is a pair $(\\mathcal{F}, \\alpha)$, where $\\mathcal{F}$ is a quasi-coherent", "$\\mathcal{O}_X$-module, and $\\alpha$ is a $\\mathcal{O}_{G \\times_B X}$-module", "map", "$$", "\\alpha : a^*\\mathcal{F} \\longrightarrow \\text{pr}_1^*\\mathcal{F}", "$$", "where $\\text{pr}_1 : G \\times_B X \\to X$ is the projection", "such that", "\\begin{enumerate}", "\\item the diagram", "$$", "\\xymatrix{", "(1_G \\times a)^*\\text{pr}_2^*\\mathcal{F} \\ar[r]_-{\\text{pr}_{12}^*\\alpha} &", "\\text{pr}_2^*\\mathcal{F} \\\\", "(1_G \\times a)^*a^*\\mathcal{F} \\ar[u]^{(1_G \\times a)^*\\alpha} \\ar@{=}[r] &", "(m \\times 1_X)^*a^*\\mathcal{F} \\ar[u]_{(m \\times 1_X)^*\\alpha}", "}", "$$", "is a commutative in the category of", "$\\mathcal{O}_{G \\times_B G \\times_B X}$-modules, and", "\\item the pullback", "$$", "(e \\times 1_X)^*\\alpha : \\mathcal{F} \\longrightarrow \\mathcal{F}", "$$", "is the identity map.", "\\end{enumerate}", "For explanation compare with the relevant diagrams of", "Equation (\\ref{equation-action})." ], "refs": [], "ref_ids": [] }, { "id": 9347, "type": "definition", "label": "spaces-groupoids-definition-groupoid", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-groupoid", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "\\begin{enumerate}", "\\item A {\\it groupoid in algebraic spaces over $B$} is a", "quintuple $(U, R, s, t, c)$ where", "$U$ and $R$ are algebraic spaces over $B$, and", "$s, t : R \\to U$ and $c : R \\times_{s, U, t} R \\to R$", "are morphisms of algebraic spaces over $B$ with the", "following property: For any scheme $T$ over $B$ the quintuple", "$$", "(U(T), R(T), s, t, c)", "$$", "is a groupoid category.", "\\item A {\\it morphism", "$f : (U, R, s, t, c) \\to (U', R', s', t', c')$", "of groupoids in algebraic spaces over $B$} is given by morphisms", "of algebraic spaces $f : U \\to U'$ and $f : R \\to R'$ over $B$", "with the following property: For any scheme", "$T$ over $B$ the maps $f$ define a functor from the", "groupoid category $(U(T), R(T), s, t, c)$ to the", "groupoid category $(U'(T), R'(T), s', t', c')$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9348, "type": "definition", "label": "spaces-groupoids-definition-groupoid-module", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-groupoid-module", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "A {\\it quasi-coherent module on $(U, R, s, t, c)$}", "is a pair $(\\mathcal{F}, \\alpha)$, where $\\mathcal{F}$ is a quasi-coherent", "$\\mathcal{O}_U$-module, and $\\alpha$ is a $\\mathcal{O}_R$-module", "map", "$$", "\\alpha : t^*\\mathcal{F} \\longrightarrow s^*\\mathcal{F}", "$$", "such that", "\\begin{enumerate}", "\\item the diagram", "$$", "\\xymatrix{", "& \\text{pr}_1^*t^*\\mathcal{F} \\ar[r]_-{\\text{pr}_1^*\\alpha} &", "\\text{pr}_1^*s^*\\mathcal{F} \\ar@{=}[rd] & \\\\", "\\text{pr}_0^*s^*\\mathcal{F} \\ar@{=}[ru] & & & c^*s^*\\mathcal{F} \\\\", "& \\text{pr}_0^*t^*\\mathcal{F} \\ar[lu]^{\\text{pr}_0^*\\alpha} \\ar@{=}[r] &", "c^*t^*\\mathcal{F} \\ar[ru]_{c^*\\alpha}", "}", "$$", "is a commutative in the category of", "$\\mathcal{O}_{R \\times_{s, U, t} R}$-modules, and", "\\item the pullback", "$$", "e^*\\alpha : \\mathcal{F} \\longrightarrow \\mathcal{F}", "$$", "is the identity map.", "\\end{enumerate}", "Compare with the commutative diagrams of Lemma \\ref{lemma-diagram}." ], "refs": [ "spaces-groupoids-lemma-diagram" ], "ref_ids": [ 9299 ] }, { "id": 9349, "type": "definition", "label": "spaces-groupoids-definition-stabilizer-groupoid", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-stabilizer-groupoid", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "The group algebraic space $j^{-1}(\\Delta_{U/B}) \\to U$ is called the", "{\\it stabilizer of the groupoid in algebraic spaces $(U, R, s, t, c)$}." ], "refs": [], "ref_ids": [] }, { "id": 9350, "type": "definition", "label": "spaces-groupoids-definition-restrict-groupoid", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-restrict-groupoid", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "Let $g : U' \\to U$ be a morphism of algebraic spaces over $B$.", "The morphism of groupoids in algebraic spaces", "$(U', R', s', t', c') \\to (U, R, s, t, c)$", "constructed in Lemma \\ref{lemma-restrict-groupoid} is called", "the {\\it restriction of $(U, R, s, t, c)$ to $U'$}.", "We sometime use the notation $R' = R|_{U'}$ in this case." ], "refs": [ "spaces-groupoids-lemma-restrict-groupoid" ], "ref_ids": [ 9312 ] }, { "id": 9351, "type": "definition", "label": "spaces-groupoids-definition-invariant-open", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-invariant-open", "contents": [ "Let $B \\to S$ as in Section \\ref{section-notation}.", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over the base $B$.", "\\begin{enumerate}", "\\item We say an open subspace $W \\subset U$ is {\\it $R$-invariant} if", "$t(s^{-1}(W)) \\subset W$.", "\\item A locally closed subspace $Z \\subset U$ is called {\\it $R$-invariant}", "if $t^{-1}(Z) = s^{-1}(Z)$ as locally closed subspaces of $R$.", "\\item A monomorphism of algebraic spaces $T \\to U$ is {\\it $R$-invariant}", "if $T \\times_{U, t} R = R \\times_{s, U} T$ as algebraic spaces over $R$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9352, "type": "definition", "label": "spaces-groupoids-definition-quotient-sheaf", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-quotient-sheaf", "contents": [ "Let $B \\to S$ and the pre-relation $j : R \\to U \\times_B U$ be as above.", "In this setting the {\\it quotient sheaf $U/R$} associated", "to $j$ is the sheafification of the presheaf", "(\\ref{equation-quotient-presheaf}) on $(\\Sch/S)_{fppf}$.", "If $j : R \\to U \\times_B U$ comes from the action of a", "group algebraic space $G$ over $B$ on $U$ as in", "Lemma \\ref{lemma-groupoid-from-action}", "then we denote the quotient sheaf $U/G$." ], "refs": [ "spaces-groupoids-lemma-groupoid-from-action" ], "ref_ids": [ 9308 ] }, { "id": 9353, "type": "definition", "label": "spaces-groupoids-definition-representable-quotient", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-representable-quotient", "contents": [ "In the situation of Definition \\ref{definition-quotient-sheaf}.", "We say that the pre-relation $j$ has a", "{\\it quotient representable by an algebraic space}", "if the sheaf $U/R$ is an algebraic space.", "We say that the pre-relation $j$ has a", "{\\it representable quotient}", "if the sheaf $U/R$ is representable by a scheme.", "We will say a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$ has a", "{\\it representable quotient}", "(resp.\\ {\\it quotient representable by an algebraic space}", "if the quotient $U/R$ with $j = (t, s)$ is representable (resp.\\ an", "algebraic space)." ], "refs": [ "spaces-groupoids-definition-quotient-sheaf" ], "ref_ids": [ 9352 ] }, { "id": 9354, "type": "definition", "label": "spaces-groupoids-definition-quotient-stack", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-definition-quotient-stack", "contents": [ "Quotient stacks. Let $B \\to S$ be as above.", "\\begin{enumerate}", "\\item Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "The {\\it quotient stack}", "$$", "p : [U/R] \\longrightarrow (\\Sch/S)_{fppf}", "$$", "of $(U, R, s, t, c)$ is the stackification (see", "Stacks, Lemma \\ref{stacks-lemma-stackify-groupoids})", "of the category fibred in groupoids $[U/_{\\!p}R]$ over", "$(\\Sch/S)_{fppf}$ associated to", "(\\ref{equation-quotient-stack}).", "\\item Let $(G, m)$ be a group algebraic space over $B$.", "Let $a : G \\times_B X \\to X$ be an action of $G$ on an algebraic space", "over $B$. The {\\it quotient stack}", "$$", "p : [X/G] \\longrightarrow (\\Sch/S)_{fppf}", "$$", "is the quotient stack associated to the groupoid in algebraic spaces", "$(X, G \\times_B X, s, t, c)$ over $B$ of", "Lemma \\ref{lemma-groupoid-from-action}.", "\\end{enumerate}" ], "refs": [ "stacks-lemma-stackify-groupoids", "spaces-groupoids-lemma-groupoid-from-action" ], "ref_ids": [ 8966, 9308 ] }, { "id": 9438, "type": "definition", "label": "spaces-descent-definition-descent-datum-quasi-coherent", "categories": [ "spaces-descent" ], "title": "spaces-descent-definition-descent-datum-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family", "of morphisms of algebraic spaces over $S$ with fixed target $X$.", "\\begin{enumerate}", "\\item A {\\it descent datum $(\\mathcal{F}_i, \\varphi_{ij})$", "for quasi-coherent sheaves} with respect to the given family", "is given by a quasi-coherent sheaf $\\mathcal{F}_i$ on $X_i$ for", "each $i \\in I$, an isomorphism of quasi-coherent", "$\\mathcal{O}_{X_i \\times_X X_j}$-modules", "$\\varphi_{ij} : \\text{pr}_0^*\\mathcal{F}_i \\to \\text{pr}_1^*\\mathcal{F}_j$", "for each pair $(i, j) \\in I^2$", "such that for every triple of indices $(i, j, k) \\in I^3$ the", "diagram", "$$", "\\xymatrix{", "\\text{pr}_0^*\\mathcal{F}_i \\ar[rd]_{\\text{pr}_{01}^*\\varphi_{ij}}", "\\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} & &", "\\text{pr}_2^*\\mathcal{F}_k \\\\", "& \\text{pr}_1^*\\mathcal{F}_j \\ar[ru]_{\\text{pr}_{12}^*\\varphi_{jk}} &", "}", "$$", "of $\\mathcal{O}_{X_i \\times_X X_j \\times_X X_k}$-modules", "commutes. This is called the {\\it cocycle condition}.", "\\item A {\\it morphism $\\psi : (\\mathcal{F}_i, \\varphi_{ij}) \\to", "(\\mathcal{F}'_i, \\varphi'_{ij})$ of descent data} is given", "by a family $\\psi = (\\psi_i)_{i\\in I}$ of morphisms of", "$\\mathcal{O}_{X_i}$-modules $\\psi_i : \\mathcal{F}_i \\to \\mathcal{F}'_i$", "such that all the diagrams", "$$", "\\xymatrix{", "\\text{pr}_0^*\\mathcal{F}_i \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\text{pr}_0^*\\psi_i}", "& \\text{pr}_1^*\\mathcal{F}_j \\ar[d]^{\\text{pr}_1^*\\psi_j} \\\\", "\\text{pr}_0^*\\mathcal{F}'_i \\ar[r]^{\\varphi'_{ij}} &", "\\text{pr}_1^*\\mathcal{F}'_j \\\\", "}", "$$", "commute.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9439, "type": "definition", "label": "spaces-descent-definition-descent-datum-effective-quasi-coherent", "categories": [ "spaces-descent" ], "title": "spaces-descent-definition-descent-datum-effective-quasi-coherent", "contents": [ "Let $S$ be a scheme.", "Let $\\{U_i \\to U\\}_{i \\in I}$ be a family of morphisms of algebraic", "spaces over $S$ with fixed target.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_U$-module.", "We call the unique descent on $\\mathcal{F}$ datum with respect to the covering", "$\\{U \\to U\\}$ the {\\it trivial descent datum}.", "\\item The pullback of the trivial descent datum to", "$\\{U_i \\to U\\}$ is called the {\\it canonical descent datum}.", "Notation: $(\\mathcal{F}|_{U_i}, can)$.", "\\item A descent datum $(\\mathcal{F}_i, \\varphi_{ij})$", "for quasi-coherent sheaves with respect to the given family", "is said to be {\\it effective} if there exists a quasi-coherent", "sheaf $\\mathcal{F}$ on $U$ such that $(\\mathcal{F}_i, \\varphi_{ij})$", "is isomorphic to $(\\mathcal{F}|_{U_i}, can)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9440, "type": "definition", "label": "spaces-descent-definition-property-morphisms-local", "categories": [ "spaces-descent" ], "title": "spaces-descent-definition-property-morphisms-local", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$.", "Let $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale\\}$.", "We say $\\mathcal{P}$ is {\\it $\\tau$ local on the base}, or", "{\\it $\\tau$ local on the target}, or", "{\\it local on the base for the $\\tau$-topology} if for any", "$\\tau$-covering $\\{Y_i \\to Y\\}_{i \\in I}$ of algebraic spaces", "and any morphism of algebraic spaces $f : X \\to Y$ we", "have", "$$", "f \\text{ has }\\mathcal{P}", "\\Leftrightarrow", "\\text{each }Y_i \\times_Y X \\to Y_i\\text{ has }\\mathcal{P}.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 9441, "type": "definition", "label": "spaces-descent-definition-property-morphisms-local-source", "categories": [ "spaces-descent" ], "title": "spaces-descent-definition-property-morphisms-local-source", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$.", "Let $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0] syntomic, \\linebreak[0]", "smooth, \\linebreak[0] \\etale\\}$. We say $\\mathcal{P}$ is", "{\\it $\\tau$ local on the source}, or", "{\\it local on the source for the $\\tau$-topology} if for", "any morphism $f : X \\to Y$ of algebraic spaces over $S$, and any", "$\\tau$-covering $\\{X_i \\to X\\}_{i \\in I}$ of algebraic spaces we have", "$$", "f \\text{ has }\\mathcal{P}", "\\Leftrightarrow", "\\text{each }X_i \\to Y\\text{ has }\\mathcal{P}.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 9442, "type": "definition", "label": "spaces-descent-definition-local-source-target", "categories": [ "spaces-descent" ], "title": "spaces-descent-definition-local-source-target", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$.", "We say $\\mathcal{P}$ is {\\it smooth local on source-and-target} if", "\\begin{enumerate}", "\\item (stable under precomposing with smooth maps)", "if $f : X \\to Y$ is smooth and $g : Y \\to Z$ has $\\mathcal{P}$,", "then $g \\circ f$ has $\\mathcal{P}$,", "\\item (stable under smooth base change)", "if $f : X \\to Y$ has $\\mathcal{P}$ and $Y' \\to Y$ is smooth, then", "the base change $f' : Y' \\times_Y X \\to Y'$ has $\\mathcal{P}$, and", "\\item (locality) given a morphism $f : X \\to Y$ the following are", "equivalent", "\\begin{enumerate}", "\\item $f$ has $\\mathcal{P}$,", "\\item for every $x \\in |X|$ there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "with smooth vertical arrows and $u \\in |U|$ with $a(u) = x$ such that", "$h$ has $\\mathcal{P}$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9443, "type": "definition", "label": "spaces-descent-definition-etale-smooth-local-source-target", "categories": [ "spaces-descent" ], "title": "spaces-descent-definition-etale-smooth-local-source-target", "contents": [ "Let $S$ be a scheme.", "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$.", "We say $\\mathcal{P}$ is {\\it \\'etale-smooth local on source-and-target} if", "\\begin{enumerate}", "\\item (stable under precomposing with \\'etale maps)", "if $f : X \\to Y$ is \\'etale and $g : Y \\to Z$ has $\\mathcal{P}$,", "then $g \\circ f$ has $\\mathcal{P}$,", "\\item (stable under smooth base change)", "if $f : X \\to Y$ has $\\mathcal{P}$ and $Y' \\to Y$ is smooth, then", "the base change $f' : Y' \\times_Y X \\to Y'$ has $\\mathcal{P}$, and", "\\item (locality) given a morphism $f : X \\to Y$ the following are", "equivalent", "\\begin{enumerate}", "\\item $f$ has $\\mathcal{P}$,", "\\item for every $x \\in |X|$ there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "with $b$ smooth and $U \\to X \\times_Y V$ \\'etale", "and $u \\in |U|$ with $a(u) = x$ such that", "$h$ has $\\mathcal{P}$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9444, "type": "definition", "label": "spaces-descent-definition-descent-datum", "categories": [ "spaces-descent" ], "title": "spaces-descent-definition-descent-datum", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces", "over $S$.", "\\begin{enumerate}", "\\item Let $V \\to Y$ be a morphism of algebraic spaces.", "A {\\it descent datum for $V/Y/X$} is an isomorphism", "$\\varphi : V \\times_X Y \\to Y \\times_X V$ of algebraic spaces over", "$Y \\times_X Y$ satisfying the {\\it cocycle condition} that the diagram", "$$", "\\xymatrix{", "V \\times_X Y \\times_X Y \\ar[rd]^{\\varphi_{01}} \\ar[rr]_{\\varphi_{02}} &", "&", "Y \\times_X Y \\times_X V\\\\", "&", "Y \\times_X Y \\times_X Y \\ar[ru]^{\\varphi_{12}}", "}", "$$", "commutes (with obvious notation).", "\\item We also say that the pair $(V/Y, \\varphi)$ is", "a {\\it descent datum relative to $Y \\to X$}.", "\\item A {\\it morphism $f : (V/Y, \\varphi) \\to (V'/Y, \\varphi')$ of", "descent data relative to $Y \\to X$} is a morphism", "$f : V \\to V'$ of algebraic spaces over $Y$ such that", "the diagram", "$$", "\\xymatrix{", "V \\times_X Y \\ar[r]_{\\varphi} \\ar[d]_{f \\times \\text{id}_Y} &", "Y \\times_X V \\ar[d]^{\\text{id}_Y \\times f} \\\\", "V' \\times_X Y \\ar[r]^{\\varphi'} & Y \\times_X V'", "}", "$$", "commutes.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9445, "type": "definition", "label": "spaces-descent-definition-descent-datum-for-family-of-morphisms", "categories": [ "spaces-descent" ], "title": "spaces-descent-definition-descent-datum-for-family-of-morphisms", "contents": [ "Let $S$ be a scheme.", "Let $\\{X_i \\to X\\}_{i \\in I}$ be a family of morphisms", "of algebraic spaces over $S$ with fixed target $X$.", "\\begin{enumerate}", "\\item A {\\it descent datum $(V_i, \\varphi_{ij})$ relative to the", "family $\\{X_i \\to X\\}$} is given by an algebraic space $V_i$ over $X_i$", "for each $i \\in I$, an isomorphism", "$\\varphi_{ij} : V_i \\times_X X_j \\to X_i \\times_X V_j$", "of algebraic spaces over $X_i \\times_X X_j$ for each pair $(i, j) \\in I^2$", "such that for every triple of indices $(i, j, k) \\in I^3$", "the diagram", "$$", "\\xymatrix{", "V_i \\times_X X_j \\times_X X_k", "\\ar[rd]^{\\text{pr}_{01}^*\\varphi_{ij}}", "\\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} &", "&", "X_i \\times_X X_j \\times_X V_k\\\\", "&", "X_i \\times_X V_j \\times_X X_k", "\\ar[ru]^{\\text{pr}_{12}^*\\varphi_{jk}}", "}", "$$", "of algebraic spaces over $X_i \\times_X X_j \\times_X X_k$ commutes", "(with obvious notation).", "\\item A {\\it morphism", "$\\psi : (V_i, \\varphi_{ij}) \\to (V'_i, \\varphi'_{ij})$", "of descent data} is given by a family $\\psi = (\\psi_i)_{i \\in I}$", "of morphisms $\\psi_i : V_i \\to V'_i$ of algebraic spaces over $X_i$", "such that all the diagrams", "$$", "\\xymatrix{", "V_i \\times_X X_j \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\psi_i \\times \\text{id}} &", "X_i \\times_X V_j \\ar[d]^{\\text{id} \\times \\psi_j} \\\\", "V'_i \\times_X X_j \\ar[r]^{\\varphi'_{ij}} & X_i \\times_X V'_j", "}", "$$", "commute.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9446, "type": "definition", "label": "spaces-descent-definition-pullback-functor", "categories": [ "spaces-descent" ], "title": "spaces-descent-definition-pullback-functor", "contents": [ "With $S, X, X', Y, Y', f, a, a', h$ as in Lemma \\ref{lemma-pullback}", "the functor", "$$", "(V, \\varphi) \\longmapsto f^*(V, \\varphi)", "$$", "constructed in that lemma is called the {\\it pullback functor} on descent data." ], "refs": [ "spaces-descent-lemma-pullback" ], "ref_ids": [ 9434 ] }, { "id": 9447, "type": "definition", "label": "spaces-descent-definition-pullback-functor-family", "categories": [ "spaces-descent" ], "title": "spaces-descent-definition-pullback-functor-family", "contents": [ "With $\\mathcal{U}' = \\{X'_i \\to X'\\}_{i \\in I'}$,", "$\\mathcal{U} = \\{X_i \\to X\\}_{i \\in I}$, $\\alpha : I' \\to I$,", "$g : X' \\to X$, and $g_i : X'_i \\to X_{\\alpha(i)}$ as in", "Lemma \\ref{lemma-pullback-family} the functor", "$$", "(V_i, \\varphi_{ij}) \\longmapsto", "(g_i^*V_{\\alpha(i)}, (g_i \\times g_j)^*\\varphi_{\\alpha(i) \\alpha(j)})", "$$", "constructed in that lemma", "is called the {\\it pullback functor} on descent data." ], "refs": [ "spaces-descent-lemma-pullback-family" ], "ref_ids": [ 9435 ] }, { "id": 9448, "type": "definition", "label": "spaces-descent-definition-effective", "categories": [ "spaces-descent" ], "title": "spaces-descent-definition-effective", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over", "$S$.", "\\begin{enumerate}", "\\item Given an algebraic space $U$ over $X$ we have the", "{\\it trivial descent datum} of $U$ relative to $\\text{id} : X \\to X$, namely", "the identity morphism on $U$.", "\\item By Lemma \\ref{lemma-pullback} we get a", "{\\it canonical descent datum} on $Y \\times_X U$", "relative to $Y \\to X$ by pulling back the trivial", "descent datum via $f$. We often", "denote $(Y \\times_X U, can)$ this descent datum.", "\\item A descent datum $(V, \\varphi)$ relative to $Y/X$", "is called {\\it effective} if $(V, \\varphi)$", "is isomorphic to the canonical descent datum", "$(Y \\times_X U, can)$ for some algebraic space $U$ over $X$.", "\\end{enumerate}" ], "refs": [ "spaces-descent-lemma-pullback" ], "ref_ids": [ 9434 ] }, { "id": 9449, "type": "definition", "label": "spaces-descent-definition-effective-family", "categories": [ "spaces-descent" ], "title": "spaces-descent-definition-effective-family", "contents": [ "Let $S$ be a scheme.", "Let $\\{X_i \\to X\\}$ be a family of morphisms of algebraic spaces over $S$", "with fixed target $X$.", "\\begin{enumerate}", "\\item Given an algebraic space $U$ over $X$", "we have a {\\it canonical descent datum} on the family of", "algebraic spaces $X_i \\times_X U$ by pulling back the trivial", "descent datum for $U$ relative to $\\{\\text{id} : S \\to S\\}$.", "We denote this descent datum $(X_i \\times_X U, can)$.", "\\item A descent datum $(V_i, \\varphi_{ij})$", "relative to $\\{X_i \\to S\\}$ is called {\\it effective}", "if there exists an algebraic space $U$ over $X$ such that", "$(V_i, \\varphi_{ij})$ is isomorphic to $(X_i \\times_X U, can)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9561, "type": "definition", "label": "decent-spaces-definition-universally-bounded", "categories": [ "decent-spaces" ], "title": "decent-spaces-definition-universally-bounded", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$, and", "let $U$ be a scheme over $S$. Let $f : U \\to X$ be a morphism over $S$.", "We say the {\\it fibres of $f$ are universally bounded}\\footnote{This is", "probably nonstandard notation.}", "if there exists an integer $n$ such that for all fields", "$k$ and all morphisms $\\Spec(k) \\to X$ the fibre", "product $\\Spec(k) \\times_X U$ is a finite scheme over $k$", "whose degree over $k$ is $\\leq n$." ], "refs": [], "ref_ids": [] }, { "id": 9562, "type": "definition", "label": "decent-spaces-definition-very-reasonable", "categories": [ "decent-spaces" ], "title": "decent-spaces-definition-very-reasonable", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item We say $X$ is {\\it decent} if for every point $x \\in X$ the equivalent", "conditions of", "Lemma \\ref{lemma-UR-finite-above-x}", "hold, in other words property $(\\gamma)$ of", "Lemma \\ref{lemma-bounded-fibres}", "holds.", "\\item We say $X$ is {\\it reasonable} if the equivalent conditions of", "Lemma \\ref{lemma-U-universally-bounded}", "hold, in other words property $(\\delta)$ of", "Lemma \\ref{lemma-bounded-fibres}", "holds.", "\\item We say $X$ is {\\it very reasonable} if the equivalent conditions of", "Lemma \\ref{lemma-characterize-very-reasonable}", "hold, i.e., property $(\\epsilon)$ of", "Lemma \\ref{lemma-bounded-fibres}", "holds.", "\\end{enumerate}" ], "refs": [ "decent-spaces-lemma-UR-finite-above-x", "decent-spaces-lemma-bounded-fibres", "decent-spaces-lemma-U-universally-bounded", "decent-spaces-lemma-bounded-fibres", "decent-spaces-lemma-characterize-very-reasonable", "decent-spaces-lemma-bounded-fibres" ], "ref_ids": [ 9463, 9466, 9464, 9466, 9465, 9466 ] }, { "id": 9563, "type": "definition", "label": "decent-spaces-definition-residue-field", "categories": [ "decent-spaces" ], "title": "decent-spaces-definition-residue-field", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $x \\in |X|$. The {\\it residue field of $X$ at $x$}", "is the unique field $\\kappa(x)$ which comes equipped with a", "monomorphism $\\Spec(\\kappa(x)) \\to X$ representing $x$." ], "refs": [], "ref_ids": [] }, { "id": 9564, "type": "definition", "label": "decent-spaces-definition-elemenary-etale-neighbourhood", "categories": [ "decent-spaces" ], "title": "decent-spaces-definition-elemenary-etale-neighbourhood", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in X$ be a point. An {\\it elementary \\'etale neighbourhood}", "is an \\'etale morphism $(U, u) \\to (X, x)$ where $U$ is a scheme,", "$u \\in U$ is a point mapping to $x$, and $\\kappa(x) \\to \\kappa(u)$", "is an isomorphism. A {\\it morphism of elementary \\'etale neighbourhoods}", "$(U, u) \\to (U', u')$ is defined as a morphism $U \\to U'$", "over $X$ mapping $u$ to $u'$." ], "refs": [], "ref_ids": [] }, { "id": 9565, "type": "definition", "label": "decent-spaces-definition-henselian-local-ring", "categories": [ "decent-spaces" ], "title": "decent-spaces-definition-henselian-local-ring", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $x \\in |X|$. The {\\it henselian local ring of $X$ at $x$}, is", "$$", "\\mathcal{O}_{X, x}^h = \\colim \\Gamma(U, \\mathcal{O}_U)", "$$", "where the colimit is over the elementary \\'etale neighbourhoods", "$(U, u) \\to (X, x)$." ], "refs": [], "ref_ids": [] }, { "id": 9566, "type": "definition", "label": "decent-spaces-definition-residual-space", "categories": [ "decent-spaces" ], "title": "decent-spaces-definition-residual-space", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$. Let $x \\in |X|$.", "The", "{\\it residual space of $X$ at $x$}\\footnote{This is nonstandard notation.}", "is the monomorphism $Z_x \\to X$ constructed in", "Lemma \\ref{lemma-find-singleton-from-point}." ], "refs": [ "decent-spaces-lemma-find-singleton-from-point" ], "ref_ids": [ 9504 ] }, { "id": 9567, "type": "definition", "label": "decent-spaces-definition-relative-conditions", "categories": [ "decent-spaces" ], "title": "decent-spaces-definition-relative-conditions", "contents": [ "Let $S$ be a scheme. We say an algebraic space $X$ over $S$", "{\\it has property $(\\beta)$} if $X$ has the corresponding property of", "Lemma \\ref{lemma-bounded-fibres}.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "\\begin{enumerate}", "\\item We say $f$ {\\it has property $(\\beta)$} if for any scheme $T$ and", "morphism $T \\to Y$ the fibre product $T \\times_Y X$ has property $(\\beta)$.", "\\item We say $f$ is {\\it decent} if for any scheme $T$ and", "morphism $T \\to Y$ the fibre product $T \\times_Y X$ is a decent", "algebraic space.", "\\item We say $f$ is {\\it reasonable} if for any scheme $T$ and", "morphism $T \\to Y$ the fibre product $T \\times_Y X$ is a reasonable", "algebraic space.", "\\item We say $f$ is {\\it very reasonable} if for any scheme $T$ and", "morphism $T \\to Y$ the fibre product $T \\times_Y X$ is a very reasonable", "algebraic space.", "\\end{enumerate}" ], "refs": [ "decent-spaces-lemma-bounded-fibres" ], "ref_ids": [ 9466 ] }, { "id": 9568, "type": "definition", "label": "decent-spaces-definition-birational", "categories": [ "decent-spaces" ], "title": "decent-spaces-definition-birational", "contents": [ "Let $S$ be a scheme. Let $X$ and $Y$ algebraic spaces over $S$.", "Assume $X$ and $Y$ are decent and that $|X|$ and $|Y|$ have finitely many", "irreducible components. We say a morphism $f : X \\to Y$ is", "{\\it birational} if", "\\begin{enumerate}", "\\item $|f|$ induces a bijection between the set of generic points", "of irreducible components of $|X|$ and the set of generic points", "of the irreducible components of $|Y|$, and", "\\item for every generic point $x \\in |X|$ of an irreducible component", "the local ring map $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$", "is an isomorphism (see clarification below).", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9569, "type": "definition", "label": "decent-spaces-definition-unibranch", "categories": [ "decent-spaces" ], "title": "decent-spaces-definition-unibranch", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $x \\in |X|$. We say that $X$ is {\\it unibranch at $x$}", "if the equivalent conditions of", "Lemma \\ref{lemma-irreducible-local-ring} hold.", "We say that $X$ is {\\it unibranch} if $X$ is", "unibranch at every $x \\in |X|$." ], "refs": [ "decent-spaces-lemma-irreducible-local-ring" ], "ref_ids": [ 9552 ] }, { "id": 9570, "type": "definition", "label": "decent-spaces-definition-number-of-geometric-branches", "categories": [ "decent-spaces" ], "title": "decent-spaces-definition-number-of-geometric-branches", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $x \\in |X|$. The {\\it number of branches of $X$ at $x$} is", "either $n \\in \\mathbf{N}$ if the equivalent conditions", "of Lemma \\ref{lemma-nr-branches-local-ring}", "hold, or else $\\infty$." ], "refs": [ "decent-spaces-lemma-nr-branches-local-ring" ], "ref_ids": [ 9553 ] }, { "id": 9571, "type": "definition", "label": "decent-spaces-definition-catenary", "categories": [ "decent-spaces" ], "title": "decent-spaces-definition-catenary", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "We say $X$ is {\\it catenary} if $|X|$ is catenary", "(Topology, Definition \\ref{topology-definition-catenary})." ], "refs": [ "topology-definition-catenary" ], "ref_ids": [ 8359 ] }, { "id": 9572, "type": "definition", "label": "decent-spaces-definition-universally-catenary", "categories": [ "decent-spaces" ], "title": "decent-spaces-definition-universally-catenary", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent and locally Noetherian", "algebraic space over $S$. We say $X$ is {\\it universally catenary}", "if for every morphism $Y \\to X$ of algebraic spaces which is ", "locally of finite type and with $Y$ decent, the algebraic space", "$Y$ is catenary." ], "refs": [], "ref_ids": [] }, { "id": 9670, "type": "definition", "label": "groupoids-definition-equivalence-relation", "categories": [ "groupoids" ], "title": "groupoids-definition-equivalence-relation", "contents": [ "Let $S$ be a scheme. Let $U$ be a scheme over $S$.", "\\begin{enumerate}", "\\item A {\\it pre-relation} on $U$ over $S$ is any morphism", "of schemes $j : R \\to U \\times_S U$. In this case we set", "$t = \\text{pr}_0 \\circ j$ and $s = \\text{pr}_1 \\circ j$, so", "that $j = (t, s)$.", "\\item A {\\it relation} on $U$ over $S$ is a monomorphism", "of schemes $j : R \\to U \\times_S U$.", "\\item A {\\it pre-equivalence relation} is a pre-relation", "$j : R \\to U \\times_S U$ such that the image of", "$j : R(T) \\to U(T) \\times U(T)$ is an equivalence relation for", "all $T/S$.", "\\item We say a morphism $R \\to U \\times_S U$ of schemes is", "an {\\it equivalence relation on $U$ over $S$}", "if and only if for every scheme $T$ over $S$ the $T$-valued", "points of $R$ define an equivalence relation", "on the set of $T$-valued points of $U$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9671, "type": "definition", "label": "groupoids-definition-restrict-relation", "categories": [ "groupoids" ], "title": "groupoids-definition-restrict-relation", "contents": [ "Let $S$ be a scheme.", "Let $U$ be a scheme over $S$.", "Let $j : R \\to U \\times_S U$ be a pre-relation.", "Let $g : U' \\to U$ be a morphism of schemes.", "The pre-relation $j' : R' \\to U' \\times_S U'$ is called", "the {\\it restriction}, or {\\it pullback} of the pre-relation $j$ to $U'$.", "In this situation we sometimes write $R' = R|_{U'}$." ], "refs": [], "ref_ids": [] }, { "id": 9672, "type": "definition", "label": "groupoids-definition-group-scheme", "categories": [ "groupoids" ], "title": "groupoids-definition-group-scheme", "contents": [ "Let $S$ be a scheme.", "\\begin{enumerate}", "\\item A {\\it group scheme over $S$} is a pair $(G, m)$, where", "$G$ is a scheme over $S$ and $m : G \\times_S G \\to G$ is", "a morphism of schemes over $S$ with the following property:", "For every scheme $T$ over $S$ the pair $(G(T), m)$", "is a group.", "\\item A {\\it morphism $\\psi : (G, m) \\to (G', m')$ of group schemes over $S$}", "is a morphism $\\psi : G \\to G'$ of schemes over $S$ such that for", "every $T/S$ the induced map $\\psi : G(T) \\to G'(T)$ is a homomorphism", "of groups.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9673, "type": "definition", "label": "groupoids-definition-closed-subgroup-scheme", "categories": [ "groupoids" ], "title": "groupoids-definition-closed-subgroup-scheme", "contents": [ "Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$.", "\\begin{enumerate}", "\\item A {\\it closed subgroup scheme} of $G$ is a closed subscheme", "$H \\subset G$ such that $m|_{H \\times_S H}$ factors through $H$ and induces a", "group scheme structure on $H$ over $S$.", "\\item An {\\it open subgroup scheme} of $G$ is an open subscheme", "$G' \\subset G$ such that $m|_{G' \\times_S G'}$ factors through $G'$", "and induces a group scheme structure on $G'$ over $S$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9674, "type": "definition", "label": "groupoids-definition-smooth-group-scheme", "categories": [ "groupoids" ], "title": "groupoids-definition-smooth-group-scheme", "contents": [ "Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$.", "\\begin{enumerate}", "\\item We say $G$ is a {\\it smooth group scheme} if the structure", "morphism $G \\to S$ is smooth.", "\\item We say $G$ is a {\\it flat group scheme} if the structure", "morphism $G \\to S$ is flat.", "\\item We say $G$ is a {\\it separated group scheme} if the structure", "morphism $G \\to S$ is separated.", "\\end{enumerate}", "Add more as needed." ], "refs": [], "ref_ids": [] }, { "id": 9675, "type": "definition", "label": "groupoids-definition-abelian-variety", "categories": [ "groupoids" ], "title": "groupoids-definition-abelian-variety", "contents": [ "Let $k$ be a field. An {\\it abelian variety} is a group scheme over", "$k$ which is also a proper, geometrically integral variety over $k$." ], "refs": [], "ref_ids": [] }, { "id": 9676, "type": "definition", "label": "groupoids-definition-action-group-scheme", "categories": [ "groupoids" ], "title": "groupoids-definition-action-group-scheme", "contents": [ "Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$.", "\\begin{enumerate}", "\\item An {\\it action of $G$ on the scheme $X/S$} is", "a morphism $a : G \\times_S X \\to X$ over $S$ such that", "for every $T/S$ the map $a : G(T) \\times X(T) \\to X(T)$", "defines the structure of a $G(T)$-set on $X(T)$.", "\\item Suppose that $X$, $Y$ are schemes over $S$ each endowed", "with an action of $G$. An {\\it equivariant} or more precisely", "a {\\it $G$-equivariant} morphism $\\psi : X \\to Y$", "is a morphism of schemes over $S$ such", "that for every $T/S$ the map $\\psi : X(T) \\to Y(T)$ is", "a morphism of $G(T)$-sets.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9677, "type": "definition", "label": "groupoids-definition-free-action", "categories": [ "groupoids" ], "title": "groupoids-definition-free-action", "contents": [ "Let $S$, $G \\to S$, and $X \\to S$ as in", "Definition \\ref{definition-action-group-scheme}.", "Let $a : G \\times_S X \\to X$ be an action of $G$ on $X/S$.", "We say the action is {\\it free} if for every scheme $T$ over $S$", "the action $a : G(T) \\times X(T) \\to X(T)$ is a free action of", "the group $G(T)$ on the set $X(T)$." ], "refs": [ "groupoids-definition-action-group-scheme" ], "ref_ids": [ 9676 ] }, { "id": 9678, "type": "definition", "label": "groupoids-definition-pseudo-torsor", "categories": [ "groupoids" ], "title": "groupoids-definition-pseudo-torsor", "contents": [ "Let $S$ be a scheme.", "Let $(G, m)$ be a group scheme over $S$.", "Let $X$ be a scheme over $S$, and let", "$a : G \\times_S X \\to X$ be an action of $G$ on $X$.", "\\begin{enumerate}", "\\item We say $X$ is a {\\it pseudo $G$-torsor} or that $X$ is", "{\\it formally principally homogeneous under $G$} if the induced", "morphism of schemes $G \\times_S X \\to X \\times_S X$,", "$(g, x) \\mapsto (a(g, x), x)$ is an isomorphism of schemes over $S$.", "\\item A pseudo $G$-torsor $X$ is called {\\it trivial} if there exists", "an $G$-equivariant isomorphism $G \\to X$ over $S$ where $G$ acts on", "$G$ by left multiplication.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9679, "type": "definition", "label": "groupoids-definition-principal-homogeneous-space", "categories": [ "groupoids" ], "title": "groupoids-definition-principal-homogeneous-space", "contents": [ "Let $S$ be a scheme.", "Let $(G, m)$ be a group scheme over $S$.", "Let $X$ be a pseudo $G$-torsor over $S$.", "\\begin{enumerate}", "\\item We say $X$ is a {\\it principal homogeneous space}", "or a {\\it $G$-torsor} if there exists a fpqc covering\\footnote{This means", "that the default type of torsor is a pseudo torsor which is trivial on an", "fpqc covering. This is the definition in \\cite[Expos\\'e IV, 6.5]{SGA3}.", "It is a little bit inconvenient for us as we most often work in the fppf", "topology.}", "$\\{S_i \\to S\\}_{i \\in I}$ such that each", "$X_{S_i} \\to S_i$ has a section (i.e., is a trivial pseudo $G_{S_i}$-torsor).", "\\item Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.", "We say $X$ is a {\\it $G$-torsor in the $\\tau$ topology}, or a", "{\\it $\\tau$ $G$-torsor}, or simply a {\\it $\\tau$ torsor}", "if there exists a $\\tau$ covering $\\{S_i \\to S\\}_{i \\in I}$", "such that each $X_{S_i} \\to S_i$ has a section.", "\\item If $X$ is a $G$-torsor, then we say that it is", "{\\it quasi-isotrivial} if it is a torsor for the \\'etale topology.", "\\item If $X$ is a $G$-torsor, then we say that it is", "{\\it locally trivial} if it is a torsor for the Zariski topology.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9680, "type": "definition", "label": "groupoids-definition-equivariant-module", "categories": [ "groupoids" ], "title": "groupoids-definition-equivariant-module", "contents": [ "Let $S$ be a scheme, let $(G, m)$ be a group scheme over $S$, and", "let $a : G \\times_S X \\to X$ be an action of the group scheme $G$", "on $X/S$. A {\\it $G$-equivariant quasi-coherent $\\mathcal{O}_X$-module},", "or simply an {\\it equivariant quasi-coherent $\\mathcal{O}_X$-module},", "is a pair $(\\mathcal{F}, \\alpha)$, where $\\mathcal{F}$ is a quasi-coherent", "$\\mathcal{O}_X$-module, and $\\alpha$ is a $\\mathcal{O}_{G \\times_S X}$-module", "map", "$$", "\\alpha : a^*\\mathcal{F} \\longrightarrow \\text{pr}_1^*\\mathcal{F}", "$$", "where $\\text{pr}_1 : G \\times_S X \\to X$ is the projection", "such that", "\\begin{enumerate}", "\\item the diagram", "$$", "\\xymatrix{", "(1_G \\times a)^*\\text{pr}_1^*\\mathcal{F} \\ar[r]_-{\\text{pr}_{12}^*\\alpha} &", "\\text{pr}_2^*\\mathcal{F} \\\\", "(1_G \\times a)^*a^*\\mathcal{F} \\ar[u]^{(1_G \\times a)^*\\alpha} \\ar@{=}[r] &", "(m \\times 1_X)^*a^*\\mathcal{F} \\ar[u]_{(m \\times 1_X)^*\\alpha}", "}", "$$", "is a commutative in the category of", "$\\mathcal{O}_{G \\times_S G \\times_S X}$-modules, and", "\\item the pullback", "$$", "(e \\times 1_X)^*\\alpha : \\mathcal{F} \\longrightarrow \\mathcal{F}", "$$", "is the identity map.", "\\end{enumerate}", "For explanation compare with the relevant diagrams of", "Equation (\\ref{equation-action})." ], "refs": [], "ref_ids": [] }, { "id": 9681, "type": "definition", "label": "groupoids-definition-groupoid", "categories": [ "groupoids" ], "title": "groupoids-definition-groupoid", "contents": [ "Let $S$ be a scheme.", "\\begin{enumerate}", "\\item A {\\it groupoid scheme over $S$}, or simply a", "{\\it groupoid over $S$} is a", "quintuple $(U, R, s, t, c)$ where", "$U$ and $R$ are schemes over $S$, and", "$s, t : R \\to U$ and $c : R \\times_{s, U, t} R \\to R$", "are morphisms of schemes over $S$ with the", "following property: For any scheme", "$T$ over $S$ the quintuple", "$$", "(U(T), R(T), s, t, c)", "$$", "is a groupoid category in the sense described above.", "\\item A {\\it morphism", "$f : (U, R, s, t, c) \\to (U', R', s', t', c')$", "of groupoid schemes over $S$} is given by morphisms", "of schemes $f : U \\to U'$ and $f : R \\to R'$ with the", "following property: For any scheme", "$T$ over $S$ the maps $f$ define a functor from the", "groupoid category $(U(T), R(T), s, t, c)$ to the", "groupoid category $(U'(T), R'(T), s', t', c')$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9682, "type": "definition", "label": "groupoids-definition-groupoid-module", "categories": [ "groupoids" ], "title": "groupoids-definition-groupoid-module", "contents": [ "Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "A {\\it quasi-coherent module on $(U, R, s, t, c)$}", "is a pair $(\\mathcal{F}, \\alpha)$, where $\\mathcal{F}$ is a quasi-coherent", "$\\mathcal{O}_U$-module, and $\\alpha$ is a $\\mathcal{O}_R$-module", "map", "$$", "\\alpha : t^*\\mathcal{F} \\longrightarrow s^*\\mathcal{F}", "$$", "such that", "\\begin{enumerate}", "\\item the diagram", "$$", "\\xymatrix{", "& \\text{pr}_1^*t^*\\mathcal{F} \\ar[r]_-{\\text{pr}_1^*\\alpha} &", "\\text{pr}_1^*s^*\\mathcal{F} \\ar@{=}[rd] & \\\\", "\\text{pr}_0^*s^*\\mathcal{F} \\ar@{=}[ru] & & & c^*s^*\\mathcal{F} \\\\", "& \\text{pr}_0^*t^*\\mathcal{F} \\ar[lu]^{\\text{pr}_0^*\\alpha} \\ar@{=}[r] &", "c^*t^*\\mathcal{F} \\ar[ru]_{c^*\\alpha}", "}", "$$", "is a commutative in the category of", "$\\mathcal{O}_{R \\times_{s, U, t} R}$-modules, and", "\\item the pullback", "$$", "e^*\\alpha : \\mathcal{F} \\longrightarrow \\mathcal{F}", "$$", "is the identity map.", "\\end{enumerate}", "Compare with the commutative diagrams of Lemma \\ref{lemma-diagram}." ], "refs": [ "groupoids-lemma-diagram" ], "ref_ids": [ 9622 ] }, { "id": 9683, "type": "definition", "label": "groupoids-definition-stabilizer-groupoid", "categories": [ "groupoids" ], "title": "groupoids-definition-stabilizer-groupoid", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid over $S$.", "The group scheme $j^{-1}(\\Delta_{U/S})\\to U$", "is called the {\\it stabilizer of the groupoid scheme", "$(U, R, s, t, c)$}." ], "refs": [], "ref_ids": [] }, { "id": 9684, "type": "definition", "label": "groupoids-definition-restrict-groupoid", "categories": [ "groupoids" ], "title": "groupoids-definition-restrict-groupoid", "contents": [ "Let $S$ be a scheme.", "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.", "Let $g : U' \\to U$ be a morphism of schemes.", "The morphism of groupoids", "$(U', R', s', t', c') \\to (U, R, s, t, c)$", "constructed in Lemma \\ref{lemma-restrict-groupoid} is called", "the {\\it restriction of $(U, R, s, t, c)$ to $U'$}.", "We sometime use the notation $R' = R|_{U'}$ in this case." ], "refs": [ "groupoids-lemma-restrict-groupoid" ], "ref_ids": [ 9642 ] }, { "id": 9685, "type": "definition", "label": "groupoids-definition-invariant-open", "categories": [ "groupoids" ], "title": "groupoids-definition-invariant-open", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid scheme over the base scheme $S$.", "\\begin{enumerate}", "\\item A subset $W \\subset U$ is {\\it set-theoretically $R$-invariant}", "if $t(s^{-1}(W)) \\subset W$.", "\\item An open $W \\subset U$ is {\\it $R$-invariant} if", "$t(s^{-1}(W)) \\subset W$.", "\\item A closed subscheme $Z \\subset U$ is called {\\it $R$-invariant}", "if $t^{-1}(Z) = s^{-1}(Z)$. Here we use the scheme theoretic inverse image, see", "Schemes, Definition \\ref{schemes-definition-inverse-image-closed-subscheme}.", "\\item A monomorphism of schemes $T \\to U$ is {\\it $R$-invariant} if", "$T \\times_{U, t} R = R \\times_{s, U} T$ as schemes over $R$.", "\\end{enumerate}" ], "refs": [ "schemes-definition-inverse-image-closed-subscheme" ], "ref_ids": [ 7749 ] }, { "id": 9686, "type": "definition", "label": "groupoids-definition-quotient-sheaf", "categories": [ "groupoids" ], "title": "groupoids-definition-quotient-sheaf", "contents": [ "Let $\\tau$, $S$, and the pre-relation $j : R \\to U \\times_S U$ be as above.", "In this setting the {\\it quotient sheaf $U/R$} associated", "to $j$ is the sheafification of the presheaf", "(\\ref{equation-quotient-presheaf}) in the $\\tau$-topology.", "If $j : R \\to U \\times_S U$ comes from the action of a group scheme", "$G/S$ on $U$ as in Lemma \\ref{lemma-groupoid-from-action} then we", "sometimes denote the quotient sheaf $U/G$." ], "refs": [ "groupoids-lemma-groupoid-from-action" ], "ref_ids": [ 9636 ] }, { "id": 9687, "type": "definition", "label": "groupoids-definition-representable-quotient", "categories": [ "groupoids" ], "title": "groupoids-definition-representable-quotient", "contents": [ "In the situation of Definition \\ref{definition-quotient-sheaf}.", "We say that the pre-relation $j$ has a", "{\\it representable quotient} if the sheaf $U/R$ is representable.", "We will say a groupoid $(U, R, s, t, c)$ has a", "{\\it representable quotient}", "if the quotient $U/R$ with $j = (t, s)$ is representable." ], "refs": [ "groupoids-definition-quotient-sheaf" ], "ref_ids": [ 9686 ] }, { "id": 9688, "type": "definition", "label": "groupoids-definition-cartesian-morphism", "categories": [ "groupoids" ], "title": "groupoids-definition-cartesian-morphism", "contents": [ "Let $S$ be a scheme. Let $f : (U', R', s', t', c') \\to (U, R, s, t, c)$ be", "a morphism of groupoid schemes over $S$. We say $f$ is {\\it cartesian}, or", "that {\\it $(U', R', s', t', c')$ is cartesian over $(U, R, s, t, c)$},", "if the diagram", "$$", "\\xymatrix{", "R' \\ar[r]_f \\ar[d]_{s'} & R \\ar[d]^s \\\\", "U' \\ar[r]^f & U", "}", "$$", "is a fibre square in the category of schemes. A {\\it morphism of groupoid", "schemes cartesian over $(U, R, s, t, c)$} is a morphism of groupoid", "schemes compatible with the structure morphisms towards $(U, R, s, t, c)$." ], "refs": [], "ref_ids": [] }, { "id": 9792, "type": "definition", "label": "local-cohomology-definition-cd", "categories": [ "local-cohomology" ], "title": "local-cohomology-definition-cd", "contents": [ "Let $I \\subset A$ be a finitely generated ideal of a ring $A$.", "The smallest integer $d \\geq -1$ satisfying the equivalent conditions", "of Lemma \\ref{lemma-cd} is called the", "{\\it cohomological dimension of $I$ in $A$} and is", "denoted $\\text{cd}(A, I)$." ], "refs": [ "local-cohomology-lemma-cd" ], "ref_ids": [ 9703 ] }, { "id": 9793, "type": "definition", "label": "local-cohomology-definition-depth-complex", "categories": [ "local-cohomology" ], "title": "local-cohomology-definition-depth-complex", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$. Let", "$K \\in D^+_{\\textit{Coh}}(A)$. We define the {\\it $I$-depth} of $K$,", "denoted $\\text{depth}_I(K)$, to be the maximal", "$m \\in \\mathbf{Z} \\cup \\{\\infty\\}$ such that $H^i_I(K) = 0$ for all $i < m$.", "If $A$ is local with maximal ideal $\\mathfrak m$", "then we call $\\text{depth}_\\mathfrak m(K)$ simply the {\\it depth} of $K$." ], "refs": [], "ref_ids": [] }, { "id": 10594, "type": "definition", "label": "more-algebra-definition-stably-free", "categories": [ "more-algebra" ], "title": "more-algebra-definition-stably-free", "contents": [ "Let $R$ be a ring. ", "\\begin{enumerate}", "\\item Two modules $M$, $N$ over $R$ are said to be", "{\\it stably isomorphic} if there exist $n, m \\geq 0$ such", "that $M \\oplus R^{\\oplus m} \\cong N \\oplus R^{\\oplus n}$", "as $R$-modules.", "\\item A module $M$ is {\\it stably free} if it is stably isomorphic", "to a free module.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10595, "type": "definition", "label": "more-algebra-definition-fitting-ideal", "categories": [ "more-algebra" ], "title": "more-algebra-definition-fitting-ideal", "contents": [ "Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $k \\geq 0$.", "The {\\it $k$th Fitting ideal} of $M$ is the ideal $\\text{Fit}_k(M)$", "constructed in Lemma \\ref{lemma-fitting-ideal}. Set $\\text{Fit}_{-1}(M) = 0$." ], "refs": [ "more-algebra-lemma-fitting-ideal" ], "ref_ids": [ 9833 ] }, { "id": 10596, "type": "definition", "label": "more-algebra-definition-zariski-pair", "categories": [ "more-algebra" ], "title": "more-algebra-definition-zariski-pair", "contents": [ "A {\\it Zariski pair} is a pair $(A, I)$ such that", "$I$ is contained in the Jacobson radical of $A$." ], "refs": [], "ref_ids": [] }, { "id": 10597, "type": "definition", "label": "more-algebra-definition-henselian-pair", "categories": [ "more-algebra" ], "title": "more-algebra-definition-henselian-pair", "contents": [ "A {\\it henselian pair} is a pair $(A, I)$ satisfying", "\\begin{enumerate}", "\\item $I$ is contained in the Jacobson radical of $A$, and", "\\item for any monic polynomial $f \\in A[T]$ and factorization", "$\\overline{f} = g_0h_0$ with $g_0, h_0 \\in A/I[T]$ monic", "generating the unit ideal in $A/I[T]$, there", "exists a factorization $f = gh$ in $A[T]$ with $g, h$ monic", "and $g_0 = \\overline{g}$ and $h_0 = \\overline{h}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10598, "type": "definition", "label": "more-algebra-definition-absolutely-integrally-closed", "categories": [ "more-algebra" ], "title": "more-algebra-definition-absolutely-integrally-closed", "contents": [ "A ring $A$ is {\\it absolutely integrally closed} if every", "monic $f \\in A[T]$ is a product of linear factors." ], "refs": [], "ref_ids": [] }, { "id": 10599, "type": "definition", "label": "more-algebra-definition-auto-ass", "categories": [ "more-algebra" ], "title": "more-algebra-definition-auto-ass", "contents": [ "A ring $R$ is said to be {\\it auto-associated} if $R$ is local and its", "maximal ideal $\\mathfrak m$ is weakly associated to $R$." ], "refs": [], "ref_ids": [] }, { "id": 10600, "type": "definition", "label": "more-algebra-definition-torsion", "categories": [ "more-algebra" ], "title": "more-algebra-definition-torsion", "contents": [ "Let $R$ be a domain. Let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item We say an element $x \\in M$ is {\\it torsion} if there exists", "a nonzero $f \\in R$ such that $fx = 0$.", "\\item We say $M$ is {\\it torsion free} if the only torsion element of $M$", "is $0$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10601, "type": "definition", "label": "more-algebra-definition-reflexive", "categories": [ "more-algebra" ], "title": "more-algebra-definition-reflexive", "contents": [ "Let $R$ be a domain. We say an $R$-module $M$ is {\\it reflexive} if", "the natural map", "$$", "j : M \\longrightarrow \\Hom_R(\\Hom_R(M, R), R)", "$$", "which sends $m \\in M$ to the map sending $\\varphi \\in \\Hom_R(M, R)$", "to $\\varphi(m) \\in R$ is an isomorphism." ], "refs": [], "ref_ids": [] }, { "id": 10602, "type": "definition", "label": "more-algebra-definition-reflexive-hull", "categories": [ "more-algebra" ], "title": "more-algebra-definition-reflexive-hull", "contents": [ "Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module.", "The module $M^{**} = \\Hom_R(\\Hom_R(M, R), R)$ is called the", "{\\it reflexive hull} of $M$." ], "refs": [], "ref_ids": [] }, { "id": 10603, "type": "definition", "label": "more-algebra-definition-content-ideal", "categories": [ "more-algebra" ], "title": "more-algebra-definition-content-ideal", "contents": [ "Let $A$ be a ring. Let $M$ be a flat $A$-module. Let $x \\in M$.", "If the set of ideals $I$ in $A$ such that $x \\in IM$ has a", "smallest element, we call it the {\\it content ideal of $x$}." ], "refs": [], "ref_ids": [] }, { "id": 10604, "type": "definition", "label": "more-algebra-definition-strict-transform", "categories": [ "more-algebra" ], "title": "more-algebra-definition-strict-transform", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal and $a \\in I$.", "Let $R[\\frac{I}{a}]$ be the affine blowup algebra, see", "Algebra, Definition \\ref{algebra-definition-blow-up}.", "Let $M$ be an $R$-module.", "The {\\it strict transform of $M$ along $R \\to R[\\frac{I}{a}]$} is", "the $R[\\frac{I}{a}]$-module", "$$", "M' = \\left(M \\otimes_R R[\\textstyle{\\frac{I}{a}}]\\right)/a\\text{-power torsion}", "$$" ], "refs": [ "algebra-definition-blow-up" ], "ref_ids": [ 1488 ] }, { "id": 10605, "type": "definition", "label": "more-algebra-definition-koszul", "categories": [ "more-algebra" ], "title": "more-algebra-definition-koszul", "contents": [ "Let $R$ be a ring. Let $\\varphi : E \\to R$ be an $R$-module map. The", "{\\it Koszul complex} $K_\\bullet(\\varphi)$ associated to $\\varphi$", "is the commutative differential graded algebra defined as follows:", "\\begin{enumerate}", "\\item the underlying graded algebra is the exterior algebra", "$K_\\bullet(\\varphi) = \\wedge(E)$,", "\\item the differential $d : K_\\bullet(\\varphi) \\to K_\\bullet(\\varphi)$", "is the unique derivation such that $d(e) = \\varphi(e)$ for all", "$e \\in E = K_1(\\varphi)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10606, "type": "definition", "label": "more-algebra-definition-koszul-complex", "categories": [ "more-algebra" ], "title": "more-algebra-definition-koszul-complex", "contents": [ "Let $R$ be a ring and let $f_1, \\ldots, f_r \\in R$. The", "{\\it Koszul complex on $f_1, \\ldots, f_r$} is the Koszul complex", "associated to the map $(f_1, \\ldots, f_r) : R^{\\oplus r} \\to R$.", "Notation $K_\\bullet(f_\\bullet)$, $K_\\bullet(f_1, \\ldots, f_r)$,", "$K_\\bullet(R, f_1, \\ldots, f_r)$, or $K_\\bullet(R, f_\\bullet)$." ], "refs": [], "ref_ids": [] }, { "id": 10607, "type": "definition", "label": "more-algebra-definition-koszul-regular-sequence", "categories": [ "more-algebra" ], "title": "more-algebra-definition-koszul-regular-sequence", "contents": [ "Let $R$ be a ring. Let $r \\geq 0$ and let $f_1, \\ldots, f_r \\in R$", "be a sequence of elements. Let $M$ be an $R$-module.", "The sequence $f_1, \\ldots, f_r$ is called", "\\begin{enumerate}", "\\item {\\it $M$-Koszul-regular} if", "$H_i(K_\\bullet(f_1, \\ldots, f_r) \\otimes_R M) = 0$ for", "all $i \\not = 0$,", "\\item {\\it $M$-$H_1$-regular} if", "$H_1(K_\\bullet(f_1, \\ldots, f_r) \\otimes_R M) = 0$,", "\\item {\\it Koszul-regular} if $H_i(K_\\bullet(f_1, \\ldots, f_r)) = 0$ for", "all $i \\not = 0$, and", "\\item {\\it $H_1$-regular} if $H_1(K_\\bullet(f_1, \\ldots, f_r)) = 0$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10608, "type": "definition", "label": "more-algebra-definition-regular-ideal", "categories": [ "more-algebra" ], "title": "more-algebra-definition-regular-ideal", "contents": [ "Let $R$ be a ring and let $I \\subset R$ be an ideal.", "\\begin{enumerate}", "\\item We say $I$ is a {\\it regular ideal} if for every", "$\\mathfrak p \\in V(I)$ there exists a $g \\in R$, $g \\not \\in \\mathfrak p$", "and a regular sequence $f_1, \\ldots, f_r \\in R_g$ such that $I_g$", "is generated by $f_1, \\ldots, f_r$.", "\\item We say $I$ is a {\\it Koszul-regular ideal} if for every", "$\\mathfrak p \\in V(I)$ there exists a $g \\in R$, $g \\not \\in \\mathfrak p$", "and a Koszul-regular sequence $f_1, \\ldots, f_r \\in R_g$ such that $I_g$", "is generated by $f_1, \\ldots, f_r$.", "\\item We say $I$ is a {\\it $H_1$-regular ideal} if for every", "$\\mathfrak p \\in V(I)$ there exists a $g \\in R$, $g \\not \\in \\mathfrak p$", "and an $H_1$-regular sequence $f_1, \\ldots, f_r \\in R_g$ such that $I_g$", "is generated by $f_1, \\ldots, f_r$.", "\\item We say $I$ is a {\\it quasi-regular ideal} if for every", "$\\mathfrak p \\in V(I)$ there exists a $g \\in R$, $g \\not \\in \\mathfrak p$", "and a quasi-regular sequence $f_1, \\ldots, f_r \\in R_g$ such that $I_g$", "is generated by $f_1, \\ldots, f_r$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10609, "type": "definition", "label": "more-algebra-definition-local-complete-intersection", "categories": [ "more-algebra" ], "title": "more-algebra-definition-local-complete-intersection", "contents": [ "A ring map $A \\to B$ is called a {\\it local complete intersection}", "if it is of finite type and for some (equivalently any) presentation", "$B = A[x_1, \\ldots, x_n]/I$ the ideal $I$ is Koszul-regular." ], "refs": [], "ref_ids": [] }, { "id": 10610, "type": "definition", "label": "more-algebra-definition-topological-ring", "categories": [ "more-algebra" ], "title": "more-algebra-definition-topological-ring", "contents": [ "\\begin{reference}", "\\cite[Sections 7.1 and 7.2]{EGA1}", "\\end{reference}", "Let $R$ be a ring and let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item We say $R$ is a {\\it topological ring} if $R$ is endowed with a topology", "such that both addition and multiplication are continuous as maps", "$R \\times R \\to R$ where $R \\times R$ has the product topology.", "In this case we say $M$ is a {\\it topological module} if $M$ is endowed", "with a topology such that addition $M \\times M \\to M$ and", "scalar multiplication $R \\times M \\to M$ are continuous.", "\\item A {\\it homomorphism of topological modules} is just a continuous", "$R$-module map. A {\\it homomorphism of topological rings} is a", "ring homomorphism which is continuous for the given topologies.", "\\item We say $M$ is {\\it linearly topologized} if $0$ has a fundamental", "system of neighbourhoods consisting of submodules. We say $R$ is", "{\\it linearly topologized} if $0$ has a fundamental system of neighbourhoods", "consisting of ideals.", "\\item If $R$ is linearly topologized, we say that $I \\subset R$ is an", "{\\it ideal of definition} if $I$ is open and if every neighbourhood", "of $0$ contains $I^n$ for some $n$.", "\\item If $R$ is linearly topologized, we say that $R$ is {\\it pre-admissible}", "if $R$ has an ideal of definition.", "\\item If $R$ is linearly topologized, we say that $R$ is {\\it admissible} if", "it is pre-admissible and", "complete\\footnote{By our conventions this includes separated.}.", "\\item If $R$ is linearly topologized, we say that $R$ is {\\it pre-adic} if", "there exists an ideal of definition $I$ such that $\\{I^n\\}_{n \\geq 0}$", "forms a fundamental system of neighbourhoods of $0$.", "\\item If $R$ is linearly topologized, we say that $R$ is {\\it adic} if", "$R$ is pre-adic and complete.", "\\end{enumerate}", "Note that a (pre)adic topological ring is the same thing as a (pre)admissible", "topological ring which has an ideal of definition $I$ such that $I^n$ is", "open for all $n \\geq 1$." ], "refs": [], "ref_ids": [] }, { "id": 10611, "type": "definition", "label": "more-algebra-definition-formally-smooth", "categories": [ "more-algebra" ], "title": "more-algebra-definition-formally-smooth", "contents": [ "Let $R \\to S$ be a homomorphism of topological rings with $R$ and $S$", "linearly topologized. We say $S$ is {\\it formally smooth over $R$} if", "for every commutative solid diagram", "$$", "\\xymatrix{", "S \\ar[r] \\ar@{-->}[rd] & A/J \\\\", "R \\ar[r] \\ar[u] & A \\ar[u]", "}", "$$", "of homomorphisms of topological rings where $A$ is a discrete ring and", "$J \\subset A$ is an ideal of square zero, a dotted arrow exists which", "makes the diagram commute." ], "refs": [], "ref_ids": [] }, { "id": 10612, "type": "definition", "label": "more-algebra-definition-formally-smooth-adic", "categories": [ "more-algebra" ], "title": "more-algebra-definition-formally-smooth-adic", "contents": [ "Let $R \\to S$ be a ring map. Let $\\mathfrak n \\subset S$ be an", "ideal. If the equivalent conditions (2)(a) and (2)(b) of", "Lemma \\ref{lemma-formally-smooth} hold, then we say", "$R \\to S$ is {\\it formally smooth for the $\\mathfrak n$-adic topology}." ], "refs": [ "more-algebra-lemma-formally-smooth" ], "ref_ids": [ 10014 ] }, { "id": 10613, "type": "definition", "label": "more-algebra-definition-regular", "categories": [ "more-algebra" ], "title": "more-algebra-definition-regular", "contents": [ "A ring map $R \\to \\Lambda$ is {\\it regular} if it is flat and", "for every prime $\\mathfrak p \\subset R$ the fibre ring", "$$", "\\Lambda \\otimes_R \\kappa(\\mathfrak p) =", "\\Lambda_\\mathfrak p/\\mathfrak p\\Lambda_\\mathfrak p", "$$", "is Noetherian and geometrically regular over $\\kappa(\\mathfrak p)$." ], "refs": [], "ref_ids": [] }, { "id": 10614, "type": "definition", "label": "more-algebra-definition-p-basis", "categories": [ "more-algebra" ], "title": "more-algebra-definition-p-basis", "contents": [ "Let $p$ be a prime number. Let $k \\to K$ be an extension of fields", "of characteristic $p$. Denote $kK^p$ the compositum of $k$ and $K^p$", "in $K$.", "\\begin{enumerate}", "\\item A subset $\\{x_i\\} \\subset K$ is called {\\it p-independent", "over $k$} if the elements $x^E = \\prod x_i^{e_i}$ where", "$0 \\leq e_i < p$ are linearly independent over $kK^p$.", "\\item A subset $\\{x_i\\}$ of $K$ is called a", "{\\it p-basis of $K$ over $k$} if the elements", "$x^E$ form a basis of $K$ over $kK^p$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10615, "type": "definition", "label": "more-algebra-definition-J", "categories": [ "more-algebra" ], "title": "more-algebra-definition-J", "contents": [ "Let $R$ be a Noetherian ring. Let $X = \\Spec(R)$.", "\\begin{enumerate}", "\\item We say $R$ is {\\it J-0} if $\\text{Reg}(X)$ contains a nonempty open.", "\\item We say $R$ is {\\it J-1} if $\\text{Reg}(X)$ is open.", "\\item We say $R$ is {\\it J-2} if any finite type $R$-algebra is J-1.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10616, "type": "definition", "label": "more-algebra-definition-G-ring", "categories": [ "more-algebra" ], "title": "more-algebra-definition-G-ring", "contents": [ "A ring $R$ is called a {\\it G-ring} if $R$ is Noetherian and for every", "prime $\\mathfrak p$ of $R$ the ring map", "$R_\\mathfrak p \\to (R_\\mathfrak p)^\\wedge$ is regular." ], "refs": [], "ref_ids": [] }, { "id": 10617, "type": "definition", "label": "more-algebra-definition-excellent", "categories": [ "more-algebra" ], "title": "more-algebra-definition-excellent", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item We say $R$ is {\\it quasi-excellent} if $R$ is Noetherian,", "a G-ring, and J-2.", "\\item We say $R$ is {\\it excellent} if $R$ is quasi-excellent", "and universally catenary.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10618, "type": "definition", "label": "more-algebra-definition-projective", "categories": [ "more-algebra" ], "title": "more-algebra-definition-projective", "contents": [ "Let $R$ be a ring. An $R$-module $J$ is {\\it injective} if and only if", "the functor $\\Hom_R(-, J) : \\text{Mod}_R \\to \\text{Mod}_R$ is", "an exact functor." ], "refs": [], "ref_ids": [] }, { "id": 10619, "type": "definition", "label": "more-algebra-definition-simple-functors", "categories": [ "more-algebra" ], "title": "more-algebra-definition-simple-functors", "contents": [ "Let $R$ be a ring.", "\\begin{enumerate}", "\\item For any $R$-module $M$ over $R$ we denote", "$M^\\vee = \\Hom(M, \\mathbf{Q}/\\mathbf{Z})$", "with its natural $R$-module structure. We think", "of {\\it $M \\mapsto M^\\vee$} as a contravariant functor", "from the category of $R$-modules to itself.", "\\item For any $R$-module $M$ we denote", "$$", "F(M) = \\bigoplus\\nolimits_{m \\in M} R[m]", "$$", "the {\\it free module} with basis given by the elements $[m]$ with", "$m \\in M$. We let $F(M)\\to M$, $\\sum f_i [m_i] \\mapsto \\sum f_i m_i$", "be the natural surjection of $R$-modules.", "We think of $M \\mapsto (F(M) \\to M)$ as a functor from", "the category of $R$-modules to the category of", "arrows in $R$-modules.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10620, "type": "definition", "label": "more-algebra-definition-K-flat", "categories": [ "more-algebra" ], "title": "more-algebra-definition-K-flat", "contents": [ "Let $R$ be a ring. A complex $K^\\bullet$ is called {\\it K-flat}", "if for every acyclic complex $M^\\bullet$ the total complex", "$\\text{Tot}(M^\\bullet \\otimes_R K^\\bullet)$ is acyclic." ], "refs": [], "ref_ids": [] }, { "id": 10621, "type": "definition", "label": "more-algebra-definition-derived-tor", "categories": [ "more-algebra" ], "title": "more-algebra-definition-derived-tor", "contents": [ "Let $R$ be a ring. Let $M^\\bullet$ be an object of $D(R)$.", "The {\\it derived tensor product}", "$$", "- \\otimes_R^{\\mathbf{L}} M^\\bullet : D(R) \\longrightarrow D(R)", "$$", "is the exact functor of triangulated categories described above." ], "refs": [], "ref_ids": [] }, { "id": 10622, "type": "definition", "label": "more-algebra-definition-tor-independent", "categories": [ "more-algebra" ], "title": "more-algebra-definition-tor-independent", "contents": [ "Let $R$ be a ring. Let $A$, $B$ be $R$-algebras. We say", "$A$ and $B$ are {\\it Tor independent over $R$} if", "$\\text{Tor}_p^R(A, B) = 0$ for all $p > 0$." ], "refs": [], "ref_ids": [] }, { "id": 10623, "type": "definition", "label": "more-algebra-definition-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-definition-pseudo-coherent", "contents": [ "Let $R$ be a ring. Denote $D(R)$ its derived category.", "Let $m \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item An object $K^\\bullet$ of $D(R)$ is {\\it $m$-pseudo-coherent}", "if there exists a bounded complex $E^\\bullet$ of finite free $R$-modules", "and a morphism $\\alpha : E^\\bullet \\to K^\\bullet$ such that", "$H^i(\\alpha)$ is an isomorphism for $i > m$ and $H^m(\\alpha)$", "is surjective.", "\\item An object $K^\\bullet$ of $D(R)$ is {\\it pseudo-coherent}", "if it is quasi-isomorphic to a bounded above complex of finite", "free $R$-modules.", "\\item An $R$-module $M$ is called {\\it $m$-pseudo-coherent}", "if $M[0]$ is an $m$-pseudo-coherent object of $D(R)$.", "\\item An $R$-module $M$ is called", "{\\it pseudo-coherent}\\footnote{This clashes with what is meant by", "a pseudo-coherent module in \\cite{Bourbaki-CA}.}", "if $M[0]$ is a pseudo-coherent object of $D(R)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10624, "type": "definition", "label": "more-algebra-definition-tor-amplitude", "categories": [ "more-algebra" ], "title": "more-algebra-definition-tor-amplitude", "contents": [ "Let $R$ be a ring. Denote $D(R)$ its derived category.", "Let $a, b \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item An object $K^\\bullet$ of $D(R)$ has", "{\\it tor-amplitude in $[a, b]$}", "if $H^i(K^\\bullet \\otimes_R^\\mathbf{L} M) = 0$ for all $R$-modules", "$M$ and all $i \\not \\in [a, b]$.", "\\item An object $K^\\bullet$ of $D(R)$ has {\\it finite tor dimension}", "if it has tor-amplitude in $[a, b]$ for some $a, b$.", "\\item An $R$-module $M$ has {\\it tor dimension $\\leq d$}", "if $M[0]$ as an object of $D(R)$ has tor-amplitude in $[-d, 0]$.", "\\item An $R$-module $M$ has {\\it finite tor dimension}", "if $M[0]$ as an object of $D(R)$ has finite tor dimension.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10625, "type": "definition", "label": "more-algebra-definition-projective-dimension", "categories": [ "more-algebra" ], "title": "more-algebra-definition-projective-dimension", "contents": [ "Let $R$ be a ring. Let $K$ be an object of $D(R)$. We say $K$ has", "{\\it finite projective dimension} if $K$ can be represented by a", "bounded complex of projective modules. We say $K$ as", "{\\it projective-amplitude in $[a, b]$} if $K$ is quasi-isomorphic", "to a complex", "$$", "\\ldots \\to 0 \\to P^a \\to P^{a + 1} \\to \\ldots \\to", "P^{b - 1} \\to P^b \\to 0 \\to \\ldots", "$$", "where $P^i$ is a projective $R$-module for all $i \\in \\mathbf{Z}$." ], "refs": [], "ref_ids": [] }, { "id": 10626, "type": "definition", "label": "more-algebra-definition-injective-dimension", "categories": [ "more-algebra" ], "title": "more-algebra-definition-injective-dimension", "contents": [ "Let $R$ be a ring. Let $K$ be an object of $D(R)$.", "We say $K$ has {\\it finite injective dimension} if $K$ can be", "represented by a finite complex of injective $R$-modules.", "We say $K$ has {\\it injective-amplitude in $[a, b]$}", "if $K$ is isomorphic to a complex", "$$", "\\ldots \\to 0 \\to I^a \\to I^{a + 1} \\to \\ldots \\to", "I^{b - 1} \\to I^b \\to 0 \\to \\ldots", "$$", "with $I^i$ an injective $R$-module for all $i \\in \\mathbf{Z}$." ], "refs": [], "ref_ids": [] }, { "id": 10627, "type": "definition", "label": "more-algebra-definition-near-projective", "categories": [ "more-algebra" ], "title": "more-algebra-definition-near-projective", "contents": [ "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module.", "We say $M$ is {\\it $I$-projective}\\footnote{This is nonstandard notation.}", "if the equivalent conditions of Lemma \\ref{lemma-near-projective} hold." ], "refs": [ "more-algebra-lemma-near-projective" ], "ref_ids": [ 10194 ] }, { "id": 10628, "type": "definition", "label": "more-algebra-definition-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-definition-perfect", "contents": [ "Let $R$ be a ring. Denote $D(R)$ the derived category of the abelian", "category of $R$-modules.", "\\begin{enumerate}", "\\item An object $K$ of $D(R)$ is {\\it perfect} if it is quasi-isomorphic", "to a bounded complex of finite projective $R$-modules.", "\\item An $R$-module $M$ is {\\it perfect} if $M[0]$ is a perfect object", "in $D(R)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10629, "type": "definition", "label": "more-algebra-definition-relatively-finitely-presented", "categories": [ "more-algebra" ], "title": "more-algebra-definition-relatively-finitely-presented", "contents": [ "Let $R \\to A$ be a finite type ring map. Let $M$ be an $A$-module.", "We say $M$ is an $A$-module {\\it finitely presented relative to $R$}", "if the equivalent conditions of", "Lemma \\ref{lemma-relatively-finitely-presented}", "hold." ], "refs": [ "more-algebra-lemma-relatively-finitely-presented" ], "ref_ids": [ 10256 ] }, { "id": 10630, "type": "definition", "label": "more-algebra-definition-relatively-pseudo-coherent", "categories": [ "more-algebra" ], "title": "more-algebra-definition-relatively-pseudo-coherent", "contents": [ "Let $R \\to A$ be a finite type ring map.", "Let $K^\\bullet$ be a complex of $A$-modules.", "Let $M$ be an $A$-module.", "Let $m \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item We say $K^\\bullet$ is {\\it $m$-pseudo-coherent relative to $R$}", "if the equivalent conditions of", "Lemma \\ref{lemma-relatively-pseudo-coherent}", "hold.", "\\item We say $K^\\bullet$ is {\\it pseudo-coherent relative to $R$}", "if $K^\\bullet$ is $m$-pseudo-coherent relative to $R$ for all", "$m \\in \\mathbf{Z}$.", "\\item We say $M$ is {\\it $m$-pseudo-coherent relative to $R$}", "if $M[0]$ is $m$-pseudo-coherent relative to $R$.", "\\item We say $M$ is {\\it pseudo-coherent relative to $R$}", "if $M[0]$ is pseudo-coherent relative to $R$.", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-relatively-pseudo-coherent" ], "ref_ids": [ 10267 ] }, { "id": 10631, "type": "definition", "label": "more-algebra-definition-pseudo-coherent-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-definition-pseudo-coherent-perfect", "contents": [ "Let $A \\to B$ be a ring map.", "\\begin{enumerate}", "\\item We say $A \\to B$ is a {\\it pseudo-coherent ring map} if it is of finite", "type and $B$, as a $B$-module, is pseudo-coherent relative to $A$.", "\\item We say $A \\to B$ is a {\\it perfect ring map} if it is a", "pseudo-coherent ring map such that $B$ as an $A$-module has finite", "tor dimension.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10632, "type": "definition", "label": "more-algebra-definition-relatively-perfect", "categories": [ "more-algebra" ], "title": "more-algebra-definition-relatively-perfect", "contents": [ "Let $R \\to A$ be a flat ring map of finite presentation.", "An object $K$ of $D(A)$ is {\\it $R$-perfect} or {\\it perfect relative to $R$}", "if $K$ is pseudo-coherent", "(Definition \\ref{definition-pseudo-coherent})", "and has finite tor dimension over $R$", "(Definition \\ref{definition-tor-amplitude})." ], "refs": [ "more-algebra-definition-pseudo-coherent", "more-algebra-definition-tor-amplitude" ], "ref_ids": [ 10623, 10624 ] }, { "id": 10633, "type": "definition", "label": "more-algebra-definition-f-power-torsion", "categories": [ "more-algebra" ], "title": "more-algebra-definition-f-power-torsion", "contents": [ "Let $R$ be a ring. Let $M$ be an $R$-module.", "\\begin{enumerate}", "\\item Let $I \\subset R$ be an ideal. We say $M$ is an", "{\\it $I$-power torsion module} if for every $m \\in M$ there exists an $n > 0$", "such that $I^n m = 0$.", "\\item Let $f \\in R$. We say $M$ is", "{\\it an $f$-power torsion module} if for each", "$m \\in M$, there exists an $n > 0$ such that $f^n m = 0$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10634, "type": "definition", "label": "more-algebra-definition-derived-complete", "categories": [ "more-algebra" ], "title": "more-algebra-definition-derived-complete", "contents": [ "Let $A$ be a ring. Let $K \\in D(A)$. Let $I \\subset A$ be an ideal.", "We say $K$ is {\\it derived complete with respect to $I$}", "if for every $f \\in I$ we have $T(K, f) = 0$.", "If $M$ is an $A$-module, then we say $M$ is", "{\\it derived complete with respect to $I$}", "if $M[0] \\in D(A)$ is derived complete with respect to $I$." ], "refs": [], "ref_ids": [] }, { "id": 10635, "type": "definition", "label": "more-algebra-definition-weakly-etale", "categories": [ "more-algebra" ], "title": "more-algebra-definition-weakly-etale", "contents": [ "A ring $A$ is called {\\it absolutely flat} if every $A$-module is flat over", "$A$. A ring map $A \\to B$ is {\\it weakly \\'etale} or {\\it absolutely flat}", "if both $A \\to B$ and $B \\otimes_A B \\to B$ are flat." ], "refs": [], "ref_ids": [] }, { "id": 10636, "type": "definition", "label": "more-algebra-definition-weak-dimension", "categories": [ "more-algebra" ], "title": "more-algebra-definition-weak-dimension", "contents": [ "Let $A$ be a ring. Let $d \\geq 0$ be an integer.", "We say that $A$ has {\\it weak dimension $\\leq d$}", "if every $A$-module has tor dimension $\\leq d$." ], "refs": [], "ref_ids": [] }, { "id": 10637, "type": "definition", "label": "more-algebra-definition-unibranch", "categories": [ "more-algebra" ], "title": "more-algebra-definition-unibranch", "contents": [ "\\begin{reference}", "\\cite[Chapter 0 (23.2.1)]{EGA4}", "\\end{reference}", "Let $A$ be a local ring. We say $A$ is {\\it unibranch}", "if the reduction $A_{red}$ is a domain and if the integral closure", "$A'$ of $A_{red}$ in its field of fractions is local.", "We say $A$ is {\\it geometrically unibranch} if $A$ is unibranch", "and moreover the residue field of $A'$ is purely inseparable over", "the residue field of $A$." ], "refs": [], "ref_ids": [] }, { "id": 10638, "type": "definition", "label": "more-algebra-definition-number-of-branches", "categories": [ "more-algebra" ], "title": "more-algebra-definition-number-of-branches", "contents": [ "Let $A$ be a local ring with henselization $A^h$ and", "strict henselization $A^{sh}$. The {\\it number of branches of $A$}", "is the number of minimal primes of $A^h$ if finite and $\\infty$", "otherwise. The {\\it number of geometric branches of $A$}", "is the number of minimal primes of $A^{sh}$ if finite and $\\infty$", "otherwise." ], "refs": [], "ref_ids": [] }, { "id": 10639, "type": "definition", "label": "more-algebra-definition-formally-catenary", "categories": [ "more-algebra" ], "title": "more-algebra-definition-formally-catenary", "contents": [ "A Noetherian local ring $A$ is {\\it formally catenary}", "if for every minimal prime $\\mathfrak p \\subset A$ the spectrum of", "$A^\\wedge/\\mathfrak p A^\\wedge$ is equidimensional." ], "refs": [], "ref_ids": [] }, { "id": 10640, "type": "definition", "label": "more-algebra-definition-extension-discrete-valuation-rings", "categories": [ "more-algebra" ], "title": "more-algebra-definition-extension-discrete-valuation-rings", "contents": [ "We say that $A \\to B$ or $A \\subset B$ is an", "{\\it extension of discrete valuation rings} if $A$ and $B$ are", "discrete valuation rings and $A \\to B$ is injective and local.", "In particular, if $\\pi_A$ and $\\pi_B$ are uniformizers of", "$A$ and $B$, then $\\pi_A = u \\pi_B^e$ for some $e \\geq 1$ and unit", "$u$ of $B$. The integer $e$ does not depend on the choice of", "the uniformizers as it is also the unique integer $\\geq 1$ such that", "$$", "\\mathfrak m_A B = \\mathfrak m_B^e", "$$", "The integer $e$ is called the {\\it ramification index} of $B$ over $A$.", "We say that $B$ is {\\it weakly unramified} over $A$ if $e = 1$.", "If the extension of residue fields", "$\\kappa_A = A/\\mathfrak m_A \\subset \\kappa_B = B/\\mathfrak m_B$", "is finite, then we set $f = [\\kappa_B : \\kappa_A]$ and we", "call it the {\\it residual degree} or {\\it residue degree}", "of the extension $A \\subset B$." ], "refs": [], "ref_ids": [] }, { "id": 10641, "type": "definition", "label": "more-algebra-definition-types-of-extensions", "categories": [ "more-algebra" ], "title": "more-algebra-definition-types-of-extensions", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$. Let $L \\supset K$", "be a finite separable extension. With $B$ and", "$\\mathfrak m_i$, $i = 1, \\ldots, n$", "as in Remark \\ref{remark-finite-separable-extension} we say the extension", "$L/K$ is", "\\begin{enumerate}", "\\item {\\it unramified with respect to $A$} if $e_i = 1$ and the extension", "$\\kappa_A \\subset \\kappa(\\mathfrak m_i)$ is separable for all $i$,", "\\item {\\it tamely ramified with respect to $A$}", "if either the characteristic of $\\kappa_A$", "is $0$ or the characteristic of $\\kappa_A$ is $p > 0$, the field extensions", "$\\kappa_A \\subset \\kappa(\\mathfrak m_i)$ are separable,", "and the ramification indices $e_i$ are prime to $p$, and", "\\item {\\it totally ramified with respect to $A$}", "if $n = 1$ and the residue field extension", "$\\kappa_A \\subset \\kappa(\\mathfrak m_1)$ is trivial.", "\\end{enumerate}", "If the discrete valuation ring $A$ is clear from context, then we sometimes", "say $L/K$ is unramified, totally ramified, or tamely ramified for short." ], "refs": [ "more-algebra-remark-finite-separable-extension" ], "ref_ids": [ 10676 ] }, { "id": 10642, "type": "definition", "label": "more-algebra-definition-decomposition-inertia", "categories": [ "more-algebra" ], "title": "more-algebra-definition-decomposition-inertia", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "Let $L/K$ be a finite Galois extension with Galois group $G$.", "Let $B$ be the integral closure of $A$ in $L$.", "Let $\\mathfrak m \\subset B$ be a maximal ideal.", "\\begin{enumerate}", "\\item The {\\it decomposition group of $\\mathfrak m$}", "is the subgroup $D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak m) = \\mathfrak m\\}$.", "\\item The {\\it inertia group of $\\mathfrak m$} is the kernel $I$ of the map", "$D \\to \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa_A)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10643, "type": "definition", "label": "more-algebra-definition-wild-inertia", "categories": [ "more-algebra" ], "title": "more-algebra-definition-wild-inertia", "contents": [ "With assumptions and notation as in Lemma \\ref{lemma-galois-inertia}.", "\\begin{enumerate}", "\\item The {\\it wild inertia group of $\\mathfrak m$} is the subgroup $P$.", "\\item The {\\it tame inertia group of $\\mathfrak m$} is the", "quotient $I \\to I_t$.", "\\end{enumerate}", "We denote $\\theta : I \\to \\mu_e(\\kappa(\\mathfrak m))$ the surjective map", "(\\ref{equation-inertia-character}) whose kernel is $P$ and which", "induces the isomorphism $I_t \\to \\mu_e(\\kappa(\\mathfrak m))$." ], "refs": [ "more-algebra-lemma-galois-inertia" ], "ref_ids": [ 10501 ] }, { "id": 10644, "type": "definition", "label": "more-algebra-definition-mixed", "categories": [ "more-algebra" ], "title": "more-algebra-definition-mixed", "contents": [ "Let $A$ be a discrete valuation ring. We say $A$ has {\\it mixed characteristic}", "if the characteristic of the residue field of $A$ is $p > 0$ and the", "characteristic of the fraction field of $A$ is $0$.", "In this case we obtain an extension of discrete valuation rings", "$\\mathbf{Z}_{(p)} \\subset A$ and the {\\it absolute ramification index}", "of $A$ is the ramification index of this extension." ], "refs": [], "ref_ids": [] }, { "id": 10645, "type": "definition", "label": "more-algebra-definition-solution", "categories": [ "more-algebra" ], "title": "more-algebra-definition-solution", "contents": [ "Let $A \\to B$ be an extension of discrete valuation rings with fraction", "fields $K \\subset L$.", "\\begin{enumerate}", "\\item We say a finite field extension $K \\subset K_1$ is a", "{\\it weak solution for $A \\subset B$} if all the extensions", "$(A_1)_{\\mathfrak m_i} \\subset (B_1)_{\\mathfrak m_{ij}}$ of", "Remark \\ref{remark-construction} are weakly unramified.", "\\item We say a finite field extension $K \\subset K_1$ is a", "{\\it solution for $A \\subset B$} if each extension", "$(A_1)_{\\mathfrak m_i} \\subset (B_1)_{\\mathfrak m_{ij}}$ of", "Remark \\ref{remark-construction} is formally smooth in", "the $\\mathfrak m_{ij}$-adic topology.", "\\end{enumerate}", "We say a solution $K \\subset K_1$ is a {\\it separable solution}", "if $K \\subset K_1$ is separable." ], "refs": [ "more-algebra-remark-construction", "more-algebra-remark-construction" ], "ref_ids": [ 10679, 10679 ] }, { "id": 10646, "type": "definition", "label": "more-algebra-definition-invertible", "categories": [ "more-algebra" ], "title": "more-algebra-definition-invertible", "contents": [ "Let $R$ be a ring. An $R$-module $M$ is {\\it invertible} if the functor", "$$", "\\text{Mod}_R \\longrightarrow \\text{Mod}_R,\\quad", "N \\longmapsto M \\otimes_R N", "$$", "is an equivalence of categories. An invertible $R$-module is said to be", "{\\it trivial} if it is isomorphic to $A$ as an $A$-module." ], "refs": [], "ref_ids": [] }, { "id": 10647, "type": "definition", "label": "more-algebra-definition-extension-valuation-rings", "categories": [ "more-algebra" ], "title": "more-algebra-definition-extension-valuation-rings", "contents": [ "We say that $A \\to B$ or $A \\subset B$ is an", "{\\it extension of valuation rings} if $A$ and $B$ are", "valuation rings and $A \\to B$ is injective and local.", "Such an extension induces a commutative diagram", "$$", "\\xymatrix{", "A \\setminus \\{0\\} \\ar[r] \\ar[d]_v & B \\setminus \\{0\\} \\ar[d]^v \\\\", "\\Gamma_A \\ar[r] & \\Gamma_B", "}", "$$", "where $\\Gamma_A$ and $\\Gamma_B$ are the value groups.", "We say that $B$ is {\\it weakly unramified} over $A$ if", "the lower horizontal arrow is a bijection.", "If the extension of residue fields", "$\\kappa_A = A/\\mathfrak m_A \\subset \\kappa_B = B/\\mathfrak m_B$", "is finite, then we set $f = [\\kappa_B : \\kappa_A]$ and we", "call it the {\\it residual degree} or {\\it residue degree}", "of the extension $A \\subset B$." ], "refs": [], "ref_ids": [] }, { "id": 10648, "type": "definition", "label": "more-algebra-definition-bezout", "categories": [ "more-algebra" ], "title": "more-algebra-definition-bezout", "contents": [ "Let $R$ be a domain.", "\\begin{enumerate}", "\\item We say $R$ is a {\\it B\\'ezout domain} if every finitely generated", "ideal of $R$ is principal.", "\\item We say $R$ is an {\\it elementary divisor domain} if for", "all $n , m \\geq 1$ and every $n \\times m$ matrix $A$, there", "exist invertible matrices $U, V$ of size $n \\times n, m \\times m$", "such that", "$$", "U A V =", "\\left(", "\\begin{matrix}", "f_1 & 0 & 0 & \\ldots \\\\", "0 & f_2 & 0 & \\ldots \\\\", "0 & 0 & f_3 & \\ldots \\\\", "\\ldots & \\ldots & \\ldots & \\ldots", "\\end{matrix}", "\\right)", "$$", "with $f_1, \\ldots, f_{\\min(n, m)} \\in R$ and $f_1 | f_2 | \\ldots$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10734, "type": "definition", "label": "etale-definition-unramified-rings", "categories": [ "etale" ], "title": "etale-definition-unramified-rings", "contents": [ "Let $A$, $B$ be Noetherian local rings. A local homomorphism $A \\to B$", "is said to be {\\it unramified homomorphism of local rings} if", "\\begin{enumerate}", "\\item $\\mathfrak m_AB = \\mathfrak m_B$,", "\\item $\\kappa(\\mathfrak m_B)$ is a finite separable extension of", "$\\kappa(\\mathfrak m_A)$, and", "\\item $B$ is essentially of finite type over $A$ (this means", "that $B$ is the localization of a finite type $A$-algebra at a prime).", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10735, "type": "definition", "label": "etale-definition-unramified-schemes", "categories": [ "etale" ], "title": "etale-definition-unramified-schemes", "contents": [ "(See Morphisms, Definition \\ref{morphisms-definition-unramified}", "for the definition in the general case.)", "Let $Y$ be a locally Noetherian scheme.", "Let $f : X \\to Y$ be locally of finite type.", "Let $x \\in X$.", "\\begin{enumerate}", "\\item We say $f$ is {\\it unramified at $x$} if", "$\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$", "is an unramified homomorphism of local rings.", "\\item The morphism $f : X \\to Y$ is said to be {\\it unramified}", "if it is unramified at all points of $X$.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-unramified" ], "ref_ids": [ 5566 ] }, { "id": 10736, "type": "definition", "label": "etale-definition-flat-rings", "categories": [ "etale" ], "title": "etale-definition-flat-rings", "contents": [ "Flatness of modules and rings.", "\\begin{enumerate}", "\\item A module $N$ over a ring $A$ is said to be {\\it flat}", "if the functor $M \\mapsto M \\otimes_A N$ is exact.", "\\item If this functor is also faithful, we say that", "$N$ is {\\it faithfully flat} over $A$.", "\\item A morphism of rings $f : A \\to B$ is said to be", "{\\it flat (resp. faithfully flat)}", "if the functor $M \\mapsto M \\otimes_A B$ is exact", "(resp. faithful and exact).", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10737, "type": "definition", "label": "etale-definition-flat-schemes", "categories": [ "etale" ], "title": "etale-definition-flat-schemes", "contents": [ "(See Morphisms, Definition \\ref{morphisms-definition-flat}).", "Let $f : X \\to Y$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item Let $x \\in X$. We say $\\mathcal{F}$ is", "{\\it flat over $Y$ at $x \\in X$} if $\\mathcal{F}_x$", "is a flat $\\mathcal{O}_{Y, f(x)}$-module.", "This uses the map $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ to", "think of $\\mathcal{F}_x$ as a $\\mathcal{O}_{Y, f(x)}$-module.", "\\item Let $x \\in X$. We say $f$ is {\\it flat at $x \\in X$}", "if $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is flat.", "\\item We say $f$ is {\\it flat} if it is flat at all points of $X$.", "\\item A morphism $f : X \\to Y$ that is flat and surjective is sometimes", "said to be {\\it faithfully flat}.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-flat" ], "ref_ids": [ 5557 ] }, { "id": 10738, "type": "definition", "label": "etale-definition-etale-ring", "categories": [ "etale" ], "title": "etale-definition-etale-ring", "contents": [ "Let $A$, $B$ be Noetherian local rings.", "A local homomorphism $f : A \\to B$ is said to be an", "{\\it \\'etale homomorphism of local rings}", "if it is flat and an unramified homomorphism of local rings", "(please see Definition \\ref{definition-unramified-rings})." ], "refs": [ "etale-definition-unramified-rings" ], "ref_ids": [ 10734 ] }, { "id": 10739, "type": "definition", "label": "etale-definition-etale-schemes-1", "categories": [ "etale" ], "title": "etale-definition-etale-schemes-1", "contents": [ "(See Morphisms, Definition \\ref{morphisms-definition-etale}.)", "Let $Y$ be a locally Noetherian scheme.", "Let $f : X \\to Y$ be a morphism of schemes which is locally of finite type.", "\\begin{enumerate}", "\\item Let $x \\in X$. We say $f$ is {\\it \\'etale at $x \\in X$} if", "$\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is an", "\\'etale homomorphism of local rings.", "\\item The morphism is said to be {\\it \\'etale} if it is \\'etale at all its", "points.", "\\end{enumerate}" ], "refs": [ "morphisms-definition-etale" ], "ref_ids": [ 5567 ] }, { "id": 10740, "type": "definition", "label": "etale-definition-strict-normal-crossings", "categories": [ "etale" ], "title": "etale-definition-strict-normal-crossings", "contents": [ "Let $X$ be a locally Noetherian scheme. A", "{\\it strict normal crossings divisor}", "on $X$ is an effective Cartier divisor $D \\subset X$ such that", "for every $p \\in D$ the local ring $\\mathcal{O}_{X, p}$ is regular", "and there exists a regular system of parameters", "$x_1, \\ldots, x_d \\in \\mathfrak m_p$ and $1 \\leq r \\leq d$", "such that $D$ is cut out by $x_1 \\ldots x_r$ in $\\mathcal{O}_{X, p}$." ], "refs": [], "ref_ids": [] }, { "id": 10741, "type": "definition", "label": "etale-definition-normal-crossings", "categories": [ "etale" ], "title": "etale-definition-normal-crossings", "contents": [ "Let $X$ be a locally Noetherian scheme. A {\\it normal crossings divisor}", "on $X$ is an effective Cartier divisor $D \\subset X$ such that for", "every $p \\in D$ there exists an \\'etale morphism $U \\to X$ with", "$p$ in the image and $D \\times_X U$ a", "strict normal crossings divisor on $U$." ], "refs": [], "ref_ids": [] }, { "id": 10797, "type": "definition", "label": "crystalline-definition-divided-power-envelope", "categories": [ "crystalline" ], "title": "crystalline-definition-divided-power-envelope", "contents": [ "Let $(A, I, \\gamma)$ be a divided power ring.", "Let $A \\to B$ be a ring map. Let $J \\subset B$ be an ideal", "with $IB \\subset J$. The divided power algebra $(D, \\bar J, \\bar\\gamma)$", "constructed in Lemma \\ref{lemma-divided-power-envelope}", "is called the {\\it divided power envelope of $J$ in $B$", "relative to $(A, I, \\gamma)$} and is denoted $D_B(J)$ or $D_{B, \\gamma}(J)$." ], "refs": [ "crystalline-lemma-divided-power-envelope" ], "ref_ids": [ 10744 ] }, { "id": 10798, "type": "definition", "label": "crystalline-definition-compatible", "categories": [ "crystalline" ], "title": "crystalline-definition-compatible", "contents": [ "Let $(A, I, \\gamma)$ and $(B, J, \\delta)$ be divided power rings.", "Let $A \\to B$ be a ring map. We say", "{\\it $\\delta$ is compatible with $\\gamma$}", "if there exists a divided power structure $\\bar\\gamma$ on", "$J + IB$ such that", "$$", "(A, I, \\gamma) \\to (B, J + IB, \\bar \\gamma)\\quad\\text{and}\\quad", "(B, J, \\delta) \\to (B, J + IB, \\bar \\gamma)", "$$", "are homomorphisms of divided power rings." ], "refs": [], "ref_ids": [] }, { "id": 10799, "type": "definition", "label": "crystalline-definition-affine-thickening", "categories": [ "crystalline" ], "title": "crystalline-definition-affine-thickening", "contents": [ "In Situation \\ref{situation-affine}.", "\\begin{enumerate}", "\\item A {\\it divided power thickening} of $C$ over $(A, I, \\gamma)$", "is a homomorphism of divided power algebras $(A, I, \\gamma) \\to (B, J, \\delta)$", "such that $p$ is nilpotent in $B$ and a ring map $C \\to B/J$ such that", "$$", "\\xymatrix{", "B \\ar[r] & B/J \\\\", "& C \\ar[u] \\\\", "A \\ar[uu] \\ar[r] & A/I \\ar[u]", "}", "$$", "is commutative.", "\\item A {\\it homomorphism of divided power thickenings}", "$$", "(B, J, \\delta, C \\to B/J) \\longrightarrow (B', J', \\delta', C \\to B'/J')", "$$", "is a homomorphism $\\varphi : B \\to B'$ of divided power $A$-algebras such", "that $C \\to B/J \\to B'/J'$ is the given map $C \\to B'/J'$.", "\\item We denote $\\text{CRIS}(C/A, I, \\gamma)$ or simply $\\text{CRIS}(C/A)$", "the category of divided power thickenings of $C$ over $(A, I, \\gamma)$.", "\\item We denote $\\text{Cris}(C/A, I, \\gamma)$ or simply $\\text{Cris}(C/A)$", "the full subcategory consisting of $(B, J, \\delta, C \\to B/J)$ such that", "$C \\to B/J$ is an isomorphism. We often denote such an object", "$(B \\to C, \\delta)$ with $J = \\Ker(B \\to C)$ being understood.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10800, "type": "definition", "label": "crystalline-definition-derivation", "categories": [ "crystalline" ], "title": "crystalline-definition-derivation", "contents": [ "Let $A$ be a ring. Let $(B, J, \\delta)$ be a divided power ring.", "Let $A \\to B$ be a ring map. Let $M$ be an $B$-module.", "A {\\it divided power $A$-derivation} into $M$ is a map", "$\\theta : B \\to M$ which is additive, annihilates the elements", "of $A$, satisfies the Leibniz rule", "$\\theta(bb') = b\\theta(b') + b'\\theta(b)$ and satisfies", "$$", "\\theta(\\delta_n(x)) = \\delta_{n - 1}(x)\\theta(x)", "$$", "for all $n \\geq 1$ and all $x \\in J$." ], "refs": [], "ref_ids": [] }, { "id": 10801, "type": "definition", "label": "crystalline-definition-divided-power-structure", "categories": [ "crystalline" ], "title": "crystalline-definition-divided-power-structure", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}$ be a sheaf of rings", "on $\\mathcal{C}$. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a", "sheaf of ideals. A {\\it divided power structure $\\gamma$} on $\\mathcal{I}$", "is a sequence of maps $\\gamma_n : \\mathcal{I} \\to \\mathcal{I}$, $n \\geq 1$", "such that for any object $U$ of $\\mathcal{C}$ the triple", "$$", "(\\mathcal{O}(U), \\mathcal{I}(U), \\gamma)", "$$", "is a divided power ring." ], "refs": [], "ref_ids": [] }, { "id": 10802, "type": "definition", "label": "crystalline-definition-divided-power-scheme", "categories": [ "crystalline" ], "title": "crystalline-definition-divided-power-scheme", "contents": [ "A {\\it divided power scheme} is a triple $(S, \\mathcal{I}, \\gamma)$", "where $S$ is a scheme, $\\mathcal{I}$ is a quasi-coherent sheaf of", "ideals, and $\\gamma$ is a divided power structure on $\\mathcal{I}$.", "A {\\it morphism of divided power schemes}", "$(S, \\mathcal{I}, \\gamma) \\to (S', \\mathcal{I}', \\gamma')$ is", "a morphism of schemes $f : S \\to S'$ such that", "$f^{-1}\\mathcal{I}'\\mathcal{O}_S \\subset \\mathcal{I}$ and such that", "$$", "(\\mathcal{O}_{S'}(U'), \\mathcal{I}'(U'), \\gamma')", "\\longrightarrow", "(\\mathcal{O}_S(f^{-1}U'), \\mathcal{I}(f^{-1}U'), \\gamma)", "$$", "is a homomorphism of divided power rings for all $U' \\subset S'$ open." ], "refs": [], "ref_ids": [] }, { "id": 10803, "type": "definition", "label": "crystalline-definition-divided-power-thickening", "categories": [ "crystalline" ], "title": "crystalline-definition-divided-power-thickening", "contents": [ "A triple $(U, T, \\gamma)$ as above is called a {\\it divided power thickening}", "if $U \\to T$ is a thickening." ], "refs": [], "ref_ids": [] }, { "id": 10804, "type": "definition", "label": "crystalline-definition-divided-power-thickening-X", "categories": [ "crystalline" ], "title": "crystalline-definition-divided-power-thickening-X", "contents": [ "In Situation \\ref{situation-global}.", "\\begin{enumerate}", "\\item A {\\it divided power thickening of $X$ relative to", "$(S, \\mathcal{I}, \\gamma)$} is given by a divided power thickening", "$(U, T, \\delta)$ over $(S, \\mathcal{I}, \\gamma)$", "and an $S$-morphism $U \\to X$.", "\\item A {\\it morphism of divided power thickenings of $X$", "relative to $(S, \\mathcal{I}, \\gamma)$} is defined in the obvious", "manner.", "\\end{enumerate}", "The category of divided power thickenings of $X$ relative to", "$(S, \\mathcal{I}, \\gamma)$ is denoted $\\text{CRIS}(X/S, \\mathcal{I}, \\gamma)$", "or simply $\\text{CRIS}(X/S)$." ], "refs": [], "ref_ids": [] }, { "id": 10805, "type": "definition", "label": "crystalline-definition-big-crystalline-site", "categories": [ "crystalline" ], "title": "crystalline-definition-big-crystalline-site", "contents": [ "In Situation \\ref{situation-global}.", "\\begin{enumerate}", "\\item A family of morphisms $\\{(U_i, T_i, \\delta_i) \\to (U, T, \\delta)\\}$", "of divided power thickenings of $X/S$ is a", "{\\it Zariski, \\'etale, smooth, syntomic, or fppf covering}", "if and only if", "\\begin{enumerate}", "\\item $U_i = U \\times_T T_i$ for all $i$ and", "\\item $\\{T_i \\to T\\}$ is a Zariski, \\'etale, smooth, syntomic, or fppf covering.", "\\end{enumerate}", "\\item The {\\it big crystalline site} of $X$ over $(S, \\mathcal{I}, \\gamma)$,", "is the category $\\text{CRIS}(X/S)$ endowed with the Zariski topology.", "\\item The topos of sheaves on $\\text{CRIS}(X/S)$ is denoted", "$(X/S)_{\\text{CRIS}}$ or sometimes", "$(X/S, \\mathcal{I}, \\gamma)_{\\text{CRIS}}$\\footnote{This clashes with", "our convention to denote the topos associated to a site $\\mathcal{C}$", "by $\\Sh(\\mathcal{C})$.}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10806, "type": "definition", "label": "crystalline-definition-crystalline-site", "categories": [ "crystalline" ], "title": "crystalline-definition-crystalline-site", "contents": [ "In Situation \\ref{situation-global}.", "\\begin{enumerate}", "\\item The (small) {\\it crystalline site} of $X$ over", "$(S, \\mathcal{I}, \\gamma)$, denoted $\\text{Cris}(X/S, \\mathcal{I}, \\gamma)$", "or simply $\\text{Cris}(X/S)$ is the full subcategory of $\\text{CRIS}(X/S)$", "consisting of those $(U, T, \\delta)$ in $\\text{CRIS}(X/S)$ such that", "$U \\to X$ is an open immersion. It comes endowed with the Zariski topology.", "\\item The topos of sheaves on $\\text{Cris}(X/S)$ is denoted", "$(X/S)_{\\text{cris}}$ or sometimes", "$(X/S, \\mathcal{I}, \\gamma)_{\\text{cris}}$\\footnote{This clashes with", "our convention to denote the topos associated to a site $\\mathcal{C}$", "by $\\Sh(\\mathcal{C})$.}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 10807, "type": "definition", "label": "crystalline-definition-modules", "categories": [ "crystalline" ], "title": "crystalline-definition-modules", "contents": [ "In Situation \\ref{situation-global}.", "Let $\\mathcal{C} = \\text{CRIS}(X/S)$ or $\\mathcal{C} = \\text{Cris}(X/S)$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_{X/S}$-modules on $\\mathcal{C}$.", "\\begin{enumerate}", "\\item We say $\\mathcal{F}$ is {\\it locally quasi-coherent} if for every", "object $(U, T, \\delta)$ of $\\mathcal{C}$ the restriction $\\mathcal{F}_T$", "is a quasi-coherent $\\mathcal{O}_T$-module.", "\\item We say $\\mathcal{F}$ is {\\it quasi-coherent} if it is quasi-coherent", "in the sense of", "Modules on Sites, Definition \\ref{sites-modules-definition-site-local}.", "\\item We say $\\mathcal{F}$ is a {\\it crystal in $\\mathcal{O}_{X/S}$-modules}", "if all the comparison maps (\\ref{equation-comparison-modules}) are", "isomorphisms.", "\\end{enumerate}" ], "refs": [ "sites-modules-definition-site-local" ], "ref_ids": [ 14289 ] }, { "id": 10808, "type": "definition", "label": "crystalline-definition-crystal-quasi-coherent-modules", "categories": [ "crystalline" ], "title": "crystalline-definition-crystal-quasi-coherent-modules", "contents": [ "If $\\mathcal{F}$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-crystal-quasi-coherent-modules}, then", "we say that $\\mathcal{F}$ is a", "{\\it crystal in quasi-coherent modules}.", "We say that $\\mathcal{F}$ is a {\\it crystal in finite locally free modules}", "if, in addition, $\\mathcal{F}$ is finite locally free." ], "refs": [ "crystalline-lemma-crystal-quasi-coherent-modules" ], "ref_ids": [ 10766 ] }, { "id": 10809, "type": "definition", "label": "crystalline-definition-global-derivation", "categories": [ "crystalline" ], "title": "crystalline-definition-global-derivation", "contents": [ "In Situation \\ref{situation-global} let", "$\\mathcal{F}$ be a sheaf of $\\mathcal{O}_{X/S}$-modules on", "$\\text{Cris}(X/S)$. An", "{\\it $S$-derivation $D : \\mathcal{O}_{X/S} \\to \\mathcal{F}$}", "is a map of sheaves such that for every object $(U, T, \\delta)$ of", "$\\text{Cris}(X/S)$ the map", "$$", "D : \\Gamma(T, \\mathcal{O}_T) \\longrightarrow \\Gamma(T, \\mathcal{F})", "$$", "is a divided power $\\Gamma(V, \\mathcal{O}_V)$-derivation where $V \\subset S$", "is any open such that $T \\to S$ factors through $V$." ], "refs": [], "ref_ids": [] }, { "id": 10810, "type": "definition", "label": "crystalline-definition-F-crystal", "categories": [ "crystalline" ], "title": "crystalline-definition-F-crystal", "contents": [ "In Situation \\ref{situation-F-crystal} an {\\it $F$-crystal on $X/S$", "(relative to $\\sigma$)} is a pair $(\\mathcal{E}, F_\\mathcal{E})$", "given by a crystal in finite locally free $\\mathcal{O}_{X/S}$-modules", "$\\mathcal{E}$ together with a map", "$$", "F_\\mathcal{E} : (F_X)_{\\text{cris}}^*\\mathcal{E} \\longrightarrow \\mathcal{E}", "$$", "An $F$-crystal is called {\\it nondegenerate} if there exists an integer", "$i \\geq 0$ a map $V : \\mathcal{E} \\to (F_X)_{\\text{cris}}^*\\mathcal{E}$", "such that $V \\circ F_{\\mathcal{E}} = p^i \\text{id}$." ], "refs": [], "ref_ids": [] }, { "id": 11141, "type": "definition", "label": "varieties-definition-variety", "categories": [ "varieties" ], "title": "varieties-definition-variety", "contents": [ "Let $k$ be a field. A {\\it variety} is a scheme $X$ over $k$", "such that $X$ is integral and the structure morphism", "$X \\to \\Spec(k)$ is separated and of finite type." ], "refs": [], "ref_ids": [] }, { "id": 11142, "type": "definition", "label": "varieties-definition-geometrically-reduced", "categories": [ "varieties" ], "title": "varieties-definition-geometrically-reduced", "contents": [ "Let $k$ be a field.", "Let $X$ be a scheme over $k$.", "\\begin{enumerate}", "\\item Let $x \\in X$ be a point.", "We say $X$ is {\\it geometrically reduced at $x$}", "if for any field extension $k \\subset k'$", "and any point $x' \\in X_{k'}$ lying over $x$", "the local ring $\\mathcal{O}_{X_{k'}, x'}$ is reduced.", "\\item We say $X$ is {\\it geometrically reduced} over $k$", "if $X$ is geometrically reduced at every point of $X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11143, "type": "definition", "label": "varieties-definition-geometrically-connected", "categories": [ "varieties" ], "title": "varieties-definition-geometrically-connected", "contents": [ "Let $X$ be a scheme over the field $k$. We say $X$ is", "{\\it geometrically connected} over $k$ if the scheme $X_{k'}$ is connected", "for every field extension $k'$ of $k$." ], "refs": [], "ref_ids": [] }, { "id": 11144, "type": "definition", "label": "varieties-definition-geometrically-irreducible", "categories": [ "varieties" ], "title": "varieties-definition-geometrically-irreducible", "contents": [ "Let $X$ be a scheme over the field $k$.", "We say $X$ is {\\it geometrically irreducible} over $k$ if the scheme", "$X_{k'}$ is irreducible\\footnote{An irreducible space is nonempty.}", "for any field extension $k'$ of $k$." ], "refs": [], "ref_ids": [] }, { "id": 11145, "type": "definition", "label": "varieties-definition-geometrically-integral", "categories": [ "varieties" ], "title": "varieties-definition-geometrically-integral", "contents": [ "Let $X$ be a scheme over the field $k$.", "\\begin{enumerate}", "\\item Let $x \\in X$. We say $X$ is", "{\\it geometrically pointwise integral at $x$} if for every", "field extension $k \\subset k'$ and every $x' \\in X_{k'}$ lying over $x$", "the local ring $\\mathcal{O}_{X_{k'}, x'}$ is integral.", "\\item We say $X$ is {\\it geometrically pointwise integral} if $X$", "is geometrically pointwise integral at every point.", "\\item We say $X$ is {\\it geometrically integral} over $k$ if the scheme", "$X_{k'}$ is integral for every field extension $k'$ of $k$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11146, "type": "definition", "label": "varieties-definition-geometrically-normal", "categories": [ "varieties" ], "title": "varieties-definition-geometrically-normal", "contents": [ "Let $X$ be a scheme over the field $k$.", "\\begin{enumerate}", "\\item Let $x \\in X$. We say $X$ is", "{\\it geometrically normal at $x$} if for every", "field extension $k \\subset k'$ and every $x' \\in X_{k'}$ lying over $x$", "the local ring $\\mathcal{O}_{X_{k'}, x'}$ is normal.", "\\item We say $X$ is {\\it geometrically normal} over $k$ if $X$", "is geometrically normal at every $x \\in X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11147, "type": "definition", "label": "varieties-definition-geometrically-regular", "categories": [ "varieties" ], "title": "varieties-definition-geometrically-regular", "contents": [ "Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$.", "\\begin{enumerate}", "\\item Let $x \\in X$. We say $X$ is {\\it geometrically regular at $x$}", "over $k$ if for every finitely generated field extension $k \\subset k'$", "and any $x' \\in X_{k'}$ lying over $x$ the local ring", "$\\mathcal{O}_{X_{k'}, x'}$ is regular.", "\\item We say $X$ is {\\it geometrically regular over $k$} if", "$X$ is geometrically regular at all of its points.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11148, "type": "definition", "label": "varieties-definition-dual-numbers", "categories": [ "varieties" ], "title": "varieties-definition-dual-numbers", "contents": [ "For any ring $R$ the {\\it dual numbers} over $R$ is the", "$R$-algebra denoted $R[\\epsilon]$. As an $R$-module it is free with", "basis $1$, $\\epsilon$ and the $R$-algebra structure comes from setting", "$\\epsilon^2 = 0$." ], "refs": [], "ref_ids": [] }, { "id": 11149, "type": "definition", "label": "varieties-definition-tangent-space", "categories": [ "varieties" ], "title": "varieties-definition-tangent-space", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$. The set of", "dotted arrows making (\\ref{equation-tangent-space}) commute with", "its canonical $\\kappa(x)$-vector space structure is called", "the {\\it tangent space of $X$ over $S$ at $x$} and we denote it $T_{X/S, x}$.", "An element of this space is called a {\\it tangent vector} of $X/S$ at $x$." ], "refs": [], "ref_ids": [] }, { "id": 11150, "type": "definition", "label": "varieties-definition-algebraic-scheme", "categories": [ "varieties" ], "title": "varieties-definition-algebraic-scheme", "contents": [ "Let $k$ be a field. An {\\it algebraic $k$-scheme} is a scheme $X$ over $k$", "such that the structure morphism $X \\to \\Spec(k)$ is of", "finite type. A {\\it locally algebraic $k$-scheme} is a scheme $X$ over $k$", "such that the structure morphism $X \\to \\Spec(k)$ is", "locally of finite type." ], "refs": [], "ref_ids": [] }, { "id": 11151, "type": "definition", "label": "varieties-definition-variety-type", "categories": [ "varieties" ], "title": "varieties-definition-variety-type", "contents": [ "Let $k$ be a field. Let $X$ be a variety over $k$.", "\\begin{enumerate}", "\\item We say $X$ is an {\\it affine variety} if $X$ is an affine scheme.", "This is equivalent to requiring $X$ to be isomorphic to a closed", "subscheme of $\\mathbf{A}^n_k$ for some $n$.", "\\item We say $X$ is a {\\it projective variety} if the", "structure morphism $X \\to \\Spec(k)$ is projective. By", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-locally-projective}", "this is true if and only if $X$ is isomorphic to a closed", "subscheme of $\\mathbf{P}^n_k$ for some $n$.", "\\item We say $X$ is a {\\it quasi-projective variety} if", "the structure morphism $X \\to \\Spec(k)$ is quasi-projective. By", "Morphisms, Lemma \\ref{morphisms-lemma-characterize-locally-quasi-projective}", "this is true if and only if $X$ is isomorphic to a", "locally closed subscheme of $\\mathbf{P}^n_k$ for some $n$.", "\\item A {\\it proper variety} is a variety such that the", "morphism $X \\to \\Spec(k)$ is proper.", "\\item A {\\it smooth variety} is a variety such that the", "morphism $X \\to \\Spec(k)$ is smooth.", "\\end{enumerate}" ], "refs": [ "morphisms-lemma-characterize-locally-projective", "morphisms-lemma-characterize-locally-quasi-projective" ], "ref_ids": [ 5421, 5403 ] }, { "id": 11152, "type": "definition", "label": "varieties-definition-euler-characteristic", "categories": [ "varieties" ], "title": "varieties-definition-euler-characteristic", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{F}$", "be a coherent $\\mathcal{O}_X$-module. In this situation the", "{\\it Euler characteristic of $\\mathcal{F}$} is the integer", "$$", "\\chi(X, \\mathcal{F}) = \\sum\\nolimits_i (-1)^i \\dim_k H^i(X, \\mathcal{F}).", "$$", "For justification of the formula see below." ], "refs": [], "ref_ids": [] }, { "id": 11153, "type": "definition", "label": "varieties-definition-regularity", "categories": [ "varieties" ], "title": "varieties-definition-regularity", "contents": [ "Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent", "sheaf on $\\mathbf{P}^n_k$. We say $\\mathcal{F}$ is {\\it $m$-regular}", "if", "$$", "H^i(\\mathbf{P}^n_k, \\mathcal{F}(m - i)) = 0", "$$", "for $i = 1, \\ldots, n$." ], "refs": [], "ref_ids": [] }, { "id": 11154, "type": "definition", "label": "varieties-definition-hilbert-polynomial", "categories": [ "varieties" ], "title": "varieties-definition-hilbert-polynomial", "contents": [ "Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent sheaf", "on $\\mathbf{P}^n_k$. The function", "$d \\mapsto \\chi(\\mathbf{P}^n_k, \\mathcal{F}(d))$ is called the", "{\\it Hilbert polynomial} of $\\mathcal{F}$." ], "refs": [], "ref_ids": [] }, { "id": 11155, "type": "definition", "label": "varieties-definition-absolute-frobenius", "categories": [ "varieties" ], "title": "varieties-definition-absolute-frobenius", "contents": [ "Let $p$ be a prime number. Let $X$ be a scheme in characteristic $p$.", "The {\\it absolute frobenius of $X$} is the morphism $F_X : X \\to X$", "given by the identity on the underlying topological space and", "with $F_X^\\sharp : \\mathcal{O}_X \\to \\mathcal{O}_X$ given by $g \\mapsto g^p$." ], "refs": [], "ref_ids": [] }, { "id": 11156, "type": "definition", "label": "varieties-definition-relative-frobenius", "categories": [ "varieties" ], "title": "varieties-definition-relative-frobenius", "contents": [ "Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$.", "Let $X$ be a scheme over $S$. We define", "$$", "X^{(p)} = X^{(p/S)} = X \\times_{S, F_S} S", "$$", "viewed as a scheme over $S$. Applying", "Lemma \\ref{lemma-frobenius-endomorphism-identity}", "we see there is a unique morphism $F_{X/S} : X \\longrightarrow X^{(p)}$", "over $S$ fitting into the commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_{F_{X/S}} \\ar[rrd] \\ar@/^1em/[rrrr]^{F_X}", "& & X^{(p)} \\ar[rr] \\ar[d] & & X \\ar[d] \\\\", "& & S \\ar[rr]^{F_S} & & S", "}", "$$", "where the right square is cartesian. The morphism $F_{X/S}$ is called the", "{\\it relative Frobenius morphism of $X/S$}." ], "refs": [ "varieties-lemma-frobenius-endomorphism-identity" ], "ref_ids": [ 11047 ] }, { "id": 11157, "type": "definition", "label": "varieties-definition-delta-invariant-algebra", "categories": [ "varieties" ], "title": "varieties-definition-delta-invariant-algebra", "contents": [ "Let $A$ be a reduced Nagata local ring of dimension $1$.", "The {\\it $\\delta$-invariant of $A$} is $\\text{length}_A(A'/A)$", "where $A'$ is as in Lemma \\ref{lemma-pre-delta-invariant}." ], "refs": [ "varieties-lemma-pre-delta-invariant" ], "ref_ids": [ 11078 ] }, { "id": 11158, "type": "definition", "label": "varieties-definition-delta-invariant", "categories": [ "varieties" ], "title": "varieties-definition-delta-invariant", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.", "Let $x \\in X$ be a point such that $\\mathcal{O}_{X, x}$", "is reduced and $\\dim(\\mathcal{O}_{X, x}) = 1$.", "The {\\it $\\delta$-invariant of $X$ at $x$} is the", "$\\delta$-invariant of $\\mathcal{O}_{X, x}$ as defined in", "Definition \\ref{definition-delta-invariant-algebra}." ], "refs": [ "varieties-definition-delta-invariant-algebra" ], "ref_ids": [ 11157 ] }, { "id": 11159, "type": "definition", "label": "varieties-definition-wedge", "categories": [ "varieties" ], "title": "varieties-definition-wedge", "contents": [ "Let $A$ and $A_i$, $1 \\leq i \\leq n$ be local rings. We say", "{\\it $A$ is a wedge of $A_1, \\ldots, A_n$}", "if there exist isomorphisms", "$$", "\\kappa_{A_1} \\to \\kappa_{A_2} \\to \\ldots \\to \\kappa_{A_n}", "$$", "and $A$ is isomorphic to the ring consisting of $n$-tuples", "$(a_1, \\ldots, a_n) \\in A_1 \\times \\ldots \\times A_n$ which map to the", "same element of $\\kappa_{A_n}$." ], "refs": [], "ref_ids": [] }, { "id": 11160, "type": "definition", "label": "varieties-definition-curve", "categories": [ "varieties" ], "title": "varieties-definition-curve", "contents": [ "Let $k$ be a field. A {\\it curve} is a variety of dimension $1$ over $k$." ], "refs": [], "ref_ids": [] }, { "id": 11161, "type": "definition", "label": "varieties-definition-degree-invertible-sheaf", "categories": [ "varieties" ], "title": "varieties-definition-degree-invertible-sheaf", "contents": [ "Let $k$ be a field, let $X$ be a proper scheme of dimension $\\leq 1$", "over $k$, and let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "The {\\it degree} of $\\mathcal{L}$ is defined by", "$$", "\\deg(\\mathcal{L}) = \\chi(X, \\mathcal{L}) - \\chi(X, \\mathcal{O}_X)", "$$", "More generally, if $\\mathcal{E}$ is a locally free sheaf of rank $n$", "we define the {\\it degree} of $\\mathcal{E}$ by", "$$", "\\deg(\\mathcal{E}) = \\chi(X, \\mathcal{E}) - n\\chi(X, \\mathcal{O}_X)", "$$" ], "refs": [], "ref_ids": [] }, { "id": 11162, "type": "definition", "label": "varieties-definition-intersection-number", "categories": [ "varieties" ], "title": "varieties-definition-intersection-number", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let", "$i : Z \\to X$ be a closed subscheme of dimension $d$. Let", "$\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible", "$\\mathcal{O}_X$-modules. We define the {\\it intersection number}", "$(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$", "as the coefficient of $n_1 \\ldots n_d$ in the numerical polynomial", "$$", "\\chi(X, i_*\\mathcal{O}_Z \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes", "\\ldots \\otimes \\mathcal{L}_d^{\\otimes n_d}) =", "\\chi(Z, \\mathcal{L}_1^{\\otimes n_1} \\otimes", "\\ldots \\otimes \\mathcal{L}_d^{\\otimes n_d}|_Z)", "$$", "In the special", "case that $\\mathcal{L}_1 = \\ldots = \\mathcal{L}_d = \\mathcal{L}$", "we write $(\\mathcal{L}^d \\cdot Z)$." ], "refs": [], "ref_ids": [] }, { "id": 11163, "type": "definition", "label": "varieties-definition-degree", "categories": [ "varieties" ], "title": "varieties-definition-degree", "contents": [ "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let", "$\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.", "For any closed subscheme the {\\it degree of $Z$ with respect to", "$\\mathcal{L}$}, denoted $\\deg_\\mathcal{L}(Z)$, is", "the intersection number $(\\mathcal{L}^d \\cdot Z)$", "where $d = \\dim(Z)$." ], "refs": [], "ref_ids": [] }, { "id": 11164, "type": "definition", "label": "varieties-definition-embed-dim", "categories": [ "varieties" ], "title": "varieties-definition-embed-dim", "contents": [ "Let $k$ be an algebraically closed field. Let $X$ be a locally algebraic", "$k$-scheme and let $x \\in X$ be a closed point. The", "{\\it embedding dimension of $X$ at $x$} is", "$\\dim_k \\mathfrak m_x/\\mathfrak m_x^2$." ], "refs": [], "ref_ids": [] }, { "id": 11165, "type": "definition", "label": "varieties-definition-embedding-dimension", "categories": [ "varieties" ], "title": "varieties-definition-embedding-dimension", "contents": [ "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.", "Let $x \\in X$ be a point. The {\\it embedding dimension of $X/k$ at $x$}", "is $\\dim_{\\kappa(x)}(T_{X/k, x})$." ], "refs": [], "ref_ids": [] }, { "id": 11244, "type": "definition", "label": "cotangent-definition-standard-resolution", "categories": [ "cotangent" ], "title": "cotangent-definition-standard-resolution", "contents": [ "Let $A \\to B$ be a ring map. The {\\it standard resolution of $B$ over $A$}", "is the augmentation $\\epsilon : P_\\bullet \\to B$ with terms", "$$", "P_0 = A[B],\\quad P_1 = A[A[B]],\\quad \\ldots", "$$", "and maps as constructed above." ], "refs": [], "ref_ids": [] }, { "id": 11245, "type": "definition", "label": "cotangent-definition-cotangent-complex-ring-map", "categories": [ "cotangent" ], "title": "cotangent-definition-cotangent-complex-ring-map", "contents": [ "The {\\it cotangent complex} $L_{B/A}$ of a ring map $A \\to B$", "is the complex of $B$-modules associated to the simplicial $B$-module", "$$", "\\Omega_{P_\\bullet/A} \\otimes_{P_\\bullet, \\epsilon} B", "$$", "where $\\epsilon : P_\\bullet \\to B$ is the standard resolution", "of $B$ over $A$." ], "refs": [], "ref_ids": [] }, { "id": 11246, "type": "definition", "label": "cotangent-definition-biderivation", "categories": [ "cotangent" ], "title": "cotangent-definition-biderivation", "contents": [ "Let $A \\to B$ be a ring map. Let $M$ be a $(B, B)$-bimodule", "over $A$. An {\\it $A$-biderivation} is an $A$-linear map $\\lambda : B \\to M$", "such that $\\lambda(xy) = x\\lambda(y) + \\lambda(x)y$." ], "refs": [], "ref_ids": [] }, { "id": 11247, "type": "definition", "label": "cotangent-definition-atiyah-class", "categories": [ "cotangent" ], "title": "cotangent-definition-atiyah-class", "contents": [ "Let $A \\to B$ be a ring map. Let $M$ be a $B$-module.", "The map $M \\to L_{B/A} \\otimes_B^\\mathbf{L} M[1]$", "in (\\ref{equation-atiyah}) is called the {\\it Atiyah class} of $M$." ], "refs": [], "ref_ids": [] }, { "id": 11248, "type": "definition", "label": "cotangent-definition-standard-resolution-sheaves-rings", "categories": [ "cotangent" ], "title": "cotangent-definition-standard-resolution-sheaves-rings", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings", "on $\\mathcal{C}$. The {\\it standard resolution of $\\mathcal{B}$ over", "$\\mathcal{A}$} is the augmentation", "$\\epsilon : \\mathcal{P}_\\bullet \\to \\mathcal{B}$", "with terms", "$$", "\\mathcal{P}_0 = \\mathcal{A}[\\mathcal{B}],\\quad", "\\mathcal{P}_1 = \\mathcal{A}[\\mathcal{A}[\\mathcal{B}]],\\quad \\ldots", "$$", "and maps as constructed above." ], "refs": [], "ref_ids": [] }, { "id": 11249, "type": "definition", "label": "cotangent-definition-cotangent-complex-morphism-sheaves-rings", "categories": [ "cotangent" ], "title": "cotangent-definition-cotangent-complex-morphism-sheaves-rings", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings", "on $\\mathcal{C}$.", "The {\\it cotangent complex} $L_{\\mathcal{B}/\\mathcal{A}}$", "is the complex of $\\mathcal{B}$-modules associated to the", "simplicial module", "$$", "\\Omega_{\\mathcal{P}_\\bullet/\\mathcal{A}}", "\\otimes_{\\mathcal{P}_\\bullet, \\epsilon} \\mathcal{B}", "$$", "where $\\epsilon : \\mathcal{P}_\\bullet \\to \\mathcal{B}$", "is the standard resolution of $\\mathcal{B}$ over", "$\\mathcal{A}$. We usually think of $L_{\\mathcal{B}/\\mathcal{A}}$", "as an object of $D(\\mathcal{B})$." ], "refs": [], "ref_ids": [] }, { "id": 11250, "type": "definition", "label": "cotangent-definition-atiyah-class-general", "categories": [ "cotangent" ], "title": "cotangent-definition-atiyah-class-general", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{B}$-modules.", "The map $\\mathcal{F} \\to", "L_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B}^\\mathbf{L} \\mathcal{F}[1]$", "in (\\ref{equation-atiyah-general}) is called the {\\it Atiyah class} of", "$\\mathcal{F}$." ], "refs": [], "ref_ids": [] }, { "id": 11251, "type": "definition", "label": "cotangent-definition-cotangent-complex-morphism-ringed-spaces", "categories": [ "cotangent" ], "title": "cotangent-definition-cotangent-complex-morphism-ringed-spaces", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$ be a morphism of", "ringed spaces. The {\\it cotangent complex} $L_f$ of $f$ is", "$L_f = L_{\\mathcal{O}_X/f^{-1}\\mathcal{O}_S}$.", "We will also use the notation", "$L_f = L_{X/S} = L_{\\mathcal{O}_X/\\mathcal{O}_S}$." ], "refs": [], "ref_ids": [] }, { "id": 11252, "type": "definition", "label": "cotangent-definition-cotangent-complex-morphism-ringed-topoi", "categories": [ "cotangent" ], "title": "cotangent-definition-cotangent-complex-morphism-ringed-topoi", "contents": [ "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.", "The {\\it cotangent complex} $L_f$ of $f$ is", "$L_f = L_{\\mathcal{O}_\\mathcal{C}/f^{-1}\\mathcal{O}_\\mathcal{D}}$.", "We sometimes write $L_f = L_{\\mathcal{O}_\\mathcal{C}/\\mathcal{O}_\\mathcal{D}}$." ], "refs": [], "ref_ids": [] }, { "id": 11253, "type": "definition", "label": "cotangent-definition-cotangent-morphism-schemes", "categories": [ "cotangent" ], "title": "cotangent-definition-cotangent-morphism-schemes", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes. The {\\it cotangent complex", "$L_{X/Y}$ of $X$ over $Y$} is the cotangent complex of $f$ as a", "morphism of ringed spaces", "(Definition \\ref{definition-cotangent-complex-morphism-ringed-spaces})." ], "refs": [ "cotangent-definition-cotangent-complex-morphism-ringed-spaces" ], "ref_ids": [ 11251 ] }, { "id": 11254, "type": "definition", "label": "cotangent-definition-cotangent-morphism-spaces", "categories": [ "cotangent" ], "title": "cotangent-definition-cotangent-morphism-spaces", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. The {\\it cotangent complex $L_{X/Y}$ of $X$ over $Y$} is the", "cotangent complex of the morphism of ringed topoi $f_{small}$", "between the small \\'etale sites of $X$ and $Y$", "(see", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-morphism-ringed-topoi}", "and", "Definition \\ref{definition-cotangent-complex-morphism-ringed-topoi})." ], "refs": [ "spaces-properties-lemma-morphism-ringed-topoi", "cotangent-definition-cotangent-complex-morphism-ringed-topoi" ], "ref_ids": [ 11882, 11252 ] }, { "id": 11347, "type": "definition", "label": "spaces-cohomology-definition-alternating-cech-complex", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-definition-alternating-cech-complex", "contents": [ "Let $S$ be a scheme. Let $f : U \\to X$ be a surjective \\'etale morphism", "of algebraic spaces over $S$. Let $\\mathcal{F}$ be an object of", "$\\textit{Ab}(X_\\etale)$. The", "{\\it alternating {\\v C}ech complex}\\footnote{This may be nonstandard notation}", "$\\check{\\mathcal{C}}^\\bullet_{alt}(f, \\mathcal{F})$", "associated to $\\mathcal{F}$ and $f$ is the complex", "$$", "\\Hom(K^0, \\mathcal{F}) \\to \\Hom(K^1, \\mathcal{F}) \\to", "\\Hom(K^2, \\mathcal{F}) \\to \\ldots", "$$", "with Hom groups computed in $\\textit{Ab}(X_\\etale)$." ], "refs": [], "ref_ids": [] }, { "id": 11348, "type": "definition", "label": "spaces-cohomology-definition-coherent", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-definition-coherent", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.", "A quasi-coherent module $\\mathcal{F}$ on $X$ is called {\\it coherent}", "if $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module on the site", "$X_\\etale$ in the sense of", "Modules on Sites, Definition \\ref{sites-modules-definition-site-local}." ], "refs": [ "sites-modules-definition-site-local" ], "ref_ids": [ 14289 ] }, { "id": 11417, "type": "definition", "label": "artin-definition-RS", "categories": [ "artin" ], "title": "artin-definition-RS", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{Z}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$. We say $\\mathcal{Z}$", "satisfies {\\it condition (RS)} if for every pushout", "$$", "\\xymatrix{", "X \\ar[r] \\ar[d] & X' \\ar[d] \\\\", "Y \\ar[r] & Y' = Y \\amalg_X X'", "}", "$$", "in the category of schemes over $S$ where", "\\begin{enumerate}", "\\item $X$, $X'$, $Y$, $Y'$ are spectra of local Artinian rings,", "\\item $X$, $X'$, $Y$, $Y'$ are of finite type over $S$, and", "\\item $X \\to X'$ (and hence $Y \\to Y'$) is a closed immersion", "\\end{enumerate}", "the functor of fibre categories", "$$", "\\mathcal{Z}_{Y'}", "\\longrightarrow", "\\mathcal{Z}_Y \\times_{\\mathcal{Z}_X} \\mathcal{Z}_{X'}", "$$", "is an equivalence of categories." ], "refs": [], "ref_ids": [] }, { "id": 11418, "type": "definition", "label": "artin-definition-formal-objects", "categories": [ "artin" ], "title": "artin-definition-formal-objects", "contents": [ "Let $S$ be a locally Noetherian scheme. Let", "$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.", "\\begin{enumerate}", "\\item A {\\it formal object} $\\xi = (R, \\xi_n, f_n)$ of $\\mathcal{X}$ consists", "of a Noetherian complete local $S$-algebra $R$, objects $\\xi_n$ of", "$\\mathcal{X}$ lying over $\\Spec(R/\\mathfrak m_R^n)$, and morphisms", "$f_n : \\xi_n \\to \\xi_{n + 1}$ of $\\mathcal{X}$ lying over", "$\\Spec(R/\\mathfrak m^n) \\to \\Spec(R/\\mathfrak m^{n + 1})$", "such that $R/\\mathfrak m$ is a field of finite type over $S$.", "\\item A {\\it morphism of formal objects}", "$a : \\xi = (R, \\xi_n, f_n) \\to \\eta = (T, \\eta_n, g_n)$", "is given by morphisms $a_n : \\xi_n \\to \\eta_n$ such that for every $n$", "the diagram", "$$", "\\xymatrix{", "\\xi_n \\ar[r]_{f_n} \\ar[d]_{a_n} & \\xi_{n + 1} \\ar[d]^{a_{n + 1}} \\\\", "\\eta_n \\ar[r]^{g_n} & \\eta_{n + 1}", "}", "$$", "is commutative. Applying the functor $p$ we obtain a compatible collection", "of morphisms $\\Spec(R/\\mathfrak m_R^n) \\to \\Spec(T/\\mathfrak m_T^n)$ and", "hence a morphism $a_0 : \\Spec(R) \\to \\Spec(T)$ over $S$. We say that", "$a$ {\\it lies over} $a_0$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11419, "type": "definition", "label": "artin-definition-effective", "categories": [ "artin" ], "title": "artin-definition-effective", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$. A formal object", "$\\xi = (R, \\xi_n, f_n)$ of $\\mathcal{X}$ is called {\\it effective}", "if it is in the essential image of the functor", "(\\ref{equation-approximation})." ], "refs": [], "ref_ids": [] }, { "id": 11420, "type": "definition", "label": "artin-definition-limit-preserving", "categories": [ "artin" ], "title": "artin-definition-limit-preserving", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred in groupoids", "over $(\\Sch/S)_{fppf}$. We say $\\mathcal{X}$ is {\\it limit preserving}", "if for every affine scheme $T$ over $S$ which is a limit $T = \\lim T_i$", "of a directed inverse system of affine schemes $T_i$ over $S$, we have", "an equivalence", "$$", "\\colim \\mathcal{X}_{T_i} \\longrightarrow \\mathcal{X}_T", "$$", "of fibre categories." ], "refs": [], "ref_ids": [] }, { "id": 11421, "type": "definition", "label": "artin-definition-versal-formal-object", "categories": [ "artin" ], "title": "artin-definition-versal-formal-object", "contents": [ "Let $S$ be a locally Noetherian scheme. Let", "$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.", "Let $\\xi = (R, \\xi_n, f_n)$ be a formal object. Set $k = R/\\mathfrak m$ and", "$x_0 = \\xi_1$. We will say that $\\xi$ is {\\it versal} if $\\xi$", "as a formal object of $\\mathcal{F}_{\\mathcal{X}, k, x_0}$", "(Remark \\ref{remark-formal-objects-match}) is versal in the sense", "of Formal Deformation Theory, Definition \\ref{formal-defos-definition-versal}." ], "refs": [ "artin-remark-formal-objects-match", "formal-defos-definition-versal" ], "ref_ids": [ 11428, 3521 ] }, { "id": 11422, "type": "definition", "label": "artin-definition-versal", "categories": [ "artin" ], "title": "artin-definition-versal", "contents": [ "Let $S$ be a locally Noetherian scheme.", "Let $\\mathcal{X}$ be fibred in groupoids over $(\\Sch/S)_{fppf}$.", "Let $U$ be a scheme locally of finite type over $S$.", "Let $x$ be an object of $\\mathcal{X}$ lying over $U$.", "Let $u_0$ be finite type point of $U$.", "We say $x$ is {\\it versal} at $u_0$ if the morphism $\\hat x$", "(\\ref{equation-hat-x}) is smooth, see Formal Deformation Theory, Definition", "\\ref{formal-defos-definition-smooth-morphism}." ], "refs": [ "formal-defos-definition-smooth-morphism" ], "ref_ids": [ 3520 ] }, { "id": 11423, "type": "definition", "label": "artin-definition-openness-versality", "categories": [ "artin" ], "title": "artin-definition-openness-versality", "contents": [ "Let $S$ be a locally Noetherian scheme.", "\\begin{enumerate}", "\\item Let $\\mathcal{X}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$. We say $\\mathcal{X}$ satisfies", "{\\it openness of versality} if given a scheme $U$ locally of finite type", "over $S$, an object $x$ of $\\mathcal{X}$ over $U$, and a finite type point", "$u_0 \\in U$ such that $x$ is versal at $u_0$, then there exists an open", "neighbourhood $u_0 \\in U' \\subset U$ such that $x$ is versal at every finite", "type point of $U'$.", "\\item Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. We say $f$ satisfies", "{\\it openness of versality} if given a scheme $U$ locally of finite type", "over $S$, an object $y$ of $\\mathcal{Y}$ over $U$, openness", "of versality holds for", "$(\\Sch/U)_{fppf} \\times_\\mathcal{Y} \\mathcal{X}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11424, "type": "definition", "label": "artin-definition-RS-star", "categories": [ "artin" ], "title": "artin-definition-RS-star", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$. We say $\\mathcal{X}$", "satisfies {\\it condition (RS*)} if given a fibre product diagram", "$$", "\\xymatrix{", "B' \\ar[r] & B \\\\", "A' = A \\times_B B' \\ar[u] \\ar[r] & A \\ar[u]", "}", "$$", "of $S$-algebras, with $B' \\to B$ surjective with square zero kernel,", "the functor of fibre categories", "$$", "\\mathcal{X}_{\\Spec(A')}", "\\longrightarrow", "\\mathcal{X}_{\\Spec(A)} \\times_{\\mathcal{X}_{\\Spec(B)}} \\mathcal{X}_{\\Spec(B')}", "$$", "is an equivalence of categories." ], "refs": [], "ref_ids": [] }, { "id": 11425, "type": "definition", "label": "artin-definition-obstruction-theory", "categories": [ "artin" ], "title": "artin-definition-obstruction-theory", "contents": [ "Let $S$ be a locally Noetherian base. Let $\\mathcal{X}$ be a category fibred", "in groupoids over $(\\Sch/S)_{fppf}$. An {\\it obstruction theory} is ", "given by the following data", "\\begin{enumerate}", "\\item for every $S$-algebra $A$ such that $\\Spec(A) \\to S$", "maps into an affine open and every object $x$ of $\\mathcal{X}$ over", "$\\Spec(A)$ an $A$-linear functor", "$$", "\\mathcal{O}_x : \\text{Mod}_A \\to \\text{Mod}_A", "$$", "of {\\it obstruction modules},", "\\item for $(x, A)$ as in (1), a ring map $A \\to B$,", "$M \\in \\text{Mod}_A$, $N \\in \\text{Mod}_B$, and an $A$-linear", "map $M \\to N$ an induced $A$-linear map $\\mathcal{O}_x(M) \\to \\mathcal{O}_y(N)$", "where $y = x|_{\\Spec(B)}$, and", "\\item for every deformation situation $(x, A' \\to A)$ an", "{\\it obstruction} element", "$o_x(A') \\in \\mathcal{O}_x(I)$ where $I = \\Ker(A' \\to A)$.", "\\end{enumerate}", "These data are subject to the following conditions", "\\begin{enumerate}", "\\item[(i)] the functoriality maps turn the obstruction modules into a functor", "from the category of triples $(x, A, M)$ to sets,", "\\item[(ii)] for every morphism of deformation situations", "$(y, B' \\to B) \\to (x, A' \\to A)$ the element $o_x(A')$ maps", "to $o_y(B')$, and", "\\item[(iii)] we have", "$$", "\\text{Lift}(x, A') \\not = \\emptyset", "\\Leftrightarrow", "o_x(A') = 0", "$$", "for every deformation situation $(x, A' \\to A)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11426, "type": "definition", "label": "artin-definition-naive-obstruction-theory", "categories": [ "artin" ], "title": "artin-definition-naive-obstruction-theory", "contents": [ "Let $S$ be a locally Noetherian base. Let $\\mathcal{X}$ be a category fibred", "in groupoids over $(\\Sch/S)_{fppf}$. Assume that $\\mathcal{X}$", "satisfies (RS*). A {\\it naive obstruction theory} is ", "given by the following data", "\\begin{enumerate}", "\\item", "\\label{item-map}", "for every $S$-algebra $A$ such that $\\Spec(A) \\to S$", "maps into an affine open $\\Spec(\\Lambda) \\subset S$ and every object $x$", "of $\\mathcal{X}$ over $\\Spec(A)$ we are given an object $E_x \\in D^-(A)$", "and a map $\\xi_x : E \\to \\NL_{A/\\Lambda}$,", "\\item", "\\label{item-inf}", "given $(x, A)$ as in (\\ref{item-map}) there are transformations of", "functors", "$$", "\\text{Inf}_x( - ) \\to \\Ext^{-1}_A(E_x, -)", "\\quad\\text{and}\\quad", "T_x(-) \\to \\Ext^0_A(E_x, -)", "$$", "\\item", "\\label{item-functoriality}", "for $(x, A)$ as in (\\ref{item-map}) and a ring map $A \\to B$", "setting $y = x|_{\\Spec(B)}$ there is a functoriality map", "$E_x \\to E_y$ in $D(A)$.", "\\end{enumerate}", "These data are subject to the following conditions", "\\begin{enumerate}", "\\item[(i)]", "in the situation of (\\ref{item-functoriality}) the diagram", "$$", "\\xymatrix{", "E_y \\ar[r]_{\\xi_y} & \\NL_{B/\\Lambda} \\\\", "E_x \\ar[u] \\ar[r]^{\\xi_x} & \\NL_{A/\\Lambda} \\ar[u]", "}", "$$", "is commutative in $D(A)$,", "\\item[(ii)]", "given $(x, A)$ as in (\\ref{item-map}) and $A \\to B \\to C$", "setting $y = x|_{\\Spec(B)}$ and $z = x|_{\\Spec(C)}$ the", "composition of the functoriality maps $E_x \\to E_y$ and $E_y \\to E_z$ is", "the functoriality map $E_x \\to E_z$,", "\\item[(iii)]", "the maps of (\\ref{item-inf}) are isomorphisms", "compatible with the functoriality", "maps and the maps of Remark \\ref{remark-functoriality},", "\\item[(iv)]", "the composition $E_x \\to \\NL_{A/\\Lambda} \\to \\Omega_{A/\\Lambda}$", "corresponds to the canonical element of", "$T_x(\\Omega_{A/\\Lambda}) = \\Ext^0(E_x, \\Omega_{A/\\Lambda})$, see", "Remark \\ref{remark-canonical-element},", "\\item[(v)]", "given a deformation situation $(x, A' \\to A)$ with $I = \\Ker(A' \\to A)$", "the composition $E_x \\to \\NL_{A/\\Lambda} \\to \\NL_{A/A'}$ is zero in", "$$", "\\Hom_A(E_x, \\NL_{A/\\Lambda}) = \\Ext^0_A(E_x, \\NL_{A/A'}) =", "\\Ext^1_A(E_x, I)", "$$", "if and only if $x$ lifts to $A'$.", "\\end{enumerate}" ], "refs": [ "artin-remark-functoriality", "artin-remark-canonical-element" ], "ref_ids": [ 11431, 11435 ] }, { "id": 11525, "type": "definition", "label": "obsolete-definition-epsilon", "categories": [ "obsolete" ], "title": "obsolete-definition-epsilon", "contents": [ "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Assume $X$ integral and $\\dim_\\delta(X) = n$.", "Let $D_1, D_2$ be two effective Cartier divisors in $X$.", "Let $Z \\subset X$ be an integral closed subscheme", "with $\\dim_\\delta(Z) = n - 1$. The {\\it $\\epsilon$-invariant}", "of this situation is", "$$", "\\epsilon_Z(D_1, D_2) = n_Z \\cdot m_Z", "$$", "where $n_Z$, resp.\\ $m_Z$ is the coefficient of", "$Z$ in the $(n - 1)$-cycle $[D_1]_{n - 1}$, resp.\\ $[D_2]_{n - 1}$." ], "refs": [], "ref_ids": [] }, { "id": 11526, "type": "definition", "label": "obsolete-definition-locally-finite-sum-effective-Cartier-divisors", "categories": [ "obsolete" ], "title": "obsolete-definition-locally-finite-sum-effective-Cartier-divisors", "contents": [ "Let $X$ be a scheme.", "Let $\\{D_i\\}_{i \\in I}$ be a locally finite collection", "of effective Cartier divisors on $X$.", "Suppose given a function", "$I \\to \\mathbf{Z}_{\\geq 0}$, $i \\mapsto n_i$.", "The {\\it sum of the effective Cartier divisors}", "$D = \\sum n_i D_i$, is the unique effective Cartier divisor", "$D \\subset X$ such that on any quasi-compact open $U \\subset X$", "we have $D|_U = \\sum_{D_i \\cap U \\not = \\emptyset} n_iD_i|_U$", "is the sum as in Divisors,", "Definition \\ref{divisors-definition-sum-effective-Cartier-divisors}." ], "refs": [ "divisors-definition-sum-effective-Cartier-divisors" ], "ref_ids": [ 8090 ] }, { "id": 11622, "type": "definition", "label": "stacks-sheaves-definition-presheaves", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-definition-presheaves", "contents": [ "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in", "groupoids.", "\\begin{enumerate}", "\\item A {\\it presheaf on $\\mathcal{X}$} is a presheaf on the", "underlying category of $\\mathcal{X}$.", "\\item A {\\it morphism of presheaves on $\\mathcal{X}$} is a morphism of", "presheaves on the underlying category of $\\mathcal{X}$.", "\\end{enumerate}", "We denote $\\textit{PSh}(\\mathcal{X})$ the category of presheaves on", "$\\mathcal{X}$." ], "refs": [], "ref_ids": [] }, { "id": 11623, "type": "definition", "label": "stacks-sheaves-definition-inherited-topologies", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-definition-inherited-topologies", "contents": [ "Let $\\mathcal{X}$ be a category fibred in groupoids over", "$(\\Sch/S)_{fppf}$.", "\\begin{enumerate}", "\\item The {\\it associated Zariski site}, denoted $\\mathcal{X}_{Zar}$,", "is the structure of site on $\\mathcal{X}$ inherited from", "$(\\Sch/S)_{Zar}$.", "\\item The {\\it associated \\'etale site}, denoted $\\mathcal{X}_\\etale$,", "is the structure of site on $\\mathcal{X}$ inherited from", "$(\\Sch/S)_\\etale$.", "\\item The {\\it associated smooth site}, denoted $\\mathcal{X}_{smooth}$,", "is the structure of site on $\\mathcal{X}$ inherited from", "$(\\Sch/S)_{smooth}$.", "\\item The {\\it associated syntomic site}, denoted $\\mathcal{X}_{syntomic}$,", "is the structure of site on $\\mathcal{X}$ inherited from", "$(\\Sch/S)_{syntomic}$.", "\\item The {\\it associated fppf site}, denoted $\\mathcal{X}_{fppf}$,", "is the structure of site on $\\mathcal{X}$ inherited from", "$(\\Sch/S)_{fppf}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11624, "type": "definition", "label": "stacks-sheaves-definition-sheaves", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-definition-sheaves", "contents": [ "Let $\\mathcal{X}$ be a category fibred in groupoids over", "$(\\Sch/S)_{fppf}$. Let $\\mathcal{F}$ be a presheaf on $\\mathcal{X}$.", "\\begin{enumerate}", "\\item We say $\\mathcal{F}$ is a {\\it Zariski sheaf}, or a", "{\\it sheaf for the Zariski topology} if $\\mathcal{F}$", "is a sheaf on the associated Zariski site $\\mathcal{X}_{Zar}$.", "\\item We say $\\mathcal{F}$ is an {\\it \\'etale sheaf}, or a", "{\\it sheaf for the \\'etale topology} if $\\mathcal{F}$", "is a sheaf on the associated \\'etale site $\\mathcal{X}_\\etale$.", "\\item We say $\\mathcal{F}$ is a {\\it smooth sheaf}, or a", "{\\it sheaf for the smooth topology} if $\\mathcal{F}$", "is a sheaf on the associated smooth site $\\mathcal{X}_{smooth}$.", "\\item We say $\\mathcal{F}$ is a {\\it syntomic sheaf}, or a", "{\\it sheaf for the syntomic topology} if $\\mathcal{F}$", "is a sheaf on the associated syntomic site $\\mathcal{X}_{syntomic}$.", "\\item We say $\\mathcal{F}$ is an {\\it fppf sheaf}, or a {\\it sheaf},", "or a {\\it sheaf for the fppf topology} if $\\mathcal{F}$", "is a sheaf on the associated fppf site $\\mathcal{X}_{fppf}$.", "\\end{enumerate}", "A morphism of sheaves is just a morphism of presheaves. We denote", "these categories of sheaves", "$\\Sh(\\mathcal{X}_{Zar})$,", "$\\Sh(\\mathcal{X}_\\etale)$,", "$\\Sh(\\mathcal{X}_{smooth})$,", "$\\Sh(\\mathcal{X}_{syntomic})$, and", "$\\Sh(\\mathcal{X}_{fppf})$." ], "refs": [], "ref_ids": [] }, { "id": 11625, "type": "definition", "label": "stacks-sheaves-definition-morphism", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-definition-morphism", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of categories", "fibred in groupoids over $(\\Sch/S)_{fppf}$. We denote", "$$", "f = (f^{-1}, f_*) :", "\\Sh(\\mathcal{X}_{fppf})", "\\longrightarrow", "\\Sh(\\mathcal{Y}_{fppf})", "$$", "the {\\it associated morphism of fppf topoi} constructed above.", "Similarly for the associated Zariski, \\'etale, smooth, and syntomic topoi." ], "refs": [], "ref_ids": [] }, { "id": 11626, "type": "definition", "label": "stacks-sheaves-definition-structure-sheaf", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-definition-structure-sheaf", "contents": [ "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category", "fibred in groupoids. The", "{\\it structure sheaf of $\\mathcal{X}$} is the sheaf of rings", "$\\mathcal{O}_\\mathcal{X} = p^{-1}\\mathcal{O}$." ], "refs": [], "ref_ids": [] }, { "id": 11627, "type": "definition", "label": "stacks-sheaves-definition-modules", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-definition-modules", "contents": [ "Let $\\mathcal{X}$ be a category fibred in groupoids over", "$(\\Sch/S)_{fppf}$.", "\\begin{enumerate}", "\\item A {\\it presheaf of modules on $\\mathcal{X}$} is a", "presheaf of $\\mathcal{O}_\\mathcal{X}$-modules. The category of", "presheaves of modules is denoted $\\textit{PMod}(\\mathcal{O}_\\mathcal{X})$.", "\\item We say a presheaf of modules $\\mathcal{F}$ is an", "{\\it $\\mathcal{O}_\\mathcal{X}$-module}, or more precisely a", "{\\it sheaf of $\\mathcal{O}_\\mathcal{X}$-modules} if $\\mathcal{F}$", "is an fppf sheaf. The category of $\\mathcal{O}_\\mathcal{X}$-modules", "is denoted $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11628, "type": "definition", "label": "stacks-sheaves-definition-pullback", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-definition-pullback", "contents": [ "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred", "in groupoids. Let $x \\in \\Ob(\\mathcal{X})$ lying over $U = p(x)$.", "Let $\\mathcal{F}$ be a presheaf on $\\mathcal{X}$.", "\\begin{enumerate}", "\\item The {\\it pullback $x^{-1}\\mathcal{F}$ of $\\mathcal{F}$} is the", "restriction $\\mathcal{F}|_{(\\mathcal{X}/x)}$ viewed as a presheaf on", "$(\\Sch/U)_{fppf}$ via the equivalence", "$\\mathcal{X}/x \\to (\\Sch/U)_{fppf}$ of", "Lemma \\ref{lemma-localizing}.", "\\item The {\\it restriction of $\\mathcal{F}$ to $U_\\etale$}", "is $x^{-1}\\mathcal{F}|_{U_\\etale}$, abusively written", "$\\mathcal{F}|_{U_\\etale}$.", "\\end{enumerate}" ], "refs": [ "stacks-sheaves-lemma-localizing" ], "ref_ids": [ 11574 ] }, { "id": 11629, "type": "definition", "label": "stacks-sheaves-definition-quasi-coherent", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-definition-quasi-coherent", "contents": [ "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred", "in groupoids. A {\\it quasi-coherent module on $\\mathcal{X}$}, or a", "{\\it quasi-coherent $\\mathcal{O}_\\mathcal{X}$-module} is a", "quasi-coherent module on the ringed site", "$(\\mathcal{X}_{fppf}, \\mathcal{O}_\\mathcal{X})$ as in", "Modules on Sites, Definition \\ref{sites-modules-definition-site-local}.", "The category of quasi-coherent sheaves on $\\mathcal{X}$", "is denoted $\\QCoh(\\mathcal{O}_\\mathcal{X})$." ], "refs": [ "sites-modules-definition-site-local" ], "ref_ids": [ 14289 ] }, { "id": 11630, "type": "definition", "label": "stacks-sheaves-definition-locally-quasi-coherent", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-definition-locally-quasi-coherent", "contents": [ "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category", "fibred in groupoids. Let $\\mathcal{F}$", "be a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules.", "We say $\\mathcal{F}$ is {\\it locally quasi-coherent}\\footnote{This is", "nonstandard notation.} if", "$\\mathcal{F}$ is a sheaf for the \\'etale topology and", "for every object $x$ of $\\mathcal{X}$ the restriction", "$x^*\\mathcal{F}|_{U_\\etale}$ is a quasi-coherent", "sheaf. Here $U = p(x)$." ], "refs": [], "ref_ids": [] }, { "id": 11712, "type": "definition", "label": "resolve-definition-normalized-blowup", "categories": [ "resolve" ], "title": "resolve-definition-normalized-blowup", "contents": [ "Let $X$ be a scheme such that every quasi-compact open has finitely", "many irreducible components. Let $x \\in X$ be a closed point.", "The {\\it normalized blowup of $X$ at $x$} is the composition", "$X'' \\to X' \\to X$ where $X' \\to X$ is the blowup", "of $X$ in $x$ and $X'' \\to X'$ is the normalization of $X'$." ], "refs": [], "ref_ids": [] }, { "id": 11713, "type": "definition", "label": "resolve-definition-reduce-to-rational", "categories": [ "resolve" ], "title": "resolve-definition-reduce-to-rational", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain", "of dimension $2$.", "\\begin{enumerate}", "\\item We say $A$ {\\it defines a rational singularity} if for every", "normal modification $X \\to \\Spec(A)$ we have $H^1(X, \\mathcal{O}_X) = 0$.", "\\item We say that {\\it reduction to rational singularities", "is possible for $A$} if the length of the $A$-modules", "$$", "H^1(X, \\mathcal{O}_X)", "$$", "is bounded for all modifications $X \\to \\Spec(A)$ with $X$ normal.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11714, "type": "definition", "label": "resolve-definition-resolution", "categories": [ "resolve" ], "title": "resolve-definition-resolution", "contents": [ "Let $Y$ be a Noetherian integral scheme. A {\\it resolution of singularities}", "of $Y$ is a modification $f : X \\to Y$ such that $X$ is regular." ], "refs": [], "ref_ids": [] }, { "id": 11715, "type": "definition", "label": "resolve-definition-resolution-surface", "categories": [ "resolve" ], "title": "resolve-definition-resolution-surface", "contents": [ "Let $Y$ be a $2$-dimensional Noetherian integral scheme.", "We say $Y$ has a {\\it resolution of singularities by normalized blowups}", "if there exists a sequence", "$$", "Y_n \\to Y_{n - 1} \\to \\ldots \\to Y_1 \\to Y_0 \\to Y", "$$", "where", "\\begin{enumerate}", "\\item $Y_i$ is proper over $Y$ for $i = 0, \\ldots, n$,", "\\item $Y_0 \\to Y$ is the normalization,", "\\item $Y_i \\to Y_{i - 1}$ is a normalized blowup for $i = 1, \\ldots, n$, and", "\\item $Y_n$ is regular.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11719, "type": "definition", "label": "exercises-definition-directed-poset", "categories": [ "exercises" ], "title": "exercises-definition-directed-poset", "contents": [ "A {\\it directed set} is a nonempty set $I$ endowed with a preorder $\\leq$", "such that given any pair $i, j \\in I$ there exists a $k \\in I$ such that", "$i \\leq k$ and $j \\leq k$. A {\\it system of rings} over $I$ is given by a", "ring $A_i$ for each $i \\in I$ and a map of rings $\\varphi_{ij} : A_i \\to A_j$", "whenever $i \\leq j$ such that the composition $A_i \\to A_j \\to A_k$ is equal to", "$A_i \\to A_k$ whenever $i \\leq j \\leq k$." ], "refs": [], "ref_ids": [] }, { "id": 11720, "type": "definition", "label": "exercises-definition-colimit", "categories": [ "exercises" ], "title": "exercises-definition-colimit", "contents": [ "The ring $A$ constructed in Exercise \\ref{exercise-directed-colimit}", "is called the {\\it colimit} of the system. Notation $\\colim A_i$." ], "refs": [], "ref_ids": [] }, { "id": 11721, "type": "definition", "label": "exercises-definition-finite-presentation", "categories": [ "exercises" ], "title": "exercises-definition-finite-presentation", "contents": [ "A module $M$ over $R$ is said to be of {\\it finite presentation} over", "$R$ if it is isomorphic to the cokernel of a map of finite free modules", "$ R^{\\oplus n} \\to R^{\\oplus m}$." ], "refs": [], "ref_ids": [] }, { "id": 11722, "type": "definition", "label": "exercises-definition-quasi-compact", "categories": [ "exercises" ], "title": "exercises-definition-quasi-compact", "contents": [ "A topological space $X$ is called {\\it quasi-compact}", "if for any open covering $X = \\bigcup_{i\\in I} U_i$ there is a finite", "subset $\\{i_1, \\ldots, i_n\\}\\subset I$ such that $X = U_{i_1}\\cup\\ldots", "U_{i_n}$." ], "refs": [], "ref_ids": [] }, { "id": 11723, "type": "definition", "label": "exercises-definition-Hausdorff", "categories": [ "exercises" ], "title": "exercises-definition-Hausdorff", "contents": [ "A topological space $X$ is said to verify the separation axiom $T_0$", "if for any pair of points $x, y\\in X$, $x\\not = y$ there is an open", "subset of $X$ containing one but not the other.", "We say that $X$ is {\\it Hausdorff} if for any pair $x, y\\in X$, $x\\not = y$", "there are disjoint open subsets $U, V$ such that $x\\in U$", "and $y\\in V$." ], "refs": [], "ref_ids": [] }, { "id": 11724, "type": "definition", "label": "exercises-definition-irreducible", "categories": [ "exercises" ], "title": "exercises-definition-irreducible", "contents": [ "A topological space $X$ is called {\\it irreducible} if $X$ is not empty", "and if $X = Z_1\\cup Z_2$ with $Z_1, Z_2\\subset X$ closed, then either", "$Z_1 = X$ or $Z_2 = X$. A subset $T\\subset X$ of a topological space", "is called {\\it irreducible} if it is an irreducible", "topological space with the topology induced from $X$.", "This definition implies $T$ is irreducible if and only", "if the closure $\\bar T$ of $T$ in $X$ is irreducible." ], "refs": [], "ref_ids": [] }, { "id": 11725, "type": "definition", "label": "exercises-definition-generic-point", "categories": [ "exercises" ], "title": "exercises-definition-generic-point", "contents": [ "A point $x$ of an irreducible topological space $X$ is called", "a {\\it generic point} of $X$ if $X$ is equal to the closure of", "the subset $\\{x\\}$." ], "refs": [], "ref_ids": [] }, { "id": 11726, "type": "definition", "label": "exercises-definition-Noetherian-space", "categories": [ "exercises" ], "title": "exercises-definition-Noetherian-space", "contents": [ "A topological space $X$ is called {\\it Noetherian} if any", "decreasing sequence $Z_1\\supset Z_2 \\supset Z_3\\supset \\ldots$", "of closed subsets of $X$ stabilizes.", "(It is called {\\it Artinian} if any increasing sequence of closed", "subsets stabilizes.)" ], "refs": [], "ref_ids": [] }, { "id": 11727, "type": "definition", "label": "exercises-definition-irreducible-component", "categories": [ "exercises" ], "title": "exercises-definition-irreducible-component", "contents": [ "A maximal irreducible subset $T\\subset X$ is called an", "{\\it irreducible component} of the space $X$. Such an irreducible", "component of $X$ is automatically a closed subset of $X$." ], "refs": [], "ref_ids": [] }, { "id": 11728, "type": "definition", "label": "exercises-definition-closed", "categories": [ "exercises" ], "title": "exercises-definition-closed", "contents": [ "A point $x\\in X$ is called {\\it closed} if $\\overline{\\{x\\}} = \\{ x\\}$.", "Let $x, y$ be points of $X$. We say that $x$ is a {\\it specialization}", "of $y$, or that $y$ is a {\\it generalization} of $x$ if", "$x\\in \\overline{\\{y\\}}$." ], "refs": [], "ref_ids": [] }, { "id": 11729, "type": "definition", "label": "exercises-definition-connected-component", "categories": [ "exercises" ], "title": "exercises-definition-connected-component", "contents": [ "A topological space $X$ is called {\\it connected} if it is nonempty and not the", "union of two nonempty disjoint open subsets. A {\\it connected component}", "of $X$ is a maximal connected subset. Any point of $X$ is contained", "in a connected component of $X$ and any connected component of $X$ is", "closed in $X$. (But in general a connected component need not be open in $X$.)" ], "refs": [], "ref_ids": [] }, { "id": 11730, "type": "definition", "label": "exercises-definition-length", "categories": [ "exercises" ], "title": "exercises-definition-length", "contents": [ "Let $A$ be a ring. Let $M$ be an $A$-module. The", "{\\it length} of $M$ as an $R$-module is", "$$", "\\text{length}_A(M)", "=", "\\sup", "\\{", "n", "\\mid", "\\exists\\ 0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M,", "\\text{ }M_i \\not = M_{i + 1}", "\\}.", "$$", "In other words, the supremum of the lengths of chains of submodules." ], "refs": [], "ref_ids": [] }, { "id": 11731, "type": "definition", "label": "exercises-definition-catenary", "categories": [ "exercises" ], "title": "exercises-definition-catenary", "contents": [ "A Noetherian ring $A$ is said to be {\\it catenary}", "if for any triple of prime ideals", "${\\mathfrak p}_1 \\subset {\\mathfrak p}_2 \\subset {\\mathfrak p}_3$", "we have", "$$", "ht({\\mathfrak p}_3 / {\\mathfrak p}_1) = ht({\\mathfrak p}_3/{\\mathfrak p}_2) +", "ht({\\mathfrak p}_2/{\\mathfrak p}_1).", "$$", "Here $ht(\\mathfrak p/\\mathfrak q)$ means the height of", "$\\mathfrak p/\\mathfrak q$ in the ring $A/\\mathfrak q$.", "In a formula", "$$", "ht(\\mathfrak p/\\mathfrak q) =", "\\dim(A_\\mathfrak p/\\mathfrak qA_\\mathfrak p) =", "\\dim((A/\\mathfrak q)_\\mathfrak p) =", "\\dim((A/\\mathfrak q)_{\\mathfrak p/\\mathfrak q})", "$$", "A topological space $X$ is {\\it catenary}, if given $T \\subset T' \\subset X$", "with $T$ and $T'$ closed and irreducible, then there exists a maximal chain", "of irreducible closed subsets", "$$", "T = T_0 \\subset T_1 \\subset \\ldots \\subset T_n = T'", "$$", "and every such chain has the same (finite) length." ], "refs": [], "ref_ids": [] }, { "id": 11732, "type": "definition", "label": "exercises-definition-finite-locally-free", "categories": [ "exercises" ], "title": "exercises-definition-finite-locally-free", "contents": [ "Let $A$ be a ring. Recall that a {\\it finite locally free} $A$-module", "$M$ is a module such that for every ${\\mathfrak p} \\in \\Spec(A)$", "there exists an", "$f\\in A$, $f \\not \\in {\\mathfrak p}$ such that $M_f$ is a finite free", "$A_f$-module. We say $M$ is an {\\it invertible module} if", "$M$ is finite locally free of rank $1$, i.e., for every", "${\\mathfrak p} \\in \\Spec(A)$ there exists an", "$f\\in A$, $f \\not \\in \\mathfrak p$ such that $M_f \\cong A_f$", "as an $A_f$-module." ], "refs": [], "ref_ids": [] }, { "id": 11733, "type": "definition", "label": "exercises-definition-class-group", "categories": [ "exercises" ], "title": "exercises-definition-class-group", "contents": [ "Let $A$ be a ring. The {\\it class group of $A$}, sometimes called", "the {\\it Picard group of $A$} is the set $\\Pic(A)$", "of isomorphism classes of invertible $A$-modules endowed with", "a group operation defined by tensor product (see", "Exercise \\ref{exercise-tensor-finite-locally-free})." ], "refs": [], "ref_ids": [] }, { "id": 11734, "type": "definition", "label": "exercises-definition-GU-GD", "categories": [ "exercises" ], "title": "exercises-definition-GU-GD", "contents": [ "Let $\\phi : A \\to B$ be a homomorphism of rings. We say", "that the {\\it going-up theorem} holds for $\\phi$ if the", "following condition is satisfied:", "\\begin{itemize}", "\\item[(GU)] for any ${\\mathfrak p}, {\\mathfrak p}' \\in \\Spec(A)$ such that", "${\\mathfrak p} \\subset {\\mathfrak p}'$, and for any $P \\in \\Spec(B)$ lying", "over ${\\mathfrak p}$, there exists $P'\\in \\Spec(B)$ lying", "over ${\\mathfrak p}'$ such that $P \\subset P'$.", "\\end{itemize}", "Similarly, we say that the {\\it going-down theorem} holds for $\\phi$", "if the following condition is satisfied:", "\\begin{itemize}", "\\item[(GD)] for any ${\\mathfrak p}, {\\mathfrak p}' \\in \\Spec(A)$ such that", "${\\mathfrak p} \\subset {\\mathfrak p}'$, and for any", "$P' \\in \\Spec(B)$ lying", "over ${\\mathfrak p}'$, there exists $P\\in \\Spec(B)$ lying", "over ${\\mathfrak p}$ such that $P \\subset P'$.", "\\end{itemize}" ], "refs": [], "ref_ids": [] }, { "id": 11735, "type": "definition", "label": "exercises-definition-numerical-polynomial", "categories": [ "exercises" ], "title": "exercises-definition-numerical-polynomial", "contents": [ "A {\\it numerical polynomial} is a polynomial $f(x) \\in {\\mathbf Q}[x]$", "such that $f(n) \\in {\\mathbf Z}$ for every integer $n$." ], "refs": [], "ref_ids": [] }, { "id": 11736, "type": "definition", "label": "exercises-definition-graded-module", "categories": [ "exercises" ], "title": "exercises-definition-graded-module", "contents": [ "A {\\it graded module} $M$ over a ring $A$ is an $A$-module $M$", "endowed with a direct sum decomposition", "$", "\\bigoplus\\nolimits_{n \\in {\\mathbf Z}} M_n", "$", "into $A$-submodules. We will say that $M$ is {\\it locally finite} if all of", "the $M_n$ are finite $A$-modules. Suppose that $A$ is a Noetherian ring and", "that $\\varphi$ is a {\\it Euler-Poincar\\'e function} on finite $A$-modules.", "This means that for every finitely generated $A$-module $M$ we are given an", "integer $\\varphi(M) \\in {\\mathbf Z}$ and for every short exact sequence", "$$", "0", "\\longrightarrow", "M'", "\\longrightarrow", "M", "\\longrightarrow", "M''", "\\longrightarrow", "0", "$$", "we have $\\varphi(M) = \\varphi(M') + \\varphi(M')$. The {\\it Hilbert function}", "of a locally finite graded module $M$ (with respect to $\\varphi$) is the", "function $\\chi_\\varphi(M, n) = \\varphi(M_n)$. We say that $M$ has a", "{\\it Hilbert polynomial} if there is some numerical polynomial", "$P_\\varphi$ such that $\\chi_\\varphi(M, n) = P_\\varphi(n)$ for all sufficiently", "large integers $n$." ], "refs": [], "ref_ids": [] }, { "id": 11737, "type": "definition", "label": "exercises-definition-graded-algebra", "categories": [ "exercises" ], "title": "exercises-definition-graded-algebra", "contents": [ "A {\\it graded $A$-algebra} is a graded $A$-module", "$B = \\bigoplus_{n \\geq 0} B_n$ together with an $A$-bilinear map", "$$", "B \\times B \\longrightarrow B, \\ (b, b') \\longmapsto bb'", "$$", "that turns $B$ into an $A$-algebra so that $B_n \\cdot B_m \\subset B_{n + m}$.", "Finally, a {\\it graded module $M$ over a graded $A$-algebra $B$} is given", "by a graded $A$-module $M$ together with a (compatible) $B$-module structure", "such that $B_n \\cdot M_d \\subset M_{n + d}$. Now you can define", "{\\it homomorphisms of graded modules/rings}, {\\it graded submodules},", "{\\it graded ideals}, {\\it exact sequences of graded modules}, etc, etc." ], "refs": [], "ref_ids": [] }, { "id": 11738, "type": "definition", "label": "exercises-definition-homogeneous-ideal", "categories": [ "exercises" ], "title": "exercises-definition-homogeneous-ideal", "contents": [ "Let $R$ be a graded ring. A {\\it homogeneous} ideal is simply an ideal", "$I \\subset R$ which is also a graded submodule of $R$. Equivalently,", "it is an ideal generated by homogeneous elements. Equivalently, if", "$f \\in I$ and", "$$", "f = f_0 + f_1 + \\ldots + f_n", "$$", "is the decomposition of $f$ into homogeneous pieces in $R$ then $f_i \\in I$", "for each $i$." ], "refs": [], "ref_ids": [] }, { "id": 11739, "type": "definition", "label": "exercises-definition-Proj-R", "categories": [ "exercises" ], "title": "exercises-definition-Proj-R", "contents": [ "We define the {\\it homogeneous spectrum $\\text{Proj}(R)$}", "of the graded ring $R$ to be the set of homogeneous, prime ideals", "${\\mathfrak p}$ of $R$ such that $R_{+} \\not \\subset {\\mathfrak p}$.", "Note that $\\text{Proj}(R)$ is a subset of $\\Spec(R)$ and hence has a", "natural induced topology." ], "refs": [], "ref_ids": [] }, { "id": 11740, "type": "definition", "label": "exercises-definition-Dplus-Vplus", "categories": [ "exercises" ], "title": "exercises-definition-Dplus-Vplus", "contents": [ "Let $R = \\oplus_{d \\geq 0} R_d$ be a graded ring, let $f\\in R_d$ and", "assume that $d \\geq 1$. We define {\\it $R_{(f)}$} to be the subring of", "$R_f$ consisting of elements of the form $r/f^n$ with $r$ homogeneous and", "$\\deg(r) = nd$. Furthermore, we define", "$$", "D_{+}(f) = \\{ {\\mathfrak p} \\in \\text{Proj}(R) | f \\not\\in {\\mathfrak p} \\}.", "$$", "Finally, for a homogeneous ideal $I \\subset R$ we define", "$V_{+}(I) = V(I) \\cap \\text{Proj}(R)$." ], "refs": [], "ref_ids": [] }, { "id": 11741, "type": "definition", "label": "exercises-definition-CM", "categories": [ "exercises" ], "title": "exercises-definition-CM", "contents": [ "A Noetherian local ring $A$ is said to be {\\it Cohen-Macaulay}", "of dimension $d$ if it has dimension $d$ and there exists a system", "of parameters $x_1, \\ldots, x_d$ for $A$ such that $x_i$ is a nonzerodivisor", "in $A/(x_1, \\ldots, x_{i-1})$ for $i = 1, \\ldots, d$." ], "refs": [], "ref_ids": [] }, { "id": 11742, "type": "definition", "label": "exercises-definition-injective-filtered", "categories": [ "exercises" ], "title": "exercises-definition-injective-filtered", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $I$ be a filtered object of $\\mathcal{A}$.", "Assume the filtration on $I$ is finite.", "We say $I$ is {\\it filtered injective} if each $\\text{gr}^p(I)$ is", "an injective object of $\\mathcal{A}$." ], "refs": [], "ref_ids": [] }, { "id": 11743, "type": "definition", "label": "exercises-definition-finite-filtration-category", "categories": [ "exercises" ], "title": "exercises-definition-finite-filtration-category", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "We denote {\\it $\\text{Fil}^f(\\mathcal{A})$} the full subcategory", "of $\\text{Fil}(\\mathcal{A})$ whose objects consist of", "those $A \\in \\Ob(\\text{Fil}(\\mathcal{A}))$", "whose filtration is finite." ], "refs": [], "ref_ids": [] }, { "id": 11744, "type": "definition", "label": "exercises-definition-filtered-quasi-isomorphism", "categories": [ "exercises" ], "title": "exercises-definition-filtered-quasi-isomorphism", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $\\alpha : K^\\bullet \\to L^\\bullet$ be a morphism of", "complexes of $\\text{Fil}(\\mathcal{A})$. We say that", "$\\alpha$ is a {\\it filtered quasi-isomorphism} if", "for each $p \\in \\mathbf{Z}$ the morphism", "$\\text{gr}^p(K^\\bullet) \\to \\text{gr}^p(L^\\bullet)$ is", "a quasi-isomorphism." ], "refs": [], "ref_ids": [] }, { "id": 11745, "type": "definition", "label": "exercises-definition-filtered-acyclic", "categories": [ "exercises" ], "title": "exercises-definition-filtered-acyclic", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $K^\\bullet$ be a complex of $\\text{Fil}^f(\\mathcal{A})$.", "We say that $K^\\bullet$ is {\\it filtered acyclic} if", "for each $p \\in \\mathbf{Z}$ the complex $\\text{gr}^p(K^\\bullet)$ is", "acyclic." ], "refs": [], "ref_ids": [] }, { "id": 11746, "type": "definition", "label": "exercises-definition-integral", "categories": [ "exercises" ], "title": "exercises-definition-integral", "contents": [ "A scheme $X$ is called {\\it integral} if $X$ is nonempty and", "for every nonempty affine open $U \\subset X$ the ring", "$\\Gamma(U, \\mathcal{O}_X) = \\mathcal{O}_X(U)$ is a domain." ], "refs": [], "ref_ids": [] }, { "id": 11747, "type": "definition", "label": "exercises-definition-dual-numbers", "categories": [ "exercises" ], "title": "exercises-definition-dual-numbers", "contents": [ "For any ring $R$ we denote $R[\\epsilon]$ the ring", "of {\\it dual numbers}. As an $R$-module it is free with", "basis $1$, $\\epsilon$. The ring structure comes from setting", "$\\epsilon^2 = 0$." ], "refs": [], "ref_ids": [] }, { "id": 11748, "type": "definition", "label": "exercises-definition-tangent-space", "categories": [ "exercises" ], "title": "exercises-definition-tangent-space", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $x \\in X$. We dub the set of dotted arrows", "of Exercise \\ref{exercise-tangent-space-Zariski}", "the {\\it tangent space of $X$ over $S$}", "and we denote it $T_{X/S, x}$. An element of this", "space is called a {\\it tangent vector} of $X/S$ at $x$." ], "refs": [], "ref_ids": [] }, { "id": 11749, "type": "definition", "label": "exercises-definition-quasi-coherent", "categories": [ "exercises" ], "title": "exercises-definition-quasi-coherent", "contents": [ "Let $X$ be a scheme.", "A sheaf $\\mathcal{F}$ of $\\mathcal{O}_X$-modules is {\\it quasi-coherent}", "if for every affine open $\\Spec(R) = U \\subset X$ the restriction", "$\\mathcal{F}|_U$ is of the form $\\widetilde M$ for some $R$-module", "$M$." ], "refs": [], "ref_ids": [] }, { "id": 11750, "type": "definition", "label": "exercises-definition-specialization", "categories": [ "exercises" ], "title": "exercises-definition-specialization", "contents": [ "Let $X$ be a topological space. Let $x, x' \\in X$.", "We say $x$ is a {\\it specialization} of $x'$", "if and only if $x \\in \\overline{\\{x'\\}}$." ], "refs": [], "ref_ids": [] }, { "id": 11751, "type": "definition", "label": "exercises-definition-Noetherian-scheme", "categories": [ "exercises" ], "title": "exercises-definition-Noetherian-scheme", "contents": [ "A scheme $X$ is called {\\it locally Noetherian} if and only if", "for every point $x \\in X$ there exists an affine open", "$\\Spec(R) = U \\subset X$ such that $R$ is Noetherian.", "A scheme is {\\it Noetherian} if it is locally Noetherian and quasi-compact." ], "refs": [], "ref_ids": [] }, { "id": 11752, "type": "definition", "label": "exercises-definition-coherent", "categories": [ "exercises" ], "title": "exercises-definition-coherent", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf of", "$\\mathcal{O}_X$-modules. We say $\\mathcal{F}$ is {\\it coherent}", "if for every point $x \\in X$ there exists an affine open", "$\\Spec(R) = U \\subset X$ such that $\\mathcal{F}|_U$", "is isomorphic to $\\widetilde M$ for some finite $R$-module $M$." ], "refs": [], "ref_ids": [] }, { "id": 11753, "type": "definition", "label": "exercises-definition-invertible-sheaf", "categories": [ "exercises" ], "title": "exercises-definition-invertible-sheaf", "contents": [ "Let $X$ be a locally ringed space.", "An {\\it invertible ${\\mathcal O}_X$-module} on $X$", "is a sheaf of ${\\mathcal O}_X$-modules ${\\mathcal L}$ such that every point", "has an open neighbourhood $U \\subset X$ such that ${\\mathcal L}|_U$", "is isomorphic to ${\\mathcal O}_U$ as ${\\mathcal O}_U$-module.", "We say that ${\\mathcal L}$ is trivial if it is isomorphic to", "${\\mathcal O}_X$ as a ${\\mathcal O}_X$-module." ], "refs": [], "ref_ids": [] }, { "id": 11754, "type": "definition", "label": "exercises-definition-invertible-module", "categories": [ "exercises" ], "title": "exercises-definition-invertible-module", "contents": [ "Let $R$ be a ring. An {\\it invertible module $M$} is an $R$-module", "$M$ such that $\\widetilde M$ is an invertible sheaf on the", "spectrum of $R$. We say $M$ is {\\it trivial} if $M \\cong R$ as", "an $R$-module." ], "refs": [], "ref_ids": [] }, { "id": 11755, "type": "definition", "label": "exercises-definition-picard-group", "categories": [ "exercises" ], "title": "exercises-definition-picard-group", "contents": [ "Let $X$ be a locally ringed space.", "The {\\it Picard group of $X$} is the set $\\Pic(X)$", "of isomorphism classes of invertible $\\mathcal{O}_X$-modules", "with addition given by tensor product.", "See Modules, Definition \\ref{modules-definition-pic}.", "For a ring $R$ we set $\\Pic(R) = \\Pic(\\Spec(R))$." ], "refs": [ "modules-definition-pic" ], "ref_ids": [ 13352 ] }, { "id": 11756, "type": "definition", "label": "exercises-definition-delta", "categories": [ "exercises" ], "title": "exercises-definition-delta", "contents": [ "(Definition of delta.) Suppose that", "$$", "0 \\to {\\mathcal F}_1 \\to {\\mathcal F}_2 \\to {\\mathcal F}_3 \\to 0", "$$", "is a short exact sequence of abelian sheaves on any topological space $X$.", "The boundary map", "$\\delta : H^0(X, {\\mathcal F}_3) \\to {\\check H}^1(X, {\\mathcal F}_1)$", "is defined as follows. Take an element $\\tau \\in H^0(X, {\\mathcal F}_3)$.", "Choose an open covering ${\\mathcal U} : X = \\bigcup_{i\\in I} U_i$ such", "that for each $i$ there exists a section $\\tilde \\tau_i \\in {\\mathcal F}_2$", "lifting the restriction of $\\tau$ to $U_i$. Then consider the assignment", "$$", "(i_0, i_1) \\longmapsto", "\\tilde \\tau_{i_0}|_{U_{i_0i_1}} - \\tilde \\tau_{i_1}|_{U_{i_0i_1}}.", "$$", "This is clearly a 1-coboundary in the {\\v C}ech complex", "${\\check C}^\\ast({\\mathcal U}, {\\mathcal F}_2)$. But we observe that", "(thinking of ${\\mathcal F}_1$ as a subsheaf of ${\\mathcal F}_2$) the RHS", "always is a section of ${\\mathcal F}_1$ over $U_{i_0i_1}$. Hence we", "see that the assignment defines a 1-cochain in the complex", "${\\check C}^\\ast({\\mathcal U}, {\\mathcal F}_2)$. The cohomology", "class of this 1-cochain is by definition {\\it $\\delta(\\tau)$}." ], "refs": [], "ref_ids": [] }, { "id": 11757, "type": "definition", "label": "exercises-definition-divisor", "categories": [ "exercises" ], "title": "exercises-definition-divisor", "contents": [ "Throughout, let $S$ be any scheme and let", "$X$ be a Noetherian, integral scheme.", "\\begin{enumerate}", "\\item A {\\it Weil divisor} on $X$ is a formal linear combination", "$\\Sigma n_i[Z_i]$ of prime divisors $Z_i$ with integer coefficients.", "\\item A {\\it prime divisor} is a closed subscheme $Z \\subset X$,", "which is integral with generic point $\\xi \\in Z$ such that", "${\\mathcal O}_{X, \\xi}$ has dimension $1$. We will use the notation", "${\\mathcal O}_{X, Z} = {\\mathcal O}_{X, \\xi}$", "when $\\xi \\in Z \\subset X$ is as above. Note that ${\\mathcal O}_{X, Z} \\subset", "K(X)$ is a subring of the function field of $X$.", "\\item The {\\it Weil divisor associated to a rational function", "$f \\in K(X)^\\ast$} is the sum $\\Sigma v_Z(f)[Z]$. Here $v_Z(f)$ is", "defined as follows", "\\begin{enumerate}", "\\item If $f \\in {\\mathcal O}_{X, Z}^\\ast$ then $v_Z(f) = 0$.", "\\item If $f \\in {\\mathcal O}_{X, Z}$ then", "$$", "v_Z(f) = \\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(f)).", "$$", "\\item If $f = \\frac{a}{b}$ with $a, b \\in {\\mathcal O}_{X, Z}$", "then", "$$", "v_Z(f) = \\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(a)) -", "\\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(b)).", "$$", "\\end{enumerate}", "\\item An {\\it effective Cartier divisor} on a scheme $S$", "is a closed subscheme $D \\subset S$ such that every point $d\\in D$", "has an affine open neighbourhood $\\Spec(A) = U \\subset S$ in $S$", "so that $D \\cap U = \\Spec(A/(f))$ with $f \\in A$ a nonzerodivisor.", "\\item The {\\it Weil divisor $[D]$ associated to an effective", "Cartier divisor $D \\subset X$} of our Noetherian integral", "scheme $X$ is defined as the sum $\\Sigma v_Z(D)[Z]$ where", "$v_Z(D)$ is defined as follows", "\\begin{enumerate}", "\\item If the generic point $\\xi$ of $Z$ is not in $D$", "then $v_Z(D) = 0$.", "\\item If the generic point $\\xi$ of $Z$ is in $D$", "then", "$$", "v_Z(D) = \\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(f))", "$$", "where $f \\in {\\mathcal O}_{X, Z} = {\\mathcal O}_{X, \\xi}$ is the nonzerodivisor", "which defines $D$ in an affine neighbourhood of $\\xi$ (as in (4) above).", "\\end{enumerate}", "\\item Let $S$ be a scheme. The {\\it sheaf of total quotient", "rings ${\\mathcal K}_S$} is the sheaf of ${\\mathcal O}_S$-algebras which is", "the sheafification of the pre-sheaf ${\\mathcal K}'$ defined as follows.", "For $U \\subset S$ open we set ${\\mathcal K}'(U) = S_U^{-1}{\\mathcal O}_S(U)$", "where $S_U \\subset {\\mathcal O}_S(U)$ is the multiplicative subset", "consisting of sections $f \\in {\\mathcal O}_S(U)$ such that the germ", "of $f$ in ${\\mathcal O}_{S, u}$ is a nonzerodivisor for every $u\\in U$.", "In particular the elements of $S_U$ are all nonzerodivisors.", "Thus ${\\mathcal O}_S$ is a subsheaf of ${\\mathcal K}_S$, and we get a", "short exact sequence", "$$", "0 \\to {\\mathcal O}_S^\\ast \\to {\\mathcal K}_S^\\ast \\to", "{\\mathcal K}_S^\\ast/{\\mathcal O}_S^\\ast \\to 0.", "$$", "\\item A {\\it Cartier divisor} on a scheme $S$ is a global", "section of the quotient sheaf ${\\mathcal K}_S^\\ast/{\\mathcal O}_S^\\ast$.", "\\item The {\\it Weil divisor associated to a Cartier divisor}", "$\\tau \\in \\Gamma(X, {\\mathcal K}_X^\\ast/{\\mathcal O}_X^\\ast)$ over our", "Noetherian integral scheme", "$X$ is the sum $\\Sigma v_Z(\\tau)[Z]$ where $v_Z(\\tau)$ is defined", "as by the following recipe", "\\begin{enumerate}", "\\item If the germ of $\\tau$ at the generic point $\\xi$", "of $Z$ is zero -- in other words the image of $\\tau$ in the stalk", "$({\\mathcal K}^\\ast/{\\mathcal O}^\\ast)_\\xi$ is ``zero'' -- then $v_Z(\\tau) = 0$.", "\\item Find an affine open neighbourhood $\\Spec(A) = U \\subset X$", "so that $\\tau|_U$ is the image of a section $f \\in {\\mathcal K}(U)$", "and moreover $f = a/b$ with $a, b \\in A$. Then we set", "$$", "v_Z(f) = \\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(a)) -", "\\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(b)).", "$$", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11809, "type": "definition", "label": "spaces-duality-definition-dualizing-scheme", "categories": [ "spaces-duality" ], "title": "spaces-duality-definition-dualizing-scheme", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a locally Noetherian algebraic space over $S$.", "An object $K$ of $D_\\QCoh(\\mathcal{O}_X)$ is called a", "{\\it dualizing complex} if $K$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-equivalent-definitions}." ], "refs": [ "spaces-duality-lemma-equivalent-definitions" ], "ref_ids": [ 11783 ] }, { "id": 11810, "type": "definition", "label": "spaces-duality-definition-relative-dualizing-proper-flat", "categories": [ "spaces-duality" ], "title": "spaces-duality-definition-relative-dualizing-proper-flat", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper, flat morphism", "of algebraic spaces over $S$ which is of finite presentation.", "A {\\it relative dualizing complex} for $X/Y$ is a pair", "$(\\omega_{X/Y}^\\bullet, \\tau)$ consisting of a", "$Y$-perfect object $\\omega_{X/Y}^\\bullet$ of $D(\\mathcal{O}_X)$", "and a map", "$$", "\\tau : Rf_*\\omega_{X/Y}^\\bullet \\longrightarrow \\mathcal{O}_Y", "$$", "such that for any cartesian square", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "where $Y'$ is an affine scheme the pair", "$(L(g')^*\\omega_{X/Y}^\\bullet, Lg^*\\tau)$", "is isomorphic to the pair", "$(a'(\\mathcal{O}_{Y'}), \\text{Tr}_{f', \\mathcal{O}_{Y'}})$", "studied in Sections", "\\ref{section-twisted-inverse-image},", "\\ref{section-base-change-map},", "\\ref{section-base-change-II},", "\\ref{section-trace},", "\\ref{section-compare-with-pullback}, and", "\\ref{section-proper-flat}." ], "refs": [], "ref_ids": [] }, { "id": 11922, "type": "definition", "label": "spaces-properties-definition-separated", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-separated", "contents": [ "(Compare Spaces, Definition \\ref{spaces-definition-separated}.)", "Consider a big fppf site", "$\\Sch_{fppf} = (\\Sch/\\Spec(\\mathbf{Z}))_{fppf}$.", "Let $X$ be an algebraic space over", "$\\Spec(\\mathbf{Z})$. Let $\\Delta : X \\to X \\times X$", "be the diagonal morphism.", "\\begin{enumerate}", "\\item We say $X$ is {\\it separated} if $\\Delta$ is a closed immersion.", "\\item We say $X$ is {\\it locally separated}\\footnote{In the", "literature this often refers to quasi-separated and locally", "separated algebraic spaces.} if $\\Delta$ is an", "immersion.", "\\item We say $X$ is {\\it quasi-separated} if $\\Delta$ is quasi-compact.", "\\item We say $X$ is {\\it Zariski locally quasi-separated}\\footnote{", "This notion was suggested by B.\\ Conrad.} if there", "exists a Zariski covering $X = \\bigcup_{i \\in I} X_i$ (see Spaces,", "Definition \\ref{spaces-definition-Zariski-open-covering}) such that", "each $X_i$ is quasi-separated.", "\\end{enumerate}", "Let $S$ is a scheme contained in $\\Sch_{fppf}$, and let", "$X$ be an algebraic space over $S$. Then we say $X$ is {\\it separated},", "{\\it locally separated}, {\\it quasi-separated}, or", "{\\it Zariski locally quasi-separated}", "if $X$ viewed as an algebraic space over $\\Spec(\\mathbf{Z})$ (see", "Spaces, Definition \\ref{spaces-definition-base-change})", "has the corresponding property." ], "refs": [ "spaces-definition-separated", "spaces-definition-Zariski-open-covering", "spaces-definition-base-change" ], "ref_ids": [ 8181, 8179, 8183 ] }, { "id": 11923, "type": "definition", "label": "spaces-properties-definition-points", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-points", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "A {\\it point} of $X$ is an equivalence class of morphisms", "from spectra of fields into $X$.", "The set of points of $X$ is denoted $|X|$." ], "refs": [], "ref_ids": [] }, { "id": 11924, "type": "definition", "label": "spaces-properties-definition-topological-space", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-topological-space", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The underlying {\\it topological space} of $X$ is the set of points", "$|X|$ endowed with the topology constructed in", "Lemma \\ref{lemma-topology-points}." ], "refs": [ "spaces-properties-lemma-topology-points" ], "ref_ids": [ 11822 ] }, { "id": 11925, "type": "definition", "label": "spaces-properties-definition-quasi-compact", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-quasi-compact", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "We say $X$ is {\\it quasi-compact} if there exists a surjective", "\\'etale morphism $U \\to X$ with $U$ quasi-compact." ], "refs": [], "ref_ids": [] }, { "id": 11926, "type": "definition", "label": "spaces-properties-definition-type-property", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-type-property", "contents": [ "Let $\\mathcal{P}$ be a property of schemes which is", "local in the \\'etale topology.", "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "We say $X$ {\\it has property $\\mathcal{P}$}", "if any of the equivalent conditions of", "Lemma \\ref{lemma-type-property}", "hold." ], "refs": [ "spaces-properties-lemma-type-property" ], "ref_ids": [ 11837 ] }, { "id": 11927, "type": "definition", "label": "spaces-properties-definition-property-at-point", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-property-at-point", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$. Let $\\mathcal{P}$ be a property of germs of schemes which is", "\\'etale local.", "We say $X$ {\\it has property $\\mathcal{P}$ at $x$} if any of the", "equivalent conditions of", "Lemma \\ref{lemma-local-source-target-at-point}", "hold." ], "refs": [ "spaces-properties-lemma-local-source-target-at-point" ], "ref_ids": [ 11838 ] }, { "id": 11928, "type": "definition", "label": "spaces-properties-definition-locally-constructible", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-locally-constructible", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $E \\subset |X|$ be a subset. We say $E$ is", "{\\it \\'etale locally constructible} if the equivalent", "conditions of Lemma \\ref{lemma-locally-constructible} are satisfied." ], "refs": [ "spaces-properties-lemma-locally-constructible" ], "ref_ids": [ 11839 ] }, { "id": 11929, "type": "definition", "label": "spaces-properties-definition-dimension-at-point", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-dimension-at-point", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$ be a point of $X$.", "We define the {\\it dimension of $X$ at $x$} to be", "the element $\\dim_x(X) \\in \\{0, 1, 2, \\ldots, \\infty\\}$", "such that $\\dim_x(X) = \\dim_u(U)$ for any (equivalently some)", "pair $(a : U \\to X, u)$ consisting of an \\'etale morphism $a : U \\to X$", "from a scheme to $X$ and a point $u \\in U$ with $a(u) = x$.", "See", "Definition \\ref{definition-property-at-point},", "Lemma \\ref{lemma-local-source-target-at-point}, and", "Descent, Lemma \\ref{descent-lemma-dimension-at-point-local}." ], "refs": [ "spaces-properties-definition-property-at-point", "spaces-properties-lemma-local-source-target-at-point", "descent-lemma-dimension-at-point-local" ], "ref_ids": [ 11927, 11838, 14660 ] }, { "id": 11930, "type": "definition", "label": "spaces-properties-definition-dimension", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-dimension", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The {\\it dimension} $\\dim(X)$ of $X$ is defined by the rule", "$$", "\\dim(X) = \\sup\\nolimits_{x \\in |X|} \\dim_x(X)", "$$" ], "refs": [], "ref_ids": [] }, { "id": 11931, "type": "definition", "label": "spaces-properties-definition-dimension-local-ring", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-dimension-local-ring", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$", "be a point. The {\\it dimension of the local ring of $X$ at $x$} is", "the element $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$ satisfying the equivalent", "conditions of Lemma \\ref{lemma-pre-dimension-local-ring}. In this case we", "will also say {\\it $x$ is a point of codimension $d$ on $X$}." ], "refs": [ "spaces-properties-lemma-pre-dimension-local-ring" ], "ref_ids": [ 11840 ] }, { "id": 11932, "type": "definition", "label": "spaces-properties-definition-reduced-induced-space", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-reduced-induced-space", "contents": [ "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.", "Let $Z \\subset |X|$ be a closed subset.", "An {\\it algebraic space structure on $Z$} is given by a closed subspace", "$Z'$ of $X$ with $|Z'|$ equal to $Z$.", "The {\\it reduced induced algebraic space structure}", "on $Z$ is the one constructed in", "Lemma \\ref{lemma-reduced-closed-subspace}.", "The {\\it reduction $X_{red}$ of $X$} is the reduced induced algebraic", "space structure on $|X|$." ], "refs": [ "spaces-properties-lemma-reduced-closed-subspace" ], "ref_ids": [ 11846 ] }, { "id": 11933, "type": "definition", "label": "spaces-properties-definition-etale", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-etale", "contents": [ "Let $S$ be a scheme.", "A morphism $f : X \\to Y$ between algebraic spaces over $S$ is", "called {\\it \\'etale} if and only if for every \\'etale morphism", "$\\varphi : U \\to X$ where $U$ is a scheme, the composition", "$f \\circ \\varphi$ is \\'etale also." ], "refs": [], "ref_ids": [] }, { "id": 11934, "type": "definition", "label": "spaces-properties-definition-etale-site", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-etale-site", "contents": [ "Let $S$ be a scheme.", "Let $\\Sch_{fppf}$ be a big fppf site containing $S$,", "and let $\\Sch_\\etale$ be the corresponding big \\'etale site", "(i.e., having the same underlying category).", "Let $X$ be an algebraic space over $S$.", "The {\\it small \\'etale site $X_\\etale$} of $X$ is defined as follows:", "\\begin{enumerate}", "\\item An object of $X_\\etale$ is a morphism $\\varphi : U \\to X$", "where $U \\in \\Ob((\\Sch/S)_\\etale)$ is a scheme and", "$\\varphi$ is an \\'etale morphism,", "\\item a morphism $(\\varphi : U \\to X) \\to (\\varphi' : U' \\to X)$", "is given by a morphism of schemes $\\chi : U \\to U'$ such that", "$\\varphi = \\varphi' \\circ \\chi$, and", "\\item a family of morphisms $\\{(U_i \\to X) \\to (U \\to X)\\}_{i \\in I}$", "of $X_\\etale$ is a covering if and only if $\\{U_i \\to U\\}_{i \\in I}$", "is a covering of $(\\Sch/S)_\\etale$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11935, "type": "definition", "label": "spaces-properties-definition-spaces-etale-site", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-spaces-etale-site", "contents": [ "Let $S$ be a scheme.", "Let $\\Sch_{fppf}$ be a big fppf site containing $S$,", "and let $\\Sch_\\etale$ be the corresponding big \\'etale site", "(i.e., having the same underlying category).", "Let $X$ be an algebraic space over $S$.", "The site {\\it $X_{spaces, \\etale}$} of $X$ is defined as follows:", "\\begin{enumerate}", "\\item An object of $X_{spaces, \\etale}$ is a morphism", "$\\varphi : U \\to X$ where $U$ is an algebraic space over $S$ and", "$\\varphi$ is an \\'etale morphism of algebraic spaces over $S$,", "\\item a morphism $(\\varphi : U \\to X) \\to (\\varphi' : U' \\to X)$ of", "$X_{spaces, \\etale}$ is given by a morphism of algebraic spaces", "$\\chi : U \\to U'$ such that $\\varphi = \\varphi' \\circ \\chi$, and", "\\item a family of morphisms", "$\\{\\varphi_i : (U_i \\to X) \\to (U \\to X)\\}_{i \\in I}$", "of $X_{spaces, \\etale}$ is a covering if and only if", "$|U| = \\bigcup \\varphi_i(|U_i|)$.", "\\end{enumerate}", "(As usual we choose a set of coverings of this type, including at least", "the coverings in $X_\\etale$, as in", "Sets, Lemma \\ref{sets-lemma-coverings-site}", "to turn $X_{spaces, \\etale}$ into a site.)" ], "refs": [ "sets-lemma-coverings-site" ], "ref_ids": [ 8800 ] }, { "id": 11936, "type": "definition", "label": "spaces-properties-definition-etale-topos", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-etale-topos", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The {\\it \\'etale topos} of $X$, or more precisely the", "{\\it small \\'etale topos} of $X$ is the category", "$\\Sh(X_\\etale)$", "of sheaves of sets on $X_\\etale$." ], "refs": [], "ref_ids": [] }, { "id": 11937, "type": "definition", "label": "spaces-properties-definition-f-map", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-f-map", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$ and", "let $\\mathcal{G}$ be a sheaf of sets on $Y_\\etale$.", "An {\\it $f$-map $\\varphi : \\mathcal{G} \\to \\mathcal{F}$}", "is a collection of maps", "$\\varphi_{(U, V, g)} : \\mathcal{G}(V) \\to \\mathcal{F}(U)$", "indexed by commutative diagrams", "$$", "\\xymatrix{", "U \\ar[d]_g \\ar[r] & X \\ar[d]^f \\\\", "V \\ar[r] & Y", "}", "$$", "where $U \\in X_\\etale$, $V \\in Y_\\etale$ such that whenever", "given an extended diagram", "$$", "\\xymatrix{", "U' \\ar[r] \\ar[d]_{g'} & U \\ar[d]_g \\ar[r] & X \\ar[d]^f \\\\", "V' \\ar[r] & V \\ar[r] & Y", "}", "$$", "with $V' \\to V$ and $U' \\to U$ \\'etale morphisms of schemes the diagram", "$$", "\\xymatrix{", "\\mathcal{G}(V)", "\\ar[rr]_{\\varphi_{(U, V, g)}}", "\\ar[d]_{\\text{restriction of }\\mathcal{G}} & &", "\\mathcal{F}(U)", "\\ar[d]^{\\text{restriction of }\\mathcal{F}} \\\\", "\\mathcal{G}(V')", "\\ar[rr]^{\\varphi_{(U', V', g')}} & &", "\\mathcal{F}(U')", "}", "$$", "commutes." ], "refs": [], "ref_ids": [] }, { "id": 11938, "type": "definition", "label": "spaces-properties-definition-geometric-point", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-geometric-point", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item A {\\it geometric point} of $X$ is a morphism", "$\\overline{x} : \\Spec(k) \\to X$, where $k$ is an algebraically", "closed field. We often abuse notation and", "write $\\overline{x} = \\Spec(k)$.", "\\item For every geometric point $\\overline{x}$ we have the corresponding", "``image'' point $x \\in |X|$. We say that $\\overline{x}$ is a", "{\\it geometric point lying over $x$}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11939, "type": "definition", "label": "spaces-properties-definition-etale-neighbourhood", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-etale-neighbourhood", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\overline{x}$ be a geometric point of $X$.", "\\begin{enumerate}", "\\item An {\\it \\'etale neighborhood} of $\\overline{x}$", "of $X$ is a commutative diagram", "$$", "\\xymatrix{", "& U \\ar[d]^\\varphi \\\\", "{\\bar x} \\ar[r]^{\\bar x} \\ar[ur]^{\\bar u} & X", "}", "$$", "where $\\varphi$ is an \\'etale morphism of algebraic spaces over $S$.", "We will use the notation $\\varphi : (U, \\overline{u}) \\to (X, \\overline{x})$", "to indicate this situation.", "\\item A {\\it morphism of \\'etale neighborhoods}", "$(U, \\overline{u}) \\to (U', \\overline{u}')$", "is an $X$-morphism $h : U \\to U'$", "such that $\\overline{u}' = h \\circ \\overline{u}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11940, "type": "definition", "label": "spaces-properties-definition-stalk", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-stalk", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a presheaf on $X_\\etale$.", "Let $\\overline{x}$ be a geometric point of $X$.", "The {\\it stalk} of $\\mathcal{F}$ at $\\overline{x}$ is", "$$", "\\mathcal{F}_{\\bar x}", "=", "\\colim_{(U, \\overline{u})} \\mathcal{F}(U)", "$$", "where $(U, \\overline{u})$ runs over all \\'etale neighborhoods", "of $\\overline{x}$ in $X$ with $U \\in \\Ob(X_\\etale)$." ], "refs": [], "ref_ids": [] }, { "id": 11941, "type": "definition", "label": "spaces-properties-definition-support", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-support", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$.", "\\begin{enumerate}", "\\item The {\\it support of $\\mathcal{F}$} is the set of", "points $x \\in |X|$ such that $\\mathcal{F}_{\\overline{x}} \\not = 0$", "for any (some) geometric point $\\overline{x}$ lying over $x$.", "\\item Let $\\sigma \\in \\mathcal{F}(U)$ be a section.", "The {\\it support of $\\sigma$} is the closed subset $U \\setminus W$, where", "$W \\subset U$ is the largest open subset of $U$ on which $\\sigma$", "restricts to zero (see", "Lemma \\ref{lemma-zero-over-image}).", "\\end{enumerate}" ], "refs": [ "spaces-properties-lemma-zero-over-image" ], "ref_ids": [ 11878 ] }, { "id": 11942, "type": "definition", "label": "spaces-properties-definition-structure-sheaf", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-structure-sheaf", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "The {\\it structure sheaf} of $X$", "is the sheaf of rings $\\mathcal{O}_X$", "on the small \\'etale site $X_\\etale$ described in", "Lemma \\ref{lemma-sheaf-condition-holds}." ], "refs": [ "spaces-properties-lemma-sheaf-condition-holds" ], "ref_ids": [ 11881 ] }, { "id": 11943, "type": "definition", "label": "spaces-properties-definition-etale-local-rings", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-etale-local-rings", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $\\overline{x}$ be a geometric point of $X$ lying over the point", "$x \\in |X|$.", "\\begin{enumerate}", "\\item The {\\it \\'etale local ring of $X$ at $\\overline{x}$}", "is the stalk of the structure sheaf $\\mathcal{O}_X$ on $X_\\etale$", "at $\\overline{x}$.", "Notation: $\\mathcal{O}_{X, \\overline{x}}$.", "\\item The {\\it strict henselization of $X$ at $\\overline{x}$}", "is the scheme $\\Spec(\\mathcal{O}_{X, \\overline{x}})$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11944, "type": "definition", "label": "spaces-properties-definition-unibranch", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-unibranch", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$. We say that $X$ is {\\it geometrically unibranch", "at $x$} if the equivalent conditions of", "Lemma \\ref{lemma-irreducible-local-ring}", "hold. We say that $X$ is {\\it geometrically unibranch} if $X$ is", "geometrically unibranch at every $x \\in |X|$." ], "refs": [ "spaces-properties-lemma-irreducible-local-ring" ], "ref_ids": [ 11889 ] }, { "id": 11945, "type": "definition", "label": "spaces-properties-definition-number-of-geometric-branches", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-number-of-geometric-branches", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$. The {\\it number of geometric branches of $X$ at $x$} is", "either $n \\in \\mathbf{N}$ if the equivalent conditions", "of Lemma \\ref{lemma-nr-branches-local-ring}", "hold, or else $\\infty$." ], "refs": [ "spaces-properties-lemma-nr-branches-local-ring" ], "ref_ids": [ 11890 ] }, { "id": 11946, "type": "definition", "label": "spaces-properties-definition-noetherian", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-noetherian", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "We say $X$ is {\\it Noetherian} if $X$ is quasi-compact, quasi-separated", "and locally Noetherian." ], "refs": [], "ref_ids": [] }, { "id": 11947, "type": "definition", "label": "spaces-properties-definition-regular-at-point", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-regular-at-point", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$ be a point. We say {\\it $X$ is regular at $x$}", "if $\\mathcal{O}_{U, u}$ is a regular local ring for any", "(equivalently some) pair $(a : U \\to X, u)$ consisting of an", "\\'etale morphism $a : U \\to X$ from a scheme to $X$ and a point", "$u \\in U$ with $a(u) = x$." ], "refs": [], "ref_ids": [] }, { "id": 11948, "type": "definition", "label": "spaces-properties-definition-quasi-coherent", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "A {\\it quasi-coherent} $\\mathcal{O}_X$-module", "is a quasi-coherent module on the ringed site", "$(X_\\etale, \\mathcal{O}_X)$ in the sense of", "Modules on Sites,", "Definition \\ref{sites-modules-definition-site-local}.", "The category of quasi-coherent sheaves on $X$ is denoted", "$\\QCoh(\\mathcal{O}_X)$." ], "refs": [ "sites-modules-definition-site-local" ], "ref_ids": [ 14289 ] }, { "id": 11949, "type": "definition", "label": "spaces-properties-definition-locally-projective", "categories": [ "spaces-properties" ], "title": "spaces-properties-definition-locally-projective", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "We say $\\mathcal{F}$ is {\\it locally projective}", "if the equivalent conditions of", "Lemma \\ref{lemma-locally-projective}", "are satisfied." ], "refs": [ "spaces-properties-lemma-locally-projective" ], "ref_ids": [ 11913 ] }, { "id": 12003, "type": "definition", "label": "intersection-definition-proper-intersection", "categories": [ "intersection" ], "title": "intersection-definition-proper-intersection", "contents": [ "Let $X$ be a nonsingular variety.", "\\begin{enumerate}", "\\item Let $W,V \\subset X$ be closed subvarieties with", "$\\dim(W) = s$ and $\\dim(V) = r$. We say that $W$ and $V$", "{\\it intersect properly} if $\\dim(V \\cap W) \\leq r + s - \\dim(X)$.", "\\item Let $\\alpha = \\sum n_i [W_i]$ be an $s$-cycle,", "and $\\beta = \\sum_j m_j [V_j]$ be an $r$-cycle on $X$. We say", "that $\\alpha$ and $\\beta$ {\\it intersect properly} if", "$W_i$ and $V_j$ intersect properly for all $i$ and $j$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12004, "type": "definition", "label": "intersection-definition-multiplicity", "categories": [ "intersection" ], "title": "intersection-definition-multiplicity", "contents": [ "In the situation above, if $d \\geq \\dim(\\text{Supp}(M))$, then we set", "$e_I(M, d)$ equal to $0$ if $d > \\dim(\\text{Supp}(M))$", "and equal to $d!$ times the", "leading coefficient of the numerical polynomial $\\chi_{I, M}$ so that", "$$", "\\chi_{I, M}(n) \\sim e_I(M, d) \\frac{n^d}{d!} + \\text{lower order terms}", "$$", "The {\\it multiplicity of $M$ for the ideal of definition $I$}", "is $e_I(M) = e_I(M, \\dim(\\text{Supp}(M)))$." ], "refs": [], "ref_ids": [] }, { "id": 12131, "type": "definition", "label": "homology-definition-preadditive", "categories": [ "homology" ], "title": "homology-definition-preadditive", "contents": [ "A category $\\mathcal{A}$ is called {\\it preadditive} if each", "morphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed", "with the structure of an abelian group such that the", "compositions", "$$", "\\Mor(x, y) \\times \\Mor(y, z)", "\\longrightarrow", "\\Mor(x, z)", "$$", "are bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of", "preadditive categories is called {\\it additive} if and only", "if $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$", "is a homomorphism of abelian groups for all", "$x, y \\in \\Ob(\\mathcal{A})$." ], "refs": [], "ref_ids": [] }, { "id": 12132, "type": "definition", "label": "homology-definition-zero-object", "categories": [ "homology" ], "title": "homology-definition-zero-object", "contents": [ "In a preadditive category $\\mathcal{A}$ we call", "{\\it zero object}, and we denote it $0$", "any final and initial object as in Lemma \\ref{lemma-preadditive-zero} above." ], "refs": [ "homology-lemma-preadditive-zero" ], "ref_ids": [ 12008 ] }, { "id": 12133, "type": "definition", "label": "homology-definition-direct-sum", "categories": [ "homology" ], "title": "homology-definition-direct-sum", "contents": [ "Given a pair of objects $x, y$ in a preadditive category $\\mathcal{A}$,", "the {\\it direct sum} $x \\oplus y$ of $x$ and $y$ is the direct", "product $x \\times y$ endowed with the morphisms", "$i, j, p, q$ as in Lemma \\ref{lemma-preadditive-direct-sum} above." ], "refs": [ "homology-lemma-preadditive-direct-sum" ], "ref_ids": [ 12009 ] }, { "id": 12134, "type": "definition", "label": "homology-definition-additive-category", "categories": [ "homology" ], "title": "homology-definition-additive-category", "contents": [ "A category $\\mathcal{A}$ is called {\\it additive}", "if it is preadditive and finite products exist, in other", "words it has a zero object and direct sums." ], "refs": [], "ref_ids": [] }, { "id": 12135, "type": "definition", "label": "homology-definition-kernel", "categories": [ "homology" ], "title": "homology-definition-kernel", "contents": [ "Let $\\mathcal{A}$ be a preadditive category.", "Let $f : x \\to y$ be a morphism.", "\\begin{enumerate}", "\\item A {\\it kernel} of $f$ is a morphism", "$i : z \\to x$ such that (a) $f \\circ i = 0$ and (b)", "for any $i' : z' \\to x$ such that $f \\circ i' = 0$ there", "exists a unique morphism $g : z' \\to z$ such that", "$i' = i \\circ g$.", "\\item If the kernel of $f$ exists, then we denote", "this $\\Ker(f) \\to x$.", "\\item A {\\it cokernel} of $f$ is a morphism", "$p : y \\to z$ such that (a) $p \\circ f = 0$ and (b)", "for any $p' : y \\to z'$ such that $p' \\circ f = 0$ there", "exists a unique morphism $g : z \\to z'$ such that", "$p' = g \\circ p$.", "\\item If a cokernel of $f$ exists we denote this", "$y \\to \\Coker(f)$.", "\\item If a kernel of $f$ exists, then a {\\it coimage", "of $f$} is a cokernel for the morphism $\\Ker(f) \\to x$.", "\\item If a kernel and coimage exist then we denote this", "$x \\to \\Coim(f)$.", "\\item If a cokernel of $f$ exists, then the {\\it image of", "$f$} is a kernel of the morphism $y \\to \\Coker(f)$.", "\\item If a cokernel and image of $f$ exist then we denote", "this $\\Im(f) \\to y$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12136, "type": "definition", "label": "homology-definition-karoubian", "categories": [ "homology" ], "title": "homology-definition-karoubian", "contents": [ "Let $\\mathcal{C}$ be a preadditive category. We say $\\mathcal{C}$", "is {\\it Karoubian} if every idempotent endomorphism of an object", "of $\\mathcal{C}$ has a kernel." ], "refs": [], "ref_ids": [] }, { "id": 12137, "type": "definition", "label": "homology-definition-abelian-category", "categories": [ "homology" ], "title": "homology-definition-abelian-category", "contents": [ "A category $\\mathcal{A}$ is {\\it abelian} if", "it is additive, if all kernels and cokernels exist,", "and if the natural map $\\Coim(f) \\to \\Im(f)$", "is an isomorphism for all morphisms $f$ of", "$\\mathcal{A}$." ], "refs": [], "ref_ids": [] }, { "id": 12138, "type": "definition", "label": "homology-definition-injective-surjective", "categories": [ "homology" ], "title": "homology-definition-injective-surjective", "contents": [ "Let $f : x \\to y$ be a morphism in an abelian category.", "\\begin{enumerate}", "\\item We say $f$ is {\\it injective} if $\\Ker(f) = 0$.", "\\item We say $f$ is {\\it surjective} if $\\Coker(f) = 0$.", "\\end{enumerate}", "If $x \\to y$ is injective, then we say that $x$ is a {\\it subobject}", "of $y$ and we use the notation $x \\subset y$. If $x \\to y$ is", "surjective, then we say that $y$ is a {\\it quotient} of $x$." ], "refs": [], "ref_ids": [] }, { "id": 12139, "type": "definition", "label": "homology-definition-exact", "categories": [ "homology" ], "title": "homology-definition-exact", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "We say a sequence of morphisms", "$$", "\\ldots \\to x \\to y \\to z \\to \\ldots", "$$", "in $\\mathcal{A}$", "is a {\\it complex} if the composition of any two (drawn)", "arrows is zero. If $\\mathcal{A}$ is abelian then", "we say a sequence as above is {\\it exact at $y$} if", "$\\Im(x \\to y) = \\Ker(y \\to z)$. We say it is {\\it exact}", "if it is exact at every object. A {\\it short exact sequence}", "is an exact complex of the form", "$$", "0 \\to A \\to B \\to C \\to 0.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 12140, "type": "definition", "label": "homology-definition-ses-split", "categories": [ "homology" ], "title": "homology-definition-ses-split", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $i : A \\to B$ and $q : B \\to C$ be morphisms", "of $\\mathcal{A}$ such that", "$0 \\to A \\to B \\to C \\to 0$ is a short", "exact sequence. We say the short exact", "sequence is {\\it split} if there exist", "morphisms $j : C \\to B$ and $p : B \\to A$ such", "that $(B, i, j, p, q)$ is the direct sum of $A$ and $C$." ], "refs": [], "ref_ids": [] }, { "id": 12141, "type": "definition", "label": "homology-definition-extension", "categories": [ "homology" ], "title": "homology-definition-extension", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A, B \\in \\Ob(\\mathcal{A})$.", "An {\\it extension $E$ of $B$ by $A$} is a short", "exact sequence", "$$", "0 \\to A \\to E \\to B \\to 0.", "$$", "An {\\it morphism of extensions} between two", "extensions $0 \\to A \\to E \\to B \\to 0$ and", "$0 \\to A \\to F \\to B \\to 0$ means a morphism", "$f : E \\to F$ in $\\mathcal{A}$ making the diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "A \\ar[r] \\ar[d]^{\\text{id}} &", "E \\ar[r] \\ar[d]^f &", "B \\ar[r] \\ar[d]^{\\text{id}} &", "0 \\\\", "0 \\ar[r] &", "A \\ar[r] &", "F \\ar[r] &", "B \\ar[r] &", "0", "}", "$$", "commutative.", "Thus, the extensions of $B$ by $A$ form a category." ], "refs": [], "ref_ids": [] }, { "id": 12142, "type": "definition", "label": "homology-definition-ext-group", "categories": [ "homology" ], "title": "homology-definition-ext-group", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A, B \\in \\Ob(\\mathcal{A})$.", "The set of isomorphism classes of extensions", "of $B$ by $A$ is denoted", "$$", "\\Ext_\\mathcal{A}(B, A).", "$$", "This is called the {\\it $\\Ext$-group}." ], "refs": [], "ref_ids": [] }, { "id": 12143, "type": "definition", "label": "homology-definition-simple", "categories": [ "homology" ], "title": "homology-definition-simple", "contents": [ "Let $\\mathcal{A}$ be an abelian category. An object $A$ of $\\mathcal{A}$", "is said to be {\\it simple} if it is nonzero and the only subobjects", "of $A$ are $0$ and $A$." ], "refs": [], "ref_ids": [] }, { "id": 12144, "type": "definition", "label": "homology-definition-Artinian", "categories": [ "homology" ], "title": "homology-definition-Artinian", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item We say an object $A$ of $\\mathcal{A}$ is {\\it Artinian} if and only if", "it satisfies the descending chain condition for subobjects.", "\\item We say $\\mathcal{A}$ is {\\it Artinian} if every object of", "$\\mathcal{A}$ is Artinian.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12145, "type": "definition", "label": "homology-definition-Noetherian", "categories": [ "homology" ], "title": "homology-definition-Noetherian", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item We say an object $A$ of $\\mathcal{A}$ is {\\it Noetherian} if and only if", "it satisfies the ascending chain condition for subobjects.", "\\item We say $\\mathcal{A}$ is {\\it Noetherian} if every object of", "$\\mathcal{A}$ is Noetherian.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12146, "type": "definition", "label": "homology-definition-serre-subcategory", "categories": [ "homology" ], "title": "homology-definition-serre-subcategory", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item A {\\it Serre subcategory} of $\\mathcal{A}$ is a", "nonempty full subcategory $\\mathcal{C}$ of $\\mathcal{A}$", "such that given an exact sequence", "$$", "A \\to B \\to C", "$$", "with $A, C \\in \\Ob(\\mathcal{C})$, then also", "$B \\in \\Ob(\\mathcal{C})$.", "\\item A {\\it weak Serre subcategory} of $\\mathcal{A}$ is a nonempty", "full subcategory $\\mathcal{C}$ of $\\mathcal{A}$ such that given an", "exact sequence", "$$", "A_0 \\to A_1 \\to A_2 \\to A_3 \\to A_4", "$$", "with $A_0, A_1, A_3, A_4$ in $\\mathcal{C}$, then also $A_2$ in $\\mathcal{C}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12147, "type": "definition", "label": "homology-definition-kernel-category", "categories": [ "homology" ], "title": "homology-definition-kernel-category", "contents": [ "Let $\\mathcal{A}$, $\\mathcal{B}$ be abelian categories.", "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor.", "Then the full subcategory of objects $C$ of $\\mathcal{A}$", "such that $F(C) = 0$ is called the {\\it kernel of the functor $F$},", "and is sometimes denoted $\\Ker(F)$." ], "refs": [], "ref_ids": [] }, { "id": 12148, "type": "definition", "label": "homology-definition-K-zero", "categories": [ "homology" ], "title": "homology-definition-K-zero", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "We denote $K_0(\\mathcal{A})$ the", "{\\it zeroth $K$-group of $\\mathcal{A}$}.", "It is the abelian group constructed as follows.", "Take the free abelian group", "on the objects on $\\mathcal{A}$", "and for every short exact sequence", "$0 \\to A \\to B \\to C \\to 0$", "impose the relation $[B] - [A] - [C] = 0$." ], "refs": [], "ref_ids": [] }, { "id": 12149, "type": "definition", "label": "homology-definition-cohomological-delta-functor", "categories": [ "homology" ], "title": "homology-definition-cohomological-delta-functor", "contents": [ "Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories.", "A {\\it cohomological $\\delta$-functor} or simply a", "{\\it $\\delta$-functor} from $\\mathcal{A}$", "to $\\mathcal{B}$ is given by the following data:", "\\begin{enumerate}", "\\item a collection $F^n : \\mathcal{A} \\to \\mathcal{B}$, $n \\geq 0$ of additive", "functors, and", "\\item for every short exact sequence $0 \\to A \\to B \\to C \\to 0$", "of $\\mathcal{A}$", "a collection $\\delta_{A \\to B \\to C} : F^n(C) \\to F^{n + 1}(A)$, $n \\geq 0$", "of morphisms of $\\mathcal{B}$.", "\\end{enumerate}", "These data are assumed to satisfy the following axioms", "\\begin{enumerate}", "\\item for every short exact sequence as above the sequence", "$$", "\\xymatrix{", "0 \\ar[r] &", "F^0(A) \\ar[r] &", "F^0(B) \\ar[r] &", "F^0(C) \\ar[lld]^{\\delta_{A \\to B \\to C}} \\\\", " &", "F^1(A) \\ar[r] &", "F^1(B) \\ar[r] &", "F^1(C) \\ar[lld]^{\\delta_{A \\to B \\to C}} \\\\", " &", "F^2(A) \\ar[r] &", "F^2(B) \\ar[r] &", "\\ldots", "}", "$$", "is exact, and", "\\item for every morphism $(A \\to B \\to C) \\to (A' \\to B' \\to C')$", "of short exact sequences of $\\mathcal{A}$ the diagrams", "$$", "\\xymatrix{", "F^n(C) \\ar[d] \\ar[rr]_{\\delta_{A \\to B \\to C}} & & F^{n + 1}(A) \\ar[d] \\\\", "F^n(C') \\ar[rr]^{\\delta_{A' \\to B' \\to C'}} & & F^{n + 1}(A')", "}", "$$", "are commutative.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12150, "type": "definition", "label": "homology-definition-morphism-delta-functors", "categories": [ "homology" ], "title": "homology-definition-morphism-delta-functors", "contents": [ "Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories.", "Let $(F^n, \\delta_F)$ and $(G^n, \\delta_G)$ be $\\delta$-functors", "from $\\mathcal{A}$ to $\\mathcal{B}$. A {\\it morphism of $\\delta$-functors", "from $F$ to $G$} is a collection of", "transformation of functors $t^n : F^n \\to G^n$, $n \\geq 0$ such", "that for every short exact sequence $0 \\to A \\to B \\to C \\to 0$", "of $\\mathcal{A}$ the diagrams", "$$", "\\xymatrix{", "F^n(C) \\ar[d]_{t^n} \\ar[rr]_{\\delta_{F, A \\to B \\to C}} &", "& F^{n + 1}(A) \\ar[d]^{t^{n + 1}} \\\\", "G^n(C) \\ar[rr]^{\\delta_{G, A \\to B \\to C}} & & G^{n + 1}(A)", "}", "$$", "are commutative." ], "refs": [], "ref_ids": [] }, { "id": 12151, "type": "definition", "label": "homology-definition-universal-delta-functor", "categories": [ "homology" ], "title": "homology-definition-universal-delta-functor", "contents": [ "Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories.", "Let $F = (F^n, \\delta_F)$ be a $\\delta$-functor", "from $\\mathcal{A}$ to $\\mathcal{B}$.", "We say $F$ is a {\\it universal $\\delta$-functor} if and only", "if for every $\\delta$-functor $G = (G^n, \\delta_G)$ and any", "morphism of functors $t : F^0 \\to G^0$ there exists", "a unique morphism of $\\delta$-functors $\\{t^n\\}_{n \\geq 0} : F \\to G$", "such that $t = t^0$." ], "refs": [], "ref_ids": [] }, { "id": 12152, "type": "definition", "label": "homology-definition-homotopy-equivalent", "categories": [ "homology" ], "title": "homology-definition-homotopy-equivalent", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "We say a morphism $a : A_\\bullet \\to B_\\bullet$", "is a {\\it homotopy equivalence} if there exists", "a morphism $b : B_\\bullet \\to A_\\bullet$", "such that there exists a homotopy between", "$a \\circ b$ and $\\text{id}_A$", "and there exists a homotopy between $b \\circ a$ and $\\text{id}_B$.", "If there exists such a morphism between $A_\\bullet$ and $B_\\bullet$, then", "we say that $A_\\bullet$ and $B_\\bullet$ are {\\it homotopy equivalent}." ], "refs": [], "ref_ids": [] }, { "id": 12153, "type": "definition", "label": "homology-definition-quasi-isomorphism", "categories": [ "homology" ], "title": "homology-definition-quasi-isomorphism", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item A morphism of chain complexes $f : A_\\bullet \\to B_\\bullet$", "is called a {\\it quasi-isomorphism} if the induced", "map $H_i(f) : H_i(A_\\bullet) \\to H_i(B_\\bullet)$", "is an isomorphism for all $i \\in \\mathbf{Z}$.", "\\item A chain complex $A_\\bullet$ is called", "{\\it acyclic} if all of its homology objects", "$H_i(A_\\bullet)$ are zero.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12154, "type": "definition", "label": "homology-definition-homotopy-equivalent-cochain", "categories": [ "homology" ], "title": "homology-definition-homotopy-equivalent-cochain", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "We say a morphism $a : A^\\bullet \\to B^\\bullet$", "is a {\\it homotopy equivalence} if there exists", "a morphism $b : B^\\bullet \\to A^\\bullet$", "such that there exists a homotopy between", "$a \\circ b$ and $\\text{id}_A$", "and there exists a homotopy between $b \\circ a$ and $\\text{id}_B$.", "If there exists such a morphism between $A^\\bullet$ and $B^\\bullet$, then", "we say that $A^\\bullet$ and $B^\\bullet$ are {\\it homotopy equivalent}." ], "refs": [], "ref_ids": [] }, { "id": 12155, "type": "definition", "label": "homology-definition-quasi-isomorphism-cochain", "categories": [ "homology" ], "title": "homology-definition-quasi-isomorphism-cochain", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item A morphism of cochain complexes $f : A^\\bullet \\to B^\\bullet$", "of $\\mathcal{A}$ is called a {\\it quasi-isomorphism} if the induced", "maps $H^i(f) : H^i(A^\\bullet) \\to H^i(B^\\bullet)$", "is an isomorphism for all $i \\in \\mathbf{Z}$.", "\\item A cochain complex $A^\\bullet$ is called", "{\\it acyclic} if all of its cohomology objects", "$H^i(A^\\bullet)$ are zero.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12156, "type": "definition", "label": "homology-definition-shift", "categories": [ "homology" ], "title": "homology-definition-shift", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let $A_\\bullet$ be a chain complex", "with boundary maps $d_{A, n} : A_n \\to A_{n - 1}$.", "For any $k \\in \\mathbf{Z}$ we define the", "{\\it $k$-shifted chain complex $A[k]_\\bullet$}", "as follows:", "\\begin{enumerate}", "\\item we set $A[k]_n = A_{n + k}$, and", "\\item we set $d_{A[k], n} : A[k]_n \\to A[k]_{n - 1}$", "equal to $d_{A[k], n} = (-1)^k d_{A, n + k}$.", "\\end{enumerate}", "If $f : A_\\bullet \\to B_\\bullet$ is a morphism of", "chain complexes, then we let", "$f[k] : A[k]_\\bullet \\to B[k]_\\bullet$ be the", "morphism of chain complexes with", "$f[k]_n = f_{k + n}$." ], "refs": [], "ref_ids": [] }, { "id": 12157, "type": "definition", "label": "homology-definition-homology-shift", "categories": [ "homology" ], "title": "homology-definition-homology-shift", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A_\\bullet$ be a chain complex", "with boundary maps $d_{A, n} : A_n \\to A_{n - 1}$.", "For any $k \\in \\mathbf{Z}$ we identify", "{\\it $H_{i + k}(A_\\bullet) \\rightarrow H_i(A[k]_\\bullet)$}", "via the identification", "$A_{i + k} = A[k]_i$." ], "refs": [], "ref_ids": [] }, { "id": 12158, "type": "definition", "label": "homology-definition-shift-cochain", "categories": [ "homology" ], "title": "homology-definition-shift-cochain", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let $A^\\bullet$ be a cochain complex", "with boundary maps $d_A^n : A^n \\to A^{n + 1}$.", "For any $k \\in \\mathbf{Z}$ we define the", "{\\it $k$-shifted cochain complex $A[k]^\\bullet$}", "as follows:", "\\begin{enumerate}", "\\item we set $A[k]^n = A^{n + k}$, and", "\\item we set $d_{A[k]}^n : A[k]^n \\to A[k]^{n + 1}$", "equal to $d_{A[k]}^n = (-1)^k d_A^{n + k}$.", "\\end{enumerate}", "If $f : A^\\bullet \\to B^\\bullet$ is a morphism of", "cochain complexes, then we let", "$f[k] : A[k]^\\bullet \\to B[k]^\\bullet$ be the", "morphism of cochain complexes with", "$f[k]^n = f^{k + n}$." ], "refs": [], "ref_ids": [] }, { "id": 12159, "type": "definition", "label": "homology-definition-cohomology-shift", "categories": [ "homology" ], "title": "homology-definition-cohomology-shift", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A^\\bullet$ be a cochain complex", "with boundary maps $d_A^n : A^n \\to A^{n + 1}$.", "For any $k \\in \\mathbf{Z}$ we identify", "{\\it $H^{i + k}(A^\\bullet) \\longrightarrow H^i(A[k]^\\bullet)$}", "via the identification $A^{i + k} = A[k]^i$." ], "refs": [], "ref_ids": [] }, { "id": 12160, "type": "definition", "label": "homology-definition-graded", "categories": [ "homology" ], "title": "homology-definition-graded", "contents": [ "Let $\\mathcal{A}$ be an additive category. The {\\it category of graded", "objects of $\\mathcal{A}$}, denoted $\\text{Gr}(\\mathcal{A})$, is", "the category with", "\\begin{enumerate}", "\\item objects $A = (A^i)$ are families of objects $A^i$, $i \\in \\mathbf{Z}$", "of objects of $\\mathcal{A}$, and", "\\item morphisms $f : A = (A^i) \\to B = (B^i)$ are families of", "morphisms $f^i : A^i \\to B^i$ of $\\mathcal{A}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12161, "type": "definition", "label": "homology-definition-graded-shift", "categories": [ "homology" ], "title": "homology-definition-graded-shift", "contents": [ "Let $\\mathcal{A}$ be an additive category. If $A = (A^i)$ is a graded object,", "then the $k$th {\\it shift} $A[k]$ is the graded object with", "$A[k]^i = A^{k + i}$." ], "refs": [], "ref_ids": [] }, { "id": 12162, "type": "definition", "label": "homology-definition-additive-monoidal", "categories": [ "homology" ], "title": "homology-definition-additive-monoidal", "contents": [ "An {\\it additive monoidal category} is an additive category $\\mathcal{A}$", "endowed with a monoidal structure $\\otimes, \\phi$", "(Categories, Definition \\ref{categories-definition-monoidal-category})", "such that $\\otimes$ is an additive functor in each variable." ], "refs": [ "categories-definition-monoidal-category" ], "ref_ids": [ 12404 ] }, { "id": 12163, "type": "definition", "label": "homology-definition-double-complex", "categories": [ "homology" ], "title": "homology-definition-double-complex", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "A {\\it double complex} in $\\mathcal{A}$ is given", "by a system $(\\{A^{p, q}, d_1^{p, q}, d_2^{p, q}\\}_{p, q\\in \\mathbf{Z}})$,", "where each $A^{p, q}$ is an object of $\\mathcal{A}$ and", "$d_1^{p, q} : A^{p, q} \\to A^{p + 1, q}$ and", "$d_2^{p, q} : A^{p, q} \\to A^{p, q + 1}$ are morphisms of $\\mathcal{A}$", "such that the following rules hold:", "\\begin{enumerate}", "\\item $d_1^{p + 1, q} \\circ d_1^{p, q} = 0$", "\\item $d_2^{p, q + 1} \\circ d_2^{p, q} = 0$", "\\item $d_1^{p, q + 1} \\circ d_2^{p, q} = d_2^{p + 1, q} \\circ d_1^{p, q}$", "\\end{enumerate}", "for all $p, q \\in \\mathbf{Z}$." ], "refs": [], "ref_ids": [] }, { "id": 12164, "type": "definition", "label": "homology-definition-associated-simple-complex", "categories": [ "homology" ], "title": "homology-definition-associated-simple-complex", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let $A^{\\bullet, \\bullet}$ be a double complex.", "The {\\it associated simple complex}, denoted $sA^\\bullet$, also", "often called the {\\it associated total complex}, denoted", "$\\text{Tot}(A^{\\bullet, \\bullet})$, is", "given by", "$$", "sA^n = \\text{Tot}^n(A^{\\bullet, \\bullet}) =", "\\bigoplus\\nolimits_{n = p + q} A^{p, q}", "$$", "(if it exists) with differential", "$$", "d_{sA^\\bullet}^n = d_{\\text{Tot}(A^{\\bullet, \\bullet})}^n =", "\\sum\\nolimits_{n = p + q} (d_1^{p, q} + (-1)^p d_2^{p, q})", "$$" ], "refs": [], "ref_ids": [] }, { "id": 12165, "type": "definition", "label": "homology-definition-filtered", "categories": [ "homology" ], "title": "homology-definition-filtered", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item A {\\it decreasing filtration} $F$ on an object $A$", "is a family $(F^nA)_{n \\in \\mathbf{Z}}$ of subobjects of $A$ such that", "$$", "A \\supset \\ldots \\supset F^nA \\supset F^{n + 1}A \\supset \\ldots \\supset 0", "$$", "\\item A {\\it filtered object of $\\mathcal{A}$} is", "pair $(A, F)$ consisting of an object $A$ of $\\mathcal{A}$", "and a decreasing filtration $F$ on $A$.", "\\item A {\\it morphism $(A, F) \\to (B, F)$ of filtered objects}", "is given by a morphism $\\varphi : A \\to B$ of $\\mathcal{A}$", "such that $\\varphi(F^iA) \\subset F^iB$ for all $i \\in \\mathbf{Z}$.", "\\item The category of filtered objects is denoted $\\text{Fil}(\\mathcal{A})$.", "\\item Given a filtered object $(A, F)$ and a subobject $X \\subset A$ the", "{\\it induced filtration} on $X$ is the filtration with $F^nX = X \\cap F^nA$.", "\\item Given a filtered object $(A, F)$ and a surjection", "$\\pi : A \\to Y$ the {\\it quotient filtration} is the filtration with", "$F^nY = \\pi(F^nA)$.", "\\item A filtration $F$ on an object $A$ is said to be {\\it finite}", "if there exist $n, m$ such that $F^nA = A$ and $F^mA = 0$.", "\\item Given a filtered object $(A, F)$ we say $\\bigcap F^iA$ exists", "if there exists a biggest subobject of $A$ contained in all $F^iA$.", "We say $\\bigcup F^iA$ exists if there exists a smallest subobject", "of $A$ containing all $F^iA$.", "\\item The filtration on a filtered object $(A, F)$ is said to be", "{\\it separated} if $\\bigcap F^iA = 0$ and", "{\\it exhaustive} if $\\bigcup F^iA = A$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12166, "type": "definition", "label": "homology-definition-strict", "categories": [ "homology" ], "title": "homology-definition-strict", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "A morphism $f : A \\to B$ of filtered objects of $\\mathcal{A}$ is", "said to be {\\it strict} if $f(F^iA) = f(A) \\cap F^iB$ for", "all $i \\in \\mathbf{Z}$." ], "refs": [], "ref_ids": [] }, { "id": 12167, "type": "definition", "label": "homology-definition-spectral-sequence", "categories": [ "homology" ], "title": "homology-definition-spectral-sequence", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item A {\\it spectral sequence in $\\mathcal{A}$} is given by a", "system $(E_r, d_r)_{r \\geq 1}$ where each $E_r$ is an object", "of $\\mathcal{A}$, each $d_r : E_r \\to E_r$ is a morphism such", "that $d_r \\circ d_r = 0$ and $E_{r + 1} = \\Ker(d_r)/\\Im(d_r)$", "for $r \\geq 1$.", "\\item A {\\it morphism of spectral sequences}", "$f : (E_r, d_r)_{r \\geq 1} \\to (E'_r, d'_r)_{r \\geq 1}$ is", "given by a family of morphisms $f_r : E_r \\to E'_r$ such that", "$f_r \\circ d_r = d'_r \\circ f_r$ and such that $f_{r + 1}$", "is the morphism induced by $f_r$ via the identifications", "$E_{r + 1} = \\Ker(d_r)/\\Im(d_r)$", "and", "$E'_{r + 1} = \\Ker(d'_r)/\\Im(d'_r)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12168, "type": "definition", "label": "homology-definition-limit-spectral-sequence", "categories": [ "homology" ], "title": "homology-definition-limit-spectral-sequence", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $(E_r, d_r)_{r \\geq 1}$ be a spectral sequence.", "\\begin{enumerate}", "\\item If the subobjects $Z_{\\infty} = \\bigcap Z_r$", "and $B_{\\infty} = \\bigcup B_r$ of $E_1$ exist then we define", "the {\\it limit}\\footnote{This notation is not universally accepted. In some", "references an additional pair of subobjects", "$Z_\\infty$ and $B_\\infty$ of $E_1$ such that", "$0 = B_1 \\subset B_2 \\subset \\ldots \\subset B_\\infty \\subset Z_\\infty", "\\subset \\ldots \\subset Z_2 \\subset Z_1 = E_1$", "is part of the data comprising a spectral sequence!}", "of the spectral sequence to be the object", "$E_{\\infty} = Z_{\\infty}/B_{\\infty}$.", "\\item We say that the spectral sequence {\\it degenerates at $E_r$}", "if the differentials $d_r, d_{r + 1}, \\ldots$ are all zero.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12169, "type": "definition", "label": "homology-definition-exact-couple", "categories": [ "homology" ], "title": "homology-definition-exact-couple", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item An {\\it exact couple} is a datum $(A, E, \\alpha, f, g)$ where", "$A$, $E$ are objects of $\\mathcal{A}$ and $\\alpha$, $f$, $g$", "are morphisms as in the following diagram", "$$", "\\xymatrix{", "A \\ar[rr]_{\\alpha} & & A \\ar[ld]^g \\\\", "& E \\ar[lu]^f &", "}", "$$", "with the property that the kernel of each arrow is the image", "of its predecessor. So $\\Ker(\\alpha) = \\Im(f)$,", "$\\Ker(f) = \\Im(g)$, and $\\Ker(g) = \\Im(\\alpha)$.", "\\item A {\\it morphism of exact couples}", "$t : (A, E, \\alpha, f, g) \\to (A', E', \\alpha', f', g')$", "is given by morphisms $t_A : A \\to A'$ and", "$t_E : E \\to E'$ such that", "$\\alpha' \\circ t_A = t_A \\circ \\alpha$,", "$f' \\circ t_E = t_A \\circ f$, and", "$g' \\circ t_A = t_E \\circ g$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12170, "type": "definition", "label": "homology-definition-spectral-sequence-associated-exact-couple", "categories": [ "homology" ], "title": "homology-definition-spectral-sequence-associated-exact-couple", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $(A, E, \\alpha, f, g)$ be an exact couple.", "The {\\it spectral sequence associated to the exact couple}", "is the spectral sequence $(E_r, d_r)_{r \\geq 1}$ with", "$E_1 = E$, $d_1 = d$, $E_2 = E'$, $d_2 = d' = g' \\circ f'$,", "$E_3 = E''$, $d_3 = d'' = g'' \\circ f''$,", "and so on." ], "refs": [], "ref_ids": [] }, { "id": 12171, "type": "definition", "label": "homology-definition-differential-object", "categories": [ "homology" ], "title": "homology-definition-differential-object", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "A {\\it differential object} of $\\mathcal{A}$", "is a pair $(A, d)$ consisting of an", "object $A$ of $\\mathcal{A}$", "endowed with a selfmap $d$ such that $d \\circ d = 0$.", "A {\\it morphism of differential objects} $(A, d) \\to (B, d)$", "is given by a morphism $\\alpha : A \\to B$ such that", "$d \\circ \\alpha = \\alpha \\circ d$." ], "refs": [], "ref_ids": [] }, { "id": 12172, "type": "definition", "label": "homology-definition-differential-object-homology", "categories": [ "homology" ], "title": "homology-definition-differential-object-homology", "contents": [ "For a differential object $(A, d)$ we denote", "$$", "H(A, d) = \\Ker(d)/\\Im(d)", "$$", "its {\\it homology}." ], "refs": [], "ref_ids": [] }, { "id": 12173, "type": "definition", "label": "homology-definition-differential-object-selfmap", "categories": [ "homology" ], "title": "homology-definition-differential-object-selfmap", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $(A, d)$ be a differential object of $\\mathcal{A}$.", "Let $\\alpha : A \\to A$ be an injective selfmap of $A$ which", "commutes with $d$. The {\\it spectral sequence associated to", "$(A, d, \\alpha)$} is the spectral sequence", "$(E_r, d_r)_{r \\geq 0}$ described above." ], "refs": [], "ref_ids": [] }, { "id": 12174, "type": "definition", "label": "homology-definition-filtered-differential", "categories": [ "homology" ], "title": "homology-definition-filtered-differential", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "A {\\it filtered differential object} $(K, F, d)$ is a filtered object", "$(K, F)$ of $\\mathcal{A}$ endowed with an endomorphism", "$d : (K, F) \\to (K, F)$ whose square is zero: $d \\circ d = 0$." ], "refs": [], "ref_ids": [] }, { "id": 12175, "type": "definition", "label": "homology-definition-filtration-cohomology-filtered-differential", "categories": [ "homology" ], "title": "homology-definition-filtration-cohomology-filtered-differential", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $(K, F, d)$ be a filtered differential object of $\\mathcal{A}$.", "The {\\it induced filtration} on $H(K, d)$ is the filtration defined", "by $F^pH(K, d) = \\Im(H(F^pK, d) \\to H(K, d))$." ], "refs": [], "ref_ids": [] }, { "id": 12176, "type": "definition", "label": "homology-definition-filtered-differential-ss-converges", "categories": [ "homology" ], "title": "homology-definition-filtered-differential-ss-converges", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $(K, F, d)$ be a filtered differential object of $\\mathcal{A}$.", "We say the spectral sequence associated to $(K, F, d)$", "\\begin{enumerate}", "\\item {\\it weakly converges to $H(K)$} if $\\text{gr}H(K) = E_{\\infty}$", "via Lemma \\ref{lemma-compute-filtered-cohomology},", "\\item {\\it abuts to $H(K)$} if it weakly converges to $H(K)$ and", "we have $\\bigcap F^pH(K) = 0$ and $\\bigcup F^pH(K) = H(K)$,", "\\end{enumerate}" ], "refs": [ "homology-lemma-compute-filtered-cohomology" ], "ref_ids": [ 12093 ] }, { "id": 12177, "type": "definition", "label": "homology-definition-filtered-complex", "categories": [ "homology" ], "title": "homology-definition-filtered-complex", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "A {\\it filtered complex $K^\\bullet$ of $\\mathcal{A}$}", "is a complex of $\\text{Fil}(\\mathcal{A})$ (see", "Definition \\ref{definition-filtered})." ], "refs": [ "homology-definition-filtered" ], "ref_ids": [ 12165 ] }, { "id": 12178, "type": "definition", "label": "homology-definition-filtration-cohomology-filtered-complex", "categories": [ "homology" ], "title": "homology-definition-filtration-cohomology-filtered-complex", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $(K^\\bullet, F)$ be a filtered complex of $\\mathcal{A}$.", "The {\\it induced filtration} on $H^n(K^\\bullet)$ is the filtration defined", "by $F^pH^n(K^\\bullet) = \\Im(H^n(F^pK^\\bullet) \\to H^n(K^\\bullet))$." ], "refs": [], "ref_ids": [] }, { "id": 12179, "type": "definition", "label": "homology-definition-bounded-ss", "categories": [ "homology" ], "title": "homology-definition-bounded-ss", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $(E_r, d_r)_{r \\geq r_0}$", "be a spectral sequence of bigraded objects of", "$\\mathcal{A}$ with $d_r$ of bidegree $(r, -r + 1)$.", "We say such a spectral sequence is", "\\begin{enumerate}", "\\item {\\it regular} if for all $p, q \\in \\mathbf{Z}$ there is", "a $b = b(p, q)$ such that the maps", "$d_r^{p, q} : E_r^{p, q} \\to E_r^{p + r, q - r + 1}$ are zero for $r \\geq b$,", "\\item {\\it coregular} if for all $p, q \\in \\mathbf{Z}$ there is a", "$b = b(p, q)$ such that the maps", "$d_r^{p - r, q + r - 1} : E_r^{p - r, q + r - 1} \\to E_r^{p, q}$", "are zero for $r \\geq b$,", "\\item {\\it bounded} if for all $n$", "there are only a finite number of nonzero $E_{r_0}^{p, n - p}$,", "\\item {\\it bounded below} if for all $n$ there is a $b = b(n)$ such that", "$E_{r_0}^{p, n - p} = 0$ for $p \\geq b$.", "\\item {\\it bounded above} if for all $n$ there is a $b = b(n)$ such that", "$E_{r_0}^{p, n - p} = 0$ for $p \\leq b$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12180, "type": "definition", "label": "homology-definition-filtered-complex-ss-converges", "categories": [ "homology" ], "title": "homology-definition-filtered-complex-ss-converges", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a", "filtered complex of $\\mathcal{A}$. We say the spectral sequence", "associated to $(K^\\bullet, F)$", "\\begin{enumerate}", "\\item {\\it weakly converges to $H^*(K^\\bullet)$} if", "$\\text{gr}^pH^n(K^\\bullet) = E_{\\infty}^{p, n - p}$", "via Lemma \\ref{lemma-compute-cohomology-filtered-complex}", "for all $p, n \\in \\mathbf{Z}$,", "\\item {\\it abuts to $H^*(K^\\bullet)$} if it weakly converges to", "$H^*(K^\\bullet)$ and $\\bigcap_p F^pH^n(K^\\bullet) = 0$ and", "$\\bigcup_p F^p H^n(K^\\bullet) = H^n(K^\\bullet)$ for all $n$,", "\\item {\\it converges to $H^*(K^\\bullet)$} if it is regular,", "abuts to $H^*(K^\\bullet)$, and", "$H^n(K^\\bullet) = \\lim_p H^n(K^\\bullet)/F^pH^n(K^\\bullet)$.", "\\end{enumerate}" ], "refs": [ "homology-lemma-compute-cohomology-filtered-complex" ], "ref_ids": [ 12098 ] }, { "id": 12181, "type": "definition", "label": "homology-definition-ss-double-complex-converge", "categories": [ "homology" ], "title": "homology-definition-ss-double-complex-converge", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $K^{\\bullet, \\bullet}$ be a double complex.", "We say the spectral sequence $({}'E_r, {}'d_r)_{r \\geq 0}$", "{\\it weakly converges to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$},", "{\\it abuts to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$}, or", "{\\it converges to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$}", "if Definition \\ref{definition-filtered-complex-ss-converges} applies.", "Similarly we say the spectral sequence $({}''E_r, {}''d_r)_{r \\geq 0}$", "{\\it weakly converges to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$},", "{\\it abuts to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$}, or", "{\\it converges to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$}", "if Definition \\ref{definition-filtered-complex-ss-converges} applies." ], "refs": [ "homology-definition-filtered-complex-ss-converges", "homology-definition-filtered-complex-ss-converges" ], "ref_ids": [ 12180, 12180 ] }, { "id": 12182, "type": "definition", "label": "homology-definition-injective", "categories": [ "homology" ], "title": "homology-definition-injective", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "An object $J \\in \\Ob(\\mathcal{A})$ is", "called {\\it injective} if for every injection", "$A \\hookrightarrow B$ and every morphism", "$A \\to J$ there exists a morphism $B \\to J$ making", "the following diagram commute", "$$", "\\xymatrix{", "A \\ar[r] \\ar[d] & B \\ar@{-->}[ld] \\\\", "J &", "}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 12183, "type": "definition", "label": "homology-definition-enough-injectives", "categories": [ "homology" ], "title": "homology-definition-enough-injectives", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "We say $\\mathcal{A}$ has {\\it enough injectives}", "if every object $A$ has an injective morphism", "$A \\to J$ into an injective object $J$." ], "refs": [], "ref_ids": [] }, { "id": 12184, "type": "definition", "label": "homology-definition-functorial-injective-embedding", "categories": [ "homology" ], "title": "homology-definition-functorial-injective-embedding", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "We say that $\\mathcal{A}$ has {\\it functorial injective embeddings}", "if there exists a functor", "$$", "J : \\mathcal{A} \\longrightarrow \\text{Arrows}(\\mathcal{A})", "$$", "such that", "\\begin{enumerate}", "\\item $s \\circ J = \\text{id}_\\mathcal{A}$,", "\\item for any object $A \\in \\Ob(\\mathcal{A})$", "the morphism $J(A)$ is injective, and", "\\item for any object $A \\in \\Ob(\\mathcal{A})$", "the object $t(J(A))$ is an injective object of $\\mathcal{A}$.", "\\end{enumerate}", "We will denote such a functor by", "$A \\mapsto (A \\to J(A))$." ], "refs": [], "ref_ids": [] }, { "id": 12185, "type": "definition", "label": "homology-definition-projective", "categories": [ "homology" ], "title": "homology-definition-projective", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "An object $P \\in \\Ob(\\mathcal{A})$ is", "called {\\it projective} if for every surjection", "$A \\rightarrow B$ and every morphism", "$P \\to B$ there exists a morphism $P \\to A$ making", "the following diagram commute", "$$", "\\xymatrix{", "A \\ar[r] & B \\\\", "P \\ar@{-->}[u] \\ar[ru] &", "}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 12186, "type": "definition", "label": "homology-definition-enough-projectives", "categories": [ "homology" ], "title": "homology-definition-enough-projectives", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "We say $\\mathcal{A}$ has {\\it enough projectives}", "if every object $A$ has an surjective morphism", "$P \\to A$ from an projective object $P$ onto it." ], "refs": [], "ref_ids": [] }, { "id": 12187, "type": "definition", "label": "homology-definition-functorial-projective-surjections", "categories": [ "homology" ], "title": "homology-definition-functorial-projective-surjections", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "We say that $\\mathcal{A}$ has {\\it functorial projective surjections}", "if there exists a functor", "$$", "P : \\mathcal{A} \\longrightarrow \\text{Arrows}(\\mathcal{A})", "$$", "such that", "\\begin{enumerate}", "\\item $t \\circ J = \\text{id}_\\mathcal{A}$,", "\\item for any object $A \\in \\Ob(\\mathcal{A})$", "the morphism $P(A)$ is surjective, and", "\\item for any object $A \\in \\Ob(\\mathcal{A})$", "the object $s(P(A))$ is an projective object of $\\mathcal{A}$.", "\\end{enumerate}", "We will denote such a functor by", "$A \\mapsto (P(A) \\to A)$." ], "refs": [], "ref_ids": [] }, { "id": 12188, "type": "definition", "label": "homology-definition-Mittag-Leffler", "categories": [ "homology" ], "title": "homology-definition-Mittag-Leffler", "contents": [ "Let $\\mathcal{C}$ be an abelian category.", "We say the inverse system $(A_i)$", "satisfies the {\\it Mittag-Leffler condition}, or for short", "is {\\it ML}, if for every $i$ there exists a $c = c(i) \\geq i$", "such that", "$$", "\\Im(A_k \\to A_i) = \\Im(A_c \\to A_i)", "$$", "for all $k \\geq c$." ], "refs": [], "ref_ids": [] }, { "id": 12328, "type": "definition", "label": "categories-definition-category", "categories": [ "categories" ], "title": "categories-definition-category", "contents": [ "A {\\it category} $\\mathcal{C}$ consists of the following data:", "\\begin{enumerate}", "\\item A set of objects $\\Ob(\\mathcal{C})$.", "\\item For each pair $x, y \\in \\Ob(\\mathcal{C})$ a set of morphisms", "$\\Mor_\\mathcal{C}(x, y)$.", "\\item For each triple $x, y, z\\in \\Ob(\\mathcal{C})$ a composition", "map $ \\Mor_\\mathcal{C}(y, z) \\times \\Mor_\\mathcal{C}(x, y)", "\\to \\Mor_\\mathcal{C}(x, z) $, denoted $(\\phi, \\psi) \\mapsto", "\\phi \\circ \\psi$.", "\\end{enumerate}", "These data are to satisfy the following rules:", "\\begin{enumerate}", "\\item For every element $x\\in \\Ob(\\mathcal{C})$ there exists a", "morphism $\\text{id}_x\\in \\Mor_\\mathcal{C}(x, x)$ such that", "$\\text{id}_x \\circ \\phi = \\phi$ and $\\psi \\circ \\text{id}_x = \\psi $ whenever", "these compositions make sense.", "\\item Composition is associative, i.e., $(\\phi \\circ \\psi) \\circ \\chi =", "\\phi \\circ ( \\psi \\circ \\chi)$ whenever these compositions make sense.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12329, "type": "definition", "label": "categories-definition-isomorphism", "categories": [ "categories" ], "title": "categories-definition-isomorphism", "contents": [ "A morphism $\\phi : x \\to y$ is an {\\it isomorphism} of the category", "$\\mathcal{C}$ if there exists a morphism $\\psi : y \\to x$", "such that $\\phi \\circ \\psi = \\text{id}_y$ and", "$\\psi \\circ \\phi = \\text{id}_x$." ], "refs": [], "ref_ids": [] }, { "id": 12330, "type": "definition", "label": "categories-definition-groupoid", "categories": [ "categories" ], "title": "categories-definition-groupoid", "contents": [ "A {\\it groupoid} is a category where every morphism is an isomorphism." ], "refs": [], "ref_ids": [] }, { "id": 12331, "type": "definition", "label": "categories-definition-functor", "categories": [ "categories" ], "title": "categories-definition-functor", "contents": [ "A {\\it functor} $F : \\mathcal{A} \\to \\mathcal{B}$", "between two categories $\\mathcal{A}, \\mathcal{B}$ is given by the", "following data:", "\\begin{enumerate}", "\\item A map $F : \\Ob(\\mathcal{A}) \\to \\Ob(\\mathcal{B})$.", "\\item For every $x, y \\in \\Ob(\\mathcal{A})$ a map", "$F : \\Mor_\\mathcal{A}(x, y) \\to \\Mor_\\mathcal{B}(F(x), F(y))$,", "denoted $\\phi \\mapsto F(\\phi)$.", "\\end{enumerate}", "These data should be compatible with composition and identity morphisms", "in the following manner: $F(\\phi \\circ \\psi) =", "F(\\phi) \\circ F(\\psi)$ for a composable pair $(\\phi, \\psi)$ of", "morphisms of $\\mathcal{A}$ and $F(\\text{id}_x) = \\text{id}_{F(x)}$." ], "refs": [], "ref_ids": [] }, { "id": 12332, "type": "definition", "label": "categories-definition-faithful", "categories": [ "categories" ], "title": "categories-definition-faithful", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.", "\\begin{enumerate}", "\\item We say $F$ is {\\it faithful} if", "for any objects $x, y$ of $\\Ob(\\mathcal{A})$ the map", "$$", "F : \\Mor_\\mathcal{A}(x, y) \\to \\Mor_\\mathcal{B}(F(x), F(y))", "$$", "is injective.", "\\item If these maps are all bijective then $F$ is called", "{\\it fully faithful}.", "\\item", "The functor $F$ is called {\\it essentially surjective} if for any", "object $y \\in \\Ob(\\mathcal{B})$ there exists an object", "$x \\in \\Ob(\\mathcal{A})$ such that $F(x)$ is isomorphic to $y$ in", "$\\mathcal{B}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12333, "type": "definition", "label": "categories-definition-subcategory", "categories": [ "categories" ], "title": "categories-definition-subcategory", "contents": [ "A {\\it subcategory} of a category $\\mathcal{B}$ is", "a category $\\mathcal{A}$ whose objects and arrows", "form subsets of the objects and arrows", "of $\\mathcal{B}$ and such that source, target", "and composition in $\\mathcal{A}$ agree with those", "of $\\mathcal{B}$. We say $\\mathcal{A}$ is a", "{\\it full subcategory} of $\\mathcal{B}$ if $\\Mor_\\mathcal{A}(x, y)", "= \\Mor_\\mathcal{B}(x, y)$ for all $x, y \\in \\Ob(\\mathcal{A})$.", "We say $\\mathcal{A}$ is a {\\it strictly full} subcategory of $\\mathcal{B}$", "if it is a full subcategory and given $x \\in \\Ob(\\mathcal{A})$ any", "object of $\\mathcal{B}$ which is isomorphic to $x$ is also in $\\mathcal{A}$." ], "refs": [], "ref_ids": [] }, { "id": 12334, "type": "definition", "label": "categories-definition-transformation-functors", "categories": [ "categories" ], "title": "categories-definition-transformation-functors", "contents": [ "Let $F, G : \\mathcal{A} \\to \\mathcal{B}$ be functors.", "A {\\it natural transformation}, or a {\\it morphism of functors}", "$t : F \\to G$, is a collection $\\{t_x\\}_{x\\in \\Ob(\\mathcal{A})}$", "such that", "\\begin{enumerate}", "\\item $t_x : F(x) \\to G(x)$ is a morphism in the category $\\mathcal{B}$, and", "\\item for every morphism $\\phi : x \\to y$ of $\\mathcal{A}$ the following", "diagram is commutative", "$$", "\\xymatrix{", "F(x) \\ar[r]^{t_x} \\ar[d]_{F(\\phi)} & G(x) \\ar[d]^{G(\\phi)} \\\\", "F(y) \\ar[r]^{t_y} & G(y) }", "$$", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12335, "type": "definition", "label": "categories-definition-equivalence-categories", "categories": [ "categories" ], "title": "categories-definition-equivalence-categories", "contents": [ "An {\\it equivalence of categories}", "$F : \\mathcal{A} \\to \\mathcal{B}$ is a functor such that there", "exists a functor $G : \\mathcal{B} \\to \\mathcal{A}$ such that", "the compositions $F \\circ G$ and $G \\circ F$ are isomorphic to the", "identity functors $\\text{id}_\\mathcal{B}$,", "respectively $\\text{id}_\\mathcal{A}$.", "In this case we say that $G$ is a {\\it quasi-inverse} to $F$." ], "refs": [], "ref_ids": [] }, { "id": 12336, "type": "definition", "label": "categories-definition-product-category", "categories": [ "categories" ], "title": "categories-definition-product-category", "contents": [ "Let $\\mathcal{A}$, $\\mathcal{B}$ be categories.", "We define the {\\it product category}", "$\\mathcal{A} \\times \\mathcal{B}$ to be the category with", "objects", "$\\Ob(\\mathcal{A} \\times \\mathcal{B}) =", "\\Ob(\\mathcal{A}) \\times \\Ob(\\mathcal{B})$", "and", "$$", "\\Mor_{\\mathcal{A} \\times \\mathcal{B}}((x, y), (x', y'))", ":=", "\\Mor_\\mathcal{A}(x, x')\\times", "\\Mor_\\mathcal{B}(y, y').", "$$", "Composition is defined componentwise." ], "refs": [], "ref_ids": [] }, { "id": 12337, "type": "definition", "label": "categories-definition-opposite", "categories": [ "categories" ], "title": "categories-definition-opposite", "contents": [ "Given a category $\\mathcal{C}$ the {\\it opposite category}", "$\\mathcal{C}^{opp}$ is the category with the same objects", "as $\\mathcal{C}$ but all morphisms reversed." ], "refs": [], "ref_ids": [] }, { "id": 12338, "type": "definition", "label": "categories-definition-contravariant", "categories": [ "categories" ], "title": "categories-definition-contravariant", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{S}$ be categories.", "A {\\it contravariant} functor $F$", "from $\\mathcal{C}$ to $\\mathcal{S}$", "is a functor $\\mathcal{C}^{opp}\\to \\mathcal{S}$." ], "refs": [], "ref_ids": [] }, { "id": 12339, "type": "definition", "label": "categories-definition-presheaf", "categories": [ "categories" ], "title": "categories-definition-presheaf", "contents": [ "Let $\\mathcal{C}$ be a category.", "\\begin{enumerate}", "\\item A {\\it presheaf of sets on $\\mathcal{C}$}", "or simply a {\\it presheaf} is a contravariant functor", "$F$ from $\\mathcal{C}$ to $\\textit{Sets}$.", "\\item The category of presheaves is denoted $\\textit{PSh}(\\mathcal{C})$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12340, "type": "definition", "label": "categories-definition-representable-functor", "categories": [ "categories" ], "title": "categories-definition-representable-functor", "contents": [ "A contravariant functor $F : \\mathcal{C}\\to \\textit{Sets}$ is said", "to be {\\it representable} if it is isomorphic to the functor of", "points $h_U$ for some object $U$ of $\\mathcal{C}$." ], "refs": [], "ref_ids": [] }, { "id": 12341, "type": "definition", "label": "categories-definition-products", "categories": [ "categories" ], "title": "categories-definition-products", "contents": [ "Let $x, y\\in \\Ob(\\mathcal{C})$.", "A {\\it product} of $x$ and $y$ is", "an object $x \\times y \\in \\Ob(\\mathcal{C})$", "together with morphisms", "$p\\in \\Mor_{\\mathcal C}(x \\times y, x)$ and", "$q\\in\\Mor_{\\mathcal C}(x \\times y, y)$ such", "that the following universal property holds: for", "any $w\\in \\Ob(\\mathcal{C})$ and morphisms", "$\\alpha \\in \\Mor_{\\mathcal C}(w, x)$ and", "$\\beta \\in \\Mor_\\mathcal{C}(w, y)$", "there is a unique", "$\\gamma\\in \\Mor_{\\mathcal C}(w, x \\times y)$ making", "the diagram", "$$", "\\xymatrix{", "w \\ar[rrrd]^\\beta \\ar@{-->}[rrd]_\\gamma \\ar[rrdd]_\\alpha & & \\\\", "& & x \\times y \\ar[d]_p \\ar[r]_q & y \\\\", "& & x &", "}", "$$", "commute." ], "refs": [], "ref_ids": [] }, { "id": 12342, "type": "definition", "label": "categories-definition-has-products-of-pairs", "categories": [ "categories" ], "title": "categories-definition-has-products-of-pairs", "contents": [ "We say the category $\\mathcal{C}$ {\\it has products of pairs", "of objects} if a product $x \\times y$", "exists for any $x, y \\in \\Ob(\\mathcal{C})$." ], "refs": [], "ref_ids": [] }, { "id": 12343, "type": "definition", "label": "categories-definition-coproducts", "categories": [ "categories" ], "title": "categories-definition-coproducts", "contents": [ "Let $x, y \\in \\Ob(\\mathcal{C})$.", "A {\\it coproduct}, or {\\it amalgamated sum} of $x$ and $y$ is", "an object $x \\amalg y \\in \\Ob(\\mathcal{C})$", "together with morphisms", "$i \\in \\Mor_{\\mathcal C}(x, x \\amalg y)$ and", "$j \\in \\Mor_{\\mathcal C}(y, x \\amalg y)$ such", "that the following universal property holds: for", "any $w \\in \\Ob(\\mathcal{C})$ and morphisms", "$\\alpha \\in \\Mor_{\\mathcal C}(x, w)$ and", "$\\beta \\in \\Mor_\\mathcal{C}(y, w)$", "there is a unique", "$\\gamma \\in \\Mor_{\\mathcal C}(x \\amalg y, w)$ making", "the diagram", "$$", "\\xymatrix{", "& y \\ar[d]^j \\ar[rrdd]^\\beta \\\\", "x \\ar[r]^i \\ar[rrrd]_\\alpha & x \\amalg y \\ar@{-->}[rrd]^\\gamma \\\\", "& & & w", "}", "$$", "commute." ], "refs": [], "ref_ids": [] }, { "id": 12344, "type": "definition", "label": "categories-definition-has-coproducts-of-pairs", "categories": [ "categories" ], "title": "categories-definition-has-coproducts-of-pairs", "contents": [ "We say the category $\\mathcal{C}$ {\\it has coproducts of pairs", "of objects} if a coproduct $x \\amalg y$", "exists for any $x, y \\in \\Ob(\\mathcal{C})$." ], "refs": [], "ref_ids": [] }, { "id": 12345, "type": "definition", "label": "categories-definition-fibre-products", "categories": [ "categories" ], "title": "categories-definition-fibre-products", "contents": [ "Let $x, y, z\\in \\Ob(\\mathcal{C})$,", "$f\\in \\Mor_\\mathcal{C}(x, y)$", "and $g\\in \\Mor_{\\mathcal C}(z, y)$.", "A {\\it fibre product} of $f$ and $g$ is", "an object $x \\times_y z\\in \\Ob(\\mathcal{C})$", "together with morphisms", "$p \\in \\Mor_{\\mathcal C}(x \\times_y z, x)$ and", "$q \\in \\Mor_{\\mathcal C}(x \\times_y z, z)$ making the diagram", "$$", "\\xymatrix{", "x \\times_y z \\ar[r]_q \\ar[d]_p & z \\ar[d]^g \\\\", "x \\ar[r]^f & y", "}", "$$", "commute, and such that the following universal property holds: for", "any $w\\in \\Ob(\\mathcal{C})$ and morphisms", "$\\alpha \\in \\Mor_{\\mathcal C}(w, x)$ and", "$\\beta \\in \\Mor_\\mathcal{C}(w, z)$ with", "$f \\circ \\alpha = g \\circ \\beta$", "there is a unique", "$\\gamma \\in \\Mor_{\\mathcal C}(w, x \\times_y z)$ making", "the diagram", "$$", "\\xymatrix{", "w \\ar[rrrd]^\\beta \\ar@{-->}[rrd]_\\gamma \\ar[rrdd]_\\alpha & & \\\\", "& & x \\times_y z \\ar[d]^p \\ar[r]_q & z \\ar[d]^g \\\\", "& & x \\ar[r]^f & y", "}", "$$", "commute." ], "refs": [], "ref_ids": [] }, { "id": 12346, "type": "definition", "label": "categories-definition-cartesian", "categories": [ "categories" ], "title": "categories-definition-cartesian", "contents": [ "We say a commutative diagram", "$$", "\\xymatrix{", "w \\ar[r] \\ar[d] &", "z \\ar[d] \\\\", "x \\ar[r] &", "y", "}", "$$", "in a category is {\\it cartesian} if $w$ and the morphisms $w \\to x$ and", "$w \\to z$ form a fibre product of the morphisms $x \\to y$ and $z \\to y$." ], "refs": [], "ref_ids": [] }, { "id": 12347, "type": "definition", "label": "categories-definition-has-fibre-products", "categories": [ "categories" ], "title": "categories-definition-has-fibre-products", "contents": [ "We say the category $\\mathcal{C}$ {\\it has fibre products} if", "the fibre product exists for any $f\\in \\Mor_{\\mathcal C}(x, y)$", "and $g\\in \\Mor_{\\mathcal C}(z, y)$." ], "refs": [], "ref_ids": [] }, { "id": 12348, "type": "definition", "label": "categories-definition-representable-morphism", "categories": [ "categories" ], "title": "categories-definition-representable-morphism", "contents": [ "A morphism $f : x \\to y$ of a category $\\mathcal{C}$ is said to be", "{\\it representable} if for every morphism $z \\to y$", "in $\\mathcal{C}$ the fibre product $x \\times_y z$ exists." ], "refs": [], "ref_ids": [] }, { "id": 12349, "type": "definition", "label": "categories-definition-representable-map-presheaves", "categories": [ "categories" ], "title": "categories-definition-representable-map-presheaves", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $F, G : \\mathcal{C}^{opp} \\to \\textit{Sets}$", "be functors. We say a morphism $a : F \\to G$ is", "{\\it representable}, or that {\\it $F$ is relatively representable", "over $G$}, if for every $U \\in \\Ob(\\mathcal{C})$", "and any $\\xi \\in G(U)$ the functor", "$h_U \\times_G F$ is representable." ], "refs": [], "ref_ids": [] }, { "id": 12350, "type": "definition", "label": "categories-definition-pushouts", "categories": [ "categories" ], "title": "categories-definition-pushouts", "contents": [ "Let $x, y, z\\in \\Ob(\\mathcal{C})$,", "$f\\in \\Mor_\\mathcal{C}(y, x)$", "and $g\\in \\Mor_{\\mathcal C}(y, z)$.", "A {\\it pushout} of $f$ and $g$ is", "an object $x\\amalg_y z\\in \\Ob(\\mathcal{C})$", "together with morphisms", "$p\\in \\Mor_{\\mathcal C}(x, x\\amalg_y z)$ and", "$q\\in\\Mor_{\\mathcal C}(z, x\\amalg_y z)$ making the diagram", "$$", "\\xymatrix{", "y \\ar[r]_g \\ar[d]_f & z \\ar[d]^q \\\\", "x \\ar[r]^p & x\\amalg_y z", "}", "$$", "commute, and such that the following universal property holds:", "For any $w\\in \\Ob(\\mathcal{C})$ and morphisms", "$\\alpha \\in \\Mor_{\\mathcal C}(x, w)$ and", "$\\beta \\in \\Mor_\\mathcal{C}(z, w)$ with", "$\\alpha \\circ f = \\beta \\circ g$ there is a unique", "$\\gamma\\in \\Mor_{\\mathcal C}(x\\amalg_y z, w)$ making", "the diagram", "$$", "\\xymatrix{", "y \\ar[r]_g \\ar[d]_f & z \\ar[d]^q \\ar[rrdd]^\\beta & & \\\\", "x \\ar[r]^p \\ar[rrrd]^\\alpha & x \\amalg_y z \\ar@{-->}[rrd]^\\gamma & & \\\\", "& & & w", "}", "$$", "commute." ], "refs": [], "ref_ids": [] }, { "id": 12351, "type": "definition", "label": "categories-definition-cocartesian", "categories": [ "categories" ], "title": "categories-definition-cocartesian", "contents": [ "We say a commutative diagram", "$$", "\\xymatrix{", "y \\ar[r] \\ar[d] & z \\ar[d] \\\\", "x \\ar[r] & w", "}", "$$", "in a category is {\\it cocartesian} if $w$ and the morphisms $x \\to w$ and", "$z \\to w$ form a pushout of the morphisms $y \\to x$ and $y \\to z$." ], "refs": [], "ref_ids": [] }, { "id": 12352, "type": "definition", "label": "categories-definition-equalizers", "categories": [ "categories" ], "title": "categories-definition-equalizers", "contents": [ "Suppose that $X$, $Y$ are objects of a category $\\mathcal{C}$", "and that $a, b : X \\to Y$ are morphisms. We say a morphism", "$e : Z \\to X$ is an {\\it equalizer} for the pair $(a, b)$ if", "$a \\circ e = b \\circ e$ and if $(Z, e)$ satisfies the following", "universal property: For every morphism $t : W \\to X$", "in $\\mathcal{C}$ such that $a \\circ t = b \\circ t$ there exists", "a unique morphism $s : W \\to Z$ such that $t = e \\circ s$." ], "refs": [], "ref_ids": [] }, { "id": 12353, "type": "definition", "label": "categories-definition-coequalizers", "categories": [ "categories" ], "title": "categories-definition-coequalizers", "contents": [ "Suppose that $X$, $Y$ are objects of a category $\\mathcal{C}$", "and that $a, b : X \\to Y$ are morphisms. We say a morphism", "$c : Y \\to Z$ is a {\\it coequalizer} for the pair $(a, b)$ if", "$c \\circ a = c \\circ b$ and if $(Z, c)$ satisfies the following", "universal property: For every morphism $t : Y \\to W$", "in $\\mathcal{C}$ such that $t \\circ a = t \\circ b$ there exists", "a unique morphism $s : Z \\to W$ such that $t = s \\circ c$." ], "refs": [], "ref_ids": [] }, { "id": 12354, "type": "definition", "label": "categories-definition-initial-final", "categories": [ "categories" ], "title": "categories-definition-initial-final", "contents": [ "Let $\\mathcal{C}$ be a category.", "\\begin{enumerate}", "\\item An object $x$ of the category $\\mathcal{C}$ is called", "an {\\it initial} object if for every object $y$ of $\\mathcal{C}$", "there is exactly one morphism $x \\to y$.", "\\item An object $x$ of the category $\\mathcal{C}$ is called", "a {\\it final} object if for every object $y$ of $\\mathcal{C}$", "there is exactly one morphism $y \\to x$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12355, "type": "definition", "label": "categories-definition-mono-epi", "categories": [ "categories" ], "title": "categories-definition-mono-epi", "contents": [ "Let $\\mathcal{C}$ be a category and let $f : X \\to Y$ be", "a morphism of $\\mathcal{C}$.", "\\begin{enumerate}", "\\item We say that $f$ is a {\\it monomorphism} if for every object", "$W$ and every pair of morphisms $a, b : W \\to X$ such that", "$f \\circ a = f \\circ b$ we have $a = b$.", "\\item We say that $f$ is an {\\it epimorphism} if for every object", "$W$ and every pair of morphisms $a, b : Y \\to W$ such that", "$a \\circ f = b \\circ f$ we have $a = b$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12356, "type": "definition", "label": "categories-definition-limit", "categories": [ "categories" ], "title": "categories-definition-limit", "contents": [ "A {\\it limit} of the $\\mathcal{I}$-diagram $M$ in the category", "$\\mathcal{C}$ is given by an object $\\lim_\\mathcal{I} M$ in $\\mathcal{C}$", "together with morphisms $p_i : \\lim_\\mathcal{I} M \\to M_i$ such that", "\\begin{enumerate}", "\\item for $\\phi : i \\to i'$ a morphism", "in $\\mathcal{I}$ we have $p_{i'} = M(\\phi) \\circ p_i$, and", "\\item for any object $W$ in $\\mathcal{C}$ and any family of", "morphisms $q_i : W \\to M_i$ (indexed by $i \\in \\mathcal{I}$)", "such that for all $\\phi : i \\to i'$", "in $\\mathcal{I}$ we have $q_{i'} = M(\\phi) \\circ q_i$ there", "exists a unique morphism $q : W \\to \\lim_\\mathcal{I} M$ such that", "$q_i = p_i \\circ q$ for every object $i$ of $\\mathcal{I}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12357, "type": "definition", "label": "categories-definition-colimit", "categories": [ "categories" ], "title": "categories-definition-colimit", "contents": [ "A {\\it colimit} of the $\\mathcal{I}$-diagram $M$ in the category", "$\\mathcal{C}$ is given by an object $\\colim_\\mathcal{I} M$ in $\\mathcal{C}$", "together with morphisms $s_i : M_i \\to \\colim_\\mathcal{I} M$ such that", "\\begin{enumerate}", "\\item for $\\phi : i \\to i'$ a morphism", "in $\\mathcal{I}$ we have $s_i = s_{i'} \\circ M(\\phi)$, and", "\\item for any object $W$ in $\\mathcal{C}$ and any family of", "morphisms $t_i : M_i \\to W$ (indexed by $i \\in \\mathcal{I}$)", "such that for all $\\phi : i \\to i'$", "in $\\mathcal{I}$ we have $t_i = t_{i'} \\circ M(\\phi)$ there", "exists a unique morphism $t : \\colim_\\mathcal{I} M \\to W$ such that", "$t_i = t \\circ s_i$ for every object $i$ of $\\mathcal{I}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12358, "type": "definition", "label": "categories-definition-product", "categories": [ "categories" ], "title": "categories-definition-product", "contents": [ "Suppose that $I$ is a set, and suppose given for every $i \\in I$ an", "object $M_i$ of the category $\\mathcal{C}$. A {\\it product}", "$\\prod_{i\\in I} M_i$ is by definition $\\lim_\\mathcal{I} M$", "(if it exists)", "where $\\mathcal{I}$ is the category having only identities as", "morphisms and having the elements of $I$ as objects." ], "refs": [], "ref_ids": [] }, { "id": 12359, "type": "definition", "label": "categories-definition-coproduct", "categories": [ "categories" ], "title": "categories-definition-coproduct", "contents": [ "Suppose that $I$ is a set, and suppose given for every $i \\in I$ an", "object $M_i$ of the category $\\mathcal{C}$. A {\\it coproduct}", "$\\coprod_{i\\in I} M_i$ is by definition $\\colim_\\mathcal{I} M$", "(if it exists) where $\\mathcal{I}$ is the category having only", "identities as morphisms and having the elements of $I$ as objects." ], "refs": [], "ref_ids": [] }, { "id": 12360, "type": "definition", "label": "categories-definition-category-connected", "categories": [ "categories" ], "title": "categories-definition-category-connected", "contents": [ "We say that a category $\\mathcal{I}$ is {\\it connected}", "if the equivalence relation generated by", "$x \\sim y \\Leftrightarrow \\Mor_\\mathcal{I}(x, y) \\not = \\emptyset$", "has exactly one equivalence class." ], "refs": [], "ref_ids": [] }, { "id": 12361, "type": "definition", "label": "categories-definition-cofinal", "categories": [ "categories" ], "title": "categories-definition-cofinal", "contents": [ "Let $H : \\mathcal{I} \\to \\mathcal{J}$ be a functor between categories.", "We say {\\it $\\mathcal{I}$ is cofinal in $\\mathcal{J}$} or that", "$H$ is {\\it cofinal} if", "\\begin{enumerate}", "\\item for all $y \\in \\Ob(\\mathcal{J})$ there exists a", "$x \\in \\Ob(\\mathcal{I})$ and a morphism $y \\to H(x)$, and", "\\item given $y \\in \\Ob(\\mathcal{J})$, $x, x' \\in \\Ob(\\mathcal{I})$", "and morphisms $y \\to H(x)$ and $y \\to H(x')$ there exists a sequence", "of morphisms", "$$", "x = x_0 \\leftarrow x_1 \\rightarrow x_2 \\leftarrow x_3 \\rightarrow \\ldots", "\\rightarrow x_{2n} = x'", "$$", "in $\\mathcal{I}$ and morphisms $y \\to H(x_i)$ in $\\mathcal{J}$", "such that the diagrams", "$$", "\\xymatrix{", "& y \\ar[ld] \\ar[d] \\ar[rd] \\\\", "H(x_{2k}) & H(x_{2k + 1}) \\ar[l] \\ar[r] & H(x_{2k + 2})", "}", "$$", "commute for $k = 0, \\ldots, n - 1$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12362, "type": "definition", "label": "categories-definition-initial", "categories": [ "categories" ], "title": "categories-definition-initial", "contents": [ "Let $H : \\mathcal{I} \\to \\mathcal{J}$ be a functor between categories.", "We say {\\it $\\mathcal{I}$ is initial in $\\mathcal{J}$} or that", "$H$ is {\\it initial} if", "\\begin{enumerate}", "\\item for all $y \\in \\Ob(\\mathcal{J})$ there exists a", "$x \\in \\Ob(\\mathcal{I})$ and a morphism $H(x) \\to y$,", "\\item for any $y \\in \\Ob(\\mathcal{J})$, $x , x' \\in \\Ob(\\mathcal{I})$ and", "morphisms $H(x) \\to y$, $H(x') \\to y$ in $\\mathcal{J}$", "there exists a sequence of morphisms", "$$", "x = x_0 \\leftarrow x_1 \\rightarrow x_2 \\leftarrow x_3 \\rightarrow \\ldots", "\\rightarrow x_{2n} = x'", "$$", "in $\\mathcal{I}$ and morphisms $H(x_i) \\to y$ in $\\mathcal{J}$", "such that the diagrams", "$$", "\\xymatrix{", "H(x_{2k}) \\ar[rd] &", "H(x_{2k + 1}) \\ar[l] \\ar[r] \\ar[d] &", "H(x_{2k + 2}) \\ar[ld] \\\\", "& y", "}", "$$", "commute for $k = 0, \\ldots, n - 1$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12363, "type": "definition", "label": "categories-definition-directed", "categories": [ "categories" ], "title": "categories-definition-directed", "contents": [ "We say that a diagram $M : \\mathcal{I} \\to \\mathcal{C}$ is {\\it directed},", "or {\\it filtered} if the following conditions hold:", "\\begin{enumerate}", "\\item the category $\\mathcal{I}$ has at least one object,", "\\item for every pair of objects $x, y$ of $\\mathcal{I}$", "there exists an object $z$ and morphisms $x \\to z$,", "$y \\to z$, and", "\\item for every pair of objects $x, y$ of $\\mathcal{I}$", "and every pair of morphisms $a, b : x \\to y$ of $\\mathcal{I}$", "there exists a morphism $c : y \\to z$ of $\\mathcal{I}$", "such that $M(c \\circ a) = M(c \\circ b)$ as morphisms in $\\mathcal{C}$.", "\\end{enumerate}", "We say that an index category $\\mathcal{I}$ is {\\it directed}, or", "{\\it filtered} if $\\text{id} : \\mathcal{I} \\to \\mathcal{I}$ is filtered", "(in other words you erase the $M$ in part (3) above)." ], "refs": [], "ref_ids": [] }, { "id": 12364, "type": "definition", "label": "categories-definition-codirected", "categories": [ "categories" ], "title": "categories-definition-codirected", "contents": [ "We say that a diagram $M : \\mathcal{I} \\to \\mathcal{C}$ is {\\it codirected}", "or {\\it cofiltered} if the following conditions hold:", "\\begin{enumerate}", "\\item the category $\\mathcal{I}$ has at least one object,", "\\item for every pair of objects $x, y$ of $\\mathcal{I}$", "there exists an object $z$ and morphisms $z \\to x$,", "$z \\to y$, and", "\\item for every pair of objects $x, y$ of $\\mathcal{I}$", "and every pair of morphisms $a, b : x \\to y$ of $\\mathcal{I}$", "there exists a morphism $c : w \\to x$ of $\\mathcal{I}$", "such that $M(a \\circ c) = M(b \\circ c)$ as morphisms in $\\mathcal{C}$.", "\\end{enumerate}", "We say that an index category $\\mathcal{I}$ is {\\it codirected}, or", "{\\it cofiltered} if $\\text{id} : \\mathcal{I} \\to \\mathcal{I}$ is", "cofiltered (in other words you erase the $M$ in part (3) above)." ], "refs": [], "ref_ids": [] }, { "id": 12365, "type": "definition", "label": "categories-definition-directed-set", "categories": [ "categories" ], "title": "categories-definition-directed-set", "contents": [ "Let $I$ be a set and let $\\leq$ be a binary relation on $I$.", "\\begin{enumerate}", "\\item We say $\\leq$ is a {\\it preorder} if it is", "transitive (if $i \\leq j$ and $j \\leq k$ then $i \\leq k$) and", "reflexive ($i \\leq i$ for all $i \\in I$).", "\\item A {\\it preordered set} is a set endowed with a preorder.", "\\item A {\\it directed set} is a preordered set $(I, \\leq)$", "such that $I$ is not empty and such that $\\forall i, j \\in I$,", "there exists $k \\in I$ with $i \\leq k, j \\leq k$.", "\\item We say $\\leq$ is a {\\it partial order} if it is a preorder", "which is antisymmetric (if $i \\leq j$ and $j \\leq i$, then $i = j$).", "\\item A {\\it partially ordered set} is a set endowed with a partial order.", "\\item A {\\it directed partially ordered set} is a directed set", "whose ordering is a partial order.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12366, "type": "definition", "label": "categories-definition-system-over-poset", "categories": [ "categories" ], "title": "categories-definition-system-over-poset", "contents": [ "Let $(I, \\leq)$ be a preordered set. Let $\\mathcal{C}$ be a category.", "\\begin{enumerate}", "\\item A {\\it system over $I$ in $\\mathcal{C}$}, sometimes called a", "{\\it inductive system over $I$ in $\\mathcal{C}$} is given by", "objects $M_i$ of $\\mathcal{C}$ and for every $i \\leq i'$ a", "morphism $f_{ii'} : M_i \\to M_{i'}$ such that $f_{ii}", "= \\text{id}$ and such that $f_{ii''} = f_{i'i''} \\circ f_{i i'}$", "whenever $i \\leq i' \\leq i''$.", "\\item An {\\it inverse system over $I$ in $\\mathcal{C}$},", "sometimes called a {\\it projective system over $I$ in $\\mathcal{C}$}", "is given by objects $M_i$ of $\\mathcal{C}$ and for every $i' \\leq i$ a", "morphism $f_{ii'} : M_i \\to M_{i'}$ such that $f_{ii}", "= \\text{id}$ and such that $f_{ii''} = f_{i'i''} \\circ f_{i i'}$", "whenever $i'' \\leq i' \\leq i$. (Note reversal of inequalities.)", "\\end{enumerate}", "We will say $(M_i, f_{ii'})$ is a (inverse) system over $I$ to", "denote this. The maps $f_{ii'}$ are sometimes", "called the {\\it transition maps}." ], "refs": [], "ref_ids": [] }, { "id": 12367, "type": "definition", "label": "categories-definition-directed-system", "categories": [ "categories" ], "title": "categories-definition-directed-system", "contents": [ "Let $I$ be a preordered set. We say a system (resp.\\ inverse system)", "$(M_i, f_{ii'})$ is a", "{\\it directed system} (resp.\\ {\\it directed inverse system})", "if $I$ is a directed set", "(Definition \\ref{definition-directed-set}): $I$ is nonempty and", "for all $i_1, i_2 \\in I$ there exists $i\\in I$ such that", "$i_1 \\leq i$ and $i_2 \\leq i$." ], "refs": [ "categories-definition-directed-set" ], "ref_ids": [ 12365 ] }, { "id": 12368, "type": "definition", "label": "categories-definition-essentially-constant-diagram", "categories": [ "categories" ], "title": "categories-definition-essentially-constant-diagram", "contents": [ "Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram in a category", "$\\mathcal{C}$.", "\\begin{enumerate}", "\\item Assume the index category $\\mathcal{I}$ is filtered and", "let $(X, \\{M_i \\to X\\}_i)$ be a cocone for $M$, see", "Remark \\ref{remark-cones-and-cocones}. We say $M$ is", "{\\it essentially constant} with {\\it value} $X$ if there exists an", "$i \\in \\mathcal{I}$ and a morphism $X \\to M_i$ such that", "\\begin{enumerate}", "\\item $X \\to M_i \\to X$ is $\\text{id}_X$, and", "\\item for all $j$ there exist $k$ and morphisms $i \\to k$ and $j \\to k$", "such that the morphism $M_j \\to M_k$ equals the composition", "$M_j \\to X \\to M_i \\to M_k$.", "\\end{enumerate}", "\\item Assume the index category $\\mathcal{I}$ is cofiltered and let", "$(X, \\{X \\to M_i\\}_i)$ be a cone for $M$, see", "Remark \\ref{remark-cones-and-cocones}. We say", "$M$ is {\\it essentially constant} with {\\it value} $X$ if", "there exists an $i \\in \\mathcal{I}$", "and a morphism $M_i \\to X$ such that", "\\begin{enumerate}", "\\item $X \\to M_i \\to X$ is $\\text{id}_X$, and", "\\item for all $j$ there exist $k$ and morphisms $k \\to i$ and $k \\to j$", "such that the morphism $M_k \\to M_j$ equals the composition", "$M_k \\to M_i \\to X \\to M_j$.", "\\end{enumerate}", "\\end{enumerate}", "Please keep in mind Lemma \\ref{lemma-essentially-constant-is-limit-colimit}", "when using this definition." ], "refs": [ "categories-remark-cones-and-cocones", "categories-remark-cones-and-cocones", "categories-lemma-essentially-constant-is-limit-colimit" ], "ref_ids": [ 12416, 12416, 12238 ] }, { "id": 12369, "type": "definition", "label": "categories-definition-essentially-constant-system", "categories": [ "categories" ], "title": "categories-definition-essentially-constant-system", "contents": [ "Let $\\mathcal{C}$ be a category. A directed system", "$(M_i, f_{ii'})$ is an {\\it essentially constant system}", "if $M$ viewed as a functor $I \\to \\mathcal{C}$", "defines an essentially constant diagram. A directed inverse system", "$(M_i, f_{ii'})$ is an {\\it essentially constant inverse system} if", "$M$ viewed as a functor $I^{opp} \\to \\mathcal{C}$ defines an", "essentially constant inverse diagram." ], "refs": [], "ref_ids": [] }, { "id": 12370, "type": "definition", "label": "categories-definition-exact", "categories": [ "categories" ], "title": "categories-definition-exact", "contents": [ "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.", "\\begin{enumerate}", "\\item Suppose all finite limits exist in $\\mathcal{A}$.", "We say $F$ is {\\it left exact} if it commutes", "with all finite limits.", "\\item Suppose all finite colimits exist in $\\mathcal{A}$.", "We say $F$ is {\\it right exact} if it commutes", "with all finite colimits.", "\\item We say $F$ is {\\it exact} if it is both left and right", "exact.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12371, "type": "definition", "label": "categories-definition-adjoint", "categories": [ "categories" ], "title": "categories-definition-adjoint", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be categories.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ and", "$v : \\mathcal{D} \\to \\mathcal{C}$ be functors.", "We say that $u$ is a {\\it left adjoint} of $v$, or that", "$v$ is a {\\it right adjoint} to $u$ if there are bijections", "$$", "\\Mor_\\mathcal{D}(u(X), Y)", "\\longrightarrow", "\\Mor_\\mathcal{C}(X, v(Y))", "$$", "functorial in $X \\in \\Ob(\\mathcal{C})$, and", "$Y \\in \\Ob(\\mathcal{D})$." ], "refs": [], "ref_ids": [] }, { "id": 12372, "type": "definition", "label": "categories-definition-compact-object", "categories": [ "categories" ], "title": "categories-definition-compact-object", "contents": [ "Let $\\mathcal{C}$ be a big\\footnote{See Remark \\ref{remark-big-categories}.}", "category. An object $X$ of $\\mathcal{C}$ is called a {\\it categorically compact}", "if we have", "$$", "\\Mor_\\mathcal{C}(X, \\colim_i M_i) =", "\\colim_i \\Mor_\\mathcal{C}(X, M_i)", "$$", "for every filtered diagram $M : \\mathcal{I} \\to \\mathcal{C}$ such that", "$\\colim_i M_i$ exists." ], "refs": [ "categories-remark-big-categories" ], "ref_ids": [ 12410 ] }, { "id": 12373, "type": "definition", "label": "categories-definition-multiplicative-system", "categories": [ "categories" ], "title": "categories-definition-multiplicative-system", "contents": [ "Let $\\mathcal{C}$ be a category. A set of arrows $S$ of $\\mathcal{C}$ is", "called a {\\it left multiplicative system} if it has the following properties:", "\\begin{enumerate}", "\\item[LMS1] The identity of every object of $\\mathcal{C}$ is in $S$ and", "the composition of two composable elements of $S$ is in $S$.", "\\item[LMS2] Every solid diagram", "$$", "\\xymatrix{", "X \\ar[d]_t \\ar[r]_g & Y \\ar@{..>}[d]^s \\\\", "Z \\ar@{..>}[r]^f & W", "}", "$$", "with $t \\in S$ can be completed to a commutative dotted square with", "$s \\in S$.", "\\item[LMS3] For every pair of morphisms $f, g : X \\to Y$ and", "$t \\in S$ with target $X$ such that $f \\circ t = g \\circ t$", "there exists a $s \\in S$ with source $Y$ such that", "$s \\circ f = s \\circ g$.", "\\end{enumerate}", "A set of arrows $S$ of $\\mathcal{C}$ is", "called a {\\it right multiplicative system}", "if it has the following properties:", "\\begin{enumerate}", "\\item[RMS1] The identity of every object of $\\mathcal{C}$ is in $S$ and", "the composition of two composable elements of $S$ is in $S$.", "\\item[RMS2] Every solid diagram", "$$", "\\xymatrix{", "X \\ar@{..>}[d]_t \\ar@{..>}[r]_g & Y \\ar[d]^s \\\\", "Z \\ar[r]^f & W", "}", "$$", "with $s \\in S$ can be completed to a commutative dotted square with", "$t \\in S$.", "\\item[RMS3] For every pair of morphisms $f, g : X \\to Y$ and", "$s \\in S$ with source $Y$ such that $s \\circ f = s \\circ g$", "there exists a $t \\in S$ with target $X$ such that", "$f \\circ t = g \\circ t$.", "\\end{enumerate}", "A set of arrows $S$ of $\\mathcal{C}$ is called a {\\it multiplicative system}", "if it is both a left multiplicative system and a right multiplicative system.", "In other words, this means that MS1, MS2, MS3 hold, where", "MS1 $=$ LMS1 $+$ RMS1, MS2 $=$ LMS2 $+$ RMS2, and", "MS3 $=$ LMS3 $+$ RMS3. (That said, of course LMS1 $=$ RMS1", "$=$ MS1.)" ], "refs": [], "ref_ids": [] }, { "id": 12374, "type": "definition", "label": "categories-definition-left-localization-as-fraction", "categories": [ "categories" ], "title": "categories-definition-left-localization-as-fraction", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative", "system of morphisms of $\\mathcal{C}$. Given any morphism", "$f : X \\to Y'$ in $\\mathcal{C}$ and any morphism $s : Y \\to Y'$ in", "$S$, we denote by {\\it $s^{-1} f$} the equivalence class of the pair", "$(f : X \\to Y', s : Y \\to Y')$. This is a morphism from $X$ to $Y$", "in $S^{-1} \\mathcal{C}$." ], "refs": [], "ref_ids": [] }, { "id": 12375, "type": "definition", "label": "categories-definition-right-localization-as-fraction", "categories": [ "categories" ], "title": "categories-definition-right-localization-as-fraction", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative", "system of morphisms of $\\mathcal{C}$. Given any morphism", "$f : X' \\to Y$ in $\\mathcal{C}$ and any morphism $s : X' \\to X$ in", "$S$, we denote by {\\it $f s^{-1}$} the equivalence class of the pair", "$(f : X' \\to Y, s : X' \\to X)$. This is a morphism from $X$ to $Y$", "in $S^{-1} \\mathcal{C}$." ], "refs": [], "ref_ids": [] }, { "id": 12376, "type": "definition", "label": "categories-definition-saturated-multiplicative-system", "categories": [ "categories" ], "title": "categories-definition-saturated-multiplicative-system", "contents": [ "Let $\\mathcal{C}$ be a category and let $S$ be a multiplicative system.", "We say $S$ is {\\it saturated} if, in addition to MS1, MS2, MS3, we", "also have", "\\begin{enumerate}", "\\item[MS4] Given three composable morphisms $f, g, h$, if", "$fg, gh \\in S$, then $g \\in S$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12377, "type": "definition", "label": "categories-definition-horizontal-composition", "categories": [ "categories" ], "title": "categories-definition-horizontal-composition", "contents": [ "Given a diagram as in the left hand side of:", "$$", "\\xymatrix{", "\\mathcal{A}", "\\rtwocell^F_{F'}{t}", "&", "\\mathcal{B}", "\\rtwocell^G_{G'}{s}", "&", "\\mathcal{C}", "}", "\\text{ gives }", "\\xymatrix{", "\\mathcal{A}", "\\rrtwocell^{G \\circ F} _{G' \\circ F'}{\\ \\ s \\star t}", "& &", "\\mathcal{C}", "}", "$$", "we define the {\\it horizontal} composition $s \\star t$ to be the", "transformation of functors ${}_{G'}t \\circ s_F = s_{F'}\\circ {}_Gt$." ], "refs": [], "ref_ids": [] }, { "id": 12378, "type": "definition", "label": "categories-definition-2-category", "categories": [ "categories" ], "title": "categories-definition-2-category", "contents": [ "A (strict) {\\it $2$-category} $\\mathcal{C}$ consists of the following data", "\\begin{enumerate}", "\\item A set of objects $\\Ob(\\mathcal{C})$.", "\\item For each pair $x, y \\in \\Ob(\\mathcal{C})$", "a category $\\Mor_\\mathcal{C}(x, y)$. The objects of", "$\\Mor_\\mathcal{C}(x, y)$ will be called {\\it $1$-morphisms}", "and denoted $F : x \\to y$. The morphisms between these $1$-morphisms", "will be called {\\it $2$-morphisms} and denoted $t : F' \\to F$.", "The composition of $2$-morphisms in $\\Mor_\\mathcal{C}(x, y)$", "will be called {\\it vertical} composition and will be", "denoted $t \\circ t'$ for $t : F' \\to F$ and $t' : F'' \\to F'$.", "\\item For each triple $x, y, z\\in \\Ob(\\mathcal{C})$ a", "functor", "$$", "(\\circ, \\star) :", "\\Mor_\\mathcal{C}(y, z) \\times \\Mor_\\mathcal{C}(x, y)", "\\longrightarrow", "\\Mor_\\mathcal{C}(x, z).", "$$", "The image of the pair of $1$-morphisms $(F, G)$ on the left hand side", "will be called the {\\it composition} of $F$ and $G$, and denoted", "$F\\circ G$. The image of the pair of $2$-morphisms $(t, s)$ will", "be called the {\\it horizontal} composition and denoted $t \\star s$.", "\\end{enumerate}", "These data are to satisfy the following rules:", "\\begin{enumerate}", "\\item The set of objects together with the set of $1$-morphisms endowed", "with composition of $1$-morphisms forms a category.", "\\item Horizontal composition of $2$-morphisms is associative.", "\\item The identity $2$-morphism $\\text{id}_{\\text{id}_x}$", "of the identity $1$-morphism $\\text{id}_x$ is a unit for", "horizontal composition.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12379, "type": "definition", "label": "categories-definition-sub-2-category", "categories": [ "categories" ], "title": "categories-definition-sub-2-category", "contents": [ "Let $\\mathcal{C}$ be a $2$-category.", "A {\\it sub $2$-category} $\\mathcal{C}'$ of $\\mathcal{C}$, is given by a subset", "$\\Ob(\\mathcal{C}')$ of $\\Ob(\\mathcal{C})$", "and sub categories $\\Mor_{\\mathcal{C}'}(x, y)$ of the", "categories $\\Mor_\\mathcal{C}(x, y)$ for all", "$x, y \\in \\Ob(\\mathcal{C}')$ such that these, together with", "the operations $\\circ$ (composition $1$-morphisms), $\\circ$ (vertical", "composition $2$-morphisms), and $\\star$ (horizontal composition)", "form a $2$-category." ], "refs": [], "ref_ids": [] }, { "id": 12380, "type": "definition", "label": "categories-definition-equivalence", "categories": [ "categories" ], "title": "categories-definition-equivalence", "contents": [ "Two objects $x, y$ of a $2$-category are {\\it equivalent} if there exist", "$1$-morphisms $F : x \\to y$ and $G : y \\to x$ such that $F \\circ G$ is", "$2$-isomorphic to $\\text{id}_y$ and $G \\circ F$ is $2$-isomorphic to", "$\\text{id}_x$." ], "refs": [], "ref_ids": [] }, { "id": 12381, "type": "definition", "label": "categories-definition-functor-into-2-category", "categories": [ "categories" ], "title": "categories-definition-functor-into-2-category", "contents": [ "Let $\\mathcal{A}$ be a category and let $\\mathcal{C}$ be a $2$-category.", "\\begin{enumerate}", "\\item A {\\it functor} from an ordinary category into a $2$-category", "will ignore the", "$2$-morphisms unless mentioned otherwise. In other words, it will be a", "``usual'' functor into the category formed out of 2-category by forgetting", "all the 2-morphisms.", "\\item A {\\it weak functor}, or", "a {\\it pseudo functor} $\\varphi$ from $\\mathcal{A}$ into the 2-category", "$\\mathcal{C}$ is given by the following data", "\\begin{enumerate}", "\\item a map $\\varphi : \\Ob(\\mathcal{A}) \\to \\Ob(\\mathcal{C})$,", "\\item for every pair $x, y\\in \\Ob(\\mathcal{A})$, and every", "morphism $f : x \\to y$ a $1$-morphism $\\varphi(f) : \\varphi(x) \\to \\varphi(y)$,", "\\item for every $x\\in \\Ob(A)$ a $2$-morphism", "$\\alpha_x : \\text{id}_{\\varphi(x)} \\to \\varphi(\\text{id}_x)$, and", "\\item for every pair of composable morphisms $f : x \\to y$,", "$g : y \\to z$ of $\\mathcal{A}$ a $2$-morphism", "$\\alpha_{g, f} : \\varphi(g \\circ f) \\to \\varphi(g) \\circ \\varphi(f)$.", "\\end{enumerate}", "These data are subject to the following conditions:", "\\begin{enumerate}", "\\item the $2$-morphisms $\\alpha_x$ and $\\alpha_{g, f}$ are all", "isomorphisms,", "\\item for any morphism $f : x \\to y$ in $\\mathcal{A}$ we have", "$\\alpha_{\\text{id}_y, f} = \\alpha_y \\star \\text{id}_{\\varphi(f)}$:", "$$", "\\xymatrix{", "\\varphi(x)", "\\rrtwocell^{\\varphi(f)}_{\\varphi(f)}{\\ \\ \\ \\ \\text{id}_{\\varphi(f)}}", "& &", "\\varphi(y)", "\\rrtwocell^{\\text{id}_{\\varphi(y)}}_{\\varphi(\\text{id}_y)}{\\alpha_y}", "& &", "\\varphi(y)", "}", "=", "\\xymatrix{", "\\varphi(x)", "\\rrtwocell^{\\varphi(f)}_{\\varphi(\\text{id}_y) \\circ \\varphi(f)}{\\ \\ \\ \\ \\alpha_{\\text{id}_y, f}}", "& &", "\\varphi(y)", "}", "$$", "\\item for any morphism $f : x \\to y$ in $\\mathcal{A}$ we have", "$\\alpha_{f, \\text{id}_x} = \\text{id}_{\\varphi(f)} \\star \\alpha_x$,", "\\item for any triple of composable morphisms", "$f : w \\to x$, $g : x \\to y$, and $h : y \\to z$ of $\\mathcal{A}$", "we have", "$$", "(\\text{id}_{\\varphi(h)} \\star \\alpha_{g, f})", "\\circ", "\\alpha_{h, g \\circ f}", "=", "(\\alpha_{h, g} \\star \\text{id}_{\\varphi(f)})", "\\circ", "\\alpha_{h \\circ g, f}", "$$", "in other words the following diagram with objects", "$1$-morphisms and arrows $2$-morphisms commutes", "$$", "\\xymatrix{", "\\varphi(h \\circ g \\circ f)", "\\ar[d]_{\\alpha_{h, g \\circ f}}", "\\ar[rr]_{\\alpha_{h \\circ g, f}}", "& &", "\\varphi(h \\circ g) \\circ \\varphi(f)", "\\ar[d]^{\\alpha_{h, g} \\star \\text{id}_{\\varphi(f)}} \\\\", "\\varphi(h) \\circ \\varphi(g \\circ f)", "\\ar[rr]^{\\text{id}_{\\varphi(h)} \\star \\alpha_{g, f}}", "& &", "\\varphi(h) \\circ \\varphi(g) \\circ \\varphi(f)", "}", "$$", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12382, "type": "definition", "label": "categories-definition-2-1-category", "categories": [ "categories" ], "title": "categories-definition-2-1-category", "contents": [ "A (strict) {\\it $(2, 1)$-category} is a $2$-category in which all", "$2$-morphisms are isomorphisms." ], "refs": [], "ref_ids": [] }, { "id": 12383, "type": "definition", "label": "categories-definition-final-object-2-category", "categories": [ "categories" ], "title": "categories-definition-final-object-2-category", "contents": [ "A {\\it final object} of a $(2, 1)$-category", "$\\mathcal{C}$ is an object $x$ such that", "\\begin{enumerate}", "\\item for every $y \\in \\Ob(\\mathcal{C})$ there is a morphism $y \\to x$,", "and", "\\item every two morphisms $y \\to x$ are isomorphic by a unique 2-morphism.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12384, "type": "definition", "label": "categories-definition-2-fibre-products", "categories": [ "categories" ], "title": "categories-definition-2-fibre-products", "contents": [ "Let $\\mathcal{C}$ be a $(2, 1)$-category.", "Let $x, y, z\\in \\Ob(\\mathcal{C})$ and", "$f\\in \\Mor_\\mathcal{C}(x, z)$", "and $g\\in \\Mor_{\\mathcal C}(y, z)$. A", "{\\it 2-fibre product of $f$ and $g$} is", "a final object in the category of 2-commutative diagrams", "described above. If a 2-fibre product exists we", "will denote it $x \\times_z y\\in \\Ob(\\mathcal{C})$, and denote the", "required morphisms $p\\in \\Mor_{\\mathcal C}(x \\times_z y, x)$ and", "$q\\in \\Mor_{\\mathcal C}(x \\times_z y, y)$ making the diagram", "$$", "\\xymatrix{", "& x \\times_z y \\ar[r]^{p} \\ar[d]_q & x \\ar[d]^{f} \\\\", "& y \\ar[r]^{g} & z }", "$$", "2-commute and we will denote the given invertible", "2-morphism exhibiting this by $\\psi : f \\circ p \\to g \\circ q$." ], "refs": [], "ref_ids": [] }, { "id": 12385, "type": "definition", "label": "categories-definition-categories-over-C", "categories": [ "categories" ], "title": "categories-definition-categories-over-C", "contents": [ "Let $\\mathcal{C}$ be a category.", "The {\\it $2$-category of categories over $\\mathcal{C}$}", "is the $2$-category defined as follows:", "\\begin{enumerate}", "\\item Its objects will be functors $p : \\mathcal{S} \\to \\mathcal{C}$.", "\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that", "$p' \\circ G = p$.", "\\item Its $2$-morphisms $t : G \\to H$ for", "$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be morphisms of functors", "such that $p'(t_x) = \\text{id}_{p(x)}$", "for all $x \\in \\Ob(\\mathcal{S})$.", "\\end{enumerate}", "In this situation we will denote", "$$", "\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{S}, \\mathcal{S}')", "$$", "the category of $1$-morphisms between", "$(\\mathcal{S}, p)$ and $(\\mathcal{S}', p')$" ], "refs": [], "ref_ids": [] }, { "id": 12386, "type": "definition", "label": "categories-definition-fibre-category", "categories": [ "categories" ], "title": "categories-definition-fibre-category", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$.", "\\begin{enumerate}", "\\item The {\\it fibre category} over an object $U\\in \\Ob(\\mathcal{C})$", "is the category $\\mathcal{S}_U$ with objects", "$$", "\\Ob(\\mathcal{S}_U) = \\{x\\in \\Ob(\\mathcal{S}) :", "p(x) = U\\}", "$$", "and morphisms", "$$", "\\Mor_{\\mathcal{S}_U}(x, y) = \\{ \\phi \\in \\Mor_\\mathcal{S}(x, y) :", "p(\\phi) = \\text{id}_U\\}.", "$$", "\\item A {\\it lift} of an object $U \\in \\Ob(\\mathcal{C})$", "is an object $x\\in \\Ob(\\mathcal{S})$ such that $p(x) = U$, i.e.,", "$x\\in \\Ob(\\mathcal{S}_U)$. We will also sometime say", "that {\\it $x$ lies over $U$}.", "\\item Similarly, a {\\it lift} of a morphism $f : V \\to U$ in $\\mathcal{C}$", "is a morphism $\\phi : y \\to x$ in $\\mathcal{S}$ such that $p(\\phi) = f$.", "We sometimes say that {\\it $\\phi$ lies over $f$}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12387, "type": "definition", "label": "categories-definition-cartesian-over-C", "categories": [ "categories" ], "title": "categories-definition-cartesian-over-C", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$.", "A {\\it strongly cartesian morphism}, or more precisely a", "{\\it strongly $\\mathcal{C}$-cartesian morphism} is a", "morphism $\\varphi : y \\to x$ of $\\mathcal{S}$ such that", "for every $z \\in \\Ob(\\mathcal{S})$ the map", "$$", "\\Mor_\\mathcal{S}(z, y)", "\\longrightarrow", "\\Mor_\\mathcal{S}(z, x)", "\\times_{\\Mor_\\mathcal{C}(p(z), p(x))}", "\\Mor_\\mathcal{C}(p(z), p(y)),", "$$", "given by $\\psi \\longmapsto (\\varphi \\circ \\psi, p(\\psi))$", "is bijective." ], "refs": [], "ref_ids": [] }, { "id": 12388, "type": "definition", "label": "categories-definition-fibred-category", "categories": [ "categories" ], "title": "categories-definition-fibred-category", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$.", "We say $\\mathcal{S}$ is a {\\it fibred category over $\\mathcal{C}$}", "if given any $x \\in \\Ob(\\mathcal{S})$ lying over", "$U \\in \\Ob(\\mathcal{C})$ and any morphism $f : V \\to U$ of", "$\\mathcal{C}$, there exists a strongly cartesian morphism $f^*x \\to x$", "lying over $f$." ], "refs": [], "ref_ids": [] }, { "id": 12389, "type": "definition", "label": "categories-definition-pullback-functor-fibred-category", "categories": [ "categories" ], "title": "categories-definition-pullback-functor-fibred-category", "contents": [ "Assume $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category.", "\\begin{enumerate}", "\\item A {\\it choice of pullbacks}\\footnote{This is probably nonstandard", "terminology. In some texts this is called a ``cleavage'' but it conjures up", "the wrong image. Maybe a ``cleaving'' would be a better word.", "A related notion is that of a ``splitting'', but in many texts a ``splitting''", "means a choice of pullbacks such that $g^*f^* = (f \\circ g)^*$", "for any composable pair of morphisms. Compare", "also with Definition \\ref{definition-split-fibred-category}.}", "for $p : \\mathcal{S} \\to \\mathcal{C}$", "is given by a choice of a strongly cartesian morphism", "$f^\\ast x \\to x$ lying over $f$ for any morphism", "$f: V \\to U$ of $\\mathcal{C}$ and any $x \\in \\Ob(\\mathcal{S}_U)$.", "\\item Given a choice of pullbacks,", "for any morphism $f : V \\to U$ of $\\mathcal{C}$", "the functor $f^* : \\mathcal{S}_U \\to \\mathcal{S}_V$ described", "above is called a {\\it pullback functor} (associated to the choices", "$f^*x \\to x$ made above).", "\\end{enumerate}" ], "refs": [ "categories-definition-split-fibred-category" ], "ref_ids": [ 12394 ] }, { "id": 12390, "type": "definition", "label": "categories-definition-fibred-categories-over-C", "categories": [ "categories" ], "title": "categories-definition-fibred-categories-over-C", "contents": [ "Let $\\mathcal{C}$ be a category.", "The {\\it $2$-category of fibred categories over $\\mathcal{C}$}", "is the sub $2$-category of the $2$-category of categories", "over $\\mathcal{C}$ (see Definition \\ref{definition-categories-over-C})", "defined as follows:", "\\begin{enumerate}", "\\item Its objects will be fibred categories", "$p : \\mathcal{S} \\to \\mathcal{C}$.", "\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that", "$p' \\circ G = p$ and such that $G$ maps strongly cartesian", "morphisms to strongly cartesian morphisms.", "\\item Its $2$-morphisms $t : G \\to H$ for", "$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be morphisms of functors", "such that $p'(t_x) = \\text{id}_{p(x)}$", "for all $x \\in \\Ob(\\mathcal{S})$.", "\\end{enumerate}", "In this situation we will denote", "$$", "\\Mor_{\\textit{Fib}/\\mathcal{C}}(\\mathcal{S}, \\mathcal{S}')", "$$", "the category of $1$-morphisms between", "$(\\mathcal{S}, p)$ and $(\\mathcal{S}', p')$" ], "refs": [ "categories-definition-categories-over-C" ], "ref_ids": [ 12385 ] }, { "id": 12391, "type": "definition", "label": "categories-definition-inertia-fibred-category", "categories": [ "categories" ], "title": "categories-definition-inertia-fibred-category", "contents": [ "Let $\\mathcal{C}$ be a category.", "\\begin{enumerate}", "\\item Let $F : \\mathcal{S} \\to \\mathcal{S}'$ be a $1$-morphism of", "fibred categories over $\\mathcal{C}$. The {\\it relative inertia", "of $\\mathcal{S}$ over $\\mathcal{S}'$} is the fibred category", "$\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'} \\to \\mathcal{C}$ of", "Lemma \\ref{lemma-inertia-fibred-category}.", "\\item By the {\\it inertia fibred category $\\mathcal{I}_\\mathcal{S}$", "of $\\mathcal{S}$} we mean", "$\\mathcal{I}_\\mathcal{S} = \\mathcal{I}_{\\mathcal{S}/\\mathcal{C}}$.", "\\end{enumerate}" ], "refs": [ "categories-lemma-inertia-fibred-category" ], "ref_ids": [ 12292 ] }, { "id": 12392, "type": "definition", "label": "categories-definition-fibred-groupoids", "categories": [ "categories" ], "title": "categories-definition-fibred-groupoids", "contents": [ "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a functor.", "We say that $\\mathcal{S}$ is {\\it fibred in groupoids} over $\\mathcal{C}$ if", "the following two conditions hold:", "\\begin{enumerate}", "\\item For every morphism $f : V \\to U$ in $\\mathcal{C}$ and every", "lift $x$ of $U$ there is a lift $\\phi : y \\to x$ of $f$ with", "target $x$.", "\\item For every pair of morphisms $\\phi : y \\to x$ and $ \\psi : z \\to x$", "and any morphism $f : p(z) \\to p(y)$ such that $p(\\phi) \\circ f = p(\\psi)$", "there exists a unique lift $\\chi : z \\to y$ of $f$ such that", "$\\phi \\circ \\chi = \\psi$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12393, "type": "definition", "label": "categories-definition-categories-fibred-in-groupoids-over-C", "categories": [ "categories" ], "title": "categories-definition-categories-fibred-in-groupoids-over-C", "contents": [ "Let $\\mathcal{C}$ be a category.", "The {\\it $2$-category of categories fibred in groupoids over $\\mathcal{C}$}", "is the sub $2$-category of the $2$-category of fibred categories", "over $\\mathcal{C}$ (see Definition \\ref{definition-fibred-categories-over-C})", "defined as follows:", "\\begin{enumerate}", "\\item Its objects will be categories", "$p : \\mathcal{S} \\to \\mathcal{C}$ fibred in groupoids.", "\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that", "$p' \\circ G = p$ (since every morphism is strongly cartesian", "$G$ automatically preserves them).", "\\item Its $2$-morphisms $t : G \\to H$ for", "$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be morphisms of functors", "such that $p'(t_x) = \\text{id}_{p(x)}$", "for all $x \\in \\Ob(\\mathcal{S})$.", "\\end{enumerate}" ], "refs": [ "categories-definition-fibred-categories-over-C" ], "ref_ids": [ 12390 ] }, { "id": 12394, "type": "definition", "label": "categories-definition-split-fibred-category", "categories": [ "categories" ], "title": "categories-definition-split-fibred-category", "contents": [ "Let $\\mathcal{C}$ be a category.", "Suppose that $F : \\mathcal{C}^{opp} \\to \\textit{Cat}$ is a functor", "to the $2$-category of categories.", "We will write $p_F : \\mathcal{S}_F \\to \\mathcal{C}$ for the", "fibred category constructed in", "Example \\ref{example-functor-categories}.", "A {\\it split fibred category} is a fibred category isomorphic (!)", "over $\\mathcal{C}$ to one of these categories {\\it $\\mathcal{S}_F$}." ], "refs": [], "ref_ids": [] }, { "id": 12395, "type": "definition", "label": "categories-definition-split-category-fibred-in-groupoids", "categories": [ "categories" ], "title": "categories-definition-split-category-fibred-in-groupoids", "contents": [ "Let $\\mathcal{C}$ be a category.", "Suppose that $F : \\mathcal{C}^{opp} \\to \\textit{Groupoids}$ is a functor", "to the $2$-category of groupoids.", "We will write $p_F : \\mathcal{S}_F \\to \\mathcal{C}$ for the", "category fibred in groupoids constructed in", "Example \\ref{example-functor-groupoids}.", "A {\\it split category fibred in groupoids} is a", "category fibred in groupoids isomorphic (!)", "over $\\mathcal{C}$ to one of these categories {\\it $\\mathcal{S}_F$}." ], "refs": [], "ref_ids": [] }, { "id": 12396, "type": "definition", "label": "categories-definition-discrete", "categories": [ "categories" ], "title": "categories-definition-discrete", "contents": [ "A category is called {\\it discrete} if the only morphisms are the identity", "morphisms." ], "refs": [], "ref_ids": [] }, { "id": 12397, "type": "definition", "label": "categories-definition-category-fibred-sets", "categories": [ "categories" ], "title": "categories-definition-category-fibred-sets", "contents": [ "Let $\\mathcal{C}$ be a category.", "A {\\it category fibred in sets}, or a {\\it category fibred", "in discrete categories} is a category fibred in groupoids all", "of whose fibre categories are discrete." ], "refs": [], "ref_ids": [] }, { "id": 12398, "type": "definition", "label": "categories-definition-categories-fibred-in-sets-over-C", "categories": [ "categories" ], "title": "categories-definition-categories-fibred-in-sets-over-C", "contents": [ "Let $\\mathcal{C}$ be a category.", "The {\\it $2$-category of categories fibred in sets over $\\mathcal{C}$}", "is the sub $2$-category of the category of categories fibred in groupoids", "over $\\mathcal{C}$ (see", "Definition \\ref{definition-categories-fibred-in-groupoids-over-C})", "defined as follows:", "\\begin{enumerate}", "\\item Its objects will be categories", "$p : \\mathcal{S} \\to \\mathcal{C}$ fibred in sets.", "\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that", "$p' \\circ G = p$ (since every morphism is strongly cartesian", "$G$ automatically preserves them).", "\\item Its $2$-morphisms $t : G \\to H$ for", "$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be morphisms of functors", "such that $p'(t_x) = \\text{id}_{p(x)}$", "for all $x \\in \\Ob(\\mathcal{S})$.", "\\end{enumerate}" ], "refs": [ "categories-definition-categories-fibred-in-groupoids-over-C" ], "ref_ids": [ 12393 ] }, { "id": 12399, "type": "definition", "label": "categories-definition-setoid", "categories": [ "categories" ], "title": "categories-definition-setoid", "contents": [ "Let us call a category a {\\it setoid}\\footnote{A set on steroids!?}", "if it is a groupoid where every object", "has exactly one automorphism: the identity." ], "refs": [], "ref_ids": [] }, { "id": 12400, "type": "definition", "label": "categories-definition-category-fibred-setoids", "categories": [ "categories" ], "title": "categories-definition-category-fibred-setoids", "contents": [ "Let $\\mathcal{C}$ be a category. A {\\it category fibred in setoids}", "is a category fibred in groupoids all of whose fibre categories are", "setoids." ], "refs": [], "ref_ids": [] }, { "id": 12401, "type": "definition", "label": "categories-definition-categories-fibred-in-setoids-over-C", "categories": [ "categories" ], "title": "categories-definition-categories-fibred-in-setoids-over-C", "contents": [ "Let $\\mathcal{C}$ be a category.", "The {\\it $2$-category of categories fibred in setoids over $\\mathcal{C}$}", "is the sub $2$-category of the category of categories fibred in groupoids", "over $\\mathcal{C}$ (see", "Definition \\ref{definition-categories-fibred-in-groupoids-over-C})", "defined as follows:", "\\begin{enumerate}", "\\item Its objects will be categories", "$p : \\mathcal{S} \\to \\mathcal{C}$ fibred in setoids.", "\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that", "$p' \\circ G = p$ (since every morphism is strongly cartesian", "$G$ automatically preserves them).", "\\item Its $2$-morphisms $t : G \\to H$ for", "$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$", "will be morphisms of functors", "such that $p'(t_x) = \\text{id}_{p(x)}$", "for all $x \\in \\Ob(\\mathcal{S})$.", "\\end{enumerate}" ], "refs": [ "categories-definition-categories-fibred-in-groupoids-over-C" ], "ref_ids": [ 12393 ] }, { "id": 12402, "type": "definition", "label": "categories-definition-representable-fibred-category", "categories": [ "categories" ], "title": "categories-definition-representable-fibred-category", "contents": [ "Let $\\mathcal{C}$ be a category.", "A category fibred in groupoids $p : \\mathcal{S} \\to \\mathcal{C}$ is", "called {\\it representable} if there exists an object", "$X$ of $\\mathcal{C}$ and an equivalence $j : \\mathcal{S} \\to \\mathcal{C}/X$", "(in the $2$-category of groupoids over $\\mathcal{C}$)." ], "refs": [], "ref_ids": [] }, { "id": 12403, "type": "definition", "label": "categories-definition-representable-map-categories-fibred-in-groupoids", "categories": [ "categories" ], "title": "categories-definition-representable-map-categories-fibred-in-groupoids", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids", "over $\\mathcal{C}$.", "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism.", "We say $F$ is {\\it representable}, or that", "{\\it $\\mathcal{X}$ is relatively representable over $\\mathcal{Y}$},", "if for every $U \\in \\Ob(\\mathcal{C})$", "and any $G : \\mathcal{C}/U \\to \\mathcal{Y}$", "the category fibred in groupoids", "$$", "(\\mathcal{C}/U) \\times_\\mathcal{Y} \\mathcal{X}", "\\longrightarrow", "\\mathcal{C}/U", "$$", "is representable." ], "refs": [], "ref_ids": [] }, { "id": 12404, "type": "definition", "label": "categories-definition-monoidal-category", "categories": [ "categories" ], "title": "categories-definition-monoidal-category", "contents": [ "A triple $(\\mathcal{C}, \\otimes, \\phi)$ where $\\mathcal{C}$ is a category,", "$\\otimes : \\mathcal{C} \\times \\mathcal{C} \\to \\mathcal{C}$ is a functor,", "and $\\phi$ is an associativity constraint is called a {\\it monoidal category}", "if there exists a unit $\\mathbf{1}$." ], "refs": [], "ref_ids": [] }, { "id": 12405, "type": "definition", "label": "categories-definition-functor-monoidal-categories", "categories": [ "categories" ], "title": "categories-definition-functor-monoidal-categories", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{C}'$ be monoidal categories.", "A {\\it functor of monoidal categories} $F : \\mathcal{C} \\to \\mathcal{C}'$", "is given by a functor $F$ as indicated and a natural transformation", "$$", "F(X) \\otimes F(Y) \\to F(X \\otimes Y)", "$$", "such that for all objects $X, Y, Z$ the diagram", "$$", "\\xymatrix{", "F(X) \\otimes (F(Y) \\otimes F(Z)) \\ar[r] \\ar[d] &", "F(X) \\otimes F(Y \\otimes Z) \\ar[r] &", "F(X \\otimes (Y \\otimes Z)) \\ar[d] \\\\", "(F(X) \\otimes F(Y)) \\otimes F(Z) \\ar[r] &", "F(X \\otimes Y) \\otimes F(Z) \\ar[r] &", "F((X \\otimes Y) \\otimes Z)", "}", "$$", "commutes and such that $F(\\mathbf{1})$ is a unit in $\\mathcal{C}'$." ], "refs": [], "ref_ids": [] }, { "id": 12406, "type": "definition", "label": "categories-definition-invertible", "categories": [ "categories" ], "title": "categories-definition-invertible", "contents": [ "Let $\\mathcal{C}$ be a monoidal category. An object $X$ of $\\mathcal{C}$", "is called {\\it invertible} if any (or all) of the equivalent conditions of", "Lemma \\ref{lemma-invertible} hold." ], "refs": [ "categories-lemma-invertible" ], "ref_ids": [ 12324 ] }, { "id": 12407, "type": "definition", "label": "categories-definition-dual", "categories": [ "categories" ], "title": "categories-definition-dual", "contents": [ "Given a monoidal category $(\\mathcal{C}, \\otimes, \\phi)$", "and an object $X$ a {\\it left dual} is an object $Y$ together with", "morphisms $\\eta : \\mathbf{1} \\to X \\otimes Y$ and", "$\\epsilon : Y \\otimes X \\to \\mathbf{1}$", "such that the diagrams", "$$", "\\vcenter{", "\\xymatrix{", "X \\ar[rd]_1 \\ar[r]_-{\\eta \\otimes 1} &", "X \\otimes Y \\otimes X \\ar[d]^{1 \\otimes \\epsilon} \\\\", "& X", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "Y \\ar[rd]_1 \\ar[r]_-{1 \\otimes \\eta} &", "Y \\otimes X \\otimes Y \\ar[d]^{\\epsilon \\otimes 1} \\\\", "& Y", "}", "}", "$$", "commute. In this situation we say that $X$ is a {\\it right dual} of $Y$." ], "refs": [], "ref_ids": [] }, { "id": 12408, "type": "definition", "label": "categories-definition-symmetric-monoidal-category", "categories": [ "categories" ], "title": "categories-definition-symmetric-monoidal-category", "contents": [ "A quadruple $(\\mathcal{C}, \\otimes, \\phi, \\psi)$ where", "$\\mathcal{C}$ is a category,", "$\\otimes : \\mathcal{C} \\otimes \\mathcal{C} \\to \\mathcal{C}$ is a functor,", "$\\phi$ is an associativity constraint, and", "$\\psi$ is a commutativity constraint compatible with $\\phi$", "is called a {\\it symmetric monoidal category} if there exists", "a unit." ], "refs": [], "ref_ids": [] }, { "id": 12409, "type": "definition", "label": "categories-definition-functor-symmetric-monoidal-categories", "categories": [ "categories" ], "title": "categories-definition-functor-symmetric-monoidal-categories", "contents": [ "Let $\\mathcal{C}$ and $\\mathcal{C}'$ be symmetric monoidal categories.", "A {\\it functor of symmetric monoidal categories}", "$F : \\mathcal{C} \\to \\mathcal{C}'$", "is given by a functor $F$ as indicated and a natural transformation", "$$", "F(X) \\otimes F(Y) \\to F(X \\otimes Y)", "$$", "such that $F$ is a functor of monoidal categories and such that", "for all objects $X, Y$ the diagram", "$$", "\\xymatrix{", "F(X) \\otimes F(Y) \\ar[r] \\ar[d] &", "F(X \\otimes Y) \\ar[d] \\\\", "F(Y) \\otimes F(X) \\ar[r] &", "F(Y \\otimes X)", "}", "$$", "commutes." ], "refs": [], "ref_ids": [] }, { "id": 12521, "type": "definition", "label": "topologies-definition-zariski-covering", "categories": [ "topologies" ], "title": "topologies-definition-zariski-covering", "contents": [ "Let $T$ be a scheme. A {\\it Zariski covering of $T$} is a family", "of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes", "such that each $f_i$ is an open immersion and such", "that $T = \\bigcup f_i(T_i)$." ], "refs": [], "ref_ids": [] }, { "id": 12522, "type": "definition", "label": "topologies-definition-standard-Zariski", "categories": [ "topologies" ], "title": "topologies-definition-standard-Zariski", "contents": [ "Compare Schemes, Definition \\ref{schemes-definition-standard-covering}.", "Let $T$ be an affine scheme. A {\\it standard Zariski covering}", "of $T$ is a Zariski covering $\\{U_j \\to T\\}_{j = 1, \\ldots, m}$", "with each $U_j \\to T$ inducing an isomorphism with a standard affine open", "of $T$." ], "refs": [ "schemes-definition-standard-covering" ], "ref_ids": [ 7739 ] }, { "id": 12523, "type": "definition", "label": "topologies-definition-big-zariski-site", "categories": [ "topologies" ], "title": "topologies-definition-big-zariski-site", "contents": [ "A {\\it big Zariski site} is any site $\\Sch_{Zar}$ as in", "Sites, Definition \\ref{sites-definition-site} constructed as follows:", "\\begin{enumerate}", "\\item Choose any set of schemes $S_0$, and any set of Zariski coverings", "$\\text{Cov}_0$ among these schemes.", "\\item As underlying category of $\\Sch_{Zar}$", "take any category $\\Sch_\\alpha$ constructed as in", "Sets, Lemma \\ref{sets-lemma-construct-category} starting with the set $S_0$.", "\\item As coverings of $\\Sch_{Zar}$ choose any set of coverings as in", "Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the", "category $\\Sch_\\alpha$ and the class of Zariski coverings,", "and the set $\\text{Cov}_0$ chosen above.", "\\end{enumerate}" ], "refs": [ "sites-definition-site", "sets-lemma-construct-category", "sets-lemma-coverings-site" ], "ref_ids": [ 8652, 8789, 8800 ] }, { "id": 12524, "type": "definition", "label": "topologies-definition-big-small-Zariski", "categories": [ "topologies" ], "title": "topologies-definition-big-small-Zariski", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski", "site containing $S$.", "\\begin{enumerate}", "\\item The {\\it big Zariski site of $S$}, denoted", "$(\\Sch/S)_{Zar}$, is the site $\\Sch_{Zar}/S$", "introduced in Sites, Section \\ref{sites-section-localize}.", "\\item The {\\it small Zariski site of $S$}, which we denote", "$S_{Zar}$, is the full subcategory of $(\\Sch/S)_{Zar}$", "whose objects are those $U/S$ such that $U \\to S$ is an open immersion.", "A covering of $S_{Zar}$ is any covering $\\{U_i \\to U\\}$ of", "$(\\Sch/S)_{Zar}$ with $U \\in \\Ob(S_{Zar})$.", "\\item The {\\it big affine Zariski site of $S$}, denoted", "$(\\textit{Aff}/S)_{Zar}$, is the full subcategory of", "$(\\Sch/S)_{Zar}$ whose objects are affine $U/S$.", "A covering of $(\\textit{Aff}/S)_{Zar}$ is any covering", "$\\{U_i \\to U\\}$ of $(\\Sch/S)_{Zar}$ which is a", "standard Zariski covering.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12525, "type": "definition", "label": "topologies-definition-restriction-small-zariski", "categories": [ "topologies" ], "title": "topologies-definition-restriction-small-zariski", "contents": [ "In the situation of", "Lemma \\ref{lemma-at-the-bottom}", "the functor $i_S^{-1} = \\pi_{S, *}$ is often", "called the {\\it restriction to the small Zariski site}, and for a sheaf", "$\\mathcal{F}$ on the big Zariski site we denote $\\mathcal{F}|_{S_{Zar}}$", "this restriction." ], "refs": [ "topologies-lemma-at-the-bottom" ], "ref_ids": [ 12439 ] }, { "id": 12526, "type": "definition", "label": "topologies-definition-etale-covering", "categories": [ "topologies" ], "title": "topologies-definition-etale-covering", "contents": [ "Let $T$ be a scheme. An {\\it \\'etale covering of $T$} is a family", "of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes", "such that each $f_i$ is \\'etale and such that $T = \\bigcup f_i(T_i)$." ], "refs": [], "ref_ids": [] }, { "id": 12527, "type": "definition", "label": "topologies-definition-standard-etale", "categories": [ "topologies" ], "title": "topologies-definition-standard-etale", "contents": [ "Let $T$ be an affine scheme. A {\\it standard \\'etale covering}", "of $T$ is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$", "with each $U_j$ is affine and \\'etale over $T$ and", "$T = \\bigcup f_j(U_j)$." ], "refs": [], "ref_ids": [] }, { "id": 12528, "type": "definition", "label": "topologies-definition-big-etale-site", "categories": [ "topologies" ], "title": "topologies-definition-big-etale-site", "contents": [ "A {\\it big \\'etale site} is any site $\\Sch_\\etale$ as in", "Sites, Definition \\ref{sites-definition-site} constructed as follows:", "\\begin{enumerate}", "\\item Choose any set of schemes $S_0$, and any set of \\'etale coverings", "$\\text{Cov}_0$ among these schemes.", "\\item As underlying category take any category $\\Sch_\\alpha$", "constructed as in Sets, Lemma \\ref{sets-lemma-construct-category}", "starting with the set $S_0$.", "\\item Choose any set of coverings as in", "Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the", "category $\\Sch_\\alpha$ and the class of \\'etale coverings,", "and the set $\\text{Cov}_0$ chosen above.", "\\end{enumerate}" ], "refs": [ "sites-definition-site", "sets-lemma-construct-category", "sets-lemma-coverings-site" ], "ref_ids": [ 8652, 8789, 8800 ] }, { "id": 12529, "type": "definition", "label": "topologies-definition-big-small-etale", "categories": [ "topologies" ], "title": "topologies-definition-big-small-etale", "contents": [ "Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale", "site containing $S$.", "\\begin{enumerate}", "\\item The {\\it big \\'etale site of $S$}, denoted", "$(\\Sch/S)_\\etale$, is the site", "$\\Sch_\\etale/S$ introduced in", "Sites, Section \\ref{sites-section-localize}.", "\\item The {\\it small \\'etale site of $S$}, which we denote", "$S_\\etale$, is the full subcategory of", "$(\\Sch/S)_\\etale$", "whose objects are those $U/S$ such that $U \\to S$ is \\'etale.", "A covering of $S_\\etale$ is any covering $\\{U_i \\to U\\}$ of", "$(\\Sch/S)_\\etale$ with $U \\in \\Ob(S_\\etale)$.", "\\item The {\\it big affine \\'etale site of $S$}, denoted", "$(\\textit{Aff}/S)_\\etale$, is the full subcategory of", "$(\\Sch/S)_\\etale$ whose objects are affine $U/S$.", "A covering of $(\\textit{Aff}/S)_\\etale$ is any covering", "$\\{U_i \\to U\\}$ of $(\\Sch/S)_\\etale$ which is a", "standard \\'etale covering.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12530, "type": "definition", "label": "topologies-definition-restriction-small-etale", "categories": [ "topologies" ], "title": "topologies-definition-restriction-small-etale", "contents": [ "In the situation of", "Lemma \\ref{lemma-at-the-bottom-etale}", "the functor $i_S^{-1} = \\pi_{S, *}$ is often", "called the {\\it restriction to the small \\'etale site}, and for a sheaf", "$\\mathcal{F}$ on the big \\'etale site we denote", "$\\mathcal{F}|_{S_\\etale}$ this restriction." ], "refs": [ "topologies-lemma-at-the-bottom-etale" ], "ref_ids": [ 12453 ] }, { "id": 12531, "type": "definition", "label": "topologies-definition-smooth-covering", "categories": [ "topologies" ], "title": "topologies-definition-smooth-covering", "contents": [ "Let $T$ be a scheme. A {\\it smooth covering of $T$} is a family", "of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes", "such that each $f_i$ is smooth and such", "that $T = \\bigcup f_i(T_i)$." ], "refs": [], "ref_ids": [] }, { "id": 12532, "type": "definition", "label": "topologies-definition-standard-smooth", "categories": [ "topologies" ], "title": "topologies-definition-standard-smooth", "contents": [ "Let $T$ be an affine scheme. A {\\it standard smooth covering}", "of $T$ is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$", "with each $U_j$ is affine, $U_j \\to T$ standard smooth", "and $T = \\bigcup f_j(U_j)$." ], "refs": [], "ref_ids": [] }, { "id": 12533, "type": "definition", "label": "topologies-definition-big-smooth-site", "categories": [ "topologies" ], "title": "topologies-definition-big-smooth-site", "contents": [ "A {\\it big smooth site} is any site $\\Sch_{smooth}$ as in", "Sites, Definition \\ref{sites-definition-site} constructed as follows:", "\\begin{enumerate}", "\\item Choose any set of schemes $S_0$, and any set of smooth coverings", "$\\text{Cov}_0$ among these schemes.", "\\item As underlying category take any category $\\Sch_\\alpha$", "constructed as in Sets, Lemma \\ref{sets-lemma-construct-category}", "starting with the set $S_0$.", "\\item Choose any set of coverings as in", "Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the", "category $\\Sch_\\alpha$ and the class of smooth coverings,", "and the set $\\text{Cov}_0$ chosen above.", "\\end{enumerate}" ], "refs": [ "sites-definition-site", "sets-lemma-construct-category", "sets-lemma-coverings-site" ], "ref_ids": [ 8652, 8789, 8800 ] }, { "id": 12534, "type": "definition", "label": "topologies-definition-big-small-smooth", "categories": [ "topologies" ], "title": "topologies-definition-big-small-smooth", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{smooth}$ be a big smooth", "site containing $S$.", "\\begin{enumerate}", "\\item The {\\it big smooth site of $S$}, denoted", "$(\\Sch/S)_{smooth}$, is the site $\\Sch_{smooth}/S$", "introduced in Sites, Section \\ref{sites-section-localize}.", "\\item The {\\it big affine smooth site of $S$}, denoted", "$(\\textit{Aff}/S)_{smooth}$, is the full subcategory of", "$(\\Sch/S)_{smooth}$ whose objects are affine $U/S$.", "A covering of $(\\textit{Aff}/S)_{smooth}$ is any covering", "$\\{U_i \\to U\\}$ of $(\\Sch/S)_{smooth}$ which is a", "standard smooth covering.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12535, "type": "definition", "label": "topologies-definition-syntomic-covering", "categories": [ "topologies" ], "title": "topologies-definition-syntomic-covering", "contents": [ "Let $T$ be a scheme. An {\\it syntomic covering of $T$} is a family", "of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes", "such that each $f_i$ is syntomic and such", "that $T = \\bigcup f_i(T_i)$." ], "refs": [], "ref_ids": [] }, { "id": 12536, "type": "definition", "label": "topologies-definition-standard-syntomic", "categories": [ "topologies" ], "title": "topologies-definition-standard-syntomic", "contents": [ "Let $T$ be an affine scheme. A {\\it standard syntomic covering} of $T$ is", "a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ with each $U_j$ is", "affine, $U_j \\to T$ standard syntomic and $T = \\bigcup f_j(U_j)$." ], "refs": [], "ref_ids": [] }, { "id": 12537, "type": "definition", "label": "topologies-definition-big-syntomic-site", "categories": [ "topologies" ], "title": "topologies-definition-big-syntomic-site", "contents": [ "A {\\it big syntomic site} is any site $\\Sch_{syntomic}$ as in", "Sites, Definition \\ref{sites-definition-site} constructed as follows:", "\\begin{enumerate}", "\\item Choose any set of schemes $S_0$, and any set of syntomic coverings", "$\\text{Cov}_0$ among these schemes.", "\\item As underlying category take any category $\\Sch_\\alpha$", "constructed as in Sets, Lemma \\ref{sets-lemma-construct-category}", "starting with the set $S_0$.", "\\item Choose any set of coverings as in", "Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the", "category $\\Sch_\\alpha$ and the class of syntomic coverings,", "and the set $\\text{Cov}_0$ chosen above.", "\\end{enumerate}" ], "refs": [ "sites-definition-site", "sets-lemma-construct-category", "sets-lemma-coverings-site" ], "ref_ids": [ 8652, 8789, 8800 ] }, { "id": 12538, "type": "definition", "label": "topologies-definition-big-small-syntomic", "categories": [ "topologies" ], "title": "topologies-definition-big-small-syntomic", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{syntomic}$ be a big syntomic", "site containing $S$.", "\\begin{enumerate}", "\\item The {\\it big syntomic site of $S$}, denoted", "$(\\Sch/S)_{syntomic}$, is the site $\\Sch_{syntomic}/S$", "introduced in Sites, Section \\ref{sites-section-localize}.", "\\item The {\\it big affine syntomic site of $S$}, denoted", "$(\\textit{Aff}/S)_{syntomic}$, is the full subcategory of", "$(\\Sch/S)_{syntomic}$ whose objects are affine $U/S$.", "A covering of $(\\textit{Aff}/S)_{syntomic}$ is any covering", "$\\{U_i \\to U\\}$ of $(\\Sch/S)_{syntomic}$ which is a", "standard syntomic covering.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12539, "type": "definition", "label": "topologies-definition-fppf-covering", "categories": [ "topologies" ], "title": "topologies-definition-fppf-covering", "contents": [ "Let $T$ be a scheme. An {\\it fppf covering of $T$} is a family", "of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes", "such that each $f_i$ is flat, locally of finite presentation and such", "that $T = \\bigcup f_i(T_i)$." ], "refs": [], "ref_ids": [] }, { "id": 12540, "type": "definition", "label": "topologies-definition-standard-fppf", "categories": [ "topologies" ], "title": "topologies-definition-standard-fppf", "contents": [ "Let $T$ be an affine scheme. A {\\it standard fppf covering}", "of $T$ is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$", "with each $U_j$ is affine, flat and of finite presentation over $T$", "and $T = \\bigcup f_j(U_j)$." ], "refs": [], "ref_ids": [] }, { "id": 12541, "type": "definition", "label": "topologies-definition-big-fppf-site", "categories": [ "topologies" ], "title": "topologies-definition-big-fppf-site", "contents": [ "A {\\it big fppf site} is any site $\\Sch_{fppf}$ as in", "Sites, Definition \\ref{sites-definition-site} constructed as follows:", "\\begin{enumerate}", "\\item Choose any set of schemes $S_0$, and any set of fppf coverings", "$\\text{Cov}_0$ among these schemes.", "\\item As underlying category take any category $\\Sch_\\alpha$", "constructed as in Sets, Lemma \\ref{sets-lemma-construct-category}", "starting with the set $S_0$.", "\\item Choose any set of coverings as in", "Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the", "category $\\Sch_\\alpha$ and the class of fppf coverings,", "and the set $\\text{Cov}_0$ chosen above.", "\\end{enumerate}" ], "refs": [ "sites-definition-site", "sets-lemma-construct-category", "sets-lemma-coverings-site" ], "ref_ids": [ 8652, 8789, 8800 ] }, { "id": 12542, "type": "definition", "label": "topologies-definition-big-small-fppf", "categories": [ "topologies" ], "title": "topologies-definition-big-small-fppf", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{fppf}$ be a big fppf", "site containing $S$.", "\\begin{enumerate}", "\\item The {\\it big fppf site of $S$}, denoted", "$(\\Sch/S)_{fppf}$, is the site $\\Sch_{fppf}/S$", "introduced in Sites, Section \\ref{sites-section-localize}.", "\\item The {\\it big affine fppf site of $S$}, denoted", "$(\\textit{Aff}/S)_{fppf}$, is the full subcategory of", "$(\\Sch/S)_{fppf}$ whose objects are affine $U/S$.", "A covering of $(\\textit{Aff}/S)_{fppf}$ is any covering", "$\\{U_i \\to U\\}$ of $(\\Sch/S)_{fppf}$ which is a", "standard fppf covering.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12543, "type": "definition", "label": "topologies-definition-standard-ph-covering", "categories": [ "topologies" ], "title": "topologies-definition-standard-ph-covering", "contents": [ "Let $T$ be an affine scheme. A {\\it standard ph covering} is a family", "$\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ constructed from a", "proper surjective morphism $f : U \\to T$ and an affine open covering", "$U = \\bigcup_{j = 1, \\ldots, m} U_j$ by setting $f_j = f|_{U_j}$." ], "refs": [], "ref_ids": [] }, { "id": 12544, "type": "definition", "label": "topologies-definition-ph-covering", "categories": [ "topologies" ], "title": "topologies-definition-ph-covering", "contents": [ "Let $T$ be a scheme. A {\\it ph covering of $T$} is a family", "of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes such", "that $f_i$ is locally of finite type and such that for every", "affine open $U \\subset T$ there exists a standard ph covering", "$\\{U_j \\to U\\}_{j = 1, \\ldots, m}$ refining the family", "$\\{T_i \\times_T U \\to U\\}_{i \\in I}$." ], "refs": [], "ref_ids": [] }, { "id": 12545, "type": "definition", "label": "topologies-definition-big-ph-site", "categories": [ "topologies" ], "title": "topologies-definition-big-ph-site", "contents": [ "A {\\it big ph site} is any site $\\Sch_{ph}$ as in", "Sites, Definition \\ref{sites-definition-site} constructed as follows:", "\\begin{enumerate}", "\\item Choose any set of schemes $S_0$, and any set of ph coverings", "$\\text{Cov}_0$ among these schemes.", "\\item As underlying category take any category $\\Sch_\\alpha$", "constructed as in Sets, Lemma \\ref{sets-lemma-construct-category}", "starting with the set $S_0$.", "\\item Choose any set of coverings as in", "Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the", "category $\\Sch_\\alpha$ and the class of ph coverings,", "and the set $\\text{Cov}_0$ chosen above.", "\\end{enumerate}" ], "refs": [ "sites-definition-site", "sets-lemma-construct-category", "sets-lemma-coverings-site" ], "ref_ids": [ 8652, 8789, 8800 ] }, { "id": 12546, "type": "definition", "label": "topologies-definition-big-small-ph", "categories": [ "topologies" ], "title": "topologies-definition-big-small-ph", "contents": [ "Let $S$ be a scheme. Let $\\Sch_{ph}$ be a big ph site containing $S$.", "\\begin{enumerate}", "\\item The {\\it big ph site of $S$}, denoted", "$(\\Sch/S)_{ph}$, is the site $\\Sch_{ph}/S$", "introduced in Sites, Section \\ref{sites-section-localize}.", "\\item The {\\it big affine ph site of $S$}, denoted", "$(\\textit{Aff}/S)_{ph}$, is the full subcategory of", "$(\\Sch/S)_{ph}$ whose objects are affine $U/S$.", "A covering of $(\\textit{Aff}/S)_{ph}$ is any finite covering", "$\\{U_i \\to U\\}$ of $(\\Sch/S)_{ph}$ with $U_i$ and $U$ affine.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12547, "type": "definition", "label": "topologies-definition-fpqc-covering", "categories": [ "topologies" ], "title": "topologies-definition-fpqc-covering", "contents": [ "Let $T$ be a scheme. An {\\it fpqc covering of $T$} is a family", "of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes", "such that each $f_i$ is flat and such that for every affine open", "$U \\subset T$ there exists $n \\geq 0$, a map", "$a : \\{1, \\ldots, n\\} \\to I$ and affine opens", "$V_j \\subset T_{a(j)}$, $j = 1, \\ldots, n$", "with $\\bigcup_{j = 1}^n f_{a(j)}(V_j) = U$." ], "refs": [], "ref_ids": [] }, { "id": 12548, "type": "definition", "label": "topologies-definition-standard-fpqc", "categories": [ "topologies" ], "title": "topologies-definition-standard-fpqc", "contents": [ "Let $T$ be an affine scheme. A {\\it standard fpqc covering}", "of $T$ is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, n}$", "with each $U_j$ is affine, flat over $T$ and $T = \\bigcup f_j(U_j)$." ], "refs": [], "ref_ids": [] }, { "id": 12549, "type": "definition", "label": "topologies-definition-sheaf-property-fpqc", "categories": [ "topologies" ], "title": "topologies-definition-sheaf-property-fpqc", "contents": [ "Let $F$ be a contravariant functor on the category", "of schemes with values in sets.", "\\begin{enumerate}", "\\item Let $\\{U_i \\to T\\}_{i \\in I}$ be a family of morphisms", "of schemes with fixed target.", "We say that $F$ {\\it satisfies the sheaf property for the given family}", "if for any collection of elements $\\xi_i \\in F(U_i)$ such that", "$\\xi_i|_{U_i \\times_T U_j} = \\xi_j|_{U_i \\times_T U_j}$", "there exists a unique element", "$\\xi \\in F(T)$ such that $\\xi_i = \\xi|_{U_i}$ in $F(U_i)$.", "\\item We say that $F$ {\\it satisfies the sheaf property for the", "fpqc topology} if it satisfies the sheaf property for any", "fpqc covering.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12550, "type": "definition", "label": "topologies-definition-standard-V-covering", "categories": [ "topologies" ], "title": "topologies-definition-standard-V-covering", "contents": [ "Let $T$ be an affine scheme. A {\\it standard V covering} is a finite family", "$\\{T_j \\to T\\}_{j = 1, \\ldots, m}$ with $T_j$ affine", "such that for every morphism $g : \\Spec(V) \\to T$ where $V$", "is a valuation ring, there is an extension $V \\subset W$ of valuation rings", "(More on Algebra, Definition", "\\ref{more-algebra-definition-extension-valuation-rings}),", "an index $1 \\leq j \\leq m$, and a commutative diagram", "$$", "\\xymatrix{", "\\Spec(W) \\ar[r] \\ar[d] & T_j \\ar[d] \\\\", "\\Spec(V) \\ar[r]^g & T", "}", "$$" ], "refs": [ "more-algebra-definition-extension-valuation-rings" ], "ref_ids": [ 10647 ] }, { "id": 12551, "type": "definition", "label": "topologies-definition-V-covering", "categories": [ "topologies" ], "title": "topologies-definition-V-covering", "contents": [ "Let $T$ be a scheme. A {\\it V covering of $T$} is a family", "of morphisms $\\{T_i \\to T\\}_{i \\in I}$ of schemes such that for every", "affine open $U \\subset T$ there exists a standard V covering", "$\\{U_j \\to U\\}_{j = 1, \\ldots, m}$ refining the family", "$\\{T_i \\times_T U \\to U\\}_{i \\in I}$." ], "refs": [], "ref_ids": [] }, { "id": 12552, "type": "definition", "label": "topologies-definition-sheaf-property-V", "categories": [ "topologies" ], "title": "topologies-definition-sheaf-property-V", "contents": [ "Let $F$ be a contravariant functor on the category", "of schemes with values in sets. We say that", "$F$ {\\it satisfies the sheaf property for the V topology}", "if it satisfies the sheaf property for any V covering", "(see Definition \\ref{definition-sheaf-property-fpqc})." ], "refs": [ "topologies-definition-sheaf-property-fpqc" ], "ref_ids": [ 12549 ] }, { "id": 12577, "type": "definition", "label": "pic-definition-picard-functor", "categories": [ "pic" ], "title": "pic-definition-picard-functor", "contents": [ "Let $\\Sch_{fppf}$ be a big site as in", "Topologies, Definition \\ref{topologies-definition-big-small-fppf}.", "Let $f : X \\to S$ be a morphism of this site. The {\\it Picard functor}", "$\\Picardfunctor_{X/S}$ is the fppf sheafification of the functor", "$$", "(\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets},\\quad", "T \\longmapsto \\Pic(X_T)", "$$", "If this functor is representable, then we denote", "$\\underline{\\Picardfunctor}_{X/S}$ a scheme representing it." ], "refs": [ "topologies-definition-big-small-fppf" ], "ref_ids": [ 12542 ] }, { "id": 12578, "type": "definition", "label": "pic-definition-genus", "categories": [ "pic" ], "title": "pic-definition-genus", "contents": [ "Let $k$ be a field. Let $X$ be a smooth projective geometrically irreducible", "curve over $k$. The {\\it genus} of $X$ is $g = \\dim_k H^1(X, \\mathcal{O}_X)$." ], "refs": [], "ref_ids": [] }, { "id": 12655, "type": "definition", "label": "constructions-definition-relative-spec", "categories": [ "constructions" ], "title": "constructions-definition-relative-spec", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent sheaf of", "$\\mathcal{O}_S$-algebras. The {\\it relative spectrum of $\\mathcal{A}$ over", "$S$}, or simply the {\\it spectrum of $\\mathcal{A}$ over $S$} is the scheme", "constructed in Lemma \\ref{lemma-glue-relative-spec} which represents the", "functor $F$ (\\ref{equation-spec}), see", "Lemma \\ref{lemma-glueing-gives-functor-spec}.", "We denote it $\\pi : \\underline{\\Spec}_S(\\mathcal{A}) \\to S$.", "The ``universal family'' is a morphism of $\\mathcal{O}_S$-algebras", "$$", "\\mathcal{A}", "\\longrightarrow", "\\pi_*\\mathcal{O}_{\\underline{\\Spec}_S(\\mathcal{A})}", "$$" ], "refs": [ "constructions-lemma-glue-relative-spec", "constructions-lemma-glueing-gives-functor-spec" ], "ref_ids": [ 12585, 12589 ] }, { "id": 12656, "type": "definition", "label": "constructions-definition-affine-n-space", "categories": [ "constructions" ], "title": "constructions-definition-affine-n-space", "contents": [ "Let $S$ be a scheme and $n \\geq 0$.", "The scheme", "$$", "\\mathbf{A}^n_S =", "\\underline{\\Spec}_S(\\mathcal{O}_S[T_1, \\ldots, T_n])", "$$", "over $S$ is called {\\it affine $n$-space over $S$}.", "If $S = \\Spec(R)$ is affine then we also call this", "{\\it affine $n$-space over $R$} and we denote it $\\mathbf{A}^n_R$." ], "refs": [], "ref_ids": [] }, { "id": 12657, "type": "definition", "label": "constructions-definition-vector-bundle", "categories": [ "constructions" ], "title": "constructions-definition-vector-bundle", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{E}$ be a quasi-coherent", "$\\mathcal{O}_S$-module\\footnote{The reader may expect here", "the condition that $\\mathcal{E}$ is finite locally free. We do not", "do so in order to be consistent with \\cite[II, Definition 1.7.8]{EGA}.}.", "The {\\it vector bundle associated to $\\mathcal{E}$} is", "$$", "\\mathbf{V}(\\mathcal{E}) = \\underline{\\Spec}_S(\\text{Sym}(\\mathcal{E})).", "$$" ], "refs": [], "ref_ids": [] }, { "id": 12658, "type": "definition", "label": "constructions-definition-abstract-vector-bundle", "categories": [ "constructions" ], "title": "constructions-definition-abstract-vector-bundle", "contents": [ "Let $S$ be a scheme. A {\\it vector bundle $\\pi : V \\to S$ over $S$} is an", "affine morphism of schemes such that $\\pi_*\\mathcal{O}_V$ is endowed with", "the structure of a graded $\\mathcal{O}_S$-algebra", "$\\pi_*\\mathcal{O}_V = \\bigoplus\\nolimits_{n \\geq 0} \\mathcal{E}_n$", "such that $\\mathcal{E}_0 = \\mathcal{O}_S$ and such that the maps", "$$", "\\text{Sym}^n(\\mathcal{E}_1) \\longrightarrow \\mathcal{E}_n", "$$", "are isomorphisms for all $n \\geq 0$. A {\\it morphism of vector bundles", "over $S$} is a morphism $f : V \\to V'$ such that the induced map", "$$", "f^* : \\pi'_*\\mathcal{O}_{V'} \\longrightarrow \\pi_*\\mathcal{O}_V", "$$", "is compatible with the given gradings." ], "refs": [], "ref_ids": [] }, { "id": 12659, "type": "definition", "label": "constructions-definition-cone", "categories": [ "constructions" ], "title": "constructions-definition-cone", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent", "graded $\\mathcal{O}_S$-algebra. Assume that $\\mathcal{O}_S \\to \\mathcal{A}_0$", "is an isomorphism\\footnote{Often one imposes the assumption that", "$\\mathcal{A}$ is generated by $\\mathcal{A}_1$ over $\\mathcal{O}_S$. We do not", "assume this in order to be consistent with \\cite[II, (8.3.1)]{EGA}.}.", "The {\\it cone associated to $\\mathcal{A}$} or the", "{\\it affine cone associated to $\\mathcal{A}$}", "is", "$$", "C(\\mathcal{A}) = \\underline{\\Spec}_S(\\mathcal{A}).", "$$" ], "refs": [], "ref_ids": [] }, { "id": 12660, "type": "definition", "label": "constructions-definition-abstract-cone", "categories": [ "constructions" ], "title": "constructions-definition-abstract-cone", "contents": [ "Let $S$ be a scheme. A {\\it cone $\\pi : C \\to S$ over $S$} is an", "affine morphism of schemes such that $\\pi_*\\mathcal{O}_C$ is endowed with", "the structure of a graded $\\mathcal{O}_S$-algebra", "$\\pi_*\\mathcal{O}_C = \\bigoplus\\nolimits_{n \\geq 0} \\mathcal{A}_n$", "such that $\\mathcal{A}_0 = \\mathcal{O}_S$. A {\\it morphism of cones}", "from $\\pi : C \\to S$ to $\\pi' : C' \\to S$", "is a morphism $f : C \\to C'$ such that the induced map", "$$", "f^* : \\pi'_*\\mathcal{O}_{C'} \\longrightarrow \\pi_*\\mathcal{O}_C", "$$", "is compatible with the given gradings." ], "refs": [], "ref_ids": [] }, { "id": 12661, "type": "definition", "label": "constructions-definition-standard-covering", "categories": [ "constructions" ], "title": "constructions-definition-standard-covering", "contents": [ "Let $S$ be a graded ring.", "Suppose that $D_{+}(f) \\subset \\text{Proj}(S)$ is a standard", "open. A {\\it standard open covering} of $D_{+}(f)$", "is a covering $D_{+}(f) = \\bigcup_{i = 1}^n D_{+}(g_i)$,", "where $g_1, \\ldots, g_n \\in S$ are homogeneous of positive degree." ], "refs": [], "ref_ids": [] }, { "id": 12662, "type": "definition", "label": "constructions-definition-structure-sheaf", "categories": [ "constructions" ], "title": "constructions-definition-structure-sheaf", "contents": [ "Let $S$ be a graded ring.", "\\begin{enumerate}", "\\item The {\\it structure sheaf $\\mathcal{O}_{\\text{Proj}(S)}$ of the", "homogeneous spectrum of $S$} is the unique sheaf of rings", "$\\mathcal{O}_{\\text{Proj}(S)}$", "which agrees with $\\widetilde S$ on the basis of standard opens.", "\\item The locally ringed space", "$(\\text{Proj}(S), \\mathcal{O}_{\\text{Proj}(S)})$ is called", "the {\\it homogeneous spectrum} of $S$ and denoted $\\text{Proj}(S)$.", "\\item The sheaf of $\\mathcal{O}_{\\text{Proj}(S)}$-modules", "extending $\\widetilde M$ to all opens of $\\text{Proj}(S)$", "is called the sheaf of $\\mathcal{O}_{\\text{Proj}(S)}$-modules", "associated to $M$. This sheaf is denoted $\\widetilde M$ as", "well.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12663, "type": "definition", "label": "constructions-definition-twist", "categories": [ "constructions" ], "title": "constructions-definition-twist", "contents": [ "Let $S$ be a graded ring. Let $X = \\text{Proj}(S)$.", "\\begin{enumerate}", "\\item We define $\\mathcal{O}_X(n) = \\widetilde{S(n)}$.", "This is called the $n$th", "{\\it twist of the structure sheaf of $\\text{Proj}(S)$}.", "\\item For any sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ we set", "$\\mathcal{F}(n) = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(n)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12664, "type": "definition", "label": "constructions-definition-projective-space", "categories": [ "constructions" ], "title": "constructions-definition-projective-space", "contents": [ "The scheme", "$\\mathbf{P}^n_{\\mathbf{Z}} = \\text{Proj}(\\mathbf{Z}[T_0, \\ldots, T_n])$", "is called {\\it projective $n$-space over $\\mathbf{Z}$}.", "Its base change $\\mathbf{P}^n_S$ to a scheme $S$ is called", "{\\it projective $n$-space over $S$}. If $R$ is a ring the base change", "to $\\Spec(R)$ is denoted $\\mathbf{P}^n_R$ and called", "{\\it projective $n$-space over $R$}." ], "refs": [], "ref_ids": [] }, { "id": 12665, "type": "definition", "label": "constructions-definition-relative-proj", "categories": [ "constructions" ], "title": "constructions-definition-relative-proj", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent sheaf of", "graded $\\mathcal{O}_S$-algebras. The", "{\\it relative homogeneous spectrum of $\\mathcal{A}$ over $S$},", "or the {\\it homogeneous spectrum of $\\mathcal{A}$ over $S$}, or the", "{\\it relative Proj of $\\mathcal{A}$ over $S$} is the scheme", "constructed in Lemma \\ref{lemma-glue-relative-proj} which represents the", "functor $F$ (\\ref{equation-proj}), see", "Lemma \\ref{lemma-glueing-gives-functor-proj}.", "We denote it $\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$." ], "refs": [ "constructions-lemma-glue-relative-proj", "constructions-lemma-glueing-gives-functor-proj" ], "ref_ids": [ 12631, 12638 ] }, { "id": 12666, "type": "definition", "label": "constructions-definition-projective-bundle", "categories": [ "constructions" ], "title": "constructions-definition-projective-bundle", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{E}$ be a quasi-coherent", "$\\mathcal{O}_S$-module\\footnote{The reader may expect here", "the condition that $\\mathcal{E}$ is finite locally free. We do not", "do so in order to be consistent with", "\\cite[II, Definition 4.1.1]{EGA}.}.", "We denote", "$$", "\\pi :", "\\mathbf{P}(\\mathcal{E}) = \\underline{\\text{Proj}}_S(\\text{Sym}(\\mathcal{E}))", "\\longrightarrow", "S", "$$", "and we call it the {\\it projective bundle associated to $\\mathcal{E}$}.", "The symbol $\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(n)$", "indicates the invertible $\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}$-module", "of Lemma \\ref{lemma-apply-relative} and is called the $n$th", "{\\it twist of the structure sheaf}." ], "refs": [ "constructions-lemma-apply-relative" ], "ref_ids": [ 12642 ] }, { "id": 12667, "type": "definition", "label": "constructions-definition-grassmannian", "categories": [ "constructions" ], "title": "constructions-definition-grassmannian", "contents": [ "Let $0 < k < n$. The scheme $\\mathbf{G}(k, n)$ representing the functor", "$G(k, n)$ is called {\\it Grassmannian over $\\mathbf{Z}$}.", "Its base change $\\mathbf{G}(k, n)_S$ to a scheme $S$ is called", "{\\it Grassmannian over $S$}. If $R$ is a ring the base change", "to $\\Spec(R)$ is denoted $\\mathbf{G}(k, n)_R$ and called", "{\\it Grassmannian over $R$}." ], "refs": [], "ref_ids": [] }, { "id": 12803, "type": "definition", "label": "algebraization-definition-derived-complete", "categories": [ "algebraization" ], "title": "algebraization-definition-derived-complete", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{I} \\subset \\mathcal{O}$ be a sheaf of ideals.", "Let $K \\in D(\\mathcal{O})$. We say that $K$ is", "{\\it derived complete with respect to $\\mathcal{I}$}", "if for every object $U$ of $\\mathcal{C}$ and $f \\in \\mathcal{I}(U)$", "the object $T(K|_U, f)$ of $D(\\mathcal{O}_U)$ is zero." ], "refs": [], "ref_ids": [] }, { "id": 12804, "type": "definition", "label": "algebraization-definition-algebraizable", "categories": [ "algebraization" ], "title": "algebraization-definition-algebraizable", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an", "object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. We say", "{\\it $(\\mathcal{F}_n)$ extends to $X$} if there exists an object", "$(\\mathcal{G}_n)$ of $\\textit{Coh}(X, I\\mathcal{O}_X)$ whose restriction", "to $U$ is isomorphic to $(\\mathcal{F}_n)$." ], "refs": [], "ref_ids": [] }, { "id": 12805, "type": "definition", "label": "algebraization-definition-canonically-algebraizable", "categories": [ "algebraization" ], "title": "algebraization-definition-canonically-algebraizable", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an", "object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. We say", "{\\it $(\\mathcal{F}_n)$ canonically extends to $X$} if the the", "inverse system", "$$", "\\{\\widetilde{H^0(U, \\mathcal{F}_n)}\\}_{n \\geq 1}", "$$", "in $\\QCoh(\\mathcal{O}_X)$ is pro-isomorphic to an object", "$(\\mathcal{G}_n)$ of $\\textit{Coh}(X, I\\mathcal{O}_X)$." ], "refs": [], "ref_ids": [] }, { "id": 12806, "type": "definition", "label": "algebraization-definition-s-d-inequalities", "categories": [ "algebraization" ], "title": "algebraization-definition-s-d-inequalities", "contents": [ "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object", "of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $a, b$ be integers.", "Let $\\delta^Y_Z$ be as in (\\ref{equation-delta-Z}).", "We say", "{\\it $(\\mathcal{F}_n)$ satisfies the $(a, b)$-inequalities} if for", "$y \\in U \\cap Y$ and a prime $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$", "with $\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$", "\\begin{enumerate}", "\\item if $V(\\mathfrak p) \\cap V(I\\mathcal{O}_{X, y}^\\wedge) \\not =", "\\{\\mathfrak m_y^\\wedge\\}$, then", "$$", "\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\delta^Y_Z(y) \\geq a", "\\quad\\text{or}\\quad", "\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > b", "$$", "\\item if $V(\\mathfrak p) \\cap V(I\\mathcal{O}_{X, y}^\\wedge) =", "\\{\\mathfrak m_y^\\wedge\\}$, then", "$$", "\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\delta^Y_Z(y) > a", "$$", "\\end{enumerate}", "We say {\\it $(\\mathcal{F}_n)$ satisfies the strict $(a, b)$-inequalities}", "if for $y \\in U \\cap Y$ and a prime", "$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with", "$\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$", "we have", "$$", "\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\delta^Y_Z(y) > a", "\\quad\\text{or}\\quad", "\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) +", "\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > b", "$$" ], "refs": [], "ref_ids": [] }, { "id": 12883, "type": "definition", "label": "spaces-over-fields-definition-integral-algebraic-space", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-integral-algebraic-space", "contents": [ "Let $S$ be a scheme. We say an algebraic space $X$ over $S$ is", "{\\it integral} if it is reduced, decent, and $|X|$ is irreducible." ], "refs": [], "ref_ids": [] }, { "id": 12884, "type": "definition", "label": "spaces-over-fields-definition-function-field", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-function-field", "contents": [ "Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$.", "The {\\it function field}, or the {\\it field of rational functions}", "of $X$ is the field $R(X)$ of", "Lemma \\ref{lemma-integral-algebraic-space-rational-functions}." ], "refs": [ "spaces-over-fields-lemma-integral-algebraic-space-rational-functions" ], "ref_ids": [ 12824 ] }, { "id": 12885, "type": "definition", "label": "spaces-over-fields-definition-degree", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-degree", "contents": [ "Let $S$ be a scheme.", "Let $X$ and $Y$ be integral algebraic spaces over $S$.", "Let $f : X \\to Y$ be locally of finite type and dominant.", "Assume any of the equivalent conditions (1) -- (5) of", "Lemma \\ref{lemma-finite-degree}. Let $x \\in |X|$ and $y \\in |Y|$", "be the generic points. Then the positive integer", "$$", "\\text{deg}(X/Y) = [\\kappa(x) : \\kappa(y)]", "$$", "is called the {\\it degree of $X$ over $Y$}." ], "refs": [ "spaces-over-fields-lemma-finite-degree" ], "ref_ids": [ 12829 ] }, { "id": 12886, "type": "definition", "label": "spaces-over-fields-definition-Weil-divisor", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-Weil-divisor", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a locally Noetherian integral algebraic space over $S$.", "\\begin{enumerate}", "\\item A {\\it prime divisor} is an integral closed subspace $Z \\subset X$", "of codimension $1$, i.e., the generic point of $|Z|$ is a point", "of codimension $1$ on $X$.", "\\item A {\\it Weil divisor} is a formal sum $D = \\sum n_Z Z$ where", "the sum is over prime divisors of $X$ and the collection", "$\\{|Z| : n_Z \\not = 0\\}$ is locally finite in $|X|$", "(Topology, Definition \\ref{topology-definition-locally-finite}).", "\\end{enumerate}", "The group of all Weil divisors on $X$ is denoted $\\text{Div}(X)$." ], "refs": [ "topology-definition-locally-finite" ], "ref_ids": [ 8376 ] }, { "id": 12887, "type": "definition", "label": "spaces-over-fields-definition-order-vanishing", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-order-vanishing", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic", "space over $S$. Let $f \\in R(X)^*$. For every prime divisor", "$Z \\subset X$ we define the {\\it order of vanishing of $f$ along $Z$}", "as the integer", "$$", "\\text{ord}_Z(f) =", "\\text{length}_{\\mathcal{O}_{X, \\xi}^h}", "(\\mathcal{O}_{X, \\xi}^h/a \\mathcal{O}_{X, \\xi}^h) -", "\\text{length}_{\\mathcal{O}_{X, \\xi}^h}", "(\\mathcal{O}_{X, \\xi}^h/b \\mathcal{O}_{X, \\xi}^h)", "$$", "where $a, b \\in \\mathcal{O}_{X, \\xi}^h$ are nonzerodivisors", "such that the image of $f$ in $Q(\\mathcal{O}_{X, \\xi}^h)$", "(Lemma \\ref{lemma-order-vanishing}) is equal to $a/b$.", "This is well defined by", "Algebra, Lemma \\ref{algebra-lemma-ord-additive}." ], "refs": [ "spaces-over-fields-lemma-order-vanishing", "algebra-lemma-ord-additive" ], "ref_ids": [ 12832, 1043 ] }, { "id": 12888, "type": "definition", "label": "spaces-over-fields-definition-principal-divisor", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-principal-divisor", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a locally Noetherian integral algebraic space over $S$.", "Let $f \\in R(X)^*$.", "The {\\it principal Weil divisor associated to $f$} is the Weil divisor", "$$", "\\text{div}(f) = \\text{div}_X(f) = \\sum \\text{ord}_Z(f) [Z]", "$$", "where the sum is over prime divisors and $\\text{ord}_Z(f)$ is as in", "Definition \\ref{definition-order-vanishing}. This makes sense", "by Lemma \\ref{lemma-divisor-locally-finite}." ], "refs": [ "spaces-over-fields-definition-order-vanishing", "spaces-over-fields-lemma-divisor-locally-finite" ], "ref_ids": [ 12887, 12834 ] }, { "id": 12889, "type": "definition", "label": "spaces-over-fields-definition-class-group", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-class-group", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a locally Noetherian integral algebraic space over $S$. The", "{\\it Weil divisor class group} of $X$ is the quotient of", "the group of Weil divisors by the subgroup of principal Weil divisors.", "Notation: $\\text{Cl}(X)$." ], "refs": [], "ref_ids": [] }, { "id": 12890, "type": "definition", "label": "spaces-over-fields-definition-order-vanishing-meromorphic", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-order-vanishing-meromorphic", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral", "algebraic algebraic space over $S$. Let $\\mathcal{L}$ be an", "invertible $\\mathcal{O}_X$-module.", "Let $s \\in \\Gamma(X, \\mathcal{K}_X(\\mathcal{L}))$", "be a regular meromorphic section of $\\mathcal{L}$.", "For every prime divisor $Z \\subset X$ with generic point $\\xi \\in |Z|$", "we define the", "{\\it order of vanishing of $s$ along $Z$}", "as the integer", "$$", "\\text{ord}_{Z, \\mathcal{L}}(s) =", "\\text{length}_{\\mathcal{O}_{X, \\xi}^h}", "(\\mathcal{O}_{X, \\xi}^h/a \\mathcal{O}_{X, \\xi}^h) -", "\\text{length}_{\\mathcal{O}_{X, \\xi}^h}", "(\\mathcal{O}_{X, \\xi}^h/b \\mathcal{O}_{X, \\xi}^h)", "$$", "where $a, b \\in \\mathcal{O}_{X, \\xi}^h$ are nonzerodivisors", "such that the element $s/s_\\xi$ of $Q(\\mathcal{O}_{X, \\xi}^h)$", "constructed above is equal to $a/b$.", "This is well defined by the above and", "Algebra, Lemma \\ref{algebra-lemma-ord-additive}." ], "refs": [ "algebra-lemma-ord-additive" ], "ref_ids": [ 1043 ] }, { "id": 12891, "type": "definition", "label": "spaces-over-fields-definition-divisor-invertible-sheaf", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-divisor-invertible-sheaf", "contents": [ "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space", "over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "\\begin{enumerate}", "\\item For any nonzero meromorphic section $s$ of $\\mathcal{L}$", "we define the {\\it Weil divisor associated to $s$} as", "$$", "\\text{div}_\\mathcal{L}(s) =", "\\sum \\text{ord}_{Z, \\mathcal{L}}(s) [Z] \\in \\text{Div}(X)", "$$", "where the sum is over prime divisors. This is well defined by", "Lemma \\ref{lemma-divisor-meromorphic-locally-finite}.", "\\item We define {\\it Weil divisor class associated to $\\mathcal{L}$}", "as the image of $\\text{div}_\\mathcal{L}(s)$ in $\\text{Cl}(X)$", "where $s$ is any nonzero meromorphic section of $\\mathcal{L}$ over $X$.", "This is well defined by", "Lemma \\ref{lemma-divisor-meromorphic-well-defined}.", "\\end{enumerate}" ], "refs": [ "spaces-over-fields-lemma-divisor-meromorphic-locally-finite", "spaces-over-fields-lemma-divisor-meromorphic-well-defined" ], "ref_ids": [ 12836, 12837 ] }, { "id": 12892, "type": "definition", "label": "spaces-over-fields-definition-modification", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-modification", "contents": [ "Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. A", "{\\it modification of $X$} is a birational proper morphism", "$f : X' \\to X$ of algebraic spaces over $S$ with $X'$ integral." ], "refs": [], "ref_ids": [] }, { "id": 12893, "type": "definition", "label": "spaces-over-fields-definition-alteration", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-alteration", "contents": [ "Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$.", "An {\\it alteration of $X$} is a proper dominant morphism $f : Y \\to X$", "of algebraic spaces over $S$ with $Y$ integral such that $f^{-1}(U) \\to U$", "is finite for some nonempty open $U \\subset X$." ], "refs": [], "ref_ids": [] }, { "id": 12894, "type": "definition", "label": "spaces-over-fields-definition-geometrically-reduced", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-geometrically-reduced", "contents": [ "Let $k$ be a field.", "Let $X$ be an algebraic space over $k$.", "\\begin{enumerate}", "\\item Let $x \\in |X|$ be a point. We say $X$ is", "{\\it geometrically reduced at $x$} if $\\mathcal{O}_{X, \\overline{x}}$", "is geometrically reduced over $k$.", "\\item We say $X$ is {\\it geometrically reduced} over $k$", "if $X$ is geometrically reduced at every point of $X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 12895, "type": "definition", "label": "spaces-over-fields-definition-geometrically-connected", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-geometrically-connected", "contents": [ "Let $X$ be an algebraic space over the field $k$. We say $X$ is", "{\\it geometrically connected} over $k$ if the base change $X_{k'}$", "is connected for every field extension $k'$ of $k$." ], "refs": [], "ref_ids": [] }, { "id": 12896, "type": "definition", "label": "spaces-over-fields-definition-geometrically-irreducible", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-geometrically-irreducible", "contents": [ "Let $k$ be a field. ", "Let $X$ be a decent algebraic space over $k$.", "We say $X$ is {\\it geometrically irreducible} if the", "topological space $|X_{k'}|$ is", "irreducible\\footnote{An irreducible space is nonempty.}", "for any field extension $k'$ of $k$." ], "refs": [], "ref_ids": [] }, { "id": 12897, "type": "definition", "label": "spaces-over-fields-definition-geometrically-integral", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-geometrically-integral", "contents": [ "Let $X$ be an algebraic space over the field $k$. We say $X$ is", "{\\it geometrically integral} over $k$ if the algebraic space", "$X_{k'}$ is integral (Definition \\ref{definition-integral-algebraic-space})", "for every field extension $k'$ of $k$." ], "refs": [ "spaces-over-fields-definition-integral-algebraic-space" ], "ref_ids": [ 12883 ] }, { "id": 12898, "type": "definition", "label": "spaces-over-fields-definition-euler-characteristic", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-euler-characteristic", "contents": [ "Let $k$ be a field. Let $X$ be a proper algebraic over $k$. Let $\\mathcal{F}$", "be a coherent $\\mathcal{O}_X$-module. In this situation the", "{\\it Euler characteristic of $\\mathcal{F}$} is the integer", "$$", "\\chi(X, \\mathcal{F}) = \\sum\\nolimits_i (-1)^i \\dim_k H^i(X, \\mathcal{F}).", "$$", "For justification of the formula see below." ], "refs": [], "ref_ids": [] }, { "id": 12899, "type": "definition", "label": "spaces-over-fields-definition-intersection-number", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-definition-intersection-number", "contents": [ "Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let", "$i : Z \\to X$ be a closed subspace of dimension $d$. Let", "$\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible", "$\\mathcal{O}_X$-modules. We define the {\\it intersection number}", "$(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$", "as the coefficient of $n_1 \\ldots n_d$ in the numerical polynomial", "$$", "\\chi(X, i_*\\mathcal{O}_Z \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes", "\\ldots \\otimes \\mathcal{L}_d^{\\otimes n_d}) =", "\\chi(Z, \\mathcal{L}_1^{\\otimes n_1} \\otimes", "\\ldots \\otimes \\mathcal{L}_d^{\\otimes n_d}|_Z)", "$$", "In the special", "case that $\\mathcal{L}_1 = \\ldots = \\mathcal{L}_d = \\mathcal{L}$", "we write $(\\mathcal{L}^d \\cdot Z)$." ], "refs": [], "ref_ids": [] }, { "id": 13013, "type": "definition", "label": "spaces-divisors-definition-weakly-associated", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-weakly-associated", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.", "Let $x \\in |X|$.", "\\begin{enumerate}", "\\item We say $x$ is {\\it weakly associated} to $\\mathcal{F}$", "if the equivalent conditions (1), (2), and (3) of", "Lemma \\ref{lemma-associated} are satisfied.", "\\item We denote $\\text{WeakAss}(\\mathcal{F})$ the set of weakly associated", "points of $\\mathcal{F}$.", "\\item The {\\it weakly associated points of $X$} are the weakly associated", "points of $\\mathcal{O}_X$.", "\\end{enumerate}", "If $X$ is locally Noetherian we will say", "{\\it $x$ is associated to $\\mathcal{F}$}", "if and only if $x$ is weakly associated to $\\mathcal{F}$ and we set", "$\\text{Ass}(\\mathcal{F}) = \\text{WeakAss}(\\mathcal{F})$.", "Finally (still assuming $X$ is locally Noetherian),", "we will say {\\it $x$ is an associated point of $X$} if and only if", "$x$ is a weakly associated point of $X$." ], "refs": [ "spaces-divisors-lemma-associated" ], "ref_ids": [ 12902 ] }, { "id": 13014, "type": "definition", "label": "spaces-divisors-definition-locally-Noetherian-fibre", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-locally-Noetherian-fibre", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $y \\in |Y|$. We say {\\it the fibre of $f$ over $y$ is", "locally Noetherian} if the equivalent conditions (1), (2), and (3)", "of Lemma \\ref{lemma-locally-noetherian-fibre} are satisfied.", "We say {\\it the fibres of $f$ are locally Noetherian} if this", "holds for every $y \\in |Y|$." ], "refs": [ "spaces-divisors-lemma-locally-noetherian-fibre" ], "ref_ids": [ 12920 ] }, { "id": 13015, "type": "definition", "label": "spaces-divisors-definition-relative-weak-assassin", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-relative-weak-assassin", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces", "over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "The {\\it relative weak assassin of $\\mathcal{F}$ in $X$ over $Y$}", "is the set $\\text{WeakAss}_{X/Y}(\\mathcal{F}) \\subset |X|$", "consisting of those $x \\in |X|$ such that the equivalent conditions of", "Lemma \\ref{lemma-relative-assassin} are satisfied.", "If the fibres of $f$ are locally Noetherian", "(Definition \\ref{definition-locally-Noetherian-fibre})", "then we use the notation $\\text{Ass}_{X/Y}(\\mathcal{F})$." ], "refs": [ "spaces-divisors-lemma-relative-assassin", "spaces-divisors-definition-locally-Noetherian-fibre" ], "ref_ids": [ 12922, 13014 ] }, { "id": 13016, "type": "definition", "label": "spaces-divisors-definition-effective-Cartier-divisor", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-effective-Cartier-divisor", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "\\begin{enumerate}", "\\item A {\\it locally principal closed subspace} of $X$ is a closed subspace", "whose sheaf of ideals is locally generated by $1$ element.", "\\item An {\\it effective Cartier divisor} on $X$ is a closed subspace", "$D \\subset X$ such that the ideal sheaf $\\mathcal{I}_D \\subset \\mathcal{O}_X$", "is an invertible $\\mathcal{O}_X$-module.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13017, "type": "definition", "label": "spaces-divisors-definition-sum-effective-Cartier-divisors", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-sum-effective-Cartier-divisors", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Given effective Cartier divisors", "$D_1$, $D_2$ on $X$ we set $D = D_1 + D_2$ equal to the", "closed subspace of $X$ corresponding to the quasi-coherent", "sheaf of ideals", "$\\mathcal{I}_{D_1}\\mathcal{I}_{D_2} \\subset \\mathcal{O}_S$.", "We call this the {\\it sum of the effective Cartier divisors", "$D_1$ and $D_2$}." ], "refs": [], "ref_ids": [] }, { "id": 13018, "type": "definition", "label": "spaces-divisors-definition-pullback-effective-Cartier-divisor", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-pullback-effective-Cartier-divisor", "contents": [ "Let $S$ be a scheme.", "Let $f : X' \\to X$ be a morphism of algebraic spaces over $S$.", "Let $D \\subset X$", "be an effective Cartier divisor. We say the {\\it pullback of", "$D$ by $f$ is defined} if the closed subspace $f^{-1}(D) \\subset X'$", "is an effective Cartier divisor. In this case we denote it either", "$f^*D$ or $f^{-1}(D)$ and we call it the", "{\\it pullback of the effective Cartier divisor}." ], "refs": [], "ref_ids": [] }, { "id": 13019, "type": "definition", "label": "spaces-divisors-definition-invertible-sheaf-effective-Cartier-divisor", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-invertible-sheaf-effective-Cartier-divisor", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$", "and let $D \\subset X$ be an effective Cartier divisor with ideal", "sheaf $\\mathcal{I}_D$.", "\\begin{enumerate}", "\\item The {\\it invertible sheaf $\\mathcal{O}_X(D)$ associated to $D$}", "is defined by", "$$", "\\mathcal{O}_X(D) =", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{I}_D, \\mathcal{O}_X) =", "\\mathcal{I}_D^{\\otimes -1}.", "$$", "\\item The canonical section, usually denoted $1$ or $1_D$, is the", "global section of $\\mathcal{O}_X(D)$ corresponding to", "the inclusion mapping $\\mathcal{I}_D \\to \\mathcal{O}_X$.", "\\item We write", "$\\mathcal{O}_X(-D) = \\mathcal{O}_X(D)^{\\otimes -1} = \\mathcal{I}_D$.", "\\item Given a second effective Cartier divisor $D' \\subset X$ we define", "$\\mathcal{O}_X(D - D') =", "\\mathcal{O}_X(D) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(-D')$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13020, "type": "definition", "label": "spaces-divisors-definition-regular-section", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-regular-section", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{L}$ be an invertible sheaf on $X$.", "A global section $s \\in \\Gamma(X, \\mathcal{L})$ is called a", "{\\it regular section} if the map $\\mathcal{O}_X \\to \\mathcal{L}$,", "$f \\mapsto fs$ is injective." ], "refs": [], "ref_ids": [] }, { "id": 13021, "type": "definition", "label": "spaces-divisors-definition-zero-scheme-s", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-zero-scheme-s", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{L}$ be an invertible sheaf.", "Let $s \\in \\Gamma(X, \\mathcal{L})$.", "The {\\it zero scheme} of $s$ is the closed subspace $Z(s) \\subset X$", "defined by the quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_X$ which is the image of the", "map $s : \\mathcal{L}^{\\otimes -1} \\to \\mathcal{O}_X$." ], "refs": [], "ref_ids": [] }, { "id": 13022, "type": "definition", "label": "spaces-divisors-definition-relative-effective-Cartier-divisor", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-relative-effective-Cartier-divisor", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "A {\\it relative effective Cartier divisor} on $X/Y$ is an", "effective Cartier divisor $D \\subset X$ such that $D \\to Y$", "is a flat morphism of algebraic spaces." ], "refs": [], "ref_ids": [] }, { "id": 13023, "type": "definition", "label": "spaces-divisors-definition-sheaf-meromorphic-functions", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-sheaf-meromorphic-functions", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "The {\\it sheaf of meromorphic functions on $X$} is", "the sheaf {\\it $\\mathcal{K}_X$} on $X_\\etale$ associated to the presheaf", "displayed above. A {\\it meromorphic function} on $X$", "is a global section of $\\mathcal{K}_X$." ], "refs": [], "ref_ids": [] }, { "id": 13024, "type": "definition", "label": "spaces-divisors-definition-meromorphic-section", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-meromorphic-section", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules", "on $X_\\etale$.", "\\begin{enumerate}", "\\item We denote $\\mathcal{K}_X(\\mathcal{F})$ the sheaf of", "$\\mathcal{K}_X$-modules which is the sheafification of the presheaf", "$U \\mapsto \\mathcal{S}(U)^{-1}\\mathcal{F}(U)$. Equivalently", "$\\mathcal{K}_X(\\mathcal{F}) =", "\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{K}_X$ (see above).", "\\item A {\\it meromorphic section of $\\mathcal{F}$}", "is a global section of $\\mathcal{K}_X(\\mathcal{F})$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13025, "type": "definition", "label": "spaces-divisors-definition-pullback-meromorphic-sections", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-pullback-meromorphic-sections", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism", "of algebraic spaces over $S$. We say that {\\it pullbacks of meromorphic", "functions are defined for $f$} if for every commutative diagram", "$$", "\\xymatrix{", "U \\ar[r] \\ar[d] & X \\ar[d] \\\\", "V \\ar[r] & Y", "}", "$$", "with $U \\in X_\\etale$ and $V \\in Y_\\etale$ and any", "section $s \\in \\mathcal{S}_Y(V)$ the pullback", "$f^\\sharp(s) \\in \\mathcal{O}_X(U)$ is an element", "of $\\mathcal{S}_X(U)$." ], "refs": [], "ref_ids": [] }, { "id": 13026, "type": "definition", "label": "spaces-divisors-definition-regular-meromorphic-section", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-regular-meromorphic-section", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "A meromorphic section $s$ of $\\mathcal{L}$ is said to be {\\it regular}", "if the induced map $\\mathcal{K}_X \\to \\mathcal{K}_X(\\mathcal{L})$", "is injective." ], "refs": [], "ref_ids": [] }, { "id": 13027, "type": "definition", "label": "spaces-divisors-definition-relative-proj", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-relative-proj", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{A}$ be a quasi-coherent sheaf of", "graded $\\mathcal{O}_X$-algebras. The", "{\\it relative homogeneous spectrum of $\\mathcal{A}$ over $X$},", "or the {\\it homogeneous spectrum of $\\mathcal{A}$ over $X$}, or the", "{\\it relative Proj of $\\mathcal{A}$ over $X$} is the algebraic space", "$F$ over $X$ of Lemma \\ref{lemma-relative-proj}.", "We denote it $\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$." ], "refs": [ "spaces-divisors-lemma-relative-proj" ], "ref_ids": [ 12962 ] }, { "id": 13028, "type": "definition", "label": "spaces-divisors-definition-relatively-ample", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-relatively-ample", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "We say $\\mathcal{L}$ is {\\it relatively ample}, or {\\it $f$-relatively ample},", "or {\\it ample on $X/Y$}, or {\\it $f$-ample} if $f : X \\to Y$", "is representable and for every morphism $Z \\to Y$", "where $Z$ is a scheme, the pullback $\\mathcal{L}_Z$ of $\\mathcal{L}$", "to $X_Z = Z \\times_Y X$ is ample on $X_Z/Z$ as in", "Morphisms, Definition \\ref{morphisms-definition-relatively-ample}." ], "refs": [ "morphisms-definition-relatively-ample" ], "ref_ids": [ 5568 ] }, { "id": 13029, "type": "definition", "label": "spaces-divisors-definition-blow-up", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-blow-up", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf", "of ideals, and let $Z \\subset X$ be the closed subspace corresponding", "to $\\mathcal{I}$", "(Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-closed-immersion-ideals}).", "The {\\it blowing up of $X$ along $Z$}, or the", "{\\it blowing up of $X$ in the ideal sheaf $\\mathcal{I}$} is", "the morphism", "$$", "b :", "\\underline{\\text{Proj}}_X", "\\left(\\bigoplus\\nolimits_{n \\geq 0} \\mathcal{I}^n\\right)", "\\longrightarrow", "X", "$$", "The {\\it exceptional divisor} of the blowup is the inverse image", "$b^{-1}(Z)$. Sometimes $Z$ is called the {\\it center} of the blowup." ], "refs": [ "spaces-morphisms-lemma-closed-immersion-ideals" ], "ref_ids": [ 4765 ] }, { "id": 13030, "type": "definition", "label": "spaces-divisors-definition-strict-transform", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-strict-transform", "contents": [ "With $Z \\subset B$ and $f : X \\to B$ as above.", "\\begin{enumerate}", "\\item Given a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$", "the {\\it strict transform} of $\\mathcal{F}$ with respect to the blowup", "of $B$ in $Z$ is the quotient $\\mathcal{F}'$ of $\\text{pr}_X^*\\mathcal{F}$", "by the submodule of sections supported on $|\\text{pr}_{B'}^{-1}E|$.", "\\item The {\\it strict transform} of $X$ is the closed subspace", "$X' \\subset X \\times_B B'$ cut out by the quasi-coherent ideal of", "sections of $\\mathcal{O}_{X \\times_B B'}$ supported on", "$|\\text{pr}_{B'}^{-1}E|$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13031, "type": "definition", "label": "spaces-divisors-definition-admissible-blowup", "categories": [ "spaces-divisors" ], "title": "spaces-divisors-definition-admissible-blowup", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $U \\subset X$ be an open subspace. A morphism", "$X' \\to X$ is called a {\\it $U$-admissible blowup} if there exists a", "closed immersion $Z \\to X$ of finite presentation with $Z$ disjoint from", "$U$ such that $X'$ is isomorphic to the blowup of $X$ in $Z$." ], "refs": [], "ref_ids": [] }, { "id": 13134, "type": "definition", "label": "dga-definition-dga", "categories": [ "dga" ], "title": "dga-definition-dga", "contents": [ "Let $R$ be a commutative ring. A {\\it differential graded algebra over $R$}", "is either", "\\begin{enumerate}", "\\item a chain complex $A_\\bullet$ of $R$-modules endowed with", "$R$-bilinear maps $A_n \\times A_m \\to A_{n + m}$,", "$(a, b) \\mapsto ab$ such that", "$$", "\\text{d}_{n + m}(ab) = \\text{d}_n(a)b + (-1)^n a\\text{d}_m(b)", "$$", "and such that $\\bigoplus A_n$ becomes an associative and unital", "$R$-algebra, or", "\\item a cochain complex $A^\\bullet$ of $R$-modules endowed with", "$R$-bilinear maps $A^n \\times A^m \\to A^{n + m}$, $(a, b) \\mapsto ab$", "such that", "$$", "\\text{d}^{n + m}(ab) = \\text{d}^n(a)b + (-1)^n a\\text{d}^m(b)", "$$", "and such that $\\bigoplus A^n$ becomes an associative and unital $R$-algebra.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13135, "type": "definition", "label": "dga-definition-homomorphism-dga", "categories": [ "dga" ], "title": "dga-definition-homomorphism-dga", "contents": [ "A {\\it homomorphism of differential graded algebras}", "$f : (A, \\text{d}) \\to (B, \\text{d})$ is an algebra map $f : A \\to B$", "compatible with the gradings and $\\text{d}$." ], "refs": [], "ref_ids": [] }, { "id": 13136, "type": "definition", "label": "dga-definition-cdga", "categories": [ "dga" ], "title": "dga-definition-cdga", "contents": [ "A differential graded algebra $(A, \\text{d})$ is {\\it commutative} if", "$ab = (-1)^{nm}ba$ for $a$ in degree $n$ and $b$ in degree $m$.", "We say $A$ is {\\it strictly commutative} if in addition $a^2 = 0$", "for $\\deg(a)$ odd." ], "refs": [], "ref_ids": [] }, { "id": 13137, "type": "definition", "label": "dga-definition-tensor-product", "categories": [ "dga" ], "title": "dga-definition-tensor-product", "contents": [ "Let $R$ be a ring.", "Let $(A, \\text{d})$, $(B, \\text{d})$ be differential graded algebras over $R$.", "The {\\it tensor product differential graded algebra} of $A$ and $B$", "is the algebra $A \\otimes_R B$ with multiplication defined by", "$$", "(a \\otimes b)(a' \\otimes b') = (-1)^{\\deg(a')\\deg(b)} aa' \\otimes bb'", "$$", "endowed with differential $\\text{d}$ defined by the rule", "$\\text{d}(a \\otimes b) = \\text{d}(a) \\otimes b + (-1)^m a \\otimes \\text{d}(b)$", "where $m = \\deg(a)$." ], "refs": [], "ref_ids": [] }, { "id": 13138, "type": "definition", "label": "dga-definition-dgm", "categories": [ "dga" ], "title": "dga-definition-dgm", "contents": [ "Let $R$ be a ring.", "Let $(A, \\text{d})$ be a differential graded algebra over $R$.", "A (right) {\\it differential graded module} $M$ over $A$ is a right $A$-module", "$M$ which has a grading $M = \\bigoplus M^n$ and a differential $\\text{d}$", "such that $M^n A^m \\subset M^{n + m}$, such that", "$\\text{d}(M^n) \\subset M^{n + 1}$, and such that", "$$", "\\text{d}(ma) = \\text{d}(m)a + (-1)^n m\\text{d}(a)", "$$", "for $a \\in A$ and $m \\in M^n$. A", "{\\it homomorphism of differential graded modules} $f : M \\to N$", "is an $A$-module map compatible with gradings and differentials.", "The category of (right) differential graded $A$-modules is denoted", "$\\text{Mod}_{(A, \\text{d})}$." ], "refs": [], "ref_ids": [] }, { "id": 13139, "type": "definition", "label": "dga-definition-shift", "categories": [ "dga" ], "title": "dga-definition-shift", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $M$ be a differential graded module.", "For any $k \\in \\mathbf{Z}$ we define the {\\it $k$-shifted module}", "$M[k]$ as follows", "\\begin{enumerate}", "\\item as $A$-module $M[k] = M$,", "\\item $M[k]^n = M^{n + k}$,", "\\item $\\text{d}_{M[k]} = (-1)^k\\text{d}_M$.", "\\end{enumerate}", "For a morphism $f : M \\to N$ of differential graded $A$-modules", "we let $f[k] : M[k] \\to N[k]$ be the map equal to $f$ on underlying", "$A$-modules. This defines a functor", "$[k] : \\text{Mod}_{(A, \\text{d})} \\to \\text{Mod}_{(A, \\text{d})}$." ], "refs": [], "ref_ids": [] }, { "id": 13140, "type": "definition", "label": "dga-definition-homotopy", "categories": [ "dga" ], "title": "dga-definition-homotopy", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. Let", "$f, g : M \\to N$ be homomorphisms of differential graded $A$-modules.", "A {\\it homotopy between $f$ and $g$} is an $A$-module map $h : M \\to N$", "such that", "\\begin{enumerate}", "\\item $h(M^n) \\subset N^{n - 1}$ for all $n$, and", "\\item $f(x) - g(x) = \\text{d}_N(h(x)) + h(\\text{d}_M(x))$ for", "all $x \\in M$.", "\\end{enumerate}", "If a homotopy exists, then we say $f$ and $g$ are {\\it homotopic}." ], "refs": [], "ref_ids": [] }, { "id": 13141, "type": "definition", "label": "dga-definition-complexes-notation", "categories": [ "dga" ], "title": "dga-definition-complexes-notation", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "The {\\it homotopy category}, denoted $K(\\text{Mod}_{(A, \\text{d})})$, is", "the category whose objects are the objects of", "$\\text{Mod}_{(A, \\text{d})}$ and whose morphisms are homotopy classes", "of homomorphisms of differential graded $A$-modules." ], "refs": [], "ref_ids": [] }, { "id": 13142, "type": "definition", "label": "dga-definition-cone", "categories": [ "dga" ], "title": "dga-definition-cone", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $f : K \\to L$ be a homomorphism of differential graded $A$-modules.", "The {\\it cone} of $f$ is the differential graded $A$-module", "$C(f)$ given by $C(f) = L \\oplus K$ with grading", "$C(f)^n = L^n \\oplus K^{n + 1}$ and", "differential", "$$", "d_{C(f)} =", "\\left(", "\\begin{matrix}", "\\text{d}_L & f \\\\", "0 & -\\text{d}_K", "\\end{matrix}", "\\right)", "$$", "It comes equipped with canonical morphisms of complexes $i : L \\to C(f)$", "and $p : C(f) \\to K[1]$ induced by the obvious maps $L \\to C(f)$", "and $C(f) \\to K$." ], "refs": [], "ref_ids": [] }, { "id": 13143, "type": "definition", "label": "dga-definition-admissible-ses", "categories": [ "dga" ], "title": "dga-definition-admissible-ses", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "\\begin{enumerate}", "\\item A homomorphism $K \\to L$ of differential graded $A$-modules", "is an {\\it admissible monomorphism} if there exists a graded $A$-module", "map $L \\to K$ which is left inverse to $K \\to L$.", "\\item A homomorphism $L \\to M$ of differential graded $A$-modules", "is an {\\it admissible epimorphism} if there exists a graded $A$-module", "map $M \\to L$ which is right inverse to $L \\to M$.", "\\item A short exact sequence $0 \\to K \\to L \\to M \\to 0$ of differential", "graded $A$-modules is an {\\it admissible short exact sequence}", "if it is split as a sequence of graded $A$-modules.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13144, "type": "definition", "label": "dga-definition-distinguished-triangle", "categories": [ "dga" ], "title": "dga-definition-distinguished-triangle", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "\\begin{enumerate}", "\\item If $0 \\to K \\to L \\to M \\to 0$ is an admissible short exact sequence", "of differential graded $A$-modules, then the {\\it triangle associated", "to $0 \\to K \\to L \\to M \\to 0$} is the triangle ", "(\\ref{equation-triangle-associated-to-admissible-ses})", "of $K(\\text{Mod}_{(A, \\text{d})})$.", "\\item A triangle of $K(\\text{Mod}_{(A, \\text{d})})$ is called a", "{\\it distinguished triangle} if it is isomorphic to a triangle", "associated to an admissible short exact sequence", "of differential graded $A$-modules.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13145, "type": "definition", "label": "dga-definition-opposite-dga", "categories": [ "dga" ], "title": "dga-definition-opposite-dga", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ be a differential graded algebra", "over $R$. The {\\it opposite differential graded algebra} is the differential", "graded algebra $(A^{opp}, \\text{d})$ over $R$ where $A^{opp} = A$", "as a graded $R$-module, $\\text{d} = \\text{d}$, and multiplication is", "given by", "$$", "a \\cdot_{opp} b = (-1)^{\\deg(a)\\deg(b)} b a", "$$", "for homogeneous elements $a, b \\in A$." ], "refs": [], "ref_ids": [] }, { "id": 13146, "type": "definition", "label": "dga-definition-shift-graded-module", "categories": [ "dga" ], "title": "dga-definition-shift-graded-module", "contents": [ "Let $R$ be a ring. Let $A$ be a $\\mathbf{Z}$-graded $R$-algebra.", "\\begin{enumerate}", "\\item Given a right graded $A$-module $M$ we define the", "{\\it $k$th shifted $A$-module} $M[k]$ as the same as", "a right $A$-module but with grading $(M[k])^n = M^{n + k}$.", "\\item Given a left graded $A$-module $M$ we define the", "{\\it $k$th shifted $A$-module} $M[k]$ as the module", "with grading $(M[k])^n = M^{n + k}$ and multiplication", "$A^n \\times (M[k])^m \\to (M[k])^{n + m}$", "equal to $(-1)^{nk}$ times the given multiplication", "$A^n \\times M^{m + k} \\to M^{n + m + k}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13147, "type": "definition", "label": "dga-definition-unbounded-derived-category", "categories": [ "dga" ], "title": "dga-definition-unbounded-derived-category", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra.", "Let $\\text{Ac}$ and $\\text{Qis}$ be as in Lemma \\ref{lemma-acyclic}.", "The {\\it derived category of $(A, \\text{d})$} is the triangulated", "category", "$$", "D(A, \\text{d}) =", "K(\\text{Mod}_{(A, \\text{d})})/\\text{Ac} =", "\\text{Qis}^{-1}K(\\text{Mod}_{(A, \\text{d})}).", "$$", "We denote $H^0 : D(A, \\text{d}) \\to \\text{Mod}_R$ the unique functor", "whose composition with the quotient functor gives back the functor", "$H^0$ defined above." ], "refs": [ "dga-lemma-acyclic" ], "ref_ids": [ 13067 ] }, { "id": 13148, "type": "definition", "label": "dga-definition-linear-category", "categories": [ "dga" ], "title": "dga-definition-linear-category", "contents": [ "Let $R$ be a ring. An {\\it $R$-linear category $\\mathcal{A}$} is a category", "where every morphism set is given the structure of an $R$-module", "and where for $x, y, z \\in \\Ob(\\mathcal{A})$ composition law", "$$", "\\Hom_\\mathcal{A}(y, z) \\times \\Hom_\\mathcal{A}(x, y)", "\\longrightarrow", "\\Hom_\\mathcal{A}(x, z)", "$$", "is $R$-bilinear." ], "refs": [], "ref_ids": [] }, { "id": 13149, "type": "definition", "label": "dga-definition-functor-linear-categories", "categories": [ "dga" ], "title": "dga-definition-functor-linear-categories", "contents": [ "Let $R$ be a ring. A {\\it functor of $R$-linear categories}, or an", "{\\it $R$-linear} is a functor $F : \\mathcal{A} \\to \\mathcal{B}$", "where for all objects $x, y$ of $\\mathcal{A}$ the map", "$F : \\Hom_\\mathcal{A}(x, y) \\to \\Hom_\\mathcal{A}(F(x), F(y))$", "is a homomorphism of $R$-modules." ], "refs": [], "ref_ids": [] }, { "id": 13150, "type": "definition", "label": "dga-definition-graded-category", "categories": [ "dga" ], "title": "dga-definition-graded-category", "contents": [ "Let $R$ be a ring. A {\\it graded category $\\mathcal{A}$", "over $R$} is a category where every morphism set is given the structure", "of a graded $R$-module and where for", "$x, y, z \\in \\Ob(\\mathcal{A})$ composition is $R$-bilinear and induces", "a homomorphism", "$$", "\\Hom_\\mathcal{A}(y, z) \\otimes_R \\Hom_\\mathcal{A}(x, y)", "\\longrightarrow", "\\Hom_\\mathcal{A}(x, z)", "$$", "of graded $R$-modules (i.e., preserving degrees)." ], "refs": [], "ref_ids": [] }, { "id": 13151, "type": "definition", "label": "dga-definition-functor-graded-categories", "categories": [ "dga" ], "title": "dga-definition-functor-graded-categories", "contents": [ "Let $R$ be a ring. A {\\it functor of graded categories over $R$}, or a", "{\\it graded functor}", "is a functor $F : \\mathcal{A} \\to \\mathcal{B}$ where for all objects", "$x, y$ of $\\mathcal{A}$ the map", "$F : \\Hom_\\mathcal{A}(x, y) \\to \\Hom_\\mathcal{A}(F(x), F(y))$", "is a homomorphism of graded $R$-modules." ], "refs": [], "ref_ids": [] }, { "id": 13152, "type": "definition", "label": "dga-definition-H0-of-graded-category", "categories": [ "dga" ], "title": "dga-definition-H0-of-graded-category", "contents": [ "Let $R$ be a ring. Let $\\mathcal{A}$ be a graded category", "over $R$. We let {\\it $\\mathcal{A}^0$} be the category with the", "same objects as $\\mathcal{A}$ and with", "$$", "\\Hom_{\\mathcal{A}^0}(x, y) = \\Hom^0_\\mathcal{A}(x, y)", "$$", "the degree $0$ graded piece of the graded module of morphisms of", "$\\mathcal{A}$." ], "refs": [], "ref_ids": [] }, { "id": 13153, "type": "definition", "label": "dga-definition-graded-direct-sum", "categories": [ "dga" ], "title": "dga-definition-graded-direct-sum", "contents": [ "Let $R$ be a ring. Let $\\mathcal{A}$ be a graded category over $R$.", "A direct sum $(x, y, z, i, j, p, q)$ in $\\mathcal{A}$ (notation as in", "Homology, Remark \\ref{homology-remark-direct-sum})", "is a {\\it graded direct sum} if $i, j, p, q$ are homogeneous", "of degree $0$." ], "refs": [ "homology-remark-direct-sum" ], "ref_ids": [ 12189 ] }, { "id": 13154, "type": "definition", "label": "dga-definition-dga-category", "categories": [ "dga" ], "title": "dga-definition-dga-category", "contents": [ "Let $R$ be a ring. A {\\it differential graded category $\\mathcal{A}$", "over $R$} is a category where every morphism set is given the structure", "of a differential graded $R$-module and where for", "$x, y, z \\in \\Ob(\\mathcal{A})$ composition is $R$-bilinear and induces", "a homomorphism", "$$", "\\Hom_\\mathcal{A}(y, z) \\otimes_R \\Hom_\\mathcal{A}(x, y)", "\\longrightarrow", "\\Hom_\\mathcal{A}(x, z)", "$$", "of differential graded $R$-modules." ], "refs": [], "ref_ids": [] }, { "id": 13155, "type": "definition", "label": "dga-definition-functor-dga-categories", "categories": [ "dga" ], "title": "dga-definition-functor-dga-categories", "contents": [ "Let $R$ be a ring. A {\\it functor of differential graded categories over $R$}", "is a functor $F : \\mathcal{A} \\to \\mathcal{B}$ where for all objects", "$x, y$ of $\\mathcal{A}$ the map", "$F : \\Hom_\\mathcal{A}(x, y) \\to \\Hom_\\mathcal{A}(F(x), F(y))$", "is a homomorphism of differential graded $R$-modules." ], "refs": [], "ref_ids": [] }, { "id": 13156, "type": "definition", "label": "dga-definition-homotopy-category-of-dga-category", "categories": [ "dga" ], "title": "dga-definition-homotopy-category-of-dga-category", "contents": [ "Let $R$ be a ring. Let $\\mathcal{A}$ be a differential graded category", "over $R$. Then we let", "\\begin{enumerate}", "\\item the {\\it category of complexes of $\\mathcal{A}$}\\footnote{This may", "be nonstandard terminology.} be the category", "$\\text{Comp}(\\mathcal{A})$ whose objects are the same as the objects", "of $\\mathcal{A}$ and with", "$$", "\\Hom_{\\text{Comp}(\\mathcal{A})}(x, y) =", "\\Ker(d : \\Hom^0_\\mathcal{A}(x, y) \\to \\Hom^1_\\mathcal{A}(x, y))", "$$", "\\item the {\\it homotopy category of $\\mathcal{A}$} be the category", "$K(\\mathcal{A})$ whose objects are the same as the objects", "of $\\mathcal{A}$ and with", "$$", "\\Hom_{K(\\mathcal{A})}(x, y) = H^0(\\Hom_\\mathcal{A}(x, y))", "$$", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13157, "type": "definition", "label": "dga-definition-dg-direct-sum", "categories": [ "dga" ], "title": "dga-definition-dg-direct-sum", "contents": [ "Let $R$ be a ring. Let $\\mathcal{A}$ be a differential graded category over", "$R$. A direct sum $(x, y, z, i, j, p, q)$ in $\\mathcal{A}$ (notation as in", "Homology, Remark \\ref{homology-remark-direct-sum})", "is a {\\it differential graded direct sum} if $i, j, p, q$ are homogeneous", "of degree $0$ and closed, i.e., $\\text{d}(i) = 0$, etc." ], "refs": [ "homology-remark-direct-sum" ], "ref_ids": [ 12189 ] }, { "id": 13158, "type": "definition", "label": "dga-definition-bimodule", "categories": [ "dga" ], "title": "dga-definition-bimodule", "contents": [ "Bimodules. Let $R$ be a ring.", "\\begin{enumerate}", "\\item Let $A$ and $B$ be $R$-algebras. An {\\it $(A, B)$-bimodule}", "is an $R$-module $M$ equippend with $R$-bilinear maps", "$$", "A \\times M \\to M, (a, x) \\mapsto ax", "\\quad\\text{and}\\quad", "M \\times B \\to M, (x, b) \\mapsto xb", "$$", "such that the following hold", "\\begin{enumerate}", "\\item $a'(ax) = (a'a)x$ and $(xb)b' = x(bb')$,", "\\item $a(xb) = (ax)b$, and", "\\item $1 x = x = x 1$.", "\\end{enumerate}", "\\item Let $A$ and $B$ be $\\mathbf{Z}$-graded $R$-algebras. A", "{\\it graded $(A, B)$-bimodule} is an $(A, B)$-bimodule $M$ which", "has a grading $M = \\bigoplus M^n$ such that", "$A^n M^m \\subset M^{n + m}$ and $M^n B^m \\subset M^{n + m}$.", "\\item Let $A$ and $B$ be differential graded $R$-algebras. A", "{\\it differential graded $(A, B)$-bimodule} is a graded $(A, B)$-bimodule", "which comes equipped with a differential", "$\\text{d} : M \\to M$ homogeneous of degree $1$", "such that $\\text{d}(ax) = \\text{d}(a)x + (-1)^{\\deg(a)}a\\text{d}(x)$ and", "$\\text{d}(xb) = \\text{d}(x)b + (-1)^{\\deg(x)}x\\text{d}(b)$", "for homogeneous elements $a \\in A$, $x \\in M$, $b \\in B$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13216, "type": "definition", "label": "spaces-more-groupoids-definition-split-at-point", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-definition-split-at-point", "contents": [ "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$", "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.", "Let $u \\in |U|$ be a point.", "\\begin{enumerate}", "\\item We say $R$ is {\\it strongly split over $u$} if there exists an open", "subspace $P \\subset R$ such that", "\\begin{enumerate}", "\\item $(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t} P})$ is a", "groupoid in algebraic spaces over $B$,", "\\item $s|_P$, $t|_P$ are finite, and", "\\item $\\{r \\in |R| : s(r) = u, t(r) = u\\} \\subset |P|$.", "\\end{enumerate}", "The choice of such a $P$ will be called a", "{\\it strong splitting of $R$ over $u$}.", "\\item We say $R$ is {\\it split over $u$} if there exists an open", "subspace $P \\subset R$ such that", "\\begin{enumerate}", "\\item $(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t} P})$ is a", "groupoid in algebraic spaces over $B$,", "\\item $s|_P$, $t|_P$ are finite, and", "\\item $\\{g \\in |G| : g\\text{ maps to }u\\} \\subset |P|$ where", "$G \\to U$ is the stabilizer.", "\\end{enumerate}", "The choice of such a $P$ will be called a", "{\\it splitting of $R$ over $u$}.", "\\item We say $R$ is {\\it quasi-split over $u$} if there exists an open", "subspace $P \\subset R$ such that", "\\begin{enumerate}", "\\item $(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t} P})$ is a", "groupoid in algebraic spaces over $B$,", "\\item $s|_P$, $t|_P$ are finite, and", "\\item $e(u) \\in |P|$\\footnote{This condition is implied by (a).}.", "\\end{enumerate}", "The choice of such a $P$ will be called a {\\it quasi-splitting of $R$ over $u$}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13332, "type": "definition", "label": "modules-definition-globally-generated", "categories": [ "modules" ], "title": "modules-definition-globally-generated", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "We say that $\\mathcal{F}$ is {\\it generated by global", "sections} if there exist a set $I$, and", "global sections $s_i \\in \\Gamma(X, \\mathcal{F})$, $i \\in I$", "such that the map", "$$", "\\bigoplus\\nolimits_{i \\in I}", "\\mathcal{O}_X \\longrightarrow \\mathcal{F}", "$$", "which is the map associated to $s_i$ on the summand corresponding to $i$,", "is surjective. In this case we say that the sections $s_i$", "{\\it generate} $\\mathcal{F}$." ], "refs": [], "ref_ids": [] }, { "id": 13333, "type": "definition", "label": "modules-definition-generated-by-local-sections", "categories": [ "modules" ], "title": "modules-definition-generated-by-local-sections", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "Given a set $I$, and", "local sections $s_i$, $i \\in I$ of $\\mathcal{F}$", "we say that the subsheaf $\\mathcal{G}$ of", "Lemma \\ref{lemma-generated-by-local-sections}", "above is the {\\it subsheaf generated by the $s_i$}." ], "refs": [ "modules-lemma-generated-by-local-sections" ], "ref_ids": [ 13228 ] }, { "id": 13334, "type": "definition", "label": "modules-definition-support", "categories": [ "modules" ], "title": "modules-definition-support", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "\\begin{enumerate}", "\\item The {\\it support of $\\mathcal{F}$} is the set of", "points $x \\in X$ such that $\\mathcal{F}_x \\not = 0$.", "\\item We denote $\\text{Supp}(\\mathcal{F})$ the support of $\\mathcal{F}$.", "\\item Let $s \\in \\Gamma(X, \\mathcal{F})$ be a global section.", "The {\\it support of $s$} is the set of points $x \\in X$", "such that the image $s_x \\in \\mathcal{F}_x$ of $s$ is", "not zero.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13335, "type": "definition", "label": "modules-definition-locally-generated", "categories": [ "modules" ], "title": "modules-definition-locally-generated", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "We say that $\\mathcal{F}$ is {\\it locally generated by sections}", "if for every $x \\in X$ there exists an open", "neighbourhood $U$ such that $\\mathcal{F}|_U$", "is globally generated as a sheaf of $\\mathcal{O}_U$-modules." ], "refs": [], "ref_ids": [] }, { "id": 13336, "type": "definition", "label": "modules-definition-finite-type", "categories": [ "modules" ], "title": "modules-definition-finite-type", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "We say that $\\mathcal{F}$ is of {\\it finite type}", "if for every $x \\in X$ there exists an open", "neighbourhood $U$ such that $\\mathcal{F}|_U$", "is generated by finitely many sections." ], "refs": [], "ref_ids": [] }, { "id": 13337, "type": "definition", "label": "modules-definition-quasi-coherent", "categories": [ "modules" ], "title": "modules-definition-quasi-coherent", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "We say that $\\mathcal{F}$ is a {\\it quasi-coherent", "sheaf of $\\mathcal{O}_X$-modules} if for every", "point $x \\in X$ there exists an open neighbourhood", "$x\\in U \\subset X$ such that $\\mathcal{F}|_U$", "is isomorphic to the cokernel of a map", "$$", "\\bigoplus\\nolimits_{j \\in J}", "\\mathcal{O}_U", "\\longrightarrow", "\\bigoplus\\nolimits_{i \\in I}", "\\mathcal{O}_U", "$$", "The category of quasi-coherent $\\mathcal{O}_X$-modules", "is denoted $\\QCoh(\\mathcal{O}_X)$." ], "refs": [], "ref_ids": [] }, { "id": 13338, "type": "definition", "label": "modules-definition-sheaf-associated", "categories": [ "modules" ], "title": "modules-definition-sheaf-associated", "contents": [ "In the situation of Lemma \\ref{lemma-construct-quasi-coherent-sheaves}", "we say $\\mathcal{F}_M$ is the {\\it sheaf associated to the module $M$", "and the ring map $\\alpha$}. If $R = \\Gamma(X, \\mathcal{O}_X)$", "and $\\alpha = \\text{id}_R$ we simply say $\\mathcal{F}_M$ is the", "{\\it sheaf associated to the module $M$}." ], "refs": [ "modules-lemma-construct-quasi-coherent-sheaves" ], "ref_ids": [ 13245 ] }, { "id": 13339, "type": "definition", "label": "modules-definition-finite-presentation", "categories": [ "modules" ], "title": "modules-definition-finite-presentation", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "We say that $\\mathcal{F}$ is of {\\it finite presentation}", "if for every point $x \\in X$ there exists an open neighbourhood", "$x\\in U \\subset X$, and $n, m \\in \\mathbf{N}$ such that $\\mathcal{F}|_U$", "is isomorphic to the cokernel of a map", "$$", "\\bigoplus\\nolimits_{j = 1, \\ldots, m}", "\\mathcal{O}_U", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n}", "\\mathcal{O}_U", "$$" ], "refs": [], "ref_ids": [] }, { "id": 13340, "type": "definition", "label": "modules-definition-coherent", "categories": [ "modules" ], "title": "modules-definition-coherent", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "We say that $\\mathcal{F}$ is a {\\it coherent $\\mathcal{O}_X$-module}", "if the following two conditions hold:", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is of finite type, and", "\\item for every open $U \\subset X$ and every finite", "collection $s_i \\in \\mathcal{F}(U)$, $i = 1, \\ldots, n$", "the kernel of the associated map", "$\\bigoplus_{i = 1, \\ldots, n} \\mathcal{O}_U \\to \\mathcal{F}|_U$", "is of finite type.", "\\end{enumerate}", "The category of coherent $\\mathcal{O}_X$-modules is denoted", "$\\textit{Coh}(\\mathcal{O}_X)$." ], "refs": [], "ref_ids": [] }, { "id": 13341, "type": "definition", "label": "modules-definition-closed-immersion", "categories": [ "modules" ], "title": "modules-definition-closed-immersion", "contents": [ "A {\\it closed immersion of ringed spaces}\\footnote{This is", "nonstandard notation; see discussion above.} is a morphism", "$i : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$", "with the following properties:", "\\begin{enumerate}", "\\item The map $i$ is a closed immersion of topological spaces.", "\\item The associated map $\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$", "is surjective. Denote the kernel by $\\mathcal{I}$.", "\\item The $\\mathcal{O}_X$-module $\\mathcal{I}$ is locally", "generated by sections.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13342, "type": "definition", "label": "modules-definition-locally-free", "categories": [ "modules" ], "title": "modules-definition-locally-free", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "\\begin{enumerate}", "\\item We say $\\mathcal{F}$ is {\\it locally free} if for every", "point $x \\in X$ there exists a set $I$ and an open", "neighbourhood $x \\in U \\subset X$", "such that $\\mathcal{F}|_U$ is isomorphic to", "$\\bigoplus_{i \\in I} \\mathcal{O}_X|_U$ as an $\\mathcal{O}_X|_U$-module.", "\\item We say $\\mathcal{F}$ is {\\it finite locally free} if we may", "choose the index sets $I$ to be finite.", "\\item We say $\\mathcal{F}$ is {\\it finite locally free of rank $r$}", "if we may choose the index sets $I$ to have cardinality $r$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13343, "type": "definition", "label": "modules-definition-flat", "categories": [ "modules" ], "title": "modules-definition-flat", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "An $\\mathcal{O}_X$-module $\\mathcal{F}$ is {\\it flat} if the functor", "$$", "\\textit{Mod}(\\mathcal{O}_X)", "\\longrightarrow", "\\textit{Mod}(\\mathcal{O}_X), \\quad", "\\mathcal{G} \\mapsto \\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F}", "$$", "is exact." ], "refs": [], "ref_ids": [] }, { "id": 13344, "type": "definition", "label": "modules-definition-flat-at-point", "categories": [ "modules" ], "title": "modules-definition-flat-at-point", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $x \\in X$.", "An $\\mathcal{O}_X$-module $\\mathcal{F}$ is", "{\\it flat at $x$} if $\\mathcal{F}_x$ is a flat", "$\\mathcal{O}_{X, x}$-module." ], "refs": [], "ref_ids": [] }, { "id": 13345, "type": "definition", "label": "modules-definition-flat-morphism", "categories": [ "modules" ], "title": "modules-definition-flat-morphism", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Let $x \\in X$. We say $f$ is {\\it flat at $x$}", "if the map of rings $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is flat.", "We say $f$ is {\\it flat} if $f$ is flat at every $x \\in X$." ], "refs": [], "ref_ids": [] }, { "id": 13346, "type": "definition", "label": "modules-definition-flat-module", "categories": [ "modules" ], "title": "modules-definition-flat-module", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of", "ringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "\\begin{enumerate}", "\\item We say that $\\mathcal{F}$ is {\\it flat over $Y$ at a point $x \\in X$}", "if the stalk $\\mathcal{F}_x$ is a flat $\\mathcal{O}_{Y, f(x)}$-module.", "\\item We say that $\\mathcal{F}$ is {\\it flat over $Y$} if", "$\\mathcal{F}$ is flat over $Y$ at every point $x$ of $X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13347, "type": "definition", "label": "modules-definition-koszul", "categories": [ "modules" ], "title": "modules-definition-koszul", "contents": [ "Let $X$ be a ringed space. Let $\\varphi : \\mathcal{E} \\to \\mathcal{O}_X$", "be an $\\mathcal{O}_X$-module map. The", "{\\it Koszul complex} $K_\\bullet(\\varphi)$ associated to $\\varphi$", "is the sheaf of commutative differential graded algebras defined as follows:", "\\begin{enumerate}", "\\item the underlying graded algebra is the exterior algebra", "$K_\\bullet(\\varphi) = \\wedge(\\mathcal{E})$,", "\\item the differential $d : K_\\bullet(\\varphi) \\to K_\\bullet(\\varphi)$", "is the unique derivation such that $d(e) = \\varphi(e)$ for all", "local sections $e$ of $\\mathcal{E} = K_1(\\varphi)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13348, "type": "definition", "label": "modules-definition-koszul-complex", "categories": [ "modules" ], "title": "modules-definition-koszul-complex", "contents": [ "Let $X$ be a ringed space and let", "$f_1, \\ldots, f_n \\in \\Gamma(X, \\mathcal{O}_X)$. The", "{\\it Koszul complex on $f_1, \\ldots, f_r$} is the Koszul complex", "associated to the map", "$(f_1, \\ldots, f_n) : \\mathcal{O}_X^{\\oplus n} \\to \\mathcal{O}_X$.", "Notation $K_\\bullet(\\mathcal{O}_X, f_1, \\ldots, f_n)$,", "or $K_\\bullet(\\mathcal{O}_X, f_\\bullet)$." ], "refs": [], "ref_ids": [] }, { "id": 13349, "type": "definition", "label": "modules-definition-invertible", "categories": [ "modules" ], "title": "modules-definition-invertible", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. An", "{\\it invertible $\\mathcal{O}_X$-module} is a sheaf", "of $\\mathcal{O}_X$-modules $\\mathcal{L}$ such that", "the functor", "$$", "\\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\textit{Mod}(\\mathcal{O}_X),\\quad", "\\mathcal{F} \\longmapsto \\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "$$", "is an equivalence of categories. We say that $\\mathcal{L}$ is", "{\\it trivial} if it is isomorphic as an $\\mathcal{O}_X$-module", "to $\\mathcal{O}_X$." ], "refs": [], "ref_ids": [] }, { "id": 13350, "type": "definition", "label": "modules-definition-powers", "categories": [ "modules" ], "title": "modules-definition-powers", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given an invertible sheaf", "$\\mathcal{L}$ on $X$ and $n \\in \\mathbf{Z}$ we define the", "$n$th {\\it tensor power} $\\mathcal{L}^{\\otimes n}$ of $\\mathcal{L}$", "as the image of $\\mathcal{O}_X$ under applying the equivalence", "$\\mathcal{F} \\mapsto\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}$", "exactly $n$ times." ], "refs": [], "ref_ids": [] }, { "id": 13351, "type": "definition", "label": "modules-definition-gamma-star", "categories": [ "modules" ], "title": "modules-definition-gamma-star", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Given an invertible sheaf $\\mathcal{L}$ on $X$ we define", "the {\\it associated graded ring} to be", "$$", "\\Gamma_*(X, \\mathcal{L})", "=", "\\bigoplus\\nolimits_{n \\geq 0} \\Gamma(X, \\mathcal{L}^{\\otimes n})", "$$", "Given a sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ we set", "$$", "\\Gamma_*(X, \\mathcal{L}, \\mathcal{F})", "=", "\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\Gamma(X,", "\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n})", "$$", "which we think of as a graded $\\Gamma_*(X, \\mathcal{L})$-module." ], "refs": [], "ref_ids": [] }, { "id": 13352, "type": "definition", "label": "modules-definition-pic", "categories": [ "modules" ], "title": "modules-definition-pic", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "The {\\it Picard group} $\\Pic(X)$ of $X$ is the", "abelian group whose elements are isomorphism classes of", "invertible $\\mathcal{O}_X$-modules, with addition", "corresponding to tensor product." ], "refs": [], "ref_ids": [] }, { "id": 13353, "type": "definition", "label": "modules-definition-derivation", "categories": [ "modules" ], "title": "modules-definition-derivation", "contents": [ "Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings. Let $\\mathcal{F}$ be an", "$\\mathcal{O}_2$-module. A {\\it $\\mathcal{O}_1$-derivation} or more precisely", "a {\\it $\\varphi$-derivation} into $\\mathcal{F}$ is a map", "$D : \\mathcal{O}_2 \\to \\mathcal{F}$ which is additive, annihilates the image", "of $\\mathcal{O}_1 \\to \\mathcal{O}_2$, and satisfies the", "{\\it Leibniz rule}", "$$", "D(ab) = aD(b) + D(a)b", "$$", "for all $a, b$ local sections of $\\mathcal{O}_2$ (wherever they are both", "defined). We denote $\\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})$", "the set of $\\varphi$-derivations into $\\mathcal{F}$." ], "refs": [], "ref_ids": [] }, { "id": 13354, "type": "definition", "label": "modules-definition-module-differentials", "categories": [ "modules" ], "title": "modules-definition-module-differentials", "contents": [ "Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings on $X$. The {\\it module of differentials}", "of $\\varphi$ is the object representing the functor", "$\\mathcal{F} \\mapsto \\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})$", "which exists by Lemma \\ref{lemma-universal-module}.", "It is denoted $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$, and the {\\it universal", "$\\varphi$-derivation} is denoted", "$\\text{d} : \\mathcal{O}_2 \\to \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$." ], "refs": [ "modules-lemma-universal-module" ], "ref_ids": [ 13310 ] }, { "id": 13355, "type": "definition", "label": "modules-definition-differentials", "categories": [ "modules" ], "title": "modules-definition-differentials", "contents": [ "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$", "be a morphism of ringed spaces.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. An {\\it $S$-derivation}", "into $\\mathcal{F}$ is a $f^{-1}\\mathcal{O}_S$-derivation, or more", "precisely a $f^\\sharp$-derivation in the sense of", "Definition \\ref{definition-derivation}.", "We denote $\\text{Der}_S(\\mathcal{O}_X, \\mathcal{F})$", "the set of $S$-derivations into $\\mathcal{F}$.", "\\item The {\\it sheaf of differentials $\\Omega_{X/S}$ of $X$ over $S$}", "is the module of differentials $\\Omega_{\\mathcal{O}_X/f^{-1}\\mathcal{O}_S}$", "endowed with its universal", "$S$-derivation $\\text{d}_{X/S} : \\mathcal{O}_X \\to \\Omega_{X/S}$.", "\\end{enumerate}" ], "refs": [ "modules-definition-derivation" ], "ref_ids": [ 13353 ] }, { "id": 13356, "type": "definition", "label": "modules-definition-differential-operators", "categories": [ "modules" ], "title": "modules-definition-differential-operators", "contents": [ "Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings on $X$. Let $k \\geq 0$ be an integer.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of $\\mathcal{O}_2$-modules.", "A {\\it differential operator $D : \\mathcal{F} \\to \\mathcal{G}$ of order $k$}", "is an is an $\\mathcal{O}_1$-linear map such that for all local sections", "$g$ of $\\mathcal{O}_2$ the map $s \\mapsto D(gs) - gD(s)$ is a", "differential operator of order $k - 1$. For the base case $k = 0$", "we define a differential operator of order $0$ to be an", "$\\mathcal{O}_2$-linear map." ], "refs": [], "ref_ids": [] }, { "id": 13357, "type": "definition", "label": "modules-definition-module-principal-parts", "categories": [ "modules" ], "title": "modules-definition-module-principal-parts", "contents": [ "Let $X$ be a topoological space.", "Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings on $X$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_2$-modules.", "The module $\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F})$", "constructed in Lemma \\ref{lemma-module-principal-parts}", "is called the {\\it module of principal parts of order $k$} of $\\mathcal{F}$." ], "refs": [ "modules-lemma-module-principal-parts" ], "ref_ids": [ 13321 ] }, { "id": 13358, "type": "definition", "label": "modules-definition-relative-differential-operators", "categories": [ "modules" ], "title": "modules-definition-relative-differential-operators", "contents": [ "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$", "be a morphism of ringed spaces.", "Let $\\mathcal{F}$ and $\\mathcal{G}$ be $\\mathcal{O}_X$-modules.", "Let $k \\geq 0$ be an integer.", "A {\\it differential operator of order $k$ on $X/S$}", "is a differential operator $D : \\mathcal{F} \\to \\mathcal{G}$", "with respect to $f^\\sharp : f^{-1}\\mathcal{O}_S \\to \\mathcal{O}_X$", "We denote $\\text{Diff}^k_{X/S}(\\mathcal{F}, \\mathcal{G})$", "the set of these differential operators." ], "refs": [], "ref_ids": [] }, { "id": 13359, "type": "definition", "label": "modules-definition-de-rham-complex", "categories": [ "modules" ], "title": "modules-definition-de-rham-complex", "contents": [ "In the situation above, the", "{\\it de Rham complex of $\\mathcal{B}$ over $\\mathcal{A}$}", "is the unique complex", "$$", "\\Omega_{\\mathcal{B}/\\mathcal{A}}^0 \\to", "\\Omega_{\\mathcal{B}/\\mathcal{A}}^1 \\to", "\\Omega_{\\mathcal{B}/\\mathcal{A}}^2 \\to \\ldots", "$$", "of sheaves of $\\mathcal{A}$-modules whose differential in degree", "$0$ is given by $\\text{d} : \\mathcal{B} \\to \\Omega_{\\mathcal{B}/\\mathcal{A}}$", "and whose differentials in higher degrees have the following property", "\\begin{equation}", "\\label{equation-rule}", "\\text{d}\\left(b_0\\text{d}b_1 \\wedge \\ldots \\wedge \\text{d}b_p\\right) =", "\\text{d}b_0 \\wedge \\text{d}b_1 \\wedge \\ldots \\wedge \\text{d}b_p", "\\end{equation}", "where $b_0, \\ldots, b_p \\in \\mathcal{B}(U)$ are sections over a common", "open $U \\subset X$." ], "refs": [], "ref_ids": [] }, { "id": 13360, "type": "definition", "label": "modules-definition-de-rham-complex-morphism-ringed-spaces", "categories": [ "modules" ], "title": "modules-definition-de-rham-complex-morphism-ringed-spaces", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism", "of ringed spaces. The {\\it de Rham complex} of $f$ or of $X$ over $Y$", "is the complex", "$$", "\\Omega^\\bullet_{X/Y} = \\Omega^\\bullet_{\\mathcal{O}_X/f^{-1}\\mathcal{O}_Y}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 13361, "type": "definition", "label": "modules-definition-naive-cotangent-complex", "categories": [ "modules" ], "title": "modules-definition-naive-cotangent-complex", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{A} \\to \\mathcal{B}$ be a", "homomorphism of sheaves of rings. The {\\it naive cotangent complex}", "$\\NL_{\\mathcal{B}/\\mathcal{A}}$ is the chain complex", "(\\ref{equation-naive-cotangent-complex})", "$$", "\\NL_{\\mathcal{B}/\\mathcal{A}} =", "\\left(\\mathcal{I}/\\mathcal{I}^2", "\\longrightarrow", "\\Omega_{\\mathcal{A}[\\mathcal{B}]/\\mathcal{A}}", "\\otimes_{\\mathcal{A}[\\mathcal{B}]} \\mathcal{B}\\right)", "$$", "with $\\mathcal{I}/\\mathcal{I}^2$ placed in degree $-1$ and", "$\\Omega_{\\mathcal{A}[\\mathcal{B}]/\\mathcal{A}}", "\\otimes_{\\mathcal{A}[\\mathcal{B}]} \\mathcal{B}$", "placed in degree $0$." ], "refs": [], "ref_ids": [] }, { "id": 13362, "type": "definition", "label": "modules-definition-cotangent-complex-morphism-ringed-topoi", "categories": [ "modules" ], "title": "modules-definition-cotangent-complex-morphism-ringed-topoi", "contents": [ "The {\\it naive cotangent complex} $\\NL_f = \\NL_{X/Y}$ of a morphism of ringed", "spaces $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ is", "$\\NL_{\\mathcal{O}_X/f^{-1}\\mathcal{O}_Y}$." ], "refs": [], "ref_ids": [] }, { "id": 13423, "type": "definition", "label": "defos-definition-strict-morphism-thickenings", "categories": [ "defos" ], "title": "defos-definition-strict-morphism-thickenings", "contents": [ "In Situation \\ref{situation-morphism-thickenings} we say that $(f, f')$ is a", "{\\it strict morphism of thickenings}", "if the map $(f')^*\\mathcal{J} \\longrightarrow \\mathcal{I}$ is surjective." ], "refs": [], "ref_ids": [] }, { "id": 13424, "type": "definition", "label": "defos-definition-strict-morphism-thickenings-ringed-topoi", "categories": [ "defos" ], "title": "defos-definition-strict-morphism-thickenings-ringed-topoi", "contents": [ "In Situation \\ref{situation-morphism-thickenings-ringed-topoi}", "we say that $(f, f')$ is a {\\it strict morphism of thickenings}", "if the map $(f')^*\\mathcal{J} \\longrightarrow \\mathcal{I}$ is surjective." ], "refs": [], "ref_ids": [] }, { "id": 13458, "type": "definition", "label": "groupoids-quotients-definition-invariant", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-definition-invariant", "contents": [ "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.", "Let $j = (t, s) : R \\to U \\times_B U$ be a pre-relation of algebraic", "spaces over $B$. We say a morphism $\\phi : U \\to X$ of algebraic spaces", "over $B$ is {\\it $R$-invariant} if the diagram", "$$", "\\xymatrix{", "R \\ar[r]_s \\ar[d]_t & U \\ar[d]^\\phi \\\\", "U \\ar[r]^\\phi & X", "}", "$$", "is commutative. If $j : R \\to U \\times_B U$ comes from the action", "of a group algebraic space $G$ on $U$ over $B$ as in", "Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-groupoid-from-action},", "then we say that $\\phi$ is {\\it $G$-invariant}." ], "refs": [ "spaces-groupoids-lemma-groupoid-from-action" ], "ref_ids": [ 9308 ] }, { "id": 13459, "type": "definition", "label": "groupoids-quotients-definition-base-change", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-definition-base-change", "contents": [ "In the situation of Lemma \\ref{lemma-base-change-on-invariant}", "we call $j' : R' \\to U' \\times_B U'$ the {\\it base change} of the pre-relation", "$j$ to $X'$. We say it is a {\\it flat base change} if $X' \\to X$ is a flat", "morphism of algebraic spaces." ], "refs": [ "groupoids-quotients-lemma-base-change-on-invariant" ], "ref_ids": [ 13439 ] }, { "id": 13460, "type": "definition", "label": "groupoids-quotients-definition-categorical", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-definition-categorical", "contents": [ "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.", "Let $j = (t, s) : R \\to U \\times_B U$ be pre-relation in algebraic spaces", "over $B$.", "\\begin{enumerate}", "\\item We say a morphism $\\phi : U \\to X$ of algebraic spaces over $B$", "is a {\\it categorical quotient} if it is $R$-invariant, and", "for every $R$-invariant morphism $\\psi : U \\to Y$ of algebraic spaces over $B$", "there exists a unique morphism $\\chi : X \\to Y$ such that", "$\\psi = \\phi \\circ \\chi$.", "\\item Let $\\mathcal{C}$ be a full subcategory of the category of algebraic", "spaces over $B$. Assume $U$, $R$ are objects of $\\mathcal{C}$.", "In this situation we say", "a morphism $\\phi : U \\to X$ of algebraic spaces over $B$", "is a {\\it categorical quotient in $\\mathcal{C}$}", "if $X \\in \\Ob(\\mathcal{C})$, and $\\phi$ is $R$-invariant,", "and for every $R$-invariant morphism", "$\\psi : U \\to Y$ with $Y \\in \\Ob(\\mathcal{C})$", "there exists a unique morphism $\\chi : X \\to Y$ such", "that $\\psi = \\phi \\circ \\chi$.", "\\item If $B = S$ and $\\mathcal{C}$ is the category of schemes over $S$,", "then we say $U \\to X$ is a", "{\\it categorical quotient in the category of schemes}, or simply a", "{\\it categorical quotient in schemes}.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13461, "type": "definition", "label": "groupoids-quotients-definition-universal-categorical", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-definition-universal-categorical", "contents": [ "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.", "Let $\\mathcal{C}$ be a full subcategory of the category of algebraic", "spaces over $B$ closed under fibre products.", "Let $j = (t, s) : R \\to U \\times_B U$ be pre-relation in", "$\\mathcal{C}$, and let $U \\to X$ be an $R$-invariant morphism with", "$X \\in \\Ob(\\mathcal{C})$.", "\\begin{enumerate}", "\\item We say $U \\to X$ is a {\\it universal categorical quotient}", "in $\\mathcal{C}$ if for every morphism $X' \\to X$ in $\\mathcal{C}$", "the morphism $U' = X' \\times_X U \\to X'$ is the categorical quotient in", "$\\mathcal{C}$ of the base change $j' : R' \\to U'$ of $j$.", "\\item We say $U \\to X$ is a {\\it uniform categorical quotient}", "in $\\mathcal{C}$ if for every flat morphism $X' \\to X$ in $\\mathcal{C}$", "the morphism $U' = X' \\times_X U \\to X'$ is the categorical quotient in", "$\\mathcal{C}$ of the base change $j' : R' \\to U'$ of $j$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13462, "type": "definition", "label": "groupoids-quotients-definition-orbit", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-definition-orbit", "contents": [ "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-relation over $B$.", "If $u \\in |U|$, then the {\\it orbit}, or more precisely the", "{\\it $R$-orbit} of $u$ is", "$$", "O_u =", "\\left\\{", "u' \\in |U|\\ :", "\\begin{matrix}", "\\exists n \\geq 1, \\ \\exists u_0, \\ldots, u_n \\in |U|\\text{ such that }", "u_0 = u \\text{ and } u_n = u' \\\\", "\\text{and for all }i \\in \\{0, \\ldots, n - 1\\}\\text{ either }", "u_i = u_{i + 1}\\text{ or } \\\\", "\\exists r \\in |R|, \\ s(r) = u_i, t(r) = u_{i + 1}", "\\text{ or } \\\\", "\\exists r \\in |R|, \\ t(r) = u_i, s(r) = u_{i + 1}", "\\end{matrix}", "\\right\\}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 13463, "type": "definition", "label": "groupoids-quotients-definition-geometric-orbits", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-definition-geometric-orbits", "contents": [ "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-relation over $B$.", "Let $\\Spec(k) \\to B$ be a geometric point of $B$.", "\\begin{enumerate}", "\\item We say $\\overline{u}, \\overline{u}' \\in U(k)$ are", "{\\it weakly $R$-equivalent} if they are in the same equivalence class", "for the equivalence relation generated by the relation", "$j(R(k)) \\subset U(k) \\times U(k)$.", "\\item We say $\\overline{u}, \\overline{u}' \\in U(k)$ are", "{\\it $R$-equivalent} if for some overfield $k \\subset \\Omega$", "the images in $U(\\Omega)$ are weakly $R$-equivalent.", "\\item The {\\it weak orbit}, or more precisely the {\\it weak $R$-orbit}", "of $\\overline{u} \\in U(k)$ is set of all", "elements of $U(k)$ which are weakly $R$-equivalent to $\\overline{u}$.", "\\item The {\\it orbit}, or more precisely the {\\it $R$-orbit}", "of $\\overline{u} \\in U(k)$ is set of all", "elements of $U(k)$ which are $R$-equivalent to $\\overline{u}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13464, "type": "definition", "label": "groupoids-quotients-definition-set-theoretically-invariant", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-definition-set-theoretically-invariant", "contents": [ "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-relation over $B$.", "\\begin{enumerate}", "\\item We say $\\phi : U \\to X$ is {\\it set-theoretically $R$-invariant}", "if and only if the map $U(k) \\to X(k)$ equalizes the two maps", "$s, t : R(k) \\to U(k)$ for every algebraically closed field $k$", "over $B$.", "\\item We say $\\phi : U \\to X$ {\\it separates orbits}, or", "{\\it separates $R$-orbits} if it is set-theoretically $R$-invariant and", "$\\phi(\\overline{u}) = \\phi(\\overline{u}')$ in $X(k)$ implies that", "$\\overline{u}, \\overline{u}' \\in U(k)$ are in the same orbit", "for every algebraically closed field $k$ over $B$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13465, "type": "definition", "label": "groupoids-quotients-definition-set-theoretic-equivalence", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-definition-set-theoretic-equivalence", "contents": [ "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-relation over $B$.", "\\begin{enumerate}", "\\item We say $j$ is a {\\it set-theoretic pre-equivalence relation} if", "for all algebraically closed fields $k$ over $B$ the relation", "$\\sim_R$ on $U(k)$ defined by", "$$", "\\overline{u} \\sim_R \\overline{u}'", "\\Leftrightarrow", "\\begin{matrix}", "\\exists\\text{ field extension }K/k, \\ \\exists\\ r \\in R(K), \\\\", "s(r) = \\overline{u}, \\ t(r) = \\overline{u}'", "\\end{matrix}", "$$", "is an equivalence relation.", "\\item We say $j$ is a {\\it set-theoretic equivalence relation}", "if $j$ is universally injective and a set-theoretic pre-equivalence", "relation.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13466, "type": "definition", "label": "groupoids-quotients-definition-orbit-space", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-definition-orbit-space", "contents": [ "Let $B \\to S$ as in Section \\ref{section-conventions-notation}.", "Let $j : R \\to U \\times_B U$ be a pre-relation.", "We say $\\phi : U \\to X$ is an {\\it orbit space for $R$} if", "\\begin{enumerate}", "\\item $\\phi$ is $R$-invariant,", "\\item $\\phi$ separates $R$-orbits, and", "\\item $\\phi$ is surjective.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13467, "type": "definition", "label": "groupoids-quotients-definition-coarse", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-definition-coarse", "contents": [ "Let $S$ be a scheme and $B$ an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-relation.", "A morphism $\\phi : U \\to X$ of algebraic spaces over $B$", "is called a {\\it coarse quotient} if", "\\begin{enumerate}", "\\item $\\phi$ is a categorical quotient, and", "\\item $\\phi$ is an orbit space.", "\\end{enumerate}", "If $S = B$, $U$, $R$ are all schemes, then we say a morphism of schemes", "$\\phi : U \\to X$ is a {\\it coarse quotient in schemes} if", "\\begin{enumerate}", "\\item $\\phi$ is a categorical quotient in schemes, and", "\\item $\\phi$ is an orbit space.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13468, "type": "definition", "label": "groupoids-quotients-definition-topological", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-definition-topological", "contents": [ "Let $S$ be a scheme and $B$ an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-relation.", "Let $\\phi : U \\to X$ be an $R$-invariant morphism of algebraic spaces over $B$.", "\\begin{enumerate}", "\\item", "\\label{item-submersive}", "The morphism $\\phi$ is submersive.", "\\item", "\\label{item-invariant-closed}", "For any $R$-invariant closed subset $Z \\subset |U|$ the image", "$\\phi(Z)$ is closed in $|X|$.", "\\item", "\\label{item-intersect-invariant-closed}", "Condition (\\ref{item-invariant-closed}) holds and for any pair of", "$R$-invariant closed subsets $Z_1, Z_2 \\subset |U|$ we have", "$$", "\\phi(Z_1 \\cap Z_2) = \\phi(Z_1) \\cap \\phi(Z_2)", "$$", "\\item The morphism $(t, s) : R \\to U \\times_X U$ is universally submersive.", "\\label{item-strong}", "\\end{enumerate}", "For each of these properties we can also require them to hold after any", "flat base change, or after any base change, see", "Definition \\ref{definition-base-change}. In this case we say condition", "(\\ref{item-submersive}),", "(\\ref{item-invariant-closed}),", "(\\ref{item-intersect-invariant-closed}), or", "(\\ref{item-strong}) holds {\\it uniformly} or {\\it universally}." ], "refs": [ "groupoids-quotients-definition-base-change" ], "ref_ids": [ 13459 ] }, { "id": 13469, "type": "definition", "label": "groupoids-quotients-definition-functions", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-definition-functions", "contents": [ "Let $S$ be a scheme and $B$ an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-relation.", "Let $\\phi : U \\to X$ be an $R$-invariant morphism.", "Denote $\\phi' = \\phi \\circ s = \\phi \\circ t : R \\to X$.", "\\begin{enumerate}", "\\item We denote $(\\phi_*\\mathcal{O}_U)^R$ the $\\mathcal{O}_X$-sub-algebra", "of $\\phi_*\\mathcal{O}_U$ which is the equalizer of the two maps", "$$", "\\xymatrix{", "\\phi_*\\mathcal{O}_U", "\\ar@<1ex>[rr]^{\\phi_*s^\\sharp}", "\\ar@<-1ex>[rr]_{\\phi_*t^\\sharp}", "& &", "\\phi'_*\\mathcal{O}_R", "}", "$$", "on $X_\\etale$. We sometimes call this the", "{\\it sheaf of $R$-invariant functions on $X$}.", "\\item We say {\\it the functions on $X$ are the $R$-invariant functions on", "$U$} if the natural map $\\mathcal{O}_X \\to (\\phi_*\\mathcal{O}_U)^R$", "is an isomorphism.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13470, "type": "definition", "label": "groupoids-quotients-definition-good", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-definition-good", "contents": [ "Let $S$ be a scheme and $B$ an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-relation.", "A morphism $\\phi : U \\to X$ of algebraic spaces over $B$", "is called a {\\it good quotient} if", "\\begin{enumerate}", "\\item $\\phi$ is invariant,", "\\item $\\phi$ is affine,", "\\item $\\phi$ is surjective,", "\\item condition (\\ref{item-intersect-invariant-closed}) holds universally, and", "\\item the functions on $X$ are the $R$-invariant functions on $U$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13471, "type": "definition", "label": "groupoids-quotients-definition-geometric", "categories": [ "groupoids-quotients" ], "title": "groupoids-quotients-definition-geometric", "contents": [ "Let $S$ be a scheme and $B$ an algebraic space over $S$.", "Let $j : R \\to U \\times_B U$ be a pre-relation.", "A morphism $\\phi : U \\to X$ of algebraic spaces over $B$", "is called a {\\it geometric quotient} if", "\\begin{enumerate}", "\\item $\\phi$ is an orbit space,", "\\item condition (\\ref{item-submersive}) holds universally, i.e.,", "$\\phi$ is universally submersive, and", "\\item the functions on $X$ are the $R$-invariant functions on $U$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13491, "type": "definition", "label": "spaces-resolve-definition-blowup-at-point", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-definition-blowup-at-point", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.", "Let $x \\in |X|$ be a closed point. By", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-space-closed-point}", "we can represent $x$ by a closed immersion $i : \\Spec(k) \\to X$.", "The {\\it blowing up $X' \\to X$ of $X$ at $x$} means the blowing up of $X$", "in the closed subspace $Z = i(\\Spec(k)) \\subset X$." ], "refs": [ "decent-spaces-lemma-decent-space-closed-point" ], "ref_ids": [ 9510 ] }, { "id": 13492, "type": "definition", "label": "spaces-resolve-definition-normalized-blowup", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-definition-normalized-blowup", "contents": [ "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$ satisfying", "the equivalent conditions of", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-prepare-normalization}.", "Let $x \\in |X|$ be a closed point. The {\\it normalized blowup of $X$ at $x$}", "is the composition $X'' \\to X' \\to X$ where $X' \\to X$ is the blowup", "of $X$ at $x$ (Definition \\ref{definition-blowup-at-point})", "and $X'' \\to X'$ is the normalization of $X'$." ], "refs": [ "spaces-morphisms-lemma-prepare-normalization", "spaces-resolve-definition-blowup-at-point" ], "ref_ids": [ 4966, 13491 ] }, { "id": 13493, "type": "definition", "label": "spaces-resolve-definition-resolution", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-definition-resolution", "contents": [ "Let $S$ be a scheme. Let $Y$ be a Noetherian integral algebraic space over", "$S$. A {\\it resolution of singularities} of $X$ is a modification", "$f : X \\to Y$ such that $X$ is regular." ], "refs": [], "ref_ids": [] }, { "id": 13494, "type": "definition", "label": "spaces-resolve-definition-resolution-surface", "categories": [ "spaces-resolve" ], "title": "spaces-resolve-definition-resolution-surface", "contents": [ "Let $S$ be a scheme. Let $Y$ be a $2$-dimensional Noetherian integral", "algebraic space over $S$. We say $Y$ has a", "{\\it resolution of singularities by normalized blowups}", "if there exists a sequence", "$$", "Y_n \\to X_{n - 1} \\to \\ldots \\to Y_1 \\to Y_0 \\to Y", "$$", "where", "\\begin{enumerate}", "\\item $Y_i$ is proper over $Y$ for $i = 0, \\ldots, n$,", "\\item $Y_0 \\to Y$ is the normalization,", "\\item $Y_i \\to Y_{i - 1}$ is a normalized blowup for $i = 1, \\ldots, n$, and", "\\item $Y_n$ is regular.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13640, "type": "definition", "label": "duality-definition-dualizing-scheme", "categories": [ "duality" ], "title": "duality-definition-dualizing-scheme", "contents": [ "Let $X$ be a locally Noetherian scheme. An object $K$ of", "$D(\\mathcal{O}_X)$ is called a {\\it dualizing complex} if", "$K$ satisfies the equivalent conditions of", "Lemma \\ref{lemma-equivalent-definitions}." ], "refs": [ "duality-lemma-equivalent-definitions" ], "ref_ids": [ 13496 ] }, { "id": 13641, "type": "definition", "label": "duality-definition-good-dualizing", "categories": [ "duality" ], "title": "duality-definition-good-dualizing", "contents": [ "Let $S$ be a Noetherian scheme and let $\\omega_S^\\bullet$ be a dualizing", "complex on $S$. Let $X$ be a scheme of finite type over $S$.", "The complex $K$ constructed above is called the", "{\\it dualizing complex normalized relative to $\\omega_S^\\bullet$}", "and is denoted $\\omega_X^\\bullet$." ], "refs": [], "ref_ids": [] }, { "id": 13642, "type": "definition", "label": "duality-definition-gorenstein", "categories": [ "duality" ], "title": "duality-definition-gorenstein", "contents": [ "Let $X$ be a scheme. We say $X$ is {\\it Gorenstein} if $X$ is", "locally Noetherian and $\\mathcal{O}_{X, x}$ is Gorenstein for all $x \\in X$." ], "refs": [], "ref_ids": [] }, { "id": 13643, "type": "definition", "label": "duality-definition-gorenstein-morphism", "categories": [ "duality" ], "title": "duality-definition-gorenstein-morphism", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume that all the fibres $X_y$ are locally Noetherian schemes.", "\\begin{enumerate}", "\\item Let $x \\in X$, and $y = f(x)$. We say that $f$ is", "{\\it Gorenstein at $x$} if $f$ is flat at $x$, and the", "local ring of the scheme $X_y$ at $x$ is Gorenstein.", "\\item We say $f$ is a {\\it Gorenstein morphism} if $f$ is", "Gorenstein at every point of $X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 13644, "type": "definition", "label": "duality-definition-relative-dualizing-complex", "categories": [ "duality" ], "title": "duality-definition-relative-dualizing-complex", "contents": [ "Let $X \\to S$ be a morphism of schemes which is flat and", "locally of finite presentation. Let $W \\subset X \\times_S X$", "be any open such that the diagonal $\\Delta_{X/S} : X \\to X \\times_S X$", "factors through a closed immersion $\\Delta : X \\to W$.", "A {\\it relative dualizing complex} is a", "pair $(K, \\xi)$ consisting of an object $K \\in D(\\mathcal{O}_X)$", "and a map", "$$", "\\xi : \\Delta_*\\mathcal{O}_X \\longrightarrow L\\text{pr}_1^*K|_W", "$$", "in $D(\\mathcal{O}_W)$ such that", "\\begin{enumerate}", "\\item $K$ is $S$-perfect (Derived Categories of Schemes, Definition", "\\ref{perfect-definition-relatively-perfect}), and", "\\item $\\xi$ defines an isomorphism of $\\Delta_*\\mathcal{O}_X$", "with", "$R\\SheafHom_{\\mathcal{O}_W}(", "\\Delta_*\\mathcal{O}_X, L\\text{pr}_1^*K|_W)$.", "\\end{enumerate}" ], "refs": [ "perfect-definition-relatively-perfect" ], "ref_ids": [ 7119 ] }, { "id": 14106, "type": "definition", "label": "more-morphisms-definition-thickening", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-thickening", "contents": [ "Thickenings.", "\\begin{enumerate}", "\\item We say a scheme $X'$ is a {\\it thickening} of a scheme $X$ if", "$X$ is a closed subscheme of $X'$ and the underlying topological spaces", "are equal.", "\\item We say a scheme $X'$ is a {\\it first order thickening} of a scheme $X$ if", "$X$ is a closed subscheme of $X'$ and the quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_{X'}$ defining $X$ has square zero.", "\\item Given two thickenings $X \\subset X'$ and $Y \\subset Y'$ a", "{\\it morphism of thickenings} is a morphism $f' : X' \\to Y'$ such that", "$f'(X) \\subset Y$, i.e., such that $f'|_X$ factors through the closed", "subscheme $Y$. In this situation we set $f = f'|_X : X \\to Y$ and we say", "that $(f, f') : (X \\subset X') \\to (Y \\subset Y')$ is a morphism of", "thickenings.", "\\item Let $S$ be a scheme. We similarly define {\\it thickenings over $S$}, and", "{\\it morphisms of thickenings over $S$}. This means that the schemes", "$X, X', Y, Y'$ above are schemes over $S$, and that the morphisms", "$X \\to X'$, $Y \\to Y'$ and $f' : X' \\to Y'$ are morphisms over $S$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14107, "type": "definition", "label": "more-morphisms-definition-first-order-infinitesimal-neighbourhood", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-first-order-infinitesimal-neighbourhood", "contents": [ "Let $i : Z \\to X$ be an immersion of schemes. The", "{\\it first order infinitesimal neighbourhood} of $Z$ in $X$ is", "the first order thickening $Z \\subset Z'$ over $X$ described above." ], "refs": [], "ref_ids": [] }, { "id": 14108, "type": "definition", "label": "more-morphisms-definition-formally-unramified", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-formally-unramified", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "We say $f$ is {\\it formally unramified} if given any solid commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\", "S & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "where $T \\subset T'$ is a first order thickening of affine schemes over $S$", "there exists at most one dotted arrow making the diagram commute." ], "refs": [], "ref_ids": [] }, { "id": 14109, "type": "definition", "label": "more-morphisms-definition-universal-thickening", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-universal-thickening", "contents": [ "Let $h : Z \\to X$ be a formally unramified morphism of schemes.", "\\begin{enumerate}", "\\item The {\\it universal first order thickening} of $Z$ over $X$", "is the thickening $Z \\subset Z'$ constructed in", "Lemma \\ref{lemma-universal-thickening}.", "\\item The {\\it conormal sheaf of $Z$ over $X$} is the conormal sheaf", "of $Z$ in its universal first order thickening $Z'$ over $X$.", "\\end{enumerate}", "We often denote the conormal sheaf $\\mathcal{C}_{Z/X}$ in this situation." ], "refs": [ "more-morphisms-lemma-universal-thickening" ], "ref_ids": [ 13697 ] }, { "id": 14110, "type": "definition", "label": "more-morphisms-definition-formally-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-formally-etale", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "We say $f$ is {\\it formally \\'etale} if given any solid commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\", "S & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "where $T \\subset T'$ is a first order thickening of affine schemes over $S$", "there exists exactly one dotted arrow making the diagram commute." ], "refs": [], "ref_ids": [] }, { "id": 14111, "type": "definition", "label": "more-morphisms-definition-formally-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-formally-smooth", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "We say $f$ is {\\it formally smooth} if given any solid commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\", "S & T' \\ar[l] \\ar@{-->}[lu]", "}", "$$", "where $T \\subset T'$ is a first order thickening of affine schemes over $S$", "there exists a dotted arrow making the diagram commute." ], "refs": [], "ref_ids": [] }, { "id": 14112, "type": "definition", "label": "more-morphisms-definition-netherlander", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-netherlander", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "The {\\it naive cotangent complex of $f$}", "is the complex defined in Modules, Definition", "\\ref{modules-definition-cotangent-complex-morphism-ringed-topoi}.", "Notation: $\\NL_f$ or $\\NL_{X/Y}$." ], "refs": [ "modules-definition-cotangent-complex-morphism-ringed-topoi" ], "ref_ids": [ 13362 ] }, { "id": 14113, "type": "definition", "label": "more-morphisms-definition-normal", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-normal", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume that all the fibres $X_y$ are locally Noetherian schemes.", "\\begin{enumerate}", "\\item Let $x \\in X$, and $y = f(x)$. We say that $f$ is {\\it normal at $x$}", "if $f$ is flat at $x$, and the scheme $X_y$ is geometrically", "normal at $x$ over $\\kappa(y)$ (see", "Varieties, Definition \\ref{varieties-definition-geometrically-normal}).", "\\item We say $f$ is a {\\it normal morphism} if $f$ is normal", "at every point of $X$.", "\\end{enumerate}" ], "refs": [ "varieties-definition-geometrically-normal" ], "ref_ids": [ 11146 ] }, { "id": 14114, "type": "definition", "label": "more-morphisms-definition-regular", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-regular", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume that all the fibres $X_y$ are locally Noetherian schemes.", "\\begin{enumerate}", "\\item Let $x \\in X$, and $y = f(x)$. We say that $f$ is {\\it regular at $x$}", "if $f$ is flat at $x$, and the scheme $X_y$ is geometrically", "regular at $x$ over $\\kappa(y)$ (see", "Varieties, Definition \\ref{varieties-definition-geometrically-regular}).", "\\item We say $f$ is a {\\it regular morphism} if $f$ is regular", "at every point of $X$.", "\\end{enumerate}" ], "refs": [ "varieties-definition-geometrically-regular" ], "ref_ids": [ 11147 ] }, { "id": 14115, "type": "definition", "label": "more-morphisms-definition-CM", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-CM", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes.", "Assume that all the fibres $X_y$ are locally Noetherian schemes.", "\\begin{enumerate}", "\\item Let $x \\in X$, and $y = f(x)$. We say that $f$ is", "{\\it Cohen-Macaulay at $x$} if $f$ is flat at $x$, and the", "local ring of the scheme $X_y$ at $x$ is Cohen-Macaulay.", "\\item We say $f$ is a {\\it Cohen-Macaulay morphism} if $f$ is", "Cohen-Macaulay at every point of $X$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14116, "type": "definition", "label": "more-morphisms-definition-etale-neighbourhood", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-etale-neighbourhood", "contents": [ "Let $S$ be a scheme. Let $s \\in S$ be a point.", "\\begin{enumerate}", "\\item An {\\it \\'etale neighbourhood of $(S, s)$} is a", "pair $(U, u)$ together with an \\'etale morphism", "of schemes $\\varphi : U \\to S$ such that $\\varphi(u) = s$.", "\\item A {\\it morphism of \\'etale neighbourhoods} $f : (V, v) \\to (U, u)$", "of $(S, s)$ is simply a morphism of $S$-schemes $f : V \\to U$ such", "that $f(v) = u$.", "\\item An {\\it elementary \\'etale neighbourhood} is an \\'etale neighbourhood", "$\\varphi : (U, u) \\to (S, s)$ such that $\\kappa(s) = \\kappa(u)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14117, "type": "definition", "label": "more-morphisms-definition-relatively-finitely-presented-sheaf", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-relatively-finitely-presented-sheaf", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. We say", "$\\mathcal{F}$ is {\\it finitely presented relative to $S$} or", "{\\it of finite presentation relative to $S$}", "if there exists an affine open covering $S = \\bigcup V_i$ and", "for every $i$ an affine open covering", "$f^{-1}(V_i) = \\bigcup_j U_{ij}$ such that $\\mathcal{F}(U_{ij})$", "is a $\\mathcal{O}_X(U_{ij})$-module of finite presentation relative", "to $\\mathcal{O}_S(V_i)$." ], "refs": [], "ref_ids": [] }, { "id": 14118, "type": "definition", "label": "more-morphisms-definition-relative-pseudo-coherence", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-relative-pseudo-coherence", "contents": [ "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.", "Let $E$ be an object of $D(\\mathcal{O}_X)$. Let $\\mathcal{F}$ be an", "$\\mathcal{O}_X$-module. Fix $m \\in \\mathbf{Z}$.", "\\begin{enumerate}", "\\item We say $E$ is {\\it $m$-pseudo-coherent relative to $S$}", "if there exists an affine open covering $S = \\bigcup V_i$ and", "for each $i$ an affine open covering $f^{-1}(V_i) = \\bigcup U_{ij}$", "such that the equivalent conditions of", "Lemma \\ref{lemma-relatively-pseudo-coherent}", "are satisfied for each of the pairs $(U_{ij} \\to V_i, E|_{U_{ij}})$.", "\\item We say $E$ is {\\it pseudo-coherent relative to $S$}", "if $E$ is $m$-pseudo-coherent relative to $S$ for all $m \\in \\mathbf{Z}$.", "\\item We say $\\mathcal{F}$ is {\\it $m$-pseudo-coherent relative to $S$} if", "$\\mathcal{F}$ viewed as an object of $D(\\mathcal{O}_X)$ is", "$m$-pseudo-coherent relative to $S$.", "\\item We say $\\mathcal{F}$ is {\\it pseudo-coherent relative to $S$} if", "$\\mathcal{F}$ viewed as an object of $D(\\mathcal{O}_X)$ is", "pseudo-coherent relative to $S$.", "\\end{enumerate}" ], "refs": [ "more-morphisms-lemma-relatively-pseudo-coherent" ], "ref_ids": [ 13958 ] }, { "id": 14119, "type": "definition", "label": "more-morphisms-definition-pseudo-coherent", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-pseudo-coherent", "contents": [ "A morphism of schemes $f : X \\to S$ is called {\\it pseudo-coherent}", "if the equivalent conditions of", "Lemma \\ref{lemma-pseudo-coherent}", "are satisfied. In this case we also say that $X$ is pseudo-coherent", "over $S$." ], "refs": [ "more-morphisms-lemma-pseudo-coherent" ], "ref_ids": [ 13975 ] }, { "id": 14120, "type": "definition", "label": "more-morphisms-definition-perfect", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-perfect", "contents": [ "A morphism of schemes $f : X \\to S$ is called {\\it perfect}", "if the equivalent conditions of", "Lemma \\ref{lemma-perfect}", "are satisfied. In this case we also say that $X$ is perfect", "over $S$." ], "refs": [ "more-morphisms-lemma-perfect" ], "ref_ids": [ 13987 ] }, { "id": 14121, "type": "definition", "label": "more-morphisms-definition-lci", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-lci", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item Let $x \\in X$. We say that $f$ is {\\it Koszul at $x$} if $f$", "is of finite type at $x$ and there exists an open neighbourhood", "and a factorization of $f|_U$ as $\\pi \\circ i$ where $i : U \\to P$", "is a Koszul-regular immersion and $\\pi : P \\to S$ is smooth.", "\\item We say $f$ is a {\\it Koszul morphism}, or that", "$f$ is a {\\it local complete intersection morphism}", "if $f$ is Koszul at every point.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14122, "type": "definition", "label": "more-morphisms-definition-weakly-etale", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-weakly-etale", "contents": [ "A morphism of schemes $X \\to Y$ is {\\it weakly \\'etale} or", "{\\it absolutely flat} if both $X \\to Y$ and the diagonal", "morphism $X \\to X \\times_Y X$ are flat." ], "refs": [], "ref_ids": [] }, { "id": 14123, "type": "definition", "label": "more-morphisms-definition-ind-quasi-affine", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-ind-quasi-affine", "contents": [ "A scheme $X$ is {\\it ind-quasi-affine} if every quasi-compact open of", "$X$ is quasi-affine. Similarly, a morphism of schemes $X \\to Y$", "is {\\it ind-quasi-affine} if $f^{-1}(V)$ is ind-quasi-affine", "for each affine open $V$ in $Y$." ], "refs": [], "ref_ids": [] }, { "id": 14124, "type": "definition", "label": "more-morphisms-definition-affine-stratification", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-affine-stratification", "contents": [ "Let $X$ be a scheme. An {\\it affine stratification} is a", "locally finite stratification $X = \\coprod_{i \\in I} X_i$", "whose strata $X_i$ are affine and such that", "the inclusion morphisms $X_i \\to X$ are affine." ], "refs": [], "ref_ids": [] }, { "id": 14125, "type": "definition", "label": "more-morphisms-definition-affine-stratification-number", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-affine-stratification-number", "contents": [ "Let $X$ be a nonempty quasi-compact and quasi-separated scheme. The", "{\\it affine stratification number} is the smallest integer $n \\geq 0$", "such that the following equivalent conditions are satisfied", "\\begin{enumerate}", "\\item there exists a finite affine stratification", "$X = \\coprod_{i \\in I} X_i$ where $I$ has length $n$,", "\\item there exists an affine stratification", "$X = X_0 \\amalg X_1 \\amalg \\ldots \\amalg X_n$ with", "index set $\\{0, \\ldots, n\\}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14126, "type": "definition", "label": "more-morphisms-definition-weighting", "categories": [ "more-morphisms" ], "title": "more-morphisms-definition-weighting", "contents": [ "Let $f : X \\to Y$ be a locally quasi-finite morphism. A", "{\\it weighting} or a {\\it pond\\'eration} of $f$ is a map", "$w : X \\to \\mathbf{Z}$ such that for any diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & U \\ar[l]^h \\ar[d]^\\pi \\\\", "Y & V \\ar[l]_g", "}", "$$", "where $V \\to Y$ is \\'etale, $U \\subset X_V$ is open, and $U \\to V$ finite,", "the function $\\int_\\pi (w \\circ h)$ is locally constant." ], "refs": [], "ref_ids": [] }, { "id": 14277, "type": "definition", "label": "sites-modules-definition-free-abelian-presheaf-on", "categories": [ "sites-modules" ], "title": "sites-modules-definition-free-abelian-presheaf-on", "contents": [ "Let $\\mathcal{C}$ be a category. Let $\\mathcal{G}$ be a presheaf of sets.", "The {\\it free abelian presheaf} $\\mathbf{Z}_\\mathcal{G}$ on $\\mathcal{G}$", "is the abelian presheaf defined by the rule", "$$", "U \\longmapsto \\mathbf{Z}[\\mathcal{G}(U)].", "$$", "In the special case $\\mathcal{G} = h_X$ of a representable presheaf", "associated to an object $X$ of $\\mathcal{C}$", "we use the notation $\\mathbf{Z}_X = \\mathbf{Z}_{h_X}$. In other words", "$$", "\\mathbf{Z}_X(U) = \\mathbf{Z}[\\Mor_\\mathcal{C}(U, X)].", "$$" ], "refs": [], "ref_ids": [] }, { "id": 14278, "type": "definition", "label": "sites-modules-definition-free-abelian-sheaf-on", "categories": [ "sites-modules" ], "title": "sites-modules-definition-free-abelian-sheaf-on", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{G}$ be a presheaf of sets.", "The {\\it free abelian sheaf} $\\mathbf{Z}_\\mathcal{G}^\\#$", "on $\\mathcal{G}$ is the abelian sheaf $\\mathbf{Z}_\\mathcal{G}^\\#$", "which is the sheafification of the free abelian presheaf on $\\mathcal{G}$.", "In the special case $\\mathcal{G} = h_X$ of a representable presheaf", "associated to an object $X$ of $\\mathcal{C}$", "we use the notation $\\mathbf{Z}_X^\\#$." ], "refs": [], "ref_ids": [] }, { "id": 14279, "type": "definition", "label": "sites-modules-definition-ringed-site", "categories": [ "sites-modules" ], "title": "sites-modules-definition-ringed-site", "contents": [ "Ringed sites.", "\\begin{enumerate}", "\\item A {\\it ringed site} is a pair $(\\mathcal{C}, \\mathcal{O})$", "where $\\mathcal{C}$ is a site and $\\mathcal{O}$ is a sheaf of rings", "on $\\mathcal{C}$. The sheaf $\\mathcal{O}$ is called the", "{\\it structure sheaf} of the ringed site.", "\\item Let $(\\mathcal{C}, \\mathcal{O})$, $(\\mathcal{C}', \\mathcal{O}')$ be ringed", "sites. A {\\it morphism of ringed sites}", "$$", "(f, f^\\sharp) :", "(\\mathcal{C}, \\mathcal{O})", "\\longrightarrow", "(\\mathcal{C}', \\mathcal{O}')", "$$", "is given by a morphism of sites $f : \\mathcal{C} \\to \\mathcal{C}'$", "(see Sites, Definition \\ref{sites-definition-morphism-sites})", "together with a map of sheaves of rings", "$f^\\sharp : f^{-1}\\mathcal{O}' \\to \\mathcal{O}$, which by adjunction", "is the same thing as a map of sheaves of rings", "$f^\\sharp : \\mathcal{O}' \\to f_*\\mathcal{O}$.", "\\item Let", "$(f, f^\\sharp) :", "(\\mathcal{C}_1, \\mathcal{O}_1) \\to (\\mathcal{C}_2, \\mathcal{O}_2)$ and", "$(g, g^\\sharp) :", "(\\mathcal{C}_2, \\mathcal{O}_2) \\to (\\mathcal{C}_3, \\mathcal{O}_3)$", "be morphisms of ringed sites. Then we define", "the {\\it composition of morphisms of ringed sites}", "by the rule", "$$", "(g, g^\\sharp) \\circ (f, f^\\sharp) = (g \\circ f, f^\\sharp \\circ g^\\sharp).", "$$", "Here we use composition of morphisms of sites defined in", "Sites, Definition \\ref{sites-definition-composition-morphisms-sites}", "and $f^\\sharp \\circ g^\\sharp$ indicates the morphism of sheaves of", "rings", "$$", "\\mathcal{O}_3 \\xrightarrow{g^\\sharp} g_*\\mathcal{O}_2", "\\xrightarrow{g_*f^\\sharp} g_*f_*\\mathcal{O}_1 = (g \\circ f)_*\\mathcal{O}_1", "$$", "\\end{enumerate}" ], "refs": [ "sites-definition-morphism-sites", "sites-definition-composition-morphisms-sites" ], "ref_ids": [ 8665, 8666 ] }, { "id": 14280, "type": "definition", "label": "sites-modules-definition-ringed-topos", "categories": [ "sites-modules" ], "title": "sites-modules-definition-ringed-topos", "contents": [ "Ringed topoi.", "\\begin{enumerate}", "\\item A {\\it ringed topos} is a pair", "$(\\Sh(\\mathcal{C}), \\mathcal{O})$", "where $\\mathcal{C}$ is a site and $\\mathcal{O}$ is a sheaf of rings", "on $\\mathcal{C}$. The sheaf $\\mathcal{O}$ is called the", "{\\it structure sheaf} of the ringed topos.", "\\item Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$,", "$(\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be ringed topoi.", "A {\\it morphism of ringed topoi}", "$$", "(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}), \\mathcal{O})", "\\longrightarrow", "(\\Sh(\\mathcal{C}'), \\mathcal{O}')", "$$", "is given by a morphism of topoi $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$", "(see Sites, Definition \\ref{sites-definition-topos})", "together with a map of sheaves of rings", "$f^\\sharp : f^{-1}\\mathcal{O}' \\to \\mathcal{O}$, which by adjunction", "is the same thing as a map of sheaves of rings", "$f^\\sharp : \\mathcal{O}' \\to f_*\\mathcal{O}$.", "\\item Let", "$(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}_1), \\mathcal{O}_1)", "\\to (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2)$ and", "$(g, g^\\sharp) :", "(\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\to", "(\\Sh(\\mathcal{C}_3), \\mathcal{O}_3)$", "be morphisms of ringed topoi. Then we define", "the {\\it composition of morphisms of ringed topoi}", "by the rule", "$$", "(g, g^\\sharp) \\circ (f, f^\\sharp) = (g \\circ f, f^\\sharp \\circ g^\\sharp).", "$$", "Here we use composition of morphisms of topoi defined in", "Sites, Definition \\ref{sites-definition-topos}", "and $f^\\sharp \\circ g^\\sharp$ indicates the morphism of sheaves of", "rings", "$$", "\\mathcal{O}_3 \\xrightarrow{g^\\sharp} g_*\\mathcal{O}_2", "\\xrightarrow{g_*f^\\sharp} g_*f_*\\mathcal{O}_1 = (g \\circ f)_*\\mathcal{O}_1", "$$", "\\end{enumerate}" ], "refs": [ "sites-definition-topos", "sites-definition-topos" ], "ref_ids": [ 8667, 8667 ] }, { "id": 14281, "type": "definition", "label": "sites-modules-definition-2-morphism-ringed-topoi", "categories": [ "sites-modules" ], "title": "sites-modules-definition-2-morphism-ringed-topoi", "contents": [ "Let", "$f, g :", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be two morphisms of ringed topoi. A {\\it 2-morphism from $f$ to $g$}", "is given by a transformation of functors $t : f_* \\to g_*$ such that", "$$", "\\xymatrix{", "& \\mathcal{O}_\\mathcal{D}", "\\ar[ld]_{f^\\sharp}", "\\ar[rd]^{g^\\sharp} \\\\", "f_*\\mathcal{O}_\\mathcal{C} \\ar[rr]^t & &", "g_*\\mathcal{O}_\\mathcal{C}", "}", "$$", "is commutative." ], "refs": [], "ref_ids": [] }, { "id": 14282, "type": "definition", "label": "sites-modules-definition-presheaf-modules", "categories": [ "sites-modules" ], "title": "sites-modules-definition-presheaf-modules", "contents": [ "Let $\\mathcal{C}$ be a category, and", "let $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$.", "\\begin{enumerate}", "\\item A {\\it presheaf of $\\mathcal{O}$-modules}", "is given by an abelian presheaf $\\mathcal{F}$ together with a", "map of presheaves of sets", "$$", "\\mathcal{O} \\times \\mathcal{F} \\longrightarrow \\mathcal{F}", "$$", "such that for every object $U$ of $\\mathcal{C}$ the map", "$\\mathcal{O}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$", "defines the structure of an $\\mathcal{O}(U)$-module", "structure on the abelian group $\\mathcal{F}(U)$.", "\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "of presheaves of $\\mathcal{O}$-modules} is a morphism of abelian presheaves", "$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ such that", "the diagram", "$$", "\\xymatrix{", "\\mathcal{O} \\times \\mathcal{F} \\ar[r] \\ar[d]_{\\text{id} \\times \\varphi} &", "\\mathcal{F} \\ar[d]^{\\varphi} \\\\", "\\mathcal{O} \\times \\mathcal{G} \\ar[r] &", "\\mathcal{G}", "}", "$$", "commutes.", "\\item The set of $\\mathcal{O}$-module morphisms as above is", "denoted $\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$.", "\\item The category of presheaves of $\\mathcal{O}$-modules is denoted", "$\\textit{PMod}(\\mathcal{O})$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14283, "type": "definition", "label": "sites-modules-definition-sheaf-modules", "categories": [ "sites-modules" ], "title": "sites-modules-definition-sheaf-modules", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$.", "\\begin{enumerate}", "\\item A {\\it sheaf of $\\mathcal{O}$-modules} is a presheaf", "of $\\mathcal{O}$-modules $\\mathcal{F}$,", "see Definition \\ref{definition-presheaf-modules},", "such that the underlying presheaf of abelian groups $\\mathcal{F}$", "is a sheaf.", "\\item A {\\it morphism of sheaves of $\\mathcal{O}$-modules}", "is a morphism of presheaves of $\\mathcal{O}$-modules.", "\\item Given sheaves of $\\mathcal{O}$-modules", "$\\mathcal{F}$ and $\\mathcal{G}$ we denote", "$\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$", "the set of morphism of sheaves of $\\mathcal{O}$-modules.", "\\item The category of sheaves of $\\mathcal{O}$-modules", "is denoted $\\textit{Mod}(\\mathcal{O})$.", "\\end{enumerate}" ], "refs": [ "sites-modules-definition-presheaf-modules" ], "ref_ids": [ 14282 ] }, { "id": 14284, "type": "definition", "label": "sites-modules-definition-pushforward", "categories": [ "sites-modules" ], "title": "sites-modules-definition-pushforward", "contents": [ "Let", "$(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi or ringed sites.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{C}$-modules.", "We define the {\\it pushforward} of $\\mathcal{F}$ as the", "sheaf of $\\mathcal{O}_\\mathcal{D}$-modules which as a sheaf", "of abelian groups equals $f_*\\mathcal{F}$ and with", "module structure given by the restriction", "via $f^\\sharp : \\mathcal{O}_\\mathcal{D} \\to f_*\\mathcal{O}_\\mathcal{C}$", "of the module structure", "$$", "f_*\\mathcal{O}_\\mathcal{C} \\times f_*\\mathcal{F}", "\\longrightarrow", "f_*\\mathcal{F}", "$$", "from Lemma \\ref{lemma-pushforward-module}.", "\\item Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_\\mathcal{D}$-modules.", "We define the {\\it pullback} $f^*\\mathcal{G}$ to be the", "sheaf of $\\mathcal{O}_\\mathcal{C}$-modules defined by the formula", "$$", "f^*\\mathcal{G}", "=", "\\mathcal{O}_\\mathcal{C} \\otimes_{f^{-1}\\mathcal{O}_\\mathcal{D}}", "f^{-1}\\mathcal{G}", "$$", "where the ring map", "$f^{-1}\\mathcal{O}_\\mathcal{D} \\to \\mathcal{O}_\\mathcal{C}$", "is $f^\\sharp$, and where the module", "structure is given by Lemma \\ref{lemma-pullback-module}.", "\\end{enumerate}" ], "refs": [ "sites-modules-lemma-pushforward-module", "sites-modules-lemma-pullback-module" ], "ref_ids": [ 14151, 14152 ] }, { "id": 14285, "type": "definition", "label": "sites-modules-definition-g-shriek", "categories": [ "sites-modules" ], "title": "sites-modules-definition-g-shriek", "contents": [ "With $u : \\mathcal{C} \\to \\mathcal{D}$ satisfying (a), (b) above.", "For $\\mathcal{F} \\in \\textit{PAb}(\\mathcal{C})$ we define", "{\\it $g_{p!}\\mathcal{F}$} as the presheaf", "$$", "V \\longmapsto \\colim_{V \\to u(U)} \\mathcal{F}(U)", "$$", "with colimits over $(\\mathcal{I}_V^u)^{opp}$ taken in $\\textit{Ab}$. For", "$\\mathcal{F} \\in \\textit{PAb}(\\mathcal{C})$ we set", "{\\it $g_!\\mathcal{F} = (g_{p!}\\mathcal{F})^\\#$}." ], "refs": [], "ref_ids": [] }, { "id": 14286, "type": "definition", "label": "sites-modules-definition-global", "categories": [ "sites-modules" ], "title": "sites-modules-definition-global", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.", "\\begin{enumerate}", "\\item We say $\\mathcal{F}$ is a {\\it free $\\mathcal{O}$-module}", "if $\\mathcal{F}$ is isomorphic as an $\\mathcal{O}$-module", "to a sheaf of the form $\\bigoplus_{i \\in I} \\mathcal{O}$.", "\\item We say $\\mathcal{F}$ is {\\it finite free} if", "$\\mathcal{F}$ is isomorphic as an $\\mathcal{O}$-module", "to a sheaf of the form $\\bigoplus_{i \\in I} \\mathcal{O}$", "with a finite index set $I$.", "\\item We say $\\mathcal{F}$ is {\\it generated by global sections}", "if there exists a surjection", "$$", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{O} \\longrightarrow \\mathcal{F}", "$$", "from a free $\\mathcal{O}$-module onto $\\mathcal{F}$.", "\\item Given $r \\geq 0$ we say $\\mathcal{F}$ is", "{\\it generated by $r$ global sections} if there exists a surjection", "$\\mathcal{O}^{\\oplus r} \\to \\mathcal{F}$.", "\\item We say $\\mathcal{F}$ is {\\it generated by finitely many global sections}", "if it is generated by $r$ global sections for some $r \\geq 0$.", "\\item We say $\\mathcal{F}$ has a {\\it global presentation}", "if there exists an exact sequence", "$$", "\\bigoplus\\nolimits_{j \\in J} \\mathcal{O} \\longrightarrow", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{O} \\longrightarrow", "\\mathcal{F} \\longrightarrow 0", "$$", "of $\\mathcal{O}$-modules.", "\\item We say $\\mathcal{F}$ has a {\\it global finite presentation}", "if there exists an exact sequence", "$$", "\\bigoplus\\nolimits_{j \\in J} \\mathcal{O} \\longrightarrow", "\\bigoplus\\nolimits_{i \\in I} \\mathcal{O} \\longrightarrow", "\\mathcal{F} \\longrightarrow 0", "$$", "of $\\mathcal{O}$-modules with $I$ and $J$ finite sets.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14287, "type": "definition", "label": "sites-modules-definition-localize-ringed-site", "categories": [ "sites-modules" ], "title": "sites-modules-definition-localize-ringed-site", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $U \\in \\Ob(\\mathcal{C})$.", "\\begin{enumerate}", "\\item The ringed site $(\\mathcal{C}/U, \\mathcal{O}_U)$ is called the", "{\\it localization of the ringed site $(\\mathcal{C}, \\mathcal{O})$", "at the object $U$}.", "\\item The morphism of ringed topoi", "$(j_U, j_U^\\sharp) :", "(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)", "\\to", "(\\Sh(\\mathcal{C}), \\mathcal{O})$", "is called the {\\it localization morphism}.", "\\item The functor", "$j_{U*} : \\textit{Mod}(\\mathcal{O}_U) \\to \\textit{Mod}(\\mathcal{O})$", "is called the {\\it direct image functor}.", "\\item For a sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ on $\\mathcal{C}$", "the sheaf $j_U^*\\mathcal{F}$ is called the", "{\\it restriction of $\\mathcal{F}$ to $\\mathcal{C}/U$}.", "We will sometimes denote it by", "$\\mathcal{F}|_{\\mathcal{C}/U}$ or even $\\mathcal{F}|_U$.", "It is described by the simple rule $j_U^*(\\mathcal{F})(X/U) = \\mathcal{F}(X)$.", "\\item The left adjoint", "$j_{U!} : \\textit{Mod}(\\mathcal{O}_U) \\to \\textit{Mod}(\\mathcal{O})$", "of restriction is called {\\it extension by zero}. It exists and is", "exact by", "Lemmas \\ref{lemma-extension-by-zero} and", "\\ref{lemma-extension-by-zero-exact}.", "\\end{enumerate}" ], "refs": [ "sites-modules-lemma-extension-by-zero", "sites-modules-lemma-extension-by-zero-exact" ], "ref_ids": [ 14169, 14170 ] }, { "id": 14288, "type": "definition", "label": "sites-modules-definition-localize-ringed-topos", "categories": [ "sites-modules" ], "title": "sites-modules-definition-localize-ringed-topos", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.", "Let $\\mathcal{F} \\in \\Sh(\\mathcal{C})$.", "\\begin{enumerate}", "\\item The ringed topos", "$(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})$", "is called the", "{\\it localization of the ringed topos", "$(\\Sh(\\mathcal{C}), \\mathcal{O})$ at $\\mathcal{F}$}.", "\\item The morphism of ringed topoi", "$(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) :", "(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})", "\\to", "(\\Sh(\\mathcal{C}), \\mathcal{O})$ of", "Lemma \\ref{lemma-localize-ringed-topos}", "is called the {\\it localization morphism}.", "\\end{enumerate}" ], "refs": [ "sites-modules-lemma-localize-ringed-topos" ], "ref_ids": [ 14176 ] }, { "id": 14289, "type": "definition", "label": "sites-modules-definition-site-local", "categories": [ "sites-modules" ], "title": "sites-modules-definition-site-local", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.", "We will freely use the notions defined in", "Definition \\ref{definition-global}.", "\\begin{enumerate}", "\\item We say $\\mathcal{F}$ is {\\it locally free}", "if for every object $U$ of $\\mathcal{C}$ there exists a covering", "$\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction", "$\\mathcal{F}|_{\\mathcal{C}/U_i}$ is a free", "$\\mathcal{O}_{U_i}$-module.", "\\item We say $\\mathcal{F}$ is {\\it finite locally free}", "if for every object $U$ of $\\mathcal{C}$ there exists a covering", "$\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction", "$\\mathcal{F}|_{\\mathcal{C}/U_i}$ is a finite free", "$\\mathcal{O}_{U_i}$-module.", "\\item We say $\\mathcal{F}$ is {\\it locally generated by sections}", "if for every object $U$ of $\\mathcal{C}$ there exists a covering", "$\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction", "$\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an", "$\\mathcal{O}_{U_i}$-module generated by global sections.", "\\item Given $r \\geq 0$ we sat $\\mathcal{F}$ is {\\it locally generated", "by $r$ sections} if for every object $U$ of $\\mathcal{C}$ there exists", "a covering $\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each", "restriction $\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an", "$\\mathcal{O}_{U_i}$-module generated by $r$ global sections.", "\\item We say $\\mathcal{F}$ is {\\it of finite type}", "if for every object $U$ of $\\mathcal{C}$ there exists a covering", "$\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction", "$\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an", "$\\mathcal{O}_{U_i}$-module generated by finitely many global sections.", "\\item We say $\\mathcal{F}$ is {\\it quasi-coherent}", "if for every object $U$ of $\\mathcal{C}$ there exists a covering", "$\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction", "$\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an", "$\\mathcal{O}_{U_i}$-module which has a global presentation.", "\\item We say $\\mathcal{F}$ is {\\it of finite presentation}", "if for every object $U$ of $\\mathcal{C}$ there exists a covering", "$\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction", "$\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an", "$\\mathcal{O}_{U_i}$-module which has a finite global presentation.", "\\item We say $\\mathcal{F}$ is {\\it coherent} if and only if", "$\\mathcal{F}$ is of finite type, and for every object", "$U$ of $\\mathcal{C}$ and any $s_1, \\ldots, s_n \\in \\mathcal{F}(U)$", "the kernel of the map", "$\\bigoplus_{i = 1, \\ldots, n} \\mathcal{O}_U \\to \\mathcal{F}|_U$", "is of finite type on $(\\mathcal{C}/U, \\mathcal{O}_U)$.", "\\end{enumerate}" ], "refs": [ "sites-modules-definition-global" ], "ref_ids": [ 14286 ] }, { "id": 14290, "type": "definition", "label": "sites-modules-definition-flat", "categories": [ "sites-modules" ], "title": "sites-modules-definition-flat", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{O}$ be a presheaf of rings.", "\\begin{enumerate}", "\\item A presheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules is called", "{\\it flat} if the functor", "$$", "\\textit{PMod}(\\mathcal{O})", "\\longrightarrow", "\\textit{PMod}(\\mathcal{O}), \\quad", "\\mathcal{G} \\mapsto \\mathcal{G} \\otimes_{p, \\mathcal{O}} \\mathcal{F}", "$$", "is exact.", "\\item A map $\\mathcal{O} \\to \\mathcal{O}'$ of presheaves of rings", "is called {\\it flat} if $\\mathcal{O}'$ is flat as a presheaf of", "$\\mathcal{O}$-modules.", "\\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings", "and $\\mathcal{F}$ is a sheaf of $\\mathcal{O}$-modules, then we", "say $\\mathcal{F}$ is {\\it flat} if the functor", "$$", "\\textit{Mod}(\\mathcal{O})", "\\longrightarrow", "\\textit{Mod}(\\mathcal{O}), \\quad", "\\mathcal{G} \\mapsto \\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F}", "$$", "is exact.", "\\item A map $\\mathcal{O} \\to \\mathcal{O}'$ of sheaves of rings on a site", "is called {\\it flat} if $\\mathcal{O}'$ is flat as a sheaf of", "$\\mathcal{O}$-modules.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14291, "type": "definition", "label": "sites-modules-definition-flat-morphism", "categories": [ "sites-modules" ], "title": "sites-modules-definition-flat-morphism", "contents": [ "Let", "$(f, f^\\sharp) :", "(\\Sh(\\mathcal{C}), \\mathcal{O})", "\\longrightarrow", "(\\Sh(\\mathcal{C}'), \\mathcal{O}')$", "be a morphism of ringed topoi. We say $(f, f^\\sharp)$ is", "{\\it flat} if the ring map $f^\\sharp : f^{-1}\\mathcal{O}' \\to \\mathcal{O}$", "is flat. We say a morphism of ringed sites is {\\it flat}", "if the associated morphism of ringed topoi is flat." ], "refs": [], "ref_ids": [] }, { "id": 14292, "type": "definition", "label": "sites-modules-definition-flat-module", "categories": [ "sites-modules" ], "title": "sites-modules-definition-flat-module", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$", "be a morphism of ringed topoi. Let $\\mathcal{F}$ be a sheaf of", "$\\mathcal{O}$-modules. We say that $\\mathcal{F}$ is", "{\\it flat over $(\\Sh(\\mathcal{D}), \\mathcal{O}')$} if", "$\\mathcal{F}$ is flat as an $f^{-1}\\mathcal{O}'$-module." ], "refs": [], "ref_ids": [] }, { "id": 14293, "type": "definition", "label": "sites-modules-definition-invertible-sheaf", "categories": [ "sites-modules" ], "title": "sites-modules-definition-invertible-sheaf", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "\\begin{enumerate}", "\\item A finite locally free $\\mathcal{O}$-module $\\mathcal{F}$ is said", "to have {\\it rank $r$} if for every object $U$ of $\\mathcal{C}$ there", "exists a covering $\\{U_i \\to U\\}$ of $U$ such that $\\mathcal{F}|_{U_i}$", "is isomorphic to $\\mathcal{O}_{U_i}^{\\oplus r}$ as an", "$\\mathcal{O}_{U_i}$-module.", "\\item An $\\mathcal{O}$-module $\\mathcal{L}$ is {\\it invertible}", "if the functor", "$$", "\\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O}),\\quad", "\\mathcal{F} \\longmapsto \\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{L}", "$$", "is an equivalence.", "\\item The sheaf {\\it $\\mathcal{O}^*$} is the subsheaf of", "$\\mathcal{O}$ defined by the rule", "$$", "U \\longmapsto \\mathcal{O}^*(U) = \\{f \\in \\mathcal{O}(U) \\mid", "\\exists g \\in \\mathcal{O}(U)\\text{ such that }fg = 1\\}", "$$", "It is a sheaf of abelian groups with multiplication as the group law.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14294, "type": "definition", "label": "sites-modules-definition-pic", "categories": [ "sites-modules" ], "title": "sites-modules-definition-pic", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "The {\\it Picard group} $\\Pic(\\mathcal{O})$ of", "the ringed site is the", "abelian group whose elements are isomorphism classes of", "invertible $\\mathcal{O}$-modules, with addition", "corresponding to tensor product." ], "refs": [], "ref_ids": [] }, { "id": 14295, "type": "definition", "label": "sites-modules-definition-derivation", "categories": [ "sites-modules" ], "title": "sites-modules-definition-derivation", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings. Let $\\mathcal{F}$", "be an $\\mathcal{O}_2$-module. A {\\it $\\mathcal{O}_1$-derivation}", "or more precisely a {\\it $\\varphi$-derivation} into $\\mathcal{F}$", "is a map $D : \\mathcal{O}_2 \\to \\mathcal{F}$ which is additive, annihilates", "the image of $\\mathcal{O}_1 \\to \\mathcal{O}_2$, and satisfies the", "{\\it Leibniz rule}", "$$", "D(ab) = aD(b) + D(a)b", "$$", "for all $a, b$ local sections of $\\mathcal{O}_2$", "(wherever they are both defined). We denote", "$\\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})$", "the set of $\\varphi$-derivations into $\\mathcal{F}$." ], "refs": [], "ref_ids": [] }, { "id": 14296, "type": "definition", "label": "sites-modules-definition-module-differentials", "categories": [ "sites-modules" ], "title": "sites-modules-definition-module-differentials", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings. The {\\it module of differentials}", "of the ring map $\\varphi$ is the object representing the functor", "$\\mathcal{F} \\mapsto \\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})$", "which exists by Lemma \\ref{lemma-universal-module}.", "It is denoted $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$, and the {\\it universal", "$\\varphi$-derivation} is denoted", "$\\text{d} : \\mathcal{O}_2 \\to \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$." ], "refs": [ "sites-modules-lemma-universal-module" ], "ref_ids": [ 14228 ] }, { "id": 14297, "type": "definition", "label": "sites-modules-definition-sheaf-differentials", "categories": [ "sites-modules" ], "title": "sites-modules-definition-sheaf-differentials", "contents": [ "Let $X = (\\Sh(\\mathcal{C}), \\mathcal{O})$ and", "$Y = (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be ringed topoi.", "Let $(f, f^\\sharp) : X \\to Y$ be a morphism of ringed topoi.", "In this situation", "\\begin{enumerate}", "\\item for a sheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules a", "{\\it $Y$-derivation} $D : \\mathcal{O} \\to \\mathcal{F}$ is just a", "$f^\\sharp$-derivation, and", "\\item the {\\it sheaf of differentials $\\Omega_{X/Y}$ of $X$ over $Y$}", "is the module of differentials of", "$f^\\sharp : f^{-1}\\mathcal{O}' \\to \\mathcal{O}$,", "see Definition \\ref{definition-module-differentials}.", "\\end{enumerate}", "Thus $\\Omega_{X/Y}$ comes equipped with a {\\it universal $Y$-derivation}", "$\\text{d}_{X/Y} : \\mathcal{O} \\longrightarrow \\Omega_{X/Y}$. We sometimes", "write $\\Omega_{X/Y} = \\Omega_f$." ], "refs": [ "sites-modules-definition-module-differentials" ], "ref_ids": [ 14296 ] }, { "id": 14298, "type": "definition", "label": "sites-modules-definition-differential-operators", "categories": [ "sites-modules" ], "title": "sites-modules-definition-differential-operators", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$", "be a homomorphism of sheaves of rings. Let $k \\geq 0$ be an integer.", "Let $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of $\\mathcal{O}_2$-modules.", "A {\\it differential operator $D : \\mathcal{F} \\to \\mathcal{G}$ of order $k$}", "is an is an $\\mathcal{O}_1$-linear map such that for all local sections", "$g$ of $\\mathcal{O}_2$ the map $s \\mapsto D(gs) - gD(s)$ is a", "differential operator of order $k - 1$. For the base case $k = 0$", "we define a differential operator of order $0$ to be an", "$\\mathcal{O}_2$-linear map." ], "refs": [], "ref_ids": [] }, { "id": 14299, "type": "definition", "label": "sites-modules-definition-module-principal-parts", "categories": [ "sites-modules" ], "title": "sites-modules-definition-module-principal-parts", "contents": [ "Let $\\mathcal{C}$ be a site.", "Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_2$-modules.", "The module $\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F})$", "constructed in Lemma \\ref{lemma-module-principal-parts}", "is called the {\\it module of principal parts of order $k$} of $\\mathcal{F}$." ], "refs": [ "sites-modules-lemma-module-principal-parts" ], "ref_ids": [ 14237 ] }, { "id": 14300, "type": "definition", "label": "sites-modules-definition-naive-cotangent-complex", "categories": [ "sites-modules" ], "title": "sites-modules-definition-naive-cotangent-complex", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a", "homomorphism of sheaves of rings on $\\mathcal{C}$.", "The {\\it naive cotangent complex} $\\NL_{\\mathcal{B}/\\mathcal{A}}$", "is the chain complex (\\ref{equation-naive-cotangent-complex})", "$$", "\\NL_{\\mathcal{B}/\\mathcal{A}} =", "\\left(\\mathcal{I}/\\mathcal{I}^2", "\\longrightarrow", "\\Omega_{\\mathcal{A}[\\mathcal{B}]/\\mathcal{A}}", "\\otimes_{\\mathcal{A}[\\mathcal{B}]} \\mathcal{B}\\right)", "$$", "with $\\mathcal{I}/\\mathcal{I}^2$ placed in degree $-1$ and", "$\\Omega_{\\mathcal{A}[\\mathcal{B}]/\\mathcal{A}}", "\\otimes_{\\mathcal{A}[\\mathcal{B}]} \\mathcal{B}$", "placed in degree $0$." ], "refs": [], "ref_ids": [] }, { "id": 14301, "type": "definition", "label": "sites-modules-definition-cotangent-complex-morphism-ringed-topoi", "categories": [ "sites-modules" ], "title": "sites-modules-definition-cotangent-complex-morphism-ringed-topoi", "contents": [ "Let $X = (\\Sh(\\mathcal{C}), \\mathcal{O})$ and", "$Y = (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be ringed topoi.", "Let $(f, f^\\sharp) : X \\to Y$ be a morphism of ringed topoi.", "The {\\it naive cotangent complex} $\\NL_f = \\NL_{X/Y}$", "of the given morphism of ringed topoi is", "$\\NL_{\\mathcal{O}_X/f^{-1}\\mathcal{O}_Y}$.", "We sometimes write $\\NL_{X/Y} = \\NL_{\\mathcal{O}_X/\\mathcal{O}_Y}$." ], "refs": [], "ref_ids": [] }, { "id": 14302, "type": "definition", "label": "sites-modules-definition-locally-ringed", "categories": [ "sites-modules" ], "title": "sites-modules-definition-locally-ringed", "contents": [ "A ringed site $(\\mathcal{C}, \\mathcal{O})$ is said to be", "{\\it locally ringed site} if (\\ref{equation-one-is-never-zero})", "is an isomorphism, and the equivalent properties of", "Lemma \\ref{lemma-locally-ringed}", "are satisfied." ], "refs": [ "sites-modules-lemma-locally-ringed" ], "ref_ids": [ 14253 ] }, { "id": 14303, "type": "definition", "label": "sites-modules-definition-locally-ringed-topos", "categories": [ "sites-modules" ], "title": "sites-modules-definition-locally-ringed-topos", "contents": [ "A ringed topos $(\\Sh(\\mathcal{C}), \\mathcal{O})$ is said to be", "{\\it locally ringed} if the underlying ringed site", "$(\\mathcal{C}, \\mathcal{O})$ is locally ringed." ], "refs": [], "ref_ids": [] }, { "id": 14304, "type": "definition", "label": "sites-modules-definition-morphism-locally-ringed-topoi", "categories": [ "sites-modules" ], "title": "sites-modules-definition-morphism-locally-ringed-topoi", "contents": [ "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi. Assume", "$(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$", "and", "$(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "are locally ringed topoi. We say that $(f, f^\\sharp)$ is a", "{\\it morphism of locally ringed topoi} if and only if the", "diagram of sheaves", "$$", "\\xymatrix{", "f^{-1}(\\mathcal{O}^*_\\mathcal{D}) \\ar[r]_-{f^\\sharp} \\ar[d] &", "\\mathcal{O}^*_\\mathcal{C} \\ar[d] \\\\", "f^{-1}(\\mathcal{O}_\\mathcal{D}) \\ar[r]^-{f^\\sharp} &", "\\mathcal{O}_\\mathcal{C}", "}", "$$", "(see", "Lemma \\ref{lemma-locally-ringed-morphism})", "is cartesian. If $(f, f^\\sharp)$ is a morphism of ringed sites, then", "we say that it is a {\\it morphism of locally ringed sites} if", "the associated morphism of ringed topoi is a morphism of locally ringed", "topoi." ], "refs": [ "sites-modules-lemma-locally-ringed-morphism" ], "ref_ids": [ 14258 ] }, { "id": 14305, "type": "definition", "label": "sites-modules-definition-locally-constant", "categories": [ "sites-modules" ], "title": "sites-modules-definition-locally-constant", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a sheaf of sets, groups,", "abelian groups, rings, modules over a fixed ring $\\Lambda$, etc.", "\\begin{enumerate}", "\\item We say $\\mathcal{F}$ is a", "{\\it constant sheaf} of", "sets, groups, abelian groups, rings, modules over a fixed ring $\\Lambda$, etc", "if it is isomorphic as a sheaf of", "sets, groups, abelian groups, rings, modules over a fixed ring $\\Lambda$, etc", "to a constant sheaf $\\underline{E}$ as in Section \\ref{section-constant}.", "\\item We say $\\mathcal{F}$ is {\\it locally constant} if for every object", "$U$ of $\\mathcal{C}$ there exists a", "covering $\\{U_i \\to U\\}$ such that $\\mathcal{F}|_{U_i}$ is a constant sheaf.", "\\item If $\\mathcal{F}$ is a sheaf of sets or groups, then we say $\\mathcal{F}$", "is {\\it finite locally constant} if the constant values are finite sets or", "finite groups.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14378, "type": "definition", "label": "derham-definition-hodge-filtration", "categories": [ "derham" ], "title": "derham-definition-hodge-filtration", "contents": [ "Let $X \\to S$ be a morphism of schemes. The {\\it Hodge filtration}", "on $H^n_{dR}(X/S)$ is the filtration with terms", "$$", "F^pH^n_{dR}(X/S) = \\Im\\left(H^n(X, \\sigma_{\\geq p}\\Omega^\\bullet_{X/S})", "\\longrightarrow H^n_{dR}(X/S)\\right)", "$$", "where $\\sigma_{\\geq p}\\Omega^\\bullet_{X/S}$ is as in", "Homology, Section \\ref{homology-section-truncations}." ], "refs": [], "ref_ids": [] }, { "id": 14379, "type": "definition", "label": "derham-definition-local-product", "categories": [ "derham" ], "title": "derham-definition-local-product", "contents": [ "Let $X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an", "effective Cartier divisor. We say the", "{\\it de Rham complex of log poles is defined for $Y \\subset X$ over $S$}", "if for all $y \\in Y$ and local equation $f \\in \\mathcal{O}_{X, y}$", "of $Y$ we have", "\\begin{enumerate}", "\\item $\\mathcal{O}_{X, y} \\to \\Omega_{X/S, y}$, $g \\mapsto g \\text{d}f$", "is a split injection, and", "\\item $\\Omega^p_{X/S, y}$ is $f$-torsion free for all $p$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14380, "type": "definition", "label": "derham-definition-log-complex", "categories": [ "derham" ], "title": "derham-definition-log-complex", "contents": [ "Let $X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an", "effective Cartier divisor. Assume the de Rham complex of log poles", "is defined for $Y \\subset X$ over $S$. Then the complex", "$$", "\\Omega^\\bullet_{X/S}(\\log Y)", "$$", "constructed in Lemma \\ref{lemma-log-complex} is the", "{\\it de Rham complex of log poles for $Y \\subset X$ over $S$}." ], "refs": [ "derham-lemma-log-complex" ], "ref_ids": [ 14337 ] }, { "id": 14443, "type": "definition", "label": "trace-definition-geometric-frobenius", "categories": [ "trace" ], "title": "trace-definition-geometric-frobenius", "contents": [ "Let $k$ be a finite field with $q = p^f$ elements. Let $X$ be a scheme", "over $k$. The {\\it geometric frobenius} of $X$ is the morphism", "$\\pi_X : X \\to X$ over $\\Spec(k)$ which equals $F_X^f$." ], "refs": [], "ref_ids": [] }, { "id": 14444, "type": "definition", "label": "trace-definition-arithmetic-frobenius", "categories": [ "trace" ], "title": "trace-definition-arithmetic-frobenius", "contents": [ "The {\\it arithmetic frobenius} is the map", "$\\text{frob}_k : \\bar k \\to \\bar k$, $x \\mapsto x^q$ of $G_k$." ], "refs": [], "ref_ids": [] }, { "id": 14445, "type": "definition", "label": "trace-definition-geometric-frobenius-on-stalk", "categories": [ "trace" ], "title": "trace-definition-geometric-frobenius-on-stalk", "contents": [ "If $x \\in X(k)$ is a rational point and $\\bar x : \\Spec(\\bar k) \\to X$", "the geometric point lying over $x$, we let $\\pi_x : \\mathcal{F}_{\\bar x} \\to", "\\mathcal{F}_{\\bar x}$ denote the action by $\\text{frob}_k^{-1}$ and call it the", "{\\it geometric frobenius}\\footnote{This notation is not standard.", "This operator is denoted $F_x$ in \\cite{SGA4.5}. We will likely change", "this notation in the future.}" ], "refs": [], "ref_ids": [] }, { "id": 14446, "type": "definition", "label": "trace-definition-trace", "categories": [ "trace" ], "title": "trace-definition-trace", "contents": [ "The {\\it trace} of the endomorphism $a$ is the sum of the diagonal entries of", "a matrix representing it. This defines an additive map $\\text{Tr} :", "\\text{End}_\\Lambda(\\Lambda^{\\oplus m}) \\to \\Lambda^\\natural$." ], "refs": [], "ref_ids": [] }, { "id": 14447, "type": "definition", "label": "trace-definition-derived-functor", "categories": [ "trace" ], "title": "trace-definition-derived-functor", "contents": [ "Let $F: \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor and assume that", "$\\mathcal{A}$ has enough injectives. We define the {\\it total right derived", "functor of $F$} as the functor $RF: D^+(\\mathcal{A}) \\to D^+(\\mathcal{B})$", "fitting into the diagram", "$$", "\\xymatrix{", "D^+(\\mathcal{A}) \\ar[r]^{RF} & D^+(\\mathcal{B}) \\\\", "K^+(\\mathcal I) \\ar[u] \\ar[r]^F & K^+(\\mathcal{B}). \\ar[u]", "}", "$$", "This is possible since the left vertical arrow is invertible by the previous", "lemma. Similarly, let $G: \\mathcal{A} \\to \\mathcal{B}$ be a right exact", "functor and assume that $\\mathcal{A}$ has enough projectives. We define the", "{\\it total left derived functor of $G$} as the functor $LG: D^-(\\mathcal{A})", "\\to D^-(\\mathcal{B})$ fitting into the diagram", "$$", "\\xymatrix{", "D^-(\\mathcal{A}) \\ar[r]^{LG} & D^-(\\mathcal{B}) \\\\", "K^-(\\mathcal{P}) \\ar[u] \\ar[r]^G & K^-(\\mathcal{B}). \\ar[u]", "}", "$$", "This is possible since the left vertical arrow is invertible by the previous", "lemma." ], "refs": [], "ref_ids": [] }, { "id": 14448, "type": "definition", "label": "trace-definition-filtered", "categories": [ "trace" ], "title": "trace-definition-filtered", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "\\begin{enumerate}", "\\item Let $\\text{Fil}(\\mathcal{A})$ be the category of filtered objects", "$(A, F)$ of $\\mathcal{A}$, where $F$ is a filtration of the form", "$$", "A \\supset \\ldots \\supset F^n A \\supset F^{n+1}A \\supset \\ldots", "\\supset 0.", "$$", "This is an additive category.", "\\item We denote $\\text{Fil}^f(\\mathcal{A})$ the full", "subcategory of $\\text{Fil}(\\mathcal{A})$ whose objects $(A, F)$ have finite", "filtration. This is also an additive category.", "\\item An object $I \\in \\text{Fil}^f(\\mathcal{A})$ is called", "{\\it filtered injective} (respectively {\\it projective}) provided", "that $\\text{gr}^p(I) = \\text{gr}_F^p(I) = F^pI/F^{p+1}I$ is injective", "(resp. projective) in $\\mathcal{A}$ for all $p$.", "\\item The category of complexes", "$\\text{Comp}(\\text{Fil}^f(\\mathcal{A})) \\supset", "\\text{Comp}^+(\\text{Fil}^f(\\mathcal{A}))$", "and its homotopy category", "$K(\\text{Fil}^f(\\mathcal{A})) \\supset K^+(\\text{Fil}^f(\\mathcal A))$", "are defined as before.", "\\item A morphism $\\alpha : K^\\bullet \\to L^\\bullet$ of complexes in", "$\\text{Comp}(\\text{Fil}^f(\\mathcal{A}))$ is called a", "{\\it filtered quasi-isomorphism} provided that", "$$", "\\text{gr}^p(\\alpha): \\text{gr}^p(K^\\bullet) \\to \\text{gr}^p(L^\\bullet)", "$$", "is a quasi-isomorphism for all $p \\in \\mathbf{Z}$.", "\\item We define $DF(\\mathcal{A})$ (resp. $DF^+(\\mathcal{A})$)", "by inverting the filtered quasi-isomorphisms in", "$K(\\text{Fil}^f(\\mathcal{A}))$ (resp. $K^+(\\text{Fil}^f(\\mathcal{A}))$).", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14449, "type": "definition", "label": "trace-definition-filtered-derived-functors", "categories": [ "trace" ], "title": "trace-definition-filtered-derived-functors", "contents": [ "Let $T: \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor and assume that", "$\\mathcal{A}$ has enough injectives. Define $RT: DF^+(\\mathcal{A}) \\to D", "F^+(\\mathcal{B})$ to fit in the diagram", "$$", "\\xymatrix{", "DF^+(\\mathcal{A}) \\ar[r]^{RT} & DF^+(\\mathcal{B}) \\\\", "K^+(\\mathcal{I}) \\ar[u] \\ar[r]^{T \\quad} & K^+(\\text{Fil}^f(\\mathcal{B})).", "\\ar[u]}", "$$", "This is well-defined by the previous lemma. Let $G: \\mathcal{A} \\to", "\\mathcal{B}$ be a right exact functor and assume that $\\mathcal{A}$ has enough", "projectives. Define $LG: DF^+(\\mathcal{A}) \\to DF^+(\\mathcal{B})$ to fit in", "the diagram", "$$", "\\xymatrix{", "DF^-(\\mathcal{A}) \\ar[r]^{LG} & DF^-(\\mathcal{B}) \\\\", "K^-(\\mathcal{P}) \\ar[u] \\ar[r]^{G \\quad} & K^-(\\text{Fil}^f(\\mathcal{B})).", "\\ar[u]}", "$$", "Again, this is well-defined by the previous lemma.", "The functors $RT$, resp.\\ $LG$, are called the {\\it filtered derived", "functor} of $T$, resp.\\ $G$." ], "refs": [], "ref_ids": [] }, { "id": 14450, "type": "definition", "label": "trace-definition-perfect", "categories": [ "trace" ], "title": "trace-definition-perfect", "contents": [ "We denote by $K_{perf}(\\Lambda)$ the category whose objects are bounded", "complexes of finite projective $\\Lambda$-modules, and whose morphisms are", "morphisms of complexes up to homotopy. The functor $K_{perf}(\\Lambda)\\to", "D(\\Lambda)$ is fully faithful (Derived Categories, Lemma", "\\ref{derived-lemma-morphisms-from-projective-complex}).", "Denote $D_{perf}(\\Lambda)$ its essential image.", "An object of $D(\\Lambda)$ is called {\\it perfect} if it is in", "$D_{perf}(\\Lambda)$." ], "refs": [ "derived-lemma-morphisms-from-projective-complex" ], "ref_ids": [ 1862 ] }, { "id": 14451, "type": "definition", "label": "trace-definition-finite-tor-dimension", "categories": [ "trace" ], "title": "trace-definition-finite-tor-dimension", "contents": [ "Let $\\Lambda$ be a (possibly noncommutative) ring.", "An object $K\\in D(\\Lambda)$ has {\\it finite $\\text{Tor}$-dimension}", "if there exist $a, b \\in \\mathbf{Z}$ such that for any", "right $\\Lambda$-module $N$, we have", "$H^i(N \\otimes_{\\Lambda}^\\mathbf{L} K) = 0$ for all", "$i \\not \\in [a, b]$." ], "refs": [], "ref_ids": [] }, { "id": 14452, "type": "definition", "label": "trace-definition-global-lefschetz-number", "categories": [ "trace" ], "title": "trace-definition-global-lefschetz-number", "contents": [ "Let $\\Lambda$ be a finite ring, $X$ a projective curve over a finite field $k$", "and $K \\in D_{ctf}(X, \\Lambda)$ (for instance $K = \\underline\\Lambda$).", "There is a canonical map $c_K : \\pi_X^{-1}K \\to K$, and its base change", "$c_K|_{X_{\\bar k}}$ induces an action denoted $\\pi_X^*$ on the perfect", "complex $R\\Gamma(X_{\\bar k}, K|_{X_{\\bar k}})$. The", "{\\it global Lefschetz number} of $K$ is the trace", "$\\text{Tr}(\\pi_X^* |_{R\\Gamma(X_{\\bar k}, K)})$ of that action.", "It is an element of $\\Lambda^\\natural$." ], "refs": [], "ref_ids": [] }, { "id": 14453, "type": "definition", "label": "trace-definition-local-lefschetz-number", "categories": [ "trace" ], "title": "trace-definition-local-lefschetz-number", "contents": [ "With $\\Lambda, X, k, K$ as in", "Definition \\ref{definition-global-lefschetz-number}.", "Since $K\\in D_{ctf}(X, \\Lambda)$, for any geometric point $\\bar x$ of $X$,", "the complex $K_{\\bar x}$ is a perfect complex (in $D_{perf}(\\Lambda)$). As we", "have seen in Section \\ref{section-frobenii}, the Frobenius $\\pi_X$ acts on", "$K_{\\bar x}$. The {\\it local Lefschetz number} of $K$ is the sum", "$$", "\\sum\\nolimits_{x\\in X(k)} \\text{Tr}(\\pi_X |_{K_{\\overline{x}}})", "$$", "which is again an element of $\\Lambda^\\natural$." ], "refs": [ "trace-definition-global-lefschetz-number" ], "ref_ids": [ 14452 ] }, { "id": 14454, "type": "definition", "label": "trace-definition-trace-G", "categories": [ "trace" ], "title": "trace-definition-trace-G", "contents": [ "Let $f : P\\to P$ be an", "endomorphism of a finite projective $\\Lambda[G]$-module", "$P$. We define", "$$", "\\text{Tr}_{\\Lambda}^G(f; P) := \\varepsilon\\left(\\text{Tr}_{\\Lambda[G]}(f;", "P)\\right)", "$$", "to be the {\\it $G$-trace of $f$ on $P$}." ], "refs": [], "ref_ids": [] }, { "id": 14455, "type": "definition", "label": "trace-definition-l-adic-sheaf", "categories": [ "trace" ], "title": "trace-definition-l-adic-sheaf", "contents": [ "Let $X$ be a Noetherian scheme. A {\\it $\\mathbf{Z}_\\ell$-sheaf} on $X$, or", "simply an {\\it $\\ell$-adic sheaf} $\\mathcal{F}$ is an", "inverse system $\\left\\{\\mathcal{F}_n\\right\\}_{n\\geq 1}$ where", "\\begin{enumerate}", "\\item", "$\\mathcal{F}_n$ is a constructible $\\mathbf{Z}/\\ell^n\\mathbf{Z}$-module on", "$X_\\etale$, and", "\\item", "the transition maps $\\mathcal{F}_{n+1}\\to \\mathcal{F}_n$ induce isomorphisms", "$\\mathcal{F}_{n+1} \\otimes_{\\mathbf{Z}/\\ell^{n+1}\\mathbf{Z}}", "\\mathbf{Z}/\\ell^n\\mathbf{Z} \\cong \\mathcal{F}_n$.", "\\end{enumerate}", "We say that $\\mathcal{F}$ is {\\it lisse} if each $\\mathcal{F}_n$ is locally", "constant. A {\\it morphism} of such is merely a morphism of inverse systems." ], "refs": [], "ref_ids": [] }, { "id": 14456, "type": "definition", "label": "trace-definition-torsion-l-adic-sheaf", "categories": [ "trace" ], "title": "trace-definition-torsion-l-adic-sheaf", "contents": [ "A $\\mathbf{Z}_\\ell$-sheaf $\\mathcal{F}$ is {\\it torsion} if", "$\\ell^n : \\mathcal{F} \\to \\mathcal{F}$ is the zero map for some $n$.", "The abelian category", "of $\\mathbf{Q}_\\ell$-sheaves on $X$ is the quotient of the abelian category of", "$\\mathbf{Z}_\\ell$-sheaves by the Serre subcategory of torsion sheaves. In", "other words, its objects are $\\mathbf{Z}_\\ell$-sheaves on $X$, and if", "$\\mathcal{F}, \\mathcal{G}$ are two such, then", "$$", "\\Hom_{\\mathbf{Q}_\\ell} \\left(\\mathcal{F}, \\mathcal{G} \\right) =", "\\Hom_{\\mathbf{Z}_\\ell} \\left(\\mathcal{F}, \\mathcal{G}\\right)", "\\otimes_{\\mathbf{Z}_\\ell} \\mathbf{Q}_\\ell.", "$$", "We denote by $\\mathcal{F} \\mapsto \\mathcal{F} \\otimes \\mathbf{Q}_\\ell$ the", "quotient functor (right adjoint to the inclusion). If $\\mathcal{F} =", "\\mathcal{F}' \\otimes \\mathbf{Q}_\\ell$ where $\\mathcal{F}'$ is a", "$\\mathbf{Z}_\\ell$-sheaf and $\\bar x$ is a geometric point, then the", "{\\it stalk} of $\\mathcal{F}$ at $\\bar x$ is $\\mathcal{F}_{\\bar x} =", "\\mathcal{F}'_{\\bar x} \\otimes \\mathbf{Q}_\\ell$." ], "refs": [], "ref_ids": [] }, { "id": 14457, "type": "definition", "label": "trace-definition-cohomology-l-adic", "categories": [ "trace" ], "title": "trace-definition-cohomology-l-adic", "contents": [ "If $X$ is a separated scheme of finite type over an algebraically closed field", "$k$ and $\\mathcal{F} = \\left\\{\\mathcal{F}_n\\right\\}_{n\\geq 1}$ is a", "$\\mathbf{Z}_\\ell$-sheaf on $X$, then we define", "$$", "H^i(X, \\mathcal{F}) := \\lim_n H^i(X, \\mathcal{F}_n)", "\\quad\\text{and}\\quad", "H_c^i(X, \\mathcal{F}) := \\lim_n H_c^i(X, \\mathcal{F}_n).", "$$", "If $\\mathcal{F} = \\mathcal{F}'\\otimes \\mathbf{Q}_\\ell$ for a", "$\\mathbf{Z}_\\ell$-sheaf $\\mathcal{F}'$ then we set", "$$", "H_c^i(X , \\mathcal{F}) := H_c^i(X,", "\\mathcal{F}')\\otimes_{\\mathbf{Z}_\\ell}\\mathbf{Q}_\\ell.", "$$", "We call these the {\\it $\\ell$-adic cohomology} of $X$ with coefficients", "$\\mathcal{F}$." ], "refs": [], "ref_ids": [] }, { "id": 14458, "type": "definition", "label": "trace-definition-L-function-finite-ring", "categories": [ "trace" ], "title": "trace-definition-L-function-finite-ring", "contents": [ "Let $X$ be a scheme of finite type over a finite field $k$. Let $\\Lambda$ be a", "finite ring of order prime to the characteristic of $k$ and $\\mathcal{F}$ a", "constructible flat $\\Lambda$-module on $X_\\etale$. Then we set", "$$", "L(X, \\mathcal{F}) :=", "\\prod\\nolimits_{x\\in |X|}", "\\det(1 - \\pi_x^*T^{\\deg x} |_{\\mathcal{F}_{\\bar x}})^{-1} \\in \\Lambda [[ T ]]", "$$", "where $|X|$ is the set of closed points of $X$, $\\deg x = [\\kappa(x): k]$ and", "$\\bar x$ is a geometric point lying over $x$. This definition clearly", "generalizes to the case where $\\mathcal{F}$ is replaced by a", "$K \\in D_{ctf}(X, \\Lambda)$. We call this the {\\it $L$-function of", "$\\mathcal{F}$}." ], "refs": [], "ref_ids": [] }, { "id": 14459, "type": "definition", "label": "trace-definition-L-function-l-adic", "categories": [ "trace" ], "title": "trace-definition-L-function-l-adic", "contents": [ "Now assume that $\\mathcal{F}$ is a $\\mathbf{Q}_\\ell$-sheaf on $X$.", "In this case we define", "$$", "L(X, \\mathcal{F}) :=", "\\prod\\nolimits_{x \\in |X|}", "\\det(1 - \\pi_x^*T^{\\deg x} |_{\\mathcal{F}_{\\bar x}})^{-1}", "\\in \\mathbf{Q}_\\ell[[T]].", "$$", "Note that this product converges since there are finitely many points of a", "given degree. We call this the {\\it $L$-function of", "$\\mathcal{F}$}." ], "refs": [], "ref_ids": [] }, { "id": 14460, "type": "definition", "label": "trace-definition-open", "categories": [ "trace" ], "title": "trace-definition-open", "contents": [ "A subgroup of the form", "$\\text{Stab}(\\overline y\\in F_{\\overline{x}}(Y))\\subset \\pi_1(X, \\overline{x})$", "is called {\\it open}." ], "refs": [], "ref_ids": [] }, { "id": 14461, "type": "definition", "label": "trace-definition-unramified", "categories": [ "trace" ], "title": "trace-definition-unramified", "contents": [ "An {\\it unramified cusp form on $\\text{GL}_2(\\mathbf{A})$ with values in", "$\\Lambda$}\\footnote{This is likely nonstandard notation.}", "is a function", "$$", "f : \\text{GL}_2(\\mathbf{A}) \\to \\Lambda", "$$", "such that", "\\begin{enumerate}", "\\item $f(x\\gamma) = f(x)$ for all $x\\in \\text{GL}_2(\\mathbf{A})$ and all", "$\\gamma\\in \\text{GL}_2(K)$", "\\item $f(ux) = f(x)$ for all $x\\in \\text{GL}_2(\\mathbf{A})$ and all", "$u\\in \\text{GL}_2(O)$", "\\item for all $x\\in \\text{GL}_2(\\mathbf{A})$,", "$$", "\\int_{\\mathbf{A} \\mod K} f", "\\left(x", "\\left(", "\\begin{matrix}", "1 & z \\\\", "0 & 1", "\\end{matrix}", "\\right)", "\\right) dz = 0", "$$", "see \\cite[Section 4.1]{dJ-conjecture}", "for an explanation of how to make sense out", "of this for a general ring $\\Lambda$ in which $p$ is invertible.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14559, "type": "definition", "label": "sheaves-definition-presheaf", "categories": [ "sheaves" ], "title": "sheaves-definition-presheaf", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item A {\\it presheaf $\\mathcal{F}$ of sets on $X$} is a rule which", "assigns to each open $U \\subset X$ a set $\\mathcal{F}(U)$ and", "to each inclusion $V \\subset U$ a map", "$\\rho^U_V : \\mathcal{F}(U) \\to \\mathcal{F}(V)$ such that", "$\\rho^U_U = \\text{id}_{\\mathcal{F}(U)}$ and", "whenever $W \\subset V \\subset U$ we have", "$\\rho^U_W = \\rho^V_W \\circ \\rho ^U_V$.", "\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "of presheaves of sets on $X$} is a rule which assigns to each", "open $U \\subset X$ a map of sets $\\varphi : \\mathcal{F}(U)", "\\to \\mathcal{G}(U)$ compatible with restriction maps,", "i.e., whenever $V \\subset U \\subset X$ are open the", "diagram", "$$", "\\xymatrix{", "\\mathcal{F}(U) \\ar[r]^\\varphi \\ar[d]^{\\rho^U_V} &", "\\mathcal{G}(U) \\ar[d]^{\\rho^U_V} \\\\", "\\mathcal{F}(V) \\ar[r]^\\varphi & \\mathcal{G}(V)", "}", "$$", "commutes.", "\\item The category of presheaves of sets on $X$ will be denoted", "$\\textit{PSh}(X)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14560, "type": "definition", "label": "sheaves-definition-constant-presheaf", "categories": [ "sheaves" ], "title": "sheaves-definition-constant-presheaf", "contents": [ "Let $X$ be a topological space. Let $A$ be a set.", "The {\\it constant presheaf with value $A$} is the", "presheaf that assigns the set $A$ to every open", "$U \\subset X$, and such that all restriction mappings", "are $\\text{id}_A$." ], "refs": [], "ref_ids": [] }, { "id": 14561, "type": "definition", "label": "sheaves-definition-abelian-presheaves", "categories": [ "sheaves" ], "title": "sheaves-definition-abelian-presheaves", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item A {\\it presheaf of abelian groups on $X$} or an", "{\\it abelian presheaf over $X$}", "is a presheaf of sets $\\mathcal{F}$ such that for each open", "$U \\subset X$ the set $\\mathcal{F}(U)$ is endowed with", "the structure of an abelian group, and such that all restriction", "maps $\\rho^U_V$ are homomorphisms of abelian groups, see", "Lemma \\ref{lemma-abelian-presheaves} above.", "\\item A {\\it morphism of abelian presheaves over $X$}", "$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of presheaves", "of sets which induces", "a homomorphism of abelian groups $\\mathcal{F}(U) \\to \\mathcal{G}(U)$", "for every open $U \\subset X$.", "\\item The category of presheaves of abelian groups on $X$ is denoted", "$\\textit{PAb}(X)$.", "\\end{enumerate}" ], "refs": [ "sheaves-lemma-abelian-presheaves" ], "ref_ids": [ 14479 ] }, { "id": 14562, "type": "definition", "label": "sheaves-definition-presheaf-values-in-category", "categories": [ "sheaves" ], "title": "sheaves-definition-presheaf-values-in-category", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{C}$ be a category.", "\\begin{enumerate}", "\\item A {\\it presheaf $\\mathcal{F}$ on $X$ with values in $\\mathcal{C}$}", "is given by a rule which assigns to every open $U \\subset X$", "an object $\\mathcal{F}(U)$ of $\\mathcal{C}$", "and to each inclusion $V \\subset U$", "a morphism $\\rho_V^U : \\mathcal{F}(U) \\to \\mathcal{F}(V)$", "in $\\mathcal{C}$ such that whenever $W \\subset V \\subset U$", "we have $\\rho_W^U = \\rho_W^V \\circ \\rho_V^U$.", "\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "of presheaves with value in $\\mathcal{C}$} is given by a", "morphism $\\varphi : \\mathcal{F}(U) \\to \\mathcal{G}(U)$", "in $\\mathcal{C}$ compatible with restriction morphisms.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14563, "type": "definition", "label": "sheaves-definition-underlying-presheaf-sets", "categories": [ "sheaves" ], "title": "sheaves-definition-underlying-presheaf-sets", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{C}$ be a category.", "Let $F : \\mathcal{C} \\to \\textit{Sets}$ be a faithful functor.", "Let $\\mathcal{F}$ be a presheaf on $X$ with values in $\\mathcal{C}$.", "The presheaf of sets $U \\mapsto F(\\mathcal{F}(U))$", "is called the {\\it underlying presheaf of sets of $\\mathcal{F}$}." ], "refs": [], "ref_ids": [] }, { "id": 14564, "type": "definition", "label": "sheaves-definition-presheaf-modules", "categories": [ "sheaves" ], "title": "sheaves-definition-presheaf-modules", "contents": [ "Let $X$ be a topological space, and let $\\mathcal{O}$ be", "a presheaf of rings on $X$.", "\\begin{enumerate}", "\\item A {\\it presheaf of $\\mathcal{O}$-modules}", "is given by an abelian presheaf $\\mathcal{F}$ together with a", "map of presheaves of sets", "$$", "\\mathcal{O} \\times \\mathcal{F} \\longrightarrow \\mathcal{F}", "$$", "such that for every open $U \\subset X$ the map", "$\\mathcal{O}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$", "defines the structure of an $\\mathcal{O}(U)$-module", "structure on the abelian group $\\mathcal{F}(U)$.", "\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "of presheaves of $\\mathcal{O}$-modules} is a morphism of abelian presheaves", "$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ such that", "the diagram", "$$", "\\xymatrix{", "\\mathcal{O} \\times \\mathcal{F} \\ar[r] \\ar[d]_{\\text{id} \\times \\varphi} &", "\\mathcal{F} \\ar[d]^{\\varphi} \\\\", "\\mathcal{O} \\times \\mathcal{G} \\ar[r] &", "\\mathcal{G}", "}", "$$", "commutes.", "\\item The set of $\\mathcal{O}$-module morphisms as above is", "denoted $\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$.", "\\item The category of presheaves of $\\mathcal{O}$-modules", "is denoted $\\textit{PMod}(\\mathcal{O})$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14565, "type": "definition", "label": "sheaves-definition-sheaf", "categories": [ "sheaves" ], "title": "sheaves-definition-sheaf", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item A {\\it sheaf $\\mathcal{F}$ of sets on $X$} is a presheaf", "of sets which satisfies the following additional property: Given", "any open covering $U = \\bigcup_{i \\in I} U_i$ and any collection", "of sections $s_i \\in \\mathcal{F}(U_i)$, $i \\in I$ such that", "$\\forall i, j\\in I$", "$$", "s_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}", "$$", "there exists a unique section $s \\in \\mathcal{F}(U)$ such that", "$s_i = s|_{U_i}$ for all $i \\in I$.", "\\item A {\\it morphism of sheaves of sets} is simply a", "morphism of presheaves of sets.", "\\item The category of sheaves of sets on $X$ is denoted", "$\\Sh(X)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14566, "type": "definition", "label": "sheaves-definition-constant-sheaf", "categories": [ "sheaves" ], "title": "sheaves-definition-constant-sheaf", "contents": [ "Let $X$ be a topological space. Let $A$ be a set.", "The {\\it constant sheaf with value $A$} denoted $\\underline{A}$, or", "$\\underline{A}_X$ is the sheaf that assigns to an open $U \\subset X$", "the set of all locally constant maps $U \\to A$ with restriction mappings", "given by restrictions of functions." ], "refs": [], "ref_ids": [] }, { "id": 14567, "type": "definition", "label": "sheaves-definition-abelian-sheaf", "categories": [ "sheaves" ], "title": "sheaves-definition-abelian-sheaf", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item An {\\it abelian sheaf on $X$} or", "{\\it sheaf of abelian groups on $X$}", "is an abelian presheaf on $X$ such that the underlying presheaf of", "sets is a sheaf.", "\\item The category of sheaves of abelian groups", "is denoted $\\textit{Ab}(X)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14568, "type": "definition", "label": "sheaves-definition-sheaf-values-in-category", "categories": [ "sheaves" ], "title": "sheaves-definition-sheaf-values-in-category", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{C}$ be", "a category with products. A presheaf $\\mathcal{F}$ with", "values in $\\mathcal{C}$ on $X$ is a {\\it sheaf}", "if for every open covering the diagram", "$$", "\\xymatrix{", "\\mathcal{F}(U) \\ar[r]", "&", "\\prod\\nolimits_{i\\in I}", "\\mathcal{F}(U_i)", "\\ar@<1ex>[r] \\ar@<-1ex>[r]", "&", "\\prod\\nolimits_{(i_0, i_1) \\in I \\times I}", "\\mathcal{F}(U_{i_0} \\cap U_{i_1})", "}", "$$", "is an equalizer diagram in the category $\\mathcal{C}$." ], "refs": [], "ref_ids": [] }, { "id": 14569, "type": "definition", "label": "sheaves-definition-sheaf-modules", "categories": [ "sheaves" ], "title": "sheaves-definition-sheaf-modules", "contents": [ "Let $X$ be a topological space.", "Let $\\mathcal{O}$ be a sheaf of rings on $X$.", "\\begin{enumerate}", "\\item A {\\it sheaf of $\\mathcal{O}$-modules} is a presheaf", "of $\\mathcal{O}$-modules $\\mathcal{F}$,", "see Definition \\ref{definition-presheaf-modules},", "such that the underlying presheaf of abelian groups $\\mathcal{F}$", "is a sheaf.", "\\item A {\\it morphism of sheaves of $\\mathcal{O}$-modules}", "is a morphism of presheaves of $\\mathcal{O}$-modules.", "\\item Given sheaves of $\\mathcal{O}$-modules", "$\\mathcal{F}$ and $\\mathcal{G}$ we denote", "$\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$", "the set of morphism of sheaves of $\\mathcal{O}$-modules.", "\\item The category of sheaves of $\\mathcal{O}$-modules", "is denoted $\\textit{Mod}(\\mathcal{O})$.", "\\end{enumerate}" ], "refs": [ "sheaves-definition-presheaf-modules" ], "ref_ids": [ 14564 ] }, { "id": 14570, "type": "definition", "label": "sheaves-definition-separated", "categories": [ "sheaves" ], "title": "sheaves-definition-separated", "contents": [ "Let $X$ be a topological space.", "A presheaf of sets $\\mathcal{F}$ on $X$ is {\\it separated}", "if for every open $U \\subset X$ the map", "$\\mathcal{F}(U) \\to \\prod_{x \\in U} \\mathcal{F}_x$ is", "injective." ], "refs": [], "ref_ids": [] }, { "id": 14571, "type": "definition", "label": "sheaves-definition-algebraic-structure", "categories": [ "sheaves" ], "title": "sheaves-definition-algebraic-structure", "contents": [ "A {\\it type of algebraic structure} is given by a category $\\mathcal{C}$", "and a functor $F : \\mathcal{C} \\to \\textit{Sets}$ with the", "following properties", "\\begin{enumerate}", "\\item $F$ is faithful,", "\\item $\\mathcal{C}$ has limits and $F$ commutes with limits,", "\\item $\\mathcal{C}$ has filtered colimits and $F$ commutes with them, and", "\\item $F$ reflects isomorphisms.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14572, "type": "definition", "label": "sheaves-definition-injective-surjective", "categories": [ "sheaves" ], "title": "sheaves-definition-injective-surjective", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item A presheaf $\\mathcal{F}$ is called a {\\it subpresheaf} of a presheaf", "$\\mathcal{G}$ if $\\mathcal{F}(U) \\subset \\mathcal{G}(U)$ for all open", "$U \\subset X$ such that the restriction maps of $\\mathcal{G}$ induce the", "restriction maps of $\\mathcal{F}$. If $\\mathcal{F}$ and", "$\\mathcal{G}$ are sheaves, then $\\mathcal{F}$ is called a {\\it subsheaf}", "of $\\mathcal{G}$. We sometimes indicate this by the notation", "$\\mathcal{F} \\subset \\mathcal{G}$.", "\\item A morphism of presheaves of sets $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "on $X$ is called {\\it injective} if and only if", "$\\mathcal{F}(U) \\to \\mathcal{G}(U)$ is injective for all $U$ open in $X$.", "\\item A morphism of presheaves of sets $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "on $X$ is called {\\it surjective} if and only if", "$\\mathcal{F}(U) \\to \\mathcal{G}(U)$ is surjective for all $U$ open in $X$.", "\\item A morphism of sheaves of sets $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "on $X$ is called {\\it injective} if and only if", "$\\mathcal{F}(U) \\to \\mathcal{G}(U)$ is injective for all $U$ open in $X$.", "\\item A morphism of sheaves of sets $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "on $X$ is called {\\it surjective} if and only if for every open", "$U$ of $X$ and every section $s$ of $\\mathcal{G}(U)$ there exists an", "open covering $U = \\bigcup U_i$ such that $s|_{U_i}$ is in", "the image of $\\mathcal{F}(U_i) \\to \\mathcal{G}(U_i)$ for all $i$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14573, "type": "definition", "label": "sheaves-definition-f-map", "categories": [ "sheaves" ], "title": "sheaves-definition-f-map", "contents": [ "Let $f : X \\to Y$ be a continuous map.", "Let $\\mathcal{F}$ be a sheaf of sets on $X$ and", "let $\\mathcal{G}$ be a sheaf of sets on $Y$.", "An {\\it $f$-map $\\xi : \\mathcal{G} \\to \\mathcal{F}$}", "is a collection of maps", "$\\xi_V : \\mathcal{G}(V) \\to \\mathcal{F}(f^{-1}(V))$", "indexed by open subsets $V \\subset Y$ such that", "$$", "\\xymatrix{", "\\mathcal{G}(V) \\ar[r]_{\\xi_V} \\ar[d]_{\\text{restriction of }\\mathcal{G}} &", "\\mathcal{F}(f^{-1}V) \\ar[d]^{\\text{restriction of }\\mathcal{F}} \\\\", "\\mathcal{G}(V') \\ar[r]^{\\xi_{V'}} &", "\\mathcal{F}(f^{-1}V')", "}", "$$", "commutes for all $V' \\subset V \\subset Y$ open." ], "refs": [], "ref_ids": [] }, { "id": 14574, "type": "definition", "label": "sheaves-definition-composition-f-maps", "categories": [ "sheaves" ], "title": "sheaves-definition-composition-f-maps", "contents": [ "Suppose that $f : X \\to Y$ and $g : Y \\to Z$ are continuous", "maps of topological spaces. Suppose that $\\mathcal{F}$ is", "a sheaf on $X$, $\\mathcal{G}$ is a sheaf on $Y$, and", "$\\mathcal{H}$ is a sheaf on $Z$.", "Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be an $f$-map.", "Let $\\psi : \\mathcal{H} \\to \\mathcal{G}$ be an $g$-map.", "The {\\it composition of $\\varphi$ and $\\psi$} is the", "$(g \\circ f)$-map $\\varphi \\circ \\psi$ defined", "by the commutativity of the diagrams", "$$", "\\xymatrix{", "\\mathcal{H}(W) \\ar[rr]_{(\\varphi \\circ \\psi)_W}", "\\ar[rd]_{\\psi_W} & &", "\\mathcal{F}(f^{-1}g^{-1}W) \\\\", "&", "\\mathcal{G}(g^{-1}W)", "\\ar[ru]_{\\varphi_{g^{-1}W}}", "}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 14575, "type": "definition", "label": "sheaves-definition-ringed-space", "categories": [ "sheaves" ], "title": "sheaves-definition-ringed-space", "contents": [ "A {\\it ringed space} is a pair $(X, \\mathcal{O}_X)$ consisting", "of a topological space $X$ and a sheaf of rings $\\mathcal{O}_X$", "on $X$. A {\\it morphism of ringed spaces}", "$(X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ is a pair", "consisting of a continuous map $f : X \\to Y$ and an", "$f$-map of sheaves of rings", "$f^\\sharp : \\mathcal{O}_Y \\to \\mathcal{O}_X$." ], "refs": [], "ref_ids": [] }, { "id": 14576, "type": "definition", "label": "sheaves-definition-composition-maps-ringed-spaces", "categories": [ "sheaves" ], "title": "sheaves-definition-composition-maps-ringed-spaces", "contents": [ "Let", "$(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ and", "$(g, g^\\sharp) : (Y, \\mathcal{O}_Y) \\to (Z, \\mathcal{O}_Z)$", "be morphisms of ringed spaces. Then we define", "the {\\it composition of morphisms of ringed spaces}", "by the rule", "$$", "(g, g^\\sharp) \\circ (f, f^\\sharp) = (g \\circ f, f^\\sharp \\circ g^\\sharp).", "$$", "Here we use composition of $f$-maps defined in", "Definition \\ref{definition-composition-f-maps}." ], "refs": [ "sheaves-definition-composition-f-maps" ], "ref_ids": [ 14574 ] }, { "id": 14577, "type": "definition", "label": "sheaves-definition-pushforward", "categories": [ "sheaves" ], "title": "sheaves-definition-pushforward", "contents": [ "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$", "be a morphism of ringed spaces.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.", "We define the {\\it pushforward} of $\\mathcal{F}$ as the", "sheaf of $\\mathcal{O}_Y$-modules which as a sheaf", "of abelian groups equals $f_*\\mathcal{F}$ and with", "module structure given by the restriction", "via $f^\\sharp : \\mathcal{O}_Y \\to f_*\\mathcal{O}_X$", "of the module structure given", "in Lemma \\ref{lemma-pushforward-module}.", "\\item Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules.", "We define the {\\it pullback} $f^*\\mathcal{G}$ to be the", "sheaf of $\\mathcal{O}_X$-modules defined by the formula", "$$", "f^*\\mathcal{G}", "=", "\\mathcal{O}_X \\otimes_{f^{-1}\\mathcal{O}_Y} f^{-1}\\mathcal{G}", "$$", "where the ring map $f^{-1}\\mathcal{O}_Y \\to \\mathcal{O}_X$", "is the map corresponding to $f^\\sharp$, and where the module", "structure is given by Lemma \\ref{lemma-pullback-module}.", "\\end{enumerate}" ], "refs": [ "sheaves-lemma-pushforward-module", "sheaves-lemma-pullback-module" ], "ref_ids": [ 14517, 14518 ] }, { "id": 14578, "type": "definition", "label": "sheaves-definition-skyscraper-sheaf", "categories": [ "sheaves" ], "title": "sheaves-definition-skyscraper-sheaf", "contents": [ "Let $X$ be a topological space.", "\\begin{enumerate}", "\\item Let $x \\in X$ be a point. Denote $i_x : \\{x\\} \\to X$ the inclusion map.", "Let $A$ be a set and think of $A$ as a sheaf on the one point space $\\{x\\}$.", "We call $i_{x, *}A$ the {\\it skyscraper sheaf at $x$ with value $A$}.", "\\item If in (1) above $A$ is an abelian group then we think of", "$i_{x, *}A$ as a sheaf of abelian groups on $X$.", "\\item If in (1) above $A$ is an algebraic structure then we think", "of $i_{x, *}A$ as a sheaf of algebraic structures.", "\\item If $(X, \\mathcal{O}_X)$ is a ringed space, then we think", "of $i_x : \\{x\\} \\to X$ as a morphism of ringed spaces", "$(\\{x\\}, \\mathcal{O}_{X, x}) \\to (X, \\mathcal{O}_X)$", "and if $A$ is a $\\mathcal{O}_{X, x}$-module, then we think", "of $i_{x, *}A$ as a sheaf of $\\mathcal{O}_X$-modules.", "\\item We say a sheaf of sets $\\mathcal{F}$ is a {\\it skyscraper sheaf}", "if there exists an point $x$ of $X$ and a set $A$ such", "that $\\mathcal{F} \\cong i_{x, *}A$.", "\\item We say a sheaf of abelian groups $\\mathcal{F}$ is a", "{\\it skyscraper sheaf} if there exists an point $x$ of $X$", "and an abelian group $A$ such that $\\mathcal{F} \\cong i_{x, *}A$", "as sheaves of abelian groups.", "\\item We say a sheaf of algebraic structures $\\mathcal{F}$ is a", "{\\it skyscraper sheaf} if there exists an point $x$ of $X$", "and an algebraic structure $A$ such that $\\mathcal{F} \\cong i_{x, *}A$", "as sheaves of algebraic structures.", "\\item If $(X, \\mathcal{O}_X)$ is a ringed space and", "$\\mathcal{F}$ is a sheaf of $\\mathcal{O}_X$-modules, then", "we say $\\mathcal{F}$ is a {\\it skyscraper sheaf} if there", "exists a point $x \\in X$ and a $\\mathcal{O}_{X, x}$-module", "$A$ such that $\\mathcal{F} \\cong i_{x, *}A$", "as sheaves of $\\mathcal{O}_X$-modules.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14579, "type": "definition", "label": "sheaves-definition-presheaf-basis", "categories": [ "sheaves" ], "title": "sheaves-definition-presheaf-basis", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{B}$ be a", "basis for the topology on $X$.", "\\begin{enumerate}", "\\item A {\\it presheaf $\\mathcal{F}$ of sets on $\\mathcal{B}$}", "is a rule which assigns to each $U \\in \\mathcal{B}$ a set", "$\\mathcal{F}(U)$ and to each inclusion $V \\subset U$", "of elements of $\\mathcal{B}$ a map", "$\\rho^U_V : \\mathcal{F}(U) \\to \\mathcal{F}(V)$ such that", "$\\rho^U_U = \\text{id}_{\\mathcal{F}(U)}$ for all $U \\in \\mathcal{B}$", "whenever $W \\subset V \\subset U$ in $\\mathcal{B}$ we have", "$\\rho^U_W = \\rho^V_W \\circ \\rho ^U_V$.", "\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "of presheaves of sets on $\\mathcal{B}$} is a rule which assigns to each", "element $U \\in \\mathcal{B}$ a map of sets $\\varphi : \\mathcal{F}(U)", "\\to \\mathcal{G}(U)$ compatible with restriction maps.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14580, "type": "definition", "label": "sheaves-definition-sheaf-basis", "categories": [ "sheaves" ], "title": "sheaves-definition-sheaf-basis", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{B}$ be a", "basis for the topology on $X$.", "\\begin{enumerate}", "\\item A {\\it sheaf $\\mathcal{F}$ of sets on $\\mathcal{B}$} is a presheaf", "of sets on $\\mathcal{B}$ which satisfies the following additional", "property: Given any $U \\in \\mathcal{B}$, and any covering", "$U = \\bigcup_{i \\in I} U_i$ with $U_i \\in \\mathcal{B}$, and", "any coverings $U_i \\cap U_j = \\bigcup_{k \\in I_{ij}} U_{ijk}$ with", "$U_{ijk} \\in \\mathcal{B}$ the sheaf condition holds:", "\\begin{itemize}", "\\item[(**)] For any collection", "of sections $s_i \\in \\mathcal{F}(U_i)$, $i \\in I$ such that", "$\\forall i, j\\in I$, $\\forall k\\in I_{ij}$", "$$", "s_i|_{U_{ijk}} = s_j|_{U_{ijk}}", "$$", "there exists a unique section $s \\in \\mathcal{F}(U)$ such that", "$s_i = s|_{U_i}$ for all $i \\in I$.", "\\end{itemize}", "\\item A {\\it morphism of sheaves of sets on $\\mathcal{B}$} is simply a", "morphism of presheaves of sets.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14581, "type": "definition", "label": "sheaves-definition-sheaf-structures-basis", "categories": [ "sheaves" ], "title": "sheaves-definition-sheaf-structures-basis", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{B}$ be a", "basis for the topology on $X$. Let $(\\mathcal{C}, F)$ be", "a type of algebraic structure.", "\\begin{enumerate}", "\\item A {\\it presheaf $\\mathcal{F}$ with values in $\\mathcal{C}$", "on $\\mathcal{B}$} is a rule which assigns to each", "$U \\in \\mathcal{B}$ an object", "$\\mathcal{F}(U)$ of $\\mathcal{C}$ and to each inclusion $V \\subset U$", "of elements of $\\mathcal{B}$ a morphism", "$\\rho^U_V : \\mathcal{F}(U) \\to \\mathcal{F}(V)$ in $\\mathcal{C}$ such that", "$\\rho^U_U = \\text{id}_{\\mathcal{F}(U)}$ for all $U \\in \\mathcal{B}$ and", "whenever $W \\subset V \\subset U$ in $\\mathcal{B}$ we have", "$\\rho^U_W = \\rho^V_W \\circ \\rho ^U_V$.", "\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "of presheaves with values in $\\mathcal{C}$", "on $\\mathcal{B}$} is a rule which assigns to each", "element $U \\in \\mathcal{B}$ a morphism of", "algebraic structures $\\varphi : \\mathcal{F}(U) \\to \\mathcal{G}(U)$", "compatible with restriction maps.", "\\item Given a presheaf $\\mathcal{F}$ with values in $\\mathcal{C}$", "on $\\mathcal{B}$ we say that $U \\mapsto F(\\mathcal{F}(U))$ is the", "underlying presheaf of sets.", "\\item A {\\it sheaf $\\mathcal{F}$ with values in $\\mathcal{C}$", "on $\\mathcal{B}$} is a presheaf with values in $\\mathcal{C}$", "on $\\mathcal{B}$ whose underlying presheaf of sets is a sheaf.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14582, "type": "definition", "label": "sheaves-definition-sheaf-modules-basis", "categories": [ "sheaves" ], "title": "sheaves-definition-sheaf-modules-basis", "contents": [ "Let $X$ be a topological space. Let $\\mathcal{B}$ be a", "basis for the topology on $X$. Let $\\mathcal{O}$ be", "a presheaf of rings on $\\mathcal{B}$.", "\\begin{enumerate}", "\\item A {\\it presheaf of $\\mathcal{O}$-modules $\\mathcal{F}$", "on $\\mathcal{B}$} is a presheaf of abelian groups on", "$\\mathcal{B}$ together with a morphism of presheaves", "of sets $\\mathcal{O} \\times \\mathcal{F} \\to \\mathcal{F}$", "such that for all $U \\in \\mathcal{B}$ the map", "$\\mathcal{O}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$", "turns the group $\\mathcal{F}(U)$ into an $\\mathcal{O}(U)$-module.", "\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "of presheaves of $\\mathcal{O}$-modules on $\\mathcal{B}$}", "is a morphism of abelian presheaves on $\\mathcal{B}$", "which induces an $\\mathcal{O}(U)$-module homomorphism", "$\\mathcal{F}(U) \\to \\mathcal{G}(U)$ for every $U \\in \\mathcal{B}$.", "\\item Suppose that $\\mathcal{O}$ is a sheaf of rings", "on $\\mathcal{B}$. A {\\it sheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules", "on $\\mathcal{B}$} is a presheaf of $\\mathcal{O}$-modules", "on $\\mathcal{B}$ whose underlying presheaf of abelian groups", "is a sheaf.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14583, "type": "definition", "label": "sheaves-definition-restriction", "categories": [ "sheaves" ], "title": "sheaves-definition-restriction", "contents": [ "Let $X$ be a topological space.", "Let $j : U \\to X$ be the inclusion of an open subset.", "\\begin{enumerate}", "\\item Let $\\mathcal{G}$ be a presheaf of sets, abelian groups or", "algebraic structures on $X$. The presheaf $j_p\\mathcal{G}$ described", "in Lemma \\ref{lemma-j-pullback} is called", "the {\\it restriction of $\\mathcal{G}$ to $U$} and denoted $\\mathcal{G}|_U$.", "\\item Let $\\mathcal{G}$ be a sheaf of sets on $X$, abelian groups or", "algebraic structures on $X$. The sheaf $j^{-1}\\mathcal{G}$ is called", "the {\\it restriction of $\\mathcal{G}$ to $U$} and denoted $\\mathcal{G}|_U$.", "\\item If $(X, \\mathcal{O})$ is a ringed space, then the pair", "$(U, \\mathcal{O}|_U)$ is called the", "{\\it open subspace of $(X, \\mathcal{O})$ associated to $U$}.", "\\item If $\\mathcal{G}$ is a presheaf of $\\mathcal{O}$-modules", "then $\\mathcal{G}|_U$ together with the multiplication map", "$\\mathcal{O}|_U \\times \\mathcal{G}|_U \\to \\mathcal{G}|_U$", "(see Lemma \\ref{lemma-pullback-module})", "is called the {\\it restriction of $\\mathcal{G}$ to $U$}.", "\\end{enumerate}" ], "refs": [ "sheaves-lemma-j-pullback", "sheaves-lemma-pullback-module" ], "ref_ids": [ 14542, 14518 ] }, { "id": 14584, "type": "definition", "label": "sheaves-definition-j-shriek", "categories": [ "sheaves" ], "title": "sheaves-definition-j-shriek", "contents": [ "Let $X$ be a topological space.", "Let $j : U \\to X$ be the inclusion of an open subset.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be a presheaf of sets on $U$. We define", "the {\\it extension of $\\mathcal{F}$ by the empty set $j_{p!}\\mathcal{F}$}", "to be the presheaf of sets on $X$ defined by the rule", "$$", "j_{p!}\\mathcal{F}(V) =", "\\left\\{", "\\begin{matrix}", "\\emptyset & \\text{if} & V \\not \\subset U \\\\", "\\mathcal{F}(V) & \\text{if} & V \\subset U", "\\end{matrix}", "\\right.", "$$", "with obvious restriction mappings.", "\\item Let $\\mathcal{F}$ be a sheaf of sets on $U$. We define", "the {\\it extension of $\\mathcal{F}$ by the empty set $j_!\\mathcal{F}$}", "to be the sheafification of the presheaf $j_{p!}\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14585, "type": "definition", "label": "sheaves-definition-j-shriek-structures", "categories": [ "sheaves" ], "title": "sheaves-definition-j-shriek-structures", "contents": [ "Let $X$ be a topological space.", "Let $j : U \\to X$ be the inclusion of an open subset.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be an abelian presheaf on $U$.", "We define the {\\it extension $j_{p!}\\mathcal{F}$ of $\\mathcal{F}$ by $0$}", "to be the abelian presheaf on $X$ defined by the rule", "$$", "j_{p!}\\mathcal{F}(V) =", "\\left\\{", "\\begin{matrix}", "0 & \\text{if} & V \\not \\subset U \\\\", "\\mathcal{F}(V) & \\text{if} & V \\subset U", "\\end{matrix}", "\\right.", "$$", "with obvious restriction mappings.", "\\item Let $\\mathcal{F}$ be an abelian sheaf on $U$. We define", "the {\\it extension $j_!\\mathcal{F}$ of $\\mathcal{F}$ by $0$}", "to be the sheafification of the abelian presheaf $j_{p!}\\mathcal{F}$.", "\\item Let $\\mathcal{C}$ be a category having an initial object $e$.", "Let $\\mathcal{F}$ be a presheaf on $U$ with values in $\\mathcal{C}$.", "We define the {\\it extension $j_{p!}\\mathcal{F}$ of $\\mathcal{F}$ by $e$}", "to be the presheaf on $X$ with values in $\\mathcal{C}$ defined by the", "rule", "$$", "j_{p!}\\mathcal{F}(V) =", "\\left\\{", "\\begin{matrix}", "e & \\text{if} & V \\not \\subset U \\\\", "\\mathcal{F}(V) & \\text{if} & V \\subset U", "\\end{matrix}", "\\right.", "$$", "with obvious restriction mappings.", "\\item Let $(\\mathcal{C}, F)$ be a type of algebraic structure", "such that $\\mathcal{C}$ has an initial object $e$.", "Let $\\mathcal{F}$ be a sheaf of algebraic structures on $U$", "(of the give type). We define the", "{\\it extension $j_!\\mathcal{F}$ of $\\mathcal{F}$ by $e$}", "to be the sheafification of the presheaf $j_{p!}\\mathcal{F}$", "defined above.", "\\item Let $\\mathcal{O}$ be a presheaf of rings on $X$.", "Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}|_U$-modules.", "In this case we define the {\\it extension by $0$}", "to be the presheaf of $\\mathcal{O}$-modules which is equal to", "$j_{p!}\\mathcal{F}$ as an abelian presheaf endowed with", "the multiplication map", "$\\mathcal{O} \\times j_{p!}\\mathcal{F} \\to j_{p!}\\mathcal{F}$.", "\\item Let $\\mathcal{O}$ be a sheaf of rings on $X$.", "Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}|_U$-modules.", "In this case we define the {\\it extension by $0$}", "to be the $\\mathcal{O}$-module which is equal to", "$j_!\\mathcal{F}$ as an abelian sheaf endowed with", "the multiplication map $\\mathcal{O} \\times j_!\\mathcal{F} \\to j_!\\mathcal{F}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14757, "type": "definition", "label": "descent-definition-descent-datum-quasi-coherent", "categories": [ "descent" ], "title": "descent-definition-descent-datum-quasi-coherent", "contents": [ "Let $S$ be a scheme. Let $\\{f_i : S_i \\to S\\}_{i \\in I}$ be a family", "of morphisms with target $S$.", "\\begin{enumerate}", "\\item A {\\it descent datum $(\\mathcal{F}_i, \\varphi_{ij})$", "for quasi-coherent sheaves} with respect to the given family", "is given by a quasi-coherent sheaf $\\mathcal{F}_i$ on $S_i$ for", "each $i \\in I$, an isomorphism of quasi-coherent", "$\\mathcal{O}_{S_i \\times_S S_j}$-modules", "$\\varphi_{ij} : \\text{pr}_0^*\\mathcal{F}_i \\to \\text{pr}_1^*\\mathcal{F}_j$", "for each pair $(i, j) \\in I^2$", "such that for every triple of indices $(i, j, k) \\in I^3$ the", "diagram", "$$", "\\xymatrix{", "\\text{pr}_0^*\\mathcal{F}_i \\ar[rd]_{\\text{pr}_{01}^*\\varphi_{ij}}", "\\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} & &", "\\text{pr}_2^*\\mathcal{F}_k \\\\", "& \\text{pr}_1^*\\mathcal{F}_j \\ar[ru]_{\\text{pr}_{12}^*\\varphi_{jk}} &", "}", "$$", "of $\\mathcal{O}_{S_i \\times_S S_j \\times_S S_k}$-modules", "commutes. This is called the {\\it cocycle condition}.", "\\item A {\\it morphism $\\psi : (\\mathcal{F}_i, \\varphi_{ij}) \\to", "(\\mathcal{F}'_i, \\varphi'_{ij})$ of descent data} is given", "by a family $\\psi = (\\psi_i)_{i\\in I}$ of morphisms of", "$\\mathcal{O}_{S_i}$-modules $\\psi_i : \\mathcal{F}_i \\to \\mathcal{F}'_i$", "such that all the diagrams", "$$", "\\xymatrix{", "\\text{pr}_0^*\\mathcal{F}_i \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\text{pr}_0^*\\psi_i}", "& \\text{pr}_1^*\\mathcal{F}_j \\ar[d]^{\\text{pr}_1^*\\psi_j} \\\\", "\\text{pr}_0^*\\mathcal{F}'_i \\ar[r]^{\\varphi'_{ij}} &", "\\text{pr}_1^*\\mathcal{F}'_j \\\\", "}", "$$", "commute.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14758, "type": "definition", "label": "descent-definition-descent-datum-effective-quasi-coherent", "categories": [ "descent" ], "title": "descent-definition-descent-datum-effective-quasi-coherent", "contents": [ "Let $S$ be a scheme.", "Let $\\{S_i \\to S\\}_{i \\in I}$ be a family of morphisms", "with target $S$.", "\\begin{enumerate}", "\\item Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module.", "We call the unique descent on $\\mathcal{F}$ datum with respect to the covering", "$\\{S \\to S\\}$ the {\\it trivial descent datum}.", "\\item The pullback of the trivial descent datum to", "$\\{S_i \\to S\\}$ is called the {\\it canonical descent datum}.", "Notation: $(\\mathcal{F}|_{S_i}, can)$.", "\\item A descent datum $(\\mathcal{F}_i, \\varphi_{ij})$", "for quasi-coherent sheaves with respect to the given covering", "is said to be {\\it effective} if there exists a quasi-coherent", "sheaf $\\mathcal{F}$ on $S$ such that $(\\mathcal{F}_i, \\varphi_{ij})$", "is isomorphic to $(\\mathcal{F}|_{S_i}, can)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14759, "type": "definition", "label": "descent-definition-descent-datum-modules", "categories": [ "descent" ], "title": "descent-definition-descent-datum-modules", "contents": [ "Let $R \\to A$ be a ring map.", "\\begin{enumerate}", "\\item A {\\it descent datum $(N, \\varphi)$ for modules", "with respect to $R \\to A$}", "is given by an $A$-module $N$ and an isomorphism of", "$A \\otimes_R A$-modules", "$$", "\\varphi : N \\otimes_R A \\to A \\otimes_R N", "$$", "such that the {\\it cocycle condition} holds: the diagram", "of $A \\otimes_R A \\otimes_R A$-module maps", "$$", "\\xymatrix{", "N \\otimes_R A \\otimes_R A \\ar[rr]_{\\varphi_{02}}", "\\ar[rd]_{\\varphi_{01}}", "& &", "A \\otimes_R A \\otimes_R N \\\\", "& A \\otimes_R N \\otimes_R A \\ar[ru]_{\\varphi_{12}} &", "}", "$$", "commutes (see below for notation).", "\\item A {\\it morphism $(N, \\varphi) \\to (N', \\varphi')$ of descent data}", "is a morphism of $A$-modules $\\psi : N \\to N'$ such that", "the diagram", "$$", "\\xymatrix{", "N \\otimes_R A \\ar[r]_\\varphi \\ar[d]_{\\psi \\otimes \\text{id}_A} &", "A \\otimes_R N \\ar[d]^{\\text{id}_A \\otimes \\psi} \\\\", "N' \\otimes_R A \\ar[r]^{\\varphi'} &", "A \\otimes_R N'", "}", "$$", "is commutative.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14760, "type": "definition", "label": "descent-definition-descent-datum-effective-module", "categories": [ "descent" ], "title": "descent-definition-descent-datum-effective-module", "contents": [ "Let $R \\to A$ be a ring map.", "We say a descent datum $(N, \\varphi)$ is {\\it effective}", "if there exists an $R$-module $M$ and an isomorphism", "of descent data from $(M \\otimes_R A, can)$ to", "$(N, \\varphi)$." ], "refs": [], "ref_ids": [] }, { "id": 14761, "type": "definition", "label": "descent-definition-split-equalizer", "categories": [ "descent" ], "title": "descent-definition-split-equalizer", "contents": [ "A {\\it split equalizer} is a diagram (\\ref{equation-equalizer}) with", "$g_1 \\circ f = g_2 \\circ f$ for which there exist auxiliary morphisms", "$h : B \\to A$ and $i : C \\to B$ such that", "\\begin{equation}", "\\label{equation-split-equalizer-conditions}", "h \\circ f = 1_A, \\quad f \\circ h = i \\circ g_1, \\quad i \\circ g_2 = 1_B.", "\\end{equation}" ], "refs": [], "ref_ids": [] }, { "id": 14762, "type": "definition", "label": "descent-definition-universally-injective", "categories": [ "descent" ], "title": "descent-definition-universally-injective", "contents": [ "A ring map $f: R \\to S$ is {\\it universally injective}", "if it is universally injective as a morphism in $\\text{Mod}_R$." ], "refs": [], "ref_ids": [] }, { "id": 14763, "type": "definition", "label": "descent-definition-C", "categories": [ "descent" ], "title": "descent-definition-C", "contents": [ "Let $R$ be a ring. Define the contravariant functor", "{\\it $C$} $ : \\text{Mod}_R \\to \\text{Mod}_R$ by setting", "$$", "C(M) = \\Hom_{\\textit{Ab}}(M, \\mathbf{Q}/\\mathbf{Z}),", "$$", "with the $R$-action on $C(M)$ given by $rf(s) = f(rs)$." ], "refs": [], "ref_ids": [] }, { "id": 14764, "type": "definition", "label": "descent-definition-effective-descent", "categories": [ "descent" ], "title": "descent-definition-effective-descent", "contents": [ "The functor $f^*: \\text{Mod}_R \\to DD_{S/R}$", "is called {\\it base extension along $f$}. We say that $f$ is a", "{\\it descent morphism for modules} if $f^*$ is fully", "faithful. We say that $f$ is an {\\it effective descent morphism for modules}", "if $f^*$ is an equivalence of categories." ], "refs": [], "ref_ids": [] }, { "id": 14765, "type": "definition", "label": "descent-definition-pushforward", "categories": [ "descent" ], "title": "descent-definition-pushforward", "contents": [ "Define the functor {\\it $f_*$} $: DD_{S/R} \\to \\text{Mod}_R$ by taking", "$f_*(M, \\theta)$ to be the $R$-submodule of $M$ for which the diagram", "\\begin{equation}", "\\label{equation-equalizer-f}", "\\xymatrix@C=8pc{f_*(M,\\theta) \\ar[r] & M \\ar@<1ex>^{\\theta \\circ (1_M \\otimes ", "\\delta_0^1)}[r] \\ar@<-1ex>_{1_M \\otimes \\delta_1^1}[r] & ", "M \\otimes_{S, \\delta_1^1} S_2 ", "}", "\\end{equation}", "is an equalizer." ], "refs": [], "ref_ids": [] }, { "id": 14766, "type": "definition", "label": "descent-definition-structure-sheaf", "categories": [ "descent" ], "title": "descent-definition-structure-sheaf", "contents": [ "Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0]", "\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$.", "Let $S$ be a scheme.", "Let $\\Sch_\\tau$ be a big site containing $S$.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module.", "\\begin{enumerate}", "\\item The {\\it structure sheaf of the big site $(\\Sch/S)_\\tau$}", "is the sheaf of rings $T/S \\mapsto \\Gamma(T, \\mathcal{O}_T)$ which is", "denoted $\\mathcal{O}$ or $\\mathcal{O}_S$.", "\\item If $\\tau = \\etale$ the structure sheaf of the small site", "$S_\\etale$ is the sheaf of rings $T/S \\mapsto \\Gamma(T, \\mathcal{O}_T)$", "which is denoted $\\mathcal{O}$ or $\\mathcal{O}_S$.", "\\item The {\\it sheaf of $\\mathcal{O}$-modules associated to", "$\\mathcal{F}$} on the big site $(\\Sch/S)_\\tau$", "is the sheaf of $\\mathcal{O}$-modules", "$(f : T \\to S) \\mapsto \\Gamma(T, f^*\\mathcal{F})$", "which is denoted $\\mathcal{F}^a$ (and often simply $\\mathcal{F}$).", "\\item Let $\\tau = \\etale$ (resp.\\ $\\tau = Zariski$). The", "{\\it sheaf of $\\mathcal{O}$-modules associated to $\\mathcal{F}$}", "on the small site $S_\\etale$ (resp.\\ $S_{Zar}$) is the sheaf of", "$\\mathcal{O}$-modules $(f : T \\to S) \\mapsto \\Gamma(T, f^*\\mathcal{F})$", "which is denoted $\\mathcal{F}^a$ (and often simply $\\mathcal{F}$).", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14767, "type": "definition", "label": "descent-definition-parasitic", "categories": [ "descent" ], "title": "descent-definition-parasitic", "contents": [ "Let $S$ be a scheme. Let $\\tau \\in \\{Zar, \\etale,", "smooth, syntomic, fppf\\}$. Let $\\mathcal{F}$ be a presheaf", "of $\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$.", "\\begin{enumerate}", "\\item $\\mathcal{F}$ is called", "{\\it parasitic}\\footnote{This may be nonstandard notation.}", "if for every flat morphism $U \\to S$ we have $\\mathcal{F}(U) = 0$.", "\\item $\\mathcal{F}$ is called {\\it parasitic for the $\\tau$-topology}", "if for every $\\tau$-covering $\\{U_i \\to S\\}_{i \\in I}$ we have", "$\\mathcal{F}(U_i) = 0$ for all $i$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14768, "type": "definition", "label": "descent-definition-property-local", "categories": [ "descent" ], "title": "descent-definition-property-local", "contents": [ "Let $\\mathcal{P}$ be a property of schemes. Let", "$\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0] syntomic, \\linebreak[0]", "smooth, \\linebreak[0] \\etale, \\linebreak[0] Zariski\\}$.", "We say $\\mathcal{P}$ is {\\it local in the $\\tau$-topology} if for any", "$\\tau$-covering $\\{S_i \\to S\\}_{i \\in I}$ (see", "Topologies, Section \\ref{topologies-section-procedure})", "we have", "$$", "S \\text{ has }\\mathcal{P}", "\\Leftrightarrow", "\\text{each }S_i \\text{ has }\\mathcal{P}.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 14769, "type": "definition", "label": "descent-definition-germs", "categories": [ "descent" ], "title": "descent-definition-germs", "contents": [ "Germs of schemes.", "\\begin{enumerate}", "\\item A pair $(X, x)$ consisting of a scheme $X$ and a point $x \\in X$ is", "called the {\\it germ of $X$ at $x$}.", "\\item A {\\it morphism of germs} $f : (X, x) \\to (S, s)$", "is an equivalence class of morphisms of schemes $f : U \\to S$ with $f(x) = s$", "where $U \\subset X$ is an open neighbourhood of $x$. Two such", "$f$, $f'$ are said to be equivalent if and only if $f$ and $f'$", "agree in some open neighbourhood of $x$.", "\\item We define the {\\it composition of morphisms of germs}", "by composing representatives (this is well defined).", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14770, "type": "definition", "label": "descent-definition-etale-morphism-germs", "categories": [ "descent" ], "title": "descent-definition-etale-morphism-germs", "contents": [ "Let $f : (X, x) \\to (S, s)$ be a morphism of germs.", "We say $f$ is {\\it \\'etale} (resp.\\ {\\it smooth}) if there exists a", "representative $f : U \\to S$ of $f$ which is an \\'etale morphism", "(resp.\\ a smooth morphism) of schemes." ], "refs": [], "ref_ids": [] }, { "id": 14771, "type": "definition", "label": "descent-definition-local-at-point", "categories": [ "descent" ], "title": "descent-definition-local-at-point", "contents": [ "Let $\\mathcal{P}$ be a property of germs of schemes.", "We say that $\\mathcal{P}$ is {\\it \\'etale local}", "(resp.\\ {\\it smooth local}) if for any", "\\'etale (resp.\\ smooth) morphism of germs $(U', u') \\to (U, u)$", "we have $\\mathcal{P}(U, u) \\Leftrightarrow \\mathcal{P}(U', u')$." ], "refs": [], "ref_ids": [] }, { "id": 14772, "type": "definition", "label": "descent-definition-property-morphisms-local", "categories": [ "descent" ], "title": "descent-definition-property-morphisms-local", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes over a base.", "Let $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale, Zariski\\}$.", "We say $\\mathcal{P}$ is {\\it $\\tau$ local on the base}, or", "{\\it $\\tau$ local on the target}, or", "{\\it local on the base for the $\\tau$-topology} if for any", "$\\tau$-covering $\\{Y_i \\to Y\\}_{i \\in I}$ (see", "Topologies, Section \\ref{topologies-section-procedure})", "and any morphism of schemes $f : X \\to Y$ over $S$ we", "have", "$$", "f \\text{ has }\\mathcal{P}", "\\Leftrightarrow", "\\text{each }Y_i \\times_Y X \\to Y_i\\text{ has }\\mathcal{P}.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 14773, "type": "definition", "label": "descent-definition-property-morphisms-local-source", "categories": [ "descent" ], "title": "descent-definition-property-morphisms-local-source", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes.", "Let $\\tau \\in \\{Zariski, \\linebreak[0] fpqc, \\linebreak[0] fppf, \\linebreak[0]", "\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$.", "We say $\\mathcal{P}$ is", "{\\it $\\tau$ local on the source}, or", "{\\it local on the source for the $\\tau$-topology} if for", "any morphism of schemes $f : X \\to Y$ over $S$, and any", "$\\tau$-covering $\\{X_i \\to X\\}_{i \\in I}$ we", "have", "$$", "f \\text{ has }\\mathcal{P}", "\\Leftrightarrow", "\\text{each }X_i \\to Y\\text{ has }\\mathcal{P}.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 14774, "type": "definition", "label": "descent-definition-local-source-target", "categories": [ "descent" ], "title": "descent-definition-local-source-target", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes.", "We say $\\mathcal{P}$ is {\\it \\'etale local on source-and-target} if", "\\begin{enumerate}", "\\item (stable under precomposing with \\'etale maps)", "if $f : X \\to Y$ is \\'etale and $g : Y \\to Z$ has $\\mathcal{P}$,", "then $g \\circ f$ has $\\mathcal{P}$,", "\\item (stable under \\'etale base change)", "if $f : X \\to Y$ has $\\mathcal{P}$ and $Y' \\to Y$ is \\'etale, then", "the base change $f' : Y' \\times_Y X \\to Y'$ has $\\mathcal{P}$, and", "\\item (locality) given a morphism $f : X \\to Y$ the following are equivalent", "\\begin{enumerate}", "\\item $f$ has $\\mathcal{P}$,", "\\item for every $x \\in X$ there exists a commutative diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "with \\'etale vertical arrows and $u \\in U$ with $a(u) = x$ such that", "$h$ has $\\mathcal{P}$.", "\\end{enumerate}", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14775, "type": "definition", "label": "descent-definition-local-source-target-at-point", "categories": [ "descent" ], "title": "descent-definition-local-source-target-at-point", "contents": [ "Let $\\mathcal{Q}$ be a property of morphisms of germs of schemes.", "We say $\\mathcal{Q}$ is {\\it \\'etale local on the source-and-target}", "if for any commutative diagram", "$$", "\\xymatrix{", "(U', u') \\ar[d]_a \\ar[r]_{h'} & (V', v') \\ar[d]^b \\\\", "(U, u) \\ar[r]^h & (V, v)", "}", "$$", "of germs with \\'etale vertical arrows we have", "$\\mathcal{Q}(h) \\Leftrightarrow \\mathcal{Q}(h')$." ], "refs": [], "ref_ids": [] }, { "id": 14776, "type": "definition", "label": "descent-definition-descent-datum", "categories": [ "descent" ], "title": "descent-definition-descent-datum", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item Let $V \\to X$ be a scheme over $X$.", "A {\\it descent datum for $V/X/S$} is an isomorphism", "$\\varphi : V \\times_S X \\to X \\times_S V$ of schemes over", "$X \\times_S X$ satisfying the {\\it cocycle condition}", "that the diagram", "$$", "\\xymatrix{", "V \\times_S X \\times_S X \\ar[rd]^{\\varphi_{01}} \\ar[rr]_{\\varphi_{02}} &", "&", "X \\times_S X \\times_S V\\\\", "&", "X \\times_S V \\times_S X \\ar[ru]^{\\varphi_{12}}", "}", "$$", "commutes (with obvious notation).", "\\item We also say that the pair $(V/X, \\varphi)$ is", "a {\\it descent datum relative to $X \\to S$}.", "\\item A {\\it morphism $f : (V/X, \\varphi) \\to (V'/X, \\varphi')$ of", "descent data relative to $X \\to S$} is a morphism", "$f : V \\to V'$ of schemes over $X$ such that", "the diagram", "$$", "\\xymatrix{", "V \\times_S X \\ar[r]_{\\varphi} \\ar[d]_{f \\times \\text{id}_X} &", "X \\times_S V \\ar[d]^{\\text{id}_X \\times f} \\\\", "V' \\times_S X \\ar[r]^{\\varphi'} & X \\times_S V'", "}", "$$", "commutes.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14777, "type": "definition", "label": "descent-definition-descent-datum-for-family-of-morphisms", "categories": [ "descent" ], "title": "descent-definition-descent-datum-for-family-of-morphisms", "contents": [ "Let $S$ be a scheme.", "Let $\\{X_i \\to S\\}_{i \\in I}$ be a family of morphisms with target $S$.", "\\begin{enumerate}", "\\item A {\\it descent datum $(V_i, \\varphi_{ij})$ relative to the", "family $\\{X_i \\to S\\}$} is given by a scheme $V_i$ over $X_i$", "for each $i \\in I$, an isomorphism", "$\\varphi_{ij} : V_i \\times_S X_j \\to X_i \\times_S V_j$", "of schemes over $X_i \\times_S X_j$ for each pair $(i, j) \\in I^2$", "such that for every triple of indices $(i, j, k) \\in I^3$", "the diagram", "$$", "\\xymatrix{", "V_i \\times_S X_j \\times_S X_k", "\\ar[rd]^{\\text{pr}_{01}^*\\varphi_{ij}}", "\\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} &", "&", "X_i \\times_S X_j \\times_S V_k\\\\", "&", "X_i \\times_S V_j \\times_S X_k", "\\ar[ru]^{\\text{pr}_{12}^*\\varphi_{jk}}", "}", "$$", "of schemes over $X_i \\times_S X_j \\times_S X_k$ commutes", "(with obvious notation).", "\\item A {\\it morphism", "$\\psi : (V_i, \\varphi_{ij}) \\to (V'_i, \\varphi'_{ij})$", "of descent data} is given by a family", "$\\psi = (\\psi_i)_{i \\in I}$ of morphisms of", "$X_i$-schemes $\\psi_i : V_i \\to V'_i$ such that all the diagrams", "$$", "\\xymatrix{", "V_i \\times_S X_j \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\psi_i \\times \\text{id}} &", "X_i \\times_S V_j \\ar[d]^{\\text{id} \\times \\psi_j} \\\\", "V'_i \\times_S X_j \\ar[r]^{\\varphi'_{ij}} & X_i \\times_S V'_j", "}", "$$", "commute.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14778, "type": "definition", "label": "descent-definition-pullback-functor", "categories": [ "descent" ], "title": "descent-definition-pullback-functor", "contents": [ "With $S, S', X, X', f, a, a', h$ as in Lemma \\ref{lemma-pullback} the functor", "$$", "(V, \\varphi) \\longmapsto f^*(V, \\varphi)", "$$", "constructed in that lemma is called the {\\it pullback functor} on descent data." ], "refs": [ "descent-lemma-pullback" ], "ref_ids": [ 14733 ] }, { "id": 14779, "type": "definition", "label": "descent-definition-pullback-functor-family", "categories": [ "descent" ], "title": "descent-definition-pullback-functor-family", "contents": [ "With $\\mathcal{U} = \\{U_i \\to S'\\}_{i \\in I}$,", "$\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$, $\\alpha : I \\to J$, $h : S' \\to S$,", "and $g_i : U_i \\to V_{\\alpha(i)}$ as in Lemma \\ref{lemma-pullback-family}", "the functor", "$$", "(Y_j, \\varphi_{jj'}) \\longmapsto", "(g_i^*Y_{\\alpha(i)}, (g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')})", "$$", "constructed in that lemma", "is called the {\\it pullback functor} on descent data." ], "refs": [ "descent-lemma-pullback-family" ], "ref_ids": [ 14734 ] }, { "id": 14780, "type": "definition", "label": "descent-definition-effective", "categories": [ "descent" ], "title": "descent-definition-effective", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to S$ be a morphism of schemes.", "\\begin{enumerate}", "\\item Given a scheme $U$ over $S$ we have the", "{\\it trivial descent datum} of $U$ relative to", "$\\text{id} : S \\to S$, namely the identity morphism on $U$.", "\\item By Lemma \\ref{lemma-pullback} we get a", "{\\it canonical descent datum} on $X \\times_S U$", "relative to $X \\to S$ by pulling back the trivial", "descent datum via $f$. We often", "denote $(X \\times_S U, can)$ this descent datum.", "\\item A descent datum $(V, \\varphi)$ relative to $X/S$ is", "called {\\it effective} if $(V, \\varphi)$", "is isomorphic to the canonical descent datum", "$(X \\times_S U, can)$ for some scheme $U$ over $S$.", "\\end{enumerate}" ], "refs": [ "descent-lemma-pullback" ], "ref_ids": [ 14733 ] }, { "id": 14781, "type": "definition", "label": "descent-definition-effective-family", "categories": [ "descent" ], "title": "descent-definition-effective-family", "contents": [ "Let $S$ be a scheme.", "Let $\\{X_i \\to S\\}$ be a family of morphisms", "with target $S$.", "\\begin{enumerate}", "\\item Given a scheme $U$ over $S$", "we have a {\\it canonical descent datum} on the family of", "schemes $X_i \\times_S U$ by pulling back the trivial", "descent datum for $U$ relative to $\\{\\text{id} : S \\to S\\}$.", "We denote this descent datum $(X_i \\times_S U, can)$.", "\\item A descent datum $(V_i, \\varphi_{ij})$", "relative to $\\{X_i \\to S\\}$ is called {\\it effective}", "if there exists a scheme $U$ over $S$ such that", "$(V_i, \\varphi_{ij})$ is isomorphic to $(X_i \\times_S U, can)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14782, "type": "definition", "label": "descent-definition-descending-types-morphisms", "categories": [ "descent" ], "title": "descent-definition-descending-types-morphisms", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of schemes over a base.", "Let $\\tau \\in \\{Zariski, fpqc, fppf, \\etale, smooth, syntomic\\}$.", "We say", "{\\it morphisms of type $\\mathcal{P}$ satisfy descent for $\\tau$-coverings}", "if for", "any $\\tau$-covering $\\mathcal{U} : \\{U_i \\to S\\}_{i \\in I}$", "(see Topologies, Section \\ref{topologies-section-procedure}),", "any descent datum $(X_i, \\varphi_{ij})$ relative to $\\mathcal{U}$", "such that each morphism $X_i \\to U_i$ has property $\\mathcal{P}$", "is effective." ], "refs": [], "ref_ids": [] }, { "id": 14910, "type": "definition", "label": "simplicial-definition-face-degeneracy", "categories": [ "simplicial" ], "title": "simplicial-definition-face-degeneracy", "contents": [ "For any integer $n\\geq 1$, and any $0\\leq j \\leq n$ we let", "{\\it $\\delta^n_j : [n-1] \\to [n]$}", "denote the injective order preserving map skipping $j$. For any", "integer $n\\geq 0$, and any $0\\leq j \\leq n$ we denote", "{\\it $\\sigma^n_j : [n + 1] \\to [n]$}", "the surjective order preserving map with", "$(\\sigma^n_j)^{-1}(\\{j\\}) = \\{j, j + 1\\}$." ], "refs": [], "ref_ids": [] }, { "id": 14911, "type": "definition", "label": "simplicial-definition-simplicial-object", "categories": [ "simplicial" ], "title": "simplicial-definition-simplicial-object", "contents": [ "Let $\\mathcal{C}$ be a category.", "\\begin{enumerate}", "\\item A {\\it simplicial object $U$ of $\\mathcal{C}$}", "is a contravariant functor $U$ from $\\Delta$ to", "$\\mathcal{C}$, in a formula:", "$$", "U : \\Delta^{opp} \\longrightarrow \\mathcal{C}", "$$", "\\item If $\\mathcal{C}$ is the category of sets, then we call", "$U$ a {\\it simplicial set}.", "\\item If $\\mathcal{C}$ is the category of abelian groups,", "then we call $U$ a {\\it simplicial abelian group}.", "\\item A {\\it morphism of simplicial objects $U \\to U'$}", "is a transformation of functors.", "\\item The {\\it category of simplicial objects of $\\mathcal{C}$}", "is denoted $\\text{Simp}(\\mathcal{C})$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14912, "type": "definition", "label": "simplicial-definition-cosimplicial-object", "categories": [ "simplicial" ], "title": "simplicial-definition-cosimplicial-object", "contents": [ "Let $\\mathcal{C}$ be a category.", "\\begin{enumerate}", "\\item A {\\it cosimplicial object $U$ of $\\mathcal{C}$}", "is a covariant functor $U$ from $\\Delta$ to", "$\\mathcal{C}$, in a formula:", "$$", "U : \\Delta \\longrightarrow \\mathcal{C}", "$$", "\\item If $\\mathcal{C}$ is the category of sets, then we call", "$U$ a {\\it cosimplicial set}.", "\\item If $\\mathcal{C}$ is the category of abelian groups,", "then we call $U$ a {\\it cosimplicial abelian group}.", "\\item A {\\it morphism of cosimplicial objects $U \\to U'$}", "is a transformation of functors.", "\\item The {\\it category of cosimplicial objects of $\\mathcal{C}$}", "is denoted $\\text{CoSimp}(\\mathcal{C})$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14913, "type": "definition", "label": "simplicial-definition-product", "categories": [ "simplicial" ], "title": "simplicial-definition-product", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $U$ and $V$ be simplicial objects of $\\mathcal{C}$.", "Assume the products $U_n \\times V_n$ exist in $\\mathcal{C}$.", "The {\\it product of $U$ and $V$} is the simplicial object", "$U \\times V$ defined as follows:", "\\begin{enumerate}", "\\item $(U \\times V)_n = U_n \\times V_n$,", "\\item $d^n_i = (d^n_i, d^n_i)$, and", "\\item $s^n_i = (s^n_i, s^n_i)$.", "\\end{enumerate}", "In other words, $U \\times V$ is the product of the presheaves", "$U$ and $V$ on $\\Delta$." ], "refs": [], "ref_ids": [] }, { "id": 14914, "type": "definition", "label": "simplicial-definition-fibre-product", "categories": [ "simplicial" ], "title": "simplicial-definition-fibre-product", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $U, V, W$ be simplicial objects of $\\mathcal{C}$.", "Let $a : V \\to U$, $b : W \\to U$ be morphisms.", "Assume the fibre products $V_n \\times_{U_n} W_n$ exist in $\\mathcal{C}$.", "The {\\it fibre product of $V$ and $W$ over $U$} is the simplicial object", "$V \\times_U W$ defined as follows:", "\\begin{enumerate}", "\\item $(V \\times_U W)_n = V_n \\times_{U_n} W_n$,", "\\item $d^n_i = (d^n_i, d^n_i)$, and", "\\item $s^n_i = (s^n_i, s^n_i)$.", "\\end{enumerate}", "In other words, $V \\times_U W$ is the fibre product of the presheaves", "$V$ and $W$ over the presheaf $U$ on $\\Delta$." ], "refs": [], "ref_ids": [] }, { "id": 14915, "type": "definition", "label": "simplicial-definition-push-out", "categories": [ "simplicial" ], "title": "simplicial-definition-push-out", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $U, V, W$ be simplicial objects of $\\mathcal{C}$.", "Let $a : U \\to V$, $b : U \\to W$ be morphisms.", "Assume the pushouts $V_n \\amalg_{U_n} W_n$ exist in $\\mathcal{C}$.", "The {\\it pushout of $V$ and $W$ over $U$} is the simplicial object", "$V\\amalg_U W$ defined as follows:", "\\begin{enumerate}", "\\item $(V \\amalg_U W)_n = V_n \\amalg_{U_n} W_n$,", "\\item $d^n_i = (d^n_i, d^n_i)$, and", "\\item $s^n_i = (s^n_i, s^n_i)$.", "\\end{enumerate}", "In other words, $V\\amalg_U W$ is the pushout of the presheaves", "$V$ and $W$ over the presheaf $U$ on $\\Delta$." ], "refs": [], "ref_ids": [] }, { "id": 14916, "type": "definition", "label": "simplicial-definition-product-cosimplicial-objects", "categories": [ "simplicial" ], "title": "simplicial-definition-product-cosimplicial-objects", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $U$ and $V$ be cosimplicial objects of $\\mathcal{C}$.", "Assume the products $U_n \\times V_n$ exist in $\\mathcal{C}$.", "The {\\it product of $U$ and $V$} is the cosimplicial object", "$U \\times V$ defined as follows:", "\\begin{enumerate}", "\\item $(U \\times V)_n = U_n \\times V_n$,", "\\item for any $\\varphi : [n] \\to [m]$ the map", "$(U \\times V)(\\varphi) : U_n \\times V_n \\to U_m \\times V_m$", "is the product $U(\\varphi) \\times V(\\varphi)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14917, "type": "definition", "label": "simplicial-definition-fibre-product-cosimplicial-objects", "categories": [ "simplicial" ], "title": "simplicial-definition-fibre-product-cosimplicial-objects", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $U, V, W$ be cosimplicial objects of $\\mathcal{C}$.", "Let $a : V \\to U$ and $b : W \\to U$ be morphisms.", "Assume the fibre products $V_n \\times_{U_n} W_n$ exist in $\\mathcal{C}$.", "The {\\it fibre product of $V$ and $W$ over $U$} is the cosimplicial object", "$V \\times_U W$ defined as follows:", "\\begin{enumerate}", "\\item $(V \\times_U W)_n = V_n \\times_{U_n} W_n$,", "\\item for any $\\varphi : [n] \\to [m]$ the map", "$(V \\times_U W)(\\varphi) : V_n \\times_{U_n} W_n \\to V_m \\times_{U_m} W_m$", "is the product $V(\\varphi) \\times_{U(\\varphi)} W(\\varphi)$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14918, "type": "definition", "label": "simplicial-definition-terminology-simplicial-sets", "categories": [ "simplicial" ], "title": "simplicial-definition-terminology-simplicial-sets", "contents": [ "Let $U$ be a simplicial set.", "We say $x$ is an {\\it $n$-simplex of $U$} to signify that", "$x$ is an element of $U_n$. We say that $y$ is the $j$the", "{\\it face of $x$} to signify that $d^n_jx = y$. We say that", "$z$ is the $j$th {\\it degeneracy of $x$} if $z = s^n_jx$.", "A simplex is called {\\it degenerate} if it is the degeneracy", "of another simplex." ], "refs": [], "ref_ids": [] }, { "id": 14919, "type": "definition", "label": "simplicial-definition-truncated-simplicial-object", "categories": [ "simplicial" ], "title": "simplicial-definition-truncated-simplicial-object", "contents": [ "An {\\it $n$-truncated simplicial object of $\\mathcal{C}$}", "is a contravariant functor from $\\Delta_{\\leq n}$ to", "$\\mathcal{C}$. A {\\it morphism of $n$-truncated", "simplicial objects} is a transformation of functors.", "We denote the category of $n$-truncated", "simplicial objects of $\\mathcal{C}$ by", "the symbol $\\text{Simp}_n(\\mathcal{C})$." ], "refs": [], "ref_ids": [] }, { "id": 14920, "type": "definition", "label": "simplicial-definition-product-with-simplicial-set", "categories": [ "simplicial" ], "title": "simplicial-definition-product-with-simplicial-set", "contents": [ "Let $\\mathcal{C}$ be a category such that the coproduct of", "any two objects of $\\mathcal{C}$ exists. Let", "$U$ be a simplicial set. Let $V$ be a simplicial", "object of $\\mathcal{C}$. Assume that each $U_n$ is", "finite nonempty. In this case we define", "the {\\it product $U \\times V$ of $U$ and $V$}", "to be the simplicial object of $\\mathcal{C}$ whose", "$n$th term is the object", "$$", "(U \\times V)_n = \\coprod\\nolimits_{u\\in U_n} V_n", "$$", "with maps for $\\varphi : [m] \\to [n]$ given by the", "morphism", "$$", "\\coprod\\nolimits_{u\\in U_n} V_n", "\\longrightarrow", "\\coprod\\nolimits_{u'\\in U_m} V_m", "$$", "which maps the component $V_n$ corresponding to $u$ to the", "component $V_m$ corresponding to $u' = U(\\varphi)(u)$", "via the morphism $V(\\varphi)$.", "More loosely, if all of the coproducts displayed above", "exist (without assuming anything about $\\mathcal{C}$)", "we will say that the {\\it product $U \\times V$ exists}." ], "refs": [], "ref_ids": [] }, { "id": 14921, "type": "definition", "label": "simplicial-definition-hom-deltak-cosimplicial", "categories": [ "simplicial" ], "title": "simplicial-definition-hom-deltak-cosimplicial", "contents": [ "Let $\\mathcal{C}$ be a category with finite products.", "Let $V$ be a cosimplicial object of $\\mathcal{C}$.", "Let $U$ be a simplicial set such that each", "$U_n$ is finite nonempty.", "We define {\\it $\\Hom(U, V)$} to be", "the cosimplicial object of $\\mathcal{C}$ defined", "as follows:", "\\begin{enumerate}", "\\item we set $\\Hom(U, V)_n = \\prod_{u \\in U_n} V_n$,", "in other words the unique object of $\\mathcal{C}$ such", "that its $X$-valued points satisfy", "$$", "\\Mor_\\mathcal{C}(X, \\Hom(U, V)_n)", "=", "\\text{Map}(U_n, \\Mor_\\mathcal{C}(X, V_n))", "$$", "and", "\\item for $\\varphi : [m] \\to [n]$ we take the map", "$\\Hom(U, V)_m \\to \\Hom(U, V)_n$", "given by $f \\mapsto V(\\varphi) \\circ f \\circ U(\\varphi)$", "on $X$-valued points as above.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14922, "type": "definition", "label": "simplicial-definition-hom-deltak-simplicial", "categories": [ "simplicial" ], "title": "simplicial-definition-hom-deltak-simplicial", "contents": [ "Let $\\mathcal{C}$ be a category with finite products.", "Let $V$ be a simplicial object of $\\mathcal{C}$.", "Let $U$ be a cosimplicial set such that each $U_n$ is finite nonempty.", "We define {\\it $\\Hom(U, V)$} to be", "the simplicial object of $\\mathcal{C}$ defined", "as follows:", "\\begin{enumerate}", "\\item we set $\\Hom(U, V)_n = \\prod_{u \\in U_n} V_n$,", "in other words the unique object of $\\mathcal{C}$ such", "that its $X$-valued points satisfy", "$$", "\\Mor_\\mathcal{C}(X, \\Hom(U, V)_n)", "=", "\\text{Map}(U_n, \\Mor_\\mathcal{C}(X, V_n))", "$$", "and", "\\item for $\\varphi : [m] \\to [n]$ we take the map", "$\\Hom(U, V)_n \\to \\Hom(U, V)_m$", "given by $f \\mapsto V(\\varphi) \\circ f \\circ U(\\varphi)$", "on $X$-valued points as above.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14923, "type": "definition", "label": "simplicial-definition-hom-from-simplicial-set", "categories": [ "simplicial" ], "title": "simplicial-definition-hom-from-simplicial-set", "contents": [ "Let $\\mathcal{C}$ be a category such that the coproduct", "of any two objects exists.", "Let $U$ be a simplicial set, with $U_n$ finite nonempty", "for all $n \\geq 0$.", "Let $V$ be a simplicial object of $\\mathcal{C}$.", "We denote {\\it $\\Hom(U, V)$} any simplicial object of", "$\\mathcal{C}$ such that", "$$", "\\Mor_{\\text{Simp}(\\mathcal{C})}(W, \\Hom(U, V))", "=", "\\Mor_{\\text{Simp}(\\mathcal{C})}(W \\times U, V)", "$$", "functorially in the simplicial object $W$ of $\\mathcal{C}$." ], "refs": [], "ref_ids": [] }, { "id": 14924, "type": "definition", "label": "simplicial-definition-split", "categories": [ "simplicial" ], "title": "simplicial-definition-split", "contents": [ "Let $\\mathcal{C}$ be a category which admits finite nonempty coproducts.", "We say a simplicial object $U$ of $\\mathcal{C}$ is {\\it split}", "if there exist subobjects $N(U_m)$ of $U_m$, $m \\geq 0$", "with the property that", "\\begin{equation}", "\\label{equation-splitting}", "\\coprod\\nolimits_{\\varphi : [n] \\to [m]\\text{ surjective}}", "N(U_m)", "\\longrightarrow", "U_n", "\\end{equation}", "is an isomorphism for all $n \\geq 0$. If $U$ is an $r$-truncated", "simplicial object of $\\mathcal{C}$ then we say $U$ is {\\it split}", "if there exist subobjects $N(U_m)$ of $U_m$, $r \\geq m \\geq 0$", "with the property that (\\ref{equation-splitting})", "is an isomorphism for $r \\geq n \\geq 0$." ], "refs": [], "ref_ids": [] }, { "id": 14925, "type": "definition", "label": "simplicial-definition-augmentation", "categories": [ "simplicial" ], "title": "simplicial-definition-augmentation", "contents": [ "Let $\\mathcal{C}$ be a category.", "Let $U$ be a simplicial object of $\\mathcal{C}$.", "An {\\it augmentation $\\epsilon : U \\to X$ of", "$U$ towards an object $X$ of $\\mathcal{C}$}", "is a morphism from $U$ into the constant simplicial", "object $X$." ], "refs": [], "ref_ids": [] }, { "id": 14926, "type": "definition", "label": "simplicial-definition-eilenberg-maclane", "categories": [ "simplicial" ], "title": "simplicial-definition-eilenberg-maclane", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Let $A$ be an object of $\\mathcal{A}$ and", "let $k$ be an integer $\\geq 0$.", "The {\\it Eilenberg-Maclane object $K(A, k)$}", "is given by the object $K(A, k) = i_{k!}U$", "which is described in", "Lemma \\ref{lemma-eilenberg-maclane-object} above." ], "refs": [ "simplicial-lemma-eilenberg-maclane-object" ], "ref_ids": [ 14858 ] }, { "id": 14927, "type": "definition", "label": "simplicial-definition-homotopy", "categories": [ "simplicial" ], "title": "simplicial-definition-homotopy", "contents": [ "Let $\\mathcal{C}$ be a category having finite coproducts.", "Suppose that $U$ and $V$ are two simplicial objects of $\\mathcal{C}$.", "Let $a, b : U \\to V$ be two morphisms.", "\\begin{enumerate}", "\\item We say a morphism", "$$", "h : U \\times \\Delta[1] \\longrightarrow V", "$$", "is a {\\it homotopy from $a$ to $b$} if $a = h \\circ e_0$ and", "$b = h \\circ e_1$.", "\\item We say the morphisms $a$ and $b$ are {\\it homotopic} or are", "{\\it in the same homotopy class}", "if there exists a sequence of morphisms $a = a_0, a_1, \\ldots, a_n = b$", "from $U$ to $V$ such that for each $i = 1, \\ldots, n$ there either exists", "a homotopy from $a_{i - 1}$ to $a_i$ or there exists a homotopy", "from $a_i$ to $a_{i - 1}$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14928, "type": "definition", "label": "simplicial-definition-homotopy-equivalent", "categories": [ "simplicial" ], "title": "simplicial-definition-homotopy-equivalent", "contents": [ "Let $U$ and $V$ be two simplicial objects of a category $\\mathcal{C}$.", "We say a morphism $a : U \\to V$ is a {\\it homotopy equivalence}", "if there exists a morphism $b : V \\to U$ such that $a \\circ b$ is", "homotopic to $\\text{id}_V$ and $b \\circ a$ is homotopic to $\\text{id}_U$.", "We say $U$ and $V$ are {\\it homotopy equivalent} if there", "exists a homotopy equivalence $a : U \\to V$." ], "refs": [], "ref_ids": [] }, { "id": 14929, "type": "definition", "label": "simplicial-definition-homotopy-cosimplicial", "categories": [ "simplicial" ], "title": "simplicial-definition-homotopy-cosimplicial", "contents": [ "Let $\\mathcal{C}$ be a category having finite products.", "Let $U$ and $V$ be two cosimplicial objects of $\\mathcal{C}$.", "Let $a, b : U \\to V$ be two morphisms of cosimplicial objects", "of $\\mathcal{C}$.", "\\begin{enumerate}", "\\item We say a morphism", "$$", "h : U \\longrightarrow \\Hom(\\Delta[1], V)", "$$", "such that $a = e_0 \\circ h$ and $b = e_1 \\circ h$ is a", "{\\it homotopy from $a$ to $b$}.", "\\item We say $a$ and $b$ are {\\it homotopic} or are", "{\\it in the same homotopy class} if there exists a sequence", "$a = a_0, a_1, \\ldots, a_n = b$ of morphisms from $U$ to $V$", "such that for each $i = 1, \\ldots, n$ there either exists a", "homotopy from $a_i$ to $a_{i - 1}$ or there exists a homotopy", "from $a_{i - 1}$ to $a_i$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14930, "type": "definition", "label": "simplicial-definition-trivial-kan", "categories": [ "simplicial" ], "title": "simplicial-definition-trivial-kan", "contents": [ "A map $X \\to Y$ of simplicial sets is called a {\\it trivial Kan fibration}", "if $X_0 \\to Y_0$ is surjective and for all $n \\geq 1$ and any commutative", "solid diagram", "$$", "\\xymatrix{", "\\partial \\Delta[n] \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Delta[n] \\ar[r] \\ar@{-->}[ru] & Y", "}", "$$", "a dotted arrow exists making the diagram commute." ], "refs": [], "ref_ids": [] }, { "id": 14931, "type": "definition", "label": "simplicial-definition-kan", "categories": [ "simplicial" ], "title": "simplicial-definition-kan", "contents": [ "A map $X \\to Y$ of simplicial sets is called a {\\it Kan fibration}", "if for all $k, n$ with $1 \\leq n$, $0 \\leq k \\leq n$ and any commutative", "solid diagram", "$$", "\\xymatrix{", "\\Lambda_k[n] \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Delta[n] \\ar[r] \\ar@{-->}[ru] & Y", "}", "$$", "a dotted arrow exists making the diagram commute. A {\\it Kan complex}", "is a simplicial set $X$ such that $X \\to *$ is a Kan fibration, where", "$*$ is the constant simplicial set on a singleton." ], "refs": [], "ref_ids": [] }, { "id": 15003, "type": "definition", "label": "discriminant-definition-trace-element", "categories": [ "discriminant" ], "title": "discriminant-definition-trace-element", "contents": [ "Let $A \\to B$ be a flat quasi-finite map of Noetherian rings.", "The {\\it trace element} is the unique\\footnote{Uniqueness", "and existence will be justified in", "Lemmas \\ref{lemma-trace-unique} and \\ref{lemma-dualizing-tau}.}", "element", "$\\tau_{B/A} \\in \\omega_{B/A}$", "with the following property: for any Noetherian $A$-algebra $A_1$", "such that $B_1 = B \\otimes_A A_1$ comes with a", "product decomposition $B_1 = C \\times D$ with $A_1 \\to C$ finite", "the image of $\\tau_{B/A}$ in $\\omega_{C/A_1}$", "is $\\text{Trace}_{C/A_1}$.", "Here we use the base change map (\\ref{equation-bc-dualizing}) and", "Lemma \\ref{lemma-dualizing-product} to get", "$\\omega_{B/A} \\to \\omega_{B_1/A_1} \\to \\omega_{C/A_1}$." ], "refs": [ "discriminant-lemma-trace-unique", "discriminant-lemma-dualizing-tau", "discriminant-lemma-dualizing-product" ], "ref_ids": [ 14956, 14960, 14950 ] }, { "id": 15004, "type": "definition", "label": "discriminant-definition-kahler-different", "categories": [ "discriminant" ], "title": "discriminant-definition-kahler-different", "contents": [ "Let $f : Y \\to X$ be a morphism of schemes which is locally of finite type.", "The {\\it K\\\"ahler different} is the $0$th fitting ideal of $\\Omega_{Y/X}$." ], "refs": [], "ref_ids": [] }, { "id": 15005, "type": "definition", "label": "discriminant-definition-different", "categories": [ "discriminant" ], "title": "discriminant-definition-different", "contents": [ "Let $f : Y \\to X$ be a flat quasi-finite morphism of Noetherian schemes.", "Let $\\omega_{Y/X}$ be the relative dualizing module and let", "$\\tau_{Y/X} \\in \\Gamma(Y, \\omega_{Y/X})$ be the trace element", "(Remarks \\ref{remark-relative-dualizing-for-quasi-finite} and", "\\ref{remark-relative-dualizing-for-flat-quasi-finite}).", "The annihilator of", "$$", "\\Coker(\\mathcal{O}_Y \\xrightarrow{\\tau_{Y/X}} \\omega_{Y/X})", "$$", "is the {\\it different} of $Y/X$. It is a coherent ideal", "$\\mathfrak{D}_f \\subset \\mathcal{O}_Y$." ], "refs": [ "discriminant-remark-relative-dualizing-for-quasi-finite", "discriminant-remark-relative-dualizing-for-flat-quasi-finite" ], "ref_ids": [ 15006, 15007 ] }, { "id": 15025, "type": "definition", "label": "stacks-limits-definition-limit-preserving", "categories": [ "stacks-limits" ], "title": "stacks-limits-definition-limit-preserving", "contents": [ "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a", "$1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.", "We say $f$ is {\\it limit preserving} if for every directed limit", "$U = \\lim U_i$ of affine schemes over $S$ the diagram", "$$", "\\xymatrix{", "\\colim \\mathcal{X}_{U_i} \\ar[r] \\ar[d]_f & \\mathcal{X}_U \\ar[d]^f \\\\", "\\colim \\mathcal{Y}_{U_i} \\ar[r] & \\mathcal{Y}_U", "}", "$$", "of fibre categories is $2$-cartesian." ], "refs": [], "ref_ids": [] } ], "others": [ { "id": 302, "type": "other", "label": "spaces-more-morphisms-remark-weakly-radicial", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-remark-weakly-radicial", "contents": [ "Let $X \\to Y$ be a morphism of algebraic spaces.", "For some applications (of radicial morphisms)", "it is enough to require that for every", "$\\Spec(K) \\to Y$ where $K$ is a field", "\\begin{enumerate}", "\\item the space $|\\Spec(K) \\times_Y X|$ is a singleton,", "\\item there exists a monomorphism", "$\\Spec(L) \\to \\Spec(K) \\times_Y X$, and", "\\item $K \\subset L$ is purely inseparable.", "\\end{enumerate}", "If needed later we will may call such a morphism {\\it weakly radicial}.", "For example if $X \\to Y$ is a surjective weakly radicial morphism", "then $X(k) \\to Y(k)$ is surjective for every algebraically closed field $k$.", "Note that the base change", "$X_{\\overline{\\mathbf{Q}}} \\to \\Spec(\\overline{\\mathbf{Q}})$", "of the morphism in", "Example \\ref{example-universally-injective-not-radicial}", "is weakly radicial, but not radicial. The analogue of", "Lemma \\ref{lemma-when-universally-injective-radicial}", "is that if $X \\to Y$ has property ($\\beta$) and is universally", "injective, then it is weakly radicial (proof omitted)." ], "refs": [ "spaces-more-morphisms-lemma-when-universally-injective-radicial" ], "ref_ids": [ 18 ] }, { "id": 303, "type": "other", "label": "spaces-more-morphisms-remark-alternative", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-remark-alternative", "contents": [ "Now that we know that $\\Omega_{X/Y}$ is quasi-coherent we can attempt", "to construct it in another manner. For example we can use the result of", "Properties of Spaces,", "Section \\ref{spaces-properties-section-quasi-coherent-presentation}", "to construct the sheaf of differentials by glueing.", "For example if $Y$ is a scheme and if $U \\to X$ is a surjective \\'etale morphism", "from a scheme towards $X$, then we see that $\\Omega_{U/Y}$ is", "a quasi-coherent $\\mathcal{O}_U$-module, and since $s, t : R \\to U$", "are \\'etale we get an isomorphism", "$$", "\\alpha : s^*\\Omega_{U/Y} \\to \\Omega_{R/Y} \\to t^*\\Omega_{U/Y}", "$$", "by using", "Morphisms, Lemma \\ref{morphisms-lemma-triangle-differentials-smooth}.", "You check that this satisfies the cocycle condition and you're done.", "If $Y$ is not a scheme, then you define $\\Omega_{U/Y}$ as the cokernel", "of the map $(U \\to Y)^*\\Omega_{Y/S} \\to \\Omega_{U/S}$, and proceed as", "before. This two step process is a little bit ugly. Another possibility", "is to glue the sheaves $\\Omega_{U/V}$ for any diagram as in", "Lemma \\ref{lemma-localize-differentials}", "but this is not very elegant either. Both approaches will work however, and", "will give a slightly more elementary construction of the sheaf of", "differentials." ], "refs": [ "morphisms-lemma-triangle-differentials-smooth", "spaces-more-morphisms-lemma-localize-differentials" ], "ref_ids": [ 5337, 32 ] }, { "id": 304, "type": "other", "label": "spaces-more-morphisms-remark-topological-invariance-etale-site", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-remark-topological-invariance-etale-site", "contents": [ "\\begin{reference}", "Email by Lenny Taelman dated May 1, 2016.", "\\end{reference}", "A universal homeomorphism of algebraic spaces need not be representable, see", "Morphisms of Spaces,", "Example \\ref{spaces-morphisms-example-universal-homeomorphism}.", "In fact Theorem \\ref{theorem-topological-invariance} does", "not hold for universal homeomorphisms. To see this, let $k$ be an", "algebraically closed field of characteristic $0$ and let", "$$", "\\mathbf{A}^1 \\to X \\to \\mathbf{A}^1", "$$", "be as in Morphisms of Spaces,", "Example \\ref{spaces-morphisms-example-universal-homeomorphism}.", "Recall that the first morphism is \\'etale and identifies", "$t$ with $-t$ for $t \\in \\mathbf{A}^1_k \\setminus \\{0\\}$", "and that the second morphism is our universal homeomorphism.", "Since $\\mathbf{A}^1_k$ has no", "nontrivial connected finite \\'etale coverings", "(because $k$ is algebraically closed of characteristic zero; details omitted),", "it suffices to construct a nontrivial connected finite \\'etale covering", "$Y \\to X$. To do this, let $Y$ be the affine line", "with zero doubled", "(Schemes, Example \\ref{schemes-example-affine-space-zero-doubled}).", "Then $Y = Y_1 \\cup Y_2$ with $Y_i = \\mathbf{A}^1_k$ glued", "along $\\mathbf{A}^1_k \\setminus \\{0\\}$.", "To define the morphism $Y \\to X$ we use the morphisms", "$$", "Y_1 \\xrightarrow{1} \\mathbf{A}^1_k \\to X", "\\quad\\text{and}\\quad", "Y_2 \\xrightarrow{-1} \\mathbf{A}^1_k \\to X.", "$$", "These glue over $Y_1 \\cap Y_2$ by the construction of $X$ and", "hence define a morphism $Y \\to X$. In fact, we claim that", "$$", "\\xymatrix{", "Y \\ar[d] & Y_1 \\amalg Y_2 \\ar[l] \\ar[d] \\\\", "X & \\mathbf{A}^1_k \\ar[l]", "}", "$$", "is a cartesian square. We omit the details; you can use for example", "Groupoids, Lemma \\ref{groupoids-lemma-criterion-fibre-product}.", "Since $\\mathbf{A}^1_k \\to X$ is \\'etale and", "surjective, this proves that $Y \\to X$", "is finite \\'etale of degree $2$ which gives the desired example.", "\\medskip\\noindent", "More simply, you can argue as follows. The scheme $Y$ has a free", "action of the group $G = \\{+1, -1\\}$ where $-1$ acts by swapping", "$Y_1$ and $Y_2$ and changing the sign of the coordinate. Then", "$X = Y/G$ (see Spaces, Definition \\ref{spaces-definition-quotient})", "and hence $Y \\to X$ is finite \\'etale. You can also show directly", "that there exists a universal homeomorphism $X \\to \\mathbf{A}^1_k$", "by using $t \\mapsto t^2$ on affine spaces. In fact, this $X$ is", "the same as the $X$ above." ], "refs": [ "spaces-more-morphisms-theorem-topological-invariance", "groupoids-lemma-criterion-fibre-product", "spaces-definition-quotient" ], "ref_ids": [ 9, 9652, 8182 ] }, { "id": 305, "type": "other", "label": "spaces-more-morphisms-remark-tempting", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-remark-tempting", "contents": [ "It is tempting to think that in the situation of", "Lemma \\ref{lemma-etale-on-top}", "we have", "``$b$ formally smooth'' $\\Leftrightarrow$ ``$b \\circ a$ formally smooth''.", "However, this is likely not true in general." ], "refs": [ "spaces-more-morphisms-lemma-etale-on-top" ], "ref_ids": [ 65 ] }, { "id": 306, "type": "other", "label": "spaces-more-morphisms-remark-action-by-derivations", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-remark-action-by-derivations", "contents": [ "Assumptions and notation as in Lemma \\ref{lemma-action-by-derivations}.", "The action of a local section $\\theta$ on $a'$ is sometimes indicated by", "$\\theta \\cdot a'$. Note that this means nothing else than the fact", "that $(a')^\\sharp$ and $(\\theta \\cdot a')^\\sharp$ differ by a derivation", "$D$ which is related to $\\theta$ by Equation (\\ref{equation-D})." ], "refs": [ "spaces-more-morphisms-lemma-action-by-derivations" ], "ref_ids": [ 97 ] }, { "id": 307, "type": "other", "label": "spaces-more-morphisms-remark-special-case", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-remark-special-case", "contents": [ "A special case of", "Lemmas \\ref{lemma-difference-derivation},", "\\ref{lemma-action-by-derivations},", "\\ref{lemma-sheaf}, and", "\\ref{lemma-action-sheaf}", "is where $Y = Y'$. In this case the map $A$ is always zero.", "The sheaf of", "Lemma \\ref{lemma-sheaf}", "is just given by the rule", "$$", "U' \\mapsto", "\\{a' : U' \\to Y\\text{ over }B\\text{ with } a'|_U = a|_U\\}", "$$", "and we act on this by the sheaf", "$\\SheafHom_{\\mathcal{O}_X}(a^*\\Omega_{Y/B}, \\mathcal{C}_{X/X'})$." ], "refs": [ "spaces-more-morphisms-lemma-difference-derivation", "spaces-more-morphisms-lemma-action-by-derivations", "spaces-more-morphisms-lemma-sheaf", "spaces-more-morphisms-lemma-action-sheaf", "spaces-more-morphisms-lemma-sheaf" ], "ref_ids": [ 96, 97, 98, 99, 98 ] }, { "id": 308, "type": "other", "label": "spaces-more-morphisms-remark-another-special-case", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-remark-another-special-case", "contents": [ "Another special case of", "Lemmas \\ref{lemma-difference-derivation},", "\\ref{lemma-action-by-derivations},", "\\ref{lemma-sheaf}, and", "\\ref{lemma-action-sheaf}", "is where $B$ itself is a thickening $Z \\subset Z' = B$", "and $Y = Z \\times_{Z'} Y'$. Picture", "$$", "\\xymatrix{", "(X \\subset X') \\ar@{..>}[rr]_{(a, ?)} \\ar[rd]_{(g, g')} & &", "(Y \\subset Y') \\ar[ld]^{(h, h')} \\\\", "& (Z \\subset Z')", "}", "$$", "In this case the map $A : a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$", "is determined by $a$: the map", "$h^*\\mathcal{C}_{Z/Z'} \\to \\mathcal{C}_{Y/Y'}$ is surjective (because we", "assumed $Y = Z \\times_{Z'} Y'$), hence the pullback", "$g^*\\mathcal{C}_{Z/Z'} = a^*h^*\\mathcal{C}_{Z/Z'} \\to", "a^*\\mathcal{C}_{Y/Y'}$ is surjective, and the composition", "$g^*\\mathcal{C}_{Z/Z'} \\to a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$", "has to be the canonical map induced by $g'$. Thus the sheaf of", "Lemma \\ref{lemma-sheaf}", "is just given by the rule", "$$", "U' \\mapsto", "\\{a' : U' \\to Y'\\text{ over }Z'\\text{ with } a'|_U = a|_U\\}", "$$", "and we act on this by the sheaf", "$\\SheafHom_{\\mathcal{O}_X}(a^*\\Omega_{Y/Z}, \\mathcal{C}_{X/X'})$." ], "refs": [ "spaces-more-morphisms-lemma-difference-derivation", "spaces-more-morphisms-lemma-action-by-derivations", "spaces-more-morphisms-lemma-sheaf", "spaces-more-morphisms-lemma-action-sheaf", "spaces-more-morphisms-lemma-sheaf" ], "ref_ids": [ 96, 97, 98, 99, 98 ] }, { "id": 309, "type": "other", "label": "spaces-more-morphisms-remark-chow-Noetherian", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-remark-chow-Noetherian", "contents": [ "In Lemmas \\ref{lemma-blowup-to-find-embedding} and \\ref{lemma-chow-noetherian}", "the morphism $g : Z' \\to Y$ is a composition of projective morphisms.", "Presumably (by the analogue for algebraic spaces of", "Morphisms, Lemma \\ref{morphisms-lemma-ample-composition})", "there exists a $g$-ample invertible sheaf on $Z'$.", "If we ever need this, then we will state and prove this here." ], "refs": [ "spaces-more-morphisms-lemma-blowup-to-find-embedding", "spaces-more-morphisms-lemma-chow-noetherian", "morphisms-lemma-ample-composition" ], "ref_ids": [ 196, 197, 5384 ] }, { "id": 310, "type": "other", "label": "spaces-more-morphisms-remark-inverse-systems-kernel-cokernel-annihilated-by", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-remark-inverse-systems-kernel-cokernel-annihilated-by", "contents": [ "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let", "$\\mathcal{I}, \\mathcal{K} \\subset \\mathcal{O}_X$ be quasi-coherent sheaves of", "ideals. Let $\\alpha : (\\mathcal{F}_n) \\to (\\mathcal{G}_n)$ be a morphism of", "$\\textit{Coh}(X, \\mathcal{I})$.", "Given an affine scheme $U = \\Spec(A)$ and a surjective \\'etale morphism", "$U \\to X$ denote $I, K \\subset A$ the ideals corresponding to the restrictions", "$\\mathcal{I}|_U, \\mathcal{K}|_U$. Denote $\\alpha_U : M \\to N$ of finite", "$A^\\wedge$-modules which corresponds to $\\alpha|_U$ via", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-affine}.", "We claim the following are equivalent", "\\begin{enumerate}", "\\item there exists an integer $t \\geq 1$ such that", "$\\Ker(\\alpha_n)$ and $\\Coker(\\alpha_n)$", "are annihilated by $\\mathcal{K}^t$ for all $n \\geq 1$,", "\\item for any (or some) affine open $\\Spec(A) = U \\subset X$ as above", "the modules $\\Ker(\\alpha_U)$ and $\\Coker(\\alpha_U)$", "are annihilated by $K^t$ for some integer $t \\geq 1$.", "\\end{enumerate}", "If these equivalent conditions hold we will say that $\\alpha$ is a", "{\\it map whose kernel and cokernel are annihilated by a power of", "$\\mathcal{K}$}. To see the equivalence we refer to", "Cohomology of Schemes, Remark", "\\ref{coherent-remark-inverse-systems-kernel-cokernel-annihilated-by}." ], "refs": [ "coherent-lemma-inverse-systems-affine", "coherent-remark-inverse-systems-kernel-cokernel-annihilated-by" ], "ref_ids": [ 3370, 3408 ] }, { "id": 311, "type": "other", "label": "spaces-more-morphisms-remark-reformulate-existence-theorem", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-remark-reformulate-existence-theorem", "contents": [ "Let $A$ be a Noetherian ring complete with respect to an ideal $I$.", "Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$.", "Let $X \\to S$ be a morphism of algebraic spaces that is separated and", "of finite type. For $n \\geq 1$ we set $X_n = X \\times_S S_n$.", "Picture:", "$$", "\\xymatrix{", "X_1 \\ar[r]_{i_1} \\ar[d] & X_2 \\ar[r]_{i_2} \\ar[d] & X_3 \\ar[r] \\ar[d] &", "\\ldots & X \\ar[d] \\\\", "S_1 \\ar[r] & S_2 \\ar[r] & S_3 \\ar[r] & \\ldots & S", "}", "$$", "In this situation we consider systems $(\\mathcal{F}_n, \\varphi_n)$", "where", "\\begin{enumerate}", "\\item $\\mathcal{F}_n$ is a coherent $\\mathcal{O}_{X_n}$-module,", "\\item $\\varphi_n : i_n^*\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n$", "is an isomorphism, and", "\\item $\\text{Supp}(\\mathcal{F}_1)$ is proper over $S_1$.", "\\end{enumerate}", "Theorem \\ref{theorem-grothendieck-existence} says that the", "completion functor", "$$", "\\begin{matrix}", "\\text{coherent }\\mathcal{O}_X\\text{-modules }\\mathcal{F} \\\\", "\\text{with support proper over }A", "\\end{matrix}", "\\quad", "\\longrightarrow", "\\quad", "\\begin{matrix}", "\\text{systems }(\\mathcal{F}_n) \\\\", "\\text{as above}", "\\end{matrix}", "$$", "is an equivalence of categories. In the special case that $X$ is", "proper over $A$ we can omit the conditions on the supports." ], "refs": [ "spaces-more-morphisms-theorem-grothendieck-existence" ], "ref_ids": [ 16 ] }, { "id": 312, "type": "other", "label": "spaces-more-morphisms-remark-weaken-separation-axioms-question", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-remark-weaken-separation-axioms-question", "contents": [ "We can ask if in Grothendieck's algebraization theorem (in the form", "of Lemma \\ref{lemma-algebraize-morphism}), we can get", "by with weaker separation axioms on the target. Let us be more precise.", "Let $A$, $I$, $S$, $S_n$, $X$, $Y$, $X_n$, $Y_n$, and $g_n$ be as in", "the statement of Lemma \\ref{lemma-algebraize-morphism} and assume that", "\\begin{enumerate}", "\\item $X \\to S$ is proper, and", "\\item $Y \\to S$ is locally of finite type.", "\\end{enumerate}", "Does there exist a morphism of algebraic spaces $g : X \\to Y$", "over $S$ such that $g_n$ is the base change of $g$ to $S_n$?", "We don't know the answer in general; if you do please email", "\\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}.", "If $Y \\to S$ is separated, then the result", "holds by the lemma (there is an immediate reduction to the case", "where $X$ is finite type over $S$, by choosing a quasi-compact", "open containing the image of $g_1$). If we only assume $Y \\to S$", "is quasi-separated, then the result is true as well. First, as before", "we may assume $Y$ is quasi-compact as well as quasi-separated.", "Then we can use either \\cite{Bhatt-Algebraize} or from", "\\cite{Hall-Rydh-coherent} to algebraize $(g_n)$. Namely, to apply", "the first reference, we use", "$$", "D_{perf}(X) \\to \\lim D_{perf}(X_n)", "\\xrightarrow{\\lim Lg_n^*}", "\\lim D_{perf}(Y_n) = D_{perf}(Y)", "$$", "where the last step uses a Grothendieck existence result for", "the derived category of the proper algebraic space $Y$ over $R$", "(compare with", "Flatness on Spaces, Remark \\ref{spaces-flat-remark-correct-generality}).", "The paper cited shows that", "this arrow determines a morphism $Y \\to X$ as desired. To apply the", "second reference we use the same argument with coherent modules:", "$$", "\\textit{Coh}(\\mathcal{O}_X) \\to \\lim \\textit{Coh}(\\mathcal{O}_{X_n})", "\\xrightarrow{\\lim g_n^*}", "\\lim \\textit{Coh}(\\mathcal{O}_{Y_n}) = \\textit{Coh}(\\mathcal{O}_Y)", "$$", "where the final equality is a consequence of Grothendieck's", "existence theorem (Theorem \\ref{theorem-grothendieck-existence}).", "The second reference tells us that this functor corresponds to a", "morphism $Y \\to X$ over $R$. If we ever need this generalization", "we will precisely state and carefully prove the result here." ], "refs": [ "spaces-more-morphisms-lemma-algebraize-morphism", "spaces-more-morphisms-lemma-algebraize-morphism", "spaces-flat-remark-correct-generality", "spaces-more-morphisms-theorem-grothendieck-existence" ], "ref_ids": [ 211, 211, 7205, 16 ] }, { "id": 313, "type": "other", "label": "spaces-more-morphisms-remark-match-relative-pseudo-coherence", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-remark-match-relative-pseudo-coherence", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of representable", "algebraic spaces over $S$ which is locally of finite type. Let", "$f_0 : X_0 \\to Y_0$ be a morphism of schemes representing $f$", "(awkward but temporary notation). Then $f_0$ is locally of finite type.", "If $E$ is an object of $D_\\QCoh(\\mathcal{O}_X)$, then $E$", "is the pullback of a unique object $E_0$ in $D_\\QCoh(\\mathcal{O}_{X_0})$, see", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}.", "In this situation the phrase ``$E$ is $m$-pseudo-coherent relative to $Y$''", "will be taken to mean ``$E_0$ is $m$-pseudo-coherent relative to $Y_0$''", "as defined in More on Morphisms, Section", "\\ref{more-morphisms-section-relative-pseudo-coherence}." ], "refs": [ "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site" ], "ref_ids": [ 2644 ] }, { "id": 314, "type": "other", "label": "spaces-more-morphisms-remark-compare-L", "categories": [ "spaces-more-morphisms" ], "title": "spaces-more-morphisms-remark-compare-L", "contents": [ "The reader may have noticed the similarity between", "Lemma \\ref{lemma-compute-ext-rel-perfect} and", "Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-compute-ext}.", "Indeed, the pseudo-coherent complex $L$ of", "Lemma \\ref{lemma-compute-ext-rel-perfect}", "may be characterized as the unique pseudo-coherent complex", "on $Y$ such that there are functorial isomorphisms", "$$", "\\Ext^i_{\\mathcal{O}_Y}(L, \\mathcal{F}) \\longrightarrow", "\\Ext^i_{\\mathcal{O}_X}(K,", "E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*\\mathcal{F})", "$$", "compatible with boundary maps for $\\mathcal{F}$ ranging over", "$\\QCoh(\\mathcal{O}_Y)$. If we ever need this we will", "formulate a precise result here and give a detailed proof." ], "refs": [ "spaces-more-morphisms-lemma-compute-ext-rel-perfect", "spaces-perfect-lemma-compute-ext", "spaces-more-morphisms-lemma-compute-ext-rel-perfect" ], "ref_ids": [ 266, 2733, 266 ] }, { "id": 1556, "type": "other", "label": "algebra-remark-tensor-product-not-exact", "categories": [ "algebra" ], "title": "algebra-remark-tensor-product-not-exact", "contents": [ "However, tensor product does NOT preserve exact sequences in general.", "In other words, if $M_1 \\to M_2 \\to M_3$ is", "exact, then it is not necessarily true that", "$M_1 \\otimes N \\to M_2 \\otimes N \\to M_3 \\otimes N$", "is exact for arbitrary $R$-module $N$." ], "refs": [], "ref_ids": [] }, { "id": 1557, "type": "other", "label": "algebra-remark-flat-module", "categories": [ "algebra" ], "title": "algebra-remark-flat-module", "contents": [ "For $R$-modules $N$, if the", "functor $-\\otimes_R N$ is exact, i.e. tensoring", "with $N$ preserves all exact", "sequences, then $N$ is said to be {\\it flat} $R$-module.", "We will discuss this later in Section \\ref{section-flat}." ], "refs": [], "ref_ids": [] }, { "id": 1558, "type": "other", "label": "algebra-remark-fundamental-diagram", "categories": [ "algebra" ], "title": "algebra-remark-fundamental-diagram", "contents": [ "A fundamental commutative diagram associated to a ring map", "$\\varphi : R \\to S$, a prime $\\mathfrak q \\subset S$ and", "the corresponding prime $\\mathfrak p = \\varphi^{-1}(\\mathfrak q)$", "of $R$ is the following", "$$", "\\xymatrix{", "\\kappa(\\mathfrak q) = S_{\\mathfrak q}/{\\mathfrak q}S_{\\mathfrak q}", "&", "S_{\\mathfrak q} \\ar[l]", "&", "S \\ar[r] \\ar[l]", "&", "S/\\mathfrak q \\ar[r]", "&", "\\kappa(\\mathfrak q)", "\\\\", "\\kappa(\\mathfrak p) \\otimes_R S =", "S_{\\mathfrak p}/{\\mathfrak p}S_{\\mathfrak p} \\ar[u]", "&", "S_{\\mathfrak p} \\ar[u] \\ar[l]", "&", "S \\ar[u] \\ar[r] \\ar[l]", "&", "S/\\mathfrak pS \\ar[u] \\ar[r]", "&", "(R \\setminus \\mathfrak p)^{-1}S/\\mathfrak pS \\ar[u]", "\\\\", "\\kappa(\\mathfrak p) =", "R_{\\mathfrak p}/{\\mathfrak p}R_{\\mathfrak p} \\ar[u]", "&", "R_{\\mathfrak p} \\ar[u] \\ar[l]", "&", "R \\ar[u] \\ar[r] \\ar[l]", "&", "R/\\mathfrak p \\ar[u] \\ar[r]", "&", "\\kappa(\\mathfrak p) \\ar[u]", "}", "$$", "In this diagram the arrows in the outer left and outer right columns", "are identical. The horizontal maps induce on the associated spectra", "always a homeomorphism onto the image. The lower two rows", "of the diagram make sense without assuming $\\mathfrak q$ exists.", "The lower squares induce fibre squares of topological spaces.", "This diagram shows that $\\mathfrak p$ is in the image", "of the map on Spec if and only if $S \\otimes_R \\kappa(\\mathfrak p)$", "is not the zero ring." ], "refs": [], "ref_ids": [] }, { "id": 1559, "type": "other", "label": "algebra-remark-cohen-bound-cardinality", "categories": [ "algebra" ], "title": "algebra-remark-cohen-bound-cardinality", "contents": [ "Let $R$ be a ring. Let $\\kappa$ be an infinite cardinal.", "By applying", "Example \\ref{example-oka-family-bound-cardinality} and", "Proposition \\ref{proposition-oka}", "we see that any ideal maximal with respect to the property of not being", "generated by $\\kappa$ elements is prime. This result is not so", "useful because there exists a ring for which every prime ideal", "of $R$ can be generated by $\\aleph_0$ elements, but some", "ideal cannot. Namely, let $k$ be a field, let $T$ be a set whose", "cardinality is greater than $\\aleph_0$ and let", "$$", "R = k[\\{x_n\\}_{n \\geq 1}, \\{z_{t, n}\\}_{t \\in T, n \\geq 0}]/", "(x_n^2, z_{t, n}^2, x_n z_{t, n} - z_{t, n - 1})", "$$", "This is a local ring with unique prime ideal", "$\\mathfrak m = (x_n)$. But the ideal $(z_{t, n})$ cannot", "be generated by countably many elements." ], "refs": [ "algebra-proposition-oka" ], "ref_ids": [ 1404 ] }, { "id": 1560, "type": "other", "label": "algebra-remark-intersection-powers-ideal", "categories": [ "algebra" ], "title": "algebra-remark-intersection-powers-ideal", "contents": [ "Lemma \\ref{lemma-intersect-powers-ideal-module-zero} in particular implies", "that $\\bigcap_n I^n = (0)$ when $I \\subset R$ is a non-unit ideal in a", "Noetherian local ring $R$. More generally, let $R$ be a Noetherian ring and", "$I \\subset R$ an ideal. Suppose that $f \\in \\bigcap_{n \\in \\mathbf{N}} I^n$.", "Then Lemma \\ref{lemma-intersection-powers-ideal-module}", "says that for every prime ideal $I \\subset \\mathfrak p$", "there exists a $g \\in R$, $g \\not \\in \\mathfrak p$", "such that $f$ maps to zero in $R_g$. In algebraic geometry we", "express this by saying that ``$f$ is zero in an open neighbourhood", "of the closed set $V(I)$ of $\\Spec(R)$''." ], "refs": [ "algebra-lemma-intersect-powers-ideal-module-zero", "algebra-lemma-intersection-powers-ideal-module" ], "ref_ids": [ 627, 628 ] }, { "id": 1561, "type": "other", "label": "algebra-remark-period-polynomial", "categories": [ "algebra" ], "title": "algebra-remark-period-polynomial", "contents": [ "If $S$ is still Noetherian but $S$ is not generated in degree $1$,", "then the function associated to a graded $S$-module is a periodic", "polynomial (i.e., it is a numerical polynomial on the", "congruence classes of integers modulo $n$ for some $n$)." ], "refs": [], "ref_ids": [] }, { "id": 1562, "type": "other", "label": "algebra-remark-ass-reverse-functorial", "categories": [ "algebra" ], "title": "algebra-remark-ass-reverse-functorial", "contents": [ "Let $\\varphi : R \\to S$ be a ring map.", "Let $M$ be an $S$-module.", "Then it is not always the case that", "$\\Spec(\\varphi)(\\text{Ass}_S(M)) \\supset \\text{Ass}_R(M)$.", "For example, consider the ring map", "$R = k \\to S = k[x_1, x_2, x_3, \\ldots]/(x_i^2)$ and $M = S$.", "Then $\\text{Ass}_R(M)$ is not empty, but $\\text{Ass}_S(S)$ is empty." ], "refs": [], "ref_ids": [] }, { "id": 1563, "type": "other", "label": "algebra-remark-bourbaki", "categories": [ "algebra" ], "title": "algebra-remark-bourbaki", "contents": [ "Let $R \\to S$ be a ring map. Let $N$ be an $S$-module.", "Let $\\mathfrak p$ be a prime of $R$. Then", "$$", "\\text{Ass}_S(N \\otimes_R \\kappa(\\mathfrak p)) =", "\\text{Ass}_{S/\\mathfrak pS}(N \\otimes_R \\kappa(\\mathfrak p)) =", "\\text{Ass}_{S \\otimes_R \\kappa(\\mathfrak p)}(N \\otimes_R \\kappa(\\mathfrak p)).", "$$", "The first equality by", "Lemma \\ref{lemma-ass-quotient-ring}", "and the second by", "Lemma \\ref{lemma-localize-ass} part (1)." ], "refs": [ "algebra-lemma-ass-quotient-ring", "algebra-lemma-localize-ass" ], "ref_ids": [ 708, 710 ] }, { "id": 1564, "type": "other", "label": "algebra-remark-weakly-ass-not-functorial", "categories": [ "algebra" ], "title": "algebra-remark-weakly-ass-not-functorial", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module.", "Then it is not always the case that", "$\\Spec(\\varphi)(\\text{WeakAss}_S(M)) \\subset \\text{WeakAss}_R(M)$", "contrary to the case of associated primes (see", "Lemma \\ref{lemma-ass-functorial}).", "An example is to consider the ring map", "$$", "R = k[x_1, x_2, x_3, \\ldots] \\to", "S = k[x_1, x_2, x_3, \\ldots, y_1, y_2, y_3, \\ldots]/", "(x_1y_1, x_2y_2, x_3y_3, \\ldots)", "$$", "and $M = S$. In this case $\\mathfrak q = \\sum x_iS$ is a minimal prime of", "$S$, hence a weakly associated prime of $M = S$ (see", "Lemma \\ref{lemma-weakly-ass-minimal-prime-support}).", "But on the other hand, for any nonzero element of $S$ the annihilator", "in $R$ is finitely generated, and hence does not", "have radical equal to $R \\cap \\mathfrak q = (x_1, x_2, x_3, \\ldots)$", "(details omitted)." ], "refs": [ "algebra-lemma-ass-functorial", "algebra-lemma-weakly-ass-minimal-prime-support" ], "ref_ids": [ 706, 726 ] }, { "id": 1565, "type": "other", "label": "algebra-remark-ass-functorial", "categories": [ "algebra" ], "title": "algebra-remark-ass-functorial", "contents": [ "Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module.", "Denote $f : \\Spec(S) \\to \\Spec(R)$ the associated map on spectra.", "Then we have", "$$", "f(\\text{Ass}_S(M)) \\subset", "\\text{Ass}_R(M) \\subset", "\\text{WeakAss}_R(M) \\subset", "f(\\text{WeakAss}_S(M))", "$$", "see", "Lemmas \\ref{lemma-ass-functorial},", "\\ref{lemma-weakly-ass-reverse-functorial}, and", "\\ref{lemma-weakly-ass-support}.", "In general all of the inclusions may be strict, see", "Remarks \\ref{remark-ass-reverse-functorial} and", "\\ref{remark-weakly-ass-not-functorial}.", "If $S$ is Noetherian, then all the inclusions are equalities as", "the outer two are equal by", "Lemma \\ref{lemma-ass-weakly-ass}." ], "refs": [ "algebra-lemma-ass-functorial", "algebra-lemma-weakly-ass-reverse-functorial", "algebra-lemma-weakly-ass-support", "algebra-remark-ass-reverse-functorial", "algebra-remark-weakly-ass-not-functorial", "algebra-lemma-ass-weakly-ass" ], "ref_ids": [ 706, 728, 724, 1562, 1564, 727 ] }, { "id": 1566, "type": "other", "label": "algebra-remark-koszul-regular", "categories": [ "algebra" ], "title": "algebra-remark-koszul-regular", "contents": [ "In the paper \\cite{Kabele} the author introduces two more", "regularity conditions for sequences $x_1, \\ldots, x_r$ of elements", "of a ring $R$. Namely, we say the sequence is {\\it Koszul-regular}", "if $H_i(K_{\\bullet}(R, x_{\\bullet})) = 0$ for $i \\geq 1$ where", "$K_{\\bullet}(R, x_{\\bullet})$ is the Koszul complex. The sequence is", "called {\\it $H_1$-regular} if $H_1(K_{\\bullet}(R, x_{\\bullet})) = 0$.", "If $R$ is a local ring (possibly non-Noetherian) and the sequence consists", "of elements of the maximal ideal, then one has the implications", "regular $\\Rightarrow$ Koszul-regular $\\Rightarrow$ $H_1$-regular", "$\\Rightarrow$ quasi-regular. By examples the author shows that", "these implications cannot be reversed in general. We introduce", "these notions in more detail in", "More on Algebra, Section \\ref{more-algebra-section-koszul-regular}." ], "refs": [], "ref_ids": [] }, { "id": 1567, "type": "other", "label": "algebra-remark-join-quasi-regular-sequences", "categories": [ "algebra" ], "title": "algebra-remark-join-quasi-regular-sequences", "contents": [ "Let $k$ be a field. Consider the ring", "$$", "A = k[x, y, w, z_0, z_1, z_2, \\ldots]/", "(y^2z_0 - wx, z_0 - yz_1, z_1 - yz_2, \\ldots)", "$$", "In this ring $x$ is a nonzerodivisor and the image of $y$ in", "$A/xA$ gives a quasi-regular sequence. But it is not true that", "$x, y$ is a quasi-regular sequence in $A$ because $(x, y)/(x, y)^2$", "isn't free of rank two over $A/(x, y)$ due to the fact that", "$wx = 0$ in $(x, y)/(x, y)^2$ but $w$ isn't zero in $A/(x, y)$.", "Hence the analogue of", "Lemma \\ref{lemma-join-regular-sequences}", "does not hold for quasi-regular sequences." ], "refs": [ "algebra-lemma-join-regular-sequences" ], "ref_ids": [ 742 ] }, { "id": 1568, "type": "other", "label": "algebra-remark-signs-double-complex", "categories": [ "algebra" ], "title": "algebra-remark-signs-double-complex", "contents": [ "The isomorphism constructed above is the ``correct'' one only up to signs.", "A good part of homological algebra is concerned with choosing signs for", "various maps and showing commutativity of diagrams with intervention", "of suitable signs. For the moment we will simply use the isomorphism", "as given in the proof above, and worry about signs later." ], "refs": [], "ref_ids": [] }, { "id": 1569, "type": "other", "label": "algebra-remark-curiosity-signs-swap", "categories": [ "algebra" ], "title": "algebra-remark-curiosity-signs-swap", "contents": [ "An interesting case occurs when $M = N$ in the above.", "In this case we get a canonical map $\\text{Tor}_i^R(M, M)", "\\to \\text{Tor}_i^R(M, M)$. Note that this map is not the", "identity, because even when $i = 0$ this map is not the", "identity! For example, if $V$ is a vector space of dimension", "$n$ over a field, then the switch map $V \\otimes_k V \\to V \\otimes_k V$", "has $(n^2 + n)/2$ eigenvalues $+1$ and $(n^2-n)/2$ eigenvalues", "$-1$. In characteristic $2$ it is not even diagonalizable.", "Note that even changing the sign of the map will not get rid", "of this." ], "refs": [], "ref_ids": [] }, { "id": 1570, "type": "other", "label": "algebra-remark-Tor-ring-mod-ideal", "categories": [ "algebra" ], "title": "algebra-remark-Tor-ring-mod-ideal", "contents": [ "The proof of Lemma \\ref{lemma-characterize-flat} actually shows", "that", "$$", "\\text{Tor}_1^R(M, R/I)", "=", "\\Ker(I \\otimes_R M \\to M).", "$$" ], "refs": [ "algebra-lemma-characterize-flat" ], "ref_ids": [ 786 ] }, { "id": 1571, "type": "other", "label": "algebra-remark-warning", "categories": [ "algebra" ], "title": "algebra-remark-warning", "contents": [ "It is not true that a finite $R$-module which is", "$R$-flat is automatically projective. A counter", "example is where $R = \\mathcal{C}^\\infty(\\mathbf{R})$", "is the ring of infinitely differentiable functions on", "$\\mathbf{R}$, and $M = R_{\\mathfrak m} = R/I$ where", "$\\mathfrak m = \\{f \\in R \\mid f(0) = 0\\}$ and", "$I = \\{f \\in R \\mid \\exists \\epsilon, \\epsilon > 0 :", "f(x) = 0\\ \\forall x, |x| < \\epsilon\\}$." ], "refs": [], "ref_ids": [] }, { "id": 1572, "type": "other", "label": "algebra-remark-flat-ML", "categories": [ "algebra" ], "title": "algebra-remark-flat-ML", "contents": [ "Let $M$ be a flat $R$-module. By Lazard's theorem", "(Theorem \\ref{theorem-lazard})", "we can write $M = \\colim M_i$ as the colimit of a", "directed system $(M_i, f_{ij})$ where the $M_i$ are free", "finite $R$-modules. For $M$ to be Mittag-Leffler, it is enough for the inverse", "system of duals $(\\Hom_R(M_i, R), \\Hom_R(f_{ij}, R))$ to be", "Mittag-Leffler. This follows from criterion (4) of", "Proposition \\ref{proposition-ML-characterization}", "and the fact that for a free finite $R$-module $F$,", "there is a functorial isomorphism", "$\\Hom_R(F, R) \\otimes_R N \\cong \\Hom_R(F, N)$", "for any $R$-module $N$." ], "refs": [ "algebra-theorem-lazard", "algebra-proposition-ML-characterization" ], "ref_ids": [ 318, 1414 ] }, { "id": 1573, "type": "other", "label": "algebra-remark-go-up-ML-modules", "categories": [ "algebra" ], "title": "algebra-remark-go-up-ML-modules", "contents": [ "Let $R \\to S$ be a finite and finitely presented ring map.", "Let $M$ be an $S$-module which is Mittag-Leffler as an $R$-module.", "Then it is in general not the case that $M$ is Mittag-Leffler as", "an $S$-module. For example suppose that $S$ is the ring of dual numbers", "over $R$, i.e., $S = R \\oplus R\\epsilon$ with $\\epsilon^2 = 0$. Then an", "$S$-module consists of an $R$-module $M$ endowed with a square zero", "$R$-linear endomorphism $\\epsilon : M \\to M$. Now suppose that $M_0$", "is an $R$-module which is not Mittag-Leffler. Choose a presentation", "$F_1 \\xrightarrow{u} F_0 \\to M_0 \\to 0$ with $F_1$ and $F_0$ free $R$-modules.", "Set $M = F_1 \\oplus F_0$ with", "$$", "\\epsilon =", "\\left(", "\\begin{matrix}", "0 & 0 \\\\", "u & 0", "\\end{matrix}", "\\right) : M \\longrightarrow M.", "$$", "Then $M/\\epsilon M \\cong F_1 \\oplus M_0$ is not Mittag-Leffler over", "$R = S/\\epsilon S$, hence not Mittag-Leffler over $S$ (see", "Lemma \\ref{lemma-mod-ideal-ML-modules}).", "On the other hand, $M/\\epsilon M = M \\otimes_S S/\\epsilon S$ which would", "be Mittag-Leffler over $S$ if $M$ was, see", "Lemma \\ref{lemma-tensor-ML-modules}." ], "refs": [ "algebra-lemma-mod-ideal-ML-modules", "algebra-lemma-tensor-ML-modules" ], "ref_ids": [ 834, 831 ] }, { "id": 1574, "type": "other", "label": "algebra-remark-characterize-projective", "categories": [ "algebra" ], "title": "algebra-remark-characterize-projective", "contents": [ "Lemma \\ref{lemma-countgen-projective} does not hold without the countable", "generation assumption. For example, the $\\mathbf Z$-module $M =", "\\mathbf{Z}[[x]]$ is flat and Mittag-Leffler but not projective. It is", "Mittag-Leffler by Lemma \\ref{lemma-power-series-ML}. Subgroups of free abelian", "groups are free, hence a projective $\\mathbf Z$-module is in fact free and so", "are its submodules. Thus to show $M$ is not projective it suffices to produce", "a non-free submodule. Fix a prime $p$ and consider the submodule $N$", "consisting of power series $f(x) = \\sum a_i x^i$ such that for every integer $m", "\\geq 1$, $p^m$ divides $a_i$ for all but finitely many $i$. Then $\\sum a_i p^i", "x^i$ is in $N$ for all $a_i \\in \\mathbf{Z}$, so $N$ is uncountable. Thus if", "$N$ were free it would have uncountable rank and the dimension of $N/pN$ over", "$\\mathbf{Z}/p$ would be uncountable. This is not true as the elements $x^i \\in", "N/pN$ for $i \\geq 0$ span $N/pN$." ], "refs": [ "algebra-lemma-countgen-projective", "algebra-lemma-power-series-ML" ], "ref_ids": [ 850, 847 ] }, { "id": 1575, "type": "other", "label": "algebra-remark-matrices-associated-to-elements-epicenter", "categories": [ "algebra" ], "title": "algebra-remark-matrices-associated-to-elements-epicenter", "contents": [ "Let $R \\to S$ be a ring map. Sometimes the set of elements", "$g \\in S$ such that $g \\otimes 1 = 1 \\otimes g$ is called the", "{\\it epicenter} of $S$. It is an $R$-algebra. By the construction of", "Lemma \\ref{lemma-kernel-difference-projections}", "we get for each $g$ in the epicenter a matrix factorization", "$$", "(g) = Y X Z", "$$", "with $X \\in \\text{Mat}(n \\times n, R)$,", "$Y \\in \\text{Mat}(1 \\times n, S)$, and", "$Z \\in \\text{Mat}(n \\times 1, S)$. Namely, let $x_{i, j}, y_i, z_j$", "be as in part (2) of the lemma. Set $X = (x_{i, j})$, let $y$ be the", "row vector whose entries are the $y_i$ and let $z$ be the column vector", "whose entries are the $z_j$. With this notation conditions (b) and (c) of", "Lemma \\ref{lemma-kernel-difference-projections}", "mean exactly that $Y X \\in \\text{Mat}(1 \\times n, R)$,", "$X Z \\in \\text{Mat}(n \\times 1, R)$.", "It turns out to be very convenient to consider the triple of", "matrices $(X, YX, XZ)$. Given $n \\in \\mathbf{N}$ and a triple", "$(P, U, V)$ we say that $(P, U, V)$ is a {\\it $n$-triple associated to $g$}", "if there exists a matrix factorization as above such that", "$P = X$, $U = YX$ and $V = XZ$." ], "refs": [ "algebra-lemma-kernel-difference-projections", "algebra-lemma-kernel-difference-projections" ], "ref_ids": [ 957, 957 ] }, { "id": 1576, "type": "other", "label": "algebra-remark-resolution-dim-1", "categories": [ "algebra" ], "title": "algebra-remark-resolution-dim-1", "contents": [ "Suppose that $R$ is a $1$-dimensional semi-local Noetherian domain.", "If there is a maximal ideal $\\mathfrak m \\subset R$ such that", "$R_{\\mathfrak m}$ is not regular, then we may apply", "Lemma \\ref{lemma-nonregular-dimension-one} to $(R, \\mathfrak m)$", "to get a finite ring extension $R \\subset R_1$.", "(For example one can do this so that $\\Spec(R_1) \\to \\Spec(R)$", "is the blowup of $\\Spec(R)$ in the ideal $\\mathfrak m$.)", "Of course $R_1$ is a $1$-dimensional semi-local Noetherian", "domain with the same fraction field as $R$. If $R_1$ is not a", "regular semi-local ring, then we may repeat the construction to", "get $R_1 \\subset R_2$. Thus we get a sequence", "$$", "R \\subset R_1 \\subset R_2 \\subset R_3 \\subset \\ldots", "$$", "of finite ring extensions which may stop if $R_n$ is regular for", "some $n$. Resolution of singularities would be the claim", "that eventually $R_n$ is indeed regular. In reality this is not", "the case. Namely, there exists a characteristic $0$", "Noetherian local domain $A$ of dimension $1$ whose completion is nonreduced,", "see \\cite[Proposition 3.1]{Ferrand-Raynaud} or our", "Examples, Section \\ref{examples-section-local-completion-nonreduced}.", "For an example in characteristic $p > 0$ see", "Example \\ref{example-bad-dvr-char-p}.", "Since the construction of blowing up commutes with completion it", "is easy to see the sequence never stabilizes.", "See \\cite{Bennett} for a discussion (mostly in positive characteristic).", "On the other hand, if the completion of $R$ in all of its maximal", "ideals is reduced, then the procedure stops (insert future reference", "here)." ], "refs": [ "algebra-lemma-nonregular-dimension-one" ], "ref_ids": [ 1022 ] }, { "id": 1577, "type": "other", "label": "algebra-remark-suitable-systems-limits", "categories": [ "algebra" ], "title": "algebra-remark-suitable-systems-limits", "contents": [ "Suppose that $R \\to S$ is a local homomorphism", "of local rings, which is essentially of finite presentation.", "Take any system $(\\Lambda, \\leq)$, $R_\\lambda \\to S_\\lambda$", "with the properties listed in", "Lemma \\ref{lemma-limit-essentially-finite-type}.", "What may happen is that this is the ``wrong'' system, namely,", "it may happen that property (4) of", "Lemma \\ref{lemma-limit-essentially-finite-presentation} is not", "satisfied. Here is an example. Let $k$ be a field. Consider the ring", "$$", "R = k[[z, y_1, y_2, \\ldots]]/(y_i^2 - zy_{i + 1}).", "$$", "Set $S = R/zR$. As system take $\\Lambda = \\mathbf{N}$ and", "$R_n = k[[z, y_1, \\ldots, y_n]]/(\\{y_i^2 - zy_{i + 1}\\}_{i \\leq n-1})$", "and $S_n = R_n/(z, y_n^2)$. All the maps", "$S_n \\otimes_{R_n} R_{n + 1} \\to S_{n + 1}$", "are not localizations (i.e., isomorphisms in this case)", "since $1 \\otimes y_{n + 1}^2$ maps to zero.", "If we take instead $S_n' = R_n/zR_n$ then the", "maps $S'_n \\otimes_{R_n} R_{n + 1} \\to S'_{n + 1}$ are", "isomorphisms. The moral of this remark is that we do have to be", "a little careful in choosing the systems." ], "refs": [ "algebra-lemma-limit-essentially-finite-type", "algebra-lemma-limit-essentially-finite-presentation" ], "ref_ids": [ 1099, 1100 ] }, { "id": 1578, "type": "other", "label": "algebra-remark-functoriality-principal-parts", "categories": [ "algebra" ], "title": "algebra-remark-functoriality-principal-parts", "contents": [ "Suppose given a commutative diagram of rings", "$$", "\\xymatrix{", "B \\ar[r] & B' \\\\", "A \\ar[u] \\ar[r] & A' \\ar[u]", "}", "$$", "a $B$-module $M$, a $B'$-module $M'$, and a $B$-linear map $M \\to M'$.", "Then we get a compatible system of module maps", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "P^2_{B'/A'}(M') \\ar[r] &", "P^1_{B'/A'}(M') \\ar[r] &", "P^0_{B'/A'}(M') \\\\", "\\ldots \\ar[r] &", "P^2_{B/A}(M) \\ar[r] \\ar[u] &", "P^1_{B/A}(M) \\ar[r] \\ar[u] &", "P^0_{B/A}(M) \\ar[u]", "}", "$$", "These maps are compatible with further composition of maps of this type.", "The easiest way to see this is to use the description of the modules", "$P^k_{B/A}(M)$ in terms of generators and relations in the proof of", "Lemma \\ref{lemma-module-principal-parts} but it can also be seen", "directly from the universal", "property of these modules. Moreover, these maps are compatible with", "the short exact sequences of Lemma \\ref{lemma-sequence-of-principal-parts}." ], "refs": [ "algebra-lemma-module-principal-parts", "algebra-lemma-sequence-of-principal-parts" ], "ref_ids": [ 1145, 1146 ] }, { "id": 1579, "type": "other", "label": "algebra-remark-composition-homotopy-equivalent-to-zero", "categories": [ "algebra" ], "title": "algebra-remark-composition-homotopy-equivalent-to-zero", "contents": [ "Let $A \\to B$ and $\\phi : B \\to C$ be ring maps.", "Then the composition $\\NL_{B/A} \\to \\NL_{C/A} \\to \\NL_{C/B}$ is", "homotopy equivalent to zero. Namely, this composition is the functoriality", "of the naive cotangent complex for the square", "$$", "\\xymatrix{", "B \\ar[r]_\\phi & C \\\\", "A \\ar[r] \\ar[u] & B \\ar[u]", "}", "$$", "Write $J = \\Ker(B[C] \\to C)$. An explicit homotopy is given by the map", "$\\Omega_{A[B]/A} \\otimes_A B \\to J/J^2$ which maps the basis element", "$\\text{d}[b]$ to the class of $[\\phi(b)] - b$ in $J/J^2$." ], "refs": [], "ref_ids": [] }, { "id": 1580, "type": "other", "label": "algebra-remark-lemma-characterize-formally-smooth", "categories": [ "algebra" ], "title": "algebra-remark-lemma-characterize-formally-smooth", "contents": [ "Lemma \\ref{lemma-characterize-formally-smooth} holds more", "generally whenever $P$ is formally smooth over $R$." ], "refs": [ "algebra-lemma-characterize-formally-smooth" ], "ref_ids": [ 1207 ] }, { "id": 1581, "type": "other", "label": "algebra-remark-construct-sh-from-h", "categories": [ "algebra" ], "title": "algebra-remark-construct-sh-from-h", "contents": [ "We can also construct $R^{sh}$ from $R^h$. Namely, for any finite separable", "subextension $\\kappa \\subset \\kappa' \\subset \\kappa^{sep}$", "there exists a unique (up to unique isomorphism) finite \\'etale local", "ring extension $R^h \\subset R^h(\\kappa')$", "whose residue field extension reproduces the given extension, see", "Lemma \\ref{lemma-henselian-cat-finite-etale}.", "Hence we can set", "$$", "R^{sh} =", "\\bigcup\\nolimits_{\\kappa \\subset \\kappa' \\subset \\kappa^{sep}}", "R^h(\\kappa')", "$$", "The arrows in this system, compatible with the arrows on the level", "of residue fields, exist by", "Lemma \\ref{lemma-henselian-cat-finite-etale}.", "This will produce a henselian local ring by", "Lemma \\ref{lemma-colimit-henselian}", "since each of the rings", "$R^h(\\kappa')$ is henselian by", "Lemma \\ref{lemma-finite-over-henselian}.", "By construction the residue field extension induced by", "$R^h \\to R^{sh}$ is the field extension $\\kappa \\subset \\kappa^{sep}$.", "Hence $R^{sh}$ so constructed is strictly henselian.", "By Lemma \\ref{lemma-composition-colimit-etale} the $R$-algebra", "$R^{sh}$ is a colimit of \\'etale $R$-algebras. Hence the uniqueness", "of Lemma \\ref{lemma-uniqueness-henselian} shows that $R^{sh}$", "is the strict henselization." ], "refs": [ "algebra-lemma-henselian-cat-finite-etale", "algebra-lemma-henselian-cat-finite-etale", "algebra-lemma-colimit-henselian", "algebra-lemma-finite-over-henselian", "algebra-lemma-composition-colimit-etale", "algebra-lemma-uniqueness-henselian" ], "ref_ids": [ 1280, 1280, 1293, 1277, 1288, 1292 ] }, { "id": 1582, "type": "other", "label": "algebra-remark-Noetherian-complete-local-ring-universally-catenary", "categories": [ "algebra" ], "title": "algebra-remark-Noetherian-complete-local-ring-universally-catenary", "contents": [ "If $k$ is a field then the power series ring $k[[X_1, \\ldots, X_d]]$", "is a Noetherian complete local regular ring of dimension $d$.", "If $\\Lambda$ is a Cohen ring then $\\Lambda[[X_1, \\ldots, X_d]]$", "is a complete local Noetherian regular ring of dimension $d + 1$.", "Hence the Cohen structure theorem implies that any Noetherian", "complete local ring is a quotient of a regular local ring.", "In particular we see that a Noetherian complete local ring is", "universally catenary, see Lemma \\ref{lemma-CM-ring-catenary}", "and Lemma \\ref{lemma-regular-ring-CM}." ], "refs": [ "algebra-lemma-CM-ring-catenary", "algebra-lemma-regular-ring-CM" ], "ref_ids": [ 937, 941 ] }, { "id": 1583, "type": "other", "label": "algebra-remark-universally-catenary-does-not-descend", "categories": [ "algebra" ], "title": "algebra-remark-universally-catenary-does-not-descend", "contents": [ "The property of being ``universally catenary'' does not descend;", "not even along \\'etale ring maps. In", "Examples, Section \\ref{examples-section-non-catenary-Noetherian-local}", "there is a construction of a finite ring map $A \\to B$ with", "$A$ local Noetherian and not universally catenary,", "$B$ semi-local with two maximal ideals $\\mathfrak m$, $\\mathfrak n$", "with $B_{\\mathfrak m}$ and $B_{\\mathfrak n}$ regular of dimension $2$ and $1$", "respectively, and the same residue fields as that of $A$.", "Moreover, $\\mathfrak m_A$ generates the maximal ideal in both", "$B_{\\mathfrak m}$ and $B_{\\mathfrak n}$ (so $A \\to B$ is unramified", "as well as finite).", "By Lemma \\ref{lemma-etale-makes-unramified-closed}", "there exists a local \\'etale ring map", "$A \\to A'$ such that $B \\otimes_A A' = B_1 \\times B_2$ decomposes", "with $A' \\to B_i$ surjective.", "This shows that $A'$ has two minimal primes $\\mathfrak q_i$", "with $A'/\\mathfrak q_i \\cong B_i$. Since $B_i$ is regular local", "(since it is \\'etale over either $B_{\\mathfrak m}$ or $B_{\\mathfrak n}$)", "we conclude that $A'$ is universally catenary." ], "refs": [ "algebra-lemma-etale-makes-unramified-closed" ], "ref_ids": [ 1274 ] }, { "id": 1649, "type": "other", "label": "moduli-curves-remark-boundedness-aut-does-not-work-surfaces", "categories": [ "moduli-curves" ], "title": "moduli-curves-remark-boundedness-aut-does-not-work-surfaces", "contents": [ "The boundedness argument in the proof of", "Lemma \\ref{lemma-curves-diagonal-separated-fp}", "does not work for moduli of surfaces and in fact,", "the result is wrong, for example because K3 surfaces", "over fields can have infinite discrete automorphism groups.", "The ``reason'' the argument does not work is that on a", "projective surface $S$ over a field,", "given ample invertible sheaves $\\mathcal{N}$", "and $\\mathcal{L}$ with Hilbert polynomials $Q$ and $P$,", "there is no a priori bound on the Hilbert polynomial", "of $\\mathcal{N} \\otimes_{\\mathcal{O}_S} \\mathcal{L}$.", "In terms of intersection theory, if $H_1$, $H_2$ are ample effective", "Cartier divisors on $S$,", "then there is no (upper) bound on the intersection number $H_1 \\cdot H_2$", "in terms of $H_1 \\cdot H_1$ and $H_2 \\cdot H_2$." ], "refs": [ "moduli-curves-lemma-curves-diagonal-separated-fp" ], "ref_ids": [ 1590 ] }, { "id": 1702, "type": "other", "label": "dpa-remark-forgetful", "categories": [ "dpa" ], "title": "dpa-remark-forgetful", "contents": [ "The forgetful functor $(A, I, \\gamma) \\mapsto A$ does not commute with", "colimits. For example, let", "$$", "\\xymatrix{", "(B, J, \\delta) \\ar[r] & (B'', J'', \\delta'') \\\\", "(A, I, \\gamma) \\ar[r] \\ar[u] & (B', J', \\delta') \\ar[u]", "}", "$$", "be a pushout in the category of divided power rings.", "Then in general the map $B \\otimes_A B' \\to B''$ isn't an", "isomorphism. (It is always surjective.)", "An explicit example is given by", "$(A, I, \\gamma) = (\\mathbf{Z}, (0), \\emptyset)$,", "$(B, J, \\delta) = (\\mathbf{Z}/4\\mathbf{Z}, 2\\mathbf{Z}/4\\mathbf{Z}, \\delta)$,", "and", "$(B', J', \\delta') =", "(\\mathbf{Z}/4\\mathbf{Z}, 2\\mathbf{Z}/4\\mathbf{Z}, \\delta')$", "where $\\delta_2(2) = 2$ and $\\delta'_2(2) = 0$ and all higher divided powers", "equal to zero. Then $(B'', J'', \\delta'') = (\\mathbf{F}_2, (0), \\emptyset)$", "which doesn't agree with the tensor product. However, note that it is always", "true that", "$$", "B''/J'' = B/J \\otimes_{A/I} B'/J'", "$$", "as can be seen from the universal property of the pushout by considering", "maps into divided power algebras of the form $(C, (0), \\emptyset)$." ], "refs": [], "ref_ids": [] }, { "id": 1703, "type": "other", "label": "dpa-remark-divided-power-polynomial-algebra", "categories": [ "dpa" ], "title": "dpa-remark-divided-power-polynomial-algebra", "contents": [ "Let $(A, I, \\gamma)$ be a divided power ring.", "There is a variant of Lemma \\ref{lemma-divided-power-polynomial-algebra}", "for infinitely many variables. First note that if $s < t$ then there", "is a canonical map", "$$", "A\\langle x_1, \\ldots, x_s \\rangle \\to A\\langle x_1, \\ldots, x_t\\rangle", "$$", "Hence if $W$ is any set, then we set", "$$", "A\\langle x_w: w \\in W\\rangle =", "\\colim_{E \\subset W} A\\langle x_e:e \\in E\\rangle", "$$", "(colimit over $E$ finite subset of $W$)", "with transition maps as above. By the definition of a colimit we see", "that the universal mapping property of $A\\langle x_w: w \\in W\\rangle$ is", "completely analogous to the mapping property stated in", "Lemma \\ref{lemma-divided-power-polynomial-algebra}." ], "refs": [ "dpa-lemma-divided-power-polynomial-algebra", "dpa-lemma-divided-power-polynomial-algebra" ], "ref_ids": [ 1661, 1661 ] }, { "id": 1704, "type": "other", "label": "dpa-remark-adjoining-set-of-variables", "categories": [ "dpa" ], "title": "dpa-remark-adjoining-set-of-variables", "contents": [ "We can also adjoin a set (possibly infinite) of exterior or divided", "power generators in a given degree $d > 0$, rather than just one", "as in Examples \\ref{example-adjoining-odd}", "and \\ref{example-adjoining-even}. Namely, ", "following Remark \\ref{remark-divided-power-polynomial-algebra}:", "for $(A,\\gamma)$", "as above and a set $J$, let $A\\langle", "T_j:j\\in J\\rangle$ be the directed colimit of the algebras", "$A\\langle T_j:j\\in S\\rangle$ over all finite subsets $S$", "of $J$. It is immediate that this algebra has a unique divided power", "structure, compatible with the given structure on $A$ and on", "each generator $T_j$." ], "refs": [ "dpa-remark-divided-power-polynomial-algebra" ], "ref_ids": [ 1703 ] }, { "id": 1705, "type": "other", "label": "dpa-remark-no-good-ci-map", "categories": [ "dpa" ], "title": "dpa-remark-no-good-ci-map", "contents": [ "It appears difficult to define an good notion of ``local complete", "intersection homomorphisms'' for maps between general Noetherian rings.", "The reason is that, for a local Noetherian ring $A$, the fibres of", "$A \\to A^\\wedge$ are not local complete intersection rings.", "Thus, if $A \\to B$ is a local homomorphism of local Noetherian rings,", "and the map of completions $A^\\wedge \\to B^\\wedge$ is a", "complete intersection homomorphism in the sense defined above,", "then $(A_\\mathfrak p)^\\wedge \\to (B_\\mathfrak q)^\\wedge$ is in general", "{\\bf not} a complete intersection homomorphism in the sense", "defined above. A solution can be had by working exclusively with", "excellent Noetherian rings. More generally, one could work with", "those Noetherian rings whose formal fibres are complete", "intersections, see \\cite{Rodicio-ci}.", "We will develop this theory in", "Dualizing Complexes, Section \\ref{dualizing-section-formal-fibres}." ], "refs": [], "ref_ids": [] }, { "id": 1754, "type": "other", "label": "moduli-remark-quot-numerical", "categories": [ "moduli" ], "title": "moduli-remark-quot-numerical", "contents": [ "Let $f : X \\to B$ and $\\mathcal{F}$ be as in the introduction to this section.", "Let $I$ be a set and for $i \\in I$ let $E_i \\in D(\\mathcal{O}_X)$ be perfect.", "Let $P : I \\to \\mathbf{Z}$ be a function. Recall that we have a morphism", "$$", "\\Quotfunctor_{\\mathcal{F}/X/B} \\longrightarrow \\Cohstack_{X/B}", "$$", "which sends the element $\\mathcal{F}_T \\to \\mathcal{Q}$", "of $\\Quotfunctor_{\\mathcal{F}/X/B}(T)$ to the object $\\mathcal{Q}$", "of $\\Cohstack_{X/B}$ over $T$, see proof of", "Quot, Proposition \\ref{quot-proposition-quot}. Hence we can form", "the fibre product diagram", "$$", "\\xymatrix{", "\\Quotfunctor^P_{\\mathcal{F}/X/B} \\ar[r] \\ar[d] &", "\\Cohstack^P_{X/B} \\ar[d] \\\\", "\\Quotfunctor_{\\mathcal{F}/X/B} \\ar[r] &", "\\Cohstack_{X/B}", "}", "$$", "This is the defining diagram for the algebraic space in the", "upper left corner. The left vertical arrow is a", "flat closed immersion which is an open and closed immersion", "for example if $I$ is finite, or $B$ is locally Noetherian, or", "$I = \\mathbf{Z}$ and $E_i = \\mathcal{L}^{\\otimes i}$ for some", "invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ (in the last", "case we sometimes use the notation", "$\\Quotfunctor^{P, \\mathcal{L}}_{\\mathcal{F}/X/B}$).", "See Situation \\ref{situation-numerical} and", "Lemmas \\ref{lemma-open-P} and \\ref{lemma-finite-list-perfect-objects} and", "Example \\ref{example-hilbert-polynomial}." ], "refs": [ "quot-proposition-quot", "moduli-lemma-open-P", "moduli-lemma-finite-list-perfect-objects" ], "ref_ids": [ 3227, 1712, 1713 ] }, { "id": 1755, "type": "other", "label": "moduli-remark-hilb-numerical", "categories": [ "moduli" ], "title": "moduli-remark-hilb-numerical", "contents": [ "Let $f : X \\to B$ be as in the introduction to this section.", "Let $I$ be a set and for $i \\in I$ let $E_i \\in D(\\mathcal{O}_X)$ be perfect.", "Let $P : I \\to \\mathbf{Z}$ be a function. Recall that", "$\\Hilbfunctor_{X/B} = \\Quotfunctor_{\\mathcal{O}_X/X/B}$, see", "Quot, Lemma \\ref{quot-lemma-hilb-is-quot}.", "Thus we can define", "$$", "\\Hilbfunctor^P_{X/B} = \\Quotfunctor^P_{\\mathcal{O}_X/X/B}", "$$", "where $\\Quotfunctor^P_{\\mathcal{O}_X/X/B}$ is as in", "Remark \\ref{remark-quot-numerical}. The morphism", "$$", "\\Hilbfunctor^P_{X/B} \\longrightarrow \\Hilbfunctor_{X/B}", "$$", "is a flat closed immersion which is an open and closed immersion", "for example if $I$ is finite, or $B$ is locally Noetherian, or", "$I = \\mathbf{Z}$ and $E_i = \\mathcal{L}^{\\otimes i}$ for some", "invertible $\\mathcal{O}_X$-module $\\mathcal{L}$. In the last case", "we sometimes use the notation $\\Hilbfunctor^{P, \\mathcal{L}}_{X/B}$." ], "refs": [ "quot-lemma-hilb-is-quot", "moduli-remark-quot-numerical" ], "ref_ids": [ 3176, 1754 ] }, { "id": 1756, "type": "other", "label": "moduli-remark-Mor-numerical", "categories": [ "moduli" ], "title": "moduli-remark-Mor-numerical", "contents": [ "Let $B, X, Y$ be as in the introduction to this section.", "Let $I$ be a set and for $i \\in I$ let", "$E_i \\in D(\\mathcal{O}_{Y \\times_B X})$ be perfect.", "Let $P : I \\to \\mathbf{Z}$ be a function. Recall that", "$$", "\\mathit{Mor}_B(Y, X) \\subset", "\\Hilbfunctor_{Y \\times_B X/B}", "$$", "is an open subspace, see Quot, Lemma \\ref{quot-lemma-Mor-into-Hilb-open}.", "Thus we can define", "$$", "\\mathit{Mor}^P_B(Y, X) =", "\\mathit{Mor}_B(Y, X) \\cap \\Hilbfunctor^P_{Y \\times_B X/B}", "$$", "where $\\Hilbfunctor^P_{Y \\times_B X/B}$ is as in", "Remark \\ref{remark-hilb-numerical}. The morphism", "$$", "\\mathit{Mor}^P_B(Y, X) \\longrightarrow \\mathit{Mor}_B(Y, X)", "$$", "is a flat closed immersion which is an open and closed immersion", "for example if $I$ is finite, or $B$ is locally Noetherian, or", "$I = \\mathbf{Z}$, $E_i = \\mathcal{L}^{\\otimes i}$", "for some invertible $\\mathcal{O}_{Y \\times_B X}$-module $\\mathcal{L}$.", "In the last case we sometimes use the notation", "$\\mathit{Mor}^{P, \\mathcal{L}}_B(Y, X)$." ], "refs": [ "quot-lemma-Mor-into-Hilb-open", "moduli-remark-hilb-numerical" ], "ref_ids": [ 3187, 1755 ] }, { "id": 2009, "type": "other", "label": "derived-remark-special-triangles", "categories": [ "derived" ], "title": "derived-remark-special-triangles", "contents": [ "Let $\\mathcal{D}$ be an additive category with translation functors $[n]$", "as in Definition \\ref{definition-triangle}. Let us call a triangle", "$(X, Y, Z, f, g, h)$ {\\it special}\\footnote{This is nonstandard notation.}", "if for every object $W$ of $\\mathcal{D}$", "the long sequence of abelian groups", "$$", "\\ldots \\to", "\\Hom_\\mathcal{D}(W, X) \\to", "\\Hom_\\mathcal{D}(W, Y) \\to", "\\Hom_\\mathcal{D}(W, Z) \\to", "\\Hom_\\mathcal{D}(W, X[1]) \\to \\ldots", "$$", "is exact. The proof of Lemma \\ref{lemma-third-isomorphism-triangle}", "shows that if", "$$", "(a, b, c) : (X, Y, Z, f, g, h) \\to (X', Y', Z', f', g', h')", "$$", "is a morphism of special triangles and if two among $a, b, c$", "are isomorphisms so is the third. There is a dual statement for", "{\\it co-special} triangles, i.e., triangles which turn into long", "exact sequences on applying the functor $\\Hom_\\mathcal{D}(-, W)$.", "Thus distinguished triangles are special and co-special, but in", "general there are many more (co-)special triangles, than there are", "distinguished triangles." ], "refs": [ "derived-definition-triangle", "derived-lemma-third-isomorphism-triangle" ], "ref_ids": [ 1967, 1759 ] }, { "id": 2010, "type": "other", "label": "derived-remark-compute-modules", "categories": [ "derived" ], "title": "derived-remark-compute-modules", "contents": [ "To see the last displayed equality in the proof above we can argue", "with elements as follows. We have", "$s\\pi(l, k, k^{+}) = (l, 0, 0)$.", "Hence the morphism of the left hand side maps", "$(l, k, k^{+})$ to $(0, k, k^{+})$.", "On the other hand $h(l, k, k^{+}) = (0, 0, k)$ and", "$d(l, k, k^{+}) = (dl, dk + k^{+}, -dk^{+})$.", "Hence $(dh + hd)(l, k, k^{+}) =", "d(0, 0, k) + h(dl, dk + k^{+}, -dk^{+}) =", "(0, k, -dk) + (0, 0, dk + k^{+}) = (0, k, k^{+})$", "as desired." ], "refs": [], "ref_ids": [] }, { "id": 2011, "type": "other", "label": "derived-remark-make-commute", "categories": [ "derived" ], "title": "derived-remark-make-commute", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Let $0 \\to A_i^\\bullet \\to B_i^\\bullet \\to C_i^\\bullet \\to 0$, $i = 1, 2$", "be termwise split exact sequences. Suppose that", "$a : A_1^\\bullet \\to A_2^\\bullet$,", "$b : B_1^\\bullet \\to B_2^\\bullet$, and", "$c : C_1^\\bullet \\to C_2^\\bullet$ are morphisms of complexes", "such that", "$$", "\\xymatrix{", "A_1^\\bullet \\ar[d]_a \\ar[r] &", "B_1^\\bullet \\ar[r] \\ar[d]_b &", "C_1^\\bullet \\ar[d]_c \\\\", "A_2^\\bullet \\ar[r] & B_2^\\bullet \\ar[r] & C_2^\\bullet", "}", "$$", "commutes in $K(\\mathcal{A})$. In general, there does {\\bf not} exist", "a morphism $b' : B_1^\\bullet \\to B_2^\\bullet$ which is homotopic to $b$", "such that the diagram above commutes in the category of complexes.", "Namely, consider", "Examples, Equation (\\ref{examples-equation-commutes-up-to-homotopy}).", "If we could replace the middle map there by a homotopic one such that", "the diagram commutes, then we would have additivity of traces which we do not." ], "refs": [], "ref_ids": [] }, { "id": 2012, "type": "other", "label": "derived-remark-boundedness-conditions-triangulated", "categories": [ "derived" ], "title": "derived-remark-boundedness-conditions-triangulated", "contents": [ "Let $\\mathcal{A}$ be an additive category.", "Exactly the same proof as the proof of", "Proposition \\ref{proposition-homotopy-category-triangulated}", "shows that the categories", "$K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, and $K^b(\\mathcal{A})$", "are triangulated categories. Namely, the cone of a morphism between", "bounded (above, below) is bounded (above, below).", "But we prove below that these are triangulated subcategories", "of $K(\\mathcal{A})$ which gives another proof." ], "refs": [ "derived-proposition-homotopy-category-triangulated" ], "ref_ids": [ 1960 ] }, { "id": 2013, "type": "other", "label": "derived-remark-homotopy-double", "categories": [ "derived" ], "title": "derived-remark-homotopy-double", "contents": [ "Let $\\mathcal{A}$ be an additive category with countable direct sums.", "Let $\\text{DoubleComp}(\\mathcal{A})$ denote the category of double complexes", "in $\\mathcal{A}$, see", "Homology, Section \\ref{homology-section-double-complexes}.", "We can use this category to construct two triangulated categories.", "\\begin{enumerate}", "\\item We can consider an object $A^{\\bullet, \\bullet}$ of", "$\\text{DoubleComp}(\\mathcal{A})$ as a complex of complexes", "as follows", "$$", "\\ldots \\to A^{\\bullet, -1} \\to A^{\\bullet, 0} \\to A^{\\bullet, 1} \\to \\ldots", "$$", "and take the homotopy category $K_{first}(\\text{DoubleComp}(\\mathcal{A}))$", "with the corresponding triangulated structure given by", "Proposition \\ref{proposition-homotopy-category-triangulated}.", "By Homology, Remark", "\\ref{homology-remark-double-complex-complex-of-complexes-first} the functor", "$$", "\\text{Tot} :", "K_{first}(\\text{DoubleComp}(\\mathcal{A}))", "\\longrightarrow", "K(\\mathcal{A})", "$$", "is an exact functor of triangulated categories.", "\\item We can consider an object $A^{\\bullet, \\bullet}$ of", "$\\text{DoubleComp}(\\mathcal{A})$ as a complex of complexes", "as follows", "$$", "\\ldots \\to A^{-1, \\bullet} \\to A^{0, \\bullet} \\to A^{1, \\bullet} \\to \\ldots", "$$", "and take the homotopy category $K_{second}(\\text{DoubleComp}(\\mathcal{A}))$", "with the corresponding triangulated structure given by", "Proposition \\ref{proposition-homotopy-category-triangulated}.", "By Homology, Remark", "\\ref{homology-remark-double-complex-complex-of-complexes-second} the functor", "$$", "\\text{Tot} :", "K_{second}(\\text{DoubleComp}(\\mathcal{A}))", "\\longrightarrow", "K(\\mathcal{A})", "$$", "is an exact functor of triangulated categories.", "\\end{enumerate}" ], "refs": [ "derived-proposition-homotopy-category-triangulated", "homology-remark-double-complex-complex-of-complexes-first", "derived-proposition-homotopy-category-triangulated", "homology-remark-double-complex-complex-of-complexes-second" ], "ref_ids": [ 1960, 12193, 1960, 12194 ] }, { "id": 2014, "type": "other", "label": "derived-remark-double-complex-as-tensor-product-of", "categories": [ "derived" ], "title": "derived-remark-double-complex-as-tensor-product-of", "contents": [ "Let $\\mathcal{A}$, $\\mathcal{B}$, $\\mathcal{C}$ be additive categories", "and assume $\\mathcal{C}$ has countable direct sums. Suppose that", "$$", "\\otimes : \\mathcal{A} \\times \\mathcal{B} \\longrightarrow \\mathcal{C},", "\\quad", "(X, Y) \\longmapsto X \\otimes Y", "$$", "is a functor which is bilinear on morphisms. This determines a functor", "$$", "\\text{Comp}(\\mathcal{A}) \\times \\text{Comp}(\\mathcal{B})", "\\longrightarrow", "\\text{DoubleComp}(\\mathcal{C}), \\quad", "(X^\\bullet, Y^\\bullet)", "\\longmapsto", "X^\\bullet \\otimes Y^\\bullet", "$$", "See", "Homology, Example \\ref{homology-example-double-complex-as-tensor-product-of}.", "\\begin{enumerate}", "\\item For a fixed object $X^\\bullet$ of $\\text{Comp}(\\mathcal{A})$", "the functor", "$$", "K(\\mathcal{B}) \\longrightarrow K(\\mathcal{C}), \\quad", "Y^\\bullet \\longmapsto \\text{Tot}(X^\\bullet \\otimes Y^\\bullet)", "$$", "is an exact functor of triangulated categories.", "\\item For a fixed object $Y^\\bullet$ of $\\text{Comp}(\\mathcal{B})$", "the functor", "$$", "K(\\mathcal{A}) \\longrightarrow K(\\mathcal{C}), \\quad", "X^\\bullet \\longmapsto \\text{Tot}(X^\\bullet \\otimes Y^\\bullet)", "$$", "is an exact functor of triangulated categories.", "\\end{enumerate}", "This follows from Remark \\ref{remark-homotopy-double} since", "the functors", "$\\text{Comp}(\\mathcal{A}) \\to \\text{DoubleComp}(\\mathcal{C})$,", "$Y^\\bullet \\mapsto X^\\bullet \\otimes Y^\\bullet$ and", "$\\text{Comp}(\\mathcal{B}) \\to \\text{DoubleComp}(\\mathcal{C})$,", "$X^\\bullet \\mapsto X^\\bullet \\otimes Y^\\bullet$", "are immediately seen to be compatible with homotopies", "and termwise split short exact sequences and hence induce", "exact functors of triangulated categories", "$$", "K(\\mathcal{B}) \\to K_{first}(\\text{DoubleComp}(\\mathcal{C}))", "\\quad\\text{and}\\quad", "K(\\mathcal{A}) \\to K_{second}(\\text{DoubleComp}(\\mathcal{C}))", "$$", "Observe that for the first of the two the isomorphism", "$$", "\\text{Tot}(X^\\bullet \\otimes Y^\\bullet[1]) \\cong", "\\text{Tot}(X^\\bullet \\otimes Y^\\bullet)[1]", "$$", "involves signs (this goes back to the signs chosen in", "Homology, Remark \\ref{homology-remark-shift-double-complex})." ], "refs": [ "derived-remark-homotopy-double", "homology-remark-shift-double-complex" ], "ref_ids": [ 2013, 12192 ] }, { "id": 2015, "type": "other", "label": "derived-remark-existence-derived", "categories": [ "derived" ], "title": "derived-remark-existence-derived", "contents": [ "In this chapter, we consistently work with ``small'' abelian categories", "(as is the convention in the Stacks project). For a ``big'' abelian", "category $\\mathcal{A}$, it isn't clear that the derived category", "$D(\\mathcal{A})$ exists, because it isn't clear that morphisms in the", "derived category are sets. In fact, in general they aren't, see", "Examples, Lemma \\ref{examples-lemma-big-abelian-category}.", "However, if $\\mathcal{A}$ is a Grothendieck abelian category, and given", "$K^\\bullet, L^\\bullet$ in $K(\\mathcal{A})$, then by", "Injectives, Theorem \\ref{injectives-theorem-K-injective-embedding-grothendieck}", "there exists a quasi-isomorphism $L^\\bullet \\to I^\\bullet$ to a", "K-injective complex $I^\\bullet$ and Lemma \\ref{lemma-K-injective} shows that", "$$", "\\Hom_{D(\\mathcal{A})}(K^\\bullet, L^\\bullet) =", "\\Hom_{K(\\mathcal{A})}(K^\\bullet, I^\\bullet)", "$$", "which is a set. Some examples of Grothendieck abelian categories", "are the category of modules over a ring, or more generally", "the category of sheaves of modules on a ringed site." ], "refs": [ "examples-lemma-big-abelian-category", "injectives-theorem-K-injective-embedding-grothendieck", "derived-lemma-K-injective" ], "ref_ids": [ 2571, 7768, 1908 ] }, { "id": 2016, "type": "other", "label": "derived-remark-truncation-distinguished-triangle", "categories": [ "derived" ], "title": "derived-remark-truncation-distinguished-triangle", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let $K^\\bullet$ be a complex", "of $\\mathcal{A}$. Let $a \\in \\mathbf{Z}$. We claim there is a canonical", "distinguished triangle", "$$", "\\tau_{\\leq a}K^\\bullet \\to K^\\bullet \\to \\tau_{\\geq a + 1}K^\\bullet \\to", "(\\tau_{\\leq a}K^\\bullet)[1]", "$$", "in $D(\\mathcal{A})$. Here we have used the canonical truncation functors $\\tau$", "from Homology, Section \\ref{homology-section-truncations}.", "Namely, we first take the distinguished", "triangle associated by our $\\delta$-functor", "(Lemma \\ref{lemma-derived-canonical-delta-functor})", "to the short exact sequence of complexes", "$$", "0 \\to \\tau_{\\leq a}K^\\bullet \\to K^\\bullet \\to", "K^\\bullet/\\tau_{\\leq a}K^\\bullet \\to 0", "$$", "Next, we use that the map $K^\\bullet \\to \\tau_{\\geq a + 1}K^\\bullet$", "factors through a quasi-isomorphism", "$K^\\bullet/\\tau_{\\leq a}K^\\bullet \\to \\tau_{\\geq a + 1}K^\\bullet$", "by the description of cohomology groups in", "Homology, Section \\ref{homology-section-truncations}.", "In a similar way we obtain canonical distinguished triangles", "$$", "\\tau_{\\leq a}K^\\bullet \\to \\tau_{\\leq a + 1}K^\\bullet \\to", "H^{a + 1}(K^\\bullet)[-a-1] \\to (\\tau_{\\leq a}K^\\bullet)[1]", "$$", "and", "$$", "H^a(K^\\bullet)[-a] \\to \\tau_{\\geq a}K^\\bullet \\to \\tau_{\\geq a + 1}K^\\bullet", "\\to H^a(K^\\bullet)[-a + 1]", "$$" ], "refs": [ "derived-lemma-derived-canonical-delta-functor" ], "ref_ids": [ 1814 ] }, { "id": 2017, "type": "other", "label": "derived-remark-easier-proofs", "categories": [ "derived" ], "title": "derived-remark-easier-proofs", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Using the fact that $K(\\mathcal{A})$ is a triangulated category we", "may use", "Lemma \\ref{lemma-acyclic-is-zero}", "to obtain proofs of some of the lemmas below which are usually proved by", "chasing through diagrams.", "Namely, suppose that $\\alpha : K^\\bullet \\to L^\\bullet$ is a quasi-isomorphism", "of complexes. Then", "$$", "(K^\\bullet, L^\\bullet, C(\\alpha)^\\bullet, \\alpha, i, -p)", "$$", "is a distinguished triangle in $K(\\mathcal{A})$", "(Lemma \\ref{lemma-the-same-up-to-isomorphisms})", "and $C(\\alpha)^\\bullet$ is an acyclic complex", "(Lemma \\ref{lemma-acyclic}).", "Next, let $I^\\bullet$ be a bounded below complex of injective objects. Then", "$$", "\\xymatrix{", "\\Hom_{K(\\mathcal{A})}(C(\\alpha)^\\bullet, I^\\bullet) \\ar[r] &", "\\Hom_{K(\\mathcal{A})}(L^\\bullet, I^\\bullet) \\ar[r] &", "\\Hom_{K(\\mathcal{A})}(K^\\bullet, I^\\bullet) \\ar[lld] \\\\", "\\Hom_{K(\\mathcal{A})}(C(\\alpha)^\\bullet[-1], I^\\bullet)", "}", "$$", "is an exact sequence of abelian groups, see", "Lemma \\ref{lemma-representable-homological}.", "At this point", "Lemma \\ref{lemma-acyclic-is-zero}", "guarantees that the outer two groups are zero and hence", "$\\Hom_{K(\\mathcal{A})}(L^\\bullet, I^\\bullet) =", "\\Hom_{K(\\mathcal{A})}(K^\\bullet, I^\\bullet)$." ], "refs": [ "derived-lemma-acyclic-is-zero", "derived-lemma-the-same-up-to-isomorphisms", "derived-lemma-acyclic", "derived-lemma-representable-homological", "derived-lemma-acyclic-is-zero" ], "ref_ids": [ 1852, 1802, 1811, 1758, 1852 ] }, { "id": 2018, "type": "other", "label": "derived-remark-easier-projective", "categories": [ "derived" ], "title": "derived-remark-easier-projective", "contents": [ "Let $\\mathcal{A}$ be an abelian category.", "Suppose that $\\alpha : K^\\bullet \\to L^\\bullet$ is a quasi-isomorphism", "of complexes. Let $P^\\bullet$ be a bounded above complex of projectives.", "Then", "$$", "\\Hom_{K(\\mathcal{A})}(P^\\bullet, K^\\bullet)", "\\longrightarrow", "\\Hom_{K(\\mathcal{A})}(P^\\bullet, L^\\bullet)", "$$", "is an isomorphism. This is dual to", "Remark \\ref{remark-easier-proofs}." ], "refs": [ "derived-remark-easier-proofs" ], "ref_ids": [ 2017 ] }, { "id": 2019, "type": "other", "label": "derived-remark-functorial-ss", "categories": [ "derived" ], "title": "derived-remark-functorial-ss", "contents": [ "The spectral sequences of Lemma \\ref{lemma-two-ss-complex-functor}", "are functorial in the complex $K^\\bullet$. This follows from functoriality", "properties of Cartan-Eilenberg resolutions. On the other hand, they are", "both examples of a more general spectral sequence which may be associated", "to a filtered complex of $\\mathcal{A}$. The functoriality will follow from", "its construction. We will return to this in the section on the filtered", "derived category, see Remark \\ref{remark-final-functorial}." ], "refs": [ "derived-lemma-two-ss-complex-functor", "derived-remark-final-functorial" ], "ref_ids": [ 1871, 2025 ] }, { "id": 2020, "type": "other", "label": "derived-remark-big-localization", "categories": [ "derived" ], "title": "derived-remark-big-localization", "contents": [ "Suppose that $\\mathcal{A}$ is a ``big'' abelian category with enough injectives", "such as the category of abelian groups. In this case we have to be slightly", "more careful in constructing our resolution functor since we cannot use", "the axiom of choice with a quantifier ranging over a class. But note that", "the proof of the lemma does show that any two localization functors are", "canonically isomorphic. Namely, given quasi-isomorphisms", "$i : K^\\bullet \\to I^\\bullet$ and $i' : K^\\bullet \\to J^\\bullet$ of", "a bounded below complex $K^\\bullet$ into bounded below complexes of injectives", "there exists a unique(!) morphism $a : I^\\bullet \\to J^\\bullet$", "in $K^{+}(\\mathcal{I})$ such that $i' = i \\circ a$ as morphisms in", "$K^{+}(\\mathcal{I})$. Hence the only issue is existence, and we will see how", "to deal with this in the next section." ], "refs": [], "ref_ids": [] }, { "id": 2021, "type": "other", "label": "derived-remark-match", "categories": [ "derived" ], "title": "derived-remark-match", "contents": [ "Suppose $inj$ is a functor such that $s \\circ inj = \\text{id}$", "as in part (2) of", "Lemma \\ref{lemma-functorial-injective-resolutions}.", "Write $inj(K^\\bullet) = (i_{K^\\bullet} : K^\\bullet \\to j(K^\\bullet))$", "as in the proof of that lemma.", "Suppose $\\alpha : K^\\bullet \\to L^\\bullet$ is a map", "of bounded below complexes. Consider the map", "$inj(\\alpha)$ in the category $\\text{InjRes}(\\mathcal{A})$.", "It induces a commutative diagram", "$$", "\\xymatrix{", "K^\\bullet", "\\ar[rr]^-{\\alpha}", "\\ar[d]_{i_K} & &", "L^\\bullet \\ar[d]^{i_L} \\\\", "j(K)^\\bullet", "\\ar[rr]^-{inj(\\alpha)}", "& &", "j(L)^\\bullet", "}", "$$", "of morphisms of complexes.", "Hence, looking at the proof of", "Lemma \\ref{lemma-resolution-functor}", "we see that the functor $j : K^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{I})$", "is given by the rule", "$$", "j(\\alpha\\text{ up to homotopy}) = inj(\\alpha)\\text{ up to homotopy}\\in", "\\Hom_{K^{+}(\\mathcal{I})}(j(K^\\bullet), j(L^\\bullet))", "$$", "Hence we see that $j$ matches $t \\circ inj$ in this case, i.e., the", "diagram", "$$", "\\xymatrix{", "\\text{Comp}^{+}(\\mathcal{A}) \\ar[rr]_{t \\circ inj} \\ar[rd] & &", "K^{+}(\\mathcal{I}) \\\\", "& K^{+}(\\mathcal{A}) \\ar[ru]_j", "}", "$$", "is commutative." ], "refs": [ "derived-lemma-functorial-injective-resolutions", "derived-lemma-resolution-functor" ], "ref_ids": [ 1878, 1874 ] }, { "id": 2022, "type": "other", "label": "derived-remark-big-abelian-category", "categories": [ "derived" ], "title": "derived-remark-big-abelian-category", "contents": [ "Let $\\textit{Mod}(\\mathcal{O}_X)$ be the category of $\\mathcal{O}_X$-modules", "on a ringed space $(X, \\mathcal{O}_X)$ (or more generally on a", "ringed site). We will see later that $\\textit{Mod}(\\mathcal{O}_X)$ has enough", "injectives and in fact functorial injective embeddings, see", "Injectives, Theorem \\ref{injectives-theorem-sheaves-modules-injectives}.", "Note that the proof of Lemma \\ref{lemma-into-derived-category} does", "not apply to $\\textit{Mod}(\\mathcal{O}_X)$. But the proof of", "Lemma \\ref{lemma-functorial-injective-resolutions} does apply", "to $\\textit{Mod}(\\mathcal{O}_X)$. Thus we obtain", "$$", "j : K^{+}(\\textit{Mod}(\\mathcal{O}_X))", "\\longrightarrow", "K^{+}(\\mathcal{I})", "$$", "which is a resolution functor where $\\mathcal{I}$ is the additive", "category of injective $\\mathcal{O}_X$-modules. This argument also", "works in the following cases:", "\\begin{enumerate}", "\\item The category $\\text{Mod}_R$ of $R$-modules over a ring $R$.", "\\item The category $\\textit{PMod}(\\mathcal{O})$ of presheaves of", "$\\mathcal{O}$-modules on a site endowed with a presheaf of rings.", "\\item The category $\\textit{Mod}(\\mathcal{O})$ of sheaves of", "$\\mathcal{O}$-modules on a ringed site.", "\\item Add more here as needed.", "\\end{enumerate}" ], "refs": [ "injectives-theorem-sheaves-modules-injectives", "derived-lemma-into-derived-category", "derived-lemma-functorial-injective-resolutions" ], "ref_ids": [ 7766, 1875, 1878 ] }, { "id": 2023, "type": "other", "label": "derived-remark-right-derived-functor", "categories": [ "derived" ], "title": "derived-remark-right-derived-functor", "contents": [ "In the situation of", "Lemma \\ref{lemma-right-derived-functor}", "we see that we have actually lifted the right derived", "functor to an exact functor", "$F \\circ j' : D^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{B})$.", "It is occasionally useful to use such a factorization." ], "refs": [ "derived-lemma-right-derived-functor" ], "ref_ids": [ 1879 ] }, { "id": 2024, "type": "other", "label": "derived-remark-filtered-localization-big", "categories": [ "derived" ], "title": "derived-remark-filtered-localization-big", "contents": [ "We can invert the arrow of the lemma", "only if $\\mathcal{A}$ is a category in our sense,", "namely if it has a set of objects. However, suppose given a big abelian", "category $\\mathcal{A}$ with enough injectives, such as", "$\\textit{Mod}(\\mathcal{O}_X)$ for example. Then for any given set of objects", "$\\{A_i\\}_{i\\in I}$ there is an abelian subcategory", "$\\mathcal{A}' \\subset \\mathcal{A}$ containing all of them", "and having enough injectives, see", "Sets, Lemma \\ref{sets-lemma-abelian-injectives}.", "Thus we may use the lemma above for $\\mathcal{A}'$.", "This essentially means that if we use a set worth of diagrams, etc", "then we will never run into trouble using the lemma." ], "refs": [ "sets-lemma-abelian-injectives" ], "ref_ids": [ 8801 ] }, { "id": 2025, "type": "other", "label": "derived-remark-final-functorial", "categories": [ "derived" ], "title": "derived-remark-final-functorial", "contents": [ "As promised in Remark \\ref{remark-functorial-ss} we discuss the connection", "of the lemma above with the constructions using Cartan-Eilenberg resolutions.", "Namely, let $T : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor", "of abelian categories, assume $\\mathcal{A}$", "has enough injectives, and let $K^\\bullet$ be a bounded below complex", "of $\\mathcal{A}$. We give an alternative construction of the", "spectral sequences ${}'E$ and ${}''E$ of", "Lemma \\ref{lemma-two-ss-complex-functor}.", "\\medskip\\noindent", "First spectral sequence. Consider the ``stupid'' filtration on $K^\\bullet$", "obtained by setting $F^p(K^\\bullet) = \\sigma_{\\geq p}(K^\\bullet)$, see", "Homology, Section \\ref{homology-section-truncations}.", "Note that this stupid in the sense that", "$d(F^p(K^\\bullet)) \\subset F^{p + 1}(K^\\bullet)$, compare", "Homology, Lemma \\ref{homology-lemma-spectral-sequence-filtered-complex-d1}.", "Note that $\\text{gr}^p(K^\\bullet) = K^p[-p]$ with this filtration.", "According to Lemma \\ref{lemma-ss-filtered-derived} there is a spectral sequence", "with $E_1$ term", "$$", "E_1^{p, q} = R^{p + q}T(K^p[-p]) = R^qT(K^p)", "$$", "as in the spectral sequence ${}'E_r$. Observe moreover that the differentials", "$E_1^{p, q} \\to E_1^{p + 1, q}$ agree with the differentials in $'{}E_1$, see", "Homology, Lemma", "\\ref{homology-lemma-spectral-sequence-filtered-complex-d1} part (2)", "and the description of ${}'d_1$ in the proof of", "Lemma \\ref{lemma-two-ss-complex-functor}.", "\\medskip\\noindent", "Second spectral sequence. Consider the filtration on the complex $K^\\bullet$", "obtained by setting $F^p(K^\\bullet) = \\tau_{\\leq -p}(K^\\bullet)$, see", "Homology, Section \\ref{homology-section-truncations}.", "The minus sign is necessary", "to get a decreasing filtration. Note that", "$\\text{gr}^p(K^\\bullet)$ is quasi-isomorphic to $H^{-p}(K^\\bullet)[p]$", "with this filtration. According to Lemma \\ref{lemma-ss-filtered-derived}", "there is a spectral sequence with $E_1$ term", "$$", "E_1^{p, q} = R^{p + q}T(H^{-p}(K^\\bullet)[p])", "= R^{2p + q}T(H^{-p}(K^\\bullet)) = {}''E_2^{i, j}", "$$", "with $i = 2p + q$ and $j = -p$. (This looks unnatural, but note that we", "could just have well developed the whole theory of filtered complexes", "using increasing filtrations, with the end result that this then looks", "natural, but the other one doesn't.) We leave it to the reader to see", "that the differentials match up.", "\\medskip\\noindent", "Actually, given a Cartan-Eilenberg resolution", "$K^\\bullet \\to I^{\\bullet, \\bullet}$ the induced morphism", "$K^\\bullet \\to \\text{Tot}(I^{\\bullet, \\bullet})$", "into the associated total complex", "will be a filtered injective resolution for either filtration", "using suitable filtrations on $\\text{Tot}(I^{\\bullet, \\bullet})$.", "This can be used", "to match up the spectral sequences exactly." ], "refs": [ "derived-remark-functorial-ss", "derived-lemma-two-ss-complex-functor", "homology-lemma-spectral-sequence-filtered-complex-d1", "derived-lemma-ss-filtered-derived", "homology-lemma-spectral-sequence-filtered-complex-d1", "derived-lemma-two-ss-complex-functor", "derived-lemma-ss-filtered-derived" ], "ref_ids": [ 2019, 1871, 12096, 1891, 12096, 1871, 1891 ] }, { "id": 2026, "type": "other", "label": "derived-remark-uniqueness-derived-colimit", "categories": [ "derived" ], "title": "derived-remark-uniqueness-derived-colimit", "contents": [ "Let $\\mathcal{D}$ be a triangulated category.", "Let $(K_n, f_n)$ be a system of objects of $\\mathcal{D}$.", "We may think of a derived colimit as an object $K$", "of $\\mathcal{D}$ endowed with morphisms $i_n : K_n \\to K$", "such that $i_{n + 1} \\circ f_n = i_n$ and such that there", "exists a morphism $c : K \\to \\bigoplus K_n$ with the property that", "$$", "\\bigoplus K_n \\xrightarrow{1 - f_n} \\bigoplus K_n \\xrightarrow{i_n}", "K \\xrightarrow{c} \\bigoplus K_n[1]", "$$", "is a distinguished triangle. If $(K', i'_n, c')$ is a second", "derived colimit, then there exists an isomorphism", "$\\varphi : K \\to K'$ such that $\\varphi \\circ i_n = i'_n$ and", "$c' \\circ \\varphi = c$. The existence of $\\varphi$ is", "TR3 and the fact that $\\varphi$ is an isomorphism is", "Lemma \\ref{lemma-third-isomorphism-triangle}." ], "refs": [ "derived-lemma-third-isomorphism-triangle" ], "ref_ids": [ 1759 ] }, { "id": 2027, "type": "other", "label": "derived-remark-functoriality-derived-colimit", "categories": [ "derived" ], "title": "derived-remark-functoriality-derived-colimit", "contents": [ "Let $\\mathcal{D}$ be a triangulated category.", "Let $(a_n) : (K_n, f_n) \\to (L_n, g_n)$ be a morphism of systems", "of objects of $\\mathcal{D}$. Let $(K, i_n, c)$ be a derived", "colimit of the first system and let $(L, j_n, d)$ be a derived", "colimit of the second system with notation as in", "Remark \\ref{remark-uniqueness-derived-colimit}.", "Then there exists a morphism $a : K \\to L$", "such that $a \\circ i_n = j_n$ and $d \\circ a = (a_n[1]) \\circ c$.", "This follows from TR3 applied to the defining distinguished", "triangles." ], "refs": [ "derived-remark-uniqueness-derived-colimit" ], "ref_ids": [ 2026 ] }, { "id": 2028, "type": "other", "label": "derived-remark-operations-functor", "categories": [ "derived" ], "title": "derived-remark-operations-functor", "contents": [ "Let $F : \\mathcal{T} \\to \\mathcal{T}'$ be an exact functor of triangulated", "categories. Given a full subcategory $\\mathcal{A}$ of $\\mathcal{T}$ we denote", "$F(\\mathcal{A})$ the full subcategory of $\\mathcal{T}'$ whose objects", "consists of all objects $F(A)$ with $A \\in \\Ob(\\mathcal{A})$. We have", "$$", "F(\\mathcal{A}[a, b]) = F(\\mathcal{A})[a, b]", "$$", "$$", "F(smd(\\mathcal{A})) \\subset smd(F(\\mathcal{A})),", "$$", "$$", "F(add(\\mathcal{A})) \\subset add(F(\\mathcal{A})),", "$$", "$$", "F(\\mathcal{A} \\star \\mathcal{B}) \\subset F(\\mathcal{A}) \\star F(\\mathcal{B}),", "$$", "$$", "F(\\mathcal{A}^{\\star n}) \\subset F(\\mathcal{A})^{\\star n}.", "$$", "We omit the trivial verifications." ], "refs": [], "ref_ids": [] }, { "id": 2029, "type": "other", "label": "derived-remark-operations-unions", "categories": [ "derived" ], "title": "derived-remark-operations-unions", "contents": [ "Let $\\mathcal{T}$ be a triangulated category. Given full subcategories", "$\\mathcal{A}_1 \\subset \\mathcal{A}_2 \\subset \\mathcal{A}_3 \\subset \\ldots$", "and $\\mathcal{B}$ of $\\mathcal{T}$ we have", "$$", "\\left(\\bigcup \\mathcal{A}_i\\right)[a, b] = \\bigcup \\mathcal{A}_i[a, b]", "$$", "$$", "smd\\left(\\bigcup \\mathcal{A}_i\\right) = \\bigcup smd(\\mathcal{A}_i),", "$$", "$$", "add\\left(\\bigcup \\mathcal{A}_i\\right) = \\bigcup add(\\mathcal{A}_i),", "$$", "$$", "\\left(\\bigcup \\mathcal{A}_i\\right) \\star \\mathcal{B} =", "\\bigcup \\mathcal{A}_i \\star \\mathcal{B},", "$$", "$$", "\\mathcal{B} \\star \\left(\\bigcup \\mathcal{A}_i\\right) =", "\\bigcup \\mathcal{B} \\star \\mathcal{A}_i,", "$$", "$$", "\\left(\\bigcup \\mathcal{A}_i\\right)^{\\star n} =", "\\bigcup \\mathcal{A}_i^{\\star n}.", "$$", "We omit the trivial verifications." ], "refs": [], "ref_ids": [] }, { "id": 2030, "type": "other", "label": "derived-remark-check-on-generator", "categories": [ "derived" ], "title": "derived-remark-check-on-generator", "contents": [ "Let $\\mathcal{D}$ be a triangulated category. Let $E$ be an object", "of $\\mathcal{D}$. Let $T$ be a property of objects of $\\mathcal{D}$.", "Suppose that", "\\begin{enumerate}", "\\item if $K_i \\in D(A)$, $i = 1, \\ldots, r$ with", "$T(K_i)$ for $i = 1, \\ldots, r$, then $T(\\bigoplus K_i)$,", "\\item if $K \\to L \\to M \\to K[1]$ is a distinguished triangle and", "$T$ holds for two, then $T$ holds for the third object,", "\\item if $T(K \\oplus L)$ then $T(K)$ and $T(L)$, and", "\\item $T(E[n])$ holds for all $n$.", "\\end{enumerate}", "Then $T$ holds for all objects of $\\langle E \\rangle$." ], "refs": [], "ref_ids": [] }, { "id": 2261, "type": "other", "label": "cohomology-remark-daniel", "categories": [ "cohomology" ], "title": "cohomology-remark-daniel", "contents": [ "Here is a different approach to the proofs of", "Lemmas \\ref{lemma-kill-cohomology-class-on-covering} and", "\\ref{lemma-describe-higher-direct-images} above.", "Let $(X, \\mathcal{O}_X)$ be a ringed space.", "Let $i_X : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{PMod}(\\mathcal{O}_X)$", "be the inclusion functor and let $\\#$ be the sheafification functor.", "Recall that $i_X$ is left exact and $\\#$ is exact.", "\\begin{enumerate}", "\\item First prove Lemma \\ref{lemma-include} below which says that the", "right derived functors of $i_X$ are given by", "$R^pi_X\\mathcal{F} = \\underline{H}^p(\\mathcal{F})$.", "Here is another proof: The equality is clear for $p = 0$.", "Both $(R^pi_X)_{p \\geq 0}$ and $(\\underline{H}^p)_{p \\geq 0}$", "are delta functors vanishing on injectives, hence both are universal,", "hence they are isomorphic. See Homology,", "Section \\ref{homology-section-cohomological-delta-functor}.", "\\item A restatement of Lemma \\ref{lemma-kill-cohomology-class-on-covering}", "is that $(\\underline{H}^p(\\mathcal{F}))^\\# = 0$, $p > 0$ for any sheaf of", "$\\mathcal{O}_X$-modules $\\mathcal{F}$.", "To see this is true, use that ${}^\\#$ is exact so", "$$", "(\\underline{H}^p(\\mathcal{F}))^\\# =", "(R^pi_X\\mathcal{F})^\\# =", "R^p(\\# \\circ i_X)(\\mathcal{F}) = 0", "$$", "because $\\# \\circ i_X$ is the identity functor.", "\\item Let $f : X \\to Y$ be a morphism of ringed spaces.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. The presheaf", "$V \\mapsto H^p(f^{-1}V, \\mathcal{F})$ is equal to", "$R^p (i_Y \\circ f_*)\\mathcal{F}$. You can prove this by noticing that", "both give universal delta functors as in the argument of (1) above.", "Hence Lemma \\ref{lemma-describe-higher-direct-images}", "says that $R^p f_* \\mathcal{F}= (R^p (i_Y \\circ f_*)\\mathcal{F})^\\#$.", "Again using that $\\#$ is exact a that $\\# \\circ i_Y$ is the identity", "functor we see that", "$$", "R^p f_* \\mathcal{F} =", "R^p(\\# \\circ i_Y \\circ f_*)\\mathcal{F} =", "(R^p (i_Y \\circ f_*)\\mathcal{F})^\\#", "$$", "as desired.", "\\end{enumerate}" ], "refs": [ "cohomology-lemma-kill-cohomology-class-on-covering", "cohomology-lemma-describe-higher-direct-images", "cohomology-lemma-include", "cohomology-lemma-kill-cohomology-class-on-covering", "cohomology-lemma-describe-higher-direct-images" ], "ref_ids": [ 2038, 2039, 2054, 2038, 2039 ] }, { "id": 2262, "type": "other", "label": "cohomology-remark-elucidate-lemma", "categories": [ "cohomology" ], "title": "cohomology-remark-elucidate-lemma", "contents": [ "Here is a down-to-earth explanation of the meaning of", "Lemma \\ref{lemma-before-Leray}. It says that given", "$f : X \\to Y$ and $\\mathcal{F} \\in \\textit{Mod}(\\mathcal{O}_X)$", "and given an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$", "we have", "$$", "\\begin{matrix}", "R\\Gamma(X, \\mathcal{F}) & \\text{is represented by} &", "\\Gamma(X, \\mathcal{I}^\\bullet) \\\\", "Rf_*\\mathcal{F} & \\text{is represented by} & f_*\\mathcal{I}^\\bullet \\\\", "R\\Gamma(Y, Rf_*\\mathcal{F}) & \\text{is represented by} &", "\\Gamma(Y, f_*\\mathcal{I}^\\bullet)", "\\end{matrix}", "$$", "the last fact coming from Leray's acyclicity lemma", "(Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity})", "and Lemma \\ref{lemma-pushforward-injective}.", "Finally, it combines this with the trivial observation that", "$$", "\\Gamma(X, \\mathcal{I}^\\bullet)", "=", "\\Gamma(Y, f_*\\mathcal{I}^\\bullet).", "$$", "to arrive at the commutativity of the diagram of the lemma." ], "refs": [ "cohomology-lemma-before-Leray", "derived-lemma-leray-acyclicity", "cohomology-lemma-pushforward-injective" ], "ref_ids": [ 2068, 1844, 2060 ] }, { "id": 2263, "type": "other", "label": "cohomology-remark-Leray-ss-more-structure", "categories": [ "cohomology" ], "title": "cohomology-remark-Leray-ss-more-structure", "contents": [ "The Leray spectral sequence, the way we proved it in Lemma \\ref{lemma-Leray}", "is a spectral sequence of $\\Gamma(Y, \\mathcal{O}_Y)$-modules. However, it", "is quite easy to see that it is in fact a spectral sequence of", "$\\Gamma(X, \\mathcal{O}_X)$-modules. For example $f$ gives rise to", "a morphism of ringed spaces", "$f' : (X, \\mathcal{O}_X) \\to (Y, f_*\\mathcal{O}_X)$.", "By Lemma \\ref{lemma-modules-abelian} the terms $E_r^{p, q}$ of the", "Leray spectral sequence for an $\\mathcal{O}_X$-module $\\mathcal{F}$", "and $f$ are identical with those for $\\mathcal{F}$ and $f'$", "at least for $r \\geq 2$. Namely, they both agree with the terms of the Leray", "spectral sequence for $\\mathcal{F}$ as an abelian sheaf.", "And since $(f_*\\mathcal{O}_X)(Y) = \\mathcal{O}_X(X)$ we see the result.", "It is often the case", "that the Leray spectral sequence carries additional structure." ], "refs": [ "cohomology-lemma-Leray", "cohomology-lemma-modules-abelian" ], "ref_ids": [ 2070, 2069 ] }, { "id": 2264, "type": "other", "label": "cohomology-remark-explain-arrow", "categories": [ "cohomology" ], "title": "cohomology-remark-explain-arrow", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Let $\\mathcal{G}$ be an $\\mathcal{O}_Y$-module.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "Let $\\varphi$ be an $f$-map from $\\mathcal{G}$ to $\\mathcal{F}$.", "Choose a resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$", "by a complex of injective $\\mathcal{O}_X$-modules.", "Choose resolutions $\\mathcal{G} \\to \\mathcal{J}^\\bullet$ and", "$f_*\\mathcal{I}^\\bullet \\to (\\mathcal{J}')^\\bullet$ by complexes", "of injective $\\mathcal{O}_Y$-modules. By", "Derived Categories, Lemma \\ref{derived-lemma-morphisms-lift}", "there exists a map of complexes", "$\\beta$ such that the diagram", "\\begin{equation}", "\\label{equation-choice}", "\\xymatrix{", "\\mathcal{G} \\ar[d] \\ar[r] &", "f_*\\mathcal{F} \\ar[r] &", "f_*\\mathcal{I}^\\bullet \\ar[d] \\\\", "\\mathcal{J}^\\bullet \\ar[rr]^\\beta & &", "(\\mathcal{J}')^\\bullet", "}", "\\end{equation}", "commutes. Applying global section functors we see", "that we get a diagram", "$$", "\\xymatrix{", " & & \\Gamma(Y, f_*\\mathcal{I}^\\bullet) \\ar[d]_{qis} \\ar@{=}[r] &", "\\Gamma(X, \\mathcal{I}^\\bullet) \\\\", "\\Gamma(Y, \\mathcal{J}^\\bullet) \\ar[rr]^\\beta & &", "\\Gamma(Y, (\\mathcal{J}')^\\bullet) &", "}", "$$", "The complex on the bottom left represents $R\\Gamma(Y, \\mathcal{G})$", "and the complex on the top right represents $R\\Gamma(X, \\mathcal{F})$.", "The vertical arrow is a quasi-isomorphism by", "Lemma \\ref{lemma-before-Leray} which becomes invertible after", "applying the localization functor", "$K^{+}(\\mathcal{O}_Y(Y)) \\to D^{+}(\\mathcal{O}_Y(Y))$.", "The arrow (\\ref{equation-functorial-derived}) is given by the", "composition of the horizontal map by the inverse of the vertical map." ], "refs": [ "derived-lemma-morphisms-lift", "cohomology-lemma-before-Leray" ], "ref_ids": [ 1853, 2068 ] }, { "id": 2265, "type": "other", "label": "cohomology-remark-correct-version-base-change-map", "categories": [ "cohomology" ], "title": "cohomology-remark-correct-version-base-change-map", "contents": [ "The ``correct'' version of the base change map is the map", "$$", "Lg^* Rf_* \\mathcal{F}^\\bullet", "\\longrightarrow", "R(f')_* L(g')^*\\mathcal{F}^\\bullet.", "$$", "The construction of this map involves", "unbounded complexes, see Remark \\ref{remark-base-change}." ], "refs": [ "cohomology-remark-base-change" ], "ref_ids": [ 2269 ] }, { "id": 2266, "type": "other", "label": "cohomology-remark-compared-ordered-complexes", "categories": [ "cohomology" ], "title": "cohomology-remark-compared-ordered-complexes", "contents": [ "This means that if we have two total orderings $<_1$ and $<_2$ on", "the index set $I$, then we get an isomorphism of complexes", "$\\tau = \\pi_2 \\circ c_1 :", "\\check{\\mathcal{C}}_{ord\\text{-}1}(\\mathcal{U}, \\mathcal{F}) \\to", "\\check{\\mathcal{C}}_{ord\\text{-}2}(\\mathcal{U}, \\mathcal{F})$.", "It is clear that", "$$", "\\tau(s)_{i_0 \\ldots i_p} =", "\\text{sign}(\\sigma) s_{i_{\\sigma(0)} \\ldots i_{\\sigma(p)}}", "$$", "where $i_0 <_1 i_1 <_1 \\ldots <_1 i_p$ and", "$i_{\\sigma(0)} <_2 i_{\\sigma(1)} <_2 \\ldots <_2 i_{\\sigma(p)}$.", "This is the sense in which the ordered {\\v C}ech complex is independent", "of the chosen total ordering." ], "refs": [], "ref_ids": [] }, { "id": 2267, "type": "other", "label": "cohomology-remark-locally-finite-sections", "categories": [ "cohomology" ], "title": "cohomology-remark-locally-finite-sections", "contents": [ "Let $X = \\bigcup_{i \\in I} U_i$ be a locally finite open covering.", "Denote $j_i : U_i \\to X$ the inclusion map. Suppose that for each $i$", "we are given an abelian sheaf $\\mathcal{F}_i$ on $U_i$. Consider the", "abelian sheaf $\\mathcal{G} = \\bigoplus_{i \\in I} (j_i)_*\\mathcal{F}_i$.", "Then for $V \\subset X$ open we actually have", "$$", "\\Gamma(V, \\mathcal{G}) = \\prod\\nolimits_{i \\in I} \\mathcal{F}_i(V \\cap U_i).", "$$", "In other words we have", "$$", "\\bigoplus\\nolimits_{i \\in I} (j_i)_*\\mathcal{F}_i =", "\\prod\\nolimits_{i \\in I} (j_i)_*\\mathcal{F}_i", "$$", "This seems strange until you realize that the direct sum of a collection", "of sheaves is the sheafification of what you think it should be.", "See discussion in Modules, Section \\ref{modules-section-kernels}.", "Thus we conclude that in this case the complex of", "Lemma \\ref{lemma-covering-resolution} has terms", "$$", "{\\mathfrak C}^p(\\mathcal{U}, \\mathcal{F}) =", "\\bigoplus\\nolimits_{i_0 \\ldots i_p}", "(j_{i_0 \\ldots i_p})_* \\mathcal{F}_{i_0 \\ldots i_p}", "$$", "which is sometimes useful." ], "refs": [ "cohomology-lemma-covering-resolution" ], "ref_ids": [ 2097 ] }, { "id": 2268, "type": "other", "label": "cohomology-remark-shift-complex-cech-complex", "categories": [ "cohomology" ], "title": "cohomology-remark-shift-complex-cech-complex", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let", "$\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be", "an open covering. Let $\\mathcal{F}^\\bullet$ be a bounded below complex", "of $\\mathcal{O}_X$-modules. Let $b$ be an integer.", "We claim there is a commutative diagram", "$$", "\\xymatrix{", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))[b]", "\\ar[r] \\ar[d]_\\gamma &", "R\\Gamma(X, \\mathcal{F}^\\bullet)[b] \\ar[d] \\\\", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet[b]))", "\\ar[r] &", "R\\Gamma(X, \\mathcal{F}^\\bullet[b])", "}", "$$", "in the derived category where the map $\\gamma$ is the map on complexes", "constructed in Homology, Remark \\ref{homology-remark-shift-double-complex}.", "This makes sense because the double complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet[b])$", "is clearly the same as the double complex", "$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet)[0, b]$", "introduced in Homology, Remark \\ref{homology-remark-shift-double-complex}.", "To check that the diagram commutes, we may choose an injective resolution", "$\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$ as in the proof of", "Lemma \\ref{lemma-cech-complex-complex}. Chasing diagrams, we see that", "it suffices to check the diagram commutes when we replace $\\mathcal{F}^\\bullet$", "by $\\mathcal{I}^\\bullet$. Then we consider the extended diagram", "$$", "\\xymatrix{", "\\Gamma(X, \\mathcal{I}^\\bullet)[b] \\ar[r] \\ar[d] &", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet))[b]", "\\ar[r] \\ar[d]_\\gamma &", "R\\Gamma(X, \\mathcal{I}^\\bullet)[b] \\ar[d] \\\\", "\\Gamma(X, \\mathcal{I}^\\bullet[b]) \\ar[r] &", "\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet[b]))", "\\ar[r] &", "R\\Gamma(X, \\mathcal{I}^\\bullet[b])", "}", "$$", "where the left horizontal arrows are (\\ref{equation-global-sections-to-cech}).", "Since in this case the horizonal arrows are isomorphisms in the derived", "category (see proof of Lemma \\ref{lemma-cech-complex-complex}) it", "suffices to show that the left square commutes. This is true because", "the map $\\gamma$ uses the sign $1$ on the summands", "$\\check{\\mathcal{C}}^0(\\mathcal{U}, \\mathcal{I}^{q + b})$, see", "formula in Homology, Remark \\ref{homology-remark-shift-double-complex}." ], "refs": [ "homology-remark-shift-double-complex", "homology-remark-shift-double-complex", "cohomology-lemma-cech-complex-complex", "cohomology-lemma-cech-complex-complex", "homology-remark-shift-double-complex" ], "ref_ids": [ 12192, 12192, 2098, 2098, 12192 ] }, { "id": 2269, "type": "other", "label": "cohomology-remark-base-change", "categories": [ "cohomology" ], "title": "cohomology-remark-base-change", "contents": [ "The construction of unbounded derived functor $Lf^*$ and $Rf_*$", "allows one to construct the base change map in full generality.", "Namely, suppose that", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "S' \\ar[r]^g &", "S", "}", "$$", "is a commutative diagram of ringed spaces. Let $K$ be an object of", "$D(\\mathcal{O}_X)$. Then there exists a canonical base change", "map", "$$", "Lg^*Rf_*K \\longrightarrow R(f')_*L(g')^*K", "$$", "in $D(\\mathcal{O}_{S'})$. Namely, this map is adjoint to a map", "$L(f')^*Lg^*Rf_*K \\to L(g')^*K$", "Since $L(f')^*Lg^* = L(g')^*Lf^*$ we see this is the same as a map", "$L(g')^*Lf^*Rf_*K \\to L(g')^*K$", "which we can take to be $L(g')^*$ of the adjunction map", "$Lf^*Rf_*K \\to K$." ], "refs": [], "ref_ids": [] }, { "id": 2270, "type": "other", "label": "cohomology-remark-compose-base-change", "categories": [ "cohomology" ], "title": "cohomology-remark-compose-base-change", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^l \\ar[d]_{g'} & Y \\ar[d]^g \\\\", "Z' \\ar[r]^m & Z", "}", "$$", "of ringed spaces. Then the base change maps of", "Remark \\ref{remark-base-change}", "for the two squares compose to give the base", "change map for the outer rectangle. More precisely,", "the composition", "\\begin{align*}", "Lm^* \\circ R(g \\circ f)_*", "& =", "Lm^* \\circ Rg_* \\circ Rf_* \\\\", "& \\to Rg'_* \\circ Ll^* \\circ Rf_* \\\\", "& \\to Rg'_* \\circ Rf'_* \\circ Lk^* \\\\", "& = R(g' \\circ f')_* \\circ Lk^*", "\\end{align*}", "is the base change map for the rectangle. We omit the verification." ], "refs": [ "cohomology-remark-base-change" ], "ref_ids": [ 2269 ] }, { "id": 2271, "type": "other", "label": "cohomology-remark-compose-base-change-horizontal", "categories": [ "cohomology" ], "title": "cohomology-remark-compose-base-change-horizontal", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X'' \\ar[r]_{g'} \\ar[d]_{f''} & X' \\ar[r]_g \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y'' \\ar[r]^{h'} & Y' \\ar[r]^h & Y", "}", "$$", "of ringed spaces. Then the base change maps of", "Remark \\ref{remark-base-change}", "for the two squares compose to give the base", "change map for the outer rectangle. More precisely,", "the composition", "\\begin{align*}", "L(h \\circ h')^* \\circ Rf_*", "& =", "L(h')^* \\circ Lh_* \\circ Rf_* \\\\", "& \\to L(h')^* \\circ Rf'_* \\circ Lg^* \\\\", "& \\to Rf''_* \\circ L(g')^* \\circ Lg^* \\\\", "& = Rf''_* \\circ L(g \\circ g')^*", "\\end{align*}", "is the base change map for the rectangle. We omit the verification." ], "refs": [ "cohomology-remark-base-change" ], "ref_ids": [ 2269 ] }, { "id": 2272, "type": "other", "label": "cohomology-remark-cup-product", "categories": [ "cohomology" ], "title": "cohomology-remark-cup-product", "contents": [ "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of", "ringed spaces. The adjointness of $Lf^*$ and $Rf_*$ allows us to construct", "a relative cup product", "$$", "Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L", "\\longrightarrow", "Rf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)", "$$", "in $D(\\mathcal{O}_Y)$ for all $K, L$ in $D(\\mathcal{O}_X)$.", "Namely, this map is adjoint to a map", "$Lf^*(Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L) \\to", "K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ for which we can take the", "composition of the isomorphism", "$Lf^*(Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L) =", "Lf^*Rf_*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*Rf_*L$", "(Lemma \\ref{lemma-pullback-tensor-product})", "with the map", "$Lf^*Rf_*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*Rf_*L", "\\to K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$", "coming from the counit $Lf^* \\circ Rf_* \\to \\text{id}$." ], "refs": [ "cohomology-lemma-pullback-tensor-product" ], "ref_ids": [ 2118 ] }, { "id": 2273, "type": "other", "label": "cohomology-remark-spectral-sequence-filtered-object", "categories": [ "cohomology" ], "title": "cohomology-remark-spectral-sequence-filtered-object", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}^\\bullet$ be a", "filtered complex of $\\mathcal{O}_X$-modules. If $\\mathcal{F}^\\bullet$", "is bounded from below and for each $n$ the filtration on $\\mathcal{F}^n$", "is finite, then there is a construction of the spectral sequence", "in Lemma \\ref{lemma-spectral-sequence-filtered-object}", "avoiding Injectives, Lemma", "\\ref{injectives-lemma-K-injective-embedding-filtration}.", "Namely, by", "Derived Categories, Lemma", "\\ref{derived-lemma-right-resolution-by-filtered-injectives}", "there is a filtered quasi-isomorphism", "$i : \\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$", "of filtered complexes", "with $\\mathcal{I}^\\bullet$ bounded below,", "the filtration on $\\mathcal{I}^n$ is finite for all $n$,", "and with each $\\text{gr}^p\\mathcal{I}^n$ an", "injective $\\mathcal{O}_X$-module.", "Then we take the spectral sequence associated to", "$$", "\\Gamma(X, \\mathcal{I}^\\bullet)", "\\quad\\text{with}\\quad", "F^p\\Gamma(X, \\mathcal{I}^\\bullet) = \\Gamma(X, F^p\\mathcal{I}^\\bullet)", "$$", "Since cohomology can be computed by evaluating on", "bounded below complexes of injectives", "we see that the $E_1$ page is as stated in the lemma.", "The convergence and boundedness under the stated conditions", "follows from", "Homology, Lemma \\ref{homology-lemma-biregular-ss-converges}.", "In fact, this is a special case of the spectral sequence", "in Derived Categories, Lemma \\ref{derived-lemma-ss-filtered-derived}." ], "refs": [ "cohomology-lemma-spectral-sequence-filtered-object", "injectives-lemma-K-injective-embedding-filtration", "derived-lemma-right-resolution-by-filtered-injectives", "homology-lemma-biregular-ss-converges", "derived-lemma-ss-filtered-derived" ], "ref_ids": [ 2124, 7797, 1887, 12101, 1891 ] }, { "id": 2274, "type": "other", "label": "cohomology-remark-cup-with-element-map-total-cohomology", "categories": [ "cohomology" ], "title": "cohomology-remark-cup-with-element-map-total-cohomology", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, M$ be objects", "of $D(\\mathcal{O}_X)$. Set $A = \\Gamma(X, \\mathcal{O}_X)$.", "Given $\\xi \\in H^i(X, K)$ we get an associated map", "$$", "\\xi = ``\\xi \\cup -'' :", "R\\Gamma(X, M)[-i]", "\\to", "R\\Gamma(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)", "$$", "by representing $\\xi$ as a map $\\xi : A[-i] \\to R\\Gamma(X, K)$ as in the", "proof of Lemma \\ref{lemma-second-cup-equals-first}", "and then using the composition", "$$", "R\\Gamma(X, M)[-i] = A[-i] \\otimes_A^\\mathbf{L} R\\Gamma(X, M)", "\\xrightarrow{\\xi \\otimes 1}", "R\\Gamma(X, K) \\otimes_A^\\mathbf{L} R\\Gamma(X, M)", "\\to", "R\\Gamma(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)", "$$", "where the second arrow is the global cup product $\\mu$ above.", "On cohomology this recovers the cup product by $\\xi$ as is clear", "from Lemma \\ref{lemma-second-cup-equals-first} and its proof." ], "refs": [ "cohomology-lemma-second-cup-equals-first", "cohomology-lemma-second-cup-equals-first" ], "ref_ids": [ 2128, 2128 ] }, { "id": 2275, "type": "other", "label": "cohomology-remark-support-cup-product", "categories": [ "cohomology" ], "title": "cohomology-remark-support-cup-product", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$", "be the inclusion of a closed subset. Given $K$ and $M$ in", "$D(\\mathcal{O}_X)$ there is a canonical map", "$$", "K|_Z \\otimes_{\\mathcal{O}_X|_Z}^\\mathbf{L} R\\mathcal{H}_Z(M)", "\\longrightarrow", "R\\mathcal{H}_Z(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)", "$$", "in $D(\\mathcal{O}_X|_Z)$. Here $K|_Z = i^{-1}K$ is the restriction of", "$K$ to $Z$ viewed as an object of $D(\\mathcal{O}_X|_Z)$. By adjointness", "of $i_*$ and $R\\mathcal{H}_Z$ of", "Lemma \\ref{lemma-cohomology-with-support-sheaf-on-support} to construct this", "map it suffices to produce a canonical map", "$$", "i_*\\left(K|_Z \\otimes_{\\mathcal{O}_X|_Z}^\\mathbf{L} R\\mathcal{H}_Z(M)\\right)", "\\longrightarrow", "K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M", "$$", "To construct this map, we choose a K-injective complex $\\mathcal{I}^\\bullet$", "of $\\mathcal{O}_X$-modules representing $M$ and a K-flat complex", "$\\mathcal{K}^\\bullet$ of $\\mathcal{O}_X$-modules representing $K$.", "Observe that $\\mathcal{K}^\\bullet|_Z$ is a K-flat complex of", "$\\mathcal{O}_X|_Z$-modules representing $K|_Z$, see", "Lemma \\ref{lemma-pullback-K-flat}. Hence we need to produce a map", "of complexes", "$$", "i_*\\text{Tot}\\left(", "\\mathcal{K}^\\bullet|_Z \\otimes_{\\mathcal{O}_X|_Z}", "\\mathcal{H}_Z(\\mathcal{I}^\\bullet)\\right)", "\\longrightarrow", "\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{I}^\\bullet)", "$$", "of $\\mathcal{O}_X$-modules. For this it suffices to produce maps", "$$", "i_*(\\mathcal{K}^a|_Z \\otimes_{\\mathcal{O}_X|_Z}", "\\mathcal{H}_Z(\\mathcal{I}^b))", "\\longrightarrow", "\\mathcal{K}^a \\otimes_{\\mathcal{O}_X} \\mathcal{I}^b", "$$", "Looking at stalks (for example), we see that the left hand side of this", "formula is equal to", "$\\mathcal{K}^a \\otimes_{\\mathcal{O}_X} i_*\\mathcal{H}_Z(\\mathcal{I}^b)$", "and we can use the inclusion", "$\\mathcal{H}_Z(\\mathcal{I}^b) \\to \\mathcal{I}^b$ to get our map." ], "refs": [ "cohomology-lemma-cohomology-with-support-sheaf-on-support", "cohomology-lemma-pullback-K-flat" ], "ref_ids": [ 2150, 2108 ] }, { "id": 2276, "type": "other", "label": "cohomology-remark-support-cup-product-global", "categories": [ "cohomology" ], "title": "cohomology-remark-support-cup-product-global", "contents": [ "With notation as in Remark \\ref{remark-support-cup-product}", "we obtain a canonical cup product", "\\begin{align*}", "H^a(X, K) \\times H^b_Z(X, M)", "& =", "H^a(X, K) \\times H^b(Z, R\\mathcal{H}_Z(M)) \\\\", "& \\to", "H^a(Z, K|_Z) \\times H^b(Z, R\\mathcal{H}_Z(M)) \\\\", "& \\to", "H^{a + b}(Z, K|_Z \\otimes_{\\mathcal{O}_X|_Z}^\\mathbf{L} R\\mathcal{H}_Z(M)) \\\\", "& \\to", "H^{a + b}(Z, R\\mathcal{H}_Z(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)) \\\\", "& =", "H^{a + b}_Z(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)", "\\end{align*}", "Here the equal signs are given by", "Lemma \\ref{lemma-local-to-global-sections-with-support},", "the first arrow is restriction to $Z$, the second", "arrow is the cup product (Section \\ref{section-cup-product}),", "and the third arrow is the map from Remark \\ref{remark-support-cup-product}." ], "refs": [ "cohomology-remark-support-cup-product", "cohomology-lemma-local-to-global-sections-with-support", "cohomology-remark-support-cup-product" ], "ref_ids": [ 2275, 2153, 2275 ] }, { "id": 2277, "type": "other", "label": "cohomology-remark-support-functorial", "categories": [ "cohomology" ], "title": "cohomology-remark-support-functorial", "contents": [ "Let $f : (X', \\mathcal{O}_{X'}) \\to (X, \\mathcal{O}_X)$ be a morphism", "of ringed spaces. Let $Z \\subset X$ be a closed subset and $Z' = f^{-1}(Z)$.", "Denote $f|_{Z'} : (Z', \\mathcal{O}_{X'}|_{Z'}) \\to (Z, \\mathcal{O}_X|Z)$ be the", "induced morphism of ringed spaces. For any $K$ in $D(\\mathcal{O}_X)$ there", "is a canonical map", "$$", "L(f|_{Z'})^*R\\mathcal{H}_Z(K) \\longrightarrow R\\mathcal{H}_{Z'}(Lf^*K)", "$$", "in $D(\\mathcal{O}_{X'}|_{Z'})$. Denote $i : Z \\to X$ and $i' : Z' \\to X'$", "the inclusion maps. By", "Lemma \\ref{lemma-complexes-with-support-on-closed} part (2)", "applied to $i'$ it is the same thing to give a map", "$$", "i'_* L(f|_{Z'})^* R\\mathcal{H}_Z(K)", "\\longrightarrow", "i'_*R\\mathcal{H}_{Z'}(Lf^*K)", "$$", "in $D_{Z'}(\\mathcal{O}_{X'})$. The map of functors", "$Lf^* \\circ i_* \\to i'_* \\circ L(f|_{Z'})^*$ of", "Remark \\ref{remark-base-change} is an isomorphism in this case", "(follows by checking what happens on stalks using that $i_*$ and $i'_*$", "are exact and that $\\mathcal{O}_{Z, z} = \\mathcal{O}_{X, z}$", "and similarly for $Z'$). Hence it suffices to construct a the top", "horizonal arrow in the following diagram", "$$", "\\xymatrix{", "Lf^* i_* R\\mathcal{H}_Z(K) \\ar[rr] \\ar[rd] & &", "i'_* R\\mathcal{H}_{Z'}(Lf^*K) \\ar[ld] \\\\", "& Lf^*K", "}", "$$", "The complex $Lf^* i_* R\\mathcal{H}_Z(K)$ is supported on $Z'$. The south-east", "arrow comes from the adjunction mapping $i_*R\\mathcal{H}_Z(K) \\to K$", "(Lemma \\ref{lemma-cohomology-with-support-sheaf-on-support}). Since the", "adjunction mapping $i'_* R\\mathcal{H}_{Z'}(Lf^*K) \\to Lf^*K$ is universal by", "Lemma \\ref{lemma-complexes-with-support-on-closed} part (3), we find that", "the south-east arrow factors uniquely over the south-west arrow and", "we obtain the desired arrow." ], "refs": [ "cohomology-lemma-complexes-with-support-on-closed", "cohomology-remark-base-change", "cohomology-lemma-cohomology-with-support-sheaf-on-support", "cohomology-lemma-complexes-with-support-on-closed" ], "ref_ids": [ 2151, 2269, 2150, 2151 ] }, { "id": 2278, "type": "other", "label": "cohomology-remark-discuss-derived-limit", "categories": [ "cohomology" ], "title": "cohomology-remark-discuss-derived-limit", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(K_n)$ be an inverse", "system in $D(\\mathcal{O}_X)$. Set $K = R\\lim K_n$. For each $n$ and $m$", "let $\\mathcal{H}^m_n = H^m(K_n)$ be the $m$th cohomology sheaf of", "$K_n$ and similarly set $\\mathcal{H}^m = H^m(K)$. Let us denote", "$\\underline{\\mathcal{H}}^m_n$ the presheaf", "$$", "U \\longmapsto \\underline{\\mathcal{H}}^m_n(U) = H^m(U, K_n)", "$$", "Similarly we set $\\underline{\\mathcal{H}}^m(U) = H^m(U, K)$.", "By Lemma \\ref{lemma-sheafification-cohomology} we see that", "$\\mathcal{H}^m_n$ is the sheafification of", "$\\underline{\\mathcal{H}}^m_n$ and $\\mathcal{H}^m$ is the", "sheafification of $\\underline{\\mathcal{H}}^m$.", "Here is a diagram", "$$", "\\xymatrix{", "K \\ar@{=}[d] &", "\\underline{\\mathcal{H}}^m \\ar[d] \\ar[r] & ", "\\mathcal{H}^m \\ar[d] \\\\", "R\\lim K_n &", "\\lim \\underline{\\mathcal{H}}^m_n \\ar[r] & ", "\\lim \\mathcal{H}^m_n", "}", "$$", "In general it may not be the case that", "$\\lim \\mathcal{H}^m_n$ is the sheafification of", "$\\lim \\underline{\\mathcal{H}}^m_n$.", "If $U \\subset X$ is an open, then we have short exact", "sequences", "\\begin{equation}", "\\label{equation-ses-Rlim-over-U}", "0 \\to", "R^1\\lim \\underline{\\mathcal{H}}^{m - 1}_n(U) \\to", "\\underline{\\mathcal{H}}^m(U) \\to", "\\lim \\underline{\\mathcal{H}}^m_n(U) \\to 0", "\\end{equation}", "by Lemma \\ref{lemma-RGamma-commutes-with-Rlim}." ], "refs": [ "cohomology-lemma-sheafification-cohomology", "cohomology-lemma-RGamma-commutes-with-Rlim" ], "ref_ids": [ 2136, 2160 ] }, { "id": 2279, "type": "other", "label": "cohomology-remark-tensor-internal-hom", "categories": [ "cohomology" ], "title": "cohomology-remark-tensor-internal-hom", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. For $K, K', M, M'$ in", "$D(\\mathcal{O}_X)$ there is a canonical map", "$$", "R\\SheafHom(K, K') \\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "R\\SheafHom(M, M')", "\\longrightarrow", "R\\SheafHom(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M,", "K' \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M')", "$$", "Namely, by (\\ref{equation-internal-hom}) is the same thing as a map", "$$", "R\\SheafHom(K, K') \\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "R\\SheafHom(M, M') \\otimes_{\\mathcal{O}_X}^\\mathbf{L}", "K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M", "\\longrightarrow", "K' \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M'", "$$", "For this we can first flip the middle two factors", "(with sign rules as in More on Algebra, Section", "\\ref{more-algebra-section-sign-rules})", "and use the maps", "$$", "R\\SheafHom(K, K') \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K \\to K'", "\\quad\\text{and}\\quad", "R\\SheafHom(M, M') \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M \\to M'", "$$", "from Lemma \\ref{lemma-internal-hom-composition} when thinking", "of $K = R\\SheafHom(\\mathcal{O}_X, K)$ and similarly for", "$K'$, $M$, and $M'$." ], "refs": [ "cohomology-lemma-internal-hom-composition" ], "ref_ids": [ 2186 ] }, { "id": 2280, "type": "other", "label": "cohomology-remark-projection-formula-for-internal-hom", "categories": [ "cohomology" ], "title": "cohomology-remark-projection-formula-for-internal-hom", "contents": [ "Let $f : X \\to Y$ be a morphism of ringed spaces.", "Let $K, L$ be objects of $D(\\mathcal{O}_X)$. We claim there is a canonical map", "$$", "Rf_*R\\SheafHom(L, K) \\longrightarrow R\\SheafHom(Rf_*L, Rf_*K)", "$$", "Namely, by (\\ref{equation-internal-hom}) this is the same thing", "as a map", "$Rf_*R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L \\to Rf_*K$.", "For this we can use the composition", "$$", "Rf_*R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L \\to", "Rf_*(R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) \\to", "Rf_*K", "$$", "where the first arrow is the relative cup product", "(Remark \\ref{remark-cup-product}) and the second arrow is $Rf_*$ applied", "to the canonical map", "$R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L \\to K$", "coming from Lemma \\ref{lemma-internal-hom-composition}", "(with $\\mathcal{O}_X$ in one of the spots)." ], "refs": [ "cohomology-remark-cup-product", "cohomology-lemma-internal-hom-composition" ], "ref_ids": [ 2272, 2186 ] }, { "id": 2281, "type": "other", "label": "cohomology-remark-relative-cup-and-composition", "categories": [ "cohomology" ], "title": "cohomology-remark-relative-cup-and-composition", "contents": [ "Let $h : X \\to Y$ be a morphism of ringed spaces.", "Let $K, L, M$ be objects of $D(\\mathcal{O}_Y)$.", "The diagram", "$$", "\\xymatrix{", "Rf_*R\\SheafHom_{\\mathcal{O}_X}(K, M)", "\\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*M", "\\ar[r] \\ar[d] &", "Rf_*\\left(R\\SheafHom_{\\mathcal{O}_X}(K, M)", "\\otimes_{\\mathcal{O}_X}^\\mathbf{L} M\\right)", "\\ar[d] \\\\", "R\\SheafHom_{\\mathcal{O}_Y}(Rf_*K, Rf_*M) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L}", "Rf_*M \\ar[r] &", "Rf_*M", "}", "$$", "is commutative. Here the left vertical arrow comes from", "Remark \\ref{remark-projection-formula-for-internal-hom}.", "The top horizontal arrow is Remark \\ref{remark-cup-product}.", "The other two arrows are instances of the map in", "Lemma \\ref{lemma-internal-hom-composition} (with one of the entries", "replaced with $\\mathcal{O}_X$ or $\\mathcal{O}_Y$)." ], "refs": [ "cohomology-remark-projection-formula-for-internal-hom", "cohomology-remark-cup-product", "cohomology-lemma-internal-hom-composition" ], "ref_ids": [ 2280, 2272, 2186 ] }, { "id": 2282, "type": "other", "label": "cohomology-remark-prepare-fancy-base-change", "categories": [ "cohomology" ], "title": "cohomology-remark-prepare-fancy-base-change", "contents": [ "Let $h : X \\to Y$ be a morphism of ringed spaces.", "Let $K, L$ be objects of $D(\\mathcal{O}_Y)$. We claim there is a", "canonical map", "$$", "Lh^*R\\SheafHom(K, L) \\longrightarrow R\\SheafHom(Lh^*K, Lh^*L)", "$$", "in $D(\\mathcal{O}_X)$. Namely, by (\\ref{equation-internal-hom})", "proved in Lemma \\ref{lemma-internal-hom}", "such a map is the same thing as a map", "$$", "Lh^*R\\SheafHom(K, L) \\otimes^\\mathbf{L} Lh^*K \\longrightarrow Lh^*L", "$$", "The source of this arrow is $Lh^*(\\SheafHom(K, L) \\otimes^\\mathbf{L} K)$", "by Lemma \\ref{lemma-pullback-tensor-product}", "hence it suffices to construct a canonical map", "$$", "R\\SheafHom(K, L) \\otimes^\\mathbf{L} K \\longrightarrow L.", "$$", "For this we take the arrow corresponding to", "$$", "\\text{id} :", "R\\SheafHom(K, L)", "\\longrightarrow", "R\\SheafHom(K, L)", "$$", "via (\\ref{equation-internal-hom})." ], "refs": [ "cohomology-lemma-internal-hom", "cohomology-lemma-pullback-tensor-product" ], "ref_ids": [ 2183, 2118 ] }, { "id": 2283, "type": "other", "label": "cohomology-remark-fancy-base-change", "categories": [ "cohomology" ], "title": "cohomology-remark-fancy-base-change", "contents": [ "Suppose that", "$$", "\\xymatrix{", "X' \\ar[r]_h \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "S' \\ar[r]^g &", "S", "}", "$$", "is a commutative diagram of ringed spaces. Let $K, L$ be objects", "of $D(\\mathcal{O}_X)$. We claim there exists a canonical base change", "map", "$$", "Lg^*Rf_*R\\SheafHom(K, L)", "\\longrightarrow", "R(f')_*R\\SheafHom(Lh^*K, Lh^*L)", "$$", "in $D(\\mathcal{O}_{S'})$. Namely, we take the map adjoint to", "the composition", "\\begin{align*}", "L(f')^*Lg^*Rf_*R\\SheafHom(K, L)", "& =", "Lh^*Lf^*Rf_*R\\SheafHom(K, L) \\\\", "& \\to", "Lh^*R\\SheafHom(K, L) \\\\", "& \\to", "R\\SheafHom(Lh^*K, Lh^*L)", "\\end{align*}", "where the first arrow uses the adjunction mapping", "$Lf^*Rf_* \\to \\text{id}$ and the second arrow is the canonical map", "constructed in Remark \\ref{remark-prepare-fancy-base-change}." ], "refs": [ "cohomology-remark-prepare-fancy-base-change" ], "ref_ids": [ 2282 ] }, { "id": 2284, "type": "other", "label": "cohomology-remark-uniqueness", "categories": [ "cohomology" ], "title": "cohomology-remark-uniqueness", "contents": [ "With notation and assumptions as in Lemma \\ref{lemma-uniqueness}.", "Suppose that $U, V \\in \\mathcal{B}$. Let $\\mathcal{B}'$ be the set of", "elements of $\\mathcal{B}$ contained in $U \\cap V$. Then", "$$", "(\\{K_{U'}\\}_{U' \\in \\mathcal{B}'},", "\\{\\rho_{V'}^{U'}\\}_{V' \\subset U'\\text{ with }U', V' \\in \\mathcal{B}'})", "$$", "is a system on the ringed space $U \\cap V$", "satisfying the assumptions of Lemma \\ref{lemma-uniqueness}.", "Moreover, both $(K_U|_{U \\cap V}, \\rho^U_{U'})$ and", "$(K_V|_{U \\cap V}, \\rho^V_{U'})$ are solutions to this system.", "By the lemma we find a unique isomorphism", "$$", "\\rho_{U, V} : K_U|_{U \\cap V} \\longrightarrow K_V|_{U \\cap V}", "$$", "such that for every $U' \\subset U \\cap V$, $U' \\in \\mathcal{B}$ the", "diagram", "$$", "\\xymatrix{", "K_U|_{U'} \\ar[rr]_{\\rho_{U, V}|_{U'}} \\ar[rd]_{\\rho^U_{U'}} & &", "K_V|_{U'} \\ar[ld]^{\\rho^V_{U'}} \\\\", "& K_{U'}", "}", "$$", "commutes. Pick a third element $W \\in \\mathcal{B}$. We obtain isomorphisms", "$\\rho_{U, W} : K_U|_{U \\cap W} \\to K_W|_{U \\cap W}$ and", "$\\rho_{V, W} : K_U|_{V \\cap W} \\to K_W|_{V \\cap W}$ satisfying", "similar properties to those of $\\rho_{U, V}$. Finally,", "we have", "$$", "\\rho_{U, W}|_{U \\cap V \\cap W} =", "\\rho_{V, W}|_{U \\cap V \\cap W} \\circ \\rho_{U, V}|_{U \\cap V \\cap W}", "$$", "This is true by the uniqueness in the lemma", "because both sides of the equality are the unique isomorphism", "compatible with the maps $\\rho^U_{U''}$ and $\\rho^W_{U''}$", "for $U'' \\subset U \\cap V \\cap W$, $U'' \\in \\mathcal{B}$.", "Some minor details omitted.", "The collection $(K_U, \\rho_{U, V})$ is a descent datum", "in the derived category for the open covering", "$\\mathcal{U} : X = \\bigcup_{U \\in \\mathcal{B}} U$ of $X$.", "In this language we are looking for ``effectiveness of the descent datum''", "when we look for the existence of a solution." ], "refs": [ "cohomology-lemma-uniqueness", "cohomology-lemma-uniqueness" ], "ref_ids": [ 2193, 2193 ] }, { "id": 2285, "type": "other", "label": "cohomology-remark-compatible-with-diagram", "categories": [ "cohomology" ], "title": "cohomology-remark-compatible-with-diagram", "contents": [ "The map (\\ref{equation-projection-formula-map}) is compatible with the", "base change map of Remark \\ref{remark-base-change} in the following sense.", "Namely, suppose that", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "Y' \\ar[r]^g &", "Y", "}", "$$", "is a commutative diagram of ringed spaces. ", "Let $E \\in D(\\mathcal{O}_X)$ and $K \\in D(\\mathcal{O}_Y)$.", "Then the diagram", "$$", "\\xymatrix{", "Lg^*(Rf_*E \\otimes^\\mathbf{L}_{\\mathcal{O}_Y} K) \\ar[r]_p \\ar[d]_t &", "Lg^*Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} Lf^*K) \\ar[d]_b \\\\", "Lg^*Rf_*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{Y'}} Lg^*K \\ar[d]_b &", "Rf'_*L(g')^*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} Lf^*K) \\ar[d]_t \\\\", "Rf'_*L(g')^*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{Y'}} Lg^*K \\ar[rd]_p &", "Rf'_*(L(g')^*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{Y'}} L(g')^*Lf^*K) \\ar[d]_c \\\\", "& Rf'_*(L(g')^*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{Y'}} L(f')^*Lg^*K)", "}", "$$", "is commutative. Here arrows labeled $t$ are gotten by an application of", "Lemma \\ref{lemma-pullback-tensor-product}, arrows labeled $b$ by an", "application of Remark \\ref{remark-base-change}, arrows labeled $p$", "by an application of (\\ref{equation-projection-formula-map}), and", "$c$ comes from $L(g')^* \\circ Lf^* = L(f')^* \\circ Lg^*$.", "We omit the verification." ], "refs": [ "cohomology-remark-base-change", "cohomology-lemma-pullback-tensor-product", "cohomology-remark-base-change" ], "ref_ids": [ 2269, 2118, 2269 ] }, { "id": 2290, "type": "other", "label": "stacks-introduction-remark-diagonal", "categories": [ "stacks-introduction" ], "title": "stacks-introduction-remark-diagonal", "contents": [ "We have the formula", "$S \\times_{\\mathcal{M}_{1, 1}} S' =", "(S \\times S')", "\\times_{\\mathcal{M}_{1, 1} \\times \\mathcal{M}_{1, 1}}", "\\mathcal{M}_{1, 1}$.", "Hence the key fact is a property of the diagonal", "$\\Delta_{\\mathcal{M}_{1, 1}}$ of $\\mathcal{M}_{1, 1}$." ], "refs": [], "ref_ids": [] }, { "id": 2291, "type": "other", "label": "stacks-introduction-remark-quotient-stack", "categories": [ "stacks-introduction" ], "title": "stacks-introduction-remark-quotient-stack", "contents": [ "The argument sketched above actually shows that", "$\\mathcal{M}_{1, 1} = [W/H]$ is a global quotient stack.", "It is true about 50\\% of the time that an argument proving a moduli", "stack is algebraic will show that it is a global quotient stack." ], "refs": [], "ref_ids": [] }, { "id": 2448, "type": "other", "label": "restricted-remark-base-change", "categories": [ "restricted" ], "title": "restricted-remark-base-change", "contents": [ "Let $\\varphi : A_1 \\to A_2$ be a ring map and let", "$I_i \\subset A_i$ be ideals such that $\\varphi(I_1^c) \\subset I_2$", "for some $c \\geq 1$. This induces ring maps", "$A_{1, cn} = A_1/I_1^{cn} \\to A_2/I_2^n = A_{2, n}$ for all $n \\geq 1$.", "Let $\\mathcal{C}_i$ be the category (\\ref{equation-C}) for $(A_i, I_i)$.", "There is a base change functor", "\\begin{equation}", "\\label{equation-base-change-systems}", "\\mathcal{C}_1 \\longrightarrow \\mathcal{C}_2,\\quad", "(B_n) \\longmapsto (B_{cn} \\otimes_{A_{1, cn}} A_{2, n})", "\\end{equation}", "Let $\\mathcal{C}_i'$ be the category (\\ref{equation-C-prime}) for $(A_i, I_i)$.", "If $I_2$ is finitely generated, then there is a base change functor", "\\begin{equation}", "\\label{equation-base-change-complete}", "\\mathcal{C}_1' \\longrightarrow \\mathcal{C}_2',\\quad", "B \\longmapsto (B \\otimes_{A_1} A_2)^\\wedge", "\\end{equation}", "because in this case the completion is complete", "(Algebra, Lemma \\ref{algebra-lemma-hathat-finitely-generated}).", "If both $I_1$ and $I_2$ are finitely generated, then", "the two base change functors agree via the functors", "(\\ref{equation-from-complete-to-systems})", "which are equivalences by Lemma \\ref{lemma-topologically-finite-type}." ], "refs": [ "algebra-lemma-hathat-finitely-generated", "restricted-lemma-topologically-finite-type" ], "ref_ids": [ 859, 2294 ] }, { "id": 2449, "type": "other", "label": "restricted-remark-take-bar", "categories": [ "restricted" ], "title": "restricted-remark-take-bar", "contents": [ "Let $A$ be a Noetherian ring and $I \\subset A$ an ideal.", "Let $\\mathfrak a \\subset A$ be an ideal. Denote $\\bar A = A/\\mathfrak a$.", "Let $\\bar I \\subset \\bar A$ be an ideal such that", "$I^c \\bar A \\subset \\bar I$ and $\\bar I^d \\subset I\\bar A$", "for some $c, d \\geq 1$. In this case the base change functor", "(\\ref{equation-base-change-complete}) for $(A, I)$ to $(\\bar A, \\bar I)$", "is given by $B \\mapsto \\bar B = B/\\mathfrak aB$. Namely, we have", "\\begin{equation}", "\\label{equation-base-change-to-closed}", "\\bar B = (B \\otimes_A \\bar A)^\\wedge = (B/\\mathfrak a B)^\\wedge =", "B/\\mathfrak a B", "\\end{equation}", "the last equality because any finite $B$-module is $I$-adically complete by", "Algebra, Lemma \\ref{algebra-lemma-completion-tensor}", "and if annihilated by $\\mathfrak a$ also $\\bar I$-adically complete by", "Algebra, Lemma \\ref{algebra-lemma-change-ideal-completion}." ], "refs": [ "algebra-lemma-completion-tensor", "algebra-lemma-change-ideal-completion" ], "ref_ids": [ 869, 865 ] }, { "id": 2450, "type": "other", "label": "restricted-remark-linear-approximation", "categories": [ "restricted" ], "title": "restricted-remark-linear-approximation", "contents": [ "Let $A$ be a ring and $I \\subset A$ be a finitely generated ideal.", "Let $C$ be an $I$-adically complete $A$-algebra.", "Let $\\psi : A[x_1, \\ldots, x_r]^\\wedge \\to C$ be a continuous", "$A$-algebra map. Suppose given $\\delta_i \\in C$, $i = 1, \\ldots, r$.", "Then we can consider", "$$", "\\psi' : A[x_1, \\ldots, x_r]^\\wedge \\to C,\\quad", "x_i \\longmapsto \\psi(x_i) + \\delta_i", "$$", "see Formal Spaces, Remark \\ref{formal-spaces-remark-universal-property}.", "Then we have", "$$", "\\psi'(g) = \\psi(g) + \\sum \\psi(\\partial g/\\partial x_i)\\delta_i + \\xi", "$$", "with error term $\\xi \\in (\\delta_i\\delta_j)$. This follows by", "writing $g$ as a power series and working term by term. Convergence", "is automatic as the coefficients of $g$ tend to zero.", "Details omitted." ], "refs": [ "formal-spaces-remark-universal-property" ], "ref_ids": [ 4016 ] }, { "id": 2451, "type": "other", "label": "restricted-remark-improve-homomorphism", "categories": [ "restricted" ], "title": "restricted-remark-improve-homomorphism", "contents": [ "Let $A$ be a Noetherian ring and $I \\subset A$ be an ideal.", "Let $B$ be an object of (\\ref{equation-C-prime}).", "Let $C$ be an $I$-adically complete $A$-algebra.", "Let $\\psi_n : B \\to C/I^nC$ be an $A$-algebra homomorphism.", "The obstruction to lifting $\\psi_n$ to an $A$-algebra", "homomorphism into $C/I^{2n}C$ is an element", "$$", "o(\\psi_n) \\in \\Ext^1_B(\\NL_{B/A}^\\wedge, I^nC/I^{2n}C)", "$$", "as we will explain. Namely, choose a presentation", "$B = A[x_1, \\ldots, x_r]^\\wedge/J$.", "Choose a lift $\\psi : A[x_1, \\ldots, x_r]^\\wedge \\to C$ of $\\psi_n$.", "Since $\\psi(J) \\subset I^nC$ we get $\\psi(J^2) \\subset I^{2n}C$", "and hence we get a $B$-linear homomorphism", "$$", "o(\\psi) :", "J/J^2 \\longrightarrow I^nC/I^{2n}C, \\quad g \\longmapsto \\psi(g)", "$$", "which of course extends to a $C$-linear map", "$J/J^2 \\otimes_B C \\to I^nC/I^{2n}C$.", "Since $\\NL_{B/A}^\\wedge = (J/J^2 \\to \\bigoplus B \\text{d}x_i)$", "we get $o(\\psi_n)$ as the image of $o(\\psi)$ by the identification", "\\begin{align*}", "& \\Ext^1_B(\\NL_{B/A}^\\wedge, I^nC/I^{2n}C) \\\\", "& =", "\\Coker\\left(\\Hom_B(\\bigoplus B\\text{d}x_i, I^nC/I^{2n}C) \\to", "\\Hom_B(J/J^2, I^nC/I^{2n}C)\\right)", "\\end{align*}", "See More on Algebra, Lemma", "\\ref{more-algebra-lemma-map-out-of-almost-free} part (1) for the equality.", "\\medskip\\noindent", "Suppose that $o(\\psi_n)$ maps to zero in", "$\\Ext^1_B(\\NL_{B/A}^\\wedge, I^{n'}C/I^{2n'}C)$", "for some integer $n'$ with $n > n' > n/2$. We claim that this means we can", "find an $A$-algebra homomorphism $\\psi'_{2n'} : B \\to C/I^{2n'}C$", "which agrees with $\\psi_n$ as maps into $C/I^{n'}C$.", "The extreme case $n' = n$ explains why we previously said", "$o(\\psi_n)$ is the obstruction to lifting $\\psi_n$ to $C/I^{2n}C$.", "Proof of the claim: the hypothesis that $o(\\psi_n)$ maps to zero tells us", "we can find a $B$-module map", "$$", "h : \\bigoplus B\\text{d}x_i \\longrightarrow I^{n'}C/I^{2n'}C", "$$", "such that $o(\\psi)$ and $h \\circ \\text{d}$ agree as maps", "into $I^{n'}C/I^{2n'}C$. Say $h(\\text{d}x_i) = \\delta_i \\bmod I^{2n'}C$", "for some $\\delta_i \\in I^{n'}C$. Then we look at the map", "$$", "\\psi' : A[x_1, \\ldots, x_r]^\\wedge \\to C,\\quad", "x_i \\longmapsto \\psi(x_i) - \\delta_i", "$$", "A computation with power series", "shows that $\\psi'(J) \\subset I^{2n'}C$. Namely, for $g \\in J$ we get", "$$", "\\psi'(g) \\equiv", "\\psi(g) - \\sum \\psi(\\partial g/\\partial x_i)\\delta_i \\equiv", "o(\\psi)(g) - (h \\circ \\text{d})(g) \\equiv", "0 \\bmod I^{2n'}C", "$$", "See Remark \\ref{remark-linear-approximation} for the first equality.", "Hence $\\psi'$ induces an $A$-algebra homomorphism", "$\\psi'_{2n'} : B \\to C/I^{2n'}C$ as desired." ], "refs": [ "more-algebra-lemma-map-out-of-almost-free", "restricted-remark-linear-approximation" ], "ref_ids": [ 10299, 2450 ] }, { "id": 2452, "type": "other", "label": "restricted-remark-discussion", "categories": [ "restricted" ], "title": "restricted-remark-discussion", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$. Let $B$ be an object", "of (\\ref{equation-C-prime}) which is rig-smooth over $(A, I)$.", "As far as we know, it is an open question as to whether $B$", "is isomorphic to the $I$-adic completion of a finite type $A$-algebra.", "Here are some things we do know:", "\\begin{enumerate}", "\\item If $A$ is a G-ring, then the answer is yes by", "Proposition \\ref{proposition-approximate}.", "\\item If $B$ is rig-\\'etale over $(A, I)$, then the answer is", "yes by Lemma \\ref{lemma-approximate}.", "\\item If $I$ is principal, then the answer is yes by", "\\cite[III Theorem 7]{Elkik}.", "\\item In general there exists an ideal $J = (b_1, \\ldots, b_s) \\subset B$", "such that $V(J) \\subset V(IB)$ and such that the $I$-adic completion of", "each of the affine blowup algebras $B[\\frac{J}{b_i}]$ are isomorphic", "to the $I$-adic completion of a finite type $A$-algebra.", "\\end{enumerate}", "To see the last statement, choose $b_1, \\ldots, b_s$ as in", "Lemma \\ref{lemma-equivalent-with-artin-smooth} part (4)", "and use the properties mentioned there to see that", "Lemma \\ref{lemma-approximate-presentation-rig-smooth}", "applies to each completion $(B[\\frac{J}{b_i}])^\\wedge$.", "Part (4) tells us that ``rig-locally a rig-smooth formal algebraic space is", "the completion of a finite type scheme over $A$'' and it tells us that", "``there is an admissible formal blowing up of $\\text{Spf}(B)$", "which is affine locally algebraizable''." ], "refs": [ "restricted-proposition-approximate", "restricted-lemma-approximate", "restricted-lemma-equivalent-with-artin-smooth", "restricted-lemma-approximate-presentation-rig-smooth" ], "ref_ids": [ 2432, 2322, 2302, 2313 ] }, { "id": 2453, "type": "other", "label": "restricted-remark-NL-well-defined-topological", "categories": [ "restricted" ], "title": "restricted-remark-NL-well-defined-topological", "contents": [ "Let $A \\to B$ be an arrow of $\\text{WAdm}^{adic*}$ which is adic and", "topologically of finite type (see Lemma \\ref{lemma-finite-type}). Write", "$B = A\\{x_1, \\ldots, x_r\\}/J$. Then we can set\\footnote{In fact, this", "construction works for arrows of $\\text{WAdm}^{count}$ satisfying the", "equivalent conditions of Formal Spaces, Lemma", "\\ref{formal-spaces-lemma-quotient-restricted-power-series}.}", "$$", "\\NL_{B/A}^\\wedge = \\left(J/J^2 \\longrightarrow \\bigoplus B\\text{d}x_i\\right)", "$$", "Exactly as in the proof of Lemma \\ref{lemma-NL-up-to-homotopy}", "the reader can show that this complex of $B$-modules is", "well defined up to (unique isomorphism) in the homotopy category $K(B)$.", "Now, if $A$ is Noetherian and $I \\subset A$ is an ideal of definition,", "then this construction reproduces the naive cotangent complex", "of $B$ over $(A, I)$ defined by Equation (\\ref{equation-NL})", "in Section \\ref{section-naive-cotangent-complex} simply because", "$A[x_1, \\ldots, x_n]^\\wedge$ agrees with $A\\{x_1, \\ldots, x_r\\}$ by", "Formal Spaces, Remark", "\\ref{formal-spaces-remark-I-adic-completion-and-restricted-power-series}.", "In particular, we find that, still when $A$ is an adic Noetherian", "topological ring, the object $\\NL_{B/A}^\\wedge$ is independent", "of the choice of the ideal of definition $I \\subset A$." ], "refs": [ "restricted-lemma-finite-type", "formal-spaces-lemma-quotient-restricted-power-series", "restricted-lemma-NL-up-to-homotopy", "formal-spaces-remark-I-adic-completion-and-restricted-power-series" ], "ref_ids": [ 2324, 3951, 2296, 4017 ] }, { "id": 2454, "type": "other", "label": "restricted-remark-rig-surjective-more-general", "categories": [ "restricted" ], "title": "restricted-remark-rig-surjective-more-general", "contents": [ "The condition as formulated in Definition \\ref{definition-rig-surjective}", "is not right even for morphisms of finite type", "of locally adic* formal algebraic spaces.", "For example, if $A = (\\bigcup_{n \\geq 1} k[t^{1/n}])^\\wedge$", "where the completion is the $t$-adic completion, then", "there are no adic morphisms $\\text{Spf}(R) \\to \\text{Spf}(A)$", "where $R$ is a complete discrete valuation ring.", "Thus any morphism $X \\to \\text{Spf}(A)$ would be rig-surjective,", "but since $A$ is a domain and $t \\in A$ is not zero, we want to", "think of $A$ as having at least one ``rig-point'', and we do not", "want to allow $X = \\emptyset$. To cover this", "particular case, one can consider adic morphisms", "$$", "\\text{Spf}(R) \\longrightarrow Y", "$$", "where $R$ is a valuation ring complete with respect to a principal", "ideal $J$ whose radical is $\\mathfrak m_R = \\sqrt{J}$.", "In this case the value group of $R$ can be embedded into", "$(\\mathbf{R}, +)$ and one obtains the point of view used by", "Berkovich in defining an analytic space associated to $Y$, see", "\\cite{Berkovich}. Another approach is championed by Huber. In his theory,", "one drops the hypothesis that $\\Spec(R/J)$ is a singleton, see", "\\cite{Huber-continuous-valuations}." ], "refs": [ "restricted-definition-rig-surjective" ], "ref_ids": [ 2446 ] }, { "id": 2455, "type": "other", "label": "restricted-remark-diagonal-gives-diagonal", "categories": [ "restricted" ], "title": "restricted-remark-diagonal-gives-diagonal", "contents": [ "In the situation above consider the diagonal morphisms", "$\\Delta_f : X' \\to X' \\times_X X'$ and", "$\\Delta_{f_{/T}} : X'_{/T'} \\to X'_{/T'} \\times_{X_{/T}} X'_{/T'}$.", "It is easy to see that", "$$", "X'_{/T'} \\times_{X_{/T}} X'_{/T'} = (X' \\times_X X')_{/T''}", "$$", "as subfunctors of $X' \\times_X X'$ where $T'' \\subset |X' \\times_X X'|$", "is the inverse image of $T$. Hence we see that", "$\\Delta_{f_{/T}} = (\\Delta_f)_{/T''}$. We will use this below", "to show that properties of $\\Delta_f$ are inherited by $\\Delta_{f_{/T}}$." ], "refs": [], "ref_ids": [] }, { "id": 2456, "type": "other", "label": "restricted-remark-compare-formal-modification-artin", "categories": [ "restricted" ], "title": "restricted-remark-compare-formal-modification-artin", "contents": [ "In \\cite[Definition 1.7]{ArtinII} a formal modification is defined as a", "proper morphism $f : X \\to Y$ of locally Noetherian formal algebraic spaces", "satisfying the following three conditions\\footnote{We will not completely", "translate these conditions into the language developed in the Stacks", "project. We hope nonetheless the discussion here will be useful to the", "reader.}", "\\begin{enumerate}", "\\item[(\\romannumeral1)] the Cramer and Jacobian ideal of", "$f$ each contain an ideal of definition of $X$,", "\\item[(\\romannumeral2)] the ideal defining the", "diagonal map $\\Delta : X \\to X \\times_Y X$", "is annihilated by an ideal of definition of $X \\times_Y X$, and", "\\item[(\\romannumeral3)] any adic morphism $\\text{Spf}(R) \\to Y$", "lifts to $\\text{Spf}(R) \\to X$ whenever $R$ is a", "complete discrete valuation ring.", "\\end{enumerate}", "Let us compare these to our list of conditions above.", "\\medskip\\noindent", "Ad (\\romannumeral1). Property (\\romannumeral1) agrees with our condition", "that $f$ be a rig-\\'etale morphism: this follows from", "Lemma \\ref{lemma-equivalent-with-artin} part (\\ref{item-condition-artin}).", "\\medskip\\noindent", "Ad (\\romannumeral2). Assume $f$ is rig-\\'etale. Then", "$\\Delta_f : X \\to X \\times_Y X$ is rig-\\'etale as a morphism", "of locally Noetherian formal algebraic spaces which are", "rig-\\'etale over $X$ (via $\\text{id}_X$ for the first one and via", "$\\text{pr}_1$ for the second one).", "See Lemmas \\ref{lemma-base-change-rig-etale} and", "\\ref{lemma-rig-etale-permanence}.", "Hence property (\\romannumeral2) agrees with our condition", "that $\\Delta_f$ be rig-surjective by", "Lemma \\ref{lemma-closed-immersion-rig-smooth-rig-surjective}.", "\\medskip\\noindent", "Ad (\\romannumeral3). Property (\\romannumeral3) does not quite", "agree with our notion of a rig-surjective morphism, as Artin", "requires all adic morphisms $\\text{Spf}(R) \\to Y$ to lift to", "morphisms into $X$ whereas our notion of rig-surjective only", "asserts the existence of a lift after replacing $R$ by an extension.", "However, since we already have that $\\Delta_f$ is", "rig-\\'etale and rig-surjective by (\\romannumeral1) and (\\romannumeral2),", "these conditions are equivalent by", "Lemma \\ref{lemma-rig-monomorphism-rig-surjective}." ], "refs": [ "restricted-lemma-equivalent-with-artin", "restricted-lemma-base-change-rig-etale", "restricted-lemma-rig-etale-permanence", "restricted-lemma-closed-immersion-rig-smooth-rig-surjective", "restricted-lemma-rig-monomorphism-rig-surjective" ], "ref_ids": [ 2314, 2380, 2382, 2395, 2399 ] }, { "id": 2506, "type": "other", "label": "more-groupoids-remark-local-source-warning", "categories": [ "more-groupoids" ], "title": "more-groupoids-remark-local-source-warning", "contents": [ "Warning:", "Lemma \\ref{lemma-local-source}", "should be used with care.", "For example, it applies to $\\mathcal{P}=$``flat'', $\\mathcal{Q}=$``empty'',", "and $\\mathcal{R}=$``flat and locally of finite presentation''. But given a", "morphism of schemes $f : X \\to Y$ the largest open $W \\subset X$ such that", "$f|_W$ is flat is {\\it not} the set of points where $f$ is flat!" ], "refs": [ "more-groupoids-lemma-local-source" ], "ref_ids": [ 2459 ] }, { "id": 2507, "type": "other", "label": "more-groupoids-remark-local-source-apply", "categories": [ "more-groupoids" ], "title": "more-groupoids-remark-local-source-apply", "contents": [ "Notwithstanding the warning in", "Remark \\ref{remark-local-source-warning}", "there are some cases where", "Lemma \\ref{lemma-local-source}", "can be used without causing too much ambiguity.", "We give a list. In each case we omit the verification of", "assumptions (1) and (2) and we give references which imply", "(3) and (4). Here is the list:", "\\begin{enumerate}", "\\item $\\mathcal{Q} = \\mathcal{R} =$``locally of finite type'', and", "$\\mathcal{P} =$``relative dimension $\\leq d$''.", "See", "Morphisms, Definition \\ref{morphisms-definition-relative-dimension-d}", "and", "Morphisms, Lemmas \\ref{morphisms-lemma-openness-bounded-dimension-fibres} and", "\\ref{morphisms-lemma-dimension-fibre-after-base-change}.", "\\item $\\mathcal{Q} = \\mathcal{R} =$``locally of finite type'', and", "$\\mathcal{P} =$``locally quasi-finite''.", "This is the case $d = 0$ of the previous item, see", "Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}.", "\\item $\\mathcal{Q} = \\mathcal{R} =$``locally of finite type'', and", "$\\mathcal{P} =$``unramified''.", "See", "Morphisms, Lemmas \\ref{morphisms-lemma-unramified-characterize} and", "\\ref{morphisms-lemma-set-points-where-fibres-unramified}.", "\\end{enumerate}", "What is interesting about the cases listed above is that we do not", "need to assume that $s, t$ are flat to get a conclusion about the locus", "where the morphism $h$ has property $\\mathcal{P}$. We continue the", "list:", "\\begin{enumerate}", "\\item[(4)] $\\mathcal{Q} =$``locally of finite presentation'',", "$\\mathcal{R} =$``flat and locally of finite presentation'', and", "$\\mathcal{P} =$``flat''. See", "More on Morphisms, Theorem", "\\ref{more-morphisms-theorem-openness-flatness} and", "Lemma \\ref{more-morphisms-lemma-flat-locus-base-change}.", "\\item[(5)] $\\mathcal{Q} =$``locally of finite presentation'',", "$\\mathcal{R} =$``flat and locally of finite presentation'', and", "$\\mathcal{P}=$``Cohen-Macaulay''. See", "More on Morphisms, Definition \\ref{more-morphisms-definition-CM}", "and", "More on Morphisms, Lemmas \\ref{more-morphisms-lemma-base-change-CM} and", "\\ref{more-morphisms-lemma-flat-finite-presentation-CM-open}.", "\\item[(6)] $\\mathcal{Q} =$``locally of finite presentation'',", "$\\mathcal{R} =$``flat and locally of finite presentation'', and", "$\\mathcal{P}=$``syntomic'' use", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-lci}", "(the locus is automatically open).", "\\item[(7)] $\\mathcal{Q} =$``locally of finite presentation'',", "$\\mathcal{R} =$``flat and locally of finite presentation'', and", "$\\mathcal{P}=$``smooth''. See", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-smooth}", "(the locus is automatically open).", "\\item[(8)] $\\mathcal{Q} =$``locally of finite presentation'',", "$\\mathcal{R} =$``flat and locally of finite presentation'', and", "$\\mathcal{P}=$``\\'etale''. See", "Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-etale}", "(the locus is automatically open).", "\\end{enumerate}" ], "refs": [ "more-groupoids-remark-local-source-warning", "more-groupoids-lemma-local-source", "morphisms-definition-relative-dimension-d", "morphisms-lemma-openness-bounded-dimension-fibres", "morphisms-lemma-dimension-fibre-after-base-change", "morphisms-lemma-locally-quasi-finite-rel-dimension-0", "morphisms-lemma-unramified-characterize", "morphisms-lemma-set-points-where-fibres-unramified", "more-morphisms-theorem-openness-flatness", "more-morphisms-lemma-flat-locus-base-change", "more-morphisms-definition-CM", "more-morphisms-lemma-base-change-CM", "more-morphisms-lemma-flat-finite-presentation-CM-open", "morphisms-lemma-set-points-where-fibres-lci", "morphisms-lemma-set-points-where-fibres-smooth", "morphisms-lemma-set-points-where-fibres-etale" ], "ref_ids": [ 2506, 2459, 5559, 5280, 5279, 5287, 5344, 5356, 13670, 13766, 14115, 13788, 13789, 5299, 5336, 5374 ] }, { "id": 2508, "type": "other", "label": "more-groupoids-remark-warn-dimension-groupoid-on-field", "categories": [ "more-groupoids" ], "title": "more-groupoids-remark-warn-dimension-groupoid-on-field", "contents": [ "Warning:", "Lemma \\ref{lemma-groupoid-on-field-dimension-equal-stabilizer}", "is wrong without the condition that $s$ and $t$ are locally of", "finite type.", "An easy example is to start with the action", "$$", "\\mathbf{G}_{m, \\mathbf{Q}} \\times_{\\mathbf{Q}} \\mathbf{A}^1_{\\mathbf{Q}}", "\\to \\mathbf{A}^1_{\\mathbf{Q}}", "$$", "and restrict the corresponding groupoid scheme to the generic point of", "$\\mathbf{A}^1_{\\mathbf{Q}}$. In other words restrict via the morphism", "$\\Spec(\\mathbf{Q}(x)) \\to", "\\Spec(\\mathbf{Q}[x]) = \\mathbf{A}^1_{\\mathbf{Q}}$.", "Then you get a groupoid scheme", "$(U, R, s, t, c)$ with", "$U = \\Spec(\\mathbf{Q}(x))$", "and", "$$", "R = \\Spec\\left(", "\\mathbf{Q}(x)[y]\\left[", "\\frac{1}{P(xy)}, P \\in \\mathbf{Q}[T], P \\not = 0", "\\right]", "\\right)", "$$", "In this case $\\dim(R) = 1$ and $\\dim(G) = 0$." ], "refs": [ "more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer" ], "ref_ids": [ 2477 ] }, { "id": 2598, "type": "other", "label": "examples-remark-reference-existence-regular-nonexcellent-rings", "categories": [ "examples" ], "title": "examples-remark-reference-existence-regular-nonexcellent-rings", "contents": [ "Non-excellent regular rings whose residue fields have a finite $p$-basis", "can be constructed even in the function field of $\\mathbb{P}^2_k$, over a ", "characteristic $p$ field $k = \\overline{k}$. See", "\\cite[$\\mathsection 4.1$]{DS18}." ], "refs": [], "ref_ids": [] }, { "id": 2599, "type": "other", "label": "examples-remark-specialization", "categories": [ "examples" ], "title": "examples-remark-specialization", "contents": [ "Here are some remarks:", "\\begin{enumerate}", "\\item The presheaves $F$ and $F_n$ are separated presheaves.", "\\item It turns out that $F$, $F_n$ are not sheaves.", "\\item One can show that $G$, $G_n$ is actually a sheaf for the fppf topology.", "\\end{enumerate}", "We will prove these results if we need them." ], "refs": [], "ref_ids": [] }, { "id": 2600, "type": "other", "label": "examples-remark-contradict-aoki", "categories": [ "examples" ], "title": "examples-remark-contradict-aoki", "contents": [ "Proposition \\ref{proposition-nonalghomstack} contradicts", "\\cite[Theorem 1.1]{AokiHomStacks}. The problem is the non-effectivity", "of formal objects for $\\underline{\\Mor}_S(X, [S/A])$. The same problem", "is mentioned in the Erratum \\cite{AokiHomStacksErr} to", "\\cite{AokiHomStacks}. Unfortunately, the Erratum goes on", "to assert that $\\underline{\\Mor}_S(\\mathcal{Y}, \\mathcal{Z})$", "is algebraic if $\\mathcal{Z}$ is separated, which also contradicts", "Proposition \\ref{proposition-nonalghomstack} as $[S/A]$ is separated." ], "refs": [ "examples-proposition-nonalghomstack", "examples-proposition-nonalghomstack" ], "ref_ids": [ 2593, 2593 ] }, { "id": 2768, "type": "other", "label": "spaces-perfect-remark-match-total-direct-images", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-remark-match-total-direct-images", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of representable", "algebraic spaces $X$ and $Y$ over $S$. Let $f_0 : X_0 \\to Y_0$ be a", "morphism of schemes representing $f$ (awkward but temporary notation).", "Then the diagram", "$$", "\\xymatrix{", "D_\\QCoh(\\mathcal{O}_{X_0})", "\\ar@{=}[rrrrrr]_{\\text{Lemma", "\\ref{lemma-derived-quasi-coherent-small-etale-site}}}", "& & & & & &", "D_\\QCoh(\\mathcal{O}_X) \\\\", "D_\\QCoh(\\mathcal{O}_{Y_0})", "\\ar[u]^{Lf^*_0}", "\\ar@{=}[rrrrrr]^{\\text{Lemma", "\\ref{lemma-derived-quasi-coherent-small-etale-site}}}", "& & & & & &", "D_\\QCoh(\\mathcal{O}_Y) \\ar[u]_{Lf^*}", "}", "$$", "(Lemma \\ref{lemma-quasi-coherence-pullback} and", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-pullback})", "is commutative. This follows as the", "equivalences", "$D_\\QCoh(\\mathcal{O}_{X_0}) \\to D_\\QCoh(\\mathcal{O}_X)$", "and", "$D_\\QCoh(\\mathcal{O}_{Y_0}) \\to D_\\QCoh(\\mathcal{O}_Y)$", "of Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site}", "come from pulling back by the (flat) morphisms of ringed sites", "$\\epsilon : X_\\etale \\to X_{0, Zar}$ and", "$\\epsilon : Y_\\etale \\to Y_{0, Zar}$", "and the diagram of ringed sites", "$$", "\\xymatrix{", "X_{0, Zar} \\ar[d]_{f_0} & X_\\etale \\ar[l]^\\epsilon \\ar[d]^f \\\\", "Y_{0, Zar} & Y_\\etale \\ar[l]_\\epsilon", "}", "$$", "is commutative (details omitted). If $f$ is quasi-compact and", "quasi-separated, equivalently if $f_0$ is quasi-compact and", "quasi-separated, then we claim", "$$", "\\xymatrix{", "D_\\QCoh(\\mathcal{O}_{X_0})", "\\ar[d]_{Rf_{0, *}} \\ar@{=}[rrrrrr]_{\\text{Lemma", "\\ref{lemma-derived-quasi-coherent-small-etale-site}}}", "& & & & & &", "D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\", "D_\\QCoh(\\mathcal{O}_{Y_0})", "\\ar@{=}[rrrrrr]^{\\text{Lemma", "\\ref{lemma-derived-quasi-coherent-small-etale-site}}}", "& & & & & &", "D_\\QCoh(\\mathcal{O}_Y)", "}", "$$", "(Lemma \\ref{lemma-quasi-coherence-direct-image} and", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-direct-image})", "is commutative as well. This also follows from the commutative", "diagram of sites displayed above as the proof of Lemma", "\\ref{lemma-derived-quasi-coherent-small-etale-site}", "shows that the functor $R\\epsilon_*$ gives the equivalences", "$D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_{X_0})$", "and", "$D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_{Y_0})$." ], "refs": [ "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-quasi-coherence-pullback", "perfect-lemma-quasi-coherence-pullback", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "spaces-perfect-lemma-quasi-coherence-direct-image", "perfect-lemma-quasi-coherence-direct-image", "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site" ], "ref_ids": [ 2644, 2644, 2648, 6944, 2644, 2644, 2644, 2652, 6946, 2644 ] }, { "id": 2769, "type": "other", "label": "spaces-perfect-remark-how-to", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-remark-how-to", "contents": [ "How to choose the collection $\\mathcal{B}$ in", "Lemma \\ref{lemma-induction-principle-separated}?", "Here are some examples:", "\\begin{enumerate}", "\\item If $X$ is quasi-compact and separated, then we can choose", "$\\mathcal{B}$ to be the set of quasi-compact and separated objects", "of $X_{spaces, \\etale}$. Then $X \\in \\mathcal{B}$ and $\\mathcal{B}$", "satisfies (1), (2), and (3)(a). With this choice of $\\mathcal{B}$", "Lemma \\ref{lemma-induction-principle-separated} reproduces", "Lemma \\ref{lemma-induction-principle}.", "\\item If $X$ is quasi-compact with affine diagonal, then we can choose", "$\\mathcal{B}$ to be the set of objects", "of $X_{spaces, \\etale}$ which are quasi-compact and have affine", "diagonal. Again $X \\in \\mathcal{B}$ and $\\mathcal{B}$", "satisfies (1), (2), and (3)(a).", "\\item If $X$ is quasi-compact and quasi-separated, then the", "smallest subset $\\mathcal{B}$ which contains $X$ and satisfies", "(1), (2), and (3)(a) is given by the rule $W \\in \\mathcal{B}$ if and only", "if either $W$ is a quasi-compact open subspace of $X$, or", "$W$ is a quasi-compact open of an affine object of $X_{spaces, \\etale}$.", "\\end{enumerate}" ], "refs": [ "spaces-perfect-lemma-induction-principle-separated", "spaces-perfect-lemma-induction-principle-separated", "spaces-perfect-lemma-induction-principle" ], "ref_ids": [ 2671, 2671, 2670 ] }, { "id": 2770, "type": "other", "label": "spaces-perfect-remark-addendum", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-remark-addendum", "contents": [ "The proof of Lemma \\ref{lemma-lift-map-from-perfect-complex-with-support}", "shows that", "$$", "R|_W = P \\oplus P^{\\oplus n_1}[1] \\oplus \\ldots \\oplus P^{\\oplus n_m}[m]", "$$", "for some $m \\geq 0$ and $n_j \\geq 0$. Thus the highest degree cohomology sheaf", "of $R|_W$ equals that of $P$. By repeating the construction for the map", "$P^{\\oplus n_1}[1] \\oplus \\ldots \\oplus P^{\\oplus n_m}[m] \\to R|_W$, taking", "cones, and using induction we can achieve equality of cohomology sheaves", "of $R|_W$ and $P$ above any given degree." ], "refs": [ "spaces-perfect-lemma-lift-map-from-perfect-complex-with-support" ], "ref_ids": [ 2706 ] }, { "id": 2771, "type": "other", "label": "spaces-perfect-remark-pullback-generator", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-remark-pullback-generator", "contents": [ "Let $S$ be a scheme.", "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated", "algebraic spaces over $S$.", "Let $E \\in D_\\QCoh(\\mathcal{O}_Y)$ be a generator", "(see Theorem \\ref{theorem-bondal-van-den-Bergh}).", "Then the following are equivalent", "\\begin{enumerate}", "\\item for $K \\in D_\\QCoh(\\mathcal{O}_X)$ we have", "$Rf_*K = 0$ if and only if $K = 0$,", "\\item $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$", "reflects isomorphisms, and", "\\item $Lf^*E$ is a generator for $D_\\QCoh(\\mathcal{O}_X)$.", "\\end{enumerate}", "The equivalence between (1) and (2) is a formal consequence of the fact that", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ is an", "exact functor of triangulated categories. Similarly, the equivalence", "between (1) and (3) follows formally from the fact that $Lf^*$", "is the left adjoint to $Rf_*$.", "These conditions hold if $f$ is affine (Lemma \\ref{lemma-affine-morphism})", "or if $f$ is an open immersion, or if $f$ is a composition of such." ], "refs": [ "spaces-perfect-theorem-bondal-van-den-Bergh", "spaces-perfect-lemma-affine-morphism" ], "ref_ids": [ 2640, 2654 ] }, { "id": 2772, "type": "other", "label": "spaces-perfect-remark-classical-generator", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-remark-classical-generator", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $G$ be a", "perfect object of $D(\\mathcal{O}_X)$ which is a generator for", "$D_\\QCoh(\\mathcal{O}_X)$. By Theorem \\ref{theorem-bondal-van-den-Bergh}", "there is at least one of these. Combining", "Lemma \\ref{lemma-quasi-coherence-direct-sums} with", "Proposition \\ref{proposition-compact-is-perfect} and with", "Derived Categories, Proposition", "\\ref{derived-proposition-generator-versus-classical-generator}", "we see that $G$ is a classical generator for $D_{perf}(\\mathcal{O}_X)$." ], "refs": [ "spaces-perfect-theorem-bondal-van-den-Bergh", "spaces-perfect-lemma-quasi-coherence-direct-sums", "spaces-perfect-proposition-compact-is-perfect", "derived-proposition-generator-versus-classical-generator" ], "ref_ids": [ 2640, 2646, 2758, 1965 ] }, { "id": 2773, "type": "other", "label": "spaces-perfect-remark-classical-generator-with-support", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-remark-classical-generator-with-support", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.", "Let $T \\subset |X|$ be a closed subset such that $|X| \\setminus T$", "is quasi-compact. Let $G$ be a", "perfect object of $D_{\\QCoh, T}(\\mathcal{O}_X)$ which is a generator for", "$D_{\\QCoh, T}(\\mathcal{O}_X)$. By Lemma \\ref{lemma-generator-with-support}", "there is at least one of these. Combining the fact that", "$D_{\\QCoh, T}(\\mathcal{O}_X)$ has direct sums with", "Lemma \\ref{lemma-compact-is-perfect-with-support} and with", "Derived Categories, Proposition", "\\ref{derived-proposition-generator-versus-classical-generator}", "we see that $G$ is a classical generator for $D_{perf, T}(\\mathcal{O}_X)$." ], "refs": [ "spaces-perfect-lemma-generator-with-support", "spaces-perfect-lemma-compact-is-perfect-with-support", "derived-proposition-generator-versus-classical-generator" ], "ref_ids": [ 2708, 2709, 1965 ] }, { "id": 2774, "type": "other", "label": "spaces-perfect-remark-DQCoh-is-Ddga-with-support", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-remark-DQCoh-is-Ddga-with-support", "contents": [ "Let $S$ be a scheme.", "Let $X$ be a quasi-compact and quasi-separated algebraic space.", "Let $T \\subset |X|$ be a closed subset such that", "$|X| \\setminus T$ is quasi-compact.", "The analogue of Theorem \\ref{theorem-DQCoh-is-Ddga} holds", "for $D_{\\QCoh, T}(\\mathcal{O}_X)$.", "This follows from the exact same argument as in the proof", "of the theorem, using", "Lemmas \\ref{lemma-generator-with-support} and", "\\ref{lemma-compact-is-perfect-with-support}", "and a variant of Lemma \\ref{lemma-tensor-with-QCoh-complex}", "with supports.", "If we ever need this, we will precisely state the", "result here and give a detailed proof." ], "refs": [ "spaces-perfect-theorem-DQCoh-is-Ddga", "spaces-perfect-lemma-generator-with-support", "spaces-perfect-lemma-compact-is-perfect-with-support", "spaces-perfect-lemma-tensor-with-QCoh-complex" ], "ref_ids": [ 2641, 2708, 2709, 2711 ] }, { "id": 2775, "type": "other", "label": "spaces-perfect-remark-independence-choice", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-remark-independence-choice", "contents": [ "Let $X$ be a quasi-compact and quasi-separated algebraic space over a ring $R$.", "By the construction of the proof of", "Theorem \\ref{theorem-DQCoh-is-Ddga}", "there exists a differential graded algebra $(A, \\text{d})$ over $R$", "such that $D_\\QCoh(X)$ is $R$-linearly equivalent to", "$D(A, \\text{d})$ as a triangulated category.", "One may ask: how unique is $(A, \\text{d})$?", "The answer is (only) slightly better than just saying that", "$(A, \\text{d})$ is well defined up to derived equivalence.", "Namely, suppose that $(B, \\text{d})$ is a second such pair.", "Then we have", "$$", "(A, \\text{d}) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, K^\\bullet)", "$$", "and", "$$", "(B, \\text{d}) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(L^\\bullet, L^\\bullet)", "$$", "for some K-injective complexes $K^\\bullet$ and $L^\\bullet$", "of $\\mathcal{O}_X$-modules corresponding to perfect generators", "of $D_\\QCoh(\\mathcal{O}_X)$. Set", "$$", "\\Omega = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, L^\\bullet)", "\\quad", "\\Omega' = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(L^\\bullet, K^\\bullet)", "$$", "Then $\\Omega$ is a differential graded $B^{opp} \\otimes_R A$-module", "and $\\Omega'$ is a differential graded $A^{opp} \\otimes_R B$-module.", "Moreover, the equivalence", "$$", "D(A, \\text{d}) \\to D_\\QCoh(\\mathcal{O}_X) \\to", "D(B, \\text{d})", "$$", "is given by the functor $- \\otimes_A^\\mathbf{L} \\Omega'$ and", "similarly for the quasi-inverse. Thus we are in the situation", "of Differential Graded Algebra, Remark \\ref{dga-remark-hochschild-cohomology}.", "If we ever need this remark we will provide a precise statement", "with a detailed proof here." ], "refs": [ "spaces-perfect-theorem-DQCoh-is-Ddga", "dga-remark-hochschild-cohomology" ], "ref_ids": [ 2641, 13172 ] }, { "id": 2776, "type": "other", "label": "spaces-perfect-remark-explain-consequence", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-remark-explain-consequence", "contents": [ "Let $S$ be a scheme. Let $(U \\subset X, f : V \\to X)$ be an", "elementary distinguished square of algebraic spaces over $S$.", "Assume $X$, $U$, $V$ are quasi-compact and quasi-separated.", "By Lemma \\ref{lemma-better-coherator} the functors", "$DQ_X$, $DQ_U$, $DQ_V$, $DQ_{U \\times_X V}$ exist. Moreover, there is a", "canonical distinguished triangle", "$$", "DQ_X(K) \\to Rj_{U, *}DQ_U(K|_U) \\oplus Rj_{V, *}DQ_V(K|_V)", "\\to Rj_{U \\times_X V, *}DQ_{U \\times_X V}(K|_{U \\times_X V}) \\to", "$$", "for any $K \\in D(\\mathcal{O}_X)$. This follows by applying the", "exact functor $DQ_X$ to the distinguished triangle of", "Lemma \\ref{lemma-exact-sequence-j-star}", "and using Lemma \\ref{lemma-pushforward-better-coherator} three times." ], "refs": [ "spaces-perfect-lemma-better-coherator", "spaces-perfect-lemma-exact-sequence-j-star", "spaces-perfect-lemma-pushforward-better-coherator" ], "ref_ids": [ 2715, 2674, 2716 ] }, { "id": 2777, "type": "other", "label": "spaces-perfect-remark-multiplication-map", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-remark-multiplication-map", "contents": [ "With notation as in Lemma \\ref{lemma-affine-morphism-and-hom-out-of-perfect}.", "The diagram", "$$", "\\xymatrix{", "R\\Hom_X(M, Rg'_*L) \\otimes_R^\\mathbf{L} R' \\ar[r] \\ar[d]_\\mu &", "R\\Hom_{X'}(L(g')^*M, L(g')^*Rg'_*L) \\ar[d]^a \\\\", "R\\Hom_X(M, R(g')_*L) \\ar@{=}[r] &", "R\\Hom_{X'}(L(g')^*M, L)", "}", "$$", "is commutative where the top horizontal arrow is the map from the lemma,", "$\\mu$ is the multiplication map, and $a$ comes from the adjunction map", "$L(g')^*Rg'_*L \\to L$. The multiplication map is the adjunction map", "$K' \\otimes_R^\\mathbf{L} R' \\to K'$ for any $K' \\in D(R')$." ], "refs": [ "spaces-perfect-lemma-affine-morphism-and-hom-out-of-perfect" ], "ref_ids": [ 2721 ] }, { "id": 2778, "type": "other", "label": "spaces-perfect-remark-base-change-of-L", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-remark-base-change-of-L", "contents": [ "The pseudo-coherent complex $L$ of part (B) of Lemma \\ref{lemma-compute-ext}", "is canonically associated to the situation. For example,", "formation of $L$ as in (B) is compatible with base change.", "In other words, given a cartesian diagram", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "Y' \\ar[r]^g &", "Y", "}", "$$", "of schemes we have canonical functorial isomorphisms", "$$", "\\Ext^i_{\\mathcal{O}_{Y'}}(Lg^*L, \\mathcal{F}') \\longrightarrow", "\\Ext^i_{\\mathcal{O}_X}(L(g')^*E,", "(g')^*\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_{X'}} (f')^*\\mathcal{F}')", "$$", "for $\\mathcal{F}'$ quasi-coherent on $Y'$. Obsere that we do {\\bf not} use", "derived pullback on $\\mathcal{G}^\\bullet$ on the right hand side.", "If we ever need this, we will", "formulate a precise result here and give a detailed proof." ], "refs": [ "spaces-perfect-lemma-compute-ext" ], "ref_ids": [ 2733 ] }, { "id": 2779, "type": "other", "label": "spaces-perfect-remark-explain-perfect-direct-image", "categories": [ "spaces-perfect" ], "title": "spaces-perfect-remark-explain-perfect-direct-image", "contents": [ "Let $R$ be a ring. Let $X$ be an algebraic space of finite presentation over", "$R$. Let $\\mathcal{G}$ be a finitely presented $\\mathcal{O}_X$-module", "flat over $R$ with support proper over $R$. By", "Lemma \\ref{lemma-base-change-tensor-perfect}", "there exists a finite complex of finite projective $R$-modules", "$M^\\bullet$ such that we have", "$$", "R\\Gamma(X_{R'}, \\mathcal{G}_{R'}) = M^\\bullet \\otimes_R R'", "$$", "functorially in the $R$-algebra $R'$." ], "refs": [ "spaces-perfect-lemma-base-change-tensor-perfect" ], "ref_ids": [ 2736 ] }, { "id": 2934, "type": "other", "label": "dualizing-remark-matlis", "categories": [ "dualizing" ], "title": "dualizing-remark-matlis", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Let $E$ be an injective hull of $\\kappa$ over $R$. Here is an", "addendum to Matlis duality: If $N$ is an $\\mathfrak m$-power torsion module", "and $M = \\Hom_R(N, E)$ is a finite module over the completion of $R$,", "then $N$ satisfies the descending chain condition. Namely, for any", "submodules $N'' \\subset N' \\subset N$ with $N'' \\not = N'$, we can", "find an embedding $\\kappa \\subset N''/N'$ and hence a nonzero", "map $N' \\to E$ annihilating $N''$ which we can extend to a map $N \\to E$", "annihilating $N''$. Thus $N \\supset N' \\mapsto M' = \\Hom_R(N/N', E) \\subset M$", "is an inclusion preserving map from submodules of $N$ to submodules", "of $M$, whence the conclusion." ], "refs": [], "ref_ids": [] }, { "id": 2935, "type": "other", "label": "dualizing-remark-exact-support", "categories": [ "dualizing" ], "title": "dualizing-remark-exact-support", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be an ideal. Set $B = A/I$.", "In this case the functor $\\Hom_A(B, -)$ is equal to the functor", "$$", "\\text{Mod}_A \\longrightarrow \\text{Mod}_B,\\quad M \\longmapsto M[I]", "$$", "which sends $M$ to the submodule of $I$-torsion." ], "refs": [], "ref_ids": [] }, { "id": 2936, "type": "other", "label": "dualizing-remark-vanishing-for-arbitrary-modules", "categories": [ "dualizing" ], "title": "dualizing-remark-vanishing-for-arbitrary-modules", "contents": [ "Let $(A, \\mathfrak m)$ and $\\omega_A^\\bullet$ be as in", "Lemma \\ref{lemma-sitting-in-degrees}.", "By More on Algebra, Lemma \\ref{more-algebra-lemma-injective-amplitude}", "we see that $\\omega_A^\\bullet$ has injective-amplitude in $[-d, 0]$", "because part (3) of that lemma applies.", "In particular, for any $A$-module $M$ (not necessarily finite) we have", "$\\Ext^i_A(M, \\omega_A^\\bullet) = 0$ for $i \\not \\in \\{-d, \\ldots, 0\\}$." ], "refs": [ "dualizing-lemma-sitting-in-degrees", "more-algebra-lemma-injective-amplitude" ], "ref_ids": [ 2861, 10188 ] }, { "id": 2937, "type": "other", "label": "dualizing-remark-specific-injective-hull", "categories": [ "dualizing" ], "title": "dualizing-remark-specific-injective-hull", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring", "with a normalized dualizing complex $\\omega_A^\\bullet$.", "By Lemma \\ref{lemma-local-cohomology-of-dualizing}", "above we see that $R\\Gamma_Z(\\omega_A^\\bullet)$", "is an injective hull of the residue field placed in degree $0$.", "In fact, this gives a ``construction'' or ``realization''", "of the injective hull which is slightly more canonical than", "just picking any old injective hull. Namely, a normalized", "dualizing complex is unique up to isomorphism, with group", "of automorphisms the group of units of $A$, whereas an", "injective hull of $\\kappa$ is unique up to isomorphism, with", "group of automorphisms the group of units of the completion", "$A^\\wedge$ of $A$ with respect to $\\mathfrak m$." ], "refs": [ "dualizing-lemma-local-cohomology-of-dualizing" ], "ref_ids": [ 2872 ] }, { "id": 3089, "type": "other", "label": "properties-remark-normal-connected-irreducible", "categories": [ "properties" ], "title": "properties-remark-normal-connected-irreducible", "contents": [ "Let $X$ be a normal scheme. If $X$ is locally Noetherian then we see that", "$X$ is integral if and only if $X$ is connected, see", "Lemma \\ref{lemma-normal-locally-Noetherian}.", "But there exists a connected affine scheme $X$ such that", "$\\mathcal{O}_{X, x}$ is a domain for all $x \\in X$, but $X$ is not", "irreducible, see Examples, Section", "\\ref{examples-section-connected-locally-integral-not-integral}.", "This example is even a normal scheme (proof omitted), so beware!" ], "refs": [ "properties-lemma-normal-locally-Noetherian" ], "ref_ids": [ 2971 ] }, { "id": 3090, "type": "other", "label": "properties-remark-non-integral-Japanese", "categories": [ "properties" ], "title": "properties-remark-non-integral-Japanese", "contents": [ "In \\cite{Hoobler-finite} a (locally Noetherian) scheme $X$ is called", "Japanese if for every $x \\in X$ and every associated prime $\\mathfrak p$", "of $\\mathcal{O}_{X, x}$ the ring $\\mathcal{O}_{X, x}/\\mathfrak p$ is", "Japanese. We do not use this definition since there exists a one", "dimensional Noetherian domain with excellent (in particular", "Japanese) local rings whose normalization is not finite. See", "\\cite[Example 1]{Hochster-loci} or \\cite{Heinzer-Levy} or", "\\cite[Expos\\'e XIX]{Traveaux}.", "On the other hand, we could circumvent this problem by calling a scheme", "$X$ Japanese if for every affine open $\\Spec(A) \\subset X$ the ring", "$A/\\mathfrak p$ is Japanese for every associated prime $\\mathfrak p$ of $A$." ], "refs": [], "ref_ids": [] }, { "id": 3091, "type": "other", "label": "properties-remark-neurotic", "categories": [ "properties" ], "title": "properties-remark-neurotic", "contents": [ "With assumptions and notation of Lemma \\ref{lemma-ample-quasi-coherent}.", "Denote the displayed map of the lemma by $\\theta_\\mathcal{F}$.", "Note that the isomorphism $f^*\\mathcal{O}_Y(n) \\to \\mathcal{L}^{\\otimes n}$", "of Lemma \\ref{lemma-ample-gcd-is-one} is just", "$\\theta_{\\mathcal{L}^{\\otimes n}}$.", "Consider the multiplication maps", "$$", "\\widetilde{M} \\otimes_{\\mathcal{O}_Y} \\mathcal{O}_Y(n)", "\\longrightarrow", "\\widetilde{M(n)}", "$$", "see", "Constructions, Equation (\\ref{constructions-equation-multiply-more-generally}).", "Pull this back to $X$ and consider", "$$", "\\xymatrix{", "f^*\\widetilde{M} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{O}_Y(n)", "\\ar[r]", "\\ar[d]_{\\theta_\\mathcal{F} \\otimes \\theta_{\\mathcal{L}^{\\otimes n}}}", "&", "f^*\\widetilde{M(n)}", "\\ar[d]^{\\theta_{\\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}}}", "\\\\", "\\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n} \\ar[r]^{\\text{id}} &", "\\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}", "}", "$$", "Here we have used the obvious identification", "$M(n) = \\Gamma_*(X, \\mathcal{L}, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n})$.", "This diagram commutes. Proof omitted." ], "refs": [ "properties-lemma-ample-quasi-coherent", "properties-lemma-ample-gcd-is-one" ], "ref_ids": [ 3057, 3056 ] }, { "id": 3092, "type": "other", "label": "properties-remark-maximal-points-affine", "categories": [ "properties" ], "title": "properties-remark-maximal-points-affine", "contents": [ "Lemma \\ref{lemma-maximal-points-affine} above is false if $X$", "is not quasi-separated. Here is an example. Take", "$R = \\mathbf{Q}[x, y_1, y_2, \\ldots]/((x-i)y_i)$.", "Consider the minimal prime ideal $\\mathfrak p = (y_1, y_2, \\ldots)$", "of $R$. Glue two copies of $\\Spec(R)$ along the", "(not quasi-compact) open $\\Spec(R) \\setminus V(\\mathfrak p)$", "to get a scheme $X$ (glueing as in", "Schemes, Example \\ref{schemes-example-affine-space-zero-doubled}).", "Then the two maximal points of $X$ corresponding to $\\mathfrak p$", "are not contained in a common affine open. The reason is", "that any open of $\\Spec(R)$ containing $\\mathfrak p$", "contains infinitely many of the ``lines'' $x = i$, $y_j = 0$,", "$j \\not = i$ with parameter $y_i$. Details omitted." ], "refs": [ "properties-lemma-maximal-points-affine" ], "ref_ids": [ 3059 ] }, { "id": 3232, "type": "other", "label": "quot-remark-hom-base-change", "categories": [ "quot" ], "title": "quot-remark-hom-base-change", "contents": [ "In Situation \\ref{situation-hom} let $B' \\to B$ be a morphism of", "algebraic spaces over $S$. Set $X' = X \\times_B B'$ and denote", "$\\mathcal{F}'$, $\\mathcal{G}'$ the pullback of", "$\\mathcal{F}$, $\\mathcal{G}$ to $X'$. Then we obtain a functor", "$\\mathit{Hom}(\\mathcal{F}', \\mathcal{G}') : (\\Sch/B')^{opp} \\to \\textit{Sets}$", "associated to the base change $f' : X' \\to B'$. For a scheme $T$ over $B'$", "it is clear that we have", "$$", "\\mathit{Hom}(\\mathcal{F}', \\mathcal{G}')(T) =", "\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})(T)", "$$", "where on the right hand side we think of $T$ as a scheme over $B$", "via the composition $T \\to B' \\to B$. This trivial remark", "will occasionally be useful to change the base algebraic space." ], "refs": [], "ref_ids": [] }, { "id": 3233, "type": "other", "label": "quot-remark-coherent-base-change", "categories": [ "quot" ], "title": "quot-remark-coherent-base-change", "contents": [ "In Situation \\ref{situation-coherent} the rule", "$(T, g, \\mathcal{F}) \\mapsto (T, g)$ defines a $1$-morphism", "$$", "\\Cohstack_{X/B} \\longrightarrow \\mathcal{S}_B", "$$", "of stacks in groupoids", "(see Lemma \\ref{lemma-coherent-stack},", "Algebraic Stacks, Section \\ref{algebraic-section-split}, and", "Examples of Stacks, Section", "\\ref{examples-stacks-section-stack-associated-to-sheaf}).", "Let $B' \\to B$ be a morphism of", "algebraic spaces over $S$. Let $\\mathcal{S}_{B'} \\to \\mathcal{S}_B$", "be the associated $1$-morphism of stacks fibred in sets.", "Set $X' = X \\times_B B'$.", "We obtain a stack in groupoids $\\Cohstack_{X'/B'} \\to (\\Sch/S)_{fppf}$", "associated to the base change $f' : X' \\to B'$. In this situation", "the diagram", "$$", "\\vcenter{", "\\xymatrix{", "\\Cohstack_{X'/B'} \\ar[r] \\ar[d] & \\Cohstack_{X/B} \\ar[d] \\\\", "\\mathcal{S}_{B'} \\ar[r] & \\mathcal{S}_B", "}", "}", "\\quad", "\\begin{matrix}", "\\text{or in} \\\\", "\\text{another} \\\\", "\\text{notation}", "\\end{matrix}", "\\quad", "\\vcenter{", "\\xymatrix{", "\\Cohstack_{X'/B'} \\ar[r] \\ar[d] & \\Cohstack_{X/B} \\ar[d] \\\\", "\\Sch/B' \\ar[r] & \\Sch/B", "}", "}", "$$", "is $2$-fibre product square. This trivial remark", "will occasionally be useful to change the base algebraic space." ], "refs": [ "quot-lemma-coherent-stack" ], "ref_ids": [ 3162 ] }, { "id": 3234, "type": "other", "label": "quot-remark-q-base-change", "categories": [ "quot" ], "title": "quot-remark-q-base-change", "contents": [ "In Situation \\ref{situation-q} let $B' \\to B$ be a morphism of", "algebraic spaces over $S$. Set $X' = X \\times_B B'$ and denote", "$\\mathcal{F}'$ the pullback of $\\mathcal{F}$ to $X'$.", "Thus we have the functor $Q_{\\mathcal{F}'/X'/B'}$ on", "the category of schemes over $B'$. For a scheme $T$ over $B'$", "it is clear that we have", "$$", "Q_{\\mathcal{F}'/X'/B'}(T) = Q_{\\mathcal{F}/X/B}(T)", "$$", "where on the right hand side we think of $T$ as a scheme over $B$", "via the composition $T \\to B' \\to B$.", "Similar remarks apply to $\\text{Q}^{fp}_{\\mathcal{F}/X/B}$.", "These trivial remarks", "will occasionally be useful to change the base algebraic space." ], "refs": [], "ref_ids": [] }, { "id": 3235, "type": "other", "label": "quot-remark-q-sheaf", "categories": [ "quot" ], "title": "quot-remark-q-sheaf", "contents": [ "Let $S$ be a scheme, $X$ an algebraic space over $S$, and $\\mathcal{F}$", "a quasi-coherent $\\mathcal{O}_X$-module. Suppose that", "$\\{f_i : X_i \\to X\\}_{i \\in I}$", "is an fpqc covering and for each $i, j \\in I$ we are given an fpqc covering", "$\\{X_{ijk} \\to X_i \\times_X X_j\\}$. In this situation we have a bijection", "$$", "\\left\\{", "\\begin{matrix}", "\\text{quotients }\\mathcal{F} \\to \\mathcal{Q}\\text{ where } \\\\", "\\mathcal{Q}\\text{ is a quasi-coherent }\\\\", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "\\text{families of quotients }f_i^*\\mathcal{F} \\to \\mathcal{Q}_i", "\\text{ where } \\\\", "\\mathcal{Q}_i\\text{ is quasi-coherent and }", "\\mathcal{Q}_i\\text{ and }\\mathcal{Q}_j\\\\", "\\text{ restrict to the same quotient on }X_{ijk}", "\\end{matrix}", "\\right\\}", "$$", "Namely, let $(f_i^*\\mathcal{F} \\to \\mathcal{Q}_i)_{i \\in I}$", "be an element of the right hand side. Then since", "$\\{X_{ijk} \\to X_i \\times_X X_j\\}$ is an fpqc covering we see that", "the pullbacks of $\\mathcal{Q}_i$ and $\\mathcal{Q}_j$ restrict", "to the same quotient of the pullback of $\\mathcal{F}$ to $X_i \\times_X X_j$", "(by fully faithfulness in", "Descent on Spaces, Proposition", "\\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}).", "Hence we obtain a descent datum for quasi-coherent modules", "with respect to $\\{X_i \\to X\\}_{i \\in I}$. By", "Descent on Spaces, Proposition", "\\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}", "we find a map of quasi-coherent $\\mathcal{O}_X$-modules", "$\\mathcal{F} \\to \\mathcal{Q}$ whose restriction to $X_i$ recovers", "the given maps $f_i^*\\mathcal{F} \\to \\mathcal{Q}_i$.", "Since the family of morphisms $\\{X_i \\to X\\}$ is jointly surjective", "and flat, for every point $x \\in |X|$ there exists an $i$ and a point", "$x_i \\in |X_i|$ mapping to $x$. Note that the induced map on", "local rings", "$\\mathcal{O}_{X, \\overline{x}} \\to \\mathcal{O}_{X_i, \\overline{x_i}}$", "is faithfully flat, see", "Morphisms of Spaces, Section \\ref{spaces-morphisms-section-flat}.", "Thus we see that $\\mathcal{F} \\to \\mathcal{Q}$ is surjective." ], "refs": [ "spaces-descent-proposition-fpqc-descent-quasi-coherent", "spaces-descent-proposition-fpqc-descent-quasi-coherent" ], "ref_ids": [ 9437, 9437 ] }, { "id": 3236, "type": "other", "label": "quot-remark-q-obs", "categories": [ "quot" ], "title": "quot-remark-q-obs", "contents": [ "In Situation \\ref{situation-q} {\\bf assume} that $\\mathcal{F}$ is flat", "over $B$. Let $T \\subset T'$ be an first order", "thickening of schemes over $B$ with ideal sheaf $\\mathcal{J}$. Then", "$X_T \\subset X_{T'}$ is a first order thickening of algebraic spaces", "whose ideal sheaf $\\mathcal{I}$ is a quotient of $f_T^*\\mathcal{J}$.", "We will think of sheaves on $X_{T'}$, resp.\\ $T'$ as sheaves on", "$X_T$, resp.\\ $T$ using the fundamental equivalence described in", "More on Morphisms of Spaces, Section", "\\ref{spaces-more-morphisms-section-thickenings}.", "Let", "$$", "0 \\to \\mathcal{K} \\to \\mathcal{F}_T \\to \\mathcal{Q} \\to 0", "$$", "define an element $x$ of $Q_{\\mathcal{F}/X/B}(T)$. Since $\\mathcal{F}_{T'}$", "is flat over $T'$ we have a short exact sequence", "$$", "0 \\to f_T^*\\mathcal{J} \\otimes_{\\mathcal{O}_{X_T}} \\mathcal{F}_T", "\\xrightarrow{i} \\mathcal{F}_{T'} \\xrightarrow{\\pi} \\mathcal{F}_T \\to 0", "$$", "and we have", "$f_T^*\\mathcal{J} \\otimes_{\\mathcal{O}_{X_T}} \\mathcal{F}_T =", "\\mathcal{I} \\otimes_{\\mathcal{O}_{X_T}} \\mathcal{F}_T$, see", "Deformation Theory, Lemma \\ref{defos-lemma-deform-module-ringed-topoi}.", "Let us use the abbreviation", "$", "f_T^*\\mathcal{J} \\otimes_{\\mathcal{O}_{X_T}} \\mathcal{G} =", "\\mathcal{G} \\otimes_{\\mathcal{O}_T} \\mathcal{J}", "$", "for an $\\mathcal{O}_{X_T}$-module $\\mathcal{G}$.", "Since $\\mathcal{Q}$ is flat over $T$, we obtain a short exact sequence", "$$", "0 \\to", "\\mathcal{K} \\otimes_{\\mathcal{O}_T} \\mathcal{J} \\to", "\\mathcal{F}_T \\otimes_{\\mathcal{O}_T} \\mathcal{J} \\to", "\\mathcal{Q} \\otimes_{\\mathcal{O}_T} \\mathcal{J} \\to", "\\to 0", "$$", "Combining the above we obtain an canonical extension", "$$", "0 \\to \\mathcal{Q} \\otimes_{\\mathcal{O}_T} \\mathcal{J} \\to", "\\pi^{-1}(\\mathcal{K})/i(\\mathcal{K} \\otimes_{\\mathcal{O}_T} \\mathcal{J}) \\to", "\\mathcal{K} \\to 0", "$$", "of $\\mathcal{O}_{X_T}$-modules. This defines a canonical class", "$$", "o_x(T') \\in", "\\Ext^1_{\\mathcal{O}_{X_T}}(\\mathcal{K},", "\\mathcal{Q} \\otimes_{\\mathcal{O}_T} \\mathcal{J})", "$$", "If $o_x(T')$ is zero, then we obtain a splitting of the short", "exact sequence defining it, in other words, we obtain a", "$\\mathcal{O}_{X_{T'}}$-submodule", "$\\mathcal{K}' \\subset \\pi^{-1}(\\mathcal{K})$ sitting in a short", "exact sequence", "$0 \\to \\mathcal{K} \\otimes_{\\mathcal{O}_T} \\mathcal{J} \\to", "\\mathcal{K}' \\to \\mathcal{K} \\to 0$.", "Then it follows from the lemma reference above that", "$\\mathcal{Q}' = \\mathcal{F}_{T'}/\\mathcal{K}'$", "is a lift of $x$ to an element of $Q_{\\mathcal{F}/X/B}(T')$.", "Conversely, the reader sees that the existence of", "a lift implies that $o_x(T')$ is zero. Moreover, if", "$x \\in Q_{\\mathcal{F}/X/B}^{fp}(T)$, then automatically", "$x' \\in Q_{\\mathcal{F}/X/B}^{fp}(T')$ by", "Deformation Theory, Lemma \\ref{defos-lemma-deform-fp-module-ringed-topoi}.", "If we ever need this", "remark we will turn this remark into a lemma, precisely formulate", "the result and give a detailed proof (in fact, all of the above", "works in the setting of arbitrary ringed topoi)." ], "refs": [ "defos-lemma-deform-module-ringed-topoi", "defos-lemma-deform-fp-module-ringed-topoi" ], "ref_ids": [ 13397, 13398 ] }, { "id": 3237, "type": "other", "label": "quot-remark-q-defos", "categories": [ "quot" ], "title": "quot-remark-q-defos", "contents": [ "In Situation \\ref{situation-q} {\\bf assume} that $\\mathcal{F}$ is flat", "over $B$. We continue the discussion of Remark \\ref{remark-q-obs}.", "Assume $o_x(T') = 0$. Then we claim that the set of lifts", "$x' \\in Q_{\\mathcal{F}/X/B}(T')$ is a principal homogeneous space", "under the group", "$$", "\\Hom_{\\mathcal{O}_{X_T}}(\\mathcal{K},", "\\mathcal{Q} \\otimes_{\\mathcal{O}_T} \\mathcal{J})", "$$", "Namely, given any $\\mathcal{F}_{T'} \\to \\mathcal{Q}'$ flat over $T'$", "lifting the quotient $\\mathcal{Q}$ we obtain a commutative diagram", "with exact rows and columns", "$$", "\\xymatrix{", "& 0 \\ar[d] & 0 \\ar[d] & 0 \\ar[d] \\\\", "0 \\ar[r] &", "\\mathcal{K} \\otimes \\mathcal{J} \\ar[r] \\ar[d] &", "\\mathcal{F}_T \\otimes \\mathcal{J} \\ar[r] \\ar[d] &", "\\mathcal{Q} \\otimes \\mathcal{J} \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{K}' \\ar[r] \\ar[d] &", "\\mathcal{F}_{T'} \\ar[r] \\ar[d] &", "\\mathcal{Q}' \\ar[r] \\ar[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{K} \\ar[d] \\ar[r] &", "\\mathcal{F}_T \\ar[d] \\ar[r] &", "\\mathcal{Q} \\ar[d] \\ar[r] &", "0 \\\\", "& 0 & 0 & 0", "}", "$$", "(to see this use the observations made in the previous remark).", "Given a map $\\varphi : \\mathcal{K} \\to \\mathcal{Q} \\otimes \\mathcal{J}$", "we can consider the subsheaf $\\mathcal{K}'_\\varphi \\subset \\mathcal{F}_{T'}$", "consisting of those local sections $s$", "whose image in $\\mathcal{F}_T$ is a local section $k$ of $\\mathcal{K}$", "and whose image in $\\mathcal{Q}'$ is the local section $\\varphi(k)$ of", "$\\mathcal{Q} \\otimes \\mathcal{J}$. Then set", "$\\mathcal{Q}'_\\varphi = \\mathcal{F}_{T'}/\\mathcal{K}'_\\varphi$.", "Conversely, any second lift of $x$ corresponds to one of the", "qotients constructed in this manner. If we ever need this", "remark we will turn this remark into a lemma, precisely formulate", "the result and give a detailed proof (in fact, all of the above", "works in the setting of arbitrary ringed topoi)." ], "refs": [ "quot-remark-q-obs" ], "ref_ids": [ 3236 ] }, { "id": 3238, "type": "other", "label": "quot-remark-quot-via-artins-axioms", "categories": [ "quot" ], "title": "quot-remark-quot-via-artins-axioms", "contents": [ "Let $S$ be a Noetherian scheme all of whose local rings are G-rings.", "Let $X$ be an algebraic space over $S$ whose structure morphism", "$f : X \\to S$ is of finite presentation and separated.", "Let $\\mathcal{F}$ be a finitely presented quasi-coherent sheaf", "on $X$ flat over $S$. In this remark we sketch how one can", "use Artin's axioms to prove that $\\Quotfunctor_{\\mathcal{F}/X/S}$", "is an algebraic space locally of finite presentation over $S$", "and avoid using the algebraicity of the stack of coherent sheaves", "as was done in the proof of Proposition \\ref{proposition-quot}.", "\\medskip\\noindent", "We check the conditions listed in Artin's Axioms, Proposition", "\\ref{artin-proposition-spaces-diagonal-representable}.", "Representability of the diagonal of $\\Quotfunctor_{\\mathcal{F}/X/S}$", "can be seen as follows: suppose we have two quotients", "$\\mathcal{F}_T \\to \\mathcal{Q}_i$, $i = 1, 2$. Denote", "$\\mathcal{K}_1$ the kernel of the first one. Then we have", "to show that the locus of $T$ over which", "$u : \\mathcal{K}_1 \\to \\mathcal{Q}_2$ becomes zero is representable.", "This follows for example from Flatness on Spaces, Lemma", "\\ref{spaces-flat-lemma-F-zero-closed-proper}", "or from a discussion of the $\\mathit{Hom}$ sheaf earlier", "in this chapter. Axioms [0] (sheaf), [1] (limits), [2] (Rim-Schlessinger)", "follow from Lemmas \\ref{lemma-quot-sheaf},", "\\ref{lemma-q-limit-preserving}, and \\ref{lemma-q-RS-star}", "(plus some extra work to deal with the properness condition).", "Axiom [3] (finite dimensionality of tangent spaces)", "follows from the description of the infinitesimal", "deformations in Remark \\ref{remark-q-defos}", "and finiteness of cohomology of coherent sheaves on proper", "algebraic spaces over fields (Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-proper-pushforward-coherent}).", "Axiom [4] (effectiveness of formal objects)", "follows from Grothendieck's existence theorem", "(More on Morphisms of Spaces, Theorem", "\\ref{spaces-more-morphisms-theorem-grothendieck-existence}).", "As usual, the trickiest to verify is axiom [5] (openness of versality).", "One can for example use the obstruction theory described", "in Remark \\ref{remark-q-obs} and the description of", "deformations in Remark \\ref{remark-q-defos}", "to do this using the criterion in", "Artin's Axioms, Lemma \\ref{artin-lemma-get-openness-obstruction-theory}.", "Please compare with the second proof of", "Lemma \\ref{lemma-coherent-defo-thy}." ], "refs": [ "quot-proposition-quot", "artin-proposition-spaces-diagonal-representable", "spaces-flat-lemma-F-zero-closed-proper", "quot-lemma-quot-sheaf", "quot-lemma-q-limit-preserving", "quot-lemma-q-RS-star", "quot-remark-q-defos", "spaces-cohomology-lemma-proper-pushforward-coherent", "spaces-more-morphisms-theorem-grothendieck-existence", "quot-remark-q-obs", "quot-remark-q-defos", "artin-lemma-get-openness-obstruction-theory", "quot-lemma-coherent-defo-thy" ], "ref_ids": [ 3227, 11414, 7182, 3174, 3172, 3173, 3237, 11331, 16, 3236, 3237, 11390, 3168 ] }, { "id": 3239, "type": "other", "label": "quot-remark-spaces-base-change", "categories": [ "quot" ], "title": "quot-remark-spaces-base-change", "contents": [ "Let $B$ be an algebraic space over $\\Spec(\\mathbf{Z})$.", "Let $B\\textit{-Spaces}'_{ft}$ be the category consisting", "of pairs $(X \\to S, h : S \\to B)$", "where $X \\to S$ is an object of", "$\\Spacesstack'_{ft}$ and $h : S \\to B$ is a morphism.", "A morphism $(X' \\to S', h') \\to (X \\to S, h)$", "in $B\\textit{-Spaces}'_{ft}$ is a morphism $(f, g)$", "in $\\Spacesstack'_{ft}$ such that $h \\circ g = h'$.", "In this situation the diagram", "$$", "\\xymatrix{", "B\\textit{-Spaces}'_{ft} \\ar[r] \\ar[d] & \\Spacesstack'_{ft} \\ar[d] \\\\", "(\\Sch/B)_{fppf} \\ar[r] & \\Sch_{fppf}", "}", "$$", "is $2$-fibre product square. This trivial remark", "will occasionally be useful to deduce results from", "the absolute case $\\Spacesstack'_{ft}$ to the case", "of families over a given base algebraic space.", "Of course, a similar construction works for", "$B\\textit{-Spaces}'_{fp, flat, proper}$" ], "refs": [], "ref_ids": [] }, { "id": 3240, "type": "other", "label": "quot-remark-spaces-defo-thy", "categories": [ "quot" ], "title": "quot-remark-spaces-defo-thy", "contents": [ "Lemma \\ref{lemma-spaces-defo-thy} can also be shown using either", "Artin's Axioms, Lemma \\ref{artin-lemma-dual-openness}", "(as in the first proof of", "Lemma \\ref{lemma-coherent-defo-thy}), or using an obstruction theory", "as in Artin's Axioms, Lemma \\ref{artin-lemma-get-openness-obstruction-theory}", "(as in the second proof of", "Lemma \\ref{lemma-coherent-defo-thy}).", "In both cases one uses the deformation and obstruction theory developed in", "Cotangent, Section \\ref{cotangent-section-deformations-ringed-topoi}", "to translate the needed properties of deformations and obstructions", "into $\\Ext$-groups to which", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-compute-ext}", "can be applied.", "The second method (using an obstruction theory and therefore", "using the full cotangent complex) is perhaps the ``standard'' method used", "in most references." ], "refs": [ "quot-lemma-spaces-defo-thy", "artin-lemma-dual-openness", "quot-lemma-coherent-defo-thy", "artin-lemma-get-openness-obstruction-theory", "quot-lemma-coherent-defo-thy", "spaces-perfect-lemma-compute-ext" ], "ref_ids": [ 3195, 11398, 3168, 11390, 3168, 2733 ] }, { "id": 3241, "type": "other", "label": "quot-remark-polarized-base-change", "categories": [ "quot" ], "title": "quot-remark-polarized-base-change", "contents": [ "Let $B$ be an algebraic space over $\\Spec(\\mathbf{Z})$.", "Let $B\\textit{-Polarized}$ be the category consisting", "of triples $(X \\to S, \\mathcal{L}, h : S \\to B)$", "where $(X \\to S, \\mathcal{L})$ is an object of", "$\\Polarizedstack$ and $h : S \\to B$ is a morphism.", "A morphism $(X' \\to S', \\mathcal{L}', h') \\to (X \\to S, \\mathcal{L}, h)$", "in $B\\textit{-Polarized}$ is a morphism $(f, g, \\varphi)$", "in $\\Polarizedstack$ such that $h \\circ g = h'$.", "In this situation the diagram", "$$", "\\xymatrix{", "B\\textit{-Polarized} \\ar[r] \\ar[d] & \\Polarizedstack \\ar[d] \\\\", "(\\Sch/B)_{fppf} \\ar[r] & \\Sch_{fppf}", "}", "$$", "is $2$-fibre product square. This trivial remark", "will occasionally be useful to deduce results from", "the absolute case $\\Polarizedstack$ to the case", "of families over a given base algebraic space." ], "refs": [], "ref_ids": [] }, { "id": 3242, "type": "other", "label": "quot-remark-polarized-defo-thy", "categories": [ "quot" ], "title": "quot-remark-polarized-defo-thy", "contents": [ "Lemma \\ref{lemma-polarized-defo-thy} can also be shown", "using an obstruction theory as in", "Artin's Axioms, Lemma \\ref{artin-lemma-get-openness-obstruction-theory}", "(as in the second proof of Lemma \\ref{lemma-coherent-defo-thy}).", "To do this one has to generalize the deformation and obstruction theory", "developed in", "Cotangent, Section \\ref{cotangent-section-deformations-ringed-topoi}", "to the case of pairs of algebraic spaces and quasi-coherent modules.", "Another possibility is to use that the $1$-morphism", "$\\Polarizedstack \\to \\Spacesstack'_{fp, flat, proper}$", "is algebraic (Lemma \\ref{lemma-polarized-to-spaces-algebraic})", "and the fact that we know openness of versality for the target", "(Lemma \\ref{lemma-spaces-defo-thy} and", "Remark \\ref{remark-spaces-defo-thy})." ], "refs": [ "quot-lemma-polarized-defo-thy", "artin-lemma-get-openness-obstruction-theory", "quot-lemma-coherent-defo-thy", "quot-lemma-polarized-to-spaces-algebraic", "quot-lemma-spaces-defo-thy", "quot-remark-spaces-defo-thy" ], "ref_ids": [ 3206, 11390, 3168, 3199, 3195, 3240 ] }, { "id": 3243, "type": "other", "label": "quot-remark-curves-base-change", "categories": [ "quot" ], "title": "quot-remark-curves-base-change", "contents": [ "Let $B$ be an algebraic space over $\\Spec(\\mathbf{Z})$.", "Let $B\\text{-}\\Curvesstack$ be the category consisting", "of pairs $(X \\to S, h : S \\to B)$", "where $X \\to S$ is an object of", "$\\Curvesstack$ and $h : S \\to B$ is a morphism.", "A morphism $(X' \\to S', h') \\to (X \\to S, h)$", "in $B\\text{-}\\Curvesstack$ is a morphism $(f, g)$", "in $\\Curvesstack$ such that $h \\circ g = h'$.", "In this situation the diagram", "$$", "\\xymatrix{", "B\\text{-}\\Curvesstack \\ar[r] \\ar[d] & \\Curvesstack \\ar[d] \\\\", "(\\Sch/B)_{fppf} \\ar[r] & \\Sch_{fppf}", "}", "$$", "is $2$-fibre product square. This trivial remark", "will occasionally be useful to deduce results from", "the absolute case $\\Curvesstack$ to the case", "of families of curves over a given base algebraic space." ], "refs": [], "ref_ids": [] }, { "id": 3244, "type": "other", "label": "quot-remark-alternative-approach-curves", "categories": [ "quot" ], "title": "quot-remark-alternative-approach-curves", "contents": [ "Consider the $2$-fibre product", "$$", "\\xymatrix{", "\\Curvesstack \\times_{\\Spacesstack'_{fp, flat, proper}}", "\\Polarizedstack \\ar[r] \\ar[d] &", "\\Polarizedstack \\ar[d] \\\\", "\\Curvesstack \\ar[r] &", "\\Spacesstack'_{fp, flat, proper}", "}", "$$", "This fibre product parametrized polarized curves, i.e., families", "of curves endowed with a relatively ample invertible sheaf.", "It turns out that the left vertical arrow", "$$", "\\textit{PolarizedCurves} \\longrightarrow \\Curvesstack", "$$", "is algebraic, smooth, and surjective. Namely, this $1$-morphism", "is algebraic (as base change of the arrow in", "Lemma \\ref{lemma-polarized-to-spaces-algebraic}),", "every point is in the image, and", "there are no obstructions to deforming invertible sheaves on curves", "(see proof of Lemma \\ref{lemma-curves-existence}).", "This gives another approach to the algebraicity of $\\Curvesstack$.", "Namely, by Lemma \\ref{lemma-curves-open-and-closed-in-spaces}", "we see that $\\textit{PolarizedCurves}$ is an open and closed substack", "of the algebraic stack $\\Polarizedstack$ and any stack in groupoids", "which is the target of a smooth algebraic morphism from an algebraic", "stack is an algebraic stack." ], "refs": [ "quot-lemma-polarized-to-spaces-algebraic", "quot-lemma-curves-existence", "quot-lemma-curves-open-and-closed-in-spaces" ], "ref_ids": [ 3199, 3213, 3215 ] }, { "id": 3245, "type": "other", "label": "quot-remark-complexes-base-change", "categories": [ "quot" ], "title": "quot-remark-complexes-base-change", "contents": [ "In Situation \\ref{situation-complexes} the rule", "$(T, g, E) \\mapsto (T, g)$ defines a $1$-morphism", "$$", "\\Complexesstack_{X/B} \\longrightarrow \\mathcal{S}_B", "$$", "of stacks in groupoids", "(see Lemma \\ref{lemma-complexes-stack},", "Algebraic Stacks, Section \\ref{algebraic-section-split}, and", "Examples of Stacks, Section", "\\ref{examples-stacks-section-stack-associated-to-sheaf}).", "Let $B' \\to B$ be a morphism of", "algebraic spaces over $S$. Let $\\mathcal{S}_{B'} \\to \\mathcal{S}_B$", "be the associated $1$-morphism of stacks fibred in sets.", "Set $X' = X \\times_B B'$.", "We obtain a stack in groupoids", "$\\Complexesstack_{X'/B'} \\to (\\Sch/S)_{fppf}$", "associated to the base change $f' : X' \\to B'$. In this situation", "the diagram", "$$", "\\vcenter{", "\\xymatrix{", "\\Complexesstack_{X'/B'} \\ar[r] \\ar[d] &", "\\Complexesstack_{X/B} \\ar[d] \\\\", "\\mathcal{S}_{B'} \\ar[r] & \\mathcal{S}_B", "}", "}", "\\quad", "\\begin{matrix}", "\\text{or in} \\\\", "\\text{another} \\\\", "\\text{notation}", "\\end{matrix}", "\\quad", "\\vcenter{", "\\xymatrix{", "\\Complexesstack_{X'/B'} \\ar[r] \\ar[d] &", "\\Complexesstack_{X/B} \\ar[d] \\\\", "\\Sch/B' \\ar[r] & \\Sch/B", "}", "}", "$$", "is $2$-fibre product square. This trivial remark", "will occasionally be useful to change the base algebraic space." ], "refs": [ "quot-lemma-complexes-stack" ], "ref_ids": [ 3220 ] }, { "id": 3406, "type": "other", "label": "coherent-remark-chow-Noetherian", "categories": [ "coherent" ], "title": "coherent-remark-chow-Noetherian", "contents": [ "In the situation of Chow's", "Lemma \\ref{lemma-chow-Noetherian}:", "\\begin{enumerate}", "\\item The morphism $\\pi$ is actually H-projective (hence projective, see", "Morphisms, Lemma \\ref{morphisms-lemma-H-projective})", "since the morphism $X' \\to \\mathbf{P}^n_S \\times_S X = \\mathbf{P}^n_X$", "is a closed immersion (use the fact that $\\pi$ is proper, see", "Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}).", "\\item We may assume that $\\pi^{-1}(U)$ is scheme theoretically dense", "in $X'$. Namely, we can simply replace $X'$ by the scheme theoretic", "closure of $\\pi^{-1}(U)$. In this case we can think of $U$ as a", "scheme theoretically dense open subscheme of $X'$.", "See Morphisms, Section \\ref{morphisms-section-scheme-theoretic-image}.", "\\item If $X$ is reduced then we may choose $X'$ reduced. This is clear", "from (2).", "\\end{enumerate}" ], "refs": [ "coherent-lemma-chow-Noetherian", "morphisms-lemma-H-projective", "morphisms-lemma-image-proper-scheme-closed" ], "ref_ids": [ 3354, 5420, 5411 ] }, { "id": 3407, "type": "other", "label": "coherent-remark-explain-perfect-direct-image", "categories": [ "coherent" ], "title": "coherent-remark-explain-perfect-direct-image", "contents": [ "A consequence of Lemma \\ref{lemma-perfect-direct-image} is that there", "exists a finite complex of finite projective $A$-modules $M^\\bullet$ such", "that we have", "$$", "H^i(X_{A'}, \\mathcal{F}_{A'}) = H^i(M^\\bullet \\otimes_A A')", "$$", "functorially in $A'$. The condition that $\\mathcal{F}$ is", "flat over $A$ is essential, see \\cite{Hartshorne}." ], "refs": [ "coherent-lemma-perfect-direct-image" ], "ref_ids": [ 3369 ] }, { "id": 3408, "type": "other", "label": "coherent-remark-inverse-systems-kernel-cokernel-annihilated-by", "categories": [ "coherent" ], "title": "coherent-remark-inverse-systems-kernel-cokernel-annihilated-by", "contents": [ "Let $X$ be a Noetherian scheme and let", "$\\mathcal{I}, \\mathcal{K} \\subset \\mathcal{O}_X$", "be quasi-coherent sheaves of ideals. Let", "$\\alpha : (\\mathcal{F}_n) \\to (\\mathcal{G}_n)$ be a morphism of", "$\\textit{Coh}(X, \\mathcal{I})$.", "Given an affine open $\\Spec(A) = U \\subset X$ with", "$\\mathcal{I}|_U, \\mathcal{K}|_U$ corresponding to ideals $I, K \\subset A$", "denote $\\alpha_U : M \\to N$ of finite $A^\\wedge$-modules which", "corresponds to $\\alpha|_U$ via Lemma \\ref{lemma-inverse-systems-affine}.", "We claim the following are equivalent", "\\begin{enumerate}", "\\item there exists an integer $t \\geq 1$ such that", "$\\Ker(\\alpha_n)$ and $\\Coker(\\alpha_n)$", "are annihilated by $\\mathcal{K}^t$ for all $n \\geq 1$,", "\\item for any affine open $\\Spec(A) = U \\subset X$ as above", "the modules $\\Ker(\\alpha_U)$ and $\\Coker(\\alpha_U)$", "are annihilated by $K^t$ for some integer $t \\geq 1$, and", "\\item there exists a finite affine open covering $X = \\bigcup U_i$", "such that the conclusion of (2) holds for $\\alpha_{U_i}$.", "\\end{enumerate}", "If these equivalent conditions hold we will say that", "$\\alpha$ is a", "{\\it map whose kernel and cokernel are annihilated by a power of", "$\\mathcal{K}$}.", "To see the equivalence we use the following commutative algebra fact:", "suppose given an exact sequence", "$$", "0 \\to T \\to M \\to N \\to Q \\to 0", "$$", "of $A$-modules with $T$ and $Q$ annihilated by $K^t$ for some", "ideal $K \\subset A$. Then for every $f, g \\in K^t$ there exists a", "canonical map $\"fg\": N \\to M$ such that $M \\to N \\to M$ is equal to", "multiplication by $fg$. Namely, for $y \\in N$ we can pick $x \\in M$", "mapping to $fy$ in $N$ and then we can set $\"fg\"(y) = gx$. Thus it is", "clear that $\\Ker(M/JM \\to N/JN)$ and $\\Coker(M/JM \\to N/JN)$", "are annihilated by $K^{2t}$ for any ideal $J \\subset A$.", "\\medskip\\noindent", "Applying the commutative algebra fact to $\\alpha_{U_i}$ and $J = I^n$", "we see that (3) implies (1). Conversely,", "suppose (1) holds and $M \\to N$ is equal to $\\alpha_U$. Then there is", "a $t \\geq 1$ such that", "$\\Ker(M/I^nM \\to N/I^nN)$ and $\\Coker(M/I^nM \\to N/I^nN)$", "are annihilated by $K^t$ for all $n$. We obtain maps", "$\"fg\" : N/I^nN \\to M/I^nM$ which in the limit induce a map $N \\to M$", "as $N$ and $M$ are $I$-adically complete. Since the composition with", "$N \\to M \\to N$ is multiplication by $fg$ we conclude that $fg$", "annihilates $T$ and $Q$. In other words $T$ and $Q$ are annihilated by", "$K^{2t}$ as desired." ], "refs": [ "coherent-lemma-inverse-systems-affine" ], "ref_ids": [ 3370 ] }, { "id": 3409, "type": "other", "label": "coherent-remark-reformulate-existence-theorem", "categories": [ "coherent" ], "title": "coherent-remark-reformulate-existence-theorem", "contents": [ "Let $A$ be a Noetherian ring complete with respect to an ideal $I$.", "Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$.", "Let $X \\to S$ be a separated morphism of finite type.", "For $n \\geq 1$ we set $X_n = X \\times_S S_n$.", "Picture:", "$$", "\\xymatrix{", "X_1 \\ar[r]_{i_1} \\ar[d] & X_2 \\ar[r]_{i_2} \\ar[d] & X_3 \\ar[r] \\ar[d] &", "\\ldots & X \\ar[d] \\\\", "S_1 \\ar[r] & S_2 \\ar[r] & S_3 \\ar[r] & \\ldots & S", "}", "$$", "In this situation we consider systems $(\\mathcal{F}_n, \\varphi_n)$", "where", "\\begin{enumerate}", "\\item $\\mathcal{F}_n$ is a coherent $\\mathcal{O}_{X_n}$-module,", "\\item $\\varphi_n : i_n^*\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n$", "is an isomorphism, and", "\\item $\\text{Supp}(\\mathcal{F}_1)$ is proper over $S_1$.", "\\end{enumerate}", "Theorem \\ref{theorem-grothendieck-existence} says that the", "completion functor", "$$", "\\begin{matrix}", "\\text{coherent }\\mathcal{O}_X\\text{-modules }\\mathcal{F} \\\\", "\\text{with support proper over }A", "\\end{matrix}", "\\quad", "\\longrightarrow", "\\quad", "\\begin{matrix}", "\\text{systems }(\\mathcal{F}_n) \\\\", "\\text{as above}", "\\end{matrix}", "$$", "is an equivalence of categories. In the special case that $X$ is", "proper over $A$ we can omit the conditions on the supports." ], "refs": [ "coherent-theorem-grothendieck-existence" ], "ref_ids": [ 3279 ] }, { "id": 3544, "type": "other", "label": "formal-defos-remark-predeformation-functor", "categories": [ "formal-defos" ], "title": "formal-defos-remark-predeformation-functor", "contents": [ "We say that a functor $F: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$", "is a {\\it predeformation functor} if the associated cofibered set is a", "predeformation category, i.e.\\ if $F(k)$ is a one element set. Thus if", "$\\mathcal{F}$ is a predeformation category, then $\\overline{\\mathcal{F}}$ is a", "predeformation functor." ], "refs": [], "ref_ids": [] }, { "id": 3545, "type": "other", "label": "formal-defos-remark-localize-cofibered-groupoid", "categories": [ "formal-defos" ], "title": "formal-defos-remark-localize-cofibered-groupoid", "contents": [ "Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in", "groupoids, and let $x \\in \\Ob(\\mathcal{F}(k))$. We denote by", "$\\mathcal{F}_x$ the category of objects over $x$.", "An object of $\\mathcal{F}_x$ is an arrow $y \\to x$.", "A morphism $(y \\to x) \\to (z \\to x)$ in $\\mathcal{F}_x$ is a commutative", "diagram", "$$", "\\xymatrix{", "y \\ar[rr] \\ar[dr] & & z \\ar[dl] \\\\", "& x &", "}", "$$", "There is a forgetful functor $\\mathcal{F}_x \\to \\mathcal{F}$. We define", "the functor $p_x : \\mathcal{F}_x \\to \\mathcal{C}_\\Lambda$ as the", "composition", "$\\mathcal{F}_x \\to \\mathcal{F} \\xrightarrow{p} \\mathcal{C}_\\Lambda$.", "Then $p_x : \\mathcal{F}_x \\to \\mathcal{C}_\\Lambda$ is a", "predeformation category (proof omitted). In this way we can pass from an", "arbitrary category cofibered in groupoids over $\\mathcal{C}_\\Lambda$", "to a predeformation category at any $x \\in \\Ob(\\mathcal{F}(k))$." ], "refs": [], "ref_ids": [] }, { "id": 3546, "type": "other", "label": "formal-defos-remark-different-sequence-ideals", "categories": [ "formal-defos" ], "title": "formal-defos-remark-different-sequence-ideals", "contents": [ "Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in", "groupoids. Suppose that for each", "$R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$ we are given a filtration", "$\\mathcal{I}_R$ of $R$ by ideals. If", "$\\mathcal{I}_R$ induces the $\\mathfrak m_R$-adic topology on $R$ for all $R$,", "then one can define a category", "$\\widehat{\\mathcal{F}}_\\mathcal{I}$ by mimicking", "the definition of $\\widehat{\\mathcal{F}}$. This category comes equipped with a", "morphism", "$\\widehat{p}_\\mathcal{I} : \\widehat{\\mathcal{F}}_\\mathcal{I} \\to", "\\widehat{\\mathcal{C}}_\\Lambda$ making it into a category cofibered in", "groupoids such that $\\widehat{\\mathcal{F}}_\\mathcal{I}(R)$ is isomorphic to", "$\\widehat{\\mathcal{F}}_{\\mathcal{I}_R}(R)$ as defined above. The categories", "cofibered in groupoids $\\widehat{\\mathcal{F}}_\\mathcal{I}$ and", "$\\widehat{\\mathcal{F}}$ are equivalent, by using over an object", "$R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$", "the equivalence of", "Lemma \\ref{lemma-formal-objects-different-filtration}." ], "refs": [ "formal-defos-lemma-formal-objects-different-filtration" ], "ref_ids": [ 3430 ] }, { "id": 3547, "type": "other", "label": "formal-defos-remark-completion-functor", "categories": [ "formal-defos" ], "title": "formal-defos-remark-completion-functor", "contents": [ "Let $F: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a functor.", "Identifying functors with cofibered sets, the completion of $F$ is the functor", "$\\widehat{F} : \\widehat{\\mathcal{C}}_\\Lambda \\to \\textit{Sets}$", "given by $\\widehat{F}(S) = \\lim F(S/\\mathfrak{m}_S^{n})$. This agrees", "with the definition in Schlessinger's paper \\cite{Sch}." ], "refs": [], "ref_ids": [] }, { "id": 3548, "type": "other", "label": "formal-defos-remark-restrict-completion", "categories": [ "formal-defos" ], "title": "formal-defos-remark-restrict-completion", "contents": [ "Let $\\mathcal{F}$ be a category cofibred in groupoids over", "$\\mathcal{C}_\\Lambda$. We claim that there is a canonical", "equivalence", "$$", "can :", "\\widehat{\\mathcal{F}}|_{\\mathcal{C}_\\Lambda}", "\\longrightarrow", "\\mathcal{F}.", "$$", "Namely, let $A \\in \\Ob(\\mathcal{C}_\\Lambda)$ and let", "$(A, \\xi_n, f_n)$ be an object of", "$\\widehat{\\mathcal{F}}|_{\\mathcal{C}_\\Lambda}(A)$.", "Since $A$ is Artinian there is a minimal $m \\in \\mathbf{N}$", "such that $\\mathfrak m_A^m = 0$. Then $can$ sends $(A, \\xi_n, f_n)$ to $\\xi_m$.", "This functor is an equivalence of categories cofibered in groupoids by", "Categories, Lemma \\ref{categories-lemma-equivalence-fibred-categories}", "because it is an equivalence on all fibre categories by", "Lemma \\ref{lemma-formal-objects-different-filtration}", "and the fact that the $\\mathfrak m_A$-adic topology on a local", "Artinian ring $A$ comes from the zero ideal. We will frequently identify", "$\\mathcal{F}$ with a full subcategory of $\\widehat{\\mathcal{F}}$ via a", "quasi-inverse to the functor $can$." ], "refs": [ "categories-lemma-equivalence-fibred-categories", "formal-defos-lemma-formal-objects-different-filtration" ], "ref_ids": [ 12297, 3430 ] }, { "id": 3549, "type": "other", "label": "formal-defos-remark-completion-morphism", "categories": [ "formal-defos" ], "title": "formal-defos-remark-completion-morphism", "contents": [ "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of categories", "cofibered in groupoids over $\\mathcal{C}_\\Lambda$. Then there is an induced", "morphism", "$\\widehat{\\varphi}: \\widehat{\\mathcal{F}} \\to \\widehat{\\mathcal{G}}$", "of categories cofibered in groupoids over $\\widehat{\\mathcal{C}}_\\Lambda$.", "It sends an object $\\xi = (R, \\xi_n, f_n)$ of", "$\\widehat{\\mathcal{F}}$ to $(R, \\varphi(\\xi_n), \\varphi(f_n))$, and it sends a", "morphism $(a_0 : R \\to S, a_n : \\xi_n \\to \\eta_n)$ between", "objects $\\xi$ and $\\eta$ of $\\widehat{\\mathcal{F}}$ to", "$(a_0 : R \\to S, \\varphi(a_n) : \\varphi(\\xi_n) \\to \\varphi(\\eta_n))$.", "Finally, if $t : \\varphi \\to \\varphi'$ is a $2$-morphism between", "$1$-morphisms $\\varphi, \\varphi': \\mathcal{F} \\to \\mathcal{G}$ of", "categories cofibred in groupoids, then we obtain a $2$-morphism", "$\\widehat{t} : \\widehat{\\varphi} \\to \\widehat{\\varphi}'$. Namely, for", "$\\xi = (R, \\xi_n, f_n)$ as above we set", "$\\widehat{t}_\\xi = (t_{\\varphi(\\xi_n)})$. Hence completion defines a", "functor between $2$-categories", "$$", "\\widehat{~} :", "\\text{Cof}(\\mathcal{C}_\\Lambda)", "\\longrightarrow", "\\text{Cof}(\\widehat{\\mathcal{C}}_\\Lambda)", "$$", "from the $2$-category of categories cofibred in groupoids over", "$\\mathcal{C}_\\Lambda$ to the $2$-category of categories cofibred", "in groupoids over $\\widehat{\\mathcal{C}}_\\Lambda$." ], "refs": [], "ref_ids": [] }, { "id": 3550, "type": "other", "label": "formal-defos-remark-completion-restriction-adjoint", "categories": [ "formal-defos" ], "title": "formal-defos-remark-completion-restriction-adjoint", "contents": [ "We claim the completion functor of", "Remark \\ref{remark-completion-morphism}", "and the restriction functor", "$|_{\\mathcal{C}_\\Lambda} : \\text{Cof}(\\widehat{\\mathcal{C}}_\\Lambda)", "\\to \\text{Cof}(\\mathcal{C}_\\Lambda)$ of", "Remarks \\ref{remarks-cofibered-groupoids}", "(\\ref{item-definition-restricting-base-category})", "are ``2-adjoint'' in the following precise sense. Let", "$\\mathcal{F} \\in \\Ob(\\text{Cof}(\\mathcal{C}_\\Lambda))$", "and let", "$\\mathcal{G} \\in \\Ob(\\text{Cof}(\\widehat{\\mathcal{C}}_\\Lambda))$.", "Then there is an equivalence of categories", "$$", "\\Phi :", "\\Mor_{\\mathcal{C}_\\Lambda}(", "\\mathcal{G}|_{\\mathcal{C}_\\Lambda}, \\mathcal{F})", "\\longrightarrow", "\\Mor_{\\widehat{\\mathcal{C}}_\\Lambda}(\\mathcal{G}, \\widehat{\\mathcal{F}})", "$$", "To describe this equivalence, we", "define canonical morphisms", "$\\mathcal{G} \\to \\widehat{\\mathcal{G}|_{\\mathcal{C}_\\Lambda}}$ and", "$\\widehat{\\mathcal{F}}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$ as follows", "\\begin{enumerate}", "\\item Let $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda))$ and let $\\xi$", "be an object of the fiber category $\\mathcal{G}(R)$.", "Choose a pushforward $\\xi \\to \\xi_n$ of $\\xi$ to", "$R/\\mathfrak m_R^n$ for each $n \\in \\mathbf{N}$, and let", "$f_n : \\xi_{n + 1} \\to \\xi_n$ be the induced morphism.", "Then $\\mathcal{G} \\to \\widehat{\\mathcal{G}|_{\\mathcal{C}_\\Lambda}}$", "sends $\\xi$ to $(R, \\xi_n, f_n)$.", "\\item This is the equivalence", "$can : \\widehat{\\mathcal{F}}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$", "of", "Remark \\ref{remark-restrict-completion}.", "\\end{enumerate}", "Having said this, the equivalence", "$\\Phi : \\Mor_{\\mathcal{C}_\\Lambda}(", "\\mathcal{G}|_{\\mathcal{C}_\\Lambda}, \\mathcal{F}) \\to", "\\Mor_{\\widehat{\\mathcal{C}}_\\Lambda}(\\mathcal{G},", "\\widehat{\\mathcal{F}})$", "sends a morphism", "$\\varphi : \\mathcal{G}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$", "to", "$$", "\\mathcal{G} \\to \\widehat{\\mathcal{G}|_{\\mathcal{C}_\\Lambda}}", "\\xrightarrow{\\widehat{\\varphi}} \\widehat{\\mathcal{F}}", "$$", "There is a quasi-inverse", "$\\Psi :", "\\Mor_{\\widehat{\\mathcal{C}}_\\Lambda}(", "\\mathcal{G}, \\widehat{\\mathcal{F}}) \\to", "\\Mor_{\\mathcal{C}_\\Lambda}(", "\\mathcal{G}|_{\\mathcal{C}_\\Lambda}, \\mathcal{F})$", "to $\\Phi$ which sends $\\psi : \\mathcal{G} \\to \\widehat{\\mathcal{F}}$ to", "$$", "\\mathcal{G}|_{\\mathcal{C}_\\Lambda} \\xrightarrow{\\psi|_{\\mathcal{C}_\\Lambda}}", "\\widehat{\\mathcal{F}}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}.", "$$", "We omit the verification that $\\Phi$ and $\\Psi$ are quasi-inverse.", "We also do not address functoriality of $\\Phi$ (because it would", "lead into 3-category territory which we want to avoid at all cost)." ], "refs": [ "formal-defos-remark-completion-morphism", "formal-defos-remarks-cofibered-groupoids", "formal-defos-remark-restrict-completion" ], "ref_ids": [ 3549, 3585, 3548 ] }, { "id": 3551, "type": "other", "label": "formal-defos-remark-completion-restriction-cofset-adjoint", "categories": [ "formal-defos" ], "title": "formal-defos-remark-completion-restriction-cofset-adjoint", "contents": [ "For a category $\\mathcal{C}$ we denote by $\\text{CofSet}(\\mathcal{C})$ the", "category of cofibered sets over $\\mathcal{C}$. It is a $1$-category", "isomorphic the category of functors $\\mathcal{C} \\to \\textit{Sets}$.", "See Remarks \\ref{remarks-cofibered-groupoids}", "(\\ref{item-convention-cofibered-sets}).", "The completion and restriction functors restrict to functors", "$\\widehat{~} : \\text{CofSet}(\\mathcal{C}_\\Lambda) \\to", "\\text{CofSet}(\\widehat{\\mathcal{C}}_\\Lambda)$ and", "$|_{\\mathcal{C}_\\Lambda} : \\text{CofSet}(\\widehat{\\mathcal{C}}_\\Lambda) \\to", "\\text{CofSet}(\\mathcal{C}_\\Lambda)$ which we denote by the same symbols.", "As functors on the categories of cofibered sets, completion and restriction", "are adjoints in the usual 1-categorical sense: the same construction as in", "Remark \\ref{remark-completion-restriction-adjoint} defines a functorial", "bijection", "$$", "\\Mor_{\\mathcal{C}_\\Lambda}(G|_{\\mathcal{C}_\\Lambda}, F)", "\\longrightarrow", "\\Mor_{\\widehat{\\mathcal{C}}_\\Lambda}(G, \\widehat{F})", "$$", "for $F \\in \\Ob(\\text{CofSet}(\\mathcal{C}_\\Lambda))$ and", "$G \\in \\Ob(\\text{CofSet}(\\widehat{\\mathcal{C}}_\\Lambda))$.", "Again the map $\\widehat{F}|_{\\mathcal{C}_\\Lambda} \\to F$ is an", "isomorphism." ], "refs": [ "formal-defos-remarks-cofibered-groupoids", "formal-defos-remark-completion-restriction-adjoint" ], "ref_ids": [ 3585, 3550 ] }, { "id": 3552, "type": "other", "label": "formal-defos-remark-restrict-complete-continuous-functor", "categories": [ "formal-defos" ], "title": "formal-defos-remark-restrict-complete-continuous-functor", "contents": [ "Let $G : \\widehat{\\mathcal{C}}_\\Lambda \\to \\textit{Sets}$", "be a functor that commutes with limits.", "Then the map $G \\to \\widehat{G|_{\\mathcal{C}_\\Lambda}}$ described in", "Remark \\ref{remark-completion-restriction-adjoint}", "is an isomorphism. Indeed, if $S$ is an object of", "$\\widehat{\\mathcal{C}}_\\Lambda$, then we have canonical bijections", "$$", "\\widehat{G|_{\\mathcal{C}_\\Lambda}}(S) =", "\\lim_n G(S/\\mathfrak{m}_S^n) =", "G(\\lim_n S/\\mathfrak{m}_S^n) = G(S).", "$$", "In particular, if $R$ is an object of $\\widehat{\\mathcal{C}}_\\Lambda$ then", "$\\underline{R} = \\widehat{\\underline{R}|_{\\mathcal{C}_\\Lambda}}$ because", "the representable functor $\\underline{R}$ commutes with limits by definition", "of limits." ], "refs": [ "formal-defos-remark-completion-restriction-adjoint" ], "ref_ids": [ 3550 ] }, { "id": 3553, "type": "other", "label": "formal-defos-remark-formal-objects-yoneda", "categories": [ "formal-defos" ], "title": "formal-defos-remark-formal-objects-yoneda", "contents": [ "Let $R$ be an object of $\\widehat{\\mathcal{C}}_\\Lambda$. It defines a functor", "$\\underline{R}: \\widehat{\\mathcal{C}}_\\Lambda \\to \\textit{Sets}$", "as described in", "Remarks \\ref{remarks-cofibered-groupoids} (\\ref{item-definition-yoneda}).", "As usual we identify this functor with the", "associated cofibered set. If $\\mathcal{F}$ is a cofibered category over", "$\\mathcal{C}_\\Lambda$, then there is an equivalence of categories", "\\begin{equation}", "\\label{equation-formal-objects-maps}", "\\Mor_{\\mathcal{C}_\\Lambda}(", "\\underline{R}|_{\\mathcal{C}_\\Lambda}, \\mathcal{F})", "\\longrightarrow", "\\widehat{\\mathcal{F}}(R).", "\\end{equation}", "It is given by the composition", "$$", "\\Mor_{\\mathcal{C}_\\Lambda}(", "\\underline{R}|_{\\mathcal{C}_\\Lambda}, \\mathcal{F})", "\\xrightarrow{\\Phi}", "\\Mor_{\\widehat{\\mathcal{C}}_\\Lambda}(", "\\underline{R}, \\widehat{\\mathcal{F}})", "\\xrightarrow{\\sim}", "\\widehat{\\mathcal{F}}(R)", "$$", "where $\\Phi$ is as in", "Remark \\ref{remark-completion-restriction-adjoint}", "and the second equivalence comes from the 2-Yoneda lemma", "(the cofibered analogue of", "Categories, Lemma \\ref{categories-lemma-yoneda-2category}).", "Explicitly, the equivalence sends a morphism", "$\\varphi : \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$", "to the formal object", "$(R, \\varphi(R \\to R/\\mathfrak{m}_R^n), \\varphi(f_n))$ in", "$\\widehat{\\mathcal{F}}(R)$, where", "$f_n : R/\\mathfrak m_R^{n + 1} \\to R/\\mathfrak m_R^n$ is the projection.", "\\medskip\\noindent", "Assume a choice of pushforwards for $\\mathcal{F}$ has been made.", "Given any $\\xi \\in \\Ob(\\widehat{\\mathcal{F}}(R))$ we construct", "an explicit", "$\\underline{\\xi} : \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$", "which maps to $\\xi$ under (\\ref{equation-formal-objects-maps}).", "Namely, say $\\xi = (R, \\xi_n, f_n)$. An object $\\alpha$ in", "$\\underline{R}|_{\\mathcal{C}_\\Lambda}$ is the same thing as a morphism", "$\\alpha : R \\to A$ of $\\widehat{\\mathcal{C}}_\\Lambda$ with $A$", "Artinian. Let $m \\in \\mathbf{N}$ be minimal such that $\\mathfrak m_A^m = 0$.", "Then $\\alpha$ factors through a unique $\\alpha_m : R/\\mathfrak m_R^m \\to A$", "and we can set $\\underline{\\xi}(\\alpha) = \\alpha_{m, *}\\xi_m$.", "We omit the description of $\\underline{\\xi}$ on morphisms and we", "omit the proof that $\\underline{\\xi}$ maps to $\\xi$", "via (\\ref{equation-formal-objects-maps}).", "\\medskip\\noindent", "Assume a choice of pushforwards for $\\widehat{\\mathcal{F}}$ has been made.", "In this case the proof of", "Categories, Lemma \\ref{categories-lemma-yoneda-2category}", "gives an explicit quasi-inverse", "$$", "\\iota :", "\\widehat{\\mathcal{F}}(R) \\longrightarrow", "\\Mor_{\\widehat{\\mathcal{C}}_\\Lambda}(", "\\underline{R}, \\widehat{\\mathcal{F}})", "$$", "to the 2-Yoneda equivalence which takes $\\xi$ to the morphism", "$\\iota(\\xi) : \\underline{R} \\to \\widehat{\\mathcal{F}}$ sending", "$f \\in \\underline{R}(S) = \\Mor_{\\mathcal{C}_\\Lambda}(R, S)$", "to $f_*\\xi$. A quasi-inverse to (\\ref{equation-formal-objects-maps})", "is then", "$$", "\\widehat{\\mathcal{F}}(R)", "\\xrightarrow{\\iota}", "\\Mor_{\\widehat{\\mathcal{C}}_\\Lambda}(", "\\underline{R}, \\widehat{\\mathcal{F}})", "\\xrightarrow{\\Psi}", "\\Mor_{\\mathcal{C}_\\Lambda}(", "\\underline{R}|_{\\mathcal{C}_\\Lambda}, \\mathcal{F})", "$$", "where $\\Psi$ is as in", "Remark \\ref{remark-completion-restriction-adjoint}.", "Given $\\xi \\in \\Ob(\\widehat{\\mathcal{F}}(R))$ we have", "$\\Psi(\\iota(\\xi)) \\cong \\underline{\\xi}$ where $\\underline{\\xi}$", "is as in the previous paragraph, because both are mapped to $\\xi$", "under the equivalence of categories (\\ref{equation-formal-objects-maps}).", "Using $\\underline{R} = \\widehat{\\underline{R}|_{\\mathcal{C}_\\Lambda}}$", "(see Remark \\ref{remark-restrict-complete-continuous-functor})", "and unwinding the definitions of $\\Phi$ and $\\Psi$ we conclude that", "$\\iota(\\xi)$ is isomorphic to the completion of $\\underline{\\xi}$." ], "refs": [ "formal-defos-remarks-cofibered-groupoids", "formal-defos-remark-completion-restriction-adjoint", "categories-lemma-yoneda-2category", "categories-lemma-yoneda-2category", "formal-defos-remark-completion-restriction-adjoint", "formal-defos-remark-restrict-complete-continuous-functor" ], "ref_ids": [ 3585, 3550, 12318, 12318, 3550, 3552 ] }, { "id": 3554, "type": "other", "label": "formal-defos-remark-formal-objects-yoneda-map", "categories": [ "formal-defos" ], "title": "formal-defos-remark-formal-objects-yoneda-map", "contents": [ "Let $\\mathcal{F}$ be a category cofibred in groupoids over", "$\\mathcal{C}_\\Lambda$. Let $\\xi = (R, \\xi_n, f_n)$ and", "$\\eta = (S, \\eta_n, g_n)$ be formal objects of $\\mathcal{F}$.", "Let $a = (a_n) : \\xi \\to \\eta$ be a morphism of formal objects, i.e.,", "a morphism of $\\widehat{\\mathcal{F}}$. Let", "$f = \\widehat{p}(a) = a_0 : R \\to S$ be the projection of $a$ in", "$\\widehat{\\mathcal{C}}_\\Lambda$. Then we obtain a $2$-commutative", "diagram", "$$", "\\xymatrix{", "\\underline{R}|_{\\mathcal{C}_\\Lambda} \\ar[rd]_{\\underline{\\xi}} & &", "\\underline{S}|_{\\mathcal{C}_\\Lambda} \\ar[ll]^f \\ar[ld]^{\\underline{\\eta}} \\\\", "& \\mathcal{F}", "}", "$$", "where $\\underline{\\xi}$ and $\\underline{\\eta}$ are the morphisms", "constructed in", "Remark \\ref{remark-formal-objects-yoneda}.", "To see this let $\\alpha : S \\to A$ be an object of", "$\\underline{S}|_{\\mathcal{C}_\\Lambda}$ (see loc.\\ cit.).", "Let $m \\in \\mathbf{N}$ be minimal such that $\\mathfrak m_A^m = 0$.", "We get a commutative diagram", "$$", "\\xymatrix{", "R \\ar[d]^f \\ar[r] & R/\\mathfrak m_R^m \\ar[d]_{f_m} \\ar[rd]^{\\beta_m} \\\\", "S \\ar[r] & S/\\mathfrak m_S^m \\ar[r]^{\\alpha_m} & A", "}", "$$", "such that the bottom arrows compose to give $\\alpha$.", "Then $\\underline{\\eta}(\\alpha) = \\alpha_{m, *}\\eta_m$ and", "$\\underline{\\xi}(\\alpha \\circ f) = \\beta_{m, *}\\xi_m$. The morphism", "$a_m : \\xi_m \\to \\eta_m$ lies over $f_m$ hence we obtain a canonical", "morphism", "$$", "\\underline{\\xi}(\\alpha \\circ f) = \\beta_{m, *}\\xi_m", "\\longrightarrow", "\\underline{\\eta}(\\alpha) = \\alpha_{m, *}\\eta_m", "$$", "lying over $\\text{id}_A$ such that", "$$", "\\xymatrix{", "\\xi_m \\ar[r] \\ar[d]^{a_m} & \\beta_{m, *}\\xi_m \\ar[d] \\\\", "\\eta_m \\ar[r] & \\alpha_{m, *}\\eta_m", "}", "$$", "commutes by the axioms of a category cofibred in groupoids. This defines", "a transformation of functors $\\underline{\\xi} \\circ f \\to \\underline{\\eta}$", "which witnesses the 2-commutativity of the first diagram of this remark." ], "refs": [ "formal-defos-remark-formal-objects-yoneda" ], "ref_ids": [ 3553 ] }, { "id": 3555, "type": "other", "label": "formal-defos-remark-spell-out-formal-object", "categories": [ "formal-defos" ], "title": "formal-defos-remark-spell-out-formal-object", "contents": [ "According to Remark \\ref{remark-formal-objects-yoneda}, giving a formal object", "$\\xi$ of $\\mathcal{F}$ is equivalent to giving a prorepresentable functor", "$U : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ and a morphism", "$U \\to \\mathcal{F}$." ], "refs": [ "formal-defos-remark-formal-objects-yoneda" ], "ref_ids": [ 3553 ] }, { "id": 3556, "type": "other", "label": "formal-defos-remark-smoothness-2-categorical", "categories": [ "formal-defos" ], "title": "formal-defos-remark-smoothness-2-categorical", "contents": [ "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of categories", "cofibered in groupoids over $\\mathcal{C}_\\Lambda$. Let $B \\to A$ be a", "ring map in $\\mathcal{C}_\\Lambda$. Choices of pushforwards along $B", "\\to A$ for objects in the fiber categories $\\mathcal{F}(B)$ and", "$\\mathcal{G}(B)$ determine functors $\\mathcal{F}(B) \\to \\mathcal{F}(A)$", "and $\\mathcal{G}(B) \\to \\mathcal{G}(A)$ fitting into a $2$-commutative", "diagram", "$$", "\\xymatrix{", "\\mathcal{F}(B) \\ar[r]^{\\varphi} \\ar[d] & \\mathcal{G}(B) \\ar[d] \\\\", "\\mathcal{F}(A) \\ar[r]^{\\varphi} & \\mathcal{G}(A) .", "}", "$$", "Hence there is an induced functor $\\mathcal{F}(B) \\to \\mathcal{F}(A)", "\\times_{\\mathcal{G}(A)} \\mathcal{G}(B)$. Unwinding the definitions shows that", "$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is smooth if and only if this", "induced functor is essentially surjective whenever $B \\to A$ is", "surjective (or equivalently, by", "Lemma \\ref{lemma-smoothness-small-extensions},", "whenever $B \\to A$ is a small extension)." ], "refs": [ "formal-defos-lemma-smoothness-small-extensions" ], "ref_ids": [ 3431 ] }, { "id": 3557, "type": "other", "label": "formal-defos-remark-compare-smooth-schlessinger", "categories": [ "formal-defos" ], "title": "formal-defos-remark-compare-smooth-schlessinger", "contents": [ "The characterization of smooth morphisms in", "Remark \\ref{remark-smoothness-2-categorical}", "is analogous to Schlessinger's notion of", "a smooth morphism of functors, cf.\\ \\cite[Definition 2.2.]{Sch}. In", "fact, when $\\mathcal{F}$ and $\\mathcal{G}$ are cofibered in sets", "then our notion is equivalent to Schlessinger's. Namely, in this case", "let $F, G : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be the corresponding", "functors, see", "Remarks \\ref{remarks-cofibered-groupoids}", "(\\ref{item-convention-cofibered-sets}).", "Then $F \\to G$ is smooth if and only if for every surjection of rings", "$B \\to A$ in $\\mathcal{C}_\\Lambda$ the map $F(B) \\to F(A) \\times_{G(A)} G(B)$", "is surjective." ], "refs": [ "formal-defos-remark-smoothness-2-categorical", "formal-defos-remarks-cofibered-groupoids" ], "ref_ids": [ 3556, 3585 ] }, { "id": 3558, "type": "other", "label": "formal-defos-remark-smooth-to-iso-classes", "categories": [ "formal-defos" ], "title": "formal-defos-remark-smooth-to-iso-classes", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$. Then the morphism", "$\\mathcal{F} \\to \\overline{\\mathcal{F}}$ is smooth.", "Namely, suppose that $f : B \\to A$ is a ring map in $\\mathcal{C}_\\Lambda$.", "Let $x \\in \\Ob(\\mathcal{F}(A))$ and let", "$\\overline{y} \\in \\overline{\\mathcal{F}}(B)$", "be the isomorphism class of $y \\in \\Ob(\\mathcal{F}(B))$ such that", "$\\overline{f_*y} = \\overline{x}$. Then we simply take $x' = y$, the", "implied morphism $x' = y \\to x$ over $B \\to A$, and the equality", "$\\overline{x'} = \\overline{y}$ as the solution to", "the problem posed in Definition \\ref{definition-smooth-morphism}." ], "refs": [ "formal-defos-definition-smooth-morphism" ], "ref_ids": [ 3520 ] }, { "id": 3559, "type": "other", "label": "formal-defos-remark-versal-object", "categories": [ "formal-defos" ], "title": "formal-defos-remark-versal-object", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal", "C_\\Lambda$, and let $\\xi$ be a formal object of $\\mathcal{F}$. It follows", "from the definition of smoothness that versality of $\\xi$ is equivalent to the", "following condition: If", "$$", "\\xymatrix{", "& y \\ar[d] \\\\", "\\xi \\ar[r] & x", "}", "$$", "is a diagram in $\\widehat{\\mathcal{F}}$ such that $y \\to x$ lies over a", "surjective map $B \\to A$ of Artinian rings (we may assume it is a small", "extension), then there exists a morphism $\\xi \\to y$ such that", "$$", "\\xymatrix{", "& y \\ar[d] \\\\", "\\xi \\ar[r] \\ar[ur] & x", "}", "$$", "commutes. In particular, the condition that $\\xi$ be versal does not depend on", "the choices of pushforwards made in the construction of", "$\\underline{\\xi} : \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$ in", "Remark \\ref{remark-formal-objects-yoneda}." ], "refs": [ "formal-defos-remark-formal-objects-yoneda" ], "ref_ids": [ 3553 ] }, { "id": 3560, "type": "other", "label": "formal-defos-remark-smooth-on-top", "categories": [ "formal-defos" ], "title": "formal-defos-remark-smooth-on-top", "contents": [ "Suppose $\\mathcal{F}$ is a predeformation category admitting a smooth morphism", "$\\varphi : \\mathcal U \\to \\mathcal{F}$ from a predeformation category", "$\\mathcal{U}$. Then by", "Lemma \\ref{lemma-smooth-morphism-essentially-surjective}", "$\\varphi$ is essentially surjective, so by", "Lemma \\ref{lemma-smooth-properties}", "$p: \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ is smooth if and only if the", "composition $\\mathcal U \\xrightarrow{\\varphi} \\mathcal{F} \\xrightarrow{p}", "\\mathcal{C}_\\Lambda$ is smooth, i.e.\\ $\\mathcal{F}$ is smooth if and only if", "$\\mathcal{U}$ is smooth." ], "refs": [ "formal-defos-lemma-smooth-morphism-essentially-surjective", "formal-defos-lemma-smooth-properties" ], "ref_ids": [ 3434, 3433 ] }, { "id": 3561, "type": "other", "label": "formal-defos-remark-compare-S1-S2-schlessinger", "categories": [ "formal-defos" ], "title": "formal-defos-remark-compare-S1-S2-schlessinger", "contents": [ "When $\\mathcal{F}$ is cofibered in sets, conditions (S1) and (S2) are exactly", "conditions (H1) and (H2) from Schlessinger's paper \\cite{Sch}.", "Namely, for a functor $F: \\mathcal{C}_\\Lambda \\to", "\\textit{Sets}$, conditions (S1) and (S2) state:", "\\begin{enumerate}", "\\item [(S1)] If $A_1 \\to A$ and $A_2 \\to A$ are maps in", "$\\mathcal{C}_\\Lambda$ with $A_2 \\to A$ surjective, then the induced", "map $F(A_1 \\times_A A_2) \\to F(A_1) \\times_{F(A)} F(A_2)$ is", "surjective.", "\\item [(S2)] If $A \\to k$ is a map in $\\mathcal{C}_\\Lambda$, then the", "induced map", "$F(A \\times_k k[\\epsilon]) \\to F(A) \\times_{F(k)} F(k[\\epsilon])$", "is bijective.", "\\end{enumerate}", "The injectivity of the map", "$F(A \\times_k k[\\epsilon]) \\to F(A) \\times_{F(k)} F(k[\\epsilon])$", "comes from the second part of condition (S2) and the fact that morphisms", "are identities." ], "refs": [], "ref_ids": [] }, { "id": 3562, "type": "other", "label": "formal-defos-remark-linear-enriched-over-modules", "categories": [ "formal-defos" ], "title": "formal-defos-remark-linear-enriched-over-modules", "contents": [ "One can define the notion of an $R$-linearity for any functor between", "categories enriched over $\\text{Mod}_R$. We made the definition", "specifically for functors $L: \\text{Mod}^{fg}_R \\to \\text{Mod}_R$ and", "$L: \\text{Mod}_R \\to \\text{Mod}_R$", "because these are the cases that we have needed so far." ], "refs": [], "ref_ids": [] }, { "id": 3563, "type": "other", "label": "formal-defos-remark-linear-functor", "categories": [ "formal-defos" ], "title": "formal-defos-remark-linear-functor", "contents": [ "If $L: \\text{Mod}^{fg}_R \\to \\text{Mod}_R$ is an $R$-linear functor,", "then $L$ preserves finite products and sends the zero module to the zero", "module, see", "Homology, Lemma \\ref{homology-lemma-additive-additive}.", "On the other hand, if a functor $\\text{Mod}^{fg}_R \\to \\textit{Sets}$", "preserves finite products and sends the zero module to a one element set,", "then it has a unique lift to a $R$-linear functor, see", "Lemma \\ref{lemma-linear-functor}." ], "refs": [ "homology-lemma-additive-additive", "formal-defos-lemma-linear-functor" ], "ref_ids": [ 12010, 3445 ] }, { "id": 3564, "type": "other", "label": "formal-defos-remark-tangent-space-cofibered-groupoid", "categories": [ "formal-defos" ], "title": "formal-defos-remark-tangent-space-cofibered-groupoid", "contents": [ "We can globalize the notions of tangent space and differential to arbitrary", "categories cofibered in groupoids as follows. Let $\\mathcal{F}$ be a category", "cofibered in groupoids over $\\mathcal{C}_\\Lambda$, and let", "$x \\in \\Ob(\\mathcal{F}(k))$. As in", "Remark \\ref{remark-localize-cofibered-groupoid},", "we get a predeformation category $\\mathcal{F}_x$. We define", "$$", "T_x\\mathcal{F} = T\\mathcal{F}_x", "$$", "to be the {\\it tangent space of $\\mathcal{F}$ at $x$}. If", "$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of categories cofibered", "in groupoids over $\\mathcal{C}_\\Lambda$ and $x \\in \\Ob(\\mathcal{F}(k))$,", "then there is an induced morphism", "$\\varphi_x: \\mathcal{F}_x \\to \\mathcal{G}_{\\varphi(x)}$. We define the", "{\\it differential", "$d_x \\varphi : T_x \\mathcal{F} \\to T_{\\varphi(x)} \\mathcal{G}$", "of $\\varphi$ at $x$} to be the map", "$d \\varphi_x: T \\mathcal{F}_x \\to T \\mathcal{G}_{\\varphi(x)}$.", "If both $\\mathcal{F}$ and $\\mathcal{G}$ satisfy (S2) then", "all of these tangent spaces have a natural $k$-vector space structure", "and all the differentials", "$d_x \\varphi : T_x \\mathcal{F} \\to T_{\\varphi(x)} \\mathcal{G}$", "are $k$-linear (use", "Lemmas \\ref{lemma-S1-S2-localize} and \\ref{lemma-k-linear-differential})." ], "refs": [ "formal-defos-remark-localize-cofibered-groupoid", "formal-defos-lemma-S1-S2-localize", "formal-defos-lemma-k-linear-differential" ], "ref_ids": [ 3545, 3442, 3453 ] }, { "id": 3565, "type": "other", "label": "formal-defos-remark-compare-schlessinger-H3-pre", "categories": [ "formal-defos" ], "title": "formal-defos-remark-compare-schlessinger-H3-pre", "contents": [ "Let $F : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a predeformation functor", "satisfying (S1) and (S2). The condition $\\dim_k TF < \\infty$", "is precisely condition (H3) from Schlessinger's paper.", "Recall that (S1) and (S2) correspond to conditions (H1) and (H2), see", "Remark \\ref{remark-compare-S1-S2-schlessinger}.", "Thus Lemma \\ref{lemma-versal-object-existence} tells us", "$$", "(H1) + (H2) + (H3)", "\\Rightarrow", "\\text{ there exists a versal formal object}", "$$", "for predeformation functors. We will make the link with hulls in", "Remark \\ref{remark-compare-schlessinger-H3}." ], "refs": [ "formal-defos-remark-compare-S1-S2-schlessinger", "formal-defos-lemma-versal-object-existence", "formal-defos-remark-compare-schlessinger-H3" ], "ref_ids": [ 3561, 3458, 3566 ] }, { "id": 3566, "type": "other", "label": "formal-defos-remark-compare-schlessinger-H3", "categories": [ "formal-defos" ], "title": "formal-defos-remark-compare-schlessinger-H3", "contents": [ "Let $F : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a predeformation functor", "satisfying (S1) and (S2) and $\\dim_k TF < \\infty$.", "Recall that these conditions correspond to the conditions", "(H1), (H2), and (H3) from Schlessinger's paper, see", "Remark \\ref{remark-compare-schlessinger-H3-pre}.", "Now, in the classical case (or if $k' \\subset k$ is separable)", "following Schlessinger we introduce the notion of a hull: a {\\it hull}", "is a versal formal object $\\xi \\in \\widehat{F}(R)$ such that", "$d\\underline{\\xi} : T\\underline{R}|_{\\mathcal{C}_\\Lambda} \\to TF$", "is an isomorphism, i.e., (\\ref{equation-bijective}) holds.", "Thus Theorem \\ref{theorem-miniversal-object-existence} tells us", "$$", "(H1) + (H2) + (H3)", "\\Rightarrow", "\\text{ there exists a hull}", "$$", "in the classical case. In other words, our theorem", "recovers Schlessinger's theorem on the existence of hulls." ], "refs": [ "formal-defos-remark-compare-schlessinger-H3-pre", "formal-defos-theorem-miniversal-object-existence" ], "ref_ids": [ 3565, 3410 ] }, { "id": 3567, "type": "other", "label": "formal-defos-remark-compose-minimal-into-iso-classes", "categories": [ "formal-defos" ], "title": "formal-defos-remark-compose-minimal-into-iso-classes", "contents": [ "Let $\\mathcal{F}$ be a predeformation category. Recall that", "$\\mathcal{F} \\to \\overline{\\mathcal{F}}$ is smooth, see", "Remark \\ref{remark-smooth-to-iso-classes}. Hence if", "$\\xi \\in \\widehat{\\mathcal{F}}(R)$ is a versal formal object,", "then the composition", "$$", "\\underline{R}|_{\\mathcal{C}_\\Lambda} \\longrightarrow", "\\mathcal{F} \\longrightarrow \\overline{\\mathcal{F}}", "$$", "is smooth (Lemma \\ref{lemma-smooth-properties}) and we conclude", "that the image $\\overline{\\xi}$ of $\\xi$ in", "$\\overline{\\mathcal{F}}$ is a versal formal object.", "If (\\ref{equation-bijective}) holds, then $\\overline{\\xi}$", "induces an isomorphism", "$T\\underline{R}|_{\\mathcal{C}_\\Lambda} \\to T\\overline{\\mathcal{F}}$", "because $\\mathcal{F} \\to \\overline{\\mathcal{F}}$ identifies tangent spaces.", "Hence in this case $\\overline{\\xi}$", "is a hull for $\\overline{\\mathcal{F}}$, see", "Remark \\ref{remark-compare-schlessinger-H3}.", "By Theorem \\ref{theorem-miniversal-object-existence}", "we can always find such a $\\xi$ if $k' \\subset k$ is separable and", "$\\mathcal{F}$ is a predeformation category satisfying", "(S1), (S2), and $\\dim_k T\\mathcal{F} < \\infty$." ], "refs": [ "formal-defos-remark-smooth-to-iso-classes", "formal-defos-lemma-smooth-properties", "formal-defos-remark-compare-schlessinger-H3", "formal-defos-theorem-miniversal-object-existence" ], "ref_ids": [ 3558, 3433, 3566, 3410 ] }, { "id": 3568, "type": "other", "label": "formal-defos-remark-compare-schlessinger-H4", "categories": [ "formal-defos" ], "title": "formal-defos-remark-compare-schlessinger-H4", "contents": [ "When $\\mathcal{F}$ is cofibered in sets, condition (RS) is exactly condition", "(H4) from Schlessinger's paper \\cite[Theorem 2.11]{Sch}. Namely, for", "a functor $F: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$, condition", "(RS) states: If $A_1 \\to A$ and $A_2 \\to A$ are maps in", "$\\mathcal{C}_\\Lambda$ with $A_2 \\to A$ surjective, then the induced", "map $F(A_1 \\times_A A_2) \\to F(A_1) \\times_{F(A)} F(A_2)$ is", "bijective." ], "refs": [], "ref_ids": [] }, { "id": 3569, "type": "other", "label": "formal-defos-remark-deformation-functor", "categories": [ "formal-defos" ], "title": "formal-defos-remark-deformation-functor", "contents": [ "We say that a functor $F: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$", "is a {\\it deformation functor} if the associated cofibered set is a", "deformation category, i.e.\\ if $F(k)$ is a one element set and $F$ satisfies", "(RS). If $\\mathcal{F}$ is a deformation category, then", "$\\overline{\\mathcal{F}}$", "is a predeformation functor but not necessarily a deformation functor, as", "Lemma \\ref{lemma-RS-associated-functor} shows." ], "refs": [ "formal-defos-lemma-RS-associated-functor" ], "ref_ids": [ 3470 ] }, { "id": 3570, "type": "other", "label": "formal-defos-remark-omit-arrow", "categories": [ "formal-defos" ], "title": "formal-defos-remark-omit-arrow", "contents": [ "When the map $f: A' \\to A$ is clear from the context, we may write", "$\\textit{Lift}(x, A')$ and $\\text{Lift}(x, A')$ in place of", "$\\textit{Lift}(x, f)$ and $\\text{Lift}(x, f)$." ], "refs": [], "ref_ids": [] }, { "id": 3571, "type": "other", "label": "formal-defos-remark-tangent-space-lifting", "categories": [ "formal-defos" ], "title": "formal-defos-remark-tangent-space-lifting", "contents": [ "Let $\\mathcal{F}$ be a category cofibred in groupoids over", "$\\mathcal{C}_\\Lambda$. Let $x_0 \\in \\Ob(\\mathcal{F}(k))$.", "Let $V$ be a finite dimensional vector space.", "Then $\\text{Lift}(x_0, k[V])$ is the set of isomorphism classes", "of $\\mathcal{F}_{x_0}(k[V])$ where $\\mathcal{F}_{x_0}$ is the", "predeformation category of objects in $\\mathcal{F}$ lying over", "$x_0$, see", "Remark \\ref{remark-localize-cofibered-groupoid}.", "Hence if $\\mathcal{F}$ satisfies (S2), then so does", "$\\mathcal{F}_{x_0}$ (see", "Lemma \\ref{lemma-S1-S2-localize})", "and by", "Lemma \\ref{lemma-tangent-space-vector-space}", "we see that", "$$", "\\text{Lift}(x_0, k[V]) = T\\mathcal{F}_{x_0} \\otimes_k V", "$$", "as $k$-vector spaces." ], "refs": [ "formal-defos-remark-localize-cofibered-groupoid", "formal-defos-lemma-S1-S2-localize", "formal-defos-lemma-tangent-space-vector-space" ], "ref_ids": [ 3545, 3442, 3452 ] }, { "id": 3572, "type": "other", "label": "formal-defos-remark-lift-bijections", "categories": [ "formal-defos" ], "title": "formal-defos-remark-lift-bijections", "contents": [ "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal", "C_\\Lambda$ satisfying (RS). Let", "$$", "\\xymatrix{", "A_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d] \\\\", "A_1 \\ar[r] & A", "}", "$$", "be a fibre square in $\\mathcal{C}_\\Lambda$ such that either", "$A_1 \\to A$ or $A_2 \\to A$ is surjective. Let", "$x \\in \\Ob(\\mathcal{F}(A))$. Given", "lifts $x_1 \\to x$ and $x_2 \\to x$ of $x$ to $A_1$ and $A_2$, we get by", "(RS) a lift $x_1 \\times_x x_2 \\to x$ of $x$ to $A_1 \\times_A A_2$.", "Conversely, by", "Lemma \\ref{lemma-RS-fiber-square}", "any lift of $x$ to $A_1 \\times_A A_2$ is of this form.", "Hence a bijection", "$$", "\\text{Lift}(x, A_1) \\times \\text{Lift}(x, A_2)", "\\longrightarrow", "\\text{Lift}(x, A_1 \\times_A A_2).", "$$", "Similarly, if $x_1 \\to x$ is a fixed lifting of $x$ to $A_1$, then", "there is a bijection", "$$", "\\text{Lift}(x_1, A_1 \\times_A A_2)", "\\longrightarrow", "\\text{Lift}(x, A_2).", "$$", "Now let", "$$", "\\xymatrix{", "A_1' \\times_A A_2 \\ar[r] \\ar[d] & A_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2", "\\ar[d] \\\\", "A_1' \\ar[r] & A_1 \\ar[r] & A", "}", "$$", "be a composition of fibre squares in $\\mathcal{C}_\\Lambda$ with", "both $A'_1 \\to A_1$ and $A_1 \\to A$ surjective. Let $x_1 \\to x$ be a morphism", "lying over $A_1 \\to A$. Then by the above we have bijections", "\\begin{align*}", "\\text{Lift}(x_1, A_1' \\times_A A_2)", "& = \\text{Lift}(x_1, A_1') \\times \\text{Lift}(x_1, A_1 \\times_A A_2) \\\\", "& = \\text{Lift}(x_1, A_1') \\times \\text{Lift}(x, A_2).", "\\end{align*}" ], "refs": [ "formal-defos-lemma-RS-fiber-square" ], "ref_ids": [ 3466 ] }, { "id": 3573, "type": "other", "label": "formal-defos-remark-free-transitive-action-functorial", "categories": [ "formal-defos" ], "title": "formal-defos-remark-free-transitive-action-functorial", "contents": [ "The action of Lemma \\ref{lemma-free-transitive-action} is functorial.", "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of deformation", "categories. Let $A' \\to A$ be a surjective ring map whose kernel $I$", "is annihilated by $\\mathfrak m_{A'}$. Let", "$x \\in \\Ob(\\mathcal{F}(A))$.", "In this situation $\\varphi$ induces the vertical arrows", "in the following commutative diagram", "$$", "\\xymatrix{", "\\text{Lift}(x, A') \\times (T\\mathcal{F} \\otimes_k I)", "\\ar[d]_{(\\varphi, d\\varphi \\otimes \\text{id}_I)} \\ar[r] &", "\\text{Lift}(x, A') \\ar[d]^\\varphi \\\\", "\\text{Lift}(\\varphi(x), A') \\times (T\\mathcal{G} \\otimes_k I) \\ar[r] &", "\\text{Lift}(\\varphi(x), A')", "}", "$$", "The commutativity follows as each of the maps", "(\\ref{equation-two}), (\\ref{equation-one}), and (\\ref{equation-three})", "of the proof of", "Lemma \\ref{lemma-free-transitive-action}", "gives rise to a similar commutative diagram." ], "refs": [ "formal-defos-lemma-free-transitive-action", "formal-defos-lemma-free-transitive-action" ], "ref_ids": [ 3473, 3473 ] }, { "id": 3574, "type": "other", "label": "formal-defos-remark-choice-pushforward-immaterial-infinitesimal-aut", "categories": [ "formal-defos" ], "title": "formal-defos-remark-choice-pushforward-immaterial-infinitesimal-aut", "contents": [ "Up to canonical isomorphism $\\text{Inf}_{x_0}(\\mathcal{F})$", "does not depend on the choice of pushforward $x_0 \\to x_0'$", "because any two pushforwards are canonically isomorphic.", "Moreover, if $y_0 \\in \\mathcal{F}(k)$ and $x_0 \\cong y_0$ in", "$\\mathcal{F}(k)$, then", "$\\text{Inf}_{x_0}(\\mathcal{F}) \\cong \\text{Inf}_{y_0}(\\mathcal{F})$", "where the isomorphism depends (only) on the choice of an isomorphism", "$x_0 \\to y_0$. In particular, $\\text{Aut}_k(x_0)$", "acts on $\\text{Inf}_{x_0}(\\mathcal{F})$." ], "refs": [], "ref_ids": [] }, { "id": 3575, "type": "other", "label": "formal-defos-remark-trivial-aut-point", "categories": [ "formal-defos" ], "title": "formal-defos-remark-trivial-aut-point", "contents": [ "Assume $\\mathcal{F}$ is a predeformation category. Then", "\\begin{enumerate}", "\\item for $x_0 \\in \\Ob(\\mathcal{F}(k))$ the automorphism group", "$\\text{Aut}_k(x_0)$ is trivial and hence", "$\\text{Inf}_{x_0}(\\mathcal{F}) = \\text{Aut}_{k[\\epsilon]}(x'_0)$, and", "\\item for $x_0, y_0 \\in \\Ob(\\mathcal{F}(k))$ there is a unique", "isomorphism $x_0 \\to y_0$ and hence a canonical identification", "$\\text{Inf}_{x_0}(\\mathcal{F}) = \\text{Inf}_{y_0}(\\mathcal{F})$.", "\\end{enumerate}", "Since $\\mathcal{F}(k)$ is nonempty, choosing $x_0 \\in \\Ob(\\mathcal{F}(k))$", "and setting", "$$", "\\text{Inf}(\\mathcal{F}) = \\text{Inf}_{x_0}(\\mathcal{F})", "$$", "we get a well defined {\\it group of infinitesimal automorphisms", "of $\\mathcal{F}$}. With this notation we have", "$\\text{Inf}(\\mathcal{F}_{x_0}) = \\text{Inf}_{x_0}(\\mathcal{F})$.", "Please compare with the equality", "$T\\mathcal{F}_{x_0} = T_{x_0}\\mathcal{F}$", "in Remark \\ref{remark-tangent-space-cofibered-groupoid}." ], "refs": [ "formal-defos-remark-tangent-space-cofibered-groupoid" ], "ref_ids": [ 3564 ] }, { "id": 3576, "type": "other", "label": "formal-defos-remark-infaut-lifting-equalities", "categories": [ "formal-defos" ], "title": "formal-defos-remark-infaut-lifting-equalities", "contents": [ "We point out some basic relationships between infinitesimal automorphism", "groups, liftings, and tangent spaces to automorphism functors. Let", "$\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$.", "Let $x' \\to x$ be a morphism lying over a ring map $A' \\to A$.", "Then from the definitions we have an equality", "$$", "\\text{Inf}(x'/x) = \\text{Lift}(\\text{id}_x, A')", "$$", "where the liftings are of $\\text{id}_x$ as an object of", "$\\mathit{Aut}(x')$. If $x_0 \\in \\Ob(\\mathcal{F}(k))$ and $x'_0$", "is the pushforward to $\\mathcal{F}(k[\\epsilon])$, then applying this to", "$x'_0 \\to x_0$ we get", "$$", "\\text{Inf}_{x_0}(\\mathcal{F}) =", "\\text{Lift}(\\text{id}_{x_0}, k[\\epsilon]) =", "T_{\\text{id}_{x_0}} \\mathit{Aut}(x_0),", "$$", "the last equality following directly from the definitions." ], "refs": [], "ref_ids": [] }, { "id": 3577, "type": "other", "label": "formal-defos-remark-confusion-groupoids-in-functors", "categories": [ "formal-defos" ], "title": "formal-defos-remark-confusion-groupoids-in-functors", "contents": [ "A groupoid in functors on $\\mathcal{C}$ amounts to the data of a functor", "$\\mathcal{C} \\to \\textit{Groupoids}$, and a morphism of groupoids", "in functors on $\\mathcal{C}$ amounts to a morphism of the corresponding", "functors", "$\\mathcal{C} \\to \\textit{Groupoids}$ (where", "$\\textit{Groupoids}$ is regarded as a 1-category). However, for our", "purposes it is more convenient to use the terminology of groupoids in functors.", "In fact, thinking of a groupoid in functors as the corresponding functor", "$\\mathcal{C} \\to \\textit{Groupoids}$, or equivalently as the", "category cofibered in groupoids associated to that functor, can lead to", "confusion (Remark \\ref{remark-smooth-groupoid-in-functors-warning})." ], "refs": [ "formal-defos-remark-smooth-groupoid-in-functors-warning" ], "ref_ids": [ 3583 ] }, { "id": 3578, "type": "other", "label": "formal-defos-remark-identity-inverse", "categories": [ "formal-defos" ], "title": "formal-defos-remark-identity-inverse", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\\mathcal{C}$.", "There are unique morphisms $e : U \\to R$ and $i : R \\to R$ such that", "for every object $T$ of $\\mathcal{C}$, $e: U(T) \\to R(T)$ sends", "$x \\in U(T)$ to the identity morphism on $x$ and $i: R(T) \\to R(T)$ sends", "$a \\in U(T)$ to the inverse of $a$ in the groupoid category", "$(U(T), R(T), s, t, c)$. We will sometimes refer to $s$, $t$, $c$, $e$,", "and $i$ as ``source'', ``target'', ``composition'', ``identity'', and", "``inverse''." ], "refs": [], "ref_ids": [] }, { "id": 3579, "type": "other", "label": "formal-defos-remark-reason-existence-coproduct", "categories": [ "formal-defos" ], "title": "formal-defos-remark-reason-existence-coproduct", "contents": [ "Hence a representable groupoid in functors on $\\mathcal{C}$ is given by", "objects $U$ and $R$ of $\\mathcal{C}$ and morphisms $s, t : U \\to R$ and", "$c : R \\to R \\amalg_{s, U, t} R$ such that", "$(\\underline{U}, \\underline{R}, s, t, c)$ satisfies the condition of", "Definition \\ref{definition-groupoid-in-functors}. The reason for requiring", "the existence of the pushout $R \\amalg_{s, U, t} R$ is so that the composition", "morphism $c$ is defined at the level of morphisms in $\\mathcal{C}$.", "This requirement will always be satisfied below when we consider", "representable groupoids in functors on", "$\\widehat{\\mathcal{C}}_\\Lambda$, since by", "Lemma \\ref{lemma-CLambdahat-pushouts}", "the category $\\widehat{\\mathcal{C}}_\\Lambda$ admits pushouts." ], "refs": [ "formal-defos-definition-groupoid-in-functors", "formal-defos-lemma-CLambdahat-pushouts" ], "ref_ids": [ 3535, 3422 ] }, { "id": 3580, "type": "other", "label": "formal-defos-remark-simplify-terminology", "categories": [ "formal-defos" ], "title": "formal-defos-remark-simplify-terminology", "contents": [ "We will say ``{\\it let $(\\underline{U}, \\underline{R}, s, t, c)$ be a", "groupoid in functors on $\\mathcal{C}$}'' to mean that we have", "a representable groupoid in functors. Thus this means that", "$U$ and $R$ are objects of $\\mathcal{C}$, there are morphisms", "$s, t : U \\to R$, the pushout $R \\amalg_{s, U, t} R$ exists,", "there is a morphism $c : R \\to R \\amalg_{s, U, t} R$, and", "$(\\underline{U}, \\underline{R}, s, t, c)$ is a", "groupoid in functors on $\\mathcal{C}$." ], "refs": [], "ref_ids": [] }, { "id": 3581, "type": "other", "label": "formal-defos-remark-notation-restriction", "categories": [ "formal-defos" ], "title": "formal-defos-remark-notation-restriction", "contents": [ "In the situation of Definition", "\\ref{definition-restricting-groupoids-in-functors}, we often denote", "$s|_{\\mathcal{C}'}, t|_{\\mathcal{C}'}, c|_{\\mathcal{C}'}$ simply by $s, t, c$." ], "refs": [ "formal-defos-definition-restricting-groupoids-in-functors" ], "ref_ids": [ 3537 ] }, { "id": 3582, "type": "other", "label": "formal-defos-remark-groupoid-in-functors-complete-restrict", "categories": [ "formal-defos" ], "title": "formal-defos-remark-groupoid-in-functors-complete-restrict", "contents": [ "Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}_\\Lambda$.", "Then there is a canonical isomorphism", "$(U, R, s, t, c)^{\\wedge}|_{\\mathcal{C}_\\Lambda} \\cong (U, R, s, t, c)$, see", "Remark \\ref{remark-restrict-completion}.", "On the other hand, let $(U, R, s, t, c)$ be a groupoid in functors on", "$\\widehat{\\mathcal{C}}_\\Lambda$ such that", "$U, R : \\widehat{\\mathcal{C}}_\\Lambda \\to \\textit{Sets}$", "both commute with limits, e.g.\\ if $U, R$ are representable.", "Then there is a canonical isomorphism", "$((U, R, s, t, c)|_{\\mathcal{C}_\\Lambda})^{\\wedge} \\cong (U, R, s, t, c)$.", "This follows from", "Remark \\ref{remark-restrict-complete-continuous-functor}." ], "refs": [ "formal-defos-remark-restrict-completion", "formal-defos-remark-restrict-complete-continuous-functor" ], "ref_ids": [ 3548, 3552 ] }, { "id": 3583, "type": "other", "label": "formal-defos-remark-smooth-groupoid-in-functors-warning", "categories": [ "formal-defos" ], "title": "formal-defos-remark-smooth-groupoid-in-functors-warning", "contents": [ "We note that this terminology is potentially confusing:", "if $(U, R, s, t, c)$ is a smooth groupoid in functors, then the quotient", "$[U/R]$ need not be a smooth category cofibred in groupoids as defined in", "Definition \\ref{definition-cofibered-groupoid-projection-smooth}.", "However smoothness of $(U, R, s, t, c)$ does imply (in fact is equivalent to)", "smoothness of the quotient morphism $U \\to [U/R]$ as we shall", "see in", "Lemma \\ref{lemma-smooth-quotient-morphism}." ], "refs": [ "formal-defos-definition-cofibered-groupoid-projection-smooth", "formal-defos-lemma-smooth-quotient-morphism" ], "ref_ids": [ 3522, 3487 ] }, { "id": 3584, "type": "other", "label": "formal-defos-remark-smooth-power-series-prorepresentable-smooth-groupoid-in-functors", "categories": [ "formal-defos" ], "title": "formal-defos-remark-smooth-power-series-prorepresentable-smooth-groupoid-in-functors", "contents": [ "Let $(\\underline{R_0}, \\underline{R_1}, s, t, c)|_{\\mathcal{C}_\\Lambda}$", "be a prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$.", "Then $(\\underline{R_0}, \\underline{R_1}, s, t, c)|_{\\mathcal{C}_\\Lambda}$", "is smooth if and only if $R_1$ is a power series over $R_0$ via both $s$", "and $t$. This follows from", "Lemma \\ref{lemma-smooth-morphism-power-series}." ], "refs": [ "formal-defos-lemma-smooth-morphism-power-series" ], "ref_ids": [ 3432 ] }, { "id": 3585, "type": "other", "label": "formal-defos-remarks-cofibered-groupoids", "categories": [ "formal-defos" ], "title": "formal-defos-remarks-cofibered-groupoids", "contents": [ "Everything about categories fibered in groupoids translates directly to the", "cofibered setting. The following remarks are meant to fix notation.", "Let $\\mathcal{C}$ be a category.", "\\begin{enumerate}", "\\item We often omit the functor $p: \\mathcal{F} \\to \\mathcal{C}$ from the", "notation.", "\\item The fiber category over an object $U$ in $\\mathcal{C}$ is denoted by", "$\\mathcal{F}(U)$. Its objects are those of $\\mathcal{F}$ lying over $U$ and its", "morphisms are those of $\\mathcal{F}$ lying over $\\text{id}_U$.", "If $x, y$ are objects of $\\mathcal{F}(U)$, we sometimes write", "$\\Mor_U(x, y)$ for $\\Mor_{\\mathcal{F}(U)}(x, y)$.", "\\item The fibre categories $\\mathcal{F}(U)$ are groupoids, see", "Categories, Lemma \\ref{categories-lemma-fibred-groupoids}.", "Hence the morphisms in $\\mathcal{F}(U)$ are all isomorphisms.", "We sometimes write $\\text{Aut}_U(x)$ for $\\Mor_{\\mathcal{F}(U)}(x, x)$.", "\\item", "\\label{item-pushforward}", "Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}$, let $f: U \\to V$ be a morphism in $\\mathcal{C}$, and", "let $x \\in \\Ob(\\mathcal{F}(U))$.", "A {\\it pushforward} of $x$ along $f$ is a morphism", "$x \\to y$ of $\\mathcal{F}$ lying over $f$. A pushforward", "is unique up to unique isomorphism (see the discussion following", "Categories, Definition \\ref{categories-definition-cartesian-over-C}).", "We sometimes write $x \\to f_*x$ for ``the'' pushforward of $x$", "along $f$.", "\\item A {\\it choice of pushforwards for $\\mathcal{F}$} is the choice of", "a pushforward of $x$ along $f$ for every pair $(x, f)$ as above. We can make", "such a choice of pushforwards for $\\mathcal{F}$ by the axiom of choice.", "\\item Let $\\mathcal{F}$ be a category cofibered in groupoids over", "$\\mathcal{C}$. Given a choice of pushforwards for $\\mathcal{F}$, there", "is an associated pseudo-functor $\\mathcal{C} \\to \\textit{Groupoids}$.", "We will never use this construction so we give no details.", "\\item", "\\label{item-cofibered-morphism}", "A morphism of categories cofibered in groupoids over $\\mathcal{C}$ is a", "functor commuting with the projections to $\\mathcal{C}$. If $\\mathcal{F}$", "and $\\mathcal{F}'$ are categories cofibered in groupoids over", "$\\mathcal{C}$, we denote the morphisms from $\\mathcal{F}$ to $\\mathcal{F}'$", "by $\\Mor_\\mathcal{C}(\\mathcal{F}, \\mathcal{F}')$.", "\\item", "\\label{item-definition-cofibered-groupoids-2-category}", "Categories cofibered in groupoids form a $(2, 1)$-category", "$\\text{Cof}(\\mathcal{C})$. Its 1-morphisms are the morphisms described in", "(\\ref{item-cofibered-morphism}). If $p : \\mathcal{F} \\to C$ and", "$p': \\mathcal{F}' \\to \\mathcal{C}$ are categories cofibered in groupoids", "and $\\varphi, \\psi : \\mathcal{F} \\to \\mathcal{F}'$ are $1$-morphisms, then", "a 2-morphism $t : \\varphi \\to \\psi$ is a morphism of functors such that", "$p'(t_x) = \\text{id}_{p(x)}$ for all $x \\in \\Ob(\\mathcal{F})$.", "\\item", "\\label{item-construction-associated-cofibered-groupoid}", "Let $F : \\mathcal{C} \\to \\textit{Groupoids}$ be a functor. There", "is a category cofibered in groupoids $\\mathcal{F} \\to \\mathcal{C}$", "associated to $F$ as follows. An object of $\\mathcal{F}$ is a pair $(U, x)$", "where $U \\in \\Ob(\\mathcal{C})$ and $x \\in \\Ob(F(U))$. A", "morphism $(U, x) \\to (V, y)$ is a pair $(f, a)$ where", "$f \\in \\Mor_\\mathcal{C}(U, V)$ and", "$a \\in \\Mor_{F(V)}(F(f)(x), y)$.", "The functor $\\mathcal{F} \\to \\mathcal{C}$ sends $(U, x)$ to $U$. See", "Categories, Section \\ref{categories-section-presheaves-groupoids}.", "\\item", "\\label{item-associated-functor-isomorphism-classes}", "Let $\\mathcal{F}$ be cofibered in groupoids over $\\mathcal{C}$.", "For $U \\in \\Ob(\\mathcal{C})$ set $\\overline{\\mathcal{F}}(U)$ equal to", "the set of isomorphisms classes of the category $\\mathcal{F}(U)$.", "If $f : U \\to V$ is a morphism of $\\mathcal{C}$, then we obtain a", "map of sets $\\overline{\\mathcal{F}}(U) \\to \\overline{\\mathcal{F}}(V)$ by", "mapping the isomorphism class of $x$ to the isomorphism class of a pushforward", "$f_*x$ of $x$ see (\\ref{item-pushforward}). Then", "$\\overline{\\mathcal{F}} : \\mathcal{C} \\to \\textit{Sets}$ is a", "functor. Similarly, if $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a", "morphism of cofibered categories, we denote by", "$\\overline{\\varphi}: \\overline{\\mathcal{F}} \\to \\overline{\\mathcal{G}}$", "the associated morphism of functors.", "\\item", "\\label{item-convention-cofibered-sets}", "Let $F: \\mathcal{C} \\to \\textit{Sets}$ be a functor. We can think of a", "set as a discrete category, i.e., as a groupoid with only identity morphisms.", "Then the construction (\\ref{item-construction-associated-cofibered-groupoid})", "associates to $F$ a category cofibered in sets. This defines a fully", "faithful embedding of the category of functors $\\mathcal{C} \\to \\textit{Sets}$", "to the category of categories cofibered in groupoids over $\\mathcal{C}$.", "We identify the category of functors with its image under this embedding.", "Hence if $F : \\mathcal{C} \\to \\textit{Sets}$ is a functor, we denote the", "associated category cofibered in sets also by $F$; and if", "$\\varphi : F \\to G$ is a morphism of functors, we denote still by $\\varphi$", "the corresponding morphism of categories cofibered in sets, and vice-versa.", "See Categories, Section \\ref{categories-section-fibred-in-sets}.", "\\item", "\\label{item-definition-yoneda}", "Let $U$ be an object of $\\mathcal{C}$. We write $\\underline{U}$ for the", "functor", "$\\Mor_\\mathcal{C}(U, -): \\mathcal{C} \\to", "\\textit{Sets}$. This defines a fully faithful embedding of $\\mathcal", "C^{opp}$ into the category of functors $\\mathcal{C} \\to", "\\textit{Sets}$. Hence, if $f : U \\to V$ is a morphism, we are", "justified in denoting still by $f$ the induced morphism $\\underline{V}", "\\to \\underline{U}$, and vice-versa.", "\\item", "\\label{item-fibre-product}", "Fiber products of categories cofibered in groupoids: If $\\mathcal{F}", "\\to \\mathcal{H}$ and $\\mathcal{G} \\to \\mathcal{H}$ are morphisms", "of categories cofibered in groupoids over $\\mathcal{C}_\\Lambda$, then a", "construction of their 2-fiber product is given by the construction for their", "2-fiber product as categories over $\\mathcal{C}_\\Lambda$, as described in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.", "\\item", "\\label{item-product}", "Products of categories cofibered in groupoids: If $\\mathcal{F}$ and", "$\\mathcal{G}$ are categories cofibered in groupoids over", "$\\mathcal{C}_\\Lambda$ then their product is defined to be the $2$-fiber product", "$\\mathcal{F} \\times_{\\mathcal{C}_\\Lambda} \\mathcal{G}$ as described in", "Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.", "\\item", "\\label{item-definition-restricting-base-category}", "Restricting the base category: Let $p : \\mathcal{F} \\to \\mathcal{C}$ be a", "category cofibered in groupoids, and let $\\mathcal{C}'$ be a full", "subcategory of $\\mathcal{C}$. The restriction $\\mathcal{F}|_{\\mathcal{C}'}$", "is the full subcategory of $\\mathcal{F}$ whose objects lie over", "objects of $\\mathcal{C}'$. It is a category cofibered in groupoids via", "the functor", "$p|_{\\mathcal{C}'}: \\mathcal{F}|_{\\mathcal{C}'} \\to \\mathcal{C}'$.", "\\end{enumerate}" ], "refs": [ "categories-lemma-fibred-groupoids", "categories-definition-cartesian-over-C", "categories-lemma-2-product-categories-over-C", "categories-lemma-2-product-categories-over-C" ], "ref_ids": [ 12294, 12387, 12280, 12280 ] }, { "id": 3644, "type": "other", "label": "adequate-remark-settheoretic", "categories": [ "adequate" ], "title": "adequate-remark-settheoretic", "contents": [ "Consider the category $\\textit{Alg}_{fp, A}$ whose objects are $A$-algebras", "$B$ of the form $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$ and whose", "morphisms are $A$-algebra maps. Every $A$-algebra $B$ is a filtered colimit", "of finitely presented $A$-algebra, i.e., a filtered colimit of objects of", "$\\textit{Alg}_{fp, A}$. By", "Lemma \\ref{lemma-adequate-finite-presentation}", "we conclude every adequate functor $F$ is determined by its restriction to", "$\\textit{Alg}_{fp, A}$. For some questions we can therefore restrict to", "functors on $\\textit{Alg}_{fp, A}$. For example, the category of adequate", "functors does not depend on the choice of the big $\\tau$-site", "chosen in", "Section \\ref{section-conventions}." ], "refs": [ "adequate-lemma-adequate-finite-presentation" ], "ref_ids": [ 3586 ] }, { "id": 3645, "type": "other", "label": "adequate-remark-linearly-adequate", "categories": [ "adequate" ], "title": "adequate-remark-linearly-adequate", "contents": [ "Let $A$ be a ring.", "The proof of", "Lemma \\ref{lemma-extension-adequate-key}", "shows that any extension $0 \\to \\underline{M} \\to E \\to L \\to 0$", "of module-valued functors on $\\textit{Alg}_A$", "with $L$ linearly adequate splits. It uses only the following properties", "of the module-valued functor $F = \\underline{M}$:", "\\begin{enumerate}", "\\item $F(B) \\otimes_B B' \\to F(B')$ is an isomorphism", "for a flat ring map $B \\to B'$, and", "\\item", "$F(C)^{(1)} = F(p_1)(F(B)^{(1)}) \\oplus F(p_2)(F(B)^{(1)})$", "where $B = A[x_1, \\ldots, x_n]/(\\sum a_{ij}x_j)$ and", "$C = A[x_1, \\ldots, x_n, y_1, \\ldots, y_n]/", "(\\sum a_{ij}x_j, \\sum a_{ij}y_j)$.", "\\end{enumerate}", "These two properties hold for any adequate functor $F$; details omitted.", "Hence we see that $L$ is a projective object of the abelian category of", "adequate functors." ], "refs": [ "adequate-lemma-extension-adequate-key" ], "ref_ids": [ 3596 ] }, { "id": 3646, "type": "other", "label": "adequate-remark-compare", "categories": [ "adequate" ], "title": "adequate-remark-compare", "contents": [ "Let $S$ be a scheme. We have functors", "$u : \\QCoh(\\mathcal{O}_S) \\to \\textit{Adeq}(\\mathcal{O})$", "and", "$v : \\textit{Adeq}(\\mathcal{O}) \\to \\QCoh(\\mathcal{O}_S)$.", "Namely, the functor $u : \\mathcal{F} \\mapsto \\mathcal{F}^a$", "comes from taking the associated $\\mathcal{O}$-module which is", "adequate by", "Lemma \\ref{lemma-adequate-characterize}.", "Conversely, the functor $v$ comes from restriction", "$v : \\mathcal{G} \\mapsto \\mathcal{G}|_{S_{Zar}}$, see", "Lemma \\ref{lemma-same-cohomology-adequate}.", "Since $\\mathcal{F}^a$ can be described as the pullback of", "$\\mathcal{F}$ under a morphism of ringed topoi", "$((\\Sch/S)_\\tau, \\mathcal{O}) \\to (S_{Zar}, \\mathcal{O}_S)$, see", "Descent, Remark \\ref{descent-remark-change-topologies-ringed-sites}", "and since restriction is the pushforward we see that $u$ and $v$", "are adjoint as follows", "$$", "\\SheafHom_{\\mathcal{O}_S}(\\mathcal{F}, v\\mathcal{G})", "=", "\\SheafHom_\\mathcal{O}(u\\mathcal{F}, \\mathcal{G})", "$$", "where $\\mathcal{O}$ denotes the structure sheaf on the big site.", "It is immediate from the description that the adjunction mapping", "$\\mathcal{F} \\to vu\\mathcal{F}$ is an isomorphism for all quasi-coherent", "sheaves." ], "refs": [ "adequate-lemma-adequate-characterize", "adequate-lemma-same-cohomology-adequate", "descent-remark-change-topologies-ringed-sites" ], "ref_ids": [ 3614, 3616, 14793 ] }, { "id": 3647, "type": "other", "label": "adequate-remark-D-adeq-independence-topology", "categories": [ "adequate" ], "title": "adequate-remark-D-adeq-independence-topology", "contents": [ "Let $S$ be a scheme.", "Let $\\tau, \\tau' \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.", "Denote $\\mathcal{O}_\\tau$, resp.\\ $\\mathcal{O}_{\\tau'}$", "the structure sheaf $\\mathcal{O}$ viewed as a sheaf on", "$(\\Sch/S)_\\tau$, resp.\\ $(\\Sch/S)_{\\tau'}$.", "Then $D_{\\textit{Adeq}}(\\mathcal{O}_\\tau)$ and", "$D_{\\textit{Adeq}}(\\mathcal{O}_{\\tau'})$ are canonically isomorphic.", "This follows from Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-compare-topologies-derived-adequate-modules}.", "Namely, assume $\\tau$ is stronger than the topology $\\tau'$, let", "$\\mathcal{C} = (\\Sch/S)_{fppf}$, and let $\\mathcal{B}$ the collection", "of affine schemes over $S$. Assumptions (1) and (2) we've seen above.", "Assumption (3) is clear and assumption (4) follows from", "Lemma \\ref{lemma-same-cohomology-adequate}." ], "refs": [ "sites-cohomology-lemma-compare-topologies-derived-adequate-modules", "adequate-lemma-same-cohomology-adequate" ], "ref_ids": [ 4293, 3616 ] }, { "id": 3648, "type": "other", "label": "adequate-remark-D-adeq-and-D-QCoh", "categories": [ "adequate" ], "title": "adequate-remark-D-adeq-and-D-QCoh", "contents": [ "Let $S$ be a scheme. The morphism $f$ see", "(\\ref{equation-compare-big-small}) induces", "adjoint functors", "$Rf_* : D_{\\textit{Adeq}}(\\mathcal{O}) \\to D_\\QCoh(S)$", "and", "$Lf^* : D_\\QCoh(S) \\to D_{\\textit{Adeq}}(\\mathcal{O})$.", "Moreover $Rf_* Lf^* \\cong \\text{id}_{D_\\QCoh(S)}$.", "\\medskip\\noindent", "We sketch the proof. By", "Remark \\ref{remark-D-adeq-independence-topology}", "we may assume the topology $\\tau$ is the Zariski topology.", "We will use the existence of the unbounded total derived", "functors $Lf^*$ and $Rf_*$ on $\\mathcal{O}$-modules and their", "adjointness, see", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-adjoint}.", "In this case $f_*$ is just the restriction to the subcategory", "$S_{Zar}$ of $(\\Sch/S)_{Zar}$. Hence it is clear that", "$Rf_* = f_*$ induces", "$Rf_* : D_{\\textit{Adeq}}(\\mathcal{O}) \\to D_\\QCoh(S)$.", "Suppose that $\\mathcal{G}^\\bullet$ is an object of", "$D_\\QCoh(S)$. We may choose a system", "$\\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet \\to \\ldots$", "of bounded above complexes of flat $\\mathcal{O}_S$-modules whose", "transition maps are termwise split injectives and a diagram", "$$", "\\xymatrix{", "\\mathcal{K}_1^\\bullet \\ar[d] \\ar[r] &", "\\mathcal{K}_2^\\bullet \\ar[d] \\ar[r] & \\ldots \\\\", "\\tau_{\\leq 1}\\mathcal{G}^\\bullet \\ar[r] &", "\\tau_{\\leq 2}\\mathcal{G}^\\bullet \\ar[r] & \\ldots", "}", "$$", "with the properties (1), (2), (3) listed in", "Derived Categories, Lemma \\ref{derived-lemma-special-direct-system}", "where $\\mathcal{P}$ is the collection of flat $\\mathcal{O}_S$-modules.", "Then $Lf^*\\mathcal{G}^\\bullet$ is computed by", "$\\colim f^*\\mathcal{K}_n^\\bullet$, see", "Cohomology on Sites, Lemmas \\ref{sites-cohomology-lemma-pullback-K-flat} and", "\\ref{sites-cohomology-lemma-derived-base-change}", "(note that our sites have enough points by", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-points-fppf}).", "We have to see that $H^i(Lf^*\\mathcal{G}^\\bullet) =", "\\colim H^i(f^*\\mathcal{K}_n^\\bullet)$ is adequate for each $i$. By", "Lemma \\ref{lemma-abelian-adequate}", "we conclude that it suffices to show that", "each $H^i(f^*\\mathcal{K}_n^\\bullet)$ is adequate.", "\\medskip\\noindent", "The adequacy of $H^i(f^*\\mathcal{K}_n^\\bullet)$ is local on $S$, hence", "we may assume that $S = \\Spec(A)$ is affine. Because $S$ is affine", "$D_\\QCoh(S) = D(\\QCoh(\\mathcal{O}_S))$, see", "the discussion in", "Derived Categories of Schemes, Section", "\\ref{perfect-section-derived-quasi-coherent}.", "Hence there exists a quasi-isomorphism", "$\\mathcal{F}^\\bullet \\to \\mathcal{K}_n^\\bullet$", "where $\\mathcal{F}^\\bullet$ is a bounded above complex of flat", "quasi-coherent modules.", "Then $f^*\\mathcal{F}^\\bullet \\to f^*\\mathcal{K}_n^\\bullet$ is a", "quasi-isomorphism, and the cohomology sheaves of", "$f^*\\mathcal{F}^\\bullet$ are adequate.", "\\medskip\\noindent", "The final assertion", "$Rf_* Lf^* \\cong \\text{id}_{D_\\QCoh(S)}$", "follows from the explicit description of the functors above.", "(In plain English: if $\\mathcal{F}$ is quasi-coherent and $p > 0$, then", "$L_pf^*\\mathcal{F}$ is a parasitic adequate module.)" ], "refs": [ "adequate-remark-D-adeq-independence-topology", "sites-cohomology-lemma-adjoint", "derived-lemma-special-direct-system", "sites-cohomology-lemma-pullback-K-flat", "sites-cohomology-lemma-derived-base-change", "etale-cohomology-lemma-points-fppf", "adequate-lemma-abelian-adequate" ], "ref_ids": [ 3647, 4249, 1903, 4241, 4242, 6427, 3618 ] }, { "id": 3649, "type": "other", "label": "adequate-remark-conclusion", "categories": [ "adequate" ], "title": "adequate-remark-conclusion", "contents": [ "Remark \\ref{remark-D-adeq-and-D-QCoh}", "above implies we have an equivalence of derived categories", "$$", "D_{\\textit{Adeq}}(\\mathcal{O})/D_\\mathcal{C}(\\mathcal{O})", "\\longrightarrow", "D_\\QCoh(S)", "$$", "where $\\mathcal{C}$ is the category of parasitic adequate modules.", "Namely, it is clear that $D_\\mathcal{C}(\\mathcal{O})$ is the kernel", "of $Rf_*$, hence a functor as indicated. For any object $X$ of", "$D_{\\textit{Adeq}}(\\mathcal{O})$ the map $Lf^*Rf_*X \\to X$ maps", "to a quasi-isomorphism in $D_\\QCoh(S)$, hence", "$Lf^*Rf_*X \\to X$ is an isomorphism in", "$D_{\\textit{Adeq}}(\\mathcal{O})/D_\\mathcal{C}(\\mathcal{O})$.", "Finally, for $X, Y$ objects of $D_{\\textit{Adeq}}(\\mathcal{O})$", "the map", "$$", "Rf_* :", "\\Hom_{D_{\\textit{Adeq}}(\\mathcal{O})/D_\\mathcal{C}(\\mathcal{O})}(X, Y)", "\\to", "\\Hom_{D_\\QCoh(S)}(Rf_*X, Rf_*Y)", "$$", "is bijective as $Lf^*$ gives an inverse (by the remarks above)." ], "refs": [ "adequate-remark-D-adeq-and-D-QCoh" ], "ref_ids": [ 3648 ] }, { "id": 3695, "type": "other", "label": "spaces-topologies-remark-change-topologies-ringed", "categories": [ "spaces-topologies" ], "title": "spaces-topologies-remark-change-topologies-ringed", "contents": [ "The sites $(\\textit{Spaces}/X)_\\etale$ and $X_{spaces, \\etale}$", "come with structure sheaves. For the small \\'etale site we", "have seen this in Properties of Spaces, Section", "\\ref{spaces-properties-section-structure-sheaf}.", "The structure sheaf $\\mathcal{O}$ on the big \\'etale site", "$(\\textit{Spaces}/X)_\\etale$ is defined by assigning to an object", "$U$ the global sections of the structure sheaf of $U$.", "This makes sense because after all $U$ is an algebraic space", "itself hence has a structure sheaf. Since $\\mathcal{O}_U$", "is a sheaf on the \\'etale site of $U$, the presheaf $\\mathcal{O}$", "so defined satisfies the sheaf condition for coverings of $U$, i.e.,", "$\\mathcal{O}$ is a sheaf.", "We can upgrade the morphisms $i_f$, $\\pi_X$, $i_X$, $f_{small}$, and", "$f_{big}$ defined above to morphisms of ringed sites, respectively topoi.", "Let us deal with these one by one.", "\\begin{enumerate}", "\\item In Lemma \\ref{lemma-put-in-T-etale} denote $\\mathcal{O}$", "the structure sheaf on $(\\textit{Spaces}/X)_\\etale$.", "We have $(i_f^{-1}\\mathcal{O})(U/Y) = \\mathcal{O}_U(U) = \\mathcal{O}_Y(U)$", "by construction.", "Hence an isomorphism $i_f^\\sharp : i_f^{-1}\\mathcal{O} \\to \\mathcal{O}_Y$.", "\\item In Lemma \\ref{lemma-at-the-bottom-etale} it was noted", "that $i_X$ is a special case of $i_f$ with $f = \\text{id}_X$", "hence we are back in case (1).", "\\item In Lemma \\ref{lemma-at-the-bottom-etale} the morphism", "$\\pi_X$ satisfies $(\\pi_{X, *}\\mathcal{O})(U) = \\mathcal{O}(U) =", "\\mathcal{O}_X(U)$. Hence we can use this to define", "$\\pi_X^\\sharp : \\mathcal{O}_X \\to \\pi_{X, *}\\mathcal{O}$.", "\\item In Lemma \\ref{lemma-morphism-big-small-etale}", "the extension of $f_{small}$ to a morphism of ringed topoi", "was discussed in Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-morphism-ringed-topoi}.", "\\item In Lemma \\ref{lemma-morphism-big-small-etale}", "the functor $f_{big}^{-1}$ is simply the restriction", "via the inclusion functor", "$(\\textit{Spaces}/Y)_\\etale \\to (\\textit{Spaces}/X)_\\etale$.", "Let $\\mathcal{O}_1$ be the structure sheaf on $(\\textit{Spaces}/X)_\\etale$", "and let $\\mathcal{O}_2$ be the structure sheaf on $(\\textit{Spaces}/Y)_\\etale$.", "We obtain a canonical isomorphism", "$f_{big}^\\sharp : f_{big}^{-1}\\mathcal{O}_1 \\to \\mathcal{O}_2$.", "\\end{enumerate}", "Moreover, with these definitions compositions work out correctly too.", "We omit giving a detailed statement and proof." ], "refs": [ "spaces-topologies-lemma-put-in-T-etale", "spaces-topologies-lemma-at-the-bottom-etale", "spaces-topologies-lemma-at-the-bottom-etale", "spaces-topologies-lemma-morphism-big-small-etale", "spaces-properties-lemma-morphism-ringed-topoi", "spaces-topologies-lemma-morphism-big-small-etale" ], "ref_ids": [ 3654, 3655, 3655, 3657, 11882, 3657 ] }, { "id": 3845, "type": "other", "label": "proetale-remark-size-w", "categories": [ "proetale" ], "title": "proetale-remark-size-w", "contents": [ "Let $A$ be a ring. Let $\\kappa$ be an infinite cardinal bigger or", "equal than the cardinality of $A$. Then the cardinality of $A_w$", "(Lemma \\ref{lemma-make-w-local})", "is at most $\\kappa$. Namely, each $A_E$ has cardinality at most", "$\\kappa$ and the set of finite subsets of $A$ has cardinality at most $\\kappa$", "as well. Thus the result follows as $\\kappa \\otimes \\kappa = \\kappa$, see", "Sets, Section \\ref{sets-section-cardinals}." ], "refs": [ "proetale-lemma-make-w-local" ], "ref_ids": [ 3714 ] }, { "id": 3846, "type": "other", "label": "proetale-remark-slightly-stronger", "categories": [ "proetale" ], "title": "proetale-remark-slightly-stronger", "contents": [ "In each of Lemmas \\ref{lemma-construct}, \\ref{lemma-construct-profinite},", "Proposition \\ref{proposition-maps-wich-identify-local-rings}, and", "Lemma \\ref{lemma-find-Zariski-w-contractible} we find an ind-Zariski ring", "map with some properties. In the paper \\cite{BS} the authors use the notion", "of an ind-(Zariski localization) which is a filtered colimit of finite", "products of principal localizations. It is possible to replace ind-Zariski", "by ind-(Zariski localization) in each of the results listed above.", "However, we do not need this and the notion of an ind-Zariski homomorphism", "of rings as defined here has slightly better formal properties. Moreover,", "the notion of an ind-Zariski ring map is the natural analogue of the", "notion of an ind-\\'etale ring map defined in the next section." ], "refs": [ "proetale-lemma-construct", "proetale-lemma-construct-profinite", "proetale-proposition-maps-wich-identify-local-rings", "proetale-lemma-find-Zariski-w-contractible" ], "ref_ids": [ 3719, 3720, 3825, 3724 ] }, { "id": 3847, "type": "other", "label": "proetale-remark-size-T", "categories": [ "proetale" ], "title": "proetale-remark-size-T", "contents": [ "Let $A$ be a ring. Let $\\kappa$ be an infinite cardinal bigger or", "equal than the cardinality of $A$. Then the cardinality of $T(A)$", "is at most $\\kappa$. Namely, each $B_E$ has cardinality at most", "$\\kappa$ and the index set $I(A)$ has cardinality at most $\\kappa$", "as well. Thus the result follows as $\\kappa \\otimes \\kappa = \\kappa$, see", "Sets, Section \\ref{sets-section-cardinals}. It follows that the", "ring constructed in the proof of Lemma \\ref{lemma-first-construction}", "has cardinality at most $\\kappa$ as well." ], "refs": [ "proetale-lemma-first-construction" ], "ref_ids": [ 3731 ] }, { "id": 3848, "type": "other", "label": "proetale-remark-first-construction-functorial", "categories": [ "proetale" ], "title": "proetale-remark-first-construction-functorial", "contents": [ "The construction $A \\mapsto T(A)$ is functorial in the following sense:", "If $A \\to A'$ is a ring map, then we can construct a commutative diagram", "$$", "\\xymatrix{", "A \\ar[r] \\ar[d] & T(A) \\ar[d] \\\\", "A' \\ar[r] & T(A')", "}", "$$", "Namely, given $(A \\to A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n))$ in", "$S(A)$ we can use the ring map $\\varphi : A \\to A'$ to obtain a corresponding", "element $(A' \\to A'[x_1, \\ldots, x_n]/(f^\\varphi_1, \\ldots, f^\\varphi_n))$", "of $S(A')$ where $f^\\varphi$ means the polynomial obtained by applying", "$\\varphi$ to the coefficients of the polynomial $f$.", "Moreover, there is a commutative diagram", "$$", "\\xymatrix{", "A \\ar[r] \\ar[d] & A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n) \\ar[d] \\\\", "A' \\ar[r] & A'[x_1, \\ldots, x_n]/(f^\\varphi_1, \\ldots, f^\\varphi_n)", "}", "$$", "which is a in the category of rings. For $E \\subset S(A)$ finite, set", "$E' = \\varphi(E)$ and define $B_E \\to B_{E'}$ in the obvious manner.", "Taking the colimit gives the desired map $T(A) \\to T(A')$, see", "Categories, Lemma \\ref{categories-lemma-functorial-colimit}." ], "refs": [ "categories-lemma-functorial-colimit" ], "ref_ids": [ 12210 ] }, { "id": 3849, "type": "other", "label": "proetale-remark-h-limit-preserving", "categories": [ "proetale" ], "title": "proetale-remark-h-limit-preserving", "contents": [ "Let $S$ be a scheme contained in a big site $\\Sch_h$. Let $F$ be a sheaf", "of sets on $(\\Sch/S)_h$ such that $F(T) = \\colim F(T_i)$ whenever", "$T = \\lim T_i$ is a directed limit of affine schemes in $(\\Sch/S)_h$.", "In this situation $F$ extends uniquely to a contravariant functor $F'$", "on the category of all schemes over $S$ such that (a) $F'$ satisfies the", "sheaf property for the h topology and (b) $F'$ is limit preserving.", "See More on Flatness, Lemma \\ref{flat-lemma-extend-sheaf-h}.", "In this situation Lemma \\ref{lemma-h-limit-preserving}", "tells us that $F'$ satisfies", "the sheaf property for the V topology." ], "refs": [ "flat-lemma-extend-sheaf-h", "proetale-lemma-h-limit-preserving" ], "ref_ids": [ 6154, 3738 ] }, { "id": 3850, "type": "other", "label": "proetale-remark-size-w-contractible", "categories": [ "proetale" ], "title": "proetale-remark-size-w-contractible", "contents": [ "Let $A$ be a ring. Let $\\kappa$ be an infinite cardinal bigger or", "equal than the cardinality of $A$. Then the cardinality of the", "ring $D$ constructed in Proposition \\ref{proposition-find-w-contractible}", "is at most", "$$", "\\kappa^{2^{2^{2^\\kappa}}}.", "$$", "Namely, the ring map $A \\to D$ is", "constructed as a composition", "$$", "A \\to A_w = A' \\to C' \\to C \\to D.", "$$", "Here the first three steps of the construction are carried out", "in the first paragraph of the proof of", "Lemma \\ref{lemma-get-w-local-algebraic-residue-field-extensions}.", "For the first step we have $|A_w| \\leq \\kappa$ by", "Remark \\ref{remark-size-w}.", "We have $|C'| \\leq \\kappa$ by", "Remark \\ref{remark-size-T}.", "Then $|C| \\leq \\kappa$ because $C$ is a localization of $(C')_w$", "(it is constructed from $C'$ by an application of", "Lemma \\ref{lemma-localize-along-closed-profinite}", "in the proof of Lemma \\ref{lemma-w-local-algebraic-residue-field-extensions}).", "Thus $C$ has at most $2^\\kappa$ maximal ideals.", "Finally, the ring map $C \\to D$ identifies local rings and the", "cardinality of the set of maximal ideals of $D$ is at most", "$2^{2^{2^\\kappa}}$ by", "Topology, Remark \\ref{topology-remark-size-projective-cover}.", "Since $D \\subset \\prod_{\\mathfrak m \\subset D} D_\\mathfrak m$ we see", "that $D$ has at most the size displayed above." ], "refs": [ "proetale-proposition-find-w-contractible", "proetale-lemma-get-w-local-algebraic-residue-field-extensions", "proetale-remark-size-w", "proetale-remark-size-T", "proetale-lemma-localize-along-closed-profinite", "proetale-lemma-w-local-algebraic-residue-field-extensions", "topology-remark-size-projective-cover" ], "ref_ids": [ 3827, 3735, 3845, 3847, 3717, 3718, 8385 ] }, { "id": 3851, "type": "other", "label": "proetale-remark-extend-to-all", "categories": [ "proetale" ], "title": "proetale-remark-extend-to-all", "contents": [ "Let $X$ be a scheme. Because $X_\\proetale$ has enough weakly contractible", "objects for all $K$ in $D(X_\\proetale)$ we have $K = R\\lim \\tau_{\\geq -n}K$", "by", "Cohomology on Sites, Proposition", "\\ref{sites-cohomology-proposition-enough-weakly-contractibles}.", "Since $R\\Gamma$ commutes with $R\\lim$ by", "Injectives, Lemma \\ref{injectives-lemma-RF-commutes-with-Rlim}", "we see that", "$$", "R\\Gamma(X, K) = R\\lim R\\Gamma(X, \\tau_{\\geq -n}K)", "$$", "in $D(\\textit{Ab})$. This will sometimes allow us to extend results from", "bounded below complexes to all complexes." ], "refs": [ "sites-cohomology-proposition-enough-weakly-contractibles", "injectives-lemma-RF-commutes-with-Rlim" ], "ref_ids": [ 4410, 7796 ] }, { "id": 3997, "type": "other", "label": "formal-spaces-remark-mcquillan", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-mcquillan", "contents": [ "There is a variant of the construction of formal schemes due to", "McQuillan, see \\cite{McQuillan}.", "He suggests a slight weakening of the condition of admissibility.", "Namely, recall that an admissible topological ring is a complete", "(and separated by our conventions) topological ring $A$", "which is linearly topologized such that there exists an", "ideal of definition: an", "open ideal $I$ such that any neighbourhood of $0$ contains $I^n$", "for some $n \\geq 1$.", "McQuillan works with what we will call {\\it weakly admissible}", "topological rings. A weakly admissible topological ring $A$ is a", "complete (and separated by our conventions) topological ring", "which is linearly topologized such that there exists an", "{\\it weak ideal of definition}: an open ideal $I$ such that", "for all $f \\in I$ we have", "$f^n \\to 0$ for $n \\to \\infty$. Similarly to the admissible case,", "if $I$ is a weak ideal of definition and $J \\subset A$ is an", "open ideal, then $I \\cap J$ is a weak ideal of definition.", "Thus the weak ideals of definition form a fundamental system of", "open neighbourhoods of $0$ and", "one can proceed along much the same route as above", "to define a larger category of formal schemes based", "on this notion. The analogues of Lemmas \\ref{lemma-fully-faithful} and", "\\ref{lemma-formal-scheme-sheaf-fppf}", "still hold in this setting (with the same proof)." ], "refs": [ "formal-spaces-lemma-fully-faithful", "formal-spaces-lemma-formal-scheme-sheaf-fppf" ], "ref_ids": [ 3852, 3853 ] }, { "id": 3998, "type": "other", "label": "formal-spaces-remark-sheafification-of-presheaves-in-top", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-sheafification-of-presheaves-in-top", "contents": [ "\\begin{reference}", "\\cite{Gray}", "\\end{reference}", "In this remark we briefly discuss sheafification of presheaves", "of topological spaces. The exact same arguments work for", "presheaves of topological abelian groups, topological rings, and", "topological modules (over a given topological ring). In order to", "do this in the correct generality let us work over a site", "$\\mathcal{C}$. The reader who is interested in the case of (pre)sheaves", "over a topological space $X$ should think of objects of $\\mathcal{C}$", "as the opens of $X$, of morphisms of $\\mathcal{C}$ as inclusions of", "opens, and of coverings in $\\mathcal{C}$ as coverings in $X$, see", "Sites, Example \\ref{sites-example-site-topological}.", "Denote $\\Sh(\\mathcal{C}, \\textit{Top})$ the category of sheaves", "of topological spaces on $\\mathcal{C}$ and denote", "$\\textit{PSh}(\\mathcal{C}, \\textit{Top})$ the category of presheaves", "of topological spaces on $\\mathcal{C}$.", "Let $\\mathcal{F}$ be a presheaf of topological spaces on $\\mathcal{C}$.", "The sheafification $\\mathcal{F}^\\#$ should satisfy the formula", "$$", "\\Mor_{\\textit{PSh}(\\mathcal{C}, \\textit{Top})}(\\mathcal{F}, \\mathcal{G})", "=", "\\Mor_{\\Sh(\\mathcal{C}, \\textit{Top})}(\\mathcal{F}^\\#, \\mathcal{G})", "$$", "functorially in $\\mathcal{G}$ from $\\Sh(\\mathcal{C}, \\textit{Top})$.", "In other words, we are trying to construct the left adjoint", "to the inclusion functor", "$\\Sh(\\mathcal{C}, \\textit{Top}) \\to \\textit{PSh}(\\mathcal{C}, \\textit{Top})$.", "We first claim that $\\Sh(\\mathcal{C}, \\textit{Top})$ has limits", "and that the inclusion functor commutes with them.", "Namely, given a category $\\mathcal{I}$ and a functor", "$i \\mapsto \\mathcal{G}_i$ into $\\Sh(\\mathcal{C}, \\textit{Top})$", "we simply define", "$$", "(\\lim \\mathcal{G}_i)(U) = \\lim \\mathcal{G}_i(U)", "$$", "where we take the limit in the category of topological spaces", "(Topology, Lemma \\ref{topology-lemma-limits}). This defines a sheaf", "because limits commute with limits", "(Categories, Lemma \\ref{categories-lemma-colimits-commute})", "and in particular products and equalizers (which are the", "operations used in the sheaf axiom). Finally, a morphism", "of presheaves from $\\mathcal{F} \\to \\lim \\mathcal{G}_i$ is", "clearly the same thing as a compatible system of morphisms", "$\\mathcal{F} \\to \\mathcal{G}_i$. In other words, the object", "$\\lim \\mathcal{G}_i$ is the limit in the category", "of presheaves of topological spaces and a fortiori in the", "category of sheaves of topological spaces.", "Our second claim is that any morphism of presheaves", "$\\mathcal{F} \\to \\mathcal{G}$ with $\\mathcal{G}$ an object of", "$\\Sh(\\mathcal{C}, \\textit{Top})$ factors through a subsheaf", "$\\mathcal{G}' \\subset \\mathcal{G}$ whose size is bounded.", "Here we define the {\\it size} $|\\mathcal{H}|$", "of a sheaf of topological spaces $\\mathcal{H}$ to be the cardinal", "$\\sup_{U \\in \\Ob(\\mathcal{C})} |\\mathcal{H}(U)|$.", "To prove our claim we let", "$$", "\\mathcal{G}'(U) =", "\\left\\{", "\\quad", "s \\in \\mathcal{G}(U)", "\\quad \\middle| \\quad", "\\begin{matrix}", "\\text{there exists a covering }\\{U_i \\to U\\}_{i \\in I} \\\\", "\\text{such that }", "s|_{U_i} \\in \\Im(\\mathcal{F}(U_i) \\to \\mathcal{G}(U_i))", "\\end{matrix}", "\\quad", "\\right\\}", "$$", "We endow $\\mathcal{G}'(U)$ with the induced topology.", "Then $\\mathcal{G}'$ is a sheaf of topological spaces (details omitted)", "and $\\mathcal{G}' \\to \\mathcal{G}$ is a morphism through which", "the given map $\\mathcal{F} \\to \\mathcal{G}$ factors. Moreover,", "the size of $\\mathcal{G}'$ is bounded by some cardinal", "$\\kappa$ depending only on $\\mathcal{C}$ and the presheaf $\\mathcal{F}$", "(hint: use that coverings in $\\mathcal{C}$", "form a set by our conventions). Putting everything together we see", "that the assumptions of Categories, Theorem", "\\ref{categories-theorem-adjoint-functor}", "are satisfied and we obtain sheafification as the left", "adjoint of the inclusion functor from sheaves to presheaves.", "Finally, let $p$ be a point of the", "site $\\mathcal{C}$ given by a functor $u : \\mathcal{C} \\to \\textit{Sets}$,", "see Sites, Definition \\ref{sites-definition-point}.", "For a topological space $M$ the presheaf defined by the rule", "$$", "U \\mapsto \\text{Map}(u(U), M) = \\prod\\nolimits_{x \\in u(U)} M", "$$", "endowed with the product topology is a sheaf of topological spaces.", "Hence the exact same argument as given in the proof of", "Sites, Lemma \\ref{sites-lemma-point-pushforward-sheaf} shows that", "$\\mathcal{F}_p = \\mathcal{F}^\\#_p$, in other words, sheafification", "commutes with taking stalks at a point." ], "refs": [ "topology-lemma-limits", "categories-lemma-colimits-commute", "categories-theorem-adjoint-functor", "sites-definition-point", "sites-lemma-point-pushforward-sheaf" ], "ref_ids": [ 8248, 12212, 12200, 8675, 8595 ] }, { "id": 3999, "type": "other", "label": "formal-spaces-remark-compare-with-affine-formal-schemes", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-compare-with-affine-formal-schemes", "contents": [ "The classical affine formal algebraic spaces correspond to the", "affine formal schemes considered in EGA (\\cite{EGA}). To explain this", "we assume our base scheme is $\\Spec(\\mathbf{Z})$. Let", "$\\mathfrak X = \\text{Spf}(A)$ be an affine formal scheme.", "Let $h_\\mathfrak X$ be its functor of points as in", "Lemma \\ref{lemma-fully-faithful}.", "Then $h_\\mathfrak X = \\colim h_{\\Spec(A/I)}$ where the colimit", "is over the collection of ideals of definition of the admissible", "topological ring $A$. This follows from", "(\\ref{equation-morphisms-affine-formal-schemes})", "when evaluating on affine schemes and it suffices to check", "on affine schemes as both sides are fppf sheaves, see", "Lemma \\ref{lemma-formal-scheme-sheaf-fppf}.", "Thus $h_\\mathfrak X$ is an affine formal algebraic space.", "In fact, it is a classical affine formal algebraic space", "by Definition \\ref{definition-types-affine-formal-algebraic-space}.", "Thus Lemma \\ref{lemma-fully-faithful} tells us", "the category of affine formal schemes is equivalent to the category", "of classical affine formal algebraic spaces." ], "refs": [ "formal-spaces-lemma-fully-faithful", "formal-spaces-lemma-formal-scheme-sheaf-fppf", "formal-spaces-definition-types-affine-formal-algebraic-space", "formal-spaces-lemma-fully-faithful" ], "ref_ids": [ 3852, 3853, 3978, 3852 ] }, { "id": 4000, "type": "other", "label": "formal-spaces-remark-compare-with-formal-schemes", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-compare-with-formal-schemes", "contents": [ "Modulo set theoretic issues the category of formal schemes \\`a la EGA", "(see Section \\ref{section-formal-schemes-EGA}) is equivalent to a full", "subcategory of the category of formal algebraic spaces. To explain this", "we assume our base scheme is $\\Spec(\\mathbf{Z})$. By", "Lemma \\ref{lemma-formal-scheme-sheaf-fppf} the functor of points", "$h_\\mathfrak X$ associated to a formal scheme $\\mathfrak X$ is a sheaf", "in the fppf topology. By Lemma \\ref{lemma-fully-faithful}", "the assignment $\\mathfrak X \\mapsto h_\\mathfrak X$ is a fully faithful", "embedding of the category of formal schemes into the category of", "fppf sheaves. Given a formal scheme $\\mathfrak X$ we choose an open covering", "$\\mathfrak X = \\bigcup \\mathfrak X_i$ with $\\mathfrak X_i$", "affine formal schemes. Then $h_{\\mathfrak X_i}$", "is an affine formal algebraic space by", "Remark \\ref{remark-compare-with-affine-formal-schemes}.", "The morphisms $h_{\\mathfrak X_i} \\to h_\\mathfrak X$ are representable", "and open immersions. Thus $\\{h_{\\mathfrak X_i} \\to h_\\mathfrak X\\}$", "is a family as in Definition \\ref{definition-formal-algebraic-space}", "and we see that $h_\\mathfrak X$ is a formal algebraic space." ], "refs": [ "formal-spaces-lemma-formal-scheme-sheaf-fppf", "formal-spaces-lemma-fully-faithful", "formal-spaces-remark-compare-with-affine-formal-schemes", "formal-spaces-definition-formal-algebraic-space" ], "ref_ids": [ 3853, 3852, 3999, 3981 ] }, { "id": 4001, "type": "other", "label": "formal-spaces-remark-set-theoretic", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-set-theoretic", "contents": [ "Let $S$ be a scheme and let $(\\Sch/S)_{fppf}$ be a big fppf site as", "in Topologies, Definition \\ref{topologies-definition-big-small-fppf}.", "As our set theoretic condition on $X$ in", "Definitions \\ref{definition-affine-formal-algebraic-space} and", "\\ref{definition-formal-algebraic-space} we take:", "there exist objects $U, R$ of $(\\Sch/S)_{fppf}$, a", "morphism $U \\to X$ which is a surjection of fppf sheaves, and", "a morphism $R \\to U \\times_X U$ which is a surjection of fppf sheaves.", "In other words, we require our sheaf to be a coequalizer of", "two maps between representable sheaves.", "Here are some observations which imply this notion behaves", "reasonably well:", "\\begin{enumerate}", "\\item Suppose $X = \\colim_{\\lambda \\in \\Lambda} X_\\lambda$", "and the system satisfies conditions (1) and (2) of", "Definition \\ref{definition-affine-formal-algebraic-space}. Then", "$U = \\coprod_{\\lambda \\in \\Lambda} X_\\lambda \\to X$ is a surjection", "of fppf sheaves. Moreover, $U \\times_X U$ is a closed subscheme", "of $U \\times_S U$ by Lemma \\ref{lemma-diagonal-affine-formal-algebraic-space}.", "Hence if $U$ is representable by an object of $(\\Sch/S)_{fppf}$", "then $U \\times_S U$ is too (see Sets, Lemma \\ref{sets-lemma-what-is-in-it})", "and the set theoretic condition is satisfied. This is always the case", "if $\\Lambda$ is countable, see Sets, Lemma \\ref{sets-lemma-what-is-in-it}.", "\\item Sanity check. Let $\\{X_i \\to X\\}_{i \\in I}$ be as in", "Definition \\ref{definition-formal-algebraic-space}", "(with the set theoretic condition as formulated above)", "and assume that each $X_i$ is actually an affine scheme.", "Then $X$ is an algebraic space. Namely, if we choose a larger", "big fppf site $(\\Sch'/S)_{fppf}$ such that $U' = \\coprod X_i$", "and $R' = \\coprod X_i \\times_X X_j$ are representable by objects", "in it, then $X' = U'/R'$ will be an object of the category", "of algebraic spaces for this choice. Then an application of", "Spaces, Lemma \\ref{spaces-lemma-fully-faithful} shows that", "$X$ is an algebraic space for $(\\Sch/S)_{fppf}$.", "\\item Let $\\{X_i \\to X\\}_{i \\in I}$ be a family of maps of sheaves", "satisfying conditions (1), (2), (3) of", "Definition \\ref{definition-formal-algebraic-space}.", "For each $i$ we can pick $U_i \\in \\Ob((\\Sch/S)_{fppf})$", "and $U_i \\to X_i$ which is a surjection of sheaves.", "Thus if $I$ is not too large (for example countable) then", "$U = \\coprod U_i \\to X$ is a surjection of sheaves and", "$U$ is representable by an object of $(\\Sch/S)_{fppf}$.", "To get $R \\in \\Ob((\\Sch/S)_{fppf})$ surjecting onto $U \\times_X U$", "it suffices to assume the diagonal $\\Delta : X \\to X \\times_S X$ is not", "too wild, for example this always works if the diagonal of $X$ is", "quasi-compact, i.e., $X$ is quasi-separated.", "\\end{enumerate}" ], "refs": [ "topologies-definition-big-small-fppf", "formal-spaces-definition-affine-formal-algebraic-space", "formal-spaces-definition-formal-algebraic-space", "formal-spaces-definition-affine-formal-algebraic-space", "formal-spaces-lemma-diagonal-affine-formal-algebraic-space", "sets-lemma-what-is-in-it", "sets-lemma-what-is-in-it", "formal-spaces-definition-formal-algebraic-space", "spaces-lemma-fully-faithful", "formal-spaces-definition-formal-algebraic-space" ], "ref_ids": [ 12542, 3977, 3981, 3977, 3866, 8795, 8795, 3981, 8168, 3981 ] }, { "id": 4002, "type": "other", "label": "formal-spaces-remark-weak-ideals-of-definition", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-weak-ideals-of-definition", "contents": [ "Let $\\mathfrak X$ be a formal scheme in the sense of McQuillan, see", "Remark \\ref{remark-mcquillan}. An {\\it weak ideal of definition}", "for $\\mathfrak X$ is an ideal sheaf", "$\\mathcal{I} \\subset \\mathcal{O}_\\mathfrak X$ such that", "for all $\\mathfrak U \\subset \\mathfrak X$ affine formal open subscheme", "the ideal", "$\\mathcal{I}(\\mathfrak U) \\subset \\mathcal{O}_\\mathfrak X(\\mathfrak U)$", "is a weak ideal of definition of the weakly admissible topological ring", "$\\mathcal{O}_\\mathfrak X(\\mathfrak U)$.", "It suffices to check the condition on the members of an affine open covering.", "There is a one-to-one correspondence", "$$", "\\{\\text{weak ideals of definition for }\\mathfrak X\\}", "\\leftrightarrow", "\\{\\text{thickenings }i : Z \\to h_\\mathfrak X\\text{ as above}\\}", "$$", "This correspondence associates to $\\mathcal{I}$ the scheme", "$Z = (\\mathfrak X, \\mathcal{O}_\\mathfrak X/\\mathcal{I})$", "together with the obvious morphism to $\\mathfrak X$.", "A {\\it fundamental system of weak ideals of definition}", "is a collection of weak ideals of definition", "$\\mathcal{I}_\\lambda$ such that on every affine open", "formal subscheme $\\mathfrak U \\subset \\mathfrak X$ the", "ideals", "$$", "I_\\lambda = \\mathcal{I}_\\lambda(\\mathfrak U) \\subset", "A = \\Gamma(\\mathfrak U, \\mathcal{O}_\\mathfrak X)", "$$", "form a fundamental system of weak ideals of definition of the", "weakly admissible topological ring $A$. It suffices to check", "on the members of an affine open covering. We conclude that", "the formal algebraic space $h_\\mathfrak X$ associated to", "the McQuillan formal scheme $\\mathfrak X$ is a colimit of schemes as", "in Lemma \\ref{lemma-colimit-is-formal} if and only if", "there exists a fundamental system of weak ideals of definition", "for $\\mathfrak X$." ], "refs": [ "formal-spaces-remark-mcquillan", "formal-spaces-lemma-colimit-is-formal" ], "ref_ids": [ 3997, 3882 ] }, { "id": 4003, "type": "other", "label": "formal-spaces-remark-ideals-of-definition", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-ideals-of-definition", "contents": [ "Let $\\mathfrak X$ be a formal scheme \\`a la EGA.", "An {\\it ideal of definition} for $\\mathfrak X$ is an ideal sheaf", "$\\mathcal{I} \\subset \\mathcal{O}_\\mathfrak X$ such that", "for all $\\mathfrak U \\subset \\mathfrak X$ affine formal open subscheme", "the ideal", "$\\mathcal{I}(\\mathfrak U) \\subset \\mathcal{O}_\\mathfrak X(\\mathfrak U)$", "is an ideal of definition of the admissible topological ring", "$\\mathcal{O}_\\mathfrak X(\\mathfrak U)$.", "It suffices to check the condition on the members of an affine open covering.", "We do {\\bf not} get the same correspondence between ideals of definition", "and thickenings $Z \\to h_\\mathfrak X$ as in", "Remark \\ref{remark-weak-ideals-of-definition}; an example", "is given in Example \\ref{example-david-hansen}.", "A {\\it fundamental system of ideals of definition}", "is a collection of ideals of definition", "$\\mathcal{I}_\\lambda$ such that on every affine open", "formal subscheme $\\mathfrak U \\subset \\mathfrak X$ the", "ideals", "$$", "I_\\lambda = \\mathcal{I}_\\lambda(\\mathfrak U) \\subset", "A = \\Gamma(\\mathfrak U, \\mathcal{O}_\\mathfrak X)", "$$", "form a fundamental system of ideals of definition of the", "admissible topological ring $A$. It suffices to check", "on the members of an affine open covering. Suppose that $\\mathfrak X$", "is quasi-compact and that $\\{\\mathcal{I}_\\lambda\\}_{\\lambda \\in \\Lambda}$", "is a fundamental system of weak ideals of definition.", "If $A$ is an admissible topological ring then all", "sufficiently small open ideals are ideals of definition", "(namely any open ideal contained in an ideal of definition", "is an ideal of definition). Thus since we only need to check", "on the finitely many members of an affine open covering", "we see that $\\mathcal{I}_\\lambda$ is an ideal of definition", "for $\\lambda$ sufficiently large. Using the discussion in", "Remark \\ref{remark-weak-ideals-of-definition} we conclude that", "the formal algebraic space $h_\\mathfrak X$ associated to", "the quasi-compact formal scheme $\\mathfrak X$ \\`a la EGA", "is a colimit of schemes as in Lemma \\ref{lemma-colimit-is-formal}", "if and only if there exists a fundamental system of ideals of definition", "for $\\mathfrak X$." ], "refs": [ "formal-spaces-remark-weak-ideals-of-definition", "formal-spaces-remark-weak-ideals-of-definition", "formal-spaces-lemma-colimit-is-formal" ], "ref_ids": [ 4002, 4002, 3882 ] }, { "id": 4004, "type": "other", "label": "formal-spaces-remark-structure-quasi-compact-quasi-separated", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-structure-quasi-compact-quasi-separated", "contents": [ "In this remark we translate the statement and proof of", "Lemma \\ref{lemma-structure-quasi-compact-quasi-separated}", "into the language of formal schemes \\`a la EGA.", "Looking at Remark \\ref{remark-ideals-of-definition} we see", "that the lemma can be translated as follows", "\\begin{itemize}", "\\item[(*)] Every quasi-compact and quasi-separated formal", "scheme has a fundamental system of ideals of definition.", "\\end{itemize}", "To prove this we first use the induction principle (reformulated for", "quasi-compact and quasi-separated formal schemes) of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle}", "to reduce to the following situation:", "$\\mathfrak X = \\mathfrak U \\cup \\mathfrak V$", "with $\\mathfrak U$, $\\mathfrak V$ open formal subschemes,", "with $\\mathfrak V$ affine, and the result is true for $\\mathfrak U$,", "$\\mathfrak V$, and $\\mathfrak U \\cap \\mathfrak V$. Pick any ideals", "of definition $\\mathcal{I} \\subset \\mathcal{O}_\\mathfrak U$", "and $\\mathcal{J} \\subset \\mathcal{O}_\\mathfrak V$.", "By our assumption that we have a fundamental system of ideals", "of definition on $\\mathfrak U$ and $\\mathfrak V$ and because", "$\\mathfrak U \\cap \\mathfrak V$ is quasi-compact, we can find", "ideals of definition $\\mathcal{I}' \\subset \\mathcal{I}$", "and $\\mathcal{J}' \\subset \\mathcal{J}$", "such that", "$$", "\\mathcal{I}'|_{\\mathfrak U \\cap \\mathfrak V} \\subset", "\\mathcal{J}|_{\\mathfrak U \\cap \\mathfrak V}", "\\quad\\text{and}\\quad", "\\mathcal{J}'|_{\\mathfrak U \\cap \\mathfrak V} \\subset", "\\mathcal{I}|_{\\mathfrak U \\cap \\mathfrak V}", "$$", "Let $U \\to U' \\to \\mathfrak U$ and $V \\to V' \\to \\mathfrak V$ be the", "closed immersions determined by the ideals of definition", "$\\mathcal{I}' \\subset \\mathcal{I} \\subset \\mathcal{O}_\\mathfrak U$", "and", "$\\mathcal{J}' \\subset \\mathcal{J} \\subset \\mathcal{O}_\\mathfrak V$.", "Let $\\mathfrak U \\cap V$ denote the open subscheme of $V$ whose", "underlying topological space is that of $\\mathfrak U \\cap \\mathfrak V$.", "By our choice of $\\mathcal{I}'$ there is a factorization", "$\\mathfrak U \\cap V \\to U'$.", "We define similarly $U \\cap \\mathfrak V$ which factors through $V'$.", "Then we consider", "$$", "Z_U = \\text{scheme theoretic image of }", "U \\amalg (\\mathfrak U \\cap V) \\longrightarrow U'", "$$", "and", "$$", "Z_V = \\text{scheme theoretic image of }", "(U \\cap \\mathfrak V) \\amalg V \\longrightarrow V'", "$$", "Since taking scheme theoretic images of quasi-compact morphisms", "commutes with restriction to opens (Morphisms, Lemma", "\\ref{morphisms-lemma-quasi-compact-scheme-theoretic-image})", "we see that $Z_U \\cap \\mathfrak V = \\mathfrak U \\cap Z_V$.", "Thus $Z_U$ and $Z_V$ glue to a scheme $Z$ which comes equipped", "with a morphism $Z \\to \\mathfrak X$. Analogous to the discussion in", "Remark \\ref{remark-weak-ideals-of-definition}", "we see that $Z$ corresponds to a weak ideal", "of definition $\\mathcal{I}_Z \\subset \\mathcal{O}_\\mathfrak X$.", "Note that $Z_U \\subset U'$ and that", "$Z_V \\subset V'$. Thus the collection of all $\\mathcal{I}_Z$", "constructed in this manner forms a fundamental system of weak", "ideals of definition. Hence a subfamily gives a fundamental system of ideals", "of definition, see Remark \\ref{remark-ideals-of-definition}." ], "refs": [ "formal-spaces-lemma-structure-quasi-compact-quasi-separated", "formal-spaces-remark-ideals-of-definition", "coherent-lemma-induction-principle", "morphisms-lemma-quasi-compact-scheme-theoretic-image", "formal-spaces-remark-weak-ideals-of-definition", "formal-spaces-remark-ideals-of-definition" ], "ref_ids": [ 3901, 4003, 3291, 5146, 4002, 4003 ] }, { "id": 4005, "type": "other", "label": "formal-spaces-remark-warning", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-warning", "contents": [ "Lemma \\ref{lemma-representable-affine} is sharp in the following", "two senses:", "\\begin{enumerate}", "\\item If $A$ and $B$ are weakly admissible rings and $\\varphi : A \\to B$", "is a continuous map, then", "$\\text{Spf}(\\varphi) : \\text{Spf}(B) \\to \\text{Spf}(A)$ is in general", "not representable.", "\\item If $f : Y \\to X$ is a representable morphism of affine", "formal algebraic spaces and $X = \\text{Spf}(A)$ is McQuillan,", "then it does not follow that $Y$ is McQuillan.", "\\end{enumerate}", "An example for (1) is to take $A = k$ a field (with discrete topology)", "and $B = k[[t]]$ with the $t$-adic topology.", "An example for (2) is given in", "Examples, Section \\ref{examples-section-affine-formal-algebraic-space}." ], "refs": [ "formal-spaces-lemma-representable-affine" ], "ref_ids": [ 3912 ] }, { "id": 4006, "type": "other", "label": "formal-spaces-remark-warning-completion", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-warning-completion", "contents": [ "Suppose $X = \\Spec(A)$ and $T \\subset X$ is the zero locus of a", "finitely generated ideal $I \\subset A$. Let $J = \\sqrt{I}$ be", "the radical of $I$. Then from the definitions we see that", "$X_{/T} = \\text{Spf}(A^\\wedge)$ where $A^\\wedge = \\lim A/I^n$ is", "the $I$-adic completion of $A$. On the other hand, the map", "$A^\\wedge \\to \\lim A/J^n$ from the $I$-adic completion to", "the $J$-adic completion can fail to be a ring isomorphisms.", "As an example let", "$$", "A = \\bigcup\\nolimits_{n \\geq 1} \\mathbf{C}[t^{1/n}]", "$$", "and $I = (t)$. Then $J = \\mathfrak m$ is the maximal ideal", "of the valuation ring $A$ and $J^2 = J$. Thus the $J$-adic completion", "of $A$ is $\\mathbf{C}$ whereas the $I$-adic completion", "is the valuation ring described in Example \\ref{example-david-hansen}", "(but in particular it is easy to see that $A \\subset A^\\wedge$)." ], "refs": [], "ref_ids": [] }, { "id": 4007, "type": "other", "label": "formal-spaces-remark-variant-adic-star", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-variant-adic-star", "contents": [ "Let $P$ be a property of morphisms of $\\textit{WAdm}^{adic*}$.", "We say $P$ is a {\\it local property} if axioms", "(\\ref{item-axiom-1}), (\\ref{item-axiom-2}), (\\ref{item-axiom-3})", "of Situation \\ref{situation-local-property}", "hold for morphisms of $\\textit{WAdm}^{adic*}$. In exactly the same way", "we obtain a variant of Lemma \\ref{lemma-property-defines-property-morphisms}", "for morphisms between locally adic* formal algebraic spaces over $S$." ], "refs": [ "formal-spaces-lemma-property-defines-property-morphisms" ], "ref_ids": [ 3923 ] }, { "id": 4008, "type": "other", "label": "formal-spaces-remark-variant-Noetherian", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-variant-Noetherian", "contents": [ "Let $P$ be a property of morphisms of $\\textit{WAdm}^{Noeth}$.", "We say $P$ is a {\\it local property} if axioms", "(\\ref{item-axiom-1}), (\\ref{item-axiom-2}), (\\ref{item-axiom-3}),", "of Situation \\ref{situation-local-property}", "hold for morphisms of $\\textit{WAdm}^{Noeth}$. In exactly the same way", "we obtain a variant of Lemma \\ref{lemma-property-defines-property-morphisms}", "for morphisms between locally Noetherian formal algebraic spaces over $S$." ], "refs": [ "formal-spaces-lemma-property-defines-property-morphisms" ], "ref_ids": [ 3923 ] }, { "id": 4009, "type": "other", "label": "formal-spaces-remark-base-change-variant-adic-star", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-base-change-variant-adic-star", "contents": [ "Let $P$ be a local property of morphisms of $\\textit{WAdm}^{adic*}$, see", "Remark \\ref{remark-variant-adic-star}. We say $P$ is {\\it stable under", "base change} if given $B \\to A$ and $B \\to C$ in $\\textit{WAdm}^{adic*}$", "we have $P(B \\to A) \\Rightarrow P(C \\to A \\widehat{\\otimes}_B C)$.", "This makes sense as $A \\widehat{\\otimes}_B C$ is an object of", "$\\textit{WAdm}^{adic*}$ by Lemma \\ref{lemma-completed-tensor-product}.", "In exactly the same way we obtain a variant of", "Lemma \\ref{lemma-base-change-property-morphisms}", "for morphisms between locally adic* formal algebraic spaces over $S$." ], "refs": [ "formal-spaces-remark-variant-adic-star", "formal-spaces-lemma-completed-tensor-product", "formal-spaces-lemma-base-change-property-morphisms" ], "ref_ids": [ 4007, 3865, 3924 ] }, { "id": 4010, "type": "other", "label": "formal-spaces-remark-base-change-variant-Noetherian", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-base-change-variant-Noetherian", "contents": [ "Let $P$ be a local property of morphisms of $\\textit{WAdm}^{Noeth}$, see", "Remark \\ref{remark-variant-Noetherian}. We say $P$ is", "{\\it stable under base change} if given $B \\to A$ and $B \\to C$", "in $\\textit{WAdm}^{Noeth}$ the property $P(B \\to A)$", "implies both that $A \\widehat{\\otimes}_B C$ is adic", "Noetherian\\footnote{See Lemma \\ref{lemma-completed-tensor-product}", "for a criterion.} and that", "$P(C \\to A \\widehat{\\otimes}_B C)$.", "In exactly the same way we obtain a variant of", "Lemma \\ref{lemma-base-change-property-morphisms}", "for morphisms between locally Noetherian formal algebraic spaces over $S$." ], "refs": [ "formal-spaces-remark-variant-Noetherian", "formal-spaces-lemma-completed-tensor-product", "formal-spaces-lemma-base-change-property-morphisms" ], "ref_ids": [ 4008, 3865, 3924 ] }, { "id": 4011, "type": "other", "label": "formal-spaces-remark-base-change-variant-variant-Noetherian", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-base-change-variant-variant-Noetherian", "contents": [ "Let $P$ and $Q$ be local properties of morphisms of", "$\\textit{WAdm}^{Noeth}$, see Remark \\ref{remark-variant-Noetherian}.", "We say {\\it $P$ is stable under base change by $Q$}", "if given $B \\to A$ and $B \\to C$ in $\\textit{WAdm}^{Noeth}$", "satisfying $P(B \\to A)$ and $Q(B \\to C)$, then", "$A \\widehat{\\otimes}_B C$ is adic Noetherian and", "$P(C \\to A \\widehat{\\otimes}_B C)$ holds.", "Arguing exactly as in the proof of", "Lemma \\ref{lemma-base-change-property-morphisms}", "we obtain the following statement:", "given morphisms $f : X \\to Y$ and $g : Y \\to Z$ of", "locally Noetherian formal algebraic spaces over $S$", "such that", "\\begin{enumerate}", "\\item the equivalent conditions of", "Lemma \\ref{lemma-property-defines-property-morphisms}", "hold for $f$ and $P$,", "\\item the equivalent conditions of", "Lemma \\ref{lemma-property-defines-property-morphisms}", "hold for $g$ and $Q$,", "\\end{enumerate}", "then the equivalent conditions of", "Lemma \\ref{lemma-property-defines-property-morphisms}", "hold for $\\text{pr}_2 : X \\times_Y Z \\to Z$ and $P$." ], "refs": [ "formal-spaces-remark-variant-Noetherian", "formal-spaces-lemma-base-change-property-morphisms", "formal-spaces-lemma-property-defines-property-morphisms", "formal-spaces-lemma-property-defines-property-morphisms", "formal-spaces-lemma-property-defines-property-morphisms" ], "ref_ids": [ 4008, 3924, 3923, 3923, 3923 ] }, { "id": 4012, "type": "other", "label": "formal-spaces-remark-composition-variant-adic-star", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-composition-variant-adic-star", "contents": [ "Let $P$ be a local property of morphisms of $\\textit{WAdm}^{adic*}$, see", "Remark \\ref{remark-variant-adic-star}. We say $P$ is {\\it stable under", "composition} if given $B \\to A$ and $C \\to B$ in $\\textit{WAdm}^{adic*}$", "we have $P(B \\to A) \\wedge P(C \\to B) \\Rightarrow P(C \\to A)$.", "In exactly the same way we obtain a variant of", "Lemma \\ref{lemma-composition-property-morphisms}", "for morphisms between locally adic* formal algebraic spaces over $S$." ], "refs": [ "formal-spaces-remark-variant-adic-star", "formal-spaces-lemma-composition-property-morphisms" ], "ref_ids": [ 4007, 3925 ] }, { "id": 4013, "type": "other", "label": "formal-spaces-remark-composition-variant-Noetherian", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-composition-variant-Noetherian", "contents": [ "Let $P$ be a local property of morphisms of $\\textit{WAdm}^{Noeth}$, see", "Remark \\ref{remark-variant-Noetherian}. We say $P$ is", "{\\it stable under composition} if given $B \\to A$ and $C \\to B$", "in $\\textit{WAdm}^{Noeth}$ we have", "$P(B \\to A) \\wedge P(C \\to B) \\Rightarrow P(C \\to A)$.", "In exactly the same way we obtain a variant of", "Lemma \\ref{lemma-composition-property-morphisms}", "for morphisms between locally Noetherian formal algebraic spaces over $S$." ], "refs": [ "formal-spaces-remark-variant-Noetherian", "formal-spaces-lemma-composition-property-morphisms" ], "ref_ids": [ 4008, 3925 ] }, { "id": 4014, "type": "other", "label": "formal-spaces-remark-permanence-variant-adic-star", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-permanence-variant-adic-star", "contents": [ "Let $P$ be a local property of morphisms of $\\textit{WAdm}^{adic*}$, see", "Remark \\ref{remark-variant-adic-star}. We say $P$ {\\it has the cancellation", "property} if given $B \\to A$ and $C \\to B$ in $\\textit{WAdm}^{adic*}$", "we have $P(C \\to A) \\wedge P(C \\to B) \\Rightarrow P(B \\to A)$.", "In exactly the same way we obtain a variant of", "Lemma \\ref{lemma-composition-property-morphisms}", "for morphisms between locally adic* formal algebraic spaces over $S$." ], "refs": [ "formal-spaces-remark-variant-adic-star", "formal-spaces-lemma-composition-property-morphisms" ], "ref_ids": [ 4007, 3925 ] }, { "id": 4015, "type": "other", "label": "formal-spaces-remark-permanence-variant-Noetherian", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-permanence-variant-Noetherian", "contents": [ "Let $P$ be a local property of morphisms of $\\textit{WAdm}^{Noeth}$, see", "Remark \\ref{remark-variant-Noetherian}. We say $P$", "{\\it has the cancellation property} if given $B \\to A$ and $C \\to B$", "in $\\textit{WAdm}^{Noeth}$ we have", "$P(C \\to B) \\wedge P(C \\to A) \\Rightarrow P(C \\to B)$.", "In exactly the same way we obtain a variant of", "Lemma \\ref{lemma-composition-property-morphisms}", "for morphisms between locally Noetherian formal algebraic spaces over $S$." ], "refs": [ "formal-spaces-remark-variant-Noetherian", "formal-spaces-lemma-composition-property-morphisms" ], "ref_ids": [ 4008, 3925 ] }, { "id": 4016, "type": "other", "label": "formal-spaces-remark-universal-property", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-universal-property", "contents": [ "\\begin{reference}", "\\cite[Chapter 0, 7.5.3]{EGA}", "\\end{reference}", "Let $A \\to C$ be a continuous map of complete linearly topologized rings.", "Then any $A$-algebra map $A[x_1, \\ldots x_r] \\to C$ extends uniquely to a", "continuous map $A\\{x_1, \\ldots, x_r\\} \\to C$ on restricted power series." ], "refs": [], "ref_ids": [] }, { "id": 4017, "type": "other", "label": "formal-spaces-remark-I-adic-completion-and-restricted-power-series", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-I-adic-completion-and-restricted-power-series", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be an ideal. If $A$ is $I$-adically", "complete, then the $I$-adic completion $A[x_1, \\ldots, x_r]^\\wedge$ of", "$A[x_1, \\ldots, x_r]$ is the restricted power series ring over $A$ as a", "ring. However, it is not clear that $A[x_1, \\ldots, x_r]^\\wedge$ is", "$I$-adically complete. We think of the topology on $A\\{x_1, \\ldots, x_r\\}$", "as the limit topology (which is always complete) whereas we often think of", "the topology on $A[x_1, \\ldots, x_r]^\\wedge$ as the $I$-adic topology", "(not always complete). If $I$ is finitely generated, then", "$A\\{x_1, \\ldots, x_r\\} = A[x_1, \\ldots, x_r]^\\wedge$ as topological", "rings, see Algebra, Lemma \\ref{algebra-lemma-hathat-finitely-generated}." ], "refs": [ "algebra-lemma-hathat-finitely-generated" ], "ref_ids": [ 859 ] }, { "id": 4018, "type": "other", "label": "formal-spaces-remark-questions", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-questions", "contents": [ "Let $A$ be a weakly admissible topological ring and let $(I_\\lambda)$", "be a fundamental system of weak ideals of definition. Let $X = \\text{Spf}(A)$,", "in other words, $X$ is a McQuillan affine formal algebraic space.", "Let $f : Y \\to X$ be a morphism of affine formal algebraic spaces.", "In general it will not be true that $Y$ is McQuillan. More specifically,", "we can ask the following questions:", "\\begin{enumerate}", "\\item Assume that $f : Y \\to X$ is a closed immersion. Then", "$Y$ is McQuillan and $f$ corresponds to a continuous map", "$\\varphi : A \\to B$ of weakly admissible topological rings", "which is taut, whose kernel $K \\subset A$ is a closed ideal, and", "whose image $\\varphi(A)$ is dense in $B$, see", "Lemma \\ref{lemma-closed-immersion-into-McQuillan}.", "What conditions on $A$ guarantee that $B = (A/K)^\\wedge$ as in", "Example \\ref{example-closed-immersion-from-quotient}?", "\\item What conditions on $A$ guarantee that closed immersions", "$f : Y \\to X$ correspond to quotients $A/K$ of $A$ by closed ideals,", "in other words, the corresponding continuous map $\\varphi$ is surjective", "and open?", "\\item Suppose that $f : Y \\to X$ is of finite type. Then we get", "$Y = \\colim \\Spec(B_\\lambda)$ where $(B_\\lambda)$ is an object of", "$\\mathcal{C}$ by Lemma \\ref{lemma-category-affine-over}.", "In this case it is true that there exists a fixed integer $r$ such", "that $B_\\lambda$ is generated by $r$ elements over $A/I_\\lambda$ for", "all $\\lambda$ (the argument is essentially already given in the proof of", "(1) $\\Rightarrow$ (2) in", "Lemma \\ref{lemma-topologically-finite-type-finite-type}).", "However, it is not clear that the projections", "$\\lim B_\\lambda \\to B_\\lambda$ are surjective, i.e.,", "it is not clear that $Y$ is McQuillan.", "Is there an example where $Y$ is not McQuillan?", "\\item Suppose that $f : Y \\to X$ is of finite type and $Y$ is McQuillan.", "Then $f$ corresponds to a continuous map $\\varphi : A \\to B$ of weakly", "admissible topological rings. In fact $\\varphi$ is taut and", "$B$ is topologically of finite type over $A$, see", "Lemma \\ref{lemma-topologically-finite-type-finite-type}.", "In other words, $f$ factors as", "$$", "Y \\longrightarrow \\mathbf{A}^r_X \\longrightarrow X", "$$", "where the first arrow is a closed immersion of McQuillan affine", "formal algebraic spaces. However, then questions (1) and", "(2) are in force for $Y \\to \\mathbf{A}^r_X$.", "\\end{enumerate}", "Below we will answer these questions when", "$X$ is countably indexed, i.e., when $A$ has a countable fundamental", "system of open ideals. If you have answers to these questions", "in greater generality, or if you have counter examples, please email", "\\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}." ], "refs": [ "formal-spaces-lemma-closed-immersion-into-McQuillan", "formal-spaces-lemma-category-affine-over", "formal-spaces-lemma-topologically-finite-type-finite-type", "formal-spaces-lemma-topologically-finite-type-finite-type" ], "ref_ids": [ 3946, 3949, 3948, 3948 ] }, { "id": 4019, "type": "other", "label": "formal-spaces-remark-not-enough-sections", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-not-enough-sections", "contents": [ "The structure sheaf does not always have ``enough sections''.", "In Examples, Section \\ref{examples-section-affine-formal-algebraic-space}", "we have seen that there exist affine formal algebraic spaces which", "aren't McQuillan and there are even examples whose points are not", "separated by regular functions." ], "refs": [], "ref_ids": [] }, { "id": 4020, "type": "other", "label": "formal-spaces-remark-bad-quasi-coherent", "categories": [ "formal-spaces" ], "title": "formal-spaces-remark-bad-quasi-coherent", "contents": [ "Even if the structure sheaf has good properties, this does not", "mean there is a good theory of quasi-coherent modules. For example,", "in Examples, Section \\ref{examples-section-nonabelian-QCoh}", "we have seen that for almost any Noetherian affine formal algebraic spaces", "the most natural notion of a quasi-coherent module leads to a", "category of modules which is not abelian." ], "refs": [], "ref_ids": [] }, { "id": 4144, "type": "other", "label": "pione-remark-covering-surjective", "categories": [ "pione" ], "title": "pione-remark-covering-surjective", "contents": [ "Under the correspondence of Lemma \\ref{lemma-sheaves-point},", "the coverings in the small \\'etale site", "$\\Spec(K)_\\etale$ of $K$ correspond to surjective families of", "maps in $G\\textit{-Sets}$." ], "refs": [ "pione-lemma-sheaves-point" ], "ref_ids": [ 4024 ] }, { "id": 4145, "type": "other", "label": "pione-remark-colimits-commute-forgetful", "categories": [ "pione" ], "title": "pione-remark-colimits-commute-forgetful", "contents": [ "Let $X$ be a scheme. Consider the natural functors", "$F_1 : \\textit{F\\'Et}_X \\to \\Sch$ and $F_2 : \\textit{F\\'Et}_X \\to \\Sch/X$.", "Then", "\\begin{enumerate}", "\\item The functors $F_1$ and $F_2$ commute with finite colimits.", "\\item The functor $F_2$ commutes with finite limits,", "\\item The functor $F_1$ commutes with connected finite limits, i.e.,", "with equalizers and fibre products.", "\\end{enumerate}", "The results on limits are immediate from the discussion in", "the proof of Lemma \\ref{lemma-finite-etale-covers-limits-colimits}", "and Categories, Lemma \\ref{categories-lemma-connected-limit-over-X}.", "It is clear that $F_1$ and $F_2$ commute with finite coproducts.", "By the dual of Categories, Lemma", "\\ref{categories-lemma-characterize-left-exact}", "we need to show that $F_1$ and $F_2$ commute with coequalizers.", "In the proof of Lemma \\ref{lemma-finite-etale-covers-limits-colimits}", "we saw that coequalizers in $\\textit{F\\'Et}_X$ look \\'etale locally", "like this", "$$", "\\xymatrix{", "\\coprod_{j \\in J} U \\ar@<1ex>[r]^a \\ar@<-1ex>[r]_b &", "\\coprod_{i \\in I} U \\ar[r] &", "\\coprod_{t \\in \\text{Coeq}(a, b)} U", "}", "$$", "which is certainly a coequalizer in the category of schemes.", "Hence the statement follows from the fact that being a coequalizer", "is fpqc local as formulate precisely in", "Descent, Lemma \\ref{descent-lemma-coequalizer-fpqc-local}." ], "refs": [ "pione-lemma-finite-etale-covers-limits-colimits", "categories-lemma-connected-limit-over-X", "categories-lemma-characterize-left-exact", "pione-lemma-finite-etale-covers-limits-colimits", "descent-lemma-coequalizer-fpqc-local" ], "ref_ids": [ 4037, 12215, 12245, 4037, 14639 ] }, { "id": 4146, "type": "other", "label": "pione-remark-variance", "categories": [ "pione" ], "title": "pione-remark-variance", "contents": [ "In the situation of Lemma \\ref{lemma-fundamental-group-Galois-group}", "let us give a more explicit construction of the isomorphism", "$\\text{Gal}(K^{sep}/K) \\to", "\\pi_1(X, \\overline{x}) = \\text{Aut}(F_{\\overline{x}})$.", "Observe that", "$\\text{Gal}(K^{sep}/K) = \\text{Aut}(\\overline{K}/K)$", "as $\\overline{K}$ is the perfection of $K^{sep}$.", "Since $F_{\\overline{x}}(Y) = \\Mor_X(\\Spec(\\overline{K}), Y)$", "we may consider the map", "$$", "\\text{Aut}(\\overline{K}/K) \\times F_{\\overline{x}}(Y) \\to F_{\\overline{x}}(Y),", "\\quad", "(\\sigma, \\overline{y}) \\mapsto", "\\sigma \\cdot \\overline{y} = \\overline{y} \\circ \\Spec(\\sigma)", "$$", "This is an action because", "$$", "\\sigma\\tau \\cdot \\overline{y} =", "\\overline{y} \\circ \\Spec(\\sigma\\tau) =", "\\overline{y} \\circ \\Spec(\\tau) \\circ \\Spec(\\sigma) =", "\\sigma \\cdot (\\tau \\cdot \\overline{y})", "$$", "The action is functorial in $Y \\in \\textit{F\\'Et}_X$ and we", "obtain the desired map." ], "refs": [ "pione-lemma-fundamental-group-Galois-group" ], "ref_ids": [ 4040 ] }, { "id": 4147, "type": "other", "label": "pione-remark-combine", "categories": [ "pione" ], "title": "pione-remark-combine", "contents": [ "Let $(A, \\mathfrak m)$ be a complete local Noetherian ring and", "$f \\in \\mathfrak m$ nonzero. Suppose that $A_f$ is $(S_2)$ and", "every irreducible component of $\\Spec(A)$ has dimension $\\geq 4$.", "Then Lemma \\ref{lemma-essentially-surjective-general-better}", "tells us that the category", "$$", "\\colim\\nolimits_{U' \\subset U\\text{ open, }U_0 \\subset U}", "\\text{ category of schemes finite \\'etale over }U'", "$$", "is equivalent to the category of schemes finite \\'etale over $U_0$.", "For example this holds if $A$ is a normal domain of dimension $\\geq 4$!" ], "refs": [ "pione-lemma-essentially-surjective-general-better" ], "ref_ids": [ 4105 ] }, { "id": 4180, "type": "other", "label": "stacks-cohomology-remark-bousfield-colocalization", "categories": [ "stacks-cohomology" ], "title": "stacks-cohomology-remark-bousfield-colocalization", "contents": [ "Let $\\mathcal{X}$ be an algebraic stack. The results of", "Lemmas \\ref{lemma-adjoint} and \\ref{lemma-adjoint-kernel-parasitic}", "imply that", "$$", "\\QCoh(\\mathcal{O}_\\mathcal{X}) =", "\\mathcal{M}_\\mathcal{X} / \\text{Parasitic} \\cap \\mathcal{M}_\\mathcal{X}", "$$", "in words: the category of quasi-coherent modules is the category", "of locally quasi-coherent modules with the flat base change property", "divided out by the Serre subcategory consisting of parasitic objects.", "See Homology, Lemma \\ref{homology-lemma-serre-subcategory-is-kernel}.", "The existence of the inclusion functor", "$i : \\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\mathcal{M}_\\mathcal{X}$", "which is left adjoint to the quotient functor means that", "$\\mathcal{M}_\\mathcal{X} \\to \\QCoh(\\mathcal{O}_\\mathcal{X})$", "is a {\\it Bousfield colocalization} or a {\\it right Bousfield localization}", "(insert future reference here). Our next goal is to show a similar result", "holds on the level of derived categories." ], "refs": [ "stacks-cohomology-lemma-adjoint", "stacks-cohomology-lemma-adjoint-kernel-parasitic", "homology-lemma-serre-subcategory-is-kernel" ], "ref_ids": [ 4160, 4161, 12048 ] }, { "id": 4423, "type": "other", "label": "sites-cohomology-remark-before-Leray", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-before-Leray", "contents": [ "As a consequence of the results above we find that", "Derived Categories, Lemma \\ref{derived-lemma-compose-derived-functors}", "applies to a number of situations. For example, given a", "morphism $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ of ringed topoi we have", "$$", "R\\Gamma(\\mathcal{D}, Rf_*\\mathcal{F}) = R\\Gamma(\\mathcal{C}, \\mathcal{F})", "$$", "for any sheaf of $\\mathcal{O}_\\mathcal{C}$-modules $\\mathcal{F}$.", "Namely, for an injective $\\mathcal{O}_\\mathcal{X}$-module $\\mathcal{I}$", "the $\\mathcal{O}_\\mathcal{D}$-module $f_*\\mathcal{I}$ is totally acyclic by", "Lemma \\ref{lemma-direct-image-injective-sheaf}", "and a totally acyclic sheaf is acyclic for $\\Gamma(\\mathcal{D}, -)$ by", "Lemma \\ref{lemma-limp-acyclic}." ], "refs": [ "derived-lemma-compose-derived-functors", "sites-cohomology-lemma-direct-image-injective-sheaf", "sites-cohomology-lemma-limp-acyclic" ], "ref_ids": [ 1872, 4217, 4219 ] }, { "id": 4424, "type": "other", "label": "sites-cohomology-remark-base-change", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-base-change", "contents": [ "The construction of unbounded derived functor $Lf^*$ and $Rf_*$", "allows one to construct the base change map in full generality.", "Namely, suppose that", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})", "\\ar[r]_{g'} \\ar[d]_{f'} &", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^f \\\\", "(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'})", "\\ar[r]^g &", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})", "}", "$$", "is a commutative diagram of ringed topoi. Let $K$ be an object of", "$D(\\mathcal{O}_\\mathcal{C})$.", "Then there exists a canonical base change map", "$$", "Lg^*Rf_*K \\longrightarrow R(f')_*L(g')^*K", "$$", "in $D(\\mathcal{O}_{\\mathcal{D}'})$. Namely, this map is adjoint to a map", "$L(f')^*Lg^*Rf_*K \\to L(g')^*K$.", "Since $L(f')^* \\circ Lg^* = L(g')^* \\circ Lf^*$ we see this is the same", "as a map $L(g')^*Lf^*Rf_*K \\to L(g')^*K$", "which we can take to be $L(g')^*$ of the adjunction map", "$Lf^*Rf_*K \\to K$." ], "refs": [], "ref_ids": [] }, { "id": 4425, "type": "other", "label": "sites-cohomology-remark-compose-base-change", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-compose-base-change", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{B}'), \\mathcal{O}_{\\mathcal{B}'})", "\\ar[r]_k \\ar[d]_{f'} &", "(\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B}) \\ar[d]^f \\\\", "(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})", "\\ar[r]^l \\ar[d]_{g'} &", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^g \\\\", "(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'})", "\\ar[r]^m &", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\\\", "}", "$$", "of ringed topoi. Then the base change maps of", "Remark \\ref{remark-base-change}", "for the two squares compose to give the base", "change map for the outer rectangle. More precisely,", "the composition", "\\begin{align*}", "Lm^* \\circ R(g \\circ f)_*", "& =", "Lm^* \\circ Rg_* \\circ Rf_* \\\\", "& \\to Rg'_* \\circ Ll^* \\circ Rf_* \\\\", "& \\to Rg'_* \\circ Rf'_* \\circ Lk^* \\\\", "& = R(g' \\circ f')_* \\circ Lk^*", "\\end{align*}", "is the base change map for the rectangle. We omit the verification." ], "refs": [ "sites-cohomology-remark-base-change" ], "ref_ids": [ 4424 ] }, { "id": 4426, "type": "other", "label": "sites-cohomology-remark-compose-base-change-horizontal", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-compose-base-change-horizontal", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}''), \\mathcal{O}_{\\mathcal{C}''})", "\\ar[r]_{g'} \\ar[d]_{f''} &", "(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})", "\\ar[r]_g \\ar[d]_{f'} &", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^f \\\\", "(\\Sh(\\mathcal{D}''), \\mathcal{O}_{\\mathcal{D}''})", "\\ar[r]^{h'} &", "(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'})", "\\ar[r]^h &", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})", "}", "$$", "of ringed topoi. Then the base change maps of", "Remark \\ref{remark-base-change}", "for the two squares compose to give the base", "change map for the outer rectangle. More precisely,", "the composition", "\\begin{align*}", "L(h \\circ h')^* \\circ Rf_*", "& =", "L(h')^* \\circ Lh^* \\circ Rf_* \\\\", "& \\to L(h')^* \\circ Rf'_* \\circ Lg^* \\\\", "& \\to Rf''_* \\circ L(g')^* \\circ Lg^* \\\\", "& = Rf''_* \\circ L(g \\circ g')^*", "\\end{align*}", "is the base change map for the rectangle. We omit the verification." ], "refs": [ "sites-cohomology-remark-base-change" ], "ref_ids": [ 4424 ] }, { "id": 4427, "type": "other", "label": "sites-cohomology-remark-cup-product", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-cup-product", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of", "ringed topoi. The adjointness of $Lf^*$ and $Rf_*$ allows us to construct", "a relative cup product", "$$", "Rf_*K \\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L} Rf_*L", "\\longrightarrow", "Rf_*(K \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} L)", "$$", "in $D(\\mathcal{O}_\\mathcal{D})$ for all $K, L$ in $D(\\mathcal{O}_\\mathcal{C})$.", "Namely, this map is adjoint to a map", "$Lf^*(Rf_*K \\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L} Rf_*L) \\to", "K \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} L$ for which we can take the", "composition of the isomorphism", "$Lf^*(Rf_*K \\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L} Rf_*L) =", "Lf^*Rf_*K \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} Lf^*Rf_*L$", "(Lemma \\ref{lemma-pullback-tensor-product})", "with the map", "$Lf^*Rf_*K \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} Lf^*Rf_*L", "\\to K \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} L$", "coming from the counit $Lf^* \\circ Rf_* \\to \\text{id}$." ], "refs": [ "sites-cohomology-lemma-pullback-tensor-product" ], "ref_ids": [ 4244 ] }, { "id": 4428, "type": "other", "label": "sites-cohomology-remark-discuss-derived-limit", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-discuss-derived-limit", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(K_n)$ be an inverse", "system in $D(\\mathcal{O})$. Set $K = R\\lim K_n$. For each $n$ and $m$", "let $\\mathcal{H}^m_n = H^m(K_n)$ be the $m$th cohomology sheaf of", "$K_n$ and similarly set $\\mathcal{H}^m = H^m(K)$. Let us denote", "$\\underline{\\mathcal{H}}^m_n$ the presheaf", "$$", "U \\longmapsto \\underline{\\mathcal{H}}^m_n(U) = H^m(U, K_n)", "$$", "Similarly we set $\\underline{\\mathcal{H}}^m(U) = H^m(U, K)$.", "By Lemma \\ref{lemma-sheafification-cohomology} we see that", "$\\mathcal{H}^m_n$ is the sheafification of", "$\\underline{\\mathcal{H}}^m_n$ and $\\mathcal{H}^m$ is the", "sheafification of $\\underline{\\mathcal{H}}^m$.", "Here is a diagram", "$$", "\\xymatrix{", "K \\ar@{=}[d] &", "\\underline{\\mathcal{H}}^m \\ar[d] \\ar[r] & ", "\\mathcal{H}^m \\ar[d] \\\\", "R\\lim K_n &", "\\lim \\underline{\\mathcal{H}}^m_n \\ar[r] & ", "\\lim \\mathcal{H}^m_n", "}", "$$", "In general it may not be the case that", "$\\lim \\mathcal{H}^m_n$ is the sheafification of", "$\\lim \\underline{\\mathcal{H}}^m_n$.", "If $U \\in \\mathcal{C}$, then we have short exact", "sequences", "\\begin{equation}", "\\label{equation-ses-Rlim-over-U}", "0 \\to", "R^1\\lim \\underline{\\mathcal{H}}^{m - 1}_n(U) \\to", "\\underline{\\mathcal{H}}^m(U) \\to", "\\lim \\underline{\\mathcal{H}}^m_n(U) \\to 0", "\\end{equation}", "by Lemma \\ref{lemma-RGamma-commutes-with-Rlim}." ], "refs": [ "sites-cohomology-lemma-sheafification-cohomology", "sites-cohomology-lemma-RGamma-commutes-with-Rlim" ], "ref_ids": [ 4255, 4266 ] }, { "id": 4429, "type": "other", "label": "sites-cohomology-remark-set-theoretic-LC", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-set-theoretic-LC", "contents": [ "The category $\\textit{LC}$ is a ``big'' category as its objects form", "a proper class. Similarly, the coverings form a proper class.", "Let us define the {\\it size} of a topological space $X$ to be the", "cardinality of the set of points of $X$. Choose a function", "$Bound$ on cardinals, for example as in", "Sets, Equation (\\ref{sets-equation-bound}).", "Finally, let $S_0$ be an initial set of objects of $\\textit{LC}$,", "for example $S_0 = \\{(\\mathbf{R}, \\text{euclidean topology})\\}$.", "Exactly as in Sets, Lemma \\ref{sets-lemma-construct-category}", "we can choose a limit ordinal $\\alpha$ such that", "$\\textit{LC}_\\alpha = \\textit{LC} \\cap V_\\alpha$", "contains $S_0$ and is preserved under all countable limits and", "colimits which exist in $\\textit{LC}$. Moreover, if $X \\in \\textit{LC}_\\alpha$", "and if $Y \\in \\textit{LC}$ and", "$\\text{size}(Y) \\leq Bound(\\text{size}(X))$, then $Y$ is isomorphic", "to an object of $\\textit{LC}_\\alpha$.", "Next, we apply Sets, Lemma \\ref{sets-lemma-coverings-site}", "to choose set $\\text{Cov}$ of qc covering on $\\textit{LC}_\\alpha$", "such that every qc covering in $\\textit{LC}_\\alpha$ is", "combinatorially equivalent to a covering this set.", "In this way we obtain a site $(\\textit{LC}_\\alpha, \\text{Cov})$", "which we will denote $\\textit{LC}_{qc}$." ], "refs": [ "sets-lemma-construct-category", "sets-lemma-coverings-site" ], "ref_ids": [ 8789, 8800 ] }, { "id": 4430, "type": "other", "label": "sites-cohomology-remark-projection-formula-for-internal-hom", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-projection-formula-for-internal-hom", "contents": [ "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.", "Let $K, L$ be objects of $D(\\mathcal{O}_\\mathcal{C})$. We claim there is", "a canonical map", "$$", "Rf_*R\\SheafHom(L, K) \\longrightarrow R\\SheafHom(Rf_*L, Rf_*K)", "$$", "Namely, by (\\ref{equation-internal-hom}) this is the same thing", "as a map", "$Rf_*R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L} Rf_*L", "\\to Rf_*K$.", "For this we can use the composition", "$$", "Rf_*R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L} Rf_*L \\to", "Rf_*(R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} L) \\to", "Rf_*K", "$$", "where the first arrow is the relative cup product", "(Remark \\ref{remark-cup-product}) and the second arrow is $Rf_*$ applied", "to the canonical map", "$R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} L \\to K$", "coming from Lemma \\ref{lemma-internal-hom-composition}", "(with $\\mathcal{O}_\\mathcal{C}$ in one of the spots)." ], "refs": [ "sites-cohomology-remark-cup-product", "sites-cohomology-lemma-internal-hom-composition" ], "ref_ids": [ 4427, 4332 ] }, { "id": 4431, "type": "other", "label": "sites-cohomology-remark-prepare-fancy-base-change", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-prepare-fancy-base-change", "contents": [ "Let $h : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$", "be a morphism of ringed topoi. Let $K, L$ be objects of $D(\\mathcal{O}')$.", "We claim there is a canonical map", "$$", "Lh^*R\\SheafHom(K, L) \\longrightarrow R\\SheafHom(Lh^*K, Lh^*L)", "$$", "in $D(\\mathcal{O})$. Namely, by (\\ref{equation-internal-hom})", "proved in Lemma \\ref{lemma-internal-hom}", "such a map is the same thing as a map", "$$", "Lh^*R\\SheafHom(K, L) \\otimes^\\mathbf{L} Lh^*K \\longrightarrow Lh^*L", "$$", "The source of this arrow is $Lh^*(\\SheafHom(K, L) \\otimes^\\mathbf{L} K)$", "by Lemma \\ref{lemma-pullback-tensor-product}", "hence it suffices to construct a canonical map", "$$", "R\\SheafHom(K, L) \\otimes^\\mathbf{L} K \\longrightarrow L.", "$$", "For this we take the arrow corresponding to", "$$", "\\text{id} :", "R\\SheafHom(K, L)", "\\longrightarrow", "R\\SheafHom(K, L)", "$$", "via (\\ref{equation-internal-hom})." ], "refs": [ "sites-cohomology-lemma-internal-hom", "sites-cohomology-lemma-pullback-tensor-product" ], "ref_ids": [ 4328, 4244 ] }, { "id": 4432, "type": "other", "label": "sites-cohomology-remark-fancy-base-change", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-fancy-base-change", "contents": [ "Suppose that", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})", "\\ar[r]_h \\ar[d]_{f'} &", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^f \\\\", "(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'})", "\\ar[r]^g &", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})", "}", "$$", "is a commutative diagram of ringed topoi. Let $K, L$ be objects", "of $D(\\mathcal{O}_\\mathcal{C})$. We claim there exists a canonical base change", "map", "$$", "Lg^*Rf_*R\\SheafHom(K, L)", "\\longrightarrow", "R(f')_*R\\SheafHom(Lh^*K, Lh^*L)", "$$", "in $D(\\mathcal{O}_{\\mathcal{D}'})$. Namely, we take the map adjoint to", "the composition", "\\begin{align*}", "L(f')^*Lg^*Rf_*R\\SheafHom(K, L)", "& =", "Lh^*Lf^*Rf_*R\\SheafHom(K, L) \\\\", "& \\to", "Lh^*R\\SheafHom(K, L) \\\\", "& \\to", "R\\SheafHom(Lh^*K, Lh^*L)", "\\end{align*}", "where the first arrow uses the adjunction mapping", "$Lf^*Rf_* \\to \\text{id}$ and the second arrow is the canonical map", "constructed in Remark \\ref{remark-prepare-fancy-base-change}." ], "refs": [ "sites-cohomology-remark-prepare-fancy-base-change" ], "ref_ids": [ 4431 ] }, { "id": 4433, "type": "other", "label": "sites-cohomology-remark-when-derived-shriek-equal", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-when-derived-shriek-equal", "contents": [ "Warning! Let $u : \\mathcal{C} \\to \\mathcal{D}$, $g$, $\\mathcal{O}_\\mathcal{D}$,", "and $\\mathcal{O}_\\mathcal{C}$ be as in", "Lemma \\ref{lemma-existence-derived-lower-shriek}.", "In general it is {\\bf not} the case that the diagram", "$$", "\\xymatrix{", "D(\\mathcal{O}_\\mathcal{C}) \\ar[r]_{Lg_!} \\ar[d]_{forget} &", "D(\\mathcal{O}_\\mathcal{D}) \\ar[d]^{forget} \\\\", "D(\\mathcal{C}) \\ar[r]^{Lg^{Ab}_!} &", "D(\\mathcal{D})", "}", "$$", "commutes where the functor $Lg_!^{Ab}$ is the one constructed in", "Lemma \\ref{lemma-existence-derived-lower-shriek}", "but using the constant sheaf $\\mathbf{Z}$ as the structure sheaf", "on both $\\mathcal{C}$ and $\\mathcal{D}$. In general it isn't even", "the case that $g_! = g_!^{Ab}$ (see", "Modules on Sites, Remark \\ref{sites-modules-remark-when-shriek-equal}),", "but this phenomenon {\\bf can occur even if $g_! = g_!^{Ab}$}! Namely,", "the construction of $Lg_!$ in the proof of", "Lemma \\ref{lemma-existence-derived-lower-shriek}", "shows that $Lg_!$ agrees with $Lg_!^{\\textit{Ab}}$ if and only if", "the canonical maps", "$$", "Lg^{Ab}_!j_{U!}\\mathcal{O}_U \\longrightarrow j_{u(U)!}\\mathcal{O}_{u(U)}", "$$", "are isomorphisms in $D(\\mathcal{D})$ for all objects $U$ in $\\mathcal{C}$.", "In general all we can say is that there exists a natural transformation", "$$", "Lg_!^{Ab} \\circ forget \\longrightarrow forget \\circ Lg_!", "$$" ], "refs": [ "sites-cohomology-lemma-existence-derived-lower-shriek", "sites-cohomology-lemma-existence-derived-lower-shriek", "sites-modules-remark-when-shriek-equal", "sites-cohomology-lemma-existence-derived-lower-shriek" ], "ref_ids": [ 4337, 4337, 14312, 4337 ] }, { "id": 4434, "type": "other", "label": "sites-cohomology-remark-fibred-category", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-fibred-category", "contents": [ "Assumptions and notation as in Situation \\ref{situation-fibred-category}.", "Note that setting $\\mathcal{C}' = \\mathcal{D}$ and $u$ equal to the", "structure functor of $\\mathcal{C}$ gives a situation as in", "Situation \\ref{situation-morphism-fibred-categories}. Hence", "Lemma \\ref{lemma-properties-lower-shriek-fibred-category}", "tells us we have functors $\\pi_!$, $\\pi_!^{\\textit{Ab}}$, $L\\pi_!$, and", "$L\\pi_!^{\\textit{Ab}}$ such that", "$forget \\circ \\pi_! = \\pi_!^{\\textit{Ab}} \\circ forget$ and", "$forget \\circ L\\pi_! = L\\pi_!^{\\textit{Ab}} \\circ forget$." ], "refs": [ "sites-cohomology-lemma-properties-lower-shriek-fibred-category" ], "ref_ids": [ 4344 ] }, { "id": 4435, "type": "other", "label": "sites-cohomology-remark-morphism-fibred-categories", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-morphism-fibred-categories", "contents": [ "Assumptions and notation as in", "Situation \\ref{situation-morphism-fibred-categories}.", "Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}$,", "let $\\mathcal{F}'$ be an abelian sheaf on $\\mathcal{C}'$,", "and let $t : \\mathcal{F}' \\to g^{-1}\\mathcal{F}$ be a map.", "Then we obtain a canonical map", "$$", "L\\pi'_!(\\mathcal{F}') \\longrightarrow L\\pi_!(\\mathcal{F})", "$$", "by using the adjoint $g_!\\mathcal{F}' \\to \\mathcal{F}$ of $t$,", "the map $Lg_!(\\mathcal{F}') \\to g_!\\mathcal{F}'$, and the", "equality $L\\pi'_! = L\\pi_! \\circ Lg_!$." ], "refs": [], "ref_ids": [] }, { "id": 4436, "type": "other", "label": "sites-cohomology-remark-map-evaluation-to-derived", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-map-evaluation-to-derived", "contents": [ "Notation and assumptions as in Example \\ref{example-category-to-point}.", "Let $\\mathcal{F}^\\bullet$ be a bounded complex of abelian sheaves on", "$\\mathcal{C}$. For any object $U$ of $\\mathcal{C}$ there is a canonical", "map", "$$", "\\mathcal{F}^\\bullet(U) \\longrightarrow L\\pi_!(\\mathcal{F}^\\bullet)", "$$", "in $D(\\textit{Ab})$. If $\\mathcal{F}^\\bullet$ is a complex of", "$\\underline{B}$-modules then this map is in $D(B)$. To prove this, note", "that we compute $L\\pi_!(\\mathcal{F}^\\bullet)$ by taking a quasi-isomorphism", "$\\mathcal{P}^\\bullet \\to \\mathcal{F}^\\bullet$ where $\\mathcal{P}^\\bullet$", "is a complex of projectives. However, since the topology is chaotic", "this means that $\\mathcal{P}^\\bullet(U) \\to \\mathcal{F}^\\bullet(U)$", "is a quasi-isomorphism hence can be inverted in", "$D(\\textit{Ab})$, resp.\\ $D(B)$. Composing with the canonical map", "$\\mathcal{P}^\\bullet(U) \\to \\pi_!(\\mathcal{P}^\\bullet)$ coming from", "the computation of $\\pi_!$ as a colimit we obtain the desired arrow." ], "refs": [], "ref_ids": [] }, { "id": 4437, "type": "other", "label": "sites-cohomology-remark-simplicial-modules", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-simplicial-modules", "contents": [ "Let $\\mathcal{C} = \\Delta$ and let $B$ be any ring. This is a special", "case of Example \\ref{example-category-to-point} where the assumptions", "of Lemma \\ref{lemma-compute-by-cosimplicial-resolution} hold.", "Namely, let $U_\\bullet$ be the cosimplicial object of $\\Delta$ given by", "the identity functor. To verify the condition we have to show that for", "$[m] \\in \\Ob(\\Delta)$ the simplicial set", "$\\Delta[m] : n \\mapsto \\Mor_\\Delta([n], [m])$ is homotopy equivalent", "to a point. This is explained in", "Simplicial, Example \\ref{simplicial-example-simplex-contractible}.", "\\medskip\\noindent", "In this situation the category $\\textit{Mod}(\\underline{B})$", "is just the category of simplicial $B$-modules and the", "functor $L\\pi_!$ sends a simplicial $B$-module $M_\\bullet$ to its associated", "complex $s(M_\\bullet)$ of $B$-modules. Thus the results above can be", "reinterpreted in terms of results on simplicial modules. For example", "a special case of Lemma \\ref{lemma-eilenberg-zilber} is:", "if $M_\\bullet$, $M'_\\bullet$ are flat simplicial", "$B$-modules, then the complex $s(M_\\bullet \\otimes_B M'_\\bullet)$ is", "quasi-isomorphic to the total complex associated to the double complex", "$s(M_\\bullet) \\otimes_B s(M'_\\bullet)$.", "(Hint: use flatness to convert from derived tensor products to usual", "tensor products.)", "This is a special case of the Eilenberg-Zilber theorem", "which can be found in \\cite{Eilenberg-Zilber}." ], "refs": [ "sites-cohomology-lemma-compute-by-cosimplicial-resolution", "sites-cohomology-lemma-eilenberg-zilber" ], "ref_ids": [ 4348, 4351 ] }, { "id": 4438, "type": "other", "label": "sites-cohomology-remark-O-homology-B-homology-general", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-O-homology-B-homology-general", "contents": [ "Let $\\mathcal{C}$ and $B$ be as in Example \\ref{example-category-to-point}.", "Assume there exists a cosimplicial object as in", "Lemma \\ref{lemma-compute-by-cosimplicial-resolution}.", "Let $\\mathcal{O} \\to \\underline{B}$ be a map sheaf of rings on", "$\\mathcal{C}$ which induces an isomorphism", "$L\\pi_!\\mathcal{O} \\to L\\pi_!\\underline{B}$.", "In this case we obtain an exact functor of triangulated categories", "$$", "L\\pi_! : D(\\mathcal{O}) \\longrightarrow D(B)", "$$", "Namely, for any object $K$ of $D(\\mathcal{O})$ we have", "$L\\pi^{\\textit{Ab}}_!(K) =", "L\\pi^{\\textit{Ab}}_!(K \\otimes_{\\mathcal{O}}^\\mathbf{L} \\underline{B})$", "by Lemma \\ref{lemma-O-homology-qis}.", "Thus we can define the displayed functor as the composition of", "$- \\otimes^\\mathbf{L}_\\mathcal{O} \\underline{B}$ with the functor", "$L\\pi_! : D(\\underline{B}) \\to D(B)$.", "In other words, we obtain a $B$-module structure on $L\\pi_!(K)$ coming", "from the (canonical, functorial) identification of $L\\pi_!(K)$ with", "$L\\pi_!(K \\otimes_\\mathcal{O}^\\mathbf{L} \\underline{B})$ of the lemma." ], "refs": [ "sites-cohomology-lemma-compute-by-cosimplicial-resolution", "sites-cohomology-lemma-O-homology-qis" ], "ref_ids": [ 4348, 4352 ] }, { "id": 4439, "type": "other", "label": "sites-cohomology-remark-homology-augmentation", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-homology-augmentation", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$\\epsilon : \\mathcal{A}_\\bullet \\to \\mathcal{O}$ be an augmentation", "(Simplicial, Definition \\ref{simplicial-definition-augmentation})", "in the category of sheaves of rings.", "Assume $\\epsilon$ induces a quasi-isomorphism", "$s(\\mathcal{A}_\\bullet) \\to \\mathcal{O}$.", "In this case we obtain an exact functor of triangulated categories", "$$", "L\\pi_! : D(\\mathcal{A}_\\bullet) \\longrightarrow D(\\mathcal{O})", "$$", "Namely, for any object $K$ of $D(\\mathcal{A}_\\bullet)$ we have", "$L\\pi_!(K) = L\\pi_!(K \\otimes_{\\mathcal{A}_\\bullet}^\\mathbf{L} \\mathcal{O})$", "by Lemma \\ref{lemma-base-change-by-qis}.", "Thus we can define the displayed functor as the composition of", "$- \\otimes^\\mathbf{L}_{\\mathcal{A}_\\bullet} \\mathcal{O}$ with the functor", "$L\\pi_! : D(\\Delta \\times \\mathcal{C}, \\pi^{-1}\\mathcal{O}) \\to", "D(\\mathcal{O})$ of Remark \\ref{remark-fibred-category}.", "In other words, we obtain a $\\mathcal{O}$-module structure on $L\\pi_!(K)$", "coming from the (canonical, functorial) identification of $L\\pi_!(K)$ with", "$L\\pi_!(K \\otimes_{\\mathcal{A}_\\bullet}^\\mathbf{L} \\mathcal{O})$ of the lemma." ], "refs": [ "simplicial-definition-augmentation", "sites-cohomology-lemma-base-change-by-qis", "sites-cohomology-remark-fibred-category" ], "ref_ids": [ 14925, 4356, 4434 ] }, { "id": 4440, "type": "other", "label": "sites-cohomology-remark-compatible-with-diagram", "categories": [ "sites-cohomology" ], "title": "sites-cohomology-remark-compatible-with-diagram", "contents": [ "The map (\\ref{equation-projection-formula-map}) is compatible with the", "base change map of Remark \\ref{remark-base-change} in the following sense.", "Namely, suppose that", "$$", "\\xymatrix{", "(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})", "\\ar[r]_{g'} \\ar[d]_{f'} &", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^f \\\\", "(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'})", "\\ar[r]^g &", "(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})", "}", "$$", "is a commutative diagram of ringed topoi.", "Let $E \\in D(\\mathcal{O}_\\mathcal{C})$ and $K \\in D(\\mathcal{O}_\\mathcal{D})$.", "Then the diagram", "$$", "\\xymatrix{", "Lg^*(Rf_*E \\otimes^\\mathbf{L}_{\\mathcal{O}_\\mathcal{D}} K)", "\\ar[r]_p \\ar[d]_t &", "Lg^*Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_\\mathcal{C}} Lf^*K)", "\\ar[d]_b \\\\", "Lg^*Rf_*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{\\mathcal{D}'}}", "Lg^*K \\ar[d]_b &", "Rf'_*L(g')^*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_\\mathcal{C}}", "Lf^*K) \\ar[d]_t \\\\", "Rf'_*L(g')^*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{\\mathcal{D}'}}", "Lg^*K \\ar[rd]_p &", "Rf'_*(L(g')^*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{\\mathcal{D}'}}", "L(g')^*Lf^*K) \\ar[d]_c \\\\", "& Rf'_*(L(g')^*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{\\mathcal{D}'}}", "L(f')^*Lg^*K)", "}", "$$", "is commutative. Here arrows labeled $t$ are gotten by an application of", "Lemma \\ref{lemma-pullback-tensor-product}, arrows labeled $b$ by an", "application of Remark \\ref{remark-base-change}, arrows labeled $p$", "by an application of (\\ref{equation-projection-formula-map}), and", "$c$ comes from $L(g')^* \\circ Lf^* = L(f')^* \\circ Lg^*$.", "We omit the verification." ], "refs": [ "sites-cohomology-remark-base-change", "sites-cohomology-lemma-pullback-tensor-product", "sites-cohomology-remark-base-change" ], "ref_ids": [ 4424, 4244, 4424 ] }, { "id": 4663, "type": "other", "label": "spaces-limits-remark-limit-preserving", "categories": [ "spaces-limits" ], "title": "spaces-limits-remark-limit-preserving", "contents": [ "Here is an important special case of", "Proposition \\ref{proposition-characterize-locally-finite-presentation}.", "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Then $X$ is locally of finite presentation over $S$ if and only", "if $X$, as a functor $(\\Sch/S)^{opp} \\to \\textit{Sets}$,", "is limit preserving. Compare with", "Limits, Remark \\ref{limits-remark-limit-preserving}.", "In fact, we will see in Lemma \\ref{lemma-surjection-is-enough}", "below that it suffices if the map", "$$", "\\colim X(T_i) \\longrightarrow X(T)", "$$", "is surjective whenever $T = \\lim T_i$ is a directed limit of", "affine schemes over $S$." ], "refs": [ "spaces-limits-proposition-characterize-locally-finite-presentation", "limits-remark-limit-preserving", "spaces-limits-lemma-surjection-is-enough" ], "ref_ids": [ 4655, 15130, 4564 ] }, { "id": 4664, "type": "other", "label": "spaces-limits-remark-cannot-embed-in-general", "categories": [ "spaces-limits" ], "title": "spaces-limits-remark-cannot-embed-in-general", "contents": [ "We have seen in Examples, Section \\ref{examples-section-embedding-affines}", "that Lemma \\ref{lemma-embedding-into-affine-over-ls-qs}", "does not hold if we drop the assumption that $X$ be locally separated.", "This raises the question: Does", "Lemma \\ref{lemma-embedding-into-affine-over-ls-qs}", "hold if we drop the assumption that $X$ be quasi-separated?", "If you know the answer, please email", "\\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}." ], "refs": [ "spaces-limits-lemma-embedding-into-affine-over-ls-qs", "spaces-limits-lemma-embedding-into-affine-over-ls-qs" ], "ref_ids": [ 4619, 4619 ] }, { "id": 4665, "type": "other", "label": "spaces-limits-remark-finite-type-gives-well-defined-system", "categories": [ "spaces-limits" ], "title": "spaces-limits-remark-finite-type-gives-well-defined-system", "contents": [ "In Situation \\ref{situation-limit-noetherian}", "Lemmas \\ref{lemma-good-diagram}, \\ref{lemma-limit-from-good-diagram}, and", "\\ref{lemma-morphism-good-diagram}", "tell us that the category of algebraic spaces quasi-separated and", "of finite type over $B$ is equivalent to certain types of", "inverse systems of algebraic spaces over $(B_i)_{i \\in I}$, namely", "the ones produced by applying Lemma \\ref{lemma-limit-from-good-diagram}", "to a diagram of the form (\\ref{equation-good-diagram}).", "For example, given $X \\to B$ finite type and quasi-separated", "if we choose two different diagrams $X \\to V_1 \\to B_{i_1}$", "and $X \\to V_2 \\to B_{i_2}$ as in (\\ref{equation-good-diagram}), then", "applying Lemma \\ref{lemma-morphism-good-diagram} to $\\text{id}_X$", "(in two directions)", "we see that the corresponding limit descriptions of", "$X$ are canonically isomorphic (up to shrinking the", "directed set $I$). And so on and so forth." ], "refs": [ "spaces-limits-lemma-good-diagram", "spaces-limits-lemma-limit-from-good-diagram", "spaces-limits-lemma-morphism-good-diagram", "spaces-limits-lemma-limit-from-good-diagram", "spaces-limits-lemma-morphism-good-diagram" ], "ref_ids": [ 4648, 4649, 4650, 4649, 4650 ] }, { "id": 4704, "type": "other", "label": "stacks-geometry-remark-upgrade", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-remark-upgrade", "contents": [ "In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$", "be a morphism, where $k$ is a finite type field over $S$.", "Let $A$ be a versal ring to $\\mathcal{X}$ at $x_0$. By Artin's Axioms,", "Lemma \\ref{artin-lemma-effective} our versal formal object", "in fact comes from a morphism", "$$", "\\Spec(A) \\longrightarrow \\mathcal{X}", "$$", "over $S$. Moreover, the results above each can be upgraded to be compatible", "with this morphism. Here is a list:", "\\begin{enumerate}", "\\item in Lemma \\ref{lemma-versal-ring} the isomorphism", "$A \\cong A'[[t_1, \\ldots, t_r]]$ or $A' \\cong A[[t_1, \\ldots, t_r]]$", "may be chosen compatible with these morphisms,", "\\item in Lemma \\ref{lemma-versal-ring-field-extension} the", "homomorphism $A \\to A'$ may be chosen compatible with these morphisms,", "\\item in Lemma \\ref{lemma-compare-versal-ring-completion}", "the morphism $\\Spec(\\mathcal{O}_{U, u_0}^\\wedge) \\to \\mathcal{X}$", "is the composition of the canonical map", "$\\Spec(\\mathcal{O}_{U, u_0}^\\wedge) \\to U$ and the given map", "$U \\to \\mathcal{X}$,", "\\item in Lemma \\ref{lemma-Artin-approximation-by-smooth-morphism}", "the isomorphism $\\mathcal{O}_{U, u_0}^\\wedge \\cong A$ may", "be chosen so $\\Spec(A) \\to \\mathcal{X}$ corresponds to the canonical map", "in the item above.", "\\end{enumerate}", "In each case the statement follows from the fact that our maps are", "compatible with versal formal elements; we note however that the", "implied diagrams are $2$-commutative only up to a (noncanonical)", "choice of a $2$-arrow. Still, this means that the implied map $A' \\to A$", "or $A \\to A'$ in (1) is well defined up to formal homotopy, see", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-versal-unique-up-to-homotopy}." ], "refs": [ "artin-lemma-effective", "stacks-geometry-lemma-versal-ring", "stacks-geometry-lemma-versal-ring-field-extension", "stacks-geometry-lemma-compare-versal-ring-completion", "stacks-geometry-lemma-Artin-approximation-by-smooth-morphism", "formal-defos-lemma-versal-unique-up-to-homotopy" ], "ref_ids": [ 11362, 4667, 4668, 4669, 4671, 3503 ] }, { "id": 4705, "type": "other", "label": "stacks-geometry-remark-groupoid-defo", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-remark-groupoid-defo", "contents": [ "In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$", "be a morphism, where $k$ is a finite type field over $S$.", "By Lemma \\ref{lemma-deformation-category} and", "Formal Deformation Theory, Theorem", "\\ref{formal-defos-theorem-presentation-deformation-groupoid}", "we know that $\\mathcal{F}_{\\mathcal{X}, k, x_0}$ has a", "presentation by a smooth prorepresentable groupoid in", "functors on $\\mathcal{C}_\\Lambda$.", "Unwinding the definitions, this means we can choose", "\\begin{enumerate}", "\\item a Noetherian complete local $\\Lambda$-algebra $A$", "with residue field $k$ and a versal formal object $\\xi$", "of $\\mathcal{F}_{\\mathcal{X}, k, x_0}$ over $A$,", "\\item a Noetherian complete local $\\Lambda$-algebra $B$", "with residue field $k$ and an isomorphism", "$$", "\\underline{B}|_{\\mathcal{C}_\\Lambda}", "\\longrightarrow", "\\underline{A}|_{\\mathcal{C}_\\Lambda}", "\\times_{\\underline{\\xi}, \\mathcal{F}_{\\mathcal{X}, k, x_0}, \\underline{\\xi}}", "\\underline{A}|_{\\mathcal{C}_\\Lambda}", "$$", "\\end{enumerate}", "The projections correspond to formally smooth maps", "$t : A \\to B$ and $s : A \\to B$ (because $\\xi$ is versal).", "There is a map $c : B \\to B \\widehat{\\otimes}_{s, A, t} B$", "which turns $(A, B, s, t, c)$ into a cogroupoid in the category", "of Noetherian complete local $\\Lambda$-algebras with residue field $k$", "(on prorepresentable functors this map is constructed in", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-presentation-construction}).", "Finally, the cited theorem tells us that $\\xi$ induces", "an equivalence", "$$", "[\\underline{A}|_{\\mathcal{C}_\\Lambda} / \\underline{B}|_{\\mathcal{C}_\\Lambda}]", "\\longrightarrow", "\\mathcal{F}_{\\mathcal{X}, k, x_0}", "$$", "of groupoids cofibred over $\\mathcal{C}_\\Lambda$. In fact, we also", "get an equivalence", "$$", "[\\underline{A}/\\underline{B}]", "\\longrightarrow", "\\widehat{\\mathcal{F}}_{\\mathcal{X}, k, x_0}", "$$", "of groupoids cofibred over the completed category", "$\\widehat{\\mathcal{C}}_\\Lambda$ (see discussion in", "Formal Deformation Theory, Section", "\\ref{formal-defos-section-prorepresentable-groupoids-in-functors}", "as to why this works). Of course $A$ is a versal ring to", "$\\mathcal{X}$ at $x_0$." ], "refs": [ "stacks-geometry-lemma-deformation-category", "formal-defos-theorem-presentation-deformation-groupoid", "formal-defos-lemma-presentation-construction" ], "ref_ids": [ 4666, 3412, 3490 ] }, { "id": 4706, "type": "other", "label": "stacks-geometry-remark-dimension-algebraic-space", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-remark-dimension-algebraic-space", "contents": [ "In general, the dimension of the algebraic space $X$ at a point $x$", "may not coincide with the dimension of the underlying topological space", "$|X|$ at $x$. E.g.\\ if $k$ is a field of characteristic zero and", "$X = \\mathbf{A}^1_k / \\mathbf{Z}$, then $X$ has dimension $1$ (the dimension", "of $\\mathbf{A}^1_k$) at each of its points,", "while $|X|$ has the indiscrete topology, and hence is of Krull", "dimension zero. On the other hand, in", "Algebraic Spaces, Example \\ref{spaces-example-infinite-product}", "there is given an example of an algebraic space", "which is of dimension $0$ at each of its points, while $|X|$ is", "irreducible of Krull dimension $1$, and admits a generic point (so that the", "dimension of $|X|$ at any of its points is $1$); see also the discussion", "of this example in", "Properties of Spaces, Section \\ref{spaces-properties-section-dimension}.", "\\medskip\\noindent", "On the other hand, if $X$ is a {\\it decent} algebraic space, in the sense of", "Decent Spaces, Definition \\ref{decent-spaces-definition-very-reasonable}", "(in particular, if $X$ is quasi-separated; see", "Decent Spaces, Section \\ref{decent-spaces-section-reasonable-decent})", "then in fact the dimension of $X$ at $x$ does coincide with the dimension", "of $|X|$ at $x$; see", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-dimension-decent-space}." ], "refs": [ "decent-spaces-definition-very-reasonable", "decent-spaces-lemma-dimension-decent-space" ], "ref_ids": [ 9562, 9496 ] }, { "id": 4707, "type": "other", "label": "stacks-geometry-remark-relative-dimension", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-remark-relative-dimension", "contents": [ "(1)", "One easily verifies (for example, by using the invariance", "of the relative dimension of locally of finite type morphisms of schemes", "under base-change; see for example", "Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change})", "that $\\dim_t(T_x)$ is well-defined, independently of the choices", "used to compute it.", "\\medskip\\noindent", "(2)", "In the case that $\\mathcal{X}$ is also an algebraic space,", "it is straightforward to confirm that this definition agrees with", "the definition of relative dimension given in", "Morphisms of Spaces, Definition", "\\ref{spaces-morphisms-definition-dimension-fibre}." ], "refs": [ "morphisms-lemma-dimension-fibre-after-base-change", "spaces-morphisms-definition-dimension-fibre" ], "ref_ids": [ 5279, 5009 ] }, { "id": 4708, "type": "other", "label": "stacks-geometry-remark-dimension-DM", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-remark-dimension-DM", "contents": [ "For Deligne--Mumford stacks which are suitably decent", "(e.g.\\ quasi-separated),", "it will again be the case that $\\dim_x(\\mathcal{X})$ coincides with the", "topologically", "defined quantity $\\dim_x |\\mathcal{X}|$. However, for more general Artin", "stacks,", "this will typically not be the case. For example, if", "$\\mathcal{X} = [\\mathbf{A}^1/\\mathbf{G}_m]$", "(over some field, with the quotient being taken with", "respect to the usual multiplication action of $\\mathbf{G}_m$ on $\\mathbf{A}^1$),", "then $|\\mathcal{X}|$ has two points, one the specialisation of the other", "(corresponding", "to the two orbits of $\\mathbf{G}_m$ on $\\mathbf{A}^1$), and hence is of", "dimension $1$ as", "a topological space; but $\\dim_x (\\mathcal{X}) = 0$ for both points", "$x \\in |\\mathcal{X}|$.", "(An even more extreme example is given by the classifying space", "$[\\Spec k/\\mathbf{G}_m]$, whose dimension at its unique point", "is equal to $-1$.)" ], "refs": [], "ref_ids": [] }, { "id": 4709, "type": "other", "label": "stacks-geometry-remark-dimension-tangent-space-well-defined", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-remark-dimension-tangent-space-well-defined", "contents": [ "Standard manipulations show that $\\dim_t(\\mathcal{T}_x)$ is well-defined,", "independently of the choices made to compute it." ], "refs": [], "ref_ids": [] }, { "id": 4710, "type": "other", "label": "stacks-geometry-remark-negative-dimension", "categories": [ "stacks-geometry" ], "title": "stacks-geometry-remark-negative-dimension", "contents": [ "We note that in the context of the preceding lemma,", "it need not be that $\\dim \\mathcal{T} \\geq \\dim \\mathcal{Z}$; this does", "not contradict the inequality in the statement of the lemma, because", "the fibres of the morphism $f$ are again algebraic stacks, and", "so may have negative dimension. This is illustrated by taking", "$k$ to be a field, and applying the lemma to the morphism", "$[\\Spec k/\\mathbf{G}_m] \\to \\Spec k$.", "\\medskip\\noindent", "If the morphism $f$ in the statement of the lemma is assumed", "to be quasi-DM (in the sense of", "Morphisms of Stacks, Definition", "\\ref{stacks-morphisms-definition-separated}; e.g.\\ morphisms that are", "representable by algebraic spaces are quasi-DM),", "then the fibres of the morphism over points of the target", "are quasi-DM algebraic stacks, and hence are of non-negative", "dimension. In this case, the lemma implies", "that indeed $\\dim \\mathcal{T} \\geq \\dim \\mathcal{Z}$. In fact, we obtain", "the following more general result." ], "refs": [ "stacks-morphisms-definition-separated" ], "ref_ids": [ 7601 ] }, { "id": 5029, "type": "other", "label": "spaces-morphisms-remark-immersion", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-remark-immersion", "contents": [ "Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic", "spaces over $S$. Since $i$ is a monomorphism we may think of $|Z|$ as", "a subset of $|X|$; in the rest of this remark we do so.", "Let $\\partial |Z|$ be the boundary of $|Z|$ in", "the topological space $|X|$. In a formula", "$$", "\\partial |Z| = \\overline{|Z|} \\setminus |Z|.", "$$", "Let $\\partial Z$ be the reduced closed subspace of $X$ with", "$|\\partial Z| = \\partial |Z|$", "obtained by taking the reduced induced closed subspace structure, see", "Properties of Spaces,", "Definition \\ref{spaces-properties-definition-reduced-induced-space}.", "By construction we see that $|Z|$ is closed in", "$|X| \\setminus |\\partial Z| = |X \\setminus \\partial Z|$.", "Hence it is true that any immersion of algebraic spaces can be", "factored as a closed immersion followed by an open immersion", "(but not the other way in general, see", "Morphisms, Example \\ref{morphisms-example-thibaut})." ], "refs": [ "spaces-properties-definition-reduced-induced-space" ], "ref_ids": [ 11932 ] }, { "id": 5030, "type": "other", "label": "spaces-morphisms-remark-space-structure-locally-closed-subset", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-remark-space-structure-locally-closed-subset", "contents": [ "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.", "Let $T \\subset |X|$ be a locally closed subset.", "Let $\\partial T$ be the boundary of $T$ in", "the topological space $|X|$. In a formula", "$$", "\\partial T = \\overline{T} \\setminus T.", "$$", "Let $U \\subset X$ be the open subspace of $X$ with", "$|U| = |X| \\setminus \\partial T$, see", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-open-subspaces}.", "Let $Z$ be the reduced closed subspace of $U$ with", "$|Z| = T$ obtained by taking the reduced induced", "closed subspace structure, see", "Properties of Spaces,", "Definition \\ref{spaces-properties-definition-reduced-induced-space}.", "By construction $Z \\to U$ is a closed immersion of algebraic spaces", "and $U \\to X$ is an open immersion, hence", "$Z \\to X$ is an immersion of algebraic spaces over $S$ (see", "Spaces, Lemma \\ref{spaces-lemma-composition-immersions}).", "Note that $Z$ is a reduced algebraic space and that", "$|Z| = T$ as subsets of $|X|$. We sometimes say", "$Z$ is the {\\it reduced induced subspace structure} on $T$." ], "refs": [ "spaces-properties-lemma-open-subspaces", "spaces-properties-definition-reduced-induced-space", "spaces-lemma-composition-immersions" ], "ref_ids": [ 11823, 11932, 8160 ] }, { "id": 5031, "type": "other", "label": "spaces-morphisms-remark-universally-injective-not-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-remark-universally-injective-not-separated", "contents": [ "A universally injective morphism of schemes is separated, see", "Morphisms, Lemma", "\\ref{morphisms-lemma-universally-injective-separated}.", "This is not the case for morphisms of algebraic spaces.", "Namely, the algebraic space", "$X = \\mathbf{A}^1_k/\\{x \\sim -x \\mid x \\not = 0\\}$", "constructed in", "Spaces, Example \\ref{spaces-example-affine-line-involution}", "comes equipped with a morphism $X \\to \\mathbf{A}^1_k$ which maps", "the point with coordinate $x$ to the point with coordinate $x^2$.", "This is an isomorphism away from $0$, and there is a unique point", "of $X$ lying above $0$. As $X$ isn't separated this is a universally", "injective morphism of algebraic spaces which is not separated." ], "refs": [ "morphisms-lemma-universally-injective-separated" ], "ref_ids": [ 5168 ] }, { "id": 5032, "type": "other", "label": "spaces-morphisms-remark-factorization-quasi-compact-quasi-separated", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-remark-factorization-quasi-compact-quasi-separated", "contents": [ "Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and", "quasi-separated morphism of algebraic spaces over $S$. Then", "$f$ has a canonical factorization", "$$", "Y \\longrightarrow \\underline{\\Spec}_X(f_*\\mathcal{O}_Y) \\longrightarrow X", "$$", "This makes sense because $f_*\\mathcal{O}_Y$ is quasi-coherent", "by Lemma \\ref{lemma-pushforward}. The morphism", "$Y \\to \\underline{\\Spec}_X(f_*\\mathcal{O}_Y)$ comes from", "the canonical $\\mathcal{O}_Y$-algebra map", "$f^*f_*\\mathcal{O}_Y \\to \\mathcal{O}_Y$ which corresponds to", "a canonical morphism", "$Y \\to Y \\times_X \\underline{\\Spec}_X(f_*\\mathcal{O}_Y)$ over $Y$ (see", "Lemma \\ref{lemma-affine-equivalence-algebras}) whence a factorization", "of $f$ as above." ], "refs": [ "spaces-morphisms-lemma-pushforward", "spaces-morphisms-lemma-affine-equivalence-algebras" ], "ref_ids": [ 4760, 4802 ] }, { "id": 5033, "type": "other", "label": "spaces-morphisms-remark-composition-P", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-remark-composition-P", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of schemes", "which is \\'etale local on the source-and-target. Suppose that moreover", "$\\mathcal{P}$ is stable under compositions. Then the class of morphisms", "of algebraic spaces having property $\\mathcal{P}$ is stable under composition." ], "refs": [], "ref_ids": [] }, { "id": 5034, "type": "other", "label": "spaces-morphisms-remark-base-change-P", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-remark-base-change-P", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of schemes", "which is \\'etale local on the source-and-target. Suppose that moreover", "$\\mathcal{P}$ is stable under base change. Then the class of morphisms", "of algebraic spaces having property $\\mathcal{P}$ is stable under base change." ], "refs": [], "ref_ids": [] }, { "id": 5035, "type": "other", "label": "spaces-morphisms-remark-when-apply", "categories": [ "spaces-morphisms" ], "title": "spaces-morphisms-remark-when-apply", "contents": [ "We will apply", "Lemma \\ref{lemma-local-source-target-global-implies-local}", "above to all cases listed in", "Descent, Remark \\ref{descent-remark-list-local-source-target}", "except ``flat''. In each case we will do this by defining", "$f$ to have property $\\mathcal{P}$ at $x$ if $f$ has", "$\\mathcal{P}$ in a neighbourhood of $x$." ], "refs": [ "spaces-morphisms-lemma-local-source-target-global-implies-local", "descent-remark-list-local-source-target" ], "ref_ids": [ 4813, 14798 ] }, { "id": 5117, "type": "other", "label": "weil-remark-transpose", "categories": [ "weil" ], "title": "weil-remark-transpose", "contents": [ "Let $X$ and $Y$ be smooth projective schemes over $k$.", "Assume $X$ is equidimensional of dimension $d$ and", "$Y$ is equidimensional of dimension $e$. Then the isomorphism", "$X \\times Y \\to Y \\times X$ switching the factors determines", "an isomorphism", "$$", "\\text{Corr}^r(X, Y) \\longrightarrow \\text{Corr}^{d - e + r}(Y, X),\\quad", "c \\longmapsto c^t", "$$", "called the {\\it transpose}. It acts on cycles as well as cycle classes.", "An example which is sometimes useful, is the transpose", "$[\\Gamma_f]^t = [\\Gamma_f^t]$ of the graph of a morphism $f : Y \\to X$." ], "refs": [], "ref_ids": [] }, { "id": 5118, "type": "other", "label": "weil-remark-lefschetz-tate", "categories": [ "weil" ], "title": "weil-remark-lefschetz-tate", "contents": [ "Let $X = \\mathbf{P}^1_k$ and $c_2$ be as in Example \\ref{example-decompose-P1}.", "In the literature the motive $(X, c_2, 0)$ is sometimes called the", "{\\it Lefschetz motive} and depending on the reference the notation", "$L$, $\\mathbf{L}$, $\\mathbf{Q}(-1)$, or $h^2(\\mathbf{P}^1_k)$", "may be used to denote it. By Lemma \\ref{lemma-inverse-h2} the Lefschetz motive", "is isomorphic to $\\mathbf{1}(-1)$. Hence the Lefschetz motive is", "invertible (Categories, Definition \\ref{categories-definition-invertible})", "with inverse", "$\\mathbf{1}(1)$. The motive $\\mathbf{1}(1)$ is sometimes called the", "{\\it Tate motive} and depending on the reference the notation", "$L^{-1}$, $\\mathbf{L}^{-1}$, $\\mathbf{T}$, or $\\mathbf{Q}(1)$ may", "be used to denote it." ], "refs": [ "weil-lemma-inverse-h2", "categories-definition-invertible" ], "ref_ids": [ 5045, 12406 ] }, { "id": 5119, "type": "other", "label": "weil-remark-replace-cup-product-classical", "categories": [ "weil" ], "title": "weil-remark-replace-cup-product-classical", "contents": [ "Let $X$ be a smooth projective variety. We obtain maps", "$$", "H^*(X) \\otimes_F H^*(X) \\longrightarrow H^*(X \\times X)", "\\xrightarrow{\\Delta^*} H^*(X)", "$$", "where the first arrow is as in axiom (B) and $\\Delta^*$", "is pullback along the diagonal morphism $\\Delta : X \\to X \\times X$.", "The composition is the cup product as pullback is an algebra homomorphism and", "$\\text{pr}_i \\circ \\Delta = \\text{id}$.", "On the other hand, given cycles $\\alpha, \\beta$ on $X$ ", "the intersection product is defined by the formula", "$$", "\\alpha \\cdot \\beta =", "\\Delta^!(\\alpha \\times \\beta)", "$$", "In other words, $\\alpha \\cdot \\beta$ is the pullback of the", "exterior product $\\alpha \\times \\beta$ on $X \\times X$ by", "the diagonal. Note also that", "$\\alpha \\times \\beta = \\text{pr}_1^*\\alpha \\cdot \\text{pr}_2^*\\beta$", "in $\\CH^*(X \\times X)$ (we omit the proof). Hence, given axiom (C)(a),", "axiom (C)(c) is equivalent to the statement that $\\gamma$ is", "compatible with exterior product in the sense that", "$\\gamma(\\alpha \\times \\beta)$ is equal to", "$\\text{pr}_1^*\\gamma(\\alpha) \\cup \\text{pr}_2^*\\gamma(\\beta)$.", "This is how axiom (C)(c) is formulated in \\cite{Kleiman-cycles}." ], "refs": [], "ref_ids": [] }, { "id": 5120, "type": "other", "label": "weil-remark-replace-cup-product", "categories": [ "weil" ], "title": "weil-remark-replace-cup-product", "contents": [ "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C)(a).", "Let $X$ be a smooth projective scheme over $k$. We obtain maps", "$$", "H^*(X) \\otimes_F H^*(X) \\longrightarrow H^*(X \\times X)", "\\xrightarrow{\\Delta^*} H^*(X)", "$$", "where the first arrow is as in axiom (B) and $\\Delta^*$", "is pullback along the diagonal morphism $\\Delta : X \\to X \\times X$.", "The composition is the cup product as pullback is an algebra homomorphism and", "$\\text{pr}_i \\circ \\Delta = \\text{id}$.", "On the other hand, given cycles $\\alpha, \\beta$ on $X$ ", "the intersection product is defined by the formula", "$$", "\\alpha \\cdot \\beta =", "\\Delta^!(\\alpha \\times \\beta)", "$$", "In other words, $\\alpha \\cdot \\beta$ is the pullback of the", "exterior product $\\alpha \\times \\beta$ on $X \\times X$ by", "the diagonal. Note also that", "$\\alpha \\times \\beta = \\text{pr}_1^*\\alpha \\cdot \\text{pr}_2^*\\beta$", "in $\\CH^*(X \\times X)$ (we omit the proof). Hence, given axiom (C)(a),", "axiom (C)(c) is equivalent to the statement that $\\gamma$ is", "compatible with exterior product in the sense that", "$\\gamma(\\alpha \\times \\beta)$ is equal to", "$\\text{pr}_1^*\\gamma(\\alpha) \\cup \\text{pr}_2^*\\gamma(\\beta)$." ], "refs": [], "ref_ids": [] }, { "id": 5121, "type": "other", "label": "weil-remark-betti-numbers-in-some-sense", "categories": [ "weil" ], "title": "weil-remark-betti-numbers-in-some-sense", "contents": [ "Let $H^*$ be a Weil cohomology theory", "(Definition \\ref{definition-weil-cohomology-theory}).", "Let $X$ be a geometrically irreducible smooth projective scheme", "of dimension $d$ over $k'$ with $k'/k$ a finite separable extension of fields.", "Suppose that", "$$", "H^0(\\Spec(k')) = F_1 \\times \\ldots \\times F_r", "$$", "for some fields $F_i$. Then we accordingly can write", "$$", "H^*(X) = \\prod\\nolimits_{i = 1, \\ldots, r}", "H^*(X) \\otimes_{H^0(\\Spec(k'))} F_i", "$$", "Now, our final assumption in Definition \\ref{definition-weil-cohomology-theory}", "tells us that $H^0(X)$ is free of rank $1$ over $\\prod F_i$.", "In other words, each of the factors", "$H^0(X) \\otimes_{H^0(\\Spec(k'))} F_i$ has dimension $1$ over $F_i$.", "Poincar\\'e duality then tells us that the same is true for", "cohomology in degree $2d$.", "What isn't clear however is that the same holds in other degrees.", "Namely, we don't know that given $0 < n < \\dim(X)$ the integers", "$$", "\\dim_{F_i} H^n(X) \\otimes_{H^0(\\Spec(k'))} F_i", "$$", "are independent of $i$! This question is closely related to the following", "open question: given an algebraically closed base field $\\overline{k}$,", "a field of characteristic zero $F$, a classical Weil cohomology theory", "$H^*$ over $\\overline{k}$ with coefficient field $F$, and a smooth projective", "variety $X$ over $\\overline{k}$ is it true that the betti numbers of $X$", "$$", "\\beta_i = \\dim_F H^i(X)", "$$", "are independent of $F$ and the Weil cohomology theory $H^*$?" ], "refs": [ "weil-definition-weil-cohomology-theory", "weil-definition-weil-cohomology-theory" ], "ref_ids": [ 5116, 5116 ] }, { "id": 5122, "type": "other", "label": "weil-remark-trace", "categories": [ "weil" ], "title": "weil-remark-trace", "contents": [ "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).", "Let $X$ be a smooth projective scheme over $k$ which is nonempty", "and equidimensional of dimension $d$. Combining what was said in", "the proofs of Lemma \\ref{lemma-poincare-duality} and", "Homology, Lemma \\ref{homology-lemma-left-dual-graded-vector-spaces}", "we see that", "$$", "\\gamma([\\Delta]) \\in \\bigoplus\\nolimits_i H^i(X) \\otimes H^{2d - i}(X)(d)", "$$", "defines a perfect duality between $H^i(X)$ and $H^{2d - i}(X)(d)$", "for all $i$.", "In particular, the linear map $\\int_X = \\lambda : H^{2d}(X)(d) \\to F$ of", "axiom (A6) is unique! We will call the linear map $\\int_X$ the trace map", "of $X$ from now on." ], "refs": [ "weil-lemma-poincare-duality", "homology-lemma-left-dual-graded-vector-spaces" ], "ref_ids": [ 5097, 12073 ] }, { "id": 5594, "type": "other", "label": "morphisms-remark-direct-argument", "categories": [ "morphisms" ], "title": "morphisms-remark-direct-argument", "contents": [ "We can also argue directly that (2) implies (1) in", "Lemma \\ref{lemma-characterize-affine} above as follows.", "Assume $S = \\bigcup W_j$ is an affine open covering", "such that each $f^{-1}(W_j)$ is affine.", "First argue that $\\mathcal{A} = f_*\\mathcal{O}_X$ is quasi-coherent", "as in the proof above.", "Let $\\Spec(R) = V \\subset S$ be affine open.", "We have to show that $f^{-1}(V)$ is affine. Set", "$A = \\mathcal{A}(V) = f_*\\mathcal{O}_X(V) = \\mathcal{O}_X(f^{-1}(V))$.", "By Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine} there is", "a canonical morphism $\\psi : f^{-1}(V) \\to \\Spec(A)$ over", "$\\Spec(R) = V$.", "By Schemes, Lemma \\ref{schemes-lemma-good-subcover} there exists", "an integer $n \\geq 0$, a standard open covering", "$V = \\bigcup_{i = 1, \\ldots, n} D(h_i)$, $h_i \\in R$, and a map", "$a : \\{1, \\ldots, n\\} \\to J$ such that each $D(h_i)$ is also", "a standard open of the affine scheme $W_{a(i)}$. The inverse image", "of a standard open under a morphism of affine schemes is standard open, see", "Algebra, Lemma \\ref{algebra-lemma-spec-functorial}. Hence we see", "that $f^{-1}(D(h_i))$ is a standard open of $f^{-1}(W_{a(i)})$,", "in particular that $f^{-1}(D(h_i))$ is affine. Because $\\mathcal{A}$", "is quasi-coherent we have", "$A_{h_i} = \\mathcal{A}(D(h_i)) = \\mathcal{O}_X(f^{-1}(D(h_i)))$,", "so $f^{-1}(D(h_i))$ is the spectrum of $A_{h_i}$.", "It follows that the morphism $\\psi$ induces an isomorphism of the open", "$f^{-1}(D(h_i))$ with the open $\\Spec(A_{h_i})$ of", "$\\Spec(A)$. Since $f^{-1}(V) = \\bigcup f^{-1}(D(h_i))$", "and $\\Spec(A) = \\bigcup \\Spec(A_{h_i})$ we win." ], "refs": [ "morphisms-lemma-characterize-affine", "schemes-lemma-morphism-into-affine", "schemes-lemma-good-subcover", "algebra-lemma-spec-functorial" ], "ref_ids": [ 5172, 7655, 7676, 390 ] }, { "id": 5595, "type": "other", "label": "morphisms-remark-affine-s-opens-cover-family", "categories": [ "morphisms" ], "title": "morphisms-remark-affine-s-opens-cover-family", "contents": [ "In Properties, Lemma \\ref{properties-lemma-affine-s-opens-cover-quasi-separated}", "we see that a scheme which has an ample invertible module", "is separated. This is wrong for schemes having an ample", "family of invertible modules. Namely, let $X$ be as in", "Schemes, Example \\ref{schemes-example-affine-space-zero-doubled}", "with $n = 1$, i.e., the affine line with zero doubled. We use the notation", "of that example except that we write $x$ for $x_1$ and $y$", "for $y_1$. There is, for every integer $n$, an invertible", "sheaf $\\mathcal{L}_n$ on $X$ which is trivial on $X_1$ and", "$X_2$ and whose transition function $U_{12} \\to U_{21}$ is", "$f(x) \\mapsto y^n f(y)$. The global sections of", "$\\mathcal{L}_n$ are pairs $(f(x), g(y)) \\in k[x] \\oplus", "k[y]$ such that $y^n f(y) = g(y)$. The sections $s = (1,", "y)$ of $\\mathcal{L}_1$ and $t = (x, 1)$ of", "$\\mathcal{L}_{-1}$ determine an open affine cover because", "$X_s = X_1$ and $X_t = X_2$. Therefore $X$ has an", "ample family of invertible modules but it is not separated." ], "refs": [ "properties-lemma-affine-s-opens-cover-quasi-separated" ], "ref_ids": [ 3045 ] }, { "id": 5596, "type": "other", "label": "morphisms-remark-flattening", "categories": [ "morphisms" ], "title": "morphisms-remark-flattening", "contents": [ "The results above are a first step towards more refined flattening techniques", "for morphisms of schemes. The article \\cite{GruRay} by Raynaud and Gruson", "contains many wonderful results in this direction." ], "refs": [], "ref_ids": [] }, { "id": 5597, "type": "other", "label": "morphisms-remark-differentials-glue", "categories": [ "morphisms" ], "title": "morphisms-remark-differentials-glue", "contents": [ "The lemma above gives a second way of constructing the module of", "differentials. Namely, let $f : X \\to S$ be a morphism of schemes.", "Consider the collection of all affine opens $U \\subset X$ which", "map into an affine open of $S$. These form a basis for the topology", "on $X$. Thus it suffices to define $\\Gamma(U, \\Omega_{X/S})$", "for such $U$. We simply set $\\Gamma(U, \\Omega_{X/S}) = \\Omega_{A/R}$ if", "$A$, $R$ are as in Lemma \\ref{lemma-differentials-affine} above.", "This works, but it takes somewhat more algebraic preliminaries", "to construct the restriction mappings and to verify the sheaf", "condition with this ansatz." ], "refs": [ "morphisms-lemma-differentials-affine" ], "ref_ids": [ 5310 ] }, { "id": 5598, "type": "other", "label": "morphisms-remark-differentials-diagonal", "categories": [ "morphisms" ], "title": "morphisms-remark-differentials-diagonal", "contents": [ "Let $X \\to S$ be a morphism of schemes. According to", "Lemma \\ref{lemma-differential-product}", "we have", "$$", "\\Omega_{X \\times_S X/S} =", "\\text{pr}_1^*\\Omega_{X/S} \\oplus \\text{pr}_2^*\\Omega_{X/S}", "$$", "On the other hand, the diagonal morphism $\\Delta : X \\to X \\times_S X$", "is an immersion, which locally has a left inverse. Hence by", "Lemma \\ref{lemma-differentials-relative-immersion-section}", "we obtain a canonical short exact sequence", "$$", "0 \\to \\mathcal{C}_{X/X \\times_S X} \\to \\Omega_{X/S} \\oplus \\Omega_{X/S}", "\\to \\Omega_{X/S} \\to 0", "$$", "Note that the right arrow is $(1, 1)$ which is indeed a split surjection.", "On the other hand, by Lemma \\ref{lemma-differentials-diagonal}", "we have an identification $\\Omega_{X/S} = \\mathcal{C}_{X/X \\times_S X}$.", "Because we chose $\\text{d}_{X/S}(f) = s_2(f) - s_1(f)$ in this", "identification it turns out that the left arrow is the map", "$(-1, 1)$\\footnote{Namely,", "the local section $\\text{d}_{X/S}(f) = 1 \\otimes f - f \\otimes 1$ of the", "ideal sheaf of $\\Delta$ maps via $\\text{d}_{X \\times_S X/X}$ to the", "local section", "$1 \\otimes 1 \\otimes 1 \\otimes f - 1 \\otimes f \\otimes 1 \\otimes 1", "-1 \\otimes 1 \\otimes f \\otimes 1 + f \\otimes 1 \\otimes 1 \\otimes 1 =", "\\text{pr}_2^*\\text{d}_{X/S}(f) - \\text{pr}_1^*\\text{d}_{X/S}(f)$.}." ], "refs": [ "morphisms-lemma-differential-product", "morphisms-lemma-differentials-diagonal" ], "ref_ids": [ 5315, 5311 ] }, { "id": 5599, "type": "other", "label": "morphisms-remark-base-change-differential-operators", "categories": [ "morphisms" ], "title": "morphisms-remark-base-change-differential-operators", "contents": [ "Let $a : X \\to S$ and $b : Y \\to S$ be morphisms of schemes.", "Denote $p : X \\times_S Y \\to X$ and $q : X \\times_S Y \\to Y$ the", "projections. In this remark, given an $\\mathcal{O}_X$-module $\\mathcal{F}$", "and an $\\mathcal{O}_Y$-module $\\mathcal{G}$ let us set", "$$", "\\mathcal{F} \\boxtimes \\mathcal{G} =", "p^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_S Y}} q^*\\mathcal{G}", "$$", "Denote $\\mathcal{A}_{X/S}$ the additive category", "whose objects are quasi-coherent $\\mathcal{O}_X$-modules and", "whose morphisms are differential operators of finite order on $X/S$.", "Similarly for $\\mathcal{A}_{Y/S}$ and $\\mathcal{A}_{X \\times_S Y/S}$.", "The construction of Lemma \\ref{lemma-base-change-differential-operators}", "determines a functor", "$$", "\\boxtimes : ", "\\mathcal{A}_{X/S} \\times \\mathcal{A}_{Y/S} \\longrightarrow", "\\mathcal{A}_{X \\times_S Y/S},", "\\quad", "(\\mathcal{F}, \\mathcal{G}) \\longmapsto \\mathcal{F} \\boxtimes \\mathcal{G}", "$$", "which is bilinear on morphisms. If $X = \\Spec(A)$, $Y = \\Spec(B)$, and", "$S = \\Spec(R)$, then via the identification of quasi-coherent sheaves", "with modules this functor is given by $(M, N) \\mapsto M \\otimes_R N$", "on objects and sends the morphism $(D, D') : (M, N) \\to (M', N')$ to", "$D \\otimes D' : M \\otimes_R N \\to M' \\otimes_R N'$." ], "refs": [ "morphisms-lemma-base-change-differential-operators" ], "ref_ids": [ 5323 ] }, { "id": 5600, "type": "other", "label": "morphisms-remark-check-val-on-open", "categories": [ "morphisms" ], "title": "morphisms-remark-check-val-on-open", "contents": [ "The assumption on uniqueness of the dotted arrows in", "Lemma \\ref{lemma-refined-valuative-criterion-universally-closed}", "is necessary (details omitted). Of course, uniqueness is guaranteed if", "$f$ is separated", "(Schemes, Lemma \\ref{schemes-lemma-separated-implies-valuative})." ], "refs": [ "morphisms-lemma-refined-valuative-criterion-universally-closed", "schemes-lemma-separated-implies-valuative" ], "ref_ids": [ 5417, 7719 ] }, { "id": 5601, "type": "other", "label": "morphisms-remark-generalize-category", "categories": [ "morphisms" ], "title": "morphisms-remark-generalize-category", "contents": [ "Here is a generalization of the category of irreducible schemes and", "dominant rational maps. For a scheme $X$ denote $X^0$ the set of", "points $x \\in X$ with $\\dim(\\mathcal{O}_{X, x}) = 0$, in other words,", "$X^0$ is the set of generic points of irreducible components of $X$.", "Then we can consider the category with", "\\begin{enumerate}", "\\item objects are schemes $X$ such that every quasi-compact open has", "finitely many irreducible components, and", "\\item morphisms from $X$ to $Y$ are rational maps $f : U \\to Y$", "from $X$ to $Y$ such that $f(U^0) = Y^0$.", "\\end{enumerate}", "If $U \\subset X$ is a dense open of a scheme, then", "$U^0 \\subset X^0$ need not be an equality, but if $X$ is an", "object of our category, then this is the case.", "Thus given two morphisms in our category, the composition", "is well defined and a morphism in our category." ], "refs": [], "ref_ids": [] }, { "id": 5602, "type": "other", "label": "morphisms-remark-pseudo-morphisms", "categories": [ "morphisms" ], "title": "morphisms-remark-pseudo-morphisms", "contents": [ "There is a variant of Definition \\ref{definition-rational-map}", "where we consider only those morphism $U \\to Y$ defined on", "scheme theoretically dense open subschemes $U \\subset X$.", "We use Lemma \\ref{lemma-intersection-scheme-theoretically-dense}", "to see that we obtain an equivalence relation.", "An equivalence class of these is called a", "{\\it pseudo-morphism from $X$ to $Y$}.", "If $X$ is reduced the two notions coincide." ], "refs": [ "morphisms-definition-rational-map", "morphisms-lemma-intersection-scheme-theoretically-dense" ], "ref_ids": [ 5579, 5153 ] }, { "id": 5603, "type": "other", "label": "morphisms-remark-quasi-finite-finite-over-dense-open", "categories": [ "morphisms" ], "title": "morphisms-remark-quasi-finite-finite-over-dense-open", "contents": [ "An alternative to", "Lemma \\ref{lemma-generically-finite}", "is the statement that a quasi-finite morphism is finite", "over a dense open of the target. This will be shown in", "More on Morphisms,", "Lemma \\ref{more-morphisms-lemma-quasi-finite-finite-over-dense-open}." ], "refs": [ "morphisms-lemma-generically-finite", "more-morphisms-lemma-quasi-finite-finite-over-dense-open" ], "ref_ids": [ 5487, 13907 ] }, { "id": 5604, "type": "other", "label": "morphisms-remark-definition-generically-finite", "categories": [ "morphisms" ], "title": "morphisms-remark-definition-generically-finite", "contents": [ "Let $f : X \\to Y$ be a morphism of schemes which is locally of finite type.", "There are (at least) two properties that we could use to define", "{\\it generically finite} morphisms. These correspond to whether you", "want the property to be local on the source or local on the target:", "\\begin{enumerate}", "\\item (Local on the target; suggested by Ravi Vakil.)", "Assume every quasi-compact open of $Y$ has finitely", "many irreducible components (for example if $Y$ is locally Noetherian).", "The requirement is that the inverse image of each generic point is finite, see", "Lemma \\ref{lemma-generically-finite}.", "\\item (Local on the source.) The requirement is that there exists", "a dense open $U \\subset X$ such that $U \\to Y$ is locally quasi-finite.", "\\end{enumerate}", "In case (1) the requirement can be formulated without the auxiliary", "condition on $Y$, but probably doesn't give the right notion for", "general schemes. Property (2) as formulated doesn't imply that the fibres", "over generic points are finite; however, if $f$ is", "quasi-compact and $Y$ is as in (1) then it does." ], "refs": [ "morphisms-lemma-generically-finite" ], "ref_ids": [ 5487 ] }, { "id": 5930, "type": "other", "label": "chow-remark-gersten-complex-milnor", "categories": [ "chow" ], "title": "chow-remark-gersten-complex-milnor", "contents": [ "For a field $k$ let us denote $K^M_*(k)$ the quotient of", "the tensor algebra on $k^*$ divided by the two-sided ideal", "generated by the elements $x \\otimes 1 - x$. Thus $K^M_0(k) = \\mathbf{Z}$,", "$K_1^M(k) = k^*$, and", "$$", "K^M_2(k) = k^* \\otimes_\\mathbf{Z} k^* / \\langle x \\otimes 1 - x \\rangle", "$$", "If $(A, \\mathfrak m)$ is a $1$-dimensional Noetherian local domain", "with fraction field $Q(A)$ and residue field $\\kappa$ there is a", "tame symbol", "$$", "\\partial_A : K_{i + 1}^M(Q(A)) \\to K_i^M(\\kappa(\\mathfrak m))", "$$", "You can use the method of Section \\ref{section-tame-symbol}", "to define these maps, provided you extend the norm map", "to $K_i^M$ for all $i$. Next, let $X$ be a Noetherian scheme with a", "dimension function $\\delta$. Then we can use these tame symbols", "to get the arrows in the following:", "$$", "\\bigoplus\\nolimits_{\\delta(x) = j + 1} K^M_{i + 1}(\\kappa(x))", "\\longrightarrow", "\\bigoplus\\nolimits_{\\delta(x) = j} K^M_i(\\kappa(x))", "\\longrightarrow", "\\bigoplus\\nolimits_{\\delta(x) = j - 1} K^M_{i - 1}(\\kappa(x))", "$$", "However, it is not clear, if you define the maps as suggested above,", "that the composition is zero. When $i = 1$ and $j$ arbitrary, this", "follows from Lemma \\ref{lemma-milnor-gersten-low-degree}.", "For excellent $X$ this follows from \\cite{Kato-Milnor-K}", "modulo the verification that Kato's maps are the same as ours." ], "refs": [ "chow-lemma-milnor-gersten-low-degree" ], "ref_ids": [ 5666 ] }, { "id": 5931, "type": "other", "label": "chow-remark-infinite-sums-rational-equivalences", "categories": [ "chow" ], "title": "chow-remark-infinite-sums-rational-equivalences", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be a scheme locally of finite type over $S$.", "Suppose we have infinite collections $\\alpha_i, \\beta_i \\in Z_k(X)$,", "$i \\in I$ of $k$-cycles on $X$. Suppose that the supports", "of $\\alpha_i$ and $\\beta_i$ form locally finite collections", "of closed subsets of $X$ so that $\\sum \\alpha_i$", "and $\\sum \\beta_i$ are defined as cycles. Moreover, assume that", "$\\alpha_i \\sim_{rat} \\beta_i$ for each $i$. Then it is not", "clear that $\\sum \\alpha_i \\sim_{rat} \\sum \\beta_i$. Namely,", "the problem is that the rational equivalences may be", "given by locally finite", "families $\\{W_{i, j}, f_{i, j} \\in R(W_{i, j})^*\\}_{j \\in J_i}$", "but the union $\\{W_{i, j}\\}_{i \\in I, j\\in J_i}$ may not", "be locally finite.", "\\medskip\\noindent", "In many cases in practice, one has a locally finite family of closed", "subsets $\\{T_i\\}_{i \\in I}$ such that $\\alpha_i, \\beta_i$", "are supported on $T_i$ and such that $\\alpha_i = \\beta_i$", "in $\\CH_k(T_i)$, in other words, the families", "$\\{W_{i, j}, f_{i, j} \\in R(W_{i, j})^*\\}_{j \\in J_i}$", "consist of subschemes $W_{i, j} \\subset T_i$. In this case it is true that", "$\\sum \\alpha_i \\sim_{rat} \\sum \\beta_i$ on $X$, simply because", "the family $\\{W_{i, j}\\}_{i \\in I, j\\in J_i}$ is automatically", "locally finite in this case." ], "refs": [], "ref_ids": [] }, { "id": 5932, "type": "other", "label": "chow-remark-good-cases-K-A", "categories": [ "chow" ], "title": "chow-remark-good-cases-K-A", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be a scheme locally of finite type over $S$.", "We will see later (in Lemma \\ref{lemma-cycles-rational-equivalence-K-group})", "that the map", "$$", "\\CH_k(X)", "\\longrightarrow", "K_0(\\textit{Coh}_{k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))", "$$", "of Lemma \\ref{lemma-from-chow-to-K} is injective.", "Composing with the canonical map", "$$", "K_0(\\textit{Coh}_{k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))", "\\longrightarrow", "K_0(\\textit{Coh}(X)/\\textit{Coh}_{\\leq k - 1}(X))", "$$", "we obtain a canonical map", "$$", "\\CH_k(X)", "\\longrightarrow", "K_0(\\textit{Coh}(X)/\\textit{Coh}_{\\leq k - 1}(X)).", "$$", "We have not been able to find a statement or conjecture in the", "literature as to whether this map is should be injective or not.", "It seems reasonable to expect the kernel of this map to be torsion.", "We will return to this question (insert future reference)." ], "refs": [ "chow-lemma-cycles-rational-equivalence-K-group", "chow-lemma-from-chow-to-K" ], "ref_ids": [ 5896, 5701 ] }, { "id": 5933, "type": "other", "label": "chow-remark-generalize-to-virtual", "categories": [ "chow" ], "title": "chow-remark-generalize-to-virtual", "contents": [ "Let $X$ be a scheme locally of finite type over $S$ as in", "Situation \\ref{situation-setup}. Let $(D, \\mathcal{N}, \\sigma)$", "be a triple consisting of a locally principal (Divisors, Definition", "\\ref{divisors-definition-effective-Cartier-divisor}) closed subscheme", "$i : D \\to X$, an invertible $\\mathcal{O}_D$-module $\\mathcal{N}$, and", "a surjection $\\sigma : \\mathcal{N}^{\\otimes -1} \\to i^*\\mathcal{I}_D$", "of $\\mathcal{O}_D$-modules\\footnote{This condition assures us that if", "$D$ is an effective Cartier divisor, then $\\mathcal{N} = \\mathcal{O}_X(D)|_D$.}.", "Here $\\mathcal{N}$ should be thought of as", "a {\\it virtual normal bundle of $D$ in $X$}. The construction of", "$i^* : Z_{k + 1}(X) \\to \\CH_k(D)$ in", "Definition \\ref{definition-gysin-homomorphism}", "generalizes to such triples, see", "Section \\ref{section-gysin-higher-codimension}." ], "refs": [ "divisors-definition-effective-Cartier-divisor", "chow-definition-gysin-homomorphism" ], "ref_ids": [ 8089, 5915 ] }, { "id": 5934, "type": "other", "label": "chow-remark-generalize-to-pseudo-divisor", "categories": [ "chow" ], "title": "chow-remark-generalize-to-pseudo-divisor", "contents": [ "Let $X$ be a scheme locally of finite type over $S$ as in", "Situation \\ref{situation-setup}. In \\cite{F} a {\\it pseudo-divisor} on $X$", "is defined as a triple $D = (\\mathcal{L}, Z, s)$ where $\\mathcal{L}$", "is an invertible $\\mathcal{O}_X$-module, $Z \\subset X$ is a closed subset,", "and $s \\in \\Gamma(X \\setminus Z, \\mathcal{L})$ is a nowhere vanishing", "section. Similarly to the above, one can define for every $\\alpha$", "in $\\CH_{k + 1}(X)$ a product $D \\cdot \\alpha$ in $\\CH_k(Z \\cap |\\alpha|)$", "where $|\\alpha|$ is the support of $\\alpha$." ], "refs": [], "ref_ids": [] }, { "id": 5935, "type": "other", "label": "chow-remark-gysin-on-cycles", "categories": [ "chow" ], "title": "chow-remark-gysin-on-cycles", "contents": [ "Let $X \\to S$, $\\mathcal{L}$, $s$, $i : D \\to X$ be as in", "Definition \\ref{definition-gysin-homomorphism} and assume", "that $\\mathcal{L}|_D \\cong \\mathcal{O}_D$. In this case we", "can define a canonical map $i^* : Z_{k + 1}(X) \\to Z_k(D)$", "on cycles, by requiring that $i^*[W] = 0$ whenever $W \\subset D$", "is an integral closed subscheme.", "The possibility to do this will be useful later on." ], "refs": [ "chow-definition-gysin-homomorphism" ], "ref_ids": [ 5915 ] }, { "id": 5936, "type": "other", "label": "chow-remark-pullback-pairs", "categories": [ "chow" ], "title": "chow-remark-pullback-pairs", "contents": [ "Let $f : X' \\to X$ be a morphism of schemes locally of finite type over $S$", "as in Situation \\ref{situation-setup}. Let $(\\mathcal{L}, s, i : D \\to X)$", "be a triple as in Definition \\ref{definition-gysin-homomorphism}.", "Then we can set $\\mathcal{L}' = f^*\\mathcal{L}$, $s' = f^*s$, and", "$D' = X' \\times_X D = Z(s')$. This gives a commutative diagram", "$$", "\\xymatrix{", "D' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\", "D \\ar[r]^i & X", "}", "$$", "and we can ask for various compatibilities between $i^*$ and $(i')^*$." ], "refs": [ "chow-definition-gysin-homomorphism" ], "ref_ids": [ 5915 ] }, { "id": 5937, "type": "other", "label": "chow-remark-when-isomorphism", "categories": [ "chow" ], "title": "chow-remark-when-isomorphism", "contents": [ "We will see later (Lemma \\ref{lemma-vectorbundle}) that if $X$ is a", "vector bundle of rank $r$ over $Y$ then the pullback map", "$\\CH_k(Y) \\to \\CH_{k + r}(X)$", "is an isomorphism. This is true whenever $X \\to Y$ satisfies", "the assumptions of Lemma \\ref{lemma-pullback-affine-fibres-surjective}, see", "\\cite[Lemma 2.2]{Totaro-group}." ], "refs": [ "chow-lemma-vectorbundle", "chow-lemma-pullback-affine-fibres-surjective" ], "ref_ids": [ 5744, 5726 ] }, { "id": 5938, "type": "other", "label": "chow-remark-restriction-bivariant", "categories": [ "chow" ], "title": "chow-remark-restriction-bivariant", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X \\to Y$", "and $Y' \\to Y$ be morphisms of schemes locally of finite type over $S$.", "Let $X' = Y' \\times_Y X$. Then there is an obvious restriction map", "$$", "A^p(X \\to Y) \\longrightarrow A^p(X' \\to Y'),\\quad", "c \\longmapsto res(c)", "$$", "obtained by viewing a scheme $Y''$ locally of finite type over $Y'$", "as a scheme locally of finite type over $Y$ and settting", "$res(c) \\cap \\alpha'' = c \\cap \\alpha''$ for any $\\alpha'' \\in \\CH_k(Y'')$.", "This restriction operation is compatible with compositions in an", "obvious manner." ], "refs": [], "ref_ids": [] }, { "id": 5939, "type": "other", "label": "chow-remark-bivariant-commute", "categories": [ "chow" ], "title": "chow-remark-bivariant-commute", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be", "locally of finite type over $S$. For $i = 1, 2$ let $Z_i \\to X$", "be a morphism of schemes locally of finite type. Let", "$c_i \\in A^{p_i}(Z_i \\to X)$, $i = 1, 2$ be bivariant classes.", "For any $\\alpha \\in \\CH_k(X)$ we can ask whether", "$$", "c_1 \\cap c_2 \\cap \\alpha = c_2 \\cap c_1 \\cap \\alpha", "$$", "in $\\CH_{k - p_1 - p_2}(Z_1 \\times_X Z_2)$. If this is true and if it holds", "after any base change by $X' \\to X$ locally of finite type, then we say", "$c_1$ and $c_2$ {\\it commute}. Of course this is the same thing as saying that", "$$", "res(c_1) \\circ c_2 = res(c_2) \\circ c_1", "$$", "in $A^{p_1 + p_2}(Z_1 \\times_X Z_2 \\to X)$. Here", "$res(c_1) \\in A^{p_1}(Z_1 \\times_X Z_2 \\to Z_2)$ is the restriction of $c_1$", "as in Remark \\ref{remark-restriction-bivariant}; similarly for $res(c_2)$." ], "refs": [ "chow-remark-restriction-bivariant" ], "ref_ids": [ 5938 ] }, { "id": 5940, "type": "other", "label": "chow-remark-more-general-bivariant", "categories": [ "chow" ], "title": "chow-remark-more-general-bivariant", "contents": [ "There is a more general type of bivariant class that doesn't seem to be", "considered in the literature. Namely, suppose we are given a diagram", "$$", "X \\longrightarrow Z \\longleftarrow Y", "$$", "of schemes locally of finite type over $(S, \\delta)$ as in", "Situation \\ref{situation-setup}. Let $p \\in \\mathbf{Z}$.", "Then we can consider a rule $c$ which assigns to every $Z' \\to Z$", "locally of finite type maps", "$$", "c \\cap - : \\CH_k(Y') \\longrightarrow \\CH_{k - p}(X')", "$$", "for all $k \\in \\mathbf{Z}$", "where $X' = X \\times_Z Z'$ and $Y' = Z' \\times_Z Y$ compatible with", "\\begin{enumerate}", "\\item proper pushforward if given $Z'' \\to Z'$ proper,", "\\item flat pullback if given $Z'' \\to Z'$ flat", "of fixed relative dimension, and", "\\item gysin maps if given $D' \\subset Z'$ as in", "Definition \\ref{definition-gysin-homomorphism}.", "\\end{enumerate}", "We omit the detailed formulations. Suppose we denote the collection", "of all such operations $A^p(X \\to Z \\leftarrow Y)$. A simple example", "of the utility of this concept is when we have a proper morphism", "$f : X_2 \\to X_1$. Then $f_*$ isn't a bivariant operation in the sense of", "Definition \\ref{definition-bivariant-class} but it is in the", "above generalized sense, namely, $f_* \\in A^0(X_1 \\to X_1 \\leftarrow X_2)$." ], "refs": [ "chow-definition-gysin-homomorphism", "chow-definition-bivariant-class" ], "ref_ids": [ 5915, 5916 ] }, { "id": 5941, "type": "other", "label": "chow-remark-pullback-cohomology", "categories": [ "chow" ], "title": "chow-remark-pullback-cohomology", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : Y' \\to Y$ be a morphism of schemes locally of finite type over $S$.", "As a special case of Remark \\ref{remark-restriction-bivariant}", "there is a canonical $\\mathbf{Z}$-algebra map $res : A^*(Y) \\to A^*(Y')$.", "This map is often denoted $f^*$ in the literature." ], "refs": [ "chow-remark-restriction-bivariant" ], "ref_ids": [ 5938 ] }, { "id": 5942, "type": "other", "label": "chow-remark-ring-loc-classes", "categories": [ "chow" ], "title": "chow-remark-ring-loc-classes", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $Z \\to X$ be a closed immersion of schemes locally of", "finite type over $S$ and let $p \\geq 0$. In this setting we define", "$$", "A^{(p)}(Z \\to X) =", "\\prod\\nolimits_{i \\leq p - 1} A^i(X) \\times", "\\prod\\nolimits_{i \\geq p} A^i(Z \\to X).", "$$", "Then $A^{(p)}(Z \\to X)$ canonically comes equipped with the structure", "of a graded algebra. In fact, more generally there is a multiplication", "$$", "A^{(p)}(Z \\to X) \\times A^{(q)}(Z \\to X)", "\\longrightarrow A^{(\\max(p, q))}(Z \\to X)", "$$", "In order to define these we define maps", "\\begin{align*}", "A^i(Z \\to X) \\times A^j(X) & \\to A^{i + j}(Z \\to X) \\\\", "A^i(X) \\times A^j(Z \\to X) & \\to A^{i + j}(Z \\to X) \\\\", "A^i(Z \\to X) \\times A^j(Z \\to X) & \\to A^{i + j}(Z \\to X)", "\\end{align*}", "For the first we use composition of bivariant classes.", "For the second we use restriction", "$A^i(X) \\to A^i(Z)$ (Remark \\ref{remark-restriction-bivariant}) and", "composition $A^i(Z) \\times A^j(Z \\to X) \\to A^{i + j}(Z \\to X)$.", "For the third, we send $(c, c')$ to $res(c) \\circ c'$", "where $res : A^i(Z \\to X) \\to A^i(Z)$ is the restriction map (see", "Remark \\ref{remark-restriction-bivariant}). We omit the", "verification that these multiplications are associative in a suitable sense." ], "refs": [ "chow-remark-restriction-bivariant", "chow-remark-restriction-bivariant" ], "ref_ids": [ 5938, 5938 ] }, { "id": 5943, "type": "other", "label": "chow-remark-res-push", "categories": [ "chow" ], "title": "chow-remark-res-push", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $Z \\to X$ be a closed immersion of schemes locally of", "finite type over $S$. Denote $res : A^p(Z \\to X) \\to A^p(Z)$", "the restriction map of Remark \\ref{remark-restriction-bivariant}.", "For $c \\in A^p(Z \\to X)$ we have", "$res(c) \\cap \\alpha = c \\cap i_*\\alpha$ for $\\alpha \\in \\CH_*(Z)$.", "Namely $res(c) \\cap \\alpha = c \\cap \\alpha$", "and compatibility of $c$ with proper pushforward", "gives $(Z \\to Z)_*(c \\cap \\alpha) = c \\cap (Z \\to X)_*\\alpha$." ], "refs": [ "chow-remark-restriction-bivariant" ], "ref_ids": [ 5938 ] }, { "id": 5944, "type": "other", "label": "chow-remark-completion-bivariant", "categories": [ "chow" ], "title": "chow-remark-completion-bivariant", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$.", "Let $X = \\coprod_{i \\in I} X_i$ and $Y = \\coprod_{j \\in J} Y_j$", "be the decomposition of $X$ and $Y$ into their connected components", "(the connected components are open as $X$ and $Y$ are locally Noetherian, see", "Topology, Lemma \\ref{topology-lemma-locally-Noetherian-locally-connected} and", "Properties, Lemma \\ref{properties-lemma-Noetherian-topology}).", "Let $a(i) \\in J$ be the index such that $f(X_i) \\subset Y_{a(i)}$.", "Then $A^p(X \\to Y) = \\prod A^p(X_i \\to Y_{a(i)})$ by", "Lemma \\ref{lemma-disjoint-decomposition-bivariant}.", "In this setting it is convenient to set", "$$", "A^*(X \\to Y)^\\wedge = \\prod\\nolimits_i A^*(X_i \\to Y_{a(i)})", "$$", "as a kind of natural completion of the graded $\\mathbf{Z}$-module", "$A^*(X \\to Y)$ of bivariant classes (we omit specifying the precise", "sense in which this is a completion). As a special case we set", "$$", "A^*(X)^\\wedge = \\prod A^*(X_i)", "$$", "If $Y \\to Z$ is a second morphism, then the", "composition $A^*(X \\to Y) \\times A^*(Y \\to Z) \\to A^*(X \\to Z)$", "extends to a composition", "$A^*(X \\to Y)^\\wedge \\times A^*(Y \\to Z)^\\wedge \\to A^*(X \\to Z)^\\wedge$", "of completions." ], "refs": [ "topology-lemma-locally-Noetherian-locally-connected", "properties-lemma-Noetherian-topology", "chow-lemma-disjoint-decomposition-bivariant" ], "ref_ids": [ 8223, 2954, 5741 ] }, { "id": 5945, "type": "other", "label": "chow-remark-equation-signs", "categories": [ "chow" ], "title": "chow-remark-equation-signs", "contents": [ "We could also rewrite equation \\ref{equation-chern-classes} as", "\\begin{equation}", "\\label{equation-signs}", "\\sum\\nolimits_{i = 0}^r", "c_1(\\mathcal{O}_P(-1))^i \\cap \\pi^*c_{r - i}", "= 0.", "\\end{equation}", "but we find it easier to work with the tautological quotient", "sheaf $\\mathcal{O}_P(1)$ instead of", "its dual." ], "refs": [], "ref_ids": [] }, { "id": 5946, "type": "other", "label": "chow-remark-extend-to-finite-locally-free", "categories": [ "chow" ], "title": "chow-remark-extend-to-finite-locally-free", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module.", "If the rank of $\\mathcal{E}$ is not constant then we can", "still define the Chern classes of $\\mathcal{E}$. Namely, in this", "case we can write", "$$", "X = X_0 \\amalg X_1 \\amalg X_2 \\amalg \\ldots", "$$", "where $X_r \\subset X$ is the open and closed subspace where", "the rank of $\\mathcal{E}$ is $r$. By ", "Lemma \\ref{lemma-disjoint-decomposition-bivariant}", "we have $A^p(X) = \\prod A^p(X_r)$.", "Hence we can define $c_i(\\mathcal{E})$ to be the", "product of the classes $c_i(\\mathcal{E}|_{X_r})$ in $A^i(X_r)$.", "Explicitly, if $X' \\to X$ is a morphism locally of finite type,", "then we obtain by pullback a corresponding decomposition of $X'$", "and we find that", "$$", "\\CH_*(X') = \\prod\\nolimits_{r \\geq 0} \\CH_*(X'_r)", "$$", "by our definitions. Then $c_i(\\mathcal{E}) \\in A^i(X)$", "is the bivariant class which preserves these direct", "product decompositions and acts by the already defined", "operations $c_i(\\mathcal{E}|_{X_r}) \\cap -$", "on the factors. Observe that in this setting it may happen", "that $c_i(\\mathcal{E})$ is nonzero for infinitely many $i$.", "In this setting we moreover define the ``rank'' of $\\mathcal{E}$", "to be the element $r(\\mathcal{E}) \\in A^0(X)$", "as the bivariant operation which sends $(\\alpha_r) \\in \\prod \\CH_*(X'_r)$", "to $(r\\alpha_r) \\in \\prod \\CH_*(X'_r)$.", "Note that it is still true that $c_i(\\mathcal{E})$ and $r(\\mathcal{E})$", "are in the center of $A^*(X)$." ], "refs": [ "chow-lemma-disjoint-decomposition-bivariant" ], "ref_ids": [ 5741 ] }, { "id": 5947, "type": "other", "label": "chow-remark-top-chern-class", "categories": [ "chow" ], "title": "chow-remark-top-chern-class", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$", "be locally of finite type over $S$. Let $\\mathcal{E}$ be a", "finite locally free $\\mathcal{O}_X$-module. In general", "we write $X = \\coprod X_r$ as in", "Remark \\ref{remark-extend-to-finite-locally-free}.", "If only a finite number of the $X_r$ are nonempty, then", "we can set", "$$", "c_{top}(\\mathcal{E}) = \\sum\\nolimits_r c_r(\\mathcal{E}|_{X_r})", "\\in A^*(X) = \\bigoplus A^*(X_r)", "$$", "where the equality is Lemma \\ref{lemma-disjoint-decomposition-bivariant}.", "If infinitely many $X_r$ are nonempty, we will use the same", "notation to denote", "$$", "c_{top}(\\mathcal{E}) = \\prod c_r(\\mathcal{E}|_{X_r})", "\\in \\prod A^r(X_r) \\subset A^*(X)^\\wedge", "$$", "see Remark \\ref{remark-completion-bivariant} for notation." ], "refs": [ "chow-remark-extend-to-finite-locally-free", "chow-lemma-disjoint-decomposition-bivariant", "chow-remark-completion-bivariant" ], "ref_ids": [ 5946, 5741, 5944 ] }, { "id": 5948, "type": "other", "label": "chow-remark-fundamental-class", "categories": [ "chow" ], "title": "chow-remark-fundamental-class", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$ satisfying the", "equivalent conditions of Lemma \\ref{lemma-locally-equidimensional}.", "Let $X = \\coprod X_n$ be the decomposition into open and closed", "subschemes such that every irreducible component of $X_n$ has", "$\\delta$-dimension $n$. In this situation we sometimes set", "$$", "[X] = \\sum\\nolimits_n [X_n]_n \\in \\CH^0(X)", "$$", "This class is a kind of ``fundamental class'' of $X$ in Chow theory." ], "refs": [ "chow-lemma-locally-equidimensional" ], "ref_ids": [ 5761 ] }, { "id": 5949, "type": "other", "label": "chow-remark-the-proof-shows-more", "categories": [ "chow" ], "title": "chow-remark-the-proof-shows-more", "contents": [ "The proof of Lemma \\ref{lemma-splitting-principle}", "shows that the morphism $\\pi : P \\to X$ has the following additional", "properties:", "\\begin{enumerate}", "\\item $\\pi$ is a finite composition of projective space bundles", "associated to locally free modules of finite constant rank, and", "\\item for every $\\alpha \\in \\CH_k(X)$ we have", "$\\alpha = \\pi_*(\\xi_1 \\cap \\ldots \\cap \\xi_d \\cap \\pi^*\\alpha)$", "where $\\xi_i$ is the first Chern class of some invertible", "$\\mathcal{O}_P$-module.", "\\end{enumerate}", "The second observation follows from the first and", "Lemma \\ref{lemma-cap-projective-bundle}.", "We will add more observations here as needed." ], "refs": [ "chow-lemma-splitting-principle", "chow-lemma-cap-projective-bundle" ], "ref_ids": [ 5762, 5742 ] }, { "id": 5950, "type": "other", "label": "chow-remark-equalities-nonconstant-rank", "categories": [ "chow" ], "title": "chow-remark-equalities-nonconstant-rank", "contents": [ "The equalities proven above remain true even when we work with", "finite locally free", "$\\mathcal{O}_X$-modules whose rank is allowed to be nonconstant.", "In fact, we can work with polynomials in the rank and the", "Chern classes as follows. Consider the graded polynomial ring", "$\\mathbf{Z}[r, c_1, c_2, c_3, \\ldots]$", "where $r$ has degree $0$ and $c_i$ has degree $i$. Let", "$$", "P \\in \\mathbf{Z}[r, c_1, c_2, c_3, \\ldots]", "$$", "be a homogeneous polynomial of degree $p$. Then for any finite locally", "free $\\mathcal{O}_X$-module $\\mathcal{E}$ on $X$ we can consider", "$$", "P(\\mathcal{E}) =", "P(r(\\mathcal{E}), c_1(\\mathcal{E}), c_2(\\mathcal{E}), c_3(\\mathcal{E}), \\ldots)", "\\in A^p(X)", "$$", "see Remark \\ref{remark-extend-to-finite-locally-free} for notation and", "conventions. To prove relations among these polynomials (for multiple", "finite locally free modules) we can work locally on $X$ and use the splitting", "principle as above. For example, we claim that", "$$", "c_2(\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}, \\mathcal{E})) =", "P(\\mathcal{E})", "$$", "where $P = 2rc_2 - (r - 1)c_1^2$.", "Namely, since $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}, \\mathcal{E}) =", "\\mathcal{E} \\otimes \\mathcal{E}^\\vee$ this follows easily from", "Lemmas \\ref{lemma-chern-classes-dual} and", "\\ref{lemma-chern-classes-tensor-product}", "above by decomposing $X$ into parts where the rank", "of $\\mathcal{E}$ is constant as in", "Remark \\ref{remark-extend-to-finite-locally-free}." ], "refs": [ "chow-remark-extend-to-finite-locally-free", "chow-lemma-chern-classes-dual", "chow-lemma-chern-classes-tensor-product", "chow-remark-extend-to-finite-locally-free" ], "ref_ids": [ 5946, 5763, 5764, 5946 ] }, { "id": 5951, "type": "other", "label": "chow-remark-extend-chern-character-to-finite-locally-free", "categories": [ "chow" ], "title": "chow-remark-extend-chern-character-to-finite-locally-free", "contents": [ "In the discussion above we have defined the Chern character", "$ch(\\mathcal{E})$ of $\\mathcal{E}$ even if the rank of $\\mathcal{E}$", "is not constant. See Remarks \\ref{remark-extend-to-finite-locally-free} and", "\\ref{remark-equalities-nonconstant-rank}." ], "refs": [ "chow-remark-extend-to-finite-locally-free", "chow-remark-equalities-nonconstant-rank" ], "ref_ids": [ 5946, 5950 ] }, { "id": 5952, "type": "other", "label": "chow-remark-splitting-principle-perfect", "categories": [ "chow" ], "title": "chow-remark-splitting-principle-perfect", "contents": [ "The Chern classes of a perfect complex, when defined, satisfy a kind of", "splitting principle. Namely, suppose that $(S, \\delta), X, E$ are as in", "Definition \\ref{definition-defined-on-perfect}", "such that the Chern classes of $E$ are defined.", "Say we want to prove a relation between the bivariant classes", "$c_p(E)$, $P_p(E)$, and $ch_p(E)$. To do this, we may choose a bounded", "complex $\\mathcal{E}^\\bullet$ of finite locally free $\\mathcal{O}_X$-modules", "representing $E$. Using the splitting principle", "(Lemma \\ref{lemma-splitting-principle}) we may assume each", "$\\mathcal{E}^i$ has a filtration whose successive", "quotients $\\mathcal{L}_{i, j}$ are invertible modules.", "Settting $x_{i, j} = c_1(\\mathcal{L}_{i, j})$ we see that", "$$", "c(E) =", "\\prod\\nolimits_{i\\text{ even}} (1 + x_{i, j})", "\\prod\\nolimits_{i\\text{ odd}} (1 + x_{i, j})^{-1}", "$$", "and", "$$", "P_p(E) = \\sum\\nolimits_{i\\text{ even}} (x_{i, j})^p -", "\\sum\\nolimits_{i\\text{ odd}} (x_{i, j})^p", "$$", "Formally taking the logarithm for the expression for $c(E)$ above", "we find that", "$$", "\\log(c(E)) = \\sum (-1)^{p - 1}\\frac{P_p(E)}{p}", "$$", "Looking at the construction of the polynomials $P_p$ in", "Example \\ref{example-power-sum} it follows that $P_p(E)$", "is the exact same expression in the Chern classes of $E$", "as in the case of vector bundles, in other words, we have", "\\begin{align*}", "P_1(E) & = c_1(E), \\\\", "P_2(E) & = c_1(E)^2 - 2c_2(E), \\\\", "P_3(E) & = c_1(E)^3 - 3c_1(E)c_2(E) + 3c_3(E), \\\\", "P_4(E) & = c_1(E)^4 - 4c_1(E)^2c_2(E) + 4c_1(E)c_3(E) + 2c_2(E)^2 - 4c_4(E),", "\\end{align*}", "and so on. On the other hand, the bivariant class $P_0(E) = r(E) = ch_0(E)$", "cannot be recovered from the Chern class $c(E)$ of $E$; the chern class", "doesn't know about the rank of the complex." ], "refs": [ "chow-definition-defined-on-perfect", "chow-lemma-splitting-principle" ], "ref_ids": [ 5923, 5762 ] }, { "id": 5953, "type": "other", "label": "chow-remark-loc-chern-classes", "categories": [ "chow" ], "title": "chow-remark-loc-chern-classes", "contents": [ "In the situation of Definition \\ref{definition-localized-chern}", "assume $E|_{X \\setminus Z}$ is finite locally free of rank $< p$.", "In this setting it is convenient to define", "$$", "c^{(p)}(Z \\to X, E) = 1 + c_1(E) + \\ldots + c_{p - 1}(E) +", "c_p(Z \\to X, E) + c_{p + 1}(Z \\to X, E) + \\ldots", "$$", "as an element of the algebra $A^{(p)}(Z \\to X)$ considered in", "Remark \\ref{remark-ring-loc-classes}." ], "refs": [ "chow-definition-localized-chern", "chow-remark-ring-loc-classes" ], "ref_ids": [ 5924, 5942 ] }, { "id": 5954, "type": "other", "label": "chow-remark-gysin-for-immersion", "categories": [ "chow" ], "title": "chow-remark-gysin-for-immersion", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be a scheme locally of finite type over $S$.", "Let $i : Z \\to X$ be an immersion of schemes.", "In this situation", "\\begin{enumerate}", "\\item the conormal sheaf $\\mathcal{C}_{Z/X}$", "of $Z$ in $X$ is defined", "(Morphisms, Definition \\ref{morphisms-definition-conormal-sheaf}),", "\\item we say a pair consisting of a finite locally free $\\mathcal{O}_Z$-module", "$\\mathcal{N}$ and a surjection $\\sigma : \\mathcal{N}^\\vee \\to \\mathcal{C}_{Z/X}$", "is a virtual normal bundle for the immersion $Z \\to X$,", "\\item choose an open subscheme $U \\subset X$ such that $Z \\to X$", "factors through a closed immersion $Z \\to U$ and set", "$c(Z \\to X, \\mathcal{N}) = c(Z \\to U, \\mathcal{N}) \\circ (U \\to X)^*$.", "\\end{enumerate}", "The bivariant class $c(Z \\to X, \\mathcal{N})$ does not depend on the choice", "of the open subscheme $U$. All of the lemmas have immediate counterparts", "for this slightly more general construction. We omit the details." ], "refs": [ "morphisms-definition-conormal-sheaf" ], "ref_ids": [ 5562 ] }, { "id": 5955, "type": "other", "label": "chow-remark-adams-derived", "categories": [ "chow" ], "title": "chow-remark-adams-derived", "contents": [ "Let $X$ be a scheme such that $2$ is invertible on $X$.", "Then the Adams operator $\\psi^2$ can be defined on the $K$-group", "$K_0(X) = K_0(D_{perf}(\\mathcal{O}_X))$", "(Derived Categories of Schemes, Definition \\ref{perfect-definition-K-group})", "in a straightforward manner.", "Namely, given a perfect complex $L$ on $X$ we get an action", "of the group $\\{\\pm 1\\}$ on $L \\otimes^\\mathbf{L} L$ by switching", "the factors. Then we can set", "$$", "\\psi^2(L) = [(L \\otimes^\\mathbf{L} L)^+] -", "[(L \\otimes^\\mathbf{L} L)^-]", "$$", "where $(-)^+$ denotes taking invariants and $(-)^-$ denotes taking", "anti-invariants (suitably defined).", "Using exactness of taking invariants and anti-invariants one can", "argue similarly to the proof of Lemma \\ref{lemma-second-adams-operator}", "to show that this is well defined.", "When $2$ is not invertible on $X$ the situation is a good deal more", "complicated and another approach has to be used." ], "refs": [ "perfect-definition-K-group", "chow-lemma-second-adams-operator" ], "ref_ids": [ 7121, 5824 ] }, { "id": 5956, "type": "other", "label": "chow-remark-chern-classes-K", "categories": [ "chow" ], "title": "chow-remark-chern-classes-K", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. The Chern class", "map defines a canonical map", "$$", "c : K_0(\\textit{Vect}(X)) \\longrightarrow \\prod\\nolimits_{i \\geq 0} A^i(X)", "$$", "by sending a generator $[\\mathcal{E}]$ on the left hand side to", "$c(\\mathcal{E}) = 1 + c_1(\\mathcal{E}) + c_2(\\mathcal{E}) + \\ldots$", "and extending multiplicatively. Thus $-[\\mathcal{E}]$ is sent to", "the formal inverse $c(\\mathcal{E})^{-1}$ which is why we have the", "infinite product on the right hand side. This is well defined by", "Lemma \\ref{lemma-additivity-chern-classes}." ], "refs": [ "chow-lemma-additivity-chern-classes" ], "ref_ids": [ 5756 ] }, { "id": 5957, "type": "other", "label": "chow-remark-chern-character-K", "categories": [ "chow" ], "title": "chow-remark-chern-character-K", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$. The Chern character", "map defines a canonical ring map", "$$", "ch : K_0(\\textit{Vect}(X)) \\longrightarrow", "\\prod\\nolimits_{i \\geq 0} A^i(X) \\otimes \\mathbf{Q}", "$$", "by sending a generator $[\\mathcal{E}]$ on the left hand side to", "$ch(\\mathcal{E})$ and extending additively. This is well defined", "by Lemma \\ref{lemma-chern-character-additive} and a ring homomorphism by", "Lemma \\ref{lemma-chern-character-multiplicative}." ], "refs": [ "chow-lemma-chern-character-additive", "chow-lemma-chern-character-multiplicative" ], "ref_ids": [ 5767, 5768 ] }, { "id": 5958, "type": "other", "label": "chow-remark-perf-Z-cohomology-K", "categories": [ "chow" ], "title": "chow-remark-perf-Z-cohomology-K", "contents": [ "Let $X$ be a locally Noetherian scheme.", "Let $Z \\subset X$ be a closed subscheme. Consider the strictly", "full, saturated, triangulated subcategory", "$$", "D_{Z, perf}(\\mathcal{O}_X) \\subset D(\\mathcal{O}_X)", "$$", "consisting of perfect complexes of $\\mathcal{O}_X$-modules", "whose cohomology sheaves are settheoretically supported on $Z$.", "Denote $\\textit{Coh}_Z(X) \\subset \\textit{Coh}(X)$", "the Serre subcategory of coherent $\\mathcal{O}_X$-modules whose set theoretic", "support is contained in $Z$. Observe that given", "$E \\in D_{Z, perf}(\\mathcal{O}_X)$ Zariski locally on $X$", "only a finite number of the cohomology sheaves $H^i(E)$ are nonzero", "(and they are all settheoretically supported on $Z$).", "Hence we can define", "$$", "K_0(D_{Z, perf}(\\mathcal{O}_X))", "\\longrightarrow", "K_0(\\textit{Coh}_Z(X)) = K'_0(Z)", "$$", "(equality by Lemma \\ref{lemma-K-coherent-supported-on-closed}) by the rule", "$$", "E \\longmapsto", "[\\bigoplus\\nolimits_{i \\in \\mathbf{Z}} H^{2i}(E)] -", "[\\bigoplus\\nolimits_{i \\in \\mathbf{Z}} H^{2i + 1}(E)]", "$$", "This works because given a distinguished triangle in", "$D_{Z, perf}(\\mathcal{O}_X)$ we have a long exact sequence of", "cohomology sheaves." ], "refs": [ "chow-lemma-K-coherent-supported-on-closed" ], "ref_ids": [ 5702 ] }, { "id": 5959, "type": "other", "label": "chow-remark-perf-Z-regular", "categories": [ "chow" ], "title": "chow-remark-perf-Z-regular", "contents": [ "Let $X$, $Z$, $D_{Z, perf}(\\mathcal{O}_X)$ be as in", "Remark \\ref{remark-perf-Z-cohomology-K}.", "Assume $X$ is Noetherian regular of finite dimension.", "Then there is a canonical map", "$$", "K_0(\\textit{Coh}(Z)) \\longrightarrow K_0(D_{Z, perf}(\\mathcal{O}_X))", "$$", "defined as follows. For any coherent $\\mathcal{O}_Z$-module", "$\\mathcal{F}$ denote $\\mathcal{F}[0]$ the object of $D(\\mathcal{O}_X)$", "which has $\\mathcal{F}$ in degree $0$ and is zero in other degrees.", "Then $\\mathcal{F}[0]$ is a perfect complex on $X$ by", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-perfect-on-regular}.", "Hence $\\mathcal{F}[0]$ is an object of $D_{Z, perf}(\\mathcal{O}_X)$.", "On the other hand, given a short exact sequence", "$0 \\to \\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{F}'' \\to 0$ of", "coherent $\\mathcal{O}_Z$-modules we obtain a distinguished triangle", "$\\mathcal{F}[0] \\to \\mathcal{F}'[0] \\to \\mathcal{F}''[0] \\to \\mathcal{F}[1]$,", "see Derived Categories, Section \\ref{derived-section-canonical-delta-functor}.", "This shows that we obtain a map", "$K_0(\\textit{Coh}(Z)) \\to K_0(D_{Z, perf}(\\mathcal{O}_X))$", "by sending $[\\mathcal{F}]$ to $[\\mathcal{F}[0]]$", "with apologies for the horrendous notation." ], "refs": [ "chow-remark-perf-Z-cohomology-K", "perfect-lemma-perfect-on-regular" ], "ref_ids": [ 5958, 6989 ] }, { "id": 5960, "type": "other", "label": "chow-remark-localized-chern-classes-K", "categories": [ "chow" ], "title": "chow-remark-localized-chern-classes-K", "contents": [ "Let $X$, $Z$, $D_{Z, perf}(\\mathcal{O}_X)$ be as in", "Remark \\ref{remark-perf-Z-cohomology-K}.", "Assume $X$ is quasi-compact, has the resolution property, and", "is of finite type over $(S, \\delta)$ as in Situation \\ref{situation-setup}.", "The localized Chern classes define a canonical map", "$$", "c(Z \\to X, -) : K_0(D_{Z, perf}(\\mathcal{O}_X)) \\longrightarrow", "A^0(X) \\times \\prod\\nolimits_{i \\geq 1} A^i(Z \\to X)", "$$", "by sending a generator $[E]$ on the left hand side to", "$$", "c(Z \\to X, E) = 1 + c_1(Z \\to X, E) + c_2(Z \\to X, E) + \\ldots", "$$", "and extending multiplicatively (with product on the right hand", "side as in Remark \\ref{remark-ring-loc-classes}). This makes sense because by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-resolution-property-perfect-complex}", "and Definition \\ref{definition-localized-chern}", "$c_i(Z \\to X, E) $ are defined for all $i \\geq 1$.", "It is well defined by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-resolution-property-map-perfect-complex}", "(every map in $D_{Z, perf}(\\mathcal{O}_X)$ can be represented", "by a map of bounded complexes of finite locally frees) and", "Lemma \\ref{lemma-additivity-loc-chern-c}." ], "refs": [ "chow-remark-perf-Z-cohomology-K", "chow-remark-ring-loc-classes", "perfect-lemma-resolution-property-perfect-complex", "chow-definition-localized-chern", "perfect-lemma-resolution-property-map-perfect-complex", "chow-lemma-additivity-loc-chern-c" ], "ref_ids": [ 5958, 5942, 7094, 5924, 7095, 5807 ] }, { "id": 5961, "type": "other", "label": "chow-remark-localized-chern-character-K", "categories": [ "chow" ], "title": "chow-remark-localized-chern-character-K", "contents": [ "Let $X$, $Z$, $D_{Z, perf}(\\mathcal{O}_X)$", "be as in Remark \\ref{remark-perf-Z-cohomology-K}.", "Assume $X$ is quasi-compact, has the resolution property, and is", "of finite type over $(S, \\delta)$ as in Situation \\ref{situation-setup}.", "The localized Chern character defines a canonical additive", "and multiplicative map", "$$", "ch(Z \\to X, -) : K_0(D_{Z, perf}(\\mathcal{O}_X)) \\longrightarrow", "\\prod\\nolimits_{i \\geq 0} A^i(Z \\to X)", "$$", "by sending a generator $[E]$ on the left hand side to", "$ch(Z \\to X, E)$ and extending additively. This makes sense because because", "$ch(Z \\to X, E)$ is defined by", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-resolution-property-perfect-complex}", "and Definition \\ref{definition-localized-chern}.", "It is well defined by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-resolution-property-map-perfect-complex}", "(every map in $D_{Z, perf}(\\mathcal{O}_X)$ can be represented", "by a map of bounded complexes of finite locally frees) and", "Lemma \\ref{lemma-additivity-loc-chern-P}. The multiplication on", "$K_0(D_{Z, perf}(X))$ is defined using derived tensor product", "(Derived Categories of Schemes, Remark \\ref{perfect-remark-perf-Z})", "hence $ch(\\alpha \\beta) = ch(\\alpha) ch(\\beta)$ by", "Lemma \\ref{lemma-loc-chern-tensor-product}." ], "refs": [ "chow-remark-perf-Z-cohomology-K", "perfect-lemma-resolution-property-perfect-complex", "chow-definition-localized-chern", "perfect-lemma-resolution-property-map-perfect-complex", "chow-lemma-additivity-loc-chern-P", "perfect-remark-perf-Z", "chow-lemma-loc-chern-tensor-product" ], "ref_ids": [ 5958, 7094, 5924, 7095, 5808, 7142, 5809 ] }, { "id": 5962, "type": "other", "label": "chow-remark-chern-classes-agree", "categories": [ "chow" ], "title": "chow-remark-chern-classes-agree", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $X$ be locally of finite type over $S$ and assume $X$", "is quasi-compact and has the resolution property.", "With $Z = X$ and notation as in", "Remarks \\ref{remark-localized-chern-classes-K} and", "\\ref{remark-localized-chern-character-K}", "we have $D_{Z, perf}(\\mathcal{O}_X) = D_{perf}(\\mathcal{O}_X)$", "and we see that", "$$", "K_0(D_{Z, perf}(\\mathcal{O}_X)) = K_0(D_{perf}(\\mathcal{O}_X)) = K_0(X)", "$$", "see ", "Derived Categories of Schemes, Definition \\ref{perfect-definition-K-group}.", "Hence we get", "$c : K_0(X) \\to \\prod A^i(X)$ and $ch : K_0(X) \\to \\prod A^i(X)$ from", "Remarks \\ref{remark-localized-chern-classes-K} and", "\\ref{remark-localized-chern-character-K}.", "Via the equality $K_0(\\textit{Vect}(X)) = K_0(X)$ of", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-K-is-old-K}", "these maps agree with the maps constructed in", "Remarks \\ref{remark-chern-classes-K} and", "\\ref{remark-chern-character-K}." ], "refs": [ "chow-remark-localized-chern-classes-K", "chow-remark-localized-chern-character-K", "perfect-definition-K-group", "chow-remark-localized-chern-classes-K", "chow-remark-localized-chern-character-K", "perfect-lemma-K-is-old-K", "chow-remark-chern-classes-K", "chow-remark-chern-character-K" ], "ref_ids": [ 5960, 5961, 7121, 5960, 5961, 7100, 5956, 5957 ] }, { "id": 5963, "type": "other", "label": "chow-remark-gysin-chern-classes", "categories": [ "chow" ], "title": "chow-remark-gysin-chern-classes", "contents": [ "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.", "Let $f : X \\to Y$ be a local complete intersection morphism", "of schemes locally of finite type over $S$. Assume the gysin", "map exists for $f$. Then", "$f^! \\circ c_i(\\mathcal{E}) = c_i(f^*\\mathcal{E}) \\circ f^!$", "and similarly for the Chern character, see", "Lemma \\ref{lemma-lci-gysin-commutes}.", "If $X$ and $Y$ satisfy the equivalent conditions of", "Lemma \\ref{lemma-locally-equidimensional} and $Y$ is Cohen-Macaulay", "(for example), then $f^![Y] = [X]$ by Lemma \\ref{lemma-lci-gysin-easy}.", "In this case we also get", "$f^!(c_i(\\mathcal{E}) \\cap [Y]) = c_i(f^*\\mathcal{E}) \\cap [X]$", "and similarly for the Chern character." ], "refs": [ "chow-lemma-lci-gysin-commutes", "chow-lemma-locally-equidimensional", "chow-lemma-lci-gysin-easy" ], "ref_ids": [ 5834, 5761, 5835 ] }, { "id": 5964, "type": "other", "label": "chow-remark-commuting-exterior", "categories": [ "chow" ], "title": "chow-remark-commuting-exterior", "contents": [ "The upshot of Lemmas \\ref{lemma-chow-cohomology-towards-point}", "and \\ref{lemma-chow-cohomology-towards-point-commutes} is the following.", "Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$.", "Let $\\alpha \\in \\CH_*(X)$. Let $Y \\to Z$ be a morphism of schemes", "locally of finite type over $k$. Let $c' \\in A^q(Y \\to Z)$. Then", "$$", "\\alpha \\times (c' \\cap \\beta) = c' \\cap (\\alpha \\times \\beta)", "$$", "in $\\CH_*(X \\times_k Y)$ for any $\\beta \\in \\CH_*(Z)$. Namely, this", "follows by taking $c = c_\\alpha \\in A^*(X \\to \\Spec(k))$ the bivariant class", "corresponding to $\\alpha$, see proof of", "Lemma \\ref{lemma-chow-cohomology-towards-point}." ], "refs": [ "chow-lemma-chow-cohomology-towards-point", "chow-lemma-chow-cohomology-towards-point-commutes", "chow-lemma-chow-cohomology-towards-point" ], "ref_ids": [ 5842, 5843, 5842 ] }, { "id": 5965, "type": "other", "label": "chow-remark-commuting-exterior-dim-1", "categories": [ "chow" ], "title": "chow-remark-commuting-exterior-dim-1", "contents": [ "The upshot of Lemmas \\ref{lemma-chow-cohomology-towards-base-dim-1}", "and \\ref{lemma-chow-cohomology-towards-base-dim-1-commutes} is the following.", "Let $(S, \\delta)$ be as above. Let $X$ be a scheme locally of finite type", "over $S$.", "Let $\\alpha \\in \\CH_*(X)$. Let $Y \\to Z$ be a morphism of schemes", "locally of finite type over $S$. Let $c' \\in A^q(Y \\to Z)$. Then", "$$", "\\alpha \\times (c' \\cap \\beta) = c' \\cap (\\alpha \\times \\beta)", "$$", "in $\\CH_*(X \\times_S Y)$ for any $\\beta \\in \\CH_*(Z)$. Namely, this", "follows by taking $c = c_\\alpha \\in A^*(X \\to S)$ the bivariant class", "corresponding to $\\alpha$, see proof of", "Lemma \\ref{lemma-chow-cohomology-towards-base-dim-1}." ], "refs": [ "chow-lemma-chow-cohomology-towards-base-dim-1", "chow-lemma-chow-cohomology-towards-base-dim-1-commutes", "chow-lemma-chow-cohomology-towards-base-dim-1" ], "ref_ids": [ 5853, 5854, 5853 ] }, { "id": 5966, "type": "other", "label": "chow-remark-explain-determinant", "categories": [ "chow" ], "title": "chow-remark-explain-determinant", "contents": [ "Let $(R, \\mathfrak m, \\kappa)$ be a local ring and assume either", "the characteristic of $\\kappa$ is zero or it is $p$ and $p R = 0$.", "Let $M_1, \\ldots, M_n$ be finite length $R$-modules.", "We will show below that there exists an", "ideal $I \\subset \\mathfrak m$ annihilating $M_i$ for $i = 1, \\ldots, n$", "and a section $\\sigma : \\kappa \\to R/I$ of the canonical surjection", "$R/I \\to \\kappa$. The restriction $M_{i, \\kappa}$ of $M_i$ via $\\sigma$", "is a $\\kappa$-vector space of dimension $l_i = \\text{length}_R(M_i)$ and", "using Lemma \\ref{lemma-determinant-quotient-ring} we see that", "$$", "\\det\\nolimits_\\kappa(M_i) = \\wedge_\\kappa^{l_i}(M_{i, \\kappa})", "$$", "These isomorphisms are compatible with the isomorphisms", "$\\gamma_{K \\to M \\to L}$ of Lemma \\ref{lemma-det-exact-sequences}", "for short exact sequences of finite length $R$-modules annihilated", "by $I$. The conclusion is that verifying a property of", "$\\det_\\kappa$ often reduces to verifying corresponding properties", "of the usual determinant on the category finite dimensional vector", "spaces.", "\\medskip\\noindent", "For $I$ we can take the annihilator", "(Algebra, Definition \\ref{algebra-definition-annihilator})", "of the module $M = \\bigoplus M_i$. In this case we see that", "$R/I \\subset \\text{End}_R(M)$ hence has finite length.", "Thus $R/I$ is an Artinian local ring with residue field $\\kappa$.", "Since an Artinian local ring is complete we see that $R/I$", "has a coefficient ring by the Cohen structure theorem", "(Algebra, Theorem \\ref{algebra-theorem-cohen-structure-theorem})", "which is a field by our assumption on $R$." ], "refs": [ "chow-lemma-determinant-quotient-ring", "chow-lemma-det-exact-sequences", "algebra-definition-annihilator", "algebra-theorem-cohen-structure-theorem" ], "ref_ids": [ 5872, 5870, 1458, 327 ] }, { "id": 5967, "type": "other", "label": "chow-remark-more-elementary", "categories": [ "chow" ], "title": "chow-remark-more-elementary", "contents": [ "Here is a more down to earth description of the determinant", "introduced above. Let $R$ be a local ring with residue field $\\kappa$.", "Let $(M, \\varphi, \\psi)$ be a $(2, 1)$-periodic complex over $R$.", "Assume that $M$ has finite length and that $(M, \\varphi, \\psi)$ is", "exact. Let us abbreviate $I_\\varphi = \\Im(\\varphi)$,", "$I_\\psi = \\Im(\\psi)$ as above.", "Assume that $\\text{length}_R(I_\\varphi) = a$ and", "$\\text{length}_R(I_\\psi) = b$, so that $a + b = \\text{length}_R(M)$", "by exactness. Choose admissible sequences", "$x_1, \\ldots, x_a \\in I_\\varphi$ and $y_1, \\ldots, y_b \\in I_\\psi$", "such that the symbol $[x_1, \\ldots, x_a]$ generates $\\det_\\kappa(I_\\varphi)$", "and the symbol $[x_1, \\ldots, x_b]$ generates $\\det_\\kappa(I_\\psi)$.", "Choose $\\tilde x_i \\in M$ such that $\\varphi(\\tilde x_i) = x_i$.", "Choose $\\tilde y_j \\in M$ such that $\\psi(\\tilde y_j) = y_j$.", "Then $\\det_\\kappa(M, \\varphi, \\psi)$ is characterized", "by the equality", "$$", "[x_1, \\ldots, x_a, \\tilde y_1, \\ldots, \\tilde y_b]", "=", "(-1)^{ab} \\det\\nolimits_\\kappa(M, \\varphi, \\psi)", "[y_1, \\ldots, y_b, \\tilde x_1, \\ldots, \\tilde x_a]", "$$", "in $\\det_\\kappa(M)$. This also explains the sign." ], "refs": [], "ref_ids": [] }, { "id": 6224, "type": "other", "label": "flat-remark-finite-presentation", "categories": [ "flat" ], "title": "flat-remark-finite-presentation", "contents": [ "Note that the $R$-algebras $B_i$ for all $i$ and $A_i$ for $i \\geq 2$", "are of finite presentation over $R$. If $S$ is of finite presentation over", "$R$, then it is also the case that $A_1$ is of finite presentation over", "$R$. In this case all the ring maps in the complete d\\'evissage are of", "finite presentation. See", "Algebra, Lemma \\ref{algebra-lemma-compose-finite-type}.", "Still assuming $S$ of finite presentation over $R$", "the following are equivalent", "\\begin{enumerate}", "\\item $M$ is of finite presentation over $S$,", "\\item $M_1$ is of finite presentation over $A_1$,", "\\item $M_1$ is of finite presentation over $B_1$,", "\\item each $M_i$ is of finite presentation both as an $A_i$-module", "and as a $B_i$-module.", "\\end{enumerate}", "The equivalences (1) $\\Leftrightarrow$ (2) and (2) $\\Leftrightarrow$ (3)", "follow from", "Algebra, Lemma \\ref{algebra-lemma-finite-finitely-presented-extension}.", "If $M_1$ is finitely presented, so is $\\Coker(\\alpha_1)$ (see", "Algebra, Lemma \\ref{algebra-lemma-extension})", "and hence $M_2$, etc." ], "refs": [ "algebra-lemma-compose-finite-type", "algebra-lemma-finite-finitely-presented-extension", "algebra-lemma-extension" ], "ref_ids": [ 333, 501, 330 ] }, { "id": 6225, "type": "other", "label": "flat-remark-same-notion", "categories": [ "flat" ], "title": "flat-remark-same-notion", "contents": [ "Let $A \\to B$ be a finite type ring map and let $N$ be a finite", "$B$-module. Let $\\mathfrak q$ be a prime of $B$ lying over the prime", "$\\mathfrak r$ of $A$. Set $X = \\Spec(B)$, $S = \\Spec(A)$ and", "$\\mathcal{F} = \\widetilde{N}$ on $X$. Let $x$ be the point corresponding", "to $\\mathfrak q$ and let $s \\in S$ be the point corresponding to", "$\\mathfrak p$. Then", "\\begin{enumerate}", "\\item if there exists a complete d\\'evissage of $\\mathcal{F}/X/S$", "over $s$ then there exists a complete d\\'evissage of", "$N/B/A$ over $\\mathfrak p$, and", "\\item there exists a complete d\\'evissage of $\\mathcal{F}/X/S$", "at $x$ if and only if there exists a complete d\\'evissage of", "$N/B/A$ at $\\mathfrak q$.", "\\end{enumerate}", "There is just a small twist in that we omitted the condition on", "the relative dimension in the formulation of ``a complete d\\'evissage of", "$N/B/A$ over $\\mathfrak p$'' which is why the implication in (1)", "only goes in one direction.", "The notion of a complete d\\'evissage at", "$\\mathfrak q$ does have this condition built in. In any case we will", "only use that existence for $\\mathcal{F}/X/S$", "implies the existence for $N/B/A$." ], "refs": [], "ref_ids": [] }, { "id": 6226, "type": "other", "label": "flat-remark-how-in-RG", "categories": [ "flat" ], "title": "flat-remark-how-in-RG", "contents": [ "Lemma \\ref{lemma-fibres-irreducible-flat-projective-nonnoetherian}", "is a key step in the development of results in this chapter. The analogue", "of this lemma in \\cite{GruRay} is \\cite[I Proposition 3.3.1]{GruRay}:", "If $R \\to S$ is smooth with geometrically integral fibres, then $S$", "is projective as an $R$-module. This is a special case of", "Lemma \\ref{lemma-fibres-irreducible-flat-projective-nonnoetherian},", "but as we will later improve on this lemma anyway, we do not gain much", "from having a stronger result at this point.", "We briefly sketch the proof of this as it is given in \\cite{GruRay}.", "\\begin{enumerate}", "\\item First reduce to the case where $R$ is Noetherian as above.", "\\item Since projectivity descends through faithfully flat ring maps, see", "Algebra, Theorem \\ref{algebra-theorem-ffdescent-projectivity}", "we may work locally in the fppf topology on $R$, hence we may assume", "that $R \\to S$ has a section $\\sigma : S \\to R$. (Just by the usual trick of", "base changing to $S$.) Set $I = \\Ker(S \\to R)$.", "\\item Localizing a bit more on $R$ we may assume that $I/I^2$ is a free", "$R$-module and that the completion $S^\\wedge$ of $S$ with respect to $I$", "is isomorphic to $R[[t_1, \\ldots, t_n]]$, see", "Morphisms, Lemma \\ref{morphisms-lemma-section-smooth-morphism}.", "Here we are using that $R \\to S$ is smooth.", "\\item To prove that $S$ is projective as an $R$-module, it suffices to", "prove that $S$ is flat, countably generated and Mittag-Leffler as an", "$R$-module, see", "Algebra, Lemma \\ref{algebra-lemma-countgen-projective}.", "The first two properties are evident. Thus it suffices to prove that $S$", "is Mittag-Leffler as an $R$-module. By", "Algebra, Lemma \\ref{algebra-lemma-power-series-ML}", "the module $R[[t_1, \\ldots, t_n]]$ is Mittag-Leffler over $R$. Hence", "Algebra, Lemma \\ref{algebra-lemma-pure-submodule-ML}", "shows that it suffices to show that the", "$S \\to S^\\wedge$ is universally injective as a map of $R$-modules.", "\\item Apply", "Lemma \\ref{lemma-base-change-universally-flat}", "to see that $S \\to S^\\wedge$ is $R$-universally injective.", "Namely, as $R \\to S$ has geometrically integral fibres, any associated", "point of any fibre ring is just the generic point of the fibre ring which", "is in the image of $\\Spec(S^\\wedge) \\to \\Spec(S)$.", "\\end{enumerate}", "There is an analogy between the proof as sketched just now, and the", "development of the arguments leading to the proof of", "Lemma \\ref{lemma-fibres-irreducible-flat-projective-nonnoetherian}.", "In both a completion plays an essential role, and both times the", "assumption of having geometrically integral fibres assures one that the", "map from $S$ to the completion of $S$ is $R$-universally injective." ], "refs": [ "flat-lemma-fibres-irreducible-flat-projective-nonnoetherian", "flat-lemma-fibres-irreducible-flat-projective-nonnoetherian", "algebra-theorem-ffdescent-projectivity", "algebra-lemma-countgen-projective", "algebra-lemma-power-series-ML", "algebra-lemma-pure-submodule-ML", "flat-lemma-base-change-universally-flat", "flat-lemma-fibres-irreducible-flat-projective-nonnoetherian" ], "ref_ids": [ 6014, 6014, 324, 850, 847, 837, 6005, 6014 ] }, { "id": 6227, "type": "other", "label": "flat-remark-complete-devissage-flat-finitely-presented-module", "categories": [ "flat" ], "title": "flat-remark-complete-devissage-flat-finitely-presented-module", "contents": [ "There is a variant of", "Lemma \\ref{lemma-complete-devissage-flat-finitely-presented-module}", "where we weaken the flatness condition by assuming only that $N$", "is flat at some given prime $\\mathfrak q$ lying over $\\mathfrak r$", "but where we strengthen the d\\'evissage condition by assuming", "the existence of a complete d\\'evissage {\\it at $\\mathfrak q$}. Compare with", "Lemma \\ref{lemma-complete-devissage-flat-finite-type-module}." ], "refs": [ "flat-lemma-complete-devissage-flat-finitely-presented-module", "flat-lemma-complete-devissage-flat-finite-type-module" ], "ref_ids": [ 6030, 6016 ] }, { "id": 6228, "type": "other", "label": "flat-remark-finite-type-flat", "categories": [ "flat" ], "title": "flat-remark-finite-type-flat", "contents": [ "Let $f : X \\to S$ be a morphism of schemes.", "Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.", "Assume", "\\begin{enumerate}", "\\item $X \\to S$ is locally of finite type,", "\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type, and", "\\item the set of weakly associated points of $S$ is locally finite in $S$.", "\\end{enumerate}", "Then $U = \\{x \\in X \\mid \\mathcal{F}\\text{ flat at }x\\text{ over }S\\}$", "is open in $X$ and $\\mathcal{F}|_U$ is flat over $S$ and locally", "finitely presented relative to $S$ (see", "More on Morphisms, Definition", "\\ref{more-morphisms-definition-relatively-finitely-presented-sheaf}).", "If we ever need this result in the Stacks project we will convert", "this remark into a lemma with a proof." ], "refs": [ "flat-theorem-finite-type-flat", "more-morphisms-definition-relatively-finitely-presented-sheaf" ], "ref_ids": [ 5968, 14117 ] }, { "id": 6229, "type": "other", "label": "flat-remark-finite-type-flat-algebra", "categories": [ "flat" ], "title": "flat-remark-finite-type-flat-algebra", "contents": [ "Let $R \\to S$ be a ring map of finite type.", "Let $M$ be a finite $S$-module.", "Assume $\\text{WeakAss}_R(R)$ is finite.", "Then", "$$", "U = \\{\\mathfrak q \\subset S \\mid M_{\\mathfrak q}\\text{ flat over }R\\}", "$$", "is open in $\\Spec(S)$ and for every $g \\in S$ such that", "$D(g) \\subset U$ the localization $M_g$ is flat over $R$ and", "an $S_g$-module finitely presented relative to $R$ (see", "More on Algebra, Definition", "\\ref{more-algebra-definition-relatively-finitely-presented}).", "If we ever need this result in the Stacks project we will convert", "this remark into a lemma with a proof." ], "refs": [ "flat-remark-finite-type-flat", "more-algebra-definition-relatively-finitely-presented" ], "ref_ids": [ 6228, 10629 ] }, { "id": 6230, "type": "other", "label": "flat-remark-discuss-finite-type", "categories": [ "flat" ], "title": "flat-remark-discuss-finite-type", "contents": [ "Let $f : X \\to S$ be a morphism which is locally of finite type", "and $\\mathcal{F}$ a quasi-coherent finite type $\\mathcal{O}_X$-module.", "In this case it is still true that (1) and (2) above are equivalent", "because the proof of", "Lemma \\ref{lemma-quasi-finite-impurity-elementary}", "does not use that $f$ is quasi-compact. It is also clear that", "(3) and (4) are equivalent. However, we don't know if (1) and (3) are", "equivalent. In this case it may sometimes be more convenient to define", "purity using the equivalent conditions (3) and (4) as is done in \\cite{GruRay}.", "On the other hand, for many applications it seems that the correct notion", "is really that of being universally pure." ], "refs": [ "flat-lemma-quasi-finite-impurity-elementary" ], "ref_ids": [ 6052 ] }, { "id": 6231, "type": "other", "label": "flat-remark-flattening-local-scheme-theoretic", "categories": [ "flat" ], "title": "flat-remark-flattening-local-scheme-theoretic", "contents": [ "Here is a scheme theoretic reformulation of", "Theorem \\ref{theorem-flattening-local}.", "Let $(X, x) \\to (S, s)$ be a morphism of pointed schemes", "which is locally of finite type. Let $\\mathcal{F}$ be a finite", "type quasi-coherent $\\mathcal{O}_X$-module.", "Assume $S$ henselian local with closed point $s$.", "There exists a closed subscheme $Z \\subset S$ with the following property:", "for any morphism of pointed schemes $(T, t) \\to (S, s)$ the following", "are equivalent", "\\begin{enumerate}", "\\item $\\mathcal{F}_T$ is flat over $T$ at all points of the fibre", "$X_t$ which map to $x \\in X_s$, and", "\\item $\\Spec(\\mathcal{O}_{T, t}) \\to S$ factors through $Z$.", "\\end{enumerate}", "Moreover, if $X \\to S$ is of finite presentation at $x$ and $\\mathcal{F}_x$", "of finite presentation over $\\mathcal{O}_{X, x}$, then $Z \\to S$", "is of finite presentation." ], "refs": [ "flat-theorem-flattening-local" ], "ref_ids": [ 5970 ] }, { "id": 6232, "type": "other", "label": "flat-remark-flattening-complete-noetherian", "categories": [ "flat" ], "title": "flat-remark-flattening-complete-noetherian", "contents": [ "Tracing the proof of", "Lemma \\ref{lemma-freebie}", "to its origins we find a long and winding road. But if we assume that", "\\begin{enumerate}", "\\item $f$ is of finite type,", "\\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module,", "\\item $E = X_s$, and", "\\item $S$ is the spectrum of a Noetherian complete local ring.", "\\end{enumerate}", "then there is a proof relying completely on more elementary algebra as", "follows: first we reduce to the case where $X$ is affine by taking", "a finite affine open cover. In this case $Z$ exists by", "More on Algebra,", "Lemma \\ref{more-algebra-lemma-flattening-complete-local-universal-property}.", "The key step in this proof is constructing the closed subscheme $Z$", "step by step inside the truncations", "$\\Spec(\\mathcal{O}_{S, s}/\\mathfrak m_s^n)$.", "This relies on the fact that flattening stratifications always exist", "when the base is Artinian, and the fact that", "$\\mathcal{O}_{S, s} = \\lim \\mathcal{O}_{S, s}/\\mathfrak m_s^n$." ], "refs": [ "flat-lemma-freebie", "more-algebra-lemma-flattening-complete-local-universal-property" ], "ref_ids": [ 6089, 9905 ] }, { "id": 6233, "type": "other", "label": "flat-remark-correct-generality", "categories": [ "flat" ], "title": "flat-remark-correct-generality", "contents": [ "The result in this section can be generalized. It is probably correct", "if we only assume $X \\to \\Spec(A)$ to be separated, of finite presentation,", "and $K_n$ pseudo-coherent relative to $A_n$ supported on a closed", "subset of $X_n$ proper over $A_n$. The outcome will be a $K$ which", "is pseudo-coherent relative to $A$ supported on a closed subset", "proper over $A$. If we ever need this, we will", "formulate a precise statement and prove it here." ], "refs": [], "ref_ids": [] }, { "id": 6234, "type": "other", "label": "flat-remark-successive-blowups", "categories": [ "flat" ], "title": "flat-remark-successive-blowups", "contents": [ "Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to S$", "be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent module on $X$.", "Let $U \\subset S$ be a quasi-compact open subscheme. Given a $U$-admissible", "blowup $S' \\to S$ we denote $X'$ the strict transform of $X$ and $\\mathcal{F}'$", "the strict transform of $\\mathcal{F}$ which we think of as a quasi-coherent", "module on $X'$ (via Divisors, Lemma \\ref{divisors-lemma-strict-transform}).", "Let $P$ be a property of $\\mathcal{F}/X/S$ which is stable under strict", "transform (as above) for $U$-admissible blowups. The general problem in", "this section is: Show (under auxiliary conditions on $\\mathcal{F}/X/S$)", "there exists a $U$-admissible blowup $S' \\to S$", "such that the strict transform $\\mathcal{F}'/X'/S'$ has $P$.", "\\medskip\\noindent", "The general strategy will be to use that a composition of", "$U$-admissible blowups is a $U$-admissible blowup, see", "Divisors, Lemma \\ref{divisors-lemma-composition-admissible-blowups}.", "In fact, we will make use of the more precise", "Divisors, Lemma \\ref{divisors-lemma-composition-finite-type-blowups}", "and combine it with", "Divisors, Lemma \\ref{divisors-lemma-strict-transform-composition-blowups}.", "The result is that it suffices to find a sequence of $U$-admissible", "blowups", "$$", "S = S_0 \\leftarrow S_1 \\leftarrow \\ldots \\leftarrow S_n", "$$", "such that, setting $\\mathcal{F}_0 = \\mathcal{F}$ and $X_0 = X$ and setting", "$\\mathcal{F}_i/X_i$ equal to the strict transform of", "$\\mathcal{F}_{i - 1}/X_{i - 1}$, we", "arrive at $\\mathcal{F}_n/X_n/S_n$ with property $P$.", "\\medskip\\noindent", "In particular, choose a finite type quasi-coherent sheaf of ideals", "$\\mathcal{I} \\subset \\mathcal{O}_S$ such that $V(\\mathcal{I}) = S \\setminus U$,", "see Properties, Lemma \\ref{properties-lemma-quasi-coherent-finite-type-ideals}.", "Let $S' \\to S$ be the blowup in $\\mathcal{I}$ and let $E \\subset S'$", "be the exceptional divisor (Divisors, Lemma", "\\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}).", "Then we see that we've reduced the", "problem to the case where there exists an effective Cartier divisor", "$D \\subset S$ whose support is $X \\setminus U$. In particular we may", "assume $U$ is scheme theoretically dense in $S$", "(Divisors, Lemma \\ref{divisors-lemma-complement-effective-Cartier-divisor}).", "\\medskip\\noindent", "Suppose that $P$ is local on $S$: If $S = \\bigcup S_i$ is a finite open", "covering by quasi-compact opens and $P$ holds for", "$\\mathcal{F}_{S_i}/X_{S_i}/S_i$ then $P$ holds for $\\mathcal{F}/X/S$.", "In this case the general problem above is local on $S$ as well, i.e.,", "if given $s \\in S$ we can find a quasi-compact open neighbourhood $W$ of $s$", "such that the problem for $\\mathcal{F}_W/X_W/W$ is solvable, then the", "problem is solvable for $\\mathcal{F}/X/S$. This follows from", "Divisors, Lemmas \\ref{divisors-lemma-extend-admissible-blowups} and", "\\ref{divisors-lemma-dominate-admissible-blowups}." ], "refs": [ "divisors-lemma-strict-transform", "divisors-lemma-composition-admissible-blowups", "divisors-lemma-composition-finite-type-blowups", "divisors-lemma-strict-transform-composition-blowups", "properties-lemma-quasi-coherent-finite-type-ideals", "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "divisors-lemma-complement-effective-Cartier-divisor", "divisors-lemma-extend-admissible-blowups", "divisors-lemma-dominate-admissible-blowups" ], "ref_ids": [ 8065, 8071, 8064, 8069, 3033, 8054, 7929, 8072, 8073 ] }, { "id": 6235, "type": "other", "label": "flat-remark-when-you-have-a-complex", "categories": [ "flat" ], "title": "flat-remark-when-you-have-a-complex", "contents": [ "Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be a perfect object such", "that $H^i(E)$ is a perfect $\\mathcal{O}_X$-module of tor dimension $\\leq 1$", "for all $i \\in \\mathbf{Z}$. This property sometimes allows one to reduce", "questions about $E$ to questions about $H^i(E)$. For example, suppose", "$$", "\\mathcal{E}^a \\xrightarrow{d^a} \\ldots", "\\xrightarrow{d^{b - 2}} \\mathcal{E}^{b - 1}", "\\xrightarrow{d^{b - 1}} \\mathcal{E}^b", "$$", "is a bounded complex of finite locally free $\\mathcal{O}_X$-modules", "representing $E$. Then $\\Im(d^i)$ and $\\Ker(d^i)$ are finite locally", "free $\\mathcal{O}_X$-modules for all $i$. Namely, suppose by induction", "we know this for all indices bigger than $i$. Then we can first use the", "short exact sequence", "$$", "0 \\to \\Im(d^i) \\to \\Ker(d^{i + 1}) \\to H^{i + 1}(E) \\to 0", "$$", "and the assumption that $H^{i + 1}(E)$ is perfect of tor dimension $\\leq 1$", "to conclude that $\\Im(d^i)$ is finite locally free.", "The same argument used again for the short exact sequence", "$$", "0 \\to \\Ker(d^i) \\to \\mathcal{E}^i \\to \\Im(d^i) \\to 0", "$$", "then gives that $\\Ker(d^i)$ is finite locally free.", "It follows that the distinguished triangles", "$$", "\\tau_{\\leq k - 1}E \\to \\tau_{\\leq k}E \\to H^k(E)[-k] \\to", "(\\tau_{\\leq k - 1}E)[1]", "$$", "are represented by the following short exact sequences of", "bounded complexes of finite locally free modules", "$$", "\\begin{matrix}", "& &", "& &", "& &", "0 \\\\", "& &", "& &", "& &", "\\downarrow \\\\", "\\mathcal{E}^a & \\to &", "\\ldots & \\to &", "\\mathcal{E}^{k - 2} & \\to &", "\\Ker(d^{k - 1}) \\\\", "\\downarrow & &", "& &", "\\downarrow & &", "\\downarrow \\\\", "\\mathcal{E}^a & \\to &", "\\ldots & \\to &", "\\mathcal{E}^{k - 2} & \\to &", "\\mathcal{E}^{k - 1} & \\to &", "\\Ker(d^k) \\\\", "& &", "& &", "& &", "\\downarrow & &", "\\downarrow \\\\", "& &", "& &", "& &", "\\Im(d^{k - 1}) & \\to &", "\\Ker(d^k) \\\\", "& &", "& &", "& &", "\\downarrow \\\\", "& &", "& &", "& &", "0", "\\end{matrix}", "$$", "Here the complexes are the rows and the ``obvious'' zeros", "are omitted from the display." ], "refs": [], "ref_ids": [] }, { "id": 6236, "type": "other", "label": "flat-remark-Leta", "categories": [ "flat" ], "title": "flat-remark-Leta", "contents": [ "Let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor", "with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.", "Let $\\mathcal{G}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules.", "By Cohomology, Lemma \\ref{cohomology-lemma-K-flat-resolution}", "there exists a quasi-isomorphism $\\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$", "such that $\\mathcal{F}^\\bullet$ is a K-flat complex whose terms are flat", "$\\mathcal{O}_X$-modules. (Even if $\\mathcal{G}^\\bullet$ is a", "complex of quasi-coherent $\\mathcal{O}_X$-modules, in general", "$\\mathcal{F}^\\bullet$ will not be so.)", "It follows that $\\mathcal{F}^i$ is", "$\\mathcal{I}$-torsion free for all $i$. In this situation we define", "$$", "L\\eta_\\mathcal{I} \\mathcal{G}^\\bullet = \\eta_\\mathcal{I} \\mathcal{F}^\\bullet", "$$", "This is independent of the choice of the K-flat resolution", "by Lemma \\ref{lemma-eta-qis}. We obtain a functor", "$L\\eta_\\mathcal{I} : D(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$.", "Beware that this functor isn't exact, i.e.,", "does not tranform distinguished triangles into distinguished triangles." ], "refs": [ "cohomology-lemma-K-flat-resolution", "flat-lemma-eta-qis" ], "ref_ids": [ 2112, 6188 ] }, { "id": 6237, "type": "other", "label": "flat-remark-complex-and-divisor-ideal", "categories": [ "flat" ], "title": "flat-remark-complex-and-divisor-ideal", "contents": [ "In Situation \\ref{situation-complex-and-divisor} for any $i \\in \\mathbf{Z}$", "there exists a finite type quasi-coherent sheaf of ideals", "$\\mathcal{J}_i \\subset \\mathcal{O}_X$ with the following property:", "for any $U \\subset X$ open such that $\\mathcal{I}|_U$,", "$\\mathcal{E}^i|_U$, and $\\mathcal{E}^{i + 1}|_U$ are", "free of ranks $1$, $r_i$, and $r_{i + 1}$, the ideal", "$\\mathcal{J}_i$ is generated by the $r_i \\times r_i$ minors of the map", "$$", "1, d^i :", "\\mathcal{I}\\mathcal{E}^i", "\\longrightarrow", "\\mathcal{E}^i \\oplus \\mathcal{I}\\mathcal{E}^{i + 1}", "$$", "with notation as in Section \\ref{section-eta}. By convention we set", "$\\mathcal{J}_i|_U = \\mathcal{O}_U$ if $r_i = 0$. Observe that", "$\\mathcal{I}^{r_i}|_U \\subset \\mathcal{J}_i|_U$ in other words,", "the closed subscheme $V(\\mathcal{J})$ is set theoretically", "contained in $D$. Formation of the", "ideal $\\mathcal{J}_i$ commutes with base change by any morphism", "$f : Y \\to X$ such that the pullback of $D$ by $f$ is defined", "(Divisors, Definition", "\\ref{divisors-definition-pullback-effective-Cartier-divisor})." ], "refs": [ "divisors-definition-pullback-effective-Cartier-divisor" ], "ref_ids": [ 8091 ] }, { "id": 6361, "type": "other", "label": "curves-remark-classical-linear-series", "categories": [ "curves" ], "title": "curves-remark-classical-linear-series", "contents": [ "Let $X$ be a smooth projective curve over an algebraically closed field $k$.", "We say two effective Cartier divisors $D, D' \\subset X$ are", "{\\it linearly equivalent} if and only if", "$\\mathcal{O}_X(D) \\cong \\mathcal{O}_X(D')$ as $\\mathcal{O}_X$-modules.", "Since $\\Pic(X) = \\text{Cl}(X)$", "(Divisors, Lemma \\ref{divisors-lemma-local-rings-UFD-c1-bijective})", "we see that $D$ and $D'$ are linearly equivalent", "if and only if the Weil divisors associated to", "$D$ and $D'$ define the same element of $\\text{Cl}(X)$.", "Given an effective Cartier divisor $D \\subset X$ of degree $d$ the", "{\\it complete linear system} or {\\it complete linear series} $|D|$ of $D$", "is the set of effective Cartier divisors $E \\subset X$", "which are linearly equivalent to $D$.", "Another way to say it is that $|D|$ is the set of closed", "points of the fibre of the morphism", "$$", "\\gamma_d :", "\\underline{\\Hilbfunctor}^d_{X/k}", "\\longrightarrow", "\\underline{\\Picardfunctor}^d_{X/k}", "$$", "(Picard Schemes of Curves, Lemma \\ref{pic-lemma-picard-pieces})", "over the closed point corresponding to $\\mathcal{O}_X(D)$.", "This gives $|D|$ a natural scheme structure and it", "turns out that $|D| \\cong \\mathbf{P}^m_k$ with", "$m + 1 = h^0(\\mathcal{O}_X(D))$. In fact, more canonically we have", "$$", "|D| = \\mathbf{P}(H^0(X, \\mathcal{O}_X(D))^\\vee)", "$$", "where $(-)^\\vee$ indicates $k$-linear dual and $\\mathbf{P}$ is as", "in Constructions, Example \\ref{constructions-example-projective-space}.", "In this language a {\\it linear system} or a {\\it linear series} on", "$X$ is a closed subvariety $L \\subset |D|$ which can be cut out by", "linear equations. If $L$ has dimension $r$, then $L = \\mathbf{P}(V^\\vee)$", "where $V \\subset H^0(X, \\mathcal{O}_X(D))$ is a linear subspace", "of dimension $r + 1$. Thus the classical linear series", "$L \\subset |D|$ corresponds to the linear series $(\\mathcal{O}_X(D), V)$", "as defined above." ], "refs": [ "divisors-lemma-local-rings-UFD-c1-bijective", "pic-lemma-picard-pieces" ], "ref_ids": [ 8029, 12571 ] }, { "id": 6362, "type": "other", "label": "curves-remark-rework-duality-locally-free", "categories": [ "curves" ], "title": "curves-remark-rework-duality-locally-free", "contents": [ "Let $X$ be a proper scheme of dimension $\\leq 1$ over a field $k$.", "Let $\\omega_X^\\bullet$ and $\\omega_X$ be as in Lemma \\ref{lemma-duality-dim-1}.", "If $\\mathcal{E}$ is a finite locally free $\\mathcal{O}_X$-module", "with dual $\\mathcal{E}^\\vee$ then we have canonical isomorphisms", "$$", "\\Hom_k(H^{-i}(X, \\mathcal{E}), k) =", "H^i(X, \\mathcal{E}^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_X^\\bullet)", "$$", "This follows from the lemma and", "Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}.", "If $X$ is Cohen-Macaulay and equidimensional of dimension $1$, then", "we have canonical isomorphisms", "$$", "\\Hom_k(H^{-i}(X, \\mathcal{E}), k) =", "H^{1 - i}(X, \\mathcal{E}^\\vee \\otimes_{\\mathcal{O}_X} \\omega_X)", "$$", "by Lemma \\ref{lemma-duality-dim-1-CM}. In particular", "if $\\mathcal{L}$ is an invertible $\\mathcal{O}_X$-module, then we have", "$$", "\\dim_k H^0(X, \\mathcal{L}) =", "\\dim_k H^1(X, \\mathcal{L}^{\\otimes -1} \\otimes_{\\mathcal{O}_X} \\omega_X)", "$$", "and", "$$", "\\dim_k H^1(X, \\mathcal{L}) =", "\\dim_k H^0(X, \\mathcal{L}^{\\otimes -1} \\otimes_{\\mathcal{O}_X} \\omega_X)", "$$" ], "refs": [ "curves-lemma-duality-dim-1", "cohomology-lemma-dual-perfect-complex", "curves-lemma-duality-dim-1-CM" ], "ref_ids": [ 6250, 2233, 6251 ] }, { "id": 6363, "type": "other", "label": "curves-remark-genus-higher-dimension", "categories": [ "curves" ], "title": "curves-remark-genus-higher-dimension", "contents": [ "Suppose that $X$ is a $d$-dimensional proper smooth variety over", "an algebraically closed field $k$.", "Then the {\\it arithmetic genus} is often defined as", "$p_a(X) = (-1)^d(\\chi(X, \\mathcal{O}_X) - 1)$ and the {\\it geometric genus}", "as $p_g(X) = \\dim_k H^0(X, \\Omega^d_{X/k})$. In this situation", "the arithmetic genus and the geometric genus no longer agree", "even though it is still true that $\\omega_X \\cong \\Omega_{X/k}^d$.", "For example, if $d = 2$, then we have", "\\begin{align*}", "p_a(X) - p_g(X) & =", "h^0(X, \\mathcal{O}_X) - h^1(X, \\mathcal{O}_X) + h^2(X, \\mathcal{O}_X) - 1", "- h^0(X, \\Omega^2_{X/k}) \\\\", "& =", "- h^1(X, \\mathcal{O}_X) + h^2(X, \\mathcal{O}_X) - h^0(X, \\omega_X) \\\\", "& =", "- h^1(X, \\mathcal{O}_X)", "\\end{align*}", "where $h^i(X, \\mathcal{F}) = \\dim_k H^i(X, \\mathcal{F})$ and", "where the last equality follows from duality.", "Hence for a surface the difference $p_g(X) - p_a(X)$ is always", "nonnegative; it is sometimes called the irregularity of the surface.", "If $X = C_1 \\times C_2$ is a product of smooth projective curves of", "genus $g_1$ and $g_2$, then the irregularity is $g_1 + g_2$." ], "refs": [], "ref_ids": [] }, { "id": 6364, "type": "other", "label": "curves-remark-quadratic-extension", "categories": [ "curves" ], "title": "curves-remark-quadratic-extension", "contents": [ "Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be a", "Noetherian local $k$-algebra. Assume that either", "$(A, \\mathfrak m, \\kappa)$ is as in Lemma \\ref{lemma-nodal-algebraic}, or", "$A$ is Nagata as in Lemma \\ref{lemma-2-branches-delta-1}, or", "$A$ is complete and as in Lemma \\ref{lemma-fitting-ideal}.", "Then $A$ defines canonically a degree $2$ separable $\\kappa$-algebra", "$\\kappa'$ as follows", "\\begin{enumerate}", "\\item let $q = ax^2 + bxy + cy^2$ be a nondegenerate quadric", "as in Lemma \\ref{lemma-nodal-algebraic} with coordinates $x, y$ chosen", "such that $a \\not = 0$ and set", "$\\kappa' = \\kappa[x]/(ax^2 + bx + c)$,", "\\item let $A' \\supset A$ be the integral closure of $A$ in its", "total ring of fractions and set $\\kappa' = A'/\\mathfrak m A'$, or", "\\item let $\\kappa'$ be the $\\kappa$-algebra such that", "$\\text{Proj}(\\bigoplus_{n \\geq 0} \\mathfrak m^n/\\mathfrak m^{n + 1}) =", "\\Spec(\\kappa')$.", "\\end{enumerate}", "The equivalence of (1) and (2) was shown in the proof of", "Lemma \\ref{lemma-2-branches-delta-1}. We omit the equivalence of", "this with (3). If $X$ is a locally Noetherian $k$-scheme and $x \\in X$", "is a point such that $\\mathcal{O}_{X, x} = A$, then (3) shows that", "$\\Spec(\\kappa') = X^\\nu \\times_X \\Spec(\\kappa)$ where $\\nu : X^\\nu \\to X$", "is the normalization morphism." ], "refs": [ "curves-lemma-nodal-algebraic", "curves-lemma-2-branches-delta-1", "curves-lemma-fitting-ideal", "curves-lemma-nodal-algebraic", "curves-lemma-2-branches-delta-1" ], "ref_ids": [ 6305, 6306, 6308, 6305, 6306 ] }, { "id": 6365, "type": "other", "label": "curves-remark-trivial-quadratic-extension", "categories": [ "curves" ], "title": "curves-remark-trivial-quadratic-extension", "contents": [ "Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be as in", "Remark \\ref{remark-quadratic-extension} and let $\\kappa'/\\kappa$", "be the associated separable algebra of degree $2$.", "Then the following are equivalent", "\\begin{enumerate}", "\\item $\\kappa' \\cong \\kappa \\times \\kappa$ as $\\kappa$-algebra,", "\\item the form $q$ of Lemma \\ref{lemma-nodal-algebraic}", "can be chosen to be $xy$,", "\\item $A$ has two branches,", "\\item the extension $A'/A$ of Lemma \\ref{lemma-2-branches-delta-1}", "has two maximal ideals, and", "\\item $A^\\wedge \\cong \\kappa[[x, y]]/(xy)$ as a $k$-algebra.", "\\end{enumerate}", "The equivalence between these conditions has been shown in the", "proof of Lemma \\ref{lemma-2-branches-delta-1}. If $X$ is a", "locally Noetherian $k$-scheme and $x \\in X$ is a point such that", "$\\mathcal{O}_{X, x} = A$, then this means exactly that", "there are two points $x_1, x_2$ of the normalization $X^\\nu$", "lying over $x$ and that $\\kappa(x) = \\kappa(x_1) = \\kappa(x_2)$." ], "refs": [ "curves-remark-quadratic-extension", "curves-lemma-nodal-algebraic", "curves-lemma-2-branches-delta-1", "curves-lemma-2-branches-delta-1" ], "ref_ids": [ 6364, 6305, 6306, 6306 ] }, { "id": 6761, "type": "other", "label": "etale-cohomology-remark-i-is-j", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-i-is-j", "contents": [ "In the last statement, it is essential not to forget the case where $i = j$", "which is in general a highly nontrivial condition (unlike in the Zariski", "topology). In fact, frequently important coverings have only one element." ], "refs": [], "ref_ids": [] }, { "id": 6762, "type": "other", "label": "etale-cohomology-remark-empty-covering", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-empty-covering", "contents": [ "For the empty covering (where $I = \\emptyset$), this implies that", "$\\mathcal{F}(\\emptyset)$ is an empty product, which is a final object in the", "corresponding category (a singleton, for both $\\textit{Sets}$ and", "$\\textit{Ab}$)." ], "refs": [], "ref_ids": [] }, { "id": 6763, "type": "other", "label": "etale-cohomology-remark-fpqc", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-fpqc", "contents": [ "The first condition corresponds to fp, which stands for", "{\\it fid\\`element plat}, faithfully flat in french, and", "the second to qc, {\\it quasi-compact}. The second part of", "the first condition is unnecessary when the second condition holds." ], "refs": [], "ref_ids": [] }, { "id": 6764, "type": "other", "label": "etale-cohomology-remark-fpqc-finest", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-fpqc-finest", "contents": [ "The fpqc is finer than the Zariski, \\'etale, smooth, syntomic, and fppf", "topologies. Hence any presheaf", "satisfying the sheaf condition for the fpqc topology will be a", "sheaf on the Zariski, \\'etale, smooth, syntomic, and fppf", "sites. In particular", "representable presheaves will be sheaves on the \\'etale site of a scheme", "for example." ], "refs": [], "ref_ids": [] }, { "id": 6765, "type": "other", "label": "etale-cohomology-remark-final-object", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-final-object", "contents": [ "In the case where $\\mathcal{C}$ has a final object, e.g.\\ $S$, it", "suffices to check the condition of the definition for", "$U = S$ in the above statement. See", "Modules on Sites, Lemma \\ref{sites-modules-lemma-local-final-object}." ], "refs": [ "sites-modules-lemma-local-final-object" ], "ref_ids": [ 14185 ] }, { "id": 6766, "type": "other", "label": "etale-cohomology-remark-presheaves-no-topology", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-presheaves-no-topology", "contents": [ "Observe that all of the preceding statements are about presheaves so we haven't", "made use of the topology yet." ], "refs": [], "ref_ids": [] }, { "id": 6767, "type": "other", "label": "etale-cohomology-remark-grothendieck-ss", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-grothendieck-ss", "contents": [ "This is a Grothendieck spectral sequence for the composition of functors", "$$", "\\textit{Ab}(\\mathcal{C}) \\longrightarrow", "\\textit{PAb}(\\mathcal{C}) \\xrightarrow{\\check H^0} \\textit{Ab}.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 6768, "type": "other", "label": "etale-cohomology-remark-refinement", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-refinement", "contents": [ "In the statement of Lemma \\ref{lemma-cech-complex} the covering $\\mathcal{U}$", "is a refinement of $\\mathcal{V}$ but not the other way around. Coverings", "of the form $\\{V \\to U\\}$ do not form an initial subcategory of the", "category of all coverings of $U$. Yet it is still true that", "we can compute {\\v C}ech cohomology $\\check H^n(U, \\mathcal{F})$ (which", "is defined as the colimit over the opposite of the category of", "coverings $\\mathcal{U}$ of $U$ of the {\\v C}ech cohomology groups of", "$\\mathcal{F}$ with respect to $\\mathcal{U}$) in terms of the coverings", "$\\{V \\to U\\}$. We will formulate a precise lemma (it only works for sheaves)", "and add it here if we ever need it." ], "refs": [ "etale-cohomology-lemma-cech-complex" ], "ref_ids": [ 6416 ] }, { "id": 6769, "type": "other", "label": "etale-cohomology-remark-right-derived-global-sections", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-right-derived-global-sections", "contents": [ "Comment on Theorem \\ref{theorem-zariski-fpqc-quasi-coherent}.", "Since $S$ is a final object in the category $\\mathcal{C}$, the cohomology", "groups on the right-hand side are merely the right derived functors of the", "global sections functor. In fact the proof shows that", "$H^p(U, f^*\\mathcal{F}) = H^p_\\tau(U, \\mathcal{F}^a)$", "for any object $f : U \\to S$ of the site $\\mathcal{C}$." ], "refs": [ "etale-cohomology-theorem-zariski-fpqc-quasi-coherent" ], "ref_ids": [ 6373 ] }, { "id": 6770, "type": "other", "label": "etale-cohomology-remark-constant-locally-constant-maps", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-constant-locally-constant-maps", "contents": [ "Let $G$ be an abstract group.", "On any of the sites $(\\Sch/S)_\\tau$ or $S_\\tau$ of", "Section \\ref{section-big-small}", "the sheafification $\\underline{G}$", "of the constant presheaf associated to $G$ in the", "{\\it Zariski topology} of the site already gives", "$$", "\\Gamma(U, \\underline{G}) =", "\\{\\text{Zariski locally constant maps }U \\to G\\}", "$$", "This Zariski sheaf is representable by the group scheme $G_S$ according to", "Groupoids, Example \\ref{groupoids-example-constant-group}.", "By", "Lemma \\ref{lemma-representable-sheaf-fpqc}", "any representable presheaf satisfies the sheaf condition for the", "$\\tau$-topology as well, and hence we conclude that the Zariski", "sheafification $\\underline{G}$ above is also the $\\tau$-sheafification." ], "refs": [ "etale-cohomology-lemma-representable-sheaf-fpqc" ], "ref_ids": [ 6401 ] }, { "id": 6771, "type": "other", "label": "etale-cohomology-remark-special-case-fpqc-cohomology-quasi-coherent", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-special-case-fpqc-cohomology-quasi-coherent", "contents": [ "In the terminology introduced above a special case of", "Theorem \\ref{theorem-zariski-fpqc-quasi-coherent}", "is", "$$", "H_{fppf}^p(X, \\mathbf{G}_a) =", "H_\\etale^p(X, \\mathbf{G}_a) =", "H_{Zar}^p(X, \\mathbf{G}_a) =", "H^p(X, \\mathcal{O}_X)", "$$", "for all $p \\geq 0$. Moreover, we could use the notation", "$H^p_{fppf}(X, \\mathcal{O}_X)$ to indicate the cohomology of the", "structure sheaf on the big fppf site of $X$." ], "refs": [ "etale-cohomology-theorem-zariski-fpqc-quasi-coherent" ], "ref_ids": [ 6373 ] }, { "id": 6772, "type": "other", "label": "etale-cohomology-remark-no-kummer-sequence-zariski", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-no-kummer-sequence-zariski", "contents": [ "Lemma \\ref{lemma-kummer-sequence} is false when ``\\'etale'' is replaced", "with ``Zariski''.", "Since the \\'etale topology is coarser than the smooth topology, see", "Topologies, Lemma \\ref{topologies-lemma-zariski-etale-smooth}", "it follows that the sequence is also exact in the smooth topology." ], "refs": [ "etale-cohomology-lemma-kummer-sequence", "topologies-lemma-zariski-etale-smooth" ], "ref_ids": [ 6419, 12459 ] }, { "id": 6773, "type": "other", "label": "etale-cohomology-remark-no-kummer-sequence-smooth-etale-zariski", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-no-kummer-sequence-smooth-etale-zariski", "contents": [ "Lemma \\ref{lemma-kummer-sequence-syntomic}", "is false for the smooth, \\'etale, or Zariski topology." ], "refs": [ "etale-cohomology-lemma-kummer-sequence-syntomic" ], "ref_ids": [ 6420 ] }, { "id": 6774, "type": "other", "label": "etale-cohomology-remark-etale-between-etale", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-etale-between-etale", "contents": [ "Since $U$ and $U'$ are \\'etale over $S$, any $S$-morphism", "between them is also \\'etale, see", "Proposition \\ref{proposition-etale-morphisms}.", "In particular all morphisms of \\'etale neighborhoods are \\'etale." ], "refs": [ "etale-cohomology-proposition-etale-morphisms" ], "ref_ids": [ 6697 ] }, { "id": 6775, "type": "other", "label": "etale-cohomology-remark-etale-neighbourhoods", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-etale-neighbourhoods", "contents": [ "Let $S$ be a scheme and $s \\in S$ a point. In", "More on Morphisms,", "Definition \\ref{more-morphisms-definition-etale-neighbourhood}", "we defined the notion of an \\'etale neighbourhood $(U, u) \\to (S, s)$", "of $(S, s)$. If $\\overline{s}$ is a geometric point of $S$ lying over", "$s$, then any \\'etale neighbourhood $(U, \\overline{u}) \\to (S, \\overline{s})$", "gives rise to an \\'etale neighbourhood $(U, u)$ of $(S, s)$ by taking", "$u \\in U$ to be the unique point of $U$ such that $\\overline{u}$", "lies over $u$. Conversely, given an \\'etale neighbourhood $(U, u)$", "of $(S, s)$ the residue field extension $\\kappa(s) \\subset \\kappa(u)$", "is finite separable (see", "Proposition \\ref{proposition-etale-morphisms})", "and hence we can find an embedding $\\kappa(u) \\subset \\kappa(\\overline{s})$", "over $\\kappa(s)$. In other words, we can find a geometric point", "$\\overline{u}$ of $U$ lying over $u$ such that $(U, \\overline{u})$", "is an \\'etale neighbourhood of $(S, \\overline{s})$.", "We will use these observations to go between the two types of", "\\'etale neighbourhoods." ], "refs": [ "more-morphisms-definition-etale-neighbourhood", "etale-cohomology-proposition-etale-morphisms" ], "ref_ids": [ 14116, 6697 ] }, { "id": 6776, "type": "other", "label": "etale-cohomology-remark-map-stalks", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-map-stalks", "contents": [ "Let $S$ be a scheme and let $\\overline{s} : \\Spec(k) \\to S$", "and $\\overline{s}' : \\Spec(k') \\to S$ be two geometric points of", "$S$. A {\\it morphism $a : \\overline{s} \\to \\overline{s}'$ of geometric points}", "is simply a morphism $a : \\Spec(k) \\to \\Spec(k')$ such that", "$a \\circ \\overline{s}' = \\overline{s}$. Given such a morphism we obtain", "a functor from the category of \\'etale neighbourhoods of $\\overline{s}'$", "to the category of \\'etale neighbourhoods of $\\overline{s}$ by the rule", "$(U, \\overline{u}') \\mapsto (U, \\overline{u}' \\circ a)$. Hence we obtain", "a canonical map", "$$", "\\mathcal{F}_{\\overline{s}'}", "=", "\\colim_{(U, \\overline{u}')} \\mathcal{F}(U)", "\\longrightarrow", "\\colim_{(U, \\overline{u})} \\mathcal{F}(U)", "=", "\\mathcal{F}_{\\overline{s}}", "$$", "from Categories, Lemma \\ref{categories-lemma-functorial-colimit}. Using the", "description of elements of stalks as triples this maps the element of", "$\\mathcal{F}_{\\overline{s}'}$ represented by the triple", "$(U, \\overline{u}', \\sigma)$ to the element of $\\mathcal{F}_{\\overline{s}}$", "represented by the triple $(U, \\overline{u}' \\circ a, \\sigma)$.", "Since the functor above is clearly an equivalence we conclude that this", "canonical map is an isomorphism of stalk functors.", "\\medskip\\noindent", "Let us make sure we have the map of stalks corresponding to $a$ pointing", "in the correct direction. Note that the above means, according to", "Sites, Definition \\ref{sites-definition-morphism-points},", "that $a$ defines a morphism $a : p \\to p'$ between the points $p, p'$ of", "the site $S_\\etale$ associated to $\\overline{s}, \\overline{s}'$ by", "Lemma \\ref{lemma-stalk-gives-point}. There are more general morphisms of", "points (corresponding to specializations of points of $S$) which we will", "describe later, and which will not be isomorphisms (insert future", "reference here)." ], "refs": [ "categories-lemma-functorial-colimit", "sites-definition-morphism-points", "etale-cohomology-lemma-stalk-gives-point" ], "ref_ids": [ 12210, 8678, 6424 ] }, { "id": 6777, "type": "other", "label": "etale-cohomology-remark-points-fppf-site", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-points-fppf-site", "contents": [ "\\begin{reference}", "This is discussed in \\cite{Schroeer}.", "\\end{reference}", "Let $S = \\Spec(A)$ be an affine scheme. Let $(p, u)$ be a point of", "the site $(\\textit{Aff}/S)_{fppf}$, see", "Sites, Sections \\ref{sites-section-points} and", "\\ref{sites-section-construct-points}. Let $B = \\mathcal{O}_p$ be the stalk", "of the structure sheaf at the point $p$. Recall that", "$$", "B = \\colim_{(U, x)} \\mathcal{O}(U) =", "\\colim_{(\\Spec(C), x_C)} C", "$$", "where $x_C \\in u(\\Spec(C))$. It can happen that", "$\\Spec(B)$ is an object of $(\\textit{Aff}/S)_{fppf}$", "and that there is an element $x_B \\in u(\\Spec(B))$ mapping to", "the compatible system $x_C$. In this case the system of neighbourhoods", "has an initial object and it follows that", "$\\mathcal{F}_p = \\mathcal{F}(\\Spec(B))$ for any sheaf $\\mathcal{F}$", "on $(\\textit{Aff}/S)_{fppf}$. It is straightforward", "to see that if $\\mathcal{F} \\mapsto \\mathcal{F}(\\Spec(B))$ defines a point", "of $\\Sh((\\textit{Aff}/S)_{fppf})$, then", "$B$ has to be a local $A$-algebra such that for every faithfully flat,", "finitely presented ring map $B \\to B'$ there is a section $B' \\to B$.", "Conversely, for any such $A$-algebra $B$ the functor", "$\\mathcal{F} \\mapsto \\mathcal{F}(\\Spec(B))$ is the stalk functor", "of a point. Details omitted. It is not clear what a general point of the", "site $(\\textit{Aff}/S)_{fppf}$ looks like." ], "refs": [], "ref_ids": [] }, { "id": 6778, "type": "other", "label": "etale-cohomology-remark-henselization-Noetherian", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-henselization-Noetherian", "contents": [ "Let $S$ be a scheme. Let $s \\in S$.", "If $S$ is locally Noetherian then $\\mathcal{O}_{S, s}^h$", "is also Noetherian and it has the same completion:", "$$", "\\widehat{\\mathcal{O}_{S, s}} \\cong \\widehat{\\mathcal{O}_{S, s}^h}.", "$$", "In particular,", "$\\mathcal{O}_{S, s} \\subset", "\\mathcal{O}_{S, s}^h \\subset", "\\widehat{\\mathcal{O}_{S, s}}$.", "The henselization of $\\mathcal{O}_{S, s}$ is in general much", "smaller than its completion and inherits many of its properties.", "For example, if $\\mathcal{O}_{S, s}$ is reduced, then so is", "$\\mathcal{O}_{S, s}^h$, but this is not true for the completion in general.", "Insert future references here." ], "refs": [], "ref_ids": [] }, { "id": 6779, "type": "other", "label": "etale-cohomology-remark-direct-image-sheaf", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-direct-image-sheaf", "contents": [ "We claim that the direct image of a sheaf is a sheaf.", "Namely, if $\\{V_j \\to V\\}$ is an \\'etale covering in $Y_\\etale$", "then $\\{X \\times_Y V_j \\to X \\times_Y V\\}$ is an \\'etale covering in", "$X_\\etale$. Hence the sheaf condition for $\\mathcal{F}$ with respect", "to $\\{X \\times_Y V_i \\to X \\times_Y V\\}$", "is equivalent to the sheaf condition for $f_*\\mathcal{F}$ with respect to", "$\\{V_i \\to V\\}$. Thus if $\\mathcal{F}$ is a sheaf, so is", "$f_*\\mathcal{F}$." ], "refs": [], "ref_ids": [] }, { "id": 6780, "type": "other", "label": "etale-cohomology-remark-functoriality-general", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-functoriality-general", "contents": [ "More generally, let $\\mathcal{C}_1, \\mathcal{C}_2$ be sites, and", "assume they have final objects and fibre products. Let", "$u: \\mathcal{C}_2 \\to \\mathcal{C}_1$ be a functor satisfying:", "\\begin{enumerate}", "\\item if $\\{V_i \\to V\\}$ is a covering of $\\mathcal{C}_2$, then", "$\\{u(V_i) \\to u(V)\\}$ is a covering of $\\mathcal{C}_1$ (we", "say that $u$ is {\\it continuous}), and", "\\item $u$ commutes with finite limits (i.e., $u$ is left exact, i.e.,", "$u$ preserves fibre products and final objects).", "\\end{enumerate}", "Then one can define", "$f_*: \\Sh(\\mathcal{C}_1) \\to \\Sh(\\mathcal{C}_2)$", "by $ f_* \\mathcal{F}(V) = \\mathcal{F}(u(V))$.", "Moreover, there exists an exact functor $f^{-1}$ which", "is left adjoint to $f_*$, see", "Sites, Definition \\ref{sites-definition-morphism-sites} and", "Proposition \\ref{sites-proposition-get-morphism}.", "Warning: It is not enough to require simply that $u$ is continuous", "and commutes with fibre products in order to get a morphism of topoi." ], "refs": [ "sites-definition-morphism-sites", "sites-proposition-get-morphism" ], "ref_ids": [ 8665, 8641 ] }, { "id": 6781, "type": "other", "label": "etale-cohomology-remark-property-C-strong", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-property-C-strong", "contents": [ "Property (C) holds if $f : X \\to Y$ is an open immersion. Namely, if", "$U \\in \\Ob(X_\\etale)$, then we can view $U$ also as an object", "of $Y_\\etale$ and $U \\times_Y X = U$. Hence property (C)", "does not imply that $f_{small, *}$ is exact as this is not", "the case for open immersions (in general)." ], "refs": [], "ref_ids": [] }, { "id": 6782, "type": "other", "label": "etale-cohomology-remark-affine-inside-equivalence", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-affine-inside-equivalence", "contents": [ "In the situation of", "Theorem \\ref{theorem-topological-invariance}", "it is also true that $V \\mapsto V_X$ induces an equivalence", "between those \\'etale morphisms $V \\to Y$ with $V$ affine and", "those \\'etale morphisms $U \\to X$ with $U$ affine.", "This follows for example from", "Limits, Proposition \\ref{limits-proposition-affine}." ], "refs": [ "etale-cohomology-theorem-topological-invariance", "limits-proposition-affine" ], "ref_ids": [ 6383, 15129 ] }, { "id": 6783, "type": "other", "label": "etale-cohomology-remark-push-pull-shriek", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-push-pull-shriek", "contents": [ "In the situation of", "Lemma \\ref{lemma-monomorphism-big-push-pull}", "it is true that the canonical map", "$\\mathcal{F} \\to f_{big}^{-1}f_{big!}\\mathcal{F}$", "is an isomorphism for any sheaf of sets $\\mathcal{F}$ on", "$(\\Sch/X)_\\tau$. The proof is the same. This also", "holds for sheaves of abelian groups. However, note", "that the functor $f_{big!}$ for sheaves of abelian groups is defined in", "Modules on Sites, Section \\ref{sites-modules-section-exactness-lower-shriek}", "and is in general different from $f_{big!}$ on sheaves of sets.", "The result for sheaves of abelian groups follows from", "Modules on Sites, Lemma \\ref{sites-modules-lemma-back-and-forth}." ], "refs": [ "etale-cohomology-lemma-monomorphism-big-push-pull", "sites-modules-lemma-back-and-forth" ], "ref_ids": [ 6462, 14166 ] }, { "id": 6784, "type": "other", "label": "etale-cohomology-remark-fppf-closed-immersion-not-closed", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-fppf-closed-immersion-not-closed", "contents": [ "In Lemma \\ref{lemma-closed-immersion-pushforward-exact} the case $\\tau = fppf$", "is missing. The reason is that given a ring $A$, an ideal $I$ and a", "faithfully flat, finitely presented ring map $A/I \\to \\overline{B}$, there", "is no reason to think that one can find {\\it any} flat finitely presented ring", "map $A \\to B$ with $B/IB \\not = 0$ such that $A/I \\to B/IB$ factors through", "$\\overline{B}$. Hence the proof of", "Lemma \\ref{lemma-closed-immersion-almost-cocontinuous}", "does not work for the fppf topology.", "In fact it is likely false that", "$f_{big, *} : \\textit{Ab}((\\Sch/X)_{fppf})", "\\to \\textit{Ab}((\\Sch/Y)_{fppf})$", "is exact when $f$ is a closed immersion.", "If you know an example, please email", "\\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}." ], "refs": [ "etale-cohomology-lemma-closed-immersion-pushforward-exact", "etale-cohomology-lemma-closed-immersion-almost-cocontinuous" ], "ref_ids": [ 6466, 6465 ] }, { "id": 6785, "type": "other", "label": "etale-cohomology-remark-colimit-variant-complexes", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-colimit-variant-complexes", "contents": [ "Many of the results above have variants for bounded below", "complexes, but one has to be careful that the bounds have", "to be uniform. We explain this in the simplest case.", "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $I$ be a directed set. Let $\\mathcal{F}_i^\\bullet$ be a", "system over $I$ of complexes of sheaves on $X_\\etale$.", "Assume there is an integer $a$ such that", "$\\mathcal{F}_i^n = 0$ for $n < a$ and all $i \\in I$.", "Then we have", "$$", "H^p_\\etale(X, \\colim \\mathcal{F}_i^\\bullet) =", "\\colim H^p_\\etale(X, \\mathcal{F}^\\bullet_i)", "$$", "If we ever need this we will state a precise lemma with", "full proof here." ], "refs": [], "ref_ids": [] }, { "id": 6786, "type": "other", "label": "etale-cohomology-remark-cohomological-descent-finite", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-cohomological-descent-finite", "contents": [ "In the situation of Lemma \\ref{lemma-cohomological-descent-finite}", "if $\\mathcal{G}$ is a sheaf of sets on $Y_\\etale$, then we have", "$$", "\\Gamma(Y, \\mathcal{G}) =", "\\text{Equalizer}(", "\\xymatrix{", "\\Gamma(X_0, f_0^{-1}\\mathcal{G})", "\\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\Gamma(X_1, f_1^{-1}\\mathcal{G})", "}", ")", "$$", "This is proved in exactly the same way, by showing that", "the sheaf $\\mathcal{G}$ is the equalizer of the two maps", "$f_{0, *}f_0^{-1}\\mathcal{G} \\to f_{1, *}f_1^{-1}\\mathcal{G}$." ], "refs": [ "etale-cohomology-lemma-cohomological-descent-finite" ], "ref_ids": [ 6483 ] }, { "id": 6787, "type": "other", "label": "etale-cohomology-remark-every-sheaf-representable", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-every-sheaf-representable", "contents": [ "Another way to state the conclusion of", "Theorem \\ref{theorem-equivalence-sheaves-point} and", "Fundamental Groups, Lemma \\ref{pione-lemma-sheaves-point}", "is to say that every sheaf on $\\Spec(K)_\\etale$ is representable", "by a scheme $X$ \\'etale over $\\Spec(K)$.", "This does not mean that every sheaf is representable in the sense of", "Sites, Definition \\ref{sites-definition-representable-sheaf}.", "The reason is that in our construction of $\\Spec(K)_\\etale$", "we chose a sufficiently large set of schemes \\'etale over $\\Spec(K)$,", "whereas sheaves on $\\Spec(K)_\\etale$ form a proper class." ], "refs": [ "etale-cohomology-theorem-equivalence-sheaves-point", "pione-lemma-sheaves-point", "sites-definition-representable-sheaf" ], "ref_ids": [ 6386, 4024, 8663 ] }, { "id": 6788, "type": "other", "label": "etale-cohomology-remark-stalk-pullback", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-stalk-pullback", "contents": [ "Let $S$ be a scheme and let $\\overline{s} : \\Spec(k) \\to S$", "be a geometric point of $S$. By definition this means that $k$", "is algebraically closed. In particular the absolute Galois group of $k$", "is trivial. Hence by", "Theorem \\ref{theorem-equivalence-sheaves-point}", "the category of sheaves on $\\Spec(k)_\\etale$ is equivalent", "to the category of sets. The equivalence is given by taking", "sections over $\\Spec(k)$. This finally provides us with an", "alternative definition of the stalk functor. Namely, the functor", "$$", "\\Sh(S_\\etale) \\longrightarrow \\textit{Sets}, \\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}_{\\overline{s}}", "$$", "is isomorphic to the functor", "$$", "\\Sh(S_\\etale)", "\\longrightarrow", "\\Sh(\\Spec(k)_\\etale) = \\textit{Sets},", "\\quad", "\\mathcal{F} \\longmapsto \\overline{s}^*\\mathcal{F}", "$$", "To prove this rigorously one can use", "Lemma \\ref{lemma-stalk-pullback} part (3)", "with $f = \\overline{s}$. Moreover, having said this the general case of", "Lemma \\ref{lemma-stalk-pullback} part (3)", "follows from functoriality of pullbacks." ], "refs": [ "etale-cohomology-theorem-equivalence-sheaves-point", "etale-cohomology-lemma-stalk-pullback", "etale-cohomology-lemma-stalk-pullback" ], "ref_ids": [ 6386, 6436, 6436 ] }, { "id": 6789, "type": "other", "label": "etale-cohomology-remark-functorial-locally-constant-on-connected", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-functorial-locally-constant-on-connected", "contents": [ "The equivalences of Lemma \\ref{lemma-locally-constant-on-connected}", "are compatible with pullbacks. More precisely, suppose $f : Y \\to X$", "is a morphism of connected schemes. Let $\\overline{y}$ be geometric", "point of $Y$ and set $\\overline{x} = f(\\overline{y})$.", "Then the diagram", "$$", "\\xymatrix{", "\\text{finite locally constant sheaves of sets on }Y_\\etale", "\\ar[r] &", "\\text{finite }\\pi_1(Y, \\overline{y})\\text{-sets} \\\\", "\\text{finite locally constant sheaves of sets on }X_\\etale", "\\ar[r] \\ar[u]_{f^{-1}} &", "\\text{finite }\\pi_1(X, \\overline{x})\\text{-sets} \\ar[u]", "}", "$$", "is commutative, where the vertical arrow on the right comes", "from the continuous homomorphism", "$\\pi_1(Y, \\overline{y}) \\to \\pi_1(X, \\overline{x})$", "induced by $f$. This follows immediately from", "the commutative diagram in", "Fundamental Groups, Theorem \\ref{pione-theorem-fundamental-group}." ], "refs": [ "etale-cohomology-lemma-locally-constant-on-connected", "pione-theorem-fundamental-group" ], "ref_ids": [ 6508, 4021 ] }, { "id": 6790, "type": "other", "label": "etale-cohomology-remark-natural-proof", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-natural-proof", "contents": [ "The ``natural'' way to prove the previous corollary is to excise $X$ from $\\bar", "X$. This is possible, we just haven't developed that theory." ], "refs": [], "ref_ids": [] }, { "id": 6791, "type": "other", "label": "etale-cohomology-remark-normalize-H1-Gm", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-normalize-H1-Gm", "contents": [ "Let $k$ be an algebraically closed field. Let $n$ be an integer prime to", "the characteristic of $k$. Recall that", "$$", "\\mathbf{G}_{m, k} = \\mathbf{A}^1_k \\setminus \\{0\\} =", "\\mathbf{P}^1_k \\setminus \\{0, \\infty\\}", "$$", "We claim there is a canonical isomorphism", "$$", "H^1_\\etale(\\mathbf{G}_{m, k}, \\mu_n) = \\mathbf{Z}/n\\mathbf{Z}", "$$", "What does this mean? This means there is an element $1_k$ in", "$H^1_\\etale(\\mathbf{G}_{m, k}, \\mu_n)$ such that for", "every morphism $\\Spec(k') \\to \\Spec(k)$ the pullback map on", "\\'etale cohomology for the map $\\mathbf{G}_{m, k'} \\to \\mathbf{G}_{m, k}$", "maps $1_k$ to $1_{k'}$. (In particular this element is", "fixed under all automorphisms of $k$.) To see this, consider the", "$\\mu_{n, \\mathbf{Z}}$-torsor", "$\\mathbf{G}_{m, \\mathbf{Z}} \\to \\mathbf{G}_{m, \\mathbf{Z}}$,", "$x \\mapsto x^n$. By the identification of torsors with", "first cohomology, this pulls back to give our canonical elements $1_k$.", "Twisting back we see that there are canonical identifications", "$$", "H^1_\\etale(\\mathbf{G}_{m, k}, \\mathbf{Z}/n\\mathbf{Z}) =", "\\Hom(\\mu_n(k), \\mathbf{Z}/n\\mathbf{Z}),", "$$", "i.e., these isomorphisms are compatible with respect to maps of", "algebraically closed fields, in particular with respect to", "automorphisms of $k$." ], "refs": [], "ref_ids": [] }, { "id": 6792, "type": "other", "label": "etale-cohomology-remark-different", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-different", "contents": [ "Objects in the derived category $D_{ctf}(X_\\etale, \\Lambda)$ in some sense have", "better global properties than the perfect objects in $D(\\mathcal{O}_X)$.", "Namely, it can happen that a complex of $\\mathcal{O}_X$-modules", "is locally quasi-isomorphic to a finite complex of finite locally free", "$\\mathcal{O}_X$-modules, without being", "globally quasi-isomorphic to a bounded complex of locally free", "$\\mathcal{O}_X$-modules. The following lemma shows this does not", "happen for $D_{ctf}$ on a Noetherian scheme." ], "refs": [], "ref_ids": [] }, { "id": 6793, "type": "other", "label": "etale-cohomology-remark-projective-each-degree", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-projective-each-degree", "contents": [ "Let $\\Lambda$ be a Noetherian ring. Let $X$ be a scheme.", "For a bounded complex $K^\\bullet$ of constructible flat $\\Lambda$-modules", "on $X_\\etale$", "each stalk $K^p_{\\overline{x}}$ is a finite projective $\\Lambda$-module.", "Hence the stalks of the complex are perfect complexes of $\\Lambda$-modules." ], "refs": [], "ref_ids": [] }, { "id": 6794, "type": "other", "label": "etale-cohomology-remark-invariance", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-invariance", "contents": [ "Let $k$ be an algebraically closed field of characteristic $p > 0$.", "In Section \\ref{section-artin-schreier} we have seen that there is", "an exact sequence", "$$", "k[x] \\to k[x] \\to", "H^1_\\etale(\\mathbf{A}^1_k, \\mathbf{Z}/p\\mathbf{Z}) \\to 0", "$$", "where the first arrow maps $f(x)$ to $f^p - f$. A set of representatives", "for the cokernel is formed by the polynomials", "$$", "\\sum\\nolimits_{p \\not | n} \\lambda_n x^n", "$$", "with $\\lambda_n \\in k$. (If $k$ is not algebraically closed", "you have to add some constants to this as well.) In particular", "when $k' \\supset k$ is an algebraically closed overfield, then", "the map", "$$", "H^1_\\etale(\\mathbf{A}^1_k, \\mathbf{Z}/p\\mathbf{Z})", "\\to", "H^1_\\etale(\\mathbf{A}^1_{k'}, \\mathbf{Z}/p\\mathbf{Z})", "$$", "is not an isomorphism in general. In particular, the map", "$\\pi_1(\\mathbf{A}^1_{k'}) \\to \\pi_1(\\mathbf{A}^1_k)$", "between \\'etale fundamental groups (insert future reference here)", "is not an isomorphism either. Thus the \\'etale homotopy type", "of the affine line depends on the algebraically closed ground field.", "From Lemma \\ref{lemma-constant-smooth-statements} above we see that", "this is a phenomenon which only happens in characteristic $p$", "with $p$-power torsion coefficients." ], "refs": [ "etale-cohomology-lemma-constant-smooth-statements" ], "ref_ids": [ 6585 ] }, { "id": 6795, "type": "other", "label": "etale-cohomology-remark-base-change-holds", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-base-change-holds", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $n$ be an integer.", "We will say $BC(f, n, q_0)$ is true if for every commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_f & X' \\ar[l] \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\", "S & S' \\ar[l] & T \\ar[l]_g", "}", "$$", "with $X' = X \\times_S S'$ and $Y = X' \\times_{S'} T$ and", "$g$ quasi-compact and quasi-separated, and every abelian sheaf", "$\\mathcal{F}$ on $T_\\etale$ annihilated by $n$ the base change map", "$$", "(f')^{-1}R^qg_*\\mathcal{F}", "\\longrightarrow", "R^qh_*e^{-1}\\mathcal{F}", "$$", "is an isomorphism for $q \\leq q_0$." ], "refs": [], "ref_ids": [] }, { "id": 6796, "type": "other", "label": "etale-cohomology-remark-define-kunneth-map", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remark-define-kunneth-map", "contents": [ "Consider a cartesian diagram in the category of schemes:", "$$", "\\xymatrix{", "X \\times_S Y \\ar[d]_p \\ar[r]_q \\ar[rd]_c & Y \\ar[d]^g \\\\", "X \\ar[r]^f & S", "}", "$$", "Let $\\Lambda$ be a ring and let $E \\in D(X_\\etale, \\Lambda)$", "and $K \\in D(Y_\\etale, \\Lambda)$. Then there is a canonical map", "$$", "Rf_*E \\otimes_\\Lambda^\\mathbf{L} Rg_*K", "\\longrightarrow", "Rc_*(p^{-1}E \\otimes_\\Lambda^\\mathbf{L} q^{-1}K)", "$$", "For example we can define this using the canonical maps", "$Rf_*E \\to Rc_*p^{-1}E$ and $Rg_*K \\to Rc_*q^{-1}K$ and", "the relative cup product defined in Cohomology on Sites,", "Remark \\ref{sites-cohomology-remark-cup-product}.", "Or you can use the adjoint to the map", "$$", "c^{-1}(Rf_*E \\otimes_\\Lambda^\\mathbf{L} Rg_*K)", "=", "p^{-1}f^{-1}Rf_*E \\otimes_\\Lambda^\\mathbf{L} q^{-1} g^{-1}Rg_*K", "\\to", "p^{-1}E \\otimes_\\Lambda^\\mathbf{L} q^{-1}K", "$$", "which uses the adjunction maps $f^{-1}Rf_*E \\to E$ and", "$g^{-1}Rg_*K \\to K$." ], "refs": [ "sites-cohomology-remark-cup-product" ], "ref_ids": [ 4427 ] }, { "id": 6797, "type": "other", "label": "etale-cohomology-remarks-theorem-modules-exactness", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remarks-theorem-modules-exactness", "contents": [ "The results on descent of modules have several applications:", "\\begin{enumerate}", "\\item The exactness of the {\\v C}ech complex in positive degrees for", "the covering $\\{\\Spec(B) \\to \\Spec(A)\\}$ where $A \\to B$ is", "faithfully flat. This will give some vanishing of cohomology.", "\\item If $(N, \\varphi)$ is a descent datum with respect to a faithfully", "flat map $A \\to B$, then the corresponding $A$-module is given by", "$$", "M = \\Ker \\left(", "\\begin{matrix}", "N & \\longrightarrow & B \\otimes_A N \\\\", "n & \\longmapsto & 1 \\otimes n - \\varphi(n \\otimes 1)", "\\end{matrix}", "\\right).", "$$", "See", "Descent, Proposition \\ref{descent-proposition-descent-module}.", "\\end{enumerate}" ], "refs": [ "descent-proposition-descent-module" ], "ref_ids": [ 14752 ] }, { "id": 6798, "type": "other", "label": "etale-cohomology-remarks-enough-points", "categories": [ "etale-cohomology" ], "title": "etale-cohomology-remarks-enough-points", "contents": [ "On points of the geometric sites.", "\\begin{enumerate}", "\\item Theorem \\ref{theorem-exactness-stalks} says that the family of points", "of $S_\\etale$ given by the geometric points of $S$", "(Lemma \\ref{lemma-stalk-gives-point}) is conservative, see", "Sites, Definition \\ref{sites-definition-enough-points}.", "In particular $S_\\etale$ has enough points.", "\\item Suppose $\\mathcal{F}$ is a sheaf on the big \\'etale site", "\\label{item-stalks-big}", "of $S$. Let $T \\to S$ be an object of the big \\'etale site of $S$,", "and let $\\overline{t}$ be a geometric point of $T$. Then we define", "$\\mathcal{F}_{\\overline{t}}$ as the stalk", "of the restriction $\\mathcal{F}|_{T_\\etale}$ of $\\mathcal{F}$", "to the small \\'etale site of $T$. In other words, we can define", "the stalk of $\\mathcal{F}$ at any geometric point of any", "scheme $T/S \\in \\Ob((\\Sch/S)_\\etale)$.", "\\item The big \\'etale site of $S$ also has enough points, by", "considering all geometric points of all objects of this site, see", "(\\ref{item-stalks-big}).", "\\end{enumerate}" ], "refs": [ "etale-cohomology-theorem-exactness-stalks", "etale-cohomology-lemma-stalk-gives-point", "sites-definition-enough-points" ], "ref_ids": [ 6376, 6424, 8679 ] }, { "id": 6880, "type": "other", "label": "equiv-remark-affine-morphism", "categories": [ "equiv" ], "title": "equiv-remark-affine-morphism", "contents": [ "Below we will use that for an affine morphism", "$h : T \\to S$ we have $h_*\\mathcal{G} \\otimes \\mathcal{H} =", "h_*(\\mathcal{G} \\otimes h^*\\mathcal{H})$ for", "$\\mathcal{G} \\in \\QCoh(\\mathcal{O}_T)$ and", "$\\mathcal{H} \\in \\QCoh(\\mathcal{O}_S)$. This follows", "immediately on translating into algebra." ], "refs": [], "ref_ids": [] }, { "id": 6881, "type": "other", "label": "equiv-remark-difficult", "categories": [ "equiv" ], "title": "equiv-remark-difficult", "contents": [ "If $F, F' : D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to \\mathcal{D}$ are siblings, $F$", "is fully faithful, and $X$ is reduced and projective over $k$ then", "$F \\cong F'$; this follows from", "Proposition \\ref{proposition-siblings-isomorphic} via the argument", "given in the proof of Theorem \\ref{theorem-fully-faithful}.", "However, in general we do not know whether siblings are isomorphic.", "Even in the situation of Lemma \\ref{lemma-exact-functor-preserving-Coh}", "it seems difficult to prove that the siblings $F$ and $F'$", "are isomorphic functors. If $X$ is smooth and proper over $k$", "and $F$ is fully faithful, then $F \\cong F'$ as is shown in", "\\cite{Noah}.", "If you have a proof or a counter example in more general situations,", "please email", "\\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}." ], "refs": [ "equiv-proposition-siblings-isomorphic", "equiv-theorem-fully-faithful", "equiv-lemma-exact-functor-preserving-Coh" ], "ref_ids": [ 6872, 6800, 6851 ] }, { "id": 6933, "type": "other", "label": "stacks-more-morphisms-remark-gerbe-of-lifts", "categories": [ "stacks-more-morphisms" ], "title": "stacks-more-morphisms-remark-gerbe-of-lifts", "contents": [ "Consider a diagram", "$$", "\\xymatrix{", "W \\ar[d]_x \\\\", "\\mathcal{X} \\ar[r] & \\mathcal{X}'", "}", "$$", "where $\\mathcal{X} \\subset \\mathcal{X}'$ is a thickening of algebraic stacks,", "$W$ is an algebraic space, and $W \\to \\mathcal{X}$ is smooth.", "We will construct a category $\\mathcal{C}$ and a functor", "$$", "p : \\mathcal{C} \\longrightarrow W_{spaces, \\etale}", "$$", "(see Properties of Spaces, Definition", "\\ref{spaces-properties-definition-spaces-etale-site} for notation)", "as follows. An object of $\\mathcal{C}$ will be a system", "$(U, U', a, i, x', \\alpha)$", "which forms a commutative diagram", "\\begin{equation}", "\\label{equation-object}", "\\vcenter{", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_i & U' \\ar[dd]^{x'} \\\\", "W \\ar[d]_x & \\\\", "\\mathcal{X} \\ar[r] & \\mathcal{X}'", "}", "}", "\\end{equation}", "with commutativity witnessed by the $2$-morphism", "$\\alpha : x \\circ a \\to x' \\circ i$ such that", "$U$ and $U'$ are algebraic spaces,", "$a : U \\to W$ is \\'etale, $x' : U' \\to \\mathcal{X}'$ is smooth,", "and such that $U = \\mathcal{X} \\times_{\\mathcal{X}'} U'$.", "In particular $U \\subset U'$ is a thickening.", "A morphism", "$$", "(U, U', a, i, x', \\alpha) \\to (V, V', b, j, y', \\beta)", "$$", "is given by $(f, f', \\gamma)$ where $f : U \\to V$ is a morphism", "over $W$, $f' : U' \\to V'$ is a morphism whose restriction", "to $U$ gives $f$, and $\\gamma : x' \\circ f' \\to y'$ is a $2$-morphism", "witnessing the commutativity in right triangle of the diagram below", "\\begin{equation}", "\\label{equation-morphism}", "\\vcenter{", "\\xymatrix{", "& V \\ar[ld]_f \\ar[ldd]^b \\ar[rr]_j & & V' \\ar[ld]_{f'} \\ar[lddd]^{y'} \\\\", "U \\ar[d]_a \\ar[rr]_i & & U' \\ar[dd]_{x'} \\\\", "W \\ar[d]_x & \\\\", "\\mathcal{X} \\ar[rr] & & \\mathcal{X}'", "}", "}", "\\end{equation}", "Finally, we require that $\\gamma$ is compatible with $\\alpha$ and $\\beta$:", "in the calculus of $2$-categories of Categories, Sections", "\\ref{categories-section-formal-cat-cat} and", "\\ref{categories-section-2-categories} this reads", "$$", "\\beta = (\\gamma \\star \\text{id}_j) \\circ (\\alpha \\star \\text{id}_f)", "$$", "(more succinctly: $\\beta = j^*\\gamma \\circ f^*\\alpha$).", "Another formulation is that objects are commutative diagrams", "(\\ref{equation-object}) with some additional properties and", "morphisms are commutative diagrams", "(\\ref{equation-morphism}) in the category $\\textit{Spaces}/\\mathcal{X}'$", "introduced in Properties of Stacks, Remark", "\\ref{stacks-properties-remark-representable-over}.", "This makes it clear that $\\mathcal{C}$ is a category", "and that the rule $p : \\mathcal{C} \\to W_{spaces, \\etale}$", "sending $(U, U', a, i, x', \\alpha)$ to $a : U \\to W$", "is a functor." ], "refs": [ "spaces-properties-definition-spaces-etale-site", "stacks-properties-remark-representable-over" ], "ref_ids": [ 11935, 8927 ] }, { "id": 7122, "type": "other", "label": "perfect-remark-support-c-equations", "categories": [ "perfect" ], "title": "perfect-remark-support-c-equations", "contents": [ "Let $X$ be a scheme. Let $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$.", "Denote $Z \\subset X$ the closed subscheme cut out by $f_1, \\ldots, f_c$.", "For $0 \\leq p < c$ and $1 \\leq i_0 < \\ldots < i_p \\leq c$ we denote", "$U_{i_0 \\ldots i_p} \\subset X$ the open subscheme where", "$f_{i_0} \\ldots f_{i_p}$ is invertible. For any $\\mathcal{O}_X$-module", "$\\mathcal{F}$ we set", "$$", "\\mathcal{F}_{i_0 \\ldots i_p} =", "(U_{i_0 \\ldots i_p} \\to X)_*(\\mathcal{F}|_{U_{i_0 \\ldots i_p}})", "$$", "In this situation the {\\it extended alternating {\\v C}ech complex}", "is the complex of $\\mathcal{O}_X$-modules", "\\begin{equation}", "\\label{equation-extended-alternating}", "0 \\to \\mathcal{F} \\to", "\\bigoplus\\nolimits_{i_0} \\mathcal{F}_{i_0} \\to", "\\ldots \\to", "\\bigoplus\\nolimits_{i_0 < \\ldots < i_p} \\mathcal{F}_{i_0 \\ldots i_p} \\to", "\\ldots \\to \\mathcal{F}_{1 \\ldots c} \\to 0", "\\end{equation}", "where $\\mathcal{F}$ is put in degree $0$. The maps are constructed as", "follows. Given", "$1 \\leq i_0 < \\ldots < i_{p + 1} \\leq c$ and $0 \\leq j \\leq p + 1$ we", "have the canonical map", "$$", "\\mathcal{F}_{i_0 \\ldots \\hat i_j \\ldots i_{p + 1}} \\to", "\\mathcal{F}_{i_0 \\ldots i_p}", "$$", "coming from the inclusion", "$U_{i_0 \\ldots i_p} \\subset U_{i_0 \\ldots \\hat i_j \\ldots i_{p + 1}}$.", "The differentials in the extended alternating complex use these", "canonical maps with sign $(-1)^j$." ], "refs": [], "ref_ids": [] }, { "id": 7123, "type": "other", "label": "perfect-remark-extended-alternating-map-to-support", "categories": [ "perfect" ], "title": "perfect-remark-extended-alternating-map-to-support", "contents": [ "Let $X$, $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$, and", "$\\mathcal{F}$ be as in Remark \\ref{remark-support-c-equations}.", "Denote $\\mathcal{F}^\\bullet$ the complex", "(\\ref{equation-extended-alternating}). By", "Lemma \\ref{lemma-extended-alternating-zero}", "the cohomology sheaves of $\\mathcal{F}^\\bullet$", "are supported on $Z$ hence $\\mathcal{F}^\\bullet$ is an object of", "$D_Z(\\mathcal{O}_X)$. On the other hand, the equality", "$\\mathcal{F}^0 = \\mathcal{F}$ determines a canonical map", "$\\mathcal{F}^\\bullet \\to \\mathcal{F}$ in $D(\\mathcal{O}_X)$.", "As $i_* \\circ R\\mathcal{H}_Z$ is a right adjoint to the", "inclusion functor $D_Z(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$, see", "Cohomology, Lemma \\ref{cohomology-lemma-complexes-with-support-on-closed},", "we obtain a canonical commutative diagram", "$$", "\\xymatrix{", "\\mathcal{F}^\\bullet \\ar[rd] \\ar[rr] & & \\mathcal{F} \\\\", "& i_*R\\mathcal{H}_Z(\\mathcal{F}) \\ar[ru]", "}", "$$", "in $D(\\mathcal{O}_X)$ functorial in the $\\mathcal{O}_X$-module $\\mathcal{F}$." ], "refs": [ "perfect-remark-support-c-equations", "perfect-lemma-extended-alternating-zero", "cohomology-lemma-complexes-with-support-on-closed" ], "ref_ids": [ 7122, 6957, 2151 ] }, { "id": 7124, "type": "other", "label": "perfect-remark-supported-map-c-equations", "categories": [ "perfect" ], "title": "perfect-remark-supported-map-c-equations", "contents": [ "With $X$, $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$, and", "$\\mathcal{F}$ as in Remark \\ref{remark-support-c-equations}.", "There is a canonical $\\mathcal{O}_X|_Z$-linear map", "$$", "c_{f_1, \\ldots, f_c} :", "i^*\\mathcal{F}", "\\longrightarrow", "\\mathcal{H}^c_Z(\\mathcal{F})", "$$", "functorial in $\\mathcal{F}$. Namely, denoting $\\mathcal{F}^\\bullet$ the", "extended alternating {\\v C}ech complex (\\ref{equation-extended-alternating})", "we have the canonical map", "$\\mathcal{F}^\\bullet \\to i_*R\\mathcal{H}_Z(\\mathcal{F})$", "of Remark \\ref{remark-extended-alternating-map-to-support}.", "This determines a canonical map", "$$", "\\Coker\\left(\\bigoplus \\mathcal{F}_{1 \\ldots \\hat i \\ldots c} \\to", "\\mathcal{F}_{1 \\ldots c}\\right)", "\\longrightarrow", "i_*\\mathcal{H}^c_Z(\\mathcal{F})", "$$", "on cohomology sheaves in degree $c$.", "Given a local section $s$ of $\\mathcal{F}$ we can consider the", "local section", "$$", "\\frac{s}{f_1 \\ldots f_c}", "$$", "of $\\mathcal{F}_{1 \\ldots c}$. The class of this section in the cokernel", "displayed above depends only on $s$ modulo the image of", "$(f_1, \\ldots, f_c) : \\mathcal{F}^{\\oplus c} \\to \\mathcal{F}$.", "Since $i_*i^*\\mathcal{F}$ is equal to the cokernel of", "$(f_1, \\ldots, f_c) : \\mathcal{F}^{\\oplus c} \\to \\mathcal{F}$", "we see that we get an $\\mathcal{O}_X$-module map", "$i_*i^*\\mathcal{F} \\to i_*\\mathcal{H}_Z^c(\\mathcal{F})$.", "As $i_*$ is fully faithful we get the map $c_{f_1, \\ldots, f_c}$." ], "refs": [ "perfect-remark-support-c-equations", "perfect-remark-extended-alternating-map-to-support" ], "ref_ids": [ 7122, 7123 ] }, { "id": 7125, "type": "other", "label": "perfect-remark-supported-functorial", "categories": [ "perfect" ], "title": "perfect-remark-supported-functorial", "contents": [ "Let $g : X' \\to X$ be a morphism of schemes. Let", "$f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$.", "Set $f'_i = g^\\sharp(f_i) \\in \\Gamma(X', \\mathcal{O}_{X'})$.", "Denote $Z \\subset X$, resp.\\ $Z' \\subset X'$ the closed", "subscheme cut out by $f_1, \\ldots, f_c$, resp.\\ $f'_1, \\ldots, f'_c$.", "Then $Z' = Z \\times_X X'$. Denote $h : Z' \\to Z$ the induced morphism", "of schemes.", "Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.", "Set $\\mathcal{F}' = g^*\\mathcal{F}$.", "In this setting, if $\\mathcal{F}$ is quasi-coherent, then the diagram", "$$", "\\xymatrix{", "(i')^{-1}\\mathcal{O}_{X'} \\otimes_{h^{-1}i^{-1}\\mathcal{O}_X}", "h^{-1}\\mathcal{H}^c_Z(\\mathcal{F}) \\ar[r] &", "\\mathcal{H}_{Z'}^c(\\mathcal{F}') \\\\", "h^*i^*\\mathcal{F} \\ar[r] \\ar[u]_-{c_{f_1, \\ldots, f_c}} &", "(i')^*\\mathcal{F}' \\ar[u]^-{c_{f'_1, \\ldots, f'_c}}", "}", "$$", "is commutative where the top horizonal arrow is", "the map of Cohomology, Remark \\ref{cohomology-remark-support-functorial}", "on cohomology sheaves in degree $c$. Namely,", "denote $\\mathcal{F}^\\bullet$, resp.\\ $(\\mathcal{F}')^\\bullet$", "the extended alternating {\\v C}ech complex constructed in", "Remark \\ref{remark-support-c-equations}", "using $\\mathcal{F}, f_1, \\ldots, f_c$,", "resp.\\ $\\mathcal{F}', f'_1, \\ldots, f'_c$.", "Note that $(\\mathcal{F}')^\\bullet = g^*\\mathcal{F}^\\bullet$.", "Then, without assuming $\\mathcal{F}$ is quasi-coherent, the diagram", "$$", "\\xymatrix{", "i'_* L(g|_{Z'})^* R\\mathcal{H}_Z(\\mathcal{F}) \\ar[r] \\ar@{=}[d] &", "i'_*R\\mathcal{H}_{Z'}(Lg^*\\mathcal{F}) \\ar[d] \\\\", "Lg^*i_*R\\mathcal{H}_Z(\\mathcal{F}) &", "i'_*R\\mathcal{H}_{Z'}(\\mathcal{F}') \\\\", "Lg^*(\\mathcal{F}^\\bullet) \\ar[u] \\ar[r] &", "(\\mathcal{F}')^\\bullet \\ar[u]", "}", "$$", "is commutative where $g|_{Z'} : (Z', (i')^{-1}\\mathcal{O}_{X'}) \\to", "(Z, i^{-1}\\mathcal{O}_X)$ is the induced morphism of ringed spaces.", "Here the top horizontal arrow is given in", "Cohomology, Remark \\ref{cohomology-remark-support-functorial}", "as is the explanation for the equal sign.", "The arrows pointing up are from", "Remark \\ref{remark-extended-alternating-map-to-support}.", "The lower horizonal arrow is the map $Lg^*\\mathcal{F}^\\bullet", "\\to g^*\\mathcal{F}^\\bullet = (\\mathcal{F}')^\\bullet$ and the arrow", "pointing down is induced by", "$Lg^*\\mathcal{F} \\to g^*\\mathcal{F} = \\mathcal{F}'$.", "The diagram commutes because going around the diagram both ways", "we obtain two arrows $Lg^*\\mathcal{F}^\\bullet \\to", "i'_*R\\mathcal{H}_{Z'}(\\mathcal{F}')$ whose composition with", "$i'_*R\\mathcal{H}_{Z'}(\\mathcal{F}') \\to \\mathcal{F}'$", "is the canonical map $Lg^*\\mathcal{F}^\\bullet \\to \\mathcal{F}'$.", "Some details omitted. Now the commutativity of the first diagram", "follows by looking at this diagram on cohomology sheaves in degree", "$c$ and using that the construction of the map", "$i^*\\mathcal{F} \\to", "\\Coker(\\bigoplus \\mathcal{F}_{1 \\ldots \\hat i \\ldots c} \\to", "\\mathcal{F}_{1 \\ldots c})$", "used in Remark \\ref{remark-supported-map-c-equations}", "is compatible with pullbacks." ], "refs": [ "cohomology-remark-support-functorial", "perfect-remark-support-c-equations", "cohomology-remark-support-functorial", "perfect-remark-extended-alternating-map-to-support", "perfect-remark-supported-map-c-equations" ], "ref_ids": [ 2277, 7122, 2277, 7123, 7124 ] }, { "id": 7126, "type": "other", "label": "perfect-remark-warning-coherator", "categories": [ "perfect" ], "title": "perfect-remark-warning-coherator", "contents": [ "Let $X$ be a quasi-compact scheme with affine diagonal. Even though we know", "that $D(\\QCoh(\\mathcal{O}_X)) = D_\\QCoh(\\mathcal{O}_X)$ by", "Proposition \\ref{proposition-quasi-compact-affine-diagonal}", "strange things can", "happen and it is easy to make mistakes with this material. One pitfall", "is to carelessly assume that this equality means derived functors are the same.", "For example, suppose we have a quasi-compact open $U \\subset X$. Then we can", "consider the higher right derived functors", "$$", "R^i(\\QCoh)\\Gamma(U, -) : \\QCoh(\\mathcal{O}_X) \\to \\textit{Ab}", "$$", "of the left exact functor $\\Gamma(U, -)$. Since this is a universal", "$\\delta$-functor, and since the functors $H^i(U, -)$ (defined for all", "abelian sheaves on $X$) restricted to $\\QCoh(\\mathcal{O}_X)$ form", "a $\\delta$-functor, we obtain canonical tranformations", "$$", "t^i : R^i(\\QCoh)\\Gamma(U, -) \\to H^i(U, -).", "$$", "These transformations aren't in general isomorphisms even if $X = \\Spec(A)$", "is affine! Namely, we have $R^1(\\QCoh)\\Gamma(U, \\widetilde{I}) = 0$", "if $I$ an injective $A$-module by construction of right derived functors", "and the equivalence of $\\QCoh(\\mathcal{O}_X)$ and $\\text{Mod}_A$.", "But Examples, Lemma \\ref{examples-lemma-nonvanishing}", "shows there exists $A$, $I$, and $U$ such that", "$H^1(U, \\widetilde{I}) \\not = 0$." ], "refs": [ "perfect-proposition-quasi-compact-affine-diagonal", "examples-lemma-nonvanishing" ], "ref_ids": [ 7107, 2554 ] }, { "id": 7127, "type": "other", "label": "perfect-remark-addendum", "categories": [ "perfect" ], "title": "perfect-remark-addendum", "contents": [ "The proof of Lemma \\ref{lemma-lift-map-from-perfect-complex-with-support}", "shows that", "$$", "R|_U = P \\oplus P^{\\oplus n_1}[1] \\oplus \\ldots \\oplus P^{\\oplus n_m}[m]", "$$", "for some $m \\geq 0$ and $n_j \\geq 0$. Thus the highest degree cohomology sheaf", "of $R|_U$ equals that of $P$. By repeating the construction for the map", "$P^{\\oplus n_1}[1] \\oplus \\ldots \\oplus P^{\\oplus n_m}[m] \\to R|_U$, taking", "cones, and using induction we can achieve equality of cohomology sheaves", "of $R|_U$ and $P$ above any given degree." ], "refs": [ "perfect-lemma-lift-map-from-perfect-complex-with-support" ], "ref_ids": [ 7005 ] }, { "id": 7128, "type": "other", "label": "perfect-remark-pullback-generator", "categories": [ "perfect" ], "title": "perfect-remark-pullback-generator", "contents": [ "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated schemes.", "Let $E \\in D_\\QCoh(\\mathcal{O}_Y)$ be a generator", "(see Theorem \\ref{theorem-bondal-van-den-Bergh}).", "Then the following are equivalent", "\\begin{enumerate}", "\\item for $K \\in D_\\QCoh(\\mathcal{O}_X)$ we have", "$Rf_*K = 0$ if and only if $K = 0$,", "\\item $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$", "reflects isomorphisms, and", "\\item $Lf^*E$ is a generator for $D_\\QCoh(\\mathcal{O}_X)$.", "\\end{enumerate}", "The equivalence between (1) and (2) is a formal consequence of the fact that", "$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ is an", "exact functor of triangulated categories. Similarly, the equivalence", "between (1) and (3) follows formally from the fact that $Lf^*$", "is the left adjoint to $Rf_*$.", "These conditions hold if $f$ is affine (Lemma \\ref{lemma-affine-morphism})", "or if $f$ is an open immersion, or if $f$ is a composition of such.", "We conclude that", "\\begin{enumerate}", "\\item if $X$ is a quasi-affine scheme then $\\mathcal{O}_X$ is a generator", "for $D_\\QCoh(\\mathcal{O}_X)$,", "\\item if $X \\subset \\mathbf{P}^n_A$ is a quasi-compact", "locally closed subscheme, then", "$\\mathcal{O}_X \\oplus \\mathcal{O}_X(-1) \\oplus \\ldots \\oplus \\mathcal{O}_X(-n)$", "is a generator for $D_\\QCoh(\\mathcal{O}_X)$ by", "Lemma \\ref{lemma-generator-P1}.", "\\end{enumerate}" ], "refs": [ "perfect-theorem-bondal-van-den-Bergh", "perfect-lemma-affine-morphism", "perfect-lemma-generator-P1" ], "ref_ids": [ 6935, 6952, 7014 ] }, { "id": 7129, "type": "other", "label": "perfect-remark-classical-generator", "categories": [ "perfect" ], "title": "perfect-remark-classical-generator", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Let $G$ be a", "perfect object of $D(\\mathcal{O}_X)$ which is a generator for", "$D_\\QCoh(\\mathcal{O}_X)$. By Theorem \\ref{theorem-bondal-van-den-Bergh}", "there is at least one of these. Combining", "Lemma \\ref{lemma-quasi-coherence-direct-sums} with", "Proposition \\ref{proposition-compact-is-perfect} and with", "Derived Categories, Proposition", "\\ref{derived-proposition-generator-versus-classical-generator}", "we see that $G$ is a classical generator for $D_{perf}(\\mathcal{O}_X)$." ], "refs": [ "perfect-theorem-bondal-van-den-Bergh", "perfect-lemma-quasi-coherence-direct-sums", "perfect-proposition-compact-is-perfect", "derived-proposition-generator-versus-classical-generator" ], "ref_ids": [ 6935, 6937, 7111, 1965 ] }, { "id": 7130, "type": "other", "label": "perfect-remark-classical-generator-with-support", "categories": [ "perfect" ], "title": "perfect-remark-classical-generator-with-support", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $T \\subset X$ be a closed subset such that $X \\setminus T$", "is quasi-compact. Let $G$ be a", "perfect object of $D_{\\QCoh, T}(\\mathcal{O}_X)$ which is a generator for", "$D_{\\QCoh, T}(\\mathcal{O}_X)$. By Lemma \\ref{lemma-generator-with-support}", "there is at least one of these. Combining the fact that", "$D_{\\QCoh, T}(\\mathcal{O}_X)$ has direct sums with", "Lemma \\ref{lemma-compact-is-perfect-with-support} and with", "Derived Categories, Proposition", "\\ref{derived-proposition-generator-versus-classical-generator}", "we see that $G$ is a classical generator for $D_{perf, T}(\\mathcal{O}_X)$." ], "refs": [ "perfect-lemma-generator-with-support", "perfect-lemma-compact-is-perfect-with-support", "derived-proposition-generator-versus-classical-generator" ], "ref_ids": [ 7011, 7015, 1965 ] }, { "id": 7131, "type": "other", "label": "perfect-remark-DQCoh-is-Ddga-with-support", "categories": [ "perfect" ], "title": "perfect-remark-DQCoh-is-Ddga-with-support", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \\subset X$", "be a closed subset such that $X \\setminus T$ is quasi-compact.", "The analogue of Theorem \\ref{theorem-DQCoh-is-Ddga} holds", "for $D_{\\QCoh, T}(\\mathcal{O}_X)$.", "This follows from the exact same argument as in the proof", "of the theorem, using", "Lemmas \\ref{lemma-generator-with-support} and", "\\ref{lemma-compact-is-perfect-with-support}", "and a variant of Lemma \\ref{lemma-tensor-with-QCoh-complex}", "with supports.", "If we ever need this, we will precisely state the", "result here and give a detailed proof." ], "refs": [ "perfect-theorem-DQCoh-is-Ddga", "perfect-lemma-generator-with-support", "perfect-lemma-compact-is-perfect-with-support", "perfect-lemma-tensor-with-QCoh-complex" ], "ref_ids": [ 6936, 7011, 7015, 7017 ] }, { "id": 7132, "type": "other", "label": "perfect-remark-independence-choice", "categories": [ "perfect" ], "title": "perfect-remark-independence-choice", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme over a ring $R$.", "By the construction of the proof of", "Theorem \\ref{theorem-DQCoh-is-Ddga}", "there exists a differential graded algebra $(A, \\text{d})$ over $R$", "such that $D_\\QCoh(X)$ is $R$-linearly equivalent to", "$D(A, \\text{d})$ as a triangulated category.", "One may ask: how unique is $(A, \\text{d})$?", "The answer is (only) slightly better than just saying that", "$(A, \\text{d})$ is well defined up to derived equivalence.", "Namely, suppose that $(B, \\text{d})$ is a second such pair.", "Then we have", "$$", "(A, \\text{d}) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, K^\\bullet)", "$$", "and", "$$", "(B, \\text{d}) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(L^\\bullet, L^\\bullet)", "$$", "for some K-injective complexes $K^\\bullet$ and $L^\\bullet$", "of $\\mathcal{O}_X$-modules corresponding to perfect generators", "of $D_\\QCoh(\\mathcal{O}_X)$. Set", "$$", "\\Omega = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, L^\\bullet)", "\\quad", "\\Omega' = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(L^\\bullet, K^\\bullet)", "$$", "Then $\\Omega$ is a differential graded $B^{opp} \\otimes_R A$-module", "and $\\Omega'$ is a differential graded $A^{opp} \\otimes_R B$-module.", "Moreover, the equivalence", "$$", "D(A, \\text{d}) \\to D_\\QCoh(\\mathcal{O}_X) \\to", "D(B, \\text{d})", "$$", "is given by the functor $- \\otimes_A^\\mathbf{L} \\Omega'$ and", "similarly for the quasi-inverse. Thus we are in the situation", "of Differential Graded Algebra, Remark \\ref{dga-remark-hochschild-cohomology}.", "If we ever need this remark we will provide a precise statement", "with a detailed proof here." ], "refs": [ "perfect-theorem-DQCoh-is-Ddga", "dga-remark-hochschild-cohomology" ], "ref_ids": [ 6936, 13172 ] }, { "id": 7133, "type": "other", "label": "perfect-remark-explain-consequence", "categories": [ "perfect" ], "title": "perfect-remark-explain-consequence", "contents": [ "Let $X$ be a quasi-compact and quasi-separated scheme.", "Let $X = U \\cup V$ with $U$ and $V$ quasi-compact open.", "By Lemma \\ref{lemma-better-coherator} the functors", "$DQ_X$, $DQ_U$, $DQ_V$, $DQ_{U \\cap V}$ exist. Moreover, there is a", "canonical distinguished triangle", "$$", "DQ_X(K) \\to Rj_{U, *}DQ_U(K|_U) \\oplus Rj_{V, *}DQ_V(K|_V)", "\\to Rj_{U \\cap V, *}DQ_{U \\cap V}(K|_{U \\cap V}) \\to", "$$", "for any $K \\in D(\\mathcal{O}_X)$. This follows by applying the", "exact functor $DQ_X$ to the distinguished triangle of", "Cohomology, Lemma \\ref{cohomology-lemma-exact-sequence-j-star}", "and using Lemma \\ref{lemma-pushforward-better-coherator} three times." ], "refs": [ "perfect-lemma-better-coherator", "cohomology-lemma-exact-sequence-j-star", "perfect-lemma-pushforward-better-coherator" ], "ref_ids": [ 7022, 2144, 7023 ] }, { "id": 7134, "type": "other", "label": "perfect-remark-multiplication-map", "categories": [ "perfect" ], "title": "perfect-remark-multiplication-map", "contents": [ "With notation as in Lemma \\ref{lemma-affine-morphism-and-hom-out-of-perfect}.", "The diagram", "$$", "\\xymatrix{", "R\\Hom_X(M, Rg'_*L) \\otimes_R^\\mathbf{L} R' \\ar[r] \\ar[d]_\\mu &", "R\\Hom_{X'}(L(g')^*M, L(g')^*Rg'_*L) \\ar[d]^a \\\\", "R\\Hom_X(M, R(g')_*L) \\ar@{=}[r] &", "R\\Hom_{X'}(L(g')^*M, L)", "}", "$$", "is commutative where the top horizontal arrow is the map from the lemma,", "$\\mu$ is the multiplication map, and $a$ comes from the adjunction map", "$L(g')^*Rg'_*L \\to L$. The multiplication map is the adjunction map", "$K' \\otimes_R^\\mathbf{L} R' \\to K'$ for any $K' \\in D(R')$." ], "refs": [ "perfect-lemma-affine-morphism-and-hom-out-of-perfect" ], "ref_ids": [ 7029 ] }, { "id": 7135, "type": "other", "label": "perfect-remark-annoying-compatibility", "categories": [ "perfect" ], "title": "perfect-remark-annoying-compatibility", "contents": [ "Let $S = \\Spec(A)$ be an affine scheme. Let $a : X \\to S$", "and $b : Y \\to S$ be morphisms of schemes. Let $\\mathcal{F}$, $\\mathcal{G}$", "be quasi-coherent $\\mathcal{O}_X$-modules and let $\\mathcal{E}$", "be a quasi-coherent $\\mathcal{O}_Y$-module. Let", "$\\xi \\in H^i(X, \\mathcal{G})$ with pullback", "$p^*\\xi \\in H^i(X \\times_S Y, p^*\\mathcal{G})$.", "Then the following diagram is commutative", "$$", "\\xymatrix{", "R\\Gamma(X, \\mathcal{F})[-i] \\otimes_A^\\mathbf{L} R\\Gamma(Y, \\mathcal{E})", "\\ar[d] \\ar[rr]_-{\\xi \\otimes \\text{id}} & &", "R\\Gamma(X, \\mathcal{G} \\otimes_{\\mathcal{O}_X} \\mathcal{F})", "\\otimes_A^\\mathbf{L} R\\Gamma(Y, \\mathcal{E}) \\ar[d] \\\\", "R\\Gamma(X \\times_S Y, p^*\\mathcal{F} \\otimes q^*\\mathcal{E})[-i]", "\\ar[rr]^-{p^*\\xi} & &", "R\\Gamma(X \\times_S Y,", "p^*(\\mathcal{G} \\otimes_{\\mathcal{O}_X} \\mathcal{F}) \\otimes q^*\\mathcal{E})", "}", "$$", "where the unadorned tensor products are over $\\mathcal{O}_{X \\times_S Y}$.", "The horizontal arrows are from Cohomology, Remark", "\\ref{cohomology-remark-cup-with-element-map-total-cohomology}", "and the vertical arrows are (\\ref{equation-kunneth-global})", "hence given by pulling back followed by cup product on $X \\times_S Y$.", "The diagram commutes because the global cup product (on $X \\times_S Y$", "with the sheaves $p^*\\mathcal{G}$, $p^*\\mathcal{F}$, and $q^*\\mathcal{E}$)", "is associative, see", "Cohomology, Lemma \\ref{cohomology-lemma-cup-product-associative}." ], "refs": [ "cohomology-remark-cup-with-element-map-total-cohomology", "cohomology-lemma-cup-product-associative" ], "ref_ids": [ 2274, 2131 ] }, { "id": 7136, "type": "other", "label": "perfect-remark-base-change-of-L", "categories": [ "perfect" ], "title": "perfect-remark-base-change-of-L", "contents": [ "The pseudo-coherent complex $L$ of part (B) of Lemma \\ref{lemma-compute-ext}", "is canonically associated to the situation. For example,", "formation of $L$ as in (B) is compatible with base change.", "In other words, given a cartesian diagram", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} &", "X \\ar[d]^f \\\\", "S' \\ar[r]^g &", "S", "}", "$$", "of schemes we have canonical functorial isomorphisms", "$$", "\\Ext^i_{\\mathcal{O}_{S'}}(Lg^*L, \\mathcal{F}') \\longrightarrow", "\\Ext^i_{\\mathcal{O}_X}(L(g')^*E,", "(g')^*\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_{X'}} (f')^*\\mathcal{F}')", "$$", "for $\\mathcal{F}'$ quasi-coherent on $S'$. Obsere that we do {\\bf not} use", "derived pullback on $\\mathcal{G}^\\bullet$ on the right hand side.", "If we ever need this, we will", "formulate a precise result here and give a detailed proof." ], "refs": [ "perfect-lemma-compute-ext" ], "ref_ids": [ 7049 ] }, { "id": 7137, "type": "other", "label": "perfect-remark-explain-perfect-direct-image", "categories": [ "perfect" ], "title": "perfect-remark-explain-perfect-direct-image", "contents": [ "Let $R$ be a ring. Let $X$ be a scheme of finite presentation over", "$R$. Let $\\mathcal{G}$ be a finitely presented $\\mathcal{O}_X$-module", "flat over $R$ with support proper over $R$. By", "Lemma \\ref{lemma-base-change-tensor-perfect}", "there exists a finite complex of finite projective $R$-modules", "$M^\\bullet$ such that we have", "$$", "R\\Gamma(X_{R'}, \\mathcal{G}_{R'}) = M^\\bullet \\otimes_R R'", "$$", "functorially in the $R$-algebra $R'$." ], "refs": [ "perfect-lemma-base-change-tensor-perfect" ], "ref_ids": [ 7052 ] }, { "id": 7138, "type": "other", "label": "perfect-remark-compare-L", "categories": [ "perfect" ], "title": "perfect-remark-compare-L", "contents": [ "The reader may have noticed the similarity between", "Lemma \\ref{lemma-compute-ext-rel-perfect} and", "Lemma \\ref{lemma-compute-ext}.", "Indeed, the pseudo-coherent complex $L$ of", "Lemma \\ref{lemma-compute-ext-rel-perfect}", "may be characterized as the unique pseudo-coherent complex", "on $S$ such that there are functorial isomorphisms", "$$", "\\Ext^i_{\\mathcal{O}_S}(L, \\mathcal{F}) \\longrightarrow", "\\Ext^i_{\\mathcal{O}_X}(K,", "E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*\\mathcal{F})", "$$", "compatible with boundary maps for $\\mathcal{F}$ ranging over", "$\\QCoh(\\mathcal{O}_S)$. If we ever need this we will", "formulate a precise result here and give a detailed proof." ], "refs": [ "perfect-lemma-compute-ext-rel-perfect", "perfect-lemma-compute-ext", "perfect-lemma-compute-ext-rel-perfect" ], "ref_ids": [ 7084, 7049, 7084 ] }, { "id": 7139, "type": "other", "label": "perfect-remark-discuss-rel-perfect", "categories": [ "perfect" ], "title": "perfect-remark-discuss-rel-perfect", "contents": [ "Our Definition \\ref{definition-relatively-perfect} of a", "relatively perfect complex is equivalent to the one given", "in \\cite{lieblich-complexes} whenever our definition applies\\footnote{To", "see this, use Lemma \\ref{lemma-affine-locally-rel-perfect} and", "More on Algebra, Lemma \\ref{more-algebra-lemma-structure-relatively-perfect}.}.", "Next, suppose that $f : X \\to S$ is only assumed to be locally", "of finite type (not necessarily flat, nor locally of finite", "presentation). The definition in the paper cited above is that", "$E \\in D(\\mathcal{O}_X)$ is relatively perfect if", "\\begin{enumerate}", "\\item[(A)] locally on $X$ the object $E$ should be", "quasi-isomorphic to a finite complex of", "$S$-flat, finitely presented $\\mathcal{O}_X$-modules.", "\\end{enumerate}", "On the other hand, the natural generalization of our", "Definition \\ref{definition-relatively-perfect} is", "\\begin{enumerate}", "\\item[(B)] $E$ is pseudo-coherent relative to $S$", "(More on Morphisms, Definition", "\\ref{more-morphisms-definition-relative-pseudo-coherence})", "and $E$ locally has finite tor dimension as an object of", "$D(f^{-1}\\mathcal{O}_S)$", "(Cohomology, Definition \\ref{cohomology-definition-tor-amplitude}).", "\\end{enumerate}", "The advantage of condition (B) is that it clearly defines a triangulated", "subcategory of $D(\\mathcal{O}_X)$, whereas we suspect this is not", "the case for condition (A). The advantage of condition (A)", "is that it is easier to work with in particular in regards to limits." ], "refs": [ "perfect-definition-relatively-perfect", "perfect-lemma-affine-locally-rel-perfect", "more-algebra-lemma-structure-relatively-perfect", "perfect-definition-relatively-perfect", "more-morphisms-definition-relative-pseudo-coherence", "cohomology-definition-tor-amplitude" ], "ref_ids": [ 7119, 7077, 10290, 7119, 14118, 2259 ] }, { "id": 7140, "type": "other", "label": "perfect-remark-K-ring", "categories": [ "perfect" ], "title": "perfect-remark-K-ring", "contents": [ "Let $X$ be a scheme. The K-group $K_0(X)$ is canonically a commutative ring.", "Namely, using the derived tensor product", "$$", "\\otimes = \\otimes^\\mathbf{L}_{\\mathcal{O}_X} :", "D_{perf}(\\mathcal{O}_X) \\times D_{perf}(\\mathcal{O}_X)", "\\longrightarrow", "D_{perf}(\\mathcal{O}_X)", "$$", "and Derived Categories, Lemma \\ref{derived-lemma-bilinear-map-K}", "we obtain a bilinear multiplication. Since $K \\otimes L \\cong L \\otimes K$", "we see that this product is commutative. Since", "$(K \\otimes L) \\otimes M = K \\otimes (L \\otimes M)$", "we see that this product is associative.", "Finally, the unit of $K_0(X)$ is the element $1 = [\\mathcal{O}_X]$.", "\\medskip\\noindent", "If $\\textit{Vect}(X)$ and $K_0(\\textit{Vect}(X))$ are as above, then", "it is clearly the case that $K_0(\\textit{Vect}(X))$ also has a", "ring structure: if $\\mathcal{E}$ and $\\mathcal{F}$ are finite locally free", "$\\mathcal{O}_X$-modules, then we set", "$$", "[\\mathcal{E}] \\cdot [\\mathcal{F}] =", "[\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{F}]", "$$", "The reader easily verifies that this indeed defines a bilinear", "commutative, associative product. Details omitted. The map", "$$", "K_0(\\textit{Vect}(X)) \\longrightarrow K_0(X)", "$$", "constructed above is a ring map with these definitions.", "\\medskip\\noindent", "Now assume $X$ is Noetherian. The derived tensor product also produces", "a map", "$$", "\\otimes = \\otimes^\\mathbf{L}_{\\mathcal{O}_X} :", "D_{perf}(\\mathcal{O}_X) \\times D^b_{\\textit{Coh}}(\\mathcal{O}_X)", "\\longrightarrow", "D^b_{\\textit{Coh}}(\\mathcal{O}_X)", "$$", "Again using Derived Categories, Lemma \\ref{derived-lemma-bilinear-map-K}", "we obtain a bilinear multiplication $K_0(X) \\times K'_0(X) \\to K'_0(X)$", "since $K'_0(X) = K_0(D^b_{\\textit{Coh}}(\\mathcal{O}_X))$ by", "Lemma \\ref{lemma-Noetherian-Kprime}.", "The reader easily shows that this gives $K'_0(X)$ the structure", "of a module over the ring $K_0(X)$." ], "refs": [ "derived-lemma-bilinear-map-K", "derived-lemma-bilinear-map-K", "perfect-lemma-Noetherian-Kprime" ], "ref_ids": [ 1902, 1902, 7097 ] }, { "id": 7141, "type": "other", "label": "perfect-remark-pushforward-K", "categories": [ "perfect" ], "title": "perfect-remark-pushforward-K", "contents": [ "Let $f : X \\to Y$ be a proper morphism of locally Noetherian schemes.", "There is a map", "$$", "f_* : K'_0(X) \\longrightarrow K'_0(Y)", "$$", "which sends $[\\mathcal{F}]$ to", "$$", "[\\bigoplus\\nolimits_{i \\geq 0} R^{2i}f_*\\mathcal{F}] -", "[\\bigoplus\\nolimits_{i \\geq 0} R^{2i + 1}f_*\\mathcal{F}]", "$$", "This is well defined because the sheaves $R^if_*\\mathcal{F}$", "are coherent (Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-locally-projective-pushforward}), because locally", "only a finite number are nonzero, and because", "a short exact sequence of coherent sheaves on $X$ produces a long", "exact sequence of $R^if_*$ on $Y$. If $Y$ is quasi-compact (the only", "case most often used in practice), then we can rewrite the above as", "$$", "f_*[\\mathcal{F}] = \\sum (-1)^i[R^if_*\\mathcal{F}] = [Rf_*\\mathcal{F}]", "$$", "where we have used the equality $K'_0(Y) = K_0(D^b_{\\textit{Coh}}(Y))$ from", "Lemma \\ref{lemma-Noetherian-Kprime}." ], "refs": [ "coherent-lemma-locally-projective-pushforward", "perfect-lemma-Noetherian-Kprime" ], "ref_ids": [ 3345, 7097 ] }, { "id": 7142, "type": "other", "label": "perfect-remark-perf-Z", "categories": [ "perfect" ], "title": "perfect-remark-perf-Z", "contents": [ "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme. Consider the", "strictly full, saturated, triangulated subcategory", "$$", "D_{Z, perf}(\\mathcal{O}_X) \\subset D(\\mathcal{O}_X)", "$$", "consisting of perfect complexes of $\\mathcal{O}_X$-modules", "whose cohomology sheaves are settheoretically supported on $Z$.", "The zeroth $K$-group $K_0(D_{Z, perf}(\\mathcal{O}_X))$", "of this triangulated category is sometimes denoted", "$K_Z(X)$ or $K_{0, Z}(X)$. Using derived tensor product exactly", "as in Remark \\ref{remark-K-ring} we see that $K_0(D_{Z, perf}(\\mathcal{O}_X))$", "has a multiplication which is associative and commutative,", "but in general $K_0(D_{Z, perf}(\\mathcal{O}_X))$ doesn't have a unit." ], "refs": [ "perfect-remark-K-ring" ], "ref_ids": [ 7140 ] }, { "id": 7143, "type": "other", "label": "perfect-remark-functorial-det", "categories": [ "perfect" ], "title": "perfect-remark-functorial-det", "contents": [ "The construction of Lemma \\ref{lemma-determinant-two-term-complexes}", "is compatible with pullbacks. More precisely, given a morphism", "$f : X \\to Y$ of schemes and a perfect object $K$ of $D(\\mathcal{O}_Y)$", "of tor-amplitude in $[-1, 0]$ then $Lf^*K$ is a", "perfect object $K$ of $D(\\mathcal{O}_X)$", "of tor-amplitude in $[-1, 0]$ and we have a canonical identification", "$$", "f^*\\det(K) \\longrightarrow \\det(Lf^*K)", "$$", "Moreover, if $K$ has rank $0$, then $\\delta(K)$ pulls back to", "$\\delta(Lf^*K)$ via this map. This is clear from the affine local", "construction of the determinant." ], "refs": [ "perfect-lemma-determinant-two-term-complexes" ], "ref_ids": [ 7102 ] }, { "id": 7205, "type": "other", "label": "spaces-flat-remark-correct-generality", "categories": [ "spaces-flat" ], "title": "spaces-flat-remark-correct-generality", "contents": [ "The result in this section can be generalized. It is probably correct", "if we only assume $X \\to \\Spec(A)$ to be separated, of finite presentation,", "and $K_n$ pseudo-coherent relative to $A_n$ supported on a closed", "subset of $X_n$ proper over $A_n$. The outcome will be a $K$ which", "is pseudo-coherent relative to $A$ supported on a closed subset", "proper over $A$. If we ever need this, we will", "formulate a precise statement and prove it here." ], "refs": [], "ref_ids": [] }, { "id": 7297, "type": "other", "label": "spaces-chow-remark-sober", "categories": [ "spaces-chow" ], "title": "spaces-chow-remark-sober", "contents": [ "In Situation \\ref{situation-setup} if $X/B$ is good, then", "$|X|$ is a sober topological space. See", "Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quasi-separated-sober}", "or Decent Spaces, Proposition \\ref{decent-spaces-proposition-reasonable-sober}.", "We will use this without further mention", "to choose generic points of irreducible closed subsets of $|X|$." ], "refs": [ "spaces-properties-lemma-quasi-separated-sober", "decent-spaces-proposition-reasonable-sober" ], "ref_ids": [ 11852, 9559 ] }, { "id": 7298, "type": "other", "label": "spaces-chow-remark-integral", "categories": [ "spaces-chow" ], "title": "spaces-chow-remark-integral", "contents": [ "In Situation \\ref{situation-setup} if $X/B$ is good, then", "$X$ is integral (Spaces over Fields, Definition", "\\ref{spaces-over-fields-definition-integral-algebraic-space})", "if and only if $X$ is reduced and $|X|$ is irreducible.", "Moreover, for any point $\\xi \\in |X|$ there is a unique integral closed", "subspace $Z \\subset X$ such that $\\xi$ is the generic point", "of the closed subset $|Z| \\subset |X|$, see", "Spaces over Fields, Lemma", "\\ref{spaces-over-fields-lemma-decent-irreducible-closed}." ], "refs": [ "spaces-over-fields-definition-integral-algebraic-space", "spaces-over-fields-lemma-decent-irreducible-closed" ], "ref_ids": [ 12883, 12828 ] }, { "id": 7299, "type": "other", "label": "spaces-chow-remark-irreducible-component", "categories": [ "spaces-chow" ], "title": "spaces-chow-remark-irreducible-component", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $Y \\subset X$ be a closed subspace. By", "Remarks \\ref{remark-sober} and \\ref{remark-integral}", "there are $1$-to-$1$ correspondences between", "\\begin{enumerate}", "\\item irreducible components $T$ of $|Y|$,", "\\item generic points of irreducible components of $|Y|$, and", "\\item integral closed subspaces $Z \\subset Y$ with the property that", "$|Z|$ is an irreducible component of $|Y|$.", "\\end{enumerate}", "In this chapter we will call $Z$ as in (3) an", "{\\it irreducible component of $Y$}", "and we will call $\\xi \\in |Z|$ its {\\it generic point}." ], "refs": [ "spaces-chow-remark-sober", "spaces-chow-remark-integral" ], "ref_ids": [ 7297, 7298 ] }, { "id": 7300, "type": "other", "label": "spaces-chow-remark-residue-field", "categories": [ "spaces-chow" ], "title": "spaces-chow-remark-residue-field", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good. Every $x \\in |X|$", "can be represented by a (unique) monomorphism $\\Spec(k) \\to X$ where", "$k$ is a field, see", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-points-monomorphism}.", "Then $k$ is the {\\it residue field of} $x$ and is denoted $\\kappa(x)$.", "Recall that $X$ has a dense open subscheme $U \\subset X$", "(Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}).", "If $x \\in U$, then $\\kappa(x)$ agrees with the residue", "field of $x$ on $U$ as a scheme. See Decent Spaces, Section", "\\ref{decent-spaces-section-residue-fields-henselian-local-rings}." ], "refs": [ "decent-spaces-lemma-decent-points-monomorphism", "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme" ], "ref_ids": [ 9486, 11917 ] }, { "id": 7301, "type": "other", "label": "spaces-chow-remark-function-field", "categories": [ "spaces-chow" ], "title": "spaces-chow-remark-function-field", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good. Assume $X$ is integral.", "In this case the {\\it function field} $R(X)$ of $X$ is defined and", "is equal to the residue field of $X$ at its generic point.", "See Spaces over Fields, Definition", "\\ref{spaces-over-fields-definition-function-field}.", "Combining this with Remark \\ref{remark-integral}", "we find that for any $x \\in X$ the residue field", "$\\kappa(x)$ is the function field of the", "unique integral closed subspace $Z \\subset X$", "whose generic point is $x$." ], "refs": [ "spaces-over-fields-definition-function-field", "spaces-chow-remark-integral" ], "ref_ids": [ 12884, 7298 ] }, { "id": 7302, "type": "other", "label": "spaces-chow-remark-infinite-sums-rational-equivalences", "categories": [ "spaces-chow" ], "title": "spaces-chow-remark-infinite-sums-rational-equivalences", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Suppose we have infinite collections $\\alpha_i, \\beta_i \\in Z_k(X)$,", "$i \\in I$ of $k$-cycles on $X$. Suppose that the supports", "of $\\alpha_i$ and $\\beta_i$ form locally finite collections", "of closed subsets of $X$ so that $\\sum \\alpha_i$", "and $\\sum \\beta_i$ are defined as cycles. Moreover, assume that", "$\\alpha_i \\sim_{rat} \\beta_i$ for each $i$. Then it is not", "clear that $\\sum \\alpha_i \\sim_{rat} \\sum \\beta_i$. Namely,", "the problem is that the rational equivalences may be", "given by locally finite", "families $\\{W_{i, j}, f_{i, j} \\in R(W_{i, j})^*\\}_{j \\in J_i}$", "but the union $\\{W_{i, j}\\}_{i \\in I, j\\in J_i}$ may not", "be locally finite.", "\\medskip\\noindent", "In many cases in practice, one has a locally finite family of closed", "subsets $\\{T_i\\}_{i \\in I}$ of $|X|$ such that $\\alpha_i, \\beta_i$", "are supported on $T_i$ and such that $\\alpha_i \\sim_{rat} \\beta_i$", "``on'' $T_i$. More precisely, the families", "$\\{W_{i, j}, f_{i, j} \\in R(W_{i, j})^*\\}_{j \\in J_i}$", "consist of integral closed subspaces $W_{i, j}$", "with $|W_{i, j}| \\subset T_i$. In this case it is true that", "$\\sum \\alpha_i \\sim_{rat} \\sum \\beta_i$ on $X$, simply because", "the family $\\{W_{i, j}\\}_{i \\in I, j\\in J_i}$ is automatically", "locally finite in this case." ], "refs": [], "ref_ids": [] }, { "id": 7303, "type": "other", "label": "spaces-chow-remark-on-cycles", "categories": [ "spaces-chow" ], "title": "spaces-chow-remark-on-cycles", "contents": [ "Let $S$, $B$, $X$, $\\mathcal{L}$, $s$, $i : D \\to X$ be as in", "Definition \\ref{definition-gysin-homomorphism} and assume", "that $\\mathcal{L}|_D \\cong \\mathcal{O}_D$. In this case we", "can define a canonical map $i^* : Z_{k + 1}(X) \\to Z_k(D)$", "on cycles, by requiring that $i^*[W] = 0$ whenever $W \\subset D$.", "The possibility to do this will be useful later on." ], "refs": [ "spaces-chow-definition-gysin-homomorphism" ], "ref_ids": [ 7292 ] }, { "id": 7304, "type": "other", "label": "spaces-chow-remark-pullback-pairs", "categories": [ "spaces-chow" ], "title": "spaces-chow-remark-pullback-pairs", "contents": [ "Let $f : X' \\to X$ be a morphism of good algebraic spaces over $B$", "as in Situation \\ref{situation-setup}. Let $(\\mathcal{L}, s, i : D \\to X)$", "be a triple as in Definition \\ref{definition-gysin-homomorphism}.", "Then we can set $\\mathcal{L}' = f^*\\mathcal{L}$, $s' = f^*s$, and", "$D' = X' \\times_X D = Z(s')$. This gives a commutative diagram", "$$", "\\xymatrix{", "D' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\", "D \\ar[r]^i & X", "}", "$$", "and we can ask for various compatibilities between $i^*$ and $(i')^*$." ], "refs": [ "spaces-chow-definition-gysin-homomorphism" ], "ref_ids": [ 7292 ] }, { "id": 7305, "type": "other", "label": "spaces-chow-remark-pullback-cohomology", "categories": [ "spaces-chow" ], "title": "spaces-chow-remark-pullback-cohomology", "contents": [ "In Situation \\ref{situation-setup} let $f : X \\to Y$ be a morphism of", "good algebraic spaces over $B$.", "Then there is a canonical $\\mathbf{Z}$-algebra map $A^*(Y) \\to A^*(X)$.", "Namely, given $c \\in A^p(Y)$ and $X' \\to X$, then we can let $f^*c$", "be defined by the map $c \\cap - : \\CH_k(X') \\to \\CH_{k - p}(X')$ which is", "given by thinking of $X'$ as an algebraic space over $Y$." ], "refs": [], "ref_ids": [] }, { "id": 7306, "type": "other", "label": "spaces-chow-remark-extend-to-finite-locally-free", "categories": [ "spaces-chow" ], "title": "spaces-chow-remark-extend-to-finite-locally-free", "contents": [ "In Situation \\ref{situation-setup} let $X/B$ be good.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module.", "If the rank of $\\mathcal{E}$ is not constant then we can", "still define the Chern classes of $\\mathcal{E}$. Namely, in this", "case we can write", "$$", "X = X_0 \\amalg X_1 \\amalg X_2 \\amalg \\ldots", "$$", "where $X_r \\subset X$ is the open and closed subspace where", "the rank of $\\mathcal{E}$ is $r$. If $X' \\to X$ is a morphism", "of good algebraic spaces over $B$, then we obtain by", "pullback a corresponding decomposition of $X'$ and we find that", "$$", "\\CH_*(X') = \\prod\\nolimits_{r \\geq 0} \\CH_*(X'_r)", "$$", "by our definitions. Then we simply define $c_i(\\mathcal{E})$", "to be the bivariant class which preserves these direct", "product decompositions and acts by the already defined", "operations $c_i(\\mathcal{E}|_{X_r}) \\cap -$", "on the factors. Observe that in this setting it may happen", "that $c_i(\\mathcal{E})$ is nonzero for infinitely many $i$." ], "refs": [], "ref_ids": [] }, { "id": 7383, "type": "other", "label": "sdga-remark-functoriality-ga", "categories": [ "sdga" ], "title": "sdga-remark-functoriality-ga", "contents": [ "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi. We have", "\\begin{enumerate}", "\\item Let $\\mathcal{A}$ be a graded $\\mathcal{O}_\\mathcal{C}$-algebra.", "The multiplication maps of $\\mathcal{A}$ induce multiplication maps", "$f_*\\mathcal{A}^n \\times f_*\\mathcal{A}^m \\to f_*\\mathcal{A}^{n + m}$", "and via $f^\\sharp$ we may view these as $\\mathcal{O}_\\mathcal{D}$-bilinear", "maps. We will denote $f_*\\mathcal{A}$ the graded", "$\\mathcal{O}_\\mathcal{D}$-algebra we so obtain.", "\\item Let $\\mathcal{B}$ be a graded", "$\\mathcal{O}_\\mathcal{D}$-algebra. ", "The multiplication maps of $\\mathcal{B}$ induce multiplication maps", "$f^*\\mathcal{B}^n \\times f^*\\mathcal{B}^m \\to f^*\\mathcal{B}^{n + m}$", "and using $f^\\sharp$ we may view these as $\\mathcal{O}_\\mathcal{C}$-bilinear", "maps. We will denote $f^*\\mathcal{B}$", "the graded $\\mathcal{O}_\\mathcal{C}$-algebra we so obtain.", "\\item The set of homomorphisms $f^*\\mathcal{B} \\to \\mathcal{A}$", "of graded $\\mathcal{O}_\\mathcal{C}$-algebras is in", "$1$-to-$1$ correspondence with the set of homomorphisms", "$\\mathcal{B} \\to f_*\\mathcal{A}$ of graded $\\mathcal{O}_\\mathcal{C}$-algebras.", "\\end{enumerate}", "Part (3) follows immediately from the usual adjunction between $f^*$ and $f_*$", "on sheaves of modules." ], "refs": [], "ref_ids": [] }, { "id": 7384, "type": "other", "label": "sdga-remark-functoriality-dga", "categories": [ "sdga" ], "title": "sdga-remark-functoriality-dga", "contents": [ "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})", "\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$", "be a morphism of ringed topoi.", "\\begin{enumerate}", "\\item Let $(\\mathcal{A}, \\text{d})$ be a differential graded", "$\\mathcal{O}_\\mathcal{C}$-algebra. The pushforward will be", "the differential graded $\\mathcal{O}_\\mathcal{D}$-algebra", "$(f_*\\mathcal{A}, \\text{d})$ where $f_*\\mathcal{A}$ is as in", "Remark \\ref{remark-functoriality-ga} and", "$\\text{d} = f_*\\text{d}$ as maps $f_*\\mathcal{A}^n \\to f_*\\mathcal{A}^{n + 1}$.", "We omit the verification that the Leibniz rule is satisfied.", "\\item Let $\\mathcal{B}$ be a differential graded", "$\\mathcal{O}_\\mathcal{D}$-algebra. The pullback will be the", "differential graded $\\mathcal{O}_\\mathcal{C}$-algebra", "$(f^*\\mathcal{B}, \\text{d})$ where $f^*\\mathcal{B}$ is as in", "Remark \\ref{remark-functoriality-ga} and", "$\\text{d} = f^*\\text{d}$ as maps $f^*\\mathcal{B}^n \\to f^*\\mathcal{B}^{n + 1}$.", "We omit the verification that the Leibniz rule is satisfied.", "\\item The set of homomorphisms $f^*\\mathcal{B} \\to \\mathcal{A}$", "of differential graded $\\mathcal{O}_\\mathcal{C}$-algebras is in", "$1$-to-$1$ correspondence with the set of homomorphisms", "$\\mathcal{B} \\to f_*\\mathcal{A}$ of differential graded", "$\\mathcal{O}_\\mathcal{D}$-algebras.", "\\end{enumerate}", "Part (3) follows immediately from the usual adjunction between $f^*$ and $f_*$", "on sheaves of modules." ], "refs": [ "sdga-remark-functoriality-ga", "sdga-remark-functoriality-ga" ], "ref_ids": [ 7383, 7383 ] }, { "id": 7385, "type": "other", "label": "sdga-remark-cone-identity", "categories": [ "sdga" ], "title": "sdga-remark-cone-identity", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{A}$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. Let $C = C(\\text{id}_\\mathcal{A})$", "be the cone on the identity map $\\mathcal{A} \\to \\mathcal{A}$ viewed", "as a map of differential graded $\\mathcal{A}$-modules.", "Then", "$$", "\\Hom_{\\text{Mod}_{(\\mathcal{A}, \\text{d})}}(C, \\mathcal{M}) =", "\\{(x, y) \\in", "\\Gamma(\\mathcal{C}, \\mathcal{M}^0) \\times", "\\Gamma(\\mathcal{C}, \\mathcal{M}^{-1}) \\mid", "x = \\text{d}(y)\\}", "$$", "where the map from left to right sends $f$ to the pair $(x, y)$", "where $x$ is the image of the global section $(0, 1)$ of", "$C^{-1} = \\mathcal{A}^{-1} \\oplus \\mathcal{A}^0$ and where", "$y$ is the image of the global section $(1, 0)$ of", "$C^0 = \\mathcal{A}^0 \\oplus \\mathcal{A}^1$." ], "refs": [], "ref_ids": [] }, { "id": 7386, "type": "other", "label": "sdga-remark-sheaf-graded-sets", "categories": [ "sdga" ], "title": "sdga-remark-sheaf-graded-sets", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. A", "{\\it sheaf of graded sets} on $\\mathcal{C}$ is a sheaf", "of sets $\\mathcal{S}$ endowed with a map", "$\\deg : \\mathcal{S} \\to \\underline{\\mathbf{Z}}$", "of sheaves of sets. Let us denote $\\mathcal{O}[\\mathcal{S}]$", "the graded $\\mathcal{O}$-module which is the", "free $\\mathcal{O}$-module on the graded sheaf of sets $\\mathcal{S}$.", "More precisely, the $n$th graded part of", "$\\mathcal{O}[\\mathcal{S}]$ is the sheafification of the rule", "$$", "U \\longmapsto", "\\bigoplus\\nolimits_{s \\in \\mathcal{S}(U),\\ \\deg(s) = n} s \\cdot \\mathcal{O}(U)", "$$", "With zero differential we also may consider this as a", "differential graded $\\mathcal{O}$-module.", "Let $\\mathcal{A}$ be a sheaf of graded $\\mathcal{O}$-algebras", "Then we similarly define $\\mathcal{A}[\\mathcal{S}]$ to be the", "graded $\\mathcal{A}$-module whose $n$th graded part is the", "sheafification of the rule", "$$", "U \\longmapsto", "\\bigoplus\\nolimits_{s \\in \\mathcal{S}(U)} s \\cdot \\mathcal{A}^{n - \\deg(s)}(U)", "$$", "If $\\mathcal{A}$ is a differential graded $\\mathcal{O}$-algebra, the", "we turn this into a differential graded $\\mathcal{O}$-module", "by setting $\\text{d}(s) = 0$ for all $s \\in \\mathcal{S}(U)$", "and sheafifying." ], "refs": [], "ref_ids": [] }, { "id": 7387, "type": "other", "label": "sdga-remark-why-graded-injective", "categories": [ "sdga" ], "title": "sdga-remark-why-graded-injective", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras", "on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{I}$ be a graded injective", "diffential graded $\\mathcal{A}$-module. Let", "$$", "0 \\to \\mathcal{M}_1 \\to \\mathcal{M}_2 \\to \\mathcal{M}_3 \\to 0", "$$", "be a short exact sequence of differential graded $\\mathcal{A}$-modules.", "Since $\\mathcal{I}$ is graded injective", "we obtain a short exact sequence of complexes", "$$", "0 \\to", "\\Hom_{\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})}}(\\mathcal{M}_3, \\mathcal{I})", "\\to", "\\Hom_{\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})}}(\\mathcal{M}_2, \\mathcal{I})", "\\to", "\\Hom_{\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})}}(\\mathcal{M}_1, \\mathcal{I})", "\\to 0", "$$", "of $\\Gamma(\\mathcal{C}, \\mathcal{O})$-modules. Taking cohomology we", "obtain a long exact sequence", "$$", "\\xymatrix{", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}_3, \\mathcal{I})", "\\ar[d] &", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}_3, \\mathcal{I})[1]", "\\ar[d] \\\\", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}_2, \\mathcal{I})", "\\ar[d] &", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}_2, \\mathcal{I})[1]", "\\ar[d] \\\\", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}_1, \\mathcal{I})", "\\ar[ruu]", "&", "\\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}_1, \\mathcal{I})[1]", "}", "$$", "of groups of homomorphisms in the homotopy category. The point is that", "we get this even though we didn't assume that our short exact sequence", "is admissible (so the short exact sequence in general does not define", "a distinguished triangle in the homotopy category)." ], "refs": [], "ref_ids": [] }, { "id": 7632, "type": "other", "label": "stacks-morphisms-remark-inertia-is-group-in-spaces", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-remark-inertia-is-group-in-spaces", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. In", "Properties of Stacks, Remark \\ref{stacks-properties-remark-representable-over}", "we have seen that the $2$-category of morphisms", "$\\mathcal{Z} \\to \\mathcal{X}$ representable by algebraic spaces", "with target $\\mathcal{X}$ forms a category.", "In this category the inertia stack of $\\mathcal{X}/\\mathcal{Y}$ is", "a {\\it group object}. Recall that an object of", "$\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$", "is just a pair $(x, \\alpha)$ where $x$ is an object of $\\mathcal{X}$", "and $\\alpha$ is an automorphism of $x$ in the fibre category of $\\mathcal{X}$", "that $x$ lives in with $f(\\alpha) = \\text{id}$. The composition", "$$", "c :", "\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}", "\\times_\\mathcal{X} \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}", "\\longrightarrow", "\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}", "$$", "is given by the rule on objects", "$$", "((x, \\alpha), (x', \\alpha'), \\beta) \\mapsto", "(x, \\alpha \\circ \\beta^{-1} \\circ \\alpha' \\circ \\beta)", "$$", "which makes sense as $\\beta : x \\to x'$ is an isomorphism in the fibre", "category by our definition of fibre products. The neutral element", "$e : \\mathcal{X} \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$ is given by the", "functor $x \\mapsto (x, \\text{id}_x)$. We omit the proof that the", "axioms of a group object hold." ], "refs": [ "stacks-properties-remark-representable-over" ], "ref_ids": [ 8927 ] }, { "id": 7633, "type": "other", "label": "stacks-morphisms-remark-composition", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-remark-composition", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces", "which is smooth local on the source-and-target and stable under composition.", "Then the property of morphisms of algebraic stacks defined in", "Definition \\ref{definition-P}", "is stable under composition. Namely, let $f : \\mathcal{X} \\to \\mathcal{Y}$", "and $g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks", "having property $\\mathcal{P}$. Choose an algebraic space $W$ and a", "surjective smooth morphism $W \\to \\mathcal{Z}$. Choose an algebraic space", "$V$ and a surjective smooth morphism $V \\to \\mathcal{Y} \\times_\\mathcal{Z} W$.", "Finally, choose an algebraic space $U$ and a surjective and smooth morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$. Then the morphisms", "$V \\to W$ and $U \\to V$ have property $\\mathcal{P}$ by definition.", "Whence $U \\to W$ has property $\\mathcal{P}$ as we assumed that", "$\\mathcal{P}$ is stable under composition. Thus, by definition again,", "we see that $g \\circ f : \\mathcal{X} \\to \\mathcal{Z}$ has", "property $\\mathcal{P}$." ], "refs": [ "stacks-morphisms-definition-P" ], "ref_ids": [ 7613 ] }, { "id": 7634, "type": "other", "label": "stacks-morphisms-remark-base-change", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-remark-base-change", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces", "which is smooth local on the source-and-target and stable under base change.", "Then the property of morphisms of algebraic stacks defined in", "Definition \\ref{definition-P}", "is stable under base change. Namely, let $f : \\mathcal{X} \\to \\mathcal{Y}$", "and $g : \\mathcal{Y}' \\to \\mathcal{Y}$ be morphisms of algebraic stacks", "and assume $f$ has property $\\mathcal{P}$. Choose an algebraic space $V$", "and a surjective smooth morphism $V \\to \\mathcal{Y}$. Choose an algebraic", "space $U$ and a surjective smooth morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$. Finally, choose an algebraic space", "$V'$ and a surjective and smooth morphism", "$V' \\to \\mathcal{Y}' \\times_\\mathcal{Y} V$. Then the morphism", "$U \\to V$ has property $\\mathcal{P}$ by definition.", "Whence $V' \\times_V U \\to V'$ has property $\\mathcal{P}$ as we assumed that", "$\\mathcal{P}$ is stable under base change. Considering the diagram", "$$", "\\xymatrix{", "V' \\times_V U \\ar[r] \\ar[d] &", "\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\ar[r] \\ar[d] &", "\\mathcal{X} \\ar[d] \\\\", "V' \\ar[r] & \\mathcal{Y}' \\ar[r] & \\mathcal{Y}", "}", "$$", "we see that the left top horizontal arrow is smooth and surjective,", "whence by definition we see that the projection", "$\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Y}'$ has", "property $\\mathcal{P}$." ], "refs": [ "stacks-morphisms-definition-P" ], "ref_ids": [ 7613 ] }, { "id": 7635, "type": "other", "label": "stacks-morphisms-remark-implication", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-remark-implication", "contents": [ "Let $\\mathcal{P}, \\mathcal{P}'$ be properties of morphisms of algebraic spaces", "which are smooth local on the source-and-target.", "Suppose that we have $\\mathcal{P} \\Rightarrow \\mathcal{P}'$ for morphisms", "of algebraic spaces. Then we also have $\\mathcal{P} \\Rightarrow \\mathcal{P}'$", "for the properties of morphisms of algebraic stacks defined in", "Definition \\ref{definition-P}", "using $\\mathcal{P}$ and $\\mathcal{P}'$. This is clear from the definition." ], "refs": [ "stacks-morphisms-definition-P" ], "ref_ids": [ 7613 ] }, { "id": 7636, "type": "other", "label": "stacks-morphisms-remark-property-automorphism-groups", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-remark-property-automorphism-groups", "contents": [ "Let $P$ be a property of algebraic spaces over fields which is invariant", "under ground field extensions. Given an algebraic stack $\\mathcal{X}$", "and $x \\in |\\mathcal{X}|$, we say the automorphism group of $\\mathcal{X}$", "at $x$ has $P$ if the equivalent conditions of", "Lemma \\ref{lemma-property-automorphism-groups} are satisfied.", "For example, we say {\\it the automorphism group of $\\mathcal{X}$", "at $x$ is finite}, if $G_x \\to \\Spec(k)$ is finite whenever", "$x : \\Spec(k) \\to \\mathcal{X}$ is a representative of $x$.", "Similarly for smooth, proper, etc.", "(There is clearly an abuse of language going on here, but we", "believe it will not cause confusion or imprecision.)" ], "refs": [ "stacks-morphisms-lemma-property-automorphism-groups" ], "ref_ids": [ 7473 ] }, { "id": 7637, "type": "other", "label": "stacks-morphisms-remark-identify-automorphism-groups", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-remark-identify-automorphism-groups", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $x \\in |\\mathcal{X}|$ be a point. To indicate the equivalent", "conditions of Lemma \\ref{lemma-iso-automorphism-groups}", "are satisfied for $f$ and $x$ in the literature the terminology", "{\\it $f$ is stabilizer preserving at $x$} or", "{\\it $f$ is fixed-point reflecting at $x$} is used.", "We prefer to say {\\it $f$ induces an isomorphism between", "automorphism groups at $x$ and $f(x)$}." ], "refs": [ "stacks-morphisms-lemma-iso-automorphism-groups" ], "ref_ids": [ 7474 ] }, { "id": 7638, "type": "other", "label": "stacks-morphisms-remark-order-type", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-remark-order-type", "contents": [ "We can wonder about the order type of the canonical stratifications which", "occur as output of the stratifications of type (a) constructed in", "Lemma \\ref{lemma-every-point-in-a-stratum}.", "A natural guess is that the well-ordered set $I$ has", "{\\it cardinality} at most $\\aleph_0$. We have no idea if this is true", "or false. If you do please email", "\\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}." ], "refs": [ "stacks-morphisms-lemma-every-point-in-a-stratum" ], "ref_ids": [ 7528 ] }, { "id": 7639, "type": "other", "label": "stacks-morphisms-remark-etale-smooth-composition", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-remark-etale-smooth-composition", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces", "which is \\'etale-smooth local on the source-and-target and", "stable under composition. Then the property of DM morphisms of algebraic stacks", "defined in Definition \\ref{definition-etale-smooth-P}", "is stable under composition. Namely, let $f : \\mathcal{X} \\to \\mathcal{Y}$", "and $g : \\mathcal{Y} \\to \\mathcal{Z}$ be DM morphisms of algebraic stacks", "having property $\\mathcal{P}$. By Lemma \\ref{lemma-composition-separated}", "the composition $g \\circ f$ is DM. Choose an algebraic space $W$ and a", "surjective smooth morphism $W \\to \\mathcal{Z}$. Choose an algebraic space", "$V$ and a surjective \\'etale morphism $V \\to \\mathcal{Y} \\times_\\mathcal{Z} W$", "(Lemma \\ref{lemma-DM}).", "Choose an algebraic space $U$ and a surjective \\'etale morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$. Then the morphisms", "$V \\to W$ and $U \\to V$ have property $\\mathcal{P}$ by definition.", "Whence $U \\to W$ has property $\\mathcal{P}$ as we assumed that", "$\\mathcal{P}$ is stable under composition. Thus, by definition again,", "we see that $g \\circ f : \\mathcal{X} \\to \\mathcal{Z}$ has", "property $\\mathcal{P}$." ], "refs": [ "stacks-morphisms-definition-etale-smooth-P", "stacks-morphisms-lemma-composition-separated", "stacks-morphisms-lemma-DM" ], "ref_ids": [ 7623, 7404, 7481 ] }, { "id": 7640, "type": "other", "label": "stacks-morphisms-remark-etale-smooth-base-change", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-remark-etale-smooth-base-change", "contents": [ "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces", "which is \\'etale-smooth local on the source-and-target and", "stable under base change. Then the property of", "DM morphisms of algebraic stacks defined in", "Definition \\ref{definition-etale-smooth-P}", "is stable under arbitrary base change.", "Namely, let $f : \\mathcal{X} \\to \\mathcal{Y}$", "be a DM morphism of algebraic stacks", "and $g : \\mathcal{Y}' \\to \\mathcal{Y}$ be a morphism of algebraic stacks", "and assume $f$ has property $\\mathcal{P}$.", "Then the base change", "$\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Y}'$", "is a DM morphism by Lemma \\ref{lemma-base-change-separated}.", "Choose an algebraic space $V$", "and a surjective smooth morphism $V \\to \\mathcal{Y}$. Choose an algebraic", "space $U$ and a surjective \\'etale morphism", "$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ (Lemma \\ref{lemma-DM}).", "Finally, choose an algebraic space", "$V'$ and a surjective and smooth morphism", "$V' \\to \\mathcal{Y}' \\times_\\mathcal{Y} V$. Then the morphism", "$U \\to V$ has property $\\mathcal{P}$ by definition.", "Whence $V' \\times_V U \\to V'$ has property $\\mathcal{P}$ as we assumed that", "$\\mathcal{P}$ is stable under base change. Considering the diagram", "$$", "\\xymatrix{", "V' \\times_V U \\ar[r] \\ar[d] &", "\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\ar[r] \\ar[d] &", "\\mathcal{X} \\ar[d] \\\\", "V' \\ar[r] & \\mathcal{Y}' \\ar[r] & \\mathcal{Y}", "}", "$$", "we see that the left top horizontal arrow is surjective and", "$$", "V' \\times_V U \\to V' \\times_\\mathcal{Y}", "(\\mathcal{Y}' \\times_{\\mathcal{Y}'} \\mathcal{X}) =", "V' \\times_V (\\mathcal{X} \\times_\\mathcal{Y} V)", "$$", "is \\'etale as a base change of $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$,", "whence by definition we see that the projection", "$\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Y}'$ has", "property $\\mathcal{P}$." ], "refs": [ "stacks-morphisms-definition-etale-smooth-P", "stacks-morphisms-lemma-base-change-separated", "stacks-morphisms-lemma-DM" ], "ref_ids": [ 7623, 7398, 7481 ] }, { "id": 7641, "type": "other", "label": "stacks-morphisms-remark-etale-smooth-implication", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-remark-etale-smooth-implication", "contents": [ "Let $\\mathcal{P}, \\mathcal{P}'$ be properties of morphisms of algebraic spaces", "which are \\'etale-smooth local on the source-and-target.", "Suppose that we have $\\mathcal{P} \\Rightarrow \\mathcal{P}'$ for morphisms", "of algebraic spaces. Then we also have $\\mathcal{P} \\Rightarrow \\mathcal{P}'$", "for the properties of morphisms of algebraic stacks defined in", "Definition \\ref{definition-etale-smooth-P}", "using $\\mathcal{P}$ and $\\mathcal{P}'$. This is clear from the definition." ], "refs": [ "stacks-morphisms-definition-etale-smooth-P" ], "ref_ids": [ 7623 ] }, { "id": 7642, "type": "other", "label": "stacks-morphisms-remark-get-property-auts-from-diagonal", "categories": [ "stacks-morphisms" ], "title": "stacks-morphisms-remark-get-property-auts-from-diagonal", "contents": [ "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.", "Let $U \\to \\mathcal{X}$ be a morphism whose source is an algebraic space.", "Let $G \\to H$ be the pullback of the morphism", "$\\mathcal{I}_\\mathcal{X} \\to", "\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$", "to $U$. If $\\Delta_f$ is unramified, \\'etale, etc, so", "is $G \\to H$. This is true because", "$$", "\\xymatrix{", "U \\times_\\mathcal{X} U \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d]^{\\Delta_f} \\\\", "U \\times_\\mathcal{Y} U \\ar[r] & \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}", "}", "$$", "is cartesian and the morphism $G \\to H$ is the base change of the", "left vertical arrow by the diagonal $U \\to U \\times U$.", "Compare with the proof of Lemma \\ref{lemma-separated-implies-isom}." ], "refs": [ "stacks-morphisms-lemma-separated-implies-isom" ], "ref_ids": [ 7421 ] }, { "id": 7758, "type": "other", "label": "schemes-remark-not-reverse-open-closed", "categories": [ "schemes" ], "title": "schemes-remark-not-reverse-open-closed", "contents": [ "If $f : X \\to Y$ is an immersion of schemes, then it is in general", "not possible to factor $f$ as an open immersion followed", "by a closed immersion. See Morphisms, Example \\ref{morphisms-example-thibaut}." ], "refs": [], "ref_ids": [] }, { "id": 7759, "type": "other", "label": "schemes-remark-intersection-affine-opens", "categories": [ "schemes" ], "title": "schemes-remark-intersection-affine-opens", "contents": [ "In general the intersection of two affine opens in $X$", "is not affine open. See Example \\ref{example-affine-space-zero-doubled}." ], "refs": [], "ref_ids": [] }, { "id": 7760, "type": "other", "label": "schemes-remark-reduced-induced-locally-closed", "categories": [ "schemes" ], "title": "schemes-remark-reduced-induced-locally-closed", "contents": [ "Let $X$ be a scheme. Let $T \\subset X$ be a locally closed subset.", "In this situation we sometimes also use the phrase", "``reduced induced scheme structure on $T$''. It refers", "to the reduced induced scheme structure from", "Definition \\ref{definition-reduced-induced-scheme}", "when we view $T$ as a closed subset of the open subscheme ", "$X \\setminus \\partial T$ of $X$. Here", "$\\partial T = \\overline{T} \\setminus T$ is the ``boundary'' of $T$", "in the topological space of $X$." ], "refs": [ "schemes-definition-reduced-induced-scheme" ], "ref_ids": [ 7745 ] }, { "id": 7761, "type": "other", "label": "schemes-remark-representable-locally-ringed", "categories": [ "schemes" ], "title": "schemes-remark-representable-locally-ringed", "contents": [ "Suppose the functor $F$ is defined on all locally ringed spaces,", "and if conditions of Lemma \\ref{lemma-glue-functors} are replaced by", "the following:", "\\begin{enumerate}", "\\item $F$ satisfies the sheaf property on the category of locally ringed", "spaces,", "\\item there exists a set $I$ and a collection of subfunctors", "$F_i \\subset F$ such that", "\\begin{enumerate}", "\\item each $F_i$ is representable by a scheme,", "\\item each $F_i \\subset F$ is representable by open immersions", "on the category of locally ringed spaces, and", "\\item the collection $(F_i)_{i \\in I}$ covers $F$", "as a functor on the category of locally ringed spaces.", "\\end{enumerate}", "\\end{enumerate}", "We leave it to the reader to spell this out further.", "Then the end result is that the functor $F$ is", "representable in the category of locally ringed spaces", "and that the representing object is a scheme." ], "refs": [ "schemes-lemma-glue-functors" ], "ref_ids": [ 7688 ] }, { "id": 7762, "type": "other", "label": "schemes-remark-fibre-product-schemes-locally-ringed", "categories": [ "schemes" ], "title": "schemes-remark-fibre-product-schemes-locally-ringed", "contents": [ "Using Remark \\ref{remark-representable-locally-ringed}", "you can show that the fibre product of morphisms of schemes", "exists in the category of locally ringed spaces and is a", "scheme." ], "refs": [ "schemes-remark-representable-locally-ringed" ], "ref_ids": [ 7761 ] }, { "id": 7763, "type": "other", "label": "schemes-remark-quasi-compact-and-quasi-separated", "categories": [ "schemes" ], "title": "schemes-remark-quasi-compact-and-quasi-separated", "contents": [ "The category of quasi-compact and quasi-separated schemes $\\mathcal{C}$", "has the following properties. If $X, Y \\in \\Ob(\\mathcal{C})$, then any", "morphism of schemes $f : X \\to Y$ is quasi-compact and quasi-separated by", "Lemmas \\ref{lemma-quasi-compact-permanence} and", "\\ref{lemma-compose-after-separated}", "with $Z = \\Spec(\\mathbf{Z})$. Moreover, if $X \\to Y$ and $Z \\to Y$", "are morphisms $\\mathcal{C}$, then $X \\times_Y Z$ is an object of $\\mathcal{C}$", "too. Namely, the projection $X \\times_Y Z \\to Z$ is quasi-compact and", "quasi-separated as a base change of the morphism $Z \\to Y$, see", "Lemmas \\ref{lemma-separated-permanence} and", "\\ref{lemma-quasi-compact-preserved-base-change}.", "Hence the composition $X \\times_Y Z \\to Z \\to \\Spec(\\mathbf{Z})$", "is quasi-compact and quasi-separated, see", "Lemmas \\ref{lemma-separated-permanence} and", "\\ref{lemma-composition-quasi-compact}." ], "refs": [ "schemes-lemma-quasi-compact-permanence", "schemes-lemma-compose-after-separated", "schemes-lemma-separated-permanence", "schemes-lemma-quasi-compact-preserved-base-change", "schemes-lemma-separated-permanence", "schemes-lemma-composition-quasi-compact" ], "ref_ids": [ 7716, 7715, 7714, 7698, 7714, 7699 ] }, { "id": 7812, "type": "other", "label": "injectives-remark-embedding", "categories": [ "injectives" ], "title": "injectives-remark-embedding", "contents": [ "The Freyd-Mitchell embedding theorem says there exists a fully faithful", "exact functor from any abelian category $\\mathcal{A}$", "to the category of modules over a ring.", "Lemma \\ref{lemma-embedding}", "is not quite as strong. But the result is suitable for the", "Stacks project as we have to understand sheaves of abelian groups on", "sites in detail anyway. Moreover, ``diagram chasing'' works in the category", "of abelian sheaves on $\\mathcal{C}$, for example by working with sections over", "objects, or by working on the level of stalks using that $\\mathcal{C}$ has", "enough points. To see how to deduce the Freyd-Mitchell embedding theorem from", "Lemma \\ref{lemma-embedding}", "see", "Remark \\ref{remark-embedding-freyd}." ], "refs": [ "injectives-lemma-embedding", "injectives-lemma-embedding", "injectives-remark-embedding-freyd" ], "ref_ids": [ 7783, 7783, 7814 ] }, { "id": 7813, "type": "other", "label": "injectives-remark-embedding-big", "categories": [ "injectives" ], "title": "injectives-remark-embedding-big", "contents": [ "If $\\mathcal{A}$ is a ``big'' abelian category, i.e., if $\\mathcal{A}$", "has a class of objects, then", "Lemma \\ref{lemma-embedding}", "does not work. In this case, given any set of objects", "$E \\subset \\Ob(\\mathcal{A})$ there exists an abelian full subcategory", "$\\mathcal{A}' \\subset \\mathcal{A}$ such that", "$\\Ob(\\mathcal{A}')$ is a set and $E \\subset \\Ob(\\mathcal{A}')$.", "Then one can apply", "Lemma \\ref{lemma-embedding}", "to $\\mathcal{A}'$. One can use this to prove that results depending on", "a diagram chase hold in $\\mathcal{A}$." ], "refs": [ "injectives-lemma-embedding", "injectives-lemma-embedding" ], "ref_ids": [ 7783, 7783 ] }, { "id": 7814, "type": "other", "label": "injectives-remark-embedding-freyd", "categories": [ "injectives" ], "title": "injectives-remark-embedding-freyd", "contents": [ "Let $\\mathcal{C}$ be a site.", "Note that $\\textit{Ab}(\\mathcal{C})$ has enough injectives, see", "Theorem \\ref{theorem-sheaves-injectives}.", "(In the case that $\\mathcal{C}$ has enough points this is straightforward", "because $p_*I$ is an injective sheaf if $I$ is an injective", "$\\mathbf{Z}$-module and $p$ is a point.)", "Also, $\\textit{Ab}(\\mathcal{C})$ has a cogenerator (details omitted).", "Hence", "Lemma \\ref{lemma-embedding}", "proves that we have a fully faithful, exact embedding", "$\\mathcal{A} \\to \\mathcal{B}$ where $\\mathcal{B}$ has a", "cogenerator and enough injectives.", "We can apply this to $\\mathcal{A}^{opp}$ and we get a", "fully faithful exact functor", "$i : \\mathcal{A} \\to \\mathcal{D} = \\mathcal{B}^{opp}$", "where $\\mathcal{D}$ has enough projectives and a generator. Hence", "$\\mathcal{D}$ has a projective generator $P$.", "Set $R = \\Mor_\\mathcal{D}(P, P)$. Then", "$$", "\\mathcal{A} \\longrightarrow \\text{Mod}_R, \\quad", "X \\longmapsto \\Hom_\\mathcal{D}(P, X).", "$$", "One can check this is a fully faithful, exact functor.", "In other words, one retrieves the", "Freyd-Mitchell theorem mentioned in", "Remark \\ref{remark-embedding}", "above." ], "refs": [ "injectives-theorem-sheaves-injectives", "injectives-lemma-embedding", "injectives-remark-embedding" ], "ref_ids": [ 7765, 7783, 7812 ] }, { "id": 7815, "type": "other", "label": "injectives-remark-embed-exact-category", "categories": [ "injectives" ], "title": "injectives-remark-embed-exact-category", "contents": [ "The arguments proving", "Lemmas \\ref{lemma-site-abelian-category} and", "\\ref{lemma-embedding}", "work also for {\\it exact categories}, see", "\\cite[Appendix A]{Buhler} and", "\\cite[1.1.4]{BBD}.", "We quickly review this here and we add more details if we ever", "need it in the Stacks project.", "\\medskip\\noindent", "Let $\\mathcal{A}$ be an additive category.", "A {\\it kernel-cokernel} pair is a pair $(i, p)$", "of morphisms of $\\mathcal{A}$ with", "$i : A \\to B$, $p : B \\to C$ such that $i$ is the kernel of", "$p$ and $p$ is the cokernel of $i$.", "Given a set $\\mathcal{E}$ of kernel-cokernel pairs we say", "$i : A \\to B$ is an {\\it admissible monomorphism}", "if $(i, p) \\in \\mathcal{E}$ for some morphism $p$.", "Similarly we say a morphism $p : B \\to C$ is an {\\it admissible epimorphism}", "if $(i, p) \\in \\mathcal{E}$ for some morphism $i$.", "The pair $(\\mathcal{A}, \\mathcal{E})$ is said to be an", "{\\it exact category} if the following axioms hold", "\\begin{enumerate}", "\\item $\\mathcal{E}$ is closed under isomorphisms of kernel-cokernel", "pairs,", "\\item for any object $A$ the morphism $1_A$ is both an admissible epimorphism", "and an admissible monomorphism,", "\\item admissible monomorphisms are stable under composition,", "\\item admissible epimorphisms are stable under composition,", "\\item the push-out of an admissible monomorphism $i : A \\to B$ via", "any morphism $A \\to A'$ exist and the induced morphism $i' : A' \\to B'$", "is an admissible monomorphism, and", "\\item the base change of an admissible epimorphism $p : B \\to C$ via", "any morphism $C' \\to C$ exist and the induced morphism $p' : B' \\to C'$", "is an admissible epimorphism.", "\\end{enumerate}", "Given such a structure let $\\mathcal{C} = (\\mathcal{A}, \\text{Cov})$", "where coverings (i.e., elements of $\\text{Cov}$) are given by", "admissible epimorphisms. The axioms listed above", "immediately imply that this is a site. Consider the functor", "$$", "F : \\mathcal{A} \\longrightarrow \\textit{Ab}(\\mathcal{C}), \\quad", "X \\longmapsto h_X", "$$", "exactly as in", "Lemma \\ref{lemma-embedding}.", "It turns out that this functor is fully faithful, exact, and reflects", "exactness. Moreover, any extension of objects in the essential image", "of $F$ is in the essential image of $F$." ], "refs": [ "injectives-lemma-site-abelian-category", "injectives-lemma-embedding", "injectives-lemma-embedding" ], "ref_ids": [ 7782, 7783, 7783 ] }, { "id": 7816, "type": "other", "label": "injectives-remark-existence-D", "categories": [ "injectives" ], "title": "injectives-remark-existence-D", "contents": [ "In the chapter on derived categories we consistently work with", "``small'' abelian categories (as is the convention in the Stacks", "project). For a ``big'' abelian category $\\mathcal{A}$ it isn't clear", "that the derived category $D(\\mathcal{A})$ exists because it isn't", "clear that morphisms in the derived category are sets. In general this", "isn't true, see", "Examples, Lemma \\ref{examples-lemma-big-abelian-category}.", "However, if $\\mathcal{A}$ is a Grothendieck abelian category, and given", "$K^\\bullet, L^\\bullet$ in $K(\\mathcal{A})$, then by", "Theorem \\ref{theorem-K-injective-embedding-grothendieck}", "there exists a quasi-isomorphism $L^\\bullet \\to I^\\bullet$ to a", "K-injective complex $I^\\bullet$ and", "Derived Categories, Lemma \\ref{derived-lemma-K-injective} shows that", "$$", "\\Hom_{D(\\mathcal{A})}(K^\\bullet, L^\\bullet) =", "\\Hom_{K(\\mathcal{A})}(K^\\bullet, I^\\bullet)", "$$", "which is a set. Some examples of Grothendieck abelian categories", "are the category of modules over a ring, or more generally", "the category of sheaves of modules on a ringed site." ], "refs": [ "examples-lemma-big-abelian-category", "injectives-theorem-K-injective-embedding-grothendieck", "derived-lemma-K-injective" ], "ref_ids": [ 2571, 7768, 1908 ] }, { "id": 7817, "type": "other", "label": "injectives-remark-direct-sum-product-derived", "categories": [ "injectives" ], "title": "injectives-remark-direct-sum-product-derived", "contents": [ "Let $R$ be a ring. Suppose that $M_n$, $n \\in \\mathbf{Z}$ are $R$-modules.", "Denote $E_n = M_n[-n] \\in D(R)$. We claim that $E = \\bigoplus M_n[-n]$ is", "{\\it both} the direct sum and the product of the objects $E_n$ in $D(R)$.", "To see that it is the direct sum, take a look at the proof of", "Lemma \\ref{lemma-derived-products}.", "To see that it is the direct product, take injective resolutions", "$M_n \\to I_n^\\bullet$. By the proof of", "Lemma \\ref{lemma-derived-products}", "we have", "$$", "\\prod E_n = \\prod I_n^\\bullet[-n]", "$$", "in $D(R)$. Since products in $\\text{Mod}_R$ are exact, we see that", "$\\prod I_n^\\bullet$ is quasi-isomorphic to $E$. This works more generally", "in $D(\\mathcal{A})$ where $\\mathcal{A}$ is a Grothendieck abelian", "category with Ab4*." ], "refs": [ "injectives-lemma-derived-products", "injectives-lemma-derived-products" ], "ref_ids": [ 7795, 7795 ] }, { "id": 7818, "type": "other", "label": "injectives-remark-ext-into-filtered-complex", "categories": [ "injectives" ], "title": "injectives-remark-ext-into-filtered-complex", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category.", "Let $K^\\bullet$ be a filtered complex of $\\mathcal{A}$, see", "Homology, Definition \\ref{homology-definition-filtered-complex}.", "For ease of notation denote $K$, $F^pK$, $\\text{gr}^pK$ the", "object of $D(\\mathcal{A})$ represented by $K^\\bullet$,", "$F^pK^\\bullet$, $\\text{gr}^pK^\\bullet$. Let $M \\in D(\\mathcal{A})$.", "Using Lemma \\ref{lemma-K-injective-embedding-filtration}", "we can construct a spectral sequence $(E_r, d_r)_{r \\geq 1}$", "of bigraded objects of $\\mathcal{A}$ with $d_r$ of bidgree", "$(r, -r + 1)$ and", "with", "$$", "E_1^{p, q} = \\Ext^{p + q}(M, \\text{gr}^pK)", "$$", "If for every $n$ we have", "$$", "\\Ext^n(M, F^pK) = 0 \\text{ for } p \\gg 0", "\\quad\\text{and}\\quad", "\\Ext^n(M, F^pK) = \\Ext^n(M, K) \\text{ for } p \\ll 0", "$$", "then the spectral sequence is bounded and converges to $\\Ext^{p + q}(M, K)$.", "Namely, choose any complex $M^\\bullet$ representing $M$, choose", "$j : K^\\bullet \\to J^\\bullet$ as in the lemma, and consider the complex", "$$", "\\Hom^\\bullet(M^\\bullet, I^\\bullet)", "$$", "defined exactly as in", "More on Algebra, Section \\ref{more-algebra-section-hom-complexes}.", "Setting $F^p\\Hom^\\bullet(M^\\bullet, I^\\bullet) =", "\\Hom^\\bullet(M^\\bullet, F^pI^\\bullet)$ we obtain a filtered complex.", "The spectral sequence of", "Homology, Section \\ref{homology-section-filtered-complex}", "has differentials and terms as described above; details omitted.", "The boundedness and convergence follows from", "Homology, Lemma \\ref{homology-lemma-ss-converges-trivial}." ], "refs": [ "homology-definition-filtered-complex", "injectives-lemma-K-injective-embedding-filtration", "homology-lemma-ss-converges-trivial" ], "ref_ids": [ 12177, 7797, 12103 ] }, { "id": 7819, "type": "other", "label": "injectives-remark-spectral-sequences-ext", "categories": [ "injectives" ], "title": "injectives-remark-spectral-sequences-ext", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category.", "Let $M, K$ be objects of $D(\\mathcal{A})$.", "For any choice of complex $K^\\bullet$ representing $K$ we", "can use the filtration $F^pK^\\bullet = \\tau_{\\leq -p}K^\\bullet$", "and the discussion in Remark \\ref{remark-ext-into-filtered-complex}", "to get a spectral sequence with", "$$", "E_1^{p, q} = \\Ext^{2p + q}(M, H^{-p}(K))", "$$", "This spectral sequence is independent of the choice of", "complex $K^\\bullet$ representing $K$. After renumbering", "$p = -j$ and $q = i + 2j$ we find a spectral sequence", "$(E'_r, d'_r)_{r \\geq 2}$ with $d'_r$ of bidegree $(r, -r + 1)$, with", "$$", "(E'_2)^{i, j} = \\Ext^i(M, H^j(K))", "$$", "If $M \\in D^-(\\mathcal{A})$ and $K \\in D^+(\\mathcal{A})$ then", "both $E_r$ and $E'_r$ are bounded and converge to $\\Ext^{p + q}(M, K)$.", "If we use the filtration $F^pK^\\bullet = \\sigma_{\\geq p}K^\\bullet$", "then we get", "$$", "E_1^{p, q} = \\Ext^q(M, K^p)", "$$", "If $M \\in D^-(\\mathcal{A})$ and $K^\\bullet$ is bounded below, then", "this spectral sequence is bounded and converges to $\\Ext^{p + q}(M, K)$." ], "refs": [ "injectives-remark-ext-into-filtered-complex" ], "ref_ids": [ 7818 ] }, { "id": 7820, "type": "other", "label": "injectives-remark-ext-from-filtered-complex", "categories": [ "injectives" ], "title": "injectives-remark-ext-from-filtered-complex", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let", "$K \\in D(\\mathcal{A})$. Let $M^\\bullet$ be a filtered complex of", "$\\mathcal{A}$, see Homology, Definition", "\\ref{homology-definition-filtered-complex}.", "For ease of notation denote $M$, $M/F^pM$, $\\text{gr}^pM$ the", "object of $D(\\mathcal{A})$ represented by $M^\\bullet$,", "$M^\\bullet/F^pM^\\bullet$, $\\text{gr}^pM^\\bullet$.", "Dually to Remark \\ref{remark-ext-into-filtered-complex}", "we can construct a spectral sequence $(E_r, d_r)_{r \\geq 1}$", "of bigraded objects of $\\mathcal{A}$ with $d_r$ of bidgree", "$(r, -r + 1)$ and", "with", "$$", "E_1^{p, q} = \\Ext^{p + q}(\\text{gr}^{-p}M, K)", "$$", "If for every $n$ we have", "$$", "\\Ext^n(M/F^pM, K) = 0 \\text{ for } p \\ll 0", "\\quad\\text{and}\\quad", "\\Ext^n(M/F^pM, K) = \\Ext^n(M, K) \\text{ for } p \\gg 0", "$$", "then the spectral sequence is bounded and converges to $\\Ext^{p + q}(M, K)$.", "Namely, choose a K-injective complex $I^\\bullet$ with injective terms", "representing $K$, see Theorem \\ref{theorem-K-injective-embedding-grothendieck}.", "Consider the complex", "$$", "\\Hom^\\bullet(M^\\bullet, I^\\bullet)", "$$", "defined exactly as in", "More on Algebra, Section \\ref{more-algebra-section-hom-complexes}.", "Setting", "$$", "F^p\\Hom^\\bullet(M^\\bullet, I^\\bullet) =", "\\Hom^\\bullet(M^\\bullet/F^{-p + 1}M^\\bullet, I^\\bullet)", "$$", "we obtain a filtered complex (note sign and shift in filtration).", "The spectral sequence of", "Homology, Section \\ref{homology-section-filtered-complex}", "has differentials and terms as described above; details omitted.", "The boundedness and convergence follows from", "Homology, Lemma \\ref{homology-lemma-ss-converges-trivial}." ], "refs": [ "homology-definition-filtered-complex", "injectives-remark-ext-into-filtered-complex", "injectives-theorem-K-injective-embedding-grothendieck", "homology-lemma-ss-converges-trivial" ], "ref_ids": [ 12177, 7818, 7768, 12103 ] }, { "id": 7821, "type": "other", "label": "injectives-remark-spectral-sequences-ext-variant", "categories": [ "injectives" ], "title": "injectives-remark-spectral-sequences-ext-variant", "contents": [ "Let $\\mathcal{A}$ be a Grothendieck abelian category.", "Let $M, K$ be objects of $D(\\mathcal{A})$.", "For any choice of complex $M^\\bullet$ representing $M$ we", "can use the filtration $F^pM^\\bullet = \\tau_{\\leq -p}M^\\bullet$", "and the discussion in Remark \\ref{remark-ext-into-filtered-complex}", "to get a spectral sequence with", "$$", "E_1^{p, q} = \\Ext^{2p + q}(H^p(M), K)", "$$", "This spectral sequence is independent of the choice of complex $M^\\bullet$", "representing $M$. After renumbering $p = -j$ and $q = i + 2j$ we find a", "spectral sequence $(E'_r, d'_r)_{r \\geq 2}$ with $d'_r$ of bidegree", "$(r, -r + 1)$, with", "$$", "(E'_2)^{i, j} = \\Ext^i(H^{-j}(M), K)", "$$", "If $M \\in D^-(\\mathcal{A})$ and $K \\in D^+(\\mathcal{A})$", "then $E_r$ and $E'_r$ are bounded and converge to $\\Ext^{p + q}(M, K)$.", "If we use the filtration $F^pM^\\bullet = \\sigma_{\\geq p}M^\\bullet$", "then we get", "$$", "E_1^{p, q} = \\Ext^q(M^{-p}, K)", "$$", "If $K \\in D^+(\\mathcal{A})$ and $M^\\bullet$ is bounded above, then", "this spectral sequence is bounded and converges to $\\Ext^{p + q}(M, K)$." ], "refs": [ "injectives-remark-ext-into-filtered-complex" ], "ref_ids": [ 7818 ] }, { "id": 8115, "type": "other", "label": "divisors-remark-base-change-relative-assassin", "categories": [ "divisors" ], "title": "divisors-remark-base-change-relative-assassin", "contents": [ "With notation and assumptions as in", "Lemma \\ref{lemma-base-change-relative-assassin}", "we see that it is always the case that", "$(g')^{-1}(\\text{Ass}_{X/S}(\\mathcal{F})) \\supset", "\\text{Ass}_{X'/S'}(\\mathcal{F}')$.", "If the morphism $S' \\to S$ is locally quasi-finite, then we actually have", "$$", "(g')^{-1}(\\text{Ass}_{X/S}(\\mathcal{F}))", "=", "\\text{Ass}_{X'/S'}(\\mathcal{F}')", "$$", "because in this case the field extensions $\\kappa(s) \\subset \\kappa(s')$", "are always finite. In fact, this holds more generally for any morphism", "$g : S' \\to S$ such that all the field extensions", "$\\kappa(s) \\subset \\kappa(s')$ are algebraic, because in this case all", "prime ideals of $\\kappa(s') \\otimes_{\\kappa(s)} \\kappa(x)$ are", "maximal (and minimal) primes, see", "Algebra, Lemma \\ref{algebra-lemma-integral-over-field}." ], "refs": [ "divisors-lemma-base-change-relative-assassin", "algebra-lemma-integral-over-field" ], "ref_ids": [ 7890, 497 ] }, { "id": 8116, "type": "other", "label": "divisors-remark-different-reflexive", "categories": [ "divisors" ], "title": "divisors-remark-different-reflexive", "contents": [ "If $X$ is a scheme of finite type over a field, then sometimes a different", "notion of reflexive modules is used (see for example", "\\cite[bottom of page 5 and Definition 1.1.9]{HL}).", "This other notion uses $R\\SheafHom$ into a dualizing complex", "$\\omega_X^\\bullet$ instead of into $\\mathcal{O}_X$ and", "should probably have a different name because it can be different", "when $X$ is not Gorenstein. For example, if", "$X = \\Spec(k[t^3, t^4, t^5])$, then a computation shows the dualizing", "sheaf $\\omega_X$ is not reflexive in our sense, but it is reflexive in the", "other sense as", "$\\omega_X \\to \\SheafHom(\\SheafHom(\\omega_X, \\omega_X), \\omega_X)$", "is an isomorphism." ], "refs": [], "ref_ids": [] }, { "id": 8117, "type": "other", "label": "divisors-remark-tensor", "categories": [ "divisors" ], "title": "divisors-remark-tensor", "contents": [ "Let $X$ be an integral locally Noetherian scheme. Thanks to", "Lemma \\ref{lemma-dual-reflexive} we know that the reflexive", "hull $\\mathcal{F}^{**}$ of a coherent $\\mathcal{O}_X$-module", "is coherent reflexive. Consider the category $\\mathcal{C}$", "of coherent reflexive $\\mathcal{O}_X$-modules. Taking", "reflexive hulls gives a left adjoint to the inclusion functor", "$\\mathcal{C} \\to \\textit{Coh}(\\mathcal{O}_X)$.", "Observe that $\\mathcal{C}$ is an additive category", "with kernels and cokernels. Namely, given", "$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ in $\\mathcal{C}$, the", "usual kernel $\\Ker(\\varphi)$ is reflexive", "(Lemma \\ref{lemma-sequence-reflexive}) and the reflexive hull", "$\\Coker(\\varphi)^{**}$ of the usual cokernel", "is the cokernel in $\\mathcal{C}$. Moreover $\\mathcal{C}$ inherits", "a tensor product", "$$", "\\mathcal{F} \\otimes_\\mathcal{C} \\mathcal{G} =", "(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G})^{**}", "$$", "which is associative and symmetric. There is an internal Hom", "in the sense that for any three objects", "$\\mathcal{F}, \\mathcal{G}, \\mathcal{H}$ of", "$\\mathcal{C}$ we have the identity", "$$", "\\Hom_\\mathcal{C}(\\mathcal{F} \\otimes_\\mathcal{C} \\mathcal{G}, \\mathcal{H}) =", "\\Hom_\\mathcal{C}(\\mathcal{F},", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{H}))", "$$", "see Modules, Lemma \\ref{modules-lemma-internal-hom}. In $\\mathcal{C}$", "every object $\\mathcal{F}$ has a {\\it dual object}", "$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X)$.", "Without further conditions on $X$ it can happen that", "$$", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) \\not \\cong", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X)", "\\otimes_\\mathcal{C} \\mathcal{G}", "\\quad\\text{and}\\quad", "\\mathcal{F} \\otimes_\\mathcal{C}", "\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X)", "\\not \\cong \\mathcal{O}_X", "$$", "for $\\mathcal{F}, \\mathcal{G}$ of rank $1$ in $\\mathcal{C}$.", "To make an example let $X = \\Spec(R)$ where $R$ is as in", "More on Algebra, Example \\ref{more-algebra-example-ring-not-S2}", "and let $\\mathcal{F}, \\mathcal{G}$ be the modules corresponding to $M$.", "Computation omitted." ], "refs": [ "divisors-lemma-dual-reflexive", "divisors-lemma-sequence-reflexive", "modules-lemma-internal-hom" ], "ref_ids": [ 7920, 7919, 13294 ] }, { "id": 8118, "type": "other", "label": "divisors-remark-ses-regular-section", "categories": [ "divisors" ], "title": "divisors-remark-ses-regular-section", "contents": [ "Let $X$ be a scheme, $\\mathcal{L}$ an invertible $\\mathcal{O}_X$-module,", "and $s$ a regular section of $\\mathcal{L}$. Then the zero scheme", "$D = Z(s)$ is an effective Cartier divisor on $X$ and there are", "short exact sequences", "$$", "0 \\to \\mathcal{O}_X \\to \\mathcal{L} \\to i_*(\\mathcal{L}|_D) \\to 0", "\\quad\\text{and}\\quad", "0 \\to \\mathcal{L}^{\\otimes -1} \\to \\mathcal{O}_X \\to i_*\\mathcal{O}_D \\to 0.", "$$", "Given an effective Cartier divisor $D \\subset X$ using", "Lemmas \\ref{lemma-characterize-OD} and", "\\ref{lemma-conormal-effective-Cartier-divisor}", "we get", "$$", "0 \\to \\mathcal{O}_X \\to \\mathcal{O}_X(D) \\to i_*(\\mathcal{N}_{D/X}) \\to 0", "\\quad\\text{and}\\quad", "0 \\to \\mathcal{O}_X(-D) \\to \\mathcal{O}_X \\to i_*(\\mathcal{O}_D) \\to 0", "$$" ], "refs": [ "divisors-lemma-characterize-OD", "divisors-lemma-conormal-effective-Cartier-divisor" ], "ref_ids": [ 7944, 7938 ] }, { "id": 8119, "type": "other", "label": "divisors-remark-affine-punctured-spectrum-standard-proof", "categories": [ "divisors" ], "title": "divisors-remark-affine-punctured-spectrum-standard-proof", "contents": [ "If $(A, \\mathfrak m)$ is a Noetherian local normal domain of", "dimension $\\geq 2$ and $U$", "is the punctured spectrum of $A$, then $\\Gamma(U, \\mathcal{O}_U) = A$.", "This algebraic version of Hartogs's theorem follows from the fact that", "$A = \\bigcap_{\\text{height}(\\mathfrak p) = 1} A_\\mathfrak p$", "we've seen in Algebra, Lemma", "\\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}.", "Thus in this case $U$ cannot be affine (since it would force $\\mathfrak m$", "to be a point of $U$). This is often used as the starting point of", "the proof of Lemma \\ref{lemma-affine-punctured-spec}.", "To reduce the case of a general Noetherian local ring to this case,", "we first complete (to get a Nagata local ring),", "then replace $A$ by $A/\\mathfrak q$ for a suitable minimal prime,", "and then normalize. Each of these steps does not change the", "dimension and we obtain a contradiction.", "You can skip the completion step, but then the normalization in", "general is not a Noetherian domain. However, it is still a", "Krull domain of the same dimension (this is proved using", "Krull-Akizuki) and one can apply the same argument." ], "refs": [ "algebra-lemma-normal-domain-intersection-localizations-height-1", "divisors-lemma-affine-punctured-spec" ], "ref_ids": [ 1313, 7958 ] }, { "id": 8120, "type": "other", "label": "divisors-remark-affine-puctured-spectrum-general", "categories": [ "divisors" ], "title": "divisors-remark-affine-puctured-spectrum-general", "contents": [ "It is not clear how to characterize the non-Noetherian local", "rings $(A, \\mathfrak m)$ whose punctured spectrum is affine.", "Such a ring has a finitely generated ideal $I$ with", "$\\mathfrak m = \\sqrt{I}$. Of course if we can take $I$", "generated by $1$ element, then $A$ has an affine puncture", "spectrum; this gives lots of non-Noetherian examples.", "Conversely, it follows from the argument in the proof of", "Lemma \\ref{lemma-affine-punctured-spec}", "that such a ring cannot possess a nonzerodivisor $f \\in \\mathfrak m$", "with $H^0_I(A/fA) = 0$ (so $A$ cannot have a regular sequence", "of length $2$). Moreover, the same holds for any ring $A'$ which is", "the target of a local homomorphism of local rings $A \\to A'$ such that", "$\\mathfrak m_{A'} = \\sqrt{\\mathfrak mA'}$." ], "refs": [ "divisors-lemma-affine-punctured-spec" ], "ref_ids": [ 7958 ] }, { "id": 8121, "type": "other", "label": "divisors-remark-not-always-extra-permanence", "categories": [ "divisors" ], "title": "divisors-remark-not-always-extra-permanence", "contents": [ "In the situation of", "Lemma \\ref{lemma-extra-permanence-regular-immersion-noetherian}", "parts (1), (2), (3) are {\\bf not} equivalent to", "``$j \\circ i$ and $j$ are regular immersions at $z$ and $y$''.", "An example is $X = \\mathbf{A}^1_k = \\Spec(k[x])$,", "$Y = \\Spec(k[x]/(x^2))$ and $Z = \\Spec(k[x]/(x))$." ], "refs": [ "divisors-lemma-extra-permanence-regular-immersion-noetherian" ], "ref_ids": [ 7996 ] }, { "id": 8122, "type": "other", "label": "divisors-remark-relative-regular-immersion-elements", "categories": [ "divisors" ], "title": "divisors-remark-relative-regular-immersion-elements", "contents": [ "The codimension of a relative quasi-regular immersion,", "if it is constant, does not change after a base change.", "In fact, if we have a ring map $A \\to B$ and a quasi-regular", "sequence $f_1, \\ldots, f_r \\in B$ such that $B/(f_1, \\ldots, f_r)$", "is flat over $A$, then for any ring map $A \\to A'$", "we have a quasi-regular sequence ", "$f_1 \\otimes 1, \\ldots, f_r \\otimes 1$ in $B' = B \\otimes_A A'$", "by More on Algebra, Lemma", "\\ref{more-algebra-lemma-relative-regular-immersion-algebra}", "(which was used in the proof of", "Lemma \\ref{lemma-relative-regular-immersion} above).", "Now the proof of", "Lemma \\ref{lemma-relative-regular-immersion-flat-in-neighbourhood}", "shows that if $A \\to B$ is flat and locally of finite", "presentation, then for every prime ideal $\\mathfrak q' \\subset B'$", "the sequence", "$f_1 \\otimes 1, \\ldots, f_r \\otimes 1$ is even a", "regular sequence in the local ring $B'_{\\mathfrak q'}$." ], "refs": [ "more-algebra-lemma-relative-regular-immersion-algebra", "divisors-lemma-relative-regular-immersion", "divisors-lemma-relative-regular-immersion-flat-in-neighbourhood" ], "ref_ids": [ 9992, 7999, 8001 ] }, { "id": 8123, "type": "other", "label": "divisors-remark-structure-sheaf-Xs", "categories": [ "divisors" ], "title": "divisors-remark-structure-sheaf-Xs", "contents": [ "Let $A$ be a Noetherian normal domain. Let $M$ be a rank $1$ finite reflexive", "$A$-module. Let $s \\in M$ be nonzero. Let $\\mathfrak p_1, \\ldots, \\mathfrak p_r$", "be the height $1$ primes of $A$ in the support of $M/As$.", "Then the open $U$ of Lemma \\ref{lemma-structure-sheaf-Xs} is", "$$", "U = \\Spec(A) \\setminus", "\\left(V(\\mathfrak p_1) \\cup \\ldots \\cup \\mathfrak p_r)\\right)", "$$", "by Lemma \\ref{lemma-Xs-codim-complement}. Moreover, if $M^{[n]}$", "denotes the reflexive hull of $M \\otimes_A \\ldots \\otimes_A M$", "($n$-factors), then", "$$", "\\Gamma(U, \\mathcal{O}_U) = \\colim M^{[n]}", "$$", "according to Lemma \\ref{lemma-structure-sheaf-Xs}." ], "refs": [ "divisors-lemma-structure-sheaf-Xs", "divisors-lemma-Xs-codim-complement", "divisors-lemma-structure-sheaf-Xs" ], "ref_ids": [ 8037, 8038, 8037 ] }, { "id": 8184, "type": "other", "label": "spaces-remark-list-properties-stable-base-change", "categories": [ "spaces" ], "title": "spaces-remark-list-properties-stable-base-change", "contents": [ "Here is a list of properties/types of morphisms", "which are {\\it stable under arbitrary base change}:", "\\begin{enumerate}", "\\item closed, open, and locally closed immersions, see", "Schemes, Lemma \\ref{schemes-lemma-base-change-immersion},", "\\item quasi-compact, see", "Schemes, Lemma \\ref{schemes-lemma-quasi-compact-preserved-base-change},", "\\item universally closed, see", "Schemes, Definition \\ref{schemes-definition-universally-closed},", "\\item (quasi-)separated, see", "Schemes, Lemma \\ref{schemes-lemma-separated-permanence},", "\\item monomorphism, see", "Schemes, Lemma \\ref{schemes-lemma-base-change-monomorphism}", "\\item surjective, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-surjective},", "\\item universally injective, see", "Morphisms, Lemma \\ref{morphisms-lemma-universally-injective},", "\\item affine, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-affine},", "\\item quasi-affine, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-quasi-affine},", "\\item (locally) of finite type, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-type},", "\\item (locally) quasi-finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-quasi-finite},", "\\item (locally) of finite presentation, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-presentation},", "\\item locally of finite type of relative dimension $d$, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-relative-dimension-d},", "\\item universally open, see", "Morphisms, Definition \\ref{morphisms-definition-open},", "\\item flat, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat},", "\\item syntomic, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-syntomic},", "\\item smooth, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-smooth},", "\\item unramified (resp.\\ G-unramified), see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-unramified},", "\\item \\'etale, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-etale},", "\\item proper, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-proper},", "\\item H-projective, see", "Morphisms, Lemma \\ref{morphisms-lemma-H-projective-base-change},", "\\item (locally) projective, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-projective},", "\\item finite or integral, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite},", "\\item finite locally free, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-locally-free},", "\\item universally submersive, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-universally-submersive},", "\\item universal homeomorphism, see", "Morphisms, Lemma \\ref{morphisms-lemma-base-change-universal-homeomorphism}.", "\\end{enumerate}", "Add more as needed." ], "refs": [ "schemes-lemma-base-change-immersion", "schemes-lemma-quasi-compact-preserved-base-change", "schemes-definition-universally-closed", "schemes-lemma-separated-permanence", "schemes-lemma-base-change-monomorphism", "morphisms-lemma-base-change-surjective", "morphisms-lemma-universally-injective", "morphisms-lemma-base-change-affine", "morphisms-lemma-base-change-quasi-affine", "morphisms-lemma-base-change-finite-type", "morphisms-lemma-base-change-quasi-finite", "morphisms-lemma-base-change-finite-presentation", "morphisms-lemma-base-change-relative-dimension-d", "morphisms-definition-open", "morphisms-lemma-base-change-flat", "morphisms-lemma-base-change-syntomic", "morphisms-lemma-base-change-smooth", "morphisms-lemma-base-change-unramified", "morphisms-lemma-base-change-etale", "morphisms-lemma-base-change-proper", "morphisms-lemma-H-projective-base-change", "morphisms-lemma-base-change-projective", "morphisms-lemma-base-change-finite", "morphisms-lemma-base-change-finite-locally-free", "morphisms-lemma-base-change-universally-submersive", "morphisms-lemma-base-change-universal-homeomorphism" ], "ref_ids": [ 7695, 7698, 7754, 7714, 7724, 5165, 5167, 5176, 5187, 5200, 5233, 5240, 5284, 5555, 5265, 5291, 5327, 5346, 5361, 5409, 5425, 5426, 5440, 5473, 5257, 5451 ] }, { "id": 8185, "type": "other", "label": "spaces-remark-list-properties-stable-composition", "categories": [ "spaces" ], "title": "spaces-remark-list-properties-stable-composition", "contents": [ "Of the properties of morphisms which are stable under base change", "(as listed in", "Remark \\ref{remark-list-properties-stable-base-change})", "the following are also {\\it stable under compositions}:", "\\begin{enumerate}", "\\item closed, open and locally closed immersions, see", "Schemes, Lemma \\ref{schemes-lemma-composition-immersion},", "\\item quasi-compact, see", "Schemes, Lemma \\ref{schemes-lemma-composition-quasi-compact},", "\\item universally closed, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-proper},", "\\item (quasi-)separated, see", "Schemes, Lemma \\ref{schemes-lemma-separated-permanence},", "\\item monomorphism, see", "Schemes, Lemma \\ref{schemes-lemma-composition-monomorphism},", "\\item surjective, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-surjective},", "\\item universally injective, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-universally-injective},", "\\item affine, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-affine},", "\\item quasi-affine, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-quasi-affine},", "\\item (locally) of finite type, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-type},", "\\item (locally) quasi-finite, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-quasi-finite},", "\\item (locally) of finite presentation, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-presentation},", "\\item universally open, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-open},", "\\item flat, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-flat},", "\\item syntomic, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-syntomic},", "\\item smooth, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-smooth},", "\\item unramified (resp.\\ G-unramified), see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-unramified},", "\\item \\'etale, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-etale},", "\\item proper, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-proper},", "\\item H-projective, see", "Morphisms, Lemma \\ref{morphisms-lemma-H-projective-composition},", "\\item finite or integral, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-finite},", "\\item finite locally free, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-locally-free},", "\\item universally submersive, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-universally-submersive},", "\\item universal homeomorphism, see", "Morphisms, Lemma \\ref{morphisms-lemma-composition-universal-homeomorphism}.", "\\end{enumerate}", "Add more as needed." ], "refs": [ "spaces-remark-list-properties-stable-base-change", "schemes-lemma-composition-immersion", "schemes-lemma-composition-quasi-compact", "morphisms-lemma-composition-proper", "schemes-lemma-separated-permanence", "schemes-lemma-composition-monomorphism", "morphisms-lemma-composition-surjective", "morphisms-lemma-composition-universally-injective", "morphisms-lemma-composition-affine", "morphisms-lemma-composition-quasi-affine", "morphisms-lemma-composition-finite-type", "morphisms-lemma-composition-quasi-finite", "morphisms-lemma-composition-finite-presentation", "morphisms-lemma-composition-open", "morphisms-lemma-composition-flat", "morphisms-lemma-composition-syntomic", "morphisms-lemma-composition-smooth", "morphisms-lemma-composition-unramified", "morphisms-lemma-composition-etale", "morphisms-lemma-composition-proper", "morphisms-lemma-H-projective-composition", "morphisms-lemma-composition-finite", "morphisms-lemma-composition-finite-locally-free", "morphisms-lemma-composition-universally-submersive", "morphisms-lemma-composition-universal-homeomorphism" ], "ref_ids": [ 8184, 7732, 7699, 5408, 7714, 7723, 5163, 5170, 5175, 5186, 5199, 5232, 5239, 5253, 5263, 5290, 5326, 5345, 5360, 5408, 5424, 5439, 5472, 5258, 5452 ] }, { "id": 8186, "type": "other", "label": "spaces-remark-list-properties-fpqc-local-base", "categories": [ "spaces" ], "title": "spaces-remark-list-properties-fpqc-local-base", "contents": [ "Of the properties mentioned which are stable under base change", "(as listed in Remark \\ref{remark-list-properties-stable-base-change})", "the following are also {\\it fpqc local on the base}", "(and a fortiori fppf local on the base):", "\\begin{enumerate}", "\\item for immersions we have this for", "\\begin{enumerate}", "\\item closed immersions, see", "Descent, Lemma \\ref{descent-lemma-descending-property-closed-immersion},", "\\item open immersions, see", "Descent, Lemma \\ref{descent-lemma-descending-property-open-immersion}, and", "\\item quasi-compact immersions, see", "Descent,", "Lemma \\ref{descent-lemma-descending-property-quasi-compact-immersion},", "\\end{enumerate}", "\\item quasi-compact, see", "Descent, Lemma \\ref{descent-lemma-descending-property-quasi-compact},", "\\item universally closed, see", "Descent, Lemma", "\\ref{descent-lemma-descending-property-universally-closed},", "\\item (quasi-)separated, see", "Descent, Lemmas", "\\ref{descent-lemma-descending-property-quasi-separated}, and", "\\ref{descent-lemma-descending-property-separated},", "\\item monomorphism, see", "Descent, Lemma \\ref{descent-lemma-descending-property-monomorphism},", "\\item surjective, see", "Descent, Lemma \\ref{descent-lemma-descending-property-surjective},", "\\item universally injective, see", "Descent, Lemma \\ref{descent-lemma-descending-property-universally-injective},", "\\item affine, see", "Descent, Lemma \\ref{descent-lemma-descending-property-affine},", "\\item quasi-affine, see", "Descent, Lemma \\ref{descent-lemma-descending-property-quasi-affine},", "\\item (locally) of finite type, see", "Descent,", "Lemmas \\ref{descent-lemma-descending-property-locally-finite-type}, and", "\\ref{descent-lemma-descending-property-finite-type},", "\\item (locally) quasi-finite, see", "Descent, Lemma \\ref{descent-lemma-descending-property-quasi-finite},", "\\item (locally) of finite presentation, see", "Descent, Lemmas", "\\ref{descent-lemma-descending-property-locally-finite-presentation}, and", "\\ref{descent-lemma-descending-property-finite-presentation},", "\\item locally of finite type of relative dimension $d$, see", "Descent,", "Lemma \\ref{descent-lemma-descending-property-relative-dimension-d},", "\\item universally open, see", "Descent, Lemma \\ref{descent-lemma-descending-property-universally-open},", "\\item flat, see", "Descent, Lemma \\ref{descent-lemma-descending-property-flat},", "\\item syntomic, see", "Descent, Lemma \\ref{descent-lemma-descending-property-syntomic},", "\\item smooth, see", "Descent, Lemma \\ref{descent-lemma-descending-property-smooth},", "\\item unramified (resp.\\ G-unramified), see", "Descent, Lemma \\ref{descent-lemma-descending-property-unramified},", "\\item \\'etale, see", "Descent, Lemma \\ref{descent-lemma-descending-property-etale},", "\\item proper, see", "Descent, Lemma \\ref{descent-lemma-descending-property-proper},", "\\item finite or integral, see", "Descent, Lemma \\ref{descent-lemma-descending-property-finite},", "\\item finite locally free, see", "Descent, Lemma \\ref{descent-lemma-descending-property-finite-locally-free},", "\\item universally submersive, see", "Descent, Lemma \\ref{descent-lemma-descending-property-universally-submersive},", "\\item universal homeomorphism, see", "Descent, Lemma \\ref{descent-lemma-descending-property-universal-homeomorphism}.", "\\end{enumerate}", "Note that the property of being an ``immersion'' may not be fpqc local", "on the base, but in", "Descent, Lemma \\ref{descent-lemma-descending-fppf-property-immersion}", "we proved that it is fppf local on the base." ], "refs": [ "spaces-remark-list-properties-stable-base-change", "descent-lemma-descending-property-closed-immersion", "descent-lemma-descending-property-open-immersion", "descent-lemma-descending-property-quasi-compact-immersion", "descent-lemma-descending-property-quasi-compact", "descent-lemma-descending-property-universally-closed", "descent-lemma-descending-property-quasi-separated", "descent-lemma-descending-property-separated", "descent-lemma-descending-property-monomorphism", "descent-lemma-descending-property-surjective", "descent-lemma-descending-property-universally-injective", "descent-lemma-descending-property-affine", "descent-lemma-descending-property-quasi-affine", "descent-lemma-descending-property-locally-finite-type", "descent-lemma-descending-property-finite-type", "descent-lemma-descending-property-quasi-finite", "descent-lemma-descending-property-locally-finite-presentation", "descent-lemma-descending-property-finite-presentation", "descent-lemma-descending-property-relative-dimension-d", "descent-lemma-descending-property-universally-open", "descent-lemma-descending-property-flat", "descent-lemma-descending-property-syntomic", "descent-lemma-descending-property-smooth", "descent-lemma-descending-property-unramified", "descent-lemma-descending-property-etale", "descent-lemma-descending-property-proper", "descent-lemma-descending-property-finite", "descent-lemma-descending-property-finite-locally-free", "descent-lemma-descending-property-universally-submersive", "descent-lemma-descending-property-universal-homeomorphism", "descent-lemma-descending-fppf-property-immersion" ], "ref_ids": [ 8184, 14684, 14681, 14686, 14666, 14668, 14667, 14671, 14696, 14672, 14673, 14683, 14685, 14675, 14677, 14689, 14676, 14678, 14690, 14669, 14680, 14691, 14692, 14693, 14694, 14679, 14688, 14695, 14670, 14674, 14698 ] }, { "id": 8187, "type": "other", "label": "spaces-remark-warning", "categories": [ "spaces" ], "title": "spaces-remark-warning", "contents": [ "Consider the property $\\mathcal{P}=$``surjective''.", "In this case there could be some ambiguity if we say", "``let $F \\to G$ be a surjective map''.", "Namely, we could mean the notion defined", "in Definition \\ref{definition-relative-representable-property}", "above, or we could mean a surjective map of presheaves, see", "Sites, Definition \\ref{sites-definition-presheaves-injective-surjective},", "or, if both $F$ and $G$ are sheaves,", "we could mean a surjective map of sheaves, see", "Sites, Definition \\ref{sites-definition-sheaves-injective-surjective},", "If not mentioned otherwise when discussing morphisms of algebraic spaces", "we will always mean the first. See", "Lemma \\ref{lemma-surjective-flat-locally-finite-presentation}", "for a case where surjectivity implies surjectivity as a map of sheaves." ], "refs": [ "spaces-definition-relative-representable-property", "sites-definition-presheaves-injective-surjective", "sites-definition-sheaves-injective-surjective", "spaces-lemma-surjective-flat-locally-finite-presentation" ], "ref_ids": [ 8173, 8648, 8660, 8137 ] }, { "id": 8381, "type": "other", "label": "topology-remark-quasi-components", "categories": [ "topology" ], "title": "topology-remark-quasi-components", "contents": [ "\\begin{reference}", "\\cite[Example 6.1.24]{Engelking}", "\\end{reference}", "Let $X$ be a topological space and $x \\in X$. Let $Z \\subset X$ be the", "connected component of $X$ passing through $x$. Consider the intersection", "$E$ of all open and closed subsets of $X$ containing $x$. It is clear that", "$Z \\subset E$. In general $Z \\not = E$. For example, let", "$X = \\{x, y, z_1, z_2, \\ldots\\}$ with the topology with the following", "basis of opens, $\\{z_n\\}$, $\\{x, z_n, z_{n + 1}, \\ldots\\}$, and", "$\\{y, z_n, z_{n + 1}, \\ldots\\}$ for all $n$. Then $Z = \\{x\\}$ and", "$E = \\{x, y\\}$. We omit the details." ], "refs": [], "ref_ids": [] }, { "id": 8382, "type": "other", "label": "topology-remark-lemma-literature", "categories": [ "topology" ], "title": "topology-remark-lemma-literature", "contents": [ "Lemma \\ref{lemma-characterize-quasi-compact} is a combination of", "\\cite[I, p. 75, Lemme 1]{Bourbaki} and", "\\cite[I, p. 76, Corollaire 1]{Bourbaki}." ], "refs": [ "topology-lemma-characterize-quasi-compact" ], "ref_ids": [ 8273 ] }, { "id": 8383, "type": "other", "label": "topology-remark-proof-literature", "categories": [ "topology" ], "title": "topology-remark-proof-literature", "contents": [ "Here are some references to the literature.", "In \\cite[I, p. 75, Theorem 1]{Bourbaki} you can find:", "(2) $\\Leftrightarrow$ (4).", "In \\cite[I, p. 77, Proposition 6]{Bourbaki} you can find:", "(2) $\\Rightarrow$ (1).", "Of course, trivially we have (1) $\\Rightarrow$ (4).", "Thus (1), (2) and (4) are equivalent.", "Fan Zhou claimed and proved that (3) and (4) are equivalent;", "let me know if you find a reference in the literature." ], "refs": [], "ref_ids": [] }, { "id": 8384, "type": "other", "label": "topology-remark-obstruction-to-dimension-function", "categories": [ "topology" ], "title": "topology-remark-obstruction-to-dimension-function", "contents": [ "Combining Lemmas \\ref{lemma-dimension-function-unique} and", "\\ref{lemma-locally-dimension-function} we see that on a catenary,", "locally Noetherian, sober topological space the obstruction to", "having a dimension function is an element of", "$H^1(X, \\mathbf{Z})$." ], "refs": [ "topology-lemma-dimension-function-unique", "topology-lemma-locally-dimension-function" ], "ref_ids": [ 8292, 8293 ] }, { "id": 8385, "type": "other", "label": "topology-remark-size-projective-cover", "categories": [ "topology" ], "title": "topology-remark-size-projective-cover", "contents": [ "Let $X$ be a quasi-compact Hausdorff space. Let $\\kappa$ be an infinite", "cardinal bigger or equal than the cardinality of $X$. Then the cardinality", "of the minimal quasi-compact, Hausdorff, extremally disconnected cover", "$X' \\to X$ (Lemma \\ref{lemma-existence-projective-cover})", "is at most $2^{2^\\kappa}$. Namely, choose a subset $S \\subset X'$", "mapping bijectively to $X$. By minimality of $X'$ the set $S$ is dense", "in $X'$. Thus $|X'| \\leq 2^{2^\\kappa}$ by Lemma \\ref{lemma-dense-image}." ], "refs": [ "topology-lemma-existence-projective-cover", "topology-lemma-dense-image" ], "ref_ids": [ 8332, 8325 ] }, { "id": 8386, "type": "other", "label": "topology-remark-locally-finite-stratification", "categories": [ "topology" ], "title": "topology-remark-locally-finite-stratification", "contents": [ "Given a locally finite stratification $X = \\coprod X_i$ of a", "topological space $X$, we obtain a family of closed subsets", "$Z_i = \\bigcup_{j \\leq i} X_j$ of $X$ indexed by $I$ such that", "$$", "Z_i \\cap Z_j = \\bigcup\\nolimits_{k \\leq i, j} Z_k", "$$", "Conversely, given closed subsets $Z_i \\subset X$ indexed by a", "partially ordered set $I$ such that $X = \\bigcup Z_i$, such that every point", "has a neighbourhood meeting only finitely many $Z_i$, and such that", "the displayed formula holds, then we obtain a locally finite", "stratification of $X$ by setting $X_i = Z_i \\setminus \\bigcup_{j < i} Z_j$." ], "refs": [], "ref_ids": [] }, { "id": 8427, "type": "other", "label": "hypercovering-remark-hypercoverings-really-set", "categories": [ "hypercovering" ], "title": "hypercovering-remark-hypercoverings-really-set", "contents": [ "The lemma does not just say that there is a cofinal", "system of choices of hypercoverings that is a set,", "but that really the hypercoverings form a set." ], "refs": [], "ref_ids": [] }, { "id": 8428, "type": "other", "label": "hypercovering-remark-P-covering", "categories": [ "hypercovering" ], "title": "hypercovering-remark-P-covering", "contents": [ "A useful special case of Lemmas \\ref{lemma-add-simplices} and", "\\ref{lemma-degeneracy-maps-coverings} is the following.", "Suppose we have a category $\\mathcal{C}$ having fibre products.", "Let $P \\subset \\text{Arrows}(\\mathcal{C})$ be a subset", "stable under base change, stable under composition,", "and containing all isomorphisms. Then one says a", "{\\it $P$-hypercovering} is an augmentation $a : U \\to X$", "from a simplicial object of $\\mathcal{C}$ such that", "\\begin{enumerate}", "\\item $U_0 \\to X$ is in $P$,", "\\item $U_1 \\to U_0 \\times_X U_0$ is in $P$,", "\\item $U_{n + 1} \\to (\\text{cosk}_n\\text{sk}_n U)_{n + 1}$", "is in $P$ for $n \\geq 1$.", "\\end{enumerate}", "The category $\\mathcal{C}/X$ has all finite limits, hence the", "coskeleta used in the formulation above exist", "(see Categories, Lemma \\ref{categories-lemma-finite-limits-exist}).", "Then we claim that the morphisms $U_n \\to X$ and $d^n_i : U_n \\to U_{n - 1}$", "are in $P$. This follows from the aforementioned", "lemmas by turning $\\mathcal{C}$ into a site whose coverings", "are $\\{f : V \\to U\\}$ with $f \\in P$ and taking $K$ given by", "$K_n = \\{U_n \\to X\\}$." ], "refs": [ "hypercovering-lemma-add-simplices", "hypercovering-lemma-degeneracy-maps-coverings", "categories-lemma-finite-limits-exist" ], "ref_ids": [ 8408, 8409, 12224 ] }, { "id": 8429, "type": "other", "label": "hypercovering-remark-contractible-category", "categories": [ "hypercovering" ], "title": "hypercovering-remark-contractible-category", "contents": [ "Note that the crux of the proof is to use", "Lemma \\ref{lemma-add-simplices}. This lemma", "is completely general and does not care about the", "exact shape of the simplicial sets (as long as they", "have only finitely many nondegenerate simplices).", "It seems altogether reasonable to expect a result", "of the following kind:", "Given any morphism $a : K \\times \\partial \\Delta[k]", "\\to L$, with $K$ and $L$ hypercoverings, there", "exists a morphism of hypercoverings $c : K' \\to K$", "and a morphism $g : K' \\times \\Delta[k] \\to L$", "such that", "$g|_{K' \\times \\partial \\Delta[k]} =", "a \\circ (c \\times \\text{id}_{\\partial \\Delta[k]})$.", "In other words, the category of hypercoverings is in", "a suitable sense contractible." ], "refs": [ "hypercovering-lemma-add-simplices" ], "ref_ids": [ 8408 ] }, { "id": 8430, "type": "other", "label": "hypercovering-remark-not-covering-set", "categories": [ "hypercovering" ], "title": "hypercovering-remark-not-covering-set", "contents": [ "One feature of this description is that if one of the multiple", "intersections $U_{i_0} \\cap \\ldots \\cap U_{i_{n + 1}}$ is empty then", "the covering on the right hand side may be the empty covering.", "Thus it is not automatically the case that the maps", "$I_{n + 1} \\to (\\text{cosk}_n\\text{sk}_n I)_{n + 1}$ are surjective.", "This means that the geometric realization of $I$ may be an interesting", "(non-contractible) space.", "\\medskip\\noindent", "In fact, let $I'_n \\subset I_n$ be the subset", "consisting of those simplices $i \\in I_n$ such that", "$U_i \\not = \\emptyset$. It is easy to see that $I' \\subset I$", "is a subsimplicial set, and that $(I', \\{U_i\\})$ is a hypercovering.", "Hence we can always refine a hypercovering to a hypercovering where", "none of the opens $U_i$ is empty." ], "refs": [], "ref_ids": [] }, { "id": 8431, "type": "other", "label": "hypercovering-remark-repackage-into-simplicial-space", "categories": [ "hypercovering" ], "title": "hypercovering-remark-repackage-into-simplicial-space", "contents": [ "Let us repackage this information in yet another way.", "Namely, suppose that $(I, \\{U_i\\})$ is a hypercovering of", "the topological space $X$. Given this data we can construct", "a simplicial topological space $U_\\bullet$ by setting", "$$", "U_n = \\coprod\\nolimits_{i \\in I_n} U_i,", "$$", "and where for given $\\varphi : [n] \\to [m]$ we let", "morphisms $U(\\varphi) : U_n \\to U_m$ be the morphism", "coming from the inclusions $U_i \\subset U_{\\varphi(i)}$", "for $i \\in I_n$. This simplicial topological space comes", "with an augmentation $\\epsilon : U_\\bullet \\to X$.", "With this morphism the simplicial space $U_\\bullet$ becomes", "a hypercovering of $X$ along which one has cohomological descent", "in the sense of \\cite[Expos\\'e Vbis]{SGA4}.", "In other words, $H^n(U_\\bullet, \\epsilon^*\\mathcal{F}) = H^n(X, \\mathcal{F})$.", "(Insert future reference here to cohomology over simplicial", "spaces and cohomological descent formulated in those terms.)", "Suppose that $\\mathcal{F}$ is an abelian sheaf on $X$.", "In this case the spectral sequence of Lemma \\ref{lemma-cech-spectral-sequence}", "becomes the spectral sequence with $E_1$-term", "$$", "E_1^{p, q} = H^q(U_p, \\epsilon_q^*\\mathcal{F})", "\\Rightarrow", "H^{p + q}(U_\\bullet, \\epsilon^*\\mathcal{F}) = H^{p + q}(X, \\mathcal{F})", "$$", "comparing the total cohomology of $\\epsilon^*\\mathcal{F}$", "to the cohomology groups of $\\mathcal{F}$ over the pieces", "of $U_\\bullet$. (Insert future reference to this spectral sequence", "here.)" ], "refs": [ "hypercovering-lemma-cech-spectral-sequence" ], "ref_ids": [ 8398 ] }, { "id": 8432, "type": "other", "label": "hypercovering-remark-taking-disjoint-unions", "categories": [ "hypercovering" ], "title": "hypercovering-remark-taking-disjoint-unions", "contents": [ "Let $\\mathcal{C}$ be a site. Let", "$K$ and $L$ be objects of $\\text{SR}(\\mathcal{C})$.", "Write $K = \\{U_i\\}_{i \\in I}$ and $L = \\{V_j\\}_{j \\in J}$.", "Assume $U = \\coprod_{i \\in I} U_i$ and $V = \\coprod_{j \\in J} V_j$", "exist. Then we get", "$$", "\\Mor_{\\text{SR}(\\mathcal{C})}(K, L) \\longrightarrow \\Mor_\\mathcal{C}(U, V)", "$$", "as follows. Given $f : K \\to L$ given by $\\alpha : I \\to J$", "and $f_i : U_i \\to V_{\\alpha(i)}$ we obtain a transformation of functors", "$$", "\\Mor_\\mathcal{C}(V, -) =", "\\prod\\nolimits_{j \\in J} \\Mor_\\mathcal{C}(V_j, -)", "\\to", "\\prod\\nolimits_{i \\in I} \\Mor_\\mathcal{C}(U_i, -) =", "\\Mor_\\mathcal{C}(U, -)", "$$", "sending $(g_j)_{j \\in J}$ to", "$(g_{\\alpha(i)} \\circ f_i)_{i \\in I}$. Hence the Yoneda lemma", "produces the corresponding map $U \\to V$. Of course, $U \\to V$", "maps the summand $U_i$ into the summand $V_{\\alpha(i)}$ via", "the morphism $f_i$." ], "refs": [], "ref_ids": [] }, { "id": 8433, "type": "other", "label": "hypercovering-remark-take-unions-hypercovering", "categories": [ "hypercovering" ], "title": "hypercovering-remark-take-unions-hypercovering", "contents": [ "Let $\\mathcal{C}$ be a site. Assume $\\mathcal{C}$ has", "fibre products and equalizers and let $K$ be a hypercovering.", "Write $K_n = \\{U_{n, i}\\}_{i \\in I_n}$. Suppose that", "\\begin{enumerate}", "\\item[(a)] $U_n = \\coprod_{i \\in I_n} U_{n, i}$ exists, and", "\\item[(b)] $\\coprod_{i \\in I_n} h_{U_{n, i}} \\to h_{U_n}$ induces", "an isomorphism on sheafifications.", "\\end{enumerate}", "Then we get another simplicial object $L$ of $\\text{SR}(\\mathcal{C})$", "with $L_n = \\{U_n\\}$, see", "Remark \\ref{remark-taking-disjoint-unions}.", "Now we claim that $L$ is a hypercovering.", "To see this we check conditions (1), (2), (3) of", "Definition \\ref{definition-hypercovering-variant}.", "Condition (1) follows from (b) and (1) for $K$.", "Condition (2) follows in exactly the same way.", "Condition (3) follows because", "\\begin{align*}", "F((\\text{cosk}_n \\text{sk}_n L)_{n + 1})^\\#", "& =", "((\\text{cosk}_n \\text{sk}_n F(L)^\\#)_{n + 1}) \\\\", "& =", "((\\text{cosk}_n \\text{sk}_n F(K)^\\#)_{n + 1}) \\\\", "& =", "F((\\text{cosk}_n \\text{sk}_n K)_{n + 1})^\\#", "\\end{align*}", "for $n \\geq 1$ and hence the condition for $K$ implies the condition for", "$L$ exactly as in (1) and (2).", "Note that $F$ commutes with connected limits and sheafification is exact", "proving the first and last equality; the middle equality follows as", "$F(K)^\\# = F(L)^\\#$ by (b)." ], "refs": [ "hypercovering-remark-taking-disjoint-unions", "hypercovering-definition-hypercovering-variant" ], "ref_ids": [ 8432, 8426 ] }, { "id": 8434, "type": "other", "label": "hypercovering-remark-take-unions-hypercovering-X", "categories": [ "hypercovering" ], "title": "hypercovering-remark-take-unions-hypercovering-X", "contents": [ "Let $\\mathcal{C}$ be a site. Let $X \\in \\Ob(\\mathcal{C})$.", "Assume $\\mathcal{C}$ has fibre products and let $K$ be a hypercovering of $X$.", "Write $K_n = \\{U_{n, i}\\}_{i \\in I_n}$. Suppose that", "\\begin{enumerate}", "\\item[(a)] $U_n = \\coprod_{i \\in I_n} U_{n, i}$ exists,", "\\item[(b)] given morphisms", "$(\\alpha, f_i) : \\{U_i\\}_{i \\in I} \\to \\{V_j\\}_{j \\in J}$ and", "$(\\beta, g_k) : \\{W_k\\}_{k \\in K} \\to \\{V_j\\}_{j \\in J}$", "in $\\text{SR}(\\mathcal{C})$ such that", "$U = \\coprod U_i$, $V = \\coprod V_j$, and $W = \\coprod W_j$", "exist, then $U \\times_V W =", "\\coprod_{(i, j, k), \\alpha(i) = j = \\beta(k)} U_i \\times_{V_j} W_k$,", "\\item[(c)] if $(\\alpha, f_i) : \\{U_i\\}_{i \\in I} \\to \\{V_j\\}_{j \\in J}$", "is a covering in the sense of", "Definition \\ref{definition-covering-SR}", "and $U = \\coprod U_i$ and $V = \\coprod V_j$ exist,", "then the corresponding morphism $U \\to V$", "of Remark \\ref{remark-taking-disjoint-unions}", "is a covering of $\\mathcal{C}$.", "\\end{enumerate}", "Then we get another simplicial object $L$ of $\\text{SR}(\\mathcal{C})$", "with $L_n = \\{U_n\\}$, see", "Remark \\ref{remark-taking-disjoint-unions}.", "Now we claim that $L$ is a hypercovering of $X$.", "To see this we check conditions (1), (2) of", "Definition \\ref{definition-hypercovering}.", "Condition (1) follows from (c) and (1) for $K$", "because (1) for $K$ says $K_0 = \\{U_{0, i}\\}_{i \\in I_0}$", "is a covering of $\\{X\\}$ in the sense of", "Definition \\ref{definition-covering-SR}.", "Condition (2) follows because $\\mathcal{C}/X$ has", "all finite limits hence $\\text{SR}(\\mathcal{C}/X)$", "has all finite limits, and condition (b) says the", "construction of ``taking disjoint unions'' commutes", "with these fimite limits. Thus the morphism", "$$", "L_{n + 1} \\longrightarrow (\\text{cosk}_n \\text{sk}_n L)_{n + 1}", "$$", "is a covering as it is the consequence of applying our", "``taking disjoint unions'' functor to the morphism", "$$", "K_{n + 1} \\longrightarrow (\\text{cosk}_n \\text{sk}_n K)_{n + 1}", "$$", "which is assumed to be a covering in the sense of", "Definition \\ref{definition-covering-SR} by (2) for $K$.", "This makes sense because property (b) in particular assures", "us that if we start with a finite diagram of", "semi-representable objects over $X$", "for which we can take disjoint unions, then", "the limit of the diagram in $\\text{SR}(\\mathcal{C}/X)$", "still is a semi-representable object over $X$ for which", "we can take disjoint unions." ], "refs": [ "hypercovering-definition-covering-SR", "hypercovering-remark-taking-disjoint-unions", "hypercovering-remark-taking-disjoint-unions", "hypercovering-definition-hypercovering", "hypercovering-definition-covering-SR", "hypercovering-definition-covering-SR" ], "ref_ids": [ 8423, 8432, 8432, 8424, 8423, 8423 ] }, { "id": 8491, "type": "other", "label": "algebraic-remark-flat-fp-presentation", "categories": [ "algebraic" ], "title": "algebraic-remark-flat-fp-presentation", "contents": [ "If the morphism $f : \\mathcal{S}_U \\to \\mathcal{X}$ of", "Lemma \\ref{lemma-stack-presentation}", "is only assumed surjective, flat and locally of finite presentation, then", "it will still be the case that $f_{can} : [U/R] \\to \\mathcal{X}$ is an", "equivalence. In this case the morphisms $s$, $t$ will be flat and", "locally of finite presentation, but of course not smooth in general." ], "refs": [ "algebraic-lemma-stack-presentation" ], "ref_ids": [ 8474 ] }, { "id": 8701, "type": "other", "label": "sites-remark-big-presheaves", "categories": [ "sites" ], "title": "sites-remark-big-presheaves", "contents": [ "As already pointed out we may consider the category of", "presheaves with values in any of the ``big'' categories", "listed in Categories, Remark \\ref{categories-remark-big-categories}.", "These will be ``big'' categories as well and they will be", "listed in the above mentioned remark as we go along." ], "refs": [ "categories-remark-big-categories" ], "ref_ids": [ 12410 ] }, { "id": 8702, "type": "other", "label": "sites-remark-functoriality-presheaves-values", "categories": [ "sites" ], "title": "sites-remark-functoriality-presheaves-values", "contents": [ "Suppose that $\\mathcal{A}$ is a category such that", "any diagram $\\mathcal{I}_Y \\to \\mathcal{A}$ has a", "colimit in $\\mathcal{A}$. In this case it is clear", "that there are functors $u^p$ and $u_p$, defined in", "exactly the same way as above, on the categories", "of presheaves with values in $\\mathcal{A}$.", "Moreover, the adjointness of the pair", "$u^p$ and $u_p$ continues to hold in this setting." ], "refs": [], "ref_ids": [] }, { "id": 8703, "type": "other", "label": "sites-remark-no-big-sites", "categories": [ "sites" ], "title": "sites-remark-no-big-sites", "contents": [ "(On set theoretic issues -- skip on a first reading.)", "The main reason for introducing sites is to study the", "category of sheaves on a site, because it is the generalization", "of the category of sheaves on a topological space that has", "been so important in algebraic geometry. In order to avoid thinking", "about things like ``classes of classes'' and so on, we will", "not allow sites to be ``big'' categories, in contrast to what", "we do for categories and $2$-categories.", "\\medskip\\noindent", "Suppose that $\\mathcal{C}$ is a category and that", "$\\text{Cov}(\\mathcal{C})$ is a proper class of coverings", "satisfying (1), (2) and (3) above. We will not allow this as a", "site either, mainly because we are going to take limits over coverings.", "However, there are several natural", "ways to replace $\\text{Cov}(\\mathcal{C})$ by a set of coverings", "or a slightly different structure", "that give rise to the same category of sheaves. For example:", "\\begin{enumerate}", "\\item In Sets, Section \\ref{sets-section-coverings-site}", "we show how to pick a suitable set of", "coverings that gives the same category of sheaves.", "\\item Another thing we can do is to take the associated topology", "(see Definition \\ref{definition-topology-associated-site}).", "The resulting topology on $\\mathcal{C}$ has the same category of sheaves.", "Two topologies have the same categories of sheaves if and only if", "they are equal, see Theorem \\ref{theorem-topology-and-topos}.", "A topology on a category is given by a choice of sieves on objects.", "The collection of all possible sieves and even all possible", "topologies on $\\mathcal{C}$ is a set.", "\\item We could also slightly modify the notion of a site, see", "Remark \\ref{remark-shrink-coverings} below, and end up with a", "canonical set of coverings.", "\\end{enumerate}", "Each of these solutions has some minor drawback. For the first, one has", "to check that constructions later on do not depend on the choice", "of the set of coverings. For the second, one has to learn about topologies", "and redo many of the arguments for sites. For the third, see", "the last sentence of Remark \\ref{remark-shrink-coverings}.", "\\medskip\\noindent", "Our approach will be to work with sites as in Definition \\ref{definition-site}", "above. Given a category $\\mathcal{C}$ with a proper class of coverings", "as above, we will replace this by a set of coverings producing a site using", "Sets, Lemma \\ref{sets-lemma-coverings-site}. It is shown in", "Lemma \\ref{lemma-choice-set-coverings-immaterial} below that the resulting", "category of sheaves (the topos) is independent of this choice. We leave it to", "the reader to use one of the other two strategies to deal with these issues if", "he/she so desires." ], "refs": [ "sites-definition-topology-associated-site", "sites-theorem-topology-and-topos", "sites-remark-shrink-coverings", "sites-remark-shrink-coverings", "sites-definition-site", "sets-lemma-coverings-site", "sites-lemma-choice-set-coverings-immaterial" ], "ref_ids": [ 8697, 8494, 8723, 8723, 8652, 8800, 8507 ] }, { "id": 8704, "type": "other", "label": "sites-remark-sheaf-condition-empty-covering", "categories": [ "sites" ], "title": "sites-remark-sheaf-condition-empty-covering", "contents": [ "If the covering $\\{U_i \\to U\\}_{i \\in I}$ is the empty family (this means", "that $I = \\emptyset$), then the sheaf condition signifies that", "$\\mathcal{F}(U) = \\{*\\}$ is a singleton set. This is because", "in (\\ref{equation-sheaf-condition}) the second and third sets", "are empty products in the category of sets, which are final objects", "in the category of sets, hence singletons." ], "refs": [], "ref_ids": [] }, { "id": 8705, "type": "other", "label": "sites-remark-both-refine-same-H0", "categories": [ "sites" ], "title": "sites-remark-both-refine-same-H0", "contents": [ "In particular this lemma shows that if $\\mathcal{U}$ is", "a refinement of $\\mathcal{V}$, and if $\\mathcal{V}$ is a", "refinement of $\\mathcal{U}$, then there is a canonical", "identification $H^0(\\mathcal{U}, \\mathcal{F}) =", "H^0(\\mathcal{V}, \\mathcal{F})$." ], "refs": [], "ref_ids": [] }, { "id": 8706, "type": "other", "label": "sites-remark-quasi-continuous", "categories": [ "sites" ], "title": "sites-remark-quasi-continuous", "contents": [ "(Skip on first reading.)", "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let us", "use the definition of tautologically equivalent families of maps,", "see Definition \\ref{definition-combinatorial-tautological}", "to (slightly) weaken the conditions defining continuity.", "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.", "Let us call $u$ {\\it quasi-continuous} if for every", "$\\mathcal{V} = \\{V_i \\to V\\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$", "we have the following", "\\begin{enumerate}", "\\item[(1')] the family of maps", "$\\{u(V_i) \\to u(V)\\}_{i\\in I}$ is tautologically equivalent", "to an element of $\\text{Cov}(\\mathcal{D})$, and", "\\item[(2)] for any morphism $T \\to V$ in $\\mathcal{C}$ the morphism", "$u(T \\times_V V_i) \\to u(T) \\times_{u(V)} u(V_i)$ is an isomorphism.", "\\end{enumerate}", "We are going to see that Lemmas \\ref{lemma-pushforward-sheaf}", "and \\ref{lemma-adjoint-sheaves} hold in case", "$u$ is quasi-continuous as well.", "\\medskip\\noindent", "We first remark that the morphisms $u(V_i) \\to u(V)$ are representable, since", "they are isomorphic to representable morphisms (by the first condition).", "In particular, the family $u(\\mathcal{V}) = \\{u(V_i) \\to u(V)\\}_{i\\in I}$", "gives rise to a zeroth {\\v C}ech cohomology group", "$H^0(u(\\mathcal{V}), \\mathcal{F})$ for any presheaf $\\mathcal{F}$ on", "$\\mathcal{D}$.", "Let $\\mathcal{U} = \\{U_j \\to u(V)\\}_{j \\in J}$ be an element", "of $\\text{Cov}(\\mathcal{D})$ tautologically", "equivalent to $\\{u(V_i) \\to u(V)\\}_{i \\in I}$. Note that $u(\\mathcal{V})$", "is a refinement of $\\mathcal{U}$ and vice versa. Hence by Remark", "\\ref{remark-both-refine-same-H0} we see that", "$H^0(u(\\mathcal{V}), \\mathcal{F}) = H^0(\\mathcal{U}, \\mathcal{F})$.", "In particular, if $\\mathcal{F}$ is a sheaf, then", "$\\mathcal{F}(u(V)) = H^0(u(\\mathcal{V}), \\mathcal{F})$ because", "of the sheaf property expressed in terms of zeroth {\\v C}ech cohomology", "groups. We conclude that $u^p\\mathcal{F}$ is a sheaf if $\\mathcal{F}$", "is a sheaf, since $H^0(\\mathcal{V}, u^p\\mathcal{F}) =", "H^0(u(\\mathcal{V}), \\mathcal{F})$ which we just observed is", "equal to $\\mathcal{F}(u(V)) = u^p\\mathcal{F}(V)$. Thus Lemma", "\\ref{lemma-pushforward-sheaf} holds. Lemma \\ref{lemma-adjoint-sheaves}", "follows immediately." ], "refs": [ "sites-definition-combinatorial-tautological", "sites-lemma-pushforward-sheaf", "sites-lemma-adjoint-sheaves", "sites-remark-both-refine-same-H0", "sites-lemma-pushforward-sheaf", "sites-lemma-adjoint-sheaves" ], "ref_ids": [ 8657, 8521, 8522, 8705, 8521, 8522 ] }, { "id": 8707, "type": "other", "label": "sites-remark-explain-left-exact", "categories": [ "sites" ], "title": "sites-remark-explain-left-exact", "contents": [ "The conditions of Proposition \\ref{proposition-get-morphism} above", "are equivalent to saying that $u$ is left exact, i.e., commutes", "with finite limits. See", "Categories, Lemmas", "\\ref{categories-lemma-finite-limits-exist} and", "\\ref{categories-lemma-characterize-left-exact}.", "It seems more natural to phrase it in terms of final objects", "and fibre products since this seems to have more geometric meaning", "in the examples." ], "refs": [ "sites-proposition-get-morphism", "categories-lemma-finite-limits-exist", "categories-lemma-characterize-left-exact" ], "ref_ids": [ 8641, 12224, 12245 ] }, { "id": 8708, "type": "other", "label": "sites-remark-quasi-continuous-morphism-sites", "categories": [ "sites" ], "title": "sites-remark-quasi-continuous-morphism-sites", "contents": [ "(Skip on first reading.)", "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Analogously to", "Definition \\ref{definition-morphism-sites} we say that", "a {\\it quasi-morphism of sites $f : \\mathcal{D} \\to \\mathcal{C}$}", "is given by a quasi-continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$", "(see Remark \\ref{remark-quasi-continuous}) such that $u_s$ is exact.", "The analogue of Proposition \\ref{proposition-get-morphism} in this", "setting is obtained by replacing the word ``continuous''", "by the word ``quasi-continuous'', and replacing the word", "``morphism'' by ``quasi-morphism''. The proof is literally the", "same." ], "refs": [ "sites-definition-morphism-sites", "sites-remark-quasi-continuous", "sites-proposition-get-morphism" ], "ref_ids": [ 8665, 8706, 8641 ] }, { "id": 8709, "type": "other", "label": "sites-remark-pt-topos", "categories": [ "sites" ], "title": "sites-remark-pt-topos", "contents": [ "There are many sites that give rise to the topos $\\Sh(pt)$.", "A useful example is the following. Suppose that $S$ is a set (of sets)", "which contains at least one nonempty element. Let $\\mathcal{S}$ be the", "category whose objects are elements of $S$ and whose morphisms are", "arbitrary set maps. Assume that $\\mathcal{S}$ has fibre products.", "For example this will be the case if $S = \\mathcal{P}(\\text{infinite set})$", "is the power set of any infinite set (exercise in set theory).", "Make $\\mathcal{S}$ into a site by declaring", "surjective families of maps to be coverings (and choose", "a suitable sufficiently large set of covering families as in", "Sets, Section \\ref{sets-section-coverings-site}).", "We claim that $\\Sh(\\mathcal{S})$ is equivalent to the category of", "sets.", "\\medskip\\noindent", "We first prove this in case $S$ contains $e \\in S$ which is a singleton.", "In this case, there is an equivalence of topoi", "$i : \\Sh(pt) \\to \\Sh(\\mathcal{S})$ given by", "the functors", "\\begin{equation}", "\\label{equation-sheaves-pt-sets}", "i^{-1}\\mathcal{F} = \\mathcal{F}(e), \\quad", "i_*E = (U \\mapsto \\Mor_{\\textit{Sets}}(U, E))", "\\end{equation}", "Namely, suppose that $\\mathcal{F}$ is a sheaf on $\\mathcal{S}$.", "For any $U \\in \\Ob(\\mathcal{S}) = S$ we can find a covering", "$\\{\\varphi_u : e \\to U\\}_{u \\in U}$, where $\\varphi_u$", "maps the unique element of $e$ to $u \\in U$. The sheaf condition", "implies in this case that", "$\\mathcal{F}(U) = \\prod_{u \\in U} \\mathcal{F}(e)$.", "In other words", "$\\mathcal{F}(U) = \\Mor_{\\textit{Sets}}(U, \\mathcal{F}(e))$.", "Moreover, this rule is compatible with restriction mappings. Hence", "the functor", "$$", "i_* :", "\\textit{Sets} = \\Sh(pt)", "\\longrightarrow", "\\Sh(\\mathcal{S}), \\quad", "E \\longmapsto (U \\mapsto \\Mor_{\\textit{Sets}}(U, E))", "$$", "is an equivalence of categories, and its inverse is the functor", "$i^{-1}$ given above.", "\\medskip\\noindent", "If $\\mathcal{S}$ does not contain a singleton, then the functor", "$i_*$ as defined above still makes sense. To show that it is still", "an equivalence in this case, choose any nonempty $\\tilde e \\in S$", "and a map $\\varphi : \\tilde e \\to \\tilde e$ whose image is a singleton.", "For any sheaf $\\mathcal{F}$ set", "$$", "\\mathcal{F}(e) :=", "\\Im(", "\\mathcal{F}(\\varphi) :", "\\mathcal{F}(\\tilde e)", "\\longrightarrow", "\\mathcal{F}(\\tilde e)", ")", "$$", "and show that this is a quasi-inverse to $i_*$. Details omitted." ], "refs": [], "ref_ids": [] }, { "id": 8710, "type": "other", "label": "sites-remark-morphism-topoi-big", "categories": [ "sites" ], "title": "sites-remark-morphism-topoi-big", "contents": [ "(Set theoretical issues related to morphisms of topoi. Skip", "on a first reading.)", "A morphism of topoi as defined above is not a set but a class.", "In other words it is given by a mathematical formula rather", "than a mathematical object. Although we may contemplate", "the collection of all morphisms between two given topoi,", "it is not a good idea to introduce it as a mathematical object.", "On the other hand, suppose $\\mathcal{C}$ and $\\mathcal{D}$ are", "given sites. Consider a functor", "$\\Phi : \\mathcal{C} \\to \\Sh(\\mathcal{D})$.", "Such a thing is a set, in other words, it is a mathematical object.", "We may, in succession, ask the following questions on $\\Phi$.", "\\begin{enumerate}", "\\item Is it true, given a sheaf $\\mathcal{F}$ on $\\mathcal{D}$,", "that the rule", "$U \\mapsto \\Mor_{\\Sh(\\mathcal{D})}(\\Phi(U), \\mathcal{F})$", "defines a sheaf on $\\mathcal{C}$? If so, this defines a functor", "$\\Phi_* : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$.", "\\item Is it true that $\\Phi_*$ has a left adjoint? If so,", "write $\\Phi^{-1}$ for this left adjoint.", "\\item Is it true that $\\Phi^{-1}$ is exact?", "\\end{enumerate}", "If the last question still has the answer ``yes'', then we obtain", "a morphism of topoi $(\\Phi_*, \\Phi^{-1})$. Moreover, given any", "morphism of topoi $(f_*, f^{-1})$ we may set", "$\\Phi(U) = f^{-1}(h_U^\\#)$ and obtain a functor $\\Phi$ as above", "with $f_* \\cong \\Phi_*$ and $f^{-1} \\cong \\Phi^{-1}$ (compatible", "with adjoint property).", "The upshot is that by working with the collection of $\\Phi$", "instead of morphisms of topoi, we (a) replaced the notion of", "a morphism of topoi by a mathematical object, and (b)", "the collection of $\\Phi$ forms a class (and not a collection", "of classes). Of course, more can be said, for example one can work", "out more precisely the significance of conditions (2) and (3) above;", "we do this in the case of points of topoi in Section \\ref{section-points}." ], "refs": [], "ref_ids": [] }, { "id": 8711, "type": "other", "label": "sites-remark-quasi-continuous-morphism-topoi", "categories": [ "sites" ], "title": "sites-remark-quasi-continuous-morphism-topoi", "contents": [ "(Skip on first reading.)", "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.", "A quasi-morphism of sites $f : \\mathcal{D} \\to \\mathcal{C}$", "(see Remark \\ref{remark-quasi-continuous-morphism-sites})", "gives rise to a morphism of topoi $f$ from", "$\\Sh(\\mathcal{D})$ to $\\Sh(\\mathcal{C})$", "exactly as in Lemma \\ref{lemma-morphism-sites-topoi}." ], "refs": [ "sites-remark-quasi-continuous-morphism-sites", "sites-lemma-morphism-sites-topoi" ], "ref_ids": [ 8708, 8528 ] }, { "id": 8712, "type": "other", "label": "sites-remark-cartesian-cocontinuous", "categories": [ "sites" ], "title": "sites-remark-cartesian-cocontinuous", "contents": [ "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor between categories.", "Given morphisms $g : u(U) \\to V$ and $f : W \\to V$ in $\\mathcal{D}$", "we can consider the functor", "$$", "\\mathcal{C}^{opp} \\longrightarrow \\textit{Sets},\\quad", "T \\longmapsto", "\\Mor_\\mathcal{C}(T, U)", "\\times_{\\Mor_\\mathcal{D}(u(T), V)}", "\\Mor_\\mathcal{D}(u(T), W)", "$$", "If this functor is representable, denote $U \\times_{g, V, f} W$", "the corresponding object of $\\mathcal{C}$.", "Assume that $\\mathcal{C}$ and $\\mathcal{D}$ are sites.", "Consider the property $P$: for every covering $\\{f_j : V_j \\to V\\}$", "of $\\mathcal{D}$ and any morphism $g : u(U) \\to V$ we have", "\\begin{enumerate}", "\\item $U \\times_{g, V, f_i} V_i$ exists for all $i$, and", "\\item $\\{U \\times_{g, V, f_i} V_i \\to U\\}$ is a covering of $\\mathcal{C}$.", "\\end{enumerate}", "Please note the similarity with the definition of continuous", "functors. If $u$ has $P$ then $u$ is cocontinuous (details omitted).", "Many of the cocontinuous functors we will encounter satisfy $P$." ], "refs": [], "ref_ids": [] }, { "id": 8713, "type": "other", "label": "sites-remark-localize-presheaves", "categories": [ "sites" ], "title": "sites-remark-localize-presheaves", "contents": [ "Localization and presheaves. Let $\\mathcal{C}$ be a category.", "Let $U$ be an object of $\\mathcal{C}$. Strictly speaking the functors", "$j_U^{-1}$, $j_{U*}$ and $j_{U!}$ have not been defined for presheaves.", "But of course, we can think of a presheaf as a sheaf for the", "chaotic topology on $\\mathcal{C}$ (see Example \\ref{example-indiscrete}).", "Hence we also obtain a functor", "$$", "j_U^{-1} :", "\\textit{PSh}(\\mathcal{C})", "\\longrightarrow", "\\textit{PSh}(\\mathcal{C}/U)", "$$", "and functors", "$$", "j_{U*}, j_{U!} :", "\\textit{PSh}(\\mathcal{C}/U)", "\\longrightarrow", "\\textit{PSh}(\\mathcal{C})", "$$", "which are right, left adjoint to $j_U^{-1}$. By", "Lemma \\ref{lemma-describe-j-shriek}", "we see that $j_{U!}\\mathcal{G}$ is the presheaf", "$$", "V \\longmapsto", "\\coprod\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{G}(V \\xrightarrow{\\varphi} U)", "$$", "In addition the functor $j_{U!}$ commutes with fibre products and", "equalizers." ], "refs": [ "sites-lemma-describe-j-shriek" ], "ref_ids": [ 8553 ] }, { "id": 8714, "type": "other", "label": "sites-remark-localization-cartesian-cocontinuous", "categories": [ "sites" ], "title": "sites-remark-localization-cartesian-cocontinuous", "contents": [ "Let $\\mathcal{C}$ be a site. Let $U \\to V$ be a morphism of $\\mathcal{C}$.", "The cocontinuous functors $\\mathcal{C}/U \\to \\mathcal{C}$ and", "$j : \\mathcal{C}/U \\to \\mathcal{C}/V$ (Lemma \\ref{lemma-relocalize})", "satisfy property $P$ of Remark \\ref{remark-cartesian-cocontinuous}.", "For example, if we have objects $(X/U)$, $(W/V)$, a morphism", "$g : j(X/U) \\to (W/V)$, and a covering $\\{f_i : (W_i/V) \\to (W/V)\\}$ then", "$(X \\times_W W_i/U)$ is an avatar of $(X/U) \\times_{g, (W/V), f_i} (W_i/V)$", "and the family $\\{(X \\times_W W_i/U) \\to (X/U)\\}$ is a covering", "of $\\mathcal{C}/U$." ], "refs": [ "sites-lemma-relocalize", "sites-remark-cartesian-cocontinuous" ], "ref_ids": [ 8559, 8712 ] }, { "id": 8715, "type": "other", "label": "sites-remark-morphism-topoi-comes-from-morphism-sites", "categories": [ "sites" ], "title": "sites-remark-morphism-topoi-comes-from-morphism-sites", "contents": [ "Notation and assumptions", "as in Lemma \\ref{lemma-morphism-topoi-comes-from-morphism-sites}.", "If the site $\\mathcal{D}$ has a final object and fibre products", "then the functor $u : \\mathcal{D} \\to \\mathcal{C}'$ satisfies", "all the assumptions of Proposition \\ref{proposition-get-morphism}.", "Namely, in addition to the properties mentioned in the lemma $u$", "also transforms the final object of $\\mathcal{D}$ into the final", "object of $\\mathcal{C}'$. This is clear from the construction of $u$.", "Hence, if we first apply", "Lemmas \\ref{lemma-topos-good-site}", "to $\\mathcal{D}$", "and then", "Lemma \\ref{lemma-morphism-topoi-comes-from-morphism-sites}", "to the resulting morphism of topoi", "$\\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D}')$", "we obtain the following statement:", "Any morphism of topoi", "$f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$", "fits into a commutative diagram", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C}) \\ar[d]_g \\ar[r]_f &", "\\Sh(\\mathcal{D}) \\ar[d]^e \\\\", "\\Sh(\\mathcal{C}') \\ar[r]^{f'} &", "\\Sh(\\mathcal{D}')", "}", "$$", "where the following properties hold:", "\\begin{enumerate}", "\\item the morphisms $e$ and $g$ are equivalences given by", "special cocontinuous functors $\\mathcal{C} \\to \\mathcal{C}'$ and", "$\\mathcal{D} \\to \\mathcal{D}'$,", "\\item the sites $\\mathcal{C}'$ and $\\mathcal{D}'$ have fibre products, final", "objects and have subcanonical topologies,", "\\item the morphism $f' : \\mathcal{C}' \\to \\mathcal{D}'$ comes from a", "morphism of sites corresponding to a functor", "$u : \\mathcal{D}' \\to \\mathcal{C}'$ to which", "Proposition \\ref{proposition-get-morphism}", "applies, and", "\\item given any set of sheaves $\\mathcal{F}_i$ (resp.\\ $\\mathcal{G}_j$)", "on $\\mathcal{C}$ (resp.\\ $\\mathcal{D}$) we may assume each of these is", "a representable sheaf on $\\mathcal{C}'$ (resp.\\ $\\mathcal{D}'$).", "\\end{enumerate}", "It is often useful to replace $\\mathcal{C}$ and $\\mathcal{D}$ by", "$\\mathcal{C}'$ and $\\mathcal{D}'$." ], "refs": [ "sites-lemma-morphism-topoi-comes-from-morphism-sites", "sites-proposition-get-morphism", "sites-lemma-topos-good-site", "sites-lemma-morphism-topoi-comes-from-morphism-sites", "sites-proposition-get-morphism" ], "ref_ids": [ 8582, 8641, 8581, 8582, 8641 ] }, { "id": 8716, "type": "other", "label": "sites-remark-equivalence-topoi-comes-from-morphism-sites", "categories": [ "sites" ], "title": "sites-remark-equivalence-topoi-comes-from-morphism-sites", "contents": [ "Notation and assumptions", "as in Lemma \\ref{lemma-morphism-topoi-comes-from-morphism-sites}.", "Suppose that in addition the original morphism of topoi", "$\\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ is an equivalence.", "Then the construction in the proof of", "Lemma \\ref{lemma-morphism-topoi-comes-from-morphism-sites}", "gives two functors", "$$", "\\mathcal{C} \\rightarrow \\mathcal{C}' \\leftarrow \\mathcal{D}", "$$", "which are both special cocontinuous functors.", "Hence in this case we can actually", "factor the morphism of topoi as a composition", "$$", "\\Sh(\\mathcal{C}) \\rightarrow", "\\Sh(\\mathcal{C}') =", "\\Sh(\\mathcal{D}') \\leftarrow", "\\Sh(\\mathcal{D})", "$$", "as in Remark \\ref{remark-morphism-topoi-comes-from-morphism-sites}, but", "with the middle morphism an identity." ], "refs": [ "sites-lemma-morphism-topoi-comes-from-morphism-sites", "sites-lemma-morphism-topoi-comes-from-morphism-sites", "sites-remark-morphism-topoi-comes-from-morphism-sites" ], "ref_ids": [ 8582, 8582, 8715 ] }, { "id": 8717, "type": "other", "label": "sites-remark-improve-proposition-points-limits", "categories": [ "sites" ], "title": "sites-remark-improve-proposition-points-limits", "contents": [ "In fact, let $\\mathcal{C}$ be a site. Assume $\\mathcal{C}$ has a final object", "$X$ and fibre products. Let $p = u: \\mathcal{C} \\to \\textit{Sets}$ be a", "functor such that", "\\begin{enumerate}", "\\item $u(X) = \\{*\\}$ a singleton, and", "\\item for every pair of morphisms $U \\to W$ and $V \\to W$ with", "the same target the map", "$u(U \\times_W V) \\to u(U) \\times_{u(W)} u(V)$ is surjective.", "\\item for every covering $\\{U_i \\to U\\}$ the map", "$\\coprod u(U_i) \\to u(U)$ is surjective.", "\\end{enumerate}", "Then, in general, $p$ is {\\bf not} a point of $\\mathcal{C}$.", "An example is the category $\\mathcal{C}$ with two objects $\\{U, X\\}$", "and exactly one non-identity arrow, namely $U \\to X$. We endow $\\mathcal{C}$", "with the trivial topology, i.e., the only coverings are $\\{U \\to U\\}$ and", "$\\{X \\to X\\}$. A sheaf $\\mathcal{F}$ is the same thing as a presheaf and", "consists of a triple $(A, B, A \\to B)$: namely $A = \\mathcal{F}(X)$,", "$B = \\mathcal{F}(U)$ and $A \\to B$ is the restriction mapping corresponding", "to $U \\to X$. Note that $U \\times_X U = U$ so fibre products exist.", "Consider the functor $u = p$ with $u(X) = \\{*\\}$ and $u(U) = \\{*_1, *_2\\}$.", "This satisfies (1), (2), and (3), but the corresponding stalk functor", "(\\ref{equation-stalk}) is the functor", "$$", "(A, B, A \\to B) \\longmapsto B \\amalg_A B", "$$", "which isn't exact. Namely, consider", "$(\\emptyset, \\{1\\}, \\emptyset \\to \\{1\\}) \\to (\\{1\\}, \\{1\\}, \\{1\\} \\to \\{1\\})$", "which is an injective map of sheaves, but is transformed into the noninjective", "map of sets", "$$", "\\{1\\} \\amalg \\{1\\} \\longrightarrow \\{1\\} \\amalg_{\\{1\\}} \\{1\\}", "$$", "by the stalk functor." ], "refs": [], "ref_ids": [] }, { "id": 8718, "type": "other", "label": "sites-remark-not-pushforward", "categories": [ "sites" ], "title": "sites-remark-not-pushforward", "contents": [ "Warning: The result of", "Lemma \\ref{lemma-stalk-j-shriek}", "has no analogue for $j_{U, *}$." ], "refs": [ "sites-lemma-stalk-j-shriek" ], "ref_ids": [ 8607 ] }, { "id": 8719, "type": "other", "label": "sites-remark-no-pullback-presheaves", "categories": [ "sites" ], "title": "sites-remark-no-pullback-presheaves", "contents": [ "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.", "Let $u : \\mathcal{D} \\to \\mathcal{C}$ be a continuous functor", "which gives rise to a morphism of sites $\\mathcal{C} \\to \\mathcal{D}$.", "Note that even in the case of abelian groups we have not defined", "a pullback functor for presheaves of abelian groups.", "Since all colimits are representable in", "the category of abelian groups, we certainly may define", "a functor $u_p^{ab}$ on abelian presheaves by the same colimits", "as we have used to define $u_p$ on presheaves of sets.", "It will also be the case that $u_p^{ab}$ is adjoint to", "$u^p$ on the categories of abelian presheaves.", "However, it will not always be the case that $u_p^{ab}$", "agrees with $u_p$ on the underlying presheaves of sets." ], "refs": [], "ref_ids": [] }, { "id": 8720, "type": "other", "label": "sites-remark-compose-base-change", "categories": [ "sites" ], "title": "sites-remark-compose-base-change", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "\\Sh(\\mathcal{B}') \\ar[r]_k \\ar[d]_{f'} &", "\\Sh(\\mathcal{B}) \\ar[d]^f \\\\", "\\Sh(\\mathcal{C}') \\ar[r]^l \\ar[d]_{g'} &", "\\Sh(\\mathcal{C}) \\ar[d]^g \\\\", "\\Sh(\\mathcal{D}') \\ar[r]^m &", "\\Sh(\\mathcal{D})", "}", "$$", "of topoi. Then the base change maps for the two squares compose to give the", "base change map for the outer rectangle. More precisely, the composition", "\\begin{align*}", "m^{-1} \\circ (g \\circ f)_*", "& =", "m^{-1} \\circ g_* \\circ f_* \\\\", "& \\to g'_* \\circ l^{-1} \\circ f_* \\\\", "& \\to g'_* \\circ f'_* \\circ k^{-1} \\\\", "& = (g' \\circ f')_* \\circ k^{-1}", "\\end{align*}", "is the base change map for the rectangle. We omit the verification." ], "refs": [], "ref_ids": [] }, { "id": 8721, "type": "other", "label": "sites-remark-compose-base-change-horizontal", "categories": [ "sites" ], "title": "sites-remark-compose-base-change-horizontal", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C}'') \\ar[r]_{g'} \\ar[d]_{f''} &", "\\Sh(\\mathcal{C}') \\ar[r]_g \\ar[d]_{f'} &", "\\Sh(\\mathcal{C}) \\ar[d]^f \\\\", "\\Sh(\\mathcal{D}'') \\ar[r]^{h'} &", "\\Sh(\\mathcal{D}') \\ar[r]^h &", "\\Sh(\\mathcal{D})", "}", "$$", "of ringed topoi. Then the base change maps", "for the two squares compose to give the base", "change map for the outer rectangle. More precisely,", "the composition", "\\begin{align*}", "(h \\circ h')^{-1} \\circ f_*", "& =", "(h')^{-1} \\circ h^{-1} \\circ f_* \\\\", "& \\to (h')^{-1} \\circ f'_* \\circ g^{-1} \\\\", "& \\to f''_* \\circ (g')^{-1} \\circ g^{-1} \\\\", "& = f''_* \\circ (g \\circ g')^{-1}", "\\end{align*}", "is the base change map for the rectangle. We omit the verification." ], "refs": [], "ref_ids": [] }, { "id": 8722, "type": "other", "label": "sites-remark-enlarge-coverings", "categories": [ "sites" ], "title": "sites-remark-enlarge-coverings", "contents": [ "Enlarging the class of coverings.", "Clearly, if $\\text{Cov}(\\mathcal{C})$", "defines the structure of a site on $\\mathcal{C}$ then we may", "add to $\\mathcal{C}$ any set of families of morphisms with fixed target", "tautologically equivalent", "(see Definition \\ref{definition-combinatorial-tautological})", "to elements of $\\text{Cov}(\\mathcal{C})$ without changing the topology." ], "refs": [ "sites-definition-combinatorial-tautological" ], "ref_ids": [ 8657 ] }, { "id": 8723, "type": "other", "label": "sites-remark-shrink-coverings", "categories": [ "sites" ], "title": "sites-remark-shrink-coverings", "contents": [ "Shrinking the class of coverings. Let $\\mathcal{C}$ be a site.", "Consider the set", "$$", "\\mathcal{S} = P(\\text{Arrows}(\\mathcal{C})) \\times \\Ob(\\mathcal{C})", "$$", "where $P(\\text{Arrows}(\\mathcal{C}))$ is the power set of the set of morphisms,", "i.e., the set of all sets of morphisms.", "Let $\\mathcal{S}_\\tau \\subset \\mathcal{S}$", "be the subset consisting of those $(T, U) \\in \\mathcal{S}$ such that", "(a) all $\\varphi \\in T$ have target $U$,", "(b) the collection $\\{\\varphi\\}_{\\varphi \\in T}$ is tautologically", "equivalent (see Definition \\ref{definition-combinatorial-tautological})", "to some covering in $\\text{Cov}(\\mathcal{C})$.", "Clearly, considering the elements of $\\mathcal{S}_\\tau$ as", "the coverings, we do not get exactly the notion of a site", "as defined in Definition \\ref{definition-site}.", "The structure $(\\mathcal{C}, \\mathcal{S}_\\tau)$", "we get satisfies slightly modified conditions. The modified", "conditions are:", "\\begin{enumerate}", "\\item[(0')] $\\text{Cov}(\\mathcal{C}) \\subset", "P(\\text{Arrows}(\\mathcal{C})) \\times \\Ob(\\mathcal{C})$,", "\\item[(1')] If $V \\to U$ is an isomorphism then", "$(\\{V \\to U\\}, U) \\in \\text{Cov}(\\mathcal{C})$.", "\\item[(2')] If $(T, U) \\in \\text{Cov}(\\mathcal{C})$", "and for $f : U' \\to U$ in $T$ we are given", "$(T_f, U') \\in \\text{Cov}(\\mathcal{C})$,", "then setting $T' = \\{f \\circ f' \\mid f \\in T,\\ f' \\in T_f\\}$,", "we get $(T', U) \\in \\text{Cov}(\\mathcal{C})$.", "\\item[(3')] If $(T, U) \\in \\text{Cov}(\\mathcal{C})$ and $g : V \\to U$", "is a morphism of $\\mathcal{C}$ then", "\\begin{enumerate}", "\\item $U' \\times_{f, U, g} V$ exists for $f : U' \\to U$ in $T$, and", "\\item setting", "$T' = \\{\\text{pr}_2 : U' \\times_{f, U, g} V \\to V \\mid f : U' \\to U \\in T\\}$", "for some choice of fibre products we get $(T', V) \\in \\text{Cov}(\\mathcal{C})$.", "\\end{enumerate}", "\\end{enumerate}", "And it is easy to verify that, given a structure satisfying", "(0') -- (3') above, then after suitably enlarging", "$\\text{Cov}(\\mathcal{C})$ (compare Sets,", "Section \\ref{sets-section-coverings-site}) we get a site.", "Obviously there is little difference between this notion and the", "actual notion of a site, at least from the point of view of the", "topology. There are two benefits:", "because of condition (0') above the coverings automatically", "form a set, and because of (0') the totality of all structures", "of this type forms a set as well.", "The price you pay for this is that you have to keep writing", "``tautologically equivalent'' everywhere." ], "refs": [ "sites-definition-combinatorial-tautological", "sites-definition-site" ], "ref_ids": [ 8657, 8652 ] }, { "id": 8803, "type": "other", "label": "sets-remark-how-to-use-reflection", "categories": [ "sets" ], "title": "sets-remark-how-to-use-reflection", "contents": [ "The lemma above can also be proved using the reflection principle.", "However, one has to be careful. Namely, suppose the sentence", "$\\phi_{scheme}(X)$ expresses the property ``$X$ is a scheme'', then", "what does the formula $\\phi_{scheme}^{V_\\alpha}(X)$ mean?", "It is true that the reflection principle says we can find $\\alpha$ such that", "for all $X \\in V_\\alpha$ we have", "$\\phi_{scheme}(X) \\leftrightarrow \\phi_{scheme}^{V_\\alpha}(X)$", "but this is entirely useless. It is only by combining two such", "statements that something interesting happens. For example suppose", "$\\phi_{red}(X, Y)$ expresses the property ``$X$, $Y$ are schemes,", "and $Y$ is the reduction of $X$'' (see", "Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme}).", "Suppose we apply the reflection principle to the pair of", "formulas $\\phi_1(X, Y) = \\phi_{red}(X, Y)$,", "$\\phi_2(X) = \\exists Y, \\phi_1(X, Y)$. Then it is easy to see that", "any $\\alpha$ produced by the reflection principle has the property that", "given $X \\in \\Ob(\\Sch_\\alpha)$ the reduction of", "$X$ is also an object of $\\Sch_\\alpha$ (left as an exercise)." ], "refs": [ "schemes-definition-reduced-induced-scheme" ], "ref_ids": [ 7745 ] }, { "id": 8804, "type": "other", "label": "sets-remark-what-is-not-in-it", "categories": [ "sets" ], "title": "sets-remark-what-is-not-in-it", "contents": [ "Let $R$ be a ring. Suppose we consider the ring", "$\\prod_{\\mathfrak p \\in \\Spec(R)} \\kappa(\\mathfrak p)$.", "The cardinality of this ring is bounded by $|R|^{2^{|R|}}$,", "but is not bounded by $|R|^{\\aleph_0}$ in general.", "For example if $R = \\mathbf{C}[x]$ it is not bounded by", "$|R|^{\\aleph_0}$ and if $R = \\prod_{n \\in \\mathbf{N}} \\mathbf{F}_2$", "it is not bounded by $|R|^{|R|}$.", "Thus the ``And so on'' of Lemma \\ref{lemma-what-is-in-it} above", "should be taken with a grain of salt. Of course, if it ever becomes", "necessary to consider these rings in arguments pertaining to", "fppf/\\'etale cohomology, then we can change the function", "$Bound$ above into the function $\\kappa \\mapsto \\kappa^{2^\\kappa}$." ], "refs": [ "sets-lemma-what-is-in-it" ], "ref_ids": [ 8795 ] }, { "id": 8805, "type": "other", "label": "sets-remark-better", "categories": [ "sets" ], "title": "sets-remark-better", "contents": [ "It is likely the case that, for some limit ordinal $\\alpha$,", "the set of coverings $\\text{Cov}(\\mathcal{C})_\\alpha$ satisfies", "the conditions of the lemma. This is after all what an application", "of the reflection principle would appear to give (modulo caveats as", "described at the end of Section \\ref{section-reflection-principle}", "and in Remark \\ref{remark-how-to-use-reflection})." ], "refs": [ "sets-remark-how-to-use-reflection" ], "ref_ids": [ 8803 ] }, { "id": 8849, "type": "other", "label": "more-etale-remark-covariance-f-shriek-separated", "categories": [ "more-etale" ], "title": "more-etale-remark-covariance-f-shriek-separated", "contents": [ "Let $f : X \\to Y$ be morphism of schemes which is separated and", "locally of finite type. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$.", "Let $X' \\subset X$ be an open subscheme. Denote $f' : X' \\to Y$", "the restriction of $f$. There is a canonical injective map", "$$", "f'_!(\\mathcal{F}|_{X'}) \\longrightarrow f_!\\mathcal{F}", "$$", "Namely, let $V \\in Y_\\etale$ and consider a section", "$s' \\in f'_*(\\mathcal{F}|_{X'})(V) = \\mathcal{F}(X' \\times_Y V)$", "with support $Z'$ proper over $V$. Then $Z'$ is closed in $X \\times_Y V$", "as well, see Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-functoriality-closed-proper-over-base}.", "Thus there is a unique section", "$s \\in \\mathcal{F}(X \\times_Y V) = f_*\\mathcal{F}(V)$", "whose restriction to $X' \\times_Y V$ is $s'$ and whose restriction", "to $X \\times_Y V \\setminus Z'$ is zero, see", "Lemma \\ref{lemma-section-support-in-locally-closed}. This construction is", "compatible with restriction maps and hence induces the desired map of", "sheaves $f'_!(\\mathcal{F}|_{X'}) \\to f_!\\mathcal{F}$ which is clearly", "injective. By construction we obtain a commutative diagram", "$$", "\\xymatrix{", "f'_!(\\mathcal{F}|_{X'}) \\ar[r] \\ar[d] &", "f_!\\mathcal{F} \\ar[d] \\\\", "f'_*(\\mathcal{F}|_{X'}) &", "f_*\\mathcal{F} \\ar[l]", "}", "$$", "functorial in $\\mathcal{F}$. It is clear that for $X'' \\subset X'$ open", "with $f'' = f|_{X''} : X'' \\to Y$ the composition of the canonical maps", "$f''_!\\mathcal{F}|_{X''} \\to f'_!\\mathcal{F}|_{X'} \\to f_!\\mathcal{F}$", "just constructed is the canonical map", "$f''_!\\mathcal{F}|_{X''} \\to f_!\\mathcal{F}$." ], "refs": [ "coherent-lemma-functoriality-closed-proper-over-base" ], "ref_ids": [ 3389 ] }, { "id": 8850, "type": "other", "label": "more-etale-remark-covariance-compact-support", "categories": [ "more-etale" ], "title": "more-etale-remark-covariance-compact-support", "contents": [ "Let $X$ be a separated scheme locally of finite type over a field $k$.", "Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$.", "Exactly as in Remark \\ref{remark-covariance-f-shriek-separated}", "there are injective maps", "$$", "H^0_c(X', \\mathcal{F}|_{X'}) \\longrightarrow H^0_c(X, \\mathcal{F})", "$$", "which turn $H^0_c$ into a ``cosheaf'' on the Zariski site of $X$." ], "refs": [ "more-etale-remark-covariance-f-shriek-separated" ], "ref_ids": [ 8849 ] }, { "id": 8851, "type": "other", "label": "more-etale-remark-f-shriek-base-change-composition", "categories": [ "more-etale" ], "title": "more-etale-remark-f-shriek-base-change-composition", "contents": [ "The isomorphisms between functors", "constructed above satisfy the following two properties:", "\\begin{enumerate}", "\\item Let $f : X \\to Y$, $g : Y \\to Z$, and $h : Z \\to T$ be composable", "morphisms of schemes which are separated and locally of finite type.", "Then the diagram", "$$", "\\xymatrix{", "(h \\circ g \\circ f)_! \\ar[r] \\ar[d] &", "(h \\circ g)_! \\circ f_! \\ar[d] \\\\", "h_! \\circ (g \\circ f)_! \\ar[r] &", "h_! \\circ g_! \\circ f_!", "}", "$$", "commutes where the arrows are those of Lemma \\ref{lemma-f-shriek-composition}.", "\\item Suppose that we have a diagram of schemes", "$$", "\\xymatrix{", "X' \\ar[d]_{f'} \\ar[r]_c & X \\ar[d]^f \\\\", "Y' \\ar[d]_{g'} \\ar[r]_b & Y \\ar[d]^g \\\\", "Z' \\ar[r]^a & Z", "}", "$$", "with both squares cartesian and $f$ and $g$ separated and", "locally of finite type. Then the diagram", "$$", "\\xymatrix{", "a^{-1} \\circ (g \\circ f)_! \\ar[d] \\ar[rr] & &", "(g' \\circ f')_! \\circ c^{-1} \\ar[d] \\\\", "a^{-1} \\circ g_! \\circ f_! \\ar[r] &", "g'_! \\circ b^{-1} \\circ f_! \\ar[r] &", "g'_! \\circ f'_! \\circ c^{-1}", "}", "$$", "commutes where the horizontal arrows are those of", "Lemma \\ref{lemma-base-change-f-shriek-separated}", "the arrows are those of Lemma \\ref{lemma-f-shriek-composition}.", "\\end{enumerate}", "Part (1) holds true because we have a similar commutative", "diagram for pushforwards. Part (2) holds by the very general", "compatibility of base change maps for pushforwards", "(Sites, Remark \\ref{sites-remark-compose-base-change})", "and the fact that the isomorphisms in", "Lemmas \\ref{lemma-base-change-f-shriek-separated} and", "\\ref{lemma-f-shriek-composition}", "are constructed using the corresponding maps fo pushforwards." ], "refs": [ "more-etale-lemma-f-shriek-composition", "more-etale-lemma-base-change-f-shriek-separated", "more-etale-lemma-f-shriek-composition", "sites-remark-compose-base-change", "more-etale-lemma-base-change-f-shriek-separated", "more-etale-lemma-f-shriek-composition" ], "ref_ids": [ 8816, 8815, 8816, 8720, 8815, 8816 ] }, { "id": 8852, "type": "other", "label": "more-etale-remark-covariance-lqf-f-shriek", "categories": [ "more-etale" ], "title": "more-etale-remark-covariance-lqf-f-shriek", "contents": [ "Let $f : X \\to Y$ be locally quasi-finite morphism of schemes. Let", "$\\mathcal{F}$ be an abelian sheaf on $X_\\etale$.", "Let $X' \\subset X$ be an open subscheme and denote $f' : X' \\to Y$", "the restriction of $f$.", "We claim there is a canonical map", "$$", "f'_!(\\mathcal{F}|_{X'}) \\longrightarrow f_!\\mathcal{F}", "$$", "Namely, this map will be the sheafification of a canonical map", "$$", "f'_{p!}(\\mathcal{F}|_{X'}) \\to f_{p!}\\mathcal{F}", "$$", "constructed as follows. Let $V \\in Y_\\etale$ and consider a section", "$s' = \\sum_{i = 1, \\ldots, n} (Z'_i, s'_i)$ as in", "(\\ref{equation-formal-sum}) defining an element of", "$f'_{p!}(\\mathcal{F}|_{X'})(V)$.", "Then $Z'_i \\subset X'_V$ may also be viewed as a locally closed subscheme", "of $X_V$ and we have $H_{Z'_i}(\\mathcal{F}|_{X'}) = H_{Z'_i}(\\mathcal{F})$.", "We will map $s'$ to the exact same sum", "$s = \\sum_{i = 1, \\ldots, n} (Z'_i, s'_i)$", "but now viewed as an element of $f_{p!}\\mathcal{F}(V)$.", "We omit the verification that this construction is compatible with", "restriction mappings and functorial in $\\mathcal{F}$.", "This construction has the following properties:", "\\begin{enumerate}", "\\item The maps $f'_{p!}\\mathcal{F}' \\to f_{p!}\\mathcal{F}$ and", "$f'_!\\mathcal{F}' \\to f_!\\mathcal{F}$ are compatible with", "the description of stalks given in Lemmas", "\\ref{lemma-finite-support-stalk} and \\ref{lemma-lqf-f-shriek-stalk}.", "\\item If $f$ is separated, then the map", "$f'_{p!}\\mathcal{F}' \\to f_{p!}\\mathcal{F}$ is the same as the map", "constructed in Remark \\ref{remark-covariance-f-shriek-separated}", "via the isomorphism in Lemma \\ref{lemma-finite-support-f-shriek-separated}.", "\\item If $X'' \\subset X'$ is another open, then the composition of", "$f''_{p!}(\\mathcal{F}|_{X''}) \\to f'_{p!}(\\mathcal{F}|_{X'}) \\to", "f_{p!}\\mathcal{F}$ is the map", "$f''_{p!}(\\mathcal{F}|_{X''}) \\to f_{p!}\\mathcal{F}$ for the", "inclusion $X'' \\subset X$. Sheafifying we conclude", "the same holds true for", "$f''_!(\\mathcal{F}|_{X''}) \\to f'_!(\\mathcal{F}|_{X'}) \\to f_!\\mathcal{F}$.", "\\item The map $f'_!\\mathcal{F}' \\to f_!\\mathcal{F}$ is injective", "because we can check this on stalks.", "\\end{enumerate}", "All of these statements are easily proven by representing elements", "as finite sums as above and considering what happens to these elements." ], "refs": [ "more-etale-lemma-finite-support-stalk", "more-etale-lemma-lqf-f-shriek-stalk", "more-etale-remark-covariance-f-shriek-separated", "more-etale-lemma-finite-support-f-shriek-separated" ], "ref_ids": [ 8821, 8823, 8849, 8820 ] }, { "id": 8853, "type": "other", "label": "more-etale-remark-alternative-lqf-f-shriek", "categories": [ "more-etale" ], "title": "more-etale-remark-alternative-lqf-f-shriek", "contents": [ "Lemma \\ref{lemma-lqf-colimit-f-shriek}", "gives an alternative construction of the functor $f_!$", "for locally quasi-finite morphisms $f$.", "Namely, given a locally quasi-finite morphism $f : X \\to Y$ of schemes", "we can choose an open covering $X = \\bigcup_{i \\in I} X_i$", "such that each $f_i : X_i \\to Y$ is separated. For example choose", "an affine open covering of $X$. Then we", "can define $f_!\\mathcal{F}$ as the cokernel of the penultimate map", "of the complex of the lemma, i.e.,", "$$", "f_!\\mathcal{F} = \\Coker\\left(", "\\bigoplus\\nolimits_{i_0, i_1} f_{i_0i_1, !} \\mathcal{F}|_{X_{i_0i_1}} \\to", "\\bigoplus\\nolimits_{i_0} f_{i_0, !} \\mathcal{F}|_{X_{i_0}}", "\\right)", "$$", "where we can use the construction of $f_{i_0, !}$ and", "$f_{i_0i_1, !}$ in Section \\ref{section-compact-support}", "because the morphisms $f_{i_0}$ and $f_{i_0 i_1}$ are separated.", "One can then compute the stalks of $f_!$ (using the separated", "case, namely Lemma \\ref{lemma-lqf-f-shriek-separated-colimits})", "and obtain the result of Lemma \\ref{lemma-lqf-f-shriek-stalk}.", "Having done so all the other results of this section can be", "deduced from this as well." ], "refs": [ "more-etale-lemma-lqf-colimit-f-shriek", "more-etale-lemma-lqf-f-shriek-separated-colimits", "more-etale-lemma-lqf-f-shriek-stalk" ], "ref_ids": [ 8824, 8819, 8823 ] }, { "id": 8854, "type": "other", "label": "more-etale-remark-construct-map-presheaves-downstairs", "categories": [ "more-etale" ], "title": "more-etale-remark-construct-map-presheaves-downstairs", "contents": [ "Let $g : Y' \\to Y$ be a morphism of schemes.", "For an abelian presheaf $\\mathcal{G}'$ on $Y'_\\etale$ let us denote", "$g_*\\mathcal{G}'$ the presheaf $V \\mapsto \\mathcal{G}'(Y' \\times_Y V)$.", "If $\\alpha : \\mathcal{G} \\to g_*\\mathcal{G}'$ is a map of abelian presheaves", "on $Y_\\etale$, then there is a unique map", "$\\alpha^\\# : \\mathcal{G}^\\# \\to g_*((\\mathcal{G}')^\\#)$", "of abelian sheaves on $Y_\\etale$ such that the diagram", "$$", "\\xymatrix{", "\\mathcal{G} \\ar[d] \\ar[r]_\\alpha & g_*\\mathcal{G}' \\ar[d] \\\\", "\\mathcal{G}^\\# \\ar[r]^-{\\alpha^\\#} & g_*((\\mathcal{G}')^\\#)", "}", "$$", "is commutative where the vertical maps come from the canonical maps", "$\\mathcal{G} \\to \\mathcal{G}^\\#$ and $\\mathcal{G}' \\to (\\mathcal{G}')^\\#$. If", "$\\alpha' : g^{-1}\\mathcal{G}^\\# \\to (\\mathcal{G}')^\\#$", "is the map adjoint to $\\alpha^\\#$, then for a geometric point", "$\\overline{y}' : \\Spec(k) \\to Y'$ with image", "$\\overline{y} = g \\circ \\overline{y}'$ in $Y$, the map", "$$", "\\alpha'_{\\overline{y}'} :", "\\mathcal{G}_{\\overline{y}} =", "(\\mathcal{G}^\\#)_{\\overline{y}} =", "(g^{-1}\\mathcal{G}^\\#)_{\\overline{y}'}", "\\longrightarrow", "(\\mathcal{G}')^\\#_{\\overline{y}'} =", "\\mathcal{G}'_{\\overline{y}'}", "$$", "is given by mapping the class in the stalk of a section $s$ of $\\mathcal{G}$", "over an \\'etale neighbourhood $(V, \\overline{v})$ to the class of the section", "$\\alpha(s)$ in $g_*\\mathcal{G}'(V) = \\mathcal{G}'(Y' \\times_Y V)$", "over the \\'etale neighbourhood $(Y' \\times_Y V, (\\overline{y}', \\overline{v}))$", "in the stalk of $\\mathcal{G}'$ at $\\overline{y}'$." ], "refs": [], "ref_ids": [] }, { "id": 8855, "type": "other", "label": "more-etale-remark-pointed-sets", "categories": [ "more-etale" ], "title": "more-etale-remark-pointed-sets", "contents": [ "The material in this section can be generalized to sheaves of pointed sets.", "Namely, for a site $\\mathcal{C}$ denote $\\Sh^*(\\mathcal{C})$ the category of", "sheaves of pointed sets. The constructions in this and the preceding section", "apply, mutatis mutandis, to sheaves of pointed sets. Thus given a locally", "quasi-finite morphism $f : X \\to Y$ of schemes we obtain", "an adjoint pair of functors", "$$", "f_! : \\Sh^*(X_\\etale) \\longrightarrow \\Sh^*(Y_\\etale)", "\\quad\\text{and}\\quad", "f^! : \\Sh^*(Y_\\etale) \\longrightarrow \\Sh^*(X_\\etale)", "$$", "such that for every geometric point $\\overline{y}$ of $Y$ there are", "isomorphisms", "$$", "(f_!\\mathcal{F})_{\\overline{y}} =", "\\coprod\\nolimits_{f(\\overline{x}) = \\overline{y}}", "\\mathcal{F}_{\\overline{x}}", "$$", "(coproduct taken in the category of pointed sets) functorial in", "$\\mathcal{F} \\in \\Sh^*(X_\\etale)$ and isomorphisms", "$$", "f^!(\\overline{y}_*S) =", "\\prod\\nolimits_{f(\\overline{x}) = \\overline{y}}", "\\overline{x}_*S", "$$", "functorial in the pointed set $S$. If", "$F : \\textit{Ab}(X_\\etale) \\to \\Sh^*(X_\\etale)$ and", "$F : \\textit{Ab}(Y_\\etale) \\to \\Sh^*(Y_\\etale)$", "denote the forgetful functors, compatibility between the constructions", "will guarantee the existence of canonical maps", "$$", "f_!F(\\mathcal{F}) \\longrightarrow F(f_!\\mathcal{F})", "$$", "functorial in $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ and", "$$", "F(f^!\\mathcal{G}) \\longrightarrow f^!F(\\mathcal{G})", "$$", "functorial in $\\mathcal{G} \\in \\textit{Ab}(Y_\\etale)$", "which produce the obvious maps on stalks, resp.\\ skyscraper sheaves.", "In fact, the transformation $F \\circ f^! \\to f^! \\circ F$ is an isomorphism", "(because $f^!$ commutes with products)." ], "refs": [], "ref_ids": [] }, { "id": 8856, "type": "other", "label": "more-etale-remark-going-around", "categories": [ "more-etale" ], "title": "more-etale-remark-going-around", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X'' \\ar[r]_{k'} \\ar[d]_{f''} & X' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y'' \\ar[r]^{l'} \\ar[d]_{g''} & Y' \\ar[r]^l \\ar[d]_{g'} & Y \\ar[d]^g \\\\", "Z'' \\ar[r]^{m'} & Z' \\ar[r]^m & Z", "}", "$$", "of schemes whose vertical arrows are proper and whose horizontal", "arrows are separated and locally quasi-finite.", "Let us label the squares of the diagram $A$, $B$, $C$, $D$", "as follows", "$$", "\\begin{matrix}", "A & B \\\\", "C & D", "\\end{matrix}", "$$", "Then the maps of Lemma \\ref{lemma-shriek-proper-and-open}", "for the squares are (where we use $Rf_* = f_*$, etc)", "$$", "\\begin{matrix}", "\\gamma_A : l'_! \\circ f''_* \\to f'_* \\circ k'_! &", "\\gamma_B : l_! \\circ f'_* \\to f_* \\circ k_! \\\\", "\\gamma_C : m'_! \\circ g''_* \\to g'_* \\circ l'_! &", "\\gamma_D : m_! \\circ g'_* \\to g_* \\circ l_!", "\\end{matrix}", "$$", "For the $2 \\times 1$ and $1 \\times 2$ rectangles we have four further", "maps", "$$", "\\begin{matrix}", "\\gamma_{A + B} :", "(l \\circ l')_! \\circ f''_* \\to f_* \\circ (k \\circ k')_* \\\\", "\\gamma_{C + D} :", "(m \\circ m')_! \\circ g''_* \\to g_* \\circ (l \\circ l')_! \\\\", "\\gamma_{A + C} :", "m'_! \\circ (g'' \\circ f'')_* \\to (g' \\circ f')_* \\circ k'_! \\\\", "\\gamma_{B + D} :", "m_! \\circ (g' \\circ f')_* \\to (g \\circ f)_* \\circ k_!", "\\end{matrix}", "$$", "By Lemma \\ref{lemma-shriek-proper-and-open-compose-horizontal} we have", "$$", "\\gamma_{A + B} = \\gamma_B \\circ \\gamma_A, \\quad", "\\gamma_{C + D} = \\gamma_D \\circ \\gamma_C", "$$", "and by Lemma \\ref{lemma-shriek-proper-and-open-compose} we have", "$$", "\\gamma_{A + C} = \\gamma_A \\circ \\gamma_C, \\quad", "\\gamma_{B + D} = \\gamma_B \\circ \\gamma_D", "$$", "Here it would be more correct to write", "$\\gamma_{A + B} = (\\gamma_B \\star \\text{id}_{k'_!}) \\circ", "(\\text{id}_{l_!} \\star \\gamma_A)$ with notation as in", "Categories, Section \\ref{categories-section-formal-cat-cat}", "and similarly for the others.", "Having said all of this we find (a priori) two transformations", "$$", "m_! \\circ m'_! \\circ g''_* \\circ f''_*", "\\longrightarrow", "g_* \\circ f_* \\circ k_! \\circ k'_!", "$$", "namely", "$$", "\\gamma_B \\circ \\gamma_D \\circ \\gamma_A \\circ \\gamma_C =", "\\gamma_{B + D} \\circ \\gamma_{A + C}", "$$", "and", "$$", "\\gamma_B \\circ \\gamma_A \\circ \\gamma_D \\circ \\gamma_C =", "\\gamma_{A + B} \\circ \\gamma_{C + D}", "$$", "The point of this remark is to point out that these transformations", "are equal. Namely, to see this it suffices to show that", "$$", "\\xymatrix{", "m_! \\circ g'_* \\circ l'_! \\circ f''_* \\ar[r]_{\\gamma_D} \\ar[d]_{\\gamma_A} &", "g_* \\circ l_! \\circ l'_! \\circ f''_* \\ar[d]^{\\gamma_A} \\\\", "m_! \\circ g'_* \\circ f'_* \\circ k'_! \\ar[r]^{\\gamma_D} &", "g_* \\circ l_! \\circ f'_* \\circ k'_!", "}", "$$", "commutes. This is true because the squares $A$ and $D$ meet in only", "one point, more precisely by", "Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats}", "or more simply the discussion preceding", "Categories, Definition \\ref{categories-definition-horizontal-composition}." ], "refs": [ "more-etale-lemma-shriek-proper-and-open", "more-etale-lemma-shriek-proper-and-open-compose-horizontal", "more-etale-lemma-shriek-proper-and-open-compose", "categories-lemma-properties-2-cat-cats", "categories-definition-horizontal-composition" ], "ref_ids": [ 8834, 8836, 8835, 12269, 12377 ] }, { "id": 8927, "type": "other", "label": "stacks-properties-remark-representable-over", "categories": [ "stacks-properties" ], "title": "stacks-properties-remark-representable-over", "contents": [ "Let $\\mathcal{Y}$ be an algebraic stack. Consider the following", "$2$-category:", "\\begin{enumerate}", "\\item An object is a morphism $f : \\mathcal{X} \\to \\mathcal{Y}$", "which is representable by algebraic spaces,", "\\item a $1$-morphism", "$(g, \\beta) :", "(f_1 : \\mathcal{X}_1 \\to \\mathcal{Y})", "\\to", "(f_2 : \\mathcal{X}_2 \\to \\mathcal{Y})$", "consists of a morphism $g : \\mathcal{X}_1 \\to \\mathcal{X}_2$ and a", "$2$-morphism $\\beta : f_1 \\to f_2 \\circ g$, and", "\\item a $2$-morphism between", "$(g, \\beta), (g', \\beta') :", "(f_1 : \\mathcal{X}_1 \\to \\mathcal{Y})", "\\to", "(f_2 : \\mathcal{X}_2 \\to \\mathcal{Y})$", "is a $2$-morphism $\\alpha : g \\to g'$ such that", "$(\\text{id}_{f_2} \\star \\alpha) \\circ \\beta = \\beta'$.", "\\end{enumerate}", "Let us denote this $2$-category $\\textit{Spaces}/\\mathcal{Y}$ by", "analogy with the notation of", "Topologies on Spaces, Section \\ref{spaces-topologies-section-procedure}.", "Now we claim that in this $2$-category the morphism categories", "$$", "\\Mor_{\\textit{Spaces}/\\mathcal{Y}}(", "(f_1 : \\mathcal{X}_1 \\to \\mathcal{Y}),", "(f_2 : \\mathcal{X}_2 \\to \\mathcal{Y}))", "$$", "are all setoids. Namely, a $2$-morphism $\\alpha$ is a rule which to each", "object $x_1$ of $\\mathcal{X}_1$ assigns an isomorphism", "$\\alpha_{x_1} : g(x_1) \\longrightarrow g'(x_1)$", "in the relevant fibre category of $\\mathcal{X}_2$ such that the diagram", "$$", "\\xymatrix{", "& f_2(x_1) \\ar[ld]_{\\beta_{x_1}} \\ar[rd]^{\\beta'_{x_1}} \\\\", "f_2(g(x_1)) \\ar[rr]^{f_2(\\alpha_{x_1})} & &", "f_2(g'(x_1))", "}", "$$", "commutes. But since $f_2$ is faithful (see", "Algebraic Stacks,", "Lemma \\ref{algebraic-lemma-characterize-representable-by-algebraic-spaces})", "this means that if $\\alpha_{x_1}$ exists, then it is unique! In other words the", "$2$-category $\\textit{Spaces}/\\mathcal{Y}$ is very close to", "being a category. Namely, if we replace $1$-morphisms by isomorphism", "classes of $1$-morphisms we obtain a category. We will often", "perform this replacement without further mention." ], "refs": [ "algebraic-lemma-characterize-representable-by-algebraic-spaces" ], "ref_ids": [ 8469 ] }, { "id": 8928, "type": "other", "label": "stacks-properties-remark-more-general-presentation", "categories": [ "stacks-properties" ], "title": "stacks-properties-remark-more-general-presentation", "contents": [ "The result of", "Lemma \\ref{lemma-points-presentation}", "can be generalized as follows.", "Let $\\mathcal{X}$ be an algebraic stack.", "Let $U$ be an algebraic space and let $f : U \\to \\mathcal{X}$ be a surjective", "morphism (which makes sense by", "Section \\ref{section-properties-morphisms}).", "Let $R = U \\times_\\mathcal{X} U$, let $(U, R, s, t, c)$ be the groupoid", "in algebraic spaces, and let $f_{can} : [U/R] \\to \\mathcal{X}$ be the", "canonical morphism as constructed in", "Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack}.", "Then the image of $|R| \\to |U| \\times |U|$ is an equivalence relation", "and $|\\mathcal{X}| = |U|/|R|$. The proof of", "Lemma \\ref{lemma-points-presentation}", "works without change. (Of course in general $[U/R]$ is not an algebraic", "stack, and in general $f_{can}$ is not an isomorphism.)" ], "refs": [ "stacks-properties-lemma-points-presentation", "algebraic-lemma-map-space-into-stack", "stacks-properties-lemma-points-presentation" ], "ref_ids": [ 8866, 8473, 8866 ] }, { "id": 8929, "type": "other", "label": "stacks-properties-remark-list-properties-local-smooth-topology", "categories": [ "stacks-properties" ], "title": "stacks-properties-remark-list-properties-local-smooth-topology", "contents": [ "Here is a list of properties which are local for the smooth topology", "(keep in mind that the fpqc, fppf, and syntomic topologies are", "stronger than the smooth topology):", "\\begin{enumerate}", "\\item locally Noetherian, see", "Descent, Lemma \\ref{descent-lemma-Noetherian-local-fppf},", "\\item Jacobson, see", "Descent, Lemma \\ref{descent-lemma-Jacobson-local-fppf},", "\\item locally Noetherian and $(S_k)$, see", "Descent, Lemma \\ref{descent-lemma-Sk-local-syntomic},", "\\item Cohen-Macaulay, see", "Descent, Lemma \\ref{descent-lemma-CM-local-syntomic},", "\\item reduced, see", "Descent, Lemma \\ref{descent-lemma-reduced-local-smooth},", "\\item normal, see", "Descent, Lemma \\ref{descent-lemma-normal-local-smooth},", "\\item locally Noetherian and $(R_k)$, see", "Descent, Lemma \\ref{descent-lemma-Rk-local-smooth},", "\\item regular, see", "Descent, Lemma \\ref{descent-lemma-regular-local-smooth},", "\\item Nagata, see", "Descent, Lemma \\ref{descent-lemma-Nagata-local-smooth}.", "\\end{enumerate}" ], "refs": [ "descent-lemma-Noetherian-local-fppf", "descent-lemma-Jacobson-local-fppf", "descent-lemma-Sk-local-syntomic", "descent-lemma-CM-local-syntomic", "descent-lemma-reduced-local-smooth", "descent-lemma-normal-local-smooth", "descent-lemma-Rk-local-smooth", "descent-lemma-regular-local-smooth", "descent-lemma-Nagata-local-smooth" ], "ref_ids": [ 14648, 14649, 14651, 14652, 14653, 14654, 14655, 14656, 14657 ] }, { "id": 8930, "type": "other", "label": "stacks-properties-remark-local-source-warning", "categories": [ "stacks-properties" ], "title": "stacks-properties-remark-local-source-warning", "contents": [ "Warning:", "Lemma \\ref{lemma-local-source}", "should be used with care. For example, it applies to", "$\\mathcal{P}=$``flat'', $\\mathcal{Q}=$``empty'',", "and $\\mathcal{R}=$``flat and locally of finite presentation''. But given a", "morphism of algebraic spaces $f : X \\to Y$ the largest open", "subspace $W \\subset X$ such that $f|_W$ is flat is {\\it not} the set of points", "where $f$ is flat!" ], "refs": [ "stacks-properties-lemma-local-source" ], "ref_ids": [ 8896 ] }, { "id": 8931, "type": "other", "label": "stacks-properties-remark-local-source-apply", "categories": [ "stacks-properties" ], "title": "stacks-properties-remark-local-source-apply", "contents": [ "Notwithstanding the warning in", "Remark \\ref{remark-local-source-warning}", "there are some cases where", "Lemma \\ref{lemma-local-source}", "can be used without causing ambiguity.", "We give a list. In each case we omit the verification of", "assumptions (1) and (2) and we give references which imply", "(3) and (4). Here is the list:", "\\begin{enumerate}", "\\item", "\\label{item-rel-dim-leq-d}", "$\\mathcal{Q} = $``locally of finite type'', $\\mathcal{R} = \\emptyset$,", "and $\\mathcal{P} =$``relative dimension $\\leq d$''.", "See", "Morphisms of Spaces,", "Definition \\ref{spaces-morphisms-definition-relative-dimension}", "and", "Morphisms of Spaces, Lemmas", "\\ref{spaces-morphisms-lemma-openness-bounded-dimension-fibres} and", "\\ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}.", "\\item", "\\label{item-loc-quasi-finite}", "$\\mathcal{Q} =$``locally of finite type'', $\\mathcal{R} = \\emptyset$,", "and $\\mathcal{P} =$``locally quasi-finite''.", "This is the case $d = 0$ of the previous item, see", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0}.", "On the other hand, properties (3) and (4) are spelled out in", "Morphisms of Spaces, Lemma", "\\ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part}.", "\\item", "\\label{item-unramified}", "$\\mathcal{Q} = $``locally of finite type'', $\\mathcal{R} = \\emptyset$,", "and $\\mathcal{P} =$``unramified''. This is", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-where-unramified}.", "\\item", "\\label{item-flat}", "$\\mathcal{Q} =$``locally of finite presentation'',", "$\\mathcal{R} =$``flat and locally of finite presentation'', and", "$\\mathcal{P} =$``flat''. See", "More on Morphisms of Spaces, Theorem", "\\ref{spaces-more-morphisms-theorem-openness-flatness} and", "Lemma \\ref{spaces-more-morphisms-lemma-flat-locus-base-change}.", "Note that here $W(\\mathcal{P}, f)$ is always exactly the set of points", "where the morphism $f$ is flat because we only consider this open", "when $f$ has $\\mathcal{Q}$ (see loc.cit.).", "\\item", "\\label{item-etale}", "$\\mathcal{Q} =$``locally of finite presentation'',", "$\\mathcal{R} =$``flat and locally of finite presentation'', and", "$\\mathcal{P}=$``\\'etale''. This follows on combining", "(\\ref{item-unramified}) and (\\ref{item-flat}) because an unramified", "morphism which is flat and locally of finite presentation is \\'etale, see", "Morphisms of Spaces,", "Lemma \\ref{spaces-morphisms-lemma-unramified-flat-lfp-etale}.", "\\item Add more here as needed (compare with the longer list at", "More on Groupoids, Remark \\ref{more-groupoids-remark-local-source-apply}).", "\\end{enumerate}" ], "refs": [ "stacks-properties-remark-local-source-warning", "stacks-properties-lemma-local-source", "spaces-morphisms-definition-relative-dimension", "spaces-morphisms-lemma-openness-bounded-dimension-fibres", "spaces-morphisms-lemma-dimension-fibre-after-base-change", "spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0", "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part", "spaces-morphisms-lemma-where-unramified", "spaces-more-morphisms-theorem-openness-flatness", "spaces-more-morphisms-lemma-flat-locus-base-change", "spaces-morphisms-lemma-unramified-flat-lfp-etale", "more-groupoids-remark-local-source-apply" ], "ref_ids": [ 8930, 8896, 5010, 4873, 4872, 4875, 4876, 4903, 10, 129, 4915, 2507 ] }, { "id": 8932, "type": "other", "label": "stacks-properties-remark-stack-structure-locally-closed-subset", "categories": [ "stacks-properties" ], "title": "stacks-properties-remark-stack-structure-locally-closed-subset", "contents": [ "Let $X$ be an algebraic stack.", "Let $T \\subset |\\mathcal{X}|$ be a locally closed subset.", "Let $\\partial T$ be the boundary of $T$ in", "the topological space $|\\mathcal{X}|$. In a formula", "$$", "\\partial T = \\overline{T} \\setminus T.", "$$", "Let $\\mathcal{U} \\subset \\mathcal{X}$ be the open substack of $X$ with", "$|\\mathcal{U}| = |\\mathcal{X}| \\setminus \\partial T$, see", "Lemma \\ref{lemma-open-substacks}.", "Let $\\mathcal{Z}$ be the reduced closed substack of $\\mathcal{U}$ with", "$|\\mathcal{Z}| = T$ obtained by taking the reduced induced", "closed subspace structure, see", "Definition \\ref{definition-reduced-induced-stack}.", "By construction $\\mathcal{Z} \\to \\mathcal{U}$ is a closed immersion of", "algebraic stacks and $\\mathcal{U} \\to \\mathcal{X}$ is an open immersion,", "hence $\\mathcal{Z} \\to \\mathcal{X}$ is an immersion of algebraic stacks by", "Lemma \\ref{lemma-composition-immersion}.", "Note that $\\mathcal{Z}$ is a reduced algebraic stack and that", "$|\\mathcal{Z}| = T$ as subsets of $|X|$. We sometimes say", "$\\mathcal{Z}$ is the {\\it reduced induced substack structure} on $T$." ], "refs": [ "stacks-properties-lemma-open-substacks", "stacks-properties-definition-reduced-induced-stack", "stacks-properties-lemma-composition-immersion" ], "ref_ids": [ 8890, 8922, 8883 ] }, { "id": 8933, "type": "other", "label": "stacks-properties-remark-dimension-empty-stack", "categories": [ "stacks-properties" ], "title": "stacks-properties-remark-dimension-empty-stack", "contents": [ "If $\\mathcal{X}$ is a nonempty stack of finite type over a field,", "then $\\dim(\\mathcal{X})$ is an integer. For an arbitrary", "locally Noetherian algebraic stack $\\mathcal{X}$,", "$\\dim(\\mathcal{X})$ is in $Z\\cup \\{\\pm \\infty\\}$,", "and $\\dim(\\mathcal{X}) = -\\infty$ if and only if $\\mathcal{X}$", "is empty." ], "refs": [], "ref_ids": [] }, { "id": 9007, "type": "other", "label": "stacks-remark-alternative", "categories": [ "stacks" ], "title": "stacks-remark-alternative", "contents": [ "Suppose that $p : \\mathcal{S} \\to \\mathcal{C}$ is fibred in groupoids.", "In this case we can prove", "Lemma \\ref{lemma-painful}", "using", "Categories, Lemma \\ref{categories-lemma-fibred-strict}", "which says that $\\mathcal{S} \\to \\mathcal{C}$ is equivalent to the", "category associated to a contravariant functor", "$F : \\mathcal{C} \\to \\textit{Groupoids}$.", "In the case of the fibred category associated to $F$", "we have $g^* \\circ f^* = (f \\circ g)^*$ on the nose", "and there is no need to use the maps $\\alpha_{g, f}$.", "In this case the lemma is (even more) trivial. Of course then", "one uses that the $\\mathit{Mor}(x, y)$ presheaf is", "unchanged when passing to an equivalent fibred category which follows from", "Lemma \\ref{lemma-presheaf-mor-map-fibred-categories}." ], "refs": [ "stacks-lemma-painful", "categories-lemma-fibred-strict", "stacks-lemma-presheaf-mor-map-fibred-categories" ], "ref_ids": [ 8934, 12307, 8935 ] }, { "id": 9008, "type": "other", "label": "stacks-remark-stack-make-small", "categories": [ "stacks" ], "title": "stacks-remark-stack-make-small", "contents": [ "(Cutting down a ``big'' stack to get a stack.)", "Let $\\mathcal{C}$ be a site. Suppose that $p : \\mathcal{S} \\to \\mathcal{C}$", "is functor from a ``big'' category to $\\mathcal{C}$, i.e., suppose", "that the collection of objects of $\\mathcal{S}$ forms a proper class.", "Finally, suppose that $p : \\mathcal{S} \\to \\mathcal{C}$ satisfies", "conditions (1), (2), (3) of", "Definition \\ref{definition-stack}.", "In general there is no way to replace $p : \\mathcal{S} \\to \\mathcal{C}$", "by a equivalent category such that we obtain a stack. The reason is that", "it can happen that a fibre categories $\\mathcal{S}_U$ may have a proper", "class of isomorphism classes of objects.", "On the other hand, suppose that", "\\begin{enumerate}", "\\item[(4)] for every $U \\in \\Ob(\\mathcal{C})$ there exists a set", "$S_U \\subset \\Ob(\\mathcal{S}_U)$ such that every object of", "$\\mathcal{S}_U$ is isomorphic in $\\mathcal{S}_U$ to an element of $S_U$.", "\\end{enumerate}", "In this case we can find a full subcategory $\\mathcal{S}_{small}$", "of $\\mathcal{S}$ such that, setting $p_{small} = p|_{\\mathcal{S}_{small}}$,", "we have", "\\begin{enumerate}", "\\item[(a)] the functor $p_{small} : \\mathcal{S}_{small} \\to \\mathcal{C}$", "defines a stack, and", "\\item[(b)] the inclusion $\\mathcal{S}_{small} \\to \\mathcal{S}$", "is fully faithful and essentially surjective.", "\\end{enumerate}", "(Hint: For every $U \\in \\Ob(\\mathcal{C})$", "let $\\alpha(U)$ denote the smallest ordinal such that", "$\\Ob(\\mathcal{S}_U) \\cap V_{\\alpha(U)}$ surjects onto the set", "of isomorphism classes of $\\mathcal{S}_U$, and set", "$\\alpha = \\sup_{U \\in \\Ob(\\mathcal{C})} \\alpha(U)$.", "Then take", "$\\Ob(\\mathcal{S}_{small}) = \\Ob(\\mathcal{S}) \\cap V_\\alpha$.", "For notation used see Sets, Section \\ref{sets-section-sets-hierarchy}.)" ], "refs": [ "stacks-definition-stack" ], "ref_ids": [ 8996 ] }, { "id": 9009, "type": "other", "label": "stacks-remarks-definition-descent-datum", "categories": [ "stacks" ], "title": "stacks-remarks-definition-descent-datum", "contents": [ "Two remarks on Definition \\ref{definition-descent-data} are in order.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category.", "Let $\\{f_i : U_i \\to U\\}_{i \\in I}$, and $(X_i, \\varphi_{ij})$", "be as in Definition \\ref{definition-descent-data}.", "\\begin{enumerate}", "\\item There is a diagonal morphism $\\Delta : U_i \\to U_i \\times_U U_i$.", "We can pull back $\\varphi_{ii}$ via this morphism to get an automorphism", "$\\Delta^\\ast \\varphi_{ii} \\in \\text{Aut}_{U_i}(x_i)$.", "On pulling back the cocycle condition for the triple $(i, i, i)$", "by $\\Delta_{123} : U_i \\to U_i \\times_U U_i \\times_U U_i$ we deduce that", "$\\Delta^\\ast \\varphi_{ii} \\circ \\Delta^\\ast \\varphi_{ii} =", "\\Delta^\\ast \\varphi_{ii}$; thus $\\Delta^\\ast \\varphi_{ii} =", "\\text{id}_{x_i}$.", "\\item There is a morphism", "$\\Delta_{13}: U_i \\times_U U_j \\to U_i \\times_U U_j \\times_U U_i$", "and we can pull back the", "cocycle condition for the triple $(i, j, i)$ to get the", "identity $(\\sigma^\\ast \\varphi_{ji}) \\circ \\varphi_{ij} =", "\\text{id}_{\\text{pr}_0^\\ast x_i}$, where", "$\\sigma : U_i \\times_U U_j \\to U_j \\times_U U_i$ is the switching morphism.", "\\end{enumerate}" ], "refs": [ "stacks-definition-descent-data", "stacks-definition-descent-data" ], "ref_ids": [ 8993, 8993 ] }, { "id": 9152, "type": "other", "label": "spaces-simplicial-remark-augmentation-site", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-remark-augmentation-site", "contents": [ "In Situation \\ref{situation-simplicial-site} an", "{\\it augmentation $a_0$ towards a site $\\mathcal{D}$} will mean", "\\begin{enumerate}", "\\item[(A)] $a_0 : \\mathcal{C}_0 \\to \\mathcal{D}$ is a morphism of sites", "given by a continuous functor $u_0 : \\mathcal{D} \\to \\mathcal{C}_0$", "such that for all $\\varphi, \\psi : [0] \\to [n]$ we have", "$u_\\varphi \\circ u_0 = u_\\psi \\circ u_0$.", "\\item[(B)] $a_0 : \\Sh(\\mathcal{C}_0) \\to \\Sh(\\mathcal{D})$ is a morphism", "of topoi given by a cocontinuous functor $u_0 : \\mathcal{C}_0 \\to \\mathcal{D}$", "such that for all $\\varphi, \\psi : [0] \\to [n]$ we have", "$u_0 \\circ u_\\varphi = u_0 \\circ u_\\psi$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 9153, "type": "other", "label": "spaces-simplicial-remark-morphism-simplicial-sites", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-remark-morphism-simplicial-sites", "contents": [ "Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi$ and", "$\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi$ be as in", "Situation \\ref{situation-simplicial-site}. A", "{\\it morphism $h$ between simplicial sites} will mean", "\\begin{enumerate}", "\\item[(A)] Morphisms of sites", "$h_n : \\mathcal{C}_n \\to \\mathcal{C}'_n$", "such that $f'_\\varphi \\circ h_n = h_m \\circ f_\\varphi$", "as morphisms of sites for all $\\varphi : [m] \\to [n]$.", "\\item[(B)] Cocontinuous functors", "$v_n : \\mathcal{C}_n \\to \\mathcal{C}'_n$", "inducing morphisms of topoi $h_n : \\Sh(\\mathcal{C}_n) \\to \\Sh(\\mathcal{C}'_n)$", "such that $u'_\\varphi \\circ v_n = v_m \\circ u_\\varphi$", "as functors for all $\\varphi : [m] \\to [n]$.", "\\end{enumerate}", "In both cases we have", "$f'_\\varphi \\circ h_n = h_m \\circ f_\\varphi$", "as morphisms of topoi, see", "Sites, Lemma \\ref{sites-lemma-composition-cocontinuous}", "for case B and Sites,", "Definition \\ref{sites-definition-composition-morphisms-sites}", "for case A." ], "refs": [ "sites-lemma-composition-cocontinuous", "sites-definition-composition-morphisms-sites" ], "ref_ids": [ 8544, 8666 ] }, { "id": 9154, "type": "other", "label": "spaces-simplicial-remark-morphism-augmentation-simplicial-sites", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-remark-morphism-augmentation-simplicial-sites", "contents": [ "Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi$ and", "$\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi$ be as in", "Situation \\ref{situation-simplicial-site}.", "Let $a_0$, resp.\\ $a'_0$ be an augmentation", "towards a site $\\mathcal{D}$, resp.\\ $\\mathcal{D}'$", "as in Remark \\ref{remark-augmentation-site}.", "Let $h$ be a morphism between simplicial sites as in", "Remark \\ref{remark-morphism-simplicial-sites}.", "We say a morphism of topoi $h_{-1} : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{D}')$", "is {\\it compatible with $h$, $a_0$, $a'_0$} if", "\\begin{enumerate}", "\\item[(A)] $h_{-1}$ comes from a morphism of sites", "$h_{-1} : \\mathcal{D} \\to \\mathcal{D}'$", "such that $a'_0 \\circ h_0 = h_{-1} \\circ a_0$", "as morphisms of sites.", "\\item[(B)] $h_{-1}$ comes from a cocontinuous functor", "$v_{-1} : \\mathcal{D} \\to \\mathcal{D}'$", "such that $u'_0 \\circ v_0 = v_{-1} \\circ u_0$", "as functors.", "\\end{enumerate}", "In both cases we have $a'_0 \\circ h_0 = h_{-1} \\circ a_0$", "as morphisms of topoi, see", "Sites, Lemma \\ref{sites-lemma-composition-cocontinuous}", "for case B and Sites,", "Definition \\ref{sites-definition-composition-morphisms-sites}", "for case A." ], "refs": [ "spaces-simplicial-remark-augmentation-site", "spaces-simplicial-remark-morphism-simplicial-sites", "sites-lemma-composition-cocontinuous", "sites-definition-composition-morphisms-sites" ], "ref_ids": [ 9152, 9153, 8544, 8666 ] }, { "id": 9155, "type": "other", "label": "spaces-simplicial-remark-morphism-simplicial-sites-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-remark-morphism-simplicial-sites-modules", "contents": [ "Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi$ and", "$\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi$ be as in", "Situation \\ref{situation-simplicial-site}.", "Let $\\mathcal{O}$ and $\\mathcal{O}'$", "be a sheaf of rings on $\\mathcal{C}_{total}$ and $\\mathcal{C}'_{total}$.", "We will say that $(h, h^\\sharp)$ is a", "{\\it morphism between ringed simplicial sites}", "if $h$ is a morphism between simplicial sites as in", "Remark \\ref{remark-morphism-simplicial-sites}", "and $h^\\sharp : h_{total}^{-1}\\mathcal{O}' \\to \\mathcal{O}$", "or equivalently $h^\\sharp : \\mathcal{O}' \\to h_{total, *}\\mathcal{O}$", "is a homomorphism of sheaves of rings." ], "refs": [ "spaces-simplicial-remark-morphism-simplicial-sites" ], "ref_ids": [ 9153 ] }, { "id": 9156, "type": "other", "label": "spaces-simplicial-remark-warning-cartesian-modules", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-remark-warning-cartesian-modules", "contents": [ "Lemma \\ref{lemma-Serre-subcat-cartesian-modules} notwithstanding, it", "can happen that the category of cartesian $\\mathcal{O}$-modules is", "abelian without being a Serre subcategory of $\\textit{Mod}(\\mathcal{O})$.", "Namely, suppose that we only know that", "$f_{\\delta_1^1}$ and $f_{\\delta_0^1}$ are flat.", "Then it follows easily from", "Lemma \\ref{lemma-characterize-cartesian-modules}", "that the category of cartesian $\\mathcal{O}$-modules is abelian.", "But if $f_{\\delta_0^2}$ is not flat (for example),", "there is no reason for the inclusion functor", "from the category of cartesian $\\mathcal{O}$-modules", "to all $\\mathcal{O}$-modules to be exact." ], "refs": [ "spaces-simplicial-lemma-Serre-subcat-cartesian-modules", "spaces-simplicial-lemma-characterize-cartesian-modules" ], "ref_ids": [ 9056, 9055 ] }, { "id": 9157, "type": "other", "label": "spaces-simplicial-remark-semi-representable-over-object", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-remark-semi-representable-over-object", "contents": [ "Let $\\mathcal{C}$ be a site. Let $X \\in \\Ob(\\mathcal{C})$.", "The category $\\text{SR}(\\mathcal{C}, X)$", "of {\\it semi-representable objects over $X$}", "is defined by the formula", "$\\text{SR}(\\mathcal{C}, X) = \\text{SR}(\\mathcal{C}/X)$.", "See Hypercoverings, Definition \\ref{hypercovering-definition-SR}.", "Thus we may apply the above discussion to the site", "$\\mathcal{C}/X$. Briefly, the constructions above give", "\\begin{enumerate}", "\\item a site $\\mathcal{C}/K$ for $K$ in $\\text{SR}(\\mathcal{C}, X)$,", "\\item a decomposition", "$\\Sh(\\mathcal{C}/K) = \\prod \\Sh(\\mathcal{C}/U_i)$ if $K = \\{U_i/X\\}$,", "\\item a localization functor $j : \\mathcal{C}/K \\to \\mathcal{C}/X$,", "\\item a morphism $f : \\Sh(\\mathcal{C}/K) \\to \\Sh(\\mathcal{C}/L)$", "for $f : K \\to L$ in $\\text{SR}(\\mathcal{C}, X)$.", "\\end{enumerate}", "All results of this section hold in this situation by replacing", "$\\mathcal{C}$ everywhere by $\\mathcal{C}/X$." ], "refs": [ "hypercovering-definition-SR" ], "ref_ids": [ 8421 ] }, { "id": 9158, "type": "other", "label": "spaces-simplicial-remark-semi-representable-ringed", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-remark-semi-representable-ringed", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_\\mathcal{C}$", "be a sheaf of rings on $\\mathcal{C}$. In this case, for any", "semi-representable object $K$ of $\\mathcal{C}$ the site", "$\\mathcal{C}/K$ is a ringed site with sheaf", "of rings $\\mathcal{O}_K = j^{-1}\\mathcal{O}_\\mathcal{C}$.", "The constructions above give", "\\begin{enumerate}", "\\item a ringed site $(\\mathcal{C}/K, \\mathcal{O}_K)$", "for $K$ in $\\text{SR}(\\mathcal{C})$,", "\\item a decomposition", "$\\textit{Mod}(\\mathcal{O}_K) =", "\\prod \\textit{Mod}(\\mathcal{O}_{U_i})$ if $K = \\{U_i\\}$,", "\\item a localization morphism", "$j : (\\Sh(\\mathcal{C}/K), \\mathcal{O}_K) \\to", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$", "of ringed topoi,", "\\item a morphism", "$f : (\\Sh(\\mathcal{C}/K), \\mathcal{O}_K) \\to", "(\\Sh(\\mathcal{C}/L), \\mathcal{O}_L)$ of ringed topoi", "for $f : K \\to L$ in $\\text{SR}(\\mathcal{C})$.", "\\end{enumerate}", "Many of the results above hold in this setting. For example, the", "functor $j^*$ has an exact left adjoint", "$$", "j_! : \\textit{Mod}(\\mathcal{O}_K) \\to \\textit{Mod}(\\mathcal{O}_\\mathcal{C}),", "$$", "which in terms of the product decomposition given in (2) sends", "$(\\mathcal{F}_i)_{i \\in I}$ to $\\bigoplus j_{i, !}\\mathcal{F}_i$.", "Similarly, given $f : K \\to L$ as above, the functor $f^*$ has", "an exact left adjoint", "$f_! : \\textit{Mod}(\\mathcal{O}_K) \\to \\textit{Mod}(\\mathcal{O}_L)$.", "Thus the functors $j^*$ and $f^*$ are exact, i.e.,", "$j$ and $f$ are flat morphisms of ringed topoi (also follows", "from the equalities $\\mathcal{O}_K = j^{-1}\\mathcal{O}_\\mathcal{C}$", "and $\\mathcal{O}_K = f^{-1}\\mathcal{O}_L$)." ], "refs": [], "ref_ids": [] }, { "id": 9159, "type": "other", "label": "spaces-simplicial-remark-semi-representable-ringed-over-object", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-remark-semi-representable-ringed-over-object", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_\\mathcal{C}$", "be a sheaf of rings on $\\mathcal{C}$. Let $X \\in \\Ob(\\mathcal{C})$", "and denote $\\mathcal{O}_X = \\mathcal{O}_\\mathcal{C}|_{\\mathcal{C}/U}$.", "Then we can combine the constructions given in", "Remarks \\ref{remark-semi-representable-over-object}", "and \\ref{remark-semi-representable-ringed} to get", "\\begin{enumerate}", "\\item a ringed site $(\\mathcal{C}/K, \\mathcal{O}_K)$", "for $K$ in $\\text{SR}(\\mathcal{C}, X)$,", "\\item a decomposition", "$\\textit{Mod}(\\mathcal{O}_K) =", "\\prod \\textit{Mod}(\\mathcal{O}_{U_i})$ if $K = \\{U_i\\}$,", "\\item a localization morphism", "$j : (\\Sh(\\mathcal{C}/K), \\mathcal{O}_K) \\to", "(\\Sh(\\mathcal{C}/X), \\mathcal{O}_X)$", "of ringed topoi,", "\\item a morphism", "$f : (\\Sh(\\mathcal{C}/K), \\mathcal{O}_K) \\to", "(\\Sh(\\mathcal{C}/L), \\mathcal{O}_L)$ of ringed topoi", "for $f : K \\to L$ in $\\text{SR}(\\mathcal{C}, X)$.", "\\end{enumerate}", "Of course all of the results mentioned in", "Remark \\ref{remark-semi-representable-ringed}", "hold in this setting as well." ], "refs": [ "spaces-simplicial-remark-semi-representable-over-object", "spaces-simplicial-remark-semi-representable-ringed", "spaces-simplicial-remark-semi-representable-ringed" ], "ref_ids": [ 9157, 9158, 9158 ] }, { "id": 9160, "type": "other", "label": "spaces-simplicial-remark-augmentation-over-object", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-remark-augmentation-over-object", "contents": [ "Let $\\mathcal{C}$ be a site. Let $X \\in \\Ob(\\mathcal{C})$.", "Recall that we have a category", "$\\text{SR}(\\mathcal{C}, X) = \\text{SR}(\\mathcal{C}/X)$", "of semi-representable objects over $X$,", "see Remark \\ref{remark-semi-representable-over-object}.", "We may apply the above discussion to the site", "$\\mathcal{C}/X$. Briefly, the constructions above give", "\\begin{enumerate}", "\\item a site $(\\mathcal{C}/K)_{total}$ for a simplicial $K$ object", "of $\\text{SR}(\\mathcal{C}, X)$,", "\\item a localization functor", "$j_{total} : (\\mathcal{C}/K)_{total} \\to \\mathcal{C}/X$,", "\\item localization functors $j_n : \\mathcal{C}/K_n \\to \\mathcal{C}/X$,", "\\item a morphism of topoi", "$a : \\Sh((\\mathcal{C}/K)_{total}) \\to \\Sh(\\mathcal{C}/X)$,", "\\item morphisms of topoi", "$a_n : \\Sh(\\mathcal{C}/K_n) \\to \\Sh(\\mathcal{C}/X)$,", "\\item a functor", "$a^{Sh}_! : \\Sh((\\mathcal{C}/K)_{total}) \\to \\Sh(\\mathcal{C}/X)$", "left adjoint to $a^{-1}$, and", "\\item a functor", "$a_! : \\textit{Ab}((\\mathcal{C}/K)_{total}) \\to \\textit{Ab}(\\mathcal{C}/X)$", "left adjoint to $a^{-1}$.", "\\end{enumerate}", "All of the results of this section hold in this setting.", "To prove this one replaces", "the site $\\mathcal{C}$ everywhere by $\\mathcal{C}/X$." ], "refs": [ "spaces-simplicial-remark-semi-representable-over-object" ], "ref_ids": [ 9157 ] }, { "id": 9161, "type": "other", "label": "spaces-simplicial-remark-augmentation-ringed", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-remark-augmentation-ringed", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.", "Given a simplicial semi-representable object $K$ of $\\mathcal{C}$", "we set $\\mathcal{O} = a^{-1}\\mathcal{O}_\\mathcal{C}$, where $a$", "is as in Lemmas \\ref{lemma-augmentation-simplicial-semi-representable} and", "\\ref{lemma-comparison}.", "The constructions above, keeping track of the sheaves of rings", "as in Remark \\ref{remark-semi-representable-ringed}, give", "\\begin{enumerate}", "\\item a ringed site $((\\mathcal{C}/K)_{total}, \\mathcal{O})$", "for a simplicial $K$ object of $\\text{SR}(\\mathcal{C})$,", "\\item a morphism of ringed topoi", "$a : (\\Sh((\\mathcal{C}/K)_{total}), \\mathcal{O}) \\to", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$,", "\\item morphisms of ringed topoi", "$a_n : (\\Sh(\\mathcal{C}/K_n), \\mathcal{O}_n) \\to", "(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$,", "\\item a functor", "$a_! : \\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_\\mathcal{C})$", "left adjoint to $a^*$.", "\\end{enumerate}", "The functor $a_!$ exists (but in general is not exact)", "because $a^{-1}\\mathcal{O}_\\mathcal{C} = \\mathcal{O}$", "and we can replace the use of", "Modules on Sites, Lemma \\ref{sites-modules-lemma-g-shriek-adjoint}", "in the proof of Lemma \\ref{lemma-comparison}", "by Modules on Sites, Lemma \\ref{sites-modules-lemma-lower-shriek-modules}.", "As discussed in Remark \\ref{remark-semi-representable-ringed}", "there are exact functors", "$a_{n!} : \\textit{Mod}(\\mathcal{O}_n) \\to", "\\textit{Mod}(\\mathcal{O}_\\mathcal{C})$", "left adjoint to $a_n^*$. Consequently, the morphisms $a$ and $a_n$ are flat.", "Remark \\ref{remark-semi-representable-ringed}", "implies the morphism of ringed topoi", "$f_\\varphi : (\\Sh(\\mathcal{C}/K_n), \\mathcal{O}_n) \\to", "(\\Sh(\\mathcal{C}/K_m), \\mathcal{O}_m)$", "for $\\varphi : [m] \\to [n]$ is flat and there exists an exact functor", "$f_{\\varphi !} : \\textit{Mod}(\\mathcal{O}_n) \\to \\textit{Mod}(\\mathcal{O}_m)$", "left adjoint to $f_\\varphi^*$. This in turn implies that for", "the flat morphism of ringed topoi", "$g_n : (\\Sh(\\mathcal{C}/K_n), \\mathcal{O}_n) \\to", "(\\Sh((\\mathcal{C}/K)_{total}), \\mathcal{O})$", "the functor $g_{n!} : \\textit{Mod}(\\mathcal{O}_n) \\to", "\\textit{Mod}(\\mathcal{O})$ left adjoint to $g_n^*$ is exact, see", "Lemma \\ref{lemma-exactness-g-shriek-modules}." ], "refs": [ "spaces-simplicial-lemma-augmentation-simplicial-semi-representable", "spaces-simplicial-lemma-comparison", "spaces-simplicial-remark-semi-representable-ringed", "sites-modules-lemma-g-shriek-adjoint", "spaces-simplicial-lemma-comparison", "sites-modules-lemma-lower-shriek-modules", "spaces-simplicial-remark-semi-representable-ringed", "spaces-simplicial-remark-semi-representable-ringed", "spaces-simplicial-lemma-exactness-g-shriek-modules" ], "ref_ids": [ 9074, 9075, 9158, 14164, 9075, 14262, 9158, 9158, 9033 ] }, { "id": 9162, "type": "other", "label": "spaces-simplicial-remark-augmentation-ringed-over-object", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-remark-augmentation-ringed-over-object", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.", "Let $X \\in \\Ob(\\mathcal{C})$ and denote", "$\\mathcal{O}_X = \\mathcal{O}_\\mathcal{C}|_{\\mathcal{C}/X}$.", "Then we can combine the constructions given in", "Remarks \\ref{remark-augmentation-over-object} and", "\\ref{remark-augmentation-ringed}", "to get", "\\begin{enumerate}", "\\item a ringed site $((\\mathcal{C}/K)_{total}, \\mathcal{O})$", "for a simplicial $K$ object of $\\text{SR}(\\mathcal{C}, X)$,", "\\item a morphism of ringed topoi", "$a : (\\Sh((\\mathcal{C}/K)_{total}), \\mathcal{O}) \\to", "(\\Sh(\\mathcal{C}/X), \\mathcal{O}_X)$,", "\\item morphisms of ringed topoi", "$a_n : (\\Sh(\\mathcal{C}/K_n), \\mathcal{O}_n) \\to", "(\\Sh(\\mathcal{C}/X), \\mathcal{O}_X)$,", "\\item a functor", "$a_! : \\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_X)$", "left adjoint to $a^*$.", "\\end{enumerate}", "Of course, all the results mentioned in", "Remark \\ref{remark-augmentation-ringed}", "hold in this setting as well." ], "refs": [ "spaces-simplicial-remark-augmentation-over-object", "spaces-simplicial-remark-augmentation-ringed", "spaces-simplicial-remark-augmentation-ringed" ], "ref_ids": [ 9160, 9161, 9161 ] }, { "id": 9163, "type": "other", "label": "spaces-simplicial-remark-compare-cohomology-hypercovering-presheaf", "categories": [ "spaces-simplicial" ], "title": "spaces-simplicial-remark-compare-cohomology-hypercovering-presheaf", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{G}$ be a presheaf of sets on", "$\\mathcal{C}$. If $\\mathcal{C}$ has equalizers and fibre products, then", "we've defined the notion of a hypercovering of $\\mathcal{G}$ in", "Hypercoverings, Definition \\ref{hypercovering-definition-hypercovering-variant}.", "We claim that all the results in this section have a", "valid counterpart in this setting.", "To see this,", "define the localization $\\mathcal{C}/\\mathcal{G}$", "of $\\mathcal{C}$ at $\\mathcal{G}$ exactly as in", "Sites, Lemma \\ref{sites-lemma-localize-topos-site}", "(which is stated only for sheaves; the topos", "$\\Sh(\\mathcal{C}/\\mathcal{G})$ is equal to the localization", "of the topos $\\Sh(\\mathcal{C})$ at the sheaf $\\mathcal{G}^\\#$).", "Then the reader easily shows that the site", "$\\mathcal{C}/\\mathcal{G}$ has fibre products and equalizers", "and that a hypercovering of $\\mathcal{G}$ in $\\mathcal{C}$", "is the same thing as a hypercovering for the site $\\mathcal{C}/\\mathcal{G}$.", "Hence replacing the site $\\mathcal{C}$ by $\\mathcal{C}/\\mathcal{G}$", "in the lemmas on hypercoverings above we obtain proofs of the", "corresponding results for hypercoverings of $\\mathcal{G}$.", "Example: for a hypercovering $K$ of $\\mathcal{G}$ we have", "$$", "R\\Gamma(\\mathcal{C}/\\mathcal{G}, E) =", "R\\Gamma((\\mathcal{C}/K)_{total}, a^{-1}E)", "$$", "for $E \\in D^+(\\mathcal{C}/\\mathcal{G})$ where", "$a : \\Sh((\\mathcal{C}/K)_{total}) \\to \\Sh(\\mathcal{C}/\\mathcal{G})$", "is the canonical augmentation. This is", "Lemma \\ref{lemma-compare-cohomology-hypercovering}.", "Let $R\\Gamma(\\mathcal{G}, -) : D(\\mathcal{C}) \\to D(\\textit{Ab})$", "be defined as the derived functor of the functor", "$H^0(\\mathcal{G}, -) = H^0(\\mathcal{G}^\\#, -)$", "discussed in Hypercoverings, Section", "\\ref{hypercovering-section-hypercoverings-verdier} and", "Cohomology on Sites, Section \\ref{sites-cohomology-section-limp}.", "We have", "$$", "R\\Gamma(\\mathcal{G}, E) = R\\Gamma(\\mathcal{C}/\\mathcal{G}, j^{-1}E)", "$$", "by the analogue of Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-cohomology-of-open}", "for the localization fuctor $j : \\mathcal{C}/\\mathcal{G} \\to \\mathcal{C}$.", "Putting everything together we obtain", "$$", "R\\Gamma(\\mathcal{G}, E) =", "R\\Gamma((\\mathcal{C}/K)_{total}, a^{-1}j^{-1}E) =", "R\\Gamma((\\mathcal{C}/K)_{total}, g^{-1}E)", "$$", "for $E \\in D^+(\\mathcal{C})$ where", "$g : \\Sh((\\mathcal{C}/K)_{total}) \\to \\Sh(\\mathcal{C})$", "is the composition of $a$ and $j$." ], "refs": [ "hypercovering-definition-hypercovering-variant", "sites-lemma-localize-topos-site", "spaces-simplicial-lemma-compare-cohomology-hypercovering", "sites-cohomology-lemma-cohomology-of-open" ], "ref_ids": [ 8426, 8585, 9080, 4186 ] }, { "id": 9185, "type": "other", "label": "examples-stacks-remark-higher-rank", "categories": [ "examples-stacks" ], "title": "examples-stacks-remark-higher-rank", "contents": [ "Note that the whole discussion in this section works", "if we want to consider those", "quasi-coherent sheaves which are locally generated by at most $\\kappa$", "sections, for some infinite cardinal $\\kappa$, e.g., $\\kappa = \\aleph_0$." ], "refs": [], "ref_ids": [] }, { "id": 9186, "type": "other", "label": "examples-stacks-remark-stack-spaces", "categories": [ "examples-stacks" ], "title": "examples-stacks-remark-stack-spaces", "contents": [ "Ignoring set theoretical difficulties\\footnote{The difficulty is not", "that $\\Spacesstack$ is a proper class, since by our definition of", "an algebraic space over $S$ there is only a set worth of isomorphism", "classes of algebraic spaces over $S$. It is rather that arbitrary disjoint", "unions of algebraic spaces may end up being too large, hence lie outside", "of our chosen ``partial universe'' of sets.}", "$\\Spacesstack$ also satisfies", "descent for objects and hence is a stack. Namely, we have to show that", "given", "\\begin{enumerate}", "\\item an fppf covering $\\{U_i \\to U\\}_{i \\in I}$,", "\\item for each $i \\in I$ an algebraic space $X_i/U_i$, and", "\\item for each $i, j \\in I$ an isomorphism", "$\\varphi_{ij} : X_i \\times_U U_j \\to U_i \\times_U X_j$ of algebraic spaces", "over $U_i \\times_U U_j$ satisfying the cocycle condition over", "$U_i \\times_U U_j \\times_U U_k$,", "\\end{enumerate}", "there exists an algebraic space $X/U$ and isomorphisms", "$X_{U_i} \\cong X_i$ over $U_i$ recovering the isomorphisms $\\varphi_{ij}$.", "First, note that by", "Sites, Lemma \\ref{sites-lemma-glue-sheaves}", "there exists a sheaf $X$ on $(\\Sch/U)_{fppf}$ recovering", "the $X_i$ and the $\\varphi_{ij}$. Then by", "Bootstrap, Lemma \\ref{bootstrap-lemma-locally-algebraic-space}", "we see that $X$ is an algebraic space (if we ignore the set theoretic", "condition of that lemma).", "We will use this argument in the next section to show that", "if we consider only algebraic spaces of finite type, then we obtain", "a stack." ], "refs": [ "sites-lemma-glue-sheaves", "bootstrap-lemma-locally-algebraic-space" ], "ref_ids": [ 8564, 2625 ] }, { "id": 9187, "type": "other", "label": "examples-stacks-remark-higher-cardinality-spaces", "categories": [ "examples-stacks" ], "title": "examples-stacks-remark-higher-cardinality-spaces", "contents": [ "Note that the whole discussion in this section works", "if we want to consider those algebraic spaces $X/U$ which are", "locally of finite type such that the inverse image in $X$ of an affine open", "of $U$ can be covered by countably many affines.", "If needed we can also introduce the notion of a morphism of", "$\\kappa$-type (meaning some bound on the number of generators of", "ring extensions and some bound on the cardinality of the affines over", "a given affine in the base) where $\\kappa$ is a cardinal, and then", "we can produce a stack", "$$", "\\Spacesstack_\\kappa \\longrightarrow (\\Sch/S)_{fppf}", "$$", "in exactly the same manner as above (provided we make sure that", "$\\Sch$ is large enough depending on $\\kappa$)." ], "refs": [], "ref_ids": [] }, { "id": 9188, "type": "other", "label": "examples-stacks-remark-principal-stack-in-groupoids", "categories": [ "examples-stacks" ], "title": "examples-stacks-remark-principal-stack-in-groupoids", "contents": [ "We conjecture that up to a replacement as in", "Stacks, Remark \\ref{stacks-remark-stack-make-small}", "the functor", "$$", "p : G\\textit{-Principal} \\longrightarrow (\\Sch/S)_{fppf}", "$$", "defines a stack in groupoids over $(\\Sch/S)_{fppf}$. This would", "follow if one could show that given", "\\begin{enumerate}", "\\item a covering $\\{U_i \\to U\\}_{i \\in I}$ of $(\\Sch/S)_{fppf}$,", "\\item an group algebraic space $H$ over $U$,", "\\item for every $i$ a principal homogeneous $H_{U_i}$-space $X_i$", "over $U_i$, and", "\\item $H$-equivariant isomorphisms", "$\\varphi_{ij} : X_{i, U_i \\times_U U_j} \\to X_{j, U_i \\times_U U_j}$", "satisfying the cocycle condition,", "\\end{enumerate}", "there exists a principal homogeneous $H$-space $X$ over $U$", "which recovers $(X_i, \\varphi_{ij})$. The technique of the proof of", "Bootstrap, Lemma \\ref{bootstrap-lemma-descent-torsor}", "reduces this to a set theoretical question, so the reader who ignores", "set theoretical questions will ``know'' that the result is true. In", "\\url{https://math.columbia.edu/~dejong/wordpress/?p=591}", "there is a suggestion as to how to approach this problem." ], "refs": [ "stacks-remark-stack-make-small", "bootstrap-lemma-descent-torsor" ], "ref_ids": [ 9008, 2632 ] }, { "id": 9189, "type": "other", "label": "examples-stacks-remark-X-mod-G-group", "categories": [ "examples-stacks" ], "title": "examples-stacks-remark-X-mod-G-group", "contents": [ "Let $S$ be a scheme.", "Let $G$ be an abstract group.", "Let $X$ be an algebraic space over $S$.", "Let $G \\to \\text{Aut}_S(X)$ be a group homomorphism.", "In this setting we can define $[[X/G]]$ similarly", "to the above as follows:", "\\begin{enumerate}", "\\item An object of $[[X/G]]$ consists of a triple", "$(U, P, \\varphi : P \\to X)$ where", "\\begin{enumerate}", "\\item $U$ is an object of $(\\Sch/S)_{fppf}$,", "\\item $P$ is a sheaf on $(\\Sch/U)_{fppf}$ which comes", "with an action of $G$ that turns it into a torsor under the constant", "sheaf with value $G$, and", "\\item $\\varphi : P \\to X$ is a $G$-equivariant map of sheaves.", "\\end{enumerate}", "\\item A morphism", "$(f, g) : (U, P, \\varphi) \\to (U', P', \\varphi')$", "is given by a morphism of schemes $f : T \\to T'$", "and a $G$-equivariant isomorphism", "$g : P \\to f^{-1}P'$ such that $\\varphi = \\varphi' \\circ g$.", "\\end{enumerate}", "In exactly the same manner as above we obtain a functor", "$$", "[[X/G]] \\longrightarrow (\\Sch/S)_{fppf}", "$$", "which turns $[[X/G]]$ into a stack in groupoids over $(\\Sch/S)_{fppf}$.", "The constant sheaf $\\underline{G}$ is (provided the cardinality of $G$ is", "not too large) representable by $G_S$ on $(\\Sch/S)_{fppf}$", "and this version of $[[X/G]]$ is equivalent to the stack", "$[[X/G_S]]$ introduced above." ], "refs": [], "ref_ids": [] }, { "id": 9281, "type": "other", "label": "models-remark-genus-equality", "categories": [ "models" ], "title": "models-remark-genus-equality", "contents": [ "Let $n, m_i, a_{ij}, w_i, g_i$ be a minimal numerical type with $n > 1$.", "Equality $g = g_{top}$ can hold in Lemma \\ref{lemma-genus-nonnegative}.", "For example, if $m_i = w_i = 1$ and $g_i = 0$ for all $i$ and", "$a_{ij} \\in \\{0, 1\\}$ for $i < j$." ], "refs": [ "models-lemma-genus-nonnegative" ], "ref_ids": [ 9204 ] }, { "id": 9282, "type": "other", "label": "models-remark-numerical-type-not-from-model", "categories": [ "models" ], "title": "models-remark-numerical-type-not-from-model", "contents": [ "Not every numerical type comes from a model for the silly reason", "that there exist numerical types whose genus is negative.", "There exist a minimal numerical types of positive genus which", "are not the numerical type associated to a model (over some dvr)", "of a smooth projective geometrically irreducible curve (over the", "fraction field of the dvr). A simple example is", "$n = 1$, $m_1 = 1$, $a_{11} = 0$, $w_1 = 6$, $g_1 = 1$.", "Namely, in this case the special fibre $X_k$ would not be", "geometrically connected because it would live over an extension", "$\\kappa$ of $k$ of degree $6$. This is a contradiction with the", "fact that the generic fibre is geometrically connected (see", "More on Morphisms, Lemma", "\\ref{more-morphisms-lemma-geometrically-connected-fibres-towards-normal}).", "Similarly, $n = 2$, $m_1 = m_2 = 1$, $-a_{11} = -a_{22} = a_{12} = a_{21} = 6$,", "$w_1 = w_2 = 6$, $g_1 = g_2 = 1$ would be an example", "for the same reason (details omitted). But if the gcd of", "the $w_i$ is $1$ we do not have an example." ], "refs": [ "more-morphisms-lemma-geometrically-connected-fibres-towards-normal" ], "ref_ids": [ 13945 ] }, { "id": 9283, "type": "other", "label": "models-remark-genus-change", "categories": [ "models" ], "title": "models-remark-genus-change", "contents": [ "In the situation of Lemma \\ref{lemma-blowdown-regular-model}", "we can also say exactly how the genus $g_i$ of $C_i$ and the genus", "$g'_i$ of $C'_i$ are related. The formula is", "$$", "g'_i = \\frac{w_i}{w'_i}(g_i - 1) + 1 +", "\\frac{(C_i \\cdot C_n)^2 - w_n(C_i \\cdot C_n)}{2w'_iw_n}", "$$", "where $w_i = [\\kappa_i : k]$, $w_n = [\\kappa_n : k]$, and", "$w'_i = [\\kappa'_i : k]$.", "To prove this we consider the short exact sequence", "$$", "0 \\to \\mathcal{O}_{X'}(-C'_i) \\to \\mathcal{O}_{X'} \\to", "\\mathcal{O}_{C'_i} \\to 0", "$$", "and its pullback to $X$ which reads", "$$", "0 \\to \\mathcal{O}_X(-C'_i - e_iC_n) \\to \\mathcal{O}_X \\to", "\\mathcal{O}_{C_i + e_i C_n} \\to 0", "$$", "with $e_i$ as in the proof of Lemma \\ref{lemma-blowdown-regular-model}.", "Since $Rf_*f^*\\mathcal{L} = \\mathcal{L}$ for any invertible module", "$\\mathcal{L}$ on $X'$ (details omitted), we conclude that", "$$", "Rf_*\\mathcal{O}_{C_i + e_i C_n} = \\mathcal{O}_{C'_i}", "$$", "as complexes of coherent sheaves on $X'_k$.", "Hence both sides have the same Euler characteristic and this", "agrees with the Euler characteristic of $\\mathcal{O}_{C_i + e_i C_n}$", "on $X_k$. Using the exact sequence", "$$", "0 \\to \\mathcal{O}_{C_i + e_i C_n} \\to", "\\mathcal{O}_{C_i} \\oplus \\mathcal{O}_{e_iC_n} \\to", "\\mathcal{O}_{C_i \\cap e_iC_n} \\to 0", "$$", "and further filtering $\\mathcal{O}_{e_iC_n}$ (details omitted) we find", "$$", "\\chi(\\mathcal{O}_{C'_i}) =", "\\chi(\\mathcal{O}_{C_i}) - {e_i + 1 \\choose 2}(C_n \\cdot C_n)", "- e_i(C_i \\cdot C_n)", "$$", "Since $e_i = -(C_i \\cdot C_n)/(C_n \\cdot C_n)$ and", "$(C_n \\cdot C_n) = -w_n$ this leads to the formula", "stated at the start of this remark. If we ever need", "this we will formulate this as a lemma and", "provide a detailed proof." ], "refs": [ "models-lemma-blowdown-regular-model", "models-lemma-blowdown-regular-model" ], "ref_ids": [ 9255, 9255 ] }, { "id": 9284, "type": "other", "label": "models-remark-compare-contractions", "categories": [ "models" ], "title": "models-remark-compare-contractions", "contents": [ "Let $f : X \\to X'$ be as in Lemma \\ref{lemma-blowdown-regular-model}.", "Let $n, m_i, a_{ij}, w_i, g_i$ be the numerical type associated to $X$ and", "let $n', m'_i, a'_{ij}, w'_i, g'_i$ be the numerical type associated to $X'$.", "It is clear from Lemma \\ref{lemma-blowdown-regular-model} and", "Remark \\ref{remark-genus-change}", "that this agrees with the contraction of numerical types in", "Lemma \\ref{lemma-contract}", "except for the value of $w'_i$.", "In the geometric situation $w'_i$ is some positive integer", "dividing both $w_i$ and $w_n$. In the numerical case", "we chose $w'_i$ to be the largest possible integer dividing", "$w_i$ such that $g'_i$ (as given by the formula) is an integer.", "This works well in the numerical setting", "in that it helps compare the Picard groups", "of the numerical types, see Lemma \\ref{lemma-contract-picard-group}", "(although only injectivity is every used in the following and this", "injectivity works as well for smaller $w'_i$)." ], "refs": [ "models-lemma-blowdown-regular-model", "models-lemma-blowdown-regular-model", "models-remark-genus-change", "models-lemma-contract", "models-lemma-contract-picard-group" ], "ref_ids": [ 9255, 9255, 9283, 9201, 9208 ] }, { "id": 9285, "type": "other", "label": "models-remark-improving-bound", "categories": [ "models" ], "title": "models-remark-improving-bound", "contents": [ "Results in the literature suggest that one can improve the bound", "given in the statement of Theorem \\ref{theorem-semistable-reduction}.", "For example, in \\cite{DM} it is shown that semistable reduction", "of $C$ and its Jacobian are the same thing if the residue field is perfect", "and presumably this is true for general residue fields as well.", "For an abelian variety we have semistable reduction if the action of Galois", "on the $\\ell$-torsion is trivial for any $\\ell \\geq 3$ not equal to the", "residue characteristic. Thus we can presumably choose $\\ell = 5$", "in the formula (\\ref{equation-bound}) for $B_g$", "(but the proof would take a lot more work; if we ever need this", "we will make a precise statement and provide a proof here)." ], "refs": [ "models-theorem-semistable-reduction" ], "ref_ids": [ 9190 ] }, { "id": 9355, "type": "other", "label": "spaces-groupoids-remark-quotient-variant", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-remark-quotient-variant", "contents": [ "A variant of the construction above would have been to sheafify", "the functor", "$$", "\\begin{matrix}", "(\\textit{Spaces}/B)^{opp}_{fppf} &", "\\longrightarrow &", "\\textit{Sets}, \\\\", "X &", "\\longmapsto &", "U(X)/\\sim_X", "\\end{matrix}", "$$", "where now $\\sim_X \\subset U(X) \\times U(X)$ is the equivalence relation", "generated by the image of $j : R(X) \\to U(X) \\times U(X)$.", "Here of course $U(X) = \\Mor_B(X, U)$ and $R(X) = \\Mor_B(X, R)$.", "In fact, the result would have been the same, via the identifications", "of (insert future reference in Topologies of Spaces here)." ], "refs": [], "ref_ids": [] }, { "id": 9356, "type": "other", "label": "spaces-groupoids-remark-fundamental-square", "categories": [ "spaces-groupoids" ], "title": "spaces-groupoids-remark-fundamental-square", "contents": [ "In future chapters we will use the ambiguous notation where", "instead of writing $\\mathcal{S}_X$ for the stack in sets associated", "to $X$ we simply write $X$. Using this notation the diagram of", "Lemma \\ref{lemma-quotient-stack-2-arrow}", "becomes the familiar diagram", "$$", "\\xymatrix{", "R \\ar[r]_s \\ar[d]_t & U \\ar[d]^\\pi \\\\", "U \\ar[r]^-\\pi & [U/R]", "}", "$$", "In the following sections we will show that this diagram has", "many good properties. In particular we will show that it is", "a $2$-fibre product", "(Section \\ref{section-quotient-stack-2-cartesian})", "and that it is close to being a $2$-coequalizer of $s$ and $t$", "(Section \\ref{section-quotient-stacks-2-coequalize})." ], "refs": [ "spaces-groupoids-lemma-quotient-stack-2-arrow" ], "ref_ids": [ 9320 ] }, { "id": 9450, "type": "other", "label": "spaces-descent-remark-list-local-source-target", "categories": [ "spaces-descent" ], "title": "spaces-descent-remark-list-local-source-target", "contents": [ "Using", "Lemma \\ref{lemma-smooth-local-source-target}", "and the work done in the earlier sections of this chapter it is easy", "to make a list of types of morphisms which are smooth local on the", "source-and-target. In each case we list the lemma which implies", "the property is smooth local on the source and the lemma which implies", "the property is smooth local on the target. In each case the third assumption", "of", "Lemma \\ref{lemma-smooth-local-source-target}", "is trivial to check, and we omit it. Here is the list:", "\\begin{enumerate}", "\\item flat, see", "Lemmas \\ref{lemma-flat-fpqc-local-source} and", "\\ref{lemma-descending-property-flat},", "\\item locally of finite presentation, see", "Lemmas \\ref{lemma-locally-finite-presentation-fppf-local-source} and", "\\ref{lemma-descending-property-locally-finite-presentation},", "\\item locally finite type, see", "Lemmas \\ref{lemma-locally-finite-type-fppf-local-source} and", "\\ref{lemma-descending-property-locally-finite-type},", "\\item universally open, see", "Lemmas \\ref{lemma-universally-open-fppf-local-source} and", "\\ref{lemma-descending-property-universally-open},", "\\item syntomic, see", "Lemmas \\ref{lemma-syntomic-syntomic-local-source} and", "\\ref{lemma-descending-property-syntomic},", "\\item smooth, see", "Lemmas \\ref{lemma-smooth-smooth-local-source} and", "\\ref{lemma-descending-property-smooth},", "\\item add more here as needed.", "\\end{enumerate}" ], "refs": [ "spaces-descent-lemma-smooth-local-source-target", "spaces-descent-lemma-smooth-local-source-target", "spaces-descent-lemma-flat-fpqc-local-source", "spaces-descent-lemma-descending-property-flat", "spaces-descent-lemma-locally-finite-presentation-fppf-local-source", "spaces-descent-lemma-descending-property-locally-finite-presentation", "spaces-descent-lemma-locally-finite-type-fppf-local-source", "spaces-descent-lemma-descending-property-locally-finite-type", "spaces-descent-lemma-universally-open-fppf-local-source", "spaces-descent-lemma-descending-property-universally-open", "spaces-descent-lemma-syntomic-syntomic-local-source", "spaces-descent-lemma-descending-property-syntomic", "spaces-descent-lemma-smooth-smooth-local-source", "spaces-descent-lemma-descending-property-smooth" ], "ref_ids": [ 9429, 9429, 9417, 9393, 9418, 9390, 9419, 9389, 9421, 9384, 9422, 9405, 9423, 9406 ] }, { "id": 9451, "type": "other", "label": "spaces-descent-remark-list-etale-smooth-local-source-target", "categories": [ "spaces-descent" ], "title": "spaces-descent-remark-list-etale-smooth-local-source-target", "contents": [ "Using Lemma \\ref{lemma-etale-smooth-local-source-target}", "and the work done in the earlier sections of this chapter it is easy", "to make a list of types of morphisms which are smooth local on the", "source-and-target. In each case we list the lemma which implies", "the property is etale local on the source and the lemma which implies", "the property is smooth local on the target. In each case the third assumption", "of Lemma \\ref{lemma-etale-smooth-local-source-target}", "is trivial to check, and we omit it. Here is the list:", "\\begin{enumerate}", "\\item \\'etale, see", "Lemmas \\ref{lemma-etale-etale-local-source} and", "\\ref{lemma-descending-property-etale},", "\\item locally quasi-finite, see", "Lemmas \\ref{lemma-locally-quasi-finite-etale-local-source} and", "\\ref{lemma-descending-property-quasi-finite},", "\\item unramified, see", "Lemmas \\ref{lemma-unramified-etale-local-source} and", "\\ref{lemma-descending-property-unramified}, and", "\\item add more here as needed.", "\\end{enumerate}", "Of course any property listed in", "Remark \\ref{remark-list-local-source-target}", "is a fortiori an example that could be listed here." ], "refs": [ "spaces-descent-lemma-etale-smooth-local-source-target", "spaces-descent-lemma-etale-smooth-local-source-target", "spaces-descent-lemma-etale-etale-local-source", "spaces-descent-lemma-descending-property-etale", "spaces-descent-lemma-locally-quasi-finite-etale-local-source", "spaces-descent-lemma-descending-property-quasi-finite", "spaces-descent-lemma-unramified-etale-local-source", "spaces-descent-lemma-descending-property-unramified", "spaces-descent-remark-list-local-source-target" ], "ref_ids": [ 9432, 9432, 9424, 9408, 9425, 9404, 9426, 9407, 9450 ] }, { "id": 9452, "type": "other", "label": "spaces-descent-remark-easier", "categories": [ "spaces-descent" ], "title": "spaces-descent-remark-easier", "contents": [ "Let $S$ be a scheme.", "Let $Y \\to X$ be a morphism of algebraic spaces over $S$.", "Let $(V/Y, \\varphi)$ be a descent datum relative to $Y \\to X$.", "We may think of the isomorphism $\\varphi$ as an isomorphism", "$$", "(Y \\times_X Y) \\times_{\\text{pr}_0, Y} V", "\\longrightarrow", "(Y \\times_X Y) \\times_{\\text{pr}_1, Y} V", "$$", "of algebraic spaces over $Y \\times_X Y$. So loosely speaking one may", "think of $\\varphi$ as a map", "$\\varphi : \\text{pr}_0^*V \\to \\text{pr}_1^*V$\\footnote{Unfortunately,", "we have chosen the ``wrong'' direction for our arrow here. In", "Definitions \\ref{definition-descent-datum} and", "\\ref{definition-descent-datum-for-family-of-morphisms}", "we should have the opposite direction to what was done in", "Definition \\ref{definition-descent-datum-quasi-coherent}", "by the general principle that ``functions'' and ``spaces'' are dual.}.", "The cocycle condition then says that", "$\\text{pr}_{02}^*\\varphi =", "\\text{pr}_{12}^*\\varphi \\circ \\text{pr}_{01}^*\\varphi$.", "In this way it is very similar to the case of a descent datum on", "quasi-coherent sheaves." ], "refs": [ "spaces-descent-definition-descent-datum", "spaces-descent-definition-descent-datum-for-family-of-morphisms", "spaces-descent-definition-descent-datum-quasi-coherent" ], "ref_ids": [ 9444, 9445, 9438 ] }, { "id": 9453, "type": "other", "label": "spaces-descent-remark-easier-family", "categories": [ "spaces-descent" ], "title": "spaces-descent-remark-easier-family", "contents": [ "Let $S$ be a scheme.", "Let $\\{X_i \\to X\\}_{i \\in I}$ be a family of morphisms", "of algebraic spaces over $S$ with fixed target $X$.", "Let $(V_i, \\varphi_{ij})$ be a descent datum relative to", "$\\{X_i \\to X\\}$. We may think of the isomorphisms $\\varphi_{ij}$", "as isomorphisms", "$$", "(X_i \\times_X X_j) \\times_{\\text{pr}_0, X_i} V_i", "\\longrightarrow", "(X_i \\times_X X_j) \\times_{\\text{pr}_1, X_j} V_j", "$$", "of algebraic spaces over $X_i \\times_X X_j$. So loosely speaking one may", "think of $\\varphi_{ij}$ as an isomorphism", "$\\text{pr}_0^*V_i \\to \\text{pr}_1^*V_j$ over $X_i \\times_X X_j$.", "The cocycle condition then says that", "$\\text{pr}_{02}^*\\varphi_{ik} =", "\\text{pr}_{12}^*\\varphi_{jk} \\circ \\text{pr}_{01}^*\\varphi_{ij}$.", "In this way it is very similar to the case of a descent datum on", "quasi-coherent sheaves." ], "refs": [], "ref_ids": [] }, { "id": 9573, "type": "other", "label": "decent-spaces-remark-recall", "categories": [ "decent-spaces" ], "title": "decent-spaces-remark-recall", "contents": [ "Before we give the proof of the next lemma let us recall some facts", "about \\'etale morphisms of schemes:", "\\begin{enumerate}", "\\item An \\'etale morphism is flat and hence generalizations lift along", "an \\'etale morphism", "(Morphisms, Lemmas \\ref{morphisms-lemma-etale-flat}", "and \\ref{morphisms-lemma-generalizations-lift-flat}).", "\\item An \\'etale morphism is unramified, an unramified morphism is locally", "quasi-finite, hence fibres are discrete", "(Morphisms, Lemmas \\ref{morphisms-lemma-flat-unramified-etale},", "\\ref{morphisms-lemma-unramified-quasi-finite}, and", "\\ref{morphisms-lemma-quasi-finite-at-point-characterize}).", "\\item A quasi-compact \\'etale morphism is quasi-finite and in particular", "has finite fibres", "(Morphisms, Lemmas \\ref{morphisms-lemma-quasi-finite-locally-quasi-compact} and", "\\ref{morphisms-lemma-quasi-finite}).", "\\item An \\'etale scheme over a field $k$ is a disjoint union of spectra", "of finite separable field extension of $k$", "(Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}).", "\\end{enumerate}", "For a general discussion of \\'etale morphisms, please see", "\\'Etale Morphisms, Section \\ref{etale-section-etale-morphisms}." ], "refs": [ "morphisms-lemma-etale-flat", "morphisms-lemma-generalizations-lift-flat", "morphisms-lemma-flat-unramified-etale", "morphisms-lemma-unramified-quasi-finite", "morphisms-lemma-quasi-finite-at-point-characterize", "morphisms-lemma-quasi-finite-locally-quasi-compact", "morphisms-lemma-quasi-finite", "morphisms-lemma-etale-over-field" ], "ref_ids": [ 5369, 5266, 5373, 5351, 5226, 5229, 5230, 5364 ] }, { "id": 9574, "type": "other", "label": "decent-spaces-remark-reasonable", "categories": [ "decent-spaces" ], "title": "decent-spaces-remark-reasonable", "contents": [ "Reasonable algebraic spaces are technically easier to work with than very", "reasonable algebraic spaces. For example, if $X \\to Y$ is a quasi-compact", "\\'etale surjective morphism of algebraic spaces and $X$ is reasonable, then", "so is $Y$, see", "Lemma \\ref{lemma-descent-conditions}", "but we don't know if this is true for the property ``very reasonable''.", "Below we give another technical property enjoyed by reasonable", "algebraic spaces." ], "refs": [ "decent-spaces-lemma-descent-conditions" ], "ref_ids": [ 9519 ] }, { "id": 9575, "type": "other", "label": "decent-spaces-remark-functoriality-henselian-local-ring", "categories": [ "decent-spaces" ], "title": "decent-spaces-remark-functoriality-henselian-local-ring", "contents": [ "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of decent algebraic spaces", "over $S$. Let $x \\in |X|$ with image $y \\in |Y|$. Choose an elementary", "\\'etale neighbourhood $(V, v) \\to (Y, y)$ (possible by", "Lemma \\ref{lemma-decent-space-elementary-etale-neighbourhood}).", "Then $V \\times_Y X$ is an algebraic space \\'etale over $X$", "which has a unique point $x'$ mapping to $x$ in $X$ and to $v$ in $V$.", "(Details omitted; use that all points can be represented by", "monomorphisms from spectra of fields.)", "Choose an elementary \\'etale neighbourhood $(U, u) \\to (V \\times_Y X, x')$.", "Then we obtain the following commutative diagram", "$$", "\\xymatrix{", "\\Spec(\\mathcal{O}_{X, \\overline{x}}) \\ar[r] \\ar[d] &", "\\Spec(\\mathcal{O}_{X, x}^h) \\ar[r] \\ar[d] &", "\\Spec(\\mathcal{O}_{U, u}) \\ar[r] \\ar[d] &", "U \\ar[r] \\ar[d] &", "X \\ar[d] \\\\", "\\Spec(\\mathcal{O}_{Y, \\overline{y}}) \\ar[r] &", "\\Spec(\\mathcal{O}_{Y, y}^h) \\ar[r] &", "\\Spec(\\mathcal{O}_{V, v}) \\ar[r] &", "V \\ar[r] &", "Y", "}", "$$", "This comes from the identifications", "$\\mathcal{O}_{X, \\overline{x}} = \\mathcal{O}_{U, u}^{sh}$,", "$\\mathcal{O}_{X, x}^h = \\mathcal{O}_{U, u}^h$,", "$\\mathcal{O}_{Y, \\overline{y}} = \\mathcal{O}_{V, v}^{sh}$,", "$\\mathcal{O}_{Y, y}^h = \\mathcal{O}_{V, v}^h$", "see in", "Lemma \\ref{lemma-describe-henselian-local-ring}", "and", "Properties of Spaces, Lemma", "\\ref{spaces-properties-lemma-describe-etale-local-ring}", "and the functoriality of the (strict) henselization", "discussed in Algebra, Sections \\ref{algebra-section-ind-etale} and", "\\ref{algebra-section-henselization}." ], "refs": [ "decent-spaces-lemma-decent-space-elementary-etale-neighbourhood", "decent-spaces-lemma-describe-henselian-local-ring", "spaces-properties-lemma-describe-etale-local-ring" ], "ref_ids": [ 9488, 9490, 11884 ] }, { "id": 9576, "type": "other", "label": "decent-spaces-remark-one-point-decent-scheme", "categories": [ "decent-spaces" ], "title": "decent-spaces-remark-one-point-decent-scheme", "contents": [ "We will see in", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-reduction-scheme}", "that an algebraic space", "whose reduction is a scheme is a scheme." ], "refs": [ "spaces-limits-lemma-reduction-scheme" ], "ref_ids": [ 4627 ] }, { "id": 9577, "type": "other", "label": "decent-spaces-remark-very-reasonable", "categories": [ "decent-spaces" ], "title": "decent-spaces-remark-very-reasonable", "contents": [ "An informal description of the properties $(\\beta)$, decent, reasonable,", "very reasonable was given in Section \\ref{section-reasonable-decent}.", "A morphism has one of these properties if (very) loosely speaking the", "fibres of the morphism have the corresponding properties.", "Being decent is useful to prove things about specializations of", "points on $|X|$. Being reasonable is a bit stronger and technically", "quite easy to work with." ], "refs": [], "ref_ids": [] }, { "id": 9689, "type": "other", "label": "groupoids-remark-warning-group-scheme-geometrically-irreducible", "categories": [ "groupoids" ], "title": "groupoids-remark-warning-group-scheme-geometrically-irreducible", "contents": [ "Warning: The result of", "Lemma \\ref{lemma-group-scheme-field-geometrically-irreducible}", "does not mean that every irreducible component of $G/k$ is", "geometrically irreducible. For example the group scheme", "$\\mu_{3, \\mathbf{Q}} = \\Spec(\\mathbf{Q}[x]/(x^3 - 1))$", "over $\\mathbf{Q}$ has two irreducible components corresponding", "to the factorization $x^3 - 1 = (x - 1)(x^2 + x + 1)$.", "The first factor corresponds to the irreducible component", "passing through the identity, and the second irreducible component", "is not geometrically irreducible over $\\Spec(\\mathbf{Q})$." ], "refs": [ "groupoids-lemma-group-scheme-field-geometrically-irreducible" ], "ref_ids": [ 9590 ] }, { "id": 9690, "type": "other", "label": "groupoids-remark-easy", "categories": [ "groupoids" ], "title": "groupoids-remark-easy", "contents": [ "If $G$ is a group scheme over a field, is there always a quasi-compact", "open and closed subgroup scheme? By", "Proposition \\ref{proposition-connected-component}", "this question is only interesting if $G$ has infinitely many connected", "components (geometrically)." ], "refs": [ "groupoids-proposition-connected-component" ], "ref_ids": [ 9667 ] }, { "id": 9691, "type": "other", "label": "groupoids-remark-when-reduced", "categories": [ "groupoids" ], "title": "groupoids-remark-when-reduced", "contents": [ "Any group scheme over a field of characteristic $0$ is reduced, see", "\\cite[I, Theorem 1.1 and I, Corollary 3.9, and II, Theorem 2.4]{Perrin-thesis}", "and also", "\\cite[Proposition 4.2.8]{Perrin}.", "This was a question raised in", "\\cite[page 80]{Oort}.", "We have seen in", "Lemma \\ref{lemma-group-scheme-characteristic-zero-smooth}", "that this holds when the group scheme is locally of finite type." ], "refs": [ "groupoids-lemma-group-scheme-characteristic-zero-smooth" ], "ref_ids": [ 9600 ] }, { "id": 9692, "type": "other", "label": "groupoids-remark-reduced-smooth-not-true-general", "categories": [ "groupoids" ], "title": "groupoids-remark-reduced-smooth-not-true-general", "contents": [ "Let $k$ be a field of characteristic $p > 0$.", "Let $\\alpha \\in k$ be an element which is not a $p$th power.", "The closed subgroup scheme", "$$", "G = V(x^p + \\alpha y^p) \\subset \\mathbf{G}_{a, k}^2", "$$", "is reduced and irreducible but not smooth (not even normal)." ], "refs": [], "ref_ids": [] }, { "id": 9693, "type": "other", "label": "groupoids-remark-fun-with-torsors", "categories": [ "groupoids" ], "title": "groupoids-remark-fun-with-torsors", "contents": [ "Let $(G, m)$ be a group scheme over the scheme $S$.", "In this situation we have the following natural types of questions:", "\\begin{enumerate}", "\\item If $X \\to S$ is a pseudo $G$-torsor and $X \\to S$ is surjective,", "then is $X$ necessarily a $G$-torsor?", "\\item Is every $\\underline{G}$-torsor on $(\\Sch/S)_{fppf}$", "representable? In other words, does every $\\underline{G}$-torsor", "come from a fppf $G$-torsor?", "\\item Is every $G$-torsor an", "fppf (resp.\\ smooth, resp.\\ \\'etale, resp.\\ Zariski) torsor?", "\\end{enumerate}", "In general the answers to these questions is no. To get a positive answer", "we need to impose additional conditions on $G \\to S$.", "For example:", "If $S$ is the spectrum of a field, then the answer to (1) is yes", "because then $\\{X \\to S\\}$ is a fpqc covering trivializing $X$.", "If $G \\to S$ is affine, then the answer to (2) is yes", "(insert future reference here).", "If $G = \\text{GL}_{n, S}$ then the answer to (3) is yes", "and in fact any $\\text{GL}_{n, S}$-torsor is locally trivial", "(insert future reference here)." ], "refs": [], "ref_ids": [] }, { "id": 9794, "type": "other", "label": "local-cohomology-remark-upshot", "categories": [ "local-cohomology" ], "title": "local-cohomology-remark-upshot", "contents": [ "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a", "subset stable under specialization.", "The upshot of the discussion above is that", "$R\\Gamma_T : D^+(A) \\to D_T^+(A)$ is the right adjoint", "to the inclusion functor $D_T^+(A) \\to D^+(A)$.", "If $\\dim(A) < \\infty$, then", "$R\\Gamma_T : D(A) \\to D_T(A)$ is the right adjoint", "to the inclusion functor $D_T(A) \\to D(A)$.", "In both cases we have", "$$", "H^i_T(K) = H^i(R\\Gamma_T(K)) = R^iH^0_T(K) =", "\\colim_{Z \\subset T\\text{ closed}} H^i_Z(K)", "$$", "This follows by combining", "Lemmas \\ref{lemma-adjoint}, \\ref{lemma-adjoint-ext},", "\\ref{lemma-equal-plus}, and \\ref{lemma-equal-full}." ], "refs": [ "local-cohomology-lemma-adjoint", "local-cohomology-lemma-adjoint-ext", "local-cohomology-lemma-equal-plus", "local-cohomology-lemma-equal-full" ], "ref_ids": [ 9714, 9715, 9716, 9717 ] }, { "id": 9795, "type": "other", "label": "local-cohomology-remark-closure", "categories": [ "local-cohomology" ], "title": "local-cohomology-remark-closure", "contents": [ "Let $j : U \\to X$ be an open immersion of locally Noetherian schemes.", "Let $x \\in U$. Let $i_x : W_x \\to U$ be the integral closed subscheme", "with generic point $x$ and let $\\overline{\\{x\\}}$ be the closure in $X$.", "Then we have a commutative diagram", "$$", "\\xymatrix{", "W_x \\ar[d]_{i_x} \\ar[r]_{j'} & \\overline{\\{x\\}} \\ar[d]^i \\\\", "U \\ar[r]^j & X", "}", "$$", "We have $j_*i_{x, *}\\mathcal{O}_{W_x} = i_*j'_*\\mathcal{O}_{W_x}$.", "As the left vertical arrow is a closed immersion we see that", "$j_*i_{x, *}\\mathcal{O}_{W_x}$ is coherent if and only if", "$j'_*\\mathcal{O}_{W_x}$ is coherent." ], "refs": [], "ref_ids": [] }, { "id": 9796, "type": "other", "label": "local-cohomology-remark-no-finiteness-pushforward", "categories": [ "local-cohomology" ], "title": "local-cohomology-remark-no-finiteness-pushforward", "contents": [ "Let $X$ be a locally Noetherian scheme. Let $j : U \\to X$ be the inclusion of", "an open subscheme with complement $Z$. Let $\\mathcal{F}$ be a coherent", "$\\mathcal{O}_U$-module. If there exists an $x \\in \\text{Ass}(\\mathcal{F})$ and", "$z \\in Z \\cap \\overline{\\{x\\}}$ such that", "$\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) \\leq 1$, then $j_*\\mathcal{F}$ is not", "coherent. To prove this we can do a flat base change to the spectrum", "of $\\mathcal{O}_{X, z}$. Let $X' = \\overline{\\{x\\}}$.", "The assumption implies $\\mathcal{O}_{X' \\cap U} \\subset \\mathcal{F}$.", "Thus it suffices to see that $j_*\\mathcal{O}_{X' \\cap U}$ is not", "coherent. This is clear because $X' = \\{x, z\\}$, hence", "$j_*\\mathcal{O}_{X' \\cap U}$ corresponds to $\\kappa(x)$ as an", "$\\mathcal{O}_{X, z}$-module which cannot be finite as $x$ is not", "a closed point.", "\\medskip\\noindent", "In fact, the converse of Lemma \\ref{lemma-sharp-finiteness-pushforward}", "holds true: given an open immersion $j : U \\to X$ of integral Noetherian", "schemes and there exists a $z \\in X \\setminus U$ and an associated prime", "$\\mathfrak p$ of the completion $\\mathcal{O}_{X, z}^\\wedge$", "with $\\dim(\\mathcal{O}_{X, z}^\\wedge/\\mathfrak p) = 1$,", "then $j_*\\mathcal{O}_U$ is not coherent. Namely, you can pass to", "the local ring, you can enlarge $U$ to the punctured spectrum,", "you can pass to the completion, and then the argument above gives", "the nonfiniteness." ], "refs": [ "local-cohomology-lemma-sharp-finiteness-pushforward" ], "ref_ids": [ 9731 ] }, { "id": 9797, "type": "other", "label": "local-cohomology-remark-astute-reader", "categories": [ "local-cohomology" ], "title": "local-cohomology-remark-astute-reader", "contents": [ "The astute reader will have realized that we can get away with a", "slightly weaker condition on the formal fibres of the local rings", "of $A$. Namely, in the situation of Theorem \\ref{theorem-finiteness}", "assume $A$ is universally catenary but make no assumptions on", "the formal fibres. Suppose we have an $n$ and we want to prove that", "$H^i_Z(M)$ are finite for $i \\leq n$. Then the exact same proof", "shows that it suffices that $s_{A, I}(M) > n$ and that", "the formal fibres of local rings of $A$ are $(S_n)$.", "On the other hand, if we want to show that $H^s_Z(M)$", "is not finite where $s = s_{A, I}(M)$, then our arguments prove", "this if the formal fibres are $(S_{s - 1})$." ], "refs": [ "local-cohomology-theorem-finiteness" ], "ref_ids": [ 9694 ] }, { "id": 9798, "type": "other", "label": "local-cohomology-remark-higher-order-operators", "categories": [ "local-cohomology" ], "title": "local-cohomology-remark-higher-order-operators", "contents": [ "We can upgrade Lemmas \\ref{lemma-derivation} and \\ref{lemma-etale-derivation}", "to include higher order differential operators.", "If we ever need this we will state and prove a", "precise lemma here." ], "refs": [ "local-cohomology-lemma-derivation", "local-cohomology-lemma-etale-derivation" ], "ref_ids": [ 9766, 9768 ] }, { "id": 9799, "type": "other", "label": "local-cohomology-remark-better-bound", "categories": [ "local-cohomology" ], "title": "local-cohomology-remark-better-bound", "contents": [ "The paper \\cite{AHS} shows, besides many other things, that if $A$ is local,", "then Proposition \\ref{proposition-uniform-artin-rees} also holds", "with $e = t$ replaced by $e = \\dim(A)$. Looking at", "Lemma \\ref{lemma-cd-sequence-Koszul} it is natural to ask whether", "Proposition \\ref{proposition-uniform-artin-rees}", "holds with $e = t$ replaced with $e = \\text{cd}(A, I)$. We don't know." ], "refs": [ "local-cohomology-proposition-uniform-artin-rees", "local-cohomology-lemma-cd-sequence-Koszul", "local-cohomology-proposition-uniform-artin-rees" ], "ref_ids": [ 9791, 9772, 9791 ] }, { "id": 9800, "type": "other", "label": "local-cohomology-remark-strict-pro-isomorphism", "categories": [ "local-cohomology" ], "title": "local-cohomology-remark-strict-pro-isomorphism", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$. Say $I = (f_1, \\ldots, f_r)$.", "Denote $K_n^\\bullet$ the Koszul complex on $f_1^n, \\ldots, f_r^n$ as in", "More on Algebra, Situation \\ref{more-algebra-situation-koszul} and", "denote $K_n \\in D(A)$ the corresponding object.", "Let $M^\\bullet$ be a bounded complex of finite $A$-modules", "and denote $M \\in D(A)$ the corresponding object.", "Consider the following inverse systems in $D(A)$:", "\\begin{enumerate}", "\\item $M^\\bullet/I^nM^\\bullet$, i.e., the complex whose terms are $M^i/I^nM^i$,", "\\item $M \\otimes_A^\\mathbf{L} A/I^n$,", "\\item $M \\otimes_A^\\mathbf{L} K_n$, and", "\\item $M \\otimes_P^\\mathbf{L} P/J^n$ (see below).", "\\end{enumerate}", "All of these inverse systems are isomorphic as pro-objects:", "the isomorphism between (2) and (3) follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-sequence-Koszul-complexes}.", "The isomorphism between (1) and (2) is given in", "More on Algebra, Lemma", "\\ref{more-algebra-lemma-derived-completion-plain-completion}.", "For the last one, see below.", "\\medskip\\noindent", "However, we can ask if these isomorphisms of pro-systems are ``strict'';", "this terminology and question is related to the discussion in", "\\cite[pages 61, 62]{quillenhomology}. Namely, given a category $\\mathcal{C}$", "we can define a ``strict pro-category'' whose objects are inverse systems", "$(X_n)$ and whose morphisms $(X_n) \\to (Y_n)$ are given by tuples", "$(c, \\varphi_n)$ consisting of a $c \\geq 0$ and morphisms", "$\\varphi_n : X_n \\to Y_{n - c}$ for all $n \\geq c$ satisfying", "an obvious compatibility condition and up to a certain equivalence", "(given essentially by increasing $c$). Then we ask whether the above", "inverse systems are isomorphic in this strict pro-category.", "\\medskip\\noindent", "This clearly cannot be the case for (1) and (3) even when $M = A[0]$.", "Namely, the system $H^0(K_n) = A/(f_1^n, \\ldots, f_r^n)$ is not strictly", "pro-isomorphic in the category of modules to the system $A/I^n$ in general.", "For example, if we take $A = \\mathbf{Z}[x_1, \\ldots, x_r]$ and $f_i = x_i$,", "then $H^0(K_n)$ is not annihilated by $I^{r(n - 1)}$.\\footnote{Of", "course, we can ask whether these pro-systems are isomorphic in", "a category whose objects are inverse systems and where maps are given", "by tuples $(r, c, \\varphi_n)$ consisting of $r \\geq 1$, $c \\geq 0$", "and maps $\\varphi_n : X_{rn} \\to Y_{n - c}$ for $n \\geq c$.}", "\\medskip\\noindent", "It turns out that the results above show that the natural map from", "(2) to (1) discussed in More on Algebra, Lemma", "\\ref{more-algebra-lemma-derived-completion-plain-completion}", "is a strict pro-isomorphism. We will sketch the proof.", "Using standard arguments involving stupid truncations, we first reduce", "to the case where $M^\\bullet$ is given by a single finite $A$-module", "$M$ placed in degree $0$. Pick $N, c \\geq 0$ as in", "Proposition \\ref{proposition-uniform-artin-rees}.", "The proposition implies that for $n \\geq N$ we get factorizations", "$$", "M \\otimes_A^\\mathbf{L} A/I^n", "\\to", "\\tau_{\\geq -t}(M \\otimes_A^\\mathbf{L} A/I^n)", "\\to", "M \\otimes_A^\\mathbf{L} A/I^{n - c}", "$$", "of the transition maps in the system (2). On the other hand, by", "More on Algebra, Lemma \\ref{more-algebra-lemma-tor-strictly-pro-zero},", "we can find another constant $c' = c'(M) \\geq 0$ such that the maps", "$\\text{Tor}_i^A(M, A/I^{n'}) \\to \\text{Tor}_i(M, A/I^{n' - c'})$", "are zero for $i = 1, 2, \\ldots, t$ and $n' \\geq c'$. Then it follows from", "Derived Categories, Lemma \\ref{derived-lemma-trick-vanishing-composition}", "that the map", "$$", "\\tau_{\\geq -t}(M \\otimes_A^\\mathbf{L} A/I^{n + tc'})", "\\to", "\\tau_{\\geq -t}(M \\otimes_A^\\mathbf{L} A/I^n)", "$$", "factors through $M \\otimes_A^\\mathbf{L}A/I^{n + tc'} \\to M/I^{n + tc'}M$.", "Combined with the previous result we obtain a factorization", "$$", "M \\otimes_A^\\mathbf{L}A/I^{n + tc'} \\to M/I^{n + tc'}M", "\\to M \\otimes_A^\\mathbf{L} A/I^{n - c}", "$$", "which gives us what we want. If we ever need this result, we will carefully", "state it and provide a detailed proof.", "\\medskip\\noindent", "For number (4) suppose we have a Noetherian ring $P$,", "a ring homomorphism $P \\to A$, and an ideal $J \\subset P$ such that $I = JA$.", "By More on Algebra, Section \\ref{more-algebra-section-derived-base-change}", "we get a functor $M \\otimes_P^\\mathbf{L} - : D(P) \\to D(A)$ and we get", "an inverse system $M \\otimes_P^\\mathbf{L} P/J^n$ in $D(A)$ as in (4).", "If $P$ is Noetherian, then the system in (4) is pro-isomorphic", "to the system in (1) because we can compare with Koszul complexes.", "If $P \\to A$ is finite, then the system (4) is strictly pro-isomorphic", "to the system (2) because the inverse system $A \\otimes_P^\\mathbf{L} P/J^n$", "is strictly pro-isomorphic to the inverse system $A/I^n$", "(by the discussion above) and because we have", "$$", "M \\otimes_P^\\mathbf{L} P/J^n = M \\otimes_A^\\mathbf{L}", "(A \\otimes_P^\\mathbf{L} P/J^n)", "$$", "by More on Algebra, Lemma \\ref{more-algebra-lemma-derived-base-change}.", "\\medskip\\noindent", "A standard example in (4) is to take $P = \\mathbf{Z}[x_1, \\ldots, x_r]$,", "the map $P \\to A$ sending $x_i$ to $f_i$, and $J = (x_1, \\ldots, x_r)$.", "In this case one shows that", "$$", "M \\otimes_P^\\mathbf{L} P/J^n =", "M \\otimes_{A[x_1, \\ldots, x_r]}^\\mathbf{L}", "A[x_1, \\ldots, x_r]/(x_1, \\ldots, x_r)^n", "$$", "and we reduce to one of the cases discussed above (although this case", "is strictly easier as $A[x_1, \\ldots, x_r]/(x_1, \\ldots, x_r)^n$ has", "tor dimension at most $r$ for all $n$ and hence the step using", "Proposition \\ref{proposition-uniform-artin-rees} can be avoided).", "This case is discussed in the proof of \\cite[Proposition 3.5.1]{BS}." ], "refs": [ "more-algebra-lemma-sequence-Koszul-complexes", "more-algebra-lemma-derived-completion-plain-completion", "more-algebra-lemma-derived-completion-plain-completion", "local-cohomology-proposition-uniform-artin-rees", "more-algebra-lemma-tor-strictly-pro-zero", "derived-lemma-trick-vanishing-composition", "more-algebra-lemma-derived-base-change", "local-cohomology-proposition-uniform-artin-rees" ], "ref_ids": [ 10391, 10421, 10421, 9791, 9954, 1817, 10136, 9791 ] }, { "id": 9801, "type": "other", "label": "local-cohomology-remark-duals", "categories": [ "local-cohomology" ], "title": "local-cohomology-remark-duals", "contents": [ "Given a pair $(M, n)$ consisting of an integer $n \\geq 0$", "and a finite $A/I^n$-module $M$ we set $M^\\vee = \\Hom_{A/I^n}(M, A/I^n)$.", "Given a pair $(\\mathcal{F}, n)$ consisting of an integer $n$ and", "a coherent $\\mathcal{O}_{Y_n}$-module $\\mathcal{F}$ we set", "$$", "\\mathcal{F}^\\vee =", "\\SheafHom_{\\mathcal{O}_{Y_n}}(\\mathcal{F}, \\mathcal{O}_{Y_n})", "$$", "Given $(M, n)$ as above, there is a canonical map", "$$", "can : p^*(M^\\vee) \\longrightarrow (p^*M)^\\vee", "$$", "Namely, if we choose a presentation", "$(A/I^n)^{\\oplus s} \\to (A/I^n)^{\\oplus r} \\to M \\to 0$", "then we obtain a presentation", "$\\mathcal{O}_{Y_n}^{\\oplus s} \\to \\mathcal{O}_{Y_n}^{\\oplus r} \\to", "p^*M \\to 0$. Taking duals we obtain exact sequences", "$$", "0 \\to M^\\vee \\to (A/I^n)^{\\oplus r} \\to (A/I^n)^{\\oplus s}", "$$", "and", "$$", "0 \\to (p^*M)^\\vee \\to", "\\mathcal{O}_{Y_n}^{\\oplus r} \\to", "\\mathcal{O}_{Y_n}^{\\oplus s}", "$$", "Pulling back the first sequence by $p$ we find the desired map $can$.", "The construction of this map is functorial in the finite", "$A/I^n$-module $M$. The kernel and cokernel of $can$", "are scheme theoretically supported", "on $Y_c$ if $M$ is an $(A, n, c)$-module. Namely, in that case for", "$a \\in I^c$ the map $a : M \\to M$ factors through a finite free", "$A/I^n$-module for which $can$ is an isomorphism. Hence $a$ annihilates", "the kernel and cokernel of $can$." ], "refs": [], "ref_ids": [] }, { "id": 10649, "type": "other", "label": "more-algebra-remark-relative-modules-over-fibre-product", "categories": [ "more-algebra" ], "title": "more-algebra-remark-relative-modules-over-fibre-product", "contents": [ "In Situation \\ref{situation-relative-module-over-fibre-product}.", "Assume $B' \\to D'$ is of finite presentation and", "suppose we are given a $D'$-module $L'$.", "We claim there is a bijective correspondence between", "\\begin{enumerate}", "\\item surjections of $D'$-modules $L' \\to Q'$ with $Q'$ of finite presentation", "over $D'$ and flat over $B'$, and", "\\item pairs of surjections of modules", "$(L' \\otimes_{D'} D \\to Q_1, L' \\otimes_{D'} C' \\to Q_2)$", "with", "\\begin{enumerate}", "\\item $Q_1$ of finite presentation over $D$ and flat over $B$,", "\\item $Q_2$ of finite presentation over $C'$ and flat over $A'$,", "\\item $Q_1 \\otimes_D C = Q_2 \\otimes_{C'} C$ as quotients of", "$L' \\otimes_{D'} C$.", "\\end{enumerate}", "\\end{enumerate}", "The correspondence between these is given by $Q \\mapsto (Q_1, Q_2)$ with", "$Q_1 = Q \\otimes_{D'} D$ and $Q_2 = Q \\otimes_{D'} C'$. And for the converse", "we use $Q = Q_1 \\times_{Q_{12}} Q_2$ where $Q_{12}$ the common quotient", "$Q_1 \\otimes_D C = Q_2 \\otimes_{C'} C$ of $L' \\otimes_{D'} C$. As quotient", "map we use", "$$", "L' \\longrightarrow", "(L' \\otimes_{D'} D) \\times_{(L' \\otimes_{D'} C)} (L' \\otimes_{D'} C')", "\\longrightarrow Q_1 \\times_{Q_{12}} Q_2 = Q", "$$", "where the first arrow is surjective by", "Lemma \\ref{lemma-module-over-fibre-product-bis}", "and the second by Lemma \\ref{lemma-surjection-module-over-fibre-product}.", "The claim follows by", "Lemmas \\ref{lemma-relative-flat-module-over-fibre-product} and", "\\ref{lemma-relative-finitely-presented-module-over-fibre-product}." ], "refs": [ "more-algebra-lemma-module-over-fibre-product-bis", "more-algebra-lemma-surjection-module-over-fibre-product", "more-algebra-lemma-relative-flat-module-over-fibre-product", "more-algebra-lemma-relative-finitely-presented-module-over-fibre-product" ], "ref_ids": [ 9821, 9822, 9829, 9830 ] }, { "id": 10650, "type": "other", "label": "more-algebra-remark-when-does-condition-hold", "categories": [ "more-algebra" ], "title": "more-algebra-remark-when-does-condition-hold", "contents": [ "Let $R$ be a ring. When does $R$ satisfy the condition mentioned in", "Lemmas \\ref{lemma-flat-finite-type-finite-presentation-local-module},", "\\ref{lemma-flat-finite-type-finite-presentation-local}, and", "\\ref{lemma-flat-graded-finite-type-finite-presentation}?", "This holds if", "\\begin{enumerate}", "\\item $R$ is local,", "\\item $R$ is Noetherian,", "\\item $R$ is a domain,", "\\item $R$ is a reduced ring with finitely many minimal primes, or", "\\item $R$ has finitely many weakly associated primes, see", "Algebra, Lemma \\ref{algebra-lemma-zero-at-weakly-ass-zero}.", "\\end{enumerate}", "Thus these lemmas hold in all cases listed above." ], "refs": [ "more-algebra-lemma-flat-finite-type-finite-presentation-local-module", "more-algebra-lemma-flat-finite-type-finite-presentation-local", "more-algebra-lemma-flat-graded-finite-type-finite-presentation", "algebra-lemma-zero-at-weakly-ass-zero" ], "ref_ids": [ 9943, 9944, 9946, 733 ] }, { "id": 10651, "type": "other", "label": "more-algebra-remark-what-does-it-mean", "categories": [ "more-algebra" ], "title": "more-algebra-remark-what-does-it-mean", "contents": [ "The assertion of Lemma \\ref{lemma-lift-fs} is quite strong. Namely,", "suppose that we have a diagram", "$$", "\\xymatrix{", "& B \\\\", "A \\ar[r] & A' \\ar[u]", "}", "$$", "of local homomorphisms of Noetherian complete local rings where", "$A \\to A'$ induces an isomorphism of residue fields", "$k = A/\\mathfrak m_A = A'/\\mathfrak m_{A'}$ and with", "$B \\otimes_{A'} k$ formally smooth over $k$.", "Then we can extend this to a commutative diagram", "$$", "\\xymatrix{", "C \\ar[r] & B \\\\", "A \\ar[r] \\ar[u] & A' \\ar[u]", "}", "$$", "of local homomorphisms of Noetherian complete local rings", "where $A \\to C$ is formally smooth in the $\\mathfrak m_C$-adic", "topology and where $C \\otimes_A k \\cong B \\otimes_{A'} k$.", "Namely, pick $A \\to C$ as in Lemma \\ref{lemma-lift-fs}", "lifting $B \\otimes_{A'} k$ over $k$. By formal smoothness we", "can find the arrow $C \\to B$, see", "Lemma \\ref{lemma-lift-continuous}.", "Denote $C \\otimes_A^\\wedge A'$ the completion of", "$C \\otimes_A A'$ with respect to the ideal $C \\otimes_A \\mathfrak m_{A'}$.", "Note that $C \\otimes_A^\\wedge A'$ is a Noetherian complete local", "ring (see Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian})", "which is flat over $A'$ (see", "Algebra, Lemma \\ref{algebra-lemma-flat-module-powers}).", "We have moreover", "\\begin{enumerate}", "\\item $C \\otimes_A^\\wedge A' \\to B$ is surjective,", "\\item if $A \\to A'$ is surjective, then $C \\to B$ is surjective,", "\\item if $A \\to A'$ is finite, then $C \\to B$ is finite, and", "\\item if $A' \\to B$ is flat, then $C \\otimes_A^\\wedge A' \\cong B$.", "\\end{enumerate}", "Namely, by Nakayama's lemma for nilpotent ideals (see", "Algebra, Lemma \\ref{algebra-lemma-NAK}) we see that", "$C \\otimes_A k \\cong B \\otimes_{A'} k$ implies that", "$C \\otimes_A A'/\\mathfrak m_{A'}^n \\to B/\\mathfrak m_{A'}^nB$", "is surjective for all $n$. This proves (1). Parts (2) and (3) follow", "from part (1). Part (4) follows from", "Algebra, Lemma \\ref{algebra-lemma-mod-injective}." ], "refs": [ "more-algebra-lemma-lift-fs", "more-algebra-lemma-lift-fs", "more-algebra-lemma-lift-continuous", "algebra-lemma-completion-Noetherian", "algebra-lemma-flat-module-powers", "algebra-lemma-NAK", "algebra-lemma-mod-injective" ], "ref_ids": [ 10033, 10033, 10016, 873, 893, 401, 883 ] }, { "id": 10652, "type": "other", "label": "more-algebra-remark-G-does-not-survive-completion", "categories": [ "more-algebra" ], "title": "more-algebra-remark-G-does-not-survive-completion", "contents": [ "Let $R$ be a G-ring and let $I \\subset R$ be an ideal.", "In general it is not the case that the $I$-adic completion $R^\\wedge$", "is a G-ring. An example was given by Nishimura in \\cite{Nishimura}.", "A generalization and, in some sense, clarification of this example can", "be found in the last section of \\cite{Dumitrescu}." ], "refs": [], "ref_ids": [] }, { "id": 10653, "type": "other", "label": "more-algebra-remark-P-resolution", "categories": [ "more-algebra" ], "title": "more-algebra-remark-P-resolution", "contents": [ "In fact, we can do better than Lemma \\ref{lemma-K-flat-resolution}.", "Namely, we can find a quasi-isomorphism", "$P^\\bullet \\to M^\\bullet$ where $P^\\bullet$ is a complex of $R$-modules", "endowed with a filtration", "$$", "0 = F_{-1}P^\\bullet \\subset F_0P^\\bullet \\subset", "F_1P^\\bullet \\subset \\ldots \\subset P^\\bullet", "$$", "by subcomplexes such that", "\\begin{enumerate}", "\\item $P^\\bullet = \\bigcup F_pP^\\bullet$,", "\\item the inclusions $F_iP^\\bullet \\to F_{i + 1}P^\\bullet$", "are termwise split injections,", "\\item the quotients $F_{i + 1}P^\\bullet/F_iP^\\bullet$ are isomorphic to direct", "sums of shifts $R[k]$ (as complexes, so differentials are zero).", "\\end{enumerate}", "This will be shown in", "Differential Graded Algebra, Lemma \\ref{dga-lemma-resolve}.", "Moreover, given such a complex we obtain a distinguished triangle", "$$", "\\bigoplus F_iP^\\bullet \\to \\bigoplus F_iP^\\bullet \\to M^\\bullet", "\\to \\bigoplus F_iP^\\bullet[1]", "$$", "in $D(R)$. Using this we can sometimes reduce statements about general", "complexes to statements about $R[k]$ (this of course only works if the", "statement is preserved under taking direct sums). More precisely, let", "$T$ be a property of objects of $D(R)$. Suppose that", "\\begin{enumerate}", "\\item if $K_i \\in D(R)$, $i \\in I$ is a family of objects with", "$T(K_i)$ for all $i \\in I$, then $T(\\bigoplus K_i)$,", "\\item if $K \\to L \\to M \\to K[1]$ is a distinguished triangle and", "$T$ holds for two, then $T$ holds for the third object,", "\\item $T(R[k])$ holds for all $k$.", "\\end{enumerate}", "Then $T$ holds for all objects of $D(R)$." ], "refs": [ "more-algebra-lemma-K-flat-resolution", "dga-lemma-resolve" ], "ref_ids": [ 10131, 13062 ] }, { "id": 10654, "type": "other", "label": "more-algebra-remark-warning-compute-base-change", "categories": [ "more-algebra" ], "title": "more-algebra-remark-warning-compute-base-change", "contents": [ "Let $R \\to A$ be a ring map, and let $N$ and $N'$ be $A$-modules.", "Denote $N_R$ and $N'_R$ the restriction of $N$ and $N'$ to $R$-modules,", "see Algebra, Section \\ref{algebra-section-base-change}.", "In this situation, the objects $N_R \\otimes_R^\\mathbf{L} N'$", "and $N \\otimes_R^\\mathbf{L} N'_R$ of $D(A)$ are in general", "not isomorphic! In other words, one has to pay careful attention", "as to which of the two sides is being used to provide the", "$A$-module structure.", "\\medskip\\noindent", "For a specific example, set $R = k[x, y]$, $A = R/(xy)$, $N = R/(x)$", "and $N' = A = R/(xy)$. The resolution", "$0 \\to R \\xrightarrow{xy} R \\to N'_R \\to 0$", "shows that $N \\otimes_R^\\mathbf{L} N'_R = N[1] \\oplus N$ in $D(A)$.", "The resolution", "$0 \\to R \\xrightarrow{x} R \\to N_R \\to 0$", "shows that $N_R \\otimes_R^\\mathbf{L} N'$ is represented by", "the complex $A \\xrightarrow{x} A$. To see these two complexes", "are not isomorphic, one can show that the second complex is", "not isomorphic in $D(A)$ to the direct sum of its cohomology groups,", "or one can show that the first complex is not a perfect object of $D(A)$", "whereas the second one is. Some details omitted." ], "refs": [], "ref_ids": [] }, { "id": 10655, "type": "other", "label": "more-algebra-remark-sign-explanation", "categories": [ "more-algebra" ], "title": "more-algebra-remark-sign-explanation", "contents": [ "In the yoga of super vector spaces the sign used in the proof", "of Lemma \\ref{lemma-evaluate-and-more} above can be explained", "as follows. A super vector space is just a finite dimensional", "vector space $V$ which comes with a direct sum decomposition", "$V = V^+ \\oplus V^-$. Here we think of the elements of $V^+$", "as the even elements and the elements of $V^-$ as the odd ones.", "Given two super vector spaces $V$ and $W$ we set", "$$", "(V \\otimes W)^+ = (V^+ \\otimes W^+) \\oplus (V^- \\otimes W^-)", "$$", "and similarly for the odd part. In the category of super vector", "spaces the isomorphism", "$$", "\\psi : V \\otimes W \\longrightarrow W \\otimes V", "$$", "is defined to be the usual one, except that on the summand", "$V^- \\otimes W^-$ we use the negative of the usual identification.", "In this way we obtain a symmetric monoidal category, see", "Categories, Section \\ref{categories-section-monoidal}. An object $V$ of the", "category of super vector spaces has a left dual which we denote $V^\\vee$", "which comes equipped with an identity $\\eta : \\mathbf{1} \\to V \\otimes V^\\vee$", "and an evaluation map $\\epsilon : V^\\vee \\otimes V \\to \\mathbf{1}$", "which induce canonical isomorphisms $\\Hom(V, W) = W \\otimes V^\\vee$", "and $\\Hom(V^\\vee, U) = V \\otimes U$, see", "Categories, Lemma \\ref{categories-lemma-left-dual}.", "Given three super vector spaces", "$U$, $V$, $W$ we can try to construct the analogue", "$$", "c : \\Hom(V, W) \\otimes U \\longrightarrow \\Hom(\\Hom(U, V), W)", "$$", "of the maps $c_{p, r, s}$ which occur in the lemma above.", "Using the formulae given above (which do not involve signs)", "this becomes a map", "$$", "W \\otimes V^\\vee \\otimes U", "\\longrightarrow", "W \\otimes (V \\otimes U^\\vee)^\\vee =", "W \\otimes (U^\\vee)^\\vee \\otimes V^\\vee", "$$", "To find this arrow in a canonical fashion we need to do two things:", "\\begin{enumerate}", "\\item we need to use the commutativity constraint", "$\\psi : V^\\vee \\otimes U \\to U \\otimes V^\\vee$ which introduces", "a sign on $(V^\\vee)^- \\otimes U^-$, and", "\\item we need to use the canonical isomorphism", "$U \\to (U^\\vee)^\\vee$ which comes from the", "identification of $U^\\vee$ as the {\\bf right} dual of", "$U$ using $\\psi$ as in", "Categories, Lemma \\ref{categories-lemma-dual-symmetric}.", "This differs from the usual identification by $-1$ on", "the odd part of $U$.", "\\end{enumerate}", "Part (1) explains the sign $(-1)^{qr}$ in the proof of the lemma", "and part (2) explains the sign $(-1)^r$ in the proof of the lemma." ], "refs": [ "more-algebra-lemma-evaluate-and-more", "categories-lemma-left-dual", "categories-lemma-dual-symmetric" ], "ref_ids": [ 10202, 12325, 12327 ] }, { "id": 10656, "type": "other", "label": "more-algebra-remark-smoothness-ext-1-zero", "categories": [ "more-algebra" ], "title": "more-algebra-remark-smoothness-ext-1-zero", "contents": [ "The following two statements follow from Lemma \\ref{lemma-ext-1-zero},", "Algebra, Definition \\ref{algebra-definition-smooth}, and", "Algebra, Proposition \\ref{algebra-proposition-characterize-formally-smooth}.", "\\begin{enumerate}", "\\item A ring map $A \\to B$ is smooth if and only if $A \\to B$ is", "of finite presentation and $\\Ext^1_B(\\NL_{B/A}, N) = 0$", "for every $B$-module $N$.", "\\item A ring map $A \\to B$ is formally smooth if and only if", "$\\Ext^1_B(\\NL_{B/A}, N) = 0$ for every $B$-module $N$.", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-ext-1-zero", "algebra-definition-smooth", "algebra-proposition-characterize-formally-smooth" ], "ref_ids": [ 10297, 1534, 1425 ] }, { "id": 10657, "type": "other", "label": "more-algebra-remark-Rlim-cohomology", "categories": [ "more-algebra" ], "title": "more-algebra-remark-Rlim-cohomology", "contents": [ "Consider the category $\\mathbf{N}$ whose objects are natural numbers and", "whose morphisms are unique arrows $i \\to j$ if $j \\geq i$. Endow $\\mathbf{N}$", "with the chaotic topology (Sites, Example \\ref{sites-example-indiscrete}) so", "that a sheaf $\\mathcal{F}$ is the same thing as an inverse system", "$$", "\\mathcal{F}_1 \\leftarrow \\mathcal{F}_2 \\leftarrow \\mathcal{F}_3", "\\leftarrow \\ldots", "$$", "of sets over $\\mathbf{N}$. Note that", "$\\Gamma(\\mathbf{N}, \\mathcal{F}) = \\lim \\mathcal{F}_n$. For an inverse", "system of abelian groups $\\mathcal{F}_n$ we have", "$$", "R^p\\lim \\mathcal{F}_n = H^p(\\mathbf{N}, \\mathcal{F})", "$$", "because both sides are the higher right derived functors of", "$\\mathcal{F} \\mapsto \\lim \\mathcal{F}_n = H^0(\\mathbf{N}, \\mathcal{F})$.", "Thus the existence of $R\\lim$ also follows from the general material in", "Cohomology on Sites, Sections", "\\ref{sites-cohomology-section-cohomology-sheaves} and", "\\ref{sites-cohomology-section-unbounded}." ], "refs": [], "ref_ids": [] }, { "id": 10658, "type": "other", "label": "more-algebra-remark-compare-derived-limit", "categories": [ "more-algebra" ], "title": "more-algebra-remark-compare-derived-limit", "contents": [ "Let $(K_n)$ be an inverse system of objects of $D(\\textit{Ab})$.", "Let $K = R\\lim K_n$ be a derived limit of this system (see", "Derived Categories, Section \\ref{derived-section-derived-limit}). Such", "a derived limit exists because $D(\\textit{Ab})$ has countable products", "(Derived Categories, Lemma \\ref{derived-lemma-products}).", "By Lemma \\ref{lemma-lift-to-system-complexes-Ab} we can also lift", "$(K_n)$ to an object $M$ of $D(\\mathbf{N})$.", "Then $K \\cong R\\lim M$ where $R\\lim$ is the functor (\\ref{equation-Rlim})", "because $R\\lim M$ is also a derived limit of the system $(K_n)$", "by Lemma \\ref{lemma-distinguished-triangle-Rlim}.", "Thus, although there may be many isomorphism classes of lifts $M$", "of the system $(K_n)$, the isomorphism type of", "$R\\lim M$ is independent of the choice because it is isomorphic", "to the derived limit $K = R\\lim K_n$ of the system. Thus we may", "apply results on $R\\lim$ proved in this section to derived limits.", "For example, for every $p \\in \\mathbf{Z}$ there is a", "canonical short exact sequence", "$$", "0 \\to R^1\\lim H^{p - 1}(K_n) \\to H^p(K) \\to \\lim H^p(K_n) \\to 0", "$$", "because we may apply Lemma \\ref{lemma-distinguished-triangle-Rlim} to $M$.", "This can also been seen directly, without invoking the existence of $M$,", "by applying the argument of the proof of", "Lemma \\ref{lemma-distinguished-triangle-Rlim} to the (defining)", "distinguished triangle $K \\to \\prod K_n \\to \\prod K_n \\to K[1]$." ], "refs": [ "derived-lemma-products", "more-algebra-lemma-lift-to-system-complexes-Ab", "more-algebra-lemma-distinguished-triangle-Rlim", "more-algebra-lemma-distinguished-triangle-Rlim", "more-algebra-lemma-distinguished-triangle-Rlim" ], "ref_ids": [ 1925, 10319, 10317, 10317, 10317 ] }, { "id": 10659, "type": "other", "label": "more-algebra-remark-Rlim-cohomology-modules", "categories": [ "more-algebra" ], "title": "more-algebra-remark-Rlim-cohomology-modules", "contents": [ "This remark is a continuation of Remark \\ref{remark-Rlim-cohomology}.", "A sheaf of rings on $\\mathbf{N}$ is just an inverse system of rings", "$(A_n)$. A sheaf of modules over $(A_n)$ is exactly the same thing", "as an object of the category $\\textit{Mod}(\\mathbf{N}, (A_n))$", "defined above. The derived functor $R\\lim$ of", "Lemma \\ref{lemma-compute-Rlim-modules}", "is simply $R\\Gamma(\\mathbf{N}, -)$ from the derived category of", "modules to the derived category of modules over the global sections", "of the structure sheaf. It", "is true in general that cohomology of groups and modules agree, see", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-cohomology-modules-abelian-agree}." ], "refs": [ "more-algebra-remark-Rlim-cohomology", "more-algebra-lemma-compute-Rlim-modules", "sites-cohomology-lemma-cohomology-modules-abelian-agree" ], "ref_ids": [ 10657, 10324, 4210 ] }, { "id": 10660, "type": "other", "label": "more-algebra-remark-how-unique", "categories": [ "more-algebra" ], "title": "more-algebra-remark-how-unique", "contents": [ "With assumptions as in Lemma \\ref{lemma-lift-to-system-complexes}.", "A priori there are many isomorphism classes of objects $M$ of", "$D(\\textit{Mod}(\\mathbf{N}, (A_n)))$ which give rise to the system", "$(K_n, \\varphi_n)$ of the lemma. For each such $M$ we can consider the", "complex $R\\lim M \\in D(A)$ where $A = \\lim A_n$. By", "Lemma \\ref{lemma-distinguished-triangle-Rlim-modules}", "we see that $R\\lim M$ is a derived limit of the inverse system", "$(K_n)$ of $D(A)$. Hence we see that the isomorphism class of $R\\lim M$ in", "$D(A)$ is independent of the choices made in constructing $M$.", "In particular, we may apply results on $R\\lim$ proved in this section to", "derived limits of inverse systems in $D(A)$.", "For example, for every $p \\in \\mathbf{Z}$ there is a", "canonical short exact sequence", "$$", "0 \\to R^1\\lim H^{p - 1}(K_n) \\to H^p(R\\lim K_n) \\to \\lim H^p(K_n) \\to 0", "$$", "because we may apply Lemma \\ref{lemma-distinguished-triangle-Rlim-modules}", "to $M$. This can also been seen directly, without invoking the existence of $M$,", "by applying the argument of the proof of", "Lemma \\ref{lemma-distinguished-triangle-Rlim-modules} to the (defining)", "distinguished triangle", "$R\\lim K_n \\to \\prod K_n \\to \\prod K_n \\to (R\\lim K_n)[1]$", "of the derived limit." ], "refs": [ "more-algebra-lemma-lift-to-system-complexes", "more-algebra-lemma-distinguished-triangle-Rlim-modules", "more-algebra-lemma-distinguished-triangle-Rlim-modules", "more-algebra-lemma-distinguished-triangle-Rlim-modules" ], "ref_ids": [ 10327, 10325, 10325, 10325 ] }, { "id": 10661, "type": "other", "label": "more-algebra-remark-constructing-tensor-with-limits-functorially", "categories": [ "more-algebra" ], "title": "more-algebra-remark-constructing-tensor-with-limits-functorially", "contents": [ "Let $A$ be a ring. Let $(E_n)$ be an inverse system of objects", "of $D(A)$. We've seen above that a derived limit $R\\lim E_n$", "exists. Thus for every object $K$ of $D(A)$ also the derived", "limit $R\\lim( K \\otimes_A^\\mathbf{L} E_n )$ exists.", "It turns out that we can construct these derived limits", "functorially in $K$ and obtain an exact functor", "$$", "R\\lim(- \\otimes_A^\\mathbf{L} E_n) : D(A) \\longrightarrow D(A)", "$$", "of triangulated categories. Namely, we first lift $(E_n)$ to an object $E$", "of $D(\\mathbf{N}, A)$, see Lemma \\ref{lemma-lift-to-system-complexes}.", "(The functor will depend on the choice of this lift.)", "Next, observe that there is a ``diagonal'' or ``constant'' functor", "$$", "\\Delta : D(A) \\longrightarrow D(\\mathbf{N}, A)", "$$", "mapping the complex $K^\\bullet$ to the constant inverse system of", "complexes with value $K^\\bullet$. Then we simply define", "$$", "R\\lim(K \\otimes_A^\\mathbf{L} E_n) = R\\lim(\\Delta(K)\\otimes^\\mathbf{L} E)", "$$", "where on the right hand side we use the functor $R\\lim$ of", "Lemma \\ref{lemma-compute-Rlim-modules}", "and the functor $- \\otimes^\\mathbf{L} -$ of", "Lemma \\ref{lemma-derived-tensor-product-systems}." ], "refs": [ "more-algebra-lemma-lift-to-system-complexes", "more-algebra-lemma-compute-Rlim-modules", "more-algebra-lemma-derived-tensor-product-systems" ], "ref_ids": [ 10327, 10324, 10330 ] }, { "id": 10662, "type": "other", "label": "more-algebra-remark-glueing-data", "categories": [ "more-algebra" ], "title": "more-algebra-remark-glueing-data", "contents": [ "In this remark we define a category of glueing data.", "Let $R \\to S$ be a ring map.", "Let $f_1, \\ldots, f_t \\in R$ and $I = (f_1, \\ldots, f_t)$.", "Consider the category $\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$", "as the category whose", "\\begin{enumerate}", "\\item objects are systems $(M', M_i, \\alpha_i, \\alpha_{ij})$, where", "$M'$ is an $S$-module, $M_i$ is an $R_{f_i}$-module,", "$\\alpha_i : (M')_{f_i} \\to M_i \\otimes_R S$ is an isomorphism, and", "$\\alpha_{ij} : (M_i)_{f_j} \\to (M_j)_{f_i}$ are isomorphisms", "such that", "\\begin{enumerate}", "\\item $\\alpha_{ij} \\circ \\alpha_i = \\alpha_j$ as maps", "$(M')_{f_if_j} \\to (M_j)_{f_i}$, and", "\\item $\\alpha_{jk} \\circ \\alpha_{ij} = \\alpha_{ik}$ as maps", "$(M_i)_{f_jf_k} \\to (M_k)_{f_if_j}$ (cocycle condition).", "\\end{enumerate}", "\\item morphisms", "$(M', M_i, \\alpha_i, \\alpha_{ij}) \\to (N', N_i, \\beta_i, \\beta_{ij})$", "are given by maps $\\varphi' : M' \\to N'$ and $\\varphi_i : M_i \\to N_i$", "compatible with the given maps $\\alpha_i, \\beta_i, \\alpha_{ij}, \\beta_{ij}$.", "\\end{enumerate}", "There is a canonical functor", "$$", "\\text{Can} : \\text{Mod}_R", "\\longrightarrow", "\\text{Glue}(R \\to S, f_1, \\ldots, f_t),", "\\quad", "M \\longmapsto (M \\otimes_R S, M_{f_i}, \\text{can}_i, \\text{can}_{ij})", "$$", "where $\\text{can}_i : (M \\otimes_R S)_{f_i} \\to M_{f_i} \\otimes_R S$", "and $\\text{can}_{ij} : (M_{f_i})_{f_j} \\to (M_{f_j})_{f_i}$", "are the canonical isomorphisms. For any object", "$\\mathbf{M} = (M', M_i, \\alpha_i, \\alpha_{ij})$ of the category", "$\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$ we define", "$$", "H^0(\\mathbf{M}) =", "\\{(m', m_i) \\mid \\alpha_i(m') = m_i \\otimes 1, \\alpha_{ij}(m_i) = m_j\\}", "$$", "in other words defined by the exact sequence", "$$", "0 \\to H^0(\\mathbf{M}) \\to", "M' \\times \\prod M_i \\to", "\\prod M'_{f_i}", "\\times", "\\prod (M_i)_{f_j}", "$$", "similar to (\\ref{equation-glueing-complex}).", "We think of $H^0(\\mathbf{M})$ as an $R$-module. Thus we also get a functor", "$$", "H^0 :", "\\text{Glue}(R \\to S, f_1, \\ldots, f_t)", "\\longrightarrow", "\\text{Mod}_R", "$$", "Our next goal is to show that the functors", "$\\text{Can}$ and $H^0$ are sometimes quasi-inverse to each other." ], "refs": [], "ref_ids": [] }, { "id": 10663, "type": "other", "label": "more-algebra-remark-formal-glueing-algebras", "categories": [ "more-algebra" ], "title": "more-algebra-remark-formal-glueing-algebras", "contents": [ "The equivalences of", "Proposition \\ref{proposition-equivalence},", "Theorem \\ref{theorem-formal-glueing}, and", "Proposition \\ref{proposition-formal-glueing}", "preserve properties of modules. For example if", "$M$ corresponds to $\\mathbf{M} = (M', M_i, \\alpha_i, \\alpha_{ij})$", "then $M$ is finite, or finitely presented, or flat, or projective over $R$", "if and only if $M'$ and $M_i$ have the corresponding property", "over $S$ and $R_{f_i}$. This follows from the fact that", "$R \\to S \\times \\prod R_{f_i}$ is faithfully flat and", "descend and ascent of these properties along faithfully flat maps, see", "Algebra, Lemma \\ref{algebra-lemma-descend-properties-modules} and", "Theorem \\ref{algebra-theorem-ffdescent-projectivity}.", "These functors also preserve the $\\otimes$-structures on either side.", "Thus, it defines equivalences of various categories", "built out of the pair $(\\text{Mod}_R, \\otimes)$, such as the category of", "algebras." ], "refs": [ "more-algebra-proposition-equivalence", "more-algebra-theorem-formal-glueing", "more-algebra-proposition-formal-glueing", "algebra-lemma-descend-properties-modules", "algebra-theorem-ffdescent-projectivity" ], "ref_ids": [ 10587, 9803, 10588, 819, 324 ] }, { "id": 10664, "type": "other", "label": "more-algebra-remark-topological-analogue", "categories": [ "more-algebra" ], "title": "more-algebra-remark-topological-analogue", "contents": [ "Given a differential manifold $X$ with a compact closed submanifold $Z$", "having complement $U$, specifying a sheaf on $X$ is the same as specifying", "a sheaf on $U$, a sheaf on an unspecified tubular neighbourhood $T$ of $Z$ in", "$X$, and an isomorphism between the two resulting sheaves along $T \\cap U$.", "Tubular neighbourhoods do not exist in algebraic geometry as such, but", "results such as", "Proposition \\ref{proposition-equivalence},", "Theorem \\ref{theorem-formal-glueing}, and", "Proposition \\ref{proposition-formal-glueing}", "allow us to work with formal neighbourhoods instead." ], "refs": [ "more-algebra-proposition-equivalence", "more-algebra-theorem-formal-glueing", "more-algebra-proposition-formal-glueing" ], "ref_ids": [ 10587, 9803, 10588 ] }, { "id": 10665, "type": "other", "label": "more-algebra-remark-not-descent", "categories": [ "more-algebra" ], "title": "more-algebra-remark-not-descent", "contents": [ "While $R \\to R_f$ is always flat, $R \\to R^\\wedge$ is typically not flat", "unless $R$ is Noetherian (see", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}", "and the discussion in", "Examples, Section \\ref{examples-section-nonflat}).", "Consequently, we cannot in general apply faithfully flat descent", "as discussed in Descent, Section", "\\ref{descent-section-descent-modules}", "to the morphism $R \\to R^\\wedge \\oplus R_f$.", "Moreover, even in the Noetherian case, the usual", "definition of a descent datum for this morphism", "refers to the ring $R^\\wedge \\otimes_R R^\\wedge$, which we will", "avoid considering in this section." ], "refs": [ "algebra-lemma-completion-flat" ], "ref_ids": [ 870 ] }, { "id": 10666, "type": "other", "label": "more-algebra-remark-BL-special-case", "categories": [ "more-algebra" ], "title": "more-algebra-remark-BL-special-case", "contents": [ "Suppose that $f$ is a nonzerodivisor. Then", "Algebra, Lemma \\ref{algebra-lemma-completion-differ-by-torsion}", "shows that $f$ is a nonzerodivisor in $R^\\wedge$.", "Hence $(R, f)$ is a glueing pair." ], "refs": [ "algebra-lemma-completion-differ-by-torsion" ], "ref_ids": [ 860 ] }, { "id": 10667, "type": "other", "label": "more-algebra-remark-noetherian-case", "categories": [ "more-algebra" ], "title": "more-algebra-remark-noetherian-case", "contents": [ "If $R \\to R^\\wedge$ is flat, then for each positive integer $n$ tensoring", "the sequence $0 \\to R[f^n] \\to R \\to R$ with $R^\\wedge$ gives the sequence", "$0 \\to R[f^n] \\otimes_R R^\\wedge \\to R^\\wedge \\to R^\\wedge$.", "Combined with Lemma \\ref{lemma-torsion-completion}", "we conclude that $R[f^n] \\to R^\\wedge[f^n]$ is an isomorphism.", "Thus $(R, f)$ is a glueing pair.", "This holds in particular if $R$ is Noetherian, see", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}." ], "refs": [ "more-algebra-lemma-torsion-completion", "algebra-lemma-completion-flat" ], "ref_ids": [ 10354, 870 ] }, { "id": 10668, "type": "other", "label": "more-algebra-remark-glueable", "categories": [ "more-algebra" ], "title": "more-algebra-remark-glueable", "contents": [ "Let $(R \\to R', f)$ be a glueing pair and let $M$ be an $R$-module.", "Here are some observations which can be used to determine whether", "$M$ is glueable for $(R \\to R', f)$.", "\\begin{enumerate}", "\\item By Lemma \\ref{lemma-same-f-torsion-module} we see that $M$", "is glueable for $(R \\to R^\\wedge, f)$", "if and only if $M[f^\\infty] \\to M \\otimes_R R^\\wedge$ is injective.", "This holds if $M[f] \\to M^\\wedge$ is injective,", "i.e., when $M[f] \\cap \\bigcap_{n = 1}^\\infty f^n M = 0$.", "\\item If $\\text{Tor}_1^R(M, R'_f) = 0$, then $M$ is glueable for", "$(R \\to R', f)$", "(use Algebra, Lemma \\ref{algebra-lemma-long-exact-sequence-tor}).", "This is equivalent to saying that $\\text{Tor}_1^R(M, R')$ is", "$f$-power torsion. In particular, any flat $R$-module", "is glueable for $(R \\to R', f)$.", "\\item If $R \\to R'$ is flat, then $\\text{Tor}_1^R(M, R') = 0$", "for every $R$-module so every $R$-module is glueable for", "$(R \\to R', f)$. This holds", "in particular when $R$ is Noetherian and $R' = R^\\wedge$, see", "Algebra, Lemma \\ref{algebra-lemma-completion-flat}", "\\end{enumerate}" ], "refs": [ "more-algebra-lemma-same-f-torsion-module", "algebra-lemma-long-exact-sequence-tor", "algebra-lemma-completion-flat" ], "ref_ids": [ 10359, 782, 870 ] }, { "id": 10669, "type": "other", "label": "more-algebra-remark-what-you-get-for-general-modules", "categories": [ "more-algebra" ], "title": "more-algebra-remark-what-you-get-for-general-modules", "contents": [ "Let $(R \\to R', f)$ be a glueing pair.", "Let $M$ be an $R$-module that is not necessarily glueable", "for $(R \\to R', f)$. Setting $M' = M \\otimes_R R'$ and $M_1 = M_f$", "we obtain the glueing datum $\\text{Can}(M) = (M', M_1, \\text{can})$.", "Then $\\tilde M = H^0(M', M_1, \\text{can})$ is an $R$-module that is", "glueable for $(R \\to R', f)$", "and the canonical map $M \\to \\tilde M$ gives isomorphisms", "$M \\otimes_R R' \\to \\tilde M \\otimes_R R'$ and", "$M_f \\to \\tilde M_f$, see Theorem \\ref{theorem-BL-glueing}.", "From the exactness of the sequences", "$$", "M \\to (M \\otimes_R R' )\\oplus M_f \\to M \\otimes_R (R')_f \\to 0", "$$", "and", "$$", "0 \\to \\tilde M \\to (\\tilde M \\otimes_R R') \\oplus \\tilde M_f \\to", "\\tilde M \\otimes_R (R')_f \\to 0", "$$", "we conclude that the map $M \\to \\tilde M$ is surjective." ], "refs": [ "more-algebra-theorem-BL-glueing" ], "ref_ids": [ 9804 ] }, { "id": 10670, "type": "other", "label": "more-algebra-remark-compare-BL", "categories": [ "more-algebra" ], "title": "more-algebra-remark-compare-BL", "contents": [ "In \\cite{Beauville-Laszlo} it is assumed that $f$ is a nonzerodivisor", "in $R$ and $R' = R^\\wedge$, which gives a glueing pair by", "Lemma \\ref{lemma-same-f-torsion}.", "Even in this setting Theorem \\ref{theorem-BL-glueing}", "says something new: the", "results of \\cite{Beauville-Laszlo} only apply to modules on which", "$f$ is a nonzerodivisor (and hence glueable in our sense, see", "Lemma \\ref{lemma-same-f-torsion-module}).", "Lemma \\ref{lemma-BL-properties} also provides a slight extension", "of the results of \\cite{Beauville-Laszlo}: not only can we allow $M$", "to have nonzero $f$-power torsion, we do not even require it to be glueable." ], "refs": [ "more-algebra-lemma-same-f-torsion", "more-algebra-theorem-BL-glueing", "more-algebra-lemma-same-f-torsion-module", "more-algebra-lemma-BL-properties" ], "ref_ids": [ 10358, 9804, 10359, 10364 ] }, { "id": 10671, "type": "other", "label": "more-algebra-remark-derived-completion", "categories": [ "more-algebra" ], "title": "more-algebra-remark-derived-completion", "contents": [ "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.", "The left adjoint to the inclusion functor $D_{comp}(A, I) \\to D(A)$", "which exists by Lemma \\ref{lemma-derived-completion} is called the", "{\\it derived completion}. To indicate this we will say", "``let $K^\\wedge$ be the derived completion of $K$''. Please keep in mind", "that the unit of the adjunction is a functorial map $K \\to K^\\wedge$." ], "refs": [ "more-algebra-lemma-derived-completion" ], "ref_ids": [ 10372 ] }, { "id": 10672, "type": "other", "label": "more-algebra-remark-Leta", "categories": [ "more-algebra" ], "title": "more-algebra-remark-Leta", "contents": [ "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor.", "Suppose that $M^\\bullet$ is a complex of $A$-modules.", "By Lemma \\ref{lemma-K-flat-resolution} we can choose a quasi-isomorphism", "$K^\\bullet \\to M^\\bullet$ such that $K^\\bullet$ is K-flat and", "consists of flat $A$-modules. In particular, $f$ is a nonzerodivisor", "on $K^i$ for all $i$ and we have $\\eta_fK^\\bullet$ defined above.", "In this situation we define", "$$", "L\\eta_f M^\\bullet = \\eta_fK^\\bullet", "$$", "This is independent of the choice of the K-flat resolution", "by Lemma \\ref{lemma-eta-qis}. We obtain a functor", "$L\\eta_f : D(A) \\to D(A)$. Beware that this functor isn't exact, i.e.,", "does not tranform distinguished triangles into distinguished triangles." ], "refs": [ "more-algebra-lemma-K-flat-resolution", "more-algebra-lemma-eta-qis" ], "ref_ids": [ 10131, 10398 ] }, { "id": 10673, "type": "other", "label": "more-algebra-remark-eta-BZ", "categories": [ "more-algebra" ], "title": "more-algebra-remark-eta-BZ", "contents": [ "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let $M^\\bullet$ be", "a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$.", "For every $i$ set $\\overline{M}^i = M^i/fM^i$. Denote", "$B^i \\subset Z^i \\subset \\overline{M}^i$ the boundaries and", "cocycles for the differentials on the complex", "$\\overline{M}^\\bullet = M^\\bullet \\otimes_A A/fA$.", "We claim that there exists a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "B^{i + 1} \\ar[r] \\ar@{=}[d] &", "B^{i + 1} \\oplus B^i \\ar[r] \\ar[d]^{s, s'} &", "B^i \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "B^{i + 1} \\ar[r]^-s &", "(\\eta_fM)^i /f(\\eta_fM)^i \\ar[r]^-t &", "Z^i \\ar[r] & 0", "}", "$$", "with exact rows. Here are the constructions of the maps", "\\begin{enumerate}", "\\item If $x \\in (\\eta_fM)^i$ then $x = f^ix'$", "with $d^i(x') = 0$ in $\\overline{M}^{i + 1}$. Hence we can define the map", "$t$ by sending $x$ to the class of $x'$.", "\\item If $y \\in M^{i + 1}$ has class $\\overline{y}$ in", "$B^{i + 1} \\subset \\overline{M}^{i + 1}$ then we", "can write $y = fy' + d^i(x)$ for $y' \\in M^{i + 1}$ and $x \\in M^i$.", "Hence we can define the map $s$ sending $\\overline{y}$ to the class of", "$f^{i + 1}x$ in $(\\eta_fM)^i /f(\\eta_fM)^i$; we omit the verification that", "this is well defined.", "\\item If $x \\in M^i$ has class $\\overline{x}$ in $B^i \\subset \\overline{M}^i$", "then we can write $x = fx' + d^{i - 1}(z)$ for $x' \\in M^i$ and", "$z \\in M^{i - 1}$. We define the map $s'$ by sending $\\overline{x}$ to the", "class of $f^i d^{i - 1}(z)$ in $(\\eta_fM)^i/f(\\eta_fM)^i$. This is well defined", "because if $fx' + d^{i - 1}(z) = 0$, then $f^ix'$ is in $(\\eta_fM)^i$", "and consequently $f^id^{i - 1}(z)$ is in $f(\\eta_fM)^i$.", "\\end{enumerate}", "We omit the verification that the lower row in the displayed", "diagram is a short exact sequence of modules.", "It is immediately clear from these constructions that we have commutative", "diagrams", "$$", "\\xymatrix{", "B^{i + 1} \\oplus B^i \\ar[d]^{s, s'} \\ar[r] &", "B^{i + 2} \\oplus B^{i + 1} \\ar[d]^{s, s'} \\\\", "(\\eta_fM)^i /f(\\eta_fM)^i \\ar[r] &", "(\\eta_fM)^{i + 1} /f(\\eta_fM)^{i + 1}", "}", "$$", "where the upper horizontal arrow is given by the identification", "of the summands $B^{i + 1}$ in source and target. In other words,", "we have found an acyclic subcomplex of", "$\\eta_fM^\\bullet / f(\\eta_fM^\\bullet) = \\eta_fM^\\bullet \\otimes_A A/fA$", "and the quotient by this subcomplex is a complex whose terms $Z^i/B^i$", "are the cohomology modules of the complex", "$\\overline{M}^\\bullet = M^\\bullet \\otimes_A A/fA$." ], "refs": [], "ref_ids": [] }, { "id": 10674, "type": "other", "label": "more-algebra-remark-weird-systems", "categories": [ "more-algebra" ], "title": "more-algebra-remark-weird-systems", "contents": [ "Let $I$ be an ideal of a Noetherian ring $A$. Set $A_n = A/I^n$ for $n \\geq 1$.", "Consider the following category:", "\\begin{enumerate}", "\\item An object is a sequence $\\{E_n\\}_{n \\geq 1}$ where $E_n$ is a finite", "$A_n$-module.", "\\item A morphism $\\{E_n\\} \\to \\{E'_n\\}$ is given by maps", "$$", "\\varphi_n : I^cE_n \\longrightarrow E'_n/E'_n[I^c]", "\\quad\\text{for }n \\geq c", "$$", "where $E'_n[I^c]$ is the torsion submodule (Section \\ref{section-torsion})", "up to equivalence: we say $(c, \\varphi_n)$ is the same as", "$(c + 1, \\overline{\\varphi}_n)$ where", "$\\overline{\\varphi}_n : I^{c + 1}E_n \\longrightarrow E'_n/E'_n[I^{c + 1}]$", "is the induced map.", "\\end{enumerate}", "Composition of $(c, \\varphi_n) : \\{E_n\\} \\to \\{E'_n\\}$", "and $(c', \\varphi'_n) : \\{E'_n\\} \\to \\{E''_n\\}$", "is defined by the obvious compositions", "$$", "I^{c + c'}E_n \\to I^{c'}E'_n/E'_n[I^{c}] \\to E''_n/E''_n[I^{c + c'}]", "$$", "for $n \\geq c + c'$. We omit the verification that this is a category." ], "refs": [], "ref_ids": [] }, { "id": 10675, "type": "other", "label": "more-algebra-remark-awkward", "categories": [ "more-algebra" ], "title": "more-algebra-remark-awkward", "contents": [ "The awkwardness in the statement of Lemma \\ref{lemma-dejong-kollar-kovacs}", "is partly due to the fact that there are no", "obvious maps between the modules $\\Ext^i_{A_n}(M_n, N_n)$", "for varying $n$. What we may conclude from the", "lemma is that there exists a $c \\geq 0$ such that", "for $m \\gg n \\gg 0$ there are (canonical) maps", "$$", "I^c\\Ext^i_{A_n}(M_m, N_m)/I^n\\Ext^i_{A_n}(M_m, N_m) \\to", "\\Ext^i_{A_n}(M_n, N_n)/\\Ext^i_{A_n}(M_n, N_n)[I^c]", "$$", "whose kernel and cokernel are annihilated by $I^c$.", "This is the (weak) sense in which we get a system of modules." ], "refs": [ "more-algebra-lemma-dejong-kollar-kovacs" ], "ref_ids": [ 10425 ] }, { "id": 10676, "type": "other", "label": "more-algebra-remark-finite-separable-extension", "categories": [ "more-algebra" ], "title": "more-algebra-remark-finite-separable-extension", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "Let $L/K$ be a finite separable field extension.", "Let $B \\subset L$ be the integral closure of $A$ in $L$.", "Picture:", "$$", "\\xymatrix{", "B \\ar[r] & L \\\\", "A \\ar[u] \\ar[r] & K \\ar[u]", "}", "$$", "By Algebra, Lemma", "\\ref{algebra-lemma-Noetherian-normal-domain-finite-separable-extension}", "the ring extension $A \\subset B$ is finite, hence $B$ is Noetherian.", "By Algebra, Lemma \\ref{algebra-lemma-integral-sub-dim-equal}", "the dimension of $B$ is $1$, hence $B$ is a Dedekind domain, see", "Algebra, Lemma \\ref{algebra-lemma-characterize-Dedekind}.", "Let $\\mathfrak m_1, \\ldots, \\mathfrak m_n$ be the maximal ideals", "of $B$ (i.e., the primes lying over $\\mathfrak m_A$). We obtain", "extensions of discrete valuation rings", "$$", "A \\subset B_{\\mathfrak m_i}", "$$", "and hence ramification indices $e_i$ and residue degrees $f_i$. We have", "$$", "[L : K] = \\sum\\nolimits_{i = 1, \\ldots, n} e_i f_i", "$$", "by Algebra, Lemma \\ref{algebra-lemma-finite-extension-dim-1}", "applied to a uniformizer in $A$.", "We observe that $n = 1$ if $A$ is henselian (by", "Algebra, Lemma \\ref{algebra-lemma-finite-over-henselian}), e.g.\\ if", "$A$ is complete." ], "refs": [ "algebra-lemma-Noetherian-normal-domain-finite-separable-extension", "algebra-lemma-integral-sub-dim-equal", "algebra-lemma-characterize-Dedekind", "algebra-lemma-finite-extension-dim-1", "algebra-lemma-finite-over-henselian" ], "ref_ids": [ 1338, 985, 1041, 1047, 1277 ] }, { "id": 10677, "type": "other", "label": "more-algebra-remark-tower-of-rings", "categories": [ "more-algebra" ], "title": "more-algebra-remark-tower-of-rings", "contents": [ "Let $A$ be a discrete valuation ring with fraction field $K$.", "Let $L/K$ be a finite Galois extension. Let $\\mathfrak m \\subset B$", "be a maximal ideal of the integral closure of $A$ in $L$.", "Let", "$$", "P \\subset I \\subset D \\subset G", "$$", "be the wild inertia, inertia, decomposition group of $\\mathfrak m$.", "Consider the diagram", "$$", "\\xymatrix{", "\\mathfrak m \\ar@{-}[d] \\ar@{-}[r] &", "\\mathfrak m^P \\ar@{-}[d] \\ar@{-}[r] &", "\\mathfrak m^I \\ar@{-}[d] \\ar@{-}[r] &", "\\mathfrak m^D \\ar@{-}[d] \\ar@{-}[r] &", "A \\cap \\mathfrak m \\ar@{-}[d] \\\\", "B & B^P \\ar[l] & B^I \\ar[l] & B^D \\ar[l] & A \\ar[l]", "}", "$$", "Observe that $B^P, B^I, B^D$ are the integral closures of", "$A$ in the fields $L^P$, $L^I$, $L^D$. Thus we also see that", "$B^P$ is the integral closure of $B^I$ in $L^P$ and so on.", "Observe that $\\mathfrak m^P = \\mathfrak m \\cap B^P$,", "$\\mathfrak m^I = \\mathfrak m \\cap B^I$, and", "$\\mathfrak m^D = \\mathfrak m \\cap B^D$. Hence the", "top line of the diagram corresponds to the images", "of $\\mathfrak m \\in \\Spec(B)$ under the induced maps of", "spectra. Having said all of this we have the following", "\\begin{enumerate}", "\\item the extension $L^I/L^D$ is Galois with group $D/I$,", "\\item the extension $L^P/L^I$ is Galois with group $I_t = I/P$,", "\\item the extension $L^P/L^D$ is Galois with group $D/P$,", "\\item $\\mathfrak m^I$ is the unique prime of $B^I$ lying over $\\mathfrak m^D$,", "\\item $\\mathfrak m^P$ is the unique prime of $B^P$ lying over $\\mathfrak m^I$,", "\\item $\\mathfrak m$ is the unique prime of $B$ lying over $\\mathfrak m^P$,", "\\item $\\mathfrak m^P$ is the unique prime of $B^P$ lying over $\\mathfrak m^D$,", "\\item $\\mathfrak m$ is the unique prime of $B$ lying over $\\mathfrak m^I$,", "\\item $\\mathfrak m$ is the unique prime of $B$ lying over $\\mathfrak m^D$,", "\\item $A \\to B^D_{\\mathfrak m^D}$ is \\'etale and induces a", "trivial residue field extension,", "\\item $B^D_{\\mathfrak m^D} \\to B^I_{\\mathfrak m^I}$ is \\'etale", "and induces a Galois extension of residue fields with Galois", "group $D/I$,", "\\item $A \\to B^I_{\\mathfrak m^I}$ is \\'etale,", "\\item $B^I_{\\mathfrak m^I} \\to B^P_{\\mathfrak m^P}$", "has ramification index $|I/P|$ prime to $p$ and induces a", "trivial residue field extension,", "\\item $B^D_{\\mathfrak m^D} \\to B^P_{\\mathfrak m^P}$", "has ramification index $|I/P|$ prime to $p$ and induces a", "separable residue field extension,", "\\item $A \\to B^P_{\\mathfrak m^P}$", "has ramification index $|I/P|$ prime to $p$ and induces a", "separable residue field extension.", "\\end{enumerate}", "Statements (1), (2), and (3) are immediate from Galois theory", "(Fields, Section \\ref{fields-section-galois-theory})", "and Lemma \\ref{lemma-galois-inertia}.", "Statements (4) -- (9) are clear from", "Lemma \\ref{lemma-galois}.", "Part (12) is Lemma \\ref{lemma-inertial-invariants-unramified}.", "Since we have the factorization", "$A \\to B^D_{\\mathfrak m^D} \\to B^I_{\\mathfrak m^I}$", "we obtain the \\'etaleness in (10) and (11) as a consequence.", "The residue field extension in (10) must be trivial because", "it is separable and $D/I$ maps onto", "$\\text{Aut}(\\kappa(\\mathfrak m)/\\kappa_A)$ as shown in", "Lemma \\ref{lemma-galois-galois}. The same argument", "provides the proof of the statement on residue fields in (11).", "To see (13), (14), and (15) it suffices to prove (13).", "By the above, the extension $L^P/L^I$ is Galois", "with a cyclic Galois group of order prime to $p$,", "the prime $\\mathfrak m^P$ is the unique prime lying over", "$\\mathfrak m^I$ and the action of $I/P$ on the residue", "field is trivial. Thus we can apply Lemma \\ref{lemma-galois-inertia}", "to this extension and the discrete valuation ring", "$B^I_{\\mathfrak m^I}$ to see that (13) holds." ], "refs": [ "more-algebra-lemma-galois-inertia", "more-algebra-lemma-galois", "more-algebra-lemma-inertial-invariants-unramified", "more-algebra-lemma-galois-galois", "more-algebra-lemma-galois-inertia" ], "ref_ids": [ 10501, 10498, 10503, 10500, 10501 ] }, { "id": 10678, "type": "other", "label": "more-algebra-remark-canonical-inertia-character", "categories": [ "more-algebra" ], "title": "more-algebra-remark-canonical-inertia-character", "contents": [ "In order to use the inertia character", "$\\theta : I \\to \\mu_e(\\kappa(\\mathfrak m))$", "for infinite Galois extensions, it is convenient", "to scale it. Let $A, K, L, B, \\mathfrak m, G, P, I, D, e, \\theta$", "be as in Lemma \\ref{lemma-galois-inertia} and", "Definition \\ref{definition-wild-inertia}.", "Then $e = q |I_t|$ with $q$ is a power of the characteristic $p$", "of $\\kappa(\\mathfrak m)$ if positive or $1$ if zero.", "Note that $\\mu_e(\\kappa(\\mathfrak m)) = \\mu_{|I_t|}(\\kappa(\\mathfrak m))$", "because the characteristic of $\\kappa(\\mathfrak m)$ is $p$. Consider", "the map", "$$", "\\theta_{can} = q\\theta : I \\longrightarrow \\mu_{|I_t|}(\\kappa(\\mathfrak m))", "$$", "This map induces an isomorphism", "$\\theta_{can} : I_t \\to \\mu_{|I_t|}(\\kappa(\\mathfrak m))$.", "We have $\\theta_{can}(\\tau \\sigma \\tau^{-1}) = \\tau(\\theta_{can}(\\sigma))$", "for $\\tau \\in D$ and $\\sigma \\in I$", "by Lemma \\ref{lemma-inertia-character}.", "Finally, if $M/L$ is an extension such that $M/K$ is Galois", "and $\\mathfrak m'$ is a prime of the integral closure of $A$ in $M$", "lying over $\\mathfrak m$, then we get the commutative diagram", "$$", "\\xymatrix{", "I' \\ar[r]_-{\\theta'_{can}} \\ar[d] &", "\\mu_{|I'_t|}(\\kappa(\\mathfrak m')) \\ar[d]^{(-)^{|I'_t|/|I_t|}} \\\\", "I \\ar[r]^-{\\theta_{can}} &", "\\mu_{|I_t|}(\\kappa(\\mathfrak m))", "}", "$$", "by Lemma \\ref{lemma-compare-inertia}." ], "refs": [ "more-algebra-lemma-galois-inertia", "more-algebra-definition-wild-inertia", "more-algebra-lemma-inertia-character", "more-algebra-lemma-compare-inertia" ], "ref_ids": [ 10501, 10643, 10502, 10504 ] }, { "id": 10679, "type": "other", "label": "more-algebra-remark-construction", "categories": [ "more-algebra" ], "title": "more-algebra-remark-construction", "contents": [ "Let $A \\to B$ be an extension of discrete valuation rings with fraction", "fields $K \\subset L$. Let $K \\subset K_1$ be a finite extension of", "fields. Let $A_1 \\subset K_1$ be the integral closure of $A$ in $K_1$.", "On the other hand, let $L_1 = (L \\otimes_K K_1)_{red}$. Then $L_1$ is a", "nonempty finite product of finite field extensions of $L$. Let $B_1$ be", "the integral closure of $B$ in $L_1$. We obtain compatible commutative", "diagrams", "$$", "\\vcenter{", "\\xymatrix{", "L \\ar[r] & L_1 \\\\", "K \\ar[u] \\ar[r] & K_1 \\ar[u]", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "B \\ar[r] & B_1 \\\\", "A \\ar[u] \\ar[r] & A_1 \\ar[u]", "}", "}", "$$", "In this situation we have the following", "\\begin{enumerate}", "\\item By Algebra, Lemma \\ref{algebra-lemma-integral-closure-Dedekind}", "the ring $A_1$ is a Dedekind domain and $B_1$ is a finite product of", "Dedekind domains.", "\\item Note that $L \\otimes_K K_1 = (B \\otimes_A A_1)_\\pi$ where $\\pi \\in A$", "is a uniformizer and that $\\pi$ is a nonzerodivisor on $B \\otimes_A A_1$. ", "Thus the ring map $B \\otimes_A A_1 \\to B_1$ is integral with kernel", "consisting of nilpotent elements. Hence $\\Spec(B_1) \\to \\Spec(B \\otimes_A A_1)$", "is surjective on spectra", "(Algebra, Lemma \\ref{algebra-lemma-integral-overring-surjective}).", "The map $\\Spec(B \\otimes_A A_1) \\to \\Spec(A_1)$ is surjective as", "$A_1/\\mathfrak m_A A_1 \\to", "B/\\mathfrak m_AB \\otimes_{\\kappa_A} A_1/\\mathfrak m_A A_1$", "is an injective ring map with $A_1/\\mathfrak m_A A_1$ Artinian.", "We conclude that $\\Spec(B_1) \\to \\Spec(A_1)$ is surjective.", "\\item Let $\\mathfrak m_i$, $i = 1, \\ldots n$ with $n \\geq 1$ be the", "maximal ideals of $A_1$. For each $i = 1, \\ldots, n$ let", "$\\mathfrak m_{ij}$, $j = 1, \\ldots, m_i$ with $m_i \\geq 1$", "be the maximal ideals of $B_1$ lying over $\\mathfrak m_i$. We obtain diagrams", "$$", "\\xymatrix{", "B \\ar[r] & (B_1)_{\\mathfrak m_{ij}} \\\\", "A \\ar[u] \\ar[r] & (A_1)_{\\mathfrak m_i} \\ar[u]", "}", "$$", "of extensions of discrete valuation rings.", "\\item If $A$ is henselian (for example complete), then $A_1$ is a", "discrete valuation ring, i.e., $n = 1$.", "Namely, $A_1$ is a union of finite extensions of $A$ which are domains,", "hence local by Algebra, Lemma \\ref{algebra-lemma-finite-over-henselian}.", "\\item If $B$ is henselian (for example complete), then $B_1$", "is a product of discrete valuation rings, i.e., $m_i = 1$ for", "$i = 1, \\ldots, n$.", "\\item If $K \\subset K_1$ is purely inseparable, then $A_1$ and $B_1$", "are both discrete valuation rings, i.e., $n = 1$ and $m_1 = 1$.", "This is true because for every $b \\in B_1$ a $p$-power power of $b$", "is in $B$, hence $B_1$ can only have one maximal ideal.", "\\item If $K \\subset K_1$ is finite separable, then $L_1 = L \\otimes_K K_1$", "and is a finite product of finite separable extensions too. Hence", "$A \\subset A_1$ and $B \\subset B_1$ are finite by", "Algebra, Lemma", "\\ref{algebra-lemma-Noetherian-normal-domain-finite-separable-extension}.", "\\item If $A$ is Nagata, then $A \\subset A_1$ is finite.", "\\item If $B$ is Nagata, then $B \\subset B_1$ is finite.", "\\end{enumerate}" ], "refs": [ "algebra-lemma-integral-closure-Dedekind", "algebra-lemma-integral-overring-surjective", "algebra-lemma-finite-over-henselian", "algebra-lemma-Noetherian-normal-domain-finite-separable-extension" ], "ref_ids": [ 1042, 495, 1277, 1338 ] }, { "id": 10680, "type": "other", "label": "more-algebra-remark-determinant-as-socle", "categories": [ "more-algebra" ], "title": "more-algebra-remark-determinant-as-socle", "contents": [ "Let $R$ be a ring. Let $M$ be a finite projective $R$-module.", "Then we can consider the graded commutative $R$-algebra", "exterior algebra $\\wedge^*_R(M)$ on $M$ over $R$.", "A formula for $\\det(M)$ is that $\\det(M) \\subset \\wedge^*_R(M)$", "is the annihilator of $M \\subset \\wedge^*_R(M)$.", "This is sometimes useful as it does not refer to the", "decomposition of $R$ into a product. Of course, to", "prove this satisfies the desired properties one has to", "either decompose $R$ into a product (as above), or one", "has to look at the localizations at primes of $R$." ], "refs": [], "ref_ids": [] }, { "id": 10742, "type": "other", "label": "etale-remark-technicality-needed", "categories": [ "etale" ], "title": "etale-remark-technicality-needed", "contents": [ "The result on ``reducedness'' does not hold with a weaker", "definition of \\'etale local ring maps $A \\to B$ where one", "drops the assumption that $B$ is essentially of finite type over $A$.", "Namely, it can happen that a Noetherian local domain $A$ has nonreduced", "completion $A^\\wedge$, see", "Examples, Section \\ref{examples-section-local-completion-nonreduced}.", "But the ring map $A \\to A^\\wedge$ is flat, and $\\mathfrak m_AA^\\wedge$", "is the maximal ideal of $A^\\wedge$ and of course $A$ and $A^\\wedge$ have", "the same residue fields. This is why it is important to consider", "this notion only for ring extensions which are essentially of finite type", "(or essentially of finite presentation if $A$ is not Noetherian)." ], "refs": [], "ref_ids": [] }, { "id": 10811, "type": "other", "label": "crystalline-remark-completed-affine-site", "categories": [ "crystalline" ], "title": "crystalline-remark-completed-affine-site", "contents": [ "In Situation \\ref{situation-affine} we denote", "$\\text{Cris}^\\wedge(C/A)$ the category whose objects are", "pairs $(B \\to C, \\delta)$ such that", "\\begin{enumerate}", "\\item $B$ is a $p$-adically complete $A$-algebra,", "\\item $B \\to C$ is a surjection of $A$-algebras,", "\\item $\\delta$ is a divided power structure on $\\Ker(B \\to C)$,", "\\item $A \\to B$ is a homomorphism of divided power rings.", "\\end{enumerate}", "Morphisms are defined as in Definition \\ref{definition-affine-thickening}.", "Then $\\text{Cris}(C/A) \\subset \\text{Cris}^\\wedge(C/A)$ is the full", "subcategory consisting of those $B$ such that $p$ is nilpotent in $B$.", "Conversely, any object $(B \\to C, \\delta)$ of $\\text{Cris}^\\wedge(C/A)$", "is equal to the limit", "$$", "(B \\to C, \\delta) = \\lim_e (B/p^eB \\to C, \\delta)", "$$", "where for $e \\gg 0$ the object $(B/p^eB \\to C, \\delta)$ lies", "in $\\text{Cris}(C/A)$, see", "Divided Power Algebra, Lemma \\ref{dpa-lemma-extend-to-completion}.", "In particular, we see that $\\text{Cris}^\\wedge(C/A)$ is a full subcategory", "of the category of pro-objects of $\\text{Cris}(C/A)$, see", "Categories, Remark \\ref{categories-remark-pro-category}." ], "refs": [ "crystalline-definition-affine-thickening", "dpa-lemma-extend-to-completion", "categories-remark-pro-category" ], "ref_ids": [ 10799, 1660, 12420 ] }, { "id": 10812, "type": "other", "label": "crystalline-remark-filtration-differentials", "categories": [ "crystalline" ], "title": "crystalline-remark-filtration-differentials", "contents": [ "Let $A \\to B$ be a ring map and let $(J, \\delta)$ be a divided", "power structure on $B$. The universal module $\\Omega_{B/A, \\delta}$", "comes with a little bit of extra structure, namely the $B$-submodule", "$N$ of $\\Omega_{B/A, \\delta}$ generated by $\\text{d}_{B/A, \\delta}(J)$.", "In terms of the isomorphism given in", "Lemma \\ref{lemma-diagonal-and-differentials}", "this corresponds to the image of", "$K \\cap J(1)$ in $\\Omega_{B/A, \\delta}$. Consider the $A$-algebra", "$D = B \\oplus \\Omega^1_{B/A, \\delta}$ with ideal $\\bar J = J \\oplus N$", "and divided powers $\\bar \\delta$ as in the proof of the lemma.", "Then $(D, \\bar J, \\bar \\delta)$ is a divided power ring", "and the two maps $B \\to D$ given by $b \\mapsto b$ and", "$b \\mapsto b + \\text{d}_{B/A, \\delta}(b)$", "are homomorphisms of divided power rings over $A$. Moreover, $N$", "is the smallest submodule of $\\Omega_{B/A, \\delta}$ such that this is true." ], "refs": [ "crystalline-lemma-diagonal-and-differentials" ], "ref_ids": [ 10757 ] }, { "id": 10813, "type": "other", "label": "crystalline-remark-divided-powers-de-rham-complex", "categories": [ "crystalline" ], "title": "crystalline-remark-divided-powers-de-rham-complex", "contents": [ "Let $A \\to B$ be a ring map and let $(J, \\delta)$ be a divided power", "structure on $B$. Set", "$\\Omega_{B/A, \\delta}^i = \\wedge^i_B \\Omega_{B/A, \\delta}$", "where $\\Omega_{B/A, \\delta}$ is the target of the universal divided power", "$A$-derivation $\\text{d} = \\text{d}_{B/A} : B \\to \\Omega_{B/A, \\delta}$.", "Note that $\\Omega_{B/A, \\delta}$ is the quotient of $\\Omega_{B/A}$ by the", "$B$-submodule generated by the elements", "$\\text{d}\\delta_n(x) - \\delta_{n - 1}(x)\\text{d}x$ for $x \\in J$.", "We claim Algebra, Lemma \\ref{algebra-lemma-de-rham-complex} applies.", "To see this it suffices to verify the elements", "$\\text{d}\\delta_n(x) - \\delta_{n - 1}(x)\\text{d}x$", "of $\\Omega_B$ are mapped to zero in $\\Omega^2_{B/A, \\delta}$.", "We observe that", "$$", "\\text{d}(\\delta_{n - 1}(x)) \\wedge \\text{d}x", "= \\delta_{n - 2}(x) \\text{d}x \\wedge \\text{d}x = 0", "$$", "in $\\Omega^2_{B/A, \\delta}$ as desired. Hence we obtain a", "{\\it divided power de Rham complex}", "$$", "\\Omega^0_{B/A, \\delta} \\to \\Omega^1_{B/A, \\delta} \\to", "\\Omega^2_{B/A, \\delta} \\to \\ldots", "$$", "which will play an important role in the sequel." ], "refs": [ "algebra-lemma-de-rham-complex" ], "ref_ids": [ 1143 ] }, { "id": 10814, "type": "other", "label": "crystalline-remark-connection", "categories": [ "crystalline" ], "title": "crystalline-remark-connection", "contents": [ "Let $A \\to B$ be a ring map. Let $\\Omega_{B/A} \\to \\Omega$", "be a quotient satisfying the assumptions of", "Algebra, Lemma \\ref{algebra-lemma-de-rham-complex}.", "Let $M$ be a $B$-module. A {\\it connection} is an additive map", "$$", "\\nabla : M \\longrightarrow M \\otimes_B \\Omega", "$$", "such that $\\nabla(bm) = b \\nabla(m) + m \\otimes \\text{d}b$", "for $b \\in B$ and $m \\in M$. In this situation we can define maps", "$$", "\\nabla : M \\otimes_B \\Omega^i \\longrightarrow M \\otimes_B \\Omega^{i + 1}", "$$", "by the rule $\\nabla(m \\otimes \\omega) = \\nabla(m) \\wedge \\omega +", "m \\otimes \\text{d}\\omega$. This works because if $b \\in B$, then", "\\begin{align*}", "\\nabla(bm \\otimes \\omega) - \\nabla(m \\otimes b\\omega)", "& =", "\\nabla(bm) \\wedge \\omega + bm \\otimes \\text{d}\\omega", "- \\nabla(m) \\wedge b\\omega - m \\otimes \\text{d}(b\\omega) \\\\", "& =", "b\\nabla(m) \\wedge \\omega + m \\otimes \\text{d}b \\wedge \\omega", "+ bm \\otimes \\text{d}\\omega \\\\", "& \\ \\ \\ \\ \\ \\ - b\\nabla(m) \\wedge \\omega - bm \\otimes \\text{d}(\\omega)", "- m \\otimes \\text{d}b \\wedge \\omega = 0", "\\end{align*}", "As is customary we say the connection is {\\it integrable} if and", "only if the composition", "$$", "M \\xrightarrow{\\nabla} M \\otimes_B \\Omega^1", "\\xrightarrow{\\nabla} M \\otimes_B \\Omega^2", "$$", "is zero. In this case we obtain a complex", "$$", "M \\xrightarrow{\\nabla} M \\otimes_B \\Omega^1", "\\xrightarrow{\\nabla} M \\otimes_B \\Omega^2", "\\xrightarrow{\\nabla} M \\otimes_B \\Omega^3", "\\xrightarrow{\\nabla} M \\otimes_B \\Omega^4 \\to \\ldots", "$$", "which is called the de Rham complex of the connection." ], "refs": [ "algebra-lemma-de-rham-complex" ], "ref_ids": [ 1143 ] }, { "id": 10815, "type": "other", "label": "crystalline-remark-base-change-connection", "categories": [ "crystalline" ], "title": "crystalline-remark-base-change-connection", "contents": [ "Consider a commutative diagram of rings", "$$", "\\xymatrix{", "B \\ar[r]_\\varphi & B' \\\\", "A \\ar[u] \\ar[r] & A' \\ar[u]", "}", "$$", "Let $\\Omega_{B/A} \\to \\Omega$ and $\\Omega_{B'/A'} \\to \\Omega'$", "be quotients satisfying the assumptions of", "Algebra, Lemma \\ref{algebra-lemma-de-rham-complex}.", "Assume there is a map $\\varphi : \\Omega \\to \\Omega'$ which", "fits into a commutative diagram", "$$", "\\xymatrix{", "\\Omega_{B/A} \\ar[r] \\ar[d] &", "\\Omega_{B'/A'} \\ar[d] \\\\", "\\Omega \\ar[r]^{\\varphi} &", "\\Omega'", "}", "$$", "where the top horizontal arrow is the canonical map", "$\\Omega_{B/A} \\to \\Omega_{B'/A'}$ induced by $\\varphi : B \\to B'$.", "In this situation, given any pair $(M, \\nabla)$ where $M$ is a $B$-module", "and $\\nabla : M \\to M \\otimes_B \\Omega$ is a connection", "we obtain a {\\it base change} $(M \\otimes_B B', \\nabla')$ where", "$$", "\\nabla' :", "M \\otimes_B B'", "\\longrightarrow", "(M \\otimes_B B') \\otimes_{B'} \\Omega' = M \\otimes_B \\Omega'", "$$", "is defined by the rule", "$$", "\\nabla'(m \\otimes b') =", "\\sum m_i \\otimes b'\\text{d}\\varphi(b_i) + m \\otimes \\text{d}b' ", "$$", "if $\\nabla(m) = \\sum m_i \\otimes \\text{d}b_i$. If $\\nabla$ is integrable,", "then so is $\\nabla'$, and in this case there is a canonical map of", "de Rham complexes (Remark \\ref{remark-connection})", "\\begin{equation}", "\\label{equation-base-change-map-complexes}", "M \\otimes_B \\Omega^\\bullet", "\\longrightarrow", "(M \\otimes_B B') \\otimes_{B'} (\\Omega')^\\bullet =", "M \\otimes_B (\\Omega')^\\bullet", "\\end{equation}", "which maps $m \\otimes \\eta$ to $m \\otimes \\varphi(\\eta)$." ], "refs": [ "algebra-lemma-de-rham-complex", "crystalline-remark-connection" ], "ref_ids": [ 1143, 10814 ] }, { "id": 10816, "type": "other", "label": "crystalline-remark-functoriality-big-cris", "categories": [ "crystalline" ], "title": "crystalline-remark-functoriality-big-cris", "contents": [ "Let $p$ be a prime number.", "Let $(S, \\mathcal{I}, \\gamma) \\to (S', \\mathcal{I}', \\gamma')$ be a", "morphism of divided power schemes over $\\mathbf{Z}_{(p)}$.", "Set $S_0 = V(\\mathcal{I})$ and $S'_0 = V(\\mathcal{I}')$.", "Let", "$$", "\\xymatrix{", "X \\ar[r]_f \\ar[d] & Y \\ar[d] \\\\", "S_0 \\ar[r] & S'_0", "}", "$$", "be a commutative diagram of morphisms of schemes and assume $p$ is", "locally nilpotent on $X$ and $Y$. Then we get a continuous and", "cocontinuous functor", "$$", "\\text{CRIS}(X/S) \\longrightarrow \\text{CRIS}(Y/S')", "$$", "by letting $(U, T, \\delta)$ correspond to $(U, T, \\delta)$", "with $U \\to X \\to Y$ as the $S'$-morphism from $U$ to $Y$.", "Hence we get a morphism of topoi", "$$", "f_{\\text{CRIS}} : (X/S)_{\\text{CRIS}} \\longrightarrow (Y/S')_{\\text{CRIS}}", "$$", "see Sites, Section \\ref{sites-section-cocontinuous-morphism-topoi}." ], "refs": [], "ref_ids": [] }, { "id": 10817, "type": "other", "label": "crystalline-remark-compare-big-zariski", "categories": [ "crystalline" ], "title": "crystalline-remark-compare-big-zariski", "contents": [ "In Situation \\ref{situation-global}.", "The functor (\\ref{equation-forget}) is cocontinuous (details omitted) and", "commutes with products and fibred products", "(Lemma \\ref{lemma-divided-power-thickening-fibre-products}).", "Hence we obtain a morphism of topoi", "$$", "U_{X/S} : (X/S)_{\\text{CRIS}} \\longrightarrow \\Sh((\\Sch/X)_{Zar})", "$$", "from the big crystalline topos of $X/S$ to the big Zariski topos of $X$.", "See Sites, Section \\ref{sites-section-cocontinuous-morphism-topoi}." ], "refs": [ "crystalline-lemma-divided-power-thickening-fibre-products" ], "ref_ids": [ 10762 ] }, { "id": 10818, "type": "other", "label": "crystalline-remark-big-structure-morphism", "categories": [ "crystalline" ], "title": "crystalline-remark-big-structure-morphism", "contents": [ "In Situation \\ref{situation-global}.", "Consider the closed subscheme $S_0 = V(\\mathcal{I}) \\subset S$.", "If we assume that $p$ is locally nilpotent on $S_0$ (which is always", "the case in practice) then we obtain a situation as in", "Definition \\ref{definition-divided-power-thickening-X} with $S_0$ instead", "of $X$. Hence we get a site $\\text{CRIS}(S_0/S)$. If $f : X \\to S_0$ is", "the structure morphism of $X$ over $S$, then we get a commutative diagram", "of morphisms of ringed topoi", "$$", "\\xymatrix{", "(X/S)_{\\text{CRIS}}", "\\ar[r]_{f_{\\text{CRIS}}} \\ar[d]_{U_{X/S}} &", "(S_0/S)_{\\text{CRIS}} \\ar[d]^{U_{S_0/S}} \\\\", "\\Sh((\\Sch/X)_{Zar}) \\ar[r]^{f_{big}} & \\Sh((\\Sch/S_0)_{Zar}) \\ar[rd] \\\\", "& & \\Sh((\\Sch/S)_{Zar})", "}", "$$", "by Remark \\ref{remark-functoriality-big-cris}. We think of the composition", "$(X/S)_{\\text{CRIS}} \\to \\Sh((\\Sch/S)_{Zar})$ as the structure morphism of", "the big crystalline site. Even if $p$ is not locally nilpotent on $S_0$", "the structure morphism", "$$", "(X/S)_{\\text{CRIS}} \\longrightarrow \\Sh((\\Sch/S)_{Zar})", "$$", "is defined as we can take the lower route through the diagram above. Thus it", "is the morphism of topoi corresponding to the cocontinuous", "functor $\\text{CRIS}(X/S) \\to (\\Sch/S)_{Zar}$ given by the rule", "$(U, T, \\delta)/S \\mapsto U/S$, see", "Sites, Section \\ref{sites-section-cocontinuous-morphism-topoi}." ], "refs": [ "crystalline-definition-divided-power-thickening-X", "crystalline-remark-functoriality-big-cris" ], "ref_ids": [ 10804, 10816 ] }, { "id": 10819, "type": "other", "label": "crystalline-remark-compatibilities-big-cris", "categories": [ "crystalline" ], "title": "crystalline-remark-compatibilities-big-cris", "contents": [ "The morphisms defined above satisfy numerous compatibilities. For example,", "in the situation of Remark \\ref{remark-functoriality-big-cris}", "we obtain a commutative diagram of ringed topoi", "$$", "\\xymatrix{", "(X/S)_{\\text{CRIS}} \\ar[d] \\ar[r] & (Y/S')_{\\text{CRIS}} \\ar[d] \\\\", "\\Sh((\\Sch/S)_{Zar}) \\ar[r] & \\Sh((\\Sch/S')_{Zar})", "}", "$$", "where the vertical arrows are the structure morphisms." ], "refs": [ "crystalline-remark-functoriality-big-cris" ], "ref_ids": [ 10816 ] }, { "id": 10820, "type": "other", "label": "crystalline-remark-functoriality-cris", "categories": [ "crystalline" ], "title": "crystalline-remark-functoriality-cris", "contents": [ "Let $p$ be a prime number.", "Let $(S, \\mathcal{I}, \\gamma) \\to (S', \\mathcal{I}', \\gamma')$", "be a morphism of divided power schemes over $\\mathbf{Z}_{(p)}$.", "Let", "$$", "\\xymatrix{", "X \\ar[r]_f \\ar[d] & Y \\ar[d] \\\\", "S_0 \\ar[r] & S'_0", "}", "$$", "be a commutative diagram of morphisms of schemes and assume $p$ is", "locally nilpotent on $X$ and $Y$. By analogy with", "Topologies, Lemma \\ref{topologies-lemma-morphism-big-small} we define", "$$", "f_{\\text{cris}} : (X/S)_{\\text{cris}} \\longrightarrow (Y/S')_{\\text{cris}}", "$$", "by the formula $f_{\\text{cris}} = \\pi_Y \\circ f_{\\text{CRIS}} \\circ i_X$", "where $i_X$ and $\\pi_Y$ are as in Lemma \\ref{lemma-compare-big-small}", "for $X$ and $Y$ and where $f_{\\text{CRIS}}$ is as in", "Remark \\ref{remark-functoriality-big-cris}." ], "refs": [ "topologies-lemma-morphism-big-small", "crystalline-lemma-compare-big-small", "crystalline-remark-functoriality-big-cris" ], "ref_ids": [ 12441, 10764, 10816 ] }, { "id": 10821, "type": "other", "label": "crystalline-remark-compare-zariski", "categories": [ "crystalline" ], "title": "crystalline-remark-compare-zariski", "contents": [ "In Situation \\ref{situation-global}.", "The functor (\\ref{equation-forget-small}) is continuous, cocontinuous, and", "commutes with products and fibred products.", "Hence we obtain a morphism of topoi", "$$", "u_{X/S} : (X/S)_{\\text{cris}} \\longrightarrow \\Sh(X_{Zar})", "$$", "relating the small crystalline topos of $X/S$ with", "the small Zariski topos of $X$.", "See Sites, Section \\ref{sites-section-cocontinuous-morphism-topoi}." ], "refs": [], "ref_ids": [] }, { "id": 10822, "type": "other", "label": "crystalline-remark-structure-morphism", "categories": [ "crystalline" ], "title": "crystalline-remark-structure-morphism", "contents": [ "In Situation \\ref{situation-global}.", "Consider the closed subscheme $S_0 = V(\\mathcal{I}) \\subset S$.", "If we assume that $p$ is locally nilpotent on $S_0$ (which is always", "the case in practice) then we obtain a situation as in", "Definition \\ref{definition-divided-power-thickening-X} with $S_0$ instead", "of $X$. Hence we get a site $\\text{Cris}(S_0/S)$. If $f : X \\to S_0$", "is the structure morphism of $X$ over $S$, then we get a", "commutative diagram of ringed topoi", "$$", "\\xymatrix{", "(X/S)_{\\text{cris}}", "\\ar[r]_{f_{\\text{cris}}} \\ar[d]_{u_{X/S}} &", "(S_0/S)_{\\text{cris}} \\ar[d]^{u_{S_0/S}} \\\\", "\\Sh(X_{Zar}) \\ar[r]^{f_{small}} & \\Sh(S_{0, Zar}) \\ar[rd] \\\\", "& & \\Sh(S_{Zar})", "}", "$$", "see Remark \\ref{remark-functoriality-cris}. We think of the composition", "$(X/S)_{\\text{cris}} \\to \\Sh(S_{Zar})$ as the structure morphism of the", "crystalline site. Even if $p$ is not locally nilpotent on $S_0$", "the structure morphism", "$$", "\\tau_{X/S} : (X/S)_{\\text{cris}} \\longrightarrow \\Sh(S_{Zar})", "$$", "is defined as we can take the lower route through the diagram above." ], "refs": [ "crystalline-definition-divided-power-thickening-X", "crystalline-remark-functoriality-cris" ], "ref_ids": [ 10804, 10820 ] }, { "id": 10823, "type": "other", "label": "crystalline-remark-compatibilities", "categories": [ "crystalline" ], "title": "crystalline-remark-compatibilities", "contents": [ "The morphisms defined above satisfy numerous compatibilities. For example,", "in the situation of Remark \\ref{remark-functoriality-cris}", "we obtain a commutative diagram of ringed topoi", "$$", "\\xymatrix{", "(X/S)_{\\text{cris}} \\ar[d] \\ar[r] & (Y/S')_{\\text{cris}} \\ar[d] \\\\", "\\Sh((\\Sch/S)_{Zar}) \\ar[r] & \\Sh((\\Sch/S')_{Zar})", "}", "$$", "where the vertical arrows are the structure morphisms." ], "refs": [ "crystalline-remark-functoriality-cris" ], "ref_ids": [ 10820 ] }, { "id": 10824, "type": "other", "label": "crystalline-remark-crystal", "categories": [ "crystalline" ], "title": "crystalline-remark-crystal", "contents": [ "To formulate the general notion of a crystal we use the language", "of stacks and strongly cartesian morphisms, see", "Stacks, Definition \\ref{stacks-definition-stack} and", "Categories, Definition \\ref{categories-definition-cartesian-over-C}.", "In Situation \\ref{situation-global} let", "$p : \\mathcal{C} \\to \\text{Cris}(X/S)$ be a stack.", "A {\\it crystal in objects of $\\mathcal{C}$ on $X$ relative to $S$}", "is a {\\it cartesian section} $\\sigma : \\text{Cris}(X/S) \\to \\mathcal{C}$,", "i.e., a functor $\\sigma$ such that $p \\circ \\sigma = \\text{id}$", "and such that $\\sigma(f)$ is strongly cartesian for all", "morphisms $f$ of $\\text{Cris}(X/S)$. Similarly for the big crystalline site." ], "refs": [ "stacks-definition-stack", "categories-definition-cartesian-over-C" ], "ref_ids": [ 8996, 12387 ] }, { "id": 10825, "type": "other", "label": "crystalline-remark-first-order-thickening", "categories": [ "crystalline" ], "title": "crystalline-remark-first-order-thickening", "contents": [ "In Situation \\ref{situation-global}.", "Let $(U, T, \\delta)$ be an object of $\\text{Cris}(X/S)$.", "Write $\\Omega_{T/S, \\delta} = (\\Omega_{X/S})_T$, see", "Lemma \\ref{lemma-module-of-differentials}.", "We explicitly describe a first order thickening $T'$ of", "$T$. Namely, set", "$$", "\\mathcal{O}_{T'} = \\mathcal{O}_T \\oplus \\Omega_{T/S, \\delta}", "$$", "with algebra structure such that $\\Omega_{T/S, \\delta}$ is an", "ideal of square zero. Let $\\mathcal{J} \\subset \\mathcal{O}_T$", "be the ideal sheaf of the closed immersion $U \\to T$. Set", "$\\mathcal{J}' = \\mathcal{J} \\oplus \\Omega_{T/S, \\delta}$.", "Define a divided power structure on $\\mathcal{J}'$ by setting", "$$", "\\delta_n'(f, \\omega) = (\\delta_n(f), \\delta_{n - 1}(f)\\omega),", "$$", "see Lemma \\ref{lemma-divided-power-first-order-thickening}.", "There are two ring maps", "$$", "p_0, p_1 : \\mathcal{O}_T \\to \\mathcal{O}_{T'}", "$$", "The first is given by $f \\mapsto (f, 0)$ and the second by", "$f \\mapsto (f, \\text{d}_{T/S, \\delta}f)$. Note that both are compatible", "with the divided power structures on $\\mathcal{J}$ and $\\mathcal{J}'$", "and so is the quotient map $\\mathcal{O}_{T'} \\to \\mathcal{O}_T$.", "Thus we get an object $(U, T', \\delta')$ of $\\text{Cris}(X/S)$", "and a commutative diagram", "$$", "\\xymatrix{", "& T \\ar[ld]_{\\text{id}} \\ar[d]^i \\ar[rd]^{\\text{id}} \\\\", "T & T' \\ar[l]_{p_0} \\ar[r]^{p_1} & T", "}", "$$", "of $\\text{Cris}(X/S)$ such that $i$ is a first order thickening whose ideal", "sheaf is identified with $\\Omega_{T/S, \\delta}$ and such that", "$p_1^* - p_0^* : \\mathcal{O}_T \\to \\mathcal{O}_{T'}$", "is identified with the universal derivation $\\text{d}_{T/S, \\delta}$", "composed with the inclusion $\\Omega_{T/S, \\delta} \\to \\mathcal{O}_{T'}$." ], "refs": [ "crystalline-lemma-module-of-differentials", "crystalline-lemma-divided-power-first-order-thickening" ], "ref_ids": [ 10768, 10750 ] }, { "id": 10826, "type": "other", "label": "crystalline-remark-second-order-thickening", "categories": [ "crystalline" ], "title": "crystalline-remark-second-order-thickening", "contents": [ "In Situation \\ref{situation-global}.", "Let $(U, T, \\delta)$ be an object of $\\text{Cris}(X/S)$.", "Write $\\Omega_{T/S, \\delta} = (\\Omega_{X/S})_T$, see", "Lemma \\ref{lemma-module-of-differentials}.", "We also write $\\Omega^2_{T/S, \\delta}$ for its second exterior", "power. We explicitly describe a second order thickening $T''$ of $T$.", "Namely, set", "$$", "\\mathcal{O}_{T''} =", "\\mathcal{O}_T \\oplus \\Omega_{T/S, \\delta} \\oplus \\Omega_{T/S, \\delta}", "\\oplus \\Omega^2_{T/S, \\delta}", "$$", "with algebra structure defined in the following way", "$$", "(f, \\omega_1, \\omega_2, \\eta) \\cdot", "(f', \\omega_1', \\omega_2', \\eta') =", "(ff', f\\omega_1' + f'\\omega_1, f\\omega_2' + f'\\omega_2,", "f\\eta' + f'\\eta + \\omega_1 \\wedge \\omega_2' + \\omega_1' \\wedge \\omega_2).", "$$", "Let $\\mathcal{J} \\subset \\mathcal{O}_T$", "be the ideal sheaf of the closed immersion $U \\to T$. Let", "$\\mathcal{J}''$ be the inverse image of $\\mathcal{J}$ under the", "projection $\\mathcal{O}_{T''} \\to \\mathcal{O}_T$.", "Define a divided power structure on $\\mathcal{J}''$ by setting", "$$", "\\delta_n''(f, \\omega_1, \\omega_2, \\eta) =", "(\\delta_n(f), \\delta_{n - 1}(f)\\omega_1, \\delta_{n - 1}(f)\\omega_2,", "\\delta_{n - 1}(f)\\eta + \\delta_{n - 2}(f)\\omega_1 \\wedge \\omega_2)", "$$", "see Lemma \\ref{lemma-divided-power-second-order-thickening}.", "There are three ring maps", "$q_0, q_1, q_2 : \\mathcal{O}_T \\to \\mathcal{O}_{T''}$", "given by", "\\begin{align*}", "q_0(f) & = (f, 0, 0, 0), \\\\", "q_1(f) & = (f, \\text{d}f, 0, 0), \\\\", "q_2(f) & = (f, \\text{d}f, \\text{d}f, 0)", "\\end{align*}", "where $\\text{d} = \\text{d}_{T/S, \\delta}$.", "Note that all three are compatible with the divided power structures", "on $\\mathcal{J}$ and $\\mathcal{J}''$. There are three ring maps", "$q_{01}, q_{12}, q_{02} : \\mathcal{O}_{T'} \\to \\mathcal{O}_{T''}$", "where $\\mathcal{O}_{T'}$ is as in Remark \\ref{remark-first-order-thickening}.", "Namely, set", "\\begin{align*}", "q_{01}(f, \\omega) & = (f, \\omega, 0, 0), \\\\", "q_{12}(f, \\omega) & =", "(f, \\text{d}f, \\omega, \\text{d}\\omega), \\\\", "q_{02}(f, \\omega) & = (f, \\omega, \\omega, 0)", "\\end{align*}", "These are also compatible with the given divided power", "structures. Let's do the verifications for $q_{12}$: Note", "that $q_{12}$ is a ring homomorphism as", "\\begin{align*}", "q_{12}(f, \\omega)q_{12}(g, \\eta) & =", "(f, \\text{d}f, \\omega, \\text{d}\\omega)(g, \\text{d}g, \\eta, \\text{d}\\eta) \\\\", "& =", "(fg, f\\text{d}g + g \\text{d}f, f\\eta + g\\omega,", "f\\text{d}\\eta + g\\text{d}\\omega + \\text{d}f \\wedge \\eta +", "\\text{d}g \\wedge \\omega) \\\\", "& = q_{12}(fg, f\\eta + g\\omega) = q_{12}((f, \\omega)(g, \\eta))", "\\end{align*}", "Note that $q_{12}$ is compatible with divided powers because", "\\begin{align*}", "\\delta_n''(q_{12}(f, \\omega)) & =", "\\delta_n''((f, \\text{d}f, \\omega, \\text{d}\\omega)) \\\\", "& =", "(\\delta_n(f), \\delta_{n - 1}(f)\\text{d}f, \\delta_{n - 1}(f)\\omega,", "\\delta_{n - 1}(f)\\text{d}\\omega + \\delta_{n - 2}(f)\\text{d}(f) \\wedge \\omega)", "\\\\", "& = q_{12}((\\delta_n(f), \\delta_{n - 1}(f)\\omega)) =", "q_{12}(\\delta'_n(f, \\omega))", "\\end{align*}", "The verifications for $q_{01}$ and $q_{02}$ are easier.", "Note that $q_0 = q_{01} \\circ p_0$, $q_1 = q_{01} \\circ p_1$,", "$q_1 = q_{12} \\circ p_0$, $q_2 = q_{12} \\circ p_1$,", "$q_0 = q_{02} \\circ p_0$, and $q_2 = q_{02} \\circ p_1$.", "Thus $(U, T'', \\delta'')$ is an object of $\\text{Cris}(X/S)$", "and we get morphisms", "$$", "\\xymatrix{", "T''", "\\ar@<2ex>[r]", "\\ar@<0ex>[r]", "\\ar@<-2ex>[r]", "&", "T'", "\\ar@<1ex>[r]", "\\ar@<-1ex>[r]", "&", "T", "}", "$$", "of $\\text{Cris}(X/S)$ satisfying the relations described above.", "In applications we will use $q_i : T'' \\to T$ and", "$q_{ij} : T'' \\to T'$ to denote the morphisms associated to the", "ring maps described above." ], "refs": [ "crystalline-lemma-module-of-differentials", "crystalline-lemma-divided-power-second-order-thickening", "crystalline-remark-first-order-thickening" ], "ref_ids": [ 10768, 10751, 10825 ] }, { "id": 10827, "type": "other", "label": "crystalline-remark-equivalence-more-general", "categories": [ "crystalline" ], "title": "crystalline-remark-equivalence-more-general", "contents": [ "The equivalence of Proposition \\ref{proposition-crystals-on-affine}", "holds if we start with a surjection $P \\to C$ where $P/A$ satisfies the", "strong lifting property of", "Algebra, Lemma \\ref{algebra-lemma-smooth-strong-lift}.", "To prove this we can argue as in the proof of", "Lemma \\ref{lemma-crystals-on-affine-smooth}.", "(Details will be added here if we ever need this.)", "Presumably there is also a direct proof of this result, but the advantage", "of using polynomial rings is that the rings $D(n)$ are $p$-adic completions", "of divided power polynomial rings and the algebra is simplified." ], "refs": [ "crystalline-proposition-crystals-on-affine", "algebra-lemma-smooth-strong-lift", "crystalline-lemma-crystals-on-affine-smooth" ], "ref_ids": [ 10793, 1216, 10777 ] }, { "id": 10828, "type": "other", "label": "crystalline-remark-vanishing", "categories": [ "crystalline" ], "title": "crystalline-remark-vanishing", "contents": [ "The proof of Proposition \\ref{proposition-compare-with-de-Rham}", "shows that the conclusion", "$$", "Ru_{X/S, *}(\\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^i_{X/S}) = 0", "$$", "for $i > 0$ is true for any $\\mathcal{O}_{X/S}$-module", "$\\mathcal{F}$ which satisfies conditions (1) and (2) of", "Proposition \\ref{proposition-compute-cohomology}.", "This applies to the following non-crystals:", "$\\Omega^i_{X/S}$ for all $i$, and any sheaf of the form", "$\\underline{\\mathcal{F}}$, where $\\mathcal{F}$ is a quasi-coherent", "$\\mathcal{O}_X$-module. In particular, it applies to the", "sheaf $\\underline{\\mathcal{O}_X} = \\underline{\\mathbf{G}_a}$.", "But note that we need something like Lemma \\ref{lemma-automatic-connection}", "to produce a de Rham complex which requires $\\mathcal{F}$ to be a crystal.", "Hence (currently) the collection of sheaves of modules for which the full", "statement of Proposition \\ref{proposition-compare-with-de-Rham} holds", "is exactly the category of crystals in quasi-coherent modules." ], "refs": [ "crystalline-proposition-compare-with-de-Rham", "crystalline-proposition-compute-cohomology", "crystalline-lemma-automatic-connection", "crystalline-proposition-compare-with-de-Rham" ], "ref_ids": [ 10796, 10794, 10772, 10796 ] }, { "id": 10829, "type": "other", "label": "crystalline-remark-compute-direct-image", "categories": [ "crystalline" ], "title": "crystalline-remark-compute-direct-image", "contents": [ "Let $p$ be a prime number. Let", "$(S, \\mathcal{I}, \\gamma) \\to (S', \\mathcal{I}', \\gamma')$ be", "a morphism of divided power schemes over $\\mathbf{Z}_{(p)}$. Let", "$$", "\\xymatrix{", "X \\ar[r]_f \\ar[d] & X' \\ar[d] \\\\", "S_0 \\ar[r] & S'_0", "}", "$$", "be a commutative diagram of morphisms of schemes and assume $p$ is", "locally nilpotent on $X$ and $X'$. Let $\\mathcal{F}$ be an", "$\\mathcal{O}_{X/S}$-module on $\\text{Cris}(X/S)$. Then", "$Rf_{\\text{cris}, *}\\mathcal{F}$ can be computed as follows.", "\\medskip\\noindent", "Given an object $(U', T', \\delta')$ of $\\text{Cris}(X'/S')$ set", "$U = X \\times_{X'} U' = f^{-1}(U')$ (an open subscheme of $X$). Denote", "$(T_0, T, \\delta)$ the divided power scheme over $S$ such that", "$$", "\\xymatrix{", "T \\ar[r] \\ar[d] & T' \\ar[d] \\\\", "S \\ar[r] & S'", "}", "$$", "is cartesian in the category of divided power schemes, see", "Lemma \\ref{lemma-fibre-product}. There is an", "induced morphism $U \\to T_0$ and we obtain a morphism", "$(U/T)_{\\text{cris}} \\to (X/S)_{\\text{cris}}$, see", "Remark \\ref{remark-functoriality-cris}.", "Let $\\mathcal{F}_U$ be the pullback of $\\mathcal{F}$.", "Let $\\tau_{U/T} : (U/T)_{\\text{cris}} \\to T_{Zar}$ be the structure morphism.", "Then we have", "\\begin{equation}", "\\label{equation-identify-pushforward}", "\\left(Rf_{\\text{cris}, *}\\mathcal{F}\\right)_{T'} =", "R(T \\to T')_*\\left(R\\tau_{U/T, *} \\mathcal{F}_U \\right)", "\\end{equation}", "where the left hand side is the restriction (see", "Section \\ref{section-sheaves}).", "\\medskip\\noindent", "Hints: First, show that $\\text{Cris}(U/T)$ is the localization (in the sense", "of Sites, Lemma \\ref{sites-lemma-localize-topos-site}) of $\\text{Cris}(X/S)$", "at the sheaf of sets $f_{\\text{cris}}^{-1}h_{(U', T', \\delta')}$. Next, reduce", "the statement to the case where $\\mathcal{F}$ is an injective module", "and pushforward of modules using that the pullback of an injective", "$\\mathcal{O}_{X/S}$-module is an injective $\\mathcal{O}_{U/T}$-module on", "$\\text{Cris}(U/T)$. Finally, check the result holds for plain pushforward." ], "refs": [ "crystalline-lemma-fibre-product", "crystalline-remark-functoriality-cris", "sites-lemma-localize-topos-site" ], "ref_ids": [ 10761, 10820, 8585 ] }, { "id": 10830, "type": "other", "label": "crystalline-remark-mayer-vietoris", "categories": [ "crystalline" ], "title": "crystalline-remark-mayer-vietoris", "contents": [ "In the situation of Remark \\ref{remark-compute-direct-image}", "suppose we have an open covering $X = X' \\cup X''$. Denote", "$X''' = X' \\cap X''$. Let $f'$, $f''$, and $f''$ be the restriction of $f$", "to $X'$, $X''$, and $X'''$. Moreover, let $\\mathcal{F}'$, $\\mathcal{F}''$,", "and $\\mathcal{F}'''$ be the restriction of $\\mathcal{F}$ to the crystalline", "sites of $X'$, $X''$, and $X'''$. Then there exists a distinguished triangle", "$$", "Rf_{\\text{cris}, *}\\mathcal{F}", "\\longrightarrow", "Rf'_{\\text{cris}, *}\\mathcal{F}' \\oplus Rf''_{\\text{cris}, *}\\mathcal{F}''", "\\longrightarrow", "Rf'''_{\\text{cris}, *}\\mathcal{F}'''", "\\longrightarrow", "Rf_{\\text{cris}, *}\\mathcal{F}[1]", "$$", "in $D(\\mathcal{O}_{X'/S'})$.", "\\medskip\\noindent", "Hints: This is a formal consequence of the fact that the subcategories", "$\\text{Cris}(X'/S)$, $\\text{Cris}(X''/S)$, $\\text{Cris}(X'''/S)$ correspond", "to open subobjects of the final sheaf on $\\text{Cris}(X/S)$ and that the", "last is the intersection of the first two." ], "refs": [ "crystalline-remark-compute-direct-image" ], "ref_ids": [ 10829 ] }, { "id": 10831, "type": "other", "label": "crystalline-remark-cech-complex", "categories": [ "crystalline" ], "title": "crystalline-remark-cech-complex", "contents": [ "Let $p$ be a prime number. Let $(A, I, \\gamma)$ be a divided power", "ring with $A$ a $\\mathbf{Z}_{(p)}$-algebra. Set $S = \\Spec(A)$ and", "$S_0 = \\Spec(A/I)$. Let $X$ be a separated\\footnote{This assumption is", "not strictly necessary, as using hypercoverings the construction of the", "remark can be extended to the general case.} scheme over", "$S_0$ such that $p$ is locally nilpotent on $X$. Let $\\mathcal{F}$ be a", "crystal in quasi-coherent $\\mathcal{O}_{X/S}$-modules.", "\\medskip\\noindent", "Choose an affine open covering", "$X = \\bigcup_{\\lambda \\in \\Lambda} U_\\lambda$ of $X$.", "Write $U_\\lambda = \\Spec(C_\\lambda)$. Choose a polynomial algebra", "$P_\\lambda$ over $A$ and a surjection $P_\\lambda \\to C_\\lambda$.", "Having fixed these choices we can construct a {\\v C}ech complex which", "computes $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$.", "\\medskip\\noindent", "Given $n \\geq 0$ and $\\lambda_0, \\ldots, \\lambda_n \\in \\Lambda$", "write $U_{\\lambda_0 \\ldots \\lambda_n} = U_{\\lambda_0} \\cap \\ldots", "\\cap U_{\\lambda_n}$. This is an affine scheme by assumption. Write", "$U_{\\lambda_0 \\ldots \\lambda_n} = \\Spec(C_{\\lambda_0 \\ldots \\lambda_n})$.", "Set", "$$", "P_{\\lambda_0 \\ldots \\lambda_n} =", "P_{\\lambda_0} \\otimes_A \\ldots \\otimes_A P_{\\lambda_n}", "$$", "which comes with a canonical surjection onto $C_{\\lambda_0 \\ldots \\lambda_n}$.", "Denote the kernel $J_{\\lambda_0 \\ldots \\lambda_n}$ and set", "$D_{\\lambda_0 \\ldots \\lambda_n}$", "the $p$-adically completed divided power envelope of", "$J_{\\lambda_0 \\ldots \\lambda_n}$ in $P_{\\lambda_0 \\ldots \\lambda_n}$", "relative to $\\gamma$. Let $M_{\\lambda_0 \\ldots \\lambda_n}$ be the", "$P_{\\lambda_0 \\ldots \\lambda_n}$-module corresponding", "to the restriction of $\\mathcal{F}$ to", "$\\text{Cris}(U_{\\lambda_0 \\ldots \\lambda_n}/S)$ via", "Proposition \\ref{proposition-crystals-on-affine}.", "By construction we obtain a cosimplicial divided power ring $D(*)$", "having in degree $n$ the ring", "$$", "D(n) =", "\\prod\\nolimits_{\\lambda_0 \\ldots \\lambda_n}", "D_{\\lambda_0 \\ldots \\lambda_n}", "$$", "(use that divided power envelopes are functorial and the trivial", "cosimplicial structure on the ring $P(*)$ defined similarly).", "Since $M_{\\lambda_0 \\ldots \\lambda_n}$ is the ``value'' of $\\mathcal{F}$", "on the objects $\\Spec(D_{\\lambda_0 \\ldots \\lambda_n})$ we see that", "$M(*)$ defined by the rule", "$$", "M(n) = \\prod\\nolimits_{\\lambda_0 \\ldots \\lambda_n}", "M_{\\lambda_0 \\ldots \\lambda_n}", "$$", "forms a cosimplicial $D(*)$-module. Now we claim that we have", "$$", "R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) = s(M(*))", "$$", "Here $s(-)$ denotes the cochain complex associated to a cosimplicial", "module (see", "Simplicial, Section \\ref{simplicial-section-dold-kan-cosimplicial}).", "\\medskip\\noindent", "Hints: The proof of this is similar to the proof of", "Proposition \\ref{proposition-compute-cohomology} (in particular", "the result holds for any module satisfying the assumptions of", "that proposition)." ], "refs": [ "crystalline-proposition-crystals-on-affine", "crystalline-proposition-compute-cohomology" ], "ref_ids": [ 10793, 10794 ] }, { "id": 10832, "type": "other", "label": "crystalline-remark-alternating-cech-complex", "categories": [ "crystalline" ], "title": "crystalline-remark-alternating-cech-complex", "contents": [ "Let $p$ be a prime number. Let $(A, I, \\gamma)$ be a divided power", "ring with $A$ a $\\mathbf{Z}_{(p)}$-algebra. Set $S = \\Spec(A)$ and", "$S_0 = \\Spec(A/I)$. Let $X$ be a separated quasi-compact scheme", "over $S_0$ such that $p$ is locally nilpotent on $X$. Let", "$\\mathcal{F}$ be a crystal in quasi-coherent $\\mathcal{O}_{X/S}$-modules.", "\\medskip\\noindent", "Choose a finite affine open covering", "$X = \\bigcup_{\\lambda \\in \\Lambda} U_\\lambda$ of $X$", "and a total ordering on $\\Lambda$.", "Write $U_\\lambda = \\Spec(C_\\lambda)$. Choose a polynomial algebra", "$P_\\lambda$ over $A$ and a surjection $P_\\lambda \\to C_\\lambda$.", "Having fixed these choices we can construct an alternating", "{\\v C}ech complex which computes $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$.", "\\medskip\\noindent", "We are going to use the notation introduced in", "Remark \\ref{remark-cech-complex}.", "Denote $\\Omega_{\\lambda_0 \\ldots \\lambda_n}$", "the $p$-adically completed module of differentials of", "$D_{\\lambda_0 \\ldots \\lambda_n}$ over $A$ compatible with the divided power", "structure. Let $\\nabla$ be the integrable connection on", "$M_{\\lambda_0 \\ldots \\lambda_n}$ coming from", "Proposition \\ref{proposition-crystals-on-affine}.", "Consider the double complex $M^{\\bullet, \\bullet}$ with", "terms", "$$", "M^{n, m} =", "\\bigoplus\\nolimits_{\\lambda_0 < \\ldots < \\lambda_n}", "M_{\\lambda_0 \\ldots \\lambda_n}", "\\otimes^\\wedge_{D_{\\lambda_0 \\ldots \\lambda_n}}", "\\Omega^m_{D_{\\lambda_0 \\ldots \\lambda_n}}.", "$$", "For the differential $d_1$ (increasing $n$) we use the usual", "{\\v C}ech differential and for the differential $d_2$ we use", "the connection, i.e., the differential of the de Rham complex.", "We claim that", "$$", "R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) = \\text{Tot}(M^{\\bullet, \\bullet})", "$$", "Here $\\text{Tot}(-)$ denotes the total complex associated to a", "double complex, see", "Homology, Definition \\ref{homology-definition-associated-simple-complex}.", "\\medskip\\noindent", "Hints: We have", "$$", "R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) = R\\Gamma(\\text{Cris}(X/S),", "\\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega_{X/S}^\\bullet)", "$$", "by Proposition \\ref{proposition-compare-with-de-Rham}.", "The right hand side of the formula is simply the alternating {\\v C}ech complex", "for the covering $X = \\bigcup_{\\lambda \\in \\Lambda} U_\\lambda$", "(which induces an open covering of the final sheaf of $\\text{Cris}(X/S)$)", "and the complex $\\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega_{X/S}^\\bullet$,", "see Proposition \\ref{proposition-compute-cohomology-crystal}.", "Now the result follows from a general result in cohomology on sites,", "namely that the alternating {\\v C}ech complex computes the cohomology", "provided it gives the correct answer on all the pieces (insert future", "reference here)." ], "refs": [ "crystalline-remark-cech-complex", "crystalline-proposition-crystals-on-affine", "homology-definition-associated-simple-complex", "crystalline-proposition-compare-with-de-Rham", "crystalline-proposition-compute-cohomology-crystal" ], "ref_ids": [ 10831, 10793, 12164, 10796, 10795 ] }, { "id": 10833, "type": "other", "label": "crystalline-remark-quasi-coherent", "categories": [ "crystalline" ], "title": "crystalline-remark-quasi-coherent", "contents": [ "In the situation of Remark \\ref{remark-compute-direct-image}", "assume that $S \\to S'$ is quasi-compact and quasi-separated and", "that $X \\to S_0$ is quasi-compact and quasi-separated. Then for a crystal", "in quasi-coherent $\\mathcal{O}_{X/S}$-modules $\\mathcal{F}$", "the sheaves $R^if_{\\text{cris}, *}\\mathcal{F}$ are locally quasi-coherent.", "\\medskip\\noindent", "Hints: We have to show that the restrictions to $T'$ are quasi-coherent", "$\\mathcal{O}_{T'}$-modules, where $(U', T', \\delta')$ is any object of", "$\\text{Cris}(X'/S')$. It suffices to do this when $T'$ is affine.", "We use the formula (\\ref{equation-identify-pushforward}),", "the fact that $T \\to T'$ is quasi-compact and quasi-separated (as $T$", "is affine over the base change of $T'$ by $S \\to S'$), and", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images}", "to see that it suffices to show that the sheaves", "$R^i\\tau_{U/T, *}\\mathcal{F}_U$ are quasi-coherent.", "Note that $U \\to T_0$ is also quasi-compact and quasi-separated, see", "Schemes, Lemmas \\ref{schemes-lemma-quasi-compact-permanence} and", "\\ref{schemes-lemma-quasi-compact-permanence}.", "\\medskip\\noindent", "This reduces us to proving that $R^i\\tau_{X/S, *}\\mathcal{F}$", "is quasi-coherent on $S$ in the case that $p$ locally nilpotent on $S$. Here", "$\\tau_{X/S}$ is the structure morphism, see", "Remark \\ref{remark-structure-morphism}.", "We may work locally on $S$, hence we may assume $S$ affine", "(see Lemma \\ref{lemma-localize}). Induction on the number", "of affines covering $X$ and Mayer-Vietoris", "(Remark \\ref{remark-mayer-vietoris}) reduces the question to", "the case where $X$ is also affine (as in the proof of", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images}).", "Say $X = \\Spec(C)$ and $S = \\Spec(A)$ so that $(A, I, \\gamma)$ and", "$A \\to C$ are as", "in Situation \\ref{situation-affine}. Choose a polynomial algebra", "$P$ over $A$ and a surjection $P \\to C$ as in", "Section \\ref{section-quasi-coherent-crystals}.", "Let $(M, \\nabla)$ be the module corresponding to $\\mathcal{F}$, see", "Proposition \\ref{proposition-crystals-on-affine}.", "Applying ", "Proposition \\ref{proposition-compute-cohomology-crystal}", "we see that $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$ is represented by", "$M \\otimes_D \\Omega_D^*$. Note that completion isn't necessary", "as $p$ is nilpotent in $A$! We have to show that this is compatible", "with taking principal opens in $S = \\Spec(A)$. Suppose that $g \\in A$.", "Then we conclude that similarly $R\\Gamma(\\text{Cris}(X_g/S_g), \\mathcal{F})$", "is computed by $M_g \\otimes_{D_g} \\Omega_{D_g}^*$ (again this uses that", "$p$-adic completion isn't necessary). Hence we conclude because localization", "is an exact functor on $A$-modules." ], "refs": [ "crystalline-remark-compute-direct-image", "coherent-lemma-quasi-coherence-higher-direct-images", "schemes-lemma-quasi-compact-permanence", "schemes-lemma-quasi-compact-permanence", "crystalline-remark-structure-morphism", "crystalline-lemma-localize", "crystalline-remark-mayer-vietoris", "coherent-lemma-quasi-coherence-higher-direct-images", "crystalline-proposition-crystals-on-affine", "crystalline-proposition-compute-cohomology-crystal" ], "ref_ids": [ 10829, 3295, 7716, 7716, 10822, 10765, 10830, 3295, 10793, 10795 ] }, { "id": 10834, "type": "other", "label": "crystalline-remark-bounded-cohomology", "categories": [ "crystalline" ], "title": "crystalline-remark-bounded-cohomology", "contents": [ "In the situation of Remark \\ref{remark-compute-direct-image}", "assume that $S \\to S'$ is quasi-compact and quasi-separated and", "that $X \\to S_0$ is of finite type and quasi-separated. Then there exists", "an integer $i_0$ such that for any crystal", "in quasi-coherent $\\mathcal{O}_{X/S}$-modules $\\mathcal{F}$", "we have $R^if_{\\text{cris}, *}\\mathcal{F} = 0$ for all $i > i_0$.", "\\medskip\\noindent", "Hints: Arguing as in Remark \\ref{remark-quasi-coherent} (using", "Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-quasi-coherence-higher-direct-images})", "we reduce to proving that $H^i(\\text{Cris}(X/S), \\mathcal{F}) = 0$ for $i \\gg 0$", "in the situation of Proposition \\ref{proposition-compute-cohomology-crystal}", "when $C$ is a finite type algebra over $A$. This is clear as we can", "choose a finite polynomial algebra and we see that $\\Omega^i_D = 0$", "for $i \\gg 0$." ], "refs": [ "crystalline-remark-compute-direct-image", "crystalline-remark-quasi-coherent", "coherent-lemma-quasi-coherence-higher-direct-images", "crystalline-proposition-compute-cohomology-crystal" ], "ref_ids": [ 10829, 10833, 3295, 10795 ] }, { "id": 10835, "type": "other", "label": "crystalline-remark-bounded-cohomology-over-point", "categories": [ "crystalline" ], "title": "crystalline-remark-bounded-cohomology-over-point", "contents": [ "In Situation \\ref{situation-global} let $\\mathcal{F}$ be a crystal in", "quasi-coherent $\\mathcal{O}_{X/S}$-modules. Assume that $S_0$", "has a unique point and that $X \\to S_0$ is of finite presentation.", "\\begin{enumerate}", "\\item If $\\dim X = d$ and $X/S_0$ has embedding dimension $e$, then", "$H^i(\\text{Cris}(X/S), \\mathcal{F}) = 0$ for $i > d + e$.", "\\item If $X$ is separated and can be covered by $q$ affines, and", "$X/S_0$ has embedding dimension $e$, then", "$H^i(\\text{Cris}(X/S), \\mathcal{F}) = 0$ for $i > q + e$.", "\\end{enumerate}", "Hints: In case (1) we can use that", "$$", "H^i(\\text{Cris}(X/S), \\mathcal{F}) = H^i(X_{Zar}, Ru_{X/S, *}\\mathcal{F})", "$$", "and that $Ru_{X/S, *}\\mathcal{F}$ is locally calculated by a de Rham", "complex constructed using an embedding of $X$ into a smooth scheme", "of dimension $e$ over $S$", "(see Lemma \\ref{lemma-compute-cohomology-crystal-smooth}).", "These de Rham complexes are zero in all degrees $> e$. Hence (1)", "follows from Cohomology, Proposition", "\\ref{cohomology-proposition-vanishing-Noetherian}.", "In case (2) we use the alternating {\\v C}ech complex (see", "Remark \\ref{remark-alternating-cech-complex}) to reduce to the case", "$X$ affine. In the affine case we prove the result using the de Rham complex", "associated to an embedding of $X$ into a smooth scheme of dimension $e$", "over $S$ (it takes some work to construct such a thing)." ], "refs": [ "crystalline-lemma-compute-cohomology-crystal-smooth", "cohomology-proposition-vanishing-Noetherian", "crystalline-remark-alternating-cech-complex" ], "ref_ids": [ 10787, 2246, 10832 ] }, { "id": 10836, "type": "other", "label": "crystalline-remark-base-change", "categories": [ "crystalline" ], "title": "crystalline-remark-base-change", "contents": [ "In the situation of Remark \\ref{remark-compute-direct-image}", "assume $S = \\Spec(A)$ and $S' = \\Spec(A')$ are affine.", "Let $\\mathcal{F}'$ be an $\\mathcal{O}_{X'/S'}$-module.", "Let $\\mathcal{F}$ be the pullback of $\\mathcal{F}'$.", "Then there is a canonical base change map", "$$", "L(S' \\to S)^*R\\tau_{X'/S', *}\\mathcal{F}'", "\\longrightarrow", "R\\tau_{X/S, *}\\mathcal{F}", "$$", "where $\\tau_{X/S}$ and $\\tau_{X'/S'}$ are the structure morphisms, see", "Remark \\ref{remark-structure-morphism}. On global sections this", "gives a base change map", "\\begin{equation}", "\\label{equation-base-change-map}", "R\\Gamma(\\text{Cris}(X'/S'), \\mathcal{F}') \\otimes^\\mathbf{L}_{A'} A", "\\longrightarrow", "R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})", "\\end{equation}", "in $D(A)$.", "\\medskip\\noindent", "Hint: Compose the very general base change map of", "Cohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change}", "with the canonical map", "$Lf_{\\text{cris}}^*\\mathcal{F}' \\to", "f_{\\text{cris}}^*\\mathcal{F}' = \\mathcal{F}$." ], "refs": [ "crystalline-remark-compute-direct-image", "crystalline-remark-structure-morphism", "sites-cohomology-remark-base-change" ], "ref_ids": [ 10829, 10822, 4424 ] }, { "id": 10837, "type": "other", "label": "crystalline-remark-base-change-isomorphism", "categories": [ "crystalline" ], "title": "crystalline-remark-base-change-isomorphism", "contents": [ "The map (\\ref{equation-base-change-map}) is an isomorphism provided", "all of the following conditions are satisfied:", "\\begin{enumerate}", "\\item $p$ is nilpotent in $A'$,", "\\item $\\mathcal{F}'$ is a crystal in quasi-coherent", "$\\mathcal{O}_{X'/S'}$-modules,", "\\item $X' \\to S'_0$ is a quasi-compact, quasi-separated morphism,", "\\item $X = X' \\times_{S'_0} S_0$,", "\\item $\\mathcal{F}'$ is a flat $\\mathcal{O}_{X'/S'}$-module,", "\\item $X' \\to S'_0$ is a local complete intersection morphism (see", "More on Morphisms, Definition \\ref{more-morphisms-definition-lci}; this", "holds for example if $X' \\to S'_0$ is syntomic or smooth),", "\\item $X'$ and $S_0$ are Tor independent over $S'_0$ (see", "More on Algebra, Definition \\ref{more-algebra-definition-tor-independent};", "this holds for example if either $S_0 \\to S'_0$ or $X' \\to S'_0$ is flat).", "\\end{enumerate}", "Hints: Condition (1) means that in the arguments below $p$-adic completion", "does nothing and can be ignored.", "Using condition (3) and Mayer Vietoris (see", "Remark \\ref{remark-mayer-vietoris}) this reduces to the case", "where $X'$ is affine. In fact by condition (6), after shrinking", "further, we can assume that $X' = \\Spec(C')$ and we are given a presentation", "$C' = A'/I'[x_1, \\ldots, x_n]/(\\bar f'_1, \\ldots, \\bar f'_c)$", "where $\\bar f'_1, \\ldots, \\bar f'_c$ is a Koszul-regular sequence in $A'/I'$.", "(This means that smooth locally $\\bar f'_1, \\ldots, \\bar f'_c$ forms", "a regular sequence, see More on Algebra,", "Lemma \\ref{more-algebra-lemma-Koszul-regular-flat-locally-regular}.)", "We choose a lift of", "$\\bar f'_i$ to an element $f'_i \\in A'[x_1, \\ldots, x_n]$. By (4) we see that", "$X = \\Spec(C)$ with $C = A/I[x_1, \\ldots, x_n]/(\\bar f_1, \\ldots, \\bar f_c)$", "where $f_i \\in A[x_1, \\ldots, x_n]$ is the image of $f'_i$.", "By property (7) we see that $\\bar f_1, \\ldots, \\bar f_c$ is a Koszul-regular", "sequence in $A/I[x_1, \\ldots, x_n]$. The divided power envelope of", "$I'A'[x_1, \\ldots, x_n] + (f'_1, \\ldots, f'_c)$ in $A'[x_1, \\ldots, x_n]$", "relative to $\\gamma'$ is", "$$", "D' = A'[x_1, \\ldots, x_n]\\langle \\xi_1, \\ldots, \\xi_c \\rangle/(\\xi_i - f'_i)", "$$", "see Lemma \\ref{lemma-describe-divided-power-envelope}. Then you check that", "$\\xi_1 - f'_1, \\ldots, \\xi_n - f'_n$ is a Koszul-regular sequence in the", "ring $A'[x_1, \\ldots, x_n]\\langle \\xi_1, \\ldots, \\xi_c\\rangle$.", "Similarly the divided power envelope of", "$IA[x_1, \\ldots, x_n] + (f_1, \\ldots, f_c)$ in $A[x_1, \\ldots, x_n]$", "relative to $\\gamma$ is", "$$", "D = A[x_1, \\ldots, x_n]\\langle \\xi_1, \\ldots, \\xi_c\\rangle/(\\xi_i - f_i)", "$$", "and $\\xi_1 - f_1, \\ldots, \\xi_n - f_n$ is a Koszul-regular sequence in the", "ring $A[x_1, \\ldots, x_n]\\langle \\xi_1, \\ldots, \\xi_c\\rangle$.", "It follows that $D' \\otimes_{A'}^\\mathbf{L} A = D$. Condition (2)", "implies $\\mathcal{F}'$ corresponds to a pair $(M', \\nabla)$", "consisting of a $D'$-module with connection, see", "Proposition \\ref{proposition-crystals-on-affine}.", "Then $M = M' \\otimes_{D'} D$ corresponds to the pullback $\\mathcal{F}$.", "By assumption (5) we see that $M'$ is a flat $D'$-module, hence", "$$", "M = M' \\otimes_{D'} D = M' \\otimes_{D'} D' \\otimes_{A'}^\\mathbf{L} A", "= M' \\otimes_{A'}^\\mathbf{L} A", "$$", "Since the modules of differentials $\\Omega_{D'}$ and $\\Omega_D$", "(as defined in Section \\ref{section-quasi-coherent-crystals})", "are free $D'$-modules on the same generators we see that", "$$", "M \\otimes_D \\Omega^\\bullet_D =", "M' \\otimes_{D'} \\Omega^\\bullet_{D'} \\otimes_{D'} D =", "M' \\otimes_{D'} \\Omega^\\bullet_{D'} \\otimes_{A'}^\\mathbf{L} A", "$$", "which proves what we want by", "Proposition \\ref{proposition-compute-cohomology-crystal}." ], "refs": [ "more-morphisms-definition-lci", "more-algebra-definition-tor-independent", "crystalline-remark-mayer-vietoris", "more-algebra-lemma-Koszul-regular-flat-locally-regular", "crystalline-lemma-describe-divided-power-envelope", "crystalline-proposition-crystals-on-affine", "crystalline-proposition-compute-cohomology-crystal" ], "ref_ids": [ 14121, 10622, 10830, 9988, 10746, 10793, 10795 ] }, { "id": 10838, "type": "other", "label": "crystalline-remark-rlim", "categories": [ "crystalline" ], "title": "crystalline-remark-rlim", "contents": [ "Let $p$ be a prime number. Let $(A, I, \\gamma)$ be a divided power", "ring with $A$ an algebra over $\\mathbf{Z}_{(p)}$ with $p$ nilpotent", "in $A/I$. Set $S = \\Spec(A)$ and $S_0 = \\Spec(A/I)$.", "Let $X$ be a scheme over $S_0$ with $p$ locally", "nilpotent on $X$. Let $\\mathcal{F}$ be any", "$\\mathcal{O}_{X/S}$-module. For $e \\gg 0$ we have $(p^e) \\subset I$", "is preserved by $\\gamma$, see", "Divided Power Algebra, Lemma \\ref{dpa-lemma-extend-to-completion}.", "Set $S_e = \\Spec(A/p^eA)$ for $e \\gg 0$.", "Then $\\text{Cris}(X/S_e)$ is a full subcategory of $\\text{Cris}(X/S)$", "and we denote $\\mathcal{F}_e$ the restriction of $\\mathcal{F}$ to", "$\\text{Cris}(X/S_e)$. Then", "$$", "R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) =", "R\\lim_e R\\Gamma(\\text{Cris}(X/S_e), \\mathcal{F}_e)", "$$", "\\medskip\\noindent", "Hints: Suffices to prove this for $\\mathcal{F}$ injective.", "In this case the sheaves $\\mathcal{F}_e$ are injective", "modules too, the transition maps", "$\\Gamma(\\mathcal{F}_{e + 1}) \\to \\Gamma(\\mathcal{F}_e)$ are", "surjective, and we have", "$\\Gamma(\\mathcal{F}) = \\lim_e \\Gamma(\\mathcal{F}_e)$ because", "any object of $\\text{Cris}(X/S)$ is locally an object of one", "of the categories $\\text{Cris}(X/S_e)$ by definition of", "$\\text{Cris}(X/S)$." ], "refs": [ "dpa-lemma-extend-to-completion" ], "ref_ids": [ 1660 ] }, { "id": 10839, "type": "other", "label": "crystalline-remark-comparison", "categories": [ "crystalline" ], "title": "crystalline-remark-comparison", "contents": [ "Let $p$ be a prime number. Let $(A, I, \\gamma)$ be a divided power", "ring with $p$ nilpotent in $A$. Set $S = \\Spec(A)$ and", "$S_0 = \\Spec(A/I)$. Let $Y$ be a smooth scheme over $S$ and set", "$X = Y \\times_S S_0$. Let", "$\\mathcal{F}$ be a crystal in quasi-coherent $\\mathcal{O}_{X/S}$-modules.", "Then", "\\begin{enumerate}", "\\item $\\gamma$ extends to a divided power structure on the ideal", "of $X$ in $Y$ so that $(X, Y, \\gamma)$ is an object of $\\text{Cris}(X/S)$,", "\\item the restriction $\\mathcal{F}_Y$ (see Section \\ref{section-sheaves})", "comes endowed with a canonical integrable connection", "$\\nabla : \\mathcal{F}_Y \\to", "\\mathcal{F}_Y \\otimes_{\\mathcal{O}_Y} \\Omega_{Y/S}$, and", "\\item we have", "$$", "R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) =", "R\\Gamma(Y, \\mathcal{F}_Y \\otimes_{\\mathcal{O}_Y} \\Omega^\\bullet_{Y/S})", "$$", "in $D(A)$.", "\\end{enumerate}", "Hints: See Divided Power Algebra, Lemma \\ref{dpa-lemma-gamma-extends} for (1).", "See Lemma \\ref{lemma-automatic-connection} for (2).", "For Part (3) note that there is a map, see", "(\\ref{equation-restriction}). This map is an isomorphism when", "$X$ is affine, see", "Lemma \\ref{lemma-compute-cohomology-crystal-smooth}.", "This shows that $Ru_{X/S, *}\\mathcal{F}$ and", "$\\mathcal{F}_Y \\otimes \\Omega^\\bullet_{Y/S}$ are quasi-isomorphic", "as complexes on $Y_{Zar} = X_{Zar}$.", "Since $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) =", "R\\Gamma(X_{Zar}, Ru_{X/S, *}\\mathcal{F})$ the result follows." ], "refs": [ "dpa-lemma-gamma-extends", "crystalline-lemma-automatic-connection", "crystalline-lemma-compute-cohomology-crystal-smooth" ], "ref_ids": [ 1657, 10772, 10787 ] }, { "id": 10840, "type": "other", "label": "crystalline-remark-perfect", "categories": [ "crystalline" ], "title": "crystalline-remark-perfect", "contents": [ "Let $p$ be a prime number. Let $(A, I, \\gamma)$ be a divided power", "ring with $p$ nilpotent in $A$. Set $S = \\Spec(A)$ and", "$S_0 = \\Spec(A/I)$. Let $X$ be a proper smooth scheme over $S_0$.", "Let $\\mathcal{F}$ be a crystal in finite locally free", "quasi-coherent $\\mathcal{O}_{X/S}$-modules.", "Then $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$ is a", "perfect object of $D(A)$.", "\\medskip\\noindent", "Hints: By Remark \\ref{remark-base-change-isomorphism} we have", "$$", "R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) \\otimes_A^\\mathbf{L} A/I", "\\cong", "R\\Gamma(\\text{Cris}(X/S_0), \\mathcal{F}|_{\\text{Cris}(X/S_0)})", "$$", "By Remark \\ref{remark-comparison} we have", "$$", "R\\Gamma(\\text{Cris}(X/S_0), \\mathcal{F}|_{\\text{Cris}(X/S_0)}) =", "R\\Gamma(X, \\mathcal{F}_X \\otimes \\Omega^\\bullet_{X/S_0})", "$$", "Using the stupid filtration on the de Rham complex we see that", "the last displayed complex is perfect in $D(A/I)$ as soon as the complexes", "$$", "R\\Gamma(X, \\mathcal{F}_X \\otimes \\Omega^q_{X/S_0})", "$$", "are perfect complexes in $D(A/I)$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-two-out-of-three-perfect}.", "This is true by standard arguments", "in coherent cohomology using that $\\mathcal{F}_X \\otimes \\Omega^q_{X/S_0}$", "is a finite locally free sheaf and $X \\to S_0$ is proper and flat", "(insert future reference here). Applying", "More on Algebra, Lemma \\ref{more-algebra-lemma-perfect-modulo-nilpotent-ideal}", "we see that", "$$", "R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) \\otimes_A^\\mathbf{L} A/I^n", "$$", "is a perfect object of $D(A/I^n)$ for all $n$. This isn't quite enough", "unless $A$ is Noetherian. Namely, even though $I$ is locally nilpotent", "by our assumption that $p$ is nilpotent, see", "Divided Power Algebra, Lemma \\ref{dpa-lemma-nil},", "we cannot conclude that $I^n = 0$ for some $n$. A counter example", "is $\\mathbf{F}_p\\langle x \\rangle$. To prove it in general when", "$\\mathcal{F} = \\mathcal{O}_{X/S}$ the argument of", "\\url{https://math.columbia.edu/~dejong/wordpress/?p=2227}", "works. When the coefficients $\\mathcal{F}$ are non-trivial the", "argument of \\cite{Faltings-very} seems to be as follows. Reduce to the", "case $pA = 0$ by More on Algebra, Lemma", "\\ref{more-algebra-lemma-perfect-modulo-nilpotent-ideal}.", "In this case the Frobenius map $A \\to A$, $a \\mapsto a^p$ factors", "as $A \\to A/I \\xrightarrow{\\varphi} A$ (as $x^p = 0$ for $x \\in I$). Set", "$X^{(1)} = X \\otimes_{A/I, \\varphi} A$. The absolute Frobenius morphism", "of $X$ factors through a morphism $F_X : X \\to X^{(1)}$ (a kind of", "relative Frobenius). Affine locally if $X = \\Spec(C)$ then", "$X^{(1)} = \\Spec( C \\otimes_{A/I, \\varphi} A)$", "and $F_X$ corresponds to $C \\otimes_{A/I, \\varphi} A \\to C$,", "$c \\otimes a \\mapsto c^pa$. This defines morphisms of ringed topoi", "$$", "(X/S)_{\\text{cris}}", "\\xrightarrow{(F_X)_{\\text{cris}}}", "(X^{(1)}/S)_{\\text{cris}}", "\\xrightarrow{u_{X^{(1)}/S}}", "\\Sh(X^{(1)}_{Zar})", "$$", "whose composition is denoted $\\text{Frob}_X$. One then shows that", "$R\\text{Frob}_{X, *}\\mathcal{F}$ is representable by a", "perfect complex of $\\mathcal{O}_{X^{(1)}}$-modules(!)", "by a local calculation." ], "refs": [ "crystalline-remark-base-change-isomorphism", "crystalline-remark-comparison", "more-algebra-lemma-two-out-of-three-perfect", "more-algebra-lemma-perfect-modulo-nilpotent-ideal", "dpa-lemma-nil", "more-algebra-lemma-perfect-modulo-nilpotent-ideal" ], "ref_ids": [ 10837, 10839, 10214, 10249, 1653, 10249 ] }, { "id": 10841, "type": "other", "label": "crystalline-remark-complete-perfect", "categories": [ "crystalline" ], "title": "crystalline-remark-complete-perfect", "contents": [ "Let $p$ be a prime number. Let $(A, I, \\gamma)$ be a divided power", "ring with $A$ a $p$-adically complete ring and $p$ nilpotent in $A/I$. Set", "$S = \\Spec(A)$ and $S_0 = \\Spec(A/I)$. Let $X$ be a proper", "smooth scheme over $S_0$. Let $\\mathcal{F}$ be a crystal in", "finite locally free quasi-coherent $\\mathcal{O}_{X/S}$-modules.", "Then $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$ is a", "perfect object of $D(A)$.", "\\medskip\\noindent", "Hints: We know that $K = R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$", "is the derived limit $K = R\\lim K_e$ of the cohomologies over $A/p^eA$,", "see Remark \\ref{remark-rlim}.", "Each $K_e$ is a perfect complex of $D(A/p^eA)$ by", "Remark \\ref{remark-perfect}.", "Since $A$ is $p$-adically complete the result", "follows from", "More on Algebra, Lemma \\ref{more-algebra-lemma-Rlim-perfect-gives-complete}." ], "refs": [ "crystalline-remark-rlim", "crystalline-remark-perfect", "more-algebra-lemma-Rlim-perfect-gives-complete" ], "ref_ids": [ 10838, 10840, 10410 ] }, { "id": 10842, "type": "other", "label": "crystalline-remark-complete-comparison", "categories": [ "crystalline" ], "title": "crystalline-remark-complete-comparison", "contents": [ "Let $p$ be a prime number. Let $(A, I, \\gamma)$ be a divided power", "ring with $A$ a Noetherian $p$-adically complete ring and $p$ nilpotent", "in $A/I$. Set $S = \\Spec(A)$ and", "$S_0 = \\Spec(A/I)$. Let $Y$ be a proper smooth scheme over $S$ and set", "$X = Y \\times_S S_0$. Let $\\mathcal{F}$ be a finite type crystal in", "quasi-coherent $\\mathcal{O}_{X/S}$-modules. Then", "\\begin{enumerate}", "\\item there exists a coherent $\\mathcal{O}_Y$-module $\\mathcal{F}_Y$", "endowed with integrable connection", "$$", "\\nabla :", "\\mathcal{F}_Y", "\\longrightarrow", "\\mathcal{F}_Y \\otimes_{\\mathcal{O}_Y} \\Omega_{Y/S}", "$$", "such that $\\mathcal{F}_Y/p^e\\mathcal{F}_Y$ is the module with connection", "over $A/p^eA$ found in Remark \\ref{remark-comparison}, and", "\\item we have", "$$", "R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) =", "R\\Gamma(Y, \\mathcal{F}_Y \\otimes_{\\mathcal{O}_Y} \\Omega^\\bullet_{Y/S})", "$$", "in $D(A)$.", "\\end{enumerate}", "Hints: The existence of $\\mathcal{F}_Y$ is Grothendieck's existence theorem", "(insert future reference here). The isomorphism of cohomologies follows", "as both sides are computed as $R\\lim$ of the versions modulo $p^e$", "(see Remark \\ref{remark-rlim} for the left hand side; use the theorem", "on formal functions, see", "Cohomology of Schemes, Theorem \\ref{coherent-theorem-formal-functions}", "for the right hand side).", "Each of the versions modulo $p^e$ are isomorphic by", "Remark \\ref{remark-comparison}." ], "refs": [ "crystalline-remark-comparison", "crystalline-remark-rlim", "coherent-theorem-formal-functions", "crystalline-remark-comparison" ], "ref_ids": [ 10839, 10838, 3278, 10839 ] }, { "id": 10843, "type": "other", "label": "crystalline-remark-F-crystal-variants", "categories": [ "crystalline" ], "title": "crystalline-remark-F-crystal-variants", "contents": [ "Let $(\\mathcal{E}, F)$ be an $F$-crystal as in", "Definition \\ref{definition-F-crystal}.", "In the literature the nondegeneracy condition is often part of the", "definition of an $F$-crystal. Moreover, often it is also assumed that", "$F \\circ V = p^n\\text{id}$. What is needed for the result below is", "that there exists an integer $j \\geq 0$ such that $\\Ker(F)$ and", "$\\Coker(F)$ are killed by $p^j$. If the rank of $\\mathcal{E}$", "is bounded (for example if $X$ is quasi-compact), then both of these", "conditions follow from the nondegeneracy condition as formulated in", "the definition. Namely, suppose $R$ is a ring, $r \\geq 1$ is an integer and", "$K, L \\in \\text{Mat}(r \\times r, R)$ are matrices with", "$K L = p^i 1_{r \\times r}$. Then $\\det(K)\\det(L) = p^{ri}$. ", "Let $L'$ be the adjugate matrix of $L$, i.e.,", "$L' L = L L' = \\det(L)$. Set $K' = p^{ri} K$ and $j = ri + i$.", "Then we have $K' L = p^j 1_{r \\times r}$ as $K L = p^i$ and", "$$", "L K' = L K \\det(L) \\det(M) = L K L L' \\det(M) = L p^i L' \\det(M) =", "p^j 1_{r \\times r}", "$$", "It follows that if $V$ is as in Definition \\ref{definition-F-crystal}", "then setting $V' = p^N V$ where $N > i \\cdot \\text{rank}(\\mathcal{E})$", "we get $V' \\circ F = p^{N + i}$ and $F \\circ V' = p^{N + i}$." ], "refs": [ "crystalline-definition-F-crystal", "crystalline-definition-F-crystal" ], "ref_ids": [ 10810, 10810 ] }, { "id": 10899, "type": "other", "label": "spaces-pushouts-remark-essentially-constant", "categories": [ "spaces-pushouts" ], "title": "spaces-pushouts-remark-essentially-constant", "contents": [ "The meaning of Lemma \\ref{lemma-essentially-constant}", "is the system $X_1 \\to X_2 \\to X_3 \\to \\ldots$ is essentially", "constant with value $X$. See Categories, Definition", "\\ref{categories-definition-essentially-constant-diagram}." ], "refs": [ "spaces-pushouts-lemma-essentially-constant", "categories-definition-essentially-constant-diagram" ], "ref_ids": [ 10889, 12368 ] }, { "id": 11166, "type": "other", "label": "varieties-remark-exact-sequence-induction", "categories": [ "varieties" ], "title": "varieties-remark-exact-sequence-induction", "contents": [ "Let $k$ be an infinite field. Let $n \\geq 1$. Given a finite number", "of coherent modules $\\mathcal{F}_i$ on $\\mathbf{P}^n_k$ we can choose", "a single $s \\in \\Gamma(\\mathbf{P}^n_k, \\mathcal{O}(1))$ such", "that the statement of Lemma \\ref{lemma-exact-sequence-induction}", "works for each of them.", "To prove this, just apply the lemma to $\\bigoplus \\mathcal{F}_i$." ], "refs": [ "varieties-lemma-exact-sequence-induction" ], "ref_ids": [ 11038 ] }, { "id": 11167, "type": "other", "label": "varieties-remark-exact-sequence-induction-cohomology", "categories": [ "varieties" ], "title": "varieties-remark-exact-sequence-induction-cohomology", "contents": [ "In the situation of Lemmas \\ref{lemma-hyperplane} and", "\\ref{lemma-exact-sequence-induction}", "we have $H \\cong \\mathbf{P}^{n - 1}_k$ with Serre twists", "$\\mathcal{O}_H(d) = i^*\\mathcal{O}_{\\mathbf{P}^n_k}(d)$.", "For every $d \\in \\mathbf{Z}$ we have a short exact sequence", "$$", "0 \\to \\mathcal{F}(d - 1) \\to \\mathcal{F}(d) \\to i_*(\\mathcal{G}(d)) \\to 0", "$$", "Namely, tensoring by $\\mathcal{O}_{\\mathbf{P}^n_k}(d)$ is an", "exact functor and by the projection formula", "(Cohomology, Lemma \\ref{cohomology-lemma-projection-formula})", "we have", "$i_*(\\mathcal{G}(d)) = i_*\\mathcal{G} \\otimes \\mathcal{O}_{\\mathbf{P}^n_k}(d)$.", "We obtain corresponding long exact sequences", "$$", "H^i(\\mathbf{P}^n_k, \\mathcal{F}(d - 1)) \\to", "H^i(\\mathbf{P}^n_k, \\mathcal{F}(d)) \\to", "H^i(H, \\mathcal{G}(d)) \\to", "H^{i + 1}(\\mathbf{P}^n_k, \\mathcal{F}(d - 1))", "$$", "This follows from the above and the fact that we have", "$H^i(\\mathbf{P}^n_k, i_*\\mathcal{G}(d)) = H^i(H, \\mathcal{G}(d))$ by", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology}", "(closed immersions are affine)." ], "refs": [ "varieties-lemma-hyperplane", "varieties-lemma-exact-sequence-induction", "cohomology-lemma-projection-formula", "coherent-lemma-relative-affine-cohomology" ], "ref_ids": [ 11037, 11038, 2243, 3284 ] }, { "id": 11168, "type": "other", "label": "varieties-remark-n-fold-relative-frobenius", "categories": [ "varieties" ], "title": "varieties-remark-n-fold-relative-frobenius", "contents": [ "Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$.", "Let $X$ be a scheme over $S$. For $n \\geq 1$", "$$", "X^{(p^n)} = X^{(p^n/S)} = X \\times_{S, F_S^n} S", "$$", "viewed as a scheme over $S$. Observe that $X \\mapsto X^{(p^n)}$", "is a functor. Applying", "Lemma \\ref{lemma-frobenius-endomorphism-identity}", "we see $F_{X/S, n} = (F_X^n, \\text{id}_S) : X \\longrightarrow X^{(p^n)}$", "is a morphism over $S$ fitting into the commutative diagram", "$$", "\\xymatrix{", "X \\ar[rr]_{F_{X/S, n}} \\ar[rrd] \\ar@/^1em/[rrrr]^{F_X^n}", "& & X^{(p^n)} \\ar[rr] \\ar[d] & & X \\ar[d] \\\\", "& & S \\ar[rr]^{F_S^n} & & S", "}", "$$", "where the right square is cartesian. The morphism $F_{X/S, n}$", "is sometimes called the", "{\\it $n$-fold relative Frobenius morphism of $X/S$}.", "This makes sense because we have the formula", "$$", "F_{X/S, n} =", "F_{X^{(p^{n - 1})}/S} \\circ \\ldots \\circ F_{X^{(p)}/S} \\circ F_{X/S}", "$$", "which shows that $F_{X/S, n}$ is the composition of $n$ relative", "Frobenii. Since we have", "$$", "F_{X^{(p^m)}/S} = F_{X^{(p^{m - 1})}/S}^{(p)} = \\ldots = F_{X/S}^{(p^m)}", "$$", "(details omitted) we get also that", "$$", "F_{X/S, n} =", "F_{X/S}^{(p^{n - 1})} \\circ \\ldots \\circ F_{X/S}^{(p)} \\circ F_{X/S}", "$$" ], "refs": [ "varieties-lemma-frobenius-endomorphism-identity" ], "ref_ids": [ 11047 ] }, { "id": 11169, "type": "other", "label": "varieties-remark-conductor", "categories": [ "varieties" ], "title": "varieties-remark-conductor", "contents": [ "Let $A$ be a reduced ring. Let $I, J$ be ideals of $A$", "such that $V(I) \\cup V(J) = \\Spec(A)$. Set $B = A/J$.", "Then $I \\to IB$ is an isomorphism of $A$-modules. Namely, we", "have $IB = I + J/J = I/(I \\cap J)$ and $I \\cap J$ is zero because", "$A$ is reduced and $\\Spec(A) = V(I) \\cup V(J) = V(I \\cap J)$.", "Thus for any projective $A$-module $P$ we also have $IP = I(P/JP)$." ], "refs": [], "ref_ids": [] }, { "id": 11170, "type": "other", "label": "varieties-remark-useless-generalization", "categories": [ "varieties" ], "title": "varieties-remark-useless-generalization", "contents": [ "In fact, if $X$ is a scheme whose reduction is a Noetherian", "separated scheme of dimension $1$, then $X$ has an ample invertible", "sheaf. The argument to prove this is the same as the proof of", "Proposition \\ref{proposition-dim-1-noetherian-separated-has-ample}", "except one uses", "Limits, Lemma \\ref{limits-lemma-ample-on-reduction}", "instead of", "Cohomology of Schemes, Lemma \\ref{coherent-lemma-ample-on-reduction}." ], "refs": [ "varieties-proposition-dim-1-noetherian-separated-has-ample", "limits-lemma-ample-on-reduction", "coherent-lemma-ample-on-reduction" ], "ref_ids": [ 11138, 15084, 3350 ] }, { "id": 11255, "type": "other", "label": "cotangent-remark-variant-cotangent-complex", "categories": [ "cotangent" ], "title": "cotangent-remark-variant-cotangent-complex", "contents": [ "Let $A \\to B$ be a ring map. Let $\\mathcal{A}$ be the category of", "arrows $\\psi : C \\to B$ of $A$-algebras and let $\\mathcal{S}$ be", "the category of maps $E \\to B$ where $E$ is a set. There are adjoint", "functors $V : \\mathcal{A} \\to \\mathcal{S}$ (the forgetful functor)", "and $U : \\mathcal{S} \\to \\mathcal{A}$ which sends $E \\to B$ to", "$A[E] \\to B$. Let $X_\\bullet$ be the simplicial object of", "$\\text{Fun}(\\mathcal{A}, \\mathcal{A})$ constructed in", "Simplicial, Section \\ref{simplicial-section-standard}.", "The diagram", "$$", "\\xymatrix{", "\\mathcal{A} \\ar[d] \\ar[r] & \\mathcal{S} \\ar@<1ex>[l] \\ar[d] \\\\", "\\textit{Alg}_A \\ar[r] & \\textit{Sets} \\ar@<1ex>[l]", "}", "$$", "commutes. It follows that $X_\\bullet(\\text{id}_B : B \\to B)$", "is equal to the standard resolution of $B$ over $A$." ], "refs": [], "ref_ids": [] }, { "id": 11256, "type": "other", "label": "cotangent-remark-resolution", "categories": [ "cotangent" ], "title": "cotangent-remark-resolution", "contents": [ "Let $A \\to B$ be any ring map. Let us call an augmented simplicial $A$-algebra", "$\\epsilon : P_\\bullet \\to B$ a {\\it resolution of $B$ over $A$} if", "each $P_n$ is a polynomial algebra and $\\epsilon$ is a trivial Kan fibration", "of simplicial sets. If $P_\\bullet \\to B$ is an augmentation of a simplicial", "$A$-algebra with each $P_n$ a polynomial algebra surjecting onto $B$, then", "the following are equivalent", "\\begin{enumerate}", "\\item $\\epsilon : P_\\bullet \\to B$ is a resolution of $B$ over $A$,", "\\item $\\epsilon : P_\\bullet \\to B$ is a quasi-isomorphism on", "associated complexes,", "\\item $\\epsilon : P_\\bullet \\to B$ induces a homotopy equivalence", "of simplicial sets.", "\\end{enumerate}", "To see this use Simplicial, Lemmas", "\\ref{simplicial-lemma-trivial-kan-homotopy},", "\\ref{simplicial-lemma-homotopy-equivalence}, and", "\\ref{simplicial-lemma-qis-simplicial-abelian-groups}.", "A resolution $P_\\bullet$ of $B$ over $A$ gives a cosimplicial object", "$U_\\bullet$ of $\\mathcal{C}_{B/A}$ as in Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}", "and it follows that", "$$", "L\\pi_!\\mathcal{F} = \\mathcal{F}(P_\\bullet)", "$$", "functorially in $\\mathcal{F}$, see Lemma \\ref{lemma-identify-pi-shriek}.", "The (formal part of the) proof of Proposition \\ref{proposition-polynomial}", "shows that resolutions exist. We also have seen in the first proof of", "Lemma \\ref{lemma-pi-shriek-standard} that the standard resolution of $B$", "over $A$ is a resolution (so that this terminology doesn't lead to a conflict).", "However, the argument in the proof of Proposition \\ref{proposition-polynomial}", "shows the existence of resolutions without appealing to the simplicial", "computations in Simplicial, Section \\ref{simplicial-section-standard}.", "Moreover, for {\\it any} choice of resolution we have a canonical isomorphism", "$$", "L_{B/A} = \\Omega_{P_\\bullet/A} \\otimes_{P_\\bullet, \\epsilon} B", "$$", "in $D(B)$ by Lemma \\ref{lemma-compute-cotangent-complex}. The freedom to", "choose an arbitrary resolution can be quite useful." ], "refs": [ "simplicial-lemma-trivial-kan-homotopy", "simplicial-lemma-homotopy-equivalence", "simplicial-lemma-qis-simplicial-abelian-groups", "sites-cohomology-lemma-compute-by-cosimplicial-resolution", "cotangent-lemma-identify-pi-shriek", "cotangent-proposition-polynomial", "cotangent-lemma-pi-shriek-standard", "cotangent-proposition-polynomial", "cotangent-lemma-compute-cotangent-complex" ], "ref_ids": [ 14892, 14900, 14899, 4348, 11173, 11241, 11174, 11241, 11175 ] }, { "id": 11257, "type": "other", "label": "cotangent-remark-homotopy-triangle", "categories": [ "cotangent" ], "title": "cotangent-remark-homotopy-triangle", "contents": [ "Suppose that we are given a square (\\ref{equation-commutative-square})", "such that there exists an arrow $\\kappa : B \\to A'$ making the diagram", "commute:", "$$", "\\xymatrix{", "B \\ar[r]_\\beta \\ar[rd]_\\kappa & B' \\\\", "A \\ar[u] \\ar[r]^\\alpha & A' \\ar[u]", "}", "$$", "In this case we claim the functoriality map $P_\\bullet \\to P'_\\bullet$", "is homotopic to the composition $P_\\bullet \\to B \\to A' \\to P'_\\bullet$.", "Namely, using $\\kappa$ the functoriality map factors as", "$$", "P_\\bullet \\to P_{A'/A', \\bullet} \\to P'_\\bullet", "$$", "where $P_{A'/A', \\bullet}$ is the standard resolution of $A'$ over $A'$.", "Since $A'$ is the polynomial algebra on the empty set over $A'$ we", "see from Simplicial, Lemma \\ref{simplicial-lemma-standard-simplicial-homotopy}", "that the augmentation $\\epsilon_{A'/A'} : P_{A'/A', \\bullet} \\to A'$", "is a homotopy equivalence of simplicial rings. Observe that the homotopy", "inverse map $c : A' \\to P_{A'/A', \\bullet}$ constructed in the proof of", "that lemma is just the structure morphism, hence", "we conclude what we want because the two compositions", "$$", "\\xymatrix{", "P_\\bullet \\ar[r] &", "P_{A'/A', \\bullet} \\ar@<1ex>[rr]^{\\text{id}}", "\\ar@<-1ex>[rr]_{c \\circ \\epsilon_{A'/A'}} & &", "P_{A'/A', \\bullet} \\ar[r] &", "P'_\\bullet", "}", "$$", "are the two maps discussed above and these are homotopic", "(Simplicial, Remark \\ref{simplicial-remark-homotopy-pre-post-compose}).", "Since the second map $P_\\bullet \\to P'_\\bullet$ induces the zero", "map $\\Omega_{P_\\bullet/A} \\to \\Omega_{P'_\\bullet/A'}$ we conclude", "that the functoriality map $L_{B/A} \\to L_{B'/A'}$ is homotopic", "to zero in this case." ], "refs": [ "simplicial-lemma-standard-simplicial-homotopy", "simplicial-remark-homotopy-pre-post-compose" ], "ref_ids": [ 14909, 14942 ] }, { "id": 11258, "type": "other", "label": "cotangent-remark-triangle", "categories": [ "cotangent" ], "title": "cotangent-remark-triangle", "contents": [ "We sketch an alternative, perhaps simpler, proof of the existence of", "the fundamental triangle.", "Let $A \\to B \\to C$ be ring maps and assume that $B \\to C$ is injective.", "Let $P_\\bullet \\to B$ be the standard resolution of $B$ over $A$ and", "let $Q_\\bullet \\to C$ be the standard resolution of $C$ over $B$.", "Picture", "$$", "\\xymatrix{", "P_\\bullet : &", "A[A[A[B]]] \\ar[d]", "\\ar@<2ex>[r]", "\\ar@<0ex>[r]", "\\ar@<-2ex>[r]", "&", "A[A[B]] \\ar[d]", "\\ar@<1ex>[r]", "\\ar@<-1ex>[r]", "\\ar@<1ex>[l]", "\\ar@<-1ex>[l]", "&", "A[B] \\ar[d] \\ar@<0ex>[l] \\ar[r] &", "B \\\\", "Q_\\bullet : &", "A[A[A[C]]]", "\\ar@<2ex>[r]", "\\ar@<0ex>[r]", "\\ar@<-2ex>[r]", "&", "A[A[C]]", "\\ar@<1ex>[r]", "\\ar@<-1ex>[r]", "\\ar@<1ex>[l]", "\\ar@<-1ex>[l]", "&", "A[C] \\ar@<0ex>[l] \\ar[r] &", "C", "}", "$$", "Observe that since $B \\to C$ is injective, the ring $Q_n$ is a", "polynomial algebra over $P_n$ for all $n$. Hence we obtain a cosimplicial", "object in $\\mathcal{C}_{C/B/A}$ (beware reversal arrows).", "Now set $\\overline{Q}_\\bullet = Q_\\bullet \\otimes_{P_\\bullet} B$.", "The key to the proof of Proposition \\ref{proposition-triangle}", "is to show that $\\overline{Q}_\\bullet$ is a resolution of $C$ over $B$.", "This follows from Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-O-homology-qis}", "applied to $\\mathcal{C} = \\Delta$, $\\mathcal{O} = P_\\bullet$,", "$\\mathcal{O}' = B$, and $\\mathcal{F} = Q_\\bullet$ (this uses that $Q_n$", "is flat over $P_n$; see Cohomology on Sites, Remark", "\\ref{sites-cohomology-remark-simplicial-modules} to relate simplicial modules", "to sheaves). The key fact implies that the distinguished triangle of", "Proposition \\ref{proposition-triangle}", "is the distinguished triangle associated to the short exact sequence", "of simplicial $C$-modules", "$$", "0 \\to", "\\Omega_{P_\\bullet/A} \\otimes_{P_\\bullet} C \\to", "\\Omega_{Q_\\bullet/A} \\otimes_{Q_\\bullet} C \\to", "\\Omega_{\\overline{Q}_\\bullet/B} \\otimes_{\\overline{Q}_\\bullet} C \\to 0", "$$", "which is deduced from the short exact sequences", "$0 \\to \\Omega_{P_n/A} \\otimes_{P_n} Q_n \\to \\Omega_{Q_n/A} \\to", "\\Omega_{Q_n/P_n} \\to 0$ of", "Algebra, Lemma \\ref{algebra-lemma-ses-formally-smooth}.", "Namely, by Remark \\ref{remark-resolution} and the key fact the complex on the", "right hand side represents $L_{C/B}$ in $D(C)$.", "\\medskip\\noindent", "If $B \\to C$ is not injective, then we can use the above to get a", "fundamental triangle for $A \\to B \\to B \\times C$. Since", "$L_{B \\times C/B} \\to L_{B/B} \\oplus L_{C/B}$ and", "$L_{B \\times C/A} \\to L_{B/A} \\oplus L_{C/A}$", "are quasi-isomorphism in $D(B \\times C)$", "(Lemma \\ref{lemma-cotangent-complex-product})", "this induces the desired distinguished triangle in $D(C)$", "by tensoring with the flat ring map $B \\times C \\to C$." ], "refs": [ "cotangent-proposition-triangle", "sites-cohomology-lemma-O-homology-qis", "sites-cohomology-remark-simplicial-modules", "cotangent-proposition-triangle", "algebra-lemma-ses-formally-smooth", "cotangent-remark-resolution", "cotangent-lemma-cotangent-complex-product" ], "ref_ids": [ 11242, 4352, 4437, 11242, 1209, 11256, 11188 ] }, { "id": 11259, "type": "other", "label": "cotangent-remark-explicit-map", "categories": [ "cotangent" ], "title": "cotangent-remark-explicit-map", "contents": [ "Let $A \\to B \\to C$ be ring maps with $B \\to C$ injective.", "Recall the notation $P_\\bullet$, $Q_\\bullet$, $\\overline{Q}_\\bullet$ of", "Remark \\ref{remark-triangle}.", "Let $R_\\bullet$ be the standard resolution of $C$ over $B$.", "In this remark we explain how to get the canonical identification", "of $\\Omega_{\\overline{Q}_\\bullet/B} \\otimes_{\\overline{Q}_\\bullet} C$", "with $L_{C/B} = \\Omega_{R_\\bullet/B} \\otimes_{R_\\bullet} C$.", "Let $S_\\bullet \\to B$ be the standard resolution of $B$ over $B$.", "Note that the functoriality map $S_\\bullet \\to R_\\bullet$ identifies", "$R_n$ as a polynomial algebra over $S_n$ because $B \\to C$ is injective.", "For example in degree $0$ we have the map $B[B] \\to B[C]$, in degree", "$1$ the map $B[B[B]] \\to B[B[C]]$, and so on. Thus", "$\\overline{R}_\\bullet = R_\\bullet \\otimes_{S_\\bullet} B$", "is a simplicial polynomial algebra", "over $B$ as well and it follows (as in Remark \\ref{remark-triangle}) from", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-O-homology-qis}", "that $\\overline{R}_\\bullet \\to C$ is a resolution. Since we have", "a commutative diagram", "$$", "\\xymatrix{", "Q_\\bullet \\ar[r] & R_\\bullet \\\\", "P_\\bullet \\ar[u] \\ar[r] & S_\\bullet \\ar[u] \\ar[r] & B", "}", "$$", "we obtain a canonical map", "$\\overline{Q}_\\bullet = Q_\\bullet \\otimes_{P_\\bullet} B \\to", "\\overline{R}_\\bullet$. Thus the maps", "$$", "L_{C/B} = \\Omega_{R_\\bullet/B} \\otimes_{R_\\bullet} C", "\\longrightarrow", "\\Omega_{\\overline{R}_\\bullet/B} \\otimes_{\\overline{R}_\\bullet} C", "\\longleftarrow", "\\Omega_{\\overline{Q}_\\bullet/B} \\otimes_{\\overline{Q}_\\bullet} C", "$$", "are quasi-isomorphisms (Remark \\ref{remark-resolution}) and composing", "one with the inverse of the other gives the desired identification." ], "refs": [ "cotangent-remark-triangle", "cotangent-remark-triangle", "sites-cohomology-lemma-O-homology-qis", "cotangent-remark-resolution" ], "ref_ids": [ 11258, 11258, 4352, 11256 ] }, { "id": 11260, "type": "other", "label": "cotangent-remark-make-map", "categories": [ "cotangent" ], "title": "cotangent-remark-make-map", "contents": [ "Let $A \\to B$ be a ring map. Working on $\\mathcal{C}_{B/A}$ as in", "Section \\ref{section-compute-L-pi-shriek} let", "$\\mathcal{J} \\subset \\mathcal{O}$ be the kernel of", "$\\mathcal{O} \\to \\underline{B}$. Note that $L\\pi_!(\\mathcal{J}) = 0$ by", "Lemma \\ref{lemma-apply-O-B-comparison}. Set", "$\\Omega = \\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B}$", "so that", "$L_{B/A} = L\\pi_!(\\Omega)$ by Lemma \\ref{lemma-compute-cotangent-complex}.", "It follows that $L\\pi_!(\\mathcal{J} \\to \\Omega) = L\\pi_!(\\Omega) = L_{B/A}$.", "Thus, for any object $U = (P \\to B)$ of $\\mathcal{C}_{B/A}$ we obtain a map", "\\begin{equation}", "\\label{equation-comparison-map-A}", "(J \\to \\Omega_{P/A} \\otimes_P B) \\longrightarrow L_{B/A}", "\\end{equation}", "where $J = \\Ker(P \\to B)$ in $D(A)$, see", "Cohomology on Sites, Remark", "\\ref{sites-cohomology-remark-map-evaluation-to-derived}.", "Continuing in this manner, note that", "$L\\pi_!(\\mathcal{J} \\otimes_\\mathcal{O}^\\mathbf{L} \\underline{B}) =", "L\\pi_!(\\mathcal{J}) = 0$ by", "Lemma \\ref{lemma-O-homology-B-homology}.", "Since $\\text{Tor}_0^\\mathcal{O}(\\mathcal{J}, \\underline{B}) =", "\\mathcal{J}/\\mathcal{J}^2$", "the spectral sequence", "$$", "H_p(\\mathcal{C}_{B/A}, \\text{Tor}_q^\\mathcal{O}(\\mathcal{J}, \\underline{B}))", "\\Rightarrow ", "H_{p + q}(\\mathcal{C}_{B/A},", "\\mathcal{J} \\otimes_\\mathcal{O}^\\mathbf{L} \\underline{B}) = 0", "$$", "(dual of", "Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor})", "implies that", "$H_0(\\mathcal{C}_{B/A}, \\mathcal{J}/\\mathcal{J}^2) = 0$", "and $H_1(\\mathcal{C}_{B/A}, \\mathcal{J}/\\mathcal{J}^2) = 0$.", "It follows that the complex of $\\underline{B}$-modules", "$\\mathcal{J}/\\mathcal{J}^2 \\to \\Omega$ satisfies", "$\\tau_{\\geq -1}L\\pi_!(\\mathcal{J}/\\mathcal{J}^2 \\to \\Omega) =", "\\tau_{\\geq -1}L_{B/A}$.", "Thus, for any object $U = (P \\to B)$ of $\\mathcal{C}_{B/A}$ we obtain a map", "\\begin{equation}", "\\label{equation-comparison-map}", "(J/J^2 \\to \\Omega_{P/A} \\otimes_P B) \\longrightarrow \\tau_{\\geq -1}L_{B/A}", "\\end{equation}", "in $D(B)$, see", "Cohomology on Sites, Remark", "\\ref{sites-cohomology-remark-map-evaluation-to-derived}." ], "refs": [ "cotangent-lemma-apply-O-B-comparison", "cotangent-lemma-compute-cotangent-complex", "sites-cohomology-remark-map-evaluation-to-derived", "cotangent-lemma-O-homology-B-homology", "derived-lemma-two-ss-complex-functor", "sites-cohomology-remark-map-evaluation-to-derived" ], "ref_ids": [ 11184, 11175, 4436, 11183, 1871, 4436 ] }, { "id": 11261, "type": "other", "label": "cotangent-remark-explicit-comparison-map", "categories": [ "cotangent" ], "title": "cotangent-remark-explicit-comparison-map", "contents": [ "We can make the comparison map of", "Lemma \\ref{lemma-relation-with-naive-cotangent-complex}", "explicit in the following way.", "Let $P_\\bullet$ be the standard resolution of $B$", "over $A$.", "Let $I = \\Ker(A[B] \\to B)$.", "Recall that $P_0 = A[B]$. The map of the", "lemma is given by the commutative diagram", "$$", "\\xymatrix{", "L_{B/A} \\ar[d] & \\ldots \\ar[r] &", "\\Omega_{P_2/A} \\otimes_{P_2} B", "\\ar[r] \\ar[d] &", "\\Omega_{P_1/A} \\otimes_{P_1} B", "\\ar[r] \\ar[d] &", "\\Omega_{P_0/A} \\otimes_{P_0} B", "\\ar[d] \\\\", "\\NL_{B/A} & \\ldots \\ar[r] &", "0 \\ar[r] & ", "I/I^2 \\ar[r] &", "\\Omega_{P_0/A} \\otimes_{P_0} B", "}", "$$", "We construct the downward arrow with target $I/I^2$", "by sending $\\text{d}f \\otimes b$ to the class of", "$(d_0(f) - d_1(f))b$ in $I/I^2$. Here $d_i : P_1 \\to P_0$,", "$i = 0, 1$ are the two face maps of the simplicial structure.", "This makes sense as $d_0 - d_1$ maps $P_1$ into $I = \\Ker(P_0 \\to B)$.", "We omit the verification that this rule is well defined.", "Our map is compatible with the differential", "$\\Omega_{P_1/A} \\otimes_{P_1} B \\to \\Omega_{P_0/A} \\otimes_{P_0} B$", "as this differential maps $\\text{d}f \\otimes b$ to", "$\\text{d}(d_0(f) - d_1(f)) \\otimes b$. Moreover, the differential", "$\\Omega_{P_2/A} \\otimes_{P_2} B \\to \\Omega_{P_1/A} \\otimes_{P_1} B$", "maps $\\text{d}f \\otimes b$ to $\\text{d}(d_0(f) - d_1(f) + d_2(f)) \\otimes b$", "which are annihilated by our downward arrow. Hence a map of complexes.", "We omit the verification that this is the same as the map of", "Lemma \\ref{lemma-relation-with-naive-cotangent-complex}." ], "refs": [ "cotangent-lemma-relation-with-naive-cotangent-complex", "cotangent-lemma-relation-with-naive-cotangent-complex" ], "ref_ids": [ 11204, 11204 ] }, { "id": 11262, "type": "other", "label": "cotangent-remark-surjection", "categories": [ "cotangent" ], "title": "cotangent-remark-surjection", "contents": [ "Adopt notation as in Remark \\ref{remark-make-map}. The arguments given", "there show that the differential", "$$", "H_2(\\mathcal{C}_{B/A}, \\mathcal{J}/\\mathcal{J}^2)", "\\longrightarrow", "H_0(\\mathcal{C}_{B/A}, \\text{Tor}_1^\\mathcal{O}(\\mathcal{J}, \\underline{B}))", "$$", "of the spectral sequence is an isomorphism. Let $\\mathcal{C}'_{B/A}$", "denote the full subcategory of $\\mathcal{C}_{B/A}$ consisting of surjective", "maps $P \\to B$. The agreement of the cotangent complex with the naive", "cotangent complex (Lemma \\ref{lemma-relation-with-naive-cotangent-complex})", "shows that we have an exact sequence of sheaves", "$$", "0 \\to \\underline{H_1(L_{B/A})} \\to", "\\mathcal{J}/\\mathcal{J}^2 \\xrightarrow{\\text{d}} \\Omega \\to", "\\underline{H_2(L_{B/A})} \\to 0", "$$", "on $\\mathcal{C}'_{B/A}$. It follows that $\\Ker(d)$ and", "$\\Coker(d)$ on the whole category $\\mathcal{C}_{B/A}$ have", "vanishing higher homology groups, since", "these are computed by the homology groups of constant simplicial abelian", "groups by Lemma \\ref{lemma-identify-pi-shriek}. Hence we conclude", "that", "$$", "H_n(\\mathcal{C}_{B/A}, \\mathcal{J}/\\mathcal{J}^2) \\to H_n(L_{B/A})", "$$", "is an isomorphism for all $n \\geq 2$. Combined with the remark above", "we obtain the formula", "$H_2(L_{B/A}) =", "H_0(\\mathcal{C}_{B/A}, \\text{Tor}_1^\\mathcal{O}(\\mathcal{J}, \\underline{B}))$." ], "refs": [ "cotangent-remark-make-map", "cotangent-lemma-relation-with-naive-cotangent-complex", "cotangent-lemma-identify-pi-shriek" ], "ref_ids": [ 11260, 11204, 11173 ] }, { "id": 11263, "type": "other", "label": "cotangent-remark-first-homology-symmetric-power", "categories": [ "cotangent" ], "title": "cotangent-remark-first-homology-symmetric-power", "contents": [ "In the situation of Lemma \\ref{lemma-vanishing-symmetric-powers}", "one can show that", "$H_k(\\mathcal{C}, \\text{Sym}^k(\\mathcal{F})) =", "\\wedge^k_B(H_1(\\mathcal{C}, \\mathcal{F}))$.", "Namely, it can be deduced from the proof that", "$H_k(\\mathcal{C}, \\text{Sym}^k(\\mathcal{F}))$ is the $S_k$-coinvariants", "of", "$$", "H^{-k}(L\\pi_!(\\mathcal{F}) \\otimes_B^\\mathbf{L}", "L\\pi_!(\\mathcal{F}) \\otimes_B^\\mathbf{L}", "\\ldots \\otimes_B^\\mathbf{L} L\\pi_!(\\mathcal{F})) =", "H_1(\\mathcal{C}, \\mathcal{F})^{\\otimes k}", "$$", "Thus our claim is that this action is given by the usual action", "of $S_k$ on the tensor product multiplied by the sign character.", "To prove this one has to work through the sign conventions", "in the definition of the total complex associated to a", "multi-complex. We omit the verification." ], "refs": [ "cotangent-lemma-vanishing-symmetric-powers" ], "ref_ids": [ 11205 ] }, { "id": 11264, "type": "other", "label": "cotangent-remark-elucidate-ss", "categories": [ "cotangent" ], "title": "cotangent-remark-elucidate-ss", "contents": [ "In the situation of Theorem \\ref{theorem-quillen-spectral-sequence}", "let $I = \\Ker(A \\to B)$. Then", "$H^{-1}(L_{B/A}) = H_1(\\mathcal{C}_{B/A}, \\Omega) = I/I^2$, see", "Lemma \\ref{lemma-surjection}.", "Hence $H_k(\\mathcal{C}_{B/A}, \\text{Sym}^k(\\Omega)) = \\wedge^k_B(I/I^2)$ by", "Remark \\ref{remark-first-homology-symmetric-power}. Thus the", "$E_1$-page looks like", "$$", "\\begin{matrix}", "B \\\\", "0 \\\\", "0 & I/I^2 \\\\", "0 & H^{-2}(L_{B/A}) \\\\", "0 & H^{-3}(L_{B/A}) & \\wedge^2(I/I^2) \\\\", "0 & H^{-4}(L_{B/A}) & H_3(\\mathcal{C}_{B/A}, \\text{Sym}^2(\\Omega)) \\\\", "0 & H^{-5}(L_{B/A}) & H_4(\\mathcal{C}_{B/A}, \\text{Sym}^2(\\Omega)) &", "\\wedge^3(I/I^2)", "\\end{matrix}", "$$", "with horizontal differential. Thus we obtain edge maps", "$\\text{Tor}_i^A(B, B) \\to H^{-i}(L_{B/A})$, $i > 0$ and", "$\\wedge^i_B(I/I^2) \\to \\text{Tor}_i^A(B, B)$. Finally, we have", "$\\text{Tor}_1^A(B, B) = I/I^2$ and there is a", "five term exact sequence", "$$", "\\text{Tor}_3^A(B, B) \\to H^{-3}(L_{B/A}) \\to \\wedge^2_B(I/I^2) \\to", "\\text{Tor}_2^A(B, B) \\to H^{-2}(L_{B/A}) \\to 0", "$$", "of low degree terms." ], "refs": [ "cotangent-theorem-quillen-spectral-sequence", "cotangent-lemma-surjection", "cotangent-remark-first-homology-symmetric-power" ], "ref_ids": [ 11171, 11203, 11263 ] }, { "id": 11265, "type": "other", "label": "cotangent-remark-elucidate-degree-two", "categories": [ "cotangent" ], "title": "cotangent-remark-elucidate-degree-two", "contents": [ "Let $A \\to B$ be a ring map. Let $P_\\bullet$ be a resolution of", "$B$ over $A$ (Remark \\ref{remark-resolution}).", "Set $J_n = \\Ker(P_n \\to B)$. Note that", "$$", "\\text{Tor}_2^{P_n}(B, B) = ", "\\text{Tor}_1^{P_n}(J_n, B) =", "\\Ker(J_n \\otimes_{P_n} J_n \\to J_n^2).", "$$", "Hence $H_2(L_{B/A})$ is canonically equal to", "$$", "\\Coker(\\text{Tor}_2^{P_1}(B, B) \\to \\text{Tor}_2^{P_0}(B, B))", "$$", "by Remark \\ref{remark-surjection}. To make this more explicit we choose", "$P_2$, $P_1$, $P_0$ as in Example \\ref{example-resolution-length-two}.", "We claim that", "$$", "\\text{Tor}_2^{P_1}(B, B) =", "\\wedge^2(\\bigoplus\\nolimits_{t \\in T} B)\\ \\oplus", "\\ \\bigoplus\\nolimits_{t \\in T} J_0\\ \\oplus", "\\ \\text{Tor}_2^{P_0}(B, B)", "$$", "Namely, the basis elements $x_t \\wedge x_{t'}$ of the first summand", "corresponds to the element $x_t \\otimes x_{t'} - x_{t'} \\otimes x_t$", "of $J_1 \\otimes_{P_1} J_1$. For $f \\in J_0$ the element $x_t \\otimes f$", "of the second summand corresponds to the element", "$x_t \\otimes s_0(f) - s_0(f) \\otimes x_t$ of $J_1 \\otimes_{P_1} J_1$.", "Finally, the map $\\text{Tor}_2^{P_0}(B, B) \\to \\text{Tor}_2^{P_1}(B, B)$", "is given by $s_0$. The map", "$d_0 - d_1 : \\text{Tor}_2^{P_1}(B, B) \\to \\text{Tor}_2^{P_0}(B, B)$", "is zero on the last summand, maps $x_t \\otimes f$ to", "$f \\otimes f_t - f_t \\otimes f$, and maps $x_t \\wedge x_{t'}$", "to $f_t \\otimes f_{t'} - f_{t'} \\otimes f_t$. All in all we conclude", "that there is an exact sequence", "$$", "\\wedge^2_B(J_0/J_0^2) \\to \\text{Tor}_2^{P_0}(B, B) \\to H^{-2}(L_{B/A}) \\to 0", "$$", "In this way we obtain a direct proof of a consequence of Quillen's spectral", "sequence discussed in Remark \\ref{remark-elucidate-ss}." ], "refs": [ "cotangent-remark-resolution", "cotangent-remark-surjection", "cotangent-remark-elucidate-ss" ], "ref_ids": [ 11256, 11262, 11264 ] }, { "id": 11266, "type": "other", "label": "cotangent-remark-functoriality-lichtenbaum-schlessinger", "categories": [ "cotangent" ], "title": "cotangent-remark-functoriality-lichtenbaum-schlessinger", "contents": [ "Consider a commutative square", "$$", "\\xymatrix{", "A' \\ar[r] & B' \\\\", "A \\ar[u] \\ar[r] & B \\ar[u]", "}", "$$", "of ring maps. Choose a factorization", "$$", "\\xymatrix{", "A' \\ar[r] & P' \\ar[r] & B' \\\\", "A \\ar[u] \\ar[r] & P \\ar[u] \\ar[r] & B \\ar[u]", "}", "$$", "with $P$ a polynomial algebra over $A$ and $P'$ a polynomial algebra over $A'$.", "Choose generators $f_t$, $t \\in T$ for $\\Ker(P \\to B)$.", "For $t \\in T$ denote $f'_t$ the image of $f_t$ in $P'$.", "Choose $f'_s \\in P'$ such that the elements $f'_t$ for", "$t \\in T' = T \\amalg S$ generate the kernel", "of $P' \\to B'$. Set $F = \\bigoplus_{t \\in T} P$ and", "$F' = \\bigoplus_{t' \\in T'} P'$. Let $Rel = \\Ker(F \\to P)$", "and $Rel' = \\Ker(F' \\to P')$ where the maps are given", "by multiplication by $f_t$, resp.\\ $f'_t$ on the coordinates.", "Finally, set $TrivRel$, resp.\\ $TrivRel'$ equal to the submodule", "of $Rel$, resp.\\ $TrivRel$ generated by the elements", "$(\\ldots, f_{t'}, 0, \\ldots, 0, -f_t, 0, \\ldots)$", "for $t, t' \\in T$, resp.\\ $T'$. Having made these choices we obtain a", "canonical commutative diagram", "$$", "\\xymatrix{", "L' : &", "Rel'/TrivRel' \\ar[r] &", "F' \\otimes_{P'} B' \\ar[r] &", "\\Omega_{P'/A'} \\otimes_{P'} B' \\\\", "L : \\ar[u] &", "Rel/TrivRel \\ar[r] \\ar[u] &", "F \\otimes_P B \\ar[r] \\ar[u] &", "\\Omega_{P/A} \\otimes_P B \\ar[u]", "}", "$$", "Moreover, tracing through the choices made in the proof of", "Lemma \\ref{lemma-compare-higher}", "the reader sees that one obtains a commutative diagram", "$$", "\\xymatrix{", "L_{B'/A'} \\ar[r] & L' \\\\", "L_{B/A} \\ar[r] \\ar[u] & L \\ar[u]", "}", "$$" ], "refs": [ "cotangent-lemma-compare-higher" ], "ref_ids": [ 11208 ] }, { "id": 11267, "type": "other", "label": "cotangent-remark-map-sections-over-U", "categories": [ "cotangent" ], "title": "cotangent-remark-map-sections-over-U", "contents": [ "It is clear from the proof of", "Lemma \\ref{lemma-compute-L-morphism-sheaves-rings}", "that for any $U \\in \\Ob(\\mathcal{C})$ there is a canonical map", "$L_{\\mathcal{B}(U)/\\mathcal{A}(U)} \\to L_{\\mathcal{B}/\\mathcal{A}}(U)$", "of complexes of $\\mathcal{B}(U)$-modules. Moreover, these maps", "are compatible with restriction maps and the complex", "$L_{\\mathcal{B}/\\mathcal{A}}$", "is the sheafification of the rule $U \\mapsto L_{\\mathcal{B}(U)/\\mathcal{A}(U)}$." ], "refs": [ "cotangent-lemma-compute-L-morphism-sheaves-rings" ], "ref_ids": [ 11216 ] }, { "id": 11268, "type": "other", "label": "cotangent-remark-compute-L-pi-shriek", "categories": [ "cotangent" ], "title": "cotangent-remark-compute-L-pi-shriek", "contents": [ "In the situation above, for every $U \\subset X$ open let", "$P_{\\bullet, U}$ be the standard resolution of $\\mathcal{O}_X(U)$", "over $\\Lambda$. Set $\\mathbf{A}_{n, U} = \\Spec(P_{n, U})$. Then", "$\\mathbf{A}_{\\bullet, U}$", "is a cosimplicial object of the fibre category", "$\\mathcal{C}_{\\mathcal{O}_X(U)/\\Lambda}$ of", "$\\mathcal{C}_{X/\\Lambda}$ over $U$. Moreover, as discussed", "in Remark \\ref{remark-resolution} we have that $\\mathbf{A}_{\\bullet, U}$", "is a cosimplicial object of $\\mathcal{C}_{\\mathcal{O}_X(U)/\\Lambda}$", "as in Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}.", "Since the construction $U \\mapsto \\mathbf{A}_{\\bullet, U}$ is functorial", "in $U$, given any (abelian) sheaf $\\mathcal{F}$ on $\\mathcal{C}_{X/\\Lambda}$", "we obtain a complex of presheaves", "$$", "U \\longmapsto \\mathcal{F}(\\mathbf{A}_{\\bullet, U})", "$$", "whose cohomology groups compute the homology of $\\mathcal{F}$ on the fibre", "category. We conclude by", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-compute-left-derived-pi-shriek}", "that the sheafification computes $L_n\\pi_!(\\mathcal{F})$.", "In other words, the complex of sheaves whose term in degree $-n$ is", "the sheafification of $U \\mapsto \\mathcal{F}(\\mathbf{A}_{n, U})$ computes", "$L\\pi_!(\\mathcal{F})$." ], "refs": [ "cotangent-remark-resolution", "sites-cohomology-lemma-compute-by-cosimplicial-resolution", "sites-cohomology-lemma-compute-left-derived-pi-shriek" ], "ref_ids": [ 11256, 4348, 4354 ] }, { "id": 11269, "type": "other", "label": "cotangent-remark-compute-L-pi-shriek-spaces", "categories": [ "cotangent" ], "title": "cotangent-remark-compute-L-pi-shriek-spaces", "contents": [ "In the situation above, for every object $U \\to X$ of $X_\\etale$", "let $P_{\\bullet, U}$ be the standard resolution of $\\mathcal{O}_X(U)$", "over $\\Lambda$. Set $\\mathbf{A}_{n, U} = \\Spec(P_{n, U})$.", "Then $\\mathbf{A}_{\\bullet, U}$", "is a cosimplicial object of the fibre category", "$\\mathcal{C}_{\\mathcal{O}_X(U)/\\Lambda}$ of", "$\\mathcal{C}_{X/\\Lambda}$ over $U$. Moreover, as discussed", "in Remark \\ref{remark-resolution} we have that $\\mathbf{A}_{\\bullet, U}$", "is a cosimplicial object of $\\mathcal{C}_{\\mathcal{O}_X(U)/\\Lambda}$", "as in Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}.", "Since the construction $U \\mapsto \\mathbf{A}_{\\bullet, U}$ is functorial", "in $U$, given any (abelian) sheaf $\\mathcal{F}$ on $\\mathcal{C}_{X/\\Lambda}$", "we obtain a complex of presheaves", "$$", "U \\longmapsto \\mathcal{F}(\\mathbf{A}_{\\bullet, U})", "$$", "whose cohomology groups compute the homology of $\\mathcal{F}$ on the fibre", "category. We conclude by", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-compute-left-derived-pi-shriek}", "that the sheafification computes $L_n\\pi_!(\\mathcal{F})$.", "In other words, the complex of sheaves whose term in degree $-n$ is", "the sheafification of $U \\mapsto \\mathcal{F}(\\mathbf{A}_{n, U})$ computes", "$L\\pi_!(\\mathcal{F})$." ], "refs": [ "cotangent-remark-resolution", "sites-cohomology-lemma-compute-by-cosimplicial-resolution", "sites-cohomology-lemma-compute-left-derived-pi-shriek" ], "ref_ids": [ 11256, 4348, 4354 ] }, { "id": 11349, "type": "other", "label": "spaces-cohomology-remark-variant", "categories": [ "spaces-cohomology" ], "title": "spaces-cohomology-remark-variant", "contents": [ "In Lemmas \\ref{lemma-check-separated-dvr} and \\ref{lemma-check-proper-dvr}", "it suffices to consider complete discrete valuation rings.", "To be precise in Lemma \\ref{lemma-check-separated-dvr} we can replace", "condition (3) by the following condition: Given any commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y", "}", "$$", "where $A$ is a complete discrete valuation ring with fraction field $K$", "there exists at most one dotted arrow making the diagram commute. Namely, given", "any diagram as in Lemma \\ref{lemma-check-separated-dvr} (3)", "the completion $A^\\wedge$ is a discrete valuation ring", "(More on Algebra, Lemma \\ref{more-algebra-lemma-completion-dvr})", "and the uniqueness of the arrow $\\Spec(A^\\wedge) \\to X$", "implies the uniqueness of the arrow $\\Spec(A) \\to X$", "for example by Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-sheaf-fpqc}.", "Similarly in Lemma \\ref{lemma-check-proper-dvr}", "we can replace condition (3) by the following condition:", "Given any commutative diagram", "$$", "\\xymatrix{", "\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\", "\\Spec(A) \\ar[r] & Y", "}", "$$", "where $A$ is a complete discrete valuation ring with fraction field $K$", "there exists an extension $A \\subset A'$ of complete discrete valuation rings", "inducing a fraction field extension $K \\subset K'$ such that there exists a", "unique arrow $\\Spec(A') \\to X$ making the diagram", "$$", "\\xymatrix{", "\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\", "\\Spec(A') \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] & Y", "}", "$$", "commute. Namely, given any diagram as in Lemma \\ref{lemma-check-proper-dvr}", "part (3) the existence of any commutative diagram", "$$", "\\xymatrix{", "\\Spec(L) \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\", "\\Spec(B) \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] & Y", "}", "$$", "for {\\it any} extension $A \\subset B$ of discrete valuation rings", "will imply there exists an arrow $\\Spec(A) \\to X$ fitting into", "the diagram. This was shown in", "Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-push-down-solution}.", "In fact, it follows from these considerations that it suffices to look", "for dotted arrows in diagrams for any class of discrete valuation rings", "such that, given any discrete valuation ring, there is an extension of it", "that is in the class. For example, we could take complete discrete valuation", "rings with algebraically closed residue field." ], "refs": [ "spaces-cohomology-lemma-check-separated-dvr", "spaces-cohomology-lemma-check-proper-dvr", "spaces-cohomology-lemma-check-separated-dvr", "spaces-cohomology-lemma-check-separated-dvr", "more-algebra-lemma-completion-dvr", "spaces-properties-proposition-sheaf-fpqc", "spaces-cohomology-lemma-check-proper-dvr", "spaces-cohomology-lemma-check-proper-dvr", "spaces-morphisms-lemma-push-down-solution" ], "ref_ids": [ 11328, 11329, 11328, 11328, 10046, 11919, 11329, 11329, 4926 ] }, { "id": 11427, "type": "other", "label": "artin-remark-deformation-category-implies", "categories": [ "artin" ], "title": "artin-remark-deformation-category-implies", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be fibred", "in groupoids over $(\\Sch/S)_{fppf}$. Let $k$ be a field of finite type over", "$S$ and $x_0$ an object", "of $\\mathcal{X}$ over $k$. Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$", "be as in (\\ref{equation-predeformation-category}). If $\\mathcal{F}$", "is a deformation category, i.e., if $\\mathcal{F}$ satisfies the", "Rim-Schlessinger condition (RS), then we see that $\\mathcal{F}$ satisfies", "Schlessinger's conditions (S1) and (S2) by", "Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-RS-implies-S1-S2}.", "Let $\\overline{\\mathcal{F}}$ be the functor of isomorphism classes, see", "Formal Deformation Theory, Remarks", "\\ref{formal-defos-remarks-cofibered-groupoids}", "(\\ref{formal-defos-item-associated-functor-isomorphism-classes}).", "Then $\\overline{\\mathcal{F}}$ satisfies (S1) and (S2) as well, see", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-S1-S2-associated-functor}.", "This holds in particular in the situation of", "Lemma \\ref{lemma-deformation-category}." ], "refs": [ "formal-defos-lemma-RS-implies-S1-S2", "formal-defos-remarks-cofibered-groupoids", "formal-defos-lemma-S1-S2-associated-functor", "artin-lemma-deformation-category" ], "ref_ids": [ 3469, 3585, 3441, 11357 ] }, { "id": 11428, "type": "other", "label": "artin-remark-formal-objects-match", "categories": [ "artin" ], "title": "artin-remark-formal-objects-match", "contents": [ "Let $S$ be a locally Noetherian scheme. Let", "$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.", "Let $\\xi = (R, \\xi_n, f_n)$ be a formal object. Set $k = R/\\mathfrak m$ and", "$x_0 = \\xi_1$. The formal object $\\xi$ defines a formal object", "$\\xi$ of the predeformation category $\\mathcal{F}_{\\mathcal{X}, k, x_0}$.", "This follows immediately from", "Definition \\ref{definition-formal-objects} above,", "Formal Deformation Theory, Definition", "\\ref{formal-defos-definition-formal-objects},", "and our construction of the predeformation category", "$\\mathcal{F}_{\\mathcal{X}, k, x_0}$ in", "Section \\ref{section-predeformation-categories}." ], "refs": [ "artin-definition-formal-objects", "formal-defos-definition-formal-objects" ], "ref_ids": [ 11418, 3518 ] }, { "id": 11429, "type": "other", "label": "artin-remark-strong-effectiveness", "categories": [ "artin" ], "title": "artin-remark-strong-effectiveness", "contents": [ "Let $S$ be a locally Noetherian scheme.", "Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$.", "Assume we have", "\\begin{enumerate}", "\\item an affine open $\\Spec(\\Lambda) \\subset S$,", "\\item an inverse system $(R_n)$ of $\\Lambda$-algebras", "with surjective transition maps whose kernels are locally nilpotent,", "\\item a system $(\\xi_n)$ of objects of $\\mathcal{X}$ lying", "over the system $(\\Spec(R_n))$.", "\\end{enumerate}", "In this situation, set $R = \\lim R_n$. We say that", "$(\\xi_n)$ is {\\it effective} if there exists an object", "$\\xi$ of $\\mathcal{X}$ over $\\Spec(R)$ whose restriction", "to $\\Spec(R_n)$ gives the system $(\\xi_n)$." ], "refs": [], "ref_ids": [] }, { "id": 11430, "type": "other", "label": "artin-remark-trade-openness-versality-diagonal-with-strong-effectiveness", "categories": [ "artin" ], "title": "artin-remark-trade-openness-versality-diagonal-with-strong-effectiveness", "contents": [ "There is a way to deduce openness of versality of the diagonal", "of an category fibred in groupoids from a strong formal effectiveness", "axiom.", "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred", "in groupoids over $(\\Sch/S)_{fppf}$. Assume", "\\begin{enumerate}", "\\item $\\Delta_\\Delta : \\mathcal{X} \\to", "\\mathcal{X} \\times_{\\mathcal{X} \\times \\mathcal{X}} \\mathcal{X}$", "is representable by algebraic spaces,", "\\item $\\mathcal{X}$ has (RS*),", "\\item $\\mathcal{X}$ is limit preserving,", "\\item given an inverse system $(R_n)$ of $S$-algebras", "as in Remark \\ref{remark-strong-effectiveness}", "where $\\Ker(R_m \\to R_n)$ is an ideal of square zero for all $m \\geq n$", "the functor", "$$", "\\mathcal{X}_{\\Spec(\\lim R_n)} \\longrightarrow", "\\lim_n \\mathcal{X}_{\\Spec(R_n)}", "$$", "is fully faithful.", "\\end{enumerate}", "Then $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$", "satisfies openness of versality. This follows by applying", "Lemma \\ref{lemma-SGE-implies-openness-versality}", "to fibre products of the form", "$\\mathcal{X} \\times_{\\Delta, \\mathcal{X} \\times \\mathcal{X}, y}", "(\\Sch/V)_{fppf}$ for any affine scheme $V$ locally", "of finite presentation over $S$ and object $y$ of", "$\\mathcal{X} \\times \\mathcal{X}$ over $V$.", "If we ever need this, we will change this remark into", "a lemma and provide a detailed proof." ], "refs": [ "artin-remark-strong-effectiveness", "artin-lemma-SGE-implies-openness-versality" ], "ref_ids": [ 11429, 11386 ] }, { "id": 11431, "type": "other", "label": "artin-remark-functoriality", "categories": [ "artin" ], "title": "artin-remark-functoriality", "contents": [ "Assumptions and notation as in Lemma \\ref{lemma-properties-lift-RS-star}.", "Suppose $A \\to B$ is a ring map and $y = x|_{\\Spec(B)}$.", "Let $M \\in \\text{Mod}_A$, $N \\in \\text{Mod}_B$", "and let $M \\to N$ an $A$-linear map. Then there are canonical maps", "$\\text{Inf}_x(M) \\to \\text{Inf}_y(N)$ and", "$T_x(M) \\to T_y(N)$ simply because there is a pullback functor", "$$", "\\textit{Lift}(x, A[M]) \\to \\textit{Lift}(y, B[N])", "$$", "coming from the ring map $A[M] \\to B[N]$. Similarly, given a morphism of", "deformation situations $(y, B' \\to B) \\to (x, A' \\to A)$ we obtain a pullback", "functor $\\textit{Lift}(x, A') \\to \\textit{Lift}(y, B')$. Since the", "construction of the action, the addition, and the scalar multiplication", "on $\\text{Inf}_x$ and $T_x$ use only morphisms in the categories of lifts", "(see proof of", "Formal Deformation Theory, Lemma", "\\ref{formal-defos-lemma-linear-functor})", "we see that the constructions above are functorial. In other words we", "obtain $A$-linear maps", "$$", "\\text{Inf}_x(M) \\to \\text{Inf}_y(N)", "\\quad\\text{and}\\quad", "T_x(M) \\to T_y(N)", "$$", "such that the diagrams", "$$", "\\vcenter{", "\\xymatrix{", "\\text{Inf}_y(J) \\ar[r] & \\text{Inf}(y'/y) \\\\", "\\text{Inf}_x(I) \\ar[r] \\ar[u] & \\text{Inf}(x'/x) \\ar[u]", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "T_y(J) \\times \\text{Lift}(y, B') \\ar[r] & \\text{Lift}(y, B') \\\\", "T_x(I) \\times \\text{Lift}(x, A') \\ar[r] \\ar[u] & \\text{Lift}(x, A') \\ar[u]", "}", "}", "$$", "commute. Here $I = \\Ker(A' \\to A)$, $J = \\Ker(B' \\to B)$,", "$x'$ is a lift of $x$ to $A'$ (which may not always exist) and", "$y' = x'|_{\\Spec(B')}$." ], "refs": [ "artin-lemma-properties-lift-RS-star", "formal-defos-lemma-linear-functor" ], "ref_ids": [ 11388, 3445 ] }, { "id": 11432, "type": "other", "label": "artin-remark-automorphisms", "categories": [ "artin" ], "title": "artin-remark-automorphisms", "contents": [ "Assumptions and notation as in Lemma \\ref{lemma-properties-lift-RS-star}.", "Let $x', x''$ be lifts of $x$ to $A'$. Then we have a composition", "map", "$$", "\\text{Inf}(x'/x) \\times", "\\Mor_{\\textit{Lift}(x, A')}(x', x'') \\times \\text{Inf}(x''/x)", "\\longrightarrow", "\\Mor_{\\textit{Lift}(x, A')}(x', x'').", "$$", "Since $\\textit{Lift}(x, A')$ is a groupoid, if", "$\\Mor_{\\textit{Lift}(x, A')}(x', x'')$ is nonempty, then this defines", "a simply transitive left action of $\\text{Inf}(x'/x)$ on", "$\\Mor_{\\textit{Lift}(x, A')}(x', x'')$ and a simply transitive", "right action by $\\text{Inf}(x''/x)$. Now the lemma says that", "$\\text{Inf}(x'/x) = \\text{Inf}_x(I) = \\text{Inf}(x''/x)$.", "We claim that the two actions described above agree via these identifications.", "Namely, either $x' \\not \\cong x''$ in which the claim is clear, or", "$x' \\cong x''$ and in that case we may assume that $x'' = x'$ in which", "case the result follows from the fact that $\\text{Inf}(x'/x)$ is", "commutative. In particular, we obtain a well defined action", "$$", "\\text{Inf}_x(I) \\times \\Mor_{\\textit{Lift}(x, A')}(x', x'')", "\\longrightarrow", "\\Mor_{\\textit{Lift}(x, A')}(x', x'')", "$$", "which is simply transitive as soon as $\\Mor_{\\textit{Lift}(x, A')}(x', x'')$", "is nonempty." ], "refs": [ "artin-lemma-properties-lift-RS-star" ], "ref_ids": [ 11388 ] }, { "id": 11433, "type": "other", "label": "artin-remark-short-exact-sequence-thickenings", "categories": [ "artin" ], "title": "artin-remark-short-exact-sequence-thickenings", "contents": [ "Let $S$ be a scheme. Let $\\mathcal{X}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $A$ be an $S$-algebra. There", "is a notion of a {\\it short exact sequence}", "$$", "(x, A_1' \\to A) \\to (x, A_2' \\to A) \\to (x, A_3' \\to A)", "$$", "of deformation situations: we ask the corresponding maps between", "the kernels $I_i = \\Ker(A_i' \\to A)$ give a short exact sequence", "$$", "0 \\to I_3 \\to I_2 \\to I_1 \\to 0", "$$", "of $A$-modules. Note that in this case the map $A_3' \\to A_1'$", "factors through $A$, hence there is a canonical isomorphism", "$A_1' = A[I_1]$." ], "refs": [], "ref_ids": [] }, { "id": 11434, "type": "other", "label": "artin-remark-compare-deformation-spaces", "categories": [ "artin" ], "title": "artin-remark-compare-deformation-spaces", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred", "in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$ has (RS*).", "Let $k$ be a field of finite type over $S$ and let $x_0$ be an object of", "$\\mathcal{X}$ over $\\Spec(k)$. Then we have equalities of", "$k$-vector spaces", "$$", "T\\mathcal{F}_{\\mathcal{X}, k, x_0} = T_{x_0}(k)", "\\quad\\text{and}\\quad", "\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0}) =", "\\text{Inf}_{x_0}(k)", "$$", "where the spaces on the left hand side of the equality signs are", "given in (\\ref{equation-tangent-space}) and", "(\\ref{equation-infinitesimal-automorphisms})", "and the spaces on the right hand side are given by", "Lemma \\ref{lemma-properties-lift-RS-star}." ], "refs": [ "artin-lemma-properties-lift-RS-star" ], "ref_ids": [ 11388 ] }, { "id": 11435, "type": "other", "label": "artin-remark-canonical-element", "categories": [ "artin" ], "title": "artin-remark-canonical-element", "contents": [ "Assumptions and notation as in Lemma \\ref{lemma-properties-lift-RS-star}.", "Choose an affine open $\\Spec(\\Lambda) \\subset S$ such that $\\Spec(A) \\to S$", "corresponds to a ring map $\\Lambda \\to A$. Consider the ring map", "$$", "A \\longrightarrow A[\\Omega_{A/\\Lambda}],", "\\quad", "a \\longmapsto (a, \\text{d}_{A/\\Lambda}(a))", "$$", "Pulling back $x$ along the corresponding morphism", "$\\Spec(A[\\Omega_{A/\\Lambda}]) \\to \\Spec(A)$ we obtain a", "deformation $x_{can}$ of $x$ over $A[\\Omega_{A/\\Lambda}]$. We call this", "the {\\it canonical element}", "$$", "x_{can} \\in T_x(\\Omega_{A/\\Lambda}) = \\text{Lift}(x, A[\\Omega_{A/\\Lambda}]).", "$$", "Next, assume that $\\Lambda$ is Noetherian and $\\Lambda \\to A$", "is of finite type. Let", "$k = \\kappa(\\mathfrak p)$ be a residue field at a finite type point $u_0$", "of $U = \\Spec(A)$. Let $x_0 = x|_{u_0}$. By (RS*) and the fact that", "$A[k] = A \\times_k k[k]$ the space $T_x(k)$ is the tangent space to the", "deformation functor $\\mathcal{F}_{\\mathcal{X}, k, x_0}$. Via", "$$", "T\\mathcal{F}_{U, k, u_0} =", "\\text{Der}_\\Lambda(A, k) = \\Hom_A(\\Omega_{A/\\Lambda}, k)", "$$", "(see Formal Deformation Theory, Example", "\\ref{formal-defos-example-tangent-space-prorepresentable-functor})", "and functoriality of $T_x$ the canonical element produces the map", "on tangent spaces induced by the object $x$ over $U$. Namely,", "$\\theta \\in T\\mathcal{F}_{U, k, u_0}$ maps to $T_x(\\theta)(x_{can})$", "in $T_x(k) = T\\mathcal{F}_{\\mathcal{X}, k, x_0}$." ], "refs": [ "artin-lemma-properties-lift-RS-star" ], "ref_ids": [ 11388 ] }, { "id": 11436, "type": "other", "label": "artin-remark-canonical-isomorphism", "categories": [ "artin" ], "title": "artin-remark-canonical-isomorphism", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category", "fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$ satisfies", "condition (RS*). Let $A$ be an $S$-algebra such that", "$\\Spec(A) \\to S$ maps into an affine open and let $x, y$ be objects of", "$\\mathcal{X}$ over $\\Spec(A)$. Further, let $A \\to B$ be a ring map and", "let $\\alpha : x|_{\\Spec(B)} \\to y|_{\\Spec(B)}$ be a morphism of", "$\\mathcal{X}$ over $\\Spec(B)$. Consider the ring map", "$$", "B \\longrightarrow B[\\Omega_{B/A}],", "\\quad", "b \\longmapsto (b, \\text{d}_{B/A}(b))", "$$", "Pulling back $\\alpha$ along the corresponding morphism", "$\\Spec(B[\\Omega_{B/A}]) \\to \\Spec(B)$ we obtain a", "morphism $\\alpha_{can}$ between the pullbacks of $x$ and $y$ over", "$B[\\Omega_{B/A}]$. On the other hand, we can pullback $\\alpha$", "by the morphism $\\Spec(B[\\Omega_{B/A}]) \\to \\Spec(B)$ corresponding", "to the injection of $B$ into the first summand of $B[\\Omega_{B/A}]$.", "By the discussion of Remark \\ref{remark-automorphisms}", "we can take the difference", "$$", "\\varphi(x, y, \\alpha) = \\alpha_{can} - \\alpha|_{\\Spec(B[\\Omega_{B/A}])} \\in", "\\text{Inf}_{x|_{\\Spec(B)}}(\\Omega_{B/A}).", "$$", "We will call this the {\\it canonical automorphism}. It depends", "on all the ingredients $A$, $x$, $y$, $A \\to B$ and $\\alpha$." ], "refs": [ "artin-remark-automorphisms" ], "ref_ids": [ 11432 ] }, { "id": 11437, "type": "other", "label": "artin-remark-no-fibre-products", "categories": [ "artin" ], "title": "artin-remark-no-fibre-products", "contents": [ "The site $(\\textit{Noetherian}/S)_\\tau$ does not have fibre products.", "Hence we have to be careful in working with sheaves. For example,", "the continuous inclusion functor", "$(\\textit{Noetherian}/S)_\\tau \\to (\\Sch/S)_\\tau$", "does not define a morphism of sites. See", "Examples, Section \\ref{examples-section-sheaves-locally-Noetherian}", "for an example in case $\\tau = fppf$." ], "refs": [], "ref_ids": [] }, { "id": 11438, "type": "other", "label": "artin-remark-G-rings", "categories": [ "artin" ], "title": "artin-remark-G-rings", "contents": [ "In particular, we cannot prove that the desired result is true for", "every Situation \\ref{situation-contractions} because we will need to", "assume the local rings of $S$ are G-rings. If you can prove the", "result in general or if you have a counter example, please let", "us know at", "\\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}." ], "refs": [], "ref_ids": [] }, { "id": 11439, "type": "other", "label": "artin-remark-how-to-think-compatibility", "categories": [ "artin" ], "title": "artin-remark-how-to-think-compatibility", "contents": [ "In Situation \\ref{situation-contractions} let $V$ be a locally Noetherian", "scheme over $S$. Let $(Z, u', \\hat x)$ be a triple satisfying (1), (2), and", "(3) above. We want to explain a way to think about the compatibility", "condition (4). It will not be mathematically precise as we are going use", "a fictitious category $\\textit{An}_S$ of analytic spaces over $S$", "and a fictitious analytification functor", "$$", "\\left\\{", "\\begin{matrix}", "\\text{locally Noetherian formal} \\\\", "\\text{algebraic spaces over }S", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\textit{An}_S,", "\\quad\\quad", "Y \\longmapsto Y^{an}", "$$", "For example if $Y = \\text{Spf}(k[[t]])$ over $S = \\Spec(k)$, then $Y^{an}$", "should be thought of as an open unit disc. If $Y = \\Spec(k)$, then $Y^{an}$", "is a single point. The category $\\textit{An}_S$ should have open and", "closed immersions and we should be able to take the open complement", "of a closed. Given $Y$ the morphism $Y_{red} \\to Y$ should induces a", "closed immersion $Y_{red}^{an} \\to Y^{an}$. We set", "$Y^{rig} = Y^{an} \\setminus Y_{red}^{an}$ equal to its open complement.", "If $Y$ is an algebraic space and if $Z \\subset Y$ is closed, then", "the morphism $Y_{/Z} \\to Y$ should induce an open immersion", "$Y_{/Z}^{an} \\to Y^{an}$ which in turn should induce an open immersion", "$$", "can : (Y_{/Z})^{rig} \\longrightarrow (Y \\setminus Z)^{an}", "$$", "Also, given a formal modification $g : Y' \\to Y$ of locally Noetherian formal", "algebraic spaces, the induced morphism $g^{rig} : (Y')^{rig} \\to Y^{rig}$", "should be an isomorphism. Given $\\text{An}_S$ and the analytification", "functor, we can consider the requirement that", "$$", "\\xymatrix{", "(V_{/Z})^{rig} \\ar[rr]_{can} \\ar[d]_{(g^{rig})^{-1} \\circ \\hat x^{an}} & &", "(V \\setminus Z)^{an} \\ar[d]^{(u')^{an}} \\\\", "(X'_{/T'})^{rig} \\ar[rr]^{can} & & (X' \\setminus T')^{an}", "}", "$$", "commutes. This makes sense as $g^{rig} : (X'_{T'})^{rig} \\to W^{rig}$", "is an isomorphism and $U' = X' \\setminus T'$. Finally, under some assumptions", "of faithfulness of the analytification functor, this requirement will", "be equivalent to the compatibility condition formulated above.", "We hope this will motivate the reader to think of the compatibility", "of $u'$ and $\\hat x$ as the requirement that some maps be equal,", "rather than asking for the existence of a certain commutative diagram." ], "refs": [], "ref_ids": [] }, { "id": 11440, "type": "other", "label": "artin-remark-diagonal", "categories": [ "artin" ], "title": "artin-remark-diagonal", "contents": [ "In Situation \\ref{situation-contractions}.", "Let $V$ be a locally Noetherian scheme over $S$.", "Let $(Z_i, u'_i, \\hat x_i) \\in F(V)$ for $i = 1, 2$. Let $V'_i \\to V$,", "$\\hat x'_i$ and $x'_i$ witness the compatibility between $u'_i$ and", "$\\hat x_i$ for $i = 1, 2$.", "\\medskip\\noindent", "Set $V' = V'_1 \\times_V V'_2$. Let $E' \\to V'$ denote the equalizer", "of the morphisms", "$$", "V' \\to V'_1 \\xrightarrow{x'_1} X'", "\\quad\\text{and}\\quad", "V' \\to V'_2 \\xrightarrow{x'_2} X'", "$$", "Set $Z = Z_1 \\cap Z_2$. Let $E_W \\to V_{/Z}$", "be the equalizer of the morphisms", "$$", "V_{/Z} \\to V_{/Z_1} \\xrightarrow{\\hat x_1} W", "\\quad\\text{and}\\quad", "V_{/Z} \\to V_{/Z_2} \\xrightarrow{\\hat x_2} W", "$$", "Observe that $E' \\to V$ is separated and locally of finite type", "and that $E_W$ is a locally Noetherian formal algebraic space", "separated over $V$.", "The compatibilities between the various morphisms involved show that", "\\begin{enumerate}", "\\item $\\Im(E' \\to V) \\cap (Z_1 \\cup Z_2)$", "is contained in $Z = Z_1 \\cap Z_2$, ", "\\item the morphism $E' \\times_V (V \\setminus Z) \\to V \\setminus Z$", "is a monomorphism and is equal to the equalizer of the restrictions", "of $u'_1$ and $u'_2$ to $V \\setminus (Z_1 \\cup Z_2)$,", "\\item the morphism $E'_{/Z} \\to V_{/Z}$ factors through $E_W$", "and the diagram", "$$", "\\xymatrix{", "E'_{/Z} \\ar[r] \\ar[d] & X'_{/T'} \\ar[d]^g \\\\", "E_W \\ar[r] & W", "}", "$$", "is cartesian. In particular, the morphism $E'_{/Z} \\to E_W$", "is a formal modification as the base change of $g$,", "\\item $E'$, $(E' \\to V)^{-1}Z$, and $E'_{/Z} \\to E_W$", "is a triple as in Situation \\ref{situation-contractions}", "with base scheme the locally Noetherian scheme $V$,", "\\item given a morphism $\\varphi : A \\to V$", "of locally Noetherian schemes, the following are equivalent", "\\begin{enumerate}", "\\item $(Z_1, u'_1, \\hat x_1)$ and $(Z_2, u'_2, \\hat x_2)$", "restrict to the same element of $F(A)$,", "\\item $A \\setminus \\varphi^{-1}(Z) \\to V \\setminus Z$", "factors through $E' \\times_V (V \\setminus Z)$", "and $A_{/\\varphi^{-1}(Z)} \\to V_{/Z}$", "factors through $E_W$.", "\\end{enumerate}", "\\end{enumerate}", "We conclude, using", "Lemmas \\ref{lemma-solution} and \\ref{lemma-functor-is-solution},", "that if there is a solution $E \\to V$", "for the triple in (4), then $E$ represents", "$F \\times_{\\Delta, F \\times F} V$ on the category of", "locally Noetherian schemes over $V$." ], "refs": [ "artin-lemma-solution", "artin-lemma-functor-is-solution" ], "ref_ids": [ 11403, 11404 ] }, { "id": 11441, "type": "other", "label": "artin-remark-separated-needed", "categories": [ "artin" ], "title": "artin-remark-separated-needed", "contents": [ "The proof of Theorem \\ref{theorem-contractions} uses that $X'$ and $W$", "are separated over $S$ in two places. First, the proof uses this in showing", "$\\Delta : F \\to F \\times F$ is representable by algebraic spaces.", "This use of the assumption can be entirely avoided by proving", "that $\\Delta$ is representable by applying the theorem in the", "separated case to the triples", "$E'$, $(E' \\to V)^{-1}Z$, and $E'_{/Z} \\to E_W$", "found in Remark \\ref{remark-diagonal} (this is the usual bootstrap", "procedure for the diagonal). Thus the proof of", "Lemma \\ref{lemma-formal-object-effective} is the only", "place in our proof of Theorem \\ref{theorem-contractions}", "where we really need to use that $X' \\to S$ is separated.", "The reader checks that we use the assumption only to obtain", "the morphism $x' : V' \\to X'$. The existence of $x'$ can be shown,", "using results in the literature, if $X' \\to S$ is quasi-separated, see", "More on Morphisms of Spaces, Remark", "\\ref{spaces-more-morphisms-remark-weaken-separation-axioms-question}.", "We conclude the theorem holds as stated with", "``separated'' replaced by ``quasi-separated''. If we ever need this", "we will precisely state and carefully prove this here." ], "refs": [ "artin-theorem-contractions", "artin-remark-diagonal", "artin-lemma-formal-object-effective", "artin-theorem-contractions", "spaces-more-morphisms-remark-weaken-separation-axioms-question" ], "ref_ids": [ 11350, 11440, 11411, 11350, 312 ] }, { "id": 11527, "type": "other", "label": "obsolete-remark-composition-of-adjoints-isomorphic-to-identity", "categories": [ "obsolete" ], "title": "obsolete-remark-composition-of-adjoints-isomorphic-to-identity", "contents": [ "The information which used to be contained in this remark is now", "subsumed in the combination of", "Categories, Lemmas \\ref{categories-lemma-adjoint-fully-faithful} and", "\\ref{categories-lemma-left-adjoint-composed-fully-faithful}." ], "refs": [ "categories-lemma-adjoint-fully-faithful", "categories-lemma-left-adjoint-composed-fully-faithful" ], "ref_ids": [ 12248, 12247 ] }, { "id": 11528, "type": "other", "label": "obsolete-remark-weak-serre-subcategory", "categories": [ "obsolete" ], "title": "obsolete-remark-weak-serre-subcategory", "contents": [ "The following remarks are obsolete as they are subsumed in", "Homology, Lemmas \\ref{homology-lemma-biregular-ss-converges} and", "\\ref{homology-lemma-first-quadrant-ss}.", "Let $\\mathcal{A}$ be an abelian category.", "Let $\\mathcal{C} \\subset \\mathcal{A}$", "be a weak Serre subcategory (see", "Homology, Definition \\ref{homology-definition-serre-subcategory}).", "Suppose that $K^{\\bullet, \\bullet}$ is a double complex to which", "Homology, Lemma \\ref{homology-lemma-first-quadrant-ss}", "applies such that for some $r \\geq 0$ all the objects", "${}'E_r^{p, q}$ belong to $\\mathcal{C}$. Then all the cohomology groups", "$H^n(sK^\\bullet)$ belong to $\\mathcal{C}$. Namely, the assumptions imply", "that the kernels and images of ${}'d_r^{p, q}$ are in $\\mathcal{C}$.", "Whereupon we see that each ${}'E_{r + 1}^{p, q}$ is in $\\mathcal{C}$.", "By induction we see that each ${}'E_\\infty^{p, q}$ is in $\\mathcal{C}$.", "Hence each $H^n(sK^\\bullet)$ has a finite filtration whose subquotients", "are in $\\mathcal{C}$. Using that $\\mathcal{C}$ is closed under extensions", "we conclude that $H^n(sK^\\bullet)$ is in $\\mathcal{C}$ as claimed.", "The same result holds for the second spectral sequence associated", "to $K^{\\bullet, \\bullet}$. Similarly, if $(K^\\bullet, F)$ is a filtered", "complex to which", "Homology, Lemma \\ref{homology-lemma-biregular-ss-converges}", "applies and for some $r \\geq 0$ all the objects $E_r^{p, q}$", "belong to $\\mathcal{C}$, then each $H^n(K^\\bullet)$ is", "an object of $\\mathcal{C}$." ], "refs": [ "homology-lemma-biregular-ss-converges", "homology-lemma-first-quadrant-ss", "homology-definition-serre-subcategory", "homology-lemma-first-quadrant-ss", "homology-lemma-biregular-ss-converges" ], "ref_ids": [ 12101, 12105, 12146, 12105, 12101 ] }, { "id": 11529, "type": "other", "label": "obsolete-remark-projective-resolution", "categories": [ "obsolete" ], "title": "obsolete-remark-projective-resolution", "contents": [ "Let $R$ be a ring.", "For any set $S$ we let $F(S)$ denote the free $R$-module on $S$.", "Then any left $R$-module has the following two step resolution", "$$", "F(M \\times M) \\oplus F(R \\times M) \\to F(M) \\to M \\to 0.", "$$", "The first map is given by the rule", "$$", "[m_1, m_2] \\oplus [r, m] \\mapsto [m_1 + m_2] - [m_1] - [m_2] + [rm] - r[m].", "$$" ], "refs": [], "ref_ids": [] }, { "id": 11530, "type": "other", "label": "obsolete-remark-section-colimits", "categories": [ "obsolete" ], "title": "obsolete-remark-section-colimits", "contents": [ "This reference/tag used to refer to a Section in", "the chapter Smoothing Ring Maps, but the material has", "since been subsumed in Algebra, Section \\ref{algebra-section-colimits-flat}." ], "refs": [], "ref_ids": [] }, { "id": 11531, "type": "other", "label": "obsolete-remark-algebra", "categories": [ "obsolete" ], "title": "obsolete-remark-algebra", "contents": [ "Let $R$ be a ring. Suppose that we have $F \\in R[X, Y]_d$", "and $G \\in R[X, Y]_e$ such that, setting $S = R[X, Y]/(F)$", "we have (1) $S_n$ is finite locally free of rank $d$ for", "all $n \\geq d$, and (2) multiplication by $G$ defines", "isomorphisms $S_n \\to S_{n + e}$ for all $n \\geq d$. In this", "case we may define a finite, locally free $R$-algebra", "$A$ as follows:", "\\begin{enumerate}", "\\item as an $R$-module $A = S_{ed}$, and", "\\item multiplication $A \\times A \\to A$ is given by", "the rule that $H_1 H_2 = H_3$ if and only if $G^d H_3 = H_1 H_2$", "in $S_{2ed}$.", "\\end{enumerate}", "This makes sense because multiplication by $G^d$", "induces a bijective map $S_{de} \\to S_{2de}$.", "It is easy to see that this defines a ring structure.", "Note the confusing fact that the element $G^d$", "defines the unit element of the ring $A$." ], "refs": [], "ref_ids": [] }, { "id": 11532, "type": "other", "label": "obsolete-remark-equation-derivatives", "categories": [ "obsolete" ], "title": "obsolete-remark-equation-derivatives", "contents": [ "This tag used to refer to an equation in the proof of", "Algebraization of Formal Spaces, Proposition", "\\ref{restricted-proposition-approximate}", "which became unused because of a rearrangement of the material." ], "refs": [ "restricted-proposition-approximate" ], "ref_ids": [ 2432 ] }, { "id": 11533, "type": "other", "label": "obsolete-remark-equation-ci", "categories": [ "obsolete" ], "title": "obsolete-remark-equation-ci", "contents": [ "This tag used to refer to an equation in the proof of", "Algebraization of Formal Spaces, Proposition", "\\ref{restricted-proposition-approximate}", "which became unused because of a rearrangement of the material." ], "refs": [ "restricted-proposition-approximate" ], "ref_ids": [ 2432 ] }, { "id": 11534, "type": "other", "label": "obsolete-remark-equation-in-ideal", "categories": [ "obsolete" ], "title": "obsolete-remark-equation-in-ideal", "contents": [ "This tag used to refer to an equation in the proof of", "Algebraization of Formal Spaces, Proposition", "\\ref{restricted-proposition-approximate}", "which became unused because of a rearrangement of the material." ], "refs": [ "restricted-proposition-approximate" ], "ref_ids": [ 2432 ] }, { "id": 11535, "type": "other", "label": "obsolete-remark-equation-derivatives-analogue", "categories": [ "obsolete" ], "title": "obsolete-remark-equation-derivatives-analogue", "contents": [ "This tag used to refer to an equation in the proof of", "Algebraization of Formal Spaces, Proposition", "\\ref{restricted-proposition-approximate}", "which became unused because of a rearrangement of the material." ], "refs": [ "restricted-proposition-approximate" ], "ref_ids": [ 2432 ] }, { "id": 11536, "type": "other", "label": "obsolete-remark-equation-go-down", "categories": [ "obsolete" ], "title": "obsolete-remark-equation-go-down", "contents": [ "This tag used to refer to an equation in the proof of", "Algebraization of Formal Spaces, Lemma", "\\ref{restricted-lemma-lift-approximation}", "which became unused because of a rearrangement of the material." ], "refs": [ "restricted-lemma-lift-approximation" ], "ref_ids": [ 2320 ] }, { "id": 11537, "type": "other", "label": "obsolete-remark-from-shriek-to-star", "categories": [ "obsolete" ], "title": "obsolete-remark-from-shriek-to-star", "contents": [ "Let $U$ be an object of $\\mathcal{C}$. For any abelian sheaf", "$\\mathcal{G}$ on $\\mathcal{C}/U$ one may wonder whether", "there is a canonical map", "$$", "c : j_{U!}\\mathcal{G} \\longrightarrow j_{U*}\\mathcal{G}", "$$", "To construct such a thing is the same as constructing a map", "$j_U^{-1}j_{U!}\\mathcal{G} \\to \\mathcal{G}$.", "Note that restriction commutes with sheafification.", "Thus we can use the presheaf of", "Modules on Sites, Lemma \\ref{sites-modules-lemma-extension-by-zero}.", "Hence it suffices to define for $V/U$ a map", "$$", "\\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{G}(V \\xrightarrow{\\varphi} U)", "\\longrightarrow", "\\mathcal{G}(V/U)", "$$", "compatible with restrictions. It looks like we can take the", "which is zero on all summands except for the one where $\\varphi$", "is the structure morphism $\\varphi_0 : V \\to U$ where we take $1$.", "However, this isn't compatible with restriction mappings: namely,", "if $\\alpha : V' \\to V$ is a morphism of $\\mathcal{C}$, then", "denote $V'/U$ the object of $\\mathcal{C}/U$ with structure", "morphism $\\varphi'_0 = \\varphi_0 \\circ \\alpha$.", "We need to check that the diagram", "$$", "\\xymatrix{", "\\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{G}(V \\xrightarrow{\\varphi} U)", "\\ar[d] \\ar[r] &", "\\mathcal{G}(V/U) \\ar[d] \\\\", "\\bigoplus\\nolimits_{\\varphi' \\in \\Mor_\\mathcal{C}(V', U)}", "\\mathcal{G}(V' \\xrightarrow{\\varphi'} U)", "\\ar[r] &", "\\mathcal{G}(V'/U)", "}", "$$", "commutes. The problem here is that there", "may be a morphism $\\varphi : V \\to U$ different from $\\varphi_0$", "such that $\\varphi \\circ \\alpha = \\varphi'_0$.", "Thus the left vertical arrow will send the summand corresponding", "to $\\varphi$ into the summand on which the lower horizontal arrow is", "equal to $1$ and almost surely the diagram doesn't commute." ], "refs": [ "sites-modules-lemma-extension-by-zero" ], "ref_ids": [ 14169 ] }, { "id": 11538, "type": "other", "label": "obsolete-remark-pullback-K-flat", "categories": [ "obsolete" ], "title": "obsolete-remark-pullback-K-flat", "contents": [ "This remark used to discuss what we know about pullbacks of K-flat complexes", "being K-flat or not, but is now obsoleted by", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-pullback-K-flat}." ], "refs": [ "sites-cohomology-lemma-pullback-K-flat" ], "ref_ids": [ 4241 ] }, { "id": 11539, "type": "other", "label": "obsolete-remark-cohomology-topics", "categories": [ "obsolete" ], "title": "obsolete-remark-cohomology-topics", "contents": [ "This tag used to refer to a section of the chapter on cohomology", "listing topics to be treated." ], "refs": [], "ref_ids": [] }, { "id": 11540, "type": "other", "label": "obsolete-remark-sites-cohomology-topics", "categories": [ "obsolete" ], "title": "obsolete-remark-sites-cohomology-topics", "contents": [ "This tag used to refer to a section of the chapter on cohomology", "listing topics to be treated." ], "refs": [], "ref_ids": [] }, { "id": 11541, "type": "other", "label": "obsolete-remark-V-implies-C", "categories": [ "obsolete" ], "title": "obsolete-remark-V-implies-C", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-C-general}", "pertaining to the situation described in", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-compare-qc-zar}." ], "refs": [ "sites-cohomology-lemma-V-implies-C-general", "sites-cohomology-lemma-compare-qc-zar" ], "ref_ids": [ 4297, 4311 ] }, { "id": 11542, "type": "other", "label": "obsolete-remark-V-implies-cohomology", "categories": [ "obsolete" ], "title": "obsolete-remark-V-implies-cohomology", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-V-implies-cohomology-general}", "pertaining to the situation described in", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-compare-qc-zar}." ], "refs": [ "sites-cohomology-lemma-V-implies-cohomology-general", "sites-cohomology-lemma-compare-qc-zar" ], "ref_ids": [ 4298, 4311 ] }, { "id": 11543, "type": "other", "label": "obsolete-remark-induction-step-V-C", "categories": [ "obsolete" ], "title": "obsolete-remark-induction-step-V-C", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-induction-step-V-C-general}", "pertaining to the situation described in", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-compare-qc-zar}." ], "refs": [ "sites-cohomology-lemma-induction-step-V-C-general", "sites-cohomology-lemma-compare-qc-zar" ], "ref_ids": [ 4301, 4311 ] }, { "id": 11544, "type": "other", "label": "obsolete-remark-V-implies-C-etale-fppf", "categories": [ "obsolete" ], "title": "obsolete-remark-V-implies-C-etale-fppf", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-C-general}", "pertaining to the situation described in", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-fppf-etale}." ], "refs": [ "sites-cohomology-lemma-V-implies-C-general", "etale-cohomology-lemma-compare-fppf-etale" ], "ref_ids": [ 4297, 6662 ] }, { "id": 11545, "type": "other", "label": "obsolete-remark-V-implies-cohomology-etale-fppf", "categories": [ "obsolete" ], "title": "obsolete-remark-V-implies-cohomology-etale-fppf", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-V-implies-cohomology-general}", "pertaining to the situation described in", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-fppf-etale}." ], "refs": [ "sites-cohomology-lemma-V-implies-cohomology-general", "etale-cohomology-lemma-compare-fppf-etale" ], "ref_ids": [ 4298, 6662 ] }, { "id": 11546, "type": "other", "label": "obsolete-remark-induction-step-V-C-etale-fppf", "categories": [ "obsolete" ], "title": "obsolete-remark-induction-step-V-C-etale-fppf", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-induction-step-V-C-general}", "pertaining to the situation described in", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-fppf-etale}." ], "refs": [ "sites-cohomology-lemma-induction-step-V-C-general", "etale-cohomology-lemma-compare-fppf-etale" ], "ref_ids": [ 4301, 6662 ] }, { "id": 11547, "type": "other", "label": "obsolete-remark-V-implies-C-etale-ph", "categories": [ "obsolete" ], "title": "obsolete-remark-V-implies-C-etale-ph", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-C-general}", "pertaining to the situation described in", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-ph-etale}." ], "refs": [ "sites-cohomology-lemma-V-implies-C-general", "etale-cohomology-lemma-compare-ph-etale" ], "ref_ids": [ 4297, 6671 ] }, { "id": 11548, "type": "other", "label": "obsolete-remark-V-implies-cohomology-etale-ph", "categories": [ "obsolete" ], "title": "obsolete-remark-V-implies-cohomology-etale-ph", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-V-implies-cohomology-general}", "pertaining to the situation described in", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-ph-etale}." ], "refs": [ "sites-cohomology-lemma-V-implies-cohomology-general", "etale-cohomology-lemma-compare-ph-etale" ], "ref_ids": [ 4298, 6671 ] }, { "id": 11549, "type": "other", "label": "obsolete-remark-V-implies-cohomology-etale-ph-extra", "categories": [ "obsolete" ], "title": "obsolete-remark-V-implies-cohomology-etale-ph-extra", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-V-implies-cohomology-extra-general}", "pertaining to the situation described in", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-ph-etale}." ], "refs": [ "sites-cohomology-lemma-V-implies-cohomology-extra-general", "etale-cohomology-lemma-compare-ph-etale" ], "ref_ids": [ 4299, 6671 ] }, { "id": 11550, "type": "other", "label": "obsolete-remark-make-class-zero", "categories": [ "obsolete" ], "title": "obsolete-remark-make-class-zero", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-make-class-zero-general}", "pertaining to the situation described in", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-ph-etale}." ], "refs": [ "sites-cohomology-lemma-make-class-zero-general", "etale-cohomology-lemma-compare-ph-etale" ], "ref_ids": [ 4300, 6671 ] }, { "id": 11551, "type": "other", "label": "obsolete-remark-induction-step-V-C-etale-ph", "categories": [ "obsolete" ], "title": "obsolete-remark-induction-step-V-C-etale-ph", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-induction-step-V-C-general}", "pertaining to the situation described in", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-ph-etale}." ], "refs": [ "sites-cohomology-lemma-induction-step-V-C-general", "etale-cohomology-lemma-compare-ph-etale" ], "ref_ids": [ 4301, 6671 ] }, { "id": 11552, "type": "other", "label": "obsolete-remark-V-implies-C-etale-h", "categories": [ "obsolete" ], "title": "obsolete-remark-V-implies-C-etale-h", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-C-general}", "pertaining to the situation described in", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-h-etale}." ], "refs": [ "sites-cohomology-lemma-V-implies-C-general", "etale-cohomology-lemma-compare-h-etale" ], "ref_ids": [ 4297, 6678 ] }, { "id": 11553, "type": "other", "label": "obsolete-remark-V-implies-cohomology-etale-h", "categories": [ "obsolete" ], "title": "obsolete-remark-V-implies-cohomology-etale-h", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-V-implies-cohomology-general}", "pertaining to the situation described in", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-h-etale}." ], "refs": [ "sites-cohomology-lemma-V-implies-cohomology-general", "etale-cohomology-lemma-compare-h-etale" ], "ref_ids": [ 4298, 6678 ] }, { "id": 11554, "type": "other", "label": "obsolete-remark-V-implies-cohomology-etale-h-extra", "categories": [ "obsolete" ], "title": "obsolete-remark-V-implies-cohomology-etale-h-extra", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-V-implies-cohomology-extra-general}", "pertaining to the situation described in", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-h-etale}." ], "refs": [ "sites-cohomology-lemma-V-implies-cohomology-extra-general", "etale-cohomology-lemma-compare-h-etale" ], "ref_ids": [ 4299, 6678 ] }, { "id": 11555, "type": "other", "label": "obsolete-remark-induction-step-V-C-etale-h", "categories": [ "obsolete" ], "title": "obsolete-remark-induction-step-V-C-etale-h", "contents": [ "This tag used to refer to the special case of", "Cohomology on Sites, Lemma", "\\ref{sites-cohomology-lemma-induction-step-V-C-general}", "pertaining to the situation described in", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-h-etale}." ], "refs": [ "sites-cohomology-lemma-induction-step-V-C-general", "etale-cohomology-lemma-compare-h-etale" ], "ref_ids": [ 4301, 6678 ] }, { "id": 11556, "type": "other", "label": "obsolete-remark-how-used", "categories": [ "obsolete" ], "title": "obsolete-remark-how-used", "contents": [ "This tag used to be in the chapter on \\'etale cohomology, but is no", "longer suitable there because of a reorganization. The content of", "the tag was the following:", "\\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-when-ctf}", "can be used to prove that if $f : X \\to Y$ is a separated, finite type", "morphism of schemes and $Y$ is Noetherian, then $Rf_!$ induces a functor", "$D_{ctf}(X_\\etale, \\Lambda) \\to D_{ctf}(Y_\\etale, \\Lambda)$.", "An example of this argument, when $Y$ is the spectrum of a field and", "$X$ is a curve is given in The Trace Formula,", "Proposition \\ref{trace-proposition-projective-curve-constructible-cohomology}." ], "refs": [ "etale-cohomology-lemma-when-ctf", "trace-proposition-projective-curve-constructible-cohomology" ], "ref_ids": [ 6560, 14438 ] }, { "id": 11557, "type": "other", "label": "obsolete-remark-proof-works-when", "categories": [ "obsolete" ], "title": "obsolete-remark-proof-works-when", "contents": [ "This remark used to discuss to what extend the original proof of", "Lemma \\ref{lemma-sheaf-fpqc-quasi-separated} (of December 18, 2009)", "generalizes." ], "refs": [ "obsolete-lemma-sheaf-fpqc-quasi-separated" ], "ref_ids": [ 11485 ] }, { "id": 11558, "type": "other", "label": "obsolete-remark-very-reasonable-Zariski-locally-quasi-separated", "categories": [ "obsolete" ], "title": "obsolete-remark-very-reasonable-Zariski-locally-quasi-separated", "contents": [ "Very reasonable algebraic spaces form a strictly larger collection than", "Zariski locally quasi-separated algebraic spaces. Consider", "an algebraic space of the form $X = [U/G]$ (see", "Spaces, Definition \\ref{spaces-definition-quotient})", "where $G$ is a finite group acting without fixed points on a", "non-quasi-separated scheme $U$. Namely, in this case", "$U \\times_X U = U \\times G$ and clearly both projections to $U$ are", "quasi-compact, hence $X$ is very reasonable. On the other hand, the diagonal", "$U \\times_X U \\to U \\times U$ is not quasi-compact, hence this", "algebraic space is not quasi-separated. Now, take $U$ the infinite", "affine space over a field $k$ of characteristic $\\not = 2$ with", "zero doubled, see", "Schemes, Example \\ref{schemes-example-not-quasi-separated}.", "Let $0_1, 0_2$ be the two zeros of $U$. Let $G = \\{+1, -1\\}$, and", "let $-1$ act by $-1$ on all coordinates, and by switching", "$0_1$ and $0_2$. Then $[U/G]$ is very reasonable but not Zariski locally", "quasi-separated (details omitted)." ], "refs": [ "spaces-definition-quotient" ], "ref_ids": [ 8182 ] }, { "id": 11559, "type": "other", "label": "obsolete-remark-different-topologies", "categories": [ "obsolete" ], "title": "obsolete-remark-different-topologies", "contents": [ "We obtain a second topology $\\tau_Y$ on $\\mathcal{C}_{X/Y}$", "by taking the topology inherited from $Y_{Zar}$.", "There is a third topology $\\tau_{X \\to Y}$ where a family of morphisms", "$\\{(U_i \\to A_i) \\to (U \\to A)\\}$ is a covering if and only", "if $U = \\bigcup U_i$, $V = \\bigcup V_i$ and $A_i \\cong V_i \\times_V A$.", "This is the topology inherited from the topology on the site", "$(X/Y)_{Zar}$ whose underlying category is the category of pairs", "$(U, V)$ as in Lemma \\ref{lemma-category-fibred} part (3). The coverings", "of $(X/Y)_{Zar}$ are families $\\{(U_i, V_i) \\to (U, V)\\}$ such that", "$U = \\bigcup U_i$ and $V = \\bigcup V_i$. There are morphisms of topoi", "$$", "\\xymatrix{", "\\Sh(\\mathcal{C}_{X/Y})", "= \\Sh(\\mathcal{C}_{X/Y}, \\tau_X) &", "\\Sh(\\mathcal{C}_{X/Y}, \\tau_{X \\to Y}) \\ar[l] \\ar[r] &", "\\Sh(\\mathcal{C}_{X/Y}, \\tau_Y)", "}", "$$", "(recall that $\\tau_X$ is our ``default'' topology). The pullback functors", "for these arrows are sheafification and pushforward is the identity on", "underlying presheaves. The diagram of topoi", "$$", "\\xymatrix{", "\\Sh(X_{Zar}) \\ar[d]^f & \\Sh(\\mathcal{C}_{X/Y}) \\ar[l]^\\pi &", "\\Sh(\\mathcal{C}_{X/Y}, \\tau_{X \\to Y}) \\ar[l] \\ar[d] \\\\", "\\Sh(Y_{Zar}) & & \\Sh(\\mathcal{C}_{X/Y}, \\tau_Y) \\ar[ll]", "}", "$$", "is {\\bf not} commutative. Namely, the pullback of a nonzero abelian sheaf on", "$Y$ is a nonzero abelian sheaf on $(\\mathcal{C}_{X/Y}, \\tau_{X \\to Y})$,", "but we can certainly find examples where such a sheaf pulls back to zero", "on $X$. Note that any presheaf $\\mathcal{F}$ on", "$Y_{Zar}$ gives a sheaf $\\underline{\\mathcal{F}}$ on $\\mathcal{C}_{Y/X}$", "by the rule which assigns to $(U \\to A/V)$ the set $\\mathcal{F}(V)$.", "Even if $\\mathcal{F}$ happens to be a sheaf it isn't true in general that", "$\\underline{\\mathcal{F}} = \\pi^{-1}f^{-1}\\mathcal{F}$. This is related", "to the noncommutativity of the diagram above, as we can describe", "$\\underline{\\mathcal{F}}$ as the pushforward of the pullback", "of $\\mathcal{F}$ to $\\Sh(\\mathcal{C}_{X/Y}, \\tau_{X \\to Y})$ via", "the lower horizontal and right vertical arrows. An", "example is the sheaf $\\underline{\\mathcal{O}}_Y$.", "But what is true is that there is a map", "$\\underline{\\mathcal{F}} \\to \\pi^{-1}f^{-1}\\mathcal{F}$", "which is transformed (as we shall see later)", "into an isomorphism after applying $\\pi_!$." ], "refs": [ "obsolete-lemma-category-fibred" ], "ref_ids": [ 11495 ] }, { "id": 11560, "type": "other", "label": "obsolete-remark-construction-E", "categories": [ "obsolete" ], "title": "obsolete-remark-construction-E", "contents": [ "Let $S$ be a scheme. Let $f : X \\to B$ be a", "morphism of algebraic spaces over $S$. Let $U$ be another algebraic", "space over $B$. Denote $q : X \\times_B U \\to U$ the second projection.", "Consider the distinguished triangle", "$$", "Lq^*L_{U/B} \\to L_{X \\times_B U/B} \\to E \\to Lq^*L_{U/B}[1]", "$$", "of Cotangent, Section \\ref{cotangent-section-fibre-product}.", "For any sheaf $\\mathcal{F}$ of", "$\\mathcal{O}_{X \\times_B U}$-modules we have the Atiyah class", "$$", "\\mathcal{F} \\to", "L_{X \\times_B U/B}", "\\otimes_{\\mathcal{O}_{X \\times_B U}}^\\mathbf{L} \\mathcal{F}[1]", "$$", "see Cotangent, Section \\ref{cotangent-section-atiyah-general}.", "We can compose this with the map to $E$ and choose a distinguished", "triangle", "$$", "E(\\mathcal{F}) \\to \\mathcal{F} \\to", "\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_B U}}^\\mathbf{L} E[1] \\to", "E(\\mathcal{F})[1]", "$$", "in $D(\\mathcal{O}_{X \\times_B U})$.", "By construction the Atiyah class lifts to a map", "$$", "e_\\mathcal{F} :", "E(\\mathcal{F})", "\\longrightarrow", "Lq^*L_{U/B} \\otimes_{\\mathcal{O}_{X \\times_B U}}^\\mathbf{L} \\mathcal{F}[1]", "$$", "fitting into a morphism of distinguished triangles", "$$", "\\xymatrix{", "\\mathcal{F} \\otimes^\\mathbf{L} Lq^*L_{U/B}[1] \\ar[r] &", "\\mathcal{F} \\otimes^\\mathbf{L} L_{X \\times_B U/B}[1] \\ar[r] &", "\\mathcal{F} \\otimes^\\mathbf{L} E[1] \\\\", "E(\\mathcal{F}) \\ar[r] \\ar[u]^{e_\\mathcal{F}} &", "\\mathcal{F} \\ar[r] \\ar[u]^{Atiyah} &", "\\mathcal{F} \\otimes^\\mathbf{L} E[1] \\ar[u]^{=}", "}", "$$", "Given $S, B, X, f, U, \\mathcal{F}$ we fix a choice of $E(\\mathcal{F})$", "and $e_\\mathcal{F}$." ], "refs": [], "ref_ids": [] }, { "id": 11561, "type": "other", "label": "obsolete-remark-construction-ob", "categories": [ "obsolete" ], "title": "obsolete-remark-construction-ob", "contents": [ "With notation as in Remark \\ref{remark-construction-E} let $i : U \\to U'$ be a", "first order thickening of $U$ over $B$. Let", "$\\mathcal{I} \\subset \\mathcal{O}_{U'}$ be the quasi-coherent sheaf of", "ideals cutting out $B$ in $B'$. The fundamental triangle", "$$", "Li^*L_{U'/B} \\to L_{U/B} \\to L_{U/U'} \\to Li^*L_{U'/B}[1]", "$$", "together with the map $L_{U/U'} \\to \\mathcal{I}[1]$ determine a", "map $e_{U'} : L_{U/B} \\to \\mathcal{I}[1]$. Combined with the map", "$e_\\mathcal{F}$ of the previous remark we obtain", "$$", "(\\text{id}_\\mathcal{F} \\otimes Lq^*e_{U'}) \\cup e_\\mathcal{F} :", "E(\\mathcal{F})", "\\longrightarrow", "\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_B U}} q^*\\mathcal{I}[2]", "$$", "(we have also composed with the map from the derived tensor product to", "the usual tensor product). In other words, we obtain an element", "$$", "\\xi_{U'} \\in", "\\Ext^2_{\\mathcal{O}_{X \\times_B U}}(", "E(\\mathcal{F}),", "\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_B U}} q^*\\mathcal{I})", "$$" ], "refs": [ "obsolete-remark-construction-E" ], "ref_ids": [ 11560 ] }, { "id": 11562, "type": "other", "label": "obsolete-remark-not-true-not-quasi-compact", "categories": [ "obsolete" ], "title": "obsolete-remark-not-true-not-quasi-compact", "contents": [ "This remark used to say that it wasn't clear whether the arrows", "of Chow Homology, Lemma \\ref{chow-lemma-cycles-k-group} were isomorphisms", "in general. However, we've now found a proof of this fact." ], "refs": [ "chow-lemma-cycles-k-group" ], "ref_ids": [ 5699 ] }, { "id": 11563, "type": "other", "label": "obsolete-remark-tangent-spaces", "categories": [ "obsolete" ], "title": "obsolete-remark-tangent-spaces", "contents": [ "You got here because of a duplicate tag. Please see", "Formal Deformation Theory, Section \\ref{formal-defos-section-tangent-spaces}", "for the actual content." ], "refs": [], "ref_ids": [] }, { "id": 11564, "type": "other", "label": "obsolete-remark-examples-formal-defos", "categories": [ "obsolete" ], "title": "obsolete-remark-examples-formal-defos", "contents": [ "This tag used to point to a section describing several examples", "of deformation problems. These now each have their own section.", "See Deformation Problems, Sections", "\\ref{examples-defos-section-finite-projective-modules},", "\\ref{examples-defos-section-representations},", "\\ref{examples-defos-section-continuous-representations}, and", "\\ref{examples-defos-section-graded-algebras}." ], "refs": [], "ref_ids": [] }, { "id": 11631, "type": "other", "label": "stacks-sheaves-remark-ambiguity", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-remark-ambiguity", "contents": [ "We only use this notation when the symbol $\\mathcal{X}$ refers to a", "category fibred in groupoids, and not a scheme, an algebraic space, etc.", "In this way we will avoid confusion with the small \\'etale site of a", "scheme, or algebraic space which is denoted $X_\\etale$ (in which", "case we use a roman capital instead of a calligraphic one)." ], "refs": [], "ref_ids": [] }, { "id": 11632, "type": "other", "label": "stacks-sheaves-remark-flat", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-remark-flat", "contents": [ "In the situation of", "Lemma \\ref{lemma-functoriality-structure-sheaf}", "the morphism of ringed topoi", "$f : \\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$", "is flat as is clear from the equality", "$f^{-1}\\mathcal{O}_\\mathcal{X} = \\mathcal{O}_\\mathcal{Y}$.", "This is a bit counter intuitive, for example because a closed", "immersion of algebraic stacks is typically not flat (as a morphism of", "algebraic stacks).", "However, exactly the same thing happens when taking a closed", "immersion $i : X \\to Y$ of schemes: in this case the associated", "morphism of big $\\tau$-sites", "$i : (\\Sch/X)_\\tau \\to (\\Sch/Y)_\\tau$", "also is flat." ], "refs": [ "stacks-sheaves-lemma-functoriality-structure-sheaf" ], "ref_ids": [ 11571 ] }, { "id": 11633, "type": "other", "label": "stacks-sheaves-remark-cech-complex-presheaves", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-remark-cech-complex-presheaves", "contents": [ "We can define the complex $\\mathcal{K}^\\bullet(f, \\mathcal{F})$", "also if $\\mathcal{F}$ is a presheaf, only we cannot use the reference to", "Sites, Section \\ref{sites-section-pullback}", "to define the pullback maps. To explain the pullback maps, suppose", "given a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{V} \\ar[rd]_g \\ar[rr]_h & & \\mathcal{U} \\ar[ld]^f \\\\", "& \\mathcal{X}", "}", "$$", "of categories fibred in groupoids over $(\\Sch/S)_{fppf}$", "and a presheaf $\\mathcal{G}$ on $\\mathcal{U}$", "we can define the pullback map $f_*\\mathcal{G} \\to g_*h^{-1}\\mathcal{G}$", "as the composition", "$$", "f_*\\mathcal{G} \\longrightarrow", "f_*h_*h^{-1}\\mathcal{G} = g_*h^{-1}\\mathcal{G}", "$$", "where the map comes from the adjunction map", "$\\mathcal{G} \\to h_*h^{-1}\\mathcal{G}$. This works because in our situation", "the functors $h_*$ and $h^{-1}$ are adjoint in presheaves (and agree with", "their counter parts on sheaves). See", "Sections \\ref{section-presheaves} and \\ref{section-sheaves}." ], "refs": [], "ref_ids": [] }, { "id": 11634, "type": "other", "label": "stacks-sheaves-remark-cech-complex-sections", "categories": [ "stacks-sheaves" ], "title": "stacks-sheaves-remark-cech-complex-sections", "contents": [ "Let us ``compute'' the value of the relative {\\v C}ech complex on an", "object $x$ of $\\mathcal{X}$. Say $p(x) = U$.", "Consider the $2$-fibre product diagram (which serves to introduce", "the notation $g : \\mathcal{V} \\to \\mathcal{Y}$)", "$$", "\\xymatrix{", "\\mathcal{V} \\ar@{=}[r] \\ar[d]_g &", "(\\Sch/U)_{fppf} \\times_{x, \\mathcal{X}} \\mathcal{U} \\ar[r] \\ar[d] &", "\\mathcal{U} \\ar[d]^f \\\\", "\\mathcal{Y} \\ar@{=}[r] &", "(\\Sch/U)_{fppf} \\ar[r]^-x & \\mathcal{X}", "}", "$$", "Note that the morphism $\\mathcal{V}_n \\to \\mathcal{U}_n$ of the proof of", "Lemma \\ref{lemma-generalities}", "induces an equivalence", "$\\mathcal{V}_n =", "(\\Sch/U)_{fppf} \\times_{x, \\mathcal{X}} \\mathcal{U}_n$.", "Hence we see from", "(\\ref{equation-pushforward})", "that", "$$", "\\Gamma(x, \\mathcal{K}^\\bullet(f, \\mathcal{F})) =", "\\check{\\mathcal{C}}^\\bullet(\\mathcal{V} \\to \\mathcal{Y}, x^{-1}\\mathcal{F})", "$$", "In words: The value of the relative {\\v C}ech complex on an object $x$ of", "$\\mathcal{X}$ is the {\\v C}ech complex of the base change of $f$ to", "$\\mathcal{X}/x \\cong (\\Sch/U)_{fppf}$. This implies for example that", "Lemma \\ref{lemma-homotopy}", "implies", "Lemma \\ref{lemma-homotopy-sheafified}", "and more generally that results on the (usual) {\\v C}ech complex imply", "results for the relative {\\v C}ech complex." ], "refs": [ "stacks-sheaves-lemma-generalities", "stacks-sheaves-lemma-homotopy", "stacks-sheaves-lemma-homotopy-sheafified" ], "ref_ids": [ 11598, 11599, 11601 ] }, { "id": 11716, "type": "other", "label": "resolve-remark-compare-Garel", "categories": [ "resolve" ], "title": "resolve-remark-compare-Garel", "contents": [ "Let $\\mathbf{F}_p \\subset \\Lambda \\subset R \\subset S$ and $\\text{Tr}$", "be as in Lemma \\ref{lemma-trace-higher}. By", "de Rham Cohomology, Proposition \\ref{derham-proposition-Garel}", "there is a canonical map of complexes", "$$", "\\Theta_{S/R} :", "\\Omega_{S/\\Lambda}^\\bullet", "\\longrightarrow", "\\Omega_{R/\\Lambda}^\\bullet", "$$", "The computation in de Rham Cohomology, Example \\ref{derham-example-Garel}", "shows that $\\Theta_{S/R}(x^i \\text{d}x) = \\text{Tr}_x(x^i\\text{d}x)$", "for all $i$. Since $\\text{Trace}_{S/R} = \\Theta^0_{S/R}$", "is identically zero and since", "$$", "\\Theta_{S/R}(a \\wedge b) = a \\wedge \\Theta_{S/R}(b)", "$$", "for $a \\in \\Omega^i_{R/\\Lambda}$ and $b \\in \\Omega^j_{S/\\Lambda}$", "it follows that $\\text{Tr} = \\Theta_{S/R}$. The advantage of using $\\text{Tr}$", "is that it is a good deal more elementary to construct." ], "refs": [ "resolve-lemma-trace-higher", "derham-proposition-Garel" ], "ref_ids": [ 11637, 14374 ] }, { "id": 11717, "type": "other", "label": "resolve-remark-dualizing-setup", "categories": [ "resolve" ], "title": "resolve-remark-dualizing-setup", "contents": [ "Let $X$ be an integral Noetherian normal scheme of dimension $2$.", "In this case the following are equivalent", "\\begin{enumerate}", "\\item $X$ has a dualizing complex $\\omega_X^\\bullet$,", "\\item there is a coherent $\\mathcal{O}_X$-module $\\omega_X$ such that", "$\\omega_X[n]$ is a dualizing complex, where $n$ can be any integer.", "\\end{enumerate}", "This follows from the fact that $X$ is Cohen-Macaulay", "(Properties, Lemma \\ref{properties-lemma-normal-dimension-2-Cohen-Macaulay}) and", "Duality for Schemes, Lemma \\ref{duality-lemma-dualizing-module-CM-scheme}.", "In this situation we will say that $\\omega_X$ is a {\\it dualizing module}", "in accordance with", "Duality for Schemes, Section \\ref{duality-section-dualizing-module}.", "In particular, when $A$ is a Noetherian normal local domain of dimension", "$2$, then we say {\\it $A$ has a dualizing module $\\omega_A$}", "if the above is true. In this case, if $X \\to \\Spec(A)$ is a normal", "modification, then $X$ has a dualizing module too, see", "Duality for Schemes, Example \\ref{duality-example-proper-over-local}.", "In this situation we always denote $\\omega_X$ the dualizing", "module normalized with respect to $\\omega_A$, i.e., such that", "$\\omega_X[2]$ is the dualizing complex normalized relative to", "$\\omega_A[2]$. See Duality for Schemes, Section \\ref{duality-section-glue}." ], "refs": [ "properties-lemma-normal-dimension-2-Cohen-Macaulay", "duality-lemma-dualizing-module-CM-scheme" ], "ref_ids": [ 2991, 13586 ] }, { "id": 11718, "type": "other", "label": "resolve-remark-pic-blowup", "categories": [ "resolve" ], "title": "resolve-remark-pic-blowup", "contents": [ "Let $b : X \\to X'$ be the contraction of an", "exceptional curve of the first kind $E \\subset X$.", "From Lemma \\ref{lemma-pic-blowup} we obtain an identification", "$$", "\\Pic(X) = \\Pic(X') \\oplus \\mathbf{Z}", "$$", "where $\\mathcal{L}$ corresponds to the pair $(\\mathcal{L}', n)$ if and only if", "$\\mathcal{L} = (b^*\\mathcal{L}')(-nE)$, i.e.,", "$\\mathcal{L}(nE) = b^*\\mathcal{L}'$. In fact the proof of", "Lemma \\ref{lemma-pic-blowup} shows that $\\mathcal{L}' = b_*\\mathcal{L}(nE)$.", "Of course the assignment $\\mathcal{L} \\mapsto \\mathcal{L}'$ is", "a group homomorphism." ], "refs": [ "resolve-lemma-pic-blowup", "resolve-lemma-pic-blowup" ], "ref_ids": [ 11703, 11703 ] }, { "id": 11758, "type": "other", "label": "exercises-remark-simple-geometric", "categories": [ "exercises" ], "title": "exercises-remark-simple-geometric", "contents": [ "Of course the idea of this exercise is to find", "a simple argument in each case rather than applying a ``big'' theorem.", "Nonetheless it is good to be guided by general principles." ], "refs": [], "ref_ids": [] }, { "id": 11759, "type": "other", "label": "exercises-remark-flat-not-projective", "categories": [ "exercises" ], "title": "exercises-remark-flat-not-projective", "contents": [ "If $M$ is of finite presentation and flat over $A$,", "then $M$ is projective over $A$. Thus your example will have to", "involve a ring $A$ which is not Noetherian. I know of an example", "where $A$ is the ring of ${\\mathcal C}^\\infty$-functions on ${\\mathbf R}$." ], "refs": [], "ref_ids": [] }, { "id": 11760, "type": "other", "label": "exercises-remark-flat-given-residue-field-extension-general", "categories": [ "exercises" ], "title": "exercises-remark-flat-given-residue-field-extension-general", "contents": [ "The same result holds for arbitrary field extensions $k \\subset K$." ], "refs": [], "ref_ids": [] }, { "id": 11761, "type": "other", "label": "exercises-remark-not-hausdorff", "categories": [ "exercises" ], "title": "exercises-remark-not-hausdorff", "contents": [ "Usually the word compact is reserved for quasi-compact and", "Hausdorff spaces." ], "refs": [], "ref_ids": [] }, { "id": 11762, "type": "other", "label": "exercises-remark-singularities", "categories": [ "exercises" ], "title": "exercises-remark-singularities", "contents": [ "A singularity on a curve over a field $k$ is called an", "ordinary double point if the complete local ring of the curve at the", "point is of the form $k'[[x, y]]/(f)$, where (a) $k'$ is a finite separable", "extension of $k$, (b) the initial term of $f$ has degree two, i.e., it", "looks like $q = ax^2 + bxy + cy^2$ for some $a, b, c\\in k'$ not all zero, and", "(c) $q$ is a nondegenerate quadratic form over $k'$ (in char 2 this means that", "$b$ is not zero). In general there is one isomorphism class of such rings for", "each isomorphism class of pairs $(k', q)$." ], "refs": [], "ref_ids": [] }, { "id": 11763, "type": "other", "label": "exercises-remark-HNSS", "categories": [ "exercises" ], "title": "exercises-remark-HNSS", "contents": [ "Let $k$ be a field. Then for every integer $n\\in {\\mathbf N}$ and", "every maximal ideal ${\\mathfrak m} \\subset k[x_1, \\ldots, x_n]$", "the quotient $k[x_1, \\ldots, x_n]/{\\mathfrak m}$ is a finite field", "extension of $k$. This will be shown later in the course. Of course", "(please check this) it implies a similar statement for maximal ideals", "of finitely generated $k$-algebras. The exercise above proves", "it in the case $k = {\\mathbf C}$." ], "refs": [], "ref_ids": [] }, { "id": 11764, "type": "other", "label": "exercises-remark-Hilbert-Nullstellensatz", "categories": [ "exercises" ], "title": "exercises-remark-Hilbert-Nullstellensatz", "contents": [ "This is the Hilbert Nullstellensatz. Namely it says", "that the closed subsets of $\\Spec(k[x_1, \\ldots, x_n])$", "(which correspond to radical ideals by a previous exercise)", "are determined by the closed points contained in them." ], "refs": [], "ref_ids": [] }, { "id": 11765, "type": "other", "label": "exercises-remark-cover", "categories": [ "exercises" ], "title": "exercises-remark-cover", "contents": [ "In algebraic geometric language this means that the property", "of ``being finitely generated'' or ``being flat'' is local for the Zariski", "topology (in a suitable sense). You can also show this for the property", "``being of finite presentation''." ], "refs": [], "ref_ids": [] }, { "id": 11766, "type": "other", "label": "exercises-remark-elimination-theory", "categories": [ "exercises" ], "title": "exercises-remark-elimination-theory", "contents": [ "Finding the image as above usually is done by using elimination theory." ], "refs": [], "ref_ids": [] }, { "id": 11767, "type": "other", "label": "exercises-remark-continuous-proj-spec", "categories": [ "exercises" ], "title": "exercises-remark-continuous-proj-spec", "contents": [ "There is a continuous map $ \\text{Proj}(R) \\longrightarrow \\Spec(R_0) $." ], "refs": [], "ref_ids": [] }, { "id": 11768, "type": "other", "label": "exercises-remark-CM-dim-1-embedding-dim-2", "categories": [ "exercises" ], "title": "exercises-remark-CM-dim-1-embedding-dim-2", "contents": [ "This suggests that a local Noetherian Cohen-Macaulay ring of dimension 1", "and embedding dimension 2 is of the form $B/FB$, where $B$ is a 2-dimensional", "regular local ring. This is more or less true (under suitable ``niceness''", "properties of the ring)." ], "refs": [], "ref_ids": [] }, { "id": 11769, "type": "other", "label": "exercises-remark-strange-fp", "categories": [ "exercises" ], "title": "exercises-remark-strange-fp", "contents": [ "Let $h \\in {\\mathbf Z}[y]$ be a monic polynomial of degree $d$.", "Then:", "\\begin{enumerate}", "\\item The map $A = {\\mathbf Z}[x] \\to B ={\\mathbf Z}[y]$,", "$x \\mapsto h$ is finite locally free of rank $d$.", "\\item For all primes $p$ the map", "$A_p = {\\mathbf F}_p[x]\\to B_p = {\\mathbf F}_p[y]$,", "$y \\mapsto h(y) \\bmod p$ is finite locally free of rank $d$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11770, "type": "other", "label": "exercises-remark-direct-sum-stalk-abelian", "categories": [ "exercises" ], "title": "exercises-remark-direct-sum-stalk-abelian", "contents": [ "Let $X$ be a topological space.", "In the category of abelian sheaves the direct sum of", "a family of sheaves $\\{{\\mathcal F}_i\\}_{i\\in I}$ is the sheaf associated to", "the presheaf $U \\mapsto \\oplus {\\mathcal F}_i(U)$. Consequently the stalk of", "the direct sum at a point $x$ is the direct sum of the stalks of the", "${\\mathcal F}_i$ at $x$." ], "refs": [], "ref_ids": [] }, { "id": 11771, "type": "other", "label": "exercises-remark-open-immersion", "categories": [ "exercises" ], "title": "exercises-remark-open-immersion", "contents": [ "When $(X, {\\mathcal O}_X)$ is a ringed space and $U \\subset X$", "is an open subset then $(U, {\\mathcal O}_X|_U)$ is a ringed space. Notation:", "${\\mathcal O}_U = {\\mathcal O}_X|_U$. There is a canonical morphisms", "of ringed spaces", "$$", "j : (U, {\\mathcal O}_U) \\longrightarrow (X, {\\mathcal O}_X).", "$$", "If $(X, {\\mathcal O}_X)$ is a locally ringed space, so is", "$(U, {\\mathcal O}_U)$ and", "$j$ is a morphism of locally ringed spaces. If $(X, {\\mathcal O}_X)$", "is a scheme", "so is $(U, {\\mathcal O}_U)$ and $j$ is a morphism of schemes. We say", "that", "$(U, {\\mathcal O}_U)$ is an {\\it open subscheme} of $(X, {\\mathcal O}_X)$", "and that", "$j$ is an {\\it open immersion}. More generally, any morphism", "$j' : (V, {\\mathcal O}_V) \\to (X, {\\mathcal O}_X)$ that is {\\it isomorphic}", "to a", "morphism $j : (U, {\\mathcal O}_U) \\to (X, {\\mathcal O}_X)$ as above is", "called an", "open immersion." ], "refs": [], "ref_ids": [] }, { "id": 11772, "type": "other", "label": "exercises-remark-separated-base-fibre-product-affines-affine", "categories": [ "exercises" ], "title": "exercises-remark-separated-base-fibre-product-affines-affine", "contents": [ "It turns out this cannot happen with $S$ separated. Do you know why?" ], "refs": [], "ref_ids": [] }, { "id": 11773, "type": "other", "label": "exercises-remark-affine-dimension", "categories": [ "exercises" ], "title": "exercises-remark-affine-dimension", "contents": [ "If your scheme is affine then dimension is the", "same as the Krull dimension of the underlying ring. So you can", "use last semesters results to compute dimension." ], "refs": [], "ref_ids": [] }, { "id": 11774, "type": "other", "label": "exercises-remark-tsen", "categories": [ "exercises" ], "title": "exercises-remark-tsen", "contents": [ "Exercise \\ref{exercise-has-rational-section}", "is a special case of ``Tsen's theorem''.", "Exercise \\ref{exercise-no-section-curve} shows that the", "method is limited to low degree equations (conics when the base and", "fibre have dimension 1)." ], "refs": [], "ref_ids": [] }, { "id": 11775, "type": "other", "label": "exercises-remark-interpretation-skolem-noether", "categories": [ "exercises" ], "title": "exercises-remark-interpretation-skolem-noether", "contents": [ "The interpretation of the results of", "Exercise \\ref{exercise-for-number-theorists} and \\ref{exercise-quasi-section}", "is that given the morphism $X \\to S$ all of whose fibres are nonempty,", "there exists a finite surjective morphism $S' \\to S$ such that", "the base change $X_{S'} \\to S'$ does have a section.", "This is not a general fact, but it holds if the base is the spectrum of a", "dedekind ring with finite residue fields at closed points, and", "the morphism $X \\to S$ is flat with geometrically irreducible generic", "fibre. See Exercise \\ref{exercise-no-quasi-section} below for an example", "where it doesn't work." ], "refs": [], "ref_ids": [] }, { "id": 11776, "type": "other", "label": "exercises-remark-tangent-space-relative", "categories": [ "exercises" ], "title": "exercises-remark-tangent-space-relative", "contents": [ "Exercise \\ref{exercise-compute-TS} explains why it is necessary", "to consider the tangent space of $X$ over $S$ to get a good notion." ], "refs": [], "ref_ids": [] }, { "id": 11777, "type": "other", "label": "exercises-remark-extend-off-open", "categories": [ "exercises" ], "title": "exercises-remark-extend-off-open", "contents": [ "If $U \\to X$ is a quasi-compact immersion then any", "quasi-coherent sheaf on $U$ is the restriction of a", "quasi-coherent sheaf on $X$.", "If $X$ is a Noetherian scheme, and $U \\subset X$ is open,", "then any coherent sheaf on $U$ is the restriction of a", "coherent sheaf on $X$.", "Of course the exercise above is easier, and shouldn't use these general facts." ], "refs": [], "ref_ids": [] }, { "id": 11778, "type": "other", "label": "exercises-remark-fitting-omega-not-sings", "categories": [ "exercises" ], "title": "exercises-remark-fitting-omega-not-sings", "contents": [ "The $k$th Fitting ideal of $\\Omega_{X/S}$ is commonly used", "to define the singular scheme of the morphism $X \\to S$ when $X$ has relative", "dimension $k$ over $S$. But as part (a) shows, you have to be careful doing", "this when your family does not have ``constant'' fibre dimension, e.g., when", "it is not flat. As part (b) shows, flatness doesn't guarantee it works either", "(and yes this is a flat family). In ``good cases'' -- such as in (c) -- for", "families of curves you expect the $0$-th Fitting ideal to be zero and", "the $1$st Fitting ideal to define (scheme-theoretically) the singular locus." ], "refs": [], "ref_ids": [] }, { "id": 11779, "type": "other", "label": "exercises-remark-invertible-projective-space", "categories": [ "exercises" ], "title": "exercises-remark-invertible-projective-space", "contents": [ "Let $k$ be a field.", "Let $\\mathbf{P}^2_k = \\text{Proj}(k[X_0, X_1, X_2])$.", "Any invertible sheaf on $\\mathbf{P}^2_k$ is isomorphic to", "$\\mathcal{O}_{\\mathbf{P}^2_k}(n)$ for some $n \\in \\mathbf{Z}$.", "Recall that", "$$", "\\Gamma(\\mathbf{P}^2_k, \\mathcal{O}_{\\mathbf{P}^2_k}(n)) =", "k[X_0, X_1, X_2]_n", "$$", "is the degree $n$ part of the polynomial ring.", "For a quasi-coherent sheaf $\\mathcal{F}$ on $\\mathbf{P}^2_k$ set", "$\\mathcal{F}(n) =", "\\mathcal{F}", "\\otimes_{\\mathcal{O}_{\\mathbf{P}^2_k}}", "\\mathcal{O}_{\\mathbf{P}^2_k}(n)$", "as usual." ], "refs": [], "ref_ids": [] }, { "id": 11780, "type": "other", "label": "exercises-remark-recall-dimension-theory", "categories": [ "exercises" ], "title": "exercises-remark-recall-dimension-theory", "contents": [ "Freely use the following facts on dimension theory", "(and add more if you need more).", "\\begin{enumerate}", "\\item The dimension of a scheme is the supremum of the length of chains", "of irreducible closed subsets.", "\\item The dimension of a finite type scheme over a field is the maximum", "of the dimensions of its affine opens.", "\\item The dimension of a Noetherian scheme is the maximum of the dimensions", "of its irreducible components.", "\\item The dimension of an affine scheme", "coincides with the dimension of the corresponding ring.", "\\item Let $k$ be a field and let $A$ be a finite type $k$-algebra.", "If $A$ is a domain, and $x \\not = 0$, then $\\dim(A) = \\dim(A/xA) + 1$.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11781, "type": "other", "label": "exercises-remark-chi", "categories": [ "exercises" ], "title": "exercises-remark-chi", "contents": [ "Given a projective scheme $X$ over a field $k$ and", "a coherent sheaf $\\mathcal{F}$ on $X$ we set", "$$", "\\chi(X, \\mathcal{F}) =", "\\sum\\nolimits_{i \\geq 0} (-1)^i\\dim_k H^i(X, \\mathcal{F}).", "$$" ], "refs": [], "ref_ids": [] }, { "id": 11782, "type": "other", "label": "exercises-remarks-divisors", "categories": [ "exercises" ], "title": "exercises-remarks-divisors", "contents": [ "Here are some trivial remarks.", "\\begin{enumerate}", "\\item On a Noetherian integral scheme $X$ the", "sheaf ${\\mathcal K}_X$ is constant with value the function field $K(X)$.", "\\item To make sense out of the definitions above one needs", "to show that", "$$", "\\text{length}_{\\mathcal O}({\\mathcal O}/(ab)) =", "\\text{length}_{\\mathcal O}({\\mathcal O}/(a)) +", "\\text{length}_{\\mathcal O}({\\mathcal O}/(b))", "$$", "for any pair $(a, b)$ of nonzero elements of a Noetherian 1-dimensional", "local domain ${\\mathcal O}$. This will be done in the lectures.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 11811, "type": "other", "label": "spaces-duality-remark-iso-on-RSheafHom", "categories": [ "spaces-duality" ], "title": "spaces-duality-remark-iso-on-RSheafHom", "contents": [ "In the situation of Lemma \\ref{lemma-iso-on-RSheafHom} we have", "$$", "DQ_Y(Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K))) =", "Rf_* DQ_X(R\\SheafHom_{\\mathcal{O}_X}(L, a(K)))", "$$", "by Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-pushforward-better-coherator}.", "Thus if $R\\SheafHom_{\\mathcal{O}_X}(L, a(K)) \\in D_\\QCoh(\\mathcal{O}_X)$,", "then we can ``erase'' the $DQ_Y$ on the left hand side of the arrow.", "On the other hand, if we know that", "$R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K) \\in D_\\QCoh(\\mathcal{O}_Y)$,", "then we can ``erase'' the $DQ_Y$ from the right hand side of the arrow.", "If both are true then we see that (\\ref{equation-sheafy-trace})", "is an isomorphism. Combining this with", "Derived Categories of Spaces, Lemma", "\\ref{spaces-perfect-lemma-quasi-coherence-internal-hom}", "we see that $Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K)) \\to", "R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K)$ is an isomorphism if", "\\begin{enumerate}", "\\item $L$ and $Rf_*L$ are perfect, or", "\\item $K$ is bounded below and $L$ and $Rf_*L$ are pseudo-coherent.", "\\end{enumerate}", "For (2) we use that $a(K)$ is bounded below if $K$", "is bounded below, see Lemma \\ref{lemma-twisted-inverse-image-bounded-below}." ], "refs": [ "spaces-duality-lemma-iso-on-RSheafHom", "spaces-perfect-lemma-pushforward-better-coherator", "spaces-perfect-lemma-quasi-coherence-internal-hom", "spaces-duality-lemma-twisted-inverse-image-bounded-below" ], "ref_ids": [ 11790, 2716, 2700, 11789 ] }, { "id": 11812, "type": "other", "label": "spaces-duality-remark-going-around", "categories": [ "spaces-duality" ], "title": "spaces-duality-remark-going-around", "contents": [ "Let $S$ be a scheme. Consider a commutative diagram", "$$", "\\xymatrix{", "X'' \\ar[r]_{k'} \\ar[d]_{f''} & X' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y'' \\ar[r]^{l'} \\ar[d]_{g''} & Y' \\ar[r]^l \\ar[d]_{g'} & Y \\ar[d]^g \\\\", "Z'' \\ar[r]^{m'} & Z' \\ar[r]^m & Z", "}", "$$", "of quasi-compact and quasi-separated algebraic spaces over $S$ where", "all squares are cartesian and where", "$(f, l)$, $(g, m)$, $(f', l')$, $(g', m')$ are", "Tor independent pairs of maps.", "Let $a$, $a'$, $a''$, $b$, $b'$, $b''$ be the", "right adjoints of Lemma \\ref{lemma-twisted-inverse-image}", "for $f$, $f'$, $f''$, $g$, $g'$, $g''$.", "Let us label the squares of the diagram $A$, $B$, $C$, $D$", "as follows", "$$", "\\begin{matrix}", "A & B \\\\", "C & D", "\\end{matrix}", "$$", "Then the maps (\\ref{equation-base-change-map})", "for the squares are (where we use $k^* = Lk^*$, etc)", "$$", "\\begin{matrix}", "\\gamma_A : (k')^* \\circ a' \\to a'' \\circ (l')^* &", "\\gamma_B : k^* \\circ a \\to a' \\circ l^* \\\\", "\\gamma_C : (l')^* \\circ b' \\to b'' \\circ (m')^* &", "\\gamma_D : l^* \\circ b \\to b' \\circ m^*", "\\end{matrix}", "$$", "For the $2 \\times 1$ and $1 \\times 2$ rectangles we have four further", "base change maps", "$$", "\\begin{matrix}", "\\gamma_{A + B} : (k \\circ k')^* \\circ a \\to a'' \\circ (l \\circ l')^* \\\\", "\\gamma_{C + D} : (l \\circ l')^* \\circ b \\to b'' \\circ (m \\circ m')^* \\\\", "\\gamma_{A + C} : (k')^* \\circ (a' \\circ b') \\to (a'' \\circ b'') \\circ (m')^* \\\\", "\\gamma_{A + C} : k^* \\circ (a \\circ b) \\to (a' \\circ b') \\circ m^*", "\\end{matrix}", "$$", "By Lemma \\ref{lemma-compose-base-change-maps-horizontal} we have", "$$", "\\gamma_{A + B} = \\gamma_A \\circ \\gamma_B, \\quad", "\\gamma_{C + D} = \\gamma_C \\circ \\gamma_D", "$$", "and by Lemma \\ref{lemma-compose-base-change-maps} we have", "$$", "\\gamma_{A + C} = \\gamma_C \\circ \\gamma_A, \\quad", "\\gamma_{B + D} = \\gamma_D \\circ \\gamma_B", "$$", "Here it would be more correct to write", "$\\gamma_{A + B} = (\\gamma_A \\star \\text{id}_{l^*}) \\circ", "(\\text{id}_{(k')^*} \\star \\gamma_B)$ with notation as in", "Categories, Section \\ref{categories-section-formal-cat-cat}", "and similarly for the others. However, we continue the", "abuse of notation used in the proofs of", "Lemmas \\ref{lemma-compose-base-change-maps} and", "\\ref{lemma-compose-base-change-maps-horizontal}", "of dropping $\\star$ products with identities as one can figure", "out which ones to add as long as the source and target of the", "transformation is known.", "Having said all of this we find (a priori) two transformations", "$$", "(k')^* \\circ k^* \\circ a \\circ b", "\\longrightarrow", "a'' \\circ b'' \\circ (m')^* \\circ m^*", "$$", "namely", "$$", "\\gamma_C \\circ \\gamma_A \\circ \\gamma_D \\circ \\gamma_B =", "\\gamma_{A + C} \\circ \\gamma_{B + D}", "$$", "and", "$$", "\\gamma_C \\circ \\gamma_D \\circ \\gamma_A \\circ \\gamma_B =", "\\gamma_{C + D} \\circ \\gamma_{A + B}", "$$", "The point of this remark is to point out that these transformations", "are equal. Namely, to see this it suffices to show that", "$$", "\\xymatrix{", "(k')^* \\circ a' \\circ l^* \\circ b \\ar[r]_{\\gamma_D} \\ar[d]_{\\gamma_A} &", "(k')^* \\circ a' \\circ b' \\circ m^* \\ar[d]^{\\gamma_A} \\\\", "a'' \\circ (l')^* \\circ l^* \\circ b \\ar[r]^{\\gamma_D} &", "a'' \\circ (l')^* \\circ b' \\circ m^*", "}", "$$", "commutes. This is true by", "Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats}", "or more simply the discussion preceding", "Categories, Definition \\ref{categories-definition-horizontal-composition}." ], "refs": [ "spaces-duality-lemma-twisted-inverse-image", "spaces-duality-lemma-compose-base-change-maps-horizontal", "spaces-duality-lemma-compose-base-change-maps", "spaces-duality-lemma-compose-base-change-maps", "spaces-duality-lemma-compose-base-change-maps-horizontal", "categories-lemma-properties-2-cat-cats", "categories-definition-horizontal-composition" ], "ref_ids": [ 11788, 11794, 11793, 11793, 11794, 12269, 12377 ] }, { "id": 11950, "type": "other", "label": "spaces-properties-remark-list-properties-local-etale-topology", "categories": [ "spaces-properties" ], "title": "spaces-properties-remark-list-properties-local-etale-topology", "contents": [ "Here is a list of properties which are local for the \\'etale topology", "(keep in mind that the fpqc, fppf, syntomic, and smooth topologies are", "stronger than the \\'etale topology):", "\\begin{enumerate}", "\\item locally Noetherian, see", "Descent, Lemma \\ref{descent-lemma-Noetherian-local-fppf},", "\\item Jacobson, see", "Descent, Lemma \\ref{descent-lemma-Jacobson-local-fppf},", "\\item locally Noetherian and $(S_k)$, see", "Descent, Lemma \\ref{descent-lemma-Sk-local-syntomic},", "\\item Cohen-Macaulay, see", "Descent, Lemma \\ref{descent-lemma-CM-local-syntomic},", "\\item Gorenstein, see", "Duality for Schemes, Lemma \\ref{duality-lemma-gorenstein-local-syntomic},", "\\item reduced, see", "Descent, Lemma \\ref{descent-lemma-reduced-local-smooth},", "\\item normal, see", "Descent, Lemma \\ref{descent-lemma-normal-local-smooth},", "\\item locally Noetherian and $(R_k)$, see", "Descent, Lemma \\ref{descent-lemma-Rk-local-smooth},", "\\item regular, see", "Descent, Lemma \\ref{descent-lemma-regular-local-smooth},", "\\item Nagata, see", "Descent, Lemma \\ref{descent-lemma-Nagata-local-smooth}.", "\\end{enumerate}" ], "refs": [ "descent-lemma-Noetherian-local-fppf", "descent-lemma-Jacobson-local-fppf", "descent-lemma-Sk-local-syntomic", "descent-lemma-CM-local-syntomic", "duality-lemma-gorenstein-local-syntomic", "descent-lemma-reduced-local-smooth", "descent-lemma-normal-local-smooth", "descent-lemma-Rk-local-smooth", "descent-lemma-regular-local-smooth", "descent-lemma-Nagata-local-smooth" ], "ref_ids": [ 14648, 14649, 14651, 14652, 13593, 14653, 14654, 14655, 14656, 14657 ] }, { "id": 11951, "type": "other", "label": "spaces-properties-remark-list-properties-local-ring-local-etale-topology", "categories": [ "spaces-properties" ], "title": "spaces-properties-remark-list-properties-local-ring-local-etale-topology", "contents": [ "Let $P$ be a property of local rings. Assume that for any", "\\'etale ring map $A \\to B$ and $\\mathfrak q$ is a prime of $B$ lying over", "the prime $\\mathfrak p$ of $A$, then", "$P(A_\\mathfrak p) \\Leftrightarrow P(B_\\mathfrak q)$.", "Then we obtain an \\'etale local property of germs $(U, u)$ of schemes", "by setting $\\mathcal{P}(U, u) = P(\\mathcal{O}_{U, u})$.", "In this situation we will use the terminology", "``the local ring of $X$ at $x$ has $P$'' to mean", "$X$ has property $\\mathcal{P}$ at $x$.", "Here is a list of such properties $P$:", "\\begin{enumerate}", "\\item Noetherian, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-Noetherian-etale-extension},", "\\item dimension $d$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-dimension-etale-extension},", "\\item regular, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-regular-etale-extension},", "\\item discrete valuation ring, follows from (2), (3), and", "Algebra, Lemma \\ref{algebra-lemma-characterize-dvr},", "\\item reduced, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-reduced},", "\\item normal, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-normal},", "\\item Noetherian and depth $k$, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-depth},", "\\item Noetherian and Cohen-Macaulay, see", "More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-CM},", "\\item Noetherian and Gorenstein, see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-flat-under-gorenstein}.", "\\end{enumerate}", "There are more properties for which this holds, for example G-ring and", "Nagata. If we every need these we will add them here", "as well as references to detailed proofs of the corresponding", "algebra facts." ], "refs": [ "more-algebra-lemma-Noetherian-etale-extension", "more-algebra-lemma-dimension-etale-extension", "more-algebra-lemma-regular-etale-extension", "algebra-lemma-characterize-dvr", "more-algebra-lemma-henselization-reduced", "more-algebra-lemma-henselization-normal", "more-algebra-lemma-henselization-depth", "more-algebra-lemma-henselization-CM", "dualizing-lemma-flat-under-gorenstein" ], "ref_ids": [ 10051, 10052, 10053, 1023, 10058, 10060, 10062, 10063, 2885 ] }, { "id": 11952, "type": "other", "label": "spaces-properties-remark-cannot-decide-yet", "categories": [ "spaces-properties" ], "title": "spaces-properties-remark-cannot-decide-yet", "contents": [ "Lemma \\ref{lemma-point-like-spaces} holds for decent algebraic spaces, see", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-point-like-spaces}.", "In fact a decent algebraic space with one point is a scheme, see", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-when-field}.", "This also holds when $X$ is locally separated, because a", "locally separated algebraic space is decent, see", "Decent Spaces, Lemma \\ref{decent-spaces-lemma-locally-separated-decent}." ], "refs": [ "spaces-properties-lemma-point-like-spaces", "decent-spaces-lemma-decent-point-like-spaces", "decent-spaces-lemma-when-field", "decent-spaces-lemma-locally-separated-decent" ], "ref_ids": [ 11854, 9499, 9507, 9512 ] }, { "id": 11953, "type": "other", "label": "spaces-properties-remark-explain-equivalence", "categories": [ "spaces-properties" ], "title": "spaces-properties-remark-explain-equivalence", "contents": [ "Let us explain the meaning of Lemma \\ref{lemma-compare-etale-sites}.", "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.", "Let $\\mathcal{F}$ be a sheaf on the small \\'etale site $X_\\etale$ of", "$X$. The lemma says that there exists a unique sheaf $\\mathcal{F}'$ on", "$X_{spaces, \\etale}$ which restricts back to $\\mathcal{F}$ on the", "subcategory $X_\\etale$. If $U \\to X$ is an \\'etale morphism of", "algebraic spaces, then how do we compute $\\mathcal{F}'(U)$? Well, by definition", "of an algebraic space there exists a scheme $U'$ and a surjective", "\\'etale morphism $U' \\to U$. Then $\\{U' \\to U\\}$ is a covering in", "$X_{spaces, \\etale}$ and hence we get an equalizer diagram", "$$", "\\xymatrix{", "\\mathcal{F}'(U) \\ar[r] &", "\\mathcal{F}(U') \\ar@<1ex>[r] \\ar@<-1ex>[r] &", "\\mathcal{F}(U' \\times_U U').", "}", "$$", "Note that $U' \\times_U U'$ is a scheme, and hence we may", "write $\\mathcal{F}$ and not $\\mathcal{F}'$.", "Thus we see how to compute $\\mathcal{F}'$", "when given the sheaf $\\mathcal{F}$." ], "refs": [ "spaces-properties-lemma-compare-etale-sites" ], "ref_ids": [ 11862 ] }, { "id": 11954, "type": "other", "label": "spaces-properties-remark-stalk-pullback", "categories": [ "spaces-properties" ], "title": "spaces-properties-remark-stalk-pullback", "contents": [ "This remark is the analogue of", "\\'Etale Cohomology, Remark \\ref{etale-cohomology-remark-stalk-pullback}.", "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $\\overline{x} : \\Spec(k) \\to X$ be a geometric point of $X$.", "By", "\\'Etale Cohomology,", "Theorem \\ref{etale-cohomology-theorem-equivalence-sheaves-point}", "the category of sheaves on $\\Spec(k)_\\etale$ is", "equivalent to the category of sets (by taking a sheaf to its global sections).", "Hence it follows from", "Lemma \\ref{lemma-stalk-pullback} part (4)", "applied to the morphism $\\overline{x}$ that the functor", "$$", "\\Sh(X_\\etale) \\longrightarrow \\textit{Sets}, \\quad", "\\mathcal{F} \\longmapsto \\mathcal{F}_{\\overline{x}}", "$$", "is isomorphic to the functor", "$$", "\\Sh(X_\\etale)", "\\longrightarrow", "\\Sh(\\Spec(k)_\\etale) = \\textit{Sets},", "\\quad", "\\mathcal{F} \\longmapsto \\overline{x}^*\\mathcal{F}", "$$", "Hence we may view the stalk functors as pullback functors along", "geometric morphisms (and not just some abstract morphisms of topoi", "as in the result of", "Lemma \\ref{lemma-stalk-gives-point})." ], "refs": [ "etale-cohomology-remark-stalk-pullback", "etale-cohomology-theorem-equivalence-sheaves-point", "spaces-properties-lemma-stalk-pullback", "spaces-properties-lemma-stalk-gives-point" ], "ref_ids": [ 6788, 6386, 11875, 11873 ] }, { "id": 11955, "type": "other", "label": "spaces-properties-remark-map-stalks", "categories": [ "spaces-properties" ], "title": "spaces-properties-remark-map-stalks", "contents": [ "Let $S$ be a scheme.", "Let $X$ be an algebraic space over $S$.", "Let $x \\in |X|$.", "We claim that for any pair of geometric points $\\overline{x}$ and", "$\\overline{x}'$ lying over $x$ the stalk functors are isomorphic.", "By definition of $|X|$ we can find a third geometric point", "$\\overline{x}''$ so that there exists a commutative diagram", "$$", "\\xymatrix{", "\\overline{x}'' \\ar[r] \\ar[d] \\ar[rd]^{\\overline{x}''} &", "\\overline{x}' \\ar[d]^{\\overline{x}'} \\\\", "\\overline{x} \\ar[r]^{\\overline{x}} & X.", "}", "$$", "Since the stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$", "is given by pullback along the morphism $\\overline{x}$ (and similarly for", "the others) we conclude by functoriality of pullbacks." ], "refs": [], "ref_ids": [] }, { "id": 12005, "type": "other", "label": "intersection-remark-trivial-generalization", "categories": [ "intersection" ], "title": "intersection-remark-trivial-generalization", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.", "Let $M$ be a finite $A$-module. Let $I \\subset A$ be an ideal.", "The following are equivalent", "\\begin{enumerate}", "\\item $I' = I + \\text{Ann}(M)$ is an ideal of definition", "(Algebra, Definition \\ref{algebra-definition-ideal-definition}),", "\\item the image $\\overline{I}$ of $I$ in $\\overline{A} = A/\\text{Ann}(M)$", "is an ideal of definition,", "\\item $\\text{Supp}(M/IM) \\subset \\{\\mathfrak m\\}$,", "\\item $\\dim(\\text{Supp}(M/IM)) \\leq 0$, and", "\\item $\\text{length}_A(M/IM) < \\infty$.", "\\end{enumerate}", "This follows from Algebra, Lemma \\ref{algebra-lemma-support-point}", "(details omitted). If this is the case we have $M/I^nM = M/(I')^nM$", "for all $n$ and $M/I^nM = M/\\overline{I}^nM$ for all $n$", "if $M$ is viewed as an $\\overline{A}$-module.", "Thus we can define", "$$", "\\chi_{I, M}(n) = \\text{length}_A(M/I^nM) =", "\\sum\\nolimits_{p = 0, \\ldots, n - 1} \\text{length}_A(I^pM/I^{p + 1}M)", "$$", "and we get", "$$", "\\chi_{I, M}(n) = \\chi_{I', M}(n) = \\chi_{\\overline{I}, M}(n)", "$$", "for all $n$ by the equalities above.", "All the results of Algebra, Section \\ref{algebra-section-Noetherian-local}", "and all the results in this section, have analogues in this setting.", "In particular we can define multiplicities $e_I(M, d)$ for", "$d \\geq \\dim(\\text{Supp}(M))$ and we have", "$$", "\\chi_{I, M}(n) \\sim e_I(M, d) \\frac{n^d}{d!} + \\text{lower order terms}", "$$", "as in the case where $I$ is an ideal of definition." ], "refs": [ "algebra-definition-ideal-definition", "algebra-lemma-support-point" ], "ref_ids": [ 1475, 693 ] }, { "id": 12006, "type": "other", "label": "intersection-remark-Serre-conjectures", "categories": [ "intersection" ], "title": "intersection-remark-Serre-conjectures", "contents": [ "Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring.", "Let $M$ and $N$ be nonzero finite $A$-modules such that $M \\otimes_A N$", "is supported in $\\{\\mathfrak m\\}$. Then", "$$", "\\chi(M, N) = \\sum (-1)^i \\text{length}_A \\text{Tor}_i^A(M, N)", "$$", "is finite. Let $r = \\dim(\\text{Supp}(M))$ and $s = \\dim(\\text{Supp}(N))$.", "In \\cite{Serre_algebre_locale} it is shown that $r + s \\leq \\dim(A)$", "and the following conjectures are made:", "\\begin{enumerate}", "\\item if $r + s < \\dim(A)$, then $\\chi(M, N) = 0$, and", "\\item if $r + s = \\dim(A)$, then $\\chi(M, N) > 0$.", "\\end{enumerate}", "The arguments that prove Lemma \\ref{lemma-tor-sheaf} and", "Proposition \\ref{proposition-positivity} can be leveraged", "(as is done in Serre's text) to show that (1) and (2) are", "true if $A$ contains a field. Currently, conjecture (1) is known", "in general and it is known that $\\chi(M, N) \\geq 0$ in general (Gabber).", "Positivity is, as far as we know, still an open problem." ], "refs": [ "intersection-lemma-tor-sheaf", "intersection-proposition-positivity" ], "ref_ids": [ 11984, 12002 ] }, { "id": 12007, "type": "other", "label": "intersection-remark-quasi-projective", "categories": [ "intersection" ], "title": "intersection-remark-quasi-projective", "contents": [ "Lemma \\ref{lemma-moving-move} and", "Theorem \\ref{theorem-well-defined}", "also hold for nonsingular quasi-projective varieties with", "the same proof. The only change is that one needs to prove the following", "version of the moving Lemma \\ref{lemma-moving}: Let $X \\subset \\mathbf{P}^N$", "be a closed subvariety. Let $n = \\dim(X)$ and $0 \\leq d, d' < n$. Let", "$X^{reg} \\subset X$ be the open subset of nonsingular points. Let", "$Z \\subset X^{reg}$ be a closed subvariety of dimension $d$ and", "$T_i \\subset X^{reg}$, $i \\in I$ be a finite collection of closed subvarieties", "of dimension $d'$. Then there exists a subvariety $C \\subset \\mathbf{P}^N$", "such that $C$ intersects $X$ properly and such that", "$$", "(C \\cdot X)|_{X^{reg}} = Z + \\sum\\nolimits_{j \\in J} m_j Z_j", "$$", "where $Z_j \\subset X^{reg}$ are irreducible of dimension $d$, distinct", "from $Z$, and", "$$", "\\dim(Z_j \\cap T_i) \\leq \\dim(Z \\cap T_i)", "$$", "with strict inequality if $Z$ does not intersect $T_i$ properly in $X^{reg}$." ], "refs": [ "intersection-lemma-moving-move", "intersection-theorem-well-defined", "intersection-lemma-moving" ], "ref_ids": [ 11999, 11957, 11997 ] }, { "id": 12189, "type": "other", "label": "homology-remark-direct-sum", "categories": [ "homology" ], "title": "homology-remark-direct-sum", "contents": [ "Note that the proof of Lemma \\ref{lemma-preadditive-direct-sum}", "shows that given $p$ and $q$ the morphisms $i$, $j$ are uniquely", "determined by the rules $p \\circ i = \\text{id}_x$,", "$q \\circ j = \\text{id}_y$, $p \\circ j = 0$, $q \\circ i = 0$.", "Moreover, we automatically have", "$i \\circ p + j \\circ q = \\text{id}_{x \\oplus y}$.", "Similarly, given $i$, $j$ the morphisms $p$ and $q$ are uniquely determined.", "Finally, given objects $x, y, z$ and morphisms", "$i : x \\to z$, $j : y \\to z$, $p : z \\to x$ and", "$q : z \\to y$ such that $p \\circ i = \\text{id}_x$,", "$q \\circ j = \\text{id}_y$, $p \\circ j = 0$, $q \\circ i = 0$", "and $i \\circ p + j \\circ q = \\text{id}_z$, then $z$", "is the direct sum of $x$ and $y$ with the four morphisms", "equal to $i, j, p, q$." ], "refs": [ "homology-lemma-preadditive-direct-sum" ], "ref_ids": [ 12009 ] }, { "id": 12190, "type": "other", "label": "homology-remark-direct-sums-not-exact", "categories": [ "homology" ], "title": "homology-remark-direct-sums-not-exact", "contents": [ "There are abelian categories $\\mathcal{A}$ having countable direct sums", "but where countable direct sums are not exact. An example", "is the opposite of the category of abelian sheaves on $\\mathbf{R}$.", "Namely, the category of abelian sheaves on $\\mathbf{R}$ has", "countable products, but countable products are not exact.", "For such a category the functor $\\text{Gr}(\\mathcal{A}) \\to \\mathcal{A}$,", "$(A^i) \\mapsto \\bigoplus A^i$", "described above is not exact. It is still true that", "$\\text{Gr}(\\mathcal{A})$ is equivalent to the category of", "graded objects $(A, k)$ of $\\mathcal{A}$, but the kernel in the category", "of graded objects of a map $\\varphi : (A, k) \\to (B, k)$ is not equal to", "$\\Ker(\\varphi)$ endowed with a direct sum decomposition, but rather it is", "the direct sum of the kernels of the maps $k^iA \\to k^iB$." ], "refs": [], "ref_ids": [] }, { "id": 12191, "type": "other", "label": "homology-remark-triple-complex", "categories": [ "homology" ], "title": "homology-remark-triple-complex", "contents": [ "Let $\\mathcal{A}$ be an additive category. Let $A^{\\bullet, \\bullet, \\bullet}$", "be a triple complex. The associated total complex is the complex with", "terms", "$$", "\\text{Tot}^n(A^{\\bullet, \\bullet, \\bullet}) =", "\\bigoplus\\nolimits_{p + q + r = n} A^{p, q, r}", "$$", "and differential", "$$", "d^n_{\\text{Tot}(A^{\\bullet, \\bullet, \\bullet})} =", "\\sum\\nolimits_{p + q + r = n}", "d_1^{p, q, r} + (-1)^pd_2^{p, q, r} + (-1)^{p + q}d_3^{p, q, r}", "$$", "With this definition a simple calculation shows that the associated total", "complex is equal to", "$$", "\\text{Tot}(A^{\\bullet, \\bullet, \\bullet}) =", "\\text{Tot}(\\text{Tot}_{12}(A^{\\bullet, \\bullet, \\bullet})) =", "\\text{Tot}(\\text{Tot}_{23}(A^{\\bullet, \\bullet, \\bullet}))", "$$", "In other words, we can either first combine the first two of the variables", "and then combine sum of those with the last, or we can first combine the", "last two variables and then combine the first with the sum of the last two." ], "refs": [], "ref_ids": [] }, { "id": 12192, "type": "other", "label": "homology-remark-shift-double-complex", "categories": [ "homology" ], "title": "homology-remark-shift-double-complex", "contents": [ "Let $\\mathcal{A}$ be an additive category. Let $A^{\\bullet, \\bullet}$", "be a double complex with differentials $d_1^{p, q}$ and $d_2^{p, q}$.", "Denote $A^{\\bullet, \\bullet}[a, b]$ the double complex with", "$$", "(A^{\\bullet, \\bullet}[a, b])^{p, q} = A^{p + a, q + b}", "$$", "and differentials", "$$", "d_{A^{\\bullet, \\bullet}[a, b], 1}^{p, q} = (-1)^a d_1^{p + a, q + b}", "\\quad\\text{and}\\quad", "d_{A^{\\bullet, \\bullet}[a, b], 2}^{p, q} = (-1)^b d_2^{p + a, q + b}", "$$", "In this situation there is a well defined isomorphism", "$$", "\\gamma :", "\\text{Tot}(A^{\\bullet, \\bullet})[a + b]", "\\longrightarrow", "\\text{Tot}(A^{\\bullet, \\bullet}[a, b])", "$$", "which in degree $n$ is given by the map", "$$", "\\xymatrix{", "(\\text{Tot}(A^{\\bullet, \\bullet})[a + b])^n =", "\\bigoplus_{p + q = n + a + b} A^{p, q}", "\\ar[d]^{\\epsilon(p, q, a, b)\\text{id}_{A^{p, q}}} \\\\", "\\text{Tot}(A^{\\bullet, \\bullet}[a, b])^n =", "\\bigoplus_{p' + q' = n} A^{p' + a, q' + b}", "}", "$$", "for some sign $\\epsilon(p, q, a, b)$. Of course the summand $A^{p, q}$", "maps to the summand $A^{p' + a, q' + b}$ when $p = p' + a$ and $q = q' + b$.", "To figure out the conditions on these signs observe that on the source we have", "$$", "d|_{A^{p, q}} = (-1)^{a + b}\\left(d_1^{p, q} + (-1)^pd_2^{p, q}\\right)", "$$", "whereas on the target we have", "$$", "d|_{A^{p' + a, q' + b}} =", "(-1)^ad_1^{p' + a, q' + b} + (-1)^{p'}(-1)^bd_2^{p' + a, q' + b}", "$$", "Thus our constraints are that", "$$", "(-1)^a \\epsilon(p, q, a, b) = \\epsilon(p + 1, q, a, b)(-1)^{a + b}", "\\Leftrightarrow", "\\epsilon(p + 1, q, a, b) = (-1)^b \\epsilon(p, q, a, b)", "$$", "and", "$$", "(-1)^{p' + b}\\epsilon(p, q, a, b) =", "\\epsilon(p, q + 1, a, b) (-1)^{a + b + p}", "\\Leftrightarrow", "\\epsilon(p, q, a, b) = \\epsilon(p, q + 1, a, b)", "$$", "Thus we choose $\\epsilon(p, q, a, b) = (-1)^{pb}$." ], "refs": [], "ref_ids": [] }, { "id": 12193, "type": "other", "label": "homology-remark-double-complex-complex-of-complexes-first", "categories": [ "homology" ], "title": "homology-remark-double-complex-complex-of-complexes-first", "contents": [ "Let $\\mathcal{A}$ be an additive category with countable direct sums.", "Let $\\text{DoubleComp}(\\mathcal{A})$ denote the category of double complexes.", "We can consider an object $A^{\\bullet, \\bullet}$ of", "$\\text{DoubleComp}(\\mathcal{A})$ as a complex of complexes", "as follows", "$$", "\\ldots \\to A^{\\bullet, -1} \\to A^{\\bullet, 0} \\to A^{\\bullet, 1} \\to \\ldots", "$$", "For the variant where we switch the role of the indices, see", "Remark \\ref{remark-double-complex-complex-of-complexes-second}.", "In this remark we show that taking the associated total complex", "is compatible with all the structures on complexes we have studied", "in the chapter so far.", "\\medskip\\noindent", "First, observe that the shift functor on double complexes viewed", "as complexes of complexes in the manner given above is the functor", "$[0, 1]$ defined in Remark \\ref{remark-shift-double-complex}.", "By Remark \\ref{remark-shift-double-complex} the functor", "$$", "\\text{Tot} : \\text{DoubleComp}(\\mathcal{A}) \\to \\text{Comp}(\\mathcal{A})", "$$", "is compatible with shift functors, in the sense that we have a functorial", "isomorphism $\\gamma : \\text{Tot}(A^{\\bullet, \\bullet})[1] \\to", "\\text{Tot}(A^{\\bullet, \\bullet}[0, 1])$.", "\\medskip\\noindent", "Second, if", "$$", "f, g : A^{\\bullet, \\bullet} \\to B^{\\bullet, \\bullet}", "$$", "are homotopic when $f$ and $g$ are viewed as morphisms of complexes", "of complexes in the manner given above, then", "$$", "\\text{Tot}(f), \\text{Tot}(g) :", "\\text{Tot}(A^{\\bullet, \\bullet}) \\to \\text{Tot}(B^{\\bullet, \\bullet})", "$$", "are homotopic maps of complexes. Indeed, let $h = (h^q)$", "be a homotopy between $f$ and $g$. If we denote", "$h^{p, q} : A^{p, q} \\to B^{p, q - 1}$ the component in degree $p$ of $h^q$,", "then this means that", "$$", "f^{p, q} - g^{p, q} = d_2^{p, q - 1} \\circ h^{p, q} +", "h^{p, q + 1} \\circ d_2^{p, q}", "$$", "The fact that $h^q : A^{\\bullet, q} \\to B^{\\bullet, q - 1}$ is a map of", "complexes means that", "$$", "d_1^{p, q - 1} \\circ h^{p, q} = h^{p + 1, q} \\circ d_1^{p, q}", "$$", "Let us define $h' = ((h')^n)$ the homotopy given by the maps", "$(h')^n : \\text{Tot}^n(A^{\\bullet, \\bullet}) \\to", "\\text{Tot}^{n - 1}(B^{\\bullet, \\bullet})$", "using $(-1)^ph^{p, q}$ on the summand $A^{p, q}$ for $p + q = n$.", "Then we see that", "$$", "d_{\\text{Tot}(B^{\\bullet, \\bullet})} \\circ h' +", "h' \\circ d_{\\text{Tot}(A^{\\bullet, \\bullet})}", "$$", "restricted to the summand $A^{p, q}$ is equal to", "$$", "d_1^{p, q - 1} \\circ (-1)^p h^{p, q} +", "(-1)^p d_2^{p, q - 1} \\circ (-1)^p h^{p, q} +", "(-1)^{p + 1} h^{p + 1, q} \\circ d_1^{p, q} +", "(-1)^p h^{p, q + 1} \\circ (-1)^p d_2^{p, q}", "$$", "which evaluates to $f^{p, q} - g^{p, q}$ by the equations given above.", "This proves the second compatibility.", "\\medskip\\noindent", "Third, suppose that in the paragraph above we have $f = g$.", "Then the assignment $h \\leadsto h'$ above is compatible with", "the identification of Lemma \\ref{lemma-homotopy-shift-cochain}.", "More precisely, if we view $h$ as a morphism of complexes", "of complexes $A^{\\bullet, \\bullet} \\to B^{\\bullet, \\bullet}[0, -1]$", "via this lemma then", "$$", "\\text{Tot}(A^{\\bullet, \\bullet})", "\\xrightarrow{\\text{Tot}(h)}", "\\text{Tot}(B^{\\bullet, \\bullet}[0, -1])", "\\xrightarrow{\\gamma^{-1}}", "\\text{Tot}(B^{\\bullet, \\bullet})[-1]", "$$", "is equal to $h'$ viewed as a morphism of complexes via the lemma.", "Here $\\gamma$ is the identification of", "Remark \\ref{remark-shift-double-complex}.", "The verification of this third point is immediate.", "\\medskip\\noindent", "Fourth, let", "$$", "0 \\to A^{\\bullet, \\bullet} \\to B^{\\bullet, \\bullet} \\to", "C^{\\bullet, \\bullet} \\to 0", "$$", "be a complex of double complexes and suppose we are given splittings", "$s^q : C^{\\bullet, q} \\to B^{\\bullet, q}$ and", "$\\pi^q : B^{\\bullet, q} \\to A^{\\bullet, q}$", "of this as in Lemma \\ref{lemma-ses-termwise-split-cochain}", "when we view double complexes", "as complexes of complexes in the manner given above.", "This on the one hand produces a map", "$$", "\\delta : C^{\\bullet, \\bullet} \\longrightarrow A^{\\bullet, \\bullet}[0, 1]", "$$", "by the procedure in Lemma \\ref{lemma-ses-termwise-split-cochain}.", "On the other hand taking $\\text{Tot}$ we obtain a complex", "$$", "0 \\to \\text{Tot}(A^{\\bullet, \\bullet}) \\to", "\\text{Tot}(B^{\\bullet, \\bullet}) \\to", "\\text{Tot}(C^{\\bullet, \\bullet}) \\to 0", "$$", "which is termwise split (see below) and hence comes with a morphism", "$$", "\\delta' :", "\\text{Tot}(C^{\\bullet, \\bullet})", "\\longrightarrow", "\\text{Tot}(A^{\\bullet, \\bullet})[1]", "$$", "well defined up to homotopy by Lemmas \\ref{lemma-ses-termwise-split-cochain}", "and \\ref{lemma-ses-termwise-split-homotopy-cochain}. Claim:", "these maps agree in the sense that", "$$", "\\text{Tot}(C^{\\bullet, \\bullet})", "\\xrightarrow{\\text{Tot}(\\delta)}", "\\text{Tot}(A^{\\bullet, \\bullet}[0, 1]) \\xrightarrow{\\gamma^{-1}}", "\\text{Tot}(A^{\\bullet, \\bullet})[1]", "$$", "is equal to $\\delta'$ where $\\gamma$ is as in", "Remark \\ref{remark-shift-double-complex}. To see this denote", "$s^{p, q} : C^{p, q} \\to B^{\\bullet, q}$ and", "$\\pi^{p, q} : B^{p, q} \\to A^{p, q}$ the components of $s^q$ and $\\pi^q$.", "As splittings", "$(s')^n : \\text{Tot}^n(C^{\\bullet, \\bullet}) \\to", "\\text{Tot}^n(B^{\\bullet, \\bullet})$", "and", "$(\\pi')^n : \\text{Tot}^n(B^{\\bullet, \\bullet}) \\to", "\\text{Tot}^n(A^{\\bullet, \\bullet})$", "we use the maps whose components are $s^{p, q}$ and $\\pi^{p, q}$", "for $p + q = n$. We recall that", "$$", "(\\delta')^n =", "(\\pi')^{n + 1} \\circ d_{\\text{Tot}(B^{\\bullet, \\bullet})}^n \\circ (s')^n :", "\\text{Tot}^n(C^{\\bullet, \\bullet}) \\to", "\\text{Tot}^{n + 1}(A^{\\bullet, \\bullet})", "$$", "The restriction of this to the summand $C^{p, q}$ is equal to", "$$", "\\pi^{p + 1, q} \\circ", "d_1^{p, q} \\circ", "s^{p, q} +", "\\pi^{p, q + 1} \\circ", "(-1)^p d_2^{p, q} \\circ", "s^{p, q} =", "\\pi^{p, q + 1} \\circ", "(-1)^p d_2^{p, q} \\circ", "s^{p, q}", "$$", "The equality holds because $s^q$ is a morphism of complexes (with $d_1$", "as differential) and because $\\pi^{p + 1, q} \\circ s^{p + 1, q} = 0$", "as $s$ and $\\pi$ correspond to a direct sum decomposition of $B$", "in every bidegree. On the other hand, for $\\delta$ we have", "$$", "\\delta^q = \\pi^q \\circ d_2 \\circ s^q :", "C^{\\bullet, q} \\to A^{\\bullet, q + 1}", "$$", "whose restriction to the summand $C^{p, q}$ is equal to", "$\\pi^{p, q + 1} \\circ d_2^{p, q} \\circ s^{p, q}$.", "The difference in signs is exactly canceled out by the sign", "of $(-1)^p$ in the isomorphism $\\gamma$ and the fourth claim is proven." ], "refs": [ "homology-remark-double-complex-complex-of-complexes-second", "homology-remark-shift-double-complex", "homology-remark-shift-double-complex", "homology-lemma-homotopy-shift-cochain", "homology-remark-shift-double-complex", "homology-lemma-ses-termwise-split-cochain", "homology-lemma-ses-termwise-split-cochain", "homology-lemma-ses-termwise-split-cochain", "homology-lemma-ses-termwise-split-homotopy-cochain", "homology-remark-shift-double-complex" ], "ref_ids": [ 12194, 12192, 12192, 12066, 12192, 12067, 12067, 12067, 12069, 12192 ] }, { "id": 12194, "type": "other", "label": "homology-remark-double-complex-complex-of-complexes-second", "categories": [ "homology" ], "title": "homology-remark-double-complex-complex-of-complexes-second", "contents": [ "Let $\\mathcal{A}$ be an additive category with countable direct sums.", "Let $\\text{DoubleComp}(\\mathcal{A})$ denote the category of double complexes.", "We can consider an object $A^{\\bullet, \\bullet}$ of", "$\\text{DoubleComp}(\\mathcal{A})$ as a complex of complexes", "as follows", "$$", "\\ldots \\to A^{-1, \\bullet} \\to A^{0, \\bullet} \\to A^{1, \\bullet} \\to \\ldots", "$$", "For the variant where we switch the role of the indices, see", "Remark \\ref{remark-double-complex-complex-of-complexes-first}.", "In this remark we show that taking the associated total complex", "is compatible with all the structures on complexes we have studied", "in the chapter so far.", "\\medskip\\noindent", "First, observe that the shift functor on double complexes viewed", "as complexes of complexes in the manner given above is the functor", "$[1, 0]$ defined in Remark \\ref{remark-shift-double-complex}.", "By Remark \\ref{remark-shift-double-complex} the functor", "$$", "\\text{Tot} : \\text{DoubleComp}(\\mathcal{A}) \\to \\text{Comp}(\\mathcal{A})", "$$", "is compatible with shift functors, in the sense that we have a functorial", "isomorphism $\\gamma : \\text{Tot}(A^{\\bullet, \\bullet})[1] \\to", "\\text{Tot}(A^{\\bullet, \\bullet}[1, 0])$.", "\\medskip\\noindent", "Second, if", "$$", "f, g : A^{\\bullet, \\bullet} \\to B^{\\bullet, \\bullet}", "$$", "are homotopic when $f$ and $g$ are viewed as morphisms of complexes", "of complexes in the manner given above, then", "$$", "\\text{Tot}(f), \\text{Tot}(g) :", "\\text{Tot}(A^{\\bullet, \\bullet}) \\to \\text{Tot}(B^{\\bullet, \\bullet})", "$$", "are homotopic maps of complexes. Indeed, let $h = (h^p)$", "be a homotopy between $f$ and $g$. If we denote", "$h^{p, q} : A^{p, q} \\to B^{p - 1, q}$ the component in degree $p$ of $h^q$,", "then this means that", "$$", "f^{p, q} - g^{p, q} = d_1^{p - 1, q} \\circ h^{p, q} +", "h^{p + 1, q} \\circ d_1^{p, q}", "$$", "The fact that $h^p : A^{p, \\bullet} \\to B^{p - 1, \\bullet}$ is a map of", "complexes means that", "$$", "d_2^{p - 1, q} \\circ h^{p, q} = h^{p, q + 1} \\circ d_2^{p, q}", "$$", "Let us define $h' = ((h')^n)$ the homotopy given by the maps", "$(h')^n : \\text{Tot}^n(A^{\\bullet, \\bullet}) \\to", "\\text{Tot}^{n - 1}(B^{\\bullet, \\bullet})$", "using $h^{p, q}$ on the summand $A^{p, q}$ for $p + q = n$.", "Then we see that", "$$", "d_{\\text{Tot}(B^{\\bullet, \\bullet})} \\circ h' +", "h' \\circ d_{\\text{Tot}(A^{\\bullet, \\bullet})}", "$$", "restricted to the summand $A^{p, q}$ is equal to", "$$", "d_1^{p - 1, q} \\circ h^{p, q} +", "(-1)^{p - 1} d_2^{p - 1, q} \\circ h^{p, q} +", "h^{p + 1, q} \\circ d_1^{p, q} +", "h^{p, q + 1} \\circ (-1)^p d_2^{p, q}", "$$", "which evaluates to $f^{p, q} - g^{p, q}$ by the equations given above.", "This proves the second compatibility.", "\\medskip\\noindent", "Third, suppose that in the paragraph above we have $f = g$.", "Then the assignment $h \\leadsto h'$ above is compatible with", "the identification of Lemma \\ref{lemma-homotopy-shift-cochain}.", "More precisely, if we view $h$ as a morphism of complexes", "of complexes $A^{\\bullet, \\bullet} \\to B^{\\bullet, \\bullet}[-1, 0]$", "via this lemma then", "$$", "\\text{Tot}(A^{\\bullet, \\bullet})", "\\xrightarrow{\\text{Tot}(h)}", "\\text{Tot}(B^{\\bullet, \\bullet}[-1, 0])", "\\xrightarrow{\\gamma^{-1}}", "\\text{Tot}(B^{\\bullet, \\bullet})[-1]", "$$", "is equal to $h'$ viewed as a morphism of complexes via the lemma.", "Here $\\gamma$ is the identification of", "Remark \\ref{remark-shift-double-complex}.", "The verification of this third point is immediate.", "\\medskip\\noindent", "Fourth, let", "$$", "0 \\to A^{\\bullet, \\bullet} \\to B^{\\bullet, \\bullet} \\to", "C^{\\bullet, \\bullet} \\to 0", "$$", "be a complex of double complexes and suppose we are given splittings", "$s^p : C^{p, \\bullet} \\to B^{p, \\bullet}$ and", "$\\pi^p : B^{p, \\bullet} \\to A^{p, \\bullet}$", "of this as in Lemma \\ref{lemma-ses-termwise-split-cochain}", "when we view double complexes", "as complexes of complexes in the manner given above.", "This on the one hand produces a map", "$$", "\\delta : C^{\\bullet, \\bullet} \\longrightarrow A^{\\bullet, \\bullet}[0, 1]", "$$", "by the procedure in Lemma \\ref{lemma-ses-termwise-split-cochain}.", "On the other hand taking $\\text{Tot}$ we obtain a complex", "$$", "0 \\to \\text{Tot}(A^{\\bullet, \\bullet}) \\to", "\\text{Tot}(B^{\\bullet, \\bullet}) \\to", "\\text{Tot}(C^{\\bullet, \\bullet}) \\to 0", "$$", "which is termwise split (see below) and hence comes with a morphism", "$$", "\\delta' :", "\\text{Tot}(C^{\\bullet, \\bullet})", "\\longrightarrow", "\\text{Tot}(A^{\\bullet, \\bullet})[1]", "$$", "well defined up to homotopy by Lemmas \\ref{lemma-ses-termwise-split-cochain}", "and \\ref{lemma-ses-termwise-split-homotopy-cochain}. Claim:", "these maps agree in the sense that", "$$", "\\text{Tot}(C^{\\bullet, \\bullet})", "\\xrightarrow{\\text{Tot}(\\delta)}", "\\text{Tot}(A^{\\bullet, \\bullet}[1, 0]) \\xrightarrow{\\gamma^{-1}}", "\\text{Tot}(A^{\\bullet, \\bullet})[1]", "$$", "is equal to $\\delta'$ where $\\gamma$ is as in", "Remark \\ref{remark-shift-double-complex}. To see this denote", "$s^{p, q} : C^{p, q} \\to B^{\\bullet, q}$ and", "$\\pi^{p, q} : B^{p, q} \\to A^{p, q}$ the components of $s^q$ and $\\pi^q$.", "As splittings", "$(s')^n : \\text{Tot}^n(C^{\\bullet, \\bullet}) \\to", "\\text{Tot}^n(B^{\\bullet, \\bullet})$", "and", "$(\\pi')^n : \\text{Tot}^n(B^{\\bullet, \\bullet}) \\to", "\\text{Tot}^n(A^{\\bullet, \\bullet})$", "we use the maps whose components are $s^{p, q}$ and $\\pi^{p, q}$", "for $p + q = n$. We recall that", "$$", "(\\delta')^n =", "(\\pi')^{n + 1} \\circ d_{\\text{Tot}(B^{\\bullet, \\bullet})}^n \\circ (s')^n :", "\\text{Tot}^n(C^{\\bullet, \\bullet}) \\to", "\\text{Tot}^{n + 1}(A^{\\bullet, \\bullet})", "$$", "The restriction of this to the summand $C^{p, q}$ is equal to", "$$", "\\pi^{p + 1, q} \\circ", "d_1^{p, q} \\circ", "s^{p, q} +", "\\pi^{p, q + 1} \\circ", "(-1)^p d_2^{p, q} \\circ", "s^{p, q} =", "\\pi^{p + 1, q} \\circ", "d_1^{p, q} \\circ", "s^{p, q}", "$$", "The equality holds because $s^p$ is a morphism of complexes (with $d_2$", "as differential) and because $\\pi^{p, q + 1} \\circ s^{p, q + 1} = 0$", "as $s$ and $\\pi$ correspond to a direct sum decomposition of $B$", "in every bidegree. On the other hand, for $\\delta$ we have", "$$", "\\delta^p = \\pi^p \\circ d_1 \\circ s^p :", "C^{p, \\bullet} \\to A^{p + 1, \\bullet}", "$$", "whose restriction to the summand $C^{p, q}$ is equal to", "$\\pi^{p + 1, q} \\circ d_1^{p, q} \\circ s^{p, q}$.", "Thus we get the same as before which matches with the fact that", "the isomorphism", "$\\gamma : \\text{Tot}(A^{\\bullet, \\bullet})[1] \\to", "\\text{Tot}(A^{\\bullet, bullet}[1, 0])$", "is defined without the intervetion of signs." ], "refs": [ "homology-remark-double-complex-complex-of-complexes-first", "homology-remark-shift-double-complex", "homology-remark-shift-double-complex", "homology-lemma-homotopy-shift-cochain", "homology-remark-shift-double-complex", "homology-lemma-ses-termwise-split-cochain", "homology-lemma-ses-termwise-split-cochain", "homology-lemma-ses-termwise-split-cochain", "homology-lemma-ses-termwise-split-homotopy-cochain", "homology-remark-shift-double-complex" ], "ref_ids": [ 12193, 12192, 12192, 12066, 12192, 12067, 12067, 12067, 12069, 12192 ] }, { "id": 12195, "type": "other", "label": "homology-remark-allow-translation-functors", "categories": [ "homology" ], "title": "homology-remark-allow-translation-functors", "contents": [ "It is often the case that the terms of a spectral sequence have", "additional structure, for example a grading or a bigrading.", "To accomodate this (and to get around certain technical issues)", "we introduce the following notion. Let $\\mathcal{A}$ be an", "abelian category. Let $(T_r)_{r \\geq 1}$ be a", "sequence of {\\it translation} or {\\it shift} functors, i.e.,", "$T_r : \\mathcal{A} \\to \\mathcal{A}$ is an isomorphism of categories.", "In this setting a {\\it spectral sequence} is given by a system", "$(E_r, d_r)_{r \\geq 1}$ where each $E_r$ is an object of", "$\\mathcal{A}$, each $d_r : E_r \\to T_rE_r$", "is a morphism such that $T_rd_r \\circ d_r = 0$ so that", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "T_r^{-1}E_r \\ar[r]^-{T_r^{-1}d_r} &", "E_r \\ar[r]^-{d_r} &", "T_rE_r \\ar[r]^{T_r d_r} &", "T_r^2E_r \\ar[r] & \\ldots", "}", "$$", "is a complex and $E_{r + 1} = \\Ker(d_r)/\\Im(T_r^{-1}d_r)$ for $r \\geq 1$.", "It is clear what a {\\it morphism of spectral sequences}", "means in this setting. In this setting we can still define", "$$", "0 = B_1 \\subset B_2 \\subset \\ldots \\subset B_r \\subset \\ldots", "\\subset Z_r \\subset \\ldots \\subset Z_2 \\subset Z_1 = E_1", "$$", "and $Z_\\infty$ and $B_\\infty$ (if they exist) as above." ], "refs": [], "ref_ids": [] }, { "id": 12196, "type": "other", "label": "homology-remark-shifted-exact-couple", "categories": [ "homology" ], "title": "homology-remark-shifted-exact-couple", "contents": [ "Let $\\mathcal{A}$ be an abelian category. Let", "$S, T : \\mathcal{A} \\to \\mathcal{A}$ be shift", "functors, i.e., isomorphisms of categories. We will indicate", "the $n$-fold compositions by $S^nA$ and $T^nA$ for", "$A \\in \\Ob(\\mathcal{A})$ and $n \\in \\mathbf{Z}$.", "In this situation an {\\it exact couple} is a datum $(A, E, \\alpha, f, g)$", "where $A$, $E$ are objects of $\\mathcal{A}$ and $\\alpha : A \\to T^{-1}A$,", "$f : E \\to A$, $g : A \\to SE$ are morphisms such that", "$$", "\\xymatrix{", "TE \\ar[r]^-{Tf} &", "TA \\ar[r]^-{T\\alpha} &", "A \\ar[r]^-{g} &", "SE \\ar[r]^{Sf} & SA", "}", "$$", "is an exact complex. Let's visualize this as follows", "$$", "\\xymatrix{", "TA \\ar[rrr]_{T\\alpha} & & &", "A \\ar[ld]^g \\ar[rrr]_\\alpha & & &", "T^{-1}A \\ar[ld]^{T^{-1}g} \\\\", "& TE \\ar[lu]^{Tf} \\ar@{..}[r] & SE & &", "E \\ar[lu]^f \\ar@{..}[r] & T^{-1}SE", "}", "$$", "We set $d = g \\circ f : E \\to SE$. Then $d \\circ S^{-1}d =", "g \\circ f \\circ S^{-1}g \\circ S^{-1}f = 0$ because $f \\circ S^{-1}g = 0$.", "Set $E' = \\Ker(d)/\\Im(S^{-1}d)$. Set $A' = \\Im(T\\alpha)$.", "Let $\\alpha' : A' \\to T^{-1}A'$ induced by $\\alpha$.", "Let $f' : E' \\to A'$ be induced by $f$ which works because", "$f(\\Ker(d)) \\subset \\Ker(g) = \\Im(T\\alpha)$.", "Finally, let $g' : A' \\to TSE'$ induced by", "``$Tg \\circ (T\\alpha)^{-1}$''\\footnote{This works because", "$TSE' = \\Ker(TSd)/\\Im(Td)$ and", "$Tg(\\Ker(T\\alpha)) = Tg(\\Im(Tf)) = \\Im(T(d))$", "and $TS(d)(\\Im(Tg)) = \\Im(TSg \\circ TSf \\circ Tg) = 0$.}.", "\\medskip\\noindent", "In exactly the same way as above we find", "\\begin{enumerate}", "\\item $\\Ker(d) = f^{-1}(\\Ker(g)) = f^{-1}(\\Im(T\\alpha))$,", "\\item $\\Im(d) = g(\\Im(f)) = g(\\Ker(\\alpha))$,", "\\item $(A', E', \\alpha', f', g')$ is an exact couple", "for the shift functors $TS$ and $T$.", "\\end{enumerate}", "We obtain a spectral sequence", "(as in Remark \\ref{remark-allow-translation-functors})", "with $E_1 = E$, $E_2 = E'$, etc, with $d_r : E_r \\to T^{r - 1}SE_r$", "for all $r \\geq 1$. Lemma \\ref{lemma-spectral-sequence-associated-exact-couple}", "tells us that", "$$", "SB_{r + 1} =", "g(\\Ker(T^{-r + 1}\\alpha \\circ \\ldots \\circ T^{-1}\\alpha \\circ \\alpha))", "$$", "and", "$$", "Z_{r + 1} = f^{-1}(\\Im(T\\alpha \\circ T^2\\alpha \\circ \\ldots \\circ T^r\\alpha))", "$$", "in this situation. The description of the map $d_{r + 1}$ is similar", "to that given in the lemma. (It may be easier to use these explicit", "descriptions to prove one gets a spectral sequence from such an exact", "couple.)" ], "refs": [ "homology-remark-allow-translation-functors", "homology-lemma-spectral-sequence-associated-exact-couple" ], "ref_ids": [ 12195, 12088 ] }, { "id": 12197, "type": "other", "label": "homology-remark-differential-object-selfmap", "categories": [ "homology" ], "title": "homology-remark-differential-object-selfmap", "contents": [ "Let $\\mathcal{A}$ be an abelian category and let", "$S, T : \\mathcal{A} \\to \\mathcal{A}$ be shift functors, i.e.,", "isomorphisms of categories. Assume that $TS = ST$ as functors.", "Consider pairs $(A, d)$ consisting of an object $A$ of $\\mathcal{A}$", "and a morphism $d : A \\to SA$ such that $d \\circ S^{-1}d = 0$.", "The category of these objects is abelian.", "We define $H(A, d) = \\Ker(d)/\\Im(S^{-1}d)$ and we observe that", "$H(SA, Sd) = SH(A, d)$ (canonical isomorphism).", "Given a short exact sequence", "$$", "0 \\to (A, d) \\to (B, d) \\to (C, d) \\to 0", "$$", "we obtain a long exact homology sequence", "$$", "\\ldots \\to S^{-1}H(C, d) \\to", "H(A, d) \\to H(B, d) \\to H(C, d) \\to SH(A, d) \\to \\ldots", "$$", "(note the shifts in the boundary maps). Since $ST = TS$ the functor", "$T$ defines a shift functor on pairs by setting $T(A, d) = (TA, Td)$.", "Next, let $\\alpha : (A, d) \\to T^{-1}(A, d)$ be injective with", "cokernel $(Q, d)$. Then we get an exact couple as in", "Remark \\ref{remark-shifted-exact-couple} with shift functors $TS$ and", "$T$ given by", "$$", "(H(A, d), S^{-1}H(Q, d), \\overline{\\alpha}, f, g)", "$$", "where $\\overline{\\alpha} : H(A, d) \\to T^{-1}H(A, d)$ is induced by $\\alpha$,", "the map $f : S^{-1}H(Q, d) \\to H(A, d)$ is the boundary", "map and $g : H(A, d) \\to TH(Q, d) = TS(S^{-1}H(Q, d))$ is induced by", "the quotient map $A \\to TQ$. Thus we get a spectral sequence as above", "with $E_1 = S^{-1}H(Q, d)$ and differentials $d_r : E_r \\to T^rSE_r$.", "As above we set $E_0 = S^{-1}Q$ and $d_0 : E_0 \\to SE_0$ given by", "$S^{-1}d : S^{-1}Q \\to Q$. If according to our conventions we define", "$B_r \\subset Z_r \\subset E_0$, then we have for $r \\geq 1$ that", "\\begin{enumerate}", "\\item $SB_r$ is the image of", "$$", "(T^{-r + 1}\\alpha \\circ \\ldots \\circ T^{-1}\\alpha)^{-1}", "\\Im(T^{-r}S^{-1}d)", "$$", "under the natural map $T^{-1}A \\to Q$,", "\\item $Z_r$ is the image of", "$$", "(S^{-1}T^{-1}d)^{-1}", "\\Im(\\alpha \\circ \\ldots \\circ T^{r - 1}\\alpha)", "$$", "under the natural map $S^{-1}T^{-1}A \\to S^{-1}Q$.", "\\end{enumerate}", "The differentials can be described as follows: if $x \\in Z_r$, then", "pick $x' \\in S^{-1}T^{-1}A$ mapping to $x$. Then $S^{-1}T^{-1}d(x')$", "is $(\\alpha \\circ \\ldots \\circ T^{r - 1}\\alpha)(y)$ for some", "$y \\in T^{r - 1}A$. Then $d_r(x) \\in T^rSE_r$ is represented by the", "class of the image of $y$ in $T^rSE_0 = T^rQ$ modulo $T^rSB_r$." ], "refs": [ "homology-remark-shifted-exact-couple" ], "ref_ids": [ 12196 ] }, { "id": 12198, "type": "other", "label": "homology-remark-need-left-exactness", "categories": [ "homology" ], "title": "homology-remark-need-left-exactness", "contents": [ "Let $\\mathcal{A}$, $\\mathcal{B}$, $u : \\mathcal{A} \\to \\mathcal{B}$ and", "$v : \\mathcal{B} \\to \\mathcal{A}$ be as in", "Lemma \\ref{lemma-adjoint-preserve-injectives}.", "In the presence of assumption (1) assumption (2) is equivalent to requiring", "that $v$ is exact. Moreover, condition (2) is necessary. Here is an example.", "Let $A \\to B$ be a ring map.", "Let $u : \\text{Mod}_B \\to \\text{Mod}_A$ be $u(N) = N_A$", "and let $v : \\text{Mod}_A \\to \\text{Mod}_B$ be", "$v(M) = M \\otimes_A B$. Then $u$ is right adjoint to $v$, and $u$ is", "exact and $v$ is right exact, but $v$ does not transform injective maps into", "injective maps in general (i.e., $v$ is not left exact).", "Moreover, it is {\\bf not} the case that $u$ transforms injective", "$B$-modules into injective $A$-modules. For example, if", "$A = \\mathbf{Z}$ and $B = \\mathbf{Z}/p\\mathbf{Z}$, then", "the injective $B$-module $\\mathbf{Z}/p\\mathbf{Z}$ is not", "an injective $\\mathbf{Z}$-module. In fact, the lemma applies to this", "example if and only if the ring map $A \\to B$ is flat." ], "refs": [ "homology-lemma-adjoint-preserve-injectives" ], "ref_ids": [ 12116 ] }, { "id": 12199, "type": "other", "label": "homology-remark-faithfulness-needed", "categories": [ "homology" ], "title": "homology-remark-faithfulness-needed", "contents": [ "Let $\\mathcal{A}$, $\\mathcal{B}$, $u : \\mathcal{A} \\to \\mathcal{B}$ and", "$v : \\mathcal{B} \\to \\mathcal{A}$ be as", "In Lemma \\ref{lemma-adjoint-enough-injectives}.", "In the presence of conditions (1) and (2) condition (4) is equivalent", "to $v$ being faithful. Moreover, condition (4) is needed.", "An example is to consider the", "case where the functors $u$ and $v$ are both the zero functor." ], "refs": [ "homology-lemma-adjoint-enough-injectives" ], "ref_ids": [ 12117 ] }, { "id": 12410, "type": "other", "label": "categories-remark-big-categories", "categories": [ "categories" ], "title": "categories-remark-big-categories", "contents": [ "Big categories. In some texts a category is allowed to have a proper", "class of objects. We will allow this as well in these notes but only", "in the following list of cases (to be updated as we go along).", "In particular, when we say: ``Let $\\mathcal{C}$ be a category''", "then it is understood that $\\Ob(\\mathcal{C})$ is a set.", "\\begin{enumerate}", "\\item The category $\\textit{Sets}$ of sets.", "\\item The category $\\textit{Ab}$ of abelian groups.", "\\item The category $\\textit{Groups}$ of groups.", "\\item Given a group $G$ the category $G\\textit{-Sets}$ of", "sets with a left $G$-action.", "\\item Given a ring $R$ the category $\\text{Mod}_R$ of $R$-modules.", "\\item Given a field $k$ the category of vector spaces over $k$.", "\\item The category of rings.", "\\item The category of schemes.", "\\item The category $\\textit{Top}$ of topological spaces.", "\\item Given a topological space $X$ the category", "$\\textit{PSh}(X)$ of presheaves of sets over $X$.", "\\item Given a topological space $X$ the category", "$\\Sh(X)$ of sheaves of sets over $X$.", "\\item Given a topological space $X$ the category", "$\\textit{PAb}(X)$ of presheaves of abelian groups over $X$.", "\\item Given a topological space $X$ the category", "$\\textit{Ab}(X)$ of sheaves of abelian groups over $X$.", "\\item Given a small category $\\mathcal{C}$ the category of functors", "from $\\mathcal{C}$ to $\\textit{Sets}$.", "\\item Given a category $\\mathcal{C}$ the category of presheaves of sets", "over $\\mathcal{C}$.", "\\item Given a site $\\mathcal{C}$ the category of sheaves", "of sets over $\\mathcal{C}$.", "\\end{enumerate}", "One of the reason to enumerate these here is to try and avoid", "working with something like the ``collection'' of ``big'' categories", "which would be like working with the collection of all classes", "which I think definitively is a meta-mathematical object." ], "refs": [], "ref_ids": [] }, { "id": 12411, "type": "other", "label": "categories-remark-unique-identity", "categories": [ "categories" ], "title": "categories-remark-unique-identity", "contents": [ "It follows directly from the definition that any two identity morphisms", "of an object $x$ of $\\mathcal{A}$ are the same. Thus we may and will", "speak of {\\it the} identity morphism $\\text{id}_x$ of $x$." ], "refs": [], "ref_ids": [] }, { "id": 12412, "type": "other", "label": "categories-remark-functor-into-sets", "categories": [ "categories" ], "title": "categories-remark-functor-into-sets", "contents": [ "Suppose that $\\mathcal{A}$ is a category.", "A functor $F$ from $\\mathcal{A}$ to $\\textit{Sets}$", "is a mathematical object (i.e., it is a set not a class or a formula", "of set theory, see", "Sets, Section \\ref{sets-section-sets-everything})", "even though the category of sets is ``big''.", "Namely, the range of $F$ on objects will be", "a set $F(\\Ob(\\mathcal{A}))$ and then we", "may think of $F$ as a functor between", "$\\mathcal{A}$ and the full subcategory", "of the category of sets whose", "objects are elements of $F(\\Ob(\\mathcal{A}))$." ], "refs": [], "ref_ids": [] }, { "id": 12413, "type": "other", "label": "categories-remark-functors-sets-sets", "categories": [ "categories" ], "title": "categories-remark-functors-sets-sets", "contents": [ "This is one instance where the same thing does not hold if", "$\\mathcal{A}$ is a ``big'' category. For example consider", "functors $\\textit{Sets} \\to \\textit{Sets}$. As we have currently", "defined it such a functor is a class and not a set. In other", "words, it is given by a formula in set theory (with some variables", "equal to specified sets)! It is not a good idea to try to consider", "all possible formulae of set theory as part of the definition of", "a mathematical object. The same problem presents itself when", "considering sheaves on the category of schemes for example.", "We will come back to this point later." ], "refs": [], "ref_ids": [] }, { "id": 12414, "type": "other", "label": "categories-remark-diagram-small", "categories": [ "categories" ], "title": "categories-remark-diagram-small", "contents": [ "The index category of a (co)limit will never be allowed to have", "a proper class of objects. In this project it means that", "it cannot be one of the categories listed in", "Remark \\ref{remark-big-categories}" ], "refs": [ "categories-remark-big-categories" ], "ref_ids": [ 12410 ] }, { "id": 12415, "type": "other", "label": "categories-remark-limit-colim", "categories": [ "categories" ], "title": "categories-remark-limit-colim", "contents": [ "We often write $\\lim_i M_i$, $\\colim_i M_i$,", "$\\lim_{i\\in \\mathcal{I}} M_i$, or $\\colim_{i\\in \\mathcal{I}} M_i$", "instead of the versions indexed by $\\mathcal{I}$.", "Using this notation, and using the description of", "limits and colimits of sets in Section \\ref{section-limit-sets}", "below, we can say the following.", "Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram.", "\\begin{enumerate}", "\\item The object $\\lim_i M_i$ if it exists satisfies the following property", "$$", "\\Mor_\\mathcal{C}(W, \\lim_i M_i)", "=", "\\lim_i \\Mor_\\mathcal{C}(W, M_i)", "$$", "where the limit on the right takes place in the category of sets.", "\\item The object $\\colim_i M_i$ if it", "exists satisfies the following property", "$$", "\\Mor_\\mathcal{C}(\\colim_i M_i, W)", "=", "\\lim_{i\\in \\mathcal{I}^\\text{opp}} \\Mor_\\mathcal{C}(M_i, W)", "$$", "where on the right we have the limit over the opposite category", "with value in the category of sets.", "\\end{enumerate}", "By the Yoneda lemma (and its dual) this formula completely determines the", "limit, respectively the colimit." ], "refs": [], "ref_ids": [] }, { "id": 12416, "type": "other", "label": "categories-remark-cones-and-cocones", "categories": [ "categories" ], "title": "categories-remark-cones-and-cocones", "contents": [ "Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram. In this setting a", "{\\it cone} for $M$ is given by an object $W$ and a family of morphisms", "$q_i : W \\to M_i$, $i \\in \\Ob(\\mathcal{I})$ such that for all morphisms", "$\\phi : i \\to i'$ of $\\mathcal{I}$ the diagram", "$$", "\\xymatrix{", "& W \\ar[dl]_{q_i} \\ar[dr]^{q_{i'}} \\\\", "M_i \\ar[rr]^{M(\\phi)} & & M_{i'}", "}", "$$", "is commutative. The collection of cones forms a category with an obvious", "notion of morphisms. Clearly, the limit of $M$, if it exists, is a final", "object in the category of cones. Dually, a {\\it cocone} for $M$ is given", "by an object $W$ and a family of morphisms $t_i : M_i \\to W$ such that for", "all morphisms $\\phi : i \\to i'$ in $\\mathcal{I}$ the diagram", "$$", "\\xymatrix{", "M_i \\ar[rr]^{M(\\phi)} \\ar[dr]_{t_i} & & M_{i'} \\ar[dl]^{t_{i'}} \\\\", "& W", "}", "$$", "commutes. The collection of cocones forms a category with an obvious notion", "of morphisms. Similarly to the above the colimit of $M$ exists", "if and only if the category of cocones has an initial object." ], "refs": [], "ref_ids": [] }, { "id": 12417, "type": "other", "label": "categories-remark-preorder-versus-partial-order", "categories": [ "categories" ], "title": "categories-remark-preorder-versus-partial-order", "contents": [ "Let $I$ be a preordered set. From $I$ we can construct a canonical", "partially ordered set $\\overline{I}$ and an order preserving map", "$\\pi : I \\to \\overline{I}$. Namely, we can define an equivalence", "relation $\\sim$ on $I$ by the rule", "$$", "i \\sim j \\Leftrightarrow (i \\leq j\\text{ and }j \\leq i).", "$$", "We set $\\overline{I} = I/\\sim$ and we let $\\pi : I \\to \\overline{I}$", "be the quotient map. Finally, $\\overline{I}$ comes with a unique", "partial ordering such that", "$\\pi(i) \\leq \\pi(j) \\Leftrightarrow i \\leq j$.", "Observe that if $I$ is a directed set, then $\\overline{I}$", "is a directed partially ordered set.", "Given an (inverse) system $N$ over $\\overline{I}$ we obtain an", "(inverse) system $M$ over $I$ by setting $M_i = N_{\\pi(i)}$.", "This construction defines a functor between the category", "of inverse systems over $I$ and $\\overline{I}$.", "In fact, this is an equivalence.", "The reason is that if $i \\sim j$, then for any system", "$M$ over $I$ the maps $M_i \\to M_j$ and $M_j \\to M_i$ are", "mutually inverse isomorphisms. More precisely, choosing", "a section $s : \\overline{I} \\to I$ of $\\pi$ a quasi-inverse", "of the functor above sends $M$ to $N$ with", "$N_{\\overline{i}} = M_{s(\\overline{i})}$.", "Finally, this correspondence is compatible with colimits of systems:", "if $M$ and $N$ are related as above and", "if either $\\colim_{\\overline{I}} N$ or $\\colim_I M$ exists", "then so does the other and", "$\\colim_{\\overline{I}} N = \\colim_I M$.", "Similar results hold for inverse systems and limits of inverse systems." ], "refs": [], "ref_ids": [] }, { "id": 12418, "type": "other", "label": "categories-remark-trick-needed", "categories": [ "categories" ], "title": "categories-remark-trick-needed", "contents": [ "Note that a finite directed set $(I, \\geq)$ always has a greatest object", "$i_\\infty$. Hence any colimit of a system $(M_i, f_{ii'})$ over such a set", "is trivial in the sense that the colimit equals $M_{i_\\infty}$. In contrast,", "a colimit indexed by a finite filtered category need not", "be trivial. For instance, let $\\mathcal{I}$ be the category with a single object", "$i$ and a single non-trivial morphism $e$ satisfying $e = e \\circ e$. The", "colimit of a diagram $M : \\mathcal{I} \\to Sets$ is the image of the", "idempotent $M(e)$. This illustrates that something like the trick of passing", "to $\\mathcal{I}\\times \\omega$ in the proof of", "Lemma \\ref{lemma-directed-category-system} is essential." ], "refs": [ "categories-lemma-directed-category-system" ], "ref_ids": [ 12236 ] }, { "id": 12419, "type": "other", "label": "categories-remark-ind-category", "categories": [ "categories" ], "title": "categories-remark-ind-category", "contents": [ "Let $\\mathcal{C}$ be a category. There exists a big category", "$\\text{Ind-}\\mathcal{C}$ of {\\it ind-objects of} $\\mathcal{C}$.", "Namely, if $F : \\mathcal{I} \\to \\mathcal{C}$ and", "$G : \\mathcal{J} \\to \\mathcal{C}$ are filtered diagrams in $\\mathcal{C}$,", "then we can define", "$$", "\\Mor_{\\text{Ind-}\\mathcal{C}}(F, G) =", "\\lim_i \\colim_j \\Mor_\\mathcal{C}(F(i), G(j)).", "$$", "There is a canonical functor $\\mathcal{C} \\to \\text{Ind-}\\mathcal{C}$", "which maps $X$ to the {\\it constant system} on $X$. This is a fully", "faithful embedding. In this language one sees that a diagram $F$ is", "essentially constant if and only if $F$ is isomorphic to a constant system.", "If we ever need this material, then we will formulate this into a lemma", "and prove it here." ], "refs": [], "ref_ids": [] }, { "id": 12420, "type": "other", "label": "categories-remark-pro-category", "categories": [ "categories" ], "title": "categories-remark-pro-category", "contents": [ "Let $\\mathcal{C}$ be a category. There exists a big category", "$\\text{Pro-}\\mathcal{C}$ of {\\it pro-objects} of $\\mathcal{C}$.", "Namely, if $F : \\mathcal{I} \\to \\mathcal{C}$ and", "$G : \\mathcal{J} \\to \\mathcal{C}$ are cofiltered diagrams in $\\mathcal{C}$,", "then we can define", "$$", "\\Mor_{\\text{Pro-}\\mathcal{C}}(F, G) =", "\\lim_j \\colim_i \\Mor_\\mathcal{C}(F(i), G(j)).", "$$", "There is a canonical functor $\\mathcal{C} \\to \\text{Pro-}\\mathcal{C}$", "which maps $X$ to the {\\it constant system} on $X$. This is a fully", "faithful embedding. In this language one sees that a diagram $F$ is", "essentially constant if and only if $F$ is isomorphic to a constant system.", "If we ever need this material, then we will formulate this into a lemma", "and prove it here." ], "refs": [], "ref_ids": [] }, { "id": 12421, "type": "other", "label": "categories-remark-pro-category-copresheaves", "categories": [ "categories" ], "title": "categories-remark-pro-category-copresheaves", "contents": [ "Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{I} \\to \\mathcal{C}$ and", "$G : \\mathcal{J} \\to \\mathcal{C}$ be cofiltered diagrams in $\\mathcal{C}$.", "Consider the functors $A, B : \\mathcal{C} \\to \\textit{Sets}$ defined by", "$$", "A(X) = \\colim_i \\Mor_\\mathcal{C}(F(i), X)", "\\quad\\text{and}\\quad", "B(X) = \\colim_j \\Mor_\\mathcal{C}(G(j), X)", "$$", "We claim that a morphism of pro-systems from $F$ to $G$ is the same thing", "as a transformation of functors $t : B \\to A$. Namely, given $t$", "we can apply $t$ to the class of $\\text{id}_{G(j)}$ in $B(G(j))$", "to get a compatible system of elements", "$\\xi_j \\in A(G(j)) = \\colim_i \\Mor_\\mathcal{C}(F(i), G(j))$", "which is exactly our definition of a morphism in $\\text{Pro-}\\mathcal{C}$ in", "Remark \\ref{remark-pro-category}. We omit the construction of a", "transformation $B \\to A$ given a morphism of pro-objects from $F$ to $G$." ], "refs": [ "categories-remark-pro-category" ], "ref_ids": [ 12420 ] }, { "id": 12422, "type": "other", "label": "categories-remark-how-to-use-it", "categories": [ "categories" ], "title": "categories-remark-how-to-use-it", "contents": [ "The lemma above is often used to construct the free something on something.", "For example the free abelian group on a set, the free group on a set, etc.", "The idea, say in the case of the free group on a set $E$ is to", "consider the functor", "$$", "F : \\textit{Groups} \\to \\textit{Sets},\\quad", "G \\longmapsto \\text{Map}(E, G)", "$$", "This functor commutes with limits. As our family of objects", "we can take a family $E \\to G_i$ consisting of groups $G_i$", "of cardinality at most $\\max(\\aleph_0, |E|)$ and set maps", "$E \\to G_i$ such that every isomorphism class of such a structure", "occurs at least once. Namely, if $E \\to G$ is a map from $E$ to", "a group $G$, then the subgroup $G'$ generated by the image has", "cardinality at most $\\max(\\aleph_0, |E|)$. The lemma tells us", "the functor is representable, hence there exists a group", "$F_E$ such that $\\Mor_{\\textit{Groups}}(F_E, G) = \\text{Map}(E, G)$.", "In particular, the identity morphism of $F_E$ corresponds to", "a map $E \\to F_E$ and one can show that $F_E$ is generated by", "the image without imposing any relations.", "\\medskip\\noindent", "Another typical application is that we can use the lemma to construct", "colimits once it is known that limits exist. We illustrate it using", "the category of topological spaces which has limits by", "Topology, Lemma \\ref{topology-lemma-limits}. Namely, suppose", "that $\\mathcal{I} \\to \\textit{Top}$, $i \\mapsto X_i$ is a functor.", "Then we can consider", "$$", "F : \\textit{Top} \\longrightarrow \\textit{Sets},\\quad", "Y \\longmapsto \\lim_\\mathcal{I} \\Mor_{\\textit{Top}}(X_i, Y)", "$$", "This functor commutes with limits. Moreover, given any topological space", "$Y$ and an element $(\\varphi_i : X_i \\to Y)$ of $F(Y)$, there is", "a subspace $Y' \\subset Y$ of cardinality at most $|\\coprod X_i|$", "such that the morphisms $\\varphi_i$ map into $Y'$. Namely, we can", "take the induced topology on the union of the images of the $\\varphi_i$.", "Thus it is clear that the hypotheses of the lemma are satisfied and we find a", "topological space $X$", "representing the functor $F$, which precisely means that $X$ is", "the colimit of the diagram $i \\mapsto X_i$." ], "refs": [ "topology-lemma-limits" ], "ref_ids": [ 8248 ] }, { "id": 12423, "type": "other", "label": "categories-remark-motivation-localization", "categories": [ "categories" ], "title": "categories-remark-motivation-localization", "contents": [ "The motivation for the construction of $S^{-1} \\mathcal{C}$ is to", "``force'' the morphisms in $S$ to be invertible by artificially", "creating inverses to them (at the cost of some existing morphisms", "possibly becoming identified with each other). This is similar to", "the localization of a commutative ring at a multiplicative subset,", "and more generally to the localization of a noncommutative ring", "at a right denominator set (see \\cite[Section 10A]{Lam}). This is", "more than just a similarity: The construction of", "$S^{-1} \\mathcal{C}$ (or, more precisely, its version for", "additive categories $\\mathcal{C}$) actually generalizes the", "latter type of localization. Namely, a noncommutative ring can be", "viewed as a pre-additive category with a single object (the morphisms", "being the elements of the ring); a multiplicative subset of this", "ring then becomes a set $S$ of morphisms satisfying LMS1 (aka", "RMS1). Then, the conditions RMS2 and RMS3 for this category and", "this subset $S$ translate into the two conditions", "(``right permutable'' and ``right reversible'') of a right", "denominator set (and similarly for LMS and left denominator sets),", "and $S^{-1} \\mathcal{C}$ (with a properly defined additive", "structure) is the one-object category corresponding to the", "localization of the ring." ], "refs": [], "ref_ids": [] }, { "id": 12424, "type": "other", "label": "categories-remark-left-localization-morphisms-colimit", "categories": [ "categories" ], "title": "categories-remark-left-localization-morphisms-colimit", "contents": [ "Let $\\mathcal{C}$ be a category. Let $S$ be a left multiplicative system.", "Given an object $Y$ of $\\mathcal{C}$ we denote $Y/S$ the category whose", "objects are $s : Y \\to Y'$ with $s \\in S$ and whose morphisms are", "commutative diagrams", "$$", "\\xymatrix{", "& Y \\ar[ld]_s \\ar[rd]^t & \\\\", "Y' \\ar[rr]^a & & Y''", "}", "$$", "where $a : Y' \\to Y''$ is arbitrary. We claim that the category", "$Y/S$ is filtered (see", "Definition \\ref{definition-directed}).", "Namely, LMS1 implies that $\\text{id}_Y : Y \\to Y$", "is in $Y/S$; hence $Y/S$ is nonempty. LMS2 implies that given", "$s_1 : Y \\to Y_1$ and $s_2 : Y \\to Y_2$ we can find a diagram", "$$", "\\xymatrix{", "Y \\ar[d]_{s_1} \\ar[r]_{s_2} & Y_2 \\ar[d]^t \\\\", "Y_1 \\ar[r]^a & Y_3", "}", "$$", "with $t \\in S$. Hence $s_1 : Y \\to Y_1$ and $s_2 : Y \\to Y_2$", "both have maps to $t \\circ s_2 : Y \\to Y_3$ in $Y/S$. Finally, given", "two morphisms $a, b$ from $s_1 : Y \\to Y_1$ to $s_2 : Y \\to Y_2$", "in $Y/S$ we see that $a \\circ s_1 = b \\circ s_1$; hence by LMS3", "there exists a $t : Y_2 \\to Y_3$ in $S$ such that", "$t \\circ a = t \\circ b$.", "Now the combined results of", "Lemmas \\ref{lemma-morphisms-left-localization} and", "\\ref{lemma-equality-morphisms-left-localization}", "tell us that", "\\begin{equation}", "\\label{equation-left-localization-morphisms-colimit}", "\\Mor_{S^{-1}\\mathcal{C}}(X, Y) =", "\\colim_{(s : Y \\to Y') \\in Y/S} \\Mor_\\mathcal{C}(X, Y')", "\\end{equation}", "This formula expressing morphism sets in $S^{-1}\\mathcal{C}$ as a filtered", "colimit of morphism sets in $\\mathcal{C}$ is occasionally useful." ], "refs": [ "categories-definition-directed", "categories-lemma-morphisms-left-localization", "categories-lemma-equality-morphisms-left-localization" ], "ref_ids": [ 12363, 12256, 12257 ] }, { "id": 12425, "type": "other", "label": "categories-remark-right-localization-morphisms-colimit", "categories": [ "categories" ], "title": "categories-remark-right-localization-morphisms-colimit", "contents": [ "Let $\\mathcal{C}$ be a category. Let $S$ be a right multiplicative system.", "Given an object $X$ of $\\mathcal{C}$ we denote $S/X$ the category whose", "objects are $s : X' \\to X$ with $s \\in S$ and whose morphisms are", "commutative diagrams", "$$", "\\xymatrix{", "X' \\ar[rd]_s \\ar[rr]_a & & X'' \\ar[ld]^t \\\\", "& X", "}", "$$", "where $a : X' \\to X''$ is arbitrary. The category", "$S/X$ is cofiltered (see", "Definition \\ref{definition-codirected}).", "(This is dual to the corresponding statement in", "Remark \\ref{remark-left-localization-morphisms-colimit}.)", "Now the combined results of", "Lemmas \\ref{lemma-morphisms-right-localization} and", "\\ref{lemma-equality-morphisms-right-localization}", "tell us that", "\\begin{equation}", "\\label{equation-right-localization-morphisms-colimit}", "\\Mor_{S^{-1}\\mathcal{C}}(X, Y) =", "\\colim_{(s : X' \\to X) \\in (S/X)^{opp}} \\Mor_\\mathcal{C}(X', Y)", "\\end{equation}", "This formula expressing morphisms in $S^{-1}\\mathcal{C}$ as a filtered", "colimit of morphisms in $\\mathcal{C}$ is occasionally useful." ], "refs": [ "categories-definition-codirected", "categories-remark-left-localization-morphisms-colimit", "categories-lemma-morphisms-right-localization", "categories-lemma-equality-morphisms-right-localization" ], "ref_ids": [ 12364, 12424, 12262, 12263 ] }, { "id": 12426, "type": "other", "label": "categories-remark-big-2-categories", "categories": [ "categories" ], "title": "categories-remark-big-2-categories", "contents": [ "Big $2$-categories.", "In many texts a $2$-category is allowed to have a class of", "objects (but hopefully a ``class of classes'' is not allowed).", "We will allow these ``big'' $2$-categories as well, but only", "in the following list of cases (to be updated as we go along):", "\\begin{enumerate}", "\\item The $2$-category of categories $\\textit{Cat}$.", "\\item The $(2, 1)$-category of categories $\\textit{Cat}$.", "\\item The $2$-category of groupoids $\\textit{Groupoids}$;", "this is a $(2, 1)$-category.", "\\item The $2$-category of fibred categories over a fixed category.", "\\item The $(2, 1)$-category of fibred categories over a fixed category.", "\\end{enumerate}", "See Definition \\ref{definition-2-1-category}.", "Note that in each case the class of objects of the $2$-category", "$\\mathcal{C}$ is a proper class, but for all objects $x, y \\in \\Ob(C)$", "the category $\\Mor_\\mathcal{C}(x, y)$ is ``small'' (according to", "our conventions)." ], "refs": [ "categories-definition-2-1-category" ], "ref_ids": [ 12382 ] }, { "id": 12427, "type": "other", "label": "categories-remark-other-2-categories", "categories": [ "categories" ], "title": "categories-remark-other-2-categories", "contents": [ "Thus there are variants of the construction of", "Example \\ref{example-2-1-category-of-categories}", "above where we look at the $2$-category of groupoids,", "or categories fibred in groupoids over a fixed", "category, or stacks. And so on." ], "refs": [], "ref_ids": [] }, { "id": 12428, "type": "other", "label": "categories-remark-other-description-2-fibre-product", "categories": [ "categories" ], "title": "categories-remark-other-description-2-fibre-product", "contents": [ "Let $\\mathcal{A}$, $\\mathcal{B}$, and $\\mathcal{C}$ be categories.", "Let $F : \\mathcal{A} \\to \\mathcal{C}$ and $G : \\mathcal{B} \\to \\mathcal{C}$", "be functors. Another, slightly more symmetrical, construction of a $2$-fibre", "product $\\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}$ is as follows.", "An object is a quintuple $(A, B, C, a, b)$ where $A, B, C$ are objects", "of $\\mathcal{A}, \\mathcal{B}, \\mathcal{C}$ and where $a : F(A) \\to C$", "and $b : G(B) \\to C$ are isomorphisms. A morphism", "$(A, B, C, a, b) \\to (A', B', C', a', b')$ is given by a triple", "of morphisms $A \\to A', B \\to B', C \\to C'$ compatible with the morphisms", "$a, b, a', b'$. We can prove directly that this leads to a $2$-fibre", "product. However, it is easier to observe that the functor", "$(A, B, C, a, b) \\mapsto (A, B, b^{-1} \\circ a)$ gives an equivalence", "from the category of quintuples to the category constructed in", "Example \\ref{example-2-fibre-product-categories}." ], "refs": [], "ref_ids": [] }, { "id": 12429, "type": "other", "label": "categories-remark-alternative-fibred-groupoids-strict", "categories": [ "categories" ], "title": "categories-remark-alternative-fibred-groupoids-strict", "contents": [ "We can use the $2$-Yoneda lemma to give an alternative proof of", "Lemma \\ref{lemma-fibred-groupoids-strict}.", "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids.", "We define a contravariant functor $F$ from $\\mathcal{C}$ to the", "category of groupoids as follows: for $U\\in \\Ob(\\mathcal{C})$", "let", "$$", "F(U) = \\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{C}/U, \\mathcal{S}).", "$$", "If $f : U \\to V$ the induced functor $\\mathcal{C}/U \\to \\mathcal{C}/V$", "induces the morphism $F(f) : F(V) \\to F(U)$. Clearly $F$ is a functor.", "Let $\\mathcal{S}'$ be the associated category fibred in groupoids from", "Example \\ref{example-functor-groupoids}.", "There is an obvious functor $G : \\mathcal{S}' \\to \\mathcal{S}$", "over $\\mathcal{C}$ given by taking the pair $(U, x)$, where", "$U \\in \\Ob(\\mathcal{C})$ and $x \\in F(U)$, to", "$x(\\text{id}_U) \\in \\mathcal{S}$. Now", "Lemma \\ref{lemma-yoneda-2category}", "implies that for each $U$,", "$$", "G_U : \\mathcal{S}'_U = F(U)=", "\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{C}/U, \\mathcal{S})", "\\to", "\\mathcal{S}_U", "$$", "is an equivalence, and thus $G$ is an equivalence between $\\mathcal{S}$ and", "$\\mathcal{S}'$ by Lemma \\ref{lemma-equivalence-fibred-categories}." ], "refs": [ "categories-lemma-fibred-groupoids-strict", "categories-lemma-yoneda-2category", "categories-lemma-equivalence-fibred-categories" ], "ref_ids": [ 12308, 12318, 12297 ] }, { "id": 12430, "type": "other", "label": "categories-remark-left-dual-adjoint", "categories": [ "categories" ], "title": "categories-remark-left-dual-adjoint", "contents": [ "Lemma \\ref{lemma-left-dual} says in particular that $Z \\mapsto Z \\otimes Y$", "is the right adjoint of $Z' \\mapsto Z' \\otimes X$. In particular, uniqueness", "of adjoint functors guarantees that a left dual of $X$, if it exists, is", "unique up to unique isomorphism.", "Conversely, assume the functor $Z \\mapsto Z \\otimes Y$ is a right adjoint of", "the functor $Z' \\mapsto Z' \\otimes X$, i.e., we're given a bijection", "$$", "\\Mor(Z' \\otimes X, Z) \\longrightarrow \\Mor(Z', Z \\otimes Y)", "$$", "functorial in both $Z$ and $Z'$. The unit of the adjunction produces", "maps", "$$", "\\eta_Z : Z \\to Z \\otimes X \\otimes Y", "$$", "functorial in $Z$ and the counit of the adjoint produces maps", "$$", "\\epsilon_{Z'} : Z' \\otimes Y \\otimes X \\to Z'", "$$", "functorial in $Z'$. In particular, we find", "$\\eta = \\eta_\\mathbf{1} : \\mathbf{1} \\to X \\otimes Y$ and", "$\\epsilon = \\epsilon_\\mathbf{1} : Y \\otimes X \\to \\mathbf{1}$.", "As an exercise in the relationship between units, counits, and", "the adjunction isomorphism, the reader can show that we have", "$$", "(\\epsilon \\otimes \\text{id}_Y) \\circ \\eta_Y = \\text{id}_Y", "\\quad\\text{and}\\quad", "\\epsilon_X \\circ (\\eta \\otimes \\text{id}_X) = \\text{id}_X", "$$", "However, this isn't enough to show that", "$(\\epsilon \\otimes \\text{id}_Y) \\circ (\\text{id}_Y \\otimes \\eta) =", "\\text{id}_Y$ and", "$(\\text{id}_X \\otimes \\epsilon) \\circ (\\eta \\otimes \\text{id}_X) =", "\\text{id}_X$, because we don't know in general that", "$\\eta_Y = \\text{id}_Y \\otimes \\eta$ and we don't know that", "$\\epsilon_X = \\epsilon \\otimes \\text{id}_X$. For this it would suffice", "to know that our adjunction isomorphism has the following property:", "for every $W, Z, Z'$ the diagram", "$$", "\\xymatrix{", "\\Mor(Z' \\otimes X, Z) \\ar[r] \\ar[d]_{\\text{id}_W \\otimes -} &", "\\Mor(Z', Z \\otimes Y) \\ar[d]^{\\text{id}_W \\otimes -} \\\\", "\\Mor(W \\otimes Z' \\otimes X, W \\otimes Z) \\ar[r] &", "\\Mor(W \\otimes Z', W \\otimes Z \\otimes Y)", "}", "$$", "If this holds, we will say {\\it the adjunction is compatible with", "the given tensor structure}. Thus the requirement that", "$Z \\mapsto Z \\otimes Y$ be the right adjoint of $Z' \\mapsto Z' \\otimes X$", "compatible with the given tensor structure is an equivalent formulation of the", "property of being a left dual." ], "refs": [ "categories-lemma-left-dual" ], "ref_ids": [ 12325 ] }, { "id": 12553, "type": "other", "label": "topologies-remark-choice-sites", "categories": [ "topologies" ], "title": "topologies-remark-choice-sites", "contents": [ "Take any category $\\Sch_\\alpha$ constructed as in", "Sets, Lemma \\ref{sets-lemma-construct-category}", "starting with the set of schemes $\\{X, Y, S\\}$. Choose any set of", "coverings $\\text{Cov}_{fppf}$ on $\\Sch_\\alpha$ as in", "Sets, Lemma \\ref{sets-lemma-coverings-site}", "starting with the category $\\Sch_\\alpha$ and the class of fppf", "coverings. Let $\\Sch_{fppf}$ denote the big fppf site so", "obtained. Next, for $\\tau \\in \\{Zariski, \\etale, smooth, syntomic\\}$", "let $\\Sch_\\tau$ have the same underlying category as", "$\\Sch_{fppf}$ with coverings", "$\\text{Cov}_\\tau \\subset \\text{Cov}_{fppf}$ simply the subset of", "$\\tau$-coverings. It is straightforward to check that this gives rise", "to a big site $\\Sch_\\tau$." ], "refs": [ "sets-lemma-construct-category", "sets-lemma-coverings-site" ], "ref_ids": [ 8789, 8800 ] }, { "id": 12579, "type": "other", "label": "pic-remark-when-proposition-applies", "categories": [ "pic" ], "title": "pic-remark-when-proposition-applies", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. The assumption of", "Proposition \\ref{proposition-hilb-d-representable} and", "hence the conclusion holds in each of the following cases:", "\\begin{enumerate}", "\\item $X$ is quasi-affine,", "\\item $f$ is quasi-affine,", "\\item $f$ is quasi-projective,", "\\item $f$ is locally projective,", "\\item there exists an ample invertible sheaf on $X$,", "\\item there exists an $f$-ample invertible sheaf on $X$, and", "\\item there exists an $f$-very ample invertible sheaf on $X$.", "\\end{enumerate}", "Namely, in each of these cases, every finite set of points of", "a fibre $X_s$ is contained in a quasi-compact open $U$ of $X$", "which comes with an ample invertible sheaf, is isomorphic", "to an open of an affine scheme, or is isomorphic to an open", "of $\\text{Proj}$ of a graded ring (in each case this follows", "by unwinding the definitions). Thus the existence of suitable", "affine opens by", "Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-affine}." ], "refs": [ "pic-proposition-hilb-d-representable", "properties-lemma-ample-finite-set-in-affine" ], "ref_ids": [ 12574, 3062 ] }, { "id": 12580, "type": "other", "label": "pic-remark-universal-object-hilb-d", "categories": [ "pic" ], "title": "pic-remark-universal-object-hilb-d", "contents": [ "Let $X$ be a geometrically irreducible smooth proper curve over a field $k$", "as in Proposition \\ref{proposition-hilb-d}. Let $d \\geq 0$. The universal", "closed object is a relatively effective divisor", "$$", "D_{univ} \\subset \\underline{\\Hilbfunctor}^{d + 1}_{X/k} \\times_k X", "$$", "over $\\underline{\\Hilbfunctor}^{d + 1}_{X/k}$ by", "Lemma \\ref{lemma-divisors-on-curves}.", "In fact, $D_{univ}$ is isomorphic as a scheme to", "$\\underline{\\Hilbfunctor}^d_{X/k} \\times_k X$, see proof of", "Lemma \\ref{lemma-universal-object}.", "In particular, $D_{univ}$ is an effective Cartier divisor and", "we obtain an invertible module", "$\\mathcal{O}(D_{univ})$. If $[D] \\in \\underline{\\Hilbfunctor}^{d + 1}_{X/k}$", "denotes the $k$-rational point corresponding to the effective", "Cartier divisor $D \\subset X$ of degree $d + 1$, then the restriction", "of $\\mathcal{O}(D_{univ})$ to the fibre $[D] \\times X$ is", "$\\mathcal{O}_X(D)$." ], "refs": [ "pic-proposition-hilb-d", "pic-lemma-divisors-on-curves", "pic-lemma-universal-object" ], "ref_ids": [ 12575, 12559, 12562 ] }, { "id": 12668, "type": "other", "label": "constructions-remark-relative-glueing-functorial", "categories": [ "constructions" ], "title": "constructions-remark-relative-glueing-functorial", "contents": [ "There is a functoriality property for the constructions explained", "in Lemmas \\ref{lemma-relative-glueing} and", "\\ref{lemma-relative-glueing-sheaves}. Namely, suppose given", "two collections of data $(f_U : X_U \\to U, \\rho^U_V)$ and", "$(g_U : Y_U \\to U, \\sigma^U_V)$ as in Lemma \\ref{lemma-relative-glueing}.", "Suppose for every $U \\in \\mathcal{B}$ given", "a morphism $h_U : X_U \\to Y_U$ over $U$ compatible with", "the restrictions $\\rho^U_V$ and $\\sigma^U_V$. Functoriality", "means that this gives rise to a morphism of schemes", "$h : X \\to Y$ over $S$ restricting back to the morphisms $h_U$,", "where $f : X \\to S$ is obtained from", "the datum $(f_U : X_U \\to U, \\rho^U_V)$ and $g : Y \\to S$", "is obtained from the datum $(g_U : Y_U \\to U, \\sigma^U_V)$.", "\\medskip\\noindent", "Similarly, suppose given", "two collections of data", "$(f_U : X_U \\to U, \\mathcal{F}_U, \\rho^U_V, \\theta^U_V)$ and", "$(g_U : Y_U \\to U, \\mathcal{G}_U, \\sigma^U_V, \\eta^U_V)$", "as in Lemma \\ref{lemma-relative-glueing-sheaves}.", "Suppose for every $U \\in \\mathcal{B}$ given", "a morphism $h_U : X_U \\to Y_U$ over $U$ compatible with", "the restrictions $\\rho^U_V$ and $\\sigma^U_V$, and a morphism", "$\\tau_U : h_U^*\\mathcal{G}_U \\to \\mathcal{F}_U$ compatible with", "the maps $\\theta^U_V$ and $\\eta^U_V$. Functoriality", "means that these give rise to a morphism of schemes", "$h : X \\to Y$ over $S$ restricting back to the morphisms $h_U$,", "and a morphism $h^*\\mathcal{G} \\to \\mathcal{F}$ restricting back", "to the maps $h_U$", "where $(f : X \\to S, \\mathcal{F})$ is obtained from the datum", "$(f_U : X_U \\to U, \\mathcal{F}_U, \\rho^U_V, \\theta^U_V)$ and", "where $(g : Y \\to S, \\mathcal{G})$ is obtained from the datum", "$(g_U : Y_U \\to U, \\mathcal{G}_U, \\sigma^U_V, \\eta^U_V)$.", "\\medskip\\noindent", "We omit the verifications and we omit a suitable formulation of", "``equivalence of categories'' between relative glueing data", "and relative objects." ], "refs": [ "constructions-lemma-relative-glueing", "constructions-lemma-relative-glueing-sheaves", "constructions-lemma-relative-glueing", "constructions-lemma-relative-glueing-sheaves" ], "ref_ids": [ 12581, 12582, 12581, 12582 ] }, { "id": 12669, "type": "other", "label": "constructions-remark-global-sections-not-isomorphism", "categories": [ "constructions" ], "title": "constructions-remark-global-sections-not-isomorphism", "contents": [ "The map from $M_0$ to the global sections of $\\widetilde M$", "is generally far from being an isomorphism. A trivial", "example is to take $S = k[x, y, z]$ with $1 = \\deg(x) = \\deg(y) = \\deg(z)$", "(or any number of variables) and to take $M = S/(x^{100}, y^{100}, z^{100})$.", "It is easy to see that $\\widetilde M = 0$, but $M_0 = k$." ], "refs": [], "ref_ids": [] }, { "id": 12670, "type": "other", "label": "constructions-remark-not-isomorphism", "categories": [ "constructions" ], "title": "constructions-remark-not-isomorphism", "contents": [ "In general the map constructed in Lemma \\ref{lemma-widetilde-tensor}", "above is not an isomorphism. Here is an example. Let $k$", "be a field. Let $S = k[x, y, z]$ with $k$ in degree $0$ and", "$\\deg(x) = 1$, $\\deg(y) = 2$, $\\deg(z) = 3$.", "Let $M = S(1)$ and $N = S(2)$, see", "Algebra, Section \\ref{algebra-section-graded}", "for notation. Then $M \\otimes_S N = S(3)$.", "Note that", "\\begin{eqnarray*}", "S_z", "& = &", "k[x, y, z, 1/z] \\\\", "S_{(z)}", "& = &", "k[x^3/z, xy/z, y^3/z^2]", "\\cong", "k[u, v, w]/(uw - v^3) \\\\", "M_{(z)} & = & S_{(z)} \\cdot x + S_{(z)} \\cdot y^2/z \\subset S_z \\\\", "N_{(z)} & = & S_{(z)} \\cdot y + S_{(z)} \\cdot x^2 \\subset S_z \\\\", "S(3)_{(z)} & = & S_{(z)} \\cdot z \\subset S_z", "\\end{eqnarray*}", "Consider the maximal ideal $\\mathfrak m = (u, v, w) \\subset S_{(z)}$.", "It is not hard to see that both $M_{(z)}/\\mathfrak mM_{(z)}$", "and $N_{(z)}/\\mathfrak mN_{(z)}$ have dimension $2$ over", "$\\kappa(\\mathfrak m)$. But", "$S(3)_{(z)}/\\mathfrak mS(3)_{(z)}$ has dimension $1$.", "Thus the map $M_{(z)} \\otimes N_{(z)} \\to S(3)_{(z)}$ is not", "an isomorphism." ], "refs": [ "constructions-lemma-widetilde-tensor" ], "ref_ids": [ 12601 ] }, { "id": 12671, "type": "other", "label": "constructions-remark-missing-finite-type", "categories": [ "constructions" ], "title": "constructions-remark-missing-finite-type", "contents": [ "What's missing in the list of properties above? Well to be sure the property", "of being of finite type. The reason we do not list this here is that we have", "not yet defined the notion of finite type at this point. (Another property", "which is missing is ``smoothness''. And I'm sure there are many more you can", "think of.)" ], "refs": [], "ref_ids": [] }, { "id": 12672, "type": "other", "label": "constructions-remark-not-in-invertible-locus", "categories": [ "constructions" ], "title": "constructions-remark-not-in-invertible-locus", "contents": [ "Assumptions as in Lemma \\ref{lemma-invertible-map-into-proj} above.", "The image of the morphism $r_{\\mathcal{L}, \\psi}$ need not be", "contained in the locus where the sheaf $\\mathcal{O}_X(1)$", "is invertible.", "Here is an example.", "Let $k$ be a field.", "Let $S = k[A, B, C]$ graded by $\\deg(A) = 1$, $\\deg(B) = 2$, $\\deg(C) = 3$.", "Set $X = \\text{Proj}(S)$.", "Let $T = \\mathbf{P}^2_k = \\text{Proj}(k[X_0, X_1, X_2])$.", "Recall that $\\mathcal{L} = \\mathcal{O}_T(1)$ is invertible", "and that $\\mathcal{O}_T(n) = \\mathcal{L}^{\\otimes n}$.", "Consider the composition $\\psi$ of the maps", "$$", "S \\to k[X_0, X_1, X_2] \\to \\Gamma_*(T, \\mathcal{L}).", "$$", "Here the first map is $A \\mapsto X_0$, $B \\mapsto X_1^2$,", "$C \\mapsto X_2^3$ and the second map is (\\ref{equation-global-sections}).", "By the lemma this corresponds to a morphism", "$r_{\\mathcal{L}, \\psi} : T \\to X = \\text{Proj}(S)$", "which is easily seen to be surjective. On the other hand, in", "Remark \\ref{remark-not-isomorphism} we showed that the sheaf", "$\\mathcal{O}_X(1)$ is not invertible at all points of $X$." ], "refs": [ "constructions-lemma-invertible-map-into-proj", "constructions-remark-not-isomorphism" ], "ref_ids": [ 12628, 12670 ] }, { "id": 12807, "type": "other", "label": "algebraization-remark-compare-with-completion", "categories": [ "algebraization" ], "title": "algebraization-remark-compare-with-completion", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of", "ideals. Let $K \\mapsto K^\\wedge$ be the derived completion functor", "of Proposition \\ref{proposition-derived-completion}.", "For any $n \\geq 1$ the object", "$K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n$", "is derived complete as it is annihilated by powers of", "local sections of $\\mathcal{I}$. Hence there is a canonical factorization", "$$", "K \\to K^\\wedge \\to K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n", "$$", "of the canonical map", "$K \\to K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n$.", "These maps are compatible for varying $n$ and we obtain a comparison map", "$$", "K^\\wedge", "\\longrightarrow", "R\\lim \\left(K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n\\right)", "$$", "The right hand side is more recognizable as a kind of completion.", "In general this comparison map is not an isomorphism." ], "refs": [ "algebraization-proposition-derived-completion" ], "ref_ids": [ 12790 ] }, { "id": 12808, "type": "other", "label": "algebraization-remark-localization-and-completion", "categories": [ "algebraization" ], "title": "algebraization-remark-localization-and-completion", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of", "ideals. Let $K \\mapsto K^\\wedge$ be the derived completion functor", "of Proposition \\ref{proposition-derived-completion}. It follows", "from the construction in the proof of the proposition that $K^\\wedge|_U$", "is the derived completion of $K|_U$ for any $U \\in \\Ob(\\mathcal{C})$.", "But we can also prove this as follows. From the definition", "of derived complete objects it follows that $K^\\wedge|_U$ is derived complete.", "Thus we obtain a canonical map $a : (K|_U)^\\wedge \\to K^\\wedge|_U$.", "On the other hand, if $E$ is a derived complete object of", "$D(\\mathcal{O}_U)$, then $Rj_*E$ is a derived complete object of", "$D(\\mathcal{O})$ by Lemma \\ref{lemma-pushforward-derived-complete}.", "Here $j$ is the localization morphism", "(Modules on Sites, Section \\ref{sites-modules-section-localize}).", "Hence we also obtain a canonical", "map $b : K^\\wedge \\to Rj_*((K|_U)^\\wedge)$. We omit the (formal) verification", "that the adjoint of $b$ is the inverse of $a$." ], "refs": [ "algebraization-proposition-derived-completion", "algebraization-lemma-pushforward-derived-complete" ], "ref_ids": [ 12790, 12699 ] }, { "id": 12809, "type": "other", "label": "algebraization-remark-completed-tensor-product", "categories": [ "algebraization" ], "title": "algebraization-remark-completed-tensor-product", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let", "$\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of ideals. ", "Denote $K \\mapsto K^\\wedge$ the adjoint of", "Proposition \\ref{proposition-derived-completion}.", "Then we set", "$$", "K \\otimes^\\wedge_\\mathcal{O} L = (K \\otimes_\\mathcal{O}^\\mathbf{L} L)^\\wedge", "$$", "This {\\it completed tensor product} defines a functor", "$D_{comp}(\\mathcal{O}) \\times D_{comp}(\\mathcal{O}) \\to D_{comp}(\\mathcal{O})$", "such that we have", "$$", "\\Hom_{D_{comp}(\\mathcal{O})}(K, R\\SheafHom_\\mathcal{O}(L, M))", "=", "\\Hom_{D_{comp}(\\mathcal{O})}(K \\otimes_\\mathcal{O}^\\wedge L, M)", "$$", "for $K, L, M \\in D_{comp}(\\mathcal{O})$. Note that", "$R\\SheafHom_\\mathcal{O}(L, M) \\in D_{comp}(\\mathcal{O})$ by", "Lemma \\ref{lemma-derived-complete-internal-hom}." ], "refs": [ "algebraization-proposition-derived-completion", "algebraization-lemma-derived-complete-internal-hom" ], "ref_ids": [ 12790, 12697 ] }, { "id": 12810, "type": "other", "label": "algebraization-remark-local-calculation-derived-completion", "categories": [ "algebraization" ], "title": "algebraization-remark-local-calculation-derived-completion", "contents": [ "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.", "Let $\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of", "ideals. Let $K \\mapsto K^\\wedge$ be the derived completion of", "Proposition \\ref{proposition-derived-completion}.", "Let $U \\in \\Ob(\\mathcal{C})$ be an object such that $\\mathcal{I}$", "is generated as an ideal sheaf by $f_1, \\ldots, f_r \\in \\mathcal{I}(U)$.", "Set $A = \\mathcal{O}(U)$ and $I = (f_1, \\ldots, f_r) \\subset A$.", "Warning: it may not be the case that $I = \\mathcal{I}(U)$.", "Then we have", "$$", "R\\Gamma(U, K^\\wedge) = R\\Gamma(U, K)^\\wedge", "$$", "where the right hand side is the derived completion of", "the object $R\\Gamma(U, K)$ of $D(A)$ with respect to $I$.", "This is true because derived completion commutes with localization", "(Remark \\ref{remark-localization-and-completion}) and", "Lemma \\ref{lemma-formal-functions-general}." ], "refs": [ "algebraization-proposition-derived-completion", "algebraization-remark-localization-and-completion", "algebraization-lemma-formal-functions-general" ], "ref_ids": [ 12790, 12808, 12708 ] }, { "id": 12811, "type": "other", "label": "algebraization-remark-references", "categories": [ "algebraization" ], "title": "algebraization-remark-references", "contents": [ "Here are some references to discussions of related material the literature.", "It seems that a ``derived formal functions theorem'' for proper maps", "goes back to \\cite[Theorem 6.3.1]{lurie-thesis}.", "There is the discussion in \\cite{dag12}, especially", "Chapter 4 which discusses the affine story, see", "More on Algebra, Section \\ref{more-algebra-section-derived-completion}.", "In \\cite[Section 2.9]{G-R} one finds a discussion of proper base change and", "derived completion using (ind) coherent modules.", "An analogue of (\\ref{equation-formal-functions})", "for complexes of quasi-coherent modules can be found as", "\\cite[Theorem 6.5]{HL-P}" ], "refs": [], "ref_ids": [] }, { "id": 12812, "type": "other", "label": "algebraization-remark-interesting-case-variant", "categories": [ "algebraization" ], "title": "algebraization-remark-interesting-case-variant", "contents": [ "In Lemma \\ref{lemma-algebraization-principal-variant}", "if $A$ is universally catenary with Cohen-Macaulay", "formal fibres (for example if $A$ has a dualizing complex), then", "the condition that", "$H^1_\\mathfrak a(A/fA)$ and $H^2_\\mathfrak a(A/fA)$", "are finite $A$-modules, is equivalent with", "$$", "\\text{depth}((A/f)_\\mathfrak q) + \\dim((A/\\mathfrak q)_\\mathfrak p) > 2", "$$", "for all $\\mathfrak q \\in V(f) \\setminus V(\\mathfrak a)$", "and $\\mathfrak p \\in V(\\mathfrak q) \\cap V(\\mathfrak a)$", "by Local Cohomology, Theorem \\ref{local-cohomology-theorem-finiteness}.", "\\medskip\\noindent", "For example, if $A/fA$ is $(S_2)$ and if every irreducible", "component of $Z = V(\\mathfrak a)$ has codimension $\\geq 3$", "in $Y = \\Spec(A/fA)$, then we get the finiteness of", "$H^1_\\mathfrak a(A/fA)$ and $H^2_\\mathfrak a(A/fA)$.", "This should be contrasted with the slightly weaker conditions", "found in Lemma \\ref{lemma-algebraization-principal}", "(see also Remark \\ref{remark-interesting-case})." ], "refs": [ "algebraization-lemma-algebraization-principal-variant", "local-cohomology-theorem-finiteness", "algebraization-lemma-algebraization-principal", "algebraization-remark-interesting-case" ], "ref_ids": [ 12754, 9694, 12766, 12814 ] }, { "id": 12813, "type": "other", "label": "algebraization-remark-discussion", "categories": [ "algebraization" ], "title": "algebraization-remark-discussion", "contents": [ "Let $Y$ be a Noetherian scheme and let $Z \\subset Y$ be a closed subset.", "By Lemma \\ref{lemma-discussion} we have", "$$", "\\delta_Z(y) \\leq \\min", "\\left\\{ k \\middle|", "\\begin{matrix}", "\\text{ there exist specializations in }Y \\\\", "y_0 \\leftarrow y'_0 \\rightarrow y_1 \\leftarrow y'_1 \\rightarrow \\ldots", "\\leftarrow y'_{k - 1} \\rightarrow y_k = y \\\\", "\\text{ with }y_0 \\in Z\\text{ and }y_i' \\leadsto y_i", "\\text{ immediate}", "\\end{matrix}", "\\right\\}", "$$", "We claim that if $Y$ is of finite type over a field,", "then equality holds. If we ever need this result we", "will formulate a precise result and prove it here.", "However, in general if we define $\\delta_Z$", "by the right hand side of this inequality, then we don't", "know if Lemma \\ref{lemma-change-distance-function} remains true." ], "refs": [ "algebraization-lemma-discussion", "algebraization-lemma-change-distance-function" ], "ref_ids": [ 12758, 12759 ] }, { "id": 12814, "type": "other", "label": "algebraization-remark-interesting-case", "categories": [ "algebraization" ], "title": "algebraization-remark-interesting-case", "contents": [ "Let $(A, \\mathfrak m)$ be a complete Noetherian normal local domain", "of dimension $\\geq 4$ and let $f \\in \\mathfrak m$ be nonzero.", "Then assumptions (1), (2), (3), (5), and (6) of", "Lemma \\ref{lemma-algebraization-principal}", "are satisfied. Thus vectorbundles", "on the formal completion of $U$ along $U \\cap V(f)$", "can be algebraized. In Lemma \\ref{lemma-algebraization-principal-bis}", "we will generalize this to more general coherent formal modules;", "please also compare with Remark \\ref{remark-interesting-case-bis}." ], "refs": [ "algebraization-lemma-algebraization-principal", "algebraization-lemma-algebraization-principal-bis", "algebraization-remark-interesting-case-bis" ], "ref_ids": [ 12766, 12768, 12815 ] }, { "id": 12815, "type": "other", "label": "algebraization-remark-interesting-case-bis", "categories": [ "algebraization" ], "title": "algebraization-remark-interesting-case-bis", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring which has a", "dualizing complex and is complete with respect to $f \\in \\mathfrak m$.", "Let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, f\\mathcal{O}_U)$", "where $U$ is the punctured spectrum of $A$.", "Set $Y = V(f) \\subset X = \\Spec(A)$.", "If for $y \\in U \\cap V(f)$ closed in $U$, i.e., with", "$\\dim(\\overline{\\{y\\}}) = 1$, we assume the", "$\\mathcal{O}_{X, y}^\\wedge$-module $\\mathcal{F}_y^\\wedge$", "satisfies the following two conditions", "\\begin{enumerate}", "\\item $\\mathcal{F}_y^\\wedge[1/f]$ is $(S_2)$ as a", "$\\mathcal{O}_{X, y}^\\wedge[1/f]$-module, and", "\\item for $\\mathfrak p \\in \\text{Ass}(\\mathcal{F}_y^\\wedge[1/f])$", "we have $\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) \\geq 3$.", "\\end{enumerate}", "Then $(\\mathcal{F}_n)$ is the completion of a coherent module on $U$.", "This follows from Lemmas \\ref{lemma-algebraization-principal-bis}", "and \\ref{lemma-unwinding-conditions}." ], "refs": [ "algebraization-lemma-algebraization-principal-bis", "algebraization-lemma-unwinding-conditions" ], "ref_ids": [ 12768, 12769 ] }, { "id": 12816, "type": "other", "label": "algebraization-remark-question", "categories": [ "algebraization" ], "title": "algebraization-remark-question", "contents": [ "We are unable to prove or disprove the analogue of", "Proposition \\ref{proposition-d-generators}", "where the assumption that $I$ has $d$ generators", "is replaced with the assumption $\\text{cd}(A, I) \\leq d$.", "If you know a proof or have a counter example, please email", "\\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}.", "Another obvious question is to what extend the conditions in", "Proposition \\ref{proposition-d-generators}", "are necessary." ], "refs": [ "algebraization-proposition-d-generators", "algebraization-proposition-d-generators" ], "ref_ids": [ 12794, 12794 ] }, { "id": 12817, "type": "other", "label": "algebraization-remark-interesting-case-ter", "categories": [ "algebraization" ], "title": "algebraization-remark-interesting-case-ter", "contents": [ "In the situation of", "Proposition \\ref{proposition-algebraization-regular-sequence}", "if we assume $A$ has a dualizing complex, then", "the condition that $H^0(U, \\mathcal{F}_1)$ and", "$H^1(U, \\mathcal{F}_1)$ are finite is equivalent to", "$$", "\\text{depth}(\\mathcal{F}_{1, y}) +", "\\dim(\\mathcal{O}_{\\overline{\\{y\\}}, z}) > 2", "$$", "for all $y \\in U \\cap Y$ and $z \\in Z \\cap \\overline{\\{y\\}}$.", "See Local Cohomology, Lemma \\ref{local-cohomology-lemma-finiteness-Rjstar}.", "This holds for example if $\\mathcal{F}_1$ is a finite locally free", "$\\mathcal{O}_{U \\cap Y}$-module, $Y$ is $(S_2)$, and", "$\\text{codim}(Z', Y') \\geq 3$ for every pair of irreducible components", "$Y'$ of $Y$, $Z'$ of $Z$ with $Z' \\subset Y'$." ], "refs": [ "algebraization-proposition-algebraization-regular-sequence", "local-cohomology-lemma-finiteness-Rjstar" ], "ref_ids": [ 12795, 9743 ] }, { "id": 12818, "type": "other", "label": "algebraization-remark-interesting-case-quater", "categories": [ "algebraization" ], "title": "algebraization-remark-interesting-case-quater", "contents": [ "Proposition \\ref{proposition-algebraization-flat} is a local version", "of \\cite[Theorem 2.10 (i)]{Baranovsky}. It is straightforward to deduce", "the global results from the local one; we will sketch the argument.", "Namely, suppose $(R, \\mathfrak m)$", "is a complete Noetherian local ring and $X \\to \\Spec(R)$ is a proper morphism.", "For $n \\geq 1$ set $X_n = X \\times_{\\Spec(R)} \\Spec(R/\\mathfrak m^n)$.", "Let $Z \\subset X_1$ be a closed subset of the special fibre.", "Set $U = X \\setminus Z$ and denote $j : U \\to X$ the inclusion morphism.", "Suppose given an object", "$$", "(\\mathcal{F}_n) \\text{ of } \\textit{Coh}(U, \\mathfrak m\\mathcal{O}_U)", "$$", "which is flat over $R$ in the sense that $\\mathcal{F}_n$ is flat over", "$R/\\mathfrak m^n$ for all $n$.", "Assume that $j_*\\mathcal{F}_1$ and $R^1j_*\\mathcal{F}_1$ are coherent", "modules. Then affine locally on $X$ we get a canonical extension", "of $(\\mathcal{F}_n)$ by", "Proposition \\ref{proposition-algebraization-flat}", "and formation of this extension commutes with localization", "(by Lemma \\ref{lemma-algebraization-principal-variant}).", "Thus we get a canonical global object $(\\mathcal{G}_n)$ of", "$\\textit{Coh}(X, \\mathfrak m\\mathcal{O}_X)$", "whose restriction of $U$ is $(\\mathcal{F}_n)$.", "By Grothendieck's existence theorem", "(Cohomology of Schemes, Proposition", "\\ref{coherent-proposition-existence-proper})", "we see there exists a coherent $\\mathcal{O}_X$-module", "$\\mathcal{G}$ whose completion is $(\\mathcal{G}_n)$.", "In this way we see that $(\\mathcal{F}_n)$ is algebraizable, i.e.,", "it is the completion of a coherent $\\mathcal{O}_U$-module.", "\\medskip\\noindent", "We add that the coherence of $j_*\\mathcal{F}_1$ and $R^1j_*\\mathcal{F}_1$", "is a condition on the special fibre. Namely, if we denote", "$j_1 : U_1 \\to X_1$ the special fibre of $j : U \\to X$, then we can", "think of $\\mathcal{F}_1$ as a coherent sheaf on $U_1$ and we have", "$j_*\\mathcal{F}_1 = j_{1, *}\\mathcal{F}_1$ and", "$R^1j_*\\mathcal{F}_1 = R^1j_{1, *}\\mathcal{F}_1$.", "Hence for example if $X_1$ is $(S_2)$ and irreducible, we have", "$\\dim(X_1) - \\dim(Z) \\geq 3$, and $\\mathcal{F}_1$ is a locally free", "$\\mathcal{O}_{U_1}$-module, then $j_{1, *}\\mathcal{F}_1$ and", "$R^1j_{1, *}\\mathcal{F}_1$ are coherent modules." ], "refs": [ "algebraization-proposition-algebraization-flat", "algebraization-proposition-algebraization-flat", "algebraization-lemma-algebraization-principal-variant", "coherent-proposition-existence-proper" ], "ref_ids": [ 12796, 12796, 12754, 3402 ] }, { "id": 12819, "type": "other", "label": "algebraization-remark-compare-SGA2", "categories": [ "algebraization" ], "title": "algebraization-remark-compare-SGA2", "contents": [ "In SGA2 we find the following result. Let $(A, \\mathfrak m)$ be a", "Noetherian local ring. Let $f \\in \\mathfrak m$. Assume $A$", "is a quotient of a regular ring, the element", "$f$ is a nonzerodivisor, and", "\\begin{enumerate}", "\\item[(a)] if $\\mathfrak p \\subset A$ is a prime ideal with", "$\\dim(A/\\mathfrak p) = 1$, then $\\text{depth}(A_\\mathfrak p) \\geq 2$, and", "\\item[(b)] $\\text{depth}(A/fA) \\geq 3$, or equivalently", "$\\text{depth}(A) \\geq 4$.", "\\end{enumerate}", "Let $U$, resp.\\ $U_0$ be the punctured spectrum of $A$, resp.\\ $A/fA$. Then", "the map", "$$", "\\Pic(U) \\to \\Pic(U_0)", "$$", "is injective. This is \\cite[Exposee XI, Lemma 3.16]{SGA2}\\footnote{Condition", "(a) follows from condition (b), see", "Algebra, Lemma \\ref{algebra-lemma-depth-localization}.}. This result", "from SGA2 follows from Proposition \\ref{proposition-injective-pic}", "because", "\\begin{enumerate}", "\\item a quotient of a regular ring has a dualizing complex (see", "Dualizing Complexes, Lemma \\ref{dualizing-lemma-regular-gorenstein} and", "Proposition \\ref{dualizing-proposition-dualizing-essentially-finite-type}), and", "\\item if $\\text{depth}(A) \\geq 4$ then $\\text{depth}(A_\\mathfrak p) \\geq 2$", "for all primes $\\mathfrak p$ with $\\dim(A/\\mathfrak p) = 2$, see", "Algebra, Lemma \\ref{algebra-lemma-depth-localization}.", "\\end{enumerate}" ], "refs": [ "algebra-lemma-depth-localization", "algebraization-proposition-injective-pic", "dualizing-lemma-regular-gorenstein", "dualizing-proposition-dualizing-essentially-finite-type", "algebra-lemma-depth-localization" ], "ref_ids": [ 777, 12798, 2880, 2926, 777 ] }, { "id": 12820, "type": "other", "label": "algebraization-remark-surjective-Pic-second", "categories": [ "algebraization" ], "title": "algebraization-remark-surjective-Pic-second", "contents": [ "Let $(A, \\mathfrak m)$ be a Noetherian local ring and $f \\in \\mathfrak m$.", "The conclusion of Lemma \\ref{lemma-surjective-Pic-first} holds if we assume", "\\begin{enumerate}", "\\item $A$ has a dualizing complex,", "\\item $A$ is $f$-adically complete,", "\\item $f$ is a nonzerodivisor,", "\\item one of the following is true", "\\begin{enumerate}", "\\item $A_f$ is $(S_2)$ and for $\\mathfrak p \\subset A$,", "$f \\not \\in \\mathfrak p$ minimal we have $\\dim(A/\\mathfrak p) \\geq 4$, or", "\\item if $\\mathfrak p \\not \\in V(f)$ and", "$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then", "$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.", "\\end{enumerate}", "\\item $H^3_{\\mathfrak m}(A/fA) = 0$.", "\\end{enumerate}", "The proof is exactly the same as the proof of", "Lemma \\ref{lemma-surjective-Pic-first}", "using Lemma \\ref{lemma-equivalence-better} instead of", "Lemma \\ref{lemma-equivalence}.", "Two points need to be made here: (a)", "it seems hard to find examples where one knows", "$H^3_{\\mathfrak m}(A/fA) = 0$ without assuming", "$\\text{depth}(A/fA) \\geq 4$, and", "(b) the proof of Lemma \\ref{lemma-equivalence-better} is a", "good deal harder than the proof of Lemma \\ref{lemma-equivalence}." ], "refs": [ "algebraization-lemma-surjective-Pic-first", "algebraization-lemma-surjective-Pic-first", "algebraization-lemma-equivalence-better", "algebraization-lemma-equivalence", "algebraization-lemma-equivalence-better", "algebraization-lemma-equivalence" ], "ref_ids": [ 12784, 12784, 12777, 12778, 12777, 12778 ] }, { "id": 12900, "type": "other", "label": "spaces-over-fields-remark-alternate-proof-scheme-codim-1", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-remark-alternate-proof-scheme-codim-1", "contents": [ "Here is a sketch of a proof of", "Lemma \\ref{lemma-codim-1-point-in-schematic-locus}", "which avoids using", "More on Groupoids, Lemma", "\\ref{more-groupoids-lemma-find-affine-codimension-1}.", "\\medskip\\noindent", "Step 1. We may assume $X$ is a reduced Noetherian separated algebraic space", "(for example by Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian}", "or by", "Limits of Spaces, Lemma \\ref{spaces-limits-lemma-reduction-scheme})", "and we may choose a finite surjective morphism", "$Y \\to X$ where $Y$ is a Noetherian scheme (by", "Limits of Spaces, Proposition", "\\ref{spaces-limits-proposition-there-is-a-scheme-finite-over}).", "\\medskip\\noindent", "Step 2. After replacing $X$ by an open neighbourhood of $x$, there", "exists a birational finite morphism $X' \\to X$ and a closed subscheme", "$Y' \\subset X' \\times_X Y$ such that $Y' \\to X'$ is surjective", "finite locally free. Namely, because $X$ is reduced there is a dense", "open subspace $U \\subset X$ over which $Y$ is flat (Morphisms of Spaces,", "Proposition \\ref{spaces-morphisms-proposition-generic-flatness-reduced}).", "Then we can choose a $U$-admissible blowup $b : \\tilde X \\to X$ such", "that the strict transform $\\tilde Y$ of $Y$ is flat over $\\tilde X$, see", "More on Morphisms of Spaces, Lemma", "\\ref{spaces-more-morphisms-lemma-flat-after-blowing-up}.", "(An alternative is to use Hilbert schemes if one wants to avoid using", "the result on blowups).", "Then we let $X' \\subset \\tilde X$ be the scheme theoretic", "closure of $b^{-1}(U)$ and $Y' = X' \\times_{\\tilde X} \\tilde Y$.", "Since $x$ is a codimension $1$ point, we see that $X' \\to X$ is finite over a", "neighbourhood of $x$ (Lemma \\ref{lemma-finite-in-codim-1}).", "\\medskip\\noindent", "Step 3. After shrinking $X$ to a smaller neighbourhood of $x$ we get that", "$X'$ is a scheme. This holds because $Y'$ is a scheme and $Y' \\to X'$", "being finite locally free and because every finite set of codimension $1$", "points of $Y'$ is contained in an affine open. Use", "Properties of Spaces, Proposition", "\\ref{spaces-properties-proposition-finite-flat-equivalence-global}", "and", "Varieties, Proposition", "\\ref{varieties-proposition-finite-set-of-points-of-codim-1-in-affine}.", "\\medskip\\noindent", "Step 4. There exists an affine open $W' \\subset X'$ containing all points", "lying over $x$ which is the inverse image of an open subspace of $X$.", "To prove this let $Z \\subset X$ be the closure of the set of points", "where $X' \\to X$ is not an isomorphism. We may assume $x \\in Z$ otherwise", "we are already done. Then $x$ is a generic point of an irreducible", "component of $Z$ and after shrinking $X$ we may assume $Z$ is an affine scheme", "(Lemma \\ref{lemma-generic-point-in-schematic-locus}).", "Then the inverse image $Z' \\subset X'$ is an affine scheme as well.", "Say $x_1, \\ldots, x_n \\in Z'$ are the points mapping to $x$.", "Then we can find an affine open $W'$ in $X'$ whose intersection with", "$Z'$ is the inverse image of a principal open of $Z$ containing $x$.", "Namely, we first pick an affine open $W' \\subset X'$ containing", "$x_1, \\ldots, x_n$ using Varieties, Proposition", "\\ref{varieties-proposition-finite-set-of-points-of-codim-1-in-affine}.", "Then we pick a principal open $D(f) \\subset Z$ containing $x$", "whose inverse image $D(f|_{Z'})$ is contained in $W' \\cap Z'$.", "Then we pick $f' \\in \\Gamma(W', \\mathcal{O}_{W'})$ restricting", "to $f|_{Z'}$ and we replace $W'$ by $D(f') \\subset W'$.", "Since $X' \\to X$ is an isomorphism away from $Z' \\to Z$ the choice", "of $W'$ guarantees that the image $W \\subset X$ of $W'$ is open", "with inverse image $W'$ in $X'$.", "\\medskip\\noindent", "Step 5. Then $W' \\to W$ is a finite surjective morphism and $W$ is a scheme by", "Cohomology of Spaces, Lemma", "\\ref{spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian}", "and the proof is complete." ], "refs": [ "spaces-over-fields-lemma-codim-1-point-in-schematic-locus", "more-groupoids-lemma-find-affine-codimension-1", "spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian", "spaces-limits-lemma-reduction-scheme", "spaces-limits-proposition-there-is-a-scheme-finite-over", "spaces-morphisms-proposition-generic-flatness-reduced", "spaces-more-morphisms-lemma-flat-after-blowing-up", "spaces-over-fields-lemma-finite-in-codim-1", "spaces-properties-proposition-finite-flat-equivalence-global", "varieties-proposition-finite-set-of-points-of-codim-1-in-affine", "spaces-over-fields-lemma-generic-point-in-schematic-locus", "varieties-proposition-finite-set-of-points-of-codim-1-in-affine", "spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian" ], "ref_ids": [ 12845, 2502, 11326, 4627, 4659, 4981, 190, 12822, 11918, 11139, 12844, 11139, 11326 ] }, { "id": 12901, "type": "other", "label": "spaces-over-fields-remark-when-does-the-argument-work", "categories": [ "spaces-over-fields" ], "title": "spaces-over-fields-remark-when-does-the-argument-work", "contents": [ "Let $k$ be finite field. Let $K \\supset k$ be a geometrically", "irreducible field extension. Then $K$ is the limit of geometrically", "irreducible finite type $k$-algebras $A$. Given $A$ the estimates", "of Lang and Weil \\cite{LW}, show that for $n \\gg 0$ there exists", "an $k$-algebra homomorphism $A \\to k'$ with $k'/k$ of degree $n$.", "Analyzing the argument given in the proof of", "Lemma \\ref{lemma-scheme-after-purely-transcendental-base-change}", "we see that if $X$ is a quasi-separated algebraic space over $k$", "and $X_K$ is a scheme, then $X$ is a scheme. If we ever need this", "result we will precisely formulate it and prove it here." ], "refs": [ "spaces-over-fields-lemma-scheme-after-purely-transcendental-base-change" ], "ref_ids": [ 12850 ] }, { "id": 13159, "type": "other", "label": "dga-remark-evaluation-map-left", "categories": [ "dga" ], "title": "dga-remark-evaluation-map-left", "contents": [ "Let $R$ be a ring. Let $A$ be a differential graded $R$-algebra.", "Let $M$ be a left differential graded $A$-module. Let", "$N^\\bullet$ be a complex of $R$-modules. The constructions above", "produce a right differential graded $A$-module $\\Hom(M, N^\\bullet)$", "and then a leftt differential graded $A$-module", "$\\Hom(\\Hom(M, N^\\bullet), N^\\bullet)$. We claim there is an", "evaluation map", "$$", "ev : M \\longrightarrow \\Hom(\\Hom(M, N^\\bullet), N^\\bullet)", "$$", "in the category of left differential graded $A$-modules. To define it, by", "Lemma \\ref{lemma-characterize-hom} it suffices to construct an", "$A$-bilinear pairing", "$$", "\\Hom(M, N^\\bullet) \\times M \\longrightarrow N^\\bullet", "$$", "compatible with grading and differentials. For this we take", "$$", "(f, x) \\longmapsto f(x)", "$$", "We leave it to the reader to verify this is compatible with grading,", "differentials, and $A$-bilinear. The map $ev$ on underlying complexes", "of $R$-modules is More on Algebra, Item (\\ref{more-algebra-item-evaluation})." ], "refs": [ "dga-lemma-characterize-hom" ], "ref_ids": [ 13051 ] }, { "id": 13160, "type": "other", "label": "dga-remark-evaluation-map-right", "categories": [ "dga" ], "title": "dga-remark-evaluation-map-right", "contents": [ "Let $R$ be a ring. Let $A$ be a differential graded $R$-algebra.", "Let $M$ be a right differential graded $A$-module. Let", "$N^\\bullet$ be a complex of $R$-modules. The constructions above", "produce a left differential graded $A$-module $\\Hom(M, N^\\bullet)$", "and then a right differential graded $A$-module", "$\\Hom(\\Hom(M, N^\\bullet), N^\\bullet)$. We claim there is an evaluation map", "$$", "ev : M \\longrightarrow \\Hom(\\Hom(M, N^\\bullet), N^\\bullet)", "$$", "in the category of right differential graded $A$-modules. To define it, by", "Lemma \\ref{lemma-characterize-hom} it suffices to construct an", "$A$-bilinear pairing", "$$", "M \\times \\Hom(M, N^\\bullet) \\longrightarrow N^\\bullet", "$$", "compatible with grading and differentials. For this we take", "$$", "(x, f) \\longmapsto (-1)^{\\deg(x)\\deg(f)}f(x)", "$$", "We leave it to the reader to verify this is compatible with grading,", "differentials, and $A$-bilinear. The map $ev$ on underlying complexes", "of $R$-modules is More on Algebra, Item (\\ref{more-algebra-item-evaluation})." ], "refs": [ "dga-lemma-characterize-hom" ], "ref_ids": [ 13051 ] }, { "id": 13161, "type": "other", "label": "dga-remark-shift-dual", "categories": [ "dga" ], "title": "dga-remark-shift-dual", "contents": [ "Let $R$ be a ring. Let $A$ be a differential graded $R$-algebra.", "Let $M^\\bullet$ and $N^\\bullet$ be complexes of $R$-modules.", "Let $k \\in \\mathbf{Z}$ and consider the isomorphism", "$$", "\\Hom^\\bullet(M^\\bullet, N^\\bullet)[-k]", "\\longrightarrow", "\\Hom^\\bullet(M^\\bullet[k], N^\\bullet)", "$$", "of complexes of $R$-modules defined in", "More on Algebra, Item (\\ref{more-algebra-item-shift-hom}).", "If $M^\\bullet$ has the structure of a left, resp.\\ right", "differential graded $A$-module, then this is a map of", "right, resp.\\ left differential graded $A$-modules (with the", "module structures as defined in this section).", "We omit the verification; we warn the reader that the", "$A$-module structure on the shift of a left graded $A$-module", "is defined using a sign, see", "Definition \\ref{definition-shift-graded-module}." ], "refs": [ "dga-definition-shift-graded-module" ], "ref_ids": [ 13146 ] }, { "id": 13162, "type": "other", "label": "dga-remark-P-resolution", "categories": [ "dga" ], "title": "dga-remark-P-resolution", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ be a differential graded $R$-algebra.", "Using P-resolutions we can sometimes reduce statements about general", "objects of $D(A, \\text{d})$ to statements about $A[k]$. Namely, let", "$T$ be a property of objects of $D(A, \\text{d})$ and assume that", "\\begin{enumerate}", "\\item if $K_i$, $i \\in I$ is a family of objects of $D(A, \\text{d})$", "and $T(K_i)$ holds for all $i \\in I$, then $T(\\bigoplus K_i)$,", "\\item if $K \\to L \\to M \\to K[1]$ is a distinguished triangle of", "$D(A, \\text{d})$ and $T$ holds for two, then $T$", "holds for the third object, and", "\\item $T(A[k])$ holds for all $k \\in \\mathbf{Z}$.", "\\end{enumerate}", "Then $T$ holds for all objects of $D(A, \\text{d})$. This is clear from", "Lemmas \\ref{lemma-property-P-sequence} and \\ref{lemma-resolve}." ], "refs": [ "dga-lemma-property-P-sequence", "dga-lemma-resolve" ], "ref_ids": [ 13059, 13062 ] }, { "id": 13163, "type": "other", "label": "dga-remark-graded-shift-functors", "categories": [ "dga" ], "title": "dga-remark-graded-shift-functors", "contents": [ "Let $R$ be a ring. Let $\\mathcal{D}$ be an $R$-linear category endowed with a", "collection of $R$-linear functors $[n] : \\mathcal{D} \\to \\mathcal{D}$,", "$x \\mapsto x[n]$ indexed by $n \\in \\mathbf{Z}$ such that", "$[n] \\circ [m] = [n + m]$ and $[0] = \\text{id}_\\mathcal{D}$ (equality as", "functors). This allows us to construct a graded category $\\mathcal{D}^{gr}$", "over $R$ with the same objects of $\\mathcal{D}$ setting", "$$", "\\Hom_{\\mathcal{D}^{gr}}(x, y) =", "\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\Hom_\\mathcal{D}(x, y[n])", "$$", "for $x, y$ in $\\mathcal{D}$. Observe that $(\\mathcal{D}^{gr})^0 = \\mathcal{D}$", "(see Definition \\ref{definition-H0-of-graded-category}). Moreover, the graded", "category $\\mathcal{D}^{gr}$ inherits $R$-linear graded functors $[n]$", "satisfying $[n] \\circ [m] = [n + m]$ and $[0] = \\text{id}_{\\mathcal{D}^{gr}}$", "with the property that", "$$", "\\Hom_{\\mathcal{D}^{gr}}(x, y[n]) = \\Hom_{\\mathcal{D}^{gr}}(x, y)[n]", "$$", "as graded $R$-modules compatible with composition of morphisms.", "\\medskip\\noindent", "Conversely, suppose given a graded category $\\mathcal{A}$ over $R$ endowed", "with a collection of $R$-linear graded functors $[n]$", "satisfying $[n] \\circ [m] = [n + m]$ and $[0] = \\text{id}_\\mathcal{A}$", "which are moreover equipped with isomorphisms", "$$", "\\Hom_\\mathcal{A}(x, y[n]) = \\Hom_\\mathcal{A}(x, y)[n]", "$$", "as graded $R$-modules compatible with composition of morphisms. Then", "the reader easily shows that $\\mathcal{A} = (\\mathcal{A}^0)^{gr}$.", "\\medskip\\noindent", "Here are two examples of the relationship", "$\\mathcal{D} \\leftrightarrow \\mathcal{A}$ we established above:", "\\begin{enumerate}", "\\item Let $\\mathcal{B}$ be an additive category. If", "$\\mathcal{D} = \\text{Gr}(\\mathcal{B})$, then", "$\\mathcal{A} = \\text{Gr}^{gr}(\\mathcal{B})$ as in", "Example \\ref{example-graded-category-graded-objects}.", "\\item If $A$ is a graded ring and $\\mathcal{D} = \\text{Mod}_A$", "is the category of graded right $A$-modules, then", "$\\mathcal{A} = \\text{Mod}^{gr}_A$, see Example \\ref{example-gm-gr-cat}.", "\\end{enumerate}" ], "refs": [ "dga-definition-H0-of-graded-category" ], "ref_ids": [ 13152 ] }, { "id": 13164, "type": "other", "label": "dga-remark-shift-tensor-no-sign", "categories": [ "dga" ], "title": "dga-remark-shift-tensor-no-sign", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$", "be differential graded algebras over $R$. Let $N$ be a", "differential graded $(A, B)$-bimodule. Let $M$ be a right differential", "graded $A$-module. Then for every $k \\in \\mathbf{Z}$ there", "is an isomorphism", "$$", "(M \\otimes_A N)[k] \\longrightarrow", "M[k] \\otimes_A N", "$$", "of right differential graded $B$-modules defined without the intervention", "of signs, see More on Algebra, Section \\ref{more-algebra-section-sign-rules}." ], "refs": [], "ref_ids": [] }, { "id": 13165, "type": "other", "label": "dga-remark-shift-hom-no-sign", "categories": [ "dga" ], "title": "dga-remark-shift-hom-no-sign", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$", "be differential graded algebras over $R$. Let $N$ be a", "differential graded $(A, B)$-bimodule. Let $N'$ be a right differential", "graded $B$-module. Then for every $k \\in \\mathbf{Z}$ there", "is an isomorphism", "$$", "\\Hom_{\\text{Mod}^{gr}_B}(N, N')[k]", "\\longrightarrow", "\\Hom_{\\text{Mod}^{gr}_B}(N, N'[k])", "$$", "of right differential graded $A$-modules defined without the intervention", "of signs, see More on Algebra, Section \\ref{more-algebra-section-sign-rules}." ], "refs": [], "ref_ids": [] }, { "id": 13166, "type": "other", "label": "dga-remark-source-graded-projective", "categories": [ "dga" ], "title": "dga-remark-source-graded-projective", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. Is there a", "characterization of those differential graded $A$-modules $P$", "for which we have", "$$", "\\Hom_{K(A, \\text{d})}(P, M) = \\Hom_{D(A, \\text{d})}(P, M)", "$$", "for all differential graded $A$-modules $M$? Let", "$\\mathcal{D} \\subset K(A, \\text{d})$ be the full subcategory", "whose objects are the objects $P$ satisfying the above. Then $\\mathcal{D}$", "is a strictly full saturated triangulated subcategory of $K(A, \\text{d})$.", "If $P$ is projective as a graded $A$-module, then to see where $P$", "is an object of $\\mathcal{D}$ it is enough to check that", "$\\Hom_{K(A, \\text{d})}(P, M) = 0$ whenever $M$ is acyclic.", "However, in general it is not enough to assume that $P$ is projective as", "a graded $A$-module. Example: take $A = R = k[\\epsilon]$ where $k$ is", "a field and $k[\\epsilon] = k[x]/(x^2)$ is the ring of dual numbers.", "Let $P$ be the object with $P^n = R$ for all $n \\in \\mathbf{Z}$", "and differential given by multiplication by $\\epsilon$. Then", "$\\text{id}_P \\in \\Hom_{K(A, \\text{d})}(P, P)$ is a nonzero element", "but $P$ is acyclic." ], "refs": [], "ref_ids": [] }, { "id": 13167, "type": "other", "label": "dga-remark-graded-projective-is-compact", "categories": [ "dga" ], "title": "dga-remark-graded-projective-is-compact", "contents": [ "Let $(A, \\text{d})$ be a differential graded algebra. Let us say a", "differential graded $A$-module $M$ is {\\it finite} if $M$ is generated,", "as a right $A$-module, by finitely many elements. If $P$ is a", "differential graded $A$-module which is finite graded projective,", "then we can ask: Does $P$ give a compact object of $D(A, \\text{d})$?", "Presumably, this is not true in general, but we do not know a", "counter example. However, if $P$ is also an object of the category", "$\\mathcal{D}$ of Remark \\ref{remark-source-graded-projective},", "then this is the case (this follows from the fact that direct sums", "in $D(A, \\text{d})$ are given by direct sums of modules; details omitted)." ], "refs": [ "dga-remark-source-graded-projective" ], "ref_ids": [ 13166 ] }, { "id": 13168, "type": "other", "label": "dga-remark-tilting-equivalence", "categories": [ "dga" ], "title": "dga-remark-tilting-equivalence", "contents": [ "In Lemma \\ref{lemma-tilting-equivalence} we can replace", "condition (2) by the condition that $N$ is a classical", "generator for $D_{compact}(B, d)$, see", "Derived Categories, Proposition", "\\ref{derived-proposition-generator-versus-classical-generator}.", "Moreover, if we knew that $R\\Hom(N, B)$ is a compact object", "of $D(A, \\text{d})$, then it suffices to check that $N$", "is a weak generator for $D_{compact}(B, \\text{d})$.", "We omit the proof; we will add it here if we ever", "need it in the Stacks project." ], "refs": [ "dga-lemma-tilting-equivalence", "derived-proposition-generator-versus-classical-generator" ], "ref_ids": [ 13124, 1965 ] }, { "id": 13169, "type": "other", "label": "dga-remark-lift-equivalence-to-dga", "categories": [ "dga" ], "title": "dga-remark-lift-equivalence-to-dga", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential", "graded $R$-algebras. Suppose given an $R$-linear equivalence", "$$", "F : D(A, \\text{d}) \\longrightarrow D(B, \\text{d})", "$$", "of triangulated categories. Set $N = F(A)$. Then $N$ is a differential", "graded $B$-module. Since $F$ is an equivalence and $A$ is a compact", "object of $D(A, \\text{d})$, we conclude that $N$ is a compact object", "of $D(B, \\text{d})$. Since $A$ generates $D(A, \\text{d})$ and", "$F$ is an equivalence, we see that $N$ generates $D(B, \\text{d})$.", "Finally, $H^k(A) = \\Hom_{D(A, \\text{d})}(A, A[k])$ and as $F$ an equivalence", "we see that $F$ induces an isomorphism", "$H^k(A) = \\Hom_{D(B, \\text{d})}(N, N[k])$ for all $k$.", "In order to conclude that there is an equivalence", "$D(A, \\text{d}) \\longrightarrow D(B, \\text{d})$ which", "arises from the construction in", "Lemma \\ref{lemma-tilting-equivalence}", "all we need is a left $A$-module structure on $N$", "compatible with derivation and commuting", "with the given right $B$-module structure. In fact, it", "suffices to do this after replacing $N$ by a quasi-isomorphic", "differential graded $B$-module.", "The module structure can be constructed in certain cases.", "For example, if we assume that $F$ can be lifted to a", "differential graded functor", "$$", "F^{dg} :", "\\text{Mod}^{dg}_{(A, \\text{d})}", "\\longrightarrow", "\\text{Mod}^{dg}_{(B, \\text{d})}", "$$", "(for notation see Example \\ref{example-dgm-dg-cat})", "between the associated differential graded categories,", "then this holds. Another case is discussed in the proposition below." ], "refs": [ "dga-lemma-tilting-equivalence" ], "ref_ids": [ 13124 ] }, { "id": 13170, "type": "other", "label": "dga-remark-rickard", "categories": [ "dga" ], "title": "dga-remark-rickard", "contents": [ "Let $A, B, F, N$ be as in Proposition \\ref{proposition-rickard}.", "It is not clear that $F$ and the functor", "$G(-) = - \\otimes_A^\\mathbf{L} N$ are isomorphic.", "By construction there is an isomorphism", "$N = G(A) \\to F(A)$ in $D(B, \\text{d})$.", "It is straightforward to extend this to a functorial isomorphism", "$G(M) \\to F(M)$ for $M$ is a differential graded $A$-module which", "is graded projective (e.g., a sum of shifts of $A$).", "Then one can conclude that $G(M) \\cong F(M)$ when $M$ is a cone", "of a map between such modules. We don't know whether more is true", "in general." ], "refs": [ "dga-proposition-rickard" ], "ref_ids": [ 13133 ] }, { "id": 13171, "type": "other", "label": "dga-remark-centers", "categories": [ "dga" ], "title": "dga-remark-centers", "contents": [ "Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras.", "If $D(A)$ and $D(B)$ are equivalent as $R$-linear triangulated", "categories, then the centers of $A$ and $B$ are isomorphic", "as $R$-algebras. In particular, if $A$ and $B$ are commutative,", "then $A \\cong B$. The rather tricky proof can be found in", "\\cite[Proposition 9.2]{Rickard} or \\cite[Proposition 6.3.2]{KZ}.", "Another approach might be to use Hochschild cohomology (see", "remark below)." ], "refs": [], "ref_ids": [] }, { "id": 13172, "type": "other", "label": "dga-remark-hochschild-cohomology", "categories": [ "dga" ], "title": "dga-remark-hochschild-cohomology", "contents": [ "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential", "graded $R$-algebras which are derived equivalent, i.e., such that there", "exists an $R$-linear equivalence $D(A, \\text{d}) \\to D(B, \\text{d})$", "of triangulated categories. We would like to show that certain invariants", "of $(A, \\text{d})$ and $(B, \\text{d})$ coincide. In many situations", "one has more control of the situation. For example, it may happen", "that there is an equivalence of the form", "$$", "- \\otimes_A \\Omega : D(A, \\text{d}) \\longrightarrow D(B, \\text{d})", "$$", "for some differential graded $(A, B)$-bimodule", "$\\Omega$ (this happens in the situation of", "Proposition \\ref{proposition-rickard} and is often true", "if the equivalence comes from a geometric construction).", "If also the quasi-inverse of our functor is given as", "$$", "- \\otimes_B^\\mathbf{L} \\Omega' : D(B, \\text{d}) \\longrightarrow D(A, \\text{d})", "$$", "for a differential graded $(B, A)$-bimodule $\\Omega'$", "(and as before such a module $\\Omega'$ often exists in practice).", "In this case we can consider the functor", "$$", "D(A^{opp} \\otimes_R A, \\text{d})", "\\longrightarrow", "D(B^{opp} \\otimes_R B, \\text{d}),\\quad", "M \\longmapsto \\Omega' \\otimes^\\mathbf{L}_A M \\otimes_A^\\mathbf{L} \\Omega", "$$", "on derived categories of bimodules (use", "Lemma \\ref{lemma-bimodule-over-tensor} to turn bimodules into", "right modules).", "Observe that this functor sends the $(A, A)$-bimodule $A$ to", "the $(B, B)$-bimodule $B$. Under suitable conditions", "(e.g., flatness of $A$, $B$, $\\Omega$ over $R$, etc)", "this functor will be an equivalence as well.", "If this is the case, then it follows that we have isomorphisms", "of Hochschild cohomology groups", "$$", "HH^i(A, \\text{d}) =", "\\Hom_{D(A^{opp} \\otimes_R A, \\text{d})}(A, A[i])", "\\longrightarrow", "\\Hom_{D(B^{opp} \\otimes_R B, \\text{d})}(B, B[i]) =", "HH^i(B, \\text{d}).", "$$", "For example, if $A = H^0(A)$, then $HH^0(A, \\text{d})$", "is equal to the center of $A$, and this gives a conceptual proof", "of the result mentioned in Remark \\ref{remark-centers}.", "If we ever need this remark we will provide a precise statement", "with a detailed proof here." ], "refs": [ "dga-proposition-rickard", "dga-lemma-bimodule-over-tensor", "dga-remark-centers" ], "ref_ids": [ 13133, 13092, 13171 ] }, { "id": 13217, "type": "other", "label": "spaces-more-groupoids-remark-finite-monoid", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-remark-finite-monoid", "contents": [ "Let $f : X \\to Y$ be a separated morphism of algebraic spaces.", "The sheaf $(X/Y)_{fin}$ comes with a natural map", "$(X/Y)_{fin} \\to Y$ by mapping the pair $(a, Z) \\in (X/Y)_{fin}(T)$", "to the element $a \\in Y(T)$. We can use", "Lemma \\ref{lemma-finite-separated}", "to define operations", "$$", "\\star_i : (X/Y)_{fin} \\times_Y (X/Y)_{fin} \\longrightarrow (X/Y)_{fin}", "$$", "by the rules", "\\begin{align*}", "\\star_1 : ((a, Z_1), (a, Z_2)) & \\longmapsto (a, Z_1 \\cup Z_2) \\\\", "\\star_2 : ((a, Z_1), (a, Z_2)) & \\longmapsto (a, Z_1 \\cap Z_2) \\\\", "\\star_3 : ((a, Z_1), (a, Z_2)) & \\longmapsto (a, Z_1 \\setminus Z_2) \\\\", "\\star_4 : ((a, Z_1), (a, Z_2)) & \\longmapsto (a, Z_2 \\setminus Z_1).", "\\end{align*}", "The reason this works is that $Z_1 \\cap Z_2$ is both open and closed", "inside $Z_1$ and $Z_2$ (which also implies that $Z_1 \\cup Z_2$ is", "the disjoint union of the other three pieces).", "Thus we can think of $(X/Y)_{fin}$ as an $\\mathbf{F}_2$-algebra", "(without unit) over $Y$ with multiplication given by", "$ss' = \\star_2(s, s')$, and addition given by", "$$", "s + s' = \\star_1(\\star_3(s, s'), \\star_4(s, s'))", "$$", "which boils down to taking the symmetric difference.", "Note that in this sheaf of algebras $0 = (1_Y, \\emptyset)$", "and that indeed $s + s = 0$ for any local section $s$.", "If $f : X \\to Y$ is finite, then this algebra has a unit namely", "$1 = (1_Y, X)$ and $\\star_3(s, s') = s(1 + s')$, and", "$\\star_4(s, s') = (1 + s)s'$." ], "refs": [ "spaces-more-groupoids-lemma-finite-separated" ], "ref_ids": [ 13197 ] }, { "id": 13218, "type": "other", "label": "spaces-more-groupoids-remark-finite-quasi-finite-separated-morphism-schemes", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-remark-finite-quasi-finite-separated-morphism-schemes", "contents": [ "Let $f : X \\to Y$ be a separated, locally quasi-finite", "morphism of schemes. In this case the sheaf $(X/Y)_{fin}$", "is closely related to the sheaf $f_!\\mathbf{F}_2$", "(insert future reference here) on $Y_\\etale$.", "Namely, if $V \\to Y$ is \\'etale, and $s \\in \\Gamma(V, f_!\\mathbf{F}_2)$,", "then $s \\in \\Gamma(V \\times_Y X, \\mathbf{F}_2)$ is a section", "with proper support $Z = \\text{Supp}(s)$ over $V$. Since $f$ is", "also locally quasi-finite we see that the projection $Z \\to V$ is actually", "finite. Since the support of a section of a constant abelian sheaf is open", "we see that the pair $(V \\to Y, \\text{Supp}(s))$ satisfies", "\\ref{equation-finite-conditions}.", "In fact, $f_!\\mathbf{F}_2 \\cong (X/Y)_{fin}|_{Y_\\etale}$", "in this case which also explains the $\\mathbf{F}_2$-algebra structure", "introduced in Remark \\ref{remark-finite-monoid}." ], "refs": [ "spaces-more-groupoids-remark-finite-monoid" ], "ref_ids": [ 13217 ] }, { "id": 13219, "type": "other", "label": "spaces-more-groupoids-remark-warning", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-remark-warning", "contents": [ "The condition that $f$ be separated cannot be dropped from", "Proposition \\ref{proposition-finite-algebraic-space}.", "An example is to take $X$ the affine line with zero doubled, see", "Schemes, Example \\ref{schemes-example-affine-space-zero-doubled},", "$Y = \\mathbf{A}^1_k$ the affine line, and $X \\to Y$ the obvious map.", "Recall that over $0 \\in Y$ there are two points $0_1$ and $0_2$", "in $X$. Thus $(X/Y)_{fin}$ has four points over $0$, namely", "$\\emptyset, \\{0_1\\}, \\{0_2\\}, \\{0_1, 0_2\\}$.", "Of these four points only three can be lifted to an open", "subscheme of $U \\times_Y X$ finite over $U$ for $U \\to Y$ \\'etale,", "namely $\\emptyset, \\{0_1\\}, \\{0_2\\}$. This shows that $(X/Y)_{fin}$", "if representable by an algebraic space is not \\'etale over $Y$.", "Similar arguments show that $(X/Y)_{fin}$ is really not an algebraic", "space. Details omitted." ], "refs": [ "spaces-more-groupoids-proposition-finite-algebraic-space" ], "ref_ids": [ 13215 ] }, { "id": 13220, "type": "other", "label": "spaces-more-groupoids-remark-not-scheme", "categories": [ "spaces-more-groupoids" ], "title": "spaces-more-groupoids-remark-not-scheme", "contents": [ "Let $Y = \\mathbf{A}^1_{\\mathbf{R}}$ be the affine line over the real", "numbers, and let $X = \\Spec(\\mathbf{C})$ mapping to the", "$\\mathbf{R}$-rational point $0$ in $Y$. In this case the morphism", "$f : X \\to Y$ is finite, but it is not the case that $(X/Y)_{fin}$", "is a scheme. Namely, one can show that in this case the algebraic", "space $(X/Y)_{fin}$ is isomorphic to the algebraic space of", "Spaces, Example \\ref{spaces-example-non-representable-descent}", "associated to the extension $\\mathbf{R} \\subset \\mathbf{C}$.", "Thus it is really necessary to leave the category of schemes", "in order to represent the sheaf $(X/Y)_{fin}$, even when $f$", "is a finite morphism." ], "refs": [], "ref_ids": [] }, { "id": 13363, "type": "other", "label": "modules-remark-sections-support-in-closed", "categories": [ "modules" ], "title": "modules-remark-sections-support-in-closed", "contents": [ "Let $X$ be a topological space. Let $Z \\subset X$ be a closed subset.", "Let $\\mathcal{F}$ be an abelian sheaf on $X$. For $U \\subset X$ open set", "$$", "\\mathcal{H}_Z(\\mathcal{F})(U) =", "\\{s \\in \\mathcal{F}(U) \\mid", "\\text{ the support of }s\\text{ is contained in }Z \\cap U\\}", "$$", "Then $\\mathcal{H}_Z(\\mathcal{F})$ is an abelian subsheaf of $\\mathcal{F}$.", "It is the largest abelian subsheaf of $\\mathcal{F}$ whose support is", "contained in $Z$. By Lemma \\ref{lemma-i-star-exact} we may (and we do)", "view $\\mathcal{H}_Z(\\mathcal{F})$ as an abelian sheaf on $Z$.", "In this way we obtain a left exact functor", "$$", "\\textit{Ab}(X) \\longrightarrow \\textit{Ab}(Z),\\quad", "\\mathcal{F} \\longmapsto \\mathcal{H}_Z(\\mathcal{F})", "\\text{ viewed as abelian sheaf on }Z", "$$", "All of the statements made above follow directly from", "Lemma \\ref{lemma-support-section-closed}." ], "refs": [ "modules-lemma-i-star-exact" ], "ref_ids": [ 13232 ] }, { "id": 13364, "type": "other", "label": "modules-remark-i-star-right-adjoint", "categories": [ "modules" ], "title": "modules-remark-i-star-right-adjoint", "contents": [ "In Sheaves, Remark \\ref{sheaves-remark-i-star-not-exact}", "we showed that $i_*$ as a functor", "on the categories of sheaves of sets", "does not have a right adjoint simply because", "it is not exact. However, it is very close to being", "true, in fact, the functor $i_*$ is exact on sheaves", "of pointed sets, sections with support in $Z$ can", "be defined for sheaves of pointed sets, and $\\mathcal{H}_Z$", "makes sense and is a right adjoint to $i_*$." ], "refs": [ "sheaves-remark-i-star-not-exact" ], "ref_ids": [ 14588 ] }, { "id": 13365, "type": "other", "label": "modules-remark-infinite-direct-sum-quasi-coherent-not", "categories": [ "modules" ], "title": "modules-remark-infinite-direct-sum-quasi-coherent-not", "contents": [ "Warning: It is not true in general that an infinite", "direct sum of quasi-coherent $\\mathcal{O}_X$-modules", "is quasi-coherent. For more esoteric behaviour of quasi-coherent", "modules see Example \\ref{example-quasi-coherent}." ], "refs": [], "ref_ids": [] }, { "id": 13366, "type": "other", "label": "modules-remark-condition-necessary", "categories": [ "modules" ], "title": "modules-remark-condition-necessary", "contents": [ "In the lemma above some condition beyond the condition that $X$", "is quasi-compact is necessary. See", "Sheaves, Example \\ref{sheaves-example-conditions-needed-colimit}." ], "refs": [], "ref_ids": [] }, { "id": 13367, "type": "other", "label": "modules-remark-sections-support-in-closed-modules", "categories": [ "modules" ], "title": "modules-remark-sections-support-in-closed-modules", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $Z \\subset X$ be", "a closed subset. For an $\\mathcal{O}_X$-module $\\mathcal{F}$ we can", "consider the {\\it submodule of sections with support in $Z$}, denoted", "$\\mathcal{H}_Z(\\mathcal{F})$, defined by the rule", "$$", "\\mathcal{H}_Z(\\mathcal{F})(U) =", "\\{s \\in \\mathcal{F}(U) \\mid \\text{Supp}(s) \\subset U \\cap Z\\}", "$$", "Observe that $\\mathcal{H}_Z(\\mathcal{F})(U)$ is a module over", "$\\mathcal{O}_X(U)$, i.e., $\\mathcal{H}_Z(\\mathcal{F})$ is an", "$\\mathcal{O}_X$-module. By construction $\\mathcal{H}_Z(\\mathcal{F})$", "is the largest $\\mathcal{O}_X$-submodule of $\\mathcal{F}$ whose support is", "contained in $Z$.", "Applying Lemma \\ref{lemma-i-star-equivalence} to", "the morphism of ringed spaces $(Z, \\mathcal{O}_X|_Z) \\to (X, \\mathcal{O}_X)$ we", "may (and we do) view", "$\\mathcal{H}_Z(\\mathcal{F})$ as an $\\mathcal{O}_X|_Z$-module on $Z$.", "Thus we obtain a functor", "$$", "\\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\textit{Mod}(\\mathcal{O}_X|_Z),", "\\quad", "\\mathcal{F} \\longmapsto \\mathcal{H}_Z(\\mathcal{F})", "\\text{ viewed as an }\\mathcal{O}_X|_Z\\text{-module on }Z", "$$", "This functor is left exact, but in general not exact.", "All of the statements made above follow directly from", "Lemma \\ref{lemma-support-section-closed}.", "Clearly the construction is compatible with the construction in", "Remark \\ref{remark-sections-support-in-closed}." ], "refs": [ "modules-lemma-i-star-equivalence", "modules-remark-sections-support-in-closed" ], "ref_ids": [ 13260, 13363 ] }, { "id": 13368, "type": "other", "label": "modules-remark-functoriality-principal-parts", "categories": [ "modules" ], "title": "modules-remark-functoriality-principal-parts", "contents": [ "Let $X$ be a topological space. Suppose given a commutative diagram of", "sheaves of rings", "$$", "\\xymatrix{", "\\mathcal{B} \\ar[r] & \\mathcal{B}' \\\\", "\\mathcal{A} \\ar[u] \\ar[r] & \\mathcal{A}' \\ar[u]", "}", "$$", "on $X$, a $\\mathcal{B}$-module $\\mathcal{F}$,", "a $\\mathcal{B}'$-module $\\mathcal{F}'$, and", "a $\\mathcal{B}$-linear map $\\mathcal{F} \\to \\mathcal{F}'$.", "Then we get a compatible system of module maps", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "\\mathcal{P}^2_{\\mathcal{B}'/\\mathcal{A}'}(\\mathcal{F}') \\ar[r] &", "\\mathcal{P}^1_{\\mathcal{B}'/\\mathcal{A}'}(\\mathcal{F}') \\ar[r] &", "\\mathcal{P}^0_{\\mathcal{B}'/\\mathcal{A}'}(\\mathcal{F}') \\\\", "\\ldots \\ar[r] &", "\\mathcal{P}^2_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{F}) \\ar[r] \\ar[u] &", "\\mathcal{P}^1_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{F}) \\ar[r] \\ar[u] &", "\\mathcal{P}^0_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{F}) \\ar[u]", "}", "$$", "These maps are compatible with further composition of maps of this type.", "The easiest way to see this is to use the description of the modules", "$\\mathcal{P}^k_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{M})$ in terms of", "(local) generators and relations in the proof of", "Lemma \\ref{lemma-module-principal-parts} but it can also be seen", "directly from the universal", "property of these modules. Moreover, these maps are compatible with", "the short exact sequences of Lemma \\ref{lemma-sequence-of-principal-parts}." ], "refs": [ "modules-lemma-module-principal-parts", "modules-lemma-sequence-of-principal-parts" ], "ref_ids": [ 13321, 13323 ] }, { "id": 13425, "type": "other", "label": "defos-remark-trivial-thickening", "categories": [ "defos" ], "title": "defos-remark-trivial-thickening", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. A first order thickening", "$i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$ is said", "to be {\\it trivial} if there exists a morphism of ringed spaces", "$\\pi : (X', \\mathcal{O}_{X'}) \\to (X, \\mathcal{O}_X)$ which is a", "left inverse to $i$. The choice of such a morphism", "$\\pi$ is called a {\\it trivialization} of the first order thickening.", "Given $\\pi$ we obtain a splitting", "\\begin{equation}", "\\label{equation-splitting}", "\\mathcal{O}_{X'} = \\mathcal{O}_X \\oplus \\mathcal{I}", "\\end{equation}", "as sheaves of algebras on $X$ by using $\\pi^\\sharp$ to split the surjection", "$\\mathcal{O}_{X'} \\to \\mathcal{O}_X$. Conversely, such a splitting determines", "a morphism $\\pi$. The category of trivialized first order thickenings of", "$(X, \\mathcal{O}_X)$ is equivalent to the category of ", "$\\mathcal{O}_X$-modules." ], "refs": [], "ref_ids": [] }, { "id": 13426, "type": "other", "label": "defos-remark-trivial-extension", "categories": [ "defos" ], "title": "defos-remark-trivial-extension", "contents": [ "Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$", "be a trivial first order thickening of ringed spaces", "and let $\\pi : (X', \\mathcal{O}_{X'}) \\to (X, \\mathcal{O}_X)$", "be a trivialization. Then given any triple", "$(\\mathcal{F}, \\mathcal{K}, c)$ consisting of a pair of", "$\\mathcal{O}_X$-modules and a map", "$c : \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{K}$", "we may set", "$$", "\\mathcal{F}'_{c, triv} = \\mathcal{F} \\oplus \\mathcal{K}", "$$", "and use the splitting (\\ref{equation-splitting}) associated to $\\pi$", "and the map $c$ to define the $\\mathcal{O}_{X'}$-module structure", "and obtain an extension (\\ref{equation-extension}). We will call", "$\\mathcal{F}'_{c, triv}$ the {\\it trivial extension} of $\\mathcal{F}$", "by $\\mathcal{K}$ corresponding", "to $c$ and the trivialization $\\pi$. Given any extension", "$\\mathcal{F}'$ as in (\\ref{equation-extension}) we can use", "$\\pi^\\sharp : \\mathcal{O}_X \\to \\mathcal{O}_{X'}$ to think of $\\mathcal{F}'$", "as an $\\mathcal{O}_X$-module extension, hence a class $\\xi_{\\mathcal{F}'}$", "in $\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K})$.", "Lemma \\ref{lemma-inf-ext} assures that", "$\\mathcal{F}' \\mapsto \\xi_{\\mathcal{F}'}$", "induces a bijection", "$$", "\\left\\{", "\\begin{matrix}", "\\text{isomorphism classes of extensions}\\\\", "\\mathcal{F}'\\text{ as in (\\ref{equation-extension}) with }c = c_{\\mathcal{F}'}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K})", "$$", "Moreover, the trivial extension $\\mathcal{F}'_{c, triv}$ maps to the zero class." ], "refs": [ "defos-lemma-inf-ext" ], "ref_ids": [ 13376 ] }, { "id": 13427, "type": "other", "label": "defos-remark-extension-functorial", "categories": [ "defos" ], "title": "defos-remark-extension-functorial", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let", "$(X, \\mathcal{O}_X) \\to (X'_i, \\mathcal{O}_{X'_i})$, $i = 1, 2$", "be first order thickenings with ideal sheaves $\\mathcal{I}_i$.", "Let $h : (X'_1, \\mathcal{O}_{X'_1}) \\to (X'_2, \\mathcal{O}_{X'_2})$", "be a morphism of first order thickenings of $(X, \\mathcal{O}_X)$.", "Picture", "$$", "\\xymatrix{", "& (X, \\mathcal{O}_X) \\ar[ld] \\ar[rd] & \\\\", "(X'_1, \\mathcal{O}_{X'_1}) \\ar[rr]^h & & ", "(X'_2, \\mathcal{O}_{X'_2})", "}", "$$", "Observe that $h^\\sharp : \\mathcal{O}_{X'_2} \\to \\mathcal{O}_{X'_1}$", "in particular induces an $\\mathcal{O}_X$-module map", "$\\mathcal{I}_2 \\to \\mathcal{I}_1$.", "Let $\\mathcal{F}$ be an", "$\\mathcal{O}_X$-module. Let $(\\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair", "consisting of an $\\mathcal{O}_X$-module $\\mathcal{K}_i$ and a map", "$c_i : \\mathcal{I}_i \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "\\mathcal{K}_i$. Assume furthermore given a map", "of $\\mathcal{O}_X$-modules $\\mathcal{K}_2 \\to \\mathcal{K}_1$", "such that", "$$", "\\xymatrix{", "\\mathcal{I}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "\\ar[r]_-{c_2} \\ar[d] &", "\\mathcal{K}_2 \\ar[d] \\\\", "\\mathcal{I}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "\\ar[r]^-{c_1} &", "\\mathcal{K}_1", "}", "$$", "is commutative. Then there is a canonical functoriality", "$$", "\\left\\{", "\\begin{matrix}", "\\mathcal{F}'_2\\text{ as in (\\ref{equation-extension}) with }\\\\", "c_2 = c_{\\mathcal{F}'_2}\\text{ and }\\mathcal{K} = \\mathcal{K}_2", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "\\mathcal{F}'_1\\text{ as in (\\ref{equation-extension}) with }\\\\", "c_1 = c_{\\mathcal{F}'_1}\\text{ and }\\mathcal{K} = \\mathcal{K}_1", "\\end{matrix}", "\\right\\}", "$$", "Namely, thinking of all sheaves $\\mathcal{O}_X$, $\\mathcal{O}_{X'_i}$,", "$\\mathcal{F}$, $\\mathcal{K}_i$, etc as sheaves on $X$, we set", "given $\\mathcal{F}'_2$ the sheaf $\\mathcal{F}'_1$ equal to the", "pushout, i.e., fitting into the following diagram of extensions", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{K}_2 \\ar[r] \\ar[d] &", "\\mathcal{F}'_2 \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar@{=}[d] \\ar[r] & 0 \\\\", "0 \\ar[r] &", "\\mathcal{K}_1 \\ar[r] &", "\\mathcal{F}'_1 \\ar[r] &", "\\mathcal{F} \\ar[r] & 0", "}", "$$", "We omit the construction of the $\\mathcal{O}_{X'_1}$-module structure", "on the pushout (this uses the commutativity of the diagram", "involving $c_1$ and $c_2$)." ], "refs": [], "ref_ids": [] }, { "id": 13428, "type": "other", "label": "defos-remark-trivial-extension-functorial", "categories": [ "defos" ], "title": "defos-remark-trivial-extension-functorial", "contents": [ "Let $(X, \\mathcal{O}_X)$, $(X, \\mathcal{O}_X) \\to (X'_i, \\mathcal{O}_{X'_i})$,", "$\\mathcal{I}_i$, and", "$h : (X'_1, \\mathcal{O}_{X'_1}) \\to (X'_2, \\mathcal{O}_{X'_2})$", "be as in Remark \\ref{remark-extension-functorial}. Assume that we are", "given trivializations $\\pi_i : X'_i \\to X$ such that", "$\\pi_1 = h \\circ \\pi_2$. In other words, assume $h$ is a morphism", "of trivialized first order thickening of $(X, \\mathcal{O}_X)$. Let", "$(\\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair consisting of an", "$\\mathcal{O}_X$-module $\\mathcal{K}_i$ and a map", "$c_i : \\mathcal{I}_i \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "\\mathcal{K}_i$. Assume furthermore given a map", "of $\\mathcal{O}_X$-modules $\\mathcal{K}_2 \\to \\mathcal{K}_1$", "such that", "$$", "\\xymatrix{", "\\mathcal{I}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "\\ar[r]_-{c_2} \\ar[d] &", "\\mathcal{K}_2 \\ar[d] \\\\", "\\mathcal{I}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "\\ar[r]^-{c_1} &", "\\mathcal{K}_1", "}", "$$", "is commutative. In this situation the construction of", "Remark \\ref{remark-trivial-extension} induces", "a commutative diagram", "$$", "\\xymatrix{", "\\{\\mathcal{F}'_2\\text{ as in (\\ref{equation-extension}) with }", "c_2 = c_{\\mathcal{F}'_2}\\text{ and }\\mathcal{K} = \\mathcal{K}_2\\}", "\\ar[d] \\ar[rr] & &", "\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K}_2) \\ar[d] \\\\", "\\{\\mathcal{F}'_1\\text{ as in (\\ref{equation-extension}) with }", "c_1 = c_{\\mathcal{F}'_1}\\text{ and }\\mathcal{K} = \\mathcal{K}_1\\}", "\\ar[rr] & &", "\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K}_1)", "}", "$$", "where the vertical map on the right is given by functoriality of $\\Ext$", "and the map $\\mathcal{K}_2 \\to \\mathcal{K}_1$ and the vertical map on the left", "is the one from Remark \\ref{remark-extension-functorial}." ], "refs": [ "defos-remark-extension-functorial", "defos-remark-trivial-extension", "defos-remark-extension-functorial" ], "ref_ids": [ 13427, 13426, 13427 ] }, { "id": 13429, "type": "other", "label": "defos-remark-short-exact-sequence-thickenings", "categories": [ "defos" ], "title": "defos-remark-short-exact-sequence-thickenings", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. We define a sequence of morphisms", "of first order thickenings", "$$", "(X'_1, \\mathcal{O}_{X'_1}) \\to", "(X'_2, \\mathcal{O}_{X'_2}) \\to", "(X'_3, \\mathcal{O}_{X'_3})", "$$", "of $(X, \\mathcal{O}_X)$ to be a {\\it complex}", "if the corresponding maps between", "the ideal sheaves $\\mathcal{I}_i$", "give a complex of $\\mathcal{O}_X$-modules", "$\\mathcal{I}_3 \\to \\mathcal{I}_2 \\to \\mathcal{I}_1$", "(i.e., the composition is zero). In this case the composition", "$(X'_1, \\mathcal{O}_{X'_1}) \\to (X_3', \\mathcal{O}_{X'_3})$ factors through", "$(X, \\mathcal{O}_X) \\to (X'_3, \\mathcal{O}_{X'_3})$, i.e.,", "the first order thickening $(X'_1, \\mathcal{O}_{X'_1})$ of", "$(X, \\mathcal{O}_X)$ is trivial and comes with", "a canonical trivialization", "$\\pi : (X'_1, \\mathcal{O}_{X'_1}) \\to (X, \\mathcal{O}_X)$.", "\\medskip\\noindent", "We say a sequence of morphisms of first order thickenings", "$$", "(X'_1, \\mathcal{O}_{X'_1}) \\to", "(X'_2, \\mathcal{O}_{X'_2}) \\to", "(X'_3, \\mathcal{O}_{X'_3})", "$$", "of $(X, \\mathcal{O}_X)$ is {\\it a short exact sequence} if the", "corresponding maps between ideal sheaves is a short exact sequence", "$$", "0 \\to \\mathcal{I}_3 \\to \\mathcal{I}_2 \\to \\mathcal{I}_1 \\to 0", "$$", "of $\\mathcal{O}_X$-modules." ], "refs": [], "ref_ids": [] }, { "id": 13430, "type": "other", "label": "defos-remark-complex-thickenings-and-ses-modules", "categories": [ "defos" ], "title": "defos-remark-complex-thickenings-and-ses-modules", "contents": [ "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be an", "$\\mathcal{O}_X$-module. Let", "$$", "(X'_1, \\mathcal{O}_{X'_1}) \\to", "(X'_2, \\mathcal{O}_{X'_2}) \\to", "(X'_3, \\mathcal{O}_{X'_3})", "$$", "be a complex first order thickenings of $(X, \\mathcal{O}_X)$, see", "Remark \\ref{remark-short-exact-sequence-thickenings}.", "Let $(\\mathcal{K}_i, c_i)$, $i = 1, 2, 3$ be pairs consisting of", "an $\\mathcal{O}_X$-module $\\mathcal{K}_i$ and a map", "$c_i : \\mathcal{I}_i \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to", "\\mathcal{K}_i$. Assume given a short exact sequence", "of $\\mathcal{O}_X$-modules", "$$", "0 \\to \\mathcal{K}_3 \\to \\mathcal{K}_2 \\to \\mathcal{K}_1 \\to 0", "$$", "such that", "$$", "\\vcenter{", "\\xymatrix{", "\\mathcal{I}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "\\ar[r]_-{c_2} \\ar[d] &", "\\mathcal{K}_2 \\ar[d] \\\\", "\\mathcal{I}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "\\ar[r]^-{c_1} &", "\\mathcal{K}_1", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "\\mathcal{I}_3 \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "\\ar[r]_-{c_3} \\ar[d] &", "\\mathcal{K}_3 \\ar[d] \\\\", "\\mathcal{I}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{F}", "\\ar[r]^-{c_2} &", "\\mathcal{K}_2", "}", "}", "$$", "are commutative. Finally, assume given an extension", "$$", "0 \\to \\mathcal{K}_2 \\to \\mathcal{F}'_2 \\to \\mathcal{F} \\to 0", "$$", "as in (\\ref{equation-extension}) with $\\mathcal{K} = \\mathcal{K}_2$", "of $\\mathcal{O}_{X'_2}$-modules with $c_{\\mathcal{F}'_2} = c_2$.", "In this situation we can apply the functoriality of", "Remark \\ref{remark-extension-functorial} to obtain an extension", "$\\mathcal{F}'_1$ on $X'_1$ (we'll describe $\\mathcal{F}'_1$", "in this special case below). By", "Remark \\ref{remark-trivial-extension}", "using the canonical splitting", "$\\pi : (X'_1, \\mathcal{O}_{X'_1}) \\to (X, \\mathcal{O}_X)$ of", "Remark \\ref{remark-short-exact-sequence-thickenings}", "we obtain", "$\\xi_{\\mathcal{F}'_1} \\in", "\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K}_1)$.", "Finally, we have the obstruction", "$$", "o(\\mathcal{F}, \\mathcal{K}_3, c_3) \\in", "\\Ext^2_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K}_3)", "$$", "see Lemma \\ref{lemma-inf-obs-ext}.", "In this situation we {\\bf claim} that the canonical map", "$$", "\\partial :", "\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K}_1)", "\\longrightarrow", "\\Ext^2_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K}_3)", "$$", "coming from the short exact sequence", "$0 \\to \\mathcal{K}_3 \\to \\mathcal{K}_2 \\to \\mathcal{K}_1 \\to 0$", "sends $\\xi_{\\mathcal{F}'_1}$", "to the obstruction class $o(\\mathcal{F}, \\mathcal{K}_3, c_3)$.", "\\medskip\\noindent", "To prove this claim choose an embedding $j : \\mathcal{K}_3 \\to \\mathcal{K}$", "where $\\mathcal{K}$ is an injective $\\mathcal{O}_X$-module.", "We can lift $j$ to a map $j' : \\mathcal{K}_2 \\to \\mathcal{K}$.", "Set $\\mathcal{E}'_2 = j'_*\\mathcal{F}'_2$ equal to the pushout", "of $\\mathcal{F}'_2$ by $j'$ so that $c_{\\mathcal{E}'_2} = j' \\circ c_2$.", "Picture:", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{K}_2 \\ar[r] \\ar[d]_{j'} &", "\\mathcal{F}'_2 \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "\\mathcal{K} \\ar[r] &", "\\mathcal{E}'_2 \\ar[r] &", "\\mathcal{F} \\ar[r] & 0", "}", "$$", "Set $\\mathcal{E}'_3 = \\mathcal{E}'_2$ but viewed as an", "$\\mathcal{O}_{X'_3}$-module via $\\mathcal{O}_{X'_3} \\to \\mathcal{O}_{X'_2}$.", "Then $c_{\\mathcal{E}'_3} = j \\circ c_3$.", "The proof of Lemma \\ref{lemma-inf-obs-ext} constructs", "$o(\\mathcal{F}, \\mathcal{K}_3, c_3)$", "as the boundary of the class of the extension of $\\mathcal{O}_X$-modules", "$$", "0 \\to", "\\mathcal{K}/\\mathcal{K}_3 \\to", "\\mathcal{E}'_3/\\mathcal{K}_3 \\to", "\\mathcal{F} \\to 0", "$$", "On the other hand, note that $\\mathcal{F}'_1 = \\mathcal{F}'_2/\\mathcal{K}_3$", "hence the class $\\xi_{\\mathcal{F}'_1}$ is the class", "of the extension", "$$", "0 \\to \\mathcal{K}_2/\\mathcal{K}_3 \\to \\mathcal{F}'_2/\\mathcal{K}_3", "\\to \\mathcal{F} \\to 0", "$$", "seen as a sequence of $\\mathcal{O}_X$-modules using $\\pi^\\sharp$", "where $\\pi : (X'_1, \\mathcal{O}_{X'_1}) \\to (X, \\mathcal{O}_X)$", "is the canonical splitting.", "Thus finally, the claim follows from the fact that we have", "a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{K}_2/\\mathcal{K}_3 \\ar[r] \\ar[d] &", "\\mathcal{F}'_2/\\mathcal{K}_3 \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "\\mathcal{K}/\\mathcal{K}_3 \\ar[r] &", "\\mathcal{E}'_3/\\mathcal{K}_3 \\ar[r] &", "\\mathcal{F} \\ar[r] & 0", "}", "$$", "which is $\\mathcal{O}_X$-linear (with the $\\mathcal{O}_X$-module", "structures given above)." ], "refs": [ "defos-remark-short-exact-sequence-thickenings", "defos-remark-extension-functorial", "defos-remark-trivial-extension", "defos-remark-short-exact-sequence-thickenings", "defos-lemma-inf-obs-ext", "defos-lemma-inf-obs-ext" ], "ref_ids": [ 13429, 13427, 13426, 13429, 13377, 13377 ] }, { "id": 13431, "type": "other", "label": "defos-remark-trivial-thickening-ringed-topoi", "categories": [ "defos" ], "title": "defos-remark-trivial-thickening-ringed-topoi", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. A first order", "thickening $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to", "(\\Sh(\\mathcal{D}), \\mathcal{O}')$ is said", "to be {\\it trivial} if there exists a morphism of ringed topoi", "$\\pi : (\\Sh(\\mathcal{D}), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$", "which is a left inverse to $i$. The choice of such a morphism", "$\\pi$ is called a {\\it trivialization} of the first order thickening.", "Given $\\pi$ we obtain a splitting", "\\begin{equation}", "\\label{equation-splitting-ringed-topoi}", "\\mathcal{O}' = \\mathcal{O} \\oplus \\mathcal{I}", "\\end{equation}", "as sheaves of algebras on $\\mathcal{C}$ by using $\\pi^\\sharp$", "to split the surjection $\\mathcal{O}' \\to \\mathcal{O}$.", "Conversely, such a splitting determines", "a morphism $\\pi$. The category of trivialized first order thickenings of", "$(\\Sh(\\mathcal{C}), \\mathcal{O})$ is equivalent to the category of ", "$\\mathcal{O}$-modules." ], "refs": [], "ref_ids": [] }, { "id": 13432, "type": "other", "label": "defos-remark-trivial-extension-ringed-topoi", "categories": [ "defos" ], "title": "defos-remark-trivial-extension-ringed-topoi", "contents": [ "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$", "be a trivial first order thickening of ringed topoi", "and let $\\pi : (\\Sh(\\mathcal{D}), \\mathcal{O}') \\to", "(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a trivialization. Then given any triple", "$(\\mathcal{F}, \\mathcal{K}, c)$ consisting of a pair of", "$\\mathcal{O}$-modules and a map", "$c : \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{K}$", "we may set", "$$", "\\mathcal{F}'_{c, triv} = \\mathcal{F} \\oplus \\mathcal{K}", "$$", "and use the splitting (\\ref{equation-splitting-ringed-topoi})", "associated to $\\pi$ and the map $c$ to define the $\\mathcal{O}'$-module", "structure and obtain an extension (\\ref{equation-extension-ringed-topoi}).", "We will call $\\mathcal{F}'_{c, triv}$ the {\\it trivial extension} of", "$\\mathcal{F}$ by $\\mathcal{K}$ corresponding", "to $c$ and the trivialization $\\pi$. Given any extension", "$\\mathcal{F}'$ as in (\\ref{equation-extension-ringed-topoi}) we can use", "$\\pi^\\sharp : \\mathcal{O} \\to \\mathcal{O}'$ to think of $\\mathcal{F}'$", "as an $\\mathcal{O}$-module extension, hence a class $\\xi_{\\mathcal{F}'}$", "in $\\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K})$.", "Lemma \\ref{lemma-inf-ext-ringed-topoi} assures that", "$\\mathcal{F}' \\mapsto \\xi_{\\mathcal{F}'}$", "induces a bijection", "$$", "\\left\\{", "\\begin{matrix}", "\\text{isomorphism classes of extensions}\\\\", "\\mathcal{F}'\\text{ as in (\\ref{equation-extension-ringed-topoi}) with }", "c = c_{\\mathcal{F}'}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K})", "$$", "Moreover, the trivial extension $\\mathcal{F}'_{c, triv}$ maps to the zero class." ], "refs": [ "defos-lemma-inf-ext-ringed-topoi" ], "ref_ids": [ 13394 ] }, { "id": 13433, "type": "other", "label": "defos-remark-extension-functorial-ringed-topoi", "categories": [ "defos" ], "title": "defos-remark-extension-functorial-ringed-topoi", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. Let", "$(\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}_i), \\mathcal{O}'_i)$,", "$i = 1, 2$ be first order thickenings with ideal sheaves $\\mathcal{I}_i$.", "Let $h : (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to", "(\\Sh(\\mathcal{D}_2), \\mathcal{O}'_2)$", "be a morphism of first order thickenings of $(\\Sh(\\mathcal{C}), \\mathcal{O})$.", "Picture", "$$", "\\xymatrix{", "& (\\Sh(\\mathcal{C}), \\mathcal{O}) \\ar[ld] \\ar[rd] & \\\\", "(\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\ar[rr]^h & & ", "(\\Sh(\\mathcal{D}_2), \\mathcal{O}'_2)", "}", "$$", "Observe that $h^\\sharp : \\mathcal{O}'_2 \\to \\mathcal{O}'_1$", "in particular induces an $\\mathcal{O}$-module map", "$\\mathcal{I}_2 \\to \\mathcal{I}_1$.", "Let $\\mathcal{F}$ be an $\\mathcal{O}$-module.", "Let $(\\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair", "consisting of an $\\mathcal{O}$-module $\\mathcal{K}_i$ and a map", "$c_i : \\mathcal{I}_i \\otimes_\\mathcal{O} \\mathcal{F} \\to", "\\mathcal{K}_i$. Assume furthermore given a map", "of $\\mathcal{O}$-modules $\\mathcal{K}_2 \\to \\mathcal{K}_1$", "such that", "$$", "\\xymatrix{", "\\mathcal{I}_2 \\otimes_\\mathcal{O} \\mathcal{F}", "\\ar[r]_-{c_2} \\ar[d] &", "\\mathcal{K}_2 \\ar[d] \\\\", "\\mathcal{I}_1 \\otimes_\\mathcal{O} \\mathcal{F}", "\\ar[r]^-{c_1} &", "\\mathcal{K}_1", "}", "$$", "is commutative. Then there is a canonical functoriality", "$$", "\\left\\{", "\\begin{matrix}", "\\mathcal{F}'_2\\text{ as in (\\ref{equation-extension-ringed-topoi}) with }\\\\", "c_2 = c_{\\mathcal{F}'_2}\\text{ and }\\mathcal{K} = \\mathcal{K}_2", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "\\mathcal{F}'_1\\text{ as in (\\ref{equation-extension-ringed-topoi}) with }\\\\", "c_1 = c_{\\mathcal{F}'_1}\\text{ and }\\mathcal{K} = \\mathcal{K}_1", "\\end{matrix}", "\\right\\}", "$$", "Namely, thinking of all sheaves $\\mathcal{O}$, $\\mathcal{O}'_i$,", "$\\mathcal{F}$, $\\mathcal{K}_i$, etc as sheaves on $\\mathcal{C}$, we set", "given $\\mathcal{F}'_2$ the sheaf $\\mathcal{F}'_1$ equal to the", "pushout, i.e., fitting into the following diagram of extensions", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{K}_2 \\ar[r] \\ar[d] &", "\\mathcal{F}'_2 \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar@{=}[d] \\ar[r] & 0 \\\\", "0 \\ar[r] &", "\\mathcal{K}_1 \\ar[r] &", "\\mathcal{F}'_1 \\ar[r] &", "\\mathcal{F} \\ar[r] & 0", "}", "$$", "We omit the construction of the $\\mathcal{O}'_1$-module structure", "on the pushout (this uses the commutativity of the diagram", "involving $c_1$ and $c_2$)." ], "refs": [], "ref_ids": [] }, { "id": 13434, "type": "other", "label": "defos-remark-trivial-extension-functorial-ringed-topoi", "categories": [ "defos" ], "title": "defos-remark-trivial-extension-functorial-ringed-topoi", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$,", "$(\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}_i), \\mathcal{O}'_i)$,", "$\\mathcal{I}_i$, and $h : (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to", "(\\Sh(\\mathcal{D}_2), \\mathcal{O}'_2)$ be as in", "Remark \\ref{remark-extension-functorial-ringed-topoi}.", "Assume that we are given trivializations", "$\\pi_i : (\\Sh(\\mathcal{D}_i), \\mathcal{O}'_i) \\to", "(\\Sh(\\mathcal{C}), \\mathcal{O})$ such that", "$\\pi_1 = h \\circ \\pi_2$. In other words, assume $h$ is a morphism", "of trivialized first order thickenings of $(\\Sh(\\mathcal{C}), \\mathcal{O})$.", "Let $(\\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair consisting of an", "$\\mathcal{O}$-module $\\mathcal{K}_i$ and a map", "$c_i : \\mathcal{I}_i \\otimes_\\mathcal{O} \\mathcal{F} \\to", "\\mathcal{K}_i$. Assume furthermore given a map", "of $\\mathcal{O}$-modules $\\mathcal{K}_2 \\to \\mathcal{K}_1$", "such that", "$$", "\\xymatrix{", "\\mathcal{I}_2 \\otimes_\\mathcal{O} \\mathcal{F}", "\\ar[r]_-{c_2} \\ar[d] &", "\\mathcal{K}_2 \\ar[d] \\\\", "\\mathcal{I}_1 \\otimes_\\mathcal{O} \\mathcal{F}", "\\ar[r]^-{c_1} &", "\\mathcal{K}_1", "}", "$$", "is commutative. In this situation the construction of", "Remark \\ref{remark-trivial-extension-ringed-topoi} induces", "a commutative diagram", "$$", "\\xymatrix{", "\\{\\mathcal{F}'_2\\text{ as in (\\ref{equation-extension-ringed-topoi}) with }", "c_2 = c_{\\mathcal{F}'_2}\\text{ and }\\mathcal{K} = \\mathcal{K}_2\\}", "\\ar[d] \\ar[rr] & &", "\\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_2) \\ar[d] \\\\", "\\{\\mathcal{F}'_1\\text{ as in (\\ref{equation-extension-ringed-topoi}) with }", "c_1 = c_{\\mathcal{F}'_1}\\text{ and }\\mathcal{K} = \\mathcal{K}_1\\}", "\\ar[rr] & &", "\\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_1)", "}", "$$", "where the vertical map on the right is given by functoriality of $\\Ext$", "and the map $\\mathcal{K}_2 \\to \\mathcal{K}_1$ and the vertical map on the left", "is the one from Remark \\ref{remark-extension-functorial-ringed-topoi}." ], "refs": [ "defos-remark-extension-functorial-ringed-topoi", "defos-remark-trivial-extension-ringed-topoi", "defos-remark-extension-functorial-ringed-topoi" ], "ref_ids": [ 13433, 13432, 13433 ] }, { "id": 13435, "type": "other", "label": "defos-remark-obstruction-extension-functorial-ringed-topoi", "categories": [ "defos" ], "title": "defos-remark-obstruction-extension-functorial-ringed-topoi", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$,", "$(\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}_i), \\mathcal{O}'_i)$,", "$\\mathcal{I}_i$, and $h : (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to", "(\\Sh(\\mathcal{D}_2), \\mathcal{O}'_2)$ be as in", "Remark \\ref{remark-extension-functorial-ringed-topoi}.", "Observe that $h^\\sharp : \\mathcal{O}'_2 \\to \\mathcal{O}'_1$", "in particular induces an $\\mathcal{O}$-module map", "$\\mathcal{I}_2 \\to \\mathcal{I}_1$.", "Let $\\mathcal{F}$ be an $\\mathcal{O}$-module.", "Let $(\\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair", "consisting of an $\\mathcal{O}$-module $\\mathcal{K}_i$ and a map", "$c_i : \\mathcal{I}_i \\otimes_\\mathcal{O} \\mathcal{F} \\to", "\\mathcal{K}_i$. Assume furthermore given a map", "of $\\mathcal{O}$-modules $\\mathcal{K}_2 \\to \\mathcal{K}_1$", "such that", "$$", "\\xymatrix{", "\\mathcal{I}_2 \\otimes_\\mathcal{O} \\mathcal{F}", "\\ar[r]_-{c_2} \\ar[d] &", "\\mathcal{K}_2 \\ar[d] \\\\", "\\mathcal{I}_1 \\otimes_\\mathcal{O} \\mathcal{F}", "\\ar[r]^-{c_1} &", "\\mathcal{K}_1", "}", "$$", "is commutative. Then we {\\bf claim} the map", "$$", "\\Ext^2_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_2)", "\\longrightarrow", "\\Ext^2_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_1)", "$$", "sends $o(\\mathcal{F}, \\mathcal{K}_2, c_2)$ to", "$o(\\mathcal{F}, \\mathcal{K}_1, c_1)$.", "\\medskip\\noindent", "To prove this claim choose an embedding", "$j_2 : \\mathcal{K}_2 \\to \\mathcal{K}_2'$", "where $\\mathcal{K}_2'$ is an injective $\\mathcal{O}$-module.", "As in the proof of Lemma \\ref{lemma-inf-obs-ext-ringed-topoi}", "we can choose an extension of $\\mathcal{O}_2$-modules", "$$", "0 \\to \\mathcal{K}_2' \\to \\mathcal{E}_2 \\to \\mathcal{F} \\to 0", "$$", "such that $c_{\\mathcal{E}_2} = j_2 \\circ c_2$.", "The proof of Lemma \\ref{lemma-inf-obs-ext-ringed-topoi} constructs", "$o(\\mathcal{F}, \\mathcal{K}_2, c_2)$", "as the Yoneda extension class (in the sense of", "Derived Categories, Section \\ref{derived-section-ext})", "of the exact sequence of $\\mathcal{O}$-modules", "$$", "0 \\to", "\\mathcal{K}_2 \\to \\mathcal{K}_2' \\to", "\\mathcal{E}_2/\\mathcal{K}_2 \\to", "\\mathcal{F} \\to 0", "$$", "Let $\\mathcal{K}_1'$ be the cokernel of", "$\\mathcal{K}_2 \\to \\mathcal{K}_1 \\oplus \\mathcal{K}_2'$.", "There is an injection $j_1 : \\mathcal{K}_1 \\to \\mathcal{K}_1'$", "and a map $\\mathcal{K}_2' \\to \\mathcal{K}_1'$ forming", "a commutative square. We form the pushout:", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{K}_2' \\ar[r] \\ar[d] &", "\\mathcal{E}_2 \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "\\mathcal{K}_1' \\ar[r] &", "\\mathcal{E}_1 \\ar[r] &", "\\mathcal{F} \\ar[r] & 0", "}", "$$", "There is a canonical $\\mathcal{O}_1$-module structure on", "$\\mathcal{E}_1$ and for this structure we have", "$c_{\\mathcal{E}_1} = j_1 \\circ c_1$ (this uses the commutativity", "of the diagram involving $c_1$ and $c_2$ above).", "The procedure of Lemma \\ref{lemma-inf-obs-ext-ringed-topoi}", "tells us that $o(\\mathcal{F}, \\mathcal{K}_1, c_1)$", "is the Yoneda extension class of the exact sequence", "of $\\mathcal{O}$-modules", "$$", "0 \\to", "\\mathcal{K}_1 \\to", "\\mathcal{K}_1' \\to", "\\mathcal{E}_1/\\mathcal{K}_1 \\to", "\\mathcal{F} \\to 0", "$$", "Since we have maps of exact sequences", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{K}_2 \\ar[d] \\ar[r] &", "\\mathcal{K}_2' \\ar[d] \\ar[r] &", "\\mathcal{E}_2/\\mathcal{K}_2 \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[r] \\ar@{=}[d] &", "0 \\\\", "0 \\ar[r] &", "\\mathcal{K}_2 \\ar[r] &", "\\mathcal{K}_2' \\ar[r] &", "\\mathcal{E}_2/\\mathcal{K}_2 \\ar[r] &", "\\mathcal{F} \\ar[r] &", "0", "}", "$$", "we conclude that the claim is true." ], "refs": [ "defos-remark-extension-functorial-ringed-topoi", "defos-lemma-inf-obs-ext-ringed-topoi", "defos-lemma-inf-obs-ext-ringed-topoi", "defos-lemma-inf-obs-ext-ringed-topoi" ], "ref_ids": [ 13433, 13395, 13395, 13395 ] }, { "id": 13436, "type": "other", "label": "defos-remark-short-exact-sequence-thickenings-ringed-topoi", "categories": [ "defos" ], "title": "defos-remark-short-exact-sequence-thickenings-ringed-topoi", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.", "We define a sequence of morphisms of first order thickenings", "$$", "(\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to", "(\\Sh(\\mathcal{D}_2), \\mathcal{O}'_2) \\to", "(\\Sh(\\mathcal{D}_3), \\mathcal{O}'_3)", "$$", "of $(\\Sh(\\mathcal{C}), \\mathcal{O})$ to be a {\\it complex}", "if the corresponding maps between", "the ideal sheaves $\\mathcal{I}_i$", "give a complex of $\\mathcal{O}$-modules", "$\\mathcal{I}_3 \\to \\mathcal{I}_2 \\to \\mathcal{I}_1$", "(i.e., the composition is zero). In this case the composition", "$(\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to", "(\\Sh(\\mathcal{D}_3), \\mathcal{O}'_3)$ factors through", "$(\\Sh(\\mathcal{C}), \\mathcal{O}) \\to", "(\\Sh(\\mathcal{D}_3), \\mathcal{O}'_3)$, i.e.,", "the first order thickening", "$(\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1)$ of", "$(\\Sh(\\mathcal{C}), \\mathcal{O})$ is trivial and comes with", "a canonical trivialization", "$\\pi : (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to", "(\\Sh(\\mathcal{C}), \\mathcal{O})$.", "\\medskip\\noindent", "We say a sequence of morphisms of first order thickenings", "$$", "(\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to", "(\\Sh(\\mathcal{D}_2), \\mathcal{O}'_2) \\to", "(\\Sh(\\mathcal{D}_3), \\mathcal{O}'_3)", "$$", "of $(\\Sh(\\mathcal{C}), \\mathcal{O})$ is {\\it a short exact sequence} if the", "corresponding maps between ideal sheaves is a short exact sequence", "$$", "0 \\to \\mathcal{I}_3 \\to \\mathcal{I}_2 \\to \\mathcal{I}_1 \\to 0", "$$", "of $\\mathcal{O}$-modules." ], "refs": [], "ref_ids": [] }, { "id": 13437, "type": "other", "label": "defos-remark-complex-thickenings-and-ses-modules-ringed-topoi", "categories": [ "defos" ], "title": "defos-remark-complex-thickenings-and-ses-modules-ringed-topoi", "contents": [ "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.", "Let $\\mathcal{F}$ be an $\\mathcal{O}$-module. Let", "$$", "(\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to", "(\\Sh(\\mathcal{D}_2), \\mathcal{O}'_2) \\to", "(\\Sh(\\mathcal{D}_3), \\mathcal{O}'_3)", "$$", "be a complex first order thickenings of $(\\Sh(\\mathcal{C}), \\mathcal{O})$, see", "Remark \\ref{remark-short-exact-sequence-thickenings-ringed-topoi}.", "Let $(\\mathcal{K}_i, c_i)$, $i = 1, 2, 3$ be pairs consisting of", "an $\\mathcal{O}$-module $\\mathcal{K}_i$ and a map", "$c_i : \\mathcal{I}_i \\otimes_\\mathcal{O} \\mathcal{F} \\to", "\\mathcal{K}_i$. Assume given a short exact sequence", "of $\\mathcal{O}$-modules", "$$", "0 \\to \\mathcal{K}_3 \\to \\mathcal{K}_2 \\to \\mathcal{K}_1 \\to 0", "$$", "such that", "$$", "\\vcenter{", "\\xymatrix{", "\\mathcal{I}_2 \\otimes_\\mathcal{O} \\mathcal{F}", "\\ar[r]_-{c_2} \\ar[d] &", "\\mathcal{K}_2 \\ar[d] \\\\", "\\mathcal{I}_1 \\otimes_\\mathcal{O} \\mathcal{F}", "\\ar[r]^-{c_1} &", "\\mathcal{K}_1", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "\\mathcal{I}_3 \\otimes_\\mathcal{O} \\mathcal{F}", "\\ar[r]_-{c_3} \\ar[d] &", "\\mathcal{K}_3 \\ar[d] \\\\", "\\mathcal{I}_2 \\otimes_\\mathcal{O} \\mathcal{F}", "\\ar[r]^-{c_2} &", "\\mathcal{K}_2", "}", "}", "$$", "are commutative. Finally, assume given an extension", "$$", "0 \\to \\mathcal{K}_2 \\to \\mathcal{F}'_2 \\to \\mathcal{F} \\to 0", "$$", "as in (\\ref{equation-extension-ringed-topoi})", "with $\\mathcal{K} = \\mathcal{K}_2$", "of $\\mathcal{O}'_2$-modules with $c_{\\mathcal{F}'_2} = c_2$.", "In this situation we can apply the functoriality of", "Remark \\ref{remark-extension-functorial-ringed-topoi}", "to obtain an extension $\\mathcal{F}'_1$ of $\\mathcal{O}'_1$-modules", "(we'll describe $\\mathcal{F}'_1$ in this special case below). By", "Remark \\ref{remark-trivial-extension-ringed-topoi}", "using the canonical splitting", "$\\pi : (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to", "(\\Sh(\\mathcal{C}), \\mathcal{O})$ of", "Remark \\ref{remark-short-exact-sequence-thickenings-ringed-topoi}", "we obtain", "$\\xi_{\\mathcal{F}'_1} \\in", "\\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_1)$.", "Finally, we have the obstruction", "$$", "o(\\mathcal{F}, \\mathcal{K}_3, c_3) \\in", "\\Ext^2_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_3)", "$$", "see Lemma \\ref{lemma-inf-obs-ext-ringed-topoi}.", "In this situation we {\\bf claim} that the canonical map", "$$", "\\partial :", "\\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_1)", "\\longrightarrow", "\\Ext^2_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_3)", "$$", "coming from the short exact sequence", "$0 \\to \\mathcal{K}_3 \\to \\mathcal{K}_2 \\to \\mathcal{K}_1 \\to 0$", "sends $\\xi_{\\mathcal{F}'_1}$", "to the obstruction class $o(\\mathcal{F}, \\mathcal{K}_3, c_3)$.", "\\medskip\\noindent", "To prove this claim choose an embedding $j : \\mathcal{K}_3 \\to \\mathcal{K}$", "where $\\mathcal{K}$ is an injective $\\mathcal{O}$-module.", "We can lift $j$ to a map $j' : \\mathcal{K}_2 \\to \\mathcal{K}$.", "Set $\\mathcal{E}'_2 = j'_*\\mathcal{F}'_2$ equal to the pushout", "of $\\mathcal{F}'_2$ by $j'$ so that $c_{\\mathcal{E}'_2} = j' \\circ c_2$.", "Picture:", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{K}_2 \\ar[r] \\ar[d]_{j'} &", "\\mathcal{F}'_2 \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "\\mathcal{K} \\ar[r] &", "\\mathcal{E}'_2 \\ar[r] &", "\\mathcal{F} \\ar[r] & 0", "}", "$$", "Set $\\mathcal{E}'_3 = \\mathcal{E}'_2$ but viewed as an", "$\\mathcal{O}'_3$-module via $\\mathcal{O}'_3 \\to \\mathcal{O}'_2$.", "Then $c_{\\mathcal{E}'_3} = j \\circ c_3$.", "The proof of Lemma \\ref{lemma-inf-obs-ext-ringed-topoi} constructs", "$o(\\mathcal{F}, \\mathcal{K}_3, c_3)$", "as the boundary of the class of the extension of $\\mathcal{O}$-modules", "$$", "0 \\to", "\\mathcal{K}/\\mathcal{K}_3 \\to", "\\mathcal{E}'_3/\\mathcal{K}_3 \\to", "\\mathcal{F} \\to 0", "$$", "On the other hand, note that $\\mathcal{F}'_1 = \\mathcal{F}'_2/\\mathcal{K}_3$", "hence the class $\\xi_{\\mathcal{F}'_1}$ is the class", "of the extension", "$$", "0 \\to \\mathcal{K}_2/\\mathcal{K}_3 \\to \\mathcal{F}'_2/\\mathcal{K}_3", "\\to \\mathcal{F} \\to 0", "$$", "seen as a sequence of $\\mathcal{O}$-modules using $\\pi^\\sharp$", "where $\\pi : (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to", "(\\Sh(\\mathcal{C}), \\mathcal{O})$ is the canonical splitting.", "Thus finally, the claim follows from the fact that we have", "a commutative diagram", "$$", "\\xymatrix{", "0 \\ar[r] &", "\\mathcal{K}_2/\\mathcal{K}_3 \\ar[r] \\ar[d] &", "\\mathcal{F}'_2/\\mathcal{K}_3 \\ar[r] \\ar[d] &", "\\mathcal{F} \\ar[r] \\ar[d] & 0 \\\\", "0 \\ar[r] &", "\\mathcal{K}/\\mathcal{K}_3 \\ar[r] &", "\\mathcal{E}'_3/\\mathcal{K}_3 \\ar[r] &", "\\mathcal{F} \\ar[r] & 0", "}", "$$", "which is $\\mathcal{O}$-linear (with the $\\mathcal{O}$-module", "structures given above)." ], "refs": [ "defos-remark-short-exact-sequence-thickenings-ringed-topoi", "defos-remark-extension-functorial-ringed-topoi", "defos-remark-trivial-extension-ringed-topoi", "defos-remark-short-exact-sequence-thickenings-ringed-topoi", "defos-lemma-inf-obs-ext-ringed-topoi", "defos-lemma-inf-obs-ext-ringed-topoi" ], "ref_ids": [ 13436, 13433, 13432, 13436, 13395, 13395 ] }, { "id": 13645, "type": "other", "label": "duality-remark-iso-on-RSheafHom", "categories": [ "duality" ], "title": "duality-remark-iso-on-RSheafHom", "contents": [ "In the situation of Lemma \\ref{lemma-iso-on-RSheafHom} we have", "$$", "DQ_Y(Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K))) =", "Rf_* DQ_X(R\\SheafHom_{\\mathcal{O}_X}(L, a(K)))", "$$", "by Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-pushforward-better-coherator}.", "Thus if $R\\SheafHom_{\\mathcal{O}_X}(L, a(K)) \\in D_\\QCoh(\\mathcal{O}_X)$,", "then we can ``erase'' the $DQ_Y$ on the left hand side of the arrow.", "On the other hand, if we know that", "$R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K) \\in D_\\QCoh(\\mathcal{O}_Y)$,", "then we can ``erase'' the $DQ_Y$ from the right hand side of the arrow.", "If both are true then we see that (\\ref{equation-sheafy-trace})", "is an isomorphism. Combining this with", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-quasi-coherence-internal-hom}", "we see that $Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K)) \\to", "R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K)$ is an isomorphism if", "\\begin{enumerate}", "\\item $L$ and $Rf_*L$ are perfect, or", "\\item $K$ is bounded below and $L$ and $Rf_*L$ are pseudo-coherent.", "\\end{enumerate}", "For (2) we use that $a(K)$ is bounded below if $K$", "is bounded below, see Lemma \\ref{lemma-twisted-inverse-image-bounded-below}." ], "refs": [ "duality-lemma-iso-on-RSheafHom", "perfect-lemma-pushforward-better-coherator", "perfect-lemma-quasi-coherence-internal-hom", "duality-lemma-twisted-inverse-image-bounded-below" ], "ref_ids": [ 13505, 7023, 6981, 13504 ] }, { "id": 13646, "type": "other", "label": "duality-remark-going-around", "categories": [ "duality" ], "title": "duality-remark-going-around", "contents": [ "Consider a commutative diagram", "$$", "\\xymatrix{", "X'' \\ar[r]_{k'} \\ar[d]_{f''} & X' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y'' \\ar[r]^{l'} \\ar[d]_{g''} & Y' \\ar[r]^l \\ar[d]_{g'} & Y \\ar[d]^g \\\\", "Z'' \\ar[r]^{m'} & Z' \\ar[r]^m & Z", "}", "$$", "of quasi-compact and quasi-separated schemes where", "all squares are cartesian and where", "$(f, l)$, $(g, m)$, $(f', l')$, $(g', m')$ are", "Tor independent pairs of maps.", "Let $a$, $a'$, $a''$, $b$, $b'$, $b''$ be the", "right adjoints of Lemma \\ref{lemma-twisted-inverse-image}", "for $f$, $f'$, $f''$, $g$, $g'$, $g''$.", "Let us label the squares of the diagram $A$, $B$, $C$, $D$", "as follows", "$$", "\\begin{matrix}", "A & B \\\\", "C & D", "\\end{matrix}", "$$", "Then the maps (\\ref{equation-base-change-map})", "for the squares are (where we use $k^* = Lk^*$, etc)", "$$", "\\begin{matrix}", "\\gamma_A : (k')^* \\circ a' \\to a'' \\circ (l')^* &", "\\gamma_B : k^* \\circ a \\to a' \\circ l^* \\\\", "\\gamma_C : (l')^* \\circ b' \\to b'' \\circ (m')^* &", "\\gamma_D : l^* \\circ b \\to b' \\circ m^*", "\\end{matrix}", "$$", "For the $2 \\times 1$ and $1 \\times 2$ rectangles we have four further", "base change maps", "$$", "\\begin{matrix}", "\\gamma_{A + B} : (k \\circ k')^* \\circ a \\to a'' \\circ (l \\circ l')^* \\\\", "\\gamma_{C + D} : (l \\circ l')^* \\circ b \\to b'' \\circ (m \\circ m')^* \\\\", "\\gamma_{A + C} : (k')^* \\circ (a' \\circ b') \\to (a'' \\circ b'') \\circ (m')^* \\\\", "\\gamma_{B + D} : k^* \\circ (a \\circ b) \\to (a' \\circ b') \\circ m^*", "\\end{matrix}", "$$", "By Lemma \\ref{lemma-compose-base-change-maps-horizontal} we have", "$$", "\\gamma_{A + B} = \\gamma_A \\circ \\gamma_B, \\quad", "\\gamma_{C + D} = \\gamma_C \\circ \\gamma_D", "$$", "and by Lemma \\ref{lemma-compose-base-change-maps} we have", "$$", "\\gamma_{A + C} = \\gamma_C \\circ \\gamma_A, \\quad", "\\gamma_{B + D} = \\gamma_D \\circ \\gamma_B", "$$", "Here it would be more correct to write", "$\\gamma_{A + B} = (\\gamma_A \\star \\text{id}_{l^*}) \\circ", "(\\text{id}_{(k')^*} \\star \\gamma_B)$ with notation as in", "Categories, Section \\ref{categories-section-formal-cat-cat}", "and similarly for the others. However, we continue the", "abuse of notation used in the proofs of", "Lemmas \\ref{lemma-compose-base-change-maps} and", "\\ref{lemma-compose-base-change-maps-horizontal}", "of dropping $\\star$ products with identities as one can figure", "out which ones to add as long as the source and target of the", "transformation is known.", "Having said all of this we find (a priori) two transformations", "$$", "(k')^* \\circ k^* \\circ a \\circ b", "\\longrightarrow", "a'' \\circ b'' \\circ (m')^* \\circ m^*", "$$", "namely", "$$", "\\gamma_C \\circ \\gamma_A \\circ \\gamma_D \\circ \\gamma_B =", "\\gamma_{A + C} \\circ \\gamma_{B + D}", "$$", "and", "$$", "\\gamma_C \\circ \\gamma_D \\circ \\gamma_A \\circ \\gamma_B =", "\\gamma_{C + D} \\circ \\gamma_{A + B}", "$$", "The point of this remark is to point out that these transformations", "are equal. Namely, to see this it suffices to show that", "$$", "\\xymatrix{", "(k')^* \\circ a' \\circ l^* \\circ b \\ar[r]_{\\gamma_D} \\ar[d]_{\\gamma_A} &", "(k')^* \\circ a' \\circ b' \\circ m^* \\ar[d]^{\\gamma_A} \\\\", "a'' \\circ (l')^* \\circ l^* \\circ b \\ar[r]^{\\gamma_D} &", "a'' \\circ (l')^* \\circ b' \\circ m^*", "}", "$$", "commutes. This is true by", "Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats}", "or more simply the discussion preceding", "Categories, Definition \\ref{categories-definition-horizontal-composition}." ], "refs": [ "duality-lemma-twisted-inverse-image", "duality-lemma-compose-base-change-maps-horizontal", "duality-lemma-compose-base-change-maps", "duality-lemma-compose-base-change-maps", "duality-lemma-compose-base-change-maps-horizontal", "categories-lemma-properties-2-cat-cats", "categories-definition-horizontal-composition" ], "ref_ids": [ 13503, 13511, 13510, 13510, 13511, 12269, 12377 ] }, { "id": 13647, "type": "other", "label": "duality-remark-check-over-affines", "categories": [ "duality" ], "title": "duality-remark-check-over-affines", "contents": [ "Consider a cartesian diagram", "$$", "\\xymatrix{", "X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\", "Y' \\ar[r]^g & Y", "}", "$$", "of quasi-compact and quasi-separated schemes with $(g, f)$ Tor independent.", "Let $V \\subset Y$ and $V' \\subset Y'$ be affine opens with", "$g(V') \\subset V$. Form the cartesian diagrams", "$$", "\\vcenter{", "\\xymatrix{", "U \\ar[r] \\ar[d] & X \\ar[d] \\\\", "V \\ar[r] & Y", "}", "}", "\\quad\\text{and}\\quad", "\\vcenter{", "\\xymatrix{", "U' \\ar[r] \\ar[d] & X' \\ar[d] \\\\", "V' \\ar[r] & Y'", "}", "}", "$$", "Assume (\\ref{equation-sheafy}) with respect to $K$", "and the first diagram and (\\ref{equation-sheafy})", "with respect to $Lg^*K$ and the second diagram are isomorphisms.", "Then the restriction of the base change map (\\ref{equation-base-change-map})", "$$", "L(g')^*a(K) \\longrightarrow a'(Lg^*K)", "$$", "to $U'$ is isomorphic to the base change map", "(\\ref{equation-base-change-map}) for $K|_V$ and the", "cartesian diagram", "$$", "\\xymatrix{", "U' \\ar[r] \\ar[d] & U \\ar[d] \\\\", "V' \\ar[r] & V", "}", "$$", "This follows from the fact that (\\ref{equation-sheafy})", "is a special case of the base change map (\\ref{equation-base-change-map})", "and that the base change maps compose correctly if we stack squares", "horizontally, see Lemma \\ref{lemma-compose-base-change-maps-horizontal}.", "Thus in order to check the base change map restricted to $U'$", "is an isomorphism it suffices to work with the last diagram." ], "refs": [ "duality-lemma-compose-base-change-maps-horizontal" ], "ref_ids": [ 13511 ] }, { "id": 13648, "type": "other", "label": "duality-remark-trace-map-finite", "categories": [ "duality" ], "title": "duality-remark-trace-map-finite", "contents": [ "If $f : Y \\to X$ is a finite morphism of Noetherian schemes, then the diagram", "$$", "\\xymatrix{", "Rf_*a(K) \\ar[r]_-{\\text{Tr}_{f, K}} \\ar@{=}[d] & K \\ar@{=}[d] \\\\", "R\\SheafHom_{\\mathcal{O}_X}(f_*\\mathcal{O}_Y, K) \\ar[r] & K", "}", "$$", "is commutative for $K \\in D_\\QCoh^+(\\mathcal{O}_X)$. This follows", "from Lemma \\ref{lemma-finite-twisted}. The lower horizontal", "arrow is induced by the map $\\mathcal{O}_X \\to f_*\\mathcal{O}_Y$ and the", "upper horizontal arrow is the trace map discussed in", "Section \\ref{section-trace}." ], "refs": [ "duality-lemma-finite-twisted" ], "ref_ids": [ 13532 ] }, { "id": 13649, "type": "other", "label": "duality-remark-relative-dualizing-complex", "categories": [ "duality" ], "title": "duality-remark-relative-dualizing-complex", "contents": [ "Let $Y$ be a quasi-compact and quasi-separated scheme.", "Let $f : X \\to Y$ be a proper, flat morphism of finite presentation.", "Let $a$ be the adjoint of Lemma \\ref{lemma-twisted-inverse-image} for $f$.", "In this situation, $\\omega_{X/Y}^\\bullet = a(\\mathcal{O}_Y)$", "is sometimes called the {\\it relative dualizing complex}. By", "Lemma \\ref{lemma-compare-with-pullback-flat-proper}", "there is a functorial isomorphism", "$a(K) = Lf^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_{X/Y}^\\bullet$", "for $K \\in D_\\QCoh(\\mathcal{O}_Y)$. Moreover, the trace map", "$$", "\\text{Tr}_{f, \\mathcal{O}_Y} : Rf_*\\omega_{X/Y}^\\bullet \\to \\mathcal{O}_Y", "$$", "of Section \\ref{section-trace} induces the trace map for all $K$", "in $D_\\QCoh(\\mathcal{O}_Y)$. More precisely the diagram", "$$", "\\xymatrix{", "Rf_*a(K) \\ar[rrr]_{\\text{Tr}_{f, K}} \\ar@{=}[d] & & &", "K \\ar@{=}[d] \\\\", "Rf_*(Lf^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_{X/Y}^\\bullet)", "\\ar@{=}[r] &", "K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*\\omega_{X/Y}^\\bullet", "\\ar[rr]^-{\\text{id}_K \\otimes \\text{Tr}_{f, \\mathcal{O}_Y}} & & K", "}", "$$", "where the equality on the lower right is", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-cohomology-base-change}.", "If $g : Y' \\to Y$ is a", "morphism of quasi-compact and quasi-separated schemes", "and $X' = Y' \\times_Y X$, then by", "Lemma \\ref{lemma-proper-flat-base-change} we have", "$\\omega_{X'/Y'}^\\bullet = L(g')^*\\omega_{X/Y}^\\bullet$ where $g' : X' \\to X$", "is the projection and by Lemma \\ref{lemma-trace-map-and-base-change}", "the trace map", "$$", "\\text{Tr}_{f', \\mathcal{O}_{Y'}} :", "Rf'_*\\omega_{X'/Y'}^\\bullet \\to \\mathcal{O}_{Y'}", "$$", "for $f' : X' \\to Y'$ is the base change of $\\text{Tr}_{f, \\mathcal{O}_Y}$", "via the base change isomorphism." ], "refs": [ "duality-lemma-twisted-inverse-image", "duality-lemma-compare-with-pullback-flat-proper", "perfect-lemma-cohomology-base-change", "duality-lemma-proper-flat-base-change", "duality-lemma-trace-map-and-base-change" ], "ref_ids": [ 13503, 13535, 7025, 13536, 13513 ] }, { "id": 13650, "type": "other", "label": "duality-remark-relative-dualizing-complex-relative-cup-product", "categories": [ "duality" ], "title": "duality-remark-relative-dualizing-complex-relative-cup-product", "contents": [ "Let $f : X \\to Y$, $\\omega^\\bullet_{X/Y}$, and $\\text{Tr}_{f, \\mathcal{O}_Y}$", "be as in Remark \\ref{remark-relative-dualizing-complex}.", "Let $K$ and $M$ be in $D_\\QCoh(\\mathcal{O}_X)$ with", "$M$ pseudo-coherent (for example perfect). Suppose given a map", "$K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M \\to \\omega^\\bullet_{X/Y}$", "which corresponds to an isomorphism", "$K \\to R\\SheafHom_{\\mathcal{O}_X}(M, \\omega^\\bullet_{X/Y})$", "via Cohomology, Equation (\\ref{cohomology-equation-internal-hom}).", "Then the relative cup product", "(Cohomology, Remark \\ref{cohomology-remark-cup-product})", "$$", "Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*M", "\\to", "Rf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)", "\\to", "Rf_*\\omega^\\bullet_{X/Y}", "\\xrightarrow{\\text{Tr}_{f, \\mathcal{O}_Y}}", "\\mathcal{O}_Y", "$$", "determines an isomorphism", "$Rf_*K \\to R\\SheafHom_{\\mathcal{O}_Y}(Rf_*M, \\mathcal{O}_Y)$.", "Namely, since $\\omega^\\bullet_{X/Y} = a(\\mathcal{O}_Y)$", "the canonical map (\\ref{equation-sheafy-trace})", "$$", "Rf_*R\\SheafHom_{\\mathcal{O}_X}(M, \\omega^\\bullet_{X/Y}) \\to", "R\\SheafHom_{\\mathcal{O}_Y}(Rf_*M, \\mathcal{O}_Y)", "$$", "is an isomorphism by", "Lemma \\ref{lemma-iso-on-RSheafHom} and", "Remark \\ref{remark-iso-on-RSheafHom}", "and the fact that $M$ and $Rf_*M$ are pseudo-coherent, see", "Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-flat-proper-pseudo-coherent-direct-image-general}.", "To see that the relative cup product", "induces this isomorphism use the commutativity of the diagram in", "Cohomology, Remark \\ref{cohomology-remark-relative-cup-and-composition}." ], "refs": [ "duality-remark-relative-dualizing-complex", "cohomology-remark-cup-product", "duality-lemma-iso-on-RSheafHom", "duality-remark-iso-on-RSheafHom", "perfect-lemma-flat-proper-pseudo-coherent-direct-image-general", "cohomology-remark-relative-cup-and-composition" ], "ref_ids": [ 13649, 2272, 13505, 13645, 7055, 2281 ] }, { "id": 13651, "type": "other", "label": "duality-remark-van-den-bergh", "categories": [ "duality" ], "title": "duality-remark-van-den-bergh", "contents": [ "Lemma \\ref{lemma-van-den-bergh} means our relative dualizing", "complex is {\\it rigid} in a sense analogous to the notion introduced", "in \\cite{vdB-rigid}. Namely, since the functor on the right of", "(\\ref{equation-rigid})", "is ``quadratic'' in $\\omega_{X/Y}^\\bullet$ and the functor on the left", "of (\\ref{equation-rigid})", "is ``linear'' this ``pins down'' the complex $\\omega_{X/Y}^\\bullet$", "to some extent. There is an approach to duality theory using", "``rigid'' (relative) dualizing complexes, see for example", "\\cite{Neeman-rigid}, \\cite{Yekutieli-rigid}, and \\cite{Yekutieli-Zhang}.", "We will return to this in Section \\ref{section-relative-dualizing-complexes}." ], "refs": [ "duality-lemma-van-den-bergh" ], "ref_ids": [ 13538 ] }, { "id": 13652, "type": "other", "label": "duality-remark-local-calculation-shriek", "categories": [ "duality" ], "title": "duality-remark-local-calculation-shriek", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$. Using the lemmas above we can compute", "$f^!$ locally as follows. Suppose that we are given affine opens", "$$", "\\xymatrix{", "U \\ar[r]_j \\ar[d]_g & X \\ar[d]^f \\\\", "V \\ar[r]^i & Y", "}", "$$", "Since $j^! \\circ f^! = g^! \\circ i^!$", "(Lemma \\ref{lemma-upper-shriek-composition})", "and since $j^!$ and $i^!$ are given by restriction", "(Lemma \\ref{lemma-shriek-open-immersion})", "we see that", "$$", "(f^!E)|_U = g^!(E|_V)", "$$", "for any $E \\in D^+_\\QCoh(\\mathcal{O}_X)$. Write", "$U = \\Spec(A)$ and $V = \\Spec(R)$ and let $\\varphi : R \\to A$", "be the finite type ring map corresponding to $g$.", "Choose a presentation $A = P/I$ where $P = R[x_1, \\ldots, x_n]$", "is a polynomial algebra in $n$ variables over $R$. Choose an", "object $K \\in D^+(R)$ corresponding to $E|_V$", "(Derived Categories of Schemes, Lemma", "\\ref{perfect-lemma-affine-compare-bounded}).", "Then we claim that $f^!E|_U$ corresponds to", "$$", "\\varphi^!(K) = R\\Hom(A, K \\otimes_R^\\mathbf{L} P)[n]", "$$", "where $R\\Hom(A, -) : D(P) \\to D(A)$ is the functor of", "Dualizing Complexes, Section \\ref{dualizing-section-trivial}", "and where $\\varphi^! : D(R) \\to D(A)$ is the functor of", "Dualizing Complexes, Section", "\\ref{dualizing-section-relative-dualizing-complex-algebraic}.", "Namely, the choice of presentation", "gives a factorization", "$$", "U \\rightarrow \\mathbf{A}^n_V \\to \\mathbf{A}^{n - 1}_V \\to \\ldots \\to", "\\mathbf{A}^1_V \\to V", "$$", "Applying Lemma \\ref{lemma-shriek-affine-line} exactly $n$ times we see that", "$(\\mathbf{A}^n_V \\to V)^!(E|_V)$ corresponds to", "$K \\otimes_R^\\mathbf{L} P[n]$. By Lemmas", "\\ref{lemma-sheaf-with-exact-support-quasi-coherent} and", "\\ref{lemma-shriek-closed-immersion} the last step corresponds to", "applying $R\\Hom(A, -)$." ], "refs": [ "duality-lemma-upper-shriek-composition", "duality-lemma-shriek-open-immersion", "perfect-lemma-affine-compare-bounded", "duality-lemma-shriek-affine-line", "duality-lemma-sheaf-with-exact-support-quasi-coherent", "duality-lemma-shriek-closed-immersion" ], "ref_ids": [ 13552, 13555, 6941, 13557, 13523, 13558 ] }, { "id": 13653, "type": "other", "label": "duality-remark-independent-omega-S", "categories": [ "duality" ], "title": "duality-remark-independent-omega-S", "contents": [ "Let $S$ be a Noetherian scheme which has a dualizing complex.", "Let $f : X \\to Y$ be a morphism of schemes of finite type", "over $S$. Then the functor", "$$", "f_{new}^! : D^+_{Coh}(\\mathcal{O}_Y) \\to D^+_{Coh}(\\mathcal{O}_X)", "$$", "is independent of the choice of the dualizing complex $\\omega_S^\\bullet$", "up to canonical isomorphism. We sketch the proof. Any second dualizing complex", "is of the form $\\omega_S^\\bullet \\otimes_{\\mathcal{O}_S}^\\mathbf{L} \\mathcal{L}$", "where $\\mathcal{L}$ is an invertible object of $D(\\mathcal{O}_S)$, see", "Lemma \\ref{lemma-dualizing-unique-schemes}.", "For any separated morphism $p : U \\to S$ of finite type we have", "$p^!(\\omega_S^\\bullet \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{L}) =", "p^!(\\omega_S^\\bullet) \\otimes^\\mathbf{L}_{\\mathcal{O}_U} Lp^*\\mathcal{L}$", "by Lemma \\ref{lemma-compare-with-pullback-perfect}.", "Hence, if $\\omega_X^\\bullet$ and $\\omega_Y^\\bullet$ are the", "dualizing complexes normalized relative to $\\omega_S^\\bullet$ we see that", "$\\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} La^*\\mathcal{L}$ and", "$\\omega_Y^\\bullet \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Lb^*\\mathcal{L}$", "are the dualizing complexes normalized relative to", "$\\omega_S^\\bullet \\otimes_{\\mathcal{O}_S}^\\mathbf{L} \\mathcal{L}$", "(where $a : X \\to S$ and $b : Y \\to S$ are the structure morphisms).", "Then the result follows as", "\\begin{align*}", "& R\\SheafHom_{\\mathcal{O}_X}(Lf^*R\\SheafHom_{\\mathcal{O}_Y}(K,", "\\omega_Y^\\bullet \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Lb^*\\mathcal{L}),", "\\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} La^*\\mathcal{L}) \\\\", "& = R\\SheafHom_{\\mathcal{O}_X}(Lf^*R(\\SheafHom_{\\mathcal{O}_Y}(K,", "\\omega_Y^\\bullet) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Lb^*\\mathcal{L}),", "\\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} La^*\\mathcal{L}) \\\\", "& = R\\SheafHom_{\\mathcal{O}_X}(Lf^*R\\SheafHom_{\\mathcal{O}_Y}(K,", "\\omega_Y^\\bullet) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} La^*\\mathcal{L},", "\\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} La^*\\mathcal{L}) \\\\", "& = R\\SheafHom_{\\mathcal{O}_X}(Lf^*R\\SheafHom_{\\mathcal{O}_Y}(K,", "\\omega_Y^\\bullet), \\omega_X^\\bullet)", "\\end{align*}", "for $K \\in D^+_{Coh}(\\mathcal{O}_Y)$.", "The last equality because $La^*\\mathcal{L}$ is invertible in", "$D(\\mathcal{O}_X)$." ], "refs": [ "duality-lemma-dualizing-unique-schemes", "duality-lemma-compare-with-pullback-perfect" ], "ref_ids": [ 13500, 13515 ] }, { "id": 13654, "type": "other", "label": "duality-remark-dualizing-finite", "categories": [ "duality" ], "title": "duality-remark-dualizing-finite", "contents": [ "Let $S$ be a Noetherian scheme and let $\\omega_S^\\bullet$ be a dualizing", "complex. Let $f : X \\to Y$ be a finite morphism between schemes of finite", "type over $S$. Let $\\omega_X^\\bullet$ and $\\omega_Y^\\bullet$ be", "dualizing complexes normalized relative to $\\omega_S^\\bullet$.", "Then we have", "$$", "f_*\\omega_X^\\bullet = R\\SheafHom(f_*\\mathcal{O}_X, \\omega_Y^\\bullet)", "$$", "in $D_\\QCoh^+(f_*\\mathcal{O}_X)$ by Lemmas \\ref{lemma-finite-twisted} and", "\\ref{lemma-proper-map-good-dualizing-complex}", "and the trace map of Example \\ref{example-trace-proper} is the map", "$$", "\\text{Tr}_f : Rf_*\\omega_X^\\bullet = f_*\\omega_X^\\bullet =", "R\\SheafHom(f_*\\mathcal{O}_X, \\omega_Y^\\bullet) \\longrightarrow", "\\omega_Y^\\bullet", "$$", "which often goes under the name ``evaluation at $1$''." ], "refs": [ "duality-lemma-finite-twisted", "duality-lemma-proper-map-good-dualizing-complex" ], "ref_ids": [ 13532, 13574 ] }, { "id": 13655, "type": "other", "label": "duality-remark-relative-dualizing-complex-shriek", "categories": [ "duality" ], "title": "duality-remark-relative-dualizing-complex-shriek", "contents": [ "Let $f : X \\to Y$ be a flat proper morphism of finite type", "schemes over a pair $(S, \\omega_S^\\bullet)$ as in", "Situation \\ref{situation-dualizing}. The relative dualizing complex", "(Remark \\ref{remark-relative-dualizing-complex}) is", "$\\omega_{X/Y}^\\bullet = a(\\mathcal{O}_Y)$. By", "Lemma \\ref{lemma-proper-map-good-dualizing-complex}", "we have the first canonical isomorphism in", "$$", "\\omega_X^\\bullet = a(\\omega_Y^\\bullet) =", "Lf^*\\omega_Y^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_{X/Y}^\\bullet", "$$", "in $D(\\mathcal{O}_X)$. The second canonical isomorphism follows from the", "discussion in Remark \\ref{remark-relative-dualizing-complex}." ], "refs": [ "duality-remark-relative-dualizing-complex", "duality-lemma-proper-map-good-dualizing-complex", "duality-remark-relative-dualizing-complex" ], "ref_ids": [ 13649, 13574, 13649 ] }, { "id": 13656, "type": "other", "label": "duality-remark-the-same-is-true", "categories": [ "duality" ], "title": "duality-remark-the-same-is-true", "contents": [ "Let $S$ be a Noetherian scheme endowed with a dualizing complex", "$\\omega_S^\\bullet$. In this case", "Lemmas \\ref{lemma-shriek}, \\ref{lemma-flat-shriek},", "\\ref{lemma-flat-quasi-finite-shriek}, and \\ref{lemma-CM-shriek}", "are true for any morphism $f : X \\to Y$ of finite type schemes over $S$", "but with $f^!$ replaced by $f_{new}^!$. This is clear because in each", "case the proof reduces immediately to the affine case", "and then $f^! = f_{new}^!$ by Lemma \\ref{lemma-duality-bootstrap}." ], "refs": [ "duality-lemma-shriek", "duality-lemma-flat-shriek", "duality-lemma-flat-quasi-finite-shriek", "duality-lemma-CM-shriek", "duality-lemma-duality-bootstrap" ], "ref_ids": [ 13578, 13579, 13581, 13582, 13575 ] }, { "id": 13657, "type": "other", "label": "duality-remark-CM-morphism-compare-dualizing", "categories": [ "duality" ], "title": "duality-remark-CM-morphism-compare-dualizing", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism", "of $\\textit{FTS}_S$. Assume $f$ is a Cohen-Macaulay morphism of", "relative dimension $d$. Let $\\omega_{X/Y} = H^{-d}(f^!\\mathcal{O}_Y)$", "be the unique nonzero cohomology sheaf of $f^!\\mathcal{O}_Y$, see", "Lemma \\ref{lemma-CM-shriek}.", "Then there is a canonical isomorphism", "$$", "f^!K = Lf^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_{X/Y}[d]", "$$", "for $K \\in D^+_\\QCoh(\\mathcal{O}_Y)$, see", "Lemma \\ref{lemma-perfect-comparison-shriek}. In particular, if", "$S$ has a dualizing complex $\\omega_S^\\bullet$,", "$\\omega_Y^\\bullet = (Y \\to S)^!\\omega_S^\\bullet$, and", "$\\omega_X^\\bullet = (X \\to S)^!\\omega_S^\\bullet$", "then we have", "$$", "\\omega_X^\\bullet =", "Lf^*\\omega_Y^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_{X/Y}[d]", "$$", "Thus if further $X$ and $Y$ are connected and Cohen-Macaulay and", "if $\\omega_Y$ and $\\omega_X$ denote the unique nonzero cohomology", "sheaves of $\\omega_Y^\\bullet$ and $\\omega_X^\\bullet$, then we", "have", "$$", "\\omega_X = f^*\\omega_Y \\otimes_{\\mathcal{O}_X} \\omega_{X/Y}.", "$$", "Similar results hold for $X$ and $Y$ arbitrary finite type schemes", "over $S$ (i.e., not necessarily separated over $S$)", "with dualizing complexes normalized with respect to", "$\\omega_S^\\bullet$ as in Section \\ref{section-glue}." ], "refs": [ "duality-lemma-CM-shriek", "duality-lemma-perfect-comparison-shriek" ], "ref_ids": [ 13582, 13562 ] }, { "id": 13658, "type": "other", "label": "duality-remark-duality-proper-over-field", "categories": [ "duality" ], "title": "duality-remark-duality-proper-over-field", "contents": [ "Let $k$, $X$, and $\\omega_X^\\bullet$", "be as in Lemma \\ref{lemma-duality-proper-over-field}.", "The identity on the complex $\\omega_X^\\bullet$ corresponds, via", "the functorial isomorphism in part (5), to a map", "$$", "t : H^0(X, \\omega_X^\\bullet) \\longrightarrow k", "$$", "For an arbitrary $K$ in $D_\\QCoh(\\mathcal{O}_X)$ the identification", "$\\Hom(K, \\omega_X^\\bullet)$ with $H^0(X, K)^\\vee$ in part (5)", "corresponds to the pairing", "$$", "\\Hom_X(K, \\omega_X^\\bullet) \\times H^0(X, K) \\longrightarrow k,\\quad", "(\\alpha, \\beta) \\longmapsto t(\\alpha(\\beta))", "$$", "This follows from the functoriality of the isomorphisms in (5). Similarly", "for any $i \\in \\mathbf{Z}$ we get the pairing", "$$", "\\Ext^i_X(K, \\omega_X^\\bullet) \\times H^{-i}(X, K) \\longrightarrow k,\\quad", "(\\alpha, \\beta) \\longmapsto t(\\alpha(\\beta))", "$$", "Here we think of $\\alpha$ as a morphism $K[-i] \\to \\omega_X^\\bullet$", "and $\\beta$ as an element of $H^0(X, K[-i])$ in order to define", "$\\alpha(\\beta)$. Observe that if $K$ is general, then we only know", "that this pairing is nondegenerate on one side: the pairing induces an", "isomorphism of $\\Hom_X(K, \\omega_X^\\bullet)$,", "resp.\\ $\\Ext^i_X(K, \\omega_X^\\bullet)$ with the $k$-linear dual of $H^0(X, K)$,", "resp.\\ $H^{-i}(X, K)$ but in general not vice versa. If $K$", "is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$, then", "$\\Hom_X(K, \\omega_X^\\bullet)$, $\\Ext_X(K, \\omega_X^\\bullet)$,", "$H^0(X, K)$, and $H^i(X, K)$ are finite dimensional $k$-vector spaces (by", "Derived Categories of Schemes, Lemmas", "\\ref{perfect-lemma-coherent-internal-hom} and", "\\ref{perfect-lemma-direct-image-coherent-bdd-below})", "and the pairings are perfect in the usual sense." ], "refs": [ "duality-lemma-duality-proper-over-field", "perfect-lemma-coherent-internal-hom", "perfect-lemma-direct-image-coherent-bdd-below" ], "ref_ids": [ 13606, 6986, 6985 ] }, { "id": 13659, "type": "other", "label": "duality-remark-coherent-duality-proper-over-field", "categories": [ "duality" ], "title": "duality-remark-coherent-duality-proper-over-field", "contents": [ "We continue the discussion in Remark \\ref{remark-duality-proper-over-field}", "and we use the same notation $k$, $X$, $\\omega_X^\\bullet$, and $t$.", "If $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module we obtain", "perfect pairings", "$$", "\\langle -, - \\rangle :", "\\Ext^i_X(\\mathcal{F}, \\omega_X^\\bullet) \\times H^{-i}(X,\\mathcal{F})", "\\longrightarrow k,\\quad", "(\\alpha, \\beta) \\longmapsto t(\\alpha(\\beta))", "$$", "of finite dimensional $k$-vector spaces. These pairings satisfy the", "following (obvious) functoriality: if $\\varphi : \\mathcal{F} \\to \\mathcal{G}$", "is a homomorphism of coherent $\\mathcal{O}_X$-modules, then we have", "$$", "\\langle \\alpha \\circ \\varphi, \\beta \\rangle =", "\\langle \\alpha, \\varphi(\\beta) \\rangle", "$$", "for $\\alpha \\in \\Ext^i_X(\\mathcal{G}, \\omega_X^\\bullet)$ and", "$\\beta \\in H^{-i}(X, \\mathcal{F})$. In other words, the $k$-linear map", "$\\Ext^i_X(\\mathcal{G}, \\omega_X^\\bullet) \\to", "\\Ext^i_X(\\mathcal{F}, \\omega_X^\\bullet)$ induced by $\\varphi$", "is, via the pairings, the $k$-linear dual of the $k$-linear map", "$H^{-i}(X, \\mathcal{F}) \\to H^{-i}(X, \\mathcal{G})$ induced", "by $\\varphi$. Formulated in this manner, this still works if", "$\\varphi$ is a homomorphism of quasi-coherent $\\mathcal{O}_X$-modules." ], "refs": [ "duality-remark-duality-proper-over-field" ], "ref_ids": [ 13658 ] }, { "id": 13660, "type": "other", "label": "duality-remark-rework-duality-locally-free-CM", "categories": [ "duality" ], "title": "duality-remark-rework-duality-locally-free-CM", "contents": [ "Let $X$ be a proper Cohen-Macaulay scheme over a field $k$", "which is equidimensional of dimension $d$.", "Let $\\omega_X^\\bullet$ and $\\omega_X$ be as in", "Lemma \\ref{lemma-duality-proper-over-field}.", "By Lemma \\ref{lemma-duality-proper-over-field-CM}", "we have $\\omega_X^\\bullet = \\omega_X[d]$.", "Let $t : H^d(X, \\omega_X) \\to k$ be the map of", "Remark \\ref{remark-duality-proper-over-field}.", "Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module", "with dual $\\mathcal{E}^\\vee$. Then we have perfect pairings", "$$", "H^i(X, \\omega_X \\otimes_{\\mathcal{O}_X} \\mathcal{E}^\\vee)", "\\times", "H^{d - i}(X, \\mathcal{E})", "\\longrightarrow", "k,\\quad", "(\\xi, \\eta) \\longmapsto t(1 \\otimes \\epsilon)(\\xi \\cup \\eta))", "$$", "where $\\cup$ is the cup-product and", "$\\epsilon : \\mathcal{E}^\\vee \\otimes_{\\mathcal{O}_X} \\mathcal{E}", "\\to \\mathcal{O}_X$ is the evaluation map.", "This is a special case of Lemma \\ref{lemma-duality-proper-over-field-perfect}." ], "refs": [ "duality-lemma-duality-proper-over-field", "duality-lemma-duality-proper-over-field-CM", "duality-remark-duality-proper-over-field", "duality-lemma-duality-proper-over-field-perfect" ], "ref_ids": [ 13606, 13608, 13658, 13607 ] }, { "id": 13661, "type": "other", "label": "duality-remark-relative-dualizing-complex-bis", "categories": [ "duality" ], "title": "duality-remark-relative-dualizing-complex-bis", "contents": [ "Let $X \\to S$ be a morphism of schemes which is flat, proper, and", "of finite presentation. By Lemma \\ref{lemma-existence-relative-dualizing}", "there exists a relative dualizing complex $(\\omega_{X/S}^\\bullet, \\xi)$", "in the sense of Definition \\ref{definition-relative-dualizing-complex}.", "Consider any morphism $g : S' \\to S$ where $S'$ is quasi-compact and", "quasi-separated (for example an affine open of $S$).", "By Lemma \\ref{lemma-base-change-relative-dualizing}", "we see that $(L(g')^*\\omega_{X/S}^\\bullet, L(g')^*\\xi)$ is", "a relative dualizing complex for the base change $f' : X' \\to S'$", "in the sense of Definition \\ref{definition-relative-dualizing-complex}.", "Let $\\omega_{X'/S'}^\\bullet$ be the relative dualizing complex", "for $X' \\to S'$ in the sense of Remark \\ref{remark-relative-dualizing-complex}.", "Combining Lemmas \\ref{lemma-flat-proper-relative-dualizing} and", "\\ref{lemma-uniqueness-relative-dualizing}", "we see that there is a unique isomorphism", "$$", "\\omega_{X'/S'}^\\bullet \\longrightarrow L(g')^*\\omega_{X/S}^\\bullet", "$$", "compatible with (\\ref{equation-pre-rigid}) and $L(g')^*\\xi$.", "These isomorphisms are compatible with morphisms between", "quasi-compact and quasi-separated schemes over $S$", "and the base change isomorphisms of Lemma \\ref{lemma-proper-flat-base-change}", "(if we ever need this compatibility we will carefully state and prove", "it here)." ], "refs": [ "duality-lemma-existence-relative-dualizing", "duality-definition-relative-dualizing-complex", "duality-lemma-base-change-relative-dualizing", "duality-definition-relative-dualizing-complex", "duality-remark-relative-dualizing-complex", "duality-lemma-flat-proper-relative-dualizing", "duality-lemma-uniqueness-relative-dualizing", "duality-lemma-proper-flat-base-change" ], "ref_ids": [ 13613, 13644, 13614, 13644, 13649, 13615, 13612, 13536 ] }, { "id": 13662, "type": "other", "label": "duality-remark-extension-by-zero", "categories": [ "duality" ], "title": "duality-remark-extension-by-zero", "contents": [ "Let $j : U \\to X$ be an open immersion of Noetherian schemes.", "Sending $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_U)$ to a Deligne", "system whose restriction to $U$ is $K$ determines a functor", "$$", "Rj_! :", "D^b_{\\textit{Coh}}(\\mathcal{O}_U)", "\\longrightarrow", "\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_X)", "$$", "which is ``exact'' by Lemma \\ref{lemma-extension-by-zero-triangle} and", "which is", "``left adjoint'' to the functor", "$j^* : D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to", "D^b_{\\textit{Coh}}(\\mathcal{O}_U)$ by Lemma \\ref{lemma-lift-map}." ], "refs": [ "duality-lemma-extension-by-zero-triangle", "duality-lemma-lift-map" ], "ref_ids": [ 13624, 13621 ] }, { "id": 13663, "type": "other", "label": "duality-remark-extension-by-zero-linear-pro-system", "categories": [ "duality" ], "title": "duality-remark-extension-by-zero-linear-pro-system", "contents": [ "Let $(A_n)$ and $(B_n)$ be inverse systems of a category $\\mathcal{C}$.", "Let us say a linear-pro-morphism from $(A_n)$ to $(B_n)$ is given", "by a compatible family of morphisms $\\varphi_n : A_{cn + d} \\to B_n$", "for all $n \\geq 1$ for some fixed integers $c, d \\geq 1$.", "We'll say $(\\varphi_n : A_{cn + d} \\to B_n)$ and", "$(\\psi_n : A_{c'n + d'} \\to B_n)$ determine the same morphism", "if there exist $c'' \\geq \\max(c, c')$ and $d'' \\geq \\max(d, d')$", "such that the two induced morphisms $A_{c'' n + d''} \\to B_n$ are the same", "for all $n$. It seems likely that Deligne systems $(K_n)$ with given value on", "$U$ are well defined up to linear-pro-isomorphisms. If we ever need this", "we will carefully formulate and prove this here." ], "refs": [], "ref_ids": [] }, { "id": 13664, "type": "other", "label": "duality-remark-covariance-open-j-lower-shriek", "categories": [ "duality" ], "title": "duality-remark-covariance-open-j-lower-shriek", "contents": [ "Let $X \\supset U \\supset U'$ be open subschemes of a Noetherian scheme $X$.", "Denote $j : U \\to X$ and $j' : U' \\to X$ the inclusion morphisms.", "We claim there is a canonical map", "$$", "Rj'_!(K|_{U'}) \\longrightarrow Rj_!K", "$$", "functorial for $K$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_U)$. Namely, by", "Lemma \\ref{lemma-lift-map} we have for any $L$ in", "$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ the map", "\\begin{align*}", "\\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_X)}(Rj_!K, L)", "& =", "\\Hom_U(K, L|_U) \\\\", "& \\to", "\\Hom_{U'}(K|_{U'}, L|_{U'}) \\\\", "& =", "\\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_X)}(Rj'_!(K|_{U'}), L)", "\\end{align*}", "functorial in $L$ and $K'$. The functoriality in $L$ shows by", "Categories, Remark \\ref{categories-remark-pro-category-copresheaves}", "that we obtain a canonical map $Rj'_!(K|_{U'}) \\to Rj_!K$ which is", "functorial in $K$ by the functoriality of the arrow above in $K$.", "\\medskip\\noindent", "Here is an explicit construction of this arrow. Namely, suppose", "that $\\mathcal{F}^\\bullet$ is a bounded complex of coherent", "$\\mathcal{O}_X$-modules whose restriction to $U$ represents $K$", "in the derived category. We have seen in the proof of", "Lemma \\ref{lemma-extension-by-zero}", "that such a complex always exists. Let $\\mathcal{I}$, resp.\\ $\\mathcal{I}'$", "be a quasi-coherent sheaf of ideals on $X$ with", "$V(\\mathcal{I}) = X \\setminus U$, resp.\\ $V(\\mathcal{I}') = X \\setminus U'$.", "After replacing $\\mathcal{I}$ by $\\mathcal{I} + \\mathcal{I}'$", "we may assume $\\mathcal{I}' \\subset \\mathcal{I}$.", "By construction $Rj_!K$, resp.\\ $Rj'_!(K|_{U'})$ is represented by the", "inverse system $(K_n)$, resp.\\ $(K'_n)$ of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$", "with", "$$", "K_n = \\mathcal{I}^n\\mathcal{F}^\\bullet", "\\quad\\text{resp.}\\quad", "K'_n = (\\mathcal{I}')^n\\mathcal{F}^\\bullet", "$$", "Clearly the map constructed above is given by the maps", "$K'_n \\to K_n$ coming from the inclusions", "$(\\mathcal{I}')^n \\subset \\mathcal{I}^n$." ], "refs": [ "duality-lemma-lift-map", "categories-remark-pro-category-copresheaves", "duality-lemma-extension-by-zero" ], "ref_ids": [ 13621, 12421, 13623 ] }, { "id": 13665, "type": "other", "label": "duality-remark-compose-inverse-systems", "categories": [ "duality" ], "title": "duality-remark-compose-inverse-systems", "contents": [ "Let $\\mathcal{C}$ be a category. Suppose given an inverse system", "$$", "\\ldots \\xrightarrow{\\alpha_4} (M_{3, n}) \\xrightarrow{\\alpha_3} (M_{2, n})", "\\xrightarrow{\\alpha_2} (M_{1, n})", "$$", "of inverse systems in the category of pro-objects of $\\mathcal{C}$.", "In other words, the arrows $\\alpha_i$ are morphisms of pro-objects. By", "Categories, Example \\ref{categories-example-pro-morphism-inverse-systems}", "we can represent each $\\alpha_i$ by a pair $(m_i, a_i)$ where", "$m_i : \\mathbf{N} \\to \\mathbf{N}$ is an increasing function and", "$a_{i, n} : M_{i, m_i(n)} \\to M_{i - 1, n}$ is a morphism of $\\mathcal{C}$", "making the diagrams", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "M_{i, m_i(3)} \\ar[d]^{a_{i, 3}} \\ar[r] &", "M_{i, m_i(2)} \\ar[d]^{a_{i, 2}} \\ar[r] &", "M_{i, m_i(1)} \\ar[d]^{a_{i, 1}} \\\\", "\\ldots \\ar[r] &", "M_{i - 1, 3} \\ar[r] &", "M_{i - 1, 2} \\ar[r] &", "M_{i - 1, 1}", "}", "$$", "commute. By replacing $m_i(n)$ by $\\max(n, m_i(n))$ and adjusting", "the morphisms $a_i(n)$ accordingly (as in the example referenced)", "we may assume that $m_i(n) \\geq n$. In this situation consider the", "inverse system", "$$", "\\ldots \\to", "M_{4, m_4(m_3(m_2(4)))} \\to", "M_{3, m_3(m_2(3))} \\to", "M_{2, m_2(2)} \\to", "M_{1, 1}", "$$", "with general term", "$$", "M_k = M_{k, m_k(m_{k - 1}(\\ldots (m_2(k))\\ldots))}", "$$", "For any object $N$ of $\\mathcal{C}$ we have", "$$", "\\colim_i \\colim_n \\Mor_\\mathcal{C}(M_{i, n}, N) =", "\\colim_k \\Mor_\\mathcal{C}(M_k, N) ", "$$", "We omit the details. In other words, we see that the inverse system $(M_k)$", "has the property", "$$", "\\colim_i \\Mor_{\\text{Pro-}\\mathcal{C}}((M_{i, n}), N) =", "\\Mor_{\\text{Pro-}\\mathcal{C}}((M_k), N)", "$$", "This property determines the inverse system $(M_k)$ up to pro-isomorphism", "by the discussion in", "Categories, Remark \\ref{categories-remark-pro-category-copresheaves}.", "In this way we can turn certain inverse systems in $\\text{Pro-}\\mathcal{C}$", "into pro-objects with countable index categories." ], "refs": [ "categories-remark-pro-category-copresheaves" ], "ref_ids": [ 12421 ] }, { "id": 13666, "type": "other", "label": "duality-remark-composition-lower-shriek", "categories": [ "duality" ], "title": "duality-remark-composition-lower-shriek", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ and $g : Y \\to Z$", "be composable morphisms of $\\textit{FTS}_S$. Let us define the composition", "$$", "Rg_! \\circ Rf_! :", "D^b_{\\textit{Coh}}(\\mathcal{O}_X)", "\\longrightarrow", "\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Z)", "$$", "Namely, by the very construction of $Rf_!$", "for $K$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$", "the output $Rf_!K$ is the pro-isomorphism class", "of an inverse system $(M_n)$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$.", "Then, since $Rg_!$ is constructed similarly, we see that", "$$", "\\ldots \\to Rg_!M_3 \\to Rg_!M_2 \\to Rg_!M_1", "$$", "is an inverse system of $\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$.", "By the discussion in Remark \\ref{remark-compose-inverse-systems}", "there is a unique pro-isomorphism class, which we will denote", "$Rg_! Rf_! K$, of inverse systems in $D^b_{\\textit{Coh}}(\\mathcal{O}_Z)$", "such that", "$$", "\\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Z)}(Rg_!Rf_!K, L) =", "\\colim_n \\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Z)}(Rg_!M_n, L)", "$$", "We omit the discussion necessary to see that this construction", "is functorial in $K$ as it will immediately follow from the next lemma." ], "refs": [ "duality-remark-compose-inverse-systems" ], "ref_ids": [ 13665 ] }, { "id": 13667, "type": "other", "label": "duality-remark-covariance-open-lower-shriek", "categories": [ "duality" ], "title": "duality-remark-covariance-open-lower-shriek", "contents": [ "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of", "$\\textit{FTS}_S$ and let $U \\subset X$ be an open. Set", "$g = f|_U : U \\to Y$. Then there is a canonical morphism", "$$", "Rg_!(K|_U) \\longrightarrow Rf_!K", "$$", "functorial in $K$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$", "which can be defined in at least 3 ways.", "\\begin{enumerate}", "\\item Denote $i : U \\to X$ the inclusion morphism. We have", "$Rg_! = Rf_! \\circ Ri_!$ by", "Lemma \\ref{lemma-composition-lower-shriek}", "and we can use $Rf_!$ applied to the map $Ri_!(K|_U) \\to K$", "which is a special case of", "Remark \\ref{remark-covariance-open-j-lower-shriek}.", "\\item Choose a compactification $j : X \\to \\overline{X}$", "of $X$ over $Y$ with structure morphism $\\overline{f} : \\overline{X} \\to Y$.", "Set $j' = j \\circ i : U \\to \\overline{X}$. We can", "use that $Rf_! = R\\overline{f}_* \\circ Rj_!$ and", "$Rg_! = R\\overline{f}_* \\circ Rj'_!$", "and we can use $R\\overline{f}_*$ applied to the map", "$Rj'_!(K|_U) \\to Rj_!K$ of", "Remark \\ref{remark-covariance-open-j-lower-shriek}.", "\\item We can use", "\\begin{align*}", "\\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)}(Rf_!K, L)", "& =", "\\Hom_X(K, f^!L) \\\\", "& \\to", "\\Hom_U(K|_U, f^!L|_U) \\\\", "& =", "\\Hom_U(K|_U, g^!L) \\\\", "& =", "\\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)}(Rg_!(K|_U), L)", "\\end{align*}", "functorial in $L$ and $K$. Here we have used", "Proposition \\ref{proposition-duality-compactly-supported}", "twice and the construction of upper shriek functors which", "shows that $g^! = i^* \\circ f^!$. The functoriality in $L$ shows by", "Categories, Remark \\ref{categories-remark-pro-category-copresheaves}", "that we obtain a canonical map $Rg_!(K|_U) \\to Rf_!K$ in", "$\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ which is", "functorial in $K$ by the functoriality of the arrow above in $K$.", "\\end{enumerate}", "Each of these three constructions gives the same arrow; we omit the", "details." ], "refs": [ "duality-lemma-composition-lower-shriek", "duality-remark-covariance-open-j-lower-shriek", "duality-remark-covariance-open-j-lower-shriek", "duality-proposition-duality-compactly-supported", "categories-remark-pro-category-copresheaves" ], "ref_ids": [ 13634, 13664, 13664, 13639, 12421 ] }, { "id": 13668, "type": "other", "label": "duality-remark-covariance-etale-lower-shriek", "categories": [ "duality" ], "title": "duality-remark-covariance-etale-lower-shriek", "contents": [ "Let us generalize the covariance of compactly supported cohomology", "given in Remark \\ref{remark-covariance-open-lower-shriek}", "to \\'etale morphisms. Namely, in Situation \\ref{situation-shriek}", "suppose given a commutative diagram", "$$", "\\xymatrix{", "U \\ar[rr]_h \\ar[rd]_g & & X \\ar[ld]^f \\\\", "& Y", "}", "$$", "of $\\textit{FTS}_S$ with $h$ \\'etale. Then there is a canonical morphism", "$$", "Rg_!(h^*K) \\longrightarrow Rf_!K", "$$", "functorial in $K$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$. We define", "this transformation using the sequence of maps", "\\begin{align*}", "\\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)}(Rf_!K, L)", "& =", "\\Hom_X(K, f^!L) \\\\", "& \\to", "\\Hom_U(h^*K, h^*(f^!L)) \\\\", "& =", "\\Hom_U(h^*K, h^!f^!L) \\\\", "& =", "\\Hom_U(h^*K, g^!L) \\\\", "& =", "\\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)}(Rg_!(h^*K), L)", "\\end{align*}", "functorial in $L$ and $K$. Here we have used", "Proposition \\ref{proposition-duality-compactly-supported}", "twice, we have used the equality $h^* = h^!$ of", "Lemma \\ref{lemma-shriek-etale}, and we have used the equality", "$h^! \\circ f^! = g^!$ of Lemma \\ref{lemma-upper-shriek-composition}.", "The functoriality in $L$ shows by", "Categories, Remark \\ref{categories-remark-pro-category-copresheaves}", "that we obtain a canonical map $Rg_!(h^*K) \\to Rf_!K$ in", "$\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ which is", "functorial in $K$ by the functoriality of the arrow above in $K$." ], "refs": [ "duality-remark-covariance-open-lower-shriek", "duality-proposition-duality-compactly-supported", "duality-lemma-shriek-etale", "duality-lemma-upper-shriek-composition", "categories-remark-pro-category-copresheaves" ], "ref_ids": [ 13667, 13639, 13566, 13552, 12421 ] }, { "id": 13669, "type": "other", "label": "duality-remark-covariance-lower-shriek", "categories": [ "duality" ], "title": "duality-remark-covariance-lower-shriek", "contents": [ "In Remarks \\ref{remark-covariance-open-lower-shriek} and", "\\ref{remark-covariance-etale-lower-shriek} we have seen", "that the construction of compactly supported cohomology is", "covariant with respect to open immersions and \\'etale morphisms.", "In fact, the correct generality is that given a commutative diagram", "$$", "\\xymatrix{", "U \\ar[rr]_h \\ar[rd]_g & & X \\ar[ld]^f \\\\", "& Y", "}", "$$", "of $\\textit{FTS}_S$ with $h$ flat and quasi-finite there exists a", "canonical transformation", "$$", "Rg_! \\circ h^* \\longrightarrow Rf_!", "$$", "As in Remark \\ref{remark-covariance-etale-lower-shriek}", "this map can be constructed using a transformation of functors", "$h^* \\to h^!$ on $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$. Recall that", "$h^!K = h^*K \\otimes \\omega_{U/X}$ where $\\omega_{U/X} = h^!\\mathcal{O}_X$", "is the relative dualizing sheaf of the flat quasi-finite morphism $h$ (see", "Lemmas \\ref{lemma-perfect-comparison-shriek} and", "\\ref{lemma-flat-quasi-finite-shriek}).", "Recall that $\\omega_{U/X}$ is the same as the relative dualizing", "module which will be constructed in Discriminants, Remark", "\\ref{discriminant-remark-relative-dualizing-for-quasi-finite}", "by", "Discriminants, Lemma \\ref{discriminant-lemma-compare-dualizing}.", "Thus we can use the trace element", "$\\tau_{U/X} : \\mathcal{O}_U \\to \\omega_{U/X}$", "which will be constructed in Discriminants, Remark", "\\ref{discriminant-remark-relative-dualizing-for-flat-quasi-finite}", "to define our transformation.", "If we ever need this, we will precisely formulate", "and prove the result here." ], "refs": [ "duality-remark-covariance-open-lower-shriek", "duality-remark-covariance-etale-lower-shriek", "duality-remark-covariance-etale-lower-shriek", "duality-lemma-perfect-comparison-shriek", "duality-lemma-flat-quasi-finite-shriek", "discriminant-remark-relative-dualizing-for-quasi-finite", "discriminant-lemma-compare-dualizing", "discriminant-remark-relative-dualizing-for-flat-quasi-finite" ], "ref_ids": [ 13667, 13668, 13668, 13562, 13581, 15006, 14999, 15007 ] }, { "id": 14127, "type": "other", "label": "more-morphisms-remark-action-by-derivations", "categories": [ "more-morphisms" ], "title": "more-morphisms-remark-action-by-derivations", "contents": [ "Assumptions and notation as in Lemma \\ref{lemma-action-by-derivations}.", "The action of a local section $\\theta$ on $a'$ is sometimes indicated by", "$\\theta \\cdot a'$. Note that this means nothing else than the fact", "that $(a')^\\sharp$ and $(\\theta \\cdot a')^\\sharp$ differ by a derivation", "$D$ which is related to $\\theta$ by Equation (\\ref{equation-D})." ], "refs": [ "more-morphisms-lemma-action-by-derivations" ], "ref_ids": [ 13717 ] }, { "id": 14128, "type": "other", "label": "more-morphisms-remark-special-case", "categories": [ "more-morphisms" ], "title": "more-morphisms-remark-special-case", "contents": [ "A special case of", "Lemmas \\ref{lemma-difference-derivation},", "\\ref{lemma-action-by-derivations},", "\\ref{lemma-sheaf}, and", "\\ref{lemma-action-sheaf}", "is where $Y = Y'$. In this case the map $A$ is always zero.", "The sheaf of", "Lemma \\ref{lemma-sheaf}", "is just given by the rule", "$$", "U' \\mapsto", "\\{a' : U' \\to Y\\text{ over }S\\text{ with } a'|_U = a|_U\\}", "$$", "and we act on this by the sheaf", "$\\SheafHom_{\\mathcal{O}_X}(a^*\\Omega_{Y/S}, \\mathcal{C}_{X/X'})$." ], "refs": [ "more-morphisms-lemma-difference-derivation", "more-morphisms-lemma-action-by-derivations", "more-morphisms-lemma-sheaf", "more-morphisms-lemma-action-sheaf", "more-morphisms-lemma-sheaf" ], "ref_ids": [ 13716, 13717, 13718, 13719, 13718 ] }, { "id": 14129, "type": "other", "label": "more-morphisms-remark-another-special-case", "categories": [ "more-morphisms" ], "title": "more-morphisms-remark-another-special-case", "contents": [ "Another special case of", "Lemmas \\ref{lemma-difference-derivation},", "\\ref{lemma-action-by-derivations},", "\\ref{lemma-sheaf}, and", "\\ref{lemma-action-sheaf}", "is where $S$ itself is a thickening $Z \\subset Z' = S$", "and $Y = Z \\times_{Z'} Y'$. Picture", "$$", "\\xymatrix{", "(X \\subset X') \\ar@{..>}[rr]_{(a, ?)} \\ar[rd]_{(g, g')} & &", "(Y \\subset Y') \\ar[ld]^{(h, h')} \\\\", "& (Z \\subset Z')", "}", "$$", "In this case the map $A : a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$", "is determined by $a$: the map", "$h^*\\mathcal{C}_{Z/Z'} \\to \\mathcal{C}_{Y/Y'}$ is surjective (because", "we assumed $Y = Z \\times_{Z'} Y'$),", "hence the pullback $g^*\\mathcal{C}_{Z/Z'} = a^*h^*\\mathcal{C}_{Z/Z'} \\to", "a^*\\mathcal{C}_{Y/Y'}$ is surjective, and the composition", "$g^*\\mathcal{C}_{Z/Z'} \\to a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$", "has to be the canonical map induced by $g'$. Thus the sheaf of", "Lemma \\ref{lemma-sheaf}", "is just given by the rule", "$$", "U' \\mapsto", "\\{a' : U' \\to Y'\\text{ over }Z'\\text{ with } a'|_U = a|_U\\}", "$$", "and we act on this by the sheaf", "$\\SheafHom_{\\mathcal{O}_X}(a^*\\Omega_{Y/Z}, \\mathcal{C}_{X/X'})$." ], "refs": [ "more-morphisms-lemma-difference-derivation", "more-morphisms-lemma-action-by-derivations", "more-morphisms-lemma-sheaf", "more-morphisms-lemma-action-sheaf", "more-morphisms-lemma-sheaf" ], "ref_ids": [ 13716, 13717, 13718, 13719, 13718 ] }, { "id": 14130, "type": "other", "label": "more-morphisms-remark-tiny-improvement", "categories": [ "more-morphisms" ], "title": "more-morphisms-remark-tiny-improvement", "contents": [ "Lemma \\ref{lemma-action-by-derivations-etale-localization}", "can be improved in the following way.", "Suppose that we have commutative diagrams as in", "Lemma \\ref{lemma-action-by-derivations-etale-localization}", "but we do not assume that $X_2 \\to X_1$", "and $S_2 \\to S_1$ are \\'etale. Next, suppose we have", "$\\theta_1 : a_1^*\\Omega_{X_1/S_1} \\to \\mathcal{C}_{T_1/T'_1}$", "and", "$\\theta_2 : a_2^*\\Omega_{X_2/S_2} \\to \\mathcal{C}_{T_2/T'_2}$", "such that", "$$", "\\xymatrix{", "f_*\\mathcal{O}_{X_2} \\ar[rr]_{f_*D_2} & &", "f_*a_{2, *}\\mathcal{C}_{T_2/T_2'} \\\\", "\\mathcal{O}_{X_1} \\ar[rr]^{D_1} \\ar[u]^{f^\\sharp} & &", "a_{1, *}\\mathcal{C}_{T_1/T_1'} \\ar[u]_{\\text{induced by }(h')^\\sharp}", "}", "$$", "is commutative where $D_i$ corresponds to $\\theta_i$ as in", "Equation (\\ref{equation-D}). Then we have the conclusion of", "Lemma \\ref{lemma-action-by-derivations-etale-localization}.", "The importance of the condition that both $X_2 \\to X_1$ and", "$S_2 \\to S_1$ are \\'etale is that it allows us to construct a $\\theta_2$", "from $\\theta_1$." ], "refs": [ "more-morphisms-lemma-action-by-derivations-etale-localization", "more-morphisms-lemma-action-by-derivations-etale-localization", "more-morphisms-lemma-action-by-derivations-etale-localization" ], "ref_ids": [ 13722, 13722, 13722 ] }, { "id": 14131, "type": "other", "label": "more-morphisms-remark-geometric-proof-bertini-irreducible", "categories": [ "more-morphisms" ], "title": "more-morphisms-remark-geometric-proof-bertini-irreducible", "contents": [ "Let us sketch a ``geometric'' proof of a special case of", "Lemma \\ref{lemma-bertini-irreducible}. Namely, say $k$", "is an algebraically closed field and $X \\subset \\mathbf{P}^n_k$ is smooth and", "irreducible of dimension $\\geq 2$. Then we claim there is a", "hyperplane $H \\subset \\mathbf{P}^n_k$ such that $X \\cap H$ is", "smooth and irreducible. Namely, by", "Varieties, Lemma \\ref{varieties-lemma-bertini}", "for a general $v \\in V = kT_0 \\oplus \\ldots \\oplus kT_n$", "the corresponding hyperplane section $X \\cap H_v$ is smooth.", "On the other hand, by Enriques-Severi-Zariski the scheme", "$X \\cap H_v$ is connected, see", "Varieties, Lemma \\ref{varieties-lemma-connectedness-ample-divisor}.", "Hence $X \\cap H_v$ is smooth and irreducible." ], "refs": [ "more-morphisms-lemma-bertini-irreducible", "varieties-lemma-bertini", "varieties-lemma-connectedness-ample-divisor" ], "ref_ids": [ 13847, 11132, 11135 ] }, { "id": 14132, "type": "other", "label": "more-morphisms-remark-necessary-condition-slice-smooth", "categories": [ "more-morphisms" ], "title": "more-morphisms-remark-necessary-condition-slice-smooth", "contents": [ "The second condition in", "Lemma \\ref{lemma-slice-smooth-once}", "is necessary even if $x$ is a closed point of a positive", "dimensional fibre. An example is the following: Let $k$ be a field", "of characteristic $p > 0$ which is imperfect. Let $a \\in k$ be an", "element which is not a $p$th power. Let", "$\\mathfrak m = (x, y^p - a) \\subset k[x, y]$. This corresponds to a closed", "point $w$ of $X = \\mathbf{A}^2_k$. Set $S = \\mathbf{A}^1_k$ and", "let $f : X \\to S$ be the morphism corresponding to $k[x] \\to k[x, y]$.", "Then there does not exist any commutative diagram", "$$", "\\xymatrix{", "S' \\ar[rr]_h \\ar[rd]_g & & X \\ar[ld]^f \\\\", "& S", "}", "$$", "with $g$ \\'etale and $w$ in the image of $h$. This is clear as the residue", "field extension $\\kappa(f(w)) \\subset \\kappa(w)$ is purely inseparable,", "but for any $s' \\in S'$ with $g(s') = f(w)$ the extension", "$\\kappa(f(w)) \\subset \\kappa(s')$ would be separable." ], "refs": [ "more-morphisms-lemma-slice-smooth-once" ], "ref_ids": [ 13876 ] }, { "id": 14133, "type": "other", "label": "more-morphisms-remark-topologies", "categories": [ "more-morphisms" ], "title": "more-morphisms-remark-topologies", "contents": [ "In terms of topologies", "Lemmas \\ref{lemma-dominate-etale-neighbourhood-finite-flat} and", "\\ref{lemma-dominate-etale-affine-finite-flat} mean the following.", "Let $S$ be any scheme. Let $\\{f_i : U_i \\to S\\}$ be an \\'etale covering", "of $S$. There exists a Zariski open covering $S = \\bigcup V_j$,", "for each $j$ a finite locally free, surjective morphism", "$W_j \\to V_j$, and for each $j$ a Zariski open covering", "$\\{W_{j, k} \\to W_j\\}$ such that the family", "$\\{W_{j, k} \\to S\\}$ refines the given \\'etale covering", "$\\{f_i : U_i \\to S\\}$. What does this mean in practice?", "Well, for example, suppose we have a descent problem which we", "know how to solve for Zariski coverings and for fppf coverings", "of the form $\\{\\pi : T \\to S\\}$ with $\\pi$ finite locally free", "and surjective. Then this descent problem has an affirmative", "answer for \\'etale coverings as well. This trick was used by", "Gabber in his proof that $\\text{Br}(X) = \\text{Br}'(X)$", "for an affine scheme $X$, see \\cite{Hoobler}." ], "refs": [ "more-morphisms-lemma-dominate-etale-neighbourhood-finite-flat", "more-morphisms-lemma-dominate-etale-affine-finite-flat" ], "ref_ids": [ 13890, 13891 ] }, { "id": 14134, "type": "other", "label": "more-morphisms-remark-change-topologies", "categories": [ "more-morphisms" ], "title": "more-morphisms-remark-change-topologies", "contents": [ "As a consequence of Lemma \\ref{lemma-fppf-ph} we obtain a comparison morphism", "$$", "\\epsilon : (\\Sch/S)_{ph} \\longrightarrow (\\Sch/S)_{fppf}", "$$", "This is the morphism of sites given by the identity functor", "on underlying categories (with suitable choices of sites", "as in Topologies, Remark \\ref{topologies-remark-choice-sites}).", "The functor $\\epsilon_*$ is the identity on underlying presheaves", "and the functor $\\epsilon^{-1}$ associated to an fppf sheaf", "its ph sheafification.", "By composition we can in addition compare the ph topology", "with the syntomic, smooth, \\'etale, and Zariski topologies." ], "refs": [ "more-morphisms-lemma-fppf-ph", "topologies-remark-choice-sites" ], "ref_ids": [ 13927, 12553 ] }, { "id": 14135, "type": "other", "label": "more-morphisms-remark-full-specialization-sequence", "categories": [ "more-morphisms" ], "title": "more-morphisms-remark-full-specialization-sequence", "contents": [ "The proof of", "Lemma \\ref{lemma-closed-point-nearby-fibre}", "actually shows that there exists a sequence of specializations", "$$", "x \\leadsto x_1 \\leadsto x_2 \\leadsto \\ldots \\leadsto x_d \\leadsto x'", "$$", "where all $x_i$ are in the fibre $X_s$, each specialization is", "immediate, and $x_d$ is a closed point of $X_s$. The integer", "$d = \\text{trdeg}_{\\kappa(s)}(\\kappa(x)) = \\dim(\\overline{\\{x\\}})$", "where the closure is taken in $X_s$. Moreover, the points", "$x_i$ can be chosen to avoid any closed subset of $X_s$ which", "does not contain the point $x$." ], "refs": [ "more-morphisms-lemma-closed-point-nearby-fibre" ], "ref_ids": [ 13938 ] }, { "id": 14136, "type": "other", "label": "more-morphisms-remark-alternative-closed-point-nearby-fibre", "categories": [ "more-morphisms" ], "title": "more-morphisms-remark-alternative-closed-point-nearby-fibre", "contents": [ "We can use", "Lemma \\ref{lemma-quasi-finite-quasi-section-meeting-nearby-open}", "or its variant", "Lemma \\ref{lemma-quasi-finite-quasi-section-meeting-nearby-open-X}", "to give an alternative proof of", "Lemma \\ref{lemma-closed-point-nearby-fibre}", "in case $S$ is locally Noetherian.", "Here is a rough sketch.", "Namely, first replace $S$ by", "the spectrum of the local ring at $s'$. Then we may use induction", "on $\\dim(S)$. The case $\\dim(S) = 0$ is trivial because then $s' = s$.", "Replace $X$ by the reduced induced scheme structure on $\\overline{\\{x\\}}$.", "Apply", "Lemma \\ref{lemma-quasi-finite-quasi-section-meeting-nearby-open-X}", "to $X \\to S$ and $x' \\mapsto s'$ and any nonempty", "open $U \\subset X$ containing $x$. This gives us a closed subscheme", "$x' \\in Z \\subset X$ a point $z \\in Z$", "such that $Z \\to S$ is quasi-finite at $x'$ and such that $f(z) \\not = s'$.", "Then $z$ is a closed point of $X_{f(z)}$, and $z \\leadsto x'$.", "As $f(z) \\not = s'$ we see $\\dim(\\mathcal{O}_{S, f(z)}) < \\dim(S)$.", "Since $x$ is the generic point of $X$ we see $x \\leadsto z$, hence", "$s = f(x) \\leadsto f(z)$.", "Apply the induction hypothesis to $s \\leadsto f(z)$ and $z \\mapsto f(z)$", "to win." ], "refs": [ "more-morphisms-lemma-closed-point-nearby-fibre" ], "ref_ids": [ 13938 ] }, { "id": 14137, "type": "other", "label": "more-morphisms-remark-perfect-permanence", "categories": [ "more-morphisms" ], "title": "more-morphisms-remark-perfect-permanence", "contents": [ "It is not true that a morphism between schemes $X, Y$ perfect over a base $S$", "is perfect. An example is $S = \\Spec(k)$, $X = \\Spec(k)$,", "$Y = \\Spec(k[x]/(x^2)$ and $X \\to Y$ the unique $S$-morphism." ], "refs": [], "ref_ids": [] }, { "id": 14306, "type": "other", "label": "sites-modules-remark-no-extension", "categories": [ "sites-modules" ], "title": "sites-modules-remark-no-extension", "contents": [ "In general the functor $g_!$ cannot be extended to categories of modules", "in case $g$ is (part of) a morphism of ringed topoi. Namely, given any", "ring map $A \\to B$ the functor $M \\mapsto B \\otimes_A M$ has a right adjoint", "(restriction) but not in general a left adjoint (because its existence", "would imply that $A \\to B$ is flat). We will see in", "Section \\ref{section-localize}", "below that it is possible to define $j_!$ on sheaves of modules", "in the case of a localization of sites.", "We will discuss this in greater generality in", "Section \\ref{section-lower-shriek-modules} below." ], "refs": [], "ref_ids": [] }, { "id": 14307, "type": "other", "label": "sites-modules-remark-localize-shriek-equal", "categories": [ "sites-modules" ], "title": "sites-modules-remark-localize-shriek-equal", "contents": [ "In the situation of Lemma \\ref{lemma-extension-by-zero}", "the diagram", "$$", "\\xymatrix{", "\\textit{Mod}(\\mathcal{O}_U) \\ar[r]_{j_{U!}} \\ar[d]_{forget} &", "\\textit{Mod}(\\mathcal{O}_\\mathcal{C}) \\ar[d]^{forget} \\\\", "\\textit{Ab}(\\mathcal{C}/U) \\ar[r]^{j^{Ab}_{U!}} &", "\\textit{Ab}(\\mathcal{C})", "}", "$$", "commutes. This is clear from the explicit description of the functor", "$j_{U!}$ in the lemma." ], "refs": [ "sites-modules-lemma-extension-by-zero" ], "ref_ids": [ 14169 ] }, { "id": 14308, "type": "other", "label": "sites-modules-remark-localize-presheaves", "categories": [ "sites-modules" ], "title": "sites-modules-remark-localize-presheaves", "contents": [ "Localization and presheaves of modules; see", "Sites, Remark \\ref{sites-remark-localize-presheaves}.", "Let $\\mathcal{C}$ be a category.", "Let $\\mathcal{O}$ be a presheaf of rings.", "Let $U$ be an object of $\\mathcal{C}$.", "Strictly speaking the functors $j_U^*$, $j_{U*}$ and $j_{U!}$", "have not been defined for presheaves of $\\mathcal{O}$-modules.", "But of course, we can think of a presheaf as a sheaf for the", "chaotic topology on $\\mathcal{C}$ (see", "Sites, Examples \\ref{sites-example-indiscrete}).", "Hence we also obtain a functor", "$$", "j_U^* :", "\\textit{PMod}(\\mathcal{O})", "\\longrightarrow", "\\textit{PMod}(\\mathcal{O}_U)", "$$", "and functors", "$$", "j_{U*}, j_{U!} :", "\\textit{PMod}(\\mathcal{O}_U)", "\\longrightarrow", "\\textit{PMod}(\\mathcal{O})", "$$", "which are right, left adjoint to $j_U^*$. Inspecting the proof of", "Lemma \\ref{lemma-extension-by-zero} we see that $j_{U!}\\mathcal{G}$", "is the presheaf", "$$", "V \\longmapsto", "\\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}", "\\mathcal{G}(V \\xrightarrow{\\varphi} U)", "$$", "In addition the functor $j_{U!}$ is exact (by", "Lemma \\ref{lemma-extension-by-zero-exact} in the", "case of the discrete topologies). Moreover, if $\\mathcal{C}$", "is actually a site, and $\\mathcal{O}$ is actually a sheaf of rings,", "then the diagram", "$$", "\\xymatrix{", "\\textit{Mod}(\\mathcal{O}_U) \\ar[r]_{j_{U!}} \\ar[d]_{forget} &", "\\textit{Mod}(\\mathcal{O}) \\\\", "\\textit{PMod}(\\mathcal{O}_U) \\ar[r]^{j_{U!}} &", "\\textit{PMod}(\\mathcal{O}) \\ar[u]_{(\\ )^\\#}", "}", "$$", "commutes." ], "refs": [ "sites-remark-localize-presheaves", "sites-modules-lemma-extension-by-zero", "sites-modules-lemma-extension-by-zero-exact" ], "ref_ids": [ 8713, 14169, 14170 ] }, { "id": 14309, "type": "other", "label": "sites-modules-remark-j-shriek-tensor", "categories": [ "sites-modules" ], "title": "sites-modules-remark-j-shriek-tensor", "contents": [ "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a sheaf of", "sets on $\\mathcal{C}$ and consider the localization morphism", "$j : \\Sh(\\mathcal{C})/\\mathcal{F} \\to \\Sh(\\mathcal{C})$.", "See Sites, Definition \\ref{sites-definition-localize-topos}.", "We claim that (a) $j_!\\mathbf{Z} = \\mathbf{Z}_\\mathcal{F}^\\#$", "and (b) $j_!(j^{-1}\\mathcal{H}) = j_!\\mathbf{Z} \\otimes_\\mathbf{Z} \\mathcal{H}$", "for any abelian sheaf $\\mathcal{H}$ on $\\mathcal{C}$.", "Let $\\mathcal{G}$ be an abelian on $\\mathcal{C}$.", "Part (a) follows from the Yoneda lemma because", "$$", "\\Hom(j_!\\mathbf{Z}, \\mathcal{G}) =", "\\Hom(\\mathbf{Z}, j^{-1}\\mathcal{G}) =", "\\Hom(\\mathbf{Z}_\\mathcal{F}^\\#, \\mathcal{G})", "$$", "where the second equality holds because both sides of", "the equality evaluate to the set of maps from $\\mathcal{F} \\to \\mathcal{G}$", "viewed as an abelian group. For (b) we use the Yoneda lemma and", "\\begin{align*}", "\\Hom(j_!(j^{-1}\\mathcal{H}), \\mathcal{G})", "& =", "\\Hom(j^{-1}\\mathcal{H}, j^{-1}\\mathcal{G}) \\\\", "& =", "\\Hom(\\mathbf{Z}, \\SheafHom(j^{-1}\\mathcal{H}, j^{-1}\\mathcal{G})) \\\\", "& =", "\\Hom(\\mathbf{Z}, j^{-1}\\SheafHom(\\mathcal{H}, \\mathcal{G})) \\\\", "& =", "\\Hom(j_!\\mathbf{Z}, \\SheafHom(\\mathcal{H}, \\mathcal{G})) \\\\", "& =", "\\Hom(j_!\\mathbf{Z} \\otimes_\\mathbf{Z} \\mathcal{H}, \\mathcal{G})", "\\end{align*}", "Here we use adjunction, the fact that taking $\\SheafHom$ commutes", "with localization, and Lemma \\ref{lemma-internal-hom-adjoint-tensor}." ], "refs": [ "sites-definition-localize-topos", "sites-modules-lemma-internal-hom-adjoint-tensor" ], "ref_ids": [ 8673, 14194 ] }, { "id": 14310, "type": "other", "label": "sites-modules-remark-functoriality-principal-parts", "categories": [ "sites-modules" ], "title": "sites-modules-remark-functoriality-principal-parts", "contents": [ "Let $\\mathcal{C}$ be a site. Suppose given a commutative diagram of", "sheaves of rings", "$$", "\\xymatrix{", "\\mathcal{B} \\ar[r] & \\mathcal{B}' \\\\", "\\mathcal{A} \\ar[u] \\ar[r] & \\mathcal{A}' \\ar[u]", "}", "$$", "a $\\mathcal{B}$-module $\\mathcal{F}$, a $\\mathcal{B}'$-module $\\mathcal{F}'$,", "and a $\\mathcal{B}$-linear map $\\mathcal{F} \\to \\mathcal{F}'$.", "Then we get a compatible system of module maps", "$$", "\\xymatrix{", "\\ldots \\ar[r] &", "\\mathcal{P}^2_{\\mathcal{B}'/\\mathcal{A}'}(\\mathcal{F}') \\ar[r] &", "\\mathcal{P}^1_{\\mathcal{B}'/\\mathcal{A}'}(\\mathcal{F}') \\ar[r] &", "\\mathcal{P}^0_{\\mathcal{B}'/\\mathcal{A}'}(\\mathcal{F}') \\\\", "\\ldots \\ar[r] &", "\\mathcal{P}^2_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{F}) \\ar[r] \\ar[u] &", "\\mathcal{P}^1_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{F}) \\ar[r] \\ar[u] &", "\\mathcal{P}^0_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{F}) \\ar[u]", "}", "$$", "These maps are compatible with further composition of maps of this type.", "The easiest way to see this is to use the description of the modules", "$\\mathcal{P}^k_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{M})$ in terms of", "(local) generators and relations in the proof of", "Lemma \\ref{lemma-module-principal-parts} but it can also be seen", "directly from the universal", "property of these modules. Moreover, these maps are compatible with", "the short exact sequences of Lemma \\ref{lemma-sequence-of-principal-parts}." ], "refs": [ "sites-modules-lemma-module-principal-parts", "sites-modules-lemma-sequence-of-principal-parts" ], "ref_ids": [ 14237, 14239 ] }, { "id": 14311, "type": "other", "label": "sites-modules-remark-not-pushforward", "categories": [ "sites-modules" ], "title": "sites-modules-remark-not-pushforward", "contents": [ "Warning: The result of", "Lemma \\ref{lemma-stalk-j-shriek}", "has no analogue for $j_{U, *}$." ], "refs": [ "sites-modules-lemma-stalk-j-shriek" ], "ref_ids": [ 14248 ] }, { "id": 14312, "type": "other", "label": "sites-modules-remark-when-shriek-equal", "categories": [ "sites-modules" ], "title": "sites-modules-remark-when-shriek-equal", "contents": [ "Warning! Let $u : \\mathcal{C} \\to \\mathcal{D}$, $g$, $\\mathcal{O}_\\mathcal{D}$,", "and $\\mathcal{O}_\\mathcal{C}$ be as in Lemma \\ref{lemma-lower-shriek-modules}.", "In general it is {\\bf not} the case that the diagram", "$$", "\\xymatrix{", "\\textit{Mod}(\\mathcal{O}_\\mathcal{C}) \\ar[r]_{g_!} \\ar[d]_{forget} &", "\\textit{Mod}(\\mathcal{O}_\\mathcal{D}) \\ar[d]^{forget} \\\\", "\\textit{Ab}(\\mathcal{C}) \\ar[r]^{g^{Ab}_!} &", "\\textit{Ab}(\\mathcal{D})", "}", "$$", "commutes (here $g^{Ab}_!$ is the one from", "Lemma \\ref{lemma-g-shriek-adjoint}). There is a transformation", "of functors", "$$", "g_!^{Ab} \\circ forget \\longrightarrow forget \\circ g_!", "$$", "From the proof of Lemma \\ref{lemma-lower-shriek-modules}", "we see that this is an isomorphism if and only if", "$g^{Ab}_!j_{U!}\\mathcal{O}_U \\to g_!j_{U!}\\mathcal{O}_U$", "is an isomorphism for all objects $U$ of $\\mathcal{C}$.", "Since we have $g_!j_{U!}\\mathcal{O}_U = j_{u(U)!}\\mathcal{O}_{u(U)}$", "this holds if and only if", "$$", "g^{Ab}_!j_{U!}\\mathcal{O}_U \\longrightarrow j_{u(U)!}\\mathcal{O}_{u(U)}", "$$", "is an isomorphism for all objects $U$ of $\\mathcal{C}$. Note that for such a", "$U$ we obtain a commutative diagram", "$$", "\\xymatrix{", "\\mathcal{C}/U \\ar[r]_-{u'} \\ar[d]_{j_U} & \\mathcal{D}/u(U) \\ar[d]^{j_{u(U)}} \\\\", "\\mathcal{C} \\ar[r]^u & \\mathcal{D}", "}", "$$", "of cocontinuous functors of sites, see", "Sites, Lemma \\ref{sites-lemma-localize-cocontinuous}", "and therefore $g^{Ab}_!j_{U!} = j_{u(U)!}(g')^{Ab}_!$ where", "$g' : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{D}/u(U))$ is the morphism", "of topoi induced by the cocontinuous functor $u'$.", "Hence we see that $g_! = g^{Ab}_!$ if the canonical map", "\\begin{equation}", "\\label{equation-compare-on-localizations}", "(g')^{Ab}_!\\mathcal{O}_U \\longrightarrow \\mathcal{O}_{u(U)}", "\\end{equation}", "is an isomorphism for all objects $U$ of $\\mathcal{C}$." ], "refs": [ "sites-modules-lemma-lower-shriek-modules", "sites-modules-lemma-g-shriek-adjoint", "sites-modules-lemma-lower-shriek-modules", "sites-lemma-localize-cocontinuous" ], "ref_ids": [ 14262, 14164, 14262, 8574 ] }, { "id": 14381, "type": "other", "label": "derham-remark-relative-cup-product", "categories": [ "derham" ], "title": "derham-remark-relative-cup-product", "contents": [ "Let $p : X \\to S$ be a morphism of schemes. Then we can think of", "$\\Omega^\\bullet_{X/S}$ as a sheaf of differential graded", "$p^{-1}\\mathcal{O}_S$-algebras, see", "Differential Graded Sheaves, Definition \\ref{sdga-definition-dga}.", "In particular, the discussion in", "Differential Graded Sheaves, Section \\ref{sdga-section-misc}", "applies. For example, this means that for any commutative diagram", "$$", "\\xymatrix{", "X \\ar[d]_p \\ar[r]_f & Y \\ar[d]^q \\\\", "S \\ar[r]^h & T", "}", "$$", "of schemes there is a canonical relative cup product", "$$", "\\mu :", "Rf_*\\Omega^\\bullet_{X/S}", "\\otimes_{q^{-1}\\mathcal{O}_T}^\\mathbf{L}", "Rf_*\\Omega^\\bullet_{X/S}", "\\longrightarrow", "Rf_*\\Omega^\\bullet_{X/S}", "$$", "in $D(Y, q^{-1}\\mathcal{O}_T)$ which is associative and", "which on cohomology reproduces the cup product discussed above." ], "refs": [ "sdga-definition-dga" ], "ref_ids": [ 7372 ] }, { "id": 14382, "type": "other", "label": "derham-remark-cup-product-as-a-map", "categories": [ "derham" ], "title": "derham-remark-cup-product-as-a-map", "contents": [ "Let $f : X \\to S$ be a morphism of schemes. Let $\\xi \\in H_{dR}^n(X/S)$.", "According to the discussion", "Differential Graded Sheaves, Section \\ref{sdga-section-misc}", "there exists a canonical morphism", "$$", "\\xi' : \\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}[n]", "$$", "in $D(f^{-1}\\mathcal{O}_S)$ uniquely characterized by", "(1) and (2) of the following list of properties:", "\\begin{enumerate}", "\\item $\\xi'$ can be lifted to a map in the derived category of right", "differential graded $\\Omega^\\bullet_{X/S}$-modules, and", "\\item $\\xi'(1) = \\xi$ in", "$H^0(X, \\Omega^\\bullet_{X/S}[n]) = H^n_{dR}(X/S)$,", "\\item the map $\\xi'$ sends $\\eta \\in H^m_{dR}(X/S)$", "to $\\xi \\cup \\eta$ in $H^{n + m}_{dR}(X/S)$,", "\\item the construction of $\\xi'$ commutes with restrictions to", "opens: for $U \\subset X$ open the restriction $\\xi'|_U$ is", "the map corresponding to the image $\\xi|_U \\in H^n_{dR}(U/S)$,", "\\item for any diagram as in Remark \\ref{remark-relative-cup-product}", "we obtain a commutative diagram", "$$", "\\xymatrix{", "Rf_*\\Omega^\\bullet_{X/S}", "\\otimes_{q^{-1}\\mathcal{O}_T}^\\mathbf{L}", "Rf_*\\Omega^\\bullet_{X/S} \\ar[d]_{\\xi' \\otimes \\text{id}}", "\\ar[r]_-\\mu &", "Rf_*\\Omega^\\bullet_{X/S} \\ar[d]^{\\xi'} \\\\", "Rf_*\\Omega^\\bullet_{X/S}[n]", "\\otimes_{q^{-1}\\mathcal{O}_T}^\\mathbf{L}", "Rf_*\\Omega^\\bullet_{X/S}", "\\ar[r]^-\\mu &", "Rf_*\\Omega^\\bullet_{X/S}[n]", "}", "$$", "in $D(Y, q^{-1}\\mathcal{O}_T)$.", "\\end{enumerate}" ], "refs": [ "derham-remark-relative-cup-product" ], "ref_ids": [ 14381 ] }, { "id": 14383, "type": "other", "label": "derham-remark-truncations", "categories": [ "derham" ], "title": "derham-remark-truncations", "contents": [ "Here is a reformulation of the calculations above in more abstract terms.", "Let $p : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an", "invertible $\\mathcal{O}_X$-module. If we view $\\text{d}\\log$ as a map", "$$", "\\mathcal{O}_X^*[-1] \\to \\sigma_{\\geq 1}\\Omega^\\bullet_{X/S}", "$$", "then using $\\Pic(X) = H^1(X, \\mathcal{O}_X^*)$ as above we find a", "cohomology class", "$$", "\\gamma_1(\\mathcal{L}) \\in H^2(X, \\sigma_{\\geq 1}\\Omega^\\bullet_{X/S})", "$$", "The image of $\\gamma_1(\\mathcal{L})$ under the map", "$\\sigma_{\\geq 1}\\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}$", "recovers $c_1^{dR}(\\mathcal{L})$. In particular we see that", "$c_1^{dR}(\\mathcal{L}) \\in F^1H^2_{dR}(X/S)$, see", "Section \\ref{section-hodge-filtration}. The image of $\\gamma_1(\\mathcal{L})$", "under the map $\\sigma_{\\geq 1}\\Omega^\\bullet_{X/S} \\to \\Omega^1_{X/S}[-1]$", "recovers $c_1^{Hodge}(\\mathcal{L})$. Taking the cup product", "(see Section \\ref{section-hodge-filtration}) we obtain", "$$", "\\xi = \\gamma_1(\\mathcal{L}_1) \\cup \\ldots \\cup \\gamma_1(\\mathcal{L}_a) \\in", "H^{2a}(X, \\sigma_{\\geq a}\\Omega^\\bullet_{X/S})", "$$", "The commutative diagrams in Section \\ref{section-hodge-filtration}", "show that $\\xi$ is mapped to", "$c_1^{dR}(\\mathcal{L}_1) \\cup \\ldots \\cup c_1^{dR}(\\mathcal{L}_a)$", "in $H^{2a}_{dR}(X/S)$ by the map", "$\\sigma_{\\geq a}\\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}$.", "Also, it follows", "$c_1^{dR}(\\mathcal{L}_1) \\cup \\ldots \\cup c_1^{dR}(\\mathcal{L}_a)$", "is contained in $F^a H^{2a}_{dR}(X/S)$. Similarly, the map", "$\\sigma_{\\geq a}\\Omega^\\bullet_{X/S} \\to \\Omega^a_{X/S}[-a]$", "sends $\\xi$ to", "$c_1^{Hodge}(\\mathcal{L}_1) \\cup \\ldots \\cup c_1^{Hodge}(\\mathcal{L}_a)$", "in $H^a(X, \\Omega^a_{X/S})$." ], "refs": [], "ref_ids": [] }, { "id": 14384, "type": "other", "label": "derham-remark-log-forms", "categories": [ "derham" ], "title": "derham-remark-log-forms", "contents": [ "Let $p : X \\to S$ be a morphism of schemes. For $i > 0$", "denote $\\Omega^i_{X/S, log} \\subset \\Omega^i_{X/S}$ the abelian subsheaf", "generated by local sections of the form", "$$", "\\text{d}\\log(u_1) \\wedge \\ldots \\wedge \\text{d}\\log(u_i)", "$$", "where $u_1, \\ldots, u_n$ are invertible local sections of $\\mathcal{O}_X$.", "For $i = 0$ the subsheaf $\\Omega^0_{X/S, log} \\subset \\mathcal{O}_X$", "is the image of $\\mathbf{Z} \\to \\mathcal{O}_X$. For every $i \\geq 0$ we", "have a map of complexes", "$$", "\\Omega^i_{X/S, log}[-i] \\longrightarrow \\Omega^\\bullet_{X/S}", "$$", "because the derivative of a logarithmic form is zero. Moreover, wedging", "logarithmic forms gives another, hence we find bilinear maps", "$$", "\\wedge : \\Omega^i_{X/S, log} \\times", "\\Omega^j_{X/S, log} \\longrightarrow \\Omega^{i + j}_{X/S, log}", "$$", "compatible with (\\ref{equation-wedge}) and the maps above.", "Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.", "Using the map of abelian sheaves", "$\\text{d}\\log : \\mathcal{O}_X^* \\to \\Omega^1_{X/S, log}$", "and the identification $\\Pic(X) = H^1(X, \\mathcal{O}_X^*)$", "we find a canonical cohomology class", "$$", "\\tilde \\gamma_1(\\mathcal{L}) \\in H^1(X, \\Omega^1_{X/S, log})", "$$", "These classes have the following properties", "\\begin{enumerate}", "\\item the image of $\\tilde \\gamma_1(\\mathcal{L})$ under the canonical", "map $\\Omega^1_{X/S, log}[-1] \\to \\sigma_{\\geq 1}\\Omega^\\bullet_{X/S}$", "sends $\\tilde \\gamma_1(\\mathcal{L})$ to the class", "$\\gamma_1(\\mathcal{L}) \\in ", "H^2(X, \\sigma_{\\geq 1}\\Omega^\\bullet_{X/S})$", "of Remark \\ref{remark-truncations},", "\\item the image of $\\tilde \\gamma_1(\\mathcal{L})$ under the canonical", "map $\\Omega^1_{X/S, log}[-1] \\to \\Omega^\\bullet_{X/S}$", "sends $\\tilde \\gamma_1(\\mathcal{L})$ to $c_1^{dR}(\\mathcal{L})$ in", "$H^2_{dR}(X/S)$,", "\\item the image of $\\tilde \\gamma_1(\\mathcal{L})$ under the canonical", "map $\\Omega^1_{X/S, log} \\to \\Omega^1_{X/S}$", "sends $\\tilde \\gamma_1(\\mathcal{L})$ to $c_1^{Hodge}(\\mathcal{L})$ in", "$H^1(X, \\Omega^1_{X/S})$,", "\\item the construction of these classes is compatible with pullbacks,", "\\item add more here.", "\\end{enumerate}" ], "refs": [ "derham-remark-truncations" ], "ref_ids": [ 14383 ] }, { "id": 14385, "type": "other", "label": "derham-remark-de-rham-complex-graded", "categories": [ "derham" ], "title": "derham-remark-de-rham-complex-graded", "contents": [ "Let $G$ be an abelian monoid written additively with neutral element $0$.", "Let $R \\to A$ be a ring map and assume $A$ comes with a grading", "$A = \\bigoplus_{g \\in G} A_g$ by $R$-modules such that $R$ maps into $A_0$", "and $A_g \\cdot A_{g'} \\subset A_{g + g'}$. Then the module of differentials", "comes with a grading", "$$", "\\Omega_{A/R} = \\bigoplus\\nolimits_{g \\in G} \\Omega_{A/R, g}", "$$", "where $\\Omega_{A/R, g}$ is the $R$-submodule of $\\Omega_{A/R}$", "generated by $a_0 \\text{d}a_1$ with $a_i \\in A_{g_i}$ such that", "$g = g_0 + g_1$. Similarly, we obtain", "$$", "\\Omega^p_{A/R} = \\bigoplus\\nolimits_{g \\in G} \\Omega^p_{A/R, g}", "$$", "where $\\Omega^p_{A/R, g}$ is the $R$-submodule of $\\Omega^p_{A/R}$", "generated by $a_0 \\text{d}a_1 \\wedge \\ldots \\wedge \\text{d}a_p$", "with $a_i \\in A_{g_i}$ such that $g = g_0 + g_1 + \\ldots + g_p$.", "Of course the differentials preserve the grading and the wedge", "product is compatible with the gradings in the obvious manner." ], "refs": [], "ref_ids": [] }, { "id": 14386, "type": "other", "label": "derham-remark-gauss-manin", "categories": [ "derham" ], "title": "derham-remark-gauss-manin", "contents": [ "In Lemma \\ref{lemma-spectral-sequence-smooth} consider the cohomology sheaves", "$$", "\\mathcal{H}^q_{dR}(X/Y) = H^q(Rf_*\\Omega^\\bullet_{X/Y}))", "$$", "If $f$ is proper in addition to being smooth and $S$ is a scheme over", "$\\mathbf{Q}$ then $\\mathcal{H}^q_{dR}(X/Y)$ is finite locally free (insert", "future reference here). If we only assume $\\mathcal{H}^q_{dR}(X/Y)$", "are flat $\\mathcal{O}_Y$-modules, then we obtain (tiny argument omitted)", "$$", "E_1^{p, q} =", "\\Omega^p_{Y/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{H}^q_{dR}(X/Y)", "$$", "and the differentials in the spectral sequence are maps", "$$", "d_1^{p, q} :", "\\Omega^p_{Y/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{H}^q_{dR}(X/Y)", "\\longrightarrow", "\\Omega^{p + 1}_{Y/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{H}^q_{dR}(X/Y)", "$$", "In particular, for $p = 0$ we obtain a map", "$d_1^{0, q} : \\mathcal{H}^q_{dR}(X/Y) \\to", "\\Omega^1_{Y/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{H}^q_{dR}(X/Y)$", "which turns out to be an integrable connection", "$\\nabla$ (insert future reference here)", "and the complex", "$$", "\\mathcal{H}^q_{dR}(X/Y) \\to", "\\Omega^1_{Y/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{H}^q_{dR}(X/Y) \\to", "\\Omega^2_{Y/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{H}^q_{dR}(X/Y) \\to \\ldots", "$$", "with differentials given by $d_1^{\\bullet, q}$", "is the de Rham complex of $\\nabla$.", "The connection $\\nabla$ is known as the {\\it Gauss-Manin connection}." ], "refs": [ "derham-lemma-spectral-sequence-smooth" ], "ref_ids": [ 14334 ] }, { "id": 14387, "type": "other", "label": "derham-remark-projective-space-bundle-formula", "categories": [ "derham" ], "title": "derham-remark-projective-space-bundle-formula", "contents": [ "In the situation of", "Proposition \\ref{proposition-projective-space-bundle-formula}", "we get moreover that the map", "$$", "\\tilde \\xi :", "\\bigoplus\\nolimits_{t = 0, \\ldots, r - 1}", "\\Omega^\\bullet_{X/S}[-2t]", "\\longrightarrow", "Rp_*\\Omega^\\bullet_{P/S}", "$$", "is an isomorphism in $D(X, (X \\to S)^{-1}\\mathcal{O}_X)$ as follows", "immediately from the application of", "Proposition \\ref{proposition-global-generation-on-fibres}.", "Note that the arrow for $t = 0$ is simply the canonical map", "$c_{P/X} : \\Omega^\\bullet_{X/S} \\to Rp_*\\Omega^\\bullet_{P/S}$", "of Section \\ref{section-de-rham-complex}.", "In fact, we can pin down this map further in this particular case.", "Namely, consider the canonical map", "$$", "\\xi' : \\Omega^\\bullet_{P/S} \\to \\Omega^\\bullet_{P/S}[2]", "$$", "of Remark \\ref{remark-cup-product-as-a-map} corresponding to", "$c_1^{dR}(\\mathcal{O}_P(1))$. Then", "$$", "\\xi'[2(t - 1)] \\circ \\ldots \\circ \\xi'[2] \\circ \\xi' : ", "\\Omega^\\bullet_{P/S} \\to \\Omega^\\bullet_{P/S}[2t]", "$$", "is the map of Remark \\ref{remark-cup-product-as-a-map} corresponding to", "$c_1^{dR}(\\mathcal{O}_P(1))^t$. Tracing through the choices made in the", "proof of Proposition \\ref{proposition-global-generation-on-fibres}", "we find the value", "$$", "\\tilde \\xi|_{\\Omega^\\bullet_{X/S}[-2t]} =", "Rp_*\\xi'[-2] \\circ \\ldots \\circ Rp_*\\xi'[-2(t - 1)] \\circ", "Rp_*\\xi'[-2t] \\circ c_{P/X}[-2t]", "$$", "for the restriction of our isomorphism to the summand", "$\\Omega^\\bullet_{X/S}[-2t]$. This has the following simple", "consequence we will use below: let", "$$", "M = \\bigoplus\\nolimits_{t = 1, \\ldots, r - 1} \\Omega^\\bullet_{X/S}[-2t]", "\\quad\\text{and}\\quad", "K = \\bigoplus\\nolimits_{t = 0, \\ldots, r - 2} \\Omega^\\bullet_{X/S}[-2t]", "$$", "viewed as subcomplexes of the source of the arrow $\\tilde \\xi$.", "It follows formally from the discussion above that", "$$", "c_{P/X} \\oplus", "\\tilde \\xi|_M :", "\\Omega^\\bullet_{X/S} \\oplus M \\longrightarrow", "Rp_*\\Omega^\\bullet_{P/S}", "$$", "is an isomorphism and that the diagram", "$$", "\\xymatrix{", "K \\ar[d]_{\\tilde \\xi|_K} \\ar[r]_{\\text{id}} &", "M[2] \\ar[d]^{(\\tilde \\xi|_M)[2]} \\\\", "Rp_*\\Omega^\\bullet_{P/S} \\ar[r]^{Rp_*\\xi'} &", "Rp_*\\Omega^\\bullet_{P/S}[2]", "}", "$$", "commutes where $\\text{id} : K \\to M[2]$ identifies the summand", "corresponding to $t$ in the deomposition of $K$ to the summand", "corresponding to $t + 1$ in the decomposition of $M$." ], "refs": [ "derham-proposition-projective-space-bundle-formula", "derham-proposition-global-generation-on-fibres", "derham-remark-cup-product-as-a-map", "derham-remark-cup-product-as-a-map", "derham-proposition-global-generation-on-fibres" ], "ref_ids": [ 14372, 14371, 14382, 14382, 14371 ] }, { "id": 14388, "type": "other", "label": "derham-remark-check-log-completion-1", "categories": [ "derham" ], "title": "derham-remark-check-log-completion-1", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $X$ be locally of finite", "type over $S$. Let $Y \\subset X$ be an effective Cartier divisor.", "If the map", "$$", "\\mathcal{O}_{X, y}^\\wedge \\longrightarrow \\mathcal{O}_{Y, y}^\\wedge", "$$", "has a section for all $y \\in Y$, then", "the de Rham complex of log poles is defined for $Y \\subset X$ over $S$.", "If we ever need this result we will formulate a precise statement and", "add a proof here." ], "refs": [], "ref_ids": [] }, { "id": 14389, "type": "other", "label": "derham-remark-check-log-completion-2", "categories": [ "derham" ], "title": "derham-remark-check-log-completion-2", "contents": [ "Let $S$ be a locally Noetherian scheme. Let $X$ be locally of finite", "type over $S$. Let $Y \\subset X$ be an effective Cartier divisor.", "If for every $y \\in Y$ we can find a diagram of schemes over $S$", "$$", "X \\xleftarrow{\\varphi} U \\xrightarrow{\\psi} V", "$$", "with $\\varphi$ \\'etale and $\\psi|_{\\varphi^{-1}(Y)} : \\varphi^{-1}(Y) \\to V$", "\\'etale, then the de Rham complex of log poles is defined for", "$Y \\subset X$ over $S$. A special case is when the pair $(X, Y)$", "\\'etale locally looks like $(V \\times \\mathbf{A}^1, V \\times \\{0\\})$.", "If we ever need this result we will formulate", "a precise statement and add a proof here." ], "refs": [], "ref_ids": [] }, { "id": 14390, "type": "other", "label": "derham-remark-local-description", "categories": [ "derham" ], "title": "derham-remark-local-description", "contents": [ "Let $A$ be a ring. Let $P = A[x_1, \\ldots, x_n]$. Let", "$f_1, \\ldots, f_n \\in P$ and set $B = P/(f_1, \\ldots, f_n)$.", "Assume $A \\to B$ is quasi-finite. Then", "$B$ is a relative global complete intersection over $A$ (Algebra, Definition", "\\ref{algebra-definition-relative-global-complete-intersection}) and", "$(f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2$ is free with generators", "the classes $\\overline{f}_i$ by Algebra, Lemma", "\\ref{algebra-lemma-relative-global-complete-intersection-conormal}.", "Consider the following diagram", "$$", "\\xymatrix{", "\\Omega_{A/\\mathbf{Z}} \\otimes_A B \\ar[r] &", "\\Omega_{P/\\mathbf{Z}} \\otimes_P B \\ar[r] &", "\\Omega_{P/A} \\otimes_P B \\\\", "&", "(f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2 \\ar[u] \\ar@{=}[r] &", "(f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2 \\ar[u]", "}", "$$", "The right column represents $\\NL_{B/A}$ in $D(B)$ hence has cohomology", "$\\Omega_{B/A}$ in degree $0$. The top row is the split short exact sequence", "$0 \\to \\Omega_{A/\\mathbf{Z}} \\otimes_A B \\to", "\\Omega_{P/\\mathbf{Z}} \\otimes_P B \\to \\Omega_{P/A} \\otimes_P B \\to 0$.", "The middle column has cohomology $\\Omega_{B/\\mathbf{Z}}$ in degree $0$", "by Algebra, Lemma \\ref{algebra-lemma-differential-seq}.", "Thus by Lemma \\ref{lemma-funny-map} we obtain canonical $B$-module maps", "$$", "\\Omega^p_{B/\\mathbf{Z}} \\longrightarrow", "\\Omega^p_{A/\\mathbf{Z}} \\otimes_A \\det(\\NL_{B/A})", "$$", "whose composition with", "$\\Omega^p_{A/\\mathbf{Z}} \\to \\Omega^p_{B/\\mathbf{Z}}$", "is multiplication by $\\delta(\\NL_{B/A})$." ], "refs": [ "algebra-definition-relative-global-complete-intersection", "algebra-lemma-relative-global-complete-intersection-conormal", "algebra-lemma-differential-seq", "derham-lemma-funny-map" ], "ref_ids": [ 1533, 1183, 1135, 14350 ] }, { "id": 14391, "type": "other", "label": "derham-remark-splitting-principle", "categories": [ "derham" ], "title": "derham-remark-splitting-principle", "contents": [ "The analogues of Weil Cohomology Theories, Lemmas", "\\ref{weil-lemma-splitting-principle} (splitting principle) and", "\\ref{weil-lemma-chern-classes-E-tensor-L} (chern classes of tensor products)", "hold for de Rham Chern classes on quasi-compact and quasi-separated schemes.", "This is clear as we've shown in the proof of", "Lemma \\ref{lemma-chern-classes}", "that all the axioms of Weil Cohomology Theories, Section", "\\ref{weil-section-chern} are satisfied." ], "refs": [ "weil-lemma-splitting-principle", "weil-lemma-chern-classes-E-tensor-L", "derham-lemma-chern-classes" ], "ref_ids": [ 5089, 5090, 14357 ] }, { "id": 14392, "type": "other", "label": "derham-remark-hodge-cohomology-is-weil", "categories": [ "derham" ], "title": "derham-remark-hodge-cohomology-is-weil", "contents": [ "In exactly the same manner as above one can show that", "Hodge cohomology $X \\mapsto H_{Hodge}^*(X/k)$ equipped", "with $c_1^{Hodge}$ determines a Weil", "cohomology theory. If we ever need this, we will precisely", "formulate and prove this here. This leads to the following", "amusing consequence: If the betti numbers of a Weil cohomology", "theory are independent of the chosen Weil cohomology theory", "(over our field $k$ of characteristic $0$), then", "the Hodge-to-de Rham spectral sequence", "degenerates at $E_1$! Of course, the degeneration of", "the Hodge-to-de Rham spectral sequence is known", "(see for example \\cite{Deligne-Illusie} for a marvelous algebraic proof),", "but it is by no means an easy result! This suggests that proving", "the independence of betti numbers is a hard problem as well", "and as far as we know is still an open problem. See", "Weil Cohomology Theories, Remark", "\\ref{weil-remark-betti-numbers-in-some-sense} for a related question." ], "refs": [ "weil-remark-betti-numbers-in-some-sense" ], "ref_ids": [ 5121 ] }, { "id": 14393, "type": "other", "label": "derham-remark-gysin-equations", "categories": [ "derham" ], "title": "derham-remark-gysin-equations", "contents": [ "Let $X \\to S$ be a morphism of schemes. Let", "$f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$. Let $Z \\subset X$", "be the closed subscheme cut out by $f_1, \\ldots, f_c$. Below we will", "study the {\\it gysin map}", "\\begin{equation}", "\\label{equation-gysin}", "\\gamma^p_{f_1, \\ldots, f_c} :", "\\Omega^p_{Z/S}", "\\longrightarrow", "\\mathcal{H}_Z^c(\\Omega^{p + c}_{X/S})", "\\end{equation}", "defined as follows. Given a local section $\\omega$ of $\\Omega^p_{Z/S}$", "which is the restriction of a section $\\tilde \\omega$ of $\\Omega^p_{X/S}$", "we set", "$$", "\\gamma^p_{f_1, \\ldots, f_c}(\\omega) =", "c_{f_1, \\ldots, f_c}(\\tilde \\omega|_Z) \\wedge", "\\text{d}f_1 \\wedge \\ldots \\wedge \\text{d}f_c", "$$", "where $c_{f_1, \\ldots, f_c} : \\Omega^p_{X/S} \\otimes \\mathcal{O}_Z \\to", "\\mathcal{H}_Z^c(\\Omega^p_{X/S})$ is the map constructed in", "Derived Categories of Schemes, Remark", "\\ref{perfect-remark-supported-map-c-equations}.", "This is well defined: given $\\omega$ we can change our choice of", "$\\tilde \\omega$ by elements of the form", "$\\sum f_i \\omega'_i + \\sum \\text{d}(f_i) \\wedge \\omega''_i$", "which are mapped to zero by the construction." ], "refs": [ "perfect-remark-supported-map-c-equations" ], "ref_ids": [ 7124 ] }, { "id": 14394, "type": "other", "label": "derham-remark-how-to-use", "categories": [ "derham" ], "title": "derham-remark-how-to-use", "contents": [ "Let $X \\to S$, $i : Z \\to X$, and $c \\geq 0$ be as in", "Lemma \\ref{lemma-gysin-global}.", "Let $p \\geq 0$ and assume that $\\mathcal{H}^i_Z(\\Omega^{p + c}_{X/S}) = 0$", "for $i = 0, \\ldots, c - 1$. This vanishing holds if $X \\to S$ is smooth", "and $Z \\to X$ is a Koszul regular immersion, see", "Derived Categories of Schemes, Lemma \\ref{perfect-lemma-supported-vanishing}.", "Then we obtain a map", "$$", "\\gamma^{p, q} :", "H^q(Z, \\Omega^p_{Z/S})", "\\longrightarrow", "H^{q + c}(X, \\Omega^{p + c}_{X/S})", "$$", "by first using", "$\\gamma^p : \\Omega^p_{Z/S} \\to \\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S})$", "to map into", "$$", "H^q(Z, \\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S})) =", "H^q(Z, R\\mathcal{H}_Z(\\Omega^{p + c}_{X/S})[c]) =", "H^q(X, i_*R\\mathcal{H}_Z(\\Omega^{p + c}_{X/S})[c])", "$$", "and then using the adjunction map", "$i_*R\\mathcal{H}_Z(\\Omega^{p + c}_{X/S}) \\to \\Omega^{p + c}_{X/S}$", "to continue on to the desired Hodge cohomology module." ], "refs": [ "derham-lemma-gysin-global", "perfect-lemma-supported-vanishing" ], "ref_ids": [ 14360, 6960 ] }, { "id": 14462, "type": "other", "label": "trace-remark-may-be-confusing", "categories": [ "trace" ], "title": "trace-remark-may-be-confusing", "contents": [ "It may or may not be the case that $F^f_U$ equals $\\pi_U$." ], "refs": [], "ref_ids": [] }, { "id": 14463, "type": "other", "label": "trace-remark-compute-degree-lifting", "categories": [ "trace" ], "title": "trace-remark-compute-degree-lifting", "contents": [ "The computation of the degrees can be done by lifting (in some obvious sense)", "to characteristic 0 and considering the situation with complex coefficients.", "This method almost never works, since lifting is in general impossible for", "schemes which are not projective space." ], "refs": [], "ref_ids": [] }, { "id": 14464, "type": "other", "label": "trace-remarks-derived-categories", "categories": [ "trace" ], "title": "trace-remarks-derived-categories", "contents": [ "Notes on derived categories.", "\\begin{enumerate}", "\\item", "There are some set-theoretical problems when $\\mathcal{A}$ is somewhat", "arbitrary, which we will happily disregard.", "\\item", "The categories $K(A)$ and $D(A)$ are endowed with the structure of a", "triangulated category.", "\\item", "The categories $\\text{Comp}(\\mathcal{A})$ and $K(\\mathcal{A})$ can also be", "defined when $\\mathcal{A}$ is an additive category.", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14465, "type": "other", "label": "trace-remark-cohomology-of-derived-functor", "categories": [ "trace" ], "title": "trace-remark-cohomology-of-derived-functor", "contents": [ "In these cases, it is true that $R^iF(K^\\bullet) = H^i(RF(K^\\bullet))$, where", "the left hand side is defined to be $i$th homology of the complex", "$F(K^\\bullet)$." ], "refs": [], "ref_ids": [] }, { "id": 14466, "type": "other", "label": "trace-remark-variant", "categories": [ "trace" ], "title": "trace-remark-variant", "contents": [ "A variant of this lemma is to consider a Noetherian scheme $X$", "and the category $D_{perf}(\\mathcal{O}_X)$ of complexes which are locally", "quasi-isomorphic to a finite complex of finite locally free", "$\\mathcal{O}_X$-modules. Objects $K$ of $D_{perf}(\\mathcal{O}_X)$", "can be characterized by having coherent cohomology sheaves and", "bounded tor dimension." ], "refs": [], "ref_ids": [] }, { "id": 14467, "type": "other", "label": "trace-remark-content-trivial-trace", "categories": [ "trace" ], "title": "trace-remark-content-trivial-trace", "contents": [ "Let us try to illustrate the content of the formula of", "Lemma \\ref{lemma-weak-trace}.", "Suppose that $\\Lambda$, viewed as a trivial $\\Gamma$-module, admits a finite", "resolution", "$", "0\\to P_r\\to \\ldots \\to P_1 \\to P_0\\to \\Lambda\\to 0", "$", "by some $\\Lambda[\\Gamma]$-modules $P_i$ which are finite and projective as", "$\\Lambda[G]$-modules. In that case", "$$", "H_*\\left(\\left(P_\\bullet\\right)_G\\right) =", "\\text{Tor}_*^{\\Lambda[G]}\\left(\\Lambda, \\Lambda\\right) = H_*(G, \\Lambda)", "$$", "and", "$$", "\\text{Tr}_\\Lambda^{Z_\\gamma}\\left(\\gamma, P_\\bullet\\right) =\\frac{1}{\\#", "Z_\\gamma}\\text{Tr}_\\Lambda(\\gamma, P_\\bullet)=\\frac{1}{\\#", "Z_\\gamma}\\text{Tr}(\\gamma, \\Lambda) = \\frac{1}{\\# Z_\\gamma}.", "$$", "Therefore, Lemma \\ref{lemma-weak-trace} says", "$$", "\\text{Tr}_\\Lambda (1 , P_G)", "= \\text{Tr}\\left(1 |_{H_*(G, \\Lambda)}\\right)", "= {\\sum_{\\gamma\\mapsto 1}}'\\frac{1}{\\# Z_\\gamma}.", "$$", "This can be interpreted as a point count on the stack $BG$. If", "$\\Lambda = \\mathbf{F}_\\ell$ with $\\ell$ prime to $\\# G$, then", "$H_*(G, \\Lambda)$ is $\\mathbf{F}_\\ell$ in degree 0 (and $0$ in", "other degrees) and the formula reads", "$$", "1 =", "\\sum\\nolimits_{", "\\frac{\\sigma\\text{-conjugacy}}{\\text{classes}\\langle\\gamma\\rangle}", "}", "\\frac{1}{\\# Z_\\gamma} \\mod \\ell.", "$$", "This is in some sense a ``trivial'' trace formula for $G$.", "Later we will see that (\\ref{equation-trace-formula}) can in", "some cases be viewed as a highly nontrivial trace formula for a", "certain type of group, see", "Section \\ref{section-abstract-trace-formula}." ], "refs": [ "trace-lemma-weak-trace", "trace-lemma-weak-trace" ], "ref_ids": [ 14427, 14427 ] }, { "id": 14468, "type": "other", "label": "trace-remark-on-trace-formula-again", "categories": [ "trace" ], "title": "trace-remark-on-trace-formula-again", "contents": [ "Remarks on Theorem \\ref{theorem-trace-formula-again}.", "\\begin{enumerate}", "\\item", "This formula holds in any dimension. By a d\\'evissage lemma (which uses proper", "base change etc.) it reduces to the current statement -- in that generality.", "\\item", "The complex $R\\Gamma_c(X_{\\bar k}, K)$ is defined by choosing an open immersion", "$j : X \\hookrightarrow \\bar X$ with $\\bar X$ projective over $k$ of dimension at", "most 1 and setting", "$$", "R\\Gamma_c(X_{\\bar k}, K) := R\\Gamma(\\bar X_{\\bar k}, j_!K).", "$$", "This is independent of the choice of $\\bar X$ follows from", "(insert reference here). We define $H^i_c(X_{\\bar k}, K)$", "to be the $i$th cohomology group of $R\\Gamma_c(X_{\\bar k}, K)$.", "\\end{enumerate}" ], "refs": [ "trace-theorem-trace-formula-again" ], "ref_ids": [ 14399 ] }, { "id": 14469, "type": "other", "label": "trace-remark-stronger", "categories": [ "trace" ], "title": "trace-remark-stronger", "contents": [ "Even though all we did are reductions and mostly algebra, the trace formula", "Theorem \\ref{theorem-trace-formula-again} is much stronger than", "Weil's geometric trace formula (Theorem \\ref{theorem-weil-trace-formula})", "because it applies to coefficient", "systems (sheaves), not merely constant coefficients." ], "refs": [ "trace-theorem-trace-formula-again", "trace-theorem-weil-trace-formula" ], "ref_ids": [ 14399, 14398 ] }, { "id": 14470, "type": "other", "label": "trace-remark-stalk-l-adic-sheaf", "categories": [ "trace" ], "title": "trace-remark-stalk-l-adic-sheaf", "contents": [ "If $\\mathcal{F} = \\left\\{\\mathcal{F}_n\\right\\}_{n\\geq 1}$ is a", "$\\mathbf{Z}_\\ell$-sheaf on $X$ and $\\bar x$ is a geometric point then", "$M_n = \\left\\{\\mathcal{F}_{n, \\bar x}\\right\\}$ is an inverse system of finite", "$\\mathbf{Z}/\\ell^n\\mathbf{Z}$-modules such that $M_{n+1}\\to M_n$ is surjective", "and $M_n = M_{n+1}/\\ell^n M_{n+1}$. It follows that", "$$", "M = \\lim_n M_n = \\lim \\mathcal{F}_{n, \\bar x}", "$$", "is a finite $\\mathbf{Z}_\\ell$-module. Indeed, $M/\\ell M= M_1$ is finite over", "$\\mathbf{F}_\\ell$, so by Nakayama $M$ is finite over $\\mathbf{Z}_\\ell$.", "Therefore, $M\\cong \\mathbf{Z}_\\ell^{\\oplus r} \\oplus \\oplus_{i = 1}^t", "\\mathbf{Z}_\\ell/\\ell^{e_i}\\mathbf{Z}_\\ell$ for some $r, t\\geq 0$, $e_i\\geq 1$.", "The module $M = \\mathcal{F}_{\\bar x}$ is called the {\\it stalk} of", "$\\mathcal{F}$ at $\\bar x$." ], "refs": [], "ref_ids": [] }, { "id": 14471, "type": "other", "label": "trace-remark-torsion-stalks", "categories": [ "trace" ], "title": "trace-remark-torsion-stalks", "contents": [ "Since a $\\mathbf{Z}_\\ell$-sheaf is only defined on a Noetherian scheme, it is", "torsion if and only if its stalks are torsion." ], "refs": [], "ref_ids": [] }, { "id": 14472, "type": "other", "label": "trace-remark-T", "categories": [ "trace" ], "title": "trace-remark-T", "contents": [ "Intuitively, $T$ should be thought of as $T = t^f$ where $p^f = \\# k$. The", "definitions are then independent of the size of the ground field." ], "refs": [], "ref_ids": [] }, { "id": 14473, "type": "other", "label": "trace-remark-which-is-harder", "categories": [ "trace" ], "title": "trace-remark-which-is-harder", "contents": [ "Since we have only developed some theory of traces and not of determinants,", "Theorem \\ref{theorem-A}", "is harder to prove than", "Theorem \\ref{theorem-B}.", "We will only prove the latter, for the former see \\cite{SGA4.5}.", "Observe also that there is no version of this theorem more general for", "$\\mathbf{Z}_\\ell$ coefficients since there is no $\\ell$-torsion." ], "refs": [ "trace-theorem-A", "trace-theorem-B" ], "ref_ids": [ 14400, 14401 ] }, { "id": 14474, "type": "other", "label": "trace-remark-projective", "categories": [ "trace" ], "title": "trace-remark-projective", "contents": [ "Thus we conclude that if $X$ is also projective then", "we have functorially in the representation $\\rho$", "the identifications", "$$", "H^0(X_{\\overline{k}}, \\mathcal{F}_\\rho) =", "M^{\\pi_1(X_{\\overline{k}}, \\overline\\eta)}", "$$", "and", "$$", "H_c^2(X_{\\overline{k}}, \\mathcal{F}_\\rho) =", "M_{\\pi_1(X_{\\overline{k}}, \\overline \\eta)}(-1)", "$$", "Of course if $X$ is not projective, then", "$H^0_c(X_{\\overline{k}}, \\mathcal{F}_\\rho) = 0$." ], "refs": [], "ref_ids": [] }, { "id": 14475, "type": "other", "label": "trace-remark-poincare-groups", "categories": [ "trace" ], "title": "trace-remark-poincare-groups", "contents": [ "By the proposition and Trivial duality then you get", "$$", "H^{2-i}_c(X_{\\overline{k}}, \\mathcal{F}_\\rho)", "\\times", "H^i(X_{\\overline{k}}, \\mathcal{F}_\\rho^\\wedge(1))", "\\to", "\\mathbf{Q}/\\mathbf{Z}", "$$", "a perfect pairing. If $X$ is projective then this is Poincare duality." ], "refs": [], "ref_ids": [] }, { "id": 14476, "type": "other", "label": "trace-remark-abstract-trace-formula", "categories": [ "trace" ], "title": "trace-remark-abstract-trace-formula", "contents": [ "Here are some observations concerning this notion.", "\\begin{enumerate}", "\\item If modeling projective curves then we can use cohomology and we", "don't need factor $q^n$.", "\\item The only examples I know are $\\Gamma = \\pi_1(X, \\overline \\eta)$", "where $X$ is smooth, geometrically irreducible and $K(\\pi, 1)$ over finite", "field. In this case $q = (\\# k)^{\\dim X}$. Modulo the proposition, we proved", "this for curves in this course.", "\\item Given the integer $q$ then the sets $S_d$ are uniquely", "determined. (You can multiple $q$ by an integer $m$ and then replace $S_d$ by", "$m^d$ copies of $S_d$ without changing the formula.)", "\\end{enumerate}" ], "refs": [], "ref_ids": [] }, { "id": 14477, "type": "other", "label": "trace-remark-lafforgue", "categories": [ "trace" ], "title": "trace-remark-lafforgue", "contents": [ "We now have, thanks to Lafforgue and many other mathematicians,", "complete theorems like this two above for $\\text{GL}_n$", "and allowing ramification!", "In other words, the full global Langlands correspondence for $\\text{GL}_n$", "is known for function fields of curves over finite fields. At the same", "time this does not mean there aren't a lot of interesting questions left", "to answer about the fundamental groups of curves over finite fields, as", "we shall see below." ], "refs": [], "ref_ids": [] }, { "id": 14586, "type": "other", "label": "sheaves-remark-confusion", "categories": [ "sheaves" ], "title": "sheaves-remark-confusion", "contents": [ "There is always a bit of confusion as to whether it is", "necessary to say something about the set of sections of", "a sheaf over the empty set $\\emptyset \\subset X$.", "It is necessary, and we already did if you read the", "definition right. Namely, note that the empty set is", "covered by the empty open covering, and hence the ``collection", "of sections $s_i$'' from the definition above actually form", "an element of the empty product which is the final object", "of the category the sheaf has values in. In other words,", "if you read the definition right you automatically deduce", "that $\\mathcal{F}(\\emptyset) = \\textit{a final object}$,", "which in the case of a sheaf of sets is a singleton.", "If you do not like this argument, then you can just require", "that $\\mathcal{F}(\\emptyset) = \\{*\\}$.", "\\medskip\\noindent", "In particular, this condition will then ensure that if", "$U, V \\subset X$ are open and {\\it disjoint} then", "$$", "\\mathcal{F}(U \\cup V) = \\mathcal{F}(U) \\times \\mathcal{F}(V).", "$$", "(Because the fibre product over a final object is a product.)" ], "refs": [], "ref_ids": [] }, { "id": 14587, "type": "other", "label": "sheaves-remark-j-shriek-not-exact", "categories": [ "sheaves" ], "title": "sheaves-remark-j-shriek-not-exact", "contents": [ "Let $j : U \\to X$ be an open immersion of topological spaces as above.", "Let $x \\in X$, $x \\not \\in U$. Let $\\mathcal{F}$ be a sheaf of sets", "on $U$. Then $j_!\\mathcal{F}_x = \\emptyset$ by Lemma \\ref{lemma-j-shriek}.", "Hence $j_!$ does not transform a final object of $\\Sh(U)$", "into a final object of $\\Sh(X)$ unless $U = X$.", "According to our conventions in", "Categories, Section \\ref{categories-section-exact-functor}", "this means that the functor $j_!$ is not left exact", "as a functor between the categories of sheaves of sets.", "It will be shown later that $j_!$ on abelian sheaves is exact,", "see Modules, Lemma \\ref{modules-lemma-j-shriek-exact}." ], "refs": [ "sheaves-lemma-j-shriek", "modules-lemma-j-shriek-exact" ], "ref_ids": [ 14543, 13224 ] }, { "id": 14588, "type": "other", "label": "sheaves-remark-i-star-not-exact", "categories": [ "sheaves" ], "title": "sheaves-remark-i-star-not-exact", "contents": [ "Let $i : Z \\to X$ be a closed immersion of topological spaces as above.", "Let $x \\in X$, $x \\not \\in Z$. Let $\\mathcal{F}$ be a sheaf of sets", "on $Z$. Then $(i_*\\mathcal{F})_x = \\{ * \\}$", "by Lemma \\ref{lemma-stalks-closed-pushforward}.", "Hence if $\\mathcal{F} = * \\amalg *$, where", "$*$ is the singleton sheaf, then", "$i_*\\mathcal{F}_x = \\{*\\} \\not = i_*(*)_x \\amalg i_*(*)_x$", "because the latter is a two point set.", "According to our conventions in", "Categories, Section \\ref{categories-section-exact-functor}", "this means that the functor $i_*$ is not right exact", "as a functor between the categories of sheaves of sets.", "In particular, it cannot have a right adjoint, see", "Categories, Lemma \\ref{categories-lemma-exact-adjoint}.", "\\medskip\\noindent", "On the other hand, we will see later (see", "Modules, Lemma \\ref{modules-lemma-i-star-right-adjoint})", "that $i_*$ on abelian sheaves is exact, and does have a right", "adjoint, namely the functor that associates to an abelian sheaf on $X$", "the sheaf of sections supported in $Z$." ], "refs": [ "sheaves-lemma-stalks-closed-pushforward", "categories-lemma-exact-adjoint", "modules-lemma-i-star-right-adjoint" ], "ref_ids": [ 14551, 12250, 13233 ] }, { "id": 14589, "type": "other", "label": "sheaves-remark-closed-immersion-spaces", "categories": [ "sheaves" ], "title": "sheaves-remark-closed-immersion-spaces", "contents": [ "We have not discussed the relationship between closed immersions", "and ringed spaces. This is because the notion of a closed immersion", "of ringed spaces is best discussed in the setting of quasi-coherent", "sheaves, see Modules, Section \\ref{modules-section-closed-immersion}." ], "refs": [], "ref_ids": [] }, { "id": 14783, "type": "other", "label": "descent-remark-standard-covering", "categories": [ "descent" ], "title": "descent-remark-standard-covering", "contents": [ "Let $R$ be a ring. Let $f_1, \\ldots, f_n\\in R$ generate the", "unit ideal. The ring $A = \\prod_i R_{f_i}$ is a faithfully flat", "$R$-algebra. We remark that the cosimplicial ring $(A/R)_\\bullet$", "has the following ring in degree $n$:", "$$", "\\prod\\nolimits_{i_0, \\ldots, i_n} R_{f_{i_0}\\ldots f_{i_n}}", "$$", "Hence the results above recover", "Algebra, Lemmas \\ref{algebra-lemma-standard-covering},", "\\ref{algebra-lemma-cover-module} and \\ref{algebra-lemma-glue-modules}.", "But the results above actually say more because of exactness", "in higher degrees. Namely, it implies that {\\v C}ech cohomology of", "quasi-coherent sheaves on affines is trivial. Thus we get a second", "proof of Cohomology of Schemes, Lemma", "\\ref{coherent-lemma-cech-cohomology-quasi-coherent-trivial}." ], "refs": [ "algebra-lemma-standard-covering", "algebra-lemma-cover-module", "algebra-lemma-glue-modules", "coherent-lemma-cech-cohomology-quasi-coherent-trivial" ], "ref_ids": [ 414, 413, 417, 3281 ] }, { "id": 14784, "type": "other", "label": "descent-remark-homotopy-equivalent-cosimplicial-algebras", "categories": [ "descent" ], "title": "descent-remark-homotopy-equivalent-cosimplicial-algebras", "contents": [ "Let $R$ be a ring. Let $A_\\bullet$ be a cosimplicial $R$-algebra.", "In this setting a descent datum corresponds to an cosimplicial", "$A_\\bullet$-module $M_\\bullet$ with the property that for", "every $n, m \\geq 0$ and every $\\varphi : [n] \\to [m]$ the", "map $M(\\varphi) : M_n \\to M_m$ induces an isomorphism", "$$", "M_n \\otimes_{A_n, A(\\varphi)} A_m \\longrightarrow M_m.", "$$", "Let us call such a cosimplicial module a {\\it cartesian module}.", "In this setting, the proof of Proposition \\ref{proposition-descent-module}", "can be split in the following steps", "\\begin{enumerate}", "\\item If $R \\to R'$ and $R \\to A$ are faithfully flat,", "then descent data for $A/R$ are effective if", "descent data for $(R' \\otimes_R A)/R'$ are effective.", "\\item Let $A$ be an $R$-algebra. Descent data for $A/R$ correspond", "to cartesian $(A/R)_\\bullet$-modules.", "\\item If $R \\to A$ has a section then $(A/R)_\\bullet$ is homotopy", "equivalent to $R$, the constant cosimplicial", "$R$-algebra with value $R$.", "\\item If $A_\\bullet \\to B_\\bullet$ is a homotopy equivalence of", "cosimplicial $R$-algebras then the functor", "$M_\\bullet \\mapsto M_\\bullet \\otimes_{A_\\bullet} B_\\bullet$", "induces an equivalence of categories between cartesian", "$A_\\bullet$-modules and cartesian $B_\\bullet$-modules.", "\\end{enumerate}", "For (1) see Lemma \\ref{lemma-descent-descends}.", "Part (2) uses Lemma \\ref{lemma-descent-datum-cosimplicial}.", "Part (3) we have seen in the proof of Lemma \\ref{lemma-with-section-exact}", "(it relies on Simplicial,", "Lemma \\ref{simplicial-lemma-push-outs-simplicial-object-w-section}).", "Moreover, part (4) is a triviality if you think about it right!" ], "refs": [ "descent-proposition-descent-module", "descent-lemma-descent-descends", "descent-lemma-descent-datum-cosimplicial" ], "ref_ids": [ 14752, 14600, 14595 ] }, { "id": 14785, "type": "other", "label": "descent-remark-reflects", "categories": [ "descent" ], "title": "descent-remark-reflects", "contents": [ "Any functor $F : \\mathcal{A} \\to \\mathcal{B}$ of abelian categories", "which is exact and takes nonzero objects to nonzero objects reflects", "injections and surjections. Namely, exactness implies that", "$F$ preserves kernels and cokernels (compare with", "Homology, Section \\ref{homology-section-functors}).", "For example, if $f : R \\to S$ is a ", "faithfully flat ring homomorphism, then", "$\\bullet \\otimes_R S: \\text{Mod}_R \\to \\text{Mod}_S$ has these properties." ], "refs": [], "ref_ids": [] }, { "id": 14786, "type": "other", "label": "descent-remark-adjunction", "categories": [ "descent" ], "title": "descent-remark-adjunction", "contents": [ "We will use frequently the standard adjunction between $\\Hom$ and tensor ", "product, in the form of the natural isomorphism of contravariant functors", "\\begin{equation}", "\\label{equation-adjunction}", "C(\\bullet_1 \\otimes_R \\bullet_2) \\cong \\Hom_R(\\bullet_1, C(\\bullet_2)): ", "\\text{Mod}_R \\times \\text{Mod}_R \\to \\text{Mod}_R", "\\end{equation}", "taking $f: M_1 \\otimes_R M_2 \\to \\mathbf{Q}/\\mathbf{Z}$ to the map $m_1 \\mapsto ", "(m_2 \\mapsto f(m_1 \\otimes m_2))$. See", "Algebra, Lemma \\ref{algebra-lemma-hom-from-tensor-product-variant}.", "A corollary of this observation is that if", "$$", "\\xymatrix@C=9pc{", "C(M) \\ar@<1ex>[r] \\ar@<-1ex>[r] & C(N) \\ar[r] & C(P)", "}", "$$", "is a split coequalizer diagram in $\\text{Mod}_R$, then so is", "$$", "\\xymatrix@C=9pc{", "C(M \\otimes_R Q) \\ar@<1ex>[r] \\ar@<-1ex>[r] & C(N \\otimes_R Q) \\ar[r] & C(P ", "\\otimes_R Q)", "}", "$$", "for any $Q \\in \\text{Mod}_R$." ], "refs": [ "algebra-lemma-hom-from-tensor-product-variant" ], "ref_ids": [ 376 ] }, { "id": 14787, "type": "other", "label": "descent-remark-functorial-splitting", "categories": [ "descent" ], "title": "descent-remark-functorial-splitting", "contents": [ "Let $f: M \\to N$ be a universally injective morphism in $\\text{Mod}_R$. By ", "choosing a splitting", "$g$ of $C(f)$, we may construct a functorial splitting of $C(1_P \\otimes f)$ ", "for each $P \\in \\text{Mod}_R$.", "Namely, by (\\ref{equation-adjunction}) this amounts to splitting $\\Hom_R(P, ", "C(f))$ functorially in $P$,", "and this is achieved by the map $g \\circ \\bullet$." ], "refs": [], "ref_ids": [] }, { "id": 14788, "type": "other", "label": "descent-remark-descent-lemma", "categories": [ "descent" ], "title": "descent-remark-descent-lemma", "contents": [ "If $f$ is a split injection in $\\text{Mod}_R$, one can simplify the argument by ", "splitting $f$ directly,", "without using $C$. Things are even simpler if $f$ is faithfully flat; in this ", "case,", "the conclusion of Lemma \\ref{lemma-descent-lemma} ", "is immediate because tensoring over $R$ with $S$ preserves all equalizers." ], "refs": [ "descent-lemma-descent-lemma" ], "ref_ids": [ 14607 ] }, { "id": 14789, "type": "other", "label": "descent-remark-when-locally-split", "categories": [ "descent" ], "title": "descent-remark-when-locally-split", "contents": [ "It would make things easier to have a faithfully", "flat ring homomorphism $g: R \\to T$ for which $T \\to S \\otimes_R T$ has some ", "extra structure.", "For instance, if one could ensure that $T \\to S \\otimes_R T$ is split in ", "$\\textit{Rings}$,", "then it would follow that every property of a module or algebra which is stable ", "under base extension", "and which descends along faithfully flat morphisms also descends along ", "universally injective morphisms.", "An obvious guess would be to find $g$ for which $T$ is not only faithfully flat ", "but also injective in $\\text{Mod}_R$,", "but even for $R = \\mathbf{Z}$ no such homomorphism can exist." ], "refs": [], "ref_ids": [] }, { "id": 14790, "type": "other", "label": "descent-remark-locally-free-descends", "categories": [ "descent" ], "title": "descent-remark-locally-free-descends", "contents": [ "Being locally free is a property of quasi-coherent modules which", "does not descend in the fpqc topology. Namely, suppose that", "$R$ is a ring and that $M$ is a projective $R$-module which is", "a countable direct sum $M = \\bigoplus L_n$ of rank 1 locally", "free modules, but not locally free, see", "Examples, Lemma \\ref{examples-lemma-projective-not-locally-free}.", "Then $M$ becomes free on making the faithfully flat base change", "$$", "R \\longrightarrow", "\\bigoplus\\nolimits_{m \\geq 1}", "\\bigoplus\\nolimits_{(i_1, \\ldots, i_m) \\in \\mathbf{Z}^{\\oplus m}}", "L_1^{\\otimes i_1} \\otimes_R \\ldots \\otimes_R L_m^{\\otimes i_m}", "$$", "But we don't know what happens for fppf coverings. In other words,", "we don't know the answer to the following question:", "Suppose $A \\to B$ is a faithfully", "flat ring map of finite presentation. Let $M$ be an $A$-module", "such that $M \\otimes_A B$ is free. Is $M$ a locally free", "$A$-module? It turns out that if $A$ is Noetherian, then the answer", "is yes. This follows from the results of \\cite{Bass}. But in general", "we don't know the answer. If you know the answer, or have a reference,", "please email", "\\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}." ], "refs": [ "examples-lemma-projective-not-locally-free" ], "ref_ids": [ 2538 ] }, { "id": 14791, "type": "other", "label": "descent-remark-Zariski-site-space", "categories": [ "descent" ], "title": "descent-remark-Zariski-site-space", "contents": [ "In Topologies, Lemma \\ref{topologies-lemma-Zariski-usual}", "we have seen that the small Zariski site of a scheme $S$ is", "equivalent to $S$ as a topological space in the sense that the", "categories of sheaves are naturally equivalent. Now that $S_{Zar}$", "is also endowed with a structure sheaf $\\mathcal{O}$ we see", "that sheaves of modules on the ringed site $(S_{Zar}, \\mathcal{O})$", "agree with sheaves of modules on the ringed space $(S, \\mathcal{O}_S)$." ], "refs": [ "topologies-lemma-Zariski-usual" ], "ref_ids": [ 12437 ] }, { "id": 14792, "type": "other", "label": "descent-remark-change-topologies-ringed", "categories": [ "descent" ], "title": "descent-remark-change-topologies-ringed", "contents": [ "Let $f : T \\to S$ be a morphism of schemes.", "Each of the morphisms of sites $f_{sites}$ listed in", "Topologies, Section \\ref{topologies-section-change-topologies}", "becomes a morphism of ringed sites. Namely, each of these morphisms of sites", "$f_{sites} : (\\Sch/T)_\\tau \\to (\\Sch/S)_{\\tau'}$, or", "$f_{sites} : (\\Sch/S)_\\tau \\to S_{\\tau'}$ is given by the continuous", "functor $S'/S \\mapsto T \\times_S S'/S$. Hence, given $S'/S$ we let", "$$", "f_{sites}^\\sharp :", "\\mathcal{O}(S'/S)", "\\longrightarrow", "f_{sites, *}\\mathcal{O}(S'/S) =", "\\mathcal{O}(S \\times_S S'/T)", "$$", "be the usual map", "$\\text{pr}_{S'}^\\sharp : \\mathcal{O}(S') \\to \\mathcal{O}(T \\times_S S')$.", "Similarly, the morphism", "$i_f : \\Sh(T_\\tau) \\to \\Sh((\\Sch/S)_\\tau)$", "for $\\tau \\in \\{Zar, \\etale\\}$, see", "Topologies, Lemmas \\ref{topologies-lemma-put-in-T} and", "\\ref{topologies-lemma-put-in-T-etale},", "becomes a morphism of ringed topoi because $i_f^{-1}\\mathcal{O} = \\mathcal{O}$.", "Here are some special cases:", "\\begin{enumerate}", "\\item The morphism of big sites", "$f_{big} : (\\Sch/X)_{fppf} \\to (\\Sch/Y)_{fppf}$,", "becomes a morphism of ringed sites", "$$", "(f_{big}, f_{big}^\\sharp) :", "((\\Sch/X)_{fppf}, \\mathcal{O}_X)", "\\longrightarrow", "((\\Sch/Y)_{fppf}, \\mathcal{O}_Y)", "$$", "as in Modules on Sites, Definition \\ref{sites-modules-definition-ringed-site}.", "Similarly for the big syntomic, smooth, \\'etale and Zariski sites.", "\\item The morphism of small sites", "$f_{small} : X_\\etale \\to Y_\\etale$", "becomes a morphism of ringed sites", "$$", "(f_{small}, f_{small}^\\sharp) :", "(X_\\etale, \\mathcal{O}_X)", "\\longrightarrow", "(Y_\\etale, \\mathcal{O}_Y)", "$$", "as in Modules on Sites, Definition \\ref{sites-modules-definition-ringed-site}.", "Similarly for the small Zariski site.", "\\end{enumerate}" ], "refs": [ "topologies-lemma-put-in-T", "topologies-lemma-put-in-T-etale", "sites-modules-definition-ringed-site", "sites-modules-definition-ringed-site" ], "ref_ids": [ 12438, 12452, 14279, 14279 ] }, { "id": 14793, "type": "other", "label": "descent-remark-change-topologies-ringed-sites", "categories": [ "descent" ], "title": "descent-remark-change-topologies-ringed-sites", "contents": [ "Remark \\ref{remark-change-topologies-ringed}", "and", "Lemma \\ref{lemma-compare-sites}", "have the following applications:", "\\begin{enumerate}", "\\item Let $S$ be a scheme.", "The construction $\\mathcal{F} \\mapsto \\mathcal{F}^a$ is", "the pullback under the morphism of ringed sites", "$\\text{id}_{\\tau, Zar} : ((\\Sch/S)_\\tau, \\mathcal{O})", "\\to (S_{Zar}, \\mathcal{O})$", "or the morphism", "$\\text{id}_{small, \\etale, Zar} :", "(S_\\etale, \\mathcal{O}) \\to (S_{Zar}, \\mathcal{O})$.", "\\item Let $f : X \\to Y$ be a morphism of schemes.", "For any of the morphisms $f_{sites}$ of ringed sites of", "Remark \\ref{remark-change-topologies-ringed}", "we have", "$$", "(f^*\\mathcal{F})^a = f_{sites}^*\\mathcal{F}^a.", "$$", "This follows from (1) and the fact that pullbacks are compatible with", "compositions of morphisms of ringed sites, see", "Modules on Sites,", "Lemma \\ref{sites-modules-lemma-push-pull-composition-modules}.", "\\end{enumerate}" ], "refs": [ "descent-remark-change-topologies-ringed", "descent-lemma-compare-sites", "descent-remark-change-topologies-ringed", "sites-modules-lemma-push-pull-composition-modules" ], "ref_ids": [ 14792, 14622, 14792, 14156 ] }, { "id": 14794, "type": "other", "label": "descent-remark-smooth-permanence", "categories": [ "descent" ], "title": "descent-remark-smooth-permanence", "contents": [ "With the assumptions (1) and $p$ smooth in", "Lemma \\ref{lemma-smooth-permanence}", "it is not automatically the case that $X \\to Y$ is smooth.", "A counter example is $S = \\Spec(k)$, $X = \\Spec(k[s])$,", "$Y = \\Spec(k[t])$ and $f$ given by $t \\mapsto s^2$.", "But see also Lemma \\ref{lemma-syntomic-permanence}", "for some information on the structure of $f$." ], "refs": [ "descent-lemma-smooth-permanence", "descent-lemma-syntomic-permanence" ], "ref_ids": [ 14644, 14645 ] }, { "id": 14795, "type": "other", "label": "descent-remark-descending-properties-standard", "categories": [ "descent" ], "title": "descent-remark-descending-properties-standard", "contents": [ "In Lemma \\ref{lemma-descending-properties} above if", "$\\tau = smooth$ then in condition (3) we may assume that", "the morphism is a (surjective) standard smooth morphism.", "Similarly, when $\\tau = syntomic$ or $\\tau = \\etale$." ], "refs": [ "descent-lemma-descending-properties" ], "ref_ids": [ 14647 ] }, { "id": 14796, "type": "other", "label": "descent-remark-descending-properties-morphisms-standard", "categories": [ "descent" ], "title": "descent-remark-descending-properties-morphisms-standard", "contents": [ "(This is a repeat of Remark \\ref{remark-descending-properties-standard} above.)", "In Lemma \\ref{lemma-descending-properties-morphisms} above if", "$\\tau = smooth$ then in condition (3) we may assume that", "the morphism is a (surjective) standard smooth morphism.", "Similarly, when $\\tau = syntomic$ or $\\tau = \\etale$." ], "refs": [ "descent-remark-descending-properties-standard", "descent-lemma-descending-properties-morphisms" ], "ref_ids": [ 14795, 14665 ] }, { "id": 14797, "type": "other", "label": "descent-remark-properties-morphisms-local-source-standard", "categories": [ "descent" ], "title": "descent-remark-properties-morphisms-local-source-standard", "contents": [ "(This is a repeat of", "Remarks \\ref{remark-descending-properties-standard}", "and \\ref{remark-descending-properties-morphisms-standard} above.)", "In Lemma \\ref{lemma-properties-morphisms-local-source} above if", "$\\tau = smooth$ then in condition (4) we may assume that", "the morphism is a (surjective) standard smooth morphism.", "Similarly, when $\\tau = syntomic$ or $\\tau = \\etale$." ], "refs": [ "descent-remark-descending-properties-standard", "descent-remark-descending-properties-morphisms-standard", "descent-lemma-properties-morphisms-local-source" ], "ref_ids": [ 14795, 14796, 14707 ] }, { "id": 14798, "type": "other", "label": "descent-remark-list-local-source-target", "categories": [ "descent" ], "title": "descent-remark-list-local-source-target", "contents": [ "Using", "Lemma \\ref{lemma-etale-local-source-target}", "and the work done in the earlier sections of this chapter it is easy", "to make a list of types of morphisms which are \\'etale local on the", "source-and-target. In each case we list the lemma which implies", "the property is \\'etale local on the source and the lemma which implies", "the property is \\'etale local on the target. In each case the third assumption", "of", "Lemma \\ref{lemma-etale-local-source-target}", "is trivial to check, and we omit it. Here is the list:", "\\begin{enumerate}", "\\item flat, see", "Lemmas \\ref{lemma-flat-fpqc-local-source} and", "\\ref{lemma-descending-property-flat},", "\\item locally of finite presentation, see", "Lemmas \\ref{lemma-locally-finite-presentation-fppf-local-source} and", "\\ref{lemma-descending-property-locally-finite-presentation},", "\\item locally finite type, see", "Lemmas \\ref{lemma-locally-finite-type-fppf-local-source} and", "\\ref{lemma-descending-property-locally-finite-type},", "\\item universally open, see", "Lemmas \\ref{lemma-universally-open-fppf-local-source} and", "\\ref{lemma-descending-property-universally-open},", "\\item syntomic, see", "Lemmas \\ref{lemma-syntomic-syntomic-local-source} and", "\\ref{lemma-descending-property-syntomic},", "\\item smooth, see", "Lemmas \\ref{lemma-smooth-smooth-local-source} and", "\\ref{lemma-descending-property-smooth},", "\\item \\'etale, see", "Lemmas \\ref{lemma-etale-etale-local-source} and", "\\ref{lemma-descending-property-etale},", "\\item locally quasi-finite, see", "Lemmas \\ref{lemma-locally-quasi-finite-etale-local-source} and", "\\ref{lemma-descending-property-quasi-finite},", "\\item unramified, see", "Lemmas \\ref{lemma-unramified-etale-local-source} and", "\\ref{lemma-descending-property-unramified},", "\\item G-unramified, see", "Lemmas \\ref{lemma-unramified-etale-local-source} and", "\\ref{lemma-descending-property-unramified}, and", "\\item add more here as needed.", "\\end{enumerate}" ], "refs": [ "descent-lemma-etale-local-source-target", "descent-lemma-etale-local-source-target", "descent-lemma-flat-fpqc-local-source", "descent-lemma-descending-property-flat", "descent-lemma-locally-finite-presentation-fppf-local-source", "descent-lemma-descending-property-locally-finite-presentation", "descent-lemma-locally-finite-type-fppf-local-source", "descent-lemma-descending-property-locally-finite-type", "descent-lemma-universally-open-fppf-local-source", "descent-lemma-descending-property-universally-open", "descent-lemma-syntomic-syntomic-local-source", "descent-lemma-descending-property-syntomic", "descent-lemma-smooth-smooth-local-source", "descent-lemma-descending-property-smooth", "descent-lemma-etale-etale-local-source", "descent-lemma-descending-property-etale", "descent-lemma-locally-quasi-finite-etale-local-source", "descent-lemma-descending-property-quasi-finite", "descent-lemma-unramified-etale-local-source", "descent-lemma-descending-property-unramified", "descent-lemma-unramified-etale-local-source", "descent-lemma-descending-property-unramified" ], "ref_ids": [ 14721, 14721, 14708, 14680, 14710, 14676, 14711, 14675, 14713, 14669, 14714, 14691, 14715, 14692, 14716, 14694, 14717, 14689, 14718, 14693, 14718, 14693 ] }, { "id": 14799, "type": "other", "label": "descent-remark-compare-definitions", "categories": [ "descent" ], "title": "descent-remark-compare-definitions", "contents": [ "At this point we have three possible definitions of what it means for a", "property $\\mathcal{P}$ of morphisms to be ``\\'etale local on the source and", "target'':", "\\begin{enumerate}", "\\item[(ST)] $\\mathcal{P}$ is \\'etale local on the source and $\\mathcal{P}$ is", "\\'etale local on the target,", "\\item[(DM)] (the definition in the paper \\cite[Page 100]{DM} by", "Deligne and Mumford) for every diagram", "$$", "\\xymatrix{", "U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\", "X \\ar[r]^f & Y", "}", "$$", "with surjective \\'etale vertical arrows we have", "$\\mathcal{P}(h) \\Leftrightarrow \\mathcal{P}(f)$, and", "\\item[(SP)] $\\mathcal{P}$ is \\'etale local on the source-and-target.", "\\end{enumerate}", "In this section we have seen that (SP) $\\Rightarrow$ (DM) $\\Rightarrow$ (ST).", "The", "Examples \\ref{example-silly-one} and \\ref{example-silly-two}", "show that neither implication can be reversed. Finally,", "Lemma \\ref{lemma-etale-local-source-target}", "shows that the difference disappears when looking at properties of", "morphisms which are stable under postcomposing with open immersions, which", "in practice will always be the case." ], "refs": [ "descent-lemma-etale-local-source-target" ], "ref_ids": [ 14721 ] }, { "id": 14800, "type": "other", "label": "descent-remark-easier", "categories": [ "descent" ], "title": "descent-remark-easier", "contents": [ "Let $X \\to S$ be a morphism of schemes. Let $(V/X, \\varphi)$ be", "a descent datum relative to $X \\to S$. We may think of the", "isomorphism $\\varphi$ as an isomorphism", "$$", "(X \\times_S X) \\times_{\\text{pr}_0, X} V", "\\longrightarrow", "(X \\times_S X) \\times_{\\text{pr}_1, X} V", "$$", "of schemes over $X \\times_S X$. So loosely speaking one may", "think of $\\varphi$ as a map", "$\\varphi : \\text{pr}_0^*V \\to \\text{pr}_1^*V$\\footnote{Unfortunately,", "we have chosen the ``wrong'' direction for our arrow here. In", "Definitions \\ref{definition-descent-datum} and", "\\ref{definition-descent-datum-for-family-of-morphisms}", "we should have the opposite direction to what was done in", "Definition \\ref{definition-descent-datum-quasi-coherent}", "by the general principle that ``functions'' and ``spaces'' are dual.}.", "The cocycle condition then says that", "$\\text{pr}_{02}^*\\varphi =", "\\text{pr}_{12}^*\\varphi \\circ \\text{pr}_{01}^*\\varphi$.", "In this way it is very similar to the case of a descent datum on", "quasi-coherent sheaves." ], "refs": [ "descent-definition-descent-datum", "descent-definition-descent-datum-for-family-of-morphisms", "descent-definition-descent-datum-quasi-coherent" ], "ref_ids": [ 14776, 14777, 14757 ] }, { "id": 14801, "type": "other", "label": "descent-remark-easier-family", "categories": [ "descent" ], "title": "descent-remark-easier-family", "contents": [ "Let $S$ be a scheme.", "Let $\\{X_i \\to S\\}_{i \\in I}$ be a family of morphisms with target $S$.", "Let $(V_i, \\varphi_{ij})$ be a descent datum relative to", "$\\{X_i \\to S\\}$. We may think of the isomorphisms $\\varphi_{ij}$", "as isomorphisms", "$$", "(X_i \\times_S X_j) \\times_{\\text{pr}_0, X_i} V_i", "\\longrightarrow", "(X_i \\times_S X_j) \\times_{\\text{pr}_1, X_j} V_j", "$$", "of schemes over $X_i \\times_S X_j$. So loosely speaking one may", "think of $\\varphi_{ij}$ as an isomorphism", "$\\text{pr}_0^*V_i \\to \\text{pr}_1^*V_j$ over $X_i \\times_S X_j$.", "The cocycle condition then says that", "$\\text{pr}_{02}^*\\varphi_{ik} =", "\\text{pr}_{12}^*\\varphi_{jk} \\circ \\text{pr}_{01}^*\\varphi_{ij}$.", "In this way it is very similar to the case of a descent datum on", "quasi-coherent sheaves." ], "refs": [], "ref_ids": [] }, { "id": 14802, "type": "other", "label": "descent-remark-morphisms-of-schemes-satisfy-fpqc-descent", "categories": [ "descent" ], "title": "descent-remark-morphisms-of-schemes-satisfy-fpqc-descent", "contents": [ "Lemma \\ref{lemma-refine-coverings-fully-faithful}", "says that morphisms of schemes satisfy fpqc descent.", "In other words, given a scheme $S$ and schemes $X$, $Y$ over $S$", "the functor", "$$", "(\\Sch/S)^{opp} \\longrightarrow \\textit{Sets},", "\\quad", "T \\longmapsto \\Mor_T(X_T, Y_T)", "$$", "satisfies the sheaf condition for the fpqc topology.", "The simplest case of this is the following. Suppose that $T \\to S$", "is a surjective flat morphism of affines. Let $\\psi_0 : X_T \\to Y_T$", "be a morphism of schemes over $T$ which is compatible with the", "canonical descent data. Then there exists a unique morphism", "$\\psi : X \\to Y$ whose base change to $T$ is $\\psi_0$. In fact this", "special case follows in a straightforward manner from", "Lemma \\ref{lemma-fully-faithful}.", "And, in turn, that lemma is a formal consequence of the following", "two facts:", "(a) the base change functor by a faithfully flat morphism is faithful, see", "Lemma \\ref{lemma-ff-base-change-faithful}", "and (b) a scheme satisfies the sheaf condition for the fpqc topology, see", "Lemma \\ref{lemma-fpqc-universal-effective-epimorphisms}." ], "refs": [ "descent-lemma-refine-coverings-fully-faithful", "descent-lemma-fully-faithful", "descent-lemma-ff-base-change-faithful", "descent-lemma-fpqc-universal-effective-epimorphisms" ], "ref_ids": [ 14745, 14738, 14736, 14638 ] }, { "id": 14803, "type": "other", "label": "descent-remark-what-product-means", "categories": [ "descent" ], "title": "descent-remark-what-product-means", "contents": [ "In the statement of Lemma \\ref{lemma-descent-data-sheaves} the condition that", "$h_{S_i} \\times F$ is representable is equivalent to", "the condition that the restriction of $F$ to", "$(\\Sch/S_i)_\\tau$ is representable." ], "refs": [ "descent-lemma-descent-data-sheaves" ], "ref_ids": [ 14751 ] }, { "id": 14932, "type": "other", "label": "simplicial-remark-relations", "categories": [ "simplicial" ], "title": "simplicial-remark-relations", "contents": [ "By abuse of notation we sometimes write $d_i : U_n \\to U_{n - 1}$", "instead of $d^n_i$, and similarly for $s_i : U_n \\to U_{n + 1}$.", "The relations among the morphisms $d^n_i$ and $s^n_i$", "may be expressed as follows:", "\\begin{enumerate}", "\\item If $i < j$, then $d_i \\circ d_j = d_{j - 1} \\circ d_i$.", "\\item If $i < j$, then $d_i \\circ s_j = s_{j - 1} \\circ d_i$.", "\\item We have $\\text{id} = d_j \\circ s_j = d_{j + 1} \\circ s_j$.", "\\item If $i > j + 1$, then $d_i \\circ s_j = s_j \\circ d_{i - 1}$.", "\\item If $i \\leq j$, then $s_i \\circ s_j = s_{j + 1} \\circ s_i$.", "\\end{enumerate}", "This means that whenever the compositions on both the left and the", "right are defined then the corresponding equality should hold." ], "refs": [], "ref_ids": [] }, { "id": 14933, "type": "other", "label": "simplicial-remark-relations-cosimplicial", "categories": [ "simplicial" ], "title": "simplicial-remark-relations-cosimplicial", "contents": [ "By abuse of notation we sometimes write $\\delta_i : U_{n - 1} \\to U_n$", "instead of $\\delta^n_i$, and similarly for $\\sigma_i : U_{n + 1} \\to U_n$.", "The relations among the morphisms $\\delta^n_i$ and $\\sigma^n_i$", "may be expressed as follows:", "\\begin{enumerate}", "\\item If $i < j$, then", "$\\delta_j \\circ \\delta_i = \\delta_i \\circ \\delta_{j - 1}$.", "\\item If $i < j$, then", "$\\sigma_j \\circ \\delta_i = \\delta_i \\circ \\sigma_{j - 1}$.", "\\item We have", "$\\text{id} = \\sigma_j \\circ \\delta_j = \\sigma_j \\circ \\delta_{j + 1}$.", "\\item If $i > j + 1$, then", "$\\sigma_j \\circ \\delta_i = \\delta_{i - 1} \\circ \\sigma_j$.", "\\item If $i \\leq j$, then", "$\\sigma_j \\circ \\sigma_i = \\sigma_i \\circ \\sigma_{j + 1}$.", "\\end{enumerate}", "This means that whenever the compositions on both the left and the", "right are defined then the corresponding equality should hold." ], "refs": [], "ref_ids": [] }, { "id": 14934, "type": "other", "label": "simplicial-remark-explicit-face-degeneracy", "categories": [ "simplicial" ], "title": "simplicial-remark-explicit-face-degeneracy", "contents": [ "Let $U$, and $U_{n + 1}$ be as in Lemma \\ref{lemma-work-out}.", "On $T$-valued points we can easily describe the face", "and degeneracy maps of $\\tilde U$.", "Explicitly, the maps $d^{n + 1}_i : U_{n + 1} \\to U_n$", "are given by", "$$", "(f_0, \\ldots, f_{n + 1}) \\longmapsto f_i.", "$$", "And the maps $s^n_j : U_n \\to U_{n + 1}$ are given by", "\\begin{eqnarray*}", "f & \\longmapsto & (", "s^{n - 1}_{j - 1} \\circ d^{n - 1}_0 \\circ f, \\\\", "& &", "s^{n - 1}_{j - 1} \\circ d^{n - 1}_1 \\circ f, \\\\", "& &", "\\ldots\\\\", "& &", "s^{n - 1}_{j - 1} \\circ d^{n - 1}_{j - 1} \\circ f, \\\\", "& &", "f, \\\\", "& &", "f, \\\\", "& &", "s^{n - 1}_j \\circ d^{n - 1}_{j + 1} \\circ f, \\\\", "& &", "s^{n - 1}_j \\circ d^{n - 1}_{j + 2} \\circ f, \\\\", "& &", "\\ldots\\\\", "& &", "s^{n - 1}_j \\circ d^{n - 1}_n \\circ f", ")", "\\end{eqnarray*}", "where we leave it to the reader to verify that the RHS", "is an element of the displayed set of Lemma \\ref{lemma-work-out}.", "For $n = 0$ there is one map, namely $f \\mapsto (f, f)$.", "For $n = 1$ there are two maps, namely", "$f \\mapsto (f, f, s_0d_1f)$ and", "$f \\mapsto (s_0d_0f, f, f)$.", "For $n = 2$ there are three maps, namely", "$f \\mapsto (f, f, s_0d_1f, s_0d_2f)$,", "$f \\mapsto (s_0d_0f, f, f, s_1d_2f)$, and", "$f \\mapsto (s_1d_0f, s_1d_1f, f, f)$.", "And so on and so forth." ], "refs": [ "simplicial-lemma-work-out", "simplicial-lemma-work-out" ], "ref_ids": [ 14839, 14839 ] }, { "id": 14935, "type": "other", "label": "simplicial-remark-cosk-simplicial-sets", "categories": [ "simplicial" ], "title": "simplicial-remark-cosk-simplicial-sets", "contents": [ "The construction of Lemma \\ref{lemma-work-out}", "above in the case of simplicial", "sets is the following. Given an $n$-truncated simplicial", "set $U$, we make a canonical $(n + 1)$-truncated simplicial", "set $\\tilde U$ as follows. We add a set of $(n + 1)$-simplices", "$U_{n + 1}$ by the formula of the lemma. Namely,", "an element of $U_{n + 1}$ is a numbered collection of", "$(f_0, \\ldots, f_{n + 1})$ of $n$-simplices,", "with the property that they glue", "as they would in a $(n + 1)$-simplex. In other words,", "the $i$th face of $f_j$ is the $(j-1)$st face of $f_i$", "for $i < j$. Geometrically it is obvious how to define the", "face and degeneracy maps for $\\tilde U$.", "If $V$ is an $(n + 1)$-truncated simplicial set,", "then its $(n + 1)$-simplices give rise to compatible collections", "of $n$-simplices $(f_0, \\ldots, f_{n + 1})$ with $f_i \\in V_n$.", "Hence there is a natural map", "$\\Mor(\\text{sk}_nV, U) \\to \\Mor(V, \\tilde U)$", "which is inverse to the canonical restriction mapping", "the other way.", "\\medskip\\noindent", "Also, it is enough to do the combinatorics of the", "construction in the case of truncated simplicial sets.", "Namely, for any object $T$ of the category $\\mathcal{C}$,", "and any $n$-truncated simplicial object $U$ of $\\mathcal{C}$", "we can consider the $n$-truncated simplicial set", "$\\Mor(T, U)$. We may apply the construction to this,", "and take its set of $(n + 1)$-simplices, and require this to be", "representable. This is a good way to think about", "the result of Lemma \\ref{lemma-work-out}." ], "refs": [ "simplicial-lemma-work-out", "simplicial-lemma-work-out" ], "ref_ids": [ 14839, 14839 ] }, { "id": 14936, "type": "other", "label": "simplicial-remark-inductive-coskeleton", "categories": [ "simplicial" ], "title": "simplicial-remark-inductive-coskeleton", "contents": [ "{\\it Inductive construction of coskeleta.}", "Suppose that $\\mathcal{C}$ is a category with", "finite limits. Suppose that $U$ is an $m$-truncated", "simplicial object in $\\mathcal{C}$. Then we can", "inductively construct $n$-truncated objects $U^n$ as", "follows:", "\\begin{enumerate}", "\\item To start, set $U^m = U$.", "\\item Given $U^n$ for $n \\geq m$ set $U^{n + 1} = \\tilde U^n$,", "where $\\tilde U^n$ is constructed from $U^n$ as in Lemma", "\\ref{lemma-work-out}.", "\\end{enumerate}", "Since the construction of Lemma \\ref{lemma-work-out} has", "the property that it leaves the $n$-skeleton of $U^n$", "unchanged, we can then define $\\text{cosk}_m U$ to be", "the simplicial object with", "$(\\text{cosk}_m U)_n = U^n_n = U^{n + 1}_n = \\ldots$.", "And it follows formally from Lemma \\ref{lemma-work-out}", "that $U^n$ satisfies the formula", "$$", "\\Mor_{\\text{Simp}_n(\\mathcal{C})}(V, U^n)", "=", "\\Mor_{\\text{Simp}_m(\\mathcal{C})}(\\text{sk}_mV, U)", "$$", "for all $n \\geq m$. It also then follows formally", "from this that", "$$", "\\Mor_{\\text{Simp}(\\mathcal{C})}(V, \\text{cosk}_mU)", "=", "\\Mor_{\\text{Simp}_m(\\mathcal{C})}(\\text{sk}_mV, U)", "$$", "with $\\text{cosk}_mU$ chosen as above." ], "refs": [ "simplicial-lemma-work-out", "simplicial-lemma-work-out", "simplicial-lemma-work-out" ], "ref_ids": [ 14839, 14839, 14839 ] }, { "id": 14937, "type": "other", "label": "simplicial-remark-existence-cosk", "categories": [ "simplicial" ], "title": "simplicial-remark-existence-cosk", "contents": [ "We do not need all finite limits in order to be able to define", "the coskeleton functors. Here are some remarks", "\\begin{enumerate}", "\\item We have seen in Examples \\ref{example-cosk0} that if $\\mathcal{C}$", "has products of pairs of objects then $\\text{cosk}_0$ exists.", "\\item For $k > 0$ the functor $\\text{cosk}_k$ exists if", "$\\mathcal{C}$ has finite connected limits.", "\\end{enumerate}", "This is clear from the inductive procedure of constructing coskeleta", "(Remarks \\ref{remark-cosk-simplicial-sets} and", "\\ref{remark-inductive-coskeleton}) but it also follows from the fact that", "the categories $(\\Delta/[n])_{\\leq k}$ for $k \\geq 1$ and", "$n \\geq k + 1$ used in Lemma \\ref{lemma-existence-cosk}", "are connected. Observe that we do not need the categories", "for $n \\leq k$ by Lemma \\ref{lemma-trivial-cosk} or", "Lemma \\ref{lemma-recover-cosk}. (As $k$ gets higher the categories", "$(\\Delta/[n])_{\\leq k}$ for $k \\geq 1$ and $n \\geq k + 1$ are more", "and more connected in a topological sense.)" ], "refs": [ "simplicial-remark-cosk-simplicial-sets", "simplicial-remark-inductive-coskeleton", "simplicial-lemma-existence-cosk", "simplicial-lemma-trivial-cosk", "simplicial-lemma-recover-cosk" ], "ref_ids": [ 14935, 14936, 14835, 14836, 14837 ] }, { "id": 14938, "type": "other", "label": "simplicial-remark-augmentation", "categories": [ "simplicial" ], "title": "simplicial-remark-augmentation", "contents": [ "Let $\\mathcal{C}$ be a category with fibre products.", "Let $V$ be a simplicial object.", "Let $\\epsilon : V \\to X$ be an augmentation.", "Let $U$ be the simplicial object whose $n$th term", "is the $(n + 1)$st fibred product of $V_0$ over $X$.", "By a simple combination of", "Lemmas \\ref{lemma-augmentation-howto} and \\ref{lemma-cosk-minus-one}", "we obtain a canonical morphism", "$V \\to U$." ], "refs": [ "simplicial-lemma-augmentation-howto", "simplicial-lemma-cosk-minus-one" ], "ref_ids": [ 14845, 14846 ] }, { "id": 14939, "type": "other", "label": "simplicial-remark-sk-literature", "categories": [ "simplicial" ], "title": "simplicial-remark-sk-literature", "contents": [ "In some texts the composite functor", "$$", "\\text{Simp}(\\mathcal{C})", "\\xrightarrow{\\text{sk}_m}", "\\text{Simp}_m(\\mathcal{C})", "\\xrightarrow{i_{m!}}", "\\text{Simp}(\\mathcal{C})", "$$", "is denoted $\\text{sk}_m$. This makes sense for simplicial sets,", "because then Lemma \\ref{lemma-n-skeleton-sets} says", "that $i_{m!} \\text{sk}_m V$ is just the sub simplicial set", "of $V$ consisting of all $i$-simplices of $V$, $i \\leq m$", "and their degeneracies. In those texts it is also customary", "to denote the composition", "$$", "\\text{Simp}(\\mathcal{C})", "\\xrightarrow{\\text{sk}_m}", "\\text{Simp}_m(\\mathcal{C})", "\\xrightarrow{\\text{cosk}_m}", "\\text{Simp}(\\mathcal{C})", "$$", "by $\\text{cosk}_m$." ], "refs": [ "simplicial-lemma-n-skeleton-sets" ], "ref_ids": [ 14851 ] }, { "id": 14940, "type": "other", "label": "simplicial-remark-degenerate-subcomplex", "categories": [ "simplicial" ], "title": "simplicial-remark-degenerate-subcomplex", "contents": [ "In the situation of Lemma \\ref{lemma-decompose-associated-complexes}", "the subcomplex $D(U) \\subset s(U)$ can also be defined as the", "subcomplex with terms", "$$", "D(U)_n = \\Im\\left(", "\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]\\text{ surjective}, \\ m < n} U_m", "\\xrightarrow{\\bigoplus U(\\varphi)}", "U_n\\right)", "$$", "Namely, since $U_m$ is the direct sum of the subobject $N(U_m)$", "and the images of $N(U_k)$ for surjections $[m] \\to [k]$ with $k < m$", "this is clearly the same as the definition of $D(U)_n$ given in", "the proof of Lemma \\ref{lemma-decompose-associated-complexes}.", "Thus we see that if $U$ is a simplicial abelian group, then", "elements of $D(U)_n$ are exactly the sums of degenerate $n$-simplices." ], "refs": [ "simplicial-lemma-decompose-associated-complexes", "simplicial-lemma-decompose-associated-complexes" ], "ref_ids": [ 14866, 14866 ] }, { "id": 14941, "type": "other", "label": "simplicial-remark-homotopy-better", "categories": [ "simplicial" ], "title": "simplicial-remark-homotopy-better", "contents": [ "Let $\\mathcal{C}$ be any category (no assumptions whatsoever). Let", "$U$ and $V$ be simplicial objects of $\\mathcal{C}$. Let $a, b : U \\to V$", "be morphisms of simplicial objects of $\\mathcal{C}$. A", "{\\it homotopy from $a$ to $b$} is given by", "morphisms\\footnote{In the literature, often the maps", "$h_{n + 1, i} \\circ s_i : U_n \\to V_{n + 1}$ are used instead", "of the maps $h_{n, i}$. Of course the relations these maps satisfy", "are different from the ones in Lemma \\ref{lemma-relations-homotopy}.}", "$h_{n, i} : U_n \\to V_n$, for $n \\geq 0$, $i = 0, \\ldots, n + 1$", "satisfying the relations of Lemma \\ref{lemma-relations-homotopy}.", "As in Definition \\ref{definition-homotopy} we say the morphisms $a$ and $b$", "are {\\it homotopic} if there exists a sequence of morphisms", "$a = a_0, a_1, \\ldots, a_n = b$ from $U$ to $V$ such that for each", "$i = 1, \\ldots, n$ there either exists a homotopy from $a_{i - 1}$ to $a_i$", "or there exists a homotopy from $a_i$ to $a_{i - 1}$.", "Clearly, if $F : \\mathcal{C} \\to \\mathcal{C}'$ is any functor", "and $\\{h_{n, i}\\}$ is a homotopy from $a$ to $b$, then", "$\\{F(h_{n, i})\\}$ is a homotopy from $F(a)$ to $F(b)$.", "Similarly, if $a$ and $b$ are homotopic, then $F(a)$ and $F(b)$", "are homotopic.", "Since the lemma says that the newer notion is the same", "as the old one in case finite coproduct exist, we deduce", "in particular that functors preserve the original notion", "whenever both categories have finite coproducts." ], "refs": [ "simplicial-lemma-relations-homotopy", "simplicial-lemma-relations-homotopy", "simplicial-definition-homotopy" ], "ref_ids": [ 14872, 14872, 14927 ] }, { "id": 14942, "type": "other", "label": "simplicial-remark-homotopy-pre-post-compose", "categories": [ "simplicial" ], "title": "simplicial-remark-homotopy-pre-post-compose", "contents": [ "Let $\\mathcal{C}$ be any category. Suppose two morphisms $a, a' : U \\to V$", "of simplicial objects are homotopic. Then for any morphism $b : V \\to W$", "the two maps $b \\circ a, b \\circ a' : U \\to W$ are homotopic.", "Similarly, for any morphism $c : X \\to U$ the two maps", "$a \\circ c, a' \\circ c : X \\to V$ are homotopic. In fact the maps", "$b \\circ a \\circ c, b \\circ a' \\circ c : X \\to W$ are homotopic.", "Namely, if the maps $h_{n, i} : U_n \\to V_n$ define a homotopy from", "$a$ to $a'$ then the maps $b \\circ h_{n, i} \\circ c$ define a homotopy", "from $b \\circ a \\circ c$ to $b \\circ a' \\circ c$. In this way we see that", "we obtain a new category $\\text{hSimp}(\\mathcal{C})$ with the same", "objects as $\\text{Simp}(\\mathcal{C})$ but whose morphisms are", "homotopy classes of of morphisms of $\\text{Simp}(\\mathcal{C})$.", "Thus there is a canonical functor", "$$", "\\text{Simp}(\\mathcal{C})", "\\longrightarrow", "\\text{hSimp}(\\mathcal{C})", "$$", "which is essentially surjective and surjective on sets of morphisms." ], "refs": [], "ref_ids": [] }, { "id": 14943, "type": "other", "label": "simplicial-remark-homotopy-cosimplicial-better", "categories": [ "simplicial" ], "title": "simplicial-remark-homotopy-cosimplicial-better", "contents": [ "Let $\\mathcal{C}$ be any category (no assumptions whatsoever). Let", "$U$ and $V$ be cosimplicial objects of $\\mathcal{C}$. Let $a, b : U \\to V$", "be morphisms of cosimplicial objects of $\\mathcal{C}$. A", "{\\it homotopy from $a$ to $b$} is given by morphisms", "$h_{n, \\alpha} : U_n \\to V_n$, for $n \\geq 0$, $\\alpha \\in \\Delta[1]_n$", "satisfying (\\ref{equation-property-homotopy-cosimplicial})", "for all morphisms $f$ of $\\Delta$ and such that", "$a_n = h_{n, 0 : [n] \\to [1]}$ and $b_n = h_{n, 1 : [n] \\to [1]}$", "for all $n \\geq 0$.", "As in Definition \\ref{definition-homotopy-cosimplicial}", "we say the morphisms $a$ and $b$", "are {\\it homotopic} if there exists a sequence of morphisms", "$a = a_0, a_1, \\ldots, a_n = b$ from $U$ to $V$ such that for each", "$i = 1, \\ldots, n$ there either exists a homotopy from $a_{i - 1}$ to $a_i$", "or there exists a homotopy from $a_i$ to $a_{i - 1}$.", "Clearly, if $F : \\mathcal{C} \\to \\mathcal{C}'$ is any functor", "and $\\{h_{n, i}\\}$ is a homotopy from $a$ to $b$, then", "$\\{F(h_{n, i})\\}$ is a homotopy from $F(a)$ to $F(b)$.", "Similarly, if $a$ and $b$ are homotopic, then $F(a)$ and $F(b)$", "are homotopic. This new notion is the same", "as the old one in case finite products exist. We deduce", "in particular that functors preserve the original notion", "whenever both categories have finite products." ], "refs": [ "simplicial-definition-homotopy-cosimplicial" ], "ref_ids": [ 14929 ] }, { "id": 15006, "type": "other", "label": "discriminant-remark-relative-dualizing-for-quasi-finite", "categories": [ "discriminant" ], "title": "discriminant-remark-relative-dualizing-for-quasi-finite", "contents": [ "Let $f : Y \\to X$ be a locally quasi-finite morphism of locally Noetherian", "schemes. It is clear from Lemma \\ref{lemma-localize-dualizing}", "that there is a unique coherent $\\mathcal{O}_Y$-module", "$\\omega_{Y/X}$ on $Y$ such that for every pair of affine opens", "$\\Spec(B) = V \\subset Y$, $\\Spec(A) = U \\subset X$ with $f(V) \\subset U$", "there is a canonical isomorphism", "$$", "H^0(V, \\omega_{Y/X}) = \\omega_{B/A}", "$$", "and where these isomorphisms are compatible with restriction maps." ], "refs": [ "discriminant-lemma-localize-dualizing" ], "ref_ids": [ 14946 ] }, { "id": 15007, "type": "other", "label": "discriminant-remark-relative-dualizing-for-flat-quasi-finite", "categories": [ "discriminant" ], "title": "discriminant-remark-relative-dualizing-for-flat-quasi-finite", "contents": [ "Let $f : Y \\to X$ be a flat locally quasi-finite morphism of locally", "Noetherian schemes. Let $\\omega_{Y/X}$ be as in", "Remark \\ref{remark-relative-dualizing-for-quasi-finite}.", "It is clear from the uniqueness, existence, and compatibility with", "localization of trace elements", "(Lemmas \\ref{lemma-trace-unique}, \\ref{lemma-dualizing-tau}, and", "\\ref{lemma-trace-base-change})", "that there exists a global section", "$$", "\\tau_{Y/X} \\in \\Gamma(Y, \\omega_{Y/X})", "$$", "such that for every pair of affine opens", "$\\Spec(B) = V \\subset Y$, $\\Spec(A) = U \\subset X$ with $f(V) \\subset U$", "that element $\\tau_{Y/X}$ maps to $\\tau_{B/A}$ under the", "canonical isomorphism", "$H^0(V, \\omega_{Y/X}) = \\omega_{B/A}$." ], "refs": [ "discriminant-remark-relative-dualizing-for-quasi-finite", "discriminant-lemma-trace-unique", "discriminant-lemma-dualizing-tau", "discriminant-lemma-trace-base-change" ], "ref_ids": [ 15006, 14956, 14960, 14958 ] }, { "id": 15008, "type": "other", "label": "discriminant-remark-construction-pairing", "categories": [ "discriminant" ], "title": "discriminant-remark-construction-pairing", "contents": [ "Let $A \\to B$ be a quasi-finite homomorphism of Noetherian rings.", "Let $J$ be the annihilator of $\\Ker(B \\otimes_A B \\to B)$.", "There is a canonical $B$-bilinear pairing", "\\begin{equation}", "\\label{equation-pairing-noether}", "\\omega_{B/A} \\times J \\longrightarrow B", "\\end{equation}", "defined as follows. Choose a factorization $A \\to B' \\to B$", "with $A \\to B'$ finite and $B' \\to B$ inducing an open immersion", "of spectra. Let $J'$ be the annihilator of $\\Ker(B' \\otimes_A B' \\to B')$.", "We first define", "$$", "\\Hom_A(B', A) \\times J' \\longrightarrow B',\\quad", "(\\lambda, \\sum b_i \\otimes c_i) \\longmapsto \\sum \\lambda(b_i)c_i", "$$", "This is $B'$-bilinear exactly because for $\\xi \\in J'$ and $b \\in B'$", "we have $(b \\otimes 1)\\xi = (1 \\otimes b)\\xi$. By", "Lemma \\ref{lemma-noether-different-localization}", "and the fact that $\\omega_{B/A} = \\Hom_A(B', A) \\otimes_{B'} B$", "we can extend this to a $B$-bilinear pairing as displayed above." ], "refs": [ "discriminant-lemma-noether-different-localization" ], "ref_ids": [ 14964 ] }, { "id": 15009, "type": "other", "label": "discriminant-remark-universal-finite-syntomic-smooth-top", "categories": [ "discriminant" ], "title": "discriminant-remark-universal-finite-syntomic-smooth-top", "contents": [ "Let $\\pi_d : Y_d \\to X_d$ be as in", "Example \\ref{example-universal-finite-syntomic}.", "Let $U_d \\subset X_d$ be the maximal open over which", "$V_d = \\pi_d^{-1}(U_d)$ is finite syntomic as in", "Lemma \\ref{lemma-universal-finite-syntomic}.", "Then it is also true that $V_d$ is smooth over $\\mathbf{Z}$.", "(Of course the morphism $V_d \\to U_d$ is not smooth when $d \\geq 2$.)", "Arguing as in the proof of Lemma \\ref{lemma-universal-finite-syntomic-smooth}", "this corresponds to the following deformation", "problem: given a small extension $C' \\to C$ and", "a finite syntomic $C$-algebra $B$ with a section $B \\to C$,", "find a finite syntomic $C'$-algebra $B'$ and a section $B' \\to C'$", "whose tensor product with $C$ recovers $B \\to C$.", "By Lemma \\ref{lemma-syntomic-finite} we may write", "$B = C[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$ as", "a relative global complete intersection.", "After a change of coordinates with may assume", "$x_1, \\ldots, x_n$ are in the kernel of $B \\to C$.", "Then the polynomials $f_i$ have vanishing constant terms.", "Choose any lifts $f'_i \\in C'[x_1, \\ldots, x_n]$ of $f_i$", "with vanishing constant terms. Then ", "$B' = C'[x_1, \\ldots, x_n]/(f'_1, \\ldots, f'_n)$", "with section $B' \\to C'$ sending $x_i$ to zero works." ], "refs": [ "discriminant-lemma-universal-finite-syntomic", "discriminant-lemma-universal-finite-syntomic-smooth", "discriminant-lemma-syntomic-finite" ], "ref_ids": [ 14988, 14989, 14987 ] }, { "id": 15010, "type": "other", "label": "discriminant-remark-local-description-delta", "categories": [ "discriminant" ], "title": "discriminant-remark-local-description-delta", "contents": [ "Let $Y \\to X$ be a locally quasi-finite syntomic morphism of schemes.", "What does the pair $(\\det(\\NL_{Y/X}), \\delta(\\NL_{Y/X}))$ look", "like locally? Choose affine opens $V = \\Spec(B) \\subset Y$,", "$U = \\Spec(A) \\subset X$ with $f(V) \\subset U$ and an integer $n$ and", "$f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$ such that", "$B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$. Then", "$$", "\\NL_{B/A} = \\left(", "(f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2", "\\longrightarrow", "\\bigoplus\\nolimits_{i = 1, \\ldots, n} B \\text{d} x_i\\right)", "$$", "and $(f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2$ is free with generators", "the classes $\\overline{f}_i$. See proof of", "Lemma \\ref{lemma-syntomic-quasi-finite}.", "Thus $\\det(L_{B/A})$ is free on the generator", "$$", "\\text{d}x_1 \\wedge \\ldots \\wedge \\text{d}x_n", "\\otimes", "(\\overline{f}_1 \\wedge \\ldots \\wedge \\overline{f}_n)^{\\otimes -1}", "$$", "and the section $\\delta(\\NL_{B/A})$ is the element", "$$", "\\delta(\\NL_{B/A}) =", "\\det(\\partial f_j/ \\partial x_i) \\cdot", "\\text{d}x_1 \\wedge \\ldots \\wedge \\text{d}x_n", "\\otimes", "(\\overline{f}_1 \\wedge \\ldots \\wedge \\overline{f}_n)^{\\otimes -1}", "$$", "by definition." ], "refs": [ "discriminant-lemma-syntomic-quasi-finite" ], "ref_ids": [ 14981 ] }, { "id": 15011, "type": "other", "label": "discriminant-remark-different-generalization", "categories": [ "discriminant" ], "title": "discriminant-remark-different-generalization", "contents": [ "We can generalize Definition \\ref{definition-different}.", "Suppose that $f : Y \\to X$ is a quasi-finite morphism of Noetherian schemes", "with the following properties", "\\begin{enumerate}", "\\item the open $V \\subset Y$ where $f$ is flat contains", "$\\text{Ass}(\\mathcal{O}_Y)$ and $f^{-1}(\\text{Ass}(\\mathcal{O}_X))$,", "\\item the trace element $\\tau_{V/X}$ comes from a section", "$\\tau \\in \\Gamma(Y, \\omega_{Y/X})$.", "\\end{enumerate}", "Condition (1) implies that $V$ contains the associated points of", "$\\omega_{Y/X}$ by Lemma \\ref{lemma-dualizing-associated-primes}.", "In particular, $\\tau$ is unique if it exists", "(Divisors, Lemma \\ref{divisors-lemma-restriction-injective-open-contains-ass}).", "Given $\\tau$ we can define the different $\\mathfrak{D}_f$ as the annihilator of", "$\\Coker(\\tau : \\mathcal{O}_Y \\to \\omega_{Y/X})$. This agrees with the", "Dedekind different in many cases (Lemma \\ref{lemma-agree-dedekind}).", "However, for non-flat maps between non-normal rings, this generalization", "no longer measures ramification of the morphism, see", "Example \\ref{example-no-different}." ], "refs": [ "discriminant-definition-different", "discriminant-lemma-dualizing-associated-primes", "divisors-lemma-restriction-injective-open-contains-ass", "discriminant-lemma-agree-dedekind" ], "ref_ids": [ 15005, 14951, 7861, 14998 ] }, { "id": 15012, "type": "other", "label": "discriminant-remark-collect-results-qf-gorenstein", "categories": [ "discriminant" ], "title": "discriminant-remark-collect-results-qf-gorenstein", "contents": [ "Let $f : Y \\to X$ be a quasi-finite Gorenstein morphism of Noetherian schemes.", "Let $\\mathfrak D_f \\subset \\mathcal{O}_Y$ be the different and let", "$R \\subset Y$ be the closed subscheme cut out by $\\mathfrak D_f$.", "Then we have", "\\begin{enumerate}", "\\item $\\mathfrak D_f$ is a locally principal ideal,", "\\item $R$ is a locally principal closed subscheme,", "\\item $\\mathfrak D_f$ is affine locally the same as the Noether different,", "\\item formation of $R$ commutes with base change,", "\\item if $f$ is finite, then the norm of $R$ is the discriminant of $f$, and", "\\item if $f$ is \\'etale in the associated points of $Y$, then", "$R$ is an effective Cartier divisor and $\\omega_{Y/X} = \\mathcal{O}_Y(R)$.", "\\end{enumerate}", "This follows from Lemmas \\ref{lemma-flat-gorenstein-agree-noether},", "\\ref{lemma-base-change-different}, and", "\\ref{lemma-norm-different-is-discriminant}." ], "refs": [ "discriminant-lemma-flat-gorenstein-agree-noether", "discriminant-lemma-base-change-different", "discriminant-lemma-norm-different-is-discriminant" ], "ref_ids": [ 14976, 14977, 14980 ] }, { "id": 15013, "type": "other", "label": "discriminant-remark-collect-results-qf-gorenstein-two", "categories": [ "discriminant" ], "title": "discriminant-remark-collect-results-qf-gorenstein-two", "contents": [ "Let $S$ be a Noetherian scheme endowed with a dualizing complex", "$\\omega_S^\\bullet$. Let $f : Y \\to X$ be a quasi-finite Gorenstein", "morphism of compactifyable schemes over $S$. Assume moreover", "$Y$ and $X$ Cohen-Macaulay and $f$ \\'etale at the generic", "points of $Y$. Then we can combine", "Duality for Schemes, Remark", "\\ref{duality-remark-CM-morphism-compare-dualizing} and", "Remark \\ref{remark-collect-results-qf-gorenstein}", "to see that we have a canonical isomorphism", "$$", "\\omega_Y = f^*\\omega_X \\otimes_{\\mathcal{O}_Y} \\omega_{Y/X} =", "f^*\\omega_X \\otimes_{\\mathcal{O}_Y} \\mathcal{O}_Y(R)", "$$", "of $\\mathcal{O}_Y$-modules. If further $f$ is finite,", "then the isomorphism $\\mathcal{O}_Y(R) = \\omega_{Y/X}$ comes", "from the global section $\\tau_{Y/X} \\in H^0(Y, \\omega_{Y/X})$", "which corresponds via duality to the map", "$\\text{Trace}_f : f_*\\mathcal{O}_Y \\to \\mathcal{O}_X$, see", "Lemma \\ref{lemma-compare-trace}." ], "refs": [ "duality-remark-CM-morphism-compare-dualizing", "discriminant-remark-collect-results-qf-gorenstein", "discriminant-lemma-compare-trace" ], "ref_ids": [ 13657, 15012, 15000 ] }, { "id": 15130, "type": "other", "label": "limits-remark-limit-preserving", "categories": [ "limits" ], "title": "limits-remark-limit-preserving", "contents": [ "Let $S$ be a scheme. Let us say that a functor", "$F : (\\Sch/S)^{opp} \\to \\textit{Sets}$ is", "{\\it limit preserving} if for every directed inverse system", "$\\{T_i\\}_{i \\in I}$ of affine schemes with limit $T$ we have", "$F(T) = \\colim_i F(T_i)$. Let $X$ be a scheme over $S$, and", "let $h_X : (\\Sch/S)^{opp} \\to \\textit{Sets}$ be its", "functor of points, see", "Schemes, Section \\ref{schemes-section-representable}.", "In this terminology", "Proposition \\ref{proposition-characterize-locally-finite-presentation}", "says that a scheme $X$ is locally of finite presentation over", "$S$ if and only if $h_X$ is limit preserving." ], "refs": [ "limits-proposition-characterize-locally-finite-presentation" ], "ref_ids": [ 15127 ] }, { "id": 15131, "type": "other", "label": "limits-remark-cannot-do-better", "categories": [ "limits" ], "title": "limits-remark-cannot-do-better", "contents": [ "We cannot do better than this if we do not assume", "more on $S$ and the morphism $f : X \\to S$.", "For example, in general it will not be possible to", "find a {\\it closed} immersion $X \\to X'$ as in the lemma.", "The reason is that this would imply that $f$ is quasi-compact which", "may not be the case. An example is to take $S$ to be infinite", "dimensional affine space with $0$ doubled and $X$ to be one of", "the two infinite dimensional affine spaces." ], "refs": [], "ref_ids": [] }, { "id": 15132, "type": "other", "label": "limits-remark-more-general-modification", "categories": [ "limits" ], "title": "limits-remark-more-general-modification", "contents": [ "The lemma above can be generalized as follows. Let $S$ be a scheme and", "let $T \\subset S$ be a closed subset. Assume there exists a cofinal", "system of open neighbourhoods $T \\subset W_i$ such that", "(1) $W_i \\setminus T$ is quasi-compact and", "(2) $W_i \\subset W_j$ is an affine morphism.", "Then $W = \\lim W_i$ is a scheme which contains $T$", "as a closed subscheme. Set $U = X \\setminus T$ and $V = W \\setminus T$.", "Then the base change functor", "$$", "\\left\\{", "\\begin{matrix}", "f : X \\to S\\text{ of finite presentation} \\\\", "f^{-1}(U) \\to U\\text{ is an isomorphism}", "\\end{matrix}", "\\right\\}", "\\longrightarrow", "\\left\\{", "\\begin{matrix}", "g : Y \\to W\\text{ of finite presentation} \\\\", "g^{-1}(V) \\to V\\text{ is an isomorphism}", "\\end{matrix}", "\\right\\}", "$$", "is an equivalence of categories. If we ever need this we will", "change this remark into a lemma and provide a detailed proof." ], "refs": [], "ref_ids": [] }, { "id": 15133, "type": "other", "label": "limits-remark-finite-type-gives-well-defined-system", "categories": [ "limits" ], "title": "limits-remark-finite-type-gives-well-defined-system", "contents": [ "In Situation \\ref{situation-limit-noetherian}", "Lemmas \\ref{lemma-good-diagram}, \\ref{lemma-limit-from-good-diagram}, and", "\\ref{lemma-morphism-good-diagram}", "tell us that the category of schemes quasi-separated and", "of finite type over $S$ is equivalent to certain types of", "inverse systems of schemes over $(S_i)_{i \\in I}$, namely", "the ones produced by applying Lemma \\ref{lemma-limit-from-good-diagram}", "to a diagram of the form (\\ref{equation-good-diagram}).", "For example, given $X \\to S$ finite type and quasi-separated", "if we choose two different diagrams $X \\to V_1 \\to S_{i_1}$", "and $X \\to V_2 \\to S_{i_2}$ as in (\\ref{equation-good-diagram}), then", "applying Lemma \\ref{lemma-morphism-good-diagram} to $\\text{id}_X$", "(in two directions)", "we see that the corresponding limit descriptions of", "$X$ are canonically isomorphic (up to shrinking the", "directed set $I$). And so on and so forth." ], "refs": [ "limits-lemma-good-diagram", "limits-lemma-limit-from-good-diagram", "limits-lemma-morphism-good-diagram", "limits-lemma-limit-from-good-diagram", "limits-lemma-morphism-good-diagram" ], "ref_ids": [ 15119, 15120, 15121, 15120, 15121 ] } ], "retrieval_examples": [ 0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 59, 64, 65, 67, 68, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 105, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 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